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author:
- Yujun Zheng
title: 'Comment on “Coherent Control of a V-Type Three-Level System in a Single Quantum Dot”'
---
Wang [*et al.*]{} studied the coherent control of single Quantum Dots recently [@wang]. In Ref. [@wang], the authors model the single Quantum Dots as the V-type three level system with two orthogonal transition dipole moments $\bm{\mu}_x$ and $\bm{\mu}_y$. Wang [*et al.*]{} took the theoretical calculations via the three level system Bloch equation under the RWA. The equation they used is written as $$\small{
\label{eq:bloch}
\dot{\overrightarrow{S}}(t)=M(t) \overrightarrow{S}(t) -\Gamma \overrightarrow{S} -\overrightarrow{\Lambda} ,
}$$ where $M(t)$ is the coefficient matrix related to external laser field. The $\Gamma$ and $\overrightarrow{\Lambda}$, in Ref. [@wang], are as following
$$\small{
\Gamma=
\left(
\begin{array}{cccccccc}
\frac{1}{2}\gamma_x & \frac{1}{2}\gamma_{xy} & 0 & \delta_x & 0 & 0 & 0 & 0 \\
\frac{1}{2}\gamma_{xy} & \frac{1}{2} \gamma_y & 0 & 0 & \delta_y & 0 & 0 & 0 \\
0 & 0 & \frac{1}{2}(\gamma_x+\gamma_y) & 0 & 0 & -\Delta & \frac{1}{3}\gamma_{xy} & \frac{1}{3}\gamma_{xy} \\
-\delta_x & 0 & 0 & \frac{1}{2}\gamma_x & \frac{1}{2}\gamma_{xy} & 0 & 0 & 0 \\
0 & -\delta_y & 0 & \frac{1}{2}\gamma_{xy} & \frac{1}{2}\gamma_y & 0 & 0 & 0 \\
0 & 0 & \Delta & 0 & 0 & \frac{1}{2}(\gamma_x+\gamma_y) & 0 & 0 \\
0 & 0 & \frac{1}{2}\gamma_{xy} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \frac{1}{2}\gamma_{xy} & 0 & 0 & 0 & 0 & 0
\end{array}
\right),
}
\label{eq:gamma}$$
and $$\small{
\overrightarrow{\Lambda} =\left( 0,0,\frac{2}{3}\gamma_{xy}, 0,0,0,
\frac{1}{3}\gamma_x, \frac{1}{3}\gamma_y \right).
}
\label{eq:lambda}$$
Their expressions of the coefficient matrices $\Gamma$ and $\overrightarrow{\Lambda}$ are wrong. In general, the expressions of the coefficient matrices $\Gamma$ and $\overrightarrow{\Lambda}$ for the V-type three level system are [@peng; @ficek]
$$\small{
\Gamma=
\left(
\begin{array}{cccccccc}
\frac{1}{2}\gamma_x & \frac{1}{2}\gamma_{xy} & 0 & \delta_x & 0 & 0 & 0 & 0 \\
\frac{1}{2}\gamma_{xy} & \frac{1}{2} \gamma_y & 0 & 0 & \delta_y & 0 & 0 & 0 \\
0 & 0 & \frac{1}{2}(\gamma_x+\gamma_y) & 0 & 0 & -\Delta & \frac{1}{3}\gamma_{xy} & \frac{1}{3}\gamma_{xy} \\
-\delta_x & 0 & 0 & \frac{1}{2}\gamma_x & \frac{1}{2}\gamma_{xy} & 0 & 0 & 0 \\
0 & -\delta_y & 0 & \frac{1}{2}\gamma_{xy} & \frac{1}{2}\gamma_y & 0 & 0 & 0 \\
0 & 0 & \Delta & 0 & 0 & \frac{1}{2}(\gamma_x+\gamma_y) & 0 & 0 \\
0 & 0 & \frac{3}{2}\gamma_{xy} & 0 & 0 & 0 & \frac{1}{3}(4 \gamma_x-\gamma_y) & -\frac{2}{3} (\gamma_x-\gamma_y)\ \\
0 & 0 & \frac{3}{2}\gamma_{xy} & 0 & 0 & 0 & \frac{2}{3}(\gamma_x-\gamma_y) & -\frac{1}{3}(\gamma_x - 4 \gamma_y)
\end{array}
\right),
}
\label{eq:gamma1}$$
and $$\small{
\overrightarrow{\Lambda} =\left( 0,0,\frac{2}{3}\gamma_{xy}, 0,0,0,\frac{1}{3}(2\gamma_x+\gamma_y), \frac{1}{3} ( \gamma_x+2\gamma_y) \right) ^\dag.
}
\label{eq:lambda1}$$
For the V-type three level system with the orthogonal excited states $|x \rangle$ and $|y \rangle$, $\gamma_{xy}=0$ [@ficek]. However the authors of Ref. [@wang] did not mentioned this in their theoretical calculations. Unfortunately, the authors of Ref. [@wang] did not give the parameters they used in their theoretical calculations, we can not compare the numerical results.
I thank Y. Peng for valuable comments and suggestions. This work was supported by the National Science Foundation of China (grant no. 10674083). Partial financial supports from the Science Foundation of Shandong Province, China.
[99]{} Q. Q. Wang, A. Muller [*et al.*]{}, Phys. Rev. Lett. [**95**]{}, 187404 (2005). Y. Peng [*et al.*]{}, J. Chem. Phys. [**126**]{}, 104303 (2007). S. Swain, P. Zhou [*et al.*]{}, Phys. Rev. A [**61**]{}, 043410 (2000).
|
---
abstract: 'We describe a reaction mechanism which is consistent with all available experimental information of high energy three-body breakup processes. The dominating channels are removal of one of the three halo particles leaving the other two either undisturbed or absorbed. We compare with the commonly used deceptive assumption of a decay through two-body resonance states. Our predictions can be tested by measuring neutron-neutron invariant mass spectra.'
address:
- 'Instituto de Estructura de la Materia, CSIC, Serrano 123, E-28006 Madrid, Spain'
- 'Institute of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark'
author:
- 'E. Garrido'
- 'D.V. Fedorov, A.S. Jensen and K. Riisager'
title: 'Reaction mechanisms for two-neutron halo breakup'
---
[2]{}
#### Introduction. {#introduction. .unnumbered}
Halo states are basically characterized as spatially extended weakly bound systems. Two-body halos interacting with a target present a three-body scattering problem, which in practice is much more complicated due to the intrinsic structure of the three constituent particles. Three-body halos present analogously at least a four-body problem, which has to be approximated preferentially by using physical insights. Reactions for high beam energies allow a separation of the degrees of freedom related to the fast relative projectile-target coordinates and the slow intrinsic halo motion.
The properties of halo systems have been discussed intensely over the last decade both in dripline nuclei [@rii94; @han95; @tan96; @jon98] and in molecular systems [@esr96; @nie98]. Precise definitions, classification and occurrence conditions were recently attempted [@rii00; @jen00] although different from an earlier definition [@goy95]. Few-body concepts and techniques are successfully applied in the descriptions [@gar98], which to a large extent focussed on nuclear three-body halos. Much efforts have been devoted to two-neutron Borromean halos like $^6$He (n+n+$^4$He) and $^{11}$Li (n+n+$^{9}$Li) where two neutrons surround a core [@zhu93]. The basic structure is essentially agreed upon while reaction descriptions and analyses of measurements still are controversial.
Halo physics is a substantial part of experimental programs with radioactive beams and it is urgent to root out widespread misconceptions and clarify how the reactions proceed. Furthermore these questions are of general interest as basic few-body reaction problems. Since the halo concept now is applied and exploited in molecular physics we also may anticipate similar implications of properly formulated reaction models.
The purpose of this letter is to (i) establish the reaction mechanism for two-neutron halo breakup in high energy collisions with light targets and in passing clarify the differences to the entirely different mechanism due to the large charges of heavy targets, (ii) investigate the validity of the erroneous but commonly used assumption of breakup through resonances in the two-body subsystems. These questions are crucial and answers urgently needed for understanding reactions with halo nuclei.
#### Reaction mechanisms. {#reaction-mechanisms. .unnumbered}
The dominating reaction channels for two-neutron halo breakup on light targets are experimentally established [@ale98; @aum99] and theoretically described [@gar98; @ber98] as removal of one neutron or destruction of the core, thereby leaving the final state with the core and the other neutron or with the two neutrons.
The decisive question in this context is which reaction mechanism is responsible for the observed behavior? The reaction time for light targets is short compared to the time scale of the intrinsic halo motion. For spatially extended systems the target can then remove one of the halo particles instantaneously without disturbing the motion of the other two particles. This means that the sudden approximation basically is valid as accepted in several previous publications [@ale98; @aum99; @zin97; @chu97].
The implication is that the remaining two-body system is left in its initial state which, as unbound for Borromean systems, falls apart influenced by the corresponding two-body interaction. This decaying two-body system is thus formed as a wave packet consisting of those parts of the relative two-body wave function present within the original three-body system, which precisely lead to the dominating reaction products [@gar98]. The surviving wave packet then has a large component describing the tail of the two-body wave function. The short distance parts lead to a large extent to removal of more than one particle at a time. All other breakup reactions are analogously described in this participant-spectator model (PSM).
#### $R$-matrix formulation. {#r-matrix-formulation. .unnumbered}
The observed invariant two-body mass spectra and the momentum distributions are routinely analyzed as arising from the decays of low-lying two-body resonances or virtual $s$-states [@ale98; @aum99; @zin97; @chu97; @sim98]. These assumptions are in direct contradiction to the short reaction time and the sudden approximation. There is not sufficient time for the remaining two halo particles to adjust their relative motion and populate corresponding resonance states. This requires at least a reaction time comparable to the intrinsic halo time scale.
Thus these analyses apparently invoke both the sudden approximation and decay through resonances or virtual $s$-states. These assumptions are strictly incompatible except when these two-body states are populated within the initial three-body system. This is clearly seen by constructing a Borromean system by adding a neutron to a neutron-core resonance state. The overlap of this and the real bound state wave function may still be substantial, but rearrangements are necessary to reach the bound three-body state, i.e. a novel few-body system carrying otherwise inaccessible information about the off-shell behavior of the nucleon-nucleon interaction.
The pertinent questions are what we can learn from measured two-body invariant spectra and what information is in fact obtained by the analyses using resonances or virtual $s$-states as intermediate states. The analyses, often erroneously claiming to use Breit-Wigner shapes [@ale98; @aum99; @zin97; @sim98], are in fact based on $R$-matrix theory [@lan58; @mcv94], where a complete basis of two-body continuum states are used after removal of one of the three particles. This basis could consist of “correct” low-lying resonance states supplemented with a discretized or continuum higher lying set of states.
In practice one proceeds by reducing the unspecified and unknown basis to a few terms, i.e. usually one or two states. This reduction of model space may be allowed if the basis consistently is renormalized, i.e. the new basis includes properties of the excluded states. Thus fitting in this context by use of a small basis seems to prohibit interpretation in terms of the “correct” two-body resonance states. In principle maintaining the basis without renormalization would be correct but this presupposes exactly that knowledge about these states, which is the very aim of the analyses. This problem cannot be solved by increasing the employed model space until convergence is reached and no renormalization is needed. A larger space implies more parameters in the fitting procedure and reproduction of the data is not unique. The problem becomes overdetermined and the extracted parameters inaccurate or directly unreliable.
The analyses using two-body resonances or virtual $s$-states therefore assume (i) that no renormalization due to truncation of model space is needed, (ii) a reaction mechanism where only the “clean” two-body resonance states or virtual $s$-states are populated and (iii) no other (known or unexpected) reaction channel contribute. These assumptions are at least inaccurate. The difficulties are enlarged when more than one resonance or more than one reaction channel contribute. When the assumptions are fairly well fulfilled the interpretation would also be approximately correct.
#### Computations. {#computations. .unnumbered}
We shall concentrate on $^6$He and $^{11}$Li colliding with light targets. We shall use the PSM formulation, where one halo particle (the participant) interacts with the target while the other two halo particles (the spectators) are left undisturbed [@gar98]. The participant-target interaction is described by the phenomenological optical model while the spectators are treated in the black sphere approximation, i.e. they are absorbed within a given radius from the target and otherwise they continue undisturbed. This model faithfully exploits the consequences of a short reaction time. We compute the population of the two-body continuum states after the instantaneous removal of the third particle.
The $R$-matrix expressions of the invariant mass spectrum $d \sigma
/dE$ of the spectator system and the relative spectator momentum distribution $P_{long}$ are [@ale98; @zin97; @sim98] $$\begin{aligned}
\frac{d \sigma}{dE} = \frac{\sigma_l}{2 \pi} \;
\frac{ \Gamma(E)}{(E-E_{r})^2 +
0.25 \Gamma^2(E) }, \;
\Gamma=\Gamma_0 \frac{E^{l+0.5}}{E_r^{l+0.5}} \; ,
\label{e1} \\
P_{long}(p) = \int^{\infty}_{E_{min}} \frac{d E} {\sqrt{E}}
\frac{d \sigma}{dE}, \hspace*{0.3cm} E_{min} = p^2/ 2 \mu
\label{e3}\end{aligned}$$ where $\sigma_l$ is the total cross section, $l$ is the orbital angular momentum, $\mu$ is the reduced mass, $E_r$ and $\Gamma_0$ are position and width parameters. The distributions are correlated and should not be fitted independently. Precisely the same procedure applies both when the core and a neutron are the participant, i.e. the final state consists of two neutrons or a neutron-core system, respectively.
The chosen observables amplify the effects of the assumed reaction mechanism. We can then compare the experimental distributions both with the PSM predictions [@gar98] and the $R$-matrix results obtained by the decay through resonance assumption. This provides evidence about the basic reaction mechanism.
#### Neutron removal. {#neutron-removal. .unnumbered}
Absorption of one neutron from $^6$He produces a neutron-$^4$He continuum state, which mainly is of $p_{3/2}$-character, since the $p_{1/2}$ neutron-core state has a higher energy and the $s_{1/2}$-wave is repulsive. The corresponding invariant mass spectrum and the relative momentum distribution are shown in Fig. \[fig1\]. The PSM computation agrees fairly well with the measured invariant mass spectrum [@ale98]. The peak position reflects the energy of 0.77 MeV of the neutron-$^4$He $p_{3/2}$-resonance with the width of 0.5 MeV used in the PSM computation. Any value of $\Gamma_0$, see Eq.(\[e1\]), from 0.4 MeV to 0.8 MeV also reproduce the experiment fairly well.
For consistency the momentum distribution should now follow with the same parameters. Indeed we see in Fig. \[fig1\] that the PSM results resemble the (almost) width independent $R$-matrix fits confirming that the width parameter for the n–$^4$He resonance can not be determined in this way. We also note the characteristic flat maximum of $p$–waves.
The $^{11}$Li system is different due to the core spin of $3/2$ and the mixture of $s$ and $p$-waves in the subsystems. The computed three–body wave function contains around 60% of $s$–wave and 40% of $p^2$–wave neutron-core configurations. The neutron-$^{9}$Li system has a low lying virtual $s$–state at 240 keV and a $p$–resonance at 0.5 MeV [@gar98]. Neutron removal results in the distributions shown in Fig. \[fig2\]. The contribution to the invariant mass spectrum from $s$-waves peaks at a very low energy determined entirely from the phase space constraint and independent of the position of the virtual $s$-state [@mcv94]. In contrast the $p$-wave contribution peaks at the two-body resonance energy. The measured spectrum is fairly well reproduced by the PSM computation again supporting the assumed initial three-body structure and the reaction mechanism.
We compare in Fig. \[fig2\] with two different $R$-matrix fits. In the first the computed $s$ and $p$ contributions are fitted separately thereby maintaining the same $p$–wave content. In the second fit we use the parameters in [@zin97], which also reproduces rather well the experimental (and PSM) data. The $^{10}$Li structures underlying these fits differ substantially as expressed clearly through the different widths. The $p$–wave contents also differ substantially, i.e. about 35% for the first and 70% for the second fit [@zin97].
In the lower part of Fig. \[fig2\] we show the corresponding relative momentum distribution. The PSM computation and the first fit produce a very similar momentum distribution, while the second fit differs in the central part due to the large fraction of $p$–waves that creates the plateau at low relative momentum. Therefore different fits of invariant mass spectra of similar accuracy can produce rather different momentum distributions due to emphasis of different features of the distribution.
#### Core destruction. {#core-destruction. .unnumbered}
The reaction assumptions can be tested by similar investigations of the other important channel corresponding to destruction of the core and leaving the two neutrons as spectators. The final states then consists of two neutrons for both $^6$He and $^{11}$Li. Thus we can separate effects of initial and final state structures. However, this assumes that fragments from the core interacting strongly with neutrons are excluded from the data. In the PSM computations, discussed in connection with Figs. \[fig1\] and \[fig2\], the spectra are sensitive to the initial three–body structure. The root mean square distance between the neutrons is more than 6 fm for $^{11}$Li and less than 4.5 fm for $^6$He. The neutron–neutron invariant mass spectrum and the corresponding momentum distribution are then both expected to be substantially narrower for $^{11}$Li than for $^6$He. The same consistent PSM model has been tested on many other, relative and absolute values of observables for neutron removal and core breakup reactions for both projectiles [@gar98]. The agreement with available experimental data is overall very convincing.
Further tests of the PSM model (and the $R$-matrix analyses) would be measurements comparing to the predictions presented in Fig. \[fig3\]. The neutron-neutron relative $s$-waves are completely dominating for both $^6$He and $^{11}$Li and consequently the invariant mass spectra have very low-lying peaks. Both spectra and momentum distributions are qualitatively similar for the two cases, but quantitatively the $^{11}$Li results are much narrower than those of $^6$He. The $R$-matrix distributions fitting the two PSM curves in Fig. \[fig3\] correspond to very different energy and width parameters without any connection to the known neutron-neutron scattering properties. The PSM model predicts different neutron-neutron spectra for $^6$He and $^{11}$Li after core breakup. A reaction mechanism populating final state two-body resonances independent of the initial structure must predict identical neutron-neutron invariant mass spectra for both projectiles. Experimental data could distinguish between these models.
#### Conclusions. {#conclusions. .unnumbered}
The dominating reaction channels for high energy breakup of Borromean three-body halos on light targets are one-particle removal and subsequent decays of the wave packets created in these processes. The reaction time is short and any resonance structure of the remaining two-body system is populated with the amount already present in the initial three-body wave function. All available experimental data for high energy three-body breakup on light nuclei are consistent with this reaction mechanism. For heavy targets the reaction mechanism for the dominating channel is quite different proceeding through a gentle excitation of the three-body continuum by the Coulomb interaction.
Analyses assuming instantaneous removal of either a neutron or the core, while populating resonances in the remaining two-body system, are conceptually inconsistent. An invariant mass spectrum reproducing the data only reflects that the corresponding energy distribution was present immediately after the final state two-body system was isolated. The problem is especially enlarged when more than one resonance or virtual state are important for the two-body subsystems. The inconsistency is highlighted in spectra obtained after core breakup, where the final states are identical (two neutrons). Then the resulting distributions should also be identical for different two-neutron halo projectiles, even when the initial three-body structure differs substantially. This is in clear disagreement with elaborated consistent model computations reproducing essentially all available data. In any case, the neutron–neutron invariant mass spectra provide direct evidence of the breakup reaction mechanism.
We conclude that correct interpretation of the invariant spectra almost inevitably require a consistent model, i.e. the three-body structure, the two-body interactions and the reaction mechanisms must be (approximately) correct. Even then interplay between the different ingredients may produce misleading results. The $R$-matrix analyses rely on assumptions or computations of the initial steps producing an isolated two-body system.
#### Acknowledgement. {#acknowledgement. .unnumbered}
We thank L.V. Chulkov and H. Emling for illuminating discussions.
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|
---
abstract: 'In this paper a general multi-phase-field model is presented which is an extension and modification of the model proposed by Folch and Plapp for three phase fields \[R. Folch and M. Plapp, Phys. Rev. E 72 011602 (2005)\] to an arbitrary number of phases. In the model a physical constraint requiring that the sum of all phase fields in the system is equal to one is resolved by the method of Lagrange multipliers. In fact, the thermodynamic driving force is reduced to its projection on the plane of the constraint. The general model functions in a $N$-dimensional phase-field space are derived which justify the requirements for the stability of the total free energy functional on dual interfaces and hence the absence of “ghost” phases. Furthermore, the case of the different interface energies and mobility parameters on the individual interfaces is resolved in a comprehensive manner. It is shown that the static equilibrium fulfils Young’s law for contact angles with good accuracy. Then the model is verified by the quantitative simulation of the solidification in an Al-Cu-Ni alloy in the case of the four-phase transformation reaction. As a result, we found the way to control the dynamic of new phase nucleation using thermal noise in free energy functional.'
author:
- 'E. Pogorelov'
- 'J. Kundin'
- 'H. Emmerich'
title: 'General Phase-Field Model with Stability Requirements on Interfaces in $N$-Dimensional Phase-Field Space'
---
Introduction
============
Multi-phase-field approaches developed in the recent years were practically applied to the simulation of the three-phase transformation in eutectic and peritectic alloys. The wide range of such realistic microstructure studies was reported (see the review article [@Singer08] and the references therein). However, the investigation of four- and multi-phase transformation reactions is not fully covered. One of the reason is the complexity of the models and the emerging challenges in terms of the understanding of multi-phase interactions.
In the last decade, two main concepts of the multi-phase modelling have been established. The first one is the multi-phase concept of Steinbach, which considers the change of any phase $i$ as the sum of other phase contributions due to the interaction between the phase $i$ and other phases on all individual interfaces, whereas the interaction between more than two phases are neglected. Therefore, the kinetic of individual interfaces can be considered separately with different interface energies and mobility parameters [@Steinbach96]. The physical constraint on phase fields, that the sum of all phase fields is equal to one, is kept in this model automatically. The main works in this area are [@Tiaden98; @Eiken06; @Kim04], where the main focus is the coupling of the kinetic equations to the diffusion equations in the multi-component systems.
The second concept supposes that the physical constraint on phase fields can be resolved by the formal application of the method of Lagrange multipliers to the free-energy functional, it works as the geometric projection of the driving force vector onto the plane of constraint. The main representatives of this concept are Folch and Plapp[@Folch05]. They developed a qualitative phase-field model for three phases with a smooth designed model function in the free energy functional that should ensure the stability of the solution and the absence of “ghost” phases on the interfaces between two phases. In this model the equations of motion for each interface can be mapped to the standard phase-field model of pure substances, where the thin-interface asymptotic analysis was applied [@Karma98]. Moreover, it exploits the idea of second order expansion of the free energy functional[@Greenwood08; @Echebarria04], that simplifies the structure of the phase-field equations. Later, in the work of Kundin and Siquieri [@Kundin11], the total free energy functional proposed by Folch and Plapp was extended by using different thermodynamic factors of phases, whereas the evolution of the phase field was modeled according to the multi-phase concept of Steinbach. Then the model was refined by the inclusion of cross terms in the thermodynamic factor matrix in multi-component systems [@Kundin12b]. The same authors [@Kundin12] applied the original model of Folch and Plapp to investigate the kinetics and morphology of the eutectic growth, where the difference in the thermodynamic factors was taken into account, too. They also shown that the nucleation of new lamellae on the boundary of the partner solid phase is rather an physical phenomena than an “artifact” of the model and the probability of this nucleation is defined by the the undercooling and the surface tension that is in agreement with the classical nucleation theory.
The second concept was further exploited in the field of multicomponent alloy solidification [@Garcke04; @Nestler05]. The authors used the Lagrange multiplier method and wrote the model in common form leaving out of consideration the stability requirements. To prevent the existence of the third phase on the individual interfaces the authors added the third order term to the obstacle potential which hardly can be applied in more general cases. Moreover, no thin-interface analysis of such models is available.
An intermediate approach between the first and second concept was proposed, which combines the different interface kinetics with the formal account for the change of the all phases in the multiple junctions [@Steinbach99]. In the study [@Guo10] it was shown that the reformulated model predicts the angles between the phases close to the analytical values except of small deviation in the 3D simulations. But by that approach the authors cannot overcome the criticism of Folch and Plapp about the instability of the solution for the chosen model functions [@Folch05].
In this paper we use the method of Lagrange multipliers and the idea of flatness and stability for model functions suggested by Folch and Plapp [@Folch05] and then extend it to the $N$-dimensional phase-field model (see Section \[Section\_Formulation\]). We construct the phase-field model functions step by step and show the implementation of different interface energies and mobility parameters for each individual interface. The numerical tests presented in Section \[Test\_Young\] were carried out to show how the model fulfills Young’s law and how the mobility parameters influence the dynamic evolution of the corresponding individual interfaces. In Section \[Al-Cu-Ni\] we verify the model by the qualitative simulations of the solidification in an Al-Cu-Ni alloy with a four-phase transformation reaction. In this section we explained the physics of four-phase solidification and write a full system of the model equations. We present the numerical simulations of the microstructure evolution and show the nucleation effects which can be observed. Finally, in conclusion we shortly summarize the main result of this paper.
Formulation of general phase-field model {#Section_Formulation}
========================================
Evolution equation for phase fields
-----------------------------------
We assume that our physical system can be fully described by $N$ phase fields $ p_{i}\in[0,1]$, $i=1\dots N$ and by $n$ concentration fields $c^A\in[0,1]$, $A=1\dots n$. We identify a vector of phase fields as $\bm{p}=(p_{1},\dots,p_{N})$, where every phase field $p_i$ means the volume fraction of $i$-th phase. Therefore, we demand that in every time our system should follow the physical constraint, that is the sum of all phase fields should be equal to one $$\sum_{i=1}^{N}p_{i}=1.$$
The total free energy functional of the system is written as $$F=\int_{V}f\, dV,$$ where a total free energy density is expanded into the following terms $$f(\bm{p},\bm{\nabla p},\bm{c},T)=Kf_{g}(\bm{\nabla p})+Hf_{b}(\bm{p})+f_{c}(\bm{p},\bm{c},T).$$ Here, $f_{g}$ sets a free energy cost depending on gradients of phase fields, forcing interfaces to have finite width. A constant $K$ has the dimension of energy per unit length and a constant $H$ has the dimension of energy per volume. $f_{b}$ is a dimensionless barrier function, which is analog to the double well potential in the theory for two phases. $f_c$ has the dimension of energy per volume and is the chemical part of the free energy which depends on concentration vector $\bm{c}=(c^{A},c^B\dots)$ and the temperature $T$.
For the evolution equations we chose Model C according to the classification given in Ref. [@HalperinPRB1974; @HalperinRevModPhys1977]. We used the Ginzburg-Landau equation for non-conserved field modified by Lagrange multiplier and the diffusion equation for conserved field $$\begin{aligned}
&\tau(\bm{p})\frac{\partial p_{i}}{\partial t}=-\frac{1}{H}\frac{\delta F}{\delta p_{i}}\biggm|_{\sum_{j}p_{j}=1}\\
&=-\frac{1}{H}\biggl(\frac{\delta F}{\delta p_{i}}-\frac{1}{N}\sum_{j}\frac{\delta F}{\delta p_{j}}\biggr),\;i=1,\dots N,\\
&\frac{\partial \bm{c}}{\partial t} = \nabla\biggl[\mathbf{\hat{M}}(\bm{p})\nabla\frac{\delta F}{\delta \bm{c}}-\bm{J}_{at}(\bm{p})\biggr].
\end{aligned}
\label{EqLagrangeMulti}$$ Here, $\tau(\bm{p})$ is a system relaxation time depending on the phase fields, $\mathbf{\hat{M}}(\bm{p})$ is a mobility matrix and $\bm{J}_{at}(\bm{p})$ is the anti-trapping current.
By using Lagrange multiplier method one get the projection of driving force onto the Gibbs simplex $S=\{\sum_{i}p_{i}=1\, p_{i}\in[0,1]\}$. This automatically ensures $\sum_i \partial p_i/\partial t = 0$.
Construction of model functions with stability and flatness requirement
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The first formulation of stability and flatness requirements for free energy model functions was made by Folch and Plapp in [@Folch05] as $$\frac{\delta^2 F}{\delta p_k^2}\biggm|_{\sum_i p_i=1,p_k=0,1}>0\;\forall k,$$ and $$\frac{\delta F}{\delta p_k}\biggm|_{\sum_i p_i=1,p_k=0,1}=0\;\forall k,$$ respectively. In the following, we will use requirements which are equivalent to Folch and Plapp [@Folch05; @Wheeler92] in three dimensional phase-field space but weaker in general $N$-dimensional space. Namely, we require on all interfaces $I_{ij}=\{p_j=1-p_i,\,p_{k\ne i\ne j}=0\}$ the stability condition $$\frac{\delta^2 F}{\delta p_k^2}\biggm|_{\bm{p}\in I_{ij}}>0\, \forall i,j,k,\label{Stability1}$$ and the flatness condition $$\frac{\delta F}{\delta p_k}\biggm|_{\bm{p}\in I_{ij}}=0\, \forall i,j,k, \label{Flatness1}$$ and analogical conditions on vertexes $V_i=\{p_i=1,p_{k\ne i}=0\}$ $$\begin{gathered}
\frac{\delta^2 F}{\delta p_k^2}\biggm|_{\bm{p}\in V_i}\geqslant 0\;\forall i,k;\label{Stability2}\\
\frac{\delta F}{\delta p_k}\biggm|_{\bm{p}\in V_i}=0\;\forall i,k.\label{Flatness2}\end{gathered}$$
We start the construction of model functions with choosing the free energy gradient term in the form $$f_g(\nabla \bm{p})=\frac{1}{2}\sum_i (\nabla p_i)^2.\label{fg}$$ It is easy to check that $f_g$ satisfies the flatness conditions (\[Flatness1\]),(\[Flatness2\]). And the flatness requirement is the main reason why we are limited to such simple form (\[fg\]).
Then, we construct the barrier function $f_b$ in such a way that we have an arbitrary positive interface energy $\sigma_{ij}$ for any individual interface $I_{ij}$. Moreover, $f_b$ should satisfy our stability and flatness conditions (\[Stability1\]-\[Flatness2\]). For this aim we define a set of barrier functions $$\begin{gathered}
f_{b,ij}=\frac{1}{2}(z_{ij}|_{\phi=p_i}+z_{ij}|_{\phi=p_j}),\; z_{ij}=\phi^3(1-\phi)^3\\
-3(1-\phi)\phi^3\sum_{k\ne i,j}p_k^2+2\phi^3\sum_{k\ne i,j}p_k^3\,.\end{gathered}$$ It is the polynomial of minimum power which follows stability and flatness conditions and $f_{b,ij}(\bm{p})=0$ $\forall \bm{p}\in I_{kl}\ne I_{ij}$. Also if $\bm{p}\in I_{ij}$ and $p_i=\varphi$, $p_j=1-\varphi$, then $f_{b,ij}=\varphi^3(1-\varphi)^3$. Finally, we can represent $$f_b=\sum_{i<j}q_{ij}f_{b, ij}, \label{fb}$$ where $q_{ij}$ will provide us the surface energy $\sigma_{ij}$ for each individual interface $I_{ij}$.
To determine constants $q_{ij}$ we should consider the evolution equation of phase fields (\[EqLagrangeMulti\]) on an interface $I_{ij}$. Using (\[Stability1\]-\[Flatness2\]) it can be shown that only $i$-th and $j$-th component of driving force are non zero. Then, taking into account that all terms $f_{b,kl}=0$ on $I_{ij}$ except $f_{b,ij}$, we will see that only $f_{b,ij}$ has a contribution in Eq. (\[EqLagrangeMulti\]). Analogically we can analyze the $f_g$ term. Therefore, without the chemical free energy term $f_c$ we can write for the static solution of Eq. (\[EqLagrangeMulti\]) $$\begin{gathered}
\frac{\partial \varphi}{\partial t} = \frac{K}{H} \frac{\partial^2 \varphi}{\partial x^2}-q_{ij}\frac{\partial \varphi^3(1-\varphi)^3}{\partial \varphi}\rightarrow 0, \text{ for } t\rightarrow \infty\\
\Rightarrow\frac{\partial^2 \varphi}{\partial x^2}=\frac{q_{ij}H}{K}\frac{\partial \varphi^3(1-\varphi)^3}{\partial\varphi}\,.\label{EqStatic}\end{gathered}$$ In equilibrium the total free energy will turn to $$F = \frac{K}{2}\biggl(\frac{\partial\varphi}{\partial x}\biggr)^2 + Hq_{ij} \varphi^3(1-\varphi)^3,$$ where we use only one dimension for the sake of simplicity. The solution of Eq. (\[EqStatic\]) can be expressed as $$\begin{gathered}
x(\varphi)=\int_{1/2}^\varphi \frac{\sqrt{K}}{\sqrt{2Hq_{ij}\phi^3(1-\phi)^3}}\, d\phi\,,\\
x\in(-\infty,\infty),\varphi\in[0,1].\end{gathered}$$ Then we have $$\begin{gathered}
\frac{\partial x}{\partial \varphi}=\frac{\sqrt{K}}{\sqrt{2Hq_{ij}\varphi^3(1-\varphi)^3}}\Rightarrow\\
\frac{\partial\varphi}{\partial x}=\sqrt{\frac{Hq_{ij}}{K}}\sqrt{2\varphi^3(1-\varphi)^3}\end{gathered}$$ and can express the surface energy as $$\begin{gathered}
\sigma_{ij}=\int_{-\infty}^\infty F\,dx=
\frac{K}{2}\int_0^1\frac{\partial\varphi}{\partial x}\,d\varphi\\
+Hq_{ij}\int_0^1\varphi^3(1-\varphi)^3
\frac{\partial x}{\partial\varphi}\,d\varphi\\
=\sqrt{HKq_{ij}} \int_0^1 \sqrt{2\varphi^3(1-\varphi)^3} \, d\varphi = a_1\sqrt{HKq_{ij}}.\end{gathered}$$ Then, $q_{ij}$ can be written as $$q_{ij}=\frac{\sigma_{ij}^2}{HKa_1^2},\, \text{ where }
a_1=\frac{3\sqrt2}{128}\pi.$$ and we rewrite Eq. (\[fb\]) as $$f_b=\frac{1}{HKa_1^2}\sum_{i<j}\sigma_{ij}^2f_{b, ij}.$$ Then constants $H$ and $K$ can be determined through a model interface width $W=\sqrt{K/H}$ and a maximal interface energy $\sigma_{\max}=\max_{ij}\sigma_{ij}=a_1\sqrt{KH}$. That is $K=W\sigma_{\max}/a_1$ and $H=\sigma_{\max}/(Wa_1)$. Using the above definitions we can write the barrier function as $$f_b=\sum_{i<j}\frac{\sigma_{ij}^2}{\sigma_{\max}^2}f_{b,ij}.$$ Therefore in our model we have different numerical interface widths $W_{ij}\sim 1/\sigma_{ij}$. If we change the interface energies $\sigma_{ij}$, then each particular interface width $W_{ij}$ will be changed automatically. We can also change the model interface width $W$ in accordance to the need of the numerical method.
Finally, we construct model functions $g_i(\bm{p})$ which will be used in the formulation of $f_c$. These functions should be equal to one on a vertex $V_i$ and 0 on other vertexes. They also should satisfy stability and flatness requirements (\[Stability1\]-\[Flatness2\]). Then we add a condition $g_i(0,\dots p_i,\dots,p_j,\dots 0)=1-g_j(0,\dots p_i,\dots,p_j,\dots 0)$ for $\bm{p}\in I_{ij}$, which comes from the thin-interface analyses.
We have found the model functions as polynomials of minimal power $$\begin{gathered}
g_i(\bm{p})=\frac{1}{2} p_i^2 \biggl( 15-25p_i+15p_i^2-3p_i^3\\
-15(1-p_i)\sum_{j\ne i} p_j^2\biggr).\end{gathered}$$ On all individual interfaces $I_{ij}$ the functions $g_i$ reduce to $p_i^3(10-15p_i + 6p_i^2)$ by taking into account the constraint $\sum_{k}p_k=1$.
To construct model functions $f_{b,ij}(\bm{p})$ (12) and $g_{i}(\bm{p})$ (23) in $N$ dimensional phase-field space we suggest to reduce the flatness (7-10) requirement to finite number of conditions. Therefore we assume that all our model functions should be respective symmetric polynomials of minimal power, where the symmetry is taken only such $p_k$, that $k\ne i$ for $g_i$ and $k\ne i,j$ for $f_{b,ij}$. We used the fundamental theorem for symmetric polynomial. That is, for every fixed power $S$ of polynomials $g_i$ and $f_{b,ij}$ we can represent them as finite expansion in terms of basic symmetric polynomials of power $s\leqslant S$. Due to the symmetry of respective derivatives we write Lagrange multiplier in a finite form. Then, from the flatness conditions (7-10) we get just a few equations instead of an undetermine number $N$. Thus solving the linear system of equations for respective coefficients of expansion, we were able to find $g_i$ and $f_{b,ij}$ which satisfy all requirements (7-10).
Evaluation of the smooth function for the mobility parameter
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Here, we propose a system mobility parameter $\tau^{-1}(\bm{p})$ which takes constant values $\tau_{ij}^{-1}$ on any individual interface $I_{ij}$ and smoothly varies on the Gibbs simplex $S$ and in neighborhood. The use of the mobility parameter $\tau^{-1}$ allows to consider any immobile interface $I_{ij}$ with a mobility parameter equal to zero $\tau^{-1}_{ij}=0$ instead of a relaxation time going to infinity $\tau_{ij} \rightarrow \infty$.
Let us identify a distance $s_{ij}$ between a point inside the Gibbs simplex $S$ and an interface $I_{ij}$ as $$\begin{aligned}
s_{ij}^2=\sum_{k\ne i,j} p_k^2+\frac{(p_i+p_j-1)^2}{2}.\end{aligned}$$ Then we can write the mobility parameter of the system as a function of $s_{ij}$ in the form $$\begin{aligned}
\tau^{-1}(\bm{p})=\sum_{i<j} \tau^{-1}_{ij} s^{-1}_{ij}/\sum_{i<j} s^{-1}_{ij}. \label{EqTau}\end{aligned}$$ The result is plotted in Fig. 1, where it can be seen that the function (\[EqTau\]) has prime lines on each dual interface as local minima. It will preserves the stability in the vicinity of interfaces.
The alternative simpler formula, which provides the similar simulation result, can be the following $$\tau^{-1}(\bm{p})=\sum_{i<j} \tau^{-1}_{ij}p^2_i p^2_j/\sum_{i<j} p^2_i p^2_j\,.$$
Evolution equation for concentration fields coupling with equation for phase fields
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To evaluate the equations for the concentration fields we consider a multi-component system, which contains $n$ chemical components ($A$, $B$, …) excluding the solvent. We identify an equilibrium composition vector as $\mathbf{A}_{i}$ with components $A_{i}^A(T)$ and equilibrium chemical free energies of phases $B_{i}^A(\mathbf{A}_{i},T)=f_{c,i}(\mathbf{A}_{i},T)$. These parameters can be defined by the common plane construction to the free energy functions of individual phases. If there are more then one equilibrium composition for a phase $i$ with respect to all other phases $j\neq i$ we will take a mean value of them.
For the definition of the driving force of the phase transformation we can write the mixture chemical free energy of a multi-component system as an interpolation between free energy functions of pure phases $f_{c,i}$ by using the second order Taylor expansion around the mixture equalibrium composition $c^{A,eq}=\sum_{i}A_{i}^Ag_{i}$ $$\begin{gathered}
f_c=\sum_i B_i g_i+\sum_A^n \mu^{A,eq}\bigl(c^A-c^{A,eq}\bigr)\\
+\sum_{A,B}^n\frac{X^{AB}}{2}\bigl(c^A-c^{A,eq}\bigr)\bigl(c^B-c^{B,eq}\bigr).
\label{LagrangeMulti}\end{gathered}$$ Here $\mu^{A,eq}$ are the components of the equilibrium diffusion potential vector of the system. These parameters are important only for derivation of model, but are not included in final evolution equations. $X^{AB}$ are components of the mixture thermodynamic factor matrix which can be defined through the thermodynamic factor matrix of phases $\mathbf{\hat{X}_{i}}$ as $$\mathbf{\hat{X}}^{-1}=\sum_{i}^N \mathbf{\hat{X}_{i}}^{-1}g_{i}.
\label{TF_Multi0}$$ The derivation of this relation is given in [@Kundin12b].
In the following we will use the mixture diffusion potential vector whose components are defined from the mixture chemical free energy as $$\mu^A=\frac{\partial f_c}{\partial c^A}=\mu^{A,eq}+\sum_{B}^nX^{AB}\left(c^B-c^{B,eq}\right). \label{TaylorMuMulti}$$ The mixture chemical free energy (\[LagrangeMulti\]) gives the thermodynamic driving force of the phase transformation. The phase-field evolution equation can be written in an explicit form $$\begin{gathered}
\tau(\bm{p})\frac{\partial p_i}{\partial t}=
W^2 \biggl(\nabla^2 p_i-\frac{1}{N}\sum_k^N \nabla^2 p_k\biggr)\\
-\biggl(\frac{\partial f_b(\bm{p})}{\partial p_i}-\frac{1}{N}\sum_k^N \frac{\partial f_b(\bm{p})}{\partial p_k}\biggr)\\
+\frac{1}{H}\sum_j^N\frac{\partial g_{j}}{\partial p_i}\biggm|_{\sum_k p_k=1} \left(\sum_A^n\mu^{A}A_{j}^A-B_{j}\right).\label{PhasefieldEq0}\end{gathered}$$
Using the mixture diffusion potential vector the diffusion equations for all chemical components transform to the following form $$\frac{\partial c^A}{\partial t}=\nabla \cdot \left[\sum_B^n M^{AB}(\bm{p})\nabla \mu^B-\bm{J}_{at}^A(\bm{p})\right],\label{B4}$$ where $M^{AB}$ are the components of the mobility matrix $ \mathbf{\hat{M}}=\mathbf{\hat{D}}\cdot \mathbf{\hat{X}}^{-1}$. The components of the diffusion matrix are defined as $D^{AB}(\bm{p})=\sum_i^N D_i^{AB}g_i$, where $D_i^{AB}$ are the terms of the diffusion matrix in a phase $i$. The values $\bm{J}_{at}^A$ are the anti-trapping currents for all components. Then Eq. (\[B4\]) can be modified by the multiplication with $X^{AB}$ and the summation over all components as the equation in terms of the diffusion potential $$\begin{gathered}
\frac{\partial \mu^A}{\partial t}=\sum_B^n X^{AB}\nabla \cdot \left[\sum_C^n M^{BC}\nabla \mu^C-\bm{J}_{at}^B(\bm{p})\right]\\
-\sum_B^n X^{AB}\sum_j^N\left(\frac{\partial g_{j}}{\partial t} A_{j}^B\right).\label{B5}\end{gathered}$$ Eqs. (30) and (32) are the evolution equations of the model.
Evaluation of the derivatives for model functions $g_i$
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The full derivatives of the model functions $g_{i}$ according to (\[EqLagrangeMulti\]a) are the following $$\begin{gathered}
\frac{\partial g_i}{\partial p_i}\biggm|_{\sum p_k=1}=\frac{\partial g_i}{\partial p_i}-\frac{1}{N}\sum_j\frac{\partial g_i}{\partial p_j},
\label{Deriv0}\\
\frac{\partial g_j}{\partial p_i}\biggm|_{\sum p_k=1}=\frac{\partial g_j}{\partial p_i}-\frac{1}{N}\sum_k\frac{\partial g_j}{\partial p_k},
\label{Deriv2}\end{gathered}$$ where $$\begin{gathered}
\frac{\partial g_i}{\partial p_i}=\frac{15p_i}{2}\Bigl((3p_i-2)\sum_{j\neq i} p_j^2-( 1-p_i)^2(p_i-2)\Bigr),\label{Deriv3a}\\
\frac{\partial g_i}{\partial p_j}=- 15p_i^2 (1-p_i)p_j,\label{Deriv3b}\\
\sum_{j\neq i}\frac{\partial g_i}{\partial p_j}=-15p_i^2(1-p_i)\sum_{j\neq i}p_j=-15p_i^2(1-p_i)^2. \label{Deriv3c}\end{gathered}$$ That yields $$\frac{\partial g_{i}}{\partial p_i}\biggm|_{\sum p_k=1}= \frac{N-1}{N}\frac{\partial g_i}{\partial p_i}+ \frac{15}{N}p_i^2 (1-p_i)^2,$$ $$\begin{gathered}
\frac{\partial g_{j}}{\partial p_i}\biggm|_{\sum p_k=1} = -\frac{1}{N}\frac{\partial g_j}{\partial p_j}+ \frac{15}{N}p_j^2 (1-p_j)^2\\
- 15p_j^2 (1-p_j)p_i,\end{gathered}$$ After substitution of (\[Deriv3a\]) and rearranging we have
$$\begin{gathered}
\label{Deriv4}
\frac{\partial g_i}{\partial p_i}\biggm|_{\sum p_k=1}=\frac{15(N-1)}{2N}p_i
\Bigl((3p_i-2)\sum_{j\neq i} p_j^2\\
+(3p_i+2)(1-p_i^2)\Bigr)-\frac{15(2N-3)}{N} p_i^2 (1-p_i)^2,\end{gathered}$$
$$\begin{gathered}
\label{Deriv5}
\frac{\partial g_j}{\partial p_i}\biggm|_{\sum p_k=1}=-\frac{15}{2N}p_j
\Bigl((3p_j-2)\sum_{k\neq j}p_k^2
+(3p_j+2)(1-p_j^2)\Bigr)\\
+15\frac{3}{N}p_j^2(1-p_j)^2-15p_j^2(1-p_j)p_i,\end{gathered}$$
Using Eqs. (\[Deriv4\]) and (\[Deriv5\]) it can be derived that for any individual interface $I_{ij}$ these derivatives are independent of $N$ and will be reduced to $15\, p_i^2 (1-p_i^2)$ and $-15\, p_j^2 (1-p_j^2)$, respectively.
Relation to the previous model
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Let us notice that if two polynomials are equal to each other at the Gibbs simplex $S$, then they are equivalent in our model, because they have the same derivatives projected at $S$. Keeping it in mind, we found that our functions $g_i$ are equivalent to the analogical functions suggested by Folch and Plapp in [@Folch05] for a 3-phase system. Moreover, the derivatives $\frac{\partial g_{i}}{\partial p_i}\vert_{\sum p_k=1}$ completely coincide with the derivatives of functions $g_i$ in Ref. [@Folch05] . The derivatives $\frac{\partial g_{j}}{\partial p_i}\vert_{\sum p_k=1}$ can be written in the form $$\begin{gathered}
\frac{\partial g_j}{\partial p_i}\biggm|_{\sum p_k=1}=-\frac{1}{(N-1)}
\frac{\partial g_j}{\partial p_j}\biggm|_{\sum p_k=1}\\
+\frac{15}{(N-1)}p_j^2(1-p_j)\Bigl(\sum_{k\neq i,j}p_k-(n-2)p_i\Bigr).\end{gathered}$$ For a three-phase system it will be reduced to $$\frac{\partial g_j}{\partial p_i}\biggm|_{\sum p_k=1}=-\frac{1}{2}\frac{\partial g_j}{\partial p_j}\biggm|_{\sum p_k=1}\!\!\!\!\!\!\!\!
+\frac{15}{2}p_j^2(1-p_j)(p_k-p_i),$$ and is similar to the derivatives of functions $g_j$ in Ref. [@Folch05], too. There is the typo in the Eq. (3.25) of the Ref. [@Folch05], where instead of $\frac{\partial g_j}{\partial p_j}$ it is written $\frac{\partial g_i}{\partial p_i}$. If one will use this equation with typo, then the wrong unexpected behavior of Folch and Plapp model can be observed. It can lead to the additional nucleation of a phase $p_{i\ne j,k}$ on the individual interface $I_{jk}$.
Numerical tests to check Young’s law, influence of mobility parameters, and absence of ghost phases {#Test_Young}
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The phase-field model should asymptotically convert towards a sharp interface model when the thickness of interface goes to zero. The corresponding sharp-interface model should fulfill the Young’s law. The aim of this section is to test how does our model fulfill the Young’s law in the case of different interface energies and mobility parameters.
In all simulations we used the chemical free energy $f_c=0$ to test the influence of interface energies and mobility parameters and to test whether “ghost” phases exist or not. Here, all simulations were done in a two dimensional coordinate space having 128x128 discrete points with the fixed boundary conditions. During the tests we measured the position of the interfaces as a mixture of two phases and the position of triple points as the mixture of three phases. For simulations in 2D space we did not find points having the mixture of four phases. In Fig. 2 the initial states for all tests are shown.
The initial state of test 1 is shown in Fig. 2(a). We investigated the evolution of three phases determined in the three-dimensional phase-field space with similar interface energies and mobility parameters $\sigma_{12}=\sigma_{13}=\sigma_{23}$, $\tau_{12}=\tau_{13}=\tau_{23}$. Then we carried out test 2 for four different phases determined in the four-dimensional phase-field space with similar initial state and the same parameters Fig. 2(b). The results of the simulation are shown in Fig. 3. We found that both systems have the same dynamic evolution with a good accuracy as it was expected. In the static equilibrium we got in the three-phase junction angles of 120$^\circ$ with a high accuracy, which satisfies the Young’s law. We can also see clearly in these figures that the numerical interface widths are constant and we do not have “ghost” phases on each individual interface.
In Fig. 4 the dynamic of triple points is shown for tests 1 and 2. We have found that interfaces (mixture of two phases) in these tests are very close to straight lines. In this figure we show numerical assymptotic limits which are very closed (within 0.1% of simulation box) to the analytical limits calculated by using Young’s law.
\[Fig2\]
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 (a)  (b)
 (c)  (d)
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\[Fig3\]
In tests 3 and 4 we have simulated three different phases defined in a three dimensional phase-field space. The system has different interface energies with ratios $\sigma_{12}:\sigma_{13}:\sigma_{23}=0.5:0.75:1$. In test 3 we have equal mobility parameters $\tau_{12}=\tau_{13}=\tau_{23}$ and in test 4 we have different mobility parameters for each interface with the following ratios $\tau_{12}:\tau_{13}:\tau_{23}=2:1:(4/3)$. Therefore for tests 3 and 4 we have a different evolution of the systems, but the same static equilibrium state. In the initial state of both tests the angles between the interfaces in the triple point are $\gamma_{12}=\gamma_{13}=\gamma_{23}=120^\circ$ as shown in Fig. 2(c). According to Young’s law the respective angles in the static equilibrium should follow the ratios $\sin\gamma_{12}:\sin\gamma_{13}:\sin\gamma_{23}=\sigma_{12}:\sigma_{13}:\sigma_{23}$. From this ratio we can calculate the equilibrium angles as $\gamma_{12}=151^\circ$, $\gamma_{13}=133.5^\circ$, $\gamma_{23}=75.5^\circ$. Therefore we got different numerical interface widths $W_{ij}$, which are inverse proportional to the corresponding interface energies $\sigma_{ij}$, as it was expected. Test 4 is shown in Fig. 5 in a similar way to Fig. 3, except of a fourth column where the shifted mobility parameter $(\tau^{-1}(\bm{p})-0.5)/2$ is shown. From this figure we can see that the Eq. (\[EqTau\]) gives us $\tau^{-1}(\bm{p})=\tau_{ij}^{-1}$ on every individual interface $I_{ij}$ with very good accuracy.
To check the difference in the evolution for tests 3 and 4 more precisily, we show the evolution of the triple point coordinates in Fig. 6 with numerical assymptotic limits which are very close (within 0.1% of simulation box) to analytical limits calculated by using Young’s law.
\[Fig4\]
\[Fig5\]
In tests 5, 6 (Figs. 7, 8) we checked the influence of mobility parameters on the evolution of the system. That is how quickly an interface will be straightened and what is the evolution of the triple point. The initial configuration is shown in Fig. 2(d). All initial interfaces have the central symmetry and the same length. We used an equal interface energies $\sigma_{12}=\sigma_{13}=\sigma_{23}$ in both tests. For test 5 we used an equal inverse mobility $\tau_{12}=\tau_{13}=\tau_{23}$ and for test 6 we have $\tau_{12}:\tau_{13}:\tau_{23}=2:1:4/3$. Therefore in test 5 we got the evolution with the central symmetry in the triple point, whereas in test 6 the symmetry is broken as it was expected. We show the evolution of triple points for both tests in Fig. 7 and the deviance from straight lines for different interfaces in Fig. 8. We have shown that numerical assymptotic limits for both tests are very close (within 0.1% of simulation box) to predicted values by the Young’s law.
The triple point in test 5 have a small deviation from the initial configuration due to numerical errors. We also got similar deviance from straight lines for different interfaces $D_{12}=D_{13}=D_{23}$ with high accuracy in this test. The picture is different for test 6 due to assymetric mobility parameters.
\[Fig7\]
Numerical test for a four-phase reaction {#Al-Cu-Ni}
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Alloy system and model parameters
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For the numerical test we have chosen a ternary Al-Cu-Ni alloy in the Al reach corner of the phase diagram. A zoomed view of the Al reach corner in Fig. 9 shows the liquidus surface, the boundary curves and the regions where NiAl$_3$, Ni$_2$ Al$_3$ and (Al) phases solidify firstly, which we identify as $\alpha$, $\beta$ and $\gamma$ phases respectively. Between the $\alpha$ and $\beta$ phases there is the peritectic line $p_5$. Between $\alpha$ and $\gamma$, $\beta$ and $\gamma$ phases there are eutectic lines. The three-phase peritectic reaction ($p_5$) and four-phase reactions ($U_5$) and ($U_7$) are indicated.
\[Fig8\]
As an example we consider the solidification of an alloys with the initial concentration of the liquid of 11 at$\%$Ni - 4.5 at$\%$Cu identified as an orange point 1 in the phase diagram in Fig. 9. In this alloy crystals of primary $\alpha$-phase will begin to precipitate at 750$^\circ$C as the temperature is lowered. If cooling continues, the composition of the liquid will change towards the boundary curve $p_5$. When the composition reaches the boundary curve at 605$^\circ$C (the point 2 in phase diagram), crystals of $\beta$-phase will precipitate along with crystals of $\alpha$-phase. With further cooling, the liquid will change its composition along the boundary curve $e_3$ towards the point $U_5$ while the liquid produce crystals of the $\gamma$-phase.
The main interest is the four-phase reaction in the point $U_5$ (604$^\circ$C) where crystals of $\gamma$ will begin to precipitate along with the $\beta$ phase while the liquid reacts with some of the crystals of the $\alpha$-phase $$\text{L} + \text{Ni}\text{Al}_3(\alpha)\rightarrow \text{Ni}_2\text{Al}_3(\beta) + \text{(Al)}(\gamma).\label{Reaction}$$ Four various phases coexist in this point. This type of reaction (which in many ways is equivalent to the peritectic point on binary diagrams) is known as the tributary reaction point (because it looks like a point where two tributaries of a river meet). The product phase $\beta$ can precipitate along with another product phase $\gamma$ on the primary phase $\alpha$, or $\beta$ may be a potent nucleant for $\gamma$, too. The microstructure formation of such a ternary alloy during the directional solidification is of great interest.
The material and model parameters considered in the simulations are listed in Table \[Tabel1\].
Parameter value used
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$\tau$ (system time scale) $1 \times 10^{-6}$ s
$l_0$ (system length scale) 1.3$\times 10^{-8}$ m
$\Delta x/l_0$ (grid discretization size) $1$
$\Delta t/\tau$ (time step) $0.025$
$W/l_0$ (interface width) $1.1 $
$D_L^{Ni}$ (diffusion in liquid phase) 1.2 $\times10^{-9}$ m$^2$/s
$D_L^{Cu}$ (diffusion in liquid phase) 0.8 $\times10^{-9}$ m$^2$/s
$D_S$ (diffusion in solid phase) $0.01 D_L$
$T_{U_5}$(4-phase reaction temperature) 604 $^\circ$C
$\sigma$ (surface energy) 0.30 J/m$^2$
: Material parameters and phase-field model parameters used in the simulation.[]{data-label="Tabel1"}
From the Gibbs free energy functions of phases we estimated the values of the equilibrium concentrations $A^{A}_{i}$, the equilibrium energies $B_{i}$ and the thermodynamic factors $X_{i}^{AB}$ at the temperature 600$^\circ$C which is below the reaction point $U_5$. The thermodynamic parameters of the alloy system are presented in Table 2.
----------- --------------- --------------- ---------- ----------------- ------------------ ----------------- ------------------
Phase $A^{Ni}_{i}$, $A^{Cu}_{i}$, $B_{i}$, $X_{i}^{Ni}$, $X_{i}^{NiCu}$, $X_{i}^{Cu}$, $X_{i}^{CuNi}$,
at$\%$ at$\%$ J/mol-at J/mol-at J/mol-at J/mol-at J/mol-at
$L$ 5.0898 6.5227 -46045 $ 4.7\cdot10^5$ $ 1.87\cdot10^5$ $ 5.0\cdot10^5$ $ 1.87\cdot10^5$
$\alpha $ 25.0 0.0 -72800 $4.0\cdot10^7$ - $4.0 \cdot10^7$ -
$\beta $ 20.332 18.768 -76910 $7.1 \cdot10^5$ $6.0 \cdot10^5$ $6.3 \cdot10^5$ $6.8 \cdot10^5$
$\gamma $ 0.2 0.2 -74775 $7.2\cdot10 ^5$ - $7.2 \cdot10^5$ -
----------- --------------- --------------- ---------- ----------------- ------------------ ----------------- ------------------
\[Table2\]
Simulation results
------------------
The four phase reaction were simulated at the constant temperature 600$^\circ$. Equations (\[PhasefieldEq0\]) and (\[B5\]) were solved numerically using the Euler method in the cubic 2D simulation box of size $200\,\Delta x$. The derivatives of the model functions $g_i$ were calculated according to Eqs. (\[Deriv4\]) and (\[Deriv5\]) with $N=4$. The simulations were started with an initial crystal of the $\alpha$-phase of radius $12\, \Delta x$. After 120 steps a nuclei of the $\beta$-phase was inserted at a random site on the solid-liquid boundary of the parent $\alpha$-phase and after 150 steps a nuclei of the $\gamma$-phase was inserted in the triple point of the $\alpha$-, $\beta$- and liquid phases.
Results of the evolution of the microstructure at various time steps are shown in Fig. 10 (a-d). Three solid phases grow from an initial multi-phase nuclei forming the two-phase boundaries. The growth velocity of the $\alpha$-phase is slower than the growth velocity of the $\beta$- and $\gamma$-phases due to the higher Gibbs free energy, so that with increasing time the product phases overgrow the crystal of the parent $\alpha$-phase. The chosen model functions serve the stability of the solution and the absence of a third phase on individual interfaces. The nucleation of the $\gamma$-phase occurs in the triple point of phases $\alpha$, $\beta$ and liquid. No nucleation of the third phase on individual interfaces can be observed even for the larger undercooling. The lamellar-like structure forms by the overgrowing of one phase over another one.
\[Fig9\]
The time evolution of the Ni and Cu concentration is shown in Fig. 10 (e-l). The composition is initially homogenious until the evolution of the phase field causes the redistribution of the alloy components between the phases. It can be seen that the concentration of Ni and Cu in $\alpha$-phase is larger near the $\alpha$/$\gamma$, $\alpha$/$\beta$ boundaries and smaller at the $\alpha$-liquid boundary. This phenomenon can be clear explained by the growth of the $\alpha$-phase in the direction of the liquid-phase. The same inhomogeneities in the composition can be observed in the $\beta$-phase by the comparison of the $\beta$-liquid boundary and $\beta$-$\gamma$ boundaries.
To proof the ability of the $N$-phase model to produce the nucleation of the third phase we carried out the simulation with an additional thermal noise [@Karma99] in the kinetic equation: $$\xi_i=r\,\frac{RT}{H}\frac{15}{(N-1)}\sum_{k,j\neq k}^n\,p_k^2(1-p_k) p_j.$$ where $r\in [-0.5;0.5]$ is a random value, $RT$ is the amlitude of thermal fluctuation, and $H$ is responsible for the surface energy. Due to this term a phase $i$ can nucleate heterogeneously on the interface between phases $j$ and $k$ and grow if the energetic conditions i.e. the concnetration distribution and the surface energy are favorable. The physical meaning is that the nucleation barrier can be overcome if the driving force is large enough. The results of the simulation are shown in Fig. 11. It can be observed that new thin $\beta$- and $\gamma$-phases form on the $\beta$-liquid and $\gamma$-liquid boundaries, respectively. After increasing time, if the $\alpha$ phase is closed and only $\beta$ and $\gamma$-phases grow, the resulting microstructure is similar to the eutectic lamellae structure. The same microstructure evolution were produced by the simulation of the eutectic reaction by means of the three-phase model of Folch and Plapp. The corresponding examples can be found in the works [@Kundin12; @Ebrahimi12].
\[Fig10\]
The comparison of the time evolution of the phase fractions for two simulated case is presented in Fig. 12. The thermal noise triggers the nucleation and produces the stable and uniform growth of the $\beta$- and $\gamma$-phases with increasing growth velocity. It can be shown in the figure that without the nucleation the lamellae of one phase overgrow the lamellae of the other phase and proceed to grow along the solid/liquid boundary where the concentration values are favorable. So far the system should wait for the moment when one phase grow enough to go around the partner phase. In the case of the additional thermal noise new thin lamellae nucleate at the the solid/liquid boundaries of the partner phase immediately after reaching the favorable concentration conditions. The amplitude of the nucleation can be adjusted in accordance to the experimental microstructure.
\[Fig11\]
---------------------
---------------------
Notice that we can insert a nuclei arbitrary on a solid/liquid interface and the nucleation will occur if the driving force for the nucleation is large enough. But in the present model the nucleation occurs at the right place and at the favorable energetic conditions. Moreover, the additional terms in the model allow to reduce or increase the nucleation barrier, in particular the barrier can be increased in the triple point and prevent the nucleation of the fourth phase in this points as it was shown in numerical tests above.
Conclusion
==========
In this work we have formulated a general phase-field model in $N$-dimensional phase-field space. The main point of the model is a spatial constructed smooth total free energy functional for an arbitrary number of phases which satisfies the requirements of the stability and flatness on all individual interfaces and vertexes. For this aim the special model functions being responsible for the interface energy barrier and for the chemical driving force of the transformation are proposed which satisfy these requirements and allow to take into account the anisotropy of the interface energies and mobility parameters.
The ability of the model to follow the Young’s law and the dynamics of the system evolution is tested by the investigation of the phase-field interactions at the interfaces and the multiple junctions. It was found that the nucleation of new phases can be controlled by the additional terms in the phase-field evolution equations. The applicability of the model to multicomponent and multiphase systems was verified by the quantitative simulation of the microstructure evolution in a ternary Al-Cu-Ni alloy in the presence of the four-phase peritectic-like reaction. It was shown that after the four-phase reaction the three-phase reaction occures and a morphology similar to the lammelar structure is developed. Furthermore, the type of morphology and growth rate of the crystals can be controlled by the thermal noise term added to the phase-field evolution equation.
In our future work the presented model will be used for the investugation of the microstructure evolution for various alloy compositions during cooling with various cooling rates and temperature gradients.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thanks M. Flack and D. Pilipenko from MPS of University Bayreuth for valuable discussions. The research was supported by German Research Foundation in the scope of SPP 1296.
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|
---
author:
- |
E. A. Paschos\
Theoretische Physik III, University of Dortmund, D-44221 Dortmund, Germany\
E-mail:
- |
I. Schienbein\
Laboratoire de Physique Subatomique et de Cosmologie,\
Université Joseph Fourier/CNRS-IN2P3,\
53 Avenue des Martyrs, F-38026 Grenoble, France\
E-mail:
- |
J.-Y. Yu\
Southern Methodist University, Dallas, Texas 75275, USA\
E-mail:
title: 'Single pion electro– and neutrinoproduction on heavy targets'
---
Introduction
============
Neutrino interactions at low and medium energies are attracting attention because they will be measured accurately in the new generation of experiments [@Drakoulakos:2004gn; @Mahn:2006ac; @*Gallagher:2006ab; @*Giacomelli:2006xn]. One aim of the experiments is to measure the precise form of the cross sections and their dependence on the input parameters. This way we check their couplings and compare the functional dependence of the form factors, where deviations from the dipole dependence have already been established (see e.g. figure 1 in [@Paschos:2003qr] and references therein). Deviation from the standard model predictions can arise either from properties of the neutrinos or from new couplings of the gauge bosons to the particles in the target. Another aim of the experiments is to establish the properties of neutrinos including their masses, mixings and their fermionic nature (Dirac or Majorana particles). This program requires a good understanding of the cross sections, which motivated a new generation of calculations. Since the experiments use nuclear targets, like ${\rm C}^{12},\, {\rm O}^{16},\, {\rm Ar}^{40},\, {\rm Fe}^{56}, ...$ it is necessary to understand the modifications brought about by the targets.
The very old calculations for quasi-elastic scattering and resonance excitation on free nucleons [@Rein:1981wg; @Schreiner:1973mj] have been replaced by new results where couplings and form factors are now better determined. For the vector couplings comparisons with electroproduction data have been very useful [@Paschos:2003qr; @Lalakulich:2005cs; @*Sato:2003rq]. Axial couplings are frequently determined by PCAC. There are already improvements and checks of the earlier quark models [@Lalakulich:2006sw]. Comparisons with experimental data are also available even though the experimental results are not always consistent with each other [@Grabosch:1989gw; @Barish:1979pj; @*Radecky:1982fn; @Kitagaki:1986ct] but there are plans for improvements that will resolve the differences [@Drakoulakos:2004gn; @Mahn:2006ac].
For reactions on nuclear targets there are modifications brought about by the propagation of the produced particles in the nuclear medium. They involve absorption of particles, restrictions from Pauli blocking, Fermi motion and charge-exchange rescatterings. One group of papers uses nuclear potentials for the propagation of the particles [@Alvarez-Ruso:2003gj]. Others use a transport theory of the final particles including channels coupled to each other [@Leitner:2006sp]. These groups gained experience by analyzing reactions with electron beams (electroproduction) and adopted their methods to neutrino reactions [@Leitner:2006sp].
Our group investigated 1-$\pi$ pion production on medium and heavy targets employing the pion multiple scattering model by Adler, Nussinov and Paschos [@Adler:1974qu] that was developed in order to understand neutral current neutrino interactions with nuclei. This model was useful in the discovery of neutral currents and has been applied to predict neutrino-induced single pion production on Oxygen, Argon and Iron targets [@Paschos:2000be; @Paschos:2001np; @Paschos:2004qh] which are used in long baseline(LBL) experiments. Among its characteristics is the importance of charge-exchange reactions that modify the $\pi^+:\pi^0:\pi^-$ ratios of the original neutrino-nucleon interaction through their scatterings within the nuclei. The presence of this effect has been confirmed by experiments [@Musset:1978gf]. We note here that our results are valid for isoscalar targets. For non-isoscalar targets like lead, used in the OPERA experiment, it is possible to extend the ANP model [@Adler:1974wu], which can be done in the future.
In this article we take an inverse route and use our calculation in neutrino reactions to go back to the electroproduction of pions on free nucleons and heavy nuclei. The plan of the paper is as follows. In section \[sec:sec2\] we summarize the neutrino production cross sections on free nucleons and in the $\Delta$ resonance region. This topic has been described by several groups in the past few years. We present cross sections differential in several variables $E_\pi,\, Q^2$ and $W$. We pay special attention to the spectrum $d\sigma/dE_\pi$, where we correct an error we found in our earlier calculation [@Paschos:2000be]. Then we obtain the electroproduction cross section by setting the axial coupling equal to zero and rescaling, appropriately, the vector current contribution.
The main content of the article appears in section \[sec:sec3\] where we describe the salient features and results of the ANP model. This model has the nice property that it can be written in analytic form including charge exchange and absorption of pions. This way we can trace the origin of the effects and formulate quantities which test specific terms and parameters. As we mentioned above several features have been tested already, and we wish to use electroproduction data in order to determine the accuracy of the predictions. We present numerical results for different target materials, and study the quality of the averaging approximation and uncertainties of the ANP model due to pion absorption effects. We discuss how the shape of the pion absorption cross section (per nucleon), an important and almost unconstrained ingredient of the ANP model, can be delineated from a measurement of the total fraction of absorbed pions. Finally, in Sec. \[sec:summary\] we summarize the main results. Averaged rescattering matrices for carbon, oxygen, argon, and iron targets and for different amounts of pion absorption have been collected in the appendices and are useful for simple estimates of the rescattering effects.
Free nucleon cross sections {#sec:sec2}
===========================
In the following sections, leptonic pion production on nuclear targets is regraded as a two step process. In the first step, the pions are produced from constituent nucleons in the target with free lepton-nucleon cross sections [@Adler:1974qu]. In the second step the produced pions undergo a nuclear interaction described by a transport matrix. Of course, the resonances themselves propagate in the nuclear medium before they decay, an effect that we will investigate in the future.
The leptonic production of pions in the $\Delta$-resonance region is theoretically available and rather well understood as described in articles for both electro- and neutrino production, where comparisons with available data are in good agreement [@Paschos:2003qr; @Lalakulich:2005cs; @*Sato:2003rq; @Lalakulich:2006sw; @Alvarez-Ruso:1998hi; @Leitner:2006sp]. The available data is described accurately with the proposed parameterizations. The vector form factors are modified dipoles [@Paschos:2003qr] which reproduce the helicity amplitudes measured in electroproduction experiments at Jefferson Laboratory [@Lalakulich:2006sw]. The coupling in the axial form factors are determined by PCAC and data. Their functional dependence in $Q^2$ is determined by fitting the $\frac{{\ensuremath{{\operatorname{d}}}}\sigma}{{\ensuremath{{\operatorname{d}}}}Q^2}$ distributions. For the vector form factors the magnetic dipole dominance for $C_3^V(q^2)$ and $C_4^V(q^2)$ gives an accurate description of the data. However, deviations with a non-zero $C_5^V(q^2)$ have also been established [@Lalakulich:2006sw]. This way a small (5%) isoscalar amplitude is reproduced.
For the propose of this article we shall use a scaling relation connecting neutrino- to electroproduction. The weak vector current is in the same isospin multiplied with the electromagnetic current and the two are related as follows: $$\begin{aligned}
<\Delta^{++}| V |p> = \sqrt{3}<\Delta^{+}|J_{em}^{I=1}|p>
= \sqrt{3}<\Delta^{0}|J_{em}^{I=1}|n> \ . \nonumber\end{aligned}$$ Taking into account the isospin Clebsch-Gordan factors for the $\Delta \rightarrow N\pi$ branchings one finds the following contributions of the $\Delta$-resonance to the cross sections for $e p \to e p \pi^0$, $e p \to e n \pi^+$, $e n \to e p \pi^-$ and $e n \to e n \pi^0$ $$\begin{aligned}
\frac{{\ensuremath{{\operatorname{d}}}}\sigma^{em,I =1}}{{\ensuremath{{\operatorname{d}}}}Q^2 {\ensuremath{{\operatorname{d}}}}W} =
\frac{8}{3}\frac{\pi^2}{G_F^2}\frac{\alpha^2}{Q^4}
\frac{{\ensuremath{{\operatorname{d}}}}V^{\nu}}{{\ensuremath{{\operatorname{d}}}}Q^2 {\ensuremath{{\operatorname{d}}}}W} \times
\begin{cases}
\frac{2}{3} &: p \pi^0 \\
\frac{1}{3} &: n \pi^+ \\
\frac{1}{3} &: p \pi^- \\
\frac{2}{3} &: n \pi^0
\end{cases}
\label{eq:em}\end{aligned}$$ where $\tfrac{\rm{d} V^{\nu}}{\rm{d} Q^2 \rm{d} W}$ denotes the cross section for the vector contribution alone to the reaction $\nu p \to \mu^- p \pi^+$. The free nucleon cross sections in Eq. will be used in our numerical analysis. We shall call this the reduced electromagnetic formula. Its accuracy was tested in figure (5) of ref. [@Paschos:2003qr]. Further comparisons can be found in [@Paschos:2004md].
For studies of the pion angular distributions (or what is the same of the pion energy spectrum in the laboratory frame) we begin with the triple differential cross section for neutrino production $$\begin{aligned}
\frac{{{\ensuremath{{\operatorname{d}}}}} \sigma}{{{\ensuremath{{\operatorname{d}}}}}Q^2 {{\ensuremath{{\operatorname{d}}}}}W {{\ensuremath{{\operatorname{d}}}}}\cos{\theta}_\pi^\star}
&=& \frac{W G_F^2}{16\pi M_N^2} \sum_{i=1}^3\big(K_i \widetilde{W}_i -\frac{1}{2}
K_i D_i (3\cos^2\theta_\pi^\star-1)\big)
\label{eq:electro}\end{aligned}$$ with $K_i$ being kinematic factors of $W$ and $Q^2$ and the structure functions $\widetilde{W}_i(Q^2,W)$ and $D_i(Q^2,W)$ representing the dynamics for the process. All of them are found in ref. [@Schreiner:1973mj]. The angle $\theta_\pi^\star$ is the polar angle of the pion in the CM frame with $$\cos{\theta}_\pi^\star = \frac{-\gamma E_\pi^{\rm CMS}
+E_\pi}{\beta\gamma|\vec{p}_\pi^{\ \rm CMS}|}$$ where $$|\vec{p}_\pi^{\ \rm CMS}| = \sqrt{(E_\pi^{\rm CMS} )^2 - m_\pi^2}
\quad {\text with} \quad
E_\pi^{\rm CMS} = \frac{W^2 + m_\pi^2 - M_N^2}{2 W}$$ and the rest of the variables defined as $$\quad \nu = \frac{W^2 + Q^2 - M_N^2}{2 M_N}\, ,\,
\gamma = \frac{\nu + M_N}{W}\, , \,
\beta \gamma = \frac{\sqrt{\nu^2 + Q^2}}{W}.
\label{eq:gam}$$ It is now straight-forward to convert the cross section differential in the solid angle to the one differential in the laboratory energy of the pion, $E_\pi$, $$\frac{d \sigma}{d E_\pi} = \frac{1}{\gamma\beta|\vec{p}_\pi^{\ \rm CMS}|}\frac
{d \sigma}{d\cos\theta_\pi^\star}\, .$$ Having expressed all quantities in and in terms of $W,\, Q^2$ and $E_\pi$ it is possible to compute the pion energy spectrum $$\frac{d \sigma}{d E_\pi} =
\int_{W_{\rm min}}^{W_{\rm max}} dW \
\int _{Q^2_{\rm min}}^{Q^2_{\rm max}} dQ^2 \
\frac{d \sigma}{dQ^2 dW dE_\pi}\ \theta(phys). \label{eq:pi}$$ The limits of integration are given as $$\begin{aligned}
Q^2_{\rm min} &=& 0\, , \quad Q^2_{\rm max} = \frac{(S-W^2)(S-M_N^2)}{S},
\nonumber\\
W_{\rm min} &=& M_N + m_\pi
\, , \quad
W_{\rm max} \simeq 1.6\ {\rm GeV}
\label{eq:q2}\end{aligned}$$ where $S=M_N^2 + 2 M_N E_1$ is the center-of-mass energy squared with $E_1$ the energy of the incoming lepton in the LAB system. The $\theta$-function takes care of the constraints from the phase space. We integrated the cross section for $E_\nu = 1\ {\rm GeV}$ and show the spectrum in figures \[fig:figoxygen\]–\[fig:figiron\]. In our earlier publication [@Paschos:2000be] the spectrum for $E_\pi$ was incorrect because we did not impose the phase space constraints correctly. The pion spectrum for charged current reactions is correctly reported in figure (4) in ref. [@Paschos:2003ej]. The discrepancy in ref. [@Paschos:2000be] has been pointed out for neutral currents in ref [@Leitner:2006sp]. The neutrino–nucleon and electron–nucleon cross sections will be used in the rest of this article in order to compute and test effects of nuclear corrections. We deduce the electroproduction cross sections from neutrino production as in Eq. . For the triple differential cross section we follow the same procedure by setting the axial form factors to zero and using the relation $$\begin{aligned}
\frac{{\ensuremath{{\operatorname{d}}}}\sigma^{em,I =1}}{{\ensuremath{{\operatorname{d}}}}Q^2 {\ensuremath{{\operatorname{d}}}}W {\ensuremath{{\operatorname{d}}}}E_\pi} =
\frac{8}{3}\frac{\pi^2}{G_F^2}\frac{\alpha^2}{Q^4}
\frac{{\ensuremath{{\operatorname{d}}}}V^{\nu}}{{\ensuremath{{\operatorname{d}}}}Q^2 {\ensuremath{{\operatorname{d}}}}W {\ensuremath{{\operatorname{d}}}}E_\pi} \times
\begin{cases}
\frac{2}{3} &: e p \to e p \pi^0 \\
\frac{1}{3} &: e p \to e n \pi^+ \\
\frac{1}{3} &: e n \to e p \pi^- \\
\frac{2}{3} &: e n \to e n \pi^0
\end{cases}
\label{eq:em1}\end{aligned}$$ A small isoscalar part in the electromagnetic cross section is omitted since it does not contribute to the $\Delta$-resonance but only to the background, which for $W<1.3\ {\rm GeV}$ is small and contributes for $1.3\ {\rm GeV}<W<1.4\ {\rm GeV}$.
Cross sections for heavy targets {#sec:sec3}
================================
In the following we will deal with single pion resonance production in the scattering of a lepton $l$ off a nuclear target $T$ ($_6C^{12},\,_8O^{16},\, _{18} Ar^{40},\, _{26} Fe^{56}$), i.e., with the reactions $$l +T\rightarrow l' +T^\prime +\pi^{\pm,0}
\label{eq:reaction}$$ where $l'$ is the outgoing lepton and $T^\prime$ a final nuclear state. Furthermore, in our analysis of nuclear rescattering effects we will restrict ourselves to the region of the $\Delta(1232)$ resonance, $1.1\ \gev < W < 1.4\ \gev$, and to isoscalar targets with equal number of protons and neutrons.
Pion rescattering in the ANP model
----------------------------------
According to the ANP model [@Adler:1974qu; @Schienbein:2003xy] the final cross sections for pions ${\ensuremath{(\pi^+,\pi^0,\pi^-)_f}}$ can be related to the initial cross sections ${\ensuremath{(\pi^+,\pi^0,\pi^-)_i}}$ for a [*free nucleon*]{} target in the simple form $$\left(\begin{array}{c}\displaystyle
{{\ensuremath{{\operatorname{d}}}}\sigma(_ZT^A;{\pi^+})\over {\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}\\
\displaystyle{{\ensuremath{{\operatorname{d}}}}\sigma(_ZT^A;{\pi^0})\over
{\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}\\
\displaystyle{{\ensuremath{{\operatorname{d}}}}\sigma(_ZT^A;{\pi^-})\over
{\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}
\end{array}\right)_{\rm f}
= M[T; Q^2,W]\
\left(\begin{array}{c}\displaystyle
{{\ensuremath{{\operatorname{d}}}}\sigma(N_T;{\pi^+})\over {\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}\\
\displaystyle{{\ensuremath{{\operatorname{d}}}}\sigma(N_T;{\pi^0})\over {\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}\\
\displaystyle{{\ensuremath{{\operatorname{d}}}}\sigma(N_T;{\pi^-})\over {\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}
\end{array}\right)_{\rm i}
\label{eq:fac}$$ with $${{\ensuremath{{\operatorname{d}}}}\sigma({N_T};\pm 0)\over {\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}
= {{Z}{{\ensuremath{{\operatorname{d}}}}\sigma({p};\pm 0)
\over {\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}
+ {(A-Z)}{{\ensuremath{{\operatorname{d}}}}\sigma({n};\pm 0)
\over {\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}}
\label{eq:free}$$ where the free nucleon cross sections are averaged over the Fermi momentum of the nucleons.[^1] For an isoscalar target the matrix $M$ is described by three independent parameters $A_p$, $d$, and $c$ in the following form [@Adler:1974qu] $$M = A_p
\left(\begin{array}{ccc}
1-c-d & d & c
\\
d & 1-2 d & d
\\
c & d & 1-c-d
\end{array}\right)\ , \label{eq:M}$$ where $A_p(Q^2,W) = g(Q^2,W)\times f(1,W)$. Here, $g(Q^2,W)$ is the Pauli suppression factor and $f(1,W)$ is a transport function for equal populations of $\pi^+, \pi^0, \pi^-$ which depends on the absorption cross section of pions in the nucleus. The parameters $c$ and $d$ describe the charge exchange contribution. The final yields of $\pi$’s depend on the target material and the final state kinematic variables, i.e., $M = M[T; Q^2,W]$.
In order to simplify the problem it is helpful to integrate the doubly differential cross sections of Eq. over $W$ in the $(3,3)$ resonance region, say, $m_p + m_\pi \le W \le 1.4\ \gev$. In this case Eq. can be replaced by an equation of identical form $${{\left(\begin{array}{c}\displaystyle
{{\ensuremath{{\operatorname{d}}}}\sigma(_ZT^A;{\pi^+})\over {\ensuremath{{\operatorname{d}}}}Q^2}\\
\displaystyle{{\ensuremath{{\operatorname{d}}}}\sigma(_ZT^A;{\pi^0})\over
{\ensuremath{{\operatorname{d}}}}Q^2}\\
\displaystyle{{\ensuremath{{\operatorname{d}}}}\sigma(_ZT^A;{\pi^-})\over
{\ensuremath{{\operatorname{d}}}}Q^2}
\end{array}\right)}_{\rm f}}
= {\ensuremath{\overline{M}}}[T; Q^2]\
{{\left(\begin{array}{c}\displaystyle
{{\ensuremath{{\operatorname{d}}}}\sigma(N_T;{\pi^+})\over {\ensuremath{{\operatorname{d}}}}Q^2}\\
\displaystyle{{\ensuremath{{\operatorname{d}}}}\sigma(N_T;{\pi^0})\over {\ensuremath{{\operatorname{d}}}}Q^2}\\
\displaystyle{{\ensuremath{{\operatorname{d}}}}\sigma(N_T;{\pi^-})\over {\ensuremath{{\operatorname{d}}}}Q^2}
\end{array}\right)}_{\rm i}}
\label{eq:appx1}$$ where the matrix ${\ensuremath{\overline{M}}}[T; Q^2]$ can be obtained by averaging the matrix $M[T; Q^2,W]$ over $W$ with the leading $W$-dependence coming from the $\Delta$ resonance contribution. Moreover, we expect the matrix $M$ to be a slowly varying function of $Q^2$ (for $Q^2 \gtrsim 0.3\ \gevsq$). For this reason we introduce a second averaging over $Q^2$ and define the double averaged matrix ${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}[T]$ which is particularly useful for giving a simple description of charge exchange effects in different nuclear targets. In the double-averaging approximation (AV2) the final cross sections including nuclear corrections are expressed as follows: $${{\left(\begin{array}{c}\displaystyle
{{\ensuremath{{\operatorname{d}}}}\sigma(_ZT^A;{\pi^+})\over {\ensuremath{{\operatorname{d}}}}Q^2 {\ensuremath{{\operatorname{d}}}}W}\\
\displaystyle{{\ensuremath{{\operatorname{d}}}}\sigma(_ZT^A;{\pi^0})\over
{\ensuremath{{\operatorname{d}}}}Q^2 {\ensuremath{{\operatorname{d}}}}W}\\
\displaystyle{{\ensuremath{{\operatorname{d}}}}\sigma(_ZT^A;{\pi^-})\over
{\ensuremath{{\operatorname{d}}}}Q^2 {\ensuremath{{\operatorname{d}}}}W}
\end{array}\right)}_{\rm f}}
= {\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}[T]\
{{\left(\begin{array}{c}\displaystyle
{{\ensuremath{{\operatorname{d}}}}\sigma(N_T;{\pi^+})\over {\ensuremath{{\operatorname{d}}}}Q^2 {\ensuremath{{\operatorname{d}}}}W}\\
\displaystyle{{\ensuremath{{\operatorname{d}}}}\sigma(N_T;{\pi^0})\over {\ensuremath{{\operatorname{d}}}}Q^2 {\ensuremath{{\operatorname{d}}}}W}\\
\displaystyle{{\ensuremath{{\operatorname{d}}}}\sigma(N_T;{\pi^-})\over {\ensuremath{{\operatorname{d}}}}Q^2 {\ensuremath{{\operatorname{d}}}}W}
\end{array}\right)}_{\rm i}}\, .
\label{eq:av2}$$ We note that the cross sections are differential in two variables while the matrix ${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}[T]$ is the average over these variables.
The above discussion will be used for a phenomenological description of nuclear rescattering effects. On the other hand, in Ref. [@Adler:1974qu] a dynamical model has been developed to calculate the charge exchange matrix $M$. As an example, for oxygen the resulting matrix in the double-averaging approximation is given by $${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}(_8O^{16}) = \overline{\overline{A_p}}\left(\begin{array}{ccc}
{0.788} & {0.158} & {0.0537} \\
{0.158} & {0.684} & {0.158} \\
{0.0537} & {0.158} & {0.788}
\end{array}\right).
\label{eq:M1}$$ with $\overline{\overline{A_p}}=0.766$, which contains the averaged Pauli suppression factor and absorption of pions in the nucleus. There are various absorption models described in the original article. Two of them are distinguished by the energy dependence of the absorption cross section beyond the $\Delta$ region. In model \[A\] the absorption increases as $W$ increases while in \[B\] it decreases for large $W$’s (beyond the $\Delta$ region). A comparison of the two absorption models (A) and (B) can be found in [@Schienbein:2003xy]. Since the fraction of absorbed pions is still rather uncertain we provide in the appendices ANP matrices for different amounts of absorption. These matrices are useful to obtain an uncertainty band for the expected nuclear corrections.
Results for various targets
---------------------------
In this section we present numerical results for 1-pion leptoproduction differential cross sections including nuclear corrections using the ANP model outlined in the preceding section.
### Neutrinoproduction
We begin with a discussion of the nuclear corrections to the pion energy spectra in neutrino scattering shown in Figs. \[fig:figoxygen\]–\[fig:figiron\], where the curves are neutral current reactions. The dotted lines are the spectra for the free nucleon cross sections. The dashed lines include the effect of the Pauli suppression (in step one of the two step process), whereas the solid line in addition takes into account the pion multiple scattering. These curves correct Figs. 8–16 in Ref. [@Paschos:2000be]. Similar curves have been obtained recently by Leitner et al. [@Leitner:2006sp] who also noticed the error in [@Paschos:2000be]. Even though the models differ in the transport matrix, they both include charge exchange effects. For example, they both find that for reactions where the charge of the pions is the same with the charge of the current the pion yield shows a substantial decrease.
![Differential cross section per nucleon for single pion spectra of $\pi^+,\ \pi^0, \ \pi^-$ for oxygen with $E_\nu = 1\ {\rm GeV}$ in dependence of pion energy $E_\pi$. The curves correspond to neutral current reactions.[]{data-label="fig:figoxygen"}](depi_ppip_nuc_o_v2.eps "fig:"){width="8.9cm"} ![Differential cross section per nucleon for single pion spectra of $\pi^+,\ \pi^0, \ \pi^-$ for oxygen with $E_\nu = 1\ {\rm GeV}$ in dependence of pion energy $E_\pi$. The curves correspond to neutral current reactions.[]{data-label="fig:figoxygen"}](depi_ppi0_nuc_o_v2.eps "fig:"){width="8.9cm"} ![Differential cross section per nucleon for single pion spectra of $\pi^+,\ \pi^0, \ \pi^-$ for oxygen with $E_\nu = 1\ {\rm GeV}$ in dependence of pion energy $E_\pi$. The curves correspond to neutral current reactions.[]{data-label="fig:figoxygen"}](depi_ppin_nuc_o_v2.eps "fig:"){width="8.9cm"}
![The same as in fig. \[fig:figoxygen\] for argon.[]{data-label="fig:figargon"}](depi_ppip_nuc_ar_v2.eps "fig:"){width="8.9cm"} ![The same as in fig. \[fig:figoxygen\] for argon.[]{data-label="fig:figargon"}](depi_ppi0_nuc_ar_v2.eps "fig:"){width="8.9cm"} ![The same as in fig. \[fig:figoxygen\] for argon.[]{data-label="fig:figargon"}](depi_ppin_nuc_ar_v2.eps "fig:"){width="8.9cm"}
![The same as in fig. \[fig:figoxygen\] for iron.[]{data-label="fig:figiron"}](depi_ppip_nuc_fe_v2.eps "fig:"){width="8.9cm"} ![The same as in fig. \[fig:figoxygen\] for iron.[]{data-label="fig:figiron"}](depi_ppi0_nuc_fe_v2.eps "fig:"){width="8.9cm"} ![The same as in fig. \[fig:figoxygen\] for iron.[]{data-label="fig:figiron"}](depi_ppin_nuc_fe_v2.eps "fig:"){width="8.9cm"}
### Electroproduction
We now turn to the electroproduction. To be specific, our analysis will be done under the conditions of the Cebaf Large Acceptance Spectrometer (CLAS) at Jefferson Lab (JLAB). The CLAS detector [@Mecking:2003zu] covers a large fraction of the full solid angle with efficient neutral and charged particle detection. Therefore it is very well suited to perform a high statistics measurement on various light and heavy nuclear targets and to test the ideas of pion multiple scattering models. In the future these measurements can be compared with results in neutrinoproduction from the Minerva experiment [@Drakoulakos:2004gn] using the high intensity Numi neutrino beam. If not stated otherwise we use an electron energy $E_e = 2.7\ \gev$ in order to come as close as possible to the relevant low energy range of the LBL experiments. For the momentum transfer we take the values $Q^2 = 0.4, 0.8\ \gevsq$ in order to avoid the experimentally and theoretically more problematic region at very low $Q^2$. Results for larger $Q^2$ and larger energies, say $E_e = 10\ \gev$, are qualitatively very similar.
![Double differential cross sections for single-pion electroproduction for an oxygen target in dependence of $W$. Spectra for $\pi^0$ and $\pi^+$ production are shown for $Q^2 = 0.4\ \gevsq$ and $Q^2 = 0.8\ \gevsq$ using an electron energy $E_e = 2.7\ \gev$. The solid and dotted lines have been obtained according to using the exact ANP matrix $M(W,Q^2)$ and utilizing the double-averaged ANP matrix ${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}$ in , respectively. The dashed lines show the free nucleon cross section .[]{data-label="fig:2"}](dq2dw_o_0.4_pi0.eps "fig:"){width="7.0cm"} ![Double differential cross sections for single-pion electroproduction for an oxygen target in dependence of $W$. Spectra for $\pi^0$ and $\pi^+$ production are shown for $Q^2 = 0.4\ \gevsq$ and $Q^2 = 0.8\ \gevsq$ using an electron energy $E_e = 2.7\ \gev$. The solid and dotted lines have been obtained according to using the exact ANP matrix $M(W,Q^2)$ and utilizing the double-averaged ANP matrix ${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}$ in , respectively. The dashed lines show the free nucleon cross section .[]{data-label="fig:2"}](dq2dw_o_0.4_pip.eps "fig:"){width="7.0cm"}
![Double differential cross sections for single-pion electroproduction for an oxygen target in dependence of $W$. Spectra for $\pi^0$ and $\pi^+$ production are shown for $Q^2 = 0.4\ \gevsq$ and $Q^2 = 0.8\ \gevsq$ using an electron energy $E_e = 2.7\ \gev$. The solid and dotted lines have been obtained according to using the exact ANP matrix $M(W,Q^2)$ and utilizing the double-averaged ANP matrix ${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}$ in , respectively. The dashed lines show the free nucleon cross section .[]{data-label="fig:2"}](dq2dw_o_0.8_pi0.eps "fig:"){width="7.0cm"} ![Double differential cross sections for single-pion electroproduction for an oxygen target in dependence of $W$. Spectra for $\pi^0$ and $\pi^+$ production are shown for $Q^2 = 0.4\ \gevsq$ and $Q^2 = 0.8\ \gevsq$ using an electron energy $E_e = 2.7\ \gev$. The solid and dotted lines have been obtained according to using the exact ANP matrix $M(W,Q^2)$ and utilizing the double-averaged ANP matrix ${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}$ in , respectively. The dashed lines show the free nucleon cross section .[]{data-label="fig:2"}](dq2dw_o_0.8_pip.eps "fig:"){width="7.0cm"}
Figure \[fig:2\] shows the double differential cross section ${\ensuremath{{\operatorname{d}}}}\sigma/{\ensuremath{{\operatorname{d}}}}Q^2 {\ensuremath{{\operatorname{d}}}}W$ for $\pi^+$ and $\pi^0$ production versus $W$ for an oxygen target. The solid lines have been obtained with help of Eq. including the nuclear corrections. The dashed lines show the result of the double-averaging approximation according to Eq. using the ANP matrix in Eq. . The dotted line is the free cross section in Eq. . One sees, the double-averaging approximation and the exact calculation give very similar results such that the former is well-suited for simple estimates to an accuracy of $10\%$ of pion rescattering effects. We observe that the cross sections for $\pi^0$ production are largely reduced by about $40\%$ due to the nuclear corrections. This can be understood since the larger $\pi^0$ cross sections are reduced by absorption effects and charge exchange effects. On the other hand, the $\pi^+$ cross sections are even slightly enlarged, because the reduction due to pion absorption is compensated by an increase due to charge exchange. The compensation is substantial since the $\pi^0$ yields are dominant. In Fig. \[fig:targets\] double differential cross sections per nucleon for different target materials are presented. The electron energy and the momentum transfer have been chosen as $E_e = 2.7\ \gev$ and $Q^2= 0.4\ \gevsq$, respectively. The results for the pion rescattering corrections have been obtained within the double-averaging approximation which allows for a simple comparison of the dependence on the target material in terms of the matrices ${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}[T]$ which can be found in Eq. and App. \[app:M1\]. For comparison the free nucleon cross section (isoscalar $\tfrac{p+n}{2}$) is also shown. As expected, the nuclear corrections become larger with increasing atomic number from carbon to iron.
![Double differential cross sections per nucleon for single-pion electroproduction for different target materials. $W$-spectra for $\pi^0$ and $\pi^+$ production are shown for $Q^2 = 0.4\ \gevsq$ using an electron energy $E_e = 2.7\ \gev$. The pion rescattering corrections have been calculated in the double-averaging approximation using the ANP matrices in and App. \[app:M1\]. For comparison, the free nucleon cross section is shown.[]{data-label="fig:targets"}](dq2dw_0.4_pi0_nom.eps "fig:"){width="7.0cm"} ![Double differential cross sections per nucleon for single-pion electroproduction for different target materials. $W$-spectra for $\pi^0$ and $\pi^+$ production are shown for $Q^2 = 0.4\ \gevsq$ using an electron energy $E_e = 2.7\ \gev$. The pion rescattering corrections have been calculated in the double-averaging approximation using the ANP matrices in and App. \[app:M1\]. For comparison, the free nucleon cross section is shown.[]{data-label="fig:targets"}](dq2dw_0.4_pip_nom.eps "fig:"){width="7.0cm"}
One of the input quantities for calculating the transport function $f(\lambda)$ in the ANP model is the pion absorption cross section ${\ensuremath{\sigma_{\rm abs}}}(W)$ describing the probability that the pion is absorbed in a single rescattering process. For ${\ensuremath{\sigma_{\rm abs}}}(W)$ the ANP article reported results for two parameterizations, models A and B, taken from Refs. [@Sternheim:1972ad; @Silbar:1973em] which have very different $W$-dependence and normalization. However, the predictions of the ANP model in the double-averaging approximation are primarily sensitive to the normalization of the pion absorption cross section at $W \simeq m_\Delta$ [@Schienbein:2003xy]. Using data by Merenyi et al. [@Merenyi:1992gf] for a neon target it was found that about $25\% \pm 5\%$ of pions are absorbed making possible the determination of the normalization of ${\ensuremath{\sigma_{\rm abs}}}(W)$ with a $20\%$ accuracy.
In order to investigate the theoretical uncertainty due to pion absorption effects we show in Fig. \[fig:abs1\] double differential cross sections ${\ensuremath{{\operatorname{d}}}}\sigma/{\ensuremath{{\operatorname{d}}}}Q^2 {\ensuremath{{\operatorname{d}}}}W$ for $\pi^+$ and $\pi^0$ production vs $W$ for different amounts of pion absorption in oxygen: $25\%$ (solid line), $20\%$ (dashed line), $30\%$ (dotted line). The $\pi^0$ and $\pi^+$ spectra have been calculated in the double-averaging approximation utilizing the matrices in App. \[app:M2\]. The three curves represent the theoretical uncertainty due to pion absorption effects. For comparison, the free nucleon cross section is shown as well.
![Double differential cross sections per nucleon for single-pion electroproduction for oxygen with $20\%$ (dashed line), $25\%$ (solid line) and $30\%$ (dotted line) pion absorption. Furthermore, $Q^2 = 0.8\ \gevsq$ and $E_e = 2.7\ \gev$. The $\pi^0$ and $\pi^+$ spectra have been calculated in the double-averaging approximation utilizing the matrices in App. \[app:M2\]. For comparison, the free nucleon cross section is shown as well.[]{data-label="fig:abs1"}](dq2dw_o_0.8_abs_pi0.eps "fig:"){width="7.0cm"} ![Double differential cross sections per nucleon for single-pion electroproduction for oxygen with $20\%$ (dashed line), $25\%$ (solid line) and $30\%$ (dotted line) pion absorption. Furthermore, $Q^2 = 0.8\ \gevsq$ and $E_e = 2.7\ \gev$. The $\pi^0$ and $\pi^+$ spectra have been calculated in the double-averaging approximation utilizing the matrices in App. \[app:M2\]. For comparison, the free nucleon cross section is shown as well.[]{data-label="fig:abs1"}](dq2dw_o_0.8_abs_pip.eps "fig:"){width="7.0cm"}
Although the predictions of the ANP model are mainly sensitive to ${\ensuremath{\sigma_{\rm abs}}}(W \simeq m_\Delta)$ it would be interesting to obtain more information on the detailed $W$-shape. The fraction of absorbed pions can be determined by measuring the inclusive pion production cross sections for a nuclear target divided by the free nucleon cross sections, $${\ensuremath{\rm Abs}}(Q^2,W) = 1 -
\frac{\sum_{k=0,\pm} \frac{{\ensuremath{{\operatorname{d}}}}\sigma(_ZT^A;\pi^k)}{{\ensuremath{{\operatorname{d}}}}Q^2 {\ensuremath{{\operatorname{d}}}}W}}
{\sum_{j=0,\pm} \frac{{\ensuremath{{\operatorname{d}}}}\sigma(N_T;\pi^j)}{{\ensuremath{{\operatorname{d}}}}Q^2 {\ensuremath{{\operatorname{d}}}}W}}
= 1 - A_p(Q^2,W)
\, ,
\label{abs}$$ where $A_p$ has been introduced in . This quantity is related to ${\ensuremath{\sigma_{\rm abs}}}(W)$ as can be seen by linearizing the transport function $f(\lambda,W)$ [@Paschos:2004qh; @Schienbein:2003xy] $${\ensuremath{\rm Abs}}(Q^2,W) \simeq \frac{1}{2} \bar{L} \rho_0 \times {\ensuremath{\sigma_{\rm abs}}}(W)\ .
\label{eq:sigabs}$$ Here $\bar{L}$ is the effective length of the nucleus averaged over impact parameters and $\rho_0$ the charge density in the center. As an example, for oxygen one finds $\bar{L} \simeq 1.9 R$ with radius $R \simeq 1.833\ {\rm fm}$ and $\rho_0 = 0.141\ {\rm fm}^{-3}$. Therefore, the $W$-dependence of ${\ensuremath{\sigma_{\rm abs}}}(W)$ can be reconstructed from the fraction of absorbed pions, i.e. ${\ensuremath{\rm Abs}}(Q^2,W)$. Summing over the three charged pions eliminates charge exchange effects. In order to verify the linearized approximation in Eq. , we show in Fig. \[fig:sigabs\] the ANP model prediction for ${\ensuremath{\rm Abs}}(Q^2,W)$ for oxygen and iron targets with $Q^2 = 0.3\ \gevsq$. This prediction strongly depends on the shape of the cross section ${\ensuremath{\sigma_{\rm abs}}}(W)$ for which we use model B from Refs. [@Silbar:1973em]. ${\ensuremath{\sigma_{\rm abs}}}(W)$ multiplied by a free normalization factors for oxygen and iron, respectively, is depicted by the dashed lines. Obviously, Eq. is quite well satisfied for oxygen and still reasonably good for iron. Finally, the dotted line shows the result of the averaging approximation. We conclude that ${\ensuremath{\sigma_{\rm abs}}}(W)$ can be extracted with help of Eqs. and .
![The fraction of absorbed pions, ${\ensuremath{\rm Abs}}(Q^2,W)$, in dependence of $W$ for oxygen and iron targets for $Q^2=0.3\ \gevsq$. Also shown is the cross section ${\ensuremath{\sigma_{\rm abs}}}(W)$ (model B) multiplied by free normalization factors (dashed lines). The dotted lines are the result for ${\ensuremath{\rm Abs}}(Q^2,W)$ in the averaging approximation.[]{data-label="fig:sigabs"}](abs.eps){width="7.9cm"}
For completeness, we mention that the pion absorption in nuclei is reported in various articles [@Ashery:1987nt; @*Ingram:2001xs; @*Ransome:2005vb]. For comparisons one should be careful because the absorption cross sections in pi-nucleus and in neutrino-nucleus reactions are different, in the former case it is a surface effect while in the latter it occurs everywhere in the nucleus.
A useful test of charge exchange effects is provided by the double ratio $${\rm DR}(Q^2,W) = \left(\frac{\pi^0}{\pi^+ + \pi^-}\right)_A /
\left(\frac{\pi^0}{\pi^+ + \pi^-}\right)_p
\label{eq:doubleratio}$$ where $(\pi^i)_A$ represents the doubly differential cross section ${\ensuremath{{\operatorname{d}}}}\sigma/{\ensuremath{{\operatorname{d}}}}Q^2 {\ensuremath{{\operatorname{d}}}}W$ for the production of a pion $\pi^i$ in $eA$ scattering. This observable is expected to be rather robust with respect to radiative corrections and acceptance differences between neutral and charged pions.[^2] In Fig. \[fig:doubleratio\] we show the double ratio for a carbon target in dependence of $W$ for a fixed $Q^2 = 0.4\ \gevsq$. The dependence on $Q^2$ is weak and results for other values of $Q^2$ are very similar. The solid line shows the exact result, whereas the dotted lines have been obtained in the double averaging approximation with minimal and maximal amounts of pion absorption. As can be seen, the results are rather insensitive to the exact amount of pion absorption. Without charge exchange effects (and assuming similar absorption of charged and neutral pions) the double ratio would be close to unity. As can be seen, the ANP model predicts a double ratio smaller than $0.6$ in the region $W \simeq 1.2\ \gev$. A confirmation of this expectation would be a clear signal of pion charge exchange predominantly governed by isospin symmetry. In this case it would be interesting to go a step further and to study similar ratios for pion angular distributions.
![Double ratio of single pion electroproduction cross sections in dependence of $W$ for fixed $Q^2=0.4\ \gevsq$ as defined in Eq. . The dotted lines show results in the double averaging approximation with varying amounts of absorption.[]{data-label="fig:doubleratio"}](double_ratio_c_pi0p_q2_0.4.eps){width="7.9cm"}
Summary {#sec:summary}
=======
Lepton induced reactions on medium and heavy nuclei include the rescattering of produced pions inside the nuclei. This is especially noticeable in the $\Delta$-resonance region, where the produced resonance decays into a nucleon and a pion. In the introduction and section \[sec:sec2\] we reviewed the progress that has been made in the calculations of neutrino-induced reactions on free protons and neutrons, because we needed them for following calculations. For several resonances the vector form factors have been recently determined by using electroproduction results in Jefferson Laboratory [@Lalakulich:2006sw]. For the axial form factors modified dipoles give an accurate description of the data. For the purposes of this article (studies of nuclear corrections) it suffices to deduce the electroproduction cross sections through Eqs. and . The main contribution of this article is contained in section \[sec:sec3\], where we describe important features of the ANP model and define single- and double averaged transport matrices. Two important aspects of rescattering are emphasized: (i) the absorption of the pions and (ii) charge exchange occurring in the multiple scattering, where we have shown that special features of the data are attributed to each of them. Finally we propose specific ratios of electroproduction reactions that are sensitive to the absorption cross section and to charge exchange effects. Using the model we calculate the transport matrix for various absorption cross sections and nuclei and present the results in appendix \[app:M1\]. We also calculated the pion energy spectra with and without nuclear corrections. The results appear in figures \[fig:figoxygen\]–\[fig:figiron\] and can be compared with other calculations [@Leitner:2006sp]. Comparison of the double averaged approximation with the exact ANP calculation shows small differences (figure \[fig:2\]). As mentioned already, electroproduction data are very useful in testing several aspects of the model and its predictions. For the absorption cross section we propose in Eq. a ratio that depends only on ${\rm A_p}(Q^2,W)=g(Q^2,W) f(1,W)$. Since we consider isoscalar targets and sum over the charges of the pions, charge exchange terms are eliminated. This leaves over the dependence on charge independent effects, like the Pauli factor and the average absorption; this is indeed the average absorption of pions and even includes the absorption of the $\Delta$-resonance itself. Another ratio $({\rm DR}(Q^2,W))$ is sensitive to charge exchange effects. In the double ratio the dependence on ${\rm A_p}(Q^2,W)$ drops out and the surviving terms are isospin dependent. Our calculation shows that the ratio depends on $W$ with the largest reduction occurring in the region $1.1 < W <1.25$ GeV. Finally, the $\Delta(1232)$ is a sharply peaked resonance, where the resonant interaction, takes place over small ranges of the kinematic variables, so that averaging over them gives accurate approximations. This is analogous to a narrow width approximation. Several comparisons in this article confirm the expectation that averaged quantities give rather accurate approximations of more extensive calculations.
[**Acknowledgments**]{}
We wish to thank W. Brooks and S. Manly for many useful discussions, their interest and encouragement. The work of J. Y. Yu is supported by the Deutsche Forschungsgemeinschaft (DFG) through Grant No. YU 118/1-1.
[**Appendix**]{}
Charge exchange matrices in the double averaging approximation {#app:M1}
==============================================================
\
$${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}(_{6}C^{12}) = \overline{\overline{A_p}}\left(\begin{array}{ccc}
{0.826}& {0.136} & {0.038} \\
{0.136}& {0.728} & {0.136} \\
{0.038}& {0.136} & {0.826}
\end{array}\right)
\label{eq:MC}$$ with $\overline{\overline{A_p}} =0.791$ .\
\
$${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}(_{18}Ar^{40}) = \overline{\overline{A_p}}\left(\begin{array}{ccc}
{0.733}& {0.187} & {0.080} \\
{0.187}& {0.626} & {0.187} \\
{0.080} & {0.187} & {0.733}
\end{array}\right)
\label{eq:MAr}$$ with $\overline{\overline{A_p}} =0.657$ .
\
$${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}(_{26}Fe^{56}) = \overline{\overline{A_p}}\left(\begin{array}{ccc}
{0.720} & {0.194}& {0.086} \\
{0.194} & {0.613} & {0.194} \\
{0.086} & {0.194} & {0.720}
\end{array}\right)
\label{eq:MFe}$$ with $\overline{\overline{A_p}} =0.631$ .
Charge exchange matrices for various amounts of pion absorption {#app:M2}
===============================================================
\
15% absorption $${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}(_{6}O^{12}) = \overline{\overline{A_p}}\left(\begin{array}{ccc}
{0.817} & {0.141} & {0.041} \\
{0.141} & {0.718} & {0.141} \\
{0.041} & {0.141} & {0.817}
\end{array}\right)
\label{eq:MC15}$$ with $\overline{\overline{A_p}} =0.831$ .\
20% absorption $${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}(_{6}C^{12}) = \overline{\overline{A_p}}\left(\begin{array}{ccc}
{0.829}& {0.134} & {0.037} \\
{0.134}& {0.731} & {0.134} \\
{0.037}& {0.134} & {0.829}
\end{array}\right)
\label{eq:MC20}$$ with $\overline{\overline{A_p}} =0.782$ .\
25% absorption $${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}(_{6}C^{12}) = \overline{\overline{A_p}}\left(\begin{array}{ccc}
{0.840}& {0.127} & {0.032} \\
{0.127}& {0.745} & {0.127} \\
{0.032}& {0.127} & {0.840}
\end{array}\right)
\label{eq:MC25}$$ with $\overline{\overline{A_p}} =0.734$ .\
\
15% absorption $${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}(_{8}O^{16}) = \overline{\overline{A_p}}\left(\begin{array}{ccc}
{0.771} & {0.167} & {0.062} \\
{0.167} & {0.665} & {0.167} \\
{0.062} & {0.167} & {0.771}
\end{array}\right)
\label{eq:MO15}$$ with $\overline{\overline{A_p}} =0.833$ .\
20% absorption $${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}(_{8}O^{16}) = \overline{\overline{A_p}}\left(\begin{array}{ccc}
{0.783} & {0.161} & {0.056} \\
{0.161} & {0.679} & {0.161} \\
{0.056} & {0.161} & {0.783}
\end{array}\right)
\label{eq:MO20}$$ with $\overline{\overline{A_p}} =0.784$ .\
25% absorption $${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}(_{8}O^{16}) = \overline{\overline{A_p}}\left(\begin{array}{ccc}
{0.797} & {0.153} & {0.050} \\
{0.153} & {0.693} & {0.153} \\
{0.050} & {0.153} & {0.797}
\end{array}\right)
\label{eq:MO25}$$ with $\overline{\overline{A_p}} =0.735$ .\
30% absorption $${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}(_{8}O^{16}) = \overline{\overline{A_p}}\left(\begin{array}{ccc}
{0.810} & {0.146} & {0.044} \\
{0.146} & {0.709} & {0.146} \\
{0.044} & {0.146} & {0.810}
\end{array}\right)
\label{eq:MO30}$$ with $\overline{\overline{A_p}} =0.687$ .
Forward- and backward charge exchange matrices {#app:M3}
==============================================
\
15% absorption $${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}_+(_{6}C^{12}) = \overline{\overline{A_{p+}}}\left(\begin{array}{ccc}
{0.870} & {0.100} & {0.029} \\
{0.100} & {0.799} & {0.100} \\
{0.029} & {0.100} & {0.870}
\end{array}\right),
{\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}_-(_{6}C^{12})=\overline{\overline{A_{p-}}}\left(\begin{array}{ccc}
{0.675} & {0.251}& {0.074} \\
{0.251} & {0.498} & {0.251} \\
{0.074} & {0.251} & {0.675}
\end{array}\right)
\label{eq:MC15m}$$ with $\overline{\overline{A_{p+}}} =0.606$ and $\overline{\overline{A_{p-}}} =0.225$.\
20% absorption $${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}_+(_{6}C^{12}) = \overline{\overline{A_{p+}}}\left(\begin{array}{ccc}
{0.880} & {0.094} & {0.026} \\
{0.094} & {0.811} & {0.094} \\
{0.026} & {0.094} & {0.880}
\end{array}\right),
{\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}_-(_{6}C^{12})=\overline{\overline{A_{p-}}}\left(\begin{array}{ccc}
{0.685} & {0.247}& {0.068} \\
{0.247} & {0.505} & {0.247} \\
{0.068} & {0.247} & {0.685}
\end{array}\right)
\label{eq:MC20m}$$ with $\overline{\overline{A_{p+}}} =0.578$ and $\overline{\overline{A_{p-}}} =0.204$.\
25% absorption $${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}_+(_{6}C^{12}) = \overline{\overline{A_{p+}}}\left(\begin{array}{ccc}
{0.889} & {0.088} & {0.022} \\
{0.088} & {0.823} & {0.088} \\
{0.022} & {0.088} & {0.889}
\end{array}\right),
{\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}_-(_{6}C^{12})=\overline{\overline{A_{p-}}}\left(\begin{array}{ccc}
{0.695} & {0.243}& {0.062} \\
{0.243} & {0.513} & {0.243} \\
{0.062} & {0.243} & {0.695}
\end{array}\right)
\label{eq:MC25m}$$ with $\overline{\overline{A_{p+}}} = 0.549$ and $\overline{\overline{A_{p-}}} = 0.184$.\
\
15% absorption $${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}_+(_{8}O^{16}) = \overline{\overline{A_{p+}}}\left(\begin{array}{ccc}
{0.829} & {0.125} & {0.046} \\
{0.125} & {0.750} & {0.125} \\
{0.046} & {0.125} & {0.829}
\end{array}\right),
{\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}_-(_{8}O^{16})=\overline{\overline{A_{p-}}}\left(\begin{array}{ccc}
{0.635} & {0.265}& {0.100} \\
{0.265} & {0.470} & {0.265} \\
{0.100} & {0.265} & {0.635}
\end{array}\right)
\label{eq:MO15m}$$ with $\overline{\overline{A_{p+}}} =0.581$ and $\overline{\overline{A_{p-}}} =0.252$.\
20% absorption $${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}_+(_{8}O^{16}) = \overline{\overline{A_{p+}}}\left(\begin{array}{ccc}
{0.840} & {0.119} & {0.041} \\
{0.119} & {0.762} & {0.119} \\
{0.041} & {0.119} & {0.840}
\end{array}\right),
{\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}_-(_{8}O^{16})=\overline{\overline{A_{p-}}}\left(\begin{array}{ccc}
{0.646} & {0.262}& {0.092} \\
{0.262} & {0.477} & {0.262} \\
{0.092} & {0.262} & {0.646}
\end{array}\right)
\label{eq:MO20m}$$ with $\overline{\overline{A_{p+}}} =0.554$ and $\overline{\overline{A_{p-}}} =0.23$.\
25% absorption $${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}_+(_{8}O^{16}) = \overline{\overline{A_{p+}}}\left(\begin{array}{ccc}
{0.852} & {0.112}& {0.036} \\
{0.112} & {0.776} & {0.112} \\
{0.036} & {0.112} & {0.852}
\end{array}\right),
{\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}_-(_{8}O^{16})=\overline{\overline{A_{p-}}}\left(\begin{array}{ccc}
{0.657} & {0.258} & {0.085} \\
{0.258} & {0.485} & {0.257} \\
{0.085} & {0.258} & {0.657}
\end{array}\right)
\label{eq:MO25m}$$ with $\overline{\overline{A_{p+}}} =0.527$ and $\overline{\overline{A_{p-}}} =0.208$.\
30% absorption $${\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}_+(_{8}O^{16}) = \overline{\overline{A_{p+}}}\left(\begin{array}{ccc}
{0.863} & {0.105} & {0.031} \\
{0.105} & {0.789} & {0.105} \\
{0.031} & {0.105} & {0.863}
\end{array}\right),
{\ensuremath{\overline{{\ensuremath{\overline{M}}}}}}_-(_{8}O^{16})=\overline{\overline{A_{p-}}}\left(\begin{array}{ccc}
{0.669} & {0.253} & {0.078} \\
{0.253} & {0.493} & {0.253} \\
{0.078} & {0.253} & {0.669}
\end{array}\right)
\label{eq:MO30m}$$ with $\overline{\overline{A_{p+}}} =0.499$ and $\overline{\overline{A_{p-}}} =0.187$.
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[^1]: However, the Fermi motion has a very small effect on the $W$ distribution and we neglect it in our numerical analysis. On the other hand, effects of the Pauli exclusion principle have been absorbed into the matrix $M$ and are taken into account.
[^2]: We are grateful to S. Manly for drawing our attention to the double ratio.
|
---
address: 'NIKHEF-Amsterdam The Netherlands'
author:
- |
Sven Schagen\
on behalf of the ZEUS Collaboration
title: Measurement of charm production in deep inelastic scattering with the ZEUS detector
---
Introduction {#sec:intro}
============
For deep inelastic electron-proton scattering perturbative QCD predicts that heavy quarks will mainly be produced by the boson-gluon fusion process: a photon emitted by the electron interacts with a gluon inside the proton to produce a $q\overline{q}$-pair, i.e. $\gamma g \to c \overline{c}$. The HVQDIS[@hvqdis] program uses NLO calculations in the DGLAP scheme at fixed order in $\alpha_s$, assuming three active flavours in the proton[@el]. The charm-quark is then only produced by the boson-gluon fusion.
Analysis of the decay chain $D^{*+} \to D^{0}\pi_{s}^{+} \to K^{-}\pi^{+}\pi_{s}^{+} (+ c.c.)$ {#sec:dstar}
==============================================================================================
For the analysis of the $D^*$ decay DIS events are selected that with $Q^{2}$ $>$ 10 GeV$^2$. By combining all available fully reconstructed tracks $D^0$ and $D^{*}$ candidates are reconstructed. A clean sample of $D^*$’s can be extracted from the data by constraining the reconstructed $D^0$ mass ($D^0$: $1.80 < \mathit{M} < 1.92$ GeV) and by cutting on the reconstructed $D^{*}$ kinematics ($D^*$: $1.5 < p_T < 15.0$ GeV and $|\eta| < 1.5$). For the full analysis, data with an integrated luminosity of 83 pb$^{-1}$ was used. The events were collected during the 1995-1997 running period, at which HERA was operated with a 27.5 GeV positron beam and a 820 GeV proton beam, and during the 1999-2000 running period when the beam energies were 27.5 and 920 GeV, respectively. The resulting differential cross sections are shown in Fig. \[pic:dstar\].
Semi-leptonic decay of charmed hadrons ($\overline{c}q \to e^-\overline{\nu}_e X$) {#sec:sle}
==================================================================================
To study the semi-leptonic decay of charmed hadrons events with $1<Q^2<1000$ GeV$^2$ and $0.03<y<0.7$ are selected. The electron candidates are identified based on the properties of the calorimeter cluster that is associated with them. We then consider the $dE/dx$ of all candidates (the electron-enriched sample in Fig. \[pic:sle-signal\] ([*left*]{})) . The (large) hadronic background that is still within this sample is determined using the $dE/dx$ distribution of a sample containing only hadronic tracks. After subtracting this hadronic background from the electron-enriched sample, a clean electron signal is visible (Fig. \[pic:sle-signal\] ([*right*]{})). This distribution still contains electrons coming from photon conversions, Dalitz decay of the $\pi^0$ and semi-leptonic decay of beauty. These contributions are all subtracted from the sample. For the 1996-1997 data from ZEUS (integrated luminosity of 34 pb$^{-1}$), the differential cross sections, as shown in Fig. \[pic:sle-xsecs\] are obtained. The measurement shows good agreement with the predictions from the HVQDIS program. In addition, the charm structure function has been unfolded from the cross section differential in $Q^2$ and $x_{BJ}$. These results are compared to previously published ZEUS results from the $D^*$ analysis of the 1996-1997 data [@dstar]. Good agreement between the two analysis can be observed.
Conclusions {#sec:conc}
===========
Results on charm production in DIS using the $D^*$ meson or the semi leptonic decay into electrons have been reported. The data show good agreement with NLO DGLAP predictions for charm production through the boson-gluon fusion process as calculated by the HVQDIS program.
[99]{}
|
---
abstract: 'A probabilistic frame is a Borel probability measure with finite second moment whose support spans ${\mathbb R}^d$. A Parseval probabilistic frame is one for which the associated matrix of second moment is the identity matrix in ${\mathbb R}^d$. Each probabilistic frame is canonically associated to a Parseval probabilistic frame. In this paper, we show that this canonical Parseval probabilistic frame is the closest Parseval probabilistic frame to a given probabilistic frame in the $2-$Wasserstein distance. Our proof is based on two main ingredients. On the one hand, we show that a probabilistic frame can be approximated in the $2-$Wasserstein metric with (compactly supported) finite frames whose bounds can be controlled. On the other hand we establish some fine continuity properties of the function that maps a probabilistic frame to its canonical Parseval probabilistic frame. Our results generalize similar ones for finite frames and their associated Parseval frames.'
address:
- 'Department of Mathematics, University of Missouri, Columbia, MO 65211-4100'
- 'Department of Mathematics and Norbert Wiener Center, University of Maryland, College Park, MD 20742'
author:
- 'Desai Cheng and Kasso A. Okoudjou'
bibliography:
- 'PFW2\_bib.bib'
title: Optimal properties of the canonical tight probabilistic frame
---
Introduction {#sec1}
============
The notion of probabilistic frames was first introduced in [@EhlerRTF] in the setting of probability measures on the unit sphere, and was later generalized to probability measures on ${\mathbb R}^d$ in [@EhlOko2012]. In essence, this theory is a generalization of the theory of finite frames which has seen a wealth of activities in recent year, [@CasKut2013; @chri2003; @KovChe1; @KovChe2; @OkoudjouFiniteFrame].
Review of finite frame theory {#subsec1.1}
-----------------------------
Before we give the definition and some elementary properties of probabilistic frames, we recall that a set $\Phi=\{\varphi_i\}_{i=1}^N \subset {\mathbb R}^d$ is a frame for ${\mathbb R}^d$ if and only if there exist $0<A\leq B<\infty$ such that $$A\|x\|^2\leq \sum_{i=1}^N{\left\langle x,\varphi_i\right\rangle}^2\leq B\|x\|^2\qquad \forall\, x\in {\mathbb R}^2.$$ The frame $\Phi$ is a *tight frame* if we can choose $A = B$. Furthermore, if $A=B=1$, $\Phi$ is called a Parseval frame. In the sequel the set of frames for ${\mathbb R}^d$ with $N$ vectors will be denoted by ${\mathcal{F}}(N,d)$, and simply ${\mathcal{F}}$ when the context is clear. The subset of frames with frame bounds $0<A\leq B<\infty$ will be denoted ${\mathcal{F}}_{A, B}(N,d),$ or simply ${\mathcal{F}}_{A,B}$. We equip the set ${\mathcal{F}}(N, d)$ with the metric $$\label{frame-metric}
d(\Phi, \Psi)= \sqrt{\sum_{i=1}^{N}\|\varphi_i - \psi_i\|^2}=\sqrt{\sum_{i=1}^d\|R_i-P_i\|^2}$$ where $\Phi=\{\varphi_i\}_{i=1}^{N}, \Psi=\{\psi_i\}_{i=1}^{N }) \in {\mathcal{F}}(M,d),$ $\{R_i\}_{i=1}^d, \{P_i\}_{i=1}^d \subset {\mathbb R}^N$ denote the rows of $\Phi$, and those of $\Psi$, respectively.
Let $\Phi=\{\varphi_i\}_{i=1}^N$ be a frame for ${\mathbb R}^d$. Throughout the paper we shall abuse notation and denote the *synthesis matrix* of the frame by $\Phi$, the $d\times N$ whose $i^{th}$ column is $\varphi_i.$ The matrix $$S:=S_{\Phi}=\Phi \Phi^{T}=\sum_{i=1}^N {\left\langle \cdot,\varphi_i\right\rangle}\varphi_i$$ is the *frame matrix*. It is known that $\Phi=\{\varphi_i\}_{i=1}^N$ is a frame for ${\mathbb R}^d$ if and only if $S$ is a positive definite matrix. Moreover, the smallest eigenvalue of $S$ is the optimal lower frame bound, while its largest eigenvalue is the optimal upper frame bound. $\Phi$ is a tight frame if and only if $S$ is a multiple of the $d\times d$ identity matrix. In particular, $\Phi$ is a Parseval frame if and only if $S=I$.
If $\Phi$ is a frame, then $S$ is positive definite and thus invertible. Consequently, $$\Phi^{\dag}=\{\varphi_i^\dag\}_{i=1}^N= \{S^{-1/2}\varphi_i\}_{i=1}^N$$ is a Parseval frame, leading to following reconstruction formula:
$$x=\sum_{i=1}^N{\left\langle x,\varphi_i^{\dag}\right\rangle}\varphi_i=\sum_{i=1}^N{\left\langle x,\varphi_i\right\rangle} \varphi_i^{\dag}\, \forall\, x\in {\mathbb R}^d.$$
In addition, $\Phi^\dag$ is the unique Parseval frame which solves the following problem [@CasKut07 Theorem 3.1]:
$$\label{min-clos-pf}
\min\{d(\Phi, \Psi)^2=\sum_{i=1}^N\|\varphi_i-\psi_i\|^2: \Psi=\{\psi_i\}_{i=1}^N\subset {\mathbb R}^d, \, \, {\textrm Parseval \, frame}\}.$$
To be specific,
\[uniqueness\_prob\_1\][@CasKut07 Theorem 3.1]
If $\Phi=\{\varphi_i\}_{i=1}^N$ is a frame for ${\mathbb R}^d$, then $\Phi^{\dag}=\{\varphi_i^\dag\}_{i=1}^N= \{S^{-1/2}\varphi_i\}_{i=1}^N$ is the unique solution to .
In Section \[sec2\], and for the sake of completeness, we give a new and simple proof of this result and we refer to [@Balan99; @BodCas10; @CahillCasazza13] for related results.
Probabilistic frames {#subsec1.2}
--------------------
The main goal of this paper is to characterize the minimizers of an optimal problem analog of for probabilistic frames. To motivate the definition of a probabilistic frame, we note that given a frame $\Phi = \{\varphi_i\}_{i=1}^N \subset {\mathbb R}^{d}$, then the discrete probability measure $$\mu_{\Phi}=\tfrac{1}{N}\sum_{k=1}^N\delta_{\varphi_k}$$ has the property that its support ($\{\varphi_k\}_{k=1}^N$) spans ${\mathbb R}^d$ and that it has finite second moment, i.e., $$\int_{{\mathbb R}^d}\|x\|^2d\mu_{\Phi}(x)=\tfrac{1}{N}\sum_{k=1}^N\|\varphi_k\|^2< \infty.$$ The probability measure $\mu_\Phi$ is an example of a probabilistic frame that was introduced in [@EhlerRTF; @EhlOko2012].
More specifically, a Borel probability measure $\mu$ is a *probabilistic frame* if there exist $0<A\leq B < \infty$ such that for all $x\in{\mathbb R}^d$ we have $$\label{pfineq}
A\|x\|^2 \leq \int_{{\mathbb R}^{d}} |\langle x,y\rangle |^2 d\mu (y) \leq B\|x\|^2.$$ The constants $A$ and $B$ are called *lower and upper probabilistic frame bounds*, respectively. When $A=B,$ $\mu$ is called a *tight probabilistic frame*. In particular, when $A=B=1$, $\mu$ is called a *Parseval probabilistic frame*.
A special class of probabilistic frames that will be considered in the sequel consists of discrete measures $\mu_{\Phi, w}= \sum_{i=1}^{N}w_i\delta_{\varphi_i}$ where $\Phi=\{\varphi_i\}_{i=1}^N\subset {\mathbb R}^d$, and $w=\{w_i\}_{i=1}^N\subset [0, \infty)$ is a set of weights such that $\sum_{i=1}^Nw_i=1$. A probability measure such as $\mu_{\Phi, w}$ will be termed *finite probabilistic frame*, if and only if it is a probabilistic frame for ${\mathbb R}^d$. When the context is clear we will simply write $\mu$ for $\mu_{\Phi, w}$. We shall also identify a finite probabilistic frame $\mu_{\Phi, w}$ with the frame $\Phi_w=\{\sqrt{w_i}\varphi_i\}_{i=1}^{N}$, as both have the same frame bounds. We refer to the surveys [@EhlOko2013; @KOPrecondPF2016] for an overview of the theory of probabilistic frames.
We shall prove an analog of Theorem \[uniqueness\_prob\_1\] by endowing the set of probabilistic frames with the Wasserstein metric. Let $\mathcal{P}:=\mathcal{P}(\mathcal{B},{\mathbb R}^d)$ denote the collection of probability measures on ${\mathbb R}^d$ with respect to the Borel $\sigma$-algebra $\mathcal{B}$. Let $$\mathcal{P}_{2}:= \mathcal{P}_{2}({\mathbb R}^{d})=\bigg\{ \mu \in \mathcal{P}: M_{2}^{2}(\mu):=\int_{{\mathbb R}^{d}}{\|{x}\|}{}^{2}d\mu(x) < \infty\bigg\}$$ be the set of all probability measures with finite second moments. For $\mu, \nu \in \mathcal{P}_2$, let $\Gamma(\mu, \nu)$ be the set of all Borel probability measures $\gamma$ on ${\mathbb R}^d \times {\mathbb R}^d$ whose marginals are $\mu$ and $\nu$, respectively, i.e., $\gamma(A\times {\mathbb R}^d)=\mu(A)$ and $\gamma({\mathbb R}^d \times B) = \nu(B)$ for all Borel subset $A, B$ in ${\mathbb R}^d$. The space $\mathcal{P}_{2}$ is equipped with the $2$-*Wasserstein metric* given by $$\label{wmetric}
W_{2}^{2}(\mu, \nu):=\min\bigg\{\int_{{\mathbb R}^d \times {\mathbb R}^d}{\|{x-y}\|}{}^{2}d\gamma(x, y), \gamma \in \Gamma(\mu, \nu)\bigg\}.$$ The minimum defined by is achieved at a measure $\gamma_0 \in \Gamma(\mu, \nu)$, that is: $$W_{2}^{2}(\mu, \nu)=\int_{{\mathbb R}^d \times {\mathbb R}^d}{\|{x-y}\|}{}^{2}d\gamma_0(x, y).$$ We refer to [@AGS2005 Chapter 7], and [@Villani2009 Chapter 6] for more details on the Wasserstein spaces.
Our contributions {#subsec1.3}
-----------------
The investigation of probabilistic frames is still at its initial stage. For example, in [@WCKO16] the authors introduced the notion of transport duals and used the setting of the Wasserstein metric to investigate the properties of such probabilistic frames. In particular, this setting offers the flexibility to find (non-discrete) probabilistic frames which are duals to a given probabilistic frame. Transport duals are the probabilistic analogues of alternate duals in frame theory [@chri2003; @Larsen]. The main contribution of this paper (Theorem \[maintheorem\]) is to investigate the properties of the canonical Parseval probabilistic frame associated to a given probabilistic frame, see Section \[sec2\] for definitions. To prove this result we approximate a given probabilistic frame with one that is compactly supported and whose frame bounds are controlled in a precise way (Theorem \[density-dpf\]). In the process of proving our main result, we prove a number of results that are of interest on their own right. For example, in Section \[sec2\] we establish a number of new results about the canonical Parseval frame $\Phi^{\dag}$ associated to a frame $\Phi$.
Optimal Parseval probabilistic frames {#sec2}
=====================================
Before proving our main result in Section \[subsec2.3\], we revisit the canonical Parseval frame $\Phi^{\dag}$ associated to a given frame $\Phi=\{\varphi_k\}_{k=1}^N\subset {\mathbb R}^d$. In particular, Section \[subsec2.1\] considers the continuity properties of the map $F(\Phi)=\Phi^\dag$. In Section \[subsec2.2\] we show how a probabilistic frame can be approximated in the $2$-Wasserstein metric by a sequence of finite frames whose bounds are controlled by those of the initial probabilistic frame. While such approximation for probability measures in the $2$-Wasserstein metric is well known [@Villani2009 Theorem 6.18], our key contribution here is the control of the frame bounds of the approximating sequence.
Continuity properties of the canonical Parseval frame {#subsec2.1}
-----------------------------------------------------
In this section we revisited the canonical Parseval frame $\Phi^{\dag}$ associated to a given frame $\Phi=\{\varphi_k\}_{k=1}^N\subset {\mathbb R}^d$. First, we give a new and elementary proof of Theorem \[uniqueness\_prob\_1\].
[Proof of Theorem \[uniqueness\_prob\_1\]]{} We first note that a frame $\Psi\subset {\mathbb R}^d$ is Parseval if the rows of its synthesis matrix are orthonormal. Furthermore, $\Psi \subset {\mathbb R}^d$ is a Parseval frame if and only if $U\Psi$ is a Parseval frame for any $d\times d$ orthogonal matrix $U$.
Now let $\Phi=\{\varphi_i\}_{i=1}^N$ be a frame for ${\mathbb R}^d$. Write $S=\Phi\Phi^T= UDU^T$ for some orthogonal matrix $U$. Observe that $U^T\Phi$ is the matrix of $\Phi$ written with respect to the orthonormal basis given by the rows of $U^T$. In addition, the rows of $U^T\Phi$ are pairwise orthogonal. Let $\Psi=\{\psi_i\}_{i=1}^N\subset {\mathbb R}^d$ be any Parseval frame, then
$$d^2(\Phi, \Psi)=d^2(U^T\Phi, U^T\Psi)=\sum_{i=1}^d\|R_i-P_i\|^2,$$ where $\{R_i\}_{i=1}^d\subset {\mathbb R}^N$ and $\{P_i\}_{i=1}^d\subset {\mathbb R}^N$ denote respectively the rows of $U^T\Phi$ and $U^T\Psi$. Consequently, finding $$\min\{d(\Phi, \Psi)^2=\sum_{i=1}^N\|\varphi_i-\psi_i\|^2: \Psi=\{\psi_i\}_{i=1}^N\subset {\mathbb R}^d, \, \, {\textrm Parseval \, frame}\}$$ is equivalent to finding $$\min \{\sum_{i=1}^d\|R_i-P_i\|^2: \{P_i\}_{i=1}^N\subset {\mathbb R}^N, \, \, {\textrm orthonormal \, set }\}$$ where $\{R_i\}_{i=1}^d$ form an orthogonal set of vectors in ${\mathbb R}^N$.
But $\Phi^\dag=\{S^{-1/2}\varphi_i\}_{i=1}^N$ is a Parseval frame, so its rows form an orthonormal set in ${\mathbb R}^N$. Consequently, $\Phi^\dag$ is a solution to . The uniqueness follows by observing that the (unique) closest orthonormal set to a given orthogonal vectors $\{u_i\}_{i=1}^d\subset {\mathbb R}^N$ is $\{\tfrac{u_i}{\|u_i\|}\}_{i=1}^d.$
Consequently, $$\min\{d(\Phi, \Psi)^2=\sum_{i=1}^N\|\varphi_i-\psi_i\|^2: \Psi=\{\psi_i\}_{i=1}^N\subset {\mathbb R}^d, \, \, {\textrm Parseval \, frame}\}=\sum_{k=1}^d(1-\lambda_k^{-1/2})^2$$ where $\{\lambda_k\}_{k=1}^d\subset (0, \infty)$ are the eigenvalues of $S=\Phi\Phi^T$.
In the remaining part of section we study the continuity properties of the functions that maps a given frame to its canonical Parseval frame. This map $$F: {\mathcal{F}}(N, d) \rightarrow {\mathcal{F}}(N,d)$$ given by $$\label{func-F}
F(\Phi)=F(\{\varphi_i\}_{i=1}^{N}) = S_{\Phi}^{-1/2}(\{\varphi_i\}_{i=1}^{N})=\{S_{\Phi}^{-1/2}\varphi_i\}_{i=1}^N.$$ In fact, our results show that for $0< A\leq B$, $F$ is uniformly continuous on ${\mathcal{F}}_{A, B}$, the set of frames with frame bounds between $A$ and $B$. More specifically,
\[cont-F\] Let $0<A\leq B <\infty$, and $\delta > 0$ be given. Then there exists $\epsilon > 0$ such that given any frame $\Phi=\{\varphi_i\}_{i=1}^N$, with frame bounds between $A$ and $B$, and $N:=N_{\Phi}\geq 2$, for any frame $\Psi=\{\psi_i\}_{i=1}^N$ such that $d(\Phi, \Psi)< \epsilon$ we have $d(F(\Phi), F(\Psi))< \delta.$
Before proving this theorem, we establish a number of preliminary results and make the following remark that will be used in the sequel.
\[rowsframe\] Let $\Phi=\{\varphi_i\}_{i=1}^{N}\in {\mathcal{F}}(N, d)$ be a frame. Then, $S=\Phi \Phi^{T}= ODO^{T}$ where $O$ is a $d\times d$ orthogonal matrix and $D$ is a positive definite diagonal matrix. Fix the orthonormal basis of ${\mathbb R}^d$ whose columns form the matrix $O$ and write each frame vector $\varphi_i$ in this basis. The synthesis matrix of the frame $\Phi$ in the basis $O$ is $$[\Phi]_{O}=O^{T}\Phi.$$ Let $\{R_i\}_{i=1}^d$ be the rows of $[\Phi]_{O}$. We shall refer to $\{R_i\}_{i=1}^d$ as simply the rows of $\Phi$.
\[rows-prop\] Let $\Phi=\{\varphi_i\}_{i=1}^{N} \in {\mathcal{F}}(N, d)$. Denote by $\{R_i\}_{i=1}^{d}$ the rows of $\Phi$ as described by Remark \[rowsframe\]. Let $\epsilon>0$ and $ \Psi=\{\psi_i\}_{i=1}^{N} \in {\mathcal{F}}(N, d)$ be such that $d(\Phi, \Psi) < \epsilon $. Denote by $\{P_i\}_{i=1}^d$ the rows of $\Psi$ when written in the orthonormal basis $O$. Then
1. $\big|\|\ R_i\|-\|P_i\|\big|< \epsilon.$ Furthermore, $\sqrt{A}-\epsilon <\|P_i\|<\sqrt{B}+\epsilon$ for each $i=1, 2, \hdots, d.$
2. $$d(\Phi,F(\Phi)) \geq \sqrt{\sum_{i=1}^{d}\|R_i - \dfrac{R_i}
{\|R_i\|}\|^2}.$$
3. For each $i\in \{1, 2\, \hdots, d\}$ we have $$\bigg\|\dfrac{P_i}{\|P_i\|} - \dfrac{R_i}{\|R_i\|}\bigg\| < \dfrac{2\epsilon}{\sqrt{A}}.$$
4. For each $i\in \{1, 2\, \hdots, d\}$ we have $$0\leq \|P_i-\tfrac{R_i}{\|R_i\|}\|^2-\|P_i-\tfrac{P_i}{\|P_i\|}\|^2\leq \tfrac{4\epsilon}{\sqrt{A}}c+\tfrac{4\epsilon^2}{A},$$ where $c=max(1-\sqrt{A}+\epsilon, \sqrt{B} + \epsilon - 1)$.
<!-- -->
1. This is trivial so we omit it.
2. This follows immediately from the fact that the rows of a Parseval frame are an orthonormal set when written with respect to any orthonormal basis and $\dfrac{R_i}{\|R_i\|}$ is the closest unit norm vector to $R_i$.
3. Since, $d(\Phi, \Psi)<\epsilon,$ we know that $\big|\|P_i\| - \|R_i\|\big| < \epsilon$. Hence $$\bigg\|\dfrac{P_i}{\|P_i\|}\cdot\|R_i\| - R_i\bigg\| \leq \bigg\|\dfrac{P_i}{\|P_i\|}\cdot\|R_i\| - P_i\bigg\| +\|P_i-R_i\|= \big|\|P_i\| - \|R_i\|\big|+\|P_i-R_i\|< 2\epsilon.$$ The result follows by recalling that $\|R_i\|\geq \sqrt{A}$.
4. It is clear that $\|P_i -\dfrac{P_i}{\|P_i\|}\| =|\|P_i\|-1| \leq max(1-\sqrt{A}+\epsilon, \sqrt{B} + \epsilon - 1) = c$. By part (c) we know that $\|\dfrac{P_i}{\|P_i\|} - \dfrac{R_i}{\|R_i\|}\| < \dfrac{2\epsilon}{\sqrt{A}}$. Using the fact hat $\dfrac{P_i}{\|P_i\|}$ is the closest unit norm vector to $P_i$, we see that $$\|P_i - \dfrac{P_i}{\|P_i\|}\| \leq \|P_i - \dfrac{R_i}{\|R_i\|}\| \leq \|P_i - \dfrac{P_i}{\|P_i\|}\| + \dfrac{2\epsilon}{\sqrt{A}}.$$ The result follows by squaring the last inequality.
Finally, we have the following technical lemma, that contains the key argument in the proof of Theorem \[cont-F\].
\[contradiction-unifcont\] Given $0<A\leq B<0$, fix $\Phi=\{\varphi_i\}_{i=1}^N\in {\mathcal{F}}_{A,B}$. Let $\epsilon, \delta>0$ be such that $\dfrac{\delta}{\sqrt{d}} - \dfrac{2\epsilon}{\sqrt{A}}>0$ and $\sqrt{A} - \epsilon>0$. Let $\Psi=\{\psi_i\}_{i=1}^N$ be such that $d(\Phi, \Psi) < \epsilon$, and $d(S_{\Phi}^{-1/2}\Phi, S^{-1/2}_{\Psi}\Psi) =d(\Phi^{\dag}, \Psi^{\dag})> \delta$. Then, $$\sum_{i=1}^{d}(\|P_i - R_{i}'\|^2 - \|P_i - \dfrac{P_i}{\|P_{i}\|}\|^2) \geq min(Cd'^2,C^2),$$ where $d' = \dfrac{\delta}{\sqrt{d}} - \dfrac{2\epsilon}{\sqrt{A}}$, $C = \min(\sqrt{A} - \epsilon, 1)$, and $\{R_i'\}_{i=1}^d\subset {\mathbb R}^d$ is the set of the rows of $S_{\Psi}^{-1/2}\Psi$.
We first show that there exists $k$ then $$\|P_k- R_{k}'\|^2 - \|P_k - \dfrac{P_k}{\|P_{k}\|}\|^2 \geq min(\|R'_{k} - \frac{P_k}{\|P_k\|}\|^2 \cdot min(\|P_k\|,1),\|P_k\|^2).$$
Since $d(S_{\Phi}^{-1/2} \Phi,S_{\Psi}^{-1/2} \Psi) \geq \delta$, then $\|\dfrac{R_k}{\|R_k\|} - R_{k}'\| \geq \dfrac{\delta}{\sqrt{d}}$ for some $k$. By Lemma \[rows-prop\] we know that $\|\dfrac{P_k}{\|P_k\|} - \dfrac{R_k}{\|R_k\|}\| < \dfrac{2\epsilon}{\sqrt{A}}$. It follows from the triangle inequality that $$\|\dfrac{P_k}{\|P_k\|} - R_{k}'\| \geq \dfrac{\delta}{\sqrt{d}} - \dfrac{2\epsilon}{\sqrt{A}} = d'.$$
Suppose that $C =\min(\sqrt{A} - \epsilon, 1)= 1$, or equivalently, $\sqrt{A} - \epsilon \geq 1$. Hence, by Lemma \[rows-prop\] we have $\|P_i\|\geq 1$ for each for all $i$.
Since the angle $\widehat{R_k'\tfrac{P_k}{\|P_k\|}P_i}> \pi/2$, it follows that $$\|P_k - R_{k}'\|^2 > \|P_k - \dfrac{P_k}{\|P_k\|}\|^2 + \|\dfrac{P_k}{\|P_k\|} - R_{k}'\|^2.$$ But since, $\|\dfrac{P_k}{\|P_k\|} - R_{k}'\| \geq \dfrac{\delta}{\sqrt{d}} - \dfrac{2\epsilon}{\sqrt{A}}$, we conclude that $$\|P_k- R_{k}'\|^2 - \|P_k - \dfrac{P_k}{\|P_k\|}\|^2> \|\dfrac{P_k}{\|P_k\|} - R_{k}'\|^2 \geq d'^2 = Cd'^2$$ and we are done.
Assume now $C=\sqrt{A}-\epsilon < 1$ and $\|P_k\|+\eta \leq 1$, where $\eta$ is defined in Figure \[fig:figure1\].
![ $Q$ is the orthogonal projection of $R'_k$ onto $P_k$, and $\eta= \|Q - \frac{P_k}{\|P_k\|}\|$. []{data-label="fig:figure1"}](Ball1.png)
Then, $$\begin{aligned}
\bigg\|P_k - R_{k}'\|^2 - \|P_k - \dfrac{P_k}{\|P_{k}\|}\bigg\|^2 &=
(1 - ( \|P_k\|+ \eta))^2 + 2\eta - \eta^2 - (1-\|P_k\|)^2 \\
&= 2\eta \|P_k\| \\
&= \bigg\|\dfrac{P_k}{\|P_k\|} - R_{k}'\bigg\|^{2}\|P_k\|.\end{aligned}$$ The the conclusion follows from $\|\dfrac{P_k}{\|P_k\|} - R_{k}'\|^{2} \geq d'^2$.
Now assume $\|P_k\|+\eta > 1$ and $\eta \leq 1$, where $\eta$ is defined in Figure \[fig:figure2\].
![$Q$ is the orthogonal projection of $R'_k$ onto $P_k$, and $\eta= \|Q - \frac{P_k}{\|P_k\|}\|$.[]{data-label="fig:figure2"}](Ball2.png)
$$\bigg\|P_k - R_{k}'\|^2 - \|P_k - \dfrac{P_k}{\|P_{k}\|}\bigg\|^2 = ((\|P_k\| + \eta) - 1)^2 + 2\eta - \eta^2 - (1-\|P_k\|)^2 = 2\eta \|P_k\|$$ and the rest of the proof is similar to the one given above.
If $\eta > 1$ where where $\eta$ is defined in Figure \[fig:figure3\], then the angle $\angle P_{k}0R'_{k} > \frac{\pi}{2}$ hence $\|P_k - R_{k}'\|^2 > \|P_k\|^2 + 1$. We know $\|P_k - \dfrac{P_k}{\|P_{k}\|}\|^2 \leq 1$ hence $$\|P_k - R_{k}'\|^2 - \|P_k - \dfrac{P_k}{\|P_{k}\|}\|^2 > \|P_k\|^2 \geq C^2$$
![$Q$ is the orthogonal projection of $R'_k$ onto $P_k$, and $\eta= \|Q - \frac{P_k}{\|P_k\|}\|$.[]{data-label="fig:figure3"}](Ball3.png)
We are now ready to prove Theorem \[cont-F\].
Assume by way of contradiction that there exists $\delta > 0$ such that for all $\epsilon > 0$ there exist $\Phi_{\epsilon}=\{\varphi_{i,\epsilon}\}_{i=1}^{N_{\epsilon}}, \in {\mathcal{F}}_{A, B}.$ and $\Psi_\epsilon=\{\psi_{i,\epsilon}\}_{i=1}^{N_{\epsilon}}$
such that $$d(\Phi_{\epsilon}, \Psi_{\epsilon})< \epsilon$$ and $$d(S_{\Phi_{\epsilon}}^{-1/2}\Phi_{\epsilon}, S_{\Psi_{\epsilon}}^{-1/2}\Psi_{\epsilon}) > \delta.$$ Furthermore, choose $\epsilon$ small enough so that $\dfrac{\delta}{\sqrt{d}} - \dfrac{2\epsilon}{\sqrt{A}}>0$ and $\sqrt{A} - \epsilon>0$ and $$\sum_{i=1}^{d}(\|P_i - \dfrac{R_i}{\|R_i\|}\|^2 - \|P_i - \dfrac{P_i}{\|P_i\|}\|^2) < min(Cd'^2,C^2)$$ where $C$ and $d'^2$ are as in Lemma \[contradiction-unifcont\](such $\epsilon$ exists by Lemma \[rows-prop\]).
Hence $$\sum_{i=1}^{d}(\|P_i - \dfrac{R_i}{\|R_i\|}\|^2 - \|P_i - \dfrac{P_i}{\|P_i\|}\|^2) < \sum_{i=1}^{d}(\|P_i - R_{i}'\|^2 - \|P_i - \dfrac{P_i}{\|P_{i}\|}\|^2)$$ Consequently, $\sum_{i=1}^{d}\|P_i - \dfrac{R_i}{\|R_i\|}\|^2 < \sum_{i=1}^{d}\|P_i - R_{i}'\|^2$ contradicting that $R_{i}'$ are the rows of the closest Parseval frame to $\Psi_{\epsilon}=\{\psi_{i,\epsilon}\}_{i=1}^{N_{\epsilon}}$.
Approximation of probabilistic frames in the $2-$ Wasserstein metric {#subsec2.2}
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In this section we prove some of the technical results needed to establish our main result. The key idea is that a probabilistic frame $\mu$ with frame bounds $A, B$ can be approximated in the Wasserstein metric by a finite probabilistic frame whose bounds are arbitrarily close to $A, B$. We prove this statement in Proposition \[density-dpf\] and point out that it is a refinement of a well-known result, e.g., [@Villani2009 Theorem 6.18]. But first, we prove a few new results about finite probabilistic frames that are of interest in their own right. In particular, Lemma \[frame\_modi\] will be a very useful technical tool that we shall often use. It shows that given a finite frame we may replace any frame vector by a finite number of new vectors so as to leave unchanged the frame operator. More specifically,
\[frame\_modi\] Given a frame $\Phi=\{\varphi_{i}\}_{i=1}^N$ with frame operator $S_\Phi$. Fix $i\in \{1, 2, \hdots, N\}$ and consider the new set of vectors $$\Phi_i=\{\varphi_k\}_{k=1, k\neq i}^N \cup\{a_{j}\varphi_i\}_{j=1}^{p}= \{\varphi_k'\}_{k=1}^{N+p-1}$$ where $\sum_{j=1}^{p}a_{j}^{2} = 1$. Then, $\Phi_i\in {\mathcal{F}}(N+p-1, d)$, that is, $\Phi_i$ is a frame for ${\mathbb R}^d$ and its frame operator is $S_{\Phi}$. Furthermore, $$\sum_{k=1}^{N}\|\varphi_k- \varphi_k^{\dag}\|^2 = \sum_{k=1}^{N+p-1}\|\varphi_k' - \varphi_k^{' \dag} \|^2$$ where $\varphi_k^{\dag}=S^{-1/2}\varphi_k$ and $\varphi_k^{'\dag}=S^{-1/2}\varphi'_k$
It is easy to see that for each $x\in {\mathbb R}^d$ we have $$\sum_{k=1}^{N}|{\left\langle x,\varphi_k\right\rangle}|^2 = \sum_{k=1}^{i-1}|{\left\langle x,\varphi_k\right\rangle}|^2 + \sum_{j=1}^{p}a_j^2|{\left\langle x,\varphi_i\right\rangle}|^2 + \sum_{k=i+1}^{N}|{\left\langle x,\varphi_k\right\rangle}|^2 .$$
We now use Lemma \[frame\_modi\] and Theorem \[uniqueness\_prob\_1\] to find the closest Parseval frame to a finite probabilistic frame in the $2$-Wasserstein metric.
\[dists-to-parseval\] Let $\mu_{\Phi, w}$ be a finite probabilistic frame with bounds $A$ and $B$, where $\Phi=\{\varphi_{i}\}_{i=1}^N \subset {\mathbb R}^{d}$ and $w=\{w_i\}_{i=1}^N \subset [0, \infty)$. Then the closest finite Parseval probabilistic frame to $\Phi$ is $\Phi^\dag=\{S^{-1/2}\varphi_i\}_{i=1}^N$ and it satisfies
$$W_{2}(\mu_{\Phi, w}, \mu_{\Phi^{\dag}, w})=\sqrt{\sum_{i=1}^{N}w_{i}\|\varphi_i - \tilde{\varphi}_i\|^2} \leq \sqrt{d\, \max((\sqrt{A} - 1)^2,(\sqrt{B} - 1)^2)}$$ where $\tilde{\varphi}_i=S^{-1/2}\varphi_i$.
We first prove that $W_{2}^2(\mu_{\Phi, w}, \mu_{\Phi^{\dag}, w}) \leq d\, \max((\sqrt{A} - 1)^2,(\sqrt{B} - 1)^2).$
Let $\Phi_w=\{\sqrt{w_i}\varphi_{i}\}_{i=1}^N$. Let $S=\Phi_w\Phi^T_w=ODO^T$ be the frame operator of $\Phi_w$. Consider the columns of $O$ as an orthonormal basis for ${\mathbb R}^d$. Writing the vectors $\sqrt{w}_k\varphi_k$ with respect to this basis leads to $\Phi_w'=O^T\Phi_w$ where $$\Phi_w = \left( \begin{array}{ccc}
| & ... & | \\
\sqrt{w_1}\varphi_1 & ... & \sqrt{w_n}\varphi_m \\
| & ... & | \end{array} \right)$$ Let $\{P_{k, w}\}_{k=1}^d$ and $\{R_{k, w}\}_{k=1}^d$ respectively denote the rows of $\Phi_w'$ and $\Phi_w$. Notice that $$\sqrt{A}\leq \|P_{k,w}\|\leq \sqrt{B}, \quad \forall\, k=1, 2, \hdots, d.$$ It is easily seen that
$$\min_{u\in {\mathbb R}^d, \|u\|=1}\|P_{k,w}-u\|^2=\|P_{k,w}-\tfrac{P_{k,w}}{\|P_{k,w}\|}\|^2=|\|P_{k,w}\|-1|^2\leq \max((\sqrt{A} - 1)^2,(\sqrt{B} - 1)^2).$$ But by construction, ${\left\langle P_{k,w},P_{\ell,w}\right\rangle}=0$ for $k\neq \ell$, and $ \tfrac{P_{k,w}}{\|P_{k,w}\|}=\lambda_k^{-1/2}P_{k,w}$ where $\lambda_k$ is the $k^{th}$ eigenvalue of $S$. Consequently, $\{\lambda_k^{-1/2}P_{k,w}\}_{k=1}^d$ represents the rows of the canonical tight frame $S^{-1/2}\Phi_w$ written in the orthonormal basis $O$. Therefore,
$$d(\Phi_w, S^{-1/2}\Phi_w)^2=\sum_{k=1}^d\|P_{k,w}-\lambda_k^{-1/2}P_{k,w}\|^2\leq d \max((\sqrt{A} - 1)^2,(\sqrt{B} - 1)^2).$$
Clearly, $$W_{2}^2(\mu_{\Phi, w}, \mu_{S^{-1/2}\Phi, w}) \leq \sum_{i=1}^{N}w_{i}\|\varphi_{i} - S^{-1/2}\varphi_{i}\|^2=d(\Phi_w, S^{-1/2}\Phi_w)^2\leq d \max((\sqrt{A} - 1)^2,(\sqrt{B} - 1)^2) .$$ Suppose there exists a finite probabilistic Parseval frame $\mu_{\Psi, v}$ where $\Psi =\{\psi_i\}_{i=1}^{M}\subset {\mathbb R}^d$, $v=\{v_i\}_{i=1}^M\subset [0, \infty)$ such that $$W_{2}^2(\mu_{\Phi, w}, \mu_{\Psi, v}) < \sum_{i=1}^{N}w_{i}\|\varphi_{i} - S^{-1/2}\varphi_{i}\|^2.$$ Let $\gamma \in \Gamma (\mu_{\Phi, w}, \mu_{\Psi, v})$ be such that $$W_2^2(\mu_{\Phi, w}, \mu_{\Psi, v}) =\iint_{{\mathbb R}^{2d}} \|x-y\|^2d\gamma(x,y).$$ Note that $\gamma$ is a discrete measure with $\gamma(x,y)=\sum_{i, j}w'_{i,j}\delta_{\varphi_{i}}(x)\delta_{\psi_{i}}(y)$ with $\sum_{j}w'_{i,j} = w_i$ and $\sum_{i}w'_{i,j} = v_j$.
Furthermore, by assumption $$W_{2}^2(\mu_{\Phi, w}, \mu_{\Psi, v})=\sum_{i,j}w'_{i,j}\|\varphi_i - \psi_j\|^2 < \sum_{i=1}^{N}w_{i}\|\varphi_{i} - S^{-1/2}\varphi_{i}\|^2.$$ Notice since $\sum_{i}w'_{i,j} = v_j$ the frame $\Psi'=\{\sqrt{w'_{i,j}}\psi_j\}_{i,j}$ is a Parseval frame. Since $\sum_{j}w'_{i,j} = w_i$, it easy to see that $\sum_{j} \tfrac{w'_{i,j}}{w_i} =1$. We now use Lemma \[frame\_modi\]. For each $i,$ replace $\sqrt{w_i}\varphi_i$ with $\{\sqrt{w_{i,j}'}\varphi_i\}_{j}$. This results in a frame $\Phi'=\{\sqrt{w_{i,j}'}\varphi_i\}_{i, j}$. Consequently, $d(\Phi', \Psi')=d(\Phi_w, \Psi_v)<d(\Phi_w, \Phi_w^{\dag})$ where $\Psi_v$ is a Parseval frame. This is a contradiction.
The next result is one of our key technical results. It allows us to approximate a probabilistic frame in the $2$-Wasserstein metric with a compactly supported finite probabilistic frame whose bounds are controlled by those of the original probabilistic frame.
\[density-dpf\] Let $\mu$ be a probabilistic frame with frame bounds $A$ and $B$, and $\epsilon > 0$. Then, there exists a finite probabilistic $\mu_{\Phi}$ with frame bounds $A', B'$ such that $A'\geq A-\epsilon$, $B'\leq B+\epsilon$ and $$\|\mu - \mu_{\Phi}\|_{W_2}<\epsilon.$$
To establish this result we first prove the following two Lemmas.
\[prop-2\] Let $\mu$ be a probabilistic frame with frame bound $A$ and $B$. Given $\epsilon>0$, there exists a probabilistic frames $\nu$ with compact support and frame bounds $A', B'$ such that
1. $W_2^2(\mu, \nu)< \epsilon$,
2. $A'\geq A-\epsilon$, and $B'=B$.
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1. Let $\mu$ be a probabilistic frame with frame bound $A$ and $B$. Given $\epsilon>0$, there exists $R_1>0$ such that $$\int_{{\mathbb R}^{d}\setminus B(0, R_1)} \|x\|^{2}d\mu(x) < \epsilon.$$
Let $\nu$ be the measure defined for each Borel set $A\subset {\mathbb R}^d$ by $$\nu(A) = \mu(A\bigcap B(0,R_1)+ \mu({\mathbb R}^d\setminus B(0,R_1))\delta_0.$$ Clearly, $\nu$ is a probabilistic measure with compact support.
We consider the marginal $\gamma$ of $\mu$ and $\nu$ defined for each Borel sets $A, B\subset {\mathbb R}^d$ by $$\gamma(A\times B)= \left\{ \begin{array} {r@{\quad {\textrm if} \quad}l}
\mu(A\bigcap B(0,R_1) \bigcap B) + \mu(A\bigcap B^{c}(0,R_1) & 0\in B\\
\mu(A\bigcap B(0,R_1) \bigcap B) & 0\not\in B \end{array}\right.$$ Since $\nu $ is supported in $B(0,R_1)$ $$\begin{aligned}
\iint_{{\mathbb R}^{2d}}\|x-y\|^{2}d\gamma(x,y) &= \iint_{{\mathbb R}^d \times B(0,R_1)}\|x-y\|^{2}d\gamma(x,y)\\
&=\iint_{B(0,R_1) \times B(0,R_1)}\|x-y\|^{2}d\gamma(x,y) \\
&+ \iint_{B^{c}(0,R_1)\times B(0,R_1)}\|x-y\|^{2}d\gamma(x,y).
\end{aligned}$$ However, we know $$\int_{B(0,R_1)\times B(0,R_1)}\|x-y\|^{2}d\gamma(x,y)=0$$ since, when restricted to $B(0,R_1)\times B(0,R_1)$, $\gamma$ is supported only on the diagonal where $\|x-y\| = 0.$ Moreover, $$\begin{aligned}
\int_{B^{c}(0,R_1)\times B(0,R_1)}\|x-y\|^{2}d\gamma(x,y) &=\iint_{B^{c}(0,R_1) \times B(0,R_1)\setminus \{0\}}\|x-y\|^{2}d\gamma(x,y)\\
& + \iint_{B^{c}(0,R_1)\times \{0\}}\|x-y\|^{2}d\gamma(x,y)\\
&=0+\iint_{B^{c}(0,R_1)\times \{0\}}\|x-y\|^{2}d\gamma(x,y)\\
&<\epsilon.\end{aligned}$$ Therefore, $W_2^2(\mu, \nu)<\epsilon.$
2. The upper bound $B$ is obtained trivially as $\nu$ is $\mu$ restricted to $B(0,R_1)$.
For $x\in {\mathbb R}^d$ we have $\int|\left\langle x,y\right\rangle|^{2}d\nu(y) = \int_{B(0,R_1)}|\left\langle x,y\right\rangle|^{2}d\mu(y).$ From the fact that $\int_{{\mathbb R}^d \setminus B(0,R_1)}\|x\|^2d\mu(x) \leq \epsilon$ it follows that $$\int_{{\mathbb R}^d \setminus B(0,R_1)}|\left\langle x,y\right\rangle|^{2}d\mu(y)\leq \|x\|^2\epsilon.$$
Suppose that $\mu$ is a probabilistic frame supported in a ball $B(0, R)$. Let $r>0$ and consider $Q=[0, r)^d$. Choose points $\{c_k\}_{k=1}^M\subset {\mathbb R}^d$ with $c_1=0$ such that $B(0, R)=\cup_{k=0}^M Q_k$ where $Q_k=c_k+Q$. Observe that $Q_k\cap Q_\ell=\emptyset$ whenever $k\neq \ell$. Let $\mu_{1,Q}=\sum_{k=1}^M\mu(Q_k)\delta_{c_k}$.
Next partition each cube $Q_k$ uniformly into cube of size $r/2$ and construct the probability measure $\mu_{2,Q}$ as above. Iterate this process to construct a sequence of probability measures $\mu_{n, Q}$.
\[prop-3\] Let $\mu$ be a probabilistic frame with bounds $A$ and $B$, which supported in a ball $B(0, R)$. For $r>0$ let $\{\mu_{n,Q}\}_{n=1}^{\infty}$ be a sequence of probability measures as constructed above. Then, $$lim_{n\to \infty}W_2(\mu, \mu_{n,Q})=0.$$ Furthermore, there exists $N$ such that for all $n\geq N$, $\mu_{n, Q}$ is a finite probabilistic frame whose bounds are arbitrarily close to those of $\mu$.
Let $d=\max_{x\in Q_k}\|x-c_k\|$. Given, $x\in Q_k$, $x=c_k + a_k$, where $\|a_k\|\leq d$.
For any $x\in {\mathbb R}^d$, $$\begin{aligned}
\bigg|\int_{B(0, R)}{\left\langle x,y\right\rangle}^2d\mu(y)-\sum_{k=1}^M{\left\langle x,c_k\right\rangle}^2\mu(Q_k)\bigg|&= \bigg|\sum_{k=1}^M\int_{Q_k}{\left\langle x,y\right\rangle}^2 d\mu(y)-\sum_{k=1}^M{\left\langle x,c_k\right\rangle}^2\mu(Q_k)\bigg|\\
&=\bigg|\sum_{k=1}^M\int_{Q_k}({\left\langle x,y\right\rangle}^2-{\left\langle x,c_k\right\rangle}^2)d\mu(y)\bigg|\\
&\leq \sum_{k=1}^M\int_{Q_k}\big|{\left\langle x,y\right\rangle}^2-{\left\langle x,c_k\right\rangle}^2\big|d\mu(y)\\
&=\sum_{k=1}^M\int_{Q_k}|{\left\langle x,c_k+a_k\right\rangle}^2-{\left\langle x,c_k\right\rangle}^2|d\mu(y)\\
&=\sum_{k=1}^M\int_{Q_k}|{\left\langle x,a_k\right\rangle}^2+2{\left\langle x,c_k\right\rangle}{\left\langle x,a_k\right\rangle}|d\mu(y)\\
&\leq \|x\|^2\sum_{k=1}^M\mu(Q_k)(\|a_k\|^2+2\|c_k\|\|a_k\|)\\
&\leq (d^2+2d(R+d))\|x\|^2.\end{aligned}$$
Note that by the iterative construction of $\mu_{n, Q}$ we get that for each $x\in {\mathbb R}^d$ $$\bigg|\int_{{\mathbb R}^{d}} {\left\langle x,y\right\rangle}^2d\mu(y)-\int_{{\mathbb R}^{d}} {\left\langle x,y\right\rangle}^2d\mu_{n, Q}(y)\bigg|\leq (d_{n}^2+2d_{n}(R+d_{n}))\|x\|^2$$ where $\lim_{n\to \infty}d_n=0$. It follows that given $\epsilon>0$, we can find $N>1$ such that for all $n\geq N$,
$$\int_{{\mathbb R}^{d}} {\left\langle x,y\right\rangle}^2d\mu_{n, Q}(y)>\int_{{\mathbb R}^{d}} {\left\langle x,y\right\rangle}^2d\mu(y)-\epsilon\|x\|^2>\|x\|^2(A-\epsilon)$$ which concludes that $\mu_{n, Q}$ is a a finite probabilistic frame whose lower bound is at least $A-\epsilon$. Furthermore, $$\int_{{\mathbb R}^{d}} {\left\langle x,y\right\rangle}^2d\mu_{n, Q}(y)<\int_{{\mathbb R}^{d}} {\left\langle x,y\right\rangle}^2d\mu(y)+\epsilon\|x\|^2\leq \|x\|^2(B+\epsilon)$$ which implies that the upper frame bound $\mu_{n, Q}$ is at most $B+\epsilon$.
Next, fix $n\geq N$ and let $\gamma_n(x,y)$ be the measure on ${\mathbb R}^d\times {\mathbb R}^d$ be defined for any Borel sets $ A, B \subset {\mathbb R}^d$ by:
$$\gamma_n(A\times B)=\sum_{k: c_k\in B}\mu(A\cap Q_k)=\sum_{k=1}^M\mu_{|_{Q_k}}\times \delta_{c_{k}}(A\times B)$$ where $A, B$ $c_k$ denoting the centers of the cubes $Q_{k}$. It is easy to see that $\gamma_n \in \Gamma(\mu, \mu_{n, Q})$ and so $$\begin{aligned}
W_2^2(\mu, \mu_{n, Q})&\leq \iint\|x-y\|^2d\gamma_n(x, y)\\
&=\sum_{k=1}^{M}\iint \|x-y\|^2d(\mu_{|_{Q_{k}}}\times \delta_{c_{k}})(x,y)\\
&=\sum_{k=1}^M \int_{Q_k}\|x-c_k\|^2d\mu(x)\\
&\leq \sum_{k=1}^M\mu(Q_k)\int_{Q_k}d_n^2d\mu(x)\\
&\leq d_n^2\end{aligned}$$ and the result follows from the fact that $\lim_{n\to \infty}d_n=0$.
We can now give a proof of Theorem \[density-dpf\].
Let $\mu$ be a probabilistic frame with frame bounds $A$ and $B$, and $\epsilon > 0$. By Lemma \[prop-2\] let $\nu$ be a compactly supported probabilistic frame with frame bounds between $A-\epsilon/2$ and $B$ and such that $W_2(\mu, \nu)< \epsilon/2$.
By Lemma \[prop-3\] we know there exists a finite probabilistic frame $\mu_{\Phi, w}$ whose frame bounds are within $\epsilon/2$ of that of $\nu$ and such that $W_2(\nu, \mu_{\Phi, w})< \epsilon/2 $. Consequently, $W_2(\mu, \mu_{\Phi, w})<\epsilon$ which concludes the proof.
\[appro-prob\] Let $\mu$ be a probabilistic Parseval frame and $\epsilon > 0.$ Then, there exists a finite Parseval probabilistic frame $\mu_{\Phi, w}$ with $$W_{2}(\mu, \mu_{\Phi, w})< \epsilon.$$
This follows from Proposition \[dists-to-parseval\] and Theorem \[density-dpf\].
Since the set of finite Parseval frames is dense in the set of all Parseval frames in the Wasserstein metric, by Proposition 2.6 since there is no finite Parseval frame closer to $\Phi$ than $\Phi^\dag=\{S^{-1/2}\varphi_i\}_{i=1}^N$, there are no Parseval frame closer to $\Phi$ than $\Phi^\dag$.
The closest Parseval frame in the $2-$Wasserstein distance {#subsec2.3}
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In this section we prove and state of our main result, Theorem \[maintheorem\]. We recall that if $\mu$ is a probabilistic frame for ${\mathbb R}^d$, then its probabilistic frame operator (equivalently, the matrix of second moments associated to $\mu$)
$$S_\mu:{\mathbb R}^d\rightarrow {\mathbb R}^d,\qquad S_\mu (x) = \int_{{\mathbb R}^d} {\left\langle x,y\right\rangle} y d\mu(y)$$ is positive definite, and thus $S_\mu^{-1/2}$ exists. We define the push-forward of $\mu$ through $S_{\mu}^{-1/2}$ by $$\mu^{\dagger} (B) =\mu(S^{1/2} B)$$ for each Borel set in ${\mathbb R}^d$. Alternatively, if $f$ is a continuous bounded function on ${\mathbb R}^d$, $$\int_{{\mathbb R}^d} f(y)d \mu^{\dag}(y) = \int_{{\mathbb R}^d} f(S_{\mu}^{-1/2}y) d\mu(y).$$
It then follows that $$x=S_{\mu}^{-1/2}S_{\mu} S_{\mu}^{-1/2}(x)= \int_{{\mathbb R}^d} {\left\langle S_{\mu}^{-1/2} x,y\right\rangle} \, S_{\mu}^{-1/2}y \, d\mu(y)= \int_{{\mathbb R}^d} {\left\langle x, y\right\rangle}\, y \, d\mu^{\dagger} (y)$$ implying that $\mu^{\dag}$ is a Parseval probabilistic frame [@EhlOko2013; @KOPrecondPF2016]. In particular, $S_{\mu^{\dag}}=I$ where $I$ is the identity matrix on ${\mathbb R}^d$. As was the case with the canonical Parseval frame $\Phi^{\dag}$ of a given frame $\Phi$, $\mu^{\dag}$ is the (unique) closest Parseval probabilistic frame to $\mu$.
\[maintheorem\] Let $\mu$ be a probabilistic frame on ${\mathbb R}^d$ with probabilistic frame operator $S_\mu$. Then $\mu^{\dag}$ is the (unique) closest probabilistic Parseval frame to $\mu$ in the $2-$Wasserstein metric, that is $$\label{optimum}
\mu^{\dag}=\textrm{arg} \min W_{2}^{2}(\mu, \nu)$$ where $\nu $ ranges over all Parseval probabilistic frames.
Before proving this theorem, we need to establish a few preliminary results. We start by extending Theorem \[cont-F\] to finite probabilistic frames in the Wasserstein metric. In particular, this extension allows use to deal with finite probabilistic frames of different cardinalities.
\[continuityFPF\] Let $0<A\leq B <\infty$, and $\delta > 0$ be given. Then there exists $\epsilon > 0$ such that given any finite probabilistic frame $\mu_{\Phi, w}=\sum_{i=1}^Nw_i\delta_{\varphi_i}$ with frame bounds between $A$ and $B$, $N:=N_{\Phi}\geq 2$, $\Phi=\{\varphi_i\}_{i=1}^N\subset {\mathbb R}^d$, and weights $w=\{w_i\}_{i=1}^N \subset [0, \infty)$, for any finite probabilistic frame $\mu_{\Psi, \eta}=\sum_{i=1}^M\eta_i\delta_{\psi_i},$ $M:=M_{\Psi}\geq 2$, where $\Psi=\{\psi_i\}_{i=1}^M\subset {\mathbb R}^d$, and weights $\eta=\{\eta_i\}_{i=1}^N \subset [0, \infty)$ if $W_2(\mu_{\Phi, w}, \mu_{\Psi, \eta})< \epsilon,$ then we have $$W_2(F(\mu_{\Phi, w}), F(\mu_{\Psi, \eta})) < \delta.$$
Fix $\delta>0$. By Theorem \[cont-F\] we know that there exists $\epsilon$ such that given a frame $X=\{x_i\}_{i=1}^M$ ($M\geq 2$ is arbitrary) with frame bounds between $A$ and $B$, and $Y=\{y_{i}\}_{i=1}^M$ is a frame such that $$d(X, Y)=\sqrt{\sum_{i=1}^{M}\|x_{i} - y_{i}\|^2} < \epsilon$$ then $$d(F(X), F(Y))=d(S^{-1/2}_{X}X, S^{-1/2}_{Y}Y) < \delta.$$
Let $\mu_{\Phi, w}=\sum_{i=1}^N w_i\delta_{\varphi_i}$ be a finite probabilistic frame with frame bounds between $A$ and $B$, $N \geq 2$, $\Phi=\{\varphi_i\}_{i=1}^N\subset {\mathbb R}^d$, and weights $w=\{w_i\}_{i=1}^N \subset [0, \infty)$. Then by Theorem \[dists-to-parseval\], $\mu_{\Phi^{\dag}, w}$ where $\Phi^{\dag}=\{S^{-1/2}_{\Phi}\varphi_i\}_{i=1}^N$ is the closest Parseval frame to $\mu_{\Phi, w}$.
Let $\mu_{\Psi, v}$ where $\Psi=\{\psi_i\}_{i=1}^M$, $M\geq 2$ such that $W_2(\mu_{\Phi, w}, \mu_{\Psi, \eta})< \epsilon$. Choose $\gamma \in \Gamma(\mu_{\Phi, w}, \mu_{\Psi, v})$ such that $$W_2(\mu_{\Phi, w}, \mu_{\Psi, \eta})^2=\iint_{{\mathbb R}^d \times {\mathbb R}^d}\|x-y\|^2d\gamma(x,y)< \epsilon^2.$$ Identify $\gamma$ with $\{w_{i,j}\}_{i, j=1}^{N, M}$. Then,
$$W_2(\mu_{\Phi, w}, \mu_{\Psi, \eta})^2=\iint_{{\mathbb R}^d \times {\mathbb R}^d}\|x-y\|^2d\gamma(x,y)=\sum_{i=1}^M\sum_{j=1}^Nw_{i,j}\|\varphi_i-\psi_j\|^2< \epsilon^2.$$
Observe that $\Phi'=\{\sqrt{w_{i,j}}\varphi_i\}_{i, j=1}^{M,N}$ is a frame whose frame bounds are the same as those for $\mu_{\Phi,w}$. Similarly, $\Psi'=\{\sqrt{w_{i,j}}\psi_j\}_{i, j=1}^{M,N}$ is a frame whose frame bounds are the same as those for $\mu_{\Psi,\eta}.$ Furthermore,
$$d(\Phi', \Psi')=W_2(\mu_{\Phi, w}, \mu_{\Psi, \eta})<\epsilon$$ which implies that
$$d(F(\Phi'), F(\Psi'))^2= \sum_{i, j=1}^{M,N}\|S^{-1/2}_{\Phi}(\sqrt{w_{i,j}}\varphi_i) - S^{-1/2}_{\Psi}(\sqrt{w_{i,j}}\psi_j)\|^2< \delta^2.$$
However, $$\sum_{i,j}\|S^{-1/2}_{\Phi}(\sqrt{w_{i,j}}\varphi_i) - S^{-1/2}_{\Psi}(\sqrt{w_{i,j}}\psi_j)\|^2 = \sum_{i,j}w_{i,j}\|S^{-1/2}_{\Phi}\varphi_i - S^{-1/2}_{\Psi}\psi_j\|^2$$
But since $w_{i,j} = \gamma(\{\varphi_i\}, \{\psi_j\})$ we have $\sum_{j}w_{i,j} = w_i$ and $\sum_{i}w_{i,j} = v_j$ we see that $$W^2_2(F(\mu_{\Phi, w}), F(\mu_{\Psi, \eta}))=W_2^2(\mu_{\Phi^{\dag}, w}, \mu_{\Psi^{\dag}, v}) \leq \sum_{i,j}w_{i,j}\|S^{-1/2}_{\Phi}\varphi_i - S^{-1/2}_{\Psi}\psi_j\|^2.$$
Let $DPF(A, B)$ denote the set of all discrete (finite) probabilistic frames in ${\mathbb R}^d$ whose lower frame bounds are less than or equal to $A$ and whose upper bounds are greater or equals to $B$. It follows from the above result that $F$ is uniformly continuous from $DPF(A, B)$ into itself when equipped with the Wasserstein metric. Consequently, we can prove the following result.
\[well-defined-F\] Let $\mu$ be a probabilistic frame with frame bounds $A$ and $B$. Let $\mu_k:=\mu_{\Phi_{k}, w_k}$, where $\Phi_{k}:=\Phi_{k, w_{k}}=\{\varphi_{k}\}_{k=1}^{N_k}$ and $\nu_k:=\mu_{\Psi_{k}, v_{k}}$, where $\Psi_{k}:=\Psi_{k, v_{k}}=\{\psi_{k}\}_{k=1}^{M_k}$ be two sequences of finite probabilistic frames in ${\mathbb R}^d$ such that $\lim_{k\to \infty}W_{2}(\mu, \mu_{\Phi_{k}})=\lim_{k\to \infty}W_2(\mu, \mu_{\Psi_{k}})=0$. Furthermore, suppose that the frame bounds of $\mu_{\Phi_{k}}$ are between $A/2$ and $B+A/2$. Then $$\lim_{k\to \infty}F(\mu_{\Phi_{k}})=\lim_{k\to \infty}F(\mu_{\Psi_{k}}).$$
Theorem \[density-dpf\] ensures the existence of the finite probabilistic frames $\mu_{\Phi_{k}}$.
Let $\delta>0$ be given. By Theorem \[continuityFPF\] there exists $\epsilon >0$ such that for any finite probabilistic frame $\nu$ and any $k\geq 1$, $$W_2(\mu_{\Phi_{k}}, \nu)< \epsilon \implies W_2(F(\nu), F(\mu_{\Phi_{k}}))< \delta.$$
Choose $N_\epsilon>1$ such that for all $k>N_{\epsilon}$, $W_{2}(\mu,\mu_{\Phi_{k}}) < \frac{\epsilon}{2}$ and $W_{2}(\mu,\mu_{\Psi_{k}}) < \frac{\epsilon}{2}$. Thus, for $k\geq N_\epsilon$, $W_{2}(\mu_{\Phi_{k}},\mu_{\Psi_{k}}) < \epsilon$, which implies that for all $k\geq N_{\epsilon}$, $W_{2}(F(\mu_{\Phi_{k}}),F(\mu_{\Psi_{k}})) < \delta$. It easily follows that $\lim_{k\to \infty}F(\mu_{\Phi_{k}})=\lim_{k\to \infty}F(\mu_{\Psi_{k}})$.
We can now use this proposition to extend the definition of the map $F$ to all probabilistic frames. Let $\mu$ be a probabilistic frame with bounds $0<A\leq B<\infty.$ Let $\{\mu_{\Phi_{k}}\}_{k=1}^{\infty}$ be a sequence of finite probabilistic frames with bounds between $A/2$ and $B+A/2$ such that $lim_{k\to \infty}W_2(\mu_{\Phi_{k}}, \mu)=0$. Then, $$F(\mu)=\lim_{k\to \infty}F(\mu_{\Phi_{k}})$$ is well-defined. Before proving Theorem \[maintheorem\] we first identify the minimizer of with $F(\mu)$.
\[optimizer1\] Let $\mu$ be a probabilistic frame on ${\mathbb R}^d$ with probabilistic frame operator $S_\mu$. Then $F(\mu)$ is the unique closest probabilistic Parseval frame to $\mu$ in the $2-$Wasserstein metric, that is $F(\mu)$ is the unique solution to .
Set $Q=\min W_{2}(\mu, \nu)$ where $\nu $ ranges over all Parseval probabilistic frames.
Let $\delta>0$, and $\mu$ be a probabilistic frame wth frame bounds $A$ and $B$. By Theorem \[density-dpf\], there exists a sequence of finite probabilistic frame $\mu_{\Phi_{k}}$ with frame bounds between $\frac{A}{2}$ and $B + \frac{A}{2}$ where $\Phi_{k}:=\Phi_{k, w(k)}=\{\varphi_{k}\}_{k=1}^{N_k} \subset {\mathbb R}^d$, $w(k)=\{w_n\}_{n=1}^{N_k}\subset (0, \infty)$, and $N_k\geq 2$ such that $\lim_{k\to \infty}W_2(\mu, \mu_{\Phi_{k}})=0$.
Observe that for all $k\geq 1$, $$W_{2}(\mu, F(\mu_{\Phi_{k}}))\leq W_2(\mu, F(\mu)) + W_2(F(\mu), F(\mu_{\Phi_{k}})).$$ Choose $\epsilon>0$ as in Theorem \[continuityFPF\] and pick $K\geq 1$ such that $W_2(\mu, \mu_{\Phi_{K}})< \epsilon.$ Thus, $W_2(F(\mu), F(\mu_{\Phi_{K}}))< \delta.$ Consequently,
$$W_{2}(\mu, F(\mu_{\Phi_{K}}))\leq W_2(\mu, F(\mu)) + W_2(F(\mu), F(\mu_{\Phi_{K}}))< W_2(\mu, F(\mu)) + \delta.$$ Since $F(\mu_{\Phi_{K}})$ is a Parseval frame we conclude that $F(\mu)$ minimizes .
We now prove that $F(\mu)$ is the unique minimizer of by considering three cases.
[**Case 1.**]{} If $\mu$ is a finite frame $\Phi=\{\varphi_i\}_{i=1}^N\subset {\mathbb R}^d$, it is known that $S^{-1/2}\Phi$ is the (unique) closest Parseval frame to $\Phi$, see Theorem \[uniqueness\_prob\_1\], and [@CasKut07 Theorem 3.1].
[**Case 2.**]{} If $\mu=\mu_{\Phi, w}$, where $\Phi=\{\varphi_i\}_{i=1}^N\subset {\mathbb R}^d$, and $w=\{w_i\}_{i=1}^N \subset [0, \infty)$. Then, $\mu_{\Phi^{\dag}, w}$ where $\Phi^{\dag}=S^{-1/2}\Phi$ is the unique closest Parseval probabilistic frame to $\Phi$. Indeed, we already know that $\mu_{\Phi^{\dag}, w}$ achieves the minimum distance Proposition \[dists-to-parseval\]. We now prove that it is unique. We argue by contradiction and assume that there exists another Parseval probabilistic frame $\nu$ that achieves this distance.
First, we assume that $\nu=\mu_{\kappa, v}$ where $\kappa =$ $\{\kappa'_i\}_{i=1}^{M}\subset {\mathbb R}^d$ with weights $v=\{v_i\}_{i=1}^M\subset [0, \infty)$. Let $\gamma \in \Gamma(\mu, \nu)$ such that $$W_2(\mu, \nu)^2=\iint\|x-y\|^2d\gamma(x,y).$$ For all $i$, $j$ let $w_{i,j} = \gamma(\varphi_i,\kappa'_j)$. Let $Q = \sum_{i=1}^{N}w_i\|\varphi_i - \varphi_i^{\dag}\|^2$, where $\varphi_i^{\dag}=S^{-1/2}\varphi_i$. Since $\kappa$ also achieved this distance we clearly have $Q = \sum_{i,j}w_{i,j}\|\varphi_i - \kappa'_j\|^2$.
We now use Lemma \[frame\_modi\]. For each $i$, we replace the vector $\varphi_i$ and its weight $w_i$ by $M$ copies of itself (i.e., $\varphi_i$ ) each weighted by $w_{i,j}$. Apply the same procedure to $\Phi^{\dag}$, and to $\kappa$, except that for the latter we break each vector $\kappa_j'$ into $N$ copies of itself with weights $w_{i,j}$. Denote by $F_1, F_2,$ and $F_3$ the three resulting frames. We note that the vectors in each of these frames can be considered to have weight $1$.
It follows from Theorem \[uniqueness\_prob\_1\] that the finite frame $F_3=\{\sqrt{w_{i,j}}\kappa'_j\}_{i,j}$ is the (unique) closest Parseval frame to $F_1=\{\sqrt{w_{i,j}}\varphi_i\}_{i,j}$, which we also know is $F_2=\{\sqrt{w_{i,j}}\varphi_i^{\dag}\}_{i,j}$. Therefore, $\mu_{\kappa, v}=\mu_{\Phi^{\dag}, w}$.
Next, we assume that $\nu$ is not discrete. Choose a sequence of finite Parseval frames $\{\nu_n\}_{n}^\infty$ such that $\lim_{n \to \infty}W_2(\nu_n, \nu)=0$. Hence, $$Q=W_{2}(\mu, F(\mu))=W_2(\mu_{\Phi, w}, \nu)=\lim_{n \to \infty}W_2(\mu_{\Phi, w}, \nu_n).$$ We now prove that $$\lim_{n\to \infty}W_2(\nu_n, \mu_{\Phi^{\dag}, w})=0.$$
Let $\delta > 0$ and choose $N\geq 1$ such that for all $n>N$ $$W_2(\nu_n, \mu_{\Phi, w})< Q + \delta.$$
Suppose by contradiction that $\lim_{n\to \infty}W_2(\nu_n, \mu_{\Phi^{\dag}, w})> 0$. Thus, there is $\epsilon>0$ such for all $k\geq 1$, there exists $n>\max(k, N)$ such that $$W_2(\nu_n, \mu_{\Phi^{\dag}, w})>\epsilon.$$
For $n$ given above, let $\gamma_n \in \Gamma(\nu_n, \mu_{\Phi, w})$ be such that $$W_2^2(\nu_n, \mu_{\Phi, w})=\iint_{{\mathbb R}^d}\|x-y\|^2d\gamma_n(x,y).$$ Since $\nu_n$ is a finite probabilistic frame we may assume further that $\nu_n=\mu_{u_{n}, v}$ where $u_n= \{\psi_i\}_{i=1}^M\subset {\mathbb R}^d$ and $v=\{v_i\}_{i=1}^M \subset [0, \infty)$. For the sake of simplicity in notations, we omit the dependence of both $\psi_i$ and $v_i$ on $n$. Let $w_{n,j,k} = \gamma_{n}(\varphi_j,\psi_k)$.
Now consider the finite frames $\{u'_j\}_{j}=\{\sqrt{w_{n,j,k}}\psi_k\}_{j,k}$ and $\Phi' =$ $\{\sqrt{w_{n,j,k}}\varphi_j\}_{j,k}$.
Note that $W_2(\mu_{\Phi'}, \mu_{\Phi'^{\dag}})=Q$. Now we consider the rows of these frames written with respect to the eigenbasis of the frame operator $S:=S_{\Phi'}$ of $\Phi'$.
Because, $W_2(\nu_n, \mu_{\Phi^{\dag}, w})>\epsilon,$ then $\sum_{j,k}w_{n,j,k}\|\psi_k - S^{-1/2}\varphi_j\|^2 > \epsilon$.
Using this and Lemma \[contradiction-unifcont\] we have the following estimates: $$W_2^2(\mu_{\Phi, w}, \nu_{n})\geq W_2^2(\mu_{\Phi, w}, \nu)+ \min( \tfrac{\epsilon^2}{d} \cdot M,M^2)$$ where $A$ is the lower frame bound of $\Phi$ and $M = \min(1,\sqrt{A})$.
Consequently,
$$W_2^2(\mu_{\Phi, w}, \nu_{n})- Q^2\geq \min( \tfrac{\epsilon^2}{d} \cdot M,M^2)>0 .$$ But, this contradicts the fact that $Q=W_2(\mu_{\Phi, w}, \nu)=\lim_{n \to \infty}W_2(\nu_{\Phi, w},\nu_n).$ Hence, $\lim_{n\to \infty}W_2(\nu_n, \mu_{S^{-1/2}\Phi, w})= 0$, and $\nu=\mu_{\Phi^{\dag}, w}.$
[**Case 3:**]{} Next, we suppose that $\mu$ is non discrete probabilistic frame with frame bounds $A,$ and $B$. Let $\{\mu_n\}_{n=1}^\infty=\{\mu_{\Phi_{n}, w(n)}\}_{n=1}^\infty$ be a sequence of finite probabilistic frames with bounds between $A/2$ and $B+A/2$ such that $\lim_{n\to \infty}W_2(\mu_n, \mu)=0$. Then $F(\mu)=\lim_{n\to \infty}F(\mu_{n})$ is such that $Q=W_{2}(F(\mu), \mu).$ Suppose there exists another Parseval frame $\nu$ such that $Q=W_2(\nu, \mu)$. Choose a sequence of finite Parseval $\{\nu_n\}_{n=1}^\infty$ such that $\lim_{n\to \infty}\nu_n=\nu.$
Observe that $Q=\lim_{n\to \infty}W_2(\mu_n, F(\mu_n))=\lim_{n\to \infty}W_2(\nu_n, \mu_n)$. Write $\Phi_n = \{\varphi_{n,j}\}_{j=1}^{M}$ and $w(n)=\{w_{j}\}_{j=1}^M$, where for simplicity we omit the dependence of $M$ on $n$. Similarly, $\{\nu_n\}_{n=1}^{\infty}= \{\psi_{n,j}\}_{j=i}^{M'}$ with weights $v(n)= \{v_{j}\}_{j=1}^{M'}$.
Let $\gamma_n\in \Gamma(\mu_n, \nu_n)$ be such that $$W_2^2(\mu_{n},\nu_n) =\iint \|x-y\|^2d\gamma_n(x,y).$$ Set $$w_{j,k} = \gamma_n(\varphi_{n,j},\psi_{n,k})$$ We know that $$\begin{aligned}
W_2^2(\mu_n, F(\mu_n)) &= \sum_{j= 1}^{M}w_j\|\varphi_{n,j} - \varphi^{\dag}_{n,j}\|^2 = \sum_{j,k}w_{j,k}\|\varphi_{n,j} - \varphi^{\dag}_{n,j}\|^2 \\
&=\sum_{j,k}\|\sqrt{w_{j,k}}\varphi_{n,j} - \sqrt{w_{j,k}}\varphi^{\dag}_{n,j}\|^2\end{aligned}$$
We also know that $$W_2^2(\mu_{n},\nu_n) = \sum_{j,k}w_{j,k}\|\varphi_{n,j} - \psi_{n,k}\|^2 =\sum_{j,k}\|\sqrt{w_{j,k}}\varphi_{n,j} - \sqrt{w_{j,k}}\psi_{n,k}\|^2$$
Suppose that $\lim_{n\to \infty} W_2(F(\mu_n), \nu_n)>0$. Thus, there exists $\epsilon>0$ and and integer $n>1$ such that $W_2(F(\mu_n),\nu_n) > \epsilon$. Consequently, $$\epsilon< \sum_{j,k}w_{j,k}\|\varphi^{\dag}_{n,j} - \psi_{n,k}\|^2 = \sum_{j,k}\|\sqrt{w_{j,k}}\varphi^{\dag}_{n,j} - \sqrt{w_{j,k}}\psi_{n,k}\|^2$$
Hence $$d(\Phi'^{\dag }_n, \Psi_n')>\epsilon$$ where $\Psi'_n = \{\sqrt{w_{j,k}}\psi_{n,k}\}$.
By the same argument as in Lemma \[contradiction-unifcont\] we conclude that $W_2^2(\mu_{n},\nu_n)- W_2^2(\mu_n, F(\mu_n))\geq min(M \frac{\epsilon^2}{d},M^2).$ where $M = \min(1,\sqrt{\frac{A}{2}})$
This contradicts the fact that Since $\lim_{n\to \infty}W_2(\mu_n,\nu_n) = Q = \lim_{n\to \infty}W_2(\mu_n,F(\mu_n))$. Thus $\lim_{n\to \infty} W_2(F(\mu_n), \nu_n)=0$ and so $F(\mu)=\nu$.
By Proposition \[well-defined-F\] it follows that given a probabilistic frame $\mu$ and any sequence $\Phi_{k}:=\Phi_{k, w_{k}}=\{\varphi_{k}\}_{k=1}^{N_k}$ of finite probabilistic frames in ${\mathbb R}^d$ such that $\lim_{k\to \infty} W_{2}(\mu, \mu_{\Phi_{k}})=0$, then $F(\mu)=\lim_{k\to \infty}F(\mu_{\Phi_{k}}).$ Furthermore, it is proved in [@WCKO17] that if $\{\mu_n\}_{n\geq 1} \subset \mathcal{P}_2$ converges in the Wassertein metric to $\mu \in \mathcal{P}_2$, then $$\|S_{\mu}-S_{\mu_{n}}\|\leq CW_2(\mu_{n}, \mu).$$
All that is needed to prove Theorem \[maintheorem\] is to show that $F(\mu)=\mu^{\dag}$.
Let $\mu$ be a probabilistic frame with bounds $A, B$. Let $0< \epsilon<A/2$ and choose a compactly supported probabilistic frame $\nu_{\epsilon}$ as in Lemma \[prop-2\]. In particular $\nu_{\epsilon}$ is supported on $B(0,R_{\epsilon})$ with frame bounds between $A/2$ and $B+A/2$, where $R_{\epsilon}>0$ is such that $$\int_{{\mathbb R}^d\setminus B(0,R_{\epsilon})}\|x\|^2dx< \epsilon/3.$$
Choose a finite probabilistic frame $\mu_{\epsilon}$ with bounds between $\frac{A}{2}$ and $B + \frac{A}{2}$ such that $W_{2}(\mu_{\epsilon}, \nu_{\epsilon}) < \frac{\epsilon}{3}$. By taking a sequence $\{\epsilon_n\}_{n=1}^{\infty}\subset [0, \infty)$ with $\lim_{n\to \infty}\epsilon_n=0$, we can pick $\{\mu_n\}_{n\geq 1}:=\{\mu_{\epsilon_n}\}_{n\geq 1}$ such that $\lim_{n\to \infty}W_{2}(\mu_{n}, \mu)=0$. Consequently, $\lim_{n\to \infty}S_{\mu_{n}}=S_{\mu}$, and $\lim_{n\to \infty}S^{-1/2}_{\mu_{n}}=S^{-1/2}_{\mu}$ in the operator norm.
We recall that $\lim_{n\to \infty}W_{2}(\mu_{n}, \mu)=0$ is equivalent to
$$\lim_{n\to \infty}\int f\, d\mu_{n}(x) =\int f\, d\mu(x) \\$$ for all continuous function $f$ such that $|f(x)|\leq C(1+\|x-x_0\|^2)$ for some $x_0\in {\mathbb R}^d$ [@Villani2009 Theorem 6.9]
We know that $\lim_{n\to \infty} F(\mu_n)=\lim_{n\to \infty} \mu_{n}^{\dag}=F(\mu)$ in the Wasserstein metric. We would like to show that $\lim_{n\to \infty} F(\mu_n)=\lim_{n\to \infty}\mu_{n}^{\dag}=\mu^{\dag}$.
We show that for all continuous function $f$ such that $|f(x)|\leq C(1+\|x-x_0\|^2)$ for some $x_0\in {\mathbb R}^d$ $$\lim_{n\to \infty}\int f\, d\mu_{n}^{\dag}(x) =\int f\, d\mu^{\dag}(x).$$
$$\begin{aligned}
|\int f\, d\mu_{n}^{\dag}(x) -\int f\, d\mu^{\dag}(x)|& = |\int f(S^{-1/2}_{\mu_{n}} x)\, d\mu_{n}(x) -\int f(S^{-1/2}_{\mu} x)\, d\mu(x)|\\
&\leq \int |f(S^{-1/2}_{\mu_{n}} x) - f(S^{-1/2}_{\mu} x)|\, d\mu_{n}(x) +\\
&|\int f(S^{-1/2}_{\mu} x)\, d\mu_{n}(x) -\int f(S^{-1/2}_{\mu} x)\, d\mu(x)|\\\end{aligned}$$
Let $f$ be continuous with $|f(x)|\leq C(1+\|x-x_0\|^2)$ for some $x_0\in {\mathbb R}^d$. Then, $f(S^{-1/2}_\mu)$ is continuous and satisfies $$|f(S^{-1/2}_\mu x)|\leq C(1+\|x_0-S^{-1/2}_\mu x\|^2)\leq C(1+\|S^{-1/2}_\mu\|^2\|x-S^{1/2}_{\mu}x_0\|^2)\leq C' (1+\|x-S^{1/2}_{\mu}x_0\|^2)).$$ Consequently, we can find $N_1$ such that for all $n\geq N_1$, $$|\int f(S^{-1/2}_{\mu} x)\, d\mu_{n}(x) -\int f(S^{-1/2}_{\mu} x)\, d\mu(x)|< \epsilon/3.$$
Since $f$ is continuous, there exists $\delta>0$ such that for all $x, y \in B(0,R')$, $\|x-y\|< \delta $ implies that $|f(x)-f(y)|< \epsilon/3$, where $R'>0$ is chosen so as to guarantee that for large $n$, and $x\in B(0,R)$, $S^{-1/2}_{\mu_{n}}x, S^{-1/2}_{\mu}x \in B(0, R)$. Since, $\lim_{n\to \infty}S^{-1/2}_{\mu_{n}}=S^{-1/2}_{\mu}$ , there exists $N_2$ such that for all $n\geq N_2$, $$\|S^{-1/2}_{\mu_{n}}x-S^{-1/2}_{\mu}x\|\leq \|S^{-1/2}_{\mu_{n}}-S^{-1/2}_{\mu}\| \|x\|\leq R \|S^{-1/2}_{\mu_{n}}-S^{-1/2}_{\mu}\|< \delta.$$
Therefore, for $n\geq N_2$, $ |f(S^{-1/2}_{\mu_{n}} x) - f(S^{-1/2}_{\mu} x)|< \epsilon/3$ for all $x\in B(0, R)$. Consequently, $$\begin{aligned}
\int |f(S^{-1/2}_{\mu_{n}} x) - f(S^{-1/2}_{\mu} x)|\, d\mu_{n}(x) &= \int_{B(0,R)} |f(S^{-1/2}_{\mu_{n}} x) - f(S^{-1/2}_{\mu} x)|\, d\mu_{n}(x)\\&+\int_{{\mathbb R}^d\setminus B(0,R)} |f(S^{-1/2}_{\mu_{n}} x) - f(S^{-1/2}_{\mu} x)|\, d\mu_{n}(x) \\
&< \epsilon/3+ \int_{{\mathbb R}^d\setminus B(0,R)} |f(S^{-1/2}_{\mu_{n}} x) - f(S^{-1/2}_{\mu} x)|\, d\mu_{n}(x)\\
&< \epsilon/3 + M \int_{{\mathbb R}^d\setminus B(0,R)} \|x\|^2\, d\mu_{n}(x)\\
&<2\epsilon/3\end{aligned}$$ where $M>0$ is a constant that depends only on $f$, and $\mu$.
It follows that for all $n\geq \max(N_1, N_2),$ we have $$|\int f\, d\mu_{n}^{\dag}(x) -\int f\, d\mu^{\dag}(x)|< \epsilon$$ which implies that $\lim_{n\to \infty}\int f\, d\mu_{n}^{\dag}(x) =\int f\, d\mu^{\dag}(x).$
Acknowledgment {#acknowledgment .unnumbered}
==============
Both authors were partially supported by ARO grant W911NF1610008. K. A. Okoudjou was also partially supported by a grant from the Simons Foundation $\# 319197$. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while K. A. Okoudjou was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.
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abstract: 'Inference of online social network users’ attributes and interests has been an active research topic. Accurate identification of users’ attributes and interests is crucial for improving the performance of personalization and recommender systems. Most of the existing works have focused on textual content generated by the users and have successfully used it for predicting users’ interests and other identifying attributes. However, little attention has been paid to user generated visual content (images) that is becoming increasingly popular and pervasive in recent times. We posit that images posted by users on online social networks are a reflection of topics they are interested in and propose an approach to infer user attributes from images posted by them. We analyze the content of individual images and then aggregate the image-level knowledge to infer user-level interest distribution. We employ image-level similarity to propagate the label information between images, as well as utilize the image category information derived from the user created organization structure to further propagate the category-level knowledge for all images. A real life social network dataset created from Pinterest is used for evaluation and the experimental results demonstrate the effectiveness of our proposed approach.'
author:
- Quanzeng You
- |
Jiebo Luo\
Department of Computer Science\
University of Rochester\
Rochester, NY 14623\
{qyou, jluo}@cs.rochester.edu\
Sumit Bhatia\
IBM Almaden Research Centre\
650 Harry Rd, San Jose, CA 95120\
{sumit.bhatia}@us.ibm.com\
bibliography:
- 'aaai\_2015.bib'
title: |
A Picture Tells a Thousand Words – About You!\
User Interest Profiling from User Generated Visual Content
---
Introduction {#intro}
============
Online Social Networks (OSNs) such as Facebook, Twitter, Pinterest, Instagram, etc. have become a part and parcel of modern lifestyle. A study by Pew Research centre[^1] reveals that three out of every four adult internet users use at least one social networking site. Such large scale adoption of OSNs and active participation of users have led to research efforts studying relationship between users’ digital behavior and their demographic attributes (such as age, interests, and preferences) that are of particular interest to social science, psychology, and marketing. A large scale study of about 58,000 Facebook users performed by Kosinski et al. reveals that digital records of human activity can be used to accurately predict a range of personal attributes such as age, gender, sexual orientation, political orientation, etc. Likewise, there have been numerous works that study variations in language used in social media with age, gender, personality, etc. [@burger2011discriminating; @gender_lexical_variation; @facebook_plos_one]. While most of the popular OSNs studied in literature are mostly text based, some of them (e.g., Facebook, Twitter) also allow people to post images and videos. Recently, OSNs such as Instagram and Pinterest that are majorly image based have gained popularity with almost 20 billion photos already been shared on Instagram and an average of 60 million photos being shared daily[^2].
![Example pinboards from one typical Pinterest user.[]{data-label="fig:example:pinboards"}](./Visio-example_pinboards2-crop){width=".47\textwidth"}
The most appealing aspect of image based OSNs is that visual content is *universal* in nature and thus, not restricted by the *barriers of language*. Users from different cultural backgrounds, nationalites, and speaking different languages can easily use the same *visual language* to express their feelings. Hence, analyzing the content of user posted images is an appealing idea with diverse applications. Some recent research efforts also provide support for the hypothesis that images posted by users on OSNs may prove to be useful for learning various personal and social attributes of users. Lovato et al. proposed a method to learn users’ latent visual preferences by extracting aesthetics and visual features from images favorited by users on Flickr. The learned models can be used to predict images likely to be favorited by the user on Flickr with reasonable accuracy. Cristani et al. infer personalities of users by extracting visual patterns and features from images marked as favorites by users on Flickr. Can et al. utilize the visual cues of tweeted images in addition to textual and structure-based features to predict the retweet count of the posted image. Motivated by these works, we investigate *if the images posted by users on online social networks can be used to predict their fine-grained interests or preferences about different topics*. To understand this better, \[fig:example:pinboards\] shows several randomly selected pinboards (collection of images, as they are called in Pinterest) for a typical Pinterest user as an example. We observe that different *pins* (a pin corresponds to an image in Pinterest) and pinboards are indicative of the user’s interests in different topics such as sports, art and food. We posit that the visual content of the images posted by a user in an OSN is a reflection of her interests and preferences. Therefore, an analysis of such posted images can be used to create an interest profile of the user by analyzing the content of individual images posted by the user and then aggregating the image-level knowledge to infer user-level preference distribution at a fine-grained level.
Problem Formulation and Overview of Proposed Approach
-----------------------------------------------------
**Problem Statement:** Given a set $\mathcal{I}$ of images posted by the user *u* on an OSN, and a set $\mathcal{C}$ of interest categories, output a probability distribution over categories in $\mathcal{C}$ as the interest distribution for the user. In order to solve this problem, we first need to understand the relationships between different categories (topics) and underlying characteristics (or features) of user posted images in the training phase. These learned relationships can then be used to predict distribution over different interest categories by analyzing images posted by a new user. Even though state-of-the-art machine learning algorithms, especially developments in deep learning, have achieved significant results for individual image classification [@krizhevsky2012imagenet], we believe that incorporating OSN and user specific information can provide further performance gains. Different image based OSNs offer users capability to group together similar images in the form of albums/pinboards, etc. In Pinterest, users create pinboards for a given topic and collect similar images in the pinboard. Given this human created group information, it is reasonable to assume that strong correlations exist between objects belonging to the same curated group or categorization. For example, a user may have two pinboards belonging to the *Sports* category, one for images related to soccer and one for images related to basketball. Therefore, even though all images in the two pinboards will share some common characteristics, images within each pinboard will share some additional similarities. Motivated by these observations, we employ image level and group level label propagation in order to build more accurate learning models. We employ image-level similarity to propagate the label information between images and category correlations are employed to further propagate the category-level knowledge for all images.
Related Work
============
User Profiling from Online Social Networks
------------------------------------------
It is discovered that in social network people’s relationship follows the rule *birds of a feather flock together* [@mcpherson2001birds]. Similarly, people in online social network also exhibit such kind of patterns. Online social network users connected with other users may be due to very different reasons. For instance, they may try to rebuild their real world social networks on the online social network. However, most of the time, people hope to show part of themselves to the rest of the world. In this case, the content generated by online social network users may help us to infer their characteristics. Therefore, we are able to build more accurate and more targeted systems for prediction and recommendation.
There are many related works on profiling different online users’ attributes. Location is quite important for advertisement targeting, local customer personalization and many novel location based applications. In Cheng et al. , the authors proposed an algorithm to estimate a city level location for Twitter users. More recently, Li et al. proposed a unified discriminative influence model to estimate the home location of online social users. They unified both the social network and user-centric signals into a probabilistic framework. In this way, they are able to more accurately estimate the home locations. Meanwhile, since most social network users tend to have multiple social network accounts, where they exhibit different behaviors on these platforms. Reza and Huan proposed MOBIUS for finding a mapping of social network users across different social platforms according to people’s behavioral patterns. The work in Mislove et al. also proposed an approach trying to inferring user profiles by employing the social network graph. In a recent work, Kosinski et al. revealed the power of using Facebook to predict private traits and attributes for Facebook volunteer users. Their results indicated that simple human social network behaviors are able to predict a wide range of human attributes, including sexual orientation, ethnic origin, political views, religion, intelligence and so on. Li et al. defined *discriminative correlation* between attributes and social connections, where they tried to infer different attributes from different circles of relations.
Visual Content Analysis
-----------------------
Visual content becomes increasingly popular in all online social networks. Recent research works have indicated the possibility of using online user generated visual content to learn personal attributes. In Kosinsky et al. , a total of $58,000$ volunteers provided their Facebook likes as well as detailed demographic profiles. Their results suggest that digital records of human activities can be used to accurately predict a range of personal attributes such as age, gender, sexual orientation, and political orientation. Meanwhile, the work in Lovato et al. tried to build the connection between cognitive effects and the consumption of multimedia content. Image features, including both aesthetics and content, are employed to predict online users’ personality traits. The findings may suggest new opportunities for both multimedia technologies and cognitive science. More recently, Lovato et al. proposed to learn users’ biometrics from their collections of favorite images. Various perceptual and content visual features are proven to be effective in terms of predicting users’ preference of images. You, Bhatia and Luo exploited visual features to determine the gender of online users from their posted images.
Proposed Approach
=================
As discussed in Section \[intro\], the problem of inferring user interests can be considered as an image classification problem. However, in contrast to the classical image classification problem where the objective is to maximize classification performance at individual level, we are focused more on learning the overall user-level *image category distribution*, which in turn yields users’ interests distribution. In our work, we use data crawled from pinterest.com, which is one of the most popular image based social networks. In Pinterest, users can share/save images that are known as *pins*. Users can categorize these pins into different *pinboards* such that a pinboard is collection of related pins (images). Also, while creating a pinboard the user has to chose a descriptive category label for the pinboard from a pre-defined list specified by Pinterest. There are a total of 34 available categories for users to chose from (listed in \[tab:pin:cat\]). A typical Pinterest manages/owns many different pinboards belonging to different categories and each pinboard will contain closely related images of the same category. Pinterest users mainly use pinboards to attract other users and also to organize the images of interest for themselves. Therefore, often times they will choose interesting and high-quality pins to add to their pinboards and one can use these carefully selected and well organized high quality images to infer the interests of the user.
[\*[7]{}[|l]{}|]{} Animals & Architecture & Art & Cars & Motorcycles & Celebrities & Design & DIY & Crafts\
Education & Film, Music & Books & Food & Drink & Gardening & Geek & Hair & Beauty & Health & Fitness\
History & Holidays & Events & Home Decor & Humor & Illustrations & Posters & Kids & Men’s Fashion\
Outdoors & Photography & Products & Quotes & Science & Nature & Sports & Tattoos\
Technology & Travel & Weddings & Women’s Fashion & Other & &\
{width=".8\textwidth"}
Training Image-level Classification Model
-----------------------------------------
We employ the Convolutional Neural Network (CNN) to train our image classifier. The main architecture comes from the Convolutional Neural Network proposed by Krizhevsky et al. , that has achieved state of the art performance in the challenging ImageNet classification problem. We train our model on top of the model of Krizhevsky et al. ). For a detailed description of the architecture of the model, we direct the reader to the original paper. In particular, we fine-tune our model by employing images and labels from Pinterest. For each image, we assign the label of the image to be the same as the label of the pinboard it belongs to. More details on the preparation of training data for our model will be discussed in the experimental section. From the trained deep CNN model, it is possible to extract deep features for each image. These features, also known as high-level features, have shown to be more appropriate for performing various other image related tasks, such as image content analysis and semantic analysis [@bengio2009learning]. Specifically, we extract the deep CNN features from the last fully connected layer of our trained CNN model and employ these deep features for label propagation as described in the next section. It is noteworthy that this work differs from image annotation, which deals with individual images and have been extensively studied. As will become more clear, the approach we take in the form of label propagation, is designed to exploit the strong connection between the images curated within and across collections by users. For the same reason, the typical noise in user data is suppressed in Pinterest due the same user curation process.
Image and Group Level Label Propagation for Prediction
------------------------------------------------------
During the prediction stage, we try to solve a more general problem, where we are given a collection of images without group information. However, we want to predict the users’ interests from these unorganized collection of images. We propose to use label propagation for image labels. The work in Yu et al. ) also tried to use collection information for label propagation. However, differently from their work, where collection information is employed to extract sparsity constraint to the original label propagation algorithm, we propose to impose the additional group-level similarity to further propagate the image labels for the same user.
Meanwhile, we observe that for most of the categories in \[tab:pin:cat\], the average distance between images in the same pinboard have closer distance than images in the same category. \[fig:pin:dist\] shows the average distance between images in the same pinboards and the average distance between images in different pinboards but in the same categories. The distance is represented by the squared Euclidean distance between deep features from last layer of Convolutional Neural Network. Intuitively, this can be explained by the fact that most of the categories are quite diverse in that they can contain quite different sub-categories. On the other hand, users generally create pinboards to collect similar images into the same group. These images are more similar in that they are more likely to be in the same sub-category of the category label chosen by the user. Hence, the motivation for implementing label propagation at user level.
Let $n$ be the number of categories, we define matrix $G\in R^{n\times n}$ to be the affinity matrix between these $n$ categories ($G$ is normalized such that the columns of $G$ sum 1). The intuitive idea is that in general, there exists some correlation between a person’s interest. If one user likes sports, then he is likely to be interested in Health & Fitness. This kind of information may help us to predict users’ interest more accurately, especially for users with very few images in their potential interest categories. Therefore, during the label propagation, we also consider the propagation of category-level information.
To incorporate the group level information into our model, we define the following iterative process for image label propagation. $$Y^{t+1} = ( 1 - \Lambda) WY^t G+ \Lambda Y^0
\label{eqn:lp}$$ where $Y^0$ is the initial prediction of image labels from the trained CNN model, $W$ is the normalized similarity matrix between images and $\Lambda$ is a diagonal matrix. Following Yu et al. ), $\Lambda$ is defined as $$\Lambda_{i,i} = \max_{j}\frac{Y^0_{i,j}}{\sum_k{Y^0_{i,k}}}.
\label{eqn:lambda}$$ We can consider the above process as two stages. In the first stage $( 1 - \Lambda) WY^t$, we use the similarity between the images to propagate the labels at image level. In the second stage, the group relationship matrix $G$ is employed to further propagate the group relationship to all images, i.e. we multiply matrix $G$ with the results of $( 1 - \Lambda) WY^t$. Next, we present the convergence analysis of the proposed label propagation framework.
$X=\{x_1,x_2,\dots,x_N\}$ a collection of images\
$M$: Fine-tuned Imagenet CNN model\
$G$: Correlation matrix between the labels Predict the categories $Y^0\in R^{N\times K}$ ($K$ is the number of categories) of $X$ using the trained CNN model $M$ Extract deep features from $M$ for all $X$. Calculate the similarity matrix $W'$ between $X$ using Gaussian kernel function $W'(i,j) = \exp(-\frac{\parallel x_i - x_j\parallel^2}{2\delta^2})$, where $x_i$ and $x_j$ are the deep features for image $i$ and $j$ respectively. \[alg:state:w\] Normalize $W'$ to get $W=D^{-1}W'$, where $D$ is diagonal matrix with $D_{ii} = \sum_j W'_{i,j}$. Calculate the diagonal matrix $\Lambda$ according to Eqn.(\[eqn:lambda\]). Calculate the affinity matrix between $G$ between different categories. Initialize, iteration index $t = 0$ Employ Eqn.(\[eqn:lp\]) to update $Y^{t+1}$ according to $Y^t$ Normalize rows of $Y^t\in R^{N \times K}$ to get $Y'^{t}\in R^{N \times K}$. $Y'^{t}$.
### Convergence Analysis
From Eqn. (\[eqn:lp\]), we have the following formula for $Y^{t+1}$. $$Y^{t+1} = \left(\left(1 - \Lambda\right)W\right)^{t+1} Y^0 G^{t+1} + \sum^{t}_{i = 0} \left(\left(1- \Lambda\right)W\right)^i\Lambda Y^0 G^i$$ Since we have $0 \leq \lambda_{i,j}, w_{i,j}, G_{i,j} < 1$ for all $i,j$, therefore $\lim_{t \rightarrow \infty} \left(\left(1 - \Lambda\right)W\right)^{t+1} Y^0 G^{t+1} = 0$. It follows that $$\lim_{t \rightarrow \infty} Y^{t} = \sum^{t-1}_{i = 0} \left(\left(1- \Lambda\right)W\right)^i\Lambda Y^0 G^i .$$ According to the theorem that *the product of two converging sequences converges* [@rosenlicht1968introduction], we define two different matrix series $A^{t} = \left(\left(1- \Lambda\right)W\right)^t\Lambda Y^0$ and $B^{t} = G^t$. For matrix series $A^t$, it converges [@yu2011collection]. For matrix series $B^t$, it also converges since $\rho (B) < 1$. Therefore, we conclude that the product of the two matrix series $A^t$ and $B^t$ also converges. More importantly, we have $\sum_{i} A_i = \left( I - \Lambda\right) \Lambda Y^0 $ and $\sum_{i} B_i = (I - Q)^{-1}$. All the elements of $A_i$ and $B_i$ are non-negative, which leads to the fact that $\sum_i {A_iB_i} <_p (\sum_i A_i)(\sum_i B_i)$. We use $<_p$ to represent the element-wise comparison between two matrix. In other words, we have a upper bound for $Y^t$, together with the fact that $A_iB_i \ge 0$, we conclude that $Y^t$ converges.
Meanwhile, since we implement label propagation in each user’s collection of images, thus the main computational and storage complexity is in the order of $O(n_i^2)$, where $n_i$ is the number of images for user $i$.
Algorithm \[alg:glp\] summarizes the main steps using the proposed group constrained label propagation framework. Note that in step \[alg:state:w\], we use a Gaussian kernel function to calculate the similarity between two images using the deep features. However, one can employ other techniques such as locally linear embedding (LLE) [@donoho2003hessian] to calculate the similarity between different instances.
Eventually, we obtain users’ interests distribution by aggregating the label distribution of the collections of their images. In other words, we simply sum up the label distribution of individual images and then normalize it to produce the users’ interest prediction results.
Experiments and Evaluations
===========================
To evaluate the proposed algorithm, we crawl data from Pinterest according to a randomly chosen user list consisting of 748 users. \[tab:data:stat\] gives the statistics of our crawled dataset. This dataset is used to train our own deep convolutional neural network. We use $80\%$ of these images as the training data and the remaining $20\%$ as validation data. We also downloaded another $77$ users’ data as our testing data. Since each pinboard has one category, we use the category label as the label for all the images in that pinboard.
Training and Validating Testing
------------------ ------------------------- ---------
Num of Users 748 77
Num of Pinboards 30,213 1,126
Num of Pins 1,586,947 66,050
: Statistics of our dataset.[]{data-label="tab:data:stat"}
We train the CNN on top of the ImageNet convolutional neural model [@krizhevsky2012imagenet]. We use the publicly available implementation Caffe [@Jia13caffe] to train our model. All of our experiments are evaluated on a Linux X86\_64 machine with 32G RAM and two NVIDIA GTX Titan GPUs. We finish the training of CNN after a total of $200,000$ iterations.
In the proposed group constrained label propagation, we need to know the similarity matrix $G$ between all categories. We propose to learn the similarity matrix $G$ from pinterest users’ behaviors of building their collections of pinboards. No additional textual semantic or visual semantic information is employed to build the similarity matrix. In our experiments, we employ Jaccard index to calculate the similarity coefficient between different categories in \[tab:pin:cat\]. Specifically, the entry of $G_{ij}$ is defined as the Jaccard index between category $i$ and $j$, which is the ratio between the number of users who have pinboards of both categories $i$ and $j$ and the number of users who have pinboards of category $i$ or $j$. \[fig:group\] shows the coefficients between all different categories. It shows that users prefer to choose some of the categories together, which may suggest potential category recommendations for Pinterest users.
![Jaccard similarity coefficients between different categories in \[tab:pin:cat\].[]{data-label="fig:group"}](./group-crop){width="45.00000%"}
Evaluation Criteria
-------------------
To evaluate the performance of different models, we employ two different criteria. 1) **Normalized Discounted Cumulative Gain (NDCG) score** NDCG is a popular measure for ranking tasks [@croft2010search]. Discounted Cumulative Gain at position $n$ is defined as $$DCG_n = p_1 + \sum_{i=1}^n \frac{p_i}{\log_2(i)},$$ where $p_i$is the relevance score at position $i$. In our case, we use the predicted probability as the relevance score. NDCG is the normalized DCG, where the value is between $0$ and $1$. To calculate the NDCG, for a user $u_i$, we first get the ground truth distribution of his interest according to the category labels of his pinboards and then rank the ground truth categories in descending order of their distribution probabilities. Note that the pinboard labels of $u_i$ may not contain all the categories and our trained classifier may give a class distribution over all the categories. Therefore, when calculating the NDCG score we add a prior to the interest distribution of each user. We first calculate the prior category distribution $p_0$ from all the training and validating data set according to their labels. Then, for each testing user $i$ we first calculate the category distribution of $p_i$ according to the ground truth labels. Next, we smooth the distribution by updating $p_i = p_i + 0.1*p_0$ and then normalize it to get the ground truth interest distribution for user $i$. In this way, we are able to calculate the NDCG score for each user according to the prediction of the model. 2) **Recall@Rank K** In this metric, we firstly build the ground truth for one user $u_i$, we rank all of his categories in descending order according to the distribution probabilities. Then, we calculate the shared top ranked categories with the ground truth. Users’ data in Pinterest may be incomplete. For example, user $i$ may be interested in Sports and Women’s Fashion. However, she only has created pinboards belonging to Women’s Fashion. She has not created Sports related pinboards. However, it does not mean that she is not interested in Sports. Thus, it makes sense to evaluate different algorithms’ performance using the Recall@Rank k.
![Filters of the first convolutional layer.[]{data-label="fig:filters"}](./conv1_weight_iter_200000-crop){width=".4\textwidth"}
Experimental Results
--------------------
![Distribution of NDCG scores for three different approaches.[]{data-label="fig:ndcg"}](./ndcg_bar_plot){width=".45\textwidth"}
We report the results for three different models. First one is the results from our fine-tuned CNN model, i.e. the initial prediction for label propagation. Second model is label propagation [@yu2011collection] (LP) and the last one is the proposed group constrained label propagation (GLP).
We first train our CNN model using the data in \[tab:data:stat\]. \[fig:filters\] shows the learned filters in the first layer of the CNN model, which is consistent with the filters learned in other image related deep learning tasks [@zeiler2013visualizing]. These Gabor-like filters can be employed to replicate the properties of human brain V1 area [@lee2008sparse], which can detect low-level texture features from raw input images. Next, we employ this model to predict the labels of the testing data, which is going to be the initial label prediction for LP and GLP models.
\[tab:ndcg:mean:std\]
Model Mean STD
------- ------- -------
CNN 0.692 0.179
LP 0.818 0.138
GLP 0.826 0.138
: Mean and Standard deviation of NDCG score
For both LP and GLP models, we choose the $\delta$ in the Gaussian kernel to be the variance of distance between each pair of deep features [@von2007tutorial]. We set the maximum number of iterations to be $100$ for both models[^3]. \[fig:ndcg\] shows the performance of using NDCG for the three models. The results shows that both LP and GLP try to move the distribution to the right, which suggests that both models can improve the performance in terms of NDCG score for most of the users. To quantitatively analyze the results, we give the mean and the standard derivation of the NDCG scores in \[tab:ndcg:mean:std\]. Both LP and GLP improve the NDCG score from about $0.69$ to over $0.80$ and reduce the standard deviation. GLP shows slight advantage over LP in terms of both mean and standard deviation of NDCG score.
![Recall@$K$ for different $K$ for the three models respectively.[]{data-label="fig:recall"}](./recall_at_10-crop){width=".45\textwidth"}
The evaluation results of using Recall@K are shown in \[fig:recall\]. Differently from the NDCG score, CNN and GLP show better recall performance then LP for all different values of $K$s. Meanwhile, GLP consistently outperforms the CNN model. The results suggest that without the propagation of category or group similarity, label propagation may cause the increase of irrelevant categories, which leads to poor recall.
Besides the user-level interests prediction, we are also interested in *whether or not label propagation can improve the performance of image-level classification performance*. Since we have the pinboard labels as the ground-truth labels for each testing image, we report the accuracy of the three models on the testing images. \[tab:img:ac\] summarizes the accuracy. Indeed, the performance of these three models is quite similar. These results may suggest that the label propagation have limited impact on the overall label distribution of individual image, i.e. the max index in the probability distribution of labels. However, they may change the probability distribution, which may benefits the overall user-level interests estimation.
Model Accuracy
------- ----------
CNN 0.431
LP 0.434
GLP 0.434
: Accuracy on Image-level classification of different models.[]{data-label="tab:img:ac"}
Conclusions
===========
We addressed the problem of user interest distribution by analyzing user generated visual content. We framed the problem as an image classification problem and trained a CNN model on training images spanning over 748 users’ photo albums. Taking advantage of the human intelligence incorporated through the user curation of the organized visual content, we used image-level similarity to propagate the label information between images, as well as utilized the image category information derived from the user created organization structure to further propagate the category-level knowledge for all images. Experimental evaluation using data derived from Pinterest provided support for the effectiveness of the proposed method. In this work, our focus was on using image content for user interest prediction. We plan to extend our work to also incorporate user generated textual content in the form of comments and user profile information for improved performance.
[^1]: http://www.pewinternet.org/2013/12/30/social-media-update-2013/
[^2]: http://instagram.com/press/
[^3]: In our experiments, the increase of iteration number over 100 leads to similar results with the reported results in this work.
|
---
abstract: 'We give a construction of an absolutely normal real number $x$ such that for every integer $b $ greater than or equal to $2$, the discrepancy of the first $N$ terms of the sequence $(b^n x \mod 1)_{n\geq 0}$ is of asymptotic order $\mathcal{O}(N^{-1/2})$. This is below the order of discrepancy which holds for almost all real numbers. Even the existence of absolutely normal numbers having a discrepancy of such a small asymptotic order was not known before.'
author:
- |
Christoph Aistleitner\
Institute of Analysis and Number Theory\
Graz University of Technology\
A-8010 Graz, Austria\
E-mail: [email protected]
- |
Verónica Becher\
Departamento de Computación, Facultad de Ciencias Exactas y Naturales\
Universidad de Buenos Aires & ICC, CONICET\
Pabellón I, Ciudad Universitaria, C1428EGA Buenos Aires, Argentina\
E-mail: [email protected]
- |
Adrian-Maria Scheerer\
Institute of Analysis and Number Theory\
Graz University of Technology\
A-8010 Graz, Austria\
E-mail: [email protected]
- |
Theodore A. Slaman\
Department of Mathematics\
University of California Berkeley\
719 Evans Hall \#3840, Berkeley, CA 94720-3840 USA\
E-mail: [email protected]
bibliography:
- 'ed.bib'
date: 'July 8, 2017'
---
[^1]
[^2]
Introduction and statement of results
=====================================
For a sequence $(x_j)_{j\geq 0}$ of real numbers in the unit interval, the discrepancy of the first $N$ elements is $$D_N((x_j)_{j\geq 0})= \sup_{0\leq \alpha_1 < \alpha_2 \leq 1} \left| \frac{1}{N} \#\{ j : 0 \leq j \leq N-1 \text{ and } \alpha_1 \leq x_j < \alpha_2 \} - (\alpha_2-\alpha_1)\ \right|.$$ A sequence $(x_j)_{j\geq 0}$ of real numbers in the unit interval is uniformly distributed if and only if $\lim_{N\to \infty} D_N((x_j)_{j\geq 0}) = 0$.
The property of Borel normality can be defined in terms of uniform distribution. For a real number $x$, we write $\{x\} = x-{\lfloor x \rfloor }$ to denote the fractional part of $x$. A real number $x$ is normal with respect to an integer base $b$ greater than or equal to $2$ if the sequence $(\{b^j x\})_{j\geq 0}$ is uniformly distributed in the unit interval. The numbers which are normal to all integer bases are called absolutely normal. In this paper we prove the following theorem.
\[th:main\] There is an absolutely normal number $x$ such that for each integer $b \geq 2$, there are numbers $N_0(b)$ and $C_b$ such that for all $N \geq N_0(b)$, $$\begin{aligned}
D_N( (\{b^j x\})_{j\geq 0}) \leq \frac{C_b}{ \sqrt{N}}.\end{aligned}$$ For the constant $C_b$ we can choose $C_b = 3433 \cdot b$. Moreover, there is an algorithm that computes the first $N$ digits of the expansion of $x$ in base $2$ after performing exponential in $N$ mathematical operations.
It follows from the work of Gál and Gál [@GalGal1964] that for almost all real numbers (in the sense of Lebesgue measure) and for all integer bases $b$ greater than or equal to $2$ the discrepancy of the sequence $(\{b^j x\})_{j\geq 0}$ obeys the law of iterated logarithm. Philipp [@Philipp1975] gave explicit constants and Fukuyama [@Fukuyama2008 Corollary]) sharpened the result. He proved that for every real $\theta>1$ there is a constant $C_\theta$ such that for almost all real $x$ we have $$\limsup_{N \to \infty} \frac{ \sqrt{N} D_N( (\{\theta^j x\})_{j\geq 0} )}{\sqrt{\log \log N}} = C_\theta.$$ In case $\theta$ is an integer greater than or equal to $2$, for $C_\theta$ one has the values $$C_\theta=\left\{
\begin{array}{ll}\medskip
\sqrt{84}/9, & \text{ if } \theta=2,
\\\medskip
\sqrt{(\theta+1)/(\theta-1)}/\sqrt{2}, & \text{ if $\theta$ is odd,}
\\\medskip
\sqrt{(\theta+1)\theta(\theta-2)/(\theta-1)^3}/\sqrt{2}, & \text{ if } \theta\geq 4 \text{ is even}.
\end{array}
\right.$$
To prove Theorem \[th:main\] we give a construction of a real number $x$ such that, for every integer $b$ greater than or equal to $2$, $D_N( (\{b^j x\})_{j\geq 0} )$ is of asymptotic order $\mathcal{O}(N^{-1/2})$, hence, below the order of discrepancy that holds for almost all real numbers. The existence of absolutely normal numbers having a discrepancy of such a small asymptotic order was not known before.
To prove Theorem \[th:main\], we define a computable sequence of nested binary intervals $(\Omega_k)_{k\geq 1}$ such that for all elements of $\Omega_k$ the discrepancy $D_N( (\{b^j x\})_{j\geq 0} )$ is sufficiently small for some range of $b$ and $N$. This argument uses methods going back to Gál and Gál [@GalGal1964] and Philipp [@Philipp1975]. The unique point in the intersection $\bigcap_{k\geq 1} \Omega_k$ is a computable number which satisfies the discrepancy estimate in the conclusion of the theorem. This is the number we obtain. The construction uses just discrete mathematics and yields directly the binary expansion of the computed number. Unfortunately, the algorithm that computes the first $N$ digits performs exponential in $N$ many operations.
In view of the method used to prove Theorem \[th:main\], the appearance of a bound of square-root order for the discrepancy is very natural. Note that the discrepancy is exactly the same as the Kolmogorov–Smironov statistic, applied to the case of the uniform distribution on $[0,1]$. By Kolmogorov’s limit theorem the Kolmogorov–Smirnov statistic of a system of independent, identically distributed (i.i.d.) random variables has a limit distribution when normalized by $\sqrt{N}$ (somewhat similar to the case of the central limit theorem). Since it is well-known that so-called lacunary function systems (such as the system $(\{b^j x\})_{j \geq 0}$ for $b \geq 2$) exhibit properties which are very similar to those of independent random systems, we can expect a similar behavior for the discrepancy of $(\{b^j x\})_{j \geq 0}$. In other words, we can find a set of values of $x$ which has positive measure, and whose discrepancy is below some appropriate constant times the square-root normalizing factor (see [@A-B-distr] for more details). Since the sequence $(b^j)_{j \geq 0}$ is very quickly increasing, we can iterate this argument and find a “good” set of values of $x$ (which we call $\Omega_k$) which gives the desired discrepancy bound and which has positive measure *within* the previously constructed set $\Omega_{k-1}$. These remarks show why a discrepancy bound of order $N^{-1/2}$ is a kind of barrier when constructing the absolutely normal number $x$ using probabilistic methods. Accordingly, any further improvement of Theorem \[th:main\] would require some truly novel ideas.
As reported in [@scheerer], prior to the present work the construction of an absolutely normal number with the smallest discrepancy bound was due to Levin [@Levin1979]. Given a countable set $L$ of reals greater than $1$, Levin constructs a real number $x$ such that for every $\theta$ in $L$, $$D_N((\{\theta^j x\})_{j\geq 0}) < \frac{C_\theta (\log N)^3}{\sqrt{N}},$$ for a constant $C_\theta$ for every $N\geq N_0(\theta)$. His construction does not produce directly the binary expansion of the defined number $x$. Instead it produces a computable sequence of real numbers that converge to $x$ and the computation of the $N$-th term requires double-exponential (in $N$) many operations including trigonometric operations, see [@AlvarezBecher2016].
It is possible to prove a version of Theorem \[th:main\] replacing the set of integer bases by any subset of computable reals greater than $1$. The proof would remain essentially the same except for a suitable version of Lemma \[lemma:bound\]. In contrast, we do not know if it is possible obtain a version of Theorem \[th:main\] where the exponential computational complexity is replaced with polynomial computational complexity as in .
Theorem \[th:main\] does not supersede the discrepancy bound obtained by Levin [@Levin:1999] for the discrepancy of a normal number with respect to one fixed base. For a fixed integer $b\geq 2$, Levin constructed a real number $x$ such that $$D_N(\{b^j x\}_{j\geq 0})< \frac{C_\theta (\log N)^2}{N}.$$ One should compare this upper bound with the lower bound obtained by Schmidt [@Schmidt:1972], who proved that there is a constant $C$ for [*every*]{} sequence $(x_j)_{j\geq 0}$ of real numbers in the unit interval there are infinitely many $N$s such that $$D_N((x_j)_{j\geq 0}) >C \frac{\log N}{ N}.$$ This lower bound is achieved by some so-called low-discrepancy sequences, (see [@Bugeaud:2012] and the references there), but it remains an important open problem whether this optimal order of discrepancy can also be achieved by a sequence of the form $(\{b^j x\})_{j\geq 0}$ for a real number $x$.
Accordingly, two central questions in this field remain open:
- Asked by Korobov [@Korobov:1955]: For a *fixed* integer $b\geq 2$, what is the function $\psi(N)$ with maximal speed of decrease to zero such that there is a real number $x$ for which $$D_N(\{b^j x\}_{j\geq 0})= \mathcal{O} \left(\psi(N)\right) \qquad \text{as $N \to \infty$?}$$
- Asked by Bugeaud (personal communication, 2017): Is there a number $x$ satisfying the minimal discrepancy estimate for normality not only in one fixed base, but in all bases at the same time? More precisely, let $\psi$ be Korobov’s function from above. Is there a real number $x$ such that for *all* integer bases $b\geq 2$, $$D_N(\{b^j x\}_{j\geq 0}) = \mathcal{O} \left( \psi(N) \right) \qquad \text{as $N \to \infty$?}$$
Definitions and lemmas {#sec:1}
======================
We use some tools from [@GalGal1964; @Philipp1975]. For non-negative integers $M$ and $N$, for a sequence of real numbers $(x_j)_{j\geq 0}$ and for real numbers $\alpha_1, \alpha_2$ such that $0\leq \alpha_1<\alpha_2\leq 1$, we define $$\begin{aligned}
F\left(M,N,\alpha_1, \alpha_2, (x_j)_{j\geq 1}\right) =
\Big|& \#\{j: M\leq j < M+N: \alpha_1\leq x_j< \alpha_2 \} - (\alpha_2-\alpha_1) N \Big|.\end{aligned}$$ To shorten notations we will write $\{b^j x\}_{j\geq 0}$ to denote $(\{b^j x\})_{j\geq 0} $. Throughout the paper we will use the fact that $$F\left(M,N, \alpha_1, \alpha_2, \{b^j x\}_{j\geq 0}\right) = F\left(0,N, \alpha_1, \alpha_2, \{b^{j+M} x\}_{j\geq 0}\right)$$ for every non-negative integer $M$.
The following lemma is a classical result from probability theory called Bernstein’s inequality (see for example [@sw Lemma 2.2.9]). We write $\mu$ for the Lebesgue measure and occasionally we write $\exp(x)$ for $e^x$.
\[lemma:bern\] Let $X_1, \dots, X_n$ be i.i.d. random variables having zero mean and variance $\sigma^2$, and assume that their absolute value is at most $1$. Then for every $\varepsilon > 0$ $$\mathbb{P} \left( \left| \sum_{k=1}^n X_k \right| > \varepsilon \sqrt{n} \right) \leq 2 \exp\left( \frac{-\varepsilon^2}{2\sigma^2 + 2/3 \varepsilon n^{-1/2}} \right).$$
\[lemma:bound\] Let $b \geq 2$ be an integer, let $h$ and $N$ be positive integers such that $N \geq h$, and let $\varepsilon$ be a positive real. Then for all integers $M \geq 0$ and $a$ satisfying $0\leq a<b^h$, $$\mu \left(\left\{x\in (0,1):
F \left(M,N,a b^{-h},(a+1) b^{-h} ,\{b^j x\}_{j\geq 0} \right) > \varepsilon \sqrt{h N} \right\} \right)$$ is at most $$2 h \exp\left( \frac{-\varepsilon^2}{2b^{-h} (1-b^{-h}) + 2/3 \varepsilon \lfloor N / h \rfloor^{-1/2}} \right).$$
We split the index set $\{M, M+1, \dots, M+N-1\}$ into $h$ classes, according to the remainder of an index when it is reduced modulo $h$. Then each of these classes contains either $\lfloor N/h \rfloor$ or $\lceil N/h \rceil$ elements. Let $\mathbf{1}_{[a b^{-h},(a+1) b^{-h})} (x)$ denote the indicator function of the interval $[a b^{-h},(a+1) b^{-h})$. Let $\mathcal{M}_0$ denote the class of all indices in $\{M, \dots, M+N-1\}$ which leave remainder zero when being reduced modulo $h$. Set $n_0 = \# \mathcal{M}_0$. Then it is an easy exercise to check that the system of functions $\big(\mathbf{1}_{[a b^{-h},(a+1) b^{-h})} (\{b^j x\}) - b^{-h}\big)_{j \in \mathcal{M}_0}$ is a system of i.i.d. random variables over the unit interval, equipped with Borel sets and Lebesgue measure.[^3] The absolute value of these random variables is trivially bounded by $1$, they have mean zero, and their variance is $b^{-h} (1-b^{-h})$. Thus by Lemma \[lemma:bern\] we have $$\begin{aligned}
& & \mu \left( \left\{x \in (0,1): \left|\sum_{j \in \mathcal{M}_0} \left( \mathbf{1}_{[a b^{-h},(a+1) b^{-h})} (\{b^j x\}) - b^{-h} \right) \right| > \varepsilon \sqrt{n_0} \right\} \right) \\
& \leq & 2 \exp\left( \frac{-\varepsilon^2}{2b^{-h} (1-b^{-h}) + 2/3 \varepsilon n_0^{-1/2}} \right).\end{aligned}$$ Clearly similar estimates hold for the indices in the other residue classes. Let $n_1, \dots, n_{h-1}$ denote the cardinalities of these other residue classes. By assumption $n_0 + \dots + n_{h-1} = N$. Note that by the Cauchy-Schwarz inequality we have $\sqrt{n_0} + \dots + \sqrt{n_{h-1}} \leq \sqrt{h}\sqrt{N}$. Thus, summing up, we obtain $$\begin{aligned}
& & \mu \left( \left\{ x \in (0,1): \left|\left(\sum_{j=M}^{M+N-1} \mathbf{1}_{[a b^{-h},(a+1) b^{-h})} (\{b^j x\})\right) - N b^{-h} \right| > \varepsilon \sqrt{h N} \right\} \right) \\
& \leq & 2 h \exp\left( \frac{-\varepsilon^2}{2b^{-h} (1-b^{-h}) + 2/3 \varepsilon \lfloor N / h \rfloor^{-1/2}} \right).\end{aligned}$$ This proves the lemma.
We will use a modified version of Lemma \[lemma:bound\], which works on any subinterval $A$ of $[0,1]$.
\[lemma:bound:mod\] Let $b \geq 2$ be an integer, let $h$ and $N$ be positive integers such that $N \geq h$, and let $\varepsilon$ be a positive real. Then for all integers $M \geq 0$ and $a$ satisfying $0\leq a<b^h$, for any subinterval $A$ of $[0,1]$ and for any positive integer $j_0$, $$\mu \left(\left\{x\in A:
F(M+j_0,N,a b^{-h},(a+1) b^{-h} ,\{b^j x\}_{j\geq 0}) > \varepsilon \sqrt{h N} \right\} \right)$$ is at most $$2 \mu(A) h\exp\left( \frac{-\varepsilon^2}{2b^{-h} (1-b^{-h}) + 2/3 \varepsilon \lfloor N / h \rfloor^{-1/2}} \right) + 2b^{-j_0}.$$
Let $B$ denote the largest interval contained in $A$ which has the property that both of its endpoints are integer multiples of $b^{-j_0}$. Then $\mu(A \backslash B) \leq 2b^{-j_0}$. Furthermore, by periodicity we have $$\begin{aligned}
&& \mu \left(\{x\in B: F \left(M + j_0,N, a b^{-m}, (a+1) b^{-m}, \{b^{j} x\}_{j\geq 0} \right) >\varepsilon \sqrt{hN} \} \right) \\
& = & \mu(B) \cdot \mu \left(\{x\in (0,1): F \left(M,N, a b^{-m}, (a+1) b^{-m}, \{b^{j} x\}_{j\geq 0} \right) >\varepsilon \sqrt{hN} \}\right),\end{aligned}$$ for which we can apply the conclusion of Lemma \[lemma:bound\]. Note that $\mu(B) \leq \mu(A)$. This proves Lemma \[lemma:bound:mod\].
The following corollary follows easily from Lemma \[lemma:bound:mod\].
\[lemma:bound:cor\] Let $b \geq 2$ be an integer, let $h$ and $N$ be positive integers such that $N \geq h$, and assume that $\varepsilon$ satisfies $$\begin{aligned}
\label{eps:criterion}
2/3 \varepsilon \lfloor N / h \rfloor^{-1/2} \leq \frac{1}{b h^5}.\end{aligned}$$ Then for all integers $M \geq 0$ and $a$ satisfying $0\leq a<b^h$, for any subinterval $A$ of $[0,1]$ and for any positive integer $j_0$ we have $$\mu \left(\left\{x\in A:
F \left(M+j_0,N,a b^{-h},(a+1) b^{-h} ,\{b^j x\}_{j\geq 0} \right) > \varepsilon \sqrt{h N} \right\} \right)$$ is at most $$\mu(A) 2 h \exp\left( \frac{-\varepsilon^2 b h^5}{529} \right) + 2b^{-j_0}.$$
The corollary follows from Lemma \[lemma:bound:mod\] and the fact that $$2b^{-h} (1-b^{-h}) \leq 2 b^{-h} \leq 528 b^{-1} h^{-5}$$ for all $b \geq 2$ and $h \geq 1$ (for the second inequality in the displayed formula it is sufficient to check that $2^{-h+2} \leq 528 h^{-5}$ for integers $h \geq 1$, which can be done numerically). Together with assumption this implies that $2b^{-h} (1-b^{-h}) + 2/3 \varepsilon \lfloor N / h \rfloor^{-1/2} \leq 529 b^{-1} h^{-5}$.
\[rk:basic\] For any two reals $\alpha_1, \alpha_2$ such that $0\leq \alpha_1<\alpha_2<1$, and for any sequence $(x_j)_{j\geq 1}$ of reals, a trivial bound yields $$F(0,N,\alpha_1,\alpha_2,(x_j)_{j\geq 1}) \leq 2 \sup_{\alpha \in [0,1)} F(0,N,0,\alpha,(x_j)_{j\geq 1}).$$ And, for any real number $\alpha \in (0,1)$, for any sequence of real numbers $(x_j)_{j\geq 1}$, and for any non-negative integers $N$ and $k$ we have $$F(0,N,0,\alpha,(x_j)_{j\geq 1}) \leq N/b^{k} +
\sum_{h=1}^k (b-1) \max_{ 0\leq a < b^h} F(0,N, a b^{-h}, (a+1) b^{-h},(x_j)_{j\geq 1}).$$ This observation follows from the fact that every interval $[0,\alpha)$ can be covered by at most $(b-1)$ intervals of length $b^{-1}$, at most $(b-1)$ intervals of length $b^{-2}$, and so on, at most $(b-1)$ intervals of length $b^{-k}$, and finally one additional interval of length $b^{-k}$. This decomposition can be easily derived from the digital representation of $\alpha$ in base $b$.
The index set can be decomposed in intervals between powers of $2$, and every possible initial segment of the index set can be written as a disjoint union of such sets. This fact is expressed in the following lemma.
\[lemma:philipp\] Let $b \geq 2$ be an integer, let $N$ be a positive integer and let $n$ be such that $2^{n-1} < N \leq 2^{n}$, and let $M$ be a non-negative integer. Then, there are non-negative integers $m_1,\ldots, m_n$ such that $m_\ell 2^\ell + 2^{\ell-1} \leq N$ for $\ell=1, \ldots, n$, and such that for any positive integer $h$ and any $a$, with $0\leq a<b^h$, $$\begin{aligned}
& & F(M,N,a b^{-h}, (a+1) b^{-h},\{b^j x\}_{j\geq 0}) \\
& \leq & N^{1/2} + \sum_{n/2 \leq \ell \leq n} F(M+ m_\ell 2^\ell, 2^{\ell-1},a b^{-h}, (a+1) b^{-h}, \{b^j x\}_{j\geq 0}).\end{aligned}$$
For the proof of Theorem \[th:main\] we proceed by induction, and define a sequence of nested binary intervals $(\Omega_k)_{k \geq 1}$ which gives us the binary digits of the absolutely normal number which we want to construct. Set $\Omega_1 = \Omega_2 = \dots = \Omega_{99} = (0,1)$ for the start of the induction. (We start the induction at $k=100$ in order to avoid trivial notational problems with small values of $k$.) We will always assume that $b \leq k$, so in step $k$ only bases $b$ from $2$ up to $k$ are considered. Different bases are added gradually as the induction steps forward.
For integers $k \geq 100$ and $b$ such that $2 \leq b \leq k$ we set $$N_k^{(b)} = \left\lceil 2^{k} \frac{\log 2}{\log b} \right\rceil.$$ We define sets $$\mathcal{N}_k^{b} = \left\{N \in \mathbb{N}:~N_k^{(b)} + 4k < N \leq N_{k+1}^{(b)} \right\}, \qquad k \geq 100, ~2 \leq b \leq k,$$ and $$\mathcal{R}_k^{b} = \left\{N \in \mathbb{N}:~ N_k^{(b)} < N \leq N_k^{(b)} + 4k \right\}, \qquad k \geq 100, ~2 \leq b \leq k.$$ The indices in $\bigcup_k \mathcal{R}_k^b$ are the “remainder”, and do not give a relevant contribution. Their purpose is to separate the elements of $\mathcal{N}_k^b$ from those of $\mathcal{N}_{k+1}^b$, so that $b^{j_2}$ is significantly larger than $b^{j_1}$ whenever $j_2 \in \mathcal{N}_{k+1}^b$ and $j_1 \in \mathcal{N}_k^{(b)}$. The sets $\mathcal{N}_k^b$ and $\mathcal{R}_k^b$ are constructed in such a way that they form a partition of $\mathbb{N}$, except for finitely many initial elements of $\mathbb{N}$. Precisely, one can check that these sets form a partition of $\mathbb{N} \backslash \left\{1, \dots, N_{\max\{100,b\}}^b \right\}$.
For the induction step, assume that $k \geq 100$ and that the interval $\Omega_{k-1}$ is already defined, and that the length of $\Omega_{k-1}$ is bounded below by $$\label{omega_measure}
\mu(\Omega_{k-1}) \geq 2^{-2^{k} - k}.$$ Set $$n_{k}^{(b)} = \left\lceil \log_2 \big(N_{k+1}^{(b)} - N_k^{(b)} - 4k\big) \right\rceil.$$ and $$T_b(k)=\left\lceil \frac{n_k^{(b)} \log 2}{2 \log b} \right\rceil.$$ For non-negative integers $b, a, h, \ell$ such that $$\begin{aligned}
2 \leq b \leq k, \quad 0\leq a< b^h, \quad
1\leq h\leq T_b(k), \quad n_k^{(b)}/2\leq \ell\leq n_k^{(b)},\end{aligned}$$ and non-negative integers $m_\ell$ such that $$\label{mell}
N_k^{(b)} + 4k + m_\ell 2^\ell + 2^{\ell-1} \leq N_{k+1}^{(b)}$$ we define the sets $$\begin{aligned}
H(b,k,a, h, \ell, m_\ell)
& = & \left\{x\in \Omega_{k-1}: F \left(N_k^{(b)} + 4k + m_\ell 2^{\ell},2^{\ell-1},a b^{-h},(a+1) b^{-h},\{b^j x\}_{j\geq 0} \right) \right. \\
& & \qquad \qquad \qquad\qquad\qquad \qquad \left. > 46 \cdot 2^{(\ell-1)/2} h^{-3/2} (n_k^{(b)}-\ell +1 )^{1/2} \right\}.\end{aligned}$$ Furthermore, set $$H_{b,k}=\bigcup_{h=1}^{T_b(k)} ~
\bigcup_{a=0}^{b^{h}-1} ~
\bigcup_{n_k^{(b)}/2 \leq \ell \leq n_k^{(b)}} ~ \bigcup_{m_\ell} ~
H(b,k,a, h, \ell, m_\ell).$$ where the last union is over those $m_\ell \geq 0$ satisfying .
The following lemma gives an upper bound for the measure of the set $H_{b,k}$. The proof of the lemma will be given in Section \[sec:prooflemmah\] below.
\[lemma:measureH\] For $k \geq 100$ and $2 \leq b \leq k$ we have $$\frac{\mu(H_{b,k})}{\mu(\Omega_{k-1})} \leq \frac{1}{2^{b}}.$$
As in the proof of Lemma \[lemma:bound\], let $\mathbf{1}_{[a b^{-h},(a+1)b^{-h})} (x)$ denote the indicator function of the interval $[a b^{-h},(a+1)b^{-h})$. For the function $F$ appearing in the definition of $H(b,k,a, h, \ell, m_\ell) $, we can write $$\begin{aligned}
&& F \left(N_k^{(b)} + 4k + m2^\ell,2^{\ell-1},ab^{-h},(a+1)b^{-h},(\{b^j x\})_{j \geq 0} \right) \label{funct_f} \\
& = & \left| \sum_{j=N_k^{(b)} + 4k + m2^\ell}^{N_k^{(b)} + 4k + m2^\ell+2^{\ell-1} -1} \left( \mathbf{1}_{[a b^{-h},(a+1)b^{-h})} (\{b^j x\}) - b^{-h} \right) \right|. \nonumber\end{aligned}$$ Note that the function $\mathbf{1}_{[a b^{-h},(a+1)b^{-h})} (x)$ is a step function which is constant on intervals ranging from one integer multiple of $b^{-h}$ to the next (it is zero everywhere, except from $ab^{-h}$ to $(a+1)b^{-h}$, where it is one). Accordingly, for some $j$, the function $$\mathbf{1}_{[a b^{-h},(a+1)b^{-h})} (\{b^j x\})$$ is a step function which is constant on intervals ranging from one integer multiple of $b^{-h} b^{-j}$ to the next. Thus the function in line is constant on all intervals ranging from one integer multiple of $b^{-h} b^{-(N_k^{(b)} + 4k + m2^\ell+2^{\ell-1} -1)}$ to the next, and thus by $h \leq T_b(k)$ and by it is also constant on all intervals ranging from one integer multiple of $b^{-T_b(k)} b^{-N_{k+1}^{(b)}}$ to the next.
As a consequence, the set $H_{b,k}$ consists of intervals whose left and right endpoints are integer multiples of $$\label{hbkform}
b^{-T_b(k)} b^{-N_{k+1}^{(b)}} = b^{- \left\lceil \frac{n_k^{(b)} \log 2}{2 \log b} \right\rceil} b^{-\left \lceil \frac{2^{k+1} \log 2}{\log b} \right\rceil}.$$ We call these intervals “elementary intervals”. We have $$b^{- \left\lceil \frac{n_k^{(b)} \log 2}{2 \log b} \right\rceil} \geq 2^{-n_k^{(b)}/2} b^{-1} \geq 2^{-(\log_2 N_{k+1}^{(b)})/2 - 1} b^{-1} \geq 2^{-k/2-2} b^{-1},$$ and $$b^{-\left \lceil \frac{2^{k+1} \log 2}{\log b} \right\rceil} \geq 2^{-2^{k+1}} b^{-1},$$ So the length of these elementary intervals of $H_{b,k}$ is at least $2^{-2^{k+1}-k/2-2} b^{-2}$.
Let $H_{b,k}^*$ denote the collection of all those intervals of the form $$\label{hbk:dyad}
\left[a 2^{-2^{k+1} - k}, (a+1) 2^{-2^{k+1} - k} \right) \qquad \text{for some integer $a$}$$ which have non-empty intersection with $H_{b,k}$. Note that by the calculations in the previous paragraph the intervals of the form are much shorter than the elementary intervals of $H_{b,k}$, and thus the total measure of $H_{b,k}^*$ is just a little bit larger than that of $H_{b,k}$. In particular, it is true that $$\mu (H_{b,k}^*) \leq \frac{11}{10} \mu (H_{b,k}).$$ Consequently, by Lemma \[lemma:measureH\] we have $$\mu \left( \Omega_{k-1} \backslash \bigcup_{b=2}^{k} H_{b,k}^* \right) \geq \left(1 - \frac{11}{10} \sum_{b=2}^{k} \frac{1}{2^b}\right) \mu(\Omega_{k-1}) \geq \frac{9}{20} \mu(\Omega_{k-1}).$$ Thus, there exists an interval of the form which is contained in $\Omega_{k-1}$, but has empty intersection with all the sets $H_{b,k}$ for $b=2, \dots, k$. We define $\Omega_k$ as this interval, and note that the length of $\Omega_k$ is $$\label{omega_k_measure_2}
\mu(\Omega_k) = 2^{-2^{k+1} - k}.$$ Now we can make the induction step $k \mapsto k+1$, where guarantees that the induction hypothesis is met.
Proof of Lemma \[lemma:measureH\] {#sec:prooflemmah}
=================================
We use Corollary \[lemma:bound:cor\] to estimate the measure of the sets $H(b,k,a, h, \ell, m_\ell)$. More precisely, we apply the corollary with the choice of $$j_0 = N_k^{(b)} + 4k, \qquad M = m_\ell 2^\ell, \qquad N = 2^{\ell-1}, \qquad A = \Omega_{k-1}, \qquad \varepsilon = 46 (n_k^{(b)}-\ell+1)^{1/2} h^{-2},$$ where $$1\leq h\leq T_b(k), \qquad n_k^{(b)}/2\leq \ell\leq n_k^{(b)},$$ and $m_\ell$ satisfies . So, $$\varepsilon \sqrt{hN} = 46\cdot 2^{(\ell-1)/2} h^{-3/2} (n_k^{(b)} -\ell+1 )^{1/2}.$$ For the corollary to be applicable, we have to check whether $N \geq h$ and hold for our choice of variables. However, both conditions are easily seen to be satisfied, since by assumption we have $N \geq 2^{n_k^{(b)}/2-1}$, which depends on $k$ exponentially, while $h \leq T_b(k) \leq \left\lceil \frac{n_k^{(b)} \log 2}{2 \log b} \right\rceil$ and $\varepsilon \leq 46 \sqrt{n_k^{(b)}}$ grow in $k$ at most linearly (remember that we assumed $k \geq 100$). Thus, we can apply Corollary \[lemma:bound:cor\], and we obtain $$\begin{aligned}
\mu(H(b,k,a, h, \ell, m_\ell)) & \leq & \mu(\Omega_{k-1}) 2 h \exp\left(-\frac{46^2 b (n_k^{(b)}-\ell+1) h^{-4} h^5}{529} \right) + 2b^{-j_0} \\
& = & \mu(\Omega_{k-1}) 2 h \exp\left(-4 b (n_k^{(b)}-\ell+1) h \right) + 2b^{-j_0}.\end{aligned}$$ Note that by , $$2b^{-j_0} = 2 b^{-N_k^{(b)}-4k} \leq 2 b^{-\frac{2^{k} \log 2}{\log b}+1-4k} \leq 2b2^{-2^k}b^{-4k} \leq 2b^{-3k+1} \mu(\Omega_{k-1}).$$ Using the facts that $T_b(h) \leq k/2+1$, that $n_{k}^{(b)} \leq k$ for all $b$, and that implies that there are at most $2^{n_k^{(b)}-\ell} \leq 2^k$ different values for $m_\ell$, we obtain $$\begin{aligned}
& & \sum_{h=1}^{T_b(k)} ~\sum_{a=0}^{b^{h}-1} ~\sum_{n_k^{(b)}/2 \leq \ell \leq n_k^{(b)}} ~\sum_{m_\ell} 2b^{-3k+1} \\
& \leq & (k/2+1) b^{k/2 +1} k 2^k 2b^{-3k+1} \\
& \leq & \frac{1}{10} b^{-k},\end{aligned}$$ where for the last inequality we use the fact that $k \geq 100$ (by assumption).
Furthermore, using the fact that $e^{-xy} \leq e^{-x} e^{-y}$ for $x,y \geq 2$, we have $$\begin{aligned}
& & \sum_{h=1}^{T_b(k)} ~\sum_{a=0}^{b^h-1} ~ \sum_{n_k^{(b)}/2
\leq \ell \leq n_k^{(b)}} ~\sum_{m_\ell} ~2 h \exp\left(-4 bh (n_k^{(b)}-\ell+1) \right) \nonumber\\
& \leq & \sum_{h=1}^{T_b(k)} b^h ~\sum_{n_k^{(b)}/2 \leq \ell \leq n_k^{(b)}} 2^{n_k^{(b)}-\ell} ~2 h \ \exp({-2bh})\ \exp\big({-2 \big(n_k^{(b)}-\ell+1\big)}\big) \nonumber\\
& \leq & \underbrace{\left( \sum_{h=1}^{\infty} 2h b^h \exp({-2bh}) \right)}_{\leq 11/10 e^{-b}} \underbrace{\sum_{n_k^{(b)}/2 \leq \ell \leq n_k^{(b)}} \exp\big({-\big(n_k^{(b)}-\ell+1\big)}\big)}_{\leq \sum_{r=1}^\infty e^{-r} \leq 6/10} \nonumber\\
& \leq & \frac{7}{10} e^{-b}, \nonumber\end{aligned}$$ where we used that $b - \log b \geq 1.3$ for $b \geq 2$ and consequently $$\sum_{h=1}^{\infty} 2h b^h e^{-2bh} =
\sum_{h=1}^\infty 2 h e^{- h (b -\log b)} e^{-bh} \leq e^{-b} \underbrace{\sum_{h=1}^\infty 2 h e^{- 1.3h}}_{\leq 11/10} \leq \frac{11}{10} e^{-b}.$$ Thus, we have $$\begin{aligned}
\mu (H_{b,k}) & \leq & \frac{7}{10} e^{-b} \mu(\Omega_{k-1}) + \frac{1}{10} b^{-k} \mu(\Omega_{k-1}) \\
& \leq & 2^{-b} \mu(\Omega_{k-1}),\end{aligned}$$ where we used the assumption that $b \leq k$. This proves the lemma.
Proof of Theorem \[th:main\]
============================
The proof of Theorem \[th:main\] now follows using well-known arguments, which allow to turn the estimates for subsums over dyadic subsets of the index set and over dyadic subintervals of the unit interval into a result which holds uniformly over all subintervals in the unit interval, and for all initial segments of the full index set.
Let $b \geq 2$ be given, and assume that $N$ is “large” (depending on $b$). Then there is a number $k$ such that $N$ is contained in either $\mathcal{N}_k^b$ or $\mathcal{R}_k^b$. Let $x$ be a real number which is contained in $\bigcap_{j \geq 1} \Omega_j$. Such a number exists, since $(\Omega_j)_{j\geq 1}$ is a sequence of non-empty nested intervals. Then for this $x$ we have, for arbitrary $0 \leq \alpha_1 < \alpha_2 \leq 1$,
$$\begin{aligned}
\MoveEqLeft F(0,N,\alpha_1, \alpha_2, \{b^j x\}_{j\geq 1}) &\leq \nonumber\\
& & F(0,N_{\lfloor k/2 \rfloor }^{(b)},\alpha_1,\alpha_2,(\{b^j x\})_{j \geq 0}) \label{line:1}\\
&&+\ \sum_{r=\lfloor k/2 \rfloor}^{k-1} F(N_r^{(b)}+4r, N_{r+1}^{(b)} - (N_r^{(b)} + 4r) , \alpha_1, \alpha_2, \{b^j x\}_{j \geq 0}) \label{line:2}\\
&&+\ F(N_{k}^{(b)},N-N_{k}^{(b)}-4k, \alpha_1, \alpha_2, \{b^j x\}_{j \geq 0}) \label{line:3}\\
&&+\ \# \left\{j:~j \in \bigcup_k \mathcal{R}_k^b,~j \leq N \right\}. \label{line:4}\end{aligned}$$
The term in line is bounded by $N_{\lfloor k/2 \rfloor }^{(b)} \leq 2^{k/2}{\frac{ \log 2}{\log b}} +1 \leq 2\sqrt{N}$, since $N >N_k^{(b)} \geq 2^k{\frac{ \log 2}{\log b}}$ by assumption. Now we bound the term in line . By Remark \[rk:basic\] and Lemma \[lemma:philipp\] and using the definition of the sets $H_{k,b}$, for every $r$ such that $\lfloor k/2\rfloor\leq r \leq k-1$, we have $$\begin{aligned}
&& F(N_r^{(b)}+4r, N_{r+1}^{(b)} - (N_r^{(b)} + 4r) , \alpha_1, \alpha_2, \{b^j x\}_{j \geq 0}) \\
&\leq & \sqrt{N_{r+1}^{(b)}} +
2 (b-1) \sum_{n^{(b)}_{r}/2 \leq \ell \leq n_r^{(b)}}
\sum_{h=1}^\infty \max_{ 0\leq a < b^h} F(N_r^{(b)}+4r + m_\ell 2^\ell, 2^{\ell-1}, a b^{-h}, (a+1)b^{-h}, \{b^j x\}_{j \geq 0})
\\&& \qquad \qquad \qquad \qquad \qquad\qquad \qquad \qquad
\text{for integers $m_1,m_2, \ldots m_{n_r^{(b)}}$ established in Lemma \ref{lemma:philipp},}
\\
& \leq & \sqrt{N_{r+1}^{(b)}} + 2(b-1) ~\sum_{n^{(b)}_{r}/2 \leq \ell \leq n_r^{(b)}} ~\sum_{h=1}^\infty h^{-3/2} 46 \cdot 2^{(\ell-1)/2} (n_r^{(b)}-\ell+1)^{1/2} \\
& \leq & \sqrt{N_{r+1}^{(b)}} + 2(b-1)\cdot 46 \underbrace{\left(\sum_{h=1}^\infty h^{-3/2} \right)}_{\leq 2.62} \underbrace{\left(\sum_{n_r^{(b)}/2 \leq \ell \leq n_r^{(b)}} 2^{(\ell-1)/2} (n_r^{(b)}-\ell+1)^{1/2}\right)}_{\leq 2^{n_r^{(b)}/2} \sum_{u=1}^\infty 2^{-u/2} u^{1/2} \leq 4.15 \cdot 2^{n_r^{(b)}/2} \leq 2.94 \sqrt{N_{r+1}^{(b)}}} \nonumber\\
& \leq & \sqrt{N_{r+1}^{(b)}} + 709\cdot b \sqrt{N_{r+1}^{(b)}} \\
& \leq & 710 \cdot b \cdot 2^{(r+1-k)/2} \sqrt{N_{k}^{(b)}}.\end{aligned}$$ Consequently, for the term in line we get $$\begin{aligned}
\sum_{r=\lfloor k/2 \rfloor}^{k-1}
F(N_r^{(b)}+4r, N_{r+1}^{(b)} - (N_r^{(b)} + 4r) , \alpha_1, \alpha_2, \{b^j x\}_{j \geq 0})
& \leq &711\cdot b \sum_{r=\lfloor k/2 \rfloor}^{k-1} 2^{(r+1-k)/2} \sqrt{N_{k}^{(b)}} \\
& \leq & 2425\cdot b \sqrt{N_{k}^{(b)}} \\
& \leq &2425 \cdot b \sqrt{N}.\end{aligned}$$
Similarly, for the term in line we get $$\begin{aligned}
F(N_{k}^{(b)},N-N_k^{(b)}-4k, \alpha_1, \alpha_2, \{b^j x\}_{j \geq 0}) & \leq & 711\cdot b\ \sqrt{N_{k+1}^{(b)}} \\
& \leq & 1005 \cdot b \sqrt{N}, \end{aligned}$$ where we used the fact that $2N \geq N_{k+1}^{(b)}$.
Finally, the term in line is bounded by $4k \lceil k/2 \rceil \leq \sqrt{N}$ for sufficiently large $N$.
Concluding our estimates for the lines –, we finally get $$F(0,N,\alpha_1, \alpha_2, \{b^j x\}_{j\geq 0}) \leq
(2 + 2425 + 1005 + 1) \ b \sqrt{N} = 3433\cdot b \sqrt{N},$$ for all sufficiently large $N$. This can be written in the form $$D_N((\{b^j x\})_{j \geq 0}) \leq 3433\cdot b\ N^{-1/2}$$ for sufficiently large $N$, which proves the theorem.
Computational complexity
------------------------
The real number determined by our construction is the unique element $x$ in $\bigcap_{k\geq 1} \Omega_k$. The definition of $(\Omega_k)_{k\geq 1}$ is inductive. Assume that $\Omega_{k-1}$ is given. The interval $\Omega_{k}$ is the leftmost interval of the form $$\left[ a 2^{-2^{k+1}-k}, (a+1) 2^{-2^{k+1}-k}\right)$$ that lies in $$\Omega_{k-1}\ \setminus \ \bigcup_{b=2}^{k} H_{b,k}.$$ So, $\Omega_k$ is one element of the partition of $\Omega_{k-1}$ in $2^{2^{k}+1}$ subintervals. Let us count first how many mathematical operations suffice to test if one of these subintervals is outside $\bigcup_{b=2}^{k} H_{b,k}$. Recall that $$\bigcup_{b=2}^{k} H_{b,k}=\bigcup_{b=2}^{k} ~\bigcup_{h=1}^{T_b(k)} ~
\bigcup_{a=0}^{b^{h}-1} ~
\bigcup_{n_k^{(b)}/2 \leq \ell \leq n_k^{(b)}} ~ \bigcup_{m_\ell} ~
H(b,k,a, h, \ell, m_\ell).$$ where the last union is over those $m_\ell \geq 0$ satisfying $N_k^{(b)} + 4k + m_\ell 2^\ell + 2^{\ell-1} \leq N_{k+1}^{(b)}$. There are at most $ 2^{n^{(b)}_k-\ell}$ values of $m_\ell$, and for each of them the evaluation of $$F(N_k^{(b)} + 4k + m2^{\ell},2^{\ell-1},a b^{-h},(a+1) b^{-h},\{b^j x\}_{j\geq 0}),$$ for any fixed $x$ and fixed $k,b,a,h$, requires the inspection at $2^{\ell-1} $ indices. Hence, the total number of indices involved in the inspection is $ 2^{n^{(b)}_k-\ell} \cdot 2^{\ell-1} = 2^{n^{(b)}_k}$. Observe that $b\leq k$, $T_b(k)<k/2$, $n_k^{(b)}\leq k$, and that there are $k/2$ values of $\ell$. Thus, ignoring the operations needed to perform base change, the number of mathematical operations to test if one candidate subinterval is outside $\bigcup_{b=2}^{k} H_{b,k}$ is at most $$k \cdot k/2 \cdot k^{k/2} 2^{k}.$$ In the worst case we need to test all the subintervals (except the last one), and there are $$2^{2^k +1}$$ of them. This last factor dominates the total number of mathematical operations that should be performed in the worst case at step $k$, which consequently is the order at most $\mathcal{O}\left(2^{2^{k+1}}\right)$, say. Since this is doubly exponential in $k$, the number of mathematical operations performed from step $1$ up to step $k$ is also at most of order $\mathcal{O}\left(2^{2^{k+1}}\right)$.
At step $k$ the construction determines $2^{k}+1$ new digits in the binary expansion of the defined number $x$. Thus, at the end of step $k$ the first $2^{k+1}$ digits will be determined. Then, to compute the $N$-th digit in the binary expansion of $x$ it suffices to compute up to step $\lceil\log_2 N\rceil$. This entails a number of mathematical operations that is at most $\mathcal{O}\left(2^{2^{\log N}}\right) =\mathcal{O}\left(2^N\right) $. This proves that there is an algorithm that computes the first $N$ digits of the binary expansion of $x$ after performing a number of operations that is exponential in $N$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank Yann Bugeaud and Katusi Fukuyama for comments concerning this paper. The first author is supported by the Austrian Science Fund FWF, project Y-901. The second author is supported by ANPCyT project PICT 2014-3260. The third author is supported by FWF projects I 1751-N26; W1230, Doctoral Program “Discrete Mathematics” and SFB F 5510-N26. The fourth author is partially supported by the National Science Foundation grant DMS-1600441. This work was initiated during a workshop on normal numbers at the Erwin Schrödinger Institute in Vienna, Austria, which was held in November 2016 and in which all four authors participated.
[^1]: 2010 *Mathematics Subject Classification*: Primary 11K16; Secondary 11-Y16,68-04.
[^2]: *Key words and phrases*: normal numbers, uniform distribution, discrepancy.
[^3]: These functions are Rademacher functions, just in base $b^h$ instead of the usual base $2$. See for example [@strook Section 1.1.3] for more details.
|
---
abstract: 'We locate the $\gamma$-ray and lower frequency emission in flares of the BL Lac object AO 0235+164 at ${\ {\raise-.5ex\hbox{$\buildrel>\over\sim$}}\ }12$pc in the jet of the source from the central engine. We employ time-dependent multi-spectral-range flux and linear polarization monitoring observations, as well as ultra-high resolution ($\sim0.15$ milliarcsecond) imaging of the jet structure at $\lambda=7$mm. The time coincidence in the end of 2008 of the propagation of the brightest superluminal feature detected in AO 0235+164 (Qs) with an extreme multi-spectral-range ($\gamma$-ray to radio) outburst, and an extremely high optical and 7mm (for Qs) polarization degree provides strong evidence supporting that all these events are related. This is confirmed at high significance by probability arguments and Monte-Carlo simulations. These simulations show the unambiguous correlation of the $\gamma$-ray flaring state in the end of 2008 with those in the optical, millimeter, and radio regime, as well as the connection of a prominent X-ray flare in October 2008, and of a series of optical linear polarization peaks, with the set of events in the end of 2008. The observations are interpreted as the propagation of an extended moving perturbation through a re-collimation structure at the end of the jet’s acceleration and collimation zone.'
address: |
$^{1}$Instituto de Astrofísica de Andalucía (CSIC), Apartado 3004, E-18080 Granada, Spain\
$^{2}$IAR, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA\
$^{3}$St. Petersburg University, Universitetskij Pr. 28, Petrodvorets, 198504 St. Petersburg, Russia\
$^{4}$Isaac Newton Institute of Chile, St. Petersburg Branch, St. Petersburg, Russia\
$^{5}$Aalto Univ. Metsähovi Radio Observatory, Metsähovintie 114, FIN-02540 Kylmälä, Finland\
$^{6}$Steward Observatory, University of Arizona, Tucson, AZ 85721-0065, USA\
$^{7}$FINCA, University of Turku, Väisäläntie 20, FIN-21500 Piikkiö, Finland\
$^{8}$CCAA, California Institute of Technology, Mail Code 222, Pasadena, CA 91125, USA\
$^{9}$DA, University of Michigan, 817 Dennison Building, Ann Arbor, MI 48 109, USA\
$^{10}$ZAH, Landessternwarte Heidelberg, Königstuhl, 69117 Heidelberg, Germany\
$^{11}$Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA\
$^{12}$IRAM, 300 Rue de la Piscine, 38406 St. Martin dÕH‘eres, France\
$^{13}$Space Science Institute, Boulder, CO 80301, USA\
$^{14}$Abastumani Observatory, Mt. Kanobili, 0301 Abastumani, Georgia
author:
- 'I. AGUDO$^{1,2}$, A. P. MARSCHER$^{2}$, S. G. JORSTAD$^{2,3}$,V. M. LARIONOV$^{3,4}$, J. L. GÓMEZ$^{1}$, A. LÄHTEENMÄKI$^{5}$, P. S. SMITH$^{6}$, K. NILSSON$^{7}$, A. C. S. READHEAD$^{8}$, M. F. ALLER$^{9}$, J. HEIDT$^{10}$, M. GURWELL$^{11}$, C. THUM$^{12}$, A. E. WEHRLE$^{13}$, O. M. KURTANIDZE$^{14}$'
title: 'LOCATION OF THE $\gamma$-RAY FLARING EMISSION IN THE PARSEC-SCALE JET OF THE BL LAC OBJECT AO 0235+164'
---
Introduction {#intr}
============
The physical information about high energy emission mechanisms provided by blazars depends on where those $\gamma$-rays were originated, which is currently subject of debate.[@Marscher:2010p14415] Two main locations of the site of $\gamma$-ray emission in blazars are claimed in the literature. The first one is a region close to the central black hole (BH), at $\lesssim0.1{\rm{-}}1$pc, which is convenient to explain short time-scales of variability of a few hours reported in some $\gamma$-ray observations of blazars.[@Ackermann:2010p13506; @Tavecchio:2010p14858] Although short time scales of variability do not necessarily imply short distances to the BH, but only small sizes of the emitting region, this scenario has the advantage that optical-UV photons from the broad line region are available for scattering to $\gamma$-ray energies by relativistic electrons in the jet. However, locating the $\gamma$-ray emission so close to the central engine implies that the $\gamma$-ray and radio-mm emission sites are far from each other and hence their emission events non-coincident, which seems unlikely for an increasing number of blazars.[@Jorstad:2010p11830][@Agudo:2011p15946] This problem is overcome if the $\gamma$-rays are emitted from a region much further away from the central engine (at $>>1$pc), more concretely where the jet starts to be visible at millimeter wavelengths with VLBI. Supporting this scenario, Ref. unambiguously locate the region of $\gamma$-ray flaring emission at ${\ {\raise-.5ex\hbox{$\buildrel>\over\sim$}}\ }14$pc from the central engine in the jet of [OJ287]{} through correlation of the millimeter light curves with the $\gamma$-ray ones and direct ultrahigh-resolution 7mm VLBI imaging. Similar results are obtained by Refs. and for [PKS 1510$-$089]{} and [3C 454.3]{}, respectively. Ref. have recently proposed a model that reconciles both the evidences in favor of the location of $\gamma$-ray flare emitting regions $>>1$pc from the BH with the “few-hour” time scales of $\gamma$-ray variability through a turbulence model with frequency-dependent filling factor. Following Ref. for OJ287, we use here time series of total flux and polarization multiwavelength observations (including monthly VLBI images) to investigate the location of the $\gamma$-ray emitting region in the BL Lac object AO 0235+164 ($z=0.94$).
Observations {#obs}
============
Our polarimetric monitoring observations of AO 0235+164 (Figs. \[maps\]-\[pol\]) include (1) 7mm images with the Very Long Baseline Array (VLBA) from the Boston University monthly blazar-monitoring program, (2) 3.5mm observations with the IRAM 30m Telescope, and (3) optical measurements from several observatories. These include Calar Alto (2.2m Telescope), Steward (2.3 and 1.54m Telescopes), Lowell (1.83m Perkins Telescope), San Pedro Mártir (0.84m Telescope), Crimean Astrophysical (0.7m Telescope), and the St. Petersburg State University (0.4m Telescope) observatories. Our total flux light curves (Fig. \[tflux\]) include data from the *Fermi* $\gamma$-ray (0.1-200GeV) and *Swift* X-ray (2.4-10keV) observatories available from the archives of these missions, from *RXTE* (2.4-10keV), from the Tuorla Blazar Monitoring Program, the Yale University SMARTS program and Maria Mitchell Observatory in the optical $R$-band, by the Submillimeter Array (SMA) at 1.3mm and 850$\mu$m, by the IRAM 30m Telescope at 1.3mm, by the Metsähovi 14m Telescope at 8mm, and by the Owens Valley Radio Observatory and the University of Michigan Radio Astronomy Observatory 26m Telescope at 2cm. For the data reduction we followed the procedures used in Ref. (see also ).
Superluminal Jet Ejection from the 2008 Millimeter Flare {#ejection}
========================================================
Our 7mm VLBA maps of AO 0235+164 (Fig. \[maps\]) show a compact total intensity distribution that is fitted by one or two circular Gaussian components in most of the observing epochs, when preserving consistency from epoch to epoch and flat residual maps. We identify the core of emission in our images with the innermost visible mm emitting region in the jet. Between mid-2008 and mid-2009, the jet structure also shows a second emission region (Qs, the brightest 7mm jet feature detected so far) that propagates at a superluminal apparent speed $<{\beta_{\rm{app}}}>=(12.6\pm1.2)\,c$.
The ejection of Qs, that we estimate in $2008.30\pm0.08$, coincides with the start of an extreme millimeter outburst (${08}_{\rm{mm}}$) started in early 2008, and peaking (at $\sim6.5$Jy at 3mm) on October 10$^{\rm{th}}$, 2008 (see Fig. \[tflux\]). Figure \[tflux\] shows that ${08}_{\rm{rad}}$ and ${08}_{\rm{mm}}$ radio and millimeter outbursts are produced by the contribution of both the VLBI core and Qs, that peak around October 20$^{\rm{th}}$ and November 16$^{\rm{th}}$ 2008, respectively. Their coherent -although delayed- co-evolution suggests that the disturbance responsible for the ejection of Qs is wide and extends from the location of the core to Qs in the frame of the observer via light travel time delay effects.[@1997ApJ...482L..33G; @Agudo:2001p460] Qs is the only strong superluminal ejection during our observations and the ${08}_{\rm{rad}}$ and ${08}_{\rm{mm}}$ flares are the only ones starting essentially after such ejection, which allow us to unambiguously relate both kinds of events.
Flaring Correlation from $\gamma$-rays to Radio Wavelengths {#corr}
===========================================================
Figure \[tflux\] reveals that the ${08}_{\rm{rad}}$ and ${08}_{\rm{mm}}$ flares were accompanied by sharp optical, X-ray, and $\gamma$-ray counterparts (${08}_{\rm{opt}}$, ${08}_{\rm{X}}$, and ${08}_{\gamma}$ flares, respectively). Our formal correlation study of the light curves in Fig. \[tflux\][@Agudo:2011p15946] confirms the correlation of the $\gamma$-ray variability with the one at 2cm, 8mm, 1mm, and optical with confidence $>99.7$%. Even the integrated flux evolution in the 7mm VLBI-core region is also correlated with the $\gamma$-ray light curve at $>$99.7% confidence. The optical linear polarization degree ($p_{\rm{opt}}$) evolution and X-ray light curve are also correlated with the optical $R$-band, 1mm, and 2cm light curves at $>99.7$% confidence, hence pointing out that the extreme flaring activity shown in our light curves is related at all wavebands from radio to $\gamma$-rays.
Note however that there is not a common variability pattern at all spectral ranges on short time scales ($\lesssim2$ months), although there is clear correlation on the long time scales (of $\sim$years), hence reflecting different variability properties on short and long time scales.
Relation with the Linear Polarization Variability {#polvar}
=================================================
Figure \[pol\] show extremely high optical polarization fraction ($p_{\rm{opt}}\gtrsim30$%) and $p_{\rm{opt}}$ variability during the sharp ${08}_{\rm{opt}}$ optical peaks. Whereas the integrated millimeter linear-polarization-degree ($p_{\rm{mm}}$), and the one of the core are observed to lie at moderate values $\lesssim5$%, for Qs, $p_{\rm{mm,Qs}}$ shows an extremely high peak at $\sim16$% close to the time of the second sharp optical sub-flare. Such an extraordinary $p_{\rm{mm,Qs}}$ peak -by far the brightest superluminal feature ever detected in AO 0235+164- has never been observed in other VLBI jet feature in the source. Its coincidence with the high optical flux and $p_{\rm{opt}}$ state, and with the flaring states at the remaining spectral ranges (including the 7mm-VLBI-core one), evidences that the ejection and propagation of Qs in AO 0235+164’s millimeter jet is physically tied to the contemporaneous multi-spectral-range flaring state and the extreme $p_{\rm{opt}}$ activity reported here.
Discussion and Conclusions
==========================
The time coincidence of the ejection and propagation of Qs -the brightest superluminal feature reported in AO 0235+164 so far- with the extreme $\gamma$-ray to radio outbursts in 2008 reported here and the extremely high $p_{\rm{opt}}$ and $p_{\rm{mm,Qs}}$, provides extremely strong evidence in favor of the physical relation of all these events. This is confirmed by simple probability arguments[@Agudo:2011p15946] -that give $p_{\gamma,{\rm{opt}},{\rm{mm}}}=5\times10^{-4}$ for observing, by chance, a contemporaneous $\gamma$-ray, an optical, and a radio-millimeter flare- and by our formal correlation study.[@Agudo:2011p15946] The latter shows the unambiguous correlation of the $\gamma$-ray outburst in the end of 2008 with those in the optical, millimeter (including the 7mm VLBI core), and radio spectral ranges. The location of the 7mm VLBI-core $\gtrsim12$pc from the jet vertex in AO 0235+164[@Agudo:2011p15946], together with the high-confidence correlation of the 7mm core flare in 2008 with the multi-wavelength flare at all other spectral ranges[@Agudo:2011p15946], evidences that the emission from the flares at these different spectral ranges was also located $\gtrsim12$pc from the central engine.
We identify the 7mm VLBI-core as a stationary re-collimation structure at the end of the acceleration and collimation zone of the jet’s acceleration and collimation zone.[@Marscher:2010p11374; @Marscher:2008p15675] Qs superluminal feature is rather related to a moving plane-perpendicular shock, given the extremely high $p_{\rm{mm,Qs}}$, and $\chi_{\rm{mm}}^{\rm{Qs}}$ parallel to Qs’s direction of propagation. The core shows a flux evolution closely tied to the Qs one, and its light curve is correlated at high confidence with those at $\gamma$-rays, optical, and millimeter frequencies. This suggests that Qs is the head of an extended structure stretched by light travel time-delays in the observer’s frame, as e.g. the front-back structure reported by Ref. .
Multi-zone SSC scenarios, e.g. as those involving emission from different turbulence cells[@Marscher:2010p14415], are likely for the case of AO 0235+164 given its good multi-spectral-range correlation on the long time scales, but poorer short time-scale correlation (see also Ref. ).
Acknowledgements
================
This research was partly funded by NASA grants NNX08AJ64G, NNX08AU02G, NNX08AV61G, and NNX08AV65G, NSF grant AST-0907893, NRAO award GSSP07-0009; RFBR grant 09-02-00092; MICIIN grant AYA2010-14844; CEIC grant P09-FQM-4784; and GNSF grant ST08/4-404. The VLBA is an instrument of the NRAO, a facility of the NSF operated under cooperative agreement by AUI. The PRISM camera at Lowell Observatory was developed by Janes et al. The Calar Alto Observatory is jointly operated by MPIA and IAA-CSIC. The IRAM 30m Telescope is supported by INSU/CNRS, MPG, and IGN. The SMA is a joint project between the SAO and the Academia Sinica.
[00]{} A. P. Marscher and S. G. Jorstad, [*in Fermi Meets Jansky-AGN at Radio and Gamma-rays, ed. T. Savolainen et al.*]{}, 171 (2010). M. Ackermann [*et al.*]{}, [*Astrophys. J.*]{} [**721**]{}, 1383 (2010). F. Tavecchio [*et al.*]{}, [*Mon. Not. R. Astron. Soc.*]{} [**405**]{}, L94 (2010). S. G. Jorstad [*et al.*]{}, [*Astrophys. J.*]{} [**715**]{}, 362 (2010). A. P. Marscher [*et al.*]{}, [*Astrophys. J.*]{} [**710**]{}, L126 (2010). I. Agudo [*et al.*]{}, [*Astrophys. J.*]{} [**726**]{}, L13 (2011a). I. Agudo [*et al.*]{}, [*Astrophys. J.*]{} [**735**]{}, L10 (2011b). I. Agudo [*et al.*]{}, [*Astron. Astrophys.*]{} [**456**]{}, 117 (2006). I. Agudo [*et al.*]{}, [*Astrophys. J. Sup. Ser.*]{} [**189**]{}, 1 (2010). V. M. Larionov [*et al.*]{}, [*Astron. Astrophys.*]{} [**492**]{}, 389 (2008). J. L. Gómez [*et al.*]{}, [*Astrophys. J.*]{} [**482**]{}, L33 (2010). I. Agudo [*et al.*]{}, [*Astrophys. J.*]{} [**549**]{}, L183 (2001). A. P. Marscher [*et al.*]{}, [*Nature*]{} [**452**]{}, 966 (2008). M.-[Á]{}. Aloy [*et al.*]{}, [*Astrophys. J.*]{} [**585**]{}, L109 (2003).
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---
abstract: 'The momentum distributions, natural orbits, spectroscopic factors and quasi-hole wave functions of the [$^{12}$C ]{},[$^{16}$O ]{},[$^{40}$Ca ]{},[$^{48}$Ca ]{}, and [$^{208}$Pb ]{} doubly closed shell nuclei, have been calculated in the framework of the Correlated Basis Function theory, by using the Fermi hypernetted chain resummation techniques. The calculations have been done by using the realistic Argonne $v''_8$ nucleon-nucleon potential, together with the Urbana IX three-body interaction. Operator dependent correlations, which consider channels up to the tensor ones, have been used. We found noticeable effects produced by the correlations. For high momentum values, the momentum distributions show large enhancements with respect to the independent particle model results. Natural orbits occupation numbers are depleted by about the 10% with respect to the independent particle model values. The effects of the correlations on the spectroscopic factors are larger on the more deeply bound states.'
author:
- 'C. Bisconti$^{\,1,2,3}$, F. Arias de Saavedra$^{\,2}$ and G. Co’$^{\,1}$'
title: 'Momentum distributions and spectroscopic factors of doubly-closed shell nuclei in correlated basis function theory'
---
INTRODUCTION {#sec:int}
============
One of the major achievements of nuclear structure studies in the last ten years is the consolidation of the validity of the non relativistic many-body approach. The idea is to describe the nucleus with a Hamiltonian of the type: $$H=-\frac{\hbar^2}{2m}\sum_{i}^A\nabla_i^2+
\sum_{i<j=1}^A v_{ij}+\sum_{i<j<k=1}^A v_{ijk}
\,\,,
\label{eq:hamiltonian}$$ where the two- and three-body interactions, $v_{ij}$ and $v_{ijk}$ respectively, are fixed to reproduce the properties of the two- and three-body nuclear systems. The Schrödinger equation has been solved without approximations for few body systems [@kam01] and light nuclei [@pud97], up to A=12 [@pie05]. The obtained results provide good descriptions, not only of the energies of these nuclei, but also of other observables.
The difficulties in extending to medium and heavy nuclei the techniques used in few body systems and light nuclei, favored the development of models, and of effective theories. The basic idea of the effective theories is to work in a restricted space of the many-body wave functions. Usually, one works with many-body wave functions which are Slater determinants of single particle wave functions. The idea of single particle wave functions implies the hypothesis of a mean-field where each nucleon move independently from the other ones. This Independent Particle Model (IPM) is quite far from the picture outlined by the microscopic calculations quoted above, describing the nucleus as a many-body system of strongly interacting nucleons. In the Hartree-Fock theory, which provides the microscopic foundation of the IPM, the Hamiltonian is not any more that of Eq. (\[eq:hamiltonian\]), but it is an effective Hamiltonian built to take into account, obviously in an effective manner, the many-body effects that the microscopic calculations explicitly consider. The construction of effective interactions starting from the microscopic ones, covers a wide page of the nuclear physics history, starting from the Brueckner G-matrix effective interactions [@bru55; @nak86], up to the recent $V_{low k}$ interaction [@bog03; @cor06] and the interaction obtained by applying the unitary correlation operator method [@rot04; @rot06].
The application of the IPM is quite successful, but there are evidences of the intrinsic limitations in its applicability. For example, the measured spectroscopic factors are systematically smaller than one [@lap93; @kra01; @van01t], which is the value predicted by the IPM. The (e,e’p) cross sections in the quasi-elastic region need a consistent reduction of the IPM hole strength to be explained [@qui88t; @bof96]. The same holds for the electromagnetic form factors of the low-lying states, especially those having large angular momentum [@lic79; @hyd87]. The emission of two like nucleons in photon and electron scattering process cannot be described by the IPM [@ond97; @ond98]. Also the charge density distributions extracted by elastic electron scattering data are, in the nuclear interior, smaller than those predicted by the IPM [@cav82; @pap86]. These examples indicate the presence of physics effects, commonly called correlations, which are not described by the IPM.
It is common practice to distinguish between long- and short-range correlations since they have different physical sources. The long-range correlations are related to collective excitations of the system, such as the giant resonances. The short-range correlations (SRC) are instead connected to the strongly repulsive core of the microscopic nucleon-nucleon interaction. The repulsive core reduces the possibility that two nucleons can approach each other, and this modifies the IPM picture where, by definition, the motion of each nucleon does not depend on the presence of the other ones.
Even though most of the calculations of medium heavy nuclei are based on the IPM, and on the effective theories, various techniques, aiming to attack the problem by using the microscopic Hamiltonian (\[eq:hamiltonian\]), have been developed. The Brueckner-Hartree-Fock approach has been recently applied to the [$^{16}$O ]{}nucleus [@dic04]. No core-shell model calculations have been done for nuclei lighter than [$^{12}$C ]{}[@nav00; @for05]. The coupled cluster method has been used to evaluate [$^{16}$O ]{}properties [@hei99; @mih99].
About fifteen years ago [@co92], we started a project aimed to apply to the description of medium and heavy nuclei the Correlated Basis Function (CBF) theory, successfully used to describe the nuclear and neutron matter properties [@wir88; @akm98]. We solve the many-body Schrödinger equation by using the variational principle: $$\delta E[\Psi]=
\delta \frac{<\Psi|H|\Psi>}{<\Psi|\Psi>} = 0 \,\,.
\label{eq:varprin}$$ The search for the minimum of the energy functional is done within a subspace of the full Hilbert space spanned by the A-body wave functions which can be expressed as: $$\Psi(A)={\cal F}(1,...,A)\Phi(1,...,A) \,\,,
\label{eq:psi}$$ where ${\cal F}(1,...,A)$ is a many-body correlation operator and $\Phi(1,...,A)$ is a Slater determinant composed by single particle wave functions, $\phi_{\alpha}(i)$. In our calculations, we used two-body interactions of Argonne and Urbana type, and we considered all the interaction channels up to the spin-orbit ones. Together with these two-body interactions, we used the appropriated three-body forces of Urbana type. The complexity of the interaction required the use of operator dependent correlations. We consider correlations of the type: $${\cal F}={\cal S}\left( \prod_{i<j=1}^{A}F_{ij} \right) \,\,,
\label{eq:cor1}$$ where ${\cal S}$ is a symmetrizer operator and $F_{ij}$ is expressed in terms of two-body correlation functions $f_p$ as: $$F_{ij}=\sum_{p=1}^6 f_p(r_{ij})O^p_{ij} \,\,.
\label{eq:cor2}$$ In the above equation we have adopted the nomenclature commonly used in this field, by defining the operators as: $$O^{p=1,6}_{ij}=[1,{\mbox{\boldmath $\sigma$}}_i\cdot{\mbox{\boldmath $\sigma$}}_j,S_{ij}]\otimes
[1,{\mbox{\boldmath $\tau$}}_i\cdot{\mbox{\boldmath $\tau$}}_j] \,\,,$$ where ${\mbox{\boldmath $\sigma$}}_i$ and ${\mbox{\boldmath $\tau$}}_i$ indicate the usual Pauli spin and isospin operators, and $S_{ij}=(3\hat{{{\bf r}}}_{ij}\cdot{\mbox{\boldmath $\sigma$}}_i\hat{{{\bf r}}}_{ij}\cdot{\mbox{\boldmath $\sigma$}}_j
-{\mbox{\boldmath $\sigma$}}_i\cdot{\mbox{\boldmath $\sigma$}}_j)$ is the tensor operator.
We recently succeeded in formulating the Fermi Hypernetted Chain (FHNC) equations, in Single Operator Chain (SOC) approximation, for nuclei non saturated in isospin, and with single particle basis described in a $jj$ coupling scheme. We presented in Ref. [@bis06] the binding energies and the charge distributions of [$^{12}$C ]{}, [$^{16}$O ]{}, [$^{40}$Ca ]{}, [$^{48}$Ca ]{}and [$^{208}$Pb ]{}doubly closed shell nuclei obtained by using the minimization procedure (\[eq:varprin\]). These calculations have the same accuracy of the best variational calculations done in nuclear and neutron matter [@wir88; @akm98].
In the present article, we show the results of our study, done in the FHNC/SOC computational scheme, on some ground state quantities related to observables. They are momentum distributions, natural orbits and their occupation numbers, quasi-hole wave functions and spectroscopic factors. We used the many-body wave functions obtained in Ref. [@bis06] by solving Eq. (\[eq:varprin\]) with the Argonne $v'_{8}$ two-nucleon potential, together with the Urbana IX three-body force. We have calculated momentum distributions also with the wave functions produced by another interaction, the Urbana $v_{14}$ truncated up to the spin-orbit terms, implemented with the Urbana VII three-body force. The results obtained with this last interaction do not show relevant differences with those obtained with the $v'_{8}$ and UIX interaction, therefore we do not present them.
The paper is organized as follows. In Sect. \[sec:obdm\] we present the results of the One-Body Density Matrix (OBDM) and of the momentum distribution. In Sect. \[sec:no\] we discuss the natural orbits, i.e. the single particle wave functions forming the basis where the OBDM is diagonal. In Sect. \[sec:specf\] we present our results about the quasi-hole wave functions and in Sect. \[sec:summary\] we summarize our results and draw our conclusions.
ONE-BODY DENSITY MATRIX AND MOMENTUM DISTRIBUTION {#sec:obdm}
=================================================
We define the one-body density matrix, (OBDM), of a system of $A$ nucleons as: $$\begin{aligned}
\nonumber
\rho(x_1,x'_{1}) & \equiv &
\sum_{s,s',t} \rho^{s,s';t}({{\bf r}}_1,{{\bf r}}'_{1}) \,
\chi^{\dagger}_s(1) \chi^{\dagger}_t(1)
\chi_{s'}(1') \chi_t(1') \\
& \equiv & \frac{A}{<\Psi|\Psi>}
\int dx_2 \ldots dx_A
\Psi^{\dagger}(x_1,x_2,\ldots,x_A)
\Psi(x'_1,x_2,\ldots,x_A) \,\,\,.
\label{eq:obdm}\end{aligned}$$ In the above expression, the variable $x_i$ indicates the position (${{\bf r}}_i$) and the third components of the spin ($s_i$) and of the isospin ($t_i$) of the single nucleon. The $\chi(i)$ functions represent the Pauli spinors. With the integral sign we understand that also the sum on spin and isospin third components of all the particles from 2 up to A, is done. In our calculations we are interested in the quantity: $$\rho^t({{\bf r}}_1,{{\bf r}}'_{1}) =
\sum_{s=\pm 1/2} \left[
\rho^{s,s;t}({{\bf r}}_1,{{\bf r}}'_{1}) + \rho^{s,-s;t}({{\bf r}}_1,{{\bf r}}'_{1})
\right] \,\,,
\label{eq:obdm1}$$ whose diagonal part (${{\bf r}}'_1={{\bf r}}_1$) represents the one-body density of neutrons or protons.
We obtain the momentum distributions of protons ($t$=1/2) or neutrons ($t$=-1/2) as: $$n^t(k)= \frac {1}{(2 \pi)^3}
\frac {1} {{\cal N}_t} \int d{{\bf r}}_1 d{{\bf r}}'_1 \,
e^{i {\bf k} \cdot ({{\bf r}}_1-{{\bf r}}'_1)}
\rho^t({{\bf r}}_1,{{\bf r}}'_1) \,\,,
\label{eq:md}$$ where we have indicated with ${\cal N}_t$ the number of protons or neutrons. The above definitions imply the following normalization of $n(k)$: $$\int d{\bf k} \,n^t(k)= 1 \,\,.
\label{eq:nor}$$ We describe doubly closed shell nuclei, with different numbers of proton and neutrons, in a $jj$ coupling scheme. The most efficient single particle basis to be used is constructed by a set of single particle wave functions expressed as: $$\phi^t_{nljm}({{\bf r}}_i) = R^t_{nlj}(r_i) \sum_{\mu,s} <l \mu {\frac{1}{2}}s| j m>
Y_{l\mu}(\Omega_i) \chi_s(i)\chi_t(i)= R^t_{nlj}(r_i)
{\bf Y}^m_{lj} (\Omega_i)\chi_t(i) \,\,.
\label{eq:spwf}$$ In the above expression we have indicated with $Y_{l\mu}$ the spherical harmonics, with $< | >$ the Clebsch-Gordan coefficient, with $R^t_{nlj}(r_i)$ the radial part of the wave function, and with ${\bf
Y}^m_{lj}$ the spin spherical harmonics [@edm57].
The uncorrelated OBDMs, those of the IPM, are obtained by substituting in Eq. (\[eq:obdm\]) the correlated function $|\Psi>$ with the Slater determinant $|\Phi>$ formed by the single particle wave functions (\[eq:spwf\]). We obtain the following expressions: $$\nonumber
\rho_o^t({{\bf r}}_1,{{\bf r}}'_1) = \sum_{s} \left[ \rho_o^{s,s;t}({{\bf r}}_1,{{\bf r}}'_1)
+\rho_{o}^{s,-s;t}({{\bf r}}_1,{{\bf r}}'_1) \right] \,\,,
\label{eq:obdm0}$$ $$\begin{aligned}
\rho_o^{s,s;t}({{\bf r}}_1,{{\bf r}}'_1)
&=&
\frac{1}{8\pi}\sum_{nlj}(2j+1)R^t_{nlj}(r_1)
R^t_{nlj}(r'_1)P_{l}(\cos \theta_{11'}) \,\,,
\label{eq:obdmp}
\\
\rho_{o}^{s,-s;t}({{\bf r}}_1,{{\bf r}}'_1)
&=&\frac{1}{4\pi}
\sum_{nlj}(-1)^{j-l-1/2}R^t_{nlj}(r_1)R^t_{nlj}(r'_1)
\sin\theta_{11'}P'_{l}(\cos \theta_{11'}) \,\,.
\label{eq:obdma}\end{aligned}$$ In the above equations $\theta_{11'}$ indicates the angle between ${{\bf r}}_1$ and ${{\bf r}}'_1$, and $P_l$ and $P'_l$ the Legendre polynomials and their first derivative respectively. The presence of the second term of Eq. (\[eq:obdm0\]), the antiparallel spin terms given in Eq. (\[eq:obdma\]), is due the $jj$ coupling scheme required to describe heavy nuclei.
The correlated OBDM is obtained by using the ansatz (\[eq:psi\]) in Eq. (\[eq:obdm\]). This calculation is done by using the cluster expansion techniques as indicated in [@co94] and [@ari96], where only scalar correlations have been used, and in [@fab01], where the state dependent correlations have been used, but in a $ls$ coupling scheme. We followed the lines of Ref. [@fab01] and, in addition, we consider the presence of the antiparallel spin terms and we distinguish proton and neutron contributions. The explicit expression of the OBDM, in terms of FHNC/SOC quantities, such as two-body density distributions, vertex corrections, nodal diagrams etc., is given in Appendix \[app:A\]. The diagonal part of the OBDM is the one-body density, normalized to the number of nucleons. Because of this, the momentum distribution satisfies the following sum rule: $$S^t_2 = \frac {\hbar^2} {2 m}
\int d {\bf k} \, k^2 \, n^t(k) / T_{FHNC}^t = 1 \,\,,
\label{eq:sr2}$$ where we have indicated with $T_{FHNC}^t$ the kinetic energy of the protons or of the neutrons. We have verified the accuracy of our calculations by testing the normalization (\[eq:nor\]) and the exhaustion of the above sum rule for every $n^t(k)$ calculated. We found that these quantities are always satisfied at the level of few parts on a thousand in a full FHNC/SOC calculation, and even better when only scalar correlations are used.
![The difference $\rho_{o}(r_1,r'_1)-\rho(r_1,r'_1)$, between the proton IPM one-body density matrix of the [$^{208}$Pb ]{}nucleus, and that obtained with our FHNC/SOC calculations. The two density matrices, have been calculated for $\theta_{11'}$=0. Note that the scale here is ten times larger than that of Fig. \[fig:obdm\]. []{data-label="fig:obdmdiff"}](rho11.ps)
![The difference $\rho_{o}(r_1,r'_1)-\rho(r_1,r'_1)$, between the proton IPM one-body density matrix of the [$^{208}$Pb ]{}nucleus, and that obtained with our FHNC/SOC calculations. The two density matrices, have been calculated for $\theta_{11'}$=0. Note that the scale here is ten times larger than that of Fig. \[fig:obdm\]. []{data-label="fig:obdmdiff"}](rho11diff.ps)
The surface shown in Fig.\[fig:obdm\] represents the proton OBDM of the [$^{208}$Pb ]{}nucleus, for $\theta_{11'}$=0. We have shown in [@bis06] that the SRC lower the one-body proton distribution, and also that of the neutrons, in the nuclear center. In order to highlight the effects of the correlations on the density matrix, we show in Fig. \[fig:obdmdiff\] the quantity $\rho_{o}(r_1,r'_1)-\rho(r_1,r'_1)$. Note that the $z$-axis scale of Fig. \[fig:obdmdiff\] is ten times larger than that of Fig.\[fig:obdm\]. It is interesting to notice that the major differences between the OBDMs are not in the diagonal part, but just beside it. The consequences of these, small, differences between the OBDMs on the momentum distributions, are shown in Fig. \[fig:md\]. In this figure, we compare the [$^{12}$C ]{}, [$^{16}$O ]{}, [$^{40}$Ca ]{}, [$^{48}$Ca ]{}and [$^{208}$Pb ]{}proton momentum distributions calculated in the IPM model, with those obtained by using scalar and operator dependent correlations.
![The proton momentum distributions of the [$^{12}$C ]{}, [$^{16}$O ]{}, [$^{40}$Ca ]{}, [$^{48}$Ca ]{}and [$^{208}$Pb ]{}nuclei calculated in the IPM model, by using the scalar correlations only ($f_1$) and the full operator dependent correlations ($f_6$). []{data-label="fig:md"}](nk.ps)
![The proton momentum distributions of the [$^{16}$O ]{}and [$^{208}$Pb ]{}nuclei multiplied by $k^2$. The full lines show the IPM results, the dotted lines have been obtained by using scalar correlations only, and the dashed lines with the complete correlation. []{data-label="fig:mdlin"}](nk_lin.ps)
![In the panels (a) and (b) we show the protons (full lines) and neutrons (dashed lines) momentum distributions of the [$^{48}$Ca ]{}and [$^{208}$Pb ]{}. The thick lines show the results of our calculations, the thin lines the IPM results. In the panels (c) and (d), we show the weighted difference (\[eq:delta\]) between uncorrelated and correlated momentum distributions. As in the upper part of the figure, the full lines show the protons results and the dashed lines those of the neutrons. []{data-label="fig:mdpn"}](nkpn.ps)
![ Proton momentum distribution of [$^{16}$O ]{}in various approximations. The thick lines are those of the analogous panel of Fig. \[fig:md\]. The thin lines have been obtained by using the first-order expansion method of Ref. [@ari97]. []{data-label="fig:mdcomp"}](nk_comp.ps)
The general behavior of the momentum distributions, is very similar for all the nuclei we have considered. Correlated and IPM distributions almost coincide in the low momentum region up to a precise value, when they start to deviate. The correlated distributions show high momentum tails, which are orders of magnitude larger than the IPM results. The value of $k$ where uncorrelated and correlated momentum distributions start to deviate, is smaller the heavier is the nucleus. It is about 1.9 fm$^{-1}$ for [$^{12}$C ]{}, and 1.5 fm$^{-1}$ for [$^{208}$Pb ]{}. We notice that the value of the Fermi momentum of symmetric nuclear matter at the saturation density is 1.36 fm$^{-1}$.
The results presented in Fig. \[fig:md\] clearly show that the effects of the scalar correlations are smaller than those obtained by including the operator dependent terms. We shall see in the following that this is a common feature of our results.
Since relatively small differences are compressed in logarithmic scale, we use the linear scale in Fig. \[fig:mdlin\] to show, as examples, the proton momentum distributions for [$^{16}$O ]{}and [$^{208}$Pb ]{}nuclei, multiplied by $k^2$. This quantity, multiplied by a factor $4 \pi$, is the probability of finding a proton with momentum $k$. We observe that the effects of the SRC on the quantity shown in Fig. \[fig:mdlin\] are basically two. The first one is the already mentioned enhancement at large values of $k$. This effect is less evident here than in Fig. \[fig:md\]. The second effect, hardly visible in Fig. \[fig:md\], is a reduction of the maxima which appear approximately at $k$=1 fm$^{-1}$ in both nuclei. These two effects are obviously related, since all the momentum distributions are normalized as indicated by Eq. (\[eq:nor\]), therefore reductions and increases must compensate.
We found that the proton and neutron momentum distributions for nuclei with $N=Z$ are very similar. For this reason we do not show the neutron momentum distributions of the [$^{12}$C ]{}, [$^{16}$O ]{}and [$^{40}$Ca ]{}nuclei. We compare in the panels (a) and (b) of Fig. \[fig:mdpn\] the proton and neutron momentum distributions of [$^{48}$Ca ]{}and [$^{208}$Pb ]{}. The thicker lines show the results of our FHNC/SOC calculation, the thinner lines the IPM momentum distributions.
The figure shows that, in our calculations, the differences between protons and neutrons momentum distributions are more related to the different single particle structure than to the correlation effects. The main differences in the two distributions is in the zone where the $n(k)$ values drops of orders of magnitudes. This corresponds to the discontinuity region of the momentum distribution in the infinite systems, which is related to the Fermi momentum. In a finite system, the larger number of neutrons implies that the neutron Fermi energy is larger than that of the protons, and, consequently, the effective Fermi momentum. For this reason, the neutrons momentum distributions drops at larger values of $k$ than the proton distributions. In the panels (c) and (d) of Fig. \[fig:mdpn\], we show the quantity $$\Delta = \frac{n^t_0(k)-n^t(k)}{n^t_0(k)+n^t(k)} \,\,,
\label{eq:delta}$$ where $n^t_0(k)$ indicates the uncorrelated momentum distribution. This quantity is useful to point out the effects of correlations. We see that in the low $k$ region $\Delta$ is almost zero. After the discontinuity region $\Delta$ reaches an almost constant value around minus one. This behavior indicates that in the low $k$ region the momentum distribution is dominated by single particle dynamics. The differences between protons and neutrons at low $k$ are due to different single particle wave functions. In the higher $k$ region the correlation plays an important role. We observe that protons and neutrons $\Delta$ are very similar, and this indicate that the effect of the SRC is essentially the same for both kinds of nucleons. Our results are in agreement with the findings of Ref. [@boz04], where $n(k)$ of asymmetric nuclear matter is presented. There is however a disagreement with the results of Ref. [@fri05], where, always in asymmetric nuclear matter, correlations effects between protons were found to be stronger than those between neutrons.
The increase of the momentum distribution at large $k$ values, induced by the SRC is a well known result in the literature, see for example the review of Ref. [@ant88]. The momentum distributions of medium-heavy nuclei, have been usually obtained by using approximated descriptions of the cluster expansion, which is instead considered at all orders in our treatment. The importance of a complete description of the cluster expansion is exemplified in Fig. \[fig:mdcomp\], where, together with our results, we also show the results of Ref. [@ari97], obtained by truncating the cluster expansion to the first order in the correlation lines. In both calculations the same correlation functions and single particle basis, those of Ref. [@bis06], have been used. The results obtained with the first order approximation, provide only a qualitative description of the correlation effects. They show high-momentum enhancements which, however, underestimate the correct results by orders of magnitude.
NATURAL ORBITS {#sec:no}
==============
The natural orbits are defined as those single particle wave functions forming the basis where the OBDM is diagonal: $$\rho^t({{\bf r}}_1,{{\bf r}}'_{1})=\sum_{nlj}c_{nlj}^t\phi_{nlj}^{*\,t,NO}({{\bf r}}_1)
\phi_{nlj}^{t,NO}({{\bf r}}'_{1}) \,\,.
\label{eq:no}$$ In the above equation the $c^t_{nlj}$ coefficients, called occupation numbers, are real numbers. In the IPM, the natural orbits correspond to the mean-field wave functions of Eq. (\[eq:spwf\]), and the $c^t_{nlj}$ numbers are 1, for the states below the Fermi surface, and 0 for those above it.
In order to obtain the natural orbits, we found convenient to express the OBDM of Eq. (\[eq:no\]) as: $$\rho^t({{\bf r}}_1,{{\bf r}}_{1'})=A^t({{\bf r}}_1,{{\bf r}}_{1'})
\rho_o^t({{\bf r}}_1,{{\bf r}}_{1'})+B^t({{\bf r}}_1,{{\bf r}}_{1'}) \,\,,
\label{eq:rhosplit}$$ where $\rho^t_{o}({{\bf r}}_1,{{\bf r}}_{1'})$ is the uncorrelated OBDM of Eq. (\[eq:obdm0\]), and the other two quantities are defined as: $$\begin{aligned}
\nonumber
A^{t}({{\bf r}}_1,{{\bf r}}'_1)&=&2C_{\omega,SOC}^{t}({{\bf r}}_1)
C_{\omega,SOC}^{t}({{\bf r}}'_1)
\exp[{N_{\omega\omega}^{t}({{\bf r}}_1,{{\bf r}}'_1)}]
+\\ &&
2C_{\omega}^{t}({{\bf r}}_1)C_{\omega}^{t}({{\bf r}}'_1)
\exp[{N_{\omega\omega}^{t}({{\bf r}}_1,{{\bf r}}'_1)}]\sum_{p>1}
A^k\Delta^{k}N_{\omega\omega,p}^{t}({{\bf r}}_1,{{\bf r}}'_1) \,\,,
\label{eq:arho}\\
\nonumber
B^{t}({{\bf r}}_1,{{\bf r}}'_1)&=&-2C_{\omega,SOC}^{t}({{\bf r}}_1)
C_{\omega,SOC}^{t}({{\bf r}}'_1)
\exp[{N_{\omega\omega}^{t}({{\bf r}}_1,{{\bf r}}'_1)}]
N_{\omega_c\omega_c}^{t}({{\bf r}}_1,{{\bf r}}'_1)
- \\
\nonumber &&
2C_{\omega}^{t}({{\bf r}}_1)C_{\omega}^{t}({{\bf r}}'_1)
\exp{[N_{\omega\omega}^{t}({{\bf r}}_1,{{\bf r}}'_1)]}\\
&&\times\sum_{p>1}
A^k\Delta^{k} \Big\{ N_{\omega\omega,p}^{t}({{\bf r}}_1,{{\bf r}}'_1)
N_{\omega_c\omega_c}^{t}({{\bf r}}_1,{{\bf r}}'_1)+
N_{\omega_c\omega_c,p}^{t}({{\bf r}}_1,{{\bf r}}'_1)\Big\} \,\,.
\label{eq:brho}\end{aligned}$$ The meaning of the $\omega\omega$, $\omega_c\omega_c$ labels used in the above equations have been defined in [@fab01] where the index $k$ has been defined as $p=2k+l-1$ with $l=0,1$ and $p=1,\ldots,6$. The detailed expressions of the vertex corrections $C$ and of the nodal functions $N$ are given in Appendix \[app:A\].
We expand the OBDM on a basis of spin spherical harmonics ${\bf Y}^m_{lj}$ , Eq. (\[eq:spwf\]), $$\rho^t({{\bf r}}_1,{{\bf r}}'_{1}) = \sum_{ljm} \frac {1}{2j+1}
\left[{\cal A}_{lj}^t(r_1,r'_1) + {\cal B}_{lj}^t(r_1,r'_1)
\right]
{\bf Y}^{* m}_{lj}(\Omega_1) {\bf Y}^m_{lj}(\Omega'_1)
\label{eq:rhoexp}$$ where $\Omega_1$ and $\Omega'_1$ indicate the polar angles identifying ${{\bf r}}_1$ and ${{\bf r}}'_1$. The explicit expressions of the ${\cal A}$ and ${\cal B}$ coefficients are: $$\begin{aligned}
\nonumber
{\cal A}_{lj}^t(r_1,r'_1) &=&
(2l+1)\sum_{n_2l_2j_2l}(2l_2+1)(2j_2+1)
\left( \begin{array}{ccc}
l & l_1 & l_2\\
0 & 0 & 0
\end{array}\right)^2
\left\{ \begin{array}{ccc}
j_2 & l_1 & j\\
l & 1/2 & l_2
\end{array}\right\}^2
\nonumber
\\
&~&R^t_{nl_2j_2}(r_1)R^t_{nl_2j_2}(r_2) A^t_{l_1}(r_1,r'_1)
\label{eq:cala}\end{aligned}$$ with $$A_l^t(r_1,r'_1)=\frac 2 {2l+1}
\int d\Omega A^t({{\bf r}}_1,{{\bf r}}'_1)P_{l}(\cos\theta_{11'})$$ and $${\cal B}_{lj}^t(r_1,r'_1)
=\frac{4\pi}{2l+1} \int
d (\cos\theta_{11'}) B^t({{\bf r}}_1,{{\bf r}}'_1)P_{l}(\cos\theta_{11'})
\label{eq:calb}$$ In the above equations we have used the 3j and 6j Wigner symbols [@edm57]. The term ${\cal A}$ depends on both orbital and total angular momenta of the single particle, $l$ and $j$ respectively, and the term ${\cal B}$ depends only on the orbital angular momentum $l$.
As it has been done in Refs. [@lew88] and [@pol95] we identify the various natural orbit with a number, $\alpha$, ordering them with respect to the decreasing value of the occupation probability. The general behavior of our results is analogous to that described in Ref. [@lew88] where a system of $^3$He drops composed by 70 atoms have been studied. The orbits corresponding to states below the Fermi level in the IPM picture, have occupation numbers very close to unity for $\alpha=1$, and very small in all the other cases.
![Occupation numbers of the proton natural orbits of the [$^{48}$Ca ]{}nucleus, having $\alpha=1$. The dashed line indicates the IPM values. The black bars show the values obtained with the scalar correlation and the gray bars those values obtained with the full correlation. []{data-label="fig:natorb1"}](istca48p.ps)
![The same as Fig. \[fig:natorb1\] for the occupation numbers of the neutron natural orbits of the [$^{48}$Ca ]{}nucleus. []{data-label="fig:natorb2"}](istca48n.ps)
As example of our results, we show in Figs. \[fig:natorb1\] and \[fig:natorb2\] the proton and neutron occupation numbers for the natural orbits with$\alpha=1$ of the [$^{48}$Ca ]{}nucleus. In the figures, the IPM results are indicated by the dashed lines. The black bars show the values obtained by using scalar correlations only, the gray bars those obtained with the complete operator dependent correlations.
The correlated occupation numbers are smaller than one for orbits below the Fermi surface, and larger than zero for those orbits above the Fermi surface. This effect is enhanced by the operator dependent correlations. We observe that, for the states above the Fermi surface, the gray bars are larger than the black ones, indicating that also for these states the operator dependent correlations, produce larger effects than the scalar ones.
We show in Fig. \[fig:natorb3\] some $\alpha=1$ natural orbits for three neutron states in [$^{48}$Ca ]{}. In this figure, we compare the IPM results (full lines) with those obtained with scalar correlation only (dotted lines), and with the full operator dependent correlation (dashed lines). The effect of the correlations is a lowering of the peak and a small widening of the function. Despite the small effect, it is interesting to point out that this is the only case where we found that the inclusion of operator dependent terms diminishes the effect of the scalar correlation. This fact is coherent with the results on the density distributions shown in Ref. [@bis06].
State $\alpha=1$ $\alpha=2$ $\alpha=3$
---------------- ------- ------------ ------- ------- ------------ ------- ------- ------------ -------
$f_1$ $f_6$ PMD $f_1$ $f_6$ PMD $f_1$ $f_6$ PMD
$1s_{1/2}$ (p) 0.956 0.873 0.921 0.011 0.038 0.013 0.002 0.007 0.002
(n) 0.957 0.873 0.012 0.039 0.003 0.008
$1p_{3/2}$ (p) 0.973 0.921 0.947 0.004 0.013 0.007 0.001 0.003 0.001
(n) 0.973 0.924 0.004 0.014 0.002 0.004
$1p_{1/2}$ (p) 0.970 0.923 0.930 0.003 0.012 0.008 0.001 0.003 0.002
(n) 0.970 0.922 0.004 0.013 0.002 0.003
$1d_{5/2}$ (p) 0.001 0.005 0.016 0.013 0.003 0.003 0.000 0.000 0.000
(n) 0.001 0.005 0.001 0.003 0.000 0.000
$1d_{3/2}$ (p) 0.002 0.005 0.019 0.001 0.003 0.005 0.000 0.000 0.001
(n) 0.001 0.005 0.001 0.003 0.000 0.000
: Protons (p) and neutrons (n) natural orbits occupation numbers for [$^{16}$O ]{}. The PMD values are those of Ref. [@pol95]. []{data-label="tab:ocn"}
In Tab. \[tab:ocn\] we show the occupation numbers of the [$^{16}$O ]{}protons and neutrons natural orbits also for $\alpha > 1$, and we make a direct comparison with the results of Ref. [@pol95]. As already said in the discussion of the [$^{48}$Ca ]{}results, the inclusion of the state dependent correlations increases the differences with respect to the IPM. The occupation numbers of the orbits below the Fermi surface are smaller than those obtained with scalar correlations only. The situation is reversed for the orbits with $\alpha > 1$ or above the Fermi level. For the states below the Fermi surface, our full calculations produce correlation effects slightly larger than those found in [@pol95], whose results are closer to those we obtain with scalar correlations only. For orbits above the IPM Fermi surface, our occupation numbers are always smaller than those of Ref. [@pol95].
QUASI-HOLE WAVE FUNCTIONS AND THE SPECTROSCOPIC FACTORS {#sec:specf}
=======================================================
The quasi-hole wave function is defined as: $$\psi_{nljm}^{t}(x)=
\sqrt{A}\frac{<\Psi_{nljm}(A-1)|\delta(x-x_A)P^{t}_{A}|\Psi(A)>}
{\left[<\Psi_{nljm}(A-1)|\Psi_{nljm}(A-1)>
<\Psi(A)|\Psi(A)>\right]^{1/2}} \,\,,
\label{eq:qhfun}$$ where $|\Psi_{nljm}(1,...,A-1)>$ and $|\Psi(1,...,A)>$ are the states of the nuclei formed by $A-1$ and $A$ nucleons respectively, and $P^t_A$ is the isospin projector. In analogy to the ansatz (\[eq:psi\]), we assume that the state of the nucleus with $A-1$ nucleons can be described as: $$\Psi_{nljm}(A-1)=
{\cal F}(1,...,A-1)\Phi_{nljm}(1,...,A-1) \,\,,
\label{eq:psiam1}$$ where $\Phi_{nljm}(1,...,A-1)$ is the Slater determinant obtained by removing from $\Phi(1,...,A)$ a particle characterized by the quantum numbers $nljm$, and the correlation function ${\cal F}$ is formed, as indicated in Eq. (\[eq:cor2\]), by the two-body correlation functions $f_p$ obtained by minimizing the $A$ nucleon system. In an uncorrelated system the quasi-hole wave functions coincide with the hole mean-field wave functions (\[eq:spwf\]).
We are interested in the radial part of the quasi-hole wave function. We obtain this quantity first by multiplying equation (\[eq:qhfun\]) by the vector spherical harmonics ${\bf Y}^{*m}_{lj}(\Omega)$, then, by integrating over the angular coordinate $\Omega$, and, finally, by summing over $m$. It is useful to rewrite the radial part of the quasi-hole wave function as [@fab01]: $$\psi_{nlj}^{t}(r)= \frac 1 {2j+1}
\sum_m \int d\Omega \,
{\bf Y}^m_{lj}(\Omega)
\, \psi_{nljm}^{t}(x)
= \frac 1 {2j+1} \sum_m {\cal X}_{nljm}^t(r)[{\cal N}_{nlj}^t]^{1/2} \,\,,
\label{eq:qhprod}$$ where we have defined: $${\cal{X}}_{nljm}^t(r)=
\sqrt{A}\frac{<\Psi_{nljm}^t(A-1)|{\bf Y}^{*m}_{lj}(\Omega) \,
\delta({{\bf r}}-{{\bf r}}_{A})P^{t}_{A}|\Psi^t(A)>}{<\Psi_{nljm}^t(A-1)|
\Psi_{nljm}^t(A-1)>} \,\, ,$$ and $${\cal N}_{nljm}^t=<\frac{\Psi_{nljm}^t(A-1)|\Psi_{nljm}^t(A-1)>}
{<\Psi^t(A)|\Psi^t(A)>} \,\, .$$
------------- ------- ------------------- ------- ------- ------------------- ------- ------- -------------------- ------- ------- -------------------- ------- ------- --------------------- -------
$nlj$ [[$^{12}$C ]{}]{} [[$^{16}$O ]{}]{} [[$^{40}$Ca ]{}]{} [[$^{48}$Ca ]{}]{} [[$^{208}$Pb ]{}]{}
$f_1$ $f_4$ $f_6$ $f_1$ $f_4$ $f_6$ $f_1$ $f_4$ $f_6$ $f_1$ $f_4$ $f_6$ $f_1$ $f_4$ $f_6$
$1s_{1/2}$ 0.96 0.95 0.91 0.95 0.90 0.85 0.93 0.84 0.78 0.94 0.85 0.78 0.93 0.83 0.77
$1p_{3/2}$ 0.96 0.96 0.94 0.96 0.93 0.89 0.95 0.87 0.82 0.95 0.87 0.81 0.94 0.83 0.77
$1p_{1/2}$ 0.96 0.93 0.89 0.95 0.87 0.81 0.95 0.83 0.80 0.94 0.83 0.77
$1d_{5/2}$ 0.96 0.90 0.86 0.96 0.90 0.85 0.94 0.84 0.79
$2s_{1/2}$ 0.96 0.92 0.87 0.94 0.92 0.86 0.94 0.86 0.80
$1d_{3/2}$ 0.95 0.90 0.85 0.96 0.90 0.84 0.94 0.84 0.79
$1f_{7/2}$ 0.94 0.86 0.81
$2p_{3/2}$ 0.95 0.87 0.82
$1f_{5/2}$ 0.95 0.86 0.80
$2p_{1/2}$ 0.95 0.87 0.82
$1g_{9/2}$ 0.95 0.88 0.83
$1g_{7/2}$ 0.94 0.88 0.82
$2d_{5/2}$ 0.95 0.89 0.83
$1h_{11/2}$ 0.94 0.90 0.86
$2d_{3/2}$ 0.95 0.89 0.83
$3s_{1/2}$ 0.95 0.90 0.85
------------- ------- ------------------- ------- ------- ------------------- ------- ------- -------------------- ------- ------- -------------------- ------- ------- --------------------- -------
: Proton spectroscopic factors of the [$^{12}$C ]{}, [$^{16}$O ]{}, [$^{40}$Ca ]{}, [$^{48}$Ca ]{}and [$^{208}$Pb ]{}nuclei. We present the results obtained by using the scalar correlation only ($f_1$), the first four operator channels of the correlation ($f_4$) and the full correlation operator ($f_6$). []{data-label="tab:sfp"}
------------- ------- ------------------- ------- ------- ------------------- ------- ------- -------------------- ------- ------- -------------------- ------- ------- --------------------- -------
$nlj$ [[$^{12}$C ]{}]{} [[$^{16}$O ]{}]{} [[$^{40}$Ca ]{}]{} [[$^{48}$Ca ]{}]{} [[$^{208}$Pb ]{}]{}
$f_1$ $f_4$ $f_6$ $f_1$ $f_4$ $f_6$ $f_1$ $f_4$ $f_6$ $f_1$ $f_4$ $f_6$ $f_1$ $f_4$ $f_6$
$1s_{12}$ 0.96 0.95 0.91 0.95 0.90 0.85 0.93 0.84 0.78 0.93 0.86 0.80 0.92 0.85 0.80
$1p_{3/2}$ 0.96 0.96 0.94 0.96 0.93 0.89 0.95 0.87 0.82 0.94 0.88 0.83 0.93 0.85 0.80
$1p_{1/2}$ 0.96 0.93 0.89 0.95 0.87 0.81 0.94 0.88 0.82 0.93 0.85 0.80
$1d_{5/2}$ 0.96 0.90 0.86 0.95 0.90 0.86 0.93 0.86 0.82
$2s_{1/2}$ 0.96 0.92 0.87 0.95 0.92 0.87 0.93 0.88 0.84
$1d_{3/2}$ 0.95 0.90 0.85 0.95 0.90 0.86 0.93 0.86 0.82
$1f_{7/2}$ 0.95 0.94 0.91 0.94 0.88 0.84
$2p_{3/2}$ 0.94 0.89 0.85
$1f_{5/2}$ 0.93 0.88 0.84
$2p_{1/2}$ 0.94 0.89 0.85
$1g_{9/2}$ 0.94 0.90 0.86
$1g_{7/2}$ 0.94 0.90 0.86
$2d_{5/2}$ 0.94 0.91 0.87
$1h_{11/2}$ 0.94 0.93 0.89
$2d_{3/2}$ 0.94 0.90 0.87
$3s_{1/2}$ 0.94 0.92 0.88
$2f_{7/2}$ 0.95 0.93 0.90
$1h_{9/2}$ 0.94 0.92 0.88
$2f_{5/2}$ 0.95 0.93 0.90
$3p_{3/2}$ 0.95 0.94 0.90
$1i_{13/2}$ 0.94 0.93 0.90
$3p_{1/2}$ 0.95 0.94 0.90
------------- ------- ------------------- ------- ------- ------------------- ------- ------- -------------------- ------- ------- -------------------- ------- ------- --------------------- -------
: The same as Tab. \[tab:sfp\] for neutron states. []{data-label="tab:sfn"}
Following the procedure outlined in Ref. [@fab01], we consider separately the cluster expansions of the two terms ${\cal N}_\alpha^t$ and ${\cal X}_\alpha^t$, where we have indicated with $\alpha$ the set of the $nljm$ quantum numbers. We obtain for ${\cal X}_\alpha^t$ the expression: $$\begin{aligned}
\nonumber
{\cal X}_{\alpha}^{t}(r)&=&
C^{t,\alpha}_{\omega,SOC}({{\bf r}})
\Bigg(
R^{t}_{nlj}(r)+
\int d^{3}r_1R^{t}_{nlj}(r_1)P_{l}(\cos{\theta}) \\
\nonumber &~& \times
\Bigg\{
g_{\omega d}^{tt,\alpha}({{\bf r}},{{\bf r}}_1)C_{d,pq}^{t,\alpha}({{\bf r}}_1)
\Big[
-\rho_{o}^{t,\alpha}({{\bf r}},{{\bf r}}_1)+N_{\omega_cc}^{t,\alpha}({{\bf r}},{{\bf r}}_1)
\Big]
\\
&~&
+\rho_{o}^{t,\alpha}({{\bf r}},{{\bf r}}_1)
-N^{t,\alpha}_{\omega_c\rho}({{\bf r}},{{\bf r}}_1)-N^{t,\alpha}_{\rho\rho}({{\bf r}},{{\bf r}}_1)
+{\cal X}_{SOC}^{t}({{\bf r}},{{\bf r}}_1)
\Bigg\}
\Bigg) \,\,,\end{aligned}$$ and for ${\cal N}_\alpha^t$ the expression: $$\begin{aligned}
\nonumber
\Big[{\cal N}_{\alpha}^{t}\Big]^{-1}&=&\int
d^3rC_{d,pq}^{t,\alpha}({{\bf r}})
\Bigg(
|\phi^{t}_{\alpha}({{\bf r}})|^2+
\int d^3r_1\phi^{t *}_{\alpha}({{\bf r}})\phi^{t}_{\alpha}({{\bf r}}_1)
\\ \nonumber &~& \times
2
\Big\{
g_{dd}^{tt,\alpha}({{\bf r}},{{\bf r}}_1)C_{d,pq}^{t,\alpha}({{\bf r}}_1)
\Big[-\rho_{o}^{t,\alpha}({{\bf r}},{{\bf r}}_1)+N^{t,\alpha}_{cc}
({{\bf r}},{{\bf r}}_1)\Big]\\
&& +\rho_{o}^{t,\alpha}({{\bf r}},{{\bf r}}_1)-N_{x\rho}^{t,\alpha}({{\bf r}},{{\bf r}}_1)
-N_{\rho\rho}^{t,\alpha}({{\bf r}},{{\bf r}}_1)
+{\cal N}_{SOC}^{t}({{\bf r}},{{\bf r}}_1)\Big\}
\Bigg) \,\,,\end{aligned}$$ The expressions of the functions ${\cal N}_{SOC}^{t}({{\bf r}},{{\bf r}}_1)$, ${\cal X}_{SOC}^{t}({{\bf r}},{{\bf r}}_1)$, are: $$\begin{aligned}
\nonumber
{\cal X}_{SOC}^{t_{1}}({{\bf r}},{{\bf r}}_1)&=&
\sum_{k=1}^{3}A^k\sum_{t_{2}=p,n}
\Big[(1-\delta_{k,1}){\cal X}_{2k-1,2k-1}^{t_{1}t_{2}}({{\bf r}},{{\bf r}}_1)\\
&&+\chi_{1}^{t_{1}t_{2}}\Big({\cal X}_{2k-1,2k}^{t_{1}t_{2}}({{\bf r}},{{\bf r}}_1)
+{\cal X}_{2k,2k-1}^{t_{1}t_{2}}({{\bf r}},{{\bf r}}_1)\Big)+
\chi_{2}^{t_{1}t_{2}}{\cal X}_{2k,2k}^{t_{1}t_{2}}({{\bf r}},{{\bf r}}_1)\Big]
\,\,,\\
\nonumber
{\cal N}_{SOC}^{t_{1}}({{\bf r}},{{\bf r}}_1)&=&
\sum_{k=1}^{3}A^k\sum_{t_{2}=p,n}
\Big[(1-\delta_{k,1}){\cal N}_{2k-1,2k-1}^{t_{1}t_{2}}({{\bf r}},{{\bf r}}_1)\\
&&+\chi_{1}^{t_{1}t_{2}}\Big({\cal N}_{2k-1,2k}^{t_{1}t_{2}}({{\bf r}},{{\bf r}}_1)
+{\cal X}_{2k,2k-1}^{t_{1}t_{2}}({{\bf r}},{{\bf r}}_1)\Big)+
\chi_{2}^{t_{1}t_{2}}{\cal
N}_{2k,2k}^{t_{1}t_{2}}({{\bf r}},{{\bf r}}_1)\Big]
\,\,.\end{aligned}$$ where the indexes $t$ refer to the isospin, and we have defined: $$\begin{aligned}
\nonumber
{\cal X}_{pq}^{t_{1}t_{2}}({{\bf r}},{{\bf r}}_1)&=&\frac{1}{2}
\Big[h_{\omega,p}^{t_{1}t_{2} ,\alpha}({{\bf r}},{{\bf r}}_1)
g_{\omega d}^{t_{1}t_{2} ,\alpha}({{\bf r}},{{\bf r}}_1)
C_{d}^{t_{2},\alpha}({{\bf r}}_1)\Big(-\rho_{o}^{t_{2},\alpha}({{\bf r}},{{\bf r}}_1)+
N_{\omega_cc}^{t_{2},\alpha}({{\bf r}},{{\bf r}}_1)\Big)\\
&&+g_{\omega d}^{t_{1}t_{2},\alpha}({{\bf r}},{{\bf r}}_1)C^{t_{2},\alpha}_{d}({{\bf r}}_1)
N_{\omega_cc,p}^{t_{2},\alpha}({{\bf r}},{{\bf r}}_1)-
N_{\omega_c\rho,p}^{t_{2},\alpha}({{\bf r}},{{\bf r}}_1)-
N_{\rho\rho,p}^{t_{2},\alpha}({{\bf r}},{{\bf r}}_1)\Big]\Delta^{k_2}
\,\,, \\
\nonumber
{\cal N}_{pq}^{t_{1}t_{2}}({{\bf r}},{{\bf r}}_1)&=&
\Big[h_{d,p}^{t_{1}t_{2} ,\alpha}({{\bf r}},{{\bf r}}_1)g_{dd}^{t_{1}t_{2} ,\alpha}
({{\bf r}},{{\bf r}}_1)
C_{d}^{t_{2},\alpha}({{\bf r}}_1)\Big(-\rho_{o}^{t_{2},\alpha}({{\bf r}},{{\bf r}}_1)+
N_{cc}^{t_{2},\alpha}({{\bf r}},{{\bf r}}_1)\Big)\\
&&+g_{dd}^{t_{1}t_{2}, \alpha}({{\bf r}},{{\bf r}}_1)C_{d}^{t_{2},\alpha}({{\bf r}}_1)
N^{t_{2},\alpha}_{cc,p}({{\bf r}},{{\bf r}}_1)-
N_{x\rho,p}^{t_{2},\alpha}({{\bf r}},{{\bf r}}_1)-
N_{\rho\rho,p}^{t_{2},\alpha}({{\bf r}},{{\bf r}}_1)\Big]\Delta^{k_2}
\,\, ,\end{aligned}$$ where $k_2=1,2,3$ for $q=1,3,5$. The expressions of the other terms are given in Appendix \[app:A\]. All the quantities used in the above expressions depend on the set of quantum numbers $\alpha$ characterizing the quasi-hole state, since we have written the various equations by using [@ari01]: $$\rho_{o}^{t,\alpha}({{\bf r}},{{\bf r}}_1)=\rho_{o}^{t}({{\bf r}},{{\bf r}}_1)-
\phi^{t *}_{\alpha}({{\bf r}})\phi^{t}_{\alpha}({{\bf r}}_1) \, \,.$$ The knowledge of the quasi-hole functions allows us to calculate the spectroscopic factors: $$S_{nlj}^t=\int dr\,r_1^2\,|\psi_{nlj}^t(r)|^2
\,\,.
\label{eq:sf}$$
![Natural orbits for some neutron states in [$^{48}$Ca ]{}. The full lines show the IPM orbits, the dotted lines those obtained with scalar correlations only and the dashed lines those obtained with the complete operator dependent correlation. []{data-label="fig:natorb3"}](natorb.ps)
![ Proton $3s_{1/2}$ and neutron $3p_{1/2}$ quasi-hole functions, squared, of the [$^{208}$Pb ]{}nucleus. The various lines show the results obtained by using different type of correlations. []{data-label="fig:qh_fun"}](psih.ps)
![Differences between charge density distributions of $^{206}$Pb and $^{205}$Tl. See the text for the explanation of the various lines. []{data-label="fig:205tl"}](205tl_exp.ps)
The proton and neutron spectroscopic factors for the [$^{12}$C ]{}, [$^{16}$O ]{}, [$^{40}$Ca ]{}, [$^{48}$Ca ]{}and [$^{208}$Pb ]{}nuclei are given in Tabs. \[tab:sfp\] and \[tab:sfn\] for each single hole state. In these tables we compare the results obtained by using scalar correlations ($f_1$), with those obtained with the four central channels ($f_4$) and with the full correlation ($f_6$). The inclusion of the correlations produce spectroscopic factors smaller than one, the mean-field value. The $f_6$ results are smaller than those of $f_4$, which are smaller than those obtained with $f_1$.
We found that the effect of the correlations becomes larger the more bound is the state. This fact emerges by observing that for a fixed set of $lj$ quantum numbers the spectroscopic factors increase with $n$, and, at the same time, that the values of the spectroscopic factors become larger when $n$ and the $lj$ values increase.
The values of the spectroscopic factors depend on the choice of the single particle basis. As we have already said in the introduction, our results have been obtained by using the Woods-Saxon single particle bases given in Ref. [@bis06]. This basis has been chosen in order to reproduce the single particle energies around the Fermi surface and the charge distribution of each nucleus considered. The correlation function has been fixed by the minimization procedure (\[eq:varprin\]). We tested the sensitivity of our results to different single particle basis, by calculating [$^{16}$O ]{}and [$^{40}$Ca ]{}spectroscopic factors by using with the Harmonic Oscillator and Woods-Saxon single particle wave functions of Ref. [@fab00], fixed by a global minimization of the energy. Despite the remarkable differences between the various single particle basis, we found that the maximum variations in the values of the spectroscopic factors is of about the 5%. This value is smaller than the variations produced by the different terms of the correlations, shown in Tabs. \[tab:sfp\] and \[tab:sfn\]. This indicates that our results are more sensitive to the SRC than to the choice of the single particle basis.
As example of correlation effects on the quasi-hole wave functions, we show in Fig. \[fig:qh\_fun\] the squares of the proton $3s_{1/2}$ and neutron $3p_{1/2}$ quasi-hole wave functions for the [$^{208}$Pb ]{} nucleus. The correlations lower the wave function in the nuclear interior. Also in this case, the effect of the correlations increases together with the number of operator channels considered.
In Fig. \[fig:205tl\] we show with a gray band the difference between the empirical charge distributions of $^{206}$Pb and $^{205}$Tl [@cav82]. The dashed dotted line, labeled as IPM, has been obtained by considering that the difference between the two charge distributions can be described as a single $3s_{1/2}$ proton hole in the core of the lead nucleus. This curve has been obtained by folding the IPM result of Fig. \[fig:qh\_fun\] with the electric proton form factor in its dipole form. In a slightly more elaborated picture, the ground state of the $^{205}$Tl is composed by the $3s_{1/2}$ proton hole in the $^{206}$Pb ground state, plus the coupling of the $2d_{5/2}$ and $2d_{3/2}$ proton levels with the first $2^+$ excited state of $^{206}$Pb [@zam75; @kle76]. This description of the $^{205}$Tl charge distribution, shown by the dotted line in the figure, is still within the IPM framework. The dashed line has been obtained by adding to the dotted line the core polarization effects produced by long-range correlations. These effects have been calculated by following the Random Phase Approximation approach of Refs. [@co87c; @ang01]. The full line has been obtained when our SRC effects are also included.
The various effects presented in Fig. \[fig:205tl\] have been obtained in different theoretical frameworks, and the final result does not have any pretense of being a well grounded and coherent description of the empirical charge differences. The point we want to make by showing this figure is that the effects of the SRC are of the same order of magnitude of those commonly considered in traditional nuclear structure calculations.
SUMMARY AND CONCLUSIONS {#sec:summary}
=======================
In this work we have extended the FHNC/SOC scheme in order to calculate the OBDM’s, the natural orbits and the quasi-hole wave functions of finite nuclear systems non saturated in isospin, and in $jj$ coupling representation of the single particle wave function basis. Our results have been obtained by using the many-body wave functions obtained by minimizing the nuclear hamiltonian with the two-body realistic interaction Argonne $v'_8$ and the associated three-body interaction Urbana IX. The calculations have been done by using operator dependent correlations which include terms up to the tensor ones.
We found that the correlations enhance by orders of magnitude the high-energy tail of the nucleon momentum distribution. The occupation numbers of the natural orbits below the Fermi level, are depleted, and the opposite happens for those above the Fermi level. Also the values of the spectroscopic factors are depleted with respect to the IPM. A reliable comparison between our spectroscopic factors with the empirical ones requires the description of the reactions used to extract them, and this is part of our future projects.
We have shown that the results of models considering expansions up to the first order correlation lines, provide only qualitative descriptions of the SRC effects. In the description of the charge density difference between $^{206}$Pb and $^{205}$Tl, the SRC effects are of comparable size of those commonly considered in traditional nuclear structure calculations based on effective theories.
A general outcome of our study, is that the effects of the correlations increase with the complexity of the correlation function. This means that operator dependent correlations enhance the effects produced by the scalar correlations. This not obvious result, is valid in general, not always. We have shown in Ref. [@bis06], that scalar and operator dependent correlations have destructive interference effects on the density distributions. We found in the present study an analogous behavior regarding the natural orbits. These quantities are related to the density distributions. It seems that the effects of the SRC are rather straightforward on quantities which involve two-nucleons, while they are more difficult to predict on quantities related to single nucleon dynamics. On these last quantities, however, these SRC effects are very small, usually negligible.
In our calculations the nuclear interaction acts only in defining the many-body wave functions by means of the variational principle (\[eq:varprin\]), more specifically, in selecting the correlation function (\[eq:cor2\]). It is therefore difficult to disentangle the role played by the various parts of the interaction, e.g. the tensor part of the three-body force, on the quantities we have studied in this article. We have instead evaluated the effects of the various parts of the correlation function.
In this work, we have highlighted a set of effects that cannot be described by mean field based effective theories. The description of the nucleus in kinematics regimes where these effects are relevant, requires the use of microscopic theories.
ACKNOWLEDGMENTS
===============
This work has been partially supported by the agreement INFN-CICYT, by the Spanish Ministerio de Educación y Ciencia (FIS2005-02145) and by the MURST through the PRIN: [*Teoria della struttura dei nuclei e della materia nucleare*]{}.
{#app:A}
For sake of completeness, we give in this appendix the detailed expression of the OBDM for finite nuclear systems not saturated in isospin, and in $jj$ coupling scheme of the single particle wave function basis (\[eq:spwf\]). The notation for the nodal functions $N$ and for the vertex corrections $C$ is that used in Ref. [@fab01]. The indexes $t_{1},t_{2},t_{3}$ indicate protons and neutrons, and the subscript $j$ is related to the antiparallel spin states.
For the correlated OBDM we obtain the expression: $$\begin{aligned}
\rho^{t}({{\bf r}}_1,{{\bf r}}_{1'})&=&2C_{\omega,SOC}^{t}({{\bf r}}_1)
C_{\omega,SOC}^{t}({{\bf r}}_{1'})
e^{N_{\omega\omega}^{t}({{\bf r}}_1,{{\bf r}}_{1'})}
\Big[\rho_{o}^{t}({{\bf r}}_1,{{\bf r}}_{1'})-
N_{\omega_c\omega_c}^{t}({{\bf r}}_1,{{\bf r}}_{1'})\Big]
+ \\
\nonumber &&
2C_{\omega}^{t}({{\bf r}}_1)C_{\omega}^{t}({{\bf r}}_{1'})
e^{N_{\omega\omega}^{t}({{\bf r}}_1,{{\bf r}}_{1'})}\\
\nonumber
&&\times\sum_{p>1}
A^k\Delta^{k} \Big\{ N_{\omega\omega,p}^{t}({{\bf r}}_1,{{\bf r}}_{1'})
\Big[\rho_{o}^{t}({{\bf r}}_1,{{\bf r}}_{1'})-
N_{\omega_c\omega_c}^{t}({{\bf r}}_1,{{\bf r}}_{1'})\Big]-
N_{\omega_c\omega_c,p}^{t}({{\bf r}}_1,{{\bf r}}_{1'})\Big\} \,\,.
\label{eq:obdm2}\end{aligned}$$ In the above equation, $k$ has been defined as in Eq. (\[eq:brho\]), and we have used $\Delta^{k=1,2,3}=1-\delta_{k,3}$, and $A^{k=1,2,3}=1,3,6$.
In the following we shall calculate the expectation value of the isospin operator sequence: $$\chi_n^{{t_1}{t_2}} = \chi_{{t_1}}^* (1)\chi_{{t_2}}^* (2)
\left({\mbox{\boldmath $\tau$}}_1 \cdot {\mbox{\boldmath $\tau$}}_2 \right)^n \chi_{{t_1}} (1)
\chi_{{t_2}} (2) \,\,,$$ by considering that $$\left({\mbox{\boldmath $\tau$}}_i \cdot {\mbox{\boldmath $\tau$}}_j \right)^n = a_n +
(1-a_n) {\mbox{\boldmath $\tau$}}_i \cdot {\mbox{\boldmath $\tau$}}_j \,\,,$$ with $$a_{n+1}=3(1-a_n) \hspace {1.0 cm} {\rm and} \hspace {1.0 cm} a_0=1
\,\,.$$ By using the above equations we have that: $$\chi_0^{{t_1}{t_2}}= 1
\hspace {0.5 cm} {\rm ,} \hspace {0.5 cm}
\chi_1^{{t_1}{t_2}}= 2\delta_{{t_1}{t_2}}-1
\hspace {0.5 cm} {\rm and} \hspace {0.5 cm}
\chi_n^{{t_1}{t_2}} = 2a_n-1 +2(1-a_n) \delta_{{t_1}{t_2}}
\,\,.$$
The expressions of the vertex corrections are: $$C_{\omega,SOC}^{t_{1}}({{\bf r}}_1)=C_{\omega}^{t_{1}}({{\bf r}}_1)\Big[1+
U_{\omega,SOC}^{t_{1}}({{\bf r}}_1)\Big]
\,\,,$$ $$\begin{aligned}
\nonumber
U^{t_{1}}_{\omega,SOC}({{\bf r}}_1)&=&\sum_{k=1}^{3}A^k\sum_{t_{2}=p,n}
\Big[(1-\delta_{k,1})U_{\omega,2k-1,2k-1}^{t_{1}t_{2}}({{\bf r}}_1)\\
&&+\chi_{1}^{t_{1}t_{2}}\Big(U_{\omega,2k-1,2k}^{t_{1}t_{2}}({{\bf r}}_1)
+U_{\omega,2k,2k-1}^{t_{1}t_{2}}({{\bf r}}_1)\Big)+
\chi_{2}^{t_{1}t_{2}}U_{\omega,2k,2k}^{t_{1}t_{2}}({{\bf r}}_1)\Big]
\,\,,\end{aligned}$$ where we have defined $$\begin{aligned}
\nonumber
U^{t_{1}t_{2}}_{\omega,pq}({{\bf r}}_1)&=&\int
d{{\bf r}}_2 h_{\omega,p}^{t_1 t_2}(r_{12})\Bigg\{\Big[g_{\omega
d}^{t_{1}t_{2}}({{\bf r}}_1,{{\bf r}}_2)C_{d,pq}^{t_{2}}({{\bf r}}_2)+g_{\omega
e}^{t_{1}t_{2}}({{\bf r}}_1,{{\bf r}}_2)
C_{e,pq}^{t_{2}}({{\bf r}}_2)\Big]N^{t_{1}t_{2}}_{\omega d,q}({{\bf r}}_1,{{\bf r}}_2)\\
&& +g^{t_{1}t_{2}}_{\omega d}({{\bf r}}_1,{{\bf r}}_2)C_{e,pq}^{t_{2}}({{\bf r}}_2)
N_{\omega e,q}^{t_{1}t_{2}}({{\bf r}}_1,{{\bf r}}_2)\Bigg\}
\,\,,
\\
h^{t_{1}t_{2}}_{\omega,p}({{\bf r}}_1,{{\bf r}}_2)& = &
\frac{f_{p}(r_{12})}{f_1(r_{12})}+
(1-\delta_{p,1})N_{\omega d,p}^{t_{1}t_{2}}({{\bf r}}_1,{{\bf r}}_2)
\,\,.\end{aligned}$$
The expressions of the two-body distribution functions for $p>1$ are: $$\begin{aligned}
g_{\omega\omega,p}^{t_{1}}({{\bf r}}_1,{{\bf r}}_{1'})&=&
g^{t_{1}}_{\omega\omega}({{\bf r}}_1,{{\bf r}}_{1'})
N^{t_{1}}_{\omega\omega,p}({{\bf r}}_1,{{\bf r}}_{1'}) \,\,,
\\
g_{\omega d,p}^{t_{1}t_{2}}({{\bf r}}_1,{{\bf r}}_2)&=&
\nonumber
h^{t_{1}t_{2}}_{\omega,p}({{\bf r}}_1,{{\bf r}}_2)g_{\omega
d}^{t_{1}t_{2}}({{\bf r}}_1,{{\bf r}}_2) \\
&=&N^{t_{1}t_{2}}_{\omega d,p}({{\bf r}}_1,{{\bf r}}_2)+X^{t_{1}t_{2}}_{\omega
d,p}({{\bf r}}_1,{{\bf r}}_2) \,\,,
\\
\nonumber
g^{t_{1}t_{2}}_{\omega e,p}({{\bf r}}_1,{{\bf r}}_2)&=&
h^{t_{1}t_{2}}_{\omega,p}({{\bf r}}_1,{{\bf r}}_2)
g_{\omega e}^{t_{1}t_{2}}({{\bf r}}_1,{{\bf r}}_2)+g_{\omega
d}^{t_{1}t_{2}}({{\bf r}}_1,{{\bf r}}_2)N^{t_{1}t_{2}}_{\omega
e,p}({{\bf r}}_1,{{\bf r}}_2) \\
&=&X^{t_{1}t_{2}}_{\omega e,p}({{\bf r}}_1,{{\bf r}}_2)+N^{t_{1}t_{2}}_{\omega
e,p}({{\bf r}}_1,{{\bf r}}_2)
\,\,,
\\
\nonumber
g_{\omega_cc,p}^{t_{1}}({{\bf r}}_1,{{\bf r}}_2)&=&
h^{t_{1}t_{1}}_{\omega,p}({{\bf r}}_1,{{\bf r}}_2)
g_{\omega_cc}^{t_{1}}({{\bf r}}_1,{{\bf r}}_2)
+g_{\omega d}^{t_{1}t_{1}}({{\bf r}}_1,{{\bf r}}_2)N_{\omega_cc,p}^{t_{1}}
({{\bf r}}_1,{{\bf r}}_2) \\
&=&X^{t_{1}}_{\omega_c c,p}({{\bf r}}_1,{{\bf r}}_2)+N^{t_{1}}_{\omega_c c,p}
({{\bf r}}_1,{{\bf r}}_2)
\,\,.\end{aligned}$$
Finally the nodals functions are expressed as: $$\begin{aligned}
N_{mn(j),p}^{t_{1}}(1,2)&=&N_{mnx(j),p}^{t_{1}}(1,2)+
N_{m\rho(j),p}^{t_{1}}(1,2)
+N_{\rho n(j),p}^{t_{1}}(1,2)+N_{\rho,p}^{t_{1}}(1,2)
\,\,,\end{aligned}$$ with $m,n=c,w_c$. The separation of the above nodal diagrams in four terms, corresponds to the classification in the $xx$, $x \rho$, $\rho
x$ and $\rho \rho$ parts [@bis06], and it has been applied to the quantities $N_{mn(j),pqr}^{t_{1}t_{3}}(1,2)$ defined in the following. $$\begin{aligned}
\nonumber
N_{mm,2k_{1}-1}^{t_{1}}(1,1')&=&
\sum_{k_{2}k_{3}=1}^{3}\sum_{t_{3}=p,n}
\Big[N_{mm,2k_{1}-1,2k_{2}-1,2k_{3}-1}^{t_{1}t_{3}}(1,1')\\
&+& \chi_{1}^{t_{1}t_{3}}\big[
N_{mm,2k_{1}-1,2k_{2},2k_{3}-1}^{t_{1}t_{3}}(1,1')+
N_{mm,2k_{1}-1,2k_{2}-1,2k_{3}}^{t_{1}t_{3}}(1,1')\big]\Big]
\,\,,
\\
N_{mm,2k_{1}}^{t_{1}t_{2}}(1,1')&=&\sum_{k_{2},k_{3}=1}^{3}
\sum_{t_{3}=p,n}\chi^{t_{1}t_{3}}_2
N_{mm,2k_{1},2k_{2},2k_{3}}^{t_{1}t_{3}}(1,1')
\,\,,
\\
N_{\omega n,2k_{1}-1}^{t_{1}t_{2}}(1,2)&=&
\sum_{k_{2}k_{3}=1}^{3}\sum_{t_{3}=p,n}
\Big[N_{\omega n,2k_{1}-1,2k_{2}-1,2k_{3}-1}^{t_{1}t_{2}t_{3}}(1,2)\\
\nonumber &+ &\chi_{1}^{t_{1}t_{3}}
N_{\omega n,2k_{1}-1,2k_{2},2k_{3}-1}^{t_{1}t_{2}t_{3}}(1,2)
+\chi_{1}^{t_{2}t_{3}}
N_{\omega n,2k_{1}-1,2k_{2}-1,2k_{3}}^{t_{1}t_{2}t_{3}}(1,2)\Big]
\,\,,
\\
N_{\omega n,2k_{1}}^{t_{1}t_{2}}(1,2)&=&\sum_{k_{2},k_{3}=1}^{3}
\sum_{t_{3}=p,n}N_{\omega
n,2k_{1},2k_{2},2k_{3}}^{t_{1}t_{2}t_{3}}(1,2)
\,\,,
\\
N_{\omega_cc(j),2k_{1}-1}^{t_1}(1,2)&=&
\sum_{k_{2},k_{3}=1}^{3}\sum_{t_{3}=p,n}
\Big[N_{\omega_cc(j),2k_{1}-1,2k_{2}-1,2k_{3}-1}^{t_{1}t_{3}}(1,2)\\
\nonumber &+&\chi_{1}^{t_{1}t_{3}}
\big[N_{\omega_cc(j),2k_{1}-1,2k_{2},2k_{3}-1}^{t_{1}t_{3}}(1,2)+
N_{\omega_cc(j),2k_{1}-1,2k_{2}-1,2k_{3}}^{t_{1}t_{3}}(1,2)\big]\Big]\\
N_{\omega_cc(j),2k_{1}}^{t_{1}}(1,2)&=&
\sum_{k_{2},k_{3}=1}^{3}\sum_{t_{3}=p,n}
N_{\omega_cc(j),2k_{1},2k_{2},2k_{3}}^{t_{1}t_{3}}(1,2)
\,\,.\end{aligned}$$ with $m=w,w_c$ and $n=d,e$. $$\begin{aligned}
N^{t_{1}t_{3}}_{\omega\omega,pqr}({{\bf r}}_1,{{\bf r}}_{1'})&=&
\Big[X^{t_{1}t_{3}}_{\omega d,q}({{\bf r}}_1,{{\bf r}}_2)
\xi^{k_2k_3k_1}_{121'}C_{d,qr}^{t_{3}}({{\bf r}}_2)|
X_{d\omega,r}^{t_{3}t_{1}}({{\bf r}}_2,{{\bf r}}_{1'})+
N_{d\omega,r}^{t_{3}t_{1}}({{\bf r}}_2,{{\bf r}}_{1'})\Big]+\\
\nonumber
&&\Big[X^{t_{1}t_{3}}_{\omega e,q}({{\bf r}}_1,{{\bf r}}_2)
\xi^{k_2k_3k_1}_{121'}C_{e,qr}^{t_{3}}({{\bf r}}_2)|
X_{d\omega,r}^{t_{3}t_{1}}({{\bf r}}_2,{{\bf r}}_{1'})+
N_{d\omega,r}^{t_{3}t_{1}}({{\bf r}}_2,{{\bf r}}_{1'})\Big]+\\
\nonumber
&&\Big[X^{t_{1}t_{3}}_{\omega d,q}({{\bf r}}_1,{{\bf r}}_2)
\xi^{k_2k_3k_1}_{121'}C_{e,qr}^{t_{3}}({{\bf r}}_2)|
X_{e\omega,sr}^{t_{3}t_{1}}({{\bf r}}_2,{{\bf r}}_{1'})+
N_{e\omega,r}^{t_{3}t_{1}}({{\bf r}}_2,{{\bf r}}_{1'})\Big]
\,\,,
\\
N^{t_{1}t_{2}t_{3}}_{\omega n,pqr}({{\bf r}}_1,{{\bf r}}_{2})&=&
\Big[X^{t_{1}t_{3}}_{\omega d,q}({{\bf r}}_1,{{\bf r}}_3)
\xi^{k_2k_3k_1}_{132}C_{d,qr}^{t_{3}}({{\bf r}}_3)|
X_{dn,r}^{t_{3}t_{2}}({{\bf r}}_3,{{\bf r}}_{2})+
N_{dn,r}^{t_{3}t_{2}}({{\bf r}}_3,{{\bf r}}_{2})\Big]+\\
\nonumber
&&\Big[X^{t_{1}t_{3}}_{\omega e,q}({{\bf r}}_1,{{\bf r}}_3)
\xi^{k_2k_3k_1}_{132}C_{e,qr}^{t_{3}}({{\bf r}}_3)|
X_{dn,r}^{t_{3}t_{2}}({{\bf r}}_3,{{\bf r}}_{2})+
N_{dn,r}^{t_{3}t_{2}}({{\bf r}}_3,{{\bf r}}_{2})\Big]+\\
\nonumber
&&\Big[X^{t_{1}t_{3}}_{\omega d,q}({{\bf r}}_1,{{\bf r}}_3)
\xi^{k_2k_3k_1}_{132}C_{e,qr}^{t_{3}}({{\bf r}}_3)|
X_{en,r}^{t_{3}t_{2}}({{\bf r}}_3,{{\bf r}}_{2})+
N_{en,r}^{t_{3}t_{2}}({{\bf r}}_3,{{\bf r}}_{2})\Big]\\
\,\,\end{aligned}$$ also in the above equations we used $n=d,e$. In the following equations we have that $m,n=c,w_c$. $$\begin{aligned}
\nonumber
N^{t_{1}t_{3}}_{mnx,pqr}(1,{2})&=&
\Big[X^{t_{1}}_{mc,q}(1,3)
\xi^{k_2k_3k_1}_{132}\frac{\Delta^{k_3}}{2}
C_{e,qr}^{t_{3}}(3)|
X_{cn}^{t_{3}}(3,{2})+N_{cnx}^{t_{3}}(3,{2})+
N_{\rho n}^{t_{3}}(3,{2}) \Big]+\\
&&(1-\delta_{r,1}) \times \\
\nonumber & &
\Big[X^{t_{1}}_{mc}(1,3)
\xi^{k_2k_3k_1}_{132}\frac{\Delta^{k_2}}{2}
C_{e,qr}^{t_{3}}(3)|
X_{cn,r}^{t_{3}}(3,{2})+
N_{cnx,r}^{t_{3}}(3,{2})+
N_{\rho n,r}^{t_{3}}(3,{2}) \Big]-\\
\nonumber
&& \hspace*{-3mm}
\delta_{k_1 1}\delta_{k_2 1}\delta_{k_3 1} \Big\{
\Big[X^{t_{1}}_{mcj,q}(1,3)
\frac{1}{2}
C_{e,qr}^{t_{3}}(3) |
X_{cnj}^{t_{3}}(3,{2})+
N_{cnxj}^{t_{3}}(3,{2})+
N_{\rho nj}^{t_{3}}(3,{2})\Big]\\
\nonumber
&&\hspace*{-3mm}
+(1-\delta_{r,1})\Big[X^{t_{1}}_{mcj}(1,3)
\frac{1}{2}
C_{e,qr}^{t_{3}}(3) |
X_{cnj,r}^{t_{3}}(3,{2})+
N_{cnxj,r}^{t_{3}}(3,{2})+
N_{\rho nj,r}^{t_{3}}(3,{2})\Big] \Big\}
\,\,,
\\
\nonumber
N^{t_{1}t_{3}}_{m\rho,pqr}(1,{2})&=&
\Big[X^{t_{1}}_{mc,q}(1,3)
\xi^{k_2k_3k_1}_{132}\frac{\Delta^{k_3}}{2}
C_{e,qr}^{t_{3}}(3)|
-\rho_{o}^{t_{3}}(3,{2})+
N_{c\rho}^{t_{3}}(3,{2})+
N_{\rho}^{t_{3}}(3,2)\Big]\\
\nonumber
&&+(1-\delta_{r,1})
\Big[X^{t_{1}}_{mc}(1,3)
\xi^{k_2k_3k_1}_{132}\frac{\Delta^{k_2}}{2}
C_{e,qr}^{t_{3}}(3)|
N_{c\rho,r}^{t_{3}}(3,{2})+
N_{\rho,r}^{t_{3}}(3,2)\Big]-\\
\nonumber
&&\delta_{k_1 1}\delta_{k_2 1}\delta_{k_3 1} \Big\{
\Big[X^{t_{1}}_{mcj,q}(1,3)
\frac{1}{2}
C_{e,qr}^{t_{3}}(3) |
-\rho_{oj}^{t_{3}}(3,{2})+
N_{c\rho j}^{t_{3}}(3,{2})+
N_{\rho j}^{t_{3}}(3,2)\Big] \\
&&+(1-\delta_{r,1})\Big[X^{t_{1}}_{mcj}(1,3)
\frac{1}{2}
C_{e,qr}^{t_{3}}(3)|
N_{c\rho j,r}^{t_{3}}(3,{2})+
N_{\rho j,r}^{t_{3}}(3,2)\Big] \Big\}
\,\,,
\\
N^{t_{1}t_{3}}_{\rho,pqr}(1,{2})&=&
-\Big[\rho^{t_{1}}_{0}(1,3)
\xi^{k_2k_3k_1}_{132}\frac{\Delta^{k_2}}{2}
C_{e,qr}^{t_{3}}(3)|
N_{c\rho,r}^{t_{3}}(3,{2})\Big]\\ \nonumber & &
-\Big[\rho^{t_{1}}_{0}(1,3)
\xi^{k_2k_3k_1}_{132}\frac{\Delta^{k_2}}{2}
(C_{e,qr}^{t_{3}}(3)-1)|
N_{\rho,r}^{t_{3}}(3,{2})-\delta_{r,1}\rho^{t_{3}}_{0}(3,2)\Big]\\
\nonumber
&&+\delta_{k_1 1}\delta_{k_2 1}\delta_{k_3 1} \Big\{
\Big[\rho^{t_{1}}_{0j}(1,3)
\frac{1}{2}
C_{e,qr}^{t_{3}}(3)|
N_{c\rho j,r}^{t_{3}}(3,{2})\Big]\\ \nonumber & &
+\Big[\rho^{t_{1}}_{0j}(1,3)
\frac{1}{2}
(C_{e,qr}^{t_{3}}(3)-1)|
N_{\rho j,r}^{t_{3}}(3,{2})-
\delta_{r,1}\rho^{t_{3}}_{0j}(3,2)\Big] \Big\}
\,\,,
\\
\nonumber
N^{t_{1}t_{3}}_{mnxj,pqr}(1,{2})&=&
\delta_{k_1 1}\delta_{k_2 1}\delta_{k_3 1}
\Big\{ \Big[X^{t_{1}}_{mc,q}(1,3)
\frac{1}{2}
C_{e,qr}^{t_{3}}(3)|
X_{cnj}^{t_{3}}(3,{2})+
N_{cnxj}^{t_{3}}(3,{2})+
N_{\rho nj}^{t_{3}}(3,2)\Big]+\\
\nonumber
&&(1-\delta_{r,1}) \Big[X^{t_{1}}_{mc}(1,3)
\frac{1}{2}
C_{e,qr}^{t_{3}}(3)|
X_{cnj,r}^{t_{3}}(3,{2})+
N_{cnxj,r}^{t_{3}}(3,{2})+
N_{\rho nj,r}^{t_{3}}(3,2)\Big]+\\
\nonumber
&&
\Big[X^{t_{1}}_{mcj,q}(1,3)
\frac{1}{2}
C_{e,qr}^{t_{3}}(3)|
X_{cn}^{t_{3}}(3,{2})+
N_{cnx}^{t_{3}}(3,{2})+
N_{\rho n}^{t_{3}}(3,2)\Big]+\\
\nonumber
&&(1-\delta_{r,1}) \Big[X^{t_{1}}_{mcj}(1,3)
\frac{1}{2}
C_{e,qr}^{t_{3}}(3)|
X_{cn,r}^{t_{3}}(3,{2})+
N_{cnx,r}^{t_{3}}(3,{2})+
N_{\rho n,r}^{t_{3}}(3,2)\Big] \Big\}
\,\,,
\\ \\
\nonumber
N^{t_{1}t_{3}}_{m \rho j,pqr}(1,{2})&=&
\delta_{k_1 1}\delta_{k_2 1}\delta_{k_3 1}
\Big\{\Big[X^{t_{1}}_{mc,q}(1,3)
\frac{1}{2}
C_{e,qr}^{t_{3}}(3)|
-\rho_{oj}^{t_{3}}(3,{2})+
N_{c\rho j}^{t_{3}}(3,{2})+
N_{\rho j}^{t_{3}}(3,2)\Big]\\
\nonumber
&&+(1-\delta_{r,1})
\Big[X^{t_{1}}_{mc}(1,3)
\frac{1}{2}
C_{e,qr}^{t_{3}}(3)|
N_{c\rho j,r}^{t_{3}}(3,{2})+
N_{\rho j,r}^{t_{3}}(3,2)\Big]\\
\nonumber
&&+\Big[X^{t_{1}}_{mcj,q}(1,3)
\frac{1}{2}
C_{e,qr}^{t_{3}}(3)|
-\rho_{o}^{t_{3}}(3,{2})+
N_{c\rho }^{t_{3}}(3,{2})+
N_{\rho }^{t_{3}}(3,2)\Big]+\\
&&(1-\delta_{r,1})
\Big[X^{t_{1}}_{mcj}(1,3)
\frac{1}{2}
C_{e,qr}^{t_{3}}(3)|
N_{c\rho,r}^{t_{3}}(3,{2})+
N_{\rho ,r}^{t_{3}}(3,2)\Big]\Big\}
\,\,,
\\
\nonumber
N^{t_{1}t_{3}}_{\rho j,pqr}(1,{2})&=&
-\delta_{k_1 1}\delta_{k_2 1}\delta_{k_3 1} \Big\{
\Big[\rho^{t_{1}}_{0}(1,3)
\frac{1}{2}
C_{e,qr}^{t_{3}}(3)|
N_{c\rho j,r}^{t_{3}}(3,{2})\Big]\\ \nonumber & &
+\Big[\rho^{t_{1}}_{0}(1,3)
\frac{1}{2}
(C_{e,qr}^{t_{3}}(3)-1)|
N_{\rho j,r}^{t_{3}}(3,{2})-\delta_{r,1}\rho^{t_{3}}_{0j}(3,2)\Big]\\
\nonumber
&&+
\Big[\rho^{t_{1}}_{0j}(1,3)
\frac{1}{2}
C_{e,qr}^{t_{3}}(3)|
N_{c\rho ,r}^{t_{3}}(3,{2})\Big]\\
&~&
+\Big[\rho^{t_{1}}_{0j}(1,3)
\frac{1}{2}
(C_{e,qr}^{t_{3}}(3)-1)|
N_{\rho ,r}^{t_{3}}(3,{2})-\delta_{r,1}\rho^{t_{3}}_{0}(3,2)\Big] \Big\}
\,\,. \end{aligned}$$ The values of the $\xi^{k_1,k_2,k_3}_{ijk}$ coefficients are given in Ref. [@pan79].
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---
abstract: 'We present a systematic procedure to obtain all necessary and sufficient (quantum) constraints on the expectation values for any set of qudit’s operators. These constraints—arise form Hermiticity, normalization, and positivity of a statistical operator and through Born’s rule—analytically define an allowed region. A point outside the admissible region does not correspond to any quantum state, whereas every point in it come from a quantum state. For a set of observables, the allowed region is a compact and convex set in a real space, and all its extreme points come from pure quantum states. By defining appropriate concave functions on the permitted region and then finding their absolute minimum at the extreme points, we obtain different tight uncertainty relations for qubit’s and spin observables. In addition, quantum constraints are explicitly given for the Weyl operators and the spin observables.'
author:
- Arun Sehrawat
title: Deriving quantum constraints and tight uncertainty relations
---
Introduction {#sec:Intro}
============
Von Neumann described a state for a quantum system with a density (statistical) operator on the system’s Hilbert space [@von-Neumann27; @von-Neumann55; @Fano57]. A valid density operator must be *Hermitian*, *positive semi-definite*, and of *unit trace*. Born provided a rule [@Born26; @Wheeler83] to compute the expectation values for any set of operators from a given statistical operator. Naturally, all necessary and sufficient constraints—called *quantum constraints* (QCs)—on the expectation values emerge from the three conditions on a density operator.
In Sec. \[sec:QC\], a systematic procedure to derive the QCs is presented, where a result from [@Kimura03; @Byrd03] is used for the positivity of a statistical operator (or simply a state). To transfer the conditions from a state onto the expectation values, one needs the Born rule and an operator-basis to represent operators. One can choose any basis, the procedure in Sec. \[sec:QC\] is basis independent.
In [@Kimura03; @Byrd03], generators of the special unitary group—that with the identity operator constitute an orthogonal operator-basis—are utilized, and the QCs on their average values are achieved by applying the Lie algebra. Alternatively, one can start with an orthonormal basis of the system’s Hilbert space, and with all possible “ket-bra” pairs one can assemble a *standard* operator-basis. Then, one can exploit the matrix mechanics—developed by Heisenberg, Born, Jordan, and Dirac [@Heisenberg25; @Born25; @Born25-b; @Dirac25; @Waerden68]—to reach the QCs as demonstrated in Sec. \[sec:QC\].
The QCs and uncertainty relations (URs) are two main strands of this paper. Heisenberg pioneered the first UR [@Heisenberg27; @Wheeler83] for the position and momentum operators. A general version of Heisenberg’s relation for a pair of operators is introduced by Robertson [@Robertson29] that is then improved by Schrödinger [@Schrodinger32]. Deutsch [@Deutsch83], Kraus [@Kraus87], Maassen and Uffink [@Maassen88] formulated URs by employing entropy—rather than the standard deviation that is exercised in [@Robertson29; @Schrodinger32]—as a measure of uncertainty. For an overview, we point to [@Wehner10; @Bialynicki11; @Coles17] for entropy URs and [@Folland97; @Busch07; @Busch14] are more in the spirit of Heisenberg’s UR.
Throughout the article, we are considering a $d$-level quantum system (qudit). For a set of <span style="font-variant:small-caps;">n</span> observables (Hermitian operators), the QCs bound an *allowed* region $\mathcal{E}$ of the expectation values in the real space $\mathbb{R}^{\textsc{n}}$. If one defines a suitable concave function on $\mathcal{E}$ to measure a combined uncertainty as described in Sec. \[sec:QC\], then creating a *tight* UR becomes an optimization problem where at most ${2(d-1)}$ parameters are involved (for example, see [@Sehrawat17; @Riccardi17]). A UR is called tight if there exists a quantum state that saturates it. With this, we close Sec. \[sec:QC\] and try its results in the subsequent sections.
In Sec. \[sec:unitary-basis\], we apply the general methodology of Sec. \[sec:QC\] to the *unitary* operator basis, which is known due to Weyl and Schwinger [@Weyl32; @Schwinger60]. In the case of a prime (power) dimension $d$, the unitary-basis can be divided into ${d+1}$ disjoint subsets such that all the operators in each subset possess a common eigenbasis [@Bandyopadhyay02; @Englert01]. These ${d+1}$ eigenbases form a maximal set of mutually unbiased bases (MUBs) [@Ivanovic81; @Wootters89; @Durt10] of the Hilbert space. In Sec. \[sec:unitary-basis\], QCs for the Weyl operators as well as for MUBs are presented. There we arrive at the same *quadratic* QC that is conceived in [@Larsen90; @Ivanovic92; @Klappenecker05]. Using the quadratic QC, tight URs for the MUBs are achieved in [@Sanchez-Ruiz95; @Ballester07], and their minimum uncertainty states are reported in [@Wootters07; @Appleby14].
In the case of ${d\geq3}$, there also exists a *cubic* QC. In Sec. \[sec:qutrit\], ${d=3}$, QCs are explicitly given for the Weyl operators of a qutrit and for a set of spin-1 operators. In addition, a number of tight URs and certainty relations (CRs) are delivered for the spin operators. By the way, the QCs for the spin-1 operators can also be achieved from [@Kimura03; @Byrd03]. In Sec. \[sec:spin-j\], tight URs and CRs are obtained for the angular momentum operators ${J_x,J_y,}$ and $J_z$, where the quantum number $\mathsf{j}$ can be ${\tfrac{1}{2}, 1,\tfrac{3}{2},2,\cdots\,}$. The paper is concluded in Sec. \[sec:conclusion\], where a list of our main contributions is prepared.
Appendix \[sec:qubit\] offers a comprehensive analysis for a qubit ${(d=2)}$ that includes the Schrödinger UR [@Schrodinger32], and URs for the symmetric informationally complete positive operator valued measure (SIC-POVM) [@Rehacek04; @Appleby09] are presented there. In the case of a qubit, it is a known result that $\mathcal{E}$ will be an ellipsoidal region for any number of observables (measurement settings) [@Kaniewski14], and it is also manifested here. Appendixes \[subsec:2settings\] and \[subsec:3settings\] separately deal with two and three measurement settings. In the case of two settings, the ellipsoid transfigures into an ellipse, which also appears in [@Lenard72; @Larsen90; @Kaniewski14; @Abbott16; @Sehrawat17]. It is revealed in [@Sehrawat17] that several tight CRs and URs known from [@Larsen90; @Busch14-b; @Garrett90; @Sanchez-Ruiz98; @Ghirardi03; @Bosyk12; @Vicente05; @Zozor13; @Deutsch83; @Maassen88; @Rastegin12] can be achieved by exploiting the ellipse. In this article, we deal with Hilbert space $\mathscr{H}_d$ of kets and Hilbert-Schmidt space $\mathscr{B}(\mathscr{H}_d)$ of operators, and their bases are differently symbolized by $\mathcal{B}$ and $\mathfrak{B}$, respectively, to avoid any confusion.
Quantum constraints, allowed region, and uncertainty measures {#sec:QC}
=============================================================
Quantum state for a qudit can be described by a statistical operator $\rho$ [@von-Neumann27; @von-Neumann55; @Fano57; @Bengtsson06], on the system’s Hilbert space $\mathscr{H}_d$, such that $$\begin{aligned}
\label{Herm-rho}
\rho&=&\rho^\dagger\quad\ \; \text{(Hermiticity)}\,,\\
\label{norm-rho}
\text{tr}(\rho)&=&1\qquad \text{(normalization)}\,,\qquad\text{and}\\
\label{pos-rho}
0&\leq& \rho\qquad\text{(positivity)}\,.\end{aligned}$$ The dagger $\dagger$ denotes the adjoint. It has been shown in [@Kimura03; @Byrd03] that an operator $\rho$ fulfills if and only if it obeys $$\begin{aligned}
\label{0<=S_n}
0&\leq&S_n \quad \mbox{for all} \quad
1\leq n\leq d\,,
\qquad\quad\text{where}\\
\label{S_n}
S_n&=&\tfrac{1}{n}
\sum_{m=1}^{n}(-1)^{m-1}\,
\text{tr}(\rho^{m})\,S_{n-m}\end{aligned}$$ commencing with ${S_1=\text{tr}(\rho),}$ and ${S_0:=1}$. It is advantageous to use inequalities between real numbers than a single operator-inequality ; see also [@+veMat]. Due to normalization , the first condition ${0\leq S_1=1}$ holds naturally. In a nutshell, an operator $\rho$ on $\mathscr{H}_d$ represents a legitimate quantum state if and only if it complies with , , and for ${2\leq n\leq d}$.
The set of all bounded operators on $\mathscr{H}_d$ form a $d^2$-dimensional Hilbert-Schmidt space $\mathscr{B}(\mathscr{H}_d)$ endowed with the inner product $$\label{HS-inner-pro}
\lgroup A,B\,\rgroup_\textsc{hs}=
\text{tr}(A^\dagger B)\,,
\quad\mbox{where}\quad
A,B\in\mathscr{B}(\mathscr{H}_d)\,.$$ Suppose $$\label{HS-basis}
\mathfrak{B}:=
\big\{\Gamma_\gamma\big\}%
_{\gamma=1}^{d^2}\,,\qquad
\lgroup \Gamma_{\gamma'},\Gamma_\gamma\,\rgroup_\textsc{hs}=
\delta_{\gamma,\gamma'}\,,$$ is an orthonormal basis of $\mathscr{B}(\mathscr{H}_d)$ [@Schwinger60; @Kimura03; @Byrd03], where ${\delta_{\gamma,\gamma'}}$ is the Kronecker delta function. Now we can resolve every operator ${A\in\mathscr{B}(\mathscr{H}_d)}$ in the basis $\mathfrak{B}$ as [@Fano57] $$\label{A-resolution}
A=\sum_{\gamma=1}^{d^2}
\texttt{a}_\gamma\,\Gamma_\gamma\,,
\quad\mbox{where}\quad
\texttt{a}_\gamma=\lgroup \Gamma_\gamma,A\,\rgroup_\textsc{hs}$$ are complex numbers. In this way, we also have the resolution of $$\label{rho-resolution}
\rho=\sum_{\gamma=1}^{d^2}
\texttt{r}_\gamma\,\Gamma_\gamma\,,
\quad\mbox{where}\quad
\texttt{r}_\gamma=\lgroup \Gamma_\gamma,\rho\,\rgroup_\textsc{hs}\,.$$
Born introduced the rule [@Born26] (see also [@von-Neumann55]) $$\begin{aligned}
\label{Born rule-1}
\langle A\rangle_{\rho}&=&
\text{tr}(\rho\, A)=\lgroup A^\dagger,\rho\rgroup_\textsc{hs}\\
\label{Born rule-2}
&=&\lgroup\rho,A\rgroup_\textsc{hs}\end{aligned}$$ to calculate the average value of an operator $A$ by taking the statistical operator $\rho$. Definition of the inner product is exploited to reach the last term in , and through Hermiticity , we get . By the rule, , one can realize $$\label{r}
\texttt{r}_\gamma=
\langle \Gamma_\gamma^{\,\dagger} \rangle_\rho
=\overline{\langle \Gamma_\gamma \rangle}_\rho\,,$$ where the last equality is due to the conjugate symmetry ${\lgroup \Gamma,\rho\rgroup_\textsc{hs}=
\overline{\lgroup \rho,\Gamma\rgroup}_\textsc{hs}}$ and . The overline designates the complex conjugation. The set of equations ${\langle \Gamma_\gamma^{\,\dagger} \rangle=\overline{\langle \Gamma_\gamma \rangle}}$ for every $\gamma$, or ${\langle A^{\dagger} \rangle=\overline{\langle A \rangle}}$ for every ${A\in\mathscr{B}(\mathscr{H}_d)}$, is equivalent to Hermiticity of $\rho$.
Using and , we can express as the standard inner product $$\label{<A>-1}
\langle A\rangle_\rho
=\sum_{\gamma=1}^{d^2}\,
\overline{\texttt{r}}_{\gamma}\;\texttt{a}_\gamma
=\texttt{R}^\dagger\texttt{A}$$ between ${\texttt{R}:=(\texttt{r}_1,\cdots,\texttt{r}_{d^2})^\intercal}$ and ${\texttt{A}:=(\texttt{a}_1,\cdots,\texttt{a}_{d^2})^\intercal}$ [@Fano57; @von-Neumann27], where $\intercal$ stands for the transpose. The column vectors ${\texttt{A},\texttt{R}\in\mathbb{C}^{d^2}}$ are the numerical representations of ${A,\rho\in\mathscr{B}(\mathscr{H}_d)}$ in basis , whereas expectation value does not depend on the basis $\mathfrak{B}$ [@unitary-eq].
Suppose ${A=A^\dagger}$ (depicts an observable) is a Hermitian operator, and $$\label{A}
A=\sum_{l=1}^{d}
a_l\,
|a_l\rangle\langle a_l|$$ is its spectral decomposition. Its expectation value \[via \] $$\label{<A>-pro}
\langle A\rangle_\rho=
\sum_{l=1}^{d}a_l\,p_l\,,
\qquad
p_l=
\big\langle\,|a_l\rangle\langle a_l|\,\big\rangle_\rho\,,$$ can be estimated by performing measurements in its eigenbasis ${\{|a_l\rangle\}_{l=1}^{d}}$. $p_l$ is the probability of getting the outcome, eigenvalue, $a_l$. Due to and , one can realize $$\begin{aligned}
\label{p-const1}
1&=&\sum_{l=1}^{d}
p_l=\text{tr}(\rho)\quad \mbox{and} \\
\label{p-const2}
0&\leq&p_l=
\langle a_l|\,\rho\,|a_l\rangle
\quad \mbox{for all} \quad
1\leq l\leq d\,.\end{aligned}$$ In , the completeness relation ${\textstyle\sum\nolimits_{l=1}^{d}|a_l\rangle\langle a_l|=I}$ plays a role, where $I$ is the identity operator. The set of all probability vectors ${\vec{p}:=(p_1,\cdots,p_d)}$ constitutes a probability space $\Omega_a$, that is—defined by and —the standard ${(d-1)}$-simplex in the $d$-dimensional real vector space $\mathbb{R}^d$ [@Bengtsson06; @Sehrawat17]. One can perceive ${\langle A\rangle_\rho}$ in as a linear function from $\Omega_a$ into $\mathbb{R}$ and then can recognize $$\label{<A>in}
\langle A\rangle\in[a_\text{min}\,,\,a_\text{max}]\,,$$ where endpoints of the interval are the smallest $a_\text{min}$ and the largest $a_\text{max}$ eigenvalues of $A$.
Every classical (discrete) probability distribution also follows and [@Bengtsson06]. The QCs become evident when we take two or more *incompatible* observables (measurements), see below. It is one of the most striking features of quantum physics that—has no classical analog—physically distinct measurements do exist, and one cannot estimate all the expectation values listed in $\overline{\texttt{R}}$ in by using a single setting for projective measurements [@SIC-POVM]. One requires at least ${d+1}$ settings. Moreover, two measurement settings can be so different that if one always gets a definite outcome in one setting, (s)he can get totally random results in the other setting [@Ivanovic81; @Wootters89]. Such settings correspond to *complementary* operators [@Schwinger60; @Kraus87] that are building blocks of the unitary-basis presented in Sec. \[sec:unitary-basis\].
Now let us take <span style="font-variant:small-caps;">n</span> number of operators: ${A,B,\cdots,C}$. We can build a single matrix equation $$\label{expt-values}
\underbrace{\begin{pmatrix}
\langle A\rangle_\rho\\
\langle B\rangle_\rho\\
\vdots\\
\langle C\rangle_\rho\\
\end{pmatrix}}_{\displaystyle\textsf{E}}
=
\underbrace{\begin{pmatrix}
\texttt{a}_{\scriptscriptstyle 1} &
\texttt{a}_{\scriptscriptstyle 2} &
\cdots &
\texttt{a}_{\scriptscriptstyle d^2} \\
\texttt{b}_{\scriptscriptstyle 1} &
\texttt{b}_{\scriptscriptstyle 2} &
\cdots &
\texttt{b}_{\scriptscriptstyle d^2} \\
\vdots & \vdots & \ddots & \vdots \\
\texttt{c}_{\scriptscriptstyle 1} &
\texttt{c}_{\scriptscriptstyle 2} &
\cdots &
\texttt{c}_{\scriptscriptstyle d^2} \\
\end{pmatrix}}_{\displaystyle \textbf{M} }
\underbrace{\begin{pmatrix}
\overline{\texttt{r}}_{\scriptscriptstyle 1} \\
\overline{\texttt{r}}_{\scriptscriptstyle 2} \\
\vdots\\
\overline{\texttt{r}}_{\scriptscriptstyle d^2} \\
\end{pmatrix}}_{\displaystyle\overline{\texttt{R}}}$$ by combining equations such as . Equation is nothing but the numerical representation of Born’s rule in basis .
We present this article by keeping the experimental scenario, $$\label{expt-situ}
\parbox{0.85\columnwidth}
{
a finite number of independent qudits are identically prepared in a quantum state $\rho$, and then individual qudits are measured using different settings for ${A,B,\cdots,C}$,
}$$ in mind, where every expectation value is drawn from a same $\rho$. Thus the subscript $\rho$ is omitted from $\langle \ \rangle_{\rho}$ at some places for simplicity of notation. In other experimental situations—(*i*) where one wants to entangle the qudit of interest to an ancillary system and then wants to perform a joint measurement or (*ii*) where one desires to execute sequential measurements on the same qudit [@Busch14]—one can also adopt the above formalism. There one may need to keep track of how the initial qudit’s state gets transformed after an entangling operation or a measurement. At each stage of an experiment, a $\rho$ must respect , , and , and the mean values can be obtained by .
Matrix equation has three parts $\displaystyle\overline{\texttt{R}}$, **M**, and :
- Conditions , , and on a density operator $\rho$ enter through $\displaystyle\overline{\texttt{R}}$ and emerge as the QCs on the expectation values listed in . In experiment situation , all the knowledge about state preparation goes into the column $\displaystyle\overline{\texttt{R}}$.
- From top to bottom, rows in the ${\textsc{n}\times d^{\,2}}$ matrix **M** completely specify ${A,B,\cdots,C}$. So **M** holds all, and only, the information about measurement settings.
- Conditions , , and as well as the mean values in do not depend on the choice of basis [@unitary-eq]. Therefore, the QCs on $\langle A\rangle,\langle B\rangle,\cdots,\langle C\rangle$ will be independent of the basis $\mathfrak{B}$. So one can adopt any basis that suits him or her best. A basis only facilitates the transfer of constraints from a quantum state $\rho$ onto the expectation values in .
Basically, one can achieve the QCs via a two-step procedure:
1. We need to express conditions , , and for ${2\leq n\leq d}$ in terms of ${\{\overline{\texttt{r}}_\gamma\}_{\gamma=1}^{d^2}}$. This delivers the QCs on mean values of the basis elements.
2. Then, we acquire the QCs on $\langle A\rangle,\langle B\rangle,\cdots,\langle C\rangle$ by matrix equation .
Let us focus on Step 1. We already have condition in terms of ${\langle\Gamma_\gamma\rangle_\rho}$, see . To write the remaining conditions and for ${2\leq n\leq d}$ in ${\langle\Gamma_\gamma\rangle_\rho}$ terms, we need to compute $$\label{tr(rho^m)}
\text{tr}(\rho^m)=
\sum_{\gamma_1}\cdots
\sum_{\gamma_m}\,
\texttt{r}_{\gamma_1}\cdots\texttt{r}_{\gamma_m}
\text{tr}\left(\,\Gamma_{\gamma_1}\cdots\Gamma_{\gamma_m}\right)$$ for every ${1\leq m\leq d}$. One can view ${\text{tr}(\rho^m)}$ as a homogeneous polynomial of degree $m$, where average values are variables, and the constants ${\text{tr}\left(\,\Gamma_{\gamma_1}\cdots\Gamma_{\gamma_m}\right)}$ are determined by basis only. Hence $S_n$ of is a $n$-degree polynomial, and ${0\leq S_n}$ \[see \] leads to a $n$-degree QC.
In [@Kimura03; @Byrd03], generators of the special unitary group $SU(d)$—that with the identity operator compose an orthogonal basis of $\mathscr{B}(\mathscr{H}_d)$—are taken, and $\text{tr}(\rho^m)$ is obtained by using the Lie algebra of $SU(d)$. The generators are ${d^{\,2}-1}$ traceless Hermitian operators, thus we call this basis the *Hermitian*-basis \[for ${d=2,3}$, see Appendix \[sec:qubit\] and Sec. \[sec:qutrit\]\]. If all the <span style="font-variant:small-caps;">n</span> operators $A,B,\cdots,C$ are Hermitian operators, then it is better to choose a Hermitian-basis because every number in will be a real number. Since the state space $$\label{set-of-states}
\mathcal{S}=\big\{\rho\in\mathscr{B}(\mathscr{H}_d)\ |\
\rho\text{ obeys \eqref{Herm-rho}, \eqref{norm-rho}, and
\eqref{0<=S_n}} \big\}$$ is a compact and convex set [@Bengtsson06], the corresponding collection of ${\texttt{R}=\overline{\texttt{R}}}$ forms a compact and convex set in $\mathbb{R}^{d^2-1}$ as the mapping ${\rho\leftrightarrow\texttt{R}}$ is a homeomorphisms [@Rudin91]. Every qudit’s state $\rho$ is completely specified by ${d^{\,2}-1}$ real numbers in ${\texttt{R}}$ [@Kimura03], where one of its components is fixed by normalization condition , that is, $\textstyle\sum\nolimits_{\gamma=1}^{d^2}
\texttt{r}_\gamma\,\text{tr}(\Gamma_\gamma)=1$.
Next one can view as a linear transformation from $\mathbb{R}^{d^2-1}$ to $\mathbb{R}^{\textsc{n}}$. Such a transformation is always continuous, and it maps a compact and convex set in $\mathbb{R}^{d^2-1}$ to a compact and convex set in $\mathbb{R}^{\textsc{n}}$ [@Rudin76; @con-to-con]. Therefore, for <span style="font-variant:small-caps;">n</span> observables (Hermitian operators), the set of expectation values $$\label{set-of-expt}
\mathcal{E}:=\big\{\,\textsf{E}\ |\
\rho\in\mathcal{S}\, \big\}$$ will be a *compact* and *convex* set \[for example, see Figs. \[fig:region-2-pojs\], \[fig:regions-MUBs\], \[fig:regions\], \[fig:ellip-SUR\], and \[fig:regions-SIC-POVM\]\] in a hyperrectangle $$\label{hyperrectangle}
\mathcal{H}:=
[a_\text{min}, a_\text{max}]\times
[b_\text{min}, b_\text{max}]\times\cdots\times
[c_\text{min}, c_\text{max}]\subset\mathbb{R}^{\textsc{n}}$$ described by the Cartesian product of the closed intervals, whose endpoints are the minimum and maximum eigenvalues of the operators. $\mathcal{E}$ is also known as the quantum convex support [@Weis11]. Furthermore, each extreme point of $\mathcal{E}$ corresponds to a pure state that is an extreme point of $\mathcal{S}$. Note that Eq. does (map $\mathcal{S}$ *onto* $\mathcal{E}$ via $\rho\leftrightarrow\texttt{R}\rightarrow \textsf{E}$) not provide a one-to-one correspondence between the state space $\mathcal{S}$ and $\mathcal{E}$ unless there are $d^{\,2}$ linearly independent operators in the set ${\{A,B,\cdots,C,I\}}$.
In summary, $\mathcal{S}$ is an abstract set, we observe its *image* $\mathcal{E}$ through an experiment scheme such as . The QCs—originate from , , and via matrix equation —bound the region $\mathcal{E}$. As the QCs are necessary and sufficient restrictions on the expectation values, any point outside $\mathcal{E}$ does not come from a quantum state, whereas every point in $\mathcal{E}$ corresponds to at least one quantum state. So as a whole $\mathcal{E}$ is the only *allowed* region in the space of expectation values. Obviously, one cannot achieve a region smaller than $\mathcal{E}$ without sacrificing a subset of quantum states.
Now we present all the above material by taking a *standard* operator-basis. With an orthonormal basis $\mathcal{B}$ of the Hilbert space $\mathscr{H}_d$, where $$\begin{aligned}
\label{B_i}
\mathcal{B}&:=&
\big\{|j\rangle\,:\,j\in\mathbb{Z}_d\big\}\,,\\
\label{Z_d}
\mathbb{Z}_d&:=&\{\,j\,\}_{j=0}^{d-1}\,,
\qquad\mbox{and}\qquad\\
\label{orth-|j>}
\text{tr}\big(|j\rangle\langle k|\big)&=&
\langle k|j\rangle=\delta_{j,k}\,,\end{aligned}$$ one can construct the standard operator-basis $$\label{Stnd-Basis}
\mathfrak{B}_\text{st}:=\big\{|j\rangle\langle k|\,:\,j,k\in\mathbb{Z}_d\big\}$$ of $\mathscr{B}(\mathscr{H}_d)$. Instead of a single index $\gamma$ that runs from $1$ to ${d^{\,2}}$, here we have two indices $j$ and $k$ for a basis element, each of them runs from $0$ to ${d-1}$. The orthonormality condition $$\label{orth-|j><k|}
\big\lgroup\, |j'\rangle\langle k'|\,,\,|j\rangle\langle k|\,\big\rgroup_\textsc{hs}
=\langle j'|j\rangle\langle k|k'\rangle
=\delta_{j,j'}\,\delta_{k,k'}$$ for $\mathfrak{B}_\text{st}$ is ensured by orthonormality relation of $\mathcal{B}$. In basis , the resolution of an operator $A$ and of a qudit’s state $\rho$ are $$\begin{aligned}
\label{A-in-stnd-basis}
A&=&
\sum_{j,k\,\in\,\mathbb{Z}_d}
\texttt{a}_{jk}\,|j\rangle\langle k|
\quad\mbox{with}\quad
\texttt{a}_{jk}=\langle j|\,A\,|k\rangle\quad\mbox{and}\qquad\\
\label{rho-in-stnd-basis}
\rho&=&
\sum_{j,k\,\in\,\mathbb{Z}_d}
\texttt{r}_{jk}\,|j\rangle\langle k|
\quad\mbox{with}\quad
\texttt{r}_{jk}=\langle j|\,\rho\,|k\rangle\,,\end{aligned}$$ respectively. The above coefficients $\texttt{a}$ and $\texttt{r}$ are obtained through and , correspondingly.
Numerical representation of Born’s rule now becomes $$\label{<A>-in-stnd-basis-2}
\langle A\rangle
=
\sum_{j,k}\,
\overline{\texttt{r}}_{jk}\,\texttt{a}_{jk}\\
=
\sum_{j,k}\,
\texttt{r}_{kj}\,\texttt{a}_{jk}\,,$$ where the second equality is due to the Hermiticity: $$\label{r_kj}
\texttt{r}_{jk}=
\big\langle\, |k\rangle\langle j|\, \big\rangle_\rho=
\overline{\texttt{r}}_{kj}
\quad\mbox{for all}\quad
j,k\in\mathbb{Z}_d$$ is a manifestation of . In standard basis , matrix equation transpires as $$\label{expt-values-std}
\underbrace{\begin{pmatrix}
\langle A\rangle_{\rho}\\
\langle B\rangle_{\rho}\\
\vdots\\
\langle C\rangle_{\rho}
\end{pmatrix}}_{\displaystyle\textsf{E}}
=
\underbrace{\begin{pmatrix}
\texttt{a}_{\scriptscriptstyle 0,0} &
\texttt{a}_{\scriptscriptstyle 0,1} & \cdots &
\texttt{a}_{\scriptscriptstyle d-1,d-1} \\
\texttt{b}_{\scriptscriptstyle 0,0} &
\texttt{b}_{\scriptscriptstyle 0,1} & \cdots &
\texttt{b}_{\scriptscriptstyle d-1,d-1} \\
\vdots & \vdots & \ddots & \vdots \\
\texttt{c}_{\scriptscriptstyle 0,0} &
\texttt{c}_{\scriptscriptstyle 0,1} & \cdots &
\texttt{c}_{\scriptscriptstyle d-1,d-1} \\
\end{pmatrix}}_{\displaystyle \textbf{M} }
\underbrace{\begin{pmatrix}
\overline{\texttt{r}}_{\scriptscriptstyle 0,0} \\
\overline{\texttt{r}}_{\scriptscriptstyle 0,1}\\
\vdots\\
\overline{\texttt{r}}_{\scriptscriptstyle d-1,d-1}
\end{pmatrix}}_{\overline{\displaystyle\texttt{R}}_d}.\qquad$$
Next, to express conditions and for ${2\leq n\leq d}$ in ${\texttt{r}_{jk}}$ terms, we need to represent $\text{tr}(\rho^m)$ for every ${1\leq m\leq d}$ as a function of $\{\texttt{r}_{jk}\}$. Orthonormality relation also yields the rule for composition $$\label{comp}
|j'\rangle\langle k'|\,|j\rangle\langle k|=
\delta_{j,k'}\,|j'\rangle\langle k|\,,$$ which gives rise to matrix multiplication in the matrix mechanics [@Heisenberg25; @Born25; @Born25-b; @Dirac25; @Waerden68]. Particularly here it is very easy to obtain $$\label{rho^m-stnd}
\rho^m=
\sum_{j_1}\cdots
\sum_{j_{m+1}}
\texttt{r}_{j_1j_2}\,\texttt{r}_{j_2j_3}\cdots\texttt{r}_{j_mj_{m+1}}
\,|j_1\rangle\langle j_{m+1}|\,.\\$$ Then, through and the linearity of trace, we secure $$\label{tr(rho^m)-stnd}
\text{tr}(\rho^m)=
\sum_{j_1}\cdots
\sum_{j_m}\,
\texttt{r}_{j_1j_2}\,\texttt{r}_{j_2j_3}\cdots\texttt{r}_{j_mj_{1}}\,.$$ One can compare with its general form . Let us explicitly write conditions and for ${n=2,3,4}$ [@Kimura03; @Fano57]: $$\begin{aligned}
\label{1=S_1}
\sum_{j}
\texttt{r}_{jj}=\text{tr}(\rho)&=& 1\,,\\
\label{0<=S_2}
\texttt{R}^\dagger \texttt{R}=
\sum_{jk}
|\texttt{r}_{jk}|^2=\text{tr}(\rho^2)&\leq& 1\,,\\
\label{0<=S_3}
3\,\text{tr}(\rho^2)-2\,\text{tr}(\rho^3)&\leq& 1\,,\\
\label{0<=S_4}
6\,\text{tr}(\rho^2)-8\,\text{tr}(\rho^3)-3\,\big(\text{tr}(\rho^2)\big)^2+6\,\text{tr}(\rho^4)&\leq& 1\qquad\end{aligned}$$ deliver linear, quadratic, cubic, and quartic QCs. In and , and the column vector `R` from are used.
As a pure state ${\rho=\rho^2}$ is an extreme point of the state space $\mathcal{S}$ \[defined in \], it saturates inequalities for all ${n=2,\cdots,d}$ [@Byrd03]. A pure state corresponds to a ket, and a qudit’s ket can be parametrized by a set of ${2(d-1)}$ real numbers by ignoring an overall phase factor (for example, see [@Arvind97]): $$\begin{aligned}
\label{|psi>}
|\psi\rangle&=&
|0\rangle \cos\theta_0+\nonumber\\
&&|1\rangle \sin\theta_0\cos\theta_1\,e^{\text{i}\phi_1}+\nonumber\\
&&|2\rangle \sin\theta_0\sin\theta_1\cos\theta_2\,e^{\text{i}\phi_2}+\nonumber\\
&&\qquad\cdots+\nonumber\\
&&|d-2\rangle \sin\theta_0\sin\theta_1\cdots\cos\theta_{d-2}\,e^{\text{i}\phi_{d-2}}+
\nonumber\\
&&|d-1\rangle \sin\theta_0\sin\theta_1\cdots\sin\theta_{d-2}\,e^{\text{i}\phi_{d-1}}\,,\end{aligned}$$ where ${\text{i}=\sqrt{-1}}$, $\theta_l\in[0,\tfrac{\pi}{2}]$ for all ${l=0,\cdots,d-2}$, and $\phi_{l'}\in[0,2\pi)$ for every ${l'=1,\cdots,d-1}$. Thus the pure state ${\rho_\text{pure}=|\psi\rangle\langle\psi|}$ and the corresponding column vector ${\texttt{R}^{(\text{pure})}_d}$ \[see for its complex conjugate\] are specified by the ${2(d-1)}$ real numbers [@Bengtsson06], for instance, $$\begin{aligned}
\label{R-pure-2}
&\texttt{R}^{(\text{pure})}_2=\begin{pmatrix}
(\cos\theta_0)^2 \\
\cos\theta_0\sin\theta_0\,e^{-\text{i}\phi_1}\\
\cos\theta_0\sin\theta_0\,e^{\text{i}\phi_1}\\
(\sin\theta_0)^2
\end{pmatrix}&
\mbox{and}\qquad\quad
\\
\label{R-pure-3}
&\texttt{R}^{(\text{pure})}_3=\begin{pmatrix}
(\cos\theta_0)^2 \\
\cos\theta_0\sin\theta_0\cos\theta_1\,e^{-\text{i}\phi_1}\\
\cos\theta_0\sin\theta_0\sin\theta_1\,e^{-\text{i}\phi_2}\\
\cos\theta_0\sin\theta_0\cos\theta_1\,e^{\text{i}\phi_1}\\
{(\sin\theta_0\cos\theta_1)}^2\\
(\sin\theta_0)^2\cos\theta_1\sin\theta_1\,e^{\text{i}(\phi_1-\phi_2)}\\
\cos\theta_0\sin\theta_0\sin\theta_1\,e^{\text{i}\phi_2}\\
(\sin\theta_0)^2\cos\theta_1\sin\theta_1\,e^{-\text{i}(\phi_1-\phi_2)}\\
{(\sin\theta_0\sin\theta_1)}^2
\end{pmatrix}.&\end{aligned}$$ By plugging ${\texttt{R}^{(\text{pure})}_d}$ in Eq. , one can reach all those points in $\mathcal{E}$ \[defined in \] that correspond to pure states in $\mathcal{S}$. All the extreme points of $\mathcal{E}$ will be a subset of these points.
In the following, we demonstrate a procedure to built a combined uncertainty measure on $\mathcal{E}$ for Hermitian operators ${A,B,\cdots,C}$. In the case of a non-Hermitian operator, considering [@Adagger], one can talk about uncertainty measures for the two Hermitian operators $A^{\scriptscriptstyle(+)}=\tfrac{1}{2}(A+A^\dagger)$ and $A^{\scriptscriptstyle(-)}=\tfrac{1}{2\text{i}}(A-A^\dagger)$. Note that $A^{\scriptscriptstyle(+)}$ and $A^{\scriptscriptstyle(-)}$ commutes if and only if $A$—is a normal operator—commutes with $A^\dagger$.
The standard deviation $$\begin{aligned}
\label{std-dev}
\Delta A&=&\sqrt{\langle A^2\rangle-\langle A\rangle^2}\nonumber\\
&=&\sqrt{\sum_{l=1}^{d}a_l^2\,p_l
-\left(\sum_{l=1}^{d}a_l\,p_l\right)^2}\end{aligned}$$ can be viewed—through the first equality—as a concave function on the allowed region for $\{A,A^2\}$, which is the convex hull of ${\{(a_l,a_l^2)\}_{l=1}^d}$, where $a_l$ is an eigenvalue of $A$ \[see \]. By finding the absolute minimum of ${\Delta A+\Delta B+\cdots+\Delta C}$ on the permitted region for $\{A,A^2,B,B^2,\cdots,C,C^2\}$, one can have a tight UR based on the standard deviations \[for example, see –\].
If one wants to built a UR in the case of two projective measurements described by ${\{|a_l\rangle\langle a_l|\}_{l=1}^d}$ and ${\{|b_k\rangle\langle b_k|\}_{k=1}^d}$, then one can consider the permissible region of the two probability vectors ${\vec{p}=(p_1,\cdots,p_d)}$ and ${\vec{q}=(q_1,\cdots,q_d)}$, where ${p_l=\big\langle |a_l\rangle\langle a_l|\big\rangle_\rho}$ \[see \] and ${q_k=\big\langle |b_k\rangle\langle b_k|\big\rangle_\rho}$. There are many uncertainty measures for $\vec{p}$ (and $\vec{q}\,$)—thanks to Shannon [@Shannon48], Rényi [@Renyi61], and Tsallis [@Tsallis88]—and many associated URs [@Coles17; @Maassen88]. Moreover, with the probability vector $\vec{p}$, we can calculate the expectation value of any function of the Hermitian operator ${A}$ \[given in \] as well as its standard deviation . Now suppose we have no access to the individual probabilities $p_l$, but only to the expectation value ${\langle A\rangle}$, then we can construct uncertainty or certainty measures as follows.
Let us recall from that ${\langle A\rangle\in[a_\text{min}\,,\,a_\text{max}]}$, and we are interested in the case ${a_\text{min}\neq\,a_\text{max}}$. We call $\rho$ an *eigenstate* corresponding to an eigenvalue $a$ of $A$ if and only if ${A\rho=a\rho=\rho A}$. If ${\langle A\rangle_\rho=a_\text{min}}$ then we can say for sure: ($i$) qudits are prepared in a minimum-eigenvalue-state of ${A}$ and ($ii$) every outcome ${a_l\neq a_\text{min}}$ will never occur in a future projective measurement ${\{|a_l\rangle\langle a_l|\}_{l=1}^d}$ for $A$. So, only in the two cases ${\langle A\rangle_\rho=a_\text{min},a_\text{max}}$, we have a minimum possible uncertainty about $\rho$ (if it is unknown) in which the individual qudits are identically prepared in and about the results of a future measurement for $A$. Therefore, for an uncertainty measure, we require a continuous function on the interval $[a_\text{min}\,,\,a_\text{max}]$ that reaches its absolute minimum at both the endpoints. Furthermore, mixing states, ${w\rho+(1-w)\rho'=\rho_{\text{mix}}}$ with ${0\leq w\leq1}$, yields the convex sum ${w\langle A\rangle_\rho+(1-w)\langle A\rangle_{\rho'}=\langle A\rangle_{\rho_{\text{mix}}}}$, and it does not decrease uncertainty (or increase certainty). A suitable concave (convex) function can be taken as a measure of uncertainty (certainty) because it does not decrease (increase) under such mixing.
The two positive semi-definite operators $$\label{Adot}
\dot{A}:=\frac{a_\text{max}\,I-A}{a_\text{max}-a_\text{min}}
\quad\text{and}\quad
\mathring{A}:=\frac{A-a_\text{min}\,I}{a_\text{max}-a_\text{min}}\,,$$ are such that ${\dot{A}+\mathring{A}}$ is the identity operator ${I\in\mathscr{B}(\mathscr{H}_d)}$, and we only need ${\langle A\rangle}$ to compute both ${\langle \dot{A}\rangle,\langle \mathring{A}\rangle\in[0,1]}$. Now we can define concave and convex functions of ${\langle A\rangle}$ that fulfill the above requirements: $$\begin{aligned}
\label{H(A)}
H(\langle A\rangle)&=&-(\langle \dot{A}\rangle\ln\,\langle \dot{A}\rangle+\langle \mathring{A}\rangle\ln\,\langle \mathring{A}\rangle)\,,
\\
\label{u(A)}
u_\kappa(\langle A\rangle)&=&{\langle \dot{A}\rangle}^\kappa+{\langle \mathring{A}\rangle}^\kappa\,,\quad 0<\kappa<\infty\,,
\quad\text{and}\qquad\\
\label{umax(A)}
u_\text{max}(\langle A\rangle)&=&\max\,\{\,
\langle \dot{A}\rangle\,,\,\langle \mathring{A}\rangle\,\}\,.\end{aligned}$$ One can easily show that $H$ and $u_\kappa$ for all ${0<\kappa<1}$ are concave functions, whereas $u_\kappa$ for all ${1<\kappa<\infty}$ and $u_\text{max}$ are convex functions. For ${\kappa=1}$, ${u_\kappa(\langle A\rangle)=1}$ for every $\langle A\rangle$, and thus it is neither a genuine measure of uncertainty nor of certainty.
With $u_\kappa$ one can create quantities like Rényi’s and Tsallis’ entropies, and $H$ of is like the Shannon entropy but, in general, it is different from ${-\sum_{l=1}^{d}p_l\ln p_l}$. If $A$ only has two distinct eigenvalues, then $\dot{A}$ and $\mathring{A}$ become mutually orthogonal projectors, and turns into the standard form of Shannon entropy \[for example, see Appendix \[sec:qubit\]\]. Note that Shannon’s and Tsallis’ entropies are concave functions but not all Rényi’s entropies are.
The ranges of the above functions are ${H\in[0,\ln2]}$, ${u_\kappa\in[1,2^{1-\kappa}]}$ for ${0<\kappa<1}$ and ${u_\kappa\in[2^{1-\kappa},1]}$ for ${1<\kappa<\infty}$, and ${u_\text{max}\in[\tfrac{1}{2},1]}$. As desired, all the above concave (convex) functions reach their absolute minimum (maximum) when ${\langle A\rangle=a_\text{min},a_\text{max}}$. In the case of a non-degenerate eigenvalue $a_\text{min}$, we will be even more certain that there is only one (pure) eigenstate state ${|a_\text{min}\rangle\langle a_\text{min}|}$ that can provide ${\langle A\rangle=a_\text{min}}$, and similarly for a non-degenerate $a_\text{max}$. Like the standard deviation , all the concave (convex) functions in – attain their absolute maximum (minimum) when ${\langle A\rangle=\tfrac{1}{2}(a_\text{min}+a_\text{max})}$. Both a ket ${\tfrac{1}{\sqrt{2}}(|a_\text{min}\rangle+e^{\text{i}\phi}|a_\text{max}\rangle)}$, ${\phi}$ is a real number, and a state that is the equal mixture of $|a_\text{min}\rangle\langle a_\text{min}|$ and $|a_\text{max}\rangle\langle a_\text{max}|$ provide the expectation value ${\langle A\rangle=\tfrac{1}{2}(a_\text{min}+a_\text{max})}$. Since the equal superposition ket gives the maximum standard deviation of $A$, the ket plays an important role in the quantum metrology [@Giovannetti06] and to determine a fundamental limit on the speed of unitary evolution generated by $A$ [@Mandelstam45; @Margolus98; @Levitin09].
The sum of concave functions is a concave function, for example, $$\label{Habc}
H(\textsf{E}):=
H(\langle A\rangle)+H(\langle B\rangle)+\cdots+H(\langle C\rangle)\,,$$ where every $H$ is defined according to . One can view as a measure of combined uncertainty on the allowed region $\mathcal{E}$. Its global minimum, say, $\mathfrak{h}$ will occur at the extreme points of $\mathcal{E}$ (see Theorem ${3.4.7}$ and Appendix A.3 in [@Niculescu93]). As every extreme point of $\mathcal{E}$ is related to a pure state, one can find the minimum by changing at most ${2(d-1)}$ parameters that appear in and then can enjoy the *tight* UR ${\mathfrak{h}\leq H(\textsf{E})}$. If a vertex of hyperrectangle is a part of $\mathcal{E}$ only then the lower bound $\mathfrak{h}$ becomes (trivial) 0. It only happens when there exists a ket ${|\text{e}\rangle}$ that is a maximum- or minimum-eigenvalue-ket of every operator in ${\{A,B,\cdots,C\}}$. There are examples in [@Sehrawat18] where all ${A,B,\cdots,C}$ share a common eigenket, thus usual URs—based on probabilities associated with projective measurements for ${A,B,\cdots,C}$ or based on the standard deviations ${\Delta A,\Delta B,\cdots,\Delta C}$—become trivial while ${0<\mathfrak{h}\leq H(\textsf{E})}$. Like , one can built combined uncertainty or certainty measures (and relations) by picking concave or convex functions from and . If one chooses a measure that is neither a concave nor convex function then its absolute extremum can occur inside $\mathcal{E}$. The above technique is applied to derive tight URs and CRs in [@Riccardi17; @Sehrawat17] and in the subsequent sections.
Apart from a few exceptions, it is not clear to us whether we can interpret a QC as a bound on a combined uncertainty or certainty. On the other hand, a UR puts a lower limit on a combined uncertainty, and it can also be perceived as a constraint on mean values as every uncertainty measure is (not necessarily concave or convex but) their function. Suppose we identify a region in hyperrectangle with a UR, for example, $$\label{R_H}
\mathcal{R}_H:=
\big\{ (\textbf{a},\cdots,\textbf{c})\in\mathcal{H}
\ |\
{\mathfrak{h}\leq H(\textbf{a})+\cdots+H(\textbf{c})}
\big\}\,,$$ where $H(\textbf{a})$ is obtained by replacing ${\langle A\rangle}$ with **a** in ${\langle \dot{A}\rangle}$, ${\langle \mathring{A}\rangle}$, and then in ; likewise, ${H(\textbf{c})}$ has the same functional form as $H(\langle C\rangle)$. One can easily prove that $\mathcal{R}_H$ is a convex set. Obviously, ${\mathcal{E}}$ will be contained in $\mathcal{R}_H$, there will be no $\rho$ for ${(\textbf{a},\cdots,\textbf{c})\in\mathcal{R}\setminus\mathcal{E}}$ such that ${(\textbf{a},\cdots,\textbf{c})=(\langle A\rangle_{\rho},\cdots,\langle C\rangle_{\rho})}$ holds, and such points cannot be realized experimentally in scheme . The relative complement of $\mathcal{E}$ in $\mathcal{R}$ is denoted by $\mathcal{R}\setminus\mathcal{E}$. One can also observe that if ${(\textbf{a},\textbf{b},\cdots,\textbf{c})}$ belongs to $\mathcal{R}_H$ then ${(\textbf{a}',\textbf{b},\cdots,\textbf{c})}$, where ${\textbf{a}'=a_{\text{min}}+a_{\text{max}}-\textbf{a}}$, will also belong to $\mathcal{R}_H$ because ${H(\textbf{a})=H(\textbf{a}')}$. In the case of ${\textbf{a}'\neq\textbf{a}}$, only one of the two points can be allowed, because a single quantum state cannot provide two different expectation values of $A$. By taking a few examples in this paper, the gap $\mathcal{R}\setminus\mathcal{E}$ between the two regions is exhibited in Figs. \[fig:region-2-pojs\], \[fig:regions-MUBs\], \[fig:regions\], \[fig:ellip-SUR\], and \[fig:regions-SIC-POVM\].
![The permitted region $\mathcal{E}$—of the expectation values of two projectors $P$ and $Q$ described by the matrices in —is bounded by the (blue) closed-curve. $\mathcal{E}$ is the convex hull of ${(0,0)}$ and the ellipse obtained by and with ${|\langle a|b\rangle|^2=\text{tr}(PQ)=\tfrac{169}{675}}$. Clearly, the (red) point ${(0.8,0.8)}$ does not belong to the allowed region. In this example, hyperrectangle is the square ${[0,1]^{\times 2}}$.[]{data-label="fig:region-2-pojs"}](fig1){width="30.00000%"}
If we have to provide a yes/no answer to a question such as: can ${0.8}$ and ${0.8}$ be the expectation values ${\langle P\rangle_\rho}$ and ${\langle Q\rangle_\rho}$, where $P$ and $Q$ are rank-1 projectors represented by $$\label{PQ-mat}
\begin{pmatrix}
\frac{1}{75} & -\frac{\text{i}}{15} & \frac{7}{15}\\[0.4em]
\frac{\text{i}}{15} & \hphantom{-}\frac{1}{3} & \frac{7\,\text{i}}{15}\\[0.4em]
\frac{7}{75} & -\frac{7\,\text{i}}{15} & \frac{49}{75}
\end{pmatrix}
\quad\mbox{and}\quad
\frac{1}{9}\begin{pmatrix}
\hphantom{-}4 & \hphantom{-}2 & -4\\[0.4em]
\hphantom{-}2 & \hphantom{-}1 & -2\\[0.4em]
-4 & -2 & \hphantom{-}4
\end{pmatrix},$$ respectively, in some orthonormal basis of $\mathscr{H}_3$? Then, a *clear* answer can be given with the allowed region. Suppose ${P=|a\rangle\langle a|}$ and ${Q=|b\rangle\langle b|}$ are two rank-1 projectors on a $d$-dimensional Hilbert space $\mathscr{H}_d$ such that ${0<|\langle a|b\rangle|<1}$ (non-commuting). For ${d=2}$, their allowed region $\mathcal{E}$ is determined by $$\begin{aligned}
\label{ellipse-PQ}
&&
\textsf{E}^\intercal\, \textbf{G}^{-1}\textsf{E} \leq 1\,,
\quad \mbox{where} \quad
\textsf{E}=\begin{pmatrix}
2\,\langle P\rangle-1\\
2\,\langle Q\rangle-1
\end{pmatrix}
\ \mbox{and}\qquad
\\
%
&&
\label{G-PQ}
\textbf{G}=
\begin{pmatrix}
1 & {\scriptstyle 2|\langle a|b\rangle|^2-1}\\
{\scriptstyle 2|\langle a|b\rangle|^2-1}& 1
\end{pmatrix}.
\qquad\end{aligned}$$ One can see through that and are the same for a qubit. In the case of ${d>2}$, the allowed region will be the convex hull of the elliptic region specified by the inequality in and the point $(0,0)$ [@Lenard72]; see also [@Sehrawat17]. This point is given by all those states that lie in the orthogonal complement of ${\{P,Q\}}$. These states are the common eigenstates of $P$ and $Q$. By the way, a UR become a trivial statement in this case.
Answer to the above question is “no” because the point ${(0.8,0.8)}$ falls outside the allowed region as shown in Fig. \[fig:region-2-pojs\]. If one asks a similar question for a set of commuting operators ${\{A,B,\cdots,C\}}$, then the permitted region will be the convex hull of $\{(\langle \text{e}_l|A|\text{e}_l\rangle,
\langle \text{e}_l|B|\text{e}_l\rangle,\cdots,
\langle \text{e}_l|C|\text{e}_l\rangle)\}_{l=1}^{d}$, where ${\{|\text{e}_l\rangle \}_{l=1}^{d}}$ is their common eigenbasis.
The unitary operator basis {#sec:unitary-basis}
==========================
With orthonormal basis of the Hilbert space $\mathscr{H}_d$, we can built a pair of (complementary) unitary operators $$\begin{aligned}
\label{X_i}
X&:=&\sum_{j\,\in\, \mathbb{Z}_d}
|j+1\rangle\langle j|
\qquad\quad\qquad(X^d=I)\quad\mbox{and}\qquad\\
\label{Z_i}
Z&:=&\sum_{j\,\in\, \mathbb{Z}_d}
\omega^{\,j}\,|j\rangle\langle j|
\qquad\quad\qquad\ (Z^d=I)\end{aligned}$$ thanks to Weyl [@Weyl32] and Schwinger [@Schwinger60], where ${j+1}$ is the modulo-$d$ addition, ${\omega=\exp(\text{i}\tfrac{2\pi}{d})}$, and $\mathbb{Z}_d$ is defined in . Under the operator multiplication, $X$ and $Z$ generate the discrete Heisenberg-Weyl group [@Weyl32; @Durt10]. The group members follow the Weyl commutation relation [@Weyl32] $$\label{Weyl commutation}
Z^zX^x=
\omega^{\,xz}\, X^xZ^z \quad\text{for every}\quad x,z\in\mathbb{Z}_d\,,$$ and the property $$\label{traceless}
\text{tr}(X^xZ^z)= d\,\delta_{x,0}\delta_{z,0}\,.$$
A subset of the Weyl group $$\label{Unitary-Basis}
\mathfrak{B}_{\text{uni}}:=\big\{X^xZ^z:x,z\in\mathbb{Z}_d\big\}$$ forms an orthogonal basis of $\mathscr{B}(\mathscr{H}_d)$, where the orthogonality relation $$\label{orth-XZ}
\big\lgroup X^{x'}Z^{z'},X^xZ^z\big\rgroup_\textsc{hs}=
\text{tr}\big(X^{x-x'}Z^{z-z'}\big)=
d\,\delta_{x,x'}\delta_{z,z'}\qquad$$ is a consequence of [@Schwinger60]. All the elements in basis are unitary operators and traceless \[see \] except the identity operator that corresponds to ${x=0=z}$. Basis is called the *unitary*-basis.
According to and , a statistical operator can be represented as $$\label{rho-in-XZ}
\rho=\tfrac{1}{d}\sum_{x,z\,\in\,\mathbb{Z}_d}
\overline{\langle X^xZ^z\rangle}_\rho\ X^xZ^z$$ in the basis $\mathfrak{B}_{\text{uni}}$. Here, the conditions for normalization and for Hermiticity become ${\langle X^0Z^0\rangle=1}$ and $$\label{<XxZz>}
\overline{\langle X^xZ^z\rangle}=\langle(X^xZ^z)^\dagger\rangle=
\omega^{\,xz}\langle X^{-x}Z^{-z}\rangle\,,$$ respectively. The second equality in is obtained by the virtue of . The inverse of a basis element, ${(X^xZ^z)^\dagger}$, does not always belong to basis but to the Weyl group. Whereas both ${X^xZ^z}$ and ${X^{-x}Z^{-z}}$ are members of $\mathfrak{B}_{\text{uni}}$, and their mean values are related through (in this regard, see also [@Adagger]).
Taking the general form, , one can easily express $\text{tr}(\rho^m)$ in the unitary-basis by using , , and , for example, $$\begin{aligned}
\label{tr_rho2-exp}
\text{tr}(\rho^2)
&=&\tfrac{1}{d}\sum_{x,z}
{|\langle X^xZ^z \rangle|}^{\,2}\quad\mbox{and}\\
\label{tr_rho3-exp}
\text{tr}(\rho^3)&=&\tfrac{1}{d^2}
\sum_{x_1,z_1}\sum\limits_{x_2,z_2}
\langle X^{-x_1}Z^{-z_1} \rangle\,
\langle X^{-x_2}Z^{-z_2} \rangle\times
\nonumber\\
&&\qquad
\langle X^{ x_1+x_2}Z^{ z_1+z_2} \rangle\;
\omega^{z_1(x_1+x_2)+z_2x_2}\,.\quad\quad\end{aligned}$$ Then, one can draw QCs on the expectation values of the Weyl operators from .
In the case of a prime dimensional $d$, the basis $\mathfrak{B}_{\text{uni}}$—without the identity operator—can be divided into ${d+1}$ disjoint subsets $$\begin{aligned}
\label{d+1 subsets}
&&\big\{\mathcal{C}^{(1,z)}\,|\,z\in\mathbb{Z}_d\big\}
\cup\big\{\mathcal{C}^{(0,1)}\big\}
\,,\quad \mbox{where} \\
\label{Cxz}
&&\mathcal{C}^{(x,z)}:=
\big\{ X^{kx} Z^{kz}\,|\,
k\in\mathbb{Z}_d\ \ \mbox{and}\ \ k\neq0\big\}\end{aligned}$$ carries ${d-1}$ pairwise commuting operators [@Bandyopadhyay02; @Englert01]. Hence, one can find a common eigenbasis of the operators in $\mathcal{C}^{(x,z)}$. In fact, there exists a complete set of ${d+1}$ MUBs of $\mathscr{H}_{d}$ [@Ivanovic81; @Wootters89; @Bandyopadhyay02]: $$\label{d+1 bases}
\big\{\mathcal{B}^{\scriptscriptstyle(z)}\,|\,z\in\mathbb{Z}_d\big\}
\cup\big\{\,\mathcal{B}\,\big\}$$ are eigenbases for the subsets in . Our original basis $\mathcal{B}$ in is an eigenbasis of ${Z\in\mathcal{C}^{(0,1)}}$ \[see \]. Let us define the remaining bases as [@Bandyopadhyay02; @Englert01] $$\label{B^z}
\mathcal{B}^{\scriptscriptstyle(z)}:=\big\{\,|z,j\rangle
\,|\,j\in\mathbb{Z}_d\big\}\,,
\ \mbox{where}\ \
XZ^z|z,j\rangle=\omega^j\,|z,j\rangle\,.$$ Eigenvalues of every non-identity ${X^xZ^z}$ are distinct powers of $\omega$ [@Englert01; @Schwinger60].
With an integral power \[obtained by repeatedly using \] $$\label{(X^xZ^z)^k}
(XZ^z)^k=
\omega^{\frac{k(k-1)}{2}z} X^{k}Z^{kz}$$ and the eigenvalue equation in , one can arrive at the spectral decomposition $$\label{X^{k}Z^{kz}}
X^{k}Z^{kz}=\omega^{-\frac{k(k-1)}{2}z}
\sum_{j\,\in\,\mathbb{Z}_d}\,
\omega^{kj}\,
|z,j\rangle\langle z,j|$$ of every operator in the subset $\mathcal{C}^{(1,z)}$. Now taking and , we can pronounce the average values as $$\begin{aligned}
\label{<X^{k}Z^{kz}>}
\langle X^{k}Z^{kz}\rangle_\rho&=&
\omega^{-\frac{k(k-1)}{2}z}
\sum_{j\,\in\,\mathbb{Z}_d}\,
\omega^{kj}\,
p^{\scriptscriptstyle(z)}_j
\quad\mbox{and}\\
\label{<Z^{k}>}
\langle Z^k\rangle_\rho&=&
\sum_{j\,\in\,\mathbb{Z}_d}\,
\omega^{kj}\,p_j\,,
\quad\mbox{where}\\
\label{pj}
p^{\scriptscriptstyle(z)}_j&=&\langle z, j|\,\rho\,|z,j\rangle
\quad\mbox{and}\quad
p_j=\langle j|\,\rho\,|j\rangle\qquad\end{aligned}$$ are the probabilities for projective measurements in ${d+1}$ MUBs . Next, we can rewrite as $$\begin{aligned}
\label{tr_rho2-exp2}
\text{tr}(\rho^2)
&=&\tfrac{1}{d}\Big[1+\sum\limits_{z\,\in\,\mathbb{Z}_d}
\underbrace{\sum\limits_{k=1}^{d-1}
{|\langle X^{k}Z^{kz} \rangle|}^{\,2}}%
_{\textstyle d \sum_{j}\big(p^{\scriptscriptstyle(z)}_j\big)^2-1}+
\underbrace{\sum\limits_{k=1}^{d-1}
{|\langle Z^k \rangle|}^{2}}%
_{\textstyle d\sum_{j}(p_j)^2-1}
\Big]\quad
\nonumber\\
&=&
\label{tr_rho2-p}
\sum_{z\,\in\,\mathbb{Z}_d}\,
\sum_{j\,\in\,\mathbb{Z}_d}
\big(p^{\scriptscriptstyle(z)}_j\big)^2
+
\sum_{j\,\in\,\mathbb{Z}_d}
(p_j)^2-1\,.\end{aligned}$$ Expression is achieved with the help of –, $$\label{sum pj=1}
\sum_{j\,\in\,\mathbb{Z}_d}
p^{\scriptscriptstyle(z)}_j=1=
\sum_{j\,\in\,\mathbb{Z}_d}
p_j$$ \[due to \] for every $z$, and ${\textstyle\sum\nolimits_{k=0}^{d-1} \omega^{k(j-j')}=d\,\delta_{j,j'}}$.
Owing to ${\text{tr}(\rho^2)\leq1}$ \[see \], we reach the *quadratic* QC for the Weyl operators in and thus $$\label{p^2<=1}
\sum_{z\,\in\,\mathbb{Z}_d}\,
\sum_{j\,\in\,\mathbb{Z}_d}
\big(p^{\scriptscriptstyle(z)}_j\big)^2
+
\sum_{j\,\in\,\mathbb{Z}_d}
(p_j)^2\,\leq2$$ for the MUB-probabilities. In [@Larsen90; @Ivanovic92], inequality is achieved from ${\text{tr}(\rho^2)\leq1}$ via a different method (see also [@Klappenecker05]). Using their result, that is , two tight URs are obtained in [@Sanchez-Ruiz95; @Ballester07] for ${d+1}$ MUBs. In the case of ${d=2}$, these relations become and . For the cubic QC due to , we need to express in terms of the probabilities. In the next section, is explicitly given for a qutrit.
Higher degree QCs for the Weyl operators and for the MUBs can be achieved—from —by adopting the general formalism of Sec. \[sec:QC\] like above. The Weyl group exists for every $d$ [@Weyl32; @Durt10; @Englert06], whereas a maximal set of ${d+1}$ MUBs is only known for a prime power dimension [@Wootters89; @Bandyopadhyay02; @Durt10]. MUBs are *optimal* for the quantum state estimation [@Ivanovic81; @Wootters89], where the QCs can be employed for the validation of an estimated state.
Qutrit and spin-1 system {#sec:qutrit}
========================
In the case of ${d\geq3}$, there is a *cubic* QC as a result of . For a qutrit (${d=3}$), let us first express $\text{tr}(\rho^m)$ of for ${m=1,2,3}$: $$\begin{aligned}
\label{tr(rho)}
\text{tr}(\rho)&=&
\texttt{r}_{00}+\texttt{r}_{11}+\texttt{r}_{22}\,,\\
\label{tr(rho^2)}
\text{tr}(\rho^2)&=&
{\texttt{r}_{00}}^2+{\texttt{r}_{11}}^2+{\texttt{r}_{22}}^2+\nonumber\\
&& 2\,\big(\,|\texttt{r}_{01}|^2+|\texttt{r}_{02}|^2+|\texttt{r}_{12}|^2\,\big)\,,\quad\mbox{and}\quad
\\
\label{tr(rho^3)}
\text{tr}(\rho^3)&=&
{\texttt{r}_{00}}^3+{\texttt{r}_{11}}^3+{\texttt{r}_{22}}^3+\nonumber\\
&& 3\,\texttt{r}_{00}\,\big(\,|\texttt{r}_{01}|^2+|\texttt{r}_{02}|^2\,\big)+\nonumber\\
&& 3\,\texttt{r}_{11}\,\big(\,|\texttt{r}_{01}|^2+|\texttt{r}_{12}|^2\,\big)+\nonumber\\
&& 3\,\texttt{r}_{22}\,\big(\,|\texttt{r}_{02}|^2+|\texttt{r}_{12}|^2\,\big)+\nonumber\\
&& 3\,\big(\,\texttt{r}_{01}\,\texttt{r}_{12}\,\texttt{r}_{20}+ \overline{\texttt{r}}_{01}\,\overline{\texttt{r}}_{12}\,\overline{\texttt{r}}_{20}\,\big)\end{aligned}$$ \[for $\texttt{r}_{jk}$, see \]. Here we consider two sets of operators: set of the Weyl operators for a qutrit and a set of spin-1 operators. In the following, we demonstrate: how to achieve $\text{tr}(\rho^m)$, straight from –, in terms of the expectation values of operators in a given set without exploiting their algebraic properties. Then, one gains automatically all the QCs from –.
In and , the Weyl operators are expressed in the linear combinations of operators belong to standard basis . Now we write $$\begin{aligned}
\label{proj-op}
|j\rangle\langle k|
&=&X^j\,|0\rangle\langle 0|\,X^{-k}
=X^j\left[\tfrac{1}{d}
\sum_{z\,\in\,\mathbb{Z}_d} Z^z\,
\right ] X^{-k}\nonumber\\
&=& \tfrac{1}{d}
\sum_{z\,\in\,\mathbb{Z}_d}\, \omega^{-kz}\, X^{j-k}\,Z^z\end{aligned}$$ by using , , and ; see also [@Durt10]. According to Born’s rule , the mean value is a linear function of an operator, so we own every $\texttt{r}_{kj}$ of as a linear sum of ${\langle X^xZ^z\rangle_\rho}$ through . This constitutes a matrix equation such as . By substituting $\texttt{r}_{kj}$ with the associated linear combination in –, one can achieve $\text{tr}(\rho^m)$ in terms of ${\langle X^x\,Z^z\rangle}$ for a qutrit: $$\begin{aligned}
\label{tr_rho3-exp-qutrit}
&&\text{tr}(\rho^3)=\nonumber\\
&&\tfrac{1}{9}\,\big[\,
1+\langle X\rangle^3+ \langle X^2\rangle^3+\langle XZ\rangle^3+
\langle X^2Z^2\,\rangle^3+\nonumber\\
&&
\qquad\quad
\langle XZ^2\rangle^3+ \langle X^2Z\rangle^3+\langle Z\rangle^3+
\langle Z^2\,\rangle^3+\nonumber\\
&&\qquad
6\,\big(\,|\langle X\rangle|^2+|\langle XZ\rangle|^2
+|\langle XZ^2\rangle|^2+|\langle Z\rangle|^2\,\big)
\nonumber\\
&&
-\,3\,\big(
\langle X\rangle\langle XZ\rangle\langle XZ^2\rangle+
\langle X^2\rangle\langle X^2Z\rangle\langle X^2Z^2\rangle+\nonumber\\
&&\qquad
\langle Z\rangle\langle XZ\rangle\langle X^2Z\rangle+
\langle Z^2\rangle\langle XZ^2\rangle\langle X^2Z^2\rangle+\nonumber\\
&&\qquad
\omega\,\langle Z\rangle\langle X^2\rangle\langle XZ^2\rangle+
\omega\,\langle Z^2\rangle\langle X\rangle\langle X^2Z\rangle+\nonumber\\
&&\qquad
\omega^2\langle Z\rangle\langle X\rangle\langle X^2Z^2\rangle+
\omega^2\langle Z^2\rangle\langle X^2\rangle\langle XZ\rangle
\big)
\big]\,,\qquad\quad\end{aligned}$$ where $\omega=\exp(\text{i}\tfrac{2\pi}{3})$, and the term $6(\cdots)$ is $3(3\text{tr}(\rho^2)-1)$. In Sec. \[sec:unitary-basis\], we get and from by exploiting algebraic properties and . One can compare that both the methods deliver the same items.
The next example, a spin-1 particle is a ${d=3}$ levels quantum system (qutrit) if we consider only the spin degree of freedom. Here we take a set of three Hermitian operators from Chap. 7 in [@Peres95]: $$\begin{aligned}
\label{Jx}
J_x&:=&-\text{i}\big(|0\rangle\langle 1|-|1\rangle\langle 0|\big)\,,
\\
\label{Jy}
J_y&:=&-\text{i}\big(|0\rangle\langle 2|-|2\rangle\langle 0|\big)\,,
\quad\mbox{and}
\\
\label{Jz}
J_z&:=&-\text{i}\big(|1\rangle\langle 2|-|2\rangle\langle 1|\big)\,. \end{aligned}$$ They obey the commutation relation ${J_xJ_y-J_yJ_x=\text{i}J_z}$ plus those obtained by the cyclic permutations of ${x,y,z}$, and thus they represent ${\text{spin-}1}$ observables. One can check that ${J_x, J_y,J_z}$ with ${J_x^{\,2},J_y^{\,2},J_z^{\,2}}$ and the anticommutators $$\label{anti-commutators}
K_{xy}=J_xJ_y+J_yJ_x\,,\quad
K_{yz}\quad\mbox{and}\quad\,K_{zx}$$ (attain by the cyclic permutations) constitute a set of nine linearly independent operators, hence they form a Hermitian-basis of $\mathscr{B}(\mathscr{H}_3)$. Though it is not an orthonormal basis with respect to inner product .
One can recognize that ${J_x,J_y,J_z}$ and ${K_{xy},K_{yz},K_{zx}}$ are the Gell-Mann operators [@Gell-Mann61], but ${J_x^{\,2},J_y^{\,2},J_z^{\,2}}$ are not. We want to emphasize that the QCs on their average values can be derived from [@Kimura03; @Byrd03]. So the following analysis is merely an alternative procedure that does not require the Lie algebra of $SU(3)$.
After expressing the elements of standard basis in terms of the spin operators, we can write the average values as $$\begin{aligned}
\label{<J>}
\texttt{r}_{00}&=&\tfrac{1}{2}\left(\hphantom{-}\langle J_x^{\,2}\rangle+
\langle J_y^{\,2}\rangle-\langle J_z^{\,2}\rangle\right)\,,
\nonumber\\
\texttt{r}_{11}&=&\tfrac{1}{2}\left(\hphantom{-}\langle J_x^{\,2}\rangle-
\langle J_y^{\,2}\rangle+\langle J_z^{\,2}\rangle\right)\,,
\nonumber\\
\texttt{r}_{22}&=&\tfrac{1}{2}\left(-\langle J_x^{\,2}\rangle+
\langle J_y^{\,2}\rangle+\langle J_z^{\,2}\rangle\right)\,,
\\
\texttt{r}_{01}&=&\tfrac{1}{2}\left(\hphantom{-}\langle K_{yz}\rangle-
\text{i}\,\langle J_x\rangle\right)=\overline{\texttt{r}}_{10}\,,
\nonumber\\
\texttt{r}_{02}&=&\tfrac{1}{2}\left(-\langle K_{zx}\rangle-
\text{i}\,\langle J_y\rangle\right)=\overline{\texttt{r}}_{20}\,,
\quad\mbox{and}
\nonumber\\
\texttt{r}_{12}&=&\tfrac{1}{2}\left(\hphantom{-}\langle K_{xy}\rangle-
\text{i}\,\langle J_z\rangle\right)=\overline{\texttt{r}}_{21}\,.
\nonumber\end{aligned}$$ This set of equations frames a matrix equation of the kind in . Employing Eqs. , we can rephrase – as $$\begin{aligned}
\label{tr(rho)-J}
\text{tr}(\rho)&=&\tfrac{1}{2}\left(\langle J_x^{\,2}\rangle+
\langle J_y^{\,2}\rangle+\langle J_z^{\,2}\rangle\right)\,,
\\
\label{tr(rho^2)-J}
\text{tr}(\rho^2)&=&-1+
\langle J_x^{\,2}\rangle^2+
\langle J_y^{\,2}\rangle^2+\langle J_z^{\,2}\rangle^2+\nonumber\\
&&\quad
\tfrac{1}{2}\big(\langle J_x\rangle^2+\langle J_y\rangle^2+
\langle J_z\rangle^2+\nonumber\\
&&\qquad
\langle K_{xy}\rangle^2+\langle K_{yz}\rangle^2+\langle K_{zx}\rangle^2
\big)\,,\ \mbox{and}\qquad\
\\
\label{tr(rho^3)-J}
\text{tr}(\rho^3)&=&1-
3\,\langle J_x^{\,2}\rangle\langle J_y^{\,2}\rangle\langle J_z^{\,2}\rangle+
\nonumber\\
&&\
\tfrac{3}{4}\,\big[\,
\left(\langle J_x\rangle^2+\langle K_{yz}\rangle^2\right)\langle J_x^{\,2}\rangle+
\nonumber\\
&&\quad\ \
\left(\langle J_y\rangle^2+\langle K_{zx}\rangle^2\right)\langle J_y^{\,2}\rangle+
\nonumber\\
&&\quad\ \
\left(\langle J_z\rangle^2+\langle K_{xy}\rangle^2\right)\langle J_z^{\,2}\rangle
\nonumber\\
&&\
-\,\langle K_{xy}\rangle\langle K_{yz}\rangle\langle K_{zx}\rangle+
\langle J_{x}\rangle\langle K_{xy}\rangle\langle J_{y}\rangle
\nonumber\\
&&\quad
+\,\langle J_{y}\rangle\langle K_{yz}\rangle\langle J_{z}\rangle+
\langle J_{z}\rangle\langle K_{zx}\rangle\langle J_{x}\rangle
\,\big]\,.\end{aligned}$$ Here, in each example, one can clearly perceive $\text{tr}(\rho^2)$ and $\text{tr}(\rho^3)$ as quadratic and cubic polynomials of the mean values. Plugging – in –, one captures all the—linear, quadratic, and cubic—QCs for the spin-1 operators. The linear constraint ${\langle J_x^{\,2}+J_y^{\,2}+J_z^{\,2}\rangle=\langle 2I\rangle}$ is used to get and in the above forms.
Now, let us call $J_x,J_y,J_z,K_{yz},K_{zx},K_{xy},
J_x^{\,2},J_y^{\,2},J_z^{\,2}$ as ${A_1,\cdots,A_9}$, respectively. In this case, every pure state $\rho_\text{pure}=|\psi\rangle\langle\psi|$ \[for ${|\psi\rangle}$, see \] of a qutrit delivers an extreme point of the allowed region $\mathcal{E}$, and the extreme points can be parameterized as $$\begin{aligned}
\label{J-pure}
\langle A_1\rangle_{\rho_\text{pure}}&=&
\sin 2\theta_0 \cos\theta_1 \sin\phi_1\,,
\nonumber\\
\langle A_2\rangle_{\rho_\text{pure}}&=&
\sin 2\theta_0 \sin\theta_1 \sin\phi_2\,,
\nonumber\\
\langle A_3\rangle_{\rho_\text{pure}}&=&
-(\sin\theta_0)^2 \sin 2\theta_1 \sin(\phi_1-\phi_2)\,,
\nonumber\\
%
\langle A_4\rangle_{\rho_\text{pure}}&=&
\sin 2\theta_0 \cos\theta_1 \cos\phi_1\,,
\nonumber\\
\langle A_5\rangle_{\rho_\text{pure}}&=&
-\sin 2\theta_0 \sin\theta_1 \cos\phi_2\,,
\\
\langle A_6\rangle_{\rho_\text{pure}}&=&
(\sin\theta_0)^2 \sin 2\theta_1 \cos(\phi_1-\phi_2)\,,
\nonumber\\
%
\langle A_7\rangle_{\rho_\text{pure}}&=&
(\cos\theta_0)^2+(\sin\theta_0)^2(\cos\theta_1)^2\,,
\nonumber\\
\langle A_8\rangle_{\rho_\text{pure}}&=&
(\cos\theta_0)^2+(\sin\theta_0)^2(\sin\theta_1)^2\,,
\quad\mbox{and}\qquad
\nonumber\\
\langle A_9\rangle_{\rho_\text{pure}}&=&
(\sin\theta_0)^2\,,
\nonumber\end{aligned}$$ where ${\theta_0,\theta_1\in[0,\tfrac{\pi}{2}]}$ and ${\phi_1,\phi_2\in[0,2\pi)}$. By putting expectation values in –, one can verify that ${\text{tr}(\rho_\text{pure}^{\,m})=1}$ for all $m=1,2,$ and 3.
The minimum and maximum eigenvalues of everyone in ${\{A_1,\cdots,A_6\}}$ are $-1$ and $+1$ and of each one in ${\{A_7,A_8,A_9\}}$ are 0 and $1$, respectively. Taking –, we formulate uncertainty or certainty measures for $\{A_i\}_{i=1}^9$, and a few combined measures are listed in $$\begin{aligned}
\label{H-J}
6\ln 2 &\leq& \sum_{i=1}^{9}H(\langle A_i\rangle)\,,
\\
\label{u1/2-J}
3 + 6 \sqrt{2} &\leq& \sum_{i=1}^{9}u_{\sfrac{1}{2}}(\langle A_i\rangle)\,,
\\
\label{u2-J}
&& \sum_{i=1}^{9}u_{2}(\langle A_i\rangle)\leq 6\,,\quad\mbox{and}
\\
\label{umax-J}
&& \sum_{i=1}^{9}u_{\text{max}}(\langle A_i\rangle)\leq 6.51702\,.\end{aligned}$$ As described in Sec. \[sec:QC\], we find the absolute minimum of a concave function and maximum of a convex function by putting in the above functions and changing the four parameters $\theta$’s and $\phi$’s. As a result, we achieve tight URs and and CRs and for the nine spin-1 observables. The basis ${\mathcal{B}=\{|0\rangle,|1\rangle,|2\rangle\}}$ in is a common eigenbasis of ${\{A_7,A_8,A_9\}}$, a qutrit’s state $\rho=|j\rangle\langle j|$ that corresponds to a ket in $\mathcal{B}$ saturates inequalities –. One pure state that saturates CR , the corresponding parameters are $$\begin{aligned}
\label{umax-ang}
\theta_0 = 0.482720\,,&&\quad
\theta_1 = 0.785398\,,\nonumber\\
\phi_1 = 2.520428\,,&&\quad
\phi_2 = 3.762757\,.\end{aligned}$$
Since the square of every operator in the set ${\{A_i\}_{i=1}^9}$ lies in the set, $$\begin{aligned}
\label{A^2}
(A_1)^2&=&(A_4)^2=(A_7)^2=A_7\,,\nonumber\\
(A_2)^2&=&(A_5)^2=(A_8)^2=A_8\,,\quad\mbox{and}\\
(A_3)^2&=&(A_6)^2=(A_9)^2=A_9\,,\nonumber\end{aligned}$$ a sum of (the square of) the standard deviations ${\Delta A_i}$ \[see \] acts as a concave function on the allowed region for the set. As above we reach the global minima and thus establish the tight URs $$\begin{aligned}
\label{std-J 9}
4 &\leq& \sum_{i=1}^{9}\Delta A_i\,,\\
\label{std-J 6}
1 + 2\sqrt{2} &\leq& \sum_{i=1}^{6}\Delta A_i\,,
\\
\label{aq-std-J}
\tfrac{10}{3} &\leq& \sum_{i=1}^{9}\big(\Delta A_i\big)^2\,,
\quad\mbox{and}\quad
\tfrac{8}{3} \leq \sum_{i=1}^{6}\big(\Delta A_i\big)^2\,.
\qquad\quad\end{aligned}$$ URs and are saturated by the eigenstates of $A_i$, $i=1,\cdots,6$, associated with 0 and the non-zero eigenvalues, respectively. The null-space (eigenspace associated with 0) of $A_i$ is the linear span of a ket in $\mathcal{B}$. The equal superposition kets ${\tfrac{1}{\sqrt{3}}(|0\rangle+e^{\text{i}\phi_1}|1\rangle+e^{\text{i}\phi_2}|2\rangle)}$ provide the minimum uncertainty (pure) states for both the URs in .
Spin-$\mathsf{j}$ operators {#sec:spin-j}
===========================
A spin- particle is a quantum system of ${d=2\,\mathsf{j}+1}$ levels provided we consider only the spin degree of freedom, and can be ${\tfrac{1}{2}, 1,\tfrac{3}{2},2,\cdots\,}$. Let us take the spin- operators ${J_x=\tfrac{1}{2}(J_++J_-)}$, ${J_y=\tfrac{1}{2\text{i}}(J_+-J_-)}$, and $J_z$ whose actions on the eigenbasis ${\{|\mathsf{m}\rangle : \mathsf{m=j,j}-1,\cdots,-\mathsf{j}\}}$ of $J_z$ are described as $$\begin{aligned}
\label{JpmJz}
J_\pm\,|\mathsf{m}\rangle&=&
\sqrt{\mathsf{(j\mp m)(j\pm m}+1)}\;
|\mathsf{m}\pm 1\rangle\quad\mbox{and}\quad\\
J_z\,|\mathsf{m}\rangle&=&\mathsf{m}\,|\mathsf{m}\rangle\,.\end{aligned}$$ For ${\mathsf{j}=\tfrac{1}{2}}$, the vector operator ${\vec{J}:=(J_x,J_y,J_z)}$ is the same as the Pauli vector operator ${\vec{\sigma}:=(X,Y,Z)}$ in Appendix up to a factor $\tfrac{1}{2}$. In –, the spin-1 operators are represented in the common eigenbasis $\mathcal{B}$ of ${\{J_x^{\,2},J_y^{\,2},J_z^{\,2}\}}$.
The permitted region $\mathcal{E}$ for the three spin-observables is bounded by the QC $$\label{JxJyJz,QC}
{\langle J_x\rangle}^2+{\langle J_y\rangle}^2+{\langle J_z\rangle}^2\leq \mathsf{j}^{\,2}\,,$$ which says that the length of the vector ${(\langle J_x\rangle,\langle J_y\rangle,\langle J_z\rangle)}$ cannot be more than $\mathsf{j}$ [@Atkins71]. So $\mathcal{E}$ is the closed ball of radius $\mathsf{j}$ in hyperrectangle that is the cube $[-\mathsf{j},\mathsf{j}]^{\times 3}$ here. Note that, except ${\mathsf{j}=\tfrac{1}{2}}$, an interior point of $\mathcal{E}$ corresponds to *not one but many* (pure as well as mixed) quantum states. However, every extreme point of $\mathcal{E}$ comes from a unique pure state ${\chi(\alpha,\beta)=|\alpha,\beta\rangle\langle\alpha,\beta|}$, where $$\label{bloch-ket}
|\alpha,\beta\rangle=\sum_{\mathsf{m=-j}}^{\mathsf{j}}
{\scriptstyle\sqrt{\tfrac{(2\mathsf{j})!}{(\mathsf{j+m})!\,(\mathsf{j-m})!}}
\left(\cos\tfrac{\alpha}{2}\right)^{\mathsf{j+m}}
\left(\sin\tfrac{\alpha}{2}\right)^{\mathsf{j-m}}
e^{-\text{i}\mathsf{m}\beta}}|\mathsf{m}\rangle$$ is known as the angular momentum (or atomic) coherent state-vector [@Atkins71; @Arecchi72]. With ${J_x^{\,2}+J_y^{\,2}+J_z^{\,2}=\mathsf{j(j}+1)I}$, QC can be turned into a tight UR $$\label{JxJyJz,UR}
\mathsf{j}\leq
(\Delta J_x)^2+(\Delta J_y)^2+(\Delta J_z)^2\,,$$ for which all the coherent states are the minimum uncertainty states (see Chap. 10 in [@Peres95]). UR is also captured in [@Larsen90; @Abbott16; @Hofmann03; @Dammeier15]. In fact, can also be interpreted as CR because on the left-hand-side there is a convex function of the expectation values. In [@Dammeier15], $(\Delta\, \widehat{\eta}.\vec{J}\,)^2$ is studied as a function of the unit vector ${\widehat{\eta}\in\mathbb{R}^3}$ for a fixed state $\rho$, and then the uncertainty regions of ${((\Delta J_x)^2,(\Delta J_y)^2,(\Delta J_z)^2)}$ are plotted by taking all $\rho$’s. Various URs are also obtained there for the three operators ${J_x,J_y,}$ and $J_z$. Our regions $\mathcal{E}$ and $\mathcal{R}$’s are different from the uncertainty regions: $\mathcal{E}$ and $\mathcal{R}$ are in the space of expectation values, and both are convex sets.
We can parametrize the extreme points of $\mathcal{E}$ as $$\begin{aligned}
\label{JxJyJz-para}
\langle\alpha,\beta|J_x|\alpha,\beta\rangle&=&\mathsf{j}\,\sin\alpha\cos\beta\,,
\nonumber\\
\langle\alpha,\beta|J_y|\alpha,\beta\rangle&=&\mathsf{j}\,\sin\alpha\sin\beta\,,
\\
\langle\alpha,\beta|J_z|\alpha,\beta\rangle&=&\mathsf{j}\,\cos\alpha\,,
\nonumber\end{aligned}$$ where ${\alpha\in[0,\pi]}$ and ${\beta\in[0,2\pi)}$, and can define different uncertainty or certainty measures on $\mathcal{E}$ using –. Since the minimum and maximum eigenvalues of $J_i$ for every ${i=x,y,z}$ are $-\mathsf{j}$ and $+\mathsf{j}$, respectively, $$\label{Jdot}
\langle\dot{J}_i\,\rangle=
\tfrac{1}{2}\big(1-\tfrac{\langle J_i\rangle}{\mathsf{j}}\big)
\quad\mbox{and}\quad
\langle\mathring{J}_i\,\rangle=
\tfrac{1}{2}\big(1+\tfrac{\langle J_i\rangle}{\mathsf{j}}\big)\,,$$ which are functions of $\alpha$ and $\beta$ on the sphere specified by . By varying the two angles we reach the tight lower and upper bounds of the uncertainty and certainty measures presented as follows $$\begin{aligned}
\label{H-UR-JxJyJz}
2\ln 2&\leq&\sum_{i=x,y,z}H(\langle J_i\rangle)\,,\\
\label{H2-UR-JxJyJz}
3\ln(\tfrac{3}{2})&\leq&\sum_{i=x,y,z}H_2(\langle J_i\rangle)\,,\\
\label{u-UR-JxJyJz}
1+2\sqrt{2}&\leq& \sum_{i=x,y,z}
u_{\sfrac{1}{2}}(\langle J_i\rangle)\,,\\
\label{u2-UR-JxJyJz}
&& \sum_{i=x,y,z}u_2(\langle J_i\rangle)\leq 2 \,,
\quad \mbox{and}\qquad\\
\label{umax-UR-JxJyJz}
&& \sum_{i=x,y,z}u_\text{max}(\langle J_i\rangle)\leq
\tfrac{1}{2}(3+\sqrt{3}) \,,\end{aligned}$$ where ${H_2=-\ln(u_2)}$ is like the Rényi entropy [@Renyi61] of order 2.
![From top-left to bottom-right, along the rows, the first region is the permissible region $\mathcal{E}$ bounded by QC and the second one is $\mathcal{R}^{\text{spin}}_H$. The third and fourth regions are $\mathcal{R}^{\text{spin}}_{H_2}$ and $\mathcal{R}^{\text{spin}}_{u_{\text{max}}}$, respectively. Although these regions are plotted for ${\mathsf{j}=2}$, they will be of the same shapes in the cube $[-\mathsf{j},\mathsf{j}]^{\times 3}$ for other $\mathsf{j}$-values.[]{data-label="fig:regions-MUBs"}](fig2){width="45.00000%"}
All – hold for every ${\mathsf{j}=\tfrac{1}{2}, 1,\tfrac{3}{2},2,\cdots\,}$ and hence in every dimension ${d=2\,\mathsf{j}+1}$, and they are saturated by some angular momentum coherent states $\chi(\alpha,\beta)$. Like , the regions characterized by URs , , and CR are denoted here by $\mathcal{R}^{\text{spin}}_H$, $\mathcal{R}^{\text{spin}}_{H_2}$, and $\mathcal{R}^{\text{spin}}_{u_{\text{max}}}$, respectively. Along with $\mathcal{E}$, they are displayed in Fig. \[fig:regions-MUBs\] for ${\mathsf{j}=2}$. $\mathcal{E}$ resides in every $\mathcal{R}$, and one can also perceive that ${\mathcal{R}^{\text{spin}}_{H_2}\subset\mathcal{R}^{\text{spin}}_{u_{\text{max}}}}$. We can not right away say which of the tight URs, or , is superior because neither $\mathcal{R}^{\text{spin}}_H$ is completely contained in $\mathcal{R}^{\text{spin}}_{H_2}$ nor vice versa. Similarly, it is difficult to compare and as ${\mathcal{R}^{\text{spin}}_H\nsubset\mathcal{R}^{\text{spin}}_{u_{\text{max}}}}$ and ${\mathcal{R}^{\text{spin}}_H\nsupset\mathcal{R}^{\text{spin}}_{u_{\text{max}}}}$. If one region is not a subset of other then one can take the area of a region as a figure of merit to compare different CRs and/or URs. However, in the paper, mostly those cases are reported where one region is completely submerged in another.
Since and are the same, every angular momentum coherent state saturates . $\mathcal{E}$ touches the periphery of $\mathcal{R}^{\text{spin}}_H$ at six different points that are related to eigenstates of $J_x,J_y,J_z$ corresponding to their extreme-eigenvalues $\pm\,\mathsf{j}$. These six pure states are only the minimum uncertainty states for UR as well as UR . The eight coherent states ${\chi(\alpha,\beta)}$—for which $\alpha=\arccos(\tfrac{1}{\sqrt{3}})$ and ${\beta=\tfrac{\pi}{4},\tfrac{3\pi}{4},\tfrac{5\pi}{4},\tfrac{7\pi}{4}}$, and the remaining four can be obtained by changing $\alpha$ into ${\pi-\alpha}$ and $\beta$ into ${\pi+\beta\, (\text{mod}\, 2\pi)}$—saturate inequalities and . The permitted region $\mathcal{E}$ touches the boundary of $\mathcal{R}^{\text{spin}}_{H_2}$ and $\mathcal{R}^{\text{spin}}_{u_{\text{max}}}$ at the associated eight points. The six cross sections in $\mathcal{R}^{\text{spin}}_{H_2}$ and $\mathcal{R}^{\text{spin}}_{u_{\text{max}}}$ are due to ${-\mathsf{j}\leq\langle J_i\rangle\leq\mathsf{j}}$ required for every ${i=x,y,z}$.
In the case of ${\mathsf{j}=\tfrac{1}{2}}$, and are equal, $\mathcal{E}$ is the Bloch ball, and all the coherent states become qubit’s pure states. Corresponding to the eight minimum uncertainty states for UR , the Bloch vectors are ${\{\pm\widehat{a}_i\}_{i=1}^{4}}$ [@Wootters07; @Appleby14], where $$\label{a1a2a3-v1v2v3}
\begin{pmatrix}
\widehat{a}_1\\
\widehat{a}_2\\
\widehat{a}_3
\end{pmatrix}
= \tfrac{1}{\sqrt{3}}
\begin{pmatrix}
\hphantom{-}1 & -1 & -1 \\
-1 & \hphantom{-}1 & -1 \\
-1 & -1 & \hphantom{-}1
\end{pmatrix}
\begin{pmatrix}
\widehat{v}_1\ \\
\widehat{v}_2\ \\
\widehat{v}_3\
\end{pmatrix}$$ and ${\widehat{a}_4=
-\textstyle\sum\nolimits_{i=1}^{3}\widehat{a}_i}$ are given in the $v$-coordinate system \[see Appendix\]. One can easily deduce that both ${\{\widehat{\mathsf{a}}_i\}_{i=1}^{3}}$ presented in Appendix \[subsec:3settings\] and ${\{\widehat{a}_i\}_{i=1}^{3}}$ share the same Gram matrix, . The two sets of vectors are related by an invertible linear transformation that can be obtained by $\textbf{M}$ in and the square matrix in . Each of ${\{\widehat{\mathsf{a}}_i\}_{i=1}^{4}}$ and ${\{\widehat{a}_i\}_{i=1}^{4}}$ constitutes a SIC-POVM via for a qubit [@Rehacek04], and the later one is known as the Weyl-Heisenberg covariant SIC-POVM [@Appleby09].
Conclusion and outlook {#sec:conclusion}
======================
There are three primary contributions from this article. First, we provided a basis-independent systematic procedure to obtain the QCs for any set of operators that act on a qudit’s Hilbert space. The QCs are necessary and sufficient restrictions that analytically specify the permitted region $\mathcal{E}$ of the expectation values. Second, we showed how to define uncertainty and certainty measures on the allowed region $\mathcal{E}$, and their properties are discussed. With a straightforward mechanism—that is also employed in [@Riccardi17; @Sehrawat17]—we achieved tight CRs and URs. Third, we bounded a regions $\mathcal{R}$ by a tight CR or UR in the space of expectation values and exhibited the gap $\mathcal{R}\setminus\mathcal{E}$ between $\mathcal{R}$ and the allowed region $\mathcal{E}$ through figures. Our additional contributions are: (*i*) the QCs for the Weyl operators and the spin observables are reported. (*ii*) Various tight URs and CRs are obtained for the spin-1 observables as well as for ${\{J_x,J_y,J_z\}}$ in the case of an arbitrary spin ${\mathsf{j}=\tfrac{1}{2}, 1,\tfrac{3}{2},2,\cdots\,}$. Since all the extreme points of the permissible region for ${\{J_x,J_y,J_z\}}$ come from the angular momentum coherent states, always a coherent state is a minimum uncertainty state for the UR formulated for the three observables. (*iii*) The case of a single qubit is thoroughly investigated in Appendix \[sec:qubit\] that includes Schrödinger’s UR, and tight URs CRs are presented there for the SIC-POVM.
Choice of an uncertainty measure to get a UR is a user’s choice. We have not yet found a single certainty or uncertainty measure that is better than others in the sense that it always provides a smaller region $\mathcal{R}$. In some examples, one behaves better, whereas in another example there is another. To compare different CRs and/or URs, the area (or volume) of $\mathcal{R}$ can be a figure of merit, particularly when one region is not contained in another. Although, it is not easy to compute such an area.
Naturally, $\mathcal{E}$ lies in all such $\mathcal{R}$’s, however it is not a primary objective of a UR to put a constraint on the mean values but on a combined uncertainty. To draw a comparison between the QCs and URs, first, we have to put them on an equal footing. That may or may not be possible because a QC is primarily a bound on expectation values not, generally, on a combined uncertainty.
URs play very important roles in different branches of physics and mathematics, recently they are applied in the field of quantum information (see Sec. VI in [@Coles17]). One can employ the QCs for those purposes as well as for the quantum state estimation [@Paris04], where one can directly appoint the QCs for the validation of an estimated state.
I am very grateful to Titas Chanda for crosschecking the numerical results.
Qubit {#sec:qubit}
=====
For a qubit (${d=2}$), the Pauli operators ${X,Y,Z}$ [@Pauli27] with the identity operator $I$ constitute the Hermitian-basis of $\mathscr{B}(\mathscr{H}_2)$ [@Kimura03; @Byrd03]. The operators $X$ and $Z$ are defined in and , respectively, and ${Y=\text{i}XZ}$. In this basis, we can express qubit’s state as $$\label{rho-in-IXYZ}
\rho=\tfrac{1}{2}\big(I+\langle X\rangle\, X +\langle Y\rangle\, Y
+\langle Z\rangle\, Z\big)\,,$$ where ${\vec{r}:=(\langle X\rangle, \langle Y\rangle,\langle Z\rangle)\in\mathbb{R}^3}$ is the well-known Bloch vector [@Bloch46; @Bengtsson06] that is the mean value of the Pauli vector operator ${\vec{\sigma}=(X,Y,Z)}$. Conditions and now become $\langle I\rangle_\rho=1$ and $$\label{|r|^2<=1}
r^2:=|\vec{r}\,|^2=\langle X\rangle_\rho^2+\langle Y\rangle_\rho^2+\langle Z\rangle_\rho^2
\leq 1\,,$$ respectively.
A projective measurement on a qubit can be completely specified by a three-component real unit vector [@A]. So, we begin with three linearly independent unit vectors ${\widehat{a},\widehat{b},\widehat{c}\in\mathbb{R}^3}$, and define three Hermitian operators $$\begin{aligned}
\label{A-qubit}
A&\,:=\,&\widehat{a}\cdot\vec{\sigma}\,:=\,2\,|a\rangle\langle a|-I\,,
\\
\label{B-qubit}
B&\,:=\,&\widehat{b}\cdot\vec{\sigma}\,:=\,2\,|b\rangle\langle b|-I\,,
\quad\mbox{and}
\\
\label{C-qubit}
C&\,:=\,&\widehat{c}\cdot\vec{\sigma}\,:=\,2\,|c\rangle\langle c|-I\,.\end{aligned}$$ One can check that ${A^2=I}$ \[with \], hence its eigenvalues are $\pm1$, and then ${\langle A\rangle_\rho\in[-1,1]}$ is due to . By definition , ${|a\rangle}$ and ${|a^\perp\rangle}$ (such that ${\langle a|a^\perp\rangle=0}$) are eigenkets of $A$ corresponding to the eigenvalues ${+1}$ and ${-1}$, respectively, and similarly for $B$ and $C$.
One can verify that the inner product between a pair of such operators is $$\label{angle-bt-axis}
\lgroup A,B\,\rgroup_\textsc{hs}=
2\;\widehat{a}\cdot\widehat{b}=
4\;{|\langle a|b\rangle|}^2-2$$ by using $$\begin{aligned}
\label{AB}
&&AB=(\widehat{a}\cdot\vec{\sigma})(\widehat{b}\cdot\vec{\sigma})
=(\widehat{a}\cdot\widehat{b})\,I+\text{i}(\widehat{a}\times\widehat{b})
\cdot\vec{\sigma}\,,\quad\\
\label{tr(pauli)}
&&\text{tr}(X)=\text{tr}(Y)=\text{tr}(Z)=0\,,
\quad\mbox{and}\quad
\text{tr}(I)=d=2\,,\qquad\end{aligned}$$ where ${\widehat{a}\cdot\widehat{b}}$ and ${\widehat{a}\times\widehat{b}}$ are the dot and cross product. Taking the statistical operator from and applying and to the Born rule, , one can get the mean values $$\begin{aligned}
\label{<A>-qubit}
\langle A\rangle_\rho&=&\widehat{a}\cdot\vec{r}=2p-1\,,
\\
\label{<B>-qubit}
\langle B\rangle_\rho&=&\widehat{b}\cdot\vec{r}=2q-1\,,
\quad\mbox{and}
\\
\label{<C>-qubit}
\langle C\rangle_\rho&=&\widehat{c}\cdot\vec{r}=2s-1\,,\end{aligned}$$ where $$\label{pqs}
p=\langle a|\rho|a\rangle\,,\quad
q=\langle b|\rho|b\rangle\,,\quad\mbox{and}\quad
s=\langle c|\rho|c\rangle\,\quad$$ are the probabilities \[see and \] associated (with $+1$ eigenvalue) to the three projective measurements. The probabilities $p,q,$ and $s$ are the mean values of three rank-1 projectors $$\label{proj}
|a\rangle\langle a|=:P\,,\quad
|b\rangle\langle b|=:Q\,,\quad\mbox{and}\quad
|c\rangle\langle c|\,.\quad$$
![() depicts linearly independent unit vectors ${\widehat{a},\widehat{b},\widehat{c}}$ and the orthonormal set ${\{\widehat{v}_1,\widehat{v}_2,\widehat{v}_3\}\in\mathbb{R}^3}$. There, a dotted line illustrates an orthogonal projection of one vector onto another, which is an integral part of the Gram-Schmidt process. () exhibits the Bloch vector $\vec{r}$, a line segment parallel to $\widehat{v}_3$ in the Bloch sphere, and the great circle in $v_1v_2$-plane. []{data-label="fig:abc-vectors"}](fig3){width="45.00000%"}
By applying the Gram-Schmidt orthonormalization process, we can turn the linearly independent set ${\{\widehat{a},\widehat{b},\widehat{c}\,\}}$ into an orthonormal set ${\{\widehat{v}_1,\widehat{v}_2,\widehat{v}_3\}}$ of vectors; they are portrayed in Fig. \[fig:abc-vectors\] (). The two sets are related through the transformation $$\label{abc-v1v2v3}
\begin{pmatrix}
\widehat{v}_1\\
\widehat{v}_2\\
\widehat{v}_3
\end{pmatrix}
=
\underbrace{\begin{pmatrix}
1 & 0 & 0 \\
-\tfrac{\widehat{a}.\widehat{b}}
{\scriptstyle\sqrt{1-(\widehat{a}.\widehat{b})^2}}&
\tfrac{1}
{\scriptstyle\sqrt{1-(\widehat{a}.\widehat{b})^2}} & 0 \\
-\tfrac{e}{g} & -\tfrac{f}{g} & \tfrac{1}{g} \\
\end{pmatrix}}_{\displaystyle\textbf{M}^{-1}}
\begin{pmatrix}
\widehat{a}\\
\widehat{b}\\
\widehat{c}
\end{pmatrix}\,,$$ where $$\begin{aligned}
\label{e}
e&=&\tfrac{\widehat{a}.\widehat{c}-
(\widehat{a}.\widehat{b})(\widehat{b}.\widehat{c})}
{1-(\widehat{a}.\widehat{b})^2}\,,
\\
\label{f}
f&=&\tfrac{\widehat{b}.\widehat{c}-
(\widehat{a}.\widehat{b})(\widehat{a}.\widehat{c})}
{1-(\widehat{a}.\widehat{b})^2}\,,\quad\mbox{and}
\\
\label{g}
g&=&\sqrt{
\tfrac{1-(\widehat{a}.\widehat{b})^2-(\widehat{a}.\widehat{c})^2-
(\widehat{b}.\widehat{c})^2+
2\,(\widehat{a}.\widehat{b})(\widehat{a}.\widehat{c})
(\widehat{b}.\widehat{c})}
{1-(\widehat{a}.\widehat{b})^2}}\,.\end{aligned}$$ We can convert into $$\label{ABC-v1v2v3}
\underbrace{\begin{pmatrix}
\langle A\rangle\\
\langle B\rangle\\
\langle C\rangle
\end{pmatrix}}_{\displaystyle\textsf{E}}
=
\underbrace{\begin{pmatrix}
1 & 0 & 0 \\
{\scriptstyle\widehat{a}.\widehat{b}}&
{\scriptstyle\sqrt{1-(\widehat{a}.\widehat{b})^2}} & 0 \\
e+f({\scriptstyle\widehat{a}.\widehat{b}}) &
f{\scriptstyle\sqrt{1-(\widehat{a}.\widehat{b})^2}} &
g \\
\end{pmatrix}}_{\displaystyle\textbf{M}}
\underbrace{\begin{pmatrix}
\widehat{v}_1.\vec{r}\ \\
\widehat{v}_2.\vec{r}\ \\
\widehat{v}_3.\vec{r}\
\end{pmatrix}}_{\displaystyle\texttt{R}}\,,$$ which is like Eq. . One can perceive that $\texttt{R}$ is real and it is the representation of Bloch vector $\vec{r}$ in the $v$-coordinate system (made of ${\widehat{v}_1,\widehat{v}_2,\widehat{v}_3}$) \[see Fig. \[fig:abc-vectors\] ()\]. From top to bottom, the rows in $\textbf{M}$ are the representations of $\widehat{a}$, $\widehat{b}$, and $\widehat{c}$ in the $v$-coordinate system. Next, one can verify that $$\label{Gram-matrix}
\textbf{M}\,\textbf{M}^\intercal=
\begin{pmatrix}
1 & {\scriptstyle\widehat{a}.\widehat{b}} & {\scriptstyle\widehat{a}.\widehat{c}}
\\
{\scriptstyle\widehat{a}.\widehat{b}}&
1 & {\scriptstyle\widehat{b}.\widehat{c}} \\
{\scriptstyle\widehat{a}.\widehat{c}} & {\scriptstyle\widehat{b}.\widehat{c}} & 1 \\
\end{pmatrix}
=:\textbf{G}$$ is the Gram matrix. Recall that $\intercal$ symbolizes the transpose.
After associating the Pauli operators with the orthonormal vectors as $$\label{v-sigma}
\widehat{v}_1\cdot\vec{\sigma}:=X\,,\quad
\widehat{v}_2\cdot\vec{\sigma}:=Y\,,\quad\mbox{and}\quad
\widehat{v}_3\cdot\vec{\sigma}:=Z\,,$$ condition emerges as $$\label{|r|^2<=1-V}
r^2=
(\widehat{v}_1\cdot\vec{r}\,)^2+
(\widehat{v}_2\cdot\vec{r}\,)^2+(\widehat{v}_3\cdot\vec{r}\,)^2
=\texttt{R}^\intercal\texttt{R}\leq 1\,.$$ And, with the matrix equation ${\textbf{M}^{-1}\textsf{E}=\texttt{R}}$—gained from or —we achieve the *quadratic* QC $$\begin{aligned}
\label{EGE<=1}
\textsf{E}^\intercal
\underbrace{(\textbf{M}^{-1})^\intercal\,\textbf{M}^{-1}}_{\displaystyle\textbf{G}^{-1}}
\textsf{E}
=\texttt{R}^\intercal\texttt{R}\leq 1\,,\end{aligned}$$ where $$\label{G^-1}
\textbf{G}^{-1}=
\tfrac{1}{\text{det}(\textbf{G})}
\begin{pmatrix}
{\scriptstyle 1\,-\,(\widehat{b}.\widehat{c})^2} & {\scriptstyle(\widehat{a}.\widehat{c})(\widehat{b}.\widehat{c})-
\widehat{a}.\widehat{b}}&
{\scriptstyle(\widehat{a}.\widehat{b})(\widehat{b}.\widehat{c})-\widehat{a}.\widehat{c}}
\\
{\scriptstyle(\widehat{a}.\widehat{c})(\widehat{b}.\widehat{c})
-\widehat{a}.\widehat{b}}&
{\scriptstyle 1\,-\,(\widehat{a}.\widehat{c})^2} & {\scriptstyle(\widehat{a}.\widehat{b})(\widehat{a}.\widehat{c})\,-\,\widehat{b}.\widehat{c}}
\\
{\scriptstyle(\widehat{a}.\widehat{b})(\widehat{b}.\widehat{c})-\widehat{a}.\widehat{c}} &
{\scriptstyle(\widehat{a}.\widehat{b})(\widehat{a}.\widehat{c})\,-\,\widehat{b}.\widehat{c}} &
{\scriptstyle 1\,-\,(\widehat{a}.\widehat{b})^2}
\end{pmatrix}$$ and ${\text{det}(\textbf{G})=({\scriptstyle 1\,-\,(\widehat{a}.\widehat{b})^2})\,g^2}$ \[for $g$, see \]. $\textbf{G}^{-1}$ does exist for linearly independent vectors ${\widehat{a},\widehat{b},\widehat{c}}$, otherwise see Appendix \[subsec:2settings\]. One can observe that the matrices **M** and **G** are independent of $\rho$ and only depend on the three operators (measurement settings).
The quadratic QC in characterizes the permissible region $\mathcal{E}$ \[defined in \] of expectation values –. The linear transformation in maps the Bloch sphere identified by the equality in onto an ellipsoid [@Meyer00]. So, for a qubit, the allowed region $\mathcal{E}$ will always be an ellipsoid with its interior [@Kaniewski14]. We want to emphasize that all the material between and is given in a general form in [@Kaniewski14; @Kaniewski; @Meyer00]. It is shown in [@Kaniewski14] that there is a one-to-one correspondence between a qubit’s state $\rho\in\mathcal{S}$ \[defined in \] and a point in $\mathcal{E}$ as long as **M** is full rank. That can be witnessed through Eq. .
The ellipsoid can be parametrized by putting $$\label{Rpure}
\texttt{R}^\intercal_{\text{pure}}=(\sin2\theta\cos\phi\,,\
\sin2\theta\sin\phi\,,\
\cos2\theta)$$ in , where ${\theta\in[0,\tfrac{\pi}{2}]}$ and ${\phi\in[0,2\pi)}$. If we put ${r\texttt{R}^\intercal_{\text{pure}}}$ in —where ${r\in[0,1]}$ is given in —then we can also reach its interior points. The column vector $\texttt{R}_{\text{pure}}$ is associated with $\texttt{R}^{(\text{pure})}_2$ of . For this section, the subscripts of ${\theta_0}$ and ${\phi_1}$ are dropped.
The real symmetric matrix $\textbf{G}$ can be diagonalized with an orthogonal matrix **O**, hence ${\textbf{O}^\intercal \textbf{G}\, \textbf{O}}$ will be a diagonal matrix with entries $\lambda_1$, $\lambda_2$, and $\lambda_3$ at its main diagonal, which are the eigenvalues of **G**. The same **O** also diagonalizes $\textbf{G}^{-1}$, and $\lambda^{-1}_l$ (${l=1,2,3}$) will be its eigenvalues. With the orthogonal matrix, we can recast condition as $$\begin{aligned}
\label{ellipsoid-f}
&&
\qquad\quad\frac{{t_1}^2}{\lambda_1}+\frac{{t_2}^2}{\lambda_2}+
\frac{{t_3}^2}{\lambda_3}\leq 1\,,\quad\mbox{where}\\
&&
\label{Ogf}
\textbf{O}^\intercal\,\textsf{E}=
\begin{pmatrix}
t_1\\
t_2\\
t_3
\end{pmatrix}
:=\begin{pmatrix}
\sqrt{\lambda_1}\,\sin\mu\,\cos\nu\\
\sqrt{\lambda_2}\,\sin\mu\,\sin\nu\\
\sqrt{\lambda_3}\,\cos\mu
\end{pmatrix}.\end{aligned}$$ Through the last equality in , one can enjoy an alternative parameterization of the ellipsoid, where the parameters ${\mu\in[0,\pi]}$ and ${\nu\in[0,2\pi)}$. By this technique one can easily find the orientation of the ellipsoid [@Meyer00]: the eigenvectors (that are columns in **O**) and the eigenvalues $\lambda_i$ of **G** characterize the semi-principal axes of the ellipsoid.
Two measurement settings {#subsec:2settings}
------------------------
In the above investigation, we assume ${\{\widehat{a},\widehat{b},\widehat{c}\}}$ is a set of linearly independent vectors. Now suppose $\widehat{c}$ is linearly dependent on $\widehat{a}$ and $\widehat{b}$, say $\widehat{c}=\vartheta_a \widehat{a} +
\vartheta_b \widehat{b} $, whereas $\widehat{a}$ and $\widehat{b}$ are still linearly independent. Then, we can discard all the items related to $\widehat{c}$ in , and thus achieve an elliptic region $\mathcal{E}$ identified by $$\begin{aligned}
&&\label{AB-v1v2}
\begin{pmatrix}
2p-1\\
2q-1
\end{pmatrix}=
\underbrace{\begin{pmatrix}
\langle A\rangle\\
\langle B\rangle
\end{pmatrix}}_{\displaystyle\textsf{E}}
=
\begin{pmatrix}
1 & 0 & 0 \\
{\scriptstyle\widehat{a}.\widehat{b}}&
{\scriptstyle\sqrt{1-(\widehat{a}.\widehat{b})^2}} & 0 \\
\end{pmatrix}
\underbrace{\begin{pmatrix}
\widehat{v}_1.\vec{r}\ \\
\widehat{v}_2.\vec{r}\ \\
\widehat{v}_3.\vec{r}\
\end{pmatrix}}_{\displaystyle\texttt{R}}\qquad\\
&&
\label{v1v2<1}
\mbox{with}\quad (\widehat{v}_1.\vec{r}\,)^2+(\widehat{v}_2.\vec{r}\,)^2\leq1 \quad \mbox{or}\\
\label{ellipse}
&&\textsf{E}^\intercal\, \textbf{G}^{-1}\textsf{E} \leq 1
\quad \mbox{with}\quad
\textbf{G}=
\begin{pmatrix}
1 & {\scriptstyle\widehat{a}.\widehat{b}}\\
{\scriptstyle\widehat{a}.\widehat{b}}& 1
\end{pmatrix}.\end{aligned}$$ We owe and to and , respectively. The average value $\langle C\rangle=\vartheta_a \langle A\rangle + \vartheta_b \langle B\rangle $ is now just a linear function, and the QC, presented by –, has no effect of $C$. To present the QCs, it is sufficient to consider only (linearly) independent operators [@Adagger]. So we are ignoring $C$ until Appendix \[subsec:3settings\].
One can notice two things with Eq. . First, a whole line segment—that is in the Bloch sphere and parallel to $\widehat{v}_3$ \[displayed in Fig. \[fig:abc-vectors\] ()\]—gets mapped onto a single point in $\mathcal{E}$ under the transformation in . Second, extreme points—that constitute the ellipse—of $\mathcal{E}$ come from the pure states that lie on the great circle \[illustrated in Fig. \[fig:abc-vectors\] ()\] of the Bloch sphere in the $v_1v_2$-plane.
![Moving horizontally from top-left to the bottom, the (blue) shaded regions are $\mathcal{R}_\Delta$ \[defined in \], $\mathcal{R}_H$, $\mathcal{R}_{u_{\sfrac{1}{2}}}$, $\mathcal{R}_{u_2}$, and $\mathcal{R}_{u_\text{max}}$, in that order. For each plot, ${\epsilon=\tfrac{3}{4}}$ (that is, ${\widehat{a}\cdot\widehat{b}=\tfrac{1}{2}}$) is taken, and the horizontal and vertical axes represent ${p\in[0,1]}$ and ${q\in[0,1]}$, respectively. All these $\mathcal{R}$’s contain permissible region , which is bounded by the ellipse displayed in each plot. The ellipse touches the boundary of a region $\mathcal{R}$ at certain points, some of them correspond to those pure states that saturate the associated CR or UR. []{data-label="fig:regions"}](fig4){width="45.00000%"}
Equivalently, one can take the projectors $P$ and $Q$ from at the places of $A$ and $B$ and then present everything in terms of the probabilities $p$ and $q$ given in , , and . In the case of projectors, hyperrectangle becomes the square ${[0,1]^{\times 2}}$, and the allowed region can be described as $$\label{ellipse-pq}
\mathcal{E}=\{(p,q)\ |\ 0\leq p,q\leq1\ \ \text{obeys}\ \
\eqref{ellipse}\}\,.$$ One can check that ${P=\mathring{A}}$ here \[for $\mathring{A}$, see \], thus ${H(p)=H(\langle A\rangle)}$, which is in fact true for all the uncertainty and certainty measures in –.
It is shown in [@Sehrawat17] that many tight CRs and URs known from [@Larsen90; @Busch14-b; @Garrett90; @Sanchez-Ruiz98; @Ghirardi03; @Bosyk12; @Vicente05; @Zozor13; @Deutsch83; @Maassen88; @Rastegin12] can be derived by using ellipse , and the same ellipse emerges in [@Lenard72; @Larsen90; @Kaniewski14; @Abbott16; @Sehrawat17] through different methods. A few such relations are $$\begin{aligned}
\label{std-UR}
\sqrt{1-(2\epsilon-1)^2}&\leq&
\Delta P+\Delta Q\,,\\
\label{entropy-UR}
\mbox{if}\ 0.7\leq\epsilon\ \mbox{then}\quad
2\,h(\tfrac{1+\sqrt{\epsilon}}{2})&\leq&H(p)+H(q)\,,
\\
\label{u-UR}
1+\sqrt{\epsilon}+\sqrt{1-\epsilon}
&\leq&u_{\sfrac{1}{2}}(p)+
u_{\sfrac{1}{2}}(q)\,,\\
\label{u2-CR}
\max\{2-\epsilon,1+\epsilon\}&\geq&u_2(p)+u_2(q)\,,
\ \mbox{and}\quad \\
\label{umax-CR}
\max\big\{1+\sqrt{1-\epsilon}\,,\,1+\sqrt{\epsilon}\,\big\}
&\geq&u_{\textrm{max}}(p)+u_{\textrm{max}}(q)\,,\qquad\quad\end{aligned}$$ where all the above functions are defined according to – for $P$ and $Q$, $$\label{entropy}
h(p):=-(p\ln p+({1-p})\ln(1-p))\,,\\$$ and ${\widehat{a}\cdot\widehat{b}=2\epsilon-1}$. The standard deviation $\Delta$, the Shannon entropy $H$ [@Shannon48], and $u_{\sfrac{1}{2}}$ are concave functions that provide tight URs –, whereas the convex functions $u_2$ and $u_{\textrm{max}}$ give tight CRs and [@Sehrawat17].
Following , one can define a region $$\begin{aligned}
\label{region-std-UR}
\mathcal{R}_\Delta=\{(p,q)\ |\ 0\leq p,q\leq1\ \ \text{obeys}\ \
\eqref{std-UR}\}\qquad\end{aligned}$$ that is limited by UR . Similarly, one can bound $\mathcal{R}_H$, $\mathcal{R}_{u_{\sfrac{1}{2}}}$, $\mathcal{R}_{u_2}$, and $\mathcal{R}_{u_\text{max}}$ by tight relations –. Taking ${\epsilon=\tfrac{3}{4}}$, we display these regions in Fig. \[fig:regions\] and realize that ${\mathcal{R}_\Delta\subset\mathcal{R}_{u_{\sfrac{1}{2}}}}$ and ${\mathcal{R}_H\subset \mathcal{R}_{u_2}\subset
\mathcal{R}_{u_\text{max}}}$, which may or may not hold for other $\epsilon$’s. Whereas neither $\mathcal{R}_\Delta$ is a subset of $\mathcal{R}_H$ nor vice versa.
One can also observe that ${(\epsilon,1)\in\mathcal{E}}$ while ${(1-\epsilon,1)\notin\mathcal{E}}$ in Fig. \[fig:regions\]. In these points, $\epsilon$ and ${1-\epsilon}$ are associated with the two distinct probability-vectors ${\vec{p}=(\epsilon,1-\epsilon)}$ and ${\vec{p}\,'=(1-\epsilon,\epsilon)}$, respectively. After the permutation, ${\vec{p}}$ turns into $\vec{p}\,'$ that is forbidden. It is a distinguish feature of a *quantum* probability ${p_l=\langle a_l|\rho |a_l\rangle}$ \[see and \] that ${p_l}$ is not only associated with the measurement setting $a$ but also with the label $l$ for an outcome.
Three measurement settings {#subsec:3settings}
--------------------------
Let us start with Schrödinger’s UR [@Schrodinger32] $$\begin{aligned}
\label{Schrodinger-UR}
&&
{\scriptstyle 0\,\leq\, \left(\langle A^2\rangle-\langle A\rangle^2\right)\left(
\langle B^2\rangle-\langle B\rangle^2\right)
-|\langle \widetilde{C}\rangle|^2
-|\langle \widetilde{D}\rangle-\langle A\rangle\langle B\rangle|^2}\,,
\nonumber\\
&&
\mbox{where}\quad
{\scriptstyle\widetilde{C}\,:=\frac{AB-BA}{2\,\text{i}}\quad\mbox{and}\quad
\widetilde{D}\,:=\frac{AB+BA}{2}}\quad\end{aligned}$$ are related to the commutator and the anticommutator, respectively, of $A$ and $B$. For qubit’s operators and , one can realize through that $$\label{c=a cros b}
\widetilde{C}=(\widehat{a}\times\widehat{b})\cdot\vec{\sigma}=
|\widehat{a}\times\widehat{b}|
\underbrace{\widehat{c}\cdot\vec{\sigma}}_{C}\,,\quad
\widehat{c}=\frac{\widehat{a}\times\widehat{b}}{|\widehat{a}\times\widehat{b}|}\,,$$ ${\widetilde{D}=(\widehat{a}\cdot\widehat{b})\,I}$, and ${A^2=I=B^2}$. Considering these and ${\scriptstyle|\widehat{a}\times\widehat{b}|=\sqrt{1-(\widehat{a}\cdot\widehat{b})^2}}$, we can rewrite Schrödinger’s UR for a qubit as $$\label{Schrodinger-UR-qubit}
0\leq (1-\langle A\rangle^2)(
1-\langle B\rangle^2)
-({\scriptstyle 1-(\widehat{a}\cdot\widehat{b})^2})\langle C\rangle^2
-({\scriptstyle\widehat{a}\cdot\widehat{b}}\,-\langle
A\rangle\langle B\rangle)^2.$$ To test in experimental scenario , one requires three measurement settings ${\widehat{a},\widehat{b},\widehat{c}}$. One can choose $\widehat{a}$ and $\widehat{b}$, and then $\widehat{c}$ is fixed by the cross product in . If one takes $\widehat{a}$ and $\widehat{b}$ collinear, then turns into the trivial statement ${0=0}$. So we are taking $\widehat{a}$ and $\widehat{b}$ linearly independent.
![The region of ${\langle A\rangle,\langle B\rangle,\langle C\rangle\in[-1,1]}$, which is restricted by Schrödinger’s UR . It is, ellipsoidal in shape, presented by picking ${\widehat{a}\cdot\widehat{b}=\tfrac{1}{2}}$, and $C$ is determined by . The ellipsoid turns into the Bloch sphere for orthogonal $\widehat{a}$ and $\widehat{b}$. []{data-label="fig:ellip-SUR"}](fig5){width="30.00000%"}
One can check that Schrödinger’s UR and QC with the Gram matrix $$\label{SR-Gram-matrix}
\textbf{G}_{\text{Sch}}=
\begin{pmatrix}
1 & {\scriptstyle\widehat{a}\cdot\widehat{b}} & 0 \\[1mm]
{\scriptstyle\widehat{a}\cdot\widehat{b}} & 1 & 0 \\[1mm]
0 &0 & 1 \\
\end{pmatrix}$$ are the same thing, and the UR is saturated by every pure state for a qubit. Without the last term in , Schrödinger’s UR becomes Robertson’s UR [@Robertson29], which will form a bigger region than the allowed region here characterized by .
Taking ${\widehat{a}\cdot\widehat{b}=\tfrac{1}{2}}$, the ellipsoid is displayed in Fig. \[fig:ellip-SUR\]. Orthogonal projection of the ellipsoid onto the ${\langle A\rangle\langle B\rangle}$–plane produces the same elliptic region that is identified by and shown in Fig. \[fig:regions\]. The parametric forms \[obtained via and with \] of the ellipsoid are $$\begin{aligned}
\label{S-UR-ellipsoid-para}
\underbrace{\begin{pmatrix}
\langle A\rangle\\
\langle B\rangle\\
\langle C\rangle
\end{pmatrix}}_{\displaystyle\textsf{E}}
&=&
\underbrace{\begin{pmatrix}
\tfrac{1}{\sqrt{2}} & \hphantom{-}\tfrac{1}{\sqrt{2}} & 0 \\[1mm]
\tfrac{1}{\sqrt{2}} & -\tfrac{1}{\sqrt{2}} & 0 \\[1mm]
0 & \hphantom{-}0 & 1 \\
\end{pmatrix}}_{\displaystyle\textbf{O}}
\begin{pmatrix}
{\scriptstyle\sqrt{1+\widehat{a}\cdot\widehat{b}}}\,\sin\mu\,\cos\nu\\
{\scriptstyle\sqrt{1-\widehat{a}\cdot\widehat{b}}}\,\sin\mu\,\sin\nu\\
\quad \cos\mu
\end{pmatrix}\qquad\\
&=&
\underbrace{\begin{pmatrix}
1 & 0 & 0 \\[1mm]
{\scriptstyle\widehat{a}\cdot\widehat{b}} & {\scriptstyle\sqrt{1-(\widehat{a}\cdot\widehat{b})^2}} & 0 \\[1mm]
0 & 0 & 1 \\
\end{pmatrix}}_{\displaystyle\textbf{M}}
\underbrace{\begin{pmatrix}
\sin2\theta\,\cos\phi\\
\sin2\theta\,\sin\phi\\
\cos2\theta
\end{pmatrix}}_{\displaystyle\texttt{R}_\text{pure}}\ .\end{aligned}$$ One can easily recognize the semi-principal axes in Fig. \[fig:ellip-SUR\] with .
For the next example, we consider three linearly-independent unit vectors ${\widehat{\mathsf{a}}_1,\widehat{\mathsf{a}}_2,\widehat{\mathsf{a}}_3}$ such that their Gram matrix is $$\label{SIC-POVM-Gram-matrix}
\textbf{G}_{\textsc{sic}}=
\begin{pmatrix}
\hphantom{-}1 & -\tfrac{1}{3} & -\tfrac{1}{3} \\[1mm]
-\tfrac{1}{3} & \hphantom{-}1 & -\tfrac{1}{3} \\[1mm]
-\tfrac{1}{3} & -\tfrac{1}{3} & \hphantom{-}1 \\
\end{pmatrix}\,.$$ It implies that there is an equal angle, $\arccos(-\tfrac{1}{3})$, between every pair of the vectors. There exists one more such unit vector ${\widehat{\mathsf{a}}_4=-\textstyle\sum\nolimits_{i=1}^{3}
\widehat{\mathsf{a}}_i}$. The set of four vectors ${\{\widehat{\mathsf{a}}_i\}_{i=1}^{4}}$ yields a SIC-POVM for a qubit [@Rehacek04; @Appleby09], whose elements are the positive semi-definite operators $$\label{Pi}
\varPi_i=\tfrac{1}{4}(I+\widehat{\mathsf{a}}_i\cdot\vec{\sigma})\,,
\quad\mbox{and}\quad
\sum_{i=1}^{4}\varPi_i=I$$ is because $\textstyle\sum\nolimits_{i=1}^{4}\widehat{\mathsf{a}}_i$ is a null vector.
![From top-left to bottom-right, moving horizontally, the first one is the allowed (ellipsoidal) region $\mathcal{E}$ restrained by QCs and . The second, third, and fourth regions are $\mathcal{R}^{\textsc{sic}}_\Delta$, $\mathcal{R}^{\textsc{sic}}_h$, and $\mathcal{R}^{\textsc{sic}}_{\mathsf{u}_{\sfrac{1}{2}}}$, respectively, which are described in the main text. $\mathcal{E}$ lies within every $\mathcal{R}^{\textsc{sic}}$ and shares a set of boundary points, which come from the minimum uncertainty states that saturate the associated UR. One can also observe that ${\mathcal{R}^{\textsc{sic}}_\Delta\subset\mathcal{R}^{\textsc{sic}}_{\mathsf{u}_{\sfrac{1}{2}}}}$ and ${\mathcal{R}^{\textsc{sic}}_h\subset\mathcal{R}^{\textsc{sic}}_{\mathsf{u}_{\sfrac{1}{2}}}}$. []{data-label="fig:regions-SIC-POVM"}](fig6){width="45.00000%"}
Since eigenvalues of every $\varPi_i$ are $0$ and $\tfrac{1}{2}$, its mean value ${\langle \varPi_i\rangle:=\mathsf{p}_i\in[0,\tfrac{1}{2}]}$ according to . Moreover, for ${(\varPi_1,\varPi_2,\varPi_3)}$, hyperrectangle is the cube ${[0,\tfrac{1}{2}]^{\times 3}}$ in which regions are exhibited in Fig. \[fig:regions-SIC-POVM\]. The linear QC $$\label{Pi=1}
\sum_{i=1}^{4}\,
\langle\widehat{\mathsf{a}}_i\cdot\vec{\sigma}\rangle=0
\quad\Leftrightarrow\quad
\sum_{i=1}^{4}\mathsf{p}_i=1$$ is due to normalization of a state, that is, ${\langle I\rangle=1}$, where the identity operator is given in . To estimate ${\langle\widehat{\mathsf{a}}_i\cdot\vec{\sigma}\rangle=4\,\mathsf{p}_i-1}$, ${i=1,\cdots,4}$, one can either choose three projective measurements along $\widehat{\mathsf{a}}_1, \widehat{\mathsf{a}}_2$, and $\widehat{\mathsf{a}}_3$ or the single POVM ${\{\varPi_i\}_{i=1}^{4}}$ that can be realized by the scheme in [@Rehacek04]. In either case, the permitted region $\mathcal{E}$ is identified by and the quadratic QC $$\label{Pi<=1/3}
\sum_{i=1}^{4}\,
\langle\widehat{\mathsf{a}}_i\cdot\vec{\sigma}\rangle^2\leq\tfrac{4}{3}
\quad\Leftrightarrow\quad
\sum_{i=1}^{4}\mathsf{p}_i^2
\leq\tfrac{1}{3}\,.$$ The right-hand-side inequality is given in [@Rehacek04], and it can also be derived from by using Gram matrix . Employing and with , we can have parametric forms $$\begin{aligned}
\label{SIC-POVM-ellipsoid-para-O}
\underbrace{\begin{pmatrix}
4\,\mathsf{p}_1-1\\
4\,\mathsf{p}_2-1\\
4\,\mathsf{p}_3-1
\end{pmatrix}}_{\displaystyle\textsf{E}}
&=&
\underbrace{\begin{pmatrix}
\tfrac{1}{\sqrt{3}} & \hphantom{-}\tfrac{1}{\sqrt{2}} & -\tfrac{1}{\sqrt{6}} \\[1mm]
\tfrac{1}{\sqrt{3}} & -\tfrac{1}{\sqrt{2}} & -\tfrac{1}{\sqrt{6}} \\[1mm]
\tfrac{1}{\sqrt{3}} & \hphantom{-}0 & \hphantom{-}\tfrac{2}{\sqrt{6}} \\
\end{pmatrix}}_{\displaystyle\textbf{O}}
\begin{pmatrix}
\tfrac{1}{\sqrt{3}}\,\sin\mu\,\cos\nu\\
\tfrac{2}{\sqrt{3}}\,\sin\mu\,\sin\nu\\
\tfrac{2}{\sqrt{3}}\,\cos\mu
\end{pmatrix}\qquad\ \ \\
\label{SIC-POVM-ellipsoid-para-M}
&=&
\label{M^-1 SIC}
\underbrace{\begin{pmatrix}
\hphantom{-}1 & \hphantom{-}0 & 0 \\[1mm]
-\tfrac{1}{3} & \hphantom{-}\tfrac{2\sqrt{2}}{3} & 0 \\[1mm]
-\tfrac{1}{3} & -\tfrac{\sqrt{2}}{3} & \tfrac{\sqrt{2}}{\sqrt{3}} \\
\end{pmatrix}}_{\displaystyle\textbf{M}}
\underbrace{\begin{pmatrix}
\sin2\theta\,\cos\phi\\
\sin2\theta\,\sin\phi\\
\cos2\theta
\end{pmatrix}}_{\displaystyle\texttt{R}_\text{pure}}.\end{aligned}$$ of the ellipsoid that is the boundary of $\mathcal{E}$. Taking , one can check orientation of the ellipsoid exhibited in Fig. \[fig:regions-SIC-POVM\] (top-left).
![Contour plot of the entropy $-\textstyle\sum\nolimits_{i=1}^{4}\mathsf{p}_i\ln\mathsf{p}_i$, which reaches its global minimum ${\ln 3}$ at the four (red) points that are associated with ${\{-\widehat{\mathsf{a}}_i\}_{i=1}^{4}}$. []{data-label="fig:entropy-plot-SIC"}](fig7){width="35.00000%"}
[c@ | @c@ c ]{}
------------------------------------------------------------------------
& $2\theta$ & $\phi$\
------------------------------------------------------------------------
$-\,\widehat{\mathsf{a}}_1$ & $\tfrac{\pi}{2}$ & $\pi$\
$-\,\widehat{\mathsf{a}}_2$ & $\tfrac{\pi}{2}$ & $\pi+\arccos(-\tfrac{1}{3})$\
$-\,\widehat{\mathsf{a}}_3$ & $\pi-\arccos(\tfrac{\sqrt{2}}{\sqrt{3}})$ & $\hphantom{\pi+}\arccos(\tfrac{1}{\sqrt{3}})$\
$-\,\widehat{\mathsf{a}}_4$ & $\hphantom{\pi-}\arccos(\tfrac{\sqrt{2}}{\sqrt{3}})$ & $\hphantom{\pi+}\arccos(\tfrac{1}{\sqrt{3}})$\
\[tab:theta-phi-for(-a)\]
To measure a combined uncertainty, if one picks a suitable concave function of ${\{\mathsf{p}_i\}_{i=1}^{4}}$, for example, the standard Shannon entropy $-\textstyle\sum\nolimits_{i=1}^{4}\mathsf{p}_i\ln\mathsf{p}_i$, then its absolute minimum will occur on the ellipsoid parametrized by $\theta$ and $\phi$ in . By plotting the entropy as a function of ${\theta\in[0,\tfrac{\pi}{2}]}$ and ${\phi\in[0,2\pi)}$ in Fig. \[fig:entropy-plot-SIC\], we observe that the entropy reaches its absolute minimum ${\ln3}$ when the Bloch vector ${\vec{r}\in\{-\widehat{\mathsf{a}}_i\}_{i=1}^{4}}$; the corresponding $\theta$’s and $\phi$’s are registered in Table \[tab:a-theta-phi\]. In this way, we establish three tight URs $$\begin{aligned}
\label{std-SIC}
2\sqrt{2}&\,\leq\,&
\sum_{i=1}^{4}
\sqrt{1-\langle\widehat{\mathsf{a}}_i\cdot\vec{\sigma}\rangle^2}
=\sum_{i=1}^{4}\sqrt{1-(4\mathsf{p}_i-1)^2}\,,\qquad\ \ \\
\label{entropy-SIC}
\ln3&\,\leq\,&
-\sum_{i=1}^{4}\mathsf{p}_i\ln\mathsf{p}_i =
h(\mathsf{p}_1,\cdots,\mathsf{p}_4)\,,
\quad\mbox{and}\\
\label{u-SIC}
\sqrt{3}&\,\leq\,&
\sum_{i=1}^{4}\sqrt{\mathsf{p}_i} =:
\mathsf{u}_{\sfrac{1}{2}}(\mathsf{p}_1,\cdots,\mathsf{p}_4)\,.\end{aligned}$$ The right-hand-side is the sum of standard deviations in UR , which is saturated by eight pure states, whose Block vectors ${\vec{r}\in\{\pm\widehat{\mathsf{a}}_i\}_{i=1}^{4}}$. Whereas, both URs and are saturated by four pure states that are related to ${\{-\widehat{\mathsf{a}}_i\}_{i=1}^{4}}$. Note that the uncertainty measures $h$ and $\mathsf{u}_{\sfrac{1}{2}}$ in and are different from and . Since ${\textstyle\sum\nolimits_{i=1}^{4}\mathsf{p}_i^2}$ is a convex function [@Sehrawat17], the right-hand-side inequality in can be seen as a tight CR. While the left-hand-side inequality delivers a tight UR for the sum of squared standard deviations ${1-\langle\widehat{\mathsf{a}}_i\cdot\vec{\sigma}\rangle^2}$, and the sum is bounded by $\tfrac{8}{3}$ from below. Both these relations are saturated by every pure state.
As before, we can restrict a set of ${(\mathsf{p}_1,\cdots,\mathsf{p}_3)}$ by one of the above URs, for instance, $$\begin{aligned}
\label{R-std-SIC}
&&\mathcal{R}^{\textsc{sic}}_\Delta:=\{(\mathsf{p}_1,\cdots,\mathsf{p}_3)\ |\ 0\leq \mathsf{p}_1,\cdots,\mathsf{p}_4\leq \tfrac{1}{2}\ \nonumber\\
&&
\qquad\qquad\qquad\qquad\qquad \text{obey}\
\eqref{Pi=1}\ \text{and}\ \eqref{std-SIC}\}\,.\qquad\end{aligned}$$ Replacing UR in by and , we define the regions $\mathcal{R}^{\textsc{sic}}_h$ and $\mathcal{R}^{\textsc{sic}}_{\mathsf{u}_{\sfrac{1}{2}}}$, respectively. One can see cross sections in $\mathcal{R}^{\textsc{sic}}_h$ and $\mathcal{R}^{\textsc{sic}}_{\mathsf{u}_{\sfrac{1}{2}}}$ caused by ${\mathsf{p}_i=\tfrac{1}{2}}$ ${(i=1,\cdots,4)}$, which shows a significance of .
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Instead of $A$ defined in , one can take more general Hermitian operator $\widetilde{A}:=a_0I+\vec{a}.\vec{\sigma}$, where $\widehat{a}$ is the unit vector of $\vec{a}$. Nevertheless, the expectation value of $\widetilde{A}$ can be obtained from ${\langle A\rangle}$: ${\langle \widetilde{A}\rangle=a_0+\tfrac{1}{|\vec{a}|}\langle A\rangle}$. So, we do not lose generality if we take $A$ rather than $\widetilde{A}$.
C. D. Meyer, *Matrix Analysis and Applied Linear Algebra* (Society for Industrial and Applied Mathematics, Philadelphia, 2000), Chap. 5, Sec. 12.
The matrix inequality ${\textsf{E}\,\textsf{E}^\intercal\leq \textbf{G}}$ is given in [@Kaniewski14], which can be transformed into ${\textsf{E}^\intercal\textbf{G}^{-1}\textsf{E}\leq 1}$ of . I am grateful to Jdrzej Kaniewski for providing me this explanation.
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---
abstract: 'We propose a model of parameter learning for signal transduction, where the objective function is defined by signal transmission efficiency. We apply this to learn kinetic rates as a form of evolutionary learning, and look for parameters which satisfy the objective. This is a novel approach compared to the usual technique of adjusting parameters only on the basis of experimental data. The resulting model is self-organizing, i.e. perturbations in protein concentrations or changes in extracellular signaling will automatically lead to adaptation. We systematically perturb protein concentrations and observe the response of the system. We find compensatory or co-regulation of protein expression levels. In a novel experiment, we alter the distribution of extracellular signaling, and observe adaptation based on optimizing signal transmission. We also discuss the relationship between signaling with and without transients. Signaling by transients may involve maximization of signal transmission efficiency for the peak response, but a minimization in steady-state responses. With an appropriate objective function, this can also be achieved by concentration adjustment. Self-organizing systems may be predictive of unwanted drug interference effects, since they aim to mimic complex cellular adaptation in a unified way.'
author:
- |
Gabriele Scheler\
Carl Correns Foundation for Mathematical Biology\
1030 Judson Drive\
Mountain View, Ca. 94040\
title: 'Self-organization of signal transduction'
---
Introduction
============
Signal transduction systems are often modeled as networks of biochemical kinetic equations implemented as continuous-time dynamical models using differential equations [@Bhalla99; @Hoops2006]. If we regard a subset of species as inputs, and make sure that the system always converges to equilibrium values by using weakly reversible equations [@Deng2011; @Akle2011; @vanderSchaft2011], we may transform these models into a set of matrices fulfilling the role of input-output transfer functions, i.e. a mapping from sustained input signal levels to steady-state concentrations for all target species [@Scheler2012]. Protein signaling functions (psfs) are a systemic generalization of individual dose-response functions, which are usually described by Hill equations [@Barlow89]. In contrast to Hill equations, which are not available for enzymatic reactions, which only calculate relative concentrations, and which only work for one reaction in isolation, the psf system calculates enzymatic and complex formation reactions in a complex systemic environment using absolute concentrations [@Scheler2012]. In addition, the reaction times to equilibrium are calculated as delay values, and the dynamic shape (’transients’) is also available for further analysis.
In this paper, we want to ask the question of optimality of signal transduction. From an evolutionary standpoint, we assume that any biological signal transduction system is constructed with optimized efficiency of signal transmission. Furthermore, we assume that cells have the ability to adapt to perturbations of protein concentrations and changes in extracellular signaling by reinstating signal transmission efficiency. In the following, we investigate this question using a biologically realistic system - beta-adrenergic signaling in a submembrane compartment of a mouse embryonic fibroblast- for a single input scenario, focusing on a selected target species as relevant output or actuator of the system (Fig. \[fig1a\] A).
Experimental analysis of signal transduction systems has shown fold-change responses to changes in input [@Ferrell2009; @Buijsman2012]. Accordingly, input-output transfer functions usually follow the shape of hyperbolic (saturating) curves, which are equivalent to sigmoids for logarithmically scaled input [@Scheler2012]. In Fig. \[fig1a\] B we show the effect of a knock-out (KO) for a RGS protein in an experimental assay in yeast [@Yi2003], and compare this with the effect in the model system [@Blackman2011]. We see that the effects of the RGS KO on dose-response signaling efficiency are robust across very different cellular systems.
Usually, when we use a computational model to investigate perturbations, we only study the effects as reflected in the simulation. By utilizing optimization in terms of signaling efficiency, we can make the system itself adjust to the perturbation. In this way, we are studying signal transduction as a self-organizing system, which uses objective functions to adapt. This basic idea is extremely powerful, and could be used with different kinds of constraints on parameters, reaction times, etc. and with different, multiple input scenarios for larger systems. To explore this question further is of significant importance in assessing cellular health and functioning.
Methods
=======
Example System: GPCR signaling in a submembrane compartment
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Fig. \[fig1a\]A shows the example system, a submembrane compartment with a GPCR (G-protein coupled receptor) pathway from a mouse embryonic fibroblast, with ISO as input (extracellular ligand to $\beta(2)$-adrenergic receptors) and the phosphorylation of a protein VASP as output. This model was implemented as an ODE model with 23 reactions and 27 molecular species, derived from 12 initial concentrations (cf. Table \[table-rates\], Table \[concentrations\]). The parameters were adapted to experimental biological data (not shown, [@Blackman2011]). In this subsystem, the central cAMP response often follows a plateau curve, i.e. a rise to steady-state, but cAMP transients which are typically observed in cytoplasm may also occur [@Blackman2011]. The dose-response transfer functions were derived as in [@Scheler2012].
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$k_{\mbox{\scriptsize\em on}}$ $k_{\mbox{\scriptsize\em off}}$ $k_{\mbox{\scriptsize\em cat}}$
$ b2 + L \leftrightarrow b2L$ 0.0003 0.1
$ b2L + PKAc \leftrightarrow b2LPKAc \rightarrow pb2L + PKAc$ 0.00026 1 5.4
$ pb2L \rightarrow b2L$ 0.1
$ GsaGDP + b2L \leftrightarrow b2LGsaGDP \rightarrow GsaGTP + b2L$ 0.006 0.8 0.2
$ GsaGTP + RGS \leftrightarrow RGSGsaGTP \rightarrow GsaGDP + RGS$ 0.0008 1.2 16
$ GiGDP + pb2L \leftrightarrow pb2LGiGDP \rightarrow GiGTP + pb2L$ 1.2 0.8 16
$ GiGTP + RGS \leftrightarrow RGSGiGTP \rightarrow GiGDP + RGS$ 1.2 0.8 16
$ GsaGTP + AC6 \leftrightarrow AC6Gsa$ 0.00385 3
$ GiGTP + AC6Gsa \leftrightarrow AC6GsaGi$ 0.00385 10
$ AC6Gsa + PKAc \leftrightarrow AC6Gsa\_PKAc \rightarrow pAC6Gsa + PKAc$ 0.00026 1.5 30.4
$ pAC6Gsa + PP1 \leftrightarrow pAC6PP1 \rightarrow AC6Gsa + PP1$ 0.0026 3 54
$ ATP + AC6Gsa\leftrightarrow AC6Gsa\_ATP \rightarrow cAMP + AC6Gsa$ 6e-05 10 80.42
$ ATP + pAC6Gsa \leftrightarrow pAC6Gsa\_ATP \rightarrow cAMP + pAC6Gsa$ 6e-05 10 8.042
$ ATP + AC6 \leftrightarrow AC6\_ATP \rightarrow cAMP + AC6$ 0.0001 120 0.142
$ cAMP + PDE4B \leftrightarrow PDE4BcAMP \rightarrow AMP + PDE4B$ 0.03 77.44 19.36
$ 1 PKA + 2 cAMP \leftrightarrow 1 PKAr2c2cAMP2$ 3.5e-08 0.06
$ 1 PKAr2c2cAMP2 + 2 cAMP \leftrightarrow 1 PKAr2c2cAMP4$ 2.7e-07 0.28
$ 1 PKAr2cAMP4 + 2 PKAc \leftrightarrow 1 PKAr2c2cAMP4$ 8.5e-08 0.05
$ VASP + PKAc \leftrightarrow VASPPKAc \rightarrow pVASP + PKAc$ 0.00026 1.5 30.4
$ pVASP + PP1 \leftrightarrow pVASPPP1 \rightarrow VASP + PP1$ 0.0026 3 54
$ AMP \rightarrow ATP$ 1
$ ATP + AC6GsaGi \leftrightarrow AC6GsaGi\_ATP \rightarrow cAMP + AC6GsaGi$ 6e-05 10 1
$ b2L + bARR \leftrightarrow b2LbARR$ 0.0006 0.1
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: Kinetic rates of the sample system, adapted from Sabio-Rk [@Sabio-Rk] and Brenda [@BRENDA], adjusted to experimental cAMP time-series data [@Blackman2011][]{data-label="table-rates"}
---------- ----- --------- -------
$PDE4B$ 200 $b2$ 100
$bARR$ 500 $PKA$ 500
$cAMP$ 100 $AC6$ 1000
$GsaGDP$ 200 $GiGDP$ 200
$RGS$ 100 $VASP$ 200
$PP1$ 100 $ATP$ 1e+06
---------- ----- --------- -------
: Initial concentrations (in nM) of species in the sample system[]{data-label="concentrations"}
Objective Function {#methods2}
------------------
A biological signal transduction system is defined by its state variable vector $x$, the set of all kinetic rate and initial concentration parameters.
We hypothesize that an efficient signal transmission would maximize the response coefficient $R_{C,S}$ (the response of species $C$ to input $S$) defined as $$R_{C,S}= {{C_{t}\over C_{0}}-1 \over {S_{t}\over S_{0}}-1}$$ with concentration change of target $C$ and input $S$ from baseline (t=0) to signal time t. For $R_{C,S}$, values $<1$ show signal loss, with 1 for perfect transmission, and values $>1$ showing signal amplification. We may also optimize for the slope $s$ of the sigmoid at half-maximum concentration. This is equivalent to maximizing $R_{C,S}$, provided that the input signal remains entirely between the upper and lower boundaries of the sigmoid (Fig. \[fig1a\]). By optimizing for $s$, additional to $R_{C,S}$, we may force the system to implement a switch-like function instead of a more linear function. However, shifting the sigmoid function to the left or to the right is more important as the slope in our models.
In addition, we optimize for reaction time (delay to steady-state). Steady-state is defined, pragmatically, as relative change of less than 2% over 100s. The delay ($d_S$) is computed for 90% (EC90) of steady-state. We may now define an aggregate objective function: $$f(x)= max_x [R_{C,S}(x), -d_S(x)]$$ to select the system state variables that minimize delay and maximize response.
In addition to signal transmission from extracellular concentration changes onto steady-state concentrations, such as they typically occur for temporally integrating proteins like transcription factors, we also look at signal transmission by transients, i.e. peak concentration, in response to extracellular signals. In this case, we minimize the delay to peak value, maximize the response at peak value, and minimize the response at steady-state value. $$f_{transient}(x)= max_x [R_{C,S}^p(x), -d_{S}^p(x), -R_{C,S}(x), d_{S}(x)]$$
Results
=======
Delay vs. Efficiency Trade-off
------------------------------
The computation of signal transmission efficiency will be explained here for a single input-output pathway of cAMP/PKA-mediated transmission in a cellular membrane compartment. It is clear that a complex signaling system may have several inputs, and a large number of outputs or target proteins, and this is especially the case for cAMP-mediated signaling. Nonetheless we will focus on the simple case here to explain the basic principle. The input is a membrane receptor, a G-protein coupled receptor (GPCR), $\beta(2)AR$, which is activated by an extracellular pharmacological agonist ISO (isoproterenol); the output is a membrane protein, VASP, which promotes actin filament elongation. Activation of the receptor selectively inhibits VASP by phosphorylating VASP to pVASP via protein kinase A (PKA) activation. In Fig. \[fig1a\]B the original ISO/pVASP transfer function is shown, which we use here for further optimization. The system was trained for a signal distribution between 10nM and 1 $\mu$M ISO adjusting both kinetic rate and concentration parameters (Fig. \[fig1a\]A). The optimization method used is a simplex algorithm [@Press2007]. Fig. \[fig1a\]B shows the transfer function before and after adjustment, maximizing for $R_{C,S}$, $d_S$ or the combined function $f$. The results are summarized in Table \[exp1\]. We see that optimizing for the response coefficient alone shifts the function to the right to better cover the input range. Optimizing for the delay alone shifts it to the left, speeding up signals in the lower range (which are slower). Both objectives together (equally weighted) are fairly close to the original, biologically validated curve, with an improved $R_{C,S}$.
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$R_{C,S}$ $d$
original 0.272 423
optimized for $R_{C,S}$ 0.41 594
optimized for $d_S$ 0.11 169
optimized for $f$ 0.34 481
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: Reaction times ($d_S$) and response efficiency ($R_{C,S}$) in a biological system and under optimization[]{data-label="exp1"}
In general, biochemical reactions are faster at higher substrate concentrations, but the relative concentration change in response to an increase in the enzyme or the binding partner is less. This is a fundamental trade-off between delay and transmission efficiency that may define an optimal operating range for a signaling system and be of relevance in disease processes [@Scheler2012]. The results obtained with this experiment are simple, intuitive and encourage continuing to explore the basic idea.
Evolutionary Learning of Kinetic Rate Parameters
------------------------------------------------
In principle, we may use all parameters in a system, concentrations or kinetic rates, to maximize signal transmission. But the evolution of protein structure and interactions shows that it is fine-tuning of molecular kinetic parameters which is subject to evolutionary learning, while concentrations are often regulated adaptively in each cell. Mass-action kinetics approximate molecular kinetic parameters, even though there are significant sources of uncertainty, such as the stochastics of molecular interactions. In the following, we explore the idea that kinetic rate learning operates on evolutionary time-scales, and that biologically attested signal transduction pathways contain reaction rates which are optimal in terms of signal transmission efficiency. We use known concentration ranges, specific by cell type, together with kinetic rate optimization.
To explore the parameter space, we drew 1662 values for all kinetic rate parameters (kon, koff, kcat) from a distribution of 20% to 500% of the original parameters, and calculated $R_{C,S}$ and $d_S$ for the corresponding models, relative to an improved signal distribution from $1nM$ to $1\mu M$ (Fig. \[fig:param\]A). We find that there are parameter combinations which greatly improve efficiency of the signal transmission function. The basic distribution of a uniform low value of $R_{C,S}$ for fast $d$ and a wide variability in $R_{C,S}$ above a certain threshold in $d$ is robust against different types of signaling input (cf. also Fig. \[transient\]A). This may, however, be highly dependent on the reaction network that underlies the transfer function. We have not further explored this question. To test for robustness of these systems against variability of concentration, we repeated experiments for 100 systems with 20% variation of original concentrations, which corresponds to generally accepted noise levels (cf. [@Sigal2006; @Newman2006]). As expected, this low variation did not significantly affect the quality of a set of kinetic rates (supplemental table).
We further analysed the parameter combinations with different signal transmission efficiency. In Fig. \[fig:param\]B and C, we distinguish low and high efficiency signal transmission. Interestingly, we find, with respect to signal efficiency of the transfer function, that all parameters are ’sloppy’ (allow a wide variation), there are no ’stiff’ (low variation) parameters [@Gutenkunst2007]. Standard deviation for all parameters in our case is similar (Fig. \[fig:param\]B). Since it has been argued that optimization to experimental data yields reactions which allow more variance than others, as an indication of their influence on the signal transmission pathway, this analysis seems to contradict this effect. Possibly, these results pertain mostly to parameter variation that results from matching a networked system with many species to selected time-series data for only few species, which may behave differently from general optimization.
Co-regulation of Protein Concentration as an Adaptive Response to Perturbation
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Co-regulation of protein expression in cellular systems is important in disease progression and often a problem in targeted interventions. Here we are exploring the question of self-organization of protein concentration after a perturbation that reduces one protein to only 10% of its previous concentration. Keeping kinetic rates fixed, all concentrations in the system are allowed to adjust until optimality of signal transmission and delay is reinstated. There is a number of interesting observations here (cf. Fig. \[fig:fig\_conc\_reduct\]A), which relate to the biological reality. For instance, reducing PDE4B causes much regulation in other proteins, but it is almost never targeted. In contrast, reducing PP1 has little effect, but PP1 is frequently responsive to other proteins. Reducing PKA, RGS and AC6 leads to widespread down-regulations, to maintain sensitivity of signaling, but reducing the receptor beta-2 leads to up-regulation. There are many individual adaptations which can be interpreted to maintain the sensitivity and responsivity of the small molecule cAMP. The results show that protein regulation is highly sensitive to positions and roles of individual proteins in the reaction network in transmitting the signal. This problem may also be amenable to a more principled mathematical analysis [@Steuer2006]. Another idea would be to rank reaction systems defined by kinetic parameters as in Fig. \[fig:param\] by how well they adapt to perturbations.
In our example, the quality of the readjustment is not always the same, but the simple optimization scheme that we use may easily produce suboptimal solutions (local minima). Concentration changes may be caused by genetic up- or down- regulation, secretion and re-uptake, increased degradation, RNA interference, etc. and are therefore not easy to model from the standpoint of mechanistic biological modeling, which would need modules for all biologically attested processes. A unified perspective by a set of constraints and a set of objectives, such as has been envisaged here, may lead to better predictive results and may also be used as a guiding principle in constructing mechanistic models. Since there are intricate biological processes of adjusting concentrations, we may assume that in the cell optimal solutions are more easily found.
Optimal signal transmission depends on the signaling level
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From the standpoint of disease modeling, an unusual protein concentration may be an adaptive response where a still functioning cell in a dysfunctional external signaling environment struggles to keep signal transmission efficient to support cellular function. In such a case, targeting this protein by pharmacological intervention will lead to co-regulation on other proteins. In general, as well as in real biological signaling systems, it is the localization of the input signaling range and the distribution of signals that the system transmits which are important for optimization. If we select the signal set that we optimize for in the right way, the input range will move towards the loglinear range, i.e. between the lower and upper input boundaries (Fig. \[fig:fig\_conc\_reduct\]B). As a result, a number of internal concentrations will change. Fig. \[fig:fig\_conc\_reduct\]C shows the relative concentration changes that result from a shift in input range. Interestingly, the low shift requires a strong reduction in RGS, and upregulation for PDE4B, among other effects. GsaGDP (the activating G protein) is strongly increased, and GiGDP (the inhibitory G protein) is decreased. In the high shift, GiGDP and PP1 (the phosphatase which decreases pVASP) are stronger, and the beta-2 receptor density is much increased. In the future, it will be interesting not only to calculate these effects based on different optimization measures, and in multiple input-output scenarios, but also to compare this with attested cases of biological adaptation. By reverse engineering, we may infer an optimization measure from a sufficient number of attested co-regulations. For instance in addiction, protein co-regulation as a form of sensitization is well-attested [@Robison2011], but also in cancer where intercellular signaling (e.g. by cytokines [@OShea2008]) is affected. Gene expression data may then be mined not only for evidence of the mechanics of genetic regulatory pathways but also for evidence of shifts in extracellular signaling, which cause altered protein expression.
Role of transients
------------------
The appearance of a transient vs. a plateau signal (or even a dampened oscillation) in response to sustained signaling depends on the construction of the biochemical reaction network (negative feedback interactions) and its parameters. We show that we can also train a system for the appearance of a transient response to a sustained signal. This means to search for high response at peak, low delay to peak, but also a low response at steady-state (cf. \[methods2\], Fig. \[transient\]A). This requirement of invariance for the steady-state is sometimes considered a form of ’robustness’ or ’homeostatic regulation’ of the cellular response [@Shinar2010; @Kitano2007], and signaling by transients is widely regarded as an important mechanism. Here we found that concentration adjustment is quite sufficient to acquire a switch from transient to plateau response, with a high variability of response shape dependent on concentration, but also on the size of the signal (Fig. \[transient\]A). In Fig. \[transient\]B we show the distribution of concentrations that achieve high or low propensity for transients, as indicated by $f_{transient}$. We focus on concentrations with fairly uniform up- or down-regulation to create an experimental graph in Fig. \[transient\]C. The simplicity of Fig. \[transient\]C undermines the notion, as proven by the random search optimization, that deviations from up- or down-regulation for individual concentration may not be irrelevant noise, but rather part of a guided adaptation process that has many different solutions. We may have to consider this complexity to understand concentration adjustments in biological cells.
By using different objective functions for each target, it will be quite possible to combine different goals for targets in a multiple-output signaling system. Interestingly, a self-organizing cellular signaling system may also incorporate adaptive controls as objective functions, in particular when ion channels or membrane transporters are targeted. In this case, the goal of the system is to respond to an input (e.g. ISO at beta-2 receptors) such that the magnitude of another input such as calcium (ion channels as targets) is controlled. Signal transmission efficiency for such a target is not to be optimized to a maximum value, but instead to the appropriate ratio between the inputs. We have not followed up on this idea, but it may be worth to further investigate as an example of natural computation for adaptive control.
Discussion
==========
The idea to look for parametric optimization as the basis of realistic cellular properties has been applied with success to metabolic fluxes [@Edwards2001]. In that case, optimization of growth is usually regarded as the single objective function. Here we use another objective function, signal transmission efficiency, to study the adaptive response of a signaling system to perturbation in concentrations. This equals maximization of concentration change in a target species in response to input signal. Optimization of growth and optimization of signal transmission are therefore related.
Signal transmission in a cellular system may have multiple functions: transcription factor activation, which may be related to the cell cycle, to morphological change or to adjustment of protein concentrations, cytosolic kinase/phosphatase activation with multiple cellular targets, membrane protein activation such as ion channels or receptors, etc. We assume that signal transduction has been optimized by evolution, and that concentration adjustment exists to maintain effective signal transmission. We have shown how to optimize a single-input single-output system for both speed and signal efficiency due to the basic properties of kinetic equations [@Scheler2012]. It is easy to extend the present discussion to optimize for multiple outputs in parallel, and a system could also be optimized for a number of I/O functions. In that case other measures, such as mutual information, may also be employed as objectives. We used optimization in a two-step process: (a) in evolutionary learning, in order to find kinetic rates for estimated concentrations (b) in cellular adaptation, in order to re-calibrate the model in response to perturbations in concentrations, changes in extracellular signaling, or to effect a transient vs. plateau-like response. This also means that we have transitioned from a biologically defined system that is built bottom-up from available parameters and data to a self-organizing, learning system that adjusts to changes on evolutionary or individual time-scales.
Matching models to experimental time-series data is an under-constrained problem that often yields large ranges of suitable parameters [@Secrier2009; @Gutenkunst2007]. Analysing signals and targets to find optimal transmission parameters may be used to further constrain and investigate the parametric space. Signal transmission efficiency in a biological system can be measured directly [@Cheong2011; @Brennan2012] and be compared to what is theoretically possible given a set of equations. During evolution, new protein subtypes develop with a different set of interactions and regulations. This corresponds to an adjustment of the available set of reactions, a type of structural learning to overcome the bottlenecks that are a result of tightly specified molecular kinetics and limited adjustment of concentrations.
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![A. Biochemical Reaction System with Selected Input(red) and Output(blue) B. (Top) ISO-pVASP transfer function (with RGS KO,red)) in the MEF model [@Blackman2011] (Bottom) Experimental Response to RGS KO for GPCR signal-response in a yeast model [@Yi2003] C. (Top) Distribution of Extracellular Signals Used to Calculate Signal Transmission. From a baseline signal $S_0$ \[10nM-900nM\] the extracellular signaling level $S_t$ rises to 110%,150% and 200%, but not above 1$\mu M$. (Bottom) Transfer Functions for ISO $\rightarrow$ pVASP. Shown is the original model, optimization by delay, by $R_{C,S}$, and by both. []{data-label="fig1a"}](R7o_graph_comp1c.eps "fig:"){width="2.5in" height="1.8in"}
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![A. Biochemical Reaction System with Selected Input(red) and Output(blue) B. (Top) ISO-pVASP transfer function (with RGS KO,red)) in the MEF model [@Blackman2011] (Bottom) Experimental Response to RGS KO for GPCR signal-response in a yeast model [@Yi2003] C. (Top) Distribution of Extracellular Signals Used to Calculate Signal Transmission. From a baseline signal $S_0$ \[10nM-900nM\] the extracellular signaling level $S_t$ rises to 110%,150% and 200%, but not above 1$\mu M$. (Bottom) Transfer Functions for ISO $\rightarrow$ pVASP. Shown is the original model, optimization by delay, by $R_{C,S}$, and by both. []{data-label="fig1a"}](R7o_RGS_exp1_norm.eps "fig:"){width="1.7in"}
![A. Biochemical Reaction System with Selected Input(red) and Output(blue) B. (Top) ISO-pVASP transfer function (with RGS KO,red)) in the MEF model [@Blackman2011] (Bottom) Experimental Response to RGS KO for GPCR signal-response in a yeast model [@Yi2003] C. (Top) Distribution of Extracellular Signals Used to Calculate Signal Transmission. From a baseline signal $S_0$ \[10nM-900nM\] the extracellular signaling level $S_t$ rises to 110%,150% and 200%, but not above 1$\mu M$. (Bottom) Transfer Functions for ISO $\rightarrow$ pVASP. Shown is the original model, optimization by delay, by $R_{C,S}$, and by both. []{data-label="fig1a"}](yeast.eps "fig:"){width="1.5in" height="1in"}
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![A. Biochemical Reaction System with Selected Input(red) and Output(blue) B. (Top) ISO-pVASP transfer function (with RGS KO,red)) in the MEF model [@Blackman2011] (Bottom) Experimental Response to RGS KO for GPCR signal-response in a yeast model [@Yi2003] C. (Top) Distribution of Extracellular Signals Used to Calculate Signal Transmission. From a baseline signal $S_0$ \[10nM-900nM\] the extracellular signaling level $S_t$ rises to 110%,150% and 200%, but not above 1$\mu M$. (Bottom) Transfer Functions for ISO $\rightarrow$ pVASP. Shown is the original model, optimization by delay, by $R_{C,S}$, and by both. []{data-label="fig1a"}](R7o_signals.eps "fig:"){width="1.5in"}
![A. Biochemical Reaction System with Selected Input(red) and Output(blue) B. (Top) ISO-pVASP transfer function (with RGS KO,red)) in the MEF model [@Blackman2011] (Bottom) Experimental Response to RGS KO for GPCR signal-response in a yeast model [@Yi2003] C. (Top) Distribution of Extracellular Signals Used to Calculate Signal Transmission. From a baseline signal $S_0$ \[10nM-900nM\] the extracellular signaling level $S_t$ rises to 110%,150% and 200%, but not above 1$\mu M$. (Bottom) Transfer Functions for ISO $\rightarrow$ pVASP. Shown is the original model, optimization by delay, by $R_{C,S}$, and by both. []{data-label="fig1a"}](R7oc50_rates.eps "fig:"){width="1.5in"}
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![A. Distribution of systems according to R and d values. $\sim$ 1600 different systems states were randomly generated with k-parameter variations (20-500%), $R_{C,S}$ and $d$ values were measured with the signal set shown on top. B. Kd and KM Parameters for high efficiency (red, $f<2$) and low efficiency (green, $f>5$) systems. C. Cross-correlation of Kd values.[]{data-label="fig:param"}](signals.eps "fig:"){width="2in" height="0.5in"}
![A. Distribution of systems according to R and d values. $\sim$ 1600 different systems states were randomly generated with k-parameter variations (20-500%), $R_{C,S}$ and $d$ values were measured with the signal set shown on top. B. Kd and KM Parameters for high efficiency (red, $f<2$) and low efficiency (green, $f>5$) systems. C. Cross-correlation of Kd values.[]{data-label="fig:param"}](MC_all.eps "fig:"){width="2in"}
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![A. Distribution of systems according to R and d values. $\sim$ 1600 different systems states were randomly generated with k-parameter variations (20-500%), $R_{C,S}$ and $d$ values were measured with the signal set shown on top. B. Kd and KM Parameters for high efficiency (red, $f<2$) and low efficiency (green, $f>5$) systems. C. Cross-correlation of Kd values.[]{data-label="fig:param"}](MC_good_bad_KmKd_1.eps){width="2.5in" height="2in"}
![A. Distribution of systems according to R and d values. $\sim$ 1600 different systems states were randomly generated with k-parameter variations (20-500%), $R_{C,S}$ and $d$ values were measured with the signal set shown on top. B. Kd and KM Parameters for high efficiency (red, $f<2$) and low efficiency (green, $f>5$) systems. C. Cross-correlation of Kd values.[]{data-label="fig:param"}](MC_good_bad_scatter_Kd.eps){width="2.2in" height="1.8in"}
![A. Concentration Changes in Response to Total Protein Concentration Reduction. Selected concentrations were reduced to 10% (shown on the x-axis). Learning was applied until $f$ values were improved (shown on top for each experiment). Colors show the relative adjustment of all concentrations on the y-axis. B. Transfer Function Response to Shifts in Input Signal Range (shown on top). C. Concentrations Changes in Response to Extracellular Signal Shift. Matrix is constructed as in A.[]{data-label="fig:fig_conc_reduct"}](fig_conc_reduct.eps){width="2.4in" height="1.6in"}
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![A. Concentration Changes in Response to Total Protein Concentration Reduction. Selected concentrations were reduced to 10% (shown on the x-axis). Learning was applied until $f$ values were improved (shown on top for each experiment). Colors show the relative adjustment of all concentrations on the y-axis. B. Transfer Function Response to Shifts in Input Signal Range (shown on top). C. Concentrations Changes in Response to Extracellular Signal Shift. Matrix is constructed as in A.[]{data-label="fig:fig_conc_reduct"}](signals_low.eps "fig:"){width="0.8in" height="0.46in"} ![A. Concentration Changes in Response to Total Protein Concentration Reduction. Selected concentrations were reduced to 10% (shown on the x-axis). Learning was applied until $f$ values were improved (shown on top for each experiment). Colors show the relative adjustment of all concentrations on the y-axis. B. Transfer Function Response to Shifts in Input Signal Range (shown on top). C. Concentrations Changes in Response to Extracellular Signal Shift. Matrix is constructed as in A.[]{data-label="fig:fig_conc_reduct"}](signals_high.eps "fig:"){width="0.8in" height="0.46in"}
![A. Concentration Changes in Response to Total Protein Concentration Reduction. Selected concentrations were reduced to 10% (shown on the x-axis). Learning was applied until $f$ values were improved (shown on top for each experiment). Colors show the relative adjustment of all concentrations on the y-axis. B. Transfer Function Response to Shifts in Input Signal Range (shown on top). C. Concentrations Changes in Response to Extracellular Signal Shift. Matrix is constructed as in A.[]{data-label="fig:fig_conc_reduct"}](low-high-psf-norm.eps "fig:"){width="2in" height="1in"}
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![A. Concentration Changes in Response to Total Protein Concentration Reduction. Selected concentrations were reduced to 10% (shown on the x-axis). Learning was applied until $f$ values were improved (shown on top for each experiment). Colors show the relative adjustment of all concentrations on the y-axis. B. Transfer Function Response to Shifts in Input Signal Range (shown on top). C. Concentrations Changes in Response to Extracellular Signal Shift. Matrix is constructed as in A.[]{data-label="fig:fig_conc_reduct"}](low-high-matrix.eps){width="2.1in" height="1.6in"}
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![ A. (top) Dynamics of selected systems with low ($f_{transient}<3$,blue) or high ($f_{transient}>10$,red) propensity for a transient response. (bottom) Distribution of concentration parameters according to $d^p$,$R^p_{C,S}$ (grey) or $R_{C,S}$,$d$ (red). B. Concentration changes in response to optimizing for transients, with plateau signaling (left) and strong transients (right), sorted by $f_{transient}$. C. Concentration shifts (orange=high, green=low) in the biochemical reaction network for a high transient system, as averaged from B. Conforming to intuition, we see that the earlier, driver complex (G-protein) is high for transients, which increase the on-slope for cAMP production.[]{data-label="transient"}](dyn-good-bad.eps "fig:"){width="2in" height="0.8in"}
![ A. (top) Dynamics of selected systems with low ($f_{transient}<3$,blue) or high ($f_{transient}>10$,red) propensity for a transient response. (bottom) Distribution of concentration parameters according to $d^p$,$R^p_{C,S}$ (grey) or $R_{C,S}$,$d$ (red). B. Concentration changes in response to optimizing for transients, with plateau signaling (left) and strong transients (right), sorted by $f_{transient}$. C. Concentration shifts (orange=high, green=low) in the biochemical reaction network for a high transient system, as averaged from B. Conforming to intuition, we see that the earlier, driver complex (G-protein) is high for transients, which increase the on-slope for cAMP production.[]{data-label="transient"}](rcs_delayT_NT01-4.eps "fig:"){width="2in" height="1in"}
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![ A. (top) Dynamics of selected systems with low ($f_{transient}<3$,blue) or high ($f_{transient}>10$,red) propensity for a transient response. (bottom) Distribution of concentration parameters according to $d^p$,$R^p_{C,S}$ (grey) or $R_{C,S}$,$d$ (red). B. Concentration changes in response to optimizing for transients, with plateau signaling (left) and strong transients (right), sorted by $f_{transient}$. C. Concentration shifts (orange=high, green=low) in the biochemical reaction network for a high transient system, as averaged from B. Conforming to intuition, we see that the earlier, driver complex (G-protein) is high for transients, which increase the on-slope for cAMP production.[]{data-label="transient"}](heatmap_scaled_NT01.eps){width="2.4in" height="1.6in"}
![ A. (top) Dynamics of selected systems with low ($f_{transient}<3$,blue) or high ($f_{transient}>10$,red) propensity for a transient response. (bottom) Distribution of concentration parameters according to $d^p$,$R^p_{C,S}$ (grey) or $R_{C,S}$,$d$ (red). B. Concentration changes in response to optimizing for transients, with plateau signaling (left) and strong transients (right), sorted by $f_{transient}$. C. Concentration shifts (orange=high, green=low) in the biochemical reaction network for a high transient system, as averaged from B. Conforming to intuition, we see that the earlier, driver complex (G-protein) is high for transients, which increase the on-slope for cAMP production.[]{data-label="transient"}](R7oTc10_delta.eps){width="2.5in" height="1.8in"}
System no basic $R_{C,S}$ mean $R_{C,S}$ std $R_{C,S}$ basic $d_{S}$ mean $d_{S}$ std $d_{S}$
----------- ----------------- ---------------- --------------- --------------- -------------- -------------
1 0.578 0.582 0.0311 757 779 41.8
2 0.623 0.623 0.0419 723 722 35.4
3 0.575 0.575 0.0483 732 740 34.89
4 0.782 0.792 0.0571 749 768 50.45
5 0.609 0.622 0.099 578 590 46.82
6 0.609 0.615 0.033 728 737 37.05
7 0.612 0.628 0.0636 659 663 18.43
8 0.672 0.666 0.061 658 662 25.36
9 0.703 0.715 0.049 782 812 44.66
10 0.655 0.651 0.0511 761 760 29.3
11 0.576 0.551 0.0698 649 638 44.94
12 0.54 0.56 0.108 664 669 20.91
13 0.694 0.695 0.043 692 695 22.5
14 0.504 0.501 0.0312 726 731 35.70
15 0.522 0.526 0.0594 620 621 24.49
16 0.8 0.806 0.0369 726 742 50
17 0.693 0.7 0.0341 717 724 24.03
18 1.24 1.29 0.221 761 779 59.1
19 0.651 0.63 0.0512 668 666 11.64
20 0.529 0.5 0.045 777 762 43.25
21 0.537 0.534 0.044 752 754 33.42
: (Supplemental) For 21 systems with $R_{C,s}>0.5$ und $d_S<800$, we re-calculate $R_{C,s}$ and $d_S<800$ from 20 variations on concentration values within a 20% interval. Shown are the original, the mean and std values.[]{data-label="robustness"}
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---
abstract: 'We consider an epidemic process on adaptive activity-driven temporal networks, with adaptive behaviour modelled as a change in activity and attractiveness due to infection. By using a mean-field approach, we derive an analytical estimate of the epidemic threshold for SIS and SIR epidemic models for a general adaptive strategy, which strongly depends on the correlations between activity and attractivenesses in the susceptible and infected states. We focus on a strong adaptive behaviour, implementing two types of quarantine inspired by recent real case studies: an active quarantine, in which the population compensates the loss of links rewiring the ineffective connections towards non-quarantining nodes, and an inactive quarantine, in which the links with quarantined nodes are not rewired. Both strategies feature the same epidemic threshold but they strongly differ in the dynamics of active phase. We show that the active quarantine is extremely less effective in reducing the impact of the epidemic in the active phase compared to the inactive one, and that in SIR model a late adoption of measures requires inactive quarantine to reach containment.'
author:
- Marco Mancastroppa
- Raffaella Burioni
- Vittoria Colizza
- Alessandro Vezzani
bibliography:
- 'BiblioAdaptive.bib'
title: 'Active and inactive quarantine in epidemic spreading on adaptive activity-driven networks'
---
*Introduction.* Understanding the effects of changes in the behaviour of individuals during epidemics is essential to improve response strategies and to foster containment [@satorras2015epidemic; @fraser2004; @hollingsworth2011mitigation; @anderson2020individual; @masuda2013predicting]. This problem is nowadays particularly relevant, due to the extraordinary measures, massively introduced to limit recent disease spreading [@who_declaration_public; @who_situation_report; @quarantine_Italy2; @quarantine_China]. In the presence of epidemics, people adapt their behaviour, modifying their actions in several ways. Infected individuals partially or totally reduce their activity, due to the appearance of symptoms or, if they are asymptomatic, due to limitations. Analogously, infected individuals can experience a reduction in their attractiveness, due to the self-protective behaviour of other individuals, who try to avoid contacts with the infected ones. All these adaptive behaviours lead to a change in activity, which can be realized in different ways, due to the peculiar structure of the society or to the perception of severity. As an example, as seen in recent real case studies [@anderson2020individual; @active_Italy], depending on the possibility to implement strong containment measures, limitation of activity in a population can be implemented in an [*active*]{} or [*inactive*]{} way. In the former, individuals avoid contacts with infected nodes, but they can readdress their activity towards non infected individuals, only shifting their focus. In the latter, individuals really become less active, in the sense that if a link is to be established with an infected node, the individual decide not to take the action, therefore reducing the overall activity of the system. A modelling framework of these effects could help in quantifying the relevance of quarantine measures [@gross2006adaptive; @lagorio2011], and to assess how important is the rapidity of adoption of adaptive measures to reach containment in the population. Simplified models, amenable to analytic control but still including the relevant dynamics, can be of some help here, as they allow testing the relevant parameters best-affecting disease containment and they can lead to uncovering counterintuitive effects, related to feedback processes in the dynamics. The natural framework to model epidemics is that of temporal networks, where links between nodes evolve in time on the same time scale of the dynamical process [@satorras2015epidemic; @holme2013temporal; @valdano2018epidemic; @holme2014birth; @perra2012activity; @liu2014controlling; @ping2018epidemic; @rizzo2014effect; @moinet2018effect; @starnini2013topological; @masuda2013predicting].
In this paper we focus on the susceptible-infected-susceptible (SIS) and susceptible-infected-recovered (SIR) models on activity-driven networks [@perra2012activity; @starnini2013topological; @ubaldi2016asymptotic; @ubaldi2017burstiness; @burioni2017asymptotic; @tizzani2018memory; @mancastroppa2019; @kim2018memory] with adaptive behaviour [@funk2010review; @feniche2011adaptive]. In activity-driven networks, the propensity to engage an interaction is modelled by assigning to each node an activity potential, measuring the typical number of activations (link formations) per time performed by the agent, plus an attractiveness, describing how likely it is for a node to be contacted by others [@pozzana2017attractiveness]. The adaptive behaviour results in a change in activity and attractiveness due to infection, expressed through a general distribution function for activity and attractiveness in infected and susceptible nodes. By using an *activity-attractiveness based mean-field* approach, we derive an analytical estimate of the epidemic threshold for SIS and SIR epidemic models, holding for all active and inactive adaptive strategies. We also obtain analytically the epidemic prevalence of the SIS process (endemic state), while for the SIR active phase we perform numerical simulations. Interestingly, the threshold strongly depends on the correlations between the activities and attractivenesses in the susceptible and infected states. In particular, we focus on *active* and *inactive quarantine*, showing that the two containment measures have the same epidemic threshold, but they strongly differ in the dynamic of the active phase. As a result, the active quarantine is extremely less effective in reducing the impact of the epidemic compared to the inactive one, near to the epidemic threshold. We also uncover the strong effects of early adoption of quarantine measures. We show that early adoption can potentially allow to contain the epidemic with an active quarantine, without affecting the activity of the healthy, while a late adoption requires a strong reduction of the overall activity to reach an effective containment.\
*The model.* We consider the SIS model on adaptive activity-driven networks [@perra2012activity; @ubaldi2016asymptotic; @pozzana2017attractiveness]: each node can be susceptible ($S$) or infected ($I$) and it is characterized by two activity parameters $a_S,a_I$ and two attractiveness parameters $b_S,b_I$ drawn form the joint distribution $\rho(a_S,a_I,b_S,b_I)$. Node activations occur with a Poisson process, with activation rate $a_S$ or $a_I$, according to the node’s status. Initially all $N$ nodes are disconnected and when a node activates it creates $m$ links with $m$ randomly selected nodes (hereafter we fix $m=1$): the probability to contact a node with attractiveness $b_i$ (with $i=S,I$) is given by $p_{b_i}=b_i/\alpha$, where $\alpha$ is a normalization factor and depends on the adaptive behaviour as we will discuss. All links are deleted after a time step and the procedure is iterated. If a link connects an infected $I$ and a susceptible $S$ node, a contagion occurs with probability $\lambda$: $S+I \xrightarrow[]{\lambda} 2I$, otherwise nothing happens. Infected nodes recover with rate $\mu$, through a Poissonian process: $S \xrightarrow[]{\mu} I$. We call $P_{a_S,a_I,b_S,b_I}(t)$ the probability that a node with activities and attractivenesses $(a_S,a_I,b_S,b_I)$ is infected at time $t$.
The adaptive behaviour can be modelled in two ways, which we will call *active* and *inactive*. In the active case, an active node connects for sure with one of the other nodes. In this case, the normalization factor $\alpha$ is the average attractiveness of the system at time $t$: $\alpha=\langle b(t)\rangle=\int da_S \, da_I \, db_S \, db_I \, \rho(a_S,a_I,b_S,b_I) [b_S (1-P_{a_S,a_I,b_S,b_I}(t)) + b_I P_{a_S,a_I,b_S,b_I}(t)]=
\overline{b_S} + \overline{b_IP}(t)-\overline{b_S P}(t) $ where we define $\overline{f}(t)=\int da_S \, da_I \, db_S \, db_I \, \rho(a_S,a_I,b_S,b_I) f_{a_S,a_I,b_S,b_I}(t)$. $P_{a_S,a_I,b_S,b_I}(t)$ evolves in time according to the following equation:
$$\partial_t P_{a_S,a_I,b_S,b_I}(t) = -\mu P_{a_S,a_I,b_S,b_I}(t)+\lambda (1-P_{a_S,a_I,b_S,b_I}(t))\frac{a_S \overline{b_IP}(t)+b_S \overline{a_I P}(t)}{\overline{b_S} + \overline{b_IP}(t)-\overline{b_S P}(t)}.
\label{eq:EQ_active}$$
In the inactive case, an active node may not connect to any of the nodes due to the reduction of the average attractiveness of the system caused by the infection process. In this case, at each time $t$ we set $\alpha=\overline{b_S}$ i.e. the average attractiveness when all sites are susceptible. The probability for an active node not to connect is: $(\overline{b_S}-\langle b(t)\rangle)/\overline{b_S}$ and the evolution of $P_{a_S,a_I,b_S,b_I}(t)$ is given by:
$$\partial_t P_{a_S,a_I,b_S,b_I}(t) = -\mu P_{a_S,a_I,b_S,b_I}(t)+\lambda (1-P_{a_S,a_I,b_S,b_I}(t))\frac{a_S \overline{b_IP}(t)+b_S \overline{a_I P}(t)}{\overline{b_S}}.
\label{eq:EQ_inactive}$$
Eq.s (\[eq:EQ\_active\],\[eq:EQ\_inactive\]) are exact due to the mean-field nature of the model, since local correlations are destroyed at each time step and since we do not consider memory [@tizzani2018memory; @kim2018memory] or burstiness effects [@mancastroppa2019; @ubaldi2017burstiness; @burioni2017asymptotic].
The SIS model features a phase transition between an absorbing and an active phase: the control parameter is $r = \lambda/\mu$ (effective infection rate). Through a linear stability analysis, we obtain the epidemic threshold $r_C$, which remarkably is the same for the active and inactive adaptive behaviour: $$r_C=\left. \frac{\lambda}{\mu}\right|_C=\frac{2 \overline{b_S}}{
\overline{a_S b_I} + \overline{a_I b_S} +
\sqrt {(\overline{a_S b_I} - \overline{a_I b_S})^2 + 4
\overline{a_S a_I} \, \overline{b_S b_I}}}.$$
The threshold strongly depends on the correlations between the activities and attractivenesses in the susceptible and infected states; moreover it reproduces the non-adaptive case without attractiveness [@perra2012activity] (fixing $a_I=a_S$ and $b_I=b_S=1$) $r_C^{NAD}=( \overline{a_S} + \sqrt{\overline{a_S^2}})^{-1}$ and the non-adaptive case with attractiveness (NADwA) [@pozzana2017attractiveness] (fixing $b_I=b_S$ and $a_I=a_S$) $r_C^{NADwA}=\overline{b_S} (\overline{a_S b_S} + \sqrt{\overline{a_S^2} \cdot \overline{b_S^2}})^{-1}$. If $b_S=c a_S$ ($c$ arbitrary constant), as observed in real systems [@pozzana2017attractiveness], we obtain $r_C^{NADwA}=\overline{a_S}/[2 \overline{a_S^2}]$.
From Eq.s (\[eq:EQ\_active\],\[eq:EQ\_inactive\]), the stationary infection probability $P_{a_S,a_I,b_S,b_I}^0=\lim\limits_{t \to \infty} P_{a_S,a_I,b_S,b_I}(t)$ in the active (Eq. ) and inactive (Eq. ) cases is: $$P_{a_S,a_I,b_S,b_I}^0=\frac{a_S \overline{b_IP}+b_S\overline{a_IP}}{\frac{\mu}{\lambda}(\overline{b_S} + \overline{b_I P} - \overline{b_S P})+a_S \overline{b_IP}+b_S\overline{a_IP}},
\label{eq:P_active}$$ $$P_{a_S,a_I,b_S,b_I}^0=\frac{a_S \overline{b_IP}+b_S\overline{a_IP}}{\frac{\mu}{\lambda} \overline{b_S} +a_S \overline{b_IP}+b_S\overline{a_IP}}.
\label{eq:P_inactive}$$
Contrarily to the epidemic threshold, the active phase depends on the implementations of the adaptive behaviour. This suggests that it is not enough to investigate the epidemic threshold, or the basic reproductive number $R_0$: the same population with two different dynamics feature a different epidemic active phase.
We remark that the proposed model allows studying several adaptive behaviours; in particular by setting the functional form of $\rho(a_S,a_I,b_S,b_I)$ one can model a mild social distancing, like the sick-leave practice where only activity is reduced in infected nodes [@ariza2018healthcareseeking]; a targeted adaptive prescription where $a_I$ and $b_I$ vary only for some activity classes, more exposed to risk [@anderson2020individual; @mossong2008social]; moreover one can consider different distributions of $a_S$ and $b_S$ modelling different social systems [@pozzana2017attractiveness; @perra2012activity]. Here we will focus on a strong social distancing approach like *quarantine*.\
*Active and inactive quarantine.* Nowadays the effects of quarantine on epidemic spreading are of enormous interest given the extraordinary isolation measures to limit the spread of COVID-19 taken by several countries [@who_declaration_public; @who_situation_report; @quarantine_China; @quarantine_Italy2]. In our model we consider that a fraction $\delta$ of the nodes, when infected, goes to quarantine by setting both $a_I$ and $b_I$ to zero. Indeed, we expect that a fraction of the infected nodes does not perform quarantine since they may not be aware of the infection (no symptoms) or they do not follow the prescriptions of social distancing: thus $\delta$ takes into account the uncertainty in the application of the containment measures. Given the recent implementations of extended quarantine [@quarantine_China; @quarantine_Italy2] we set $\delta$ in the range $0.7-0.9$, considering a population which responds effectively to recommendations and control measures.
Our approach naturally distinguishes an active quarantine from an inactive one. In the active quarantine an active node selects randomly its contact among the non-quarantining nodes: links directed to quarantining nodes are effectively rewired towards potentially susceptible or non-quarantining infected nodes. Dynamical link rewiring can produce non-trivial effects on epidemic spreading, as observed in [@gross2006adaptive] for static network. This behaviour creates a temporally connected cluster of not-quarantining nodes and this can strengthen the epidemic [@scarpino2016effect; @gross2006adaptive]. In the inactive case the population does not compensate the ineffective links towards nodes in quarantine, by forming links with other nodes, and their activation in that case will not be effective.
We consider a system with $b_S = c a_S$, accordingly to observations on real data [@pozzana2017attractiveness], which denote a linear correlation between the two parameters, identifying the presence of hubs which are very active, generating many links, and very attractive, receiving just as many. If a fraction $\delta$ of the population performs quarantine ($a_I=b_I=0$), while the remaining $1-\delta$ keeps $a_I=a_S$ and $b_I=b_S$, we get $\rho(a_S,a_I,b_S,b_I)= \rho_S(a_S) \delta(b_S-c a_S)[(1-\delta) \delta(a_I-a_S) \delta(b_I-b_S)+ \delta \cdot \delta(a_I) \delta(b_I)]$, where $\delta(x)$ is the Dirac-delta function. The epidemic threshold is: $$r_C^{quarantine}=\frac{1}{1-\delta}\frac{\overline{a_S}}{2 \overline{a_S^2}}=\frac{r_C^{NADwA}}{1-\delta}.
\label{eq:soglia}$$ The quarantine increases the epidemic threshold of the non adaptive case by a factor $(1-\delta)^{-1}$ which is particularly significant when $\delta \sim 1$.
Although the active and inactive quarantine display the same epidemic threshold, the active phase is different. The stationary infection probability in the active quarantine is: $$P_{a_S,a_I,b_S,b_I}^0=\frac{2 a_S \overline{a_SP}}{\frac{\mu}{\lambda} \left( \frac{\overline{a_S}-\overline{a_SP}}{1-\delta}+\overline{a_SP} \right) + 2 a_S \overline{a_S P}},
\label{eq:P_SIS_active}$$ while, in the inactive case it is: $$P_{a_S,a_I,b_S,b_I}^0=\frac{2 a_S \overline{a_SP}}{\frac{\mu}{\lambda}\frac{\overline{a_S}}{1-\delta}+ 2 a_S \overline{a_SP}}.
\label{eq:P_SIS_inactive}$$
![Effects of quarantine on SIS active phase. In panel (a) we plot the epidemic prevalence $\overline{P}$ as a function of $r/r_C$ (with $r_C$ the threshold for the adaptive case), for the NADwA case and for the active/inactive quarantines, fixing $\delta=0.9$. In panel (b) we plot the ratio $\overline{P}^{ACT}/\overline{P}^{INACT}$ between the epidemic prevalence in the active and inactive case as a function of $r/r_C$ for several values of $\delta$. In both panels, $\nu=0.5$, $a_m=10^{-3}$, $a_M=1$.[]{data-label="fig:SIS_prev"}](figure1.eps){width="45.00000%"}
Eq.s (\[eq:P\_SIS\_active\],\[eq:P\_SIS\_inactive\]) can be solved self consistently fixing $\rho_S(a_S)$: we consider a power-law distribution $\rho_S(a_S) \sim a_S^{-(\nu+1)}$ with $a_S \in [a_m,a_M]$ [@perra2012activity; @ubaldi2016asymptotic], modelling the presence of heterogeneities and large activity fluctuations. Indeed, many real human systems feature a broad power-law distribution of $a_S$ with exponent $\nu \sim 0.3-1.5$ [@ubaldi2016asymptotic; @perra2012activity; @karsai2014time]. Heterogeneities account for different social propensity in engaging social interactions (e.g. different works, social roles), producing different number of contacts over time [@perra2012activity; @ubaldi2016asymptotic]: in the following we will fix $\nu=0.5$.
In Fig. \[fig:SIS\_prev\](a) we compare the epidemic prevalence $\overline{P}=\int da_S \, da_I \, db_S \, db_I \, \rho(a_S,a_I,b_S,b_I) P_{a_S,a_I,b_S,b_I}^0$ in the NADwA case ($\delta=0$), with that of the active/inactive quarantine at $\delta=0.9$. The inactive quarantine deeply lowers the epidemic prevalence in the active phase compared to the NADwA case. On the contrary, the active quarantine produces a much higher epidemic prevalence which is similar to the non-adaptive one, making this strategies extremely less effective than the inactive one. Indeed, in active quarantine all connections are created among nodes which do not perform quarantine, giving rise to temporal clusters of connected susceptible and infected nodes, where all links are potentially contagious. In Fig. \[fig:SIS\_prev\](b) we plot the ratio between the epidemic prevalence of the active and inactive case as a function of $r/r_C$. The difference between the two quarantines is maximized when $r/r_C \sim 5-10$ (depending on $\delta$) and it increases with higher $\delta$ values. Also for more realistic $r/r_C \sim 1$ values, where the quarantine is effective in moving the systems near to the critical point, the active quarantine produce an epidemic prevalence which is about twice the inactive one.\
*Effects of quarantine on SIR epidemic model.* In the SIR model, a recovered node is no more susceptible but enters at the recovery rate $\mu$ into the immune state $R$ where it cannot be infected: $I \xrightarrow[]{\mu} R$. We consider the SIR model on our activity-driven model: each node is described by six parameters $(a_S,a_I,a_R,b_S,b_I,b_R)$ extracted from the joint distribution $\rho_{SIR}(a_S,a_I,a_R,b_S,b_I,b_R)$. Hereafter, we consider the case in which all recovered nodes behave as if they were susceptible $a_R=a_S$ and $b_R=b_S$: $\rho_{SIR}(a_S,a_I,a_R,b_S,b_I,b_R)=\rho(a_S,a_I,b_S,b_I) \delta(b_R-b_S) \delta(a_R-a_S)$, with $\rho(a_S,a_I,b_S,b_I)$ equal to the previous distribution for quarantine.
Due to the mean-field nature of the model, the epidemic threshold is the same of the SIS model independently of $a_R$ and $b_R$ and it is equal for active and inactive quarantine (Eq. \[eq:soglia\]). On the contrary, the dynamics in the active phase is known to be different. In particular, the SIR model lacks an endemic steady state, since the dynamics always halts reaching a state in which no infected nodes are present. The final fraction of recovered nodes $R_\infty$ is used as an order parameter, however the stationary condition on the dynamical equation does not provide a solution for $R_\infty$ and numerical simulation are necessary to obtain insight on the dynamics.
In this perspective, we implement the SIR model with a Gillespie-like algorithm [@gillespie1976general; @mancastroppa2019], in a system of $N$ nodes and we average over a number of realizations of the dynamical evolution and of the disorder, so that the error on $R_{\infty}$ is lower than 1%. The initial conditions are imposed, by infecting the node with the highest activity $a_I$ [@boguna2013], immediately implementing the quarantine measures.
![Effects of quarantine on the SIR active phase. In panel (a) we plot the epidemic final-size $R_{\infty}$ as a function of $r/r_C$ (with $r_C$ the threshold for the adaptive case), for the NADwA case and for the active/inactive quarantines. In panel (b) we plot the temporal evolution of the average infection probability $\overline{P}(t)$ for the NADwA case and for the active/inactive quarantines, fixing $r/r_C=1.4$. In both panels, $\delta=0.9$, $N=10^3$, $\nu=0.5$, $a_m=10^{-3}$, $a_M=1$[]{data-label="fig:SIR_prev"}](figure2.eps){width="45.00000%"}
In Fig. \[fig:SIR\_prev\](a) we compare $R_{\infty}$ of the NADwA case with the active/inactive quarantines, computed as a function of $r/r_C$, fixing the same $\delta=0.9$. Both quarantines deeply lowers the epidemic final-size compared to the non-adaptive case, however the active quarantine produces an higher $R_{\infty}$ than the inactive one. The epidemic final-size of the active case is about 10% higher than the inactive one for small infectivity $r/r_C \sim 1$, i.e. the significant regime for an effective quarantine. Moreover, both quarantine strategies greatly impact on the SIR dynamics: in Fig. \[fig:SIR\_prev\](b) we plot the temporal evolution of the average infection probability $\overline{P}(t)$ for $\delta=0.9$ and $r/r_C = 1.4$. The infection peak is strongly flattened compared to the NADwA case: it occurs approximatively at the same time and in general its height for the active quarantine is 10%-20% larger than the inactive one for $r/r_C \sim 1$ (depending on $\delta$ value). These estimates outlines a difference between the two strategies even if the effect is much smaller compared to the SIS model.
Thus far we considered quarantine measures implemented at the beginning of the epidemic, however typically the containment measures are applied only after a fraction $\beta$ of the population has been infected. This is relevant for the SIR model, since the dynamics and $R_\infty$ depend strongly on the initial conditions, contrary to the SIS. Therefore, we let the epidemic process evolve on the NADwA network, without adaptive behaviours, until a fraction $\beta$ of the population has been infected. Then, the quarantine measures are implemented and are kept effective until the end of the epidemic.
![Effects of timing in quarantine implementations. It is plotted the epidemic final-size $R_{\infty}$ as a function of $\beta$ fraction of ever infected nodes when quarantine measures are implemented. We consider the active and inactive quarantines, fixing $\delta=0.9$ and we compare them with the NADwA case. In the insert we plot the ratio $R_{\infty}^{ACT}/R_{\infty}^{INACT}$ between the epidemic final-size in the active and inactive case as a function of $\beta$ for several $\delta$ values. In both panels, $r/r_C=1.4$ (with $r_C$ the threshold for the adaptive case), $N=10^3$, $\nu=0.5$, $a_m=10^{-3}$, $a_M=1$.[]{data-label="fig:SIR_fraz_Rinf"}](figure3.eps){width="45.00000%"}
![Effects of timing in quarantine implementations. In panel (a) and (b) we plot the temporal evolution of the average infection probability $\overline{P}(t)$ for the NADwA case and for the active/inactive quarantine, fixing $\delta=0.9$ and respectively $\beta=0.0025$ and $\beta=0.02$. In panel (c) we plot the ratio $\overline{P}_{max}^{ACT}/\overline{P}_{max}^{INACT}$ between the height of the infection peak in the active and inactive case as a function of $\beta$, for several values of $\delta$. In all panels, $r/r_C=1.4$ (with $r_C$ the threshold for the adaptive case), $N=10^3$, $\nu=0.5$, $a_m=10^{-3}$, $a_M=1$.[]{data-label="fig:SIR_fraz_picco"}](figure4.eps){width="45.00000%"}
In Fig. \[fig:SIR\_fraz\_Rinf\] we plotted $R_{\infty}$ as a function of $\beta$, fixing $\delta=0.9$ and $r/r_C=1.4$: both strategies show the importance of an early adoption of quarantine, since $R_{\infty}$ increases significantly with $\beta$, despite in a different way for the two strategies. In the inactive quarantine approximatively we get $R_{\infty}(\beta) \sim R_{\infty}(\beta=0)+\beta$, i.e. the epidemic final-size for the quarantine at $\beta = 0$ is summed to the fraction $\beta$ of infected nodes before the quarantine adoption. On the contrary, in active quarantine, $R_{\infty}$ grows very rapidly with $\beta$ and the difference between the two strategies in weakening the epidemic becomes very different. In the insert of Fig. \[fig:SIR\_fraz\_Rinf\], we show the fast growth with $\beta$ of the ratio between the epidemic final-size of the active and inactive case. If the quarantine is implemented immediately ($\beta=0$), the difference is about 10%; if the containment measures are applied when about 2% of the population has been infected $R_{\infty}$ for the active case can be up to $2.5$ times the inactive one (for $\delta =0.9$).
Finally, we investigate the effects of the timing in the quarantine adoption on the SIR temporal dynamics: in Fig. \[fig:SIR\_fraz\_picco\](a)-(b) we plot the infection peak respectively for $\beta=0.0025$ and $\beta=0.02$, fixing $\delta=0.9$ and $r/r_C=1.4$. If the fraction $\beta$ increases, both quarantines are extremely less effective in flattening the infection peak, showing the importance of early adoption of measures. Comparing these results with Fig. \[fig:SIR\_prev\](b) ($\beta=0$), for $\beta>0$ the inactive infection peak is significantly flattened compared to the active one. This is highlighted in Fig. \[fig:SIR\_fraz\_picco\](c) where we plot the ratio between the height of the infection peak in the active and inactive case. If the containment measures are applied immediately, the difference is about 10%; if they are applied when a small fraction between 0.25% and 4% of the population has been infected, the active peak is about twice larger than the inactive one at $\delta = 0.9$; a smaller but significant difference is observed also for $\delta =0.7$.\
Our results hold for a realistically heterogeneous population [@perra2012activity] with extensive containment measures ($\delta=0.7-0.9$), such as those recently implemented [@quarantine_China; @quarantine_Italy2], able to move the system just above the threshold ($r/r_C=1.4$, consistent with realistic estimates of $R_0$ [@who_declaration_public]). In particular, in this range of parameters, our findings support the crucial role of timing in the adoption of containment measures. If the measures are implemented immediately, both active and inactive strategies are effective in reducing the infection peak of 75%-85% (depending on $\delta$) compared to the NADwA case. However, any delay in the adoption of quarantine measures produces a drastic reduction in the effectiveness of the strategies. More interestingly, the differences between the two strategies increases in the case of a late adoption. This means that if containment measures are not immediately taken, an inactive quarantine, of a more stringent type, also binding the healthy to decrease their activity, is required to produce an effective result. With an early adoption, of course an inactive quarantine is still the best choice, but if this strong social distancing cannot be implemented for social reasons, an active strategy, focusing only on infected individuals, is still able to produce strong containment.
|
---
abstract: 'We study a 1D semilinear wave equation modeling the dynamic of an elastic string interacting with a rigid substrate through an adhesive layer. The constitutive law of the adhesive material is assumed elastic up to a finite critical state, beyond such a value the stress discontinuously drops to zero. Therefore the semilinear equation is characterized by a source term presenting jump discontinuity. Well-posedness of the initial boundary value problem of Neumann type, as well as qualitative properties of the solutions are studied and the evolution of different initial conditions are numerically investigated.'
address:
- ' Dipartimento di Matematica, Università di Bari, Via E. Orabona 4,I–70125 Bari, Italy.'
- ' Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via E. Orabona 4,I–70125 Bari, Italy, INFN, Sezione di Bari, I–70126 Bari, Italy.'
- ' Dipartimento di Matematica, Università di Bari, Via E. Orabona 4,I–70125 Bari, Italy.'
- ' Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via E. Orabona 4,I–70125 Bari, Italy.'
author:
- 'G. M. Coclite'
- 'G. Florio'
- 'M. Ligabò'
- 'F. Maddalena'
title: Nonlinear waves in adhesive strings
---
[^1]
Introduction {#sec:intro}
============
Adhesion, capillarity and wetting phenomena (see [@DBQ; @KKR]) constitute a challenging arena for mathematical problems due to the complexity of physical mechanisms involved. A rational understanding in the format of analytical descriptions of such problems, in addition to being in itself interesting, is relevant for both life sciences and manufacturing engineering. In some recent papers (see, e.g., [@MP; @MPPT; @MPT; @MPT1]) one of the authors has studied the static problem of adhesion of elastic thin structures under various constitutive assumptions on the adhesive material. The main goal of those works relies in characterizing, with the tools of the calculus of variations, the interplay of the occurrence of debonding with other constitutive properties. The study of the evolution problem related to these physical manifestations require the analysis of multidimensional hyperbolic problems involving mathematical issues not yet well understood. In this paper we address a prototypical dynamical problem by studying the adhesion of an elastic string glued to a rigid substrate, assuming a discontinuous softening behavior of the adhesive material, i.e. the adhesive stress jumps to zero when a critical value of the displacement is reached. We consider the mechanical system with the following energy density: $$\label{def:energydensity}
e[u]= \frac{1}{2}\rho ({\partial_t}u)^2+ \frac{1}{2} K_e ({\partial_x }u)^2 + \Phi(u),$$ where $\rho>0$ denotes the mass density, $K_e$ denotes the elastic stiffness of the string, and $\Phi(u)$ denotes the adhesion potential modeling the energetic contribution of the glue layer. To taking into account the possibility of debonding we assume for the potential $\Phi$ a behavior like in Fig. \[fig:phipotential\], for example $$\label{eq:Phi_intro}
\Phi(u)=\begin{cases}
u^2,&\qquad \text{if $|u|\le u^* $},\\
(u^*)^2,&\qquad \text{if $|u|> u^*$},
\end{cases}$$ where $u^*$ denotes the threshold beyond which the glue cannot sustain further stress.
![(Color online) Potential $\Phi(u)$ in Eq. \[eq:Phi\_intro\]. []{data-label="fig:phipotential"}](phipotential.pdf){height="35.00000%"}
We are interested in the qualitative properties of the Euler equations associated to the above energy density given by $$\label{eq:e0}
\rho {\partial_{tt}^2}u=K_e {\partial_{xx}^2}u-\Phi'\left(u\right),$$ equipped with Neumann boundary conditions.
The paper is organized as follows. In Section \[sec:pb\] we introduce the problem, the main assumptions and the associated energy. In Section \[sec:existence\] we give the definition of dissipative solution, prove existence (cf. Theorem \[th:exist\]), regularity (cf. Theorem \[th:regularity\]), and non-uniqueness for the solutions of initial boundary value problem related to (cf. Examples \[ex:1\], \[ex:2\], \[ex:3\]). In Section \[disc\] we focus on the first order formulation of the problem and investigate the interplay between debonding and propagation of singularities along characteristics (cf. Theorem \[th:sing\]). Finally, in Section \[sec:numerics\] we consider several initial conditions (in different classes of regularity), numerically investigate the evolutions and highlight peculiar behaviors of the propagation along the characteristics.
Statement of the problem {#sec:pb}
========================
Let us consider a one dimensional material body, i.e. a [*string*]{}, whose rest configuration at the initial time $t=0$ coincides with the interval $[0,L]$ and the displacement field is denoted by $$u:[0,\infty)\times[0,L]\rightarrow {\mathbb{R}}.$$ The material is assumed linear elastic and, for sake of notational simplicity, the mass density $\rho$ and the extensional stiffness $K_e$ are assumed both equal to 1. The string interacts with an underlying rigid support through an infinitesimal layer of adhesive material characterized by an internal energy $u\mapsto\Phi(u)$ with the threshold $u^*$ set to 1.
The balance of momentum delivers the initial boundary value problem $$\label{eq:el}
\begin{cases}
{\partial_{tt}^2}u={\partial_{xx}^2}u-\Phi'\left(u\right),&\quad t>0,0<x<L,\\
{\partial_x }u(t,0)={\partial_x }u(t,L)=0,&\quad t>0,\\
u(0,x)=u_0(x),&\quad 0<x<L,\\
{\partial_t}u(0,x)=u_1(x),&\quad 0<x<L.
\end{cases}$$ We shall assume that
\[ass:phi\] $\Phi\in C({\mathbb{R}})\cap C^1({\mathbb{R}}\setminus\{1,-1\})$, $\Phi$ is constant in $(-\infty,-1]$ and in $[1,\infty)$, convex in $[-1,1]$, decreasing in $[-1,0]$ and increasing in $[0,1]$;
\[ass:init\] $u_0\in H^1(0,L)$, $u_1\in L^2(0,L)$.
As a consequence of \[ass:phi\], $\Phi'$ has a jump discontinuity in $u=\pm1$ and $$\begin{aligned}
u\in(-\infty,-1)\cup(1,\infty)&\Rightarrow \Phi'(u)=0,\\
0<u<1&\Rightarrow 0<\Phi'(u)\le \lim_{u\to1^-}\Phi'(u),\\
-1<u<0&\Rightarrow 0>\Phi'(u)\ge \lim_{u\to-1^+}\Phi'(u).\end{aligned}$$ Assumption \[ass:phi\] characterizes the constitutive behavior of the adhesive material, i.e. when $\vert u\vert =1$ the loss of adhesion manifests through the jump discontinuity of the stress $\Phi'$, hence debonding of the string occurs.
To fix ideas, a function satisfying such assumption is $$\label{eq:Phi}
\Phi(u)=\begin{cases}
u^2,&\qquad \text{if $|u|\le 1$},\\
1,&\qquad \text{if $|u|> 1$}.
\end{cases}$$ In particular we have $$\label{eq:Phi'}
\Phi'(u)=\begin{cases}
2u,&\qquad \text{if $|u|\le 1$},\\
0,&\qquad \text{if $|u|>1$}.
\end{cases}$$
The natural energy associated to the problem is $$\label{en}
E(t)=\int_0^L\left(\frac{({\partial_t}u(t,x))^2+({\partial_x }u(t,x))^2}{2}+\Phi(u(t,x))\right)dx.$$ Due to the lack of Lipschitz continuity in the nonlinear term $\Phi'$ we cannot expect the existence of conservative solutions, i.e., solutions that preserve the energy. This is coherent with the physic behind the problem, when our material is ungluing, indeed in [@MPPT Sec. 3.3] the authors describe the hysteresis cycles and the dissipation associated with the maximum delay strategy corresponding to the quasistatic evolution for a discrete system where the macroscopic limit (obtained by $\Gamma$-convergence) could be viewed as the system here analyzed. Moreover, even mathematically the dissipation of energy is natural. Indeed, when we study the compactness of some approximate solutions we cannot have bounds on the second derivatives because we cannot differentiate the equation in . Therefore, we have to live with bounds on the first derivatives and then we can have only weak convergence in $H^1$.
Well-posedness and regularity of weak solutions {#sec:existence}
===============================================
This section is dedicated to the well-posedness and regularity analysis of . We show the existence of Lipshitz continuous dissipative solutions. Some examples show that those solutions are not unique and do not depend continuously on the initial conditions. Indeed, in the following section we shall focus on a qualitative analysis of the discontinuity curves of the first derivatives of the solutions. These are the loci where the dissipation of energy occurs. Therefore, it seems quite natural to introduce the concept of *dissipative solution*:
\[def:sol\] We say that a function $u:[0,\infty)\times[0,L]\to{\mathbb{R}}$ is a dissipative solution of if
- $u\in C([0,\infty)\times[0,L])$;
- ${\partial_t}u,\,{\partial_x }u\in L^\infty(0,\infty;L^2(0,L))$;
- for every test function ${\varphi}\in C^\infty({\mathbb{R}}^2)$ with compact support $$\label{eq:weak}
\begin{split}
\int_0^\infty\int_0^L& \left(u{\partial_{tt}^2}{\varphi}+{\partial_x }u{\partial_x }{\varphi}+\Phi'\left(u\right){\varphi}\right)dtdx\\
&-\int_0^L u_1(x){\varphi}(0,x)dx+\int_{\mathbb{R}}u_0(x){\partial_t}{\varphi}(0,x)dx=0;
\end{split}$$
- (energy dissipation) for almost every $t>0$ $$\label{eq:energydissip}
\begin{split}
\int_0^L&\left(\frac{({\partial_t}u(t,x))^2+({\partial_x }u(t,x))^2}{2}+\Phi(u(t,x))\right)dx\\
&\qquad\qquad\le \int_0^L\left(\frac{(u_1(x))^2+(u_0'(x))^2}{2}+\Phi(u_0(x))\right)dx.
\end{split}$$
Existence {#subsec:existence}
---------
The main result of this subsection is the following.
\[th:exist\] Let $u_0$ and $u_1$ be given and assume \[ass:phi\], \[ass:init\]. Then admits a weak solution in the sense of Definition \[def:sol\].
Our argument is based on the approximation of the Neumann problem with a sequence of Neumann problems with smooth source terms and smooth initial data.
Let $\{u_{0,n}\}_{n\in{\mathbb{N}}},\,\{u_{1,n}\}_{n\in{\mathbb{N}}}\subset C^\infty([0,L]),\,\{\Phi_n\}_{n\in{\mathbb{N}}}\subset C^\infty({\mathbb{R}})$ be sequences of smooth approximations of $u_0,\,u_1$, and $\Phi$ such that
$$\label{eq:assn}
\begin{split}
&u_{0,n}\to u_0\quad\text{in $H^1(0,L)$},\quad u_{1,n}\to u_1\quad\text{in $L^2(0,L)$},\quad \Phi_n\to \Phi \quad \text{uniformly in ${\mathbb{R}}$},\\
&\Phi_n'\to \Phi' \quad \text{pointwise in ${\mathbb{R}}$ and uniformly in ${\mathbb{R}}\setminus\left\{(-1-{\varepsilon},-1+{\varepsilon})\cup (1-{\varepsilon},1+{\varepsilon})\right\}$ for every ${\varepsilon}$},\\
&|u|\ge1+{\varepsilon}\Rightarrow \Phi_n'(u)=0,\qquad {\varepsilon}>0,\> n\in{\mathbb{N}},\\
&{\left\|u_{0,n}\right\|}_{H^1(0,L)}\le C,\quad {\left\|u_{1,n}\right\|}_{L^2(0,L)}\le C, \quad 0\le \Phi_n,\,\Phi_n'\le C,\qquad n\in{\mathbb{N}},\\
&u_{0,n}'(0)=u_{0,n}'(L)=u_{1,n}(0)=u_{1,n}(L)=0,\qquad n\in{\mathbb{N}},
\end{split}$$
where $C>0$ denotes some constant independent on $n$.
Let $u_n$ be the unique classical solution of the initial boundary value problem $$\label{eq:eln}
\begin{cases}
{\partial_{tt}^2}u_n={\partial_{xx}^2}u_n-\Phi_n'(u_n),&\quad t>0,0<x<L,\\
{\partial_x }u_n(t,0)={\partial_x }u_n(t,L)=0,&\quad t>0,\\
u_n(0,x)=u_{0,n}(x),&\quad 0<x<L,\\
{\partial_t}u_n(0,x)=u_{1,n}(x),&\quad 0<x<L.
\end{cases}$$
The well-posedness of is guaranteed for short time by the Cauchy-Kowaleskaya Theorem [@taylor]. The solutions are indeed global in time thanks to the following a priori estimates.
\[lm:energy\] The function $$t\mapsto E_n(t)=\int_0^L\left(\frac{({\partial_t}u_n(t,x))^2+({\partial_x }u_n(t,x))^2}{2}+\Phi_n(u_n(t,x))\right)dx$$ is constant for every $n$. In particular, $\{{\partial_t}u_n\}_{n\in{\mathbb{N}}}$ and $\{{\partial_x }u_n\}_{n\in{\mathbb{N}}}$ are bounded in $L^\infty(0,\infty;L^2(0,L))$.
We have that $$\begin{aligned}
E_n'(t)=&\frac{d}{dt}\int_0^L\left(\frac{({\partial_t}u_n)^2+({\partial_x }u_n)^2}{2}+\Phi_n(u_n)\right)dx\\
=&\int_0^L\left({\partial_t}u_n{\partial_{tt}^2}u_n+{\partial_x }u_n{\partial_{tx}^2}u_n+\Phi_n'(u_n){\partial_t}(u_n)\right)dx\\
=&\int_0^L{\partial_t}u_n\underbrace{\left({\partial_{tt}^2}u_n-{\partial_{xx}^2}u_n+\Phi_n'(u_n)\right)}_{=0}dx=0.\end{aligned}$$
\[lm:l2\] The sequence $\{u_n\}_{n\in{\mathbb{N}}}$ is bounded in $L^\infty(0,T;L^2(0,L))$, for every $T>0$.
Since $$\begin{aligned}
\int_0^L u_n^2(t,x)dx=&\int_0^L\left(u_{0,n}(x)+\int_0^t {\partial_s}u_n(s,x)ds\right)^2dx\\
\le&2\int_0^L u_{0,n}^2(x)dx+2 \int_0^L\left(\int_0^t |{\partial_s}u_n(s,x)|ds\right)^2dx\\
\le&2\int_0^L u_{0,n}^2(x)dx+2t \int_0^t\int_0^L ({\partial_s}u_n(s,x))^2dsdx\\
\le&2\int_0^L u_{0,n}^2(x)dx+2t^2 \sup_{s\ge0}\int_0^L ({\partial_s}u_n(s,x))^2dx,\end{aligned}$$ the claim follows from Lemma \[lm:energy\].
\[lm:linfty\] The sequence $\{u_n\}_{n\in{\mathbb{N}}}$ is bounded in $L^\infty((0,T)\times(0,L))$, for every $T>0$.
Fix $0<t<T$ and $0<x<L$. Lemmas \[lm:energy\] and \[lm:l2\] imply that $\{u_n\}_{n\in{\mathbb{N}}}$ is bounded in $L^\infty(0,T;H^1(0,L))$. Since $H^1(0,L)\subset L^\infty(0,L)$ we have $$|u_n(t,x)|\le {\left\|u_n(t,\cdot)\right\|}_{L^\infty(0,L)}\le c{\left\|u_n(t,\cdot)\right\|}_{H^1(0,L)}
\le c{\left\|u_n\right\|}_{L^\infty(0,T;H^1(0,L))},$$ for some constant $c>0$ dependeing only on $L$. Therefore $${\left\|u_n\right\|}_{L^\infty((0,T)\times(0,L))}\le c{\left\|u_n\right\|}_{L^\infty(0,T;H^1(0,L))},$$ that gives the claim.
Thanks to Lemmas \[lm:energy\], \[lm:l2\] and [@S Theorem 5] there exists a function $u$ satisfying ($i$) and ($ii$) of Definition \[def:sol\] such that, passing to a subsequence, $$\label{eq:conv}
\begin{split}
& u_n{\rightharpoonup}u \quad \text{in $H^1((0,T)\times(0,L))$, for each $T\ge0$},\\
& u_n \to u \quad \text{in $L^\infty((0,T)\times(0,L))$, for each $T\ge0$}.
\end{split}$$
We have to verify that $u$ is a weak solution of . Let ${\varphi}\in C^\infty({\mathbb{R}}^2)$ be a test function with compact support. From , for every $n$ we have $$\begin{aligned}
\int_0^\infty\int_0^L& \left(u_n{\partial_{tt}^2}{\varphi}+{\partial_x }u_n{\partial_x }{\varphi}+\Phi_n'(u_n){\varphi}\right)dtdx\\
&-\int_0^L u_{1,n}(x){\varphi}(0,x)dx+\int_{\mathbb{R}}u_{0,n}(x){\partial_t}{\varphi}(0,x)dx=0.\end{aligned}$$ As $n\to\infty$, using and , we get .
Finally, follows from Lemma \[lm:energy\], , and .
Uniqueness {#subsec:uniqueness}
----------
The dissipative solutions of are not unique. This is made clear form the following three examples. In the first example, we show that different regularizations of the discontinuous nonlinear term $\Phi'$ may lead to different dissipative solutions of . In the second example, we use only one regularization of $\Phi'$ and approximate the initial conditions in two different ways. Lastly, the third example shows that the solutions of do not continuously depend on the initial data. Moreover, it seem quite difficult to identify a common asymptotic behavior as $t\to\infty$.
In all the following examples we assume that $\Phi$ is the one defined in .
\[ex:1\] Let ${\varepsilon}>0$. Consider the functions $$\begin{aligned}
\widetilde\Phi_{\varepsilon}(u)&=\begin{cases}
u^2,&\quad \text{if $|u|\le 1-{\varepsilon},$}\\
\frac{2u-u^2}{{\varepsilon}}-(1-{\varepsilon})\left({\varepsilon}+\frac{1}{{\varepsilon}}\right),&\quad \text{if $1-{\varepsilon}\le u\le 1,$}\\
-\frac{2u+u^2}{{\varepsilon}}-(1-{\varepsilon})\left({\varepsilon}+\frac{1}{{\varepsilon}}\right),&\quad \text{if $-1\le u\le -1+{\varepsilon},$}\\
1+{\varepsilon}^2-{\varepsilon},&\quad \text{if $|u|\ge 1,$}
\end{cases}\\
\overline{\Phi}_{\varepsilon}(u)&=\begin{cases}
u^2,&\quad \text{if $|u|\le 1,$}\\
\frac{2(1+{\varepsilon})u-u^2}{{\varepsilon}}-\left(1+\frac{1}{{\varepsilon}}\right),&\quad \text{if $1\le u\le 1+{\varepsilon},$}\\
-\frac{2(1+{\varepsilon})u+u^2}{{\varepsilon}}-\left(1+\frac{1}{{\varepsilon}}\right),&\quad \text{if $-1-{\varepsilon}\le u\le -1,$}\\
1+{\varepsilon},&\quad \text{if $|u|\ge 1+{\varepsilon}.$}
\end{cases}\end{aligned}$$ We have $$\begin{aligned}
\widetilde\Phi_{\varepsilon}'(u)&=\begin{cases}
2u,&\quad \text{if $|u|\le 1-{\varepsilon},$}\\
2\frac{1-u}{{\varepsilon}},&\quad \text{if $1-{\varepsilon}\le u\le 1,$}\\
-2\frac{1+u}{{\varepsilon}},&\quad \text{if $-1\le u\le -1+{\varepsilon},$}\\
0,&\quad \text{if $|u|\ge 1,$}
\end{cases}\\
\overline{\Phi}_{\varepsilon}'(u)&=\begin{cases}
2u,&\quad \text{if $|u|\le 1,$}\\
2\frac{1+{\varepsilon}-u}{{\varepsilon}},&\quad \text{if $1\le u\le 1+{\varepsilon},$}\\
-2\frac{1+{\varepsilon}+u}{{\varepsilon}},&\quad \text{if $-1-{\varepsilon}\le u\le -1,$}\\
0,&\quad \text{if $|u|\ge 1+{\varepsilon}.$}
\end{cases}\end{aligned}$$
The functions $$\widetilde u_{\varepsilon}(t,x)=1,\qquad \overline{u}_{\varepsilon}(t,x)=\cos\left(\sqrt{2}\,t\right),$$ solve $$\begin{aligned}
\label{eq:ex.1.1}
&\begin{cases}
{\partial_{tt}^2}\widetilde u_{\varepsilon}={\partial_{xx}^2}\widetilde u_{\varepsilon}-\widetilde \Phi_{\varepsilon}'(\widetilde u_{\varepsilon}),&\quad t>0,0<x<L,\\
{\partial_x }\widetilde u_{\varepsilon}(t,0)={\partial_x }\widetilde u_{\varepsilon}(t,L)=0,&\quad t>0,\\
\widetilde u_{\varepsilon}(0,x)=1,&\quad 0<x<L,\\
{\partial_t}\widetilde u_{\varepsilon}(0,x)=0,&\quad 0<x<L,
\end{cases}
\\&
\label{eq:ex.1.2}
\begin{cases}
{\partial_{tt}^2}\overline{u}_{\varepsilon}={\partial_{xx}^2}\overline{u}_{\varepsilon}-\overline{\Phi}_{\varepsilon}'(\overline{u}_{\varepsilon}),&\quad t>0,0<x<L,\\
{\partial_x }\overline{u}_{\varepsilon}(t,0)={\partial_x }\overline{u}_{\varepsilon}(t,L)=0,&\quad t>0,\\
\overline{u}_{\varepsilon}(0,x)=1,&\quad 0<x<L,\\
{\partial_t}\overline{u}_{\varepsilon}(0,x)=0,&\quad 0<x<L.
\end{cases}\end{aligned}$$ As ${\varepsilon}\to0$ we have $$\widetilde u_{\varepsilon}(t,x)\to \widetilde u(t,x)=1,\qquad \overline{u}_{\varepsilon}(t,x)\to \overline{u}(t,x)=\cos\left(\sqrt{2}\,t\right),$$ and $\widetilde u$ and $\overline{u}$ provide two different solutions of in correspondence of the initial data $$u_0(x)=1,\qquad u_1(x)=0.$$
The energies associated to and are $$\begin{aligned}
\widetilde E_{\varepsilon}(t)&=\int_0^L\left(\frac{({\partial_t}\widetilde u_{\varepsilon}(t,x))^2+({\partial_x }\widetilde u_{\varepsilon}(t,x))^2}{2}+\widetilde \Phi_{\varepsilon}(\widetilde u_{\varepsilon}(t,x))\right)dx=(1+{\varepsilon}^2-{\varepsilon})L,\\
\overline{E}_{\varepsilon}(t)&=\int_0^L\left(\frac{({\partial_t}\overline{u}_{\varepsilon}(t,x))^2+({\partial_x }\overline{u}_{\varepsilon}(t,x))^2}{2}+\overline{\Phi}_{\varepsilon}(\overline{u}_{\varepsilon}(t,x))\right)dx=L,\end{aligned}$$ respectively.
\[ex:2\] Let ${\varepsilon}>0$. Consider the function $$\Phi_{\varepsilon}(u)=\begin{cases}
\frac{2-{\varepsilon}}{2}u^2,&\quad \text{if $|u|\le 1,$}\\
\frac{2-{\varepsilon}}{{\varepsilon}}\left((1+{\varepsilon})\left(u-\frac{1}{2}\right)-\frac{u^2}{2}\right),&\quad \text{if $1\le u\le 1+{\varepsilon},$}\\
\frac{{\varepsilon}-2}{{\varepsilon}}\left((1+{\varepsilon})\left(u+\frac{1}{2}\right)+\frac{u^2}{2}\right),&\quad \text{if $-1-{\varepsilon}\le u\le -1,$}\\
\frac{(2-{\varepsilon})(1+{\varepsilon})}{2},&\quad \text{if $ |u|\ge 1+{\varepsilon}.$}
\end{cases}$$ We have $$\Phi_{\varepsilon}'(u)=\begin{cases}
(2-{\varepsilon})u,&\quad \text{if $|u|\le 1,$}\\
\frac{2-{\varepsilon}}{{\varepsilon}}(1+{\varepsilon}-u),&\quad \text{if $1\le u\le 1+{\varepsilon},$}\\
\frac{{\varepsilon}-2}{{\varepsilon}}(1+{\varepsilon}+u),&\quad \text{if $-1-{\varepsilon}\le u\le -1,$}\\
0,&\quad \text{if $|u|\ge 1+{\varepsilon}.$}
\end{cases}$$ The functions $$u_{\varepsilon}(t,x)=(1-{\varepsilon})\cos\left(\sqrt{2-{\varepsilon}}\,t\right),\qquad v_{\varepsilon}(t,x)=1+{\varepsilon}$$ solve $$\begin{aligned}
\label{eq:ex.2.1}
&\begin{cases}
{\partial_{tt}^2}u_{\varepsilon}={\partial_{xx}^2}u_{\varepsilon}-\Phi_{\varepsilon}'(u_{\varepsilon}),&\quad t>0,0<x<L,\\
{\partial_x }u_{\varepsilon}(t,0)={\partial_x }u_{\varepsilon}(t,L)=0,&\quad t>0,\\
u_{\varepsilon}(0,x)=1-{\varepsilon},&\quad 0<x<L,\\
{\partial_t}u_{\varepsilon}(0,x)=0,&\quad 0<x<L,
\end{cases}
\\&
\label{eq:ex.2.2}
\begin{cases}
{\partial_{tt}^2}v_{\varepsilon}={\partial_{xx}^2}v_{\varepsilon}-\Phi_{\varepsilon}'(v_{\varepsilon}),&\quad t>0,0<x<L,\\
{\partial_x }v_{\varepsilon}(t,0)={\partial_x }v_{\varepsilon}(t,L)=0,&\quad t>0,\\
v_{\varepsilon}(0,x)=1+{\varepsilon},&\quad 0<x<L,\\
{\partial_t}v_{\varepsilon}(0,x)=0,&\quad 0<x<L.
\end{cases}\end{aligned}$$ As ${\varepsilon}\to0$ we have $$u_{\varepsilon}(t,x)\to u(t,x)=\cos\left(\sqrt{2}\,t\right),\qquad v_{\varepsilon}(t,x)\to v(t,x)=1,$$ and $u$ and $v$ provides two different solutions of in correspondence of the initial data $$u_0(x)=1,\qquad u_1(x)=0.$$ The energies associated to and are $$\begin{aligned}
E_{\varepsilon}(t)&=\int_0^L\left(\frac{({\partial_t}u_{\varepsilon}(t,x))^2+({\partial_x }u_{\varepsilon}(t,x))^2}{2}+ \Phi_{\varepsilon}(u_{\varepsilon}(t,x))\right)dx=\frac{(2-{\varepsilon})(1-{\varepsilon})^2}{2}L,\\
\mathcal{E}_{\varepsilon}(t)&=\int_0^L\left(\frac{({\partial_t}v_{\varepsilon}(t,x))^2+({\partial_x }v_{\varepsilon}(t,x))^2}{2}+\Phi_{\varepsilon}(v_{\varepsilon}(t,x))\right)dx=\frac{(2-{\varepsilon})(1+{\varepsilon})}{2}L,\end{aligned}$$ respectively.
\[ex:3\] For every ${\varepsilon}>0$, the solutions $u_{\varepsilon}$ and $v_{\varepsilon}$ of the two following problems $$\begin{aligned}
\label{eq:ex.3.1}
&\begin{cases}
{\partial_{tt}^2}u_{\varepsilon}={\partial_{xx}^2}u_{\varepsilon}-\Phi'(u_{\varepsilon}),&\quad t>0,0<x<L,\\
{\partial_x }u_{\varepsilon}(t,0)={\partial_x }u_{\varepsilon}(t,L)=0,&\quad t>0,\\
u_{\varepsilon}(0,x)=1+{\varepsilon},&\quad 0<x<L,\\
{\partial_t}u_{\varepsilon}(0,x)={\varepsilon},&\quad 0<x<L,
\end{cases}
\\&
\label{eq:ex.3.2}
\begin{cases}
{\partial_{tt}^2}v_{\varepsilon}={\partial_{xx}^2}v_{\varepsilon}-\Phi'(v_{\varepsilon}),&\quad t>0,0<x<L,\\
{\partial_x }v_{\varepsilon}(t,0)={\partial_x }v_{\varepsilon}(t,L)=0,&\quad t>0,\\
v_{\varepsilon}(0,x)=1-{\varepsilon},&\quad 0<x<L,\\
{\partial_t}v_{\varepsilon}(0,x)=0,&\quad 0<x<L,
\end{cases}\end{aligned}$$ are $$u_{\varepsilon}(t,x)={\varepsilon}t+1+{\varepsilon},\qquad
v_{\varepsilon}(t,x)=(1-{\varepsilon})\cos(\sqrt{2}t).
$$ We have $$\begin{aligned}
&{\left\|u_{\varepsilon}(0,\cdot)-v_{\varepsilon}(0,\cdot)\right\|}_{L^2(0,L)}+{\left\|{\partial_t}u_{\varepsilon}(0,\cdot)-{\partial_t}v_{\varepsilon}(0,\cdot)\right\|}_{L^2(0,L)}=3{\varepsilon}\sqrt{L},\\
&\lim_{t\to\infty}u_{\varepsilon}(t,x)=\infty,\qquad \limsup_{t\to\infty}v_{\varepsilon}(t,x)=1-{\varepsilon}.\end{aligned}$$ Moreover, as ${\varepsilon}\to0$, $$u_{\varepsilon}(t,x)\to 1,\qquad
v_{\varepsilon}(t,x)\to \cos(\sqrt{2}t).
$$ The energies associated to and are $$\begin{aligned}
E_{\varepsilon}(t)&=\int_0^L\left(\frac{({\partial_t}u_{\varepsilon}(t,x))^2+({\partial_x }u_{\varepsilon}(t,x))^2}{2}+ \Phi(u_{\varepsilon}(t,x))\right)dx=\frac{{\varepsilon}^2+2}{2}L,\\
\mathcal{E}_{\varepsilon}(t)&=\int_0^L\left(\frac{({\partial_t}v_{\varepsilon}(t,x))^2+({\partial_x }v_{\varepsilon}(t,x))^2}{2}+\Phi(v_{\varepsilon}(t,x))\right)dx=(1-{\varepsilon})^2L,\end{aligned}$$ respectively.
Regularity {#subsec:reg}
----------
This subsection is devoted to the maximal regularity we can expect for the dissipative solutions of . We show that if the $t$ and $x$ derivative of the solutions at time $t=0$ are bounded then we have locally Lipshitz continuous solutions. In the following section, using a first order formulation of we will show that we cannot expct more regularity even if we consider more regular inital data.
\[th:regularity\] Let $u_0$ and $u_1$ be given and assume \[ass:phi\], \[ass:init\]. If $u$ is a dissipative solution of and $$\label{eq:regass}
u_0\in W^{1,\infty}(0,L),\qquad u_1\in L^{\infty}(0,L),$$ then $$\label{eq:regclaim}
u\in C([0,\infty)\times[0,L])\cap W^{1,\infty}((0,T)\times(0,L)),$$ for every $T>0$.
Let $u$ be a solution of . Consider the function $\widetilde u:[0,\infty)\times{\mathbb{R}}\to{\mathbb{R}}$ defined as the $2L-$periodic (in space) extension of the function $(t,x)\in[0,\infty)\times[-L,L]\mapsto u(t,|x|)$. $\widetilde u$ is the unique solution of the Cauchy Problem $$\label{eq:elu}
\begin{cases}
{\partial_{tt}^2}v={\partial_{xx}^2}v-\Phi'(\widetilde u),&\quad t>0,x\in{\mathbb{R}},\\
v(0,x)=\widetilde u_0(x),&\quad x\in{\mathbb{R}},\\
{\partial_t}v(0,x)=\widetilde u_1(x),&\quad x\in{\mathbb{R}},
\end{cases}$$ where $\widetilde u_0$ and $\widetilde u_1$ are the $2L-$periodic extensions of the functions $x\in[-L,L]\mapsto u_0(|x|)$ and $x\in[-L,L]\mapsto u_1(|x|)$, respectively. Therefore, the following representation formula holds $$\label{eq:u}
\widetilde u(t,x)=\frac{\widetilde u_0(x+t)+\widetilde u_0(x-t)}{2}+\frac{1}{2}\int_{x-t}^{x+t}\widetilde u_1(s)ds
+\frac{1}{2}\int_0^t\int_{x-(t-s)}^{x+(t-s)}\Phi'(\widetilde u(s,y))dsdy.$$
We have that $$\begin{aligned}
|\widetilde u(t,x)&-\widetilde u(t',x')|\\
=&\frac{|\widetilde u_0(x+t)-\widetilde u_0(x'+t')|+|\widetilde u_0(x-t)-\widetilde u_0(x'-t')|}{2}\\
&+\frac{1}{2}\left|\int_{x-t}^{x+t}\widetilde u_1(s)ds-\int_{x'-t}^{x'+t}\widetilde u_1(s)ds\right|
+\frac{1}{2}\left|\int_{x'-t}^{x'+t}\widetilde u_1(s)ds-\int_{x'-t'}^{x'+t'}\widetilde u_1(s)ds\right|\\
&+\frac{1}{2}\left|\int_0^t\int_{x-(t-s)}^{x+(t-s)}\Phi'(\widetilde u(s,y))dsdy-\int_0^{t'}\int_{x-(t-s)}^{x+(t-s)}\Phi'(\widetilde u(s,y))dsdy\right|\\
&+\frac{1}{2}\left|\int_0^{t'}\int_{x-(t-s)}^{x+(t-s)}\Phi'(\widetilde u(s,y))dsdy-\int_0^{t'}\int_{x-(t'-s)}^{x+(t'-s)}\Phi'(\widetilde u(s,y))dsdy\right|\\
&+\frac{1}{2}\left|\int_0^{t'}\int_{x-(t'-s)}^{x+(t'-s)}\Phi'(\widetilde u(s,y))dsdy-\int_0^{t'}\int_{x'-(t'-s)}^{x'+(t'-s)}\Phi'(\widetilde u(s,y))dsdy\right|\\
\le&\left({\left\|\widetilde u_0'\right\|}_{L^\infty({\mathbb{R}})}+\frac{{\left\|\widetilde u_1\right\|}_{L^\infty({\mathbb{R}})}}{2}+\frac{3}{2}{\left\|\Phi'\right\|}_{L^\infty({\mathbb{R}})}(t+t')\right)\left(|x-x'|+|t-t'|\right).
$$ Thanks to we have $$\label{eq:regclaim.1}
\widetilde u\in C([0,\infty)\times {\mathbb{R}})\cap W^{1,\infty}((0,T)\times{\mathbb{R}}),\qquad T>0,$$ and then .
The following simple example shows that we cannot expect $C^2$ regularity on the solutions. More precisely, we start with constant initial data and we explicitly construct conservative solutions exhibiting a singularity in the second derivative. The insurgence of such singularity is due to the lack of continuity of the nonlinear source $\Phi'$.
\[ex:reg\] Consider the function $$\label{eq:reg.1}
u(t,x)=
\begin{cases}
\sqrt{2}\sin(\sqrt{2}t),&\qquad \text{if $0\le t\le \frac{\pi}{4\sqrt{2}}$},\\
\sqrt{2} t+1-\frac{\pi}{4},&\qquad \text{if $t\ge \frac{\pi}{4\sqrt{2}}$}.
\end{cases}$$ Clearly, $u$ solves the problem $$\begin{cases}
{\partial_{tt}^2}u={\partial_{xx}^2}u-\Phi'(u),&\quad t>0,x\in{\mathbb{R}},\\
u(0,x)=0,&\quad x\in{\mathbb{R}},\\
{\partial_t}u(0,x)=2,&\quad x\in{\mathbb{R}},
\end{cases}$$ but $$u\in C^1\setminus C^2.$$ Indeed $$\begin{aligned}
\lim_{t \to \frac{\pi}{4\sqrt{2}}^- }u\left(t,x\right)=1,\qquad & \lim_{t \to \frac{\pi}{4\sqrt{2}}^+}u\left(t,x\right)=1,\\
\lim_{t \to \frac{\pi}{4\sqrt{2}}^-}{\partial_t}u\left(t,x\right)=\sqrt{2},\qquad & \lim_{t \to \frac{\pi}{4\sqrt{2}}^+} {\partial_t}u\left(t,x\right)=\sqrt{2},\\
\lim_{t \to \frac{\pi}{4\sqrt{2}}^-}{\partial_{tt}^2}u\left(t,x\right)=-2,\qquad & \lim_{t \to\frac{\pi}{4\sqrt{2}}^+ } {\partial_{tt}^2}u\left(t,x\right)=0.\end{aligned}$$ The energy associated to is $$E(t)=\int_0^L\left(\frac{({\partial_t}u_{\varepsilon}(t,x))^2+({\partial_x }u_{\varepsilon}(t,x))^2}{2}+ \Phi(u_{\varepsilon}(t,x))\right)dx=2L.$$
Discontinuities, debonding and propagation of singularities {#disc}
===========================================================
In this section we shall focus on some qualitative analysis aimed to investigate the occurence of singularities in the solutions of and the interplay of such singularities with debonding process. Based on a first order system associated to , we give a qualitative description of the discontinuity curves of the first derivatives of the solutions. These are the loci where the dissipation of energy occurs. Moreover, we show that we cannot expct more regularity even if we consider more regular inital data.
We can rewrite the equation in as a first order system in the following way $$\label{eq:system}
{\partial_t}Z+A{\partial_x }Z=B(Z),$$ where $$Z=\left(\begin{matrix}z_1\\z_2\\z_3\end{matrix}\right)=\left(\begin{matrix}{\partial_t}u\\{\partial_x }u\\u\end{matrix}\right),\qquad
A=\left(\begin{matrix}0&-1&0\\-1&0&0\\0&0&0\end{matrix}\right), \qquad B(Z)=\left(\begin{matrix}-\Phi'(z_3)\\0\\z_1\end{matrix}\right).$$
Since $Z\mapsto B(Z)$ is discontinuous the solution $Z$ of may develop discontinuities. Let $t\mapsto(t,\gamma(t))$ be a discontinuity curve for $Z$. Thanks to the qualitative analysis of [@B Chapter 10] $\gamma$ is locally Lipschitz continuous and the Rankine-Hugoniot condition [@B Section 4.2] holds $$A\left(Z(t,\gamma(t)^+)-Z(t,\gamma(t)^-)\right)=\gamma'(t)\left(Z(t,\gamma(t)^+)-Z(t,\gamma(t)^-)\right),\qquad\text{a.e. $t$},$$ where $$Z(t,\gamma(t)^{\pm})=\lim_{s \to \gamma(t)^\pm} Z(t,s).$$ Since the eigenvalues of the matrix $A$ are $-1,\,0$ and $1$, we must have $$\gamma'(t)\in\{-1,0,1\},\qquad\text{a.e. $t$},$$ namely $t\mapsto(t,\gamma(t))$ is a polygonal of the plane $(t,x)$ with slopes $-1,\,0$ and $1$.
Several remarks are needed. In addition to the propagation velocities $1,\ -1$ of the wave equation here we have one more characteristic speed. This feature is coherent with the one obtained in [@Be]. There the appearance of the stationary characteristics was generated by a third order hyperbolic operator and a smooth nonlinear source term $f(u)$ in one spatial dimension. In [@RR] the authors completed the picture showing that if the operator is of the second order and the nonlinear source term $f(u)$ is smooth we can only have two characteristic speeds. Here, we are able to obtain the third characteristic speed even with a second order wave operator because our nonlinear source term $\Phi'(u)$ is discontinuous.
System admits the following entropy/entropy flux pair $$\label{eq:entr}
\eta(Z)=\frac{|Z|^2}{2},\qquad q(Z)=-z_1z_2,\qquad Z=\left(\begin{matrix}z_1\\z_2\\z_3\end{matrix}\right)\in {\mathbb{R}}^3.$$ Coherently with Definition \[def:sol\], the solutions of satisfy the following entropy inequality $$\label{eq:entr-in}
{\partial_t}\eta(Z)+{\partial_x }q(Z)\le \eta'(Z)B(Z),$$ in the sense of distributions. When a shock occurs the inequality in becomes strict. Indeed we consider dissipative solutions [@Se].
The interplay between the propagation of singularities and debonding is described by the following necessary condition relating the singular points in space-time with the occurrence of attachment-debonding in the characteristic cone.
\[th:sing\] Let $u$ be a dissipative solution of and $(t_0,x_0)\in (0,\infty)\times(0,L)$. We have that if $u$ is not $C^1$ in $(t_0,x_0)$ then for all ${\varepsilon}>0$ there exist $(t_1,x_1),\,(t_2,x_2)\in \mathcal{T}_{\varepsilon}(t_0,x_0)$ such that $|u(t_1,x_1)|<1<|u(t_2,x_2)|$, where $$\mathcal{T}_{\varepsilon}(t_0,x_0)=\bigcup_{\max\{t_0-{\varepsilon},0\}\le t\le t_0}\Big(\max\{0,x_0-{\varepsilon}+(t-t_0)\},\min\{x_0+{\varepsilon}-(t-t_0),L\}\Big).$$
We argue by contradiction, namely we prove that if there exists ${\varepsilon}>0$ such that for all $(t,x)\in \mathcal{T}_{\varepsilon}(t_0,x_0), \,|u(t,x)|\le1$ or for all $(t,x)\in \mathcal{T}_{\varepsilon}(t_0,x_0),\, |u(t,x)|\ge1$ then $u$ is $C^1$ in $(t_0,x_0)$. We can always choose ${\varepsilon}$ so small such that $$\label{eq:C^1proof2}
t_0-{\varepsilon}>0,\qquad 0<x_0-2{\varepsilon}< x_0+2{\varepsilon}<L,$$ in this way $$\label{eq:C^1proof3}
\mathcal{T}_{\varepsilon}(t_0,x_0)=\bigcup_{t_0-{\varepsilon}\le t\le t_0}\Big(x_0-{\varepsilon}+(t-t_0),x_0+{\varepsilon}-(t-t_0)\Big).$$
Assume that $$|u(t,x)|\le1, \qquad(t,x)\in \mathcal{T}_{\varepsilon}(t_0,x_0),$$ and consider a $C^1$ function $\overline{\Phi}$ such that $$|u|\le1\Longrightarrow \overline{\Phi}(u)=\Phi(u).$$ Let $\overline{u}$ be the solution of the Cauchy problem $$\begin{cases}
{\partial_{tt}^2}\overline{u}={\partial_{xx}^2}\overline{u}-\overline{\Phi}'(\overline{u}),&\quad t>t_0-{\varepsilon},x\in{\mathbb{R}},\\
\overline{u}(t_0-{\varepsilon},x)={u}(t_0-{\varepsilon},x)\chi_{[x_0-2{\varepsilon},x_0+2{\varepsilon}]}(x),&\quad x\in{\mathbb{R}},\\
{\partial_t}\overline{u}(t_0-{\varepsilon},x)={u}(t_0-{\varepsilon},x)\chi_{[x_0-2{\varepsilon},x_0+2{\varepsilon}]}(x),&\quad x\in{\mathbb{R}}.
\end{cases}$$ Due to the finite speed of propagation of the wave operator we have $$u=\overline{u}\qquad\text{in $\mathcal{T}_{\varepsilon}(t_0,x_0)$}.$$ Using the first order reformulation of the equation for $\overline{u}$ we see that $\overline{u}$ is not developing any singularity in $\mathcal{T}_{\varepsilon}(t_0,x_0)$. In the case $$|u(t,x)|\ge1, \qquad (t,x)\in \mathcal{T}_{\varepsilon}(t_0,x_0),$$ we have only to consider a $C^1$ function $\overline{\Phi}$ such that $$|u|\ge1\Longrightarrow \overline{\Phi}(u)=\Phi(u),$$ and use the same argument.
We conclude this Section by considering the Cauchy problem associated to . The motivations behind this analysis are:
- due to the finite speed of propagation, the Cauchy and Neumann problem share the same solution for short time and compactly supported initial data;
- explicit formulas for the the solutions can be obtained for the Cauchy problem and those formulas do not rely on Fourier series the regularity issue is more clear.
We begin with the Cauchy problem $$\label{eq:systemCauchy}
\begin{cases}{\partial_t}Z+A{\partial_x }Z=0 & {} \\
Z(0,x)=Z_{0}(x) & {}
\end{cases}$$ where $$Z=\left(\begin{matrix}z_1\\z_2\\z_3\end{matrix}\right)=\left(\begin{matrix}{\partial_t}u\\{\partial_x }u\\u\end{matrix}\right),\qquad
A=\left(\begin{matrix}0&-1&0\\-1&0&0\\0&0&0\end{matrix}\right)$$ and $$Z_0=\left(\begin{matrix}z_{1,0}\\z_{2,0}\\z_{3,0}\end{matrix}\right): \mathbb{R} \to \mathbb{R}^3,$$ is the vector of the initial conditions. If we diagonalize the matrix $A$ we obtain that $A=P^{-1}DP$, where $$D=\left(\begin{matrix}1&0&0\\0&0&0\\0&0&-1\end{matrix}\right), \qquad P=\left(\begin{matrix}1&0&1\\-1&0&1\\0&1&0\end{matrix}\right), \qquad P^{-1}=\frac{1}{2}\left(\begin{matrix}1&-1&0\\0&0&2\\1&1&0\end{matrix}\right).$$ If we define $W=PZ$ and $W_0=PZ_0$ we obtain that $$\label{eq:systemCauchyW}
\begin{cases}{\partial_t}W+D{\partial_x }W=0 &{}\\
W(0,x)=W_{0}(x)&{}
\end{cases}$$ that can be easily solved as $$\label{eq:systemCauchyw}
\begin{cases}w_1(t,x)=w_{1,0}(x-t),&{} \\
w_2(t,x)=w_{2,0}(x),&{} \\
w_3(t,x)=w_{3,0}(x+t),&{}
\end{cases}$$ Now, since $Z=P^{-1}W$, it results that $Z=S_t Z_0$, where $$S_t Z_0:= \frac{1}{2}\left(\begin{matrix}z_{1,0}(x-t)+z_{3,0}(x-t)+z_{1,0}(x)-z_{3,0}(x)\\2z_{2,0}(x+t)\\z_{1,0}(x-t)+z_{3,0}(x-t)-z_{1,0}(x)+z_{3,0}(x)\end{matrix}\right)$$ Now if we consider the Cauchy problem $$\label{eq:systemCauchymod}
\begin{cases}{\partial_t}Z+A{\partial_x }Z=B(Z),&{}\\
Z(0,x)=Z_{0}(x),&{}\\
\end{cases}$$ where $$B(Z)=\left(\begin{matrix}-\Phi'(z_3)\\0\\z_1\end{matrix}\right)$$ we have that $$\begin{aligned}
Z(t,x)&=S_t Z_0 + \int_{0}^{t} S_{t-s}B(Z(s,x))\; ds \\
&=S_t Z_0 + \frac{1}{2} \int_{0}^{t} \left( \begin{matrix}-\Phi'(z_3(s,x-(t-s)))+z_1(s,x-(t-s))-\Phi'(z_3(s,x))-z_1(s,x)\\ 0 \\-\Phi'(z_3(s,x-(t-s)))+z_1(s,x-(t-s))+\Phi'(z_3(s,x))+z_1(s,x)\end{matrix}\right)\; ds.\end{aligned}$$ The fact that the evolution of $z_2={\partial_x }u$ only involves $S_t$ implies that the new singularities of $u$ may occur only in ${\partial_t}u(t, \cdot)$. From the physical point of view this means that the stretching ${\partial_x }u$ is not sensitive to debonding-attachment phenomena.
Numerical examples {#sec:numerics}
==================
In this Section we provide some numerical examples of solutions of the system described in \[eq:el\]. As we will see, they exhibit, with proper initial conditions, a rich phenomenology. We will consider both cases with smooth and non-smooth initial conditions in order to obtain solutions with different behaviors in their derivative that explicitly show propagation along the characteristics.
\[ex:1.1\] Let us take initial value such that $u(0,x)=u_0(x)$ is in $C^2([0,L])$: $$\begin{aligned}
\label{eq:initC2}
\begin{cases}
u_0(x)=\xi_0 \left(\frac{x^3}{3}-L\frac{x^2}{2}\right),&\quad 0\le x \le L,\\
u_1(x)=\xi_1,&\quad 0 \le x \le L.
\end{cases}\end{aligned}$$
In Figures \[fig:uregolare\] and \[fig:uregolareder\] we plot, respectively, the solution of \[eq:el\] with initial conditions (\[eq:initC2\]) and its first order derivatives with respect to time and space coordinates. In the simulation we have used the values $\xi_0=0.006$, $\xi_1=1.2$, $L=10$ and a total time evolution $T=10$. In Figure \[fig:uregolare\] we have also inserted the plans with $u=\pm 1$ in order to show the regions where the solution has reached and exceeded the critical values. From the results, it is evident that the initial conditions allow a part of the system to pass $u=1$. It is possible to see that these two values depend on $\xi_0$ and $\xi_1$. On the other hand, the simulations show that the system also exhibits another feature: the debonding process is reversed and $u$ takes values less than $1$. The values $t^*$ (time) and $x^*$ (position) where again $u=1$ and the debonding is reversed are more evident by inspecting the behavior of ${\partial_t}u$ and ${\partial_x }u$. Interestingly, ${\partial_x }u$ is not sensible to the debonding process. This is consistent with the results discussed at the end of Section \[disc\]. Moreover, it is evident that the inversion point act as a source for the explicit observation of the propagation along the characteristic curves. These features will appear also in other examples in the following.
![(Color online) Solution of (\[eq:el\]) with initial conditions (\[eq:initC2\]). See text for the numerical values used in the simulation. []{data-label="fig:uregolare"}](uregolare.pdf){height="40.00000%"}
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![(Color online) Left: derivative with respect to time of $u$ shown in Figure \[fig:uregolare\]. Right: derivative with respect to space of $u$ shown in Figure \[fig:uregolare\]. See text for the numerical values used in the simulation. []{data-label="fig:uregolareder"}](uregolaredert.pdf "fig:"){height="40.00000%"} ![(Color online) Left: derivative with respect to time of $u$ shown in Figure \[fig:uregolare\]. Right: derivative with respect to space of $u$ shown in Figure \[fig:uregolare\]. See text for the numerical values used in the simulation. []{data-label="fig:uregolareder"}](uregolarederx.pdf "fig:"){height="40.00000%"}
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\[ex:1.2\] We consider initial values such that $u$ is in $C^1([0,L])$ for $t=0$: $$\begin{aligned}
\label{eq:initC1}
\begin{cases}
u_0(x)=\xi_0 b\; c(x),&\quad 0\le x \le L,\\
u_1(x)=\xi_1,&\quad 0 \le x \le L.
\end{cases}\end{aligned}$$ where $b$ is a constant defined as $$\label{eq:initC1funca}
b=\frac{1}{-\frac{1}{4} L \sqrt{4 a^2-L^2}+a^2
\left(-\tan ^{-1}\left(\frac{L}{\sqrt{4
a^2-L^2}}\right)\right)+a L},$$ and $$\begin{aligned}
\label{eq:initC1funcb1}
c(x)=&-\frac{1}{2} x \sqrt{a^2-x^2}+\frac{1}{2} a^2 \tan
^{-1}\left(\frac{x
\sqrt{a^2-x^2}}{x^2-a^2}\right)+a x ,\quad 0\le x \le L/2,\\
\label{eq:initC1funcb2}
c(x)=&-\frac{1}{4} L \sqrt{4 a^2-L^2}+a^2 \left(-\tan
^{-1}\left(\frac{L}{\sqrt{4
a^2-L^2}}\right)\right)\notag\\
&+\frac{1}{2} \left((L-x)
\sqrt{a^2-(L-x)^2}+a^2 \tan
^{-1}\left(\frac{L-x}{\sqrt{a^2-(L-x)^2}}\right)+2
a x\right),\quad L/2 < x \le L,\end{aligned}$$ with $a>L/2$.
We notice that the second space derivative of $c$ is not continuous in $x=L/2$. We have performed two different simulations. In Figures \[fig:uC1\] and \[fig:uC1der\] we plot, respectively, the solution of with initial conditions (\[eq:initC1\]) and its first order derivatives with respect to time and space coordinates. In the simulation we have used the values $\xi_0=0.7$, $\xi_1=1.1$, $a=6$, $L=10$ and a total time evolution $T=3$. In Figures \[fig:uC1b\] and \[fig:uC1bder\] we find the solution for the same system but with $\xi_1=1.4$. As in the previous example, we observe that the inversion point (where the debonding is reversed) acts as a source for the direct observation of propagation along the characteristics. Moreover, we explicitly observe propagation along the characteristics from the initial point $(t=0,x=L/2)$ where ${\partial_x }u$ is not $C^2$. From other numerical experiments (not shown in the paper) we can deduce that this phenomenon is ubiquitous whenever there are two (or more) points in the $x$-domain at $t=0$ with ${\partial_x }u$ not in $C^2$.
![(Color online) Solution of (\[eq:el\]) with initial conditions (\[eq:initC1\]). See text for the numerical values used in the simulation. []{data-label="fig:uC1"}](uC1.pdf){height="40.00000%"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(Color online) Left: derivative with respect to time of $u$ shown in Figure \[fig:uC1\]. Right: derivative with respect to space of $u$ shown in Figure \[fig:uC1\]. See text for the numerical values used in the simulation. []{data-label="fig:uC1der"}](uC1dert.pdf "fig:"){height="40.00000%"} ![(Color online) Left: derivative with respect to time of $u$ shown in Figure \[fig:uC1\]. Right: derivative with respect to space of $u$ shown in Figure \[fig:uC1\]. See text for the numerical values used in the simulation. []{data-label="fig:uC1der"}](uC1derx.pdf "fig:"){height="40.00000%"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(Color online) Solution of (\[eq:el\]) with initial conditions (\[eq:initC1\]). See text for the numerical values used in the simulation. []{data-label="fig:uC1b"}](uC1b.pdf){height="40.00000%"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(Color online) Left: derivative with respect to time of $u$ shown in Figure \[fig:uC1b\]. Right: derivative with respect to space of $u$ shown in Figure \[fig:uC1b\]. See text for the numerical values used in the simulation. []{data-label="fig:uC1bder"}](uC1bdert.pdf "fig:"){height="40.00000%"} ![(Color online) Left: derivative with respect to time of $u$ shown in Figure \[fig:uC1b\]. Right: derivative with respect to space of $u$ shown in Figure \[fig:uC1b\]. See text for the numerical values used in the simulation. []{data-label="fig:uC1bder"}](uC1bderx.pdf "fig:"){height="40.00000%"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\[ex:3.1\] We consider now the initial conditions: $$\begin{aligned}
\label{eq:initC1middle}
\begin{cases}
u_0(x)=\xi_0 (-\frac{1}{2}+b\; c(x))+1,&\quad 0\le x \le L,\\
u_1(x)=\xi_1,&\quad 0 \le x \le L.
\end{cases}\end{aligned}$$ where we have defined $b$ and $c$ in Equations (\[eq:initC1funca\])-(\[eq:initC1funcb2\]).
Thus, we fix the discontinuity point of the second derivative of $u$ with respect to $x$ (at $t=0$) when $u(0,L/2)=1$. Moreover, we set $\xi_0=0.7$, $\xi_1=-1.2$ (the initial velocity is reversed), $a=6$, $L=10$ and a total time evolution $T=3$. As in the previous cases, in Figures \[fig:uC1middle\] and \[fig:uC1middleder\] we plot, respectively, the solution of (\[eq:el\]) with initial conditions (\[eq:initC1middle\]) and its first order derivatives with respect to time and space coordinates. We notice that the point $(t=0, x=L/2)$ is now both an inversion point and a discontinuity point for the initial second space derivative of $u$. Thus we again observe the explicit propagation along the characteristics. Moreover, the value of $\xi_1$ is large enough to observe a complete debonding phenomenon with $u<-1$ after some time. This is evident form the behavior of ${\partial_t}u$ in Figure \[eq:initC1funcb2\]. We also notice that there are not new sources of characteristics.
![(Color online) Solution of (\[eq:el\]) with initial conditions (\[eq:initC1middle\]). See text for the numerical values used in the simulation. []{data-label="fig:uC1middle"}](uC1middle.pdf){height="40.00000%"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(Color online) Left: derivative with respect to time of $u$ shown in Figure \[fig:uC1middle\]. Right: derivative with respect to space of $u$ shown in Figure \[fig:uC1middle\]. See text for the numerical values used in the simulation. []{data-label="fig:uC1middleder"}](uC1middledert.pdf "fig:"){height="40.00000%"} ![(Color online) Left: derivative with respect to time of $u$ shown in Figure \[fig:uC1middle\]. Right: derivative with respect to space of $u$ shown in Figure \[fig:uC1middle\]. See text for the numerical values used in the simulation. []{data-label="fig:uC1middleder"}](uC1middlederx.pdf "fig:"){height="40.00000%"}
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We stress that we have numerically tested that separating the inversion point and the discontinuity point gives rise to two separated characteristics sets. This is evident from the results obtained in the following example.
\[ex:4\] Let us consider the initial conditions $$\begin{aligned}
\label{eq:initC1middleb}
\begin{cases}
u_0(x)=\xi_0 (\frac{1}{2}+b\; c(x)),&\quad 0\le x \le L,\\
u_1(x)=\xi_1,&\quad 0 \le x \le L.
\end{cases}\end{aligned}$$
In Figures \[fig:uC1middleb\] and \[fig:uC1middlebder\] we show, respectively, the solution of (\[eq:el\]) with initial conditions (\[eq:initC1middleb\]) and its first order derivatives with respect to time and space coordinates \[same parameters used for the simulations with the conditions in Equation (\[eq:initC1middle\])\]. Moreover, in this case there is not a complete debonding, even in the example where a large negative value of $\xi_1$ has been chosen. As a consequence, we can observe a new set of characteristics in the plots of ${\partial_t}u$ and ${\partial_x }u$.
![(Color online) Solution of (\[eq:el\]) with initial conditions (\[eq:initC1middleb\]). See text for the numerical values used in the simulation. []{data-label="fig:uC1middleb"}](uC1middleb.pdf){height="40.00000%"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(Color online) Left: derivative with respect to time of $u$ shown in Figure \[fig:uC1middleb\]. Right: derivative with respect to space of $u$ shown in Figure \[fig:uC1middle\]. See text for the numerical values used in the simulation. []{data-label="fig:uC1middlebder"}](uC1middlebdert.pdf "fig:"){height="40.00000%"} ![(Color online) Left: derivative with respect to time of $u$ shown in Figure \[fig:uC1middleb\]. Right: derivative with respect to space of $u$ shown in Figure \[fig:uC1middle\]. See text for the numerical values used in the simulation. []{data-label="fig:uC1middlebder"}](uC1middlebderx.pdf "fig:"){height="40.00000%"}
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![(Color online) Solution of (\[eq:el\]) with initial conditions (\[eq:initC1double\]). See text for the numerical values used in the simulation. []{data-label="fig:uC1double"}](uC1double.pdf){height="40.00000%"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(Color online) Left: derivative with respect to time of $u$ shown in Figure \[fig:uC1double\]. Right: derivative with respect to space of $u$ shown in Figure \[fig:uC1double\]. See text for the numerical values used in the simulation. []{data-label="fig:uC1doubleder"}](uC1doubledert.pdf "fig:"){height="40.00000%"} ![(Color online) Left: derivative with respect to time of $u$ shown in Figure \[fig:uC1double\]. Right: derivative with respect to space of $u$ shown in Figure \[fig:uC1double\]. See text for the numerical values used in the simulation. []{data-label="fig:uC1doubleder"}](uC1doublederx.pdf "fig:"){height="40.00000%"}
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\[ex:5\] We have also considered the case of initial condition where ${\partial_t}u(0,x)=u_1(x)$ is in $C([0,L])$: $$\begin{aligned}
\label{eq:initC1double}
\begin{cases}
u_0(x)=\xi_0 (-\frac{1}{2}+b\; c(x))+1,&\quad 0\le x \le L,\\
u_1(x)=\xi_1 b\;\frac{d}{dx}[c(x)],&\quad 0 \le x \le L.
\end{cases}\end{aligned}$$
The discontinuity point of the second derivative of $u$ with respect to $x$ (at $t=0$) appears for $x=L/2$ when $u(0,L/2)=1$. We have fixed $\xi_0=0.7$, $\xi_1=0.8$, $a=6$, $L=10$ and a total time evolution $T=3$. In Figures \[fig:uC1double\] and \[fig:uC1doubleder\] we show, respectively, the solution of (\[eq:el\]) with initial conditions (\[eq:initC1double\]) and its first order derivatives with respect to time and space coordinates. We observe the propagation of the discontiuity in both partial derivatives.
![(Color online) Solution of (\[eq:el\]) with initial conditions (\[eq:initCmoll\]). See text for the numerical values used in the simulation. []{data-label="fig:uCmoll"}](uCmoll.pdf){height="40.00000%"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(Color online) Left: derivative with respect to time of $u$ shown in Figure \[fig:uCmoll\]. Right: derivative with respect to space of $u$ shown in Figure \[fig:uCmoll\]. See text for the numerical values used in the simulation. []{data-label="fig:uCmollder"}](uCmolldert.pdf "fig:"){height="40.00000%"} ![(Color online) Left: derivative with respect to time of $u$ shown in Figure \[fig:uCmoll\]. Right: derivative with respect to space of $u$ shown in Figure \[fig:uCmoll\]. See text for the numerical values used in the simulation. []{data-label="fig:uCmollder"}](uCmollderx.pdf "fig:"){height="40.00000%"}
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\[ex:1.3\] Finally, we have considered the case of initial condition where $u$ at $t=0$ is in $C([0,L])$. In order to perform the numerical simulation we have mollified the initial data in the following way: $$\begin{aligned}
\label{eq:initCmoll}
\begin{cases}
u_0(x)=\xi_0 f_\eta(x),&\quad 0\le x \le L,\\
u_1(x)=\xi_1,&\quad 0 \le x \le L,
\end{cases}\end{aligned}$$ where $$\begin{aligned}
\label{eq:feta}
f_\eta(x)=\frac{2}{\left(\frac{L}{2}-\eta\right) L}\times
\begin{cases}
x^2,&\quad 0\leq x<\frac{L}{2}-\eta,\\
-\frac{L/2-\eta}{\eta}x^2+L\frac{L/2-\eta}{\eta}x-L\frac{(L/2-\eta)^2}{2\eta},&\frac{L}{2}-\eta\leq x<\frac{L}{2}+\eta,\\
(x-L)^2,&\frac{L}{2}+\eta\le x\le L,
\end{cases}\end{aligned}$$ and $\eta$ is the mollification parameter.
We have fixed $\xi_0=0.5$, $\xi_1=1.4$, $\eta=0.3$, $L=10$ and a total time evolution $T=3$. In Figures \[fig:uCmoll\] and \[fig:uCmollder\] we show, respectively, the solution of (\[eq:el\]) with initial conditions (\[eq:initCmoll\]) and its first order derivatives with respect to time and space coordinates. Also in this case, we directly observe the presence of characteristics due to debonding and an hint of the propagation due to the jump of the value of the first derivative in $t=0, x=L/2$. We have verified an analogous behavior of the solution $u$ when the value of $\eta$ is reduced.
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[^1]: G. M. Coclite and F. Maddalena are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). G. Florio and M. Ligabò are supported by the Gruppo Nazionale per la Fisica Matematica (GNFM) of the Istituto Nazionale di Alta Matematica (INdAM). G> Florio is supported by MIUR through the project VirtualMurgia.
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abstract: 'Topological Majorana fermion (MF) quasiparticles have been recently suggested to exist in semiconductor quantum wires with proximity induced superconductivity and a Zeeman field. Although the experimentally observed zero bias tunneling peak and a fractional ac-Josephson effect can be taken as necessary signatures of MFs, neither of them constitutes a sufficient “smoking gun" experiment. Since one pair of Majorana fermions share a single conventional fermionic degree of freedom, MFs are in a sense fractionalized excitations. Based on this fractionalization we propose a tunneling experiment that furnishes a nearly unique signature of end state MFs in semiconductor quantum wires. In particular, we show that a “teleportation"-like experiment is not enough to distinguish MFs from pairs of MFs, which are equivalent to conventional zero energy states, but our proposed tunneling experiment, in principle, can make this distinction.'
author:
- 'Jay D. Sau$^1$'
- Brian Swingle$^2$
- 'Sumanta Tewari$^{3}$'
title: 'A proposal to probe quantum non-locality of Majorana fermions in tunneling experiments'
---
#### Introduction:
Majorana fermions [@Majorana] are localized particle-like neutral zero energy states that occur at topological defects and boundaries in superconductors. Zero energy simply means that they occur at the chemical potential. A MF creation operator is a hermitian second quantized operator $\gamma^{\dagger}=\gamma$ which anti-commutes with other fermion operators. The hermiticity of MF operators implies that they can be construed as particles which are their own anti-particles [@Majorana; @Wilczek; @Franz; @Read-Green]. The key issues at this time in the condensed matter context are two fold, first, we must predict and characterize materials supporting Majorana fermions and second, we must detect them experimentally. In this paper we address the second issue of experimental detection by proposing a nearly sufficent experimental signature for Majorana fermions.
MFs have recently been proposed to exist in the topologically superconducting (TS) phase of a spin-orbit (SO) coupled cold atomic gases [@zhang_c], semiconductor 2D thin film [@Sau; @Long-PRB] or 1D nanowire [@Long-PRB; @Roman; @Oreg] with proximity induced $s$-wave superconductivity and Zeeman splitting from a sufficiently large magnetic field. In principle, the MFs in such systems may be detected either by measuring the zero-bias conductance peak (ZBCP) from tunneling electrons into the end MFs [@tanaka; @Long-PRB; @Sengupta-2001; @R1; @flensberg], by detecting the predicted fractional ac Josephson effect [@Kitaev-1D; @Roman; @Oreg; @Kwon; @Fu-Frac]. The semiconductor Majorana wire structure, which will be the system of our focus, is of particular present interest since there is experimental evidence for both the ZBCP [@Mourik; @Deng; @Rokhinson; @Weizman] and the fractional ac Josephson effect in the form of doubled Shapiro steps [@Rokhinson].
Despite their conceptual simplicity, neither the ZBCP nor the fractional ac-Josephson effect experiments constitute a sufficient proof of MFs at the ends of topological superconducting wires. A non-quantized ($2e^2/h$) zero bias peak, such as that observed in the recent experiments [@Mourik; @Deng; @Weizman], can in principle arise even without end state MFs [@Liu; @Kells; @Beenakker-Weak]. Similarly, while the observation of doubled Shapiro steps are consistent with the TS state [@platero; @meyer; @nazarov; @aguado], a fractional ac-Josephson effect can exist even in Josephson junctions made of ordinary quasi-1D $p$-wave superconductors such as organic superconductors [@Kwon] or the non-topological phase of the semiconductor nanowire [@sau_bert]. Given these caveats as well as the considerable complexity of existing experiments, there have been several alternative proposals to detect the presence of MFs [@gil; @demler; @zoller; @beri]. Based on the inherent quantum non-locality of MFs, in this work we propose an alternative tunneling experiment on semiconductor Majorana wires that furnishes a nearly sufficient signature of end-state MFs. We discuss in detail why only topological systems would show such quantum non-locality, which would even be absent for systems with a pair of MFs at each end.
#### Non-local electron transfer
Non-locality arises in MFs because they differ from conventional complex (Dirac) fermions in that they have no occupation number associated with them. To define a quantum state of a system with MFs we must consider a pair of MFs. The pair of MFs $\gamma_a$ and $\gamma_b$ at the ends $a$ and $b$ of a nanowire (NW) shown in Fig. \[Fig2\] can be combined into a zero-energy complex fermion operator $d^\dagger=\frac{1}{2}[\gamma_a+i\gamma_b]$ associated with the pair of MFs $\gamma_a$ and $\gamma_b$ [@Kitaev-1D]. The quantum state of the system is then determined by the eigenvalue of $n_d=d^\dagger d= 0,1$. Since the fermion parity $F_P=(-1)^{n_d}$ associated with the operator $d^\dagger$ is related to the MFs by $$F_P=(-1)^{n_d}=i\gamma_a\gamma_b,$$ we see that the fermion parity of the whole system is determined by non-local correlations between the fractionalized MFs $\gamma_a$ and $\gamma_b$. In fact, the fractionalization of the $F_P$ into a pair of spatially separated operators $\gamma_{a,b}$ in one-dimensional systems with localized fermion excitations, is a unique characterization of the topological state of the system [@berg]. Our central concern is how to probe this non-locality to provide a robust and sufficient criterion of MFs.
An immediate idea involves trying to inject an electron into $\gamma_a$ and retrieve it from $\gamma_b$ [@sumanta; @bolech; @semenoff; @fu]. By connecting leads to the left and the right ends of the TS wire in Fig. \[Fig2\], one could imagine that an electron injected into the end $a$ flips the occupation number $n_d$ from $n_d=0$ to $n_d=1$. The injected electron can then escape from the end $b$ flipping the state back from $n_d=1$ to $n_d=0$. Such a process where an electron can enter from one end $a$ and exit at the other lead $b$, can be interpreted as a transfer of an electron, which we will refer to as Majorana-assisted electron tunneling. However, as has been discussed in previous works [@sumanta; @bolech], such a transfer occurs in a way so as to not violate locality and causality.
![(Color online) Schematic experiment to detect Majorana-assisted electron tunneling between the MFs $a$ and $b$ along the dashed lines. The electrons in the ring are transported via the Majorana-assisted tunneling process along the dashed line, which enclosed the flux $\Phi$. The dark green metallic loop is weakly proximity coupled to a superconductor underneath so as to have a small superconducting gap and a correspondingly large coherence length. Similar to the Aharonov-Bohm oscillations in a mesoscopic ring, the interference can be detected as a $2\Phi_0=hc/e$ periodicity of the excitation spectrum as a function of $\Phi$ that may be measured by the tunnel probe [@manucharyan]. The nanowire $NW$ is placed on a supercondcuting island (shown in light blue) which has a charging energy $E_C$. []{data-label="Fig2"}](MF_teleportation_Fig.pdf)
The amplitude for the Majorana-assisted electron tunneling [@sumanta; @semenoff] can be written in terms of the retarded Green function as $$G_{R}(\tau;ab)={\langle {g|\gamma_a(\tau)\gamma_b(0)|g} \rangle}\Theta(\tau),\label{eq:A}$$ where ${\left\vert {g} \right\rangle}$ is a ground state of the Majorana wire, $\tau$ is the time-interval between the tunneling events, $\Theta(\tau)$ is the Heaviside step function and $\gamma_{a,b}$ are the Majorana fermion operators at the left and right end of the wire. The amplitude $G_R(\tau;ab)$ contains contributions from Majorana-assisted normal electron tunneling as well as Majorana-assisted non-local Andreev processes where the electron entering at the end $a$ exits as a hole at the end $b$. When the state ${\left\vert {g} \right\rangle}$ is a ground state of the TS system with a definite fermion-parity $F_P\equiv i{\langle {g|\gamma_a(0)\gamma_b(0)|g} \rangle}$ the amplitude $$G_{R}(\tau;ab)=-i F_P\label{eq:GR}$$ is non-zero since $\gamma_a$ is a zero-energy mode. Since $G_R(\tau;ab)$ is directly related to the fermion parity $F_P$, the detection of such a non-vanishing amplitude for a non-local Green function $G_R(\tau;ab)$ is a signature of the fractionalization associated with MFs. In equilibrium, the degeneracy of the different fermion parity states characteristic of a TS system lead to fluctuations in $F_P$, that would result in a vanishing average for the tunneling amplitude $G_R(\tau;ab)$. This is remedied [@fu] by introducing a charging energy $E_C$ on the superconducting island supporting the $NW$, which makes one of the fermion parities energetically favorable over the other.
#### Coincidence probability:
The amplitude $G_{R}(\tau;ab)$ can lead to a so-called coincidence probability for the absence of an electron on the left lead and the presence of an electron on the right lead after the tunneling event [@sumanta]. The coincidence probability $P_c(\tau)$ maybe measured by using a joint measurement by two point contact detectors [@gong]. Alternatively, the non-local transfer of electron in the presence of charging energy [@fu], which fixes the fermion parity $F_P$, can also be measured by a non-local conductance or transconductance $\frac{dI_b}{d V_a}$ between the ends $a$ and $b$ in Fig. \[Fig2\]. This measurement does not require the loop SC in Fig. \[Fig2\] and would require adding a lead to the end $a$. In such a set-up, a voltage $V_a$ applied to the left-lead $a$ (relative to the SC) results in a current $I_b$ in the right lead $b$. Using results of Ref. for symmetric $t_a=t_b=t$ we find that $$\frac{dI_b}{d V_a}=\delta \frac{32 V_a}{16 \Gamma^2+(\delta^2-V_a^2)^2+8\Gamma^2(\delta^2+V_a^2)},
\label{Paraconductance}$$ which clearly vanishes for $\delta\rightarrow 0$ (energy separation between $F_P=\pm 1$). Here $\Gamma\propto t^2$ is the lead-induced broadening of the MFs.
#### Topological versus non-topological systems:
However, a coincidence measurement does not directly imply a non-zero $G_R(\tau;ab)$ in more general situations. The amplitude $G_R(\tau;ab)$ in Eq. \[eq:A\], reflects the amplitude for being able to transfer an electron from $a$ to $b$ while leaving the state ${\left\vert {g} \right\rangle}$ invariant. On the other hand, the measurement of the coincidence probability, $P_c$, does not keep track of the internal state of the system. For a general system (i.e. one that may be topological or non-topological), $P_c$ for an electron entering at $a$ and exiting at $b$ can be written more generally as $$P_{c}(\tau)=\sum_{g_1,g_2}|{\langle {g_2|\gamma_b(0) \gamma_a^\dagger(\tau) |g_1} \rangle}|^2,\label{eq:P}$$ where $g_1,g_2$ are the internal states of the wire, which are not necessarily identical. Here for a non-topological system $\gamma_{a,b}^\dagger$ are conventional fermionic operators. While TS systems with MFs have a non-degenerate ground state in a given fermion parity sector, more general systems with zero-energy Dirac end states may have multiple allowed values for $g_1,g_2$. Therefore, the coincidence probability $P_c$ cannot be considered a unique signature for a topological system.
An important example of the inequivalence of $P_c(\tau)$ and $G_R(\tau;ab)$ is a non-topological superconductor with a complex fermion zero mode at each end (such a system could arise because we have an even number of weakly coupled Majorana fermions at each end). The quantum state is characterized by the occupancy $n_a,n_b$ of the two conventional zero energy end modes. We can easily have $P_c \neq 0$ in this non-topological setup. Suppose the initial state is $g_1\equiv(n_a=n_b=0)$, then the sum for $P_{c}$ in Eq. \[eq:P\] would have a non-zero contribution from $g_2\equiv (n_a=n_b=1)$. The tunneling of an electron from the lead into the zero-mode at $a$ changes the occupation from $n_a=0$ to $n_a=1$. On the other hand, the electron required to change the occupation of the state $b$ from $n_b=0$ to $n_b=1$ comes from breaking of a Cooper pair. The other electron from the broken Cooper pair is emitted into the lead in the vicinity near $b$. Note that the process conserves the number of electrons within the system and cannot be eliminated even by the introduction of a finite charging energy [@fu]. Therefore in order to clearly distinguish this case from the process of Majorana-assisted electron tunneling (that also returns a non-zero $P_c$), we require $G_R(\tau;ab)$ given in Eq. \[eq:A\] itself to be non-zero. In other words, we require that the system return to the same state $g$ after the tunneling process, so the same electron that enters at $a$ leaves at $b$. In this paper we focus on how $G_R(\tau;ab)$ can be directly measured to result in a unique signature of end state MFs [^1].
#### Proposed set-up:
As we now argue, the interference experiment shown in Fig. \[Fig2\] can be used to measure the amplitude of coherent non-local quasiparticle transmission between the ends $a,b$ by measuring $G_R(\tau;ab)$. Unlike previous interference schemes [@interferometry0] for MFs, this scheme will avoid the use of chiral edge states. The setup in Fig. \[Fig2\] consists of an external superconducting electrode (SC) that is connected to the ends $a$ and $b$ through tunnel barriers. The superconducting electrode is assumed to be shorter than its coherence length $\xi_{SC}=v_F/\Delta_{loop}$ so that quasiparticles with energies below the SC gap have a finite amplitude of tunneling from end $a$ to end $b$ through the superconducting wire. The coherence length $\xi_{SC}$ can be made large by weakly proximity coupling the loop to an external superconductor so that $\Delta_{SC}$ is small.
In the presence of a finite Majorana-assisted non-local electron transfer amplitude $G_R(\tau;ab)$ a quasiparticle tunneling from $a$ to $b$ through the loop (SC) can complete the circuit by tunneling back from $b$ to $a$ through the TS nanowire. The Berry phase associated with such a tunneling around the loop is sensitive to the flux $\Phi$ through the loop with a periodicity $2\Phi_0=hc/e$. Therefore, similar to the Aharonov-Bohm effect in mesoscopic rings [@AB], the energy-level of such a quasiparticle excitation spectrum $\varepsilon(\Phi)$ in the ring is expected to develop a $2\Phi_0$ periodic dependence on $\Phi$ for TS systems where there is a finite electron transfer amplitude across the $NW$. Other contributions to the Aharonov-Bohm effect that may result from either Cooper-pair transport around the ring have a periodicity of the superconducting flux quantum $\Phi_0$. Therefore, the contribution from Majorana-assisted non-local quasiparticle transfer around the ring, in the topological case, can be extracted from the $2\Phi_0$ periodicity of the quasiparticle excitation spectrum $\varepsilon(\Phi)$.
#### Numerical calculation:
To calculate the quasiparticle excitation gap $\varepsilon(\Phi)$ we consider the Hamiltonian of the system $$H=E_C(i\partial_\phi+\hat{N}/2-n_g)^2+H_{BCS}(\phi,\Phi),\label{eq:H}$$ where $H_{BCS}(\phi,\Phi)$ is the BCS Hamiltonian for the system, $\phi$ is the phase of the superconducting island supporting the $NW$ and $\hat{N}$ is the number of electrons in the $NW$. In the above $E_C$ is the charging energy of the superconducting island together with the $NW$ and $2 e n_g$ is the gate charge [@vanheck; @interferometry]. In the limit $E_C\ll E_g$ (where $E_g$ is the gap of two-particle excitation), the lowest energy eigenstates of $H$ can be written in the form $$\begin{aligned}
&{\left\vert {\Psi} \right\rangle}=\int d\phi \psi(\phi){\left\vert {\xi(\phi)} \right\rangle}\otimes {\left\vert {\phi} \right\rangle},\label{eq:Psi}\end{aligned}$$ where ${\left\vert {\xi(\phi)} \right\rangle}$ is the instantaneous ground state of $H_{BCS}(\phi,\Phi)$ so that $H_{BCS}(\phi,\Phi){\left\vert {\xi(\phi)} \right\rangle}=E_J(\phi){\left\vert {\xi(\phi)} \right\rangle}$ and $\psi(\phi)$ is the probability amplitude for the superconducting phase to be at $\phi$. Computing the expectation value of ${\langle {\Psi|H|\Psi} \rangle}$ using Eq. \[eq:H\] and Eq. \[eq:Psi\], t he variational Schrodinger equation satisfied by $\psi(\phi)$ is found to be $$H_{eff}=E_C(i\partial_\phi-\Gamma+{\langle {\xi(\phi)|\hat{N}|\xi(\phi)} \rangle}/2-n_g)^2+E_{J}(\phi),\label{eq:Heff}$$ where $$\begin{aligned}
&\Gamma=-i\int_0^{2\pi}\frac{d\phi}{2\pi}{\langle {\xi(\phi)|\partial_\phi|\xi(\phi)} \rangle},\end{aligned}$$ is the Berry flux and can be computed numerically from ${\left\vert {\xi(\phi)} \right\rangle}$ [@stone-chung]. As a model, we have considered a spin-orbit coupled semiconductor nanowire [@Roman; @Oreg] for the $NW$. In order to model the contribution of conventional Andreev tunneling at the ends $a$ and $b$ we have added a term $-2 E_{J,0}\cos{(\pi\Phi/\Phi_0)}\cos{(\phi+\pi\Phi/\Phi_0)}$, which ensures that the wave-function $\psi(\phi)$ is localized near $\phi\sim -(\pi\Phi/\Phi_0)$. The details of the model and the derivation of Eq. \[eq:Heff\] are presented in the Supplementary material [@Supplementary2].
#### Result:
Since a superconducting system conserves fermion parity, one can compute a ground state with energy $\varepsilon_{e,o}(\Phi)$ in each of the fermion-parity sectors i.e. even and odd. The quasiparticle excitation gap that can be measured using tunneling using the configuration in Fig. \[Fig2\] is given by $$\begin{aligned}
\varepsilon(\Phi)=|\varepsilon_{e}(\Phi)-\varepsilon_o(\Phi)|\end{aligned}$$ is plotted as a function of $\Phi$ in Fig. \[Fig3\] for both the cases where $NW$ is in the topological and non-topological phase. The result in Fig. \[Fig3\], shows that despite the $\Phi_0$ periodicity of the excitation spectrum of $H_{BCS}(\phi,\Phi)$, the excitation spectrum including the charging energy shows a $2\Phi_0$ periodicity in the topological case, as expected from non-local Majorana-induced quasiparticle transfer across the wire.
![ Quasiparticle excitation spectrum $\varepsilon$ as a function of flux $\Phi/\Phi_0$ in the loop in Fig. \[Fig2\] in both the topological phase and the nontopological phase. Each spectrum is plotted with an even and odd multiple of flux $\Phi_0=hc/2e$ added to the loop. The excitation gap for even and odd flux quanta coincide with each other in the non-topological case, while the spectra in the topological case shows a dependence on the parity of the flux quantum near $\Phi\sim\Phi_0/2$. The change in the energy excitation spectrum from inserting a flux quantum $\Phi_0$ is shown to arise from the change in fermion parity, which is a unique character of the TS phase. []{data-label="Fig3"}](totspectrum122413-eps-converted-to.pdf)
Since the presence of a finite tunneling amplitude $G_R(\tau;ab)$ depends on the charging energy $E_C$, which is competes by the effective Josephson energy $E_{J,eff}(\Phi)=2 E_{J,0}|\cos{(\pi\Phi/\Phi_0)}|$, the $2\Phi_0$ periodic modulation in the tunneling gap is substantial in the TS phase only when $E_{J,eff}(\Phi)\lesssim E_C$. Thus the set-up in Fig. \[Fig3\] can be used to separate out Majorana-assisted electron transfer from direct transfer by tunneling of quasiparticles through $NW$ since the former would be strongly peaked at $\Phi\sim \Phi_0/2$, where $E_C$ dominates over the Josephson energy.
#### Comparison with the fractional Josephson effect:
The signature of a TS phase in Fig. \[Fig3\] appears as a $2\Phi_0$ periodicity of the tunneling spectrum. Formally, this resembles the $2\Phi_0$ periodicity of the ABS spectrum in the fractional Josephson effect in TS systems[@Kitaev-1D; @Sengupta-2001]. However, the ABS spectrum of the Josephson junction in a TS system is $2\Phi_0$-periodic as a result of zero-energy crossings in the ABS spectrum that are protected by fermion parity. The fermion parity protection is typically accomplished by requiring a non-equilibrium AC Josephson measurement [@Sengupta-2001].
In contrast, the measurement proposed in this work is an equilibrium measurement. This is surprising given that the set-up in Fig. \[Fig2\] might be viewed as a Josephson junction formed between the two MFs at the ends (a) and (b) of a TS nanowire and therefore as essentially identical to the set-up for the AC fractional Josephson effect. The additional ingredient that allows us to turn an otherwise non-equilibrium measurement into a more accessible DC measurement is the charging energy in the Hamiltonian in Eq. \[eq:H\]. The characterizing property of a TS system with periodic boundary conditions as in Fig. \[Fig2\], which also is ultimately responsible for the AC fractional Josephson effect, is the $2\Phi_0$ periodic dependence of the ground state fermion parity [@Read-Green]. The charging energy in Eq. \[eq:H\] leads to an additional ${\langle {N} \rangle}$-dependence in the Hamiltonian that ultimately leads to a $2\Phi_0$-periodicity in the excitation spectrum that can be measured by tunneling even in equilibirum.
#### Summary and Conclusion:
In this paper we have proposed a scheme for uniquely identifying the Majorana assisted non-local electron tunneling between two MFs at the ends of a wire in the TS phase. In principle, such a non-local transfer of electrons may be observable by a coincidence measurement [@sumanta; @semenoff; @gong]. However, as we have shown here that the Majorana assisted electron tunneling process using either a coincidence detection [@sumanta; @gong] or by measuring the transconductance with a charging energy [@fu], while interesting, cannot be taken as a definitive signature of MF modes because even conventional near-zero energy states trapped near the spatially separated leads can also produce such non-local signature. Instead we have proposed an interferometry experiment [@fu] appropriately generalized to geometries without edge modes. We have shown that such a measurement can distinguish conventional and Majorana zero modes. Our proposed non-local correlation experiment in terms of tunneling, which requires the inclusion of charging energy to fix the fermion parity, provides a direct verification of the non-locality of MFs in TS wires. We emphasize that the non-locality of the end state MFs arises from the non-locality of the fermion parity, which is unique to topological systems and cannot be emulated by conventional systems [@berg].
J. D. S acknowledges support from the Harvard Quantum Optics Center. S. T. would like to thank DARPAMTO, Grant No. FA9550-10-1-0497 and NSF, Grant No. PHY-1104527 for support. B. G. S. is supported by a Simons Fellowship through Harvard University. We acknowledge valuable conversations with Bertrand Halperin, Liang Fu, Charlie Marcus and Brian Swingle.
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Absence of non-local correlations in the non-topological case
=============================================================
We mention in the main text that the non-local correlation $G_R(\tau;ab)$ must vanish in the non-topological case. A particularly counter-intuitive case is the non-topological situation with two MFs at each end of a wire (as in Ref. ) where the wires are not coupled to each other. Such a system can in principle be prepared in a non-local initial state so that our current proposal gives a positive result by preparing each wire in a definite fermion parity state. However, as shown in the rest of the section, decoherence induced by coupling to the leads disfavors this non-local initial state and washes out any signature of non-locality in the present experiment. Hence, although a weakly coupled pair of MFs (or a conventional zero-energy state) at each end may still show a ZBCP at or near zero bias at finite temperature, our interference measurement will correctly demonstrate that the system is smoothly connected to a non-topological phase.
In the present section of the appendix, we show that coupling to a Fermionic bath leads to such desctruction of non-local correlations. We accomplish this in two sub-sections. In the first part, we show that when placed in contact with a fermion bath in thermal equilibrium, the system reaches a unique thermal equilibrium state. In the second part, we show that the system in thermal equilibrium cannot have any non-local correlations of the type that might lead to the measured interference or non-local tunneling amplitude.
Let us now discuss equilibration of the ground state in the simple case of non-interacting fermions. The non-interacting fermion problem can be solved in terms of $$\begin{aligned}
&\rho_{ab}(t)=i{\langle {\gamma_a(t)\gamma_b(t)} \rangle}.\end{aligned}$$ The Hamiltonian in terms of Majorana operators is written as $$H=i\sum_{a,b}h_{a,b}\gamma_a \gamma_b.$$ The equation of motion is written as $$\partial_t \gamma_a = h_{a,b}\gamma_b,$$ with $h_{a,b}$ being anti-symmetric. The time-evolution of $\gamma_a(t)$ is written as $$\gamma(t)=G^{(R)}(t)\gamma(0),$$ where $G^{(R)}(t)=e^{h t}$ for $t>0$. Expanding $G^{(R)}(t)$ in eigenstates $$\begin{aligned}
&G^{(R)}(t)=\sum_n \psi_n \psi_n^\dagger e^{i\varepsilon_n t}\nonumber\\
&=\sum_n \psi_n \psi_n^\dagger \int_{-\infty}^{\infty}\frac{e^{i\omega t}}{\omega-\varepsilon_n-i\delta}=\int_{-\infty}^{\infty}\frac{e^{i\omega t}}{\omega-i h-i\delta}.\end{aligned}$$ The time evolution of the density matrix is written as $$\rho(t)=G^{(R)}(t)\rho(0) G^{(A)}(t).$$
Assuming that we start with a density matrix that is diagonal between system $(s)$ and bath $(b)$ (i.e. lead), the system density matrix $$\rho_s(t)=G^{(R)}_{s}(t)\rho_s(0)G^{(A)}_{s}(t)+G^{(R)}_{sb}(t)\rho_b(0)G^{(A)}_{bs}(t).\label{eq:rhos}$$
Assuming the system-bath structure for the Hamiltonian and defining $g_b^{(R,A)}=(\omega\mp i\delta-H_{b})^{-1}$, one can write the final system Green function as $$\begin{aligned}
&G^{(R,A)}_s(\omega)=(\omega\mp i\delta -H_{sb}g^{(R,A)}_b(\omega)H_{bs})^{-1}.\end{aligned}$$ The system bath interaction Green function can be written as $$\begin{aligned}
&G^{(R,A)}_{sb}(\omega)=G_s^{(R,A)}(\omega)H_{sb}g^{(R,A)}_b(\omega)\nonumber\\
&G^{(R,A)}_{bs}(\omega)=g^{(R,A)}_b(\omega)H_{bs}G_s^{(R,A)}(\omega).\label{eq:Gsb}\end{aligned}$$ Provided the anti-hermitean part of the fermion-self-energy $$\Sigma^{(R,A)}_s(\omega)=Im[H_{sb}g^{(R,A)}_b(\omega)H_{bs}]$$ does not have a null-space (i.e. a space of zero-eigenvalues) the Green function $G^{(R,A)}_s(t)$ can be assumed to decay exponentially in time. Therefore it follows from Eq. \[eq:rhos\] that the first term must vanish. To evaluate the second term in Eq. \[eq:rhos\], we note that from Eq. \[eq:Gsb\], it is clear that $$G^{(R,A)}_{sb}(t)=\int d\tau G_s(t-\tau)H_{sb}g_b(\tau).$$ Since $G_s(t-\tau)$ is exponentially decaying, the $\Theta(\tau)$ component of $g_b$ is not relevant and one can approximate $$\begin{aligned}
&G^{(R,A)}_{sb}(t)\approx \int d\tau G_s(t-\tau)H_{sb}e^{-h_b\tau}\nonumber\\
&=\int_{0}^{\infty} d\tau G_s(\tau)H_{sb}e^{-h_b(t-\tau)}\end{aligned}$$ up to exponential factors. Applying this identity to Eq. \[eq:rhos\] we obtain $$\begin{aligned}
&\rho_s(t)\nonumber\\
&\approx \int_0^\infty d\tau_1 d\tau_2 G_s^{(R)}(\tau_1)H_{sb}e^{-h_b(t-\tau_1)}\rho_b(0)e^{h_b(t-\tau_2)}H_{bs}G_s^{(A)}(\tau_2)\nonumber\\
&=\int_{-\infty}^\infty d\tau_1 d\tau_2 G_s^{(R)}(\tau_1)H_{sb}\rho_b(0)e^{h_b(\tau_1-\tau_2)}H_{bs}G_s^{(A)}(\tau_2),\end{aligned}$$ which is a $t$-independent asymptotic value. Transforming to frequency space, $$\begin{aligned}
&\rho_s(t)\approx \int_{-\infty}^\infty d\omega f(\omega)G_s^{(R)}(\omega)H_{sb}A_b(\omega)H_{bs}G_s^{(A)}(\omega).\end{aligned}$$ Considering the integrand at $\omega$ away from a pole of $g_s(\omega)$, $$\begin{aligned}
&G_s^{(R)}(\omega)H_{sb}A_b(\omega)H_{bs}G_s^{(A)}(\omega)\nonumber\\
&=i G_s^{(R)}(\omega)H_{sb}\{g^{(R)}_b(\omega)-g^{(A)}_b(\omega)\}H_{bs}G_s^{(A)}(\omega)\nonumber\\
&=i G_s^{(R)}(\omega)\{\Sigma^{(R)}_s(\omega)-\Sigma^{(A)}_s(\omega)\}G_s^{(A)}(\omega)\nonumber\\
&=i \{G_s^{(R)}(\omega)-G_s^{(A)}(\omega)\},\end{aligned}$$ which can be checked using Dyson’s equation. Therefore, in the absence of a protected null-space in the dissipative part of the self-energy $\Sigma_s(\omega)$, the density matrix equilibrates to the grand-canonical thermal equilibrium value $$\begin{aligned}
&\rho_s(t)\approx i\int_{-\infty}^\infty d\omega f(\omega)\{G_s^{(R)}(\omega)-G_s^{(A)}(\omega)\}.\end{aligned}$$
Let us now discuss correlations in the grand-canonical thermal state. First we prove a general lemma: Consider the grand-canonical thermal partition function of a system of Majorana fermions (or fermions) which can be partitioned into two parts $L$ and $R$, so that the Hamiltonian has a symmetry $\gamma_{L,a}\rightarrow -\gamma_{L,a}$ for all MFs $\gamma_{L,a}$ on the left half of the system $L$. Then all correlation functions ${\langle {\gamma_{L,a}\gamma_{R,b}} \rangle}=0$. To prove this simply apply the transformation $\gamma_{L,a}\rightarrow -U\gamma_{L,a}U^\dagger$ to $$\begin{aligned}
&{\langle {\gamma_{L,a}\gamma_{R,b}} \rangle}\propto Tr[\gamma_{L,a}\gamma_{R,b}e^{-\beta H}]=Tr[U\gamma_{L,a}U^\dagger\gamma_{R,b} Ue^{-\beta H}U^\dagger]\nonumber\\
&=-Tr[\gamma_{L,a}\gamma_{R,b}e^{-\beta H}]=0.\end{aligned}$$ Therefore all such non-local fermionic correlations must vanish in the thermal state.
As discussed before, in the topological state, the charging energy gives rise to a fermion parity dependent term in the Hamiltonian $$H_{nl}\propto i\gamma_a\gamma_b,$$ which violates the conditions for the above lemma and leaves a non-zero value for this non-local correlator. Therefore the presence of the Majorana fermion term is crucial for the appearance of a non-local correlation function that can contribute to the interference. Such an interference will not happen even if the conventional end zero mode is made of a pair of Majorana fermions. This is because, as discussed in the previous sub-section, coupling Majorana fermions to leads causes them to thermalize into the grand-canonical ensemble. Once they have thermalized in this ensemble, $H_{nl}$ violates the conditions of the lemma and generates a non-local fermion correlation. The long-range Coulomb interaction for conventional fermionic modes does not lead to the violation of the conditions of the lemma and does not lead to any long range correlation. Following the rest of the argument in the text, it is clear that once non-local Green functions are absent, there can be no non-local correlation.
Technical details of calculating the excitation spectrum
========================================================
The quasiparticle degrees of freedom in the Hamiltonian in Eq. \[eq:H\] is taken to be the BCS Hamiltonian for a superconducting proximity coupled semiconductor nanowire given by $$\begin{aligned}
&H_{BCS}(\phi,\Phi)=\sum_{\sigma}\int dx \frac{\hbar^2}{2 m^*}|(i\partial_x -a(x))c_{\sigma}(x)|^2\nonumber\\
&+(V(x)-\mu)c_\sigma^\dagger(x)c_\sigma(x)+V_{Z,z}\sigma c_\sigma^\dagger(x)c_\sigma(x)+V_{Z,x} c_{-\sigma}^\dagger(x)c_\sigma(x)\nonumber\\
&+\alpha c^\dagger_{-\sigma}(x)(i\hbar\partial_x -a(x))c_{\sigma}(x) +(\Delta(x)\sigma c_\sigma^\dagger(x)c_{-\sigma}^\dagger(x)+h.c),\end{aligned}$$ where $c^\dagger_\sigma(x)$ is the electron creation operator in the nanowire with spin $\sigma$, $m^*$ is the effective mass, $a(x)$ is the vector potential from the flux in the loop and $V(x)$ is the gate controlled potential. The chemical potential of the electrons is represented by $\mu$, $V_Z$ is the Zeeman potential set by a magnetic field, $\alpha$ is the Rashba spin-orbit coupling strength and $\Delta(x)$ is the proximity-induced superconducting pair potential. To represent the geometry in Fig. \[Fig2\], the electrons are chosen to have periodic boundary conditions with the end $a$ represented by $x=0$ and the superconducting pair potential $\Delta(x)=\Delta_0 e^{i\phi}$ in $NW$ and $\Delta(x)=\Delta_1$ in the SC loop. The vector potential $a(x)=2\pi \delta(x)\Phi/\Phi_0$ is determined by the flux $\Phi$ in the loop. For the result in Fig. \[Fig3\] we have chosen $\mu=2$ K, $V(x)=0$ K in the $NW$, $V(x)=1$ K in the SC loop, the spin-orbit coupling strength is $\alpha=0.5\, \textrm{eV}-\AA$, the pair potential in the $NW$ is $\Delta_0=3$ K, and the pair potential in the SC loop is $\Delta_1=0.2$ K. The Zeeman potentials are taken to be $V_{Z,z}=4.5$ K and $V_{Z,x}= 0.6$ K. The effective mass is taken to be $m^*=0.015\, m_e$ for InSb, and the length of the $NW$ and the SC loop is taken to be $L=5\,\mu m$. The charging energy $E_C$ in the Hamiltonian $H$ in Eq. \[eq:H\] is taken to be $E_C=0.7$ K, while the $\phi$-dependence of the other electrons in the quasi one-dimensional system is modelled as a Josephson energy of amplitude $E_{J,0}=0.4$ K.
The Hamiltonian in Eq. \[eq:H\] is complicated by the term $\hat{N}/2$ in the charging energy term proportional to $E_C$. This term can be eliminated by a unitary transformation $U=\int d\phi {\left\vert {\phi} \right\rangle}{\left\langle {\phi} \right \vert}e^{i\phi\hat{N}/2}$. Such a unitary transformation transforms the BCS Hamiltonian $H_{BCS}(\phi)$ to $\tilde{H}_{BCS}(\phi)=U H_{BCS}(\phi)U^\dagger$ and the eigenstate ${\left\vert {\xi(\phi)} \right\rangle}$ to ${\left\vert {\tilde{xi}(\phi)} \right\rangle}=U{\left\vert {\xi(\phi)} \right\rangle}$. Defining a variational ansatz for the eigenstate for the transformed Hamiltonian $\tilde{H}=U H U^\dagger$ as $$\begin{aligned}
&{\left\vert {\tilde{\Psi}} \right\rangle}=\int d\phi \psi(\phi){\left\vert {\tilde{\xi}(\phi)} \right\rangle}\otimes {\left\vert {\phi} \right\rangle},\end{aligned}$$ we obtain an effective Hamiltonian for $\tilde{\psi}$ which is written as $$H_{eff}=E_C(i\partial_\phi-\tilde{\Gamma}-n_g)^2+\tilde{E}_{J}(\phi),$$ where $\tilde{\Gamma}=-i{\langle {\tilde{\xi}(\phi)|\partial_\phi|\tilde{\xi}(\phi)} \rangle}=\Gamma+{\langle {\xi(\phi)|\hat{N}|\xi(\phi)} \rangle}/2$, and $\Gamma=-i{\langle {\xi(\phi)|\partial_\phi|\xi(\phi)} \rangle}$. The eigenstate of the original Hamiltonian $H$ is defined in terms of $\psi(\phi)$ as $$\begin{aligned}
&{\left\vert {\Psi} \right\rangle}=U{\left\vert {\tilde{\Psi}} \right\rangle}=\int d\phi \psi(\phi){\left\vert {\xi(\phi)} \right\rangle}\otimes {\left\vert {\phi} \right\rangle},\end{aligned}$$ where $\psi(\phi)$ satisfies Eq. \[eq:Heff\].
[^1]: One may be concerned about a case where one has a pair of isolated topological wires with a pair of MFs at each end. Strictly speaking, this system would show a non-local correlation if each wire is in a definite fermion parity state. However, as we discuss in the Supplementary material (See first section of Supplementary material), this correlation is fragile and will be destroyed in the presence of dissipation. Note that, even though any residual gapless electronic conducting states in the quantum wire are also expected to yield a non-zero $G_R(\tau;ab)$. However, unlike the topological contribution this contribution survives in the presence of charging energy.
|
---
author:
- |
Xuefeng Peng, MSc.$^{1}$, Yi Ding, MSc.$^{2}$, David Wihl, A.L.B.$^{1}$, Omer Gottesman, PhD$^{1}$,\
Matthieu Komorowski, MD$^{3}$, Li-wei H. Lehman, PhD$^{4}$, Andrew Ross, MSE$^{1}$,\
Aldo Faisal, PhD$^{3}$, Finale Doshi-Velez, PhD$^{1}$
bibliography:
- 'reference.bib'
title: |
Improving Sepsis Treatment Strategies by Combining\
Deep and Kernel-Based Reinforcement Learning
---
[**Abstract**]{}
*Sepsis is the leading cause of mortality in the ICU. It is challenging to manage because individual patients respond differently to treatment. Thus, tailoring treatment to the individual patient is essential for the best outcomes. In this paper, we take steps toward this goal by applying a mixture-of-experts framework to personalize sepsis treatment. The mixture model selectively alternates between neighbor-based (kernel) and deep reinforcement learning (DRL) experts depending on patient’s current history. On a large retrospective cohort, this mixture-based approach outperforms physician, kernel only, and DRL-only experts.*
Introduction {#introduction .unnumbered}
============
Sepsis is a medical emergency requiring rapid treatment. [@seymour2017time] It is the cause of 6.0% of hospital admissions but 15.0% of hospital mortality.[@rhee2017incidence] It is also costly: in 2011 alone, the US spent \$20.3 billion dollars on hospital care for patients with sepsis.[@pfuntner2006costs] Managing sepsis remains challenging, in part because there exists large variation in patient response to existing sepsis management strategies.[@waechter2014interaction]
In this work, we focus on two interventions in the context of sepsis management: intravenous (IV) fluid (adjusted for fluid tonicity) and vasopressors (VP). These two drugs are respectively used to correct the hypovolemia and counteract sepsis-induced vasodilation. While hypovolemia and vasodilation are common among patients with sepsis, there exists little clinical consensus about when and how these should be treated.[@marik2015demise] However, these choices can have large implications for patient mortality:[@waechter2014interaction] vasopressors are known to have harmful effects in certain patients, and recent studies have also demonstrated the association between fluid-overload and negative outcomes in the ICUs.[@kelm2015]
Thus, it is essential to identify ways to personalize treatment. The availability of large observational critical care data sets [@johnson2016mimic] has made it possible to hypothesize improved sepsis management strategies, and prior studies [@DBLP:journals/corr/RaghuKCSG17; @komorowski2018intensive] have used this resource to suggest optimal treatment strategies for patients with sepsis. As with those earlier works, we personalize strategies by using reinforcement learning (RL), a technique for optimizing sequences of decisions given patient context. However, we use a mixture-of-experts approach to combine two very different RL techniques with very different strengths—a model-free deep RL approach (DRL) and a model-based kernel RL approach (KRL)—to improve the quality of the recommended treatment policy. Specifically, our work extends prior efforts in three important ways:
1. *Recurrent encoding of the patient’s history.* To date, work in this domain has assumed that the patient’s current measurements are sufficient to summarize their history. To retain potentially decision-relevant information from the patient’s past, we use a recurrent autoencoder to represent the patient’s entire history.
2. *Safe-guards on the Deep RL.* DRL-based approaches can be particularly poor at extrapolating, and even in areas of dense data, they can suggest non-sensical actions. We explicitly restrict the DRL to only suggest actions commonly taken by clinicians, moving us toward more clinically-credible policies.
3. *Combining DRL with Kernel RL.* Finally, we use a mixture-of-experts (MoE) to combine the restricted DRL approach with a kernel RL approach selectively based on the context. DRL is more flexible but can be prone to various pathologies; KRL is guaranteed to stay close to the data but as a result can extrapolate poorly. The MoE combines their strengths.
Related Work {#related-work .unnumbered}
============
Reinforcement Learning has been applied to a number of applications in healthcare, ranging from emergency decision support,[@thapa2005agent] treating malaria, [@rajpurkar2017malaria] and managing HIV.[@parbhoo2014reinforcement] Prasad et al.[@prasad2017reinforcement] use RL to identify when to the wean patients from mechanical ventilation in ICUs.
With respect to fluid and vaospressor use in sepsis management, Komorowski et al.[@komorowski2018intensive] model a discrete Markov decision process from data and then utilize it to learn a treatment strategy. Raghu et al.[@DBLP:journals/corr/RaghuKCSG17] extend this work by considering a much more expressive continuous representation of patient state. They use a traditional, non-recurrent autoencoder to first compress measurements from each time step into a continuous state representation, and then they learn a mapping from the state representation to an appropriate treatment via a Dueling Double-Deep Q Network (Dueling DDQN). Our work uses an even more expressive state representation that represents the patient’s entire history, and we also add safe-guards against inappropriate actions and develop richer policies through our mixture of experts.
The mixture of experts aspect of our work builds from ideas developed by Parbhoo et al.[@parbhoo2017combining] in the context of HIV management. In their case, they switch between a kernel-based policy and a discrete Bayesian Partially Observable Markov Decision Process (POMDP). We follow the idea of combining experts, but use the DDQN as an expert rather than a discrete POMDP, as Raghu et al.[@DBLP:journals/corr/RaghuKCSG17] have already demonstrated that a continuous expressive state representation is valuable for the sepsis management task. Because we use a recurrent encoding to summarize the entire patient history, our state can be thought of as a sufficient statistic, much like the POMDP belief state in Parbhoo et al.[@parbhoo2017combining]
Background {#background .unnumbered}
==========
The reinforcement learning framework models a sequence of decisions as an agent interacting with an environment over time. At each time step $t$, the RL agent observes a state $s$ from the state space $S$ and selects an action $a$ from the action space $A$ based on some policy $\pi(s,a)$, which assigns a probability to action in each state. Upon taking the action, the agent receives some reward $r$ and transitions to a new state $s^\prime$. The agent’s goal is to maximize their expected longterm discounted return $\mathbb{E} [\sum_t \gamma^t r_t]$. The optimal value function is defined as $V^{*}(s)=\max_{\pi} \mathbb{E} [\sum_t \gamma^t r_t]|s_0=s, \pi]$, and the optimal state-action value function $Q^{*}(s,a)=\max_{\pi}\mathbb{E} [\sum_t \gamma^t r_t|s_0 = s,a_0 = a,\pi]$. The latter satisfies the Bellman equation $Q^{*}(s,a)= r(s,a)+\gamma \max_{a^{'}} \mathbb{E}[Q^{*}(s^{'},a^{'})]$, where $\gamma$ is the discount factor determines the trade-off between immediate and future rewards. Q-learning methods aim to learn an optimal policy by minimizing the temporal difference (TD) error, defined as $r(s,a) + \gamma Q^{*}(s^{'}, a^{'}) - Q^{*}(s,a)$.
Cohort and Data Processing {#cohort-and-data-processing .unnumbered}
==========================
*Cohort.* We used the same patient set as in Raghu et al. [@DBLP:journals/corr/RaghuKCSG17] which applied the Sepsis-3 criteria to the Multi-parameter Intelligent Monitoring in Intensive Care (MIMIC-III v1.4) database. [@johnson2016mimic] Our cohort consisted of 15,415 adults (age range of 18 to 91), summarized in Table \[table:demographics\].
% Female Mean Age Total Population
--------------- ---------- ---------- ------------------
Survivors 44.1% 63.9 13,535
Non-survivors 44.3% 67.4 1,880
: Comparison of cohort statistics for subjects that fulfilled the Sepsis-3 criteria[]{data-label="table:demographics"}
*Cleaning and Preprocessing.* As in Raghu et al. [@DBLP:journals/corr/RaghuKCSG17], patient histories were partitioned into 4-hour windows each containing fifty attributes, ranging from vitals (e.g. heart rate, mean blood pressure) to clinician assessments of the patient’s conditions (e.g. sequential organ failure assessment (SOFA) score). Patients with missing values were excluded. The observations, which range from demographics, lab values and vital signs, all have different scales. Following Raghu et al., [@DBLP:journals/corr/RaghuKCSG17] we performed log transformations of observations with large values and standardized the remaining values. After the standardization and log transformation, all values were rescaled into $[0-1]$. The details of attributes and preprocessing is shown in Table \[table:preprocessing\].The data set was split into a fixed 75% training and validation set and a 25% test set.
[|l|l|]{} Preprocessing & Attributes\
Standardization &
----------------------------------------------------------------------
`age,Weight_kg,GCS,HR,SysBP,MeanBP,DiaBP,RR,Temp_C,FiO2_1,`
`Potassium,Sodium,Chloride,Glucose,Magnesium,Calcium,Hb,`
`WBC_count,Platelets_count,PTT,PT,Arterial_pH,paO2,paCO2,`
`Arterial_BE,HCO3,Arterial_lactate,SOFA,SIRS,Shock_Index,PaO2_FiO2,`
`cumulated_balance_tev, Elixhauser, Albumin, CO2_mEqL, Ionised_Ca`
----------------------------------------------------------------------
: Physiological attributes, treatments and corresponding preprocessing methods[]{data-label="table:preprocessing"}
\
Log transformation &
[@l@]{}`max_dose_vaso,SpO2,BUN,Creatinine,SGOT,SGPT,Total_bili,`\
`INR,input_total_tev,input_4hourly_tev,output_total,output_4hourly`
\
*Treatment Discretization.* In this work, we focus on administrating two drugs: intravenous (IV) fluid and vasopressor (VP). In the cohort, the usage of IV and VP for each patient are recorded at each 4-hour window. Following Raghu et al.[@DBLP:journals/corr/RaghuKCSG17], the dosages for each drug are discretized into 5 bins, resulting in a $5\times5$ action space indexed from 0 to 24. Note that the first action ($a=0$) means “no action"—neither IV nor VP are prescribed.
Method Overview {#method-overview .unnumbered}
===============
Applying RL to the sepsis management problem involves several pieces. The first is defining our inputs and treatments (sections above). Next we describe how we compress patient histories into a state via a recurrent autoencoder, how we attribute rewards to each state, and also how we determine the quality of some policy given observational data. With these pieces in place, we finally describe how to derive treatment policies that optimize for our rewards, including our mixture-of-experts (MoE) approach.
Compressing Patient Histories {#compressing-patient-histories .unnumbered}
-----------------------------
Prior efforts [@komorowski2018intensive; @DBLP:journals/corr/RaghuKCSG17] assumed that the patient’s current measures were sufficient to summarize their history; however, past and trend patient information is often valuable to deciding the appropriate course of action. To capture more of this temporal information, we encoded patient states recurrently using an LSTM autoencoder representing the cumulative history for each patient. The LSTM had a single layer of 128 hidden units for both encoder and decoder—resulting in a state $s$ that consisted of 128 real-valued vector. The autoencoder was trained to minimize the reconstruction loss (MSE) between the original measurements and the decoded measurements. We trained the model with mini-batches of 128 and the Adam optimizer for 50 epochs until its convergence.
Reward Formulation {#subsec:rewardformulation .unnumbered}
------------------
Broadly, we are interested in reducing mortality among patients with sepsis. However, mortality is a challenging objective to optimize for because it is only observed after a long sequence of decisions; it can be hard to ascertain which action was responsible for a good or bad outcome. Thus, for the purposes of our training, we introduce an *intermediate* reward that can give a preliminary signal for whether our sequence of treatment decisions is likely to reduce mortality.
Specifically, we first train a regressor that predicts the probability of mortality given a patient’s current observations (implications of this choice in ). Next, we define the reward as the change in the negative mortality log-odds of mortality between the current observations and the next observations. (Log-odds were used because the probabilities of mortality actually vary over a relatively small range.) Let $f(o)$ be the probability of mortality given current observations $o$. Then we define the reward $r(o,a,o^\prime)$ as $$\begin{aligned}
\label{eqn:network}
r(o,a,o^\prime) = - \log \frac{f(o^\prime)}{1-f(o^\prime)} f(o^\prime) + \log \frac{f(o)}{1-f(o)}\end{aligned}$$ For a sense of scale, among those over 186K patient state transitions from both training and testing sets, the rewards are in the interval $[-3, 3]$.
The mortality predictor $f(o)$ itself was a two-layer neural network with $64$ and $32$ units for each layer and L1-regularized gradients (see Ross et al. [@nips2017timl] for details). The L1 regularization encourages sparse local linear approximations, and thus makes its behaviors more interpretable. We resampled balanced batches (between survivors and non-survivors) during training with batch sizes of $128$ observations for $50$ epochs, and our predictor $f(o)$ achieved a test accuracy of $73.1\%$. Figure \[fig:mortality-log-odds\] shows the log-odds distributions for each class.
![\[fig:mortality-log-odds\] Mortality log-odds distribution for mortality and survivor classes.](figures/log-odds.png){width="0.4\linewidth"}
Off-Policy Evaluation via the WDR Estimator {#subsec:wdr .unnumbered}
-------------------------------------------
The natural question is how to evaluate the quality of a proposed policy $\pi_e$ given only retrospective data collected according to a clinician policy $\pi_b$. The weighted doubly robust (WDR) estimator[@thomas2016data] is widely used for off-policy evaluation in RL. It uses estimated value $\hat{V}$ and action-value $\hat{Q}$ functions as control variates to reduce the variance of the off-policy estimation. In the following, we used the value function of the DRL policy as our control variate; we also explored using the value of the clinician policy, estimated as the mean, instead of the max, of the DRL Q-values over action space.
$$\begin{aligned}
\label{eq:wdr}
\text{WDR}(D) := \sum_{i=1}^I\sum_{t=0}^{T} \gamma^tw_i^tr_t^{H_i} - \sum_{i=1}^I\sum_{t=0}^{T}\gamma^t(w_t^i\hat{Q}^{\pi_e}(S_t^{H_i}, A_t^{H_i}) - w_{t-1}^i\hat{V}^{\pi_e}(S_t^{H_i}))\end{aligned}$$
Here, $I$ is the number of patients, $t$ is the time step, and $H_{i}$ refers to the $i_{th}$ patient’s ICU-stay state trajectory. The importance weight of the state of the patient $i$ at time step $t$ is defined as $w^{i}_{t}=\frac{\rho^{i}_{t}}{\sum^{I}_{j=1}\rho^{j}_{t}}$, where $\rho^{i}_{t} = \prod^{t}_{t'=0}\frac{\pi_{e}(A^{H_i}_{t'}|S^{H_i}_{t'})}{\pi_{b}(A^{H_i}_{t'}|S^{H_i}_{t'})}$. The reward $r$, value function $\hat{V}$, and action-value function $\hat{Q}$ are as defined above. Finally, we estimate the clinician policy $\pi_b$ as the empirical distribution over actions of the 300 neighbors in the training set with states closest to $s$, as research shows that clinicians typically make decisions based on their experience treating similar patients [@norman2005research].
Deriving Policies {#subsec:dp .unnumbered}
=================
With a means of representing patient history, rewards, and a metric for evaluating policies, we can now start optimizing treatment strategies. Below we describe each expert—the DRL and the KRL—and how we combine them.
*Kernel Policy.* One simple way to derive a treatment decision rule is to look at the nearest neighbors to the current state $s$, identify the survivors, and choose actions that correspond to the distribution of treatments performed on these nearby survivors (see cartoon in Figure \[fig:kernel\_policy\]). Specifically, we
1. Get the encoding $s$ from the patient’s history $h$ up to time $t$ via the LSTM autoencoder.
2. Search *k* nearest neighbors in the training set in this encoded representation space using Euclidean distance.
3. The kernel policy $\pi_k$ is the distribution of actions taken over the surviving nearest neighbors.
We cross-validated the *k* values ranging from 200 to 500 using WDR, and proceeded with $k=300$.
![\[fig:kernel\_policy\] The circle in the left shows an example of the neighborhoods of a new state $s$, red and green marks the mortality and surviving states respectively, and each of these states is associated with a physician action $A_{i}$.](figures/kernel.png){width="70.00000%"}
*DQN Policy.* Double DQN (DDQN) with dueling structure[@wang2015dueling], which is a variant of DQN, has been applied to derive a policy that outperforms the physician policy[@DBLP:journals/corr/RaghuKCSG17]. The structure of dueling DDQN is particularly suitable for sepsis treatment strategy learning, as it differentiates the value function $V$ into the value of the patient’s underlying physiological condition, called the *Value* stream, and the value of the treatment given, called the *Advantage* stream.
We train the dueling DDQN for 200,000 steps with $batch\,size=30$ to minimize the TD-error. At each given state, the agent is trained to take an action with the highest Q-value, in order to achieve the ultimate goal of improving the overall survival rate. To stabilize the training process and improve the performance, we applied regularization term $\lambda$ to the Q-network loss to penalize output Q-values which exceeded the maximum observed rewards $r_{max}=3$ and used prioritized experience replay[@schaul2015prioritized] to balance the training sets with high-value and high-error states. $$\label{eq:qnetwork}
\textit{\L}(\theta) = E[(Q_{double-target}-Q(s,a;\theta))^{2}] + \lambda\,\max(|Q(s,a;\theta)-r_{max}|, 0)$$ where $$Q_{double-target} = r + \gamma Q(s,\operatorname*{arg\,max}_{a^\prime} Q(s,a^\prime;\theta^\prime))$$ and $\theta, \theta'$ are parameters of the DQN networks.
Finally, given a set of action-values $Q(s,a)$, we must still define a policy $\pi_d$. Typically, these actions are chosen by the $\max Q$-value, but this ignores the fact that two actions may have very similar values—and given the limitations of our learning, it may not be possible which is actually the best. We define the DRL policy $\pi_d$ as the softmax of the action-value or *Advantage* stream, giving higher probability to actions with higher values but not forcing us to take (what might be a brittle) best value.
Mixture-of-Experts (MoE) {#mixture-of-experts-moe .unnumbered}
========================
The two approaches above—KRL and DRL—have different strengths. For patient states which are atypical, i.e. farther Euclidean distance away from any neighbors, the kernel policy may end up relying on neighbors that are not really that similar to the patient. In contrast, DQN, in trying to fit a value function to the whole state space at once, may still underfit in regions with plentiful data. Our mixture-of-experts (MoE) uses properties of the patient’s current state, and the relationship between the patient’s current state and states observed during training, to switch between the kernel and DQN policies (see Figure \[fig:moe\] for a cartoon).
![\[fig:moe\] The architecture of MoE, it produces a mixed policy via combining kernel.](figures/MOE){width="70.00000%"}
*Action Restriction.* The DDQN, as a complex function approximator, comes with relatively few guarantees. Sometimes, it can place high value on actions that were rarely or never performed by clinicians. To safe guard against these rare (and likely dangerous) actions, we restricted the DDQN to actions only taken more than 1% of the time by the physicians among its 300 nearest neighbors. Specifically, let $\pi_{d}(s,a)$ be the DDQN policy, and $\pi_{b}(s,a)$ be the physician policy. If $\pi_{b}(s,a) < 0.01$, then we set $\pi_{d}(s,a) \leftarrow 0$ and then normalize $\pi_d(s,a)$ to be a valid probability distribution. (We note that the kernel policy, which is derived directly from clinician actions, cannot deviate in this way; once we restrict the DDQN actions, the MoE will also never take rare actions.)
*Choice of Gating Function.* We examined several medical sources [@jones2009sequential; @beier2011elevation; @tamion2010albumin] to determine which features might be most useful for selecting between experts. Our final set of features were: age, Elixhauser, SOFA, $FiO_{2}$, BUN, GCS, Albumin, trajectory length, and max distance from neighbors. For our MoE gating function we combined these features $x$ linearly via weights $w$, along with a bias term $b$, and passed them through a logit to get the probability of choosing each policy: $$\begin{aligned}
\label{eq:moe}
p_k = \text{sigmoid}(w \cdot x + b) \quad \textrm{and} \quad p_{d} = 1 - p_{k}\end{aligned}$$ where $p_{k}$ and $p_{d}$ denote the assigned probability for choosing the kernel and DQN policy respectively.
*Optimizing the Gating Function.* Of course, the core question is how to choose the gating parameters so as to maximize long-term rewards. Given a set of weights $w$ and bias $b$, the MoE policy $\pi_m$ is defined as $\pi_{m}(s,a)=p_{k}\pi_{k}(s,a) + p_{d}\pi_{d}(s,a)$. We can estimate the expected discounted return of the policy $\pi_m$ WDR from above; we again perform gradient descent on the gating parameters (with a minibatch of 256 samples). Due to the nonconvexity of the WDR-based objective, we take the best of $1000$ random restarts.
Results {#results .unnumbered}
=======
The estimates of the discounted expected return for each policy are presented in Table \[tbl:wdr\]. We provide two columns for the mixture of experts policy because it is challenging to derive accurate value estimates $\hat{V}$ and $\hat{Q}$ for the WDR estimator in this case. Thus, we consider two sensible options: using the value estimates $\hat{V}$ and $\hat{Q}$ from the clinician policy, and using the estimates from the DDQN policy.
Regardless of the choice of evaluation covariate, both kernel and DQN policy improve over the physician policy, and the MoE policy projects a further improvement. Using a recurrent state representation that compresses the entire history results in a further improvement for all RL policies (we used a sparse autoencoder[@ng2011cs294a] for the non-recurrent encoding). The Figure \[fig:gradients\] includes boostrap intervals for these values[@DBLP:journals/corr/abs-1805-12298].
![The distribution of the WDR estimator difference between MoE and three other policies based on 1000 bootstrapped test datasets. The red vertical lines indicate the difference calculated for the original test dataset.\[fig:gradients\]](figures/boostrap){width="60.00000%"}
On average, the WDR estimator predicts the MoE policy to outperform all other proposed policies. We note, however, that for some bootstraped datasets the difference is negative. This is to be expected, as Gottesman et al. \[24\] demonstrate that high variability of IS based estimators is common in healthcare-data. Gottesman et al. \[24\] further demonstrate that IS based estimators can have a high selection bias when evaluating policies which are significantly different from the behavior policy.
Physician Kernel DQN $MoE_{V_{d}, Q_{d}}$ $MoE_{V_{b}, Q_{b}}$
----------------------- ----------- -------- ------ ---------------------- ----------------------
non-recurrent encoded 3.76 3.73 4.06 3.93 4.31
recurrent encoded 3.76 4.46 4.23 5.03 5.72
: Estimate of the discounted expected return for policies over test set, $\gamma=0.99$. $V_{d}$ indicates approximating the MoE $V$ by DQN $V$ function, $V_{b}$ indicates approximating the MoE $V$ by behavioral policy, namely, physician $V$ function[]{data-label="tbl:wdr"}
*Analysis of discovered policies.* Figure \[fig:expert\_actions\] shows the action distributions for the KRL, DRL, clinician, and MoE policies over the test set. The no treatment $a=0$ action dominates policies. Actions which are favored by physicians, such as IV but no vasopressor are also (as expected) favored by the kernel policy. That said, the kernel policy tends to be more conservative than the clinicians as it suggests nonaction at approximately twice the clinicians’ frequency—perhaps reflecting a bias toward the fact that those patients who were not treated were somehow healthier and thus survived. Perhaps in a similar vein, the kernel expert prescribes more fluid alone than the clinicians: while both suggest high fluids and no vasopressor almost in the same frequency, the kernel expert very rarely suggests actions with vasopressor.
The DRL policy, like clinician and the kernel policies, favors giving actions a range of fluid values. But, over the test set, DQN expert prescribes more extreme values than clinicians. Clinicians frequently prescribe high fluids and low vasopressor; however, the DQN policy also tends to give more high vasopressor dosage actions in addition to fluids.
Over the test set, the MoE policy is closer to the kernel policy. But influenced by the DQN policy, MoE prescribes more high dosage actions.Table \[tbl:overlap\] shows how actions suggested by the experts overlap; the rate is high for the kernel and MoE policies again reflecting the fact that most of the time, patients can find similar neighbors. In 4.4% of circumstances, the gating results in a MoE policy follows neither that of kernel nor that of DQN policy.
kernel DQN MoE
----------- -------- ------- ------- --
physician 0.305 0.151 0.296
kernel - 0.182 0.871
DQN - - 0.258
: Percent similarity of different policies over test set patient states[]{data-label="tbl:overlap"}
*Evaluation Quality Assessment.* The WDR estimator of policy quality relies on having a large enough collection of patient histories in the evaluation set having non-zero weight $w_t^i$. For the MoE policy, $90\%$ of the importance sampling weights are non-zero and $86\%$ final weights in the sequences are non-zero. These high numbers of non-zero importance weights indicates that nearly all of our data was used in the evaluation of the policy. We plot the full distribution of weights in Figure \[fig:WDR\_weights\]. A significant number of weights lie in the range of $[10^{-4},10^{-3}]$ and only very few observations have weights significantly larger than that range (the samples with significantly smaller weights are unlikely to have a significant influence on the estimate). However, the few observations with weights on the order of $10^{-1}$ could potentially have large influence (see Figure \[fig:gradients\] for variances computed via boostrap).
*Running Time.* Besides the quality of policy, the ability to make recommendations quickly is also important in the ICU (note that it is less important for the initial training time to be fast). We measured the computational time for all components in our framework on a dual-core Intel Core i5 processor. The 2-layer NN for the reward function $f(o)$ took 56.8s to train by 128-sized mini-batch for 50 epochs on dataset with $39856 \times 45$ dimension. The recurrent autoencoder for the state space took 491s for 50 epochs with $128$ patients per mini-batch.
The kernel policy required 909s to identify policies for the $150720$ samples in the training set. Training DQN by sampling $30$ transitions for $200,000$ times required $5.15\times 10^4$s. Finally, MoE gating function took 62s to train by a 256-sized mini-batch, for 1 epoch using $1e-4$ learning rate, on dataset with $150720 \times 9$ dimension; however, since MoE cannot guarantee global maxima, we conducted $1000$ random restarts over the initial parameters, and trained each for $50$ epochs.
Most importantly, with regard to prediction at test time, for a patient with $10$ timesteps in the ICU, it took only 0.162s to encode all the observations, compute the kernel and DRL policies, and compute the gating function for the final MoE policy.
[0.5]{} {width="\linewidth"}
[0.5]{} {width="\linewidth"}
Discussion {#section:discussion .unnumbered}
==========
Overall, the DQN policy recommended a treatment strategy with more aggressive use of both vasopressors and fluids. In comparison to the physician policy, DQN recommended 70% more actions involving medium-to-high fluid volume and vasopressor dosage (actions 18,19,23, and 24). Most notably, frequency for the DQN action corresponding to maximum levels of both fluid and vasopressor (action 24) increased by 3.8 fold from the physician policy. These results suggest that despite the recent advances in deep reinforcement learning, further investigations are required, and careful clinical judgment should be exercised to guard against potentially high-risk actions introduced from pathologies in non-linear function approximation.
The proposed kernel policy displayed a different kind of bias. It recommended far fewer actions involving vasopressors in comparison to both the physician policy and DQN, perhaps because amongst a patient’s neighbors, the survivors were relatively healthier and thus treated less aggressively. By focusing on survivors the kernel policy also focuses on patients who did not just receive a good treatment or were potentially healthier now, but also patients that received good treatments and remained healthy in the future. If healthier patients are easier to treat in general, then we might expect a bias toward less aggressive treatment from the kernel policy as well.
More broadly, while it appears that our MoE policy significantly outperforms the clinician policy (as well as each individual expert), and we have ensured that the actions it suggests are at least sensible (that is, often taken by clinicians), there still exist a number of limitations. When encoding the patient clinical course in a recurrent representation, in spite of our high prediction accuracy, we cannot be certain with only these 50 measures that there are no hidden confounding factors; aside from pre-ICU fluid balance, we have no information from prior to their ICU stay. The choice of representation also influences the quality of our off-policy evaluation, as the WDR estimator assumes that the system in Markov in the state. More generally, the WDR estimator requires either the behavior policy estimate $\pi_b$ to be accurate or the control variate estimates $\hat{V}$ and $\hat{Q}$ to be accurate to be unbiased; something that we could not guarantee. That said, we do demonstrate that our results were at least insenstive to different choices of $\hat{V}$ and $\hat{Q}$.
Our work also focused on a very specific reward structure. To apply off-policy evaluation (WDR) in a statistically credible way, we considered the accumulation of low mortality risk as the objective, rather than mortality itself (as using the latter, the policies could not be evaluated reliably). To maximize the interpretability of the reward to clinicians, this risk was calculated only from the current observations and not the patient’s entire history. Creating reward functions that can both be checked by human experts and accurately convey clinical goals is a direction for future work.
Finally, with respect to sepsis management, there also exist many other interventions, such as antibiotics use and mechanical ventilation, that also affect patient outcomes. Future work remains to investigate policies that incorporated a broader scope of patient history as well as a larger variety of interventions.
Conclusion {#conclusion .unnumbered}
==========
We presented a MoE framework to learn improved fluid and vasopressor administration strategies for sepsis patients in ICUs using observational data. We demonstrated that the proposed mixture model approach can automatically adapt to patient states at each time step, and dynamically switch between a conservative kernel policy and a more aggressive deep-RL policy to achieve, under our reward measure, better expected outcomes than clinician, kernel policy, and deep-RL policy. While much further investigation is required to truly validate the efficacy of derived policies, the proposed MoE framework represents a novel approach to take advantage of the strengths of different treatment policies.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank the other students in Harvard CS282R - Reinforcement Learning for Healthcare, Fall 2017 for their insights, encouragement and feedback. Omer Gottesman was supported by the Harvard Data Science Initiative. L. Lehman was supported by NIH grant 2RO1GM104987.
|
---
abstract: |
We embark on a detailed analysis of the close relations between combinatorial and geometric aspects of the scalar parabolic PDE $$\label{eq:*}
u_t = u_{xx} + f(x,u,u_x) \, ,\tag{$*$}$$ on the unit interval $0 < x<1$ with Neumann boundary conditions. We assume $f$ to be dissipative at infinity with $N$ hyperbolic equilibria $v\in\mathcal{E}$. The *Thom-Smale complex* $\mathcal{C}$ of consists of the unstable manifolds of all equilibria $v$, as cells. Together these form a signed regular cell decomposition of the global attractor $\mathcal{A}$ of , also called the *Sturm global attractor*.\
Given that signed cell decomposition only, we derive the resulting boundary orders $h_\iota:\{1,...,N\}\rightarrow\mathcal{E}$ of the equilibrium values $v(x)$ at the Neumann boundaries $\iota=x=0,1,$ respectively. In previous work we have already established how the resulting Sturm permutation $$\sigma:=h_{0}^{-1} \circ h_1,$$ conversely, determines the global attractor $\mathcal{A}$ uniquely, up to topological conjugacy.
author:
- |
\
[ ]{}\
Bernold Fiedler\* and Carlos Rocha\*\*\
date: version of
---
\*\
Institut für Mathematik\
Freie Universität Berlin\
Arnimallee 3\
14195 Berlin, Germany\
\
\*\*\
Center for Mathematical Analysis, Geometry and Dynamical Systems\
Instituto Superior Técnico\
Universidade de Lisboa\
Avenida Rovisco Pais\
1049–001 Lisbon, Portugal\
Introduction {#sec1}
============
For our general introduction we first follow [@firo3d-1; @firo3d-2; @firo3d-3] and the references there. *Sturm global attractors* $\mathcal{A}_f$ are the global attractors of scalar parabolic equations $$u_t = u_{xx} + f(x,u,u_x)
\label{eq:1.1}$$ on the unit interval $0<x<1$. Just to be specific we consider Neumann boundary conditions $u_x=0$ at $x \in \{0,1\}$. Standard semigroup theory provides local solutions $u(t,x)$ for $t \geq 0$ and given initial data at time $t=0$, in suitable Sobolev spaces $u(t, \cdot) \in X \subseteq C^1 ([0,1], \mathbb{R})$. Under suitable dissipativeness assumptions on $f \in C^2$, any solution eventually enters a fixed large ball in $X$. For large times $t$, in fact, that large ball of initial conditions itself limits onto the maximal compact and invariant subset $\mathcal{A}=\mathcal{A}_f$ of $X$ which is called the global attractor. See [@he81; @pa83; @ta79] for a general PDE background, and [@bavi92; @chvi02; @edetal94; @ha88; @haetal02; @la91; @ra02; @seyo02; @te88] for global attractors in general.
For the convenience of the reader, we give a rather complete background on our current understanding of the global attractors of . It is not required, and would in fact be pedantic, to read all the references given. Rather, the present paper is elementary, although nontrivial, given the background facts mentioned.
Equilibria $u(t,x) = v(x)$ are time-independent solutions, of course, and hence satisfy the ODE $$0 = v_{xx} + f(x,v,v_x)
\label{eq:1.2}$$ for $0\leq x \leq 1$, again with Neumann boundary. Here and below we assume that all equilibria $v$ of , are *hyperbolic*, i.e. without eigenvalues (of) zero (real part) of their linearization. Let $\mathcal{E} = \mathcal{E}_f \subseteq \mathcal{A}_f$ denote the set of equilibria. Our generic hyperbolicity assumption and dissipativeness of $f$ imply that $N$:= $|\mathcal{E}_f|$ is odd.
It is known that possesses a Lyapunov function, alias a variational or gradient-like structure, under separated boundary conditions; see [@ze68; @ma78; @mana97; @hu11; @fietal14; @lafi18]. In particular, the global attractor consists of equilibria and of solutions $u(t, \cdot )$, $t \in \mathbb{R}$, with forward and backward limits, i.e. $$\underset{t \rightarrow -\infty}{\mathrm{lim}} u(t, \cdot ) = v\,,
\qquad
\underset{t \rightarrow +\infty}{\mathrm{lim}} u(t, \cdot ) = w\,.
\label{eq:1.3}$$ In other words, the $\alpha$- and $\omega$-limit sets of $u(t,\cdot )$ are two distinct equilibria $v$ and $w$. We call $u(t, \cdot )$ a *heteroclinic* or *connecting* orbit, or *instanton*, and write $v \leadsto w$ for such heteroclinically connected equilibria. See fig. \[fig:1.0\](a) for a simple 3-ball example with $N=9$ equilibria.
![ *Example of a Sturm 3-ball global attractor $\mathcal{A}_f = clos W^u(\mathcal{O})$. Equilibria are labeled as $\mathcal{E}=\{1,\ldots,9\}$. The previous papers [@firo3d-1; @firo3d-2] established the equivalence of the viewpoints (a)–(d). (a) The Sturm global attractor $\mathcal{A}$, 3d view, including the location of the poles $\textbf{N}$, $\textbf{S}$, the (green) meridians $\textbf{WE}$, $\textbf{EW}$, the central equilibrium $\mathcal{O}=4$ and the hemispheres $\textbf{W}$ (green), $\textbf{E}$. (b) The Thom-Smale complex $\mathcal{C}_f$ of the boundary sphere $\Sigma^2 = \partial c_\mathcal{O}$, including the Hamiltonian SZS-pair of paths $(h_0,h_1)$, (red/blue). The right and left boundaries denote the same $\textbf{EW}$ meridian and have to be identified. (c) The Sturm meander $\mathcal{M}$ of the global attractor $\mathcal{A}$. The meander $\mathcal{M}$ is the curve $a \mapsto (v,v_x)$, at $x=1$, which results from Neumann initial conditions $(v,v_x)=(a,0)$ at $x=0$ by shooting via the equilibrium ODE . Intersections of the meander with the horizontal $v$-axis indicate equilibria. (d) Spatial profiles $x\mapsto v(x)$ of the equilibria $v \in \mathcal{E}$. Note the different orderings of $v(x)$, by $h_0 = \mathrm{id}$ at the left boundary $x=0$, and by the Sturm permutation $\sigma = h_1 = (1\ 8\ 3\ 4\ 7\ 6\ 5\ 2\ 9)$ at the right boundary $x=1$. The same orderings define the meander $\mathcal{M}$ in (c) and the Hamiltonian SZS-pair $(h_0,h_1)$ in the Thom-Smale complex (b).* []{data-label="fig:1.0"}](figs/{fig1_0}.pdf){width="90.00000%"}
We attach the name of *Sturm* to the PDE , and to its global attractor $\mathcal{A}_f$. This refers to a crucial nodal property of its solutions, which we express by the *zero number* $z$. Let $0 \leq z (\varphi) \leq \infty$ count the number of (strict) sign changes of $\varphi : [0,1] \rightarrow \mathbb{R}, \, \varphi \not\equiv 0$. Then $$t \quad \longmapsto \quad z(u^1(t, \cdot ) - u^2(t, \cdot ))\,
\label{eq:1.4}$$ is finite and nonincreasing with time $t$, for $t>0$ and any two distinct solutions $u^1$, $u^2$ of . Moreover $z$ drops strictly with increasing $t$, at any multiple zero of $x \mapsto u^1(t_0 ,x) - u^2(t_0 ,x)$; see [@an88]. See Sturm [@st1836] for a linear autonomous version. For a first introduction see also [@ma82; @brfi88; @fuol88; @mp88; @brfi89; @ro91; @fisc03; @ga04] and the many references there. As a convenient notational variant of the zero number $z$, we also write $$z(\varphi) = j_{\pm}
\label{eq:1.5}$$ to indicate $j$ strict sign changes of $\varphi$, by $j$, and $\pm \varphi (0) >0$, by the index $\pm$. For example $z(\pm \varphi_j) = j_{\pm}$, for the $j$-th Sturm-Liouville eigenfunction $\varphi_j$.
The dynamic consequences of the Sturm structure are enormous. In a series of papers, we have given a combinatorial description of Sturm global attractors $\mathcal{A}_f$; see [@firo96; @firo99; @firo00]. Define the two *boundary orders* $h_0, h_1$: $\lbrace 1, \ldots, N \rbrace \rightarrow \mathcal{E}$ of the equilibria such that $$h_\iota (1) < h_\iota (2) < \ldots < h_\iota (N) \qquad \text{at}
\qquad x=\iota \in \{0,1\}\,.
\label{eq:1.6}$$ See fig. \[fig:1.0\](d) for an illustration with $N=9$ equilibrium profiles, $\mathcal{E} = \{1,\ldots,9\}, \ h_0 = \mathrm{id},\ h_1 = (1\ 8\ 3\ 4\ 7\ 6\ 5\ 2\ 9)$.
The combinatorial description is based on the *Sturm permutation* $\sigma \in S_N$ which was introduced by Fusco and Rocha in [@furo91] and is defined as $$\sigma:= h_0^{-1} \circ h_1\,.
\label{eq:1.7a}$$ Already in [@furo91], the following explicit recursions have been derived for the Morse indices $i_k:=i(h_0(k))$: $$\begin{aligned}
i_1 &:= i_N := 0\,,\\
i_{k+1} &:= i_k+(-1)^{k+1}\,
\text{sign} (\sigma^{-1}(k+1)-\sigma^{-1}(k))\,.\\
\end{aligned}
\label{eq:1.7b}$$ Similarly, the (unsigned) zero numbers $z_{jk} := z(v_j-v_k)$ are given recursively, for $j\geq k$, as $$\begin{aligned}
z_{kk} &:= i_k\,,\\
z_{j+1,k} &:= z_{jk} + \tfrac{1}{2}(-1)^{j+1}
\cdot\\
&\phantom{:= z_{jk} + \tfrac{1}{2}}\cdot
\left[ \text{sign}\left(\sigma^{-1}(j+1)-
\sigma^{-1}(k)\right)-
\text{sign}\left(\sigma^{-1}(j)-\sigma^{-1}(k)\right)\right]\,.
\end{aligned}
\label{eq:1.7c}$$
Using a shooting approach to the ODE boundary value problem , the Sturm permutations $\sigma \in S_N$ have been characterized, purely combinatorially, as *dissipative Morse meanders* in [@firo99]. Here *dissipative* requires fixed $\sigma(1)=1$ and $\sigma(N)=N$. *Morse* requires nonnegative Morse indices $i_k\geq0$ in , for all k. The *meander* property, finally, requires the formal path $\mathcal{M}$ of alternating upper and lower half-circles defined by the permutation $\sigma$, as in fig. \[fig:1.0\](c), to be Jordan, i.e. non-selfintersecting. See [@ka17] for ample additional material on many aspects of meanders.
In [@firo96] we have shown how to determine which equilibria $v, w$ possess a heteroclinic orbit connection , explicitly and purely combinatorially from dissipative Morse meanders $\sigma$. This was based, in particular, on the results and of [@furo91].
More geometrically, global Sturm attractors $\mathcal{A}_f$ and $\mathcal{A}_g$ of nonlinearities $f, g$ with the same Sturm permutation $\sigma_f = \sigma_g$ are $C^0$ orbit-equivalent [@firo00]. For $C^1$-small perturbations, from $f$ to $g$, this global rigidity result is based on the $C^0$ structural stability of Morse-Smale systems; see e.g. [@pasm70] and [@pame82]. It is the Sturm property which implies the Morse-Smale property, for hyperbolic equilibria. Stable and unstable manifolds $W^u(v_-)$, $W^s(v_+)$, which intersect precisely along heteroclinic orbits $v_- \leadsto v_+$, are in fact automatically transverse: $W^u(v_-) {\mathrel{\text{{ \vbox{
\baselineskip\z@skip
\lineskip-.52ex
\lineskiplimit\maxdimen
\m@th
\ialign{##\crcr\hidewidth\smash{$-$}\hidewidth\crcr$\pitchfork$\crcr}
}}}}}W^s(v_+)$. See [@he85; @an86]. In the Morse-Smale setting, Henry already observed, that a heteroclinic orbit $v_- \leadsto v_+$ is equivalent to $v_+$ belonging to the boundary $\partial W^u(v_-)$ of the unstable manifold $W^u(v_-)$; see [@he85].
More recently, we have pursued a more explicitly geometric approach. Let us consider *finite regular* CW-*complexes* $$\mathcal{C} = \bigcup\limits_{v\in \mathcal{E}} c_v\,,
\label{eq:1.8a}$$ i.e. finite disjoint unions of *cell interiors* $c_v$ with additional gluing properties. We think of the labels $v\in \mathcal{E}$ as *barycenter* elements of $c_v$. For CW-complexes we require the closures $\overline{c}_v$ in $\mathcal{C}$ to be the continuous images of closed unit balls $\overline{B}_v$ under *characteristic maps*. We call $\mathrm{dim}\,\overline{B}_v$ the dimension of the (open) cell $c_v$. For positive dimensions of $\overline{B}_v$ we require $c_v$ to be the homeomorphic images of the interiors $B_v$. For dimension zero we write $B_v := \overline{B}_v$ so that any 0-cell $c_v= B_v$ is just a point. The *m-skeleton* $\mathcal{C}^m$ of $\mathcal{C}$ consists of all cells of dimension at most $m$. We require $\partial c_v := \overline{c}_v \setminus c_v \subseteq \mathcal{C}^{m-1}$ for any $m$-cell $c_v$. Thus, the boundary $(m-1)$-sphere $S_v := \partial B_v = \overline{B}_v \setminus B_v$ of any $m$-ball $B_v$, $m>0$, maps into the $(m-1)$-skeleton, $$\partial B_v \quad \longrightarrow \quad \partial c_v \subseteq \mathcal{C}^{m-1}\,,
\label{eq:1.9a}$$ for the $m$-cell $c_v$, by restriction of the characteristic map. The continuous map is called the *attaching* (or *gluing*) *map*. For *regular* CW-complexes, in contrast, the characteristic maps $ \overline{B}_v \rightarrow \overline{c}_v $ are required to be homeomorphisms, up to and including the *attaching* (or *gluing*) *homeomorphism*. We require the $(m-1)$-sphere $\partial{c_v}$ to be a sub-complex of $\mathcal{C}^{m-1}$. See [@frpi90] for some further background on this terminology.
The disjoint dynamic decomposition $$\mathcal{A}_f = \bigcup\limits_{v \in \mathcal{E}_f} W^u(v) =: \mathcal{C}_f
\label{eq:1.10a}$$ of the global attractor $\mathcal{A}_f$ into unstable manifolds $W^u$ of equilibria $v$ is called the *Thom-Smale complex* or *dynamic complex*; see for example [@fr79; @bo88; @bizh92]. In our Sturm setting with hyperbolic equilibria $v \in \mathcal{E}_f$, the Thom-Smale complex is a finite regular CW-complex. The open cells $c_v$ are the unstable manifolds $W^ u (v)$ of the equilibria $v \in \mathcal{E}_f$. The proof follows from the Schoenflies result of [@firo13]; see [@firo14] for a summary.
We can therefore define the *Sturm complex* $\mathcal{C}_f$ to be the regular Thom-Smale complex $\mathcal{C}$ of the Sturm global attractor $\mathcal{A}=\mathcal{A}_f$, provided all equilibria $v \in \mathcal{E}_f$ are hyperbolic. Again we call the equilibrium $v \in \mathcal{E}_f$ the *barycenter* of the cell $c_v=W^u(v)$. The dimension $i(v)$ of $c_v$ is called the *Morse index* of $v$. A planar Sturm complex $\mathcal{C}_f$, for example, is the regular Thom-Smale complex of a planar $\mathcal{A}_f$, i.e. of a Sturm global attractor for which all equilibria $v \in \mathcal{E}_f$ have Morse indices $i(v) \leq 2$. See fig. \[fig:1.0\](b) for the Sturm complex $\mathcal{C}_f$ of the Sturm global attractor $\mathcal{A}_f$ sketched in fig. \[fig:1.0\](a). With this identification we may henceforth omit the explicit subscripts $f$, when the context is clear.
We can now formulate the *main task* of this paper:\
*Let the Thom-Smale complex $\mathcal{E}$ of a Sturm global attractor $\mathcal{A}$ be given, as an abstract complex. Derive the orders $h_\iota:\{1, \ldots, N\}\rightarrow \mathcal{E}$ of the equilibria $v\in \mathcal{E}$, evaluated at the boundaries $x=\iota=0,1$.*
So far, we have solved this task for Sturm global attractors $\mathcal{A}$ of dimension $$\dim \mathcal{A} = \max\{i(v)\mid v \in \mathcal{E}\}
\label{eq:1.7}$$ equal to two; see the planar trilogy [@firo08; @firo3d-2; @firo3d-3]. For Sturm 3-balls $\mathcal{A}=\bar{c}_ \mathcal{O}$, which are the closure of the unstable manifold cell $c_\mathcal{O}$ of a single equilibrium $\mathcal{O}$ of maximal Morse index $i (\mathcal{O})=3$, our solution has been presented in the 3-ball trilogy [@firo3d-1; @firo3d-2; @firo3d-3]. The present paper solves the general case.
Our results are crucially based on the disjoint *signed hemisphere decomposition* $$\partial W^u(v) =
\bigcup\limits_{0\leq j< i(v)}^\centerdot
\Sigma_\pm^j(v)
\label{eq:1.8}$$ of the topological boundary $\partial W^u= \partial c_v = \overline{c}_v \smallsetminus c_v$ of the unstable manifold $W^u(v)=c_v$, for any equilibrium $v$. As in [@firo3d-2 (1.19)] we define the hemispheres by their Thom-Smale cell decompositions $$\Sigma_ \pm^j(v) :=
\bigcup\limits_{w\in \mathcal{E}_\pm^j(v)}^\centerdot
W^u(w)
\label{eq:1.9}$$ with the equilibrium sets $$\mathcal{E}_\pm^j(v) := \lbrace w\in \mathcal{E}_f\,|\,z(w-v)=j_\pm \ \text{and} \
v\leadsto w\rbrace\,,
\label{eq:1.10}$$ for $0\leq j<i(v)$. Equivalently, we may define the hemisphere decompositions, inductively, via the topological boundary $j$-spheres $$\Sigma^j(v):=
\bigcup\limits_{0\leq k<j}^\centerdot \Sigma_\pm^k(v)
\label{eq:1.11}$$ of the fast unstable manifolds $W^{j+1}(v)$. Here $W^{j+1}(v)$ is tangent to the eigenvectors $\varphi_0, \ldots ,\varphi_j$ of the first $j+1$ unstable eigenvalues $\lambda_0 > \ldots > \lambda_j >0$ of the linearization at the equilibrium $v$. See [@firo3d-1] for details.
For 3-ball Sturm attractors $\mathcal{A}=\bar{c}_\mathcal{O}$, for example, the signed hemisphere decomposition reads $$\Sigma^2 = \partial W^u (\mathcal{O}):=\text{clos }W^ u(\mathcal{O})\setminus W^u(\mathcal{O})=
\bigcup\limits_{j=0}^2 \Sigma_\pm^j\ (\mathcal{O})
\label{eq:1.12}$$ at $v=\mathcal{O}$ with Morse index $i(\mathcal{O})=3$. Here the North pole $\mathbf{N}=\Sigma^0_- (\mathcal{O})$ and the South pole $\mathbf{S}=\Sigma^0_+ (\mathcal{O})$ denote the boundary of the one-dimensional fastest unstable manifold $W^1 = W^1(\mathcal{O})$, tangent to the positive eigenfunction $\varphi_0$ of the largest eigenvalue $\lambda_0$ at $\mathcal{O}$. Indeed, solutions $t \mapsto u(t,x)$ in $W^1$ are monotone in $t$, for any fixed $x$.Accordingly $$\begin{aligned}
z(\mathbf{N}- \mathcal{O}) = 0_-\,,
\quad z(\mathbf{S}- \mathcal{O}) = 0_+\,,
\end{aligned}
\label{eq:1.13}$$ i.e. $\mathbf{N} < \mathcal{O} < \mathbf{S}$ for all $0\leq x\leq 1$. The *poles* $\mathbf{N},\mathbf{S}$ split the circle boundary $\Sigma^1 = \partial W^2 (\mathcal{O})$ of the 2-dimensional fast unstable manifold into the two *meridian* half-circles $\mathbf{EW}= \Sigma^1_- (\mathcal{O})$ and $\mathbf{WE}=\Sigma^1_+(\mathcal{O})$. The circle $\Sigma^1$, in turn, splits the boundary sphere $\Sigma^2 = \partial W^u(\mathcal{O})$ of the whole unstable manifold $W^u$ of $\mathcal{O}$ into the Western hemisphere $\mathbf{W}=\Sigma^2_-(\mathcal{O})$ and the Eastern hemisphere $\mathbf{E}=\Sigma^2_+(\mathcal{O})$. Omitting the explicit references to the central equilibrium $\mathcal{O}$, the hemisphere translation table becomes: $$\begin{aligned}
(\Sigma_-^0, \Sigma_+^0) \quad &\mapsto \quad (\mathbf{N}, \mathbf{S})\\
(\Sigma_-^1, \Sigma_+^1) \quad &\mapsto \quad
(\mathbf{EW}, \mathbf{WE})\\
(\Sigma_-^2, \Sigma_+^2) \quad &\mapsto \quad (\mathbf{W}, \mathbf{E}).
\end{aligned}
\label{eq:1.14}$$
To address our main task, let us fix any unstable equilibrium $\mathcal{O}\in \mathcal{A}$ of Morse index $n:=i (\mathcal{O})$. Without loss of generality we may assume $\mathcal{O}\equiv 0$ is the zero solution of , i.e. $f(x, 0, 0)\equiv0$. Else we substract $\mathcal{O}$ from the solutions $u(t, x)$. It is our task to identify the *predecessors* and *successors* $$\begin{aligned}
w^\iota_\pm:= h_\iota(h^{-1}_\iota (\mathcal{O})\pm1)
\end{aligned}
\label{eq:1.15}$$ of $\mathcal{O}$, along the boundary orders $h_\iota$ at $x=\iota=0,1$.
Already implies adjacency of the Morse indices $i(w_\pm^\iota)$ of the $\iota$-neighbors $w^\iota_\pm$ with $i(\mathcal{O})$: $$\begin{aligned}
&i(h_\iota(1)) = i(h_\iota(N))=0;\\
&i(w^\iota_\pm)= i(\mathcal{O})\pm(-1)^{i(\mathcal{O})}\mathrm{sign}(h^{-1}_{1-\iota}(w^\iota_\pm)-h^{-1}_{1-\iota}(\mathcal{O})).\\
\end{aligned}
\label{eq:1.16}$$ To determine the $\iota$-neighbors $w^\iota_\pm$ of $\mathcal{O}$ geometrically, we develop the notion of descendants next. See [@firo3d-2] for the special case $n=3$.
\[def:1.1\] For fixed $n:=i(\mathcal{O})>0$, let $\mathbf{s}= s_{n-1} ... s_0$ denote any sequence of $n$ symbols $s_j\in\{\pm\}$. Let $$\begin{aligned}
v^j(\mathbf{s})\in \mathcal{E}^j_{s_j}(\mathcal{O})\subseteq \Sigma^j_{s_j}(\mathcal{O}), \qquad j=1, ..., n-1
\end{aligned}
\label{eq:1.18}$$ be defined, recursively for increasing $j$, as the unique equilibrium in the signed hemisphere $\Sigma^j_{s_j}(\mathcal{O})$ such that $$\begin{aligned}
v^j(\mathbf{s}) &\in \Sigma_{s_j}^j \,,\\
v^{j-1}(\mathbf{s}) &\in \partial W^u(v^j(\mathbf{s}))=\partial c_{v^j(\mathbf{s})} \, .
\end{aligned}
\label{eq:1.19a}$$ For $j=0$ we start the recursion with the unique polar equilibria $$\{v^0(\mathbf{s})\}:= \Sigma^0_{s_0} (\mathcal{O})
\label{eq:1.17}$$ at the two endpoints of the one-dimensional fastest unstable manifold $W^0(\mathcal{O})$. We call the sequence $v^j(\mathbf{s}),\ j=n-1, ..., 0,$ the $\mathbf{s}$-*descendants* of $\mathcal{O}$. For constant sequences $s_j=+$, we call $v^j(++ ...)$ the $+$descendants of $\mathcal{O}$.
In section 2 we show that the descendants $v^j(\mathbf{s})$ are in fact defined uniquely. We also determine the Morse indices $i(v^j(\mathbf{s}))=j$ and show that the descendants define a sequence of heteroclinic orbits between equilibria of descending adjacent Morse indices: $$\begin{aligned}
\mathcal{O}\leadsto v^{n-1}(\mathbf{s})\leadsto ...\leadsto v^0(\mathbf{s});
\end{aligned}
\label{eq:1.19b}$$ see –.
Clearly, the notion – of descendants is purely geometric: it is based on the signed hemisphere decomposition $\Sigma^j_\pm(\mathcal{O})$, only, and does not involve any more explicit data on the boundary orderings $h_\iota$. In fact, we will only use alternating and constant symbol sequences $s_j$. We therefore abbreviate these sequences as follows $$\begin{aligned}
&\mathbf{s}=+-...:\qquad s_j:=(-1)^{n-1-j} \, ;\\
&\mathbf{s}=-+...:\qquad s_j:=(-1)^{n-j} \, ;\\
&\mathbf{s}=++...:\qquad s_j:=+ \, ;\\
&\mathbf{s}=--...:\qquad s_j:=- \, .\\
\end{aligned}
\label{eq:1.20}$$
With this notation we can now formulate our main result.
\[thm:1.2\] Consider any unstable equilibrium $\mathcal{O}$ with unstable dimension $n= i(\mathcal{O})>0$. Assume that any of the boundary successors $w^\iota_+$ or predecessors $w^\iota_-$ of $\mathcal{O}$ at $x=\iota=0,1$, as defined in , is more stable than $\mathcal{O}$, i.e. $$\begin{aligned}
i(w^\iota_\pm)=n-1.
\end{aligned}
\label{eq:1.21}$$
Then, that $\iota$-neighbor $w^\iota_\pm$ of $\mathcal{O}$ is given by the leading descendant $v^{n-1}(\mathbf{s})$ of $\mathcal{O}$, according to the following list: $$\begin{aligned}
w^0_-&=& v^{n-1}(-+...); \label{eq:1.22}\\
w^0_+&=& v^{n-1}(+-...); \label{eq:1.23}\\
w^1_-&=&
\begin{cases}
v^{n-1}(++...), \qquad \text{for even n;}\\
v^{n-1}(--...), \qquad \text{for odd n;}
\end{cases} \label{eq:1.24}\\
w^1_+&=&
\begin{cases}
v^{n-1}(--...), \qquad\text{for even n;}\\
v^{n-1}(++...), \qquad\text{for odd n.}\\
\end{cases} \label{eq:1.25}
\end{aligned}$$
The above result determines the boundary orders $h_\iota$ of the equilibria at the boundaries $x=\iota=0,1$ uniquely. Indeed any two equlibria $v_1$ and $v_2$ which are adjacent in the order $h_\iota$, say at $\iota=x=0$, possess adjacent Morse indices $i(v_2)=i(v_1)\pm1$ by . The more unstable equilibrium therefore qualifies as $\mathcal{O}$, in theorem 1.2, and the other equilibrium qualifies as the predecessor $w^0_-$, or as the successor $w^0_+$, of $\mathcal{O}$. Let us therefore start at the top level barycenters of maximal cell dimension $\dim c_\mathcal{O}=i(\mathcal{O})=\dim\mathcal{A}$, but without any a priori knowledge of $h_\iota$ or $\sigma$ in , . By , all $h^\iota_-$ neighbors $w^\iota_\pm$ of such $\mathcal{O}$ can then be identified, purely geometrically, as top descendants $v^{n-1}(\mathbf{s})$ of $\mathcal{O}$ with Morse index $i(w_{\pm}^\iota)=\dim\mathcal{A}-1$. Next, consider all barycenters of cell dimension $\dim \mathcal{A}-1$. Unless their $h_\iota$-neighbors possess higher Morse index, and those adjacencies have already been taken care of, we may apply theorem 1.2 again to determine their remaining $h_\iota$-neighbors, this time of Morse index $\dim \mathcal{A}-2$. Iterating this procedure we eventually determine all $h_\iota$-adjacencies and our main task is complete.
In sections \[sec2\], \[sec3\] we prove Theorem \[thm:1.2\] by the following strategy. First we reduce the four cases – to the single case $$\begin{aligned}
\mathbf{s}=++...
\end{aligned}
\label{eq:1.26}$$ by four *trivial equivalences*. Indeed, the class of Sturm attractors $\mathcal{A}$ remains invariant under the transformations $$\begin{aligned}
x\mapsto1-x, \quad u\mapsto-u,
\end{aligned}
\label{eq:1.27}$$ separately. Since the two involutions commute, they generate the Klein 4-group $\mathbb{Z}_2\times\mathbb{Z}_2$ of trivial equivalences. Since this group acts transitively on the four constant and alternating symbol sequences , as considered in theorem \[thm:1.2\], it is sufficient to consider the case $\mathbf{s}=++...$ of . The remaining cases of – then follow by application of the trivial equivalences. For even $n$, for example, maps to under $x\mapsto 1-x$, to under $u\mapsto -u$ and to under the combination of both. We henceforth restrict to the $+$case $\mathbf{s}=(++...)$ of . We also restrict to the case of odd $n$, the even case being analogous.
In section 2 we study the descendants of $\mathcal{O}$ for $\mathbf{s}=++...$. We abbreviate $$\begin{aligned}
v^j:= v^j(++...),
\end{aligned}
\label{eq:1.28}$$ for $0\leq j < n= i(\mathcal{O})$. In sections 3 and 4 we study the additional elements $$\begin{aligned}
&\underline{v}^k:&\textrm{the equilibrium } v \in \mathcal{E}^k_+(\mathcal{O})\subseteq \Sigma^k_+(\mathcal{O})
\textrm { which is closest to } \mathcal{O} \textrm{ at }x =1, \label{eq:1.29}\\
&\overline{v}^k:&\textrm{the equilibrium } v\in \mathcal{E}^k_+(\mathcal{O})\subseteq \Sigma^k_+(\mathcal{O})
\textrm { which is maximal at } x =0, \label{eq:1.30}
\end{aligned}$$
In section 3 we show $\underline{v}^k = v^k$, for all $0 \leqslant k<n= i(\mathcal{O})$; see theorem 3.1. As a corollary, for $k=n-1$, this proves theorem (\[thm:1.2\]) and completes our task.
In section 4 we show, in addition to $\underline{v}^j=v^j$, that $\underline{v}^j=\overline{v}^{j}$, for all $0\leqslant j<n= i(\mathcal{O})$; see theorem \[thm:4.3\]. Strictly speaking, this result is not required by the identification task to derive the $h_\iota$-neigbors $w^\iota_\pm$ from $\mathcal{O}$. However, it simplifies the task to identify the equilibria $\mathcal{E}^j_\pm(\mathcal{O})$ in the hemispheres $\Sigma^j_\pm(\mathcal{O})$ from the Sturm meander $\mathcal{M}$, directly. See for example the Thom-Smale complex of fig. \[fig:1.0\](b), where $ \mathcal{O}=4$, $w^\iota_- = 3$, $w^0_+=5$, $ w^1_+=7$.
Indeed, the equilibria $v\in \mathcal{E}^j_s(\mathcal{O})$ in the hemispheres $\Sigma^j_ s(\mathcal{O})$, for $s = \pm$, are given by the table
$$\label{eq:1.36}$$
$s$ $-$ $+$
------- --------------------- --------------------------
$j=0$ $1= \mathbf{N}$ $9=\mathbf{S}$
$j=1$ $2 \in \mathbf{EW}$ $8 \in \mathbf{WE}$
$j=2$ $3 \in \mathbf{W}$ $5, 6, 7 \in \mathbf{E}$
Therefore the descendants $v^j(\mathbf{s})$ of $\mathcal{O}= 4$ are given by the table
$$\label{eq:1.37}$$
$\mathbf{s}= s_2s_1s_0$ $-+-$ $+-+$ $---$ $+++$
------------------------- --------------------------- --------------------------- ------------------------- --------------------------
$j=0$ $1= \mathbf{N}$ $9= \mathbf{S}$ $1= \mathbf{N}$ $9= \mathbf{S}$
$j=1$ $8 \in \mathbf{WE}$ $2 \in \mathbf{EW}$ $2 \in \mathbf{EW}$ $8 \in \mathbf{WE}$
$j=2$ $3 = w^0_-\in \mathbf{W}$ $5= w^0_+ \in \mathbf{E}$ $3=w^1_-\in \mathbf{W}$ $7= w^1_+\in \mathbf{E}$
The $+$descendants $v^j=v^j(+ + +)$ of $\mathcal{O}=4$ for example, are constructed as $v^0=9=\mathbf{S}$ because $\Sigma^0_+=\mathcal{E}^0_+=\{\mathbf{S}\}$, and $v^1=8$ because $\mathbf{WE}=\Sigma^1_+\supseteq \mathcal{E}^1_+=\{8\}$. Finally, $v^2\in \mathcal{E}^2_+= \{5,6,7\}\subseteq \Sigma^2_+= \mathbf{E}$ must satisfy $8=v^1\in c_{v^1}\subseteq \partial c_{v^2}$, by recursion , and therefore $v^2=7$. Since $\underline{v}^2= v^2$, by theorem 3.1, we conclude that the successor $w^1_+$ of $\mathcal{O}$, with odd Morse index $n=i(\mathcal{O})=3$, is given by $w^1_+=\underline{v}^2= v^2= v^2(+++)=7$. This agrees with the equilibrium profiles in fig \[fig:1.0\](d). Note that $\underline{v}^2=7\in \mathbf{E}= \Sigma^2_+$ is in fact the $\mathcal{O}$-closest equilibrium in $\mathcal{E}^2_+\subseteq \Sigma^2_+$, at the right boundary $x=1$. At the same time, $7=\underline{v}^2=\overline{v}^2$ is also the maximal equilibrium in $\mathcal{E}^2_+=\{5,6,7\}$ above $\mathcal{O}$, at the left boundary $x=0$. The red meander $\mathcal{M}$, in fig. \[fig:1.0\](c), therefore traverses all equilibria $5,6,7$ in the hemisphere $\mathbf{E}= \Sigma^2_+$, after $\mathcal{O}$, with 7 last, before it leaves that hemisphere forever. See also fig. \[fig:1.0\](b), where the red meander path $h_0$ of the ordering at $x=0$ leaves $\mathbf{E}= \Sigma^2_+$ at 7, where the blue path $h_1$ of the ordering at $x=1$ enters $\mathbf{E}= \Sigma^2_+$. Similarly, the blue path $h_1$ leaves $\mathbf{E}$ at 5, where $h_0$ enters. This illustrates theorem \[thm:4.3\]. For many more examples see the discussion in section 5, most of which is instructive even before reading the other sections.
The companion paper [@rofi18] gives a direct proof of theorem \[thm:4.3\], only based on a detailed analysis of the Surm meander. The property $v^{n-1}=\underline{v}^{n-1}$ of theorem \[thm:3.1\], which holds independently of theorem \[thm:4.3\], then allows us to identify, conversely, the geometric location of predecessors, successors, and signed hemispheres in the associated Thom-Smale complex. These results combined, can therefore be viewed as first steps towards the still elusive goal of a complete geometric characterization of the Thom-Smale complexes for all Sturm global attractors.
**Acknowledgments.** Extended mutually delightful hospitality by the authors is gratefully acknowledged. In addition, Clodoaldo Grotta-Ragazzo, Sergio Oliva, and Waldyr Oliva provided an inspiring and cheerful 24/7 environment at IME-USP: viva! Anna Karnauhova has contributed the illustrations with her inimitable artistic touch. Original typesetting was patiently accomplished by Patricia Hăbăşescu. This work was partially supported by DFG/Germany through SFB 910 project A4, and by FCT/Portugal through project UID/MAT/04459/2013.
Descendants {#sec2}
===========
In this section we fix any unstable hyperbolic equilibrium $\mathcal{O}$ of positive Morse index $n:=i(\mathcal{O})>0$, in a Sturm global attractor. Let $$\begin{aligned}
\Sigma^{n-1}= \partial W^u (\mathcal{O})= \bigcup\limits_{0\leq j< n}^\centerdot \Sigma^j_\pm
\end{aligned}
\label{eq:2.1}$$ be the disjoint signed decomposition of the $(n-1)$-sphere boundary of the $n$-dimensional unstable manifold $W^u(\mathcal{O})$, i.e. we abbreviate $\Sigma^j_\pm := \Sigma^j_\pm(\mathcal{O})$. Let $\mathcal{E}^j_\pm := \mathcal{E}^j_\pm (\mathcal{O})$ abbreviate the equilibria in hemisphere $\Sigma^j_\pm \subseteq \partial W^{j+1}(\mathcal{O})$. From we recall $$\begin{aligned}
\mathcal{E}^j_\pm = \{v\in \mathcal{E}\mid z(v- \mathcal{O})= j_\pm \textrm{ and } \mathcal{O}\leadsto v \}.
\end{aligned}
\label{eq:2.2}$$ Concerning the descendants $v^j=v^j(\mathbf{s})$ of $\mathcal{O}$, according to definition \[def:1.1\], we also fix any sequence $\mathbf{s}=s_{n-1}...s_0$ of $n$ signs $s_j = \pm$, for $0\leqslant j < n$. We first explain why the descendants $v^j$ are well-defined. After a pigeon-hole proposition \[eq:2.1\], we collect some elementary properties of descendants in lemma \[lem:2.2\].
Although we continue working in the general setting and notation of section \[sec1\], we emphasize that we do not restrict our general analysis of descendants to the case of theorem \[thm:1.2\]. Only there, the descendant $v^{n-1}$ coincides with an immediate successor or predeccessor $w_\pm^\iota$ of $\mathcal{O}$ on a boundary $x=\iota=0,1$.
Let us examine the recursive definition \[def:1.1\] first. For $s_0= \pm$, the equilibrium $\{v^0\} :=\Sigma^0_{s_0}(\mathcal{O})$ is defined uniquely by . Now consider $1 \leqslant j <n$ and assume $v^0, ..., v^{j-1}$ are well-defined already. By the Schoenflies result [@firo13] on the $j$-sphere boundary $\Sigma^j= \partial W^{j+1}= \text{ clos } W^{j+1} \setminus W^{j}$ of the $(j+1)$-dimensional fast unstable manifold $W^{j+1}=W^{j+1}(\mathcal{O})$ of $\mathcal{O}$, we have the disjoint decomposition $\text {clos } \Sigma^j_{s_j} = \Sigma^j_{s_j} \dot{\cup} \Sigma^{j-1} $, and hence $$\begin{aligned}
v^{j-1}\in \Sigma^{j-1}_{s_{j-1}} \subseteq \Sigma^{j-1}= \partial \Sigma^j_{s_j} \,.
\end{aligned}
\label{eq:2.3}$$
We claim that there exists a unique cell $c_{v^j}=W^u(v^j)$ in $$\begin{aligned}
\Sigma^j_{s_j} = \bigcup \limits_{v \in \mathcal{E}^j_{s_j}} W^u(v),
\end{aligned}
\label{eq:2.4}$$ such that holds, i.e. such that $$\begin{aligned}
v^{j-1}\in \partial c_{v^j}\, .
\end{aligned}
\label{eq:2.5}$$ This follows again from [@firo13], which asserts that the eigenprojection $P^j$ projects the closed $j$-dimensional hemisphere $\text{clos}\ \Sigma^j_{s_j}$ into the $j$-dimensional tangent space $T_\mathcal{O}W^j=\big \langle \varphi_0,...,\varphi_{j-1} \big \rangle$ at $\mathcal{O}$, homeomorphically onto a topological $j$-dimensional ball with Schoenflies $(j-1)$-sphere boundary. This homeomorphic projection preserves the regular Thom-Smale cell decomposition of clos $\Sigma^j_{s_j}$. In particular, any $(j-1)$-cell in $\Sigma^j_{s_j}$ possesses precisely two $j$-cell neighbors in $\Sigma^j_{s_j}$, separating them as a shared boundary. Any $(j-1)$-cell $c_{v^{j-1}}\subseteq \Sigma ^ {j-1}= \partial \Sigma^j_{s_j}$, however, possesses a *unique* $j$-cell neighbor $v^j\in \Sigma^j_{s_j}$ such that $$\begin{aligned}
c_{v^{j-1}}\subseteq \partial c_{v^j}.
\end{aligned}
\label{eq:2.6}$$ This proves that defines $v^j$ uniquely, and explains why all descendants $v^j$ are well-defined, by definition \[def:1.1\].
Since $c_{v^j}= W^u(v^j)$ is a $j$-cell, in our construction of descendants, we immediately obtain the Morse indices $$\begin{aligned}
i(v^j)=j,
\end{aligned}
\label{eq:2.7}$$ for all $0\leqslant j<n$. Also we recall from in the introduction how alias $v^{j-1}\in \partial W^u(v^j)$, implies $v^j\leadsto v^{j-1}$: $$\begin{aligned}
\mathcal{O}\leadsto v^{n-1}\leadsto ...\leadsto v^1 \leadsto v^o.
\end{aligned}
\label{eq:2.8}$$ This heteroclinic chain with Morse indices descending by 1, stepwise, motivates the name “descendants” for the equilibria $v^j$. Note that Sturm transversality of stable and unstable manifolds implies transitivity of the relation “$\leadsto$”. In particular, not only does $\mathcal{O}$ connect to any $v^j\in \Sigma^j_{s_j}(\mathcal{O})$, but also $$\begin{aligned}
0\leqslant j < k < n \quad \Rightarrow \quad v^k \leadsto v^j.
\end{aligned}
\label{eq:2.9}$$ Any heteroclinic orbit $v^j\leadsto v^{j-1}$ in the chain of descendants, from Morse index $j$ to adjacent Morse index $j-1$, is also known to be unique; [[@brfi89 Lemma 3.5]]{}.
The $y$-map, first constructed in [@brfi88] by a topological argument, is an alternative possibility to construct the descendant heteroclinic chain , directly. It allows to identify at least one solution $u(t,x)$ with initial condition $u(0, \cdot)$ in a small sphere around $\mathcal{O}$ in $W^u(\mathcal{O})$, with prescribed signed zero numbers $$\begin{aligned}
z(u(t, \cdot)- \mathcal{O})= j_{s_{j-1}}
\end{aligned}
\label{eq:2.9a}$$ for $t_j<t<t_{j-1},\ 1\leqslant j <n= i (\mathcal{O})$. Here $t_{n-1}:= -\infty$, $t_0= +\infty$, and the remaining partition $t_j$ can be chosen arbitrarily. Consider sequences $t_j$ such that the lenghts of all finite intervals $(t_j, t_{j-1})$ tend to infinity. Passing to convergent subsequences, then, suitably time-shifted trajectory pieces, starting at the strict dropping finite times $t_j$ tend to the desired heteroclinic orbits $$\begin{aligned}
u_j(t, \cdot): \quad \mathcal{E}^j_{s_j}\ni v_j\leadsto v_{j-1} \in \mathcal{E}^{j-1}_{s_{j-1}},
\end{aligned}
\label{eq:2.9b}$$ for $1\leqslant j< n$. By construction of $u(0, \cdot)\in W^u (\mathcal{O})$, we also have $\mathcal{O}\leadsto v^{n-1}$. This argument with convergent subsequences is very similar to the argument in Henry’s paper [@he85] on transversality. Uniqueness of the chain, however, is not obtained by this topological argument.
Before we collect the more specific properties of the $+$descendants, in lemma \[lem:2.2\], we mention a useful pigeon hole triviality which we invoke repeatedly below.
\[prop:2.1\] Let $\zeta_j$ be a strictly increasing sequence of m integers, $0\leq j<m$, which satisfy $$\begin{aligned}
0\leqslant \zeta_j<m, \qquad ~for~ all~ j.
\end{aligned}
\label{eq:2.10}$$ Then $$\begin{aligned}
\zeta_j=j, \quad ~for ~all~ j.
\end{aligned}
\label{eq:2.11}$$
For example, the descending heteroclinic chain , with $\zeta_j:=i(v^j)$ and $m:=n=i(\mathcal{O})$, reaffirms $i(v^j)=\zeta_j=j$, as already stated in .
In the following we call $v^j$, with j even, the *even descendants. Odd descendants* $v^j$ refer to odd $j$. We occasionally use the abbreviations $$\begin{aligned}
v_1<_0 v_2, \ \text{and}\ v_1 < _1 v_2,
\end{aligned}
\label{eq:2.12}$$ to indicate that $v_1< v_2$ holds at $x=0$, and at $x=1$, respectively.
\[lem:2.2\] Fix the symbol sequence $\mathbf{s}= ++...$ and consider the $+$descendants $v^j=v^j(\mathbf{s})$ of $\mathcal{O}$. Then, for any $0\leqslant j, k<n=i(\mathcal{O})$ and the $+$descendants $v^j$ of $\mathcal{O}$, the following statements hold:
- $j<k \Rightarrow v^j> v^k, \quad at~ x=0$;
- $\mathcal{O}< v^{n-1}< ...< v^0, \quad at~ x=0$;
- for even k and even descendants, $\mathcal{O}<v^k<...<v^2<v^0, \quad at~ x=1$;
- for odd k and odd descendants, $\mathcal{O}> v^k>...>v^3>v^1, \quad at~ x=1$
- $z(v^j-v^k)=\min\{j,k\}, \quad for ~ j\neq k$;
- $+$descendants of $+$descendants are $+$descendants.
To prove (i), indirectly, suppose $v^j<_0 v^k$. For the $+$descendant $v^j\in \Sigma^j_+(\mathcal{O})$ we also have $\mathcal{O} <_0 v^j$. Therefore $v^j$ is between $\mathcal{O}$ and $v^k$, at $x=0$. For the heteroclinic orbit $u(t, \cdot)$ from $\mathcal{O}$ to $v^k$ this implies the strict dropping $$\begin{aligned}
j=z(\mathcal{O}-v^j)= \lim_{t\rightarrow -\infty} z(u(t, \cdot)-v^j)> \lim_{t\rightarrow+\infty}z(u(t, \cdot)-v^j)= z(v^k-v^j);
\end{aligned}
\label{eq:2.13}$$ see , . Indeed, for $u(t_0, 0)=v^j(0)$, a multiple zero of $x\mapsto u(t_0,x)- v^j(x)$ occurs at the left Neumann boundary $x=0$.
On the other hand, the $z$-inequalities $$\begin{aligned}
u\in W^u (v)\Rightarrow \quad z(u-v)< i(v)
\end{aligned}
\label{eq:2.14}$$ $$\begin{aligned}
u\in W^s (v) \setminus \{v\}\Rightarrow \quad z(u-v)\geq i(v) \
\end{aligned}
\label{eq:2.15}$$ were already observed in [@brfi86]. Hence the heteroclinic orbit $u(t, \cdot)\in W^s(v_j)\setminus\{v_j\}$ from $v^k$ to $v^j$, for $k>j$, induced by the +descending heteroclinic chain , implies $$\begin{aligned}
z(v^k-v^j)=\lim_{t\mapsto -\infty} z(u(t, \cdot)-v^j)\geqslant i(v^j)=j,
\end{aligned}
\label{eq:2.16}$$ in view of the Morse indices . The contradiction between and proves claim (i).
Claim (ii) is an immediate consequence of $\mathcal{O}< _0 v^{n-1}\in \Sigma^{n-1}_+ (\mathcal{O})$ and property (i).
To prove claim (iii), we address even $k$ first. Consider nonnegative even $j<k$ and suppose, indirectly, that $v^j<_1 v^k$. Since $v^j\in \Sigma^j_+(\mathcal{O})$ we have $\mathcal{O}<_0 v^j$ and $z(v^j-\mathcal{O})=j$. Since $j$ is even, we also have $\mathcal{O}<_1 v^j$. As in , strict dropping of $z$ at $v^j$ for $\mathcal{O}\leadsto v^k$, this time at $x=1$, then implies $$\begin{aligned}
j=z(\mathcal{O}-v^j)> z(v^k-v^j).
\end{aligned}
\label{eq:2.17}$$ As in , , transitive $v^k\leadsto v^j$ on the other hand implies $$\begin{aligned}
z(v^k-v^j)\geqslant i(v^j)=j.
\end{aligned}
\label{eq:2.18}$$ This contradiction proves claim (iii), for even $k$. The case (iv) of odd $k$ is analogous, arguing indirectly, for odd $j<k$ and $v^k<_1 v^j$, via $\Sigma^j_+(\mathcal{O})\ni v^j<_1\mathcal{O}$.
To prove claim (v), consider $0\leq j<k<n=i(\mathcal{O})$. To show $0\leq \zeta_j := z(v^j-v^k)=j$, for those $j$, we invoke the pigeon hole proposition \[prop:2.1\]. Assumption holds, for $m:=k$, because $v^k\leadsto v^j$ and imply $z(v^j-v^k)<i(v_k)=k$. To show that the sequence $\zeta_j$ increases strictly, with $j$, we compare $\zeta_{j-1}$ and $\zeta_j$ for $1\leq j< k$. Since $j-1$ and $j$ are of opposite parity, mod 2, they lie on opposite sides of $v^k, k>j$, at $x=1$; see (iii), (iv). Therefore $v^j\leadsto v^{j-1}$ implies strict dropping of $z$ $$\begin{aligned}
\zeta_j= z(v^j-v^k)> z(v^{j-1}-v^k)=\zeta_{j-1} \,.
\end{aligned}
\label{eq:2.19}$$ Hence pigeon hole proposition \[prop:2.1\] proves claim (iv).
It remains to prove claim (vi). Consider the $+$descendants $v^k\in\Sigma^k_+(\mathcal{O})$. In view of definition \[eq:2.1\], , and , the $+$descendants of $\mathcal{O}$ are uniquely characterized by the descendant heteroclinic chain $$\begin{aligned}
\mathcal{O}\leadsto v^{n-1}\leadsto ... \leadsto v^k \leadsto ... \leadsto v^j\leadsto ... \leadsto v^0
\end{aligned}
\label{eq:2.20}$$ together with the conditions $$\begin{aligned}
z(v^j-\mathcal{O})=j_+ \, .
\end{aligned}
\label{eq:2.21}$$ To show that the unique $+$descendants $\tilde{v}^j\in\Sigma^j_+(v^k)$ of $v^k$, for $0\leq j<k$, coincide with the $+$descendants $v^j$ of $\mathcal{O}$, it only remains to show $$\begin{aligned}
z(v^j-v^k)= j_+ \, .
\end{aligned}
\label{eq:2.22}$$ Property (v) asserts $z(v^j-v^k)=j$, since $0 \leq j < k$. Ordering (i) asserts $v^j-v^k>_00$. This proves , claim (vi), and the lemma.
We conclude this section with an illustration of the action on lemma 2.2 (ii)–(iv) of the four trivial equivalences generated by $u\mapsto -u$ and $x\mapsto 1-x$; see . The trivial equivalence $u\mapsto -u$ flips (ii) into the opposite adjacent order $\mathcal{O}>v^{n-1}>...>v^0$, at $x=0$, which corresponds to constant $s_j=-$. The trivial equivalence $x\mapsto 1-x$ makes the adjacent order (ii) and its opposite appear at $x=1$, respectively. Therefore, the four trivial equivalences are characterized by the unique one of the four half axes of $u$, at $x=0$ and $x=1$, where the descendants are ordered adjacently. The alternating orders appear on the $x$-opposite $u$-axis, respectively. In particular $x \mapsto 1-x$ interchanges constant and alternating sign sequences $\mathbf{s}$.
First descendants and nearest neighbors {#sec3}
=======================================
In this section we prove our main result, theorem \[thm:1.2\]. As explained in the introduction, the trivial equivalences reduce the four cases – to the single case $\mathbf{s}=++...$ of $+$descendants $v^k=v^k(++...)$ with $k=n-1,\ n:=i(\mathcal{O})$; see , . We also recall the notation $v=\underline{v}^k$ of for the equilibrium $v\in \mathcal{E}^k_+(\mathcal{O})\subseteq \Sigma^k_+(\mathcal{O})$ which is closest to $\mathcal{O}$ at $x=1$. In theorem \[thm:3.1\] below, we show $\underline{v}^k= v^k$, for all $0\leq k<n$. Invoking theorem \[thm:3.1\] for the special case $k=n-1$, we then prove theorem \[thm:1.2\].
\[thm:3.1\] With the above notation, and in the setting of the introduction, $$\begin{aligned}
\underline{v}^k=v^k
\end{aligned}
\label{eq:3.1}$$ holds for all $0\leq k< n = i(\mathcal{O})$.
To prove (\[thm:3.1\]), indirectly, suppose $\underline{v}^k \neq v^k$. Because $\underline{v}^k, v^k\in \mathcal{E}^k_+(\mathcal{O})$, the definition of $\underline{v}^k$ implies that $\underline{v}^k$ is strictly closer to $\mathcal{O}$ than $v^k$, at $x=1$: $$\begin{aligned}
\mathcal{O}<_1(-1)^k \underline{v}^k<_1 (-1)^k v^k,
\end{aligned}
\label{eq:3.2}$$ if we normalize $\mathcal{O}\equiv 0$, without loss of generality. Consider any $1\leq j\leq k<n$. Then the part $$\begin{aligned}
v^k\leadsto ... \leadsto v^j \leadsto v^{j-1}\leadsto ... \leadsto v^0
\end{aligned}
\label{eq:3.3}$$ of the descending heteroclinic chain of the $\mathcal{O}$-descendants implies $v^j\leadsto v^{j-1}$, and hence $$\begin{aligned}
\zeta_j:=z(v^j-\underline{v}^k)>z(v^{j-1}- \underline{v}^k)=\zeta_{j-1}\geq0.
\end{aligned}
\label{eq:3.4}$$
Indeed, lemma \[lem:2.2\](iii),(iv) and imply that $\underline{v}^k$ is between $v^j$ and $v^{j-1}$ at $x=1$, due to the opposite parity of $j-1$ and $j$, mod 2. (Recall how we have used arguments of this type in our proof of lemma \[lem:2.2\], repeatedly.) In [@firo3d-1 Proposition 3.1 (iv)], on the other hand, we have already observed that $$\begin{aligned}
0 \leq \zeta_k = z(v^k-\underline{v}^k)\leq k-1,
\end{aligned}
\label{eq:3.5}$$ for any two distinct equilibria $v^k, \underline{v}^k \in \Sigma^k_+ (\mathcal{O})$. By the standard pigeon hole argument, however, the $k+1$ distinct numbers $\zeta_0<...<\zeta_k$ of cannot fit into the $k$ available positions ${0, ..., k-1}$ of . This contradiction proves the theorem.
In case $n=i(\mathcal{O})>0$ is even, and for $\mathbf{s}= ++...$, we may assume $i(w^1_-)=n-1$ for the predecessor $w^1_-$ of $\mathcal{O}$ at $x=1$, see . We have to show assertion , i.e. $$\begin{aligned}
w^1_-=v^{n-1} (++...)=v^{n-1},
\end{aligned}
\label{eq:3.6}$$ in the notation of the present section.
We first claim $\mathcal{O}\leadsto w^1_-.$ To prove the claim we recall Wolfrum’s Lemma; see [@wo02] and [@firo3d-2 Appendix]: for equilibria $v_1, v_2$ with $i(v_1)>i(v_2)$ we have $v_1\leadsto v_2$ if, and only if, there does not exist any equilibrium $w$ with boundary values strictly between $v_1$ and $v_2$, at $x=0$ and $x=1$, such that $$\begin{aligned}
z(v_1-w)=z(v_2-w)=z(v_1-v_2).
\end{aligned}
\label{eq:3.7}$$ For $v_1:=\mathcal{O}$ and $v_2:=w^1_-$, by definition of $w^1_-$ as the predecessor of $\mathcal{O}$ at $x=1$, there do not exist any equilibria at all between $v_1$ and $v_2$, at $x=1$. Therefore $i(\mathcal{O})=n>n-1=i(w^1_-)$ implies $\mathcal{O}\leadsto w^1_-,$ as claimed.
We claim $w^1_-\in \mathcal{E}^{n-1}_+(\mathcal{O})$ next. Since $\mathcal{O}\leadsto w^1_-$ with adjacent Morse indices $n$ and $n-1$, properties , of the zero numbers on unstable and stable manifolds imply $$\label{eq:3.8}
n= i(\mathcal{O})> z(\mathcal{O}-w^1_-)\geq i(w^1_-)= n-1,$$ i.e. $z(\mathcal{O}- w^1_-)=n-1$. Since $w^1_-$, by definition, is the predecessor of $\mathcal{O}$, at $x=1$, with even $n$ and odd $n-1$, we also have $w^1_-> \mathcal{O}$ at $x=0$. In view of this proves $w^1_-\in \mathcal{E}^{n-1}_+(\mathcal{O})$. Again by definition of $w^1_-$, as the predecessor of $\mathcal{O}$ at $x=1$, the fact $w^1_-\in\mathcal{E}^{n-1}_+(\mathcal{O})$ implies $w^1_-=\underline{v}^{n-1}$. Invoking theorem \[thm:3.1\] for $k=n-1$ shows $$\label{eq:3.9}
w^1_-=\underline{v}^{n-1}= v^{n-1},$$ as claimed in , for even $n$.
For odd $n$, and even $n-1$, we can repeat the exact same steps for the successor $w^1_+$ of $\mathcal{O}$ at $x=1$ replacing the predecessor $w^1_-$. This proves theorem \[thm:1.2\].
Minimax: the range of hemispheres {#sec4}
=================================
For the $+$descendants $v^k=v^k(++...)$ of $\mathcal{O}$, with $0\leq k<n:= i(\mathcal{O})$, we have shown $$\begin{aligned}
v^k=\underline{v}^k
\end{aligned}
\label{eq:4.1}$$ in the previous section. See theorem \[thm:3.1\], where $\underline{v}^k$ denoted the equilibrium closest to $\mathcal{O}$, at $x=1$, in the hemisphere $\Sigma^k_+(\mathcal{O})$. In theorem \[thm:4.3\] of the present section we show $$\begin{aligned}
v^k=\overline{v}^k,
\end{aligned}
\label{eq:4.2}$$ where $\overline{v}^k$ denotes the equilibrium farthest from $\mathcal{O}$ at the opposite boundary $x=0$, in the same hemisphere $\Sigma^k_+(\mathcal{O})$. In particular $$\begin{aligned}
\underline{v}^k=v^k=\overline{v}^k,
\end{aligned}
\label{eq:4.3}$$ shows how minimal distance from $\mathcal{O}$, along the meander axis $h_1$ of $x=1$, coincides with maximal distance from $\mathcal{O}$, along the meander $h_0$ of $x=0$ itself.
Throughout this section we fix $k$. In lemma \[lem:4.1\] we show $$\begin{aligned}
i(\overline{v}^k)=k,
\end{aligned}
\label{eq:4.4}$$ in analogy to $i({v}^k)=k$. We then study the $+$descendants $w^j\in \Sigma^j_+(\overline{v}^k)$ of $\overline{v}^k$, for $0\leq j<k$. In lemma \[lem:4.2\], in particular, we show $$\begin{aligned}
z(w^{k-1}- \mathcal{O})=k-1.
\end{aligned}
\label{eq:4.5}$$ Combining and will then prove the claim of theorem \[thm:4.3\].
We conclude the section, in corollary \[cor:4.4\], with a summary of our results for all four cases of constant and alternating descendants.
\[lem:4.1\] For any $0\leq k < n = i(\mathcal{O})$, claim holds true.
We first show $i(\overline{v}^k)= \dim W^u(\overline{v}^k)\leq k.$ Indeed $$\begin{aligned}
\overline{v}^k\in \mathcal{E}^k_+(\mathcal{O})\subseteq \Sigma^k_+(\mathcal{O}) = \bigcup \limits_{v\in \mathcal{E}^k_+(\mathcal{O})}W^u(v)
\end{aligned}
\label{eq:4.6}$$ implies $W^u(\overline{v}^k)\subseteq \Sigma^k_+(\mathcal{O})$, and hence $\dim W^u(\overline{v}^k) \leq \dim \Sigma^k_+(\mathcal{O}) = k$.
To prove $i(\overline{v}^k)=k$, indirectly, suppose $i(\overline{v}^k)<k$. The eigenprojection $P^k$ projects the $k$-dimensional hemisphere disc $\mathrm{clos} \, \Sigma^k_+(\mathcal{O})$ homeomorphically into the tangent space $\langle \varphi_0, ..., \varphi_{k-1}\rangle$ of the fast unstable manifold $W^k(\mathcal{O})$, at $\mathcal{O}$; see also – above. Therefore the (projected) interior cell $c_{\overline{v}^k}$, of dimension $k'$ less than $k$, possesses at least one neighboring cell $c_{\tilde{v}}$ in $\Sigma^k_+(\mathcal{O})$ of dimension $k'+1\leq k$, such that $\tilde{v}\leadsto \overline{v}^k$ and $\tilde{v}>_0 \overline{v}^k >_0 \mathcal{O}$. Here we use local surjectivity of the projection $P^k$, and positivity of the first eigenfunction $\varphi_0$. This contradicts maximality of $\overline{v}^k$ in $\Sigma^k_+(\mathcal{O})$, at $x=0$, and proves the lemma.
We consider the $+$descendants $w^j\in \Sigma^j_+(\overline{v}^k)$ of $\overline{v}^k$ next, for $0\leq j< k$: $$\begin{aligned}
\overline{v}^k \leadsto w^{k-1} \leadsto ... \leadsto \mathcal{O}.
\end{aligned}
\label{eq:4.7}$$ By construction, $z(w^j- \overline{v}^k)= j_+$. However, this does not yet determine $z(w^{k-1}-\mathcal{O})$ to be $k-1$, as claimed in .
\[lem:4.2\] For any $1\leq k<n= i(\mathcal{O})$, claim holds true for the first $+$descendant $w^{k-1}$ of $\overline{v}^k$. In particular $$\begin{aligned}
w^{k-1}\in\Sigma^{k-1}_+(\mathcal{O}).
\end{aligned}
\label{eq:4.8a}$$
By construction of the $+$descendant $w^{k-1}$ of $\overline{v}^k$, we have $$\begin{aligned}
i(w^{k-1})=k-1;
\end{aligned}
\label{eq:4.8b}$$ see . Since $\Sigma^k_+(\mathcal{O})\ni \overline{v}^k \leadsto w^{k-1},$ by , we have $$\begin{aligned}
k=z(\overline{v}^k-\mathcal{O})\geqslant z(w^{k-1}- \mathcal{O})
\end{aligned}
\label{eq:4.9}$$
On the other hand, $\mathcal{O}\leadsto \overline{v}^k \in \Sigma^k_+ (\mathcal{O}), \ \overline{v}^k \leadsto w^{k-1}$, and transitivity of $\leadsto$, imply $\mathcal{O} \leadsto w^{k-1}$. Therefore , for $v:=w^{k-1}$ and yield $$\begin{aligned}
k-1=i(w^{k-1})\leq z(\mathcal{O}- w^{k-1})= z(w^{k-1}- \mathcal{O}).
\end{aligned}
\label{eq:4.10}$$ Together, and leave us with the options $z(w^{k-1}- \mathcal{O})\in \{k-1,k\}$.
Suppose, indirectly, that the bad option $z(w^{k-1}-\mathcal{O})= k$ holds true. Then $\mathcal{O}<_0 \overline{v}^k< _0w^{k-1}$, by $\overline{v}^k\in \Sigma^k_+(\mathcal{O})$ and $w^{k-1}\in \Sigma^{k-1}_+(\overline{v}^k)$, implies $w^{k-1}\in \Sigma^k_+(\mathcal{O}). $ This contradicts the maximality of $\overline{v}^k \in \Sigma^k_+(\mathcal{O})$, at $x=0$, and proves $z(w^{k-1}- \mathcal{O})=k-1$, as claimed in .
Since $\mathcal{O}<_0 \overline{v}^k< _0w^{k-1}$ still holds, we also obtain $z(w^{k-1}-\mathcal{O})= (k-1)_+$. Moreover we recall $\mathcal{O}\leadsto w^{k-1}$. Together this establishes $w^{k-1}\in\Sigma^{k-1}_+(\mathcal{O})$, as claimed in , and the lemma is proved.
\[thm:4.3\] With the above notation, and in the setting of the introduction, $$\begin{aligned}
\overline{v}^k=\underline{v}^k
\end{aligned}
\label{eq:4.11}$$ holds for all $0\leq k< n = i(\mathcal{O})$.
For $k=0$, where $\Sigma^0_+(\mathcal{O})= \{v^0\}$ consists of a single equilibrium anyway, there is nothing to prove. Therefore consider $1\leq k < n$. We proceed indirectly and suppose $\overline{v}^k\neq \underline{v}^k$. To reach a contradiction we prove the following three contradictory claims, separately: $$\begin{aligned}
z(w^{k-1}-\underline{v}^k) &=& k-1, \label{eq:4.12}\\
z(w^{k-1}-\underline{v}^k) &<& z(\overline{v}^k- \underline{v}^k),\text{ and} \label{eq:4.13}\\
z(\overline{v}^k-\underline{v}^k) &\leq& k-1. \label{eq:4.14}\end{aligned}$$ Here $w^j\in \Sigma^j_+(\overline{v}^k)$ denote the $+$descendants of $\overline{v}^k$, as in and in lemma \[lem:4.2\].
We first recall that $\underline{v}^k$ is closest to $\mathcal{O}$ in $\mathcal{E}^k_+(\mathcal{O})\subseteq \Sigma^k_+(\mathcal{O})$, at $x=1$, and $\overline{v}^k\in \mathcal{E}^k_+(\mathcal{O})\subseteq \Sigma^k_+(\mathcal{O})$ is maximal at $x=0$; see definitions and . In particular, $\underline{v}^k$ is strictly between $\mathcal{O}$ and $\overline{v}^k$, both, at $x=0$ and $x=1$, by our indirect assumption $\underline{v}^k \neq \overline{v}^k$. By [@firo3d-1], Proposition 3.1(iv), $\underline{v}^k, \overline{v}^k\in \Sigma^k_+(\mathcal{O})$ implies $z(\overline{v}^k-\underline{v}^k)\leq k-1$, as claimed in .
By lemma \[lem:2.2\](iii),(iv), the $+$descendants $w^j$ of $\overline{v}^k$ satisfy $$\begin{aligned}
\overline{v}^k< w^j < ... < w^2 <w^0, &\quad& \text{for even}~ j<k, \label{eq:4.15}\\
\overline{v}^k> w^j > ... > w^3 >w^1, &\quad& \text{for odd}~ j<k, \label{eq:4.16}\end{aligned}$$ at $x=1$. Moreover $\overline{v}^k\in \Sigma^k_+(\mathcal{O})$, and, by lemma \[lem:4.2\], $w^{k-1}\in\Sigma^{k-1}_+(\mathcal{O})$ lie on opposite sides of $\mathcal{O}$, at $x=1$, by opposite parities of $k, k-1$ mod 2. Because $\underline{v}^k$ is closest to $\mathcal{O}$ in $\mathcal{E}^k_+(\mathcal{O})\subseteq \Sigma^k_+(\mathcal{O})$, at $x=1$, it lies strictly between $\mathcal{O}$ and $\overline{v}^k\in \mathcal{E}^k_+(\mathcal{O})\subseteq \Sigma^k_+(\mathcal{O}) $ there. In particular, and hold for $\underline{v}^k$ as well: $$\begin{aligned}
\underline{v}^k< w^j < ... < w^2 < w^0, &\quad& \text{for even}~ j<k, \label{eq:4.17}\\
\underline{v}^k> w^j > ... > w^3 > w^1, &\quad& \text{for odd}~ j<k. \label{eq:4.18}\end{aligned}$$ Moreover, $\underline{v}^k$ lies strictly between $\overline{v}^k$ and $w^{k-1}$, at $x=1$. Therefore $\overline{v}^k\leadsto w^{k-1}$ implies $z(\overline{v}^k- \underline{v}^k)> z(w^{k-1}- \underline{v}^k)$, as claimed in .
To prove claim , finally, let $\zeta_j:= z(w^j- \underline{v}^k)$. We apply the pigeon hole proposition \[eq:2.1\]. First, we note $0\leq\zeta_{j-1}< \zeta_j$, for all $j=1, ..., k-1,$ because $w^j\leadsto w^{j-1}$ and $w^{j-1}, w^j$ are on opposite sides of $\underline{v}^k$, by , . Since , and slightly weakened , imply $\zeta_{k-1}\leq k-1$, claim follows from proposition \[eq:2.1\] which asserts $\zeta_j=j$ for all $0\leq j \leq k-1$. This proves the theorem by the contradictions – .
So far, we have only considered descendants $\underline{v}^k=v^k=\overline{v}^k$ based on the constant sign sequence $\mathbf{s}=++\ldots$ of . The four trivial equivalences provide the following variant of theorem \[thm:4.3\].
\[cor:4.4\] Let $0\leq k< n = i(\mathcal{O})$ and $\iota=0,1$. Then the equilibrium in $\Sigma_\pm^k(\mathcal{O})$ closest to $\mathcal{O}$, at $x=\iota$, coincides with the equilibrium in $\Sigma_\pm^k(\mathcal{O})$ most distant from $\mathcal{O}$, at the opposite boundary $x=1-\iota$.
In other words, the $h_\iota$ closest equilibrium to $\mathcal{O}$ is $h_{1-\iota}$ most distant, in the same hemisphere $\Sigma_\pm^k(\mathcal{O})$.
Discussion {#sec5}
==========
In this final section we explore what our main theorem \[thm:1.2\] does, and does not, say. We first review the most celebrated Sturm global attractor, the $n$-dimensional Chafee-Infante attractor [@chin74]. Contrary to the common approach, which starts from an explicit cubic nonlinearity, an ODE discussion of equilibria, and the time map of their pendulum boundary value problem, we start from an abstract description of the associated PDE Thom-Smale complex. We then apply theorem \[thm:1.2\] to derive the well-known associated shooting meander, and the Sturm permutation, in this much more general context. In the second part of our discussion, we present three examples of abstract signed regular complexes which are 3-balls. We first adapt the general recipe of theorem \[thm:1.2\] for the construction of the associated boundary orders $h_0, h_1$ to the special case of 3-balls, in the spirit of [@firo3d-1; @firo3d-2]. See theorem \[thm:5.2\] and definitions \[def:5.1\], \[def:5.3\], \[def:5.4\]. Our first example, in fig. \[fig:5.2\], then constructs $h_0, h_1$ and the permutation $\sigma = h_0^{-1}\circ h_1$ for the unique Sturm solid tetrahedron with two faces in each hemisphere. The locations of the poles $\mathbf{N}, \mathbf{S}$ turn out to be edge-adjacent, necessarily, along the meridian circle. In our second example, fig. \[fig:5.3\], we deviate from the unique associated signed Thom-Smale complex, by changing the position of the poles. The new locations of the poles are not edge-adjacent, along the same meridian circle. Still, our recipe succeeds to construct a Sturm permutation $\sigma$ which, however, necessarily fails to describe that non-Sturm modification of the signed regular solid tetrahedron. In our third example, fig. \[fig:5.4\], we start from a signed regular solid octahedron complex with antipodal pole locations. It was first observed in [@firo14] that such antipodal octahedra cannot be of Sturm type. Our construction of the permutation $\sigma$ still succeeds, in that case, but fails to define a meander. We conclude with comments on the still elusive goal of a geometric characterization of all Sturm global attractors.
The $n$-dimensional *Chafee-Infante attractor* $\mathcal{CI}_n$ is the Sturm global attractor of $$u_t = u_{xx} + \lambda^2 u(1-u^2)
\label{eq:CI.1}$$ on the unit interval $0<x<1$, with parameter $0<(n-1)\pi < \lambda < n\pi$, cubic nonlinearity, and for Neumann boundary conditions. See [@chin74] for the closely related original Dirichlet setting. Geometrically, $\mathcal{CI}_0$ can be thought of as the single trivial equilibrium $\mathcal{O}=0$, and $\mathcal{CI}_n$ is the one-dimensionally unstable double cone suspension of $\mathcal{CI}_{n-1}$, recursively for $n>0$. See [@he85; @fi94]. The double cone suspension is a generalization of the passage to a sphere $\Sigma^n$ from its equator $\Sigma^{n-1}$, of course. The Chafee-Infante attractor $\mathcal{CI}_n$ can also be characterized as the $n$-dimensional Sturm attractor with minimal number $N=2n+1$ of equilibria. Equivalently, $\mathcal{CI}_n$ is the Sturm attractor of maximal dimension $n=(N-1)/2$, for any (necessarily odd) number $N$ of equilibria. See [@fi94].
The signed Thom-Smale complex of the Chafee-Infante attractor $\mathcal{CI}_n$ is given as follows. Let $\Sigma_\pm^k=\Sigma_\pm^k(\mathcal{O})$ denote the signed hemisphere decomposition of $\Sigma^{n-1}=\partial W^u(\mathcal{O})$ into $2n$ hemispheres, $0 \leq k < n$. Each hemisphere has to contain at least one nontrivial equilibrium. By minimality $2n$ of their number we may enumerate them as $v_\pm^k$ such that $$\label{eq:CI.2}
\Sigma_\pm^k(\mathcal{O}) = W^u(v_\pm^k)\,;$$ see . By construction, we obtain the signed zero numbers $z(v_\pm^k-\mathcal{O})=k_\pm$ and Morse indices $i(v_\pm^k)=k$; see . More generally, the successive pitchfork bifurcations of at the bifurcation points $\lambda=n\pi$ provide all zero numbers and Morse indices for $0\leq j < k<n\,$ as $$\label{eq:CI.3}
z(v_\pm^j-v_\pm^k) = z(v_\pm^j-v_\mp^k)= z(v_\pm^j-\mathcal{O})=j_\pm \,, \qquad i(v_\pm^j)=j\,.$$
![ *The meander $h_0$ (red) of the $n$-dimensional Chafee-Infante attractor . Note the two collections of nested arcs, one above and one below the horizontal $h_1$ axis (blue). The outermost arcs begin and terminate at the poles $v_\pm^0$, respectively, and the innermost arcs involve $\mathcal{O}$. Because the two nests are shifted with respect to each other, their arcs join to form a meander curve, in fact a double spiral, with an inflection at the center $\mathcal{O}$. The neighbors $v_\pm^{n-1}$ of $\mathcal{O}$ are $v_+^{n-1}$ to the left and $v_-^{n-1}$ to the right, if $n-1$ is odd. For even $n$ the subscript signs are reversed.* []{data-label="fig:CI.1"}](figs/{figCI_1}.pdf){width="100.00000%"}
Following theorem \[thm:1.2\], we can now derive the meanders and Sturm permutations of the Chafee-Infante attractors $\mathcal{CI}_n$, directly from the above abstract description of the Chafee-Infante signed Thom-Smale complex as an abstract signed regular cell complex. Of course our derivation is for illustration purposes only: we carefully avoid any further reference to the common ODE derivation of the Sturm permutation, directly from the shooting approach, via the integrable equilibrium ODE $v_{xx} + \lambda^2 v(1-v^2) = 0$ and monotonicity of the time map.
The $\mathbf{s}$-descendants $v^j(\mathbf{s})$ of $\mathcal{O}$, with $\mathbf{s}=s_j\ldots s_0$, are easily identified as $$\label{eq:CI.4}
v^j(\mathbf{s})=v_{s_j}^j\,.$$ The same statement holds true for the $\mathbf{s}$-descendants $v^j(\mathbf{s})$ of $v_\pm^k$ and $v_\mp^k$, with $0\leq j<k<n$. Indeed , imply $$\label{eq:CI.3a}
\Sigma_\pm^j(v_\pm^k) = \Sigma_\pm^j(v_\mp^k) =\Sigma_\pm^j(\mathcal{O}).$$ We may now apply theorem \[thm:1.2\], successively by descending order of Morse indices, to determine the Hamiltonian boundary orders $h_0, h_1$. This identifies the $h_0$-predecessor $w_-^0$ and $h_0$-successor $w_+^0$ of $\mathcal{O}$ to be $w_\pm^0 = v_\pm^{n-1}$, at $x=0$; see , , and . The $h_1$-predecessor $w_-^1$ and $h_1$-successor $w_+^1$ of $\mathcal{O}$ depend on the even/odd parity of $n=i(\mathcal{O})$, due to , . For odd $n$, implies $w_\pm^1 = v_\pm^{n-1}$ at $x=1$, as at the left boundary $x=0$, because $n-1$ is even. For even $n$, alias odd $n-1$, we obtain the reversed order $w_\pm^1 = v_\mp^{n-1}$ at $x=1$. Analogous remarks identify the predecessors and successors of all remaining $v_\pm^k$. In view of , we simply replace $\mathcal{O}$ by $v_\pm^k$, and $n$ by $k$, in our previous remarks, and respect the parity of $k$, appropriately.
Alternatively, and perhaps more directly, the unique enumeration of the equilibria $\mathcal{E}_\pm^k(\mathcal{O})=\{v_\pm^k\}$ in the hemispheres $\Sigma_\pm^k(\mathcal{O})$ allows us to invoke lemma \[lem:2.2\](ii) and conclude the $h_0$ order at $x=0$ to be $$\label{eq:CI.5}
v_-^0 < \ldots < v_-^{n-1} < \mathcal{O} < v_+^{n-1} < \ldots < v_+^0 \,.$$ Here we have applied the trivial equivalence $u\mapsto-u$ to also derive the ordering of $v_-^k$ at $x=0$. Analogously, lemma \[lem:2.2\](iii),(iv) imply the $h_1$ order at $x=1$ to be $$\label{eq:CI.6}
v_-^0 < v_+^1 < v_-^2 < v_+^3 < \ldots < \mathcal{O} < \ldots v_-^3 < v_+^2 < v_-^1 < v_+^0 \,.$$ Here $v_+^k$ appear below $\mathcal{O}$ for odd $k$, in increasing order, and above $\mathcal{O}$ for even $k$, in decreasing order. For $v_-^k$ the parities of $k$ are reversed. The interlacing of $v_\pm^k$ follows directly from .
See fig. \[fig:CI.1\] for the resulting meander, based on the orders , . Notably the meander $h_0$ consists of two collections of nested arcs, one above and one below the horizontal $h_1$ axis. The outermost arcs begin and terminate at the poles $v_\pm^0$, respectively, and the innermost arcs involve the inflection point $\mathcal{O}$. In cycle notation, the Chafee-Infante Sturm permutation $\sigma=h_0^{-1}\circ h_1$ of the $N=2n+1$ equilibria follows easily from , to be the involution $$\label{ eq:CI.7}
\sigma = (2\ \ 2n)\ (4\ \ 2n-2) \ldots (2[\tfrac{n}{2}]\ \ 2[\tfrac{n+3}{2}])\,.$$ Here $[\cdot]$ denotes the integer part.
A priori knowledge of all signed zero numbers $z(v_j-v_k)$, as defined in , determines the Sturm permutation $\sigma=h_0^{-1}\circ h_1$, in any Thom-Smale complex. Indeed, the signs of $z(v_j-v_k)$ immediately determine the total order $h_0$ of all equilibria $v_k$, at $x=0$. Keeping the even/odd parity of $k$ in mind, the same signs determine the total order $h_1$ of all equilibria $v_k$, at $x=1$.
For the abstract Chafee-Infante signed regular complex , the signed zero numbers $j_\pm$ in therefore provide a third, completely elementary, approach to the determination of the boundary orders $h_0, h_1$, as in , , and hence of the underlying Sturm permutation $\sigma$.
For general abstract signed regular complexes, however, matters are not that simple. The prescribed hemisphere signs do not keep track of the relative boundary orders of all barycenter pairs $v_j, v_k$. Rather, this information is restricted to those pairs $v_j, v_k$ for which one barycenter is in the cell boundary of the other. (A posteriori, in other words, these are the heteroclinic pairs $v_k \leadsto v_j$ in the resulting Sturm attractor.) How to extend this partial order to the different total orders $h_0, h_1$, uniquely, which turn out to be the boundary orders in the underlying Sturm setting of the originally unknown Sturm permutation $\sigma=h_0^{-1}\circ h_1$, was the main result of the present paper. See theorem \[thm:1.2\].
We turn to 3-ball Sturm attractors $\mathcal{A}_f$ next. A purely geometric characterization of their signed hemisphere decompositions – has been achieved in [@firo3d-1; @firo3d-2]; see also [@firo3d-3] for many examples. Dropping all Sturmian PDE interpretations, we defined 3-cell templates, abstractly, in the class of signed regular cell complexes $\mathcal{C}$ and without any reference to PDE or dynamics terminology. Recall fig. \[fig:1.0\](b) for a first illustration.
\[def:5.1\] A finite signed regular cell complex $\mathcal{C} = \bigcup_{v \in \mathcal{E}} c_v$ is called a *3-cell template* if the following four conditions all hold for the hemispheres $\Sigma_\pm^j=\Sigma_\pm^j(\mathcal{O})$ and descendants $v^j=v^j(\mathbf{s})$ of $\mathcal{O}$, according to definition \[def:1.1\].
- $\mathcal{C} = \text{clos } c_{\mathcal{O}}= S^2 \,\dot{\cup}\, c_{\mathcal{O}}$ is the closure of a single 3-cell $c_{\mathcal{O}}$.
- The 1-skeleton $\mathcal{C}^1$ of $\mathcal{C}$ possesses a *bipolar orientation* from a pole vertex $\mathbf{N}:=\Sigma_-^0$ (North) to a pole vertex $\mathbf{S}:=\Sigma_+^0$ (South), with two disjoint directed *meridian paths* $\mathbf{EW}:=\Sigma_-^1$ and $\mathbf{WE}:=\Sigma_+^1$ from $\mathbf{N}$ to $\mathbf{S}$. The meridians decompose the boundary sphere $S^2$ into remaining hemisphere components $\Sigma_-^2:=\mathbf{W}$ (West) and $\Sigma_+^2:=\mathbf{E}$ (East).
- Edges are directed towards the meridians, in $\Sigma_-^2$, and away from the meridians, in $\Sigma_+^2$, at end points on the meridians other than the poles $\Sigma_\pm^0$.
- For $\iota=0,1$, let $w_\pm^\iota$ denote the first descendants $v^2(\mathbf{s})$ of $\mathcal{O}$, as defined in –. Then the boundaries of the 2-cells of $w_-^\iota$ and of $w_+^{1-\iota}$ overlap in at least one edge, along the appropriate meridian $\Sigma_\pm^1$, between their respective second descendants $v^1(\mathbf{s})$.
We recall here that an edge orientation of the 1-skeleton $\mathcal{C}^1$ is called bipolar if it is without directed cycles, and with a single “source” vertex $\mathbf{N}$ and a single “sink” vertex $\mathbf{S}$ on the boundary of $\mathcal{C}$. Here “source” and “sink” are understood, not dynamically but, with respect to edge direction. The edge orientation of any 1-cell $c_v$ runs from $\Sigma_-^0(v)$ to $\Sigma_+^0(v)$. The most elementary hemi-“sphere” decomposition of 1-cells, in other words, can simply be viewed as an edge orientation. Bipolarity is a local and global compatibility condition for these orientations which, in particular, forbids directed cycles.
By definition \[def:1.1\] of descendants, the 2-cells $\mathbf{NE}$ of $w_-^0$ and $\mathbf{SW}$ of $w_+^1$ denote the unique faces in $\mathbf{W}$, $\mathbf{E}$, respectively, which contain the first, last edge of the meridian $\mathbf{WE}$ in their boundary, respectively. In definition \[def:5.1\](iv), the boundaries of $\mathbf{NE}$ and $\mathbf{SW}$ are required to overlap in at least one shared edge along the meridian $\mathbf{WE}$.
Similarly, the 2-cells $\mathbf{NW}$ of $w_-^1$ and $\mathbf{SE}$ of $w_+^0$ denote the unique faces in $\mathbf{W}$, $\mathbf{E}$, respectively, which contain the first, last edge of the meridian $\mathbf{EW}$ in their boundary, respectively. The boundaries of $\mathbf{NW}$ and $\mathbf{SE}$ are required to overlap in at least one shared edge along the meridian $\mathbf{EW}$.
The main result of [@firo3d-1; @firo3d-2], in our language of descendants, reads as follows.
\[thm:5.2\] [@firo3d-2 theorems 1.2 and 2.6]. A finite signed regular cell complex $\mathcal{C}$ coincides with the signed Thom-Smale dynamic complex $c_v= W^u(v) \in \mathcal{C}_f$ of a 3-ball Sturm attractor $\mathcal{A}_f$ if, and only if, $\mathcal{C}$ is a 3-cell template.
In [@firo3d-1 theorem 4.1] we proved that the signed Thom-Smale complex $\mathcal{C}$:= $\mathcal{C}_f$ of a Sturm 3-ball $\mathcal{A}_f$ indeed satisfies properties (i)–(iv) of definition \[def:1.1\]. In our example of fig. \[fig:1.0\] this simply means the passage (a) $\Rightarrow$ (b). In general, the 3-cell property (i) of $c_{\mathcal{O}} = W^u(\mathcal{O})$ is obviously satisfied. The bipolar orientation (ii) of the edges $c_v$ of the 1-skeleton $\mathcal{C}^1$ is a necessary condition, for Sturm signed Thom-Smale complexes $\mathcal{C}=\mathcal{C}_f$. Indeed, acyclicity of the orientation of edges $c_v$, alias the one-dimensional unstable manifolds $c_v = W^u(v)$ of $i(v)=1$ saddles $v$, simply results from the strictly monotone $z=0$ ordering of each edge $c_v$: from the lowest equilibrium vertex $\Sigma_-^0 (v)$ in the closure $\bar{c}_v$ to the highest equilibrium vertex $\Sigma_+^0(v)$. The ordering is uniform for $0 \leq x \leq 1$, and holds at $x \in \{0,1\}$, in particular. The poles $\mathbf{N}$ and $\mathbf{S}$ indicate the lowest and highest equilibrium, respectively, in that order. Again we refer to fig. \[fig:1.0\] for an illustrative example. The meridian cycle is the boundary $\Sigma^1$ of the two-dimensional fast unstable manifold. Properties (iii) and (iv) are far less obvious, at first sight.
The main result of our present paper, theorem \[thm:1.2\], determines the boundary paths $h_0, h_1$ which identify a 3-cell template $\mathcal{C}$ as a 3-ball Sturm attractor $\mathcal{A}_f$ with signed Thom-Smale complex $\mathcal{C}_f=\mathcal{C}$. In our example, this describes the passage from fig. \[fig:1.0\](b) to fig. \[fig:1.0\](a). We describe an equivalent practical simplification of this construction next, in terms of an SZS-pair $(h_0, h_1)$ of Hamiltonian paths $h_\iota$: $\lbrace 1, \ldots , N\rbrace \rightarrow \mathcal{E}$; see [@firo3d-3 section 2] for further details.
To prepare our construction, we first consider planar regular CW-complexes $\mathcal{C}$, abstractly, with a bipolar orientation of the 1-skeleton $\mathcal{C}^1$. Here bipolarity requires that the unique poles $\mathbf{N}$ and $\mathbf{S}$ of the orientation are located at the boundary of the embedded regular complex $\mathcal{C} \subseteq \mathbb{R}^2$.
To traverse the vertices $v \in \mathcal{E}$ of a planar complex $\mathcal{C}$, in two different ways, we construct a pair of directed Hamiltonian paths $$h_0, h_1: \quad \lbrace 1, \ldots , N \rbrace \rightarrow \mathcal{E}
\label{eq:5.1}$$ as follows. Let $\mathcal{O}$ indicate any source, i.e. (the barycenter of) any 2-cell face $c_{\mathcal{O}}$ in $\mathcal{C}$. (We temporarily deviate from the standard 3-ball notation, here, to emphasize analogies with the passage of $h_\iota$ through a 3-cell.) By planarity of $\mathcal{C}$ the bipolar orientation of $\mathcal{C}^1$ defines unique extrema on the boundary circle $\partial c_{\mathcal{O}}$ of the 2-cell $c_\mathcal{O}$. Let $w_-^0$ denote the barycenter on $\partial c_{\mathcal{O}}$ of the edge to the right of the minimum, and $w_+^0$ the barycenter to the left of the maximum. See fig. \[fig:5.1\]. Similarly, let $w_-^1$ be the barycenter to the left of the minimum, and $w_+^1$ to the right of the maximum. Then the following definition serves as our practical construction recipe for the pair $(h_0,h_1$).
![ *Traversing a face vertex $\mathcal{O}$ by a ZS-pair $h_0, h_1$. Note the resulting shapes “Z” of $h_0$ (red) and “S” of $h_1$ (blue). The paths $h_\iota$ may also continue into adjacent neighboring faces, beyond $w_\pm^\iota$, without turning into the face boundary $\partial c_{\mathcal{O}}$.* []{data-label="fig:5.1"}](figs/{fig5_1}.pdf){width="75.00000%"}
\[def:5.3\] The bijections $h_0, h_1$ in are called a *ZS-pair* $(h_0, h_1)$ in the finite, regular, planar and bipolar cell complex $\mathcal{C} = \bigcup_{v \in \mathcal{E}} c_v$ if the following three conditions all hold true:
- $h_0$ traverses any face $c_\mathcal{O}$ from $w_-^0$ to $w_+^0$;
- $h_1$ traverses any face $c_\mathcal{O}$ from $w_-^1$ to $w_+^1$
- both $h_\iota$ follow the same bipolar orientation of the 1-skeleton $\mathcal{C}^1$, unless defined by (i), (ii) already.
We call $(h_0,h_1)$ an *SZ-pair*, if $(h_1, h_0)$ is a ZS-pair, i.e. if the roles of $h_0$ and $h_1$ in the rules (i) and (ii) of the face traversals are reversed.
Properties (i)-(iii) of definition \[def:5.3\] of a ZS-pair $(h_0,h_1)$ are equivalent to our present theorem \[thm:1.2\], in the language of descendants. Indeed, we just have to define the signed hemisphere decomposition of each planar face $c_\mathcal{O}$ such that $\Sigma_-^1(\mathcal{O})$ appears to the right of the boundary minimum $\Sigma_-^0(\mathcal{O})$, or the boundary maximum $\Sigma_+^0(\mathcal{O})$. Similarly, $\Sigma_+^1(\mathcal{O})$ appears to the left; see fig. \[fig:5.1\]. Note how shared edges between adjacent faces receive opposite signatures, from either face. For an SZ-pair, in contrast, we have to reverse the roles of $\Sigma_\pm^1(\mathcal{O})$. The planar trilogy [@firo08; @firo09; @firo10] contains ample material and examples on the planar case.
After these preparations we can now return to the general 3-cell templates $\mathcal{C}$ of definition \[def:5.1\] and define the SZS-pair $(h_0,h_1)$ associated to $\mathcal{C}$.
\[def:5.4\] Let $\mathcal{C} = \bigcup_{v \in \mathcal{E}} c_v$ be a 3-cell template with oriented 1-skeleton $\mathcal{C}^1$, poles $\mathbf{N}, \mathbf{S}$, hemispheres $\mathbf{W}, \mathbf{E}$, and meridians $\mathbf{EW}$, $\mathbf{WE}$. A pair $(h_0, h_1)$ of bijections $h_\iota$: $ \lbrace 1, \ldots , N \rbrace \rightarrow \mathcal{E}$ is called the *SZS-pair assigned to* $\mathcal{C}$ if the following conditions hold.
- The restrictions of range $h_\iota$ to $\text{clos } \mathbf{W}$ form an SZ-pair $(h_0, h_1)$, in the closed Western hemisphere. The analogous restrictions form a ZS-pair $(h_0,h_1)$ in the closed Eastern hemisphere $\text{clos } \mathbf{E}$. See definition \[def:5.1\].
- In the notation of definition \[def:5.1\](iv) for the descendants $w_\pm^\iota$ of $\mathcal{O}$, and for each $\iota \in \{0,1\}$, the permutation $h_\iota$ traverses $w_-^\iota, \mathcal{O}, w_+^\iota$, successively.
The swapped pair $(h_1,h_0)$ is called the *ZSZ-pair of* $\mathcal{C}$.
See fig. \[fig:1.0\] for a specific example. Condition (i) identifies the closed hemispheres $\mathbf{W}=\Sigma_-^2(\mathcal{O})$ and $\mathbf{E}=\Sigma_+^2(\mathcal{O})$ as the signed Thom-Smale dynamic complexes of planar Sturm attractors. Note how opposite hemispheres receive opposite planar orientation, in fig. \[fig:1.0\](b). As a consequence, any shared meridian edge $c_v$ in $\Sigma_\pm^1(\mathcal{O})$ receives the same sign from the planar orientation of its two adjacent faces in either signed hemisphere.
Given the Sturm signed Thom-Smale complex of fig. \[fig:1.0\](b), with the orientation of the 1-skeleton induced by the poles $\mathbf{N}=1$ and $\mathbf{S}=9$, we thus arrive at the SZS-pair $(h_0,h_1)$ indicated there. The meander in fig. \[fig:1.0\](c) is based on the Sturm permutation $\sigma=h_0^{-1}\circ h_1$, as usual.
In summary, theorem \[thm:1.2\] and, for 3-cell templates equivalently, definition \[def:5.4\] reconstruct the same generating Hamiltonian paths $h_0, h_1$, and hence the same generating Sturm permutation, of any 3-cell template.
![ *The Sturm tetrahedron 3-ball with 2+2 faces in the hemispheres $\mathbf{W} = \Sigma_-^2(\mathcal{O})$ and $\mathbf{E} = \Sigma_+^2(\mathcal{O})$. (a) Equilibrium labels $\mathcal{E}=\{1,\ldots, 15\}$, bipolar orientation of the 1-skeleton with poles $\mathbf{N}=1$, $\mathbf{S}=4$ which are edge-adjacent along the meridian circle (green), and hemisphere decomposition. The face with barycenter 14 is drawn as the exterior, in the 1-point compactification of the plane. The meridians are indicated as $\mathbf{EW} = \Sigma_-^1(\mathcal{O})$ and $\mathbf{WE} = \Sigma_+^1(\mathcal{O})$. See the legend for the predecessors and successors $w_\pm^\iota$ of $\mathcal{O}=15$. (b) The SZS-pair of Hamiltonian paths $h_0$ (red) and $h_1$ (blue). Here we identify the right and left copies of the meridian $\mathbf{EW}$. See definition \[def:5.3\], for the $h_\iota$ predecessors and successors $w_\pm^\iota$ of $\mathcal{O}$, and definition \[def:5.4\], for the remaining paths in the respective hemispheres. See also for the resulting paths $h_0, h_1$. (c) The dissipative Morse meander defined by the label-independent Sturm permutation $\sigma=h_0^{-1}\circ h_1$; see also .* []{data-label="fig:5.2"}](figs/{fig5_2}.pdf){width="100.00000%"}
![ *A signed regular tetrahedron 3-ball complex which is not Sturm. We use the same decomposition into hemispheres $\mathbf{W} = \Sigma_-^2(\mathcal{O})$ and $\mathbf{E} = \Sigma_+^2(\mathcal{O})$ with 2+2 faces as in fig. \[fig:5.2\]. Only the edge-adjacent poles have been replaced by $\mathbf{N}=1$, $\mathbf{S}=3$, which are not edge-adjacent along the meridian circle (green). (a) Adapted bipolar orientation of the 1-skeleton, and hemisphere decomposition. Only the orientation condition (iii) of definition \[def:5.1\] is violated, necessarily, by the orientation of edge 10 in the hemisphere $\mathbf{E}$. (b) The SZS-pair $h_0$ (red) and $h_1$ (blue), constructed according to definitions \[def:5.3\] and \[def:5.4\], still provides Hamiltonian paths . (c) The dissipative Morse meander defined by the label-independent Sturm permutation $\sigma=h_0^{-1}\circ h_1$; see also . By [@firo3d-3], the original signed regular tetrahedron 3-ball complex is not Sturm. Therefore the Sturm permutation $\sigma$ necessarily fails to describe the original non-Sturm signed complex (a). Instead, $\sigma$ describes a Sturm signed Thom-Smale complex which is not a 3-ball.* []{data-label="fig:5.3"}](figs/{fig5_3}.pdf){width="100.00000%"}
In the general case, not restricted to 3-balls, we have assumed that the signed regular complex $\mathcal{C}=\mathcal{C}_f$ is presented as a signed Thom-Smale complex, from the start. In particular, all hemisphere signs were given by the zero number. We have then described the precise relation between that signed complex $\mathcal{C}=\mathcal{C}_f$ and the boundary orders, at $x=\iota=0,1$, of the paths $h_\iota$ traversing it. In particular we have proved that the signed Thom-Smale complex $\mathcal{C}=\mathcal{C}_f$ determines the Sturm permutation $\sigma=\sigma_f$, uniquely. Conversely, abstract Sturm permutations determine their signed Thom-Smale complex, uniquely. See [@firo96; @firo00; @firo13; @firo3d-1]. This provides a 1-1 correspondence between Sturm permutations and signed Thom-Smale complexes.
In general, however, we are still lacking a geometric characterization of those signed regular cell complexes $\mathcal{C}$ which arise as Sturm Thom-Smale complexes $\mathcal{C}=\mathcal{C}_f$. Indeed, the characterization by theorem \[thm:5.2\] covers 3-cell templates $\mathcal{O}$, only.
Three difficulties may arise in an attempt to realize a given signed regular cell complex $\mathcal{C}$ as a Sturm complex $\mathcal{C}=\mathcal{C}_f$. First, the recipe of theorem \[thm:1.2\] might fail to provide Hamiltonian paths $h_0,h_1$. For example, the same barycenter $w$ of an $(n-1)$-cell may be identified as the successor $w_+^\iota$ of the barycenters $\mathcal{O}$ and $\mathcal{O}'$ of two different $n$-cells, for the same directed path $h_\iota$. Or that “path” might turn out to contain additional cyclic connected components. Second, even if both paths turn out to be Hamiltonian, from “source” $\mathbf{N}$ to “sink” $\mathbf{S}$, the resulting permutation $\sigma = h_0^{-1}\circ h_1$ may fail to define a Morse meander – precluding any realization in the Sturm PDE setting . Third, and even if we prevail against both obstacles, we will have to prove that the lucky signed regular original complex $\mathcal{C}$ coincides, isomorphically, with the signed Thom-Smale complex $\mathcal{C}_f$ associated to the thus constructed Sturm permutation $\sigma=\sigma_f$.
Let us corroborate the above speculations by three specific examples. Our first example, fig. \[fig:5.2\], recalls the unique Sturm tetrahedron 3-ball with 2+2 faces in the hemispheres $\Sigma_\pm^2 = \Sigma_\pm^2(\mathcal{O})$, alias $\mathbf{W}$ and $\mathbf{E}$; see the detailed discussion in [@firo3d-3]. For such a hemisphere decomposition of the 3-ball tetrahedron, there exists only one signed Thom-Smale complex which complies with all requirements of definition \[def:5.1\]; see fig. \[fig:5.2\](a). In particular, both, the edge-adjacent location, along the meridian circle, of the poles $\Sigma_\pm^0 = \Sigma_\pm^0(\mathcal{O})$, alias $\mathbf{N}$ and $\mathbf{S}$, and the bipolar orientation are then determined uniquely, up to geometric automorphisms of the tetrahedral complex and trivial equivalences.
In fig. \[fig:5.2\](b) we construct the resulting SZS-pair of Hamiltonian paths $h_0,h_1$. We follow the practical recipes of definition \[def:5.3\], for the $h_\iota$ predecessors and successors $w_\pm^\iota$ of $\mathcal{O}$, and of definition \[def:5.4\], for the remaining paths in the respective hemispheres. With the labels $\mathcal{E}=\{1,\ldots,15\}$ of equilibria in fig. \[fig:5.2\], the resulting paths $h_\iota:\ \{1,\ldots,15\}\rightarrow \mathcal{E}$ are $$\begin{aligned}
h_0: \,\, 1\;\; &\text{5 11 6 12 15 14 7 2 10 13 8 3 9 4}\,;\\
h_1: \,\, 1\;\; &\text{7 2 8 12 6 3 9 11 15 13 10 14 5 4}\,.
\end{aligned}
\label{eq:5.2}$$ For the label-independent Sturm permutation $\sigma=h_0^{-1}\circ h_1$ we therefore obtain the dissipative Morse meander of fig. \[fig:5.2\](c), for the 2+2 decomposed Sturm tetrahedron 3-ball: $$\begin{aligned}
\sigma
&= \lbrace 1,\text{8, 9, 12, 5, 4, 13, 14, 3, 6, 11, 10, 7, 2, 15}\rbrace =\\
&= \text{(2 8 14) (3 9) (4 12 10 6) (7 13)}\,.
\end{aligned}
\label{eq:5.3}$$
Our second example, fig. \[fig:5.3\], starts from a minuscule variation (a) of the same signed tetrahedral 3-ball. We only move the South pole $\mathbf{S}$ away from the position 4, which is edge-adjacent to $\mathbf{N}=1$ along the meridian circle. The new, more “symmetric” location 3 of $\mathbf{S}$ is not edge-adjacent to $\mathbf{N}$ along the meridian circle. We keep the 2+2 hemisphere decomposition unchanged, and only adjust the bipolarity of the 1-skeleton accordingly. By tetrahedral symmetry our orientation of the edge 10, from 2 to 4, is not a restriction. Note however that any orientation of edge 10 now violates the orientation condition (iii) of definition \[def:5.1\] in the hemisphere $\mathbf{E}=\Sigma_+^2$. All other requirements of definition \[def:5.1\], including the overlap condition (iv), are satisfied.
In fig. \[fig:5.3\](b) we construct the resulting paths $h_0,h_1$ from the practical recipes of definitions \[def:5.3\] and \[def:5.4\], as before, with the usual labels of equilibria. This time, we obtain $$\begin{aligned}
h_0: \,\, 1\;\; &\text{5 14 7 2 10 4 9 11 6 12 15 13 8 3}\,;\\
h_1: \,\, 1\;\; &\text{7 2 8 12 6 11 15 13 10 14 5 4 9 3}\,.
\end{aligned}
\label{eq:5.4}$$ For the Sturm permutation $\sigma=h_0^{-1}\circ h_1$ we therefore obtain the dissipative Morse meander of fig. \[fig:5.2\](c): $$\begin{aligned}
\sigma
&= \lbrace 1,\text{4, 5, 14, 11, 10, 9, 12, 13, 6, 3, 2, 7, 8, 15}\rbrace =\\
&= \text{(2 4 14 8 12) (3 5 11) (6 10) (7 9 13)}\,.
\end{aligned}
\label{eq:5.5}$$
The Sturm global attractor $\mathcal{A}_f$ which results from that Sturm permutation $\sigma=\sigma_f$, however, is not a tetrahedral 3-ball. In fact, $\mathcal{A}_f$ is not a 3-ball at all. We prove this indirectly: suppose $\mathcal{A}_f$ is a 3-ball with $\mathcal{O}=15$. Consider the $h_0$-successor $w_+^0= 13$ of $\mathcal{O}=15$, of Morse index $i(13)=2$; see and fig. \[fig:5.3\](c). By corollary \[cor:4.4\] in a 3-ball, the $h_0$-successor 13 of $\mathcal{O}=15$ must coincide with the $h_1$ most distant equilibrium from $\mathcal{O}=15$, in $\mathcal{E}_+^2(\mathcal{O})$. Since 2 is even, however, that $h_1$-last equilibrium in $\mathcal{E}_+^2(\mathcal{O})$ is easily identified by its label 14. This contradiction shows that $\sigma=\sigma_f$ from is not a tetrahedral 3-ball. Alternatively to this indirect proof we could also have shown blocking of any heteroclinic orbit from $\mathcal{O}=15$ to the face equilibrium 14, based on zero numbers.
In fact we should have expected such failure: our construction of $h_0, h_1$ in theorem \[thm:1.2\] is based on a signed cell complex which is assumed to be a signed Thom-Smale complex of Sturm type.
![ *A signed regular octahedron 3-ball complex with antipodal poles $\mathbf{N}=1$ and $\mathbf{S}=6$. By [@firo14; @firo3d-1; @firo3d-3], there does not exist any Sturm octahedron complex with antipodal poles. See figs. \[fig:5.2\], \[fig:5.3\] for our general setting and notation. (a) Equilibria $\mathcal{E} =\{1,\ldots,27\}$, bipolar orientation of the 1-skeleton, and hemisphere decomposition $\mathbf{W}, \mathbf{E}$ into 4+4 faces, one exterior. Only the overlap condition (iv) of definition \[def:5.1\] is violated by the faces of the two pairs $w_-^\iota, w_+^{1-\iota}$, respctively. (b) The SZS-pair $h_0$ (red) and $h_1$ (blue), constructed according to definitions \[def:5.3\] and \[def:5.4\], provides Hamiltonian paths. See also for the resulting paths $h_0, h_1$. (c) The involutive permutation $\sigma=h_0^{-1}\circ h_1$ of is dissipative and Morse, but fails to define a meander. There are 16 self-crossings. Therefore the signed regular octahedron 3-ball complex (a) with antipodal poles fails to define a Sturm signed Thom-Smale complex.* []{data-label="fig:5.4"}](figs/{fig5_4}.pdf){width="100.00000%"}
Our third and final example, fig. \[fig:5.4\], applies our path construction to an octahedral 3-ball. I ndeed fig. \[fig:5.4\](a) prescribes a signed octahedron complex with diagonally opposite poles $\mathbf{N}=1$ and $\mathbf{S}=6$. In [@firo3d-1; @firo3d-3], however, we have shown that there does not exist any Sturm signed Thom-Smale octahedral complex with diagonally opposite poles. See also [@firo14] for this phenomenon. So our construction is asking for trouble, again. To be specific we choose a symmetric decomposition into hemispheres $\mathbf{W}, \mathbf{E}$ with 4+4 faces, as indicated in fig. \[fig:5.4\](a). All edge orientations in the bipolar 1-skeleton are then determined to satisfy conditions (i)–(iii) of definition \[def:5.1\]. Only the overlap condition (iv) has to be violated, this time. See fig. \[fig:5.4\](b).
Without difficulties, the practical recipes of definitions \[def:5.3\] and \[def:5.4\] provide Hamiltonian paths $h_0,h_1$, as before, with the equilibrium labels indicated in \[fig:5.4\](a), (b): $$\begin{aligned}
h_0: \text{1 7 21 8 3 11 2 12 22 10 19 9 20 27 23 13 26 16 25 17 4 15 5 14 24 18 6}\,;\\
h_1: \text{1 17 20 8 3 9 4 18 19 10 22 11 21 27 24 15 25 16 26 7 2 13 5 14 23 12 6}\,.
\end{aligned}
\label{eq:5.6}$$ For the permutation $\sigma=h_0^{-1}\circ h_1$ we therefore obtain the involution $$\begin{aligned}
\sigma
&= \lbrace 1,\text{ 20, 13, 4, 5, 12, 21, 26, 11, 10, 9, 6, 3, 14,}\\
&\phantom{=\lbrace 1,\ \,}\text{ 25, 22, 19, 18, 17, 2, 7, 16, 23, 24, 15, 8, 27}\rbrace =\\
&= \text{(2 20) (3 13) (6 12) (7 21) (8 26) (9 11) (15 25) (16 22) (17 19)}\,.
\end{aligned}
\label{eq:5.7}$$ This time, however, due to the violation of the overlap condition in definition \[def:5.1\](iv), the permutation $\sigma$ does not define a meander. See fig. \[fig:5.4\](c) for the 16 resulting self-crossings generated by the permutation $\sigma$.
In conclusion we see how the recipe of theorem \[thm:1.2\], for the construction of the unique Hamiltonian boundary orders $h_0,h_1$ and the unique associated Sturm permutation $\sigma=h_0^{-1}\circ h_1$, works well for signed regular complexes $\mathcal{C}$ – provided that these complexes are the signed Thom-Smale complexes of a Sturm global attractor, already. In other words, there is a 1-1 correspondence between Sturm permutations and Sturm signed Thom-Smale complexes. For non-Sturm signed regular complexes, however, the construction recipe for $h_0,h_1$ may fail to provide a Sturm permutation $\sigma=h_0^{-1}\circ h_1$. This was the case for the octahedral example of fig. \[fig:5.4\]. But even if the construction of a Sturm permutation $\sigma$ succeeds, by our recipe, the result will – and must – fail to produce the naively intended Sturm realization of the prescribed non-Sturm signed regular complex. This was the case for the second tetrahedral example of fig. \[fig:5.3\]. The goal of a complete geometric description of all Sturm signed Thom-Smale complexes, as abstract signed regular complexes, therefore requires a precise geometric characterization of the Sturm case, on the cell level. Only for planar cell complexes, and for 3-balls, has that elusive goal been reached, so far.
[999999999]{}
|
---
author:
- |
Robin Leadbeater\
\
Three Hills Observatory, UK\
[email protected]\
\
Published in proceedings of\
“Stellar Winds in Interaction”, editors T. Eversberg and J.H. Knapen.\
Full proceedings volume is available on http://www.stsci.de/pdf/arrabida.pdf
title: |
**The International\
Epsilon Aurigae Campaign 2009-2011.\
A description of the campaign\
and early results to May 2010**
---
PS. @plain[mkbothoddheadoddfoot[Workshop “Stellar Winds in Interaction” Convento da Arrábida, 2010 May 29 - June 2]{}evenheadevenfootoddfoot]{} \#1
\#1
Background
==========
In early 2009, immediately following the end of the WR140 periastron campaign (see these proceedings), I turned my telescope back to $\epsilon$Aurigae in time for the start of the eclipse. As well as being an interesting object in its own right, the Pro-Am campaign being run on $\epsilon$Aurigae during the current eclipse is a good example of how amateur spectroscopists can make a useful contribution. $\epsilon$Aurigae is a naked eye magnitude 3 star and was first noted to be variable by Johan Frisch in 1821. It was subsequently found to be an eclipsing binary with a period of 27.1 years which undergoes an approximately 2 year long flat-bottomed eclipse with approximately 0.8 magnitude drop in $V$ (Fig. \[lead1\], note also an apparent brightening around mid eclipse in this light curve from the last eclipse.)
![\[lead1\] $V$-band light curve of $\epsilon$Aurigae during the last eclipse (Hopkins 1987; R. Stencel, private communication).](lead1.jpg){height="6cm"}
All the visible light from the system appears to come from one component (an F-type star) with no visible light from the eclipsing object, which clearly cannot be a normal star given the large size and very low luminosity in the visible part of the spectrum. Since we see only one component, there is not enough data for a complete orbital solution so there has been plenty of room for speculation over the years as to the size, mass and nature of the components and the scale of the system. To make life more difficult, the F star also shows variability outside eclipse both in brightness and in its spectral lines. Every 27 years a new generation of astronomers with a new generation of instruments try to solve the mystery. This time amateurs equipped with high resolution spectrographs are joining in the fun.
Review of early results from the current eclipse
================================================
- Eclipse began - 17th Aug 2009
- “2nd contact" - 9th Jan 2009
- Mid-eclipse - 4th Aug 2010
- “3rd contact" - 19th Mar 2011
- Eclipse ends - 13th May 2011?
The star is currently (May 2010) in eclipse, with 1st contact having occurred 17th August 2009 and mid eclipse due in about 2 months time (4th August 2010). Already during this eclipse there have been some announcements which have changed our thinking on the system. The so called “high mass" model was the common consensus model up to the end of last year (2009), with an F supergiant eclipsed by a disk of opaque material with possibly at least 2 B stars hidden at the centre to take care of the mass-luminosity discrepancy (Fig. \[lead2\]).
This view has now been challenged following publication of a combined UV/Visual/IR spectrum by Hoard, Howell, & Stencel (2010; hereafter HHS) in which components from a single B5 star in the UV, a cool disk in the IR and the F star in the visible have been identified. The inference from this is that the F star is not a supergiant but a highly evolved object (post AGB?) with low mass but large diameter. The disk would then contain a single main sequence B star. The scale of the system is also reduced by about 40% compared with the high mass model (Figs. \[lead2\] and \[lead3\]).
![\[lead2\] Modifications to the consensus model following the HHS paper.](lead2.jpg){height="6cm"}
![\[lead3\] Taming the Invisible Monster: System parameter constraints for $\epsilon$Aurigae from the Far-ultraviolet to the Mid-infrared (Hoard et al. 2010).](lead3.jpg){height="6cm"}
Also just announced in Nature this April (2010) are remarkable direct observations of the disk crossing the F star made by Kloppenborg et al. (2010) using the CHARA interferometry system (Fig. \[lead4\]). The scale of the system and rate of movement measured from the interferometry is consistent with the HHS low mass model.
![\[lead4\] Infrared images of the transiting disk in the $\epsilon$Aurigae system (Kloppenborg et al. 2010).](lead4.jpg){height="4cm"}
The role of amateurs—Photometry
===============================
![\[lead5\] Hopkins et al. - International epsilon Aurigae campaign 2009-11 (http://www.hposoft.com/Campaign09.html) ](lead5.jpg){height="8cm"}
So what part are amateurs playing during this eclipse? An international campaign is running to provide information exchange and to collect amateur data. This is being organised by professional Bob Stencel and amateur Jeff Hopkins, who also collaborated during the last eclipse, Jeff Hopkins coordinating the collection of photometric data. I have joined them in the past few months to coordinate the amateur spectroscopy side of things. (There is also an international outreach/education project “Citizen Sky" being run by AAVSO, aimed at non-specialists interested in finding out how scientific research is done but I will be concentrating on the work being done by experienced amateurs in this paper.)
By filtering out the medium-timescale brightness variations which are also seen outside eclipse, an estimate of 2nd contact can be made from the $V$ mag curve for the eclipse to date (Fig. \[lead5\]).
If the CHARA images are superimposed, however, it is immediately clear that the point identified as 2nd contact is not the traditionally defined point when the disk reaches the far edge of the star. Instead it likely corresponds to the point when the bottom half of the star is completely obscured. This explains the anomalously long ingress period ($\sim$140 days from the photometry compared with $\sim$90 days based on the star diameter and the orbital velocity of the system).
The role of amateurs—Spectroscopy
=================================
What about $\epsilon$Aur spectroscopically during eclipse? There are currently 12 amateur observers from five countries contributing spectra (Table \[sources\]). Most are following specific lines at $R\sim$18000 using the LHIRES III spectrograph but two observers are providing wide wavelength coverage at $R\sim$12000 using the new eShel fibre fed echelle spectrograph (see T. Eversberg, these proceedings).
I have recently set up a web page giving access to the spectra[^1] and to date (May 2010) there are over 330. All the amateur data are made freely available for research purposes with conditions similar to those of the AAVSO, i.e., the campaign should be mentioned in papers and the observers included as co-authors if their data form a significant part of the paper. Figure \[lead7\] shows the coverage to date.
Lothar Schanne Germany LHIRES III
------------------- ---------- ----------------------
Jose Ribeiro Portugal LHIRES III
Robin Leadbeater UK LHIRES III
Christian Buil France eShel
Olivier Thizy France eShel
Benji Mauclaire France LHIRES III
Thiery Garrel France LHIRES III
Francois Teyssier France LHIRES III (low res)
Brian McCandless USA SBIG SGS
Jeff Hopkins USA LHIRES III
Stan Gorodenski USA LHIRES III
Jim Edlin USA LHIRES III
: List of amateurs involved in the $\epsilon$Aur campaign. \[sources\]
![\[lead7\] Spectroscopic coverage by amateurs during the current eclipse.](lead7.jpg){height="5cm"}
The wide coverage of the observations made using the eShel echelle spectrograph stand out compared with the narrow range of the LHIRES. The main lines covered are H$\alpha$ and Sodium D. The set of results at 7700[Å]{} are mine and I will talk about them in more detail.
My observatory (Three Hills - Fig. \[lead8\]) is located in the far north west of England and is housed in small plastic shed. The top half hinges off to reveal the telescope. It is controlled remotely from the house by wireless link. (There is no eyepiece and no room for a human observer!)
![\[lead8\] LHIRES III spectrograph at Three Hills Observatory.](lead8.jpg){height="6cm"}
The location is good for observing $\epsilon$Aur which is circumpolar from my latitude. I also have a good horizon to the north which is important as $\epsilon$Aur is only 8 deg altitude in June. The LHIRES III spectrograph is attached to the 280mm aperture Celestron C11 telescope (this is an upgrade from the WR140 campaign when I used a 200mm aperture telescope.). During eclipse the F star acts like a searchlight revealing different parts of the eclipsing object in silhouette as it moves past. This is depicted in Fig. \[lead9\].
![\[lead9\] Sketch of opaque and semi-opaque regions in the rotating eclipsing disk just after second contact (the actual distribution of material in the disk is currently unknown).](lead9.jpg){height="2cm"}
As well as the opaque regions which are responsible for producing the light curve, the eclipsing object also has semi-opaque gaseous regions which produce an absorption spectrum when in front of the F star. By measuring the intensities and radial velocities of lines in the spectrum it is potentially possible to track the distribution and velocity profile of the gas component throughout the eclipsing object as it transits in front of the F star. The absorption features from the eclipsing disk are superimposed on the spectrum of the F star. Because the F star spectrum is variable and most of the lines are common between F star and disk, the removal of the F star features can only be approximate, but the result is a spectrum of narrow lines from the disk. By measuring the radial velocities (RVs) of the disk lines it is possible to identify the rotation of the disk. This was done during the last eclipse, for example, by Lambert & Sawyer (1986; hereafter L&S) who tracked the neutral potassium line at 7699[Å]{}. The change in RV from positive to negative from ingress to egress is clear (Fig. \[lead10\]).
The H$\alpha$ line is an obvious line to follow but analysis is further complicated by variable emission wings which are seen outside eclipse, assumed to come from circumstellar material around the F star (a disk or outflowing wind?). The red and blue emission components swing up and down in intensity quasi-periodically and occasionally the absorption line has been seen to diminish significantly (Schanne 2007). Despite this, the emergence of an additional increasingly redshifted absorption component during this eclipse is clear (Fig. \[lead11\]).
(7,6) ![\[lead11\] H$\alpha$ line profile evolution during ingress.](lead10.jpg "fig:"){width="7cm"}
(7,6) ![\[lead11\] H$\alpha$ line profile evolution during ingress.](lead11.jpg "fig:"){width="7cm"}
I decided to concentrate on the K[i]{} 7699[Å]{} line that L&S had studied. The main reason for choosing this line is that it is absent from the spectrum outside eclipse, so the effect of the eclipsing object can be observed directly (there is a small constant interstellar component which can be removed). It is surrounded by O2 telluric lines, but fortunately they are far enough away not to interfere. In fact, they make useful calibration markers. To reach 7699[Å]{} at maximum resolution I had to modify the grating adjustment on the spectrograph. At this wavelength the 2400 l/mm grating is set at a very low angle which gives good dispersion but low efficiency. My aim was to look in more detail at the evolution of the line profile by taking many more spectra than L&S had been able to do (89 spectra to date, one every 4-5 days on average compared with 8 by L&S in the same period. The false colour contour plot shows the evolution of the line (Fig. \[lead12\]).
Note that the eclipsing object appeared in the spectrum over 2 months before the brightness started dropping. The line gets wider and the core of the line first moves slightly to the red and is currently (May 2010) moving to the blue. A preliminary analysis of what has been happening in this line has been published (Leadbeater & Stencel 2010). By measuring the maximum radial velocity at the red edge of the line we can measure the orbital velocity of the fastest rotating component in front of the F star during the eclipse (Fig. \[lead13\]).
(6,6) ![\[lead13\] Neutral Potassium 7699[Å]{} line profile evolution during ingress.](lead12.jpg "fig:"){width="6cm"}
(6,6) ![\[lead13\] Neutral Potassium 7699[Å]{} line profile evolution during ingress.](lead13.jpg "fig:"){width="6cm"}
As expected for a Keplerian disk, this velocity is low at the outer edge of the disk but increases as the smaller-radius parts of the disk move in front of the F star. These data can be used to estimate the central mass, giving a result which is consistent with the value published in the HHS paper (Fig. \[lead14\]).
![\[lead14\] Radial velocity of the red edge of the 7699[Å]{} line during ingress compared with that predicted for a Keplerian disk.](lead14.jpg){height="5cm"}
By plotting the equivalent width of the eclipsing object component of the line, we see that the development is not smooth but takes place in a series of steps, marked by letters (Fig. \[lead15\], the first two of which occur before the decrease in brightness).
![\[lead15\] Excess equivalent width of 7699[Å]{} line during ingress (out of eclipse component removed).](lead15.jpg){height="5cm"}
These steps were confirmed by some observations made by Apache Point Observatory using the 3.5m ARC telescope (W. Ketzeback, private communication), though they were unable to cover the initial emergence of the K[i]{} component. The steps are assumed to represent changes in density in the disk profile and it is difficult to resist speculating that the disk might have a ring-like structure, though these may be spiral arms or arcs (Fig. \[lead16\]).
![\[lead16\] Schematic of the postulated density variations in the eclipsing disk.](lead16.jpg){height="3cm"}
(6,6) ![\[lead18\] Latest H$\alpha$ spectra.](lead17.jpg "fig:"){width="6cm"}
(6,6) ![\[lead18\] Latest H$\alpha$ spectra.](lead18.jpg "fig:"){width="6cm"}
Latest results
==============
There have been differences in the behaviour of the lines on the approach to mid eclipse. The K[i]{} 7699 line started moving to the blue in March followed by the Na D lines in May but as of May 2010 the H$\alpha$ line continues to move to the red (Figs. \[lead17\] to \[lead20\]).
![\[lead20\] Latest Na D spectra.](lead20.jpg){height="6cm"}
Prospects for the rest of the eclipse
=====================================
To bring the story right up to date, there are also reports that the brightness is now increasing but caution is needed to make sure the correct extinction corrections have been applied now $\epsilon$ Aur is at low elevation (not a problem with the spectra which are normalised to the continuum.) The next month or two will be difficult observing conditions owing to the low elevation and twilight. The picture shows an actual observing run at Three Hills at solar conjunction last June (Fig. \[lead21\]). There are fewer tree branches this year!
![\[lead21\] Observing $\epsilon$Aurigae at Three Hills Observatory during solar conjunction 2009.](lead21.jpg){height="6cm"}
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to thank Dr Bob Stencel, University of Denver and Jeff Hopkins, the organisers of the international epsilon Aurigae campaign and all observers who have submitted data to the campaign. Thanks also to the epsilon Aurigae spectral monitoring team at Apache Point Observatory (W. Ketzeback, J.Barentine, et al.) for allowing us to view their 7699A line data.
Hoard, D. W., Howell, S. B., & Stencel, R. E. 2010, ApJ, 714, 549
Hopkins, J. L. 1987, IAPPP, 27, 30
Kloppenborg, B., et al. 2010, Nature, 464, 870
Lambert, D., Sawyer, S., 1986, PASP, 98, 389
Leadbeater, R., & Stencel, R., 2010, arXiv:1003.3617
Schanne, L., 2007, IBVS, 5747, 1
[^1]: www.threehillsobservatory.co.uk/espsaur\_spectra.htm
|
---
abstract: 'We present comprehensive observations and analysis of the energetic H-stripped SN2016coi (a.k.a. ASASSN-16fp), spanning the $\gamma$-ray through optical and radio wavelengths, acquired within the first hours to $\sim$420 days post explosion. Our observational campaign confirms the identification of He in the SN ejecta, which we interpret to be caused by a larger mixing of Ni into the outer ejecta layers. From the modeling of the broad bolometric light curve we derive a large ejecta mass to kinetic energy ratio ($M_{\rm{ej}}\sim 4-7\,\rm{M_{\odot}}$, $E_{\rm{k}}\sim 7-8\times 10^{51}\,\rm{erg}$). The small \[\] 7291,7324 to \[\] 6300,6364 ratio ($\sim$0.2) observed in our late-time optical spectra is suggestive of a large progenitor core mass at the time of collapse. We find that SN2016coi is a luminous source of X-rays ($L_{X}>10^{39}\,\rm{erg\,s^{-1}}$ in the first $\sim100$ days post explosion) and radio emission ($L_{8.5\,GHz}\sim7\times 10^{27}\,\rm{erg\,s^{-1}Hz^{-1}}$ at peak). These values are in line with those of relativistic SNe (2009bb, 2012ap). However, for SN2016coi we infer substantial pre-explosion progenitor mass-loss with rate $\dot M \sim (1-2)\times 10^{-4}\,\rm{M_{\odot}yr^{-1}}$ and a sub-relativistic shock velocity $v_{sh}\sim0.15c$, in stark contrast with relativistic SNe and similar to normal SNe. Finally, we find no evidence for a SN-associated shock breakout $\gamma$-ray pulse with energy $E_{\gamma}>2\times 10^{46}\,\rm{erg}$. While we cannot exclude the presence of a companion in a binary system, taken together, our findings are consistent with a massive single star progenitor that experienced large mass loss in the years leading up to core-collapse, but was unable to achieve complete stripping of its outer layers before explosion.'
author:
- 'G. Terreran'
- 'R. Margutti'
- 'D. Bersier'
- 'J. Brimacombe'
- 'D. Caprioli'
- 'P. Challis'
- 'R. Chornock'
- 'D. L. Coppejans'
- Subo Dong
- 'C. Guidorzi'
- 'K. Hurley'
- 'R. Kirshner'
- 'G. Migliori'
- 'D. Milisavljevic'
- 'D. M. Palmer'
- 'J. L. Prieto'
- 'L. Tomasella'
- 'P. Marchant'
- 'A. Pastorello'
- 'B. J. Shappee'
- 'K. Z. Stanek'
- 'M. D. Stritzinger'
- 'S. Benetti'
- Ping Chen
- 'L. DeMarchi'
- 'N. Elias-Rosa'
- 'C. Gall'
- 'J. Harmanen'
- 'S. Mattila'
bibliography:
- 'terreran.bib'
title: 'SN2016 (ASASSN-16): an energetic H-stripped core-collapse supernova from a massive stellar progenitor with large mass loss'
---
Introduction {#sec:intro}
============
{width="\textwidth"}
Hydrogen-stripped core-collapse supernovae (i.e., type Ibc SNe), also called stripped-envelope SNe [SESNe; @Clocchiatti1996], have enjoyed a surge of interest in the last two decades thanks to the association of the most energetic elements of the class with Gamma-Ray Bursts (GRBs). Yet, the stellar progenitors of type Ibc SNe have so far eluded uncontroversial detection in pre-explosion images (@Gal-Yam2005 [@Maund2005; @Elias-Rosa2013; @Eldridge2013]). Relevant in this respect is the discovery of a progenitor in pre-explosion images of the type Ib SN iPTF13bvn, interpreted to be a single Wolf-Rayet (WR) star with a mass at zero-age main-sequence (ZAMS) $\sim33$ [@Cao2013; @Groh2013]. This result was later disputed by [@Bersten14]. More recently, [@VanDyk2018], [@Kilpatrick2018] and [@Xiang2019] identified a source in archival *Hubble Space Telescope* (HST) images covering the location of the type Ic SN2017ein, with properties compatible with a WR star of $\sim$$55$ (although the presence of a companion star could not be ruled out).
The stripping of the hydrogen and helium envelope in massive stars mainly occurs through two channels: (i) line-driven winds, which dominate the mass-loss yield in single star evolution; or (ii) interaction with a companion star in a binary system. In the former scenario, the progenitor is expected to be an isolated, massive WR star [$\gtrsim20$; @Hamann2006], consistent with the inferences by [@Cao2013], [@Groh2013], [@VanDyk2018] and [@Kilpatrick2018] with typical mass-loss rate $\dot M\sim 10^{-5}\,\rm{M_{\sun}\,yr^{-1}}$ [@Maeder1981; @Woosley1995; @Begelman1986]. In the binary progenitor scenario, instead, the primary exploding star is expected to be a helium star (or a C+O star in case of type Ic SNe) with lower-mass $\gtrsim12$ [@Podsiadlowski1992; @Yoon2010a; @Eldridge2013; @Dessart2015]. The lower mass of the progenitor stars in the binary progenitor scenario naturally accounts for the discrepancy between the large inferred rate of SESNe compared to massive WR stars [@Georgy2009; @Smith2011; @Eldridge2013; @Smith2014], and for the low ejecta masses inferred from the modeling of the bolometric light-curves of type Ibc SNe [$M_{\rm{ej}}\lesssim 3$ ; e.g., @Ensman1988; @Drout2011; @Dessart2012; @Bersten2014; @Eldridge2015; @Lyman2016; @Taddia2018]. In reality, both scenarios are likely contributing in different amounts to the observed population of SESNe.
[@Yamanaka17] [@Kumar18] [@Prentice17] This Work
---------------------------------- ------------------------- ------------------------- --------------------------- -------------------------
Distance (modulus $\mu$) 17.2 Mpc (31.18 mag) 18.1 Mpc (31.29 mag) 15.8 Mpc (31 mag) $18.1$ Mpc (31.29 mag)
Color Excess $E(B-V)_{\rm{tot}}$ 0.075 mag 0.074 mag 0.205 mag 0.075 mag
Explosion Date MJD 57532.5 MJD 57533.9 MJD 57533.5 MJD 57531.9
Nickel Mass 0.15 0.10 0.14 0.15
Ejecta Mass $\Mej$ 10 $4.5$ $2.4-4$ $4-7$
Kinetic Energy $\Ek$ $3-5\times10^{52}$ erg $6.9\times10^{51}$ erg $4.5-7\times 10^{51}$ erg $7-8\times 10^{51}$ erg
He Velocity $\sim18000$ km s$^{-1}$ $\sim20000$ km s$^{-1}$ $\sim22000$ km s$^{-1}$ $\sim22000$ km s$^{-1}$
\[tab:other\_works\]
Here we present the results from an extensive multi-wavelength campaign of the H-poor (a.k.a. ASASSN-16fp) from $\gamma$-rays to radio wavelengths, from a few hrs to $\sim 420$ days post explosion. From our comprehensive analysis we infer that originated from a compact massive progenitor with large mass loss before explosion, potentially consistent with a single WR progenitor star. was discovered on 2016 May 27.55 UT [@Holoien2016 MJD 57535.55;] by the All Sky Automated Survey for SuperNovae[^1] [ASAS-SN; @Shappee2014] in the irregular galaxy UGC 11868 (Fig. \[Fig:rgb\]). was initially classified by the NOT Unbiased Transient Survey [NUTS; @Mattila2016] as a type Ic-BL SN similar to those that accompany GRBs [@Elias-Rosa2016], although it was soon realized that traces of He might have been present at early times [@Yamanaka2016]. The optical/UV properties of have been studied by [@Yamanaka17], [@Prentice17] and [@Kumar18]. These authors conclude that is an energetic SN with large ejecta mass and spectroscopic similarities to type Ic-BL SNe. In terms of SN classification, is intermediate between type Ib and Ic SNe. Unlike type Ib SNe, where He lines become more prominent with time [e.g., @Gal-Yam2017], the He features of disappear after maximum light.
This paper is organized as follows. We first describe our UV, optical and NIR photometry data analysis and derive the explosion properties through modeling of the SN bolometric emission in §\[sec:phot\]. Our spectroscopic campaign and inferences on the spectral properties of are described in §\[Sec:Spec\]. In §\[sec:Radio\] we present radio observations of , along with the modeling of the blast-wave synchrotron emission, while §\[sec:XRT\] is dedicated to the analysis of the luminous X-ray emission of and the constraints on the progenitor mass-loss history. We describe our search for a shock breakout signal in the $\gamma$-rays in §\[sec:SBO\]. We discuss our findings in the context of properties of potential stellar progenitors in §\[sec:disc\] and draw our conclusions in §\[sec:concl\].
In this paper we follow [@Kumar18] and adopt $z\simeq0.00365$, which corrected for Virgo infall corresponds to a distance of $18.1\pm1.3$ Mpc ($H_{0} = 73$kms$^{-1}$Mpc$^{-1}$, $\Omega_{\rm M} = 0.27$, $\Omega_{\rm \lambda} = 0.73$), equivalent to a distance modulus of $\mu =31.29\pm 0.15$ mag [@Mould2000]. We further adopt a total color excess in the direction of $E(B-V)_{\rm{tot}}=0.075$ mag [@Schlafly11] as in [@Yamanaka17] and [@Kumar18]. Unless otherwise stated, time is referred to the inferred time of first light (§\[sec:phot\]), which is UT May 23.9 2016 (MJD 57531.9; see §\[sec:Lbol\]). The presence of a “dark phase” with a duration of a few hours to a few days (e.g., @Piro13) has no impact on our major conclusions. Therefore we use the term “from explosion” and “from first light” interchangeably. A summary of our adopted and inferred parameters is provided in Table \[tab:other\_works\]. Uncertainties are listed at the $1\,\sigma$ confidence level (c.l.), and upper limits are provided at the $3\,\sigma$ c.l. unless otherwise noted.
UV, Optical and NIR photometry {#sec:phot}
==============================
Data Analysis {#subsec:dataUVoptical}
-------------
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Our photometric data have been obtained from several different telescopes and instruments, which are listed in Table \[tab: instr\]. UV data have been acquired with the Ultraviolet Optical Telescope [UVOT; @Roming2005], on the Neil Gehrels *Swift* Observatory [@Gehrels2004]. We measured the SN instrumental magnitudes by performing aperture photometry with the `uvotsource` task within the <span style="font-variant:small-caps;">HEAsoft</span> v6.22, and following the guidelines in [@Brown2009]. An aperture of 3 was used. We estimated the level of contamination from the host galaxy flux using late-time observations acquired at $t\sim322$ d after first light, when the SN contribution is negligible. We then subtracted the measured count-rate at the location of the SN from the count rates in the SN images following the prescriptions in [@Brown2014].
Images acquired with the Liverpool Telescope have been processed with a custom-made pipeline, while we use standard overscan, bias and flatfielding within <span style="font-variant:small-caps;">iraf</span>[^2] for the remaining optical photometry. NOTCam NIR images were reduced with a modified version of the external <span style="font-variant:small-caps;">IRAF</span> package <span style="font-variant:small-caps;">IRAF</span> (v. 2.5)[^3]. The remaining NIR data reduction has been performed through standard flat-field correction, sky background subtraction and stacking of the individual exposures for an improved signal-to-noise ratio. The photometry has been extracted using the <span style="font-variant:small-caps;">SNOoPY</span>[^4] package. We performed point-spread-function (PSF) photometry with <span style="font-variant:small-caps;">DAOPHOT</span> [@Stetson1987]. For non-detections we calculated upper limits corresponding to a S/N of 3. Zero points and color terms for each night have been estimated based on the magnitudes of field stars in the Sloan Digital Sky Survey[^5] [SDSS; @York2000] catalog (DR9). We converted the SDSS *ugriz* magnitudes to Johnson/Cousins *UBVRI* filters following [@Chonis2008]. For NIR images, we used the Two Micron All Sky Survey (2MASS) catalog[^6] [@Skrutskie2006]. We quantified the uncertainty on the instrumental magnitude injecting artificial stars [e.g., @Hu2011]. The resulting uncertainty was then added in quadrature to the fit uncertainties returned by <span style="font-variant:small-caps;">DAOPHOT</span> and the uncertainties from the photometric calibration to obtain the total uncertainty on the photometry. Our final values are reported in Tables \[fig: ugriz\]$-$\[fig: SDA\] and shown in Fig. \[Fig:snlcall\].
Our UV-to-NIR campaign densely samples the evolution of in its first $\sim400$ days post explosion, with more than 1100 observations distributed over 166 nights (the gap around 200-300 days corresponds to when was behind to the Sun). As Fig. \[Fig:snlcall\] shows, rises to peak considerably faster in the bluer bands. The UV filters also show the fastest decline post peak, before relaxing on a significantly slower decay at $t\gtrsim40$. This sharp change of decay rate is not present in the redder bands, which instead show a roughly constant decay rate after peak. The late-time $V$-band decays as 1.7 mag 100d$^{-1}$, faster than expected from the radioactive decay of , suggesting leakage of $\gamma$-rays. Our last detections of at $\sim373$ d post explosion are consistent with the temporal decay inferred from earlier observations at $50\rm{d}\lesssim t\lesssim300 {d}$ (Fig. \[Fig:snlcall\]). Finally, by using a low-order polynomial fit we measure the time of maximum light in the V band V$_{max}=18.34\pm0.16$ d after discovery, corresponding to MJD 57550.24 (2016 June 11.24 UT). The time of peak in other bands is reported in Table \[tab: max\].
Bolometric Luminosity and Explosion Parameters {#sec:Lbol}
----------------------------------------------
![Comparison of the *uvoir* bolometric light curve of with the sample of SESNe from [@Lyman2016]. The light curves have been normalized to maximum light. is among the objects with the broadest light curve, suggesting a larger than average diffusion time. The broad light curves of SNe 2004aw [which lies exactly below ; @Taubenberger2006], 2005az [@Drout2011] and 2011bm [@Valenti2012] are also marked.[]{data-label="Fig:LCcomparison"}](Ly.png){width="\columnwidth"}
Our extensive photometric coverage allows us to reconstruct the bolometric emission from from the UV to the NIR from a few days to $\sim200$ days after explosion. As a comparison, the bolometric light-curve from [@Prentice17] has similar temporal coverage but does not include the NIR and UV contributions, while [@Kumar18] and [@Yamanaka17] include either the UV emission or the NIR emission until $\delta t\le 60$ days post explosion, respectively. We build the bolometric luminosity curve of starting from extinction-corrected flux densities, and we interpolate the flux densities in each filter to estimate the SN emission at any given time of interest. In case of incomplete UV-to-NIR photometric coverage we assume constant color from the previous closest epochs. Finally, we integrate the resulting spectral energy distributions (SEDs) from the UV to the NIR with the trapezoidal rule to obtain the bolometric light-curve shown in Fig. \[Fig:LCcomparison\].
We compare the bolometric light curve of with a sample of well-observed H-stripped core-collapse SNe from [@Lyman2016] in Fig. \[Fig:LCcomparison\]. [@Lyman2016] used the parameter $\Delta_{15}$ as an estimator of the broadness of the light curve, defined as the the difference in magnitude between the luminosity at peak at the luminosity 15 d after that. The smaller the $\Delta_{15}$, the “slower” the event, i.e., the broader the light curve. The SN with the broadest light curve in the sample of [@Lyman2016] is the type-Ic SN2011bm, which has $\Delta_{15}=$0.2 mag. Other slow events are the type-Ic SNe 2004aw, 2005az (with $\Delta_{15}$=0.41 and $\Delta_{15}$=0.42 mag, respectively) and the type-Ib SNe 1999dn and 2004dk (with $\Delta_{15}$=0.32 and $\Delta_{15}$=0.41 mag, respectively). Figure \[Fig:LCcomparison\] shows that with $\Delta_{15}$=0.41 mag, is among the SNe with the broadest light-curves. [@Kumar18] did a similar analysis looking at the $\Delta_{15}$ in the single bands and reached the same conclusion.
The broad light curve indicates a large photon diffusion time scale, and hence a large ejecta mass ($M_{ej}$) to kinetic energy ($E_{k}$) ratio. Assuming standard energetics, this translates to a considerably large ejecta mass, in agreement with previous findings by [@Prentice17] and [@Kumar18]. Interestingly, shows a very slow post-peak decline with a standard time to peak $t_{rise}<20$ days (Fig. \[Fig:LCcomparison\]). This phenomenology might result from mixing of $^{56}$Ni in the outer stellar ejecta, as opposed of having all the located at the centre of the explosion. We quantify these statements below.
We model the bolometric light-curve of adopting the formalism by [@Arnett82] modified following [@Valenti08] and [@Wheeler15]. We adopt a mean opacity $\kappa_{opt}=0.07\,\rm{cm^{2}g^{-1}}$ and break the model degeneracy using a photospheric velocity $v_{phot}\sim16000\,\rm{km\,s^{-1}}$ around maximum light, as inferred from spectral lines (§\[Sec:Spec\]). We find that during the photospheric phase at $t<30$ days the light-curve is well described by a model with kinetic energy $E_{k,phot}\sim7\times 10^{51}\,\rm{erg}$, mass $M_{Ni,phot}=0.13\,\rm{M_{\sun}}$ and ejecta mass $M_{ej,phot}\sim4\,\rm{M_{\sun}}$, consistent with the findings of [@Kumar18]. However, this model significantly underestimates the bolometric emission during the nebular phase. This is a common outcome of the modeling of energetic type-Ic SN light-curves, which motivated [@Maeda03] and [@Valenti08] to consider a two-component model. In two-component models the “outer component” dominates the early-time emission during the photospheric phase, while the late-time nebular emission receives a significant contribution from a denser inner core (“inner component”). Applying this modeling we find a total ejecta mass $M_{ej}\sim (4-7)\,\rm{M_{\sun}}$, $E_{k}\sim (7-8)\times 10^{51}\,\rm{erg}$ and $M_{Ni}\sim 0.15\,\rm{M_{\sun}}$, with a larger fraction of *Ni* per unit mass in the outer component. This model also allows us to constrain the time of first light to MJD $57531.9\pm 1.5$ days (May 23.9, 2016 UT).
As a comparison, the spectral modeling by [@Prentice17] indicates $M_{ej}=2.4-4\,\rm{M_{\sun}}$, $E_k=(4.5-7)\times 10^{51}\,\rm{erg}$. Scaling the emission of to the GRB-associated SN 2006aj and SN2008D, [@Yamanaka17] find $M_{ej}\sim 10\,\rm{M_{\sun}}$, $E_k=(3-5)\times 10^{52}\,\rm{erg}$ (Table \[tab:other\_works\]). The rough agreement among the results is not surprising given the very different methods used (with different assumptions) and the fact that the modeling of [@Prentice17] is limited to optical spectra, and [@Yamanaka17] only consider the optical/NIR emission of during the early photospheric phase.
Optical Spectroscopy {#Sec:Spec}
====================
Data Analysis {#SubSec:dataAnSpec}
-------------
-0.0 true cm
-0.0 true cm
-0.0 true cm ![A zoomed-in plot of the spectral region of the \[\] 6300,6364, \[\] 7291,7323 and \] 4571 lines of the MMT+BlueChannel nebular spectrum acquired on 2017, June 28 ($\sim400$ d after explosion). Gaussian profiles have been used to reconstruct the doublet components of the emission features. The observed lack of asymmetry of the \[\] emission feature might result from spherically symmetric ejecta, or possibly from an axisymmetric explosion, viewed at an angle below $50^\circ$.[]{data-label="fig:neb"}](neb2.png "fig:"){width="\columnwidth"}
-0.0 true cm {width="\textwidth"}
We obtained optical spectroscopy of from a few days until $t>400$ days post explosion with a variety of instruments on different telescopes (Table \[tab: instr2\]). The spectroscopic log can be found in Table \[fig: spec\_log\]. We extracted our time series of optical spectra with <span style="font-variant:small-caps;">IRAF</span> following standard procedures. Comparison lamps and standard stars acquired during the same night and with the same instrumental setting have been used for the wavelength and flux calibrations, respectively. When possible, we further removed telluric bands using standard stars.
Our spectroscopic campaign comprises 65 spectra (Fig. \[Fig:spec\]). The overall evolution of is similar to that of type-Ic SNe. At early times $t\lesssim60$ d the blue part of the spectrum at $\lambda\lesssim 5500$ Å is dominated by blends of several multiplets. We identify the main spectral feature at $\sim6000$ Å as 6355. Before maximum light, we associate the absorption feature around $\sim5500$ Å to 5876, with possible contamination by D. dominates after maximum light. At $\lambda>7000$ Å the spectra of show emission associated with 7771,7774,7775 and the NIR triplet. Nebular features start to appear $\sim90$ d after explosion, when the forbidden \[\] 6300,6364 and the \[\] 7291,7323 doublets begin to emerge.
In Fig. \[fig:neb\] we show a zoomed-in plot of the nebular spectrum acquired with MMT+BlueChannel at $\sim400$ d after explosion. We plot the region of the forbidden \[\] 6300,6364, \[\] 7291,7323 doublets, and semi-forbidden \] 4571 emission line. We use gaussian profiles to model each emission line. For the doublets, we kept the separation between the two components fixed, while allowing for rigid shifts of the overall profile (this scheme will also be followed in §\[subsec:broad LC\]). This simple approach allows us to adequately reproduce the emission line profiles (Fig. \[fig:neb\]). We find that the ratio between the oxygen lines fluxes is $\sim2.6$, in reasonable agreement with the theoretical expectation of $\sim3$. However, the doublet is blue-shifted by $\sim10$ Å ($\sim400$ kms$^{-1}$). We find similar blue-shifts for the Ca and Mg lines. Blue-shifted oxygen line profiles of this kind are not uncommon in type Ibc SN nebular spectra, and several causes have been invoked to explain this observed phenomenology, including dust obscuration, internal scattering, contamination from other lines or residual opacity in the core of the ejecta [@Modjaz2008b; @Taubenberger2009; @Milisavljevic2010]. We do not observe asymmetric structures in the spectral lines, nor do we detect any sharp decrease in the light curve of , therefore we can confidently exclude the presence of dust [@Elmhamdi2003; @Elmhamdi2004]. As the \[\] forbidden doublet is fairly isolated, we disfavour contamination from other lines as the origin for the blue-shift. The fact that lines of different species show this behaviour might suggest a geometrical effect. An asymmetric explosion with a bulk of material moving towards the observer could indeed cause the blue-shift. Qualitative inferences on the geometry and distribution of the oxygen-rich ejecta in can be drawn from the line profile of the forbidden \[\] 6300,6364 [@Modjaz2008b; @Taubenberger2009; @Milisavljevic2010]. Double-peaked oxygen lines are usually interpreted to be formed in asymmetric explosions viewed at a high angle between the observer point of view and the jet direction [@Maeda08; @Taubenberger2009]. As shown in Fig. \[fig:neb\], the oxygen doublet in presents a single, symmetric profile, reproducible with simple Gaussian functions. This result is consistent with spherically symmetric ejecta. As [@Maeda08] have shown that an asymmetric profile would not develop for asymmetric explosions viewed from angles below $\sim50^\circ$, an asymmetric explosion cannot be ruled out. However, the asymmetric explosion scenario might actually be supported by the red excess visible in both the oxygen and magnesium line. Indeed, magnesium and oxygen are expected to have similar spatial distribution within the SN ejecta [e.g., @Maeda06; @Taubenberger2009]. Such an excess, visible in both features, is unlikely to be caused by line contamination, and is rather the result of ejecta asymmetries common to both line emission regions.
We conclude with a consideration on intrinsic reddening. In our highest resolution spectra acquired on November 2, 2016 UT ($\sim162$ days after first light) with LBT+MODS, we find a weak narrow D absorption at the redshift of the host galaxy, from which we infer $E(B-V)_{\rm{host}}\sim0.017$ mag [@Turatto2003b; @Poznanski2012]. However, given the large uncertainties of this method [@Phillips2013], and the lack of evidence for significant $E(B-V)_{\rm{host}}$, in the following we assume $E(B-V)_{\rm{host}}=0$ mag.[^7] This assumption has no impact on our conclusions. Following [@Schlafly11], the Milky Way color excess in the direction of is $E(B-V)_{\rm{MW}}$=0.075 mag, which we use to correct our spectro-photometric data for extinction.
SN Classification and presence of Helium in the ejecta {#sec: class}
------------------------------------------------------
initially showed spectral similarities to type Ic-BL SNe (and in particular to SN2006aj, associated with GRB060218) but later evolved to resemble a normal type Ic SN (Fig. \[Fig:spec\]). Indeed, both [@Kumar18] and [@Prentice17] identified as being an intermediate object between the two classes, while [@Yamanaka17] classified as a BL-Ib SN, because of the presence of helium in the spectra and expansion velocities larger than in normal type Ib SNe. We quantitatively explore the questions of how the ejecta velocity of compares to other H-stripped SNe, and the presence of He in its ejecta below.
We compare to the spectral templates of normal type Ic SNe and BL-Ic SNe from [@Modjaz2016] in Fig. \[fig:modjaz\] after applying the same renormalization procedure. The result is presented for three different epochs: 10 days before maximum light, around maximum light, and 20 days after maximum light. Fig. \[fig:modjaz\] demonstrates that the typically prominent H+K absorption feature of type Ic SNe spectra is almost absent in (Fig. \[fig:modjaz\], top and middle panels), in closer similarity to type Ic-BL SN spectra. Notably, shows a prominent absorption feature at $\sim6000$ Å that we identify as with $v\sim19000$ km s$^{-1}$, which is typically not present with this strength in normal type Ic SNe [e.g., @Parrent2016].
From our comparison, more closely resembles normal type Ic SNe, especially before maximum light. Compared to type Ic-BL SNe, shows more prominent peaks and troughs (upper panels in Fig. \[fig:modjaz\]), as a result of its lower ejecta velocities before maximum light, which cause less severe blending of the spectral features. Compared to normal type Ic SNe, however, shows systematically blue-shifted spectral features. [@Modjaz2016] showed that in type Ic SNe (both normal and broad-line) the “broadness” of the spectral features correlates with the blue-shift of their minima, as is expected from an expanding atmosphere [e.g., @Dessart2011]. However, with very blue-shifted absorption minima similar to type Ic-BL, but less prominent broadening, seems to deviate from this trend.
[@Modjaz2016] used the $\lambda$5169 to show this correlation between the blue-shift of the minima and the broadening of the absorption feature. We use the same fitting technique as in [@Modjaz2016] to measure the broadening of this same line for at maximum light, obtaining $v_{broad}\sim2380$ km s$^{-1}$. Comparing this value with their Fig. 7, it is possible to see how this is quite low for a BL-Ic, while the velocity inferred from the position of the minimum of the line profile is $v_{min}\sim18050$ km s$^{-1}$, well within the range of the other BL-Ic of their sample. Another event that had very blue-shifted minima but relatively low broadening was PTF12gzk [@Ben-Ami12]. These observations were interpreted as resulting from either the departure from spherical symmetry, or from a steep gradient of the density profile of the progenitor envelope. Interestingly, [@Ben-Ami12] inferred a massive ejecta of 25-35 and a large kinetic energy of $5-10\times 10^{51}$ erg for PTF12gzk, which is comparable to . thus shows spectral properties that are intermediate between type Ic-BL SNe (with which also shares the large kinetic energy $E_{k}>10^{51}\,\rm{erg}$ but lower velocities before peak) and normal type Ic SNe. These results agree with the findings by [@Kumar18] and [@Prentice17].
-0.0 true cm ![*Right panel:* Evolution of the 6355 line before maximum light in the velocity space. The position of the minimum of the absorption feature is marked with a vertical blue short-dashed line. *Left panel:* Evolution of the 5500 Åfeature, assumed to be the 5876 line. The position of the minimum of the absorption feature is marked with a red long-dashed line. The velocity of the absorption minimum of 6355 is also shown for comparison with a blue short dashed line. The match between the evolution of the two lines suggest a correct interpretation of the 5500 Åfeature as the 5876 line. By the time of maximum light, He absorption is no longer apparent in the spectra of , and the presence of He becomes hard to quantify due the possible emergence of the 5890,5896 doublet.[]{data-label="Fig:He_ev"}](He_ev.png "fig:"){width="\columnwidth"}
We next address the presence of He in the ejecta of (in Fig. \[fig:modjaz\] we marked the position of the 4472, 5876 6678 and 7065 lines). We investigate the velocity evolution of the most prominent spectral features among those associated with at 5876 Å in Fig. \[Fig:He\_ev\], and use the velocity evolution inferred from 6355 as a comparison. From Fig. \[Fig:He\_ev\] we find that He and Si show a very similar temporal evolution, with expansion velocities evolving from $v\sim20000\,\rm{km\,s^{-1}}$ at $\sim$2 weeks before maximum light, to $v\sim15000\,\rm{km\,s^{-1}}$ around peak. The identification of He might inspire a connection with type Ib SNe. However, we note that in the He features slowly subside (by the time of maximum light He absorption is no longer prominent, Fig. \[Fig:He\_ev\]), while in type Ib SNe He features develop with time [e.g., @Filippenko97; @Gal-Yam2017]. The presence of He in has been recognized as a peculiar characteristic of by [@Yamanaka17], [@Kumar18] and [@Prentice17]. [@Yamanaka17] concluded the presence of He in based on the comparison with a smoothed out and blueshifted version of the type Ib SN 2012au [@Takaki2013], finding a correspondence with the position of the main helium features. They also cross-checked this result with synthetic spectra generated with the code SYN++ [@Thomas2011]. [@Kumar18] adopted a similar strategy to the one presented in this work, performing a detailed velocity analysis of the single features. With their 1D Monte Carlo spectra synthesis code, [@Prentice17] investigated what other elements could be responsible for the absorption at $\sim5500$ Å. They showed that He is indeed the favored interpretation, and that in the absence of He, unphysical amounts of and would be necessary to reproduce the observed spectra.
Similar velocities between Si-rich and He-rich ejecta indicates a clear departure from the expectations of a homologous explosion of a stratified progenitor star where the outer He-rich layers are expected to expand significantly faster than the inner Si-rich layers of ejecta. This finding suggests a higher level of mixing of the ejecta, which might be connected with the capability to excite He (and hence the detection of He in our spectra).
Radio {#sec:Radio}
=====
-0.0 true cm ![Radio SED of at 10.5, 20.5, 45.5, 106.3 and 269.8 days after first light (see Table \[Tab:radio\]). The radio emission from is well described by a synchrotron self-absorbed spectrum (SSA) with spectral peak frequency $\nu_{pk}\propto t^{-0.97\pm0.02}$ and peak flux $F_{pk}\propto t^{-0.31 \pm 0.02}$. We find $F_{\nu}\propto \nu^{2.4 \pm 0.1}$ for the optically thick part of the spectrum, consistent with $F_{\nu}\propto \nu^{5/2}$ as expected for SSA. The optically thin part of the spectrum $F_{\nu}\propto \nu^{-(p-1)/2}$ scales as $F_{\nu}\propto \nu^{-0.96 \pm 0.05}$, from which we infer $p\sim 3$, as typically found in radio SNe (e.g., @Chevalier06). []{data-label="Fig:VLAmodeling"}](multivariate.eps "fig:"){width="\columnwidth"}
VLA Data Analysis
-----------------
We present in Fig. \[Fig:VLAmodeling\] multi-band observations of taken up to 278 days post explosion with the Karl G. Jansky Very Large Array (VLA, projects 16A-447 and 17A-167). The details of these data are given in Table \[Tab:radio\]. We used standard phase referencing mode and the standard flux density calibrators 3C48 and 3C286 were used to set the absolute flux density scale. The data were calibrated using the VLA pipeline in <span style="font-variant:small-caps;">CASA</span> version 5.4.1, and imaged in <span style="font-variant:small-caps;">CASA</span> [@McMullin2007] following standard routines. We used Briggs weighting with a robust parameter of one to image. In the epochs where was sufficiently bright, we performed phase-only self-calibration on the target. We subsequently fitted the sources in the image plane using the Python Blob Detector and Source Finder (<span style="font-variant:small-caps;">PyBDSF</span>, @Mohan15). The uncertainties listed in Table \[Tab:radio\] take into consideration the errors on the fit and a 5% uncertainty on the absolute flux density scale. The flux density evolution of at $\sim$8.5 GHz is presented in Fig. \[Fig:xraysALL\] (left panel), together with a comparison with other SESNe and GRBs at the same frequency.
Inferences on the Progenitor Properties and Mass-loss History from Radio Observations {#SubSec:radiomodeling}
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Radio emission in type Ibc SNe is well explained as synchrotron emission from relativistic electrons with a power-law distribution of Lorentz factors $\gamma$ ($N_e(\gamma)\propto \gamma^{-p}$) that gyrate in shock amplified magnetic fields (e.g., @Chevalier06). shows the characteristic “bell-shaped” spectrum of radio sources dominated by synchrotron self-absorption (SSA), with spectral peak flux $F_{pk}\propto t^{-0.31 \pm 0.02}$ and peak frequency $\nu_{pk}\propto t^{-0.97\pm0.02}$. By fitting a broken power-law to the radio data of we find that the optically thin part of the spectrum $F_{\nu}\propto \nu^{-(p-1)/2}$ scales as $F_{\nu}\propto \nu^{-0.96 \pm 0.05}$, which implies $p\sim 3$, as typically found in radio SNe (e.g., @Chevalier06). For the optically thick part of the spectrum our fits indicate $F_{\nu}\propto \nu^{2.4 \pm 0.1}$, consistent with the SSA expectation $F_{\nu}\propto \nu^{5/2}$. We find no evidence for free-free external absorption (e.g., @Weiler02), which would cause the optically thick spectrum to be steeper than $F_{\nu}\propto \nu^{5/2}$.
Using the SSA formalism by [@Chevalier98], the best fitting $F_{pk}(t)$ and $\nu_{pk}(t)$ above translate into a constrain on the outer shock radius evolution $R_{sh}(t)$, magnetic field $B(R)$ and circumstellar density profile $\rho_{CMS}(R)$. We find evidence for a slightly decelerating blastwave with $R_{sh}(t)\propto t^{0.82\pm0.02}$ and $B(R)\propto R^{-1.14 \pm 0.03}$ propagating into a medium with density profile $\rho_{CSM}(R)\propto R^{-1.84 \pm 0.04}$. The inferred $B(R)$ profile is steeper than the $B(R)\propto R^{-1}$ scaling typically observed in H-stripped SNe (e.g., @Horesh13) and causes the observed decay of $F_{pk}(t)$ with time. Normal non-decelerating type Ibc SNe typically show a constant $F_{pk}(t)$ [@Chevalier98]. The inferred $\rho_{CSM}(R)\propto R^{-1.84 \pm 0.04}$ is slightly flatter than a pure wind density profile $\rho_{wind}\propto R^{-2}$, which implies an *increasing* effective mass-loss with radius $\dot M_{eff}\propto R^2 \rho_{CSM}\propto R^{0.16 \pm 0.04}$. We find $\dot M_{eff}(R_2)\sim2\times\dot M_{eff}(R_1)$, where $R_1 \sim4\times 10^{16}\,\rm{cm}$ is the blast wave radius at $10.5$ days and $R_2\sim10^{17}\,\rm{cm}$ is the blast wave radius at the end of the radio monitoring presented here, at $\delta t\sim 280$ days. For an assumed wind velocity $v_w=1000\,\rm{km\,s^{-1}}$ (appropriate for compact massive stars like WRs; @Crowther07), these results imply that the stellar progenitor of experienced a phase of enhanced mass-loss $\ge30$ yrs before collapse. We estimate that $\sim30$ yrs before death, the stellar progenitor of was loosing twice the amount of material per unit time compared to $\sim$10 yrs before stellar demise.
According to the self-similar solutions by [@Chevalier82] the interaction of a steep SN outer ejecta profile $\rho_{SN}\propto R^{-n}$ with a shallower medium with $\rho_{CSM}\propto R^{-s}$ produces an interaction region that expands as $R_{sh}\propto t^{m}$ with $m=(n-3)/(n-s)$. For , the inferred $R_{sh}(t)\propto t^{0.82\pm0.02}$ and $s= -1.84 \pm 0.04$ thus imply $n=8.2 \pm 0.7$. This result is consistent with the theoretical calculations of the post-explosion outer-ejecta density profiles of compact stars (e.g., WRs), for which [@Matzner99] find $n\sim10$. Extended red supergiants can have steeper outer density gradients [e.g. $\gtrsim20$; @Fransson96]. We conclude that radio observations of favor a compact progenitor star at the time of collapse.
All the considerations above do not depend on the assumed shock microphysical parameters $\epsilon_B$ and $\epsilon_e$ (i.e., the fraction of post-shock energy in magnetic fields and electrons, respectively). Below we provide the best-fitting values of the shock radius $R_{sh}$, internal energy $U$, magnetic field $B$ and effective mass loss $\dot M_{eff}$ at a given reference epoch under the assumption of equipartition of energy between electrons, protons and $B$ (i.e., $\epsilon_B=\epsilon_e=0.33$). Following [@Chevalier98], we find: $$\label{Eq:Bfield}
B(10.5\,\rm{d})=(4.0\pm0.2) \Big(\frac{\epsilon_e}{0.33}\Big)^{-\frac{4}{19}}
\Big(\frac{\epsilon_B}{0.33}\Big)^{+4/19}\,\,\rm{G},$$ $$\label{Eq:Radius}
R_{sh}(10.5\,\rm{d})=(3.1 \pm 0.1)\times 10^{15} \Big(\frac{\epsilon_e}{0.33}\Big)^{-1/19} \Big(\frac{\epsilon_B}{0.33}\Big)^{+1/19}\,\,\rm{cm}.$$ The outer shock radius of Eq. \[Eq:Radius\] does not strongly depend on the assumed microphysical parameter values. From Eq. \[Eq:Radius\], we can thus derive a solid estimate of the average SN shock velocity at $t=10.5$ days $v_{sh}\sim 0.15c$. This value is similar to normal type Ibc SNe (e.g., @Chevalier06) and different from GRB-SNe and relativistic SNe, which show evidence for ultra-relativistic and mildly relativistic outflows [@Soderberg10; @Margutti14b; @Chakraborti15]. The effective mass-loss is: $$\label{Eq:masslossRadio}
\dot M_{eff}(10.5\,\rm{d})=(3.6\pm0.3)\times 10^{-5} \Big(\frac{\epsilon_e}{0.33}\Big)^{-8/19} \Big(\frac{\epsilon_B}{0.33}\Big)^{-11/19} \,\,\rm{M_{\sun}yr^{-1}},$$ and the shock internal energy is: $$U(10.5\,\rm{d})=(1.1\pm 0.1)\times 10^{47}\Big(\frac{\epsilon_e}{0.33}\Big)^{-11/19} \Big(\frac{\epsilon_B}{0.33}\Big)^{-8/19}\,\,\rm{erg}.
\label{Eq:U}$$ Under the assumption of equipartition, the internal energy value $U(10.5\,\rm{d})=(1.1\pm 0.1)\times 10^{47}\,\rm{erg}$ sets a lower limit on the true internal energy of the system at $t=10.5$ d, and on the kinetic energy of the radio emitting material. $U$ increases with time, as the shock decelerates and more kinetic energy of the shock wave is converted into internal energy. At $t\sim 280$ d we measure $v_{sh}\sim0.06\pm0.01$ c and $U(280\,\rm{d})=(7.6\pm 0.9)\times 10^{47}$ erg (in equipartition), which places among energetic shocks from normal H-stripped SNe (Fig. 2 in @Margutti14b).
Realistic values of $\epsilon_e$ and $\epsilon_B$ in SN shocks are likely $<0.33$, implying that both the equipartition $\dot M_{eff}$ and $U$ are lower limits on the true values of the system. For comparison, for more realistic values of $\epsilon_e=0.1$ and $\epsilon_B=0.01$ [typical values for relativistic shocks; @Sironi2011], we infer $\dot M_{eff}(10.5\,\rm{d})= (4.5\pm0.4)\times 10^{-4}\,\rm{M_{\odot}yr^{-1}}$ and $U(280\,\rm{d})=(6.7\pm 0.8)\times 10^{48}$ erg. Recent kinetic simulations of trans-relativistic shocks suggest values of $\epsilon_B\sim 0.01$ and $\epsilon_e\gtrsim 10^{-3}$ [@Park2015; @Crumley2019]. An $\epsilon_e\sim 3\times 10^{-3}$ would imply a very energetic explosion, with $U(280\,\rm{d})=(5.1\pm 0.6)\times 10^{49}$ erg, however it also yields an unrealistic $\dot M_{eff}(10.5\,\rm{d})= (2.0\pm0.2)\times 10^{-3}\,\rm{M_{\odot}yr^{-1}}$. Similar values of mass-loss are more typical of progenitor stars of Type IIn SNe and are likely too high for . In general, the theory of particle acceleration at strong trans-relativistic shocks not only does not explain large values of $\epsilon_e$, but also does not produce the typical $p\sim 3$ often inferred in radio SNe. The explanation of @Chevalier06 invokes shocks modified by the dynamical backreaction of the accelerated particles, which was thought to lead to concave spectra, steeper than $E^{-2}$ below a few GeV, but such an argument is at odds with observations of Galactic SN remnants [@Caprioli12].
A robust upper limit on the effective mass-loss can be inferred from the lack of free-free external absorption in the radio spectra. Indeed, the absence of a low-frequency cut-off can be used to constrain the environment density independently from the shock microphysics. From [@Weiler02], the free-free optical depth of unshocked ionized gas in a wind density profile is: $$\tau_{\rm ff} \simeq \frac{\alpha_{\rm ff}r}{3} \approx \newline 10\left(\frac{\nu}{10{\rm GHz}}\right)^{-2}\left(\frac{T_{\rm g}}{10^{4}{\rm K}}\right)^{-3/2} \dot M_{-3}^2 \left(\frac{v_{sh}}{0.1c}\right)^{-3}t_{\rm wk}^{-3}\,\,,$$ where $\dot M$ is in units of $10^{-3}\,\rm{M_{\sun}yr^{-1}}$ for $v_{w}=1000\,\rm{km\,s^{-1}}$, $T_{\rm g}$ is the temperature of the gas, normalized to a value $T_{\rm g} \gtrsim 10^{4}$ K typical of photoionized gas, and time is units of 1 week. Furthermore, we used $\kappa_{\rm es} = 0.38$ cm$^{2}$ g$^{-1}$ for fully ionized solar-composition ejecta and $\alpha_{\rm ff} \approx 0.03 n_w^{2}\nu^{-2}T_{\rm g}^{-3/2}$ cm$^{-1}$ as the free-free absorption coefficient. The lack of evidence for free-free absorption at 10.5 days at $\nu=5.9$ GHz, and at 45.5 days at $\nu=3$ GHz demands $\tau_{ff}\ll1$, which translates into $\dot M<10^{-3}\,\rm{M_{\sun}yr^{-1}}$ for $v_{w}=1000\,\rm{km\,s^{-1}}$.
X-rays {#sec:XRT}
======
Swift-XRT and XMM-Newton Data Analysis {#SubSec:Xraydata}
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-0.0 true cm {width="48.00000%"} {width="48.00000%"}
The X-Ray Telescope [XRT; @Burrows05], on board the Neil Gehrels *Swift* Observatory, started observing on May 27, 2016 ($\delta t\sim2$ days post explosion) until April 17, 2017 ($\delta t\sim326$ days), for a total exposure time of 94.4 ks. *Swift*-XRT data have been analyzed using the latest HEAsoft release v6.22 and corresponding calibration files. Standard filtering and screening criteria have been applied (see @Margutti13 for details). An X-ray source is clearly detected at the location of until $\delta t\sim100$ days post explosion. The X-ray source is located $\sim30''$ from the host galaxy nucleus (which is not detected by *Swift*-XRT and does not represent a source of contaminating X-ray emission; see Fig. \[Fig:rgb\], bottom panel) and shows a fading behavior with time, from which we conclude that the detected X-ray emission is physically associated with . The spectrum can be fit with an absorbed power-law spectral model with best fitting photon index $\Gamma= 1.78 \pm 0.18$. We find no evidence for intrinsic absorption and we place a $3\sigma$ limit for the neutral hydrogen absorption column $NH_{i}<0.4\times 10^{22}\,\rm{cm^{-2}}$. The Galactic $NH_{i}$ in the direction of is $NH_{mw}=0.056\times 10^{22}\,\rm{cm^{-2}}$ [@Kalberla05]. For this spectrum, the 0.3-10 keV count-to-flux conversion factor is $\sim 4.22\times 10^{-11}\,\rm{erg\,s^{-1}cm^{-2}ct^{-1}}$ (unabsorbed). The *Swift*-XRT count-rate and flux-calibrated light-curve is reported in Table \[Tab:xrays\] and shown in Fig. \[Fig:xraysALL\] (right panel).
We started deep X-ray observations of the field of with XMM-Newton on June 6, 2016 (PI Margutti). We obtained two epochs of observations at $\delta t\sim 11.5$ days (exposure time of 29 ks, observation ID 0782420201) and $\delta t\sim 27.5$ days (exposure of 28 ks, observation ID 0782420301). XMM data have been analyzed with SAS (v15.0). The first observation was heavily affected by proton flaring and the net exposure time of the EPIC-pn camera after filtering out the intervals of high background was reduced to 1.1 ks, whereas for the second epoch we have 13.4 ks net exposure time. is clearly detected by all three cameras in both epochs. The inferred EPIC-pn count-rate is $(4.0\pm0.7)\times 10^{-2}\,\rm{ct\,s^{-1}}$ and $(1.7\pm0.2)\times 10^{-2}\,\rm{ct\,s^{-1}}$ (0.3-10 keV) for the first and second epoch, respectively. A spectrum extracted from the first (second) epoch can be fitted with an absorbed power-law model with $\Gamma=1.9\pm 0.3$ ($\Gamma=1.8\pm 0.3$). The corresponding flux is $\sim 1.1\times 10^{-13}\,\rm{erg\,s^{-1}cm^{-2}}$ and $\sim 0.6\times 10^{-13}\,\rm{erg\,s^{-1}cm^{-2}}$ for the first and the second epoch, respectively, consistent with the results from our *Swift*-XRT monitoring (Table \[Tab:xrays\] and Fig. \[Fig:xraysALL\]).
We do not find evidence for significant spectral evolution. From a joint fit of *Swift*-XRT and XMM data we find a best-fitting $\Gamma=1.80\pm0.10$ and $NH_{i}<0.17\times 10^{22}\,\rm{cm^{-2}}$.
Inferences on the Mass-loss History of the Stellar Progenitor from X-ray Observations {#Sec:massloss}
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In normal H-poor SNe, the early time ($\delta t\lesssim 30$ days) X-ray emission is expected to be dominated by Inverse Compton (IC) scattering of optical photospheric photons by relativistic electrons accelerated at the shock fronts (e.g., @Bjornsson04 [@Chevalier06]). The non-thermal X-ray spectrum of with $\Gamma\sim 2$ and lacking evidence for intrinsic absorption is consistent with this expectation. Adopting the formalism by [@Margutti12], the IC emission depends on: (i) density profile of the SN ejecta $\rho_{ej}$; (ii) properties of the electron distribution responsible for the up-scattering $N_e(\gamma)$; (iii) blastwave velocity, which, in turns, depends on the circumstellar medium (CSM) density and explosion’s parameters (kinetic energy $E_{k}$ and ejecta mass $M_{ej}$); (iv) optical bolometric luminosity of the SN (from §\[sec:Lbol\]), which is the ultimate source of photons that are upscattered to X-ray energies $L_{X,IC}\propto L_{bol}$. We parametrize the CSM density as a wind medium $\rho_{CSM}=\dot M/4\pi v_w R^{2}$ (where $v_w$ is the progenitor wind and $\dot M$ is the mass-loss rate) and we use $\rho_{ej}\propto R^{-n}$ with $n\sim10$, as appropriate for SNe with compact progenitors (e.g., @Matzner99 [@Chevalier00]), consistently with the results from the modeling of radio data in §\[SubSec:radiomodeling\]. We further assume a power-law distribution of electrons $N_e(\gamma)\propto \gamma ^{-p}$ with $p\sim3$ and $\epsilon_e=0.1$ for consistency with the modeling of other SNe (e.g., @Chevalier06).
Considering the range of explosion parameters of §\[sec:Lbol\] and assuming a wind velocity of $v_w=1000\,\rm{km\,s^{-1}}$, for a shock velocity of $v_{sh}\sim0.1$ c (§\[SubSec:radiomodeling\]) we infer a mass-loss rate of $\dot M\sim (1-2)\times 10^{-4}\,\rm{M_{\sun}yr^{-1}}$. Our X-ray analysis thus provides independent evidence that exploded in a dense environment when compared to type Ic-BL SNe (Fig. \[Fig:massloss\]). This suggests that the stellar progenitor of experienced significant mass loss in the last years before core-collapse. This result is independent of $\epsilon_B$. A comparison to Eq. \[Eq:masslossRadio\] suggests that for $\epsilon_e=0.1$ $\epsilon_B\le0.1$.
-0.0 true cm ![Fastest ejecta velocity in the explosion vs. environmental number density of in the context of H-stripped core-collapse SNe. Type IIb SNe (blue diamonds) explode in the densest environments, while SN that accompany GRBs are associated with the lowest density environments (orange squares). Normal type Ibc SN are shown with black filled circles. SN with broad features in their spectra (Ibc/BL in the plot, orange triangles) also tend to be associated with low density media. An exception to this behavior is , which exploded in a dense environment (red star). For we show here the equipartition number density. The true number density in the environment of is $\sim10$ times the equipartition value if $\epsilon_B= 0.01$. Grey shaded regions: density in the environments of WRs and the recently discovered new type of WR stars WN3/O3 (@deJager88 [@Marshall04; @vanLoon05; @Crowther07; @Massey15]). References: [@vanDyk94; @Fransson98; @Berger02; @Weiler02; @Ryder04; @Soderberg05; @Chevalier06; @Soderberg06b; @Soderberg06c; @Soderberg08; @Roming09; @Soderberg10; @Soderberg10b; @Krauss12; @Milisavljevic13; @Margutti14b; @Kamble14; @Corsi14; @Chakraborti15; @Drout16; @Kamble16; @Margutti17]. []{data-label="Fig:massloss"}](vfast_vs_density_SN2016coi.eps "fig:"){width="\columnwidth"}
The right panel of Fig. \[Fig:xraysALL\] clearly shows that IC (dashed orange line) fails to reproduce the bright X-ray emission at $\sim$ 100 days by a large factor. At these times, synchrotron emission is expected to dominate (e.g., @Chevalier06). The extrapolation of the optically thin $F_{\nu}\propto \nu^{-0.96}$ radio spectrum to the X-ray band under-predicts the observed X-ray emission by a large factor $\sim30$. The discrepancy between the extrapolation of synchrotron spectrum that best fits the radio and the observed X-ray data is even larger when we consider that the synchrotron cooling frequency $\nu_c=18\pi m_{e} c q/(t^2\,B^3\,\sigma_T^2)$ is $\nu_c\sim 10^{11}-10^{12}\,\rm{Hz}$ at $\sim 100$ days using $B$ from Eq. \[Eq:Bfield\] and $\epsilon_B=0.01-0.1$ (where $q$ is the electron charge and $m_e$ is the electron mass). Above $\nu_c$ the flux density steepens as $F_{\nu}\propto \nu^{-p/2}$, leading to an even lower expected X-ray flux. The conclusion is that the late time $t\ge100$ days X-ray emission from is too luminous to be explained within the standard framework of synchrotron radiation from a population of electrons accelerated into a simple power-law distribution $N_e(\gamma)\propto \gamma^{-p}$.
The problem of having very luminous X-ray emission from H-stripped core-collapse SNe at late times is not new and was explored in detail by [@Chevalier06]. These authors favor an interpretation where the particle spectrum is modified and becomes flatter for $\gamma\ge1000$. The net effect is an increase of the X-ray synchrotron emission, while the effect on the radio synchrotron emission is minor (see their Fig. 1). At the time of writing it is unclear what physical effect might produce this shape of the particle spectrum, as the cosmic-ray dominated shocks invoked by [@Chevalier06] have not been confirmed by recent particle-in-cell (PIC) simulations [@Park2015]. We end by noting that at the large mass-loss rates inferred for in the case of deviation from equipartition $\dot M_{eff}\sim5\times 10^{-4}\rm{M_{\sun}\,yr^{-1}}$, the X-rays are likely to receive a contribution from free-free emission. From [@Chevalier06], their Eq. 30, for this mass-loss rate we estimate $L_{x,ff}\sim5\times 10^{38}\,\rm{erg\,s^{-1}}$ at $t\sim100$ days, which is a factor $\sim2$ lower than the observed X-ray emission at this epoch.
Search for shock breakout emission at high energies {#sec:SBO}
===================================================
\[Sec:SBO\] For compact massive H-stripped stars that are progenitors of (some) type Ibc SNe, the very first electromagnetic signal able to escape from the explosion site and reach the observer (i.e., the breakout pulse) is expected to peak at X-ray and $\gamma$-ray energies (e.g., SN2008D; @Soderberg08). We searched for a high-energy pulse associated with the shock breakout of using data collected by the InterPlanetary Network (IPN), which includes Mars Odyssey, Konus-*Wind*, RHESSI, INTEGRAL (SPI-ACS; SPectrometer on INTEGRAL-Anti-Coincidence System), *Swift*-BAT (Burst Alert Telescope) and *Fermi*-GBM (Gamma-Ray Burst Monitor). The IPN observes the entire sky with temporal duty cycle $\sim100$% when all the experiments are considered. A total of 6 bursts were detected by the spacecraft of the IPN between May 22.4, 2016 and May 25.6, 2016, which covers the most likely explosion date window May $23.9 \pm 1.5$ days that we inferred in §\[sec:Lbol\]. None has a localization region consistent with the position of . We thus conclude that there is no evidence for a SN-associated shock breakout pulse down to the IPN sensitivity threshold with fluence $F_\gamma \sim 6\times 10^{-7}\,\rm{erg\,cm^{-2}}$ ($E_{\gamma}\sim 2\times 10^{46}\,\rm{erg}$ at the distance of ).
Comparison to Breakout Models
-----------------------------
Following [@Katz12] (and references therein), the expected shock breakout energy is $E_{BO}=8 \pi R_*^{2} v_0 c \kappa^{-1}$ (their Eq. 40), where $R_*$ is the stellar radius, $\kappa$ is the opacity and $v_0$ is the shock velocity at breakout. For we assume $\kappa\sim0.4\,\rm{cm^2\, g^{-1}}$ and adopt $v_0 \approx 0.3\,\rm{c}$ (i.e., a shock velocity at breakout similar to the maximum ejecta velocity as inferred from the X-ray observations, see @Katz12, their Eq. 25). For compact progenitors like WR stars with $R_*=10^{11}\,\rm{cm}$ we find $E_{BO}\approx 10^{44}\,\rm{erg}$, significantly below the IPN sensitivity. *Fermi*-GBM and *Swift*-BAT are more sensitive and reach fluence limits of $\sim4\times 10^{-8}\,\rm{erg\,cm^{-2}}$ and $\sim6\times 10^{-9}\,\rm{erg\,cm^{-2}}$, respectively, corresponding to $E_{\gamma}\sim 2\times 10^{45}\,\rm{erg}$ and $E_{\gamma}\sim 2\times 10^{44}\,\rm{erg}$. The *Swift*-BAT threshold for detection is comparable to the expected $E_{BO}$. *Swift*-BAT observes $\sim1/6$ of the sky with $\sim90$% temporal duty cycle. It is thus possible that *Swift*-BAT missed the breakout pulse, or that the breakout pulse lies just below the *Swift*-BAT threshold of detection. For comparison the breakout pulse in SN2008D showed $L_x\sim10^{44}\,\rm{erg\,s^{-1}}$ (0.3-10 keV) at peak with a duration of $\sim5$ minutes.
More extended progenitors with radii $R_*=10^{13}\,\rm{cm}$ would lead to inferred $E_{BO}\approx 10^{48}\,\rm{erg}$. In this case however the spectrum of breakout pulse is expected to peak at lower frequencies $<1$ keV which are not probed by the hard X-ray/$\gamma$-ray observations presented here. A similar reasoning and conclusion apply if the radiation breakout occurred in a thick medium outside the star.
Discussion {#sec:disc}
==========
Our data analysis and modeling characterize as an energetic H-stripped SN with (i) He in the ejecta, (ii) a broad bolometric light-curve, and (iii) luminous X-ray and radio emission. These three observables distinguish from the rest of the population of known H-stripped SNe and directly map into properties of its progenitor star: a massive, well-mixed star that experienced substantial mass loss in the years preceding core-collapse. We discuss below the implications of these findings in the broader context of stellar progenitors of H-stripped SNe.
Broad Bolometric Light-Curve and Nebular Spectroscopy Indicate a Massive Progenitor {#subsec:broad LC}
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Among H-stripped SNe, shows one of the broadest bolometric light-curves (Fig. \[Fig:LCcomparison\]), from which we infer $M_{ej}\sim5-7\,\rm{M_{\sun}}$ [§\[sec:Lbol\]; @Kumar18; @Prentice17]. This value is larger than the typical ejecta mass $M_{ej}\sim2-3\,\rm{M_{\sun}}$ inferred for H-stripped SNe [e.g., @Drout2011; @Bianco2014; @Lyman2016; @Taddia2018].
Other type Ic SNe with broad light-curves are SNe 2004aw [@Taubenberger2006] and 2011bm [the broadest light curve in Fig. \[Fig:LCcomparison\]; @Valenti2012], for which the inferred ejecta mass is $3.5-8.0$ and $7-17$ , respectively. Additionally, SN2004aw also displayed relatively high ejecta velocities ($v\sim12000$ km/s$^{-1}$) around maximum light, similar to . Interestingly, a tentative identification of He in the ejecta of SN2004aw has also been reported based on NIR spectroscopy [@Taubenberger2006].
Nebular spectroscopy of provides additional constraints on the mass of its stellar progenitor. The relative abundances of different elements in a stellar envelope depend on the core mass. In particular, in the models by [@Fransson1989] the ratio of the integrated fluxes of the \[\] 7291,7324 doublet and the \[\] 6300,6364 doublet can be used as an indicator of the progenitor core mass, with lower values signifying of more massive cores. This fact mainly results from two factors: first, in the models by [@Fransson1989] the relative abundance Ca/O is lower for progenitors with a more massive core, producing a lower \[\]/\[\] flux ratio; second, in stars with a smaller core mass and with a stratified envelope like the ones in [@Fransson1989] models, the Ca is more mixed within the oxygen layers. Since Ca is a significantly more efficient coolant, this translates into more prominent \[\]/\[\] ratio. However, several other factors can play a role in determining the observed \[\]/\[\] flux ratio. [@Fransson1989], for example, showed that higher densities of the ejecta (i.e., lower kinetic energies) would also lead to lower \[\]/\[\] flux ratios. Additionally, a high degree of mixing of the entire stellar envelope, where oxygen is more centrally located, also leads to a lower \[\]/\[\] flux ratio (as in this case cooling through oxygen would act as a competitor to calcium in the inner envelope). These factors make the observed \[\]/\[\] flux ratio a non-monotonic tracer of the stellar core-mass.
With these caveats in mind, the \[\]/\[\] flux ratio has been used in the past as a diagnostic for the progenitor star in core-collapse SNe [e.g., @Fransson1987; @Fransson1989; @Elmhamdi2011; @Kuncarayakti2015]. We therefore compute this ratio as a function of time in and show the results in Fig. \[Fig:Ca\_O\]. To build a homogeneous comparison sample, we retrieved late-time spectra of stripped-envelope SNe available from the literature[^8]. We select all the SNe with observations at $t>$100 days after explosion, and with at least one spectrum covering both the \[\] and the \[\] region. We focus on those SNe with a \[\]/\[\] ratio below 1.3. Our sample comprises 83 spectra from 29 different SNe. We then fit the line doublets with two Gaussian profiles (see Fig. \[fig:neb\]). The position of the first centroid is kept as a free parameter, but the separation between the two lines of each doublet is kept fixed at the expected value.
-0.0 true cm ![Comparison between the evolution of the \[\] 7291,7324 to \[\] 6300,6364 ratio for and other type Ic and BL-Ic SNe. is characterised by low \[\]/\[\] ratio, suggesting a high progenitor, with a high level of mixing at the time of explosion. Other events with a low ratio are SNe 1985F, 1997B and 2004aw. Measurements were performed depending on availability of late-time spectra retrievable from the literature [@Gaskell1986; @Filippenko1986; @Filippenko1995; @Barbon1999; @Patat2001; @Foley2003; @Elmhamdi2004; @Taubenberger2006; @Tanaka09; @Taubenberger2009; @Milisavljevic2010; @Valenti2011; @Silverman2012; @Valenti2012; @Benetti2011; @Ben-Ami2014; @Modjaz2014; @Ergon2015; @Kuncarayakti2015; @Milisavljevic15c; @Milisavljevic15; @Smartt2015; @Drout16; @Nicholl2016; @Kangas2017; @Taddia2019].[]{data-label="Fig:Ca_O"}](Ca_O.png "fig:"){width="\columnwidth"}
From Fig. \[Fig:Ca\_O\] it is clear that at very late phases occupies the lower part of the plot, and at $t>400$ d has the lowest \[\]/\[\] ratio, close to $\sim0.2$. For reference, [@Fransson1989] found ratios of $\sim0.6$ and $\sim5.6$ for their 8 and 4 He-core progenitor models, respectively. This analysis independently supports the idea that originated from a stellar progenitor with larger mass than the average progenitor of H-stripped core-collapse SNe. However, as described above, other factors can contribute to the observed \[\]/\[\] ratio, like mixing and ejecta densities. Indeed, the values measured by [@Fransson1989] would become $\sim0.3$ and $\sim1.6$ for the same progenitor models as above, with lower explosion kinetic energies ($\sim$8 times higher densities).
Other SNe with low flux ratios ($<0.3$) are the type Ib SN 1985F [@Schlegel1989; @Elmhamdi2004], and the two type Ic SNe 1997B (spectra retrieved from the Asiago Supernova Archive) and 2004aw [@Taubenberger2006]. Interestingly, at least two of these SNe also have broad light curves. The bolometric light-curve of SN2004aw is very similar to (Fig. \[Fig:LCcomparison\]), while SN1985F has an even broader light curve, with a $\Delta_{15}$ in $B$-band of 0.52 mag [@Tsvetkov1986] ($\Delta_{15}$=1.01 mag for @Kumar18). Unfortunately, SN1997B was discovered after peak. Consistent with the caveats above, some SNe with broad light curve have large \[\]/\[\] ratio (e.g., SN2011bm, @Lyman2016).
Based on the estimated $M_{ej}\sim 4-7$ (§\[sec:Lbol\]), and assuming a fiducial mass for the remnant compact object between 1.5 (for a neutron star) and 3 (for a black hole), we estimate a mass of the C+O stellar progenitor of at the time of collapse of $\sim$6-10 . Similar values have been inferred for the progenitor of SN2004aw, for which [@Mazzali2017] estimated a ZAMS mass of $\sim23-30$ .[^9] The actual value of the inferred ZAMS mass strongly dependends on the adopted mass loss prescriptions [e.g., @Smith2014].
Luminous Radio and X-ray Emission from Large Progenitor Mass-loss before Explosion {#SubSec:MassLossDisc}
----------------------------------------------------------------------------------
-0.0 true cm ![Constraints on the recent mass-loss history of (red shaded area) in the context of observed mass-loss rates and wind velocities in massive stars. Here we conservatively plot the equipartition $\dot M$. The true $\dot M$ is $\sim10$ times larger if $\epsilon_B= 0.01$. Galactic WR stars from @Crowther07, WN3/O3 stars from @Massey15, red supergiants (RSGs) winds from @deJager88 [@Marshall04; @vanLoon05]. Typical locations of Luminous Blue Variable (LBV) winds and eruptions are from @Smith2014 and @Smith06. Black, blue and dotted green lines mark the sample of type Ic-BL, Ibc and IIb SNe from [@Drout16]. Inferred mass-loss rates for type-IIn SNe are from [@Kiewe12].[]{data-label="Fig:massloss2"}](masslossplot16coi.pdf "fig:"){width="\columnwidth"}
The recent mass-loss history of the progenitor star in the centuries leading up to the explosion can be constrained with radio and X-ray observations, which sample the emission originating from the interaction between the fastest SN ejecta and the CSM. The resulting luminosity mainly depends on the shock velocity and on the environment density, with faster shocks and denser environments powering the most luminous radio and X-ray displays.
We compare the properties of that we inferred in §\[SubSec:radiomodeling\] and \[Sec:massloss\] to a sample of H-stripped SNe in Fig. \[Fig:massloss\]. SNe with fast ejecta velocities like Ic-BL SNe (orange squares and triangles in Fig. \[Fig:massloss\]) tend to be associated with low-density environments, while type IIb SNe are located within the densest circumstellar media. From the radio we inferred a shock velocity of $v_{sh}\sim0.15c$ for . This sub-relativistic shock can be caused either by a lower shock velocity at breakout, or by a denser-than-average environment surrounding the progenitor. In the latter case, the CSM was sculpted by a prolonged enhanced mass-loss phase of the stellar progenitor in the years before stellar death. The environment of is among the densest in the sample of type Ibc SNe. Assuming equipartion of energy, from the radio data we obtained a lower limit for the mass-loss of of $\dot M \sim 3-4 \times 10^{-5}$ yr$^{-1}$ (for $v_w=1000$ km s$^{-1}$). X-ray analysis pushed this value even higher, with $\dot M \sim 1-2 \times 10^{-4}$ yr$^{-1}$. Such a large mass-loss rate is consistent with those associated with extreme line-driven winds in WR stars [@Crowther07], as we show in Fig. \[Fig:massloss\]-\[Fig:massloss2\]. This result is also supported by the radio modelling, which showed that the post-explosion density profile of the outer ejecta is consistent with having originated from a compact object like a WR star. The inferred $\dot M$, *if* sustained for the entire $\sim10^{5}\rm{yr}$ duration of the WR phase, implies a total mass-loss of several $M_{\sun}$ possibly sufficient to strip the progenitor star of a large fraction of its helium envelope even in the absence of interactions with a binary companion [^10]. This scenario is consistent with the indication of a massive stellar progenitor of §\[subsec:broad LC\].
He Spectral Features as a Signature of Asymmetries? {#SubSec:Hedisc}
---------------------------------------------------
The presence of He in is supported by the comparison with the velocity profile of 6355 (§\[sec: class\]), and by the detailed spectral modeling of [@Prentice17]. lines are formed through non-thermal excitation and ionization [@Lucy1991], for example by $\gamma$-rays produced by the radioactive decay of and . For this excitation channel to be effective, plumes of -rich material must have been able to reach the outer He-rich layers of the progenitor star of , consistent with the results from our two-zone modeling of the bolometric emission from in §\[sec:Lbol\], and consistent with the results from recent 3D simulations of stellar explosions (e.g., @Wongwathanarat2015) and studies of SN remnants [@Milisavljevic15b]. Indeed, models of [@Dessart2012] showed that a single plume of -rich material injected into the outer layers (e.g., in the form of a jet) is capable of producing weak He features. Alternatively, 3D simulations of neutrino-driven explosions have shown mixing instabilities that are capable of injecting - and Si-rich plumes to the higher velocities layers of the ejecta [e.g., @Hammer2010]. Indeed, we observed a very similar velocity evolution for and , supporting this scenario. In their spectral modelling, [@Prentice17] also showed that at early phases heavy ions were travelling at a very high speed; was the fastest species at 26 000 km s$^{-1}$. Also the showed similar high velocities, possibly hinting to some high velocity material coming from the progenitor core, as seen in GRB-SNe [e.g. @Bufano12; @Toy2016; @Ashall2019].
Among type Ic SNe with broad lines, SNe 2009bb and 2012ap have been reported to have signatures of He in their spectra[^11] [@Pignata2011; @Milisavljevic15]. Interestingly, SNe 2009bb and 2012ap are currently the only two cases of SNe with mildly relativistic ejecta not associated with a GRB [@Soderberg10; @Margutti14b; @Chakraborti15]. This phenomenology has been suggested to be the result of a jet-driven stellar explosion where the jet fails to break through the stellar envelope [@Morsony07; @Lazzati12; @Margutti14b]. In this picture, relativistic SNe and GRBs are intrinsically different types of explosions, as opposed to similar explosions viewed from different perspectives. In relativistic SNe the jet is possibly choked by the more extended stellar envelope, but manages to “transport” some -rich material outwards and then excites some residual He that the stellar progenitor failed to shed before stellar death [@Maeda2002; @Suzuki2018; @Izzo2019].
We speculate that a similar scenario applies to , for which the lack of evidence for mildly relativistic ejecta can be explained as the result of a jet that died deep inside the star, leaving no imprint on the dynamics of the fastest ejected material, yet accelerating the inner layers to velocities larger than in normal type Ic SNe (Fig. \[fig:modjaz\]), and at the same time injecting metal-rich material into the outer layer of the ejecta. In this context, the difference between normal type Ibc SNe and those with large ejecta velocities (including type Ic-BL and ) would be ascribed to the absence/presence of a jet at the time of core-collapse [@Khokhlov1999; @Granot04; @Wheeler10; @Lazzati12; @Nagakura12; @Margutti14b; @Soker2016].
Summary and Conclusions {#sec:concl}
=======================
We present the results of a multi-wavelength, $\gamma$-rays to radio campaign on the peculiar [@Yamanaka17; @Kumar18; @Prentice17] during its first 420 days of evolution. Our findings can be summarized as follows:
- From extensive UV/optical/NIR photometry we derive a broad bolometric light-curve (Fig. \[Fig:LCcomparison\]), which is suggestive of a larger-than-average explosion ejecta mass. From our two-zone modeling we infer $M_{ej,tot}\sim 4-7\,\rm{M_{\sun}}$ (consistent with previous findings by @Kumar18), with a larger fraction of per unit mass in the outer part of the ejecta. We also constrain a total kinetic energy of $E_{k}\sim (7-8)\times 10^{51}\,\rm{erg}$.
- Our spectroscopic analysis supports the presence of He in the SN ejecta, confirming the previous findings by [@Yamanaka17], [@Kumar18] and [@Prentice17]. We furthermore find a low \[\] 7291,7324 to \[\] 6300,6364 ratio, suggestive of a large progenitor core mass at the time of collapse.
- is a luminous source of radio and X-rays, which result from the propagation of a sub-relativistic blast wave with $v\sim 0.15c$ into a dense environment sculpted by sustained mass-loss from the progenitor star before core-collapse. We infer a lower limit on the mass-loss rate of $\dot M \sim (3-4)\times 10^{-5}\,\rm{M_{\odot}yr^{-1}}$ (for wind velocity $v_w=1000\,\rm{km\,s^{-1}}$ and assuming energy equipartion), significantly larger than in type Ic-BL SNe.
- Radio modelling also revealed a phase of higher mass-loss rate lasting until $\sim30$ years before explosion. Additionally, we inferred a post-explosion density profile of the outer ejecta compatible with the explosion of a compact star (e.g., a WR, as opposed to extended progenitors like red and yellow supergiant stars).
- We investigated the presence of a high-energy prompt pulse of emission in the $\gamma$-rays. From our analysis we can rule out a SN-associated shock breakout pulse with energy $E_{\gamma}>2\times 10^{46}\,\rm{erg}$, consistent with the theoretical expectations of shock break out from WR stars or from extended winds.
The emerging picture is that of a massive compact progenitor star that was able to retain some He until collapse, despite the heavy mass loss experienced in the years leading up to stellar demise. The combination of (i) large ejecta mass and (ii) large mass-loss in a H-stripped core-collapse SN with (iii) weak He features in the spectra, set apart from all SNe with similar data coverage and quality in the literature. We speculate that the energetic might be the result of a failed jet that was choked by the extended envelope mass of its progenitor star, in analogy with the relativistic type Ic-BL SNe 2009bb and 2012ap for which He has been identified in the ejecta. It is possible that this picture of a jet-driven explosion where the jet has been choked while trying to pierce through the He-rich stellar envelope extends to the entire class of type Ic-BL SNe that are not associated with GRBs. Future observing campaigns of type Ic-BL SNe with coordinated optical and NIR spectroscopy will reveal if traces of He in type Ic-BL SNe are more common than currently thought.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank N. Morrell for observing at du Pont telescope, and E. Falco for observing at FLWO.
ASAS-SN is supported by the Gordon and Betty Moore Foundation through grant GBMF5490 to the Ohio State University and NSF grant AST-1515927. Development of ASAS-SN has been supported by NSF grant AST-0908816, the Mt. Cuba Astronomical Foundation, the Center for Cosmology and AstroParticle Physics at the Ohio State University, the Chinese Academy of Sciences South America Center for Astronomy (CASSACA), the Villum Foundation, and George Skestos. We thank the Las Cumbres Observatory (LCOGT) and its staff for its continuing support of the ASAS-SN project. The Liverpool Telescope is operated on the island of La Palma by Liverpool John Moores University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias with financial support from the UK Science and Technology Facilities Council. Based on observations made with the Nordic Optical Telescope Scientific Association at the Observatorio Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofísica de Canarias. Partially based on observations obtained with Copernico 1.82m Telescope (Asiago) operated by INAF Osservatorio Astronomico di Padova. This work makes use of data gathered with the 6.5-m Magellan Telescopes, at Las Campanas Observatory in Chile. Based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. We acknowledge the use of public data from the Swift data archive. This work made use of the data products generated by the NYU SN group, and released under DOI:10.5281/zenodo.58766, available at <https://github.com/nyusngroup/SESNtemple/>. This work has made use of the Weizmann Interactive Supernova Data Repository (<https://wiserep.weizmann.ac.il>). This work has made use of the Berkeley Supernova Database (<http://heracles.astro.berkeley.edu/sndb/>). 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 NASA. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The RM group at Northwestern is partially supported by the National Aeronautics and Space Administration through Chandra Award Number GO6-17053A issued by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060, the Swift Guest Investigator program, NASA Grants NNX17AD82G and 80NSSC18K0575 and the XMM Guest Investigator program, NASA Grant NNX16AT58G. SD and PC acknowledge Project 11573003 supported by NSFC. This research uses data obtained through the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences, and the Special Fund for Astronomy from the Ministry of Finance. KH is grateful for support under the Fermi Guest Investigator program, NASA Grant NNX15AU74G. Support for JLP is provided in part by FONDECYT through the grant 1191038 and by the Ministry of Economy, Development, and Tourism’s Millennium Science Initiative through grant IC120009, awarded to The Millennium Institute of Astrophysics, MAS. LT and SB are partially supported by PRIN INAF 2017 “Towards the SKA and CTA era: discovery, localisation and physics of transient sources (PI M. Giroletti).” MS is supported by a generous grant (13261) from VILLUM FONDEN and a project grant (8021-00170B) from the Independent Research Fund Denmark. NER acknowledges support from the Spanish MICINN grant ESP2017-82674-R and FEDER funds. CG is supported by a VILLUM FONDEN Investigator grant (project number 16599). JH acknowledges financial support from the Finnish Cultural Foundation.
.
[c<lccr<>c<>lc]{} Code & Telescope & Instrument & Pixel size & & Filters\
ASA & 14cm Brutus[^12] & Fairchild CCD3041 & 7.8& 4.47&$\times$& 4.47& *V*\
RK & 25cm Meade SCT [^13] & Apogee AP-47 & 1.02& 17.4&$\times$& 17.4& *BVRI*\
JB & 43cm PlaneWave CDK [^14] &SBIG STL-6303 & 0.63& 34&$\times$& 22& *BVri*\
GS & 43cm PlaneWave CDK [^15]& SBIG STXL-11002 & 0.63& 42&$\times$& 28& *BV*\
T50 &T50 [^16]& ProLine PL16801 & 0.54& 36.9&$\times$&36.9& *BVri*\
Gem & Gem[^17] & SBIG 6303e & 1.08& 27.6&$\times$& 18.4& *BVgri*\
DEM & 51cm PlaneWave CDK[^18] & Fairchild CCD3041 & 0.90& 30.7&$\times$& 30.7& *BVri*\
WHO & WHO 1m telescope [^19]&Andor iKon-L DZ936-N & 0.35& 12&$\times$& 12& *BVri*\
LCOGT & Las Cumbres Observatory (LCOGT) &Sinistro & 0.40& 26&$\times$&26& *BVgri*\
1.82 & 1.82m Copernico & AFOSC& 0.52& 8.7&$\times$&8.7&*UBVRIugriz*\
&& REM-IR & 0.58& 9.1&$\times$& 9.1& *JHK*\
& & ROSS2 & 1.22& 9.9&$\times$& 9.9& *griz*\
&& IO:O & 0.30& 10&$\times$&10& *BVgriz*\
& & IO:I & 0.18& 6.3&$\times$&6.3& *H*\
& &NOTCam&0.24&4&$\times$&4&*JHK*\
& &ALFOSC &0.19&6.4&$\times$&6.4&*UBVugriz*\
MMT & Multiple Mirror Telescope (MMT)& MMTCam & 0.08& 2.7&$\times$&2.7& *gri*\
&&&&& &*UV-W1,M2,W2*\
&& & & &&& *UBV*\
\[tab: instr\]
[>m[1.2cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[3cm]{}]{} MJD & *u* & *g* & *r* & *i* & *z* & Instrument & (mag) & (mag) & (mag) & (mag) & (mag) & 57536.14 & $-$ & $-$ & 15.20 (0.04) & 15.46 (0.03) & $-$ & NOT 57536.40 & $-$ & 15.80 (0.05) & 15.17 (0.05) & 15.55 (0.10) & $-$ & Gem 57537.15 & $-$ & $-$ & 14.84 (0.15) & 15.09 (0.11) & $-$ & NOT 57537.41 & $-$ & 15.43 (0.04) & 14.87 (0.05) & 15.25 (0.10) & $-$ & Gem 57538.39 & $-$ & 15.17 (0.04) & 14.63 (0.05) & 15.03 (0.10) & $-$ & Gem 57539.39 & $-$ & 14.93 (0.04) & 14.43 (0.05) & 14.83 (0.10) & $-$ & Gem 57539.76 & $-$ & $-$ & $-$ & 14.69 (0.16) & $-$ & WHO 57540.17 & 15.68 (0.03) & 14.76 (0.03) & 14.25 (0.03) & 14.61 (0.02) & 14.56 (0.03) & LT 57540.76 & $-$ & $-$ & 14.14 (0.05) & 14.59 (0.04) & $-$ & WHO 57541.40 & $-$ & 14.59 (0.04) & 14.14 (0.05) & 14.54 (0.10) & $-$ & Gem 57541.75 & $-$ & $-$ & 14.11 (0.06) & 14.48 (0.06) & $-$ & WHO 57542.21 & 15.61 (0.04) & 14.47 (0.02) & 14.04 (0.03) & 14.39 (0.03) & 14.24 (0.02) & LT 57542.38 & $-$ & $-$ & 13.99 (0.03) & 14.39 (0.06) & $-$ & DEM 57543.37 & $-$ & 14.35 (0.05) & 13.90 (0.06) & 14.32 (0.10) & $-$ & Gem 57543.41 & $-$ & $-$ & 13.89 (0.03) & 14.30 (0.06) & $-$ & DEM 57544.37 & $-$ & 14.29 (0.04) & 13.86 (0.05) & 14.22 (0.10) & $-$ & Gem 57544.44 & $-$ & $-$ & 13.80 (0.03) & 14.23 (0.06) & $-$ & DEM 57545.34 & $-$ & $-$ & 13.74 (0.03) & 14.17 (0.06) & $-$ & DEM 57545.39 & $-$ & 14.21 (0.05) & 13.77 (0.07) & 14.15 (0.10) & $-$ & Gem 57546.22 & $-$ & $-$ & $-$ & 14.20 (0.10) & $-$ & NOT 57546.29 & $-$ & 14.22 (0.09) & 13.70 (0.06) & 14.13 (0.13) & $-$ & REM 57546.39 & $-$ & 14.19 (0.04) & 13.72 (0.06) & 14.09 (0.10) & $-$ & Gem 57546.43 & $-$ & $-$ & 13.68 (0.06) & 14.10 (0.06) & $-$ & LCOGT 57546.44 & $-$ & $-$ & 13.68 (0.03) & 14.10 (0.06) & $-$ & DEM 57547.33 & $-$ & $-$ & 13.65 (0.03) & 14.07 (0.06) & $-$ & DEM 57547.37 & $-$ & $-$ & 13.65 (0.07) & 14.06 (0.10) & $-$ & REM 57547.37 & $-$ & 14.16 (0.04) & 13.71 (0.05) & 14.08 (0.10) & $-$ & Gem 57548.38 & $-$ & $-$ & 13.62 (0.08) & 14.04 (0.11) & $-$ & REM 57549.42 & $-$ & 14.15 (0.06) & 13.59 (0.06) & 14.02 (0.11) & $-$ & REM 57549.43 & $-$ & $-$ & 13.60 (0.03) & 13.96 (0.05) & $-$ & LCOGT 57550.41 & $-$ & $-$ & 13.62 (0.03) & 13.97 (0.05) & $-$ & LCOGT 57550.44 & $-$ & $-$ & 13.59 (0.06) & 13.95 (0.12) & $-$ & REM 57551.19 & 15.92 (0.06) & 14.19 (0.02) & 13.54 (0.04) & 13.97 (0.04) & 13.69 (0.03) & LT 57551.42 & $-$ & $-$ & 13.56 (0.04) & 13.93 (0.06) & $-$ & DEM 57552.06 & 15.96 (0.06) & 14.19 (0.03) & 13.58 (0.02) & 13.95 (0.03) & 13.64 (0.03) & LT 57552.35 & $-$ & 14.22 (0.04) & 13.62 (0.05) & 13.95 (0.10) & $-$ & Gem 57553.17 & 16.10 (0.06) & 14.29 (0.05) & 13.59 (0.06) & 13.89 (0.06) & 13.55 (0.04) & LT 57553.19 & 16.10 (0.08) & $-$ & $-$ & $-$ & $-$ & NOT 57553.35 & $-$ & 14.26 (0.05) & 13.62 (0.05) & 13.95 (0.10) & $-$ & Gem 57553.37 & $-$ & $-$ & 13.57 (0.03) & 13.92 (0.06) & $-$ & DEM 57554.12 & 16.23 (0.04) & 14.32 (0.04) & 13.64 (0.07) & 13.95 (0.06) & 13.55 (0.04) & LT 57554.42 & $-$ & $-$ & 13.57 (0.03) & $-$ & $-$ & DEM 57555.24 & $-$ & 14.39 (0.25) & 13.59 (0.10) & 13.91 (0.11) & $-$ & REM 57556.37 & $-$ & 14.43 (0.04) & 13.67 (0.05) & 13.95 (0.10) & $-$ & Gem 57557.03 & 16.39 (0.03) & 14.44 (0.10) & 13.70 (0.07) & $-$ 13.54 (0.07) & 1.82 57557.13 & 16.59 (0.04) & 14.45 (0.02) & 13.58 (0.04) & 13.91 (0.03) & 13.59 (0.03) & LT 57557.22 & $-$ & $-$ & 13.62 (0.08) & 13.91 (0.12) & $-$ & REM 57557.43 & $-$ & $-$ & 13.64 (0.04) & 13.90 (0.05) & $-$ & LCOGT 57558.10 & 16.70 (0.04) & 14.55 (0.02) & 13.65 (0.03) & 13.98 (0.03) & 13.69 (0.04) & LT 57558.23 & $-$ & $-$ & 13.68 (0.06) & 13.93 (0.15) & $-$ & REM \[fig: ugriz\]
[>m[1.2cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[3cm]{}]{} MJD & *u* & *g* & *r* & *i* & *z* & Instrument & (mag) & (mag) & (mag) & (mag) & (mag) & 57558.40 & $-$ & $-$ & 13.68 (0.06) & 13.94 (0.11) & $-$ & LCOGT 57559.07 & 16.89 (0.05) & 14.60 (0.03) & 13.65 (0.03) & 13.95 (0.03) & 13.59 (0.03) & LT 57559.24 & $-$ & $-$ & 13.70 (0.08) & 13.90 (0.13) & $-$ & REM 57559.44 & $-$ & $-$ & 13.73 (0.03) & 13.97 (0.05) & $-$ & LCOGT 57560.30 & $-$ & 14.76 (0.35) & 13.74 (0.08) & 13.94 (0.11) & $-$ & REM 57560.40 & $-$ & $-$ & $-$ & 14.00 (0.05) & $-$ & LCOGT 57561.06 & 17.22 (0.06) & 14.72 (0.03) & 13.81 (0.06) & 14.04 (0.06) & 13.70 (0.03) & LT 57561.30 & $-$ & $-$ & 13.75 (0.08) & 13.92 (0.14) & $-$ & REM 57562.07 & 17.26 (0.07) & 14.81 (0.10) & 13.76 (0.10) & 14.02 (0.09) & 13.48 (0.08) & 1.82 57562.11 & 17.27 (0.05) & 14.90 (0.02) & 13.72 (0.04) & 14.05 (0.03) & 13.61 (0.03) & LT 57562.40 & $-$ & $-$ & 13.81 (0.06) & 14.00 (0.07) & $-$ & LCOGT 57563.13 & 17.47 (0.05) & 14.99 (0.03) & 13.86 (0.05) & 14.09 (0.04) & 13.90 (0.07) & LT 57563.43 & $-$ & $-$ & $-$ & 14.05 (0.05) & $-$ & LCOGT 57566.32 & $-$ & $-$ & 14.00 (0.07) & 14.12 (0.11) & $-$ & REM 57566.41 & $-$ & $-$ & 14.01 (0.03) & 14.18 (0.05) & $-$ & LCOGT 57567.33 & $-$ & $-$ & 14.03 (0.06) & 14.14 (0.13) & $-$ & REM 57567.40 & $-$ & $-$ & 14.07 (0.03) & 14.22 (0.05) & $-$ & LCOGT 57568.15 & 17.93 (0.05) & 15.34 (0.02) & 14.04 (0.03) & 14.19 (0.03) & 13.84 (0.02) & LT 57568.35 & $-$ & $-$ & 14.08 (0.09) & 14.16 (0.11) & $-$ & REM 57568.40 & $-$ & $-$ & 14.12 (0.03) & 14.24 (0.05) & $-$ & LCOGT 57569.08 & 18.05 (0.08) & 15.38 (0.01) & 14.03 (0.03) & 14.21 (0.03) & 13.90 (0.03) & LT 57569.36 & $-$ & $-$ & 14.13 (0.07) & 14.22 (0.12) & $-$ & REM 57570.17 & 18.06 (0.05) & 15.46 (0.03) & 14.15 (0.04) & 14.28 (0.03) & 13.85 (0.03) & LT 57572.98 & $-$ & $-$ & 14.31 (0.04) & 14.39 (0.06) & $-$ & LCOGT 57573.02 & 18.14 (0.07) & 15.67 (0.14) & 14.33 (0.11) & 14.37 (0.11) & $-$ & NOT 57576.09 & $-$ & $-$ & 14.47 (0.04) & 14.49 (0.06) & $-$ & LCOGT 57577.10 & 18.46 (0.05) & 15.71 (0.04) & 14.54 (0.05) & 14.51 (0.03) & 14.05 (0.03) & LT 57577.13 & $-$ & $-$ & 14.51 (0.04) & 14.53 (0.06) & $-$ & LCOGT 57578.13 & $-$ & $-$ & 14.56 (0.04) & 14.56 (0.06) & $-$ & LCOGT 57579.11 & 18.44 (0.06) & 15.79 (0.03) & 14.52 (0.03) & 14.63 (0.03) & 14.14 (0.03) & LT 57580.06 & $-$ & $-$ & 14.64 (0.04) & 14.61 (0.06) & $-$ & LCOGT 57580.11 & 18.48 (0.05) & 15.88 (0.03) & 14.59 (0.04) & 14.67 (0.03) & 14.26 (0.04) & LT 57581.10 & 18.48 (0.04) & 15.99 (0.03) & 14.58 (0.04) & 14.63 (0.05) & 14.13 (0.03) & LT 57581.13 & $-$ & $-$ & 14.66 (0.04) & 14.65 (0.06) & $-$ & LCOGT 57582.13 & $-$ & $-$ & 14.73 (0.04) & 14.70 (0.05) & $-$ & LCOGT 57583.13 & $-$ & $-$ & 14.75 (0.04) & 14.72 (0.06) & $-$ & LCOGT 57584.11 & $-$ & 15.98 (0.03) & 14.79 (0.04) & 14.75 (0.07) & $-$ & LCOGT 57585.29 & $-$ & $-$ & 14.82 (0.04) & 14.76 (0.06) & $-$ & LCOGT 57586.13 & $-$ & $-$ & 14.85 (0.05) & 14.77 (0.08) & $-$ & LCOGT 57586.13 & 18.61 (0.04) & 16.15 (0.04) & 14.80 (0.05) & 14.79 (0.03) & 14.31 (0.02) & LT 57587.34 & $-$ & $-$ & 14.89 (0.04) & 14.83 (0.07) & $-$ & LCOGT 57588.13 & 18.66 (0.04) & 16.01 (0.02) & 14.93 (0.03) & 14.87 (0.02) & 14.36 (0.03) & LT 57590.10 & 18.93 (0.15) & 16.14 (0.15) & 14.95 (0.13) & 14.84 (0.08) & $-$ & NOT 57591.32 & $-$ & $-$ & 14.99 (0.06) & 14.92 (0.08) & $-$ & LCOGT 57593.07 & $-$ & $-$ & 15.01 (0.07) & 14.97 (0.05) & $-$ &T50 57593.11 & 18.78 (0.05) & 16.16 (0.04) & 14.97 (0.03) & 14.98 (0.03) & 14.42 (0.03) & LT 57594.05 & $-$ & $-$ & 15.11 (0.08) & 15.04 (0.07) & $-$ &T50 57594.70 & $-$ & $-$ & 15.11 (0.04) & 15.00 (0.06) & $-$ & LCOGT 57596.01 & 18.86 (0.06) & 16.22 (0.05) & 15.05 (0.05) & 15.07 (0.03) & 14.52 (0.08) & LT 57596.07 & $-$ & $-$ & 15.17 (0.06) & 15.06 (0.04) & $-$ &T50
[>m[1.2cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[3cm]{}]{} MJD & *u* & *g* & *r* & *i* & *z* & Instrument & (mag) & (mag) & (mag) & (mag) & (mag) & 57597.07 & $-$ & $-$ & 15.16 (0.28) & $-$ & $-$ &T50 57597.73 & $-$ & $-$ & 15.18 (0.04) & 15.07 (0.06) & $-$ & LCOGT 57598.20 & 18.80 (0.04) & 16.29 (0.03) & 15.13 (0.04) & 15.08 (0.04) & 14.46 (0.03) & LT 57598.72 & $-$ & $-$ & 15.20 (0.04) & 15.08 (0.06) & $-$ & LCOGT 57599.70 & $-$ & $-$ & 15.24 (0.04) & 15.09 (0.06) & $-$ & LCOGT 57601.14 & 18.92 (0.07) & 16.40 (0.03) & 15.25 (0.04) & 15.14 (0.03) & $-$ & NOT 57604.99 & $-$ & $-$ & 15.37 (0.04) & 15.24 (0.06) & $-$ & LCOGT 57607.06 & $-$ & $-$ & 15.38 (0.09) & $-$ & $-$ & LCOGT 57609.00 & 19.20 (0.09) & 16.41 (0.09) & 15.44 (0.10) & 15.29 (0.09) & 14.73 (0.15) & 1.82 57610.68 & $-$ & $-$ & 15.50 (0.04) & 15.29 (0.06) & $-$ & LCOGT 57612.65 & $-$ & $-$ & 15.52 (0.04) & 15.37 (0.06) & $-$ & LCOGT 57619.04 & 19.08 (0.14) & 16.53 (0.09) & 15.61 (0.11) & 15.59 (0.08) & $-$ & NOT 57621.27 & $-$ & $-$ & 15.68 (0.04) & 15.46 (0.06) & $-$ & LCOGT 57623.06 & 19.41 (0.15) & 16.52 (0.12) & 15.76 (0.14) & 15.61 (0.11) & 14.93 (0.14) & 1.82 57623.29 & $-$ & $-$ & 15.69 (0.04) & 15.52 (0.06) & $-$ & LCOGT 57624.99 & $-$ & $-$ & 15.72 (0.04) & 15.58 (0.06) & $-$ & LCOGT 57625.92 & 19.14 (0.10) & 16.63 (0.08) & 15.86 (0.07) & 15.78 (0.04) & 15.10 (0.06) & 1.82 57627.00 & $-$ & $-$ & 15.75 (0.04) & 15.62 (0.06) & $-$ & LCOGT 57628.98 & $-$ & $-$ & 15.75 (0.05) & 15.59 (0.07) & $-$ & LCOGT 57636.22 & $-$ & $-$ & 15.91 (0.04) & 15.79 (0.06) & $-$ & LCOGT 57638.63 & $-$ & $-$ & 15.94 (0.04) & 15.82 (0.06) & $-$ & LCOGT 57640.24 & $-$ & $-$ & 15.98 (0.04) & 15.81 (0.06) & $-$ & LCOGT 57644.01 & 19.53 (0.12) & 16.91 (0.05) & 16.04 (0.05) & 15.93 (0.04) & 15.32 (0.05) & NOT 57644.17 & $-$ & $-$ & 16.03 (0.05) & 15.86 (0.08) & $-$ & LCOGT 57650.92 & $-$ & $-$ & 16.14 (0.05) & 16.00 (0.07) & $-$ & LCOGT 57652.18 & $-$ & $-$ & 16.16 (0.04) & 16.04 (0.06) & $-$ & LCOGT 57654.83 & $-$ & $-$ & 16.20 (0.04) & 16.10 (0.06) & $-$ & LCOGT 57654.90 & 19.55 (0.12) & 16.92 (0.10) & 16.22 (0.11) & 16.16 (0.05) & 15.35 (0.12) & 1.82 57655.11 & 19.71 (0.07) & 17.01 (0.03) & 16.20 (0.04) & 16.04 (0.03) & 15.43 (0.04) & NOT 57662.15 & $-$ & 17.19 (0.06) & 16.38 (0.04) & 16.18 (0.03) & 15.58 (0.03) & NOT 57663.90 & $-$ & $-$ & 16.34 (0.05) & 16.31 (0.07) & $-$ & LCOGT 57665.84 & $-$ & $-$ & 16.34 (0.04) & 16.27 (0.06) & $-$ & LCOGT 57667.86 & $-$ & $-$ & 16.33 (0.04) & 16.28 (0.07) & $-$ & LCOGT 57670.08 & $-$ & $-$ & 16.35 (0.04) & 16.30 (0.06) & $-$ & LCOGT 57672.85 & $-$ & $-$ & 16.43 (0.09) & 16.34 (0.10) & $-$ & LCOGT 57680.10 & $-$ & $-$ & 16.55 (0.04) & 16.47 (0.06) & $-$ & LCOGT 57687.84 & $-$ & $-$ & 16.58 (0.04) & 16.61 (0.06) & $-$ & LCOGT 57694.82 & $-$ & $-$ & 16.70 (0.04) & 16.71 (0.06) & $-$ & LCOGT 57699.81 & $-$ & $-$ & 16.81 (0.04) & 16.85 (0.06) & $-$ & LCOGT 57704.77 & $-$ & $-$ & 16.81 (0.05) & 16.89 (0.07) & $-$ & LCOGT 57707.79 & $-$ & $-$ & 17.11 (0.05) & 16.94 (0.06) & $-$ & LCOGT 57719.17 & $-$ & $-$ & 17.06 (0.13) & 17.38 (0.29) & $-$ & LCOGT 57727.71 & $-$ & 18.08 (0.06) & 17.30 (0.04) & 17.20 (0.09) & 17.06 (0.07) & 1.82 57728.06 & $-$ & $-$ & 17.18 (0.04) & 17.26 (0.07) & $-$ & LCOGT 57728.86 & 20.89 (0.36) & 18.34 (0.14) & 17.34 (0.10) & 17.32 (0.07) & 17.27 (0.15) & 1.82 57736.06 & $-$ & $-$ & 17.27 (0.07) & 17.32 (0.10) & $-$ & LCOGT 57906.38 & $-$ & 21.38 (0.61) & 19.57 (0.51) & 19.98 (0.53) & $-$ & MMT
[>m[1.2cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[3cm]{}]{} MJD & *U* & *B* & *V* & *R* & *I* & Instrument & (mag) & (mag) & (mag) & (mag) & (mag) & 57527.53 & $-$ & $-$ & $>$17.43 & $-$ & $-$ & ASA 57529.54 & $-$ & $-$ & $>$17.29 & $-$ & $-$ & ASA 57535.55 & $-$ & $-$ & 15.74 (0.07) & $-$ & $-$ & ASA 57535.77 & 15.86 (0.06) & 16.51 (0.15) & 15.86 (0.09) & $-$ & $-$ & UVOT 57536.14 & $-$ & 16.30 (0.03) & 15.51 (0.03) & $-$ & $-$ & NOT 57536.36 & $-$ & 16.29 (0.04) & 15.46 (0.03) & 15.10 (0.04) & 15.19 (0.07) & RK 57536.40 & $-$ & 16.27 (0.10) & 15.45 (0.06) & $-$ & $-$ & Gem 57537.14 & $-$ & 15.88 (0.14) & 15.14 (0.09) & $-$ & $-$ & NOT 57537.35 & $-$ & 15.95 (0.04) & 15.11 (0.03) & 14.78 (0.04) & 14.95 (0.07) & RK 57537.41 & $-$ & 15.95 (0.09) & 15.07 (0.06) & $-$ & $-$ & Gem 57538.37 & $-$ & $-$ & 14.80 (0.03) & 14.51 (0.04) & 14.69 (0.07) & RK 57538.39 & $-$ & 15.70 (0.09) & 14.80 (0.05) & $-$ & $-$ & Gem 57539.09 & 15.09 (0.05) & 15.61 (0.10) & 15.07 (0.06) & $-$ & $-$ & UVOT 57539.39 & $-$ & 15.49 (0.09) & 14.55 (0.05) & $-$ & $-$ & Gem 57539.57 & $-$ & $-$ & 14.56 (0.02) & $-$ & $-$ & ASA 57539.76 & $-$ & 15.42 (0.09) & 14.50 (0.08) & $-$ & $-$ & WHO 57540.76 & $-$ & 15.27 (0.08) & 14.28 (0.06) & $-$ & $-$ & WHO 57541.21 & 14.84 (0.11) & 15.16 (0.08) & 14.08 (0.06) & $-$ & $-$ & NOT 57541.36 & $-$ & 15.18 (0.04) & 14.21 (0.03) & 14.01 (0.04) & 14.24 (0.06) & RK 57541.40 & $-$ & 15.18 (0.09) & 14.19 (0.05) & $-$ & $-$ & Gem 57541.75 & $-$ & 15.14 (0.09) & 14.14 (0.07) & $-$ & $-$ & WHO 57542.36 & $-$ & 15.09 (0.04) & 14.07 (0.03) & 13.88 (0.04) & 14.14 (0.07) & RK 57542.38 & $-$ & 15.05 (0.05) & 14.06 (0.04) & $-$ & $-$ & DEM 57542.61 & 14.69 (0.06) & 15.01 (0.06) & 14.09 (0.05) & $-$ & $-$ & UVOT 57543.37 & $-$ & 15.00 (0.09) & 13.96 (0.06) & $-$ & $-$ & Gem 57543.41 & $-$ & 14.96 (0.04) & 13.93 (0.04) & $-$ & $-$ & DEM 57543.97 & 14.68 (0.06) & 14.93 (0.06) & 13.94 (0.05) & $-$ & $-$ & UVOT 57544.37 & $-$ & 14.91 (0.09) & 13.85 (0.06) & $-$ & $-$ & Gem 57544.44 & $-$ & 14.90 (0.04) & 13.83 (0.04) & $-$ & $-$ & DEM 57544.57 & $-$ & $-$ & 13.90 (0.02) & $-$ & $-$ & ASA 57545.34 & $-$ & 14.86 (0.05) & 13.78 (0.04) & $-$ & $-$ & DEM 57545.39 & $-$ & 14.87 (0.09) & 13.76 (0.06) & $-$ & $-$ & Gem 57545.94 & 14.74 (0.05) & 14.87 (0.06) & 13.81 (0.04) & $-$ & $-$ & UVOT 57546.39 & $-$ & 14.84 (0.09) & 13.73 (0.07) & $-$ & $-$ & Gem 57546.43 & $-$ & 14.84 (0.05) & 13.79 (0.05) & $-$ & $-$ & LCOGT 57546.44 & $-$ & 14.82 (0.04) & 13.73 (0.04) & $-$ & $-$ & DEM 57547.33 & $-$ & 14.83 (0.04) & 13.71 (0.04) & $-$ & $-$ & DEM 57547.37 & $-$ & 14.85 (0.09) & 13.72 (0.06) & $-$ & $-$ & Gem 57547.47 & 14.86 (0.06) & 14.82 (0.06) & 13.73 (0.04) & $-$ & $-$ & UVOT 57549.31 & 14.98 (0.06) & 14.85 (0.06) & 13.71 (0.05) & $-$ & $-$ & UVOT 57549.43 & $-$ & 14.85 (0.06) & 13.71 (0.03) & $-$ & $-$ & LCOGT 57549.50 & $-$ & $-$ & 13.76 (0.02) & $-$ & $-$ & ASA 57550.41 & $-$ & 14.89 (0.06) & 13.74 (0.03) & $-$ & $-$ & LCOGT 57550.58 & 15.04 (0.06) & 14.85 (0.06) & 13.62 (0.04) & $-$ & $-$ & UVOT 57551.42 & $-$ & 14.89 (0.04) & 13.68 (0.04) & $-$ & $-$ & DEM 57552.18 & $-$ & 14.88 (0.11) & 13.61 (0.14) & $-$ & $-$ & NOT 57552.35 & $-$ & 14.96 (0.09) & 13.72 (0.05) & $-$ & $-$ & Gem 57552.40 & 15.14 (0.06) & 14.94 (0.06) & 13.69 (0.05) & $-$ & $-$ & UVOT 57553.19 & $-$ & 14.98 (0.07) & 13.73 (0.07) & $-$ & $-$ & NOT 57553.35 & $-$ & 15.01 (0.09) & 13.75 (0.05) & $-$ & $-$ & Gem \[fig: UBVRI\]
[>m[1.2cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[3cm]{}]{} MJD & *U* & *B* & *V* & *R* & *I* & Instrument & (mag) & (mag) & (mag) & (mag) & (mag) & 57553.37 & $-$ & 14.99 (0.04) & 13.74 (0.04) & $-$ & $-$ & DEM 57554.20 & 15.45 (0.06) & 15.05 (0.06) & 13.73 (0.05) & $-$ & $-$ & UVOT 57554.30 & $-$ & 15.09 (0.05) & 13.77 (0.04) & 13.47 (0.05) & 13.68 (0.08) & RK 57554.42 & $-$ & 15.05 (0.04) & 13.76 (0.04) & $-$ & $-$ & DEM 57555.34 & $-$ & 15.12 (0.04) & 13.82 (0.03) & 13.46 (0.05) & 13.63 (0.07) & RK 57555.56 & 15.55 (0.06) & 15.07 (0.06) & 13.78 (0.04) & $-$ & $-$ & UVOT 57556.37 & $-$ & 15.18 (0.10) & 13.87 (0.06) & $-$ & $-$ & Gem 57557.03 & $-$ & 15.06 (0.08) & 13.89 (0.05) & $-$ & $-$ & 1.82 57557.34 & $-$ & 15.28 (0.10) & 13.99 (0.15) & 13.52 (0.12) & 13.60 (0.16) & JB 57557.43 & $-$ & 15.29 (0.06) & 13.89 (0.04) & $-$ & $-$ & LCOGT 57557.75 & 15.88 (0.07) & 15.32 (0.07) & 13.90 (0.05) & $-$ & $-$ & UVOT 57558.40 & $-$ & 15.38 (0.08) & 13.98 (0.09) & $-$ & $-$ & LCOGT 57558.45 & $-$ & $-$ & 13.97 (0.02) & $-$ & $-$ & ASA 57559.44 & $-$ & 15.51 (0.06) & 14.03 (0.03) & $-$ & $-$ & LCOGT 57560.57 & 16.24 (0.07) & 15.62 (0.08) & 14.08 (0.05) & $-$ & $-$ & UVOT 57561.52 & $-$ & $-$ & 14.16 (0.03) & $-$ & $-$ & ASA 57562.06 & $-$ & 15.77 (0.11) & 14.28 (0.11) & $-$ & $-$ & 1.82 57562.40 & $-$ & 15.78 (0.09) & 14.21 (0.05) & $-$ & $-$ & LCOGT 57563.43 & $-$ & 15.85 (0.06) & 14.25 (0.04) & $-$ & $-$ & LCOGT 57566.23 & 16.90 (0.09) & 16.07 (0.10) & 14.49 (0.05) & $-$ & $-$ & UVOT 57566.33 & $-$ & 16.05 (0.04) & 14.46 (0.04) & 13.83 (0.04) & 13.89 (0.07) & RK 57566.41 & $-$ & 16.11 (0.06) & 14.47 (0.03) & $-$ & $-$ & LCOGT 57567.40 & $-$ & 16.19 (0.06) & 14.55 (0.03) & $-$ & $-$ & LCOGT 57568.40 & $-$ & 16.25 (0.07) & 14.60 (0.03) & $-$ & $-$ & LCOGT 57568.53 & $-$ & $-$ & 14.64 (0.02) & $-$ & $-$ & ASA 57569.38 & $-$ & 16.29 (0.10) & 14.55 (0.13) & 13.97 (0.13) & 13.81 (0.17) & JB 57569.42 & $-$ & 16.18 (0.04) & 14.65 (0.02) & $-$ & $-$ & GS 57570.43 & $-$ & $-$ & 14.77 (0.03) & $-$ & $-$ & ASA 57572.28 & $-$ & 16.39 (0.05) & 14.79 (0.03) & 14.08 (0.04) & 14.05 (0.07) & RK 57572.98 & $-$ & $-$ & 14.83 (0.04) & $-$ & $-$ & LCOGT 57572.98 & 17.45 (0.16) & 16.45 (0.12) & 14.92 (0.07) & $-$ & $-$ & UVOT 57573.02 & $-$ & 16.46 (0.13) & 14.81 (0.14) & $-$ & $-$ & NOT 57573.35 & $-$ & 16.55 (0.08) & 14.80 (0.12) & 14.16 (0.12) & 14.03 (0.16) & JB 57574.35 & $-$ & 16.44 (0.05) & 14.90 (0.03) & $-$ & $-$ & GS 57574.52 & $-$ & $-$ & 14.92 (0.03) & $-$ & $-$ & ASA 57575.29 & $-$ & 16.50 (0.06) & 14.92 (0.13) & 14.22 (0.22) & 14.14 (0.16) & RK 57575.44 & $-$ & 16.67 (0.13) & 14.91 (0.14) & 14.25 (0.13) & 14.09 (0.19) & JB 57576.09 & $-$ & 16.69 (0.08) & 15.00 (0.04) & $-$ & $-$ & LCOGT 57576.77 & 17.58 (0.14) & 16.48 (0.12) & 15.04 (0.06) & $-$ & $-$ & UVOT 57577.13 & $-$ & $-$ & 15.04 (0.04) & $-$ & $-$ & LCOGT 57577.54 & $-$ & $-$ & 15.09 (0.03) & $-$ & $-$ & ASA 57578.13 & $-$ & $-$ & 15.07 (0.04) & $-$ & $-$ & LCOGT 57578.73 & 17.72 (0.15) & 16.65 (0.13) & 15.19 (0.06) & $-$ & $-$ & UVOT 57579.14 & $-$ & $-$ & 15.09 (0.04) & $-$ & $-$ & LCOGT 57580.06 & $-$ & $-$ & 15.17 (0.04) & $-$ & $-$ & LCOGT 57580.33 & $-$ & 16.69 (0.09) & 15.12 (0.13) & 14.40 (0.12) & 14.18 (0.17) & JB 57580.50 & $-$ & $-$ & 15.23 (0.03) & $-$ & $-$ & ASA 57581.13 & $-$ & 16.80 (0.07) & 15.19 (0.04) & $-$ & $-$ & LCOGT
[>m[1.2cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[3cm]{}]{} MJD & *U* & *B* & *V* & *R* & *I* & Instrument & (mag) & (mag) & (mag) & (mag) & (mag) & 57581.30 & 17.59 (0.14) & 16.74 (0.14) & 15.20 (0.06) & $-$ & $-$ & UVOT 57582.13 & $-$ & 16.85 (0.07) & 15.24 (0.04) & $-$ & $-$ & LCOGT 57582.34 & $-$ & 16.77 (0.10) & 15.22 (0.13) & 14.49 (0.13) & 14.28 (0.17) & JB 57583.13 & $-$ & $-$ & 15.28 (0.04) & $-$ & $-$ & LCOGT 57583.27 & $-$ & 16.66 (0.06) & 15.24 (0.06) & 14.47 (0.25) & 14.25 (0.15) & RK 57583.42 & $-$ & $-$ & 15.38 (0.04) & $-$ & $-$ & ASA 57583.45 & $-$ & 16.70 (0.04) & 15.25 (0.03) & $-$ & $-$ & GS 57584.11 & $-$ & $-$ & 15.28 (0.04) & $-$ & $-$ & LCOGT 57585.29 & $-$ & 16.84 (0.06) & 15.34 (0.04) & $-$ & $-$ & LCOGT 57586.13 & $-$ & $-$ & 15.42 (0.06) & $-$ & $-$ & LCOGT 57586.57 & 17.90 (0.16) & 16.84 (0.16) & 15.41 (0.06) & $-$ & $-$ & UVOT 57587.34 & $-$ & 16.80 (0.11) & 15.40 (0.06) & $-$ & $-$ & LCOGT 57587.41 & $-$ & $-$ & 15.38 (0.06) & $-$ & $-$ & ASA 57587.42 & $-$ & 16.73 (0.05) & 15.39 (0.03) & $-$ & $-$ & GS 57589.39 & $-$ & 16.88 (0.06) & 15.42 (0.04) & $-$ & $-$ & GS 57590.10 & $-$ & 16.86 (0.09) & 15.37 (0.09) & $-$ & $-$ & NOT 57590.15 & 17.77 (0.14) & 16.90 (0.17) & 15.53 (0.07) & $-$ & $-$ & UVOT 57591.32 & $-$ & 16.80 (0.09) & 15.49 (0.06) & $-$ & $-$ & LCOGT 57591.41 & $-$ & $-$ & 15.47 (0.03) & $-$ & $-$ & GS 57593.07 & $-$ & 16.82 (0.07) & 15.55 (0.05) & $-$ & $-$ &T50 57594.05 & $-$ & 16.92 (0.06) & 15.61 (0.10) & $-$ & $-$ &T50 57594.33 & $-$ & 16.90 (0.06) & 15.49 (0.14) & 14.83 (0.14) & 14.63 (0.17) & JB 57594.54 & 17.83 (0.15) & 16.95 (0.17) & 15.63 (0.07) & $-$ & $-$ & UVOT 57594.70 & $-$ & 17.01 (0.08) & 15.53 (0.04) & $-$ & $-$ & LCOGT 57595.46 & $-$ & $-$ & 15.65 (0.05) & $-$ & $-$ & ASA 57596.07 & $-$ & 16.82 (0.15) & 15.49 (0.13) & $-$ & $-$ &T50 57597.06 & $-$ & 16.95 (0.06) & 15.57 (0.08) & $-$ & $-$ &T50 57597.29 & $-$ & 17.11 (0.08) & 15.56 (0.12) & 14.87 (0.13) & 14.69 (0.17) & JB 57597.41 & $-$ & $-$ & 15.65 (0.04) & $-$ & $-$ & ASA 57597.73 & $-$ & $-$ & 15.61 (0.04) & $-$ & $-$ & LCOGT 57598.26 & $-$ & 17.07 (0.09) & 15.57 (0.13) & 14.88 (0.13) & 14.69 (0.17) & JB 57598.72 & $-$ & 16.86 (0.06) & 15.67 (0.04) & $-$ & $-$ & LCOGT 57598.77 & $-$ & 16.91 (0.16) & 15.71 (0.10) & $-$ & $-$ & UVOT 57599.70 & $-$ & 17.00 (0.07) & 15.65 (0.04) & $-$ & $-$ & LCOGT 57600.24 & $-$ & 17.11 (0.06) & 15.66 (0.11) & 14.95 (0.10) & 14.76 (0.15) & JB 57601.13 & $-$ & 17.07 (0.07) & 15.66 (0.04) & $-$ & $-$ & NOT 57601.44 & $-$ & $-$ & 15.77 (0.04) & $-$ & $-$ & ASA 57603.62 & 18.24 (0.26) & 17.11 (0.20) & 15.83 (0.08) & $-$ & $-$ & UVOT 57604.31 & $-$ & 17.12 (0.04) & 15.68 (0.09) & 15.15 (0.09) & 14.83 (0.14) & JB 57604.53 & $-$ & $-$ & 15.75 (0.04) & $-$ & $-$ & ASA 57604.99 & $-$ & $-$ & 15.82 (0.04) & $-$ & $-$ & LCOGT 57606.84 & $-$ & 17.11 (0.18) & 15.90 (0.10) & $-$ & $-$ & UVOT 57607.06 & $-$ & $-$ & 15.74 (0.12) & $-$ & $-$ & LCOGT 57607.50 & $-$ & $-$ & 15.85 (0.05) & $-$ & $-$ & ASA 57608.47 & $-$ & 17.11 (0.18) & 15.95 (0.10) & $-$ & $-$ & UVOT 57609.00 & $-$ & 16.95 (0.09) & 15.84 (0.09) & $-$ & $-$ & 1.82 57609.24 & $-$ & 17.12 (0.08) & 15.71 (0.15) & 15.19 (0.18) & 14.80 (0.13) & RK 57610.35 & $-$ & $-$ & 15.91 (0.02) & $-$ & $-$ & GS 57610.37 & $-$ & $-$ & 15.80 (0.05) & $-$ & $-$ & ASA 57610.68 & $-$ & $-$ & 15.88 (0.04) & $-$ & $-$ & LCOGT
[>m[1.2cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[3cm]{}]{} MJD & *U* & *B* & *V* & *R* & *I* & Instrument & (mag) & (mag) & (mag) & (mag) & (mag) & 57611.53 & $-$ & $-$ & 15.97 (0.05) & $-$ & $-$ & ASA 57612.40 & $-$ & 17.16 (0.05) & 15.89 (0.01) & $-$ & $-$ & GS 57612.65 & $-$ & $-$ & 15.91 (0.04) & $-$ & $-$ & LCOGT 57613.20 & $-$ & 17.17 (0.06) & 15.78 (0.11) & 15.24 (0.24) & 14.90 (0.18) & RK 57613.31 & $-$ & $-$ & 15.90 (0.06) & $-$ & $-$ & ASA 57613.51 & 17.96 (0.21) & 17.26 (0.20) & 15.94 (0.10) & $-$ & $-$ & UVOT 57614.36 & $-$ & 17.07 (0.05) & 15.91 (0.03) & $-$ & $-$ & GS 57614.56 & $-$ & $-$ & 15.87 (0.05) & $-$ & $-$ & ASA 57615.28 & $-$ & $-$ & 16.09 (0.10) & $-$ & $-$ & ASA 57615.74 & 18.29 (0.22) & 17.27 (0.21) & 15.97 (0.09) & $-$ & $-$ & UVOT 57618.48 & $-$ & $-$ & 15.97 (0.12) & $-$ & $-$ & ASA 57619.04 & $-$ & 17.21 (0.07) & 16.13 (0.07) & $-$ & $-$ & NOT 57621.27 & $-$ & 17.12 (0.07) & 16.03 (0.04) & $-$ & $-$ & LCOGT 57623.05 & $-$ & 17.23 (0.09) & 16.02 (0.12) & $-$ & $-$ & 1.82 57623.29 & $-$ & $-$ & 16.09 (0.04) & $-$ & $-$ & LCOGT 57624.99 & $-$ & 17.24 (0.07) & 16.09 (0.04) & $-$ & $-$ & LCOGT 57625.92 & $-$ & 17.02 (0.09) & 16.14 (0.06) & $-$ & $-$ & 1.82 57626.53 & $-$ & $-$ & 16.07 (0.06) & $-$ & $-$ & ASA 57627.00 & $-$ & $-$ & 16.12 (0.04) & $-$ & $-$ & LCOGT 57627.28 & $-$ & 17.30 (0.21) & 16.20 (0.10) & $-$ & $-$ & UVOT 57628.98 & $-$ & $-$ & 16.23 (0.06) & $-$ & $-$ & LCOGT 57629.64 & 18.34 (0.31) & 17.55 (0.26) & 16.24 (0.11) & $-$ & $-$ & UVOT 57630.48 & $-$ & $-$ & 16.33 (0.07) & $-$ & $-$ & ASA 57632.00 & $-$ & 17.12 (0.19) & 16.36 (0.15) & $-$ & $-$ & UVOT 57634.24 & $-$ & 17.26 (0.17) & 16.10 (0.11) & 15.54 (0.13) & 15.31 (0.07) & RK 57635.08 & $-$ & $-$ & 16.36 (0.08) & $-$ & $-$ & ASA 57636.08 & $-$ & $-$ & 16.25 (0.09) & $-$ & $-$ & ASA 57636.22 & $-$ & $-$ & 16.28 (0.04) & $-$ & $-$ & LCOGT 57636.28 & 18.40 (0.24) & 17.47 (0.24) & 16.38 (0.11) & $-$ & $-$ & UVOT 57637.50 & $-$ & $-$ & 16.36 (0.07) & $-$ & $-$ & ASA 57638.63 & $-$ & 17.33 (0.07) & 16.30 (0.04) & $-$ & $-$ & LCOGT 57639.40 & $-$ & $-$ & 16.35 (0.06) & $-$ & $-$ & ASA 57640.24 & $-$ & $-$ & 16.35 (0.04) & $-$ & $-$ & LCOGT 57642.16 & $-$ & $-$ & 16.29 (0.14) & $-$ & $-$ & ASA 57643.52 & 18.10 (0.18) & 17.52 (0.26) & 16.57 (0.12) & $-$ & $-$ & UVOT 57644.00 & $-$ & 17.58 (0.06) & 16.45 (0.05) & $-$ & $-$ & NOT 57644.17 & $-$ & 17.11 (0.11) & 16.36 (0.06) & $-$ & $-$ & LCOGT 57650.36 & $-$ & $-$ & $-$ & $-$ & $-$ & UVOT 57650.41 & $-$ & $-$ & 16.68 (0.13) & $-$ & $-$ & ASA 57650.92 & $-$ & $-$ & 16.53 (0.06) & $-$ & $-$ & LCOGT 57652.18 & $-$ & 17.63 (0.85) & 16.55 (0.04) & $-$ & $-$ & LCOGT 57654.83 & $-$ & 17.63 (0.09) & 16.60 (0.04) & $-$ & $-$ & LCOGT 57654.90 & $-$ & 17.32 (0.10) & 16.53 (0.06) & $-$ & $-$ & 1.82 57655.04 & $-$ & $-$ & 16.61 (0.10) & $-$ & $-$ & ASA 57655.11 & $-$ & 17.63 (0.04) & 16.50 (0.03) & $-$ & $-$ & NOT 57656.22 & $-$ & 17.49 (0.10) & 16.89 (0.14) & 15.94 (0.20) & 15.65 (0.13) & RK 57657.35 & $-$ & $-$ & 16.68 (0.07) & $-$ & $-$ & ASA 57657.96 & 18.66 (0.50) & 17.56 (0.26) & 16.87 (0.21) & $-$ & $-$ & UVOT 57660.11 & $-$ & $-$ & 16.62 (0.10) & $-$ & $-$ & ASA 57660.83 & $-$ & 17.79 (0.31) & 16.80 (0.15) & $-$ & $-$ & UVOT
[>m[1.2cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[3cm]{}]{} MJD & *U* & *B* & *V* & *R* & *I* & Instrument & (mag) & (mag) & (mag) & (mag) & (mag) & 57662.15 & $-$ & 17.73 (0.08) & 16.70 (0.03) & $-$ & $-$ & NOT 57662.69 & 18.81 (0.31) & 17.84 (0.33) & 16.93 (0.15) & $-$ & $-$ & UVOT 57663.90 & $-$ & $-$ & 16.75 (0.06) & $-$ & $-$ & LCOGT 57665.84 & $-$ & 17.96 (0.09) & 16.77 (0.04) & $-$ & $-$ & LCOGT 57667.42 & $-$ & $-$ & 16.80 (0.11) & $-$ & $-$ & ASA 57667.86 & $-$ & 17.85 (0.10) & 16.75 (0.05) & $-$ & $-$ & LCOGT 57668.33 & $-$ & $-$ & 17.00 (0.10) & $-$ & $-$ & ASA 57670.08 & $-$ & 17.81 (0.07) & 16.81 (0.04) & $-$ & $-$ & LCOGT 57671.29 & $-$ & $-$ & 16.96 (0.18) & $-$ & $-$ & ASA 57671.53 & 18.66 (0.29) & 17.93 (0.35) & 17.15 (0.19) & $-$ & $-$ & UVOT 57672.85 & $-$ & 17.76 (0.21) & 16.95 (0.13) & $-$ & $-$ & LCOGT 57675.22 & $-$ & $-$ & 16.90 (0.20) & $-$ & $-$ & ASA 57678.36 & $-$ & $-$ & 17.25 (0.27) & $-$ & $-$ & ASA 57680.10 & $-$ & 17.94 (0.09) & 16.98 (0.05) & $-$ & $-$ & LCOGT 57680.26 & $-$ & $-$ & 17.10 (0.10) & $-$ & $-$ & ASA 57681.37 & 18.81 (0.29) & 18.14 (0.43) & 17.13 (0.17) & $-$ & $-$ & UVOT 57687.36 & $-$ & $-$ & 17.17 (0.14) & $-$ & $-$ & ASA 57687.84 & $-$ & 18.03 (0.07) & 17.09 (0.04) & $-$ & $-$ & LCOGT 57690.35 & $-$ & $-$ & 17.03 (0.09) & $-$ & $-$ & ASA 57691.39 & 19.35 (0.45) & 18.36 (0.53) & 17.41 (0.21) & $-$ & $-$ & UVOT 57692.26 & $-$ & $-$ & 16.96 (0.10) & $-$ & $-$ & ASA 57694.82 & $-$ & 18.14 (0.08) & 17.20 (0.05) & $-$ & $-$ & LCOGT 57695.34 & $-$ & $-$ & 17.39 (0.15) & $-$ & $-$ & ASA 57697.33 & $-$ & $-$ & 17.11 (0.12) & $-$ & $-$ & ASA 57699.81 & $-$ & 18.32 (0.08) & 17.36 (0.05) & $-$ & $-$ & LCOGT 57702.25 & $-$ & $-$ & 17.61 (0.23) & $-$ & $-$ & ASA 57702.88 & $-$ & 18.57 (0.62) & 17.52 (0.23) & $-$ & $-$ & UVOT 57704.77 & $-$ & 18.43 (0.16) & 17.30 (0.08) & $-$ & $-$ & LCOGT 57707.21 & $-$ & $-$ & 17.78 (0.33) & $-$ & $-$ & ASA 57707.79 & $-$ & 18.66 (0.09) & 17.73 (0.06) & $-$ & $-$ & LCOGT 57710.21 & $-$ & $-$ & 17.35 (0.12) & $-$ & $-$ & ASA 57711.22 & 18.93 (0.34) & 18.58 (0.61) & 17.69 (0.27) & $-$ & $-$ & UVOT 57715.05 & $-$ & $-$ & 17.60 (0.23) & $-$ & $-$ & ASA 57718.26 & $-$ & $-$ & 18.04 (0.27) & $-$ & $-$ & ASA 57719.17 & $-$ & $-$ & 17.59 (0.77) & $-$ & $-$ & LCOGT 57721.26 & $-$ & $-$ & 18.15 (0.27) & $-$ & $-$ & ASA 57723.42 & 19.26 (0.45) & 18.98 (0.60) & 17.98 (0.60) & $-$ & $-$ & UVOT 57727.71 & $-$ & 18.77 (0.06) & 17.71 (0.07) & $-$ & $-$ & 1.82 57728.06 & $-$ & 18.68 (0.09) & 17.87 (0.06) & $-$ & $-$ & LCOGT 57728.85 & $-$ & 18.60 (0.10) & 17.81 (0.10) & $-$ & $-$ & 1.82 57730.81 & $-$ & 18.93 (0.81) & 18.02 (0.36) & $-$ & $-$ & UVOT 57736.06 & $-$ & 18.94 (0.18) & 18.00 (0.11) & $-$ & $-$ & LCOGT 57737.19 & $-$ & $-$ & 17.45 (0.34) & $-$ & $-$ & ASA 57851.63 & $-$ & $-$ & $>$18.17 & $-$ & $-$ & ASA 57860.49 & $>$19.69 & $>$19.89 & $>$18.99 & $-$ & $-$ & UVOT 57870.60 & $-$ & $-$ & $>$18.40 & $-$ & $-$ & ASA 57876.58 & $-$ & $-$ & $>$18.29 & $-$ & $-$ & ASA 57880.57 & $-$ & $-$ & $>$17.80 & $-$ & $-$ & ASA 57885.55 & $-$ & $-$ & $>$17.65 & $-$ & $-$ & ASA 57893.59 & $-$ & $-$ & $>$18.55 & $-$ & $-$ & ASA
[>m[1.2cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[3cm]{}]{} MJD & *J* & *H* & *K* & Instrument & (mag) & (mag) & (mag) & 57540.17 & $-$ & 13.78 (0.27) & $-$ & LT 57542.21 & $-$ & 13.55 (0.30) & $-$ & LT 57546.29 & 13.33 (0.03) & 13.21 (0.02) & 13.14 (0.03) & REM 57547.37 & 13.30 (0.02) & 13.12 (0.02) & 13.00 (0.04) & REM 57548.38 & 13.26 (0.04) & 13.08 (0.02) & 12.94 (0.04) & REM 57549.42 & 13.25 (0.04) & 13.09 (0.02) & 12.98 (0.06) & REM 57550.44 & 13.12 (0.03) & 13.00 (0.02) & 12.86 (0.05) & REM 57551.14 & 13.09 (0.35) & 12.97 (0.31) & 12.95 (0.33) & NOT 57551.19 & $-$ & 13.03 (0.27) & $-$ & LT 57552.06 & $-$ & 13.01 (0.25) & $-$ & LT 57553.17 & $-$ & 12.94 (0.29) & $-$ & LT 57554.12 & $-$ & 12.96 (0.26) & $-$ & LT 57554.23 & 13.12 (0.06) & 12.91 (0.03) & 12.77 (0.05) & REM 57555.24 & 13.01 (0.01) & 12.88 (0.02) & $-$ & REM 57557.13 & $-$ & 12.86 (0.25) & $-$ & LT 57557.22 & 13.04 (0.03) & 12.93 (0.02) & 12.74 (0.09) & REM 57558.10 & $-$ & 12.86 (0.29) & $-$ & LT 57558.23 & 13.02 (0.06) & 12.89 (0.03) & 12.79 (0.12) & REM 57559.07 & $-$ & 12.82 (0.30) & $-$ & LT 57559.24 & 13.01 (0.01) & 12.86 (0.02) & 12.69 (0.02) & REM 57560.30 & 13.04 (0.05) & 12.88 (0.02) & 12.76 (0.09) & REM 57561.06 & $-$ & 12.91 (0.23) & $-$ & LT 57561.30 & 13.01 (0.03) & 12.85 (0.02) & 12.69 (0.03) & REM 57562.12 & $-$ & 12.84 (0.25) & $-$ & LT 57562.31 & 13.14 (0.05) & 12.96 (0.04) & $-$ & REM 57563.13 & $-$ & 12.90 (0.32) & $-$ & LT 57566.32 & 13.14 (0.04) & 13.01 (0.02) & 12.79 (0.07) & REM 57567.33 & 13.13 (0.03) & 12.94 (0.02) & 12.78 (0.04) & REM 57568.35 & 13.23 (0.05) & 13.00 (0.02) & 12.84 (0.10) & REM 57569.08 & $-$ & 13.06 (0.20) & $-$ & LT 57569.36 & 13.19 (0.03) & 13.00 (0.03) & $-$ & REM 57570.17 & $-$ & 13.03 (0.26) & $-$ & LT 57575.35 & 13.45 (0.02) & 13.22 (0.02) & 13.03 (0.04) & REM 57577.10 & $-$ & 13.25 (0.24) & $-$ & LT 57579.11 & $-$ & 13.34 (0.20) & $-$ & LT 57580.12 & $-$ & 13.28 (0.30) & $-$ & LT 57581.10 & $-$ & 13.34 (0.27) & $-$ & LT 57583.34 & 13.78 (0.05) & 13.43 (0.02) & 13.22 (0.06) & REM 57586.13 & $-$ & 13.47 (0.29) & $-$ & LT 57587.43 & 13.92 (0.04) & 13.54 (0.03) & 13.43 (0.05) & REM 57588.14 & $-$ & 13.58 (0.23) & $-$ & LT 57591.43 & 14.14 (0.04) & 13.68 (0.03) & 13.44 (0.06) & REM 57593.12 & $-$ & 13.71 (0.23) & $-$ & LT 57596.31 & 14.23 (0.04) & 13.79 (0.03) & $-$ & REM 57600.32 & 14.55 (0.10) & 13.87 (0.05) & $-$ & REM 57604.32 & 14.55 (0.06) & $-$ & $-$ & REM 57608.36 & 14.70 (0.07) & 14.23 (0.04) & 13.92 (0.11) & REM 57612.38 & 14.80 (0.05) & 14.21 (0.03) & 14.01 (0.14) & REM 57617.22 & 14.98 (0.06) & 14.37 (0.04) & 14.15 (0.11) & REM 57622.36 & 15.28 (0.05) & 14.57 (0.05) & 14.44 (0.20) & REM
\[fig: JHK\]
[>m[1.2cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[3cm]{}]{} MJD & *J* & *H* & *K* & Instrument & (mag) & (mag) & (mag) & 57623.15 & 15.11 (0.40) & 14.69 (0.34) & 14.19 (0.39) & NOT 57629.11 & 15.47 (0.06) & 14.72 (0.07) & 14.70 (0.17) & REM 57634.96 & $-$ & 14.80 (0.28) & $-$ & LT 57636.95 & $-$ & 14.82 (0.30) & $-$ & LT 57638.94 & $-$ & 14.89 (0.28) & $-$ & LT 57642.93 & $-$ & 15.02 (0.28) & $-$ & LT 57650.92 & $-$ & 15.17 (0.29) & $-$ & LT 57654.96 & $-$ & 15.32 (0.25) & $-$ & LT 57657.92 & $-$ & 15.42 (0.23) & $-$ & LT 57663.93 & $-$ & 15.58 (0.26) & $-$ & LT 57666.87 & $-$ & 15.58 (0.22) & $-$ & LT 57669.88 & $-$ & 15.64 (0.31) & $-$ & LT 57672.86 & $-$ & 15.69 (0.30) & $-$ & LT 57675.84 & $-$ & 15.70 (0.28) & $-$ & LT 57678.87 & $-$ & 15.87 (0.30) & $-$ & LT 57681.88 & $-$ & 15.87 (0.28) & $-$ & LT 57689.87 & $-$ & 15.97 (0.27) & $-$ & LT 57693.83 & $-$ & 16.02 (0.23) & $-$ & LT 57700.84 & $-$ & 16.15 (0.26) & $-$ & LT 57708.86 & $-$ & 16.38 (0.26) & $-$ & LT 57711.85 & $-$ & 16.33 (0.27) & $-$ & LT 57732.87 & $-$ & 16.71 (0.26) & $-$ & LT 57856.24 & $-$ & 18.56 (0.35) & $-$ & LT 57863.21 & $-$ & $>$17.65 & $-$ & LT 57866.22 & $-$ & 18.61 (0.35) & $-$ & LT
[>m[1.2cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[1.8cm]{}>m[3cm]{}]{} MJD & *UV-W2* & *UV-M2* & *UV-W1* & Instrument & (mag) & (mag) & (mag) & 57535.77 & 17.74 (0.26) & 17.71 (0.22) & 16.74 (0.17) & UVOT 57539.09 & 17.11 (0.21) & 17.06 (0.14) & 16.07 (0.13) & UVOT 57542.61 & 17.01 (0.16) & 17.26 (0.16) & 15.88 (0.10) & UVOT 57543.98 & 17.02 (0.16) & 17.50 (0.18) & 15.98 (0.11) & UVOT 57545.94 & 17.25 (0.19) & 17.58 (0.20) & 16.08 (0.11) & UVOT 57547.47 & 17.31 (0.19) & 17.84 (0.23) & 16.28 (0.12) & UVOT 57549.32 & 17.37 (0.19) & 18.13 (0.29) & 16.45 (0.14) & UVOT 57550.58 & 17.43 (0.21) & 18.04 (0.27) & 16.45 (0.14) & UVOT 57552.40 & 17.75 (0.25) & 18.46 (0.37) & 16.49 (0.14) & UVOT 57554.20 & 17.81 (0.26) & 18.39 (0.35) & 16.78 (0.17) & UVOT 57555.56 & 17.91 (0.28) & 18.38 (0.38) & 17.04 (0.20) & UVOT 57557.75 & 18.20 (0.34) & $-$ & 17.20 (0.22) & UVOT 57560.58 & 18.37 (0.40) & 19.33 (0.76) & 17.47 (0.27) & UVOT 57569.56 & 19.09 (0.69) & $-$ & 18.74 (0.74) & UVOT 57579.06 & $>$19.50 & $>$19.22 & $>$18.85 & UVOT 57588.37 & $-$ & $>$19.67 & $>$18.91 & UVOT 57592.35 & $>$19.70 & $-$ & $-$ & UVOT 57596.65 & $-$ & $>$19.18 & $>$18.87 & UVOT 57603.62 & $>$19.59 & $>$19.52 & $>$19.18 & UVOT 57612.13 & $>$19.83 & $>$19.53 & $>$18.93 & UVOT 57619.20 & $-$ & $-$ & $>$19.23 & UVOT 57631.56 & $>$19.75 & $>$19.81 & $>$19.32 & UVOT 57646.94 & $>$19.88 & $>$19.73 & $>$19.31 & UVOT 57659.40 & $>$19.85 & $>$19.81 & $>$19.28 & UVOT 57668.01 & $>$19.85 & $>$19.80 & $>$19.35 & UVOT 57681.37 & $>$20.10 & $>$19.94 & $>$19.56 & UVOT 57691.39 & $>$20.14 & $>$19.98 & $>$19.62 & UVOT 57707.04 & $>$20.11 & $>$19.90 & $>$19.54 & UVOT 57726.19 & $>$19.90 & $>$19.89 & $>$19.43 & UVOT 57860.49 & $>$20.29 & $>$20.08 & $>$19.76 & UVOT \[fig: SDA\]
--------- -------------------- -------------- -------------- ------------------
Band MJD max Mag max
(from disc.) (from expl.)
*UV-W2* $57541.50\pm 4.13$ $5.95 $ $9.60 $ $16.99 \pm 0.17$
*UV-M2* $57540.13\pm 4.34$ $4.58 $ $8.23 $ $17.01 \pm 0.98$
*UV-W1* $57542.01\pm 0.95$ $6.46 $ $10.11 $ $15.90 \pm 0.07$
*u* $57543.51\pm 0.52$ $7.96 $ $11.61 $ $15.61 \pm 0.02$
*U* $57543.85\pm 2.21$ $8.30 $ $11.95 $ $14.68 \pm 0.06$
*B* $57547.87\pm 0.50$ $12.32 $ $15.97 $ $14.83 \pm 0.02$
*g* $57548.85\pm 2.83$ $13.30 $ $16.95 $ $14.15 \pm 0.09$
*V* $57550.24\pm 0.16$ $14.69 $ $18.34 $ $17.86 \pm 0.01$
*r* $57551.88\pm 0.36$ $16.33 $ $19.98 $ $13.59 \pm 0.01$
*R* $57551.10\pm 0.60$ $15.55 $ $19.20 $ $13.41 \pm 0.04$
*i* $57554.38\pm 0.86$ $18.83 $ $22.48 $ $13.93 \pm 0.02$
*I* $57553.54\pm 2.26$ $17.99 $ $21.64 $ $13.63 \pm 0.06$
*z* $57555.50\pm 0.67$ $19.95 $ $23.60 $ $13.58 \pm 0.01$
*J* $57558.74\pm 0.84$ $23.19 $ $26.84 $ $13.03 \pm 0.03$
*H* $57562.30\pm 2.75$ $26.75 $ $30.40 $ $12.85 \pm 0.09$
*K* $57560.88\pm 2.13$ $25.33 $ $28.98 $ $12.72 \pm 0.03$
--------- -------------------- -------------- -------------- ------------------
\[tab: max\]
[lccccr<>c<>l]{} Telescope & Instrument & Grism/Grating & Slit &Resolution \[R\]&\
1.22m Galileo & B&C spectrograph & 300 ln/mm&3.93&636&3800&$-$&8000 Å\
& & Gr\#4&&363&3360&$-$&7740 Å\
& & VPH6&&300&4500&$-$&10000 Å\
& & VPH7&&375&3200&$-$&7300 Å\
du Pont Telescope&B&C spectrograph&300 ln/mm&2.71&1667&3300&$-$&9500 Å\
&&Red&&2300&5500&$-$&10500 Å\
&&Blue&&1850&3200&$-$&6000 Å\
Magellan & IMACS &300 ln/mm& 0.9&1100&3650&$-$&9740 Å\
& &VPH-Blue &1.2&1600&3900&$-$&6800 Å\
& & &1.0&1600&3920&$-$&9050 Å\
& &&1.2&1500&5350&$-$&10500 Å\
Multiple Mirror Telescope (MMT)&Blue Channel&300 ln/mm&1.0&750&3300&$-$&8600 Å\
LT& SPRAT & Blue & 1.8&350&4000 &$-$& 8000 Å\
NOT &ALFOSC& Gr\#4& 1.0& 360& 3200&$-$&9600 Å\
Tillinghast 1.5m (FLWO)& FAST & 300 ln/mm & 3.0& 900 & 3530&$-$&7470 Å\
& & LR-B & & 585& 3000&$-$&8430 Å\
&&LR-R&&714&4470&$-$&10070 Å\
\[tab: instr2\]
Date obs. MJD Tel.+Inst. Slit Grism/Grating
-------------------- --------- ------------------ ------ ---------------
2016$-$05$-$28 57536.2 NOT+ALFOSC 1.0 Gr\#4
2016$-$05$-$28 57536.2 LT+SPRAT 1.8 Blue
2016$-$05$-$29 57537.2 LT+SPRAT 1.8 Blue
2016$-$05$-$29 57537.1 NOT+ALFOSC 1.0 Gr\#4
2016$-$05$-$31 57539.4 MDM+OSMOS 1.2 VPH-Blue
2016$-$06$-$01 57540.1 LT+SPRAT 1.8 Blue
2016$-$06$-$02 57541.4 Till+FAST 3.0 300 ln/mm
2016$-$06$-$02 57541.4 MDM+OSMOS 1.2 VPH-Blue
2016$-$06$-$03 57542.5 Till+FAST 3.0 300 ln/mm
2016$-$06$-$04 57543.4 Till+FAST 3.0 300 ln/mm
2016$-$06$-$05 57544.5 Till+FAST 3.0 300 ln/mm
2016$-$06$-$06 57545.4 Till+FAST 3.0 300 ln/mm
2016$-$06$-$07 57546.2 NOT+ALFOSC 1.0 Gr\#4
2016$-$06$-$08 57547.4 Till+FAST 3.0 300 ln/mm
2016$-$06$-$09 57548.5 Till+FAST 3.0 300 ln/mm
2016$-$06$-$10 57549.2 LT+SPRAT 1.8 Blue
2016$-$06$-$12 57551.2 LT+SPRAT 1.8 Blue
2016$-$06$-$13 57552.2 NOT+ALFOSC 1.0 Gr\#4
2016$-$06$-$14 57553.1 LT+SPRAT 1.8 Blue
2016$-$06$-$14 57553.4 LBT+MODS 0.6 Red+Blue
2016$-$06$-$15 57554.1 LT+SPRAT 1.8 Blue
2016$-$06$-$18 57557.0 1.82+AFOSC 1.69 VHP6+VPH7
2016$-$06$-$19 57558.1 LT+SPRAT 1.8 Blue
2016$-$06$-$22 57561.1 LT+SPRAT 1.8 Blue
2016$-$06$-$23 57562.0 1.82+AFOSC 1.69 VHP6+VPH7
2016$-$06$-$23 57562.1 LT+SPRAT 1.8 Blue
2016$-$06$-$24 57563.1 LT+SPRAT 1.8 Blue
2016$-$06$-$26 57565.0 LT+SPRAT 1.8 Blue
2016$-$07$-$02 57571.2 LT+SPRAT 1.8 Blue
2016$-$07$-$05 57574.4 Till+FAST 3.0 300 ln/mm
2016$-$07$-$07 57576.1 1.22+B&C 3.93 300 ln/mm
2016$-$07$-$10 57579.4 Till+FAST 3.0 300 ln/mm
2016$-$07$-$10 57579.3 MMT+Blue Channel 1.0 300 ln/mm
2016$-$07$-$12 57581.1 LT+SPRAT 1.8 Blue
2016$-$07$-$17 57586.1 LT+SPRAT 1.8 Blue
2016$-$07$-$18 57587.0 LT+SPRAT 1.8 Blue
2016$-$07$-$21 57590.1 NOT+ALFOSC 1.0 Gr\#4
2016$-$07$-$26 57596.0 LT+SPRAT 1.8 Blue
2016$-$08$-$01 57601.1 NOT+ALFOSC 1.0 Gr\#4
2016$-$08$-$09 57609.0 1.82+AFOSC 1.69 VHP6+VPH7
2016$-$08$-$12 57613.0 TNG+LRS 1.0 LR-B+LT-R
2016$-$08$-$22 57623.0 1.82+AFOSC 1.69 VHP6+VPH7
2016$-$08$-$24 57624.0 LT+SPRAT 1.8 Blue
2016$-$08$-$25 57625.9 1.82+AFOSC 1.69 VHP6+Gr\#4
2016$-$09$-$03 57634.8 1.82+AFOSC 1.69 VHP6+VPH7
2016$-$09$-$03 57635.0 LT+SPRAT 1.8 Blue
2016$-$09$-$08 57639.1 du Pont+B&C 2.71 300 ln/mm
2016$-$09$-$16 57647.9 LT+SPRAT 1.8 Blue
2016$-$09$-$23 57654.9 1.82+AFOSC 1.69 VHP6+VPH7
\[fig: spec\_log\]
Date obs. MJD Tel.+Inst. Slit Grism/Grating
---------------- --------- ------------------ ------ ---------------
2016$-$09$-$24 57655.1 NOT+ALFOSC 1.0 Gr\#4
2016$-$09$-$30 57662.0 LT+SPRAT 1.8 Blue
2016$-$10$-$01 57662.1 NOT+ALFOSC 1.0 Gr\#4
2016$-$10$-$11 57672.9 LT+SPRAT 1.8 Blue
2016$-$10$-$13 57674.8 LT+SPRAT 1.8 Blue
2016$-$11$-$02 57694.1 LBT+MODS 0.6 Red+Blue
2016$-$11$-$01 57693.9 LT+SPRAT 1.8 Blue
2016$-$11$-$15 57707.8 MDM+OSMOS 1.2 VPH-Red
2016$-$11$-$18 57710.8 LT+SPRAT 1.8 Blue
2016$-$12$-$05 57727.7 1.82+AFOSC 1.69 VHP6+VPH7
2016$-$12$-$06 57728.8 1.82+AFOSC 1.69 VHP6+VPH7
2016$-$12$-$10 57732.8 LT+SPRAT 1.8 Blue
2017$-$06$-$17 57921.8 MDM+OSMOS 1.0 VPH-Red
2017$-$06$-$28 57932.4 MMT+Blue Channel 1.0 300 ln/mm
2017$-$07$-$20 57954.2 Magellan+IMACS 0.9 300ln/mm
--------------- ------------------------ ----------- ------- --------------- ---------
Start Date Time since First Light Frequency Flux Density Project
(UT) (days) (GHz) (GHz) (mJy)
2016-06-03 11 5.95 2.048 $1.15\pm0.06$ 16A-477
2016-06-03 11 9.80 2.048 $3.4\pm0.2$ 16A-477
2016-06-03 11 14.75 2.048 $8.9\pm0.4$ 16A-477
2016-06-03 11 21.85 2.048 $21\pm1$ 16A-477
2016-06-13 21 5.95 2.048 $4.8\pm0.2$ 16A-477
2016-06-13 21 9.80 2.048 $11.3\pm0.6$ 16A-477
2016-06-13 21 14.75 2.048 $20\pm1$ 16A-477
2016-06-13 21 21.85 2.048 $24\pm1$ 16A-477
2016-07-08 46 3.00 2.048 $3.5\pm0.2$ 16A-477
2016-07-08 46 5.95 2.048 $12.9\pm0.6$ 16A-477
2016-07-08 46 9.80 2.048 $20\pm1$ 16A-477
2016-07-08 46 21.85 2.048 $16.7\pm0.8$ 16A-477
2016-09-07 106 3.00 2.048 $14.7\pm0.7$ 16A-477
2016-09-07 106 5.95 2.048 $13.5\pm0.7$ 16A-477
2016-09-07 106 9.80 2.048 $9.6\pm0.5$ 16A-477
2016-09-07 106 21.85 2.048 $5.3\pm0.3$ 16A-477
2017-02-17 270 5.50 2.048 $6.2\pm0.3$ 17A-167
2017-02-17 270 9.00 2.048 $3.7\pm0.2$ 17A-167
2017-02-25 278 3.00 2.048 $10.1\pm0.5$ 17A-167
\[Tab:radio\]
--------------- ------------------------ ----------- ------- --------------- ---------
--------------- ---------------------- --------------- --------------------------------- ---------------------------------- ------------
MJD Time since Explosion Time Range Count-rate Unabsorbed Flux Instrument
(days) (days) $(c\,s^{-1})$ $(\rm{erg\,s^{-1}cm^{-2}})$
57538.0 6.1 (1.8-10.0) $(3.44\pm 0.85)\times 10^{-3}$ $(1.44\pm0.36)\times 10^{-13}$ XRT
57543.4 11.5 - ($1.44\pm 0.61)\times 10^{-13}$ XMM
57545.6 13.7 (10.1-16.8) $(2.43\pm 0.67)\times 10^{-3}$ $(1.03\pm0.28)\times 10^{-13} $ XRT
57553.3 21.4 (17.9-24.0) $(2.25\pm 0.61 )\times 10^{-3}$ $(0.95\pm0.25)\times 10^{-13}$ XRT
57559.4 27.5 - $(0.64 \pm 0.19)\times 10^{-13}$ XMM
57626.0 94.1 (26.6-157.6) $(7.47\pm 0.18)\times 10^{-3}$ $(0.31\pm0.08)\times 10^{-13}$ XRT
57785.1 253.2 (168.9-326.9) $< 1.54 \times 10^{-3}$ $<0.65\times 10^{-13} $ XRT
\[Tab:xrays\]
--------------- ---------------------- --------------- --------------------------------- ---------------------------------- ------------
[^1]: [http://www.astronomy.ohio- state.edu/\~assassin/index.shtml](http://www.astronomy.ohio- state.edu/~assassin/index.shtml)
[^2]: <span style="font-variant:small-caps;">IRAF</span> is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. http://iraf.noao.edu/
[^3]: <http://www.not.iac.es/instruments/notcam/guide/observe.html#reductions>
[^4]: Cappellaro, E. (2014). <span style="font-variant:small-caps;">SNOoPY</span>: a package for SN photometry, <http://sngroup.oapd.inaf.it/snoopy.html>
[^5]: <http://www.sdss.org>
[^6]: <http://www.ipac.caltech.edu/2mass/>
[^7]: This is in agreement with the assumption by [@Yamanaka17] and [@Kumar18]. On the other hand, [@Prentice17] assumed a host extinction of $E(B-V)_{\rm{host}}$=0.125 mag.
[^8]: The spectra were retrieved from WISeREP [<https://wiserep.weizmann.ac.il> @Yaron2012] and the OSC [<https://sne.space>; @Guillochon2017].
[^9]: The field of was not serendipitously observed by HST before explosion, which prevents a constraining search for a progenitor star in pre-explosion images. The Galaxy Survey [@James2004] observed the field of on June 6th, 2000. A compact source of emission is clearly detected $\sim4.9$ from the SN location, (yellow mark in Fig. \[Fig:rgb\]). At the distance of , this angular separation corresponds to a projected distance of 0.43 kpc. [@Prentice17] estimated 0.375 kpc, using however a shorter distance to the host galaxy (see Table \[tab:other\_works\]).
[^10]: The measured mass-loss refers to the WR-phase of the progenitor, therefore little could be said about the process responsible for the stripping of its hydrogen envelope.
[^11]: There is also a disputed claim of He in SN 1998bw [@Patat2001], and in the more recent SN 2017iuk [@Izzo2019].
[^12]: Operated by the ASAS-SN team [@Shappee2014].
[^13]: Operated by R. A. Koff at Antelope Hills Observatory, Bennett, CO, USA.
[^14]: Operated by Joseph Brimacombe at New Mexico Skies, New Mexico, USA.
[^15]: Operated by Geoffrey Stone at Sierra Remote Observatories, Auberry, CA, USA.
[^16]: Operated by the Astronomical Observatory of the University of Valencia (OAUV) at Aras de los Olmos, Valencia, Spain.
[^17]: Operated by the University of Iowa at Iowa Robotic Observatory.
[^18]: Operated by DEMONEXT [@Villanueva2018].
[^19]: Operated by the Weihai Observatory of Shandong University, China.
|
---
abstract: 'We study fundamental groups of toroidal compactifications of non compact ball quotients and show that the Shafarevich conjecture on holomorphic convexity for these complex projective manifolds is satisfied in dimension $2$ provided the corresponding lattice is arithmetic and small enough. The method is to show that the Albanese mapping on an étale covering space generates jets on the interior, if the lattice is small enough. We also explore some specific examples of Picard-Eisenstein type.'
author:
- 'Philippe [Eyssidieux]{}'
date: 'April, 30 2018'
title: Orbifold Kähler Groups related to arithmetic complex hyperbolic lattices
---
[^1]
Introduction
============
Given $X$ a compact Kähler manifold, the question raised by Shafarevich whether the universal covering space of $X$ is holomorphically convex, also known as the Shafarevich Conjecture on holomorphic convexity $SC(X)$, and the study of the Serre problem of characterizing the finitely presented groups arising as fundamental groups of complex algebraic manifolds lead to consider several properties the fundamental group of $X$ may or may not satisfy:
1. $Inf(X)$: $\pi_1(X)$ is infinite.
2. $Inf_{et}(X)$: Assuming $Inf(X)$, the profinite completion $\hat \pi_1(X)$ is infinite.
3. $RF(X)$: $\pi_1(X)$ is residually finite.
4. $Q(X)$: Assuming $Inf(X)$, $\pi_1(X)$ has a finite rank representation in a complex vector space whose image is infinite.
5. $SSC(X)$: For every $f:Z\to X$ where $Z$ is a compact connected complex analytic space and $f$ is holomorphic $\#\mathrm{Im}(\pi_1(Z) \to \pi_1(X)) =+\infty \Leftrightarrow
\exists N \in {\mathbb N}^* \ \exists \rho: \pi_1(X) \to GL_N({\mathbb C}) \ \#\rho(\mathrm{Im}(\pi_1(Z) \to \pi_1(X))=+\infty. $
6. $L(X)$: $\pi_1(X)$ is a linear group.
One has the following implications, the first one being a classical result of Malčev, the last one was proved in increasing generality in [@EKPR; @CCE; @ajm]: $$L(X)\Rightarrow RF(X), SSC(X), \ \ SSC(X) \Rightarrow Q(X) \Rightarrow Inf_{et}(X), \ \ SSC(X) \Rightarrow SC(X).$$
The counterexamples to $RF(\_)$ -hence $L(\_)$- [@Tol; @Trento] do not give rise to counterexamples of $SSC(\_)$, actually all the other statements hold trivially true for the complex projective manifolds considered there.
All these properties make sense if $X$ is replaced by a compact Kähler smooth Deligne-Mumford stack, see e.g. [@ajm], a complex algebraic manifold (say quasi projective) or a smooth separated Deligne-Mumford stack with quasi projective or quasi Kähler moduli space. Using orbifold compactifications of a given quasi-Kähler manifold $U$, one can produce compact Kähler orbifolds with [**[potentially]{}**]{} interesting fundamental groups if $\pi_1(U)$ has a sufficiently rich normal subgroup lattice. These orbifold Kähler groups can sometimes be proven to be fundamental groups of related compact Kähler manifolds. This happens if the inertia morphisms are injective after passing to the profinite completion. Checking this propery requires at least [*some*]{} understanding of the structure of the orbifold fundamental group.
A first non trivial class of $U$, complements of line arrangements in projective space, was analysed in [@ajm] where $SSC(\_)$ was settled affirmatively in the equal weight case and it seems much more difficult to settle $L(\_)$. The resulting orbifolds are abelian quotients of the Hirzebruch algebraic surfaces ramified over the arrangement [@Hirz; @BHH]. A general theory of the fundamental group of these surfaces in the unequal weight case seems to be out of reach except in specific cases: in the case of $CEVA(2)$, which was investigated in depth for the construction of complex hyperbolic lattices, the theory is fascinating [@DM; @DMC; @Mos].
The present article investigates a second non-trivial class, where a lot of the most beautiful examples come precisely from the aforementioned work of Mostow and Deligne-Mostow: finite covolume non compact quotients of a complex hyperbolic space. Then $\pi_1(U)$ is a non-uniform complex hyperbolic lattice, and the manifolds we investigate are the toroidal compactifications of non compact ball quotients [@AMRT; @Mok] . These objects have attracted a lot of attention in recent years from several perspectives, construction of interesting lattices [@DPP; @Stov; @DiCSto], their position in the classification of algebraic varieties [@DiC2; @BT], Kobayashi hyperbolicity [@Cad1; @Cad2].
In contrast to the case of rank $\ge 2$ where the fundamental group of a toroidal compactification of an irreducible hermitian locally symmetric space is finite, a toroidal compactification of a ball quotient can have a large fundamental group. For instance, [@HS] constructs a Riemannian metric of non positive curvature on the toroidal compactification of a small enough lattice. These toroidal compactifications are thus $K(\pi,1)$, the universal covering space being diffeomorphic to a real affine space and the fundamental group has exponential growth. But it does not seem possible to use the methods in [@HS] to prove that the universal covering space is Stein (which what $SC$ predicts) or to construct linear representations of the fundamental group.
Let us describe the content of this article.
Given $\Gamma<PU(n,1)$ a non uniform lattice, $n\ge 2$, decorating the construction of [@Hol p. 29-30] with its natural stack structure, or interpreting [@Hol3 Ch. 4] and [@Ulu] in a more flexible language, we construct an orbifold compactifications $[\Gamma \backslash {{\mathbb H}^2_{\mathbb C}}]\subset \mathcal{X}_{\Gamma}^{tor}$ with no codimension $1$ ramification at infinity and a singular DM-stack compactification $[\Gamma \backslash \mathbb{H}^n_{\mathbb{C}}]\subset \mathcal{X}_{\Gamma}^{BBS}$. When $\Gamma$ is neat this is the usual construction from [@AMRT].
Let $C\subset \partial \mathbb{H}^n_{\mathbb{C}}$ be the set of the preimages of the cusps in $\Gamma \backslash \mathbb{H}^n_{\mathbb{C}})^{BBS}$. The finite set $\Gamma \backslash C$ is the set of cusps of $\Gamma$.
For each $c\in C$ of $\Gamma$ denote by $H_c$ (resp. $Z_c$) the intersection of $\Gamma$ with the unipotent radical of the parabolic subgroup attached to $c$ (resp. its center). Then:
1. $\pi_1( \mathcal{X}_{\Gamma}^{BBS})= \Gamma^{BBS}=\Gamma \slash < H_c, \ c \in C >$.
2. $\pi_1( \mathcal{X}_{\Gamma}^{tor})= \Gamma^{tor}=\Gamma \slash < Z_c, \ c \in C >$.
Here $<\_ >$ stands for the subgroup generated by $\_$ which is normal in the two above cases. We have not been able so far to find any piece of information on these rather natural quotients of $\Gamma$ in the litterature. In spite of the fact that the credit for this result should be given to [@LKM] and [@KaSa], we nevertheless display it in the introduction, in order to translate the problem we study here in terms familiar to complex hyperbolic geometers.
Then, we focus on constructing virtually abelian linear representations of the fundamental group $\Gamma^{tor}$, hence on studying the virtual first Betti number. We will give evidence for the following conjecture:
If $\bar X_{\Gamma}$ is the toroidal compactification of a non uniform arithmetic lattice $\Gamma<PU(n,1)$, which is torsion free and torsion free at infinity [^2] , $SSC(\bar X_{\Gamma})$ holds and the representations can be taken to be virtually abelian. Furthermore, there is a finite étale Galois covering $\bar X_{\Gamma'}\to \bar X_{\Gamma}$ with Galois group $G$ such that the Stein factorization of the quotient Albanese morphism $\bar X_{\Gamma} \to G\backslash Alb(\bar X_{\Gamma'})$ is the Shafarevich morphism of $\bar X_{\Gamma}$.
\[thm2\] If $\Gamma''<PU(2,1)$ is arithmetic, there is a finite index subgroup $\Gamma'<_{fi}\Gamma''$ such that if $\Gamma<_{fi}\Gamma'$, $\bar X_{\Gamma}$ has Albanese dimension $2$ and its image contains no translate of an elliptic curve at its generic point. It satisfies $SC, Q$.
With the notations of Theorem \[thm2\], $SSC(\bar X_{\Gamma})$ would follow from the following statement: $$\leqno{(\dagger)} \quad \exists \Gamma_* \triangleleft_{fi} \Gamma'' \quad \forall c\in C \quad H_c\cap \Gamma_* \slash Z_c \cap \Gamma_* \to H_1(\Gamma_*^{tor}, {\mathbb Q})
\quad \mathrm{is \ injective,}$$ and the universal covering space of $\bar X_{\Gamma}$ would be a Stein manifold. It is enough to look at all $c$ in a finite representative set for $\Gamma''\backslash C$. We write $<_{fi}$ to when we want to emphasize a subgroup inclusion has finite index.
\[thm3\] If $n\ge 3$ and $\Gamma''<PU(n,1)$ is arithmetic, there is a finite index subgroup $\Gamma'<\Gamma''$ such that if $\Gamma<_{fi}\Gamma'$, $\bar X_{\Gamma}$ has Albanese dimension $n$ and its image contains no translate of an abelian variety at its generic point. It satisfies $Q$.
As a corollary of our approach, we get a new proof of the following known facts:
\[thm4\] Under the assumptions of Theorems \[thm2\] \[thm3\], $\bar X_{\Gamma}$ has ample cotangent bundle modulo its boundary, is Kobayashi hyperbolic modulo its boundary, and the rational points over a number field over which it is defined are finite modulo its boundary.
One would like effective versions of these results and quantify them in terms of the ramification indices at infinity in the spirit of [@BT; @Cad1] which but our method is inherently non-effective. The examples we have studied so far, most notably the $2$-dimensional Picard-Eisenstein case where the hard work has been done by Feustel and Holzapfel [@Hol], suggest much better statements. The article finishes with a detailled discussion of the Picard-Eisenstein commensurability class from the present perspective. A very strong version of $(\dagger)$ holds in this class.
In order to streamline the discusion, we introduce in Definition \[refined\] an equivalence relation, [*refined commensurability*]{}, on the set of all lattices in a commensurability class, namely that their (orbifold) toroidal compactification are related by an étale correspondance. There is a natural partial ordering on its the quotient set, which is induced by reverse inclusion of lattices. The questions we study here depend only of the refined commensurability class and as in [@ajm] are more delicate for small classes.
When writing this article we were not sure whether the statement about rational points was new, it is not a corollary of [@Ull]. In the final stage of the redaction, Y. Brunebarbe informed us that, if $n=2$, it follows from [@Dim Theorem 0.3] a paper we were not aware of. The Kobayashi hyperbolicity statement is not new since it follows from [@Nad] and an effective, hence better, version was proved in [@BT] - and even more precise results follow from [@Dim] if $n=2$. On the other hand [@Dim] does not imply $SC$.
This article follows the same basic idea as [@Dim] where N. Fakhruddin is credited for it. [@Dim Proposition 3.8] gives an explicit $\Gamma$ in each commensurability class such that $q(X_{\Gamma})>2$. Here, with Lemma \[primitive\], we go a little further in the study of the differential geometry of the Albanese mapping, using as the only automorphic input the classical fact [@Wa] that one may achieve $q(X_{\Gamma})>0$. We did not find a reference for the analogous property in the non arithmetic case and will refrain from making any conjecture in that case. Our method however seems to be hopelessly non-effective and relies on the commensurator property of arithmetic lattices.
These results say that small covolume arithmetic lattices and, unsurprisingly, non-arithmetic lattices are the most interesting ones from the present perspective and we hope to come back to their study in future work.
The author would like to thank Y. Brunebarbe, B. Cadorel, B. Claudon, M. Deraux, B. Klingler, F. Sala and G. Wüstholz for useful conversations related to this article and the Freiburg Institute for Advanced Studies for hospitality during its preparation.
Orbifold partial compactifications
==================================
As advocated by [@No2; @art:lerman2010], we define an orbifold to be a smooth Deligne-Mumford stack with trivial generic isotropy groups relative to the category of complex analytic spaces[^3] with the classical topology: we assume that the moduli space is Hausdorff and that the inertia groups are finite. There is an analytification $2$-functor from DM-stacks over ${\mathbb C}$ to complex analytic DM-stacks and an underlying topological stack functor from complex analytic DM-stacks to topological DM-stacks [@No2].
An orbifold ${\mathcal X}$ is said to be [*developable*]{} if its universal covering stack [@No2] [^4] is an ordinary manifold, which is equivalent to the injectivity of every local inertia morphism $I_x=\pi_1^{loc}({\mathcal X},x) \to \pi_1({\mathcal X},x)$, $x$ being an orbifold point of ${\mathcal X}$. If $x_0$ is a base point of $\mathcal{X}$, we have a conjugacy class of morphisms $I_x \to \pi_1^{et}({\mathcal X}, x_0)$, also called the local inertia morphisms. We will abuse notation and drop the base point dependency of $\pi_1$ when harmless. The orbifold $\mathcal{X}$ is said to be [*uniformizable*]{} whenever the profinite completion $I_x \to \pi_1^{et}({\mathcal X})$ of every local inertia morphism is injective[^5]. When $\pi_1({\mathcal X})$ is residually finite, uniformizability and developability are equivalent properties. The condition residually finite cannot be dropped, a counterexample is given in [@ajm]. The fundamental group of a compact Kähler uniformizable orbifold occurs as the fundamental group of a compact Kähler manifold [@vjm].
The fundamental group of a weighted DCN {#wdcn}
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Let us recall the simplest examples of orbifold compactification of quasi-Kähler manifolds described in [@ajm].
### Root stacks
Let $M$ be a (Hausdorff second countable) complex analytic space and $D$ be an effective Cartier divisor. Let $r\in {\mathbb N}^*$. Then, one can construct $P \to M$ the principal ${\mathbb C}^*$-bundle attached to $\mathcal{O}_M(-D)$ and the tautological section $s_D \in H^0(M,\mathcal{O}_M(D))$ can be lifted to a holomorphic function $f_D: P \to {\mathbb C}$ satisfying $f_D(p. \lambda)= \lambda f_D(p)$ for every $\lambda\in{\mathbb C}^*, \ p\in P$. Define a complex analytic space $Y:=Y_D\subset P \times {\mathbb C}_z$ by the equation $z^r=f_D(p)$. One can define a ${\mathbb C}^*$-action on $Y$ by $(p,z). \lambda= (p\lambda^r, \lambda.z)$. The complex analytic stack $$M(\sqrt[r]{D}):=[Y_D/{\mathbb C}^*]$$ (see e.g. [@EySa; @bn]) is then a Deligne-Mumford separated complex analytic stack with trivial generic isotropy groups whose moduli space is $M$ and an orbifold if $M$ and $D$ are smooth. The non-trivial isotropy groups live over the points of $D$ and are isomorphic to $\mu_r$ the group of $r$-roots of unity. In the smooth case, the corresponding differentiable stack can be expressed as the quotient by the natural infinitesimally free $U(1)$ action on the restriction $UY$ of $Y$ to the unit subbundle for (any hermitian metric) of $P$ which is a manifold indeed. It is straightforward to see that this is an analytic version of Vistoli’s root stack construction:
If $(M, D)$ is the analytification of $(\mathcal{M}, \mathcal{D})$ a pair consisting of a ${\mathbb C}$-(separated) scheme and a Cartier divisor, then $M(\sqrt[r]{D})$ is the analytification of $\mathcal{M}_{O (\mathcal{D}), s_{\mathcal{D}}, r}$ in the notation of [@cadman2007 Section 2].
A Cartier divisor $D$ on a scheme $M$ should be thought of as a pair of an invertible sheaf $\mathcal{L}$ and a section $s_{D} : \mathcal{O} \to \mathcal{L}$ such that $[s_D]=D$. In other words a map of algebraic stacks $\mu: M \to [{\mathbb C}\slash {\mathbb C}^*]$. We have the natural $r$-th power map $.^r:[{\mathbb C}\slash {\mathbb C}^*] \to [{\mathbb C}\slash {\mathbb C}^*]$ and an equivalence $M(\sqrt[r]{D}) \to M\times_{[{\mathbb C}\slash {\mathbb C}^*] \mu, .^r} [{\mathbb C}\slash {\mathbb C}^*]$. Using this we may even promote $M$ to be a stack and allow for $s_{D}=0$. In the latter case we get a ${\mathbb Z}\slash r{\mathbb Z}$-gerbe on $M$ whose class is the reduction mod $r$ of $c_1(\mathcal{L})$. The main property is treated in the scheme-theoretic setting by [@cadman2007]:
\[morphstack\] If $S$ is a complex analytic stack $$Hom(S,M(\sqrt[r]{D})) = \{f: S\to M \ \mathrm{{\mathbb C}-analytic}, \exists D_S \ \mathrm{Cartier \ on \ } S \ \mathrm{s. t.} \ D_S=r.f^*D\}.$$
###
Let $\bar X$ be a compact Kähler manifold and $x_0\in X$ a base point, $n=\dim_{{\mathbb C}}(X)$, and let $D:=D_1+ \ldots + D_l$ be a simple normal crossing divisor whose smooth irreducible components are denoted by $D_i$. We assume for simplicity $x_0\not \in D$. For each choice of weights $d:=(d_1, \ldots, d_l)$, $d_i\in {\mathbb N}^*$, one may construct as [@cadman2007 Definition 2.2.4] does in the setting of scheme theory the compact Kähler orbifold (Compact Kähler DM stack with trivial generic isotropy) $${\mathcal X}(\bar X, D, d):= \bar X(\sqrt[d_1]{D_1} )\times_{\bar X} \ldots \times_{\bar X} \bar X(\sqrt[d_l]{D_l}).$$ In other words, ${\mathcal X}(\bar X, D, d)=[Y_{D_1}\times_{\bar X} \dots \times_{\bar X} Y_{D_l} / {\mathbb C}^{*l}]$. Denote by $X$ the quasi-Kähler manifold $X:=\bar X \setminus D$. View ${\mathcal X}(\bar X, D, d)$ as an orbifold compactification of $X$ and denote by $j_d: X \hookrightarrow {\mathcal X}(\bar X, D, d)$ the natural open immersion.
By Zariski-Van Kampen, the fundamental group $\pi_1(\bar X,x_0)$ is the quotient of $\pi_1(X,x_0)$ by the normal subgroup generated by the $\gamma_i$, where $\gamma_i$ is a meridian loop for $D_i$. Zariski-Van Kampen generalizes to orbifolds, see e.g. [@No1; @Zoo], and $\pi_1({\mathcal X}(\bar X, D, d),x_0)$ is the quotient of $\pi_1(X,x_0)$ by the normal subgroup generated by the $\gamma_i^{d_i}$.
If $(X,\Delta)$ is an orbifolde in the sense of Campana, the Campana orbifold fundamental group is the fundamental group of the root stack on $X\setminus D^1$ where $D^1$ is the log-singular set of $(X,\Delta)$. It may happens that the fundamental group of the trace of the root stack on $X\setminus D^1$ of a small ball centered on $D^1$ is finite and in this case we get a DM-stack compactification which may be singular. In dimension 2, the list of local configurations giving rise to an orbifold compactification with a smooth moduli space is given in [@Ulu].
The orbifolds constructed here are specified up to equivalence by their moduli space and ramification indices in codimension 1 by [@GerSa]. Hence, in dimension 2, these orbifolds carry the same information as Holzapfel’s orbital surfaces or Uludag’s orbifaces. One could also use orbifolds in the sense of Thurston in higher dimension. However, it is convenient to have at our disposal maps of stacks (defined as functors of fibered categories), moduli spaces, substacks, (2-)fiber products hence fibers, basic homotopy theory [@No2] and many differential geometric constructions [@EySa].
Toroidal compactifications of complex hyperbolic orbisurfaces
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Orbifold Toroidal compactifications
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###
The following constructions are well known to the experts [@BT] but we recall them in order to fix the notations. Let ${{\mathbb H}^2_{\mathbb C}}:=PU(1,2)\slash P(U(1)\times U(2))$ be the complex hyperbolic plane. Let $\Gamma < PU(1,2)$ be non uniform lattice. Then $X_{\Gamma}:=\Gamma \backslash {{\mathbb H}^2_{\mathbb C}}$ is an algebraic quasi-projective variety by Baily-Borel’s theorem in the arithmetic case, and [@Mok] for the non arithmetic case. It has a complex projective compactification $X_{\Gamma} \subset X_{\Gamma}^{BBS}$ obtained by adding a finite number of points we shall refer to as cusps.
Each cusp $c$ has some neighborhood $V_c$ such that the preimage of $U_c:=V_c\setminus\{ c \}$ is a disjoint union of horoballs $W_{\tilde c}$ of ${{\mathbb H}^2_{\mathbb C}}$ labelled by $ \partial {{\mathbb H}^2_{\mathbb C}}\ni \tilde c := \partial {{\mathbb H}^2_{\mathbb C}}\cap \overline{ W_{\tilde c}}$ where the closure and boundary are taken in the euclidean topology of the projective plane $\mathbb{P}$ dual to ${{\mathbb H}^2_{\mathbb C}}$. The set $C$ of all such $\tilde c$ is acted upon with finitely many orbits by $\Gamma$. For such a $\tilde c$, let $\Gamma_{\tilde c}$ be the stabilizer of $\tilde c$ in $\Gamma$.
Since the stabilizer $S_{\tilde c}$ of $\tilde c$ in $PU(1,2)$ has a 3 dimensional Heisenberg group of unipotent $3\times 3$ upper triangular matrices with real coefficients as its uniportent radical $H_{\tilde c}$ and ${\mathbb C}^*$ has its Levi component, see [@Par], we have an exact sequence: $$1 \to H_{\tilde c} \cap \Gamma_{\tilde c} \to \Gamma_{\tilde c} \to \mu_{k_c} \to 1$$ where $\mu_k < {\mathbb C}^*$ is group of $k_c$-roots of unity. Whenever this causes no confusion, we use a shorthand notation $k=k_c$. Then, $ H_{\tilde c} \cap \Gamma_{\tilde c} < H_{\tilde c} $ is a lattice in $H_{\tilde c}$ One has $\gamma W_{\tilde c} \cap W_{\tilde c} \not = \emptyset \Leftrightarrow \gamma \in \Gamma_{\tilde c}$. The center $Z_{\tilde c}$ of $H_{\tilde c} \cap \Gamma_{\tilde c}$ is a cyclic infinite subgroup and we have $Z_{\tilde c}=Z(H_{\tilde c})\cap \Gamma_{\tilde c}$.
The group $\Lambda_{\tilde c}:= Z_{\tilde c} \backslash H_{\tilde c} \cap \Gamma_{\tilde c} $ is a lattice in the real 2-dimensional additive group $A_{\tilde c}^\tau:=Z(H_{\tilde c}) \backslash H_{\tilde c}$ which has a natural structure of an affine complex line $A_{\tilde c}$. Hence $A^\tau_{\tilde c}$ is naturally a one dimensionnal complex additive group acting as the translation group on $A_{\tilde c}$. This complex structure comes from the fact that $Z(H_{\tilde c})$ stabilizes a unique complex geodesic having $\tilde c$ as a boundary point. All such complex geodesics are the trace of a complex projective line in $\mathbb{P}$ through $\tilde c$ and form a single orbit of $H_{\tilde c}$. In particular we have a bijection $ A_{\tilde c}^\tau \to \mathbb{P}(T_p\mathbb{P}) \setminus \{l_{\tilde c}\}$ where $l_{\tilde c}$ is the complex line tangent to $\partial {{\mathbb H}^2_{\mathbb C}}$ at $\tilde c$. The linear projection from $\tilde c$ give an equivariant holomorphic map $W_{\tilde c} \to A_{\tilde c}$ where $S_{\tilde c}$ acts through its quotient group $Z_{\tilde c} \backslash S_{\tilde c} \simeq A^{\tau}_{\tilde c} \rtimes {\mathbb C}^*$. The latter group acts as the complex affine group of $A_{\tilde c}$.
The natural map $\psi:\tilde V^1_{\tilde c}:=Z_{\tilde c}\backslash W_{\tilde c} \to A_{\tilde c}$ is a holomorphic fiber bundle whose fiber at $\lambda \in A_{\tilde c}$ is $Z_{\tilde c} \backslash\lambda \cap W_{\tilde c}$ a pointed disk. There is an effective action of $Z_{\tilde c} \backslash S_{\tilde c}$ on $\tilde V^1_{\tilde c}$ hence an action of $A_{\tilde c}^{\tau}$ such that $\psi$ is equivariant. Now, consider the genus one Riemann Surface $E_{\tilde c}:=\Lambda_{\tilde c} \backslash A_{\tilde c}$. The commutator gives a natural symplectic form on $\Lambda_{\tilde c}$ with values in $Z_{\tilde c}$ hence, with an adequate choice of a generator of $Z_{\tilde c}$, a polarization $\Theta_{\tilde c}$ of the weight $-1$ Hodge structure on $\Lambda_c$ induced by the complex structure on $\Lambda_{\tilde c} \otimes_{{\mathbb Z}}{\mathbb R}$. There is also a map $\psi': V^1_{\tilde c}:=\Lambda_{\tilde c} \backslash \tilde V^1_{\tilde c} \to E_{\tilde c}$ which is a holomorphic fiber bundle in pointed disks. A coordinate calculation enables to see that it is biholomorphic to the complement of the zero section of a unit disk bundle of a hermitian line bundle $(L_{\tilde c}, h_{\tilde c})$ on $E_{\tilde c}$ with constant curvature $c_1(L_{\tilde c})=-\Theta_{\tilde c}$. The degree of $\Theta_{\tilde c}$ is the index in $Z_{\tilde c}$ of the image of the symplectic form on $\Lambda_{\tilde c}$.
Hence there is a partial compactification $V^1_{\tilde c} \simeq U^{1, tor}_{\tilde c} \setminus D^{1, tor}_{\tilde c}$ where $U^{1, tor}_{\tilde c}$ is the full unit bundle of $(L_{\tilde c}, h_{\tilde c})$ and $D^{1, tor}_{\tilde c}$ is the zero section, a smooth divisor isomorphic to $E_{\tilde c}$.
The quotient action of the quotient a group $\mu_k$ on the hermitian line bundle $(L_{\tilde c},h_{\tilde c})$ gives an action on $U^{1, tor}_{\tilde c}$ in such a way that the natural retraction $\pi^1_{\tilde c}: (U^{1, tor}_{\tilde c}, D^{1,tor}_{\tilde c}) \to E_{\tilde c}$ is equivariant. In particular the group $\mu_k$ acts on $E_{\tilde c}$ in such a way that the action on $H^0(E_{\tilde c}, \Omega^1)$ is given by complex multiplication. Since it is an action by automorphisms and since the Lefschetz number of an automorphism is an integer it follows that $k \in \{ 1, 2, 3,4, 6 \}$[^6].
Dividing out $\pi^1_{\tilde c}$ by $\mu_k$ we get a retraction $\pi^1_{\tilde c}: U^{tor}_{\tilde c} \to D^{tor}_{\tilde c}$ in such a way that $U_c \simeq U^{tor}_{\tilde c} \setminus D^{tor}_{\tilde c}$ and a map $U_{\tilde c}^{tor} \to U_{c}$ contracting $D^{tor}_{\tilde c}$ to $\tilde c$.
Since the construction is indepedant of $\tilde c$ we may drop this dependency in our notation and redefine $(U^{1,tor}_c, D^{1, tor}_c,E_c):= (U^{1,tor}_{\tilde c},
D^{1, tor}_{\tilde c},E_{\tilde c}) $ for some $\tilde c$.
Gluing the $U^{tor}_{c}$ with $X_{\Gamma}$ along $U_c$ gives a normal surface $X^{tor}_{\Gamma}$ with a family $(D_c)_{c\in C}$ of disjoint curves with an isomorphism $X_{\Gamma} \to X^{tor}_{\Gamma} \setminus \bigcup_{c\in C} D_c$ and a map $X^{tor}_{\Gamma} \to X^{BBS}_{\Gamma}$ contracting $D_c$ to $c$. Then, $X^{tor}_{\Gamma}$ is a normal surface with quotient singularities and is projective algebraic too [@AMRT; @Mok].
It is actually better to glue the orbifold $ \mathcal{X}_{\Gamma}:=[\Gamma \backslash {{\mathbb H}^2_{\mathbb C}}]$ (stack theoretic quotient) with $[\mu_k \backslash U^{1, tor}_{\tilde c}]$ along $[\Gamma_{\tilde c} \backslash W_{\tilde c}]$ to get a compact orbifold $\mathcal{X}_{\Gamma}^{tor}$ containing $ \mathcal{X}_{\Gamma}$ as the complement of a smooth divisor $\mathcal{D}_{\Gamma}$ consisting of a finite number of disjoint smooth substacks $(\mathcal{D}_c)_{c\in C}$ whose generic points have no inertia. One also has a stack theoretic compactification $\mathcal{X}_{\Gamma}^{BBS}$ obtained by adding the disjoint union of the $(B\mu_{k_c})_{c\in C}$ and a contraction map $\mathcal{X}_{\Gamma}^{tor} \to \mathcal{X}_{\Gamma}^{BBS}$.
When the lattice is neat the stack we constructed is equivalent to the usual smooth toroidal compactification of [@AMRT; @Mok]. On the other other hand, $\mathcal{X}_{\Gamma}^{tor}$ is NOT the quotient stack of the toroidal compactification attached to a neat normal sublattice: the generic point of the boundary has trivial isotropy.
###
Some comments have to be made regarding the gluing construction we are performing. First of all, gluing DM topological stacks along open substacks is always possible thanks to [@No2 Cor. 16.11, p. 57] - we are using the class of local homeomorphisms as [**[LF]{}**]{}. In order to have a better picture of the toroidal compactifications, they are smooth complex DM-stacks, we can present the open substacks as the quotient of their frame bundles by the general linear group. The frame bundle is indeed representable, e.g. an ordinary complex manifold, and carries an [*infinitesimally*]{} free proper action of the general linear group. If this action is free the stack is equivalent to an ordinary manifold. An equivalence of smooth DM stacks then gives rise to an isomorphism of the frame bundles intertwining the infinitesimally free actions and these glue perfectly well along invariant open subsets.
Also, the construction can be performed with a non-effective finite kernel action whose image is a lattice the price being that one has to consider general smooth Deligne-Mumford stacks. This seems to be inevitable if one wants to work with lattices in $U(2,1)$ as in [@Hol].
Fundamental Groups of orbifold toroidal compactifications
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The moduli space of $\mathcal{X}_{\Gamma}^{*}$ is $X_{\Gamma}^{*}$ for $*=\emptyset , BBS, tor$. We will denote by $m:\mathcal{X}_{\Gamma}^{*}\to X_{\Gamma}^{*}$ the moduli map. Using Van Kampen [@Zoo], we get:
\[vk\]
Let $x_0$ be a base point of $\mathcal{X}_{\Gamma}$. Then $\pi_1(\mathcal{X}_{\Gamma},x_0)=\Gamma$,
- $\pi_1(\mathcal{X}^{tor}_{\Gamma},x_0)$ is the quotient $\Gamma^{tor}$ of $\Gamma$ by the subgroup normally generated by the $Z_{c}$
- $\pi_1(\mathcal{X}^{BBS}_{\Gamma},x_0)$ is the quotient $\Gamma^{BBS}$ of $\Gamma$ by the subgroup normally generated by the $H_{ c}\cap \Gamma_{ c}$.
The natural map $H^1(\Gamma^{tor}, {\mathbb Q}) \to H^1(\Gamma, {\mathbb Q})$ is an isomorphism, consequently the Deligne MHS on $H^1(\Gamma \backslash {{\mathbb H}^2_{\mathbb C}}, {\mathbb Z})$ is pure of weight one.
Dually, it is enough to show that $H_1(\Gamma^{tor}, {\mathbb Q}) \leftarrow H_1(\Gamma, {\mathbb Q})$ is an isomorphism. This amounts to proving that the image of $H_1(Z_{ c}) \to H_1(\Gamma)$ is torsion. The group $Z_{ c}$ contains the commutator subgroup of $H_{ c} \cap \Gamma_{ c}$ as a finite index subgroup. In particular it maps to the torsion subgroup of $H_1(\Gamma)=\Gamma \slash [\Gamma, \Gamma]$.
The moduli map $m$ presents the fundamental group of $X_{\Gamma}^{*}$ is the quotient of the fundamental group of $\mathcal{X}^{*}_{\Gamma}$ by the normal subgroup generated by the images of the inertia morphisms, see [@No1], and gives an isomorphism on cohomology with rational coefficients.
Lemma \[vk\] is not new, the first point is the easiest special case of [@KaSa], the second point results from [@LKM]. Note that in the rank $\ge 2$ case, Margulis’ normal subgroup theorem implies that the fundamental group of a toroidal compactification of a ${\mathbb R}$-rank $\ge 2$ irreducible locally hermitian symmetric space is finite.
Can $\Gamma^{tor}$ be finite?
Ramification of the natural Orbifold Toroidal compactifications maps
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\[ramif\] Let $\Gamma'< \Gamma$ be a finite index subgroup. Then, the finite covering map $X_{\Gamma'} \to X_{\Gamma}$ lifts to an orbifold map $\mathcal{X}^{tor}_{\Gamma'} \to \mathcal{X}^{tor}_{\Gamma'}$ which restricts over $X_{\Gamma}$ to an étale map $\mathcal{X}_{\Gamma'} \to \mathcal{X}_{\Gamma}$.
- Let $c'$, $c$ be cusps such that the mapping $\eta: X^{BBS}_{\Gamma'} \to X^{BBS}_{\Gamma}$ maps $c'$ to $c$. The ramification index of $\mathcal{D}_{c'}$ over $\mathcal{D}_c$ is $d_{c',c}:=[Z_{c'}:Z_{c}]$.
- If $\Gamma'$ is normal in $\Gamma$ and $G=\Gamma' \backslash \Gamma$ is the quotient subgroup, $G$ acts [^7] on $\mathcal{X}^{tor}_{\Gamma'}$, $d_{c',c}:=d_{c}$ depends only on $c$ and $[G\backslash \mathcal{X}^{tor}_{\Gamma'}]= \mathcal{X}^{tor}_{\Gamma}(\mathcal{D}_{\Gamma}, (d_c)_{c\in C})$.
Due to ramification, the $\Gamma^{tor}, \ \Gamma \in \mathcal{C}_d$, split into infinitely many commensurability classes.
Hermitian forms over ${\mathbb Q}(\sqrt{-d})$
---------------------------------------------
Let $d\in {\mathbb N}^*$ be a squarefree positive integer.
The imaginary quadratic field ${\mathbb Q}(\sqrt{-d})$ has the complex conjugation as its Galois isomorphism. A non degenerate hermitian form $H$ over ${{\mathbb Q}(\sqrt{-d})}$ defines a ${\mathbb Q}$-algebraic group $U(H)$ which is a form of $GL(\dim H)$. The ${{\mathbb Q}(\sqrt{-d})}$ vector space $V_H$ underlying $H$ will be denoted by $V_H$. The signature of $H$ is the signature of the corresponding hermitian form also denoted by $H$ on $V:=V_H\otimes_ {Q} R:=H\otimes_{{{\mathbb Q}(\sqrt{-d})}} {\mathbb C}$ where ${{\mathbb Q}(\sqrt{-d})}\to {\mathbb C}$ is a given embedding.
Let ${\mathcal{O}_d}\subset {\mathbb Q}(\sqrt{-d})$ be the subring of its quadratic integers. Let $L$ be a free ${\mathcal{O}_d}$ module which is a lattice in the ${{\mathbb Q}(\sqrt{-d})}$ vector space $V_H$ underlying $H$. Then $\Gamma_{H, L} = PU(H)\cap PAut(L)$ is an arithmetic subgroup and the $\Gamma_{H,L}$ all belong to a commensurability class $\mathcal{C}_d$ of lattices in $PU(H)$. Denote by $\Gamma_{{\mathbb Q}}$ be the group of ${\mathbb Q}$-points of $PU(H)$ which is dense in $PU(H_{{\mathbb R}})\simeq PU(2,1)$ with respect to the classical topology.
The study of the groups $\pi_1(\mathcal{X}^{tor}_{\Gamma})$ and $\pi_1(\mathcal{X}^{tor}_{BBS})$ for $\Gamma\in \mathcal{C}_d$ does not seem to have been carried out systematically in the litterature even in that simple case.
It is known that all non uniform commensurability classes of arithmetic lattices in $PU(2,1)$ are of the form $\mathcal{C}_d$.
Proof of the main theorems
==========================
Commensurability classes with non vanishing virtual $b_1$
---------------------------------------------------------
By commensurability, we mean commensurability in the wide sense:
Two lattices $\Gamma_1, \Gamma_2<PU(2,1)$ are commensurable if there exists a finite index torsion free lattice $\Gamma'_i < \Gamma_i$ and a holomorphic isometry $\Gamma'_1\backslash {{\mathbb H}^2_{\mathbb C}}\to \Gamma'_2\backslash {{\mathbb H}^2_{\mathbb C}}$.
Say a commensurability class $\mathcal{C}$ of non uniform lattices in $PU(2,1)$ has non vanisihing virtual $b_1$ if some member $\Gamma_0$ satisfies $b_1(\Gamma_0, {\mathbb Q})>0$.
Each commensurability class of non uniform arithmetic lattices in $PU(2,1)$ has non vanishing virtual $b_1$ [@Wa], generalizing [@Ka]. Since these lattices are finitely generated and passing to a sublattice increases $b_1(\_, {\mathbb Q})$:
If $\mathcal{C}$ has non vanisihing virtual $b_1$ every member $\Gamma$ has a finite index normal subgroup $\Gamma_1$ such that $b_1(\Gamma_1, {\mathbb Q})>0$.
Freeness of virtual Albanese Mappings
-------------------------------------
We will now fix an [**[arithmetic]{}**]{} lattice which is torsion free and has unipotent mondromies so that the toroidal compatification $\bar X$ is a complex projective manifold. Using $b_1(\Gamma, {\mathbb Q})= b_1(\Gamma^{tor}, {\mathbb Q})$ we conclude that there is a non zero closed holomorphic one-form $\alpha$ on $\bar X$. We restrict $\alpha$ to $X$ lift it to ${{\mathbb H}^2_{\mathbb C}}$ to construct a closed one form $\tilde \omega \in \Omega^1({{\mathbb H}^2_{\mathbb C}})$.
Every element of the vector space $V$ spanned by the $\Gamma_{{\mathbb Q}}.\tilde \omega$ is the lift of a holomorphic closed one form on $X_{\Gamma'}^{tor}$ for some normal finite index sugroup $\Gamma'\le \Gamma$.
Indeed $\Gamma_{{\mathbb Q}}$ is the commensurator of $\Gamma$. Here we use arithmeticity in a crucial way.
The closure $\bar V$ in the Fréchet space $\Omega^1({{\mathbb H}^2_{\mathbb C}})$ is a non zero vector space of closed holomorphic $1$-forms which is preserved by the action of $PU(H_{{\mathbb R}})$.
Indeed $\Gamma_{{\mathbb Q}}$ is dense in $PU(H_{{\mathbb R}})$.
\[coframe\] For every point $o$ of ${{\mathbb H}^2_{\mathbb C}}$ there are two elements of $V$ which gives a coframe of the tangent space at $o$.
The restriction $r:\bar V \to \Omega^1_{{{\mathbb H}^2_{\mathbb C}}, o}$ is equivariant under the stabilizer $K$ of $o$. Hence the image of this linear map is $K$-equivariant and no zero, the surjectivity of $r$ follows from the irreducibility of the isotropy action on the cotangent space.
\[unram\] There is a finite index normal subgroup $\Gamma'$ such that every $\Gamma''<\Gamma'$, has an Albanese map which is unramified on $X_{\Gamma''}$. In particular $\bar{X}^{tor}_{\Gamma''}$ satisfies the Shafarevich conjecture.
The immersivity outside the boundary is an immediate consequence of Corollary \[coframe\] using noetherian induction. I learned from [@Stov] that the fact that the Albanese map does not factor through a curve follows from [@Clo]. The application to Shafarevich conjecture is then a consequence of [@Nap] since an [**[irreducible]{}**]{} connected component of a fibre of the Albanese mapping cannot give rise to a Nori chain. Note that $SSC$ need not be satisfied.
\[primitive\] The space $W=\int \bar V \subset \mathcal{O}
({{\mathbb H}^2_{\mathbb C}})$ of primitives of the elements of $\bar V$ is infinite dimensional. Actually the map $W \to \mathcal{O}_{{{\mathbb H}^2_{\mathbb C}},o}/\mathfrak{m}^N$ is surjective for all $N>0$.
$W$ is invariant under the whole group $PU(H_{{\mathbb R}})$. The evaluation map $ev:W \to \mathcal{O}_{{{\mathbb H}^2_{\mathbb C}},o}$ is $K$-equivariant. So is the completed evaluation map: $W \to \widehat{\mathcal{O}}_{{{\mathbb H}^2_{\mathbb C}},o}$. But this is $K$ equivariantly isomorphic to ${\mathbb C}[[\mathfrak{m}\slash\mathfrak{m}^2]]=\sum_{n\in {\mathbb N}} Sym^n \mathfrak{m}\slash\mathfrak{m}^2$. To construct this isomorphism we have chosen the polynomials as a $K$-invariant subspace of holomorphic functions and the linear functions as generators of the maximal ideal $\mathfrak{m}$. The $K$ equivariance implies $ev(W)\supset ev(W)\cap Sym^n \mathfrak{m}\slash\mathfrak{m}^2$ for all $n$. If $W$ were finite there would be a finite number of nontrivial $ev(W)\cap Sym^n \mathfrak{m}\slash\mathfrak{m}^2$ which would give a direct sum decomposition of $ev(W)$. Hence $W$ would consist of polynomials. Restricting to a generic line through the origin (viewed as a complex geodesic curve isomorphic to a unit disk) there would be a non zero space of polynomials in one variable, generated by monomials, fixed by the homographies which are the automorphisms of the unit disk, which is obviously absurd.
We have already proved surjectivity of $\bar V \to \mathcal{O}_{{{\mathbb H}^2_{\mathbb C}},o}/\mathfrak{m}^2$. We now use induction on $N$ and assume the theorem is proved for some $N$. Consider the smallest integer $M>N$ such that $ev(W)\cap Sym^M \mathfrak{m}\slash\mathfrak{m}^2\not =0$. $M$ exists since $W$ is infinite dimensional. Using the irreducibilty of the isotropy representation on $Sym^M \mathfrak{m}\slash\mathfrak{m}^2$ we see that $ev(W)\cap Sym^M \mathfrak{m}\slash\mathfrak{m}^2=Sym^M \mathfrak{m}\slash\mathfrak{m}^2$. Hence there is a monomial say $z_1^M$ in $W$. Now one of the infinitesimal generators of the Lie algebra of $PU(H_{{\mathbb R}})$ takes the form $\xi=\frac{\partial}{\partial z_1} + \xi'$ where $\xi'\in \mathfrak{m} \mathcal{T}_{{{\mathbb H}^2_{\mathbb C}}}$. If $M>N+1$, $\xi . z_1^M \in W$ would contradict the minimality of $M$. Hence $M=N+1$ which is the desired conclusion.
Fix $M\in {\mathbb N}$. There is a finite index normal subgroup $\Gamma'$ such that forall $\Gamma''<\Gamma'$, $\bar{X}^{tor}_{\Gamma''}$ has an Albanese map which generates $M$ jets at all points of $X_{\Gamma''}$.
In particular, if $M=2$ the genus zero curves in the Albanese image lie on the image of the boundary and we recover special cases of classical results:
\[[@Nad; @BT]\] $\bar{X}^{tor}_{\Gamma''}$ satisfies the Green-Griffiths-Lang conjecture
Immediate corollary of [@Och Theorem D].
$X_{\Gamma''}$ has finitely many rational points on any number field of definition.
Immediate corollary of Faltings’ solution of Lang’s conjecture [@Fal], see also [@HinSil Theorem F.1.1, p. 436].
Refined commensurability classes
--------------------------------
The following definition is a slight generalization of the notion of completely étale map between negatively elliptic bounded surfaces [@Hol2 p.256]:
\[refined\] Two (commensurable) lattices $\Gamma_1, \Gamma_2<PU(2,1)$ are refined commensurable if there exists a finite index lattice $\Gamma'_i < \Gamma_i$ and a holomorphic isometry $\Gamma'_1\backslash {{\mathbb H}^2_{\mathbb C}}\to \Gamma'_2\backslash {{\mathbb H}^2_{\mathbb C}}$ such that $\mathcal{X}_{\Gamma'_i}^{tor} \to \mathcal{X}_{\Gamma_i}^{tor}$ is étale.
If $ \mathcal{X}_{\Gamma_i}^{tor}$ is uniformizable we can assume that $\Gamma'_i$ is torsion free and neat. The questions (1)-(6) (and the orbifold Kodaira dimension) in the introduction depend only on the refined commensurability class.
Two commensurable arithmetic lattices $\Gamma_1,\Gamma_2$ having a common finite index lattice are refined commensurable iff for every $\tilde c \in C \subset\partial {{\mathbb H}^2_{\mathbb C}}$ there is a common finite index subgroup $\Gamma_3$ such that $Z(H_{\tilde c})\cap \Gamma_1=Z(H_{\tilde c})\cap \Gamma_2=Z(H_{\tilde c})\cap \Gamma_3$.
There is an order on the set $\mathcal{C}_r$ of refined commensurability classes of a given commensurability class. We say $\mathcal{C}_1 \prec \mathcal{C}_2$ is there is some member of $ \mathcal{C}_2$ is a finite index subgroup of a member of $ \mathcal{C}_1$. It is clear that $(\mathcal{C}_r,\prec)$ is a filtering order and that the questions investigated in the introduction are more difficult for small elements (with the exception of $L(\_)$).
Is $(\mathcal{C}_r,\prec)$ Artinian? Is every initial segment finite?
A refined commensurability class is said to be small if the common universal covering stack of the corresponding $\mathcal{X}^{tor}_{\Gamma}$ is not a Stein manifold. Its $\Gamma$-dimension is the dimension of the Campana quotient of the moduli space of the universal covering stack by its compact complex subvarieties.
They are the most interesting classes from the present perspective.
Higher dimensions
-----------------
The only missing ingredient being [@Nap], we get if $n\ge 3$:
If $\Gamma<PU(n,1)$ is arithmetic and small enough in its commensurability class, $\bar X_{\Gamma}$ has Albanese dimension $n$ and its image contains no translate of an abelian subvariety at its generic point. It satisfies $ Q$, has big cotangent bundle, is Kobayashi hyperbolic modulo its boundary, and the rational points over a number field over which it is defined are finite modulo its boundary.
The Picard-Eisenstein commensurability class
============================================
The Picard-Eisenstein Lattice
-----------------------------
Let us look at the Picard-Eisenstein lattices using its beautiful two-generator presentation [@FP]. This group, denoted by $PU(H_0, {\mathbb Z}[\omega])$ is in $\mathcal{C}_3$ and is actually of the form $\Gamma_{H, L}$ for the hermitian form whose matrix is $H_0=
\left(
\begin{array}{ccc}
0&0&1 \\
0&1&0 \\
1&0&0
\end{array}
\right)
$ and $L=\mathcal {O}_3^{\oplus 3}={\mathbb Z}[\frac{-1+i\sqrt{3}}{2}]^{\oplus 3}$ is the standard lattice.
\[surj\] $PU(H_0, {\mathbb Z}[\omega])^{tor}$ has a surjective morphism to a $(2,3,6)$ orbifold group.
The presentation in [@FP] is $$<P,Q,R| R^2=[PQ^{-1} , R]= (RP)^3=P^3Q^{-2}=(QP^{-1})^6>
.$$ We are imposing $P^3=Q^2=1$ in this presentation by [@FP p. 258]. Further imposing $R=1$ gives $<P,Q| P^3=Q^2=(PQ^{-1})^6=1>$ as a quotient group.
It may be reassuring to get a confirmation of the:
The Picard-Eisenstein commensurability class $\mathcal{C}_3$ has non vanishing virtual $b_1$.
Hirzebruch’s example of a configuration of elliptic curves in an abelian surface whose complement is complex hyperbolic {#sec2}
-----------------------------------------------------------------------------------------------------------------------
It is a well known theorem of Holzapfel [@Hol2] that the following example is in $\mathcal{C}_3$ .
###
Let $E\simeq {\mathbb C}/ {\mathbb Z}[j]$ be the elliptic curve isomorphic to Fermat’s cubic curve and let $\bar X$ be the blow up of $(0_{E \times E})$. We denote by $D_i$ the strict transforms in $\bar X$ of the ellitic curves $E_i$ where: $$E_1=E\times \{ 0_E\}, E_2= \{ 0_E\}\times E, E_3=\Delta=Gr(id_E), E_4=Gr(-j)$$ and we obtain a SNC divisor $(\bar X, D)$ where $D=D_1+D_2+D_3+D_4$. The $D_i$ are pairwise disjoint.
(Hirzebruch, [@BHH]) $X\simeq \Gamma_{Hirz} \backslash {{\mathbb H}^2_{\mathbb C}}$ where $\Gamma_{Hirz} \subset PU(2,1)$ is a non uniform lattice. $\bar X$ is its toroidal compactification.
The notation $(\bar X, D)$ will be used throughout this section to denote this particular confinguration.
There is a neat lattice $\Gamma'\in \mathcal{C}_3$ such that for every $\Gamma''<\Gamma'$, $\bar{X}^{tor}_{\Gamma''}$ has a finite Albanese map. In particular the universal covering space of $\bar{X}^{tor}_{\Gamma''}$ is Stein.
Corollary \[unram\] implies that the only curves contracted by the Albanese map lie on the boundary. It is thus enough to replace the $\Gamma'$ in Corollary \[unram\] with $\Gamma'\cap \Gamma_{Hirz}$ for which the Albanese map does not contract any boundary curve.
\[symhirz\] [@DM Prop 15.17, P. 155] This configuration of elliptic curves has an order $72$ complex reflection group $H$ of automorphisms which is a $\mu_6$ central extension of $A_4$ acting as the symmetry group of the confinguration of $4$ points in $\mathbb{P}^1$ given by the points $0, 1, \infty, e^{\frac{2\pi i}{6}}$. $A_4$ acts as an alternating group on the set whose elements are the 4 elliptic curves.
### The Orbifold attached to a Normal subgroup of $\Gamma_{Hirz}$
Let us make Lemma \[ramif\] more explicit in the present case.
Let $\Gamma' \triangleleft_{fi} \Gamma_{Hirz}$ be a finite index normal subgroup. Then $\Gamma'$ is torsion-free and torsion-free at infinity so that the toroidal compactification $\bar X'\supset X'=\Gamma' \backslash {{\mathbb H}^2_{\mathbb C}}$ is a complex projective manifold with an effective $G=\Gamma_{Hirz} \slash \Gamma'$-action which is fixed point free on $X'$. It is clear that $G\backslash \bar X'= \bar X$ and we denote the corresponding quotient orbifold by $\mathcal{X}'= [G\backslash \bar X']$ and the corresponding quotient map by $\pi: \bar X' \to \bar X$.
\[lem22\] Let $p'_i \subset E'_i$ be a flag consisting of a point $p_i\in \pi^{-1}(E_i)$ and $E'_i$ the irreducible component of $\pi^{-1}(E_i)=\sum_j F_{ij}$ via $p_i$. Consider $$S'_i=Stab_G(p'_i)
< H_i=Stab_G(E_i') < G.$$ Then $S'_i$ is a cyclic central subgroup of $H_i$ of order $d_i=d_i(\Gamma')$, $G_i=H_i\slash S_i'$ acts effectively on $E'_i$ without fixed points, $E_i=G_i\backslash E'_i$ and $p^*E_i=\sum_{j\in H_i\backslash G} d_i F_{ij}$.
Furthermore $\mathcal{X}'$ is equivalent to $\mathcal{X}(\bar X, D, d)$ with $d=(d_1, \ldots d_4)$.
We shall adopt the notations of Lemma \[lem22\] in the rest of section \[sec2\].
Conversely every proper finite étale mapping $p: \bar Y \to \mathcal{X}(\bar X, D, d)$ comes from a finite index subgroup $\Gamma'= Im
(\pi_1(\bar Y \setminus p^{-1} (D)) \to \pi_1(\bar X \setminus D)) $ such that $p^* D= \sum_i d_i Supp(p^{-1} (E_i))$.
The support $Supp(D)$ of an effective Cartier divisor $D$ is the sum of its irreducible components with multiplicity one.
The existence of $\bar Y$ is equivalent to $\mathcal{X}(\bar X, D, d)$ being uniformizable.
\[q1\] For which $d\in {\mathbb N}_{\ge 1}^4$ is $\mathcal{X}(\bar X, D, d)$ uniformizable?
$\mathcal{X}(\bar X, D, d)$ is not developpable if $\{d_1, \ldots, d_4\}=\{1,1,1, m \}$ or $\{1,1,m, m' \}$ with $1<m<m'$.
In the listed case the exceptional curve in $\bar X$ gives a sub-orbifold equivalent to $\mathbb{P}(m.[0])$ or to $\mathbb{P}(m.[0]+m'[1])$ which are non developpable, which obstructs the developpability of the ambient orbifold.
\[q2\] Assume $d\in {\mathbb N}_{\ge 1}^4$ is such that $\mathcal{X}(\bar X, D, d)$ uniformizable. Let $Z:=Exc$ be the exceptional curve and $\mathcal{Z}=(\mathbb
P^1, d_1,d_2,d_3,d_4)$ be the natural suborbifold. $SS(\mathcal{X}(\bar X, D, d)) $ holds iff it holds for $\mathcal{Z}$.
### $d_3=d_4=1$
This corresponds to studying the pair $(\bar X, D'':=D_1+D_2)$. In order to fix notations, we denote by $U''$ the complement of $E_1+E_2$ in $E\times E$ and by $X''$ the complement of $D_1+D_2$ in $\bar X$. Plainly $U''=X''\setminus Exc$.
The fundamental group of $U''$ is a product of two finite groups on 2 generators $\pi_1(U'')=\mathbb F_2(a,b) \times \mathbb F_2(c,d)$. The fundamental group of $X''$ is the quotient of $\pi_1(U'')$ by the normal subgroup generated by $[a,b].[c,d]$.
Elementary calculation.
The natural morphism $F_2(a,b) \to \pi_1(X'')$ factors through $H_{{\mathbb Z}}$ the Heisenberg group of unipotent $3\times 3$ upper triangular matrices with integer coefficients .
Since $a$ and $b$ commute with $c$ and $d$ the relation $[a,b]=[c,d]^{-1}$ implies that $a$ and $b$ commute with $[a,b]$. But $F_2(a,b)/<<[a,[a,b]], [b,[a,b]]>>\simeq H_{{\mathbb Z}}$.
We have a non trivial central extension: $$1\to {\mathbb Z}_{[a,b]}=Z(H_{{\mathbb Z}} ) \to H_{{\mathbb Z}} \to {\mathbb Z}^2_{a,b} \to 1.$$
The geometric interpretation is clear. Consider the boundary $B$ of a regular neighborhood of $D_1$ in $\bar X$. If the base point is in $B$ $a$ and $b$ can be homotoped in $B\cap U''$. However $B\subset X''$ and $\pi_1(B)=H_{{\mathbb Z}}$ since $D_1^2=-1$.
The fundamental group $\pi_1(X'')$ is the quotient group of $H_{{\mathbb Z}} ^2$ by the diagonal subgroup of its center $\Delta_{{\mathbb Z}}: {\mathbb Z}\to {\mathbb Z}^2$. Thus, we have the central extension: $$1 \to {\mathbb Z}\to \pi_1(X'') \to {\mathbb Z}^4=H_1(X'') \to 1,$$ $[a,b]$ mapping to a generator of the center and $[c,d]$ to its opposite.
The map $H_{{\mathbb Z}} ^2 \to\pi_1(X'')$ comes from the previous lemma which can also be applied with $c,d$ and all these groups have naturally isomorphic abelianization.
\[unif1\] $\mathcal{X}(\bar X, D, n,n,1,1)$ is uniformizable.
The new relations to get $\pi_1(\mathcal{X}(\bar X, D, n,n,1,1))$ are $[a,b]^n=[c,d]^n=1$. It thus suffices to consider the quotient map $$\pi_1(\mathcal{X}(\bar X, D, n,n,1,1) \to H^2_{{\mathbb Z}/n{\mathbb Z}}\slash\Delta_{{\mathbb Z}}({\mathbb Z}/n{\mathbb Z}).$$
Thanks to lemma \[q2\] $SSC(\mathcal{X}(\bar X, D, n,n,1,1))$ is trivially true since the fundamental group of $\mathcal Z\simeq \mathbb{P}^1(\sqrt[n]{0+\infty})$ is finite.
### $d_4=1$
This corresponds to studying the pair $(\bar X, D':=D_1+D_2+D_3)$. In order to fix notations, we denote by $U'$ the complement of $E_1+E_2+E_3$ in $E\times E$ and by $X'$ the complement of $D_1+D_2+D_3$ in $\bar X$. Plainly $U'=X'\setminus Exc$.
\[relx’\] The fundamental group $G'=\pi_1(U')$ has 5 generators $a,b,c,d,e$ and is presented by the relations: $$a^{-1} ca=c, a^{-1} e a= e, a^{-1} d a = c^{-1} d c e,$$ $$b^{-1}db=d, b^{-1} e b=e, b^{-1} c b=d^{-1}c d e^{-1}.$$
Omitted. Easy calculation.
It is is a semi-direct product of $F_3(c,d,e)$ by $F_2(a,b)$.
One has to set $\alpha=c^{-1}a$ $\beta=d^{-1}b$ $e=1$ to recover the previous case.
Let $V'\subset U'$ the trace on $U'$ of a regular neighborhood of $Exc$. Then $\pi_1(V')\hookrightarrow \pi_1(U')$ is generated by $a_1,a_2,a_3$ subject to the relations: $$a_3a_2a_1=a_2a_1a_3=a_1a_3a_1$$ where $$a_3=e, \ a_1=c^{-1}d^{-1}cde^{-1}, \ a_2=\alpha^{-1}\beta^{-1}e^{-1}\alpha\beta.$$ The center $Z(\pi_1(V'))$ is infinite cyclic generated by the element $\gamma_Z=a_3a_2a_1=eaba^{-1}b^{-1}.$
Omitted. In principle the calculation is easy, but it turned out to be rather messy.
The fundamental group $G:=\pi_1(X')$ has 5 generators $a,b,c,d,e$ and is presented by the relations \[relx’\] plus: $$eaba^{-1}b^{-1}=1.$$
The fundamental group $\pi_1(\mathcal{X}(\bar X, D, d_1,d_2,d_3,1))$ has 5 generators $a,b,c,d,e$ and is presented by the relations \[relx’\] plus: $$a_3a_2a_1=a_1^{d_1}= a_2^{d_2}= a_3^{d_3}=1.$$
The map $\pi_1(\mathcal Z)=F_2(a_1,a_2)/<<a_1^{d_1}, a_2^{d_2},(a_2a_1)^{d_3}>>\to \pi_1(\mathcal{X}(\bar X, D, d_1,d_2,d_3,1))$ maps $a_1,a_2$ to their above expressions.
Hence by Lemma \[q2\] SSC holds true in that case if and only if we can find a finite index normal subgroup $H$ of $G(d_1,d_2,d_3):=\pi_1(\mathcal{X}(\bar X, D, d_1,d_2,d_3,1))$ and $\eta \in H_1(H, {\mathbb Q})$ which does not vanish on $<a_1,a_2>\cap H$. The worst possible choice is when $G(d_1,d_2,d_3)/H$ is abelian. Unfortunately since $H_1(G(d_1,d_2,d_3))$ has rank $4$ a lot of the $H$ one gets with `LowIndexSubgroups` in GAP or MAGMA have that property.
\[unif2\] $\mathcal{X}(\bar X,D,(n,n,n,1))$ is uniformizable.
Immediate consequence of Corollary \[unif1\]. Indeed, thanks to Lemma \[morphstack\], we have a map $$\mathcal{X}(\bar X,D,(n,n,n,1)) \to \mathcal{X}(\bar X,D,(n,n,1,1))$$ and a map $\mathcal{X}(\bar X,D,(n,n,n,1)) \to \mathcal{X}(\bar X,D,(1,n,n,1))$ thanks to Remark \[symhirz\] which gives a group morphism in a finite group $$\pi_1(\bar X,D,(n,n,n,1))
\to (H^2_{{\mathbb Z}/n{\mathbb Z}}\slash\Delta_{{\mathbb Z}}({\mathbb Z}/n{\mathbb Z}))^2$$ which is injective on the isotropy groups.
Hence $G_n=\pi_1(\mathcal{X}(\bar X, D, n,n,n,1))$ is the fundamental group of a complex projective surface. Let us introduce the following quotient of $G_n$: $$G_n^-:=G_n/<< a^n,b^n, c^n,d^n>>.$$ where $a,b,c,d$ denote the natural images of the generators of group $G$.
The nilpotent group[^8] of class $2$ $G_n^-/\gamma_3(G_n^-)$ is isomorphic to $H^2_{{\mathbb Z}/ n{\mathbb Z}}$.
Using MAGMA [@magma], we get an isomorphism: $$G/\gamma_3(G)\simeq H^2_{{\mathbb Z}},$$ whence the result since we are killing the $n$-th power of the lifted generators of $H_1(G)=G/\gamma_2(G)$. A file containing the MAGMA code is available on my webpage.
$SSC(\mathcal{X}(\bar X,D,(3,3,3,1))$ holds and the universal covering space is Stein.
The abelianization $A_3$ of $K_3:=\ker(\phi)$, where we denote by $\phi$ the resulting epimorphism $\phi : G_3 \to H^2_{{\mathbb Z}/ 3{\mathbb Z}}$ is free of rank $10$, thanks to MAGMA. MAGMA computes the image of $a_1a_3a_1^{-1}a_3^{-1}$ to be the row vector $(0,-1,-1,1,-1,0,0,0,-1,2)$ in MAGMA’s basis of $A_3$.
For all $k,l, m \in {\mathbb N}^*$, $SSC(\mathcal{X}(\bar X,D,(3k,3l,3m,1)))$ holds and the moduli space of the universal covering stack is Stein, hence the universal covering space is Stein provided the orbifold is indeed developable.
Use the natural map $\mathcal{X}(\bar X,D,(3k,3l,3m,1))\to \mathcal{X}(\bar X,D,(3,3,3,1))$ given by Lemma \[morphstack\] to deduce that the Albanese map is virtually finite.
### General case {#subsec:general}
We have no general results on uniformizability. It follows from the above that $\mathcal{X}(\bar X,D,(n,n,n,n))$ is uniformizable for all $n\in {\mathbb N}^*$ , and $SSC(\mathcal{X}(\bar X,D,(3k,3l,3m,3p))$ holds.
Some small refined commensurability classes in $\mathcal{C}_3$
--------------------------------------------------------------
###
Quite confusingly, the Picard-Eisenstein lattice studied in [@Hol2; @Hol] is not the same as the one studied in [@FP]. Also the relationship with $\Gamma_{Hirz}$ and other lattices in $\mathcal{C}_3$ is slightly involved. Holzapfel’s Picard modular group is (conjugate by a transposition matrix to) $\Gamma_{H_1, L_{st}}$ for the hermitian form whose matrix is $H_1$ and $L_{st}=\mathcal {O}_3^{\oplus 3}={\mathbb Z}[\frac{1+i\sqrt{3}}{2}]^{\oplus 3}$ is the standard lattice. By definition $H_0=^t{\bar g} H_1 g$ where we have used the notations $$H_1=
\left(
\begin{array}{ccc}
1&0&0 \\
0&-1&0 \\
0&0&1
\end{array}
\right), \
g=\left(
\begin{array}{ccc}
0&1&0 \\
1&0&1/2 \\
1&0&-1/2
\end{array}
\right).$$
As ${\mathbb Q}$-algebraic groups $PU(H_1)\simeq PU(H_0)$ - more precisely $gPU(H_0)g^{-1}=PU(H_1)$ but the $PU(H_i,\mathcal{O}_3)$ $i=0,1$ are different lattices. As we will see, the refined commensurability classes of these two lattices have a very similar behaviour but we did not complete the elementary but lenghty calculations to check that they are equal.
###
The article [@Hol2] uses another special elliptic configuration in $E\times E$. Let $0_E, Q_1, Q_2$ be the fixed points of the automorphism $j: E \to E$ where $\mathcal{O}_3^{\times}$ acts by multiplication on $E={\mathbb C}\slash \mathcal{O}_3$ and $\mathcal{O}_3={\mathbb Z}[j]\subset {\mathbb C}$ is the inclusion given by our choice of $i=\sqrt{-1}$ and the formula $j=-\frac{1}{2} + i \frac{\sqrt{3}}{2}$. Then $\{ 0_E, Q_1, Q_2 \}$ defines a group of translations $T:=\mathbb{{\mathbb Z}}\slash 3 \mathbb{Z}$ of $E$. Let $T$ acts diagonally on $E\times E$. It turns out that there is an isomorphism $T\backslash E\times E \simeq E \times E$ such that the inverse image of the Hirzebruch configuration is the union of 6 elliptic curves. This elliptic curve configuration has $\{ 0_E\times 0_E ,Q_1 \times Q_1, Q_2 \times Q_2 \}$ as its multiple (quadruple) points. These 6 elliptic curves $\{ S_k \}_{k=1}^6$ are the graph of the automorphisms $1,j, j^2$ and the horizontal factors $\{ E\times 0_E, E\times Q_1, E\times Q_2 \}$. Thus there is an index $3$ subgroup $\Gamma_{Holz} \subset \Gamma_{Hirz}$ which can be defined by the Galois correspondance $$\Gamma_{Holz}=Im(\pi_1(Bl_{T.(0_E\times 0_E) }(E\times E)\setminus \cup_{k=1}^6 S'_k)\to \pi_1(\bar X\setminus D)=\Gamma_{Hirz})\lhd\Gamma_{Hirz}$$ where $S'_k$ is the strict transform of $S_k$ and the notation $(\bar X,D)$ of subsection \[sec2\] still applies.
One of the facts [@Hol2] uses is that we have a composition sequence:
$$\Gamma_{Holz} \lhd P\tilde\Gamma'_{Holz}:=P(SU(H_1, \mathcal{O}_3)(1-j)) \lhd PU(H_1, \mathcal{O}_3)$$
with graded quotients $$P\tilde\Gamma'_{Holz}\slash \Gamma_{Holz}=\mu_3 \times \mu_3, \quad PU(H_1, \mathcal{O}_3)\slash P(SU(H_1, \mathcal{O}_3)(1-j))=S_4.$$
Furthermore, thanks to [@GerSa], we can interpret a crucial ingredient in [@Hol] as an equivalence $$[P\tilde\Gamma'_{Holz}\backslash {{\mathbb H}^2_{\mathbb C}}] \simeq \mathcal{X}(\mathbb{P}^2\setminus \{ 4 pts\}, {CEVA(2)},(3, 3,3,3,3,3))$$ $$[S_4 \backslash [P\tilde\Gamma'_{Holz}\backslash {{\mathbb H}^2_{\mathbb C}}]]\simeq[PU(H_1, \mathcal{O}_3)\backslash{{\mathbb H}^2_{\mathbb C}}],$$
where $CEVA(2)$ is the complete quadrangle built on the 4 marked points (in general linear position) in $\mathbb{P}^2$ and $S_4<PGL(3,{\mathbb C})$ permutes these $4$ points.
###
Using the equivalence $[\mu_3 \backslash E]\simeq \mathbb{P}^1(\sqrt[3]{0+1+\infty})$ we see easily that the quotient orbifold $[\mu_3\times \mu_3 \backslash Bl_{T.(0_E\times 0_E) }(E\times E)]$ is the following orbisurface: the moduli space is $Bl_{(0,0),(1,1), (\infty,\infty)}\mathbb{P}^1 \times \mathbb{P}^1$ and the ramification has order $3$ on the 9 curves given by the 3 exceptional curves and the strict transforms of the vertical and horizontal factors through the 3 blown up points. There is no ramification over the strict transform of the diagonal.
In particular $\mathcal{X}_{P\tilde\Gamma'_{Holz}}^{tor}$ is the orbifold whose moduli space is $Bl_{(0,0),(1,1), (\infty,\infty)}\mathbb{P}^1 \times \mathbb{P}^1$ and which ramifies at order 3 on the 6 (-1)-rational curves given by the 3 exceptional curves and the 3 strict transforms of the vertical factors. The $4$ boundary components are the strict transforms of the diagonal and the 3 strict transforms of the horizontal factors carry multiplicity $1$ since our construction of the orbifold toroidal compactifications precisely excludes orbifold behaviour at the general point of the boundary.
If $CEVA'(2)$ denotes the strict transform of $CEVA(2)$ in the blow up surface $Bl_{4 pts}(\mathbb{P}^2)$ we see using the familiar contraction of these 4 disjoint rational curves and [@GerSa] an equivalence: $$\mathcal{X}_{P\tilde\Gamma'_{Holz}}^{tor}\simeq \mathcal{X}( Bl_{4 pts}(\mathbb{P}^2), {CEVA'(2)}, (3, 3,3,3,3,3))$$
Using the language of [@BHH] we assign weight 3 to the strict transforms of the lines in $CEVA(2)$ and the weight 1 to the exceptional curves. Since the map $\mathcal{X}_{\Gamma_{Holz}}^{tor}\to[\mu_3\times \mu_3 \backslash Bl_{3 pts}(E\times E)]$ is étale, $\mathcal{X}_{\Gamma_{Holz}} \to \mathcal{X}_{P\tilde\Gamma'_{Holz}}$ is the only non-étale map in the orbifold version of the main diagram in [@Hol2]: $$\xymatrix{
& &\mathcal{X}_{\Gamma_{Holz}}^{tor}= Bl_{3 \ pts}(E\times E)\ar[dl]_ {ram} \ar[dr]^{et} & \\
&\mathcal{X}_{P\tilde\Gamma'_{Holz}}^{tor} \ar[dl]_{et}& &\mathcal{X}_{\Gamma_{Hirz}}^{tor}= Bl_{0}(E\times E) \\
\mathcal{X}_{PU(H_1,\mathcal{O}_3)}^{tor}&&&
}$$
In other words, we have in $\mathcal{C}_{3r}$: $$[PU(H_1,\mathcal{O}_3)]=[P\tilde\Gamma'_{Holz}] \prec [\Gamma_{Holz}]=[\Gamma_{Hirz}].$$ Both classes are small of $\Gamma$-dimensions $1$ and $2$.
\[fib\] $(P\tilde\Gamma'_{Holz})^{tor} \cong \pi_1(\mathbb{P}^1(3,3,3))\cong{\mathbb Z}[j]\rtimes \mu_3 $. $PU(H_1,\mathcal{O}_3)^{tor}$ is virtually abelian of rank 2 sitting in an exact sequence: $$\{1\}\to {\mathbb Z}[j]\rtimes \mu_3 \to PU(H_1,\mathcal{O}_3)^{tor} \to S_4 \to \{1\}.$$
Let us consider the linear system $|I_{4 pts}(2)|$ of conics through the four points. It defines a rational map $\phi: \mathbb{P}^2 \dashrightarrow \mathbb{P}^1$ which becomes regular on $Bl_{4pts}(\mathbb{P}^2)$, the exceptional curves are then isomorphically mapped $\phi$ and $CEVA'(2)$ has three connected components which have 2 irreducible as components and coincide to the three singular fibers of $\phi$. The generic fibre is a smooth conic with no deleted points. In other words, $$\phi: \mathbb{P}^2 \setminus CEVA'(2) \to \mathbb{P}^1 \setminus \{ 3 \ pts \}$$ is a projective smooth conic bundle. Lemma \[morphstack\] gives a map: $$\bar \phi: \mathcal{X}_{P\tilde\Gamma'_{Holz}}^{tor} \to \mathbb{P}^1(3,3,3)$$ whose general fiber is a smooth rational curve. This gives an isomorphism $$(P\tilde\Gamma'_{Holz})^{tor} \buildrel{\cong }\over\longrightarrow \pi_1(\mathbb{P}^1(3,3,3)).$$ The other statements are immediate consequences granted the geometric description above.
When $\Gamma$ lies in $[PU(H_1,\mathcal{O}_3)]$ or $ [\Gamma_{Hirz}]$, $\Gamma^{tor}$ is infinite virtually abelian and linear.
###
In the notations of [@Mos], the group $PU(H_0,\mathcal{O}_3)$ can be described as $\Gamma_{\mu, S_3}$ with $\mu=\frac{1}{6}(2,2,2,1,5)$ [@Der]. Since this ball 5-uple satisfies INT, the orbifold $[PU(H_0,\mathcal{O}_3)\backslash {{\mathbb H}^2_{\mathbb C}}]$ can be easily described.
The moduli space of $[\Gamma_{\mu}\backslash {{\mathbb H}^2_{\mathbb C}}]$ is $\mathbb{P}^2\setminus \{ P \}$ this point $P$ being a triple point of $CEVA(2)$. In $[\Gamma_{\mu}\backslash {{\mathbb H}^2_{\mathbb C}}]$ the $3$ lines through $P$ have orbifold weight 3, the three remaining lines have orbifold weight 2 and the 3 triple points have a non abelian order $36$ inertia group [@Ulu pp 392-393].
Then $\mathcal{X}_{\Gamma_{\mu}}^{tor}$ has $Bl_P(\mathbb{P}^2)$ as its moduli space and the only modification is that we should affect the weight $1$ to the exceptional curve.
When we mod out be the action of $S_3$ fixing $P$ and permuting the other triple points we observe that there is no ramification on the exceptional curve. In particular $[\mathcal{X}_{PU(H_0,\mathcal{O}_3)}^{tor}]\simeq S_3 \backslash\mathcal{X}_{\Gamma_{\mu}}^{tor}$. In terms of refined coholomogy classes $ [PU(H_0,\mathcal{O}_3)]=[\Gamma_{\mu}]$.
The central projection to $P$ defines a map $\mathcal{X}_{\Gamma_{\mu}}^{tor}\to \mathbb{P}^1(3,3,3)$ whose general fiber $F$ is a $ \mathbb{P}^1(2,2,2)$ an elliptic orbifold whose fundamental group is the Vierergruppe. This implies that $PU(H_0,\mathcal{O}_3)^{tor}$ is virtually abelian of rank $2$ and that the morphism of Proposition \[surj\] has a finite kernel. Actually, $\pi_1(F)$ injects thanks to:
There is a (split) exact sequence $$1 \to (P\tilde\Gamma'_{Holz})^{tor} \to \Gamma_{\mu} ^{tor}\to K_4 \to 1$$ and the refined commensurability classes of $PU(H_0,\mathcal{O}_3)$ and $PU(H_1,\mathcal{O}_3)$ are the same.
The group $S_4$ acts on the linear system $|I_{4 pts}(2)|$ by the projectivities preserving the 4 points. On the 3 singular members it acts as $S_3$ where $S_3=S_4/K_4$ where $K_4$ is the Vierergruppe or Klein group isomorphic to $({\mathbb Z}/2)^2$. In particular $K_4$ acts as automorphisms of the map $\phi$ (see also [@Hol I.6.2]) and its orbifold compactification $\bar \phi$ in Proposition \[fib\] . It is then easy to interpret [@Hol I.3.6.3] and see that $[K_4 \backslash \mathcal{X}_{P\tilde\Gamma'_{Holz}}^{tor}]=[\mathcal{X}_{\Gamma_\mu}^{tor}]$.
### The universal covering stack attached to $[PU(H_0,\mathcal{O}_3)]$
$[\mathcal{X}_{PU(H_0,\mathcal{O}_3)}^{tor}]$ is not developable.
We consider the étale map $e:[\mu_3\backslash E]\buildrel{\cong}\over\longrightarrow \mathbb{P}^1(3,3,3)$. Then $$E \times_{e, \bar \phi} \mathcal{X}_{P\tilde\Gamma'_{Holz}}^{tor}\simeq \mathcal{X}_{P\Gamma'_{Holz}}^{tor}$$ where $P\Gamma'_{Holz}$ is the lattice in $PU(2,1)$ corresponding to the group $\Gamma'$ in the notations of [@Hol p. 27]. The moduli space of that stack is a surface birational to $E\times\mathbb{P}^1$ with 3 $A_2$ singular points which is actually isomorphic to $\mu_3\times \{1 \} \backslash Bl_{T.0_E\times 0_E} E\times E$. The orbifold structure of $\mathcal{X}_{P\Gamma'_{Holz}}^{tor}$ a $\mu_3$ inertia group at the singular points. So there are orbifold points in the fiber of the map $\mathcal{X}_{P\Gamma'_{Holz}}^{tor} \to E$ which is an isomorphism on $\pi_1$. In particular the universal covering stack is equivalent to ${\mathbb C}\times_E \mathcal{X}_{P\Gamma'_{Holz}}^{tor}$ and has infinitely many $\mu_3$ orbifold points.
### $SSC(\mathcal{X}(\bar X, D, (n,n,n,n)))$
The results in subsection \[subsec:general\] were slightly unsatisfying but one can settle $SSC$ in the case where the $d_i$ have a common factor:
If $n\in {\mathbb N}_{\ge 2}$, $SSC(\mathcal{X}(\bar X, D, (n,n,n,n))$ holds and the universal covering space is a Stein manifold.
It is enough to prove this for the 3-1 étale cover $\mathcal{X}'$ given by the blow up at 3 points of $E\times E$ with weights $n$ on the strict transform of the Holzapfel configuration of $6$ elliptic curves. Let us take the (étale) quotient stack by the action of $\mu_3\times \mu_3$ we have already encountered. The resulting orbifold $\mathcal{Y}$ can be described in the language of [@BHH] as follows: the moduli space is $\mathbb{P}^2$ blown up at 4 points, the strict transforms of the lines in CEVA(2) have weight 3, one exceptional curve has weight $n$, the 3 other exceptional curves have weight $3n$.
Let us now consider the linear projection from this blown up $\mathbb{P}^2$ to $\mathbb{P}^1$ with center the point whose exceptional curve has weight $n$. It is a regular map which has 3 special fibers which are isomorphic to a nodal conic, the irreducible constituents carrying weights 3 and $3n$. Hence there is a map $\mathcal{Y} \to \mathbb{P}^1(3,3,3)$. Composing with the natural étale map $\mathcal{X}'\to \mathcal{Y}$, we get a map $\mathcal{X}'\to \mathbb{P}^1(3,3,3)$ which is not constant on the 3 preimages of $\mathcal{Z}$. Since $\mathbb{P}^1(3,3,3)$ is elliptic and has virtually abelian rank 2 fundamental group the proposition follows.
Concluding remarks
==================
We conclude by a discussion of some interesting examples from the litterature.
In [@Stov; @DS], bielliptic smooth toroidal compactifications of ball quotients are constructed. They satisfy $L(\_)$ since the fundamental group is virtually abelian hence linear. The Shafarevich conjecture is established for surfaces of Kodaira dimension $\le 1$ by [@GUR] and their argument gives that the fundamental group is linear.
More to the point, [@DiCSto Theorem 1.3] asserts that a smooth toroidal compactification of a ball quotient which is birational to an abelian or a bielliptic surface is the blow up in finitely many distinct points of the minimal surface. In particular, it has a finite (abelian) cover such that the connected components of the Albanese fibres are smooth hence irreducible. Hence, one cannot construct a counterexample to $SC(\_)$ by ramifying along these connected components as in subsection \[sec2\]. It is not clear whether $SSC(\_)$ is satisfied.
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[Philippe Eyssidieux]{}\
[Université de Grenoble-Alpes. Institut Fourier. 100 rue des Maths, BP 74, 38402 Saint Martin d’Hères Cedex, France]{}\
[[email protected]]{} [http://www-fourier.ujf-grenoble.fr/$\sim$eyssi/]{}\
[^1]: This research was partially supported by the ANR project Hodgefun ANR-16-CE40-0011-01.
[^2]: Actually, torsion can and will be dealt with orbifold methods.
[^3]: Except in specific cases, one should not work relatively to the category of complex manifolds, since the Yoneda functor should distinguish a complex space and the normalisation of underlying reduced complex space.
[^4]: The covering theory of [@No2] has nothing to do with the complex structure.
[^5]: The literature also uses [*good*]{} orbifold for developable and [*very good*]{} orbifold for uniformizable.
[^6]: A torsion free lattice will be called neat if $k=1$ for every cusp.
[^7]: in the sense of [@Rom]. Actually $G$ acts on the frame bundle of $\mathcal{X}^{tor}_{\Gamma'}$, an ordinary complex manifold, and the action commutes with the right action of the general linear group, from this it is easy to find an étale chart with a strict action of $G$ on the corresponding étale groupoid.
[^8]: We use the following notation for the central series of a group $G$: $\gamma_1(G)=G$, $\gamma_2(G)=[G,G]$, $\gamma_{k+1}(G)=[\gamma_k(G),G]$
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abstract: 'We explore the correlation between an asteroid’s taxonomy and photometric phase curve using the $H,\!G_{12}$ photometric phase function, with the shape of the phase function described by the single parameter $G_{12}$. We explore the usability of $G_{12}$ in taxonomic classification for individual objects, asteroid families, and dynamical groups. We conclude that the mean values of $G_{12}$ for the considered taxonomic complexes are statistically different, and also discuss the overall shape of the $G_{12}$ distribution for each taxonomic complex. Based on the values of $G_{12}$ for about half a million asteroids, we compute the probabilities of C, S, and X complex membership for each asteroid. For an individual asteroid, these probabilities are rather evenly distributed over all of the complexes, thus preventing meaningful classification. We then present and discuss the $G_{12}$ distributions for asteroid families, and predict the taxonomic complex preponderance for asteroid families given the distribution of $G_{12}$ in each family. For certain asteroid families, the probabilistic prediction of taxonomic complex preponderance can clearly be made. In particular, the C complex preponderant families are the easiest to detect, the Dora and Themis families being prime examples of such families. We continue by presenting the $G_{12}$-based distribution of taxonomic complexes throughout the main asteroid belt in the proper element phase space. The Nysa-Polana family shows two distinct regions in the proper element space with different $G_{12}$ values dominating in each region. We conclude that the $G_{12}$-based probabilistic distribution of taxonomic complexes through the main belt agrees with the general view of C complex asteroid proportion increasing towards the outer belt. We conclude that the $G_{12}$ photometric parameter cannot be used in determining taxonomic complex for individual asteroids, but it can be utilized in the statistical treatment of asteroid families and different regions of the main asteroid belt.'
author:
- |
D. A. Oszkiewicz$^{1,2,3}$, E. Bowell$^2$, L. H. Wasserman$^2$,\
K. Muinonen$^{1,4}$, A. Penttilä$^1$, T. Pieniluoma$^1$,\
D. E. Trilling$^3$, C. A. Thomas$^3$\
\
\
\
\
bibliography:
- 'biblio.bib'
title: Asteroid taxonomic signatures from photometric phase curves
---
=1
Introduction
============
The photometric phase function describes the relationship between the reduced magnitude (apparent magnitude at $1$ AU distance) and the solar phase angle (Sun-asteroid-observer angle). Previously in [@DO], we have fitted $H,\!G_1,\!G_2$ and $H,\!G_{12}$ phase functions presented in [@KM2010] for about half a million asteroids contained in the Lowell Observatory database and obtained absolute magnitudes and photometric parameter(s) for each asteroid. The absolute magnitude $H$ for an asteroid is defined as the apparent $V$ band magnitude that the object would have if it were $1$ AU from both the Sun and the observer and at zero phase angle. The absolute magnitude relates directly to asteroid size and geometric albedo. The geometric albedo of an object is the ratio of its actual brightness at zero phase angle to that of an idealized Lambertian disk having the same cross-section.
The shape of the phase curve described by the $G_1,\!G_2$ and $G_{12}$ parameters relates to the physical properties of an asteroid’s surface, such as geometric albedo, composition, porosity, roughness, and grain size distribution. For phase angles larger than 10$^\circ$, steep phase curves are characteristic of low-albedo objects with an exposed regolith, whereas flat phase curves can indicate, for example, a high-albedo object with a substantial amount of multiple scattering in its regolith.
At small phase angles, atmosphereless bodies (such as asteroids) exhibit a pronounced nonlinear surge in apparent brightness known as the opposition effect [@KM2010a]. The opposition effect was first recognized for asteroid (20) Massalia [@Gehrels]. The explanation of the opposition effect is two-fold: (1) self-shadowing arising in a rough and porous regolith, and (2) coherent backscattering; that is, constructive interference between two electromagnetic wave components propagating in opposite directions in the random medium [@KM2010a]. The width and height of the opposition surge can suggest, for example, the compaction state of the regolith and the distribution of particle sizes.
[@BelskayaShevchenko] have analyzed the opposition behavior of 33 asteroids having well-measured photometric phase curves and concluded that the surface albedo is the main factor influencing the amplitude and width of the opposition effect. Phase curves of high-albedo asteroids have been also described by [@Harrisalbedo] and [@ScaltritiZappala]. [@Harrisalbedo] have concluded that the opposition spikes of (44) Nysa and (64) Angelina can be explained as an ordinary property of moderate-to-high albedo atmosphereless surfaces. [@KaasalainenS] have presented a method for interpreting asteroid phase curves, based on empirical modeling and laboratory measurements, and emphasized that more effort could be put into laboratory studies to find a stronger connection between phase curves and surface characteristics. Laboratory measurements of meteorite phase curves have been performed, for example, by [@Capaccioni] and measurements of regolith samples by [@KaasalainenSanna].
The relationship between the phase-curve shape and taxonomy has also been explored. [@LagerkvistMagnusson] have computed absolute magnitudes and parameters for 69 asteroids using the $H,\!G$ magnitude system and computed mean values of the $G$ parameter for taxonomic classes S, M, and C. They have emphasized that the $G$ parameter varies with taxonomic class. [@GoidetDevel] have considered phase curves of about 35 individual asteroids and analogies between phase curves of asteroids belonging to different taxonomic classes. [@HarrisTaxa] have examined the mean values of slope parameters for different taxonomic classes.
In our previous study [@DO], we have fitted phase curves of about half a million asteroids using recalibrated data from the Minor Planet Center[^1]. We have found a relationship between the family-derived photometric parameters $G_1$ and $G_2$, and the median family albedo. We have showed that, in general, asteroids in families tend to have similar photometric parameters, which could in turn mean similar surface properties. We have also noticed a correlation between the photometric parameters and the Sloan Digital Sky Survey color indices (SDSS). The SDSS color indices correlate with the taxonomy, as do the photometric parameters.
In the present article, we explore the correlation of the photometric parameter $G_{12}$ with different taxonomic complexes. For about half a million individual asteroids, we compute the probabilities of C, S, and X complex membership given the distributions of their $G_{12}$ values. Based on the $G_{12}$ distributions for members of asteroid families, we investigate taxonomic preponderance in asteroid families. In Sec. \[methods\], we describe methods to compute $G_{12}$ for individual asteroids and the probability for an asteroid to belong to a taxonomic complex given $G_{12}$, and methods to determine the taxonomic preponderance in asteroid families. In Sec. \[res\], we describe our results and discuss the usability of $G_{12}$ in taxonomic classification. In Sec. \[concl\], we present our conclusions.
Taxonomy from photometric phase curves {#methods}
======================================
Fitting phase curves
--------------------
In the previous study [@DO], we made use of three photometric phase functions: the $H,\!G$; the $H,\!G_1,\!G_2$; and the $H,\!G_{12}$ phase functions. The $H,\!G$ phase function [@EB] was adopted by the International Astronomical Union in 1985. It is based on trigonometric functions and fits the vast majority of the asteroid phase curves in a satisfactory way. However, it fails to describe, for example, the opposition brightening for E class asteroids and the linear magnitude-phase relationship for F class asteroids. The $H,\!G_{1},\!G_{2}$ and the $H,\!G_{12}$ phase functions [@KM2010] are based on cubic splines and accurately fit phase curves of all asteroids. The $H,\!G_{1},\!G_{2}$ phase function is designed to fit asteroid phase curves containing large numbers of accurate observations, whereas the $H,\!G_{12}$ phase function is applicable to asteroids that have sparse or low-accuracy photometric data. Therefore, the $H,\!G_{12}$ phase function is best suited to our data [@DO]. The present study is mostly based on results obtained from the $H,\!G_{12}$ phase function. Both $H,\!G_1,\!G_2$ and $H,\!G_{12}$ phase functions are briefly described below.
### $H,\!G_1,\!G_2$ phase function
In the $H,\!G_1,\!G_2$ phase function [@KM2010], the reduced magnitudes $V(\alpha)$ can be obtained from $$\begin{split}
10^{-0.4 V(\alpha)} & = a_1 \Phi_1(\alpha) + a_2 \Phi_2(\alpha) + a_3 \Phi_3(\alpha) \\
& = 10^{-0.4H}\left[G_1 \Phi_1(\alpha) + G_2 \Phi_2(\alpha) +
(1-G_1-G_2) \Phi_3(\alpha)\right],
\end{split}$$ where $\alpha$ is the phase angle and $V(\alpha)$ is the reduced magnitude. The coefficients $a_1$, $a_2$, $a_3$ are estimated from the observations using the linear least-squares method. The basis functions $\Phi_1(\alpha)$, $\Phi_2(\alpha)$, and $\Phi_3(\alpha)$ are given in terms of cubic splines. The absolute magnitude $H$ and the photometric parameters $G_1$ and $G_2$ can then be obtained from the $a_1$, $a_2$, $a_3$ coefficients.
### $H,\!G_{12}$ phase function
In the $H,\!G_{12}$ phase function [@KM2010], the parameters $G_1$ and $G_2$ of the three-parameter phase function are replaced by a single parameter $G_{12}$ analogous to the parameter $G$ in the $H,\!G$ magnitude system (although there is no exact correspondence). The reduced flux densities can be obtained from $$10^{-0.4V(\alpha)} = L_0\left[G_1 \Phi_1(\alpha) + G_2\Phi_2(\alpha)+
(1-G_1-G_2)\Phi_3(\alpha) \right],$$ where $$\begin{split}
G_1 & = \begin{cases}
0.7527 G_{12} + 0.06164, & \mathrm{if} \ G_{12} < 0.2 \\
0.9529 G_{12} + 0.02162, & \mathrm{otherwise}
\end{cases} \\
G_2 & = \begin{cases}
-0.9612 G_{12} + 0.6270, & \mathrm{if} \ G_{12} < 0.2 \\
-0.6125 G_{12} + 0.5572, & \mathrm{otherwise}
\end{cases}\end{split}
\label{G1G2}$$ and $L_0$ is the disk-integrated brightness at zero phase angle. The basis functions are as in the $H,\!G_1,\!G_2$ phase function. The parameters $L_0$ and $G_{12}$ are estimated from the observations using the nonlinear least-squares method.
The $H,\!G_1,\!G_2$ and the $H,\!G$ phase functions are fitted to the observations in the flux-density domain using the linear least-squares method. In order to fit the $H,\!G_{12}$ phase function, downhill simplex non-linear regression [@NM] is utilized. in order to compute uncertainties in the photometric parameters, we use Monte Carlo and Markov-chain Monte Carlo methods. A detailed description of these procedures can be found in [@DO]. In the current study, we used the Asteroid Phase Curve Analyzer[^2].
Taxonomic preponderance for asteroid families {#method}
---------------------------------------------
As we discuss further in Sec. \[res\], we find correlation between $G_{12}$ and taxonomy. In Fig. \[taxaHisto\], we show $G_{12}$ histograms for different taxonomic complexes. Only the main taxonomic complexes—that is, the C \[containing classes B, C, Cb, Cg, Ch, Cgh\], S \[S, Sa, Sk, Sl, Sr, K, L, Ld\], and X \[X, Xc, Xk\] complexes—have large enough sample size for statistical treatment. The small number of objects belonging to A, D \[D, T\], E \[E, Xe\], O, Q \[Q, Sq\], R, and V complexes prevent further conclusions using $G_{12}$ statistics in those groups. We approximate the $G_{12}$ distributions for taxonomic complexes by a Gaussian distribution, and the means and standard deviations of those distributions are listed in Table \[G12taxa\]. The $G_{12}$ distributions for C and S complexes are smoother than the one for the X complex. The X complex comprises three different albedo groups, namely E, M, and P class objects. Those cannot be separated within the X complex only based on spectra, and additional albedo information is usually required. The X complex degeneracy was discussed for example by [@CT2011]. The unusual shape of the X complex can be related to the different albedo groups as $G_{12}$ correlates well with albedo [@KM2010; @DO]. Unfortunately, due to the small number of E, M, and P class objects in our sample, we cannot determine how useful $G_{12}$ could be in breaking the X complex into E, M, P class groups.
Complex Nr of objects mean std
----------- --------------- -------------- --------------
A 16 0.39 0.19
[**C**]{} [**391**]{} [**0.64**]{} [**0.16**]{}
D 23 0.47 0.14
E 26 0.39 0.16
O 3 0.57 0.05
Q 72 0.41 0.14
R 4 0.24 0.18
[**S**]{} [**584**]{} [**0.41**]{} [**0.16**]{}
V 35 0.41 0.14
[**X**]{} [**212**]{} [**0.48**]{} [**0.19**]{}
: Means and standard deviations of $G_{12}$ for asteroid taxonomic complexes.[]{data-label="G12taxa"}
Based on the approximated $G_{12}$ distributions for the different taxonomic complexes (Table \[G12taxa\]), we can compute the probability for an asteroid to belong to a given taxonomic complex as the a posteriori probability using Bayes’s rule. For example, the probability for an asteroid to belong to the C complex can be computed using $$p_C(x \in C \mid G_{12}) = A \, pr(x \in C) \, p(G_{12} \mid x \in C),
\label{prob}$$ where $A$ is a normalization constant, $p_C(x \in C \mid G_{12})$ is the a posteriori probability for an asteroid $x$ to belong to the C complex, given a particular $G_{12}$ value; $pr(x \in C)$ is the a priori probability for an asteroid $x$ to belong to the C complex; and $p(G_{12} \mid x \in C)$ is the probability for an asteroid $x$ to have a specific $G_{12}$ value, given that it belongs to the C complex.
As estimates for the probabilities $p(G_{12} \mid x \in C)$, we adopt the Gaussian approximations for the empirical $G_{12}$ distributions of different taxonomic complexes. We make use of three different a priori distributions: (1) a uniform a priori distribution; (2) an a priori distribution based on the frequency of C, S, and X complex objects among asteroids with taxonomy defined in the Planetary Data System database [PDS, see @pds]; (3) an a priori distribution based on the frequencies of C, S, and, X complexes in different parts of the main asteroid belt (inner, mid, and outer main belt) from the PDS database. Testing the results obtained with different a priori distributions is important for insuring that the results are driven by data and not by the a priori distribution.
The probabilities computed based on the different a priori distributions should agree for different a priori assumptions if the photometric parameter brings substantial information overriding the information contained in the different a priori distributions. By using choice (1), we assume no previous knowledge of asteroid taxonomy. By using choice (2), we assume that the a priori probability for an asteroid $x$ to belong to a specific complex is equal to the frequency of occurrence of asteroids of that complex in the sample of known asteroid taxonomies in the PDS database. This means that, for the C, S, and X complexes, we use the a priori probabilities equal to $0.33$, $0.49$, and $0.18$, respectively. To derive the a priori distribution for choice (3), we first divide the main asteroid belt into three regions: the inner (region I), mid (region II), and outer main belt (region III). The boundaries between the regions are based on the most prominent Kirkwood gaps. Region I lies between the 4:1 resonance ($2.06$ AU) and 3:1 resonance ($2.5$ AU). Region II continues from the end of region I out to the 5:2 resonance ($2.82$ AU). Region III extends from the outer edge of region II to the 2:1 resonance ($3.28$ AU). The frequencies derived for those regions are as follows: for the C complex, 0.19 (I), 0.38 (II), 0.45 (III); for the S complex, 0.70 (I), 0.42 (II), 0.33 (III); and, for the X complex, 0.11 (I), 0.20 (II), 0.22 (III). The regional frequencies are also computed based on the data available in the PDS database. In general, a better choice of the a priori distributions would be based on debiased ratios of taxonomic complexes, but those are not available. In general, a single asteroid can have non-zero probability for belonging to two or more complexes.
The probability for an asteroid family being dominated, for example, by the C complex can be computed as $$% Not good, in the latter the sum somehow selects only the i's that belong to C
P_{C} = \frac{\sum_{i=1}^{N_{mem}} p^{(i)}_C}{N_C+ N_S + N_X}, % = \frac{\sum_{i=1}^{N_C} p^{(i)}_C}{N_{mem}},$$ where $N_{C}$, $N_{S}$, and $N_{X}$ are the numbers of asteroids classified as belonging to the C, S, and X complexes, $N_{mem}$ is the number of members in a family, and $p^{(i)}_C$ is the probability of member $i$ belonging to the C complex. The probabilities for an asteroid family being dominated by other complexes can be computed in a similar fashion. In practice, $P_{C}$ represents the probability that a random asteroid from a given family would be of C complex.
Validation
----------
In order to validate the method described in Sec. \[method\], we have checked the number of correct taxonomic complex classifications of asteroids with known taxa via so-called N-folded tests. First, we derived the frequencies of different taxonomic complexes, skipping 50 random asteroids in each complex which we later use for testing. The general frequencies for the C, S, and X complexes were $0.33$, $0.52$, and $0.15$. In the inner, mid, and outer main belt, the numbers are, respectively, as follows: 0.20, 0.72, and 0.09; 0.37, 0.44, and 0.18; 0.46, 0.37 and 0.17. Those frequencies are then used as priors in Eq. \[prob\].
The success ratio is measured as $$R_s = \frac{N_{corr}}{N_{total}},$$ where $N_{corr}$ is the number of correct identifications among $N_{total}$ asteroids.
Using the uniform a priori distribution (1) results in a $64\%$ overall success ratio ($80\%$ for the C complex, $60\%$ for the S complex, and $52\%$ for the X complex). Using the overall frequencies (2) results in a $63\%$ overall success ratio ($98\%$ for the C complex, $68\%$ for the S complex and $22\%$ for the X complex). The last choice (3) leads to a $63 \%$ overall success ratio ($96\%$ for the C complex, $72\%$ for the S complex and $22\%$ for the X complex). We compared these success ratios with those arising from random guessing. We conclude that there is general improvement in success ratios for all the taxonomic complexes.
Results and discussion {#res}
======================
We explore the correlation of the photometric parameter $G_{12}$ with the taxonomic classification based on about half a million asteroid phase curves in the Lowell Observatory database [@DO]. In Fig. \[proper\], we present the distribution of the orbital proper elements color-coded with the $G_{12}$ values, with with a larger number of asteroids included as compared to the results in [@DO]. The updated figure strengthens our previous findings of $G_{12}$ homogeneity within asteroid families. Even though the distributions of the $G_{12}$ values in asteroid families can be broad, asteroids in families stand out and tend to have similar values of $G_{12}$ [@DO]. Asteroids having disparate $G_{12}$ values but still identified as family members can be so-called interlopers, asteroids originating from a differentiated parent body, and asteroids with differently evolved surfaces. This result is consistent with previous findings on the homogeneity of asteroid families. For example, it was previously found that asteroids within families can share similar spectral properties [@CellinoA3] and colors [@colors]. The tendency toward family homogeneity might be helpful in deriving the family membership. Note that this tendency does not support the claim that asteroid families originate from differentiated parent bodies, since objects resulting from the disruption of a differentiated parent body would show differing photometric phase curves. Therefore, the distribution of the $G_{12}$ values could contribute to the understanding the origin and evolution of asteroid families. $G_{12}$ could also be used, along with the proper elements, for asteroid family classification. The trend from smaller average $G_{12}$ values for the inner belt to larger $G_{12}$ values for the outer belt is consistent with the distribution of C and S class asteroids in the asteroid belt. We also note that the $G_{12}$ values of family members in Fig. \[proper\] correlate well with the SDSS color-color plot [@colors]. The correlation relates to the fact that both the SDSS colors and $G_{12}$ correlate with asteroid taxonomy.
![Distribution of asteroid proper elements, color-coded according to the $G_{12}$ value.[]{data-label="proper"}](ProperG12){width="\textwidth"}
In Fig. \[SDSS\], we plot the distribution of asteroids in SDSS color-color space, coded according to the $G_{12}$ value. The $x$-axis is defined as $a^* = 0.89 (g - r) + 0.45 (r - i) - 0.57$ and $y$-axis as $i\!-\!z$, where $g$, $r$, $i$, and $z$ are magnitudes in the SDSS filters. The two clouds correspond to the C and S class asteroids, and the V class asteroids are located in the lower right corner of the plot (with large $a^*$ and small $i\!-\!z$ values). C class asteroids tend to have, on average, larger values of $G_{12}$, S class smaller, and V class often very small $G_{12}$ values.
![Distribution of asteroids in $(a^*, i\!-\!z)$ SDSS color space, color-coded according to the $G_{12}$ value.[]{data-label="SDSS"}](3dSDSS_aStarIZ){width="\textwidth"}
To investigate the correlation of $G_{12}$ with taxonomy, we further extracted taxonomic classifications from PDS. The data set contains entries for 2615 objects. Each of the eight taxonomies represented produced classifications for a subset of the objects: [@Tholena; @Tholenb] – 978 objects; [@Barucci] – 438 objects; [@Tedescoa], [@Tedescob] – 357 objects; [@Howell] – 112 objects; [@Xu] – 221 objects; [@Bus] – 1447 objects; [@Lazzaro] – 820 objects; and [@DeMeo] – 371 objects. We make use of the @Bus classification, which contains the largest number of asteroids. We divide our sample into thirteen complexes: A, C \[B, C, Cb, Cg, Ch, Cgh\], D \[D, T\], E \[E, Xe\], M, P, O, Q \[Q, Sq\], R, S \[S, Sa, Sk, Sl, Sr, K, L, Ld\], V, X \[X, Xc, Xk\], and U. We produce histograms of the $G_{12}$ values for each of them (see Fig. \[taxaHisto\]). Each taxonomic complex is then approximated by a Gaussian distribution. The means and standard deviations of the $G_{12}$ values for all the complexes are listed in Table \[G12taxa\]. Most of the complexes contain too few objects for meaningful statistical treatment, except for the S, C, and X complexes. The means of the distributions for the S, C, and X complexes are clearly different.
The S complex has a mean $G_{12}$ of $0.41$, the C complex has a higher mean $G_{12}$ of $0.64$, and the X complex is intermediate having a mean of $0.48$. In general, asteroids within the same taxonomic complex could have varying surface properties (for example, different regolith porosities or grain-size distributions) leading to different $G_{12}$ values, resulting in broad $G_{12}$ histograms for a complex. An additional challenge follows from the fact that the $G_{12}$ distributions for the different taxonomic complexes partially overlap. Based on those distributions, selected priors (see Sec. \[method\]) and previously obtained photometric parameters [@DO] for each of the half a million asteroids, we computed the C, S, and X complex classification probabilities for each asteroid. Due to broad and overlapping $G_{12}$ distributions, these probabilities are often be similar enough to prevent a meaningful classification of the asteroid into any one of the complexes.
For some asteroids, $G_{12}$ can, however, be a good indicator of taxonomic complex. For example, an asteroid with $G_{12} = 0.8$ from the outer belt has a probability of 82% for being of C complex and low probabilities of being of S or X complex (5% and 13%). If we assume no knowledge on asteroid location nor on the frequency of different taxonomic complexes (uniform prior (1)), an asteroid with $G_{12} = 0.8$ would still have a chance of 70% for being of C complex. For reference, we list the probabilities for an asteroid with $G_{12} = 0.8$ being of C, S, and X complex in Table \[probEx\], assuming different priors and different locations in the belt (or no knowledge on location in the belt).
Prior $P_c$ $P_s$ $P_x$
--------------- ------- ------- -------
(1) 70% 6% 24%
(2) 76% 10% 14%
\(3) Inner MB 66% 21% 13%
\(3) Mid MB 79% 7% 14%
\(3) Outer MB 82% 5% 13%
: Example result for a single asteroid. Probabilities for an asteroid with $G_{12}=0.8$ to be of C, S, and X complex.[]{data-label="probEx"}
Asteroid families containing, for example, a large number of asteroids with high $G_{12}$ values resulting in high $P_c$ values could be identified as C-complex preponderant. As a prime example, we indicate the Dora family having the mean $G_{12}=0.7$ and standard deviation $\sigma_{G_{12}}=0.18$, which result in a high C-complex preponderance probability. The $G_{12}$ distribution (Fig. \[family\]) for the Dora family also matches nicely the C-complex distribution profile.
-------------- ------- ---------- ---------- --------- --------- --------- --------- --------- --------- --------- --------- ---------
Family or Nr of $G_{12}$ $G_{12}$ $P_{C}$ $P_{S}$ $P_{X}$ $P_{C}$ $P_{S}$ $P_{X}$ $P_{C}$ $P_{S}$ $P_{X}$
cluster mem. mean std.
Adeona 987 0.64 0.2 0.49 0.21 0.29 0.55 0.27 0.18 0.55 0.27 0.18
Aeolia 55 0.66 0.23 0.51 0.2 0.29 0.54 0.29 0.16 0.57 0.25 0.17
Agnia 472 0.58 0.21 0.42 0.27 0.31 0.47 0.34 0.19 0.47 0.34 0.19
Astrid 94 0.64 0.23 0.49 0.22 0.29 0.43 0.45 0.12 0.55 0.27 0.17
Baptistina 1966 0.53 0.18 0.36 0.31 0.33 0.46 0.32 0.22 0.26 0.62 0.12
Beagle 38 0.64 0.23 0.5 0.21 0.29 0.6 0.21 0.18 0.6 0.21 0.18
Brangäne 37 0.68 0.19 0.53 0.19 0.28 0.64 0.19 0.18 0.6 0.24 0.17
Brasilia 186 0.49 0.21 0.31 0.35 0.34 0.4 0.37 0.23 0.4 0.37 0.23
Charis 136 0.61 0.21 0.45 0.25 0.31 0.55 0.25 0.2 0.55 0.25 0.2
Chloris 200 0.63 0.17 0.48 0.22 0.3 0.59 0.22 0.19 0.55 0.28 0.18
Clarissa 41 0.64 0.22 0.49 0.22 0.29 0.42 0.46 0.12 0.42 0.46 0.12
Datura 4 0.41 0.2 0.23 0.42 0.35 0.3 0.45 0.25 0.17 0.72 0.11
Dora 528 0.7 0.18 0.56 0.16 0.28 0.67 0.16 0.17 0.63 0.2 0.16
Emma 111 0.66 0.23 0.52 0.19 0.29 0.59 0.24 0.17 0.63 0.19 0.18
Emilkowalski 2 0.56 0.07 0.39 0.28 0.33 0.45 0.35 0.2 0.45 0.35 0.2
Eos 3413 0.64 0.19 0.5 0.21 0.29 0.56 0.27 0.18 0.6 0.21 0.19
Erigone 806 0.63 0.19 0.48 0.22 0.3 0.54 0.28 0.18 0.4 0.48 0.12
Eunomia 4707 0.51 0.18 0.33 0.34 0.33 0.37 0.42 0.2 0.37 0.42 0.2
Flora 6316 0.53 0.18 0.35 0.32 0.33 0.26 0.63 0.11 0.26 0.63 0.11
Gefion 1938 0.56 0.19 0.39 0.29 0.32 0.49 0.3 0.21 0.44 0.36 0.19
Hestia 103 0.55 0.2 0.39 0.29 0.32 0.44 0.37 0.19 0.44 0.37 0.19
Hoffmeister 341 0.68 0.22 0.53 0.19 0.28 0.63 0.19 0.18 0.59 0.24 0.17
Hygiea 1729 0.66 0.21 0.51 0.2 0.29 0.61 0.2 0.18 0.61 0.2 0.18
Iannini 30 0.45 0.23 0.26 0.4 0.34 0.29 0.5 0.21 0.29 0.5 0.21
Juno 359 0.52 0.21 0.36 0.32 0.33 0.27 0.61 0.11 0.4 0.4 0.2
Karin 159 0.54 0.22 0.38 0.3 0.32 0.29 0.59 0.12 0.47 0.31 0.22
Kazvia 12 0.6 0.21 0.45 0.24 0.31 0.51 0.31 0.18 0.51 0.31 0.18
Konig 58 0.68 0.17 0.53 0.18 0.28 0.6 0.23 0.17 0.6 0.23 0.17
Koronis 2913 0.58 0.2 0.42 0.27 0.31 0.52 0.28 0.21 0.52 0.28 0.21
Lau 6 0.54 0.17 0.39 0.28 0.33 0.5 0.28 0.21 0.5 0.28 0.21
Lixiaohua 171 0.64 0.23 0.5 0.21 0.29 0.43 0.45 0.12 0.59 0.22 0.19
Lucienne 37 0.5 0.21 0.33 0.34 0.33 0.25 0.64 0.11 0.25 0.64 0.11
Maria 2009 0.5 0.2 0.33 0.34 0.33 0.37 0.42 0.2 0.37 0.42 0.2
Massalia 1911 0.56 0.2 0.39 0.29 0.32 0.44 0.37 0.19 0.3 0.58 0.12
Meliboea 40 0.68 0.14 0.55 0.17 0.28 0.62 0.21 0.17 0.67 0.16 0.17
Merxia 425 0.53 0.22 0.36 0.31 0.33 0.46 0.33 0.22 0.41 0.39 0.2
Misa 220 0.68 0.21 0.54 0.18 0.28 0.6 0.23 0.17 0.6 0.23 0.17
Naema 98 0.66 0.2 0.51 0.2 0.29 0.58 0.25 0.17 0.62 0.2 0.18
Nemesis 258 0.68 0.19 0.54 0.18 0.28 0.64 0.18 0.18 0.6 0.23 0.17
Nysa-Polana 8289 0.57 0.19 0.4 0.28 0.32 0.46 0.35 0.19 0.31 0.57 0.12
Padua 372 0.66 0.2 0.52 0.19 0.29 0.59 0.24 0.17 0.59 0.24 0.17
Rafita 477 0.55 0.2 0.39 0.29 0.32 0.48 0.3 0.21 0.44 0.37 0.19
Sulamitis 92 0.66 0.21 0.52 0.2 0.29 0.62 0.2 0.18 0.45 0.43 0.12
Sylvia 30 0.56 0.26 0.43 0.25 0.31 0.53 0.27 0.21 0.46 0.37 0.17
Telramund 240 0.58 0.23 0.42 0.27 0.31 0.47 0.34 0.19 0.51 0.28 0.21
Terentia 7 0.41 0.2 0.24 0.4 0.36 0.15 0.74 0.11 0.32 0.42 0.25
Themis 2559 0.69 0.19 0.55 0.17 0.28 0.62 0.22 0.17 0.66 0.17 0.17
Theobalda 60 0.61 0.22 0.47 0.23 0.3 0.39 0.49 0.12 0.57 0.24 0.19
Tirela 818 0.61 0.2 0.46 0.24 0.3 0.56 0.24 0.2 0.56 0.24 0.2
Veritas 291 0.68 0.21 0.54 0.18 0.28 0.61 0.22 0.17 0.65 0.18 0.17
Vesta 8445 0.5 0.18 0.32 0.35 0.34 0.36 0.44 0.2 0.22 0.66 0.11
18405 24 0.63 0.16 0.48 0.22 0.3 0.51 0.32 0.17 0.6 0.21 0.19
18466 94 0.49 0.22 0.31 0.35 0.34 0.24 0.65 0.11 0.36 0.44 0.2
-------------- ------- ---------- ---------- --------- --------- --------- --------- --------- --------- --------- --------- ---------
: The number of family members included in the study, the mean and standard deviation of $G_{12}$, and the probabilities of C, S, and X complex preponderance based on the a priori distributions (1), (2), and (3) (see above). Family classifications are from [@Nesvorny].[]{data-label="mean"}
In order to check how well we can identify asteroid families as being dominated by one of the taxonomic complexes, we produce $G_{12}$ histograms for the different asteroid families (histograms for chosen families are presented in Fig. \[family\], histograms for the remaining families can be found in supplementary materials; numerical values are in Table \[mean\]) and use methods based on Bayesian statistics (described in Sec. \[methods\]) to establish the dominant taxonomic complex. We then compare our results with the published results from other studies.
-- --
-- --
: The normalized $G_{12}$ distributions for asteroid families as listed in Table \[mean\]. The family classification is from PDS [@Nesvorny]. The dashed line indicates the distribution weighted with one over the sum of the absolute two-sided errors, and the solid line is the unweighted histogram. For comparison, we plot the a posteriori functions for the different taxonomic complexes (the C complex is indicated by the thick dashed line, the S complex by the thick dotted line, and the X complex by the thick dash-dotted line) based on prior (2). []{data-label="family"}
Deciding on the taxonomic complex preponderance based on $G_{12}$ can prove difficult and one should be careful in drawing conclusions when the resulting probabilities for the different complexes are similar. To pick asteroid families that show a preference in taxonomic complex, we set the following requirements:
1. The minimum number of asteroids in the sample must be around 100 or more.
2. The probability for preponderance must be the highest for all the assumed a priori probabilities. This is to make sure that the inference is driven by data and not by the a priori distribution.
3. The probability for the preponderant complex must be close to 50% or more for all the assumed a priori distributions.
Table \[mean\] lists the probabilities for taxonomic complex preponderance, along with the family means and standard deviations of the $G_{12}$ values. Several families have too few members to draw any conclusions. Some of the families result in very similar taxonomic complex preponderance probabilies for each complex and therefore no complex can be indicated as dominating. For some families the computed probabilities suggest different preponderant complex based on different a priori probabilies. For those families no conclusions could be made. Several families, however show clear preference of taxonomic complex. Those include:
- [**(145) Adeona**]{} (region II): The $G_{12}$ distribution for the Adeona family contains 987 asteroids and is visibly shifted towards high $G_{12}$ values. Based on the computed probabilities, the Adeona family seems to be dominated by C complex objects: the C complex probabilities for the family are 49%, 55%, and 55% based on the a priori distributions (1), (2), and (3), respectively, which agrees with the literature. The computed C complex preponderance probabilities are 20% to 37% higher than those for the other complexes. Visual inspection of the histogram also suggests that the majority of asteroids in this family must have come from the C-complex distribution (see Fig. \[family\]). The $G_{12}$ distribution for this family is smooth. In the literature, the Adeona family has 12 members with known spectroscopic classification: 9 in class Ch, 1 in C, 1 in X, 1 in D, and 1 in class Xk [@Mothe].
- [**(627) Charis**]{} (region III): The Charis cluster seems to be strongly C complex preponderant. C complex preponderance probabilities are 45%, 55% and 55% for the a priori distributions (1), (2), and (3). The $G_{12}$ distribution is smooth and clearly shifted towards large $G_{12}$ values. The profile of the $G_{12}$ distribution also matches the $G_{12}$ distribution profile of the C complex.
- [**(410) Chloris**]{} (region II): The profile of the $G_{12}$ distribution for the Chloris cluster is similar to that of the Charis family, also matching the profile of the $G_{12}$ distribution for the C complex. The computed probabilities also indicate C complex preponderance: they are 48%, 59%, and 55% for the a priori distributions (1), (2), and (3), and are about 20% larger than for any other complex. This cluster has also been spectroscopically characterized as C complex dominant by [@BusThesis].
- [**(668) Dora**]{} (region II): The Dora family is strongly C complex dominated. The probabilities of C complex preponderance are 56%, 67%, and 63% for the priori distributions (1), (2), and (3), and are 28–51% higher than those for the S and X complexes. Also, the $G_{12}$ distribution is smooth and matches better the C complex distribution than the S or X complex distributions. In the literature, Dora has 29 members with known spectra, all belonging to the C complex (24 in class Ch, 4 in C, and 1 in class B) [@Mothe; @BusThesis].
- [**(283) Emma**]{} (region III): The $G_{12}$ distribution for the Emma family is smooth and dominated by asteroids with large $G_{12}$ values. The probability of Emma being C complex preponderant is 52%, 59%, and 63% for the a priori distributions (1), (2), and (3). These probabilities are about 30% larger than those for the S and X complexes. Therefore, Emma can be classified as C complex preponderant.
- [**(1726) Hoffmeister**]{} (region II): The Hoffmeister family is C complex dominant. The C complex preponderance probability is 53%, 63%, and 59% for the a priori distributions (1), (2), and (3), and is about 25% larger than those of the S or X complex preponderance. There are 10 members of this family with known spectra: 8 in class C (4 in class C, 3 in Cb, and 1 in class B), 1 in Xc, and 1 in class Sa [@Mothe]. The $G_{12}$ distribution for the Hoffmeister family is peculiar and steadily increasing towards larger $G_{12}$ values.
- [**(10) Hygiea**]{} (region III): Similarly to the Hoffmeister family, the Hygiea family is C complex preponderant. The C complex preponderance probabilities are 51%, 61%, and 61% for the a priori distributions (1), (2), and (3). The probabilities for the other complexes are about 20–40% smaller. Most of the asteroids in the family are of class B (C complex) [@Mothe].
- [**(569) Misa**]{} (region II): The Misa family is C complex preponderant. The C complex preponderance probabilities are 54%, 60%, and 60% for the a priori distributions (1), (2), and (3), and are about 35% higher than those of being S or X complex preponderant.
- [**(845) Naema**]{} (region III): The Naema family is C complex preponderant. The C complex preponderance probabilities are 51%, 58%, and 62% for the a priori distributions (1), (2), and (3), and are about 20–40% higher than those of being S or X complex preponderant.
- [**(128) Nemesis**]{} (region II): The Nemesis family is C complex preponderant. The C complex preponderance probabilities are 54%, 64%, and 60% for the a priori distributions (1), (2), and (3), and are about 25–35% higher than those of being S or X complex preponderant.
- [**(363) Padua**]{} (region II): The $G_{12}$ distribution for the Padua family is shifted towards large values of $G_{12}$ and indicates C complex preponderance. The C complex preponderance probabilities are 52%, 59%, and 59% for the a priori distributions (1), (2), and (3). The Padua family has 9 members with spectral classification. Most of them are X class asteroids (6 in class X and 1 in Xc), and there are also 2 C class members [@BusThesis; @Mothe].
- [**(24) Themis**]{} (region III): The Themis family is C complex preponderant which agrees with the literature analyses. The C complex preponderance probabilities are 55%, 62%, and 66% for the a priori distributions (1), (2), and (3), and are about 30–50% larger than those of being S or X complex preponderant. In the literature, 43 Themis family asteroids have spectra available. The taxonomy of these asteroids is homogeneous: there are 36 asteroids from the C complex (6 in class C, 17 in B, 5 in Ch, and 8 in class Cb) and 7 asteroids from the X complex (5 in class X, 1 in Xc, and 1 in Xk) [@Mothe; @Florczak1999].
- [**(490) Veritas**]{} (region III): The Veritas family is C complex preponderant which agrees with the literature analyses. The C complex preponderance probabilities are 54%, 61%, and 65% for the a priori distributions (1), (2), and (3), and are about 25–50% higher than those of being S or X complex preponderant. In the literature, the Veritas family has 8 members with known spectra, all of them belonging to the C complex: 6 in class Ch, 1 in C, and 1 in class Cg [@Mothe].
For a number of families it was not possible to indicate preponderant taxonomic complex. Out of those a particular case is the Nysa-Polana family, which shows a clear differentiation into two separate regions.
[**(44) Nysa - (142) Polana**]{} (region I): The taxonomic preponderance probabilities for the Nysa-Polana family are similar for all the complexes, therefore no single complex can be indicated as preponderant. Additionally, the different a priori distributions result in differing preponderances. It has been previously suggested [@CellinoNP] on the basis of spectral analysis that the Nysa-Polana family is actually composed of two distinct families, which is incompatible with the hypothesis of common origin. The first one (Polana) was suggested to be composed of dark objects and the second one (Mildred) of brighter S class asteroids. [@Parker2008] has performed a statistical analysis and showed that, based on the SDSS colors, it is possible to separate the Nysa-Polana region into two families. in order to assess this suggestion, we plot the distribution of proper elements (semimajor axis and eccentricity) of the asteroids from the Nysa-Polana region in Fig. \[NysaPolana\], color coded according to the $G_{12}$ values and the SDSS $a^*$ values for comparison.
The two taxonomically different regions stand out both in $G_{12}$ and in SDSS $a^*$. The sample size for the plot color-coded according to the $G_{12}$ value is much larger than that color-coded with the SDSS $a^*$ value. The $G_{12}$ plot suggests that there is much more structure in the Nysa-Polana region and that there might be more than the two main taxonomic groups present, or that they might be more mixed. In their spectral analyses, [@CellinoNP] also found three asteroids of class X, next to the 11 Tholen F class and 8 S class asteroids. For reference, we list the $G_{12}$ values for the main members of the Nysa-Polana region in Table \[NP\]. Generally, it might turn out difficult to separate the two groups as they seem quite strongly intermixed. We carried out a $k$-means clustering operation for this region (with $k=2$). Clustering in the proper elements and in the SDSS $a^*$ parameter gave, overall, the same results as using the proper elements and the $G_{12}$ parameter.
Designation H\[mag\] $G_{12}$ taxon
------------- ---------------------------- ------------------------- -----------
44 Nysa $+6.9^{+0.05}_{-0.05}$ $+0.08^{+0.07}_{-0.07}$ Xe [@pds]
142 Polana $+10.20^{+0.013}_{-0.014}$ $+0.69^{+0.09}_{-0.09}$ B [@pds]
135 Hertha $+8.13^{+0.01}_{+0.01}$ $+0.32^{+0.08}_{-0.08}$ Xk [@pds]
878 Mildred $+14.51^{+0.03}_{-0.03}$ $+0.79^{+0.17}_{-0.17}$
: The $G_{12}$ parameters for the main members of Nysa-Polana family.[]{data-label="NP"}
Other families for which no conclusion could be made, but the shape of the $G_{12}$ distributions can still be discussed are:
- [**(847) Agnia**]{} (region II, also called (125) Liberatrix): altogether 472 members are considered for the Agnia family, with a smooth but wide $G_{12}$ distribution. The diverse $G_{12}$ values basically span through the entire range of possible $G_{12}$ values. The decision requirement (3) is not met for this family and therefore no definite conclusions can be made. However, based on the large $G_{12}$ values for many members of the family, we would suggest that the Agnia family can contain large numbers of both S and C complex asteroids. In the literature, the Agnia family has 15 members with known spectroscopic taxonomy, all belonging to the S complex (8 in class Sq, 6 in S, and 1 in Sr) [@Mothe; @BusThesis].
- [**(1128) Astrid**]{} (region II): The $G_{12}$ values for 94 members of the Astrid family result in a high probability of C complex preponderance for the a priori distributions (1) and (3) and almost equal probability of C and S complex preponderance for the a priori distribution (2). Accordingly, no conclusions can be made. In the literature, the Astrid family has 5 spectrally characterized members, all of which belong to the C complex (4 class C, 1 in Ch) [@Mothe; @BusThesis].
- [**(298) Baptistina**]{} (region I): The $G_{12}$ distribution for the Baptistina family is quite broad and results in similar probabilities for all the complexes for the a priori distributions (1) and (2). For the a priori distribution (3), the probability of S complex preponderance is the largest. In the literature, the Baptistina family has 8 spectrally characterized members. These asteroids tend to have different spectral classifications: 1 in class Xc, 1 in X, 1 in C, 1 in L, 2 in S, 1 in V, and 1 in class A [@Mothe]. Due to the lack of fulfillment of the requirements (2) and (3), we cannot evaluate the taxonomic preponderance in this family.
- [**(293) Brasilia**]{} (region III): The $G_{12}$ distribution of the Brasilia family is wide (spreading through the entire range of possible $G_{12}$ values). The resulting preponderance probabilities are similar for all the taxonomic complexes. In the literature, 4 members of the Brasilia family have known spectroscopic classification: 2 in class X, 1 in C, and 1 in class Ch [@Mothe].
- [**(221) Eos**]{} (region III): The Eos family has 92 members that have taxonomic classification. There are 26 members in class T, 17 in D, 12 in K, 8 in Ld, 13 in Xk, 4 in Xc, 5 in X, 3 in L, 2 in S, 1 in C, and 1 class B. The family has an inhomogeneous taxonomy [@Mothe]. This means that 43 asteroids originate from D complex, 25 from the S complex, 22 from the X complex, and 2 from the C complex. In our treatment, we have decided to refrain from considering complexes other than the three main ones, so indicating D complex preponderance is not possible. The $G_{12}$ histogram for Eos is shifted towards intermediate and large $G_{12}$ values, and is more indicative of C complex rather than S or X complex preponderance.
- [**(163) Erigone**]{} (region I): The $G_{12}$ distribution for this family is slightly shifted tpwards large $G_{12}$s. Erigone has 48% and 54% probabilities of C complex preponderance based on the a priori distributions (1) and (2), and a 48% probability of S complex preponderance based on the a priori distribution (3). Therefore, no particular complex can be indicated as preponderant.
- [**(15) Eunomia**]{} (region II): The Eunomia family has a smooth $G_{12}$ distribution with a profile matching the combined profile of all complexes. The probabilities of Eunomia being S complex preponderant are 34%, 42%, and 42% for the a priori distributions (1), (2), and (3) and are the largest among the different complexes, which agrees with the literature analyses. The difference between the S and the other complex preponderance probabilities are however only about 10%. Generally, the probabilities are below the required 50%, so no complex can be indicated as preponderant. In the literature, the Eunomia family has 43 members that have observed spectra, most members classified as belonging to the S complex. There are 16 members in class S (including (15) Eunomia), 2 in Sk, 10 in Sl, 1 in Sq, 7 in L, 4 in K, 1 in Cb, 1 in T, and 1 in class X [@Lazzaro1999; @Mothe].
- [**(8) Flora**]{} (region I): The $G_{12}$ distribution for the Flora family is smooth with a mean at $G_{12}=0.53$. The probability of Flora being S complex dominant is the largest and is 63% for the a priori distributions (2) and (3). Assuming a uniform a priori distribution leads to almost equal taxonomic complex preponderance probabilities. In the literature, Flora is considered S complex preponderant [@Florczak]. However, due to the lack of fulfillment of the decision requirements, we do not make a final conclusion on the taxonomic preponderance in this family.
- [**(1272) Gefion**]{} (region II, also identified as (1) Ceres or (93) Minerva): In the literature, the Gefion family has 35 members that have spectral classification. Out of these asteroids, 31 belong to the S complex (26 in class S, 2 in Sl, 2 in Sr, 1 in Sq, and 1 in L), 2 belong to the C complex (1 in class Cb class and 1 in Ch), and there is 1 X class asteroid [@Mothe]. Our complex preponderance probability computation results in similar probabilities for all three complexes and is inconclusive for this family. The $G_{12}$ distribution for Gefion spreads through all the complexes and is slightly shifted towards higher $G_{12}$ values.
- [**(46) Hestia**]{} (region II): The $G_{12}$ distribution for the Hestia family is similar to that of the Gefion family. Thus, no conclusions can be made as the probabilities of taxonomic preponderance are comparable for all the complexes.
- [**(3) Juno**]{} (region II): In the case of the Juno family, the $G_{12}$ distribution is similar to the two previous families, the taxonomic complex preponderance probabilities are similar for all the complexes. Therefore, no single complex can be indicated as preponderant.
- [**(832) Karin**]{} (region III): For the Karin family, the preponderance probabilities are similar for all the complexes for the a priori distribution (1). Therefore, no single complex can be indicated as preponderant. For the a priori distributions (2) and (3), the probabilities are discordant. The $G_12$ distribution for this family is wide, whithout a clear prefference for any of the complexes.
- [**(158) Koronis**]{} (region III): In the literature, the Koronis family has 31 members with spectral classification. There are 29 asteroids from the S complex (19 in class S, 1 in Sk, 3 in Sq, 2 in Sa, and 4 in class K), 1 from class X, and 1 from class D. The spectra of eight of these members have been analyzed by Binzel et al. who found a moderate spectral diversity among these objects [@Binzel1993; @Mothe]. In our computation, the Koronis family has the highest chance of being C complex dominated. However, the probability of being S dominant cannot be excluded as it is also quite high. Due to the lack of fulfillment of the decision requirement (3), we do not make a final conclusion on the taxonomic preponderance in this family.
- [**(3556) Lixiaohua**]{} (region III): The taxonomic preponderance probabilities for the Lixiaohua family are quite high for the C complex. The probability of C complex preponderance are 50%, 43% and 59% for priors (1), (2), (3) respectively. However for prior (2) the probability of this family being S complex dominated is 45%. Therefore, no single complex can be indicated as preponderant. $G_{12}$ distribution for this family spans though at the full range of allowed $G_{12}$ and shows surplus of high $G_{12}$s.
- [**(170) Maria**]{} (region II): Even though the Maria family has the largest probability of being S complex preponderant, which agrees with the literature, no conclusions should be made as the probabilities of the C and X complex preponderance are large . In the literature, the Maria family has 16 members which have spectral classification, all belonging to the S complex (4 in class S, 5 in L, 4 in Sl, 2 in K, and 1 class Sk) [@Mothe]. $G_{12}$ distrubution for this family matches the combine $G_{12}$ profile from all the complexes.
- [**(20) Massalia**]{} (region I): The taxonomic preponderance probabilities for the Massalia family are similar for all the complexes. Therefore, no single complex can be indicated as preponderant. Additionally, the three different a priori distributions result in differing preponderant complexes. $G_{12}$ distribution for Massalia family seems to be slightly shifted towards high $G_{12}$s.
- [**(808) Merxia**]{} (region II): The taxonomic preponderance probabilities for the Merxia family are inconclusive as none of the computed probabilities arises significantly above the rest. In the literature, the Merxia family has 8 asteroids spectrally characterized: 1 member belongs to the X complex while the remaining 7 members belong to the S complex (3 in class Sq, 2 in S, 1 in Sr, and 1 class Sl) [@Mothe; @BusThesis]. $G_{12}$ distribution for this family is wide and also matches the total $G_{12}$ distribution of all complexes combined.
- [**(1644) Rafita**]{} (region II): The taxonomic preponderance probabilities for the Rafita family are similar for all the complexes. Therefore, no single complex can be indicated as preponderant. $G_{12}$ distribution for this family is wide and matches the total $G_{12}$ distribution of all complexes combined.
- [**(752) Sulamitis**]{} (region I): The $G_{12}$ distribution for the Sulamitis family is shifted towards large values of $G_{12}$, indicating C complex predominance. The C complex preponderance probabilities based on the a priori distributions (1) and (2) are large (52 and 62%). But based on the a priori distribution (3), the preonderance probability is only 45% and, therefore, we draw no final conclusions.
- [**(9506) Telramund**]{} (region III): The taxonomic preponderance probabilities for the Telramund family are similar for all the complexes. Therefore, no single complex can be indicated as preponderant. $G_{12}$ distribution for Teramund shows slight surplus of high $G_{12}$s.
- [**(1400) Tirela**]{} (region III): Tirela family also seems to be C complex predominant with the $G_{12}$ distribution shifted towardslarge $G_{12}$ values. However, due to the lack of fulfillment of the decision criterion (3), we refrain from making a final judgment.
- [**(4) Vesta**]{} (region I): The Vesta family is dominated by the V complex asteroids. Here we are not considering V complex asteroids. $G_{12}$ distribution for this family is wide and also matches the total $G_{12}$ distribution of all complexes combined.
- [**(18466)**]{} (region II): The taxonomic preponderance probabilities for the family of asteroid (18466) are similar for all the complexes. Therefore, no single complex can be indicated as preponderant. $G_{12}$ distribution for this family is wide, matches the total $G_{12}$ distribution of all complexes combined and shows a slight surplus of low to inermediate $G_{12}$s.
Number of families have to few members in our sample to be analyzed. Those include: (396) Aeolia (region II, 55 members), (656) Beagle (region III, 38 members), (606) Brangäne (region II, 37 members), (302) Clarissa (region I, 41 members), (1270) Datura (region I, 4 members), (14627) Emilkowalski (region II, 2 members), (4652) Iannini (region II 30 members), (7353) Kazuya (region II, 12 members), (3815) König (region II, 58 members), (10811) Lau (region III, 6 members), (1892) Lucienne (region I, 37 members), (137) Meliboea (region III, 40 members), (87) Sylvia (region III, 30 members), (1189) Terentia (region III, 7 members), (778) Theobalda (region III, 60 members), (18405) (region III, 24 members). Even though conclusions for those could not be made, it is worth mentioning several of these families. The $G_{12}$ distribution for Datura and Iannini families are shifted towards small $G_{12}$ values and could be candidates for an S complex preponderant families. Asteroid (1270) Datura is spectraly identified as S class [@pds]. The $G_{12}$ distribution for the Theobalda family seems to be shifted towards C complex asteroids making it a candidate for C complex dominated. Asteroid (778) Theobalda is classified as F type [@pds]. C complex preponderance is also possible for Meliboea family, which also has been previously classified as C complex preponderant [@Mothe]. The C complex preponderance probabilities are 55%, 62% and 67% for the a priori distributions (1), (2), and (3), and are about 25–40% larger than those of being S or X complex preponderant. (137) Meliboea is spectrally classified as C class asteroid [@pds].
Overall, the strict decision criteria requirements result in C complex preponderance in the Adeona, Charis, Chloris, Dora, Emma, Hoffmeister, Hygiea, Misa, Naema, Nemesis, Padua, Themis, and Veritas families. Out of these, Adeona, Chloris, Dora, Hoffmester, Hygiea, Themis, and Veritas have spectral classifications that indicate C complex preponderance. Padua has only 9 members spectrally classified (7 of X complex and 2 of C complex). The Charis, Emma, Misa, Naema, and Nemesis families are yet to be spectrally classified.
There are no families that we can indicate as S or X complex preponderant mainly because those two are more difficult to separate. Also, there are no families which would have the distribution clearly shifted towards small $G_{12}$ values and fullfil our strict decision criteria.
We have also computed taxonomic complex probability for all asteroids having proper elements. Figure \[semiTaxa\] shows the distribution of C, S, and X complex asteroids in the main asteroid belt weighted with the probabilities of belonging to the C, S, and X complexes. Fig. \[semiTaxaFraction\] shows the weighted fraction of different taxonomic complexes in the main belt. The overall distribution agrees with the general view of more S complex asteroids in the inner main belt and C complex asteroids dominating in the outer main belt [see, e.g., @Gradie; @Zellner; @Thais; @Yoshida; @BusThesis]. On the basis of the computed C, S, and X complex probabilities for each asteroid, we modify the distributions of $G_{12}$ values for the C, S, and X complexes in Fig. \[up\]. The gap at $G_{12}=0.2$ is related to the numerical function that is used to derive the $H,\!G_{12}$ phase function, which is nondifferentiable at $G_{12}=0.2$ [for more details, see @KM2010]. This causes many asteroids to end up at $G_{12}=0.2$ in least-squares fitting. To avoid the artificial peak at $G_{12}=0.2$, we remove all asteroids with $G_{12}$ exactly equal to $0.2$. However, from the dip at $G_{12}=0.2$, it is clear that some valid solutions were thereby removed. Should the $H,\!G_{12}$ phase function be revised, we recommend that the $G_1=G_1(G_{12})$, $G_2=G_2(G_{12})$ functions are made differentiable. Figure \[up\] shows the updated $G_{12}$ distributions for the different complexes before and after correction for the $G_{12}=0.2$ artifact.
\[semiTaxa\_\]
Conclusions {#concl}
===========
We have analyzed the photometric parameter $G_{12}$ for all known asteroids as well as $G_{12}$ distributions for asteroid families. We have strengthened our previous finding of $G_{12}$ homogeneity in some asteroid families and also confirmed a correlation between $G_{12}$ and taxonomy. $G_{12}$ could be potentially used in asteroid family membership classification. We have further analyzed asteroid families for C, S, or X complex preponderance.
We conclude that, although $G_{12}$ is related to surface properties, on its own it is mostly insufficient to unambiguously assign taxonomic complex of individual asteroids. Generally, the complex separation in the $G_{12}$ space is small. The $G_{12}$ distributions also overlap and, generally, no definitive conclusions should be made for individual objects based only on $G_{12}$ values. Rather probabilities of belonging to a given complex can be computed. All classification results for individual objects based on the $G_{12}$ values should be taken with caution and used rather to confirm previous results than to derive classifications based only on the current $G_{12}$ values.
In some cases $G_{12}$ values can, however, be an indication of asteroid taxonomic complex. Particularly, the C complex is the easiest to be separated, which was also confirmed by the high success ratio in the testing. Accordingly $G_{12}$ distributions can be used in verifying taxonomic complex preponderance in some asteroid families. We found a preponderance of C complex asteroids in several families. We compared our findings to the results available in the literature, and concluded that, based on the $G_{12}$ distributions in the families, we could confirm complex preponderance for several families available.
The $G_{12}$ values in conjunction with the SDSS colors could possibly result in a better separation of the different taxonomic complexes. An increased number of taxonomy classified asteroids and better quality data potentially leading to better constrained $G_{12}$ values could improve our knowledge of the $G_{12}$-taxonomy correlation. More detailed knowledge of asteroids surface properties would also benefit the classification problem. The Gaussian approximations of complex distributions could be replaced by more sophisticated distributions, and the a priori distributions for the Bayesian analysis could be replaced by those deriving from debiased taxonomic distributions. Particularly, a continuous debiased function describing the fraction of different complexes throughout the belt could be used.
Acknowledgments {#acknowledgments .unnumbered}
===============
Research has been supported by the Magnus Ehrnrooth Foundation, Academy of Finland (project No. $127461$), Lowell Observatory, and the Spitzer Science Center. We would like to thank Dr. Michael Thomas Flanagan (University College London) for developing and maintaining the Java Scientific Library, which we have used in the Asteroid Phase Function Analyzer. DO thanks Berry Holl for help with Java plotters and Saeid Zoonemat Kermani for valuable advice on Java applets. We thank the Department of Physics of Northern Arizona University for CPU time on its Javelina open cluster allocated for our computing.
[^1]: IAU Minor Planet Center, see http://minorplanetcenter.net/iau/mpc.html
[^2]: Asteroid Phase Curve Analyzer — an online java applet, available at <http://asteroid.astro.helsinki.fi/astphase/>
|
---
abstract: 'In this paper we develop a feed-back control framework for the real-time minimization of microstructures grown within the rechargeable battery. Due to quickening nature of the branched evolution, we identify the critical ramified peaks in the early stages and based on the state we compute the relaxation time for the concentration in those branching fingers. The control parameter is a function of the maximum curvature (i.e. minimum radius) of the branched microstructure, where the higher rate dendritic evolution would lead to the more critical state to be controlled. The charging time is minimized for generating the most packed microstructures and obtained results correlate closely with those of considerably higher charging time periods. The developed framework could be utilized as a smart charging protocol for the safe and sustainable operation the rechargeable batteries, where the branching of the microstructures could be correlated to the sudden variation in the current/voltage.'
author:
- 'Asghar Aryanfar$^{\dagger}$$^{\ddagger}$[^1] , Yara Ghamlouche$^{\dagger}$, William A. Goddard III$^{\S}$'
bibliography:
- 'Refs.bib'
title: Control of Dendrites in Rechargeable Batteries using Smart Charging
---
*$\dagger$ American University of Beirut, Riad El-Solh, Lebanon 1107 2020*\
*$\ddagger$ Bahçeşehir University, 4 Çirağan Cad, Beşiktaş, Istanbul, Turkey 34353*\
*$\S$ California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125*
Introduction
============
The modern era of wireless revolution and portable electronics demands the utilization of reliable intermittent renewables and long-lasting electrical energy storage facilities [@Rugolo12; @Dunn11]. As well, the growing demand for portable computational power as well as the introduction of electric vehicles demand novel and reliable high capacity energy storage devices. Despite such impressive growth of the need in the daily lifestyle, the underlying science remains to be developed. Rechargeable batteries, which retrieve/store energy from/within the chemical bonds, have proven the be the most reliable and cleanest resource of electrical energy for the efficient management of the power. [@Marom11; @Goodenough13] Metallic electrodes such as lithium $Li$ [@Cheng17], sodium $Na$ [@Slater13], magnesium $Mg$ [@Davidson18], and zinc $Zn$ [@Pei14] are arguably highly attractive candidates for use in high-energy and high-power density rechargeable batteries. Lithium $Li$ possess the lowest mass density ($\rho_{Li}=0.53~g.cm^{-3}$) and the highest electropositivity ($E^{0}=-3.04V$ vs SHE[^2]) which provides the highest gravimetric energy density and likely the highest voltage output, making it suitable for high-power applications such as electric vehicles [@Li19Lithium; @Xu14; @Zhamu12]. Sodium has a lower cost and is more earth abundant and is operational for large-scale stationary energy storage applications [@Ellis12]. Magnesium $Mg$ possess a high specific capacity and reactivity [@Shterenberg14] whereas Zinc $Zn$ is earth-abundant, has low cost and high storage capacity [@Li14Recent].
During the charging, the fast-pace formation of microstructures with relatively low surface energy from Brownian dynamics, leads to the branched evolution with high surface to volume ratio [@Xu14]. The quickening tree-like morphologies could occupy a large volume, possibly reach the counter-electrode and short the cell. Additionally, they can also dissolve from their thinner necks during subsequent discharge period and form detached dead crystals, leading to thermal instability and capacity decay [@Xu04; @Tewari20]. The quickening tree-like morphologies could occupy a large volume, possibly reach the counter-electrode and short the cell. Additionally, they can also dissolve from their thinner necks during subsequent discharge period. Such a formation-dissolution cycle is particularly prominent for the metal electrodes due to lack of intercalation[^3], where the depositions in the surface is the only dominant formation mechanism versus the diffusion into the inner layers as the housing [@Dresselhaus81; @Li14Review] The growing amorphous crystals can pierce into the polymer electrolyte and short the cell afterwards, given their higher porosity, they could have mechanical properties comparable to the bulk form [@Aryanfar17Bulk].
Previous studies have investigated various factors on dendritic formation such as current density [@Ren15], electrode surface roughness [@Nielsen15; @Natisiavas16], impurities [@Steiger14], solvent and electrolyte chemical composition [@Schweikert13; @Younesi15], electrolyte concentration [@Jeong08], utilization of powder electrodes [@Seong08] and adhesive polymers[@Stone12], temperature [@Aryanfar15Annealing; @Aryanfar15Thermal], guiding scaffolds [@Yao19; @Qian19], capillary pressure [@Deng19], cathode morphology [@Abboud19] and mechanics [@Klinsmann19; @Xu17; @Wang19; @Liu19Piezo]. Some of conventional characterization techniques used include NMR [@Bhattacharyya10] and MRI. [@Chandrashekar12] Recent studies also have shown the necessity of stability of solid electrolyte interphase (i.e. SEI) layer for controlling the nucleation and growth of the branched medium [@Li19Energy; @Kasmaee16] as well as pulse charging [@Li01; @Chandrashekar16; @Aryanfar18].
Earlier model of dendrites had focused on the electric field and space charge as the main responsible mechanism [@Chazalviel90] while the later models focused on ionic concentration causing the diffusion limited aggregation (DLA). [@Monroe03; @Witten83; @Zhang19; @Fleury97] Both mechanisms are part of the electrochemical potential [@Bard80; @Tewari19], indicating that each could be dominant depending on the localizations of the electric potential or ionic concentration within the medium. Recent studies have explored both factors and their interplay, particularly in continuum scale and coarser time intervals, matching the scale of the experimental time and space [@Aryanfar14Dynamics]. Other simplified frameworks include phase field modeling [@Mu19; @Ely14; @Gogswell15] and analytical developments [@Akolkar13].
During charge period the ions accumulate at the dendrites tips (unfavorable) due to high electric field in convex geometry and at the same time tend to diffuse away to other less concentrated regions due to diffusion (favorable). Such dynamics typically occurs within the double layer (or stern layer [@Bazant11]) which is relatively small and comparable to the Debye length. In high charge rates, the ionic concentration is depleted on the reaction sites and could tend to zero [@Fleury97]; Nonetheless, our continuum-level study extends to larger scale, beyond the double layer region [@Aryanfar19Finite]
Dendrites instigation is rooted in the non-uniformity of electrode surface morphology at the atomic scale combined with Brownian ionic motion during electrodeposition. Any asperity in the surface provides a sharp electric field that attracts the upcoming ions as a deposition sink. Indeed the closeness of a convex surface to the counter electrode, as the source of ionic release, is another contributing factor. In fact, the same mechanism is responsible for the further semi-exponential growth of dendrites in any scale. During each pulse period the ions accumulate at the dendrites tips (unfavorable) due to high electric field in convex geometry and during each subsequent rest period the ions tend to diffuse away to other less concentrated regions (favorable). The relaxation of ionic concentration during the idle period provides a useful mechanism to achieve uniform deposition and growth during the subsequent pulse interval. Such dynamics typically occurs within the double layer (or stern layer [@Bazant11]) which is relatively small and comparable to the Debye length. In high charge rates, the ionic concentration is depleted and concentration on the depletion reaches zero [@Fleury97]; Nonetheless, our continuum-level study extends to larger scale, beyond the double layer region [@Aryanfar14Dynamics].
Pulse method has been qualitatively proved as a powerful approach for the prevention of dendrites [@Li01], which has previously been utilized for uniform electroplating [@Chandrashekar08]. In the preceding publication we have experimentally found that the optimum rest period correlates well with the relaxation time of the double layer for the blocking electrodes which is interpreted as the *RC time* of the electrochemical system [@Bazant04]. We have explained qualitatively how relatively longer pulse periods with identical duty cycles $\mathbf{D}$ will lead to longer and more quickening growing dendrites. We developed coarse grained computationally affordable algorithm that allowed us reach to the experimental time scale (*$ms$*). We have developed theoretical limit the optimal minimization of the dendrites [@Aryanfar18] and we have obtained the pulse charging parameters for individual curved peaks based on their curvature [@Aryanfar19Finite].
In this paper, we elaborate on the real-time controlling of the pulse charging parameters for the minimization of microstructures grown in the scales extending to the cell domain. We analyze the oscillatory behavior and the transition from initial to steady-state growth regimes.
Methodology
===========
![Square pulse wave.\[fig:PulseCurve\]](1DiffMigVectors){height="0.24\textheight"}
![Square pulse wave.\[fig:PulseCurve\]](2PulseRest){height="0.24\textheight"}
The electrochemical flux is generated either from the gradients of concentration ($\nabla C$) or electric potential ($\nabla V$). In the ionic scale, the regions of higher concentration tend to collide and repel more and, given enough time, diffuse to lower concentration zones, following Brownian motion. In the continuum (i.e. coarse) scale, such inter-collisions could be added-up and be represented by the diffusion length $\delta\vec{\textbf{r}}_{D}$ as: [@Aryanfar14Dynamics] [^4]
$$\delta\vec{\textbf{r}}_{D}=\sqrt{2D^{+}\delta t}\text{ }\hat{\textbf{g}}\label{eq:DiffDis}$$
where $\vec{\textbf{r}}_{D}$ is diffusion displacement of individual ion, $D^{+}$ is the ionic diffusion coefficient in the electrolyte, $\delta t$ is the coarse time interval [^5], and $\hat{\textbf{g}}$ is a normalized vector in random direction, representing the Brownian dynamics. The diffusion length represents the average progress of a diffusive wave in a given time, obtained directly from the diffusion relationship [@Philibert06].
On the other hand, ions tend to acquire drift velocity in the electrolyte medium when exposed to electric field and during the given time $\delta t$ their progress $\delta\vec{\textbf{r}}_{M}$ is given as:
$$\delta\vec{\textbf{r}}_{M}=\mu^{+}\vec{\textbf{E}}\delta t\label{eq:MigDis}$$
where $\mu^{+}$ is the mobility of cations in electrolyte, $\vec{\textbf{E}}$ is the local electric field. The voltage $V$ is obtained from the laplacian relationship in the domain as:
$$\nabla^{2}V\approx0\label{eq:Lap}$$
where the dendrite body is part of the boundary condition per see. The electric field is the gradient of electric potential as:
$$\vec{\textbf{E}}=-\nabla V\label{eq:EV}$$
Therefore the total effective displacement $\delta\vec{\textbf{r}}$ with neglecting convection[^6] would be:
$$\delta\vec{\textbf{r}}=\delta\vec{\textbf{r}}_{D}+\delta\vec{\textbf{r}}_{M}\label{eq:TotalDis}$$
as represented in the Figure \[fig:Displacements\]. The pulse charging in its simplest form consists of trains of square active period $t_{ON}$ , followed by a square rest interval $t_{OFF}$ in terms of current $I$ or voltage $V$ as shown in Figure \[fig:PulseCurve\]. The period $P=t_{ON}+t_{OFF}$ is the time lapse of a full cycle. Hence the pulse frequency $f$ is:
$$f=\frac{1}{t_{ON}+t_{OFF}}\label{eq:Frequency}$$
and the duty cycle **$\mathbf{D}$** represents the fraction of time in the period $P$ that the pulse is active :
$$\mathbf{D}=ft_{ON}\label{eq:DftON}$$
While the dendrites grow, due to random nature of the evolution of branches, they stick out randomly, which becomes a source for the quickening growth of the dendrite per see due to the concentration of electric field in the sharp interfaces as well as their closer proximity to the upcoming ions. The relaxation time allows the concentrated ions in the ramified peaks to dissipate away into the less concentrated areas and the concentration gradient is relaxed. The time required for the such relaxation depends on the curvature of the interface, where the higher curvature sites would require longer time for the concentration relaxation within the double layer region [@Aryanfar19Finite]. In fact such relaxation could occur for the larger interfacial double layer scale with the thickness of $\kappa$, spanning to the entire cell domain, requiring the most relaxation time for the highest curvature regions and least for the flat counterparts. From dimensional analysis the relaxation time of the double layer in the flat electrode is in the range of ${\displaystyle \sim\frac{\kappa^{2}}{D^{+}}}$ [@Hunter01]and for the larger domain of the cell with the representative length of $l$ would scale up to ${\displaystyle \sim\frac{l^{2}}{D^{+}}}$, where as their geometric mean has been later considered as ${\displaystyle \sim\frac{\kappa l}{D^{+}}}$ [@Bazant04]. In fact the relaxation of the concentration depends highly to the curvature of the peaks and the entire growing interface possess a wide range of radius of curvature values $r_{d}$, expanding from the flat surface to the highly-curved fingers as small as the atomic value, therefore:
$$r_{d}\in[r_{atom},\infty)\label{eq:rdRange}$$
Hence, the relaxation time for a randomly-growing interface with variation of curvature along the interfacial line, would vary as well. Hereby, we define the feedback relaxation time $t_{REL}$ a curvature dependent function $f(r_{d})$ multiplied by the geometric mean time for the concentration relaxation (i.e. $RC$ time) as:
$$t_{REL}=f(r_{d})\frac{\kappa l}{D^{+}}\label{eq:tREL}$$
As the interface grows from the initial flat state ($r_{d}\to\infty$) to creating sharp fields ($r_{d}\to r_{atom}$), the feedback relaxation time $t_{REL}$ should adapt respectively based on the most critical state of the interface, which is the location of ionic concentration. Therefore the range of acceptable value for the feedback relaxation time $t_{REL}$ should lie between the relaxation scale within the double layer to the scale of the cell domain. The variation of the feedback relaxation time $t_{REL}$ occurs from it’s minimum value during the instigation, to it’s maximum value in the atomic scale. Therefore considering the relevant time-scales from beginning (flat) to the most ramified state (atomic sclae), from Equation \[eq:tREL\]one has:
$$\begin{cases}
\lim_{r_{d}\rightarrow\infty}f(r_{d})=1 & \text{Flat}\\
\lim_{r_{d}\to r_{atom}}f(r_{d})={\displaystyle \frac{l}{\kappa}} & \text{Ramified}
\end{cases}\label{eq:BC}$$
Assuming the form of the control function an combination of linear and exponential terms as $f(r_{d})=ar_{d}+b\exp(cr_{d})$, from the boundary conditions in the Equation \[eq:ControlFunction\] one gets:
$$f(r_{d})=1+({\displaystyle \frac{l}{\kappa}}-1)\exp[-r_{d}]\label{eq:ControlFunction}$$
Thus the feedback relaxation time $t_{REL}$ is obtained as:
$$t_{REL}=\frac{\kappa l}{D^{+}}\left(1+(\frac{l}{\kappa}-1)\exp[-r_{d}]\right)\label{eq:tOpt}$$
![Square pulse wave.\[fig:tOFFControl\]](3Diagram){width="100.00000%" height="0.22\textheight"}
![Square pulse wave.\[fig:tOFFControl\]](4SchematicsofModel){height="0.22\textheight"}
The radius of curvature $r_{d}$ can be approximated via the contours of the iso-potential curvature of the electric field in the vicinity of electrodeposits, where it occurs typically within the double layer region of thickness $\kappa$. The corresponding line could be obtained locating magnitude of the isopotential contour matching with the electrode ($V=V_{electrode}$). If $(x,y)$ represents the coordinates of the curvature line, the point of the minimum radius of curvature would address the most critical state, and requires higher dissipation of ionic concentration. The radius of curvature $r_{d}$ can be calculated from Equation \[eq:rd\] as:
$$r_{d}=min\left(\Bigg|{\displaystyle \frac{{\displaystyle 1+\frac{dy}{dx}}}{{\displaystyle \frac{d^{2}y}{dx^{2}}}}}\Bigg|\right)\Bigg|_{x=0}^{x=l}\label{eq:rd}$$
This value is computed in real time and inserted into the feedback algorithm. In fact the curvature-dependent relaxation time provides a positive feedback for halting of the quickening dendrites, which is negative feedback for dendritic evolution. The Flowchart \[fig:Flowchart\] represents the control loop representing the real-time computation of the curvature and the corresponding feedback relaxation time for the minimization of the dendritic branching. Figure \[fig:tOFFControl\] schematically represents such variation where the feedback relation time $t_{REL}$ starts from the minimum value of ${\displaystyle \sim\frac{\kappa l}{D^{+}}}$ in the flat surface and varies based on the measurement of the highest curvature of the tip given in the Equation \[eq:rd\].
Var. Ref.
------------------- -- ----------------------- -- -- --
$\delta t(\mu s)$ [@Dror10]
$D(m^{2}/s)$ [@Aryanfar14Dynamics]
$\#Li^{+}$ [@Aryanfar14Dynamics]
$\#Li^{0}$ [@Aryanfar14Dynamics]
$l(nm)$ [@Aryanfar14Dynamics]
$\Delta V(mV)$ [@Aryanfar14Dynamics]
: Simulation parameters. \[tab:Sample\]
The thickness of the double layer $\kappa$can be obtained from [@Bazant04]:
$$\kappa=\sqrt{\frac{\varepsilon k_{B}T}{2z^{2}e^{2}C_{b}}}$$
where $\varepsilon$ is the permittivity of the solvent, $k_{B}$ is Boltzmann constant, $T$ is the temperature, $z$ is the valence number, $e$ is the electron charge and $C_{b}$ is the average ambient electrolyte concentration.
The computation was carried out based on the simulation parameters given in the Table \[tab:Sample\]. Figure \[fig:Densities\] illustrates the resulted morphologies of the grown dendrites based on the applied relaxation time.
The density of the electro-deposits can easily be calculated from confining the atoms in a rectangle, the height of which spans to the highest dendrite coordinates $h_{max}$. Therefore:
$$\rho=\frac{n\pi r^{2}}{h_{max}l}$$
where $n$ is the number of atoms composing the dendrite, $r$ is the atomic radius and $l$ is the scale of the domain. The density of the morphologies, sample of which is shown in the Figure \[fig:Opt\] is provided in the Figures \[fig:200\], \[fig:400\] and \[fig:800\] versus various relaxation time values as the control parameter.
As well, the variation of the highest interfacial curvature (minimum radius of curvature $r_{d}$) and their corresponding feedback relaxation time $t_{REL}$ and the density $\rho$ versus the number of deposited atoms is shown in the Figures \[fig:rd\], \[fig:tOFF\] and \[fig:Density\].
Results & Discussion
====================
The mechanism used in the pulse charging works based on the relaxation of the ionic concentration in the dendritic tips. The formation of such gradient in the concentration is, in fact, the feedback for the quickening upcoming growth regime after the instigation and therefore the applied feedback relaxation time should effectively dissipate away the accumulated ions from the accumulated regions. In fact the sharper interfaces, which have been growing faster than the rest of the interface, have higher number of concentrated ions around them and therefore they are in the most critical state, which have been used for the computation of the feedback relaxation time.
In the larger scale since the electro-migration displacement (Eq. \[eq:MigDis\]) scales with $\sim t$ and the diffusion displacement (Eq. \[eq:DiffDis\]) scales with the square of time $\sim\sqrt{t}$ . During the pulse both electro-migration and diffusion are in action whereas during the rest period only diffusion is the main drive. Therefore since the average reach for electro-migration is higher than the sole-diffusion the range of reach in the rest period should in fact be competitive with the pulse period. Therefore:
$$\sqrt{2D^{+}t_{OFF}}\geq\mu^{+}\vec{\textbf{E}}t_{ON}\pm\sqrt{2D^{+}t_{ON}}\label{eq:toffton}$$
and performing further, we get the maximum value of duty cycle $\mathbf{D}$ for effective pulse charging:
$$\mathbf{D_{max}}=max\left(\frac{1}{\left(1+{\displaystyle \frac{|\vec{\textbf{E}}|}{RT}}\sqrt{{\displaystyle \frac{D^{+}}{2f}}}\right)^{2}\pm1}\right)\leq\frac{1}{2}\label{eq:Duty}$$
where the duty cycle of the ${\displaystyle \frac{1}{2}}$ is the limiting value for the effective suppression of dendrites. The formation of local branches indicates that the concentration of ions in those specific sights is high and therefore those sites should be focus locations for the feedback relaxation time $t_{REL}$, which highly depends on the radius of curvature $r_{d}$ of the dendritic peaks [@Aryanfar19Finite]. For an individual ramified peak with the radius of curvature $r_{d}$, the time required for the concentration relaxation $t_{REL}^{DL}$within the double layer with the scale of $\sim\kappa$ is :
$$t_{REL}^{DL}\approx\frac{\kappa(\kappa+r_{d})}{D^{+}}\label{eq:tOFFInd}$$
where $D^{+}$ is the diffusivity value for the ionic transport.
which in fact shows faster relaxation relative to the flat interface. For the larger scale, extending to the entire cell domain, the feedback relaxation time $t_{REL}$ act on the uniformization of ionic concentration in the global range (i.e. $\sim l$) . Therefore the scale of transport for the time and space would lead to the following comparison: :
$$\frac{\kappa^{2}}{D}\leq t_{REL}^{DL}\leq\frac{kl}{D}\leq t_{REL}\leq\frac{l^{2}}{D}\label{eq:tOFFRange}$$
Therefore the relaxation time scale varies from ${\displaystyle \sim\frac{\kappa^{2}}{D}}$ in the individual peaks to the ${\displaystyle \sim\frac{l^{2}}{D}}$ in the larger domain of the cell, and the control relaxation time in fact varies in such range. Figure \[fig:Densities\] represents the density of the dendrites $\rho$ versus the intervals of pulse $t_{ON}$ and the rest (i.e. relaxation) $t_{OFF}$ periods. The density of the electro-deposits $\rho$correlates inversely with the pulse charing time $t_{ON}$. This is due to the exacerbated branching during the charge time which upon growing further gets more difficult to halt. On the other hand, applying finer pulse periods $t_{ON}$ provides better possibility for the suppression of dendrites. Needless to mention that such pulse period $t_{ON}$ could not indefinitely get short since the ions ultimately would require enough time to reach the dendrites during this time and react from ionic to atomic species.
As well, the density values $\rho$ correlates with the relaxation time $t_{OFF}$ until reaching a certain saturation limit. Since the length of the domain is much larger than the double layer ($l\gg\kappa$), the range of feedback relaxation time $t_{REL}$, shown by color gradient, would extremely reduce the charging time with negligible compensation in the density of electro-deposits $\rho$. The underlying reason is that the relaxation would let the ionic concentration to relax and uniform ionic distribution. On the other hand, extra relaxation period will not helpful since the ionic concentration is already relaxed and the concentration gradient has already vanished.
As well, imposing higher-than-limit relaxation time would slightly reduce the density $\rho$ since additional concentration from the ambient electrolyte could be depleted in the into the non-reacting dendritic sites. The negligible increase in the density of the dendrites $\rho$ in the span of Figures \[fig:200\], \[fig:400\] and \[fig:800\] illustrates the effective-ness of the pulse charging method for the multitudes of the charge amount $N$.
![Density. \[fig:Density\]](7rd){height="0.2\textheight"}
![Density. \[fig:Density\]](8tOFF){height="0.2\textheight"}
![Density. \[fig:Density\]](9Density){height="0.2\textheight"}
The dendritic evolution can be divided into two distinctive stages of the transient and steady-state ($S.S.$) growth regimes [@Bai16; @Gonzalez08], which has been illustrated in the Figures \[fig:rd\], \[fig:tOFF\] and \[fig:Density\]. The initial transient regime in fact is a stochastic in nature whereas the steady state regime can illustrate an effective trend. Figure \[fig:rd\] represents the variation of the radius of the curvature $r_{d}$ in the growing interface versus the deposition progress $N$ (i.e. number of the atoms). In this figure, the transition stage during the higher pulse time shows more fluctuation which indicates the non-uniform regime of growth for the augmented pulse intervals. On the other hand during the steady-state ($S.S.$) regime, radius of curvature correlates inversely with the pulse time interval $t_{ON}$, which indicates that the morphology is controlled for more finer pulse periods.
Figure \[fig:tOFF\] represents the control relaxation time $t_{REL}$ versus the progress in electrodeposition during the dendritic evolution, where the higher pulse intervals would require higher amount of control relaxation time $t_{REL}$ for the effective suppression of the dendrites. As well the higher fluctuation for the higher amount of pulse charge $t_{ON}$ shows the higher control rate due to faster dynamics of variation in the curvature.
The same trend of transition-to-steady state regimes has been observed in the Figure \[fig:Density\], where the highest fluctuation occurs for the higher pulsing time $t_{ON}$ where is leads to the lowest density $\rho$ after reaching the steady growth regime.
In fact, the quickening growth regime of the dendrites illustrates that the larger height $h$ of the electrodeposits the rate of their growth would be higher as well. This can simply be represented by the following:
$$\frac{dh}{dt}\propto h$$
where the integration leads to the exponential relationship for the growth regime as:
$$h(t)\propto\exp(bt)$$ where $b$ is the coefficient of the proportion. Setting exponential relationship causes a very high sensitivity for the control relaxation time $t_{REL}$ to act vigilantly versus smallest perturbation in the ramified peaks and the form of the control relation time $t_{REL}$ proportionally contains the exponential form in Equation \[eq:tOpt\]. Note that other forms of the relaxation time would as well could satisfy the boundary conditions given in the Equation \[eq:BC\] such as $t_{REL}^{alt}$ given as:
$$t_{REL}^{alt}\approx\frac{\kappa l}{D}\left(\frac{r_{d}+l}{r_{d}+\kappa}\right)\label{eq:AlternativetOFF}$$
which has lower sensitivity for the radius of curvature $r_{d}$ relative to proposed control relation time $t_{REL}$. This can be proven by calculating their derivative with respect to radius of curvature ( ${\displaystyle \frac{dt_{OFF}}{dr_{d}}}$ ) and show that:
$$\frac{dt_{REL}}{dr_{d}}\gg\frac{dt_{REL}^{alt}}{dr_{d}}$$
Thus from equations \[eq:tOpt\] and \[eq:AlternativetOFF\] and considering the negative value for both derivatives, one must have:
$${\displaystyle \frac{\kappa-l}{\kappa\exp(r_{d})}}\frac{\kappa l}{D}>\frac{\kappa-l}{(r_{d}+\kappa)^{2}}\frac{\kappa l}{D}\label{eq:Comparison}$$
since $\kappa\ll l$ dividing by negative value of $\kappa-l$ changes the inequality sign, therefore:
$$\exp(r_{d})>(r_{d}+\kappa)^{2}\label{eq:ExpQuad}$$
The equation \[eq:ExpQuad\] is obvious for a large values of radius of curvature $r_{d}$ since the exponential term in the denominator will surpass the quadratic term in the right side. As well for an infinitesimally small value of the radius of curvature $r_{d}$ one can use Taylor expansion as: $\exp(r_{d})\approx1+r_{d}{\displaystyle +\cancelto{0}{O(r_{d}^{2})}}$ and one has:
$$\kappa(1+r_{d})>(r_{d}+\kappa)^{2}$$
re-arranging gives:
$$r_{d}^{2}+\kappa r_{d}+\kappa^{2}-\kappa<0$$
which is a quadratic equation in terms of the radius of curvature $r_{d}$ and the root is found as:
$$r_{d}=\sqrt{\kappa-\frac{3}{4}\kappa^{2}}-\frac{\kappa}{2}$$
Considering the infinitesimal value for thickness of the double layer ($\kappa\to0$) the value for $r_{d}$ would be very small. Therefore for the most range of $r_{d}\in[r_{atom},\infty)$ the exponential relationship remains as the most sensitive to the variations in the radius of curvature $r_{d}$ and the Equation \[eq:tOpt\] as effective control relaxation time for suppression of the dendrites..
In practice, the discerning the formation of such a peak in the dendrite morphology for potentiostatic charging (constant applied voltage $V$) could be obtained by computing the sudden increase in the current density, representing the runaway process, whereas for galvanic charging (constant applied current $I$) could be the sudden drop in the potential value, where both of these events represent the runaway process (i.e jump) in time..
Conclusions
===========
In this paper we have developed an effective real-time feedback control relaxation method for minimization of dendritic grown during electrodeposition for preventing the branched evolution of the grown microstructures. The control parameter has been considered as the maximum curvature of the growing interface based on the radius of curvature of the most critical peaks. The sensitivity of the feedback relaxation time to the curvature has been analyzed to be extremely high with the exponential correlation, analogous to the growth dynamics of the branches. The methodology can be used for smart charging in rechargeable batteries for the controlling the morphology of the grown electro-deposits.
[^1]: Corresponding author; email: <[email protected]>
[^2]: $SHE$: Standard Hydrogen Electrode, taken conventionally as the reference ($E_{H^{2}}^{0}=0$)
[^3]: Intercalation: diffusion into inner layer as the housing for the charge, as opposed to depositing in the surface.
[^4]:
[^5]: $\delta t=\sum_{i=1}^{n}\delta t_{i}$ where $\delta t_{k}$ is the inter-collision time, typically in the range of $fs$.
[^6]: Since Rayleigh number $Ra$ is highly dependent to the thickness (i.e. $Ra\propto l^{3}$), for a thin layer of electrodeposition we have $Ra<1500$ and thus the convection is negligible. [@Fox16]
|
---
abstract: 'Moving object detection is critical for automated video analysis in many vision-related tasks, such as surveillance tracking, video compression coding, etc. Robust Principal Component Analysis (RPCA), as one of the most popular moving object modelling methods, aims to separate the temporally-varying (i.e., moving) foreground objects from the static background in video, assuming the background frames to be low-rank while the foreground to be spatially sparse. Classic RPCA imposes sparsity of the foreground component using $\ell_1$-norm, and minimizes the modeling error via $\ell_2$-norm. We show that such assumptions can be too restrictive in practice, which limits the effectiveness of the classic RPCA, especially when processing videos with dynamic background, camera jitter, camouflaged moving object, etc. In this paper, we propose a novel RPCA-based model, called Hyper RPCA, to detect moving objects on the fly. Different from classic RPCA, the proposed Hyper RPCA jointly applies the maximum correntropy criterion (MCC) for the modeling error, and Laplacian scale mixture (LSM) model for foreground objects. Extensive experiments have been conducted, and the results demonstrate that the proposed Hyper RPCA has competitive performance for foreground detection to the state-of-the-art algorithms on several well-known benchmark datasets.'
author:
- 'Zerui Shao, Yifei Pu, Jiliu Zhou, Bihan Wen, and Yi Zhang, [^1] [^2] [^3]'
bibliography:
- 'reference.bib'
title: 'Hyper RPCA: Joint Maximum Correntropy Criterion and Laplacian Scale Mixture Modeling On-the-Fly for Moving Object Detection'
---
Moving object detection, Background subtraction, Maximum correntropy criterion, Laplacian scale mixture model.
Introduction
============
object detection (MOD) of surveillance video frames is critical for many computer vision applications, such as video compression coding, object behavior extraction and surveillance object tracking [@paul2010video; @chen2012surveillance; @jodoin2012behavior; @cullen2012detection; @yilmaz2006object]. In the past decades, various MOD methods [@oreifej2012simultaneous; @xu2013gosus; @gao2014block; @xin2015background; @gemignani2016robust; @javed2016spatiotemporal; @bouwmans2017decomposition; @javed2017background] have been proposed. Early strategies proposed to directly distinguish background pixels from foreground through simple statistical measures, such as median, mean and histogram model [@zheng2006extracting]. Later, more elaborate methods proposed to classify the pixels by learning background and foreground models, such as the Gaussian mixture models (GMM) [@zivkovic2004improved] and local binary pattern (LBP) models [@heikkila2006texture]. However, these methods failed to exploit critical structures, such as temporal similarity between frames, or sparsity of foreground objects. When processing complex video data involving camera jitter, dynamic background and illumination, etc., the performance would degrade significantly. Recently, based on the assumption that the background is low-rank and the foreground is sparse, the RPCA-based approaches [@candes2011robust; @zhou2012moving; @feng2013online; @shakeri2016corola; @vaswani2018robust; @bouwmans2018applications] have attracted much attention for background subtraction. By exploiting the low-rank property of the background and the sparse prior of the foreground, the RPCA-based methods have achieved a remarkable success in MOD tasks.
Despite the success of the classic RPCA-based methods [@candes2011robust; @feng2013online], they suffer from two major limitations for MOD in practice: (1) The sparsity assumption, which is imposed by $\ell_1$-norm penalty, is sometimes too restrictive for large and complicated foreground in practice. (2) It is unclear whether the $\ell_2$-norm is the optimal penalty for the RPCA modeling error, which highly depends on the model accuracy and video data distribution. Various works have been proposed to tackle the limitation (1) towards more effective foreground object modeling [@javed2018moving; @zhou2012moving; @shi2018robust; @yong2017robust; @ebadi2017foreground]. For example, [@zhou2012moving] applied Markov Random Field (MRF) to constrain the sparse foreground parts, and $\ell_0$-norm is utilized to regularize the sparse component. [@ebadi2017foreground] was inspired by the concept of the group sparsity structure, and exploited the tree-structured property of the moving objects. Instead of using fixed foreground distribution, [@yong2017robust] proposed to model the foreground pixel with separate mixture of Gaussian (MoG) distribution. In [@javed2018moving], graph Laplacian was imposed in both spatial and temporal domains to explore the spatiotemporal correlation in sparse component. The Gaussian scale mixture (GSM) model was utilized in [@shi2018robust] to model each foreground pixel, which significantly improved the estimation accuracy by jointly estimating the variances of the known and unknown sparse coefficients. On the contrary, few works addressed the limitation (2) on modeling error. Very recently, [@guo2017godec+] applied the maximum correntropy criterion (MCC) [@he2011robust] for modeling error, to obtain higher-quality background extraction. However, [@guo2017godec+] is a batch algorithm, thus scales poorly for real-time video MOD.
In this paper, we propose a novel online MOD scheme via background subtraction, dubbed Hyper RPCA. The proposed scheme simultaneously tackles the two aforementioned limitations of the classic RPCA, by jointly applying the Laplacian scale mixture (LSM) and MCC models, to effectively model the complicated foreground moving objects, and approximation error, respectively. Furthermore, the proposed Hyper RPCA can be learned and applied on the fly (i.e., process video streaming sequentially) with high scalability and low latency, which is more efficient for processing video streams from surveillance cameras in practice. Our contributions in this paper are summarized as follows.
- We propose a novel Hyper RPCA formulation, which combines the LSM model and MCC respectively for complicated foreground moving objects and accurate error modeling. Compared with classic RPCA, the proposed formulation can effectively model foreground pixels and is capable of adapting a wide range of modeling error.
- A highly-efficient online algorithm to solve Hyper RPCA for MOD is derived. The proposed online algorithm avoids high computational complexity and can be used for real-time MOD applications.
- Extensive experiments over several datasets of challenging scenarios are conducted. The results demonstrate that the proposed Hyper RPCA outperforms the state-of-the-art MOD methods over several benchmark datasets.
The remainder of the paper is organized as follows. Section \[sec.related works\] reviews the related works to the Hyper RPCA. Section \[sec.Hyper RPCA\] presents the proposed Hyper RPCA learning formulation. Section \[sec.algorithm\] describes the highly-efficient Hyper RPCA algorithm for MOD. Section \[sec.experiments\] presents the experimental results on several challenging MOD cases. Section \[sec.conclusion\] concludes this paper.
Related Works {#sec.related works}
=============
Online RPCA
-----------
While the classic RPCA process batch data, recent works proposed online RPCA schemes for more efficient and scalable MOD [@zhan2016online; @javed2018moving; @shi2018robust; @feng2013online; @lois2015online]. The idea is to decompose the nuclear norm of batch RPCA [@candes2011robust] to an explicit product of two low-rank matrices, then the objective function can be solved by stochastic optimization algorithm [@feng2013online], i.e., the coefficient and sparse component of each frame are updated by the previous basis, and then the basis is updated alternately. Comparing to the batch RPCA, the online RPCA has lower latency, thus is more suitable for video MOD.
Laplacian Scale Mixture (LSM) Model
-----------------------------------
LSM model has demonstrated its potential for sparse signal modeling in recent works [@garrigues2010group; @huang2017mixed; @dong2018robust]: In [@garrigues2010group], LSM model has been used to model the dependencies among sparse coding coefficients for image coding and compressive sensing recovery. The basic idea of LSM model is to model each sparse pixel as a product of a random Laplacian variable and a positive hidden multiplier, and impose a hyperprior (i.e., the Jeffrey prior [@gep1973bayesian] used in this paper) over the positive hidden multiplier. These models allow to jointly estimate both sparse pixels and hidden multipliers from the observed data under the MAP estimation framework via alternative optimization. In [@huang2017mixed], the LSM model was used for mixed noise removal, where the impulse noise is modeled with LSM distributions. In [@dong2018robust], the tensor coefficients were modeled with LSM for multi-frame image and video denoising. By jointly estimating the variances and recovering the values of coefficients, LSM-based methods significantly improved the modeling accuracy from plain sparse coding with more flexibility and robustness.
Maximum Correntropy Criterion (MCC)
-----------------------------------
Correntropy [@liu2007correntropy] is known for more effectively modeling error or noise beyond Gaussian distribution in practice [@guo2017godec+]. It models a nonlinear similarity between two random variables $X$ and $Y$ as $V_{\sigma}(X,Y)=E[k_{\sigma}(X,Y)]=\int k_{\sigma}(x,y)dF_{XY}(x,y)$, where $E[\cdot]$ is the expectation operator, $F_{XY}(x,y)$ is the joint distribution function of $(X,Y)$, and $k_{\sigma}(\cdot)$ is the kernel function, we select Gaussian kernel as the kernel function, so $k_{\sigma}(e)=g_{\sigma}(e)=\exp(-e^{2}/(2\sigma^{2}))$, where $e=x-y$, $\sigma$ is the kernel width and controls the radial range of the Gaussian kernel function, and $\exp(\cdot)$ denotes the exponential operation on each element of the input parameters. As the joint distribution is mostly unknown in practice, the correntropy of $(X,Y)$ can be approximated by $\hat{V}_{n,\sigma}(X,Y)=(1/n)\sum\nolimits_{i=1}^{n}\left[ g_{\sigma}(x_{i}-y_{i}) \right]$, given a finite number of samples $\{ (x_{i},y_{i}) \}_{i=1}^{n}$, and accordingly the MCC is equivalent to minimizing $(1/n)\sum\nolimits_{i=1}^{n}\sigma^{2}(1-g_{\sigma}(x_{i}-y_{i}))$ [@he2019robust]. Recent works applied MCC for a wide range of important applications, including face recognition, matrix completion, and low-rank matrix decomposition [@he2011robust; @guo2017godec+; @he2019robust; @he2010maximum; @du2017robust; @lu2013correntropy], which is shown to be highly effective.
Hyper RPCA Learning Formulation {#sec.Hyper RPCA}
===============================
We first demonstrate the major limitations of classic RPCA and then introduce to the LSM and MCC, based on which we propose the Hyper RPCA formulation.
![Visual results of different methods for foreground detection. (a) Input images, (b) PCP [@candes2011robust], (c) ORPCA [@feng2013online] and (d) the proposed Hyper RPCA method. From top to bottom: the 1541th frame of WaterSurface sequence from I2R dataset [@li2004statistical], the 1248th frame of WavingTrees sequence from Wallflower dataset [@toyama1999wallflower], the 301th frame of Escalator sequence from I2R dataset, the 1220th frame of lakeSide sequence from CDnet dataset [@wang2014cdnet], the 253th frame of Camouflage sequence from Wallflower dataset. \[fig.1\] ](LSM_foreground.pdf){width=".95\linewidth"}
Preliminary
-----------
The classic online RPCA method [@feng2013online] assumes that background is low-rank and foreground is sparse simultaneously. It utilizes $\ell_{2}$-norm and $\ell_{1}$-norm for modeling approximation error, and the sparsity of the foreground, respectively. The formulation of the classic online RPCA is the following
$$\label{eq.1}
\begin{split}
(\bm{U},\bm{V},\bm{S})=&\mathop{\arg\min}_{\bm{U},\bm{V},\bm{S}}
\|\bm{D}-\bm{UV}^\top-\bm{S}\|_{F}^{2} + \\
&\eta\{ \|\bm{U}\|_F^2 + \|\bm{V}\|_F^2 \} +
\lambda\|\bm{S}\|_1\,\text{,}
\end{split}$$
where $\bm{D}=[\bm{d}_1,\bm{d}_2,\cdots,\bm{d}_T]\in\mathbb{R}^{p \times T}$ is the data matrix of $T$ frames, i.e., $\bm{d}_t\in\mathbb{R}^p$ denotes the $t$-th frame, and $p=m \times n$ denotes the frame size. The foreground and background components are denoted as $\bm{L}=\bm{UV}^\top=[\bm{l}_1,\bm{l}_2,\cdots,\bm{l}_T]\in\mathbb{R}^{p \times T}$ and $\bm{S}=[\bm{s}_1,\bm{s}_2,\cdots,\bm{s}_T]\in\mathbb{R}^{p \times T}$, respectively. As the background is assumed low-rank, $\bm{U}\in\mathbb{R}^{p \times r}$ and $\bm{V}\in\mathbb{R}^{T \times r}$are used to represent the background basis and coefficients, respectively, with $r \ll p,T$, thus, $\bm{L=UV}^\top$ is the low-rank approximation of the background. In practice, the RPCA assumption sometimes becomes too restrictive:
###
The foreground may not be sufficiently sparse in the spatial domain. Fig. \[fig.1\] shows the foreground extraction results of some example video frames from three challenging datasets [@wang2014cdnet; @li2004statistical; @toyama1999wallflower], using PCP [@candes2011robust], ORPCA [@feng2013online] and the proposed Hyper RPCA method. Comparing to the proposed Hyper RPCA using more complicated LSM, the sparse modeling based on $\ell_{1}$-norm fails to capture the complete foreground objects effectively, e.g., dynamic background (rows 1 and 2), small moving objects (rows 3 and 4) or large-area foreground objects relative to the background (row 5) in which the moving objects is not sparse spatially.
###
The real video data is often corrupted by complicated or hybrid types of noise, thus the modeling error is deviated from Gaussian distribution (which is usually modeled using $\ell_{2}$-norm). Fig. \[fig.2\] presents an analysis of the modeling error distribution of an example video [@wang2014cdnet] (a) using the classic RPCA-based method [@candes2011robust; @feng2013online]. Fig. \[fig.2\] (b) and (c) plot their empirical distribution of modeling error, respectively, which both deviate from Gaussian distribution.
Hyper RPCA for moving object detection
--------------------------------------
We propose the novel Hyper RPCA, to tackle the aforementioned limitations of the classic online RPCA-based method, by jointly applying MCC and LSM models. The batch learning formulation of Hyper RPCA is the following
$$\label{eq.2}
\begin{split}
\mathop{\arg\min}_{\bm{W},\bm{U},\bm{V},\bm{B},\bm{A}}
&\|\bm{W}^{1/2}\odot(\bm{D}-\bm{UV}^\top-\bm{B}\odot\bm{A})\|_F^2 \\
& + \eta\sigma_{w}^2 \{ \|\bm{U}\|_F^2 + \|\bm{V}\|_F^2 \} +
2\sigma_{w}^2\sum\nolimits_{t}\sum\nolimits_{i}\vert\alpha_{i,t}\vert \\
& + 4\sigma_{w}^2\sum\nolimits_{t}\sum\nolimits_{i}\log(b_{i,t}+\varepsilon)\,\text{,}
\end{split}$$
where $\bm{B}=[\bm{b}_1,\bm{b}_2,\cdots,\bm{b}_T]\in\mathbb{R}^{p \times T}$ denotes the matrix of positive hidden multipliers $b_{i,t}$, similarly, $\bm{A}=[\bm{\alpha}_1,\bm{\alpha}_2,\cdots,\bm{\alpha}_T]\in\mathbb{R}^{p \times T}$ is the matrix representation of the Laplacian variables $\alpha_{i,t}$, $\bm{W}$ is the weight matrix, $\sigma_w^2$ is the variance of the modeling error, $\varepsilon$ is a small constant for numerical stability, and $\odot$ denotes element-wise multiplication of two matrices. In addition, the online solution for the proposed method Eq. \[eq.2\] can be reformulated as
$$\label{eq.3}
\begin{split}
\mathop{\arg\min}_{\bm{W},\bm{U},\bm{V},\bm{B},\bm{A}}
&\sum\nolimits_{i=1}^{T} \{
\|\bm{w}_t\circ(\bm{d}_t-\bm{Uv}_t-\bm{b}_t\circ\bm{\alpha}_t)\|_2^2 \\
& + \eta\sigma_{w}^2\|\bm{v}_t\|_2^2
+ 2\sigma_{w}^2\sum\nolimits_{i}\vert\alpha_{i,t}\vert \\
& + 4\sigma_{w}^2\sum\nolimits_{i}\log(b_{i,t}+\varepsilon) \}
+ \eta\sigma_{w}^2\|\bm{U}\|_F^2\,\text{,}
\end{split}$$
where $\bm{b}_t=[b_{1,t},b_{2,t},\cdots,b_{p,t}]^\top\in\mathbb{R}^p$, $\bm{\alpha}_t=[\alpha_{1,t},\alpha_{2,t},\cdots,\alpha_{p,t}]^\top\in\mathbb{R}^p$, $\bm{v}_t$ and $\bm{w}_t$ are the $t$-th column of $\bm{B}$, $\bm{A}$, $\bm{V}^\top$ and $\bm{W}^{1/2}$, respectively, and $\circ$ denotes element-wise multiplication of two vectors.
LSM for foreground modeling
---------------------------
In realistic scenarios, the prior distribution of foreground $P(\bm{S})$ is unknown and it is difficult to estimate. In LSM modeling, each foreground pixel $s_{i,t}$ is expressed by $s_{i,t}=b_{i,t}\cdot\alpha_{i,t}$, where $\alpha_{i,t}$ is a random Laplacian variable, and $b_{i,t}$ is a positive hidden multiplier. $s_{i,t}$ denotes the $i$-th pixel of $\bm{s}_t$, and is modeled with a zero-mean Laplacian distribution with standard deviation $b_{i,t}$, i.e., $P(s_{i,t} \vert b_{i,t})=(1/2b_{i,t})\exp(-\vert s_{i,t} \vert / b_{i,t})$. A hyper prior $P(b_{i,t})$ is further used to model $b_{i,t}$. Then, the LSM model of $s_{i,t}$ can be expressed as $P(s_{i,t})=\int_{0}^{\infty}P(s_{i,t} \vert b_{i,t})P(b_{i,t})db_{i,t}$, which cannot be expressed in an analytical form in general. Thus, the estimation of $\bm{L}$ and $\bm{S}$ from $\bm{D}$ are considered in the maximum a posterior (MAP) estimator as
$$\label{eq.4}
\begin{split}
(\bm{L},\bm{S},\bm{B})=\mathop{\arg\max}
&\log P(\bm{D} \vert \bm{L},\bm{S}) + \log P(\bm{L}) \\
& + \log P(\bm{S} \vert \bm{B}) + \log P(\bm{B})\,\text{,}
\end{split}$$
where $P(\bm{D} \vert \bm{L},\bm{S})$ is the Gaussian likelihood term with mean of zero and variance of $\sigma_w^2$ and $\bm{B}$ denotes the matrix of positive hidden multipliers. In this paper, we use the noninformative Jeffrey’s prior $P(b_{i,})=1/b_{i,t}$ to model hidden variable $b_{i,t}$ [@gep1973bayesian], and the prior of $\bm{L}$ is modeled as $P(L)\propto\exp(-\eta \|\bm{L}\|_*)$. By assuming $b_{i,t}$ and $s_{i,t}$ are independent, and $\bm{S}$ is i.i.d., Eq. \[eq.4\] can be rewritten as
$$\label{eq.5}
\begin{split}
(\bm{L},\bm{B},\bm{A})=&\mathop{\arg\min}_{\bm{L},\bm{B},\bm{A}}
\|\bm{D}-\bm{L}-\bm{B}\odot\bm{A}\|_F^2 \\
& + 2\eta\sigma_w^2\|\bm{L}\|_*
+ 2\sigma_w^2\sum\nolimits_{t}\sum\nolimits_{i}\vert \alpha_{i,t} \vert \\
& + 4\sigma_w^2\sum\nolimits_{t}\sum\nolimits_{i}\log(b_{i,t}+\varepsilon)
\,\text{,}
\end{split}$$
where $\bm{S}=\bm{B}\odot\bm{A}$ denotes sparse component, and $\bm{A}$ is the matrix representation of the Laplacian variables.
MCC for error modeling
----------------------
To improve the performance of classic online RPCA-based background subtraction method to deal with non-Gaussian modeling errors, we model this part with correntropy instead of $\ell_2$-norm and Eq. \[eq.1\] can be rewritten as
$$\label{eq.6}
\begin{split}
(\bm{U},\bm{V},\bm{S})=&\mathop{\arg\min}_{\bm{U},\bm{V},\bm{S}}
\sigma^2[1-g_{\sigma}(\bm{D-L-S})] \\
& + \eta\{ \|\bm{U}\|_F^2 + \|\bm{V}\|_F^2 \} + \lambda\|\bm{S}\|_1\,\text{,}
\end{split}$$
where $g_{\sigma}(\cdot)$ denotes the Gaussian kernel operation on each element of the input parameters. Based on the Half-Quadratic (HQ) optimization theory [@nikolova2005analysis], Eq. \[eq.6\] becomes a weighted matrix factorization problem that can be solved by the strategy used in [@he2019robust]. Then, Eq. \[eq.6\] can be rewritten as
$$\label{eq.7}
\begin{split}
(\bm{W},\bm{U},\bm{V},\bm{S})=&\mathop{\arg\min}_{\bm{W},\bm{U},\bm{V},\bm{S}}
\|\bm{W}^{1/2}\odot(\bm{D-L-S})\|_F^2 + \\
& \eta\{ \|\bm{U}\|_F^2 + \|\bm{V}\|_F^2 \} + \lambda\|\bm{S}\|_1\,\text{,}
\end{split}$$
where $\bm{W}$ is the weight matrix, and its values can be obtained by the modeling error [@he2019robust].
Algorithm {#sec.algorithm}
=========
We propose an efficient algorithm solving Eq. \[eq.3\] by an alternating minimization, the proposed algorithm calculate frame by frame, detect moving objects and gradually ameliorate the background based on the real-time video variations. We describe each sub-problem as follows.
Solving the W-Subproblem
------------------------
Based on the optimization theory in [@he2019robust; @nikolova2005analysis], when the background $\bm{l}_t=\bm{UV}_t$ and $\bm{s}_t=\bm{b}_t \circ \bm{\alpha}_t$ are fixed, for each frame, the optimal solutions of $\bm{w}_t$ can be obtained as
$$\label{eq.8}
\bm{w}_t=sqrt(g_\sigma(\bm{d}_t-\bm{l}_t-\bm{s}_t))\,\text{,}$$
where $sqrt(\cdot)$ denotes the square root operation on each element of input parameters. For a detailed derivation process, please see [@he2019robust] for optimizing $\bm{w}_t$.
Solving the V-Subproblem
------------------------
For fixed $\bm{W}$, $\bm{S}$ and $\bm{U}$, the $\bm{V}$-subproblem can be formulated as
$$\label{eq.9}
\begin{split}
\bm{V}=\mathop{\arg\min}_{\bm{V}} \sum\nolimits_{t=1}^{T}
& \{ \|\bm{w}_t\circ(\bm{d}_t-\bm{Uv}_t-\bm{s}_t)\|_2^2 \\
& + \eta\sigma_w^2\|\bm{v}_t\|_2^2 \}\,\text{,}
\end{split}$$
For each frame, $\bm{v}_t$ can be obtained by
$$\label{eq.10}
\bm{v}_t=\mathop{\arg\min}_{\bm{v}_t}
\|\bm{w}_t\circ(\bm{d}_t-\bm{Uv}_t-\bm{s}_t)\|_2^2
+ \eta\sigma_w^2\|\bm{v}_t\|_2^2\,\text{,}$$
which has the closed-form solution as
$$\label{eq.11}
\bm{v}_t=[\bm{U}^\top diag(\bm{w}_t)^2 \bm{U} + \eta\sigma_w^2\bm{I}]^{-1}
\bm{U}^\top diag(\bm{w}_t)^2 (\bm{d}_t-\bm{s}_t)\,\text{,}$$
where $diag(\cdot)$ denotes the diagonalization operation for vector, and $\bm{I}$ is an $r \times r$ identity matrix.
Solving the S-Subproblem
------------------------
For fixed $\bm{W}$ and $\bm{L}$, $\bm{S}$ can be obtained by solving the following optimization problem
$$\label{eq.12}
\begin{split}
(\bm{B},\bm{A})=&\mathop{\arg\min}_{\bm{B},\bm{A}} \sum\nolimits_{t=1}^{T}
\{ \|\bm{w}_t\circ(\bm{d}_t-\bm{Uv}_t-\bm{s}_t)\|_2^2 \\
& + 2\sigma_w^2\sum\nolimits_{i}\vert \alpha_{i,t} \vert
+ 4\sigma_w^2\sum\nolimits_{i}\log(b_{i,t}+\varepsilon) \}
\,\text{,}
\end{split}$$
### Solving for $b_t$
For each frame, while $\bm{\alpha}_t$ is fixed, $\bm{b}_t$ can be obtained as
$$\label{eq.13}
\begin{split}
\bm{b}_t&=\mathop{\arg\min}_{\bm{b}_t}
\|\bm{w}_t\circ(\bm{d}_t-\bm{l}_t-\bm{b}_t\circ\bm{\alpha}_t)\|_2^2 \\
& \quad + 4\sigma_w^2 \sum\nolimits_{i}\log(b_{i,t}+\varepsilon) \\
&=\mathop{\arg\min}_{\bm{b}_t} \sum\nolimits_{i} \{
(\sqrt{w_{i,t}}(d_{i,t}-l_{i,t}-b_{i,t}\alpha_{i,t}))^2 \\
& \quad + 4\sigma_w^2\log(b_{i,t}+\varepsilon) \}\,\text{,}
\end{split}$$
Moreover, each $b_{i,t}$ can be solved independently by
$$\label{eq.14}
\begin{split}
b_{i,t}=&\mathop{\arg\min}_{b_{i,t}}
(\sqrt{w_{i,t}}(d_{i,t}-l_{i,t}-b_{i,t}\alpha_{i,t}))^2 \\
& + 4\sigma_w^2\log(b_{i,t}+\varepsilon)\,\text{,}
\end{split}$$
Though Eq. \[eq.14\] is nonconvex, the closed-form solution can be obtained by taking $\mathrm{d}f(b_{i,t}) / \mathrm{d}b_{i,t}=0$, where $f(b_{i,t})$ is the right side of Eq. \[eq.14\]. The solution of Eq. \[eq.14\] is given by
$$\label{eq.15}
b_{i,t}=
\begin{cases}
0&,\text{if $(2a\varepsilon+h)^2/(16a^2)-
(h\varepsilon+q)/(2a)<0$} \\
T_{i,t}&,\text{otherwise}\,\text{,}
\end{cases}$$
where $a=(\sqrt{w_{i,t}}\alpha_{i,t})^2$, $h=-2w_{i,t}(d_{i,t}-l_{i,t})\alpha_{i,t}$, $q=4\sigma_w^2$ and $T_{i,t}=\min\{f(0),f(b^{\ast})\}$, where $b^{\ast}$ is the stationary point of $f(b_{i,t})$, which is defined as
$$\label{eq.16}
b^{\ast}=-\frac{2a\varepsilon+h}{4a}\pm\
\sqrt{\frac{(2a\varepsilon+h)^2}{16a^2}-\frac{h\varepsilon+q}{2a}}
\,\text{,}$$
### Solving for $\alpha_t$
For fixed $\bm{b}_t$, $\bm{\alpha}_t$ can be solved by minimizing
$$\label{eq.17}
\begin{split}
\bm{\alpha}_t&=\mathop{\arg\min}_{\bm{\alpha}_t}
\|\bm{w}_t\circ(\bm{d}_t-\bm{l}_t-\bm{b}_t\circ\bm{\alpha}_t)\|_2^2 \\
& \quad + 2\sigma_w^2 \sum\nolimits_{i}\vert \alpha_{i,t} \vert \\
&=\mathop{\arg\min}_{\bm{\alpha}_t} \sum\nolimits_{i} \{
(\sqrt{w_{i,t}}(d_{i,t}-l_{i,t}-b_{i,t}\alpha_{i,t}))^2 \\
& \quad + 2\sigma_w^2\vert \alpha_{i,t} \vert \}
\,\text{,}
\end{split}$$
Similarly, each $\alpha_{i,t}$ can be solved independently by
$$\label{eq.18}
\alpha_{i,t}=\mathop{\arg\min}_{\alpha_{i,t}}
(\sqrt{w_{i,t}}(d_{i,t}-l_{i,t}-b_{i,t}\alpha_{i,t}))^2
+ 2\sigma_w^2\vert \alpha_{i,t} \vert
\,\text{,}$$
which admits a closed-form solution
$$\label{eq.19}
\alpha_{i,t}=S_{\tau_{i,t}}((d_{i,t}-l_{i,t})/(b_{i,t}+\varepsilon))
\,\text{,}$$
where $S_{\tau_{i,t}}(\cdot)$ is the soft-thresholding operator with threshold $\tau_{i,t}=2\sigma_w^2/(\sqrt{w_{i,t}}b_{i,t}+\varepsilon)^2$. Finally, $\bm{s}_t$ can be computed by $\bm{s}_t=\bm{b}_t\circ\bm{\alpha}_t$ once $\bm{b}_t$ and $\bm{\alpha}_t$ are obtained.
Initialization: $\bm{U}^{(0)}$ using ($\ref{eq.23}$),$\bm{L}$,$\bm{S}$.\
Solving the U-Subproblem
------------------------
Similar to [@feng2013online], we use the online learning method to update $\bm{U}$. After estimating $\bm{w}_t$, $\bm{s}_t$ and $\bm{v}_t$, the $\bm{U}$ of $t$-th frame can be updated as
$$\label{eq.20}
\begin{split}
\bm{U}^{(t)}&\triangleq\mathop{\arg\min}_{\bm{U}}
\frac{1}{t}\sum\nolimits_{i=1}^t\frac{1}{2}
\|\bm{w}_i\circ(\bm{d}_i-\bm{Uv}_i-\bm{s}_i)\|_2^2 \\
& \quad + \frac{\eta\sigma_w^2}{2t}\|\bm{U}\|_F^2 \\
&\triangleq\mathop{\arg\min}_{\bm{U}}
\frac{1}{2}Tr[\bm{U}^\top(\bm{C}_t+\eta\sigma_w^2\bm{I})\bm{U}]
- Tr(\bm{U}^\top\bm{F}_t)
\,\text{,}
\end{split}$$
where $Tr(\cdot)$ denotes the trace of matrix. $\bm{C}_t$ and $\bm{F}_t$ are updated as
$$\label{eq.21}
\begin{split}
&\bm{C}_t\leftarrow\bm{C}_{t-1}+\bm{v}_t^{'}\bm{v}_t^{'\top} \\
&\bm{F}_t\leftarrow\bm{F}_{t-1}+(\bm{d}_t^{'}-\bm{s}_t^{'})\bm{v}_t^{'\top}\,\text{,}
\end{split}$$
where $\bm{C}_0=\bm{0}$, $\bm{F}_0=\bm{0}$, $\bm{d}_t^{'}=diag(\bm{w}_t)\bm{d}_t$, $\bm{s}_t^{'}=diag(\bm{w}_t)\bm{s}_t$, and $\bm{v}_t^{'}=(\bm{U}^\top \bm{U})^{-1}\bm{U}^\top diag(\bm{w}_t)\bm{Uv}_t$. In practice, the $i$-th column $\bm{u}_i$ of $\bm{U}$ can be updated individually while keeping other columns fixed [@bertsekas1999nonlinear] as
$$\label{eq.22}
\bm{u}_i^{(t)}\leftarrow\bm{u}_i^{(t-1)}+\frac{1}{\tilde{\bm{c}}_{i,i}}
(\bm{f}_i-\bm{U}^{(t-1)}\tilde{\bm{c}}_{i})\,\text{,}$$
where $\tilde{\bm{C}}=\bm{C}_t+\eta\sigma_w^2\bm{I}$ and $\tilde{\bm{c}}_i$ and $\bm{f}_i$ are the $i$-th column of $\tilde{\bm{C}}$ and $\bm{F}_t$, respectively. Before foreground extraction, we initialize the background by taking the median of pixel values of the first several frames. Then, the basis $\bm{U}_0$ can be initialized by the bilateral random projection method proposed in [@zhou2012bilateral] as
$$\label{eq.23}
\bm{U}^{(0)}=\bm{A}_1(\bm{R}_2^{\top}\bm{A}_1)^{-1}\bm{A}_2^\top
\,\text{,}$$
where $\bm{R}_1\in\mathbb{R}^{n \times r}$ and $\bm{R}_2\in\mathbb{R}^{m \times r}$ denote two bilateral Gaussian random projections, $\bm{A}_1=\bm{AR}_1$, $\bm{A}_2=\bm{A}^{\top}\bm{R}_2$ and $\bm{A}\in\mathbb{R}^{m \times n}$ denotes matrix of the initial background. In summary, the proposed Hyper RPCA is summarized in Algorithm \[alg:1\].
[0.97]{}[ccccccccccc]{} &Corridor&DRoom&Library&Canoe&WSurface& Skating&Blizzard&TunnelExit&TimeOfDay&Fountain02\
&(26.53/0.96)&(26.35/0.95)&(20.20/0.90)&(24.20/0.63)&(26.97/0.80)& (20.78/0.96)&(46.65/0.99)&(41.97/0.99)&(39.91/0.99)&(33.52/0.93)\
[GoDec+]{}&(26.97/0.96)&(26.39/0.95)&(19.05/0.89)&(24.24/0.63)&(25.59/0.79)& (21.08/0.96)&(47.19/0.99)&(43.10/0.99)&(39.89/0.99)&(32.95/0.93)\
[GRASTA]{}&(30.82/0.90)&(23.19/0.86)&(19.44/0.75)&(21.25/0.55)&(22.50/0.71)& (20.52/0.84)&(37.90/0.95)&(34.59/0.95)&(33.74/0.94)&(27.14/0.86)\
[incPCP]{}&(33.57/0.95)&(28.76/0.94)&(23.68/0.90)&(23.28/0.57)&(27.12/0.77)& (24.74/0.95)&(45.39/0.99)&(41.54/0.98)&(40.40/0.98)&(32.33/0.91)\
[noncvx-RPCA]{}&(32.31/0.95)&(26.40/0.95)&(18.49/0.87)&(24.24/0.63)& (27.93/$\bm{0.82}$)&(23.72/0.93)&($\bm{47.85}$/$\bm{0.99}$)& (42.44/$\bm{0.99}$)&(40.30/0.99)&($\bm{33.93}$/$\bm{0.93}$)\
[ORPCA]{}&(36.07/0.94)&(28.65/0.93)&(26.93/0.89)&(26.08/0.65)&(29.09/0.76)& (26.63/0.80)&(44.39/0.97)&(36.13/0.96)&(38.37/0.97)&(29.90/0.89)\
[PRMF]{}&(36.70/0.91)&(29.74/0.96)&(16.01/0.69)&(21.09/0.62)&(24.94/0.74)& (24.38/0.86)&(35.23/0.84)&(38.51/0.97)&($\bm{45.16}$/0.95)&(33.23/0.93)\
[Hyper RPCA]{}&($\bm{46.59}$/$\bm{0.99}$)&($\bm{44.98}$/$\bm{0.98}$)& ($\bm{44.21}$/$\bm{0.98}$)&($\bm{26.38}$/$\bm{0.70}$)&($\bm{29.84}$/0.78)& ($\bm{38.45}$/$\bm{0.97}$)&(46.43/0.99)&($\bm{43.92}$/0.99)&(40.68/$\bm{0.99}$)& (33.28/0.92)\
{width=".80\linewidth"}
Experiments {#sec.experiments}
===========
The major parameters of the proposed Hyper RPCA were set as, $r=25$ and $\sigma=1 \times 10^{+3}$. $\sigma_w^2$ was set in the range of $[1,10] \times 10^{-5}$ according to the differences of pixel intensity between the foreground and background.
We tested the proposed Hyper RPCA on 16 representative video sequences, including 11 challenging clips selected from CDnet [@wang2014cdnet] (Canoe, Fountain02 and Overpass sequences in “Dynamic Background” category, Boulevard and Traffic sequences in “Camera Jitter” category, Corridor, DiningRoom (DRoom) and Library sequences in “Thermal” category, Blizzard and Skating sequences in “Bad Weather” category, TunnelExit\_0\_35fps (TunnelExit) sequence in “Low Framerate” category), 3 sequences (Hall, Lobby and WaterSurface (WSurface)) from I2R [@li2004statistical] and 2 sequences (ForegroundAperture (FAperture) and TimeOfDay) from Wallflower [@toyama1999wallflower] dataset, respectively. Since the offline methods require all the consecutive frames, which results in a large computational cost, only 200 consecutive frames from each test video were chosen for our experiment.
The proposed method was implemented on Matlab platform. For all of the competing methods, we used the publicly available codes from their official websites, with the default parameters. All the experiments in this paper were performed on a PC with 2.3GHZ Intel Core i5 processor and 8GB of RAM.
Experimental Results on Background extraction
---------------------------------------------
To verify the performance of background extraction, the proposed Hyper RPCA was compared with 7 state-of-art-methods. The utilized offline methods include: noncvx-RPCA [@kang2015robust], PRMF [@wang2012probabilistic], GoDec [@zhou2011godec] and GoDec+ [@guo2017godec+], and online methods include: incPCP [@rodriguez2016incremental], GRASTA [@he2012incremental], ORPCA [@feng2013online]. Peak signal-to-noise ratio (*PSNR*) and structural similarity index (*SSIM*) [@wang2004image] were used as the quantitative metrics for the background extraction result. Ground-truth background images of static videos are obtained by averaging all background frames which exclude the foreground part. Table \[tab.1\] shows the average *PSNR* and *SSIM* values of different methods for some video sequences without camera jitter. (In this paper, bold number shows the best result in each Table.) The proposed Hyper RPCA achieves highest scores in 7 sequences in terms of *PSNR* and 6 sequences in terms of *SSIM*, and competitive performance in the rest.
[0.97]{}[cccccccccccccc]{} &Blizzard&Corridor&DRoom&TunnelExit&Library&TimeOfDay&Boulevard& Canoe&Fountain02&Skating&Traffic&WSurface&Average\
&$\bm{0.97}$&0.98&0.93&0.83&0.98& 0.78&0.76&0.80&0.89&0.96&0.44&0.91&0.85\
[PAWCS]{}&0.77&0.97&0.95&$\bm{0.86}$&0.98& 0.44&0.57&0.86&0.91&0.97&0.46&0.85&0.80\
[WeSamBE]{}&0.76&0.75&0.61&0.77&0.66& $\bm{0.85}$&0.73&0.72&0.81&0.75&0.75&0.85&0.75\
[COROLA]{}&0.90&0.98&0.94&0.84&0.99& 0.66&0.81&$\bm{0.87}$&$\bm{0.92}$&0.97&0.74&$\bm{0.92}$&0.88\
[OPRMF]{}&0.86&0.85&0.86&0.71&0.84& 0.42&0.75&0.73&0.85&0.82&0.73&0.83&0.77\
[SOBS]{}&0.61&0.96&0.90&0.38&0.99& 0.51&0.45&0.71&0.81&0.96&0.58&0.87&0.73\
[ViBe]{}&0.58&0.94&0.83&0.53&0.98& 0.41&0.35&0.52&0.68&0.95&0.57&0.86&0.68\
[GMM]{}&0.90&0.72&0.49&0.63&0.50& 0.56&0.41&0.42&0.68&0.82&0.58&0.41&0.59\
[DECOLOR]{}&0.78&0.96&0.94&0.59&0.98& 0.51&$\bm{0.93}$&0.41&0.81&$\bm{0.98}$&0.79&0.84&0.79\
[RegL1]{}&0.94&0.93&0.73&0.76&0.94& 0.58&0.74&0.72&0.85&0.95&0.75&0.90&0.81\
[MAMR]{}&0.94&0.93&0.73&0.76&0.92& 0.58&0.74&0.70&0.85&0.95&0.75&0.90&0.80\
[MoG-RPCA]{}&0.94&0.83&0.70&0.76&0.66& 0.58&0.76&0.59&0.85&0.88&0.76&0.82&0.76\
[Hyper RPCA]{}&0.94&$\bm{0.99}$&$\bm{0.96}$&0.76&$\bm{0.99}$& 0.77&0.88&0.76&0.89&0.97&$\bm{0.84}$&0.87&$\bm{0.89}$\
{width=".95\linewidth"}
Several representative background extraction visual results from the most challenging test videos are demonstrated in Fig. \[fig.3\], from which we can see that the results of the proposed Hyper RPCA have superior performance over other methods. The extracted backgrounds by the proposed Hyper RPCA are obviously closer to the ground-truth, while other methods produce ghosting artifacts in different degrees.
Since the ground-truth is difficult to estimate for Canoe and Fountain02, which have dynamic background, the *PSNR* and *SSIM* values of different methods are close. For Blizzard, TunnelExit, TimeOfDay and Fountain02, the moving objects run fast in the scenes, all the methods can estimate the background well. However, when it comes to moving slowly objects, for example, the walking men in Corridor and running boats in Canoe, which move through an area of frame and shade background for a long time, the results from other methods contain severe ghosting artifacts. Our method generates much cleaner results than others, which demonstrates the potential of the proposed method for this situation.
Experimental Results on Moving object detection
-----------------------------------------------
While the background is extracted, the moving objects can be obtained via background subtraction. To demonstrate the effectiveness of the proposed Hyper RPCA, we compared our method with 12 state-of-the-art methods for foreground detection, including four offline methods: RegL1 [@zheng2012practical], MAMR [@ye2015foreground], MoG-RPCA [@zhao2014robust], DECOLOR [@zhou2012moving], and eight online methods: SuBSENSE [@st2014subsense], PAWCS [@st2016universal], WeSamBE [@jiang2017wesambe], COROLA [@shakeri2016corola], OPRMF [@wang2012probabilistic], SC-SOBS [@maddalena2012sobs], ViBe [@barnich2010vibe], GMM [@zivkovic2004improved]. The foreground detection result is assessed using *F-measure* which is defined as
$$\label{eq.24}
\textit{F-measure}=2 \times
\frac{\textit{precision} \times \textit{recall}}
{\textit{precision} + \textit{recall}}
\,\text{.}$$
where *precision*=*TP*/(*TP*+*FP*) and *recall*=*TP*/(*TP*+*FN*). True Positives (*TP*) denotes the number of pixels correctly classified as foreground objects, False Positives (*FP*) represents the number of pixels incorrectly classified as foreground object, and False Negatives (*FN*) is the number of pixels incorrectly classified as background. Table \[tab.2\] shows the quantitative results of different methods on some test sequences, from which we can see that our proposed Hyper RPCA obtains the highest average *F-measure*. Although in some scenarios, our method did not achieve the best score, the gaps between ours and the first places are inconspicuous.
[0.97]{}[cccccccccccc]{} Category&SuBSENSE&SGSM-BS-block&PCP&SC-SOBS&GMM&COROLA& DECOLOR&GRASTA&incPCP&OMoGMF+TV&Hyper RPCA\
Baseline&$\bm{0.95}$&0.93&0.60&0.92&0.91&0.85& 0.92&0.66&0.81&0.85&0.94\
Dynamic Background&0.81&0.83&0.34&0.65&0.54&$\bm{0.86}$& 0.70&0.35&0.71&0.76&0.76\
Camera Jitter&0.81&0.81&0.54&0.82&0.56&0.82& 0.77&0.43&0.78&0.78&$\bm{0.83}$\
Shadow&$\bm{0.89}$&0.86&0.51&0.82&0.73&0.78& 0.83&0.52&0.74&0.68&0.87\
Thermal&0.81&$\bm{0.82}$&0.34&0.68&0.65&0.80& 0.70&0.42&0.70&0.70&0.80\
Intermittent Object Motion&0.65&0.70&0.32&0.59&0.53&0.71& 0.59&0.35&0.75&0.71&$\bm{0.77}$\
Bad Weather&$\bm{0.86}$&0.80&0.76&0.66&0.54&0.78& 0.76&0.68&N/A&0.78&0.83\
Low Framerate&0.64&0.73&0.44&0.55&0.50&N/A& 0.50&N/A&N/A&N/A&$\bm{0.75}$\
Average&0.80&0.81&0.48&0.71&0.62&0.80& 0.72&0.48&0.74&0.75&$\bm{0.81}$\
{width="0.95\linewidth"}
Fig. \[fig.4\] shows several visual results of foreground detection generated by the test methods on some test videos. In Fig. \[fig.4\], it can be seen that for indoor scenarios, such as Corridor and DRoom (rows 1 and 2), where people keep moving slowly across many frames, PAWCS, DECOLOR and Hyper RPCA perform better than other methods, and other methods fail to accurately detect the objects due to the pollution of the background. For the scenarios with camera jitter, such as Boulevard and Traffic (rows 3 and 4), whose background changes all the time, only WeSamBE, MOG-RPCA and Hyper RPCA can detect the foreground objects accurately, and in contrast, other methods, such as SuBSENSE and PAWCS, misclassified the background as foreground. For the scenarios with dynamic background, such as Canoe and WSurface (rows 6 and 7), DECOLOR and WeSamBE fail to detect the complete foreground part, in contrast, the shapes of people and boat are complete in the mask extracted by the proposed Hyper RPCA. For the outdoor scenario Skating (row 5), where people keep moving from the right side of the scene to the left, WeSamBE fails to detect the moving people on the right. It can be also noticed that COROLA performed well in each scenario, but the extracted masks are not as accurate as ours and it produced some fake positions in some results. Overall, the proposed Hyper RPCA achieves best visual performance on the test video sequences in all the methods.
[0.97]{}[ccccccccccccc]{} Dataset&COROLA&DECOLOR&TVRPCA&SRPCA&GRASTA&incPCP&OMoGMF+TV&3TD&RegL1&MAMR&SGSM-BS-block&Hyper RPCA\
I2R&$\bm{0.81}$&0.74&0.69&0.80&0.63&0.62&0.77&0.72&0.63&0.75&0.78&0.79\
Wallflower&0.75&0.59&0.61&0.85&0.33&N/A&0.82&0.75&N/A&0.80&N/A&$\bm{0.86}$\
{width=".95\linewidth"}
Experimental Results on Long Testing sequences
----------------------------------------------
To further demonstrate the performance of the proposed Hyper RPCA for foreground detection, we compared Hyper RPCA with other methods on three long sequence datasets, including CDnet [@wang2014cdnet], I2R [@li2004statistical] and Wallflower [@toyama1999wallflower]. The comparison methods include six offline methods: 3TD [@oreifej2012simultaneous], PCP [@candes2011robust], DECOLOR [@zhou2012moving], RegL1 [@zheng2012practical], TVRPCA [@cao2015total], MAMR [@ye2015foreground], and night online methods: SuBSENSE [@st2014subsense], GMM [@zivkovic2004improved], SC-SOBS [@maddalena2012sobs], COROLA [@shakeri2016corola], GRASTA [@he2012incremental], incPCP [@rodriguez2016incremental], OMoGMF+TV [@yong2017robust], SRPCA [@javed2016spatiotemporal], SGSM-BS-block [@shi2018robust].
Eight categories, including “Baseline”, “Dynamic Background”, “Camera Jitter”, “Shadow”, “Thermal”, “Intermittent Object Motion”, “Bad Weather” and “Low Framerate”, from CDnet 2014 dataset were tested. Table \[tab.3\] shows the quantitative results of all the methods. (In this paper, N/A indicates that the authors did not report the performance for these categories or datasets in their original references.) As shown in Table \[tab.3\], Hyper RPCA outperforms most of other methods and is competitive with SGSM-BS-block [@shi2018robust], SuBSENSE [@st2014subsense], WeSamBE [@jiang2017wesambe] and COROLA [@shakeri2016corola] methods, which can effectively deal with complex scenes in practice. I2R and Wallflower datasets consist of nine and six videos with complex background respectively. As shown in Table \[tab.4\], Hyper RPCA achieves the best performance on the Wallflower dataset on average. The performance of Hyper RPCA on the I2R dataset outperforms the other methods, except COROLA and SRPCA, which are state-of-the-art methods for moving object detection.
The visual results of eight typical methods and Hyper RPCA are shown in Fig. \[fig.5\]. In Fig. \[fig.5\], it can be seen that for scenario Sofa (row 3), where box was kept in a fixed place for a long time, only OMoGMF+TV, SGSM-BS-block and Hyper RPCA can detect foreground accurately, and other methods fail to detect the box wholly. For indoor scenario CopyMachine (row 2) and outdoor scenario Campus (row 6), GRASTA, incPCP and OMoGMF+TV misclassified the background as foreground. It can be noticed that the shape of people, for scenario WavingTrees (row 8), are complete in the mask extracted by Hyper RPCA, and other methods produced some fake positions in results. Overall, the foreground detection results of Hyper RPCA are closer to the ground-truth images.
Ablation Study
--------------
To verify the effectiveness of the proposed MCC and LSM regularization terms, we implemented three variants of the proposed model, i.e., the LSM-based method without MCC (denoted as LSM-ORPCA), the MCC-based method without LSM (denoted as MCC-ORPCA), and Hyper RPCA based on LSM and MCC. Some representative results are shown in Table \[tab.5\], from which we can see that LSM-ORPCA and Hyper RPCA significantly outperforms PCP and ORPCA. By utilizing MCC to model the error part, Hyper RPCA performs better than LSM-ORPCA. MCC-ORPCA, which uses $\ell_1$-norm to model foreground pixels, degrades while dealing with dynamic backgrounds. Experiment results demonstrate that LSM model has the advantage of characterizing the varying sparsity of the foreground pixels, and is more suitable for foreground modeling than $\ell_1$-norm in practice.
[0.47]{}[cccccccc]{} &Corridor&DRoom&Library&Boulevard&Skating&WSurface&Average\
PCP&0.74&0.79&0.69&0.85&0.75&0.60&0.73\
ORPCA&0.89&0.78&0.84&0.80&0.73&0.74&0.79\
MCC-ORPCA&0.99&0.96&0.97&0.52&0.97&$\bm{0.90}$&0.88\
LSM-ORPCA&0.99&$\bm{0.96}$&0.99&0.86&0.97&0.87&0.94\
Hyper RPCA&$\bm{0.99}$&0.96&$\bm{0.99}$&$\bm{0.88}$& $\bm{0.97}$&0.87&$\bm{0.94}$\
[0.47]{}[ccccc]{} &FAperture&Hall&Lobby&Overpass\
PCP&(14.18/0.77)&(27.43/0.93)&(28.98/0.95)&(24.14/0.86)\
ORPCA&(20.29/0.90)&(23.77/0.87)&(28.37/0.90)&(24.43/0.83)\
LSM-ORPCA&(19.73/0.83)&(25.22/0.89)&(27.84/0.85)&(25.78/0.84)\
MCC-ORPCA&(25.12/$\bm{0.97}$)&($\bm{32.25}$/$\bm{0.96}$)& (31.88/0.97)&($\bm{28.83}$/$\bm{0.91}$)\
Hyper RPCA&($\bm{29.14}$/0.97)&(31.83/0.96)& ($\bm{32.70}$/$\bm{0.97}$)&(28.61/0.91)\
In Table \[tab.6\] and Fig. \[fig.6\], the results demonstrate that MCC can handle non-Gaussian noise well. MCC-ORPCA can gain cleaner background than PCP and ORPCA. Original test sequences were corrupted by Poisson+Salt&Pepper noise (corruption percentage is set to 20%). By introducing LSM model, Hyper RPCA obtains the higher average *PSNR* (30.57dB) and *SSIM* (0.9580) values than MCC-ORPCA (29.52dB/0.9579). Similar to PCP and ORPCA, the background extraction results of LSM-ORPCA have few noise and artifacts.
Parameters Selection
--------------------
In the proposed method, three major parameters need to be tuned, including the setting of Gaussian kernel function $\sigma$, the bilateral random projections number $r$ and the variance of the Gaussian error $\sigma_w^2$. Fig. \[fig.7\] shows the average *F-measure*, precision and recall curves as functions of $\log_{10}(\sigma)$ and $\sigma_w^2$ on 12 test video sequences used in Table \[tab.2\]. From Fig. \[fig.7\] (a), we can see that the performance of the proposed method is insensitive to $\sigma$. From Fig. \[fig.7\] (b) and (c), we can see that the precision result increases and the recall result decreases with the value of $\sigma_w^2$ increases. In our implementation, we set $\sigma=1 \times 10^{+3}$ experimentally. $\sigma_w^2=1 \times 10^{-5}$ for the sequences with similar spatial homogeneity between foreground and background, or $\sigma_w^2=1 \times 10^{-4}$ for the opposite situation.
Fig. \[fig.8\] shows the *PSNR* and *F-measure* curves as functions of $r$ on 6 test video sequences. From Fig. \[fig.8\] (a), we can see that the result of background extraction is insensitive to $r$. From Fig. \[fig.8\] (b), it can be observed that as $r$ increases the performance is improved for the sequences with dynamic background. On the other hand, the performance for the sequences with stable background is insensitive to $r$. Considering the time cost and accuracy of foreground detection, we set $r=25$ in our experiments.
Computational Complexity
------------------------
The computational complexity of the proposed method mainly depends on the costs of the optimization schemes, including 1) estimating $\bm{w}_t$; 2) estimating $\bm{v}_t$; 3) estimating $\bm{s}_t$; and 4) updating $\bm{U}$. During one iteration per frame, the complexities of estimating $\bm{w}_t$, $\bm{v}_t$, $\bm{s}_t$ and $\bm{U}$ are $O(p)$, $O(pr^2)$, $O(p)$ and $O(pr^2)$ respectively. Thus, the total time complexity of the proposed online algorithm for a video sequence with $T$ frames is $O(Tpr^2)$, which is linearly proportional to the size and number of the frames. We also compared the performing time of the proposed method with some representative methods, including SuBSENSE [@st2014subsense], PAWCS [@st2016universal], WeSamBE [@jiang2017wesambe], DECOLOR [@zhou2012moving], MoG-RPCA [@zhao2014robust], and ORPCA [@feng2013online]. Table \[tab.7\] reports the average running time of different methods for 12 test video sequences, and the proposed method is quite competitive in running time.
[0.47]{}[ccccccc]{} SuBSENSE&PAWCS&WeSamBE&DECOLOR&MoG-RPCA&ORPCA&Hyper RPCA\
3.45&0.64&5.78&2.29&1.25&3.97&1.10\
Conclusion {#sec.conclusion}
==========
Although RPCA-based models have been successfully used for moving object detection, the intrinsic shortcomings of the $\ell_1$-norm and $\ell_2$-norm still need to be solved. In this paper, we proposed an online background subtraction model by integrating maximum correntropy criterion (MCC) and Laplacian scale mixture (LSM) models for moving object detection. The proposed Hyper RPCA applies correntropy as the error measurement to improve the robustness to non-Gaussian modeling error. Moreover, the LSM model is adopted to formulate foreground pixels of the moving object. Experimental results show that our algorithm has competitive performance to the state-of-the-art moving object detection methods and demonstrates powerful abilities in characterizing non-Gaussian errors and the sparsity of sparse foreground pixels. In the future, developing an adaptive parameter selection strategy for different complex scenarios will be considered.
[^1]: Corresponding authors: Bihan Wen and Yi Zhang.
[^2]: Z. Shao, Y. Pu, J. Zhou and Y. Zhang are with the College of Computer Science, Sichuan University, Chengdu 610065, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]).
[^3]: B. Wen is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]).
|
---
abstract: 'It has been recently found that the local fluctuations of the QSO’s Ly$\alpha$ absorption spectrum transmitted flux show spiky structures. This implies that the mass fields of the intergalactic medium (IGM) is intermittent. This feature cannot be explained by the clustering evolution of cosmic mass field in the linear regimes and is also difficult to incorporate into the hierarchical clustering scenario. We calculate the structure functions and intermittent exponent of the IGM and HI for full hydrodynamical simulation samples. The result shows the intermittent features of the Ly$\alpha$ transmitted flux fluctuations as well as the mass field of the IGM. We find that within the error bars of current data, all the intermittent behavior of the simulation samples are consistent with the observation. This result is different from our earlier result (Pando et al 2002), which shows that the intermittent behavior of samples generated by pseudo-hydro simulation cannot be fitted with observed data. One difference between the pseudo-hydro and full hydro simulations is in treating the dynamical relation between the IGM (or HI) and dark matter fields. The former assumes that the IGM density distribution traces the underlying dark matter point-by-point on scales larger than the Jeans length in either the linear or nonlinear regimes. However, hydrodynamic studies have found that a statistical discrepancy between the IGM field and underlying dark matter in nonlinear regime is possible. We find that the point-by-point correlation between the IGM density perturbations and dark matter become weaker on comoving scales less than 2 h$^{-1}$ Mpc (in LCDM model), which is larger than the IGM Jeans length.'
author:
- 'Long-Long Feng, Jesus Pando, and Li-Zhi Fang'
title: 'Intermittent Features of the QSO Ly$\alpha$ Transmitted Flux: Results from Hydrodynamic Cosmological Simulations'
---
Introduction
============
In the past few years there has been tremendous progress in understanding the evolution of cosmic structures on scales from galaxies to about 100 h$^{-1}$ Mpc. This progress has occurred mainly in two distinct areas: 1.) constraining cosmological parameters such as the mass density, dark energy, power spectrum of the initial perturbations of the cosmic mass field, etc., 2.) the modeling of the origin of light-emitting objects to explain the observed mass density profiles, velocity dispersions, mass-to-light ratios, substructures, bias behavior, etc., of quasars, galaxies and galaxy clusters.
These two distinct areas coincide with two different regimes in the context of the dynamics of structure formation. The first one is based mainly on the evolution of the cosmic mass field in the linear approximation. The basic statistical tool is the power spectrum, either observed or recovered by Gaussianization, of the weakly non-linear field. The second regime is characterized by the dynamics of collapsed massive halos. The formation and evolution of luminous clumps can be modeled by the growing and merging of massive halos from initially Gaussian perturbations of the mass and velocity fields. The hierarchical clustering of halos is the most effective scenario for modeling galaxy formation.
However, the clustering behavior of the mass and velocity fields of Inter-Galactic Medium (IGM), i.e., the major component of baryonic matter in the universe, is missing from these two regimes. It is certain that the IGM field is not linear and Gaussian because the distributions of QSO’s Ly$\alpha$ forests from either observational data or simulation samples (Bi 1993, Bi, Ge & Fang 1995), have been found to be non-Gaussian even on scales as large as a few h$^{-1}$ Mpc (Pando & Fang 1996, 1998a). Neither is the evolution of the IGM field well modeled by the dynamics of collapsing halos, as the QSO’s Ly$\alpha$ forests are generally not gravitationally confined (Fang et al 1996). The success of modeling the QSO’s Ly$\alpha$ forests via a lognormal model of the IGM (Bi & Davidsen 1997) directly indicates that the IGM field probably does not (or has not yet) undergone a correlation hierarchy evolution.
The clustering behavior of the IGM mass field can play an important role in the dynamical gap between linear power spectrum and collapsed massive halos. For instance, it has recently been found that the QSO Ly$\alpha$ transmitted flux fluctuations are intermittent (Jamkhedkar, Zhan & Fang 2000, Feng, Pando & Fang 2001, Zhan, Jamkhedkar & Fang 2001). A field is intermittent if the 2$n^{th}$ power of the local differences grows faster than the $n^{th}$ power of the variance of these differences (see Section 2.) This result cannot be found using the statistical tools of the linear regime such as power spectrum. At the same time, it is unclear how to incorporate intermittent behavior with the hierarchical clustering scenario. The intermittent features have also been studied with simulated Ly$\alpha$ forests produced by a pseudo hydro technique. All the pseudo hydro samples for low density cold dark matter (LCDM) and warm dark matter (WDM) models are found to be substantially intermittent. Although the intermittent features of the simulation samples were different from those of the observations (Pando et al 2002), the results implies that intermittency is a common feature of non-linear evolution and is potentially useful for model discrimination.
This motivated us to re-do the intermittent analysis with simulated Ly$\alpha$ transmitted fluxes produced by a full hydrodynamical treatment. This motivation was strengthened when we realized that the relation between the IGM and dark matter fields in the non-linear regime is treated differently in the pseudo-hydro and full hydro simulations. In the pseudo-hydro simulation, the density of the IGM in each pixel is assigned according to the dark matter density at that pixel. That is, the pseudo-hydro simulation assumes that the density distribution of IGM traces the underlying dark matter point-by-point on scales larger than the Jeans length in either the linear or nonlinear regimes. This assumption looks reasonable when one considers that the IGM is only a passive substance in the cosmic mass field since its evolution is dominated by the gravity of dark matter.
However, hydrodynamic studies have shown that a passive substance in the system does not always behave like the underlying dominant mass field when non-linear evolution has taken place (for a review, Shraiman & Siggia 2001). The statistical properties of a passive substance can [*decouple*]{} from those of the underlying field. As we show here, the dynamical equations of the IGM within the gravitational field of dark matter are the same as that used for describing the statistical decoupling between the passive substance and underlying field. Therefore, it is possible there is a statistical discrepancy between the IGM field and underlying dark matter in nonlinear regime.
The purpose of this paper is then two-fold. First we study whether the intermittency of the QSO Ly$\alpha$ transmitted flux is matched by the full hydro simulation samples. Secondly we search for possible signs of the statistical discrepancy between the IGM field and the underlying dark matter with these samples. The paper is organized as follows. §2 presents a brief introduction to intermittent fields and the method to quantify intermittent random fields. §3 describes the observed samples that we use. §4 describes full hydrodynamic simulations. The statistical deviation of the IGM distribution from the dark matter field is shown in §5. A comparison between the model predicted and observed intermittency is performed in §6. Finally, the conclusions and discussions are presented in §7. The dynamical equations for the IGM evolution in non-linear regime is given in Appendix.
Intermittency
=============
Much of this section has been presented in our previous work (Pando et al 2002). For the paper to be self-contained, we repeat these results briefly.
The intermittent exponent
-------------------------
Intermittency is used to characterize a random field with ‘spike-gap-spike’ features, i.e., structures that are essentially strong enhancements randomly and widely scattered in space and/or time, with a low field value between the spikes. The spike-gap-spike feature is more pronounced on smaller scales. This feature was originally found in the temperature and velocity distributions in turbulence (Batchelor & Townsend 1949; Frisch 1995).
An intermittent random density field $\rho({\bf x})$ is defined by the asymptotic behavior (indicated below by $\asymp$) of the ratio between the high- and low-order moments of the field as (Gärtner & Molchanov, 1990; Zel’dovich, Ruzmaikin, & Sokoloff, 1990) $$\frac{\langle [\rho({\bf x+ r})- \rho({\bf x})]^{2n}\rangle}
{[\langle [\rho({\bf x + r})- \rho({\bf x})]^2\rangle]^n}
\asymp\left \langle \frac{r}{L} \right \rangle^{\zeta},$$ where $L$ is the size of the sample. Intermittency is measured by the $n$- and $r$-dependencies of $\zeta$ – the intermittent exponent. If the exponent $\zeta$ is negative on small scales $r$, the field is intermittent. That is, the ratio in eq. (1) for an intermittent field diverges as $r\rightarrow 0$.
This divergence cannot be measured by individual higher order correlation functions. Although three- and four-point correlation functions are useful for distinguishing a Gaussian from a non-Gaussian field, they are insensitive to the difference between an intermittent field and a non-Gaussian, but non-intermittent field. To measure intermittency, we define the structure function as $$S^{2n}_r = \langle|\Delta_r(x)|^{2n} \rangle$$ where $\Delta_r(x)\equiv \rho(x+r)-\rho(x)$, and $\langle... \rangle$ is the average over the ensemble of fields. If the field is statistically homogeneous, $S^{2n}_r$ is independent of $x$ and depends only on $r$. $S^{2n}_r$ is called the structure function. When $n=1$, we have $$S^2_r =\langle |\Delta_r(x)|^{2} \rangle.$$ $S^2_r$ is the mean of the square of the density fluctuations at wavenumber $k\simeq 2\pi/r$, and therefore, $S^2_r$ is the power spectrum of the field. With the structure function, eq.(1) can be rewritten as $$\frac{S^{2n}_r}{[S^{2}_r]^n} \propto
\left(\frac{r}{L} \right )^{\zeta}.$$ The ratio $S^{2n}_r/(S^{2}_r)^n$ is the n$^{th}$ moment $S^{2n}$ normalized by the power $S^2$. Since $S^{2}_r$ measures the width of the probability distribution function (PDF) of $|\Delta_r(x)|$, while $S^{2n}_r$ is sensitive to the tail of the PDF, the ratio eq.(4) measures the fraction of density fluctuations in the tail on different scales.
For an intermittent field the ratio of $S^{2n}_r$ to $[S^{2}_r]^n$ is larger for smaller $r$ and indicates that the field contains “abnormal” events of large density fluctuations $|\Delta_r(x)|$. That is, the PDF of $|\Delta_r(x)|$ is long-tailed. The long-tail events correspond to a sharp increase or decrease of the density $\rho({\bf x})$. They are rare events and yield high peaks or spikes in the distribution of density difference. Generally speaking, the intermittent exponent $\zeta$ measures the smoothness of the field: for positive $\zeta$, the field is smoother on smaller scales. If $\zeta$ is negative, the field is rough on small scales.
The basic quantity used for calculating the intermittent exponent $\zeta$ is the density difference $|\Delta_r(x)|$. The exponent $\zeta$ does not rely on the mean density $\bar{\rho}$ and can be measured even if the mass field is fractal, in which case the mean density is no longer available.
Examples of the intermittent exponent
-------------------------------------
In our previous work we presented several examples of intermittency in various kinds of fields. Here we summarize those results for the kinds of fields that pertain to this work.
For a Gaussian field, $$\frac{S^{2n}_r}{[S^{2}_r]^n} = (2n-1)!!.$$ This ratio is independent of scale $r$, and therefore, the intermittent exponent $\zeta=0$. A Gaussian field is not intermittent. A self-similar field is also not intermittent.
For a hierarchically clustered field, it has been shown (Feng, Pando, & Fang 2001) that $$\frac{S^{2n}_r}{[S^{2}_r]^n} \propto \left(\frac{r}{L}
\right )^{-(d-\kappa)(n-1)},$$ where $d$ is the spatial dimension, and the coefficient $\kappa$ is a constant depending on the power law index of the power spectrum. In the case where $\kappa <d$, the field is intermittent with exponent $$\zeta \simeq -(d-\kappa)(n-1).$$
Finally, for a lognormal field the PDF of $\Delta_r(x)$ is given by $$P[\Delta_r(x)] =\frac{1}{2^{3/2}\pi^{1/2}|\Delta_r(x)|\sigma(r)}
\exp\left \{ -\frac{1}{2}
\left (\frac {\ln|\Delta_r(x)|-\ln |\Delta_r(x)|_m}
{\sigma(r)}\right)^2 \right \},$$ where $|\Delta_r(x)|_m$ is the [*median*]{} of $|\Delta_r(x)|$ (Vanmarcke 1983). The variance $\sigma(r)$ of $\ln|\Delta_r(x)|$ generally can be a function of the scale $r$. Using eq.(8), we have $$\frac{S^{2n}_r}{[S^{2}_r]^n} = e^{2(n^2-n)\sigma^2(r)}.$$ The intermittent exponent of a lognormal field is then $$\zeta \simeq 2(n^2-n)\sigma^2(r)/\ln(r/L).$$ Because $r < L$, $\zeta$ is negative. Therefore, a lognormal field is intermittent.
To summarize, the structure function and intermittent exponent provide a complete and unified description of intermittent fields. The $n$- and $r$-dependencies of the structure functions and intermittent exponent $\zeta$ are sensitive to the details of the intermittency of the field. These measures are very powerful for distinguishing among fields that are Gaussian, self-similar, mono-fractal, multi-fractal, and singular.
The intermittent exponent in the wavelet basis
----------------------------------------------
The quantity $\Delta_r(x)$ or $[\rho(x+r)-\rho(x)]$ contains two variables: the position $x$ and the scale $r$, and therefore, $\rho(x+r)-\rho(x)$, can best be calculated with a space-scale decomposition. Among the mathematical tools used to decompose functions in scale space as well as physical space, the discrete wavelet transform (DWT) is an excellent choice. The DWT basis are orthogonal and complete, and therefore, the DWT decomposition does not lose information (completeness), and does not cause false correlations due to redundancy (non-orthogonality). These two points are essential in studying the higher order statistical features of random fields.
We restrict our discussion to a 1-D random field $\rho(x)$ extending in a physical or redshift spaces $L=x_2-x_1$. To apply the DWT, we first chop the spatial range $L=x_2-x_1$ into $2^j$ subintervals labeled $l=0, ...2^j-1$ where $j$ is a positive integer. Each subinterval spans a spatial range $L/2^j$ with the $l^{th}$ subinterval spanning from $x_1+ Ll/2^j$ to $x_1 + L(l+1)/2^j$. This operation decomposes the space $L$ into cells $(j,l)$, where $j$ denotes the scale $L/2^j$, and $l$ the spatial range.
Corresponding to each cell, there is a scaling function $\phi_{jl}(x)$, and a wavelet function $\psi_{jl}(x)$. The scaling function $\phi_{j,l}(x)$ is a window function on scale $j$ and at position $l$. The wavelet functions $\psi_{jl}(x)$ form the basis for the scale-space decomposition. The most important property of the DWT basis is its locality in both scale and physical spaces.
With the DWT, a density field $\rho(x)$ can be decomposed as (Fang & Thews 1998). $$\rho(x) = \sum_{l=0}^{2^j-1}\epsilon_{j,l}\phi_{jl}(x)+
\sum_{j'=j}^{J} \sum_{l=0}^{2^{j'}-1}
\tilde{\epsilon}_{j',l} \psi_{j',l}(x),$$ where $J$ is given by the finest scale (resolution) of the sample, i.e., $\Delta x=L/2^J$, and $j$ is the scale of interest. The scaling function coefficient (SFC) $\epsilon_{jl}$ in eq.(11) is given by projecting $\rho(x)$ onto $\phi_{j,l}(x)$ $$\epsilon_{j,l}=\int \rho(x)\phi_{j,l}(x)dx.$$ The SFC $\epsilon_{jl}$ describes the mean field of the mode $(j,l)$.
The wavelet function coefficient (WFC), $\tilde{\epsilon}_{j,l}$, in eq.(11) is obtained by projecting $\rho(x)$ onto $\psi_{j,l}(x)$ $$\tilde{\epsilon}_{j,l}= \int \rho(x) \psi_{j,l}(x)dx.$$ The wavelet function $\psi_{jl}(x)$ is admissible (Daubechies 1992), i.e., $\int \psi_{jl}(x)dx=0$, so the WFC is basically the difference between $\rho(x)$ and $\rho(x+r)$, where $x \simeq x_1+ lL/2^j$ and $r\simeq L/2^j$. Therefore, the WFC eq.(13) measures the fluctuation on scale $j$ and at position $l$. Because the admissibility of wavelets, eq.(13) can also be written as $$\tilde{\epsilon}_{j,l}= \int \delta(x) \psi_{j,l}(x)dx,$$ if the normalization is $\overline{\rho}=1$. Thus, the WFC can be used as the variable $\rho(x+r)-\rho(x)$ or $\delta(x+r)-\delta(x)$ in eq.(2). All compactly supported wavelet bases produce similar results. We will use Daubechies 4 (Daubechies, 1992) in the study below.
When the “fair sample hypothesis" is applicable (Peebles 1980), the structure function eq.(2) can be calculated as the spatial average $$S^{2n}_r =\frac{1}{L}\int |\Delta_r(x)|^{2n}dx,$$ where $L$ is the spatial range of the sample. In the DWT representation, eq.(15) yields $$S^{2n}_j = \langle|\tilde{\epsilon}_{j,l}|^{2n} \rangle
=\frac{1}{2^j}\sum_{l=0}^{2^j-1}|\tilde{\epsilon}_{j,l}|^{2n},$$ where $j$ plays the same role as $r$ in eq.(2). $S^{2n}_j$ is the mean of moment $|\tilde{\epsilon}_{j,l}|^{2n}$ over the position index $l$. The intermittent exponent $\zeta$ in eq.(4) can be calculated from $$\frac{S^{2n}_j}{[S^{2}_j]^n} \propto 2^{-j\zeta}.$$ Generally, $\zeta$ depends on $n$ and $j$.
The Observed Ly$\alpha$ transmitted flux
========================================
QSO’s Ly$\alpha$ absorption spectra
-----------------------------------
The observed transmitted flux at wavelength $\lambda$ is $F(\lambda)=F_c e^{-\tau(\lambda)}$, where $\tau(\lambda)$ is optical depth, and $F_c$ the continuum which is generally $\lambda$-dependent. $F_c$ is given by the emission spectrum of the QSO considered. To study the intermittent behavior of the IGM we focus on the fluctuation field $e^{-\tau(\lambda)}$. The intermittent features of transmitted flux of QSO’s Ly$\alpha$ absorption spectrum were previously studied with a set of about 30 Keck HIRES spectra (Jamkhedkar, Zhan, & Fang, 2000, Pando et al. 2002). This work will consider the two best absorption spectra in that the sample set.
[*1. HS1700+6416*]{}
The absorption spectrum of QSO HS1700+6416 (redshift $z = 2.72$) covers a wavelength range from 3727.012 Å to 5523.554 Å, for a total of 55882 pixels. On average, a pixel is about 0.028 Å. For all pixels in this data set, the ratio $\Delta \lambda/\lambda \simeq 7\times 10^{-6}$ and constant, or in terms of the local velocity, $\delta v =c\Delta \lambda/\lambda \simeq 2.1$ km/s, and therefore, the resolution is about 4 km/s. The distance between $N$ pixels in units of the local velocity scale is given by $\Delta v=2c(1-\exp{[-(1/2)N~\delta v/c]})$ km/s, or wavenumber $k=2\pi /\Delta v$ s/km. As $\delta v /c \ll 1$, we have $\Delta v \simeq N~\delta v$. In this work, we use the data from $\lambda =$3815.6 Å to 4434.3 Å, which corresponds to $z = 2.14 \sim 2.65$. The lower limit of the wavelength is set to exclude Ly$\beta$ absorption, the upper to eliminate proximity effects. The continuum $F_c$ for this spectrum was done by IRAF CONTINUUM fitting by Kirkman and Tytler (1997).
In each pixel the data contain wavelength $\lambda_i$, flux $F(\lambda_i)$ and noise $\sigma(\lambda_i)$, which accounts for the Poisson fluctuations in the photon count, the noise due to the background and the instrumentation. The S/N ratio is about 8.
This data set contains a fitted flux which smooths out all fluctuations on smallest scales, i.e. one or two pixels. The fluctuations on these scales are substantially suppressed by the fitting procedure. However, we analyze only scales greater than or equal to 16 pixels. Therefore, the smoothing has little effect on our results. Metal lines for this spectrum have been identified down to the Doppler parameters $b \sim$ 10 km/s by Wei Zheng (unpublished).
[*2. q0014+8118*]{}
This spectra has the highest S/N$\sim $ 30 among the 30 Keck QSO spectra collected by Kirkman and Tytler (1997). The redshift of q0014+8118 is 3.387. For all pixels in this data set $\Delta \lambda/\lambda
\simeq 13.8 \times 10^{-6}$, or $\delta v = c\Delta
\lambda/\lambda\simeq 4.01$ km/s, and, therefore, the resolution is about 8 km/s. In each pixel, the data also contains wavelength $\lambda_i$, flux $F(\lambda_i)$ and noise $\sigma(\lambda_i)$. The continuum of the spectrum is given by IRAF CONTINUUM fitting as well.
We use the wavelength region from the Ly$\beta$ emission to the Ly$\alpha$ emission, excluding a region of about 0.06 in redshift close to the quasar to avoid any proximity effects. This corresponds to $z=2.702$ to 3.314, which contains 12703 pixels. The distance between $N$ pixels in the units of the local velocity scale is given by $\Delta v \simeq N\times 4.01$ km/s, or wavenumber $k=2\pi /\Delta v$ s/km. Absorption features with $b <20$ km s$^{-1}$ of this spectrum have been identified.
Treatment of unwanted data
--------------------------
To study the intermittency of the transmission flux fluctuations, we should remove the effects of unwanted data, such as 1.) bad pixels (gaps without data), 2.) metal lines, and 3.) negative flux pixels. The last category generally consists of saturated absorption regions having lower $S/N$. Although the percentage of low $S/N$ data is not large, it will introduce large uncertainties in the analysis.
The conventional technique for removing unwanted data is to eliminate the unwanted bins and to smoothly rejoin the rest of the forest spectra. However, by taking advantage of the localization of wavelets we can use the algorithm of DWT de-noising by thresholding (Donoho 1995) as follows
1. Calculate the SFCs for both the transmission $F(\lambda)$ and noise $\sigma(\lambda)$, i.e. $$\epsilon^F_{jl}=\int F(x)\phi_{jl}(x)dx, \hspace{3mm}
\epsilon^N_{jl}=\int \sigma(x)\phi_{jl}(x)dx.$$
2. Identify unwanted mode $(j,l)$ using the condition $$\left| \frac{\epsilon^F_{jl}} {\epsilon^N_{jl}} \right| < f$$ where $f$ is a constant. This condition flags all modes with S/N less than $f$. We can also flag modes dominated by metal lines.
3. Since all the statistical quantities in the DWT representation are based on an average over the modes $(j,l)$, we do not count all the flagged modes when computing these averages.
This is the conditional-counting method to treat the unwanted data (Pando et al. 2002).
It should be emphasized that the condition (19) is applied at each scale $j$. For instance, if the size of a saturated absorption region is $d$, eq.(19) requires the removal of all modes $(j,l)$ on scales less than $d$ and located within this region, as these modes have very low S/N ($\epsilon^F_{jl}/\epsilon^N_{jl} < f$) and the local fluctuations in these regions are not reliable. However, condition (19) does not require the removal of the saturated absorption region when $d$ is less than the scale of modes $(j,l)$ considered. If $\epsilon^F_{jl}/\epsilon^N_{jl} > f$, the mode $(j,l)$ is counted in the statistics, regardless of whether the mode $(j,l)$ contains the saturated absorption regions. The DWT analysis allows an effective application of data containing low S/N regions. That is, with the scale-by-scale denoising, it eliminates low S/N modes on small scales and in these regions, but keeps their contributions to large scale modes. With this procedure, no eliminating and smoothly rejoining of the data is needed, and therefore, it avoids corrupting long wavelength modes.
With this method, we can still calculate the structure functions by eq.(17), but the average is not over all modes $l$, but over the un-flagged modes only. We can also flag modes $(j,l)$ one by one which are dominated by metal lines. Since the DWT calculation assumes the sample is periodized, this may cause uncertainty at the boundary. To reduce this effect, we drop 3 modes near the boundary, and also flag two modes around a unwanted mode to reduce any boundary effects of the chunks.
Power spectrum of HS1700
------------------------
To check the quality of the data, we calculate the power spectrum of the transmitted flux fluctuations, $\Delta F=F(\lambda) - \langle F(\lambda)\rangle$, of HS1700+6416. In the DWT representation, this power spectrum is given by (Pando & Fang 1998b; Fang & Feng 2000, Jamkhedkar, Bi, & Fang, 2001) $$P_j=\frac{1}{2^j}\sum_{l=0}^{2^j-1}(\tilde{\epsilon}^F_{jl})^2
- \frac{1}{2^j}\sum_{l=0}^{2^j-1}(\tilde{\epsilon}^n_{jl})^2.$$ The first term on the r.h.s. of eq.(20) is the power of $\Delta F$ on scale $j$, in which the wavelet coefficients (WFC) $\tilde{\epsilon}^F_{jl}$ are given by $$\label{eq:WFC2}
\tilde{\epsilon}^F_{jl} = \int F(x)\psi_{jl}(x)dx.$$ The second term on the r.h.s. of eq.(20) is due to the noise field $\sigma(\lambda)$ and calculated by $$(\tilde{\epsilon}^n_{jl})^2 = \int\sigma^2(x)\psi_{jl}^2(x)dx.$$
The DWT power spectrum $P_j$ of a random field $F(x)$ is related to its Fourier power spectrum $P(n)$ by $$P_j = \frac{1}{2^j} \sum_{n = - \infty}^{\infty}
|\hat{\psi}(n/2^j)|^2 P(n).$$ where $\hat{\psi}$ is the Fourier transform of the wavelet. This implies that the DWT power spectrum $P_j$ is the banded Fourier power of the flux fluctuations $F(\lambda)-\langle
F(\lambda)\rangle$. Since for D4 wavelet $|\hat{\psi}(\eta)|^2$ is peaked at $\eta \simeq 1$, the band $j$ corresponds to the wavenumber $k = 2 \pi n/L \simeq 2\pi 2^j/L$.
Figure 1 plots the results of the power spectra for HS1700+6416, in which the data is treated three ways, 1.) using the original fitted flux, 2.) using the conditional counting method eq.(19) with $f=1$, and 3.) with $f=3$.
At the first glance, the conditional-counting condition eq.(19) would seem to preferentially miss–count modes in the low transmission regions and lead to an $f$-dependence in the power spectrum. However Fig. 1 shows that the power spectrum $P_j$ is independent of $f$ on entire the scale range considered from $f=1$ to 3. The reason can be seen from eq.(20), which shows that the contribution to the power $P_j$ given by mode $(j,l)$ is $(\tilde{\epsilon}^F_{jl})^2-(\tilde{\epsilon}^n_{jl})^2$. The noise subtraction term $(\tilde{\epsilon}^n_{jl})^2$ guarantees that the contribution of modes with small ratio $S/N$ to $P_j$ is always small or negligible. In other words, all the miss–counted modes make very small or negligible contributions to $P_j$ regardless the parameter $f$.
Also in Fig. 1, on scales $ j > 10$, or $\Delta v < 32 $ km/s, the power of the fitted flux is lower than the power spectrum. This discrepancy is probably because the fitted flux is over-suppressed, i.e., not only noise, but also real fluctuations are suppressed by the fitting procedure. This discrepancy indicates that the effects of noise become significant on scales $\Delta v < 32 $ km/s. For this reason, we use only data on scales $\Delta v \geq 32 $ km/s. Sample q0014+8118 is of similar quality and for it we also only use scales $\Delta v \geq 32 $ km/s.
Hydrodynamical simulations
==========================
The simulation
--------------
For the hydrodynamic simulations, we use a Eulerian code based on a weighted essentially non-oscillatory (WENO) scheme for a hyperbolic system (Jiang & Shu, 1996; Shu, 1998). WENO realizes the idea of adaptive stencils in the reconstruction procedure based on the local smoothness of the numerical solution to automatically achieve high order accuracy and non-oscillatory property near discontinuities. It is extremely robust and stable for solutions containing strong shock and complex solution structures. This is especially important in studying intermittency. In the context of cosmological applications, we developed a hybrid N-body/hydrodynamical code that incorporates a Lagrangian particle-mesh algorithm to evolve the collision-less matter with the fifth order WENO scheme to solve the equation of gas dynamics (Feng et.al, 2002).
In the present application, we computed the cosmic evolution of the coupled system of both dark matter and baryonic matter in a flat low density CDM model ($\Lambda$CDM), which is specified by the density parameter $\Omega_m=0.3$, cosmological constant $\Omega_{\Lambda}=0.7$, Hubble constant $h=0.7$, and mass fluctuation within a sphere of radius 8h$^{-1}$Mpc, $\sigma_8=0.9$. The baryon fraction is fixed using the constraint from primordial nucleosynthesis as $\Omega_b=0.0125h^{-2}$ (Walker et.al, 1991). The linear power spectrum is taken from the fitting formulae presented by Eisenstein & Hu (1998). We perform the simulations in a 128$^3$ grid with an equal number of dark matter particles. In order to examine the effects of numerical resolution, two simulations on a periodic, cubical box of size 6h$^{-1}$Mpc and 12h$^{-1}$Mpc are generated. The simulations start at a redshift $z=49$ and the results were outputed at the average redshifts of the each observation samples (for q0014+8118, $z_m=2.915$, and HS1700+6416, $z_m=2.329$). The timestep is chosen by the minimum value among three time scales. The first is from the Courant condition given by $$\delta t \le \frac{ cfl \times a(t) \Delta x}{\hbox{max}(|u_x|+c_s,
|u_y|+c_s, |u_z|+c_s)}$$ where $\Delta x$ is the cell size, $c_s$ is the local sound speed, $u_x$, $u_y$ and $u_z$ are the local fluid velocities and $cfl$ is the Courant number, typically, we take $cfl=0.6$. The second constraint is imposed by cosmic expansion which requires that $\Delta a /a <0.02$ within single time step. The last constraint comes from the requirement that a particle move no more than a fixed fraction of cell size in one time step.
Atomic processes including ionization, radiative cooling and heating are modeled as in Cen (1996) in a primeval plasma of hydrogen and helium of composition ($X=0.76$, $Y=0.24$). The uniform UV-background of ionizing photons is assumed to have a power-law spectrum of the form $J(\nu)
=J_{21}\times
10^{-21}(\nu/\nu_{HI})^{-\alpha}$ergs$^{-1}$cm$^{-2}$sr$^{-1}
$Hz$^{-1}$, where the photoionizing flux is normalized by parameter $J_{21}$ at the Lyman limit frequency $\nu_{HI}$, and is suddenly switched on at $z\sim 6$ to heat the gas and reionize the universe.
One-dimensional fields are extracted along randomly selecting lines of sight in the simulation box. The density, temperature and velocity of the neutral gas fraction on grids are firstly Gaussian smoothed using FFT techniques which form the fundamental data set. The one-dimensional grid containing the physical quantities is further interpolated by a cubic spline. Using this one-dimensional grid, the optical depth $\tau$ at each pixel is then obtained by integrating in real space and we include the effect of the peculiar velocity and convolve with Voigt thermal broadening. To have a fair comparison with observed spectra, $\tau$ was Gaussian smoothed to match with the spectral resolutions of observation. The transmitted flux $F=\exp(-\tau)$ is normalized such that the mean flux decrement in the spectra match with observations.
Each mock spectrum is sampled on a $2^{10}$ grid with the same spectral resolution as the observation. As the corresponding comoving scale for $2^{10}$ pixels is larger than the simulation box size, we replicate periodically the sample. To achieve the greatest the statistical independence, we randomly change the direction of line of sight while crossing the boundary of the simulation box. For each observed spectrum, 1000 mock spectra are generated. In addition, we also output 100 samples of the density distribution of both dark matter and neutral hydrogen on the one-dimensional grid.
Power spectrum of hydrodynamical simulation samples
---------------------------------------------------
Figure 2 compares the DWT power spectra measured in the mock samples and the observed data of HS1700. The comoving wavelength $k$ in Fig. 2 is given by $k = 2\pi/L$, where $L = (\Delta v
/H_0)[\Omega_m(1+z_m)^3+\Omega_{\Lambda}]^{-1/2}$. The 1-$\sigma$ errors for the simulation samples are derived from the 1000 mock spectra. For the observed spectrum of HS1700, we divide the first $2^{14}$ pixels of the observed spectrum into 16 subsections, each with $2^{10}$ pixels. The DWT power spectra are obtained for each subsection respectively, and the 1-$\sigma$ error is estimated from the variance of the 16 sections. \[For an intermittent field, the error estimation of power spectrum is sensitively dependent on how one treats the intermittency (Jamkhedkar, 2002). We do not discuss the details of this point here as it will not affect the following discussion.\]
Figure 2 shows that simulation samples with the 12 Mpc box are in good agreement with observations on all scales $k \leq 15$ h Mpc$^{-1}$, which corresponds to $j=10$ in Fig. 1. The simulation samples with the 6 h$^{-1}$Mpc box have higher power than observed sample on small scales, but lower power on large scales. This is expected because the simulation in the 6 h$^{-1}$Mpc box lacks perturbations on scales $k \leq 1$ h Mpc$^{-1}$ and the normalization of the mean flux decrement will lead to the higher power on small scales to compensate for the loss of powers on large scales.
On small scales $k >15$ h Mpc$^{-1}$, the simulated power is significantly lower than the observation data. This shows again that noise contamination becomes important on scales $k >15$ h Mpc$^{-1}$, or $j > 10$. Therefore, as in §3.3, we use the simulation data only on scales $k >15$ h Mpc$^{-1}$, or $j
\leq 10$, or $\Delta v \geq 32 $ km/s.
Statistical discrepancy between baryonic gas and dark matter
============================================================
Relation between IGM and dark matter mass fields
------------------------------------------------
It is well known that in the linear approximation the density contrast of the IGM, $\delta_b({\bf x}, t)$ is equal to that of dark matter $\delta_{DM}({\bf x}, t)$ on scales larger than the Jeans length. This is true even if initially $\delta_b({\bf x}, t_0) \neq \delta_{DM}({\bf x},
t_0)$ (Bi, Börner & Chu 1993, Nusser 2000, Matarrese & Mohayee 2002). In other words, the density fluctuations of the IGM trace the dark matter fields point-by-point. Since the clustering of the IGM is governed by the gravity of dark matter, the IGM can be considered as “passive substance” with respect to the underlying dark matter. Thus, the point-by-point relations $\rho_b({\bf x}, t) \propto \rho_{DM}({\bf x}, t)$ and $\delta_b({\bf x}, t) = \delta_{DM}({\bf x}, t)$ on scales larger than the Jeans length also looks reasonable in the nonlinear regime. In this case, [*all*]{} the statistical properties of the IGM field on scales larger than the Jeans length are the same as the underlying dark matter field.
However, it is known that a passive substance in a hydrodynamic system does not always behave like the underlying dominant mass field when non-linear evolution takes place. The statistical properties of a passive substance can [*decouple*]{} from those of the underlying field in the nonlinear regime. Moreover, the evolution of the IGM driven by the gravitational field of dark matter is described by a non-linear equation which is essentially the same as that used for modeling the statistical decoupling between a passive substance and its underlying field (see Appendix A.) Hence, we expect to see a statistical discrepancy between $\delta_b({\bf x}, t)$ and $\delta_{DM}({\bf x}, t)$ on scales larger than the Jeans length.
To determine whether any statistical decoupling has occurred, we begin by assuming that $\delta_b({\bf x}, t) = \delta_{DM}({\bf x}, t)$. Then wavelet transforming yields $\tilde{\epsilon}^{b}_{jl}=\tilde{\epsilon}^{DM}_{jl}$, and therefore $$\langle \tilde{\epsilon}^{b}_{jl}\tilde{\epsilon}^{DM}_{jl}\rangle
= \langle [\tilde{\epsilon}^{b}_{jl}]^2 \rangle =
\langle[\tilde{\epsilon}^{DM}_{jl}]^2\rangle.$$ $\langle [\tilde{\epsilon}^{b}_{jl}]^2 \rangle$ and $\langle[\tilde{\epsilon}^{DM}_{jl}]^2\rangle$ are, respectively, the DWT power spectrum of the IGM and dark matter fields. Therefore, the first conclusion of the point-by-point relation $\delta_b({\bf x}, t) = \delta_{DM}({\bf x}, t)$ is that the power spectra of the IGM and dark matter should be the same. Figure 3 indeed shows that the power spectra of the IGM and dark matter fields are about the same.
With eq.(24), it is obvious that the power spectrum of the field $\delta_b({\bf x}, t) - \delta_{DM}({\bf x}, t)$ should be equal to zero, or considering noise, very small. Figure 4 gives the power spectra $\langle[\tilde{\epsilon}^{b}_{jl}- \tilde{\epsilon}^{DM}_{jl}]^2\rangle$ and $\langle[\tilde{\epsilon}^{b}_{jl}]^2\rangle$ and shows that on large scales $j< 8$ ($\Delta v > 128$ km/s) $\langle[\tilde{\epsilon}^{b}_{jl}- \tilde{\epsilon}^{DM}_{jl}]^2\rangle$ is much less than $\langle[\tilde{\epsilon}^{b}_{jl}]^2\rangle$. The ratio of $\langle[\tilde{\epsilon}^{b}_{jl}-
\tilde{\epsilon}^{DM}_{jl}]^2\rangle$ to $\langle[\tilde{\epsilon}^{b}_{jl}]^2\rangle$ is $< 0.05$, which probably is due to noise. However, on scales $j\geq 8$ ($\Delta v \geq 128$ km/s), this ratio gradually increases to $\simeq 0.4$. This difference between large and small scales cannot be explained by the noise, as can be seen from Fig. 2 in which the error bars in the range $1 < k <10$ h$^{-1}$ Mpc (corresponding to $j=7$ to 10) are about the same. Figure 4 indicates that $\langle \tilde{\epsilon}^{b}_{jl}\tilde{\epsilon}^{DM}_{jl}\rangle
< \langle [\tilde{\epsilon}^{b}_{jl}]^2 \rangle$ on small scales, i.e., the point-by-point correlation between the IGM density fields and dark matter fields is weaker on small scales. This is a sign of the statistical discrepancy between the IGM and dark matter mass fields.
We can further make the case by calculating the structure functions for the field $\delta_b({\bf x}, t) - \delta_{DM}({\bf x}, t)$. The result is plotted in Fig. 5 and shows that for $j=6$ the ratio of the moments is Gaussian, i.e. $\ln S^{2n}_j/(S^{2}_j)^n = \ln (2n-1)!!$ \[eq.(5)\]. On scales $j\geq 8$, $\ln S^{2n}_j/(S^{2}_j)^n$ is significantly larger than a Gaussian distributions which strongly indicates of the statistical discrepancy between $\delta_b({\bf x}, t)$ and $\delta_{DM}({\bf x},t)$ is not due to noise. For a given $n$, $\ln S^{2n}_j/(S^{2}_j)^n$ increases with scale $j$ so that the intermittent exponent $\zeta$ \[eq.(21)\] is non-zero and negative, and therefore, the field $\delta_b({\bf x}, t) - \delta_{DM}({\bf x}, t)$ is intermittent. This result is consistent with the dynamical equation (A14), which requires that the difference $\delta_b({\bf x}, t) - \delta_{DM}({\bf x}, t)$ generally is an intermittent field. Thus, we conclude again that there is statistical discrepancy from the point-by-point relation $\delta_b({\bf x}, t) = \delta_{DM}({\bf x}, t)$.
Relation between HI and dark matter
-----------------------------------
The statistical discrepancy in the relation between the density fields of neutral hydrogen and dark matter is important as well. For instance, the pseudo-hydro technique assumes that $\delta_{HI}({\bf x}, t) \propto \delta_{DM}^{a}({\bf x}, t)$ with $1.5 < a < 1.9$ (Hui & Gnedin, 1997). This relation yields $\langle \tilde{\epsilon}^{HI}_{jl}\tilde{\epsilon}^{DM}_{jl}\rangle >0$, and therefore the power spectrum $\langle[\tilde{\epsilon}^{HI}_{jl}+ \tilde{\epsilon}^{DM}_{jl}]^2\rangle$ should be larger than $\langle[\tilde{\epsilon}^{HI}_{jl}-
\tilde{\epsilon}^{DM}_{jl}]^2\rangle$. Figure 6 presents the power spectra $\langle [\tilde{\epsilon}^{HI}_{jl}\pm
\tilde{\epsilon}^{DM}_{jl}]^2\rangle$ and shows $\langle[\tilde{\epsilon}^{HI}_{jl}+ \tilde{\epsilon}^{DM}_{jl}]^2\rangle
>
\langle[\tilde{\epsilon}^{HI}_{jl}- \tilde{\epsilon}^{DM}_{jl}]^2\rangle $ on large scales, but this difference is significantly smaller on smaller scales.
Figure 7 gives the power spectra $\langle[\tilde{\epsilon}^{HI}_{jl}- \tilde{\epsilon}^{DM}_{jl}]^2\rangle
$ and $\langle [\tilde{\epsilon}^{b}_{jl}]^2 \rangle$ and reveals that it is similar to Fig. 4. All the results indicate that the positive correlation $\langle \tilde{\epsilon}^{HI}_{jl}\tilde{\epsilon}^{DM}_{jl}\rangle >0$ becomes weaker on scales $j \geq 8$ or $k < 3$ h Mpc$^{-1}$ or $L > 2$ h$^{-1}$ Mpc. Since intermittent features become significant on scales $j \geq 8$, the dynamical assumption that $\rho_{HI}({\bf x})\propto \rho_{DM}^{a}({\bf x})$ is applicable only for lower order statistics (like the power spectrum.)
Comparison of intermittent features between real and simulation samples
=======================================================================
Structure functions of real and simulation samples
--------------------------------------------------
We now calculate the structure functions for the transmitted flux fluctuations of the simulation samples. The results are illustrated in Fig. 8 for HS1700 and Fig. 9 for q0014. The error bars are the 1-$\sigma$ deviations estimated from the 1000 simulation samples. Comparing Figs. 8 and 9 with Fig. 2, we see that $\ln S^{2n}_j/(S^{2}_j)^4$ with $n=2$ has smaller errors than the power spectrum. Even at 6th and 8th orders, the $j$-dependence of $\ln S^{2n}_j/(S^{2}_j)^n$ can be very well fitted by a line. This implies that to measure the higher order statistical behavior of an intermittent field, the structure functions are effective and stable. For an intermittent field, the large uncertainty in the power spectrum are caused by rare and improbable high spikes in the fluctuations. However, the structure function is an ensemble average of the ratio between $S^{2n}_j$ and $(S^{2}_j)^n$ and this reduces the effect of individual high spikes.
Figures 8 and 9 also show that for a given $n$, $\ln
S^{2n}_j/(S^{2}_j)^n$ increases with scale $j$ so that the intermittent exponent $\zeta$ \[eq.(21)\] is non-zero and negative. The transmitted flux fluctuations are, hence, intermittent. In Fig. 10, we show $\ln S^{2n}_j/(S^{2}_j)^n$ vs. $j$ of HS1700 mock samples produced by both the 12 and 6 h$^{-1}$Mpc box. The 12 h$^{-1}$Mpc samples is a little higher $\ln
S^{2n}_j/(S^{2}_j)^n$, but there are no significant differences between the simulation samples on scales considered.
The structure functions measured for the real data of HS1700 is also shown in Fig. 10, in which the error bars HS1700 are found by bootstrap re-sampling. Within the error bars, the observed structure function is consistent with the 12 Mpc box simulation samples for the LCDM model. Although the consistency shown in Fig. 10 is still not perfect, it is substantially better than the results obtained from the pseudo hydro simulations. Pando et al (2002) found that the structure functions of the pseudo hydro simulation samples were generally larger than those of the real data \[see Fig. 7 of Pando et al. (2002)\].
The structure functions for both HS1700 and q0014 have similar behavior although they lie at different redshifts. The result demonstrates that one cannot see a significant redshift evolution in the range of $2 < z <3$ with the $j$-dependence of the structure functions.
$n$-dependence of the structure function
----------------------------------------
We now turn to the $n$-dependence of the structure functions. Figures 11 and 12 are, respectively, $\ln S^{2n}_j/(S^{2}_j)^n$ vs. $n$ for the simulation samples and real data of HS1700 and q0014 on scales $j=9$ and 10. The figures show again that the simulation result is consistent with observations on the scales considered.
For a Gaussian field, the $n$-dependence of structure function is $\ln S^{2n}_j/(S^{2}_j)^n =\ln (2n-1)!!$ \[eq.(5)\], which is also plotted in Figs. 11 and 12. We note, from Figs. 11 and 12, that although $\ln S^{2n}_j/(S^{2}_j)^n$ is above that of a Gaussian field, $\ln S^{2n}_j/(S^{2}_j)^n$ seems to “parallel" the Gaussian curve. This means that although the field on scale $j=9$ is non-Gaussian, the PDF of $\tilde{\epsilon}^F_{jl}$ doesn’t have a significantly long tail. On scale $j=10$, the slope of $S^{2n}_j/(S^{2}_j)^n$ becomes larger than $(2n-1)!!$, which indicates that the PDF of $\tilde{\epsilon}^F_{jl}$ tends to be long tailed.
The $n$-dependence of $\ln S^{2n}_j/(S^{2}_j)^n$ appear to be more sensitive to redshift evolution. Figure 11 shows that the relation of $S^{2n}_j/(S^{2}_j)^n$ for $j=9$ is clearly above the Gaussian field, while for Fig. 12 it is only marginally different from a Gaussian field. Thus, the intermittent behavior on scale $j=9$ probably developed at redshift $z \leq 3$.
Similar to the power spectrum (Fig. 1), the results of Figs. 8 - 12 result is $f$-independent on the entire scale range considered for $f=1$ to 3 (Pando et al 2002). To see the effect of metal lines, we plot Fig. 13, which shows the $n$-dependence of $\ln S^{2n}_j/(S^{2}_j)^n$ ($j=10$) with and without metal line removal. The result for $j=10$ is less affected by the metal line removing because most metal lines have width less than $\Delta v = 32$ km/s ($j=10$).
Intermittent exponent
---------------------
The $n$-dependence of the intermittent exponent $\zeta$ can be calculated by eq.(17), which gives $$\zeta(n) = -\frac{1}{j}\ln_2 \frac{S^{2n}_j}{(S^{2}_j)^n} + {\rm const},$$ or $$\zeta(n) - \zeta(1) = -\frac{1}{j}\ln_2 \frac{S^{2n}_j}{(S^{2}_j)^n}.$$ For the simulation samples, the relation of $\ln_2 S^{2n}_j/(S^{2}_j)^n$ vs. $n$ shown in Fig. 10 can be very well fitted by a line in the range from $j=7$ to 10 (or $2 \leq k \leq 16$ h Mpc$^{-1}$), and therefore, one can find a constant $\zeta(n)$ for each given $n$. The real data is not as well fitted over the same scales, but we nonetheless show the results in figures 14 and 15.
Again the full hydro simulation shows better result than the pseudo hydro simulation. The value of $|\zeta|$ for the real data is found to be systematically lower than that given by pseudo hydro simulation for the LCDM models (Pando at al. 2002). Although, the value $|\zeta|$ from the hydro simulation is still higher than real data, the differences are no longer larger than the error bars for all $n$. Figure 16 shows the intermittent exponent for samples with 12 and 6 h$^{-1}$Mpc boxes. Note that the conclusions drawn from Figs. 14 and 15 hold also for samples with the 6 h$^{-1}$Mpc box.
Intermittency of HI density field
---------------------------------
It has been argued that the intermittency of the Ly$\alpha$ transmitted flux field $F(\lambda)$ may not indicate intermittency in the IGM distribution $\delta_{b}(\lambda)$ or HI distribution $\delta_{HI}(\lambda)$. The reason is that in the exponential relation between the flux and optical depth $F(\lambda)=F_ce^{-\tau(\lambda)}$, and the optical depth $\tau(\lambda)$ is approximately related to $\delta_{b}(\lambda)$ by $$\tau(\lambda) =
A[\rho_b(\lambda)/\bar{\rho}_b]^a=A[1+\delta_b(\lambda)]^a,$$ where the parameter $A$ depends on the cosmic baryonic density, the photoionization rate of HI, and the temperature of IGM. Therefore, even when $\delta_b(\lambda)$ or $\delta_{HI}(\lambda)$ are Gaussian, the PDF of the flux $F(\lambda)$ might be lognormal due to the exponential relation between $F(\lambda)$ and $\delta_{HI}(\lambda)$.
However as has been emphasized in §2, the intermittency of $F(\lambda)$ field is not measured by the PDF of $F(\lambda)$, but its difference $\Delta_{\delta \lambda} F(\lambda) \equiv
F(\lambda +\delta \lambda)-F(\lambda)$. The structure functions are defined by the density difference. Thus, we have approximately $$\Delta_{\delta \lambda} F(\lambda) \simeq - e^{-\tau(\lambda)}
\Delta_{\delta \lambda}\tau(\lambda)$$ where $\Delta_{\delta \lambda}\tau(\lambda)\equiv \tau(\lambda
+\delta \lambda)-\tau(\lambda)$. Using eq.(27), we have $$\Delta_{\delta \lambda}\tau(\lambda) \simeq -A
a[1+\delta_H(\lambda)]^{a-1}
\Delta_{\delta \lambda}\delta_b(\lambda),$$ where $\Delta_{\delta \lambda}\delta_b(\lambda) \equiv
\delta_b(\lambda+\delta \lambda) - \delta_b(\lambda)$.
Eqs.(29) and (30) show that if the transmitted flux fluctuations are intermittent, the baryonic matter density field and HI field should also be intermittent (Zhan, Jamkhedkar, & Fang 2001). Figure 17 presents the structure functions for the HI field of the 12 Mpc box samples simulation and shows that the HI field is indeed intermittent. In fact, this field has larger structure functions than the flux fluctuations probably because the transmitted flux cannot trace the intermittency of optical depth at heavy absorption regions.
Discussion and conclusions
==========================
We analyzed the intermittent behavior of the Ly$\alpha$ transmitted flux fluctuations of real data and samples produced by full hydro simulations. This analysis covers scales from $\sim$ 10 h$^{-1}$ Mpc to 400 h$^{-1}$ kpc ($j= 6$ to 10), and statistical orders $2n$ from 2 to 8. The intermittent behavior is found to be significant on scales less than about $\sim$ 1 h$^{-1}$ Mpc, and redshift $z\leq 3$. Over the entire scale range and statistical orders considered, the intermittent behavior of the real data matches the samples produced by full hydrodynamic simulations in the LCDM model.
We have shown that the point-by-point correlation of the IGM and HI fields with dark matter field becomes weaker on scales less than about 2 h$^{-1}$ Mpc, which is much larger than the Jeans length at the mean density. Although the power spectrum of the IGM field is still about the same as that of dark matter at these scales, the mass density field of the IGM is no longer point-by-point proportional to the underlying dark matter field.
By using intermittency we have detected a the dynamical discrepancy between the IGM field and the dark matter distribution. The study of cosmic large scale structure has mainly concentrated on either the two-point correlation function and its scaling, or on the massive collapsed halos of dark matter. Intermittency may bridge the gap between the two approaches and provide new, physically interesting insights in cosmic clustering.
We thank Dr. D. Tytler for kindly providing the data of the Keck spectrum HS1700+64 and q0014+8118, and Dr. W. Zheng for providing the metal-line identification of HS1700+64. Dr. Jamkhedkar is acknowledged for her help, especially, Figure 1 which is taken from her thesis. We thank also Dr. D. Weinberg for comments and suggestions in his referee’s report. LLF acknowledges support from the National Science Foundation of China (NSFC) and National Key Basic Research Science Foundation.
Dynamical equations of the IGM and dark matter mass field
=========================================================
Basic equations
---------------
Let us consider a flat universe having cosmic factor $a(t)\propto t^{2/3}$ and dominated by dark matter. We describe the dark matter by a mass density field $\rho({\bf x}, t)$ and a peculiar velocity field ${\bf v}({\bf x},
t)$, where ${\bf x}$ is the comoving coordinate. In hydrodynamical descriptions, the equations of dark matter consist of the continuity, the momentum, and the gravitational potential equations as follows (Wasserman 1978) $$\frac{\partial \delta}{\partial t} +
\frac{1}{a}\nabla \cdot (1+\delta) {\bf v}=0$$ $$\frac{\partial a{\bf v}}{\partial t}+
({\bf v}\cdot \nabla){\bf v}= -\nabla \phi$$ $$\nabla^2 \phi = 4\pi G a^2\bar{\rho}\delta$$ where density perturbation $\delta({\bf x},t)=[\rho({\bf x}, t) - \bar{\rho}]/\bar{\rho}$, and the mean density $\bar {\rho} =1/6\pi Gt^2 \propto a^{-3}$. The peculiar gravitational potential $\phi$ is zero (or constant) when density perturbation $\delta=0$. The operator $\nabla$ is acting on the comoving coordinate ${\bf x}$. For the growth mode of the perturbations, the velocity is irrotational. In this case, one can define a velocity potential by $${\bf v}=- \frac {1}{a}\nabla \varphi$$ With this potential, the momentum equation (A2) can be rewritten as the Bernoulli equation $$\frac{\partial \varphi}{\partial t}-
\frac{1}{2a^2}(\nabla \varphi)^2 = \phi.$$
Since there is only gravitational interaction between dark matter and the IGM or cosmic baryonic gas, it is convenient to describe the IGM by its mass density field $\rho_b({\bf x},t)$ and velocity field ${\bf v}_b({\bf x},t)$. The hydrodynamical equations of the IGM are $$\frac{\partial \delta_b}{\partial t} +
\frac{1}{a}\nabla \cdot (1+\delta_b) {\bf v}_b=0$$ $$\frac{\partial a{\bf v}_b}{\partial t}+
({\bf v}_b\cdot \nabla){\bf v}_b=
-\frac{1}{\rho_b}\nabla p_b - \nabla \phi$$ where the density perturbation of the IGM $\delta_b({\bf x},t)= [\rho_b({\bf x},t)-\bar{\rho}_b]/\bar{\rho}_b$, and $\bar{\rho}_b$ the mean density of the IGM. The gravity of the IGM is negligible. The evolution of the IGM is governed by the gravitation of dark matter, and therefore, the gravitational potential $\phi$ in eq.(A7) is still given by eq. (A3).
To sketch the gravitational clustering of the IGM, we will not consider the details of heating and cooling. Thermal processes are generally local, and therefore, it is reasonable to describe the thermal processes by a polytropic relation $p_b({\bf x},t) \propto \rho_b^{\gamma}({\bf
x},t)$. Thus eq.(A7) becomes $$\frac{\partial a{\bf v}_b}{\partial t}+
({\bf v}_b\cdot \nabla){\bf v}_b=
-\frac{\gamma k_B T}{\mu m_p} \frac{\nabla \delta_b}{(1+\delta_b)}
- \nabla \phi$$ where the parameter $\mu$ is the mean molecular weight of IGM particles, and $m_p$ the proton mass. In this case, we don’t need the energy equation and the IGM temperature evolves as $T \propto \rho^{\gamma-1}$, or $T =T_0(1+\delta_b)^{\gamma-1}$. Eq.(A8) is different from eq.(A2) only by the term with temperature $T$. If we treat this term in the linear approximation, we have $$\frac{\partial \varphi_b}{\partial t}-
\frac{1}{2a^2}(\nabla \varphi_b)^2 - \frac{\nu_b}{a^2}\nabla^2 \varphi_b
=\phi,$$ where $\varphi_b$ is the velocity potential for IGM field defined by $${\bf v}_b= - \frac {1}{a}\nabla \varphi_b.$$ The coefficient $\nu_b$ is given by $$\nu_b=\frac{\gamma k_BT_0}{\mu m_p (d \ln D(t)/dt)},$$ where $D(t)$ describes the linear growth behavior.
Eq.(A9) shows that the clustering of the IGM field is completely controlled by the gravity of underlying mass fields of dark matter. It can then be considered as a “passive substance” with respect to the underlying dark matter. The term with $\nu_b$ in eq.(A9) acts like a viscosity which is due to thermal diffusion characterized by the Jeans length $k_J^2=(a^2/t^2)(\nu m_p/\gamma k_BT_0)$. In linear regime, eqs.(A5) and (A9) yield solution $\varphi_b = \varphi$ on scales larger than the Jeans length.
Equation of the difference between the IGM and dark matter fields
-----------------------------------------------------------------
To study the possible difference between the IGM and dark matter, we define a variable $\Theta$ by $$\Theta = \varphi_b - \varphi$$ which describes the deviation of IGM from dark matter. With eqs. (A5) and (A9), we have $$\frac{\partial \Theta}{\partial t}-
\frac{1}{2a^2}(\nabla \Theta)^2 - \frac{\nu_b}{a^2}\nabla^2 \Theta
= \frac{\nu_b}{a^2}\nabla^2 \varphi +
\frac{1}{a^2}(\nabla \Theta)(\nabla \varphi).$$ When $\nu_b=0$, this has solution $\Theta=$ const, if the initial condition is $\Theta=$ const. During linear evolution, we have $\Theta= 0$, and therefore $\Theta= 0$ is the initial condition for eq.(A13). Thus, the solution $\Theta=0 $ is correct even in the nonlinear regime if $\nu_b=0$. There is no deviation of IGM from dark matter.
In the case of $\nu_b \neq 0$, $\Theta=0 $ will no longer be the solution of eq.(A13) even if it is $\Theta=0$ initially. Considering that the potential $\varphi$ of dark matter essentially is stochastic, the r.h.s. of eq.(13) plays the role of stochastic forces. It drives $\Theta$ to be non-zero. This leads to a statistical or stochastic discrepancy of the IGM field from dark matter field. This discrepancy is not trivial even when we consider the linear approximation of eq.(A13) with respect to $\Theta$, i.e. $$\frac{\partial \Theta}{\partial t}
- \frac{1}{a^2}(\nabla \Theta)(\nabla \varphi)-
\frac{\nu_b}{a^2}\nabla^2 \Theta = \frac{\nu_b}{a^2}\nabla^2 \varphi.$$ Eq.(14) essentially is the same as that widely used to dynamically model the discrepancy, or decoupling, of the statistical properties of the passive substance from the underlying field (for a review, Shraiman & Siggia, 2001). Since the advection term $\frac{1}{a^2}(\nabla \Theta)(\nabla \varphi)$ doesn’t depend on the diffusion scales, the statistical discrepancy of the passive substance from the underlying field can appear on scales larger than the diffusion scale.
The field $\Theta$ given by eqs.(A13) and (A14) is generally intermittent. It has been shown that the field $\Theta$ given by eq.(A14) will be intermittent, even when the field $\varphi$ is Gaussian (Kraichnan, 1994). Eq.(A13) is similar to the stochastic-force-driven Burgers’ equation, or the so-called KPZ equation (Kardar, Parisi, & Zhang 1986, Berera & Fang 1994, Barabási, & Stanley, 1995, Jones 1999), which is typical of the dynamical model of intermittency (Polyakov 1995; Balkovsky et al. 1997; E et al. 1997). Therefore, the difference between the IGM and dark matter fields, $\Theta$, is probably intermittent.
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abstract: 'We generalize the directed percolation (DP) model by relaxing the strict directionality of DP such that propagation can occur in either direction but with anisotropic probabilities. We denote the probabilities as $p_{\downarrow}= p \cdot p_d$ and $p_{\uparrow}=p \cdot (1-p_d)$, with $p $ representing the average occupation probability and $p_d$ controlling the anisotropy. The Leath-Alexandrowicz method is used to grow a cluster from an active seed site. We call this model with two main growth directions [*biased directed percolation*]{} (BDP). Standard isotropic percolation (IP) and DP are the two limiting cases of the BDP model, corresponding to $p_d=1/2$ and $p_d=0,1$ respectively. In this work, besides IP and DP, we also consider the $1/2<p_d<1$ region. Extensive Monte Carlo simulations are carried out on the square and the simple-cubic lattices, and the numerical data are analyzed by finite-size scaling. We locate the percolation thresholds of the BDP model for $p_d=0.6$ and $0.8$, and determine various critical exponents. These exponents are found to be consistent with those for standard DP. We also determine the renormalization exponent associated with the asymmetric perturbation due to $p_d -1/2 \neq 0$ near IP, and confirm that such an asymmetric scaling field is relevant at IP.'
author:
- 'Zongzheng Zhou$^{1}$, Ji Yang$^{1}$, Robert M. Ziff$^2$ $\footnote{Email: [email protected]}$, Youjin Deng$^1$ $\footnote{Email: [email protected]}$'
title: Crossover from isotropic to directed percolation
---
Introduction
============
Directed percolation (DP), introduced in 1957 by Broadbent and Hammersley [@DP], is a fundamental model in non-equilibrium statistical mechanics and represents the most common dynamic universality class [@Marro-Dickman]. DP has a very wide application, including flow in a porous rock in a gravitational field, forest fires, epidemic spreading, and surface chemical reactions [@Grassberger-JSP-79-13]. The DP process can be illustrated in the simple example of bond DP on the square lattice. Along the horizontal (vertical) edges of the lattice, the propagation occurs in a particular direction only, e.g., toward the right (the up). Frequently, the preferred spreading direction is termed “temporal,” and the perpendicular one is called “spatial;” the two-dimensional DP is thus often called “(1+1)-dimensional DP.” The DP process has two distinct phases: the inactive phase for small occupation probability $p$ where the propagation quickly dies out, and the active phase for large $p<1$. Between these two phases, a transition occurs at $p_c$. As the threshold $p_c$ is approached, the temporal ($\parallel$) and the spatial ($\perp$) correlation lengths diverge but with distinct critical exponents: $\xi_{\parallel}\sim|p-p_c|^{-\nu_{\parallel}}$ and $\xi_{\perp}\sim|p-p_c|^{-\nu_{\perp}}$. The anisotropy is characterized by the so-called dynamic exponent $z=\nu_{\parallel}/\nu_{\perp}$. For $p>p_c$, the order parameter ${{\mathcal P}}_{\infty}$, defined as the probability that a randomly selected site can generate an infinite cluster, becomes non-zero and its behavior can be described as ${{\mathcal P}}_{\infty}\sim (p-p_c)^{\beta}$, with $\beta$ another critical exponent. Below the upper critical dimensionality $(d_c+1)$ with $d_c=4$, the three independent critical exponents, $\nu_{\parallel}$, $\beta$, and $z$, are sufficient to describe the DP universality class. While analytical results are scarce for DP, even in $(1+1)$ dimensions, approximation techniques like series expansion [@SE1; @SE2; @SE3; @SE4; @SE5] and Monte Carlo simulations [@Grassberger-JPA-1989; @Grassberger-Zhang; @Lubeck; @Voigt-Ziff] have produced fruitful results. Moreover, after a great deal of efforts, experimental realization of the DP process has been achieved [@Takeuchi; @errotum-Takeuchi; @extension-Takeuchi] in nematic liquid crystals, where the DP transition occurs between two turbulent states.
Analogously, standard isotropic percolation (IP) [@IP] is a fundamental model in equilibrium statistical mechanics. IP has attracted extensive research attention both in the physical and the mathematical communities, and the critical behavior is now well understood. Due to the isotropy, there exists only one spatial correlation length, which scales as $\xi\sim|p-p_c|^{-\nu}$ near $p_c$. Numerous exact results are now available in two dimensions (2D). For bond IP on the square lattice, the self-duality yields the threshold $p_c=1/2$ [@Kesten]; the values of $p_c$ are also exactly known for bond and site percolation on several other lattices [@Ziff-06-0; @Ziff-06-1; @Ziff-10-0], or have been determined to a high precision [@Feng08]. Thanks to conformal field theory and Coulomb gas theory [@Cardy; @Smirnov; @Lawler; @Kesten-1987], the critical exponents $\nu$ and $\beta$ are also exactly known as $\nu = 4/3$ and $\beta=5/36$.
![ (a) State of an edge. (b) A typical cluster in the BDP process. The seed is at site o, and the “infected” sites are denoted as solid dots. Dashed lines represent vacant bonds.[]{data-label="conf"}](conf.eps)
In this work, we introduce a generalized percolation propagation process that contains DP and IP as two special cases. On a given lattice, each edge is assigned to one of the three possible states: occupied by a directed bond along a particular direction, occupied by a directed bond against the particular direction, or unoccupied. This is illustrated in Fig. \[conf\] (a), and the associated probabilities are denoted as $p_{\downarrow}$, $p_{\uparrow}$, and $1-p_{\downarrow}-p_{\uparrow}$, respectively. As a result, the percolation process has two main growth directions. For $p_{\downarrow}=p_{\uparrow}$, the symmetry between the two opposite directions is restored, and the system reduces to standard bond IP. In the limiting case $p_{\downarrow}=0$ or $1$, propagation against or along the particular direction is forbidden, and one has standard DP. We call this percolation model with two main growth directions [*biased directed percolation*]{} (BDP). We note that the BDP model is described by the field-theoretic equation in [@Frey].
A natural question arises: in between standard DP and IP, what is the nature of phase transition for BDP? For later convenience, we replace parameters $p_{\downarrow}$ and $p_{\uparrow}$ by two new variables as $$p_{\downarrow}=p \cdot p_d \; , \hspace{8mm} p_{\uparrow}=p \cdot (1-p_d) \; .
\label{eq_p0pd}$$ The parameter $p$ is the average bond-occupation probability (irrespective of the bond direction), and $p_d$ accounts for the anisotropy. DP corresponds to $p_d=0$ or 1, while $p_d=1/2$ is for IP.
In this work, extensive Monte Carlo simulations are carried out for BDP in two and three dimensions. A dimensionless ratio is defined to locate the percolation threshold. The data are analyzed by finite-size scaling, and the critical exponents are determined. The numerical results suggest that the asymmetric perturbation due to $p_d -1/2 \neq 0$ is relevant near IP, and thus that as long as $p_d \neq 1/2$, BDP is in the DP universality class. These results further raise the following questions, remaining to be explored. For IP, is the asymmetric renormalization exponent a “new” critical exponent or related in some way to the known ones like $\nu$ and $\beta$? Particularly, can this “new” exponent be exactly obtained in two dimensions? If so, what is the exact value?
The remainder of this work is organized as follows. Section II introduces the BDP model, the sampled quantities, and the associated scaling behavior. Numerical results are presented in Secs. III and IV. A brief discussion is given in Sec. V.
Model, sampled quantities, scaling behavior
===========================================
Model
-----
We shall describe in details the BDP model on the square lattice. The generalization to higher dimensions is straightforward.
As usual in the study of DP or IP, we view the BDP model as a stochastic growth process, and use the Leath-Alexandrowicz method [@Leath; @Alex] to grow the percolation cluster starting from a seed site. Given the square lattice and the seed “o” in Fig. \[conf\](b), for each of the neighboring edges of site o, a random number is drawn to determine the edge state. If and only if the edge is occupied and the direction originates from the seed o, the neighboring site is activated and belongs to the growing cluster. For instance, in Fig. \[conf\](b), the four neighboring edges of site o are all occupied, but site c remains inactivated because of the “wrong” direction. After all the four neighboring edges have been visited, one continues the growing procedure from the newly added sites. In other words, one grows the percolation cluster shell by shell (the breadth-first scheme). The growth of the cluster continues until the procedure dies out or the maximum distance is reached, which is set at the beginning of the simulation.
Sampled quantities
------------------
In the cluster-growing process, the number of activated sites $N(s)$ is recorded as a function of the shell number $s$. Let us count the shell of site “o” to be the first shell, the configuration in Fig. \[conf\](b) has $N=3, 5, 6$ for $s=2, 3, 4$, respectively. Besides $N(s)$, one also records the Euclidean distance $r$ of each activated site to the seed “o” for IP and to the $y$ axis for the anisotropic case. The reason for using different definitions of $r$ is that, for the anisotropic case, the average center of activated sites is expected to move linearly along the preferred direction, as $s$ increases. Accordingly, we define a revised gyration radius $R(s)$ as $$\label{eq:gyration_radius}
R(s) = \left\{ \begin{array} {ll}
0 & \hspace{10mm} \mbox{if } N(s) = 0 \\
\sqrt{\sum_{i=1}^N r_i^2/N} & \hspace{10mm} \mbox{if } N(s) \geq 1 \\
\end{array}
\right.$$ The statistical averages ${{\mathcal N}}(s) \equiv \langle N(s) \rangle$ and ${{\mathcal R}}(s) \equiv \langle R(s) \rangle$ are then measured, as well as their statistical uncertainties. We also measure the survival probability ${{\mathcal P}}(s)$ that at least one site remains activated at the $s$th shell and the accumulated activated site number ${{\mathcal A}}(s) \equiv \langle\sum_{s'=1}^s N(s') \rangle $.
In Monte Carlo study of critical phenomena and phase transitions, it is found that dimensionless ratios like the Binder cumulant are very useful in locating the critical point. Therefore, we also define a dimensionless ratio $ Q_{{{\mathcal N}}} (s) = {{\mathcal N}}(2s)/{{\mathcal N}}(s)$.
Scaling behavior
----------------
Near the percolation threshold $p_c$, one expects the following scaling behavior $$\begin{aligned}
{{\mathcal P}}(s,\epsilon) & \sim & s^{-Y_P} \mathbb{P} (\epsilon s^{Y_\epsilon}) \; , \nonumber \\
{{\mathcal N}}(s,\epsilon) & \sim & s^{ Y_N} \mathbb{N} (\epsilon s^{Y_\epsilon}) \; , \nonumber \\
{{\mathcal A}}(s,\epsilon) & \sim & s^{ Y_A} \mathbb{A} (\epsilon s^{Y_\epsilon}) \; , \nonumber \\
{{\mathcal R}}(s,\epsilon) & \sim & s^{ Y_R} \mathbb{R} (\epsilon s^{Y_\epsilon}) \; , \nonumber \\
Q_{{{\mathcal N}}}(s,\epsilon) & \sim & 2^{ Y_N} \mathbb{Q} (\epsilon s^{Y_\epsilon}) \; ,
\label{eq:scaling}
\end{aligned}$$ where $\epsilon = p-p_c$ represents a small deviation from $p_c$. Symbols $Y_P, Y_N, Y_A, Y_R$, and $Y_\epsilon$ denote the associated critical exponents, and $\mathbb{P}, \mathbb{N}, \mathbb{A}$, $\mathbb{R}$, and $\mathbb{Q}$ are universal functions. For simplicity, only one scaling field, which accounts for the effect due to deviation from $p_c$, is explicitly included in Eq. (\[eq:scaling\]). Right at $p_c$, as $s$ increases, the survival probability ${{\mathcal P}}(s)$ decays to zero while the other quantities diverge, except for the ratio $Q_{{\mathcal N}}$ which goes to a constant. A trivial relation is $Y_A = Y_N +1$.
For standard DP ($p_d=0$ or 1), exponents $Y_P$ and $Y_N$ are normally denoted as $\delta$ and $\eta$, respectively ($Y_N$ is also denoted as $\theta$ in [@Hinrichsen-2000]). It can be shown that exponent $Y_\epsilon$ is $Y_\epsilon = 1/\nu_\parallel$. Further, exponent $Y_R$ relates to $\delta$ and the dynamic exponent $z$ as $Y_R = -\delta +1/z$, where $-\delta$ arises from the behavior ${{\mathcal P}}(s) \sim s^{-\delta}$. Below the upper critical dimensionality $(d_c+1)$ with $d_c=4$, there exist three independent exponents, which can be chosen as $\nu_\parallel$, $\beta$, and $z$. The others can be obtained by the scaling relations [@Hinrichsen-2000] $$\nu_\perp = \nu_\parallel/z \; , \hspace{5mm} \delta = \beta/\nu_\parallel \; , \hspace{5mm}
\eta = (d \nu_\perp - 2\beta)/\nu_\parallel \; ,$$ where the last one involves the spatial dimensionality $d$ and is called the hyperscaling relation. In $(1+1)$ dimensions, these exponents have been determined to high precision: $\nu_\parallel = 1.733\,847 (6)$, $\beta = 0.276\,486(8)$, and $z=1.580745(10)$ [@SE4]. In $(2+1)$ dimensions, these exponents are $\nu_\parallel = 1.2890(7)$, $\beta = 0.581\,2(6)$, and $z=1.7665(4)$ [@Grassberger-Zhang; @Voigt-Ziff; @Perlsman; @Junfeng].
For standard IP ($p_d=1/2$), the shell number $s$ is frequently called “chemical distance” [@Havlin], accounting for the minimum length among all the possible paths between the seed site and the activated sites on the $s$th shell. At $p_c$, the length $s$ of the chemical path relates to the Euclidean distance $r$ as $s \sim r^{d_{\rm min}}$ [@Grassberger-JPA-1992; @Grassberger-JPA-1985], with $d_{\rm min} \geq 1$ denoting the shortest-path exponent. In terms of the Euclidean distance $r$, it is known that the survival probability scales as ${{\mathcal P}}(r) \sim r^{-\beta/\nu} $, the accumulated site number ${{\mathcal A}}(r) \sim r^{\gamma/\nu}$, and the $p_c$-deviating scaling behavior $\epsilon r^{1/\nu}$. This immediately yields $Y_P = -\beta/(\nu d_{\rm min}), Y_A = \gamma/(\nu d_{\rm min}),
Y_N = \gamma/(\nu d_{\rm min})-1, Y_R = (1-\beta/\nu)/d_{\rm min}$, and $Y_\epsilon = 1/(\nu d_{\rm min})$. For IP, one has the scaling relation as $$\gamma/\nu = d -2\beta/\nu \; .$$ In 2D, $\nu$ and $\beta$ are exactly known as $\nu = 4/3$ and $\beta = 5/36$, which yield $\gamma/\nu = 43/24 \approx 1.79166\ldots$ and $\beta/\nu = 5/48 \approx 0.104166\ldots$. The shortest-path exponent $d_{\rm min}$, together with the so-called backbone exponent, is among the few critical exponents of which the exact values are not known for the 2D percolation universality class. It was conjectured to be $d_{\rm min} = 217/192 = 1.13020 \ldots$ [@Deng-PRE-81], and some recent estimates are 1.1306(3) [@Grassberger-JPA-32] and 1.13078(5) [@JY]. In three dimensions, no exact results are available, and the numerical estimates are $d_{\rm min} =1.374(4)$ [@Grassberger-JPA-25], $\beta/\nu=0.4774(1)$ and $\nu = 0.8734 (6)$ [@deng], which yield $\beta = 0.4170 (4)$.
Results
=======
In this work, we consider the BDP model on the square lattice for 2D and the simple-cubic lattice for 3D. The simulation applies the aforementioned Leath-Alexandrowicz growth method. The dimensionless ratio $Q_{{\mathcal N}}$ is used to locate the percolation threshold $p_c$. According to Eq. (\[eq:scaling\]), ratio $Q_{{\mathcal N}}$ is expected to have an approximate common intersection at $p_c$ for different shell number $s$. At the threshold $p_c$, as $s \rightarrow \infty$, the common intersection converges to a universal value $2^{Y_N}$ and the slope of $Q_{{\mathcal N}}$ increases as $s^{Y_\epsilon}$.
Standard IP
-----------
Standard IP corresponds to $p_d=1/2$. Monte Carlo simulation was carried out up to $s_{\rm max} = 8192$ for 2D and $2048$ for 3D. About $10^8$ samples were taken for each data point on each lattice. The $Q_{{\mathcal N}}$ data are shown in Fig. \[fig:IP\]. Indeed, we find an approximate common intersection near $p=1$ and $0.4976$ for 2D and 3D, respectively. This agrees with the known threshold $p_c/2=1/2 \; (2D)$ and $0.248\,812\,6 (5) \; (3D)$ [@Lorenz]. Note that, since the occupied bond can propagate the growth process only if it has the correct orientation, there is a factor of 2 difference between the bond-occupation probability $p$ here and the $p$ of the equivalent bond percolation probability.
![Ratio $Q_{{\mathcal N}}$ for IP in 2D (top) and 3D (bottom).[]{data-label="fig:IP"}](IP-QN.eps)
To have a better estimate of $p_c$, according to a least-squared criterion, the $Q_{{\mathcal N}}$ data are fitted by $$\begin{aligned}
&& Q_{{\mathcal N}}(s,\epsilon)=Q_{{{\mathcal N}},c} + \sum_{k=1}^4 q_k \epsilon^k s^{kY_\epsilon} +b_1 s^{y_1} \nonumber \\
&& +b_2 s^{-2} +c \epsilon s^{Y_\epsilon +y_1}+n \epsilon^2 s^{Y_\epsilon} + ...\; ,
\label{eq:fitQ}
\end{aligned}$$ which is obtained by Taylor-expanding Eq. (\[eq:scaling\]) and taking into account finite-size corrections due to the leading irrelevant scaling field and analytical background contribution. These are described by the two terms with amplitudes $b_1$ and $b_2$, of which the term with $n$ arises from the nonlinearity of the relevant scaling field in terms of the deviation $\epsilon$, and the one with $c$ accounts for the combined effect of the leading relevant and irrelevant scaling fields. In the fits, various formulas are tried, which correspond to different combinations of those terms in Eq. (\[eq:fitQ\]). For a given formula, the $Q_{{\mathcal N}}$ data for small $s <s_{\rm min}$ are gradually excluded from the fits to see how the residual $\chi^2$ changes with respect to $s_{\rm min}$. The results from different formulas are compared with each other to estimate the possible systematic errors. In two dimensions, we obtain $p_c = 1.000\, 000(4)$, $Q_{{{\mathcal N}},c} = 1.499\, 5(1)$, $Y_\epsilon = 0.664(3)$, and $y_1 = -0.96 (6)$. Note that the leading irrelevant thermal scaling field is $ \omega = -2$ for 2D percolation universality [@Ziff-PRE-2011]; apparently, the leading correction exponent $y_1 = -0.96$ does not correspond to $\omega$. Instead, $y_1$ should be associated with the chemical distance. From the relations $Q_{{{\mathcal N}},c} = 2^{Y_N}$, $Y_N = \gamma/(\nu d_{\rm min}) -1$, and $Y_\epsilon = 1/ (\nu d_{\rm min})$, and the exact values $\gamma/\nu =43/24$ and $1/\nu = 3/4$, we determine $d_{\rm min} = 1.130\,76 (10)$ from $Q_{{{\mathcal N}},c} = 1.499\, 5(1)$, and $d_{\rm min} = 1.130 (6)$ from $Y_\epsilon = 0.664(3)$.
In three dimensions, our results are $p_c = 0.497\,624(1)$, $Q_{{{\mathcal N}},c} = 1.400(1)$, $Y_\epsilon = 0.830(1)$, and $y_1 = -0.8(2)$. Our estimate of $p_c/2 = 0.248\,812\,0(5)$ agrees with the existing one $0.248\,812\,6(5)$ [@Lorenz], and has a comparable error margin.
$\beta$ $\nu$ $d_{\rm min}$ $p_c/2$
------ ----------- ------------------------------------------------------- ------------------------------------------------------ ---------------------------------------------------------- ------------------------------
$2D$ (known) $5/36$ [@IP; @Cardy; @Smirnov; @Lawler; @Kesten-1987] $4/3$ [@IP; @Cardy; @Smirnov; @Lawler; @Kesten-1987] $1.130\,6 (3)$ [@Grassberger-JPA-32; @Deng-PRE-81; @JY] $1/2$ [@IP; @Kesten]
(present) $0.138\,7(10)$ $1.332(6)$ $1.130\,76 (10)$ $0.500\, 000(2)$
$3D$ (known) $0.4167(4)$ $0.873\,4(6)$ [@deng] $1.374(4)$ [@Grassberger-JPA-25] $0.248\,812\,6(5)$ [@Lorenz]
(present) $0.417(1)$ $0.876(2)$ $1.375(1)$ $0.248\,812\,0(5)$
To estimate other critical exponents, we simulate right at the threshold $p/2 =1/2$ for 2D and $0.248\, 812 \, 0$ for 3D. The simulation was carried out for $s$ up to $s_{\rm max} =8192$ for 2D and 2048 for 3D. Further, to eliminate one more unknown parameter in the fits, we measure the dimensionless ratios $Q_{{\mathcal P}}(s) = {{\mathcal P}}(2s)/{{\mathcal P}}(s)$ and $Q_{{\mathcal R}}= {{\mathcal R}}(2s)/{{\mathcal R}}(s)$. These $Q$ data are fitted by $$Q(s) = Q_c + b_1 s^{y_1} + b_2 s^{-2} \; .
\label{eq:fitQc}$$ In two dimensions, the results are $Q_{{{\mathcal P}},c} = 0.9382(1)$ and $y_1 = -0.80(7)$ for $Q_{{\mathcal P}}$, and $Q_{{{\mathcal R}},c} = 1.7318(2)$ and $y_1 = -0.9(1)$ for $Q_{{\mathcal R}}$. For all these three ratios, the leading correction is described by an exponent $y_1 \approx -1$. Taking into account the exact values $\beta/\nu = 5/48$, one has $d_{\rm min} = 1.132(2)$ from $Q_{{{\mathcal P}},c}$ and $d_{\rm min} = 1.130\, 7(3)$ from $Q_{{{\mathcal R}},c}$.
In three dimensions, the results are $Q_{{{\mathcal P}},c} = 0.7865(2)$ and $y_1 = -0.7(2)$ for $Q_{{\mathcal P}}$, and $Q_{{{\mathcal R}},c} = 1.3020(3)$ and $y_1 = -0.9(2)$ for $Q_{{\mathcal R}}$. Combining the estimate $Q_{{{\mathcal P}},c}$ and $Q_{{{\mathcal R}},c}$ together, one has $\beta/\nu = 0.4765(8)$ and $d_{\rm min} = 1.375(1)$. Our result $d_{\rm min} = 1.375(1)$ agrees well with the existing result $d_{\rm min} = 1.374(4)$ [@Grassberger-JPA-25], and significantly improves the error margin.
For comparison, these results are summarized in Table \[tab:IP\].
Standard DP
-----------
We simulate standard DP by taking $p_d=1$. The simulation was carried out for $s$ up to $s_{\rm max} = 16384$ for 2D, and 2048 for 3D. The number of samples for each data point is about $8 \times 10^8$ in 2D and $1.6 \times 10^8$ in 3D.
The $Q_{{\mathcal N}}$ data are shown in Fig. \[fig:DP\]. A good intersection is observed for both 2D and 3D, which yields $p_c = 0.64470 $ for 2D and $0.38222$ for 3D, from a rough visual fitting.
![Ratio $Q_{{\mathcal N}}$ for standard DP in 2D (top) and 3D (bottom).[]{data-label="fig:DP"}](dp-QN.eps)
We fit the $Q_{{\mathcal N}}$ data more precisely using Eq. (\[eq:fitQ\]). On the square lattice, we obtain $p_c=0.644\,700\,5(8)$, $Q_{{{\mathcal N}},c}=1.242\, 9(2)$, $Y_\epsilon = 0.576(3)$, and $y_1=-0.9(1)$. The estimate of the percolation threshold agrees well with the existing more precise result $0.644\,700\,185 (5)$ [@SE4]. From the relations $Q_{{{\mathcal N}},c} = 2^{Y_N} = 2^\eta$ and $Y_\epsilon = 1/ \nu_\parallel$, we have $\eta = 0.313\,7 (2)$ and $\nu_\parallel = 1.736 (9)$. On the simple-cubic lattice, our results are $p_c=0.382\,225\,6 (5)$, $Y_\epsilon = 0.777(2)$, $Q_{{{\mathcal N}},c}=1.1738(1)$, which yield $\eta = 0.2312(1)$ and $\nu_\parallel = 1.287(4)$. Here the $y_1$ is too small to estimate since the numerical data of $s\geq 24$ can be well described even though we do not include any corrections. The agreement of $p_c$ with the existing estimate $p_c = 0.382\,224\,64 (4)$ [@Junfeng] is within two standard deviations.
Analogously, we simulate right at the percolation threshold $p_c = 0.644\,700\,185$ for 2D and $p_c = 0.382\,224\,64$ for 3D. The dimensionless ratios $Q_{{\mathcal P}}$ and $Q_{{\mathcal R}}$ are measured, and the data are fitted by Eq. (\[eq:fitQc\]). For 2D, the results are $Q_{{{\mathcal P}},c} = 0.89537(5)$, $y_1 = -0.98(5)$ and $Q_{{{\mathcal R}},c} = 1.3882(1)$, $y_1=-1.1(1)$, which yield $Y_P = \delta = 0.159\,44 (9)$ and $Y_{R} = (-\delta +1/z) = 0.47322 (10)$. Taking into account the estimates of $\nu_{\parallel}$ and $\delta$, one has $\nu_{\perp} = 1.098(6)$. For 3D, the results are $Q_{{{\mathcal P}},c} = 0.7311(4)$ and $Q_{{{\mathcal R}},c} = 1.0822(1)$, which yield that $\delta = 0.4519(8)$ and $\nu_{\perp} = 0.728(4)$.
These results are listed in Table \[tab:DP\].
BDP
---
![Ratio $Q_{{\mathcal N}}$ for BDP in 2D. The top (bottom) panel corresponds to $p_d = 0.8$ ($0.6$) case.[]{data-label="fig:BDP2D"}](2d-bdp.eps)
![Ratio $Q_{{\mathcal N}}$ for BDP in 3D. The top (bottom) panel corresponds to $p_d = 0.8$ ($0.6$) case.[]{data-label="fig:BDP3D"}](3d-bdp.eps)
For the purpose of studying BDP, we choose $p_d=0.6$ and $0.8$. The simulation was carried out for $s$ up to $s_{\rm max} = 16384$ for 2D and 2048 for 3D. About $2 \times 10^8 $ samples were taken for each data point in each case.
The $Q_{{\mathcal N}}$ data are shown in Fig. \[fig:BDP2D\] for 2D and Fig. \[fig:BDP3D\] for 3D. The transitions are also clearly observed, but the approximate common intersections are not as good as those for standard DP and IP. This suggests the existence of additional finite-size corrections.
The $Q_{{\mathcal N}}$ data are also fitted by Eq. (\[eq:fitQ\]) according to a least-squared criterion. To account for the possible existence of additional corrections, we replace the terms in Eq. (\[eq:fitQ\]), with $b_1$, $b_2$, and $c$, by $b_is^{y_i}+b_1s^{y_1}+c \epsilon s^{y_i+y_\epsilon}$. The exponent $y_1$ is fixed at $-1$, in accordance with our above estimate of $y_1$ for both standard IP and DP. Indeed, the new source of finite-size correction can be identified in the fits, which yield $y_i = -0.5 (2)$ both in 2D and 3D. The results for $p_c$, $\eta = \log_2 Q_{{{\mathcal N}},c}$, and $\nu_{\parallel} = 1/Y_\epsilon$ are summarized in Table \[tab:DP\].
D Ref. $p_d$ $p_{c}$ $\beta$ $\nu_{\parallel}$ $z$ $\eta$ $\delta$
--- ------------ ------- ---------------------- ----------------- ------------------- ------------------ ----------------- -----------------
2 [@SE4] 1 $0.644\,700\,185(5)$ $0.276\,486(8)$ $1.733\,847(6)$ $1.580\,745(10)$ $0.313\,686(8)$ $0.159\,464(6)$
1 0.6447005(8) 0.277(2) 1.736(9) 1.5806(3) 0.3137(2) 0.15944(9)
0.8 0.768708(1) 0.278(2) 1.74(1) 1.577(5) 0.3141(4) 0.1595(1)
0.6 0.929668(3) 0.279(2) 1.754(6) 1.578(5) 0.3161(8) 0.159(1)
3 [@Junfeng] 1 0.38222464(4) 0.5812(6) 1.2890(7) 1.7665(2) 0.23081(7) 0.4509(2)
1 0.3822256(5) 0.582(5) 1.287(4) 1.767(3) 0.2312(1) 0.4519(8)
0.8 0.430941(2) 0.577(5) 1.289(5) 1.77(1) 0.229(3) 0.448(2)
0.6 0.481310(2) 0.583(8) 1.292(5) 1.76(2) 0.226(9) 0.452(4)
The determination of the critical exponents $\delta$ and $z$ is obtained in an analogous way by simulating at the estimated percolation threshold, and the results are listed in Table \[tab:DP\].
The results in Table \[tab:DP\] strongly suggest that, as long as $p_d$ deviates from $1/2$, the system falls into the standard DP universality class. For an illustration, we make the log-log plot of the critical quantity ${{\mathcal N}}$ versus the shell number $s$ in Fig. \[fig:N\]. Clearly, the slope for $p_d=1/2$ is distinct from those for the other cases, which are independent of $p_d$.
![Log-log plot of ${{\mathcal N}}$ versus $s$ at $p_c$. The bottom (top) panel is for 2D (3D). It is clearly seen that the slope is identical for all the $p_d \neq 1/2$ cases, and is distinct from that of IP ($p_d=1/2$).[]{data-label="fig:N"}](N.eps)
Crossover exponent
==================
The fact that BDP for $p_d \neq 1/2$ is in the DP universality means that in the language of renormalization group theory, the operator associated with the asymmetric perturbation is relevant near the IP fixed point. To confirm this, we simulate BDP near IP with $p=p_c=1$ by varying $\epsilon_d=p_d-1/2$. The simulation is up to $s_{\rm max}=8192$, and $\epsilon_d$ is set at $0$, $10^{-3}$ and $2\times10^{-3}$. The results for $Q_{{{\mathcal N}}}$ in two dimensions are shown in Fig. \[fig:fix\_p\] versus $\epsilon_d^2$; note that BDPs for $\pm \epsilon_d$ are identical. These $Q_{{{\mathcal N}}}$ data are also analyzed by Eq. (\[eq:fitQ\]) with $Y_{\epsilon}$ being replaced by the exponent $Y_{\epsilon_d}$ for the symmetric scaling field and the odd terms with respect to $\epsilon_d$ being set zero. We obtain $Y_{\epsilon_d}=0.500(5)$, which suggests that $Y_{\epsilon_d}$ may be exactly $1/2$.
![Top: Ratio $Q_{{{\mathcal N}}}$ versus $(p_d - 1/2)^2$ with $p = 1$ on square lattice. Bottom: Log-log plot of $p_{dc}-1/2$ versus $1-p_c$ for the transition line $(p_{dc},p_c)$ near IP. The dashed line has slope $0.754$.[]{data-label="fig:fix_p"}](fix_p.eps)
According to scaling theory, the phase transition line $(p_c,p_{dc})$ approaches to the critical IP $(p_c=1,p_{dc}=1/2)$ as [@Riedel-1969] $$1 - p_c \propto |(p_{dc}-1/2)|^{1/\phi} \; , \label{eq:cross_exponent}$$ where $\phi = Y_{\epsilon_d}/Y_{\epsilon}$ is the so-called crossover exponent. We carried out some Monte Carlo simulations and determined a set of critical points near IP; they are 13 critical points with $p_{dc}\in [0.52,0.6]$. In Fig. \[fig:fix\_p\], we plot $p_{dc}-1/2$ versus $1-p_c$ in log-log scale, which indeed has slope approximately equal to $\phi = Y_{\epsilon_d}/Y_{\epsilon}=0.754$.
We also perform a similar study near the critical IP in 3D, and obtain $Y_{\epsilon_d}=0.56(1)$ and $\phi=0.67(1)$.
Discussion
==========
We introduce a biased directed percolation model, which includes standard isotropic and directed percolation as two special cases. Large-scale Monte Carlo simulations are carried out in two and three dimensions. We find that the operator associated with the anisotropy is relevant near the IP fixed point, which implies that BDP in the region $p_d\neq1/2$ is in the DP university class. On this basis, the phase diagram and the associated renormalization flows are shown in Fig. \[fig:phase\_diagram\].
![Phase diagram of the BDP model in 2D (left) and 3D (right). The $p_d=1/2$ line corresponds to isotropic percolation. The diagram for $p_d<1/2$ is drawn by symmetry. The arrows represent the direction of the renormalization flows.[]{data-label="fig:phase_diagram"}](phase.eps)
Since the upper critical dimensionality is different for standard IP and DP, it is not clear whether the similar renormalization flows would hold in higher dimensions. We mention that such crossover phenomena have attract much attention both in the fields of equilibrium and non-equilibrium statistical mechanics [@Pfeuty; @Aharony; @Lubeck-JSM; @Mendes; @Frojdh; @Janssen; @Schonmann-JSP-1986]. In retrospect, it is not surprising that the asymmetric perturbation is relevant near IP. At IP, all the directions are equivalent and “spatial” and “temporal” directions cannot be defined. However, as soon as $p_d -1/2 \neq 0$, such a symmetry is broken and the center of the activated sites moves along the “temporal” direction as the growing process continues. It is also plausible that as long as the “spatial” and “temporal” symmetry is not restored, such an asymmetric perturbation is irrelevant near DP. This is similar to the fact that asymmetric diffusion on the basic contact process is irrelevant [@Schonmann-JSP-1986]. In terms of the chemical distance $s$, the effect from the anisotropy can be asymptotically described as $\propto (p_d-1/2) s^{Y_{\epsilon_d}}$ with $Y_{\epsilon_d}(2D) = 0.500(5)$ and $Y_{\epsilon_d}(3D) = 0.589(10)$. One can also use the Euclidean distance $r$ to describe such an anisotropic effect as $\propto (p_d-1/2) r^{1/\nu_d}$ with $Y_{\epsilon_d}=1/(\nu_d d_{\rm min})$. Substituting the $d_{\rm min}$ value into $Y_{\epsilon_d} $, one obtains $\nu_d (2D) = 1.77(1)$ and $\nu_d (3D) = 1.30(2)$.
When viewing standard isotropic percolation in the framework of BDP, one observes that two independent critical exponents, e.g., $\nu$ and $\beta$, are no longer sufficient to describe the critical scaling behavior. In this case, the shortest-path exponent $d_{\rm min}$ appears naturally and becomes indispensable, and thus isotropic percolation also has three independent critical exponents. Our estimate of $d_{\rm min}$ significantly improves over the existing results both in two and three dimensions. Our result $d_{\rm min} = 1.130\,76(10)$ does not agree with the recently conjectured value $217/192$ [@Deng-PRE-81] in two dimensions. This result appears to refute the conjectured value. On the other hand, we note that, in terms of the chemical distance $s$, a new source of finite-size corrections occurs in the scaling behavior, and these corrections are not well understood yet. Further, we observe that the restored symmetry for IP can be regarded as $\nu_\parallel = \nu_\perp$ in the BDP model. In some cases, the coincidence of two critical exponents may suggest the existence of logarithmic corrections of the $\log$ or $\log \log$ form, and they can be either additive or multiplicative. In practice, logarithmic finite-size corrections have indeed been observed for standard isotropic percolation in two dimensions [@Feng08], which is in terms of Euclidean distance. In this sense, we cannot entirely exclude the possibility that the tiny difference between the present numerical result for $d_\mathrm{min}$ and the conjectured value arises from some unknown corrections that have not been taken into account in the numerical analysis. Numerical investigation of this problem seems very difficult if not impossible. Nevertheless, since the exact value of $d_{\rm min}$ is conjectured as a function of $q$ for the $q$-state Potts model [@Deng-PRE-81], one can accumulate more numerical evidence by studying the $q \neq 1$ case.
Finally, the numerical estimate of the critical exponent $Y_{\epsilon_d}$ or $\nu_d$ due to the asymmetric perturbation near IP raises a question: is it a “new” independent critical exponent or simply related in some way to the known ones like $\beta$, $\nu$, and $d_{\rm min}$? In particular, in two dimensions, one would ask whether $\nu_d$ or $Y_{\epsilon_d}$ can be exactly obtained in the framework of Stochastic Loewner Evolution (SLE), conformal field theory or Coulomb gas theory.
Acknowledgments
===============
This work was supported in part by NSFC under Grant No. 10975127 and 91024026, and the Chinese Academy of Science. RMZ acknowledges support from National Science Foundation Grant No. DMS-0553487. We also would like to thank Dr. Timothy M. Garoni in Monash University for valuable comments.
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abstract: 'A recent work of the authors on the analysis of pairwise comparison matrices that can be made consistent by the modification of a few elements is continued and extended. Inconsistency indices are defined for indicating the overall quality of a pairwise comparison matrix. It is expected that serious contradictions in the matrix imply high inconsistency and vice versa. However, in the 35-year history of the applications of pairwise comparison matrices, only one of the indices, namely $CR$ proposed by Saaty, has been associated to a general level of acceptance, by the well known ten percent rule. In the paper, we consider a wide class of inconsistency indices, including $CR$, $CM$ proposed by Koczkodaj and Duszak and $CI$ by Peláez and Lamata. Assume that a threshold of acceptable inconsistency is given (for $CR$ it can be 0.1). The aim is to find the minimal number of matrix elements, the appropriate modification of which makes the matrix acceptable. On the other hand, given the maximal number of modifiable matrix elements, the aim is to find the minimal level of inconsistency that can be achieved. In both cases the solution is derived from a nonlinear mixed-integer optimization problem. Results are applicable in decision support systems that allow real time interaction with the decision maker in order to review pairwise comparison matrices.'
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\
Sándor BOZÓKI$^{1}$\
\
János FÜLÖP\
\
Attila POESZ\
\
\
Research was supported in part by OTKA grant K77420.
**Keywords:** Multi-attribute decision making, pairwise comparison matrix, inconsistency, mixed 0-1 convex programming\
Introduction {#intro}
============
Pairwise comparison matrices (Saaty, 1977) are used in multi-attribute decision problems, where relative importance of the criteria, the evaluations of the alternatives with respect to each criterion are to be quantified. The method of pairwise comparison is also applied for determining voting powers in group decision making. One of the advantages of pairwise comparison matrices is that the decision maker is faced to a sequence of elementary questions concerning the comparison of two criteria/alternatives at a time, instead of a complex task of providing the weights of the whole set of them.
A real $n \times n$ matrix $A$ is a *pairwise comparison matrix* if it is positive and reciprocal, i.e.,$$\begin{aligned}
a_{ij}&>&0, \label{eq:positive}\\
a_{ij}&=&\frac{1}{a_{ji}} \label{eq:reciprocal}\end{aligned}$$ for all $i,j=1,\dots,n$. ${A}$ is *consistent* if the transitivity property $$\begin{aligned}
a_{ij} a_{jk} = a_{ik} \label{eq:TransitivityProperty} $$ holds for all $i,j,k=1,2,\ldots,n$; otherwise it is called *inconsistent*.
For a positive $n\times n$ matrix $A$, let $\bar A=\log A$ denote the $n\times n$ matrix with the elements $$\bar a_{ij}=\log a_{ij}, \ \ i,j=1,\dots ,n.$$ Then $A$ is consistent if and only if $$\begin{aligned}
\bar a_{ij}+\bar a_{jk}+\bar a_{ki}=0, \ \forall \, i,j,k=1,\dots ,n \label{eq:subspace}\end{aligned}$$ holds. Matrices $\bar A$ fulfilling the homogenous linear system (\[eq:subspace\]) constitute a linear subspace in $\mathbb{R}^{n\times n}$.
Let ${\cal P}_n$ denote the set of the $n\times n$ pairwise comparison matrices, and ${\cal C}_n\subset{\cal P}_n$ the set of the consistent matrices. Since the reciprocity constraint (\[eq:reciprocal\]) corresponds to $\bar a_{ij}=-\bar a_{ji}$ in the logarithmized space, the set $\log {\cal P}_n=\{\log A \mid A \in {\cal P}_n\}$ is the set of $n\times n$ skew-symmetric matrices, an $n(n-1)/2$-dimensional linear subspace of $\mathbb{R}^{n\times n}$. The set $\log
{\cal C}_n=\{\log A \mid A\in {\cal C}_n\}$ is the set of matrices fulfilling (\[eq:subspace\]), and as pointed out in Chu (1997), is an $(n-1)$-dimensional linear subspace of $\mathbb{R}^{n\times n}$. Clearly, $\log {\cal C}_n\subset \log {\cal P}_n$.
In decision problems of real life, the pairwise comparison matrices are rarely consistent. Nevertheless, decision makers are interested in the level of inconsistency of their judgements, which somehow expresses the goodness or “quality” of pairwise comparisons totally, because conflicting judgements may lead to senseless decisions. Therefore, some index is needed to measure the possible contradictions and inconsistencies of the pairwise comparison matrix.
A function $\phi_n : {\cal P}_n \to R$ is called an *inconsistency index* if $\phi_n (A)=0$ for every consistent and $\phi_n (A)>0$ for every inconsistent pairwise comparison matrix $A$. The inconsistency indices used in the practice are continuous, and the value of $\phi_n (A)>0$ indicates, more or less, how much an inconsistent matrix differs from a consistent one.
Since in the practice the consistency of a pairwise comparison matrix is not easy to assure, certain level of inconsistency is usually accepted by the decision makers. This works in the practice in such a way that for a given inconsistency index $\phi_n$ an acceptance threshold $\alpha_n \ge 0$ is chosen, and a matrix $A\in {\cal P}_n$ is kept for further use only if $\phi_n (A)\le \alpha_n$ holds; otherwise, it is rejected or the pairwise comparisons are carried out again. The carrying out of all pairwise comparisons for filling-in the matrix is often a time-consuming task. Therefore, before the total rejection of a pairwise comparison matrix with an inconsistency level above a prescribes acceptance threshold, it may be worth investigating whether it is possible to improve the inconsistency of the matrix to an acceptable level by performing fewer pairwise comparisons.
The paper will concentrate on the following problem: for a given $A\in {\cal P}_n$, inconsistency index $\phi_n$ and acceptance level $\alpha_n$, what is the minimal number of the elements of matrix $A$ that by modifying these elements, and of course their reciprocals, the pairwise comparison matrix can be made acceptable. We shall show that under a slight boundedness assumption, this can be achieved by solving a nonlinear mixed 0-1 optimization problem. If it comes out that the matrix can be turned into an acceptable one by modifying relatively few elements, then it may be a case when a more-or-less consistent evaluator was less attentive at these few elements, or a data-recording error happened. So it may be worth re-evaluating these elements. Of course, if the the evaluator insists on the previous values, or the acceptable inconsistency threshold cannot be reached with the new values, then this approach was unsuccessful: all pairwise comparisons are to be evaluated again. If however after the revision of the critical elements, the inconsistency level of the modified matrix is already acceptable, then we can continue the decision process with it.
Concerning the investigations above, when solving the nonlinear mixed 0-1 programming problems, it is very beneficial if the nonlinear optimization problems obtained after the relaxation of the 0-1 variables are convex optimization problems. In the convex case several sophisticated methods and softwares are available, while in the nonconvex case methodological and implementation difficulties may arise. Since $\log {\cal C}_n$ is a linear subspace, ${\cal C}_n$ is a nonconvex manifold in $\mathbb{R}^{n\times n}$. One can immediately conclude that it is better to investigate the convexity issues in the logarithmized space.
Several proposals of inconsistency indices are known, see the overviews of Brunelli and Fedrizzi (2011, 2013a) and Brunelli et al. (2013b) for detailed lists and properties. This paper focuses on three well-known inconsistency indices. They are $CR$ proposed by Saaty (1980), $CM$ proposed by Koczkodaj and Duszak (Koczkodaj 1993; Duszak and Koczkodaj 1994), and $CI$ proposed by Peláez and Lamata (2003). The properties and relationship of the fundamental indices $CR$ and $CM$ were also studied in Bozóki and Rapcsák (2008). In this paper we point out that for the inconsistency indices in our focus, the nonlinear mixed 0-1 optimization problems mentioned above can be formulated in the logarithmized space, and appropriate convexity properties hold on them. We show that $CR$ and $CI$ are convex function in the logarithmized space, and $CM$ is quasiconvex, but can be transformed into a convex function by applying a suitable strictly monotone univariate function on it.
This paper is in a close relation to an earlier paper of the authors (Bozóki et al. 2011b). In the latter paper we investigated the special case when the acceptance threshold $\alpha_n$ is 0, i.e. the modified pairwise comparison matrix must be consistent. No inconsistency indices were needed for this investigation, simple graph theoretic ideas were applied. Unfortunately, the technique applied for $\alpha_n=0$ cannot be extended to the general case, therefore, a new approach is proposed in this paper.
We also mention that some of the issues investigated in this paper were already considered, in Hungarian, in Bozóki et al. (2012).
Since inconsistent matrices are in the focus of this paper, and for $n=1$ and $n=2$ the pairwise comparison matrices are consistent, we shall assume in the sequel, without loss of generality, that $n\ge 3$.
In Section 2, the optimization problems to be solved are presented in a general form. The general issues are specialized and investigated for the inconsistency indices $CR$ of Saaty, $CM$ of Koczkodaj and Duszak, and $CI$ proposed by Peláez and Lamata in Sections 3 through 5, respectively. A numerical example is presented in Section 6.
The general form of the optimization problems
=============================================
Let $\phi_n$ be an inconsistency index and $\alpha_n$ be an acceptance threshold, and let $$\begin{aligned}
{\cal A}_n(\phi_n,\alpha_n)=\{A\in {\cal P}_n \mid \phi_n (A)\le \alpha_n\} \label{eq:Cal_A_n}\end{aligned}$$ denote the set of $n\times n$ pairwise comparison matrices with inconsistency $\phi_n$ not exceeding threshold $\alpha_n$. Let $A, \hat A \in {\cal P}_n$ and $$\begin{aligned}
d(A,\hat A)=\,\mid\!\{(i,j): 1\le i<j\le n, a_{ij}\ne \hat
a_{ij}\}\!\mid\label{eq:d}\end{aligned}$$ denote the number of matrix elements above the main diagonal, where matrices $A$ and $\hat A$ differ from each other. By reciprocity, the number of different elements is the same as in positions below the main diagonal.\
Consider pairwise comparison matrix $A \in {\cal P}_n$ with $\phi_n (A)> \alpha_n$ as it is not acceptable in terms of inconsistency. We want to calculate the minimal number of matrix elements above the main diagonal to be modified in order to make matrix acceptable (elements below the main diagonal are determined by the elements above the main diagonal). That is to solve the optimization problem $$\begin{array}{ll}
\min &d(A,\hat A)\\
{\rm s.t.} & \hat A\in {\cal A}_n(\phi_n,\alpha_n),
\end{array}
\label{eq:min_d}$$ where the elements above the main diagonal of $\hat A$ are variables.
We could also ask the minimal inconsistency of $A \in {\cal P}_n$ matrix can be reached by modifying at most $K$ elements and their reciprocals. The optimization problem is $$\begin{array}{ll}
\min &\alpha\\
{\rm s.t.} & d(A,\hat A)\le K,\\
&\hat A\in {\cal A}_n(\phi_n,\alpha),
\end{array}
\label{eq:min_alpha}$$ where $\alpha$ and the elements above the main diagonal of $\hat A$ are variables.
Problems (\[eq:min\_d\]) and (\[eq:min\_alpha\]) can be formulated in logarithmic space: $$\begin{aligned}
\log {\cal A}_n(\phi_n,\alpha_n)=\{X\in \log {\cal P}_n \mid \phi_n (\exp
X)\le \alpha_n\}, \label{eq:log_Cal_A_n}\end{aligned}$$ therefore (\[eq:min\_d\]) is equivalent to $$\begin{array}{ll}
\min &d(\log A, X)\\
{\rm s.t.} & X\in \log {\cal P}_n,\\
& \phi_n(\exp X)\le \alpha_n,
\end{array}
\label{eq:log_min_d}$$ where elements above the main diagonal of $X$ are variables. The first constraint in (\[eq:log\_min\_d\]) means that $X$ belongs to the subspace of skew-symmetric matrices. In this paper we show that the second, nonlinear inequality is a convex constraint in case of inconsistency indices $CR$ (Saaty 1980), $CM$ (Koczkodaj 1993; Duszak and Koczkodaj 1994) and $CI$ (Peláez and Lamata, 2003).
Problem (\[eq:min\_alpha\]) can be rewritten in the same way as above: $$\begin{array}{ll}
\min &\alpha\\
{\rm s.t.} & d(\log A,X)\le K,\\
& X\in \log {\cal P}_n,\\
& \phi_n(\exp X )\le \alpha,
\end{array}
\label{eq:log_min_alpha}$$ where $\alpha$ and elements above the main diagonal of $X$ are variables.
The objective function $d$ can be replaced by using the well-known “Big M” method. Assume that $M\ge 1$ is given as an upper bound of the values of the elements in $A\in {\cal P}_n$ and the computed $\hat A \in {\cal P}_n$ matrices, which is determined as the optimal solution of problems (\[eq:min\_d\]) and (\[eq:min\_alpha\]), i.e., $$\begin{aligned}
1/M\le a_{ij}\le M,\ 1/M\le \hat a_{ij}\le M,\ i,j=1,\dots ,n.
\label{eq:bigm1}\end{aligned}$$
We can find such an upper bound $M$ if we get a bounded interval by knowing the actual level of $\phi_n$, which contains at least one optimal solution of problems (\[eq:min\_d\]), and (\[eq:min\_alpha\]).
On the other hand, if a theoretical upper bound $M$ is not given, then a reasonable bound $M$ is usually determined on the values of the pairwise comparison matrices in every specific problem. Constraint (\[eq:bigm1\]) can be described as $$\begin{aligned}
A, \hat A\in [1/M,M]^{n\times n} \label{eq:bigm2}\end{aligned}$$ in matrix form, and if the condition (\[eq:bigm2\]) associated with $\hat A$ is attached to problems (\[eq:min\_d\]) and also (\[eq:min\_alpha\]), we get $$\begin{array}{ll}
\min &d(A,\hat A)\\
{\rm s.t.} & \hat A\in {\cal A}_n(\phi_n,\alpha_n)\cap [1/M,M]^{n\times n},
\end{array}
\label{eq:min_d2}$$ and, respectively, $$\begin{array}{ll}
\min &\alpha\\
{\rm s.t.} & d(A,\hat A)\le K,\\
&\hat A\in {\cal A}_n(\phi_n,\alpha)\cap [1/M,M]^{n\times n}.
\end{array}
\label{eq:min_alpha2}$$
Introduce $\bar M=\log M$, problems (\[eq:min\_d2\]) and (\[eq:min\_alpha2\]) become equivalent to $$\begin{array}{ll}
\min &d(\log A, X)\\
{\rm s.t.} & X\in \log {\cal P}_n \cap [-\bar M,\bar M]^{n\times n},\\
& \phi_n(\exp X)\le \alpha_n,
\end{array}
\label{eq:log_min_d2}$$ and $$\begin{array}{ll}
\min &\alpha\\
{\rm s.t.} & d(\log A,X)\le K,\\
& X\in \log {\cal P}_n \cap [-\bar M,\bar M]^{n\times n},\\
& \phi_n(\exp X )\le \alpha.
\end{array}
\label{eq:log_min_alpha2}$$ in the logarithmic space.
The “Big M” method can be applied for (\[eq:log\_min\_d2\]) and (\[eq:log\_min\_alpha2\]). Let $\bar A=\log A$, and introduce binary variables $y_{ij}\in \{0,1\},\ 1\le i <j \le n$. Problem (\[eq:log\_min\_d2\]) can be altered by using $\bar A \in [-\bar M,\bar M]^{n\times n}$ into the following mixed 0-1 programming problem: $$\begin{array}{rllllll}
\min &&\sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^{n} y_{ij}\\
\mathrm{s.t.}&& \phi_n(\exp X)\le \alpha_n, \\
&& x_{ij}=-x_{ji},&& 1\le i \le j \le n,\\
&& -\bar M\le x_{ij} \le \bar M, &&1\le i <j \le n,\\
&& -2\bar My_{ij}\le x_{ij}-\bar a_{ij} \le 2\bar My_{ij}, &&1\le i <j \le n,\\
&& y_{ij} \in \{0,1\},&&1\le i <j \le n.\\
\end{array}
\label{eq:log_min_d3}$$
The optimal value of (\[eq:log\_min\_d3\]) gives the minimal number of the matrix elements above the main diagonal to be modified in order to achieve $\phi_n \leq \alpha_n.$ In the optimal solution, $y_{ij}=1$ indicates the matrix elements that (and their reciprocal pairs) are modified, and $\exp x_{ij}$ gives a feasible value of these elements.
Problem (\[eq:log\_min\_d3\]) may have multiple optimal solutions with respect to the binary variables. If all of them are of interest, we list them one by one as follows. Assume that $L^*$ is the optimum value of the problem (\[eq:log\_min\_d3\]), $y_{ij}^*$, $1\le i<j\le n$, is an optimal solution and $I_0^*=\{(i,j)\mid y_{ij}^*=0, 1\le i<j\le n\}$. By adding the constraint $$\begin{aligned}
\sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^{n} y_{ij}=L^*
\label{eq:csat1}\end{aligned}$$ to (\[eq:log\_min\_d3\]) we can ensure, that the optimal solutions of (\[eq:log\_min\_d3\]) can only be the feasible solutions of (\[eq:log\_min\_d3\])-(\[eq:csat1\]).
The addition of constraint $$\begin{aligned}
\sum\limits_{(i,j)\in I_0^*} y_{ij}\ge 1
\label{eq:csat2}\end{aligned}$$ excludes the already known solution from further search. If problem (\[eq:log\_min\_d3\])-(\[eq:csat1\])-(\[eq:csat2\]) has no feasible solution, then all optimal solutions of (\[eq:log\_min\_d3\]) have been found. Otherwise, each recently found optimal solution brings a constraint as (\[eq:csat2\]), and resolve (\[eq:log\_min\_d3\])-(\[eq:csat1\])-(\[eq:csat2\]). The algorithm stops in a finite number of steps, resulting in all optimal solutions through binary variables (\[eq:log\_min\_d3\]).
Problem (\[eq:log\_min\_alpha2\]) can also be rewritten as in (\[eq:log\_min\_d3\]): $$\begin{array}{rllllll}
\min &&\alpha\\
\mathrm{s.t.}&& \phi_n(\exp X)\le \alpha, \\
&&\sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^{n} y_{ij}\le K, \\
&& x_{ij}=-x_{ji},&& 1\le i \le j \le n,\\
&& -\bar M\le x_{ij} \le \bar M, &&1\le i <j \le n,\\
&& -2\bar My_{ij}\le x_{ij}-\bar a_{ij} \le 2\bar My_{ij}, &&1\le i <j \le n,\\
&& y_{ij} \in \{0,1\},&&1\le i <j \le n.\\
\end{array}
\label{eq:log_min_alpha3}$$ If $\phi_n(\exp X)$ is a convex function of the elements (above the main diagonal) of $X$, then the relaxations of (\[eq:log\_min\_d3\]) and (\[eq:log\_min\_alpha3\]) are convex optimization problems, consequently, (\[eq:log\_min\_d3\]) and (\[eq:log\_min\_alpha3\]) are mixed 0-1 convex problems.
Inconsistency index $CR$ of Saaty
=================================
Saaty (1980) proposed to index the inconsistency of pairwise comparison matrix $A$ of size $n \times n$ by a positive linear transformation of its largest eigenvalue $\lambda_{\max}$. The normalized right eigenvector associated to $\lambda_{\max}$ also plays an important role, since it provides the estimation of the weights in the eigenvector method. However, in this paper weighting methods are not discussed. Saaty (1977) showed that $\lambda_{\max} \ge n$ and $\lambda_{\max} = n$ if and only if $A$ is consistent. Let us generate a large number of random pairwise comparison matrices of size $n \times n$, where each element above the main diagonal are chosen from the ratio scale $1/9 , 1/8 , 1/7 , \dots, 1/2 , 1, 2, . . . , 8, 9$ with equal probability. Take the largest eigenvalue of each matrix and let $\overline{\lambda_{\max}}$ denote their average value.\
Let $RI_n = (\overline{\lambda_{\max}}-n)/(n-1)$. Saaty defined the inconsistency of matrix $A$ as $$CR_n(A) = \frac{\frac{\lambda_{\max}(A)-n}{n-1}}{RI_n}$$ being a positive linear transformation of $\lambda_{\max}(A)$. Then $CR_n(A) \geq 0$ and $CR_n(A) = 0$ if and only if $A$ is consistent. The heuristic rule of acceptance is $CR_n \leq 0.1$ for all sizes, also known as the ten percent rule (Saaty, 1980), supported by Vargas’ (1982) statistical analysis. However, some refinements are also known: $CR_3 \leq 0.05$ for $3\times 3$ matrices $CR_4 \leq 0.08$ for $4\times 4$ matrices (Saaty, 1994). Note that any rule of acceptance is somehow heuristic.
Now we apply the results of Section 2 by setting $\phi_n = CR_n$. Let $X\in \log {\cal P}_n$ and let $\lambda_{\max}(\exp X)$ denote the largest eigenvalue of $A=\exp X$. Then $$\begin{aligned}
\phi_n(\exp X)=\frac {\lambda_{\max}(\exp X)-n}{RI_n(n-1)}. \label{eq:phinexpx}\end{aligned}$$ Bozóki et al. (2010) showed that $\lambda_{\max}(\exp X)$ is a convex function of the elements of $X$, therefore, through (\[eq:phinexpx\]), $\phi_n(\exp X)$ is a convex function of the elements of $X$, too.
It is proven that (\[eq:phinexpx\]) implies that both (\[eq:log\_min\_d3\]) and (\[eq:log\_min\_alpha3\]) are mixed 0-1 convex optimization problems. However, they are still challenging from numerical computational point of view, since $\phi_n(\exp X)$ cannot be given in an explicit form as $\lambda_{\max}$ values are themselves computed by iterative methods (Saaty, 1980). We will show that $\lambda_{\max}$ is not only a limit of an iterative process, but an optimal solution of a convex optimization problem as well. The embedded convex optimization problem can be considered together the embedding optimization problem.
Harker (1987) described the derivatives of $\lambda_{\max}$ with respect to a matrix element and recommended to change the element with the largest decrease in $\lambda_{\max}$. The theorems in this section, based on other tools, can be considered as some extensions of Harker’s idea. Reducing $CR$, being equivalent to decreasing $\lambda_{\max}$, is in the focus of Xu and Wei (1999) and Cao et al. (2008).
A special case of Frobenius theorem is applied (Saaty, 1977; Sekitani and Yamaki, 1999):
**Theorem 1.** *Let $A$ be an $n \times n$ irreducibile nonnegative matrix and $\lambda_{\max}(A)$ denote the maximal eigenvalue of A. Then the following equalities hold $$\begin{aligned}
\max_{w>0} \min_{i=1,\dots ,n} \frac{\sum\limits_{j=1}^{n} a_{ij}w_j}{w_i}
=\lambda_{\max}(A)= \min_{w>0} \max_{i=1,\dots ,n}
\frac{\sum\limits_{j=1}^{n} a_{ij}w_j}{w_i}. \label{eq:Frobenius}\end{aligned}$$*
Since the pairwise comparison matrices are positive, Theorem 1 can be applied. In order to rewrite the right-hand side of (\[eq:Frobenius\]), $\bar a_{ij}=\log a_{ij},$ $i,j=1,\dots ,n$, and $z_i=\log w_i,~i=1,\dots ,n$ are used: $$\begin{aligned}
\lambda_{\max}(A)= \min_{z} \max_{i=1,\dots ,n} \sum\limits_{j=1}^{n} e^{\bar
a_{ij}+z_j-z_i} \label{eq:Frobenius2}\end{aligned}$$
The sum of convex exponential functions in the right-hand side (\[eq:Frobenius2\]), furthermore, their maximum are also convex. Thus, $\lambda_{\max}$ can be determined as the optimum value of a convex optimization problem, and the form (\[eq:Frobenius2\]) is equivalent to the optimization problem $$\begin{aligned}
\min {~\lambda} ~~~\text{s.t}.~~~\sum_{j=1}^n e^{\bar a_{ij}+ z_j-z_i} \leq \lambda, ~ i=1, \dots,n,
\label{eq:Frobenius3}\end{aligned}$$ where $\lambda$ and $z_{i}, i=1,~\dots,n$ are variables.
Let $\alpha_n$ be given as a threshold of inconsistency index $\phi_n=CR_n$. Then the constraint $$\begin{aligned}
\phi_n(\exp X)\le \alpha_n \label{eq:phinexpx2a}\end{aligned}$$ from problem (\[eq:log\_min\_d3\]) can be transformed by using (\[eq:phinexpx\]) as $$\begin{aligned}
\lambda_{\max}(\exp X)\le n+RI_n(n-1)\alpha_n.
\label{eq:lambdamax}\end{aligned}$$ Denote $\alpha_n^*=n+RI_n(n-1)\alpha_n$. Hence, the formula (\[eq:Frobenius2\]), substituting $x_{ij}$ = $\bar a_{ij}$, implies an equivalent form $$\begin{aligned}
\sum\limits_{j=1}^{n} e^{x_{ij}+z_{j}-z_{i}}\le \alpha_n^*,\ i=1,\dots,n.
\label{eq:lambdamax2}\end{aligned}$$
Let us replace formula (\[eq:phinexpx2a\]) by (\[eq:lambdamax2\]) in problem (\[eq:log\_min\_d3\]). We get a mixed 0-1 convex programming problem: $$\begin{array}{rllllll}
\min &&\sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^{n} y_{ij}\\
\mathrm{s.t.}&& \sum\limits_{j=1}^{n} e^{x_{ij}+z_{j}-z_{i}}\le \alpha_n^*,&& i=1,\dots,n, \\
&& x_{ij}=-x_{ji},&& 1\le i \le j \le n,\\
&& -\bar M\le x_{ij} \le \bar M, &&1\le i <j \le n,\\
&& -2\bar My_{ij}\le x_{ij}-\bar a_{ij} \le 2\bar My_{ij}, &&1\le i <j \le n,\\
&& y_{ij} \in \{0,1\},&&1\le i <j \le n.\\
\end{array}
\label{eq:log_min_d4}$$
**Theorem 2.** *Let $\alpha_n$ denote the acceptance threshold of inconsistency and let $\alpha_n^*=n+RI_n(n-1)\alpha_n$. Then the optimum value of (\[eq:log\_min\_d4\]) gives the minimal number of the elements to be modified above the main diagonal in $A$ (and their reciprocals) in order to achieve that $CR_n \leq \alpha_n$.*
Problem (\[eq:log\_min\_alpha3\]) can also be rewritten in case of $\phi_n=CR_n$. In the light of (\[eq:phinexpx\]), the minimization of $\phi_n$ is equivalent to the minimization of $\lambda_{\max}$. Furthermore, program (\[eq:Frobenius3\]) depending on $\lambda_{\max}$ is used to obtain:
$$\begin{array}{rllllll}
\min &&\lambda\\
\mathrm{s.t.}&& \sum\limits_{j=1}^{n} e^{x_{ij}+z_{j}-z_{i}}\le \lambda,&& i=1,\dots,n, \\
&&\sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^{n} y_{ij}\le K, \\
&& x_{ij}=-x_{ji},&& 1\le i \le j \le n,\\
&& -\bar M\le x_{ij} \le \bar M, &&1\le i <j \le n,\\
&& -2\bar My_{ij}\le x_{ij}-\bar a_{ij} \le 2\bar My_{ij}, &&1\le i <j \le n,\\
&& y_{ij} \in \{0,1\},&&1\le i <j \le n.\\
\end{array}
\label{eq:log_min_alpha4}$$
**Theorem 3.** *Denote the optimum value of (\[eq:log\_min\_alpha4\]) by $\lambda_{\mathrm{opt}}$, and let $\alpha_{\mathrm{opt}}=\frac{\lambda_{\mathrm{opt}}-n}{RI_n(n-1)}$. Then $\alpha_{\mathrm{opt}}$ is the minimal value of inconsistency $CR_n$ which can be obtained by the modification of at most $K$ elements above the main diagonal of $A$ (and their reciprocals).*
Inconsistency index $CM$ of Koczkodaj and Duszak
================================================
The inconsistency index introduced by Koczkodaj (1993) is based on $3\times 3 $ submatrices, called *triads*. For the $3\times 3$ pairwise comparison matrix $$\begin{aligned}
\left(
\begin{array}{ccc}
1 & a & b \\
1/a& 1 & c \\
1/b & 1/c & 1 \\
\end{array}\label{eq:triad}
\right)\end{aligned}$$ let $$CM(a,b,c)=\min\left\{\frac{1}{a} \left|a-\frac{b}{c}\right|,\frac{1}{b}\left|b-ac\right|,\frac{1}{c}\left|c-\frac{b}{a}\right|\right\}.$$ $CM$ can be extended to larger sizes (Duszak and Koczkodaj, 1994): $$\begin{aligned}
CM(A)=\max\left\lbrace CM(a_{ij}, a_{ik}, a_{jk}) |~1\le i< j < k\le n \right\rbrace .
\label{eq:cma}\end{aligned}$$ Unlike $CR_n$, the construction above does not contain any parameter depending on $n$, so we dispense with the use of the notation $CM_n$. It is easy to see that $CM$ is an inconsistency index since $CM(A)\ge 0$ for any $A\in {\cal P}_n$, and $CM(A)=0$ if and only if $A$ is consistent.
For a general triad $(a,b,c)$ let $$\begin{aligned}
T(a,b,c)=
\max\left\lbrace \frac{ac}{b}, \frac{b}{ac} \right\rbrace .\label{eq:tabc}\end{aligned}$$ It can be shown (Bozóki and Rapcsák, 2008) that there exists a direct relation between $CM$ and $T$: $$\begin{aligned}
CM(a,b,c)=1-\frac{1}{T(a,b,c)}, \ \
T(a,b,c)=\frac{1}{1-CM(a,b,c)}\label{eq:cmt}.\end{aligned}$$ Since $T(a,b,c)\ge 1$, we get $0\le CM(a,b,c)< 1$, so $0\le CM(A)< 1$.
Let $(\bar a,\bar b,\bar c)$ denote the logarithmized values of the triad $(a,b,c)$, and let $$\bar T(\bar a, \bar b,\bar c)= \max\left\lbrace \bar a+\bar c-\bar b,~
-(\bar a+\bar c-\bar b) \right\rbrace .$$ Then $$\begin{aligned}
T(a,b,c)&= \exp (\bar T(\bar a, \bar b,\bar c) ),\label{eq:tabc2}\\
CM(a,b,c)&=1-\frac{1}{\exp (\bar T(\bar a, \bar b,\bar c)) }\label{eq:cmabc}.\end{aligned}$$
It is easy to check that even for triads, $CM$ is not a convex function of the logarithmized matrix elements, thus, if we choose the inconsistency index $\phi_n=CM$, then $\phi_n(\exp X)$ appearing in (\[eq:log\_min\_d3\]) and (\[eq:log\_min\_alpha3\]) is not a convex function of the element of matrix $X$. We show however that by using the univariate function $$\begin{aligned}
f(t)=\frac {1}{1-t}\label{eq:f}\end{aligned}$$ being strictly monotone increasing on the interval $(-\infty,1)$, $f(\phi_n(\exp X))=f(CM(\exp X))$ is already a convex function of the elements of matrix $X$. Then we can change the constraint $$\phi_n(\exp X)\le \alpha_n$$ of problem (\[eq:log\_min\_d3\]) to the convex constraint $$f(\phi_n(\exp X))\le f(\alpha_n).$$ Also, instead of function $\phi_n(\exp
X)$ appearing in problem (\[eq:log\_min\_alpha3\]) we can write $f(\phi_n(\exp X))$ directly, and the value $f^{-1}(\alpha^*)$ computed from the optimal value $\alpha^*$ of the modified problem is the optimal value of the original problem (\[eq:log\_min\_alpha3\]).
To show the statement above, extend the index $T$ defined in (\[eq:tabc\]) for arbitrary $n \times n$ pairwise comparison matrix $A$: $$\begin{aligned}
T(A)=\max\left\lbrace T(a_{ij}, a_{ik}, a_{jk}) |~1\le i< j < k\le n \right\rbrace .
\label{eq:ta}\end{aligned}$$ According to (\[eq:cmt\]), used there for triads, there is a strictly monotone increasing functional relationship between $CM$ and $T$. Consequently, $$\begin{aligned}
CM(A)=1-\frac{1}{T(A)}=f^{-1}(T(A)), \ \
T(A)=\frac{1}{1-CM(A)}=f(CM(A))\label{eq:cmta},\end{aligned}$$ where $f$ is the function defined in (\[eq:f\]).
By expressing $T$ in the logarithmized space, we get $$\begin{aligned}
T(\exp X)=\max\left\lbrace \max\{ e^{x_{ij}+x_{jk}+x_{ki}}, e^{-x_{ij}-x_{jk}-x_{ki}}\}
\mid 1\le i< j < k\le n \right\rbrace .
\label{eq:texpx}\end{aligned}$$ Since on the right-hand-side of (\[eq:texpx\]) the maximum of convex functions is taken, $T(\exp X)$ is convex function of the elements of matrix $X$. Consequently, if we choose the inconsistency index $\phi_n=CM$, then $f(\phi_n(\exp X))$ is already a convex function, and the problems (\[eq:log\_min\_d3\]) and (\[eq:log\_min\_alpha3\]) modified as shown above are already convex mixed 0-1 optimization problems.
Although $CM(\exp X)$ is not convex, it is quasiconvex. To prove it, we show that the lower level sets of $CM(\exp X)$ are convex. Let $\beta\in [0,1)$ an arbitrary possible value of $CM(\exp X)$. Since $f$ is strictly monotone increasing, we have $$\{X\in \mathbb{R}^{n\times n}\mid CM(\exp X)\le \beta\}=\{X\in \mathbb{R}^{n\times n}\mid f(CM(\exp X))\le f(\beta )\}.$$ Due to the convexity of $T(\exp X)=f(CM(\exp X))$ the above level set are convex, and this implies the quasiconvexity of $CM(\exp X)$.
**Theorem 4.** *$CM(\exp X)$ is quasiconvex on the set of the $n \times
n$ matrices, and $T(\exp X)=f(CM(\exp X))$ is convex, where $f$ is defined in (\[eq:f\]).*
In the following we show that problems (\[eq:log\_min\_d3\]) and (\[eq:log\_min\_alpha3\]) can be solved in an easier way, namely, by solving appropriate linear mixed 0-1 optimization problems. By exploiting the strictly monotone increasing property of the exponential function, (\[eq:texpx\]) can also be written in the following form: $$\begin{aligned}
T(\exp X)= e^{\max\left\lbrace \max\{{x_{ij}+x_{jk}+x_{ki}}, {-x_{ij}-x_{jk}-x_{ik}}\}
\mid 1\le i< j < k\le n \right\rbrace }.
\label{eq:texpx2}\end{aligned}$$ Now, (\[eq:texpx2\]) also means that $CM(A)$ can be obtained by determining the maximum of linear expressions of the elements of matrix $\bar A=\log A$ and by applying the exponential function and function $f$ once.
**Theorem 5.** (Bozóki et al. 2011a) *For any $n \times n$ pairwise comparison matrix $A$, inconsistency index $CM$ can be obtained from the optimal solution of the following univariate linear program: $$\begin{array}{rllllll}
\min && z \\
\mathrm{s.t.}&& \bar a_{ij}+\bar a_{jk}+\bar a_{ki}\le z,&&1\le i <j<k \le n, \ \\
&& -(\bar a_{ij}+\bar a_{jk}+\bar a_{ki})\le z &&1\le i <j<k \le n. \\
\end{array} \label{eq:zopt}$$ Let $z_{\mathrm{opt}}$ be the optimal value of (\[eq:zopt\]). Then $CM(A)=1-\frac{1}{\exp(z_{\mathrm{opt}})}$.*
In the following let $\alpha_n$ denote the acceptance threshold associated with the inconsistency index $\phi_n=CM$, and let $$\begin{aligned}
\alpha_n^*=\log\left( \frac{1}{1-\alpha_n}\right).\end{aligned}$$
Consider the linear mixed 0-1 optimization problem $$\begin{array}{rllllll}
\min &&\sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^{n} y_{ij}\\
\mathrm{s.t.}&& x_{ij}+x_{jk}+x_{ki}\le \alpha_n^*,&&1\le i <j<k \le n,\\
&& -(x_{ij}+x_{jk}+x_{ki})\le \alpha_n^* ,&&1\le i <j<k \le n,\\
&& x_{ij}=-x_{ji},&&1\le i \le j \le n,\\
&& -\bar M\le x_{ij} \le \bar M,&&1\le i <j \le n,\\
&& -2\bar My_{ij}\le x_{ij}-\bar a_{ij} \le 2\bar My_{ij},&&1\le i <j \le n,\\
&& y_{ij} \in \{0,1\},&&1\le i <j \le n.\\
\end{array}
\label{eq:mip2}$$
Based on the findings above, the following two theorems follow.
**Theorem 6.** *Let $\alpha_n$ denote the acceptance threshold of inconsistency and let $\alpha_n^*=\log(\frac{1}{1-\alpha_n})$. Then the optimum value of (\[eq:mip2\]) gives the minimal number of the elements to be modified above the main diagonal in $A$ (and their reciprocals) in order to achieve that $CM \leq \alpha_n$.*\
By some alterations in (\[eq:mip2\]), the following linear mixed 0-1 optimization problem can be written: $$\begin{array}{rllllll}
\min && \alpha\\
\mathrm{s.t.}&& x_{ij}+x_{jk}+x_{ki}\le \alpha,&&1\le i <j<k \le n,\\
&& -(x_{ij}+x_{jk}+x_{ki})\le \alpha ,&&1\le i <j<k \le n,\\
&& \sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^{n} y_{ij}\le K,\\
&& x_{ij}=-x_{ji},&&1\le i \le j \le n,\\
&& -\bar M\le x_{ij} \le \bar M,&&1\le i <j \le n,\\
&& -2\bar My_{ij}\le x_{ij}-\bar a_{ij} \le 2\bar My_{ij},&&1\le i <j \le n,\\
&& y_{ij} \in \{0,1\},&&1\le i <j \le n.\\
\end{array}
\label{eq:mip3}$$
**Theorem 7.** *Let $\alpha_{\mathrm{opt}}$ denote the optimum value of (\[eq:mip3\]). Then $1-\frac{1}{\exp(\alpha_{\mathrm{opt}})}$ is the minimal value of inconsistency $CM$ which can be obtained by the modification of at most $K$ elements above the main diagonal of $A$ (and their reciprocals).*
Inconsistency index $CI$ of Peláez and Lamata
=============================================
Similarly to $CM$, the inconsistency index $CI$ proposed by Peláez and Lamata (2003) is also based on triads of form (\[eq:triad\]). It is easy to see that the determinant of the triad (\[eq:triad\]) is nonnegative, and it is zero if and only if the triad is consistent. Based on this interesting property, Peláez and Lamata (2003) proposed to characterize the inconsistency of a pairwise comparison matrix $A \in {\cal P}_n$ by the average of the determinants of the triads of matrix $A$: $$CI_n(A) =
\begin{cases}
\det(A), & \textrm{for} ~n=3, \\
\frac{1}{NT(n)} \sum\limits_{i=1}^{NT(n)} \det(\Gamma_i), & \textrm{for} ~n>3,\\
\end{cases}
\label{eq:CI_def}$$ where $\Gamma_i$, $i=1,\dots,NT(n)$ denote the triads of matrix $A$, and $NT(n)={n \choose 3}$ is the number of triads in $A$.
We show that $CI$ is a convex function of the logarithmized matrix elements, thus if the inconsistency index $\phi_n=CI_n$ is chosen, then $\phi_n(\exp X)$ appearing in problems (\[eq:log\_min\_d3\]) and (\[eq:log\_min\_alpha3\]) is a convex function of the elements of matrix $X$.
The determinant of triad $\Gamma \in {\cal P}_3$ comparing objects $(i,j,k)$ can be written as $$\det(\Gamma)=\frac{a_{ik}}{a_{ij}a_{jk}} + \frac{a_{ij} a_{jk}}{a_{ik}}-2. \label{eq:sarrus}$$
Let $ X= \log \Gamma \in\log ~{\cal P}_3$, i.e., $\Gamma= \exp X$. Equation (\[eq:sarrus\]) can be reformulated as a convex function of the elements of $X$: $$\det(\exp X)=e^{x_{ik} - x_{ij} - x_{jk}} + e^{x_{ij} + x_{jk} - x_{ik}}-2. \label{eq:sarrus_log}$$
Let $\alpha_n$ be a given acceptance threshold for the inconsistency index $\phi_n=CI_n$. According to (\[eq:CI\_def\]) and (\[eq:sarrus\_log\]), the constraint $$\phi_n(\exp X)\le \alpha_n \label{eq:phinexpx2}$$ appearing in (\[eq:log\_min\_d3\]) can be expressed as $$\frac{1}{{n \choose 3}} \sum\limits_{i=1}^{n-2}\sum\limits_{j=i+1}^{n-1} \sum\limits_{k=j+1}^{n} \left( e^{x_{ik}-x_{ij}-x_{jk}} + e^{x_{ij}+x_{jk}-x_{ik}}-2 \right) \le \alpha_n.
\label{eq:CI_eq_log}$$ By denoting $\alpha_n^*=(\alpha_n+2) {n \choose 3}$, (\[eq:CI\_eq\_log\]) can be simplified as $$\sum\limits_{i=1}^{n-2}\sum\limits_{j=i+1}^{n-1} \sum\limits_{k=j+1}^{n} \left( e^{x_{ik}-x_{ij}-x_{jk}} + e^{x_{ij}+x_{jk}-x_{ik}} \right) \le \alpha_n^*,
\label{eq:CI_eq_log_2}$$ and inserting it into (\[eq:log\_min\_d3\]), we get the mixed 0-1 convex optimization problem $$\begin{array}{lllll}
\min &\sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^{n} y_{ij}\\
\mathrm{s.t.}& \sum\limits_{i=1}^{n-2}\sum\limits_{j=i+1}^{n-1} \sum\limits_{k=j+1}^{n} \left( e^{x_{ik}-x_{ij}-x_{jk}} + e^{x_{ij}+x_{jk}-x_{ik}} \right) \le \alpha_n^* ,\\
& x_{ij}=-x_{ji},\ &\hskip -3cm 1\le i <j \le n,\\
& -\bar M\le x_{ij} \le \bar M,\ &\hskip -3cm 1\le i <j \le n,\\
& -2\bar My_{ij}\le x_{ij}-\bar a_{ij} \le 2\bar My_{ij}, &\hskip -3cm 1\le i <j \le n,\\
& y_{ij} \in \{0,1\},&\hskip -3cm 1\le i <j \le n.\\
\end{array}
\label{eq:minlp1_CI}$$
**Theorem 8.** *Let $\alpha_n$ denote the acceptance threshold of inconsistency and let $\alpha_n^*=(\alpha_n+2) {n \choose 3}$. Then the optimum value of (\[eq:minlp1\_CI\]) gives the minimal number of the elements to be modified above the main diagonal in $A$ (and their reciprocals) in order to achieve that $CI \leq \alpha_n$.*
In the same way as for other inconsistency indices, the following mixed 0-1 convex optimization problem can also be considered: $$\begin{array}{lllll}
\min && \alpha\\
\mathrm{s.t.}&& \sum\limits_{i=1}^{n-2}\sum\limits_{j=i+1}^{n-1} \sum\limits_{k=j+1}^{n} \left( e^{x_{ik}-x_{ij}-x_{jk}} + e^{x_{ij}+x_{jk}-x_{ik}} \right) \le \alpha ,\\
&& x_{ij}=-x_{ji}, & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 1\le i <j \le n,\\
&& -\bar M\le x_{ij} \le \bar M,\ & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 1\le i <j \le n,\\
&& -2\bar My_{ij}\le x_{ij}-\bar a_{ij} \le 2\bar My_{ij}, & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 1\le i <j \le n,\\
&& y_{ij} \in \{0,1\}, & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 1\le i <j \le n,\\
&& \sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^{n} y_{ij} \le K.
\end{array}
\label{eq:minlp2_CI}$$
**Theorem 9.** *Let $\alpha_{\mathrm{opt}}$ denote the optimum value of (\[eq:minlp2\_CI\]). Then $ \frac{\alpha_{\mathrm{opt}}}{{n \choose 3}} -2$ is the minimal value of inconsistency $CI$ which can be obtained by the modification of at most $K$ elements above the main diagonal of $A$ (and their reciprocals).*
A numerical example
===================
Our approach is also presented on a classic numerical example from the book of Saaty (1980), for the inconsistency index $CR$. Table 1 contains pairwise comparison values of six cities concerning their distances from Philadelphia. As an example, the evaluator judged that the distance between London and Philadelphia is five times greater than that between Chicago and Philadelphia.
**Table 1.** Comparison of distances of cities from Philadelphia
Cairo Tokyo Chicago San Francisco London Montreal
--------------- ------- ------- --------- --------------- -------- ----------
Cairo 1 1/3 8 3 3 7
Tokyo 3 1 9 3 3 9
Chicago 1/8 1/9 1 1/6 1/5 2
San Francisco 1/3 1/3 6 1 1/3 6
London 1/3 1/3 5 3 1 6
Montreal 1/7 1/9 1/2 1/6 1/6 1
Let $A$ denote the pairwise comparison matrix concerning Table 1. We get that $\lambda_{\max}(A)=6.4536$, and from $RI_6=1.24$, also $CR(A)=0.0732$. Since the value of $CR(A)$ is significantly below the $10\%$ threshold, we can consider the inconsistency of $A$ acceptable.
Let $A^{(1)}$ denote the matrix obtained from $A$ by exchanging the elements $a_{1,2}$ (and $a_{2,1}$). This is a typical mistake at filling-in a pairwise comparison matrix. For the matrix $A^{(1)}$, we get $CR(A^{(1)})=0.0811$. Therefore, although the level of inconsistency of $A^{(1)}$ has increased as consequence of the data-recording error, it is still below the acceptance level of 10%. In this case the proposed methodology is not able to detect the mistake, and $A^{(1)}$ is still accepted.
Consider now the case when $a_{1,3}$ and $a_{3,1}$ are exchanged, say by accident, in the matrix $A$. Let $A^{(2)}$ denote the matrix obtained in this way. Then $CR(A^{(2)})=0.5800$, which is well over the acceptance level of 10%, and it refers to a rough inconsistency in the matrix. By solving the corresponding problem (\[eq:log\_min\_d4\]), we obtain that the inconsistency of $A^{(2)}$ can be pushed below the critical 10% by modifying a single element (and its reciprocal). This element is just in the spoilt position $a_{1,3}$. It can also be shown that this is the single optimal solution to problem (\[eq:log\_min\_d4\]) considering the 0-1 variables. Consequently, the proposed methodology has detected the single possible element for the case of correcting in a single position (and in its reciprocal). It also turned out that this single position is just the one of the values exchanged by accident.
In the previous example the spoilt matrix caused a rough increase of the inconsistency. In this view, it is not surprising that the proposed method offers a unique way of repairing. However, at smaller increase of inconsistency the situation is not that obvious.
Assume now that the element $a_{1,3}$ of matrix $A$ is changed to 2 instead of the value 1/8 of the previous example. This is a smaller difference in relation to the original value 8, the increase of the inconsistency of the modified matrix, denoted by $A^{(3)}$, is also less: $CR(A^{(3)})=0.1078$. The inconsistency of $A^{(3)}$ barely exceeds the critical level 10%, therefore, one would expect that by the modification of a single element can make the inconsistency decrease below 10%, and also that several positions are eligible for this purpose. Indeed, the optimal value of the relating problem (\[eq:log\_min\_d4\]) is 1, and by resolving the problem after adding the constraints (\[eq:csat1\]) and (\[eq:csat2\]) we find that problem (\[eq:log\_min\_d4\]) has 6 different optimal solutions according to the binary variables. Namely, the inconsistency of matrix $A^{(3)}$ decreases below 10% not only by modifying $a_{1,3}$, but also by modifying any single element of $\{a_{1,4}, a_{1,5}, a_{2,6}, a_{3,4}, a_{4,5}\}$. In the ideal case, the evaluator spots the data-recording error in position $a_{1,3}$ immediately. If not, then s/he may have to reconsider the evaluation of each of the 6 positions, but it is still fewer than the 15 possible positions in the upper triangular part of the matrix.
[.3in]{}[1]{}
**References**
Bozóki S, Fülöp J, Koczkodaj WW (2011a) LP-based consistency-driven supervision for incomplete pairwise comparison matrices. Mathematical and Computer Modelling 54(1-2):789–793
Bozóki S, Fülöp J, Poesz A (2011b) On pairwise comparison matrices that can be made consistent by the modification of a few elements. Central European Journal of Operations Research 19(2):157–175
Bozóki S, Fülöp J, Poesz A (2012) Convexity properties related to pairwise comparison matrices of acceptable inconsistency and applications, (in Hungarian, Elfogadható inkonzisztenciájú páros összehasonlítás mát-rixokkal kapcsolatos konvexitási tulajdonságok és azok alkalmazásai). In: Solymosi T, Temesi J (eds.) Egyensúly és optimum: Tanulmányok Forgó Ferenc 70. születésnapjára, Aula Kiadó, pp. 169–184
Bozóki S, Fülöp J, Rónyai L (2010) On optimal completions of incomplete pairwise comparison matrices. Mathematical and Computer Modelling 52(1-2):318–333
Bozóki S, Rapcsák T (2008) On Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices. Journal of Global Optimization 42(2):157–175
Brunelli M, Fedrizzi M (2011) Characterizing properties for inconsistency indices in the AHP. Proceedings of the 11th International Symposium on the AHP, Sorrento, Naples, Italy, June 15-18, 2011 Brunelli M, Fedrizzi M (2013a) Axiomatic properties of inconsistency indices for pairwise comparisons. Submitted, http://arxiv.org/abs/1306.6852
Brunelli M, Canal L, Fedrizzi M (2013b) Inconsistency indices for pairwise comparison matrices: a numerical study. Annals of Operations Research, published online first, DOI 10.1007/s10479-013-1329-0
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---
abstract: 'We consider an alternative to the standard picture of CV and LMXB evolution, namely the idea that most CVs (and by extension LMXBs) may not yet have had time to evolve to their theoretical minimum orbital periods. We call this the Binary Age Postulate (BAP). The observed short–period cutoff in the CV histogram emerges naturally as the shortest period yet reached in the age of the Galaxy, while the post–minimum–period space density problem is removed. The idea has similar desirable consequences for LMXBs. In both cases systems with nuclear–evolved secondary stars form a prominent part of the short–period distributions. Properties such as the existence and nature of ultrashort–period systems, and the spread in mass transfer rates at a given orbital period, are naturally reproduced.'
author:
- 'A. R. King and K. Schenker'
title: A New Evolutionary Picture for CVs and LMXBs
---
\#1 1.25in .125in .25in
Introduction
============
The current picture of CV evolution (see e.g. King, 1988 for a review) has remained essentially unchanged for more than two decades. Its main elements are the propositions that
[*1. Formation:*]{} Common–envelope (CE) evolution produces pre–CVs close to contact at all periods $P$ such that $1.5 \la P \la 12$ hr
[*2. Secular evolution:*]{} angular momentum loss brings pre–CVs rapidly into contact after the CE phase and also drives their subsequent evolution as CVs, accounting for the main features of the observed CV period histogram. The loss mechanism is gravitational radiation (GR) and some rather stronger agency, perhaps magnetic braking (MB) at periods $P \ga 3$ hr
[*3. Youth 1:*]{} CVs are significantly younger than the Galaxy, so many generations of CVs have passed through the observed CV period range
[*4. Youth 2:*]{} CV secondaries are not nuclear–evolved
The great difficulty inherent in theoretical studies of CE evolution via numerical simulations has meant that elements 1, 3 and 4 have remained largely unexamined. Until recently, most researchers have concentrated on tying down the details of 2. above.
The evolution of low–mass X–ray binaries (LMXBs) is more complex than that of CVs because of the need to form a neutron star or black hole through a supernova explosion, with a resulting potential for disruption of the binary. In particular proposition 1. is no longer obvious. Nevertheless the similarity of [*short–period*]{} LMXBs to CVs has encouraged tacit adoption of at least 2. and 4. for them also. Significant differences (cf Fig. 1) in the period histograms of the two types of binary (first pointed out by White & Mason, 1985) have received relatively little attention.
Here we suggest that for CVs there are real reasons to question the three linked propositions 1.–3.–4., and even some of the apparent successes of 2. We suggest a way out of these difficulties which appears promising for both CVs and LMXBs.
The CV period minimum as an age effect
======================================
The observed CV period histogram (Fig. 2) cuts off sharply at an orbital period of $P = P_0 \simeq 76$ min. There are respectively 0 and 12 systems in the period ranges $P_0 \pm 5$ min. The idea that $P_0$ represents a global period minimum ($\dot P = 0$ for $P_{\rm min}$) for CVs has been widely accepted for the last two decades. It is clear that such a global minimum can exist (Paczyński & Sienkiewicz, 1981; Paczyński, 1981; Rappaport, Joss & Webbink, 1982). As the mass $M_2$ of an unevolved secondary star in a CV is reduced by mass transfer, the binary period $P$ decreases also. However for very small $M_2 \la 0.1 \, {{\rm M_{\odot}}}$, the secondary’s Kelvin–Helmholtz time $t_{\rm KH}$ begins to exceed the timescale $t_{\rm M} = -M_2/\dot M_2$ for mass loss driven by angular momentum loss, e.g. by gravitational radiation. At this point the star will expand adiabatically, causing the binary period to increase rather than decrease.
Detailed calculations always predict a value $P_{\rm min}$ very close to, if slightly shorter than, the observed $P_0$. The discrepancy $P_{\rm min} < P_0$ is persistent, but may reflect uncertain or over–simple input physics (cf Kolb & Baraffe, 1999). However there is a much more serious problem with this interpretation of the observed cutoff at $P_0$. This concerns the discovery probability $$p(P) \propto {(-\dot M_2)^{\alpha}\over |\dot P|}.
\label{prob}$$
Here $\alpha$ is some (presumably positive) power describing observational selection effects (e.g. $\alpha = 3/2$ for a bolometric flux–limited sample). Since $\dot P = 0$ at $P = P_{\rm min}$, $p(P)$ must clearly have a significant maximum there unless $-\dot M_2$ declines very sharply near this period. In other words, the observed CV period histogram should show a sharp rise near a global minimum $P_{\rm min}$ unless the mass transfer rate drops there. However all evolutionary calculations show that $-\dot M_2$ changes very little as $P_{\rm min}$ is approached. We conclude that there should be a large ‘spike’ in the CV period histogram near a global minimum $P_{\rm min}$ (cf Kolb & Baraffe, 1999).
The lack of such a spike in the observed period histogram (Fig. 2) has prompted numerous theoretical investigations. Many of these propose ways in which CVs might become difficult to discover near $P_{\rm min}$. A basic problem for this type of argument is that, as we have seen, there is nothing at all unusual about the system parameters (mass transfer rate, separation etc) at this period. Further, attempts to use accretion disc properties as a way of making systems hard to discover founder on the fact that the AM Herculis systems, which have no accretion discs, have precisely the same observed short–period cutoff $P_0 \simeq 80$ min, and no spike either (cf Fig. 2).
The identification of the observed cutoff $P_0$ with the global minimum period $P_{\rm min}$ creates a second problem, particularly emphasized by Patterson (see the review in this volume, and references therein). Namely, if many generations of CVs have completed their evolution and passed the minimum period in the history of the Galaxy, the predicted space density of post–minimum CVs becomes uncomfortably high. The nearest systems should be close enough to be detectable even as bare white dwarfs; and the problem gets worse when one realises that mass transfer decreases only very slowly (timescales $\sim
10^{10}$ yr) after passing $P_{\rm min}$, so that they are definitely brighter than the bare white dwarfs. Since the orbital period also changes very little, the result should be a very large number of nearby CVs with brightness and periods very close to those at $P_0$, which are not observed.
In view of these and other difficulties, it seems reasonable to consider dropping the assumed identification of the observed cutoff $P_0$ with the global minimum $P_{\rm min}$. Thus, from now on we will instead investigate the idea that $P_{\rm min}$ might be genuinely shorter than $P_0$, or more succinctly, [*that even the oldest CVs have not yet reached $P_{\rm min}$.*]{}
The timescale $t_{\rm evol}$ for the secular evolution of CVs down to $P_{\rm min}$ is considerably shorter than the age of the Galaxy, even from the longest commonly observed periods $\sim 8-10$ hr (assuming that magnetic braking is not drastically reduced as has been recently proposed – see the article by Pinsonneault in this volume). Thus to maintain the idea that $P_{\rm min} < P_0$, we must require that most CVs came into contact only a time $< t_{\rm evol}$ ago. In the conventional picture where CE evolution produces pre–CVs close to contact at all orbital periods, this would mean that CVs emerged from CE evolution relatively recently in the age of the Galaxy, presumably as the result of a starburst. This seems unlikely, so we assume instead that for most systems [*the time $t_{contact}$ to shrink the binary enough to initiate mass transfer is at least comparable to the Galactic age $t_{\rm Gal}$*]{} (more precisely $t_{contact} \ga t_{\rm
Gal} - t_{\rm evol}$).
We call this idea the Binary Age Postulate (BAP). In the usual language, it amounts to requiring [*either*]{} that CE evolution is more efficient in removing the envelope of the white dwarf progenitor, thus leaving wider systems than usually assumed, [*or*]{} that with conventional CE evolution, orbital decay into contact by angular momentum loss is much slower than usually assumed. The second possibility would fit with the idea of drastically reduced magnetic braking (see the article by Pinsonneault in this volume). However, unless the usual value of the magnetic braking torque is restored once the system reaches contact there are obvious problems in explaining the brighter (novalike) CVs and the period gap itself. Hence while the reduced braking idea is worth bearing in mind, for the expository purposes of this paper we shall assume the first possibility, i.e. that BAP is satisfied because CE evolution is more efficient than usually assumed. Thus most CVs with secondaries massive enough to have magnetic braking would emerge from CE evolution with periods of order 12 hr or more. CVs with lower–mass secondaries could emerge with shorter orbital periods, but such that relatively few reached contact (evolving via gravitational radiation) within $t_{\rm
Gal}$. We note that this type of distribution is not in conflict with the observed pre–CV distribution (Ritter & Kolb, 1998).
Armed with these assumptions we can give two immediate consequences for CVs, and two for LMXBs.
\(i) [*The period distribution near $P_0$*]{}. The characteristic square shape of this distribution is naturally reproduced, provided only that CVs at longer periods decrease their periods more quickly than those at short periods, e.g. $-\dot P = G(P)$ with ${\rm d}G/{\rm d}P > 0$. This is of course true for the usually assumed forms of magnetic braking. Hence the observed distribution appears naturally if CVs generally come into contact with secondary masses $M_2$ large enough ($M_2 \ga 0.3 \, {{\rm M_{\odot}}}$) for their pre–contact evolution to have been driven by this mechanism. This is of course precisely the content of the BAP idea.
\(ii) [*The space density problem.*]{} This is removed, since there is no presumption that many generations of CVs have passed the observed cutoff $P_0$, and thus no presumed rate at which CVs are piling up in the Galaxy. Assuming that the formation rate of pre–CVs has decreased markedly since the early epochs of the Galaxy, there is no corresponding problem with the space density of pre–CVs. Note that if we do not make the latter assumption, the space density problem is inevitable in [*any*]{} picture: CVs pile up either as post–minimum or pre–contact systems, as we presumably cannot destroy the white dwarfs in either state.
\(iii) [*The LMXB minimum period.*]{} Magnetic braking acts more slowly on LMXBs than CVs, as the binary inertia is greater. Thus we might expect LMXBs to have longer $P_0$ cutoffs than CVs – consistent with observation (see Fig. 1) – as they will presumably be unable to reach such short periods in the age of the Galaxy.
\(iv) [*The faint transient problem.*]{} King (2000) points out that LMXBs with very low mass transfer rates, such as any which have passed the equivalent of the CV global minimum period, will be readily observable as faint transients. Although some such faint systems are observed, the total number in the Galaxy is much too low compared with the number of ‘normal’ LMXBs to interpret them as post–minimum systems in the standard picture. Again this is as expected if most LMXBs have not yet reached the global minimum. We shall see a further feature of these systems explained in the next section.
Thermal–Timescale and Nuclear Evolution
=======================================
The list (i –iv) above shows that BAP has interesting consequences for CV and LMXB evolution. Unsurprisingly there are more. The longer timescales envisaged for orbital decay via angular momentum loss open the possibility that nuclear evolution of the secondary star might bring the binary into contact instead, in direct contrast to the older proposition 4. detailed in the Introduction. This possibility becomes even more pressing when we allow for the fact that white–dwarf and neutron–star binaries can survive a phase of thermal–timescale mass transfer (TTMT) in which $M_2 \ga M_1$ (with $M_1, M_2$ the primary and secondary masses). For white dwarf systems the TTMT phase, at least in mild cases with $M_2$ not too large compared with $M_1$, is probably what drives many supersoft X–ray binaries. The realisation that some neutron–star systems, notably Cyg X–2, must have survived quite violent (highly super–Eddington) TTMT is relatively recent (King & Ritter, 1999; Podsiadlowski & Rappaport, 2000; King & Begelman, 1999). TTMT may be observable in systems such as SS433, and the ultraluminous X–ray sources recently identified in external galaxies (King, Taam & Begelman, 2000; King et al, 2001).
Figure 3 shows schematically how angular momentum loss and nuclear evolution may compete in bringing a compact binary into contact. For low (initial) secondary masses $M_2$ the nuclear evolution timescale is longer than the age of the Galaxy, so angular momentum loss automatically dominates. For large $M_2$ nuclear evolution is rapid, and wins over angular momentum loss. For intermediate secondary masses thermal–timescale mass transfer may occur if $M_2 \ga M_1$, although this phase eventually becomes dynamically unstable (the ‘delayed dynamical instability’) for $M_2$ larger than some critical value $M_{\rm 2, DDI}$. If some nuclear evolution has already occurred, this phase can shrink the binary drastically and strip the hydrogen–rich envelope from the donor, ultimately producing an ultrashort–period system with a low–mass, hydrogen–poor and probably degenerate secondary.
Figure 4 shows the situation for the specific case of CVs. The important result is that BAP hypothesis allows a numerous population of significantly nuclear–evolved CVs to coexist with the familiar unevolved CVs envisaged in the standard picture (cf assumptions 1 – 4 above). The accompanying paper (Schenker & King, this volume) considers this in more detail, and shows that the resulting distribution has several desirable properties, such as possibly explaining the spread in mass transfer rates above the CV period gap.
Figure 5 shows the situation for neutron–star LMXBs. This is qualitatively similar to the CV case. However the slower orbital decay here leads to a larger cutoff period $P_0$ (as noted earlier) and a much stronger tendency to nuclear evolution. Thus many systems evolve to long orbital periods (days), while as for CVs the short–period systems include many with significantly evolved secondaries. This agrees with the deduction by King, Kolb & Burderi (1996) that the neutron–star soft X–ray transients observed at such periods must have nuclear–evolved secondaries, as are now indeed observed in some cases (e.g. Haswell et al., 2000). As in the CV case, ultrashort–period neutron–star LMXBs may form after a TTMT phase.
The black–hole case is shown in Figure 6. The larger $M_1$ and consequently still slower orbital decay intensifies the trends towards larger $P_0$, more long–period systems, and a greater degree of chemical evolution in short–period systems noted for neutron stars. The major difference here is that the smaller mass ratio $M_2/M_1$ makes a TTMT phase unlikely. As a result very few ultrashort–period LMXBs with black hole primaries can form, at least by this channel. If as seems likely the faint transients discussed above are LMXBs with low–mass highly–evolved secondaries, either already at ultrashort periods or evolving towards them, this offers a natural explanation for the observation that almost all of them appear to contain neutron stars.
A general point emerging from this discussion is that [*short–period LMXBs are exceptional*]{}: for most LMXBs nuclear evolution wins, leading to long orbital periods $\sim 10 - 100$ d. Ironically these systems spend almost all of their lifetimes as soft X–ray transients with enormously long recurrence times (cf Ritter & King, this volume), and thus remain undiscovered. The greater observational prominence of short–period systems results from their being either persistent X–ray sources (neutron–star plus unevolved secondary) or soft X–ray transients with fairly short recurrence times (all other short–period LMXBs).
Conclusion
==========
The BAP idea that even the first generation of CVs and LMXBs have yet to complete their evolution represents a radical break with the standard picture of CV and LMXB evolution. However it appears to have some promising aspects. Given the difficulties with the standard picture, it seems worthwhile to consider it further.
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abstract: 'In this paper we present some general solution of the system of linear equations formed by Guibas and Odlyzko in Th.3.3 [@Gui]. We derive probabilities for given patterns to be first to appear in random text and the expected waiting time till one of them is observed and also till one of them occurs given it is known which pattern appears first.'
---
[On the waiting time till some patterns occur in i.i.d. sequences]{}
[Urszula Ostaszewska, Krzysztof Zajkowski]{} [^1]\
Institute of Mathematics, University of Bialystok\
Akademicka 2, 15-267 Bialystok, Poland\
[email protected]\
[email protected]
[*2010 Mathematics Subject Classification:*]{} 60E05
[*Key words: probability generating functions, waiting time, conditional expectation value*]{}
Introduction
============
It is a classical problem in probability theory to study occurrence of patterns in a random text formed by independent realizations of letters chosen from finite alphabet. Feller in [@Fel] considered randomness and recurrent patterns connected with Bernoulli trials. Solov’ev in [@Sol] found the formula on the expected waiting time for an appearance of one pattern. Penney in [@Pen] proposed a coin-flip game with a coin tossed repeatedly until one of patterns appeared, then this pattern wins. A formula for computing the odds of winning for two competing pattern was discovered by Conway and described by Gardner [@Con]. A martingale approach to the study of occurrence of patterns in repeated experiments was presented by Li [@Li].
Results dealing with the occurences of patterns have been applied in several areas of information theory including source coding, code synchronization, randomness testing, etc. They are also important in molecular biology in DNA analysis and for gene recognition.
Guibas and Odlyzko in Th.3.3 [@Gui] formed the system of linear equations which relates the generating function of probabilities that given pattern occurs before others and the generating function for tails distribution of probabilities until some pattern appears. In this paper we present some general solution of this system (Th.\[mTh\]). It has a general form but it allows us to obtain formulas on the probability that a given pattern precedes the remaining ones and a new formula on the expected waiting time until some pattern occurs in a random text (Cor.\[wn1\] and \[wn2\]). For complementary of our presentation we recall in a general context some facts obtained in [@KZ] dealt with the probability-generating functions but in this paper we focus our attention on the expected waiting time and we show how this general solution could be used to obtain conditional expected waiting times (Prop.\[stw\]). We also present on some example how it could be used to prove in a simply way some known result deal with presented topic (Rem.\[uw1\]).
Waiting time on patterns
========================
Let $(\xi_n)$ be a sequence of i.i.d. random letters from a finite alphabet $\Omega$. For a given pattern (word) $A=(a_1,a_2,...,a_l)\in\Omega^l$ by $A_{(k)}$ and $A^{(k)}$ we will denote subpatterns formed by first and last $k$ letters of $A$, respectively; $A_{(k)}=(a_1,a_2,...,a_k)$ and $A^{(k)}=(a_{l-k+1},a_{l-k+2},...,a_l)$. For two patterns $A$ and $B$ we will denote $[A=B]=1$ if $A=B$ and $[A=B]=0$ if not. Now we define a correlation polynomial $w_A^B$ of $A$ and $B$ (of the lengths $l$ and $m$, respectively) as $$w_A^B(s)=\sum_{k=1}^{\min\{l,m\}}[A_{(k)} = B^{(k)}]Pr(A^{(l-k)})s^{l-k};$$ if $l=\min\{l,m\}$ then in the above sum for the index $k=l$ we assume that $P(A^{(0)})=1$.
Let $\Omega=\{H,T\}$ and $(\xi_n)$ be a sequence of i.i.d. letters in $\Omega$ with the distribution $$Pr(\xi_n=H)=p\quad{\rm and}\quad Pr(\xi_n=T)=q=1-p.$$ Consider two patterns $A=THH$ and $B=THTH$. Then the correlation polynomials have the following forms: $$w_A^B(s)=ps,\;\; w_B^A(s)=0,\;\; w_A^A(s)=1\;\;{\rm and}\;\;w_B^B(s)=pqs^2+1.$$
Let us note that the correlation polynomials up to some differences in their definition can be used to investigate the appearances of patterns not only in Bernoulli trials but also in Markov chains (see for instance [@Reg]).
Now we briefly recall results presented by Guibas and Odlyzko in [@Gui sec.3]. Consider a set of $m$ patterns (words) $A_i$ ($1\le i \le m$) of lengths $l_i$, respectively. We assume that the set of patterns is reduced that is none of the patterns contains any other as a subpattern.
Let $\tau_i$ denote the stopping time until $A_i$ occurs and $\tau$ be the stopping time till some of considered patterns is observed, i.e. $$\tau=\min\{\tau_i:\;1\le i \le m\}.$$ Let $p_{n}$ and $p^{A_i}_n$ be probabilities $Pr(\tau=n)$ and $Pr(\tau=\tau_i=n)$, respectively. Let $g_\tau(s)$ denote $E(s^\tau)=\sum_{n=0}^\infty p_ns^n$ the probability generating function of random variable $\tau$ and $g_\tau^{A_i}$ be the generating functions $\sum_{n=0}^\infty p^{A_i}_ns^n$, $1\le i \le m$. Since $p_n=\sum_{i=1}^m p^{A_i}_n$ we have that $g_\tau=\sum_{i=1}^m g_\tau^{A_i}$. By $Q_\tau$ we denote the generating function for tails probabilities of $\tau$, i.e. $Q_\tau(s)=\sum_{n=0}^\infty q_ns^n$, where $q_n=Pr(\tau>n)=\sum_{k>n}p_k$.
Let $B_n$ be the set of sequences such that any pattern $A_i$ does not appear in the string of the first $n$ letters of these sequences. Notice that $Pr(B_n)=q_n$. In the system of $m$ patterns if we add to each initial $n$-string in $B_n$ the word $A_i$ then we must check if neither it nor other ones appear earlier. Since $Pr(B_n)=q_n$ we get the following system of equations $$\label{req1}
q_nPr(A_i)=\sum_{j=1}^m\sum_{k=1}^{\min\{l_i,l_j\}}[A_{i(k)}=A_j^{(k)}]Pr(A_i^{(l_i-k)})p^{A_j}_{n+k},$$ for each $1\le i \le m$, which is compatible with (3.2) in [@Gui]. Notice now that $$\label{req2}
q_n=q_{n+1}+\sum_{j=0}^m p_{n+1}^{A_j}$$ (see (3.1) in [@Gui]). Multiplying (\[req1\]) by $s^{n+l_i}$ and (\[req2\]) by $s^n$ and summing from $n=0$ to infinity we obtain the following system of linear equations $$\label{eq1}
\left\{
\begin{array}{ccl}
(1-s)Q_\tau(s)+\sum_{j=1}^m g_\tau^{A_j}(s) & = & 1\\
Pr(A_i)s^{l_i}Q_\tau(s)-\sum_{j=1}^m w_{A_i}^{A_j}(s)g_\tau^{A_j}(s) & = & 0\quad (1\le i \le m).
\end{array}
\right.$$
In our opinion the definition of the correlation polynomial (second line from above page 195 [@Gui]) appearing in the system of linear equation formed in Th.3.3 [@Gui] should be of the following form $$c_{GH}(z)=\sum_{r\in GH} z^{r-1}\frac{Pr(h_{r+1},...,h_{|H|})}{Pr(H)}=\sum_{r\in GH}\frac{z^{r-1}}{Pr(h_1,...,h_r)}=\sum_{r\in GH}\frac{z^{r-1}}{Pr(h_1)\cdot...\cdot Pr(h_r)}.$$ Then substituting $z=\frac{1}{s}$ we get the equivalence of the system of linear equation in Th.3.3 [@Gui] and this one described in (\[eq1\]).
\[mTh\] The solution of the system of linear equations (\[eq1\]) has the following form $$\label{pgf}
g_\tau^{A_i}(s)=\frac{\det \mathcal{B}^i(s)}{\sum_{j=1}^{m}\det \mathcal{B}^j(s)+(1-s)\det \mathcal{B}(s)}\quad(1\le i \le m)$$ and $$\label{tgf}
Q_\tau(s)=\frac{\det \mathcal{B}(s)}{\sum_{j=1}^{m}\det \mathcal{B}^j(s)+(1-s)\det \mathcal{B}(s)},$$ where $\mathcal{B}$ denotes a matrix formed by correlations polynomials $w_{A_i}^{A_j}$, i.e. $$\mathcal{B}(s)=
\begin{bmatrix}
w_{A_i}^{A_j}(s)
\end{bmatrix}
_{1\le i,j \le m}$$ and $\mathcal{B}^j(s)$ is the matrix arisen by replacing the $j$-th column of $\mathcal{B}(s)$ by the column vector $[P(A_i)s^{l_i}]_{1\le i \le m}$.
Leading $Q_\tau(s)$ out of the first equation of the system (\[eq1\]) and substituting it into the remaining ones we obtain an equivalent system of linear equations of the form $$P(A_i)s^{l_i}=\sum_{j=1}^m g_\tau^{A_j}(s)[P(A_i)s^{l_i}+(1-s)w_{A_i}^{A_j}(s)]\quad (1\le i \le m).$$ Let $\mathcal{A}$ denote the coefficient matrix of the above system, i.e. $$\mathcal{A}(s) =
\begin{bmatrix}
P(A_i)s^{l_i}+(1-s)w_{A_i}^{A_j}(s)
\end{bmatrix}
_{1\le i,j \le m}.$$ Notice that because $w_{A_i}^{A_i}(0)=1$ and $w_{A_i}^{A_j}(0)=0$ for $i\neq j$ then $\mathcal{A}(0)$ is the identity matrix. Since $\det\mathcal{A}(0)=1$ and $\det\mathcal{A}(s)$ is a polynomial, $\det\mathcal{A}(s)\neq 0$ on some neighborhood of zero. It means that on this neighborhood there exist solution of the system.
Because the determinant of matrices $m\times m$ is a $m$-linear functional with respect to columns (equivalently to rows) then one can check that $$\det \mathcal{A}(s)=(1-s)^m\det \mathcal{B}(s) +(1-s)^{m-1}\sum_{j=1}^m \det \mathcal{B}^j(s).$$ If now similarly $\mathcal{A}^j(s)$ denotes the matrix formed by replacing the $j$-th column of $\mathcal{A}(s)$ by the column vector $[P(A_i)s^{l_i}]_{1\le i \le m}$ then the determinant’s calculus gives that $\det \mathcal{A}^j(s)=(1-s)^{m-1}\det \mathcal{B}^j(s)$. By the Cramer’s rule we obtain $$g_\tau^{A_i}(s)=\frac{\det\mathcal{A}^i(s)}{\det\mathcal{A}(s)}=\frac{\det \mathcal{B}^i(s)}{\sum_{j=1}^m \det \mathcal{B}^j(s)+(1-s)\det \mathcal{B}(s) }$$ for $1\le i \le m$.
Substituting the functions $g_\tau^{A_i}$ into the first equation of the system (\[eq1\]) one can calculate that $$Q_\tau(s)=\frac{\det \mathcal{B}(s)}{\sum_{j=1}^{m}\det \mathcal{B}^j(s)+(1-s)\det \mathcal{B}(s)}.$$
Notice that the probability-generating function $g_\tau^{A_i}(s)=\sum_{n=0}^\infty p_n^{A_i}s^n$ is well define on the interval $[-1,1]$ for sure (it is an analytic function on $(-1,1)$). The right hand side of (\[pgf\]) is a rational function equal to $g_\tau^{A_i}$ on some neighborhood of zero. By the analytic extension we know that there exists the limit of the right hand side of (\[pgf\]) by $s\to 1^-$ which is equal to $g_\tau^{A_i}(1)$. Thus we obtain the following
\[wn1\] The probability $Pr(\tau=\tau_i)$ that the pattern $A_i$ precedes all the remaining $m-1$ patterns is equal to $g_X^{A_i}(1)$, that is $$\label{pro}
Pr(\tau=\tau_i)=\frac{\det \mathcal{B}^i(1)}{\sum_{j=1}^m \det \mathcal{B}^j(1)},$$ where the right hand side of the above equality we understand as the limit of (\[pgf\]) by $s\to 1^-$.
An application of the above to the generalization of the Conway’s formula one can find in [@KZ].
Since $Q_\tau(1)$ is the expected value of $\tau$, we can formulate the another
\[wn2\] The expected waiting time till one of $m$ patterns is observed is given by $$\label{evQ}
E\tau=\frac{\det \mathcal{B}(1)}{\sum_{j=1}^{m}\det \mathcal{B}^j(1)},$$ where the above right hand side is the limit of (\[tgf\]) by $s\to 1^-$.
Note that the above formula is also true for one pattern $A\in\Omega^l$ ($m=1$) since $$\label{solov}
E\tau=\frac{w^A_A(1)}{Pr(A)}=\sum_{k=1}^l\frac{[A_{(k)}=A^{(k)}]}{Pr(A_{(k)})}$$ (compare Solov’ev’s result in [@Sol]).
Now we can proceed to calculate the expected waiting time for the pattern $A_i$ knowing that it appears as a first, namely the conditional expectation of the random variable $\tau$ given the event $\{\tau=\tau_i\}$ that is we prove the following
\[stw\] $$\label{procond}
E({\tau}|\tau=\tau_i)= E\tau+\frac{1}{Pr(\tau=\tau_i)}\cdot \frac{d}{ds}\Big(\frac{\det\mathcal{B}^i}{\sum_{j=1}^{m}\det \mathcal{B}^j}\Big)(1).$$
Observe that the conditional expectation $E({\tau}|\tau=\tau_i)$ can be expressed in terms of the function $g_\tau^{A_i}$ as follows $$\begin{aligned}
\label{Contau}
E({\tau}|\tau=\tau_i) & = & \sum_{n=1}^\infty nPr(\tau=n|\tau=\tau_i)=\frac{1}{Pr(\tau=\tau_i)}\sum_{n=1}^\infty nPr(\tau=\tau_i=n) \nonumber\\
\; & = & \frac{1}{Pr(\tau=\tau_i)}\frac{d}{ds}g_\tau^{A_i}(1).\end{aligned}$$ Differentiating the function $g_\tau^{A_i}$ given by formula (\[pgf\]) and taking the value at 1 we get $$\begin{aligned}
\frac{d}{ds}g_\tau^{A_i} (1)&=& \frac{\frac{d}{ds}\det\mathcal{B}^i(1) \sum_{j=1}^{m}\det \mathcal{B}^j (1)- \det\mathcal{B}^i(1)\sum_{j=1}^{m}\frac{d}{ds}\det \mathcal{B}^j (1)}{[\sum_{j=1}^{m}\det \mathcal{B}^j (1)]^2} \nonumber \\
&&
+\frac{\det\mathcal{B}^i(1)}{\sum_{j=1}^{m}\det \mathcal{B}^j (1)} \cdot \frac{\det\mathcal{B}(1)}{\sum_{j=1}^{m}\det \mathcal{B}^j(1)}. \nonumber\end{aligned}$$ Notice that the first summand can be expressed as $\frac{d}{ds}\Big(\frac{\det\mathcal{B}^i}{\sum_{j=1}^{m}\det \mathcal{B}^j}\Big)(1)$ and the second one, by Corollaries \[wn1\] and \[wn2\], is equal to $Pr(\tau=\tau_i)E\tau$. Thus, by (\[Contau\]), we get the formula (\[procond\]) on $E({\tau}|\tau=\tau_i)$.
Before we present some examples we would like to show applications of our results to obtain some known fact contained in [@Li].
\[uw1\] Define a number $A_j \ast A_i$ as $$\frac{w_{A_i}^{A_j}(1)}{Pr(A_i)}
=\sum_{k=1}^{\min\{l_i,l_j\}}\frac{[A_{i(k)}=A_j^{(k)}]}{Pr(A_{i(k)})}.$$ Let us emphasizes that the above coincides with the notation (2.3) in [@Li]. Consider now a matrix $$\mathcal{C}=
\begin{bmatrix}
A_j \ast A_i
\end{bmatrix}
_{1\le i,j \le m}.$$ Observe that $\det\mathcal{B}(1)=\prod_{i=1}^m Pr(A_i)\det\mathcal{C}$ and $\det\mathcal{B}^j(1)=\prod_{i=1}^m Pr(A_i)\det\mathcal{C}^j$, where $\mathcal{C}^j$ is the matrix formed by replacing the $j$-th column of $\mathcal{C}$ by the column vector $\begin{bmatrix} 1\end{bmatrix}_{1\le i \le m}$. In this way we can rewrite Corollaries \[wn1\] and \[wn2\] in terms of matrix $\mathcal{C}$ as follows
$$Pr(\tau=\tau_i)=\frac{\det \mathcal{C}^i}{\sum_{j=1}^m \det \mathcal{C}^j}\quad {\rm and}\quad E\tau=\frac{\det \mathcal{C}}{\sum_{j=1}^m \det \mathcal{C}^j}.$$ By the martingale arguments Li proved in [@Li] (see Theorem 3.1) that for every $i=1,2, \ldots, m$ we have $$E\tau= \sum_{j=1}^{m} Pr(\tau=\tau_j)( A_j \ast A_i)$$ which in our notation is equivalent to the following formula $$\det \mathcal{C} = \sum_{j=1}^{m} (A_j \ast A_i) \det \mathcal{C}^j, \quad (1\leq i \leq m).$$ Now we independently prove the above determinant’s identity.
Let $\mathcal{D}$ be extended matrix $\mathcal{C}$ on an initial row and column as follows $$\mathcal{D}=
\left[\begin{array}{c|c}
a_{00}& \begin{array}{cccc} a_{01} & a_{02} & \ldots & a_{0m} \end{array} \\
\hline
\begin{array}{c} a_{10} \\ a_{20} \\ \vdots \\ a_{m0} \end{array}&\mathcal{C}
\end{array}\right],$$ where $a_{00}=1$, $(a_{01}, \ldots, a_{0m})$ is the $i$-th row of the matrix $\mathcal{C}$, i.e. $a_{0j}=A_j \ast A_i$, $1\le j \le m$, and $a_{k0}=1$ for $k\neq i$ and $a_{i0}=0$. The Laplace expansion along the zero column yields $$\det \mathcal{D} = \det \mathcal{C}$$ whereas taking the Laplace expansion along the $i$-th row we obtain $$\det \mathcal{D} = \sum_{j=1}^{m}(A_j \ast A_i) (-1)^{i+j}\det\mathcal{D}_{ij}.$$ Notice that for every matrix $\mathcal{D}_{ij}$ permuting the zero column and zero row in the place of removed ones, by the determinant’s properties, we get $$\det \mathcal{D}_{ij}=(-1)^{i+j-2}\det \mathcal{C}^j.$$ Thus $$\det \mathcal{D} = \sum_{j=1}^{m}(A_j \ast A_i)\det \mathcal{C}^j$$ and this completes the proof.
Let $\Omega=\{A,C,G,T\}$ and $(\xi_n)$ be a sequence of i.i.d. letters in $\Omega$ with the probabilities $$Pr(\xi_n=A)=p_a, \quad Pr(\xi_n=C)=p_c, \quad Pr(\xi_n=G)=p_g\quad {\rm and} \quad P(\xi_n=T)=p_t.$$ Consider the set of three patterns: $A_1=ACG, A_2=ATG$ and $A_3=AG$. Observe that $w_{A_i}^{A_j}(s)=0$ if $i\neq j $ and $w_{A_i}^{A_i}(s)=1$. So in this case the matrix $\mathcal{B}(s)= [w_{A_i}^{A_j}(s)]_{1\leq i,j \leq 3}$ is the identity matrix. For the matrices $\mathcal{B}^i(s)$ we obtain $$\det \mathcal{B}^i(s)= P(A_i)s^{l_i}.$$ Since $Pr(\tau=\tau_i)= \frac{det \mathcal{B}^i(1)}{\sum_{j=1}^{m}det \mathcal{B}^j(1)}$ the probabilities that the $i$-th pattern occurs as a first are given by $$Pr(\tau=\tau_1)=\frac{p_c}{p_c+p_t+1}, \ \ \ \ \
Pr(\tau=\tau_2)=\frac{p_t}{p_c+p_t+1}, \ \ \ \ \
Pr(\tau=\tau_3)=\frac{1}{p_c+p_t+1}.$$ Applying now Colorary \[wn2\] we obtain $$E\tau = \frac{1}{p_ap_g(p_c+p_t+1)}.$$ By Proposition \[stw\] we can calculate $$\begin{aligned}
E(\tau|\tau=\tau_1)&=& \frac{p_ap_g+1}{p_ap_g(p_c+p_t+1)}, \nonumber \\
E(\tau|\tau=\tau_2)&=& \frac{p_ap_g+1}{p_ap_g(p_c+p_t+1)}, \nonumber \\
E(\tau|\tau=\tau_3)&=& \frac{1-p_ap_g(p_c+p_t)}{p_ap_g(p_c+p_t+1)}. \nonumber\end{aligned}$$ Note that for each pattern by the Solov’ev’s formula (\[solov\]) we have $$E\tau_1=\frac{1}{p_ap_cp_g}, \ \ \ \ \
E\tau_2= \frac{1}{p_ap_gp_t}, \ \ \ \ \
E\tau_3= \frac{1}{p_ap_g}.$$ Observe that for different $p_a,p_c,p_g,p_t$ the values $E\tau_1,E\tau_2,E\tau_3$ are mostly different. But in this example we obtained that $E(\tau|\tau=\tau_1)=E(\tau|\tau=\tau_2)$ for any $p_a,p_c,p_g,p_t$.
If $\mathcal{B}(s)$ is the identity matrix then applying Proposition \[stw\] one can calculate that $$\begin{aligned}
E(\tau|\tau=\tau_i)&=& E\tau + \frac{l_i Pr(A_i)\sum_{k=1}^{m}Pr(A_k)- Pr(A_i)\sum_{k=1}^{m} l_k Pr(A_k)}{Pr(A_i)\sum_{k=1}^{m}Pr(A_k)} \nonumber \\
\; &=& E\tau + l_i - \frac{\sum_{k=1}^{m}l_k Pr(A_k)}{\sum_{k=1}^{m} Pr(A_k)}. \nonumber \end{aligned}$$ It means that if additionally patterns $A_i$ and $A_j$ are of the same length then $E(\tau|\tau=\tau_i)=E(\tau|\tau=\tau_j)$. For this reason in the above example $E(\tau|\tau=\tau_1)=E(\tau|\tau=\tau_2)$.
Let now $\Omega=\{H,T\}$ and $Pr(\xi_n=H)=p$ and $Pr(\xi_n=T)=q=1-p$. Consider three patterns: $A_1=THH, A_2=HTH$ and $A_3=HHT$. In this case $$\mathcal{B}(s)= \begin{pmatrix} w_{A_i}^{A_j}(s)\end{pmatrix}_{1\leq i,j \leq 3}=
\begin{pmatrix}
1 & ps & p^2s^2 \\
pqs^2 & pqs^2+ 1 & ps \\
pqs^2+qs & pqs^2 & 1
\end{pmatrix}$$ and $$\begin{aligned}
\det\mathcal{B}(s) & = &-p^3q^2 s^5-2p^2qs^3+pqs^2+1,\\
\det\mathcal{B}^1(s) & = & p^2qs^3(-p^2qs^3+pqs^2-ps+1),\\
\det\mathcal{B}^2(s) & = & p^2qs^3(1-ps),\\
\det\mathcal{B}^3(s) & = & p^2qs^3(-pq^2s^3-qs+1).\end{aligned}$$ By Corollary \[wn1\] we get $$Pr(\tau=\tau_1)=\frac{q(1+pq)}{1+q}, \ \
Pr(\tau=\tau_2)=\frac{q}{1+q}, \ \
Pr(\tau=\tau_3)=\frac{p(1-q^2)}{1+q}.$$ Taking into account that $q=1-p$, by virtue of Corollary \[wn2\] we obtain the formula on the expected waiting time of $\tau$ $$\begin{aligned}
E\tau &=&\frac{-p^5+2p^4+p^3-3p^2+p+1}{p^2(1-p)(2-p)}. \nonumber \end{aligned}$$ By Proposition \[stw\] one can calculate that $$\begin{aligned}
E(\tau|\tau=\tau_1)&=& E\tau+ \frac{-p^5+9p^4-12p^3-6p^2+12p-3 }{(1-p)(1+p-p^2)(2-p)}, \nonumber \\
E(\tau|\tau=\tau_2)&=& E\tau+ \frac{p^3-10p^2+11p-3}{(1-p)(2-p)}, \nonumber \\
E(\tau|\tau=\tau_3)&=& E\tau+ \frac{-p^4+ 7p^3+3p^2-10p+3}{p^2(2-p)}. \nonumber\end{aligned}$$ In the symmetric case $p=q=\frac{1}{2}$ the above quantities take values $$Pr(\tau=\tau_1)=\frac{5}{12},\quad
Pr(\tau=\tau_2)=\frac{1}{3},\quad
Pr(\tau=\tau_3)=\frac{1}{4},$$ $$E\tau=\frac{31}{6}$$ and $$E(\tau|\tau=\tau_1)=\frac{86}{15},\quad E(\tau|\tau=\tau_2)=\frac{16}{3},\quad E(\tau|\tau=\tau_3)=4.$$
[ ]{}
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S-Y.R. Li, [*A martingale approach to the study of occurrence of sequence patterns in repeated experiments*]{}, The Annals of Probability, Vol. 8. (1980), 1171-1176.
W. Penney, [*Problem 95: Penney-Ante*]{}, Journal of Recreational Mathematics 7 (1974), 321.
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[^1]: The authors are supported by the Polish National Science Center, Grant no. DEC-2011/01/B/ST1/03838
|
---
abstract: 'Let $\sigma:A\rightarrow B$ and $\rho:A\rightarrow C$ be two homomorphisms of noetherian rings such that $B\otimes_{A}C$ is a noetherian ring. we show that if $\sigma$ is a regular (resp. complete intersection, resp. Gorenstein, resp. Cohen-Macaulay, resp. $(S_{n}$), resp. almost Cohen-Macaulay) homomorphism, so is $\sigma\otimes I_{C}$ and the converse is true if $\rho$ is faithfully flat. We deduce the transfert of the previous properties of $B$ and $C$ for $B\otimes_{A}C$, and then for the completed tensor product $B\hat{\otimes}_{A}C$. If $B\otimes_{A}B$ is noetherian and $\sigma$ is flat, we give a necessary and sufficient condition to $B\otimes_{A}B$ be a regular ring.'
author:
- |
Mohamed Tabaâ\
[*Faculté des Sciences, Département de Mathématiques, B.P. 1014 Rabat Maroc* ]{}
title: 'Sur le produit tensoriel d’algèbres[^1]'
---
Introduction
============
Tous les anneaux considérés sont supposés commutatifs et unitaires. Les notations sont celles de [@Grot1].
Rappelons ([@Grot1],7.3.1) que si $\sigma :A\rightarrow B$ un homomorphisme d’anneaux noethériens, on dit que $\sigma $ est régulier s’il est plat et si pour tout idéal premier $\mathfrak{p}$ de $A$ la $k(\mathfrak{p})$-algèbre $B\otimes _{A}k(\mathfrak{p})$ est géométriquement régulière, et que $\sigma $ est d’intersection complète (resp. de Gorenstein, resp. de Cohen-Macaulay, resp.($S_{n}$), resp. presque de Cohen-Macaulay) s’il est plat et si pour tout idéal premier $\mathfrak{p}$ de $A$ l’anneau $B\otimes _{A}k(\mathfrak{p})$ est d’intersection complète (resp. de Gorenstein, resp. de Cohen-Macaulay, resp. vérifie ($S_{n}$), resp. presque de Cohen-Macaulay).
Dans ce qui suit nous montrons, que si $\sigma :A\rightarrow B$ et $\rho
:A\rightarrow C$ sont deux homomorphismes d’anneaux noethériens tels que $B\otimes _{A}C$ soit un anneau noethérien, alors $\sigma\otimes I_{C}$ est régulier (resp. d’intersection complète, resp. de Gorenstein, resp. de Cohen-Macaulay, resp. ($S_{n}$), resp. presque de Cohen-Macaulay) si $\sigma $ l’est; et que la réciproque est vraie si $\rho $ est fidèlement plat. On en déduit, en particulier, que si $\sigma$ est plat, alors $B\otimes_{A}C$ est un anneau d’intersection complète (resp. de Gorenstein, resp. de Cohen-Macaulay) si $B$ et $C$ le sont, et il est presque de Cohen-Macaulay si l’un des anneaux, $B$ ou $C$, est de Cohen-Macaulay et l’autre est presque de Cohen-Macaulay. Si $A$ est un corps, on retrouve le Théorème 2 de [@Watanabe] si $B$ et $C$ sont des anneaux de Gorenstein, le Théorème 2.1 de [@abou1] si $B$ et $C$ sont des anneaux de Cohen-Macaulay et le Théorème 6 de [@Tousi] si $B$ et $C$ sont des anneaux d’intersection complète (resp. vérifient ($S_{n}$)).
Comme application nous montrons que si $\sigma$ et $\rho$ sont deux homomorphismes locaux d’anneaux locaux noethériens, et si le corps résiduel de $C$ est de rang fini sur celui de $A$, alors le produit tensoriel complété $B\hat{\otimes}_{A}C$ est régulier si l’homomorphisme $\sigma$ est formellement lisse et $C$ est régulier, et il est d’intersection complète (resp. de Gorenstein, resp. de Cohen-Macaulay) si $\sigma$ est plat et $B$ et $C$ le sont, et il est presque de Cohen-Macaulay si $\sigma$ est plat et si l’un des anneaux, $B$ ou $C$, est de Cohen-Macaulay et l’autre est presque de Cohen-Macaulay.
Si $\sigma$ est plat et $B\otimes_{A}B$ est un anneau noethérien, nous montrons que $B\otimes_{A}B$ est régulier si et seulement si $B$ est régulier et $\sigma$ est régulier.
Dans toute la suite nous utilisons librement les résultats de [@Mat] et de [@Avramov1], et l’homologie d’André-Quillen telle qu’elle est définie dans [@andre].
Résultats
=========
Soit $\sigma:A\rightarrow B$ un homomorphisme d’anneaux noethériens. Les propriétés suivantes sont équivalentes:
1. L’homomorphisme $\sigma$ est régulier (resp. d’intersection complète).
2. L’homomorphisme $\sigma$ est plat et $H_{1}(A,B,k(\mathfrak{q}))=0$ (resp. $H_{2}(A,B,k(\mathfrak{q}))$ $=0$) pour tout idéal premier $\mathfrak{q}$ de $B$.
**Démonstration.** Cas régulier: cf. [@andre] Supplément Théorème 30.
Cas d’intersection complète: (cf. [@Marot] ) On utilise [@andre]. Soit $\mathfrak{q}$ idéal premier de $B$ et $\mathfrak{p}=\sigma ^{-1}(\mathfrak{q})$. D’après le Corollaire 5.27, la Propositon 4.54, la suite exacte associée aux homomorphismes $k(\mathfrak{p})\rightarrow B_{\mathfrak{q}}/\mathfrak{p}B_{\mathfrak{q}}\rightarrow \ k(\mathfrak{q})$ et d’après la Proposition 7.4, on a $H_{2}(A,B,k(\mathfrak{q})\cong
H_{3}(B_{\mathfrak{q}}/\mathfrak{p}B_{\mathfrak{q}},k(\mathfrak{q}),k(\mathfrak{q}))$; l’équivalence résulte donc de la Proposition 6.27.
Soient $\sigma:A\rightarrow B$ et $\rho:A\rightarrow C$ deux homomorphismes d’anneaux, $\mathfrak{Q}$ un idéal premier de $B\otimes _{A}C$ et $\mathfrak{q}=(I_{B}\otimes\rho)^{-1}(\mathfrak{Q})$. Si $\sigma$ est plat, alors on l’isomorphisme suivant: $$H_{n}(A,B,k(\mathfrak{q}))\otimes_{k(\mathfrak{q})}k(\mathfrak{Q})\cong H_{n}(C,B\otimes_{A}C,k(\mathfrak{Q}))$$
**Démonstration.** En effet, d’après le Lemme 3.20 de [@andre] on a $H_{n}(A,B,k(\mathfrak{q}))\otimes_{k(\mathfrak{q})}k(\mathfrak{Q})\cong H_{n}(A,B,k(\mathfrak{Q}))$ et d’après la Proposition 4.54 de [@andre] on a $H_{n}(A,B,k(\mathfrak{Q}))$ $ \cong H_{n}(C,B\otimes_{A}C,k(\mathfrak{Q}))$; d’où le Lemme.
Soient $\sigma:A\rightarrow B$ et $\rho:A\rightarrow C$ deux homomorphismes d’anneaux noethériens. On suppose que $B\otimes_{A}C$ est un anneau noethérien. Alors:
a\) Si $\sigma$ est régulier, il en est de même de $\sigma\otimes I_{C}:C\rightarrow$ $B\otimes_{A}C$; la réciproque est vraie si $\sigma$ est plat et $^a\rho$ est surjective.
b\) Si les fibres de $\sigma$ sont des anneaux d’intersection complète (resp. de Gorenstein, resp. de Cohen-Macaulay, resp. vérifient $(S_{n})$), il en est de même de celles de $\sigma\otimes I_{C}$; la réciproque est vraie si $^a\rho$ est surjective.
**Démonstration.** a) Supposons que $\sigma$ est un homomorphisme régulier , alors il est plat et par suite $\sigma\otimes I_{C}$ est plat. L’implication résulte alors de la Proposition précédente en tenant compte du Lemme.
Réciproquement, d’après Proposition (**I**,3.6.1) de [@Grot2], l’application $^a(\sigma\otimes I_{C})$ est surjective. La réciproque résulte aussi de la Proposition précédente en tenant compte du Lemme . b) i) Supposons d’abord que $\sigma$ est un homomorphisme d’intersection complète, le même raisonnement que dans le cas précédent montre que $\sigma\otimes I_{C}$ est un homomorphisme d’intersection complète.
ii\) Supposons maintenant que les fibres de $\sigma$ sont des anneaux d’intersection complète (resp. de Gorenstein, resp. de Cohen-Macaulay, resp. vérifient $(S_{n})$ ). Posons $D=B\otimes_{A}C$ et soit $\mathfrak{r}$ un idéal premier de $C$. L’anneau $D\otimes_{c}k(\mathfrak{r})=$ $(B\otimes_{A}C)\otimes_{C}k(\mathfrak{r})$ est isomorphe à $B\otimes_{A}k(\mathfrak{r})$. Soit $\mathfrak{p}=\rho ^{-1}(\mathfrak{r})$. Donc $D\otimes_{C}k(\mathfrak{r})$ est isomorphe à $(B\otimes _{A}k(\mathfrak{p}))\otimes_{k(\mathfrak{p})}k(\mathfrak{r})$. Comme l’homomorphisme $k(\mathfrak{p})\rightarrow k(\mathfrak{r})$ est d’intersection complète, il résulte du cas précédent appliqué aux homomorphismes $k(\mathfrak{p})\rightarrow k(\mathfrak{r})$ et $k(\mathfrak{p})\rightarrow B\otimes_{A}k(\mathfrak{p})$ que l’homomorphisme $B\otimes_{A}k(\mathfrak{p})\rightarrow (B\otimes_{A}k(\mathfrak{p}))\otimes_{k(\mathfrak{p})}k(\mathfrak{r})$ est d’intersection complète. On en déduit que l’homomorphisme $B\otimes_{A}k(\mathfrak{p})\rightarrow
D\otimes_{C}k(\mathfrak{r})$ est d’intersection complète (resp. de Gorenstein, resp. de Cohen-Macaulay, resp. $(S_{n})$) et que par suite $D\otimes_{C}k(\mathfrak{r})$ est un anneau d’intersection complète (resp. de Gorenstein, resp. de Cohen-Macaulay, resp. vérifie $(S_{n})$ ).
Réciproquement, soit $\mathfrak{p}$ un idéal premier de $A$ et soit $\mathfrak{r}$ un idéal premier de $C$ tel que $\mathfrak{p}=\rho^{-1}(\mathfrak{r})$. L’homomorphisme $k(\mathfrak{p})\rightarrow k(\mathfrak{r})$ est fidèlement plat, il en est de même de l’homomorphisme $B\otimes_{A}k(\mathfrak{p})\rightarrow
D\otimes_{C}k(\mathfrak{r})$. Donc $B\otimes_{A}k(\mathfrak{p})$ est un anneau d’intersection complète (resp. Gorenstein, resp. de Cohen-Macaulay, resp. vérifie $(S_{n})$ ).\
[*Remarques.*]{} i) Si l’homomorphisme $\rho$ est fidèlement plat alors l’application $^a\rho$ est surjective et si de plus $\sigma\otimes I_{C}$ est plat alors $\sigma$ est plat. ii) Dans [@Avramov2], [@Avramov3], [@Avramov4], on trouve des résultats sur le changement de base pour les homomorphismes qu’ils ont défini.
Soient $\sigma:A\rightarrow B$ et $\rho:A\rightarrow C$ deux homomorphismes d’anneaux noethériens. On suppose que $B\otimes_{A}C$ est un anneau noethérien et que $\sigma$ est régulier (resp. d’intersection complète, resp. de Gorenstein, resp. de Cohen-Macaulay, resp. $\ (S_{n})$). Si $C$ est un anneau régulier (resp. d’intersection complète, resp. de Gorenstein, resp. de Cohen-Macaulay, resp. vérifie $(S_{n})$ ) il en est de même de $B\otimes_{A}C$; la réciproque est vraie si $\sigma$ est fidèlement plat.
**Démonstration.** D’après le Théorème précédent, l’homomorphisme $\sigma\otimes I_{C}$ est régulier (resp. d’intersection complète, resp. de Gorenstein, resp. de Cohen-Macaulay, resp. $\ (S_{n})$); d’où l’implication. La réciproque résulte du fait que $\sigma\otimes I_{C}$ est fidèlement plat.\
On en déduit que si $k$ est un corps et si $B\otimes_{k}C$ est un anneau noethérien alors $B\otimes_{k}C$ vérifie $(S_{n}) $ si B et C la vérifient.
Soit $k$ un corps. On suppose que $B\otimes _{k}C$ est un anneau noethérien et que pour tout idéal maximal $\mathfrak{n}$ de $C$, $k(\mathfrak{n})$ est séparable sur $k$. Si $B$ et $C$ sont réguliers alors $B\otimes
_{k}C$ est régulier.
**Démonstration.** Pout tout idéal maximal $\mathfrak{n}$ de $C,\ \ k(\mathfrak{n})$ est séparable sur $k$ et $C_{\mathfrak{n}}$ est régulier, donc $C_{\mathfrak{n}}$ est géométriquement régulière sur $k$. Le résultat découle donc de la Proposition précédente puisque l’ homomorphisme $k\rightarrow C$ est régulier.\
Si $C$ est régulier, il peut se faire que $k(\mathfrak{n})$ soit séparable sur $k$ pour tout idéal maximal $\mathfrak{n}$ de $C$, sans que $k(\mathfrak{r})$ soit séparable sur $k$ pour tout idéal premier $\mathfrak{r}$ de $C$ \[6, Theorem 2.11\]. En effet, soient $k$ un corps non parfait de caractéristique $p>0$, $a$ un élément de $k-k^p$, $A$ l’anneau de polynômes $k[X,Y]$ et $C$ l’anneau local de $A$ en l’idéal maximal engendré par $X$ et $Y$. L’anneau $C$ est régulier et son corps résiduel $k(X,Y)$ est séparable sur $k$. Soit $f$ le polynôme $Y^p-aX^p$. Comme $a\notin k$, $f$ est irréductible dans $A$. Notons $\mathfrak{p}$ l’idéal premier de $A$ engendré par $f$ et $\mathfrak{r}$ l’idéal premier $\mathfrak{p}C$ de $C$. Montrons que $ k(\mathfrak{r})$ n’est pas séparable sur $k$. Le corps $ k(\mathfrak{r})$ s’identifie canoniquement à $k(\mathfrak{p})$ donc il suffit de montrer que $k(\mathfrak{p})$ n’est pas séparable sur $k$. Si $x$ et $y$ sont les images respectives de $X$ et $Y$ dans $A{/\mathfrak{p}}$, on a bien $(\frac{y}{x})^p \in k$ et $\frac{y}{x}\notin k$. Donc $k(\mathfrak{p})$ n’est pas séparable sur $k$.
Soient $\ \sigma :A\rightarrow B$ et $\rho :A\rightarrow C$ deux homomorphismes d’anneaux noethériens. On suppose que $B\otimes _{A}C$ est un anneau noethérien. Si $\sigma $ est plat alors $B\otimes _{A}C$ est un anneau d’intersection complète (resp. de Gorenstein, resp. de Cohen-Macaulay) si B et C le sont.
**Démonstration.** Si $B$ est un anneau d’intersection complète (resp. de Gorenstein, resp. de Cohen-Macaulay) alors $\sigma$ est un homomorphisme d’intersection complète (resp. de Gorenstein, resp. de Cohen-Macaulay). Le résultat découle donc de la Proposition 2.4.
Soient $\ \sigma :A\rightarrow B$ et $\rho :A\rightarrow C$ deux homomorphismes locaux d’anneaux locaux noethériens. On suppose que le corps résiduel $C/\mathfrak{n}$ de $C$ est de rang fini sur le corps résiduel $A/\mathfrak{m}$ de A.a) Si l’homomorphisme $\sigma $ est formellement lisse et $C$ est régulier alors l’anneau semi-local $B\hat{\otimes}_{A}C$ est régulier.b) Si $\sigma $ est plat alors $B\hat{\otimes}_{A}C$ est un anneau d’intersection complète (resp. de Gorenstein, resp. de Cohen-Macaulay) si $B$ et $C$ le sont.
**Démonstration.** Cas où $B$ est complet. On utilise la Proposition (**0**,7.7.10) de [@Grot2]: Posons $E=B\hat{\otimes}_{A}C$ . D’après i) $E$ est semi-local noethérien. Montrons que c’est un anneau régulier (resp. d’intersection complète , resp. de Gorenstein, resp. de Cohen-Macaulay). Soit $\mathfrak{Q}$ un idéal maximal de $E$. D’après ii) $\mathfrak{Q}$ est au dessus de $\mathfrak{n}$. Pour montrer que $E_{\mathfrak{Q}}$ est régulier (resp. d’intersection complète , resp. de Gorenstein, resp. de Cohen-Macaulay) il suffit de montrer que $(E/\mathfrak{n}E)_{\mathfrak{Q}}$ est régulier (resp. d’intersection complète , resp. de Gorenstein, resp. de Cohen-Macaulay) puisque $C$ l’est et d’après iii) $E$ est un $C$-module plat. D’après ii) $E/\mathfrak{n}E
$ est isomorphe à $B\otimes _{A}(C/\mathfrak{n}).$ Donc b) résulte de la Proposition 2.6. D’autre part $B\otimes _{A}(C/\mathfrak{n})$ est isomorphe à $(B/\mathfrak{m}B)\otimes _{A/\mathfrak{m}}(C/\mathfrak{n})$ donc a) résulte de la Proposition 2.4 puisque l’homomorphisme $A/\mathfrak{m}\longrightarrow B/\mathfrak{m}B$ est régulier.
Cas général. L’anneau $B\hat{\otimes}_{A}C$ s’identifie à $\hat{B}\hat{\otimes}_{A}C$. Il suffit d’appliquer le cas précédent aux homomorphismes $A\overset{\sigma }{\longrightarrow }B\longrightarrow \hat{B}$ et $\rho $. Dans a) l’homomorphisme $A\overset{\sigma }{\longrightarrow }B\longrightarrow \hat{B}$ est formellement lisse et dans b) $\hat{B}$ vérifie la même propriété que $B$.\
[*Exemples.*]{} Les deux exemples suivants montrent que les résultats précédents tombent en défaut si les homomorphismes ne sont pas plats.
On prend pour $A$ un anneau de valuation discrète complet, $\pi$ une uniformisante de $A$ et $k$ son corps résiduel.
i\) Si $B=k$ et $C=A[[X]]/({X^2}-\pi)$ où $X$ est indéterminée sur $A$, alors $B$ et $C$ sont des anneaux locaux réguliers et la $k$-algèbre $B\otimes_{A}k$ est géométriquement régulière mais les anneaux $B\otimes _{A}C$ et $B\hat{\otimes}_{A}C$ qui sont isomorphes d’après la Proposition (**0**,7.7.9) de [@Grot2], ne sont pas réguliers car $B\otimes _{A}C$ est isomorphe à $k[[X]]/(X^2)$.
ii\) Si $B=A[[X,Y]]/({X^2}-\pi,XY)$ où $X$ et $Y$ sont deux indéterminées sur $A$ et $C=k$, les anneaux $B$ et $C$ sont des anneaux locaux d’intersection complète, mais les anneaux $B\otimes _{A}C$ et $B\hat{\otimes}_{A}C$ ne sont pas des anneaux de Cohen-Macaulay car ils sont isomorphes à $k[[X,Y]]/(X^2,XY)$.
Soit $\sigma:A\rightarrow B$ un homomorphisme d’anneaux noethériens. On suppose que $B\otimes_{A}B$ est un anneau noethérien et que $\sigma$ plat. Alors les propriétés suivantes sont équivalentes:i) L’anneau $B$ est régulier et l’homomorphisme $\sigma$ est régulier.ii) L’anneau $B\otimes_{A}B$ est régulier.
**Démonstration.** $i)\Rightarrow ii)$ Cela résulte du Corollaire 2.2.
$ii)\Rightarrow i)$ Supposons $B\otimes_{A}B$ régulier. L’homomorphisme $\sigma$ est plat donc $\sigma\otimes I_{B}$ est fidèlememt plat et par suite $B$ est régulier. Montrons que $\sigma$ est régulier. Soit $\mathfrak{q}$ un idéal premier de B. Comme $H_{1}(A,B,k(\mathfrak{q}))\cong H_{1}(B,B\otimes_{A}B,k(\mathfrak{q}))$ il suffit de montrer que $H_{1}(B,B\otimes_{A}B,k(\mathfrak{q}))=0$. On a la suite exacte $$H_{2}(B\otimes_{A}B,B,k(\mathfrak{q}))\rightarrow H_{1}(B,B\otimes_{A}B,k(\mathfrak{q}))\rightarrow H_{1}(B,B,k(\mathfrak{q}))$$ associée à la factorisation $p\circ(\sigma\otimes_{A} I_{B})=I_{B}$, où $p:B\otimes_{A}B\rightarrow B$ est l’homomorphisme canonique défini par $p(b\otimes b^{\prime})=bb^{\prime}$. D’après [@andre Supplément, Proposition. 32] on a $H_{2}(B\otimes_{A}B,B,k(\mathfrak{q}))=0$. Donc $H_{1}(B,B\otimes_{A}B,k(\mathfrak{q}))=0$ car on a $H_{1}(B,B,k(\mathfrak{q}))=0$.
Soient $K$ un corps et $L$ une extension de $K$. On suppose que l’anneau $L\otimes_{K}L$ est noethérien. Alors $L\otimes_{K}L$ est un anneau régulier si et seulement si L est séparable sur K.
**Démonstration.** En effet, l’homomorphisme $K\rightarrow L$ est régulier si et seulement si $L$ est une extension séparable de $K$.
Cas Presque Cohen-Macaulay
==========================
Suivant ([@Kang],1.5) on dira qu’un anneau noethérien $A$ est presque de Cohen-Macaulay si $dim(A_{\mathfrak{p}})\leq{prof(A_{\mathfrak{p}}})+1$ pour tout idéal premier $\mathfrak{p}$ de $A$. Il est clair que si $A$ est presque de Cohen-Macaulay alors, pour tout idéal premier $\mathfrak{p}$ de $A$, $A_{\mathfrak{p}}$ est presque de Cohen-Macaulay; et d’après ([@Kang], 2.6), $A$ est presque de Cohen-Macaulay, si et seulement si, $dim(A_{\mathfrak{m}})\leq{prof(A_{\mathfrak{m}}})+1$ pour tout idéal maximal $\mathfrak{m}$ de $A$. Le résultat suivant est une variante de la Proposition 2.2 de [@Ion] qui distingue les anneaux presque de Cohen-Macaulay des anneaux considérés dans le paragraphe précédent.
Soient $\sigma:A\rightarrow B$ un homomorphisme local d’anneaux locaux noethériens et $\mathfrak{m}$ l’idéal maximal de $A$. On suppose que $\sigma:A\rightarrow B$ est plat. Alors les propriétés suivantes sont équivalentes:a) L’anneau $B$ est presque de Cohen-Macaulay.b) L’un des anneaux, $A$ ou $B/\mathfrak{m}B$, est de Cohen-Macaulay et l’autre est presque de Cohen-Macaulay.
**Démonstration.** L’équivalence résulte des égalités $$dimB=dimA+dim{B/\mathfrak{m}B}\newline$$ $$profB=profA+prof{B/\mathfrak{m}B}$$ et de la double inégalité $$profA\leq{dimA}\leq{profA+1}.$$
L’exemple suivant donne une réponse à la question 2.3 posée par Ionescu dans [@Ion].
[*Exemple.*]{} Soient $k$ un corps, $R$ l’anneau $k[X,Y]/(X^2,XY)$ et $S$ l’anneau $k[X,Y,U,V]/(X^2,XY,U^2,UV)$. L’homomorphisme canonique $R\rightarrow S$ est plat car L’homomorphisme $R\rightarrow R\otimes_{k}R$ est plat et $S$ s’identifie à $R\otimes_{k}R$. Soient $A$ l’anneau local de $R$ en l’idéal maximal $(x,y)$ et $B$ celui de $S$ en l’idéal maximal $(x,y,u,v)$. L’homomorphisme induit $A\rightarrow B$ est local et plat. On vérifie que $xu$ annule l’idéal $(x,y,u,v)$ et que $xu\neq0$. Donc $profB=0$ et par suite $profA=prof{B/\mathfrak{m}B}=0$. D’autre part on a $dimA=dim{B/\mathfrak{m}B}=1$ et $dimB=2$. Donc les anneaux $A$ et ${B/\mathfrak{m}B}$ sont des anneaux presque de Cohen-Macaulay mais $B$ ne l’est pas.\
Pour les homomorphismes presque de Cohen-Macaulay on a:
Soient $\sigma:A\rightarrow B$ et $\rho:A\rightarrow C$ deux homomorphismes d’anneaux noethériens. On suppose que $B\otimes_{A}C$ est un anneau noethérien. Si les fibres de $\sigma$ sont presque de Cohen-Macaulay il en est de même de celles de $\sigma\otimes I_{C}$; la réciproque est vraie si $^a\rho$ est surjective..
**Démonstration.** Soient $\mathfrak{r}$ un idéal premier de $C$ et $\mathfrak{p}=\rho ^{-1}(\mathfrak{r})$. L’homomorphisme $k(\mathfrak{p})\rightarrow k(\mathfrak{r})$ est de Cohen-Macaulay donc, d’après le Théorème, l’homomorphisme $B\otimes_{A}k(\mathfrak{p})\rightarrow
({B\otimes_{A}C})\otimes_{C}k(\mathfrak{r})$ l’est aussi; l’implication résulte alors du Lemme précédent. La réciproque en résulte aussi puisque l’homomorphisme est fidèlement plat.
Soient $\ \sigma :A\rightarrow B$ et $\rho :A\rightarrow C$ deux homomorphismes d’anneaux noethériens. On suppose que $B\otimes _{A}C$ est un anneau noethérien et que $\sigma $ est plat. Si pour tout idéal maximal $\mathfrak{N}$ de $B\otimes _{A}C$ l’un des anneaux, $B_{\mathfrak{q}}$ ou $C_{\mathfrak{r}}$, où $\mathfrak{q}$ et $\mathfrak{r}$ sont les images réciproques respectives dans $B$ et $C$, est de Cohen-Macaulay et l’autre est presque de Cohen-Macaulay alors l’anneau $B\otimes _{A}C$ est presque de Cohen-Macaulay.
**Démonstration.** L’anneau ${(B\otimes _{A}C)}_{\mathfrak{N}}$ s’identifie à un anneau de fractions de $B_{\mathfrak{q}}\otimes _{A_{\mathfrak{p}}}C_{\mathfrak{r}}$, où $\mathfrak{p}$ est l’image réciproque de $\mathfrak{N}$ dans $A$. Donc on se ramène au cas où l’un des anneaux, $B$ ou $C$, est de Cohen-Macaulay et l’autre est presque de Cohen-Macaulay.
Supposons que $B$ est un anneau de Cohen-Macaulay (resp. presque de Cohen-Macaulay ); alors dans ce cas, la conclusion découle du Lemme puisque $\sigma$ est un homomorphisme de Cohen-Macaulay (resp. presque de Cohen-Macaulay ) et par suite, d’après le Théorème (resp. la Proposition précédente ), $\sigma\otimes I_{C}$ l’est.\
Soient $\ \sigma :A\rightarrow B$ et $\rho :A\rightarrow C$ deux homomorphismes locaux d’anneaux locaux noethériens. On suppose que le corps résiduel de $C$ est de rang fini sur celui de A. Si $\sigma $ est plat alors l’anneau $B\hat{\otimes}_{A}C$ est presque de Cohen-Macaulay si l’un des anneaux, $B$ ou $C$, est de Cohen-Macaulay et l’autre est presque de Cohen-Macaulay.
**Démonstration.** On raisonne comme dans la Proposition 2.7: Si $B$ est un anneau de Cohen-Macaulay (resp. presque de Cohen-Macaulay ) on utilise la Proposition 2.6(resp. 3.3) puis le Lemme.\
L’exemple suivant montre que les deux Propositions précédentes tombent en défaut si les deux anneaux ( locaux ) sont presque de Cohen-Macaulay. Soient $k$ un corps, $B$ l’anneau local de $k[X,Y]/(X^2,XY)$ en l’idéal maximal $(x,y)$, $C$ celui de $k[U,V]/(U^2,UV)$ en l’idéal maximal $(u,v)$. Les anneaux $B$ et $C$ sont presque de Cohen-Macaulay et $B\otimes_{k}C$ est noethérien. Notons $D$ l’anneau $B\otimes_{A}C$, $\mathfrak{r}$ l’idéal maximal ${(x,y)\otimes_{k}B+C\otimes_{k}(u,v)}$ de $D$, $E$ l’anneau $B\hat{\otimes}_{A}C$ et $T$ l’anneau local de $k[X,Y,U,V]/(X^2,XY,U^2,UV)$ en l’idéal maximal $(x,y,u,v)$.
i\) L’anneau $D_{\mathfrak{r}}$ est isomorphe à $T$ donc $D$ n’est pas un anneau presque de Cohen-Macaulay.
ii\) L’anneau $E$ est complet donc $\mathfrak{r}E$ est contenu dans son radical, et $\mathfrak{r}E$ est un idéal maximal de $E$ car $E/\mathfrak{r}E$ est isomorphe à $D/\mathfrak{r}$. Donc $E$ est un anneau local et $\mathfrak{r}E$ est son idéal maximal. L’homomorphisme $D\rightarrow E$ est plat. Donc l’ homomorphisme induit $D_{\mathfrak{r}}\rightarrow E$ est local et plat. D’après le Lemme précédent, $E$ n’est pas un anneau presque de Cohen-Macaulay car $D_{\mathfrak{r}}$ ne l’est pas.\
[99]{}
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[^1]: Le théorème 2.1, le Corollaire 2.2, le Corollaire 2.4 et la Proposition 2.6 de la première version de cet article (Arxiv: 1304.5395v1 \[math.AC\]) figurent dans le projet de note intitulé “Sur le produit tensoriel d’algèbres” que l’auteur avait soumis aux CRAS de Paris (n$^o$ CRMATHEMATIQUE-D-12-00204) le 07 Juin 2012.
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abstract: 'We prove an analogue of Gutzmer’s formula for Hermite expansions. As a consequence we obtain a new proof of a characterisation of the image of $ L^2(\R^n)$ under the Hermite semigroup. We also obtain some new orthogonality relations for complexified Hermite functions.'
address: |
Department of Mathematics\
Indian Institute of Science\
Bangalore 560 012, India. [*e-mail :*]{} [[email protected]]{}
author:
- 'S. Thangavelu'
title: |
An analogue of Gutzmer’s formula for\
.5em Hermite expansions\
1.5em [by]{}
---
**Introduction**
=================
By Gutzmer’s formula we mean any analogue of the formula $$(2\pi)^{-1}\int_0^{2\pi} |f(x+iy)|^2 dx =
\sum_{k=-\infty}^\infty |\hat{f}(k)|^2
e^{-2ky}$$ valid for any $ 2\pi $ periodic holomorphic function $ f $ in a strip in the complex plane. Here $ \hat{f}(k) $ stands for the Fourier coefficients of the restriction of $ f $ to the real line. An analogue of such a formula was established by Lassalle \[9\] for holomorphic functions on the complexification of compact symmetric spaces. A similar formula for holomorphic functions on the complex crowns associated to noncompact Riemannian symmetric spaces was discovered by Faraut \[3\]. As can be seen from Faraut \[4\] and Krötz-Olafsson-Stanton \[8\] such formulas are useful in the study of Segal-Bargmann or heat kernel transforms.
Recently in \[15\] we have proved an analogue of Gutzmer’s formula on the Heisenberg groups and used them to study heat kernel transforms and Paley-Wiener theorems.
In this paper we prove an analogue of Gutzmer’s formula for Hermite expansions. Let $ H $ be the Hermite operator on $ \R^n $ having the spectral decomposition $ H = \sum_{k=0}^\infty (2k+n) P_k.$ Let $ \H^n = \R^n \times \R^n \times \R $ be the Heisenberg group whose complexification is $ \C^n \times \C^n \times \C.$ Let $ \pi(x,u) $ be the projective representation of $ \R^n \times \R^n $ related to the Schrödinger representation of $ \H^n $ and denote by $ \pi(x+iy,u+iv) $ its extension to $ \C^n \times \C^n.$ Let $ K =
Sp(n,\R)\cap O(2n,\R) $ which acts on $ \C^n \times \C^n.$ Denote by $ \varphi_k(z,w) $ the Laguerre functions of type $ (n-1) $ extended to $ \C^n \times \C^n.$ Our main result is the following.
Let $ F $ be an entire function on $ \C^n .$ Denote by $ f $ its restriction to $ \R^n.$ Then for any $ z = x+iy, w= u+iv \in \C^n $ we have $$\int_{\R^n} \int_K |\pi(\sigma.(z,w))F(\xi)|^2 d\sigma d\xi$$ $$= e^{(u\cdot y - v \cdot x)} \sum_{k=0}^\infty \frac{k!(n-1)!}{(k+n-1)!}
\varphi_k(2iy,2iv) \|P_kf\|_2^2 .$$
As an immediate corollary we obtain the following characterisation of the image of $ L^2(\R^n) $ under the Hermite semigroup $ e^{-tH}, t > 0.$ Let $$U_t(x,y) = 2^n(\sinh(4t))^{-\frac{n}{2}}e^{\tanh(2t)|x|^2 -\coth(2t)|y|^2}.$$
An entire function $ F $ on $ \C^n $ belongs to the image of $ L^2(\R^n) $ under $ e^{-tH} $ if and only if $$\int_{\R^n}\int_{\R^n} |F(x+iy)|^2 U_t(x,y) dx dy < \infty.$$
This characterisation is not new and there are several proofs available in the literature, see Byun \[1\], Karp \[6\] and Thangavelu \[14\]. In Section 4 we derive some more consequences of the Gutzmer’s fomula.
We conclude the introduction with some remarks about the methods used in proving Gutzmer formulas. As in the case of Fourier series, Lassalle \[9\] used Plancherel theorem for the Laurent expansions of holomorphic functions on the complexifications of compact symmetric spaces $ X = K/M.$ The matrix coefficients associated to class one represenations in the unitary dual of a compact Lie group $ K $ holomorphically extend to its complexification $ K_\C.$ Thus any function $ f $ whose ’Fourier coefficients’ have exponential decay can be extended to the complexification $ X_\C = K_\C/M_\C.$ Then by appealing to Plancherel theorem and using orthogonality relations the required formula was established. In \[2\] Faraut considered a general unimodular group $ G $ and proved a proposition from which Gutzmer’s formula can be deduced for noncompact Riemannian symmetric spaces \[3\] and Heisenberg groups \[15\].
Thus in all the previous settings the basic functions appearing in the Fourier series or transform are matrix coefficients of certain irreducible unitary representations of the underlying group. Contrary to this, the Hermite functions do not occur as matrix coefficients. However, the Hermite functions are used to calculate the matrix coefficients associated to Schrödinger representations of $ \H^n $ resulting in special Hermite or Laguerre functions. This explains why the representation $ \pi(z,w) $ occurs in our Gutzmer’s formula. The close relationship between Hermite and Laguerre functions are then used to derive the Gutzmer’s formula.
**Preliminaries**
=================
In this section we collect some relevant information about special Hermite functions and prove some results that are required in the next section. We closely follow the notations used in \[12\] and \[13\] and we refer the reader to these monographs for more details.
Let $ \Phi_\alpha, \alpha \in \N^n $ be the Hermite functions on $ \R^n $ normalised so that their $ L^2 $ norms are one. These are eigenfunctions of the Hermite operator $ H = -\Delta +|x|^2 $ with eigenvalues $ (2|\alpha|+n).$ On finite linear combinations of such functions we can define certain operators $ \pi(z,w) $ where $ z, w \in ~\C^n $ as follows: $$\pi(z,w)\Phi_\alpha(\xi) = e^{i(z \cdot \xi+\frac{1}{2}z \cdot w)}
\Phi_\alpha
(\xi+w)$$ where $ z \cdot \xi = \sum_{j=1}^n z_j \xi_j $ and $ z \cdot w =
\sum_{j=1}^n z_j w_j $. Note that $ \Phi_\alpha(\xi)= H_\alpha(\xi)
e^{-\frac{1}{2}|\xi|^2} $ where $ H_\alpha $ is a polynomial on $ \R^n $ and for $ z \in \C^n $ we define $ \Phi_\alpha(z) $ to be $H_\alpha(z)
e^{-\frac{1}{2}z^2} $ where $ z^2 = z \cdot z.$ The special Hermite functions $ \Phi_{\alpha,\beta}(z,w) $ are then defined by $$\Phi_{\alpha,\beta}(z,w) = (2\pi)^{-\frac{n}{2}}( \pi(z,w)\Phi_\alpha,
\Phi_\beta).$$ The restrictions of $ \Phi_{\alpha,\beta}(z,w) $ to $ \R^n\times\R^n $ are usually called the special Hermite functions and the family $\{\Phi_{\alpha,\beta}(x,u): \alpha, \beta \in \N^n \} $ forms an orthonormal basis for $ L^2(\C^n).$
As we have mentioned in the introduction the operators $ \pi(z,w) $ are related to the Schrödinger representation $ \pi_1 $ of the Heisenberg group $ \H^n.$ Recall that $ \H^n = \R^n \times \R^n \times \R $ is equipped with the group law $ (x,u,t)(x',u',t') = (x+x',u+u',t+t'+\frac{1}{2}(u\cdot x'-x\cdot u')).$ For each nonzero real number $ \lambda $ we have a representation of $ \H^n $ realised on $ L^2(\R^n) $ given by $$\pi_\lambda(x,u,t)\varphi(\xi) = e^{i\lambda t} e^{i\lambda(x\cdot \xi+\frac
{1}{2}x\cdot u)}\varphi(\xi+u).$$ Thus $ \pi(x,u) = \pi_1(x,u,0) $ and it defines a projective representation of $ \R^n \times \R^n.$
For $ (z,w) \in \C^{2n} $ the operators $ \pi(z,w) $ are not even bounded on $ L^2(\R^n). $ However, they are densely defined and satisfy $$\pi(z,w)\pi(z',w') = e^{\frac{i}{2}(z' \cdot w-z \cdot w')}\pi(z+z',w+w').$$ Moreover, $$(\pi(iy,iv)\Phi_\alpha, \Phi_\beta) = (\Phi_\alpha,\pi(iy,iv)\Phi_\beta).$$ This means that $ \pi(iy,iv) $ are self adjoint operators. We need to calculate the $ L^2 $ norms of $ \pi(z,w)\Phi_\alpha.$ Let $ L_k^{n-1}$ be Laguerre polynomials of type $ (n-1) $ and define the Laguerre functions $ \varphi_k $ by $$\varphi_k(x,u) = L_k^{n-1}(\frac{1}{2}(x^2+u^2))e^{-\frac{1}{4}
(x^2+u^2)}.$$ Then it is known that $$\varphi_k(x,u) = (2\pi)^{n/2}\sum_{|\alpha|=k} \Phi_{\alpha,\alpha}(x,u).$$ These functions have a natural holomorphic extension to $ \C^n \times \C^n $ denoted by the same symbol: $$\varphi_k(z,w) = (2\pi)^{n/2}\sum_{|\alpha|=k} \Phi_{\alpha,\alpha}(z,w).$$
For any $ z = x+iy, w = u+iv \in \C^n $ and $ \alpha \in \N^n $ we have $$\int_{\R^n} |\pi(z,w)\Phi_\alpha(\xi)|^2 d\xi = (2\pi)^{\frac{n}{2}}
e^{(u\cdot y -v\cdot x)} \Phi_{\alpha,\alpha}(2iy,2iv).$$
[**Proof:**]{} It is enough to prove the result in one dimension. Recall Mehler’s formula satisfied by the Hermite functions $ h_k $ on $ \R $: $$\sum_{k=0}^\infty h_k(\xi)h_k(\eta) r^k = \pi^{-\frac{1}{2}}
e^{-\frac{1}{2}\frac{1+r^2}{1-r^2}(\xi^2+\eta^2)+\frac{2r}{1-r^2}\xi \eta}$$ valid for all $ r $ with $ |r| <1.$ The formula is clearly valid even if $ \xi $ and $ \eta $ are complex. A simple calculation shows that $$\sum_{k=0}^\infty r^k |\pi(z,w)h_k(\xi)|^2$$ $$= \pi^{-\frac{1}{2}}(1-r^2)^{-\frac{1}{2}} e^{-(uy+vx)}e^{\frac{1+r}{1-r}
v^2}e^{-\frac{1-r}{1+r}(\xi+u)^2}e^{-2y\xi}.$$ Integrating both sides with respect to $ \xi $ we obtain $$\sum_{k=0}^\infty r^k \int_{\R} |\pi(z,w)h_k(\xi)|^2 d\xi$$ $$= (1-r)^{-1} e^{(uy-vx)} e^{\frac{1+r}{1-r}(y^2+v^2)}.$$ We now recall that the generating function for the Laguerre functions $ \varphi_k(x,u) $ when $ n =1 $ reads as $$\sum_{k=0}^\infty r^k \varphi_k(x,u) = (1-r)^{-1} e^{-\frac{1}{4}
(x^2+u^2)}.$$ A comparison with this shows that $$\int_\R |\pi(z,w)h_k(\xi)|^2 d\xi = e^{(uy-vx)} \varphi_k(2iy,2iv).$$ Since $ \Phi_{k,k}(x,u) = (2\pi)^{-\frac{1}{2}}\varphi_k(x,u) $ this proves the Lemma.
In the above lemma we have calculated the $ L^2 $ norm of $ \pi(z,w)\Phi_\alpha $ by integrating the generating function. We can also calculate the norm by expanding $ \pi(z,w)\Phi_\alpha $ in terms of the Hermite basis and appealing to the Plancherel theorem for Hermite expansions. This leads to the following identity which is crucial for our main result.
For any $ \alpha \in \N^n, z = x+iy, w = u+iv \in \C^n $ we have $$\sum_{\beta \in \N^n} |\Phi_{\alpha,\beta}(z,w)|^2 = (2\pi)^{\frac{-n}{2}}
e^{(u\cdot y -v\cdot x)} \Phi_{\alpha,\alpha}(2iy,2iv).$$
[**Proof:**]{} We just have to recall that $ (\pi(z,w)\Phi_\alpha,\Phi_\beta)
= (2\pi)^{\frac{n}{2}}\Phi_{\alpha,\beta}.$
We also need some estimates on the holomorphically extended Hermite functions on $ \C^n.$ Let us define $ \Phi_k(x,u) = \sum_{|\alpha| = k}\Phi_\alpha(x)
\Phi_\alpha(u) $ which is the kernel of the projection $ P_k.$ Note that $
\Phi_k $ extends to $ \C^n \times \C^n $ as an entire function. Using Mehler’s formula for Hermite functions and the generating function for Laguerre functions we can get the following representation of $ \Phi_k $ in terms of Laguerre functions of type $ (n/2-1).$
$$\Phi_k(z,w) = \pi^{-\frac{n}{2}}\sum_{j=0}^k (-1)^j L_j^{n/2-1}(\frac{1}{2}
(z+w)^2)L_{k-j}^{n/2-1}(\frac{1}{2}(z-w)^2) e^{-\frac{1}{2}(z^2+w^2)}$$ where $ z^2 = \sum_{j=1}^n z_j^2 $ and $ w^2 = \sum_{j=1}^n w_j^2.$
[**Proof:**]{} The Laguerre functions of type $ (n/2-1) $ are given by the generating function $$\sum_{k=0} r^k L_k^{n/2-1}(\frac{1}{2}z^2)e^{-\frac{1}{4}z^2}
= (1-r)^{-n/2}e^{-\frac{1}{4}\frac{1+r}{1-r}z^2}.$$ A simple calculation shows that $$(1-r)^{-n/2}e^{-\frac{1}{4}\frac{1+r}{1-r}(z+w)^2}(1+r)^{-n/2}
e^{-\frac{1}{4}\frac{1-r}{1+r}(z-w)^2}$$ $$= (1-r^2)^{-n/2} e^{-\frac{1}{2}\frac{1+r^2}{1-r^2}(z^2+w^2)+
\frac{2r}{1-r^2}zw}.$$ Comparing this with Mehler’s formula and rewriting the left hand side as a power series in $ r $ and then equating coefficients of $ r^k $ we obtain the lemma.
The above lemma has been already used by us in the study of Bochner-Riesz means for multiple Hermite expansions. Here we need the above in order to get the following estimate on $ \Phi_k(z,w).$
For all $ z = x+iy \in \C^n $ and $ k = 1,2,... $ we have $$|\Phi_k(z,\bar{z})| \leq C(y) k^{\frac{3}{4}(n-1)}
e^{2(k)^{\frac{1}{2}}|y|}$$ where $ C(y) $ is locally bounded.
[**Proof:**]{} From the previous lemma we have $$\Phi_k(z,\bar{z}) = \pi^{-\frac{n}{2}}\sum_{j=0}^k (-1)^j L_j^{n/2-1}
(2|x|^2)e^{-|x|^2} L_{k-j}^{n/2-1}(-2|y|^2) e^{|y|^2}.$$ We now make use of the following estimates on Laguerre functions. First of all we know that $$|L_j^{n/2-1}(2|x|^2)e^{-|x|^2}| \leq C j^{n/2-1}$$ uniformly in $ x.$ On the other hand Perron’s formula for Laguerre polynomials in the complex domain (see Theorem 8.22.3 in Szego \[11\] ) gives us $$L_{j}^{n/2-1}(-2|y|^2) e^{|y|^2} \leq C(y) j^{\frac{(n-3)}{4}}
e^{2(j)^{\frac{1}{2}}|y|}$$ valid for all $ |y| \geq 1.$ Since $
L_{j}^{n/2-1}(-2|y|^2) \leq L_{j}^{n/2-1}(-2) $ we have the same estimate for all values of $ y.$ These two estimates give the required bound on $ \Phi_k(z,
\bar{z}).$
We conclude the preliminaries with establishing some more notation. Let $ Sp(n,\R) $ stand for the symplectic group consisting of $ 2n \times 2n $ real matrices that preserve the symplectic form $ [(x,u),(y,v)] = (u\cdot y-v\cdot x)
$ on $ \R^{2n} $ and have determinant one. Let $ O(2n,\R) $ be the orthogonal group and we define $ K = Sp(n,\R)\cap O(2n,\R).$ Then there is a one to one correspondence between $ K $ and the unitary group $ U(n) .$ Let $ \sigma
= a+ib $ be an $ n \times n $ complex matrix with real and imaginary parts $ a $ and $ b.$ Then $ \sigma $ is unitary if and only if the matrix $ A =
\begin{pmatrix}a & -b \cr b & a
\end{pmatrix}$ is in $ K.$ For these facts we refer to Folland \[4\]. By $ \sigma.(x,u) $ we denote the action of the correspoding matrix $ A $ on $ (x,u).$ This action has a natural extension to $ \C^n \times \C^n $ denoted by $ \sigma.(z,w) $ and is given by $ \sigma.(z,w) = (a.z-b.w, a.w+b.z) $ where $ \sigma = a+ib.$ For example, when $ n = 1 $ and $ \sigma = e^{i\theta} $ we see that the corresponding matrix $ A $ is $
\begin{pmatrix} \cos\theta & -\sin\theta \cr
\sin\theta & \cos\theta .
\end{pmatrix}$ Given $ \theta = (\theta_1,....,\theta_n) \in
\R^n $ we denote by $ k(\theta) $ the diagonal matrix in $ U(n) $ with entries $ e^{i\theta_j}.$ We denote by $ d\sigma $ the normalised Haar measure on $ K $ and by $ d\theta $ the Lebesgue measure $ d\theta_1 d\theta_2....d\theta_n.$
**The main results**
====================
Having set up notation and collected relevant results on special Hermite functions we are now ready to prove our main results. We begin with
Let $ f \in L^2(\R^n) $ be such that $ \|P_kf\|_2 \leq C_t
e^{-2k^{\frac{1}{2}}t} $ for all $ t > 0 $ and $ k \in \N.$ Then $ f $ has a holomorphic extension $ F $ to $ \C^n $ and we have the following formula for any $ z = x+iy, w= u+iv \in \C^n $: $$\int_{\R^n} \int_K |\pi(\sigma.(z,w))F(\xi)|^2 d\sigma d\xi$$ $$= e^{(u\cdot y - v \cdot x)} \sum_{k=0}^\infty \frac{k!(n-1)!}{(k+n-1)!}
\varphi_k(2iy,2iv) \|P_kf\|_2^2 .$$
[**Proof:**]{} Consider the Hermite expansion of the function $ f $ given by $$f(x) = \sum_{k=0}^\infty \sum_{|\alpha|=k}(f,\Phi_\alpha)\Phi_\alpha(x).$$ By Cauchy-Schwarz inequality $$|\sum_{|\alpha|=k}(f,\Phi_\alpha)\Phi_\alpha(x+iy)|^2 \leq
\Phi_k(x+iy,x-iy) \|P_kf\|_2^2 .$$ In view of Lemma 2.4 the hypothesis on $ f $ allows us to conclude that the series $$\sum_{k=0}^\infty \sum_{|\alpha|=k}(f,\Phi_\alpha)\Phi_\alpha(x+iy)$$ converges uniformly over compact subsets of $ \C^n $ and hence $ f $ extends to an entire function $ F $ on $ \C^n.$
Let $ D $ be the subgroup of $ K $ consisting of $ 2n \times 2n $ matrices associated to the elements $ k(\theta) \in U(n).$ We claim that it is enough to prove $$(2\pi)^{-n} \int_{\R^n} \int_D |\pi(k(\theta).(z,w))F(\xi)|^2 d\theta d\xi$$ $$= (2\pi)^{n/2} e^{(u\cdot y - v \cdot x)} \sum_{\alpha \in \N^n}
\Phi_{\alpha,\alpha}(2iy,2iv)|(f,\Phi_\alpha)|^2.$$ To see the claim, suppose we have the above formula. Then writing $$\int_{\R^n} \int_K |\pi(\sigma.(z,w))F(\xi)|^2 d\sigma d\xi$$ $$= (2\pi)^{-n}\int_{\R^n} \int_D \int_K |\pi(k(\theta)\sigma.(z,w))
F(\xi)|^2 d\sigma d\theta d\xi$$ we get $$\int_{\R^n} \int_K |\pi(\sigma.(z,w))F(\xi)|^2 d\sigma d\xi$$ $$= (2\pi)^{n/2} e^{(u'\cdot y' - v' \cdot x')} \sum_{\alpha \in \N^n}
\Phi_{\alpha,\alpha}(2iy',2iv')|(f,\Phi_\alpha)|^2$$ where $ (z',w') = \sigma.(z,w).$ Since the action of $ \sigma $ preserves the symplectic form we have $ e^{(u\cdot y - v \cdot x)} =
e^{(u'\cdot y' - v' \cdot x')}.$ Thus we are left with proving $$(2\pi)^{n/2} \int_K \Phi_{\alpha,\alpha}(\sigma.(2iy,2iv)) d\sigma =
\frac{k!(n-1)!}{(k+n-1)!}\varphi_k(2iy,2iv)$$ whenever $ |\alpha| =k.$ But this is a well known fact. A representation theoretic proof of this can be found in Ratnakumar et al \[10\].
(Another way to see this is the following. The functions $ \Phi_{\alpha,\alpha}(x,u) $ are eigenfunctions of the special Hermite operator $ L $ with eigenvalue $ (2|\alpha|+n).$ And hence the function $ \int_K \Phi_{\alpha,\alpha}(\sigma.(x,u)) d\sigma $ is a radial eigenfunction of the same operator. But any bounded radial eigenfunction with eigenvalue $(2k+n) $ is a constant multiple of $ \varphi_k(x,u).$ This proves that $$(2\pi)^{n/2} \int_K \Phi_{\alpha,\alpha}(\sigma.(x,u)) d\sigma =
\frac{k!(n-1)!}{(k+n-1)!}\varphi_k(x,u)$$ and hence they are same on $ \C^n \times \C^n $ as well.)
We now turn our attention to prove the formula for the action of $ D.$ The idea is to expand the operator valued function $ \pi(k(\theta).(z,w)) $ into a Fourier series. Defining $$\pi_m(z,w)F(\xi) = (2\pi)^{-n} \int_D \pi(k(\theta).(z,w))F(\xi)
e^{-im\cdot \theta} d\theta$$ we have the expansion $$\pi(k(\theta).(z,w))F(\xi) = \sum_{m \in \Z^n}
\pi_m(z,w)F(\xi) e^{im\cdot \theta}.$$ By the orthogonality of the Fourier series we obtain $$(2\pi)^{-n} \int_{\R^n}\int_D |\pi(k(\theta).(z,w))F(\xi)|^2 d\theta d\xi$$ $$= \sum_{m \in \Z^n} \int_{\R^n}|\pi_m(z,w)F(\xi)|^2 d\xi.$$ In calculating the $ L^2 $ norm of $ \pi_m(z,w)F $ we make use of another property of special Hermite functions, namely that $ \Phi_{\alpha,\beta}(x,u) $ is $
(\beta-\alpha)-$ homogeneous. By this we mean $$\Phi_{\alpha,\beta}(k(\theta).(x,u)) = e^{i(\beta-\alpha)\cdot \theta}
\Phi_{\alpha,\beta}(x,u).$$ A proof of this can be found in \[12\] (see Proposition 1.4.2).
Expanding $ f $ in terms of the Hermite basis we see that $$\pi_m(z,w)F = \sum_{\alpha,\beta} (f,\Phi_\alpha)
(\pi_m(z,w)\Phi_\alpha,\Phi_\beta) \Phi_\beta.$$ But $$(\pi_m(x,u)\Phi_\alpha,\Phi_\beta) = (2\pi)^{-n/2}\int_D
\Phi_{\alpha,\beta}(k(\theta).(x,u))e^{-im\cdot \theta}d\theta = 0$$ unless $ \beta = \alpha +m $ due to the homogeneity properties of the special Hermite functions. Therefore, the expansion of $ \pi_m(z,w)F $ reduces to $$\pi_m(z,w)F = (2\pi)^{n/2} \sum_{\alpha \in \N^n} (f,\Phi_\alpha)
\Phi_{\alpha,\alpha +m}(z,w) \Phi_{\alpha +m}.$$ This leads us to $$\|\pi_m(z,w)F\|_2^2 = (2\pi)^n \sum_{\alpha \in \N^n} |(f,\Phi_\alpha)|^2
|\Phi_{\alpha,\alpha +m}(z,w)|^2.$$ Thus we have proved $$(2\pi)^{-n} \int_{\R^n}\int_D |\pi(k(\theta).(z,w))F(\xi)|^2 d\theta d\xi$$ $$= (2\pi)^n \sum_{m \in \Z^n} \sum_{\alpha \in \N^n} |(f,\Phi_\alpha)|^2
|\Phi_{\alpha,\alpha +m}(z,w)|^2.$$ This proves our claim since the sum over $ m \in \Z^n $ is precisely $ (2\pi)^{-n/2} e^{(u\cdot y - v \cdot x)} \Phi_{\alpha,\alpha}(2iy,2iv)$ in view of Lemma 2.2. Hence the proof of the theorem is complete.
The above theorem has a natural converse which we state and prove now. Together they prove Theorem 1.1 stated in the introduction. In the proof of the above theorem the hypothesis on the Hermite projections of $ f $ are used twice. First we used the estimates to conclude that $ f $ has an entire extension to $ \C^n.$ Then we used them to show that the sum and the integral appearing in the above theorem are finite. In the next theorem we begin with an entire function for which the integral is finite and obtain the estimates on the projections.
Let $ F $ be an entire function on $ \R^n $ for which the integral $$\int_{\R^n} \int_K |\pi(\sigma.(z,w))F(\xi)|^2 d\sigma d\xi$$ is finite for all $ z, w \in \C^n.$ Then $ \|P_kf\|_2 \leq C_t
e^{-2k^{\frac{1}{2}}t} $ for all $ t > 0.$
[**Proof:**]{} We proceed as in the proof of the previous theorem. Since $ F $ is holomorphic $ \pi(z,w)F $ makes sense. As before, for almost every $ \sigma \in
U(n) $ we have $$\int_{\R^n} \int_D |\pi(k(\theta)\sigma.(z,w))F(\xi)|^2 d\theta d\xi
< \infty .$$ Expanding the operator $ \pi(k(\theta).(z,w)) $ into Fourier series and proceeding exactly as in the previous theorem and noting that at each stage the resulting sums are finite we get the Gutzmer’s formula, namely the integral in the theorem is equal to $$e^{(u\cdot y-v\cdot x)} \sum_{k=0}^\infty \frac{k!(n-1)!}{(k+n-1)!}
\varphi_k(2iy,2iv)\|P_kf\|_2^2$$ and hence the sum is finite. Now Perron’s formula for Laguerre functions on the negative real axis also gives lower bounds. That is to say, the Laguerre functions $ \varphi_k(2iy,2iv) $ behave like $ e^{2(k)^{\frac{1}{2}}
(|y|^2+|v|^2)^{\frac{1}{2}}}.$ In view of this we immediately get the decay estimates on the projections $ P_kf.$
**Some consequences**
=====================
In this section we deduce some interesting consequences of our Gutzmer’s formula. First we obtain the characterisation of the image of $ L^2(\R^n) $ under the Hermite semigroup mentioned in Corollary 1.2. As we have pointed out earlier the result is not new but we give a different proof.
Consider the heat kernel $ p_t(y,v) $ associated to the special Hermite operator which is explicitly given by $$p_t(y,v) = (2\pi)^{-n} (\sinh(t))^{-n}
e^{-\frac{1}{4}\coth(t)(|y|^2+|v|^2)}.$$ We now look at the integral $$\int_{\R^n} \left( \int_{\R^{2n}} |\pi(iy,iv)f(\xi)|^2 p_{2t}(2y,2v)dydv
\right) d\xi.$$ Since the function $ p_t(y,v) $ and the Lebesgue measure $ dy dv $ are both invariant under the action of the group $ K $ we can rewrite the above integral as $$\int_{\R^{2n}}\left( \int_{\R^n} \int_K |\pi(\sigma.(iy,iv))f(\xi)|^2
d\sigma d\xi \right) p_{2t}(2y,2v)dydv .$$ In view of Gutzmer’s formula the above reduces to $$\sum_{k=0}^\infty \frac{k!(n-1)!}{(k+n-1)!} \left(\int_{\R^{2n}}
\varphi_k(2iy,2iv)p_{2t}(2y,2v) dydv \right) \|P_kf\|_2^2.$$ We now make use of the fact that $$\frac{k!(n-1)!}{(k+n-1)!}\int_{\R^{2n}}
\varphi_k(2iy,2iv)p_{2t}(2y,2v) dydv = e^{2(2k+n)t}$$ which we have established in \[15\] (see Lemma 6.3).
Therefore, replacing $ f $ by $ e^{-tH}f $ we have established $$\int_{\R^{2n}} \left( \int_{\R^n} |\pi(iy,iv)e^{-tH}f(\xi)|^2 d\xi \right)
p_{2t}(2y,2v) dy dv$$ $$= \sum_{k=0}^\infty \|P_kf\|_2^2 = \int_{\R^n} |f(\xi)|^2 d\xi .$$ Writing $ F $ for $ e^{-tH}f $ a simple calculation shows that the above integral is equal to $$(2\pi \sinh(2t))^{-n} \int_{\R^{2n}} \left( \int_{\R^n} |F(\xi+iv)|^2 e^{-2y\cdot \xi} e^{-\coth(2t)(|y|^2+|v|^2)} dy \right) d\xi dv.$$ Performing the integration with respect to $ y $ we see that the above is nothing but $$\int_{\R^{2n}}|F(\xi+iv)|^2 U_t(\xi,v) d\xi dv .$$ This completes the proof of Corollary 1.2.
We remark that if we have only assumed the estimate $ \|P_kf\|_2 \leq C
e^{-2k^{\frac{1}{2}}t} $ for some $ t > 0 $ ( not for all $ t $ as in Theorem 3.1) then the proof of Theorem 3.1 shows that $ f $ can be extended as a holomorphic function to cetain tube domain $ \Omega_t = \{ z \in \C^n:
|y| < t \} $ and still we have Gutzmer’s formula as long as $ |y|^2+|v|^2 < t^2.$ We may think of Gutzmer’s formula as a characterisation of the image of $ L^2(\R^n) $ under the Hermite-Poisson semigroup $ e^{-tH^{\frac{1}{2}}}.$ Compare this with the results of Janssen and Eijndhoven \[5\] on the growth of Hermite coefficients.
Another interesting consequence of the Gutzmer’s formula is the following orthogonality relations for Hermite functions on $ \C^n.$ Polarising Gutzmer we obtain $$\int_{\R^n} \int_K \pi(\sigma.(z,w))F(\xi)
\overline{\pi(\sigma.(z,w))G(\xi)} d\sigma d\xi$$ $$= e^{(u\cdot y - v \cdot x)} \sum_{k=0}^\infty \frac{k!(n-1)!}{(k+n-1)!}
\varphi_k(2iy,2iv) (P_kf,P_kg) .$$ Specialising to Hermite functions we get the following result which, to our knowledge, seems to be new.
For any $ z,w \in \C^n $ and $ \alpha,\beta \in \N^n$ we have $$\int_{\R^n} \int_K \pi(\sigma.(z,w))\Phi_\alpha(\xi)
\overline{\pi(\sigma.(z,w))\Phi_\beta(\xi)} d\sigma d\xi$$ $$= e^{(u\cdot y - v \cdot x)}\frac{k!(n-1)!}{(k+n-1)!}\varphi_k(2iy,2iv)
\delta_{\alpha,\beta}.$$
The above shows that in the one dimensional case the Hermite functions $ h_k $ satisfy the following relations. The choice $ z = i\eta, w = 0 $ gives $$\int_{\R} \int_0^{2\pi} e^{-2\xi \eta \cos \theta}
h_k(\xi+i\eta \sin \theta)\overline{h_j(\xi+i\eta \sin \theta)} d\theta d\xi$$ $$= (2\pi) L_k^0(-2\eta^2)e^{\eta^2} \delta_{k,j}.$$ The choice $ z = \eta, w = i\eta $ leads to $$\int_{\R} \int_0^{2\pi}e^{2\xi \eta \sin \theta - \eta^2 \cos(2\theta)}
h_k(\xi+i\eta e^{-i\theta})\overline{h_j(\xi+i\eta e^{-i\theta})}
d\theta d\xi$$ $$= (2\pi) L_k^0(-2\eta^2) \delta_{k,j}.$$ Other interesting relations in higher dimensional cases can be obtained by suitable choices of $ z, w $ and also by choosing various subgroups of $ K.$
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|
**A numerical method based on the reproducing kernel Hilbert space method for the solution of fifth-order boundary-value problems**
Mustafa Inc, Ali Akgül and Mehdi Dehghan
Department of Mathematics, Science Faculty, Firat University, 23119 Elazi ğ / Türkiye
Department of Mathematics, Education Faculty, Dicle University, 21280 Diyarbakir / Türkiye
Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran
[email protected]
**Abstract:** In this paper, we present a fast and accurate numerical scheme for the solution of fifth-order boundary-value problems. We apply the reproducing kernel Hilbert space method (RKHSM) for solving this problem. The analytical results of the equations have been obtained in terms of convergent series with easily computable components. We compare our results with spline methods, decomposition method, variational iteration method, Sinc-Galerkin method and homotopy perturbation methods. The comparison of the results with exact ones is made to confirm the validity and efficiency.
**Keywords:** Reproducing kernel method, Series solutions, fifth-order boundary-value problems, Reproducing kernel space.
**1. Introduction**
In this work we consider the numerical approximation for the fifth-order boundary-value problems of the form$$y^{\left( v\right) }=f\left( x\right) y+g\left( x\right) ,\quad x\in \left[
a,b\right] , \tag{1.1}$$with boundary conditions$$y\left( a\right) =A_{0},\quad y^{\prime }\left( a\right) =A_{1},\quad
y^{\prime \prime }\left( a\right) =A_{2},\quad y\left( b\right) =B_{0},\quad
y^{\prime }\left( b\right) =B_{1},\quad \tag{1.2}$$where the functions $f\left( x\right) $ and $g\left( x\right) $ are continuous on $\left[ a,b\right] $ and $A_{0},$ $A_{1},$ $A_{2},$ $B_{0},$ $B_{1}$ are finite real constants. For more details about computational code of boundary value problems, the reader is referred to \[1-3\].
This type of boundary-value problems arise in the mathematical modelling of viscoelastic flows and other branches of mathematical, physical and engineering sciences \[4,5\] and references therein. Theorems which list the conditions for the existence and uniqueness of solutions of such problems are thoroughly discussed in a book by Agarwal \[6\]. Khan \[7\] investigated the fifth-order boundary-value problems by using finite difference methods. Wazwaz \[8\] applied Adomian decomposition method for solution of such type of boundary-value problems. The use of spline function in the context of fifth-order boundary-value problems was studied by Fyfe \[9\], who used the quintic polynomial spline functions to develop consistency relations connecting the values of solution with fifth-order derivatives at the respective nodes. Polynomial sextic spline functions were used \[10\] to develop the smooth approximations to the solution of problems (1.1) and (1.2). Caglar et al. \[11\] have used sixth-degree B-spline functions to develop first-order accurate method for the solution two-point special fifth-order boundary-value problems. Noor and Mohyud-Din \[12,13\] applied variational iteration and homotopy perturbation methods for solving the problems (1.1) and (1.2), respectively. Khan \[14\] have used the non-polynomial sextic spline functions for the solution fifth-order boundary-value problems. El-Gamel \[15\] employed the sinc-Galerkin method to solve the problems (1.1) and (1.2). Lamnii et al. \[16\] developed and analyzed two sextic spline collocation methods for the problem. Siddiqi et al. \[17,18\] used the non-polynomial sextic spline method for special fifth-order problems (1.1) and (1.2). Wang et al. \[19\] attempted to obtain upper and lower approximate solutions of such problems by applying the sixth-degree B-spline residual correction method.
In this paper, the RKHSM \[20,21\] will be used to investigate the fifth-order boundary-value problems. In recent years, a lot of attetion has been devoted to the study of RKHSM to investigate various scientific models. The RKHSM which accurately computes the series solution is of great interest to applied sciences. The method provides the solution in a rapidly convergent series with components that can be elegantly computed. The efficiency of the method was used by many authors to investigate several scientific applications. Geng and Cui \[22\] applied the RKHSM to handle the second-order boundary value problems. Yao and Cui \[23\] and Wang et al. \[24\] investigated a class of singular boundary value problems by this method and the obtained results were good. Zhou et al. \[25\] used the RKHSM effectively to solve second-order boundary value problems. In \[26\], the method was used to solve nonlinear infinite-delay-differential equations. Wang and Chao \[27\], Li and Cui \[28\], Zhou and Cui \[29\] independently employed the RKHSM to variable-coefficient partial differential equations. Geng and Cui \[30\], Du and Cui \[31\] investigated the approximate solution of the forced Duffing equation with integral boundary conditions by combining the homotopy perturbation method and the RKHSM. Lv and Cui \[32\] presented a new algorithm to solve linear fifth-order boundary value problems. In \[33,34\], authors developed a new existence proof of solutions for nonlinear boundary value problems. Cui and Du \[35\] obtained the representation of the exact solution for the nonlinear Volterra-Fredholm integral equations by using the reproducing kernel Hilbert space method. Wu and Li \[36\] applied iterative reproducing kernel method to obtain the analytical approximate solution of a nonlinear oscillator with discontinuties. Recently, the method was apllied the fractional partial differential equations and multi-point boundary value problems \[34-37\]. For more details about RKHSM and the modified forms and its effectiveness, see \[20-43\] and the references therein.
The paper is organized as follows. Section 2 is devoted to several reproducing kernel spaces and a linear operator is introduced. S[olution represantation in]{}** **$W_{2}^{6}[a,b]$ has been presented in Section 3. It provides the main results, the exact and approximate solution of $(1.1)$ and an iterative method are developed for the kind of problems in the reproducing kernel space. We have proved that the approximate solution converges to the exact solution uniformly. Some numerical experiments are illustrated in Section 4. There are some conclusions in the last section.
**2. Preliminaries**
**2.1. Reproducing Kernel Spaces**
In this section, we define some useful reproducing kernel spaces.
**Definition 2.1.** *(Reproducing kernel)*. Let $E$ be a nonempty abstract set. A function $K:E\times E\longrightarrow C$ is a reproducing kernel of the Hilbert space $H$ if and only if$$\left\{
\begin{array}{c}
\forall t\in E,\text{ }K\left( .,t\right) \in H, \\
\forall t\in E,\text{ }\forall \varphi \in H,\text{ }\left\langle \varphi
\left( .\right) ,K\left( .,t\right) \right\rangle =\varphi \left( t\right) .\end{array}\right. \tag{2.1}$$
The last condition is called “the reproducing property”: the value of the function $\varphi $ at the point $t$ is reproduced by the inner product of $\varphi $ with $K\left( .,t\right) $
**Definition 2.2.**$$W_{2}^{6}[0,1]=\left\{
\begin{array}{c}
u(x)\mid u(x),\text{ }u^{\prime }(x),\text{ }u^{\prime \prime }(x),\text{ }u^{\prime \prime \prime }(x),\text{ }u^{(4)}(x),\text{ }u^{(5)}(x)\text{\ }
\\
\text{are absolutely continuous in }[0,1], \\
u^{\left( 6\right) }(x)\in L^{2}[0,1],\text{ }x\in \lbrack 0,1],\text{ }u(0)=u(1)=u^{\prime }(0)=u^{\prime }(1)=0=u^{\prime \prime }(0)=0\end{array}\right\} ,$$The sixth derivative of $u(x)$ exists almost everywhere since $u^{(5)}(x)$ is absolutely continuous. The inner product and the norm in $W_{2}^{6}[0,1]$ are defined respectively by$$\left\langle u(x),g(x)\right\rangle
_{W_{2}^{6}}=\sum_{i=0}^{5}u^{(i)}(0)g^{(i)}(0)+\int_{0}^{1}u^{(6)}(x)g^{(6)}(x)dx,\text{ \ }u(x),g(x)\in W_{2}^{6}[0,1],$$and
$$\left\Vert u\right\Vert _{W_{2}^{6}}=\sqrt{\left\langle u,u\right\rangle
_{_{W_{2}^{6}}}},\ u\in W_{2}^{6}[0,1].$$
The space $W_{2}^{6}[0,1]$ is a reproducing kernel space, i.e., for each fixed $y\in \lbrack 0,1]$ and any $u(x)\in W_{2}^{6}[0,1],$ there exists a function $R_{y}(x)$ such that
$$u(y)=\left\langle u(x),\ R_{y}(x)\right\rangle _{W_{2}^{6}}.$$
**Definition 2.3.**
$$W_{2}^{1}[0,1]=\left\{
\begin{array}{c}
u(x)\mid u(x)\text{ is absolutely continuous in \ }[0,1]\text{ } \\
u^{\prime }(x)\in L^{2}[0,1],\text{ }x\in \lbrack 0,1],\end{array}\right\} ,$$
The inner product and the norm in $W_{2}^{1}[0,1]$ are defined respectively by
$$\left\langle u(x),g(x)\right\rangle
_{W_{2}^{1}}=u(0)g(0)+\int_{0}^{1}u^{\prime }(x)g^{\prime }(x)dx,\text{ \ }u(x),g(x)\in W_{2}^{1}[0,1],$$
and
$$\left\Vert u\right\Vert _{W_{2}^{1}}=\sqrt{\left\langle u,u\right\rangle _{_{{\LARGE W}_{2}^{1}}}},\text{ }u\in W_{2}^{1}[0,1].$$
The space $W_{2}^{1}[0,1]$ is a reproducing kernel space and its reproducing kernel function ${\normalsize T}_{x}{\normalsize (y)}$ is given by
$${\normalsize T}_{x}{\normalsize (y)}{\LARGE =}\left\{
\begin{array}{c}
1+x,\text{ \ \ }x\leq y, \\
\\
1+y,\text{ \ \ }x>y.\end{array}\right. \tag{2.2}$$
**Theorem 2.1.** The space $W_{2}^{6}[0,1]$ is a reproducing kernel Hilbert space whose reproducing kernel function is given by,$$R_{y}\left( x\right) =\left\{
\begin{array}{c}
\sum_{i=1}^{12}c_{i}\left( y\right) x^{i-1},\quad x\leq y, \\
\\
\sum_{i=1}^{12}d_{i}\left( y\right) x^{i-1},\quad x>y,\end{array}\right.$$where
$$c_{1}(y)=0,$$
$${\normalsize c}_{2}{\normalsize (y)}{\normalsize =0,}$$
$$c_{3}(y)=0,$$
$$\begin{aligned}
c_{4}(y) &=&\frac{2461}{42301989}y^{5}+\frac{2461}{253811934}y^{6}+\frac{335}{507623868}y^{7} \\
&& \\
&&-\frac{8321}{10660101228}y^{10}-\frac{7325}{1776683538}y^{8}-\frac{11725}{84603978}y^{4} \\
&& \\
&&+\frac{12221}{169207956}y^{3}+\frac{56255}{21320202456}y^{9}+\frac{2003}{21320202456}y^{11},\end{aligned}$$
$$\begin{aligned}
c_{5}(y) &=&-\frac{158419}{1353663648}y^{5}-\frac{158419}{8121981888}y^{6}+\frac{99305}{14213468304}y^{7} \\
&& \\
&&+\frac{157481}{341123239296}y^{10}-\frac{335}{16243963776}y^{8}+\frac{728021}{2707327296}y^{4} \\
&& \\
&&-\frac{11725}{84603978}y^{3}-\frac{196265}{170561619648}y^{9}-\frac{2687}{42640404912}y^{11},\end{aligned}$$
$$\begin{aligned}
c_{6}(y) &=&\frac{4056701}{67683182400}y^{5}-\frac{2011}{126905967}y^{6}+\frac{158419}{284269366080}y^{7} \\
&& \\
&&+\frac{11843}{304574320800}y^{10}+\frac{2461}{284269366080}y^{8}-\frac{158419}{1353663648}y^{4} \\
&& \\
&&+\frac{2461}{42301989}y^{3}-\frac{168263}{1705616196480}y^{9}-\frac{8999}{1705616196480}y^{11},\end{aligned}$$
$$\begin{aligned}
c_{7}(y) &=&\frac{4056701}{406099094400}y^{5}-\frac{2011}{7614358020}y^{6}+\frac{158419}{1705616196480}y^{7} \\
&& \\
&&+\frac{11843}{1827445924800}y^{10}+\frac{2461}{1705616196480}y^{8}-\frac{158419}{8121981888}y^{4} \\
&& \\
&&+\frac{2461}{253811934}y^{3}-\frac{168263}{10233697178880}y^{9}-\frac{8999}{10233697178880}y^{11},\end{aligned}$$
$$\begin{aligned}
c_{8}(y) &=&\frac{158419}{284269366080}y^{5}+\frac{158419}{1705616196480}y^{6}-\frac{19861}{596965668768}y^{7} \\
&& \\
&&-\frac{157481}{71635880252160}y^{10}+\frac{67}{682246478592}y^{8}-\frac{104003}{81219818880}y^{4} \\
&& \\
&&+\frac{335}{507623868}y^{3}+\frac{39253}{7163588025216}y^{9}+\frac{2687}{8954485031520}y^{11},\end{aligned}$$
$$\begin{aligned}
c_{9}(y) &=&\frac{2461}{284269366080}y^{5}+\frac{2461}{1705616196480}y^{6}+\frac{67}{682246478592}y^{7} \\
&& \\
&&-\frac{8321}{71635880252160}y^{10}-\frac{1465}{2387862675072}y^{8}-\frac{335}{16243963776}y^{4} \\
&& \\
&&+\frac{12221}{1137077464320}y^{3}+\frac{11251}{28654352100864}y^{9}+\frac{2003}{143271760504320}y^{11},\end{aligned}$$
$$\begin{aligned}
c_{10}(y) &=&-\frac{168263}{1705616196480}y^{5}-\frac{168263}{10233697178880}y^{6}+\frac{39253}{7163588025216}y^{7} \\
&& \\
&&+\frac{38153}{85963056302592}y^{10}+\frac{11251}{28654352100864}y^{8}-\frac{196265}{170561619648}y^{4} \\
&& \\
&&+\frac{56255}{21320202456}y^{3}-\frac{6313}{5372691018912}y^{9}-\frac{12751}{214907640756480}y^{11}-\frac{1}{725760}y^{2},\end{aligned}$$
$$\begin{aligned}
c_{11}(y) &=&\frac{11843}{304574320800}y^{5}+\frac{11843}{1827445924800}y^{6}-\frac{157481}{71635880252160}y^{7} \\
&& \\
&&-\frac{45611}{268634550945600}y^{10}-\frac{8321}{71635880252160}y^{8}+\frac{157481}{341123239296}y^{4} \\
&& \\
&&-\frac{8321}{10660101228}y^{3}+\frac{38153}{85963056302592}y^{9}+\frac{49001}{2149076407564800}y^{11}+\frac{1}{3628800}y,\end{aligned}$$
$$\begin{aligned}
c_{12}(y) &=&-\frac{8999}{1705616196480}y^{5}-\frac{8999}{10233697178880}y^{6}+\frac{2687}{8954485031520}y^{7} \\
&& \\
&&+\frac{49001}{2149076407564800}y^{10}+\frac{2003}{143271760504320}y^{8}-\frac{2687}{42640404912}y^{4} \\
&& \\
&&+\frac{2003}{21320202456}y^{3}-\frac{12751}{214907640756480}y^{9}-\frac{725}{236398404832128}y^{11}-\frac{1}{39916800},\end{aligned}$$
$$d_{1}(y)=-\frac{1}{39916800}y^{11},$$
$$d_{2}(y)=\frac{1}{3628800}y^{10},$$
$$d_{3}(y)=-\frac{1}{725760}y^{9},$$
$$\begin{aligned}
\text{\ }d_{4}(y) &=&\frac{12221}{169207956}y^{3}+\frac{12221}{1137077464320}y^{8}+\frac{2461}{42301989}y^{5} \\
&& \\
&&+\frac{2461}{253811934}y^{6}+\frac{335}{507623868}y^{7}-\frac{8321}{10660101228}y^{10} \\
&& \\
&&-\frac{11725}{84603978}y^{4}+\frac{56255}{21320202456}y^{9}+\frac{2003}{21320202456}y^{11},\end{aligned}$$
$$\begin{aligned}
d_{5}(y) &=&\frac{728021}{2707327296}y^{4}-\frac{104003}{81219818880}y^{7}-\frac{158419}{1353663648}y^{5} \\
&& \\
&&-\frac{158419}{8121981888}y^{6}+\frac{157481}{341123239296}y^{10}-\frac{335}{16243963776}y^{8} \\
&& \\
&&-\frac{11725}{84603978}y^{3}-\frac{196265}{170561619648}y^{9}-\frac{2687}{42640404912}y^{11},\end{aligned}$$
$$\begin{aligned}
d_{6}(y) &=&\frac{4056701}{67683182400}y^{5}+\frac{4056701}{406099094400}y^{6}+\frac{158419}{284269366080}y^{7} \\
&& \\
&&+\frac{11843}{304574320800}y^{10}+\frac{2461}{284269366080}y^{8}-\frac{158419}{1353663648}y^{4} \\
&& \\
&&+\frac{2461}{42301989}y^{3}-\frac{168263}{1705616196480}y^{9}-\frac{8999}{1705616196480}y^{11},\end{aligned}$$
$$\begin{aligned}
d_{7}(y) &=&-\frac{2011}{6768318240}y^{5}-\frac{2011}{7614358020}y^{6}+\frac{158419}{1705616196480}y^{7} \\
&& \\
&&+\frac{11843}{1827445924800}y^{10}+\frac{2461}{1705616196480}y^{8}-\frac{158419}{8121981888}y^{4} \\
&& \\
&&+\frac{2461}{253811934}y^{3}-\frac{168263}{10233697178880}y^{9}-\frac{8999}{10233697178880}y^{11},\end{aligned}$$
$$\begin{aligned}
d_{8}(y) &=&\frac{158419}{284269366080}y^{5}+\frac{158419}{1705616196480}y^{6}-\frac{19861}{596965668768}y^{7} \\
&& \\
&&-\frac{157481}{71635880252160}y^{10}+\frac{67}{682246478592}y^{8}+\frac{99305}{14213468304}y^{4} \\
&& \\
&&+\frac{335}{507623868}y^{3}+\frac{39253}{7163588025216}y^{9}+\frac{2687}{8954485031520}y^{11},\end{aligned}$$
$$\begin{aligned}
d_{9}(y) &=&\frac{2461}{284269366080}y^{5}+\frac{2461}{1705616196480}y^{6}+\frac{67}{682246478592}y^{7} \\
&& \\
&&-\frac{8321}{71635880252160}y^{10}-\frac{1465}{2387862675072}y^{8}-\frac{335}{16243963776}y^{4} \\
&& \\
&&-\frac{7325}{1776683538}y^{3}+\frac{11251}{28654352100864}y^{9}+\frac{2003}{143271760504320}y^{11},\end{aligned}$$
$$\begin{aligned}
d_{10}(y) &=&-\frac{168263}{1705616196480}y^{5}-\frac{168263}{10233697178880}y^{6}+\frac{39253}{7163588025216}y^{7} \\
&& \\
&&+\frac{38153}{85963056302592}y^{10}+\frac{11251}{28654352100864}y^{8}-\frac{196265}{170561619648}y^{4} \\
&& \\
&&+\frac{56255}{21320202456}y^{3}-\frac{6313}{5372691018912}y^{9}-\frac{12751}{214907640756480}y^{11},\end{aligned}$$
$$\begin{aligned}
d_{11}(y) &=&\frac{11843}{304574320800}y^{5}+\frac{11843}{1827445924800}y^{6}-\frac{157481}{71635880252160}y^{7} \\
&& \\
&&-\frac{45611}{268634550945600}y^{10}-\frac{8321}{71635880252160}y^{8}+\frac{157481}{341123239296}y^{4} \\
&& \\
&&-\frac{8321}{10660101228}y^{3}+\frac{38153}{85963056302592}y^{9}+\frac{49001}{2149076407564800}y^{11},\end{aligned}$$
$$\begin{aligned}
d_{12}(y) &=&-\frac{2687}{42640404912}y^{4}+\frac{2003}{21320202456}y^{3}-\frac{8999}{1705616196480}y^{5} \\
&& \\
&&-\frac{8999}{10233697178880}y^{6}+\frac{2687}{8954485031520}y^{7}+\frac{2003}{143271760504320}y^{8} \\
&& \\
&&-\frac{12751}{214907640756480}y^{9}-\frac{725}{236398404832128}y^{11}+\frac{49001}{2149076407564800}y^{10},\end{aligned}$$
**Proof:**$$\begin{aligned}
\left\langle u(x),{\normalsize R}_{y}{\normalsize (x)}\right\rangle
_{_{W_{2}^{6}}} &=&\sum_{i=0}^{5}u^{(i)}(0){\normalsize R}_{y}^{(i)}{\normalsize (0)}+\int_{0}^{1}u^{(6)}(x){\normalsize R}_{y}^{(6)}{\normalsize (x)}dx,\text{ } \TCItag{2.3} \\
&&\left( u(x),\text{ }{\normalsize R}_{y}{\normalsize (x)}\in
W_{2}^{6}[0,1]\right) , \notag\end{aligned}$$Through several integrations by parts for (2.3) we have
$$\begin{aligned}
\left\langle u(x),{\normalsize R}_{y}{\normalsize (x)}\right\rangle
_{_{W_{2}^{6}}} &=&\sum_{i=0}^{5}{\normalsize u}^{(i)}{\normalsize (0)}\left[
{\normalsize R}_{y}^{(i)}{\normalsize (0)-(-1)}^{(5-i)}{\normalsize R}_{y}^{(11-i)}{\normalsize (0)}\right] \TCItag{2.4} \\
&&{\normalsize +}\sum_{i=0}^{5}{\normalsize (-1)}^{(5-i)}{\normalsize u}^{(i)}{\normalsize (1)R}_{y}^{(11-i)}{\normalsize (1)+}\int_{0}^{1}u{\normalsize (x)R}_{y}^{(12)}{\normalsize (x)dx.} \notag\end{aligned}$$
Note that property of the reproducing kernel
$$\left\langle u(x),\ R_{y}(x)\right\rangle _{W_{2}^{6}}=u(y),$$
If $$\left\{
\begin{array}{c}
R_{y}^{(5)}(0)-R_{y}^{(6)}(0)=0, \\
R_{y}^{(4)}(0)+R_{y}^{(7)}(0)=0, \\
R_{y}^{\prime \prime \prime }(0)-R_{y}^{(8)}(0)=0, \\
R_{y}^{(6)}(1)=0, \\
R_{y}^{(7)}(1)=0, \\
R_{y}^{(8)}(1)=0, \\
R_{y}^{(9)}(1)=0,\end{array}\right. \tag{2.5}$$then (2.4) implies that,
$${\normalsize R}_{y}^{(12)}{\normalsize (x)=\delta (x-y),}$$
When $x\neq y,$
$${\normalsize R}_{y}^{(12)}{\normalsize (x)=0,}$$
therefore
$${\normalsize R}_{y}{\normalsize (x)=}\left\{
\begin{array}{c}
\sum_{i=1}^{12}\ c_{i}(y)x^{i-1},\text{ \ }x\leq y, \\
\\
\ \sum_{i=1}^{12}d_{i}(y)x^{i-1},\text{ \ }x>y.\end{array}\right.$$
Since
$${\normalsize R}_{y}^{(12)}{\normalsize (x)=\delta (x-y),}$$
we have
$${\normalsize \partial }^{k}{\normalsize R}_{y^{+}}{\normalsize (y)=\partial }^{k}{\normalsize R}_{y^{-}}{\normalsize (y),}\text{ \ }k=0,1,2,3,4,5,6,7,8,9,10, \tag{2.6}$$
and
$${\normalsize \partial }^{11}{\normalsize R}_{y^{{\LARGE +}}}{\normalsize (y)-\partial }^{11}{\normalsize R}_{y^{-}}{\normalsize (y)=1.} \tag{2.7}$$
Since ${\normalsize R}_{y}{\normalsize (x)\in }W_{2}^{6}[0,1]$, it follows that
$$R_{y}(0)=0,\text{ \ }R_{y}(1)=0,\text{ }R_{y}^{\prime }(0)=0,\text{ \ }R_{y}^{\prime }(1)=0\text{, }R_{y}^{\prime \prime }(0)=0. \tag{2.8}$$
From (2.5)-(2.8), the unknown coefficients $c_{i}(y)$ ve $d_{i}(y)$ $(i=1,2,...,12)$ can be obtained. Thus for $x\leq y,$ ${\normalsize R}_{y}{\normalsize (x)}$ is given by,
$$\begin{aligned}
R_{y}(x) &=&\frac{2461}{42301989}x^{3}y^{5}+\frac{2461}{253811934}x^{3}y^{6}+\frac{335}{507623868}x^{3}y^{7}-\frac{8321}{10660101228}x^{3}y^{10} \\
&& \\
&&-\frac{7325}{1776683538}x^{3}y^{8}-\frac{11725}{84603978}x^{3}y^{4}+\frac{12221}{169207956}x^{3}y^{3}+\frac{56255}{21320202456}x^{3}y^{9} \\
&& \\
&&+\frac{2003}{21320202456}x^{3}y^{11}-\frac{158419}{1353663648}x^{4}y^{5}-\frac{158419}{8121981888}x^{4}y^{6}+\frac{99305}{14213468304}x^{4}y^{7} \\
&& \\
&&+\frac{157481}{341123239296}x^{4}y^{10}-\frac{335}{16243963776}x^{4}y^{8}+\frac{728021}{2707327296}x^{4}y^{4}-\frac{11725}{84603978}x^{4}y^{3} \\
&& \\
&&-\frac{196265}{170561619648}x^{4}y^{9}-\frac{2687}{42640404912}x^{4}y^{11}+\frac{4056701}{67683182400}x^{5}y^{5}-\frac{2011}{126905967}x^{5}y^{6} \\
&& \\
&&+\frac{158419}{284269366080}x^{5}y^{7}+\frac{11843}{304574320800}x^{5}y^{10}+\frac{2461}{284269366080}x^{5}y^{8}-\frac{158419}{135366364}x^{5}y^{4} \\
&& \\
&&+\frac{2461}{42301989}x^{5}y^{3}-\frac{168263}{1705616196480}x^{5}y^{9}-\frac{8999}{1705616196480}x^{5}y^{11}+\frac{4056701}{40609909440}x^{6}y^{5}
\\
&& \\
&&-\frac{2011}{7614358020}x^{6}y^{6}+\frac{158419}{1705616196480}x^{6}y^{7}+\frac{11843}{1827445924800}x^{6}y^{10}+\frac{2461}{170561619}x^{6}y^{8} \\
&& \\
&&-\frac{158419}{8121981888}x^{6}y^{4}+\frac{2461}{253811934}x^{6}y^{3}-\frac{168263}{10233697178880}x^{6}y^{9}-\frac{8999}{1023369717888}x^{6}y^{11}
\\
&& \\
&&+\frac{158419}{284269366080}x^{7}y^{5}+\frac{158419}{1705616196480}x^{7}y^{6}-\frac{19861}{596965668768}x^{7}y^{7}-\frac{157481}{716358802}x^{7}y^{10} \\
&& \\
&&+\frac{67}{682246478592}x^{7}y^{8}-\frac{104003}{81219818880}x^{7}y^{4}+\frac{335}{507623868}x^{7}y^{3}+\frac{39253}{7163588025216}x^{7}y^{9} \\
&& \\
&&+\frac{2687}{8954485031520}x^{7}y^{11}+\frac{2461}{284269366080}x^{8}y^{5}+\frac{2461}{1705616196480}x^{8}y^{6}+\frac{67}{682246478}x^{8}y^{7}\end{aligned}$$
$$\begin{aligned}
&&-\frac{8321}{71635880252160}x^{8}y^{10}-\frac{1465}{2387862675072}x^{8}y^{8}-\frac{335}{16243963776}x^{8}y^{4}+\frac{12221}{1137077464320}x^{8}y^{3} \\
&& \\
&&+\frac{11251}{28654352100864}x^{8}y^{9}+\frac{2003}{143271760504320}x^{8}y^{11}-\frac{168263}{1705616196480}x^{9}y^{5}-\frac{168263}{1023369717}x^{9}y^{6} \\
&& \\
&&+\frac{39253}{7163588025216}x^{9}y^{7}+\frac{38153}{85963056302592}x^{9}y^{10}+\frac{11251}{28654352100864}x^{9}y^{8}-\frac{196265}{17056161964}x^{9}y^{4} \\
&& \\
&&+\frac{56255}{21320202456}x^{9}y^{3}-\frac{6313}{5372691018912}x^{9}y^{9}-\frac{12751}{214907640756480}x^{9}y^{11}-\frac{1}{725760}x^{9}y^{2} \\
&& \\
&&+\frac{11843}{304574320800}x^{10}y^{5}+\frac{11843}{1827445924800}x^{10}y^{6}-\frac{157481}{71635880252160}x^{10}y^{7}-\frac{45611}{2686345509}x^{10}y^{10} \\
&& \\
&&-\frac{8321}{71635880252160}x^{10}y^{8}+\frac{157481}{341123239296}x^{10}y^{4}-\frac{8321}{10660101228}x^{10}y^{3}+\frac{38153}{859630563025}x^{10}y^{9} \\
&& \\
&&+\frac{49001}{2149076407564800}y^{11}+\frac{1}{3628800}y-\frac{8999}{1705616196480}x^{11}y^{5}-\frac{8999}{10233697178880}x^{11}y^{6} \\
&& \\
&&+\frac{2687}{8954485031520}x^{11}y^{7}+\frac{49001}{2149076407564800}x^{11}y^{10}+\frac{2003}{143271760504320}x^{11}y^{8} \\
&& \\
&&-\frac{2687}{42640404912}x^{11}y^{4}+\frac{2003}{21320202456}x^{11}y^{3}-\frac{12751}{214907640756480}x^{11}y^{9}-\frac{725}{236398404832128}.\end{aligned}$$
**3. Solution represantation in** $W_{2}^{6}[0,1]$
In this section, the solution of equation (1.1) is given in the reproducing kernel space $W_{2}^{6}[0,1].$
On defining the linear operator $L:W_{2}^{6}[0,1]\rightarrow W_{2}^{1}[0,1]$ as
$$Lu=u^{(5)}(x)-f(x)u(x).$$
Model problem (1.1) changes the following problem:
$$\left\{
\begin{array}{c}
Lu=K(x),\text{ }x\in \lbrack 0,1] \\
u(a)=0,\text{ \ }u^{\prime }(a)=0,\text{ \ }u^{\prime \prime }(a)=0,\text{ \
}u(b)=0,\text{ \ }u^{\prime }(b)=0.\text{\ }\end{array}\right. \text{\ } \tag{3.1}$$
**3.1. The Linear boundedness of operator** $L.$
**Lemma 3.1.** If $u(x)\in W_{2}^{6}[a,b],$ then $\left\Vert
u^{(k)}(x)\right\Vert _{L^{\infty }}\leq M_{k}\left\Vert u(x)\right\Vert
_{W_{2}^{6}},$ where $M_{k}$ $(k=0,1,...,5)~$are positive constants.
**Proof:** For any $x\in \lbrack a,b]$ it holds that
$$\left\Vert R_{x}(y)\right\Vert _{W_{2}^{6}}=\sqrt{\left\langle
R_{x}(y),R_{x}(y)\right\rangle _{W_{2}^{6}}}=\sqrt{R_{x}(x),}$$
from the continuity of $R_{x}(x),$ there exists a constant $M_{0}>0,$ such that $\left\Vert R_{x}(y)\right\Vert _{W_{2}^{6}}\leq M_{0}.$ By (2.1) one gets
$$\left\vert u(x)\right\vert =\left\vert \left\langle
u(y),R_{x}(y)\right\rangle _{_{W_{2}^{6}}}\right\vert \leq \left\Vert
R_{x}(y)\right\Vert _{W_{2}^{6}}\left\Vert u(y)\right\Vert _{W_{2}^{6}}\leq
M_{0}\left\Vert u(y)\right\Vert _{W_{2}^{6}}. \tag{3.2}$$
For any $x,$ $y\in \lbrack a,b]$, there exists $M_{k}$ $(k=1,2,...,5),$ such that
$$\left\Vert R_{x}^{(k)}(y)\right\Vert _{W_{2}^{6}}\leq M_{k}\text{ \ }(k=1,2,...,5),$$
we have$$\begin{aligned}
\left\vert u^{(k)}(x)\right\vert &=&\left\vert \left\langle
u(y),R_{x}^{(k)}(y)\right\rangle _{_{W_{2}^{6}}}\right\vert \leq \left\Vert
R_{x}^{(k)}(y)\right\Vert _{W_{2}^{6}}\left\Vert u(y)\right\Vert
_{W_{2}^{6}}\leq M_{k}\left\Vert u(y)\right\Vert _{W_{2}^{6}}\text{ }
\TCItag{3.3} \\
(k &=&1,2,...,5). \notag\end{aligned}$$Combining (3.2) and (3.3), it follows that
$$\left\Vert u^{(k)}(x)\right\Vert _{L^{\infty }}\leq M_{k}\left\Vert
u(y)\right\Vert _{W_{2}^{6}}\text{ }(k=1,2,...,5).$$
**Theorem 3.1.** Suppose $f_{i}^{\prime }\in L^{2}[a,b]$ $(i=0,1,2,3,4), $ Then $L:W_{2}^{6}[a,b]\rightarrow W_{2}^{1}[a,b]$ is a bounded linear operator.
**Proof:** (i)** **By the definition of the operator it is clear that $L$ is a linear operator.
\(ii) Due to the definiton of $W_{2}^{1}[a,b]$ , we have$$\begin{aligned}
\left\Vert (Lu)(x)\right\Vert _{W_{2}^{1}}^{2} &=&\left\langle
(Lu)(x),(Lu)(x)\right\rangle _{W_{2}^{1}} \\
&=&[(Lu)(a)]^{2}+\int_{a}^{b}[(Lu)^{\prime }(x)]^{2}dx \\
&=&\left[ \sum_{i=0}^{5}f_{i}(a)u^{(i)}(a)\right] ^{2}+\int_{a}^{b}\left[
\left( \sum_{i=0}^{5}f_{i}(x)u^{(i)}(x)\right) ^{\prime }\right] ^{2}dx.\end{aligned}$$
$$\begin{aligned}
\int_{a}^{b}[(Lu)^{\prime }(x)]^{2}dx &=&\int_{a}^{b}\left[
u^{(6)}(x)+\sum_{i=0}^{4}(f_{i}^{\prime }(x)u^{(i)}(x)+f_{i}(x)u^{(i+1)}(x))\right] ^{2}dx \\
&=&\int_{a}^{b}\left[ u^{(6)}(x)\right] ^{2}dx+2\int_{a}^{b}\left[
u^{(6)}(x)\sum_{i=0}^{4}(f_{i}^{\prime }(x)u^{(i)}(x)+f_{i}(x)u^{(i+1)}(x))\right] dx \\
&&+\int_{a}^{b}\left[ \sum_{i=0}^{4}(f_{i}^{\prime
}(x)u^{(i)}(x)+f_{i}(x)u^{(i+1)}(x))\right] ^{2}dx,\end{aligned}$$
where
$$\int_{a}^{b}\left[ u^{(6)}(x)\right] ^{2}dx\leq \left\Vert u(x)\right\Vert
_{W_{2}^{6}}^{2},$$
and
$$\begin{aligned}
&&\int_{a}^{b}\left[ u^{(6)}(x)\sum_{i=0}^{4}(f_{i}^{\prime
}(x)u^{(i)}(x)+f_{i}(x)u^{(i+1)}(x))\right] dx \\
&\leq &\left\{ \int_{a}^{b}\left[ u^{(6)}(x)\right] ^{2}dx\right\}
^{2}\left\{ \int_{a}^{b}\left[ \sum_{i=0}^{4}(f_{i}^{\prime
}(x)u^{(i)}(x)+f_{i}(x)u^{(i+1)}(x))\right] ^{2}dx\right\} ^{2}.\end{aligned}$$
By lemma $\left( 3.1\right) $ and $f_{i}^{\prime }(x)$ $\in L^{2}[a,b],$ we can obtain a constant $N>0,$ satisfying
$$\int_{a}^{b}\left[ \sum_{i=0}^{4}(f_{i}^{\prime
}(x)u^{(i)}(x)+f_{i}(x)u^{(i+1)}(x))\right] ^{2}dx\leq N(b-a)\left\Vert
u(x)\right\Vert _{W_{2}^{6}}^{2}.$$
Furthermore one gets
$$\int_{a}^{b}[(Lu)^{\prime }(x)]^{2}dx\leq \left\Vert u(x)\right\Vert
_{W_{2}^{6}}^{2}+2\sqrt{N(b-a)}\left\Vert u(x)\right\Vert
_{W_{2}^{6}}^{2}+N(b-a)\left\Vert u(x)\right\Vert _{W_{2}^{6}}^{2},$$
let $G=\left( 1+2\sqrt{N(b-a)}+N(b-a)\right) >0,$ then
$$\int_{a}^{b}[(Lu)^{\prime }(x)]^{2}dx\leq G\left\Vert u(x)\right\Vert
_{W_{2}^{6}}^{2}.$$
Therefore $L$ is a boundend operator. So we obtain the result as required. $\square $
**3.2. The normal orthogonal function system of** $W_{2}^{6}[a,b]$
We choose $\left\{ x_{i}\right\} _{i=1}^{\infty }$ as any dense set in $[a,b] $ and let $\Psi _{x}(y)=L^{\ast }T_{x}(y),$ where $L^{\ast \text{ }}$ is conjugate operator of $L$ and $T_{x}(y)$ is given by (2.2). Furthermore, for simplicity let $\Psi _{i}(x)=\Psi _{x_{i}}(x),$ namely,
$$\Psi _{i}(x)\overset{def}{=}\Psi _{x_{i}}(x)=\text{ }L^{\ast \text{ }}T_{x_{i}}(x).$$
Now several lemmas are given.
**Lemma 3.2.** $\left\{ \Psi _{i}(x)\right\} _{i=1}^{\infty }$ is complete system of $W_{2}^{6}[a,b].$
**Proof:** For $u(x)\in W_{2}^{6}[a,b]$, let $\left\langle u(x),\Psi
_{i}(x)\right\rangle =0$ $(i=1,2,...),$ that is
$$\left\langle u(x),\text{ }L^{\ast \text{ }}T_{x_{i}}(x)\right\rangle
=(Lu)(x_{i})=0.$$
Note that $\left\{ x_{i}\right\} _{i=1}^{\infty }$ is the dense set in $[a,b],$ therefore $(Lu)(x)=0.$ It follows that $u(x)=0$ from the existence of $L^{-1}.$ $\square $
**Lemma 3.3.** The following formula holds
$$\Psi _{i}(x)=\left( L\eta R_{x}(\eta )\right) \left( x_{i}\right) ,$$
where the subscript $\eta $ of operator $L\eta $ indicates that the operator $L$ applies to function of $\eta .$
**Proof:**$$\begin{aligned}
\Psi _{i}(x) &=&\left\langle \Psi _{i}(\xi ),R_{x}(\xi )\right\rangle
_{W_{2}^{6}[a,b]} \\
&=&\left\langle L^{\ast \text{ }}T_{x_{i}}\left( \xi \right) ,R_{x}(\xi
)\right\rangle _{W_{2}^{6}[a,b]} \\
&=&\left\langle \left( T_{x_{i}}\right) \left( \xi \right) ,\left( L_{\eta
}R_{x}(\eta )\right) \left( \xi \right) \right\rangle _{W_{2}^{1}[a,b]} \\
&=&\left( L_{\eta }R_{x}(\eta )\right) \left( x_{i}\right) .\end{aligned}$$This completes the proof. $\square $
**Remark 3.1.** The orthonormal system $\left\{ \overline{\Psi }_{i}(x)\right\} _{i=1}^{\infty }$ of $W_{2}^{6}[a,b]$ can be derived from Gram-Schmidt orthogonaliztion process of $\left\{ \Psi _{i}(x)\right\}
_{i=1}^{\infty },$
$$\overline{\Psi }_{i}(x)=\sum_{k=1}^{i}\beta _{ik}\Psi _{k}(x),\text{ \ }(\beta _{ii}>0,\text{ \ }i=1,2,...) \tag{3.7}$$
where $\beta _{ik}$ are orthogonal cofficients.
In the following, we will give the represantation of the exact solution of Eq.(1.1) in the reproducing kernel space $W_{2}^{6}[a,b].$
**3.3. The structure of the solution and the main results**
**Theorem 3.2.** If $u(x)$ is the exact solution of Eq.(1.1), then
$$u(x)=\sum_{i=1}^{\infty }\sum_{k=1}^{i}\beta _{ik}g(x_{k})\overline{\Psi }_{i}(x),$$
where $\left\{ x_{i}\right\} _{i=1}^{\infty }$ is a dense set in $[a,b].$
**Proof:** From the (3.7) and uniqeness of solution of (1.1) (see \[32\]), we have
$$\begin{aligned}
u(x) &=&\sum_{i=1}^{\infty }\left\langle u(x),\overline{\Psi }_{i}(x)\right\rangle _{W_{2}^{6}}\overline{\Psi }_{i}(x) \\
&=&\sum_{i=1}^{\infty }\sum_{k=1}^{i}\beta _{ik}\left\langle u(x),L^{\ast
}T_{x_{k}}(x)\right\rangle _{W_{2}^{6}}\overline{\Psi }_{i}(x) \\
&=&\sum_{i=1}^{\infty }\sum_{k=1}^{i}\beta _{ik}\left\langle
Lu(x),T_{x_{k}}(x)\right\rangle _{W_{2}^{1}}\overline{\Psi }_{i}(x) \\
&=&\sum_{i=1}^{\infty }\sum_{k=1}^{i}\beta _{ik}\left\langle
g(x),T_{x_{k}}(x)\right\rangle _{W_{2}^{1}}\overline{\Psi }_{i}(x) \\
&=&\sum_{i=1}^{\infty }\sum_{k=1}^{i}\beta _{ik}g(x_{k})\overline{\Psi }_{i}(x).\text{ \ \ \ \ \ \ }\square\end{aligned}$$
Now the approximate solution $u_{n}(x)$ can be obtained by truncating the $n- $ term of the exact solution $u(x),$
$$u_{n}(x)=\sum_{i=1}^{n}\sum_{k=1}^{i}\beta _{ik}g(x_{k})\overline{\Psi }_{i}(x).$$
**Theorem 3.3.** Assume $u(x)$ is the solution of Eq.(1.1) and $r_{n}(x)
$ is the error between the approximate solution $u_{n}(x)$ and the exact solution $u(x).$ Then the error sequence $r_{n}(x)$ is monotone decreasing in the sense of $\left\Vert .\right\Vert _{W_{2}^{6}}$ and $\left\Vert
r_{n}(x)\right\Vert _{W_{2}^{6}}\rightarrow 0$ \[23\].
**4. Numerical Results**
In this section, four numerical examples are provided to show the accuracy of the present method. All computations are performed by Maple 13. The RKHSM does not require discretization of the variables, i.e., time and space, it is not effected by computation round off errors and one is not faced with necessity of large computer memory and time. The accuracy of the RKHSM for the fifth-order boundary value problems is controllable and absolute errors are small with present choice of $x$ (see Table 1-4). The numerical results we obtained justify the advantage of this methodology.
**Example 4.1.** (\[8,11,12\]). We first consider the linear boundary value problem$$\left\{
\begin{array}{c}
y^{(5)}(x)=y-15e^{x}-10xe^{x},\text{ }0<x<1 \\
y(0)=0,\text{ }y^{\prime }(0)=1,\text{ }y^{\prime \prime }(0)=0,\text{ }y(1)=0,\text{ }y^{\prime }(1)=-e\end{array}\right. \tag{4.1}$$The exact solution of (4.1) is
$$y\left( x\right) =x(1-x)e^{x}.$$
If we homogenize the boundary conditions of (4.1), then the following (4.2) is obtained$$\left\{
\begin{array}{c}
u^{(5)}(x)-u(x)=1-5e^{x}\left[ 2-3x+3x^{2}(2-\dfrac{5}{e})+4x^{3}(\dfrac{-3}{2}+\dfrac{4}{e})\right] \\
\\
-10e^{x}\left[ -3+6x(2-\dfrac{5}{e})+12x^{2}(\dfrac{-3}{2}+\dfrac{4}{e})\right] \\
\\
-10e^{x}\left[ 6(2-\dfrac{5}{e})+24x(\dfrac{-3}{2}+\dfrac{4}{e})\right]
-5e^{x}\left[ 24(\dfrac{-3}{2}+\dfrac{4}{e})\right] \\
\\
-15e^{x}-10xe^{x},\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }0<x<1 \\
\\
u(0)=0,\text{ }u^{\prime }(0)=0,\text{ }u^{\prime \prime }(0)=0,\text{ }u(1)=0,\text{ }u^{\prime }(1)=0\end{array}\right. \tag{4.2}$$
**Example 4.2.** (\[8,12\]). We now consider the nonlinear BVP$$\left\{
\begin{array}{c}
y^{(5)}(x)=e^{-x}y^{2}(x),\text{ }0<x<1 \\
y(0)=1,\text{ }y^{\prime }(0)=1,\text{ }y^{\prime \prime }(0)=1,\text{ }y(1)=e,\text{ }y^{\prime }(1)=e\end{array}\right. \tag{4.3}$$The exact solution of (4.3) is
$$y(x)=e^{x}$$
If we homogenize the boundary conditions of (4.3), then the following (4.4) is obtained$$\left\{
\begin{array}{c}
u^{(5)}(x)-2e^{-x}\left[ 1+e^{x}(x-\dfrac{x^{2}}{2}+x^{3}(2-\dfrac{5}{e})+x^{4}(\dfrac{-3}{2}+\dfrac{4}{e}))\right] u(x) \\
\\
=e^{-x}u^{2}(x)+e^{-x}\left[ 1+e^{x}(x-\dfrac{x^{2}}{2}+x^{3}(2-\dfrac{5}{e})+x^{4}(\dfrac{-3}{2}+\dfrac{4}{e}))\right] ^{2} \\
\\
-e^{x}\left[ x-\dfrac{x^{2}}{2}+x^{3}(2-\dfrac{5}{e})+x^{4}(\dfrac{-3}{2}+\dfrac{4}{e})\right] \\
\\
-5e^{x}\left[ 1-x+3x^{2}(2-\dfrac{5}{e})+4x^{3}(\dfrac{-3}{2}+\dfrac{4}{e})\right] \\
\\
-10e^{x}\left[ -1+6x(2-\dfrac{5}{e})+12x^{2}(\dfrac{-3}{2}+\dfrac{4}{e})\right] \\
\\
-10e^{x}\left[ 6(2-\dfrac{5}{e})+24x(\dfrac{-3}{2}+\dfrac{4}{e})\right] \\
\\
-5e^{x}\left[ 24(\dfrac{-3}{2}+\dfrac{4}{e})\right] ,\text{ \ \ \ \ \ \ }0<x<1\text{\ \ \ \ \ \ \ \ } \\
\\
u(0)=0,\text{ }u^{\prime }(0)=0,\text{ }u^{\prime \prime }(0)=0,\text{ }u(1)=0,\text{ }u^{\prime }(1)=0\end{array}\right. \tag{4.4}$$
**Example 4.3.(\[15\]).** Consider the nonlinear BVP$$\left\{
\begin{array}{c}
y^{(5)}(x)=-24e^{-y(x)}+\frac{{\LARGE 48}}{{\LARGE (1+x)}^{5}},\text{ \ \ \ }0<x<1 \\
y(0)=0,\text{ }y^{\prime }(0)=1,\text{ }y^{\prime \prime }(0)=-1,\text{ }y(1)=\ln 2,\text{ }y^{\prime }(1)=0.5\end{array}\right. . \tag{4.5}$$The exact solution of (4.5) is
$$y\left( x\right) =\ln (x+1).$$
If we homogenize the boundary conditions of (4.5), then the following (4.6) is obtained
$$\left\{
\begin{array}{c}
u^{(5)}(x)=-24e^{-\left( {\LARGE u(x)}+{\LARGE x}-\dfrac{x^{2}}{2}+{\LARGE x}^{3}({\LARGE 4\ln 2-}\frac{{\LARGE 5}}{{\LARGE 2}}{\LARGE )}+{\LARGE x}^{4}{\LARGE (2-3}\ln {\LARGE 2)}\right) }+\frac{{\LARGE 48}}{{\LARGE (1+x)}^{5}}
\\
u(0)=0,\text{ }u^{\prime }(0)=0,\text{ }u^{\prime \prime }(0)=0,\text{ }u(1)=0,\text{ }u^{\prime }(1)=0\end{array}\right. \tag{4.6}$$
**Example 4.4. (\[15\]).** This is the nonlinear BVP$$\left\{
\begin{array}{c}
y^{(5)}(x)+y^{(4)}(x)+e^{-2x}y^{2}(x)=2e^{x}+1\text{ \ \ \ }0<x<1 \\
y(0)=0,\text{ }y^{\prime }(0)=1,\text{ }y^{\prime \prime }(0)=1,\text{ }y(1)=e,\text{ }y^{\prime }(1)=e\end{array}\right. . \tag{4.7}$$The exact solution of (4.7) is
$$y\left( x\right) =e^{x}.$$
If we homogenize the boundary conditions of (4.7), then the following (4.8) is obtained
$$\left\{
\begin{array}{c}
u^{(5)}(x)+u^{(4)}(x)=-e^{-2x}(u(x)+1+x+\frac{x^{2}}{2}+x^{3}(3e-8)+x^{4}(\frac{11}{2}-2e))^{2}+2e^{x}+48e-131 \\
u(0)=0,\text{ }u^{\prime }(0)=0,\text{ }u^{\prime \prime }(0)=0,\text{ }u(1)=0,\text{ }u^{\prime }(1)=0\end{array}\right. \tag{4.8}$$
**Remark 4.1.** Lamnii et al. \[16\] solved the problem (5.1) by using sextic spline collocation method. He obtained the accurate approximate solutions of this problem for the small $h$ values. Zhang \[44\] investigated approximate solution of the problem (5.1) by using variational iteration method. In addition, the same problem is solved by Noor and Mohyud-Din \[12\] previously and they got better results by using the variational iteration method.
Lv and Cui \[32\] studied only the linear fifth-order two-point boundary value problems by using reproducing kernel Hilbert space method. We use the RKHSM for the same linear problem with different boundary conditions and also use different reproducing kernel function for computations. We also use the RKHSM for the nonlinear problems.
Using our method we chose $36$ points on $[0,1].$ In Tables 1-4, we computed the absolute errors $\left\vert u\left( x,t\right) -u_{n}\left( x,t\right)
\right\vert $ at the points $\left\{ \left( x_{i}\right) :\text{ }x_{i}=i,\text{ \ }i=0.0,0.1,...,1.0\right\} .$
$$\begin{tabular}{|l|l|l|l|l|}
\hline
$x$ & Exact Sol. & RKHSM & HPM [13] & B-Spline [11] \\ \hline
$0.0$ & $0.0000$ & $0.000$ & $0.0000$ & $0.0000$ \\ \hline
$0.1$ & $0.099465382$ & $5.89\times 10^{-7}$ & $3\times 10^{-11}$ & $8.0\times 10^{-3}$ \\ \hline
$0.2$ & $0.195424441$ & $1.73\times 10^{-8}$ & $2\times 10^{-10}$ & $1.2\times 10^{-3}$ \\ \hline
$0.3$ & $0.283470349$ & $6.02\times 10^{-7}$ & $4\times 10^{-10}$ & $5.0\times 10^{-3}$ \\ \hline
$0.4$ & $0.358037927$ & $7.42\times 10^{-7}$ & $8\times 10^{-10}$ & $3.0\times 10^{-3}$ \\ \hline
$0.5$ & $0.412180317$ & $3.32\times 10^{-7}$ & $1.2\times 10^{-9}$ & $8.0\times 10^{-3}$ \\ \hline
$0.6$ & $0.437308512$ & $3.10\times 10^{-7}$ & $2\times 10^{-9}$ & $6.0\times 10^{-3}$ \\ \hline
$0.7$ & $0.422888068$ & $3.08\times 10^{-7}$ & $2.2\times 10^{-9}$ & $0.000$
\\ \hline
$0.8$ & $0.356086548$ & $4.58\times 10^{-7}$ & $1.9\times 10^{-9}$ & $9.0\times 10^{-3}$ \\ \hline
$0.9$ & $0.221364280$ & $4.30\times 10^{-7}$ & $1.4\times 10^{-9}$ & $9.0\times 10^{-3}$ \\ \hline
$1.0$ & $0.0000$ & $2.36\times 10^{-13}$ & $0.0000$ & $0.0000$ \\ \hline
\end{tabular}\begin{tabular}{|l|l|}
\hline
ADM [8] & Sinc [15] \\ \hline
$0.0000$ & $0.0000$ \\ \hline
$3\times 10^{-11}$ & $0.0000$ \\ \hline
$2\times 10^{-10}$ & $0.1\times 10^{-5}$ \\ \hline
$4\times 10^{-10}$ & $0.3\times 10^{-5}$ \\ \hline
$8\times 10^{-10}$ & $0.3\times 10^{-5}$ \\ \hline
$1.2\times 10^{-9}$ & $0.0000$ \\ \hline
$2\times 10^{-9}$ & $0.5\times 10^{-5}$ \\ \hline
$2.2\times 10^{-9}$ & $0.9\times 10^{-5}$ \\ \hline
$1.9\times 10^{-9}$ & $0.2\times 10^{-5}$ \\ \hline
$1.4\times 10^{-9}$ & $0.1\times 10^{-5}$ \\ \hline
$0.0000$ & $\ 0.0000$ \\ \hline
\end{tabular}$$
\
**Table 1.** The absolute error of Example 5.1 for boundary conditions at $0.0\leq x\leq 1.0.$
$$\begin{tabular}{|l|l|l|l|l|}
\hline
$x$ & Exact Solution & RKHSM & HPM [13] & B-Spline[11] \\ \hline
$0.0$ & $0.$ & $0.0000$ & $0.0000$ & $0.0000$ \\ \hline
$0.1$ & $1.105170918$ & $5.19\times 10^{-7}$ & $1\times 10^{-9}$ & $7.0$ $\times 10^{-4}$ \\ \hline
$0.2$ & $1.221402758$ & $0.60\times 10^{-7}$ & $2\times 10^{-9}$ & $7.2$ $\times 10^{-4}$ \\ \hline
$0.3$ & $1.349858808$ & $3.19\times 10^{-7}$ & $1\times 10^{-9}$ & $4.1$ $\times 10^{-4}$ \\ \hline
$0.4$ & $1.491824698$ & $2.50\times 10^{-7}$ & $2\times 10^{-8}$ & $4.6$ $\times 10^{-4}$ \\ \hline
$0.5$ & $1.648721271$ & $3.03\times 10^{-7}$ & $3.1\times 10^{-8}$ & $4.7$ $\times 10^{-4}$ \\ \hline
$0.6$ & $1.822118800$ & $9.60\times 10^{-7}$ & $3.7\times 10^{-8}$ & $4.8$ $\times 10^{-4}$ \\ \hline
$0.7$ & $2.013752707$ & $4.20\times 10^{-7}$ & $4.1\times 10^{-8}$ & $3.9$ $\times 10^{-4}$ \\ \hline
$0.8$ & $2.225540928$ & $4.09\times 10^{-7}$ & $3.1\times 10^{-8}$ & $3.1$ $\times 10^{-4}$ \\ \hline
$0.9$ & $2.459603111$ & $5.46\times 10^{-7}$ & $1.4\times 10^{-8}$ & $1.6$ $\times 10^{-4}$ \\ \hline
$1.0$ & $2.718281828$ & $5.34\times 10^{-7}$ & $0.0000$ & $0.0000$ \\ \hline
\end{tabular}\begin{tabular}{|l|l|}
\hline
ADM [8] & VIM [12] \\ \hline
\multicolumn{1}{|l|}{$0.0000$} & $0.0000$ \\ \hline
\multicolumn{1}{|l|}{$1\times 10^{-9}$} & $1\times 10^{-9}$ \\ \hline
\multicolumn{1}{|l|}{$2\times 10^{-9}$} & $2\times 10^{-9}$ \\ \hline
\multicolumn{1}{|l|}{$1\times 10^{-9}$} & $1\times 10^{-9}$ \\ \hline
\multicolumn{1}{|l|}{$2\times 10^{-8}$} & $2\times 10^{-8}$ \\ \hline
\multicolumn{1}{|l|}{$3.1\times 10^{-8}$} & $3.1\times 10^{-8}$ \\ \hline
\multicolumn{1}{|l|}{$3.7\times 10^{-8}$} & $3.7\times 10^{-8}$ \\ \hline
\multicolumn{1}{|l|}{$4.1\times 10^{-8}$} & $4.1\times 10^{-8}$ \\ \hline
\multicolumn{1}{|l|}{$3.1\times 10^{-8}$} & $3.1\times 10^{-8}$ \\ \hline
\multicolumn{1}{|l|}{$1.4\times 10^{-8}$} & $1.4\times 10^{-8}$ \\ \hline
\multicolumn{1}{|l|}{$0.0000$} & $0.0000$ \\ \hline
\end{tabular}$$
**Table 2.** The absolute error of Example 5.2 for boundary conditions at $0.0\leq x\leq 1.0.$
$$\begin{tabular}{|l|l|l|l|}
\hline
x & Exact Solution & RKHSM &
\begin{tabular}{l}
AE, $1.0E-8$ \\
RKHSM\end{tabular}
\\ \hline
$0.0$ & $0.0$ & $0.0$ & $0.0$ \\ \hline
$0.0806$ & $0.07751644243$ & $0.07751634304$ & $0.003$ \\ \hline
$0.1648$ & $0.1525493985$ & $0.1525493860$ & $1.25$ \\ \hline
$0.2285$ & $0.2057939130$ & $0.2057939053$ & $0.77$ \\ \hline
$0.3999$ & $0.3364008055$ & $0.3364007134$ & $9.21$ \\ \hline
$0.5$ & $0.4054651081$ & $0.4054650667$ & $4.14$ \\ \hline
$0.6923$ & $0.5260885504$ & $0.5260885142$ & $3.62$ \\ \hline
$0.7714$ & $0.5717701944$ & $0.5717701752$ & $1.92$ \\ \hline
$0.8836$ & $0.6331848394$ & $0.6331848843$ & $4.49$ \\ \hline
$0.9447$ & $0.6651077235$ & $0.6651077287$ & $0.52$ \\ \hline
$1.0$ & $0.6931471806$ & $0.6931471783$ & $0.23$ \\ \hline
\end{tabular}\begin{tabular}{|l|}
\hline
\begin{tabular}{l}
AE,$1.0E-4$ [15] \\
Sinc-Galerkin\end{tabular}
\\ \hline
$0.0$ \\ \hline
$0.0$ \\ \hline
$0.2$ \\ \hline
$0.2$ \\ \hline
$0.4$ \\ \hline
$0.1$ \\ \hline
$0.2$ \\ \hline
$0.3$ \\ \hline
$0.2$ \\ \hline
$0.5$ \\ \hline
$0.0$ \\ \hline
\end{tabular}$$
**Table 3.** The absolute error (AE) of Example 5.3 for boundary conditions at $0.0\leq x\leq 1.0.$
$$\begin{tabular}{|l|l|l|l|}
\hline
x & Exact Solution & RKHSM &
\begin{tabular}{l}
AE \\
RKHSM\end{tabular}
\\ \hline
$0.0$ & $1.0$ & $1.0$ & $0.0$ \\ \hline
$0.0100$ & $1.105170918$ & $1.105170918$ & $0.0$ \\ \hline
$0.1184$ & $1.125694299$ & $1.125694299$ & $0.0$ \\ \hline
$0.1517$ & $1.163811041$ & $1.163811041$ & $0.0$ \\ \hline
$0.2410$ & $1.272521035$ & $1.272521035$ & $0.0$ \\ \hline
$0.3604$ & $1.433902861$ & $1.433902861$ & $0.0$ \\ \hline
$0.4287$ & $1.535260387$ & $1.535260387$ & $0.0$ \\ \hline
$0.5000$ & $1.648721271$ & $1.648721271$ & $0.0$ \\ \hline
$0.6395$ & $1.895532876$ & $1.895532876$ & $0.0$ \\ \hline
$0.8482$ & $2.335439276$ & $2.335439276$ & $0.0$ \\ \hline
$0.9996$ & $2.717194733$ & $2.717194734$ & $1\times 10^{-9}$ \\ \hline
$1.0$ & $2.718281828$ & $2.718281828$ & $0.0$ \\ \hline
\end{tabular}\begin{tabular}{|l|}
\hline
\begin{tabular}{l}
AE,$1.0E-3$ [15] \\
Sinc-Galerkin\end{tabular}
\\ \hline
$0.0$ \\ \hline
$0.0$ \\ \hline
$0.0$ \\ \hline
$0.1$ \\ \hline
$0.0$ \\ \hline
$0.1$ \\ \hline
$0.0$ \\ \hline
$0.2$ \\ \hline
$0.1$ \\ \hline
$0.2$ \\ \hline
$0.2$ \\ \hline
$0.0$ \\ \hline
\end{tabular}$$
**Table 4.** The absolute error of Example 4 for boundary conditions at $0.0\leq x\leq 1.0.$
** Remark 4.2.** The RKHSM tested on four problems, one linear and three nonlinears. A comparison with decomposition method by Wazwaz \[8\], sixth B-spline method by Caglar et al. \[11\], variational iteration and homotopy perturbation methods by Noor and Mohyid-Din \[12,13\] and sinc-galerkin method by Gamel \[15\] are made and it was seen that the present method yields good results (see Tables 1-4).
**6. Conclusion**
In this paper, we introduce an algorithm for solving the fifth-order problem with boundary conditions. For illustration purposes, chose four examples which were selected to show the computational accuracy. It may be concluded that, the RKHSM is very powerful and efficient in finding exact solution for a wide class of boundary value problems. The method gives more realistic series solutions that converge very rapidly in physical problems. The approximate solution obtained by the present method is uniformly convergent.
Clearly, the series solution methodology can be applied to much more complicated nonlinear differential equations and boundary value problems \[20-43\] as well. However, if the problem becomes nonlinear, then the RKHSM does not require discretization or perturbation and it does not make closure approximation. Results of numerical examples show that the present method is an accurate and reliable analytical method for the fifth order problem with boundary conditions.
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---
abstract: 'An enhancement of localized nonlinear modes in coupled systems gives rise to a novel type of escape process. We study a spatially one dimensional set-up consisting of a linearly coupled oscillator chain of $N$ mass-points situated in a metastable nonlinear potential. The Hamilton-dynamics exhibits breather solutions as a result of modulational instability of the phonon states. These breathers localize energy by freezing other parts of the chain. Eventually this localised part of the chain grows in amplitude until it overcomes the critical elongation characterized by the transition state. Doing so, the breathers ignite an escape by pulling the remaining chain over the barrier. Even if the formation of singular breathers is insufficient for an escape, coalescence of moving breathers can result in the required concentration of energy. Compared to a chain system with linear damping and thermal fluctuations the breathers help the chain to overcome the barriers faster in the case of low damping. With larger damping, the decreasing life time of the breathers effectively inhibits the escape process.'
author:
- 'Torsten Gross[^1]'
- Dirk Hennig
- 'Lutz Schimansky-Geier'
date: 'May 10, 2013'
title: 'Self-organized escape processes of linear chains in nonlinear potentials'
---
\[ra\_ch1\]
Introduction {#intro}
============
A chain of binary interacting units is a simple model for discussing the emergence of collective phenomena. Despite its simplicity, such setup appears frequently in various physical contexts such as for the description of mechanical and electrical systems, polymers, networks of superconducting elements, chemical reactions in connected discrete boxes, to name but a few [@Sato06_Rev; @Flach98_PR; @Dauxois92_PD; @Reineker89_PLA; @Doi; @Nicolis; @MakNek97].
In this study we use the linear chain and its cooperative dynamical phenomena as a paradigm of a multidimensional dynamical system. We aim to investigate escape processes of the chain out of a metastable state[@Langer; @Hanggi90] also known as the nucleation of a kink-antikink pair [@Sodano; @HanMarRis; @Christen95; @Christen_EPL; @Christen_PRE; @Pankratov; @Gulevich] in biased sinusoidal potentials. To this end we place a chain with linear springs being responsible for the interaction between the units in a nonlinear potential modelled by polynomial of 3rd degree. As will be seen, energy along the chain will become inhomogeneously distributed and parts of the chain with large elongations will collect energy from their neighbouring regions. Such localized modes of energy are know as breather-solutions and have been studied intensively in the past in various contexts including micro-mechanical cantilever arrays [@Sato06_Rev; @cantilever1; @cantilever2], arrays of coupled Josephson junctions [@josephson1; @josephson2], coupled optical wave guides [@optical1; @optical2], Bose-Einstein condensates in optical lattices [@BEC], in coupled torsion pendula [@Jesus], electrical transmission lines [@electrical1; @electrical2], and granular crystals [@crystals].
We concentrate here on the escape process and elaborate how the localized breathers modify this process [@HeFug07; @HeLsg07]. For this purpose we consider first the pure deterministic set-up and study the properties of breathers arising on the chain whose units evolve in the nonlinear potential. In the second set-up we investigate thermally activated escape dynamics. The chain will be exposed to a thermal bath with temperature $T$. Consequently damping and noise is added to the deterministic dynamics accounting for coupled Langevin equations. Our main new findings concern the study of how a change of the friction coefficient modifies the escape process. While for stronger damping breather solutions do not play a significant role, in case of weak damping the escape times become even shorter compared to the deterministic case. Notably, the establishment of breathers along the chain helps the emergence of critical elongations from thermal fluctuations.
This work is structured as follows: In Sec. \[1d:1d-chain\_model\] we introduce the model of a linearly coupled chain situated in an external nonlinear potential describing a metastable situation. We study the critical transition state, i.e. the bottleneck configuration which the chain has to cross in order that a transition over the potential barrier takes place. In Sec. \[sec2\] we derive conditions for the modulational instability which determine the time scale for the growth of the breathers. We find two generic scenarios which govern the transition. With larger energy singular breathers achieve large elongations and can surpass the transition states alone. Differently, if the elongation of the breathers are too small, they undergo an erratic motion. On collision, breathers tend to merge. Thereby they cumulatively localize energy which can eventually cause the barrier crossing. In Sec. \[sec3\] we study the thermally activated escape of the interacting chain in the metastable potential landscape. Finally, we summarize our findings.
The one-dimensional chain model {#1d:1d-chain_model}
===============================
We study an one-dimensional chain of $N$ linearly coupled oscillators of mass $m$ with elongations $q_n(t), n=1,\ldots,N$. The chain is positioned in a cubic external potential. Every mass point experiences a nonlinear force caused by the potential $$V(q_n)=\frac{m\,\omega_0^2}{2}q^2_n-\frac{a}{3}q^3_n$$ and spring forces created from the neighbours with spring constant $\kappa$. Periodic boundary conditions are applied. Positions of particles perpendicular to the potential variation are kept constant[@HeLsg07], (for an alternative case see [@FugHe07; @martens08]). First, we assume that there is no noise and no damping, hence yielding a canonic situation with a Hamiltonian dynamics and corresponding momenta $p_n(t), n=1,\ldots,N$ canonically conjugate to the positions $q_n(t)$. Consequently the total energy of the chain is conserved.
In order to obtain dimensionless quantities we rescale units and parameters, $\widetilde{q_n}=a/(m\,\omega_0^2)\,q_n$, ${\widetilde{p}_n}\,^2=a^2/(m^4\,\omega_0^6)p_n$ and ${\widetilde{t}}\,^2=\omega_0^2\,t^2$. As a result, we remain with dimensionless Hamiltonian with one remaining parameter only, the effective coupling strength $\widetilde{\kappa}=\kappa/(m\,\omega_0^2)$. In what follows we omit the tildes.
The Hamiltonian of the considered chain reads: $$\begin{aligned}
\mathcal{H}=\sum_{n=0}^{N-1}\left[\frac{p_n^2}{2}+\frac{\kappa}{2}\left(q_n-q_{n+1}\right)^2+V\left(q_n\right)
\right], \qquad V(q_n)=\frac{q_n^2}{2}-\frac{q_n^3}{3}.\end{aligned}$$ The resulting equations of motion become $$\begin{aligned}
\label{equ_motion_1d}
\ddot{q_n}+q_n-{q_n}^2-\kappa \left(q_{n+1}+q_{n-1}-2\,q_n\right)=0,
\qquad q_{N+1}=q_1.\end{aligned}$$ In this paper we will consider for our numerical simulations chains comprising $N=100$ units. A study of the dependence of the escape process on the number of oscillators can be found in [@HeFug07].
For the study of an escape, we initially place the units of the chain close to the bottom of the external potential, that is nearby $q_{min}=0$ and provide them with energy $E$. As will be seen, the chain eventually generates critical elongations surpassing the potentials local maximum. This initiates a transition of the chain into the unbounded regime $q_n > q_{\mathrm{max}}=1,
n=1,\ldots,N$, which we refer to as an escape. For a single particle to overcome the potential barrier it needs to be supplied with an energy $\Delta E=V(1)-V(0)=1/6$.
In the following we want to illustrate that the generation of these critical states is efficient even in cases where the chain energy $E$ is small compared to $E \ll N\cdot \Delta E$. This low-energy setting is obtained through the following initial preparation of the system $$\begin{aligned}
q^{\mathrm{IC}}_i=\Delta q_i+\Delta \qquad
p^{\mathrm{IC}}_i=\Delta p_i,\end{aligned}$$ where $\Delta q_i$ and $\Delta p_i$ are small random perturbations taken from a uniform distribution within the intervals $$\begin{aligned}
\Delta q_i\in \left[-\Delta q^{\mbox{IC}},\Delta q^{\mbox{IC}}
\right] \text{ and } \Delta p_i\in \left[-\Delta
p^{\mbox{IC}},\Delta p^{\mbox{IC}} \right],\end{aligned}$$ and the coordinate shift, $0< \Delta <1$, is chosen to increase the system energy to a desired value. This procedure results in a variety of perturbed flat initial states comprising a specific energy which we view as a statistical ensemble.
Transition states
-----------------
For Hamiltonian systems the local minima of the energy surface (in phase space) are Lyapunov stable. That is, orbits in the vicinity of a local minimum never leave it as their associated energy is conserved. Therefore, orbits with an energy exceeding the energy associated with a neighbouring saddle point of first order are no longer bound to the basin. Thus the saddle point is referred to as transition state, as it separates bounded from unbounded orbits. Concerning our problem, the system’s energy has to exceed the transition state energy to make escape events possible. To determine this transition states, we have to solve $\nabla U(q_1,q_2,\ldots)=0$, where $U$ denotes the potential energy (thus the transition state is a fixed-point) and the solution must render all eigenvalues of the Hessian matrix of $U$ positive, except for a single negative one. In general, this is a non-trivial task requiring sophisticated numerical methods. Here, we used the dimer method. It is a minimum-mode following method that solely makes use of gradients of the potential surface. It was first introduced in [@TS:henkelman] and its computational effort scales favorable with the system size.
In the one-dimensional chain model the transition state configurations solve the stationary equation[^2] $$\begin{aligned}
\label{ts1d:equ_sad_point_1d}
q_n-{q_n}^2-\kappa \left(q_{n+1}+q_{n-1}-2\,q_n\right)=0\end{aligned}$$ and fulfill the condition on the eigenvalues, $\lambda^H$, of the Hessian matrix $H$ $$H_{i,j}=\delta_{i,j}(2\,\kappa+1-2\,q_i)-\kappa\,\left(\delta_{i,j+1}+\delta_{i,j-1}\right).$$
![Transition state chain configurations for different values of $\kappa$. $N=100$[]{data-label="fig:ts:configs"}](./images/1d_saddlepoint1 "fig:"){width="0.3\linewidth"}![Transition state chain configurations for different values of $\kappa$. $N=100$[]{data-label="fig:ts:configs"}](./images/1d_saddlepoint2 "fig:"){width="0.3\linewidth"}![Transition state chain configurations for different values of $\kappa$. $N=100$[]{data-label="fig:ts:configs"}](./images/1d_saddlepoint3 "fig:"){width="0.3\linewidth"}
![Activation energy (energy of transition state configurations), $N=100$[]{data-label="fig:ts:E_act"}](./images/E_act){width="0.5\linewidth"}
In the case of a vanishing coupling strength the oscillators and thus the equations of motion (\[equ\_motion\_1d\]) are no longer coupled. Consequently, the fixed points of the system consist of all the configurations where each oscillators is placed either on the maximum of the potential barrier or the potential valley, $q_n^*=\left\lbrace
0,1 \right\rbrace$. In this case also the Hessian matrix, $H$, becomes diagonal and we can directly read off its eigenvalues, $\lambda^H_n=1-2\,q_n$. Demanding all eigenvalues to be negative except for one positive the transition states are found to be all the configurations where all oscillator are positioned in the potential valley except for one that is placed on the potential barrier. The according energy reads $E_{\mathrm{act}}(\kappa=0)=\Delta E=1/6$.
In contrast, a very large coupling strength corresponds to a situation where the chain effectively becomes a single oscillator so that the transition state refers to a chain configuration where all oscillators are placed on the maximum of the potential. This can be shown by taking the limit $\kappa \rightarrow \infty $ in Eq. (\[ts1d:equ\_sad\_point\_1d\]). If we want $q_n^*$ to take on values within a bounded regime we must have $q_{n+1}^*+q_{n-1}^*-2\,q_n^*=0$ in order to satisfy Eq. (\[ts1d:equ\_sad\_point\_1d\]) in this limit. In the case of periodic boundary conditions this becomes equivalent to $q_n^*=q^*$ so that Eq. (\[ts1d:equ\_sad\_point\_1d\]) becomes $q^*(1-q^*)=0$. Which of its two roots corresponds to the transition state becomes clear from the linear stability analysis of Eq. (\[equ\_motion\_1d\]) for this effective one oscillator problem $$\ddot{q}=-q+{q}^2\approx
-q^*+{q^*}^2+\left(-1+2\,q^*\right)q=\left(-1+2\,q^*\right)q.$$ Only the case $q_n=q^*=1$ is associated with the inherent instability of a transition state. Accordingly, the transition state energy is found to be $E_{\mathrm{act}}(\kappa\rightarrow\infty)=N\,\Delta E=N/6$.
The intermediate parameter regime has been evaluated using the dimer method and the results are represented in Fig. \[fig:ts:configs\] and Fig. \[ts1d:equ\_sad\_point\_1d\]. The maximal amplitude of the hair pin-like transition state configuration grows with increasing $\kappa$ until it reaches a critical elongation from which on it decreases until the entire chain approaches the maximum of the potential barrier as described above.
The formation of breathers {#sec2}
==========================
Modulational instability of a chain in a nonlinear potential
------------------------------------------------------------
The energy that is initially homogeneously distributed along the entire chain quickly concentrates into local excitations of single oscillators. This process is governed by the formation of regularly shaped wave patterns, so-called breathers which are spatially localized and time-periodically varying solutions. Their emergence is due to a modulation instability the mechanism of which applied to our situation is described later on. We follow [@MI_1] and [@MI_2] in this paragraph.
As an approximation for small oscillation amplitudes we can neglect the nonlinear term in Eq. \[equ\_motion\_1d\]. The resulting equation in linear approximation exhibits phonon solutions with frequency $\omega$ and wave number $k=2\pi\,k_0/N$ (with $k_0\in\mathbb{Z}$ and $-N/2\leq k_o
\leq N/2$) related by the dispersion relation $$\begin{aligned}
%\label{MI1d1:disperse_linear}
\omega ^2=1+4\,\kappa\,\sin^2\left(\frac{k}{2}\right)\end{aligned}$$ We make an Ansatz that only takes into account the first harmonics (rotating wave approximation) $$\begin{aligned}
%\label{MI1d:Ansatz}
q_i=F_{1,i}(t)\,e^{-i t}+ F_{0,i}(t)+F_{2,i}(t)\,e^{-2i t}+c.c.\end{aligned}$$ The amplitudes of the harmonics are expected to be of a lower order of magnitude ($| F_{0,i}|\ll| F_{1,i}|$, $| F_{2,i}|\ll| F_{1,i}|$). Furthermore, we assume our envelope functions to vary slowly ($|\dot{F}_{m,i}|\ll |{F}_{m,i}|$) as well as the phonon band to be small ($1> 4\kappa$). Within the limits of these assumptions we obtain a discrete nonlinear Schrödinger equation (DNLS) for the amplitudes of the first harmonic. $$\begin{aligned}
\label{MI1d:DNLS}
2i\dot{F}_{1,i}=\kappa
\left(\left(F_{1,i-1}+F_{1,i+1}\right)+2F_{1,i}\right)-\frac{10}{3}\,\left|F_{1,i}\right|^2F_{1,i}\end{aligned}$$ We want to study the stability of this equation’s plane wave solutions in the presence of small perturbations $\left|\delta B_i(t)\right|\ll
1$ and $\left|\delta \Psi_i(t)\right|\ll 1$, leading to a new Ansatz for the envelope function $$\begin{aligned}
\label{MI1d:envelope_Ansatz}
{{F}_{1,n}}^{pert.}=\left(A+\delta B_n(t)\right)
e^{i\left((k\,n-\Delta\omega\,t)+\delta\Psi_n(t)\right)}.\end{aligned}$$ The perturbations are sufficiently small so that we can expand the envelope function up to the first order in $\delta$ and neglect all terms of higher order. Using the Ansatz (\[MI1d:envelope\_Ansatz\]) in Eq. (\[MI1d:DNLS\]) leads to a complex differential equation for the perturbation functions $B(t)$ and $\Psi(t)$. The real and imaginary part of this equation are independent. Hence, collecting all terms of first order in $\delta$ results in two linear relations. $$\begin{aligned}
-A\,\dot{\Psi}_i&=-\frac{\kappa}{2} \left\lbrace A\,\sin k \left(\Psi_{i-1}-\Psi_{i+1}\right)+\cos k \left(B_{i+1}+B_{i-1}\right)\right\rbrace -\frac{10}{3} \,A^2\, B_i\\
2 \dot{B}_i&=-\kappa\left\lbrace A\,\cos k \left(\Psi_{i+1}+
\Psi_{i-1}-2\Psi_i \right)+\sin k
\left(B_{i+1}-B_{i-1}\right)\right\rbrace\end{aligned}$$ Again, the solution to those coupled equations are plane waves $$\begin{aligned}
\Psi_n=\Psi ^0 e^{i(Q\,n-\Omega\,t)} \qquad
B_n=B^0e^{i(Q\,n-\Omega\,t)}\end{aligned}$$ with the dispersion relation $$\begin{aligned}
\label{MI1d:pert.-disp_rel}
\left(\Omega-\kappa\,\sin k\,\sin Q\right)^2=
\kappa\,\cos k\, \sin ^2
\left(\frac{Q}{2}\right)\left(4\,\kappa\,\cos k \, \sin ^ 2
\left(\frac{Q}{2}\right)-\frac{20}{3} A^2 \right)
%\\
% \left(\Omega-\kappa\,\sin k\,\sin Q\right)^2= \kappa\,\cos k\,
% \sin ^2 \left(\frac{Q}{2}\right)\left(4\,\kappa\,\cos k \, \sin ^
% 2 \left(\frac{Q}{2}\right)-\frac{20}{3} A^2 \right)\end{aligned}$$ which describes the stability of the $Q$-mode perturbation on the $k$-mode carrier wave. $Q$ and $k$ have a $2\,\pi$ periodicity and can therefore be chosen to be in the first Brillouin zone. Furthermore, we can restrict the range of $k$ and $Q$: $k,Q\in\lbrace 0 ,\pi \rbrace$, because negative values correspond to waves with the opposite direction of propagation.
![Fastest growing modes (solid line) - Eq. (\[MI1d:Q\_max\]) - and their growth rates (dashed line), k=0.[]{data-label="fig:MI:MI_multi"}](./images/MI_tau){width="\linewidth"}
![Fastest growing modes (solid line) - Eq. (\[MI1d:Q\_max\]) - and their growth rates (dashed line), k=0.[]{data-label="fig:MI:MI_multi"}](./images/MI_multi){width="\linewidth"}
The perturbations are stable for $\Omega\in\mathbb{R}$ which is the case when the right hand side of Eq. (\[MI1d:pert.-disp\_rel\]) is positive. Therefore, all carrier waves with $k\in\lbrace\pi/2, \pi
\rbrace$ are stable with respect to all perturbation modes. For $k\in\lbrace 0 , \pi/2\rbrace$ perturbations will grow, provided that $$\begin{aligned}
\label{MI1d:unstable_Q}
\cos k \, \sin ^2\left(\frac{Q}{2}\right)&\leq
\frac{5\,A^2}{3\,\kappa}.\end{aligned}$$ We can then find an according growth rate $$\label{MI1d:growth_rate}
\Gamma (Q)=\left |\operatorname{Im} (\Omega) \right |=\sin\left(\frac{Q}{2}\right)\sqrt{\frac{20}{3}\kappa\,\cos k\,\left({A}^2-\frac{3}{5}\kappa\sin^2 \left(\frac{Q}{2}\right)\cos k\right)}$$ which, for the case that $ A^2\leq \frac{6}{5}\kappa \cos k, $ has its maximum at $$\label{MI1d:Q_max}
Q_\mathrm{max}=2\,\arcsin\sqrt{\frac{5\,A^2}{6\,\kappa\, \cos k}}.$$ Otherwise the maximum growth rate is found at $Q=\pi$. The corresponding growth rates become $$\begin{aligned}
\label{MI1d:max_growth_rate}
\Gamma_\mathrm{max}=
\begin{cases}\Gamma(Q_\mathrm{max})=\frac{5}{3}A^2&\text{if }A^2\leq \frac{6}{5}\kappa\cos k\\
\Gamma(\pi)=\sqrt{\frac{20}{3}\kappa\cos k
(A^2-\frac{3}{5}\kappa\cos k)}<\Gamma(Q_\mathrm{max})&\text{if
}A^2> \frac{6}{5}\kappa\cos k\\
\end{cases}\end{aligned}$$
We recall from Sect. \[1d:1d-chain\_model\] that our system is initially prepared in a slightly perturbed $k=0$ mode. This is thus the only possible carrier wave mode as the amplitudes, $A$, of all other modes (which scale with the amplitude of the perturbation) are likely to be too small to generate growing modes – see inequality (\[MI1d:unstable\_Q\]) – or the arising maximal growth rates are suppressed. Evaluating the growth rate of instabilities on the $k=0$ carrier mode for different values of $\kappa$ (Fig. \[fig:MI:MI\_tau\]), we find that the modulational instability becomes more mode selective with increasing $\kappa$. Hence, for large values of $\kappa$ the only relevant unstable modes are near the fastest growing mode depicted in Fig. \[fig:MI:MI\_multi\]. In such a situation we expect the emergence of a regular wave pattern (an array of breathers) that efficiently localizes energy and thereby enhances the escape of the chain.
Optimal coupling {#1d:optimal_coupling}
----------------
According to our findings in the previous sectionthe appearing breather array becomes more regular with increasing $\kappa$ . In particular its prominent mode number, and therefore the number of breathers, gets smaller, all of which results in an efficient energy localization. However, an increase in the coupling strength comes along with an increase in the activation energy – see Fig. \[fig:ts:E\_act\] – which hinders a swift escape for a given system energy. Therefore, we can expect to find an intermediate $\kappa$ that optimizes the escape rate.
We can analytically approximate the optimal $\kappa$ by assuming that the entire system energy is evenly distributed among $N_B$ non-interacting oscillators, where $N_B$ is the number of breathers related to the prominent wave length, $Q_{\mathrm{max}}$, of the modulational instability. The ratio of such an oscillator’s energy, $E_B$, to the activation energy can be regarded as a measure of escape efficiency. It is dependent on $Q_{\mathrm{max}}$ which in turn depends on the $k=0$ phonon amplitude $A$ which we relate to the system energy via $E(A)=N\,V(A)$. Thus we can write $$\begin{aligned}
\frac{E_B(\kappa)}{E_{\mathrm{act}}}\propto \frac{1}{Q_{\mathrm{max}}(E,\kappa)\,E_{\mathrm{act}}(\kappa)}.\end{aligned}$$ This escape efficiency takes on its maximum for the optimal coupling strength, $\kappa
^*$, formally $$\begin{aligned}
\label{eq:kappa_reso_analyt}
\kappa ^*(E) = \underset{ \kappa \in \mathbb{R^+}}{\operatorname{arg\,max}} \, \frac{1}{Q_{\mathrm{max}}(E,\kappa)\,E_{\mathrm{\mathrm{act}}}(\kappa)}.\end{aligned}$$
![Average escape times (marker symbols) for 500 realizations with randomized initial conditions as described in Sec. \[1d:1d-chain\_model\]. Parameters: $\Delta q^\mathrm{IC} =0.05,
\Delta p^\mathrm{IC}=0.05,N=100$. The solid grey line represents the analytical approximation for the optimal coupling strength – Eq. (\[eq:kappa\_reso\_analyt\]) – for energy values given by the right-hand axis.[]{data-label="fig:ts:T_esc_optimal_kappa"}](./images/T_esc_optimal_kappa){width="0.65\linewidth"}
This approximation can now be compared to the numerical evaluation of average escape times in dependence of $\kappa$, see Fig. \[fig:ts:T\_esc\_optimal\_kappa\]. Equation (\[equ\_motion\_1d\]) has been integrated using a fourth order Runge-Kutta scheme. Numerical accuracy was obtained by ensuring the energy deviation to remain smaller than the order of $10^{-12}$. The average escape times were determined from $500$ realization of randomized initial conditions at a given energy according to Sec. \[1d:1d-chain\_model\] for each marker symbol. The escape time measures the time it takes from the initialization to the moment when all oscillators have surpassed the potential barrier. For all depicted values of $\kappa$ at least $95$ % of the chain realizations escaped in the maximal integration time of $5\cdot 10^5$ time units.
[0.48]{} ![Temporal evolution of the energy distribution $E_n(t)$. The localization of energy from an initially homogeneous state causes in both cases an escape at the end of the depicted time frame. While the system energy in Fig. \[fig:ts:energy\_evolution\_a\] is sufficiently high to let an individual breather of the early regular breather array surpass the potential barrier, the lower energy in Fig. \[fig:ts:energy\_evolution\_b\] necessitates a merging of breathers to cause the critical chain elongation. Parameters: $\kappa=0.15, \Delta q^\mathrm{IC} =0.05, \Delta
p^\mathrm{IC} =0.05,N=100$[]{data-label="fig:ts:energy_evolution"}](./images/1d_energy_evolution1 "fig:"){width="\linewidth"}
[0.48]{} ![Temporal evolution of the energy distribution $E_n(t)$. The localization of energy from an initially homogeneous state causes in both cases an escape at the end of the depicted time frame. While the system energy in Fig. \[fig:ts:energy\_evolution\_a\] is sufficiently high to let an individual breather of the early regular breather array surpass the potential barrier, the lower energy in Fig. \[fig:ts:energy\_evolution\_b\] necessitates a merging of breathers to cause the critical chain elongation. Parameters: $\kappa=0.15, \Delta q^\mathrm{IC} =0.05, \Delta
p^\mathrm{IC} =0.05,N=100$[]{data-label="fig:ts:energy_evolution"}](./images/1d_energy_evolution2 "fig:"){width="\linewidth"}
Figure \[fig:ts:T\_esc\_optimal\_kappa\] clearly shows the predicted resonance behaviour and also reveals a fairly good accordance of the analytical approximation of the optimal $\kappa$ with the simulation results. This seems to verify our initial assumption of a regular breather array that fully concentrates the energy into single oscillators. But this reasoning fails to explain the pronounced variation of the average escape times (ranging over several orders of magnitude) for different energies. Additionally, the equipartition of a low system energy will not allocate single breathers with an energy sufficient to trigger an escape event. E.g. for $\kappa=0.15$ and $E/(N\,\Delta E)=0.1$ we expect an array with ten or more breathers so that each one could only hold an energy $E/N_B<\Delta E$. Nevertheless, an escape takes place, eventually. This implies a further concentration of energy beyond the initial creation of the breather array. In order to study this process, we look at the snapshots of the energy distribution $$\begin{aligned}
E_n=\frac{p_n^2}{2}+V(q_n)+\frac{\kappa}{4}\left\lbrace
(q_n-q_{n+1})^2+ (q_{n-1}-q_{n})^2\right\rbrace,\end{aligned}$$ In Fig. \[fig:ts:energy\_evolution\] $E_n$ has been tracked in time (upwards) for two exemplary cases. Energy is localized in both cases starting from an initially homogeneous state. In Fig. \[fig:ts:energy\_evolution\_a\] we see the appearance of a regular breather pattern. Every breather concentrates enough energy to certain oscillators in order to trigger an escape. In contrast, the lower system energy in Fig. \[fig:ts:energy\_evolution\_b\] does not allow for a direct escape of the initial breathers. Instead, breathers start an erratic movement. After an inelastic interaction they merge and can thereby eventually result in a configuration exceeding the critical chain elongation, see also [@Peyrard1998_localization]. However, this secondary process is slow compared to the (direct) breather formation which explains the different orders of magnitude of the escape times scale in Fig. \[fig:ts:T\_esc\_optimal\_kappa\].
Thermally activated escape supported by breathers {#sec3}
=================================================
In the previous sections we have been concerned with the deterministic chain dynamics leading to an escape event. In this section we study how a thermal bath with temperature $T$ will modify the transition over the barrier. For this purpose we consider the associated Langevin equation, $$\begin{aligned}
\label{1d_Langevin_eq}
\ddot{q_n}+q_n-{q_n}^2-\kappa
\left(q_{n+1}+q_{n-1}-2\,q_n\right)+\gamma \dot{q_n}+\xi_n(t)=0,\end{aligned}$$ with the friction parameter $\gamma$ and a Gaussian white noise term $\xi_n(t)$. In order to be able to compare the deterministic situation to the thermally activated setting, the associated conserved energy $E$ in the Hamiltonian case and the average energy $\overline{E}$ transferred from the bath need to be equal. The latter is governed by the correlation function of the noisy force $\xi(t)$. Its permanent variation yields source of energy for the chain which is balanced by the dissipative friction forces.
The transferred energy is defined if the autocorrelations functions of the noise sources scale as $$\begin{aligned}
\label{fdt}
\left\langle \xi_n(t) \xi_{n^\prime}(t^\prime) \right\rangle=2\,\gamma
\,\overline{E}/N \,\delta_{n,n^\prime}\,\delta(t-t^\prime)\,.\end{aligned}$$ This relation, known as fluctuation dissipation theorem, implies that the mean energy of all particles is given by $\overline{E}$. If expressed by the bath temperature, every particle gets in averrage $k_B\,T$, [*i.e.*]{} $\overline{E} = N\, k_B\, T$ with $k_B$ being the Boltzmann constant.
In numerical simulations of the Langevin equation with fulfilled relation (\[fdt\]), we have assured that the full average energy converges to $\overline{E}$ also for the transient state of the transition. Initially after the chain has relaxed to a stationary situation around the metastable minimum of the potential $V(q_{min})$, the oscillators obey the canonic distribution in phase space near to this minimum.
[0.49]{} ![Average escape times in the thermally activated case for different values of the friction constant and their comparison to the deterministic setting for 500 realizations each. The grey area in Fig. \[fig:comp\_plot2\] sketches the average escape times for the thermally activated case for $0.005 < \gamma < 1.0$ as shown in Fig. \[fig:comp\_plot1\] where as the symbols shows results from the deterministic set-up with initial conditions given in the inset. Parameters: $\kappa=0.15, N=100$[]{data-label="fig:comp_plot"}](./images/comp_plot1 "fig:"){width="\linewidth"}
[0.49]{} ![Average escape times in the thermally activated case for different values of the friction constant and their comparison to the deterministic setting for 500 realizations each. The grey area in Fig. \[fig:comp\_plot2\] sketches the average escape times for the thermally activated case for $0.005 < \gamma < 1.0$ as shown in Fig. \[fig:comp\_plot1\] where as the symbols shows results from the deterministic set-up with initial conditions given in the inset. Parameters: $\kappa=0.15, N=100$[]{data-label="fig:comp_plot"}](./images/comp_plot2 "fig:"){width="\linewidth"}
We measure average escape times for the system described by Eq. (\[1d\_Langevin\_eq\]). The latter is numerically integrated using an Euler scheme, again with a maximal integration time of $5\cdot 10^5$ time units. The system is initialized with all oscillators set to the minimum of the potential and zero momenta. We then let the system thermalize until its energy reaches $\overline{E}$ for the first time. The time from this moment until all oscillators have surpassed the potential’s maximum counts as the escape time.
We study the system for the optimal coupling constant, $\kappa=0.15$, as described in Sec. \[1d:optimal\_coupling\]. Figure \[fig:comp\_plot1\] shows the average escape times of 500 realizations for different values of the friction constant in dependence of $\overline{E}$. Figure \[fig:comp\_plot2\] compares these times (depicted as the grey surface) to the according average escape times of the deterministic system. It additionally shows the characteristic time constant for the formation of breathers, $\Gamma_{\mathrm{max}}^{-1}$, taken from Eq. (\[MI1d:max\_growth\_rate\]), where again we relate the $k=0$ phonon amplitude, $A$, to the system energy via $E(A)=N\,V(A)$.
Especially for smaller energies the deterministic escape is considerably faster than the thermally activated one. Notably for $E/(N\,\Delta E)<0.1$ and quite contrary to the deterministic setting, escape events are practically absent during our simulations time in the thermal case. For larger energy values this picture can change to a higher efficiency of the thermal escape process when damping is weak. Also two deterministic settings with different magnitudes of the random initial perturbations have a converse behaviour for low and high energies.
[0.48]{} ![Temporal evolution of the energy distribution $E_n(t)$ in the thermally activated case. A smaller friction constant leads to a higher degree of energy localization. Parameters: $\overline{E}/(N\,\Delta E)=0.15, \quad \kappa=0.15$.[]{data-label="fig:1d:E_evo_thermal"}](./images/E_evol_thermal_gam_i005 "fig:"){width="\linewidth"}
[0.48]{} ![Temporal evolution of the energy distribution $E_n(t)$ in the thermally activated case. A smaller friction constant leads to a higher degree of energy localization. Parameters: $\overline{E}/(N\,\Delta E)=0.15, \quad \kappa=0.15$.[]{data-label="fig:1d:E_evo_thermal"}](./images/E_evol_thermal_gam_1i "fig:"){width="\linewidth"}
How can this result be explained by observations of the chain dynamics? For small values of $\gamma$ the system approaches the deterministic setting. This entails an observable tendency towards a more localized energy distribution as seen in comparing Fig. \[fig:1d:E\_evo\_thermal\_1\] and Fig. \[fig:1d:E\_evo\_thermal\_2\]. The relaxation time of the chain scales with the inverse of the damping constant. Correspondingly, the life times of local excitations grow with decreasing $\gamma$. The outcome is a more heterogeneous energetic structure where thermal fluctuations are more likely to cause critical chain elongations. This explains the faster escape comparing small with large damping constants. But even in the case of very small $\gamma$ the relaxation time is still much shorter than the time needed for the coalescence of multiple breathers (which is of the order of several hundred time units, see Fig. \[fig:ts:energy\_evolution\_b\]). Therefore, the long term cumulative concentration of energy, as described in \[1d:optimal\_coupling\], is generally inhibited in the thermal case. This explains the virtual impossibility of a thermal escape for small energies.
In the opposite case of higher energies the deterministic escape is mostly proceeded by initial breathers. The formation time of the initial breather array can be estimated by the inverse of the maximal growth rate, $\Gamma_{\mathrm{max}}$ from Eq. \[MI1d:max\_growth\_rate\]. Figure \[fig:comp\_plot2\] shows the expected convergence of $\Gamma_{\mathrm{max}}^{-1}$ to the deterministic escape times for large energies. We recall from Sec. \[sec2\] that $\Gamma_{\mathrm{max}}$ was determined starting with a linear expansion in the perturbations. We believe that this explains the better match in the case of smaller initial perturbations. The divergence of $\Gamma_{\mathrm{max}}^{-1}$ with the average escape times for small energies again gives evidence to the fact that the initial breather array does not induce an escape and other mechanisms are needed.
In the thermally activated case the large energy setting holds two characteristic scenarios. The mobility of the breathers becomes amplified if noise acts. If the breathers possess longer life times, [*i.e.*]{} for smaller damping, a few (usually not more the two) breathers can temporarily merge and thus approach a critical chain elongation. Starting with a heterogeneous energy structure after thermalization, this then leads to smaller escape times compared to the deterministic setting that first has to re-allocate energy from an initially homogeneous state. Oppositely for stronger damping, the life times of breather is too short and the energy distribution is mostly homogeneous (see Fig. \[fig:1d:E\_evo\_thermal\_2\]). The escapes then relies entirely on rare, large enough spontaneous fluctuation of the noise term and the average escape times generally become comparatively large.
Finally, we want to examine the converse behaviour for different magnitudes of the initial perturbations in the deterministic case. As the initial conditions for smaller perturbations are closer to the $k=0$ phonon mode the initial breather array emerges more quickly so that the escape times are smaller when the energy is high enough for initial breathers to ignite the escape. Contrarily, stronger perturbations lead to a higher breather mobility which accelerates the breather coalescence so that the escape times become smaller when low energy necessitates coalescence.
Summary {#sec:4}
=======
We have studied various examples of collective escape processes in many-particle systems. First we started with a noise-free escape in a chain of coupled oscillators evolving in a metastable potential. The second part of this work examined a thermally activated escape. To this end the original system has been augmented by a linear friction term and Gaussian thermal, white noise of vanishing mean satisfying the fluctuation dissipation theorem.
While a thermally activated escape becomes virtually impossible at low system energy, the deterministic system remains capable of efficiently crossing the potential barrier. In more detail, the deterministic dynamics of interacting chain units leads to the formation of breather solutions localizing energy such that the chain passes through a transition state and crosses the potential barrier. In particular at low system energy, the interaction between several formed breathers resulting in their coalescence eventually enables the chain to accumulate sufficient energy to overcome the potential barrier. Interestingly, in the thermally activated setting with a sufficiently weak damping, the breather formation and thermal fluctuation cooperate to localize energy which accomplishes effective barrier crossings.
We underline that the dynamics of interacting particles exhibiting collective behaviour such as breather formation and their interaction has a huge impact on the escape and activation dynamics in such many-body systems. Hence, our study intends to offer new perspectives on the understanding of such collective escape processes.
Acknowledgments
===============
L. Schimansky-Geier thanks for support from IRTG 1640 of the Deutsche Forschungsgemeinschaft. The authors acknowledge previous co-authors for a successful cooperation.\
Preprint of an article published in the book “First-Passage Phenomena and Their Applications”: pp. 554-570, May 2014; doi: 10.1142/9789814590297[\_]{}0022 World Scientific Publishing Company.
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[^1]: [email protected]
[^2]: An alternative approach for the one-dimensional chain model is presented in [@HeLsg07]. It casts the stationary equation into a two-dimensional map and links the localized lattice solutions to its homoclinic orbits.
|
---
author:
- |
Hsien-Chih Chang Jeff Erickson\
Department of Computer Science\
University of Illinois, Urbana-Champaign\
[{hchang17, jeffe}@illinois.edu](mailto:[email protected],[email protected])
bibliography:
- 'bib/jeffe.bib'
- 'bib/topology.bib'
- 'bib/compgeom.bib'
date: |
Submitted to *Discrete & Computational Geometry* — July 16, 2016\
Revised and resubmitted —
title: 'Untangling Planar Curves[^1]'
---
Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called *homotopy moves*. We prove that simplifying a planar closed curve with $n$ self-crossings requires $\Theta(n^{3/2})$ homotopy moves in the worst case. Our algorithm improves the best previous upper bound $O(n^2)$, which is already implicit in the classical work of Steinitz; the matching lower bound follows from the construction of closed curves with large *defect*, a topological invariant of generic closed curves introduced by Aicardi and Arnold. Our lower bound also implies that $\Omega(n^{3/2})$ are required to reduce any graph with treewidth $\Omega(\sqrt{n})$ to a single , matching known upper bounds for rectangular and cylindrical grid graphs. More generally, we prove that transforming one immersion of $k$ circles with at most $n$ self-crossings into another requires $\Theta(n^{3/2} + nk + k^2)$ homotopy moves in the worst case. Finally, we prove that transforming one noncontractible closed curve to another on any orientable surface requires $\Omega(n^2)$ homotopy moves in the worst case; this lower bound is tight if the curve is homotopic to a simple closed curve.
[0.46]{} “It’s hardly fair,” muttered Hugh, “to give us such a jumble as this to work out!”
Introduction
============
Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of the following local operations:
- : Remove an empty loop.
- : Separate two subpaths that bound an empty bigon.
- : Flip an empty triangle by moving one subpath over the opposite intersection point.
![Homotopy moves $\arc10$, $\arc20$, and $\arc33$.[]{data-label="F:homotopy"}](Fig/homotopy-moves-aligned "fig:"){width="0.9\linewidth"}\
See Figure \[F:homotopy\]. Each of these operations can be performed by continuously deforming the curve within a small neighborhood of one face; consequently, we call these operations and their inverses . Our notation is nonstandard but mnemonic; the numbers before and after each arrow indicate the number of local vertices before and after the move. Homotopy moves are “shadows” of the classical Reidemeister moves used to manipulate knot and link diagrams [@ab-tkc-26; @r-ebk-27].
We prove that $\Theta(n^{3/2})$ homotopy moves are sometimes necessary and always sufficient to simplify a closed curve in the plane with $n$ self-crossings. Before describing our results in more detail, we review several previous results.
Past Results
------------
An algorithm to simplify any planar closed curve using at most $O(n^2)$ homotopy moves is implicit in Steinitz’s proof that every 3-connected planar graph is the 1-skeleton of a convex polyhedron [@s-pr-1916; @sr-vtp-34]. Specifically, Steinitz proved that any non-simple closed curve (in fact, any 4-regular plane graph) with no empty loops contains a *bigon* (“Spindel”): a disk bounded by a pair of simple subpaths that cross exactly twice, where the endpoints of the (slightly extended) subpaths lie outside the disk. Steinitz then proved that any *minimal* bigon (“irreduzible Spindel”) can be transformed into an empty bigon using a sequence of $\arc33$ moves, each removing one triangular face from the bigon, as shown in Figure \[F:SR-spindle\]. Once the bigon is empty, it can be deleted with a single $\arc20$ move. See Grünbaum [@g-cp-67], Hass and Scott [@hs-scs-94], Colin de Verdière [@cgv-rep-96], or Nowik [@n-cpsc-09] for more modern treatments of Steinitz’s technique. The $O(n^2)$ upper bound also follows from algorithms for *regular* homotopy, which forbids $\biarc01$ moves, by Francis [@f-frtcs-69], Vegter [@v-kfdp-89] (for polygonal curves), and Nowik [@n-cpsc-09].
![Top: A minimal bigon. Bottom: $\arc33$ moves removing triangles from the side or the end of a (shaded) minimal bigon. All figures are from Steinitz and Rademacher [@sr-vtp-34].[]{data-label="F:SR-spindle"}](Fig/SR-irreducible-spindle "fig:")\
![Top: A minimal bigon. Bottom: $\arc33$ moves removing triangles from the side or the end of a (shaded) minimal bigon. All figures are from Steinitz and Rademacher [@sr-vtp-34].[]{data-label="F:SR-spindle"}](Fig/SR-spindle-side-before "fig:") ![Top: A minimal bigon. Bottom: $\arc33$ moves removing triangles from the side or the end of a (shaded) minimal bigon. All figures are from Steinitz and Rademacher [@sr-vtp-34].[]{data-label="F:SR-spindle"}](Fig/SR-spindle-side-after "fig:") ![Top: A minimal bigon. Bottom: $\arc33$ moves removing triangles from the side or the end of a (shaded) minimal bigon. All figures are from Steinitz and Rademacher [@sr-vtp-34].[]{data-label="F:SR-spindle"}](Fig/SR-spindle-end-before "fig:") ![Top: A minimal bigon. Bottom: $\arc33$ moves removing triangles from the side or the end of a (shaded) minimal bigon. All figures are from Steinitz and Rademacher [@sr-vtp-34].[]{data-label="F:SR-spindle"}](Fig/SR-spindle-end-after "fig:")
The $O(n^2)$ upper bound can also be derived from an algorithm of Feo and Provan [@fp-dtert-93] for reducing a *plane* graph to a single edge by *electrical transformations*: degree-1 reductions, series-parallel reductions, and $\Delta$Y-transformations. (We consider electrical transformations in more detail in Section \[S:electric\].) Any curve divides the plane into regions, called its *faces*. The *depth* of a face is its distance to the outer face in the dual graph of the curve. Call a homotopy move *positive* if it decreases the sum of the face depths; in particular, every $\arc10$ and $\arc20$ move is positive. of Feo and Provan implies that every non-simple curve in the plane admits a positive homotopy move [@fp-dtert-93 Theorem 1]. Thus, the sum of the face depths is an upper bound on the minimum number of moves required to simplify the curve. Euler’s formula implies that every curve with $n$ crossings has $O(n)$ faces, and each of these faces has depth $O(n)$.
Gitler [@g-dtaa-91] conjectured that a variant of Feo and Provan’s algorithm that always makes the *deepest* positive move requires only $O(n^{3/2})$ moves. Song [@s-iifpd-01] observed that if Feo and Provan’s algorithm always chooses the *shallowest* positive move, it can be forced to make $\Omega(n^2)$ moves even when the input curve can be simplified using only $O(n)$ moves.
Tight bounds are known for two special cases where some homotopy moves are forbidden. First, Nowik [@n-cpsc-09] proved a tight $\Omega(n^2)$ lower bound for regular homotopy. Second, Khovanov [@k-dg-97] defined two curves to be *doodle equivalent* if one can be transformed into the other using $\biarc10$ and $\biarc20$ moves. Khovanov [@k-dg-97] and Ito and Takimura [@it-whkp-13] independently proved that any planar curve can be transformed into its unique equivalent doodle with the smallest number of vertices, using only $\arc10$ and $\arc20$ moves. Thus, two doodle equivalent curves are connected by a sequence of $O(n)$ moves, which is obviously tight. Looser bounds are also known for the minimum number of Reidemeister moves needed to reduce a diagram of the unknot [@hn-udrqn-10; @l-pubrm-15], to separate the components of a split link [@hhn-unnrm-12], or to move between two equivalent knot diagrams [@hh-msrmd-11; @cl-ubrm-14].
New Results
-----------
In Section \[S:lower-bound\], we derive an $\Omega(n^{3/2})$ lower bound using a numerical curve invariant called *defect*, introduced by Arnold [@a-tipcc-94; @a-pctip-94] and Aicardi [@a-tc-94]. Each homotopy move changes the defect of a closed curve by at most $2$. The lower bound therefore follows from constructions of Hayashi [@hh-msrmd-11; @hhsy-musrm-12] and Even-Zohar [@ehln-irkl-14] of closed curves with defect $\Omega(n^{3/2})$. We simplify and generalize their results by computing the defect of the standard planar projection of any $p\times q$ torus knot where either $p\bmod q = 1$ or $q\bmod p = 1$. Our calculations imply that for any integer $p$, reducing the standard projection of the $p\times (p+1)$ torus knot requires at least $\smash{\binom{p+1}{3}} \ge n^{3/2}/6 - O(n)$ homotopy moves. Finally, using winding-number arguments, we prove that in the worst case, simplifying an arrangement of $k$ closed curves requires $\Omega(n^{3/2}+ nk)$ homotopy moves, with an additional $\Omega(k^2)$ term if the target configuration is specified in advance.
In Section \[S:electric\], we provide a proof, based on arguments of Truemper [@t-drpg-89] and Noble and Welsh [@nw-kg-00], that reducing a *unicursal* graph $G$— whose medial graph is the image of a single closed curve—using electrical transformations requires at least as many steps as reducing the medial graph of $G$ to a simple closed curve using homotopy moves. The homotopy lower bound from Section \[S:lower-bound\] then implies that reducing any $n$-vertex graph with treewidth $\Omega(\sqrt{n})$ requires $\Omega(n^{3/2})$ electrical transformations. This lower bound matches known upper bounds for rectangular and cylindrical grid graphs.
We develop a new algorithm to simplify any closed curve in $O(n^{3/2})$ homotopy moves in Section \[S:upper\]. First we describe an algorithm that uses $O(D)$ moves, where $D$ is the sum of the face depths of the input curve. At a high level, our algorithm can be viewed as a variant of Steinitz’s algorithm that empties and removes *loops* instead of bigons. We then extend our algorithm to *tangles*: collections of boundary-to-boundary paths in a closed disk. Our algorithm simplifies a tangle as much as possible in $O(D + ns)$ moves, where $D$ is the sum of the depths of the tangle’s faces, $s$ is the number of paths, and $n$ is the number of intersection points. Then, we prove that for any curve with maximum face depth $\Omega(\sqrt{n})$, we can find a simple closed curve whose interior tangle has . Simplifying this tangle and then recursively simplifying the resulting curve requires a total of $O(n^{3/2})$ moves. We show that this simplifying sequence of homotopy moves can be computed in $O(1)$ amortized time per move, assuming the curve is presented in an appropriate graph data structure. We conclude this section by proving that any arrangement of $k$ closed curves can be simplified in $O(n^{3/2} + nk)$ homotopy moves, or in $O(n^{3/2} + nk + k^2)$ homotopy moves if the target configuration is specified in advance, precisely matching our lower bounds for all values of $n$ and $k$.
Finally, in Section \[S:genus\], we consider curves on surfaces of higher genus. We prove that $\Omega(n^2)$ homotopy moves are required in the worst case to transform one non-contractible closed curve to another on the torus, and therefore on any orientable surface. Results of Hass and Scott [@hs-ics-85] imply that this lower bound is tight if the non-contractible closed curve is homotopic to a simple closed curve.
Definitions
-----------
A in a surface $M$ is a continuous map $\gamma \colon S^1 \to M$. In this paper, we consider only *generic* closed curves, which are injective except at a finite number of self-intersections, each of which is a transverse double point; closed curves satisfying these conditions are called *immersions* of the circle. A closed curve is if it is injective. For most of the paper, we consider only closed curves in the plane; we consider more general surfaces in Section \[S:genus\].
The image of any non-simple closed curve has a natural structure as a 4-regular plane graph. Thus, we refer to the self-intersection points of a curve as its , the maximal subpaths between vertices as , and the components of the complement of the curve as its . Two curves $\gamma$ and $\gamma'$ are *isomorphic* if their images ; we will not distinguish between isomorphic curves.
A *corner* of $\gamma$ is the intersection of a face of $\gamma$ and a small neighborhood of a vertex of $\gamma$. A in a closed curve $\gamma$ is a subpath of $\gamma$ that begins and ends at some vertex $x$, intersects itself only at $x$, and encloses exactly one corner at $x$. A in $\gamma$ consists of two simple interior-disjoint subpaths of $\gamma$ with the same endpoints and enclose one corner at each of those endpoints. A loop or bigon is if its interior does not intersect $\gamma$. Notice that a $\arc10$ move is applied to an empty loop, and a $\arc20$ move is applied on an empty bigon.
We adopt a standard sign convention for vertices first used by Gauss [@g-n1gs-00]. Choose an arbitrary basepoint $\gamma(0)$ and orientation for the curve. we define $\sgn(x) = +1$ if the first traversal through the vertex crosses the second traversal from right to left, and $\sgn(x) = -1$ otherwise. See Figure \[F:signs\].
![Gauss’s sign convention.[]{data-label="F:signs"}](Fig/vertex-signs)
A between two curves $\gamma$ and $\gamma'$ in surface $M$ is a continuous function $H\colon {S^1 \times [0,1] \to M}$ such that $H(\cdot,0) = \gamma$ and $H(\cdot,1) = \gamma'$. Any homotopy $H$ describes a continuous deformation $\gamma$ $\gamma'$, where the second argument of $H$ is “time”. Each homotopy move can be executed by a homotopy. Conversely, Alexander’s simplicial approximation theorem [@a-cas-26], together with combinatorial arguments of Alexander and Briggs [@ab-tkc-26] and Reidemeister [@r-ebk-27], imply that any generic homotopy between two closed curves can be decomposed into a finite sequence of homotopy moves. Two curves are *homotopic*, or in the same *homotopy class*, if there is a homotopy from one to the other. All closed curves in the plane are homotopic.
A is an immersion of one or more disjoint circles; in particular, a is an immersion of $k$ disjoint circles. A multicurve is *simple* if it is injective, or equivalently, if it can be decomposed into pairwise disjoint simple closed curves. The image of any multicurve in the plane is the disjoint union of simple closed curves and 4-regular plane graphs. A of a multicurve $\gamma$ is any multicurve whose image is a connected component of the image of $\gamma$. We call the individual closed curves that comprise a multicurve its ; see Figure \[F:components\]. The definition of homotopy and the decomposition of homotopies into homotopy moves extend naturally to multicurves.
![A multicurve with two components and three constituent curves, one of which is simple.[]{data-label="F:components"}](Fig/multicurve-example)
Lower Bounds {#S:lower-bound}
============
Defect {#SS:defect}
------
To prove our main lower bound, we consider a numerical invariant of closed curves in the plane introduced by Arnold [@a-tipcc-94; @a-pctip-94] and Aicardi [@a-tc-94] called . Polyak [@p-icfgd-98] proved that defect can be computed—or for our purposes, defined—as follows: $$\Defect(\gamma) \coloneqq -2 \sum_{x\between y} \sgn(x)\cdot\sgn(y).$$ Here the sum is taken over all *interleaved* pairs of vertices of $\gamma$: two vertices $x\ne y$ are interleaved, denoted , if they alternate in cyclic order—$x$, $y$, $x$, $y$—along $\gamma$. (The factor of $-2$ is a historical artifact, which we retain only to be consistent with Arnold’s original definitions [@a-tipcc-94; @a-pctip-94].) Even though the signs of individual vertices depend on the basepoint and orientation of the curve, the defect of a curve is independent of those choices. Moreover, the defect of any curve is preserved by any homeomorphism from the plane (or the sphere) to itself, including reflection.
Trivially, every simple closed curve has defect zero. Straightforward case analysis [@p-icfgd-98] implies that any single homotopy move changes the defect of a curve by at most $2$; the various cases are listed below and illustrated in Figure \[F:defect-change\].
- A $\arc10$ move leaves the defect unchanged.
- A $\arc20$ move decreases the defect by $2$ if the two disappearing vertices are interleaved, and leaves the defect unchanged otherwise.
- A $\arc33$ move increases the defect by $2$ if the three vertices before the move contain an even number of interleaved pairs, and decreases the defect by $2$ otherwise.
In light of this case analysis, the following lemma is trivial:
\[L:defect\] Simplifying any closed curve $\gamma$ in the plane requires at least $\abs{\Defect(\gamma)}/2$ homotopy moves.
$1\arcto 0$
------------- ----- ------ ------ ------
$0$ $0$ $-2$ $+2$ $+2$
Flat Torus Knots {#SS:torus-knots}
----------------
For any relatively prime positive integers $p$ and $q$, let denote the curve with the following parametrization, where $\theta$ runs from $0$ to $2\pi$: $$T(p,q)(\theta) \coloneqq \left((\cos (q\theta)+2) \cos(p\theta),~ (\cos (q\theta)+2) \sin(p\theta)\right).$$ The curve $T(p,q)$ winds around the origin $p$ times, oscillates $q$ times between two concentric circles, and crosses itself exactly $(p-1)q$ times. We call these curves .
{width="1.75in"} {width="1.75in"}
Hayashi [@hhsy-musrm-12 Proposition 3.1] proved that for any integer $q$, the flat torus knot $T(q+1,q)$ has defect $-2\binom{q}{3}$. Even-Zohar [@ehln-irkl-14] used a star-polygon representation of the curve $T(p, 2p+1)$ as the basis for a universal model of random knots; in our notation, they proved that $\Defect(T(p, 2p+1)) = 4\binom{p+1}{3}$ for any integer $p$. In this section we simplify and generalize both of these results to all flat torus knots $T(p,q)$ where either $q\bmod p = 1$ or $p\bmod q = 1$.
![Transforming $T(8,17)$ into a flat braid.[]{data-label="F:flat-braid"}](Fig/8,17-cut-braid)
\[L:braid-wide\] $\Defect(T(p, ap+1)) = 2a \binom{p+1}{3}$ for all integers $a\ge 0$ and $p \ge 1$.
The curve $T(p, 1)$ can be reduced using only $\arc10$ moves, so its defect is zero.
We can reduce $T(p, ap+1)$ to $T(p, ({a-1})p+1)$ by straightening one strand at a time. Straightening the bottom strand of block requires the following $\binom{p}{2}$ moves, as shown in Figure \[F:braid-wide\].
- $\binom{p-1}{2}$ $\arc33$ moves pull the bottom strand downward over one intersection point of every other pair of strands. Just before each $\arc33$ move, exactly one of the three pairs of the three relevant vertices is interleaved, so each move decreases the defect by $2$.
- $(p-1)$ $\arc20$ moves eliminate a pair of intersection points between the bottom strand and every other strand. Each of these moves also decreases the defect by $2$.
Altogether, straightening one strand decreases the defect by $\smash{2\binom{p}{2}}$. Proceeding similarly with the other strands, we conclude that $\Defect(T(p, ap+1)) = \Defect(T(p, {(a-1)p+1})) + 2\binom{p+1}{3}$. The lemma follows immediately by induction.
![Straightening one strand in a block of $T(8, 8a+1)$.[]{data-label="F:braid-wide"}](Fig/braid-wide-cmyk)
\[L:braid-deep\] $\Defect(T(aq+1, q)) = -2a \binom{q}{3}$ for all integers $a\ge 0$ and $q \ge 1$.
The curve $T(1,q)$ is simple, so its defect is trivially zero. For any positive integer $a$, we can transform $T(aq+1, q)$ into $T((a-1)q+1, q)$ by incrementally removing the innermost $q$ *loops*. We can remove the first loop using $\binom{q}{2}$ homotopy moves, as shown in Figure \[F:braid-deep\]. (The first transition in Figure \[F:braid-deep\] just reconnects the top left and top right endpoints of the flat braid.)
- $\binom{q-1}{2}$ $\arc33$ moves pull the left side of the loop to the right, over the crossings inside the loop. Just before each $\arc33$ move, the three relevant vertices contain two interleaved pairs, so each move *increases* the defect by $2$.
- $(q-1)$ $\arc20$ moves pull the loop over $q-1$ strands. The strands involved in each move are oriented in opposite directions, so these moves leave the defect unchanged.
- Finally, we can remove the loop with a single $\arc10$ move, which does not change the defect.
Altogether, removing one loop increases the defect by $\smash{2\binom{q-1}{2}}$. Proceeding similarly with the other loops, we conclude that $\Defect(T(aq+1, q)) = \Defect(T((a-1)q+1, q)) - 2 \binom{q}{3}$. The lemma follows immediately by induction.
![Removing one loop from the innermost block of $T(7a+1, 7)$.[]{data-label="F:braid-deep"}](Fig/braid-deep-simple)
Either of the previous lemmas imply the following lower bound, which is also implicit in the work of Hayashi [@hhsy-musrm-12].
For every positive integer $n$, there are closed curves with $n$ vertices whose defects are $n^{3/2}/3 - O(n)$ and $-n^{3/2}/3 + O(n)$, and therefore requires at least $n^{3/2}/6 - O(n)$ homotopy moves to reduce to a simple closed curve.
The lower bound follows from the previous lemmas by setting $a=1$. If $n$ is a prefect square, then the flat torus knot $T(\sqrt{n}+1, \sqrt{n})$ has $n$ vertices and defect $\smash{-2\binom{\sqrt{n}}{3}}$. If $n$ is not a perfect square, we can achieve defect ${-2\binom{\floor{\sqrt{n}}}{3}}$ by applying $\arc01$ moves to the curve $T(\floor{\sqrt{n}}+1, \floor{\sqrt{n}})$. Similarly, we obtain an $n$-vertex curve with defect $\smash{2\binom{\floor{\sqrt{n+1}}+1}{3}}$ by adding loops to the curve $T(\floor{\sqrt{n+1}}, \floor{\sqrt{n+1}}+1)$. Lemma \[L:defect\] now immediately implies the lower bound on homotopy moves.
Multicurves {#SS:multi-lower}
-----------
Our previous results immediately imply that simplifying a multicurve with $n$ vertices requires at least $\Omega(n^{3/2})$ homotopy moves; in this section we derive additional lower bounds in terms of the number of constituent curves. We distinguish between two natural variants of simplification: transforming a multicurve into an *arbitrary* set of disjoint simple closed curves, or into a *particular* set of disjoint simple closed curves.
Both lower bound proofs rely on the classical notion of . Let $\gamma$ be an arbitrary closed curve in the plane, let $p$ be any point outside the image of $\gamma$, and let $\rho$ be any ray from $p$ to infinity that intersects $\gamma$ transversely. The winding number of $\gamma$ around $p$, which we denote , is the number of times $\gamma$ crosses $\rho$ from right to left, minus the number of times $\gamma$ crosses $\rho$ from left to right. The winding number does not depend on the particular choice of ray $\rho$. All points in the same face of $\gamma$ have the same winding number. Moreover, if there is a homotopy from one curve $γ$ to another curve $γ’$, does not include $p$, then $\Wind(γ, p) = \Wind(γ', p)$ [@h-udtse-35].
Transforming a $k$-curve with $n$ vertices in the plane into $k$ arbitrary disjoint circles requires $\Omega(nk)$ homotopy moves in the worst case.
For arbitrary positive integers $n$ and $k$, we construct a multicurve with $k$ disjoint constituent , all but one of which are simple, as follows. The first $k-1$ constituent $\gamma_1, \dots, \gamma_{k-1}$ are disjoint circles inside the open unit disk centered at the origin. (The precise configuration of these circles is unimportant.) The remaining curve $\gamma_o$ is a spiral winding $n+1$ times around the closed unit disk centered at the origin, plus a line segment connecting the endpoints of the spiral; $\gamma_o$ is the simplest possible curve with winding number $n+1$ around the origin. Let $\gamma$ be the disjoint union of these $k$ curves; we claim that $\Omega(nk)$ homotopy moves are required to simplify $\gamma$. See Figure \[F:winding\].
![Simplifying this multicurve requires $\Omega(nk)$ homotopy moves.[]{data-label="F:winding"}](Fig/winding-target)
Consider the faces of the outer curve $\gamma_o$ during any homotopy of $\gamma$. Adjacent faces of $\gamma_o$ have winding numbers that differ by $1$, and the outer face has winding number $0$. Thus, for any non-negative integer $w$, as long as the maximum absolute winding number $\Abs{\max_p \Wind(\gamma_o,p)}$ is at least $w$, the curve $\gamma_o$ has at least $w+1$ faces (including the outer face) and therefore at least $w-1$ vertices, by Euler’s formula. On the other hand, if any curve $\gamma_i$ intersects a face of $\gamma_o$, no homotopy move can remove that face . Thus, before the simplification of $\gamma_o$ is complete, each curve $\gamma_i$ must intersect only faces with winding number $0$, $1$, or $-1$.
For each index $i$, let $w_i$ denote the maximum absolute winding number of $\gamma_o$ around any point of $\gamma_i$: $$w_i \coloneqq \max_\theta \Abs{\Wind\left(\gamma_o,\strut \gamma_i(\theta)\right)}.$$ Let $W = \sum_i w_i$. Initially, $W = k(n+1)$, and when $\gamma_o$ first becomes simple, we must have $W \le k$. Each homotopy move changes $W$ by at most $1$; specifically, at most one term $w_i$ changes at all, and that term either increases or decreases by $1$. The $\Omega(nk)$ lower bound now follows immediately.
\[Th:multi-lower\] Transforming a $k$-curve with $n$ vertices in the plane into an arbitrary set of $k$ simple closed curves requires $\Omega(n^{3/2} + nk)$ homotopy moves in the worst case.
We say that a collection of $k$ disjoint simple closed curves is if some point lies in the interior of every curve, and if the curves have disjoint interiors.
Transforming $k$ nested circles in the plane into $k$ unnested circles requires $\Omega(k^2)$ homotopy moves.
Let $\gamma$ and $\gamma'$ be two nested circles, with $\gamma'$ in the interior of $\gamma$ and with $\gamma$ directed counterclockwise. Suppose we apply an arbitrary homotopy to these two curves. If the curves remain disjoint during the entire homotopy, then $\gamma'$ always lies inside a face of $\gamma$ with winding number $1$; in short, the two curves remain nested. Thus, any sequence of homotopy moves that takes $\gamma$ and $\gamma'$ to two non-nested simple closed curves contains at least one $\arc{0}{2}$ move that makes the curves cross (and symmetrically at least one $\arc{2}{0}$ move that makes them disjoint again).
Consider a set of $k$ nested circles. Each of the $\smash{\binom{k}{2}}$ pairs of circles requires at least one $\arc{0}{2}$ move and one $\arc{2}{0}$ move to unnest. Because these moves involve distinct pairs of curves, at least $\smash{\binom{k}{2}}$ $\arc{0}{2}$ moves and $\smash{\binom{k}{2}}$ $\arc{2}{0}$ moves, and thus at least $k^2-k$ moves altogether, are required to unnest every pair.
\[Th:multi-lower2\] Transforming a $k$-curve with $n$ vertices in the plane into $k$ nested (or unnested) circles requires $\Omega(n^{3/2} + nk + k^2)$ homotopy moves in the worst case.
\[C:multi-lower3\] Transforming one $k$-curve with at most $n$ vertices into another $k$-curve with at most $n$ vertices requires $\Omega(n^{3/2} + nk + k^2)$ homotopy moves in the worst case.
Although our lower bound examples consist of disjoint curves, all of these lower bounds apply without modification to *connected* multicurves, because any $k$-curve can be connected with at most $k-1$ $\arc{0}{2}$ moves. On the other hand, any connected $k$-curve has at least $2k-2$ vertices, so the $\Omega(k^2)$ terms in Theorem \[Th:multi-lower2\] and Corollary \[C:multi-lower3\] are redundant.
Electrical Transformations {#S:electric}
==========================
Now we consider a related set of local operations on plane graphs, called , consisting of six operations in three dual pairs, as shown in Figure \[F:elec-dual\].
- *degree-$1$ reduction*: Contract the edge incident to a vertex of degree $1$, or delete the edge incident to a face of degree $1$
- *series-parallel reduction*: Contract either edge incident to a vertex of degree $2$, or delete either edge incident to a face of degree $2$
- *$\Delta Y$ transformation*: Delete a vertex of degree 3 and connect its neighbors with three new edges, or delete the edges bounding a face of degree 3 and join the vertices of that face to a new vertex.
![Facial electrical transformations in a plane graph $G$ and its dual graph $G^*$.[]{data-label="F:elec-dual"}](Fig/graph-moves-2){width="85.00000%"}
have been used since the end of the 19th century [@k-etscn-1899; @r-md-1904] to analyze resistor networks and other electrical circuits, but have since been applied to a number of other combinatorial problems on planar graphs, including shortest paths and maximum flows [@a-wtns-60]; multicommodity flows [@f-erpns-85]; and counting spanning trees, perfect matchings, and cuts [@cpv-nastc-95]. We refer to our earlier preprint [@defect Section 1.1] for a more detailed history and an expanded list of applications.
In light of these applications, it is natural to ask *how many* electrical transformations are required in the worst case.
Even the special case of regular grids is open and interesting. Truemper [@t-drpg-89; @t-md-92] describes a method to reduce the $p\times p$ grid in $O(p^3)$ steps. Nakahara and Takahashi [@nt-aafts-96] prove an upper bound of $O(\min\set{pq^2, p^2q})$ for the $p\times q$ cylindrical grid. Because every $n$-vertex graph is a minor of an $O(n)\times O(n)$ grid [@v-ucvc-81; @s-mncpe-84], both of these results imply an $O(n^3)$ upper bound for arbitrary plane graphs; see Lemma \[L:smoothing\]. Feo and Provan [@fp-dtert-93] claim without proof that Truemper’s algorithm actually performs only $O(n^2)$ electrical transformations. On the other hand, the smallest (cylindrical) grid containing every $n$-vertex plane graph as a minor has size $Ω(n) \times Ω(n)$ [@v-ucvc-81]. Archdeacon [@acgp-frpwg-00] asked whether the $O(n^{3/2})$ upper bound for square grids can be improved to near-linear:
> It is possible that a careful implementation and analysis of the grid-embedding schemes can lead to an $O(n\sqrt{n})$-time algorithm for the general planar case. It would be interesting to obtain a near-linear algorithm for the grid…. However, it may well be that reducing planar grids is $Ω(n\sqrt{n})$.
![Plane graphs with two terminals that cannot be further reduced using only facial electrical transformations.[]{data-label="F:bullseye"}](Fig/spine8)
Definitions
-----------
The of a plane graph $G$, which we denote , is another plane graph whose vertices correspond to the edges of $G$ and whose edges correspond to incidences (with multiplicity) between vertices of $G$ and faces of $G$. Two vertices of $G^\times$ are connected by an edge if and only if the corresponding edges in $G$ are consecutive in cyclic order around some vertex, or equivalently, around some face in $G$. Every vertex in every medial graph has degree $4$; thus, every medial graph is the image of a multicurve. The medial graphs of any plane graph $G$ and its dual $G^*$ are identical. To avoid trivial boundary cases, we define the medial graph of an isolated vertex to be a circle. electrical transformations in any plane graph $G$ correspond to local transformations in the medial graph $G^\times$ that are almost identical to homotopy moves. Each degree-$1$ reduction in $G$ corresponds to a $\arc10$ homotopy move in $G^\times$, and each $\Delta$Y transformation in $G$ corresponds to a $\arc33$ homotopy move in $G^\times$. A series-parallel reduction in $G$ contracts an empty bigon in $G^\times$ to a single vertex. Extending our earlier notation, we call this transformation a move. We collectively refer to these transformations and their inverses as ; see Figure \[F:medial-elec\].
![Medial electrical moves $\arc10$, $\arc21$, and $\arc33$.[]{data-label="F:medial-elec"}](Fig/graph-moves-3 "fig:"){width="0.9\linewidth"}\
a $γ$ at a vertex $x$ means replacing the intersection of $γ$ with a small neighborhood of $x$ with two disjoint simple paths, so that the result is another . (There are two possible smoothings at each vertex; see Figure \[F:smoothing\].) A of $γ$ is any graph obtained by smoothing zero or more vertices of $γ$, and a of $γ$ is any smoothing other than $\gamma$ itself. For any plane graph $G$, the (proper) smoothings of the medial graph $G^\times$ are precisely the medial graphs of (proper) minors of $G$.
![Smoothing a vertex.[]{data-label="F:smoothing"}](Fig/smoothing)
Electrical to Homotopy
----------------------
The main result of this section is that the number of *homotopy* moves required to simplify a closed curve is a lower bound on the number of *medial electrical moves* required to simplify the same closed curve. This result is already implicit in the work of Noble and Welsh [@nw-kg-00], and most of our proofs closely follow theirs. We include the proofs here to make the inequalities explicit and to keep the paper self-contained.
For any connected multicurve (or 4-regular graph) $\gamma$, let denote the minimum number of medial electrical moves required to reduce $γ$ to a simple closed curve, and let is the minimum number of homotopy moves required to reduce $γ$ to an arbitrary collection of disjoint simple closed curves.
The following key lemma follows from close reading of proofs by Truemper [@t-drpg-89 Lemma 4] and several others [@g-dtaa-91; @nt-aafts-96; @acgp-frpwg-00; @nw-kg-00] that every minor of a ΔY-reducible graph is also ΔY-reducible. Our proof most closely resembles an argument of Gitler [@g-dtaa-91 Lemma 2.3.3], but restated in terms of medial electrical moves to simplify the case analysis.
\[L:smoothing\] For any connected plane graph $G$, reducing any connected proper minor of $G$ to a single vertex requires strictly fewer electrical transformations than reducing $G$ to a single vertex. Equivalently, $X(\overline{γ}) < X(γ)$ for every connected proper smoothing $\overline{γ}$ of every connected multicurve $γ$.
Let $γ$ be a connected multicurve, and let $\overline{γ}$ be a connected proper smoothing of $γ$. If $γ$ is already simple, the lemma is vacuously true. Otherwise, the proof proceeds by induction on $X(γ)$.
We first consider the special case where $\overline{γ}$ is obtained from $γ$ by smoothing a single vertex $x$. Let $γ'$ be the result of the first medial electrical move in the minimum-length sequence that reduces $γ$ . We immediately have $X(γ) = X(γ')+1$. There are two nontrivial cases to consider.
First, suppose the move from $γ$ to $γ'$ does not involve the smoothed vertex $x$. Then we can apply the same move to $\overline{γ}$ to obtain a new graph $\overline{γ}'$; the same graph can also be obtained from $γ'$ by smoothing $x$. We immediately have $X(\overline{γ}) \le X(\overline{γ}') + 1$, and the inductive hypothesis implies $X(\overline{γ}')+1 < X(γ')+1 = X(γ)$.
Now suppose the first move in $Σ$ does involve $x$. In this case, we can apply at most one medial electrical move to $\overline{γ}$ to obtain a (possibly trivial) smoothing $\overline{γ}'$ of $γ'$. There are eight subcases to consider, shown in Figure \[F:smooth-moves\]. One subcase for the $\arc01$ move is impossible, because $\overline{γ}$ is connected. In the remaining $\arc01$ subcase and one $\arc21$ subcase, the curves $\overline{γ}$, $\overline{γ}'$ and $γ'$ are all isomorphic, which implies $X(\overline{γ}) = X(\overline{γ}') = X(γ') = X(γ)-1$. In all remaining subcases, $\overline{γ}'$ is a connected proper smoothing of $γ'$, so the inductive hypothesis implies $X(\overline{γ}) ≤ X(\overline{γ}')+1 < X(γ')+1 = X(γ)$.
![Cases for the proof of the Lemma \[L:smoothing\]; the circled vertex is $x$.[]{data-label="F:smooth-moves"}](Fig/smooth-move-2){width="80.00000%"}
Finally, we consider the more general case where $\overline{γ}$ is obtained from $γ$ by smoothing more than one vertex. Let $\widetilde{γ}$ be any intermediate curve, obtained from $γ$ by smoothing just one of the vertices that were smoothed to obtain $\overline{γ}$. As $\overline{γ}$ is a connected smoothing of $\widetilde{γ}$, the curve $\widetilde{γ}$ itself must be connected too. Our earlier argument implies that $X(\widetilde{γ}) < X(γ)$. Thus, the inductive hypothesis implies $X(\overline{γ}) < X(\widetilde{γ})$, which completes the proof.
\[L:monotonicity\] For every connected multicurve $γ$, there is a minimum-length sequence of medial electrical moves that reduces $γ$ and that does not contain $\arc01$ or $\arc12$ moves.
Our proof follows an argument of Noble and Welsh [@nw-kg-00 Lemma 3.2].
Consider a minimum-length sequence of medial electrical moves that reduces an arbitrary connected multicurve $γ$ . For any integer $i ≥ 0$, let $γ_i$ denote the result of the first $i$ moves in this sequence; in particular, $γ_0 = γ$ and $γ_{X(γ)}$ is a . Minimality of the reduction sequence implies that $X(γ_i) = X(γ)-i$ for all $i$. Now let $i$ be an arbitrary index such that $γ_i$ has one more vertex than $γ_{i-1}$. Then $γ_{i-1}$ is a connected proper smoothing of $γ_i$, so Lemma \[L:smoothing\] implies that $X(γ_{i-1}) < X(γ_i)$, giving us a contradiction.
\[L:homotopy\] $X(γ) ≥ H(γ)$ for every closed curve $γ$.
The proof proceeds by induction on $X(γ)$, following an argument of Noble and Welsh [@nw-kg-00 Proposition 3.3].
Let $γ$ be a closed curve. If $X(γ) = 0$, then $γ$ is already simple, so $H(γ) = 0$. Otherwise, let $Σ$ be a minimum-length sequence of medial electrical moves that reduces $γ$ to a . Lemma \[L:monotonicity\] implies that we can assume that the first move in $Σ$ is neither $\arc01$ nor $\arc12$. If the first move is $1\arcto 0$ or $3\arcto 3$, the theorem immediately follows by induction.
The only interesting first move is $2\arcto 1$. Let $γ'$ be the result of this $\arc21$ move, and let $\overline{γ}$ be the result of the corresponding $\arc20$ homotopy move. The minimality of $Σ$ implies that $X(γ) = X(γ')+1$, and we trivially have $H(γ) \le H(\overline{γ}) + 1$. Because $γ$ consists of *one* single curve, $\overline{γ}$ connected. The curve $\overline{γ}$ is also a proper smoothing of $γ'$, so the Lemma \[L:smoothing\] implies $X(\overline{γ}) < X(γ') < X(γ)$. Finally, the inductive hypothesis implies that $X(\overline{γ}) \ge H(\overline{γ})$, which completes the proof.
We call a plane graph $G$ if its medial graph $G^\times$ is the image of a single closed curve.
\[Th:lowerbound\] For every connected graph $G$ and every unicursal minor $H$ of $G$, reducing $G$ to a single vertex requires at least $\abs{\Defect(H^\times)}/2$ electrical transformations.
Either $H$ equals $G$, or Lemma \[L:smoothing\] implies that reducing a proper minor $H$ of $G$ to a single vertex requires strictly fewer electrical transformations than reducing $G$ to a single vertex. Note that electrical transformations performed on $H$ corresponds precisely to medial electrical moves performed on $H^\times$. Now because $γ \coloneqq H^\times$ is unicursal, Lemma \[L:defect\] and Lemma \[L:homotopy\] implies that $X(γ) ≥ H(γ) ≥ \abs{\Defect(\gamma)}/2$.
Cylindrical Grids
-----------------
Finally, we derive explicit lower bounds for the number of electrical transformations required to reduce any cylindrical grid to a single vertex. For any positive integers $p$ and $q$, we define two cylindrical grid graphs; see Figure \[F:cylinders\].
- is the Cartesian product of a cycle of length $q$ and a path of length $p-1$. If $q$ is odd, then the medial graph of $C(p,q)$ is the flat torus knot $T(2p, q)$.
- is obtained by connecting a new vertex to the vertices of one of the $q$-gonal faces of $C(p,q)$, or equivalently, by contracting one of the $q$-gonal faces of $C(p+1,q)$ to a single vertex. If $q$ is even, then the medial graph of $C'(p,q)$ is the flat torus knot $T(2p+1, q)$.
![The cylindrical grid graphs $C(4,7)$ and $C'(3,8)$ and (in light gray) their medial graphs $T(8,7)$ and $T(7,8)$.[]{data-label="F:cylinders"}](Fig/C47 "fig:"){width="1.75in"}![The cylindrical grid graphs $C(4,7)$ and $C'(3,8)$ and (in light gray) their medial graphs $T(8,7)$ and $T(7,8)$.[]{data-label="F:cylinders"}](Fig/Cprime38 "fig:"){width="1.75in"}
\[C:cylindrical-grid\] For all positive integers $p$ and $q$, the cylindrical grid $C(p,q)$ requires $Ω(\min\set{p^2 q, p q^2})$ electrical transformations to reduce to a single vertex.
First suppose $p \le q$. Because $C(p-1,q)$ is a minor of $C(p,q)$, we can assume without loss of generality that $p$ is even and $p<q$. Let $H$ denote the cylindrical grid $C(p/2, ap+1)$, where $a \coloneqq \floor{(q-1)/p} \ge 1$. $H$ is a minor of $C(p, q)$ (because $ap+1 \le q$), and the medial graph of $H$ is the flat torus knot $T(p, ap+1)$. Lemma \[L:braid-wide\] implies $$\Defect\!\left(T(p,\strut ap+1)\right) = 2a\binom{p+1}{3} = Ω(ap^3) = Ω(p^2q).$$ Theorem \[Th:lowerbound\] now implies that reducing $C(p,q)$ requires at least $\Omega(p^2 q)$ electrical transformations.
The symmetric case $p > q$ is similar. We can assume without loss of generality that $q$ is odd. Let $H$ denote the cylindrical grid $C'(aq, q)$, where $a \coloneqq \floor{(p-1)/q} \ge 1$. $H$ is a proper minor of $C(p, q)$ (because $aq<p$), and the medial graph of $H$ is the flat torus knot $T(2aq+1, q)$. Lemma \[L:braid-deep\] implies $$\Abs{ \Defect\!\left(T(2aq+1,\strut q)\right)}
= 4a\binom{q}{3} = Ω(aq^3) = Ω(pq^2).$$ Theorem \[Th:lowerbound\] now implies that reducing $C(p,q)$ requires at least $\Omega(p q^2)$ electrical transformations.
In particular, reducing any $\Theta(\sqrt{n})\times\Theta(\sqrt{n})$ cylindrical grid requires at least $\Omega(n^{3/2})$ electrical transformations. Our lower bound matches an $O(\min\set{pq^2, p^2q})$ upper bound by Nakahara and Takahashi [@nt-aafts-96]. Because every $p\times q$ rectangular grid contains $C(\floor{p/3}, \floor{q/3})$ as a minor, the same $Ω(\min\set{p^2 q, p q^2})$ lower bound applies to rectangular grids. In particular, Truemper’s $O(p^3) = O(n^{3/2})$ upper bound for the $p\times p$ square grid [@t-drpg-89] is tight. Finally, because every graph with treewidth $t$ contains an $Ω(t)\times Ω(t)$ grid minor [@rst-qepg-94], reducing *any* $n$-vertex graph with treewidth $t$ requires at least $Ω(t^3 + n)$ electrical transformations.
Like Gitler [@g-dtaa-91], Feo and Provan [@fp-dtert-93], and Archdeacon [@acgp-frpwg-00], we conjecture that any $n$-vertex graph can be reduced using $O(n^{3/2})$ electrical transformations. More ambitiously, we conjecture an upper bound of $O(nt)$ for any $n$-vertex graph with treewidth $t$.
Upper Bound {#S:upper}
===========
For any point $p$, let denote the minimum number of times a path from $p$ to infinity crosses $\gamma$. Any two points in the same face of $\gamma$ have the same depth, so each face $f$ has a well-defined depth, which is its distance to the outer face in the dual graph of $\gamma$; see Figure \[F:curve-depth-contract\]. The depth of the curve, denoted , is the maximum depth of the faces of $\gamma$; and the is the sum of the depths of the faces. Euler’s formula implies that any 4-regular graph with $n$ vertices has exactly $n+2$ faces; thus, for any curve $\gamma$ with $n$ vertices, we have $n+1 \le D(\gamma) \le (n+1)\cdot \Depth(\gamma)$.
Contracting Simple Loops
------------------------
\[L:contract\] Every closed curve $\gamma$ in the plane can be simplified using at most $3D(\gamma) - 3$ homotopy moves.
The lemma is trivial if $\gamma$ is already simple, so assume otherwise. Let $x \coloneqq \gamma(\theta) = \gamma(\theta')$ be the first vertex to be visited twice by $\gamma$ after the (arbitrarily chosen) basepoint $\gamma(0)$. Let $\alpha$ denote the subcurve of $\gamma$ from $\gamma(\theta)$ to $\gamma(\theta')$; our choice of $x$ implies that $\alpha$ is a simple loop. Let $m$ and $s$ denote the number of vertices and maximal subpaths of $\gamma$ in the interior of $\alpha$ respectively.
Finally, let $\gamma'$ denote the closed curve obtained from $\gamma$ by removing $\alpha$. The first stage of our algorithm transforms $\gamma$ into $\gamma'$ by contracting the loop $\alpha$ via homotopy moves.
![Transforming $\gamma$ into $\gamma'$ by contracting a simple loop. Numbers are face depths.[]{data-label="F:curve-depth-contract"}](Fig/shrink-loop)
We remove the vertices and edges from the interior of $\alpha$ one at a time as follows . If we can perform a $\arc20$ move to remove one edge of $\gamma$ from the interior of $\alpha$ and decrease $s$, we do so. Otherwise, either $\alpha$ is empty, or some vertex of $\gamma$ lies inside $\alpha$. In the latter case, at least one vertex $x$ inside $\alpha$ has a neighbor that lies on $\alpha$. We move $x$ outside $\alpha$ with a $\arc02$ move (which increases $s$ ) followed by a $\arc33$ move . Once $\alpha$ is an empty loop, we remove it with a single $\arc10$ move. Altogether, our algorithm transforms $\gamma$ into $\gamma'$ using at most $3m + s + 1$ homotopy moves. Let $M$ denote the actual number of homotopy moves used.
![Moving a loop over an interior empty bigon or an interior vertex. []{data-label="F:move-strand"}](Fig/move-strand-2)
Euler’s formula implies that $\alpha$ contains exactly faces of $\gamma$. The Jordan curve theorem implies that $\Depth(p, \gamma') \le \Depth(p, \gamma)-1$ for any point $p$ inside $\alpha$, and trivially $\Depth(p, \gamma') \le \Depth(p, \gamma)$ for any point $p$ outside $\alpha$. It follows that $D(\gamma') \le D(\gamma) - (\EDIT{m+s+1}) \le D(\gamma) - M/3$, and therefore $M \le 3D(\gamma) - 3 D(\gamma')$. The induction hypothesis implies that we can recursively simplify $\gamma'$ using at most $3D(\gamma') - 3$ moves. The lemma now follows immediately.
Our upper bound is a factor of 3 larger than Feo and Provan’s [@fp-dtert-93]; however our algorithm has the advantage that it extends to *tangles*, as described in the next subsection.
Tangles {#SS:tangles}
-------
A is a collection of boundary-to-boundary paths $\gamma_1, \gamma_2, \dots,\gamma_s$ in a closed topological disk $\Sigma$, which (self-)intersect only pairwise, transversely, and away from the boundary of $\Sigma$. This terminology is borrowed from knot theory, where a tangle usually refers to the intersection of a knot or link with a closed 3-dimensional ball [@c-eklst-70; @cdm-ivki-12]; our tangles are perhaps more properly called *flat tangles*, as they are images of tangles under appropriate projection. (Our tangles are unrelated to the obstructions to small branchwidth introduced by Robertson and Seymour [@rs-gm10-91].) Transforming a curve into a tangle is identical to (an inversion of) the *flarb* operation defined by Allen [@abil-ivd-16].
We call each individual path $\gamma_i$ a of the tangle. The of a tangle is the boundary of the disk $\Sigma$ that contains it; we usually denote the boundary by $\sigma$. By the Jordan-Schönflies theorem, we can assume without loss of generality that $\sigma$ is actually a circle. We can obtain a tangle from any closed curve $\gamma$ by considering its restriction to any closed disk whose boundary $\sigma$ intersects $\gamma$ transversely away from its vertices; we call this restriction the of $\sigma$.
The strands and boundary of any tangle define a plane graph $T$ whose boundary vertices each have degree $3$ and whose interior vertices each have degree $4$. Depths and potential are defined exactly as for closed curves: The depth of any face $f$ of $T$ is its distance to the outer face in the dual graph $T^*$; the depth of the tangle is its maximum face depth; and the potential $D(T)$ of the tangle is the sum of its face depths.
A tangle is if every pair of strands intersects at most once and otherwise. Every loose tangle contains either an empty loop or a (not necessarily empty) bigon. Thus, any tangle with $n$ vertices can be transformed into a tight tangle—or less formally, *tightened*—in $O(n^2)$ homotopy moves using Steinitz’s algorithm. On the other hand, there are infinite classes of loose tangles for which no homotopy move decreases the potential, so we cannot directly apply Feo and Provan’s algorithm to this setting.
![Tightening a tangle in two phases: First simplifying the individual strands, then removing excess crossings between pairs of strands.[]{data-label="F:tighten-tangle"}](Fig/just-tangle){width="0.9\linewidth"}
\[L:pretangle\]
Fix an arbitrary reference point on the boundary circle $\sigma$ that is not an endpoint of a strand. For each index $i$, let $\sigma_i$ be the arc of $\sigma$ between the endpoints of $\gamma_i$ that does not contain the reference point. A strand $\gamma_i$ is *extremal* if the corresponding arc $\sigma_i$ does not contain any other arc $\sigma_j$.
Choose an arbitrary extremal strand $\gamma_i$. Let $m_i$ denote the number of tangle vertices in the interior of the disk bounded by $\gamma_i$ and the boundary arc $\sigma_i$; . Let $s_i$ denote the number of intersections between $\gamma_i$ and other strands. Finally, let $\gamma'_i$ be a path inside the disk $\Sigma$ defining tangle $T$, with the same endpoints as $\gamma_i$, that intersects each other strand in $T$ at most once, such that the disk bounded by $\sigma_i$ and $\gamma'_i$ has no tangle vertices inside its interior. .
We can deform $\gamma_i$ into $\gamma'_i$ using essentially the algorithm from Lemma \[L:contract\]; . If contains an empty bigon , remove it with a $\arc20$ move . If has an interior vertex with a neighbor on $\gamma_i$, remove it using at most two homotopy moves (which increases $s_i$ ). Altogether, this deformation requires at most $3m_i + s_i \le 3n$ homotopy moves.
![Moving one strand out of the way and shrinking the tangle boundary.[]{data-label="F:move-strand-tangle"}](Fig/tangle-one-strand){width="0.9\linewidth"}
After deforming $\gamma_i$ to $\gamma_i'$, we slightly to exclude $\gamma'_i$, without creating or removing any . We emphasize that shrinking the boundary does not modify the strands and therefore does not require any homotopy moves. The resulting smaller tangle has exactly $s-1$ strands, each of which is simple. Thus, the induction hypothesis implies that we can recursively tighten this smaller tangle using at most $3n(s-1)$ homotopy moves.
\[C:tangle\] Every tangle $T$ with $n$ vertices and $s$ strands can be tightened using at most $3D(T) + 3ns$ homotopy moves.
Main Algorithm
--------------
We call a simple closed curve $\sigma$ for $\gamma$ if $\sigma$ intersects $\gamma$ transversely away from its vertices, and the interior tangle $T$ of $\sigma$ has at least $s^2$ vertices, where $s \coloneqq \abs{\sigma\cap\gamma}/2$ is the number of strands. Our main algorithm repeatedly finds a useful closed curve whose interior tangle has $O(\sqrt{n})$ depth, and tightens its interior tangle; if there are no useful closed curves, then we fall back to the loop-contraction algorithm of Lemma \[L:contract\].
\[L:useful\] Let $\gamma$ be an arbitrary non-simple closed curve in the plane with $n$ vertices. Either there is a useful simple closed curve for $\gamma$ whose interior tangle has depth $O(\sqrt{n})$, or the depth of $\gamma$ is $O(\sqrt{n})$.
To simplify notation, let $d \coloneqq \Depth(\gamma)$. For each integer $j$ between $1$ and $d$, let $R_j$ be the set of points $p$ with $\Depth(p, \gamma) \ge d+1-j$, and let $\tilde{R}_j$ denote a small open neighborhood of the closure of $R_j \cup \tilde{R}_{j-1}$, where $\tilde{R}_0$ is the empty set. Each region $\tilde{R}_j$ is the disjoint union of closed disks, whose boundary cycles intersect $\gamma$ transversely away from its vertices, if at all. In particular, $\tilde{R}_d$ is a disk containing the entire curve $\gamma$.
Fix a point $z$ such that $\Depth(z,\gamma) = d$. For each integer $j$, let $\Sigma_j$ be the unique component of $\tilde{R}_j$ that contains $z$, and let $\sigma_j$ be the boundary of $\Sigma_j$. Then $\sigma_1, \sigma_2, \dots, \sigma_d$ are disjoint, nested, simple closed curves; see Figure \[F:curve-levels\]. Let $n_j$ be the number of vertices and let $s_j \coloneqq \abs{\gamma \cap \sigma_j}/2$ be the number of strands of the interior tangle of $\sigma_j$. For notational convenience, we define $\Sigma_0 \coloneqq \varnothing$ and thus $n_0 = s_0 = 0$. We ignore the outermost curve $\sigma_d$, because it contains the entire curve $\gamma$. The next outermost curve $\sigma_{d-1}$ contains every vertex of $\gamma$, so $n_{d-1} = n$.
![Nested depth cycles around a point of maximum depth.[]{data-label="F:curve-levels"}](Fig/tangle-levels)
By construction, for each $j$, the interior tangle of $\sigma_j$ has depth $j+1$. Thus, to prove the lemma, it suffices to show that if none of the curves $\sigma_1, \sigma_2, \dots, \sigma_{d-1}$ is useful, then $d = O(\sqrt{n})$.
Fix an index $j$. Each edge of $\gamma$ crosses $\sigma_j$ at most twice. Any edge of $\gamma$ that crosses $\sigma_j$ has at least one endpoint in the annulus $\Sigma_j \setminus \Sigma_{j-1}$, and any edge that crosses $\sigma_j$ twice has both endpoints in $\Sigma_j \setminus \Sigma_{j-1}$. Conversely, each vertex in $\Sigma_j$ is incident to at most two edges that cross $\sigma_j$ and no edges that cross $\sigma_{j+1}$. It follows that $\abs{\sigma_j\cap\gamma} \le 2(n_j - n_{j-1})$, and therefore $n_j \geq n_{j-1} + s_j$. Thus, by induction, we have $$n_j \ge \sum_{i=1}^j s_i$$ for every index $j$.
Now suppose no curve $\sigma_j$ with $1\le j <d$ is useful. Then we must have $s_j^2 > n_j$ and therefore $$s_j^2 > \sum_{i=1}^j s_i$$ for all $1\le j < d$. Trivially, $s_1\ge 1$, because $\gamma$ is non-simple. A straightforward induction argument implies that $s_j \ge (j+1)/2$ and therefore $$n
~=~ n_{d-1}
~\ge~ \sum_{i = 1}^{d-1} \frac{i+1}{2}
~\ge~ \frac{1}{2} \binom{d+1}{2}
~>~ \frac{d^2}{4}.$$ We conclude that $d \le 2\sqrt{n}$, which completes the proof.
\[Th:upper\] Every closed curve in the plane with $n$ vertices can be simplified in $O(n^{3/2})$ homotopy moves.
Let $\gamma$ be an arbitrary closed curve in the plane with $n$ vertices. If $\gamma$ has depth $O(\sqrt{n})$, Lemma \[L:contract\] and the trivial upper bound $D(\gamma) \le {(n+1)}\cdot \Depth(\gamma)$ imply that we can simplify $\gamma$ in $O(n^{3/2})$ homotopy moves. For purposes of analysis, we charge $O(\sqrt{n})$ of these moves to each vertex of $\gamma$. Otherwise, let $\sigma$ be an arbitrary useful closed curve chosen according to Lemma \[L:useful\]. Suppose the interior tangle of $\sigma$ has $m$ vertices, $s$ strands, and depth $d$. Lemma \[L:useful\] implies that $d = O(\sqrt{n})$, and the definition of useful implies that $s \le \sqrt{m}$, which is $O(\sqrt{n})$. Thus, by , we can tighten the interior tangle of $\sigma$ in $O(m d + m s) = O(m \sqrt{n})$ moves. This simplification removes at least $m - s^2/2 \ge m/2$ vertices from $\gamma$, as the resulting tight tangle has at most $s^2/2$ vertices. Again, for purposes of analysis, we charge $O(\sqrt{n})$ moves to each deleted vertex. We then recursively simplify the resulting closed curve.
In either case, each vertex of $\gamma$ is charged $O(\sqrt{n})$ moves as it is deleted. Thus, simplification requires at most $O(n^{3/2})$ homotopy moves in total.
Efficient Implementation
------------------------
Here we describe how to implement our curve-simplification algorithm to run in $O(n^{3/2})$ time; in fact, our implementation spends only constant amortized time per homotopy move. We assume that the input curve is given in a data structure that allows fast exploration and modification of plane graphs, such as a quad-edge data structure [@gs-pmgsc-85] or a doubly-connected edge list [@bcko-cgaa-08]. If the curve is presented as a polygon with $m$ edges, an appropriate graph representation can be constructed in $O(m\log m + n)$ time using classical geometric algorithms [@cs-arscg-89; @m-fppa-90; @ce-oails-92]; more recent algorithms can be used for piecewise-algebraic curves [@ek-ee2aa-08].
\[Th:upper-algo\] Given a simple closed curve $\gamma$ in the plane with $n$ vertices, we can compute a sequence of $M = O(n^{3/2})$ homotopy moves that simplifies $\gamma$ in $O(M)$ time.
We begin by labeling each face of with its depth, using a breadth-first search of the dual graph in $O(n)$ time. Then we construct the depth contours of —the boundaries of the regions $\tilde{R}_j$ from the proof of Lemma \[L:useful\]—and organize them into a *contour tree* in $O(n)$ time by brute force. Another $O(n)$-time breadth-first traversal computes the number of strands and the number of interior vertices of every contour’s interior tangle; in particular, we identify which depth contours are useful. To complete the preprocessing phase, we place all the leafmost useful contours into a queue. We can charge the overall $O(n)$ preprocessing time to the $\Omega(n)$ homotopy moves needed to simplify the curve.
As long as the queue of leafmost useful contours is non-empty, we extract one contour $\sigma$ from this queue and simplify its interior tangle $T$ as follows. Suppose $T$ has $m$ interior vertices. Following the proof of Theorem \[Th:upper\], we first simplify every loop in each strands of $T$. We identify loops by traversing the strand from one endpoint to the other, marking the vertices as we go; the first time we visit a vertex that has already been marked, we have found a loop $\alpha$. We can perform each of the homotopy moves required to shrink $\alpha$ in $O(1)$ time, because each such move modifies only a constant-radius boundary of a vertex on $\alpha$. After the loop is shrunk, we continue walking along the strand starting at the most recently marked .
The second phase of the tangle-simplification algorithm proceeds similarly. We walk around the boundary of $T$, marking vertices as we go. As soon as we see the second endpoint of any strand $\gamma_i$, we pause the walk to straighten $\gamma_i$. As before, we can execute each homotopy move used to move $\gamma_i$ to $\gamma'_i$ in $O(1)$ time. We then move the boundary of the tangle over the vertices of $\gamma'_i$, and remove the endpoints of $\gamma'_i$ from the boundary curve, in $O(1)$ time per vertex.
The only portions of the running time that we have not already charged to homotopy moves are the time spent marking the vertices on each strand and the time to update the tangle boundary after moving a strand aside. Altogether, the uncharged time is $O(m)$, which is less than the number of moves used to tighten $T$, because the contour $\sigma$ is useful. Thus, tightening the interior tangle of a useful contour requires $O(1)$ amortized time per homotopy move.
Once the tangle is tight, we must update the queue of useful contours. The original contour $\sigma$ is still a depth contour in the modified curve, and tightening $T$ only changes the depths of faces that intersect $T$. Thus, we could update the contour tree in $O(m)$ time, which we could charge to the moves used to tighten $T$; but in fact, this update is unnecessary, because no contour in the interior of $\sigma$ is useful. We then walk up the contour tree from $\sigma$, updating the number of interior vertices until we find a useful ancestor contour. The total time spent traversing the contour tree for new useful contours is $O(n)$; we can charge this time to the $\Omega(n)$ moves needed to simplify the curve.
Multicurves {#multicurves}
-----------
Finally, we describe how to extend our $O(n^{3/2})$ upper bound to multicurves. Just as in Section \[SS:multi-lower\], we distinguish between two variants, depending on whether the target of the simplification is an *arbitrary* set of disjoint cycles or a *particular* set of disjoint cycles. In both cases, our upper bounds match the lower bounds proved in Section \[SS:multi-lower\].
First we extend our loop-contraction algorithm from Lemma \[L:contract\] to the multicurve setting. The main difficulty is that component of the multicurve might lie inside a face of another component, making progress on the larger component impossible. To handle this potential obstacle, we simplify the *innermost* components of the first, and we move isolated simple closed curves toward the outer face as quickly as possible. Figure \[F:multi-shrink\] sketches the basic steps of our algorithm when the input multicurve is connected.
![Simplifying a connected multicurve: shrink an arbitrary simple loop or cycle, recursively simplify any inner components, translate inner circle clusters to the outer face, and recursively simplify the remaining non-simple components.[]{data-label="F:multi-shrink"}](Fig/multi-shrink-loop){width="0.95\linewidth"}
\[L:multi-contract\] Every $n$-vertex $k$-curve $\gamma$ in the plane can be transformed into $k$ disjoint simple closed curves using at most $3D(\gamma) + 4nk$ homotopy moves.
Let $\gamma$ be an arbitrary $k$-curve with $n$ vertices. If $\gamma$ is connected, we either contract and delete a loop, exactly as in Lemma \[L:contract\], or we contract a simple constituent curve to an isolated circle, using essentially the same algorithm. In either case, the number of moves performed is at most $3D(\gamma) - 3D(\gamma')$, where $\gamma'$ is the multicurve after the contraction. The lemma now follows immediately by induction.
We call a component of $\gamma$ an if it is incident to the unbounded outer face of $\gamma$, and an otherwise. If $\gamma$ has more than one outer component, we partition $\gamma$ into subcurves, each consisting of one outer component $\gamma\!_o$ and all inner components located inside faces of $\gamma\!_o$, and we recursively simplify each subcurve independently; the lemma follows by induction. If any outer component is simple, we ignore that component and simplify the rest of $\gamma$ recursively; again, the lemma follows by induction.
Thus, we can assume without loss of generality that our multicurve $\gamma$ is disconnected but has only one outer component $\gamma\!_o$, which is non-simple. For each face $f$ of $\gamma\!_o$, let $\gamma\!_f$ denote the union of all components inside $f$. Let $n_f$ and $k_f$ respectively denote the number of vertices and constituent of $\gamma\!_f$. Similarly, let $n_o$ and $k_o$ respectively denote the number of vertices and constituent of the outer component $\gamma\!_o$.
We first recursively simplify each subcurve $\gamma\!_f$; let $\kappa_f$ denote the resulting *cluster* of $k_f$ simple closed curves. By the induction hypothesis, this simplification requires at most $3D(\gamma\!_f) + 4 n_f k_f$ homotopy moves. We *translate* each cluster $\kappa_f$ to the outer face of $\gamma\!_o$ by shrinking $\kappa_f$ to a small $\e$-ball and then moving the entire cluster along a shortest path in the dual graph of $\gamma\!_o$. This translation requires at most $4n_o k_f $ homotopy moves; each circle in $\kappa_f$ uses one $\arc{2}{0}$ move and one $\arc{0}{2}$ move to cross any edge of $\gamma\!_o$, and in the worst case, the cluster might cross all $2n_0$ edges of $\gamma\!_o$. After all circle clusters are in the outer face, we recursively simplify $\gamma\!_o$ using at most $3 D(\gamma\!_o) + 4 n_o k_o$ homotopy moves.
The total number of homotopy moves used in this case is $$\sum_f 3D(\gamma\!_f) + 3 D(\gamma\!_o)
~+~
\sum_f 4 n_f k_f + \sum_f 4 n_o k_f + 4 n_o k_o.$$ Each face of $\gamma\!_o$ has the same depth as the corresponding face of $\gamma$, and for each face $f$ of $\gamma\!_o$, each face of the subcurve $\gamma\!_f$ has lesser depth than the corresponding face of $\gamma$. It follows that $$\sum_f D(\gamma\!_f) + D(\gamma\!_o) \le D(\gamma).$$ Similarly, $\sum_f n_f + n_o = n$ and $\sum_f k_f + k_o = k$. The lemma now follows immediately.
To reduce the leading term to $O(n^{3/2})$, we extend the definition of a tangle to the intersection of a multicurve $\gamma$ with a closed disk whose boundary intersects the multicurve transversely away from its vertices, or not at all. Such a tangle can be decomposed into boundary-to-boundary paths, called *open* strands, and closed curves that do not touch the tangle boundary, called *closed* strands. A tangle is *tight* if every strand is simple, every pair of open strands intersects at most once, and otherwise all strands are disjoint.
\[Th:multi-upper\] Every $k$-curve in the plane with $n$ vertices can be transformed into a set of $k$ disjoint simple closed curves using $O(n^{3/2} + nk)$ homotopy moves.
Let $\gamma$ be an arbitrary $k$-curve with $n$ vertices. Following the proof of Lemma \[L:multi-contract\], we can assume without loss of generality that $\gamma$ has a single outer component $\gamma\!_o$, which is non-simple.
When $\gamma$ is disconnected, we follow the strategy in the previous proof. Let $\gamma\!_f$ denote the union of all components inside any face $f$ of $\gamma\!_o$. For each face $f$, we recursively simplify $\gamma\!_f$ and translate the resulting cluster of disjoint circles to the outer face; when all faces are empty, we recursively simplify $\gamma\!_o$. The theorem now follows by induction.
When $\gamma$ is non-simple and connected, we follow the useful closed curve strategy from Theorem \[Th:upper\]. We define a closed curve $\sigma$ to be useful for $\gamma$ if the interior tangle of $\sigma$ has its number of vertices at least the square of the number of *open* strands; then the proof of Lemma \[L:useful\] applies to connected multicurves with no modifications. So let $T$ be a tangle with $m$ vertices, $s \le \sqrt{m}$ open strands, $\ell$ closed strands, and depth $d = O(\sqrt{n})$. We straighten $T$ in two phases, almost exactly as in , contracting loops and simple closed strands in the first phase, and straightening open strands in the second phase.
In the first phase, contracting one loop or closed uses at most $3D(T) - 3D(T')$ homotopy moves, where $T'$ is the tangle after contraction. After each contraction, if $T'$ is disconnected—in particular, if we just contracted a closed —we simplify and extract any isolated components as follows. Let $T'_o$ denote the component of $T'$ includes the boundary cycle, and for each face $f$ of $T'_o$, let $\gamma_f$ denote the union of all components of $T'$ inside $f$. We simplify each multicurve $\gamma_f$ using the algorithm from Lemma \[L:multi-contract\]—*not recursively!*—and then translate the resulting cluster of disjoint circles *to the outer face of $\gamma$*. Altogether, simplifying and translating these subcurves requires at most $
3D(T') - 3D(T'') + 4n \sum_f k_f
$ homotopy moves, where $T''$ is the resulting tangle.
![Whenever shrinking a loop or simple closed strand disconnects the tangle, simplify each isolated component and translate the resulting cluster of circles to the outer face of the entire multicurve.[]{data-label="F:tighten-open"}](Fig/just-tangle-open){width="\linewidth"}
The total number of moves performed in the first phase is at most $3D(T) + 4m\ell = O(m\sqrt{n} + n\ell)$. The first phase ends when the tangle consists entirely of simple open strands. Thus, the second phase straightens the remaining open strands exactly as in the proof of ; the total number of moves in this phase is $O(ms) = O(m\sqrt{n})$. We charge $O(\sqrt{n})$ time to each deleted vertex and $O(n)$ time to each constituent curve that was simplified and translated outward. We then recursively simplify the remaining multicurve, ignoring any outer circle clusters.
Altogether, each vertex of $\gamma$ is charged $O(\sqrt{n})$ time as it is deleted, and each constituent curve of $\gamma$ is charged $O(n)$ time as it is translated outward.
With $O(k^2)$ additional homotopy moves, we can transform the resulting set of $k$ disjoint circles into $k$ nested or unnested circles.
\[Th:multi-upper2\] Any $k$-curve with $n$ vertices in the plane can be transformed into $k$ nested (or unnested) simple closed curves using $O(n^{3/2} + nk + k^2)$ homotopy moves.
\[C:multi-upper3\] Any $k$-curve with at most $n$ vertices in the plane can be transformed into any other $k$-curve with at most $n$ vertices using $O(n^{3/2} + nk + k^2)$ homotopy moves.
Theorems \[Th:multi-lower\] and \[Th:multi-lower2\] and Corollary \[C:multi-lower3\] imply that these upper bounds are tight in the worst case for all possible values of $n$ and $k$. As in the lower bounds, the $O(k^2)$ terms are redundant for connected multicurves.
More careful analysis implies that any $k$-curve with $n$ vertices and depth $d$ can be simplified in $O(n \min\set{d, n^{1/2}} + k\min\set{d, n})$ homotopy moves, or transformed into $k$ unnested circles using $O(n \min\set{d, n^{1/2}} + k \min\set{d, n} + k \min\set{d, k})$ homotopy moves. Moreover, these upper bounds are tight for all possible values of $n$, $k$, and $d$. We leave the details of this extension as an exercise for the reader.
Higher-Genus Surfaces {#S:genus}
=====================
Finally, we consider the natural generalization of our problem to closed curves on orientable surfaces of higher genus. Because these surfaces have non-trivial topology, not every closed curve is homotopic to a single point or even to a simple curve. A closed curve is if it is homotopic to a single point. We call a closed curve if it has the minimum number of self-intersections in its homotopy class.
Lower Bounds {#lower-bounds}
------------
Although defect was originally defined as an invariant of *planar* curves, Polyak’s formula $\Defect(\gamma) = -2\sum_{x\between y} \sgn(x)\sgn(y)$ extends naturally to closed curves on any orientable surface; homotopy moves change the resulting invariant exactly as described in Figure \[F:defect-change\]. Thus, Lemma \[L:defect\] immediately generalizes to any orientable surface as follows.
\[L:defect-surface\] Let $\gamma$ and $\gamma'$ be arbitrary closed curves that are homotopic on an arbitrary orientable surface. Transforming $\gamma$ into $\gamma'$ requires at least $\abs{\Defect(\gamma) - \Defect(\gamma')}/2$ homotopy moves.
The following construction implies a quadratic lower bound for tightening noncontractible curves on orientable surfaces with any positive genus.
For any positive integer $n$, there is a closed curve on the torus with $n$ vertices and defect $\Omega(n^2)$ that is homotopic to a simple closed curve but not contractible.
Without loss of generality, suppose $n$ is a multiple of $8$. The curve $\gamma$ is illustrated on the left in Figure \[F:bad-torus\]. The torus is represented by a rectangle with opposite edges identified. We label three points $a,b,c$ on the vertical edge of the rectangle and decompose the curve into a red path from $a$ to $b$, a green path from $b$ to $c$, and a blue path from $c$ to $a$. The red and blue paths each wind vertically around the torus, first $n/8$ times in one direction, and then $n/8$ times in the opposite direction.
![A curve $\gamma$ on the torus with defect $\Omega(n^2)$ and a simple curve homotopic to $\gamma$.[]{data-label="F:bad-torus"}](Fig/bad-torus-plain)
As in previous proofs, we compute the defect of $\gamma$ by describing a sequence of homotopy moves that the curve, while carefully tracking the changes in the defect that these moves incur. We can unwind one turn of the red path by performing one $\arc20$ move, followed by $n/8$ $\arc33$ moves, followed by one $\arc20$ move, as illustrated in Figure \[F:bad-torus-unwind\]. Repeating this sequence of homotopy moves $n/8$ times removes all intersections between the red and green paths, after which a sequence of $n/4$ $\arc20$ moves straightens the blue path, yielding the simple curve shown on the right in Figure \[F:bad-torus\]. Altogether, we perform $n^2/64 + n/2$ homotopy moves, where each $\arc33$ move increases the defect of the curve by $2$ and each $\arc20$ move decreases the defect of the curve by $2$. We conclude that $\Defect(\gamma) = -n^2/32 + n$.
![Unwinding one turn of the red path.[]{data-label="F:bad-torus-unwind"}](Fig/bad-torus-unwind){width="0.9\linewidth"}
\[Th:lower-torus\] Tightening a closed curve with $n$ crossings on a torus requires $\Omega(n^2)$ homotopy moves in the worst case, even if the curve is homotopic to a simple curve.
Upper Bounds
------------
Hass and Scott proved that any non-simple closed curve on any orientable surface that is homotopic to a simple closed curve contains either a simple (in fact empty) contractible loop or a simple contractible bigon [@hs-ics-85 Theorem 1]. It follows immediately that any such curve can be simplified in $O(n^2)$ moves using Steinitz’s algorithm; Theorem \[Th:lower-torus\] implies that the upper bound is tight for non-contractible curves.
For the most general setting, where the given curve is not necessarily homotopic to a simple closed curve, we are not even aware of a *polynomial* upper bound! Steinitz’s algorithm does not work here; there are curves with excess self-intersections but no simple contractible loops or bigons [@hs-ics-85]. Hass and Scott [@hs-scs-94] and De Graff and Schrijver [@gs-mcmcr-97] independently proved that any closed curve on any surface can be simplified using a *finite* number of homotopy moves that never increase the number of self-intersections. Both proofs use discrete variants of curve-shortening flow; for sufficiently well-behaved curves and surfaces, results of Grayson [@g-sec-89] and Angenent [@a-pecs2-91] imply a similar result for differential curvature flow. Unfortunately, without further assumptions about the precise geometries of both the curve and the underlying surface, the number of homotopy moves cannot be bounded by any function of the number of crossings; even in the plane, there are closed curves with three crossings for which curve-shortening flow alternates between a $\arc33$ move and its inverse arbitrarily many times. Paterson [@p-cails-02] describes a combinatorial algorithm to compute a tightening sequence of homotopy moves without such reversals, but she offers no analysis of her algorithm.
We conjecture that any arbitrary curves (or even multicurves) on any surface can be simplified with at most $O(n^2)$ homotopy moves.
#### Acknowledgements.
We would like to thank Nathan Dunfield, Joel Hass, Bojan Mohar, and Bob Tarjan for encouragement and helpful discussions, and Joe O’Rourke for asking about multiple curves.
[^1]: Work on this paper was partially supported by NSF grant CCF-1408763. A preliminary version of this paper was presented at the 32nd International Symposium on Computational Geometry [@tangle]. See for the most recent version of this paper.
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abstract: 'In this paper we present sets of linear integral equations which make possible to compute the finite volume expectation values of the trace of the stress energy tensor ($\Theta$) and the $U(1)$ current ($J_\mu$) in any eigenstate of the Hamiltonian of the sine-Gordon model. The solution of these equations in the large volume limit allows one to get exact analytical formulas for the expectation values in the Bethe-Yang limit. These analytical formulas are used to test an earlier conjecture for the Bethe-Yang limit of expectation values in non-diagonally scattering theories. The analytical tests have been carried out upto three particle states and gave agreement with the conjectured formula, provided the definition of polarized symmetric diagonal form-factors is modified appropriately. Nevertheless, we point out that our results provide only a partial confirmation of the conjecture and further investigations are necessary to fully determine its validity. The most important missing piece in the confirmation is the mathematical proof of the finiteness of the symmetric diagonal limit of form-factors in a non-diagonally scattering theory.'
---
[**On the finite volume expectation values of local operators in the sine-Gordon model**]{}
[Árpád Hegedűs]{}
Wigner Research Centre for Physics,\
H-1525 Budapest 114, P.O.B. 49, Hungary\
Introduction {#intro}
============
Finite volume form-factors of integrable quantum field theories play an important role in the $AdS_5/CFT_4$ correspondence [@BJsftv; @BJhhl] and in condensed matter applications [@KoEss], as well. In AdS/CFT, their knowledge is indispensable for the computation of the string field theory vertex [@BJsftv] and of the heavy-heavy-light 3-point functions [@BJhhl] of the theory. In condensed matter systems finite volume form-factors can be used to compute quantum correlation-functions describing quasi 1-dimensional quantum-magnets, Mott insulators and carbon nanotubes [@KoEss].
The systematic study of finite volume form-factors of integrable quantum field theories was initiated in [@PT08a; @PT08b], where the finite volume matrix-elements of local operators are sought in the form of a systematic large volume series. From the investigation of finite volume 2-point functions it turned out, that upto exponentially small finite volume corrections, but including all corrections in the inverse of the volume, the non-diagonal finite volume form-factors are equal to their infinite volume counterparts taken at the positions of the solutions of the Bethe-Yang equations and normalized by the square roots of the densities of the sandwiching states.
As a consequence of the Dirac-delta contact terms in the crossing relations of the form-factor axioms, the diagonal form-factors cannot be obtained from the non-diagonal ones by taking their straightforward diagonal limit. Thus, diagonal form-factors are related to the infinite volume form-factors in a bit more indirect way. According to the conjectures [@PT08b], they can be represented as density weighted linear combinations of the so-called “connected” or “symmetric” diagonal limits of the infinite volume form-factors of the theory. In [@BWu], it has been shown, that the conjecture [@PT08b] being valid for purely elastic scattering-theories, can be derived from the leading order formula for the non-diagonal finite volume form-factors by considering such non-diagonal matrix elements, in which there is one particle more in the “bra” sandwiching state and the rapidity of this additional particle is taken to infinity appropriately.
The conjectures for purely elastic scattering theories [@PT08b] went through extensive analytical and numerical tests [@PST14] providing convincing amount of evidence for their validity. So far the conjecture for the more subtle non-diagonally scattering theories [@Palmai13] has not gone through convincing amount of tests. It has been tested in the sine-Gordon model, where it was checked analytically in the whole pure soliton sector against exact [@En; @En1] and numerical [@FPT11] results and numerically against TCSA data for mixed soliton-antisoliton two particles states [@Palmai13]. Thus, analytical tests of this conjecture is still missing in the soliton-antisoliton mixed sector. In this paper we would like to fill this gap and we check the conjecture of [@Palmai13] upto 3-particle soliton-antisoliton mixed states.
Beyond the leading polynomial in the inverse of the volume terms, the exponentially small in volume corrections are also necessary. Their determination is still an open problem in general. Nevertheless, some progress has been reached in this direction, as well. For the non-diagonal form-factors in purely elastic scattering-theories there is some knowledge about the leading order exponentially small in volume corrections termed the Lüscher corrections. The so-called $\mu$-term Lüscher corrections were determined in [@Pmu] and the F-term corrections for vacuum-1-particle form factors has been determined in [@BBCL]. Unfortunately, the Lüscher corrections to form-factors in non-diagonally scattering theories and higher order exponentially small in volume corrections in any integrable quantum field theory are presently out of reach.
Nevertheless, much is known about the exact finite volume behavior of the diagonal form-factors both in purely elastic and non-diagonally scattering theories.
In [@Pozsg13; @PST14] a LeClair-Mussardo type [@LM99] series representation was conjectured to describe exactly the finite volume dependence of diagonal matrix elements of local operators in purely elastic scattering theories. In non-diagonally scattering theories the description of finite volume diagonal matrix elements is less complete. So far only the sine-Gordon model has been studied in this class of theories. There, based on computations done in the framework of its integrable lattice regularization [@ddvlc], a LeClair-Mussardo type series representation was proposed to describe the finite volume dependence of the expectation values of local operators in pure soliton states [@En]. Nevertheless, soliton-antisoliton mixed states have not been investigated so far. In this paper we partly fill this gap and derive integral equations to get any finite volume diagonal matrix elements of two important operators of the theory. These are the trace of the stress energy tensor ($\Theta$) and the $U(1)$ current ($J_\mu$). Our formulas are valid to any value of the volume and to any eigenstate of the Hamiltonian of the model.
In the paper the Bethe-Yang limit of the diagonal form-factors will play an important role. In our terminology this limit means, that the exponentially small in volume corrections are neglected from the large volume expansion of the exact result[[^1]]{}.
In the repulsive regime, where there are no breathers in the spectrum, we solve our equations in the Bethe-Yang limit and give exact formulas for the expectation values of our operators in this limit. The formulas depend on the rapidities of the physical particles and on the magnonic Bethe-roots of the Bethe-Yang equations.
With the help of these exact formulas we check the conjecture of [@Palmai13] for the Bethe-Yang limit of the diagonal matrix elements of local operators in non-diagonally scattering theories. The conjectured formula in [@Palmai13] contains the symmetric diagonal limit of the infinite volume form-factors. The determination of these symmetric diagonal form-factors becomes more and more complicated as the number of particles increases. This is why we complete the test upto three particle states. Upto 3-particle states our exact formulas give perfect agreement with the conjectured formula of [@Palmai13] for the operators $\Theta$ and $J_\mu,$ provided the sandwiching color wave function $\Psi$ is replaced by its complex conjugate in the original formulas of [@Palmai13].
However despite the success of our checks, the details of the computations shed light on some subtle points of the conjecture, which require further work to be confirmed. The most important of them is to prove that the symmetric diagonal limit of form-factors is finite for a generic sandwiching state in a non-diagonally scattering theory. Though, this statement looks intuitively quite trivial, in section \[sect9\], where we comment the computation of the symmetric diagonal limit of form-factors, we argue that this statement is not trivial at all.
The outline of the paper is as follows.
In section \[sect2\]. we summarize the most important facts about the models and about the operators of our interest. In section \[sect3\]. the equations governing the exact finite volume dependence of diagonal matrix elements of the operators $\Theta$ and $J_\mu$ are derived. The solution of the equations in the large volume limit is given in section \[sect4\].
The basic ingredients of the form-factor bootstrap program for the sine-Gordon model can be found in section \[sect5\]. In section \[PTsejt\]. we summarize the conjecture of [@Palmai13] for the Bethe-Yang limit of the diagonal matrix elements of local operators in non-diagonally scattering theories. In section \[sect7\]. we compute the symmetric diagonal form-factors of the operators under consideration upto 3-particle states. In section \[sect8\]. we perform the analytical checks of the conjecture [@Palmai13] upto 3-particle states. In section \[sect9\]. we comment on some subtle points of the conjecture of [@Palmai13]. The body of the paper is closed by our summary and conclusions in section \[sect10\].
The paper contains two appendices, as well. Appendix \[appA\] contains the detailed form of the linear integral equations governing the finite volume dependence of the expectation values of the operators $\Theta$ and $J_\mu.$ In appendix \[appB\] the diagonalization of the soliton transfer-matrix is performed by means of algebraic Bethe-Ansatz. This appendix contains the classification of Bethe-roots, as well.
The models and operators {#sect2}
========================
In this paper we investigate the sine-Gordon and the massive Thirring models. They are given by the well known Lagrangians: $$\label{sG_Lagrangian}
{\cal L}_{SG}= \displaystyle\frac{1}{2}\partial _{\nu }\Phi \partial ^{\nu }\Phi +\displaystyle \alpha_0 \left( \cos \left( \beta \Phi \right)-1 \right), \, \qquad 0<\beta^2<8 \pi,$$ and $$\label{mTh_Lagrangian}
{\cal L}_{MT}= \bar{\Psi }(i\gamma _{\nu }\partial ^{\nu }-m_{0})\Psi -\displaystyle\frac{g}{2}\bar{\Psi }\gamma^{\nu }\Psi \bar{\Psi }\gamma _{\nu }\Psi \,,$$ where $m_0$ and $g$ denote the bare mass and the coupling constant of the massive Thirring model, respectively. In (\[mTh\_Lagrangian\]) $\gamma_\mu$ stand for the $\gamma$-matrices, which satisfy the algebraic relations: $\{\gamma^\mu,\gamma^\nu\}=2 \eta^{\mu \nu}$ with $\eta^{\mu \nu}=\text{diag}(1,-1)$.
The two models are equivalent in their even $U(1)$ charge sector [@s-coleman; @klassme], provided the coupling constants of the two theories are related by the formula: $$\label{gbeta}
1+\frac{g}{4 \pi}=\frac{4 \pi}{\beta^2}.$$ In the sequel we will prefer the following parameterization of the coupling constant $\beta:$ $$\frac{\beta^2}{4 \pi}=\frac{2 \, p}{p+1}, \qquad 0<p\in \mathbb{R}.$$ The ranges $0<p<1$ and $1<p$ correspond to the attractive and repulsive regimes of the theory respectively.
The fundamental particles in the theory are the soliton ($+$) and the antisoliton ($-$) of mass ${\cal M}.$ Their exact S-matrix is well known [@ZamZam] and in terms of the coupling constant $p$ it can be written in the form as follows: $$\label{Smatr}
\begin{split}
{\cal S}_{ab}^{cd}(\theta)=S_0(\theta) \, S_{ab}^{cd}(\theta), \qquad a,b,c,d, \in \{\pm\},
\end{split}$$ where $\theta$ is the relative rapidity of the scattering particles, $S_0(\theta)$ is the soliton-soliton scattering amplitude: $$\label{CHI}
\begin{split}
S_0(\theta)=-e^{i \chi(\theta)}, \qquad \chi(\theta)=\! \int\limits_{0}^{\infty} \! d\omega \,
\frac{\sin(\omega \, \theta)}{\omega} \, \frac{\sinh(\tfrac{(p-1) \,\pi \omega}{2})}{ \cosh( \tfrac{\pi \omega}{2})
\,\sinh(\tfrac{p \, \pi \, \omega}{2})}.
\end{split}$$ The nonzero matrix elements of $S_{ab}^{cd}(\theta)$ in (\[Smatr\]) can be expressed in terms of elementary functions as follows: $$\label{Selem}
\begin{split}
S_{++}^{++}(\theta)&=S_{--}^{--}(\theta)=1, \\
S_{+-}^{+-}(\theta)&=S_{-+}^{-+}(\theta)=B_0(\theta), \\
S_{+-}^{-+}(\theta)&=S_{-+}^{+-}(\theta)=C_0(\theta),
\end{split}$$ where $$\label{B0}
\begin{split}
B_0(\theta)=\frac{\sinh \tfrac{\theta}{p}}{\sinh \tfrac{i \, \pi-\theta}{p}},
\end{split}$$ $$\label{C0}
\begin{split}
C_0(\theta)=\frac{\sinh \tfrac{i \, \pi}{p}}{\sinh \tfrac{i \, \pi-\theta}{p}}.
\end{split}$$ The S-matrix (\[Smatr\]) obeys the Yang-Baxter equation[[^2]]{}: $$\label{YBE}
\begin{split}
{\cal S}_{k_2 k_3}^{j_2 j_3}(\theta_{23}) \, {\cal S}_{k_1 i_3}^{j_1 k_3}(\theta_{13}) \,
{\cal S}_{i_1 i_2}^{k_1 k_2}(\theta_{12})=
{\cal S}_{k_1 k_2}^{j_1 j_2}(\theta_{12}) \,
{\cal S}_{i_1 k_3}^{k_1 j_3}(\theta_{13}) \,
{\cal S}_{i_2 i_3}^{k_2 k_3}(\theta_{23}),
\end{split}$$ with $ \theta_{ij}=\theta_i-\theta_j,$ for $i,j\in\{1,2,3\},$ and it satisfies the properties as follows: $$\begin{aligned}
& \bullet & \text{Parity-symmetry:} \qquad \qquad \qquad
{\cal S}_{ab}^{cd}(\theta)={\cal S}_{ba}^{dc}(\theta), \label{Bose} \\
& \bullet & \text{Time-reversal symmetry:} \qquad \quad \!
{\cal S}_{ab}^{cd}(\theta)={\cal S}_{cd}^{ab}(\theta), \label{Time} \\
& \bullet & \text{Crossing-symmetry:} \qquad \qquad \quad
{\cal S}_{ab}^{cd}(\theta)={\cal S}_{a\bar{d}}^{c\bar{b}}(i \, \pi-\theta), \label{Cross} \\
& \bullet & \text{Unitarity:} \qquad \qquad \qquad \, \,
{\cal S}_{ab}^{ef}(\theta)\, {\cal S}_{ef}^{cd}(-\theta)=\delta_a^c \, \delta_b^d, \label{Unit} \\
& \bullet & \text{Real analyticity:} \qquad \qquad \quad \quad \!
{\cal S}_{ab}^{cd}(\theta)^*={\cal S}_{ab}^{cd}(-\theta^*), \label{Real} \end{aligned}$$ where for any index $a,$ $\bar{a}$ denotes the charge conjugated particle ($\bar{a}=-a$). The charge conjugate of a soliton is an antisoliton and vice versa, thus the charge conjugation matrix acting on the two dimensional vector space spanned by the soliton and the antisoliton, is equal to the first Pauli-matrix: $$\label{Cmatr}
C=\sigma_x=\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}, \quad \text{or equivalently:} \quad C_{ab}=\delta_{a \bar{b}}.$$
In this paper we determine the finite volume expectation values of the operators as follows; the trace of the stress energy tensor: $$\label{THETA}
\Theta=2 \, \alpha_0 \, (1-\tfrac{\beta^2}{8 \pi})\, \cos(\beta \, \Phi).$$ and the $U(1)$ current of the theory: $$\label{Jmu}
J_\mu=\tfrac{\beta}{2 \pi}\, \epsilon_{\mu \nu} \partial^\nu \Phi, \qquad \mu=0,1,$$ where $\epsilon_{\mu \nu}$ denotes the antisymmetric matrix with nonzero entries: $\epsilon_{10}=-\epsilon_{01}=1.$
Both operators correspond to some conserved quantity of the theory and in the subsequent sections their finite volume expectation values will be expressed in terms of the counting-function governing the finite volume spectrum of the theory.
The two operators have different parities under charge conjugation; $\Theta$ is positive, while $J_\mu$ is negative. This property proves to be an important difference between the two operators, when the symmetric diagonal limit of their form-factors are computed.
Finite volume expectation values of $\Theta$ and $J_\mu$ {#sect3}
========================================================
In this section we give the equations, which govern the finite volume dependence of all diagonal form-factors of the trace of the stress energy tensor ($\Theta$) and of the $U(1)$ current ($J_\mu$) of the sine-Gordon theory. The equations for pure solitonic expectation values have been derived in [@En; @En1]. The derivations were based on an integrable lattice regularization of the model, on the so-called light-cone lattice regularization [@ddvlc]. In this section we extend the results of [@En; @En1] from the pure soliton sector to all excited states of the model. To keep the paper within reasonable size, instead of repeating the lattice regularization based derivations we will derive the equations in a more pragmatic way. From [@zamiz] it is well known, that the expectation values of the trace of the stress energy tensor can be computed from the finite volume dependence of the energy of the sandwiching state by the formula as follows: $$\label{Th0}
\begin{split}
\langle \Theta \rangle_L={\cal{M}} \left( \frac{E(\ell)}{\ell}+\frac{d}{d\ell}E(\ell) \right),
\end{split}$$ where $\ell={\cal M} L$ with ${\cal M}$ and $L$ being the soliton mass and the finite volume respectively. This implies that the diagonal form-factors of $\Theta$ can be expressed in terms of certain derivatives of the counting-function of the model [@En1]. In the case of $\Theta,$ the derivatives entering the equations are the derivative with respect to the spectral parameter and the derivative with respect to the dimensionless volume of the model. The computations achieved in the pure soliton sector [@En] imply, that the same derivatives describe the finite volume dependence of the expectation values of the $U(1)$ current, too.
In order to formulate the equations describing the finite volume diagonal form-factors of our interest, we have to recall how the finite volume spectrum of the theory is described in terms of the nonlinear integral equations (NLIE) [@KP1; @ddv92] satisfied by the counting-function. Since we know that the expectation values of our interest can be expressed in terms of certain derivatives of the counting function, we can skip the intermediate lattice versions of the equations, and we can formulate the problem directly in the continuum limit.
Nonlinear integral equations for the counting-function
------------------------------------------------------
The counting-function $Z(\theta)$ is a periodic function on the complex plane with period $i {\pi}(1+p).$ To describe general excited states of the model one needs to know how to determine $Z(\theta)$ for any $\theta$ lying in the whole strip $\left[ -i \frac{\pi}{2}(1+p),i \frac{\pi}{2}(1+p)\right].$ The counting-function satisfies different equations in the different domains of the periodicity strip. In the fundamental domain defined by the strip $|\mbox{Im} \theta|\leq \mbox{min}(p \pi,\pi)$ the continuum limit of the counting-function satisfies the nonlinear-integral equations as follows [@ddv97; @FRT1; @FRT2; @FRT3]: $$\label{DDV}
\begin{split}
Z(\theta)=\ell \sinh \theta \! +\! \sum\limits_{k=1}^{m_H} \chi(\theta-h_k) \! -\! \sum\limits_{k=1}^{m_C} \chi(\theta-c_k) \!
-\!\sum\limits_{k=1}^{m_S} \left(\chi(\theta-y_k+i \eta)\!+\!\chi(\theta-y_k-i \eta) \right)
\\
-\!\sum\limits_{k=1}^{m_W} \chi_{II}(\theta-w_k)\!+ \! \int\limits_{-\infty}^{\infty} \frac{d\theta'}{2 \pi i} G(\theta-\theta'-i\eta) L_{+}(\theta'+i \eta)
- \int\limits_{-\infty}^{\infty} \frac{d\theta'}{2 \pi i} G(\theta-\theta'+i\eta) L_-(\theta'-i \eta),
\end{split}$$ where $$\label{Lpm}
L_{\pm}(\theta)=\ln\left(1+(-1)^\delta \, e^{\pm i \, Z(\theta)} \right),$$ such that the parameter $\delta$ can take values $0$ or $1.$ Its value affect the quantization equations of the objects entering the source terms of the integral equation. In (\[DDV\]), $\chi(\theta)$ is the soliton-soliton scattering phase given by (\[CHI\]) and $G(\theta)$ denotes its derivative. It can be given by the Fourier-integral as follows: $$\label{G}
G(\theta)= \frac{d}{d\theta} \chi(\theta)=\! \int\limits_{-\infty}^{\infty} \! d\omega \,
e^{-i \, \omega \theta} \, \frac{\sinh(\tfrac{(p-1) \,\pi \omega}{2})}{2 \cosh( \tfrac{\pi \omega}{2})
\,\sinh(\tfrac{p \, \pi \, \omega}{2})}.$$ The equations contain the so-called second determination [@ddv97] of $\chi(\theta),$ as well. For any function $f,$ the definition of second determination is different in the attractive ($0<p<1$) and repulsive ($1<p$) regimes of the model: $$\label{fII}
\begin{split}
f_{II}(\theta)=\left\{
\begin{array}{r}
f( \theta)+f(\theta-i \, \pi \,\text{sign}(\text{Im} \theta)), \qquad
\qquad 1<p, \\
f( \theta)-f(\theta-i \, \pi \,p \, \text{sign}(\text{Im} \theta)),
\qquad 0<p<1.
\end{array}\right.
\end{split}$$ For the function $\chi(\theta)$ we provide the concrete functional forms as well [@Fevphd]: $$\label{CHI2}
\begin{split}
\chi_{II}(\theta)=\left\{
\begin{array}{r}
i \, \, \text{sign}(\text{Im} \, \theta)\left( \log\sinh\frac{\theta}{p}-\log \sinh\frac{\theta-i \, \pi \, \text{sign}\, \text{Im} \, \theta}{p} \right), \qquad
\qquad 1<p, \\
i \, \, \text{sign}(\text{Im} \, \theta)\left( \log\left(-\tanh\frac{\theta}{p}\right)+\log \tanh\frac{\theta-i \, \pi \, p \, \text{sign}\, \text{Im} \, \theta}{p} \right),
\qquad 0<p<1.
\end{array}\right.
\end{split}$$ In (\[DDV\]), $\eta$ is an arbitrary positive contour-deformation parameter, which should be in the range $[0,\text{min}(p \pi,\pi,|\mbox{Im} \, c_j|)].$ As we have already mentioned, $\ell$ denotes the dimensionless volume made out of the the soliton mass ${\cal M}$ and of the volume $L$ of the theory by the formula $\ell={\cal M} L.$ All objects entering the source terms in (\[DDV\]) satisfy the equation: $$\label{qo}
\begin{split}
1+(-1)^{\delta}e^{i Z({\frak O})}=0, \qquad {\frak O} \in \{h_k\}_{k=1}^{m_H} \cup \{c_k\}_{k=1}^{m_C} \cup \{w_k\}_{k=1}^{m_W}\cup \{y_k\}_{k=1}^{m_S}.
\end{split}$$ It is useful to classify them as follows [@ddv97]:
- holes: $h_k \in \mathbb{R}, \quad k=1,...,m_H$
- close roots: $c_k \quad k=1,...,m_C$, with $|\mbox{Im} c_k|\leq \mbox{min}(\pi,p \pi)$,
- wide roots: $w_k \quad k=1,...,m_W$, with $\mbox{min}(\pi,p \pi)<|\mbox{Im} w_k|\leq \tfrac{\pi}{2}(1+p)$,
- special objects[[^3]]{} : $y_k \in \mathbb{R}, \quad k=1,...,m_S \,\,$ defined by the equations $1\!+\!(-1)^{\delta}e^{i Z(y_k)}\!=\!0$ with $Z'(y_k)<0.$
Their numbers determine the topological charge $Q$ of the state by the so-called counting-equation: $$\label{counteq}
\begin{split}
Q=m_H-2 m_S-m_C-2 H(p-1) m_W,
\end{split}$$ where here $H(x)$ denotes the Heaviside-function. As a consequence of (\[qo\]) the source objects satisfy the quantization equations as follows: $$\begin{aligned}
&\bullet& \, \text{holes:} \qquad Z(h_k)=2\pi \, I_{h_k}, \qquad I_{h_k} \in \mathbb{Z}+\tfrac{1+\delta}{2}, \qquad k=1,..,m_H, \label{Hkvant}\\
&\bullet& \, \text{close roots:} \qquad Z(c_k)=2\pi \, I_{c_k}, \qquad I_{c_k} \in \mathbb{Z}+\tfrac{1+\delta}{2}, \qquad k=1,..,m_C, \label{Ckvant} \\
&\bullet& \, \text{wide roots:} \qquad Z(w_k)=2\pi \, I_{w_k}, \qquad I_{w_k} \in \mathbb{Z}+\tfrac{1+\delta}{2}, \qquad k=1,..,m_W, \label{Wkvant} \\
&\bullet& \, \text{special objects:} \qquad Z(y_k)=2\pi \, I_{y_k}, \qquad I_{y_k} \in \mathbb{Z}+\tfrac{1+\delta}{2}, \qquad k=1,..,m_S. \label{Skvant}\end{aligned}$$ From this list one can see that the actual value of the parameter $\delta \in \{0,1\}$ determines whether the source objects are quantized by integer or half integer quantum numbers. It was shown in [@FRT1; @FRT2; @FRT3], that not all choices of $\delta$ are possible to describe properly the states of the sine-Gordon or of the Massive Thirring model. To describe the proper states of these quantum field theories the following selection rules have to be satisfied by the parameter $\delta:$ $$\begin{aligned}
&\bullet& \quad \frac{Q+\delta+M_{sc}}{2} \in \mathbb{Z}, \qquad \text{sine-Gordon,} \qquad \qquad\qquad \qquad \qquad\qquad \label{SGkvant} \\
%\label{MTkvant}
&\bullet& \quad \frac{\delta+M_{sc}}{2} \in \mathbb{Z}, \qquad \text{massive Thirring,} \qquad \qquad\qquad \qquad \qquad\qquad \label{MTkvant}\end{aligned}$$ where here $M_{sc}$ stands for the number of self-conjugate roots, which are such wide roots, whose imaginary parts are fixed by the periodicity of $Z(\theta)$ to $i\, \tfrac{\pi}{2}(1+p).$
In order to be able to impose the quantization equations (\[Wkvant\]) for the wide roots, the integral representation of $Z(\theta)$ must be known in the strip $\text{min}(p \, \pi,\pi)<\text{Im} \, \theta\leq \tfrac{\pi}{2}(1+p),$ as well. In this “wide-root domain” $Z(\theta)$ is given by the equations as follows [@ddv97; @FRT1; @FRT2; @FRT3]: $$\label{DDVII}
\begin{split}
Z(\theta)=\ell \sinh_{II} (\theta) \! +{\cal D}_{II}(\theta)+\sum\limits_{\alpha=\pm} \, \alpha
\! \int\limits_{-\infty}^{\infty} \frac{d\theta'}{2 \pi i} G_{II}(\theta-\theta'-i \, \alpha \, \eta) L_{\alpha}(\theta'+i \, \alpha \, \eta),
\end{split}$$ where ${\cal D}_{II}(\theta)$ is the second determination (\[fII\]) of the source term function of (\[DDV\]): $$\label{calD}
\begin{split}
{\cal D}(\theta)\!\!=\!\!\sum\limits_{k=1}^{m_H} \chi(\theta\!-\!h_k) \! -\! \sum\limits_{k=1}^{m_C} \!\chi(\theta\!-\!c_k) \!
-\!\!\sum\limits_{k=1}^{m_S} \left(\chi(\theta\!-\!y_k\!+\!i \eta)\!+\!\chi(\theta\!-\!y_k\!-\!i \eta) \right)\!-\!\sum\limits_{k=1}^{m_W} \chi_{II}(\theta\!-\!w_k).
\end{split}$$ The energy and momentum of the model can be expressed in terms of the solution of the nonlinear integral equations by the following formulas [@ddv97; @FRT1; @FRT2; @FRT3]: $$\label{EL}
\begin{split}
E(L)\!\!=\!\!{\cal M} \left( \sum\limits_{k=1}^{m_H} \cosh(h_k) \! -\! \sum\limits_{k=1}^{m_C} \!\cosh(c_k) \!
-\!\!\sum\limits_{k=1}^{m_S} \left(\cosh(y_k\!+\!i \eta)\!+\!\cosh(\!y_k\!-\!i \eta) \right)\!-\!\!\sum\limits_{k=1}^{m_W} \cosh_{II}(w_k) - \right.\\
\left. \int\limits_{-\infty}^{\infty} \! \frac{d\theta}{2 \pi i} \sinh(\theta+i\eta) L_{+}(\theta+i \eta)
+ \! \int\limits_{-\infty}^{\infty} \! \frac{d\theta}{2 \pi i} \sinh(\theta-i\eta) L_-(\theta-i \eta)
\right),
\end{split}$$ $$\label{PL}
\begin{split}
P(L)\!\!=\!\!{\cal M} \left( \sum\limits_{k=1}^{m_H} \sinh(h_k) \! -\! \sum\limits_{k=1}^{m_C} \!\sinh(c_k) \!
-\!\!\sum\limits_{k=1}^{m_S} \left(\sinh(y_k\!+\!i \eta)\!+\!\sinh(\!y_k\!-\!i \eta) \right)\!-\!\!\sum\limits_{k=1}^{m_W} \sinh_{II}(w_k) - \right.\\
\left. \int\limits_{-\infty}^{\infty} \! \frac{d\theta}{2 \pi i} \cosh(\theta+i\eta) L_{+}(\theta+i \eta)
+ \! \int\limits_{-\infty}^{\infty} \! \frac{d\theta}{2 \pi i} \cosh(\theta-i\eta) L_-(\theta-i \eta)
\right).
\end{split}$$
Expectation values of $\Theta$
------------------------------
The computations of finite volume expectation values of the trace of the stress energy tensor goes analogously to the former computations done in purely elastic scattering theories [@PST14]. Formula (\[Th0\]) implies that the finite volume expectation values of the trace of the stress energy tensor $\Theta$ can be expressed in terms of the $\theta$ and $\ell$ derivatives of $Z(\theta).$ By differentiating the equations (\[DDV\])-(\[DDVII\]) it is easy to show that these derivatives satisfy linear integral equations with kernels containing the counting-equation itself [@En; @En1].
We introduce two functions with related sets of discrete variables by the definitions as follows: $$\label{Gd}
\begin{split}
{\cal G}_d(\theta)&=Z'(\theta), \qquad \\
X_{d,k}^{(h)}&=\frac{{\cal G}_d(h_k)}{Z'(h_k)}=1, \qquad k=1,...,m_H, \\
X_{d,k}^{(c)}&=\frac{{\cal G}_d(c_k)}{Z'(c_k)}=1, \qquad k=1,...,m_C, \\
X_{d,k}^{(y)}&=\frac{{\cal G}_d(y_k)}{Z'(y_k)}=1, \qquad k=1,...,m_S, \\
X_{d,k}^{(w)}&=\frac{{\cal G}_d(w_k)}{Z'(w_k)}=1, \qquad k=1,...,m_W.
\end{split}$$ and $$\label{Gl}
\begin{split}
{\cal G}_\ell(\theta)&=\frac{d}{d\ell}Z(\theta|\ell), \qquad \\
X_{\ell,k}^{(h)}&=\frac{{\cal G}_\ell(h_k)}{Z'(h_k)}=-h_k'(\ell), \qquad k=1,...,m_H, \\
X_{\ell,k}^{(c)}&=\frac{{\cal G}_\ell(c_k)}{Z'(c_k)}=-c_k'(\ell), \qquad k=1,...,m_C, \\
X_{\ell,k}^{(y)}&=\frac{{\cal G}_\ell(y_k)}{Z'(y_k)}=-y_k'(\ell), \qquad k=1,...,m_S, \\
X_{\ell,k}^{(w)}&=\frac{{\cal G}_\ell(w_k)}{Z'(w_k)}=-w_k'(\ell), \qquad k=1,...,m_W.
\end{split}$$ Taking the appropriate derivatives of the NLIE (\[DDV\])-(\[DDVII\]) it can be shown, that the variables in (\[Gd\]) and in (\[Gl\]) satisfy sets of linear integral equations. We relegated these equations to appendix \[appA\], where their explicit form is given by the formulas (\[ujset\])-(\[bfGdef\]).
With the help of (\[Th0\]) it can be shown, that he finite volume expectation value of $\Theta$ in a state described by the NLIE (\[DDV\])-(\[DDVII\]) can be expressed in terms of the variables (\[Gd\]) and (\[Gl\]) by the following formula: $$\label{THexp}
\begin{split}
\langle \Theta \rangle_L=\langle \Theta \rangle_\infty+{\cal M}^2 \, \Theta_{rest}(\ell),
\end{split}$$ where $\langle \Theta \rangle_\infty$ stands for the infinite volume “bulk” vacuum expectation value [@ddvaft; @ddv95]: $$\label{THinf}
\begin{split}
\langle \Theta \rangle_\infty=-\frac{{\cal M}^2}{4} \tan\left(\tfrac{p \pi}{4}\right),
\end{split}$$ and $\Theta_{rest}(\ell)$ denotes the dimensionless part of the rest of the expectation value. It is given by the formula: $$\label{THrest}
\begin{split}
\Theta_{rest}(\ell)\!=\!\sum\limits_{k=1}^{m_H} \!\! \left(\cosh h_k \frac{X_{d,k}^{(h)}}{\ell}\!-\!\sinh h_k X_{\ell,k}^{(h)}\right)
\!-\!\sum\limits_{k=1}^{m_C} \!\! \left(\cosh c_k \frac{X_{d,k}^{(c)}}{\ell}\!-\!\sinh c_k X_{\ell,k}^{(c)}\right)- \\
\!-\!\sum\limits_{k=1}^{m_S} \!\! \left(\left(\cosh (y_k+i \eta)+\cosh (y_k-i \eta)\right) \frac{X_{d,k}^{(y)}}{\ell}\!-\!\left(\sinh (y_k+i \eta)+\sinh (y_k-i \eta)\right) X_{\ell,k}^{(y)}\right) \\
\!-\!\sum\limits_{k=1}^{m_W} \!\! \left(\cosh_{II}(w_k) \frac{X_{d,k}^{(w)}}{\ell}\!-\!\sinh_{II} (w_k) X_{\ell,k}^{(w)}\right)+ \\
\sum\limits_{\alpha=\pm} \! \int\limits_{-\infty}^{\infty} \! \frac{d\theta}{2 \pi} \!
\left[\cosh(\theta+i \, \alpha \, \eta) \, \frac{{\cal G}_d(\theta+i \, \alpha \, \eta)}{\ell}-\sinh(\theta+i \, \alpha \, \eta) \, {\cal G}_\ell(\theta+i \, \alpha \, \eta) \right]
{\cal F}_{\alpha}(\theta+i \, \alpha \, \eta),
\end{split}$$ where ${\cal F}_\pm(\theta)$ stands for the nonlinear combinations: $$\label{calF}
\begin{split}
{\cal F}_{\pm}(\theta)=\frac{(-1)^\delta \, e^{\pm i \, Z(\theta)}}
{1+(-1)^\delta \, e^{\pm i \, Z(\theta)}}.
\end{split}$$
Expectation values of $J_\mu$
-----------------------------
The finite volume expectation values of the $U(1)$ current can be derived from the light-cone lattice regularization [@ddvlc] of the model. In this way the expectation values of $J_\mu$ between pure soliton states have been determined in [@En]. Nevertheless, the computations of [@En] can be easily extended to all excited states of the model. Here, we skip the lengthy, but quite straightforward computations and present only the final result. As the pure soliton results of [@En] suggest, the expectation values of $J_0$ and $J_1$ can be expressed in terms of the set of variables of (\[Gd\]) and of (\[Gl\]), respectively: $$\label{J0exp}
\begin{split}
\langle J_0 \rangle_L=\frac{1}{L}\left\{ \sum\limits_{j=1}^{m_H} X_{d,j}^{(h)} -
2 \sum\limits_{j=1}^{m_S} X_{d,j}^{(y)}-\sum\limits_{j=1}^{m_C} X_{d,j}^{(c)}-
2 \, H(p-1) \, \sum\limits_{j=1}^{m_W} X_{d,j}^{(w)}- \right. \\
\left. \sum\limits_{\alpha=\pm} \int\limits_{-\infty}^{\infty} \! \frac{d\theta}{2 \pi} \,
{\cal G}_d(\theta+i \, \alpha \, \eta) \, {\cal F}_\alpha(\theta+i \, \alpha \,\eta)
\right\},
\end{split}$$ $$\label{J1exp}
\begin{split}
\langle J_1 \rangle_L={\cal M}\left\{ \sum\limits_{j=1}^{m_H} X_{\ell,j}^{(h)} -
2 \sum\limits_{j=1}^{m_S} X_{\ell,j}^{(y)}-\sum\limits_{j=1}^{m_C} X_{\ell,j}^{(c)}-
2 \, H(p-1) \, \sum\limits_{j=1}^{m_W} X_{\ell,j}^{(w)}- \right. \\
\left. \sum\limits_{\alpha=\pm} \int\limits_{-\infty}^{\infty} \! \frac{d\theta}{2 \pi} \,
{\cal G}_\ell(\theta+i \, \alpha \, \eta) \, {\cal F}_\alpha(\theta+i \, \alpha \,\eta)
\right\}.
\end{split}$$ Here $H(x)$ is the Heaviside function. Using the definitions (\[Gd\]) and the counting equation (\[counteq\]), it is easy to show that formula (\[J0exp\]) gives the correct result $\langle J_0 \rangle_L=\tfrac{Q}{L}$ for the finite volume expectation values of $J_0$ in sandwiching states with topological charge $Q.$
Large volume solution {#sect4}
=====================
In this section we provide exact formulas for the Bethe-Yang limit of the expectation values of the trace of the stress energy tensor and of the $U(1)$ current in the repulsive ($1<p$) regime of the sine-Gordon model[[^4]]{}. The reason why we restrict ourselves to the repulsive regime is that in this regime the correspondence between the Bethe-roots entering the NLIE (\[DDV\]) and magnonic Bethe-roots of the Bethe-Yang equations (\[ABAE\]) is quite direct. In the attractive regime the correspondence is much more complicated and more indirect.
The first step to make a correspondence between the source objects or the Bethe-roots of the NLIE and the roots of (\[ABAE\]), is to find the relation between the holes of the NLIE (\[DDV\]) and the rapidity of physical particles entering the magnonic part of the Bethe-Yang equations (\[ABAE\]). It is well known in the literature [@ddv97] that the holes in the NLIE description describe the rapidities of the physical particles in the large volume limit ($\{h_j\}=\{\theta_j\}$). Then one has to know what kind of complexes the roots of the NLIE fall into, when the infinite volume limit is taken. In the repulsive regime these complexes are as follows [@Viallet]: $$\begin{aligned}
&\bullet& \text{2-strings:} \quad s^{(2)}_j=s_j\pm i \, \tfrac{\pi}{2}, \qquad \text{with:} \quad s_j\in{\mathbb R}, \quad j=1,...,n_2, \nonumber \\
%&\bullet& \text{quartets:} \quad q_j=\{q^{(1)}_j \!\pm\! i \tfrac{\pi}{2}, q^{(2)}_j \! \pm\! i \tfrac{\pi}{2}\}, \quad \text{with:} \quad
%|\text{Im}\, q^{(a)}_j|\leq \tfrac{\pi}{2}, \qquad a\!=\!1,2, \quad j\!=\!1,...,n_4, \nonumber \\
&\bullet& \text{quartets:} \quad q_j=\{q^{(\pm)}_j \!\pm\! i \tfrac{\pi}{2}\}, \quad \text{with:} \quad q^{(+)}_j=(q^{(-)}_j)^*, \quad
|\text{Im}\, q^{(\pm)}_j|\leq \tfrac{\pi}{2}, \quad \, j\!=\!1,...,n_4, \nonumber \\
&\bullet& \text{wide-roots:} \quad w_j, \quad \text{with:} \quad \pi<|\text{Im}\, w_j|\leq\tfrac{(1+p)\, \pi}{2}, \quad j\!=\!1,...,m_w,
\label{quartett}\end{aligned}$$ such that wide-roots either form complex conjugate pairs or they are self-conjugate roots with fixed imaginary part: $\text{Im} \, w^{(sc)}_j=\frac{(1+p) \, \pi}{2}.$ From this classification, one can see that only the close-roots fall into special configurations in the infinite volume limit ($m_C=2n_2+4n_4$). Namely, they form either quartets or 2-strings, where the latter can be thought of as degenerate quartets. Here we note that in the $\ell \to \infty$ limit there are no special objects, so they do not enter the expressions in this limit.
The counting equation (\[counteq\]) tells us how these root configurations act on the topological charge of a state:
- a 2-string decreases the charge by 2,
- a quartet decreases the charge by 4,
- a wide-root decreases the charge by 2.
On the other hand each root of the magnonic part of the Bethe-Yang equations decrease the topological charge of a state by 2 units and as it was mentioned in appendix \[appABA2\] the roots of these equations form conjugate pairs with respect to the line $\text{Im} \, z=\tfrac{\pi}{2}.$ These suggest the following identification between the $\ell \to \infty$ complexes in (\[quartett\]) and the different types of roots of the magnonic part of the Bethe-Yang equations given in (\[rootclassR\]):
- Real-roots of (\[rootclassR\]) correspond to 2-strings in (\[quartett\]), such that the real part of a real-root is equal to the center of the corresponding 2-string: $\lambda_j-i \tfrac{\pi}{2}=s_j.$
- a close-pair $\lambda^{(\pm)}_j$ in (\[rootclassR\]) corresponds to a quartet in (\[quartett\]), such that the positions of the close-pair are given by complex conjugate pair describing the quartet: $\{\lambda^{(\pm)}_j-i \tfrac{\pi}{2}\}=\{q_j^{(\pm)}\}.$
- wide-roots in (\[rootclassR\]) correspond to wide roots of (\[quartett\]), such that: $\lambda^{wide}_j-i \tfrac{\pi}{2}=w_j-i \,\text{sign}(\text{Im}\, w_j) \,\tfrac{\pi}{2}.$
With this correspondence the $\ell \to \infty$ limit of the NLIE can be mapped to the equations (\[ABAE\]). This proves to be important in finding an exact formula expressed in terms of the roots of (\[ABAE\]) for the Bethe-Yang limit of the diagonal form-factors of $\Theta$ and $J_\mu.$
To find the leading order large volume solution of the diagonal form-factors of $\Theta$ and $J_\mu,$ one should recognize that the integral terms in (\[THexp\]), (\[THrest\]), (\[J0exp\]) and (\[J1exp\]) are exponentially small in the volume, and so negligible at leading order. Consequently, the only task is to determine the Bethe-Yang limit of the discrete variables $X_{d,j}$ and $X_{\ell,j}.$ They are solutions of the equations (\[ujset\])-(\[Qyk\]). For the first sight, it does not seem to be easy to find the general solutions of these equations in the large volume limit, but the relations of these $X$-variables to the $\theta$ and $\ell$ derivatives of the counting-function given in (\[Gd\]) and in (\[Gl\]), makes it quite easy to find the required solutions.
First let us consider the variables related to the $\ell$ derivative of $Z(\theta)$ in (\[Gl\]). Then using (\[Gl\]), for a complex root[[^5]]{} $u_j$ the corresponding $X$-variable can be written as: $$\label{Xul}
\begin{split}
X^{(u)}_{\ell,j}=-u_j'(\ell)=-\sum\limits_{k=1}^{m_H} \frac{\partial u_j}{\partial h_k} \, h_k'(\ell)
\stackrel{\ell \to \infty}{\approx} \sum\limits_{k=1}^{m_H} \frac{\partial \lambda_j}{\partial h_k} \, X^{(h)}_{\ell,j},
\end{split}$$ where we used (\[Gl\]) and exploited the large volume correspondence between the Bethe-roots of the NLIE (\[DDV\]) and magnonic the Bethe-roots of (\[ABAE\]). The formula (\[Xul\]) expresses the complex root’s $X$-variables in terms of those of the holes in the large volume limit. Then the large $\ell$ solution goes as follows: One should insert (\[Xul\]) into (\[Qhk\]) taken at $\nu=\ell,$ such that the integral terms are neglected because they are exponentially small in volume. This way one gets a closed discrete set of linear equations for the variables $X^{(h)}_{\ell,j}.$
The equations through (\[Xul\]) contain the derivative matrix $\frac{\partial \lambda_j}{\partial h_k},$ which can be computed by differentiating the logarithm of the equations (\[ABAE\]). The final result takes the form: $$\label{dldh}
\begin{split}
\frac{\partial \lambda_j}{\partial h_k}=\sum\limits_{s=1}^r \psi_{js}^{-1} \, V_{s k}, \qquad
V_{s k}=(\ln B_0)'(\lambda_s-h_k), \qquad k=1,...,m_H, \quad
s=1,..,r,
\end{split}$$ where we exploited the infinite volume correspondence between the holes and the rapidities of the physical particles $\{h_j\}_{j=1}^{m_H} \leftrightarrow \{\theta_j\}_{j=1}^n$ and we introduced $\psi,$ a symmetric $r \times r$ matrix with the definition as follows: $$\label{kispsi}
\begin{split}
\psi_{jk}=z'(\lambda_j) \, \delta_{jk}+(\ln E_0)'(\lambda_j-\lambda_k), \qquad j,k=1,..,r,
\end{split}$$ with $$\label{kisz}
\begin{split}
z(\lambda)=\sum\limits_{k=1}^{m_H} \ln B_0(\lambda-h_k)-\sum\limits_{k=1}^r \ln E_0(\lambda-\lambda_k), \qquad
E_0(\lambda)=\frac{B_0(\lambda)}{B_0(-\lambda)}.
\end{split}$$ Then equations (\[Qhk\]) with $\nu=\ell$ for $X_{\ell,j}^{(h)}$ can be written in the repulsive regime as follows: $$\label{Xhdet1}
\begin{split}
\sum\limits_{k=1}^{m_H} \Phi_{jk} \,X_{\ell,k}^{(h)}=\sinh h_j, \qquad j=1,...,m_H,
\end{split}$$ where $\Phi_{jk}$ is the Gaudin-matrix of the physical particles in the state described by the magnonic roots $\{\lambda_j\}_{j=1}^{r}:$ $$\label{Phi}
\begin{split}
\Phi_{jk}=\left\{ \begin{array}{l}\ell \cosh h_j+\sum\limits_{s=1 \atop s\neq j}^{m_H} {\tilde G}_{js}, \qquad j=k, \\
-{\tilde G}_{jk}, \qquad \qquad \qquad \qquad \! \! \! j\neq k,
\end{array} \right.
\end{split}$$ $$\label{tildeG}
\begin{split}
{\tilde G}_{jk}=G_{jk}+\tfrac{1}{i} \sum\limits_{s,q=1}^{r} V_{sj}\, \psi_{sq}^{-1} \, V_{qk}, \qquad
j,k=1,..,m_H.
\end{split}$$ Here $V_{jk}$ is defined in (\[dldh\]) and we introduced the short notation: $G_{jk}=G(h_j-h_k).$ Now it is easy to solve (\[Xhdet1\]) for $X_{\ell,j}^{(h)}:$ $$\label{Xhres}
\begin{split}
X_{\ell,j}^{(h)}=\sum\limits_{k=1}^{m_H} \Phi_{jk}^{-1} \, \sinh h_k, \qquad j=1,..,m_H.
\end{split}$$ Then using (\[Xul\]) and (\[dldh\]), the $X$-variables of the complex roots can also be obtained from (\[Xhres\]): $$\label{Xures1}
\begin{split}
X^{(u)}_{\ell,j}=\sum\limits_{k=1}^{m_H} \sum\limits_{s=1}^r \psi_{js}^{-1} V_{sk} \sum\limits_{k'=1}^{m_H} \Phi_{kk'}^{-1} \, \sinh h_{k'}, \qquad j=1,..,r.
\end{split}$$
As for the $d$-type of $X$-variables, from (\[Gd\]) we know exactly that the value of each of them is exactly 1. Nevertheless, with the help of the large $\ell$ solution of (\[Qhk\]), this value can be expressed in a more complicated way, as well: $$\label{Xdhres}
\begin{split}
X_{d,j}^{(h)}=\sum\limits_{k=1}^{m_H} \Phi_{jk}^{-1} \,\ell \cosh h_k, \qquad j=1,..,m_H,
\end{split}$$ $$\label{Xdures1}
\begin{split}
X^{(u)}_{d,j}=\sum\limits_{k=1}^{m_H} \sum\limits_{s=1}^r \psi_{js}^{-1} V_{sk} \sum\limits_{k'=1}^{m_H} \Phi_{kk'}^{-1} \, \ell \, \cosh h_{k'}, \qquad j=1,..,r.
\end{split}$$ For the derivation of the second expression the following discrete set of equations should have been used as well: $$\label{identity}
\begin{split}
\sum\limits_{j=1}^{r} \sum\limits_{k=1}^{m_H} V_{jk} \sum\limits_{q=1}^r \psi_{jq}^{-1} V_{qs}=
\sum\limits_{j=1}^r V_{js}, \qquad s=1,...,m_H,
\end{split}$$ which can be derived from the logarithmic derivative of the (\[ABAE\]). The point in making the simple to complicated is that in this way the solutions for both subscript $\nu=\ell,d$ can be written on equal footing: $$\label{Xnuhres}
\begin{split}
X_{\nu,j}^{(h)}=\sum\limits_{k=1}^{m_H} \Phi_{jk}^{-1} \, f_{\nu} (h_k), \qquad
\nu\in\{d,\ell\},\qquad j=1,..,m_H,
\end{split}$$ $$\label{Xnuures}
\begin{split}
X^{(u)}_{d,j}=\sum\limits_{k=1}^{m_H} \sum\limits_{s=1}^r \psi_{js}^{-1} V_{sk} \sum\limits_{k'=1}^{m_H} \Phi_{kk'}^{-1} \, f_\nu( h_{k'}), \quad \nu\in\{d,\ell\},\qquad j=1,..,r,
\end{split}$$ where $f_\nu(\theta)$ is the source term of the linear problem (\[Gnu\]). It is given in (\[fd\]) and (\[fl\]). Inserting the large volume solutions (\[Xnuhres\]), (\[Xnuures\]) into the expectation value formulas: (\[THrest\]), (\[J0exp\]) and (\[J1exp\]), one ends up with the large volume solutions as follows: $$\label{THrest8}
\begin{split}
\Theta_{rest}(\ell)_{BY}=\sum\limits_{j,k=1}^{m_H} \left(
\cosh h_j \, \Phi_{jk}^{-1} \cosh h_k -\sinh h_j \, \Phi_{jk}^{-1} \sinh h_k\right),
\end{split}$$ $$\label{J08}
\begin{split}
\langle J_0 \rangle_{BY}={\cal M} \sum\limits_{k,s=1}^{m_H} \Phi_{sk}^{-1} \, \cosh h_k \,
\left( 1-2 \, \sum\limits_{j,q=1}^r \psi_{jq}^{-1} V_{qs}\right),
\end{split}$$ $$\label{J18}
\begin{split}
\langle J_1 \rangle_{BY}={\cal M} \sum\limits_{k,s=1}^{m_H} \Phi_{sk}^{-1} \, \sinh h_k \,
\left( 1-2 \, \sum\limits_{j,q=1}^r \psi_{jq}^{-1} V_{qs}\right).
\end{split}$$
In the computations we have done so far, it was not necessary to impose the quantization equations (\[Hkvant\]) for the holes. Thus the formulas above can be considered as analytical functions of the $m_H$ pieces of holes (rapidities). Nevertheless, if one would like to get the Bethe-Yang limit of the expectation values, formulas (\[THrest8\]), (\[J08\]) and (\[J18\]) must be taken at the solutions of Bethe-Yang limit of the quantization equations (\[Hkvant\]), which takes the well-known form: $$\label{BYhole}
\begin{split}
e^{i \ell \sinh \tilde{h}_j} \, \Lambda(\tilde{h}_j|\vec{\tilde{h}})=1, \qquad j=1,..,m_H,
\end{split}$$ where $\tilde{h}_j$ denotes the solutions of the Bethe-Yang equations and $\Lambda(\theta|\vec{\tilde{h}})$ denotes that eigenvalue (\[eival\]) of the soliton transfer matrix (\[trans\]), which corresponds to the sandwiching state.
Form-factors in the sine-Gordon theory {#sect5}
======================================
Having the exact formulas (\[THrest8\]), (\[J08\]) and (\[J18\]), for the Bethe-Yang limit of the expectation values of our operators, we would like to check analytically the conjecture of [@Palmai13] for the Bethe-Yang limit of the diagonal matrix elements of local operators in a non-diagonally scattering theory. To do so, we need the infinite volume form-factors of the theory. There are several ways to determine these form-factors. The earliest construction is written in the seminal work of Smirnov [@Smirnov]. Later other constructions arose in the literature, like Lukyanov’s free-field representation [@LUK1; @LUK2] and the off-shell Bethe-Ansatz based method of [@BABK1; @BABK2]. In this section we summarize the axiomatic equations satisfied by the form-factors of local operators in an integrable quantum field theory.
Let ${\cal O}(x,t)$ a local operator of the theory. Then its matrix elements between asymptotic multiparticle states is given by [@Smirnov]: $$\label{FFxt}
\begin{split}
{}^{(in)}\langle \gamma_1,b_1;...;\gamma_m,b_m|{\cal O}(x,t)|\beta_1,a_1;...;\beta_n,a_n \rangle^{(in)}=
e^{i \, t \, (E_\gamma-E_\beta)-i\, x\, (P_\gamma-P_\beta) }\,\times \\
F^{\cal O}_{\bar{b}_m...\bar{b}_1\, a_1...a_n}(\gamma_m+i \, \pi-i \epsilon_m,...,\gamma_1+i \, \pi-i\, \epsilon_1,
\beta_1,...,\beta_n)+\text{Dirac-delta terms},
\end{split}$$ where the form-factors of the local operator ${\cal O}$ is denoted by $F^{\cal O},$ $\epsilon_j$s are positive infinitesimal numbers and the orderings $\beta_n<...<\beta_2<\beta_1,$ $\gamma_1<\gamma_2<...<\gamma_m$ are meant in the $in$ states. The Latin and Greek letters denote the polarizations and rapidities of the sandwiching multisoliton states, respectively. Thus $a_j,b_j\in\{\pm\},$ and $E_\gamma, E_\beta, P_\gamma, P_\beta$ denote the energies and the momenta of the corresponding states: $$\label{EPgb}
\begin{split}
E_\gamma&=\sum\limits_{j=1}^m {\cal M} \cosh \gamma_j, \qquad E_\beta=\sum\limits_{j=1}^n {\cal M} \cosh \beta_j, \\
P_\gamma&=\sum\limits_{j=1}^m {\cal M} \sinh \gamma_j, \qquad P_\beta=\sum\limits_{j=1}^n {\cal M} \sinh \beta_j.
\end{split}$$ We choose the normalization for the scalar product of states in infinite volume as follows: $$\label{scalar8}
\begin{split}
{}^{(in)}\langle \gamma_1,b_1;...;\gamma_n,b_n|\beta_1,a_1;...;\beta_n,a_n \rangle^{(in)}=(2 \pi)^n \, \prod\limits_{j=1}^n
\delta_{b_j a_j} \, \delta(\gamma_j-\beta_j).
\end{split}$$ In this convention the form-factors $F^{{\cal O}}$ of the operator ${\cal O}(x,t)$ satisfy the the following axioms [@Smirnov]: $$\label{ax1}
\begin{split}
F^{\cal O}_{a_1...a_n}(\theta_1+\theta,...,\theta_n+\theta)=e^{s({\cal O}) \theta} \, F^{\cal O}_{a_1...a_n}(\theta_1,...,\theta_n),
\end{split}$$ where $s({\cal O})$ is the Lorentz-spin of ${\cal O}.$ $$\label{ax2}
\begin{split}
F^{\cal O}_{...a_j a_{j+1}...}(...,\theta_j,\theta_{j+1},...)={\cal S}_{a_j a_{j+1}}^{b_{j+1} b_j}(\theta_j-\theta_{j+1}) \, F^{\cal O}_{...b_{j} b_{j+1}...}(...,\theta_{j+1},\theta_{j},...),
\end{split}$$ $$\label{ax3}
\begin{split}
F^{\cal O}_{a_1 a_2...a_n}(\theta_1+2 \pi \, i,...,\theta_n)=e^{2 \pi \, i \,\omega({\cal O})}\,
F^{\cal O}_{a_2...a_n a_1}(\theta_2,...,\theta_n,\theta_1),
\end{split}$$ where $\omega({\cal O})$ denotes the mutual locality index between ${\cal O}$ and the asymptotic field which creates the solitons. $$\label{ax4}
\begin{split}
F^{\cal O}_{ab u_1...u_n}(\theta+i\, \pi+\epsilon,\theta, \theta_1,...,\theta_n)\stackrel{\epsilon \to 0}{\simeq}
\frac{i}{\epsilon} \bigg\{
C_{ab} \, F^{\cal O}_{u_1...u_n}( \theta_1,...,\theta_n)-\\
e^{2 \pi \, i \,\omega({\cal O})} \!
\sum\limits_{v_1,..,v_n=\pm}{\cal T}_{b}^{\bar{a}}(\theta|\theta_1,..,\theta_n)_{u_1...u_n}^{v_1...v_n} \,
F^{\cal O}_{v_1...v_n}(\theta_1,...,\theta_n)
\bigg\},
\end{split}$$ where ${\cal T}$ denotes the soliton monodromy matrix defined in (\[monodr\]), and $C_{ab}$ is the charge conjugation matrix (\[Cmatr\]). In this paper we will focus on the repulsive regime of the sine-Gordon theory, where there are no soliton-antisoliton bound states in the spectrum. This is why we skipped to present the dynamical singularity axiom, which relates the form-factors of bound states to those of its constituents.
We just remark that for the operators of our interest the mutual locality index is zero: $\omega(\Theta)=\omega(J_\mu)=0.$
The Pálmai-Takács conjecture {#PTsejt}
============================
In this section we summarize the conjecture of Pálmai and Takács [@Palmai13] for the Bethe-Yang limit of the diagonal matrix elements of local operators in a non-diagonally scattering theory. By Bethe-Yang limit we mean those terms of the large volume expansion which are polynomials in the inverse of the volume. Namely, the exponentially small in volume corrections are neglected from the exact result.
In finite volume the particle rapidities become quantized. The quantization equations which account for the polynomial in the inverse of the large volume correctons are called the Bethe-Yang equations. In a non-diagonally scattering theory at large volume, the amplitudes describing the “color part” of the multisoliton eigenstates of the Hamiltonian are eigenvectors of the soliton transfer matrix (\[trans\]). They form a complete normalized basis on the space of “color” degrees of freedom of the wave function. On an $n$-particle state it can be formulated as follows: $$\label{bazis}
\begin{split}
\tau(\theta|\vec{\theta})_{a_1...a_n}^{b_1...b_n} \, \Psi^{(t)}(\vec{\theta})_{b_1...b_n}\,
=\Lambda^{(t)}(\theta|\vec{\theta}) \,
\Psi^{(t)}(\vec{\theta})_{a_1...a_n} \qquad t=1,..,2^n,
\end{split}$$ $$\label{ortog}
\begin{split}
\sum\limits_{a_1,...,a_n=\pm}\Psi^{(s)}(\vec{\theta})_{a_1...a_n} \Psi^{(t)*}(\vec{\theta})_{a_1...a_n}&=\delta_{st}, \\
\sum\limits_{t=1}^{2^n}\Psi^{(t)}(\vec{\theta})_{a_1...a_n} \Psi^{(t)*}(\vec{\theta})_{b_1...b_n}&=
\prod\limits_{j=1}^n \delta_{a_j b_j},
\end{split}$$ where for short we use the notation $\vec{\theta}=\{\theta_1,...,\theta_n\}$ and $\Lambda^{(t)}(\theta|\vec{\theta})$ stands for the eigenvalue of the $t$th eigenstate. This eigenvalue can be obtained by the Algebraic Bethe method [@FST79] summarized in appendix \[appB\]. Its expression in terms of the magnonic Bethe-roots is given by (\[eival\]).
In the language of the soliton transfer matrix (\[trans\]), for an $n$-particle state the Bethe-Yang quantization equations take the form: $$\label{BY1}
\begin{split}
e^{i \ell \sinh \tilde{\theta}_j} \, \Lambda^{(t)}(\tilde{\theta_j}|\vec{\tilde{\theta}})=1, \qquad j=1,..,n,
\end{split}$$ where we introduced the notation that the set $\{\theta_j\}_{j=1}^n,$ means an arbitrary unquantized set of rapidities, while the set with tilde $\{\tilde{\theta}_j\}_{j=1}^n,$ denotes the set of rapidities satisfying the Bethe-Yang equations (\[BY1\]). It is more common to rephrase (\[BY1\]) in its logarithmic form. To do so, first one has to define the function: $$\label{Qt}
\begin{split}
Q^{(t)}(\theta|\theta_1,..,\theta_n)=\ell \sinh \theta+\tfrac{1}{i} \Lambda^{(t)}({\theta}|\{\theta_1,..,\theta_n\}).
\end{split}$$ Then the logarithmic form of the Bethe-Yang equations take the form: $$\label{BYln}
\begin{split}
Q^{(t)}(\tilde{\theta}_j|\tilde{\theta}_1,..,\tilde{\theta}_n)=2 \pi \, I^{(t)}_j, \qquad j=1,...,n, \quad
t=1,..,2^n,
\end{split}$$ where $I_j^{(t)} \in{\mathbb Z}$ are the quantum numbers characterizing the individual rapidities of the $t\,$th eigenstate. The function $Q^{(t)}$ in (\[Qt\]) allows one to define the density of states in the $t\,$th eigenstate $\Psi^{(t)}$ by the Jacobi determinant as follows: $$\label{Rhot}
\begin{split}
\rho^{(t)}(\theta_1,...,\theta_n)=\text{det}\left\{ \frac{\partial Q^{(t)}(\theta_j|\theta_1,..,\theta_n)}{\partial \theta_k} \right\}_{j,k=1,..,n} \qquad t=1,...,2^n.
\end{split}$$ With the help of the basis (\[bazis\]), (\[ortog\]) one can define form-factors being polarized with respect to the eigenvectors (\[bazis\]). In [@Palmai13] this quantity was defined by the formula as follows: $$\label{polform}
\begin{split}
&F^{{\cal O}}_{(s,t)}(\theta_m',...,\theta'_1|\theta_1,...,\theta_n)=
\sum\limits_{b_1,..,b_m=\pm} \, \sum\limits_{a_1,..,a_n=\pm} \Psi_{b_1...b_m}^{(s)*}(\theta'_1,..,\theta'_m) \times \\
&F^{{\cal O}}_{\bar{b}_m...\bar{b}_1 a_1...a_n}(\theta'_m+i \, \pi,...,\theta'_1+i \, \pi,\theta_1,...,\theta_n) \,
\Psi_{a_1...a_n}^{(t)}(\theta_1,..,\theta_n), \quad s=1,...,2^m, \quad t=1,..,2^n.
\end{split}$$ Based on our computations described in the forthcoming sections, we suggest the following slightly modified definition: $$\label{polformen}
\begin{split}
&F^{{\cal O}}_{(s,t)}(\theta_m',...,\theta'_1|\theta_1,...,\theta_n)=
\sum\limits_{b_1,..,b_m=\pm} \, \sum\limits_{a_1,..,a_n=\pm} \Psi_{b_1...b_m}^{(s)}(\theta'_1,..,\theta'_m) \times \\
&F^{{\cal O}}_{\bar{b}_m...\bar{b}_1 a_1...a_n}(\theta'_m+i \, \pi,...,\theta'_1+i \, \pi,\theta_1,...,\theta_n) \,
\Psi_{a_1...a_n}^{(t)*}(\theta_1,..,\theta_n), \quad s=1,...,2^m, \quad t=1,..,2^n.
\end{split}$$ The only difference between the two definitions is a $\Psi \to \Psi^*$ exchange. Or equivalently, as a consequence of the hermiticity property of the soliton transfer matrix (\[tauH\]), one can maintain the original form (\[polform\]) for the definition of polarized form-factors, but in this case the vector $\Psi$ should not be considered as a right eigenvector of the $\tau(\theta|\vec{\theta}),$ but it should be considered as a left eigenvector of the soliton transfer matrix (\[trans\]). In the rest of the paper we will keep the form of the original definition (\[polform\]), but we will consider $\Psi$ as a left eigenvector of $\tau(\theta|\vec{\theta}).$
Now we are in the position to formulate the conjecture of Pálmai and Takács for the expectation values of local operators in non-diagonally scattering theories. Let $$\label{state}
\begin{split}
|\bar{\theta}_1,..,\bar{\theta}_n \rangle^{(s)}_L,
\end{split}$$ that eigenstate of the Hamiltonian defined in finite volume $L$ of the system, which is described by the eigenstate $\Psi^{(s)}$ of the soliton transfer matrix in the large volume limit. Here $\{\bar{\theta}\}_{j=1}^n$ denote the exact finite volume rapidities, which become $\{\tilde{\theta}\}_{j=1}^n$ if the exponentially small in volume corrections are neglected in the large volume limit. Then the conjecture of [@Palmai13] states that the finite volume expectation value of a local operator in an $n$-particle state can be written as follows: $$\label{PT1}
\begin{split}
{}^{(s)}\langle \bar{\theta}_1,..,\bar{\theta}_n |{\cal O}(0,0)|\bar{\theta}_1,..,\bar{\theta}_n \rangle^{(s)}_L=
{\sc F}_n^{{\cal O},(s)}(\tilde{\theta}_1,...,\tilde{\theta}_n)+O(e^{-\ell}), \qquad s=1,..,2^n,
\end{split}$$ where according to the conjecture, the function ${\sc F}_n^{{\cal O},(s)}$, which should be taken at the positions of the roots of the Bethe-Yang equations (\[BYln\]) can be constructed from the infinite volume form-factors of the theory by the following formula: $$\label{PTfv}
\begin{split}
{\sc F}_n^{{\cal O},(s)}({\theta}_1,...,{\theta}_n)=\frac{1}{\rho^{(s)}_n(1,...,n)} \sum\limits_{A\subset\{1,..,n\}}
\sum\limits_{q,t} |C_{qt}^{(s)}\left(\{\theta_k\}|A\right)|^2 \,
{\sc F}_{2|A|,symm}^{{\cal O},(q)}(A) \, \rho^{(t)}_{|\bar{A}|}(\bar{A}),
\end{split}$$ where the first sum runs for all bipartite partitions of the set of indexes $A^{(n)}=\{1,2,...,n\}.$ Namely, $A \cup \bar{A}=A^{(n)}.$ The number of elements of $A$ is denoted by $|A|,$ then the number of elements of $\bar{A}$ is $|\bar{A}|=n-|A|.$ In the sequel we denote the elements of the sets $A$ and $\bar{A}$ as follows[[^6]]{}: $$\label{AAbar}
\begin{split}
A=\{A_1,A_2,...,A_{|A|}\}, \\
\bar{A}=\{\bar{A}_1,\bar{A}_2,...,\bar{A}_{|\bar{A}|}\}.
\end{split}$$ The second sum in (\[PTfv\]) runs for all decompositions of the $n$-particle color wave function with respect to the normalized eigenvectors[[^7]]{} of the transfer matrices acting only on the index sets $A$ and $\bar{A}:$ $$\label{decomp}
\begin{split}
\Psi^{(t)}_{a_1...a_n}(\theta_1,...,\theta_n)\!\!=\!\!\sum\limits_{q=1}^{2^{|A|}} \sum\limits_{s=1}^{2^{|\bar{A}|}}
C_{qs}^{(t)}(\{\theta_k\}|A) \,
\Psi^{(q)}_{a_{A_1}...a_{A_{|A|}}}\!(\theta_{A_1},...,\theta_{A_{|A|}}) \,
\Psi^{(s)}_{a_{\bar{A}_1}...a_{\bar{A}_{|\bar{A}|}}}\!(\theta_{\bar{A}_1},...,\theta_{\bar{A}_{|\bar{A}|}}),
\end{split}$$ where as a consequence of (\[ortog\]) the branching coefficients $C_{qs}^{(t)}(\{\theta_k\}|A)$ satisfy the normalization condition: $$\label{branchnorm}
\begin{split}
\sum\limits_{q,s}|C_{qs}^{(t)}(\{\theta_k\}|A)|^2=1.
\end{split}$$ Here we note, that the earlier discussed $\Psi \to \Psi^*$ exchange in the formulation of the problem, doesnot cause problem in the determination of these branching coefficients, since it corresponds to a simple complex conjugation. This is irrelevant from the conjecture’s point of view, since the final formula depends only on the absolute value square of these branching coefficients.
Now we have two further missing definitions in (\[PTfv\]). In accordance with [@Palmai13] we introduced some more compact notations for the densities: $$\label{densrov}
\begin{split}
\rho^{(s)}_{n}(1,2,..,n)=\rho^{(s)}(\theta_1,\theta_2,..,\theta_n), \\
\rho^{(t)}_{|\bar{A}|}(\bar{A})=\rho^{(t)}(\theta_{\bar{A}_1},\theta_{\bar{A}_2},..,\theta_{\bar{A}_{|\bar{A}|}}),
\end{split}$$ with $\rho^{(s)}$ functions in the right hand side given by (\[Rhot\]). The last so far undefined object in (\[PTfv\]) is ${\sc F}_{2|A|,symm}^{{\cal O},(q)}(A).$ It is defined as the uniform diagonal limit of a $(q,q)$ polarized form-factor of ${\cal O},$ such that the indexes of the rapidities of the sandwiching states run the set $A:$ $$\label{szimFA}
\begin{split}
{\sc F}_{2|A|,symm}^{{\cal O},(q)}(A)=\lim\limits_{\epsilon \to 0} F^{{\cal O}}_{(q,q)}(\theta_{A_{|A|}}+\epsilon,...,\theta_{A_1}+\epsilon|\theta_{A_1},...,\theta_{A_{|A|}}),
\end{split}$$ with $F^{{\cal O}}_{(q,q)}$ defined in (\[polform\]). In analogy with the terminology in purely elastic scattering theories the function ${\sc F}_{2n,symm}^{{\cal O},(q)}$ is called the the $2n$-particle $q$-polarized symmetric diagonal form-factor of the operator ${\cal O}.$
For the operators $\Theta$ and $J_\mu$ the functions ${\sc F}_n^{{\cal O},(s)}(\theta_1,...,\theta_n)$ were computed in the previous sections. Their form taken at the positions of the holes $\{h_j\}_{j=1}^{m_H}$ are given by the formulas (\[THrest8\]), (\[J08\]) and (\[J18\]). In the rest of the paper we will compare these formulas with the conjecture (\[PT1\]), (\[PTfv\]) applied to the operators $\Theta$ and $J_\mu.$ In the forthcoming sections we will do the comparison upto 3-particle states. The only missing piece to this comparison is the knowledge of the symmetric diagonal form-factors. Thus our next task is to compute them upto the required particle numbers.
Symmetric diagonal form-factors for $\Theta$ and $J_\mu$ {#sect7}
========================================================
Both the trace of the stress energy tensor and the $U(1)$ current are related to some conserved quantities of the theory. In purely elastic scattering theories the symmetric diagonal from-factors of such operators can be computed in a simple way [@LM99; @Mussbook]. The key point in the computation is that by exploiting of the corresponding conservation law, it is not necessary to find the explicit solutions of the axioms (\[ax1\])-(\[ax4\]). In this paper we use the same method to compute the symmetric diagonal form-factors upto 3-particle states. It turns out that this simple method allows one to compute the symmetric diagonal form-factors for any number of particles in the pure soliton sector, but for soliton-antisoliton mixed states it works only upto 3-particle states. For higher number of particles the explicit solution of the axioms (\[ax1\])-(\[ax4\]) is required.
The form-factor axioms allow one to compute form-factors of higher number of particles from those of lower number of particles. Thus, we should start with the computation of the 2-particle symmetric diagonal form-factors of the operators of our interest.
2-particle symmetric diagonal form-factors
------------------------------------------
[**The case of $\Theta$:**]{}
The stress energy tensor $T_{\mu \nu}$ is a conserved quantity, which implies that it can be written as appropriate derivative of some Lorentz scalar field $\phi:$ $$\label{Tmn}
\begin{split}
T_{\mu\nu}=(\partial_\mu \partial_\nu-\eta_{\mu\nu} \partial^{\tau} \partial_{\tau})\, \phi,
\end{split}$$ where $\eta_{\mu\nu}$ is the 2-dimensional Minkowski metric. In this representation the trace of the stress energy tensor take the form: $$\label{THfi}
\begin{split}
\Theta=T_{\, \mu}^{\mu}=(\partial^2_1-\partial^2_0)\, \phi.
\end{split}$$ It can be shown [@BMN97], that the Lorentz scalar field $\phi$ is not a local quantum field. Consequently, not all of its form-factors satisfy the axioms (\[ax1\])-(\[ax4\]). To be more precise from the representation (\[Tmn\]), it can be shown, that the 3- or more particle form-factors of $\phi$ satisfy the axioms (\[ax1\])-(\[ax4\]), but the 2-particle ones become more singular, than it is expected from (\[ax4\]). (See (\[TH2A\]).)
Using the space-time structure of the form-factors (\[FFxt\]), the form-factors of $\Theta$ being close to the diagonal limit can be written as follows: $$\label{THff1}
\begin{split}
F^{\Theta}(\hat{\theta}_n,...,\hat{\theta}_1,\theta_1,..,\theta_n)=-{\cal M}^2
\left[ \sum\limits_{j,k=1}^n
\epsilon_j \epsilon_k \cosh(\theta_j-\theta_k)+O(\epsilon^3) \right] F^{\phi}(\hat{\theta}_n,...,\hat{\theta}_1,\theta_1,..,\theta_n),
\end{split}$$ where $F^{\phi}$ denotes the form-factors of the scalar operator $\phi$ in (\[THfi\]) and we introduced the notation $\hat{\theta}_j=\theta_j+i\, \pi+\epsilon_j$ for all values of the index $j.$ In (\[THff1\]) the symbol $O(\epsilon^3)$ means at least cubic in $\epsilon$ terms when the uniform $\epsilon_1=...=\epsilon_n=\epsilon \to 0$ limit is taken. For the sake of simplicity we did not write out the subscripts of the form-factors.
The basic idea of computing the 2-particle form-factors near their diagonal limit is that the near diagonal matrix elements of the Hamiltonian ${\cal H}=\int dx \, T_{00}$ can be computed in two different ways. First, it can be computed directly by acting with ${\cal H}$ on the eigenstates: $$\label{HszamTH}
\begin{split}
\langle \theta+\epsilon,a|{\cal H}|\theta,b\rangle=2\, \pi \, {\cal M} \, \cosh \theta \,
\delta_{ab} \, \delta(\epsilon), \qquad a,b \in\{\pm\}.
\end{split}$$ Second, it can be computed by using the representation $\int dx \, T_{00}$ for the Hamiltonian, and the matrix element is computed by integrating the space-time dependence of the corresponding form-factor: $$\label{FszamTH1}
\begin{split}
\langle \theta\!+\!\epsilon,a|{\cal H}|\theta,b\rangle\!=\!\!\!\int \!\!dx \langle \theta\!+\!\epsilon,a|T_{00}(x,0)|\theta,b\rangle\!=\!
- 2\pi (\epsilon^2\!+\!O(\epsilon^3)) {\cal M} \cosh \theta \, \delta(\epsilon) \,
F^{\phi}_{\bar{a}b}(\theta\!+\!i\, \pi\!+\!\epsilon,\theta),
\end{split}$$ where we used (\[FFxt\]) and (\[THff1\]). Comparing the results (\[HszamTH\]) and (\[FszamTH1\]) of the two different computations allows one to compute the near diagonal limit of the scalarized form-factor: $$\label{TH2A}
\begin{split}
F^{\phi}_{ab}(\theta\!+\!i\, \pi\!+\!\epsilon,\theta)=-\frac{1}{\epsilon^2} \delta_{\bar{a}b}+O(\tfrac{1}{\epsilon})
, \qquad a,b\in\{\pm\}.
\end{split}$$ Combining (\[TH2A\]) with (\[THff1\]), the symmetric diagonal 2-particle form-factor of $\Theta$ can also be determined: $$\label{TH2F}
\begin{split}
F^{\Theta}_{ab}(\theta\!+\!i\, \pi,\theta)={\cal M}^2 \delta_{\bar{a}b}, \qquad a,b\in\{\pm\}.
\end{split}$$ The matrix structure $\delta_{\bar{a}b}$ in (\[TH2A\]) and (\[TH2F\]) accounts for the charge conjugation invariance of the operator $\Theta.$ $$\nonumber$$
[**The $J_\mu$ case:**]{}
The computation of the near diagonal limit of the 2-particle form-factors of the $U(1)$ current goes analogously to that of the operator $\Theta.$ The conservation law for the current implies the following representation: $$\label{Jmuff1}
\begin{split}
J_0=-i\, \partial_1 \psi, \qquad J_1=-i \, \partial_0 \psi,
\end{split}$$ with $\psi$ being a (non-local) Lorentz scalar operator. The form-factors of $\psi$ satisfy the same form-factor axioms as the form-factors of $\phi$ do. This together with (\[FFxt\]) gives the following representation for the near diagonal form-factors: $$\label{Jmuff2}
\begin{split}
F^{J_0}(\hat{\theta}_n,...,\hat{\theta}_1,\theta_1,..,\theta_n)=
-{\cal M} \left[ \sum\limits_{j=1}^n \cosh \theta_j \, \epsilon_j+O(\epsilon^2)\right]
F^{\psi}(\hat{\theta}_n,...,\hat{\theta}_1,\theta_1,..,\theta_n), \\
F^{J_1}(\hat{\theta}_n,...,\hat{\theta}_1,\theta_1,..,\theta_n)=
{\cal M} \left[ \sum\limits_{j=1}^n \sinh \theta_j \, \epsilon_j +O(\epsilon^2)\right]
F^{\psi}(\hat{\theta}_n,...,\hat{\theta}_1,\theta_1,..,\theta_n).
\end{split}$$ The topological charge $Q\!=\!\int \! dx J_0$ acts on one-particle states as follows: $$\label{Qact1}
\begin{split}
Q|\theta,a\rangle\!=\!\sum\limits_{b=\pm} \! q_{ab}|\theta,b\rangle, \quad a=\pm,\quad
\text{with} \quad
q_{++}\!=\!1, \quad q_{--}\!=\!-1, \quad q_{+-}\!=\!q_{-+}\!=\!0.
\end{split}$$ Using this action and the scalar product formula (\[scalar8\]) the near diagonal limit of the matrix elements of the charge can be computed directly: $$\label{Qdirect}
\begin{split}
\langle\theta+\epsilon,a|Q|\theta,b\rangle=2\pi\, q_{ba} \, \delta(\epsilon).
\end{split}$$ On the other hand this matrix element can also be computed by integrating the space-time dependence of the form-factor of $J_0:$ $$\label{Qff}
\begin{split}
\langle\theta+\epsilon,a|Q|\theta,b\rangle\!=\!\int \!\! dx \, \langle\theta+\epsilon,a|J_0|\theta,b\rangle=
\frac{2\pi}{{\cal M}\, \cosh \theta}\, \delta(\epsilon) \, F^{J_0}_{\bar{a}b}(\theta+i \pi+\epsilon,\theta).
\end{split}$$ Comparing the results of the two different computations one obtains the symmetric diagonal limit of the 2-particle form-factors of $J_0:$ $$\label{J0ffsy2}
\begin{split}
F^{J_0}_{ab}(\theta+i \, \pi,\theta)={\cal M} \cosh \theta \, q_{b\bar{a}},
\end{split}$$ with $q_{ab}$ given in (\[Qact1\]). Formula (\[J0ffsy2\]) and (\[Jmuff2\]) allows one to compute the near diagonal limit of the 2-particle scalarized form-factor $F^{\psi}_{ab}:$ $$\label{Fpsi2}
\begin{split}
F^{\psi}_{ab}(\theta+i \, \pi+\epsilon,\theta)=\frac{1}{\epsilon} \, q_{\bar{b}a}+O(\epsilon),
\end{split}$$ which together with (\[Jmuff2\]) gives the 2-particle symmetric diagonal form-factor of $J_1$ as well: $$\label{J1ffsy2}
\begin{split}
F^{J_1}_{ab}(\theta+i \, \pi,\theta)={\cal M} \sinh \theta \, q_{\bar{b}a}.
\end{split}$$ We note that the pure comparison of (\[J0ffsy2\]) and (\[Jmuff2\]) would imply that in (\[Fpsi2\]) there are $O(1)$ terms in $\epsilon$ as well. However, the Lorentz invariance (\[ax1\]), the cyclic axiom (\[ax3\]) and the charge conjugation negativity of $J_\mu,$ implies that the form-factor $F^{\psi}_{ab}(\theta+i \, \pi+\epsilon,\theta)$ is independent of $\theta$ and is an odd function of $\epsilon.$ This oddity forbids the appearance of constant in $\epsilon$ terms in the right hand side of (\[Fpsi2\]).
4-particle symmetric diagonal form-factors
------------------------------------------
The next step in solving the form-factor axioms (\[ax1\])-(\[ax4\]) in the near diagonal limit is the determination of the 4-particle form-factors. To obtain them we need to determine the singular-parts of the near diagonal 4-particle form-factors of the scalar fields $\phi$ and $\psi$ of (\[Tmn\]) and (\[Jmuff1\]).
To analyse the near diagonal limit of 4-particle form-factors, the following two useful formulas can be derived from the appropriate combination of the axioms (\[ax2\])-(\[ax4\]): $$\label{AXe1}
\begin{split}
F_{a_2 a_1 b_1 b_2}(\hat{\theta}_2,\hat{\theta}_1,\theta_1,\theta_2)\!=\!\frac{i}{\epsilon_1} \!
\left\{ C_{a_1 b_1} F_{a_2 b_2}(\hat{\theta}_2,\theta_2)\!-\!{\cal T}^{\bar{a}_1}_{b_1}(\theta_1|\theta_2,\tilde{\theta}'_2)_{b_2 a_2}^{v_1 v_2}
\, F_{v_2 v_1}(\hat{\theta}_2,\theta_2)
\right\}+O(1)_{\epsilon_1},
\end{split}$$ $$\label{AXe2}
\begin{split}
F_{a_2 a_1 b_1 b_2}(\hat{\theta}_2,\hat{\theta}_1,\theta_1,\theta_2)\!=\!-\frac{i}{\epsilon_2} \!
\left\{ C_{b_2 a_2} F_{a_1 b_1}(\hat{\theta}_1,\theta_1)\!-\!{\cal T}^{\bar{b}_2}_{a_2}(\hat{\theta}_2|\hat{\theta}_1,{\theta}_1)_{a_1 b_1}^{v_1 v_2}
\, F_{v_1 v_2}(\hat{\theta}_2,\theta_2)
\right\}+O(1)_{\epsilon_2},
\end{split}$$ where we introduced the short notation $\tilde{\theta}'_{j}=\theta_j\!-\!i \pi\!+\!\epsilon_j$ for any value of the index $j.$ The symbol $O(1)_{\epsilon_1}$ denotes terms which are of order one in $\epsilon_1.$
The application of formulas (\[AXe1\]) and (\[AXe2\]) to the 4-particle form factors of the scalar field $\phi,$ one obtains the result as follows: $$\label{aphi}
F^{\phi}_{\alpha \beta \gamma \delta}(\hat{\theta}_2,\hat{\theta}_1,\theta_1,\theta_2)=
\frac{1}{\epsilon_1 \epsilon_2} a^\phi_{\alpha \beta \gamma \delta}(\theta_1,\theta_2)+O(\tfrac{1}{\epsilon}),
\qquad \alpha,\beta,\gamma,\delta=\pm,$$ where the nonzero elements of the tensor $a^\phi(\theta_1,\theta_2)$ are as follows: $$\label{ATHmmpp}
\begin{split}
a^{\phi}_{--++}(\theta_1,\theta_2)=a^{\phi}_{++--}(\theta_1,\theta_2)=
- G(\theta_1-\theta_2),
\end{split}$$ $$\label{ATHpmpm}
\begin{split}
a^{\phi}_{+-+-}(\theta_1,\theta_2)=a^{\phi}_{-+-+}(\theta_1,\theta_2)=
- \varphi(\theta_1-\theta_2),
\end{split}$$ $$\label{ATHpmmp}
\begin{split}
a^{\phi}_{+--+}(\theta_1,\theta_2)=a^{\phi}_{-++-}(\theta_1,\theta_2)=
- \Omega(\theta_1-\theta_2).
\end{split}$$ The functions $\varphi$ and $\Omega$ are given by the formulas: $$\label{kernelsTH2}
\begin{split}
%\sigma(\theta)&=\frac{1}{2 \pi i} (\log S_0)'(\theta)=\frac{1}{2 \pi } G(\theta), \\
\varphi(\theta)&=-i \big(C_0(\theta)\, B_0'(-\theta)+B_0(\theta)\, C_0'(-\theta)\big), \\
\Omega(\theta)&=-i \big(C_0(\theta)\, C_0'(-\theta)+B_0(\theta)\, B_0'(-\theta)\big)+G(\theta),
\end{split}$$ where $G,$ $B_0$ and $C_0$ are defined in (\[G\]), (\[B0\]) and (\[C0\]) respectively. As a consequence of the unitarity of the S-matrix (\[Unit\]), all the functions of (\[kernelsTH2\]) are even in $\theta.$ Inserting (\[aphi\]) with (\[ATHmmpp\]), (\[ATHpmpm\]) and (\[ATHpmmp\]) into (\[THff1\]) and taking the uniform $\epsilon_1=\epsilon_2=\epsilon \to 0$ limit, one obtains the symmetric diagonal 4-particle form-factors of $\Theta:$ $$\label{FTHmmpp}
\begin{split}
F^{\Theta,symm}_{--++}(\theta_1,\theta_2)=F^{\Theta,symm}_{++--}(\theta_1,\theta_2)=
2 \,{\cal M}^2\, \left( 1+\cosh(\theta_1-\theta_2)\right) \, G(\theta_1-\theta_2),
\end{split}$$ $$\label{FTHpmpm}
\begin{split}
F^{\Theta,symm}_{+-+-}(\theta_1,\theta_2)=F^{\Theta,symm}_{-+-+}(\theta_1,\theta_2)=
2 \,{\cal M}^2\, \big( 1+\cosh(\theta_1-\theta_2)\big) \, \Omega(\theta_1-\theta_2),
\end{split}$$ $$\label{FTHpmmp}
\begin{split}
F^{\Theta,symm}_{+--+}(\theta_1,\theta_2)=F^{\Theta,symm}_{-++-}(\theta_1,\theta_2)=
2 \,{\cal M}^2\, \big( 1+\cosh(\theta_1-\theta_2)\big) \, \varphi(\theta_1-\theta_2).
\end{split}$$ All functions entering these formulas are even, thus these form-factors are really symmetric with respect to the exchange of the two rapidities $\theta_1 \leftrightarrow \theta_2$.
The very same procedure can be repeated for the $U(1)$ current and for the scalar operator $\psi$ associated to it by (\[Jmuff1\]). We just write down the final results below. In the near diagonal limit the 4-particle form factors of the scalar $\psi$ take the form: $$\label{Fpsimmpp}
\begin{split}
F^{\psi}_{--++}(\hat{\theta}_2,\hat{\theta}_1,\theta_1,\theta_2)=-F^{\psi}_{++--}(\hat{\theta}_2,\hat{\theta}_1,\theta_1,\theta_2)=-
2 \pi \sigma(\theta_{12}) \, \left( \frac{1}{\epsilon_1}+\frac{1}{\epsilon_2}\right)+O(1)_\epsilon,
\end{split}$$ $$\label{Fpsipmmp}
\begin{split}
F^{\psi}_{-++-}(\hat{\theta}_2,\hat{\theta}_1,\theta_1,\theta_2)\!=\!-F^{\psi}_{+--+}(\hat{\theta}_2,\hat{\theta}_1,\theta_1,\theta_2)\!=\!
\frac{{\cal G}_0(\theta_{12})}{\epsilon_1 \, \epsilon_2}\!+\!\frac{{\cal G}_1(\theta_{12})}{\epsilon_1 }\!+\!
\frac{{\cal G}_2(\theta_{12})}{ \epsilon_2}\!+\!O(1)_\epsilon,
\end{split}$$ $$\label{Fpsipmpm}
\begin{split}
F^{\psi}_{-+-+}(\hat{\theta}_2,\hat{\theta}_1,\theta_1,\theta_2)\!=\!-F^{\psi}_{+-+-}(\hat{\theta}_2,\hat{\theta}_1,\theta_1,\theta_2)\!=\!
\frac{{\cal H}_0(\theta_{12})}{\epsilon_1 \, \epsilon_2}\!+\!\frac{{\cal H}_1(\theta_{12})}{\epsilon_1 }\!+\!
\frac{{\cal H}_2(\theta_{12})}{ \epsilon_2}\!+\!O(1)_\epsilon,
\end{split}$$ where $\theta_{12}=\theta_1-\theta_2$ and $$\label{G0}
\begin{split}
{\cal G}_0(\theta)=-i \, \big(B_0(\theta) \, C_0(-\theta)-C_0(\theta) \, B_0(-\theta)\big),
\end{split}$$ $$\label{H0}
\begin{split}
{\cal H}_0(\theta)=-i \, \big(1+C_0(\theta) \, C_0(-\theta)-B_0(\theta) \, B_0(-\theta)\big),
\end{split}$$ $$\label{GH12}
\begin{split}
{\cal G}_j(\theta)=g_j(\theta)+G(\theta)\, \hat{g}_j(\theta), \quad
\quad {\cal H}_j(\theta)=h_j(\theta)+G(\theta)\, \hat{h}_j(\theta), \qquad j=1,2,
\end{split}$$ with $$\label{g1}
\begin{split}
g_1(\theta)&=-i \big( B_0(\theta) \, C_0'(-\theta)-C_0(\theta) \, B_0'(-\theta)\big), \quad g_2(\theta)=-g_1(\theta), \\
\hat{g}_1(\theta)&= B_0(\theta) \, C_0(-\theta)-C_0(\theta) \, B_0(-\theta), \qquad \quad \hat{g}_2(\theta)=-\hat{g}_1(\theta),
\end{split}$$ $$\label{h1}
\begin{split}
h_1(\theta)&=-i \big( C_0(\theta) \, C_0'(-\theta)-B_0(\theta) \, B_0'(-\theta)\big), \quad h_2(\theta)=-h_1(\theta), \\
\hat{h}_1(\theta)&= C_0(\theta) \, C_0(-\theta)-B_0(\theta) \, B_0(-\theta), \qquad \quad \hat{h}_2(\theta)=-\hat{h}_1(\theta).
\end{split}$$ Then using (\[Jmuff2\]) the symmetric diagonal 4-particle form-factors of $J_\mu$ can be computed. It turns out that only the ones which correspond to the expectation values in pure soliton or pure antisoliton states have finite uniform $\epsilon_1=\epsilon_2=\epsilon \to 0$ limit: $$\label{FJmu2sym}
\begin{split}
F^{J_0,symm}_{--++}(\theta_1,\theta_2)=-F^{J_0,symm}_{++--}(\theta_1,\theta_2)=2 \,{\cal M}\, \left(\cosh \theta_1+\cosh \theta_2\right) \, G(\theta_{12}), \\
F^{J_1,symm}_{--++}(\theta_1,\theta_2)=-F^{J_1,symm}_{++--}(\theta_1,\theta_2)=2 \,{\cal M}\, \left(\sinh \theta_1+\sinh \theta_2\right) \, G(\theta_{12}).
\end{split}$$ The other form-factors will diverge as $\frac{1}{\epsilon}$ when the symmetric diagonal limit is taken. Nevertheless it can be shown, that these divergences cancel, when according to (\[polform\]) the symmetric diagonal[[^8]]{} limit is taken between Bethe eigenvectors of the soliton transfer matrix. Simple application of the charge conjugation negativity of $J_\mu$ shows that these non-pure solitonic 4-particle symmetric diagonal form-factors are actually zero.
6-particle symmetric diagonal form-factors
------------------------------------------
If one would like to compute the symmetric diagonal limit of the 6-particle form-factors of the operators of our interest, after some computations it becomes obvious, that with fixed subscripts in general this diagonal limit does not exist. Namely, the $\epsilon \to 0$ limit becomes divergent. Nevertheless, in the Pálmai-Takács conjecture summarized in section \[PTsejt\], the symmetric diagonal limit of form-factors polarized with respect to eigenvectors of the soliton transfer matrix (\[polform\]) should be determined. To do this computation, first we rewrite the necessary form-factor axioms in the language of the eigenvectors of the soliton transfer matrix (\[trans\]). For our computations we need the appropriate versions of two axioms, the exchange (\[ax2\]) and the kinematical singularity (\[ax4\]) ones.
The kinematical pole axiom for a near diagonal settings of the rapidities can be written as follows: $$\label{ax3ind}
\begin{split}
F_{a_n...a_1 b_1...b_n}(\hat{\theta}_n,...,\hat{\theta}_1,\theta_1,...,\theta_n)\!=&\frac{i}{\epsilon_1} \!
\left\{ \delta_{b_1}^{\bar{a}_1} \prod\limits_{k=2}^n \delta_{a_k}^{\beta_k} \delta_{b_k}^{\alpha_k}\!-\!
\tau(\theta_1|\vec{\theta})_{b_1 b_2...b_n}^{l\alpha_2...\alpha_n} \, \tau^{-1}(\theta_1^\epsilon|\vec{\theta^\epsilon})_{l \bar{\beta}_2...\bar{\beta}_n}^{\bar{a}_1 \bar{a}_2...\bar{a}_n}
\right\}\!\!\times \\
& F_{\beta_n...\beta_2 \alpha_2...\alpha_n}(\hat{\theta}_n,...,\hat{\theta}_2,\theta_2,...,\theta_n)+O(1)_{\epsilon_1},
\end{split}$$ where we introduced the notations $\theta_j^\epsilon=\theta_j+\epsilon_j$ and $\vec{\theta^\epsilon}=\{\theta_1^\epsilon,...,\theta_n^\epsilon\}.$ Now, analogously to the definition (\[polform\]), one can sandwich this axiom with two color wave functions $\Psi$ and $\Psi^{(\epsilon)},$ such that they become complex conjugate to each other in the $\epsilon \to 0$ diagonal limit: $$\label{Fsand1}
\begin{split}
F_{\Psi}(\hat{\theta}_n,...,\hat{\theta}_1,\theta_1,...,\theta_n)\!=\!\!\!\!\!\!\! \sum\limits_{i_1,...,i_n=\pm} \sum\limits_{j_1,...,j_n=\pm} \!
\Psi_{j_1...j_n}^{(\epsilon)*} F_{\bar{j}_n...\bar{j}_1 i_1...i_n}(\hat{\theta}_n,...,\hat{\theta}_1,\theta_1,...,\theta_n)
\Psi^{i_1...i_n}.
\end{split}$$ Then this form-factor satisfies the kinematical pole equation as follows: $$\label{ax3Fsand}
\begin{split}
F_{\! \Psi}(\hat{\theta}_n,.,\hat{\theta}_1,\theta_1,.,\theta_n)\!\!&=\!\!\frac{i}{\epsilon_1}\!\!\left\{ \Psi^{(\epsilon)*}_{k \bar{\beta}_2...\bar{\beta}_n} \! \Psi^{k \alpha_2...\alpha_n}\!\!-\!
\Psi^{i_1...i_n }\tau(\theta_1|\vec{\theta})_{i_1 i_2...i_n}^{l\alpha_2...\alpha_n}
\tau^{-1}(\theta_1^\epsilon|\vec{\theta^\epsilon})_{l \bar{\beta}_2...\bar{\beta}_n}^{j_1 j_2...j_n} \Psi^{(\epsilon)*}_{j_1..j_n}
\right\}\!\times \\
& F_{\beta_n...\beta_2 \alpha_2...\alpha_n}(\hat{\theta}_n,...,\hat{\theta}_2,\theta_2,...,\theta_n)+O(1)_{\epsilon_1}.
\end{split}$$ It follows, that this equation can be diagonalized, if $\Psi$ is chosen to be a left eigenvector of $\tau(\theta_1|\vec{\theta})$ and $\Psi^{(\epsilon)*}$ is chosen to be a right eigenvector of $\tau(\theta_1^\epsilon|\vec{\theta^\epsilon}):$ $$\label{PPe}
\begin{split}
\Psi^{i_1...i_n }\tau(\theta_1|\vec{\theta})_{i_1 i_2...i_n}^{l\alpha_2...\alpha_n}&=\Lambda(\theta_1|\vec{\theta}) \Psi^{l\alpha_2...\alpha_n}, \\
\tau(\theta_1^\epsilon|\vec{\theta^\epsilon})_{l \bar{\beta}_2...\bar{\beta}_n}^{j_1 j_2...j_n} \Psi^{(\epsilon)*}_{j_1..j_n}&=\Lambda(\theta_1^\epsilon|\vec{\theta^\epsilon})
\Psi^{(\epsilon)*}_{l \bar{\beta}_2...\bar{\beta}_n}.
\end{split}$$ With such sandwiching states the kinematical singularity axiom in the near diagonal limit takes the form: $$\label{Fax3diag}
\begin{split}
F_{\! \Psi}(\hat{\theta}_n,.,\hat{\theta}_1,\theta_1,.,\theta_n)\!\!&=\!\frac{i}{\epsilon_1}\!\left(\!1\!-\!\frac{\Lambda(\theta_1|\vec{\theta})}{\Lambda(\theta_1^\epsilon|\vec{\theta^\epsilon})}\!\right) \Psi^{(\epsilon)*}_{k \bar{\beta}_2...\bar{\beta}_n} \! \Psi^{k \alpha_2...\alpha_n}\,
F_{\beta_n...\beta_2 \alpha_2...\alpha_n}(\hat{\theta}_n,...,\hat{\theta}_2,\theta_2,...,\theta_n)
\\&+O(1)_{\epsilon_1}
\end{split}$$ A few important comments are in order. First, we pay the attention that $\Psi^{(\epsilon)*}$ is not the complex conjugate vector of $\Psi,$ because it is an eigenvector of a transfer matrix whose inhomogeneities are shifted with $\epsilon$s with respect to those of $\tau.$ They form a conjugate pair only in the $\epsilon_j \to 0$ limit. On the other hand in [@Palmai13] the symmetric diagonal form-factors are defined by a sandwich (\[polform\]), where $\Psi$ must be a [**right**]{} eigenvector of $\tau$ (\[trans\]). Nevertheless, the near diagonal limit formulation of the kinematical singularity axiom (\[Fax3diag\]) suggest, that the diagonal limit, should be taken such that in (\[polform\]) the vector $\Psi$ must be the [**left**]{} eigenvector of the transfer matrix (\[trans\]). Actually this was the reason why we redefined the original definition of polarized form-factors (\[polform\]) by the formula (\[polformen\]). Nevertheless, in the sequel we keep the defining formula (\[polform\]), but based on the implications of formulas (\[PPe\]) and (\[Fax3diag\]), *we require $\Psi$ to be a left eigenvector of $\tau(\theta_1|\vec{\theta})$ and $\Psi^{(\epsilon)*}$ to be right eigenvector of $\tau(\theta_1^\epsilon|\vec{\theta^\epsilon}).$*
Now an important remark is in order. It is worth to analyse, what the form-factor equation (\[Fax3diag\]) tells about the symmetric diagonal limit, when $\epsilon_j$ tends to zero uniformly. The term $\tfrac{i}{\epsilon_1}\!\left(\!1\!-\!\tfrac{\Lambda(\theta_1|\vec{\theta})}{\Lambda(\theta_1^\epsilon|\vec{\theta^\epsilon})}\!\right)$ on the right hand side have a finite limiting value. The sum $ \Psi^*_{k \bar{\beta}_2...\bar{\beta}_n} \! \Psi^{k \alpha_2...\alpha_n}\,
F_{\beta_n...\beta_2 \alpha_2...\alpha_n}(\hat{\theta}_n,...,\hat{\theta}_2,\theta_2,...,\theta_n)$ contains the sum of near diagonal form-factors with all possible indexes. In the previous section we saw, that not all of them have finite $\epsilon \to 0$ limit. This implies that the existence of the symmetric diagonal limit of a form-factor is not obvious, and if eventually it exists, it must be a consequence of non-trivial cancellations between divergent terms. We will discuss this point in more detail in section \[sect9\].
We continue with writing the exchange axiom (\[ax2\]) applied to the near diagonal limit in terms of the eigenvectors of the transfer matrix. These eigenvectors can be given as actions of the off diagonal elements of the monodromy matrix (\[monodr\]) on the trivial vacuum (\[trivvac\]). Using the representations (\[Psi1R\]) and (\[Psi\*1R\]) for the Bethe-eigenvectors: $$\begin{split} \label{PsiPsi}
\Psi^{a_1...a_n}\equiv \Psi^{a_1...a_n }(\{\lambda_j\}|\vec{\theta})\sim
{}^{a_1...a_n}(\langle 0|\prod\limits_{j=1}^r {\cal C}(\lambda_j|\vec{\theta})), \\
%\sim (\prod\limits_{j=1}^r {\cal B}(\lambda_j|\vec{\theta})|0\rangle )^{a_1..a_n}, \\
\Psi^{(\epsilon)*}_{b_1...b_n}\equiv\Psi(\{\lambda_j^\epsilon\}|\vec{\theta^\epsilon})_{b_1...b_n}^*
%\sim {}_{b_1...b_n}(\langle 0|\prod\limits_{j=1}^r {\cal C}(\lambda_j^\epsilon|\vec{\theta}^\epsilon))
\sim (\prod\limits_{j=1}^r {\cal B}(\lambda_j^\epsilon|\vec{\theta^\epsilon})|0\rangle )_{b_1..b_n},
\end{split}$$ the exchange axiom in the near diagonal limit can be written as follows: $$\label{Fax2diag}
\begin{split}
\Psi(\{\lambda_j^\epsilon\}|\vec{\theta^\epsilon})_{b_1...b_n}^*
F_{\bar{b}_n...\bar{b}_1 a_1...a_n}(...,\hat{\theta}_{s+1},\hat{\theta}_s,...,\theta_s,\theta_{s+1},...)
\Psi^{a_1...a_n }(\{\lambda_j\}|\vec{\theta})\!=\! S_0(\theta_{s+1}^\epsilon-\theta_s^\epsilon) \! \times\\
S_0(\theta_{s}-\theta_{s+1}) \,
\Psi(\{\lambda_j^\epsilon\}|\vec{\theta^\epsilon}_{\!\!ex})_{b_1...b_n}^*
F_{\bar{b}_n...\bar{b}_1 a_1...a_n}(...,\hat{\theta}_{s},\hat{\theta}_{s+1},...,\theta_{s+1},\theta_{s},...)
\Psi^{a_1...a_n }(\{\lambda_j\}|\vec{\theta}_{ex}),
\end{split}$$ where the set $\{\lambda_j\}_{j=1}^r$ is the solution of the Bethe-equations (\[ABAE\]) and the set $\{\lambda_j^\epsilon\}_{j=1}^r$ also solves (\[ABAE\]) but with $\theta_j \to \theta_j^\epsilon=\theta_j+\epsilon_j$ replacement[[^9]]{}. The most important details of the formula are the vectors $\vec{\theta}$ and $\vec{\theta}_{ex}.$ In these vectors the order of the rapidity matters! The difference between them is the order of the exchanged rapidities $\theta_s$ and $\theta_{s+1}.$ Namely, $$\label{thetavec}
\begin{split}
\vec{\theta}&=\{\theta_1,...,\theta_s,\theta_{s+1},..,\theta_n\}, \qquad \vec{\theta^\epsilon}\!=\!\{\theta_1\!+\!\epsilon_1,..,\theta_s\!+\!\epsilon_s,\theta_{s+1}\!+\!\epsilon_{s+1},..,\theta_n\!+\!\epsilon_n\},\\
\vec{\theta}_{ex}&=\{\theta_1,...,\theta_{s+1},\theta_{s},..,\theta_n\}, \qquad \vec{\theta^\epsilon}_{\!\!ex}\!=\!\{\theta_1\!+\!\epsilon_1,.,\theta_{s+1}\!+\!\epsilon_{s+1},
\theta_{s}\!+\!\epsilon_{s},..,\theta_n\!+\!\epsilon_n\}.
\end{split}$$ This means that the Bethe-vectors on the left and right hand sides of the equation (\[Fax2diag\]) are different, since they are eigenvectors of different transfer matrices! We would like to explain this in a bit more detail. The rapidities $\theta_j$ are inhomogeneities of the transfer matrix. The transfer matrix is not invariant under the permutation of the inhomogeneities among the $n$ lattice sites. Nevertheless, the Bethe-equations (\[ABAE\]) and the eigenvalue expression are also invariant under the permutation of the rapidities. Thus the transfer matrices $\tau(\theta|\vec{\theta})$ and $\tau(\theta|\vec{\theta_{ex}})$ are only isospectral, but have different eigenvectors connected by a unitary transformation. This recognition has also some implication on the Pálmai-Takács conjecture (section \[PTsejt\]), since there in the wave-function decomposition (\[decomp\]) the orders of rapidities in the arguments of the wave functions matter!
### Solitonic 6-particle symmetric diagonal form-factors
If one starts to analyse the 3-particle Bethe-equations (\[ABAE\]), it becomes immediately obvious that the relevant solutions are the zero and 1-root solutions, since they account for all states in the Q=3 and Q=1 sectors. The missing Q=-3 and Q=-1 sectors can be obtained from the previous ones by the charge conjugation symmetry. The Q=3 sector is the pure soliton sector with no Bethe-root in (\[ABAE\]). In this case the complicated sum in the right hand side of the kinematical pole equation (\[Fax3diag\]) applied to the scalar operators $\phi$ and $\psi$ will contain only a single term, which includes only the pure solitonic near diagonal form-factors (\[ATHmmpp\]) and (\[Fpsimmpp\]). Since in this limit the diagonal pure solitonic matrix elements does not mix with other states, the computation of their symmetric and connected limits can be computed in exactly the same way as in a purely elastic scattering theory [@Mussbook].Their explicit form for the operators $\Theta$ and $J_\mu$ can be found in references [@En1] and [@En], respectiveley. In these papers analytical formulas describing the Bethe-Yang limit of pure solitonic expectation values of the operators $J_\mu$ and $\Theta$ can also be found. This made it possible to verify the conjecture of [@Palmai13] in this sector for any number of solitons. The pure solitonic sector is very similar to the case of a purely elastic scattering theory. As a consequence the actual form of conjecture of [@Palmai13] goes through remarkable simplifications in this sector and becomes identical with the formula conjectured for diagonally scattering theories [@Pozsg13; @PST14]. Our purpose is to check the general form of the conjecture of [@Palmai13]. Thus we will test it in a sector, where there is mixing between the states with different polarizations. This simplest such nontrivial sector is the $Q=1$ sector of the 3-particle space. In the language of the Bethe-equations (\[ABAE\]) it is described by a single Bethe-root.
### 6-particle symmetric diagonal form-factors in the $Q=1$ sector
The first step to compute the symmetric diagonal form-factors of the operators of our interest in the $Q=1$ sector, is to write down the actual form of the wave functions which should sandwich our form-factors according to (\[polform\]). Here we denote their matrix elements as follows: $$\label{vektorok}
\begin{split}
\Psi^{i_1 i_2 i_3 \,}=\frac{C^{i_1 i_2 i_3}}{N_\Psi}, \\
\Psi_{i_1 i_2 i_3}^{(\epsilon)*}=\frac{B^\epsilon_{i_1 i_2 i_3}}{N_\Psi},
\end{split}$$ where the nonzero coefficients in the $Q=1$ sector can be read off from the formulas (\[Psi1R\]), (\[Psi\*1R\]) coming from the Algebraic Bethe-Ansatz diagonalization of the soliton-transfer matrix: $$\label{BC123}
\begin{split}
C^{+--}=C_1, \qquad C^{-+-}=B_1 \, C_2, \qquad C^{--+}=B_1 \, B_2 \, C_3, \\
B^\epsilon_{+--}=C_1^\epsilon \, B_2^\epsilon \, B_3^\epsilon, \qquad B^\epsilon_{-+-}= C_2^\epsilon \, B_3^\epsilon, \qquad B^\epsilon_{--+}= C_3^\epsilon,
\end{split}$$ where for later convenience we introduced the short notations as follows: $$\label{BjCj}
\begin{split}
B_j=&B_0(\lambda_1-\theta_j), \qquad C_j=C_0(\lambda_1-\theta_j), \qquad
B_j^\epsilon=B_0(\lambda_1^\epsilon-\theta_j^\epsilon), \qquad C_j^\epsilon=C_0(\lambda_1^\epsilon-\theta_j^\epsilon), \\
&\text{with} \qquad \theta_j^\epsilon=\theta_j+\epsilon_j, \qquad \text{for} \quad j=1,2,3,
\end{split}$$ such that the single Bethe-roots $\lambda_1$ and $\lambda_1^\epsilon$ are solutions of the Bethe-equations (\[3BAE\]): $$\label{BBE}
\begin{split}
B_1 \, B_2 \, B_3=1, \qquad B_1^\epsilon \, B_2^\epsilon\, B_3^\epsilon=1.
\end{split}$$ The normalization factor $N_\Psi$ is chosen to be the Gaudin-norm (\[3NPsi\]) of the vector[[^10]]{} $\Psi.$ We note that this normalization factor is invariant under any permutations of the three rapidities $\{\theta_j\}_{j=1}^3.$
$$\nonumber$$
[**The case of $\Theta$:**]{}
Now we are in the position to compute the 6-particle symmetric diagonal form-factors of $\Theta$ in the $Q=1$ subsector. This subsector is characterized by a single Bethe-root solving the equation (\[BBE\]).
Looking at the formula (\[THff1\]) it turns out that to get the required limit of our 6-particle form-factor one needs to know the $\tfrac{1}{\epsilon^2}$ order part of the $\Psi$-sandwiched matrix element of the scalar operator $\phi$ defined in (\[Tmn\]). To compute this part, one needs to use only the equations (\[Fax3diag\]) and (\[Fax2diag\]). These equations together with the concrete forms (\[aphi\])-(\[ATHpmmp\]) of the near diagonal 4-particle form factors imply the following small $\epsilon$ series for the required form-factor of $\phi:$ $$\label{Wphi}
\begin{split}
W^\phi(\theta_1,\epsilon_1;\theta_2,\epsilon_2;\theta_3,\epsilon_3)=
\frac{1}{N_\Psi^2} B^\epsilon_{j_1 j_2 j_3}
F^\phi_{\bar{j}_3 \bar{j}_2 \bar{j}_1 i_1 i_2 i_3}(\hat{\theta}_3,\hat{\theta}_2,\hat{\theta}_1, \theta_1,\theta_2,\theta_3) \, C^{i_1 i_2 i_3},
\end{split}$$ $$\label{Wexp}
\begin{split}
W^\phi(\theta_1,\epsilon_1;\theta_2,\epsilon_2;\theta_3,\epsilon_3)\!=\!
\frac{A_{12}(\theta_1,\theta_2,\theta_3)}{\epsilon_1 \epsilon_2}+
\frac{A_{13}(\theta_1,\theta_2,\theta_3)}{\epsilon_1 \epsilon_3}+
\frac{A_{23}(\theta_1,\theta_2,\theta_3)}{\epsilon_2 \epsilon_3}+
O(\tfrac{1}{\epsilon}).
\end{split}$$ Then equation (\[Fax2diag\]) tells us how $W^\phi$ of (\[Wphi\]) changes when exchanging the pairs $(\theta_j,\epsilon_j) \leftrightarrow (\theta_k,\epsilon_k)$ in the argument. This gives the following relations among the $A_{ij}$ functions in (\[Wexp\]): $$\label{Aijrel1}
\begin{split}
A_{13}(\theta_1,\theta_2,\theta_3)=A_{12}(\theta_3,\theta_1,\theta_2), \qquad
A_{23}(\theta_1,\theta_2,\theta_3)=A_{13}(\theta_3,\theta_1,\theta_2),%=A_{12}(\theta_2,\theta_3,\theta_1),
\end{split}$$ and in addition $A_{ij}$ is invariant under the exchange of its $i$th and $j$th arguments.
The functions $A_{12}$ and $A_{13}$ can be directly computed from the $\tfrac{1}{\epsilon_1}$ pole given by equation (\[Fax3diag\]). Then $A_{23}$ can be determined from them by using (\[Aijrel1\]). Straightforward application of (\[Fax3diag\]) leads to the following expressions for $A_{12}$ and $A_{13}:$ $$\label{A12}
\begin{split}
A_{12}(\theta_1,\theta_2,\theta_3)\!=\!i\, \partial_3 \ln \Lambda(\theta_1|\vec{\theta}) \,
T^\phi(\theta_1,\theta_2,\theta_3), \\
A_{13}(\theta_1,\theta_2,\theta_3)\!=\!i\, \partial_2 \ln \Lambda(\theta_1|\vec{\theta}) \,
T^\phi(\theta_1,\theta_2,\theta_3),
\end{split}$$ where $T^\phi$ is the singularity eliminated tensorial sum part of (\[Fax3diag\]): $$\label{Tphi}
\begin{split}
T^\phi(\theta_1,\theta_2,\theta_3)\!&=\!\lim\limits_{\epsilon \to 0}
\frac{1}{N_\Psi^2} B^\epsilon_{k \bar{\beta}_2 \bar{\beta}_3} C^{k \alpha_2 \alpha_3}
a^\phi_{\beta_3 \beta_2 \alpha_2 \alpha_3}(\theta_2,\theta_3)=\\
&-\frac{1}{N_\Psi^2}\left[ \frac{C_1^2}{B_1}G(\theta_{23})
+\left(\frac{C_2^2}{B_2}+\frac{C_3^2}{B_3}\right)\,\Omega(\theta_{23})+(1+B_1)\, C_2 \, C_3 \, \varphi(\theta_{23})
\right],
\end{split}$$ with the constituent functions given in (\[kernelsTH2\]) and (\[BjCj\]). Having the explicit expression for $A_{12}$ and $A_{13},$ with the help of the exchange relation (\[Aijrel1\]) $A_{23}$ can also be obtained from them. Finally using (\[THff1\]) and (\[Wphi\]), the symmetric diagonal limit of the form-factors of $\Theta$ in a 3-particle state described by the Bethe-root $\lambda_1$ can be given by the formula as follows: $$\label{FSDTH1}
\begin{split}
F^{\Theta, (\Psi)}_{6,symm}(\theta_1,\theta_2,\theta_3)=-\frac{{\cal M}^2}{N_\Psi^2}
\left[ A_{12}(\theta_1,\theta_2,\theta_3)+A_{13}(\theta_1,\theta_2,\theta_3)+A_{23}(\theta_1,\theta_2,\theta_3)\right]
\times \\
\, \left[3+2 \cosh(\theta_{12}) +2 \cosh(\theta_{13})+2 \cosh(\theta_{23})\right].
\end{split}$$ We note that the $\lambda_1$ dependence is implicit in this expression. It is hidden in the expression of $T^\Phi$ in (\[Tphi\]) and in the derivative of the eigenvalue in (\[A12\]). A useful formula for the latter is given in (\[Lsq\]).
$$\nonumber$$
[**The case of $J_\mu$:**]{}
The computation of the 6-particle symmetric diagonal form-factors of the $U(1)$ current is a bit more subtle than that of the trace of the stress energy tensor. The method described in the previous paragraphs is the same, but one should be much more careful in the small $\epsilon_j$ expansion. In this case the linear in $\epsilon_j$ terms of the Bethe-vector $B^\epsilon_{i_1 i_2 i_3}$ (\[BC123\]) will also give relevant contributions to the symmetric form-factors. The first step is to compute the 6-particle form-factor of the scalar field $\psi$ in the near diagonal limit. Thus the quantity we compute is defined by: $$\label{Wpsi}
\begin{split}
W^\psi(\lambda^\epsilon_1,\lambda_1|1^\epsilon,2^\epsilon,3^\epsilon)=
\frac{1}{N_\Psi^2} B^\epsilon_{j_1 j_2 j_3}
F^\psi_{\bar{j}_3 \bar{j}_2 \bar{j}_1 i_1 i_2 i_3}(\hat{\theta}_3,\hat{\theta}_2,\hat{\theta}_1, \theta_1,\theta_2,\theta_3) \, C^{i_1 i_2 i_3},
\end{split}$$ where for short we introduced the symbolic notation for a pair: $\theta_j,\epsilon_j \to j^\epsilon,$ and we also indicated in the list of arguments the Bethe-root dependence of this form-factor. Using the kinematical pole equation for the $\tfrac{1}{\epsilon_1}$ singularity, the terms proportional to $\tfrac{1}{\epsilon_1}$ in the small $\epsilon$ expansion of $W^\psi$ can be computed. To facilitate this task first we do the computations in some smaller building blocks of $W^\psi.$ Let $Y$ denote the eigenvalue part of (\[Fax3diag\]): $$\label{Ydef}
\begin{split}
Y(\lambda^\epsilon_1,\lambda_1|1^\epsilon,2^\epsilon,3^\epsilon)=1-\frac{\Lambda(\theta_1|\vec{\theta})}{\Lambda(\theta^\epsilon_1|\vec{\theta^\epsilon})}=\sum\limits_{j=1}^3 \epsilon_j \, \partial_j \ln \Lambda(\theta_1|\vec{\theta})+
O(\epsilon^2),
\end{split}$$ and let denote $T^\psi$ the tensorial sum part of (\[Fax3diag\]): $$\label{Tpsi}
\begin{split}
T^\psi(\lambda_1^\epsilon,\lambda|1^\epsilon,2^\epsilon,3^\epsilon)=B^\epsilon_{k \bar{\beta}_2 \bar{\beta}_3}
C^{\, k \alpha_2 \alpha_3}
F^\psi_{\beta_3 \beta_2 \alpha_2 \alpha_3}(\hat{\theta}_3,\hat{\theta}_2,\theta_2,\theta_3) .
\end{split}$$ Taking the near diagonal limit of the 4-particle form-factors of $\psi$ given in (\[Fpsimmpp\])-(\[h1\]), one obtains the following small $\epsilon$ expansion for $T^\psi:$ $$\label{Tpsiexp}
\begin{split}
T^\psi(\lambda_1^\epsilon,\lambda_1|1^\epsilon,2^\epsilon,3^\epsilon)=
\frac{T_{23}(\vec{\theta})}{\epsilon_2 \epsilon_3}+
\frac{T_2(\vec{\theta})}{\epsilon_2}+\frac{T_3(\vec{\theta})}{\epsilon_3}+O(1),
\end{split}$$ where the functions $T_{23}(\vec{\theta}),\, T_{2}(\vec{\theta}), \,T_{3}(\vec{\theta})$ take the form: $$\label{T23}
\begin{split}
T_{23}(\vec{\theta})={\cal H}_0(\theta_{23})\left( \frac{C_3^2}{B_3}-\frac{C_2^2}{B_2}\right)\!+
{\cal G}_0(\theta_{23}) \, C_2 \, C_3 \, (B_1-1),
\end{split}$$ $$\label{T2}
\begin{split}
T_{2}(\vec{\theta})\!=&{\cal G}_0(\theta_{23})\!\!\left[\!B^{(3)}_3\!(\vec{\theta})C^{(2)}\!(\vec{\theta})\!-\!
B^{(2)}_3\!(\vec{\theta})C^{(3)}\!(\vec{\theta}) \!\right]
\!\!+\!
{\cal G}_1(\theta_{23})\!\left[\!B^{(3)}_0\!(\vec{\theta})C^{(2)}\!(\vec{\theta})\!-\!
B^{(2)}_0\!(\vec{\theta})C^{(3)}\!(\vec{\theta}) \!\right]\!+\\
&{\cal H}_0(\theta_{23})\!\!\left[\!B^{(3)}_3\!(\vec{\theta})C^{(3)}\!(\vec{\theta})\!-\!
B^{(2)}_3\!(\vec{\theta})C^{(2)}\!(\vec{\theta}) \!\right]
\!\!+\!
{\cal H}_1(\theta_{23})\!\left[\!B^{(3)}_0\!(\vec{\theta})C^{(3)}\!(\vec{\theta})\!-\!
B^{(2)}_0\!(\vec{\theta})C^{(2)}\!(\vec{\theta}) \!\right]\!+\\
& G(\theta_{23}) B^{(1)}_0\!(\vec{\theta}) C^{(1)}(\vec{\theta}),
\end{split}$$ $$\label{T3}
\begin{split}
T_{3}(\vec{\theta})\!=&{\cal G}_0(\theta_{23})\!\!\left[\!B^{(3)}_2\!(\vec{\theta})C^{(2)}\!(\vec{\theta})\!-\!
B^{(2)}_2\!(\vec{\theta})C^{(3)}\!(\vec{\theta}) \!\right]
\!\!+\!
{\cal G}_2(\theta_{23})\!\left[\!B^{(3)}_0\!(\vec{\theta})C^{(2)}\!(\vec{\theta})\!-\!
B^{(2)}_0\!(\vec{\theta})C^{(3)}\!(\vec{\theta}) \!\right]\!+\\
&{\cal H}_0(\theta_{23})\!\!\left[\!B^{(3)}_2\!(\vec{\theta})C^{(3)}\!(\vec{\theta})\!-\!
B^{(2)}_2\!(\vec{\theta})C^{(2)}\!(\vec{\theta}) \!\right]
\!\!+\!
{\cal H}_2(\theta_{23})\!\left[\!B^{(3)}_0\!(\vec{\theta})C^{(3)}\!(\vec{\theta})\!-\!
B^{(2)}_0\!(\vec{\theta})C^{(2)}\!(\vec{\theta}) \!\right]\!+\\
& G(\theta_{23}) B^{(1)}_0\!(\vec{\theta}) C^{(1)}(\vec{\theta}),
\end{split}$$ where the functions $B^{(j)}_k$ and $C^{(k)}$ are coming from the small $\epsilon$ expansion of the components of the Bethe-eigenvectors (\[BC123\]) in the following way: $$\begin{aligned}
%\label{BCeps}
B_{+--}^\epsilon&=&B^{(1)}_0(\vec{\theta})+\sum\limits_{j=1}^3 B^{(1)}_j(\theta) \epsilon_j+O(\epsilon^2),
\qquad C_{+--}=C^{(1)}(\vec{\theta}),
\nonumber \\
B_{-+-}^\epsilon&=&B^{(2)}_0(\vec{\theta})+\sum\limits_{j=1}^3 B^{(2)}_j(\theta) \epsilon_j+O(\epsilon^2),
\qquad C_{-+-}=C^{(2)}(\vec{\theta}), \label{BCeps} \\
B_{--+}^\epsilon&=&B^{(3)}_0(\vec{\theta})+\sum\limits_{j=1}^3 B^{(3)}_j(\theta) \epsilon_j+O(\epsilon^2),
\qquad C_{--+}=C^{(3)}(\vec{\theta}).
\nonumber\end{aligned}$$ Their actual form can be computed from (\[BC123\]) and (\[BjCj\]). Here we give only the ones entering (\[T2\]) and (\[T3\]): $$\begin{aligned}
B^{(1)}_0(\theta)\!\!&=&\!\!C_1 \, B_2 \, B_3, \qquad \qquad C^{(1)}(\vec{\theta})=C_1, \nonumber \\
B^{(2)}_0(\theta)&=&C_2 \, B_3, \qquad \qquad C^{(2)}(\vec{\theta})=B_1 \, C_2, \label{BC1ek} \\
B^{(3)}_0(\theta)&=&C_3, \qquad \qquad C^{(3)}(\vec{\theta})=B_1 \, B_2 \, C_3, \nonumber\end{aligned}$$ $$\label{BB1}
\begin{split}
B^{(1)}_2(\vec{\theta})&=\partial_{\lambda_1} B^{(1)}_0(\vec{\theta})\cdot \frac{\partial \lambda_1}{\partial \theta_2}-
C_1 \, B'_2 \, B_3, \\
B^{(1)}_3(\vec{\theta})&=\partial_{\lambda_1} B^{(1)}_0(\vec{\theta})\cdot \frac{\partial \lambda_1}{\partial \theta_3}-
C_1 \, B_2 \, B'_3,
\end{split}$$ $$\label{BB2}
\begin{split}
B^{(2)}_2(\vec{\theta})&=\partial_{\lambda_1} B^{(2)}_0(\vec{\theta})\cdot \frac{\partial \lambda_1}{\partial \theta_2}- C'_2 \, B_3, \\
B^{(2)}_3(\vec{\theta})&=\partial_{\lambda_1} B^{(2)}_0(\vec{\theta})\cdot \frac{\partial \lambda_1}{\partial \theta_3}- C_2 \, B'_3,
\end{split}$$ $$\label{BB3}
\begin{split}
B^{(3)}_2(\vec{\theta})&=\partial_{\lambda_1} B^{(3)}_0(\vec{\theta})\cdot \frac{\partial \lambda_1}{\partial \theta_2}, \\
B^{(3)}_3(\vec{\theta})&=\partial_{\lambda_1} B^{(3)}_0(\vec{\theta})\cdot \frac{\partial \lambda_1}{\partial \theta_3}- C'_3,
\end{split}$$ where introduced the notations: $$\label{short}
\begin{split}
B'_j=B'_0(\lambda_1-\theta_j), \qquad C'_j=C'_0(\lambda_1-\theta_j), \qquad j=1,2,3.
\end{split}$$ In the above formulas we did not write down explicitely the $\lambda_1$ dependence of the functions. Nevertheless, it is important because of the $\partial_{\lambda_1}$ partial derivatives. Here the $\lambda_1$ dependence is simply meant by the $\lambda_1$ dependence of the objects $B_j$ and $C_j$ given by (\[BjCj\]).
Now we have all ingredients to compute the 6-particle symmetric diagonal form-factors of $J_\mu$ in the $Q=1$ sector of the 3-particle subspace. Looking at the formula (\[Jmuff2\]) one can see that the symmetric diagonal limit is finite only if $F^\psi$ or equivalently $T^\psi$ has only $\tfrac{1}{\epsilon_j}$ order divergences. However, the order $\tfrac{1}{\epsilon^2}$ term in the expansion (\[Tpsiexp\]) of $T^\psi$ implies, that the symmetric diagonal limit is divergent in this case, provided the coefficient function $T_{23}$ is nonzero. Looking at its explicit form (\[T23\]) it does not seem to be zero. Nevertheless, with some work, exploiting the Yang-Baxter equations (\[YBE\]) and the Bethe-equations (\[BBE\]) for $\lambda_1$ one can show that: $$\label{T23=0}
\begin{split}
T_{23}(\vec{\theta})=0.
\end{split}$$ This nontrivial for the first sight result ensures, that the symmetric diagonal limit of the 6-particle form-factors of $J_\mu$ in the $Q=1$ sector will be well defined. Nevertheless this computation sheds light on the fact that the higher and higher $\tfrac{1}{\epsilon}$ divergences of the nondiagonal form-factors could make the symmetric diagonal limit divergent[[^11]]{}, too. On the other hand this computation might also imply that the special properties of integrability might ensure the cancellation of these (would be?) divergences.
Due to the cancellation of the $\tfrac{1}{\epsilon^2}$ divergent term in $T^\psi,$ $W^\psi$ (\[Wpsi\]) admits the following small $\epsilon$ expansion: $$\label{Wpsiexp}
\begin{split}
W^\psi(\lambda^\epsilon_1,\lambda_1|1^\epsilon,2^\epsilon,3^\epsilon)&=
\frac{W_1(\vec{\theta})}{\epsilon_1}+\frac{W_2(\vec{\theta})}{\epsilon_2}+
\frac{W_3(\vec{\theta})}{\epsilon_3}+
W^{(1)}(\vec{\theta})\frac{\epsilon_1}{\epsilon_2 \epsilon_3}
+
W^{(2)}(\vec{\theta})\frac{\epsilon_2}{\epsilon_1 \epsilon_3}
+\\
&W^{(3)}(\vec{\theta})\frac{\epsilon_3}{\epsilon_1 \epsilon_2}
+O(1),
\end{split}$$ such that the coefficient functions $W_1, \, W^{(2)}$ and $W^{(3)}$ can be computed from the kinematical pole equation (\[Fax3diag\]) by using the formulas (\[Tpsiexp\]) and (\[Ydef\]): $$\label{W1}
\begin{split}
W_1(\vec{\theta})&=\frac{i}{N_\Psi^2} \, \left[T_2(\vec{\theta}) \, \partial_2 \ln \Lambda(\theta_1|\vec{\theta}) +
T_3(\vec{\theta}) \, \partial_3 \ln \Lambda(\theta_1|\vec{\theta})\right], \\
W^{(2)}(\vec{\theta})&=\frac{i}{N_\Psi^2} \, T_3(\vec{\theta}) \, \partial_2 \ln \Lambda(\theta_1|\vec{\theta}), \\
W^{(3)}(\vec{\theta})&=\frac{i}{N_\Psi^2} \, T_2(\vec{\theta}) \, \partial_3 \ln \Lambda(\theta_1|\vec{\theta}).
\end{split}$$ The exchange equation (\[Fax2diag\]) allows one to compute from (\[W1\]) the other still unknown $W$-functions of the expansion (\[Wpsiexp\]), since (\[Fax2diag\]) implies that they are related by argument exchanges: $$\label{Wujak}
\begin{split}
W_2(\theta_1,\theta_2,\theta_3)&=W_1(\theta_2,\theta_1,\theta_3), \\
W_3(\theta_1,\theta_2,\theta_3)&=W_1(\theta_3,\theta_2,\theta_1), \\
W^{(1)}(\theta_1,\theta_2,\theta_3)&=W^{(2)}(\theta_2,\theta_1,\theta_3).
\end{split}$$ Nevertheless, (\[Fax2diag\]) gives further relations among these functions, which can be used to test the obtained result. These are as follows. The functions $W_j(\theta_1,\theta_2,\theta_3)$ and $W^{(j)}(\theta_1,\theta_2,\theta_3)$ are symmetric with respect to the exchange of the rapidities $\theta_s$ and $\theta_q$ with $s,q \neq j.$ According to (\[Fax2diag\]), $W^{(2)}$ and $W^{(3)}$ are also not independent: $$\label{W^23}
\begin{split}
W^{(2)}(\theta_1,\theta_2,\theta_3)&=W^{(3)}(\theta_3,\theta_1,\theta_2).
\end{split}$$ It can be checked that our formulas in (\[W1\]) satisfy this requirement.
With the help of (\[Jmuff2\]) the symmetric diagonal 6-particle form-factors of the current can be expressed in terms of the previously computed $W$-functions as follows: $$\label{FJmu6}
\begin{split}
F^{J_\mu,(\Psi)}_{6,symm}(\theta_1,\theta_2,\theta_3)=(-1)^{\mu+1} {\cal M}
\left(\sum\limits_{j=1}^3 v_j^{(\mu)}\right)
\sum\limits_{j=1}^3 \left[ W_j(\vec{\theta})+W^{(j)}(\vec{\theta})\right],
\end{split}$$ with the vector $$\label{vmu}
\begin{split}
v^{(\mu)}_j=\left\{\begin{array}{r} \cosh(\theta_j), \qquad \text{for} \quad \mu=0, \\
\sinh(\theta_j), \qquad \text{for} \quad \mu=1.
\end{array} \right.
\end{split}$$ The formula (\[FJmu6\]) can also be rephrased in an equivalent way, which reflects manifestly the invariance of the symmetric form-factor with respect to the permutations of the rapidities: $$\label{FJmu6equiv}
\begin{split}
F^{J_\mu,(\Psi)}_{6,symm}(\theta_1,\theta_2,\theta_3)=(-1)^{\mu+1} \frac{\cal M}{2}
\left(\sum\limits_{j=1}^3 v_j^{(\mu)}\right) \times \\
\sum\limits_{\sigma \in S^3} \left[ W_1(\theta_{\sigma(1)},\theta_{\sigma(2)},\theta_{\sigma(3)})
+W^{(2)}(\theta_{\sigma(1)},\theta_{\sigma(2)},\theta_{\sigma(3)})\right],
\end{split}$$ where the second sum runs for the six possible permutations of the indexes $\{1,2,3\}.$
Checking the Pálmai-Takács conjecture {#sect8}
=====================================
In the previous sections we computed the symmetric diagonal form-factors of the operators $\Theta$ and $J_\mu$ upto 6-particles. This makes it possible to check the conjecture of Pálmai and Takács for the diagonal matrix elements of local operators [@Palmai13] (summarized in section \[PTsejt\].) against the exact results given in (\[THrest8\])-(\[J18\]) upto 3-particle expectation values. In the pure soliton sector[[^12]]{} the validity of this conjecture have been already verified for the operators $\Theta$ and $J_\mu$ in references [@En1] and [@En], respectively. This is why in our work we will only focus on states in which soliton and antisoliton states are mixed.
As implied by (\[bazis\]), in the conjecture the eigenvectors of the multisoliton transfer matrix (\[trans\]), play an important role. To test the conjecture upto 3-particle states, one needs the complete Bethe-basis on the space of 1- and 2-particle states and one also needs the Bethe-eigenvector corresponding to the sandwiching 3-particle state. Thus, as a first step we write down these Bethe-eigenvectors.
For the one particle states the eigenvectors are simple: $$\label{1wave}
\begin{split}
\varphi^{(a)}_{i_1}=\delta_{i_1,a}, \qquad a,i_1=\pm.
\end{split}$$ In the space of 1-particle states the basis is two dimensional corresponding to the soliton and the antisoliton. The index $a$ distinguishes the two basis vectors of this space and $i_1$ is the index of the vector. Here we pay the attention to two trivial, but for later considerations important properties of this basis. First of all the vector components are independent of the particle’s rapidities. Second of all these vectors are real. The 2-particle basis is also very simple [@Palmai13]: $$\label{2wave}
\begin{split}
\Psi^{(1)}_{i_1 i_2}&=\delta_{i_1 -} \delta_{i_2 -}, \qquad \qquad \qquad \qquad \qquad
\Psi^{(2)}_{i_1 i_2}=\tfrac{1}{\sqrt{2}}\left(\delta_{i_1 +} \delta_{i_2 -}+\delta_{i_1 -} \delta_{i_2 +}\right),
\\
\Psi^{(3)}_{i_1 i_2}&=\tfrac{1}{\sqrt{2}}\left(\delta_{i_1 +} \delta_{i_2 -}-\delta_{i_1 -} \delta_{i_2 +}\right),
\qquad \quad \Psi^{(4)}_{i_1 i_2}=\delta_{i_1 +} \delta_{i_2 +}.
\end{split}$$ Here again the superscript indexes the basis vectors and the subscripts $i_1,i_2=\pm$ denotes the vector indexes in the 2-particle vector space. Here we also emphasize that this 2-particle basis is real and rapidity independent. With this remark we would like to pay the attention, that the first numerical checks of the Pálmai-Takács conjecture in [@Palmai13], which were performed upto 2-particle states, were not sensible to the difference between the two definitions (\[polform\]) and (\[polformen\]).
It is worth to discuss a bit more on the meaning of the basis vectors of (\[2wave\]). The vectors $\Psi^{(4)}$ and $\Psi^{(1)}$ correspond to the two antisoliton and two soliton states, respectively. The vector $\Psi^{(2)}$ and $\Psi^{(3)}$ describe the symmetric and antisymmetric soliton-antisoliton states, respectively. At the level of the magnonic Bethe-equations (\[ABAE\]), $\Psi^{(4)}$ and $\Psi^{(1)}$ are states without Bethe-roots, while $\Psi^{(2)}$ and $\Psi^{(3)}$ are described by a single Bethe-root. Using the terminology of appendix \[appABA2\] $\Psi^{(2)}$ is described by a real Bethe-root: $\lambda^{(2)}=\tfrac{\theta_1+\theta_2}{2}+i \tfrac{\pi}{2}$ and $\Psi^{(3)}$ is given by a self-conjugate root: $\lambda^{(3)}=\tfrac{\theta_1+\theta_2}{2}+i \tfrac{(1+p) \pi}{2},$ provided we are in the repulsive $1<p$ regime of the theory.
In the space of 3-particle states, we need the Bethe-eigenvectors only in the $Q=1$ sector. It has also a simple form: $$\label{3wave}
\begin{split}
\Psi_{i_1 i_2 i_3}=\Psi_{+--}\delta_{i_1 +} \delta_{i_2 -} \delta_{i_3 -}+
\Psi_{-+-}\delta_{i_1 -} \delta_{i_2 +} \delta_{i_3 -}+
\Psi_{+--}\delta_{i_1 -} \delta_{i_2 -} \delta_{i_3 +},
\end{split}$$ where $$\label{Psiii}
\begin{split}
\Psi_{i_1 i_2 i_3}=\frac{C^{i_1 i_2 i_3}}{N_\Psi},
\end{split}$$ such that $C^{i_1 i_2 i_3}$ is given by (\[BC123\]) with (\[BBE\]) and $N_\Psi$ is given by (\[3NPsi\]). Actually this vector stands for 3 eigenvectors, since it depends on a Bethe-root $\lambda_1,$ and the Bethe-equation (\[BBE\]) have 3 independent solutions in this sector: a real one and two self-conjugate ones. One can recognize that the vector $\Psi$ in (\[3wave\]) differs by a complex conjugation with respect to the one enters the conjecture of [@Palmai13]. The reason is that at the study of the diagonal limit of the kinematical pole axiom (\[Fax3diag\]), we recognized that one should sandwich the form-factor with the left eigenvector of the transfer-matrix instead of its right eigenvector proposed earlier in [@Palmai13]. Due to the hermiticity properties of the soliton transfer matrix (\[tauH\]), this is just a complex conjugation at the level of the eigenvectors. Here the 3-particle wave vector $\Psi$ is complex and rapidity dependent, thus all these affairs matter. In the states upto 2-particles, which was studied in [@Palmai13] to check the conjecture, this complex conjugation problem didnot arise.
Now, we know how the Bethe-eigenvectors we need are described by the roots of the magnonic Bethe-equations (\[ABAE\]). This makes possible to write down the densities of the states (\[Rhot\]) corresponding to the eigenstates under consideration. They can be simply read off from the formulas (\[dldh\])-(\[tildeG\]), which give the Bethe-Yang limit of the Gaudin-matrix. We will specialize these formulas for the zero and one root cases of the 1-, 2- and 3-particle eigenstates. First, we rewrite the matrix $\Phi$ (\[Phi\]) for the zero and one-root states by emphasizing the Bethe-root and particle number dependence better: $$\label{Phin}
\begin{split}
\Phi_{j,k}^{(n)}(\lambda)&=\left(\ell \cosh \theta_j+\sum\limits_{s=1}^n \tilde{G}_{j s}(\lambda) \right) \, \delta_{j k}-\tilde{G}_{j k}(\lambda), \\
\tilde{G}_{jk}(\lambda)&=G(\theta_j-\theta_k)+\frac{1}{i} \frac{V_j(\lambda) \, V_{k}(\lambda)}{\psi^{(n)}(\lambda)},
\quad \psi^{(n)}(\lambda)=\sum\limits_{j=1}^n V_j(\lambda), \\
V_j(\lambda)&=(\ln B_0)'(\lambda-\theta_j).
\end{split}$$ For the zero root case the $\sim \frac{V_j(\lambda) \, V_{k}(\lambda)}{\psi^{(n)}(\lambda)}$ term must be skipped[[^13]]{}.
Then the densities of the states given by the Bethe-vectors (\[1wave\]), (\[2wave\]) and (\[3wave\]) can be given by determinants of this matrix. Now, we list the necessary densities below. The densities (\[Rhot\]) for the 1-particle states (\[1wave\]) are the same as those of a free particle: $$\label{1partrho}
\begin{split}
\rho^{(\pm)}_1(1)\equiv\rho_1(1)=\ell \cosh \theta_1=\ell \, c_1,
\end{split}$$ where for short we introduce the notations $c_j=\cosh \theta_j$ and $s_j=\sinh \theta_j.$ The densities (\[Rhot\]) for the 4-dimensional basis of 2-particle states (\[1wave\]) are given by the determinants as follows: : $$\label{2rhos}
\begin{split}
\rho^{(1)}_2(1,2)&=\text{det}_{ \!\!\!\! \!\!\!\!\!\!\!{\atop 2 \times 2}} \Phi^{(2)}(\emptyset)=\ell^2 c_1 \, c_2+ \ell (c_1+c_2) G(\theta_{12}), \\
\rho^{(2)}_2(1,2)&=\text{det}_{ \!\!\!\! \!\!\!\!\!\!\!{\atop 2 \times 2}} \Phi^{(2)}(\lambda^{(2)})=\ell^2 c_1 \, c_2+ \ell (c_1+c_2) \tilde{G}_{12}(\lambda^{(2)}), \\
\rho^{(3)}_2(1,2)&=\text{det}_{ \!\!\!\! \!\!\!\!\!\!\!{\atop 2 \times 2}} \Phi^{(2)}(\lambda^{(3)})=\ell^2 c_1 \, c_2+ \ell (c_1+c_2) \tilde{G}_{12}(\lambda^{(3)}), \\
\rho^{(4)}_2(1,2)&=\text{det}_{ \!\!\!\! \!\!\!\!\!\!\!{\atop 2 \times 2}} \Phi^{(2)}(\emptyset)=\ell^2 c_1 \, c_2+ \ell (c_1+c_2) G(\theta_{12}),
\end{split}$$ with $$\label{lam23}
\begin{split}
\lambda^{(2)}=\tfrac{\theta_1+\theta_2}{2}+i \tfrac{\pi}{2}, \qquad
\lambda^{(3)}=\tfrac{\theta_1+\theta_2}{2}+i \tfrac{\pi(1+p)}{2}.
\end{split}$$
Finally the density corresponding to the $Q=1$ sector of the 3-particle states is given by: $$\label{3rho}
\begin{split}
\rho_3^{(\Psi)}(1,2,3)
%(\theta_1,\theta_2,\theta_3)
&=\text{det}_{ \!\!\!\! \!\!\!\!\!\!\!{\atop 3 \times 3}} \Phi^{(3)}(\lambda_1),
\end{split}$$ where $\lambda_1$ denotes the solution of the magnonic Bethe-equations (\[BBE\]). Now we are in the position to check the conjecture of [@Palmai13] analytically in the 2-particle sector.
Checking the conjecture for 2-particle states
---------------------------------------------
We make the test only in the $Q=0$ sector of the 2-particle space, since in the purely solitonic $Q=\pm 2$ sectors the conjecture has been verified for the operators $J_\mu$ and $\Theta$ and for any number of solitons in papers [@En] and [@En1]. In the sequel we compute the expectation values in the states described by the color wave functions $\Psi^{(2)}$ and $\Psi^{(3)}$ given in (\[2wave\]). The first step in the computation is to determine the branching coefficients of these wave-functions with respect to the 1-particle color wave-functions of (\[1wave\]). Due to the simple form of these vectors the branching coefficients of the decomposition (\[decomp\]) of $\Psi^{(2)}$ and $\Psi^{(3)}$ can be read off immediately: $$\label{branch2}
\begin{split}\left.
\begin{array}{l}
C_{+-}^{(s)}=\tfrac{1}{\sqrt{2}}, \qquad C_{+-}^{(s)}=\tfrac{r_s}{\sqrt{2}},
\qquad \\
C_{--}^{(s)}=0, \qquad \quad C_{++}^{(s)}=0, \end{array} \right\}
\text{for} \quad s=2,3 \quad \text{with} \quad r_2=1, \quad r_3=-1.
\end{split}$$ Now applying the conjectured formula (\[PTfv\]) to the $Q=0$ states of the 2-particle space, one obtains: $$\label{FOs2}
\begin{split}
F^{{{\cal O}, (s)}}_2(\theta_1,\theta_2)=\langle {\cal O} \rangle_0+\frac{1}{\rho_2^{(s)}(1,2)}\left\{
F_{4,symm}^{{\cal O}, (s)}(1,2)+\frac{1}{2} F_{2,symm}^{{\cal O}, (+)}(1) \, \rho_1^{(-)}(2)+
\right.\\
\left. \frac{1}{2} F_{2,symm}^{{\cal O}, (-)}(1) \, \rho_1^{(+)}(2)+
\frac{1}{2} F_{2,symm}^{{\cal O}, (+)}(2) \, \rho_1^{(-)}(1)+
\frac{1}{2} F_{2,symm}^{{\cal O}, (-)}(2) \, \rho_1^{(+)}(1)
\right\}, \qquad s=2,3,
\end{split}$$ where $\langle {\cal O} \rangle_0$ denotes the vacuum expectation value, the densities are given by (\[1partrho\]) and (\[2rhos\]) and the symmetric diagonal form-factors in the states $\Psi^{(2)}$ and $\Psi^{(3)}$ can be determined from the formula (\[polform\]) using the results (\[kernelsTH2\])-(\[FJmu2sym\]).
It is easier to start testing the conjecture with the operator $J_\mu.$ Since $J_\mu$ is a charge conjugation negative operator all the symmetric form-factors entering (\[FOs2\]) become zero. The vacuum expectation value is also zero because of the same reason $\langle J_\mu \rangle_0=0.$ Thus, the conjecture of [@Palmai13] suggests that $F^{ J_\mu,(s)}_{2}(\theta_1,\theta_2)=0, \, \,$ for $s=2,3.$ Due to the charge conjugation negativity of the current it is true exactly, as well. Thus for $J_\mu$ the conjecture gives the expected trivial result.
For the trace of the stress energy tensor, due to the charge conjugation positivity[[^14]]{} of the operator, (\[FOs2\]) simplifies: $$\label{FOs21}
\begin{split}
F_2^{\Theta,(s)}\!(\theta_1,\theta_2)\!=\!\!\langle {\Theta} \rangle_0\!+\!\!\frac{1}{\rho_2^{(s)}(1,\!2)}\!\!\left\{
\! F_{4,symm}^{{\Theta}\, (s)}(1,\! 2)\!+\!F_{2,symm}^{\Theta\, (+)}(1) \, \rho_1(2)
\!+ \!F_{2,symm}^{\Theta\, (+)}(2) \, \rho_1(1) \!\right\}\!,
\, \, s\!=\!2,\!3,
\end{split}$$ where the symmetric diagonal 2-particle form-factor is a constant as it can be read off from (\[TH2F\]): $$\label{TH2Fuj}
\begin{split}
F_{2,symm}^{\Theta, (\pm)}(1)={\cal M}^2,
\end{split}$$ the necessary densities are listed in (\[1partrho\]) and (\[2rhos\]) and the symmetric diagonal 4-particle form-factors can be constructed from (\[FTHpmpm\]), (\[FTHpmmp\]) by the prescription (\[polform\]) and exploiting the charge conjugation positivity: $$\label{F4STH}
\begin{split}
F_{4,symm}^{{\Theta}, (s)}(1,2)\!=\!F^{\Theta, symm}_{+-+-}(\theta_1,\theta_2)+r_s \, F^{\Theta, symm}_{+--+}(\theta_1,\theta_2),
\end{split}$$ with $r_s$ given in (\[branch2\]). Using the identity $$\label{azon}
\begin{split}
%\frac{1}{i} \frac{V_1(\lambda^{(s)}) \, V_2(\lambda^{(s)})}{\psi_0(\lambda^{(s)})}=
\tilde{G}_{12}(\lambda^{(s)})=
%2 \pi \left(
\Omega(\theta_{12})+r_s \,
\varphi(\theta_{12})
%\right)
, \qquad s=2,3,
\end{split}$$ and the formulas (\[FTHpmpm\]), (\[FTHpmmp\]) the concrete form of these form-factors can be written in the form as follows: $$\label{F4THconc}
\begin{split}
F_{4,symm}^{{\Theta}, (s)}(1,2)\!=\!2 {\cal M}^2 (1+c_1 \, c_2-s_1 \, s_2) \,\tilde{G}_{12}(\lambda^{(s)}), \qquad s=2,3.
\end{split}$$ Putting everything together one ends up with the final formula for the expectation value as follows: $$\label{FOs2TH}
\begin{split}
F_2^{\Theta,(s)}\!(\theta_1,\theta_2)\!=\!\!\langle {\Theta} \rangle_0\!+\!\!\frac{1}{\rho_2^{(s)}(1,2)}\!\!\left\{
2 {\cal M}^2 (1\!+\!c_1 \, c_2\!-\!s_1 \, s_2) \,\tilde{G}_{12}(\lambda^{(s)})\!+\!
{\cal M}^2 \ell (c_1\!+\!c_2) \right\}, \quad s=2,3,
\end{split}$$ which is exactly the same as the formula (\[THrest8\]) coming from the exact result and specified to the single Bethe-root configurations $\lambda^{(2)}$ and $\lambda^{(3)}.$
The next step is to check the conjecture of [@Palmai13] in the $Q=1$ sector of the space of 3-particle states. Here the computations are much more involved, this is why we will write down only the main steps and list the ingredients of the necessary computations.
Checking the conjecture for 3-particle states
---------------------------------------------
The first step in the computation is the decomposition (\[decomp\]) of the 3-particle wave function (\[3wave\]) in terms of 1- (\[1wave\]) and 2-particle (\[2wave\]) wave functions. For a subset $A=\{A_1\}\subset\{1,..,3\}$ with a single element, the branching coefficients can be computed by the scalar product as follows: $$\label{branchc1}
\begin{split}
C_{st}(A)=\sum\limits_{i_1,i_2,i_3=\pm} \varphi_{i_{A_1}}^{(s) \, *} \psi_{i_{\bar{A}_1} i_{\bar{A}_2}}^{(t) *} \, \Psi_{i_1 i_2 i_3}, \qquad
s\in\{+,-\}, \quad t\in\{1,2,3,4\}.
\end{split}$$ Writing the analogous formula for the case, when $A\subset\{1,..,3\}$ has two elements, one obtains the relation: $$\label{Cstrel}
\begin{split}
C_{s t}(A)=C_{t s }(\bar{A}), \qquad s\in\{+,-\}, \quad t\in\{1,2,3,4\}, \qquad A\subset\{1,..,3\}.
\end{split}$$ Thus from (\[branchc1\]) and from the “color” wave functions (\[1wave\]), (\[2wave\]) and (\[3wave\]), all the necessary branching coefficients can be determined: $$\label{Clist}
\begin{split}
C_{1+}(\{1,2\})&=C_{+1}(\{3\})=\Psi_{--+}, \quad C_{2-}(\{1,2\})=C_{-2}(\{3\})=\frac{\Psi_{+--}+\Psi_{-+-}}{\sqrt{2}}, \\
C_{3-}(\{1,2\})&=C_{-3}(\{3\})=\frac{\Psi_{+--}-\Psi_{-+-}}{\sqrt{2}}, \quad C_{1+}(\{1,3\})=C_{+1}(\{2\})=\Psi_{-+-}, \\
C_{2-}(\{1,3\})&=C_{-2}(\{2\})=\frac{\Psi_{--+}+\Psi_{+--}}{\sqrt{2}}, \quad C_{3-}(\{1,3\})=C_{-3}(\{2\})=\frac{\Psi_{+--}-\Psi_{--+}}{\sqrt{2}}, \\
C_{1+}(\{2,3\})&=C_{+1}(\{1\})=\Psi_{+--}, \quad C_{2-}(\{2,3\})=C_{-2}(\{1\})=\frac{\Psi_{--+}+\Psi_{-+-}}{\sqrt{2}}, \\
C_{3-}(\{2,3\})&=C_{-3}(\{1\})=\frac{\Psi_{-+-}-\Psi_{--+}}{\sqrt{2}},
\end{split}$$ where $\Psi_{i_1 i_2 i_3}$ is given in (\[Psiii\]). In the main formula (\[PTfv\]) for the diagonal matrix elements, the absolute value squared of these coefficients arise. They can be expressed in terms of the elements of a Hermitian matrix $M$: $$\label{Mform}
\begin{split}
M=\begin{pmatrix} \tfrac{V_1}{V_1+V_2+V_3} & \tfrac{C_1 C_2 B_3}{N_\Psi} & \tfrac{C_1 C_3}{N_\Psi} \\
\tfrac{C_1 C_2 }{N_\Psi} & \tfrac{V_2}{V_1+V_2+V_3} & \tfrac{B_1 C_2 C_3}{N_\Psi} \\
\tfrac{C_1 C_3 B_2 }{N_\Psi} & \tfrac{C_2 C_3 }{N_\Psi} & \tfrac{V_3}{V_1+V_2+V_3}
\end{pmatrix},
\end{split}$$ in the following way[[^15]]{}: $$\label{Cnormlist}
\begin{split}
|C_{1+}(\{1,2\})|^2&\!=\!|C_{+1}(\{3\})|^2\!=\!M_{33}, \quad |C_{2-}(\{1,2\})|^2\!=\!|C_{-2}(\{3\})|^2\!=\!L_{12}^{(+)}, \\
|C_{3-}(\{1,2\})|^2&\!=\!|C_{-3}(\{3\})|^2\!=\! L_{12}^{(-)}, \quad |C_{1+}(\{1,3\})|^2\!=\!|C_{+1}(\{2\})|^2\!=\!M_{22} , \\
|C_{2-}(\{1,3\})|^2&\!=\!|C_{-2}(\{2\})|^2\!=\!L_{13}^{(+)}, \quad |C_{3-}(\{1,3\})|^2\!=\!|C_{-3}(\{2\})|^2\!=\! L_{13}^{(-)}, \\
|C_{1+}(\{2,3\})|^2&\!=\!|C_{+1}(\{1\})|^2\!=\! M_{11}, \quad |C_{2-}(\{2,3\})|^2\!=\!|C_{-2}(\{1\})|^2\!\!=\!L_{23}^{(+)}, \\
|C_{3-}(\{2,3\})|^2&\!=\!|C_{-3}(\{1\})|^2\!=\!L_{23}^{(-)},
\end{split}$$ where we introduced the short notation: $L_{ij}^{(\pm)}=\tfrac{M_{ii}+M_{jj}\pm M_{ij}\pm M_{ji}}{2}.$
Now we have all the necessary ingredients to compare the conjectured formula (\[PTfv\]) to the exact ones (\[THrest8\])-(\[J18\]) for the operators $\Theta$ and $J_\mu.$ Now, the computations become quite involved, thus they were performed by the software [*Mathematica*]{}. We just write down in words the strategy of the comparison. The $\tfrac{1}{\rho^{(\Psi)}_3}$ term naturally arises in the exact formulas (\[THrest8\])-(\[J18\]), if the inverse Gaudin-matrix is expressed by the co-factor matrix ${\cal K}$: $$\label{Gaudinv}
\begin{split}
\Phi^{-1}_{jk}=\frac{{\cal K}_{kj}}{\det \Phi}=\frac{{\cal K}_{kj}}{\rho^{(\Psi)}_3(1,2,3)}.
\end{split}$$ Then only the numerator of the conjectured formula (\[PTfv\]) remains to be checked. Inserting all previously computed form factors and branching coefficients, it turns out the numerator is a second order polynomial in $\ell,$ such that the coefficients are composed of elementary functions multiplied by $G(\theta_{ij})$ transcendental terms[[^16]]{}. It is easy to see that this “transcendental” structure is the same for both the conjecture (\[PTfv\]) and the exact results: (\[THrest8\])-(\[J18\]). The coefficients of these transcendental terms are complicated combinations of elementary functions containing the single Bethe-root $\lambda_1$ in the argument. These coefficients do not seem to match for the first sight, but exploiting the Bethe-equations finally it turns out that they agree. Thus upto 3-particle states in the sine-Gordon model, the Pálmai-Takács conjecture [@Palmai13] gives the correct result for Bethe-Yang limit of the diagonal matrix elements of the $U(1)$ current and the trace of the stress energy tensor, provided one modifies the definition of polarized form-factors from the original form (\[polform\]) to (\[polformen\]). This simple modification corresponds to a $\Psi \to \Psi^*$ exchange in the original definition of [@Palmai13].
Comments on some subtle points of the conjecture of [@Palmai13] {#sect9}
===============================================================
In the previous section for the operators $\Theta$ and $J_\mu,$ we checked the conjecture of [@Palmai13] for the Bethe-Yang limit of expectation values of local operators upto 3-particle states. The agreement found between the conjectured and the exact results seems to be a convincing evidence for the correctness of this conjecture. Nevertheless, in this section we would like to pay the attention to some delicate points of the conjecture, which require further work to be confirmed. These two subtle points are the existence of the symmetric diagonal limit of form-factors and the order of rapidities in the Bethe-wave functions.
Existence of symmetric diagonal limit of form-factors in a non-diagonally scattering theory
-------------------------------------------------------------------------------------------
In this section we argue, that the existence of symmetric diagonal limit of form-factors in a non-diagonally scattering theory is not obvious at all. Apparently, we cannot prove the existence of this limit in general, thus it cannot be excluded, that this limit is divergent in most of the cases.
We start our argument by writing down the kinematical singularity axiom (\[ax4\]) in the limit, when only the rapidities of the sandwiching states are close to each other. Similarly to (\[Fax3diag\]) we formulate the axiom on the basis of the Bethe-eigenvectors. Let $\Psi$ and $\Phi^{(\epsilon)}$ be the color wave functions corresponding to the two states sandwiching the operator. Thus, $\Psi$ is a left eigenvector of the soliton transfer-matrix (\[trans\]) with rapidity parameters: $\vec{\theta}=\{\theta_1,..,\theta_n\}$ and $\Phi^{(\epsilon)*}$ is a right eigenvector of the soliton transfer-matrix with rapidity parameters[[^17]]{}: $\vec{\theta^{\epsilon}}=\{\theta^{\epsilon}_1,..,\theta^{\epsilon}_n\}.$ Then they satisfy the eigenvalue equations: $$\label{Pshe}
\begin{split}
\Psi^{i_1...i_n }\tau(\theta_1|\vec{\theta})_{i_1 i_2...i_n}^{l\alpha_2...\alpha_n}&=\Lambda_\Psi(\theta_1|\vec{\theta}) \Psi^{l\alpha_2...\alpha_n}, \\
\tau(\theta_1^\epsilon|\vec{\theta^\epsilon})_{l \bar{\beta}_2...\bar{\beta}_n}^{j_1 j_2...j_n} \Phi^{(\epsilon) *}_{j_1..j_n}&=\Lambda_\Phi(\theta_1^\epsilon|\vec{\theta^\epsilon})
\Phi^{(\epsilon) *}_{l \bar{\beta}_2...\bar{\beta}_n}.
\end{split}$$ With these sandwiching states the kinematical singularity axiom takes the form: $$\label{Fax3sh}
\begin{split}
F_{\Phi\Psi}(\hat{\theta}_n,.,\hat{\theta}_1,\theta_1,.,\theta_n)\!\!&
=\!\frac{i}{\epsilon_1}\!\!\left(\!1\!-\!\frac{\Lambda_\Psi(\theta_1|\vec{\theta})}{\Lambda_\Phi(\theta_1^\epsilon|\vec{\theta^\epsilon})}\!\right) \Phi^{(\epsilon)*}_{k \bar{\beta}_2...\bar{\beta}_n} \! \Psi^{k \alpha_2...\alpha_n}\,
\!F_{\beta_n...\beta_2 \alpha_2...\alpha_n}\!(\hat{\theta}_n,..,\hat{\theta}_2,\theta_2,..,\theta_n)
\\&+O(1)_{\epsilon_1}
\end{split}$$ where in accordance with the definition (\[polform\]), $F_{\Phi\Psi}$ denotes the form-factor polarized with the Bethe-vectors $\Phi$ and $\Psi:$ $$\label{Fsh}
\begin{split}
&F_{\Phi\Psi}(\hat{\theta}_n,...,\hat{\theta}_1,\theta_1,...,\theta_n)=
\sum\limits_{b_1,..,b_n=\pm} \, \sum\limits_{a_1,..,a_n=\pm} \Phi_{b_1...b_m}^{(\epsilon)*}(\theta^\epsilon_1,..,\theta^\epsilon_n) \times \\
&F_{\bar{b}_n...\bar{b}_1 a_1...a_n}(\theta^\epsilon_n+i \, \pi,...,\theta^\epsilon_1+i \, \pi,\theta_1,...,\theta_n) \,
\Psi_{a_1...a_n}(\theta_1,..,\theta_n) .
\end{split}$$ Again, we used the short notation: $\hat{\theta}_j=\theta_j+\epsilon_j+i \pi.$ Formula (\[Fax3sh\]) has serious implications on the existence of the symmetric diagonal limit of form-factors in a non-diagonally scattering theory.
If the theory is of purely elastic scattering, than there is no index structure in (\[Fax3sh\]). Thus $\Phi=\Psi=1$ and $\Lambda_\Phi(\theta_1^\epsilon|\vec{\theta^\epsilon})\to\Lambda_\Psi(\theta_1|\vec{\theta})
$ can be written. In this case the prefactor $\frac{i}{\epsilon_1}\!\!\left(\!1\!-\!\frac{\Lambda_\Psi(\theta_1|\vec{\theta})}
{\Lambda_\Psi(\theta_1^\epsilon|\vec{\theta^\epsilon})}\!\right)$ becomes $O(1)$ in $\epsilon,$ which imples that the symmetric diagonal limit always exist (finite). Moreover the exchange axiom (\[ax2\]) ensures, that the limiting form-factor is a symmetric function of the rapidities.
If the theory is of non-diagonally scattering the prefactor $\frac{i}{\epsilon_1}\!\!\left(\!1\!-\!\frac{\Lambda_\Psi(\theta_1|\vec{\theta})}{\Lambda_\Phi(\theta_1^\epsilon|\vec{\theta^\epsilon})}\!\right)$ in (\[Fax3sh\]) is not always $O(1)$ in $\epsilon!$ Moreover it is always divergent if the vectors $\Psi$ and $\Phi^{(\epsilon)}\big|_{\epsilon=0}$ are not equal. This means, that the near diagonal in rapidity limit of the form-factors is divergent if the matrix element is nondiagonal in the color space. On the other hand, the prefactor $\frac{i}{\epsilon_1}\!\!\left(\!1\!-\!\frac{\Lambda_\Psi(\theta_1|\vec{\theta})}{\Lambda_\Phi(\theta_1^\epsilon|\vec{
\theta^\epsilon})}\!\right)$ in (\[Fax3sh\]) has a finite value in $\epsilon \to 0$ limit, if[[^18]]{} both sandwiching states correspond to the same eigenstate of the soliton transfer-matrix. Nevertheless, this fact alone doesnot guarantee, that the symmetric diagonal limit of the form-factors would be finite. This is so, because apart form the prefactor we analyzed, there is another term in (\[Fax3sh\]), a sum of the near diagonal in rapidity limit of form-factors with all polarizations, weighted by the color wave-functions. We argued in the previous lines, that the near diagonal in rapidity limit of form-factors is divergent in general, which means that this sum is composed of divergent terms in the $\epsilon \to 0$ limit. [Actually, the degree of divergence in $\epsilon$ increases with the number of sandwiching particles.]{} To get finite result for these matrix elements very nontrivial cancellations must occur! Actually such nontrivial cancellations happened, when we computed the symmetric diagonal limit of the 3-particle form-factors of the current in section \[sect7\].
Nevertheless, our conclusion is that the symmetric diagonal limit of form-factors in a non-diagonally scattering theory is not obviously finite. Thus, to trust the conjecture of [@Palmai13] beyond 3-particle states, it would be necessary to prove that the symmetric diagonal limit of form-factors exists for generic states, as well.
The order of rapidities
-----------------------
The next delicate point in the conjecture of [@Palmai13] is the matter of the order of rapidities. Here we will not state, that there might be problems with the conjecture, but rather we would like to shed light on the fact, that the so far achieved analytical tests of this paper are still not enough to confirm certain parts of the conjecture. This unconfirmed part is how to do correctly the color-wave function decomposition (\[decomp\]). The issue here is that in general these wave functions do depend on the particle’s rapidities. What’s more they do depend on their orderings, as well. In this paper we did computations upto 3-particle matrix elements. Thus 3-particle wave functions must have been decomposed with respect to 1- and 2-particle color wave functions. But, as it is emphasized in section \[sect8\], incidentally the 1- and 2-particle color wave functions are independent of the rapidities. Consequently, our computations cannot confirm, whether the ordering of rapidities in the arguments of the wave-functions in the right hand side of the decomposion formula (\[decomp\]) is correct if more than three particle states are considered.
A possible reassuring solution to this problem could be, if one could prove that the conjectured formula of [@Palmai13] is invariant under any permutations of the rapidities of the sandwiching state.
Summary and conclusion {#sect10}
======================
In this paper we consider two important local operators of the sine-Gordon theory; the trace of the stress energy tensor and the $U(1)$ current.
We showed, that the finite volume expectation values of these operators in any eigenstate of the Hamiltonian of the model, can be expressed in terms of solutions of sets of linear integral equations (\[ujset\])-(\[bfGdef\]). The large volume solution of these equations allowed us to get analytical formulas in the repulsive regime for the Bethe-Yang limit of these diagonal matrix elements. These formulas are expressed in terms of the Bethe-roots characterizing the corresponding eigenstate of the soliton transfer matrix (\[trans\]). This analytical formula allowed us to check a former conjecture [@Palmai13] for the Bethe-Yang limit of expectation values of local operators in a non-diagonally scattering theory. We computed all expectation values upto 3-particle states both from our analytical formulas and from the conjectured formula of [@Palmai13], and we found perfect agreement between the results of the two different computations. To be more precise to get agreement we had to make a tiny modification in the conjectured formula of [@Palmai13]. Namely, we had to change slightly the definition of the symmetric diagonal form-factors, which are basic building blocks of the formula. In the conjecture of [@Palmai13] they are defined as appropriately (\[polform\]) polarized sandwiches of the form-factors with right eigenvectors of the soliton transfer matrix. However, from our computations it turns out that they should be defined as polarized sandwiches of the form-factors with left eigenvectors of the soliton transfer matrix. Since upto 2-particle states the left and right eigenvectors are the same, this issue arises first at the level of 3-particle states, which were not tested in the original paper [@Palmai13].
Despite the success of the 3-particle checks, there are still some subtle points of the conjecture, which could not be confirmed by our analytical computations. First of all, the finiteness of the symmetric diagonal limit of form-factors for a generic state in a non-diagonally scattering theory is still unproven. Second, the hereby performed analytical tests were still not sensible to some details of the conjectured formula. Namely, upto 3-particle states our computations could not check the correctness of the rapidity dependence of the eigenvectors entering the right hand side of the decomposition rule (\[decomp\]), since all 1- and 2-particle wave-functions are incidentally independent of the rapidities.
Thus our final conclusions are as follows. Our analytical checks gave very strong support for the validity of the conjectured formula of [@Palmai13] for the Bethe-Yang limit of expectation values in non-diagonally scattering theories. Our computations suggest, that the conjecture is well established upto 3-particle states, but to firmly trust it beyond 3-particle states, two further statements should be proven. First, it should be proven that the symmetric diagonal limit of form-factors is finite in a non-diagonally scattering theory, as well. Second, to get some more confidence about whether the rapidity dependence of wave functions is correctly embedded into the conjectured formula, one should also prove that the conjectured formula is invariant with respect to the permutations of the rapidities of the sandwiching state.
Nevetheless, the fact that the conjecture of [@Palmai13] was found to be correct at least upto 3-particle states, opens the door to safely apply it to compute finite temperature correlators [@PT10T], and various one-point functions [@BucTak; @Bert14; @CortCub17; @HKT], by their form-factor series representations upto 3-particle contributions.
The author would like to thank Zoltán Bajnok and János Balog for useful discussions and Gábor Takács for his useful comments on the manuscript. This work was supported by the Hungarian Science Fund OTKA (under K116505) and by an MTA-Lendület Grant.
Integral equations for the derivatives of the counting-function {#appA}
===============================================================
In this appendix we write down the linear integral equations satisfied by the $\theta-$ and $\ell-$ derivatives of the counting-function. The equations we list below can be obtained by differentiating the NLIE (\[DDV\])-(\[DDVII\]). The equations related to the derivative of $Z(\theta|\ell)$ with respect to $\theta$ (\[Gd\]) and $\ell$ (\[Gl\]) can be written in an incorporated way, because the equations for the two different derivatives differ only in a single source term. To have a more compact representation of the equations it is useful to pack all complex roots into a single set: $$\label{ujset}
\begin{split}
\{u_j\}_{j=1}^{m_K}=\{c_j\}_{j=1}^{m_C} \cup \{w_k\}_{k=1}^{m_W}, \qquad m_K=m_C+m_W,
\end{split}$$ such that $$\label{ujindex}
\begin{split}
u_j&=c_j, \qquad j=1,...,m_C, \\
u_{m_C+j}&=w_j, \qquad j=1,...,m_W.
\end{split}$$ In accordance with (\[ujindex\]), from (\[Gd\]) and (\[Gl\]) we define the corresponding $X$ variables as well: $$\label{Xu}
\begin{split}
X^{(u)}_{\nu,j}&=X^{(c)}_j, \qquad \nu\in\{d,\ell\}, \quad j=1,...,m_C, \\
X^{(u)}_{\nu,m_C+j}&=X^{(w)}_j, \qquad \nu\in\{d,\ell\}, \quad j=1,...,m_W.
\end{split}$$ Using this notation the linear integral equations take the form: $$\label{Gnu}
\begin{split}
{\cal G}_\nu(\theta)\!=\!f_\nu(\theta)\!+\!\sum\limits_{j=1}^{m_H} {\bf G}(\theta,h_j) X^{(h)}_{\nu,j}\!-\!
\sum\limits_{j=1}^{m_S} \left( {\bf G}(\theta,y_j-i \eta)+{\bf G}(\theta,y_j+i \eta)\right) X^{(y)}_{\nu,j}
- \\
\sum\limits_{j=1}^{m_K} {\bf G}(\theta,u_j) X^{(u)}_{\nu,j}\!+\!
\sum\limits_{\alpha=\pm} \! \int\limits_{-\infty}^{\infty} \!\! \frac{d \theta'}{2 \pi}
{\bf G}(\theta,\theta'-i \, \alpha \, \eta) {\cal G}_{\nu}(\theta'+i \, \alpha \, \eta) \,
{\cal F}_{\alpha}(\theta'+i\, \alpha \, \eta),
\end{split}$$ $$\label{Qhk}
\begin{split}
&\sum\limits_{k=1}^{m_H} \left[ Z'(h_j) \delta_{jk}-{\bf G}(h_j,h_k)\right]X_{\nu,k}^{(h)}\!=\!
f_\nu(h_j)\!-\!
\sum\limits_{k=1}^{m_S}\left({\bf G}(h_j,y_k+i \eta)+{\bf G}(h_j,y_k-i\eta)\right) X_{\nu,k}^{(y)}
\!-\! \\
&\sum\limits_{k=1}^{m_K}\!{\bf G}(h_j,u_k) X_{\nu,k}^{(u)}\!+\!\!
\sum\limits_{\alpha=\pm} \! \int\limits_{-\infty}^{\infty} \!\!\! \frac{d \theta'}{2 \pi}
{\bf G}(h_j,\theta'\!-\!i \, \alpha \, \eta) {\cal G}_{\nu}(\theta'\!+\!i\, \alpha \,\eta) \,
{\cal F}_{\alpha}(\theta'\!+\!i\, \alpha \, \eta), \qquad j\!=\!1,..,m_H,
\end{split}$$ $$\label{Quk}
\begin{split}
&\sum\limits_{k=1}^{m_K} \left[ Z'(u_j) \delta_{jk}+{\bf G}(u_j,u_k)\right]X_{\nu,k}^{(u)}\!=\!
f_\nu(u_j)\!-\!
\sum\limits_{k=1}^{m_S}\left({\bf G}(u_j,y_k+i \eta)+{\bf G}(u_j,y_k-i\eta)\right) X_{\nu,k}^{(y)}
\!+\! \\
&\sum\limits_{k=1}^{m_H}\!{\bf G}(u_j,h_k) X_{\nu,k}^{(h)}\!+\!\!
\sum\limits_{\alpha=\pm} \! \int\limits_{-\infty}^{\infty} \!\! \frac{d \theta'}{2 \pi}
{\bf G}(u_j,\theta'\!-\!i\, \alpha \,\eta) {\cal G}_{\nu}(\theta'\!+\!i \,\alpha \,\eta) \,
{\cal F}_{\alpha}(\theta'\!+\!i \, \alpha\, \eta), \qquad j\!=\!1,..,m_K,
\end{split}$$ $$\label{Qyk}
\begin{split}
&\sum\limits_{k=1}^{m_S} \left[ Z'(y_j) \delta_{jk}+{\bf G}(y_j,y_k+i \eta)
+{\bf G}(y_j,y_k-i \eta)\right]X_{\nu,k}^{(y)}\!=\!
f_\nu(y_j)\!+\!
\sum\limits_{k=1}^{m_H}\!{\bf G}(y_j,h_k) X_{\nu,k}^{(h)}
\!-\! \\
&\sum\limits_{k=1}^{m_K}\!{\bf G}(y_j,u_k) X_{\nu,k}^{(u)}\!+\!\!
\sum\limits_{\alpha=\pm} \! \int\limits_{-\infty}^{\infty} \!\! \frac{d \theta'}{2 \pi}
{\bf G}(y_j,\theta'\!-\!i\, \alpha \,\eta) {\cal G}_{\nu}(\theta'\!+\!i \,\alpha \,\eta) \,
{\cal F}_{\alpha}(\theta'\!+\!i \, \alpha\, \eta), \qquad j\!=\!1,..,m_S,
\end{split}$$ where $\eta$ is a positive contour deformation parameter[[^19]]{} such that $\eta<\text{min} (p , p\, \pi,|\text{Im}\, u_j|),$ ${\cal F}_\pm(\theta)$ is defined in (\[calF\]), the index $\nu$ can be either $d$ or $\ell$ telling us which derivative of $Z(\theta)$ is considered. The source term $f_\nu(\theta)$ for the two choices of the index $\nu$ is given by the formulas: $$\label{fd}
\begin{split}
f_{d}(\theta)=\left\{
\begin{array}{r}
\ell \cosh( \theta), \qquad \qquad \qquad \quad
\qquad |\text{Im} \theta|\leq\text{min}(\pi, p \pi), \\
\ell \cosh_{II}(\theta) \quad
\qquad \text{min}(\pi, p \pi)<|\text{Im} \theta|\leq \tfrac{\pi}{2}(1+p),
\end{array}\right.
\end{split}$$ $$\label{fl}
\begin{split}
f_{\ell}(\theta)=\left\{
\begin{array}{r}
\sinh( \theta), \qquad \qquad \qquad \quad
\qquad |\text{Im} \theta|\leq\text{min}(\pi, p \pi), \\
\sinh_{II}(\theta) \quad
\qquad \text{min}(\pi, p \pi)<|\text{Im} \theta|\leq \tfrac{\pi}{2}(1+p),
\end{array}\right.
\end{split}$$ where the second determination of a function is defined by (\[fII\]). The function ${\bf G}(\theta,\theta')$ in the equations (\[Gnu\])-(\[Qyk\]) agrees with $G(\theta-\theta')$ of (\[G\]) in the fundamental domain and it is equal to the appropriate second determination of $G(\theta)$ if either of its arguments goes out of the fundamental domain $|\text{Im} (\theta-\theta')|\leq\text{min}(\pi, p \pi)$. In the sequel we give the precise prescription, how to compute ${\bf G}(\theta,\theta')$ for any pair of values of its arguments. In this way we can get rid of the possible errors which can be easily committed when multiple second determination of a function should be done. The function ${\bf G}(\theta,\theta')$ will be defined as the solution of a linear integral equation. Let: $$\label{K}
\begin{split}
K(\theta)=\frac{1}{p+1} \,
\frac{\sin\tfrac{2 \pi}{p+1}}{\sinh\tfrac{\theta-i \pi}{p+1} \, \sinh\tfrac{\theta+i \pi}{p+1}}.
\end{split}$$ This function is the derivative of the scattering-phase of the elementary magnon excitations of the 6-vertex model with anisotropy parameter $\gamma=\tfrac{\pi}{p+1}.$ Then ${\bf G}(\theta,\theta')$ for arbitrary values of $\theta$ and $\theta'$ can be determined by solving the linear integral equation as follows: $$\label{bfGdef}
\begin{split}
{\bf G}(\theta,\theta')+\!\!\int\limits_{-\infty}^{\infty} \!\frac{d \theta''}{2 \pi} K(\theta-\theta'')\,
{\bf G}(\theta'',\theta')=K(\theta-\theta').
\end{split}$$ This equation can be solved by means of Fourier transformation along any horizontal lines of the complex plane. When both arguments are in the fundamental domain: $\text{max} \{|\text{Im} (\theta)|,\, |\text{Im} \theta')|\}\leq\text{min}(\pi, p \pi),$ then the solution of (\[bfGdef\]) gives the well known kernel of the NLIE of the sine-Gordon theory. Namely, ${\bf G}(\theta,\theta')=G(\theta-\theta')$ with $G(\theta)$ given in (\[G\]). The linear integral equation (\[bfGdef\]) tells us how to continuate analytically ${\bf G}(\theta,\theta')$ out of this fundamental regime. For example, if one continues one of the variables of ${\bf G}(\theta,\theta')$ out of the fundamental domain, then one gets the second determination of $G(\theta-\theta')$ defined by (\[fII\]) etc. Thus the function ${\bf G}(\theta,\theta')$ incorporates all possible second determinations which appear in the NLIE (\[DDV\])-(\[DDVII\]) of the model. This means that one does not need to take care of the subtle rules of second determination, but the solution of (\[bfGdef\]) will automatically give the functional form of ${\bf G}$ in any regime of the complex plane.
Algebraic Bethe Ansatz for the soliton transfer matrix {#appB}
======================================================
The monodromy and transfer matrices made out of the S-matrix (\[Smatr\]) of the sine-Gordon model are of central importance in this paper. They enter the form-factor axiom (\[ax4\]) and play an important role in the conjecture of [@Palmai13] for the diagonal matrix elements of local operators of the theory.
In this appendix we summarize the most important properties of the monodromy matrix and recall the Algebraic Bethe Ansatz [@FST79] diagonalization of the transfer matrix.
The basic object is the $n$-particle monodromy matrix built from the S-matrix of the model (\[Smatr\]): $$\label{monodr}
\begin{split}
{\cal T}_a^b(\theta|\theta_1,...,\theta_n)_{a_1 a_2 ... a_n}^{b_1 b_2 ...b_n}=
{\cal S}_{a \, a_1}^{k_1 \, b_1}(\theta-\theta_1)
{\cal S}_{k_1 \, a_2}^{k_2 \, b_2}(\theta-\theta_2)...
{\cal S}_{k_{n-1}\, a_n}^{b \, b_n}(\theta-\theta_n).
\end{split}$$ For the algebraic Bethe Ansatz techniques, it is generally written as a 2 by 2 matrix in the auxiliary space: $$\label{ABCD}
\begin{split}
{\cal T}(\theta|\vec{\theta})=
\begin{pmatrix} {\cal T}_-^-(\theta|\vec{\theta}) & {\cal T}_-^+(\theta|\vec{\theta}) \\
{\cal T}_+^-(\theta|\vec{\theta}) & {\cal T}_+^+(\theta|\vec{\theta})\end{pmatrix}
=\begin{pmatrix} A(\theta|\vec{\theta}) & B(\theta|\vec{\theta}) \\
C(\theta|\vec{\theta}) & D(\theta|\vec{\theta})
\end{pmatrix},
\end{split}$$ such that the entries act on the $2^n$ dimensional vector space spanned by $n$ soliton dublets ${\cal V}_n=\left({\mathbb C}^2\right)^{\otimes n} $. Here for short we introduced the notation $\vec{\theta}=\{\theta_1,\theta_2,...,\theta_n\}.$
As a consequence of the Yang-Baxter equation (\[YBE\]), the entries of the monodromy matrix satisfy the Yang-Baxter algebra relations: $$\label{YBA}
\begin{split}
{\cal S}_{a_1 \, a_2}^{k_1 \,k_2}(\theta-\theta') \,
{\cal T}_{k_1}^{b_1}(\theta|\vec{\theta}) \,
{\cal T}_{k_2}^{b_2}(\theta'|\vec{\theta})=
{\cal T}_{a_2}^{k_1}(\theta'|\vec{\theta}) \,
{\cal T}_{a_1}^{k_2}(\theta|\vec{\theta}) \,
{\cal S}_{k_1 \, k_2}^{b_2 \,b_1}(\theta-\theta').
\end{split}$$ The transfer matrix is defined as the trace of the monodromy matrix over the auxiliary space: $$\label{trans}
\begin{split}
\tau(\theta|\vec{\theta})=\sum\limits_{a=\pm} {\cal T}_a^a(\theta|\vec{\theta}).
\end{split}$$ As a consequence of (\[YBA\]) the transfer matrices form a commuting family of operators on ${\cal V}_n$: $$\label{transcom}
\begin{split}
\tau(\theta|\vec{\theta}) \, \tau(\theta'|\vec{\theta})=\tau(\theta'|\vec{\theta})\, \tau(\theta|\vec{\theta}).
\end{split}$$ This means that the eigenvectors of the transfer matrices are independent of the spectral parameter $\theta,$ but the they do depend on the inhomogeneity vector $\vec{\theta},$ such that the order of rapidities within this vector matters, as well! The transfer matrix commutes with the solitonic charge ${\cal Q}$ and the charge parity ${\cal C}$ operators, which act on a vector $V\in{\cal V}_n$ as follows: $$\label{calQ}
\begin{split}
({\cal Q} V)_{i_1 \, i_2 ....i_n}=Q \, V_{i_1 \, i_2...i_n}\, \qquad Q=\sum\limits_{k=1}^n i_k, \qquad
\end{split}$$ $$\label{calC}
\begin{split}
({\cal C}V)_{i_1\, i_2...i_n}=V_{\bar{i}_1 \, \bar{i}_2...\bar{i}_n}, \qquad \text{with} \quad \bar{i}_k=-i_k, \qquad
k=1,...,n.
\end{split}$$ The $B(\theta|\vec{\theta})$ and $C(\theta|\vec{\theta})$ elements of the monodromy matrix act as charge raising and lowering operators: $$\label{QB}
\begin{split}
[{\cal Q},B(\theta|\vec{\theta})]=2 \, B(\theta|\vec{\theta}),
\end{split}$$ $$\label{QC}
\begin{split}
[{\cal Q},C(\theta|\vec{\theta})]=-2 \, C(\theta|\vec{\theta}).
\end{split}$$ The diagonalization of the transfer matrix can be done using the usual procedure of the Algebraic Bethe Ansatz [@FST79]. There exist a trivial eigenvector of $\tau(\lambda|\vec{\theta}),$ the pure antisoliton state: $$\label{trivvac}
\begin{split}
|0\rangle_{a_1 \, a_2...a_n}=\prod\limits_{j=1}^n \delta_{a_j}^-.
\end{split}$$ Then the eigenvectors of the transfer matrix are given by acting a sequence of $B$-operators on this trivial eigenstate: $$\label{eivec1}
\begin{split}
\Psi(\{\lambda_j\}|\vec{\theta})=\frac{1}{{\cal N}_\Psi}\, B(\lambda_1|\vec{\theta})\, B(\lambda_2|\vec{\theta})...
B(\lambda_r|\vec{\theta})|0\rangle,
\end{split}$$ such that the $\lambda_j$ spectral parameters of the $B$-operators satisfy the Bethe-equations as follows: $$\label{ABAE}
\begin{split}
\prod\limits_{k=1}^n B_0(\lambda_j-\theta_k)=\prod\limits_{k\neq j}^r \frac{B_0(\lambda_k-\lambda_j)}{B_0(\lambda_j-\lambda_k)}, \qquad j=1,..,r.
\end{split}$$ The term ${\cal N}_{\Psi}$ in (\[eivec1\]) is to fix the norm of the state to the required value. In our computations the normalization condition for ${\cal N}_\Psi$ is that the norm of $\Psi$ should be $1.$ The eigenvalue of the transfer matrix on the state $\Psi(\{\lambda_j\}|\vec{\theta});$ $$\label{taupsi}
\begin{split}
\tau(\lambda|\vec{\theta}) \, \Psi(\{\lambda_j\}|\vec{\theta})=\Lambda(\lambda,\{\lambda_j\}|\vec{\theta})\, \Psi(\{\lambda_j\}|\vec{\theta}),
\end{split}$$ is given by the formula: $$\label{eival}
\begin{split}
\Lambda(\lambda,\{\lambda_j\}|\vec{\theta})=\prod\limits_{j=1}^n S_0(\lambda-\theta_k)\, \Lambda_0(\lambda,\{\lambda_j\}|\vec{\theta}),
\end{split}$$ where $S_0(\theta)$ is given in (\[CHI\]) and $\Lambda_0(\lambda,\{\lambda_j\}|\vec{\theta})$ is the eigenvalue of the transfer matrix made out of $S_{ab}^{cd}(\theta)$ and it is given by: $$\label{Lambda0}
\begin{split}
\Lambda_0(\lambda,\{\lambda_j\}|\vec{\theta})=\prod\limits_{j=1}^r \frac{1}{B_0(\lambda_j-\lambda)}+
\prod\limits_{k=1}^n B_0(\lambda-\theta_k) \,
\prod\limits_{j=1}^r \frac{1}{B_0(\lambda-\lambda_j)}.
\end{split}$$ In the computations of the paper we need an analogous to (\[eivec1\]) expression for the complex conjugate vector of $\Psi(\{\lambda_j\}|\vec{\theta}),$ too. For this reason we need the properties of the monodromy and transfer matrices under hermitian conjugation. From the properties (\[Bose\])-(\[Real\]) of the S-matrix and from the definition (\[monodr\]) one can prove the following hermitian conjugation rule for the monodromy matrix: $$\label{monoHerm}
\begin{split}
{\cal T}_{a}^b(\lambda|\vec{\theta})^\dag={\cal T}_{\bar a}^{\bar b}(\lambda^*+i \, \pi|\vec{\theta}),
\end{split}$$ which implies for the components the following rules: $$\label{ABCDH}
\begin{split}
A^\dag(\lambda|\vec{\theta})=D(\lambda^*+i \, \pi|\vec{\theta}), \qquad
D^\dag(\lambda|\vec{\theta})=A(\lambda^*+i \, \pi|\vec{\theta}), \\
B^\dag(\lambda|\vec{\theta})=C(\lambda^*+i \, \pi|\vec{\theta}), \qquad
C^\dag(\lambda|\vec{\theta})=B(\lambda^*+i \, \pi|\vec{\theta}).
\end{split}$$ It follows for the transfer matrix that: $$\label{tauH}
\begin{split}
\tau^\dag(\lambda|\vec{\theta})=\tau(\lambda^*+i \, \pi|\vec{\theta}).
\end{split}$$ Thus, the transfer matrix is a hermitian operator along the line: $\lambda=\rho+i \tfrac{\pi}{2},$ with $\rho \in {\mathbb R}.$ The hermitian conjugation relations (\[ABCDH\]) imply that the complex conjugate vector $\Psi^*(\{\lambda_j\}|\vec{\theta})$ can be represented as follows: $$\label{Psi*}
\begin{split}
\Psi^*(\{\lambda_j\}|\vec{\theta})=\langle 0|C(\lambda_1^*+i \, \pi|\vec{\theta})\,
C(\lambda_2^*+i \, \pi|\vec{\theta})...C(\lambda_r^*+i \, \pi|\vec{\theta}) \, \frac{1}{{\cal N}_\Psi}.
\end{split}$$ It can be seen that if a set $\{\lambda_j\}_{j=1}^r$ is a solution of the Bethe-equations (\[ABAE\]), then the set $\{\lambda^*_j+i \, \pi\}_{j=1}^r$ is also a solution of (\[ABAE\]). Thus for solutions which are invariant under this transformation the complex conjugate vector can be written in a simpler form: $$\label{Psi*1}
\begin{split}
\Psi^*(\{\lambda_j\}|\vec{\theta})=\langle 0|C(\lambda_1|\vec{\theta})\,
C(\lambda_2|\vec{\theta})...C(\lambda_r|\vec{\theta}) \, \frac{1}{{\cal N}_\Psi}.
\end{split}$$ Now it is easy to determine the normalization constant ${\cal N}_\Psi,$ because it is nothing but the Gaudin-norm [@Gaudin0; @Gaudin1; @Korepin] of the Bethe-state $B(\lambda_1|\vec{\theta})\, B(\lambda_2|\vec{\theta})...
B(\lambda_r|\vec{\theta})|0\rangle:$ $$\label{calN}
\begin{split}
{\cal N}_\Psi^2=\langle 0|C(\lambda_1|\vec{\theta})\,
C(\lambda_2|\vec{\theta})...C(\lambda_n|\vec{\theta}) \, B(\lambda_1|\vec{\theta})\, B(\lambda_2|\vec{\theta})...
B(\lambda_r|\vec{\theta})|0\rangle.
\end{split}$$ If one would like to apply the Algebraic Bethe Ansatz technique directly to $\tau(\lambda|\vec{\theta})$, one should carry unnecessarily a lot of $S_0(\theta)$ factors. This can be avoided, if one diagonalizes the transfer matrix constructed out of the $S_0$ removed part of the S-matrix. To be more concrete analogously to (\[monodr\]) one should define the “reduced” monodromy matrix by the formula: $$\label{monodrR}
\begin{split}
{\sc T}_a^b(\theta|\theta_1,...,\theta_n)_{a_1 a_2 ... a_n}^{b_1 b_2 ...b_n}=
{ S}_{a \, a_1}^{k_1 \, b_1}(\theta-\theta_1)
{ S}_{k_1 \, a_2}^{k_2 \, b_2}(\theta-\theta_2)...
{ S}_{k_{n-1}\, a_n}^{b \, b_n}(\theta-\theta_n),
\end{split}$$ where $S_{ab}^{cd}(\theta)$ is the matrix part of the S-matrix (\[Smatr\]) given by (\[Selem\])-(\[C0\]). Analogously to (\[ABCD\]) it can be written as a 2 by 2 matrix in the auxiliary space: $$\label{ABCDR}
\begin{split}
{\sc T}(\lambda|\vec{\theta})=
\begin{pmatrix} {\sc T}_-^-(\lambda|\vec{\theta}) & {\sc T}_-^+(\lambda|\vec{\theta}) \\
{\sc T}_+^-(\lambda|\vec{\theta}) & {\sc T}_+^+(\lambda|\vec{\theta})\end{pmatrix}
=\begin{pmatrix} {\cal A}(\lambda|\vec{\theta}) & {\cal B}(\lambda|\vec{\theta}) \\
{\cal C}(\lambda|\vec{\theta}) & {\cal D}(\lambda|\vec{\theta})
\end{pmatrix}.
\end{split}$$ Its matrix elements satisfy the same Yang-Baxter algebra (\[YBA\]) as those of ${\cal T}(\lambda|\vec{\theta}).$ The “reduced” transfer matrix ${\sc t}(\lambda|\vec{\theta})$ is defined by taking the trace in the auxiliary space: $$\label{transR}
\begin{split}
{\sc t}(\lambda|\vec{\theta})=\sum\limits_{a=\pm} {\sc T}_a^a(\lambda|\vec{\theta}).
\end{split}$$ It differs from $\tau(\lambda|\vec{\theta})$ in only a trivial scalar factor: $$\label{tautau}
\begin{split}
\tau(\lambda|\vec{\theta})=\prod\limits_{k=1}^n S_0(\lambda-\theta_k)\, {\sc t}(\lambda|\vec{\theta}).
\end{split}$$ Thus their common eigenvector $\Psi(\lambda,\{\lambda_j\}|\vec{\theta})$ (\[eivec1\]) and its complex conjugate (\[Psi\*1\]) can be expressed in terms of the elements of the “reduced” monodromy matrix completely analogously to the formulas (\[eivec1\]) and (\[Psi\*1\]): $$\label{Psi1R}
\begin{split}
\Psi(\{\lambda_j\}|\vec{\theta})=\frac{1}{N_\Psi}\, {\cal B}(\lambda_1|\vec{\theta})\, {\cal B}(\lambda_2|\vec{\theta})...
{\cal B}(\lambda_r|\vec{\theta})|0\rangle,
\end{split}$$ $$\label{Psi*1R}
\begin{split}
\Psi^*(\{\lambda_j\}|\vec{\theta})=\langle 0|{\cal C}(\lambda_1|\vec{\theta})\,
{\cal C}(\lambda_2|\vec{\theta})...{\cal C}(\lambda_n|\vec{\theta}) \, \frac{1}{ N_\Psi}.
\end{split}$$ Certainly the normalization factor is also changed compared to (\[eivec1\]) and (\[Psi\*1\]): $$\label{NPsi}
\begin{split}
{N}_\Psi^2=\langle 0|{\cal C}(\lambda_1|\vec{\theta})\,
{\cal C}(\lambda_2|\vec{\theta})...{\cal C}(\lambda_n|\vec{\theta}) \, {\cal B}(\lambda_1|\vec{\theta})\, {\cal B}(\lambda_2|\vec{\theta})...
{\cal B}(\lambda_r|\vec{\theta})|0\rangle,
\end{split}$$ which can be written as a Slavnov-determinant [@Slavnov]. The eigenvalue of ${\sc t}(\lambda|\vec{\theta})$ on $\Psi(\{\lambda_j\}|\vec{\theta})$ is exactly $\Lambda_0(\lambda,\{\lambda_j\}|\vec{\theta})$ given in (\[Lambda0\]).
We continue this appendix by specializing the main formulas of this appendix to the 3-particle case.
Formulas for the 3-particle case
--------------------------------
Due to the charge conjugation symmetry, in the 3-particle case the number of Bethe-roots in (\[ABAE\]) can be either zero or one. The zero root case corresponds to the trivial pure solitonic eigenvector (\[trivvac\]). Here we do not deal with this trivial case, but we are interested in the state described by a single Bethe-root. In this case the Bethe-equations take the simple form: $$\label{3BAE}
\begin{split}
\prod\limits_{j=1}^3 B_0(\lambda_1-\theta_k)=1.
\end{split}$$ The eigenvalue of the soliton transfer matrix (\[eival\]), when its spectral parameter takes the value of one of the rapidities, is given[[^20]]{} by: $$\label{3eival}
\begin{split}
\Lambda(\theta_j|\vec{\theta})=\prod\limits_{k=1}^3 S_0(\theta_j-\theta_k) \frac{1}{B_0(\lambda_1-\theta_j)}, \qquad j=1,2,3.
\end{split}$$ In this one-root case the normalization factor $N_\Psi$ in (\[NPsi\]) takes the form: $$\label{3NPsi}
\begin{split}
{N}_\Psi^2=\langle 0|{\cal C}(\lambda_1|\vec{\theta})\,
{\cal B}(\lambda_1|\vec{\theta})|0\rangle=p \, \sinh\left( \tfrac{i \, \pi}{p}\right) \, \sum\limits_{j=1}^3 (\ln B_0)'(\lambda_1-\theta_j).
\end{split}$$ In the computation of the symmetric form-factors some derivatives with respect to the particle’s rapidities will be important. Differentiating (\[3eival\]) with respect to $\theta_q$ one obtains: $$\label{Lsq}
\begin{split}
\partial_q \log \Lambda(\theta_s|\vec{\theta})=-i \,G(\theta_s-\theta_q)-(\ln B_0)'(\lambda_1-\theta_s) \,
\frac{\partial \lambda_1}{\partial \theta_q}, \qquad %\sigma_{sq}=\frac{1}{2 \pi } G(\theta_s-\theta_q),
\qquad s\neq q,
\end{split}$$ with $G(\theta)$ given in (\[G\]). The derivative $\frac{\partial \lambda_1}{\partial \theta_q}$ can be obtained by differentiating the Bethe-equation (\[3BAE\]): $$\label{3lamder}
\begin{split}
\frac{\partial \lambda_1}{\partial \theta_q}=
\frac{(\ln B_0)'(\lambda_1-\theta_q)}{\sum\limits_{k=1}^3 (\ln B_0)'(\lambda_1-\theta_k) }, \qquad q=1,2,3.
\end{split}$$ If we have one single root then due to the $\lambda \to \lambda^*+i \, \pi$ symmetry of the Bethe-equation the single root of the equation can be either “real” or “self-conjugate”. A solution $\lambda_1$ is called real, if it is a fixed point of the symmetry $\lambda \to \lambda^*+i \, \pi,$ i.e. $\lambda_1= \lambda_1^*+i \, \pi.$ Here we use the term “real”, because in a more convenient parameterization this type of roots would be actually a real numbers. Namely, if it is parameterized as $\lambda_1=\rho_1+i \tfrac{\pi}{2}$ then the fix point equation restricts $\rho_1$ to be real.
Due to the $i \,p \, \pi $ symmetry of the functions entering the the Bethe-equations (\[ABAE\]), they have another symmetry, as well. Namely if $\lambda_j$ is a solution of the equations then $\lambda_j+i \, \pi \, p$ is also a solution. This means that the solutions can be resticted to a fundamental domain given by the strip of width $i \, p \, \pi.$ By definition a “self-conjugate” root satisfies the combination of symmetries: $\lambda \to \lambda^*+i \, \pi$ and $\lambda \to \lambda\pm i \,p \, \pi,$ namely $$\label{lsc}
\begin{split}
\lambda_1=\lambda_1^*+i\, \pi\pm i \, p \pi.
\end{split}$$ If it is parameterized again as $\lambda_1=\rho_1+i \, \tfrac{\pi}{2},$ then $\rho_1$ has a fixed imaginary part: $\text{Im} \rho_1=\tfrac{p \, \pi}{2}.$
The numerical solution of the equation (\[3BAE\]) shows that in the repulsive regime $(1<p)$ from the 3 different solutions, two ones are self-conjugated and one is real.
Classification of the magnonic Bethe-roots {#appABA2}
------------------------------------------
As the simple discussion at the end of the previous subsection shows, there are two symmetries of the magonic Bethe-equations (\[ABAE\]): $$\label{sym12}
\begin{split}
&\bullet \qquad \qquad \{\lambda_j\}_{j=1}^r= \{\lambda_j^*+i \, \pi\}_{j=1}^r, \\
&\bullet \qquad \qquad \{\lambda_j\}_{j=1}^r= \{\lambda_j+i \,p\, \pi\}_{j=1}^r.
\end{split}$$ They imply the following classification of the roots. $$\begin{aligned}
&\bullet& \, \text{Real-roots:} \qquad \, \text{Im}\, (\lambda_j-i \, \tfrac{\pi}{2})=0, \qquad
\qquad \qquad \qquad \qquad j=1,..,n_r, \nonumber \\
&\bullet& \, \text{Close-roots:} \qquad |\text{Im}\, (\lambda_j-i \, \tfrac{\pi}{2})|\leq
\text{min} (\tfrac{\pi}{2},\tfrac{ (2 p-1)\, \pi}{2}) , \qquad \quad \, j=1,..,n_c, \label{rootclassR} \\
&\bullet& \, \text{Wide-roots:} \qquad \text{min} (\tfrac{\pi}{2},\tfrac{ (2 p-1)\, \pi}{2}) < |\text{Im}\,
(\lambda_j-i \, \tfrac{\pi}{2})|\leq \tfrac{p \, \pi}{2} , \quad j=1,..,n_w. \nonumber\end{aligned}$$ A special type of wide-root is the self-conjugate root, whose imaginary part is exactly $i \tfrac{(1+p) \,\pi}{2}.$ From the symmerties (\[sym12\]) of the asymptotic Bethe-equations it also follows that all roots, which are neither real nor self-conjugate appear in pairs being symmetric to the line $\text{Im} z=\tfrac{\pi}{2}.$ In this way we can speak about close-and wide-pairs similarly to the Bethe-roots entering the NLIE (\[DDV\]), which describes the exact finite volume spectrum of the sine-Gordon model.
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[^1]: Only the terms being polynomials in the inverse of the volume remain.
[^2]: The matrix part $S_{ab}^{cd}(\theta)$ of the S-matrix also satisfies the Yang-Baxter equation.
[^3]: With this interpretation of special objects the contour deformation parameter $\eta$ should be considered to be a positive infinitesimal number.
[^4]: Our formulas are valid for the massive Thirring model, as well. Only the value of the parameter $\delta$ should be set in accordance with (\[MTkvant\]).
[^5]: For more precise notation see (\[ujset\]) in appendix \[appA\].
[^6]: Though it was not specified clearly in [@Palmai13], we assume the following orderings within these sets: $A_i<A_j$ and $\bar{A}_i<\bar{A}_j$ if $i<j.$
[^7]: Normalized eigenvectors mean that they fullfill the conditions (\[bazis\]) and (\[ortog\]).
[^8]: Here the word diagonal means diagonality in the Bethe eigenstates, as well.
[^9]: We just note that (\[Fax2diag\]) remains valid if the sets $\{\lambda_j\}_{j=1}^r$ and $\{\lambda_j^\epsilon\}_{j=1}^r$ are not solutions of the Bethe-equations, but are arbitrary sets. Here we require them to be solutions of (\[ABAE\]) for later convenience.
[^10]: Namely, the $\epsilon \to 0 $ limit of the scalar product $C^{i_1 i_2 i_3} B^\epsilon_{i_1 i_2 i_3}.$
[^11]: Since their sum enter the right hand of the kinematical pole axiom (\[Fax3diag\]). See section \[sect9\] for a more detailed discussion.
[^12]: For arbitrary number of solitons and not only upto 3.
[^13]: We denote this case by writing symbolically $\emptyset$ instead of $\lambda$ into the argument.
[^14]: Namely, in this case form-factors are invariant with respect to conjugating their indexes.
[^15]: We just recall: $B_j=B_0(\lambda_1-\theta_j), \quad C_j=C_0(\lambda_1-\theta_j), \quad V_j=(\ln B_0)'(\lambda_1-\theta_j).$
[^16]: The same structure arose in the 2-particle case. See (\[FOs2TH\]).
[^17]: We just recall the notation used in the preceding sections: $\theta_j^{\epsilon}=\theta_j+\epsilon_j,\quad j=1,..,n.$
[^18]: In the $\epsilon \to 0$ limit.
[^19]: If $m_S\neq 0$ it is preferable to consider $\eta$ to be a positive infinitesimal parameter.
[^20]: Since here we have only one Bethe-root, for short we skipped it from the list of arguments of the eigenvalue.
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abstract: 'The Kaup-Newell (KN) hierarchy contains the derivative nonlinear Schr" odinger equation (DNLSE) amongst others interesting and important nonlinear integrable equations. In this paper, a general higher grading affine algebraic construction of integrable hierarchies is proposed and the KN hierarchy is established in terms of a $\hat{s\ell}_2$ Kac-Moody algebra and principal gradation. In this form, our spectral problem is linear in the spectral parameter. The positive and negative flows are derived, showing that some interesting physical models arise from the same algebraic structure. For instance, the DNLSE is obtained as the second positive, while the Mikhailov model as the first negative flows, respectively. The equivalence between the latter and the massive Thirring model is explicitly demonstrated also. The algebraic dressing method is employed to construct soliton solutions in a systematic manner for all members of the hierarchy. Finally, the equivalence of the spectral problem introduced in this paper with the usual one, which is quadratic in the spectral parameter, is achieved by setting a particular automorphism of the affine algebra, which maps the homogeneous into principal gradation.'
author:
- 'G. S. França$^{ab\star}$, J. F. Gomes$^{b\dagger}$ and A. H. Zimerman$^{b\ddagger}$'
bibliography:
- 'biblio.bib'
title: '**The algebraic structure behind the derivative nonlinear Schr" odinger equation** '
---
$^{a}$[*Department of Physics, Cornell University, Ithaca, NY, USA* ]{}
$^{b}$[ *Instituto de F' isica Te' orica—IFT/UNESP, S\~ ao Paulo, SP, Brazil* ]{}
(1,0)[175]{}
[$^{\star}[email protected], $^{\dagger}[email protected], $^{\ddagger}[email protected] ]{}
Introduction {#sec:intro}
============
The *derivative nonlinear Schr" odinger equation* (DNLSE-I) $$\label{dnlse1}
i\pa_t \psi + \pa_x^2 \psi \pm i \pa_x\(|\psi|^2 \psi\) = 0$$ is a well known integrable model with interesting physical applications. In particular, it describes nonlinear Alfv' en waves in plasma physics [@mio; @mjolhus; @mjolhus2; @spangler; @ruderman; @fedun] and the propagation of ultra-short pulses in nonlinear optics [@tzoar; @anderson_lisak; @degasperis].
Equation , and also other related models that will be mentioned in the following, has been extensively studied since a long time ago. Its inverse scattering transform (IST) with a vanishing boundary condition (VBC), $\psi\to0$ as $|x|\to\infty$, was first solved in [@kaup_newell]. Equation is one of the nonlinear evolution equations comprising the Kaup-Newell (KN) hierarchy.
The complete integrability of and the hierarchy of Hamiltonian structures of the KN hierarchy were constructed in [@gerdjikov_kulish]. Moreover, the Riemann-Hilbert problem was considered as well as expansions over the squared solutions [@gerdjikov_kulish]. B" acklund transformation for the KN hierarchy was obtained [@kundu_backlund], and also a nonholonomic deformation of the hierarchy was proposed [@kundu_deformation]. A specific feature of these models is that they have a Lax operator containing a *quadratic* power on the spectral parameter. More specifically, and its related models can be obtained from a zero-curvature representation where the standard spectral problem is given by $$\label{kn_lax}
A_x = \begin{pmatrix}
\l^2 & \l q \\
\l r & -\l^2
\end{pmatrix}, \qquad \(\pa_x + A_x\)\Psi = 0,$$ where $q=q(x,t)$ and $r=r(x,t)$ are the fields and $\l$ is the complex spectral parameter. Due to the quadratic power of $\l$ the original form of the IST [@zakharov_shabat] has a divergent Cauchy integral when $|\l|\to\infty$. This is the main reason for introducing the revision in the IST [@kaup_newell], through some weight functions that control such divergence. Let us recall that the Zakharov-Shabat approach [@zakharov_shabat] was initially proposed to solve the Ablowitz-Kaup-Newell-Segur (AKNS) spectral problem $$\label{akns_lax}
A_x = \begin{pmatrix}
\l & q \\
r & -\l
\end{pmatrix}.$$ Another obvious difference worth mentioning is that in the fields are associated to $\l$, unlike . This implies in a *higher grading* construction from the algebraic formalism point of view, contrary to the AKNS construction where the fields are in a *zero grade* subspace.
For a nonvanishing boundary condition (NVBC), $\psi\to\mbox{const.}$ as $|x|\to\infty$, an IST approach was also proposed [@kawata_inoue; @kawata_kobayashi_inoue], but it is a difficult procedure due to the appearance of a double-valued function of $\l$ and, therefore, the IST had to be developed on its Riemann sheets. A more straightforward method introduces a convenient affine parameter that avoids the construction of the Riemann sheets [@chen_lam]. Recently, solutions with VBC and also NVBC were constructed through Darboux/B" acklund transformations yielding explicit and useful formulas [@steudel; @xu_wang_darboux]. These works are all based on operator or the revised form of the IST [@kaup_newell].
There are also two other known types of derivative nonlinear Schr" odinger equations, namely, the DNLSE-II [@chen_lee_liu] $$\label{dnlse2}
i\pa_t\psi + \pa_x^2\psi \mp 4i |\psi|^2\pa_x\psi = 0$$ and the DNLSE-III [@gerdjikov] $$\label{dnlse3}
i\pa_t\psi + \pa_x^2\psi \pm 4i\psi^2\pa_x\psi^* + 8|\psi|^4\psi = 0.$$ The gauge equivalence between , and was analyzed for the first time in [@gerdjikov] and its Hamiltonian structures have also being extensively studied. (Regarding gauge equivalent models see also [@kundu_gauge].) A hierarchy containing within the Sato-Wilson dressing formalism was also considered and it was shown that can be reduced to the fourth Painlev' e equation [@kakei_kikuchi].
The well known *massive Thirring model* $$\label{thirr}
\begin{split}
i\pa_x v - mu + 2g |u|^2v &= 0, \\
i\pa_t u + mv - 2g |v|^2u &= 0,
\end{split}$$ was proved to be integrable and solved through the IST for the first time in [@mikhailov_thirring] (see also [@orfanidis; @thirring_kaup_newell]). The relation between and was also pointed out [@gerdjikov_kulish; @gerdjikov]. We will show this relation precisely through another model, arising naturally from the first negative flow of the KN hierarchy, namely $$\label{modPLR}
\pa_x\pa_t\vphi - \vphi \mp 2i|\vphi|^2\pa_x \vphi = 0.$$ This relativistically invariant model has attracted attention only recently although it was already proposed a long time [@gerdjikov_kulish; @gerdjikov] and is known as the *Mikhailov model*. It is already known that is equivalent to [@gerdjikov_kulish; @gerdjikov]. Equation was also called Fokas-Lenells equation in recent works and its multi-soliton solutions were obtained through Hirota’s bilinear method [@matsuno_fl], where it was pointed out that they have essentially the same form as those of if one introduces the potential $\psi=\pa_x\varphi$ and change the dispersion relation. We will explain precisely the origin of this relation. Equation was also referred as the modified Pohlmeyer-Lund-Regge model [@kikuchi_mplr] and it can be reduced to the third Painlev' e equation.
It is important to mention that there are matrix or multi-field generalizations of DNLS and Thirring like models [@tsuchida_wadati; @tsuchida_wadati2; @tsuchida].
The integrable properties of , , and , like their soliton solutions and Hamiltonian hierarchies, had already been thoroughly studied, specially in [@kaup_newell; @gerdjikov_kulish; @gerdjikov; @mikhailov_thirring]. Nevertheless, these models, and more recently , continue to attract attention and they have not being fully studied through more recent techniques. One of the approaches to obtain soliton solutions is through the dressing method proposed in the pioneer paper [@zakharov_shabat], which is connected to the Riemann-Hilbert problem. Another approach occurs in connection to $\tau$-functions and transformation groups [@date_jimbo], where solitons are obtained through vertex operators. The precise relation between these two methods were established in [@babelon_dressing] (see also [@babelon_book] for a thoroughly explanation) and it enables one to construct soliton solutions in a purely affine algebraic manner. Thus we refer to this approach as the *algebraic dressing method*.
Since then [@babelon_dressing], there has been a significant development of affine algebraic techniques [@olive_vertex; @gomes_soliton; @aratyn_symmetry_flows; @aratyn_construction] relying on the algebraic structure underlying the equations of motion, providing *general* and *systematic methods*. This algebraic approach is well understood for models that fit into the AKNS construction, where the fields are associated to *zero grade* operators. Nevertheless, there are some generalizations of this formalism, for instance, the addition of fermionic fields to *higher grading* operators [@gervais_higher_grading; @ferreira_affine_toda_matter] and a generalization of the dressing method to include NVBC for the AKNS construction [@dressing_nvc; @mkdv_nvc; @negative_even]. Recently, it was proposed a higher grading construction that includes the Wadati-Konno-Ichikawa hierarchy and the algebraic dressing formalism was supplemented with reciprocal transformations [@wki_spe]. Following this line of though, it is important to embrace other known integrable models into such algebraic formalism, to extend these techniques beyond the AKNS scheme. This is accomplished here regarding the KN hierarchy. We introduce its underlying algebraic structure and employ the algebraic dressing method to construct its soliton solutions systematically. Through gauge transformations we also obtain solutions of the other related models like , and , for instance. An important remark is in order. We construct these models starting from a spectral problem that is *not quadratic* on the spectral parameter and it simplifies the procedure to construct the solutions. We also prove that our construction is equivalent to the quadratic one .
Our work is thus organized as follows. In section \[sec:general\_hierarchy\] we introduce a *general* higher grading affine algebraic construction of integrable hierarchies. In section \[sec:kn\_hierarchy\], when the algebra $\hat{A}_1$ with *principal gradation* is chosen, this construction yields the KN hierarchy as a particular case, where appears as the second positive and as the first negative flows, respectively. We introduce the explicit transformations relating equations , and . Furthermore, we propose a transformation that takes to the Thirring model , demonstrating their equivalence. In section \[sec:dressing\] we employ the algebraic dressing method, which along with representation theory of affine Lie algebras, provides a systematic construction of the solutions for all the models within the KN hierarchy. We also obtain the explicit solutions of the other mentioned models. In section \[sec:equivalency\] we prove the equivalence between our construction, which is linear in $\l$, and the usual one . Finally, section \[sec:conclusions\] contains our concluding remarks. We refer the reader to the appendix \[sec:algebra\] for the algebraic concepts involved in this paper.
A general integrable hierarchy {#sec:general_hierarchy}
==============================
Let $\alie$ be a semi-simple Kac-Moody algebra with a grading operator $Q$. The algebra is then decomposed into graded subspaces[^1] $$\alie = \sum_{m\in\integer}\alie^{(m)}, \qquad
\alie^{(m)} = \big\{ T_a^{n} \ | \ \big[Q,T_a^{n}\big]=mT_a^{n}\big\}.$$ The parentheses in the superscript of an operator denote its grade according to $Q$ and should not be confused with the affine index $n$ without parentheses, which is the power of the spectral parameter.
Let $\E{2}$ be a semi-simple element of *grade two*. Define the kernel, $\ck$, and image, $\cm$, subspaces as follows $$\label{kernel_image}
\begin{split}
\ck &\equiv \big\{ T_a^{n} \in \alie \ | \ \big[ E^{(2)}, T_a^{n}\big] =
0 \big\}, \\
\alie &= \ck \oplus \cm.
\end{split}$$ From the Jacobi identity we conclude that $$\label{subset1}
\[\ck,\ck\]\subset\ck, \qquad \[\ck,\cm\]\subset\cm,$$ and we assume the symmetric space structure $$\label{subset2}
\[\cm,\cm\]\subset\ck.$$
Let $\F{1}[\bvphi]\in \cm^{(1)}$ be the operator containing the *fields* of the models, i.e. choose all generators of *grade one* in $\cm$, $\cm^{(1)}=\blcurl R_1^{(1)}, \dotsc, R_r^{(1)} \brcurl$, and take the linear combination $\F{1}[\bvphi]=\vphi_1R_1^{(1)}+\dotsb+
\vphi_rR_r^{(1)}$ where $\bvphi=(\vphi_1,\dotsc,\vphi_r)$ and $\vphi_i=\vphi_i(x,t)$. Now we define the integrable hierarchy of PDE’s starting from the zero curvature condition $$\label{zero_curvature}
\pa_xA_t - \pa_tA_x + \[A_x, A_t\] = 0$$ with the potentials defined as follows
\[potentials\] $$\begin{aligned}
A_x &\equiv \E{2} + \F{1}[\bvphi], \label{potential_u}\\
A_t &\equiv \sum_{m=-2M}^{2N}\D{m}[\bvphi], \label{potential_v}\end{aligned}$$
where $N$ and $M$ are arbitrary fixed *positive integers* that label each model within the hierarchy and $\D{m}[\bvphi] \in \alie^{(m)}$.
Let us now show that with the algebraic structure defined by , the zero curvature equation can be solved nontrivially, yielding a PDE for each choice of $N$ and $M$. Note that is well defined and independent of $N$ and $M$, therefore, we need to show that each $\D{m}$ in can be uniquely determined in terms of the fields. Projecting into each graded subspace we obtain the following set of equations $$\begin{aligned}
\big[\E{2},\D{2N}\big] &=0 \qquad \text{grade $2N+2$}, \nnn
\big[\E{2},\D{2N-1}\big] + \big[\F{1},\D{2N}\big] &=0
\qquad\text{grade $2N+1$}, \nnn
\pa_x\D{2N} + \big[\E{2},\D{2N-2}\big] + \big[\F{1},\D{2N-1}\big] &=0 \qquad
\text{grade $2N$},\nnn
& \ \, \vdots \nnn
\pa_x\D{2} + \big[\E{2},\D{0}\big] + \big[\F{1},\D{1}\big] &=0 \qquad
\text{grade $2$},\nnn
\pa_x\D{1} - \pa_t\F{1}+ \big[\E{2},\D{-1}\big] + \big[\F{1},\D{0}\big] &= 0
\qquad \text{grade $1$}, \label{set_of_equations} \\
\pa_x\D{0} + \big[\E{2},\D{-2}\big] + \big[\F{1},\D{-1}\big] &= 0
\qquad \text{grade 0},\nnn
& \ \, \vdots \nnn
\pa_x\D{-2M+2} + \big[\E{2},\D{-2M}\big] + \big[\F{1},\D{-2M+1}\big] &=0
\qquad\text{grade $-2M+2$},\nnn
\pa_x\D{-2M+1} + \big[\F{1},\D{-2M}\big] &= 0 \qquad \text{grade $-2M+1$},\nnn
\pa_x\D{-2M} &= 0 \qquad \text{grade $-2M$}. \nn\end{aligned}$$ These equations can be solved recursively, starting from grade $2N+2$ until grade $2$ and from grade $-2M$ until grade zero. Each equation still splits into $\ck$ and $\cm$ components according to and . We also have the $\ck$ component of the grade one equation as a constraint. Thus each $\D{m}$ is determined in terms of the fields $\bvphi$ and, consequently, the equations of motion are obtained from the $\cm$ component of the grade one projection $$\label{movement}
\pa_x\D{1}_\cm-\pa_t \F{1} + \[ \E{2}, \D{-1}_\cm \] +
\[ \F{1}, \D{0}_\ck \] = 0.$$
The potential generates *mixed flows* [@gomes_mixed]. If one is interested in *positive flows* only, yielding the positive part of the hierarchy, one must restrict the sum as $$\label{positive_flows}
A_t\equiv\sum_{m=1}^{2N}\D{m}[\bvphi]$$ and the equations of motion then simplify to $$\label{motion_2}
\pa_x\D{1}_\cm-\pa_t \F{1} = 0.$$ For *negative flows* one must restrict the sum as $$\label{negative_flows}
A_t\equiv\sum_{m=0}^{-2N}\D{m}[\bvphi]$$ and the equations of motion are then given by $$\label{movement3}
\pa_t\F{1}-\[ \E{2}, \D{-1}_\cm\]-\[ \F{1},\D{0}_\ck\]=0.$$
Therefore, we have shown that the hierarchy defined by solves the zero curvature equation . This construction is valid for an arbitrary graded affine Lie algebra $\alie$ and the hierarchy is defined through the choice of $\langle \alie,Q,\E{2}\rangle$, leading to an immediate algebraic classification. Each choice of positive integers $N$ and $M$ yields one mixed model. For positive or negative flows the models are labeled by $N$ only.
The Kaup-Newell hierarchy {#sec:kn_hierarchy}
=========================
Let us take the previous construction with the loop-algebra $\llie=\tilde{A}_1=\{\H{n},\Ea{n},\Eb{n}\}$ and *principal gradation* $Q=\tfrac{1}{2}\H{0}+2\hd$, leading to the decomposition $\alie^{(2m)}=\{\H{m}\}$ and $\alie^{(2m+1)}=\{\Ea{m}, \Eb{m+1}\}$. The semi-simple element is chosen as $E^{(2)}=\H{1}$ and then is given by $$\label{graded_sub}
\ck^{(2m)}=\big\{\H{m} \big\}, \qquad
\cm^{(2m+1)}=\big\{\Ea{m}, \Eb{m+1}\big\}.$$ The operator containing the fields must now have the form $F^{(1)} = q(x,t)\Ea{0} + r(x,t)\Eb{1}$ and then reads
\[kn\_potentials\] $$\begin{aligned}
A_x &= \H{1} + q\Ea{0} + r\Eb{1}, \label{kn_p1} \\
A_t &= \sum_{m=1}^{2N}\D{m}
\qquad \text{or} \qquad
A_t=\sum_{m=0}^{-2N}\D{m}, \label{kn_p2}\end{aligned}$$
where $\D{2m}=c_{2m}\H{m}$ and $\D{2m+1}=a_{2m+1}\Ea{m}+b_{2m+1}\Eb{m+1}$. The coefficients $a_{2m+1}$, $b_{2m+1}$ and $c_{2m}$ will be determined in terms of the fields $q$ and $r$ by solving the zero curvature equation, as explained previously in . The first equality in is valid for the *positive flows*, while the second one for the *negative flows*.
#### Comment.
If instead of the principal gradation one considers the *homogeneous* gradation $Q=\hd$, yielding $\alie^{(m)}=\lcurl \Ea{m}, \Eb{m}, \H{m}\rcurl$, and the semi-simple element $\E{2}=\H{2}$, we obtain the following Lax pair for the KN hierarchy
\[standard\_lax\_kn\] $$\begin{aligned}
A_x &= \H{2} + q\Ea{1} + r\Eb{1}, \label{standard1} \\
A_t &= \sum_{m=1}^{2N}\D{m} \qquad \mbox{or} \qquad
A_t = \sum_{m=0}^{-2N}\D{m},\end{aligned}$$
where $\D{m} = a_m\Ea{m} + b_m\Eb{m} + c_m\H{m}$. The operator is exactly the standard one found in the literature [@kaup_newell; @gerdjikov_kulish; @gerdjikov; @mikhailov_thirring] (see the matrix representation in appendix \[sec:algebra\]). With the construction we obtain precisely the same equations of motion as the construction , which will be derived in the sequel. Moreover, we will demonstrate the equivalence between both constructions in section \[sec:equivalency\]. Note, however, that *does not have a quadratic power* on the spectral parameter, contrary to . The convenience of using appears clearly when constructing the solutions of the models within the hierarchy.
Positive flows
--------------
The models within the positive part of the hierarchy are obtained from the zero curvature equation $$\label{kn_positive}
\[\pa_x+\H{1}+q\Ea{0}+r\Eb{1},
\pa_t+\D{2N}+\D{2N-1}+\dotsb+\D{1}\]=0.$$ For $N=1$ we have the trivial equations $\pa_t q=\pa_xq$ and $\pa_t r = \pa_x r$. For $N=2$, after solving each equation in , we get the following equations of motion
\[dnls\] $$\begin{aligned}
2\pa_t q + \pa_x^2q + \pa_x\(q^2r\) &= 0, \\
2\pa_t r - \pa_x^2r + \pa_x\( qr^2\) &=0,\end{aligned}$$
whose explicit Lax pair is given by
\[dnls\_lax\] $$\begin{aligned}
A_x&=\H{1}+q\Ea{0}+r\Eb{1}, \\
A_t&=\H{2}+q\Ea{1}+r\Eb{2}-\tfrac{1}{2}qr\H{1}
- \tfrac{1}{2}\( q^2r+\pa_xq\)\Ea{0}-
\tfrac{1}{2}\( qr^2-\pa_xr\)\Eb{1}.\end{aligned}$$
Taking under the transformations $x \to ix$, $t \to 2it$, $q = \psi$ and $r = \pm \psi^*$, we obtain precisely equation , $$\label{dnlse}
i\pa_t \psi + \pa_x^2 \psi \pm i\pa_x\(|\psi|^2\psi\) = 0.$$ Considering $N=3$, and after solving , we obtain the model
\[kn3\] $$\begin{aligned}
4\pa_t q - \pa_x^3 q - 3\pa_x\(qr\pa_xq\) - \tfrac{3}{2}\pa_x\(q^3r^2\) &= 0,\\
4\pa_t r - \pa_x^3 r + 3\pa_x\(qr\pa_xr\) - \tfrac{3}{2}\pa_x\(q^2r^3\) &= 0,\end{aligned}$$
together with its Lax pair
$$\begin{aligned}
A_x &= \H{1} + q\Ea{0} + r\Eb{1}, \\
A_t &= \H{3} + q\Ea{2}+r\Eb{3} - \tfrac{1}{2}qr\H{2}
-\tfrac{1}{2}\( q^2r+\pa_xq\)\Ea{1}
\nn \\ & \qquad
-\tfrac{1}{2}\( qr^2 - \pa_xr\)\Eb{2}
+\tfrac{1}{4}\( r\pa_xq - q\pa_xr + \tfrac{3}{2}q^2r^2\)\H{1}
\nn \\ & \qquad
+\tfrac{1}{4}\( \pa_x^2q + 3qr\pa_xq + \tfrac{3}{2}q^3r^2\) \Ea{0}
+\tfrac{1}{8}\( \pa_x^2r - 3qr\pa_xr + \tfrac{3}{2}q^2r^3 \) \Eb{1}.\end{aligned}$$
The system under $x\to ix$, $t\to -4it$, $q=\psi$ and $r=\pm \psi^*$ becomes $$\pa_t\psi - \pa_x^3\psi \mp 3i \pa_x\(|\psi|^2\pa_x\psi\)
+\tfrac{3}{2}\pa_x\(|\psi|^4 \psi \) = 0.$$ Continuing in this way for $N=4,5,\dotsc$ one can obtain higher order nonlinear PDE’s.
Negative flows
--------------
The negative flows of the KN hierarchy are obtained from the zero curvature equation $$\label{kn_negative}
\[\pa_x+\H{1}+q\Ea{0}+r\Eb{1},
\pa_t+\D{-2N}+\D{-2N+1}+\dotsb+\D{0}\]=0.$$ Taking $N=1$, after solving , we obtain the following nonlocal field equations
\[mpl\_nloc\] $$\begin{aligned}
\tfrac{1}{4}\pa_t q - \int_{-\infty}^x qdx' +
q\int_{-\infty}^x qdx' \int_{-\infty}^x rdx' &= 0, \\
\tfrac{1}{4}\pa_t r - \int_{-\infty}^x rdx' -
r\int_{-\infty}^x qdx'\int_{-\infty}^x rdx' &=0.\end{aligned}$$
These equations can be cast in a local form if we introduce new fields defined by $$\label{fields_derivative}
q \equiv \pa_x \phi, \qquad
r \equiv \pa_x \rho,$$ and then we obtain the relativistically invariant *Mikhailov model* [@gerdjikov_kulish; @gerdjikov]
\[mpl\] $$\begin{aligned}
\tfrac{1}{4}\pa_x\pa_t \phi - \phi + \phi\rho \pa_x\phi & = 0, \\
\tfrac{1}{4}\pa_x\pa_t \rho - \rho - \phi\rho \pa_x\rho & = 0 .\end{aligned}$$
Its explicit Lax pair is given by
\[lax\_mpl\] $$\begin{aligned}
A_x &= \H{1}+\pa_x\phi_1 \Ea{1}+\pa_x\rho\Eb{1},\\
A_t &= \H{-1}+2\phi\Ea{-1}-2\rho\Eb{0}+2\phi\rho\H{0}.\end{aligned}$$
Taking with $x \to \tfrac{i}{2}x$, $t \to -\tfrac{i}{2}t$, $\phi=\vphi$ and $\rho=\pm \vphi^*$ we obtain exactly , $$\label{mplr}
\pa_x\pa_t \vphi - \vphi \mp 2i |\vphi|^2\pa_x \vphi = 0.$$ This model can also be derived from the following Lagrangian $$\mathcal{L} = \tfrac{1}{2}\(\pa_x\vphi\pa_t\vphi^* +
\pa_x\vphi^*\pa_t \vphi\) + |\vphi|^2
\mp \tfrac{i}{2}|\vphi|^2\(\vphi\pa_x\vphi^* - \vphi^*\pa_x\vphi\).$$ If one consider $N=2,3,\dotsc$ it is possible to obtain higher order integro-differential equations like .
Relations between DNLS type equations
-------------------------------------
Let us now introduce the transformations connecting the three types of DNLS equations. From it immediately follows the continuity equation $$\label{cont_dnls}
\pa_t\(qr\) + \pa_x j = 0, \qquad
j = \tfrac{1}{2}\(r\pa_xq - q\pa_xr\) + \tfrac{3}{4}\(qr\)^2.$$ Let us define new fields through the following gauge transformation[^2] $$\label{tilde_fields}
\tq \equiv -\tfrac{1}{2}qe^{c \mathcal{J}}, \qquad
\tr \equiv \tfrac{1}{2}re^{-c \mathcal{J}}, \qquad
\mathcal{J} \equiv \int_{-\infty}^x qrdx',$$ where $c$ is a constant. Upon using the equations of motion can be written in terms of these new fields, yielding
\[tilde\_eqs\] $$\begin{aligned}
2\pa_t\tq + \pa_x^2\tq - A\tq^3\tr^2
+ B\tq^2\pa_x\tr - C\pa_x\(\tq^2\tr\) &= 0, \\
2\pa_t\tr - \pa_x^2\tr + A\tq^2\tr^3
+ B\tr^2\pa_x\tq - C\pa_x\(\tq\tr^2\) &=0,\end{aligned}$$
where $A=8c\(2c-1\)$, $B=4c$ and $C=4\(1-c\)$. Note that in we have a fifth order nonlinearity and two types of derivative nonlinear terms. Let us use the same transformation leading to equation , i.e. $x\to ix$, $t\to 2it$, $\tq=\tpsi$ and $\tr=\pm\tpsi^*$. If besides this we set $c = \tfrac{1}{2}$ we obtain the equation , $$\label{dnls2}
i\pa_t\tpsi + \pa_x^2\tpsi \mp 4 i |\tpsi|^2\pa_x\tpsi = 0.$$ On the other hand, if we set $c = 1$ we obtain the equation , $$\label{dnls3}
i\pa_t\tpsi + \pa_x^2\tpsi \pm 4 i \tpsi^2\pa_x\tpsi^* + 8|\tpsi|^4\tpsi = 0.$$ Therefore, the transformation connects explicitly the three types of DNLS equations. If one knows a solution of it is possible to obtain a solution of , which in particular yields solutions of and .
The massive Thirring model
--------------------------
The massive Thirring model is obtained from the Lagrangian $$\label{thirring_lagrangian}
\mathcal{L} = \bar{\Phi}\(i \gamma^\mu\pa_\mu - m\) \Phi
+\dfrac{g}{2}J_\mu J^{\mu}, \qquad J^\mu = \bar{\Phi}\gamma^\mu\Phi,$$ where $m$ is the mass, $g$ is the coupling constant and $$\Phi = \begin{pmatrix} u \\ v \end{pmatrix}, \qquad
\gamma^0 = \gamma_0 = \sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},
\qquad
\gamma^1 = -\gamma_1 = i\sigma_2 = \begin{pmatrix} 0 & 1 \\ -1 & 0
\end{pmatrix}.$$ The $\sigma_i$ are the usual Pauli matrices and $\bar{\Phi} \equiv \Phi^{\dagger}\gamma^0$. The equations of motion are then given by $$i\gamma^\mu\pa_\mu \Phi - m\Phi + gJ_\mu\gamma^\mu \Phi = 0,$$ or written in component form and in the light cone coordinates $x \equiv \tfrac{1}{2}\(x^1+x^0\)$ and $t\equiv\tfrac{1}{2}\(x^1-x^0\)$, we thus have
\[thirr1\] $$\begin{aligned}
i\pa_x v - m u + 2g |u|^2 v &= 0, \\
i\pa_t u + m v - 2g |v|^2 u &= 0.\end{aligned}$$
An important conclusion pointed out in [@thirring_kaup_newell] is that the solutions of cannot, in general, correspond to solutions of the sine-Gordon model. However, as will be shown below, its solutions can be obtained, in general, from the solutions of the model .
Consider under the transformations $x \to - \tfrac{i}{2} x$, $t \to \tfrac{im^2}{2} t$ and $\phi = \vphi = \rho^*$. Thus we have the equation of motion given by $$\label{mplr2}
\pa_x\pa_t \vphi = m^2 \vphi - 2i m^2 |\vphi|^2\pa_x\vphi,$$ from which it follows the continuity equation $$\label{conservation}
\pa_t|\pa_x \vphi|^2 = m^2 \pa_x|\vphi|^2.$$ Let us define new fields through the following relations $$\label{new_fields}
u \equiv \dfrac{1}{\sqrt{2g}} \( \pa_x \vphi\) e^{i \mathcal{J}}, \qquad
v \equiv -\dfrac{im}{\sqrt{2g}} \vphi e^{i \mathcal{J}}, \qquad
\mathcal{J} \equiv \int_{-\infty}^x|\pa_{x'} \vphi|^2dx'.$$ Then, calculating $\pa_x v$ and also $\pa_t u$, making use of and , after writing the result in terms of the fields $u$ and $v$ we get precisely the equations . Therefore, if $\varphi$ is a solution of , then yields a solution of the Thirring model [^3].
Dressing approach to the KN hierarchy {#sec:dressing}
=====================================
We now employ the algebraic dressing method [@babelon_dressing] to construct soliton solutions for the KN hierarchy, assuming $q\to0$ and $r\to0$ when $|x|\to\infty$. To extract the fields in the dressing formalism it is necessary to employ a highest weight representation of the algebra, so we need the full Kac-Moody algebra $\hat{A}_1$ including the central term. Thus, consider the Lax pair but with $A_x$ in the following slightly different form $$\label{lax_dress}
A_x = \H{1} + q\Ea{0} + r\Eb{1} - \(\pa_x\nu - 2t\d_{N+1}\)\hc,$$ where $\nu$ is a function to be determined. Note that the central term does not change the equations of motion, since it commutes with all other generators. The vacuum solution, obtained by setting $q\to 0$, $r\to 0$ and $\nu \to 0$, is then given by
\[vacuum\] $$\begin{aligned}
\bar{A}_x &= \H{1} + 2t\delta_{N+1,0}\hc, \label{vac1}\\
\bar{A}_t &= \H{N}, \label{vac2} \end{aligned}$$
where $N=2,3,\dotsc$ for the *positive flows* and $N=-1,-2,\dotsc$ for the *negative flows*. Note that remains the same for every model, unless for the central term, while changes according to each model. The potentials can still be written in the pure gauge form $\bar{A}_\mu = -\pa_\mu\bar{\Psi}\bar{\Psi}^{-1}$ with $$\label{psi_vacuum}
\bar{\Psi} = e^{-\H{1}x}e^{-\H{N}t}e^{-2xt\d_{N+1,0}\hc}.$$ The idea of the dressing method is to obtain the general potentials $A_\mu$, with a nontrivial field configuration, out from the vacuum $\bar{A}_\mu$, through the gauge transformations $$\label{gauge}
A_\mu = \Thpm \bar{A}_\mu \Thpm^{-1} - \pa_\mu\Thpm \Thpm^{-1}.$$ Furthermore, we have the gauge freedom $\Psi \to \Psi'=\Psi g$ where $g$ is a constant group element. Hence, the dressing operators must satisfy $\Thp\bar{\Psi} = \Thm\bar{\Psi}g$, which is the Riemann-Hilbert problem $$\label{rh}
\Thm^{-1}\Thp = \bar{\Psi} g \bar{\Psi}^{-1}.$$ Assuming a Gauss decomposition, the dressing operators can be further factorized as
\[gauss\] $$\begin{aligned}
\Thp &= e^{\A{0}}e^{\B{1}}e^{\B{2}}\dotsc \\
\Thm &= e^{\B{0}}e^{\B{-1}}e^{\B{-2}}\dotsc\end{aligned}$$
where $\A{0}$ and $\B{m}$ are graded elements. According to the principal gradation, these operators must have the following form $$\label{Bm}
\B{2m+1} = \chi_{2m+1}\Ea{m}+\psi_{2m+1}\Eb{m+1}, \qquad
\B{2m} = \phi_{2m}\H{m},$$ where the fields $\chi_{2m+1}$, $\psi_{2m+1}$ and $\phi_{2m}$ are now our unknowns.
It is enough to consider for the Lax operator $A_x$. Taking it first with the operator $\Thp$, the projection into the *grade zero* subspace yields $$\label{A0}
\A{0} = \nu\hc.$$ The projection into the *grade one* subspace yields $q\Ea{0}+r\Eb{1} = -\pa_x\B{1}$, therefore $$\label{B1}
q = -\pa_x\chi_1, \qquad r = -\pa_x\psi_1.$$ In this way, by considering higher grade projections we can determine all operators appearing in $\Thp$, but just relations are enough for our purposes. Now let us consider with the operator $\Thm$. Taking the *grade two* projection we have $\[ \H{1}, \B{0} \] = 0$, therefore $$\label{B0}
\B{0} = \phi_0\H{0}.$$ Note that we already have a central term in so we do not need to include another one in $\B{0}$. The field $\phi_0$ will be determined by the next lower grades. Considering the *grade one* projection we obtain $$\label{B-1}
q = -2\chi_{-1}e^{2\phi_0}, \qquad r = 2\psi_{-1}e^{-2\phi_0}.$$ The *grade zero* projection gives one equation for $\phi_{-2}$, which we do not need, and also an equation for $\phi_0$ which is then given by $$\label{phi0}
\phi_0 = -\tfrac{1}{2}\int_{-\infty}^x qrdx'.$$
Let us point out some important facts that came out naturally from these approach. Comparing with we see that the solutions of the first negative flow are given by $$\label{sol_neg}
\phi = -\chi_{1}, \qquad \rho = -\psi_{1},$$ while the solutions of contains an extra derivative . This explains why the solutions of are connected to those of through the potential variable $\psi=\pa_x\vphi$ [@matsuno_fl; @tsuchida]. Comparing and with the gauge transformation , we note that the form of the transformation connecting the three types of DNLS equations are already contained in the dressing operators. The same also applies to the relations . Moreover, we see from that $\mathcal{J}=-2\phi_0$. Precisely for the case $c=1$, corresponding to equation , we have from that $\tq = \chi_{-1}$ and $\tr = \psi_{-1}$. For equations and we need to include the arbitrary constants. The term $e^{\pm i\phi_0}$ is precisely the weight function introduced in the revised form of the IST [@kaup_newell].
Tau functions
-------------
We now introduce an important class of functions that contain the explicit space-time dependence of the solutions. They are directly related to the fields $\chi_{2m+1}$, $\psi_{2m+1}$ and $\phi_{2m}$ of and, consequently, to the physical fields in the equations of motion through the relations –. Consider the highest weight states[^4] $\{\ket{\l_0}, \ket{\l_1}\}$ and let us also introduce the following *notation* for convenience[^5] $$\ket{\l_2} \equiv \Ea{-1}\ket{\l_0}, \qquad
\ket{\l_3} \equiv \Eb{0}\ket{\l_1}.$$ The procedure to extract the fields in the dressing approach is to project the left hand side of between appropriate states. Thus we have $$\label{chi_fields}
\begin{split}
e^{\nu} &= \bra{\l_0} \Thm^{-1}\Thp \ket{\l_0}, \\
\psi_1e^{\nu} &= \bra{\l_0} \Thm^{-1}\Thp \ket{\l_2}, \\
\chi_{-1}e^{\nu} &= -\bra{\l_2} \Thm^{-1}\Thp \ket{\l_0},
\end{split}\qquad
\begin{split}
e^{\nu-\phi_0} &= \bra{\l_1} \Thm^{-1}\Thp \ket{\l_1}, \\
\chi_1e^{\nu-\phi_0} &= \bra{\l_1} \Thm^{-1}\Thp \ket{\l_3}, \\
\psi_{-1}e^{\nu-\phi_0} &= -\bra{\l_3} \Thm^{-1}\Thp \ket{\l_1}.
\end{split}$$ Note that the right hand side of contains the explicit space-time dependence through , so we define the $\tau$-*functions* as $$\label{tau}
\tau_{ab} \equiv \bra{\l_a} \bar{\Psi}g\bar{\Psi}^{-1} \ket{\l_b},$$ where $a,b=0,1,2,3.$ The $\tau$-functions are classified according to the arbitrary group element $g$. Combining the results of and we have $$\label{tau_ansatz}
\phi_0 = \ln\(\dfrac{\tau_{00}}{\tau_{11}}\), \quad
\psi_1 = \dfrac{\tau_{02}}{\tau_{00}}, \quad
\chi_1 = \dfrac{\tau_{13}}{\tau_{11}}, \quad
\psi_{-1} = -\dfrac{\tau_{31}}{\tau_{11}}, \quad
\chi_{-1} = -\dfrac{\tau_{20}}{\tau_{00}}.$$ From these relations we can express the solutions of all previous models in terms of $\tau$-functions, which can be algebraically calculated if we have an appropriate form for the group element $g$. For instance, from the solutions of are expressed as $$\label{sol_mpl}
\phi = -\dfrac{\tau_{13}}{\tau_{11}}, \qquad
\rho = -\dfrac{\tau_{02}}{\tau_{00}},$$ while from we have the solutions for the positive flows, like and , given by $$\label{sol_pos}
q = -\pa_x\(\dfrac{\tau_{13}}{\tau_{11}}\), \qquad
r = -\pa_x\(\dfrac{\tau_{02}}{\tau_{00}}\).$$ The integral appearing in the gauge transformations can be obtained from yielding $$\label{integral}
\mathcal{J}= 2\ln\dfrac{\tau_{11}}{\tau_{00}},$$ showing that it is indeed a local function. Then, the solutions of are given by $$\label{sol_tilde}
\tq = \dfrac{1}{2}
\(\dfrac{\tau_{11}}{\tau_{00}}\)^{2c}
\pa_x\(\dfrac{\tau_{13}}{\tau_{11}}\), \qquad
\tr = -\dfrac{1}{2}
\(\dfrac{\tau_{00}}{\tau_{11}}\)^{2c}
\pa_x\(\dfrac{\tau_{02}}{\tau_{00}}\).$$ In particular, for equation ($c=1$) it is easier to use directly thus $$\label{sol_dnls3}
\tq = -\dfrac{\tau_{20}}{\tau_{00}}, \qquad
\tr = -\dfrac{\tau_{31}}{\tau_{11}}.$$ Recall that we must further impose the transformations $x\to ix$, $t\to 2it$, and $\tau_{31}/\tau_{11}=\pm\(\tau_{20}/\tau_{00}\)^*$. For the massive Thirring model we obtain from the following solution $$\label{sol_thirr}
u = -\dfrac{1}{\sqrt{2g}}
\dfrac{\tau_{11}}{\tau_{00}}
\pa_x\(\dfrac{\tau_{13}}{\tau_{11}}\), \qquad
v = \dfrac{im}{\sqrt{2g}}\dfrac{\tau_{13}}{\tau_{00}}.$$ We must further impose the transformations $x\to -\tfrac{i}{2} x$ and $t\to \tfrac{im^2}{2} t$ in the space-time dependence of the $\tau$-functions and also $\tau_{02}/\tau_{00}=\(\tau_{13}/\tau_{11}\)^*$.
Vertex operators
----------------
In order to be able to evaluate explicitly let us assume that $g$ is given in the following form [@babelon_dressing; @olive_vertex] $$\label{vertex}
g \equiv \prod_{j=1}^{n}\exp\(\Gamma_j\), \qquad \Gamma_j = \Gamma\(\k_j\)$$ where $\Gamma_j$ is a *vertex operator* depending on a *complex* parameter $\kappa_j$. Furthermore, let us assume that the vertex operator satisfy an eigenvalue commutation relation with , $$\label{general_eigen}
\[\Gamma_j, \bar{A}_x x + \bar{A}_t t \] = \eta_j(x,t)\Gamma_j.$$ The function $\eta_j$ encodes the *dispersion relation*. From we then have $$\label{psigpsi}
\bar{\Psi} g \bar{\Psi}^{-1} = \prod_{j=1}^{n}\bar{\Psi}\exp\(\Gamma_j\)
\bar{\Psi}^{-1}
= \prod_{j=1}^{n}\exp\(\bar{\Psi}\Gamma_j\bar{\Psi}^{-1}\)
= \prod_{j=1}^{n}\exp\(e^{\eta_j}\Gamma_j\).$$ The vertex operators satisfy the nilpotency property between the states, which eliminates terms containing its powers, $\(\Gamma_j\)^m$ for $m\ge2$, and truncates the exponential series in . Therefore, the $\tau$-functions assume the following form $$\label{tau_g}
\tau_{ab} = \bra{\l_a} \prod_{j=1}^n\(1+e^{\eta_j(x,t)}\Gamma_j\) \ket{\l_b}.$$
Let us now apply these general concepts to the KN hierarchy. Consider the following two vertex operators $$\label{vertices}
\Gamma_j \equiv \sum_{n=-\infty}^{\infty}\k_j^{-n}\Ea{n-1},
\qquad
\Gamma'_j \equiv \sum_{n=-\infty}^{\infty}\k_j^{-n}\Eb{n},
$$ which satisfy the following eigenvalue commutation relations with the vacuum $$\label{eigen}
\[ \Gamma_j, \H{m} \] = -2\k_j^{m}\Gamma_j, \qquad
\[ \Gamma'_j, \H{m} \] = 2\k_j^{m}\Gamma'_j.$$ The dispersion relations of the KN hierarchy are, therefore, given by $$\label{dispersion}
\eta_j = -2\k_jx - 2\k_j^N t, \qquad
\eta'_j = 2\k_jx + 2\k_j^N t,$$ where $N$ is the integer labelling the respective flow, i.e. $N=2,3,\dotsc$ for the positive flows or $N=-1,-2,\dotsc$ for the negative flows. Thus explains the change in the dispersion relation between models and [@matsuno_fl].
Two vertices solution
---------------------
Using a single vertex operator in we get a non interesting simple exponential solution with one of the fields vanishing, which corresponds to a linearization of the equations of motion. The first nontrivial solution is obtained with two vertices in the form $$g=\exp\(\Gamma_1\)\exp\(\Gamma'_2\).$$ Then, from we obtain $$\label{tau_2vert}
\tau_{ab} = \bra{\l_a} 1 + \Gamma_1 e^{\eta_1} + \Gamma'_2 e^{\eta'_2} +
\Gamma_1\Gamma'_2 e^{\eta_1+\eta'_2} \ket{\l_b}.$$ The matrix elements are calculated in appendix \[sec:matrix1\], leading to the following $\tau$-functions $$\label{tau_2vertices}
\begin{aligned}
\tau_{00} &= 1+\dfrac{\k_2}{\(\k_1-\k_2\)^2}e^{\eta_1+\eta'_2}, & \qquad
\tau_{02} &= \dfrac{1}{\k_2}e^{\eta'_2}, & \qquad
\tau_{13} &= \dfrac{1}{\k_1}e^{\eta_1}, \\
\tau_{11} &= 1+\dfrac{\k_1}{\(\k_1-\k_2\)^2}e^{\eta_1+\eta'_2}, &
\tau_{20} &= e^{\eta_1}, &
\tau_{31} &= e^{\eta'_2}.
\end{aligned}$$ Replacing in relations – we can obtain explicit one-soliton solutions for the all the previous models considered in this paper. Let us also recall that we must pick the right dispersion relation for the corresponding flow $N$. For instance, from we have a solution of , that under the reduction $x\to\tfrac{i}{2}x$, $t\to-\tfrac{i}{2}t$ and $\k_2=\k_1^*=\k$, which is compatible with the choice $\phi=\rho^*=\varphi$, yields a solution of with the *minus sign*. Still writing $\k = \k_R + i\k_I$ we have found $$\label{mplr_square}
|\vphi|^2 = \dfrac{\(\k_R^2+\k_I^2\)^{-1}e^{\theta}}{
1-\dfrac{\k_R}{2\k_I^2}e^{\theta}+\dfrac{\k_R^2+\k_I^2}{16\k_I^4}
e^{2\theta}}, \qquad
\theta = -2\k_I\(x - \dfrac{1}{\k_R^2+\k_I^2}t\).$$ Using we have a solution of , that with the appropriate reduction yields a solution of with the *plus sign*, whose square modulus reads $$\label{dnlse_square}
|\psi|^2 = \dfrac{4e^\theta}{1-\dfrac{\k_R}{2\k_I^2}e^{\theta}+
\dfrac{\k_R^2+\k_I^2}{16\k_I^4}e^{2\theta}}, \qquad
\theta = -4\k_I\(x+4\k_Rt\).$$ Both functions and are, respectively, plotted in figure \[fig:1\]. The solution has an unusual behaviour, as can be seen from its graph. The soliton gets wider as its height and velocity increases. This behaviour does not occur for that shows the usual solitonic profile.
![Graphs of and , corresponding to one-soliton solution of and (with the first sign), respectively, for different values of $\k = \k_R + i\k_I$. Note the peculiar feature of the first graph, where the soliton becomes wider when it gets higher, contrary to the second graph that exhibits the usual soliton behaviour.[]{data-label="fig:1"}](figure1.pdf)
From we have the following one-soliton solution for the Thirring model $$\label{onesol_thirr}
u = \dfrac{i}{\sqrt{2g}}\dfrac{e^{\eta^*}}{
1+ \dfrac{\k^*}{\(\k-\k^*\)^2}e^{\eta+\eta^*}}, \qquad
v = \dfrac{im}{\sqrt{2g}}
\dfrac{\(\k^*\)^{-1}e^{\eta^*}}{1+\dfrac{\k}{\(\k-\k^*\)^2}
e^{\eta+\eta^*}},$$ where $\eta = -i\(\k x - m^2 \k^{-1}t\)$. Taking the square modulus of these solutions we obtain exactly the behaviour of for $v$ and the behaviour of for $u$, which are sketched in figure \[fig:1\]. The formulas are almost identical.
Four vertices solution
----------------------
Let us consider a more complex solution by choosing the group element with four vertices in the form $$g = \exp\(\G_1\)\exp\(\G'_2\)\exp\(\G_3\)\exp\(\G'_4\).$$ We then calculate the $\tau$-functions analogously to . After calculating the matrix elements, which are presented in appendix \[sec:matrix1\], we obtain
\[tau\_4vertices\] $$\begin{aligned}
\tau_{00} &= \bra{\l_0} 1+
\G_1\G'_2 e^{\eta_1+\eta'_2}+
\G_1\G'_4 e^{\eta_1+\eta'_4}+
\G'_2\G_3 e^{\eta'_2+\eta_3}+
\G_3\G'_4 e^{\eta_3+\eta'_4} \nn \\
&\qquad \qquad
+\G_1\G'_2\G_3\G'_4 e^{\eta_1+\eta'_2+\eta_3+\eta'_4}\ket{\l_0}, \\
\tau_{11} &= \bra{\l_1} 1+
\G_1\G'_2 e^{\eta_1+\eta'_2}+
\G_1\G'_4 e^{\eta_1+\eta'_4}+
\G'_2\G_3 e^{\eta'_2+\eta_3}+
\G_3\G'_4 e^{\eta_3+\eta'_4} \nn \\
&\qquad \qquad
+\G_1\G'_2\G_3\G'_4 e^{\eta_1+\eta'_2+\eta_3+\eta'_4}\ket{\l_1}, \\
\tau_{02} &= \bra{\l_0} \G'_2 e^{\eta'_2}+\G'_4 e^{\eta'_4}
+\G'_2\G_3\G'_4 e^{\eta'_2+\eta_3+\eta'_4}
+\G_1\G'_2\G'_4 e^{\eta_1+\eta'_2+\eta'_4} \ket{\l_2}, \\
\tau_{13} &= \bra{\l_1} \G_1 e^{\eta_1}+\G_3 e^{\eta_3}
+\G_1\G'_2\G_3 e^{\eta_1+\eta'_2+\eta_3}
+\G_1\G_3\G'_4 e^{\eta_1+\eta_3+\eta'_4} \ket{\l_3}, \\
\tau_{20} &= \bra{\l_2} \G_1 e^{\eta_1}+\G_3 e^{\eta_3}
+\G_1\G'_2\G_3 e^{\eta_1+\eta'_2+\eta_3}
+\G_1\G_3\G'_4 e^{\eta_1+\eta_3+\eta'_4} \ket{\l_0}, \\
\tau_{31} &= \bra{\l_3} \G'_2 e^{\eta'_2}+\G'_4 e^{\eta'_4}
+\G'_2\G_3\G'_4 e^{\eta'_2+\eta_3+\eta'_4}
+\G_1\G'_2\G'_4 e^{\eta_1+\eta'_2+\eta'_4} \ket{\l_1}.\end{aligned}$$
Considering special transformations, for instance, $\k_1=\k_2^*=\k$ and $\k_3=\k_4^*=\zeta$ one obtains a two-soliton solution in the same way as –. We will not write down further explicit formulas but in figure \[fig:2\] we show a graph of the two-soliton solution of the Thirring model obtained in this way from .
![Graphs of solutions involving four vertices. In the left column we have ploted the time evolution of $|u|^2$ and in the right column the time evolution of $|v|^2$. We have set the parameters as $m=1$, $g=\tfrac{1}{2}$, $\k_2=\k_1^*=\k=1+i$ and $\k_4=\k_3^*=\zeta=2+2i$. The waves travel from right to left. Note that for $|u|^2$ the small soliton is faster than the largest one, while for $|v|^2$ the higher soliton is faster.[]{data-label="fig:2"}](figure2.pdf)
The one- and two-soliton solutions, given by and , respectively, were explicitly checked against the all the previous models mentioned in this paper, with the aid of symbolic computation. We also want to stress that in spite of the technical difficulty, the procedure to compute the $\tau$-functions with $n$ vertices is well defined and systematic.
Equivalent construction {#sec:equivalency}
=======================
The standard construction of the KN hierarchy is given by the following Lax operator, with the *homogeneous gradation* [@kaup_newell; @gerdjikov_kulish; @gerdjikov; @mikhailov_thirring] $$\label{standard_kn}
A_x = \H{2} + q\Ea{1} + r\Eb{1} =
\begin{pmatrix} \l^2 & \l q \\ \l r & -\l^2 \end{pmatrix}, \qquad
Q = \hd = \l\dfrac{d}{d\l}.$$ Given an affine Lie algebra $\alie$ with generators $T_a^n$, the *conjugation* by a group element $h$, i.e. $T_a^n \mapsto hT_a^nh^{-1}$, is an *automorphism* since it preserves the commutator . Consider the algebra $\hat{A}_1$ and let us define the following group element $$\label{automorphism}
h \equiv \exp\left\{-\tfrac{1}{2}\ln\(\l\) \H{0}\right\},$$ which yields the following mapping $$\begin{split}
h\H{n}h^{-1} &= \H{n}, \\
h\hc h^{-1} &= \hc,
\end{split}\qquad
\begin{split}
h\Ea{n}h^{-1} &= \Ea{n-1}, \\
h\hd h^{-1} &= \hd + \tfrac{1}{2}\H{0},
\end{split}\qquad
\begin{split}
h\Eb{n}h^{-1} &= \Eb{n+1}.
\end{split}$$ Thus is mapped into $$\label{almost_mapping}
A_x\mapsto hA_xh^{-1} = \H{2} + q\Ea{0} +r\Eb{2}, \qquad
Q\mapsto hQh^{-1}=\tfrac{1}{2}\H{0}+\hd.$$ If we now introduce a *new spectral parameter* $\zeta\equiv\l^2$, then $\zeta\tfrac{d}{d\zeta}=\tfrac{1}{2}\l\tfrac{d}{d\l}$ and by redefining the operators in with respect to the spectral parameter $\zeta$, we map into another algebraic construction with *principal gradation* $$\label{equivalent_construction}
\begin{cases}
\begin{aligned}[t]
A_x &= \H{2}+q\Ea{1}+r\Eb{1} \\
&= \begin{pmatrix}\l^2 & \l q \\ \l r & -\l^2 \end{pmatrix} \\
Q &= \hd
\end{aligned}
\end{cases}
\begin{aligned}[c]
\xrightarrow[\zeta = \l^2]{h = e^{-1/2\ln\l\H{0}}}
\end{aligned}\quad
\begin{cases}
\begin{aligned}[t]
hA_xh^{-1} &= \H{1}+q\Ea{0}+r\Eb{1} \\
&= \begin{pmatrix}\zeta & q \\ \zeta r & -\zeta \end{pmatrix} \\
hQh^{-1} &= \tfrac{1}{2}\H{0} +2\hd
\end{aligned}
\end{cases}$$ Therefore, we have demonstrated the equivalence of the standard construction and the one proposed in this paper . We have introduced the construction because it is much simpler and natural to be treated under the dressing method than , and it eliminates the spurious quadratic power of the spectral parameter that is responsible for divergences in some complex integrals appearing in the IST.
#### Comment.
It is possible to apply the method used in this paper to the homogeneous construction and the same relations – arises, but the vacuum now stays in the form $$\label{new_vac}
\bar{\Psi} = e^{-\H{2}x}e^{-\H{2N}t}e^{-2xt\d_{N+1,0}\hc}.$$ The vertex operators still statisfy the eigenvalue relations with this vacuum but are not uniformely graded according to the homogeneous gradation. Moreover, to be able to obtain the *correct* solution we must redefine the spectral parameter in $\zeta\equiv\l^2$. Therefore, the procedure does not occur in a natural way as in the case of principal gradation and it is necessary to introduce some ingredients by hand.
Concluding remarks {#sec:conclusions}
==================
The KN hierarchy was obtained from a *higher grading* affine algebraic construction with the algebra $\hat{A}_1$ and *principal gradation*. In fact, we have proposed a general construction that can generates novel integrable models if different affine Lie algebras are employed. The results of this paper should extend naturally to these cases.
The main models within the KN hierarchy were derived. The DNLSE-I arises from the second positive flow, while the Mikhailov model is obtained from the first negative flow. The gauge transformation connects the system to , which in particular yields relations between the three kinds of DNLS equations, namely, , and . Furthermore, we have demonstrated a general relation between the model and the massive Thirring model through .
We developed the algebraic dressing method for the KN hierarchy and several relations found previously in the literature emerges naturally. For instance, the form of the gauge transformations linking the three DNLS equations and the precise connection between the solutions of and those of . Moreover, the weight function introduced in the revised IST [@kaup_newell] also arises from the algebraic dressing method. We stress that this method is general and systematic, and relies only on the algebraic structure of the hierarchy. The solutions of all models considered in this paper were expressed in terms of $\tau$-functions, which can be systematically calculated through a vertex representation theory of the algebra. We considered explicitly one- and two-soliton solutions. The solitons of the model , given by , possess an unusual behaviour were its width increases with its height, as shown in figure \[fig:1\]. Finally, we demonstrated that our construction is conjugate related to the usual construction found in the literature . However, the dressing procedure applied to our Lax pair is greatly simplified compared to . Several works on VBC and also NVBC are based on the standard Lax pair and the revised IST. We conclude that these problems can be simplified if one considers our construction instead. We plan to illustrate this fact more precisely regarding NVBC within the dressing formalism in a future opportunity.
### Acknowledgments {#acknowledgments .unnumbered}
We thank CAPES, CNPq and Fapesp for financial support. GSF thanks the support from CNPq under the “Ci\^ encia sem fronteiras” program.
Algebraic concepts {#sec:algebra}
==================
Let $\lie$ be a finite dimensional Lie algebra, with commutator $\[ T_a, T_b \]$ for $T_a,T_b \in \lie$ and with a symmetric bilinear Killing form $\inner{T_a}{T_b}$. The *infinite* dimensional loop-algebra is defined by $\llie \equiv \lie\otimes\complex(\l,\l^{-1})$, i.e. $T_a \mapsto T_a\otimes\lambda^n\equiv T_a^n \in \llie$ for $n\in \integer$ and $\l$ is the so called complex *spectral parameter*. Let us introduce the central term $\hc$, that commutes with every other generator, and also the derivative operator $\hd \equiv \lambda \tfrac{d}{d\lambda}$. The Kac-Moody algebra is then defined by $\alie \equiv \llie\oplus\complex\hc\oplus\complex\hd$ with commutator $$\label{kac_commutator}
\blb T_a^n, T_b^m \brb \equiv \blb T_a,T_b \brb \otimes \lambda^{n+m} + \hc
n\delta_{n+m,0}\inner{T_a}{T_b}.$$ If we set $\hc = 0$ we have the commutator for the loop-algebra $\llie$. For example, considering the algebra $A_1 = \blcurl E_\a, E_{-\a}, H \brcurl$, where $\blb E_{\a}, E_{-\a} \brb = H$ and $\blb H, E_{\pm\a}\brb = \pm 2 E_{\pm\a}$, we have the Kac-Moody algebra $\hat{A}_1$ with generators $\blcurl E_\a^{n}, \, E_{-\a}^{n}, \, H^n, \, \hc, \hd \brcurl$ and commutation relations $$\label{commutators}
\begin{split}
\blb H^{n}, H^{m} \brb &= 2n\d_{n+m,0}\hc, \\
\blb H^{n}, E^{m}_{\pm\a} \brb &= \pm 2 E^{n+m}_{\pm\a},
\end{split}\qquad
\begin{split}
\blb E^n_{\a}, E^{m}_{-\a} \brb &= H^{n+m} + n\d_{n+m,0}\hc, \\
\blb \hd, T^{n} \brb &= nT^{n}, \\
\end{split}\qquad
\begin{split}
\blb \hc, T^{n} \brb = 0,
\end{split}$$ where $T^{n} \in \lcurl H^{n},\,E^{n}_{\a}, \, E^{n}_{-\a}\rcurl$.
We can introduce a grading operator $Q$, that splits the algebra into graded subspaces in the following way. For $T_a^{n} \in \alie$, if $\[ Q, T_a^{n} \] = m T_a^{n}$ for $m \in \integer$, then $\alie = \bigoplus_{m\in\integer}\alie^{(m)}$ where $\alie^{(m)}=\blcurl T_a^{n} \, | \, \[Q,T_a^n\]=mT_a^{n} \brcurl$. In the case of $\alie=\hat{A}_1$ the grading operator $Q\equiv \hd$ (homogeneous) induces a natural gradation, $\alie^{(m)} = \lcurl H^m, E_{\a}^m, E_{-\a}^m \rcurl$. For $Q\equiv \tfrac{1}{2}\H{0}+2\hd$ (principal) we have $\alie^{(2m+1)} = \lcurl E_{\a}^m, E_{-\a}^{m+1}\rcurl$ and $\alie^{(2m)} = \lcurl H^m \rcurl$. The operators $\hc$ and $\hd$ have zero grade.
The highest weight states of the algebra is a set of states satisfying $T_a^n\ket{\l_a}=0$, if $T_a^n$ have grade higher than zero. Precisely for the case of $\hat{A}_1$, the highest weight states are $\lcurl \ket{\l_0}, \ket{\l_1} \rcurl$ and obey the following actions $$\begin{aligned}[c]
E_{\pm \a}^{n}\ket{\l_a} &= 0 \quad (n > 0), \\
H^{n}\ket{\l_a} &= 0 \quad (n > 0),
\end{aligned}\qquad
\begin{aligned}[c]
H^{0}\ket{\l_0} &= 0, \\
H^{0}\ket{\l_1} &= \ket{\l_1},
\end{aligned}\qquad
\begin{aligned}[c]
\Ea{0}\ket{\l_a} &= 0, \\
\hc\ket{\l_a} &= \ket{\l_a},
\end{aligned}$$ where $a=0,1$. The adjoint relations are $\(\H{n}\)^{\dagger}=\H{-n}$, $\(E_{\pm\a}^{n}\)^{\dagger}=E_{\mp\a}^{-n}$ and $\hc^{\dagger}=\hc$.
A $2\times2$ matrix representation of the $\hat{A}_1$ generators can be given as follows $$\H{n} = \begin{pmatrix} \l^n & 0 \\ 0 & -\l^n \end{pmatrix}, \quad
\Ea{n} = \begin{pmatrix} 0 & \l^n \\ 0 & 0 \end{pmatrix}, \quad
\Eb{n} = \begin{pmatrix} 0 & 0 \\ \l^n & 0 \end{pmatrix}, \quad
\hc = \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}.$$
Matrix elements {#sec:matrix1}
===============
The relevant states are $\ket{\l_0}$, $\ket{\l_1}$, $\ket{\l_2} = \Ea{-1}\ket{\l_0}$ and $\ket{\l_3} = \Eb{0}\ket{\l_1}$. The vertices for the KN hierarchy are $$\G_j = \sum_{n=-\infty}^{\infty}\k_j^{-n}\Ea{n-1}, \qquad
\G'_j = \sum_{n=-\infty}^{\infty}\k_j^{-n}\Eb{n},$$ and satisfy the eigenvalue equations . We will write only the *nonvanishing* matrix elements used in and . The *nilpotency* property of the vertices reads $\bra{\l_a} \(\Gamma_i\)^n \ket{\l_b} =
\bra{\l_a} \(\Gamma'_i\)^n \ket{\l_b} = 0$ for $n\ge2$. In addition, any matrix element having a power of a vertex vanishes, e.g. $\bra{\l_a} \(\Gamma_i\)^2\Gamma'_j \ket{\l_b} = 0$. Another useful result is that the number of $\Ea{n}$ and $\Eb{m}$ in a matrix element must be balanced in pairs, e.g. $\bra{\l_a}\G_i\G'_j\ket{\l_a}\ne 0$, while $\bra{\l_0}\G_i\G'_j\ket{\l_2}=\bra{\l_0}\G_i\G'_j\Ea{-1}\ket{\l_0}=0$. Thus, the nonvanishing matrix elements relevant for the solution with two vertices are $$\begin{aligned}
\bra{\l_2}\Gamma_1\ket{\l_0} &= 1, \\
\bra{\l_1}\Gamma_1\ket{\l_3} &= \dfrac{1}{\k_1}, \\
\bra{\l_0}\Gamma'_2\ket{\l_2} &= \dfrac{1}{\k_2}, \\
\bra{\l_3}\Gamma'_2\ket{\l_1} &= 1, \\
\bra{\l_0}\Gamma_1\Gamma'_2\ket{\l_0} &= \dfrac{\k_2}{\(\k_1-\k_2\)^{2}}, \\
\bra{\l_1}\Gamma_1\Gamma'_2\ket{\l_1} &= \dfrac{\k_1}{\(\k_1-\k_2\)^{2}}.\end{aligned}$$ Note also that $\bra{\l_a}\G'_2\G_1\ket{\l_a}=\bra{\l_a}\G_1\G'_2\ket{\l_a}$. Besides these elements, the nonvanishing elements for the solution with four vertices are $$\begin{aligned}
\bra{\l_0}\G'_2\G_3\G'_4\ket{\l_2} &=
\dfrac{\k_3^2\(\k_2-\k_4\)^2}{\k_2\k_4\(\k_2-\k_3\)^2\(\k_3-\k_4\)^2}, \\
\bra{\l_0}\G_1\G'_2\G'_4\ket{\l_2} &=
\dfrac{\k_1^2\(\k_2-\k_4\)^2}{\k_2\k_4\(\k_1-\k_2\)^2\(\k_1-\k_4\)^2}, \\
\bra{\l_1}\G_1\G'_2\G_3\ket{\l_3} &=
\dfrac{\k_2^2\(\k_1-\k_3\)^2}{\k_1\k_3\(\k_1-\k_2\)^2\(\k_2-\k_3\)^2}, \\
\bra{\l_1}\G_1\G_3\G'_4\ket{\l_3} &=
\dfrac{\k_4^2\(\k_1-\k_3\)^2}{\k_1\k_3\(\k_1-\k_4\)^2\(\k_3-\k_4\)^2}, \\
\bra{\l_2}\G_1\G'_2\G_3\ket{\l_0} &=
\dfrac{\k_2\(\k_1-\k_3\)^2}{\(\k_1-\k_2\)^2\(\k_2-\k_3\)^2}, \\
\bra{\l_2}\G_1\G_3\G'_4\ket{\l_0} &=
\dfrac{\k_4\(\k_1-\k_3\)^2}{\(\k_1-\k_4\)^2\(\k_3-\k_4\)^2}, \\
\bra{\l_3}\G'_2\G_3\G'_4\ket{\l_1} &=
\dfrac{\k_3\(\k_2-\k_4\)^2}{\(\k_2-\k_3\)^2\(\k_3-\k_4\)^2}, \\
\bra{\l_3}\G_1\G'_2\G'_4\ket{\l_1} &=
\dfrac{\k_1\(\k_2-\k_4\)^2}{\(\k_1-\k_2\)^2\(\k_1-\k_4\)^2}, \\
\bra{\l_0}\G_1\G'_2\G_3\G'_4\ket{\l_0} &=
\dfrac{\k_2\k_4\(\k_1-\k_3\)^2\(k_2-\k_4\)^2}{
\(\k_1-\k_2\)^2\(\k_1-\k_4\)^2\(\k_2-\k_3\)^2\(k_3-\k_4\)^2}, \\
\bra{\l_1}\G_1\G'_2\G_3\G'_4\ket{\l_1} &=
\dfrac{\k_1\k_3\(\k_1-\k_3\)^2\(k_2-\k_4\)^2}{
\(\k_1-\k_2\)^2\(\k_1-\k_4\)^2\(\k_2-\k_3\)^2\(k_3-\k_4\)^2}. \end{aligned}$$
[^1]: See the appendix \[sec:algebra\] for the algebraic concepts.
[^2]: It will be shown that, in fact, $\mathcal{J}$ is a local function.
[^3]: If we consider the same kind of transformation for both equations , without requiring $\rho=\phi^*$, and define $\chi_1 \equiv \tfrac{1}{\sqrt{2g}}\pa_x\phi e^{i\mathcal{J}}$, $\chi_2 \equiv \tfrac{1}{\sqrt{2g}}\pa_x\rho e^{-i\mathcal{J}}$, $\chi_3 \equiv \tfrac{im}{\sqrt{2g}}\rho e^{-i\mathcal{J}}$ and $\chi_4 \equiv -\tfrac{im}{\sqrt{2g}}\phi e^{i\mathcal{J}}$, where $\mathcal{J}\equiv \int_{-\infty}^x \pa_{x'}\phi\pa_{x'}\rho dx'$, we obtain the four component Thirring like model considered in [@tsuchida], equation $(2.31)$.
[^4]: See the appendix \[sec:algebra\].
[^5]: These are not highest weight states of the algebra.
|
---
abstract: 'We obtain the radius of convergence of the small–amplitude approximation to the period of the nonlinear oscillator $\ddot{x}+(1+\dot{x}^{2})x=0$ with the initial conditions $x(0)=A$ and $\dot{x}(0)=0$ and show that the inverted perturbation series appears to converge smoothly from below.'
address: |
INIFTA (UNLP, CCT La Plata-CONICET), División Química Teórica,\
Diag. 113 y 64 (S/N), Sucursal 4, Casilla de Correo 16,\
1900 La Plata, Argentina
author:
- 'Francisco M. Fernández'
title: 'On the small–amplitude approximation to the differential equation $\ddot{x}+(1+\dot{x}^{2})x=0$'
---
There has recently been great interest in the study of the period of the nonlinear oscillator $$\begin{aligned}
&&\ddot{x}(t)+[1+\dot{x}(t)^{2}]x(t)=0 \nonumber \\
&&x(0)=A,\;\dot{x}(0)=0 \label{eq:difeq_x}\end{aligned}$$ as a function of the amplitude $A$. Apparently, it aroused from the fact that the first–order harmonic balance method yielded the approximate frequency[@C03] $$\omega ^{HB}(A)=\frac{2}{\sqrt{4-A^{2}}} \label{eq:omega_HB}$$ that is not defined for $A>2$.
By straightforward analysis of the dynamical trajectories in the $x-y$ plane, where $y=\dot{x}$, Beatty and Mickens[@BM05] concluded that such a restriction is merely an artifact of the harmonic balance method.
Later, Mickens[@M06] derived an explicit expression for the period $$T(A)=4A\int_{0}^{1}\frac{du}{\sqrt{e^{A^{2}(1-u^{2})}-1}} \label{eq:T(A)}$$ where $u=x/A$. By means of this expression he proved that $dT/dA<0$ and obtained upper and lower bounds to the period.
Kalmár–Nagy and Erneux[@KE08] derived the behaviour of the period for small and large values of $A$$$\begin{aligned}
T(A) &\simeq &2\pi \left( 1-\frac{A^{2}}{8}\right) ,\;A\ll 1 \nonumber \\
T(A) &\simeq &\frac{2\pi }{A},\;A\gg 1 \label{eq:T(A)_asymp}\end{aligned}$$ as well as most interesting approximations to the periodic orbits in both limits. In particular, they showed that the trajectory $u(t)$ satisfies the equation $$\begin{aligned}
\ddot{u}+\frac{dV}{du} &=&0, \nonumber \\
V(u) &=&\frac{1-e^{\rho (1-u^{2})}}{2\rho },\;\rho =A^{2} \label{eq:V(u)}\end{aligned}$$ that leads to the same expression for the period (\[eq:T(A)\]) derived earlier by Mickens[@M06].
The results of those authors clearly show that the period $T(A)$ does not exhibit singular points for real values of $A$ but they do not explain why the harmonic balance fails as shown in equation (\[eq:omega\_HB\])[@C03]. A possible explanation is that the harmonic balance is reflecting a singular point in the complex $A$–plane. If it exists, then the small–amplitude expansion will have a finite radius of convergence.
In order to derive the small–amplitude expansion we change the integration variable in equation (\[eq:T(A)\]) to $u=\cos \theta $ so that the period becomes $$T(\rho )=4\int_{0}^{\pi /2}\frac{d\theta }{\sqrt{F(\rho \sin ^{2}\theta )}}
\label{eq:T(rho)}$$ where $$F(z)=\frac{e^{z}-1}{z}=\sum_{j=0}^{\infty }\frac{z^{j}}{(j+1)!}
\label{eq:F(z)}$$ If we substitute the expansion $$\frac{1}{\sqrt{F(z)}}=\sum_{j=0}^{\infty }c_{j}z^{j}=1-\frac{z}{4}+\frac{z^{2}}{96}+\frac{z^{3}}{384}+\ldots \label{eq:1/sqrt(F)_series}$$ into the equation (\[eq:T(rho)\]) we obtain as many coefficients as desired of the small–amplitude series $$\begin{aligned}
T(A) &=&T_{0}+T_{1}\rho +T_{2}\rho ^{2}+\ldots \nonumber \\
&=&2\pi \left( 1-\frac{\rho }{8}+\frac{\rho ^{2}}{256}+\ldots \right)
\label{eq:T_rho_series}\end{aligned}$$
The function $1/\sqrt{e^{z}-1}$ has two complex–conjugate singular points closest to the origin at $z=\pm 2\pi i$; therefore $$\lim_{j\rightarrow \infty }\left| \frac{c_{j}}{c_{j+1}}\right| =2\pi$$ If we take into account that $$I_{j}=\int_{0}^{\pi /2}\sin ^{2j}\theta \,d\theta =\frac{\sqrt{\pi }\Gamma
(j+1/2)}{2\Gamma (j+1)}$$ then we conclude that $$\lim_{j\rightarrow \infty }\left| \frac{c_{j}\,I_{j}}{c_{j+1}\,I_{j+1}}\right| =\lim_{j\rightarrow \infty }\left| \frac{c_{j}}{c_{j+1}}\right| =2\pi$$ In other words, the $\rho$–power series has a finite radius of convergence $R_{\rho }=2\pi $ because of a pair of complex conjugate singular points at $\rho _{c}=\pm 2\pi i$.
Fig. \[fig:PT\] shows the first partial sums $S_{T}^{[N]}(\rho
)=T_{0}+T_{1}\rho +\ldots +T_{N}\rho ^{N}$ and the accurate numerical values of $T(A)$. It dramatically illustrates the effect of the nonzero convergence radius $R_{A}=\sqrt{2\pi }$ of the small–amplitude expansion determined by the singular points of $T(A)$ closest to the origin in the complex $A$–plane.
We can provide another argument about the location of the singular points of $T(A)$. First, note that the effective potential–energy function $V(u)$ given in equation (\[eq:V(u)\]) exhibits a minimum $V(0)=(1-e^{\rho
})/(2\rho )<0$ and that the energy of the oscillatory motion is $E=\dot{u}^{2}/2+V(u)=0$ for the given initial conditions. Therefore, we expect a critical value of $\rho $ given by $V(0)=0$ that yields $\rho _{c}=\pm 2\pi
i $ in agreement with the analysis above based on the small–amplitude series.
In order to verify those exact analytical results in a numerical way we constructed Padé approximants $[N,N](\rho )$[@BO78] from the partial sums $S_{T}^{[2N]}(\rho )$ and looked for the complex zeroes of the denominator. A sequence of such zeroes appeared to converge to a limit quite close to $\pm 6.3i$ with a small real part that was negligible compared to the errors of the estimates. Besides, assuming that there is an algebraic singular point[@BO78] closest to origin of the form $(z-z_{0})^{\alpha }$ we carried out the same Padé analysis, but now on $T^{-1}dT/dA$ (as a function of $\rho$), and obtained roughly the same complex numbers that are quite close to $\pm 2\pi i$. Therefore, there appears to be no doubt that the radius of convergence of the $\rho $–power series is in fact $R_{\rho
}=2\pi $ and is due to complex conjugate singular points located on the imaginary axis of the complex $\rho $–plane at $\pm 2\pi i$.
Recently, Amore and Fernández[@AF08] investigated the possible advantages of the inverted perturbation series that in the present case takes the form $$\begin{aligned}
\rho &=&\rho _{1}\Delta T+\rho _{2}\Delta T^{2}+\ldots \nonumber \\
&=&-\frac{4\Delta T}{\pi }+\frac{\Delta T^{2}}{2\pi ^{2}}-\frac{13\Delta
T^{3}}{24\pi ^{3}}+\ldots \nonumber \\
\Delta T &=&T-2\pi \label{eq:rho_DT_series}\end{aligned}$$ Fig. \[fig:IPT\] shows that the partial sums for the inverted series $S_{\rho }^{[N]}(\Delta T)=\rho _{1}\Delta T+\rho _{2}\Delta T^{2}+\ldots
+\rho _{N}\Delta T^{N}$ converge smoothly from below towards the accurate numerical values of $T(A)$. We are presently unable to prove such most interesting feature of the inverted series rigorously.
**Summarizing**: Earlier studies on the nonlinear oscillator (\[eq:difeq\_x\])[@BM05; @M06; @KE08] have clearly shown that the period is finite for all values of the amplitude. However, they did not cast any light on the failure of the harmonic balance (with the ansatz $x^{HB}(t)=A\cos (\omega t)$) that predicts a singularity for $A=2$. In this paper we suggest that the harmonic balance may be reflecting the singular points that determine the radius of convergence of the small–amplitude series for the period. We have exactly calculated the location of those singular points and concluded that the series converge for $0<A<\sqrt{2\pi }$. This result may help to understand similar difficulties in future applications of the harmonic balance. We expect that a harmonic–balance approach with more terms will give a frecuency with only complex singular points.
In addition to what was mentioned above, we have shown that for this problem the inverted perturbation series appears to converge smoothly from below and it is therefore preferable to the original small–amplitude expansion. It is an old and well–known approach that may, in some cases, lead to surprisingly accurate results[@AF08].
[9]{} A. Chatterjee, Harmonic Balance Based Averaging: Approximate Realizations of an Asymptotic Technique, Nonlinear Dynamics 32:323-343 (2003).
J. Beatty and E. Mickens, Approximating small and large amplitude periodic orbits of the oscillator $\ddot{x}+(1+\dot{x}^{2})x=0$, Journal of Sound and Vibration 283:475-477 (2005).
E. Mickens, Investigation of the properties of the period for the nonlinear oscillator $\ddot{x}+(1+\dot{x}^{2})x=0$, Journal of Sound and Vibration 292:1031-1035 (2006).
T. Kalmár-Nagy and T. Erneux, Approximating small and large amplitude periodic orbits of the oscillator $\ddot{x}+(1+\dot{x}^{2})x=0$, Journal of Sound and Vibration 313:806-811 (2008).
P. Amore and F. M. Fernández, Inversion of the perturbation series, Journal of Physics A 41:025201 (7 pp) (2008).
C. M. Bender and S. A. Orszag, Advanced mathematical methods for scientists and engineers. McGraw-Hill, New York, 1978.
![Accurate numerical period (solid line) and partial sums for the small–amplitude approximation (dashed lines).[]{data-label="fig:PT"}](NOPT.eps){width="9cm"}
![Accurate numerical period (solid line) and partial sums for the inverse perturbation series (dashed lines).[]{data-label="fig:IPT"}](NOIPT.eps){width="9cm"}
|
---
abstract: 'We construct a family of PL triangulations of the $d$-dimensional real projective space $\mathbb{R}P^d$ on $\Theta((\frac{1+\sqrt{5}}{2})^{d+1})$ vertices for every $d\geq 1$. This improves a construction due to Kühnel on $2^{d+1}-1$ vertices.'
address:
- ' Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, GERMANY. '
- ' Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043, USA '
author:
- Lorenzo Venturello
- Hailun Zheng
bibliography:
- 'refs.bib'
title: 'A new family of triangulations of $\mathbb{R}P^d$'
---
Introduction and main results
=============================
Triangulations of topological spaces play an important role in many areas of mathematics, from the more theoretical to the applied ones. A classical problem in PL topology asks for the minimum number of vertices that a simplicial complex with a certain geometric realization can have. This invariant depends on the underlying topological space: triangulable spaces with complicated homology or homotopy groups tend to need more vertices in their vertex-minimal triangulations. However, to determine this number is in general very hard, even if we restrict our discussion to manifolds or *PL manifolds*. One of the few families for which this number is known is that of sphere bundles over the circle. Künhel [@Kuhnel86] showed that the boundary complex of the $(2d+3)$-vertex stacked $(d+1)$-manifold whose facet-ridge graph is a cycle is a PL triangulation of $\mathbb{S}^{d-1}\times \mathbb{S}^1$ for even $d$, and a PL triangulation of the twisted bundle $\mathbb{S}^{d-1}\dtimes \mathbb{S}^1$ for odd $d$. Furthermore, Kühnel’s triangulations are vertex-minimal. In the remaining cases, namely when $d$ is odd and the bundle is orientable, or when $d$ is even and the bundle is non-orientable, the minimum number of vertices is $2d+4$, and it is attained for every $d$ [@BagchiDatta-JCTA-08; @ChestnutSapirSwartz]. Novik and Swartz [@NSw-socle] (in the orientable case) and later Murai [@Mur15] proved that the number of vertices $f_0(\Delta)$ of a ${{\bf k}}$-homology $d$-manifold $\Delta$ must satisfy $\binom{f_0(\Delta)-d-1}{2}\geq\binom{d+2}{2}\widetilde{\beta}_1(\Delta; {{\bf k}})$, for $d\geq 3$ and any field ${{\bf k}}$. Equality in this formula is attained precisely by members in the *Walkup class* which are moreover $2$-neighborly, i.e, their graphs are complete. In low dimensions computational methods are a precious source. Using the program BISTELLAR [@BjLu] several vertex-minimal triangulations of $3$- and $4$-dimensional manifolds have been constructed [@LutThesis].
For a manifold whose homology (computed with coefficients in ${{\mathbb Z}}$) has a nontrivial torsion part, the Novik-Swartz-Murai lower bound is far from being tight. In this article we focus on the triangulations of the real projective space $\mathbb{R}P^d$. A lower bound on the number of vertices of a triangulation of $\mathbb{R}P^d$ was given in [@ArMa].
[@ArMa]\[thm: lower b\] Let $\Delta$ be a triangulation of $\mathbb{R}P^d$, with $d\geq 3$. Then $$f_0(\Delta)\geq\binom{d+2}{2}+1.$$
It is well known that there is a unique vertex-minimal triangulation of $\mathbb{R}P^2$ on $6$ vertices, and Walkup [@Walkup] proved that the number of vertices needed to triangulate $\mathbb{R}P^3$ is at least $11$, and constructed one such complex. More recently, an enumeration of manifolds on $11$ vertices was obtained in [@SuLu], revealing $30$ non-isomorphic such triangulations, exhibiting $5$ different $f$-vectors (see [@SuLu Table 10]). This also shows that Walkup’s construction is the *unique* $f$-vectorwise minimal triangulation of ${{\mathbb R}}P^3$. Again by computer search, a vertex-minimal triangulation of $\mathbb{R}P^4$ on $16$ vertices was found. This highly symmetric simplicial complex was studied by Balagopalan [@Bal] who described three different ways to construct this complex. Even though the $3$- and $4$-dimensional cases suggest the formula in is tight in $d=3,4$, for $d=5$ the computer search could not find triangulations of $\mathbb{R}P^5$ on less than $24$ vertices. It is now very tempting to conjecture a tight lower bound of $\binom{d+2}{2}+\lfloor\frac{d-1}{2}\rfloor$, which fits all the known cases and reflects the fact that the number of nontrivial integral homology groups of ${{\mathbb R}}P^d$ depends on the parity of $d$. Unfortunately, in higher dimensions we don’t know any triangulation of ${{\mathbb R}}P^d$ with $O(d^2)$ or even $O(d^i)$ vertices, for any $i$. The current record is due to Künhel [@Kuhnel87]. He observed that the barycentric subdivision of the boundary of the $(d+1)$-simplex possess a *free involution*, and the quotient w.r.t. the involution is PL homeomorphic to the $d$-dimensional real projective space. This construction provides a PL triangulation of $\mathbb{R}P^d$ on $2^{d+1}-1$ vertices. Indeed, a natural way to construct a triangulation of ${{\mathbb R}}P^d$ is to find a centrally symmetric triangulation of $\operatorname{\mathbb{S}}^d$, the double cover of ${{\mathbb R}}P^d$, with an additional combinatorial condition:
\[lem: from sphere to rpd\] Let $\Delta$ be a cs PL $d$-sphere with free involution $\sigma$ and with no induced cs $4$-cycle. Then $\Delta/\sigma$ is a PL triangulation of ${{\mathbb R}}P^d$.
In this article, we construct a family of PL $d$-spheres as in for every $d\geq 0$. Let $F_i$ be the $i$-th Fibonacci number, i.e., $F_0=0$, $F_1=1$ and $F_n=F_{n-1}+F_{n-2}$ for every $n\geq2$. Our main result is the following.
\[thm: exists cs with no cycle\] There exists a cs PL $d$-sphere $S_d$ with no cs induced $4$-cycles and $f_0(S_d)=3F_{d+1}+7F_d+3F_{d-1}-4$.
Via we obtain PL triangulations of $\mathbb{R}P^d$, for every $d\geq 1$.
\[thm: super main\] There exists a PL triangulation $\Delta$ of $\mathbb{R}P^d$ with $f_0(\Delta)=\frac{3}{2}F_{d+1}+\frac{7}{2}F_d+\frac{3}{2}F_{d-1}-2$.
Observe that $\frac{3}{2}F_{d+1}+\frac{7}{2}F_d+\frac{3}{2}F_{d-1}-2<2^{d+1}-1$ for every $d\geq 3$, hence improving Künhel’s construction in any dimension. Moreover the improvement of the bound is asymptotically significant, since $\frac{3}{2}F_{d+1}\leq \frac{3}{2\sqrt{5}}\left( \frac{1+\sqrt{5}}{2}\right)^{d+1}\sim \frac{3}{2\sqrt{5}}(1.61803\dots)^{d+1}$.
Definitions {#sec: 2}
===========
A *simplicial complex* $\Delta$ with vertex set $V=V(\Delta)$ is a collection of subsets of $V$ that is closed under inclusion. The elements of $\Delta$ are called [*faces*]{}. For brevity, we usually denote $\{v\}$ as $v$ and with $f_0(\Delta)$ the number $|V(\Delta)|$. The *dimension of a face* $F\in\Delta$ is $\dim F:=|F|-1$. The *dimension of $\Delta$*, $\dim\Delta$, is the maximum dimension of its faces.
If $F$ is a face of $\Delta$, then the [*star of $F$*]{} and the [*link of $F$*]{} in $\Delta$ are the simplicial complexes $$\operatorname{\mathrm{st}}_\Delta(F):=\{\sigma\in\Delta \ : \ \sigma\cup F\in\Delta\} \quad \mbox{and} \quad \operatorname{\mathrm{lk}}_\Delta(F):= \{\sigma\in \operatorname{\mathrm{st}}_\Delta(F) \ : \ \sigma\cap F=\emptyset\}.$$ If $\Delta$ and $\Gamma$ are simplicial complexes on disjoint vertex sets, then the *join* of $\Delta$ and $\Gamma$ is the simplicial complex $\Delta*\Gamma = \{\sigma \cup \tau \ : \ \sigma \in \Delta \text{ and } \tau \in \Gamma\}$. In particular, if $\Delta_1=\{\emptyset, \{v\}\}$, then $\Delta_1*\Delta_2$ is called the *cone* over $\Delta_2$ with apex $v$.
Let $\Delta$ be a simplicial complex and $W\subseteq V(\Delta)$. The *induced* subcomplex of $\Delta$ on $W$ is $\Delta_W=\{F\in\Delta: \; F\subseteq W\}$. If $\Gamma$ is a subcomplex of $\Delta$, define $$\Delta\setminus\Gamma=\Delta_{V(\Delta)\setminus V(\Gamma)}=\{F\in \Delta: F\cap V(\Gamma)=\emptyset\}.$$ The complement of $\Gamma$ in $\Delta$, denoted by $\Delta-\Gamma$, is the set of faces in $\Delta$ but not in $\Gamma$. The closure $\overline{\Delta-\Gamma}$ is the subcomplex of $\Delta$ generated by the facets of $\Delta-\Gamma$. If $f$ is an automorphism of $\Delta$, then the *quotient* of $\Delta$ w.r.t. $f$, $\Delta/f$, is the simplicial complex obtained identifying the vertices in the same orbit and all the faces with the same vertex set. Observe that in general $|\Delta/f|\ncong|\Delta|/\tilde{f}$, where $\tilde{f}: |\Delta| \to |\Delta|$ is the continuous map induced by $f$. For instance, if $\Delta$ is a square and $f$ maps every vertex $v$ to the unique vertex not adjacent to $v$ then $\Delta/f$ is the $1$-dimensional simplex, while $|\Delta|/\tilde{f}\cong \mathbb{S}^1$.
We say two simplicial complexes $\Delta_1$ and $\Delta_2$ are *PL homeomorphic*, denoted as $\Delta_1 {\overset{\mathrm{PL}}{\cong}}\Delta_2$, if there exists subdivisions $\Delta_1'$ of $\Delta_1$ and $\Delta_2'$ of $\Delta_2$ that are simplicially isomorphic. A *PL $d$-ball* is a simplicial complex PL homeomorphic to a $d$-simplex. Similarly, a *PL $d$-sphere* is a simplicial complex PL homeomorphic to the boundary complex of a $(d+1)$-simplex. A $d$-dimensional simplicial complex $\Delta$ is called a *PL $d$-manifold* if the link of every non-empty face $F$ of $\Delta$ is a $(d-|F|)$-dimensional PL ball or sphere; in the former case, we say $F$ is a *boundary face* while in the latter case $F$ is an *interior face*. The *boundary complex* $\partial \Delta$ is the subcomplex of $\Delta$ that consists of all boundary faces of $\Delta$. A PL manifold whose geometric realization is homeomorphic to a closed manifold $M$ is called a *PL triangulation* of $M$. In the literature PL manifolds as defined above are sometimes called *combinatorial manifolds* or *combinatorial triangulations of manifolds*.
For $d\geq 5$ the class of PL $d$-manifolds is strictly contained in that of triangulated $d$-manifolds. In particular, the double suspension of any homology $3$-sphere with a non-trivial fundamental group is a non-PL simplicial 5-sphere (see e.g., [@RoSa]).
PL manifolds have the following nice properties, see [@Al30] and [@Lickorish]:
\[lem: property of PL manifolds\] Let $\Delta_1$ and $\Delta_2$ be PL $d_1$- and $d_2$-manifolds, respectively.
1. If $\Delta_1$ and $\Delta_2$ are PL balls, so is $\Delta_1*\Delta_2$.
2. If $d_1=d_2=d$ and $\Gamma:=\Delta_1\cap \Delta_2=\partial\Delta_1\cap \partial \Delta_2$ is a PL $(d-1)$-manifold, then $\Delta_1\cup \Delta_2$ is a PL $d$-manifold. If furthermore $\Delta_2$ and $\Gamma$ are PL balls, then $\Delta_1\cup \Delta_2{\overset{\mathrm{PL}}{\cong}}\Delta_1$.
(Newman’s theorem)\[lem: newmann\] Let $\Delta$ be a PL $d$-sphere and $\Psi\subset\Delta$ be a PL $d$-ball. Then the closure of the complement of $\Psi$ in $\Delta$ is a PL $d$-ball.
Let $\Delta$ be a PL manifold without boundary. The *prism over $\Delta$* is the pure polyhedral complex $\Delta\times [-1,1]$, whose cells are of the form $F\times\{1\}$, $F\times\{-1\}$ or $F\times[-1,1]$, for every $F\in\Delta$. Clearly $\left|\Delta\times [-1,1] \right|\cong \left|\Delta\right|\times [-1,1]$. The boundary of the prism consists of the cells $F\times \{1\}$ and $F\times \{-1\}$, where $F\in \Delta$.
A simplicial complex $\Delta$ is *centrally symmetric* or [*cs*]{} if its vertex set is endowed with a [*free involution*]{} $\sigma: V(\Delta) \rightarrow V(\Delta)$ that induces a free involution on the set of all non-empty faces of $\Delta$. Let $\Delta$ be a centrally symmetric PL $d$-sphere and let $\sigma$ be the free involution on $\Delta$. An *induced cs $4$-cycle* in $\Delta$ is an induced subcomplex $C\subseteq\Delta$ isomorphic to a $4$-cycle, with $\sigma(C)=C$. In particular, the vertices of $C$ are $v,w,\sigma(v),\sigma(w)$, for some $v,w\in V(\Delta)$. The complex $\Delta$ has no induced cs $4$-cycle if and only if $\operatorname{\mathrm{st}}_{\Delta}(v)\cap\operatorname{\mathrm{st}}_{\Delta}(\sigma(v))=\{\{\emptyset\}\}$ for every $v\in V(\Delta)$.
In what follows we define two properties in the structure of a cs PL $d$-sphere.
\[def: P\_d\] Let $S$ be a cs PL $d$-sphere with free involution $\sigma$. Assume that there exists a PL $d$-ball $B\subset S$ satisfying the following conditions:
- $S= B\cup\sigma(B)$, where $B\cap \sigma(B)=\partial B=\partial \sigma(B)$.
- $S\setminus \partial B = D \uplus \sigma(D)$, with $D\subseteq S$ a PL $d$-ball.
- There exists $v\in V(D)$ such that $V(\operatorname{\mathrm{st}}_{S}(v))\cup V(\operatorname{\mathrm{st}}_{S}(\sigma(v)))=V(S)$ and furthermore, $D\subseteq\operatorname{\mathrm{st}}_{S}(v)\subseteq B$.
Then we say that $S$ satisfies *Property $\mathrm{P}_d$* w.r.t. the triple $(B,D,v)$.
We say that a cs PL $d$-sphere $S$ satisfies *Property $\mathrm{Q}_d$* if there exists a sequence of subcomplexes $S_0\subseteq S_2\subseteq \dots \subseteq S_d=S$ such that each $S_i$ is a cs PL $i$-sphere that satisfies Property $\mathrm{P}_i$ w.r.t. a triple $(B_i, D_i, v_i)$ with $\partial B_i=S_{i-1}$ for all $1\leq i\leq d$.
\[ex: S\_1\] The cs 6-cycle $S_1$ satisfies Property $\mathrm{Q}_{1}$.
\[ex: S\_0\] The boundary complex of the icosahedron can be realized as a cs PL $2$-sphere $S_2$ on $12$ vertices which satisfies Property $\mathrm{Q}_{2}$: indeed there is an induced cs 6-cycle $S_1$ in $S_2$ that divides $S_2$ into two antipodal 2-balls. In this case $S_2\setminus S_1$ is the disjoint union of two triangles $F, \sigma(F)$ and any vertex $v\in F$ satisfies the third condition in Definition \[def: P\_d\].
From a triangulation of $\operatorname{\mathbb{S}}^{d-1}$ to a triangulation of $\operatorname{\mathbb{S}}^{d}$
===============================================================================================================
Given a cs PL $(d-1)$-sphere $S$ with $2n$ vertices but with no induced cs 4-cycles, one may build a cs PL $d$-sphere with $4n+2$ vertices as follows: first build the prism over $S$; then triangulate the prism in such a way that no additional induced cs 4-cycles and interior vertices are created; finally cone over the boundaries of the prism (as two disjoint copies of $S$) with two new vertices. The PL $d$-sphere obtained in this way has no induced 4-cycles. In this section, we will modify the above approach to reduce the number of vertices in the construction.
In what follows, assume that $S$ is a cs PL $(d-1)$-sphere with involution $\sigma$ and it satisfies Property $\mathrm{Q}_{d-1}$; in particular, we have $S$ satisfies Property $\mathrm{P}_{d-1}$ under the triple $(B,D,p)$ and $S':=\partial B$ is a cs PL $(d-2)$-sphere that satisfies Property $\mathrm{P}_{d-2}$ under the triple $(B', D', p')$. We will first give a triangulation $\Sigma$ of $S\times [-1,1]$ such that
- $\Sigma$ is centrally symmetric;
- there exists a cs subcomplex $\Gamma\subseteq\Sigma$ with $\Gamma\cong S$ and $D\times\{1\}, \sigma(D)\times\{-1\}\subseteq \Gamma$.
This will be done in two steps. First we construct a polyhedral complex that satisfies the above conditions, see Proposition \[prop: sigma\]. Then we triangulate it while preserving these properties, see Corollary \[cor: exists triangulation prism\]. Finally, we build a cs PL $d$-sphere from $\Sigma$, see Corollary \[cor: phi is sphere\].
The prism over $S$
------------------
The main point of this section is . By *refinement* of a polyhedral complex $P$ we mean a polyhedral complex $\Delta$ with $|\Delta|\cong|P|$ obtained by subdividing $P$.
\[prop: sigma\] Let $S$ be a cs PL $(d-1)$-sphere satisfying Property $\mathrm{Q}_{d-1}$. There exists a refinement $\Sigma''$ of $S\times[-1,1]$ such that:
- $\Sigma''$ is centrally symmetric.
- There exists a cs subcomplex $\Gamma\subseteq\Sigma''$ with $\Gamma\cong S$ and $D\times\{1\}, \sigma(D)\times\{-1\}\subseteq \Gamma$.
In order to prove we first refine $S\times[-1,1]$ to a three layered prism over $S$.
We define $\Sigma'$ to be the polyhedral complex $$\Sigma'=(S\times[-1,0])\cup (S\times[0,1]).$$
In other words, $\Sigma'$ is a prism over $S$ with three layers $S\times \{-1\}$, $S\times \{0\}$ and $S\times \{1\}$. Furthermore, $\Sigma'$ is centrally symmetric with the free involution $\sigma'$ induced by $(v,i)\mapsto(\sigma(v),-i)$, for $i=-1,0,1$. We define the following maps:
$$\begin{aligned}
\psi_{+0}: V(S\times \{1\})&\rightarrow V(\Sigma')\nonumber\\
(v,1)&\mapsto \begin{cases}
(v,1) & \text{if } v\in V(D)\uplus V(\operatorname{\mathrm{st}}_{S'}(v'))\nonumber\\
(v,0) & \text{if } v\in V(\sigma(D))\uplus V(\sigma(\operatorname{\mathrm{st}}_{S'}(v'))\\
\end{cases},\\
\psi_{-0}: V(S\times \{-1\})&\rightarrow V(\Sigma')\\
(v,-1)&\mapsto \begin{cases}
(v,-1) & \text{if } v\in V(\sigma(D))\uplus V(\sigma(\operatorname{\mathrm{st}}_{S'}(v')))\nonumber\\
(v,0) & \text{if } v\in V(D)\uplus V(\operatorname{\mathrm{st}}_{S'}(v'))\\
\end{cases}.
\nonumber
\end{aligned}$$
Observe that the maps $\psi_{+0}$ and $\psi_{-0}$ induce maps from the polyhedral complex $\Sigma'$ to itself.
\[Sigma’\] We define $$\Sigma'':=\Sigma'/\sim,$$ where $v\sim w$ if and only if $w=\psi_{+0}(v)$ or $w=\psi_{-0}(v)$.
Informally speaking, we identify the subcomplexes of the middle layer of $\Sigma$ with the upper and lower layers. In this way we obtain a polyhedral complex containing an induced subcomplex isomorphic to $S$ (see ).
![The complexes $\Sigma'$ and $\Sigma''$, together with a cs triangulation $\Sigma$.[]{data-label="fig: prism"}](Triangulating_prism_2_opacity.pdf)
\[lem: property of Sigma”\] The complex $\Sigma''$ in has the following properties:
- $\left| \Sigma''\right| \cong \mathbb{S}^{d-1}\times[0,1]$.
- $\partial\Sigma''\cong S\uplus S$.
- There exists an induced cs subcomplex $\Gamma\subseteq\Sigma''$ with $\Gamma\cong S$ and the images of $D\times\{1\}$ and $\sigma(D)\times\{-1\}$ in $\Sigma''$ under the maps $\psi_{+0}$ and $\psi_{-0}$ are subcomplexes of $\Gamma$.
- $\Sigma''$ is centrally symmetric under an involution $\sigma''$ induced from $\sigma'$.
Part $i$ is clear, and for $ii$ it suffices to observe that the image of $S\times\{1\}$ and $S\times\{-1\}$ in $\Sigma''$ are disjoint and isomorphic to $S$.\
Let $\Gamma$ be the image of $S\times\{0\}$ in $\Sigma'$. Since the vertices of $S\times\{0\}$ are not subject to any identification, $\Gamma= S\times\{0\}\cong S$. Moreover $\psi_{+0}(D\times\{1\})= D\times\{1\}\subseteq\Gamma$ and $\psi_{-0}(\sigma(D)\times\{-1\})=\sigma(D)\times\{-1\}\subseteq\Gamma$. This proves $iii$.\
Finally, we observe that if $F\sim G$ in $\Sigma''$ then $\sigma(F)\sim \sigma(G)$. Equivalently, the map that assigns the equivalence class of the vertex $(v,i)$ to the class of $\sigma'((v,i))$ is well defined. Therefore, it induces a free involution of $\sigma'':\Sigma''\to \Sigma''$.
The cs triangulation $\Sigma$ of $\Sigma''$
-------------------------------------------
In this subsection we construct a centrally symmetric *simplicial* complex $\Sigma$ which refines $\Sigma''$ such that
- $V(\Sigma)=V(\Sigma'')$, i.e., no new vertex is introduced.
- $\Sigma$ is centrally symmetric with a free involution $\tau$, such that $\tau|_{\Sigma''}=\sigma''$.
Our construction is based on certain orientations of the graph of a simplicial complex called locally acyclic orientations. We refer to [@DeLoera-et-all Section 7.2] for a more detailed treatment of the subject.
A *locally acyclic orientation* (l.a.o.) of $\Delta$ is an orientation of the edges of its graph such that none of the $2$-simplices of $\Delta$ contains an oriented cycle.
\[triangulation of prism\] Let $\Delta$ be a simplicial complex. The simplicial refinements of $\Delta\times[0,1]$ are in bijection with locally acyclic orientations of $\Delta$.
Every simplicial complex has a locally acyclic orientation obtained from acyclic orientations of its graph. The bijection in Lemma \[triangulation of prism\] is easy to describe: if $\{i,j\}$ is an edge of $\Delta$ with $i\rightarrow j$, then $\{(i,0),(j,1)\}$ is an edge of $\Delta\times[0,1]$ and vice versa. This induces a triangulation of $F\times [-1,1]$ for every face $F\in \Delta$. Furthermore, the locally acyclicity guarantees that the union of triangulations of individual cells can be coherently completed to a triangulation of $\Delta\times [-1,1]$. The following observation is straightforward.
\[lem: symmetry l.a.o.\] Let $\Delta$ be a centrally symmetric simplicial complex with involution $\sigma$ and consider a refinement of $\Delta\times[-1,1]$ which is centrally symmetric with involution induced by $\widetilde{\sigma}:(v,-1) \mapsto (\sigma(v),1), \; (v, 1)\mapsto (\sigma(v), -1)$ for any vertex $v\in \Delta$. Then the corresponding locally acyclic orientation of $\Delta$ is order reversing w.r.t. the symmetry, i.e., $v\rightarrow w$ if and only if $\sigma(w) \rightarrow \sigma(v)$.
In any cs triangulation of $\Delta\times[-1,1]$ the set $\{(v,-1),(w,1)\}$ is an edge if and only if $\{\widetilde{\sigma}((v,-1)),\widetilde{\sigma}((w,1))\}=\{(\sigma(v),1),(\sigma(w),-1)\}$ is an edge, which implies that on the corresponding l.a.o. we have that $v\rightarrow w$ if and only if $\sigma(w)\rightarrow \sigma(v)$.
Recall that $S'$ is a cs PL $(d-2)$-sphere in $S$ that satisfies $\mathrm{P}_{d-2}$ under the triple $(B',D', p')$. Hence $V(S')=V(\operatorname{\mathrm{st}}_{S'}(p'))\cup V(\operatorname{\mathrm{st}}_{S'}(\sigma(p'))$ and $$\label{eq: decompose S}
V(S)=V(\sigma(D))\cup V(\sigma(\operatorname{\mathrm{st}}_{S'}(p')))\cup V(D)\cup V(\operatorname{\mathrm{st}}_{S'}(p')).$$ We define a subset $\mathbf{A}$ of the edges of $S$ as follows: $$\mathbf{A}=\{\{v,w\}:\text{ } v\in V(D)\uplus V(\operatorname{\mathrm{st}}_{S'}(p'))\text{ and }w\in V(\sigma(D))\uplus V(\sigma(\operatorname{\mathrm{st}}_{S'}(p')))\}.
$$ Observe that $\sigma(e)\in \mathbf{A}$ for every $e\in\mathbf{A}$.
\[lem: l.a.o. for S\] There exists a l.a.o. of $S$ such that:
- $v\rightarrow w$ for every edge $\{v,w\}$ in $\mathbf{A}$.
- For every edge $\{v,w\}\in S$, $v\rightarrow w$ if and only if $\sigma(w)\rightarrow \sigma(v)$.
First we choose any l.a.o. of the induced subcomplex of $S$ on ${V(D)\uplus V(\operatorname{\mathrm{st}}_{S'}(p'))}$. Following we impose on the induced subcomplex on $V(\sigma(D))\uplus V(\sigma(\operatorname{\mathrm{st}}_{S'}(p')))$ a reverse orientation. By (\[eq: decompose S\]), the remaining edges of $S$ are those in $\mathbf{A}$. We orient all edges $\{v,w\}$ of type $\mathbf{A}$ as $v\rightarrow w$. This orientation is by definition acyclic on every $2$-simplex not containing edges in $\mathbf{A}$. It suffices to check that no edge in $\mathbf{A}$ is contained in an oriented cycle. Since every $2$-simplex containing an edge in $\mathbf{A}$ also contains another edge in $\mathbf{A}$, it contains a vertex $v$ or a vertex $w$ such that either $w_1\leftarrow v\rightarrow w_2$ or $v_1\rightarrow w\leftarrow v_2$. This proves the claim.
The l.a.o. in the proof of Lemma \[lem: l.a.o. for S\] defines a centrally symmetric triangulation of $\operatorname{\mathbb{S}}^{d-1} \times [-1,1]$. The following lemma shows that this triangulation is a *PL manifold with boundary* without interior vertices. For any prism $\Delta\times [-1,1]$, we denote by $(\Delta\times [-1,1])^{\ell}$ the refinement of $\Delta\times [-1,1]$ induced by the l.a.o. $\ell$ on $\Delta$.
![Two different locally acyclic orientations of $S_1$ satisfying and the corresponding triangulations of the prism.[]{data-label="fig: l.a.o."}](locally_acyclic_orientations_opacity.pdf)
\[lem: is PL manifold\] Let $\Gamma$ be a PL $(d-1)$-sphere, $\Delta=\Gamma\times[-1,1]$ and let $\ell$ be any locally acyclic orientation on $\Gamma$. Then, the refinement $\Delta^{\ell}$ is a PL $d$-manifold with boundary and without interior vertices.
Since $\Gamma$ is a PL $(d-1)$-sphere, there exists a subdivision of $\Gamma'$ for which every simplex can be linearly embedded in ${{\mathbb R}}^N$ for some $N$. Moreover, for every locally acyclic orientation $\ell$ of $\Gamma$, consider a linear ordering of the vertices of $\Gamma'$ such that $v>w$ for every $v\in V(\Gamma')\setminus V(\Gamma)$ and $w\in V(\Gamma)$. Orient all the edges $\{u,v\}\notin \Gamma$ with $u\to v$ if $u>v$. This extends to a locally acyclic orientation $\ell'$ of $\Gamma'$ that agrees with $\ell$ when restricted to the edges of $\Gamma$. The corresponding refinement of $\Gamma'\times[-1,1]$ is a subdivision of $\Gamma\times[-1,1]$ in which every simplex is linearly embedded in ${{\mathbb R}}^{N+1}$. Therefore, $\Delta^{\ell}$ is a PL manifold with boundary. Furthermore $V(\Delta^{\ell})=V(\Gamma\times \{-1\})\cup V(\Gamma\times \{1\})=V(\partial \Delta^\ell)$ and hence $\Delta^\ell$ has no interior vertices.
\[cor: exists triangulation prism\] There exists a l.a.o. $\ell$ on the cs $(d-1)$-sphere $S$ such that
- $(S\times[-1,1])^\ell$ is a cs PL manifold with boundary that refines $\Sigma''$;
- $(S\times [-1,1])^\ell$ contains an induced cs subcomplex $\Gamma$ isomorphic to $S$ which contains $(D\times\{1\})\cup (\sigma(D)\times\{-1\})$;
- $V((S\times [-1,1])^\ell)=V(\Sigma'')$.
Choose any l.a.o. $\ell$ on $S$ that satisfies the conditions in Lemma \[lem: l.a.o. for S\]. The orientation on any edge $\{v,w\}\in \mathbf{A}$ generates a new edge $\{(v,1), (w,-1)\}\in \Sigma$. This coincides with those edges in $\Sigma''$ but not in $S\times [-1,1]$. Hence $(S\times [-1,1])^\ell$ is a refinement of $\Sigma''$. By Lemma \[lem: is PL manifold\], $(S\times [-1,1])^\ell$ is a PL manifold. The other two properties follow from Lemmas \[lem: property of Sigma”\] and \[lem: is PL manifold\].
In what follows, we fix any l.a.o. on $S$ as in and denote by $\Sigma$ the PL manifold with boundary obtained in . Although $\Sigma$ can be defined directly from the l.a.o., the second bullet point in Corollary \[cor: exists triangulation prism\] (which follows from subsection 3.1) will play a key role in our inductive construction in Section 4.
From $\Sigma$ to a PL triangulation of $\operatorname{\mathbb{S}}^d$
--------------------------------------------------------------------
In this subsection we complete the cs triangulation of the prism over a PL $(d-1)$-sphere as in to a PL $d$-sphere. We introduce two pairs of new vertices $v_{+},w_{+},v_{-},w_{-}$ and define $$\Phi':= \Sigma \cup (v_{+} * K_+) \cup (v_{-} * K_-)\cup (w_{+} * L_+)\cup (w_{-} * L_-) ,$$ where $K_{\pm}:=B\times\{\pm 1\}$ and $L_{\pm}:=(\sigma(B)\times\{\pm 1\})\cup_{\partial \sigma(B)\times\{\pm 1\}} (v_{\pm}*\partial \sigma(B)\times\{\pm 1\})$. The second complex in offers a visualization of $\Phi'$ in the $2$-dimensional case.
The simplicial complex $\Phi'$ is a cs PL triangulation of $\operatorname{\mathbb{S}}^d$.
By , $\Sigma$ is a PL manifold with boundary. By Property $\mathrm{Q}_{d-1}$, $K_{\pm}$ is a PL $(d-1)$-ball and by , $L_{\pm}$ is a PL $(d-1)$-sphere. Again by , $\Phi'$ is a PL manifold. Finally it is clear that $\Phi'$ is centrally symmetric and $|\Phi'|$ is homeomorphic to $\operatorname{\mathbb{S}}^d$.
We next contract certain edges of $\Phi'$ in order to reduce the number of vertices. A simplicial complex is obtained from $\Delta$ via an *edge contraction* of $e=\{u,v\}\in\Delta$ if it is the image of $\Delta$ w.r.t. the simplicial map identifying $u$ and $v$. The edges we will contract are those of the form $\{(v,1),(v,-1)\}$, where $v\in D\cup \sigma(D)$. In other words, we identify $D\times\{1\}$ and $\sigma(D)\times\{-1\}$ with $D\times\{-1\}$ and $\sigma(D)\times\{1\}$. It is easy to see that in this case the procedure does not depend on the order in which contractions are applied. To prove that the resulting complex is still a PL sphere we use a result by Nevo [@Nevo07].
\[lem eran\] Let $\Delta$ be a PL manifold and $\Delta'$ be the contraction of $\Delta$ at the edge $\{u,v\}$. Then $\Delta'$ is PL homeomorphic to $\Delta$ if and only if $\operatorname{\mathrm{lk}}_{\Delta}(u)\cap\operatorname{\mathrm{lk}}_{\Delta}(v)=\operatorname{\mathrm{lk}}_{\Delta}(\{u,v\})$.
\[lem: condition\] The complex $\Sigma$ satisfies the following link condition: $$\operatorname{\mathrm{lk}}_{\Sigma}((v,1))\cap\operatorname{\mathrm{lk}}_{\Sigma}((v,-1))=\operatorname{\mathrm{lk}}_{\Sigma}(\{(v,1),(v,-1)\}).$$
By the definition of $\Sigma$, a face $F=\{(w_1,t_1),\dots (w_k,t_k)\}\in\Sigma$ belongs to $\operatorname{\mathrm{lk}}_{\Sigma}((v,\pm 1))$ if and only if
- $\{w_1,\dots, w_k\}\in \operatorname{\mathrm{lk}}_{\Gamma}(v)$;
- $\{(w_i,t_i),(v,\pm 1)\}$ is an edge of $\Sigma$ for every $i=1,\dots n$.
Similarly, a face is in $\operatorname{\mathrm{lk}}(\{(v,1), (v,-1)\})$ if and only if the above two conditions hold. This proves the claim.
\[cor: phi is sphere\] The simplicial complex $\Phi$ obtained contracting $\Phi'$ along every edge of the form $\{(v,1),(v,-1)\}$ with $v\in V(D)\cup V(\sigma(D))$, is a cs PL $d$-sphere.
Let $V(D)=\{v_0, \dots, v_k\}$, $e_i=\{(v_i,1),(v_i,-1)\}$ for $i=0,1,\dots,k$ and $$\Phi'=\Phi_0\overset{e_0, \sigma(e_0)}{\longrightarrow}\Phi_1\overset{e_1, \sigma(e_1)}{\longrightarrow}\dots\Phi_{k-1}\overset{e_{k-1},\sigma(e_{k-1})}{\longrightarrow}\Phi_k=\Phi,$$ the sequence of contracting antipodal edges $e_i, \sigma(e_i)$ from $\Phi'$ to $\Phi$. For every vertex $v\in S'$ we have that $$\operatorname{\mathrm{lk}}_{\Phi'}((v,1))\cap\operatorname{\mathrm{lk}}_{\Phi'}((v,-1))=\operatorname{\mathrm{lk}}_{\Sigma}((v,1))\cap\operatorname{\mathrm{lk}}_{\Sigma}((v,-1))$$ $$\mathrm{and}\quad \operatorname{\mathrm{lk}}_{\Phi'}(\{(v,1),(v,-1)\})=\operatorname{\mathrm{lk}}_{\Sigma}(\{(v,1),(v,-1)\}).$$ By and and the fact that the links of $\{(v,1), (v,-1)\}$, and $\{(\sigma(v), 1), (\sigma(v), -1)\}$ in $\Sigma$ are disjoint, it follows that the simplicial complex $\Phi_1$ is a PL $d$-sphere. Observe that $\{(v_{i-1},-1),(v_{i-1},1)\}$ belongs to at most one of the complexes $\operatorname{\mathrm{lk}}_{\Phi_{i-1}}((v_i,1))$ and $\operatorname{\mathrm{lk}}_{\Phi_{i-1}}((v_i,-1))$. In particular, for every $i=1,\dots,k$, $$\{(v_{i-1},-1),(v_{i-1},1)\}\notin \operatorname{\mathrm{lk}}_{\Phi_{i-1}}((v_i,1))\cap\operatorname{\mathrm{lk}}_{\Phi_{i-1}}((v_i,-1))$$ $$\mathrm{and}\quad\{(v_{i-1},-1),(v_{i-1},1)\}\notin \operatorname{\mathrm{lk}}_{\Phi_{i-1}}(\{(v_i,1),(v_i,-1)\}).$$ This fact, together with , implies that $$\begin{aligned}
\operatorname{\mathrm{lk}}_{\Phi_{i-1}}((v_i,1))\cap\operatorname{\mathrm{lk}}_{\Phi_{i-1}}((v_i,-1))&= \operatorname{\mathrm{lk}}_{\Phi'}((v_i,1))\cap\operatorname{\mathrm{lk}}_{\Phi'}((v_i,-1))\\
&=\operatorname{\mathrm{lk}}_{\Phi'}(\{(v,1),(v,-1)\})\\
&= \operatorname{\mathrm{lk}}_{\Phi_{i-1}}(\{(v_i,1),(v_i,-1)\}),
\end{aligned}$$ which shows that every $\Phi_i$ is a PL $d$-sphere.
\[rem: more contractions\] In fact, we can contract more edges $\{(v,-1),(v,1)\}$ and their antipodes with $v\in V(\Gamma)$ and still obtain a cs PL $d$-sphere. However, contracting too many edges would create induced cs 4 cycles in the resulting complex.
By Corollary \[cor: exists triangulation prism\], the cs $d$-sphere $\Phi$ that we define is usually not unique. (Indeed, the number of combinatorial types of $\Phi$ is related to the number of l.a.o. on $S$ that satisfies the conditions in Lemma \[lem: l.a.o. for S\].) With a slight abuse of notation we will often write “a sphere $\Phi$" to indicate any PL $d$-sphere that could be constructed from $S$ as in this section.
The induction step
==================
In this section, we show that the sphere $\Phi$ constructed in the previous section satisfies property $\mathrm{Q}_d$ w.r.t. a certain flag of spheres $S_0\subseteq S_2\subseteq \dots \subseteq S_d=\Phi$. Finally, we show that if $S_{d-1}$ does not have cs induced $4$-cycles then the same holds for $S_d$. Using this fact, together with the initial cases being the $0$-dimensional sphere $S_0$ and the cs 6-cycle $S_1$, we prove . Assume that inductively we’ve constructed a sequence of cs $i$-spheres $S_{i}$, $0\leq i\leq d-1$, that satisfies Property $\mathrm{Q}_{i}$. In particular, the triple $(B_{i}, D_{i}, v_{i})$ is such that
- $B_i$ is a PL $i$-ball in $S_i$ with boundary $S_{i-1}$.
- $D_i\cong \{v_i, w_i\}*\operatorname{\mathrm{st}}_{S_{i-2}}(v_{i-2})$ for some $w_i\in S_i$.
- $V(\operatorname{\mathrm{st}}_{S_i}(v_{i}))=V(D_i)\cup V(\operatorname{\mathrm{st}}_{S_{i-1}}(v_{i-1}))$.
We will now show that the PL $d$-sphere $\Phi$ constructed in the previous section by setting $\Gamma:=S_{d-1}$ satisfies Property $\mathrm{Q}_d$. With the notation introduced earlier we define:
- $D_d:= \{v_{+},w_{+}\}*\operatorname{\mathrm{st}}_{S_{d-2}}(v_{d-2})$.
- $B_d$ is the closure of one of the two connected components of $\Phi-\Gamma:=\{F\in \Phi\;|\;F\cap \Gamma=\emptyset\}$.
- $v_d:=v_+$.
By Jordan’s theorem, the geometric realization of $\Phi -\Gamma$ consists of two connected components. Since $S$ and $\Gamma$ are PL spheres, it is known that $B_d$ is a simplicial ball. However, it is an open problem in PL topology (known as PL Schoenflies problem) to decide whether $B_d$ is also a PL ball. The following lemma gives a positive answer in the special case of our construction.
![An illustration of in the case $d=2$[]{data-label="fig: rp2"}](RP2_opacity2.pdf)
The triple $(B_d, D_d, v_d)$ satisfies the following properties:
- $B_d$ is a PL $d$-ball in $\Phi$ with boundary $\Gamma\cong S_{d-1}$.
- $D_d$ is isomorphic to $\{v_d, w_{+}\}* \operatorname{\mathrm{st}}_{S_{d-2}}(v_{d-2})$.
- $D_d\subseteq \operatorname{\mathrm{st}}_{\Phi}(v_d) \subseteq B_d$ and $V(\operatorname{\mathrm{st}}_{\Phi}(v_d))= V(D_d)\cup V(\operatorname{\mathrm{st}}_\Gamma(v_{d-1}))$.
In particular, $\Phi$ satisfies Property $\mathrm{P}_d$ under the triple $(B_d, D_d, v_d)$
We only need to verify the first and third bullet points. By the inductive hypothesis and the definition of $D_d$, we obtain $$\begin{split}
V(\operatorname{\mathrm{st}}_{ \Phi}(v_d))&=V(B_{d-1}\times \{1\})\cup\{v_d, w_{+}\}\\
&=\Big( V(D_{d-1}\times \{1\})\cup V(\operatorname{\mathrm{st}}_{S_{d-2}}(\sigma(v_{d-2})\times \{1\})\Big) \cup \Big( V(\operatorname{\mathrm{st}}_{S_{d-2}}(v_{d-2})\times \{1\})\cup\{v_d, w_{+}\}\Big) \\
&=V(\operatorname{\mathrm{st}}_{\Gamma}(v_{d-1}))\cup V(D_{d}).
\end{split}$$ To see that $B_d$ is a PL $d$-ball, we consider the simplicial complex $\Phi^*$ obtained from $\Phi$ by contracting all the edges of the form $\{(v,-1),(v,1)\}$ with $v\in V(\operatorname{\mathrm{st}}_{\phi}(v_d))$. By and , $\Phi^*$ is a PL $d$-sphere. The image of $\Sigma \cup (v_{+} * K_+)\cup (w_{+} * L_+)$ is a PL $d$-ball, since $K_+$ and $L_+$ are a PL $(d-1)$-ball and a $(d-2)$-sphere respectively. By the (closure of) the complement of the PL $d$-ball $\Sigma \cup (v_{+} * K_+)\cup (w_{+} * L_+)$ w.r.t. a PL $d$-sphere $\Phi^*$, $B_d$, is a PL $d$-ball.
Finally, by the definition, we have that $B_d\setminus \Gamma=D_d$ and $D_d\subseteq \operatorname{\mathrm{st}}_\Phi(v_d)\subseteq B_d$. Furthermore, $V(\operatorname{\mathrm{st}}_\Phi(v_d))\cup V(\operatorname{\mathrm{st}}_\Phi(\sigma(v_d))=V(\Phi)$ follows from the inductive assumption that $V(\operatorname{\mathrm{st}}_\Gamma(v_{d-1}))\cup V(\operatorname{\mathrm{st}}_\Gamma(\sigma(v_{d-1})))=V(\Gamma)$.
The property which motivates our construction is the content of the following lemma.
\[prop: no induced 4\] The complex $\Phi$ as constructed above has no induced cs $4$-cycle if $\Gamma\cong S_{d-1}$ has no induced cs $4$-cycle.
As we see from the construction, since $\operatorname{\mathrm{lk}}_{\Phi}(v_+)\cap \operatorname{\mathrm{lk}}_{\Phi}(v_-)=\operatorname{\mathrm{lk}}_{\Phi}(w_+)\cap \operatorname{\mathrm{lk}}_{\Phi}(w_-)=\emptyset$, it follows that none of the vertices $v_+, v_-, w_+,w_-$ belongs to any induced cs $4$-cycle. Furthermore, any vertex $(a,1)\in S_{d-2}\times\{1\}$ is only adjacent to either $v_+, w_+$, or its neighbors in $S_{d-1}\times\{1\}$, or some $(\sigma(b),-1)\in S_{d-2}\times\{-1\}$ where $\{\sigma(a),\sigma(b)\}\in S_{d-2}\times\{-1\}$. By the inductive hypothesis, $S_{d-2}$ and $S_{d-1}$ do not contain any induced cs $4$-cycle. We conclude that $(a,1)$ is also not in any induced cs $4$-cycle in $S$. As any cs $4$-cycle must include two vertices outside $D_d\cup\sigma(D_d)\in \Phi$, this prove our claim that $\Phi$ has no induced cs $4$-cycle.
Finally, we conclude with the main result.
\[thm: main\] There exists a family of cs PL $i$-spheres $S_0\subseteq S_1 \subseteq S_2\dots$ such that each $S_i$ satisfies Property $\mathrm{P}_d$ with respect to $S_{i-1}$ and each $S_i$ has no induced cs 4-cycles. Furthermore $f_0(S_{i+1})=f_0(S_i)+f_0(S_{i-1})+4$ for $i\leq 1$.
The first statement follows directly from and . The number of vertices in the prism over $S_{i-1}$ equals $2f_0(S_{i-1})$, and together with $v_+, v_-, w_+,w_-$ sums up to $2f_0(S_{i-1})+4$. Identifying vertices of $D_{i-1}\times\{\pm 1\}$ and $\sigma(D_{i-1})\times\{\pm 1\}$ decreases the number of vertices by $2f_0(D_{i-1})$. Since $f_0(S_{i-1})=2f_0(D_{i-1})+f_0(S_{i-2})$, the claim follows.
[*Proof of : *]{} Via we know there exists a PL $d$-sphere $S_d$ whose number of vertices $n_d$ is given by the sequence $n_0=2$, $n_1=6$, $n_{i+1}=n_{i}+n_{i-1}+4$. Solving the recursion we obtain the desired formula. $\square$
[*Proof of : *]{} The result follows from a direct application of and . $\square$
Our inductive method produces the minimal triangulation of ${{\mathbb R}}P^2$, the boundary of icosahedron, from the minimal triangulation of ${{\mathbb R}}P^1$, the 6-cycle; see Figure \[fig: rp2\]. Furthermore, we find PL triangulations of ${{\mathbb R}}P^3$ with the $f$-vector $(11,52,82,41)$, starting from the boundary of the icosahedron. They are vertex-minimal but not $f$-vectorwise minimal triangulations. For $d=4,5$, our construction is not vertex-minimal.
Open problems
=============
We conclude this article with a few questions. The first one is about the (asymptotic) tight lower bound on the number of vertices required for a vertex-minimal triangulation of ${{\mathbb R}}P^d$.
Does there exist a PL triangulation of $\mathbb{R}P^d$ with $\binom{d+2}{2}+\lfloor\frac{d-1}{2}\rfloor$ vertices for every $d\geq 1$? Does at least a construction with a number of vertices that is polynomial in $d$ exist?
We do not know if the PL spheres constructed in are polytopal, i.e., they can be realized as the boundary complex of a simplicial polytope. It is natural to ask the following question.
What is the minimum number of vertices required for a cs $d$-polytope with no induced cs 4-cycles?
Frequently in the literature, additional combinatorial properties are imposed on a triangulation. We focus on two properties, namely flagness and balancedness. A simplicial complex is *flag* if all minimal subsets of the vertices which do not form a face are edges. A $d$-dimensional simplicial complex is *balanced* if there exists a simplicial projection (often called coloring) to the $d$-simplex which preserves the dimension of faces. Recently in [@BOWWZZ] and [@Ven] local flips and transformations have been implemented to obtain flag and balanced triangulations of manifolds which are vertex-minimal w.r.t. these properties. In particular the authors obtained a flag and balanced vertex-minimal triangulation of $\mathbb{R}P^2$ on $11$ and $9$ vertices respectively, and a balanced vertex-minimal triangulation of $\mathbb{R}P^3$ on $16$ vertices. For higher $d$, a flag and balanced triangulation of $\mathbb{R}P^d$ can be obtained by considering the barycentric subdivision of the boundary complex of the $(d+1)$-dimensional *cross-polytope*, and identifying antipodal vertices. These simplicial complexes have $\frac{3^{d+1}-1}{2}$ vertices.
Construct flag or balanced PL triangulations of $\mathbb{R}P^d$ for every $d$ with less then $\frac{3^{d+1}-1}{2}$ vertices. Does a construction on a number of vertices that is polynomial in $d$ exist?
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Basudeb Datta and Isabella Novik for helpful comments.
|
---
abstract: 'We present an *ab initio* study of electronic correlation effects in a molecular cluster derived from the hexanuclear ferric wheel \[LiFe$_6$(OCH$_3$)$_{12}$-(dbm)$_6$\]PF$_6$. The electronic and magnetic properties of this cluster have been studied with all-electron Hartree-Fock, full-potential density functional calculations and multi-reference second-order perturbation theory. For different levels of correlation, a detailed study of the impact of the electronic correlation on the exchange parameter was feasible. As the main result, we found that the influence of the bridge oxygen atoms on the exchange parameter is less intense than the influence of the apical ligand groups, which is due to the geometry of the cluster. With respect to the cluster model approach, the experimental value of the exchange parameter was affirmed.'
author:
- 'H. Nieber'
- 'K. Doll'
- 'G. Zwicknagl'
title: 'Ab initio correlation approach to a ferric wheel-like molecular cluster'
---
[^1]
Introduction
============
In contemporary condensed-matter physics, the role of molecular magnetism is steadily growing and receives attention from both experimental and theoretical physicists and chemists [@sessoli1993; @gatteschi1994; @caneschi1999; @pilawa1999]. The most studied and therefore best known molecule in this field is the Mn-12-acetate [@regnault2002], but the class of the ferric wheels (wheel-shaped iron rings) is becoming an important subject of various studies[@schnack2004; @waldmann1999; @waldmann2001].
In the past few years, molecular magnets and some ferric wheels have been treated with *ab initio* quantum chemical methods like the Hartree-Fock approximation and density functional approaches using several functionals like the local density approximation or hybrid functionals. The usually observed results are that Hartree-Fock theory often strongly underestimates physical properties like the exchange parameter [@nieber2005; @towler1994; @ricart1995; @catti1995], while the DFT methods overestimate those values [@nieber2005; @postnikov2003; @postnikov2004; @iberio].
This paper will deal with a molecular cluster derived from the hexanuclear ferric wheel \[LiFe$_6$(OCH$_3$)$_{12}$-(dbm)$_6$\]PF$_6$ [@nieber2005; @Abbati1997]. The previous analysis based on the full molecule [@nieber2005] showed that one-determinantal Hartree-Fock theory failed to reproduce the observed exchange parameter $J$=-21 K [@Abbati1997]. Density functional calculations, on the other hand, indicated that there is an enormous dependence of the computed exchange parameter on the functional chosen. This problem was already observed earlier for other systems, e.g. [@MartinIllas1997; @iberio; @Iberio2004]. A possible solution to this discrepancy is to consider the electronic correlation, which is strongly influencing the exchange parameter, by wave function-based methods. This should lead to a more controlled description of the magnetic behavior of the complex, as it was demonstrated for various systems (e.g. [@casanovas1996; @vanoosten1996; @fink1; @fink2; @Graaf2001; @Calzado2000; @MoreiraKNiF; @GraafKuprate; @degraaf2004]).
![Geometry of the molecular cluster. The model complex was derived from the full molecule \[LiFe$_6$(OCH$_3$)$_{12}$-(dbm)$_6$\]PF$_6$ and was slightly modified to achieve $C_s(x)$-symmetry.[]{data-label="geometry"}](dritte2.eps){width="8cm"}
For antiferromagnetic systems like the ferric wheels, the magnetic exchange can be qualitatively described considering the electronic charge transfer between the magnetic centers over a bridging atom or between the magnetic centers and various ligand groups. Within a cluster approach, the electronic charge transfer can be analyzed within a multi-reference second-order perturbation theory scheme (MRPT2) [@celani2000]. MRPT2 is very similar to the complete active space second-order perturbation theory (CASPT2) that has been found to give very accurate results for the exchange parameter (see, e.g. [@degraaf2004; @Graaf2001; @iberio]). The MRPT2 method is based on a reference ground state wave function which can be chosen as a multiconfiguration self-consistent field (MCSCF) wave function. This reference wave function is the initial point for treating the electronic correlation perturbationally. In some cases, the occurrence of intruder states can cause convergence problems [@degraaf2004; @hozoi2002], and the level-shift technique proposed by Roos et al [@roos] has to be applied.
In this paper, we analyze the electronic properties of the simplified complex displayed in Figure \[geometry\]. The model is derived from the molecule \[LiFe$_6$(OCH$_3$)$_{12}$-(dbm)$_6$\]PF$_6$ whose structure has been determined by Abbati et al [@Abbati1997]. We first apply the Hartree-Fock and a hybrid functional (B3LYP) approach, in order to verify the cluster model which is used to represent a fragment of the full molecule. Then the effect of electronic correlations is studied more detailed, with second order perturbation theory at the MRPT2 level. The influence of the level shift on the MRPT2 results is determined in detail.
Method
======
\[methodsection\] The geometry of the molecular cluster is based on the measurements of the primal ferric wheel \[LiFe$_6$(OCH$_3$)$_{12}$-(dbm)$_6$\]PF$_6$ by Abbati et al [@Abbati1997]. From these data, a complex consisting of two iron atoms and some ligands was modeled (see Figure \[geometry\]), as this is the maximum what can be treated by MRPT2. The iron atoms are six-fold coordinated and thus have the proper coordination of the Fe ions like in the full molecule. Two point charges of +1 were added at the position of the neighboring iron atoms in order to restore the charge neutrality of the cluster. The C$_6$H$_5$ rings were replaced with hydrogen atoms (bonding length 0.93[Å]{}) as well as the methyl groups at the bridge oxygen atoms (0.95[Å]{}). To achieve $C_s(x)$-symmetry which is necessary to keep the MRPT2 calculations tractable, the positions of the atoms were slightly modified. The full geometry is given in Table \[coordinates\].
For a proper description of the physical properties of the molecular cluster, calculations with the codes CRYSTAL2003 [@crystal; @dovesi] and MOLPRO2002 [@molpro] were carried out. Within the scope of the CRYSTAL calculations, we employed the unrestricted Hartree-Fock (UHF) method and the hybrid functional B3LYP (a functional with admixtures, amongst others, of functionals by Becke, Lee, Yang and Parr). Note that the CRYSTAL code can treat systems of any periodicity, so that the molecular cluster was treated as a single molecule, i.e. not as a periodic system. These calculations are of broken symmetry type [@Noodleman1981; @Caballol1997; @Illas2000; @Illas2004], as the space symmetry is lowered. The state is not an eigenfunction of $\mathbf{S^2}$, but only of $S_z$.
x y z
------ ----------- ----------- -----------
Fe -1.568236 2.716264 0.000000
O(1) 0.000000 1.952334 -1.010690
O(2) 0.000000 3.375149 1.066879
H 0.000000 2.019278 -1.966749
H 0.000000 4.227152 1.505766
O -1.660933 4.316006 -1.146858
O -2.906901 3.596739 1.146858
C(3) -2.381312 5.371681 -1.023420
C(1) -3.232182 5.601224 0.000000
C(2) -3.460321 4.748871 1.023420
H -4.021835 4.973378 1.723433
H -3.551594 6.475448 0.000000
H -2.293279 5.970805 -1.723433
O -2.815456 1.465882 -0.940540
O -1.442379 1.009753 1.120700
H -3.525746 1.728168 -1.530737
H -1.465828 1.108581 2.107450
+1 -2.774296 -0.177610 0.182920
: \[coordinates\] Geometrical parameters of the molecular cluster, in [Å]{}. The mirror plane is the yz plane.
For the needs of a molecular system, we first chose the same basis set as used in ref. [@nieber2005] for the full molecule, which was found to be reliable. Henceforth this basis set is referred as basis set A. In contrast to the code MOLPRO, the possibility of adding point charges is not implemented in the CRYSTAL code. A point charge of +1 can however be achieved by a H atom with a basis function with a very high exponent (100000 a.u.), which does not allow charge transfer to the H atom and thus acts like a point charge. The idea of the CRYSTAL calculations was to verify that the results did not significantly change when the geometry was modified from the full ferric wheel with six iron atoms to the cluster with two iron atoms. This was first done with basis set A, so that an identical basis set was applied for the full molecule and the fragment.
With the code MOLPRO, calculations at the level of MCSCF and MRPT2 were performed. For these methods, a modified basis set (from now on labeled as basis set B) was used. For iron, a \[*8s5p3d*\] [@wachters] basis set was chosen, where a $f$-exponent of 2.48 was added which was optimized in a preceding calculation. For oxygen, a \[*4s3p*\] [@dunning] basis set was chosen where a $d$-exponent of 0.8 was added. The basis sets for carbon and hydrogen were chosen accordingly to basis set A. Thus, the final basis set B was of the size \[*8s5p3d1f*\] (iron), \[*3s2p*\] (carbon), \[*4s3p1d*\] (oxygen) and \[*2s*\] (hydrogen). The enlargement of the basis set for iron and oxygen in basis set B compared to basis set A is a necessary procedure to properly account for the needs of *post* Hartree-Fock calculations, which in contrast to the calculations at the UHF and B3LYP level include electronic excitations and thus require a larger virtual orbital space. To investigate the impact of the enlargement of the basis set on the results, calculations were carried out with MOLPRO using either basis set A or basis set B at the MCSCF level. Despite the slightly different basis sets, no significant changes in the results were observable, and basis set B can be considered as an extension of basis set A for wave function based correlation calculations. Subseqently all MRPT2 calculations were performed with the enlarged basis set B.
![Spin densities of the molecular cluster for the antiferromagnetic (AF) state at the UHF level (upper panel) and the B3LYP level (lower panel). Both graphs show the spin density in the plane given by the planar arrangement of the six iron atoms of the primal ferric wheel. For all figures, the contour lines range from -0.0004 to 0.0005 in steps of 0.000035 electrons$/($a.u.$)^3$. Full lines indicate positive spin density and dashed lines indicate negative spin density.[]{data-label="spindichte"}](spindensity.eps){width="7cm"}
![Magnetic exchange parameter of the molecular cluster for different values of the level shift. The upper graph (squares) represents the uncorrected values, the lower graph (triangles) the corrected values (correction performed according to Roos et al [@roos]).[]{data-label="levelshift"}](levelshiftplot.ps){width="8cm"}
The properties of the ferromagnetic (FM) state (all spins parallel, total spin 10 $\mu_B$) and of the antiferromagnetic (AF) state (spins alternating up and down, total spin 0) were computed with each method. To obtain the net charge, a Mulliken population analysis was performed at all levels of theory.
Results {#resultssection}
=======
Table \[deltaenergy\] summarizes the results for the total energies of the ferromagnetic and the antiferromagnetic ground states, their differences and the magnetic exchange parameter $J$ for the molecular cluster.
Depending on the ansatz for the ground state wave functions the magnetic coupling in the molecular cluster is derived from fits to an Ising or Heisenberg model.
The analysis in terms of a Heisenberg model
$$H_H = -J\cdot \textbf{S}_1 \cdot \textbf{S}_2$$
with
$$\textbf{S}_1\cdot \textbf{S}_2=\frac{(\textbf{S}_1+\textbf{S}_2)^2-\textbf{S}_1^2-\textbf{S}_2^2}{2}$$
is appropriate when the approximate ground state is an eigenfunction of $(\textbf{S}_1+\textbf{S}_2)=\textbf{S}^2$. This is the case for MCSCF and MRPT2. The exchange parameter is obtained from the difference between the ferromagnetic $(S=5)$ and the antiferromagnetic alignments which is given by
$$\begin{aligned}
E_{H,tot}^{FM}-E_{H,tot}^{AF} & = &-J({\textbf{S}_1\cdot \textbf{S}_2}^{FM}-{\textbf{S}_1\cdot \textbf{S}_2}^{AF}) \nonumber \\
& = & -15\cdot J \nonumber\end{aligned}$$
for quantum spins. The trial wave functions used for UHF and B3LYP calculations usually lack invariance under spin rotation. Since they are constructed as eigenfunctions of the spin projection $S_z$ the energy gain from magnetic correlations is analyzed in terms of the Ising model:
$$H_I= -J\cdot S_{1z} \cdot S_{2z}$$
The corresponding energy difference between ferromagnetic $(S_{1z}=S_{2z})$ and antiferromagnetic $(S_{1z}=-S_{2z})$ alignment is given by
$$E_{I,tot}^{FM}-E_{I,tot}^{AF}=-2\cdot J\cdot S_{1z}^2=-12.5\cdot J$$
which agrees with the Heisenberg value in the classical limit $S_i\rightarrow\infty$.
The experimental value for the exchange parameter of the primal ferric wheel was found to be $J$=-21 K [@Abbati1997]. Thus for all results in this dimension the difference in the exchange parameter between the Ising and the Heisenberg model is about a few Kelvin; with respect to this actuality we consider the Ising model approach as a valid description of the magnetic coupling in the cluster.
We first compare the results of one-determinantal methods for the model cluster and the full molecule. In a second step, we focus on the influence of correlations. In our previous studies of the full molecule [@nieber2005], we determined an exchange parameter of $J$=+7 K at the UHF level and $J$=-31 K at the B3LYP level.
The UHF result for the full molecule is perfectly reproduced by the cluster yielding $J$=+7 K. A good agreement between cluster and periodic system at the Hartree-Fock level was already observed in earlier studies, e.g. NiO[@iberio], KNiF$_3$ and K$_2$NiF$_4$ [@MoreiraKNiF] or Ca$_2$CuO$_3$ and Sr$_2$CuO$_3$ [@GraafKuprate]. At the B3LYP level, the computed exchange parameter is $J$=-20 K. The agreement with the value for the full molecule is thus not as perfect as at the UHF level, but still reasonable (a similar deviation was found, for example, when comparing B3LYP exchange couplings for K$_2$CuF$_4$ from cluster [@iberio1999] and periodic systems [@Iberio2004]; it should be mentioned that there seem to be fewer comparisons between cluster and periodic calculations at the B3LYP level). In addition, the Mulliken population analysis (Tables \[charge\] and \[fepop\]) demonstrates that the charge is practically identical at the various levels: B3LYP populations for the cluster and the full molecule [@nieber2005] are virtually identical, and similarly UHF populations agree very well. In addition, the MCSCF charge agrees with the UHF charge. As a whole, we feel that the cluster model can be considered to be physically valid.
Concerning the individual charge, we note that as a confirmation of the former findings [@nieber2005], the net charge of iron is between 1.55 (B3LYP) and 2.26 (MCSCF) and thus far away from the formal charge of +3 of the primal ferric wheel [@Abbati1997]. The charge is thus more delocalized at the B3LYP level, which can also be seen in the spin-density plot in Figure \[spindichte\]. At the B3LYP level the local magnetic moment is distributed over the cluster to a certain extent, while the level of delocalization is apparently smaller at the UHF level.
To serve as a starting point for the MRPT2 calculations, a MCSCF calculation was performed. The active space is made of the iron $d$-orbitals. The obtained value for the exchange parameter is $J$=-0.5 K. Compared to UHF, a MCSCF wave function is a better approximation to the ground state than a single determinant, which is the reason for the slight change of the exchange parameter from UHF to MCSCF towards the experimental value. A direct comparison between the calculations with the code CRYSTAL and MOLPRO is only approximatively possible, and only in case of high spin. First, the MOLPRO energy must be corrected for the fact that MOLPRO does not take the interaction of point charges into account when computing the total energy: when this is done, then the MCSCF energy with MOLPRO is -3507.90171+1/(2\*2.774296/.5291772) $E_h$=-3507.80634 $E_h$. The remaining difference to the energy computed with CRYSTAL (-3507.80720 $E_h$) is because the MOLPRO MCSCF wave function has identical orbitals for up and down spin, whereas the CRYSTAL UHF wave function has not; and because the codes use different screening parameters for the selection of the integrals.
----------- ----------------------------- -------- ----------------- ----------------- ----------------------- ---------
basis set molecule method FM total energy AF total energy difference of $J$ (K)
$(E_h)$ $(E_h)$ total energy $(mE_h)$
A full molecule [@nieber2005] UHF -13295.73287 -13295.73125 -1.62 +7
A full molecule [@nieber2005] B3LYP -13334.50676 -13334.51409 7.33 -31
A cluster UHF -3507.80720 -3507.80693 -0.27 +7
A cluster B3LYP -3514.95130 -3514.95209 0.79 -20
A cluster MCSCF -3507.90171 -3507.90175 +0.04 -1
B cluster MCSCF -3508.31083 -3508.31085 +0.02 -0.5
B cluster MRPT2 -3511.04152 -3511.04209 0.57 -14.4
----------- ----------------------------- -------- ----------------- ----------------- ----------------------- ---------
basis set molecule method O (apical) O (bridge) C(1) C(2)/C(3)
----------- ----------------------------- -------- ------------ ------------ ------- -----------
A full molecule [@nieber2005] UHF -1.10 -1.18 -0.22 0.82
A full molecule [@nieber2005] B3LYP -0.86 -0.88 -0.11 0.62
A cluster UHF -1.03 -1.13 -0.17 0.77
A cluster B3LYP -0.80 -0.89 +0.01 0.60
A cluster MCSCF -1.03 -1.14 -0.17 0.77
B cluster MCSCF -1.15 -1.22 -0.13 0.85
Within the MCSCF scheme, the wave function is built from all Slater determinants representing charge transfer configurations between the orbitals in the active space, and the orbitals are optimized. The MRPT2 calculation is based on the MCSCF orbitals. The reference configurations are made of all determinants in the active space (i. e. the iron $d$-orbitals), and subsequently second-order perturbation theory is applied. The level of correlation can additionally be varied by keeping different sets of core orbitals frozen, as described later on in this section.
While performing the MRPT2 calculations, intruder state problems appeared which are due to a near degeneracy of the ground state. One possibility to remedy those intruder states is to increase the active space, which is definitely not possible for the considered molecular cluster, or, following the proposal by Roos et al [@roos], to implement a level shift to the MRPT2 calculations. By means of this technique, a level shift parameter is added to the zeroth order Hamiltonian to avoid those intruder states. Thus, the resulting exchange parameters are influenced by the level shift (c.f. upper graph (squares) in Fig. \[levelshift\], corresponding to equation 6 in [@roos]). To approximatively correct for this effect, a correction to the second order energies can be applied afterwards, as was suggested in [@roos], equation 7. The corresponding data are displayed in the lower graph (triangles) in Fig. \[levelshift\]. The quantitative dependence of the exchange parameter on the level shift is shown in Table \[shiftlevel\]. For level shift values smaller than 0.17 $E_h$ ($E_h\equiv$ hartree), the MRPT2 results became unstable. The level shift interval (for the corrected values of the exchange parameter) from 0.20$E_h$ to 0.30$E_h$ leads to an exchange parameter of $J$=15$\pm$2 K, which is a reasonably stable result. For other systems, similar results were obtained, see e.g. de Graaf et al [@degraaf2004] and Hozoi et al [@hozoi2002].
basis set molecule method net charge *s* *p* *d*
----------- ----------------------------- -------- ------------ ------ ------- ------
A full molecule [@nieber2005] UHF 2.16 6.28 12.28 5.29
A full molecule [@nieber2005] B3LYP 1.56 6.38 12.37 5.69
A cluster UHF 2.13 6.29 12.30 5.28
A cluster B3LYP 1.55 6.39 12.40 5.66
A cluster MCSCF 2.14 6.29 12.30 5.27
B cluster MCSCF 2.26 6.37 12.07 5.31
level shift FM total energy $(E_h)$ AF total energy $(E_h)$ difference of total energy $(mE_h)$ $J$ (K)
------------- ------------------------- ------------------------- ------------------------------------- ---------
0.20 -3511.05498 -3511.05563 0.65 -16.4
0.21 -3511.05375 -3511.05439 0.64 -16.1
0.22 -3511.05251 -3511.05314 0.63 -15.9
0.23 -3511.05124 -3511.05185 0.62 -15.6
0.24 -3511.04993 -3511.05054 0.61 -15.4
0.25 -3511.04860 -3511.04921 0.60 -15.2
0.26 -3511.04724 -3511.04784 0.59 -15.0
0.27 -3511.04586 -3511.04644 0.59 -14.8
0.28 -3511.04444 -3511.04502 0.58 -14.6
0.30 -3511.04152 -3511.04209 0.57 -14.4
------- --------------------------------- ----------------- ----------------- ----------------------- ---------
level of correlation FM total energy AF total energy difference of $J$ (K)
$(E_h)$ $(E_h)$ total energy $(mE_h)$
(I) iron *d*-orbitals -3508.43623 -3508.43649 0.26 -6.4
(II) (I)+*2sp*-orbitals of O(1) -3508.63134 -3508.63154 0.19 -4.9
(III) (I)+*2sp*-orbitals of O(2) -3508.64415 -3508.64444 0.29 -7.4
(IV) (I)+*2sp*-orbitals of O(1),O(2) -3508.84371 -3508.84394 0.23 -5.8
(V) (I)+all oxygen orbitals -3510.58726 -3510.58781 0.54 -13.7
(VI) all orbitals except core -3511.04152 -3511.04209 0.57 -14.4
------- --------------------------------- ----------------- ----------------- ----------------------- ---------
The results for the MRPT2 calculations are given in table \[corrlevel\]. Considering the magnetic iron $d$-orbitals as the only orbitals to be correlated led to an exchange parameter of\
$J$=-6.4 K. The included configurations are thus the charge transfer configurations between the occupied iron $d$-orbitals and the excitations to the virtual orbitals (for the high-spin state, only excitations to the virtual orbitals are possible). Therefore, accounting for those charge transfer configurations explains the change of the exchange parameter from the MCSCF level to the MRPT2 level (-0.5K (MCSCF)$\rightarrow$ -6.4 K (MRPT2)). Adding the $2sp$-orbitals of one of the bridging oxygens to the orbitals to be correlated gave rise to an exchange parameter of $J$=-4.9 K (O(1)) and $J$=-7.4 K (O(2)). Taking both these $2sp$-orbitals into account caused only an exchange parameter of $J$=-5.8 K, i. e. the effect of correlating both orbital groups is approximatively additive.
Correlating all oxygen atoms results in a value of -13.7 K. A further increase of the orbitals to be correlated (up to a maximum where only the iron *1s-, 2sp-, 3sp*-, the oxygen *1s*- and the carbon *1s*-orbitals are kept frozen) led to a exchange parameter of $J$=-14.4 K, which is in the range of the experimental value. With respect to the geometry of the cluster, the influence of those apical ligand groups to the exchange parameter is the dominant one, whereas the oxygen bridge atoms have only little impact on the exchange parameter. As a conclusion, for this particular ferric wheel under consideration the metal $\leftrightarrow$ ligand charge transfer configurations dominate the metal $\leftrightarrow$ metal charge transfer configurations over the bridging oxygen atoms, which is basically the result of the ordering of the magnetic orbitals according to the Goodenough-Kanamori rules [@kahn]: essentially, the Fe-O-Fe angle is nearly right-angled and thus the coupling is small.
In the experiments, when comparing various ferric wheels, an approximatively linear relationship between the Fe-O-Fe angle and the value of the exchange coupling was observed [@waldmann2001] and confirmed [@Pilawa2003]. This indicates that this angle is crucial for the strength of the coupling (as long as the Fe-Fe distance is approximatively constant, otherwise this distance may also have an impact). This is not in contradiction to the findings here: essentially, the strength and the nature of the coupling (ferro- or antiferromagnetic) is strongly influenced by this angle, but still, to compute the interaction properly, the ligands must be included in the correlation treatment. This was demonstrated, for example, in [@Graaf2001] (figure 2): when the correlation treatment is not sufficient, the exchange couplings come out too small; but still, the dependence on the angle is correct. Even more striking were earlier calculations where the ligands were crucial to obtain reasonable values for the exchange couplings, e.g. for KNiF$_3$ and K$_2$NiF$_4$ [@MoreiraKNiF] or NiO [@deGraaf1997JCP].
Conclusion {#summarysection}
==========
Wave function-based correlation methods were applied to a molecular cluster derived from the hexanuclear ferric wheel \[LiFe$_6$(OCH$_3$)$_{12}$-(dbm)$_6$\]PF$_6$ [@Abbati1997]. The validity of the molecular cluster containing two iron atoms was tested by means of a one-determinantal approach with respect to the formerly calculated results for the full molecule [@nieber2005] for the exchange parameter of the primal ferric wheel. In addition, at the UHF and B3LYP level, the spin densities and the electronic population, at the MCSCF level the electronic population were calculated. The population analysis supported the validity of the cluster model approach.
The best result for the exchange coupling parameter $J$ was obtained at the MRPT2 level ($J$=15$\pm$2 K). The influence of intruder state problems on the exchange parameter was explicitly investigated, and applying the level shift technique [@roos] was found to lead to stable results. MRPT2 thus gives a more controlled approach to the importance of electronic correlations for exchange couplings, whereas the density functional results depend strongly on the functional chosen. Also, the impact of certain atom groups on the exchange parameter was determined at the MRPT2 level. Correlation of the electrons of the bridging oxygen atoms was of minor importance for the coupling strength. A strong enhancement of the computed exchange coupling was however observed by additionally correlating the electrons of the apical oxygen atoms.
Acknowledgments
===============
Most of the calculations were performed at the compute-server *cfgauss* (Compaq ES 45) of the data processing center of the TU Braunschweig. The geometry plot of the molecular cluster was performed with VMD [@vmd].
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[^1]: present address: Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, Universitätsstr. 150, 44801 Bochum
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---
abstract: 'We show that Brownian motion is spatially not symmetric for mesoscopic particles embedded in a fluid if the particle is not in thermal equilibrium and its shape is not spherical. In view of applications on molecular motors in biological cells, we sustain non-equilibrium by stopping a non-spherical particle at periodic sites along a filament. Molecular dynamics simulations in a Lennard-Jones fluid demonstrate that directed motion is possible without a ratchet potential or temperature gradients if the asymmetric non-equilibrium relaxation process is hindered by external stopping. Analytic calculations in the ideal gas limit show that motion even against a fluid drift is possible and that the direction of motion can be controlled by the shape of the particle, which is completely characterized by tensorial Minkowski functionals.'
author:
- Susan Sporer
- Christian Goll
- Klaus Mecke
title: 'Motion by Stopping: Rectifying Brownian Motion of Non-spherical Particles'
---
Mesoscopic particles dissolved in a fluid are expected to perform symmetric thermal fluctuations around a mean position [@einstein05; @smoluchowski06]. The shape of the particles is irrelevant for this so-called Brownian motion - as long as the particle is in thermal equilibrium with the fluid. Consequently, a net transport in a preferred direction is not possible without applying an external force which breaks the spatial symmetry. In the past, several models for Brownian motors have been proposed based on different methods of rectifying thermal noise; for instance, by an asymmetric external potential which is switched on and off periodically (for a review see Ref. [@reimann:2002]). Another concept is based on two heatbaths at different temperatures such as the Feynman ratchet [@feynman:63] and its simplification by Van den Broeck [@broeck-meurs-kawai:2004]. It was shown that directed Brownian motion can be achieved as long as the two reservoirs have different temperatures and spatial symmetry is broken in some way. This letter demonstrates that a Brownian motor can be built even in a single heatbath without violating the second law of thermodynamics. Hence, it is a possible theoretical model for molecular motors in biological cells. We show that a steady state motion of non-spherical particles is obtained if their relaxation towards equilibrium is prohibited by periodical stopping.
To illustrate the concept we restrict the system to two dimensions and follow Ref. [@broeck-meurs-kawai:2004]. Since the spatial symmetry breaking is a prerequisite for net transport, we consider an asymmetric motor $K$ of mass $M$ in a single heatbath at temperature T. The motion of the motor is restricted to a one-dimensional track, here the $x$-axis, so that the motor velocity can be written as ${\bf \widetilde{V}}=(V\sqrt{k_BT/M},0)$ with the normalized velocity $V$ in $x$-direction. Furthermore, elastic interactions are assumed between the motor $K$ and the surrounding fluid particles. In contrast to Refs. [@cleuren:2007; @costantini:2007] there is no dissipation involved in this collision. However, due to the confinement only the axial momentum component is conserved. In thermal equilibrium, directed motion in a single heatbath is prohibited by the second law of thermodynamics. The motor $K$ behaves like a Brownian particle with a Maxwellian velocity distribution with mean $\langle V \rangle =0$ and variance $\langle V^2 \rangle = 1$; independent of its particular shape. Hence, in order to get directed motion, we have to sustain a non-equilibrium state.
Our model is motivated by a molecular motor in a cell, e.g. kinesin, that binds to a filament after every step [@howard:1996]. Similarly, our particle is stopped at periodically spaced binding sites along the track, i.e. its kinetic energy is set to zero whenever it reaches a binding site. Thermodynamically, of course, the stopping requires a greater amount of work than the decrease in entropy; here ${1\over 2} k_BT$ from the reduction of the Maxwellian distribution for $V$ to the velocity distribution $P(V)=\delta (V)$ for the stopped particle. After binding the motor is released again and starts to relax towards thermal equilibrium. By means of collisions between fluid particles and the motor, kinetic energy is transfered from the fluid to the motor until the motor has the same effective temperature as the surrounding bath.
![An asymmetrically shaped motor (here a triangle) is built from fluid particles and placed in a Lennard-Jones fluid. Its motion is restricted to a one-dimensional track with periodically spaced binding sites along the x-axis. If the motor’s center of mass crosses a binding site, the velocity of the motor is set to zero. To test the importance of the motor’s shape for the momentum transfer during collisions with fluid particles we orient the triangle in two different ways. []{data-label="fig:sim_fig"}](figure1a.eps "fig:"){width="0.49\linewidth"} ![An asymmetrically shaped motor (here a triangle) is built from fluid particles and placed in a Lennard-Jones fluid. Its motion is restricted to a one-dimensional track with periodically spaced binding sites along the x-axis. If the motor’s center of mass crosses a binding site, the velocity of the motor is set to zero. To test the importance of the motor’s shape for the momentum transfer during collisions with fluid particles we orient the triangle in two different ways. []{data-label="fig:sim_fig"}](figure1b.eps "fig:"){width="0.49\linewidth"}
In the following, we show that this non-equilibrium relaxation process is asymmetric due to the asymmetry of the particle. This asymmetric relaxation process can be used to rectify thermal fluctuations. To test this hypothesis, we performed 2D molecular dynamics simulations of an asymmetric particle in a two-dimensional fluid. The interaction between the fluid particles is described by the Lennard-Jones (LJ) potential $\mathcal{U}(r)= 4 \mathcal{E} \left( \left( \frac{\sigma}{r} \right)^{12}
- \left( \frac{\sigma}{r}\right)^6 \right)$ where $r$ is the distance between two particles with diameter $\sigma$ and $\mathcal{E}$ is the depth of the potential well. In the following, all lengths are given in units of $\sigma$, energies in units of $\mathcal{E}$ and times are expressed in units of $t_0=\sigma \sqrt{m/\mathcal{E}}$ where $m$ denotes the mass of the fluid particles. The simulation box with edge length $100 \sigma$ contains $n=800$ fluid particles yielding a particle density $\varrho \approx 0.08 \sigma^{-2}$. The temperature is set to $T=3 \mathcal{E}/k_B$ and the overall simulation time was $2000 t_0$. With these parameters the LJ fluid is in the gas phase without any long range correlations. Periodic boundary conditions are used and also a cut-off radius of $2.5 \sigma$ in $\mathcal{U}(r)$ to limit the range of the potential.
As illustrated in Fig. \[fig:sim\_fig\] the motor is built of $N$ stiffly-linked fluid particles which interact with the surrounding fluid also by a Lennard-Jones potential. Consequently, the motor mass $M=Nm$ scales with the mass $m$ of the fluid particles. The motion of the motor is confined to a one-dimensional track with periodically spaced stopping sites. Every time the motor’s center of mass reaches one of those sites, the velocity of the motor is set to zero and the corresponding kinetic energy is re-transferred to the fluid.
![Histograms of end positions at time $t=2000t_0$ of a triangular motor where the distance between stopping sites is $\delta s$ (see Fig. \[fig:sim\_fig\]). The asymmetrically oriented motor moves preferable to positive X-values whereas the symmetric triangle diffuses around $X=0$. A discrete spacing is visible since the motors accumulate around the stopping sites. 400 realisations are used with integration step $\delta t=0.001 t_0$ and $2000000$ steps. With decreasing distances $\delta s$ between stopping sites the distribution becomes narrower, as expected for a hindered diffusion process. []{data-label="fig:endpositions"}](figure2.eps){width="0.8\linewidth"}
Consider a motor with the shape of an isosceles triangle of height $10\sigma$ and basis length $20\sigma$ which corresponds to an apex angle of $\theta=0.25268$, $N=53$ and mass $M=53 m$. The motor is placed in the beginning ($t=0$) at the position $X=0$. Figs. \[fig:endpositions\] and \[fig:sim\_results\] show the simulation results for the motion of the motor. Note, that for the chosen parameters the typical time between two successive stops of the motor is smaller than the relaxation time into equilibrium. In case of an isosceles triangle, the motor can be oriented symmetrically or asymmetrically with respect to its direction of motion (Fig. \[fig:sim\_fig\]), to test the importance of the shape. Fig. \[fig:endpositions\] presents the distribution over multiple runs of the end positions, i.e. the positions of the motors after time $2000t_0$, for different stopping distances $\delta s$. If the triangle is placed with the apex onto the track, i.e. asymmetric with respect to the direction of motion, the maximum of the distribution of the end positions is shifted to positive values $x>0$. Hence, on average, the triangle travels a certain distance towards its basis. This is in accordance with the results shown in Fig. \[fig:sim\_results\] where a net speed of about $\langle v \rangle = 0.011 \frac{\sigma}{t_0}$ in the direction of the triangle basis can be observed for asymmetric orientation. In contrast, aligning the triangle basis parallel to the $x$-axis conserves spatial symmetry, i.e. mirror-symmetry with respect to the y-axis, and therefore causes the mean position to fluctuate around zero without any net motion (see Fig. \[fig:sim\_results\]). The end positions of the motors are in this case symmetrically distributed around the origin (see Fig. \[fig:endpositions\]). This shows two things: first, the proposed model demonstrates the feasibility of rectifying Brownian motion in a single heatbath by stopping the particle at periodic sites. Second, it is sufficient to break the spatial symmetry by the shape of the particle, so that an asymmetric external potential is not necessary.
![The average position (400 runs) of an asymmetrically oriented, isosceles triangle of height $10\sigma$ and basis length $20\sigma$ moves with a constant velocity in the direction opposite to its sharp edge (i.e., in the direction of its basis). Thus, directed motion is possible when the triangle is oriented asymmetrically with respect to the direction of motion. Otherwise the motor performs symmetric fluctuations around its starting position. Inset: trajectories of individual motors. Note, that the velocity decreases for decreasing distances $\delta s$ between stopping sites if the typical stopping time is smaller than the relaxation time of the motor. []{data-label="fig:sim_results"}](figure3.eps){width="0.9\linewidth"}
To get a better insight into the microscopic mechanism of the driving force, we calculate analytically the stochastic dynamics of this non-equilibrium relaxation process for a simplified model. The interaction between the fluid and the motor is no longer a Lennard-Jones potential but an ideal hard sphere interaction with infinitesimal interaction time. The total energy is conserved as well as the momentum in $x$-direction and tangential to the particle’s surface. Details of this model can be found in Ref. [@broeck-meurs-kawai:2004].
Furthermore, we assume an infinite heatbath, so that the temperature is constant and the velocity distribution of the fluid is a Maxwell distribution $\Phi_{\boldsymbol u}({\boldsymbol v})=\frac{m}{2 \pi k_B T} \exp
\left( - \frac{m}{2 k_B T} ({\boldsymbol v}-{\boldsymbol u})^2 \right)\, $. To study motion against a drift we introduce an average drift velocity $\boldsymbol u$ of the fluid. In the beginning, we assume ${\boldsymbol u}=0$. In order to exclude multiple collisions, we use the ideal gas limit of the fluid and confine the motor shape to convex objects $K$ so that the stochastic motion is a Markov process. This allows us to describe the time dependence of the velocity distribution $P(V,t)$ of the motor by a Master equation with a transition rate [@broeck-meurs-kawai:2004] $$\begin{aligned}
\label{eqn:transitionprob}
W(V'|V)&= \int_{\partial K} d\mathcal{O} \int_{-{\infty}}^{+\infty} d{\boldsymbol v}
\; \rho \; \Phi_0({\boldsymbol v}) \, ({\bf \widetilde{V}}-{\boldsymbol v}) \cdot {\bf \hat{e}}_{\perp} \nonumber \\
& \times \Theta \big[ ({\bf \widetilde{V}}-{\boldsymbol v}) \cdot {\bf \hat{e}}_{\perp} \big]
\; \delta \big[ \widetilde{V}'-\widetilde{V} + \frac{\Delta\boldsymbol p}{M} \cdot {\bf \hat{e}}_{x} \big] \, \end{aligned}$$ ($\widetilde{V}=\sqrt{k_BT/M} \, V$) which equals the number of collision processes from $V$ to $V'$ per unit time using the dimensionless time $t = 8 \epsilon ^2 \sqrt{\frac{k_B T \rho}{2 \pi m}} \tilde{t}$. Here, ${\bf \hat{e}}_{\parallel}=(\cos \phi, \sin \phi)$ is the tangential and ${\bf \hat{e}}_{\perp}=\left( \sin \phi, - \cos \phi \right)$ the normal vector in the collision point ${\bf r} \in \partial K$ on the boundary $\partial K$ of the motor and $\phi$ is the polar angle. The momentum transfer of a single collision is then given by $\Delta{\boldsymbol p}=\frac{2 \; m\, M{\bf \hat{e}}_{\perp}}{M + m \left( {\bf \hat{e}}_{\perp} \cdot
{\bf \hat{e}}_{x} \right)^2} \,\left( {\bf \widetilde{V}}-{\boldsymbol v} \right) \cdot {\bf \hat{e}}_{\perp}$ (see Ref. [@broeck-meurs-kawai:2004]). Because the Master equation cannot be solved exactly, we apply a Kramers-Moyal expansion yielding $$\begin{aligned}
\label{eqn:kramers-moyal}
\frac{\partial P (V,t)}{\partial t} \;
& = \; \sum_{n=1}^{\infty} \frac{1}{n!} \left( - \frac{\partial}{\partial V} \right)^n
\big[ A_n(V,t) P(V,t) \big] \end{aligned}$$ with the coefficients $A_n(V,t) = \int dV' (V'-V)^n W(V|V')$. Different to Ref. [@broeck-meurs-kawai:2004], we emphasise two important points: (i) the dependence on the shape of the motor can be fully described by tensorial Minkowski functionals $M_\nu^{(r,s)}(K)$ of the motor $K$; (ii) the non-stationary solution of Eq. (\[eqn:kramers-moyal\]) is the essential ingredient to rectify Brownian motion if relaxation is prohibited by stopping sites (for details see Ref. [@sporer:2006]).
Tensorial Minkowski functionals are defined as surface integrals in $d$ dimensions [@lecturenotes02] $$\begin{aligned}
\label{eqn:def_minkowski_tensor}
M_1&^{(r,s)}(K) = \int_{\partial K} \!\!\! d\mathcal{O} \; \overbrace{{\bf r} \otimes \ldots \otimes {\bf r}}^{r-times} \otimes \overbrace{{\bf \hat{e}}_{\perp} \otimes \ldots \otimes {\bf \hat{e}}_{\perp} }^{s-times} \end{aligned}$$ over tensor products of position ${\bf r}$ and normal vector ${\bf \hat{e}}_{\perp}$ on the surface $\partial K$ of motor $K$. Using $ \left( {\bf \hat{e}}_{\perp} \cdot { \bf \hat{e}}_{x} \right)^n= \sin^n \phi$, the $x \ldots x$-components of the first tensorial Minkowski functional in two dimensions can be written as $\left( M_1^{(0,n)}(K) \right) _ {\overbrace{x \ldots x}}^{n-times}
= \int_{\partial K} d\mathcal{O} \sin^n \phi$. For convex bodies $K$ one finds $M_{\nu}^{(0,1)}(K) = 0$ so that the first two jump moments read $A_1(V)=\tau^{-1}(V+4V_{\rm max}(V^2-1))$ and $A_2(V)=2\tau^{-1}(1+12V_{\rm max}V)$. Here, we introduced the relaxation rate $\tau^{-1}=\sqrt{\rho} \left( M_1^{(0,2)} (K) \right)_{xx}/2$ and the maximum velocity $$\label{maxvel}
V_\text{max}={\epsilon \over 4} \, \sqrt{\frac{\pi}{8}}
\frac{\left(M_1^{(0,3)}(K) \right)_{xxx}}{\left(M_1^{(0,2)}(K) \right)_{xx}} \;\;.$$ The shape dependence of the stochastic dynamics enters only via these two parameters which can be expressed in terms of Minkowski tensors. These functionals are known in many cases to allow for a complete description of the shape-dependence of physical properties [@lecturenotes02]. The other Minkowski tensors in Eq. (\[eqn:def\_minkowski\_tensor\]) play a role when the collision rules in Eq. (\[eqn:transitionprob\]) are modified or the drift is non-zero, for instance [@sporer:2006].
From Eq. (\[eqn:kramers-moyal\]), one immediately obtains the time dependence of the average velocity $\partial_{t} \langle V \rangle = \langle A_1(V) \rangle$ and the average squared velocity $\partial_{t} \langle V^2 \rangle
= 2 \langle V A_1(V) \rangle + \langle A_2(V)\rangle$. The equations are coupled to higher moments of $V$ because of the dependence of $A_n(V)$ on $V$. However, assuming the mass $m$ of the fluid particles to be small against the motor mass $M$, an expansion in $\epsilon=\sqrt{\frac{m}{M}}$ leads to a decoupling of the differential equations from higher moments when terms of order $\mathcal{O}(\epsilon ^4)$ or higher are neglected $$\begin{aligned}
\label{eqn:dgl_x}
\frac{\partial \langle V \rangle }{\partial t} &=- {1\over \tau} \left[\langle V \rangle + \bigg(\langle V^2 \rangle -1 \bigg)V_\text{max} \right] + \mathcal{O} \left( \epsilon^4 \right)
\nonumber \\
\frac{\partial \langle V^2 \rangle }{\partial t} &= {2\over \tau}
\left(1-\langle V^2 \rangle \right) + \mathcal{O} \left( \epsilon^4 \right) \; .\end{aligned}$$ Hence, the solutions give readily the mean velocity $\langle V \rangle(t) = 4 V_\text{max} \left( e^{ - t/\tau} -e^{ -2 t/\tau} \right)$, $\langle V^2 \rangle(t) = 1 -e^{ -2 t/\tau}$ and the average of the travelled distance $\langle X\rangle(t)=\int_0^{t} dt' \, V(t') = 2 V_\text{max} \tau \left( 1 - e^{-t/\tau}\right)^2$. The time $\tau$ characterizes the increase of the kinetic energy ${M\over 2} \langle \widetilde{V}^2 \rangle$ towards its equilibrium value ${1\over 2} k_BT$, which is the stationary solution of Eq. (\[eqn:dgl\_x\]). The relaxation process depends on the shape of the motor only via the maximum of the averaged velocity $V_{\rm max}$, so that the shape dependence is fully described by the Minkowski functionals of the particle $K$. For instance, the Minkowski functionals of a triangular motor with the top pointing in negative $x$-direction reads $(M_1^{(0,2)}(K))_{xx}= L\left(1+ \sin \theta \right)$ and $(M_1^{(0,3)}(K))_{xxx} = L\cos ^2 \theta$, so that $V_\text{max}= {\epsilon \over 4} \sqrt{\frac{\pi}{8}} \left(1-\sin \theta \right)$ and $\tau^{-1}= L \sqrt{\rho}(1+\sin \theta)/2$ depend on the apex angle $\theta$ of the triangle (see Fig. \[fig:sim\_fig\]).
The shape dependence of this non-equilibrium Brownian motion can be perfectly demonstrated with the ’capped-triangular’ motor shown in the inset of Fig. \[fig:relaxation\_drift\](b), whose Minkowski functionals are $(M_1^{(0,2)} (K))_{xx}= L\left(\frac{\pi}{4} + \sin \theta \right)$ and $(M_1^{(0,3)} (K))_{xxx}= L\left( \frac{2}{3} - \; \sin ^2 \theta \right)$. As shown in Fig. \[fig:relaxation\_drift\](b), the relaxation time $\tau$ decreases with increasing opening angle $\theta$ whereas the maximum average velocity $V_\text{max}$ increases with $\theta$ from negative values to positive ones. Interestingly, one can change the direction of motion just by adjusting the shape of the motor. For small $\theta$, i.e. long triangle legs relative to the arc length of the semi-circle, the motor moves towards the semi-circle, whereas it travels towards the triangle if $\theta$ is close to $\frac{\pi}{2}$. Note, that there is exactly one value for $\theta$ for which the motor does not move on average ($V_\text{max}=0$ vanishes) though the motor definitely breaks spatial symmetry.
The relaxation process of the triangular motor is shown in Fig. \[fig:relaxation\_drift\]. The average velocity first increases until a maximum value $V_\text{max}$ is reached and than decays exponentially towards the equilibrium with vanishing mean velocity. Note, that in the beginning the motor travels an average distance $X_{\rm max}=2V_\text{max}\tau $ before it fluctuates around the mean position $X_{\rm max}$ in the long time limit. Shortly after releasing, the motion of the motor has a preferred direction. Thus repeated stopping yields directed motion if it occurs before the system has relaxed completely.
A simple analogy of the rectification mechanism is a piston in a cylinder separating two fluids in thermal equilibrium at the same temperature. If the piston is arrested or has an infinite mass, one finds in thermodynamic textbooks that the averaged momentum transfer on one side per unit area of its cylindrical cross section equals the pressure of the fluid on this side. Since the fluids on both sides are equilibrated at the same temperature there is no net force on the piston. The same is true if the piston can move freely and is in thermal equilibrium with the fluids on both sides. However, a piston with finite mass $M$ and not in equilibrium experiences kinematic effects due to momentum conservation when momentum is transfered by collisions, so that the net force depends on the shape of the piston.
The motor is also able to move against small fluid drifts. Repeating the analytic calculations with $\boldsymbol u \neq 0$ and confining the drift to the $x$-direction we find the mean velocity shown in Fig. \[fig:relaxation\_drift\](a) for a triangular motor with its apex pointing in negative $x$-direction. In the long time limit the motor equilibrates always to the drift velocity independent of sign and strength of the drift. For small drifts in the preferred direction - here the positive $x$-direction - the motor accelerates to a maximum average velocity $V_\text{max}$ above the equilibrium value before it relaxes to the drift velocity $\boldsymbol u$. When applying a small drift in opposite direction, it can be seen that the motor starts to move in positive direction but changes its direction of motion after a certain time. When stopping it repeatedly before the direction is changed, an average motion against a fluid drift is achieved.
Fig. \[fig:DataVsTheory\] compares our analytic results with the numerical data from the molecular dynamics simulations. Evidently, a real fluid with Lennard-Jones interactions cannot be in quantitative agreement with calculations based on an ideal gas. Nevertheless, we find the same functional form for the non-equilibrium relaxation process given by the solution of Eq. (\[eqn:dgl\_x\]), which confirms our theoretical understanding of this rectification mechanism. We showed that thermal fluctuations in a single heatbath can be used to get directed motion if (i) the motor is sustained in a non-equilibrium state by stopping it at periodically spaced sites on a one-dimensional track (or alternatively at periodic time intervals); (ii) the spatial symmetry is broken by an asymmetric shape of the motor which makes the non-stationary relaxation process also asymmetric. The shape-dependence of the non-equilibrium relaxation is used to get directed motion even against small fluid drifts. By changing the shape of the motor, it is possible to change the direction of motion and to control the average velocity of the motor.
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|
---
abstract: 'Set-based person re-identification (SReID) is a matching problem that aims to verify whether two sets are of the same identity (ID). Existing SReID models typically generate a feature representation per image and aggregate them to represent the set as a single embedding. However, they can easily be perturbed by noises – perceptually/semantically low quality images – which are inevitable due to imperfect tracking/detection systems, or overfit to trivial images. In this work, we present a novel and simple solution to this problem based on ID-aware quality that measures the perceptual and semantic quality of images guided by their ID information. Specifically, we propose an ID-aware Embedding that consists of two key components: (1) Feature learning attention that aims to learn robust image embeddings by focusing on ‘medium’ hard images. This way it can prevent overfitting to trivial images, and alleviate the influence of outliers. (2) Feature fusion attention is to fuse image embeddings in the set to obtain the set-level embedding. It ignores noisy information and pays more attention to discriminative images to aggregate more discriminative information. Experimental results on four datasets show that our method outperforms state-of-the-art approaches despite the simplicity of our approach.'
author:
- |
Xinshao Wang^1,2^, Elyor Kodirov^2^, Yang Hua^1,2^, Neil M. Robertson^1,2^\
^1^ School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, UK\
^2^ Anyvision Research Team, UK\
{xwang39, y.hua, n.robertson}@qub.ac.uk, {elyor}@anyvision.co
bibliography:
- 'references\_aaai2019.bib'
title: 'ID-aware Quality for Set-based Person Re-identification'
---
Introduction {#sec:introduction}
============
Set-based person re-identification (SReID) [@zheng2016mars; @liu2017qan; @song2017region; @li2018diversity] is a matching problem that targets identifying the same person across multiple non-overlapping cameras. Each person is represented by a set consisting of multiple images. There has been an increasing attention recently because of its critical applications in video surveillance, e.g., airport and shopping mall.
[0.47]{} ![ Problems in set-based person re-identification. Four sets of examples are shown (a-d); each set corresponds to a particular problem. They are grouped into two: perceptual quality and semantic quality problems. (a) belongs to the first one while the remaining ones belong to another. Note that some images in (b), (c) and (d) are less semantically related to its set ID due to incomplete body, heavy occlusion by a different person, and muptiple people in an image. []{data-label="fig:quality_analysis"}](Visualization/LPW_scen3_view4_202.png "fig:"){width="\textwidth"}
\[fig:quality\_blur\]
[0.47]{} ![ Problems in set-based person re-identification. Four sets of examples are shown (a-d); each set corresponds to a particular problem. They are grouped into two: perceptual quality and semantic quality problems. (a) belongs to the first one while the remaining ones belong to another. Note that some images in (b), (c) and (d) are less semantically related to its set ID due to incomplete body, heavy occlusion by a different person, and muptiple people in an image. []{data-label="fig:quality_analysis"}](Visualization/537_T0002.png "fig:"){width="\textwidth"}
\[fig:quality\_missing\_part\]
[0.47]{} ![ Problems in set-based person re-identification. Four sets of examples are shown (a-d); each set corresponds to a particular problem. They are grouped into two: perceptual quality and semantic quality problems. (a) belongs to the first one while the remaining ones belong to another. Note that some images in (b), (c) and (d) are less semantically related to its set ID due to incomplete body, heavy occlusion by a different person, and muptiple people in an image. []{data-label="fig:quality_analysis"}](Visualization/38_T0002.png "fig:"){width="\textwidth"}
\[fig:quality\_occlusion\]
[0.47]{} ![ Problems in set-based person re-identification. Four sets of examples are shown (a-d); each set corresponds to a particular problem. They are grouped into two: perceptual quality and semantic quality problems. (a) belongs to the first one while the remaining ones belong to another. Note that some images in (b), (c) and (d) are less semantically related to its set ID due to incomplete body, heavy occlusion by a different person, and muptiple people in an image. []{data-label="fig:quality_analysis"}](Visualization/38_T0010.png "fig:"){width="\textwidth"}
\[fig:quality\_multiple persons\]
Although many SReID approaches exist [@liu2015spatio; @yan2016person; @you2016top; @zhu2016video; @mclaughlin2017video; @liu2017video; @chung2017two] , they mainly follow two steps: feature representation and feature fusion. At high level, the feature representation is learned with a deep convolutional neural network (CNN), and then they are aggregated by simple average fusion of image feature representations in the set. However, since the average fusion treats all images equally in the set, it ignores the fact that some images are more informative than the others in the set. To this end, some SReID methods [@zhou2017see; @xu2017jointly; @liu2017qan; @song2017region; @li2018diversity] applied attentive aggregation in which they modify CNN such that it can generate the attention score for each image to estimate the quality, e.g., image quality estimation network [@liu2017qan]. Their main purpose is to identify low quality/non-discriminative image embeddings (features) and ignore them from the set at the fusing stage. Nevertheless, with regard to the learning strategy, the attention is learned implicitly without any extra supervision, instead only guided by the standard loss function. As a result, in this paper we argue that these quality-based methods address *perceptual quality problems*, e.g., blurry images (Fig. \[fig:quality\_blur\]), but they cannot solve *semantic quality problems* which could be ‘incomplete body’, ‘total occlusion’, and ‘mutiple people in one image’ as shown in Fig. \[fig:quality\_missing\_part\], Fig. \[fig:quality\_occlusion\], and Fig. \[fig:quality\_multiple persons\], respectively. The semantic quality problems are inevitable in real-life since we do not have perfect detection and tracking systems yet.
Estimating the perceptual and semantic quality of images in the set is non-trivial. Based on our observations from Fig. \[fig:quality\_analysis\], we find that both kinds of quality could be estimated by how much the image is related to its set ID. To this end, we define *ID-aware quality* that measures the perceptual quality and semantic quality of images guided by their ID information. To be more specific, let us have a look at two examples in Fig. \[fig:quality\_analysis\]. The leftmost image which is blurry in Fig. \[fig:quality\_blur\] is of less perceptual quality thus less related to its set ID – low ID quality. The middle image in Fig. \[fig:quality\_occlusion\] is of high perceptual quality because it is clear (no blur). However, its semantic quality is low due to the occlusion by a different person – low ID quality again. Thus we argue that ID-aware quality covers more diverse quality issues. Moreover, the ID label for the middle image in Fig. \[fig:quality\_occlusion\] is wrong with respect to set ID. The existing methods [@song2017region; @li2018diversity] which rely on ID classification loss suffer from such kind of corrupted labels.
To realise ID-aware quality, we propose to use the classification confidence score. Our intuition is that in general high quality images (i.e., trivial images) are easily classified and get high confidence scores (e.g., 0.99) while low quality images (e.g., outliers) are less related to its set ID and get low confidence scores (e.g., 0.01). Similarly, ‘medium’ quality images get medium confidence scores. Furthermore, we find that there are two attractive properties of utilising classification confidence to estimate ID-aware quality. The first is that ID-aware quality can be estimated by readily available ID information – no need for extra annotations. The next is that it is easy to obtain as ID classification loss is commonly used in most existing SReID methods.
Based on ID-aware quality, we formulate an ID-aware Embedding (IDE) to learn a robust embedding function. IDE consists of two main components: (1) Feature learning attention (FLA) that targets learning robust image embeddings by focusing on ‘medium’ quality images (i.e., medium hard images). This is because high quality images (i.e., trivial images) contribute little to the loss and very small gradients while low quality images (very hard images or outliers) contribute large but misleading gradients. In this case, it can prevent overfitting to trivial images, and alleviate the influence of outliers. (2) Feature fusion attention (FFA) is to fuse image embeddings in the set to obtain the set-level embedding. It ignores noisy information and pays more attention to discriminative images to aggregate more discriminative information. IDE is an end-to-end framework built on CNN optimised by ID classification loss and set-based verification loss jointly. It is worth mentioning that it is simple as we learn global representations of images and aggregate them into a singe embedding without applying sophisticated attentive spatiotemporal features [@li2018diversity].\
**Contributions**: (1) A novel concept named ID-aware quality based on the classification confidence is proposed for the set-based person re-identification. It can estimate not only the perceptual quality, but also the semantic quality of images with respect to the set ID. (2) To show the applicability of ID-aware quality, we formulate ID-aware Embedding (IDE) which learns robust set-level embeddings that can be used for set-based person re-identifcation. Our model can be trained in an end-to-end manner. (3) Extensive experiments are carried out on four benchmarks for set-based person re-identification: MARS [@zheng2016mars], iLIDS-VID [@wang2014person], PRID-2011 [@hirzer2011person], and LPW [@song2017region]. Our method achieves new state-of-the-art performance on all datasets. In addition, cross-dataset experiments are conducted, which also achieves state-of-art performance.
{width="\linewidth"}
Related Work {#related_work}
============
**Learning Image Representations** Common approach to learning image representation is based on CNN with a certain kind of objective function such as identification loss and verification loss. This is mainly motivated by the image-based person re-identification methods [@xiao2016learning; @zhong2017re; @cheng2016person; @zheng2016person], where each ID has a single image instead of multiple images. Some methods learn a global feature representation at an image level [@liu2017qan; @chen2018video] while others learn local image features to obtain more fine-grained representation [@song2017region; @li2018diversity]. In this work, we follow the same strategy. However, the difference is that our method focuses on ‘medium’ quality images, while discard trivial (easy) images and outliers (very hard).\
**Learning Set Representations** We review several approaches of learning set representations for SReID. We divide them into two groups in terms of whether they consider the quality of the images.
*Non-quality-based methods*. The methods belonging to this group do not consider quality, rather they focus more on temporal information [@liu2018spatial; @mclaughlin2017video; @liu2015spatio; @you2016top; @zhu2016video; @yan2016person; @mclaughlin2016recurrent; @liu2017video; @chung2017two] or designing different network architechtures based on 3D convolutional nets [@li2018multi]. For example, long short term memory (LSTM) [@hochreiter1997long] is applied in [@yan2016person] for aggregating image representations and the output at the last time stamp is taken as the final representation of each video/set; Recurrent neural network (RNN) is applied in [@mclaughlin2017video; @liu2017video; @chung2017two] for accumulating the temporal information in each sequence and spatial features of all images are fused by average pooling.
*Quality-based methods.* The methods belonging to this group consider the quality of individual images in the set when aggregating image-level feature embeddings. The quality is formulated in the form of attention paradigm. That is, if the attention score is high for a particular image, then it is assumed that it is of high quality. The methods in [@wang2014person; @huang2018video] select the most discriminative set (video) fragments to learn set representations, which can be regarded as fragment-level attention. Recently, image-level attention attracts a great deal of interest and has been widely studied [@zhou2017see; @xu2017jointly; @liu2017qan; @song2017region; @chen2018video; @li2018diversity]. In particular, in attentive spatial-temporal pooling networks [@xu2017jointly] and co-attentive embedding [@chen2018video], an attention mechanism is designed such that the computation of set representations in the gallery depends on the probe data. A similar approach is taken in [@zhou2017see] which contains an attention unit for weighted fusion of image embeddings. Score generation branch is designed for generating attention score in quality aware network [@liu2017qan], region-based quality estimation network [@song2017region], and diversity regularized spatiotemporal attention [@li2018diversity]. More recently, in order to take the parts into account, [@fu2018sta] proposed an attention mechanism by dividing the feature map into fixed set of horizontal parts.
Our work has the following key differences: (1) we only consider spatial appearance information in the image set without using the temporal information and optical flow information as compared to [@fu2018sta; @zhu2016video; @song2017region; @li2018diversity]. (2) All quality-based approaches mentioned above learn the quality scores implicitly. In contrast, we learn quality with respect to set ID explicitly – supervised by the ID information. (3) Unlike methods in [@liu2017qan; @song2017region] which only deal with perceptual quality problem, our method can cope with a diverse set of quality problems. (4) ID-aware embedding does not depend on the probe data unlike the methods in [@xu2017jointly; @chen2018video], thus being more scalable and applicable. (5) Furthermore, although region-based feature extraction [@song2017region; @li2018diversity; @fu2018sta] has achieved great success, to demonstrate the robustness and effectivenss of IDE, we simply learn a global representation for every image in the set.\
**Attention** Numerious works exist about attention, and they could be divided into two directions [@jetley2018learn; @Andrea2019]. One direction is post hoc network analysis, and an attention unit in this direction relies on fully trained classification network model [@karen; @Zhou_2016_CVPR]. The other one goes with trainable attention in which the weights of attention unit and the original network weights are learned jointly [@jetley2018learn; @seo2016]. Our work aligns with the first direction which is the post hoc network analysis. However, our problem setting is verification, in which the label space between test and train data are disjoint. Thus the attention mechanism used during training cannot be used for testing.
Methodology {#sec:proposed_method}
===========
The overall pipeline of our framework, IDE, is shown in Fig. \[fig:overall\_architecture\]. First, the given image set, which consists of several person images, goes through CNN network and outputs representation for each image in the set. It is followed by a fully connected layer and softmax normalisation to generate ID-aware qualities. Then, these qualities are used in two ways: (1) Feature learning attention (FLA) component utilises the qualities such that medium hard images get more weights. This ends with the weighted cross-entropy loss for image-based classification. (2) Feature fusion attention (FFA) component transforms the qualites so that the ones with noise are given smaller weights to aggregate more discriminative information. This is supervised by the contrastive loss for set-based verification. IDE is trained end-to-end and optimized by two losses jointly. More formally, we aim to learn an embedding function $\Phi$ that takes an image set $\mathbf{x} \in \mathbb{R}^{n\times h\times w\times c}$ as input, where $n$ is the number of images, $h,~w$ and $c$ are height, width and the number of channels respectively, and it outputs a $d$-dimensional discriminative feature vector $\mathbf{z}\in \mathbb{R}^{d}$, $\Phi: \mathbf{x} \rightarrow \mathbf{z}$.
In what follows, we present the key components in detail, i.e., ID-aware quality generation, FLA, FFA, and loss functions.
ID-aware Quality Generation {#subsec: ID_aware}
---------------------------
ID-aware quality indicates how much it is semantically related to its set ID. We propose to generate it as follows. Firstly, we obtain image representations: for the $j$-th image set in the training batch, $ (\mathbf{x}^j, y^j) = (\{\mathbf{x}^{j}_1, \mathbf{x}^{j}_2, ..., \mathbf{x}^{j}_n\}, y^{j}
) $, where $ \mathbf{x}^j_i \in \mathbb{R}^{h\times w\times c}$ and $y^j$ are an image and its identity, respectively (for brevity, hereafter we omit the superscript $j$ where appropriate), we employ a deep CNN $f$ to embed each image $ \mathbf{x}_i$ to a $d$-dimensional feature representation, i.e, $\mathbf{z}_i = f(\mathbf{x}_i) \in \mathbb{R}^d$. Then, in order to obtain quality scores we apply a fully connected layer followed by a softmax normalisation. This way, the learned parameters $\{ \mathbf{c}_k \}^{C}_{k=1}$ of the fully connected layer are ID context vectors, where $\mathbf{c}_k \in \mathbb{R}^d$ is the $k$-th ID’s context vector and $C$ is the number of identities in total. In short, $c_k$ plays a role of ID classifier. More formally, the semantic relation of an image to an ID can be measured by the compatibility between the image’s feature vector $\mathbf{z}_i$ and the ID’s context vector $\mathbf{c}_k$. We calculate the dot product between two vectors to measure their compatibility [@jetley2018learn], followed by a softmax operator which is used for normalising semantic relations over all identities. That is: $$\begin{aligned}
\label{equation:s_i}
\mathrm{s}_i &= \frac{\exp(\mathbf{z}_i^\top\mathbf{c}_y)}{\sum\limits_{k=1}^C \exp(\mathbf{z}_i^\top\mathbf{c}_k)}.\end{aligned}$$
Inherently, Eq. (\[equation:s\_i\]) computes the classification confidence (likelihood) of $\mathbf{x}_i$ with respect to its set ID $y$. Similarly, ID-aware quality is estimated by the classification confidence directly [@goldberger2016training].
Feature Learning Attention {#subsec: FLA}
--------------------------
During learning image embeddings, FLA focuses on medium hard images whose classification confidences are around the centre of distribution, i.e., 0.5, as the distribution is between 0 and 1. This is a trade-off between gradient magnitude and gradient correctness. Intuitively, high quality images are easily classified and obtain high classification confidences (e.g., 0.99), but they are trivial because they contribute little to the loss and their gradients are relatively small. Low quality images (e.g. outliers) have low classification confidences (e.g., 0.01) and large gradients, but their gradient directions are misleading. To achieve the trade-off between gradient magnitude and gradient correctness, medium hard images are given higher weights/attention in FLA. The medium hard images are those whose ID-aware qualities are around the centre of distribution – 0.5.
To achieve this intution, we propose the following Gaussian function[^1] to compute FLA scores from ID-aware qualities: $$\begin{aligned}
\label{eq:fla}
\mathrm{FLA}_i &=
\exp(
{{{ - \left( {\mathrm{s}_i - 0.5 } \right)^2 } \mathord{\left/ {\vphantom {{ - \left( {\mathrm{s}(\mathrm{x}_i) - 0.5 } \right)^2 } {2 \sigma_\mathrm{FLA} ^2 }}} \right. \kern-\nulldelimiterspace} { \sigma_\mathrm{FLA} ^2 }}}
),\end{aligned}$$ where 0.5 is a mean, and $\sigma_\mathrm{FLA}$ is a temperature parameter controlling the distribution of FLA scores. $\mathrm{FLA}_i$ is the FLA score of $\mathbf{x}_i$, indicating its weight value in the ID classification task. Note that a choice of 0.5 is due to our intution stated above. One intriguing property of this function is that we can change the distribution by merely changing the parameter $\sigma_\mathrm{FLA}$. Fig. \[fig:FLA\_motivation\] illustrates FLA with Eq. (\[eq:fla\]).
[0.47]{} ![FLA aims to learn robust image-level embeddings. High quality images and low quality images are neglected because of small gradients and large but mis-leading gradients respectively; Medium hard images, i.e., medium quality images, are emphasized, which is a trade-off between gradient magnitude and gradient correctness. []{data-label="fig:FLA_motivation"}](Figures/FLA_2 "fig:"){width="\textwidth"}
Feature Fusion Attention
------------------------
During feature fusion which is used for verification in a later stage, FFA aggregates most informative feature embeddings. In general, high quality images (ID-aware quality) are very discriminative, thus being more informative for embedding the set. This is shown in Fig. \[fig:FFA\_motivation\]. To realise this intuition, we compute FFA scores from ID-aware qualities using the following Gaussian function with $\mu=1$: $$\begin{aligned}
\mathrm{FFA}_i &=
\exp(
{{{ - \left( {\mathrm{s}_i - 1 } \right)^2 } \mathord{\left/ {\vphantom {{ - \left( {\mathrm{s}(\mathrm{x}_i) - 1 } \right)^2 } {2 \sigma_\mathrm{FFA} ^2 }}} \right. \kern-\nulldelimiterspace} { \sigma_\mathrm{FFA} ^2 }}}
),\end{aligned}$$ where $\sigma_\mathrm{FFA}$ is a temperature parameter controlling the distribution of FFA scores. $\mathrm{FFA}_i$ is the score of $\mathbf{x}_i$, indicating its importance when being aggregated into the set embedding. As a result, we obtain the set representation by weighted fusion of image representations in the set: $$\Phi\left(\mathbf{x} \right) = \frac{
\sum\limits_{i=1}^n{ \mathrm{FFA}_i \cdot \mathbf{z}_i }
}
{
\sum\limits_{i=1}^n{ \mathrm{FFA}_i }
}.$$ The term in denominator $\sum\limits_{i=1}^n{ \mathrm{FFA}_i }$ is the sum of FFA scores for normalisation. At each iteration, FFA scores of images are computed in the forward process and used as constant values for just scaling the gradient vectors during gradient back-propagation.
[0.47]{} ![ FFA fuses image-level representations to obtain the set-level embedding. More discriminative images in the set are emphasized to accumulate more discriminative information into the set representation. The images with higher ID-aware quality are more discriminative. []{data-label="fig:FFA_motivation"}](Figures/FFA_2 "fig:"){width="\textwidth"}
Loss Functions
--------------
Suppose there are $m$ image sets in each mini-batch, i.e.,$\{ (\mathbf{x}^j, y^j) \}^{m}_{j=1}$ and each set contains $n$ images, thus the mini-batch size is $mn$.\
**Weighted Cross-Entropy Loss**. To learn robust image representations based on FLA scores, we propose a weighted cross-entropy loss for the image-level ID classification task: $$L_{\mathrm{WCEL}} = - \frac{
\sum\limits_{j=1}^{m}
\sum\limits_{i=1}^{n}
({\mathrm{FLA}^j_i \cdot \log{\mathrm{s}^j_i}}
)
}
{
\sum\limits_{j=1}^{m}
\sum\limits_{i=1}^{n}
{\mathrm{FLA}^j_i }
},$$ where the term in the denominator $\sum\limits_{j=1}^{m}
\sum\limits_{i=1}^{n}
{\mathrm{FLA}^j_i }$ is for normalisation. Accordingly, the partial derivative of $L_{\mathrm{WCEL}}$ w.r.t. $\mathrm{s}^j_i$ is: $$\begin{aligned}
\label{equation: derivative}
\frac{
\partial L_{\mathrm{WCEL}}
}
{
\partial {\mathrm{s}^j_i}
}
= &
\frac{\mathrm{FLA}^j_i} {{\sum\limits_{j=1}^{m}\sum\limits_{i=1}^n {\mathrm{FLA}^j_i}}}
\cdot
(- \frac{1} {\mathrm{s}^j_i}). \end{aligned}$$ At each iteration, after being computed in the forward process, FLA scores of images are assumed as constant values for scaling the gradients during the back-propagation process. Compared with standard cross-entropy loss, we use the normalised FLA score $\frac{\mathrm{FLA}^j_i} {{\sum\limits_{j=1}^{m}\sum\limits_{i=1}^n {\mathrm{FLA}^j_i}}}$ to scale image’s gradient.\
**Contrastive Loss**. For set-based verification, on top of set-level representations, we employ contrastive loss [@hadsell2006dimensionality] using the multi-batch setting [@tadmor2016learning]. The contrastive loss is a well-known loss function used for verification task. It pulls sets from the same identity as close as possible and pushes the sets from different identities farther than a pre-defined margin $\alpha$. Specifically, we construct a pair between every two set embeddings, resulting in $m(m-1)/2$ pairs in total. For all the set embeddings $\{ (\mathbf{z}^j, y^j) \}^{m}_{j=1}$ in the mini-batch, we compute the contrastive loss per pair: $$L(\alpha; \mathbf{\Phi}^{j}, \mathbf{\Phi}^{k}, y^{jk}) = y^{jk}d_{jk}^2 + (1-y^{jk})\max(0, \alpha - d_{jk})^2,$$ where $y^{jk}=1$ if $y^{j}=y^{k}$ and $y^{jk}=0$, otherwise. $d_{jk} = \left\Vert
\mathbf{\Phi}^j - \mathbf{\Phi}^k
\right\Vert_2 $ is the distance between the set pair. The contrastive loss of the mini-batch is the average loss over all the pairs: $$\label{equation: CL}
L_{\mathrm{CL}} =
\frac{2}{m(m-1)}
\sum_{j=1}^{m-1}
\sum_{k=j+1}^{m}
L(\alpha; \mathbf{\Phi}^{j}, \mathbf{\Phi}^{k}, y^{jk}).$$ IDE is trained end-to-end by optimising the weighted cross-entropy loss and set-based contrastive loss jointly: $$L =
L_{\mathrm{WCEL}}
+
L_{\mathrm{CL}}.$$ After the training is finished, the trained CNN model can be applied to extract image features. Generally, the label spaces of the training and testing sets are disjoint in the verification problems. Therefore, we cannot estimate the ID-aware quality of testing images as their classification confidences are not available during testing. To obtain the set embedding in the test phase, we aggregate image representations in the set simply by average fusion. After that, we verify whether two image sets show the same person simply by computing their cosine distance.\
**Remarks:** 1) We would like to note that although we use the term ‘attention’ to mean the weight for images, our method is *different* from the classical attention-based methods. In our formulation, the scores of FLA and FFA are taken as a constant values after forward propogation. In contrast, standard attention-based methods would allow the gradients to flow through FFA and FLA. 2) During training we can estimate the quality, however we cannot estimate the quality at the testing stage. Our main goal during training is to learn a robust embedding function by ignoring perceptually and semantically low quality images. This way we assume that our model generalises better because it does not overfit to the training dataset. Our extensive experiments validate that this assumption is reasonable. Please see Sec. \[related\_work\] for more.\
Experiments {#sec:experiments}
===========
Datasets and Settings
---------------------
**Datasets.** We use two large-scale and two small-scale datasets in our experiments: (1) *LPW* [@song2017region] is a large-scale dataset released recently. The persons in the dataset are collected across three different scenes separately. Three cameras are placed in the first scene, while four cameras are placed in other two scenes. Each person is captured by more than one camera so that cross-camera search could be possible in each scene. There are totally 7694 image sets with about 77 images per set. Following the evaluation setting in [@song2017region], 1975 persons captured in the 2$^{nd}$ scene and 3$^{rd}$ scene are used for training, while 756 persons from the first scene are used for testing. The dataset is challenging as the evaluation protocol is close to real-world situation, that is, the training data and the testing data are different in terms of not only identities but also scenes. (2) *MARS* [@zheng2016mars] is another large-scale dataset. There are 20478 tracklets (image sets) of 1261 persons in total and each person is shot by at least two cameras. This dataset is challenging due to automatic detection and tracking errors. (3) *iLIDS-VID* [@wang2014person] is a small-scale dataset. Since it is collected in airport environment, image sequences contain significant viewpoint variations, occlusions and background clutter. There are 300 identities in total and each identity has two tracklets from two different cameras. (4) *PRID2011* [@hirzer2011person] consists of 400 tracklets for 200 persons from two cameras. The tracklet length varies from 5 to 675. Compared to the previous datasets, PRID2011 is less challenging because of few variations and rare occlusion. For LPW and MARS, we follow the evaluation setting in [@song2017region] and [@zheng2016mars] respectively. For iLIDS-VID and PRID2011, they are split into two subsets with equal size following [@wang2014person], one for training and the other for testing. In addition, the 10 random trials are fixed and the same as in [@wang2014person] for fair comparison. A summary of the datasets is shown in Table \[table:dataset\].
[lllll]{} Datasets & LPW & MARS & iLIDS-VID & PRID2011\
\#identities & 2731 & 1261 & 300 & 200\
\#boxes & 590,547 & 1,067,516 & 43,800 & 40,000\
\#tracklets & 7694 & 20,715 & 600 & 400\
\#cameras & 11 & 6 & 2 & 2\
DT failure & No & Yes & No & No\
resolution & 256x128 & 256x128 & 128x64 & 128x64\
detection & detector & detector & hand & hand\
evaluation & CMC & CMC & mAP & CMC & CMC\
**Evaluation Metrics.** We report the Cumulated Matching Characteristics (CMC) results for all the datasets. We also report the mean average precision (mAP) for MARS following the common practice.
**Implementation Details.** We use GoogLeNet with batch normalisation [@ioffe2015batch] as our backbone architecture. Every input image is resized to $224\times224$. We do not apply any data augmentation for training and testing. Each mini-batch contains 3 persons, 2 image sets per person, 9 images per set, so the batch size is 54 ($m = 6, n = 9$). Each image set in the mini-batch is randomly sampled from the complete image set during training. According to Eq. (\[equation: CL\]), we have 3 positive set pairs and 12 negative set pairs totally, thus the positive-to-negative rate is 1:4. The margin of contrastive loss is set to 1.2 ($\alpha=1.2$). Stochastic gradient descent (SGD) optimiser is applied with an initial learning rate of $1e^{-3}$. When training the model on each dataset, we initialise it by the pre-trained GoogLeNet model on ImageNet. We use Caffe [@jia2014caffe] for implementation.
For the temperature parameters $\sigma_\mathrm{FLA}$ and $\sigma_\mathrm{FFA}$, they control the variances of FLA scores and FFA scores respectively. Based on our experimental results, they are insensitive and are fixed in all the experiments ($\sigma_\mathrm{FLA} = 0.18, \sigma_\mathrm{FFA} = 0.68 $) although better results could be obtained by exploring optimal parameters for each dataset. Analysis of these parameters is reported in the supplimentary material.
--------------- --------- ----------- ---------- ---------- --
Methods Quality Backbone CMC-1 mAP
IDE+Euc No CaffeNet 58.7 40.4
IDE+XQDA No CaffeNet 65.3 47.6
IDE+XQDA+RR No CaffeNet
IDE No ResNet50 62.7 44.1
IDE+XQDA No ResNet50 70.5 55.1
IDE+XQDA+RR No ResNet50
CNN+RNN No Custom 43.0 –
CNN+RNN+XQDA No Custom 52.0 –
AMOC+EpicFlow No Custom 68.3 52.9
ASTPN Yes Custom 44.0 –
SRM+TAM Yes CaffeNet 70.6 50.7
RQEN Yes GoogleNet 73.7 51.7
RQEN+XQDA+RR Yes GoogleNet
DRSA Yes ResNet50 82.3 65.8
CAE Yes ResNet50 82.4 67.5
**Ours** Yes GoogleNet **83.3** **71.7**
**Ours**+RR Yes GoogleNet
--------------- --------- ----------- ---------- ---------- --
: Comparison with state-of-the-art methods on MARS in terms of CMC-1(%) and mAP(%). Quality-based methods are indicated by ‘Quality’ column. indicates ‘re-ranking (RR)’. ‘Custom’ means the backbone is customised.[]{data-label="table:literature_mars"}
Method Quality iLIDS-VID PRID2011 LPW
--------------- --------- ----------- ---------- ----------
STA No 44.3 64.1 –
SI$^2$DL No 48.7 76.7 –
IDE+XQDA No 53.0 77.3 –
TDL No 56.3 56.7 –
CNN+RNN No 58.0 70.0 –
AMOC+EpicFlow No 68.7 83.7 –
ASTPN Yes 62.0 77.0 –
SRM+TAM Yes 55.2 79.4 –
QAN Yes 68.0 90.3 –
RQEN Yes 76.1 92.4 57.1
DRSA Yes 80.2 93.2 –
CAE Yes 74.4 86.0 –
**Ours** Yes **81.9** **93.7** **70.9**
: Comparison with state-of-the-art methods on iLIDS-VID, PRID2011 and LPW in terms of CMC-1(%). Quality-based methods are indicated by ‘Quality’ column. LPW is a difficult cross-scene dataset and released in [@song2017region] recently.[]{data-label="table:literature_ilid_prid"}
Comparison with State-of-the-art Methods {#subsec:state_of_the_art}
----------------------------------------
**MARS Dataset.** We compare our approach with several state-of-the-art methods. Overall, we divide them into two categories: non-quality-based and attention-based methods. *The non-quality-based methods*: the softmax loss is used during training with custom/common networks, e.g., IDE(CaffeNet) [@zheng2016mars], IDE(ResNet50) [@zhong2017re], and metric learning/re-ranking strategies are applied for boosting the performance, e.g., IDE(CaffeNet)+XQDA [@zheng2016mars], IDE(CaffeNet)+XQDA+RR [@zhong2017re], IDE(ResNet50)+XQDA [@zhong2017re], IDE(ResNet50)+XQDA+RR [@zhong2017re]; the temporal information of image sequences is used by combining recurrent neural networks with convolutional networks (CNN+RNN [@mclaughlin2017video]) and again metric learning is used on top of this (CNN+RNN+XQDA); optical flow is used to capture motion information (AMOC+EpicFlow [@liu2017video]). *The quality-based methods*: combining image-level attention and temporal pooling to select informative images (ASTPN [@xu2017jointly], SRM+TAM [@zhou2017see], CAE[^2] [@chen2018video]); designing attentive spatiotemporal models to extract complementary region-based information between images in the set (RQEN [@song2017region], DRSA[@li2018diversity]). Results of our method and compared methods are shown in Table \[table:literature\_mars\]. *The following key findings are observed*: *(1)* Our method considerably outperforms all the compared methods in both metrics (CMC, mAP). Noticeably, our mAP is around 6% better than DSRA and 4% better than CAE. DSRA applies spatiotemporal attention to extract features of latent regions and is pre-trained on several large image-based person ReID datasets. CAE is a co-attentive embedding model and computationally expensive as the embeddings of sets in the gallery are dependent on the probe data. In addition, CAE combines optical flow information and RGB information. *(2)* Generally, the quality-based methods outperform non-quality-based methods except for ASTPN with a relatively shallow architecture. *(3)* When these methods are coupled with re-ranking technique, there is always a performance boost (ours achieves 82.2 % in terms of mAP).
**iLIDS-VID, PRID2011, and LPW datasets.** Grouping the methods for these datasets is the same, and most compared methods are borrowed from Table \[table:literature\_mars\]. Additional methods include: a spatial-temporal body-action model (STA [@liu2015spatio]), novel metric learning approaches (SI$^2$DL [@zhu2016video], TDL [@you2016top]). The results are shown in Table \[table:literature\_ilid\_prid\]. We find that our observations are consistent with MARS dataset. In particular: *(1)* In all datasets, our method achieves the best performance; *(2)* The margin of iLIDS-VID and LPW is larger than PRID2011. This is because PRID2011 is a much cleaner dataset and its accuracy is already high; *(3)* Noticeably the LPW dataset is the most challenging since the scenes are different for training and testing, and yet our method is around 14% higher than RQEN [@song2017region], showing the generalisation capability of our proposed method.
Model 1 5 10 20
---------- ---------- ---------- ---------- ----------
Baseline 63.2 85.4 90.6 94.8
FFA 67.9 88.8 93.3 95.8
FLA 68.8 88.1 93.5 **96.3**
FFA+FLA **70.9** **89.3** **93.9** 95.8
: Effectiveness of FLA and FFA. The ablation studies are conducted on large dataset LPW. CMC (%) results are presented. Baseline means standard cross-entropy loss and average fusion are applied. Compared with the baseline, FLA applies weighted cross-entropy loss while FFA uses weighted fusion. FLA+FFA combines weighted cross-entropy loss and weighted fusion. []{data-label="table:component_effectiveness_v1"}
Effectiveness of FLA and FFA {#sec: two_attention_unit}
----------------------------
IDE has two key components: (1) FLA aims to learn robust image representations. Based on FLA, weighted cross-entropy loss is proposed to replace standard cross-entropy loss in the baseline; (2) During training, FFA is proposed for weighted fusion of image embeddings to replace average fusion in the baseline. We use LPW for this analysis. Table \[table:component\_effectiveness\_v1\] shows that the performance improves significantly using either FLA or FFA compared to the baseline, demonstrating the effectiveness of FLA and FFA. We obtain the best performance by FFA+FLA at ranks ranging from 1 to 10.
A Variant of FFA {#sec: variant_ffa}
----------------
We employ the same idea as FLA which pays more attention to medium hard samples to see the impact. We name this as FFA~MH~. Table \[table:different\_fusion\] shows that in both settings (without FLA or with FLA) FFA~MH~ performs similar as average fusion but much worse than our proposed FFA. This validates our assumption that assigning more weight to high quality images is desirable at the verification stage.
Model 1 5 10 20
------------- ---------- ---------- ---------- ----------
Baseline 63.2 85.4 90.6 94.8
FFA **67.9** **88.8** **93.3** **95.8**
FFA~MH~ 64.5 86.8 91.5 94.3
FLA 68.8 88.1 93.5 **96.3**
FLA+FFA **70.9** **89.3** **93.9** 95.8
FLA+FFA~MH~ 68.5 89.2 92.7 95.8
: Evaluation of FFA~MH~ on LPW in terms of CMC (%). FFA~MH~ is a variant of FFA and attends more attention to medium hard images. The results of Baseline (average fusion), FLA(average fusion), FFA, and FLA+FFA are copied from Table \[table:component\_effectiveness\_v1\]. We compare them in two different settings: without FLA, and with FLA.[]{data-label="table:different_fusion"}
Cross-Dataset Evaluation
------------------------
Cross-dataset testing is a better way to evaluate a system’s real-world performance than only evaluating its performance on the same dataset used for training. Generally any public dataset only represents a small proportion of all real-world data. The model trained on A dataset could perform much worse when applied to B dataset, which indicates the model overfits to the particular scenario.
We conduct cross-dataset testing on PRID2011 to evaluate the generalisation of our method. The diverse and large MARS and iLIDS-VID are used for training. We follow the evaluation setting in CNN-RNN [@mclaughlin2017video] and ASTPN [@xu2017jointly], i.e., the model is tested on 50% of PRID2011. We use the same testing data as Table \[table:literature\_ilid\_prid\] so the cross-dataset testing results can be compared with the results in Table \[table:literature\_ilid\_prid\]. The results are shown in Table \[table:cross\_dataset\_test\].[^3]As expected, the results are worse than within-dataset testing because of dataset bias. However, our method generalises well and achieves state-of-the-art cross-dataset testing performance. For CMC-1 accuracy, our method achieves 41.4% when trained on MARS and 61.7% when trained on iLIDS-VID. Noticeably, it is comparable with the performance (64.1%) of spatial-temporal body-action model STA [@liu2015spatio] in Table \[table:literature\_ilid\_prid\]. This indicates that IDE enhances the generalisation ability significantly.\
Method Train Test 1 5 10 20
--------- ----------- ---------- ---------- ---------- ---------- ----------
CNN+Euc MARS PRID2011 7.6 24.6 39.0 51.8
CNN+RNN MARS PRID2011 18.0 46.0 61.0 74.0
Ours MARS PRID2011 **41.4** **57.5** **69.1** **81.2**
CNN+RNN iLIDS-VID PRID2011 28.0 57.0 69.0 81.0
ASTPN iLIDS-VID PRID2011 30.0 58.0 71.0 85.0
Ours iLIDS-VID PRID2011 **61.7** **88.5** **95.4** **98.8**
: Cross-dataset testing in terms of CMC (%). []{data-label="table:cross_dataset_test"}
Conclusion
==========
In this paper we propose a new concept, ID-aware quality, which measures the semantic quality of images guided by their ID information. Based on ID-aware quality, we propose ID-aware embedding which contains FLA and FFA. To learn robust image embeddings, FLA gives more weight to medium hard images. To accumulate more discriminative information when fusing image features, FFA assigns more weight to high quality images. Compared with previous state-of-the-art attentive spatiotemporal methods, our method works much better in both within-dataset testing and cross-dataset testing. Furthermore, IDE is much simpler compared with previous attentive spatiotemporal methods.
The temperature parameter of FLA: $\sigma_{\mathrm{FLA}}$
==========================================================
To study the impact of $\sigma_{\mathrm{FLA}}$, we set the temperature of FFA $\sigma_{\mathrm{FFA}}=0.68$ in all experiments. We conduct experiments on MARS and LPW and report the results in Table \[table:different\_sigma\_FLA\] and Figure \[fig:different\_sigma\_FLA\]. Firstly, we can see that the performance is not sensitive to $\sigma_{\mathrm{FLA}}$. For example, the performance difference is smaller than 1.5% on MARS when $\sigma_{\mathrm{FLA}}$ ranges from 0.12 to 0.24. Secondly, better results can be obtained by exploring the optimal $\sigma_{\mathrm{FLA}}$ on each dataset. In the main paper, we fix $\sigma_{\mathrm{FLA}}=0.18$ on all datasets. We observe that $\sigma_{\mathrm{FLA}}=0.18$ works best on LPW but not on MARS.
$\sigma_{\mathrm{FFA}}=0.68$ MARS LPW
------------------------------ ------ ------
$\sigma_{\mathrm{FLA}}=0.12$ 82.1 68.3
$\sigma_{\mathrm{FLA}}=0.15$ 83.5 68.5
$\sigma_{\mathrm{FLA}}=0.18$ 83.3 70.9
$\sigma_{\mathrm{FLA}}=0.21$ 83.5 69.7
$\sigma_{\mathrm{FLA}}=0.24$ 83.5 69.6
: The results of different $\sigma_{\mathrm{FLA}}$ on MARS and LPW in terms of CMC-1 (%). We fix $\sigma_{\mathrm{FFA}}=0.68$ in all experiments. []{data-label="table:different_sigma_FLA"}
![ The results of different $\sigma_{\mathrm{FLA}}$ on MARS and LPW in terms of CMC-1 (%). This is a line figure representation of results shown in Table \[table:different\_sigma\_FLA\]. []{data-label="fig:different_sigma_FLA"}](FLA){width="1.0\linewidth"}
The temperature parameter of FFA: $\sigma_{\mathrm{FFA}}$
==========================================================
To study the influence of $\sigma_{\mathrm{FFA}}$, we set $\sigma_{\mathrm{FLA}}=0.18$ in all experiments. The experiments are conducted on MARS and LPW and their results are presented in Table \[table:different\_sigma\_FFA\] and Figure \[fig:different\_sigma\_FFA\]. First, we observe that the performance is also insensitive to $\sigma_{\mathrm{FFA}}$. The performance gap is less than 1.0% on MARS and around 2.0% on LPW. Second, we can also obtain better results by searching optimal parameters for different datasets. We fix $\sigma_{\mathrm{FFA}}=0.68$ on all datasets in the main paper. We notice that $\sigma_{\mathrm{FFA}}=0.68$ works best on LPW while $\sigma_{\mathrm{FFA}}=0.71$ is the best on MARS.
$\sigma_{\mathrm{FLA}}=0.18$ MARS LPW
------------------------------ ------ ------
$\sigma_{\mathrm{FFA}}=0.62$ 83.1 68.8
$\sigma_{\mathrm{FFA}}=0.65$ 83.1 69.0
$\sigma_{\mathrm{FFA}}=0.68$ 83.3 70.9
$\sigma_{\mathrm{FFA}}=0.71$ 83.8 70.5
$\sigma_{\mathrm{FFA}}=0.74$ 83.6 70.6
: The results of different $\sigma_{\mathrm{FFA}}$ on MARS and LPW in terms of CMC-1 (%). We fix $\sigma_{\mathrm{FLA}}=0.18$ in all experiments. []{data-label="table:different_sigma_FFA"}
![ The results of different $\sigma_{\mathrm{FFA}}$ on MARS and LPW in terms of CMC-1 (%). This is the line figure illustration of results presented in Table \[table:different\_sigma\_FFA\]. ](FFA){width="1.0\linewidth"}
\[fig:different\_sigma\_FFA\]
[^1]: Note that different functions could be also designed to reach the same goal. Exploring different functions are left for the future work.
[^2]: For CAE, we present the results of complete sequence instead of multiple subsets so that it can be compared with other methods. Multiple subsets can be regarded as data augmentation.
[^3]: Cross-dataset experimental results of QAN and RQEN are not reported as they were tested on 100% of PRID
|
---
abstract: |
Brandão and Svore recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension $n$ of the problem and the number $m$ of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure.
We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimizations problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with $mn$ when $m\approx n$, which is the same as classical.
author:
- 'Joran van Apeldoorn[^1]'
- 'András Gilyén[^2]'
- 'Sander Gribling[^3]'
- 'Ronald de Wolf[^4]'
bibliography:
- 'qc.bib'
title: 'Quantum SDP-Solvers: Better upper and lower bounds'
---
Introduction
============
Semidefinite programs {#subsec:sdp}
---------------------
In the last decades, particularly since the work of Gr[ö]{}tschel, Lov[á]{}sz, and Schrijver [@gls:geometric], *semidefinite programs* (SDPs) have become an important tool for designing efficient optimization and approximation algorithms. SDPs generalize and strengthen the better-known *linear* programs (LPs), but (like LPs) they are still efficiently solvable. The basic form of an SDP is the following: $$\begin{aligned}
\label{eq:SDP}
\max \quad &{\mbox{\rm Tr}}(CX) \\
\text{s.t.}\ \ \ &{\mbox{\rm Tr}}(A_j X) \leq b_j \quad \text{ for all } j \in [m], \notag \\
&X \succeq 0, \notag\end{aligned}$$ where $[m]:=\{1,\ldots,m\}$. The input to the problem consists of Hermitian $n\times n$ matrices $C,A_1,\ldots,A_m$ and reals $b_1,\ldots,b_m$. For normalization purposes we assume ${\left\lVertC\right\rVert},{\left\lVertA_j\right\rVert} \leq 1$. The number of constraints is $m$ (we do not count the standard $X\succeq 0$ constraint for this). The variable $X$ of this SDP is an $n\times n$ positive semidefinite (psd) matrix. LPs correspond to the case where all matrices are diagonal.
A famous example is the algorithm of Goemans and Williamson [@GoemansWilliamson95] for approximating the size of a maximum cut in a graph $G=([n],E)$: the maximum, over all subsets $S$ of vertices, of the number of edges between $S$ and its complement $\bar{S}$. Computing MAXCUT$(G)$ exactly is NP-hard. It corresponds to the following integer program $$\begin{aligned}
\max \quad &\frac{1}{2}\sum_{\{i,j\}\in E}(1-v_iv_j)\\
\text{s.t.}\ \ \ & v_j\in\{+1,-1\}\quad \text{ for all } j \in [n],\end{aligned}$$ using the fact that $(1-v_iv_j)/2=1$ if $v_i$ and $v_j$ are different signs, and $(1-v_iv_j)/2=0$ if they are the same. We can relax this integer program by replacing the signs $v_j$ by unit vectors, and replacing the product $v_iv_j$ in the objective function by the dot product $v_i^T v_j$. We can implicitly optimize over such vectors (of unspecified dimension) by explicitly optimizing over an $n\times n$ psd matrix $X$ whose diagonal entries are 1. This $X$ is the Gram matrix of the vectors $v_1,\ldots,v_n$, so $X_{ij}=v_i^T v_j$. The resulting SDP is $$\begin{aligned}
\max \quad &\frac{1}{2}\sum_{\{i,j\}\in E}(1-X_{ij}) \\
\text{s.t.}\ \ \ &{\mbox{\rm Tr}}(E_{jj} X) =1 \quad \text{ for all } j \in [n], \\
&X \succeq 0,\end{aligned}$$ where $E_{jj}$ is the $n\times n$ matrix that has a 1 at the $(j,j)$-entry, and 0s elsewhere.
This SDP is a relaxation of a maximization problem, so it may overshoot the correct value, but Goemans and Williamson showed that an optimal solution to the SDP can be rounded to a cut in $G$ whose size is within a factor $\approx 0.878$ of MAXCUT$(G)$ (which is optimal under the Unique Games Conjecture [@kkmo:maxcutj]). This SDP can be massaged into the form of by replacing the equality ${\mbox{\rm Tr}}(E_{jj} X)=1$ by inequality ${\mbox{\rm Tr}}(E_{jj} X)\leq 1$ (so $m=n$) and letting $C$ be a properly normalized version of the Laplacian of $G$.
Classical solvers for LPs and SDPs
----------------------------------
Ever since Dantzig’s development of the simplex algorithm for solving LPs in the 1940s [@Dantzig1947], much work has gone into finding faster solvers, first for LPs and then also for SDPs. The simplex algorithm for LPs (with some reasonable pivot rule) is usually fast in practice, but has worst-case exponential runtime. Ellipsoid methods and interior-point methods can solve LPs and SDPs in polynomial time; they will typically *approximate* the optimal value to arbitrary precision. The best known general SDP-solvers [@lsw:faster] approximate the optimal value of such an SDP up to additive error ${\varepsilon}$, with complexity $${\mathcal O}(m(m^2+n^\omega + mns)\, \text{polylog}(m,n,R,1/{\varepsilon})),$$ where $\omega\in[2,2.373)$ is the (still unknown) optimal exponent for matrix multiplication; $s$ is the *sparsity*: the maximal number of non-zero entries per row of the input matrices; and $R$ is an upper bound on the trace of an optimal $X$.[^5] The assumption here is that the rows and columns of the matrices of SDP can be accessed as adjacency lists: we can query, say, the $\ell$th non-zero entry of the $k$th row of matrix $A_j$ in constant time.
Arora and Kale gave an alternative way to approximate , using a matrix version of the “multiplicative weights update” method.[^6] In Section \[sec:classicalAK\] we will describe their framework in more detail, but in order to describe our result we will start with an overly simplified sketch here. The algorithm goes back and forth between candidate solutions to the primal SDP and to the corresponding *dual* SDP, whose variables are non-negative reals $y_1,\ldots,y_m$: $$\begin{aligned}
\label{eq:SDP2}
\min \quad & b^T y \\ \notag
\text{s.t.}\ \ \ &\sum_{j=1}^m y_j A_j - C \succeq 0,\\ \notag
&y \geq 0.\end{aligned}$$ Under assumptions that will be satisfied everywhere in this paper, strong duality applies: the primal SDP and dual SDP will have the same optimal value . The algorithm does a binary search for by trying different guesses $\alpha$ for it. Suppose we have fixed some $\alpha$, and want to find out whether $\alpha$ is bigger or smaller than ${\mbox{\rm OPT}}$. Start with some candidate solution $X^{(1)}$ for the primal, for example a multiple of the identity matrix ($X^{(1)}$ has to be psd but need not be a *feasible* solution to the primal). This $X^{(1)}$ induces the following polytope: $$\begin{aligned}
\mathcal{P}_{{\varepsilon}}(X^{(1)}) := \{ y \in \mathbb R^m:\ &b^T y \leq \alpha, \\
&{{\mbox{\rm Tr}}\left(\Big(\sum_{j=1}^m y_j A_j - C\Big) X^{(1)}\right)} \geq -{\varepsilon},\\
& y \geq 0 \}.\end{aligned}$$ This polytope can be thought of as a relaxation of the feasible region of the dual SDP with the extra constraint that ${\mbox{\rm OPT}}\leq\alpha$: instead of requiring that $\sum_j y_j A_j - C$ is psd, we merely require that its inner product with the particular psd matrix $X^{(1)}$ is not too negative. The algorithm then calls an “oracle” that provides a $y^{(1)}\in \mathcal{P}_{{\varepsilon}}(X^{(1)})$, or outputs “fail” if $\mathcal{P}_{0}(X^{(1)})$ is empty (how to efficiently implement such an oracle depends on the application). In the “fail” case we know there is no dual-feasible $y$ with objective value $\leq\alpha$, so we can increase our guess $\alpha$ for , and restart. In case the oracle produced a $y^{(1)}$, this is used to define a Hermitian matrix $H^{(1)}$ and a new candidate solution $X^{(2)}$ for the primal, which is proportional to $e^{-H^{(1)}}$. Then the oracle for the polytope $\mathcal{P}_{{\varepsilon}}(X^{(2)})$ induced by this $X^{(2)}$ is called to produce a candidate $y^{(2)}\in\mathcal{P}_{{\varepsilon}}(X^{(2)})$ for the dual (or “fail”), this is used to define $H^{(2)}$ and $X^{(3)}$ proportional to $e^{-H^{(2)}}$, and so on.
Surprisingly, the average of the dual candidates $y^{(1)},y^{(2)},\ldots$ converges to a nearly-dual-feasible solution. Let $R$ be an upper bound on the trace of an optimal $X$ of the primal, $r$ be an upper bound on the sum of entries of an optimal $y$ for the dual, and $w^*$ be the “width” of the oracle for a certain SDP: the maximum of ${\left\lVert\sum_{j=1}^m y_j A_j - C\right\rVert}$ over all psd matrices $X$ and all vectors $y$ that the oracle may output for the corresponding polytope $\mathcal{P}_{{\varepsilon}}(X)$. In general we will not know the width of an oracle exactly, but only an upper bound $w \geq w^*$, that may depend on the SDP; this is, however, enough for the Arora-Kale framework. In Section \[sec:classicalAK\] we will show that without loss of generality we can assume the oracle returns a $y$ such that ${\left\lVerty\right\rVert}_1 \leq r$. Because we assumed ${\left\lVertA_j\right\rVert},{\left\lVertC\right\rVert} \leq 1$, we have $w^* \leq r+1$ as an easy width-bound. General properties of the multiplicative weights update method guarantee that after $T=\widetilde{\mathcal O}(w^2R^2/{\varepsilon}^2)$ iterations[^7], if no oracle call yielded “fail”, then the vector $\frac{1}{T}\sum_{t=1}^T y^{(t)}$ is close to dual-feasible and satisfies $b^T y\leq\alpha$. This vector can then be turned into a dual-feasible solution by tweaking its first coordinate, certifying that ${\mbox{\rm OPT}}\leq\alpha+{\varepsilon}$, and we can decrease our guess $\alpha$ for ${\mbox{\rm OPT}}$ accordingly.
The framework of Arora and Kale is really a meta-algorithm, because it does not specify how to implement the oracle. They themselves provide oracles that are optimized for special cases, which allows them to give a very low width-bound for these specific SDPs. For example for the MAXCUT SDP, they obtain a solver with near-linear runtime in the number of edges of the graph. They also observed that the algorithm can be made more efficient by not explicitly calculating the matrix $X^{(t)}$ in each iteration: the algorithm can still be made to work if instead of providing the oracle with $X^{(t)}$, we feed it good estimates of ${\mbox{\rm Tr}}(A_j X^{(t)})$ and ${\mbox{\rm Tr}}(C X^{(t)})$. Arora and Kale do not describe oracles for general SDPs, but as we show at the end of Section \[sec:runtime\] (using Appendix \[app:trace\] to estimate ${\mbox{\rm Tr}}(A_j X^{(t)})$ and ${\mbox{\rm Tr}}(C X^{(t)})$), one can get a general classical SDP-solver in their framework with complexity $$\label{eq:AKgeneralupperbound}
{\widetilde{\mathcal O}\left(nms\left(\frac{Rr}{{\varepsilon}}\right)^{4}+ns\left(\frac{Rr}{{\varepsilon}}\right)^{7}\right)}.$$ Compared to the complexity of the SDP-solver of [@lsw:faster], this has much worse dependence on $R$ and ${\varepsilon}$, but better dependence on $m$ and $n$. Using the Arora-Kale framework is thus preferable over standard SDP-solvers for the case where $Rr$ is small compared to $mn$, and a rough approximation to (say, small constant ${\varepsilon}$) is good enough. It should be noted that for many specific cases, Arora and Kale get significantly better upper bounds than by designing oracles that are specifically optimized for those cases.
Quantum SDP-solvers: the Brandão-Svore algorithm
------------------------------------------------
Given the speed-ups that *quantum* computers give over classical computers for various problems [@shor:factoring; @grover:search; @dhhm:graphproblemsj; @ambainis:edj; @hhl:lineq], it is natural to ask whether quantum computers can solve LPs and SDPs more efficiently as well. Very little was known about this, until recently Brandão and Svore discovered quantum algorithms that significantly outperform classical SDP-solvers in certain regimes. Because of the general importance of quickly solving LPs and SDPs, and the limited number of quantum algorithms that have been found so far, this is a very interesting development.
The key idea of the Brandão-Svore algorithm is to take the Arora-Kale approach and to replace two of its steps by more efficient quantum subroutines. First, given a vector $y^{(t-1)}$, it turns out one can use “Gibbs sampling” to prepare the new primal candidate $X^{(t)}\propto e^{-H^{(t-1)}}$ *as a $\log(n)$-qubit quantum state $\rho^{(t)}:=X^{(t)}/{\mbox{\rm Tr}}(X^{(t)})$* in much less time than needed to compute $X^{(t)}$ as an $n\times n$ matrix. Second, one can efficiently implement the oracle for $\mathcal{P}_{{\varepsilon}}(X^{(t)})$ based on a number of copies of $\rho^{(t)}$, using those copies to estimate ${\mbox{\rm Tr}}(A_j \rho^{(t)})$ and ${\mbox{\rm Tr}}(A_j X^{(t)})$ when needed (note that ${\mbox{\rm Tr}}(A\rho)$ is the expectation value of operator $A$ for the quantum state $\rho$). This is based on something called “Jaynes’s principle.” The resulting oracle is weaker than what is used classically, in the sense that it outputs a sample $j\sim y_j/{\left\lVerty\right\rVert}_1$ rather than the whole vector $y$. However, such sampling still suffices to make the algorithm work (it also means we can assume the vector $y^{(t)}$ to be quite sparse).
Using these ideas, Brandão and Svore obtain a quantum SDP-solver of complexity $$\widetilde{\mathcal O}(\sqrt{mn}s^2 R^{32} /\delta^{18}),$$ with *multiplicative* error $1\pm\delta$ for the special case where $b_j\geq 1$ for all $j\in[m]$, and ${\mbox{\rm OPT}}\geq 1$ (the latter assumption allows them to convert additive error ${\varepsilon}$ to multiplicative error $\delta$) . They describe a reduction to transform a general SDP of the form to this special case, but that reduction significantly worsens the dependence of the complexity on the parameters $R$, $r$, and $\delta$.
Note that compared to the runtime of our general instantiation of the original Arora-Kale framework, there are quadratic improvements in both $m$ and $n$, corresponding to the two quantum modifications made to Arora-Kale. However, the dependence on $R,r,s$ and $1/{\varepsilon}$ is much worse now than in . This quantum algorithm thus provides a speed-up only in regimes where $R,r,s,1/{\varepsilon}$ are fairly small compared to $mn$ (finding good examples of SDPs in such regimes is an open problem).
Our results
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In this paper we present two sets of results: improvements to the Brandão-Svore algorithm, and better lower bounds for the complexity of quantum LP-solvers (and hence for quantum SDP-solvers as well).
### Improved quantum SDP-solver
Our quantum SDP-solver, like the Brandão-Svore algorithm, works by quantizing some aspects of the Arora-Kale algorithm. However, the way we quantize is different and faster than theirs.
First, we give a more efficient procedure to estimate the quantities ${\mbox{\rm Tr}}(A_j\rho^{(t)})$ required by the oracle. Instead of first preparing some copies of Gibbs state $\rho^{(t)}\propto e^{-H^{(t-1)}}$ as a mixed state, we coherently prepare a purification of $\rho^{(t)}$, which can then be used to estimate ${\mbox{\rm Tr}}(A_j\rho^{(t)})$ more efficiently using amplitude-estimation techniques. Also, our purified Gibbs sampler has logarithmic dependence on the error, which is exponentially better than the Gibbs sampler of Poulin and Wocjan [@PoulinWocjan:GibbsEval] that Brandão and Svore invoke. Chowdhury and Somma [@ChowdhurySomma:GibbsHit] also gave a Gibbs sampler with logarithmic error-dependence, but assuming query access to the entries of $\sqrt{H}$ rather than $H$ itself.
Second, we have a different implementation of the oracle, without using Gibbs sampling or Jaynes’s principle (though, as mentioned above, we still use purified Gibbs sampling for approximating the ${\mbox{\rm Tr}}(A\rho)$ quantities). We observe that the vector $y^{(t)}$ can be made very sparse: *two* non-zero entries suffice.[^8] We then show how we can efficiently find such a 2-sparse vector (rather than merely sampling from it) using two applications of a new generalization of the well-known quantum minimum-finding algorithm of D[ü]{}rr and H[ø]{}yer , which is based on Grover search [@grover:search].
These modifications both simplify and speed up the quantum SDP-solver, resulting in complexity $$\widetilde{\mathcal O}(\sqrt{mn}s^2 (Rr/{\varepsilon})^{8}).$$ The dependence on $m$, $n$, and $s$ is the same as in Brandão-Svore, but our dependence on $R$, $r$, and $1/{\varepsilon}$ is substantially better. Note that each of the three parameters $R$, $r$, and $1/{\varepsilon}$ now occurs with the same 8th power in the complexity. This is no coincidence: as we show in Appendix \[app:reductions\], these three parameters can all be traded for one another, in the sense that we can massage the SDP to make each one of them small at the expense of making the others proportionally bigger. These trade-offs suggest we should actually think of $Rr/{\varepsilon}$ as *one* parameter of the primal-dual pair of SDPs, not three separate parameters. For the special case of LPs, we can improve the runtime to $$\widetilde{\mathcal O}(\sqrt{mn} (Rr/{\varepsilon})^{5}).$$
Like in Brandão-Svore, our quantum oracle produces very sparse vectors $y$, in our case even of sparsity 2. This means that after $T$ iterations, the final ${\varepsilon}$-optimal dual-feasible vector (which is a slightly tweaked version of the average of the $T$ $y$-vectors produced in the $T$ iterations) has only $\mathcal O(T)$ non-zero entries. Such sparse vectors have some advantages, for example they take much less space to store than arbitrary $y\in\mathbb{R}^m$. In fact, to get a sublinear running time in terms of $m$, this is necessary. However, this sparsity of the algorithm’s output also points to a weakness of these methods: if *every* ${\varepsilon}$-optimal dual-feasible vector $y$ has many non-zero entries, then the number of iterations needs to be large. For example, if every ${\varepsilon}$-optimal dual-feasible vector $y$ has $\Omega(m)$ non-zero entries, then these methods require $T=\Omega(m)$ iterations before they can reach an ${\varepsilon}$-optimal dual-feasible vector. Since $T = {{\mathcal O}}\left(\frac{R^2r^2}{{\varepsilon}^2} \ln(n)\right)$ this would imply that $\frac{Rr}{{\varepsilon}} = \Omega(\sqrt{m/\ln(n)})$, and hence many classical SDP-solvers would have a better complexity than our quantum SDP-solver. As we show in Section \[sec:downside\], this will actually be the case for families of SDPs that have a lot of symmetry.
### Tools that may be of more general interest
Along the way to our improved SDP-solver, we developed some new techniques that may be of independent interest. These are mostly tucked away in appendices, but here we will highlight two.
#### Implementing smooth functions of a given Hamiltonian.
In Appendix \[apx:LowWeight\] we describe a general technique to apply a function $f(H)$ of a sparse Hamiltonian $H$ to a given state ${|\phi\rangle}$. Roughly speaking, what this means is that we want a unitary circuit that maps ${|0\rangle}{|\phi\rangle}$ to ${|0\rangle}f(H){|\phi\rangle}+{|1\rangle}{|*\rangle}$. If need be, we can then combine this with amplitude amplification to boost the ${|0\rangle}f(H){|\phi\rangle}$ part of the state. If the function $f:\mathbb{R}\to\mathbb{C}$ can be approximated well by a low-degree Fourier series, then our preparation will be efficient in the sense of using few queries to $H$ and few other gates. The novelty of our approach is that we construct a good Fourier series from the polynomial that approximates $f$ (for example a truncated Taylor series for $f$). Our Theorem \[thm:Taylor\] can be easily applied to various smooth functions without using involved integral approximations, unlike previous works building on similar techniques. Our most general result Corollary \[cor:patched\] only requires that the function $f$ can be nicely approximated locally around each possible eigenvalues of $H$, improving on Theorem \[thm:Taylor\].
In this paper we mostly care about the function $f(x)=e^{-x}$, which is what we want for generating a purification of the Gibbs state corresponding to $H$; and the function $f(x)=\sqrt{x}$, which is what we use for estimating quantities like ${\mbox{\rm Tr}}(A\rho)$. However, our techniques apply much more generally than these two functions. For example, they also simplify the analysis of the improved linear-systems solver of Childs et al. [@ChildsKothariSomma:lse], where the relevant function is $f(x)=1/x$. As in their work, the Linear Combination of Unitaries technique of Childs et al. [@ChildsWiebeLCU; @BerryChilds:hamsim; @BerryChilds:hamsimFOCS] is a crucial tool for us.
#### A generalized minimum-finding algorithm.
D[ü]{}rr and H[ø]{}yer showed how to find the minimal value of a function $f:[N]\to\mathbb{R}$ using ${{\mathcal O}}(\sqrt{N})$ queries to $f$, by repeatedly using Grover search to find smaller and smaller elements of the range of $f$. In Appendix \[app:genMinFind\] we describe a more general minimum-finding procedure. Suppose we have a unitary $U$ which prepares a quantum state $U{|0\rangle}=\sum_{k=1}^{N}{|\psi_k\rangle}{|x_k\rangle}$, where the ${|\psi_k\rangle}$ are unnormalized states. Our procedure can find the minimum value $x_{k^*}$ among the $x_k$’s that have support in the second register, using roughly ${{\mathcal O}}(1/{\left\lVert\psi_{k^*}\right\rVert})$ applications of $U$ and $U^{-1}$. Also, upon finding the minimal value $k^*$ the procedure actually outputs the normalized state proportional to ${|\psi_{k^*}\rangle}{|x_{k^*}\rangle}$. This immediately gives the D[ü]{}rr-H[ø]{}yer result as a special case, if we take $U$ to produce $U{|0\rangle}=\frac{1}{\sqrt{N}}\sum_{k=1}^{N}{|k\rangle}{|f(k)\rangle}$ using one query to $f$. Unlike D[ü]{}rr-H[ø]{}yer, we need not assume direct query access to the individual values $f(k)$.
More interestingly for us, for a given $n$-dimensional Hamiltonian $H$, if we combine our minimum-finder with phase estimation using unitary $U=e^{iH}$ on one half of a maximally entangled state, then we obtain an algorithm for estimating the smallest eigenvalue of $H$ (and preparing its ground state) using roughly ${{\mathcal O}}(\sqrt{n})$ applications of phase estimation with $U$. A similar result on approximating the smallest eigenvalue of a Hamiltonian was already shown by Poulin and Wocjan [@PoulinWocjan:GroundMany], but we improve on their analysis to be able to apply it as a subroutine in our procedure to estimate ${\mbox{\rm Tr}}(A_j\rho)$.
### Lower bounds
What about lower bounds for quantum SDP-solvers? Brandão and Svore already proved that a quantum SDP-solver has to make $\Omega(\sqrt{n}+\sqrt{m})$ queries to the input matrices, for some SDPs. Their lower bound is for a family of SDPs where $s,R,r,1/{\varepsilon}$ are all constant, and is by reduction from a search problem.
In this paper we prove lower bounds that are quantitatively stronger in $m$ and $n$, but for SDPs with non-constant $R$ and $r$. The key idea is to consider a Boolean function $F$ on $N=abc$ input bits that is the composition of an $a$-bit majority function with a $b$-bit OR function with a $c$-bit majority function. The known quantum query complexities of majority and OR, combined with composition properties of the adversary lower bound, imply that every quantum algorithm that computes this function requires $\Omega(a\sqrt{b}c)$ queries. We define a family of LPs, with constant $1/{\varepsilon}$ but non-constant $r$ and $R$ (we could massage this to make $R$ or $r$ constant using the results of Appendix \[app:reductions\], but not $Rr/{\varepsilon}$), such that constant-error approximation of computes $F$. Choosing $a$, $b$, and $c$ appropriately, this implies a lower bound of $$\Omega\left(\sqrt{\max\{n,m\}} \left( \min\{n,m\} \right)^{3/2} \right)$$ queries to the entries of the input matrices for quantum LP-solvers. Since LPs are SDPs with sparsity $s=1$, we get the same lower bound for quantum SDP-solvers. If $m$ and $n$ are of the same order, this lower bound is $\Omega(mn)$, the same scaling with $mn$ as the classical general instantiation of Arora-Kale . In particular, this shows that we cannot have an $O(\sqrt{mn})$ upper bound without simultaneously having polynomial dependence on $Rr/{\varepsilon}$. The construction of our lower bound implies that for the case $m\approx n$, this polynomial dependence has to be at least $(Rr/{\varepsilon})^{1/4}$.
#### Subsequent work.
Following the first version of our paper, improvements in the running time were obtained in [@brandao2017QSDPSpeedupsLearning; @apeldoorn2018ImprovedQSDPSolving], the latter providing a runtime of ${\widetilde{\mathcal O}\left((\sqrt{m}+\sqrt{n} \frac{Rr}{\epsilon}) s \left(\frac{Rr}{\epsilon}\right)^4\right)}$. More recently, a quantum interior point method for solving SDPs and LPs was obtained by Kerenidis and Prakash [@kerenidis2018QIntPoint]. It is hard to compare the latter algorithm to the other SDP-solvers for two reasons. First, the output of their algorithm consists only of almost-feasible solutions to the primal and dual (their algorithm has a polynomial dependence on the distance to feasibility). It is therefore not clear what their output means for the optimal value of the SDPs. Secondly, the runtime of their algorithm depends polynomially on the condition number of the matrices that the interior point method encounters, and no explicit bounds for these condition numbers are given.
Our results on implementing smooth functions of a given Hamiltonian have been extended to more general input models (block-encodings) in [@gilyen2018QSingValTransf]. This recent paper builds on some of our techniques, but achieves slightly improved complexities by directly implementing the transformations without using Hamiltonian simulation as a subroutine.
Very recently van Apeldoorn et al. [@apeldoorn2018ConvexOptUsingQuantumOracles] and Chakrabarti et al. [@chakrabarti2018QuantumConvexOpt] developed quantum algorithms for general black-box convex optimization, where one optimizes over a general convex set $K$, and the access to $K$ is via membership and/or separation oracles. Since we work in a model where we are given access directly to the constraints defining the problem, our results are incomparable to theirs.
#### Organization.
The paper is organized as follows. In Section \[sec:upperbounds\] we start with a description of the Arora-Kale framework for SDP-solvers, and then we describe how to quantize different aspects of it to obtain a quantum SDP-solver with better dependence on $R$, $r$, and $1/{\varepsilon}$ (or rather, on $Rr/{\varepsilon}$) than Brandão and Svore got. In Section \[sec:downside\] we describe the limitations of primal-dual SDP-solvers using general oracles (not optimized for specific SDPs) that produce sparse dual solutions $y$: if good solutions are dense, this puts a lower bound on the number of iterations needed. In Section \[sec:lowerbounds\] we give our lower bounds. A number of the proofs are relegated to the appendices: how to classically approximate ${\mbox{\rm Tr}}(A_j\rho)$ (Appendix \[app:trace\]), how to efficiently implement smooth functions of Hamiltonians on a quantum computer (Appendix \[apx:LowWeight\]), our generalized method for minimum-finding (Appendix \[app:genMinFind\]), upper and lower bounds on how efficiently we can query entries of sums of sparse matrices (Appendix \[app:sparsematrixsum\]), how to trade off $R$, $r$, and $1/{\varepsilon}$ against each other (Appendix \[app:reductions\]), and the composition property of the adversary method that we need for our lower bounds (Appendix \[app:adversarycomposition\]).
An improved quantum SDP-solver {#sec:upperbounds}
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Here we describe our quantum SDP-solver. In Section \[sec:classicalAK\] we describe the framework designed by Arora and Kale for solving semidefinite programs. As in the recent work by Brandão and Svore, we use this framework to design an efficient quantum algorithm for solving SDPs. In particular, we show that the key subroutine needed in the Arora-Kale framework can be implemented efficiently on a quantum computer. Our implementation uses different techniques than the quantum algorithm of Brandão and Svore, allowing us to obtain a faster algorithm. The techniques required for this subroutine are developed in Sections \[sec:trCalc\] and \[sec:oracle\]. In Section \[sec:runtime\] we put everything together to prove the main theorem of this section (the notation is explained below):
[theorem]{}[upperbound]{} \[thm:upperbound\] Instantiating Meta-Algorithm \[alg:AKSDP\] using the trace calculation algorithm from Section \[sec:trCalc\] and the oracle from Section \[sec:oracle\] (with width-bound $w:=r+1$), and using this to do a binary search for ${\mbox{\rm OPT}}\in[-R,R]$ (using different guesses $\alpha$ for ${\mbox{\rm OPT}}$), gives a quantum algorithm for solving SDPs of the form , which (with high probability) produces a feasible solution $y$ to the dual program which is optimal up to an additive error ${\varepsilon}$, and uses $${\widetilde{\mathcal O}\left(\sqrt{nm} s^2\left( \frac{Rr}{{\varepsilon}}\right)^{\!\!8} \right)}$$ queries to the input matrices and the same order of other gates.
#### Notation/Assumptions.
We use $\log$ to denote the logarithm in base $2$. We denote the all-zero matrix and vector by $0$. Throughout we assume each element of the input matrices can be represented by a bitstring of size $\text{poly}(\log n,\log m)$. We use $s$ as the sparsity of the input matrices, that is, the maximum number of non-zero entries in a row (or column) of any of the matrices $C, A_1,\ldots, A_m$ is $s$. Recall that for normalization purposes we assume ${\left\lVertA_1\right\rVert}, \ldots, {\left\lVertA_m\right\rVert}, {\left\lVertC\right\rVert} \leq 1$. We furthermore assume that $A_1 = I$ and $b_1 = R$, that is, the trace of primal-feasible solutions is bounded by $R$ (and hence also the trace of primal-optimal solutions is bounded by $R$). The analogous quantity for the dual SDP , an upper bound on $\sum_{j=1}^m y_j$ for an optimal dual solution $y$, will be denoted by $r$. However, we do not add the constraint $\sum_{j=1}^m y_j \leq r$ to the dual. We will assume $r\geq 1$. For $r$ to be well-defined we have to make the explicit assumption that the optimal solution in the dual is attained. In Section \[sec:downside\] it will be necessary to work with the best possible upper bounds: we let $R^*$ be the smallest trace of an optimal solution to SDP , and we let $r^*$ be the smallest $\ell_1$-norm of an optimal solution to the dual. These quantities are well-defined; indeed, both the primal and dual optimum are attained: the dual optimum is attained by assumption, and due to the assumption $A_1 = I$, the dual SDP is strictly feasible, which means that the optimum in is attained.
Unless specified otherwise, we always consider *additive* error. In particular, an ${\varepsilon}$-optimal solution to an SDP will be a feasible solution whose objective value is within additive error ${\varepsilon}$ of the optimum.
#### Input oracles:
We assume sparse black-box access to the elements of the matrices $C, A_1,\ldots, A_m$ defined in the following way: for input $(j, k,\ell) \in (\{0\}\cup[m]) \times [n] \times [s]$ we can query the location and value of the $\ell$th non-zero entry in the $k$th row of the matrix $A_j$ (where $j=0$ would indicate the $C$ matrix).
Specifically in the quantum case, as described in [@BerryChilds:hamsimFOCS], we assume access to an oracle $O_I$ which serves the purpose of sparse access. $O_I$ calculates the $\text{index}_{A_j}: [n] \times [s] \to [n]$ function, which for input $(k,\ell)$ gives the column index of the $\ell$th non-zero element in the $k$th row of $A_j$. We assume this oracle computes the index “in place": $$O_I{|j,k,\ell\rangle} = {|j,k,\text{index}_{A_j}(k,\ell)\rangle}.
\label{eq:oracleind}$$ (In the degenerate case where the $k$th row has fewer than $\ell$ non-zero entries, $\text{index}_{A_j}(k,\ell)$ is defined to be $\ell$ together with some special symbol.) We also assume we can apply the inverse of $O_I$.
We also need another oracle $O_M$, returning a bitstring representation of $(A_j)_{ki}$ for any $j \in \{0\}\cup[m]$ and $ k,i \in [n]$: $$O_M{|j,k,i,z\rangle} = {|j,k,i,z \oplus{(A_j)_{ki}}\rangle}.
\label{eq:oraclemat}$$ The slightly unusual “in place” definition of oracle $O_I$ is not too demanding. In particular, if instead we had an oracle that computed the non-zero entries of a row in order, then we could implement both $O_I$ and its inverse using $\log(s)$ queries (we can compute $\ell$ from $k$ and $\text{index}_{A_j}(k,\ell)$ using binary search) [@BerryChilds:hamsimFOCS].
#### Computational model:
As our computational model, we assume a slight relaxation of the usual quantum circuit model: a classical control system that can run quantum subroutines. We limit the classical control system so that its number of operations is at most a polylogarithmic factor bigger than the gate complexity of the quantum subroutines, i.e., if the quantum subroutines use $C$ gates, then the classical control system may not use more than ${{\mathcal O}}(C\,\text{polylog}(C))$ operations.
When we talk about gate complexity, we count the number of two-qubit quantum gates needed for implementation of the quantum subroutines. Additionally, we assume for simplicity that there exists a unit-cost QRAM gate that allows us to store and retrieve qubits in a memory, by means of a swap of two registers indexed by another register: $$QRAM : {|i,x,r_1,\dots,r_K\rangle} \mapsto {|i,r_i,r_1,\dots,r_{i-1},x,r_{i+1},\ldots,r_K\rangle},$$ where the registers $r_1,\dots,r_K$ are only accessible through this gate. The QRAM gate can be seen as a quantum analogue of pointers in classical computing. The only place where we need QRAM is in Appendix \[app:sparsematrixsum\], for a data structure that allows efficient access to the non-zero entries of a sum of sparse matrices; for the special case of LP-solving it is not needed.
The Arora-Kale framework for solving SDPs {#sec:classicalAK}
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In this section we give a short introduction to the Arora-Kale framework for solving semidefinite programs. We refer to for a more detailed description and omitted proofs.
The key building block is the Matrix Multiplicative Weights (MMW) algorithm introduced by Arora and Kale in . The MMW algorithm can be seen as a strategy for you in a game between you and an adversary. We first introduce the game. There is a number of rounds $T$. In each round you present a density matrix $\rho$ to an adversary, the adversary replies with a loss matrix $M$ satisfying $-I \preceq M \preceq I$. After each round you have to pay ${{\mbox{\rm Tr}}\left(M \rho\right)}$. Your objective is to pay as little as possible. The MMW algorithm is a strategy for you that allows you to lose not too much, in a sense that is made precise below. In Algorithm \[alg:MMW\] we state the MMW algorithm, the following theorem shows the key property of the output of the algorithm.
Input
: Parameter $\eta \leq 1$, number of rounds $T$.
Rules
: In each round player $1$ (you) presents a density matrix $\rho$, player $2$ (the adversary) replies with a matrix $M$ satisfying $-I \preceq M \preceq I$.
Output
: A sequence of symmetric $n \times n$ matrices $M^{(1)},\ldots, M^{(T)}$ satisfying $-I \preceq M^{(t)} \preceq I$, for $t \in [T]$ and a sequence of $n \times n$ psd matrices $\rho^{(1)},\ldots, \rho^{(T)}$ satisfying ${{\mbox{\rm Tr}}\left(\rho^{(t)}\right)}=1$ for $t \in [T]$.
Strategy of player $1$:
:
Take $\rho^{(1)} := I/n$ In round $t$:
1. Show the density matrix $\rho^{(t)}$ to the adversary.
2. Obtain the loss matrix $M^{(t)}$ from the adversary.
3. Update the density matrix as follows: $$\rho^{(t+1)}:= \left.\exp\left(- \eta \sum_{\tau=1}^t M^{(\tau)}\right)\right/{{\mbox{\rm Tr}}\left(\exp\left(- \eta \sum_{\tau=1}^t M^{(\tau)}\right)\right)}$$
For every adversary, the sequence $\rho^{(1)}, \ldots, \rho^{(T)}$ of density matrices constructed using the Matrix Multiplicative Weights Algorithm satisfies $$\sum_{t=1}^T {{\mbox{\rm Tr}}\left(M^{(t)} \rho^{(t)}\right)} \leq \lambda_{\min}\left(\sum_{t=1}^T M^{(t)}\right) + \eta \sum_{t=1}^T {{\mbox{\rm Tr}}\left( (M^{(t)})^2 \rho^{(t)}\right)} + \frac{\ln(n)}{\eta}.$$
Arora and Kale use the MMW algorithm to construct an SDP-solver. For that, they construct an adversary who promises to satisfy an additional condition: in each round $t$, the adversary returns a matrix $M^{(t)}$ whose trace inner product with the density matrix $\rho^{(t)}$ is non-negative. The above theorem shows that then, after $T$ rounds, the average of the adversary’s responses satisfies the stronger condition that its smallest eigenvalue is not too negative: $\lambda_{\min}\left(\frac{1}{T} \sum_{t=1}^T M^{(t)}\right) \geq - \eta - \frac{\ln(n)}{\eta T}$. More explicitly, the MMW algorithm is used to build a vector $y \geq 0$ such that $$\frac{1}{T} \sum_{t=1}^T M^{(t)} \propto \sum_{j=1}^m y_j A_j -C$$ and $b^T y \leq \alpha$. That is, the smallest eigenvalue of the matrix $\sum_{j=1}^m y_j A_j -C$ is only slightly below zero and $y$’s objective value is at most $\alpha$. Since $A_1=I$, increasing the first coordinate of $y$ makes the smallest eigenvalue of $\sum_j y_j A_j -C$ bigger, so that this matrix becomes psd and hence dual-feasible. By the above we know how much the minimum eigenvalue has to be shifted, and with the right choice of parameters it can be shown that this gives a dual-feasible vector $\overline{y}$ that satisfies $b^T \overline{y} \leq \alpha + {\varepsilon}$. In order to present the algorithm formally, we require some definitions.
Given a candidate solution $X \succeq 0$ for the primal problem and a parameter ${\varepsilon}\geq 0$, define the polytope $$\begin{aligned}
\mathcal{P}_{{\varepsilon}}(X) := \{ y \in \mathbb R^m:\ &b^T y \leq \alpha, \\
&{{\mbox{\rm Tr}}\left(\Big(\sum_{j=1}^m y_j A_j - C\Big) X\right)} \geq -{\varepsilon},\\
& y \geq 0 \}.\end{aligned}$$ One can verify the following:
\[lem:xfeas\] If for a given candidate solution $X \succeq 0$ the polytope $\mathcal P_0(X)$ is empty, then a scaled version of $X$ is primal-feasible and of objective value at least $\alpha$.
The Arora-Kale framework for solving SDPs uses the MMW algorithm where the role of the adversary is taken by an ${\varepsilon}$-approximate oracle:
Input
: An $n \times n$ psd matrix $X$, a parameter ${\varepsilon}$, and the input matrices and reals of .
Output
: Either the $_{{\varepsilon}}$ returns a vector $y$ from the polytope $\mathcal P_{\varepsilon}(X)$ or it outputs “fail”. It may only output fail if $\mathcal P_0(X) = \emptyset$.
As we will see later, the runtime of the Arora-Kale framework depends on a property of the oracle called the *width*:
The *width* of $_{{\varepsilon}}$ for an SDP is the smallest $w^* \geq 0$ such that for every primal candidate $X \succeq 0$, the vector $y$ returned by $_{{\varepsilon}}$ satisfies ${\left\lVert\sum_{j=1}^m y_j A_j- C\right\rVert} \leq w^*$.
In practice, the width of an oracle is not always known. However, it suffices to work with an upper bound $w \geq w^*$: as we can see in Meta-Algorithm \[alg:AKSDP\], the purpose of the width is to rescale the matrix $M^{(t)}$ in such a way that it forms a valid response for the adversary in the MMW algorithm.
Input
: The input matrices and reals of SDP and trace bound $R$. The current guess $\alpha$ of the optimal value of the dual . An additive error tolerance ${\varepsilon}>0$. An $\frac{{\varepsilon}}{3}$-approximate oracle $_{{\varepsilon}/3}$ as in Algorithm \[alg:Oracle\] with width-bound $w$.
Output
: Either “Lower” and a vector $\overline{y} \in \mathbb R^{m}_{+}$ feasible for with $b^T \overline{y} \leq \alpha+{\varepsilon}$\
or “Higher” and a symmetric $n \times n$ matrix $X$ that, when scaled suitably, is primal-feasible with objective value at least $\alpha$.
$T := \left\lceil \frac{9 w^2 R^2 \ln(n)}{{\varepsilon}^2}\right\rceil$. $\eta := \sqrt{\frac{\ln (n)}{T}}$. $\rho^{(1)} := I/n$ Run $_{{\varepsilon}/3}$ with $X^{(t)} = R\rho^{(t)}$. “Higher” and a description of $X^{(t)}$. Let $y^{(t)}$ be the vector generated by $_{{\varepsilon}/3}$. Set $M^{(t)} = \frac{1}{w} \left( \sum_{j=1}^m y_j^{(t)} A_j - C\right)$. Define $H^{(t)} = \sum_{\tau=1}^t M^{(\tau)}$. Update the state matrix as follows: $\rho^{(t+1)}:= \exp\left(- \eta H^{(t)}\right)/{{\mbox{\rm Tr}}\left(\exp\left(- \eta H^{(t)}\right)\right)}$. If $_{{\varepsilon}/3}$ does not output “fail” in any of the $T$ rounds, then output the dual solution $\overline{y} = \frac{{\varepsilon}}{R} e_1 + \frac{1}{T} \sum_{t=1}^T y^{(t)}$ where $e_1 = (1,0,\ldots, 0) \in \mathbb R^m$.
The following theorem shows the correctness of the Arora-Kale primal-dual meta-algorithm for solving SDPs, stated in Meta-Algorithm \[alg:AKSDP\]:
Given an SDP of the form with input matrices $A_1 = I, A_2, \ldots, A_m$ and $C$ having operator norm at most $1$, and input reals $b_1=R, b_2, \ldots, b_m$. Assume Meta-Algorithm \[alg:AKSDP\] does not output “fail” in any of the rounds, then the returned vector $\overline{y}$ is feasible for the dual with objective value at most $\alpha+{\varepsilon}$. If $_{{\varepsilon}/3}$ outputs “fail” in the $t$-th round then a suitably scaled version of $X^{(t)}$ is primal-feasible with objective value at least $\alpha$.
The SDP-solver uses $T = \left\lceil \frac{9 w^2 R^2 \ln(n)}{{\varepsilon}^2}\right\rceil$ iterations. In each iteration several steps have to be taken. The most expensive two steps are computing the matrix exponential of the matrix $- \eta H^{(t)}$ and the application of the oracle. Note that the only purpose of computing the matrix exponential is to allow the oracle to compute the values ${{\mbox{\rm Tr}}\left(A_j X\right)}$ for all $j$ and ${{\mbox{\rm Tr}}\left(CX\right)}$, since the polytope depends on $X$ only through those values. To obtain faster algorithms it is important to note, as was done already by Arora and Kale, that the primal-dual algorithm also works if we provide a (more accurate) oracle with approximations of ${{\mbox{\rm Tr}}\left(A_j X\right)}$. Let $a_j:={{\mbox{\rm Tr}}\left(A_j\rho\right)} = {{\mbox{\rm Tr}}\left(A_jX\right)}/{{\mbox{\rm Tr}}\left(X\right)}$ and $c:={{\mbox{\rm Tr}}\left(C\rho\right)} = {{\mbox{\rm Tr}}\left(CX\right)}/{{\mbox{\rm Tr}}\left(X\right)}$. Then, given a list of reals $\tilde{a}_1, \ldots, \tilde{a}_m, \tilde{c}$ and a parameter $\theta \geq 0$, such that $|\tilde{a}_j - a_j| \leq \theta$ for all $j$, and $|\tilde{c}- c| \leq \theta$, we define the polytope $$\begin{aligned}
{\tilde{\mathcal{P}}}(\tilde{a}_1, \ldots, \tilde{a}_m,\tilde{c}-(r+1)\theta) := \{ y \in \mathbb R^m:\ & b^T y \leq \alpha,\\&\sum_{j=1}^m y_j \leq r, \\
&\sum_{j=1}^m \tilde{a}_j y_j \geq \tilde{c} - (r+1)\theta\\&y \geq 0\}.\end{aligned}$$ For convenience we will denote $\tilde{a} = (\tilde{a}_1, \ldots, \tilde{a}_m)$ and ${c^{\prime}}:= \tilde{c}- (r+1) \theta$. Notice that ${\tilde{\mathcal{P}}}$ also contains a new type of constraint: $\sum_j y_j \leq r$. Recall that $r$ is defined as a positive real such that there exists an optimal solution $y$ to SDP with ${\left\lVerty\right\rVert}_1 \leq r$. Hence, using that $\mathcal P_0(X)$ is a *relaxation* of the feasible region of the dual (with bound $\alpha$ on the objective value), we may restrict our oracle to return only such $y$: $$\mathcal P_0(X) \neq \emptyset \Rightarrow \mathcal P_0(X) \cap \{y \in \mathbb R^m: \sum_{j=1}^m y_j \leq r\} \neq \emptyset.$$ The benefit of this restriction is that an oracle that always returns a vector with bounded $\ell_1$-norm automatically has a width $w^* \leq r+1$, due to the assumptions on the norms of the input matrices. The downside of this restriction is that the analogue of Lemma \[lem:xfeas\] does not hold for $\mathcal P_0(X) \cap \{y \in \mathbb R^m: \sum_{j} y_j \leq r\}$.[^9]
The following shows that an oracle that always returns a vector $y \in {\tilde{\mathcal{P}}}(\tilde{a},c')$ if one exists, is a $4Rr\theta$-approximate oracle as defined in Algorithm \[alg:Oracle\].
\[lem:approxP\] Let $\tilde{a}_1, \ldots, \tilde{a}_m$ and $\tilde{c}$ be $\theta$-approximations of ${{\mbox{\rm Tr}}\left(A_1 \rho\right)}, \ldots, {{\mbox{\rm Tr}}\left(A_m \rho\right)}$ and ${{\mbox{\rm Tr}}\left(C\rho\right)}$, respectively, where $X = R \rho$. Then the following holds: $$\mathcal P_0(X) \cap \{y \in \mathbb R^m: \sum_{j=1}^m y_j \leq r\} \subseteq {\tilde{\mathcal{P}}}(\tilde{a},c') \subseteq \mathcal P_{4Rr\theta}(X).$$
First, suppose $y \in \mathcal P_{0}(X) \cap \{y \in \mathbb R^m: \sum_{j} y_j \leq r\}$. We now have $$\sum_{j=1}^m \tilde{a}_j y_j - c \geq - \sum_{j=1}^m |\tilde{a}_j - {{\mbox{\rm Tr}}\left(A_j \rho\right)}| y_j - |\tilde{c} -{{\mbox{\rm Tr}}\left(C \rho\right)}| \geq - \theta {\left\lVerty\right\rVert}_1 - \theta \geq - (r+1) \theta$$ which shows that $y \in {\tilde{\mathcal{P}}}(\tilde{a},c')$.
Next, suppose $y \in {\tilde{\mathcal{P}}}(\tilde{a},{c^{\prime}})$. We show that $y \in \mathcal P_{4Rr \theta}(X)$. Indeed, since $|{{\mbox{\rm Tr}}\left(A_j\rho\right)} - \tilde{a}_j| \leq \theta$ we have $${{\mbox{\rm Tr}}\left(\!\!\left(\sum_{j=1}^m y_j A_j -C\right)\rho\!\right)} \geq \left(\sum_{j=1}^m \tilde{a}_jy_j + \tilde{c} \right) -(r+1)\theta \geq -(2+r + {\left\lVerty\right\rVert}_1) \theta \geq\! -4r \theta$$ where the last inequality used our assumptions $r\geq 1$ and ${\left\lVerty\right\rVert}_1\leq r$. Hence $${{\mbox{\rm Tr}}\left(\left(\sum_{j=1}^m y_j A_j -C\right)X\right)} \geq -4r{{\mbox{\rm Tr}}\left(X\right)} \theta = -4Rr\theta.$$ For the latter inequality we use ${{\mbox{\rm Tr}}\left(X\right)} = R$.
We have now seen the Arora-Kale framework for solving SDPs. To obtain a quantum SDP-solver it remains to provide a quantum oracle subroutine. By the above discussion it suffices to set $\theta = {\varepsilon}/(12Rr)$ and to use an oracle that is based on $\theta$-approximations of ${{\mbox{\rm Tr}}\left(A\rho\right)}$ (for $A \in \{A_1, \ldots, A_m, C\}$), since with that choice of $\theta$ we have $\mathcal P_{4Rr \theta}(X)=\mathcal P_{{\varepsilon}/3}(X)$. In the section below we first give a quantum algorithm for approximating ${{\mbox{\rm Tr}}\left(A\rho\right)}$ efficiently (see also Appendix \[app:trace\] for a classical algorithm). Then, in Section \[sec:oracle\], we provide an oracle using those estimates. The oracle will be based on a simple geometric idea and can be implemented both on a quantum computer and on a classical computer (of course, resulting in different runtimes). In Section \[sec:runtime\] we conclude with an overview of the runtime of our quantum SDP-solver. We want to stress that our solver is meant to work for any SDP. In particular, our oracle does not use the structure of a specific SDP. As we will show in Section \[sec:downside\], any oracle that works for all SDPs necessarily has a large width-bound. To obtain quantum speedups for a *specific* class of SDPs it will be necessary to develop oracles tuned to that problem, we view this as an important direction for future work. Recall from the introduction that Arora and Kale also obtain fast classical algorithms for problems such as MAXCUT by developing specialized oracles.
Approximating the expectation value traces using a quantum algorithm {#sec:trCalc}
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In this section we give an efficient quantum algorithm to approximate quantities of the form ${{\mbox{\rm Tr}}\left(A\rho\right)}$. We are going to work with Hermitian matrices $A,H\in\mathbb{C}^{n\times n}$, such that $\rho$ is the Gibbs state $e^{-H}/{{\mbox{\rm Tr}}\left(e^{-H}\right)}$. Note the analogy with quantum physics: in physics terminology ${{\mbox{\rm Tr}}\left(A\rho\right)}$ is simply called the “expectation value" of $A$ for a quantum system in a thermal state corresponding to $H$.
The general approach is to separately estimate ${{\mbox{\rm Tr}}\left(A e^{-H}\right)}$ and ${{\mbox{\rm Tr}}\left(e^{-H}\right)}$, and then to use the ratio of these estimates as an approximation of ${{\mbox{\rm Tr}}\left(A\rho\right)}={{\mbox{\rm Tr}}\left(A e^{-H}\right)}/{{\mbox{\rm Tr}}\left(e^{-H}\right)}$. Both estimations are done using state preparation to prepare a pure state with a flag, such that the probability that the flag is 0 is proportional to the quantity we want to estimate, and then to use amplitude estimation to estimate that probability. Below in Section \[ssec:generalapproach\] we first describe the general approach. In Section \[sec:lptrace\] we then instantiate this for the special case where all matrices are diagonal, which is the relevant case for LP-solving. In Section \[sec:estTrArhogeneral\] we handle the general case of arbitrary matrices (needed for SDP-solving); the state-preparation part will be substantially more involved there, because in the general case we need not know the diagonalizing bases for $A$ and $H$, and $A$ and $H$ may not be simultaneously diagonalizable.
### General approach {#ssec:generalapproach}
To start, consider the following lemma about the multiplicative approximation error of a ratio of two real numbers that are given by multiplicative approximations:
\[lemma:trTogether\] Let $0\leq \theta \leq 1$ and let $\alpha, \tilde{\alpha}, Z, \tilde{Z}$ be positive real numbers such that $|\alpha - \tilde{\alpha}| \leq \alpha\theta / 3$ and $|Z - \tilde{Z}| \leq Z\theta / 3$. Then $$\left|\frac{\alpha}{Z}-\frac{\tilde{\alpha}}{\tilde{Z}}\right| \leq \theta\frac{\alpha}{Z}$$
Observe $|\tilde{Z}|\geq |Z|2/3$, thus $$\begin{aligned}
\left|\frac{\alpha}{Z}-\frac{\tilde{\alpha}}{\tilde{Z}}\right|
&= \left|\frac{\alpha\tilde{Z}-\tilde{\alpha}Z}{Z\tilde{Z}}\right|
= \left|\frac{\alpha\tilde{Z}-\alpha Z+\alpha Z-\tilde{\alpha}Z}{Z\tilde{Z}}\right|\\
&\leq \left|\frac{\alpha\tilde{Z}-\alpha Z}{Z\tilde{Z}}\right|+\left|\frac{\alpha Z-\tilde{\alpha}Z}{Z\tilde{Z}}\right|
\leq\frac{\alpha}{Z} \left|\frac{\tilde{Z}-Z}{\tilde{Z}}\right|+\left|\frac{\theta\alpha }{3\tilde{Z}}\right|\\
&\leq \frac{3}{2}\left(\frac{\alpha}{Z}\left|\frac{\tilde{Z}-Z}{Z}\right|+\frac{\theta\alpha }{3Z}\right)
\leq \frac{3}{2}\left(\frac{2\theta}{3}\frac{\alpha}{Z}\right)=\theta\frac{\alpha}{Z}.
\end{aligned}$$ -5mm
\[col:split\] Let $A$ be such that ${\left\lVertA\right\rVert}\leq 1$. A multiplicative $\theta/9$-approximation of both ${{\mbox{\rm Tr}}\left(\frac{I+A/2}{4}e^{-H}\right)}$ and ${{\mbox{\rm Tr}}\left(\frac{I}{4}e^{-H}\right)}$ suffices to get an additive $\theta$-approximation of $\frac{{{\mbox{\rm Tr}}\left(Ae^{-H}\right)}}{{{\mbox{\rm Tr}}\left(e^{-H}\right)}}$.
According to Lemma \[lemma:trTogether\] by dividing the two multiplicative approximations we get $$\frac{\theta}{3}\frac{{{\mbox{\rm Tr}}\left(\frac{I+A/2}{4}e^{-H}\right)}}{{{\mbox{\rm Tr}}\left(\frac{I}{4}e^{-H}\right)}}
= \frac{\theta}{3}\left(1+\frac{{{\mbox{\rm Tr}}\left(\frac{A}{2}e^{-H}\right)}}{{{\mbox{\rm Tr}}\left(e^{-H}\right)}}\right)
\leq \frac{\theta}{3}\left(1+\frac{{\left\lVertA\right\rVert}}{2}\right)
\leq \theta/2,$$ i.e., an additive $\theta/2$-approximation of $$1+\frac{{{\mbox{\rm Tr}}\left(\frac{A}{2}e^{-H}\right)}}{{{\mbox{\rm Tr}}\left(e^{-H}\right)}},$$ which yields an additive $\theta$-approximation to ${{\mbox{\rm Tr}}\left(A\rho\right)}$.
It thus suffices to approximate both quantities from the corollary separately. Notice that both are of the form ${{\mbox{\rm Tr}}\left(\frac{I+A/2}{4}e^{-H}\right)}$, the first with the actual $A$, the second with $A=0$. Furthermore, a multiplicative $\theta/9$-approximation to both can be achieved by approximating both up to an additive error $\theta {{\mbox{\rm Tr}}\left(e^{-H}\right)} / 72$, since ${{\mbox{\rm Tr}}\left(\frac{I}{8} e^{-H}\right)}\leq {{\mbox{\rm Tr}}\left(\frac{I+A/2}{4}e^{-H}\right)}$.
For now, let us assume we can construct a unitary $U_{A,H}$ such that if we apply it to the state ${|0\ldots 0\rangle}$ then we get a probability $\frac{{{\mbox{\rm Tr}}\left((I+A/2)e^{-H}\right)}}{4n}$ of outcome 0 when measuring the first qubit. That is: $${\left\lVert({\langle0|}\otimes I)U_{A,H}{|0\ldots 0\rangle}\right\rVert}^2 = \frac{{{\mbox{\rm Tr}}\left((I+A/2)e^{-H}\right)}}{4n}.$$ In practice we will not be able to construct such a $U_{A,H}$ exactly, instead we will construct a $\tilde{U}_{A,H}$ that yields a sufficiently close approximation of the correct probability. Since we have access to such a unitary, the following lemma allows us to use amplitude estimation to estimate the probability and hence ${{\mbox{\rm Tr}}\left(\frac{I+A/2}{4}e^{-H}\right)}$ up to the desired error.
\[lem:ampest\] Suppose we have a unitary $U$ acting on $q$ qubits such that $U{|0\ldots 0\rangle}={|0\rangle}{|\psi\rangle}+{|\Phi\rangle}$ with $({\langle0|}\otimes I){|\Phi\rangle}=0$ and ${\left\lVert\psi\right\rVert}^2 =p\geq p_{\min}$ for some known bound $p_{\min}$. Let $\mu \in(0,1]$ be the allowed multiplicative error in our estimation of $p$. Then with ${{\mathcal O}}\left(\frac{1}{\mu\sqrt{p_{\min}}}\right)$ uses of $U$ and $U^{-1}$ and using ${{\mathcal O}}\left(\frac{q}{\mu\sqrt{p_{\min}}}\right)$ gates on the $q$ qubits we obtain a $\tilde{p}$ such that $|p-\tilde{p}|\leq \mu p$ with probability at least $4/5$.
We use the amplitude-estimation algorithm of [@bhmt:countingj Theorem 12] with $M$ applications of $U$ and $U^{-1}$. This provides an estimate $\tilde{p}$ of $p$, that with probability at least $8/\pi^2>4/5$ satisfies $$|p-\tilde{p}|
\leq2\pi\frac{\sqrt{p(1-p)}}{M}+\frac{\pi^2}{M^2}
\leq\frac{\pi}{M}\left(2\sqrt{p}+\frac{\pi}{M}\right).$$ Choosing $M$ the smallest power of $2$ such that $M \geq 3\pi/(\mu\sqrt{p_{\min}})$, with probability at least $4/5$ we get $$|p-\tilde{p}|
\leq\mu\frac{\sqrt{p_{\min}}}{3}\left(2\sqrt{p}
+\mu\frac{ \sqrt{p_{\min}}}{3}\right)
\leq\mu\frac{\sqrt{p}}{3}\left(3\sqrt{p}\right)
\leq \mu p.$$ The $q$ factor in the gate complexity comes from the implementation of the amplitude amplification steps needed in amplitude-estimation. The gate complexity of the whole amplitude-estimation procedure is dominated by this contribution, proving the final gate complexity.
\[col:main22\] Suppose we are given the positive numbers $z\leq {{\mbox{\rm Tr}}\left(e^{-H}\right)}$, $\theta\in(0,1]$, and unitary circuits $\tilde{U}_{A',H}$ for $A'=0$ and $A'=A$ with ${\left\lVertA\right\rVert}\leq 1$, each acting on at most $q$ qubits such that $$\left|{\left\lVert({\langle0|}\otimes I)\tilde{U}_{A',H}{|0\ldots 0\rangle}\right\rVert}^2-\frac{{{\mbox{\rm Tr}}\left((I+A'/2)e^{-H}\right)}}{4n}\right| \leq \frac{\theta z}{144n}.$$ (To clarify the notation: if ${|\psi\rangle}$ is a 2-register state, then $({\langle0|}\otimes I){|\psi\rangle}$ is the (unnormalized) state in the 2nd register that results from projecting on ${|0\rangle}$ in the 1st register.)
Applying the procedure of Lemma \[lem:ampest\] to $\tilde{U}_{A',H}$ (both for $A'=0$ and for $A'=A$) with $p_{\min}=\frac{z}{9n}$ and $\mu=\theta/19$, and combining the results using Corollary \[col:split\] yields an additive $\theta$-approximation of ${{\mbox{\rm Tr}}\left(A\rho\right)}$ with high probability. The procedure uses $${{\mathcal O}}\left(\frac{1}{\theta}\sqrt{\frac{n}{z}}\right)$$ applications of $\tilde{U}_{A,H}$, $\tilde{U}_{0,H}$ and their inverses, and ${{\mathcal O}}\left(\frac{q}{\theta}\sqrt{\frac{n}{z}}\right)$ additional gates.
First note that since $I+A'/2 \succeq I/2$ we have $$p:=\frac{{{\mbox{\rm Tr}}\left((I+A'/2)e^{-H}\right)}}{4n} \geq \frac{{{\mbox{\rm Tr}}\left(e^{-H}\right)}}{8n}$$ and thus $$\label{eq:pDiff}
\left| {\left\lVert({\langle0|}\otimes I)\tilde{U}_{A',H}{|0\ldots 0\rangle}\right\rVert}^2 - p\right|
\leq \frac{\theta z }{144n}
\leq \frac{\theta}{18}\cdot\frac{{{\mbox{\rm Tr}}\left(e^{-H}\right)}}{8 n}
\leq \frac{\theta}{18} p
\leq \frac{p}{18}.$$ Therefore $${\left\lVert({\langle0|}\otimes I)\tilde{U}_{A',H}{|0\ldots 0\rangle}\right\rVert}^2
\geq \left(1-\frac{1}{18}\right)p
\geq \left(1-\frac{1}{18}\right)\frac{{{\mbox{\rm Tr}}\left(e^{-H}\right)}}{8n}
> \frac{{{\mbox{\rm Tr}}\left(e^{-H}\right)}}{9 n}
\geq \frac{z}{9 n}=p_{\min}.$$ Also by we have $${\left\lVert({\langle0|}\otimes I)\tilde{U}_{A',H}{|0\ldots 0\rangle}\right\rVert}^2
\leq \left(1+\frac{\theta}{18}\right) p
\leq \frac{19}{18} p.$$ Therefore using Lemma \[lem:ampest\], with $\mu=\theta/19$, with high probability we get a $\tilde{p}$ satisfying $$\label{eq:ptDiff}
\left| \tilde{p} - {\left\lVert({\langle0|}\otimes I)\tilde{U}_{A',H}{|0\ldots 0\rangle}\right\rVert}^2 \right|
\leq \frac{\theta}{19}\cdot{\left\lVert({\langle0|}\otimes I)\tilde{U}_{A',H}{|0\ldots 0\rangle}\right\rVert}^2
\leq \frac{\theta}{18} p.$$ By combining - using the triangle inequality we get $$\left| p - \tilde{p}\right|
\leq \frac{\theta}{9}p,$$ so that Corollary \[col:split\] can indeed be applied. The complexity statement follows from Lemma \[lem:ampest\] and our choices of $p_{\min}$ and $\mu$.
Notice the $1/\sqrt{z}\geq 1/\sqrt{{{\mbox{\rm Tr}}\left(e^{-H}\right)}}$ factor in the complexity statement of the last corollary. To make sure this factor is not too large, we would like to ensure ${{\mbox{\rm Tr}}\left(e^{-H}\right)}=\Omega(1)$. This can be achieved by substituting $H_+ = H-\lambda_{\min}I$, where $\lambda_{\min}$ is the smallest eigenvalue of $H$. It is easy to verify that this will not change the value ${{\mbox{\rm Tr}}\left(Ae^{-H}/{{\mbox{\rm Tr}}\left(e^{-H}\right)}\right)}$.
It remains to show how to compute $\lambda_{\min}$ and how to apply $\tilde{U}_{A,H}$. Both of these steps are considerably easier in the case where all matrices are diagonal, so we will consider this case first.
### The special case of diagonal matrices – for LP-solving {#sec:lptrace}
In this section we consider diagonal matrices, assuming oracle access to $H$ of the following form: $$O_H{|i\rangle}{|z\rangle} = {|i\rangle}{|z\oplus H_{ii}\rangle}$$ and similarly for $A$. Notice that this kind of oracle can easily be constructed from the general sparse matrix oracle that we assume access to.
\[lem:diagU\] Let $A,H \in \mathbb{R}^{n\times n}$ be diagonal matrices such that ${\left\lVertA\right\rVert}\leq 1$ and $H\succeq 0$, and let $\mu > 0$ be an error parameter. Then there exists a unitary $\tilde{U}_{A,H}$ such that $$\left| {\left\lVert ({\langle0|}\otimes I)\tilde{U}_{A,H}{|0\ldots0\rangle} \right\rVert}^2 - {{\mbox{\rm Tr}}\left(\frac{I+A/2}{4n}e^{-H}\right)}\right| \leq \mu,$$ which uses $1$ quantum query to $A$ and $H$ and ${{\mathcal O}}(\log^{{{\mathcal O}}(1)}(1/\mu) + \log(n))$ other gates.
For simplicity assume $n$ is a power of two. This restriction is not necessary, but makes the proof a bit simpler to state.
In the first step we prepare the state $\sum_{i=1}^{n}{|i\rangle}/\sqrt{n}$ using $\log(n)$ Hadamard gates on ${|0\rangle}^{\otimes\log(n)}$. Then we query the diagonal values of $H$ and $A$ to get the state $\sum_{i=1}^{n}{|i\rangle}{|H_{ii}\rangle}{|A_{ii}\rangle}/\sqrt{n}$. Using these binary values we apply a finite-precision arithmetic circuit to prepare $$\frac{1}{\sqrt{n}} \sum_{i=1}^{n} {|i\rangle}{|H_{ii}\rangle}{|A_{ii}\rangle}{|\beta_i\rangle} \text{, where } \beta_i:=\arcsin\left(\sqrt{\frac{1+A_{ii}/2}{4}e^{-H_{ii}}+\delta_i}\right)/\pi \text{, and } |\delta_i|\leq \mu.$$ Note that the error $\delta_i$ comes from writing down only a finite number of bits $b_1.b_2b_3\dots b_{\log(8/\mu)}$. Due to our choice of $A$ and $H$, we know that $\beta_i$ lies in $[0,1]$. We proceed by first adding an ancilla qubit initialized to ${|1\rangle}$ in front of the state, then we apply $\log(8/\mu)$ controlled rotations to this qubit: for each $b_j=1$ we apply a rotation by angle $\pi2^{-j}$. In other words, if $b_1 = 1$, then we rotate ${|1\rangle}$ fully to ${|0\rangle}$. If $b_2 = 1$, then we rotate halfway, and we proceed further by halving the angle for each subsequent bit. We will end up with the state: $$\frac{1}{\sqrt{n}}\sum_{i=1}^{n} \left(\sqrt{\frac{1+A_{ii}/2}{4}e^{-H_{ii}}+\delta_i}{|0\rangle}+\sqrt{1-\frac{1+A_{ii}/2}{4}e^{-H_{ii}}-\delta_i}{|1\rangle}\right) {|i\rangle}{|A_{ii}\rangle}{|H_{ii}\rangle}{|\beta_i\rangle}.$$ It is now easy to see that the squared norm of the ${|0\rangle}$-part of this state is as required: $${\left\lVert\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\sqrt{\frac{1+A_{ii}/2}{4}e^{-H_{ii}}+\delta_i}{|i\rangle}\right\rVert}^2
=\frac{1}{n}\sum_{i=1}^{n}\left(\frac{1+A_{ii}/2}{4}e^{-H_{ii}}+\delta_i\right)
=\frac{{{\mbox{\rm Tr}}\left((I+A/2)e^{-H}\right)}}{4n}+\sum_{i=1}^{n}\frac{\delta_i}{n},$$ which is an additive $\mu$-approximation since $\left| \sum_{i=1}^{n}\frac{\delta_i}{n} \right| \leq \mu$.
\[col:traceDiagCalc\] Let $A,H \in \mathbb{R}^{n\times n}$ be diagonal matrices, with ${\left\lVertA\right\rVert}\leq 1$. An additive $\theta$-approximation of $${{\mbox{\rm Tr}}\left(A\rho\right)} = \frac{{{\mbox{\rm Tr}}\left(Ae^{-H}\right)}}{{{\mbox{\rm Tr}}\left(e^{-H}\right)}}$$ can be computed using ${{\mathcal O}}\left(\frac{\sqrt{n}}{\theta}\right)$ queries to $A$ and $H$ and ${\widetilde{\mathcal O}\left(\frac{\sqrt{n}}{\theta}\right)}$ other gates.
Since $H$ is a diagonal matrix, its eigenvalues are exactly its diagonal entries. Using the quantum minimum-finding algorithm of D[ü]{}rr and H[ø]{}yer one can find (with high success probability) the minimum $\lambda_{\min}$ of the diagonal entries using ${{\mathcal O}}(\sqrt{n})$ queries to the matrix elements. Applying Lemma \[lem:diagU\] and Corollary \[col:main22\] to $H_+ = H-\lambda_{\min}I$, with $z=1$, gives the stated bound.
### General case – for SDP-solving {#sec:estTrArhogeneral}
In this section we will extend the ideas from the last section to non-diagonal matrices. There are a few complications that arise in this more general case. These mostly follow from the fact that we now do not know the eigenvectors of $H$ and $A$, which were the basis states before, and that these eigenvectors might not be the same for both matrices. For example, to find the minimal eigenvalue of $H$, we can no longer simply minimize over its diagonal entries. To solve this, in Appendix \[app:genMinFind\] we develop new techniques that generalize minimum-finding.
Furthermore, the unitary $\tilde{U}_{A,H}$ in the LP case could be seen as applying the operator $$\sqrt{\frac{I+A/2}{4}e^{-H}}$$ to a superposition of its eigenvectors. This is also more complicated in the general setting, due to the fact that the eigenvectors are no longer the basis states. In Appendix \[apx:LowWeight\] we develop general techniques to apply smooth functions of Hamiltonians to a state. Among other things, this will be used to create an efficient purified Gibbs sampler.
Our Gibbs sampler uses similar methods to the work of Chowdhury and Somma [@ChowdhurySomma:GibbsHit] for achieving logarithmic dependence on the precision. However, the result of [@ChowdhurySomma:GibbsHit] cannot be applied to our setting, because it implicitly assumes access to an oracle for $\sqrt{H}$ instead of $H$. Although their paper describes a way to construct such an oracle, it comes with a large overhead: they construct an oracle for $\sqrt{H'}=\sqrt{H + \nu I}$, where $\nu\in\mathbb{R}_+$ is some possibly large positive number. This shifting can have a huge effect on $Z'={{\mbox{\rm Tr}}\left(e^{-H'}\right)}=e^{-\nu}{{\mbox{\rm Tr}}\left(e^{-H}\right)}$, which can be prohibitive due to the $\sqrt{1/Z'}$ factor in the runtime, which blows up exponentially in $\nu$.
In the following lemma we show how to implement $\tilde{U}_{A,H}$ using the techniques we developed in Appendix \[apx:LowWeight\].
\[lemma:trPreEst\] Let $A,H\in \mathbb{C}^{n\times n}$ be Hermitian matrices such that ${\left\lVertA\right\rVert}\leq 1$ and $I\preceq H\preceq K I$ for a known $K\in \mathbb{R}_+$. Assume $A$ is $s$-sparse and $H$ is $d$-sparse with $s\leq d$. Let $\mu > 0$ be an error parameter. Then there exists a unitary $\tilde{U}_{A,H}$ such that $$\left| {\left\lVert ({\langle0|}\otimes I)\tilde{U}_{A,H}{|0\ldots0\rangle} \right\rVert}^2 - {{\mbox{\rm Tr}}\left(\frac{I+A/2}{4n}e^{-H}\right)}\right| \leq \mu$$ that uses ${\widetilde{\mathcal O}\left(Kd\right)}$ queries to $A$ and $H$, and the same order of other gates.
The basic idea is that we first prepare a maximally entangled state $\sum_{i=1}^{n}{|i\rangle}{|i\rangle}/\sqrt{n}$, and then apply the (norm-decreasing) maps $e^{-H/2}$ and $\sqrt{\frac{I+A/2}{4}}$ to the first register. Note that we can assume without loss of generality that $\mu\leq 1$, otherwise the statement is trivial.
Let $\tilde{W}_0=\left({\langle0|}\otimes I\right)\tilde{W}\left({|0\rangle}\otimes I\right)$ be a $\mu/5$-approximation of the map $e^{-H/2}$ (in operator norm) implemented by using Theorem \[thm:emH\], and let $\tilde{V}_0=\left({\langle0|}\otimes I\right)\tilde{V}\left({|0\rangle}\otimes I\right)$ be a $\mu/5$-approximation of the map $\sqrt{\frac{I+A/2}{4}}$ implemented by using Theorem \[thm:Taylor\]. We define $\tilde{U}_{A,H}:=\tilde{V}\tilde{W}$, noting that there is a hidden $\otimes I$ factor in both $\tilde{V}$ and $\tilde{W}$ corresponding to each other’s ancilla qubit. As in the linear programming case, we are interested in $p$, the probability of measuring a $00$ in the first register (i.e., the two “flag” qubits) after applying $\tilde{U}_{A,H}$. We will analyze this in terms of these operators below. We will make the final approximation step precise in the next paragraph. $$\begin{aligned}
p'&:={\left\lVert\left({\langle00|}\otimes I\right) \tilde{U}_{A,H}\left({|00\rangle}\otimes I\right)\sum_{i=1}^{n}\frac{{|i\rangle}{|i\rangle}}{\sqrt{n}}\right\rVert}^2 \nonumber\\
&={\left\lVert\tilde{V}_0\tilde{W}_0\sum_{i=1}^{n}\frac{{|i\rangle}{|i\rangle}}{\sqrt{n}}\right\rVert}^2\nonumber\\
&=\frac{1}{n}\sum_{i=1}^{n}{\langlei|}\tilde{W}_0^\dagger\tilde{V}_0^\dagger\tilde{V}_0\tilde{W}_0{|i\rangle}\nonumber\\
&=\frac{1}{n}{{\mbox{\rm Tr}}\left(\tilde{W}_0^\dagger\tilde{V}_0^\dagger\tilde{V}_0\tilde{W}_0\right)}\nonumber\\
&=\frac{1}{n}{{\mbox{\rm Tr}}\left(\tilde{V}_0^\dagger\tilde{V}_0\tilde{W}_0\tilde{W}_0^\dagger\right)}\label{eq:exactTrace}\\
&\approx\frac{1}{n}{{\mbox{\rm Tr}}\left(\frac{I+A/2}{4}e^{-H}\right)}. \label{eq:approxTrace}
\end{aligned}$$ Note that for all matrices $B,\tilde{B}$ with ${\left\lVertB\right\rVert}\leq 1$, we have $$\begin{aligned}
{\left\lVertB^\dagger B - \tilde{B}^\dagger \tilde{B}\right\rVert}& = {\left\lVert(B^\dagger-\tilde{B}^\dagger)B + B^\dagger(B-\tilde{B}) - (B^\dagger-\tilde{B}^\dagger)(B-\tilde{B})\right\rVert}\\
& \leq {\left\lVert(B^\dagger-\tilde{B}^\dagger)B\right\rVert} + {\left\lVertB^\dagger(B-\tilde{B})\right\rVert} +{\left\lVert(B^\dagger-\tilde{B}^\dagger)(B-\tilde{B})\right\rVert}\\
& \leq {\left\lVertB^\dagger-\tilde{B}^\dagger\right\rVert}{\left\lVertB\right\rVert} + {\left\lVertB^\dagger\right\rVert}{\left\lVertB-\tilde{B}\right\rVert} +{\left\lVertB^\dagger-\tilde{B}^\dagger\right\rVert}{\left\lVertB-\tilde{B}\right\rVert}\\
& \leq 2{\left\lVertB-\tilde{B}\right\rVert}+{\left\lVertB-\tilde{B}\right\rVert}^2.\end{aligned}$$ Since $\mu\leq 1$, and hence $2\mu/5 + (\mu /5)^2\leq \mu/2$, this implies (with $B=e^{-H/2}$ and $\tilde{B}=\tilde{W}_0^\dagger$) that ${\left\lVerte^{-H} - \tilde{W}_0\tilde{W}_0^\dagger\right\rVert}\leq \mu / 2$, and also (with $B=\sqrt{(I+A/2)/4}$ and $\tilde{B}=\tilde{V}_0$) ${\left\lVert(I+A/2)/4 - \tilde{V}_0^\dagger\tilde{V}_0\right\rVert}\leq \mu/2$. Let ${\left\lVert\cdot\right\rVert}_1$ denote the trace norm (a.k.a. Schatten 1-norm). Note that for all $C,D,\tilde{C},\tilde{D}$: $$\begin{aligned}
\left|{{\mbox{\rm Tr}}\left(CD\right)}-{{\mbox{\rm Tr}}\left(\tilde{C}\tilde{D}\right)}\right|
&\leq {\left\lVertCD-\tilde{C}\tilde{D}\right\rVert}_1\\
&= {\left\lVert (C-\tilde{C})D + C(D-\tilde{D})-(C-\tilde{C})(D-\tilde{D}) \right\rVert}_1\\
&\leq {\left\lVert(C-\tilde{C})D\right\rVert}_1+{\left\lVertC(D-\tilde{D})\right\rVert}_1+{\left\lVert(C-\tilde{C})(D-\tilde{D})\right\rVert}_1\\
& \leq {\left\lVertC-\tilde{C}\right\rVert}{\left\lVertD\right\rVert}_1+{\left\lVertD-\tilde{D}\right\rVert}\left({\left\lVertC\right\rVert}_1+{\left\lVertC-\tilde{C}\right\rVert}_1\right).
\end{aligned}$$ Which, in our case (setting $C=(I+A/2)/4$, $D=e^{-H}$, $\tilde{C}=\tilde{V}_0^\dagger\tilde{V}_0$, and $\tilde{D}=\tilde{W}_0\tilde{W}_0^\dagger$) implies that $$\left|{{\mbox{\rm Tr}}\left(\left(I+A/2\right) e^{-H}/4\right)}-{{\mbox{\rm Tr}}\left(\tilde{V}_0^\dagger\tilde{V}_0\tilde{W}_0\tilde{W}_0^\dagger\right)}\right| \leq (\mu/2) {{\mbox{\rm Tr}}\left(e^{-H}\right)} +(\mu/2)(1/2+\mu / 2)n.$$ Dividing both sides by $n$ and using equation then implies $$\begin{aligned}
\left|{{\mbox{\rm Tr}}\left(\left(I+A/2\right) e^{-H}\right)}/(4n)-p'\right| & \leq \frac{\mu}{2} \frac{{{\mbox{\rm Tr}}\left(e^{-H}\right)}}{n}+\frac{\mu}{2}\left(\frac{1}{2}+\frac{\mu}{2}\right) \\
&\leq \frac{\mu}{2}+ \frac{\mu}{2}\\
&= \mu.
\end{aligned}$$ This proves the correctness of $\tilde{U}_{A,H}$. It remains to show that the complexity statement holds. To show this we only need to specify how to implement the map $\sqrt{\frac{I+A/2}{4}}$ using Theorem \[thm:Taylor\], since the map $e^{H/2}$ is already dealt with in Theorem \[thm:emH\]. To use Theorem \[thm:Taylor\], we choose $x_0:=0$, $K:=1$ and $r:=1$, since ${\left\lVertA\right\rVert}\leq 1$. Observe that $\sqrt{\frac{1+x/2}{4}}=\frac{1}{2}\sum_{k=0}^{\infty}\binom{1/2}{k}\left(\frac{x}{2}\right)^k$ whenever $|x|\leq 1$. Also let $\delta=1/2$, so $r+\delta=\frac{3}{2}$ and $\frac{1}{2}\sum_{k=0}^{\infty}\left|\binom{1/2}{k}\right|\left(\frac{3}{4}\right)^k\leq 1=:B$. Recall that $\tilde{V}$ denotes the unitary that Theorem \[thm:Taylor\] constructs. Since we choose the precision parameter to be $\mu/5=\Theta(\mu)$, Theorem \[thm:Taylor\] shows $\tilde{V}$ can be implemented using ${{\mathcal O}}\left(d\log^2\left(1/\mu\right)\right)$ queries and ${{\mathcal O}}\left(d\log^2\left(1/\mu\right)\left[\log(n)+\log^{2.5}\left(1/\mu\right)\right]\right)$ gates. This cost is negligible compared to our implementation cost of $e^{-H/2}$ with $\mu/5$ precision: Theorem \[thm:emH\] uses ${{\mathcal O}}\left(Kd\log^2\left(K/\mu\right)\right)$ queries and ${{\mathcal O}}\left(Kd\log^2\left(Kd/\mu\right)\left[\log(n)+\log^{2.5}\left(Kd/\mu\right)\right]\right)$ gates to implement $\tilde{W}$.
\[col:traceCalc\] Let $A,H\in \mathbb{C}^{n\times n}$ be Hermitian matrices such that ${\left\lVertA\right\rVert}\leq 1$ and ${\left\lVertH\right\rVert}\leq K$ for a known bound $K\in \mathbb{R}_+$. Assume $A$ is $s$-sparse and $H$ is $d$-sparse with $s\leq d$. An additive $\theta$-approximation of $${{\mbox{\rm Tr}}\left(A\rho\right)} = \frac{Ae^{-H}}{{{\mbox{\rm Tr}}\left(e^{-H}\right)}}$$ can be computed using ${\widetilde{\mathcal O}\left(\frac{\sqrt{n}dK}{\theta}\right)}$ queries to $A$ and $H$, while using the same order of other gates.
Start by computing an estimate $\tilde{\lambda}_{\min}$ of $\lambda_{\min}(H)$, the minimum eigenvalue of $H$, up to additive error ${\varepsilon}= 1/2$ using Lemma \[lemma:normEst\]. We define $H_+ := H-(\tilde{\lambda}_{\min}-3/2)I$, so that $I\preceq H_+$ but $2I\nprec H_+$.
Applying Lemma \[lemma:trPreEst\] and Corollary \[col:main22\] to $H_+$ with $z=e^{-2}$ gives the stated bound.
An efficient 2-sparse oracle {#sec:oracle}
----------------------------
Recall from the end of Section \[sec:classicalAK\] that $\tilde{a}_j$ is an additive $\theta$-approximation to ${{\mbox{\rm Tr}}\left(A_j\rho\right)}$, $\tilde{c}$ is a $\theta$-approximation to ${{\mbox{\rm Tr}}\left(C\rho\right)}$ and ${c^{\prime}}= \tilde{c} - r \theta - \theta$. We first describe our quantum 2-sparse oracle assuming access to a unitary which acts as ${|j\rangle}{|0\rangle}{|0\rangle} \mapsto {|j\rangle} {|\tilde{a}_j\rangle} {|\psi_j\rangle}$, where ${|\psi_j\rangle}$ is some workspace state depending on $j$. We then briefly discuss how to modify the analysis when we are given an oracle which acts as ${|j\rangle}{|0\rangle}{|0\rangle} \mapsto {|j\rangle} \sum_i \beta_j^{i}{|\tilde{a}^{i}_j\rangle} {|\psi^{i}_j\rangle}$ (where each $\tilde{a}^{i}_j$ is an additive $\theta$-approximation to ${{\mbox{\rm Tr}}\left(A_j \rho\right)}$), since this is the output of the trace-estimation procedure of the previous section.
Our goal is to find a $y \in \tilde{\mathcal{P}}(\tilde{a},{c^{\prime}})$, i.e., a $y$ such that $$\begin{aligned}
{\left\lVerty\right\rVert}_1&\leq r\\
b^T y &\leq \alpha\\
\tilde{a}^T y &\geq {c^{\prime}}\\
y &\geq 0\end{aligned}$$ If $\alpha \geq 0$ and ${c^{\prime}}\leq 0$, then $y=0$ is a solution and our oracle can return it. If not, then we may write $y = Nq$ with $N={\left\lVerty\right\rVert}_1 > 0$ and hence ${\left\lVertq\right\rVert}_1 = 1$. So we are looking for an $N$ and a $q$ such that $$\begin{aligned}
b^T q &\leq \alpha / N\label{eq:Nbound}\\
\tilde{a}^T q &\geq {c^{\prime}}/ N\notag\\\notag
{\left\lVertq\right\rVert}_1 &= 1\\\notag
q &\geq 0\\\notag
0 &< N \leq r\end{aligned}$$ We can now view $q \in \mathbb{R}_+^m$ as the coefficients of a convex combination of the points $p_i = (b_i,\tilde{a}_i)$ in the plane. We want such a combination that lies to the upper left of $g_N = (\alpha/N,{c^{\prime}}/ N)$ for some $0 < N\leq r$. Let $\mathcal{G}_N$ denote the upper-left quadrant of the plane starting at $g_N$.
\[lem:2DArg\] If there is a $y \in {\tilde{\mathcal{P}}}(\tilde{a},{c^{\prime}})$, then there is a $2$-sparse $y^{\prime} \in {\tilde{\mathcal{P}}}(\tilde a,{c^{\prime}})$ such that ${\left\lVerty\right\rVert}_1 = {\left\lVerty^{\prime}\right\rVert}_1$.
Consider $p_i = (b_i,\tilde{a}_i)$ and $g = (\alpha / N,{c^{\prime}}/ N)$ as before, and write $y = Nq$ where $\sum_{j=1}^m q_j =1$, $q \geq 0$. The vector $q$ certifies that a convex combination of the points $p_i$ lies in $\mathcal{G}_N$. But then there exist $j,k \in \lbrack m \rbrack$ such that the line segment $\overline{p_jp_k}$ intersects $\mathcal{G}_N$. All points on this line segment are convex combinations of $p_j$ and $p_k$, hence there is a convex combination of $p_j$ and $p_k$ that lies in $\mathcal{G}_N$. This gives a $2$-sparse $q^{\prime}$, and $y' = N q' \in {\tilde{\mathcal{P}}}(\tilde{a},{c^{\prime}})$.
We can now restrict our search to $2$-sparse $y$. Let $\mathcal{G} = \bigcup_{N \in (0, r\rbrack } \mathcal{G}_N$. Then we want to find two points $p_j,p_k$ that have a convex combination in $\mathcal{G}$, since this implies that a scaled version of their convex combination gives a $y \in \tilde{\mathcal{P}} (\tilde{a},{c^{\prime}})$ and ${\left\lVerty\right\rVert}_1 \leq r$.
\[lem:oracle\] There is an oracle that returns a 2-sparse vector $y\in \tilde{\mathcal{P}}(\tilde{a},{c^{\prime}})$, if one exists, using one search and two minimizations over the $m$ points $p_j=(b_j,\tilde{a}_j)$. This gives a classical algorithm that uses ${{\mathcal O}}(m)$ calls to the subroutine that gives the entries of $\tilde{a}$ and ${{\mathcal O}}(m)$ other operations, and a quantum algorithm that (in order to solve the problems with high probability) uses ${{\mathcal O}}(\sqrt{m})$ calls to the subroutine that gives the entries of $\tilde{a}$ and ${\widetilde{\mathcal O}\left(\sqrt{m}\right)}$ other gates.
The algorithm can be summarized as follows:
1. Check if $\alpha \geq 0$ and ${c^{\prime}}\leq 0$. If so, output $y=0$. \[it:step1\]
2. Check if there is a $p_i\in \mathcal G$. If so, let $q = e_i$ and $N = \frac{{c^{\prime}}}{\tilde{a}_i}$. \[it:step2\]
3. Find $p_j,p_k$ so that the line segment $\overline{p_jp_k}$ goes through $\mathcal G$. This gives coefficients $q$ of a convex combination that can be scaled by $N = \frac{{c^{\prime}}}{\tilde{a}^T q}$ to give $y$. The main realization is that we can search separately for $p_j$ and $p_k$. \[it:step3\]
First we will need a better understanding of the shape of $\mathcal G$ (see Figure \[fig:graph\] for illustration). This depends on the sign of $\alpha$ and ${c^{\prime}}$. If we define ${\mbox{\rm sign}}(0) = 1$:
- If ${\mbox{\rm sign}}(\alpha) = -1, {\mbox{\rm sign}}({c^{\prime}}) = -1$. The corner point of $\mathcal{G}$ is $(\alpha / r,{c^{\prime}}/ r)$. One edge goes up vertically and an other follows the line segment $\lambda \cdot (\alpha,{c^{\prime}})$ for $\lambda \in [ 1/r,\infty)$ starting at the corner.
- If ${\mbox{\rm sign}}(\alpha) = -1, {\mbox{\rm sign}}({c^{\prime}}) = 1$. Here $\mathcal G_N \subseteq \mathcal G_r$ for $N\leq r$. So $\mathcal G = \mathcal G_r$. The corner point is again $(\alpha / r,{c^{\prime}}/ r)$, but now one edge goes up vertically and one goes to the left horizontally.
- If ${\mbox{\rm sign}}(\alpha) = 1, {\mbox{\rm sign}}({c^{\prime}}) = -1$. This is the case where $y=0$ is a solution, $\mathcal G$ is the whole plane and has no corner.
- If ${\mbox{\rm sign}}(\alpha) = 1, {\mbox{\rm sign}}({c^{\prime}}) = 1$. The corner point of $\mathcal G$ is again $(\alpha / r,{c^{\prime}}/ r)$. From there one edge goes to the left horizontally and one edge follows the line segment $\lambda \cdot (\alpha,{c^{\prime}})$ for $\lambda \in [ 1/r,\infty)$.
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Since $\mathcal G$ is always an intersection of at most $2$ halfspaces, steps \[it:step1\]-\[it:step2\] of the algorithm are easy to perform. In step \[it:step1\] we handle case (c) by simply returning $y=0$. For the other cases $(\alpha /r ,{c^{\prime}}/r)$ is the corner point of $\mathcal G$ and the two edges are simple lines. Hence in step \[it:step2\] we can easily search through all the points to find out if there is one lying in $\mathcal G$; since $\mathcal G$ is a very simple region, this only amounts to checking on which side of the two lines a point lies.
Now, if we cannot find a single point in $\mathcal{G}$, we need a combination of two points in step \[it:step3\]. Let $L_1, L_2$ be the edges of $\mathcal G$ and let $\ell_j$ and $\ell_k$ be the line segments from $(\alpha/r,{c^{\prime}}/r)$ to $p_j$ and $p_k$, respectively. Then, as can be seen in Figure \[fig:angles\], the line segment $\overline{p_jp_k}$ goes through $\mathcal G$ if and only if (up to relabeling $p_j$ and $p_k$) $\angle \ell_j L_1 + \angle L_1 L_2 + \angle L_2 \ell_k \leq \pi$. Since $\angle L_1 L_2$ is fixed, we can simply look for a $j$ such that $\angle \ell_j L_1$ is minimized and a $k$ such that $\angle L_2\ell_k$ is minimized. If $\overline{p_jp_k}$ does not pass through $\mathcal G$ for this pair of points, then it does not for any of the pairs of points.
Notice that these minimizations can be done separately and hence can be done in the stated complexity. Given the minimizing points $p_j$ and $p_k$, it is easy to check if they give a solution by calculating the angle between $\ell_j$ and $\ell_k$. The coefficients of the convex combination $q$ are then easy to compute. It remains to compute the scaling $N$. This is done by rewriting the constraints of : $$\frac{{c^{\prime}}}{q^T \tilde{a}}\leq N \leq \frac{\alpha}{q^T b}$$ So we can pick any value in this range for $N$. -0.5cm
The analysis above applies if we are given an oracle which acts as ${|j\rangle}{|0\rangle}{|0\rangle} \mapsto {|j\rangle} {|\tilde{a}_j\rangle} {|\psi_j\rangle}$. However, the trace estimation procedure of Corollary \[col:traceCalc\] acts as ${|j\rangle}{|0\rangle}{|0\rangle} \mapsto {|j\rangle} \sum_i \beta_j^{i}{|\tilde{a}^{i}_j\rangle} {|\psi^{i}_j\rangle}$ where each ${|\tilde{a}^{i}_j\rangle}$ is an approximation of $a_j$ and the amplitudes $\beta_j^{i}$ are such that measuring the second register with high probability returns an $\tilde{a}^{i}_j$ which is $\theta$-close to $a_j$. Since we can exponentially reduce the probability that we obtain an $\tilde{a}^{i}_j$ which is further than $\theta$ away from $a_j$, we will for simplicity assume that for all $i,j$ we have $|\tilde{a}^{i}_j - a_j| \leq \theta$; the neglected exponentially small probabilities will only affect the analysis in negligible ways. Note that while we do not allow our oracle enough time to obtain classical descriptions of all $\tilde a_j$s (we aim for a runtime of ${\widetilde{\mathcal O}\left(\sqrt{m}\right)}$), we do have enough time to compute $\tilde{c}$ once initially. Knowing $\tilde c$, we can compute the angles defined by the points $(b_j,\tilde{a}_j^{i})$ with respect to the corner point of $(\alpha/r,(\tilde{c}-\theta)/ r-\theta)$ and the lines $L_1, L_2$ (see Figure \[fig:angles\]). We now apply our generalized minimum-finding algorithm with runtime ${\widetilde{\mathcal O}\left(\sqrt{m}\right)}$ (see Theorem \[thm:genMin\]) starting with a uniform superposition over the $j$s to find $\ell,k \in [m]$ with points $(\tilde a_\ell,b_\ell)$ and $(\tilde a_k, b_k)$ approximately minimizing the respective angles to lines $L_1,L_2$. It follows from Lemma \[lem:approxP\] and Lemma \[lem:2DArg\] that if $\mathcal P_{0}(X) \cap \{y \in \mathbb R^m: \sum_{j} y_j \leq r\}$ is non-empty, then some convex combination of $(\tilde a_\ell, b_\ell)$ and $(\tilde a_k, b_k)$ lies in $\mathcal G$. On the other hand, if $\mathcal P_{4Rr\theta}(X) \cap \{y \in \mathbb R^m: \sum_{j} y_j \leq r\}$ is empty, then by the same lemmas the respective angles will be such that we correctly conclude that $\mathcal P_{0}(X) \cap \{y \in \mathbb R^m: \sum_{j} y_j \leq r\}$ is empty.
Total runtime {#sec:runtime}
-------------
We are now ready to add our quantum implementations of the trace calculations and the oracle to the classical Arora-Kale framework.
Using our implementations of the different building blocks, it remains to calculate what the total complexity will be when they are used together.
Cost of the oracle for $H^{(t)}$.
: The first problem in each iteration is to obtain access to an oracle for $H^{(t)}$. In each iteration the oracle will produce a $y^{(t)}$ that is at most $2$-sparse, and hence in the $(t+1)$th iteration, $H^{(t)}$ is a linear combination of $2t$ of the $A_j$ matrices and the $C$ matrix.
We can write down a sparse representation of the coefficients of the linear combination that gives $H^{(t)}$ in each iteration by adding the new terms coming from $y^{(t)}$. This will clearly not take longer than ${\widetilde{\mathcal O}\left(T\right)}$, since there are only a constant number of terms to add for our oracle. As we will see, this term will not dominate the complexity of the full algorithm.
Using such a sparse representation of the coefficients, one query to a sparse representation of $H^{(t)}$ will cost ${\widetilde{\mathcal O}\left(st\right)}$ queries to the input matrices and ${\widetilde{\mathcal O}\left(st\right)}$ other gates. For a detailed explanation and a matching lower bound, see Appendix \[app:sparsematrixsum\].
Cost of the oracle for ${{\mbox{\rm Tr}}\left(A_j\rho\right)}$.
: In each iteration $M^{(t)}$ is made to have operator norm at most $1$. This means that $${\left\lVert-\eta H^{(t)}\right\rVert} \leq \eta \sum_{\tau = 1}^t {\left\lVertM^{(\tau)}\right\rVert} \leq \eta t.$$ Furthermore we know that $H^{(t)}$ is at most $d:=s(2t+1)$-sparse. Calculating ${{\mbox{\rm Tr}}\left(A_j\rho\right)}$ for one index $j$ up to an additive error of $\theta := {\varepsilon}/(12Rr)$ can be done using the algorithm from Corollary \[col:traceCalc\]. This will take $${\widetilde{\mathcal O}\left(\sqrt{n}\frac{{\left\lVertH\right\rVert}d}{\theta}\right)} = {\widetilde{\mathcal O}\left(\sqrt{n} s\frac{\eta t^2 Rr}{{\varepsilon}}\right)}$$ queries to the oracle for $H^{(t)}$ and the same order of other gates. Since each query to $H^{(t)}$ takes ${\widetilde{\mathcal O}\left(st\right)}$ queries to the input matrices, this means that $${\widetilde{\mathcal O}\left(\sqrt{n}s^2\frac{\eta t^3 Rr}{{\varepsilon}}\right)}$$ queries to the input matrices will be made, and the same order of other gates, for each approximation of a ${{\mbox{\rm Tr}}\left(A_j\rho\right)}$ (and similarly for approximating ${{\mbox{\rm Tr}}\left(C\rho\right)}$).
Total cost of one iteration.
: Lemma \[lem:oracle\] tells us that we will use ${\widetilde{\mathcal O}\left(\sqrt{m}\right)}$ calculations of ${{\mbox{\rm Tr}}\left(A_j\rho\right)}$, and the same order of other gates, to calculate a classical description of a $2$-sparse $y^{(t)}$. This brings the total cost of one iteration to $${\widetilde{\mathcal O}\left(\sqrt{nm} s^2\frac{\eta t^3 Rr}{{\varepsilon}}\right)}$$ queries to the input matrices, and the same order of other gates.
Total quantum runtime for SDPs.
: Since $w\leq r+1$ we can set $T = {\widetilde{\mathcal O}\left(\frac{R^2r^2}{{\varepsilon}^2}\right)}$. With $\eta = \sqrt{\frac{\ln(n)}{T}}$, summing over all iterations in one run of the algorithm gives a total cost of $$\begin{aligned}
{\widetilde{\mathcal O}\left(\sum_{t=1}^T \sqrt{nm} s^2\frac{\eta t^3 Rr}{{\varepsilon}}\right)} &= {\widetilde{\mathcal O}\left(\sqrt{nm} s^2\frac{\eta T^4 Rr}{{\varepsilon}}\right)} \\
&= {\widetilde{\mathcal O}\left(\sqrt{nm} s^2\left( \frac{Rr}{{\varepsilon}}\right)^{\!\!8} \right)}
\end{aligned}$$ queries to the input matrices and the same order of other gates.
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#### Total quantum runtime for LPs.
The final complexity of our algorithm contains a factor ${\widetilde{\mathcal O}\left(sT\right)}$ that comes from the sparsity of the $H^{(t)}$ matrix. This assumes that when we add the input matrices together, the rows become less sparse. This need not happen for certain SDPs. For example, in the SDP relaxation of MAXCUT, the $H^{(t)}$ will always be $d$-sparse, where $d$ is the degree of the graph. A more important class of examples is that of linear programs: since LPs have diagonal $A_j$ and $C$, their sparsity is $s=1$, and even the sparsity of the $H^{(t)}$ is always $1$. This, plus the fact that the traces can be computed without a factor ${\left\lVertH\right\rVert}$ in the complexity (as shown in Corollary \[col:traceDiagCalc\] in Section \[sec:lptrace\]), means that our algorithm solves LPs with $${\widetilde{\mathcal O}\left(\sqrt{nm} \left( \frac{Rr}{{\varepsilon}}\right)^{\!\!5} \right)}$$ queries to the input matrices and the same order of other gates.
#### Total classical runtime.
Using the classical techniques for trace estimation from Appendix \[app:trace\], and the classical version of our oracle (Lemma \[lem:oracle\]), we are also able to give a general classical instantiation of the Arora-Kale framework. The final complexity will then be $${\widetilde{\mathcal O}\left(nms\left(\frac{Rr}{{\varepsilon}}\right)^{\!\!4}+ns\left(\frac{Rr}{{\varepsilon}}\right)^{\!\!7}\right)}.$$ The better dependence on $Rr/{\varepsilon}$ and $s$, compared to our quantum algorithm, comes from the fact that we now have the time to write down intermediate results explicitly. For example, we do not need to recalculate parts of $H^{(t)}$ for every new query to it, instead we can just calculate it once at the start of the iteration by adding $M^{(t)}$ to $H^{(t-1)}$ and writing down the result.
#### Further remarks.
We want to stress again that our solver is meant to work for *all* SDPs. In particular, it does not use the structure of a specific SDP. As we show in the next section, every oracle that works for all SDPs must have large width. To obtain quantum speedups for a *specific* class of SDPs, it will be necessary to develop oracles tuned to that problem. We view this as an important direction for future work. Recall from the introduction that Arora and Kale also obtain fast classical algorithms for problems such as MAXCUT by doing exactly that: they develop specialized oracles for those problems.
Downside of this method: general oracles are restrictive {#sec:downside}
========================================================
In this section we show some of the limitations of a method that uses sparse or general oracles, i.e., ones that are not optimized for the properties of specific SDPs. We will start by discussing sparse oracles in the next section. We will use a counting argument to show that sparse solutions cannot hold too much information about a problem’s solution. In Section \[sec: general width bounds\] we will show that width-bounds that do not depend on the specific structure of an SDP are for many problems not efficient. As in the rest of the paper, we will assume the notation of Section \[sec:upperbounds\], in particular of Meta-Algorithm \[alg:AKSDP\].
Sparse oracles are restrictive
------------------------------
If, for some specific SDP, every ${\varepsilon}$-optimal dual-feasible vector has at least $\ell$ non-zero elements, then the width $w$ of any $k$-sparse oracle for this SDP is such that $\frac{Rw}{{\varepsilon}} = \Omega\left(\sqrt{\frac{\ell}{k\ln(n)}}\right)$.
The vector $\bar{y}$ returned by Meta-Algorithm \[alg:AKSDP\] is, by construction, the average of $T$ vectors $y^{(t)}$ that are all $k$-sparse, plus one extra $1$-sparse term of $\frac{{\varepsilon}}{R}e_1$, and hence $\ell \leq kT+1$. The stated bound on $\frac{Rw}{{\varepsilon}}$ then follows directly by combining this inequality with $T = {{\mathcal O}}(\frac{R^2w^2}{{\varepsilon}^2}\ln(n))$.
The oracle presented in Section \[sec:oracle\] always provides a $2$-sparse vector $y$. This implies that if an SDP requires an $\ell$-sparse dual solution, we must have $\frac{Rw}{{\varepsilon}} = \Omega(\sqrt{\ell / \ln(n)})$. This in turn means that the upper bound on the runtime of our algorithm will be of order $\ell^4 \sqrt{nm} s^2$. This is clearly bad if $\ell$ is of the order $n$ or $m$.
Of course it could be the case that almost every (useful) SDP has a sparse approximate dual solution (or can easily be rewritten so that it does), and hence sparseness might be not a restriction at all. However, this does not seem to be the case. We will prove that for certain kinds of SDPs, no useful dual solution can be very sparse. Let us first define what we mean by *useful*.
\[def:problem\] A problem is defined by a function $f$ that, for every element $p$ of the problem domain $\mathcal D$, gives a subset of the solution space $\mathcal S$, consisting of the solutions that are considered correct. We say a family of SDPs, $\{SDP^{(p)}\}_{p\in \mathcal D}$, solves the problem via the dual if there is an ${\varepsilon}\geq 0$ and a function $g$ such that for every $p \in \mathcal D$ and every ${\varepsilon}$-optimal dual-feasible vector $y^{(p)}$ to $SDP^{(p)}$: $$g(y^{(p)}) \in f(p).$$ In other words, an ${\varepsilon}$-optimal dual solution can be converted into a correct solution of the original problem without more knowledge of $p$.
For these kinds of SDP families we will prove a lower bound on the sparsity of the dual solutions. The idea for this bound is as follows. If you have a lot of different instances that require different solutions, but the SDPs are equivalent up to permuting the constraints and the coordinates of $\mathbb{R}^n$, then a dual solution should have a lot of unique permutations and hence cannot be too sparse.
Consider a problem and a family of SDPs as in Definition \[def:problem\]. Let $\mathcal T \subseteq \mathcal D$ be such that for all $p,q \in \mathcal T$:
- $f(p) \cap f(q) = \emptyset$. That is, a solution to $p$ is not a solution to $q$ and vice-versa.
- Let $A_j^{(p)}$ be the constraints of $SDP^{(p)}$ and $A_j^{(q)}$ those from $SDP^{(q)}$ (and define $C^{(p)}$, $C^{(q)}$, $b_j^{(p)}$, and $b_j^{(q)}$ in the same manner). Then there exist $\sigma \in S_n$, $\pi \in S_m$ s.t. $\sigma^{-1} A^{(p)}_{\pi(j)} \sigma = A^{(q)}_j$ (and $\sigma^{-1} C^{(p)} \sigma = C^{(q)}$). That is, the SDPs are the same up to permutations of the labels of the constraints and permutations of the coordinates of $\mathbb{R}^n$.
If $y^{(p)}$ is an ${\varepsilon}$-optimal dual-feasible vector to $SDP^{(p)}$ for some $p\in \mathcal T$, then $y^{(p)}$ is at least $\frac{\log(|\mathcal T|)}{\log m}$-dense (i.e., has at least that many non-zero entries).
We first observe that, with $SDP^{(p)}$ and $SDP^{(q)}$ as in the lemma, if $y^{(p)}$ is an ${\varepsilon}$-optimal dual-feasible vector of $SDP^{(p)}$, then $y^{(q)}$ defined by $$y^{(q)}_j := y^{(p)}_{\pi(j)} = \pi(y^{(p)})_j$$ is an ${\varepsilon}$-optimal dual vector for $SDP^{(q)}$. Here we use the fact that a permutation of the $n$ coordinates in the primal does not affect the dual solutions. Since $f(p)\cap f(q) = \emptyset$ we know that $g(y^{(p)}) \neq g(y^{(q)})$ and so $y^{(p)} \neq y^{(q)}$. Since this is true for every $q$ in $\mathcal T$, there should be at least $|\mathcal T|$ different vectors $y^{(q)} = \pi (y^{(p)})$.
A $k$-sparse vector can have $k$ different non-zero entries and hence the number of possible unique permutations of that vector is at most $$\binom{m}{k} k! = \frac{m!}{(m-k)!} = \prod_{t = m-k+1}^m t \leq m^k$$ so $$\frac{\log |\mathcal{T}|}{\log m} \leq k.$$ -5mm
#### Example.
Consider the $(s,t)$-mincut problem, i.e., the dual of the $(s,t)$-maxflow. Specifically, consider a simple instance of this problem: the union of two complete graphs of size $z+1$, where $s$ is in one subgraph and $t$ in the other. Let the other vertices be labeled by $\{1,2,\dots,2z\}$. Every assignment of the labels over the two halves gives a unique mincut, in terms of which labels fall on which side of the cut. There is exactly one partition of the vertices in two sets that cuts no edges (namely the partition consists of the two complete graphs), and every other partition cuts at least $z$ edges. Hence a $z/2$-approximate cut is a mincut. This means that there are $\binom{2z}{z}$ problems that require a different output. So for every family of SDPs that is symmetric under permutation of the vertices and for which a $z/2$-approximate dual solution gives an $(s,t)$-mincut, the sparsity of a $z/2$-approximate dual solution is at least[^10] $$\frac{\log {\binom{2z}{z}}}{\log m} \geq \frac{z}{ \log{m}},$$ where we used that $\binom{2z}{z} \geq \frac{2^{2z}}{2\sqrt{z}}$.
General width-bounds are restrictive for certain SDPs {#sec: general width bounds}
-----------------------------------------------------
In this section we will show that width-bounds can be restrictive when they do not consider the specific structure of an SDP.
A function $w(n,m,s,r,R,{\varepsilon})$ is called a *general width-bound* for an oracle if $w(n,m,s,r,R,{\varepsilon})$ is a correct width-bound for that oracle, for every SDP with parameters $n,m,s,r,R,{\varepsilon}$. In particular, the function $w$ may not depend on the structure of the $A_j$, $C$, and $b$.
We will show that general width-bounds need to scale with $r^{*}$ (recall that $r^{*}$ denotes the smallest $\ell_1$-norm of an optimal solution to the dual). We then go on to show that if two SDPs in a class can be combined to get another element of that class in a natural manner, then, under some mild conditions, $r^{*}$ will be of the order $n$ and $m$ for some instances of the class.
We start by showing, for specifically constructed LPs, a lower bound on the width of any oracle. Although these LPs will not solve any useful problem, every general width-bound should also apply to these LPs. This gives a lower bound on general width-bounds.
\[lem:genisrstar\] For every $n \geq 4$, $m\geq 4$, $s\geq 1$, $R^{*}>0$, $r^{*}>0$, and ${\varepsilon}\leq 1/2$ there is an SDP with these parameters for which every oracle has width at least $\frac{1}{2} r^{*}$.
We will construct an LP for $n=m=3$. This is enough to prove the lemma since LPs are a subclass of SDPs and we can increase $n$, $m$, and $s$ by adding more dimensions and $s$-dense SDP constraints that do not influence the analysis below. For some $k> 0$, consider the following LP $$\begin{aligned}
\max \ \ \ & (1,0,0) x\\
\text{s.t.} \ \ \ & \begin{bmatrix}
1 & 1 & 1\\
1/k & 1 & 0\\
-1 & 0 & -1
\end{bmatrix} x \leq \begin{bmatrix}
R\\
0\\
-R
\end{bmatrix}\\
& x \geq 0
\end{aligned}$$ where the first row is the primal trace constraint. Notice that $x_1 = x_2 = 0$ due to the second constraint. This implies that ${\mbox{\rm OPT}}= 0$ and, due to the last constraint, that $x_3\geq R$. Notice that $(0,0,R)$ is actually an optimal solution, so $R^{*} = R$.
To calculate $r^{*}$, look at the dual of the LP: $$\begin{aligned}
\min \ \ \ & (R,0,-R) y\\
\text{s.t.} \ \ \ & \begin{bmatrix}
1 & 1/k & -1\\
1 & 1 & 0\\
1 & 0 & -1
\end{bmatrix} y \geq \begin{bmatrix}
1\\
0\\
0
\end{bmatrix}\\
& y \geq 0,
\end{aligned}$$ due to strong duality its optimal value is $0$ as well. This implies $y_1 = y_3$, so the first constraint becomes $y_2 \geq k$. This in turn implies $r^{*}\geq k$, which is actually attained (by $y = (0,k,0)$) so $r^{*} = k$.
Since the oracle and width-bound should work for every $x\in \mathbb{R}^3_+$ and every $\alpha$, they should in particular work for $x = (R,0,0)$ and $\alpha = 0$. In this case the polytope for the oracle becomes $$\begin{aligned}
\mathcal{P}_{{\varepsilon}}(x) := \{ y \in \mathbb R^m:\ & y_1 - y_3 \leq 0, \\
& y_1-y_3+y_2/k\geq 1-{\varepsilon},\\
& y \geq 0 \}.
\end{aligned}$$ since $b^T y = y_1-y_3$, $c^T x = 1$, $a_1^T x = 1$, $a_2^T x = 1/k$ and $a_3^T x = -1$. This implies that for every $y\in \mathcal{P}_{{\varepsilon}}(x)$, we have $y_2 \geq k/(1-{\varepsilon}) \geq k/2 \geq r^{*}/2$.
Notice that the term $${\left\lVert\sum^m_{j=1} y_j A_j - C\right\rVert}$$ in the definition of width for an SDP becomes $${\left\lVert A^T y - c\right\rVert}_{\infty}$$ in the case of an LP. In our case, due to the second constraint in the dual, we know that $${\left\lVert A^T y - c\right\rVert}_{\infty} \geq y_1 + y_2 \geq \frac{r^{*}}{2}$$ for every vector $y$ from $\mathcal{P}_{{\varepsilon}}(x)$. This shows that any oracle has width at least $r^{*}/2$ for this LP.
For every general width-bound $w(n,m,s,r,R,{\varepsilon})$, if $n,m\geq 3$, $s\geq 1$, $r>0$, $R>0$, and ${\varepsilon}\leq 1/2$, then $$w(n,m,s,r,R,{\varepsilon}) \geq \frac{r}{2}.$$
Note that this bound applies to both our algorithm and the one given by Brandão and Svore. It turns out that for many natural classes of SDPs, $r^{*}, R^{*}$, ${\varepsilon}$, $n$ and $m$ can grow linearly for some instances. In particular, this is the case if SDPs in a class combine in a natural manner. Take for example two SDP relaxations for the MAXCUT problem on two graphs $G^{(1)}$ and $G^{(2)}$ (on $n^{(1)}$ and $n^{(2)}$ vertices, respectively):
$$\begin{aligned}
\max \quad & {{\mbox{\rm Tr}}\left(L(G^{(1)})X^{(1)}\right)} \\
\text{s.t.}\ \ \ & {{\mbox{\rm Tr}}\left(X^{(1)}\right)} \leq n^{(1)}\\
& {{\mbox{\rm Tr}}\left(E_{jj} X^{(1)}\right)} \leq 1 \text{ for }j=1,\dots,n^{(1)}\\
&X^{(1)} \succeq 0
\end{aligned}$$
$$\begin{aligned}
\max \quad & {{\mbox{\rm Tr}}\left(L(G^{(2)})X^{(2)}\right)} \\
\text{s.t.}\ \ \ & {{\mbox{\rm Tr}}\left(X^{(2)}\right)} \leq n^{(2)}\\
& {{\mbox{\rm Tr}}\left(E_{jj} X^{(2)}\right)} \leq 1 \text{ for }j=1,\dots,n^{(2)}\\
&X^{(2)} \succeq 0
\end{aligned}$$
Where $L(G)$ is the Laplacian of a graph. Note that this is not normalized to operator norm $\leq 1$, but for simplicity we ignore this here. If we denote the direct sum of two matrices by $\oplus$, that is $$A\oplus B = \begin{bmatrix}A &0\\0&B\end{bmatrix},$$ then, for the disjoint union of the two graphs, we have $$L(G^{(1)} \cup G^{(2)}) = L(G^{(1)}) \oplus L(G^{(2)}).$$ This, plus the fact that the trace distributes over direct sums of matrices, means that the SDP relaxation for MAXCUT on $G^{(1)} \cup G^{(2)}$ is the same as a natural combination of the two separate maximizations: $$\begin{aligned}
\max \quad & {{\mbox{\rm Tr}}\left(L(G^{(1)})X^{(1)}\right)} + {{\mbox{\rm Tr}}\left(L(G^{(2)})X^{(2)}\right)} \\
\text{s.t.}\ \ \ & {{\mbox{\rm Tr}}\left(X^{(1)}\right)}+ {{\mbox{\rm Tr}}\left(X^{(2)}\right)} \leq n^{(1)}+n^{(2)}\\
& {{\mbox{\rm Tr}}\left(E_{jj} X^{(1)}\right)} \leq 1 \text{ for }j=1,\dots,n^{(1)}\\
& {{\mbox{\rm Tr}}\left(E_{jj} X^{(2)}\right)} \leq 1 \text{ for }j=1,\dots,n^{(2)}\\
&X^{(1)},X^{(2)} \succeq 0.\end{aligned}$$ It is easy to see that the new value of $n$ is $n^{(1)}+n^{(2)}$, the new value of $m$ is $m^{(1)}+m^{(2)}-1$ and the new value of $R^{*}$ is $n^{(1)}+n^{(2)} = R^{* (1)}+R^{* (2)}$. Since it is natural for the MAXCUT relaxation that the additive errors also add, it remains to see what happens to $r^{*}$, and so, for general width-bounds, what happens to $w$. As we will see later in this section, under some mild conditions, these kind of combinations imply that there are MAXCUT-relaxation SDPs for which $r^{*}$ also increases linearly, but this requires a bit more work.
\[def:combining\] We say a class of SDPs (with associated allowed approximation errors) is *combinable* if there is a $k\geq 0$ so that for every two elements in this class, $(SDP^{(a)},{\varepsilon}^{(a)})$ and $(SDP^{(b)},{\varepsilon}^{(b)})$, there is an instance in the class, $(SDP^{(c)},{\varepsilon}^{(c)})$, that is a combination of the two in the following sense:
- $C^{(c)} = C^{(a)} \oplus C^{(b)}$.
- $A^{(c)}_j = A^{(a)}_j \oplus A^{(b)}_j$ and $b_j^{(c)} = b_j^{(a)}+b_j^{(b)}$ for $j \in \lbrack k \rbrack$.
- $A^{(c)}_{j} = A^{(a)}_{j}\oplus \mathbf{0}$ and $b^{(c)}_{j} = b^{(a)}_{j}$ for $j = k+1,\dots,m^{(a})$.
- $A^{(c)}_{m^{(a)}+j-k} = \mathbf{0} \oplus A^{(b)}_{j}$ and $b^{(c)}_{m^{(a)}+j-k} = b^{(b)}_{j}$ for $j = k+1,\dots,m^{(b)}$.
- ${\varepsilon}^{(c)} \leq {\varepsilon}^{(a)}+{\varepsilon}^{(b)}$.
In other words, some fixed set of constraints are summed pairwise, and the remaining constraints get added separately.
Note that this is a natural generalization of the combining property of the MAXCUT relaxations (in that case $k=1$ to account for the trace bound).
If a class of SDPs is combinable and there is an element $SDP^{(1)}$ for which every optimal dual solution has the property that $$\sum_{j=k+1}^m y_m \geq \delta$$ for some $\delta >0$, then there is a sequence $(SDP^{(t)})_{t\in \mathbb{N}}$ in the class such that $\frac{R^{* (t)}r^{* (t)}}{{\varepsilon}^{(t)}}$ increases linearly in $n^{(t)}$, $m^{(t)}$ and $t$.
The sequence we will consider is the $t$-fold combination of $SDP^{(1)}$ with itself. If $SDP^{(1)}$ is
$$\begin{aligned}
\max \quad &{\mbox{\rm Tr}}(CX) \\
\text{s.t.}\ \ \ &{\mbox{\rm Tr}}(A_j X) \leq b_j \quad \text{ for } j \in [m^{(1)}], \\
&X \succeq 0
\end{aligned}$$
$$\begin{aligned}
\min \quad &\sum_{j=1}^{m^{(1)}} b_j y_j \\
\text{s.t.}\ \ \ &\sum_{j=1}^{m^{(1)}} y_j A_j - C \succeq 0,\\
&y \geq 0
\end{aligned}$$
then $SDP^{(t)}$ is $$\begin{aligned}
\max \quad &\ \sum_{i=1}^t {{\mbox{\rm Tr}}\left(CX_i\right)} & &\\
\text{s.t.}\ \ \ & \sum_{i=1}^t {\mbox{\rm Tr}}(A_j X_i) \leq t b_j &&\text{ for } j \in [k], \\
& {\mbox{\rm Tr}}(A_j X_i) \leq b_j &&\text{ for } j = k+1 , \dots , m^{(1)} \text{ and } i = 1 , \dots , t \\
&X_i \succeq 0 &&\text{ for all } i = 1, \dots,t
\end{aligned}$$ with dual $$\begin{aligned}
\min \quad &\sum_{j=1}^k t b_j y_j + \sum_{i=1}^t \sum_{j=k+1}^{m^{(1)}} b_j y_j^i\\
\text{s.t.}\ \ \ &\sum_{j=1}^k y_j A_j + \sum_{j=k+1}^{m^{(1)}} y_j^i A_j \succeq C \text{ for } i = 1,\dots,t\\
&y,y^i \geq 0.
\end{aligned}$$ First, let us consider the value of ${\mbox{\rm OPT}}^{(t)}$. Let $X^{(1)}$ be an optimal solution to $SDP^{(1)}$ and for all $i \in \lbrack t \rbrack$ let $X_i = X^{(1)}$. Since these $X_i$ form a feasible solution to $SDP^{(t)}$, this shows that ${\mbox{\rm OPT}}^{(t)} \geq t\cdot {\mbox{\rm OPT}}^{(1)}$. Furthermore, let $y^{(1)}$ be an optimal dual solution of $SDP^{(1)}$, then $(y^{(1)}_1,\dots,y^{(1)}_k)\oplus \left( y^{(1)}_{k+1},\cdots,y^{(1)}_{m^{(1)}}\right)^{\oplus t}$ is a feasible dual solution for $SDP^{(t)}$ with objective value $t\cdot {\mbox{\rm OPT}}^{(1)}$, so ${\mbox{\rm OPT}}^{(t)} = t \cdot {\mbox{\rm OPT}}^{(1)}$.
Next, let us consider the value of $r^{* (t)}$. Let $\tilde{y} \oplus y^1 \oplus \dots \oplus y^t$ be an optimal dual solution for $SDP^{(t)}$, split into the parts of $y$ that correspond to different parts of the combination. Then $\tilde{y} \oplus y^i$ is a feasible dual solution for $SDP^{(1)}$ and hence $b^T ( \tilde{y}\oplus y^i) \geq {\mbox{\rm OPT}}^{(1)}$. On the other hand we have $$t \cdot {\mbox{\rm OPT}}^{(1)} = {\mbox{\rm OPT}}^{(t)} = \sum_{i=1}^t b^T (\tilde{y} \oplus y^i),$$ this implies that each term in the sum is actually equal to ${\mbox{\rm OPT}}^{(1)}$. But if $(\tilde{y}\oplus y^i)$ is an optimal dual solution of $SDP^{(1)}$ then ${\left\lVert(\tilde{y}\oplus y^i)\right\rVert}_1 \geq r^{* (1)}$ by definition and ${\left\lVerty^i\right\rVert}_1 \geq \delta$. We conclude that $r^{* (t)} \geq r^{* (1)} - \delta + t\delta$.
Now we know the behavior of $r^{*}$ under combinations, let us look at the primal to find a similar statement for $R^{*(t)}$. Define a new SDP, $\widehat{SDP}^{(t)}$, in which all the constraints are summed when combining, that is, in Definition \[def:combining\] we take $k = n^{(1)}$, however, contrary to that definition, we even sum the psd constraints: $$\begin{aligned}
\max \quad &\ \sum_{i=1}^t {{\mbox{\rm Tr}}\left(CX_i\right)} \\
\text{s.t.}\ \ \ & \sum_{i=1}^t {\mbox{\rm Tr}}(A_j X_i) \leq t b_j \quad \text{ for } j \in [m^{(1)}], \\
&\sum_{i=1}^t X_i \succeq 0.
\end{aligned}$$ This SDP has the same objective function as $SDP^{(t)}$ but a larger feasible region: every feasible $X_1,\dots,X_t$ for $SDP^{(t)}$ is also feasible for $\widehat{SDP}^{(t)}$. However, by a change of variables, $X := \sum_{i=1}^t X_i$, it is easy to see that $\widehat{SDP}^{(t)}$ is simply a scaled version of $SDP^{(1)}$. So, $\widehat{SDP}^{(t)}$ has optimal value $t\cdot {\mbox{\rm OPT}}^{(1)}$. Since optimal solutions to $\widehat{SDP}^{(t)}$ are scaled optimal solutions to $SDP^{(1)}$, we have $\hat{R}^{*(t)} = t\cdot R^{* (1)}$. Combining the above, it follows that every optimal solution to $SDP^{(t)}$ is optimal to $\widehat{SDP}^{(t)}$ as well, and hence has trace at least $t\cdot R^{*(1)}$, so $R^{* (t)}\geq t\cdot R^{* (1)}$.
We conclude that $$\frac{
R^{*(t)} r^{*(t)}
}{
{\varepsilon}^{(t)}
}
\geq
\frac{
tR^{* (1)}
(r^{* (1)}+(t-1)\delta)
}{
t{\varepsilon}^{(1)}}
= \Omega\left( t \right)$$ and $n^{(t)} = t n^{(1)}$, $m^{(t)} = t ( m^{(1)}-k)+k$.
This shows that for many natural SDP formulations for combinatorial problems, such as the MAXCUT relaxation or LPs that have to do with resource management, $R^{*}r^{*}/{\varepsilon}$ increases linearly in $n$ and $m$ for some instances. Hence, using $R^{*}\leq R$ and Lemma \[lem:genisrstar\], $Rw/{\varepsilon}$ grows at least linearly when a general width-bound is used.
Lower bounds on the quantum query complexity {#sec:lowerbounds}
============================================
In this section we will show that every LP-solver (and hence every SDP-solver) that can distinguish two optimal values with high probability needs $\Omega\left(\sqrt{\max\{n,m\}} \left( \min\{n,m\} \right)^{3/2} \right)$ quantum queries in the worst case.
For the lower bound on LP-solving we will give a reduction from a composition of Majority and OR functions.
Given input bits $Z_{ij\ell} \in \{0,1\}^{a\times b\times c}$ the problem of calculating $$\begin{aligned}
MAJ_a(&\\
&OR_b(MAJ_c(Z_{111},\dots,Z_{11c}),\dots,MAJ_c(Z_{1b1},\dots,Z_{1bc})),\\
&\dots,\\
&OR_b(MAJ_c(Z_{a11},\dots,Z_{a1c}),\dots,MAJ_c(Z_{ab1},\dots,Z_{abc}))\\
)&
\end{aligned}$$ with the promise that
- Each inner ${\mbox{\rm MAJ}}_c$ is a boundary case, in other words $\sum_{\ell=1}^c Z_{ij\ell} \in \{c/2,c/2+1\}$ for all $i,j$.
- The outer ${\mbox{\rm MAJ}}_a$ is a boundary case, in other words, if $\tilde{Z} \in \{0,1\}^a$ is the bitstring that results from all the OR calculations, then $|\tilde{Z}| \in \{a/2,a/2+1\}$.
is called the promise ${{\mbox{\rm MAJ}}_a\mhyphen{\mbox{\rm OR}}_b\mhyphen{\mbox{\rm MAJ}}_c}$ problem.
\[lem:knownlow\] It takes at least $\Omega(a\sqrt{b} c)$ queries to the input to solve the promise ${{\mbox{\rm MAJ}}_a\mhyphen{\mbox{\rm OR}}_b\mhyphen{\mbox{\rm MAJ}}_c}$ problem.
The promise version of ${\mbox{\rm MAJ}}_k$ is known to require $\Omega(k)$ quantum queries. Likewise, it is known that the OR$_k$ function requires $\Omega(\sqrt{k})$ queries. Furthermore, the adversary bound tells us that query complexity is multiplicative under composition of functions (even promise functions). For completeness we include a proof of this composition property as Theorem \[thm:adv\] in Appendix \[app:adversarycomposition\]. The proof is a straightforward modification of a proof of Lee et al. [@lmrss:stateconv Section 5] for the case where the outer function is a total Boolean function; Kimmel [@Kimmel:adv] already gave a similar modification of the proof of Lee et al. for the case where a promise function is composed with itself.
\[lem:calculation\] Determining the value $$\sum_{i=1}^a \max_{j \in \lbrack b\rbrack} \sum_{\ell=1}^c Z_{ij\ell}$$ for a $Z$ from the promise ${{\mbox{\rm MAJ}}_a\mhyphen{\mbox{\rm OR}}_b\mhyphen{\mbox{\rm MAJ}}_c}$ problem up to additive error ${\varepsilon}= 1/3$, solves the promise ${{\mbox{\rm MAJ}}_a\mhyphen{\mbox{\rm OR}}_b\mhyphen{\mbox{\rm MAJ}}_c}$ problem.
Notice that due to the first promise, $\sum_{\ell=1}^c Z_{ij\ell} \in \{c/2,c/2+1\}$ for all $i\in [a],j\in[b]$. This implies that
- If the $i$th OR is $0$, then all of its inner MAJ functions are $0$ and hence $$\max_{j \in \lbrack b\rbrack} \sum_{\ell=1}^c Z_{ij\ell} = \frac{c}{2}$$
- If the $i$th OR is $1$, then at least one of its inner MAJ functions is $1$ and hence $$\max_{j \in \lbrack b\rbrack} \sum_{\ell=1}^c Z_{ij\ell} = \frac{c}{2} + 1$$
Now, if we denote the string of outcomes of the OR functions by $\tilde{Z}\in \{0,1\}^a$, then $$\sum_{i=1}^a \max_{j \in \lbrack b\rbrack} \sum_{\ell=1}^c Z_{ij\ell} = a \frac{c}{2} + |\tilde{Z}|$$ Hence determining the left-hand side will determine $|\tilde{Z}|$; this Hamming weight is either $\frac{a}{2}$ if the full function evaluates to $0$, or $\frac{a}{2}+1$ if it evaluates to $1$.
\[lem:generallp\] For an input $Z\in \{0,1\}^{a\times b\times c}$ there is an LP with $m = c+a$ and $n = c+ab$ for which the optimal value is $$\sum_{i=1}^a \max_{j \in \lbrack b\rbrack} \sum_{\ell=1}^c Z_{ij\ell}$$ Furthermore, a query to an entry of the input matrix or vector costs at most $1$ query to $Z$.
Let $Z^{(i)}$ be the matrix one gets by fixing the first index of $Z$ and putting the entries in a $c\times b$ matrix, so $Z^{(i)}_{\ell j} = Z_{ij\ell}$. We define the following LP: $$\begin{aligned}
{\mbox{\rm OPT}}= \text{max} \ \ \ & \sum_{k=1}^c w_k\\
\text{s.t.} \ \ \ &
\begin{bmatrix}
I & -Z^1 & \cdots & -Z^a \\
0 & \mathbf{1}^T & &\\
0 & & \ddots &\\
0 & & & \mathbf{1}^T
\end{bmatrix}
\begin{bmatrix}
w\\
v^{(1)}\\
\vdots\\
v^{(a)}\\
\end{bmatrix}
\leq
\begin{bmatrix}
0\\
\mathbf{1}\\
\vdots\\
\mathbf{1}
\end{bmatrix}\\
& v^1,\dots,v^a \in \mathbb{R}_{+}^b ,w \in \mathbb{R}_{+}^c
\end{aligned}$$ Notice every $Z^{(i)}$ is of size $c\times b$, so that indeed $m = c+a$ and $n = c + ab$.
For every $i \in [a]$ there is a constraint that says $$\sum_{j=1}^b v^{(i)}_j \leq 1.$$ The constraints involving $w$ say that for every $k\in[c]$ $$w_k \leq \sum_{i=1}^a \sum_{j=1}^b v^{(i)}_j Z^{(i)}_{k j} = \sum_{i=1}^a ( Z^{(i)} v^{(i)} )_k$$ where $( Z^{(i)} v^{(i)} )_k$ is the $k$th entry of the matrix-vector product $Z^{(i)} v^{(i)}$. Clearly, for an optimal solution these constraints will be satisfied with equality, since in the objective function $w_k$ has a positive weight. Summing over $k$ on both sides, we get the equality $$\begin{aligned}
{\mbox{\rm OPT}}&= \sum_{k = 1}^c w_{k}\\
&= \sum_{k = 1}^c \sum_{i=1}^a ( Z^{(i)} v^{(i)} )_{k}\\
&= \sum_{i=1}^a \sum_{k = 1}^c ( Z^{(i)} v^{(i)} )_{k}\\
&= \sum_{i=1}^a {\left\lVert Z^{(i)} v^{(i)}\right\rVert}_1
\end{aligned}$$ so in the optimum ${\left\lVert Z^{(i)} v^{(i)}\right\rVert}_1$ will be maximized. Note that we can use the $\ell_1$-norm as a shorthand for the sum over vector elements since all elements are positive. In particular, the value of ${\left\lVert Z^{(i)} v^{(i)}\right\rVert}_1$ is given by $$\begin{aligned}
\text{max} \ \ \ & {\left\lVert Z^{(i)} v^{(i)}\right\rVert}_1\\
\text{s.t.} \ \ \ & {\left\lVertv^{(i)}\right\rVert}_1\leq 1\\
& v^{(i)}\geq 0
\end{aligned}$$ Now ${\left\lVert\smash{ Z^{(i)} v^{(i)}}\right\rVert}_1$ will be maximized by putting all weight in $v^{(i)}$ on the index that corresponds to the column of $Z^{(i)}$ that has the highest Hamming weight. In particular in the optimum ${\left\lVert \smash{Z^{(i)} v^{(i)}}\right\rVert}_1 = \max_{j\in \lbrack b \rbrack} \sum_{\ell = 1}^c Z^{(i)}_{\ell j}$. Putting everything together gives: $${\mbox{\rm OPT}}= \sum_{i=1}^a {\left\lVert Z^{(i)} v^{(i)}\right\rVert}_1 = \sum_{i=1}^a \max_{j\in \lbrack b \rbrack} \sum_{\ell = 1}^c Z^{(i)}_{\ell j} = \sum_{i=1}^a \max_{j\in \lbrack b \rbrack} \sum_{\ell = 1}^c Z_{i j \ell}$$ -5mm
\[thm:lowbound\] There is a family of LPs, with $m \leq n$ and two possible integer optimal values, that require at least $\Omega(\sqrt{n}m^{3/2})$ quantum queries to the input to distinguish those two values.
Let $a = c = m/2$ and $b = \frac{n-c}{a} = \frac{2n}{m} - 1$, so that $n =c+ab$ and $m=c+a$. By Lemma \[lem:generallp\] there exists an LP with $n =c+ab$ and $m=c+a$ that calculates $$\sum_{i=1}^a \max_{j \in \lbrack b\rbrack} \sum_{\ell=1}^c Z_{ij\ell}$$ for an input $Z$ to the promise ${{\mbox{\rm MAJ}}_a\mhyphen{\mbox{\rm OR}}_b\mhyphen{\mbox{\rm MAJ}}_c}$ problem. By Lemma \[lem:calculation\], calculating this value will solve the promise ${{\mbox{\rm MAJ}}_a\mhyphen{\mbox{\rm OR}}_b\mhyphen{\mbox{\rm MAJ}}_c}$ problem. By Lemma \[lem:knownlow\] the promise ${{\mbox{\rm MAJ}}_a\mhyphen{\mbox{\rm OR}}_b\mhyphen{\mbox{\rm MAJ}}_c}$ problem takes $\Omega(a\sqrt{b}c)$ quantum queries in the worst case. This implies a lower bound of $$\Omega\left(m^2\sqrt{\frac{n}{m}}\right) = \Omega(m^{3/2}\sqrt{n})$$ quantum queries on solving these LPs.
Distinguishing two optimal values of an LP (and hence also of an SDP) with additive error ${\varepsilon}<1/2$ requires $$\Omega\left(\sqrt{\max\{n,m\}} \left( \min\{n,m\} \right)^{3/2} \right)$$ quantum queries to the input matrices in the worst case.
This follows from Theorem \[thm:lowbound\] and LP duality.
It is important to note that the parameters $R$ and $r$ from the Arora-Kale algorithm are not constant in this family of LPs ($R,r = \Theta(\min\{n,m\}^2)$ here), and hence this lower bound does not contradict the scaling with $\sqrt{mn}$ of the complexity of our SDP-solver or Brandão and Svore’s. Since we show in the appendix that one can always rewrite the LP (or SDP) so that $2$ of the parameters $R,r,{\varepsilon}$ are constant, the lower bound implies that any algorithm with a sub-linear dependence on $m$ or $n$ has to depend at least polynomially on $Rr/{\varepsilon}$. For example, the above family of LPs shows that an algorithm with a $\sqrt{mn}$ dependence has to have an $(Rr/{\varepsilon})^\kappa$ factor in its complexity with $\kappa\geq 1/4$. It remains an open question whether a lower bound of $\Omega(\sqrt{mn})$ can be proven for a family of LPs/SDPs where ${\varepsilon}$, $R$ and $r$ all constant.
Conclusion
==========
In this paper we gave better algorithms and lower bounds for quantum SDP-solvers, improving upon recent work of Brandão and Svore . Here are a few directions for future work:
- [**Better upper bounds.**]{} The runtime of our algorithm, like the earlier algorithm of Brandão and Svore, has better dependence on $m$ and $n$ than the best classical SDP-solvers, but worse dependence on $s$ and on $Rr/{\varepsilon}$. It may be possible to improve the dependence on $s$ to linear and/or the dependence on $Rr/{\varepsilon}$ to less than our current 8th power.
- [**Applications of our algorithm.**]{} As mentioned, both our and Brandão-Svore’s quantum SDP-solvers only improve upon the best classical algorithms for a specific regime of parameters, namely where $mn\gg Rr/{\varepsilon}$. Unfortunately, we don’t know particularly interesting problems in combinatorial optimization in this regime. As shown in Section \[sec:downside\], many natural SDP formulations will not fall into this regime. However, it would be interesting to find useful SDPs for which our algorithm gives a significant speed-up.
- [**New algorithms.**]{} As in the work by Arora and Kale, it might be more promising to look at oracles (now quantum) that are designed for specific SDPs. Such oracles could build on the techniques developed here, or develop totally new techniques. It might also be possible to speed up other classical SDP solvers, for example those based on interior-point methods.
- [**Better lower bounds.**]{} Our lower bounds are probably not optimal, particularly for the case where $m$ and $n$ are not of the same order. The most interesting case would be to get lower bounds that are simultaneously tight in the parameters $m$, $n$, $s$, and $Rr/{\varepsilon}$.
#### Acknowledgments.
We thank Fernando Brandão for sending us several drafts of and for answering our many questions about their algorithms, Stacey Jeffery for pointing us to [@Kimmel:adv], and Andris Ambainis and Robin Kothari for useful discussions and comments. We also thank the anonymous FOCS’17 referees for helpful comments that improved the presentation.
Classical estimation of the expectation value traces {#app:trace}
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To provide contrast to Section \[sec:trCalc\], here we describe a *classical* procedure to efficiently estimate $\mathrm{Tr}(A \rho)$ where $A$ is a Hermitian matrix such that ${\left\lVertA\right\rVert} \leq 1$, and $\rho = \exp(-H)/{{\mbox{\rm Tr}}\left(\exp(-H)\right)}$ for some Hermitian matrix $H$. The results in this section can be seen as a generalization of . The key observation is that if we are given a Hermitian matrix $B\succeq 0$, and if we take a random vector $u = (u_1, \ldots, u_n)$ where $u_i \in \{\pm 1\}$ is uniformly distributed, then, using $\mathbb E[u_i] = 0$, $\mathbb E[u_i^2] = 1$, we have $$\begin{aligned}
\mathbb E \left[ u^T \sqrt{B} A \sqrt{B} u \right] &= \mathbb E \left[ {{\mbox{\rm Tr}}\left(\sqrt{B} A \sqrt{B} u u^T\right)} \right] = {{\mbox{\rm Tr}}\left(\sqrt{B} A \sqrt{B} \mathbb E[uu^T]\right)} \\
&= {{\mbox{\rm Tr}}\left(\sqrt{B} A \sqrt{B} I\right)} = \mathrm{Tr}(A B).\end{aligned}$$ We now show that $u^T \sqrt{B} A \sqrt{B} u$ is highly concentrated around its mean by Chebyshev’s inequality.
\[lem:JLherm0\] Given a Hermitian matrix $A$, with $||A|| \leq 1$, a psd matrix $B$, and a parameter $0 < \theta \leq 1$. With probability $1-1/16$, the average of $k = {{\mathcal O}}\left(1/\theta^2\right)$ independent samples from the distribution $u^T \sqrt{B} A \sqrt{B} u$ is at most $\theta {{\mbox{\rm Tr}}\left(B\right)}$ away from ${{\mbox{\rm Tr}}\left(AB\right)}$. Here $u = (u_i)$ and each $u_i \in \{\pm 1\}$ is i.i.d. uniformly distributed.
We let $F_k$ be the random variable $\frac{1}{k} \sum_{i=1}^k (u^{(i)})^T \sqrt{B} A \sqrt{B} u^{(i)}$, where each of the vectors $u^{(i)} \in \{\pm 1\}^n$ is sampled from the distribution described above. By the above it is clear that $\mathbb{E}[F_k] = \mathrm{Tr}(AB)$.
We will use Chebyshev’s inequality which, in our setting, states that for every $t >0$ $$\label{eq:chebyshev}
\mathrm{Pr}\big( |F_k - {{\mbox{\rm Tr}}\left(AB\right)}| \geq t \sigma_k \big) \leq \frac{1}{t^2},$$ here $\sigma_k^2$ is the variance of $F_k$. We will now upper bound the variance of $F_k$. First note that $\mathrm{var}(F_k) = \frac{1}{k} \mathrm{var}(u^T \sqrt{B} A \sqrt{B} u)$. It therefore suffices to upper bound the variance $\sigma^2$ of $u^T \sqrt{B} A \sqrt{B} u$. We first write $$\begin{aligned}
\sigma^2 &= \mathrm{var}(u^T \sqrt{B} A \sqrt{B} u) = \mathbb{E} \left[(u^T \sqrt{B} A \sqrt{B} u)^2\right] - \mathbb{E}\left[u^T \sqrt{B} A \sqrt{B} u\right]^2 \\
&=\underbrace{\mathbb{E} \left[ \sum_{i,j,k,l = 1}^n u_i u_j u_k u_l (\sqrt{B} A \sqrt{B})_{ij} (\sqrt{B} A \sqrt{B})_{kl} \right]}_{(*)} - \mathrm{Tr}(\sqrt{B} A \sqrt{B})^2.
\end{aligned}$$ We then calculate $(*)$ using $\mathbb{E}[u_i] = 0$, $\mathbb{E}[u_i^2]=1$, and the independence of the $u_i$’s: $$\begin{aligned}
(*) &= \sum_{i \neq j} (\sqrt{B} A \sqrt{B})_{ij} \left((\sqrt{B} A \sqrt{B})_{ij} + (\sqrt{B} A \sqrt{B})_{ji}\right) + \sum_{i,k=1}^n (\sqrt{B} A \sqrt{B})_{ii} (\sqrt{B} A \sqrt{B})_{kk} \\
&= \sum_{i \neq j} 2(\sqrt{B} A \sqrt{B})_{ij}^2 + \mathrm{Tr}(\sqrt{B} A \sqrt{B})^2.
\end{aligned}$$ Therefore, using Cauchy-Schwarz, we have $$\begin{aligned}
\sigma^2 &= (*) - \mathrm{Tr}(\sqrt{B} A \sqrt{B})^2 = \sum_{i \neq j} 2(\sqrt{B} A \sqrt{B})_{ij}^2 \leq \sum_{i,j} 2(\sqrt{B} A \sqrt{B})_{ij}^2\\
&= 2 \mathrm{Tr}((\sqrt{B} A \sqrt{B})^2) = 2 |\langle ABA, B\rangle | \leq 2 |\langle ABA,ABA\rangle|^{1/2} |\langle B,B\rangle|^{1/2} \\
&= 2 {{\mbox{\rm Tr}}\left(A^2 B A^2 B\right)}^{1/2} {{\mbox{\rm Tr}}\left(B^2\right)}^{1/2} \leq 2 {{\mbox{\rm Tr}}\left(B A^2 B\right)}^{1/2} {{\mbox{\rm Tr}}\left(B^2\right)}^{1/2} \leq 2 {{\mbox{\rm Tr}}\left(B^2\right)} \leq 2 {{\mbox{\rm Tr}}\left(B\right)}^2,
\end{aligned}$$ where on the last line we use ${\left\lVertA\right\rVert} \leq 1$ and ${{\mbox{\rm Tr}}\left(A Y\right)} \leq {\left\lVertA\right\rVert} {{\mbox{\rm Tr}}\left(Y\right)}$ for any $Y \succeq 0$, in particular for $BA^2B$ and $B^2$.
It follows that $\sigma_k^2 \leq 2 {{\mbox{\rm Tr}}\left(B\right)}^2/k$. Chebyshev’s inequality therefore shows that for $k = \lceil 32 /\theta^2\rceil $ and $t = 4$, $$\mathrm{Pr}\big( |F_k - {{\mbox{\rm Tr}}\left(AB\right)}| \geq \theta {{\mbox{\rm Tr}}\left(B\right)} \big) \leq \frac{1}{16}.$$ -5mm
A simple computation shows that the success probability in the above lemma can be boosted to $1-\delta$ by picking the median of ${{\mathcal O}}(\log(1/\delta))$ repetitions. To show this, let $K = \lceil \log(1/\delta)\rceil$ and for each $i \in [K]$ let $(F_k)_i$ be the average of $k$ samples of $u^T \sqrt{B} A \sqrt{B} u$. Let $z_K$ denote the median of those $K$ numbers. We have $$\mathrm{Pr}\big(|z_K - {{\mbox{\rm Tr}}\left(AB\right)}| \geq \theta {{\mbox{\rm Tr}}\left(B\right)}\big) = \underbrace{\mathrm{Pr}\big( z_K \geq \theta {{\mbox{\rm Tr}}\left(B\right)} + {{\mbox{\rm Tr}}\left(AB\right)} \big)}_{(*)} + \mathrm{Pr}\big( z_K \leq - \theta {{\mbox{\rm Tr}}\left(B\right)} + {{\mbox{\rm Tr}}\left(AB\right)} \big)$$ We upper bound $(*)$: $$\begin{aligned}
(*) &\leq \sum_{I \subseteq [K]: |I| \geq K/2} \prod_{i \in I} \mathrm{Pr}\big( (F_k)_i \geq \theta {{\mbox{\rm Tr}}\left(B\right)} + {{\mbox{\rm Tr}}\left(AB\right)} \big) \\
&\leq (|\{I \subseteq [K]: |I| \geq K/2 \}| ) \left(\frac{1}{16}\right)^{K/2} \\
&= 2^{K-1}\left(\frac{1}{4}\right)^K \\
&\leq \frac{1}{2}\left(\frac{1}{2}\right)^{\log(1/\delta)} = \frac{1}{2} \delta.\end{aligned}$$ Analogously, one can show that $\mathrm{Pr}\big( z_K \leq - \theta {{\mbox{\rm Tr}}\left(B\right)} + {{\mbox{\rm Tr}}\left(AB\right)} \big) \leq \frac{1}{2} \delta$. Hence $$\mathrm{Pr}\big(|z_K - {{\mbox{\rm Tr}}\left(AB\right)}| \geq \theta {{\mbox{\rm Tr}}\left(B\right)}\big) \leq \delta.$$ This proves the following lemma:
\[lem:JLherm\] Given a Hermitian matrix $A$, with $||A|| \leq 1$, a psd matrix $B$, and parameters $0 < \delta \leq 1/2$ and $0 < \theta \leq 1$. Using $k = {{\mathcal O}}\left(\log(\frac{1}{\delta})/\theta^2\right)$ samples from the distribution $u^T \sqrt{B} A \sqrt{B} u$, one can find an estimate of ${{\mbox{\rm Tr}}\left(AB\right)}$ that, with probability $1-\delta$, has additive error at most $\theta {{\mbox{\rm Tr}}\left(B\right)}$. Here $u = (u_i)$ and the $u_i \in \{\pm 1\}$ are i.i.d. uniformly distributed.
Looking back at Meta-Algorithm \[alg:AKSDP\], we would like to apply the above lemma to $B =\exp(- H)$.[^11] Since it is expensive to compute the exponent of a matrix, it is of interest to consider samples from $v^T A v$, where $v$ is an approximation of $\sqrt{B} u$.
Say ${\left\lVert\sqrt{B} u -v\right\rVert} \leq \kappa$ and, as always, ${\left\lVertA\right\rVert} \leq 1$. Then $$\begin{aligned}
|u^T \sqrt{B} A \sqrt{B} u - v^T Av| &= | u^T \sqrt{B} A \sqrt{B} u - u^T \sqrt{B} A v + u^T \sqrt{B} A v - v^T A v| \\
&\leq | u^T \sqrt{B} A \sqrt{B} u - u^T \sqrt{B} A v| + |u^T \sqrt{B} A v - v^T A v| \\
&=| u^T \sqrt{B} A (\sqrt{B} u-v)| + |(\sqrt{B}u-v)^T A v| \\
&\leq {\left\lVertu^T \sqrt{B} A\right\rVert} {\left\lVert\sqrt{B} u -v\right\rVert} + {\left\lVert\sqrt{B} u -v\right\rVert} {\left\lVertAv\right\rVert} \\
&\leq {\left\lVert \sqrt{B} u\right\rVert} {\left\lVertA\right\rVert} \kappa + \kappa {\left\lVertA\right\rVert} {\left\lVertv\right\rVert} \\
&\leq \kappa ( {\left\lVert \sqrt{B} u\right\rVert} + {\left\lVertv\right\rVert}) \\
&\leq \kappa ( {\left\lVert \sqrt{B} u\right\rVert} + {\left\lVert\sqrt{B} u + v - \sqrt{B}u \right\rVert}) \\
&\leq \kappa \left({\left\lVert \sqrt{B} u\right\rVert}+ {\left\lVert \sqrt{B} u\right\rVert} + {\left\lVert\sqrt{B} u -v\right\rVert}\right) \\
&\leq 2\kappa {\left\lVert \sqrt{B} u\right\rVert} + \kappa^2\end{aligned}$$ Now observe that we are interested in $$\frac{{{\mbox{\rm Tr}}\left(A \exp(- H)\right)}}{{{\mbox{\rm Tr}}\left(\exp(-H)\right)}} = \frac{{{\mbox{\rm Tr}}\left(A \exp(- H + \gamma I)\right)}}{{{\mbox{\rm Tr}}\left(\exp(-H + \gamma I)\right)}}.$$ Suppose an upper bound $K$ on ${\left\lVertH\right\rVert}$ is known, then we can consider $H' = H -K I$ which satisfies $H' \preceq 0$. It follows that ${\left\lVert\exp(- H')\right\rVert} \geq 1$ and, with $B = \exp(-H')$, therefore ${\left\lVert\smash{\sqrt{B}}\right\rVert} \leq {\left\lVertB\right\rVert} \leq {{\mbox{\rm Tr}}\left(B\right)}$. Hence, taking $\kappa \leq \min\{\theta/{\left\lVertu\right\rVert},{\left\lVert\smash{\sqrt{B}}u\right\rVert}\} = \theta/ {\left\lVertu\right\rVert}$,[^12] we find $$|u^T \sqrt{B} A \sqrt{B} u - v^T Av| \leq 2\kappa {\left\lVert \sqrt{B} u\right\rVert} + \kappa^2 \leq 3 \kappa {\left\lVert\sqrt{B} u\right\rVert} \leq 3 \kappa {\left\lVert\sqrt{B}\right\rVert} {\left\lVertu\right\rVert} \leq 3 \theta {\left\lVertB\right\rVert} \leq 3 \theta {{\mbox{\rm Tr}}\left(B\right)}.$$ This shows that the additional error incurred by sampling from $v^T Av$ is proportional to $\theta {{\mbox{\rm Tr}}\left(B\right)}$. Finally, a $\kappa$-approximation of $\smash{\sqrt{B}}u$, with $\kappa = \theta/{\left\lVertu\right\rVert}$ can be obtained by using the truncated Taylor series of $\exp(-H'/2)$ of degree $p = \max\{2e {\left\lVertH'\right\rVert}, \log\left(\frac{\sqrt{n}}{\theta}\right) \}$: $$\begin{aligned}
{\left\lVert\exp(-H'/2) - \sum_{i = 0}^p \frac{(H'/2)^i}{i!} \right\rVert} &= {\left\lVert\sum_{i = p+1}^\infty \frac{(H'/2)^i}{i!} \right\rVert} \leq \sum_{j=p+1}^\infty \frac{{\left\lVertH'\right\rVert}^j}{j!} \leq \sum_{j=p+1}^\infty \left(\frac{e {\left\lVertH'\right\rVert}}{j}\right)^j \\
&\leq \left(\frac{e {\left\lVertH'\right\rVert}}{p+1}\right)^{p+1} \frac{1}{1-\left(\frac{e{\left\lVertH'\right\rVert}}{(p+1)}\right)} \leq \left(\frac{1}{2}\right)^p = \theta/\sqrt{n},\end{aligned}$$
\[lem:23\] Given a Hermitian $s$-sparse matrix $A$, with $||A|| \leq 1$, a psd matrix $B = \exp(-H)$ with $H \preceq 0$, for a $d$-sparse $H$, and parameters $0 < \delta \leq 1/2$ and $0 < \theta \leq 1$. With probability $1-\delta$, using $k = {{\mathcal O}}\left(\log(\frac{1}{\delta})/\theta^2\right)$ samples from the distribution $v^T A v$, one can find an estimate that is at most $\theta {{\mbox{\rm Tr}}\left(B\right)}$ away from ${{\mbox{\rm Tr}}\left(AB\right)}$. Here $$v = \sum_{i=0}^p \frac{(H/2)^i}{i!} u$$ where $p = {{\mathcal O}}(\max\{{\left\lVertH\right\rVert},\log\left(\frac{\sqrt{n}}{\theta}\right) \})$, and $u = (u_j)$ where the $u_j \in \{\pm 1\}$ are i.i.d. uniformly distributed.
Given $m$ Hermitian $s$-sparse $n \times n$ matrices $A_1=I,A_2, \ldots, A_m$, with ${\left\lVertA_j\right\rVert} \leq 1$ for all $j$, a Hermitian $d$-sparse $n \times n$ matrix $H$ with ${\left\lVertH\right\rVert} \leq K$, and parameters $0 < \delta \leq 1/2$ and $0 < \theta \leq 1$. With probability $1-\delta$, we can compute $\theta$-approximations $a_1, \ldots, a_m$ of ${{\mbox{\rm Tr}}\left(A_1 B\right)}/{{\mbox{\rm Tr}}\left(B\right)}, \ldots, {{\mbox{\rm Tr}}\left(A_m B\right)}/{{\mbox{\rm Tr}}\left(B\right)}$ where $B = \exp(-H)$, using $${{\mathcal O}}\left(\frac{\log(\frac{m}{\delta})}{\theta^2} \max\left\{K,\log\left(\frac{\sqrt{n}}{\theta}\right)\right\} d n + \frac{\log(\frac{m}{\delta})}{\theta^2} m s n\right)$$ queries to the entries of $A_1, \ldots, A_m$, $H$ and arithmetic operations.
As observed above, for every matrix $A$, $$\frac{{{\mbox{\rm Tr}}\left(A \exp(- H)\right)}}{{{\mbox{\rm Tr}}\left(\exp(- H)\right)}} = \frac{{{\mbox{\rm Tr}}\left(A \exp(- H + \gamma I)\right)}}{{{\mbox{\rm Tr}}\left(\exp(- H + \gamma I)\right)}}.$$ Lemma \[lem:23\] states that for $B' = \exp(- H + K I)$ and $A \in \{A_1, \ldots, A_m\}$ using $k = {{\mathcal O}}\left(\log(\frac{m}{\delta})/\theta^2\right)$ samples from the distribution $v^T A v$, one can find an estimate that is at most $\theta {{\mbox{\rm Tr}}\left(B\right)}$ away from ${{\mbox{\rm Tr}}\left(AB\right)}$ with probability $1-\delta/m$. Here $$v = \sum_{i=0}^p \frac{((H-KI)/2)^i}{i!} u$$ where $p = {{\mathcal O}}(\max\{K,\log\left(\frac{\sqrt{n}}{\theta}\right) \})$, and $u = (u_j)$ with the $u_j \in \{\pm 1\}$ i.i.d. uniformly distributed.
Observe that the $k$ samples from $v^T A v$ are really obtained from $k$ samples of vectors $u = (u_j)$ combined with some post-processing, namely obtaining $v = \sum_{i=0}^p \frac{((H-KI)/2)^i}{i!} u$ and two more sparse matrix vector products.
We can therefore obtain $k$ samples from each of $v^T A_1 v, \ldots, v^T A_m v$ by *once* calculating $k$ vectors $v = \sum_{i=0}^p \frac{((H-KI)/2)^i}{i!} u$, and then, for each of the $m$ matrices $A_j$ computing the $k$ products $v^T A_j v$. The $k$ vectors $v$ can be constructed using $${{\mathcal O}}\left(\frac{\log(\frac{m}{\delta})}{\theta^2} \max\left\{K,\log\left(\frac{\sqrt{n}}{\theta}\right)\right\}d n\right)$$ queries to the entries of $H$ and arithmetic operations. The $mk$ matrix vector products can be computed using $${{\mathcal O}}\left( \frac{\log(\frac{m}{\delta})}{\theta^2} m s n \right)$$ arithmetic operations and queries to the entries of $A_1, \ldots, A_m$ and $H$. This leads to total complexity $${{\mathcal O}}\left(\frac{\log(\frac{m}{\delta})}{\theta^2} \max\left\{K,\log\left(\frac{\sqrt{n}}{\theta}\right)\right\} d n + \frac{\log(\frac{m}{\delta})}{\theta^2} m s n\right)$$ for computing $k$ samples from each of $v^TA_1 v, \ldots, v^TA_m v$.
The results of Lemma \[lem:23\] say that for each $j$, using those $k$ samples of $v^T A_j v$ we can construct a $\theta {{\mbox{\rm Tr}}\left(B'\right)}/4$-approximation $a_j'$ of ${{\mbox{\rm Tr}}\left(A_j B'\right)}$, with probability $1- \delta/(2m)$. Therefore, by a union bound, with probability $1-\delta/2$ we can construct $\theta {{\mbox{\rm Tr}}\left(B'\right)}/4$-approximations $a_1', \ldots, a_m'$ of ${{\mbox{\rm Tr}}\left(A_1 B'\right)}, \ldots, {{\mbox{\rm Tr}}\left(A_m B'\right)}$. Therefore, for each $j$, with probability at least $1-\delta$, by Lemma \[lemma:trTogether\] we have that $a_j = a_j'/a_1'$ is a $\theta$-approximation of ${{\mbox{\rm Tr}}\left(A_j B'\right)}/{{\mbox{\rm Tr}}\left(B'\right)}$, and hence it is a $\theta$-approximation of ${{\mbox{\rm Tr}}\left(A_j B\right)}/{{\mbox{\rm Tr}}\left(B\right)}$.
Implementing smooth functions of Hamiltonians {#apx:LowWeight}
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In this appendix we show how to efficiently implement smooth functions of a given Hamiltonian. First we explain what we mean by a function of a Hamiltonian $H\in\mathbb{C}^{n\times n}$, i.e., a Hermitian matrix. Since Hermitian matrices are diagonalizable using a unitary matrix, we can write $H=U^\dagger \text{diag}(\lambda) U$, where $\lambda\in\mathbb{R}^n$ is the vector of eigenvalues. Then for a function $f:\mathbb{R}\rightarrow\mathbb{C}$ we define $f(H):=U^\dagger \text{diag}(f(\lambda)) U$ with a slight abuse of notation, where we apply $f$ to the eigenvalues in $\lambda$ one-by-one. Note that if we approximate $f$ by $\tilde{f}$, then ${\left\lVert\tilde{f}(H)-f(H)\right\rVert}={\left\lVert\text{diag}(\tilde{f}(\lambda))-\text{diag}(f(\lambda))\right\rVert}$. Suppose $D\subseteq\mathbb{R}$ is such that $\lambda\in D^n$, then we can upper bound this norm by the maximum of $|\tilde{f}(x)-f(x)|$ over $x\in D$. Finally we note that $D=[-{\left\lVertH\right\rVert},{\left\lVertH\right\rVert}]$ is always a valid choice.
The main idea of the method presented below, is to implement a map $\tilde{f}(H)$, where $\tilde{f}$ is a good (finite) Fourier approximation of $f$ for all $x\in[-{\left\lVertH\right\rVert},{\left\lVertH\right\rVert}]$. The novelty in our approach is that we construct a Fourier approximation based on some polynomial approximation. In the special case, when $f$ is analytic and ${\left\lVertH\right\rVert}$ is less than the radius of convergence of the Taylor series, we can obtain good polynomial approximation functions simply by truncating the Taylor series, with logarithmic dependence on the precision parameter. Finally we implement the Fourier series using Hamiltonian simulation and the Linear Combination of Unitaries (LCU) trick [@ChildsWiebeLCU; @BerryChilds:hamsim; @BerryChilds:hamsimFOCS].
This approach was already used in several earlier papers, particularly in [@ChildsKothariSomma:lse; @ChowdhurySomma:GibbsHit]. There the main technical difficulty was to obtain a good truncated Fourier series. This is a non-trivial task, since on top of the approximation error, one needs to optimize two other parameters of the Fourier approximation that determine the complexity of implementation, namely:
$\bullet$ the largest time parameter $t$ that appears in some Fourier term $e^{-itH}$, and
$\bullet$ the total weight of the coefficients, by which we mean the $1$-norm of the vector of coefficients.\
Earlier works used clever integral approximations and involved calculus to construct a good Fourier approximation for a specific function $f$. We are not aware of a general result.
In contrast, our Theorem \[thm:Taylor\] and Corollary \[cor:patched\] avoids the usage of any integration. It obtains a low-weight Fourier approximation function using the Taylor series. The described method is completely general, and has the nice property that the maximal time parameter $t$ depends logarithmically on the desired approximation precision. Since it uses the Taylor series, it is easy to apply to a wide range of smooth functions.
The circuit we describe for the implementation of the linear operator $f(H): \mathbb{C}^n \rightarrow \mathbb{C}^n$ is going to depend on the specific function $f$, but not on $H$; the $H$-dependence is only coming from Hamiltonian simulation. Since the circuit for a specific $f$ can be constructed in advance, we do not need to worry about the (polynomial) cost of constructing the circuit, making the analysis simpler. When we describe gate complexity, we count the number of two-qubit gates needed for a quantum circuit implementation, just as in Section \[sec:upperbounds\].
Since this appendix presents stand-alone results, here we will deviate slightly from the notation used throughout the rest of the paper, to conform to the standard notation used in the literature (for example, ${\varepsilon}$, $r$, $\theta$ and $a$ have a different meaning in this appendix). For simplicity we also assume, that the Hamiltonian $H$ acts on $\mathbb{C}^{n}$, where $n$ is a power of 2. Whenever we write $\log(\text{formula})$ in some complexity statement we actually mean $\log_2(2+\text{formula})$ in order to avoid incorrect near-$0$ or even negative expressions in complexity bounds that would appear for small values of the formula.
#### Hamiltonian simulation.
We implement each term in a Fourier series using a Hamiltonian simulation algorithm, and combine the terms using the LCU Lemma. Specifically we use [@BerryChilds:hamsimFOCS], but in fact our techniques would work with any kind of Hamiltonian simulation algorithm.[^13] The following definition describes what we mean by controlled Hamiltonian simulation.
\[def:controlledSim\] Let $M=2^J$ for some $J\in \mathbb{N}$, $\gamma\in\mathbb{R}$ and $\epsilon\geq0$. We say that the unitary $$W:=\sum_{m=-M}^{M-1}{|m\rangle\! \langle m|}\otimes e^{im\gamma H}$$ implements controlled $(M,\gamma)$-simulation of the Hamiltonian $H$, where ${|m\rangle}$ denotes a (signed) bitstring ${|b_Jb_{J-1}\ldots b_0\rangle}$ such that $m=-b_J2^J+\sum_{j=0}^{J-1}b_j2^j$. The unitary $\tilde{W}$ implements controlled $(M,\gamma,{\varepsilon})$-simulation of the Hamiltonian $H$, if $${\left\lVert\tilde{W}-W\right\rVert}\leq {\varepsilon}.$$
Note that in this definition we assume that both positive and negative powers of $e^{iH}$ are simulated. This is necessary for our Fourier series, but sometimes we use only positive powers, e.g., for phase estimation; in that case we can simply ignore the negative powers.
The following lemma is inspired by the techniques of [@ChildsKothariSomma:lse]. It calculates the cost of such controlled Hamiltonian simulation in terms of queries to the input oracles - as described in Section \[sec:upperbounds\].
\[lemma:controlledHamsin\] Let $H\in\mathbb{C}^{n\times n}$ be a $d$-sparse Hamiltonian. Suppose we know an upper bound $K\in\mathbb{R_+}$ on the norm of $H$, i.e., ${\left\lVertH\right\rVert}\leq K$, and let $\tau:=M\gamma K$. If ${\varepsilon}>0$ and $\gamma=\Omega(1/(Kd))$, then a controlled $(M,\gamma,{\varepsilon})$-simulation of $H$ can be implemented using ${{\mathcal O}}(\tau d\log(\tau /{\varepsilon})/\log\log(\tau /{\varepsilon}))$ queries and ${{\mathcal O}}\left(\tau d\log(\tau /{\varepsilon})/\log\log(\tau /{\varepsilon})\left[\log(n)+\log^{\frac{5}{2}}(\tau /{\varepsilon})\right]\right)$ gates.
We use the results of [@BerryChilds:hamsimFOCS Lemma 9-10], which tell us that a $d$-sparse Hamiltonian $H$ can be simulated for time $t$ with ${\varepsilon}$ precision in the operator norm using $$\label{eq:hamSimQueries}
{{\mathcal O}}\left(\left(t{\left\lVertH\right\rVert_{\max}}d+1\right)\frac{\log\left(t{\left\lVertH\right\rVert}/{\varepsilon}\right)}{\log\log\left(t{\left\lVertH\right\rVert}/{\varepsilon}\right)}\right)$$ queries and gate complexity $$\label{eq:hamSimGates}
{{\mathcal O}}\left(\left(t{\left\lVertH\right\rVert_{\max}}d+1\right)\frac{\log\left(t{\left\lVertH\right\rVert}/{\varepsilon}\right)}{\log\log\left(t{\left\lVertH\right\rVert}/{\varepsilon}\right)}\left[\log(n)+\log^{\frac{5}{2}}\left(t{\left\lVertH\right\rVert}/{\varepsilon}\right)\right]\right).$$
Now we use a standard trick to remove log factors from the implementation cost, and write the given unitary $W$ as the product of some increasingly precisely implemented controlled Hamiltonian simulation unitaries. For $b\in \{0,1\}$ let us introduce the projector ${|b\rangle\! \langle b|}_j:=I_{2^j}\otimes{|b\rangle\! \langle b|}\otimes I_{2^{J-j}}$, where $J=\log(M)$. Observe that $$\label{eq:cleverW}
W=\left({|1\rangle\! \langle 1|}_{J}\otimes e^{-i2^{J}\gamma H}+{|0\rangle\! \langle 0|}_{J}\otimes I\right)\prod_{j=0}^{J-1}\left({|1\rangle\! \langle 1|}_{j}\otimes e^{i2^{j}\gamma H}+{|0\rangle\! \langle 0|}_{j}\otimes I\right).$$
The $j$-th operator $e^{\pm i2^{j}\gamma H}$ in the product can be implemented with $2^{j-J-1}\epsilon$ precision using ${{\mathcal O}}\left(2^j\gamma Kd\log\left(\frac{2^j\gamma K}{{\varepsilon}2^{j-J-1}}\right)/\log\log\left(\frac{2^j\gamma K}{{\varepsilon}2^{j-J-1}}\right)\right)
={{\mathcal O}}(2^j\gamma Kd\log(\tau/\epsilon)/\log\log(\tau/\epsilon))$ queries by and using ${{\mathcal O}}(2^jd\log(\tau/\epsilon)/\log\log(\tau/\epsilon)[\log(n)+\log^{\frac{5}{2}}(\tau/\epsilon)])$ gates by . Let us denote by $\tilde{W}$ the concatenation of all these controlled Hamiltonian simulation unitaries. Adding up the costs we see that our implementation of $\tilde{W}$ uses ${{\mathcal O}}(\tau d\log(\tau/{\varepsilon})/\log\log(\tau/{\varepsilon}))$ queries and has gate complexity ${{\mathcal O}}\left(\tau d\log(\tau /{\varepsilon})/\log\log(\tau/{\varepsilon})\left[\log(n)+\log^{\frac{5}{2}}(\tau /{\varepsilon})\right]\right)$. Using the triangle inequality repeatedly, it is easy to see that ${\left\lVertW\!-\!\tilde{W}\right\rVert}\leq\sum_{j=0}^{J}2^{j-J-1}{\varepsilon}\leq {\varepsilon}$.
Implementation of smooth functions of Hamiltonians: general results
-------------------------------------------------------------------
The first lemma we prove provides the basis for our approach. It shows how to turn a polynomial approximation of a function $f$ on the interval $[-1,1]$ into a nice Fourier series in an efficient way, while not increasing the weight of coefficients. This is useful, because we can implement a function given by a Fourier series using the LCU Lemma, but only after scaling it down with the weight of the coefficients.
\[lemma:LowWeightAPX\] Let $\delta,{\varepsilon}\in\!(0,1)$ and $f:\mathbb{R}\rightarrow \mathbb{C}$ s.t. $\left|f(x)\!-\!\sum_{k=0}^K a_k x^k\right|\leq {\varepsilon}/4$ for all $x\in\![-1+\delta,1-\delta]$. Then $\exists\, c\in\mathbb{C}^{2M+1}$ such that $$\left|f(x)-\sum_{m=-M}^M c_m e^{\frac{i\pi m}{2}x}\right|\leq {\varepsilon}$$ for all $x\in\![-1+\delta,1-\delta]$, where $M=\max\left(2\left\lceil \ln\left(\frac{4{\left\lVerta\right\rVert}_1}{{\varepsilon}}\right)\frac{1}{\delta} \right\rceil,0\right)$ and ${\left\lVertc\right\rVert}_1\leq {\left\lVerta\right\rVert}_1$. Moreover $c$ can be efficiently calculated on a classical computer in time $\text{poly}(K,M,\log(1/{\varepsilon}))$.
Let us introduce the notation ${\left\lVertf\right\rVert}_\infty=\sup\{|f(x)|: x\in\![-1+\delta,1-\delta]\}$. First we consider the case when ${\left\lVerta\right\rVert}_1<{\varepsilon}/2$. Then ${\left\lVertf\right\rVert}_\infty\leq {\left\lVertf(x)-\sum_{k=0}^K a_k x^k\right\rVert}_\infty + {\left\lVert\sum_{k=0}^K a_k x^k\right\rVert}_\infty < {\varepsilon}/4 + {\varepsilon}/2 < {\varepsilon}$. So in this case the statement holds with $M=0$ and $c=0$, i.e., even with an empty sum.
From now on we assume ${\left\lVerta\right\rVert}_1\geq {\varepsilon}/2$. We are going to build up our approximation gradually. Our first approximate function $\tilde{f}_1(x):=\sum_{k=0}^K a_k x^k$ satisfies ${\left\lVertf-\tilde{f}_1\right\rVert}_\infty\!\!\leq {\varepsilon}/4$ by assumption. In order to construct a Fourier series, we will work towards a linear combination of sines. To that end, note that $\forall x\in[-1,1]$: $\tilde{f}_1(x)\!=\!\sum_{k=0}^K a_k \left(\frac{\arcsin\left(\sin(x\pi/2 )\right)}{\pi/2}\right)^{\!k}\!$. Let $b^{(k)}$ denote the series of coefficients such that $\left(\frac{\arcsin(y)}{\pi/2}\right)^k=\sum_{\ell=0}^{\infty}b_\ell^{(k)} y^\ell$ for all $y\in[-1,1]$. For $k=1$ the coefficients are just $\frac{2}{\pi}$ times the coefficients of the Taylor series of $\arcsin$ so we know that $b^{(1)}_{2\ell}=0$ while $b^{(1)}_{2\ell+1}=\binom{2\ell}{\ell}\frac{2^{-2\ell}}{2\ell+1}\frac{2}{\pi}$. Since $\left(\frac{\arcsin(y)}{\pi/2}\right)^{\!k+1}\!\!=\left(\frac{\arcsin(y)}{\pi/2}\right)^{\!k}\left(\sum_{\ell=0}^{\infty}b_\ell^{(1)} y^\ell\right)$, we obtain the formula $b^{(k+1)}_{\ell}=\sum_{\ell'=0}^{\ell}b^{(k)}_{\ell'}b^{(1)}_{\ell-\ell'}$, so one can recursively calculate each $b^{(k)}$. As $b^{(1)}\geq 0$ one can use the above identity inductively to show that $b^{(k)}\geq 0$. Therefore ${\left\lVertb^{(k)}\right\rVert}_1=\sum_{\ell=0}^{\infty}b_\ell^{(k)} 1^\ell=\left(\frac{\arcsin(1)}{\pi/2}\right)^k=1$. Using the above definitions and observations we can rewrite $$\forall x\in[-1,1]: \tilde{f}_1(x)=\sum_{k=0}^K a_k \sum_{\ell=0}^{\infty} b^{(k)}_\ell \sin^\ell(x\pi/2).$$ To obtain the second approximation function, we want to truncate the summation over $\ell$ at $L=\ln\left(\frac{4{\left\lVerta\right\rVert}_1}{{\varepsilon}}\right)\frac{1}{\delta^2}$ in the above formula. We first estimate the tail of the sum. We are going to use that for all $\delta\in [0,1]$: $\sin((1-\delta)\pi/2)\leq 1 - \delta^2$. For all $k\in\mathbb{N}$ and $x\in\![-1+\delta,1-\delta]$ we have: $$\begin{aligned}
\left|\sum_{\ell=\lceil L\rceil}^{\infty} b^{(k)}_\ell \sin^\ell(x\pi/2)\right|
&\leq \sum_{\ell=\lceil L\rceil}^{\infty} b^{(k)}_\ell \left|\sin^\ell(x\pi/2)\right| \\
&\leq \sum_{\ell=\lceil L\rceil}^{\infty} b^{(k)}_\ell \left|1-\delta^2\right|^\ell\\
&\leq \left(1-\delta^2\right)^L\sum_{\ell=\lceil L\rceil}^{\infty} b^{(k)}_\ell \\
&\leq \left(1-\delta^2\right)^L\\
&\leq e^{-\delta^2L} \\
&= \frac{{\varepsilon}}{4{\left\lVerta\right\rVert}_1}.
\end{aligned}$$ Thus we have ${\left\lVert\tilde{f}_1-\tilde{f}_2\right\rVert}_\infty\leq {\varepsilon}/4$ for $$\tilde{f}_2(x):=\sum_{k=0}^K a_k \sum_{\ell=0}^{\lfloor L \rfloor} b^{(k)}_\ell \sin^\ell(x\pi/2).$$ To obtain our third approximation function, we will approximate $\sin^\ell(x\pi/2)$. First observe that $$\label{eq:sinl}
\sin^\ell(z)=\left(\frac{e^{-iz}-e^{iz}}{-2i}\right)^{\!\ell}
=\left(\frac{i}{2}\right)^{\!\ell}\sum_{m=0}^{\ell}(-1)^{m}\binom{\ell}{m}e^{iz(2m-\ell)}$$ which, as we will show (for $M'$ much larger than $\sqrt{\ell}$) is very well approximated by $$\left(\frac{i}{2}\right)^{\!\ell}\sum_{m=\lceil\ell/2\rceil-M'}^{\lfloor\ell/2\rfloor+M'}(-1)^{m}\binom{\ell}{m}e^{iz(2m-\ell)}.$$ Truncating the summation in based on this approximation reduces the maximal time evolution parameter (i.e., the maximal value of the parameter $t$ in the $\exp(izt)$ terms) quadratically. To make this approximation precise, we use Chernoff’s inequality for the binomial distribution, or more precisely its corollary for sums of binomial coefficients, stating $$\sum_{m=\lceil\ell/2+M'\rceil}^{\ell}2^{-\ell}\binom{\ell}{m}
\leq e^{-\frac{2 (M')^2}{\ell}}.$$ Let $M'=\left\lceil \ln\left(\frac{4{\left\lVerta\right\rVert}_1}{{\varepsilon}}\right)\frac{1}{\delta} \right\rceil$ and suppose $\ell\leq L$, then this bound implies that $$\label{eq:binomBound}
\sum_{m=0}^{\lfloor\ell/2\rfloor-M'}2^{-\ell}\binom{\ell}{m}
=\sum_{m=\lceil\ell/2\rceil+M'}^{\ell}2^{-\ell}\binom{\ell}{m}
\leq e^{-\frac{2 (M')^2}{\ell}}
\leq e^{-\frac{2 (M')^2}{L}}
\leq \left(\frac{{\varepsilon}}{4{\left\lVerta\right\rVert}_1}\right)^2
\leq \frac{{\varepsilon}}{4{\left\lVerta\right\rVert}_1},$$ where for the last inequality we use the assumption ${\varepsilon}\leq 2 {\left\lVerta\right\rVert}_1$. By combining and we get that for all $\ell \leq L$ $${\left\lVert\sin^\ell(z)-\left(\frac{i}{2}\right)^{\!\ell}\sum_{m=\lceil\ell/2\rceil-M'}^{\lfloor\ell/2\rfloor+M'}(-1)^{m}\binom{\ell}{m}e^{iz(2m-\ell)}\right\rVert}_\infty\leq \frac{{\varepsilon}}{2{\left\lVerta\right\rVert}_1}.$$ Substituting $z=x\pi/2$ into this bound we can see that ${\left\lVert\tilde{f}_2-\tilde{f}_3\right\rVert}_\infty\leq {\varepsilon}/2$, for $$\label{eq:finalFourier}
\tilde{f}_3(x):=\sum_{k=0}^K a_k \sum_{\ell=0}^{\lfloor L \rfloor} b^{(k)}_\ell \left(\frac{i}{2}\right)^\ell\sum_{m=\lceil\ell/2\rceil-M'}^{\lfloor\ell/2\rfloor+M'}(-1)^{m}\binom{\ell}{m}e^{\frac{i\pi x}{2}(2m-\ell)},$$ using $\sum_{k=0}^K |a_k| \sum_{\ell=0}^{\lfloor L \rfloor} \left|b^{(k)}_\ell\right|\leq \sum_{k=0}^K |a_k|={\left\lVerta\right\rVert}_1$. Therefore we can conclude that $\tilde{f}_3$ is an ${\varepsilon}$-approximation to $f$: $${\left\lVertf-\tilde{f}_3\right\rVert}_\infty\leq {\left\lVertf-\tilde{f}_1\right\rVert}_\infty + {\left\lVert\tilde{f}_1-\tilde{f}_2\right\rVert}_\infty + {\left\lVert\tilde{f}_2-\tilde{f}_3\right\rVert}_\infty \leq {\varepsilon}.$$ Observe that in the largest value of $|m-\ell|$ in the exponent is upper bounded by $2M'=M$. So by rearranging the terms in $\tilde{f}_3$ we can write $\tilde{f}_3(x)=\sum_{m=-M}^{M}c_{m} e^{\frac{i\pi m}{2}x}$. Now let us fix a value $k$ in the first summation of . Observe that after taking the absolute value of each term, the last two summations still yield a value $\leq 1$, since ${\left\lVertb^{(k)}\right\rVert}_1=1$ and $\sum_{m=0}^{\ell}\binom{\ell}{m}=2^\ell$. It follows that ${\left\lVertc\right\rVert}_1\leq {\left\lVerta\right\rVert}_1$. From the construction of the proof, it is easy to see that (an ${\varepsilon}$-approximation of) $c$ can be calculated in time $\text{poly}(K,M,\log(1/{\varepsilon}))$.
Now we present the Linear Combination of Unitaries (LCU) Lemma [@ChildsWiebeLCU; @BerryChilds:hamsim; @BerryChilds:hamsimFOCS], which we will use for combining the Fourier terms in our quantum circuit. Since we intend to use LCU for implementing non-unitary operations, we describe a version without the final amplitude amplification step. We provide a short proof for completeness.
\[lemma:LCU\] Let $U_1,U_2,\ldots,U_m$ be unitaries on a Hilbert space $\mathcal{H}$, and $L=\sum_{i=1}^{m}a_i U_i$, where $a\in\mathbb{R}_+^{m}\setminus\{0\}$. Let $V=\sum_{i=1}^{m}{|i\rangle}\!{\langlei|}\otimes U_i$ and $A\in\mathbb{C}^{m\times m}$ be a unitary such that $A{|0\rangle}=\sum_{i=1}^{m}\sqrt{\frac{a_i}{{\left\lVerta\right\rVert}_1}}{|i\rangle}$. Then $\frac{L}{{\left\lVerta\right\rVert}_1}=\left({\langle0|}\otimes I\right) \left(A^\dagger\otimes I\right) V \left(A\otimes I\right)\left({|0\rangle}\otimes I\right)$, i.e., for every ${|\psi\rangle}\in \mathcal{H}$ we have $\left(A^\dagger\otimes I\right) V \left(A\otimes I\right){|0\rangle}{|\psi\rangle}={|0\rangle}\frac{L}{{\left\lVerta\right\rVert}_1}{|\psi\rangle}+{|\Phi^\perp\rangle}$, where $\left({|0\rangle\! \langle 0|}\otimes I\right){|\Phi^\perp\rangle}=0$.
$$\begin{aligned}
\left({\langle0|}\otimes I\right)(A^\dagger\otimes I) V \left(A\otimes I\right){|0\rangle}{|\psi\rangle}
&= \left(\left(\sum_{i=1}^{m}\sqrt{\frac{a_i}{{\left\lVerta\right\rVert}_1}}{\langlei|}\right)\otimes I\right)V \sum_{i=1}^{m}\sqrt{\frac{a_i}{{\left\lVerta\right\rVert}_1}}{|i\rangle}{|\psi\rangle}\\
&= \left(\left(\sum_{i=1}^{m}\sqrt{\frac{a_i}{{\left\lVerta\right\rVert}_1}}{\langlei|}\right)\otimes I\right) \sum_{i=1}^{m}\sqrt{\frac{a_i}{{\left\lVerta\right\rVert}_1}}{|i\rangle}U_i{|\psi\rangle}\\
&= \sum_{i=1}^{m}\frac{a_i}{{\left\lVerta\right\rVert}_1}U_i{|\psi\rangle}\\
&= \frac{L}{{\left\lVerta\right\rVert}_1}{|\psi\rangle}
\end{aligned}$$
-5mm
The next result summarizes how to efficiently implement a Fourier series of a Hamiltonian.
\[lemma:LCUApplied\] Suppose $f(x)=\sum_{m=-M}^{M-1} c_m e^{im\gamma x}$, for a given $c\in\mathbb{C}^{2M}\setminus\{0\}$. We can construct a unitary $\tilde{U}$ which implements the operator $\frac{f(H)}{{\left\lVertc\right\rVert}_1}=\sum_{m=-M}^{M-1} \frac{c_m}{{\left\lVertc\right\rVert}_1} e^{im\gamma H}$ with ${\varepsilon}$ precision, i.e., such that $${\left\lVert({\langle0|}\otimes I)\tilde{U}({|0\rangle}\otimes I)-\frac{f(H)}{{\left\lVertc\right\rVert}_1}\right\rVert}\leq{\varepsilon},$$ using ${{\mathcal O}}(M(\log(M)+1))$ two-qubit gates and a single use of a circuit implementing controlled $(M,\gamma,{\varepsilon})$-simulation of $H$.
This is a direct corollary of Lemma \[lemma:LCU\]. To work out the details, note that we can always extend $c$ with some $0$ values, so we can assume without loss of generality that $M$ is a power of $2$. This is useful, because then we can represent each $m\in [-M,M-1]$ as a $(J+1)$-bit signed integer for $J=\log(M)$.
The implementation of the operator $A$ in Lemma \[lemma:LCU\] does not need any queries and it can be constructed exactly using ${{\mathcal O}}(M(\log(M)+1))$ two-qubit gates, e.g., by the techniques of [@Grover:probabilityDistPrepare]. We sketch the basic idea which is based on induction. For $J=1$ the operator $A$ is just a two-qubit unitary. Suppose we proved the claim for bitstrings of length $J$ and want to prove the claim for length $J+1$. Let $a\in\mathbb{R}_+^{2^{J+1}}$ be such that ${\left\lVerta\right\rVert}_1=1$ and define $\tilde{a}\in\mathbb{R}_+^{2^{J}}$ such that $\tilde{a}_b=a_{b,0}+a_{b,1}$ for all bitstrings $b\in\{0,1\}^{J}$. Then we have a circuit $\tilde{A}$ that uses ${{\mathcal O}}(2^J (J+1))$ gates and satisfies $\sqrt{\tilde{a}_b}={\langleb|}\tilde{A}{|0\ldots 0\rangle}$ for all $b\in\{0,1\}^{J}$. We can add an extra ${|0\rangle}$-qubit and implement a controlled rotation gate $R_b$ on it for each $b\in\{0,1\}^{J}$. Let $R_b$ have rotation angle $\arccos\left(\sqrt{a_{b,0}/\tilde{a}_b}\right)$ and be controlled by $b$. It is easy to see that the new unitary $A$ satisfies $\sqrt{a_{b'}}={\langleb'|}A{|0\ldots 0\rangle}$ for each $b'\in\{0,1\}^{J}$. Each $R_b$ can be implemented using ${{\mathcal O}}(J)$ two-qubit gates and ancilla qubits, justifying the gate complexity and concluding the induction.
What remains is to implement the operator $V=\sum_{m=-M}^{M-1}{|m\rangle\! \langle m|}\otimes \frac{c_m}{|c_m|} e^{im\gamma H}$ from Lemma \[lemma:LCU\]. We implement $V=PW$ in two steps, where $P=\sum_{m=-M}^{M-1}{|m\rangle\! \langle m|}\otimes \frac{c_m}{|c_m|}I$. This $P$ can be implemented exactly using ${{\mathcal O}}(M(\log(M)+1))$ gates simply by building a controlled gate that adds the right phase for each individual bitstring. Since the bitstring on which we want to do a controlled operation has length $\log(M)+1$, each controlled operation can be constructed using ${{\mathcal O}}(\log(M)+1)$ gates and ancilla qubits resulting in the claimed gate complexity. We use a circuit implementing controlled $(M,\gamma,{\varepsilon})$-simulation of $H$, denoted by $\tilde{W}$, which is an ${\varepsilon}$-approximation of $W$ by definition.
Finally $\tilde{U}:=(A^\dagger\otimes I) P\tilde{W} \left(A\otimes I\right)$. This yields an ${\varepsilon}$-precise implementation, since $$\begin{aligned}
{\left\lVert({\langle0|}\otimes I)\tilde{U}({|0\rangle}\otimes I)-\frac{f(H)}{{\left\lVertc\right\rVert}_1}\right\rVert}
&= {\left\lVert({\langle0|}\otimes I)\tilde{U}({|0\rangle}\otimes I)-({\langle0|}\otimes I)(A^\dagger\otimes I) PW \left(A\otimes I\right)({|0\rangle}\otimes I)\right\rVert}\\
&\leq {\left\lVert\tilde{U}-(A^\dagger\otimes I) PW \left(A\otimes I\right)\right\rVert}\\
&= {\left\lVert(A^\dagger\otimes I) P\tilde{W} \left(A\otimes I\right)-(A^\dagger\otimes I) PW \left(A\otimes I\right)\right\rVert}\\
&= {\left\lVert\tilde{W}-W \right\rVert} \leq {\varepsilon}.
\end{aligned}$$ -0.5cm
Now we can state the main result of this appendix, which tells us how to efficiently turn a function (provided with its Taylor series) of a Hamiltonian $H$, into a quantum circuit by using controlled Hamiltonian simulation.
In the following theorem we assume that the eigenvalues of $H$ lie in a radius-$r$ ball around $x_0$. The main idea is that if even $r+\delta$ is less than the radius of convergence of the Taylor series, then we can obtain an ${\varepsilon}$-approximation of $f$ by truncating the series at logarithmically high powers. $B$ will be an upper bound on the absolute value of the function within the $r+\delta$ ball around $x_0$, in particular ${\left\lVertf(H)/B\right\rVert}\leq 1$. Therefore we can implement $f(H)/B$ as a block of some larger unitary. It turns out that apart from the norm and sparsity of $H$ and precision parameters, the complexity depends on the ratio of $\delta$ and $r$.
\[thm:Taylor\] Let $x_0\in\mathbb{R}$ and $r>0$ be such that $f(x_0+x)=\sum_{\ell=0}^{\infty} a_\ell x^\ell$ for all $x\in\![-r,r]$. Suppose $B>0$ and $\delta\in(0,r]$ are such that $\sum_{\ell=0}^{\infty}(r+\delta)^\ell|a_\ell|\leq B$. If ${\left\lVertH-x_0I\right\rVert}\leq r$ and ${\varepsilon}\in\!\left(0,\frac{1}{2}\right]$, then we can implement a unitary $\tilde{U}$ such that ${\left\lVert({\langle0|}\otimes I)\tilde{U}({|0\rangle}\otimes I)-\frac{f(H)}{B}\right\rVert}\leq{\varepsilon}$, using ${{\mathcal O}}\left(r/\delta\log\left(r/(\delta{\varepsilon})\right)\log\left(1/{\varepsilon}\right)\right)$ gates and a single use of a circuit for controlled $\left({{\mathcal O}}(r\log(1/{\varepsilon})/\delta),{{\mathcal O}}(1/r),{\varepsilon}/2\right)$-simulation of $H$.
Suppose we are given $K$ such that ${\left\lVertH\right\rVert}\leq K$ and $r={{\mathcal O}}(K)$. If, furthermore, $H$ is $d$-sparse and is accessed via oracles -, then the whole circuit can be implemented using $${{\mathcal O}}\left(\!\frac{Kd}{\delta}\log\left(\frac{K}{\delta{\varepsilon}}\right)\log\left(\frac{1}{{\varepsilon}}\right)\!\right)\text{ queries and }
{{\mathcal O}}\left(\!\frac{Kd}{\delta}\log\left(\frac{K}{\delta{\varepsilon}}\right)\log\left(\frac{1}{{\varepsilon}}\right)\!\left[\log(n)\!+\!\log^{\frac{5}{2}}\left(\frac{K}{\delta{\varepsilon}}\right)\right]\!\right)\text{ gates.}$$
The basic idea is to combine Lemma \[lemma:LowWeightAPX\] and Lemma \[lemma:LCUApplied\] and apply them to a transformed version of the function. First we define $\delta':=\delta/(r+\delta)$, which is at most $1/2$ by assumption. Then, for all $\ell\in\mathbb{N}$ let $b_\ell:=a_\ell(r+\delta)^\ell$ and define the function $g:[-1+\delta',1-\delta'] \rightarrow \mathbb{R}$ by $g(y):=\sum_{\ell=0}^{\infty}b_\ell y^\ell$ so that $$\label{eq:sclaedF}
f(x_0+x)=g(x/(r+\delta)) \qquad \text{ for all } x\in\![-r,r].$$ Now we set $L:=\left\lceil\frac{1}{\delta'}\log\left(\frac{8}{{\varepsilon}}\right)\right\rceil$. Then for all $y\in\![-1+\delta',1-\delta']$ $$\begin{aligned}
\left|g(y)-\sum_{\ell=0}^{L-1}b_\ell y^\ell\right|
&= \left|\sum_{\ell=L}^{\infty}b_\ell y^\ell\right|\\
&\leq \sum_{\ell=L}^{\infty}\left|b_\ell (1-\delta')^\ell\right|\\
&\leq (1-\delta')^L \sum_{\ell=L}^{\infty}\left|b_\ell\right|\\
&\leq \left(1-\delta'\right)^{L}B\\
&\leq e^{-\delta'L}B\\
&\leq \frac{{\varepsilon}B}{8}.
\end{aligned}$$ We would now like to obtain a Fourier-approximation of $g$ for all $y\in[-1+\delta',1-\delta']$, with precision ${\varepsilon}'=\frac{{\varepsilon}B}{2}$. Let $b':=(b_0,b_1,\ldots,b_{L-1})$ and observe that ${\left\lVertb'\right\rVert}_1\leq{\left\lVertb\right\rVert}_1\leq B$. We apply Lemma \[lemma:LowWeightAPX\] to the function $g$, using the polynomial approximation corresponding to the truncation to the first $L$ terms, i.e., using the coefficients in $b'$. Then we obtain a Fourier ${\varepsilon}'$-approximation $\tilde{g}(y):=\sum_{m=-M}^{M}\tilde{c}_m e^{\frac{i\pi m}{2}y}$ of $g$, with $$M={{\mathcal O}}\left(\frac{1}{\delta'}\log\left(\frac{{\left\lVertb'\right\rVert}_1}{{\varepsilon}'}\right)\right)={{\mathcal O}}\left(\frac{r}{\delta}\log\left(\frac{1}{{\varepsilon}}\right)\right)$$ such that the vector of coefficients $\tilde{c}\in\mathbb{C}^{2M+1}$ satisfies ${\left\lVert\tilde{c}\right\rVert}_1\leq{\left\lVertb'\right\rVert}_1 \leq {\left\lVertb\right\rVert}_1 \leq B$. Let $$\tilde{f}(x_0+x):=\tilde{g}\left(\frac{x}{r+\delta}\right)=\sum_{m=-M}^{M}\tilde{c}_m e^{\frac{i\pi m}{2(r+\delta)}x};$$ by we see that $\tilde{f}$ is an ${\varepsilon}'$-precise Fourier approximation of $f$ on the interval $[x_0-r,x_0+r]$. To transform this Fourier series to its final form, we note that $\tilde{f}(z)=\sum_{m=-M}^{M}\tilde{c}_m e^{\frac{i\pi m}{2(r+\delta)}(z-x_0)}$, so by defining $c_m:=\tilde{c}_me^{-\frac{i\pi m}{2(r+\delta)}x_0}$ we get a Fourier series in $z$, while preserving ${\left\lVertc\right\rVert}_1={\left\lVert\tilde{c}\right\rVert}_1\leq B$.
In the trivial case, when $c=0$, we choose a unitary $\tilde{U}$, such that it maps the ${|0\rangle}$ ancilla state to ${|1\rangle}$, then clearly $({\langle0|}\otimes I)\tilde{U}({|0\rangle}\otimes I)=0=\tilde{f}(H)$. Clearly such a $\tilde{U}$ can be implemented using ${{\mathcal O}}(1)$ gates and $0$ queries. Otherwise we can apply Lemma \[lemma:LCUApplied\] to this modified Fourier series to construct a unitary circuit $\tilde{V}$ implementing an $\frac{{\varepsilon}}{2}$-approximation of $\tilde{f}(H)/{\left\lVertc\right\rVert}_1$. We can further scale down the amplitude of the ${|0\rangle}$-part of the output by a factor of ${\left\lVertc\right\rVert}_1/B\leq 1$, to obtain an approximation of $\tilde{f}(H)/B$ as follows. We simply add an additional ancilla qubit initialized to ${|0\rangle}$ on which we act with the one-qubit unitary $$Rot:=\left(\begin{array}{cc}
\frac{{\left\lVertc\right\rVert}_1}{B} & \sqrt{1-\frac{{\left\lVertc\right\rVert}_1^2}{B^2}}\\
-\sqrt{1-\frac{{\left\lVertc\right\rVert}_1^2}{B^2}} & \frac{{\left\lVertc\right\rVert}_1}{B}
\end{array}\right).$$ Finally we define $\tilde{U}:=Rot\otimes \tilde{V}$, and define ${|0\rangle}{|0\rangle}$ as the new success indicator, where the first qubit is the new ancilla. We show that $\tilde{U}$ implements $f(H)/B$ with ${\varepsilon}$ precision: (if $c=0$, let us use the definition $\tilde{f}(H)/{\left\lVertc\right\rVert}_1:=0$) $$\begin{aligned}
{\left\lVert({\langle0|}{\langle0|}\otimes I)\tilde{U}({|0\rangle}{|0\rangle}\otimes I)-\frac{f(H)}{B}\right\rVert}
&\leq
{\left\lVert({\langle0|}{\langle0|}\otimes I)\tilde{U}({|0\rangle}{|0\rangle}\otimes I)-\frac{\tilde{f}(H)}{B}\right\rVert}
+ {\left\lVert\frac{\tilde{f}(H)}{B}-\frac{f(H)}{B}\right\rVert}\\
&= \frac{{\left\lVertc\right\rVert}_1}{B}{\left\lVert({\langle0|}\otimes I)\tilde{V}({|0\rangle}\otimes I)-\frac{\tilde{f}(H)}{{\left\lVertc\right\rVert}_1}\right\rVert}
+ {\left\lVert\frac{\tilde{f}(H)-f(H)}{B}\right\rVert}\\
&\leq \frac{{\left\lVertc\right\rVert}_1}{B}\frac{{\varepsilon}}{2}+\frac{{\varepsilon}'}{B}\\
&\leq {\varepsilon}.
\end{aligned}$$ Lemma \[lemma:LCUApplied\] uses ${{\mathcal O}}\left(M\log(M+1)\right)={{\mathcal O}}\left(r/\delta\log(1/{\varepsilon})\log\left(r/(\delta{\varepsilon})\right)\right)$ gates and a single use of a controlled $\left(M,\gamma=\pi/(2r+2\delta),{\varepsilon}/2\right)$-simulation of $H$. If ${\left\lVertH\right\rVert}={{\mathcal O}}(K)$, we can use Lemma \[lemma:controlledHamsin\] to conclude ${{\mathcal O}}\left(M\gamma Kd\log\left(\frac{1}{{\varepsilon}}\right)\log\left(\frac{M\gamma K}{{\varepsilon}}\log\left(\frac{1}{{\varepsilon}}\right)\right)\right)={{\mathcal O}}\left(\frac{Kd}{\delta} \log\left(\frac{K}{\delta{\varepsilon}}\right)\log\left(\frac{1}{{\varepsilon}}\right)\right)$ query and $$\begin{aligned}
{{\mathcal O}}\left(M\gamma Kd\log\left(\frac{1}{{\varepsilon}}\right)\!\log\left(\frac{M\gamma K}{{\varepsilon}}\log\left(\frac{1}{{\varepsilon}}\right)\!\right)\!\left[\log(n)+\log^{\frac{5}{2}}\!\left(\frac{M\gamma K}{{\varepsilon}}\log\left(\frac{1}{{\varepsilon}}\right)\!\right)\!\right]\right)\kern-52mm&\\ &={{\mathcal O}}\left(\frac{Kd}{\delta}\log\left(\frac{K}{\delta{\varepsilon}}\right)\log\left(\frac{1}{{\varepsilon}}\right)\!\left[\log(n)+\log^{\frac{5}{2}}\!\left(\frac{K}{\delta{\varepsilon}}\right)\!\right]\right)
\end{aligned}$$ gate complexity. Finally note that the polynomial cost of calculating $c$ that is required by Lemma \[lemma:LowWeightAPX\] does not affect the query complexity or the circuit size, it only affects the description of the circuit.
\[rem:subSpaceFun\] Note that in the above theorem we can relax the criterion ${\left\lVertH-x_0I\right\rVert}\leq r$. Suppose we have an orthogonal projector $\Pi$, which projects to eigenvectors with eigenvalues in $[x_0-r,x_0+r]$, i.e., $[H,\Pi]=0$ and ${\left\lVert\Pi\left(H-x_0I\right)\Pi\right\rVert}\leq r$. Then the circuit $\tilde{U}$ constructed in Theorem \[thm:Taylor\] satisfies $${\left\lVert\Pi\left(({\langle0|}\otimes I)\tilde{U}({|0\rangle}\otimes I)-\frac{f(H)}{B}\right)\Pi\right\rVert}\leq{\varepsilon}.$$
The following corollary shows how to implement functions piecewise in small “patches" using Remark \[rem:subSpaceFun\]. The main idea is to first estimate the eigenvalues of $H$ up to $\theta$ precision, and then implement the function using the Taylor series centered around a point close to the eigenvalue.
This approach has multiple advantages. First, the function may not have a Taylor series that is convergent over the whole domain of possible eigenvalues of $H$. Even if there is such a series, it can have very poor convergence properties, making $B$ large and therefore requiring a lot of amplitude amplification. Nevertheless, for small enough neighborhoods the Taylor series always converges quickly, overcoming this difficulty.
\[cor:patched\] Suppose $(x_\ell)\in\mathbb{R}^L$ and $r,\theta\in\mathbb{R_+}$ are such that the spectrum of $H$ lies in the domain $\bigcup_{\ell=1}^L [x_\ell-(r-2\theta),x_\ell+(r+2\theta)]$.[^14] Suppose there exist coefficients $a_k^{(\ell)}\in\mathbb{R}$ such that for all $\ell\in [L]$ and $x\in\![-r,r]$ we have $f(x_\ell+x)=\sum_{k=0}^{\infty} a_k^{(\ell)} x^k$, and $\sum_{k=0}^{\infty}(r+\delta)^k|a_k^{(\ell)}|\leq B$ for some fixed $\delta\in[0,r]$ and $B>0$. If ${\left\lVertH\right\rVert}\leq K$ and ${\varepsilon}\in\!\left(0,\frac{1}{2}\right]$, then we can implement a unitary $\tilde{U}$ such that $${\left\lVert({\langle0|}\otimes I)\tilde{U}({|0\rangle}\otimes I)-\frac{f(H)}{B}\right\rVert}\leq{\varepsilon},$$ using ${{\mathcal O}}\left(Lr/\delta\log\left(r/(\delta{\varepsilon})\right)\log\left(1/{\varepsilon}\right)+\log(K/\theta)\log\log(K/(\theta{\varepsilon}))\right)$ gates, and with ${{\mathcal O}}(\log(1/{\varepsilon}))$ uses of an $({{\mathcal O}}(1/\theta),\pi/K,\Omega({\varepsilon}^2/\log(1/{\varepsilon})))$-simulation of $H$ and a single use of a circuit for controlled $\left({{\mathcal O}}(r\log(1/{\varepsilon})/\delta),{{\mathcal O}}(1/r),{\varepsilon}/2\right)$-simulation of $H$. If $r={{\mathcal O}}(K)$, $\theta\leq r/4$, $\theta=\Omega(\delta)$, ${\left\lVertH\right\rVert}\leq K$, $H$ is $d$-sparse and is accessed via oracles -, then the circuit can be implemented using -3.5mm $${{\mathcal O}}\left(\!\frac{Kd}{\delta}\log\left(\frac{K}{\delta{\varepsilon}}\right)\log\left(\frac{1}{{\varepsilon}}\right)\!\right)\text{ queries and }
{{\mathcal O}}\left(\!\frac{Kd}{\delta}\log\left(\frac{K}{\delta{\varepsilon}}\right)\log\left(\frac{1}{{\varepsilon}}\right)\!\left[\log(n)\!+\!\log^{\frac{5}{2}}\left(\frac{K}{\delta{\varepsilon}}\right)\right]\!\right)\text{ gates.}$$
We start by performing phase estimation on $e^{iH}$ with $\approx\theta$ resolution in phase. We boost the success probability by taking the median outcome of ${{\mathcal O}}(\log(1/{\varepsilon}))$ parallel repetitions, so that we get a worse-than-$\theta$ estimation with probability at most ${{\mathcal O}}({\varepsilon}^2)$. This way the boosted phase estimation circuit is ${{\mathcal O}}({\varepsilon})$-close in operator norm to an “idealized" phase estimation unitary that never makes approximation error greater than $\theta$ (for more details on this type of argument, see the proof of Lemma \[lemma:normEst\]). Phase estimation uses controlled $({{\mathcal O}}(K/\theta),\pi/K,\Omega({\varepsilon}^2/\log(1/{\varepsilon})))$-simulation of $H$ and a Fourier transform on ${{\mathcal O}}(\log(K/\theta))$-bit numbers which can be implemented using ${{\mathcal O}}(\log(K/\theta)\log\log(K/(\theta{\varepsilon})))$ gates. The probability-boosting uses ${{\mathcal O}}(\log(1/{\varepsilon}))$ repetitions. Controlled on the phase estimate $\tilde{\lambda}$ that we obtained, we implement a $1/B$-scaled version of the corresponding function “patch” $\left.f(x)\right|_{[x_\ell-r,x_\ell+r]}$ centered around $\arg\min|x_\ell-\tilde{\lambda}|$ using Theorem \[thm:Taylor\] and Remark \[rem:subSpaceFun\]. The additional gate complexities of the “patches” add up to ${{\mathcal O}}\left(Lr/\delta\log\left(r/(\delta{\varepsilon})\right)\log\left(1/{\varepsilon}\right)\right)$, but since each “patch” uses the same controlled $\left({{\mathcal O}}(r\log(1/{\varepsilon})/\delta),{{\mathcal O}}(1/r),{\varepsilon}/2\right)$-simulation of $H$, we only need to implement that once. Finally we uncompute phase estimation.[^15] For the final complexity, note that we can assume without loss of generality that $L(r-2\theta)={{\mathcal O}}(K)$, since otherwise we can just remove some redundant intervals from the domain. Hence $Lr={{\mathcal O}}(K)$ and Lemma \[lemma:controlledHamsin\] implies the stated complexities.
This corollary is essentially as general and efficient as we can hope for. Let $D$ denote the domain of possible eigenvalues of $H$. If we want to implement a reasonably smooth function $f$, then it probably satisfies the following: there is some $r=\Omega(1)$, such that for each $x\in D$, the Taylor series in the radius-$r$ neighborhood of $x$ converges quickly, more precisely the Taylor coefficients $a_k^{(x)}$ for the $x$-centered series satisfy $\sum_{k=0}^{\infty}|a_k^{(x)}|r^k={{\mathcal O}}({\left\lVertf\right\rVert}_{\infty})$, where we define ${\left\lVertf\right\rVert}_{\infty}:=\sup_{x\in D}|f(x)|$. If this is the case, covering $D$ with radius-${{\mathcal O}}(r)$ intervals, choosing $\theta=\Theta(r)$ and $\delta=\Theta(r)$, Corollary \[cor:patched\] provides an ${\widetilde{\mathcal O}\left({\left\lVertH\right\rVert}d\right)}$ query and gate complexity implementation of $f(H)/B$, where $B={{\mathcal O}}({\left\lVertf\right\rVert}_{\infty})$. The value of $B$ is optimal up to constant factors, since $f(H)/B$ must have norm at most $1$. Also the ${\left\lVertH\right\rVert}d$ factor in the complexity is very reasonable, and we achieve the logarithmic error dependence which is the primary motivation of the related techniques. An application along the lines of this discussion can be found in Lemma \[lemma:squareGibbs\].
Also note that in the above corollary we added up the gate complexities of the different “patches.” Since these gates prepare the Fourier coefficients of the function corresponding to the different Taylor series at different points, one could use this structure to implement all coefficients with a single circuit. This can potentially result in much smaller circuit sizes, which could be beneficial when the input model allows more efficient Hamiltonian simulation (which then would no longer be the bottleneck in the complexity).
Applications of smooth functions of Hamiltonians
------------------------------------------------
In this subsection we use the input model for the $d$-sparse matrix $H$ as described at the start of Section \[sec:upperbounds\]. We calculate the implementation cost in terms of queries to the input oracles -, but it is easy to convert the results to more general statements as in the previous subsection.
The following theorem shows how to efficiently implement the function $e^{-H}$ for some $H\succeq I$. We use this result in the proof of Lemma \[lemma:trPreEst\] to estimate expectation values of the quantum state $\rho=e^{-H}/{{\mbox{\rm Tr}}\left(e^{-H}\right)}$ (for the application we ensure that $H\succeq I$ by adding some multiple of $I$).
\[thm:emH\] Suppose that $I\preceq H$ and we are given $K\in\mathbb{R}_+$ such that ${\left\lVertH\right\rVert}\leq 2K$. If ${\varepsilon}\in(0,1/3)$, then we can implement a unitary $\tilde{U}$ such that ${\left\lVert({\langle0|}\otimes I)\tilde{U}({|0\rangle}\otimes I)-e^{-H}\right\rVert}\leq{\varepsilon}$ using $${{\mathcal O}}\left(Kd\log\left(\frac{K}{{\varepsilon}}\right)\log\left(\frac{1}{{\varepsilon}}\right)\right)\text{ queries and }{{\mathcal O}}\left(Kd\log\left(\frac{K}{{\varepsilon}}\right)\log\left(\frac{1}{{\varepsilon}}\right)\left[\log(n)+\log^{\frac{5}{2}}\left(\frac{K}{{\varepsilon}}\right)\right]\right)\text{ gates.}$$
In order to use Theorem \[thm:Taylor\] we set $x_0:=K+1/2$ so that ${\left\lVertH-x_0 I\right\rVert}\leq K=:r$, and use the function $$f(x_0+x)=e^{-x_0-x}=e^{-x_0}e^{-x}=e^{-x_0}\sum_{\ell=0}^{\infty}\frac{(-x)^\ell}{\ell!}.$$ We choose $\delta:=1/2$ so that $e^{-x_0}\sum_{\ell=0}^{\infty}\frac{(r+\delta)^\ell}{\ell!}=e^{-x_0}\sum_{\ell=0}^{\infty}\frac{x_0^\ell}{\ell!}=1$, therefore we set $B:=1$. Theorem \[thm:Taylor\] tells us that we can implement a unitary $\tilde{U}$, such that $\tilde{f}(H):=({\langle0|}\otimes I)\tilde{U}({|0\rangle}\otimes I)$ is an ${\varepsilon}$-approximation of $f(H)/B= e^{-H}$, using $${{\mathcal O}}\left(Kd\log\left(\frac{K}{{\varepsilon}}\right)\log\left(\frac{1}{{\varepsilon}}\right)\right)\text{ queries and }{{\mathcal O}}\left(Kd\log\left(\frac{K}{{\varepsilon}}\right)\log\left(\frac{1}{{\varepsilon}}\right)\left[\log(n)+\log^{\frac{5}{2}}\left(\frac{K}{{\varepsilon}}\right)\right]\right) \text{ gates}.$$
To conclude this appendix, we now sketch the proofs of a few interesting consequences of the earlier results in this appendix. These will, however, not be used in the body of the paper.
First, we show how to use the above subroutine together with amplitude amplification to prepare a Gibbs state with cost depending logarithmically on the precision parameter, as shown by the following lemma. To our knowledge this is the first Gibbs sampler that achieves logarithmic dependence on the precision parameter without assuming access to the entries of $\sqrt{H}$ as in [@ChowdhurySomma:GibbsHit]. This can mean a significant reduction in complexity; for more details see the introduction of Section \[sec:estTrArhogeneral\].
\[lemma:preGibbs\] We can probabilistically prepare a purified Gibbs state ${|\tilde{\gamma}\rangle}_{AB}$ such that with high probability we have ${\left\lVert{\mbox{\rm Tr}}_B\left({|\tilde{\gamma}\rangle\! \langle \tilde{\gamma}|}_{AB}\right)-e^{-H}/{{\mbox{\rm Tr}}\left(e^{-H}\right)}\right\rVert}_1\leq {\varepsilon}$, using an expected cost ${\widetilde{\mathcal O}\left(\sqrt{n / {{\mbox{\rm Tr}}\left(e^{-H}\right)}}\right)}$ times the complexity of Theorem \[thm:emH\]. If we are given a number $z \leq {{\mbox{\rm Tr}}\left(e^{-H}\right)}$, then we can also prepare ${|\tilde{\gamma}\rangle}_{AB}$ in a unitary fashion with cost ${\widetilde{\mathcal O}\left(\sqrt{n / z}\right)}$ times the complexity of Theorem \[thm:emH\].
First we show how to prepare a purified sub-normalized Gibbs state. Then we use the exponential search algorithm of Boyer et al. [@bbht:bounds] (with exponentially decreasing guesses for the norm $a$ of the subnormalized Gibbs state, and hence exponentially increasing number of amplitude amplification steps) to postselect on this sub-normalized state in a similar fashion as in Algorithm \[alg:genMin\]. There is a possible caveat here: if we postselect on a state with norm $a$, then it gets rescaled by $1/a$ and its preparation error is rescaled by $1/a$ as well. Therefore during the rounds of the search algorithm we always increase the precision of implementation to compensate for the increased (error) amplification. Since the success of postselection in a round is upper bounded by the square of ${{\mathcal O}}\left(a\cdot\#\{\text{amplification steps in the round}\}\right)$, the probability for the postselection to succeed in any of the early rounds is small.
Now we describe how to prepare a purified sub-normalized Gibbs state. Let $H\!=\!\sum_{j=1}^{n}E_j{|\phi_j\rangle\! \langle \phi_j|}$, where $\{{|\phi_j\rangle}:j\in[n]\}$ is an orthonormal eigenbasis of $H$. Due to the invariance of maximally entangled states under transformations of the form $W\otimes W^T$ for unitary $W$, there is an orthonormal basis $\{{|\upsilon_j\rangle}:j\in[n]\}$ such that $$\label{eq:preGibbsMaxEnt}
\frac{1}{\sqrt{n}}\sum_{j=0}^{n-1}{|j\rangle}_A{|j\rangle}_B=\frac{1}{\sqrt{n}}\sum_{j=1}^{n}{|\phi_j\rangle}_A{|\upsilon_j\rangle}_B.$$ Suppose we can implement a unitary $U$ such that $({\langle0|}\otimes I)U({|0\rangle}\otimes I)=e^{-H/2}$. If we apply $U$ to the A-register of the state , then we get a state ${|\gamma\rangle}$ such that ${\mbox{\rm Tr}}_B\left(({\langle0|}\otimes I){|\gamma\rangle\! \langle \gamma|}({|0\rangle}\otimes I)\right)=e^{-H}/n$.
If we implement $e^{-H/2}$ with sufficient precision using Theorem \[thm:emH\] in the exponential search algorithm, then after ${{\mathcal O}}\left(\sqrt{n/{{\mbox{\rm Tr}}\left(e^{-H}\right)}}\right)$ rounds of amplitude amplification, with high probability we obtain a Gibbs state using the claimed expected running time.
If we also know a lower bound $z \leq {{\mbox{\rm Tr}}\left(e^{-H}\right)}$, then we have an upper bound on the expected runtime, therefore we can turn the procedure into a unitary circuit using standard techniques.
We can also recover the Gibbs sampler of Chowdhury and Somma [@ChowdhurySomma:GibbsHit]: if we apply our Corollary \[cor:patched\] to the function $e^{-x^2}$ assuming access to $\sqrt{H}$ for some psd matrix $H$, then we get a Gibbs sampler for the state $e^{-H}/{\mbox{\rm Tr}}(e^{-H})$, similar to [@ChowdhurySomma:GibbsHit]. The advantage of the presented approach is that it avoids the usage of involved integral transformations, and can be presented without writing down a single integral sign, also due to our general results the proof is significantly shorter. Before we prove the precise statement in Lemma \[lemma:squareGibbs\], we need some preparation for the application of Corollary \[cor:patched\]:
\[lemma:emx2\] For all $k\in\mathbb{N}$ we have $$\label{eq:emx2}
\partial_x^{k+1} e^{-x^2}=-2x\partial_x^{k} e^{-x^2}-2k\partial_x^{k-1} e^{-x^2}$$ and for all $x\in\mathbb{R}$ $$\label{eq:emx2n}
\left|\partial_x^{k} e^{-x^2}\right|\leq (2|x|+2k)^k e^{-x^2}.$$ Therefore $$\label{eq:emx2Taylor}
\sum_{k=0}^{\infty}\frac{\left|\partial_x^{k} e^{-x^2}\right|}{k!}\left(\frac{1}{8e}\right)^k\leq 2.$$
We prove both claims by induction. $\partial_x^{0} e^{-x^2}=e^{-x^2}$, $\partial_x^{1} e^{-x^2}=-2xe^{-x^2}$ and $\partial_x^{2} e^{-x^2}=4x^2e^{-x^2}-2e^{-x^2}$, so holds for $k=1$. Suppose holds for $k$, we prove the inductive step as follows: $$\begin{aligned}
\partial_x^{k+2} e^{-x^2}
=\partial_x\left(\partial_x^{k+1} e^{-x^2}\right)
\overset{\eqref{eq:emx2}}{=}\partial_x\left(-2x\partial_x^{k} e^{-x^2}-2k\partial_x^{k-1} e^{-x^2}\right)
=-2x\partial_x^{k+1} e^{-x^2}-2(k+1)\partial_x^{k} e^{-x^2}.
\end{aligned}$$ Similarly, observe that holds for $k=0$ and $k=1$. Suppose holds for $k$, then we show the induction step as follows: $$\begin{aligned}
\left|\partial_x^{k+1} e^{-x^2}\right|
\kern+1.2mm&\kern-1.2mm\overset{\eqref{eq:emx2}}{=}\left|-2x\partial_x^{k} e^{-x^2}-2k\partial_x^{k-1} e^{-x^2}\right|\\
&\leq\left|2x\partial_x^{k} e^{-x^2}\right|+\left|2k\partial_x^{k-1} e^{-x^2}\right|\\
\kern+1.2mm&\kern-1.2mm\overset{\eqref{eq:emx2n}}{\leq}2|x|(2|x|+2k)^k e^{-x^2}+2k(2|x|+2(k-1))^{k-1} e^{-x^2}\\
&\leq (2|x|+2(k+1))^{k+1} e^{-x^2}.
\end{aligned}$$ Finally, using the previous two statements we can prove by the following calculation: $$\begin{aligned}
\sum_{k=0}^{\infty}\frac{\left|\partial_x^{k} e^{-x^2}\right|}{k!}\left(\frac{1}{8e}\right)^k
\kern+1.2mm&\kern-1.2mm\overset{\eqref{eq:emx2n}}{\leq} \sum_{k=0}^{\infty}\frac{(2|x|+2k)^k e^{-x^2}}{k!}\left(\frac{1}{8e}\right)^k\\
&\leq \sum_{k=0}^{\infty}\frac{(4|x|)^k e^{-x^2}}{k!}\left(\frac{1}{8e}\right)^k
+\sum_{k=1}^{\infty}\frac{(4k)^k e^{-x^2}}{k!}\left(\frac{1}{8e}\right)^k\\
&\leq e^{-x^2}\left(\sum_{k=0}^{\infty}\frac{1}{k!}\left(\frac{4|x|}{8e}\right)^k
+\sum_{k=1}^{\infty}\frac{1}{k!}\left(\frac{4k}{8e}\right)^k\right)\\
&\leq e^{-x^2}\left(e^{\frac{|x|}{2e}}
+\sum_{k=1}^{\infty}\frac{1}{\sqrt{2\pi}}\left(\frac{e}{k}\right)^k\left(\frac{k}{2e}\right)^k\right)\\
&= e^{-\left(|x|-\frac{1}{4e}\right)^2}e^{\left(\frac{1}{4e}\right)^2} + \frac{e^{-x^2}}{\sqrt{2\pi}}\\
&\leq e^{\left(\frac{1}{4e}\right)^2} + \frac{1}{\sqrt{2\pi}}\\
&\leq 2.
\end{aligned}$$
\[lemma:squareGibbs\] Suppose we know a $K>1$ such that ${\left\lVertH\right\rVert}\leq K$. If ${\varepsilon}\in(0,1/3)$, then we can implement a unitary $\tilde{U}$ such that ${\left\lVert({\langle0|}\otimes I)\tilde{U}({|0\rangle}\otimes I)-e^{-H^2}/2\right\rVert}\leq{\varepsilon}$ using $${{\mathcal O}}\left(Kd\log\left(\frac{K}{{\varepsilon}}\right)\log\left(\frac{1}{{\varepsilon}}\right)\right)\text{ queries and }{{\mathcal O}}\left(Kd\log\left(\frac{K}{{\varepsilon}}\right)\log\left(\frac{1}{{\varepsilon}}\right)\left[\log(n)+\log^{\frac{5}{2}}\left(\frac{K}{{\varepsilon}}\right)\right]\right)\text{ gates.}$$
We apply Corollary \[cor:patched\] to the function $e^{-x^2}$. For this let $L_0:=\lceil 32K \rceil$, $L:=2L_0+1$ and let $x_\ell:=(\ell-1-L_0)/32$ for all $\ell\in[L]$. We choose $r:=1/32$, $\delta:=\theta:=1/128$ and $B=2$ so that the conditions of Corollary \[cor:patched\] are satisfied, as shown by Lemma \[lemma:emx2\]. Indeed $r+\delta\leq1/(8e)$, hence for all $\ell\in[L]$ we have $\sum_{k=0}^{\infty}a_k^{(\ell)}(r+\delta)^k\leq 2=B$ as we can see by . Since $\delta=\Theta(1)$, Corollary \[cor:patched\] provides the desired complexity.
We can use the above lemma to prepare Gibbs states in a similar way to Lemma \[lemma:preGibbs\]. In case we have access to $\sqrt{H}$, the advantage of this method is that the dependence on ${\left\lVertH\right\rVert}$ is reduced to $\sqrt{{\left\lVertH\right\rVert}}$.
#### Improved HHL algorithm.
Our techniques can also be used to improve the Harrow-Hassidim-Lloyd (HHL) algorithm [@hhl:lineq] in a similar manner to Childs et al. [@ChildsKothariSomma:lse]. The problem the HHL algorithm solves is the following. Suppose we have a circuit $U$ preparing a quantum state ${|b\rangle}$ (say, starting from ${|0\rangle}$), and have $d$-sparse oracle access to a non-singular Hamiltonian $H$. The task is to prepare a quantum state, that ${\varepsilon}$-approximates $H^{-1}{|b\rangle}/{\left\lVertH^{-1}{|b\rangle}\right\rVert}$. For simplicity here we only count the number of uses of $U$ and the number of queries to $H$. Childs et al. [@ChildsKothariSomma:lse] present two different methods for achieving this, one based on Hamiltonian simulation, and another directly based on quantum walks. Under the conditions ${\left\lVertH\right\rVert}\leq 1$ and ${\left\lVertH^{-1}\right\rVert}\leq\kappa$, the former makes ${{\mathcal O}}\left(\kappa\sqrt{\log(\kappa/{\varepsilon})}\right)$ uses of $U$ and has query complexity ${{\mathcal O}}(d\kappa^2\log^{2.5}(\kappa/{\varepsilon}))$. The latter makes ${{\mathcal O}}\left(\kappa\log(d\kappa/{\varepsilon})\right)$ uses of $U$ and has query complexity ${{\mathcal O}}(d\kappa^2\log^{2}(d\kappa/{\varepsilon}))$.
Now we provide a sketch of how to solve the HHL problem with ${{\mathcal O}}(\kappa)$ uses of $U$ and with query complexity ${{\mathcal O}}(d\kappa^2\log^{2}(\kappa/{\varepsilon}))$ using our techniques. The improvement on both previously mentioned results is not very large, but our proof is significantly shorter thanks to our general Theorem \[thm:Taylor\] and Corollary \[cor:patched\].
To solve the HHL problem we need to implement the function $H^{-1}$, i.e., apply the function $f(x)=1/x$ to $H$. Due to the constraints on $H$, the eigenvalues of $H$ lie in the union of the intervals $[-1,-1/\kappa]$ and $[1/\kappa,1]$. We first assume that the eigenvalues actually lie in $[1/\kappa,1]$. In this case we can easily implement the function $1/x$ by Theorem \[thm:Taylor\] using the Taylor series around $1$: $$\label{eq:1oz}
(1+z)^{-1}=\frac{1}{1+z}=\sum_{k=0}^{\infty}(-1)^kz^k.$$ As $H^{-1}=(I+(H-I))^{-1}$, we are interested in the eigenvalues of $H-I$. The eigenvalues of $H-I$ lie in the interval $[-1+1/\kappa,0]$, so we choose $r:=1-1/\kappa$ and $\delta:=1/(2\kappa)$. By substituting $z:=-1+1/(2\kappa)$ in , we can see that $B:=2\kappa$ satisfies the conditions of Theorem \[thm:Taylor\]. Let ${\varepsilon}'\in(0,1/2)$, then Theorem \[thm:Taylor\] provides an ${{\mathcal O}}\left(d\kappa \log\left(\kappa/{\varepsilon}'\right)\log\left(1/{\varepsilon}'\right)\right)$-query implementation of an ${\varepsilon}'$-approximation of the operator $H^{-1}/(2\kappa)$, since ${\left\lVertH\right\rVert}\leq 1$. We can proceed similarly when the eigenvalues of $H-I$ lie in the interval $[-1,-1/\kappa]$, and we can combine the two cases using Corollary \[cor:patched\].
Setting ${\varepsilon}':=c{\varepsilon}/\kappa$ for an appropriate constant $c$ and using amplitude amplification, we can prepare an ${\varepsilon}$-approximation of the state $H^{-1}{|b\rangle}/{\left\lVertH^{-1}{|b\rangle}\right\rVert}$ as required by HHL using ${{\mathcal O}}(\kappa)$ amplitude amplification steps. Therefore we use $U$ at most ${{\mathcal O}}(\kappa)$ times and make ${{\mathcal O}}\left(d\kappa^2 \log^2\left(\kappa/{\varepsilon}\right)\right)$ queries.
Generalized minimum-finding algorithm {#app:genMinFind}
=====================================
In this appendix we describe our generalized quantum minimum-finding algorithm, which we are going to apply to finding an approximation of the ground state energy of a Hamiltonian. This algorithm generalizes the results of D[ü]{}rr and H[ø]{}yer in a manner similar to the way amplitude amplification [@bhmt:countingj] generalizes Grover search: we do not need to assume the ability to query individual elements of the search space, we just need to be able to generate a superposition over the search space. The algorithm also has the benefit over binary search that it removes a logarithmic factor from the complexity.
The backbone of our analysis will be the meta-algorithm below from . The meta-algorithm finds the minimal element in the range of the random variable $X$ by sampling, where by “range” we mean the values which occur with non-zero probability. We assume $X$ has finite range.
Input
: A discrete random variable $X$ with finite range.
Output
: The minimal value $x_{\min}$ in the range of $X$.
**Init** $t\leftarrow 0$; $s_0\leftarrow \infty$ **Repeat** until $s_t$ is minimal in the range of $X$
1. $t\leftarrow t+1$
2. Sample a value $s_t$ according to the conditional distribution $\mathrm{Pr}(X=s_t \mid X<s_{t-1})$.
Note that the above algorithm will always find the minimum, since the obtained samples are strictly decreasing.
\[lemma:metaMin\] Let $X$ be a finite discrete random variable whose range of values is $x_1<x_2<\ldots<x_N$. Let $S(X)=\{s_1,s_2,\dots\}$ denote the random set of values obtained via sampling during a run of Meta-Algorithm \[alg:metaMin\] with input random variable $X$. If $k\in[N]$, then $$\mathrm{\mathrm{Pr}}(x_k\in S(X))=\frac{\mathrm{\mathrm{Pr}}(X=x_k)}{\mathrm{\mathrm{Pr}}(X\leq x_k)}.$$
The intuition of the proof is to show that $$\label{eq:conditional}
\mathrm{Pr}(s_t=x_k \mid t\in[N]\text{ is the first time such that } s_t\leq x_k)=\frac{\mathrm{Pr}(X=x_k)}{\mathrm{Pr}(X\leq x_k)}.$$ To formulate the statement more precisely[^16] we consider a fixed value $t\in[N]$. For notational convenience let $x:=x_k$, $x_{N+1}:=\infty$, we prove by: $$\begin{aligned}
\mathrm{Pr}\left(s_t=x \mid s_t\leq x \wedge s_{t-1}>x\right)
&=\frac{\mathrm{Pr}\left(s_t=x\right)}{\mathrm{Pr}\left(s_t\leq x \wedge s_{t-1}>x\right)}\\
&=\sum_{x_\ell>x}\frac{\mathrm{Pr}\left(s_t=x\wedge s_{t-1}=x_\ell\right)}{\mathrm{Pr}\left(s_t\leq x \wedge s_{t-1}>x\right)}\\
&=\sum_{x_\ell>x}\frac{\mathrm{Pr}\left(s_t=x\wedge s_{t-1}=x_\ell\right)}{\mathrm{Pr}\left(s_t\leq x \wedge s_{t-1}=x_\ell\right)}\frac{\mathrm{Pr}\left(s_t\leq x\wedge s_{t-1}=x_\ell\right)}{\mathrm{Pr}\left(s_t\leq x \wedge s_{t-1}>x\right)}\\
&=\sum_{x_\ell>x}\frac{\mathrm{Pr}\left(s_t=x\mid s_{t-1}=x_\ell\right)\mathrm{Pr}\left(s_{t-1}=x_\ell\right)}{\mathrm{Pr}\left(s_t\leq x \mid s_{t-1}=x_\ell\right)\mathrm{Pr}\left(s_{t-1}=x_\ell\right)}\frac{\mathrm{Pr}\left(s_t\leq x\wedge s_{t-1}=x_\ell\right)}{\mathrm{Pr}\left(s_t\leq x \wedge s_{t-1}>x\right)}\\
&=\frac{\mathrm{Pr}\left(X=x\right)}{\mathrm{Pr}\left(X\leq x\right)}\sum_{x_\ell>x}\frac{\mathrm{Pr}\left(s_t\leq x\wedge s_{t-1}=x_\ell\right)}{\mathrm{Pr}\left(s_t\leq x \wedge s_{t-1}>x\right)}\\
&=\frac{\mathrm{Pr}\left(X=x\right)}{\mathrm{Pr}\left(X\leq x\right)}.
\end{aligned}$$ This is enough to conclude the proof, since there is always a smallest $t\in[N]$ such that $s_t\leq x$, as the algorithm always finds the minimum in at most $N$ steps. So we can finish the proof by $$\begin{aligned}
\mathrm{Pr}\left(x\in S(X)\right)&=\sum_{t=1}^{N}\mathrm{Pr}\left(s_t=x \right)\\
&=\sum_{t=1}^{N}\mathrm{Pr}\left(s_t=x \mid s_t\leq x \wedge s_{t-1}>x\right)\mathrm{Pr}\left(s_t\leq x \wedge s_{t-1}>x\right)\\
&=\frac{\mathrm{Pr}\left(X=x\right)}{\mathrm{Pr}\left(X\leq x\right)}\sum_{t=1}^{N}\mathrm{Pr}\left(s_t\leq x \wedge s_{t-1}>x\right)\\
&=\frac{\mathrm{Pr}\left(X=x\right)}{\mathrm{Pr}\left(X\leq x\right)}\mathrm{Pr}\left(\exists t\in [N]\!:\,s_t\leq x \wedge s_{t-1}>x\right)\\
&=\frac{\mathrm{Pr}\left(X=x\right)}{\mathrm{Pr}\left(X\leq x\right)}.
\end{aligned}$$ -5mm
Now we describe our generalized minimum-finding algorithm which is based on Meta-Algorithm \[alg:metaMin\]. We take some unitary $U$, and replace $X$ by the distribution obtained if we measured the second register of $U{|0\rangle}$. We implement conditional sampling via amplitude amplification and the use of the exponential search algorithm of Boyer et al. [@bbht:bounds]. If a unitary prepares the state ${|0\rangle}{|\phi\rangle}+{|1\rangle}{|\psi\rangle}$ where ${\left\lVert\phi\right\rVert}^2+{\left\lVert\psi\right\rVert}^2=1$, then this exponential search algorithm built on top of amplitude amplification prepares the state ${|1\rangle}{|\psi\rangle}$ probabilistically using an expected number of $\mathcal{O}\left(1/{\left\lVert\psi\right\rVert}\right)$ applications of $U$ and $U^{-1}$ (we will skip the details here, which are straightforward modifications of [@bbht:bounds]).
Input
: A number $M$ and a unitary $U$, acting on $q$ qubits, such that $U{|0\rangle}=\sum_{k=1}^{N}{|\psi_k\rangle}{|x_k\rangle}$, where $x_k$ is a binary string representing some number and ${|\psi_k\rangle}$ is an unnormalized quantum state on the first register. Let $x_1<x_2<\ldots <x_N$ and define $X$ to be the random variable with $\mathrm{Pr}(X=x_k)={\left\lVert\psi_k\right\rVert}^2$.
Output
: Some ${|\psi_k\rangle}{|x_{k}\rangle}$ for a (hopefully) small $k$.
**Init** $t\leftarrow 0$; $s_0\leftarrow \infty$ **While** the total number of applications of $U$ and $U^{-1}$ does not exceed $M$:
1. $t\leftarrow t+1$
2. Use the exponential search algorithm with amplitude amplification on states such that $x_k< s_{t-1}$ to obtain a sample ${|\psi_k\rangle}{|x_k\rangle}$.
3. $s_t\leftarrow x_k$
\[lemma:genMinExp\] There exists $C\in\mathbb{R_+}$, such that if we run Algorithm \[alg:genMin\] indefinitely (setting $M=\infty$), then for every $U$ and $x_k$ the expected number of uses of $U$ and $\kern0.2mm U^{-1}$ before obtaining a sample $x\leq x_k$ is at most $\frac{C}{\sqrt{\mathrm{Pr}(X\leq x_k)}}$ .
Let $X_{<x_\ell}$ denote the random variable for which $\mathrm{Pr}(X_{<x_\ell}=x)=\mathrm{Pr}(X=x\mid X<x_\ell)$. The expected number of uses of $U$ and $U^{-1}$ in Algorithm \[alg:genMin\] before obtaining any value $x\leq x_k$ is $$\begin{aligned}
\mathbb{E}\left[\#\text{uses of }U\text{ before finding }x\!\leq\! x_k\right]
\!&=\!\!\sum_{\ell=k+1}^{N}\!\!\mathrm{Pr}(x_\ell\!\in\! S(X))\,\mathbb{E}\left[\#\text{uses of }U\text{ for sampling from }X_{< x_\ell}\right]\\
&=\!\!\sum_{\ell=k+1}^{N}\frac{\mathrm{Pr}(X=x_\ell)}{\mathrm{Pr}(X\leq x_\ell)}{{\mathcal O}}\left(\frac{1}{\sqrt{\mathrm{Pr}(X< x_\ell)}}\right)\\
&={{\mathcal O}}\left(\sum_{\ell=k+1}^{N}\frac{\mathrm{Pr}(X=x_\ell)}{\mathrm{Pr}(X\leq x_\ell)}\frac{1}{\sqrt{\mathrm{Pr}(X< x_\ell)}}\right)\\
&={{\mathcal O}}\left(\frac{1}{\sqrt{\mathrm{Pr}(X\leq x_k)}}\right),
\end{aligned}$$ where the last equality follows from Equation below. The constant $C$ from the lemma is the constant hidden by the ${{\mathcal O}}$. The remainder of this proof consists of proving using elementary calculus. Let us introduce the notation $p_0:=\mathrm{Pr}(X\leq x_k)$ and for all $j\in[N-k]$ let $p_j:=\mathrm{Pr}(X=x_{k+j})$. Then the expression inside the ${{\mathcal O}}$ on the second-to-last line above becomes $$\label{eq:pSimplified}
\sum_{\ell=k+1}^{N}\frac{\mathrm{Pr}(X=x_\ell)}{\mathrm{Pr}(X\leq x_\ell)}\sqrt{\frac{1}{\mathrm{Pr}(X< x_\ell)}}
=\sum_{j=1}^{N-k}\frac{p_j}{\sum_{i=0}^{j}p_i}\sqrt{\frac{1}{\sum_{i=0}^{j-1}p_i}}.$$ The basic idea is that we treat the expression on the right-hand side of as an integral approximation sum for the integral $\int_{p_0}^{1}z^{-3/2}dz$, and show that it is actually always less than the value of this integral. We proceed by showing that subdivision always increases the sum.
Let us fix some $j\in[N-k]$ and define $$p'_i =
\begin{cases}
p_i &\text{for } i\in\{0,1,\ldots, j-1\}\\
p_i/2 & \text{for } i\in\{j,j+1\}\\
p_{i-1} & \text{for } i\in\{j+2,\ldots,N-k+1\}
\end{cases}$$ and observe that $$\begin{aligned}
&\sum_{j=1}^{N-k+1}\frac{p'_j}{\sum_{i=0}^{j}p'_i}\sqrt{\frac{1}{\sum_{i=0}^{j-1}p'_i}}
-\sum_{j=1}^{N-k}\frac{p_j}{\sum_{i=0}^{j}p_i}\sqrt{\frac{1}{\sum_{i=0}^{j-1}p_i}}\nonumber\\
=&\frac{p'_j}{\sum_{i=0}^{j}p'_i}\sqrt{\frac{1}{\sum_{i=0}^{j-1}p'_i}}+\frac{p'_{j+1}}{\sum_{i=0}^{j+1}p'_i}\sqrt{\frac{1}{\sum_{i=0}^{j}p'_i}}-\frac{p_j}{\sum_{i=0}^{j}p_i}\sqrt{\frac{1}{\sum_{i=0}^{j-1}p_i}}\nonumber\\
=&\frac{p_j/2}{p_j/2+\sum_{i=0}^{j-1}p_i}\sqrt{\frac{1}{\sum_{i=0}^{j-1}p_i}}+\frac{p_j/2}{p_j+\sum_{i=0}^{j-1}p_i}\sqrt{\frac{1}{p_j/2+\sum_{i=0}^{j-1}p_i}}-\frac{p_j}{p_j+\sum_{i=0}^{j-1}p_i}\sqrt{\frac{1}{\sum_{i=0}^{j-1}p_i}}.\label{eq:abDiff}
\end{aligned}$$ We show that is $\geq 0$ after simplifying the expression by substituting $a:=\sum_{i=0}^{j-1}p_i$ and $b:=p_j/2$: $$\begin{aligned}
\frac{b}{a+b}\sqrt{\frac{1}{a}}+\frac{b}{a+2b}\sqrt{\frac{1}{a+b}}-\frac{2b}{a+2b}\sqrt{\frac{1}{a}}=\frac{\left(a+b-\sqrt{a} \sqrt{a+b}\right)b }{(a+b)^{3/2} (a+2 b)}\geq 0.
\end{aligned}$$ Let us fix some parameter $\delta>0$. Recursively applying this halving procedure for different indices, we can find some $J\in\mathbb{N}$ and $\tilde{p}\in\mathbb{R}_+^{J+1}$ such that $\sum_{j=0}^{J}\tilde{p}_j=1$, $\tilde{p}_0=p_0$ and $\tilde{p}_j\leq \delta$ for all $j\in [J]$, moreover $$\begin{aligned}
& \sum_{j=1}^{N-k}\frac{p_j}{\sum_{i=0}^{j}p_i}\sqrt{\frac{1}{\sum_{i=0}^{j-1}p_i}}
\leq \sum_{j=1}^{J}\frac{\tilde{p}_j}{\sum_{i=0}^{j}\tilde{p}_i}\sqrt{\frac{1}{\sum_{i=0}^{j-1}\tilde{p}_i}} .
\end{aligned}$$ Observe that for all $j\in [J]$ $$\begin{aligned}
\label{eq:deltaMore}
\sqrt{\frac{1}{\sum_{i=0}^{j-1}\tilde{p}_i}}
=\sqrt{\frac{1}{\sum_{i=0}^{j}\tilde{p}_i}}\sqrt{\frac{\sum_{i=0}^{j}\tilde{p}_i}{\sum_{i=0}^{j-1}\tilde{p}_i}}
=&\sqrt{\frac{1}{\sum_{i=0}^{j}\tilde{p}_i}}\sqrt{1+\frac{\tilde{p}_j}{\sum_{i=0}^{j-1}\tilde{p}_i}}\nonumber \\
\leq& \sqrt{\frac{1}{\sum_{i=0}^{j}\tilde{p}_i}}\sqrt{1+\frac{\delta}{\tilde{p}_0}}
\leq \sqrt{\frac{1}{\sum_{i=0}^{j}\tilde{p}_i}}\left(1+\frac{\delta}{p_0}\right).
\end{aligned}$$ Therefore $$\begin{aligned}
\sum_{j=1}^{N-k}\frac{p_j}{\sum_{i=0}^{j}p_i}\sqrt{\frac{1}{\sum_{i=0}^{j-1}p_i}}
\leq& \sum_{j=1}^{J}\frac{\tilde{p}_j}{\sum_{i=0}^{j}\tilde{p}_i}\sqrt{\frac{1}{\sum_{i=0}^{j-1}\tilde{p}_i}}\\
\text{(by~\eqref{eq:deltaMore}) }\leq& \sum_{j=1}^{J}\frac{\tilde{p}_j}{\sum_{i=0}^{j}\tilde{p}_i}\sqrt{\frac{1}{\sum_{i=0}^{j}\tilde{p}_i}}\left(1+\frac{\delta}{p_0}\right)\\
=& \left(1+\frac{\delta}{p_0}\right)\sum_{j=1}^{J}\frac{\tilde{p}_j}{\left(\sum_{i=0}^{j}\tilde{p}_i\right)^{3/2}}\\
\leq& \left(1+\frac{\delta}{p_0}\right)\sum_{j=1}^{J}\int_{\sum_{i=0}^{j-1}\tilde{p}_i}^{\sum_{i=0}^{j}\tilde{p}_i}z^{-\frac{3}{2}} dz\\
=& \left(1+\frac{\delta}{p_0}\right)\int_{p_0}^{1}z^{-\frac{3}{2}} dz\\
=& \left(1+\frac{\delta}{p_0}\right)\left[-2 z^{-\frac{1}{2}}\right]_{p_0}^1 \\
\leq& \left(1+\frac{\delta}{p_0}\right)\left(\frac{2}{\sqrt{p_0}}\right).
\end{aligned}$$ Since this inequality holds for every $\delta>0$, we can conclude using that $$\begin{aligned}
\label{eq:intApx}
\sum_{\ell=k+1}^{N}\frac{\mathrm{Pr}(X=x_\ell)}{\mathrm{Pr}(X\leq x_\ell)}\sqrt{\frac{1}{\mathrm{Pr}(X< x_\ell)}}
\leq\frac{2}{\sqrt{p_0}}=\frac{2}{\sqrt{\mathrm{Pr}(X\leq x_k)}}.
\end{aligned}$$ -6mm
It is not too hard to work out the constant by following the proof of [@bbht:bounds] providing something like $C\approx 25$. The following theorem works with any $C$ satisfying Lemma \[lemma:genMinExp\].
\[thm:genMin\] If we run Algorithm \[alg:genMin\] with input satisfying $M\geq 4C/\sqrt{\mathrm{Pr}(X\leq x)}$ for $C$ as in Lemma \[lemma:genMinExp\] and a unitary $U$ that acts on $q$ qubits, then at termination we obtain an $x_i$ from the range of $X$ that satisfies $x_i\leq x$ with probability at least $\frac{3}{4}$. Moreover the success probability can be boosted to at least $1-\delta$ with ${{\mathcal O}}(\log(1/\delta))$ repetitions. This uses at most $M$ applications of $U$ and $U^{-1}$ and ${{\mathcal O}}(qM)$ other gates.
Let $x_k$ be the largest value in the range of $X$ such that $x_k\leq x$. Then Lemma \[lemma:genMinExp\] says that the expected number of applications of $U$ and $U^{-1}$ before finding a value $x_i\leq x_k$ is at most $C/\sqrt{\mathrm{Pr}(X\leq x_k)}=C/\sqrt{\mathrm{Pr}(X\leq x)}$, therefore by the Markov inequality we know that the probability that we need to use $U$ and $U^{-1}$ at least $4C/\sqrt{\mathrm{Pr}(X\leq x)}$ times is at most $1/4$. The boosting of the success probability can be done using standard techniques, e.g., by repeating the whole procedure ${{\mathcal O}}(\log(1/\delta))$ times and taking the minimum of the outputs.
The number of applications of $U$ and $U^{-1}$ follows directly form the algorithms description. Then, for the number of other gates, each amplitude amplification step needs to implement a binary comparison and a reflection through the ${|0\rangle}$ state, both of which can be constructed using ${{\mathcal O}}(q)$ elementary gates, giving a total of ${{\mathcal O}}(qM)$ gates.
Note that this result is a generalization of D[ü]{}rr and H[ø]{}yer : if we can create a uniform superposition over $N$ values $x_1<x_2<\ldots<x_N$, then $\mathrm{Pr}(X\leq x_1)=1/N$ and therefore Theorem \[thm:genMin\] guarantees that we can find the minimum with high probability with ${{\mathcal O}}(\sqrt{N})$ steps.
Now we describe an application of this generalized search algorithm that we need in the paper.
This final lemma in this appendix describes how to estimate the smallest eigenvalue of a Hamiltonian. A similar result was shown by Poulin and Wocjan [@PoulinWocjan:GroundMany], but we improve on the analysis to fit our framework better. We assume sparse oracle access to the Hamiltonian $H$ as described in Section \[sec:upperbounds\], and will count queries to these oracles. We use some of the techniques introduced in Appendix \[apx:LowWeight\].
\[lemma:normEst\] If $H\!=\!\sum_{j=1}^{n}E_j{|\phi_j\rangle\! \langle \phi_j|}$, with eigenvalues $E_1\leq E_2 \leq \ldots \leq E_n$, is such that ${\left\lVertH\right\rVert}\leq K$, ${\varepsilon}\leq K/2$, and $H$ is given in $d$-sparse oracle form then we can obtain an estimate $E$ such that $\left|E_1-E\right|\leq {\varepsilon}$, with probability at least $2/3$, using $${{\mathcal O}}\left(\frac{Kd\sqrt{n}}{{\varepsilon}}\log^2\left(\frac{Kn}{{\varepsilon}}\right)\right)\text{ queries and }{{\mathcal O}}\left(\frac{Kd\sqrt{n}}{{\varepsilon}}\log^{\frac{9}{2}}\left(\frac{Kn}{{\varepsilon}}\right)\right) \text{ gates}.$$
The general idea is as follows: we prepare a maximally entangled state on two registers, and apply phase estimation [@cemm:revisited] to the first register with respect to the unitary $e^{\pi iH/K}$. We then use Theorem \[thm:genMin\] to find the minimal phase. In order to guarantee correctness we need to account for all the approximation errors coming from approximate implementations. This causes some technical difficulty, since the approximation errors can introduce phase estimates that are much less than the true minimum. We need to make sure that the minimum-finding algorithm finds these faulty estimates only with a tiny probability.
We first initialize two $\log(n)$-qubit registers in a maximally entangled state $\smash{\frac{1}{\sqrt{n}}\sum_{j=0}^{n-1}{|j\rangle}{|j\rangle}}$. This can be done using $\log(n)$ Hadamard and CNOT gates. Due to the invariance of maximally entangled states under transformations of the form $W\otimes W^T$ for unitary $W$, there is an orthonormal basis $\{{|\upsilon_j\rangle}:j\in[n]\}$ such that $$\frac{1}{\sqrt{n}}\sum_{j=0}^{n-1}{|j\rangle}{|j\rangle}=\frac{1}{\sqrt{n}}\sum_{j=1}^{n}{|\phi_j\rangle}{|\upsilon_j\rangle}.$$
Let $T:=2^{\left\lceil\log\left(\frac{K}{{\varepsilon}}\right)+2\right\rceil}$ and first assume that we have access to a perfect unitary $V$ which implements $V=\sum_{t=0}^{T-1}{|t\rangle\! \langle t|}\otimes e^{\pi t i H/K}$. Let $e_j:=E_jT/2K$. If we apply phase estimation to the quantum state ${|\phi_j\rangle}$, then we get some phase estimate ${|e\rangle}$ such that $|e-e_j|\leq 3$ with high probability. Therefore the final estimate $E:= e 2 K /T$ satisfies $|E-E_j|=|e-e_j|2K/T\leq 3{\varepsilon}/4<{\varepsilon}$. If we repeat phase estimation ${{\mathcal O}}(\log(n))$ times, and take the median of the estimates, then we obtain an $e$ such that $|e-e_j|\leq 3$ with probability at least $1-b/n$, for some $b=\Theta(1)$.
Since in our maximally entangled state ${|\phi_j\rangle}$ is entangled with ${|\upsilon_j\rangle}$ on the second register, applying phase estimation to the first register in superposition does not cause interference. Denote the above preparation-estimation-boost circuit by $U$. Define $\Pi$ to be the projector which projects to the subspace of estimation values $e$ such that there is a $j\in [n]$ with $|e-e_j|\leq {\varepsilon}$. By the non-interference argument we can see that, after applying $U$, the probability that we get an estimation $e$ such that $|e-e_j|>3$ for all $j\in [n]$, is at most $b/n$. Therefore ${\left\lVert(I-\Pi)U{|0\rangle}\right\rVert}^2\leq b/n$. Also let $\Pi_1$ denote the projector which projects to phase estimates that yield $e$ such that $|e-e_1|\leq 3$. It is easy to see that ${\left\lVert\Pi_1 U{|0\rangle}\right\rVert}^2\geq 1/n-b/n^2$.
Now let us replace $V$ by $\tilde{V}$ implemented via Lemma \[lemma:controlledHamsin\], such that ${\left\lVert\smash{V-\tilde{V}}\right\rVert}\leq c'/(n\log(n))$ for some $c'=\Theta(1)$. Let $\tilde{U}$ denote the circuit that we obtain from $U$ by replacing $V$ with $\tilde{V}$. Since in the repeated phase-estimation procedure we use $V$ in total ${{\mathcal O}}(\log(n))$ times, by using the triangle inequality we see that ${\left\lVert\smash{U-\tilde{U}}\right\rVert}\leq c/(2n)$, where $c=\Theta(1)$. We use the well-known fact that if two unitaries are $\delta$-close in operator norm, and they are applied to the same quantum state, then the measurement statistics of the resulting states are $2\delta$-close. Therefore we can upper bound the difference in probability of getting outcome $(I-\Pi)$: $${\left\lVert\smash{(I-\Pi)\tilde{U}{|0\rangle}}\right\rVert}^2-{\left\lVert(I-\Pi)U{|0\rangle}\right\rVert}^2\leq 2{\left\lVert\smash{U{|0\rangle}-\tilde{U}{|0\rangle}}\right\rVert}\leq c/n,$$ hence ${\left\lVert\smash{(I-\Pi)\tilde{U}{|0\rangle}}\right\rVert}^2\leq (b+c)/n$, and we can prove similarly that ${\left\lVert\smash{\Pi_1 \tilde{U}{|0\rangle}}\right\rVert}^2\geq 1/n-(b+c)/n$.
Now let ${|\psi\rangle}:=(I-\Pi)\tilde{U}{|0\rangle}/{\left\lVert\smash{(I-\Pi)\tilde{U}{|0\rangle}}\right\rVert}$ be the state that we would get after post-selecting on the $(I-\Pi)$-outcome of the projective measurement $\Pi$. For small enough $b,c$ we have that ${\left\lVert\smash{{|\psi\rangle}-\tilde{U}{|0\rangle}}\right\rVert}= {{\mathcal O}}(\sqrt{(b+c)/n})$ by the triangle inequality. Thus there exists an idealized unitary $U'$ such that ${|\psi\rangle}=U'{|0\rangle}$, and ${\left\lVert\smash{\tilde{U}-U'}\right\rVert}= {{\mathcal O}}(\sqrt{(b+c)/n})$. Observe that ${\left\lVert\Pi_1 U'{|0\rangle}\right\rVert}^2={\left\lVert{|\psi\rangle}\right\rVert}^2\geq {\left\lVert\smash{\Pi_1 \tilde{U}{|0\rangle}}\right\rVert}^2\geq 1/n-(b+c)/n$.
Now suppose $(b+c)\leq 1/2$ and we run the generalized minimum-finding algorithm of Theorem \[thm:genMin\] using $U'$ with $M=6C\sqrt{n}$. Since $$\mathrm{Pr}(e\leq e_1+3)\geq{\left\lVert\Pi_1 U'{|0\rangle}\right\rVert}^2 \geq (1-b-c)/n \geq 1/(2n)>4/(9n)$$ we will obtain an estimate $e$ such that $e\leq e_1+3$, with probability at least $3/4$. But since $\Pi{|\psi\rangle}={|\psi\rangle}$, we find that any estimate that we might obtain satisfies $e\geq e_1-3$. So an estimate $e\leq e_1+3$ always satisfies $|e-e_q|\leq 3$.
The problem is that we only have access to $\tilde{U}$ as a quantum circuit. Let $C_{MF}(\tilde{U})$ denote the circuit that we get from Theorem \[thm:genMin\] when using it with $\tilde{U}$ and define similarly $C_{MF}(U')$ for $U'$. Since we use $\tilde{U}$ a total of ${{\mathcal O}}(\sqrt{n})$ times in $C_{MF}(\tilde{U})$ and $${\left\lVert\tilde{U}-U'\right\rVert}= {{\mathcal O}}(\sqrt{(b+c)/n})\text{, we get that } {\left\lVertC_{MF}(\tilde{U})-C_{MF}(U')\right\rVert}= {{\mathcal O}}(\sqrt{b+c}).$$ Therefore the measurement statistics of the two circuits differ by at most ${{\mathcal O}}(\sqrt{b+c})$. Choosing $b,c$ small enough constants ensures that $C_{MF}(\tilde{U})$ outputs a proper estimate $e$ such that $|e-e_q|\leq 3$ with probability at least $2/3$. As we have shown at the beginning of the proof, such an $e$ yields an ${\varepsilon}$-approximation of $E_1$ via $E:=e 2 K /T$.
The query complexity has an ${{\mathcal O}}(Td\log(Tn))={{\mathcal O}}(Kd/{\varepsilon}\log(Kn/{\varepsilon}))$ factor coming from the implementation of $\tilde{V}$ by Lemma \[lemma:controlledHamsin\]. This gets multiplied with ${{\mathcal O}}(\log(n))$ by the boosting of phase estimation, and by ${{\mathcal O}}(\sqrt{n})$ due to the minimum-finding algorithm. The gate complexity is dominated by the cost ${{\mathcal O}}(Kd/{\varepsilon}\log^{7/2}(Kn/{\varepsilon}))$ of implementing $\tilde{V}$, multiplied with the ${{\mathcal O}}(\sqrt{n}\log(n))$ factor as for the query complexity.
Note that the minimum-finding algorithm of Theorem \[thm:genMin\] can also be used for state preparation. If we choose $2{\varepsilon}$ less than the energy-gap of the Hamiltonian, then upon finding the approximation of the ground state energy we also prepare an approximate ground state. The precision of this state preparation can be improved with logarithmic cost, as can be seen from the proof of Lemma \[lemma:normEst\].
Sparse matrix summation {#app:sparsematrixsum}
=======================
As seen in Section \[sec:upperbounds\], the Arora-Kale algorithm requires an approximation of $\exp(-\eta H^{(t)})$ where $H^{(t)}$ is a sum of matrices. To keep this section general we simplify the notation. Let $H$ be the sum of $k$ different $d$-sparse matrices $M$: $$H = \sum^{k}_{i=1} M_i$$ In this section we study the complexity of one oracle call to $H$, given access to respective oracles for the matrices $M_1, \ldots, M_k$. Here we assume that the oracles for the $M_i$ are given in sparse matrix form, as defined in Section \[sec:upperbounds\]. In particular, the goal is to construct a procedure that acts as a similar sparse matrix oracle for $H$. We will only focus on the oracle that computes the non-zero indices of $H$, since the oracle that gives element access is easy to compute by summing the separate oracles.
In the remainder of this section we only consider one row of $H$. We denote this row by $R_H$ and the corresponding rows of the matrices $M_i$ by $R_i$. Notice that such a row is given as an ordered list of integers, where the integers are the non-zero indices in $R_i$. Then $R_H$ will again be an ordered list of integers, containing all integers in the $R_i$ lists once (i.e., $R_H$ does not contain duplicates).
A lower bound
-------------
We show a lower bound on the query complexity of the oracle $O^I_H$ described above by observing that determining the number of elements in a row of $H$ solves the majority function. Notice that, given access to $O^I_H$, we can decide whether there are at least a certain number of non-zero elements in a row of $H$.
Given $k+1$ ordered lists of integers $R_0,\dots,R_k$, each of length at most $d$. Let $R_H$ be the merged list that is ordered and contains every element in the lists $R_i$ only once (i.e., we remove duplicates). Deciding whether $|R_H| \leq d+ \frac{dk}{2}$ or $|R_H| \geq d+\frac{dk}{2}+1$ takes $\Omega\left(dk\right)$ quantum queries to the input lists in general.
We prove this by a reduction from MAJ on $dk$ elements. Let $Z\in \{0,1\}^{d\times k}$ be a Boolean string. It is known that it takes at least $\Omega(dk)$ quantum queries to $Z$ to decide whether $|Z|\leq \frac{dk}{2}$ or $|Z|\geq \frac{dk}{2}+1$. Now let $R_0,R_1,\dots,R_k$ be lists of length $d$ defined as follows:
- $R_0\lbrack j \rbrack = j(k+1)$ for $j = 1, \ldots, d$.
- $R_i\lbrack j \rbrack = j(k+1)+j Z_{ij}$ for $j = 1, \ldots, r$ and $i = 1,\ldots,k$.
By construction, if $Z_{ij} = 1$, then the value of the entry $R_i \lbrack j \rbrack$ is unique in the lists $R_0, \ldots, R_k$, and if $Z_{ij} = 0$ then $R_i [ j ] = R_0\lbrack j \rbrack$. So in $R_H$ there will be one element for each element in $R_0$ and one element for each bit in $Z_{ij}$ that is one. The length of $R_H$ is therefore $d+|Z|$. Hence, distinguishing between $|R_H| \leq d + \frac{dk}{2}$ and $|R_H| \geq d + \frac{dk}{2} + 1$ would solve the MAJ problem and therefore requires at least $\Omega(d k)$ queries to the lists in general.
Implementing a query to a sparse matrix oracle $O^I_H$ for $$H = \sum_{j=i}^k M_j$$ where each $M_j$ is $d$-sparse, requires $\Omega\left(dk\right)$ queries to the $O^I_{M_j}$ in general.
An upper bound
--------------
We first show that an oracle for the non-zero indices of $H$ can be constructed efficiently classically. The important observation is that in the classical case we can write down the oracle once and store it in memory. Hence, we can create an oracle for $H$ as follows. We start from the oracle of $M_1$ and then we “add” the oracle of $M_2$, then that of $M_2$, etc. By “adding” the oracle $M_i$ to $H = \sum_{j=1}^{i-1} M_j$, we mean that, per row, we insert the non-zero indices in the list of $M_i$ into that of $H$ (if it is not already there). When an efficient data structure (for example a binary heap) is used, then such insertions can be done in polylog time. This shows that in the classical case such an oracle can be made in time ${\widetilde{\mathcal O}\left(ndk\right)}$. Note that in the application we are interested in, Meta-algorithm \[alg:AKSDP\], in each iteration $t$ only one new matrix $M^{(t)}$ ‘arrives’, hence from the oracle for $H^{(t-1)}$ an oracle for $H^{(t)}$ can be constructed in time ${\widetilde{\mathcal O}\left(nd\right)}$.
The quantum case is similar, but we need to add all the matrices together each time a query to $O^I_H$ is made, since writing down each row of $H$ in every iteration would take $\Omega(n)$ operations.
To implement one such query to $O^I_H$, in particular to the the $t$th entry of $R_H$, start with an empty heap and add all elements of $R_1$ to it. Continue with the elements of $R_2$, but this time, for each element first check if it is already present, if not, add it, if it is, just continue. Overall this will take ${{\mathcal O}}(dk)$ insertions and searches in the data structure and hence ${\widetilde{\mathcal O}\left(d k\right)}$ operations. We end up with a full description of $R_H$. We can then find the index of the $t$th non-zero element from this and uncompute the whole description. Similarly, we need to be able to compute the inverse function since we need an in-place calculation. Given an index $i$ of a non-zero element in $R_H$, we can compute all the indices for $R_H$ as above, and find where $i$ is in the heap to find the corresponding value of $t$.
Equivalence of R, r and 1 over epsilon {#app:reductions}
======================================
In this section we will prove the equivalence of the three parameters $R$, $r$ and ${\varepsilon}^{-1}$ in the Arora-Kale meta-algorithm. That is, we will show any two of the three parameters can be made constant by increasing the third. Therefore, $\frac{Rr}{{\varepsilon}}$ as a whole is the interesting parameter. This appendix is structured as a set of reductions, in each case we will denote the parameters of the new SDP with a tilde.
For every SDP with $R,r \geq 1$ and $0 < {\varepsilon}\leq 1$, there is an SDP with parameters $\tilde{R}=1$ and $\tilde{r} = r$ such that solving that SDP with precision $\tilde{{\varepsilon}} = \frac{{\varepsilon}}{R}$ solves the original SDP.
Let $\tilde{A}_j = A_j$, $\tilde{C} = C$ and $\tilde{b} = \frac{b}{R}$. Now clearly $\tilde{R} = 1$, but $\widetilde{{\mbox{\rm OPT}}} = {\mbox{\rm OPT}}/R$. Hence determining $\widetilde{{\mbox{\rm OPT}}}$ up to additive error $\tilde{{\varepsilon}}=\frac{{\varepsilon}}{R}$ will determine ${\mbox{\rm OPT}}$ up to additive error ${\varepsilon}$. Notice that the feasible region of the dual did not change, so $\tilde{r} = r$.
For every SDP with $R,r\geq 1$ and $0 < {\varepsilon}\leq 1$, there is an SDP with parameters $\tilde{R} = \frac{R}{{\varepsilon}}$ and $\tilde{r} = r$ such that solving that SDP with precision $\tilde{{\varepsilon}}=1$ solves the original SDP.
Let $\tilde{A} = A$, $\tilde{c} = c$ and $\tilde{b} = \frac{b}{{\varepsilon}}$. Now $\tilde{R} = \frac{R}{{\varepsilon}}$ and $\widetilde{{\mbox{\rm OPT}}} = {\mbox{\rm OPT}}/{\varepsilon}$. Hence determining $\widetilde{{\mbox{\rm OPT}}}$ up to additive error $\tilde{{\varepsilon}}=1$ will determine ${\mbox{\rm OPT}}$ up to additive error ${\varepsilon}$. Notice that again the feasible region of the dual did not change, so $\tilde{r} = r$.
For every SDP with $R,r \geq 1$ and $0 < {\varepsilon}\leq 1$, there is an SDP with parameters $\tilde{R} = R$ and $\tilde{r}=1$ such that solving that SDP with precision $\tilde{{\varepsilon}} = \frac{{\varepsilon}}{r}$ solves the original SDP.
Let $\tilde{A} = A$, $\tilde{b} = b$ and $\tilde{C} = \frac{1}{r} C$. Now $\tilde{r} = 1$ and $\widetilde{{\mbox{\rm OPT}}} = {\mbox{\rm OPT}}/r$. Hence determining $\widetilde{{\mbox{\rm OPT}}}$ up to additive error $\tilde{{\varepsilon}}=\frac{{\varepsilon}}{r}$ will determine ${\mbox{\rm OPT}}$ up to additive error ${\varepsilon}$. Since $r \geq 1$ and ${\left\lVertC\right\rVert}\leq 1$, we find ${\left\lVert\smash{\tilde{C}}\right\rVert} \leq 1$ as required. Notice that the feasible region of the primal did not change, so $\tilde{R} = R$.
At this point we would like to state the last reduction by setting $\tilde{C} = \frac{1}{{\varepsilon}}C$, but this would not guarantee that ${\left\lVert\smash{\tilde{C}}\right\rVert} \leq 1$. Instead we give an algorithm that performs multiple calls to an SDP-solver, each of which has constant ${\varepsilon}$ but higher $r$.
Assume an SDP-solver that costs $$C(n,m,s,R,{\varepsilon},r)$$ to solve one SDP, and assume that $C$ is non-decreasing in $r$. Every SDP with $R,r \geq 1$ and $0 < {\varepsilon}\leq 1$, can be solved with cost $$\sum_{k=1}^{\log\left(\frac{1}{{\varepsilon}}\right)} C\left(n + 1,m+ 1,s,R+ 4 \log\left(\frac{1}{{\varepsilon}}\right),1,2^k(r + 1)\right),$$ by solving $\log\left(\frac{1}{{\varepsilon}}\right)$ SDPs, where the $k$-th SDP has parameters $$\tilde{n} = n+1, \ \tilde{m} = m+1, \ \tilde{R} = {{\mathcal O}}\left(R+ 4 \log\left(\frac{1}{{\varepsilon}}\right)\right), \ \tilde{r} \leq 2^k(r + 1), \text{ and in particular } \tilde{{\varepsilon}} = 1,$$ and input matrices whose elements can be described by bitstrings of length $\text{poly}(\log n,\log m, \log\left(\frac{1}{{\varepsilon}}\right))$. Furthermore, if $C(n,m,s,R,1,r) = \text{poly}(n,m,s,R,r)$, then the above cost becomes $C(n,m,s,R,1,r/{\varepsilon})$.
The high-level idea is that we want to learn a small interval in which the optimum lies, whilst using a “big” precision of $1$. We do so as follows: given an interval $[L,U]$ with the promise that ${\mbox{\rm OPT}}\in [L,U]$, we formulate another SDP for which a $1$-approximation of the optimum learns us a new, smaller, interval $[L',U']$ such that ${\mbox{\rm OPT}}\in [L',U']$. We will moreover have $U'-L' \leq \frac{1}{2}(U-L)$. In the remainder of the proof we first show how to do this reformulation, we then use this technique to prove the lemma.
Given an SDP $p$, given in the form of Equation , of which we know an interval $[L,U]$ such that ${\mbox{\rm OPT}}\in [L,U]$ (with $0 < U-L \leq 1$), we can write down an equivalent SDP $p'$ such that an optimal solution of $p$ corresponds one-to-one to an optimal solution of $p'$, and the optimum of $p'$ lies in $[0,4]$: $$\begin{aligned}
(p') \qquad \qquad \max \quad & {{\mbox{\rm Tr}}\left(\begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix} \tilde{X}\right)} \\
\text{s.t.}\ \ \ & {{\mbox{\rm Tr}}\left(\begin{pmatrix} -C & 0 \\ 0 & \frac{U-L}{4} \end{pmatrix} \tilde{X}\right)} \leq -L,\\
&{{\mbox{\rm Tr}}\left(\begin{pmatrix} A_j & 0 \\ 0 & 0 \end{pmatrix} \tilde{X}\right)} \leq b_j \quad \text{ for all } j \in [m], \\
&\tilde{X} \succeq 0,
\end{aligned}$$ here the variable $\tilde{X}$ is of size $(n+1) \times (n+1)$, and should be thought of as $\tilde{X} = \begin{pmatrix} X & \cdot \\ \cdot & z\end{pmatrix}$, where $X$ is the variable of the original SDP: an $n \times n$ positive semidefinite matrix. Observe that by assumption, for every feasible $X$ of the original SDP, $L \leq {{\mbox{\rm Tr}}\left(CX\right)} \leq U$. Therefore, the first constraint implies $0 \leq z \leq 4$ and hence the new optimum lies between $0$ and $4$, and the new trace bound is $\tilde{R} = R + 4$. We now determine $\tilde{r}$. The dual of the above program is given by: $$\begin{aligned}
(d') \qquad \qquad \min \quad & -L y_0 + \sum_{j=1}^m b_j y_j \\
\text{s.t.}\ \ \ & \begin{pmatrix} -C & 0 \\ 0 & \frac{U-L}{4} \end{pmatrix} y_0 + \sum_{j=1}^m \begin{pmatrix} A_j & 0 \\ 0 & 0 \end{pmatrix} y_j \succeq \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \\
&y \geq 0
\end{aligned}$$
\[claim33\] An optimal solution $\tilde{y}$ to $d'$ is of the form $\tilde{y} = \frac{4}{U-L}(1, y)$ where $y$ is an optimal solution to $d$, the dual of $p$.
The proof of the claim is deferred to the end of this section. The claim implies that $\tilde{r} = \frac{4}{U-L} (1+ r)$. We also have $${\mbox{\rm OPT}}= L + {\mbox{\rm OPT}}' \frac{U-L}{4},$$ and hence, a $1$-approximation to ${\mbox{\rm OPT}}'$ gives a $\frac{U-L}{4}$-approximation to ${\mbox{\rm OPT}}$.
We now use the above technique to prove the lemma. Assume an SDP of the form is given. By assumption, ${\left\lVertC\right\rVert} \leq 1$ and therefore ${\mbox{\rm OPT}}\in [-R,R]$. Calling the SDP-solver on this problem with ${\varepsilon}= 1$ will give us an estimate of ${\mbox{\rm OPT}}$ up to additive error $1$. Call this estimate ${\mbox{\rm OPT}}_0$, then ${\mbox{\rm OPT}}\in [{\mbox{\rm OPT}}_0 - 1, {\mbox{\rm OPT}}_0 +1] =: [L_0,U_0]$. We now define a new SDP $p'$ as above, with $U = U_0, L= L_0$ (notice $U_0 - L_0 \leq 2$). By the above, solving $p'$ with $\tilde{{\varepsilon}} = 1$ determines a new interval $[L_1, U_1]$ of length at most $2 \frac{U_0 - L_0}{4}$ such that ${\mbox{\rm OPT}}\in [L_1, U_1]$. We use the interval $[L_1, U_1]$ to build a new SDP $p'$ and solve that with $\tilde{{\varepsilon}} = 1$ to get an interval $[L_2, U_2]$. Repeating this procedure $k = \log\left(\frac{1}{{\varepsilon}}\right)+1$ times determines an interval $[L_k, U_k]$ of length at most $\frac{1}{2^k}(U_0-L_0) \leq {\varepsilon}$. Hence, we have determined the optimum of $p$ up to an additive error of ${\varepsilon}$. The total time needed for this procedure is at most $$\sum_{k=1}^{\log\left(\frac{1}{{\varepsilon}}\right)} C\left(n + 1,m+ 1,s,R+ 4 \log\left(\frac{1}{{\varepsilon}}\right),1,2^k(r + 1)\right). \qedhere$$
First observe that the linear matrix inequality in $d'$ implies the inequality $y_0 \frac{U-L}{4} \geq 1$ and hence $y_0 \geq \frac{4}{U-L}$. Suppose we fix a value $y_0$ (with $y_0 \geq \frac{4}{U-L}$) in $d'$, then the resulting SDP is of the form $$\begin{aligned}
(d'') \qquad \qquad -L y_0 + \min \quad & \sum_j b_j y_j \\
\text{s.t.}\ \ \ & \sum_{j=1}^m y_j A_j \succeq y_0 C, \\
&y \geq 0.
\end{aligned}$$ and hence, an optimal solution $\tilde y$ to $d''$ is of the form $\tilde y = y_0 y$ where $y$ is an optimal solution to $d$. It follows that an optimal solution to $d'$ is of the form $(y_0, y_0 y)$ where $y$ is an optimal solution to $d$. Observe that the optimal value of $d'$ as a function of $y_0$ is of the form $y_0 \cdot (-L + {\mbox{\rm OPT}})$. Since by assumption ${\mbox{\rm OPT}}\geq L$, the objective is increasing (linearly) with $y_0$ and hence $y_0 = \frac{4}{U-L}$ is optimal.
Composing the adversary bound for multiple promise functions {#app:adversarycomposition}
============================================================
The adversary bound is a powerful tool in the study of quantum query complexity. There are a few different statements of the bound but we will restrict to the general adversary bound on Boolean functions. For a more in-depth overview, see for example [@Belovs14:thesis; @hls:madv]. For a function $f : \mathcal{D} \rightarrow \{0,1\}$ with $\mathcal{D} \subseteq \{0,1\}^n$, the adversary bound is given as an optimization problem over matrices in $\mathbb{R}^{\mathcal{D}\times\mathcal{D}}$, that is, matrices indexed by the possible inputs to $f$. For $i \in \lbrack n \rbrack$, let $\Delta_i$ be the $|\mathcal D \times \mathcal D|$-zero-one matrix that is defined as $$\Delta_i(x,y) = \begin{cases}
1 \text{ if } x_i \neq y_i\\
0 \text{ if } x_i = y_i.
\end{cases}$$ Then the adversary bound ${\mbox{\rm ADV}^{\pm}(f)}$ is given by $$\begin{aligned}
{\mbox{\rm ADV}^{\pm}(f)} = \max_{\Gamma \neq \mathbf{0}} \ \ &\frac{{\left\lVert\Gamma\right\rVert}}{\max_{i\in \lbrack n \rbrack}{\left\lVert\Gamma \circ \Delta_i\right\rVert}} \\
\text{s.t. } \ \ & \Gamma(x,y) = 0 \text{ for all } x,y \text{ where } f(x) = f(y)
\end{aligned}$$ where without loss of generality the matrix $\Gamma$ can be taken to be symmetric and real. This optimization problem can be rewritten as an SDP, and duality can be used to give upper and lower bounds on ${\mbox{\rm ADV}^{\pm}(f)}$, but here we will only need it in the form stated above. It turns out that the general adversary bound characterizes the quantum query complexity up to a constant factor. We only need the lower bound:
For every function $f: \mathcal{D} \rightarrow \{0,1\}$ with $\mathcal{D} \subseteq \{0,1\}^n$, we have $$\mathcal{Q}_{{\varepsilon}}(f) \geq \frac{1-2\sqrt{{\varepsilon}(1-{\varepsilon})}}{2} {\mbox{\rm ADV}^{\pm}(f)}$$
One of the most important results in this area is that the adversary bound, and hence the quantum query complexity, is multiplicative under composition of functions.
#### Notation
Let $f: \mathcal{D} \rightarrow \{0,1\}$ with $\mathcal{D} \subseteq \{0,1\}^n$ and $g_j: \mathcal{C}_j \rightarrow \{0,1\}$ with $\mathcal{C}_j \subseteq \{0,1\}^{k_j}$. We write $h = f \circ (g_1,\dots,g_n)$ for the function on $\sum^n_{j=1} k_j$ bits that first evaluates $g_j$ on the $j$th part of the input and then $f$ on the result. We write $\tilde{x}\in \{0,1\}^n$ for the input of $f$ corresponding to $x$, and $x^j \in \{0,1\}^{k_j}$ for the $j$th part of the input $x$. We say $\tilde{x}$ *agrees* with $x$, and write $x \simeq \tilde{x}$, if $\tilde{x} = (g_1(x^1),\dots,g_n(x^n))$. Then the domain of $h$ is $$\mathcal{H} = \left\{ x \in \{0,1\}^{\sum^n_{j=1} k_j} : \forall j : x^j \in \mathcal{C}_j \text{ and } \tilde{x} \in \mathcal{D} \right\}$$
The following composition theorem was already stated in the literature by [@hls:madv].
Let $f: \{0,1\}^n \rightarrow \{0,1\}$ and $g_j: \{0,1\}^{k_j} \rightarrow \{0,1\}$ be total functions, then for $h = f \circ (g_1,\dots,g_n)$ $${\mbox{\rm ADV}^{\pm}(h)} = \Omega\left( {\mbox{\rm ADV}^{\pm}(f)} \min_{j\in\lbrack n\rbrack} {\mbox{\rm ADV}^{\pm}(g_j)} \right)$$
Note that this theorem applies to total functions, not partial functions. A version of the theorem that applies to compositions of partial functions has been stated by [@Kimmel:adv], but only for partial functions composed with them self.
We are not aware of any statement for arbitrary partial Boolean functions, but we can obtain it by modifying the proof of [@hls:madv] slightly, as we will show in the remainder of this section.
Let $\Gamma_f$ and $\Gamma_{g_j}$ denote feasible solutions to the maximization problem for $f$ and $g_j$, not necessarily optimal. It can be helpful to think of these as block matrices with four blocks by rearranging the rows and columns so the zero-outputs come first $$\Gamma_f = \begin{bmatrix}
0 & \Gamma_f^{1,0}\\
\Gamma_f^{0,1} & 0
\end{bmatrix}.$$ Now, with $\hat{\Gamma}_{g_j} = \Gamma_{g_j} + I \cdot {\left\lVert\Gamma_{g_j}\right\rVert}$, let $$\Gamma^{\prime} = \Gamma_f \otimes \bigotimes_{j=1}^n \hat{\Gamma}_{g_j}$$ indexed by the tuples $(\tilde{x},x^1,\dots,x^n)$. Our, not necessarily optimal, candidate solution for the adversary bound of $h$ is $\Gamma_h$, element-wise defined by $$\Gamma_h(x,y) = \Gamma_f(\tilde{x},\tilde{y}) \cdot \prod^n_{j=1} \hat{\Gamma}_{g_j}(x^j,y^j).$$ This can be seen as the $|\mathcal{H}| \times |\mathcal{H}|$-submatrix of $\Gamma^{\prime}$ obtained by taking only the rows and columns where $\tilde{x}$ agrees with $x^1,\dots,x^n$. We will denote the map that constructs $\Gamma_h$ from $\Gamma_f$ and $\Gamma_{g_1},\dots,\gamma_{g_n}$ by $\gamma$, so $$\gamma : (\Gamma_f,\Gamma_{g_1},\dots,\Gamma_{g_n}) \mapsto \Gamma_h$$ Note that $\Gamma_h$ is indeed a feasible solution to the adversary SDP:
- The matrix has the right dimensions and is indexed only by the strings in $\mathcal{H}$. This is because by definition each $x^j$ is from $\mathcal{C}_j$ and $\tilde{x}$ is from $\mathcal{D}$.
- If $\Gamma_f$ and all $\Gamma_{g_j}$ are non-zero, then $\Gamma_h$ is non-zero too.
- If $h(x) = h(y)$, then $f(\tilde{x}) = f(\tilde{y})$ and hence $\Gamma_h(x,y) = 0$.
It remains to calculate the objective value of this matrix. To do so, we need the following lemma.
\[lem:mnorm\] Let $\Gamma_f$,$\Gamma_{g_j}$ be feasible solutions for the corresponding SDPs and let $\Gamma_h = \gamma\left(\Gamma_f,\Gamma_{g_1},\dots,\Gamma_{g_n}\right)$. Then $${\left\lVert\Gamma_h\right\rVert} = {\left\lVert\Gamma_f\right\rVert} \prod^n_{j=1} {\left\lVert\Gamma_{g_j}\right\rVert}$$
To begin, note that since $\Gamma_h$ is a submatrix of $\Gamma^{\prime}$ we have $${\left\lVert\Gamma_h\right\rVert} \leq {\left\lVert\Gamma^{\prime}\right\rVert} = {\left\lVert\Gamma_f\right\rVert} \prod^n_{j=1} {\left\lVert\Gamma_{g_j}\right\rVert}.$$ It will therefore suffice to find an eigenvector of $\Gamma_h$ with eigenvalue ${\left\lVert\Gamma_f\right\rVert} \prod^n_{j=1} {\left\lVert\Gamma_{g_j}\right\rVert}$. We claim that, if $\delta_f$ and $\delta_{g_j}$ are principal eigenvectors of $\Gamma_f$ and $\Gamma_{g_j}$ respectively, then $\delta_h$ defined element-wise by $$\delta_h(x) = \delta_f(\tilde{x}) \prod_{j=1}^n \delta_{g_j}(x^j)$$ is such an eigenvector. To see this, simply write out the product of $\delta_h$ with $\Gamma_h$ for one entry: $$\begin{aligned}
\left( \Gamma_h\delta_h\right)(x) &= \sum_{y\in \mathcal{H}} \Gamma_h(x,y)\delta_h(y)\\
&= \sum_{y\in \mathcal{H}} \left( \Gamma_f(\tilde{x},\tilde{y}) \cdot \prod^n_{j=1} \hat{\Gamma}_{g_j}(x^j,y^j) \right) \cdot \left( \delta_f(\tilde{y}) \prod_{j=1}^n \delta_{g_j}(y^j) \right)\\
&= \sum_{y\in \mathcal{H}} \Gamma_f(\tilde{x},\tilde{y}) \delta_f(\tilde{y}) \prod_{j=1}^n \hat{\Gamma}_{g_j}(x^j,y^j) \delta_{g_j}(y^j) \\
&= \sum_{\tilde{y}\in \mathcal{D}} \Gamma_f(\tilde{x},\tilde{y}) \delta_f(\tilde{y}) \sum_{y:y \simeq \tilde{y}}\prod_{j=1}^n \hat{\Gamma}_{g_j}(x^j,y^j) \delta_{g_j}(y^j) \\
&= \sum_{\tilde{y}\in \mathcal{D}} \Gamma_f(\tilde{x},\tilde{y}) \delta_f(\tilde{y})
\prod_{j=1}^n \sum_{y^j: g_j(y^j) = \tilde{y}_j} \hat{\Gamma}_{g_j}(x^j,y^j) \delta_{g_j}(y^j).
\end{aligned}$$ Notice that the inner sum is just the $x^j$-entry of the matrix-vector product $\hat{\Gamma}_{g_j}\delta_{g_j}$ but with $\delta_{g_j}$ restricted to the indices that agree with $\tilde{y}_j$. Since $\delta_{g_j}$ is a principal eigenvector of $\Gamma_{g_j}$ we know that $$\Gamma_{g_j} \delta_{g_j} =
\begin{bmatrix}
0 & \Gamma_{g_j}^{1,0}\\
\Gamma_{g_j}^{0,1} & 0
\end{bmatrix}
\begin{bmatrix}
\delta_{g_j}^{0}\\
\delta_{g_j}^{1}
\end{bmatrix}
=
{\left\lVert\Gamma_{g_j}\right\rVert} \begin{bmatrix}
\delta_{g_j}^{0}\\
\delta_{g_j}^{1}
\end{bmatrix}.$$ Suppose that $\tilde{y}_j = 1$, then the inner sum becomes: $$\sum_{y^j: g_j(y^j) = \tilde{y}_j} \hat{\Gamma}_{g_j}(x^j,y^j) \delta_{g_j}(y^j) =
e_{x^j}^T
\begin{bmatrix}
{\left\lVert\Gamma_{g_j}\right\rVert}I & \Gamma_{g_j}^{1,0}\\
\Gamma_{g_j}^{0,1} & {\left\lVert\Gamma_{g_j}\right\rVert}I
\end{bmatrix}
\begin{bmatrix}
0\\
\delta_{g_j}^{1}
\end{bmatrix}
=
e_{x^j}^T
{\left\lVert\Gamma_{g_j}\right\rVert} \begin{bmatrix}
\delta_{g_j}^{0}\\
\delta_{g_j}^{1}
\end{bmatrix} = {\left\lVert\Gamma_{g_j}\right\rVert} \delta_{g_j}(x^j),$$ where $e_{x^j}$ is the vector with a one on the $x^j$-entry and zeroes elsewhere. The same identity between left hand side and right hand side can be shown when $\tilde{y}_j = 0$. Combining the above shows: $$\begin{aligned}
\left( \Gamma_h\delta_h\right)(x) &= \sum_{\tilde{y}} \Gamma_f(\tilde{x},\tilde{y}) \delta_f(\tilde{y})
\prod_{j=1}^n \sum_{y^j:y^j \simeq \tilde{y}_j} \hat{\Gamma}_{g_j}(x^j,y^j) \delta_{g_j}(y^j) \\
&= \sum_{\tilde{y}} \Gamma_f(\tilde{x},\tilde{y}) \delta_f(\tilde{y})
\prod_{j=1}^n {\left\lVert\Gamma_{g_j}\right\rVert} \delta_{g_j}(x^j) \\
&= {\left\lVert\Gamma_f\right\rVert} \delta_f(\tilde{x}) \prod_{j=1}^n {\left\lVert\Gamma_{g_j}\right\rVert} \delta_{g_j}(x^j) \\
&= {\left\lVert\Gamma_f\right\rVert} \prod_{j=1}^n {\left\lVert\Gamma_{g_j}\right\rVert} \delta_{h}(x)
\end{aligned}$$ so we conclude that $\delta_h$ is an eigenvector of $\Gamma_h$ with eigenvalue ${\left\lVert\Gamma_f\right\rVert} \prod_{j=1}^n {\left\lVert\Gamma_{g_j}\right\rVert}$ and hence a principal eigenvector.
We are now ready to prove the lower bound.[^17]
\[thm:adv\] Let $f: \mathcal{D} \rightarrow \{0,1\}$ for $\mathcal{D} \subseteq \{0,1\}^n$ and $g_j: \mathcal{C}_j \rightarrow \{0,1\}$ for $\mathcal{C}_i \subseteq\{0,1\}^{k_j}$, then for $h = f \circ (g_1,\dots,g_n)$ as before $${\mbox{\rm ADV}^{\pm}(h)} \geq {\mbox{\rm ADV}^{\pm}(f)} \min_j {\mbox{\rm ADV}^{\pm}(g_j)}$$
Let $\Gamma_f^{\star}$, $\Gamma_{g_j}^{\star}$ be optimal solutions to the adversary bound for $f$ and $g_j$ and let $\Gamma_h^{\star} = \gamma\left( \Gamma_f^{\star}, \Gamma_{g_1}^{\star},\dots,\Gamma_{g_n}^{\star} \right)$ as before. We already know that $\Gamma_h^{\star}$ is feasible, so it remains to calculate its objective value. From Lemma \[lem:mnorm\] we know that $${\left\lVert\Gamma_h^{\star}\right\rVert} = {\left\lVert\Gamma_f^{\star}\right\rVert} \prod_{i=1}^n {\left\lVert\Gamma_{g_j}^{\star}\right\rVert}$$ so it remains to say something about ${\left\lVert\Gamma_h^{\star} \circ \Delta_{\ell}\right\rVert}$. Let $\ell$ be the $q$th bit in the $p$th input block. Notice that $\Gamma_f^{\star} \circ \Delta_{p}$ and $\Gamma_{g_p}^{\star} \circ \Delta_{q}$ are both either feasible solutions to the adversary SDP or they are zero. We claim that $\Gamma_h^{\star} \circ \Delta_{\ell} = \gamma\left( \Gamma_f^{\star} \circ \Delta_{p}, \Gamma_{g_1}^{\star},\dots,\Gamma_{g_p}^{\star} \circ \Delta_{q},\dots,\Gamma_{g_n}^{\star} \right)$. To see this, look at the element-wise description of this claim, $$(\Gamma^{\star}_h \circ \Delta_{\ell})(x,y) = (\Gamma^{\star}_f \circ \Delta_{p})(\tilde{x},\tilde{y})\cdot ( \Gamma^{\star}_{g_p} \circ \Delta_q + I {\left\lVert\Gamma^{\star}_{g_p} \circ \Delta_q\right\rVert} )(x^p,y^p) \cdot \prod^n_{j=1} \hat{\Gamma}^{\star}_{g_j}(x^j,y^j)$$ and consider four different cases
- $x_{\ell} \neq y_{\ell}$ and $\tilde{x}_{p} \neq \tilde{y}_{p}$. The entries of $\Delta_{\ell},\Delta_p,\Delta_q$ that are present in the product are all $1$ and $(x^p,y^p)$ is off-diagonal. This means that the element-wise description is just the definition of $\Gamma^{\star}_h(x,y)$ and hence is correct.
- $x_{\ell} \neq y_{\ell}$ and $\tilde{x}_{p} = \tilde{y}_{p}$. The right-hand side of the equation is clearly zero since $(\Gamma^{\star}_f \circ \Delta_{p})(\tilde{x},\tilde{y})$ is. For the left-hand side, since $g_p(x^p) = g_p(y^p)$ we know $\Gamma^{\star}_{g_p}(x^p,y^p)$ is zero, but $(x^p,y^p)$ is off-diagonal and therefore $(\Gamma^{\star}_{g_p}+ I{\left\lVert\Gamma^{\star}_{g_p}\right\rVert})(x^p,y^p) =0 $. We conclude that $\Gamma^{\star}_{h}(x,y) = 0$ and hence the left-hand side of the equation is zero.
- $x_{\ell} = y_{\ell}$ and $\tilde{x}_{p} \neq \tilde{y}_{p}$. The left-hand side is clearly zero. On the right-hand side $\Gamma^{\star}_{g_p}(x^p,y^p) \circ \Delta_q$ is zero. Since $\tilde{x}_p = g_p(x^p) \neq g_p(y^p) = \tilde{y}_p$ we know that $x^p\neq y^p$, so $(x^p,y^p)$ is off-diagonal. This means that $\left(\Gamma^{\star}_{g_p} \circ \Delta_q + I {\left\lVert\Gamma^{\star}_{g_p} \circ \Delta_q\right\rVert}\right)(x^p,y^p)$ is zero too, and hence the right-hand side is zero.
- $x_{\ell} = y_{\ell}$ and $\tilde{x}_{p} = \tilde{y}_{p}$. Again the left hand side is clearly zero. So is the right hand since $\left(\Gamma^{\star}_{f}\circ \Delta_p \right)(\tilde{x},\tilde{y})$ is zero.
Hence set $\Gamma_h^{\star} \circ \Delta_{\ell} = \gamma\left( \Gamma_f^{\star} \circ \Delta_{p}, \Gamma_{g_1}^{\star},\dots,\Gamma_{g_p}^{\star} \circ \Delta_{q},\dots,\Gamma_{g_n}^{\star} \right)$.
Then, by Lemma \[lem:mnorm\] $${\left\lVert\Gamma^{\star}_h \circ \Delta_{\ell}\right\rVert}= {\left\lVert\Gamma^{\star}_f \circ \Delta_{p}\right\rVert}\cdot {\left\lVert \Gamma^{\star}_{g_p} \circ \Delta_q \right\rVert} \cdot \prod^n_{j=1} {\left\lVert\Gamma^{\star}_{g_j}\right\rVert}$$ When $\Gamma^{\star}_h \circ \Delta_{\ell} \neq 0$, it follows that $$\frac{{\left\lVert\Gamma^{\star}_h\right\rVert}}{{\left\lVert\Gamma^{\star}_h \circ \Delta_{\ell}\right\rVert}} = \frac{{\left\lVert\Gamma^{\star}_f\right\rVert}}{{\left\lVert\Gamma^{\star}_f \circ \Delta_p\right\rVert}}
\cdot \frac{{\left\lVert\Gamma^{\star}_{g_p}\right\rVert}}{{\left\lVert\Gamma^{\star}_{g_p} \circ \Delta_q\right\rVert}}$$ which leads us to $$\begin{aligned}
{\mbox{\rm ADV}^{\pm}(h)} &\geq \min \left\{ \frac{{\left\lVert\Gamma^{\star}_h\right\rVert}}{{\left\lVert\Gamma^{\star}_h \circ \Delta_{\ell}\right\rVert}}: \, \Gamma^{\star}_h \circ \Delta_{\ell} \neq 0 \right\}\\
&= \min \left\{ \frac{{\left\lVert\Gamma^{\star}_f\right\rVert}}{{\left\lVert\Gamma^{\star}_f \circ \Delta_p\right\rVert}} \cdot \frac{{\left\lVert\Gamma^{\star}_{g_p}\right\rVert}}{{\left\lVert\Gamma^{\star}_{g_p} \circ \Delta_q\right\rVert}}: \, \Gamma^{\star}_f \circ \Delta_{p} \neq 0 \text{ and } \Gamma^{\star}_{g_p} \circ \Delta_q \neq 0 \right\}\\
&= \min \left\{ \frac{{\left\lVert\Gamma^{\star}_f\right\rVert}}{{\left\lVert\Gamma^{\star}_f \circ \Delta_p\right\rVert}}: \, \Gamma^{\star}_f \circ \Delta_p \neq 0 \right\} \cdot \min \left\{ \frac{{\left\lVert\Gamma^{\star}_{g_p}\right\rVert}}{{\left\lVert\Gamma^{\star}_{g_p} \circ \Delta_q\right\rVert}}: \, \Gamma^{\star}_{g_p} \circ \Delta_q \neq 0 \right\}\\
&= \min \left\{\frac{{\left\lVert\Gamma^{\star}_f\right\rVert}}{{\left\lVert\Gamma^{\star}_f \circ \Delta_p\right\rVert}} : \, \Gamma^{\star}_f \circ \Delta_p \neq 0 \right\} \cdot {\mbox{\rm ADV}^{\pm}(g_p)}\\
&\geq \min \left\{\frac{{\left\lVert\Gamma^{\star}_f\right\rVert}}{{\left\lVert\Gamma^{\star}_f \circ \Delta_p\right\rVert}} : \, \Gamma^{\star}_f \circ \Delta_p \neq 0 \right\} \cdot \left( \min_{j\in \lbrack n\rbrack} {\mbox{\rm ADV}^{\pm}(g_j)} \right)\\
&= {\mbox{\rm ADV}^{\pm}(f)} \min_{j\in \lbrack n\rbrack} {\mbox{\rm ADV}^{\pm}(g_j)}
\end{aligned}$$ -0.5cm
[^1]: QuSoft, CWI, the Netherlands. Supported by the Netherlands Organization for Scientific Research, grant number 617.001.351. [[email protected]]{}
[^2]: QuSoft, CWI, the Netherlands. Supported by ERC Consolidator Grant 615307-QPROGRESS. [[email protected]]{}
[^3]: QuSoft, CWI, the Netherlands. Supported by the Netherlands Organization for Scientific Research, grant number 617.001.351. [[email protected]]{}
[^4]: QuSoft, CWI and University of Amsterdam, the Netherlands. Partially supported by ERC Consolidator Grant 615307-QPROGRESS and QuantERA project QuantAlgo 680-91-034. [[email protected]]{}
[^5]: See Lee, Sidford, and Wong [@lsw:faster Section 10.2 of arXiv version 2], and note that our $m,n$ are their $n,m$, their $S$ is our $mns$, and their $M$ is our $R$. The bounds for other SDP-solvers that we state later also include another parameter $r$; it follows from the assumptions of [@lsw:faster Theorem 45 of arXiv version 2] that in their setting $r\leq mR$, and hence $r$ is absorbed in the $\log^{O(1)}(mnR/{\varepsilon})$ factor.
[^6]: See also [@ahk:multweights] for a subsequent survey; the same algorithm was independently discovered around the same time in the context of learning theory [@TRW:matrixexp; @WK:onlinevarmin].
[^7]: The $\widetilde{\mathcal O}(\cdot)$ notation hides polylogarithmic factors in all parameters.
[^8]: Independently of us, Ben David, Eldar, Garg, Kothari, Natarajan, and Wright (at MIT), and separately Ambainis observed that in the special case where all $b_j$ are at least 1, the oracle can even be made 1-sparse, and the one entry can be found using one Grover search over $m$ points (in both cases personal communication 2017). The same happens implicitly in our Section \[sec:oracle\] in this case. However, in general 2 non-zero entries are necessary in $y$.
[^9]: Using several transformations of the SDP, from Appendix \[app:reductions\] and Lemma 2 of , one can show that there is a way to remove the need for this restriction. Hence, after these modifications, if for a given candidate solution $X \succeq 0$ the oracle outputs that the set $\mathcal P_0(X)$ is empty, then a scaled version of $X$ is primal feasible for this new SDP, with objective value at least $\alpha$. This scaled version of $X$ can be modified to a near-feasible solution to the original SDP (it will be psd, but it might violate the linear constraints a little bit) with nearly the same objective value.
[^10]: Here $m$ is the number of constraints, not the number of edges in the graph.
[^11]: For ease of notation, we write $H$ for $\eta H$.
[^12]: Here we assume that $\theta \leq 1$. Then, since $\lambda_{\min}(B) \geq 1$, we trivially have $\theta/{\left\lVertu\right\rVert} \leq 1/\sqrt{n} \leq \sqrt{n} \leq {\left\lVert\smash{\sqrt{B}} u\right\rVert}$.
[^13]: For example there is a more recent method for Hamiltonian simulation [@LowChuang:Qubitization; @LowChuang:SignalProc] that could possibly improve on some of the log factors we get from [@BerryChilds:hamsimFOCS], but one could even consider completely different input models allowing different simulation methods.
[^14]: This way, even if we make $\theta$ error during the estimation of an eigenvalue $\lambda_i$, the closest $x_\ell$ will still contain $\lambda_i$ in its radius-$r$ neighborhood.
[^15]: Note that phase estimation on some eigenvector of $e^{iH}$ can produce a superposition of different estimates of the phase. If some intervals $[x_\ell-r,x_\ell+r]$ overlap for $\ell$ and $\ell'$, those estimates could lead to different implementations of $f(H)$ (one based on the coefficients $a_k^{(\ell)}$ and one based on $a_k^{(\ell')}$). However, this causes no difficulty; since we used the same normalization $1/B$ for all implementations, both implementations lead to essentially the same state after postselecting on the ${|0\rangle}$ ancilla state.
[^16]: Throughout this proof, whenever a fraction is $0/0$, we simply interpret it as $0$. Therefore we also interpret conditional probabilities, conditioned on events that happen with probability $0$, as $0$.
[^17]: The corresponding upper bound, with $\min_j$ replaced by $\max_j$, was proved by Reichardt [@reichardt:tight].
|
---
author:
- Dongwen Yang
- Jian Lv
- Xingang Zhao
- Qiaoling Xu
- Yuhao Fu
- Yiqiang Zhan
- Alex Zunger
- Lijun Zhang
title: '**Supplemental Material** for “Functionality-directed Screening of Pb-free Hybrid Organic-inorganic Perovskites with Desired Intrinsic Photovoltaic Functionalities”'
---
{width="3.5in"}
{width="3.5in"}
{width="3.5in"}
![ Band structures of (a) \[FM\]PbBr$_3$, (b) \[FM\]SnBr$_3$, (c) \[FM\]GeBr$_3$ calculated without (left panels) and with (right panels) the spin-orbit coupling (SOC) effect. The orbital projections onto the FM molecule are indicated by red circles.](figS4.eps){width="3.5in"}
{width="3.5in"}
![ Evaluation of the steric sizes of organic molecular cations within the idealized solid-sphere model \[see the Experimental Section (iv)\]. ](figS6.eps){width="3.5in"}
\[Table S1\]
[?c?c?c?c?c?c?c?c?c?c?p[102cm]{}?]{} $\Delta$H (eV) & PbI$_3$ & PbBr$_3$ & PbCl$_3$ & SnI$_3$ & SnBr$_3$ & SnCl$_3$ & GeI$_3$ & GeBr$_3$ & GeCl$_3$\
\[M\]$^{+}$ & -0.59 & -0.48 & -0.49 & -0.45 & -0.35 & -0.34 & -0.38 & -0.10 & –\
\[Cs\]$^{+}$ & -0.17 & -0.02 & -0.01 & -0.01 & 0.13 & 0.17 & 0.02 & 0.35 & –\
\[HA\]$^{+}$ & -0.45 & -0.26 & -0.29 & -0.32 & -0.23 & -0.23 & -0.30 & -0.03 & –\
\[DA\]$^{+}$ & -0.25 & -0.19 & -0.21 & -0.12 & -0.09 & -0.09 & -0.13 & 0.07 & –\
\[MA\]$^{+}$ & -0.02 & 0.12 & 0.12 & 0.10 & 0.21 & 0.24 & 0.03 & 0.33 & –\
\[FM\]$^{+}$ & -0.21 & -0.09 & -0.11 & -0.11 & -0.02 & -0.01 & -0.18 & 0.08 & –\
\[FA\]$^{+}$ & -0.10 & -0.01 & -0.02 & -0.03 & 0.04 & 0.08 & -0.16 & 0.10 & –\
\[EA\]$^{+}$ & 0.09 & 0.16 & 0.01 & 0.16 & 0.24 & 0.27 & 0.01 & 0.25 & –\
\[GA\]$^{+}$ & -0.14 & -0.11 & -0.24 & -0.09 & -0.06 & 0.02 & -0.29 & -0.07 & –\
\[DEA\]$^{+}$ & 0.16 & 0.20 & 0.10 & 0.22 & 0.28 & 0.30 & 0.06 & 0.34 & –\
\[Table S2\]
[?c?c?c?c?c?c?c?c?c?c?p[102cm]{}?]{} $\Delta$H (eV) & PbI$_3$ & PbBr$_3$ & PbCl$_3$ & SnI$_3$ & SnBr$_3$ & SnCl$_3$ & GeI$_3$ & GeBr$_3$ & GeCl$_3$\
\[M\]$^{+}$ & -0.59 & -0.48 & -0.49 & -0.45 & -0.35 & -0.34 & -0.38 & -0.10 & –\
\[Cs\]$^{+}$ & -0.17 & -0.02 & -0.01 & -0.01 & 0.13 & 0.17 & 0.02 & 0.35 & –\
\[HA\]$^{+}$ & -0.38 & -0.23 & -0.24 & -0.27 & -0.14 & -0.15 & -0.28 & 0.01 & –\
\[DA\]$^{+}$ & -0.23 & -0.17 & -0.20 & -0.10 & -0.07 & -0.07 & -0.13 & 0.08 & –\
\[MA\]$^{+}$ & 0.01 & 0.17 & 0.18 & 0.13 & 0.26 & 0.29 & 0.04 & 0.33 & –\
\[FM\]$^{+}$ & -0.17 & -0.01 & 0.00 & -0.06 & 0.07 & 0.10 & -0.09 & 0.18 & –\
\[FA\]$^{+}$ & -0.20 & 0.03 & -0.07 & -0.07 & 0.10 & 0.11 & -0.17 & 0.16 & –\
\[EA\]$^{+}$ & 0.08 & 0.14 & 0.06 & 0.14 & 0.24 & 0.27 & -0.01 & 0.27 & –\
\[GA\]$^{+}$ & -0.14 & -0.11 & -0.20 & -0.10 & -0.06 & -0.10 & -0.28 & -0.10 & –\
\[DEA\]$^{+}$ & 0.16 & 0.20 & 0.10 & 0.23 & 0.28 & 0.29 & 0.08 & 0.35 & –\
\[Table S3\]
[?c?c?c?c?c?c?c?c?c?c?p[102cm]{}?]{} E$_{g}^{d}$ (eV) & PbI$_3$ & PbBr$_3$ & PbCl$_3$ & SnI$_3$ & SnBr$_3$ & SnCl$_3$ & GeI$_3$ & GeBr$_3$ & GeCl$_3$\
\[M\]$^{+}$ & 1.95 & 2.71 & 3.39 & 1.21 & 1.91 & 2.68 & 1.59 & 2.01 & 2.54\
\[Cs\]$^{+}$ & 1.30 & 1.98 & 2.63 & 0.95 & 1.53 & 2.14 & 1.15 & 1.64 & 2.17\
\[HA\]$^{+}$ & 1.75 & 2.74 & 3.73 & 1.15 & 1.87 & 2.85 & 1.70 & 2.26 & 3.12\
\[DA\]$^{+}$ & 1.73 & 2.55 & 3.55 & 1.17 & 1.91 & 2.83 & 1.78 & 2.47 & 3.40\
\[MA\]$^{+}$ & 1.55 & 2.43 & 3.23 & 1.26 & 2.00 & 3.01 & 1.98 & 3.01 & 4.10\
\[FM\]$^{+}$ & 1.76 & 2.65 & 3.49 & 1.32 & 2.16 & 3.20 & 2.04 & 3.08 & 4.17\
\[FA\]$^{+}$ & 1.74 & 2.60 & 3.25 & 1.21 & 2.54 & 3.73 & 2.36 & 3.79 & 4.95\
\[EA\]$^{+}$ & 1.86 & 2.67 & 3.56 & 1.70 & 3.13 & 4.47 & 2.61 & 3.99 & 5.11\
\[GA\]$^{+}$ & 1.90 & 2.70 & 3.99 & 1.78 & 3.77 & 4.91 & 3.28 & 4.13 & 5.90\
\[DEA\]$^{+}$ & 1.81 & 2.60 & 3.44 & 1.62 & 3.19 & 4.51 & 2.82 & 4.15 & 5.07\
\[Table S4\]
[?c?c?c?c?c?c?c?c?c?c?p[102cm]{}?]{} m$_{e}^{*}$ & & & & & & & & &\
m$_{h}^{*}$ & & & & & & & & &\
& 0.49 & 0.70 & 0.92 & 0.59 & 0.70 & 0.73 & 0.35 & 0.52 & 0.57\
& 0.37 & 0.36 & 0.38 & 0.18 & 0.20 & 0.16 & 0.24 & 0.10 & 0.13\
& 0.18 & 0.23 & 0.38 & 0.24 & 0.32 & 0.38 & 0.19 & 0.26 & 0.33\
& 0.22 & 0.21 & 0.21 & 0.17 & 0.15 & 0.10 & 0.21 & 0.18 & 0.19\
& 0.46 & 0.65 & 0.85 & 0.43 & 0.55 & 0.63 & 0.33 & 0.38 & 0.41\
& 0.33 & 0.46 & 0.59 & 0.19 & 0.22 & 0.25 & 0.29 & 0.17 & 0.24\
& 0.47 & 0.67 & 0.97 & 0.42 & 0.53 & 0.62 & 0.32 & 0.37 & 0.47\
& 0.31 & 0.36 & 0.47 & 0.19 & 0.21 & 0.23 & 0.27 & 0.21 & 0.31\
& 0.42 & 0.70 & 0.88 & 0.27 & 0.50 & 0.58 & 0.38 & 0.51 & 0.70\
& 0.18 & 0.28 & 0.31 & 0.18 & 0.19 & 0.17 & 0.28 & 0.31 & 0.37\
& 0.54 & 1.12 & 8.81 & 0.56 & 0.66 & 4.50 & 0.40 & 0.57 & 7.83\
& 0.25 & 0.36 & 0.42 & 0.19 & 0.21 & 0.22 & 0.30 & 0.34 & 0.43\
& 0.44 & 0.68 & 0.54 & 0.33 & 0.73 & 0.75 & 0.38 & 0.57 & 0.78\
& 0.23 & 0.35 & 0.29 & 0.25 & 0.31 & 0.44 & 0.37 & 0.51 & 0.73\
& 0.46 & 0.60 & 0.80 & 0.38 & 0.57 & 0.82 & 0.40 & 0.58 & 0.88\
& 0.28 & 0.34 & 0.41 & 0.14 & 0.30 & 0.50 & 0.42 & 0.61 & 0.81\
& 0.39 & 0.61 & 0.68 & 0.32 & 0.85 & 0.91 & 0.51 & 0.55 & 0.77\
& 0.37 & 0.35 & 0.51 & 0.21 & 0.39 & 0.54 & 0.64 & 0.80 & 0.95\
& 0.26 & 0.40 & 0.66 & 0.30 & 0.48 & 0.61 & 0.41 & 0.56 & 0.57\
& 0.33 & 0.32 & 0.36 & 0.20 & 0.29 & 0.46 & 0.55 & 0.73 & 0.82\
\[Table S5\]
[?c?c?c?c?c?c?c?c?c?c?p[102cm]{}?]{} E$_B$ (meV) & PbI$_3$ & PbBr$_3$ & PbCl$_3$ & SnI$_3$ & SnBr$_3$ & SnCl$_3$ & GeI$_3$ & GeBr$_3$ & GeCl$_3$\
\[M\]$^{+}$ & 94.25 & 165.80 & 274.85 & 23.92 & 49.79 & 143.66 & 29.05 & 49.39 & 117.68\
\[Cs\]$^{+}$ & 43.89 & 75.87 & 68.84 & 13.73 & 26.75 & 48.69 & 14.29 & 26.40 & 47.42\
\[HA\]$^{+}$ & 73.51 & 187.52 & 347.17 & 20.93 & 48.35 & 109.24 & 46.83 & 75.02 & 146.80\
\[DA\]$^{+}$ & 67.51 & 144.02 & 287.43 & 20.75 & 48.01 & 96.39 & 48.14 & 87.14 & 183.32\
\[MA\]$^{+}$ & 52.34 & 135.85 & 223.67 & 30.05 & 45.93 & 83.12 & 61.63 & 127.55 & 247.88\
\[FM\]$^{+}$ & 33.00 & 230.12 & 412.57 & 28.93 & 67.96 & 158.42 & 64.59 & 152.64 & 460.45\
\[FA\]$^{+}$ & 56.75 & 125.80 & 192.72 & 38.09 & 101.64 & 282.48 & 94.22 & 274.09 & 607.20\
\[EA\]$^{+}$ & 109.75 & 191.90 & 357.38 & 36.26 & 115.80 & 337.07 & 144.62 & 338.33 & 632.50\
\[GA\]$^{+}$ & 83.06 & 153.76 & 313.45 & 38.22 & 200.53 & 432.95 & 202.35 & 419.56 & 772.06\
\[DEA\]$^{+}$ & 69.89 & 116.20 & 315.05 & 30.94 & 123.16 & 322.60 & 188.67 & 408.48 & 653.61\
\[Table S6\]
[?c?c?c?c?c?c?c?c?c?c?p[102cm]{}?]{} $\alpha_{ex}$ (nm) & PbI$_3$ & PbBr$_3$ & PbCl$_3$ & SnI$_3$ & SnBr$_3$ & SnCl$_3$ & GeI$_3$ & GeBr$_3$ & GeCl$_3$\
\[M\]$^{+}$ & 1.40 & 1.01 & 0.73 & 3.60 & 2.40 & 1.09 & 3.26 & 2.41 & 1.20\
\[Cs\]$^{+}$ & 2.83 & 2.07 & 2.74 & 5.56 & 3.82 & 2.82 & 5.19 & 3.72 & 2.66\
\[HA\]$^{+}$ & 1.79 & 0.91 & 0.58 & 4.06 & 2.43 & 1.44 & 2.18 & 1.79 & 1.18\
\[DA\]$^{+}$ & 1.93 & 1.14 & 0.68 & 4.12 & 2.47 & 1.60 & 2.21 & 1.61 & 0.98\
\[MA\]$^{+}$ & 2.40 & 1.19 & 0.86 & 2.91 & 2.62 & 1.99 & 1.89 & 1.26 & 0.82\
\[FM\]$^{+}$ & 3.94 & 0.72 & 0.47 & 3.17 & 1.88 & 1.08 & 1.86 & 1.08 & 0.44\
\[FA\]$^{+}$ & 2.29 & 1.30 & 1.00 & 2.74 & 1.50 & 0.72 & 1.45 & 0.70 & 0.39\
\[EA\]$^{+}$ & 1.40 & 1.00 & 0.66 & 2.85 & 1.36 & 0.62 & 1.04 & 0.60 & 0.39\
\[GA\]$^{+}$ & 1.61 & 1.12 & 0.67 & 2.78 & 0.92 & 0.52 & 0.77 & 0.50 & 0.32\
\[DEA\]$^{+}$ & 1.87 & 1.41 & 0.75 & 3.21 & 1.33 & 0.67 & 0.84 & 0.51 & 0.37\
\[Table S7\]
[?c?c?c?c?c?c?c?c?c?c?p[102cm]{}?]{} Materials & $\Delta$H & E$_{g}^{d}$ & m$_{e}^{*}$ & m$_{h}^{*}$ & E$_B$ (eV)& $\alpha_{ex}$ (nm)\
& &\
CsPb\[BF$_4$\]$_3$ & 0.31 & 8.71 & 1.60 & 1.43 & 2.70 & 0.14\
CsSn\[BF$_4$\]$_3$ & 0.32 & 8.70 & 1.63 & 1.45 & 2.58 & 0.14\
MAPb\[BF$_4$\]$_3$ & 0.71 & 9.43 & 6.97 & 7.70 & 12.07 & 0.03\
MASn\[BF4\]3 & 0.53 & 9.15 & 4.86 & 2.93 & 5.89 & 0.06\
& &\
CsPb\[SCN\]$_3$ & 0.39 & 4.42 & 1.82 & 1.12 & 0.79 & 0.26\
CsSn\[SCN\]$_3$ & 0.34 & 4.97 & 1.86 & 1.12 & 0.71 & 0.28\
MAPb\[SCN\]$_3$ & -0.33 & 4.46 & 2.38 & 1.20 & 0.93 & 0.23\
MASn\[SCN\]$_3$ & -0.17 & 4.75 & 2.59 & 1.01 & 0.72 & 0.27\
FAPb\[SCN\]$_3$ & -0.47 & 4.61 & 1.74 & 1.24 & 0.81 & 0.26\
FASn\[SCN\]$_3$ & -0.38 & 4.68 & 2.31 & 1.02 & 0.69 & 0.28\
|
---
abstract: 'We derive the Gauss-Codazzi equation in the holonomy and plane-angle representations and we use the result to write a Gauss-Codazzi equation for a discrete (2+1)-dimensional manifold, triangulated by isosceles tetrahedra. This allows us to write operators acting on spin network states in (2+1)-dimensional loop quantum gravity, representing the 3-dimensional intrinsic, 2-dimensional intrinsic, and 2-dimensional extrinsic curvatures.'
author:
- 'Seramika Ariwahjoedi$^{1,3}$, Jusak Sali Kosasih$^{3}$, Carlo Rovelli$^{1,2}$, Freddy P. Zen$^{3}$'
bibliography:
- 'library.bib'
title: 'Curvatures and discrete Gauss-Codazzi equation in (2+1)-dimensional loop quantum gravity'
---
Introduction
============
Three distinct notions of curvature are used in general relativity: the intrinsic curvature of the spacetime manifold, $M$, the intrinsic curvature of the spacial hypersurface $\Sigma$ embedded in $M$ which is utilised in the canonical framework, and the extrinsic curvature of $\Sigma$ [@key-1]. The three are related by the Gauss-Codazzi equation. On a discrete geometry, the definition of extrinsic curvature is not entirely clear [@key-2]. Even less so in loop quantum gravity (LQG), where the phase-space variables are derived from the first order formalism [@key-3], which generically does not yield a direct interpretation as spacetime geometry. But a definition of these curvatures is important in LQG, because we expect a discrete (twisted) geometry to emerge from the theory in an appropriate semi-classical limit [@key-4].
In $n$-dimensional discrete geometry, the manifold is formed by $n$-simplices [@key-2; @key-5]. The intrinsics curvature sits on the $(n-2)$-simplices, called *hinges*, and can be defined by the *angle of rotation*: a vector parallel-transported around the hinge gets rotated by this angle. The rotation is in the hyperplane dual to the hinge.
In the canonical formulation of general relativity it is convenient to use the ADM formalism [@key-6] or a generalisation [@key-8]. The phase space variables are defined on an $n-1$ surface: the “initial time", or, more generally, “boundary" surface $\Sigma$. At the core of the ADM formalism is the Gauss-Codazzi equation, relating the intrinsic curvature of the 4D spacetime with the intrinsic and extrinsic curvatures of $\Sigma$. In the context of a discrete geometry, $\Sigma$ is defined as an $(n-1)$-dimensional simplicial manifold formed by $(n-1)$-simplices. To develop a formalism analogous to the ADM one, we need an equation relating the intrinsic and extrinsic curvatures of $\Sigma$ and $M$, a ’discrete’ version of the Gauss-Codazzi equation.
In this paper, we explore a definition of extrinsic curvature for the discrete geometry which allows us to write an equation relating the extrinsic and intrinsics curvatures, which we refer as the Gauss-Codazzi equation for discrete geometry.
To get some insight into this problem, we first review the standard Gauss-Codazzi equation, both in second and first order formalism. In Section \[II\] we derive the holonomy (matrix) representation and plane-angle representation of the Gauss-Codazzi equation. We consider the discrete version of the Gauss-Codazzi equation in Section \[III\], where we define all the curvatures. In Section \[IV\], we move to the loop quantum gravity picture and study the operators corresponding to these curvatures into operators.
Gauss-Codazzi equation {#II}
======================
Standard Gauss-Codazzi equation
-------------------------------
The Gauss-Codazzi equation relates the Riemann *intrinsic* curvatures of a manifold and its submanifold with the *extrinsic* curvature of the submanifold. We will briefly review the continuous (2+1)-dimensional Gauss-Codazzi equation in this section, but the formula will be valid in $(n+1)$-dimension.
### Second order formulation
Consider a 3-dimensional manifold $M$ and a 2-dimensional surface $\Sigma$ embedded in $M$. In the second order formulation of general relativity [@key-7], the Riemann intrinsic curvature of $M$ is a $\left(\begin{array}{c}
1\\
3
\end{array}\right)$ tensor: $$\,^{3}R=\,^{3}R_{\mu\nu\beta}^{\alpha}\partial_{\alpha}\wedge dx^{\beta}\otimes dx^{\mu}\wedge dx^{\nu},\qquad\alpha,\beta,\mu,\nu=0,1,2$$ which can be thought as a map that rotates a vector $v\in T_{p}M$ parallel-transported along an infinitesimal square loop defined by unit vectors $\left\{ \partial_{\mu},\partial_{\nu}\right\} $. Next, we define the *projection tensor* as: $$q_{\mu\nu}=g_{\mu\nu}-\left\langle n,n\right\rangle n_{\mu}n_{\nu},$$ with $n_{\mu}$ is the normal to hypersurface $\Sigma\subset M$. $\left\langle n,n\right\rangle =\pm1$, depends on the signature of the metric. The projected Riemann curvature of $\,^{3}R$ on $\Sigma$ is defined as: $$\left.\,^{3}R\right|_{\Sigma}=\,^{3}\widetilde{R}_{\mu\nu\beta}^{\alpha}\partial_{\alpha}\wedge dx^{\beta}\otimes dx^{\mu}\wedge dx^{\nu}=q_{\alpha'}^{\alpha}q_{\mu}^{\mu'}q_{\nu}^{\nu'}q_{\beta}^{\beta'}\,^{3}R_{\mu'\nu'\beta'}^{\alpha'}\partial_{\alpha}\wedge dx^{\beta}\otimes dx^{\mu}\wedge dx^{\nu}.$$ Taking the projected part $\left.\,^{3}R\right|_{\Sigma}$ from the full part $\,^{3}R$, we have relation as follow: $$\,^{3}R=\left.\,^{3}R\right|_{\Sigma}+S,\qquad\left(\textrm{decomposition formula}\right),\label{eq:2.1}$$ $S$ is the ’residual’ part of $\,^{3}R.$ The projected part $\left.\,^{3}R\right|_{\Sigma}$ can be written as: $$\left.\,^{3}R\right|_{\Sigma}=\,^{2}R+\left[K,K\right]\qquad\left(\textrm{Gauss equation}\right),\label{eq:2.2}$$ which can be written in terms of components: $$\,^{3}\widetilde{R}_{\mu\nu\beta}^{\alpha}=\,^{2}R_{\mu\nu\beta}^{\alpha}+\left(K_{\mu}^{\alpha}K_{\beta\nu}-K_{\nu}^{\alpha}K_{\beta\mu}\right),\quad K_{\alpha\beta}=q_{\alpha}^{\mu}q_{\beta}^{\nu}\nabla_{\mu}n_{\nu}.\label{eq:2.3}$$ $\,^{2}R$ and $K$ are, respectively, the 2-dimensional Riemannian curvature and extrinsic curvature of hypersurface $\Sigma$. The residual part $S$ can be written as: $$S_{\mu\nu\beta}^{\alpha}=n^{\alpha}\left(\,^{2}\nabla_{\nu}K_{\mu\beta}-\,^{2}\nabla_{\mu}K_{\nu\beta}\right)\qquad\left(\textrm{Codazzi equation}\right),$$ with $\,^{2}\nabla$ is the covariant derivative on the slice $\Sigma.$
Writing (\[eq:2.1\]) and (\[eq:2.2\]) together, we obtain: $$\,^{3}R=\underset{\left.\,^{3}R\right|_{\Sigma}}{\underbrace{\,^{2}R+\left[K,K\right]}+S}.$$
### First order formulation
The gravitational field is a gauge field and can be written in a form closer to Yang-Mills theory. This way of representing gravity is known as first order formulation of general relativity [@key-3; @key-7]. Let manifold $M$ be a spacetime. Let $e$ be a local *trivialization*, a diffeomorphism map between the trivial vector bundle $M\times\mathbb{R}^{3}$ with the tangent bundle over $M$: $TM=\cup_{p}\left\{ p\times T_{p}M\right\}$, and $A$ be the connection on $M\times\mathbb{R}^{3}$. The 3-dimensional intrinsic curvature of the connection is the *curvature 2-form*: $$\,^{3}F=F_{\mu\nu J}^{I}\xi_{I}\wedge\xi^{J}\otimes dx^{\mu}\wedge dx^{\nu},\quad\mu,\nu,I,J=0,1,2.$$ which comes from the exterior covariant derivative of the connection: $$F=d_{D}A.\label{eq:2.4}$$ $\left\{ \partial_{\mu},dx^{\mu}\right\}$ and $\left\{ \xi_{I},\xi^{I}\right\}$ are local coordinate basis on $M$ and $\mathbb{R}^{3},$ respectively. In terms of components: $$F_{\mu\nu J}^{I}=\partial_{\mu}A_{\nu J}^{I}-\partial_{\nu}A_{\mu J}^{I}+A_{\mu K}^{I}A_{\nu J}^{K}-A_{\nu K}^{I}A_{\mu J.}^{K}\label{eq:2.5}$$ We use nice coordinates such that the time coordinate $\xi_{0}$ in fibre $\mathbb{R}^{3}$ is mapped by $e$ to the time coordinate $\partial_{0}$ in the basespace $M$, i.e., the map $e$ is fixed into: $$e\left(\xi^{0}\right)=e^{0}=\delta_{\mu}^{0}dx^{\mu}=dx^{0}.$$ For the foliation, we use the *time gauge*, where the normal of the hypersurface $\Sigma$ is taken to be the time direction: $$n=\partial_{0},\qquad e^{-1}\left(n\right)=\xi_{0}.$$ Then the $(2+1)$ split is simply carried by spliting the indices as $I=0,a,$ and $\mu=0,i,$ with $0$ as the temporal part: $$\begin{aligned}
\,^{3}F & = & \left(-F_{a0i0}+F_{00ia}+F_{ai00}-F_{0i0a}\right)\xi^{a}\wedge\xi^{0}\otimes dx^{i}\wedge dx^{0}+\left(F_{aij0}-F_{0ija}\right)\xi^{a}\wedge\xi^{0}\otimes dx^{i}\wedge dx^{j}+\label{eq:2.5a}\\
& & +\left(F_{ai0b}-F_{a0ib}\right)\xi^{a}\wedge\xi^{b}\otimes dx^{i}\wedge dx^{0}+F_{aijb}\xi^{a}\wedge\xi^{b}\otimes dx^{i}\wedge dx^{j}\nonumber \end{aligned}$$ The projected part of the curvature 2-form on $\Sigma$ is: $$\left.\,^{3}F\right|_{\Sigma}=F_{aijb}\xi^{a}\wedge\xi^{b}\otimes dx^{i}\wedge dx^{j},$$ and the residual part is: $$\begin{aligned}
S & = & \left(-F_{a0i0}+F_{00ia}+F_{ai00}-F_{0i0a}\right)\xi^{a}\wedge\xi^{0}\otimes dx^{i}\wedge dx^{0}+\left(F_{aij0}-F_{0ija}\right)\xi^{a}\wedge\xi^{0}\otimes dx^{i}\wedge dx^{j}\\
& & +\left(F_{ai0b}-F_{a0ib}\right)\xi^{a}\wedge\xi^{b}\otimes dx^{i}\wedge dx^{0}.\end{aligned}$$ Therefore, the *decomposition formula* is clearly: $$\,^{3}F=\left.\,^{3}F\right|_{\Sigma}+S.\label{eq:2.6}$$ Next, we take only the projected part: $$\left.\,^{3}F\right|_{\Sigma}=F_{aijb}\xi^{a}\wedge\xi^{b}\otimes dx^{i}\wedge dx^{j},$$ which can be written in terms of components as follow: $$F_{aijb}=\,^{2}F_{aijb}+A_{i0}^{a}A_{jb}^{0}-A_{j0}^{a}A_{ib}^{0}\qquad\textrm{(Gauss equation).}\label{eq:2.7}$$ The closed part of $\left.\,^{3}F\right|_{\Sigma}$ is clearly the 3D intrinsic curvature of connection in $\Sigma$ and the rest is the extrinsic curvature part: $$\,^{2}F=\,^{2}F_{aijb}\xi^{a}\wedge\xi^{b}\otimes dx^{i}\wedge dx^{j}=\left(\partial_{i}A_{ajb}-\partial_{j}A_{aib}+A_{aic}A_{jb}^{c}-A_{ajc}A_{ib.}^{c}\right)\xi^{a}\wedge\xi^{b}\otimes dx^{i}\wedge dx^{j},$$ $$K=K_{i}^{a}\xi_{\alpha}\otimes dx^{i}=A_{i0}^{a}\xi_{\alpha}\wedge\xi^{0}\otimes dx^{i}.$$ The residual part $S,$ in this special coordinates and special gauge fixing satisfies: $$F_{0ija}=\,^{2}D_{j}K_{ia}-\,^{2}D_{i}K_{ja}\qquad\textrm{(Codazzi equation),}$$ and the other temporal components are zero. It must be kept in mind that this is the Gauss-Codazzi equation for a special local coordinate and special choice of gauge fixing, for a general case, they are not this simple.
To conclude, we have the decomposition and Gauss-Codazzi equation for a fibre bundle of gravity: $$\,^{3}F=\underset{\left.\,^{3}F\right|_{\Sigma}}{\underbrace{\,^{2}F+\left[\,^{2}K,\,^{2}K\right]}+S.}$$
Gauss-Codazzi equation in holonomy representation
-------------------------------------------------
### Holonomy around a loop
In this section, we will write the Gauss-Codazzi equation in terms of holonomy. The holonomy is defined by the parallel transport of any section of a bundle, say, $s_{0}$, so that it satisfies the equation as follow: $$\begin{aligned}
D_{v}\left(s_{0}\right) & =\partial s_{0}+A\left(s_{0}\right)= & 0.\label{eq:2.8}\end{aligned}$$ Solving (\[eq:2.8\]) using recursive method [@key-3; @key-7; @key-8], we obtain the solution: $$s\left(t\right)=U\left(\gamma\left(t\right),D\right)s_{0},$$ $U\left(\gamma\left(t\right),D\right)$ is *holonomy* of connection $D$ along path $\gamma\left(t\right)$: $$U\left(\gamma\left(t\right),D\right)=U_{\gamma}\equiv{P}\exp\int A,\label{eq:2.30}$$ with ${P}$ is the *path-ordered operator* (See [@key-3; @key-8] for the details of the derivation).
Consider a square loop $\gamma$ embedded in $\Sigma$ which encloses a 2-dimensional area. The holonomy around the square loop can be written as a product of four holonomies, since holonomy is piecewise-linear: $$U_{\gamma}=U_{\gamma_{4}}U_{\gamma_{3}}U_{\gamma_{2}}U_{\gamma_{1},}$$ Taylor expanding the holonomy in (\[eq:2.30\]) up to the second order [@key-3; @key-7; @key-9], we obtain: $$U_{\gamma}=1+\frac{a^{\mu\nu}}{2}F_{\mu\nu}+\mathcal{O}^{4},$$ with $a^{\mu\nu}$ is an infinitesimal square area inside loop $\gamma.$ The formula: $$\frac{a^{\mu\nu}}{2}F_{\mu\nu}=U_{\gamma}-1-\mathcal{O}^{4}\label{eq:2.31}$$ will be used to write the Gauss-Codazzi equation in terms of holonomies.
### First order formulation in holonomy representation
Contracting (\[eq:2.5a\]) by an infinitesimal area $a^{\mu\nu}$, we obtain: $$\begin{aligned}
\frac{a^{\mu\nu}}{2}\,^{3}F_{I\mu\nu J} & =\,^{3}U_{IJ}-\delta_{IJ}-\mathcal{O}^{4}= & \frac{a^{i0}}{2}\left(-F_{a0i0}+F_{00ia}+F_{ai00}-F_{0i0a}\right)\xi^{a}\wedge\xi^{0}+\frac{a^{ij}}{2}\left(F_{aij0}-F_{0ija}\right)\xi^{a}\wedge\xi^{0}+\\
& & +\frac{a^{i0}}{2}\left(F_{ai0b}-F_{a0ib}\right)\xi^{a}\wedge\xi^{b}+\frac{a^{ij}}{2}F_{aijb}\xi^{a}\wedge\xi^{b},\end{aligned}$$ where we have used the result in (\[eq:2.31\]), relating the holonomy with the curvature of the connection. Taking the projected part (\[eq:2.7\]) contracted with $a^{ij},$ and using the relation between holonomy with the curvature 2-form, we can write the projected part in terms of 2-dimensional holonomy and the contracted extrinsic curvature as follow: $$\begin{aligned}
\frac{a^{ij}}{2}F_{aijb} & = & \frac{a^{ij}}{2}\,^{2}F_{aijb}+\frac{a^{ij}}{2}\left(K_{ai}K_{jb}-K_{aj}K_{ib}\right)=\,^{2}U_{ab}-\delta_{ab}-\mathcal{O}^{4}+\frac{a^{ij}}{2}\left(K_{ai}K_{jb}-K_{aj}K_{ib}\right).\end{aligned}$$ Finally, collecting all these result together, we obtain: $$\begin{aligned}
\underset{\left(\,^{3}U_{IJ}-\delta_{IJ}-\mathcal{O}^{4}\right)\xi^{I}\wedge\xi^{J}}{\underbrace{\frac{a^{\mu\nu}}{2}\,^{3}F_{I\mu\nu J}\xi^{I}\wedge\xi^{J}}} & = & \underset{\left(\,^{2}U_{ab}-\delta_{ab}-\mathcal{O}^{4}\right)\xi^{a}\wedge\xi^{b}}{\underbrace{\frac{a^{ij}}{2}\,^{2}F_{aijb}\xi^{a}\wedge\xi^{b}}}+\underset{\left[K,K\right]_{ab}\xi^{a}\wedge\xi^{b}}{\underbrace{\frac{a^{ij}}{2}\left(K_{ai}K_{jb}-K_{aj}K_{ib}\right)\xi^{a}\wedge\xi^{b}}}\\
& & \left.\begin{array}{c}
+\frac{a^{i0}}{2}\left(\left(-F_{a0i0}+F_{00ia}+F_{ai00}-F_{0i0a}\right)\xi^{a}\wedge\xi^{0}+\left(F_{ai0b}-F_{a0ib}\right)\xi^{a}\wedge\xi^{b}\right)\\
+\frac{a^{ij}}{2}\left(F_{aij0}-F_{0ija}\right)\xi^{a}\wedge\xi^{0}
\end{array}\right\} S,\end{aligned}$$ or simply: $$\,^{3}U=\,^{2}U+\underset{T}{\underbrace{\left[K,K\right]+S+\left(\,^{3}I-\,^{2}I\right)}},\label{eq:2.9}$$ where we have written the Gauss-Codazzi and the decomposition formula together, noting the residual terms as $S$. Remember that $U$ is the holonomy of the connection, which describes the curvature of the connection of fibre $F$, *not* the curvature of the basespace $M$.
### Second order formulation in holonomy representation
In the same manner as above, for a tangent bundle $TM$, we obtain the relation $$\begin{aligned}
\underset{\left(\,^{3}H_{\alpha\beta}-\delta_{\alpha\beta}-\mathcal{O}^{4}\right)dx^{\alpha}\wedge dx^{\beta}}{\underbrace{\frac{a^{\mu\nu}}{2}\,^{3}R_{\alpha\mu\nu\beta}dx^{\alpha}\wedge dx^{\beta}}} & = & +\underset{\left(\,^{2}H_{kl}-\delta_{kl}-\mathcal{O}^{4}\right)dx^{k}\wedge dx^{l}}{\underbrace{\frac{a^{ij}}{2}\,^{2}R_{kijl}dx^{k}\wedge dx^{l}}}+\underset{\left[K,K\right]_{kl}dx^{k}\wedge dx^{l}}{\underbrace{\frac{a^{ij}}{2}\left(K_{ki}K_{jl}-K_{kj}K_{il}\right)dx^{k}\wedge dx^{l}}}\\
& & \left.\begin{array}{c}
+\left(\frac{a^{i0}}{2}\left(-R_{j0i0}+R_{00ij}+R_{ji00}-R_{0i0j}\right)+\frac{a^{ij}}{2}\left(R_{jik0}-R_{0ikj}\right)\right)dx^{j}\wedge dx^{0}\\
+\frac{a^{i0}}{2}\left(R_{ji0k}-R_{j0ik}\right)dx^{j}\wedge dx^{k}
\end{array}\right\} S,\end{aligned}$$ or simply: $$\,^{3}H=\,^{2}H+\underset{T}{\underbrace{\left[K,K\right]+S+\left(\,^{3}I-\,^{2}I\right)}},\label{eq:2.10}$$ with $H$ is the holonomy around any loop $\gamma$ embedded in $\Sigma\subseteq M$, coming from the Riemann tensor $R$.
Gauss-Codazzi equation in plane-angle representation
----------------------------------------------------
Rotations can be represented in two ways, using the *holonomy (matrix) representation*, or using the *plane-angle representation*. For a continuous theory, the holonomy representation provides a simpler way to do calculation concerning rotation. In a discrete theory, where we would like to get rid of coordinates, the plane-angle representation is a natural way to represent rotations and curvatures. The plane-angle representation is only a representation of the rotation group using its Lie algebra for the plane of rotation and one (real) parameter group times the norm of the algebra for the angle of rotation. There exist a bijective map sending the holonomy to the plane-angle representation, known as the *exponential map* [@key-10].
In this section, we rewrite the Gauss-Codazzi equation using the plane-angle representation. We only do the calculation for the second order formalism, the first order formalism version can be obtained in a similar way. Firstly, let us define the variables; we have two equations: the decomposition formula and the Gauss-Codazzi equation written compactly in (\[eq:2.10\]). Since we are working in (2+1)-dimension, it is natural to use matrix group $SO(3)$ to represents the 3-dimensional intrinsic curvature, and the subgroup $SO(2)$ to represents the 2-dimensional intrinsic curvature. But for simplicity of the calculation, we use the unitary group $SU(2)$ –the double cover of $SO(3)$– instead of $SO(3)$. Having information of $\,^{3}H\in SU(2)$ is equivalent with having information of the plane *and* angle of the 3-dimensional rotation, which are denoted, respectively, by $\left\{ J\in\mathfrak{su(2)},\delta\theta\in\mathbb{R}\right\} $. The same way goes with $\,^{2}H\in SO(2)$, it contains same amount of information with the plane-angle of the 2-dimensional rotation $\left\{ j\in\mathfrak{so(2)},\delta\phi\in\mathbb{R}\right\}$. Using the exponential map, we can write: $$^{2}H=\exp\left(j\delta\phi\right)=I_{2\times2}\cos\delta\phi+j\sin\delta\phi,\label{eq:2-10a}$$ with $$j=\left[\begin{array}{cc}
0 & 1\\
-1 & 0
\end{array}\right],\label{eq:2.11a}$$ is the generator of $SO(2).$ Doing the same way to the element of $SU(2)$, we obtain: $$^{3}H\sim\exp\left(J\delta\theta\right)=I_{2\times2}\cos\frac{\delta\theta}{2}+J\sin\frac{\delta\theta}{2},\label{eq:2-11}$$ with $J\in\mathfrak{su(2)}$ is an element of the Lie algebra of SU(2), satisfying: $$J=J^{a}\bar{\sigma}_{a},\quad J^{a}J_{a}=1.$$ $\bar{\sigma}_{a}$ are the basis of $\mathfrak{su(2),}$ namely, the generator of $SU(2)$: $$\bar{\sigma}_{a}=i\sigma_{a},$$ satisfying the algebra structure relation as follow: $$\left[\bar{\sigma}_{a},\bar{\sigma}_{b}\right]=-2\varepsilon_{ab}^{\:\, c}\bar{\sigma}_{c},\label{eq:3}$$ with $\sigma_{a}$ are the Pauli matrices: $$\sigma_{x}=\left[\begin{array}{cc}
0 & 1\\
1 & 0
\end{array}\right],\quad\sigma_{y}=\left[\begin{array}{cc}
0 & -i\\
i & 0
\end{array}\right],\quad\sigma_{z}=\left[\begin{array}{cc}
1 & 0\\
0 & -1
\end{array}\right]$$ The $\frac{1}{2}$ factor in (\[eq:2-11\]) comes out from the normalization occuring when we write $SO(3)$ using $SU(2)$ representation, i.e., to reduce the factor 2 in relation (\[eq:3\])
Taking the trace of (\[eq:2.10\]) gives the relation between 3-dimensional and 2-dimensional rotation angle: $$\underset{2\cos\frac{\delta\theta}{2}}{\underbrace{\textrm{tr}\,^{3}H}}=\underset{2\cos\delta\phi}{\underbrace{\textrm{tr}\,^{2}H}}+\textrm{tr}T,\label{eq:2.12}$$ using the fact that the elements of the algebra are skew-symmetric. The other relation we need to have the full information contained by (\[eq:2.10\]) is the planes of rotation. Since the embedded surface is 2-dimensional, the plane of rotation for $\delta\phi$ is trivial, which is (\[eq:2.11a\]). The plane of rotation $J$ for $\delta\theta$ can be obtained by solving complex $2\times2$ matrix linear equation (\[eq:2-11\]): $$J=\frac{^{3}H-I_{2\times2}\cos\frac{\delta\theta}{2}}{\sin\frac{\delta\theta}{2}}.\label{eq:2.13}$$ In conclusion, the contracted Gauss-Codazzi equation concerning the relation of curvatures of a manifold and its submanifold in (2+1) dimension can be written using holonomy representation (\[eq:2.10\]) or using the plane-angle representation, i.e., the relation between rotation angles by (\[eq:2.12\]) and the condition for the planes of rotation, by (\[eq:2.11a\]) and (\[eq:2.13\]).
Discrete (2+1) geometry {#III}
=======================
Geometrical setting
-------------------
Using the Gauss-Codazzi equation in terms of holonomy and plane-angle representation obtained in the previous section, we can write the Gauss-Codazzi equation for discrete (2+1) geometry; this is the objective of this section. Let a portion of curved 3-dimensional manifold $M$ be discretized by flat tetrahedra, we call this discretized manifold $M_{\triangle}$. See FIG. 1.
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### Angle relation
In this section we derive the Gauss-Codazzi equation purely from the angles between simplices, i.e, the relation between geometrical objects, without reference to coordinates (this is, indeed, the reason Tullio Regge developed his *Regge calculus* and discrete general relativity, in his famous paper titled ’General relativity without coordinates’ [@key-5]). There are two kinds of relevant angles in a flat tetrahedron: the *’angle at a vertex’* between two segments of the tetrahedron meeting at a vertex, which we denote $\phi$; and the *’dihedral angle at a segment’*, namely the angle between two triangles, which we denote $\theta$. These two types of angles are related by the ’angle formula of the tetrahedron’: $$\cos\theta_{abac}=\frac{\cos\phi_{bc}-\cos\phi_{ab}\cos\phi_{ac}}{\sin\phi_{ab}\sin\phi_{ac}}.\label{eq:3-1}$$ See FIG. 2.
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### Isosceles tetrahedron
For simplicity of the derivation in the next section, we consider *isosceles* tetrahedron. An isosceles tetrahedron is build by three isosceles triangles and one equilateral triangle. In this case, all angles around one point $p$ of the tetrahedron are equal, say $\phi_{i}=\phi$, and the dihedral angle relation (\[eq:3-1\]) becomes: $$\begin{aligned}
\cos\theta & = & \frac{\cos\phi}{\left(1+\cos\phi\right)}.\end{aligned}$$ This implies that the dihedral angles between two isosceles triangles are equal as well, say $\theta_{i}=\theta$. Another geometrical property of a tetrahedron which is useful for the derivation in this section is the volume. The volume of unit isosceles parallelepiped is: $$\textrm{vol}=\left(\vec{n}_{1}\times\vec{n}_{2}\right)\cdot\vec{n}_{3}=\frac{1}{3}\left(1-\cos\phi\right)\sqrt{1+2\cos\phi}=\frac{1}{3}\left(\sin^{2}\frac{\phi}{2}\right)\sqrt{4\cos^{2}\frac{\phi}{2}-1}.\label{eq:vol}$$
Curvatures in discrete geometry
-------------------------------
In $n$-dimensional discrete geometry, the notion of curvature is represented using the plane-angle representation. The intrinsic curvature is represented by the angle of rotation, which in this case is called as *deficit angle*, and this deficit angle is located on the *hinge*, the $(n-2)$ form duals to the plane of rotation. The geometrical interpretation of deficit angle is illustrated by the FIG. 3.
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To obtain the 2-dimensional and 3-dimensional intrinsic curvatures in discrete geometry picture, firstly we need to define a discretized surface embedded in $M_{\Delta},$ say $\Sigma_{\Delta}.$ The simplest way is to take the surface of the tetrahedron as a slice (see FIG. 1). Having $\Sigma_{\Delta}$ embedded on $M_{\Delta}$, we consider a loop $\gamma$ on $\Sigma_{\Delta}.$ We consider the holonomy related to the 2D and 3D curvatures defined by this loop. The 3D curvature is defined by the 3D holonomy $^{3}H\in SO(3)\sim SU(2)$ around loop $\gamma$, which describes the intrinsic curvature of $M_{\Delta},$ while the 2D curvature is the 2D holonomy $^{2}H\in SO(2),$ describing the intrinsic curvature of slice $\Sigma_{\Delta}$, still along the same loop. Taking trace of the holonomies, we obtain the angles of rotation, i.e., *the amount of a components of a vector parallel to the plane of rotation get rotated when parallel transported along loop $\gamma$*. In discrete geometry, this rotation angle is the same as the deficit angle on the hinges.
### 2D intrinsic curvature $(SO(2))$
Using FIG. 4 as the simplest case, we take loop $\gamma$ circling point $p$ on the surface $\Sigma$, therefore the 2D curvature is the deficit angle on $p$: $$^{2}R=\delta\phi=2\pi-\sum_{i}\phi_{i},\label{eq:(aa)}$$ or in terms of the cosine function: $$\cos\delta\phi=\prod_{i=1}^{3}\cos\phi_{i}-\sum_{i=1,i\neq j,k,j<k}^{3}\cos\phi_{i}\sin\phi_{j}\sin\phi_{k}.\label{eq:bb}$$ For the isosceles tetrahedron case, (\[eq:(aa)\])-(\[eq:bb\]) become: $$^{2}R=\delta\phi=2\pi-3\phi,$$ $$\cos\delta\phi=\cos^{3}\phi-3\cos\phi\sin^{2}\phi=\cos\phi\left(4\cos^{2}\phi-3\right).$$ We could obtain the holonomy by the exponential map (\[eq:2-10a\]), using the $SO(2)$ algebra (\[eq:2.11a\]).
### 3D intrinsic curvature $(SU(2))$
We could write 3D curvature by the holonomy of SU(2) using (\[eq:2-11\]), but the 3-dimensional holonomy along loop $\gamma$ circling point $p$ is a *product* of three holonomies on the three hinges, because the loop crosses three hinges. See FIG. 4.
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$$^{3}H=\prod_{i=1}^{3}{}^{3}H_{i}\label{eq:hol}$$
Using (\[eq:2-11\]), we obtain: $$\begin{aligned}
^{3}H & = & \prod_{i=1}^{3}\left(I\cos\frac{\delta\theta_{i}}{2}+J_{i}\sin\frac{\delta\theta_{i}}{2}\right)\\
& = & I\prod_{i=1}^{3}\cos\frac{\delta\theta_{i}}{2}+\sum_{i=1,i\neq j,k,j<k}^{3}J_{i}\sin\frac{\delta\theta_{i}}{2}\cos\frac{\delta\theta_{j}}{2}\cos\frac{\delta\theta_{k}}{2}+\sum_{i=1,i<j,i,j\neq k}^{3}J_{i}\sin\frac{\delta\theta_{i}}{2}J_{j}\sin\frac{\delta\theta_{j}}{2}\cos\frac{\delta\theta_{k}}{2}+\prod_{i=1}^{3}J_{i}\sin\frac{\delta\theta_{i}}{2},\end{aligned}$$ where each hinge (segment) $i$ is dual to plane $J_{i}=\vec{J}_{i}\cdot\hat{\bar{\sigma}}$ and is attached by a deficit angle $\delta\theta_{i}$. Taking trace of the equation above gives: $$\begin{aligned}
\underset{2\cos\frac{\delta\Theta}{2}}{\underbrace{\textrm{tr}{}^{3}H}} & =\underset{2}{\underbrace{\textrm{tr}I}}\prod_{i=1}^{3}\cos\frac{\delta\theta_{i}}{2}+\sum_{i=1,i\neq j,k,j<k}^{3}\underset{0}{\underbrace{\textrm{tr}J_{i}}}\sin\frac{\delta\theta_{i}}{2}\cos\frac{\delta\theta_{j}}{2}\cos\frac{\delta\theta_{k}}{2}+\sum_{i=1,i<j,i,j\neq k}^{3}\underset{-2\left(\vec{J}_{i}\cdot\vec{J}_{j}\right)}{\underbrace{\textrm{tr}\left(J_{i}J_{j}\right)}}\sin\frac{\delta\theta_{i}}{2}\sin\frac{\delta\theta_{j}}{2}\cos\frac{\delta\theta_{k}}{2}\\
& +\underset{+2\left(\vec{J}_{1}\times\vec{J}_{2}\right)\cdot\vec{J}_{3}}{\underbrace{\textrm{tr}\left(\prod_{i=1}^{3}J_{i}\right)}}\prod_{i=1}^{3}\sin\frac{\delta\theta_{i}}{2},\end{aligned}$$ or simply: $$\cos\frac{\delta\Theta}{2}=\prod_{i=1}^{3}\cos\frac{\delta\theta_{i}}{2}-\sum_{i=1,i<j,i,j\neq k}^{3}\left(\vec{J}_{i}\cdot\vec{J}_{j}\right)\sin\frac{\delta\theta_{i}}{2}\sin\frac{\delta\theta_{j}}{2}\cos\frac{\delta\theta_{k}}{2}+\left(\vec{J}_{1}\times\vec{J}_{2}\right)\cdot\vec{J}_{3}\sin\frac{\delta\theta_{i}}{2},$$ But $\left(\vec{J}_{i}\cdot\vec{J}_{j}\right)=\cos\phi_{i}$, and $\left(\vec{J}_{1}\times\vec{J}_{2}\right)\cdot\vec{J}_{3}$ is just the volume of the parallelepiped defined by $\left\{ \vec{J}_{1},\vec{J}_{2},\vec{J}_{3}\right\} $. (Remember that $\vec{J}=\star J$, or 1-form $\vec{J}$ is the Hodge dual of the 2-form $J$ in 3-dimensional vector space. The direction of $\vec{J}$ is normal to the plane discribed by $J$). Therefore, the 3D intrinsic curvature is:[^1] $$\cos\frac{\delta\Theta}{2}=\prod_{i=1}^{3}\cos\frac{\delta\theta_{i}}{2}-\sum_{i=1,i<j,i,j\neq k}^{3}\cos\phi_{i}\sin\frac{\delta\theta_{i}}{2}\sin\frac{\delta\theta_{j}}{2}\cos\frac{\delta\theta_{k}}{2}+\left(\textrm{vol}\right)\prod_{i=1}^{3}\sin\frac{\delta\theta_{i}}{2},\label{eq:cc}$$ or $$^{3}R=\delta\Theta=2\arccos\left(\prod_{i=1}^{3}\cos\frac{\delta\theta_{i}}{2}-\sum_{i=1,i<j,i,j\neq k}^{3}\cos\phi_{i}\sin\frac{\delta\theta_{i}}{2}\sin\frac{\delta\theta_{j}}{2}\cos\frac{\delta\theta_{k}}{2}+\left(\textrm{vol}\right)\prod_{i=1}^{3}\sin\frac{\delta\theta_{i}}{2}\right),$$ with $\textrm{vol}$ is the volume form described in (\[eq:vol\]). For isosceles tetrahedron case, we have $\phi_{i}=\phi$ and $\delta\theta_{i}=\delta\theta,$ so the holonomy is: $$\begin{aligned}
^{3}H & = & I\cos^{3}\frac{\delta\theta}{2}+\left(\cos^{2}\frac{\delta\theta}{2}\,\sin\frac{\delta\theta}{2}\right)\left(\sum_{i=1}^{3}\vec{J}_{i}\right)\cdot\hat{\bar{\sigma}}+\left(\cos\frac{\delta\theta}{2}\,\sin^{2}\frac{\delta\theta}{2}\right)\sum_{i,j=1,i\neq j}^{3}\left(\vec{J}_{i}\cdot\hat{\bar{\sigma}}\right)\left(\vec{J}_{j}\cdot\hat{\bar{\sigma}}\right)+\sin^{3}\frac{\delta\theta}{2}\prod_{i=1}^{3}\left(\vec{J}_{i}\cdot\hat{\bar{\sigma}}\right),\end{aligned}$$ Therefore, the 3D intrinsic curvature for isosceles tetrahedra case is: $$^{3}R=\delta\Theta=2\arccos\left(\cos^{3}\frac{\delta\theta}{2}-3\cos\phi\,\cos\frac{\delta\theta}{2}\,\sin^{2}\frac{\delta\theta}{2}+\left(\textrm{vol}\right)\sin^{3}\frac{\delta\theta}{2}\right).$$
### 2D extrinsic curvature
Equation (\[eq:2.12\]) is a linear relation between the trace of 3D holonomy with the 2D holonomy and its ’residual’ part $T.$ Inserting (\[eq:bb\]) and (\[eq:cc\]) to (\[eq:2.12\]), we could obtain $\textrm{tr}T,$ but this is *not* the extrinsic curvature, since $T=S+\left[K,K\right]+\left(I_{3}-I_{2}\right).$ Next, we would like to obtain an angle describing the extrinsic curvature $K$. We do *not* expect the relation between this angle with $^{3}R$ and $^{2}R$ to be linear, since linearity is held on the holonomy representation of Gauss-Codazzi (\[eq:2.10\]), while the relation between their rotation angle $^{3}R$ and $^{2}R$ does *not* need to be linear.
Consider the isosceles tetrahedron case. On each segment of the tetrahedron, lies the 3D deficit angle $\delta\theta.$ This deficit angle is defined as: $$\delta\theta=2\pi-\left(\theta+\alpha\right),$$ where $\theta$ is the *internal* dihedral angle, and $\alpha$ is the *external* dihedral angle coming from the dihedral angles of *other* tetrahedra (remember that the full discretized $(2+1)$ picture is the 1-4 Pachner moves). Let’s introduce the quantity $$\theta^{'}=\alpha-\pi.\label{eq:key}$$ For the case where $M_{\Delta}$ is flat, $\delta\theta=0$. This causes $\theta+\alpha=2\pi,$ and using definition (\[eq:key\]), we obtain: $$\theta+\theta'=\pi.$$ In this flat case, it is clear that $\theta'$ is the angle between the normals of two triangles, see FIG. 5.
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Therefore, $\theta'$ is in accordance with the definition of extrinsic curvature in (\[eq:2.3\]), where $K$ is defined as the covariant derivative of the normal $n$ to the hypersurface $\Sigma$. Because of this reason, we define $\theta^{'}$ as the 2D *extrinsic curvature*, since in a general curved case, it will inherit the curvature of the 3D manifold.
Discrete Gauss-Codazzi equation in (2+1) dimension
--------------------------------------------------
A simpler but equivalent way to obtain the 3D instrinsic curvature is to use the purely geometrical picture of the discretized manifold. See FIG. 6.
![The 3D deficit angle on each segment can be thought as a ’piece of space’ inserted along the hinge. The ’size’ of this ’piece of space’ is described by the deficit angle $\delta\theta,$ which is located on the plane normal to the hinge. The total deficit angle $\delta\Theta$ obtained by adding three rotation in (\[eq:hol\]), is located in the plane normal to an ’artificial’ hinge in the direction $\hat{k}$, since we use isosceles tetrahedron. Therefore, projecting $\delta\theta$ to this plane, we obtain $\left.\delta\theta\right|_{\gamma}$.](d2){height="6cm"}
From Figure 6, we obtain the projected 3D intrinsic curvature of each segment on the ’artificial’ segment dual to the total plane of rotation as: $$\cos^{2}\frac{\left.\delta\theta\right|_{\gamma}}{2}=\frac{\sin^{2}\chi'\cos^{2}\frac{\delta\theta}{2}}{1-\cos^{2}\chi'\cos^{2}\frac{\delta\theta}{2}}.\label{eq:2.17}$$ Using $$\vec{J}_{i}=\sin\chi'\hat{k}+\cos\chi\hat{r}_{i},$$ we obtain: $$\left\langle \vec{J}_{i},\vec{J}_{j}\right\rangle =\cos\phi=\sin^{2}\chi'+\cos^{2}\chi'\cos^{2}\frac{2\pi}{3},$$ or: $$\begin{aligned}
\cos^{2}\chi' & = & \frac{2}{3}\left(1-\cos\phi\right),\label{eq:2.17a}\\
\sin^{2}\chi' & = & \frac{1}{3}\left(1+2\cos\phi\right),\label{eq:2.17b}\end{aligned}$$ since we use isosceles tetrahedron. All the curvatures: 3D intrinsic, 2D intrinsic, and 2D extrinsic curvatures are contained in (\[eq:2.17\]), through: $$\left.\delta\theta\right|_{\gamma}=\frac{\delta\Theta}{3}=\frac{^{3}R}{3},\label{eq:2.17c}$$ $$\phi=\frac{2\pi-\delta\phi}{3}=\frac{2\pi-{}^{2}R}{3}\label{eq:2.17d}$$ $$\delta\theta=\pi-\left(\theta+\theta'\right)=\pi-\left(\arccos\left(\frac{\cos\phi}{1+\cos\phi}\right)+K\right).\label{eq:2.17e}$$ Using the half-angle formula and inserting (\[eq:2.17a\])-(\[eq:2.17e\]) to (\[eq:2.17\]), we obtain the relation between angles of the discrete Gauss-Codazzi equation for a special discretization using isosceles tetrahedra: $$1-\cos\left(K+\arccos\left(\frac{\cos\frac{2\pi-{}^{2}R}{3}}{1+\cos\frac{2\pi-{}^{2}R}{3}}\right)\right)=\frac{1+\cos\frac{^{3}R}{3}}{1-\frac{1}{3}\left(1-\cos\frac{^{3}R}{3}\right)\left(1-\cos\frac{2\pi-{}^{2}R}{3}\right)}.\label{eq:GC}$$ Together with their planes of rotation: $j$ in (\[eq:2.11a\]) for $^{2}R$ and $\hat{k}\cdot\bar{\sigma}$ for $^{3}R$ (since we use isosceles tetrahedron), (\[eq:GC\]) is the Gauss-Codazzi equation in plane-angle representation, which contains same amount of information provided by the linear Gauss-Codazzi equation in holonomy representation in (\[eq:2.10\]).
Spin network states and 3D loop quantum gravity {#IV}
===============================================
2-complex and its slice
-----------------------
In this section, we apply the results obtained above to loop quantum gravity (LQG). We will briefly review the quantization of gravity in first order formalism using the loop representation. From the real space representation, we move to the (Hodge) dual space [@key-8]. (The dual space is introduced in analogy to what is usually done in higher dimension.) In LQG, the fundamental geometrical object is the *2-complex*. A 2-complex is a 2-dimensional geometrical object dual to the discrete manifold $M_{\triangle}.$ Let us call the 2-complex as $\star M_{\triangle}$. We can take a slice on a 2-complex, which is a *1-complex* called as $\star\Sigma_{\triangle}$, dual to the discrete hypersurface $\Sigma_{\triangle}$ of $M_{\triangle}.$ A 1-complex is a *graph* consisting oriented lines called *link*s $l$ and points called *nodes* $n$. The variables attached to the graphs are the variables coming from first order formalism: the holonomy $U_{l}\in SU(2)$ and algebra $J_{l}\in\mathfrak{su(2)}^{*}$ on each links $l$ of the graph. A graph with elements of $SU(2)\times\mathfrak{su(2)}^{*}$ is called a *spin network*. In the canonical quantization, we promote $\left\{ U_{l},J_{l}\right\} $ to operators $\left\{ \hat{U}_{l},\hat{J}_{l}\right\} $. The space $SU(2)\times\mathfrak{su(2)}^{*}$ is the phase-space of LQG, and the pair $\left\{ \hat{U}_{l},\hat{J}_{l}\right\} $ are the pair conjugate to each other, satisfying a quantized-algebra structure as follow [@key-3; @key-8; @key-11]: $$\begin{aligned}
\left[\hat{U}_{l},\hat{U}_{l'}\right] & = & 0\\
\left[\hat{U}_{l},\hat{J}_{l'}^{i}\right] & = & -\delta_{ll'}\bar{\sigma}^{i}\hat{U}_{l}\\
\left[\hat{J}_{l}^{i},\hat{J}_{l'}^{j}\right] & = & -\delta_{ll'}\varepsilon_{\; k}^{ij}\hat{J}_{l}^{k}.\end{aligned}$$ On each node, the gauge invariant condition must be satisfied: $$\hat{\mathcal{G}}\left|\psi\right\rangle =\sum_{l\in n}\hat{J}_{l}\left|\psi\right\rangle =0.$$ This condition selects an invariant Hilbert space $\mathcal{K}=L_{2}\left[SU(2)^{l}/SU(2)^{n}\right]\ni\left|\psi\right\rangle $ of a closed triangle formed by $\left\{ \hat{J}_{1},\hat{J}_{2},\hat{J}_{3}\right\} $ [@key-3; @key-8; @key-11]. As an example for our case, see FIG. 7.
![The figure above is a 2-complex dual to the geometric figure in FIG. 1. Suppose we have a 3-dimensional curved manifold $M_{\Delta}$ discretized by four tetrahedron as the first figure above (in flat case it is known as 1-4 Pachner move). Then we take an embedded slice $\Sigma_{\triangle}$ as the surface of one tetrahedron (the dark blue surface discretized by three triangle). Going to the dual space picture, $M_{\Delta}$ is dual to a 2-complex $\star M_{\Delta}$ called as the 3D *bubble* [@key-8] (its geometry is defined by black lines and black points), while $\Sigma_{\triangle}$ is dual to a 1-complex $\star\Sigma_{\triangle}$ called as 2D *bubble* (defined by purple lines and point). Clearly, the 2D bubble $\star\Sigma_{\triangle}$ is embedded on the 3D bubble $\star M_{\Delta}$. The bubble graph define complete curvature of the portion of space.](f){height="6cm"}
Curvature operators
-------------------
Given a 2D bubble graph with a phase space variable on each link, we can construct the operators of 3D, 2D intrinsics and 2D extrinsic curvature from these variables.
{height="2.5cm"}
We construct these operators using the basic phase-space operators, for example, the angle operator from the phase-space variables [@key-12]: $$\hat{\phi}_{i}=\arccos\left(\frac{\left|\hat{J}_{j}\right|^{2}+\left|\hat{J}_{k}\right|^{2}-\left|\hat{J}_{i}\right|^{2}}{2\left|\hat{J}_{j}\right|\left|\hat{J}_{k}\right|}\right),\quad i,j,k\in n.$$ From this angle operator, we obtain the 2D intrinsic curvature along loop $\gamma$ as: $$^{2}\hat{R}=2\pi-\sum_{i\subseteq\gamma}\hat{\phi}_{i},$$ which is clearly the deficit angle on a point in the real space representation. See FIG. 8.
The 3D intrinsic curvature can be obtained from the holonomy around a loop. Remember that the direct geometrical interpretation of spacetime can only be obtained from second order formalism, while in LQG, the fundamental variables *comes from first order formalism*: the holonomy $U$ comes from the curvature 2-form $F$, instead of the Riemannian curvature $R$ (see Section 1). The relation between $F$ and $R$, subjected to the torsionless condition is: $$R=e\left(F\right),\label{eq:send}$$ with $e$ is the local trivialization map between them. Contracting (\[eq:send\]) with an infinitesimal area inside the loop (carried by the derivation in Section 1) gives: $$H=e\left(U\right),\quad U=e^{-1}\left(H\right),$$ since $e$ is a diffeomorphism and must have inverse.
The 3D intrinsic curvature is $^{3}R\sim\textrm{tr}H$, with $H$ is the holonomy coming from the contraction of Riemanian curvature $R$ with an infinitesimal area. But since $e$ is a diffeomorphism and trace of the holonomy is *invariant* under diffeomorphism and gauge transformation, we obtain $\textrm{tr}H=\textrm{tr}U.$ Therefore, we can write the 3D intrinsic curvature operator as:
$$^{3}\hat{R}=2\arccos\frac{\textrm{tr}\hat{U}_{1}\hat{U}_{2}\hat{U}_{3}}{2}.\label{eq:r3}$$
The extrinsic curvature operator can be obtained directly from the discrete Gauss-Codazzi equation (\[eq:GC\]):
$$\hat{K}=\arccos\left(1-\frac{1+\cos\frac{^{3}\hat{R}}{3}}{1-\frac{1}{3}\left(1-\cos\frac{^{3}\hat{R}}{3}\right)\left(1-\cos\frac{2\pi-{}^{2}\hat{R}}{3}\right)}\right)-\arccos\left(\frac{\cos\frac{2\pi-{}^{2}\hat{R}}{3}}{1+\cos\frac{2\pi-{}^{2}\hat{R}}{3}}\right).$$
The semi-classical limit
------------------------
In this section, we show that the linearity of the relation between $\textrm{tr}\,^{3}H$ and $\textrm{tr}\,^{2}H$ in equation (\[eq:2.12\]) is recovered from the spin network calculation in the semi-classical limit, by dropping the Maslov phase. Let us take a simple spin network illustrated in FIG. 9.
{height="3.5cm"}
For simplicity in our calculation, we use a closed graph as an example, but the result is valid for any spin network graph. The vector and covector state of the spin network $\star\Sigma_{\Delta}$ in the algebra representation are: $$\left|\psi\right\rangle =\sum_{\binom{m_{1}\ldots m_{6}}{n_{1}..n_{6}}}i^{m_{6}m_{3}n_{2}}i^{m_{5}m_{2}n_{1}}i^{m_{4}m_{1}n_{3}}i^{n_{6}n_{5}n_{4}}\left|j_{1}m_{1}n_{1}\ldots j_{6}m_{6}n_{6}\right\rangle ,\label{eq:vect}$$ $$\left\langle \psi\right|=\sum_{\binom{m_{1'}\ldots m_{6'}}{n_{1'}..n_{6'}}}i^{m_{6}'m_{3}'n_{2}'}i^{m_{5}'m_{2}'n_{1}'}i^{m_{4}'m_{1}'n_{3}'}i^{n_{6}'n_{5}'n_{4}'}\left\langle j_{1}'m_{1}'n_{1}'\ldots j_{6}'m_{6}'n_{6}'\right|,\label{eq:covect}$$ with $i$ is the intertwinner attached on the node. See [@key-8; @key-11; @key-13] for a detail explanation about the spin network state. Changing the representation basis using: $$\left.\left\langle U\right|jmn\right\rangle =D_{mn}^{j}\left(U\right),$$ with $D_{mn}^{j}\left(U\right)$ is the matrix representation of $U\in SU(2)$ in $(2j+1)$-dimension, we can write the vector and covector states in the group representation basis: $$\left|\psi\right\rangle =\sum_{\binom{m_{1}\ldots m_{6}}{n_{1}..n_{6}}}i^{m_{6}m_{3}n_{2}}i^{m_{5}m_{2}n_{1}}i^{m_{4}m_{1}n_{3}}i^{n_{6}n_{5}n_{4}}\intop dU_{1}..dU_{6}\prod_{i=1}^{6}D_{m_{i}n_{i}}^{j_{i}}\left(U_{i}\right)\left|U_{1}..U_{6}\right\rangle ,$$ $$\left\langle \psi\right|=\sum_{\binom{m_{1'}\ldots m_{6'}}{n_{4'}..n_{6'}}}i^{m_{6}'m_{3}'n_{2}'}i^{m_{5}'m_{2}'n_{1}'}i^{m_{4}'m_{1}'n_{3}'}i^{n_{6}'n_{5}'n_{4}'}\intop dU_{1}'..dU_{6}'\prod_{i=1}^{6}D_{m_{i}'n_{i}'}^{j_{i}'}\left(U_{i}'\right)\left\langle U_{1}'..U_{6}'\right|.$$
We choose a specific closed path on spin network $\star\Sigma_{\Delta}$, say, loop $\gamma$, defined by links $\left\{ 1-2-3\right\} $ (See FIG. 9). The holonomy $U=U_{1}U_{2}U_{3}$ are attached along $\gamma$, therefore we have a well-defined operator related to the 3D intrinsic curvature operator (\[eq:r3\]) on the loop: $$\textrm{tr}\,^{3}H=\underset{2\cos\frac{^{3}\hat{R}}{2}}{\underbrace{\textrm{tr}U}}=\textrm{tr}\left(U_{1}U_{2}U_{3}\right)=\sum_{m,n}\delta_{mn}D_{mn}^{j}\left(U_{1}U_{2}U_{3}\right).\label{eq:x}$$ Since the matrix representation satisfies: $$\left\langle j,m\right|\hat{D}\left(U_{1}U_{2}U_{3}\right)\left|j,n\right\rangle =D_{mn}^{j}\left(U_{1}U_{2}U_{3}\right)=\sum_{k,l}D_{mk}^{j}\left(U_{1}\right)D_{kl}^{j}\left(U_{2}\right)D_{ln}^{j}\left(U_{3}\right),\label{eq:y}$$ we can write $$\textrm{tr}\,^{3}H=\sum_{k,l,m,n}\delta_{mn}D_{mk}^{j}\left(U_{1}\right)D_{kl}^{j}\left(U_{2}\right)D_{ln}^{j}\left(U_{3}\right).\label{eq:z}$$ The geometrical interpretation of the spin-$j$ in (\[eq:x\])-(\[eq:z\]) is the measure on the ’artificial’ hinge dual to the plane of rotation inside loop $\gamma.$ See FIG. 6 and Section III C.
Acting $\textrm{tr}\,^{3}H$ with the spin network states gives: $$\begin{aligned}
\left\langle \psi\right|\textrm{tr}\,^{3}H\left|\psi\right\rangle = & \left(\sum_{m_{6},l,m_{3},n_{2},m_{3'},n_{2'}}i^{m_{6}m_{3}n_{2}}i^{m_{6}m_{3}'n_{2}'}i^{m_{3}lm_{3}'}i^{n_{2}ln_{2}'}\right)\left(\sum_{m_{5},k,m_{2},n_{1},m_{2'},n_{5'}}i^{m_{5}m_{2}n_{1}}i^{m_{5}m_{2}'n_{1}'}i^{m_{2}km_{2}'}i^{n_{1}kn_{1}'}\right)\times..\\
& \;...\times\left(\sum_{m_{4},m,m_{1},n_{3},m_{1'},n_{3'}}i^{m_{4}m_{1}n_{3}}i^{m_{4}m_{1}'n_{3}'}i^{n_{3}mn_{3}'}i^{m_{1}mm_{1}'}\right)\left(\sum_{n_{4}..n_{6}}i^{n_{6}n_{5}n_{4}}i^{n_{6}n_{5}n_{4}}\right),\end{aligned}$$ but the first three-terms in the parantheses are only the Wigner 6j-symbols: $$\left\langle \psi\right|\textrm{tr}\,^{3}H\left|\psi\right\rangle =\underset{\prod_{i=1}^{3}\left\{ 6j\right\} _{i}}{\underbrace{\left\{ \begin{array}{ccc}
j_{6} & j_{3} & j_{2}\\
j_{3}' & j_{2}' & j
\end{array}\right\} \left\{ \begin{array}{ccc}
j_{5} & j_{2} & j_{1}\\
j_{2}' & j_{1}' & j
\end{array}\right\} \left\{ \begin{array}{ccc}
j_{4} & j_{1} & j_{3}\\
j_{1}' & j_{3}' & j
\end{array}\right\} }}\left(\sum_{n_{4}..n_{6}}i^{n_{6}n_{5}n_{4}}i^{n_{6}n_{5}n_{4}}\right).\label{eq:huhu}$$
We are ready to take the semi-classical limit by setting all the $j_{i}$’s to be large. Using the result proved by Roberts [@key-14; @key-15], for large spin $j,$ the 6j-symbol can be approximated as follow: $$\left\{ 6j\right\} =\left\{ \begin{array}{ccc}
j_{1} & j_{2} & j_{3}\\
j_{4} & j_{5} & j_{6}
\end{array}\right\} \approx\frac{1}{\sqrt{12\pi\left|V\right|}}\cos\left(\sum_{i=1}^{6}j_{i}\theta_{i}-\frac{\pi}{4}\right),$$ where the six $j_{i}$ constructs a tetrahedron, $\theta_{i}$ are the internal dihedral angle on segment $j_{i},$ and $\left|V\right|$ is the volume of the tetrahedron. The $\frac{\pi}{4}$ term is known as the Maslov phase. Using this result to our case, we obtain: $$\begin{aligned}
\left\{ \begin{array}{ccc}
j_{6} & j_{3} & j_{2}\\
j_{3}' & j_{2}' & j
\end{array}\right\} & \approx\frac{1}{\sqrt{12\pi\left|V\right|}}\cos\left(j_{6}\theta_{6}+j_{3}\theta_{3}+j_{2}\theta_{2}+j_{3}'\theta_{3}'+j_{2}'\theta_{2}'+j\theta_{l}-\frac{\pi}{4}\right)\nonumber \\
& \approx\frac{1}{\sqrt{12\pi\left|V\right|}}\cos\left(j_{6}\theta_{6}+2j_{3}\theta_{3}+2j_{2}\theta_{2}+j\theta_{l}-\frac{\pi}{4}\right).\label{eq:hu}\end{aligned}$$ where the last step we use the fact that $j_{i}=j_{i}',$ noting that $j_{i}'$ comes from the covector state (\[eq:covect\]). This fact also cause $\theta_{i}=\theta_{i}'$. See FIG. 10.
{height="3.5cm"}
Remember that $j$ is the measure on the ’artificial’ hinge inside loop $\gamma$, in other words, the spin number $j$ is the ’length’ of the artificial hinge. Along this artificial hinge, there is a point $p$ where the 2D deficit angle is located. Taking $j_{i}\ggg j,$ we have:
$$\theta_{6}\approx0,\:\theta_{3}\approx\frac{\pi}{2},\:\theta_{2}\approx\frac{\pi}{2},\:\theta_{l}\approx\phi_{23}=\phi_{6},$$ See FIG. 10. Using this approximation, (\[eq:hu\]) becomes: $$\left\{ \begin{array}{ccc}
j_{6} & j_{3} & j_{2}\\
j_{3}' & j_{2}' & j
\end{array}\right\} \approx\frac{1}{\sqrt{12\pi\left|V_{1}\left(j_{6},j_{3},j_{2},j\right)\right|}}\cos\left(\left(j_{3}+j_{2}\right)\pi+j\phi_{6}-\frac{\pi}{4}\right).$$ Doing the same way for the other two 6j-symbols, we obtain: $$\left\{ \begin{array}{ccc}
j_{5} & j_{2} & j_{1}\\
j_{2}' & j_{1}' & j
\end{array}\right\} \approx\frac{1}{\sqrt{12\pi\left|V_{2}\left(j_{5},j_{2},j_{1},j\right)\right|}}\cos\left(\left(j_{2}+j_{1}\right)\pi+j\phi_{5}-\frac{\pi}{4}\right),$$ $$\left\{ \begin{array}{ccc}
j_{4} & j_{1} & j_{3}\\
j_{1}' & j_{3}' & j
\end{array}\right\} \approx\frac{1}{\sqrt{12\pi\left|V_{3}\left(j_{4},j_{1},j_{3},j\right)\right|}}\cos\left(\left(j_{1}+j_{3}\right)\pi+j\phi_{4}-\frac{\pi}{4}\right).$$ By writing the cosine function using the Euler formula: $$\begin{aligned}
\cos x & = & \frac{1}{2}\left(\exp ix+\exp-ix\right),\end{aligned}$$ we can write the products of the three 6j-symbols as: $$\begin{aligned}
\prod_{i=1}^{3}\left\{ 6j\right\} _{i} & \approx & \frac{1}{4}\frac{1}{12\pi\sqrt{12\pi\left|V_{1}\right|\left|V_{2}\right|\left|V_{3}\right|}}\times\left(\:\cos\left(2\left(j_{1}+j_{2}+j_{3}\right)\pi+j\left(\phi_{4}+\phi_{5}+\phi_{6}\right)-\frac{3\pi}{4}\right)+...\right.\nonumber \\
& & \qquad...+\cos\left(2j_{3}\pi+j\left(\phi_{4}-\phi_{5}+\phi_{6}\right)-\frac{\pi}{4}\right)+\cos\left(2j_{2}\pi+j\left(-\phi_{4}+\phi_{5}+\phi_{6}\right)-\frac{\pi}{4}\right)+...\nonumber \\
& & \qquad...+\left.\cos\left(2j_{1}\pi+j\left(\phi_{4}+\phi_{5}-\phi_{6}\right)-\frac{\pi}{4}\right)\,\right).\label{eq:huh}\end{aligned}$$ From this point, we take the sum of $j_{1}+j_{2}+j_{3}$ to be an integer/natural number (we discard the half-integer possibility) and choose the spin number $j=1,$ then we could write: $$\begin{aligned}
\prod_{i=1}^{3}\left\{ 6j\right\} _{i} & \approx\frac{1}{4}\frac{1}{12\pi\sqrt{12\pi\left|V_{1}\right|\left|V_{2}\right|\left|V_{3}\right|}}\:\cos\left(\delta\phi-\frac{3\pi}{4}\right)+\mathcal{O},\label{eq:huhuhu}\end{aligned}$$ where we denoted the remaining terms by $\mathcal{O}$ and using $\phi_{4}+\phi_{5}+\phi_{6}=2\pi-\delta\phi$ (see FIG. 9). Inserting (\[eq:huhuhu\]) to (\[eq:huhu\]), we obtain: $$\begin{aligned}
\left\langle \cos\frac{^{3}\hat{R}}{2}\right\rangle & \sim\left\langle \cos\left(^{2}\hat{R}-\frac{3\pi}{4}\right)\right\rangle.\end{aligned}$$ The linearity relation (\[eq:2.12\]) can be recovered by dropping the Maslov phase: $$\begin{aligned}
\left\langle \textrm{tr}\,^{3}H\right\rangle & \sim\left\langle \textrm{tr}\,^{2}H\right\rangle .\end{aligned}$$
An important point is we have a freedom to choose the spin number $j,$ since this spin number comes from (\[eq:x\]), which is the dimension of the $SU(2)$ representation. If we set $j=0,$ then, the volume of the 6j tetrahedra $\left|V_{i}\right|$ will be zero and relation (\[eq:huh\]) will diverge, given any $j_{i}'$s. But since the range of spin $j$ is $j\geq\frac{1}{2}$, with $j$ multiple of half, then the spin number $j$ acts as an ultraviolet cut-off which prevents relation (\[eq:huh\]) from divergence.
Conclusion {#V}
==========
We have obtained the Gauss-Codazzi equation for a discrete (2+1)-dimensional manifold, discretized by isosceles tetrahedra. We have studied the definition of extrinsic curvature in the discretized context. With definitions of the 3-dimensional intrinsic, 2-dimensional intrinsic, and 2-dimensional extrinsic curvature in the discrete picture, we have promoted them to curvature operators, acting on the spin network states of (2+1)-dimensional loop quantum gravity. In the semi classical limit, we have shown that the linearity between $\textrm{tr}\,^{3}H$ and $\textrm{tr}\,^{2}H$ in equation (\[eq:2.12\]) is recovered in the spin network calculation, by dropping the Maslov phase. There exist a natural ultraviolet cut-off which prevents the discrete Gauss-Codazzi equation from divergences in the semi-classical limit.
[10]{}
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[^1]: (\[eq:cc\]) has a similar form with (\[eq:bb\]), except the $\cos\phi_{i}$ and the term which contains the volume form.
|
---
abstract: |
We apply the one-loop results of the $SU(3)_L\times SU(3)_R$ ChPT suplemented with the inverse amplitude method to fit the available experimental data on $\pi\pi$ and $\pi K$ scattering. With esentially only three parameters we describe accurately data corresponding to six different channels, namely $(I,J)=(0,0),
(2,0), (1,1), (1/2,0), (3/2,0)$ and $(1/2,1)$. In addition we reproduce the first resonances of the $(1,1)$ and $(1/2,1)$ channel with the right mass corresponding to the $\rho$ and the $K^*(892)$ particles.\
PACS:14.60Jj
author:
- |
[**A. Dobado and J. R. Peláez**]{}\
Departamento de Física Teórica\
Universidad Complutense de Madrid\
28040 Madrid, Spain
title: 'A global fit of $\pi\pi$ and $\pi K$ elastic scattering in ChPT with dispersion relations'
---
3.0cm FT/UCM/10/92
0.83 true cm 20 true cm
[**Introduction:**]{} In 1979 Weinberg \[1\] suggested that it is possible to summarize many previous current algebra results in a phenomenological lagrangian that incorporates all the constraints coming from the chiral symmetry of the strong interactions and QCD. This technique, also called Chiral Perturbation Theory, was developped some time later to one-loop level and in great detail in a celebrated set of papers by Gasser and Leutwyler \[2\]. In these works, the authors showed how it is possible to compute many different Green functions involving low-energy pions and kaons as functions of the lowest powers of their momenta, their masses and a few phenomenological parameters. In these and in further works it was also shown that this method provides a good parametrization of many low-energy experimental data.
More recently, the strongly interacting symmetry breaking sector of the Standard Model \[3\] has also been described phenomenologically using ChPT \[4\].
In both contexts , one of the main obstacles found when one tries to apply ChPT to higher energies relies in the issue of unitarity. ChPT, being a consistent theory, is unitary in the perturbative sense. However, as the expansion parameters are the momenta and the Nambu-Goldstone boson masses, perturbative unitarity breaks down, sometimes even at moderate energies \[5\]. Different attempts to improve this behaviour of ChPT and extend the applicability to higher energies, have been proposed in the literature. These methods include the use of unitarization procedures such as the Padé expansion \[5,6\], the inverse amplitude method \[5\], and the explicit introduction of more fields describing resonances \[7\]. All them improve the unitarity behavior of the ChPT expansion and provide a more accurate description of the data although there is some controversy about which of them is more appropriate.
In this note we show the results of the application of the inverse amplitude method to the ChPT one-loop computation of elastic $\pi\pi$ and $\pi K$ scattering \[2,8\]. The obtained amplitude converges at low energies with that of standard ChPT but it fulfills strictly elastic unitarity, since the contribution of the right cut in the corresponding dispersion relation is taken into account exactly in the inverse amplitude method. Using this approach (which incidentaly is connected with the formal \[1,1\] Padé approximant of the one-loop amplitudes) we will make a global fit of the data for elastic $\pi\pi$ and $\pi K$ scattering just varying the $L_1$, $L_2$ and $L_3$ ChPT constants. The resulting values of these parameters will not be quite different from those obtained previously, but the range of energies and the quality of the fit will be enlarged quite amazingly.
[**The partial waves in ChPT:**]{} The elastic scattering partial waves are defined from the corresponding isospin amplitude $T_I(s,t)$ as: $$t_{IJ}(s)=\frac{1}{32 K \pi}\int_{-1}^{1}d(cos \theta) P_J(cos
\theta)T_I(s,t)$$ where $K=2$ or $1$ depending on whether the two particles in the reaction are identical or not. For elastic $\pi\pi$ scattering the possible isospin channels are $I=0,1,2$ whilst for $\pi K$ we can have $I=1/2$ or $I=3/2$. In the first case the isospin amplitudes $T_I$ can be written in terms of a simple function $A(s,t,u)$ as follows; $T_0(s,t,u)=3A(s,t,u)+A(t,s,u)+A(u,t,s),
T_1(s,t,u)=A(s,t,u)-A(t,s,u)$ and $T_2(s,t,u)=A(s,t,u)+A(t,s,u)$. In the second, the $I=1/2$ amplitude can be written as $T_{1/2}(s,t,u)=(3/2)T_{3/2}(u,t,s)-(1/2)T_{3/2}(s,t,u)$. The Mandelstam variables $s,t,u$ fulfill $s+t+u=2(M_{\alpha}^2+M_{\beta}^2)$ where we use the notation $\alpha$ and $\beta=\pi$ or $K$, so that we can describe both processes with the same general formulae. For $s > s_{th}=(M_{\alpha}+M_{\beta})^2$ the partial waves $t_{IJ}$ can be parametrized as:
$$t_{IJ}(s)= e^{i\delta_{IJ}(s) }\sin \delta_{IJ}(s)/\sigma_{\alpha \beta}(s)$$
where $$\sigma_{\alpha\beta}(s) = \sqrt{ (1-\frac{(M_{\alpha}+M_{\beta})^2}{s})
(1-\frac{(M_{\alpha}-M_{\beta})^2}{s})}$$ The $t_{IJ}(s)$ amplitude fulfills the elastic unitarity condition $$Imt_{IJ}=\sigma_{\alpha \beta} \mid t_{IJ} \mid ^2$$ on the right cut.
Using standard one-loop ChPT it is possible to compute the above scattering amplitudes to order $p^4$ (as it is customary, any Mandelstam variable, $M_{\pi}^2$ and $M_{K}^2$ will be considered of the order of $p^2$). The relevant functions $A(s,t,u)$ and $T_{3/2}$ were computed in \[2,8\]. $$\begin{aligned}
A(s,t,u)&=&(s-M_{\pi}^2)/F_{\pi}^2+B(s,t,u)+C(s,t,u)+O(E^6) \\ \nonumber
B(s,t,u)&=&\frac{1}{F_{\pi}^4} (
\frac {M_{\pi}^4}{18} J_{\eta\eta}^r (s) +
\frac{1}{2} ( s^2 - M^4_{\pi}) J_{\pi\pi}^r(s) + \frac {1}{8} s^2 J_{KK}^r (s)
\\
\nonumber
&+& \frac{1}{4}( t - 2 M_{\pi}^2)^2 J_{\pi\pi}^r (t) + t(s-u) [
M_{\pi\pi}^r (t)+\frac{1}{2} M_{KK}^r (t) ]+ (t \leftrightarrow u) )\\
\nonumber
C(s,t,u)&=&\frac{4}{F_{\pi}^4} ( (2L_1^r + L_3 )( s - 2M_{\pi}^2)^2 +
L_2^r [ (t-2M_{\pi}^2)^2 + ( u - 2M_{\pi}^2)^2] +\\ \nonumber
&+& (4L_4^r+2L_5^r)M_{\pi}^2(s-2M_{\pi}^2) + (8L_6^r+4L_8^r)M_{\pi}^4 ) \\
\nonumber\end{aligned}$$ and $$\begin{aligned}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
T^{3/2}(s,t,u)&=&\frac{M_{\pi}^2+M_{K}^2-s}{2F^2_{\pi}}+T^T_4(s,t,u)+T^P_4(s,t,u)+
T^U_4(s,t,u) \\ \nonumber
T^T_4(s,t,u)&=& \frac{1}{16F^2_{\pi}}(\mu_{\pi}[10s-7M_{\pi}^2-13M_{K}^2] \\
\nonumber
&+& \mu_{K}[2M_{\pi}^2+6M_{K}^2-4s]
+\mu_{\eta}[5M_{\pi}^2+7M_{K}^2-6s] ) \\ \nonumber
T^P_4(s,t,u)&=& \frac{2}{F^2_{\pi}F^2_{K}}( 4L_1^r(t-2M_{\pi}^2 )(t-2M_{K}^2 )
\\ \nonumber
&+& 2L_2^r [ (s - M_{\pi}^2-M_{K}^2)^2 + (u - M_{\pi}^2-M_{K}^2)^2] \\
\nonumber
&+& L_3^r[(u - M_{\pi}^2-M_{K}^2)^2 + (t-2M_{\pi}^2 )(t-2M_{K}^2)] \\
\nonumber
&+& 4L_4^r[t(M_{\pi}^2+M_{K}^2) - 4M_{\pi}^2M_{K}^2]\\ \nonumber
&+& 2L_5^rM_{\pi}^2(M_{\pi}^2-M_{K}^2-s)+8(2L_6^r+L_8^r)M_{\pi}^2M_{K}^2 )\\
\nonumber
T^U_4(s,t,u)&=& \frac{1}{4F^2_{\pi}F^2_{K}}(t(u-s)[2M_{\pi\pi}^r(t)+M_{KK}^r(t)
]\\ \nonumber
&+&\frac{3}{2}[(s-t)(L_{\pi K}(u)+L_{K\eta}(u)-u(M_{\pi
K}^r(u)+M_{K\eta}^r(u))) \\ \nonumber
&+& (M_{K}^2 - M_{\pi}^2)(M_{\pi K}^r(u)+M_{K \eta}^r(u))] +
J_{\pi K}^r(s)(s-M_{K}^2-M_{\pi}^2)^2 \\ \nonumber
&+& \frac{1}{2} (M_{K}^2 - M_{\pi}^2) [ K_{\pi K}(u)(5u-2M_{K}^2-2M_{\pi}^2)+
K_{K \eta}(u)(3u-2M_{K}^2-2M_{\pi}^2)] \\ \nonumber
&+&\frac{1}{8}J_{\pi K}^r(u)[ 11u^2 -
12u(M_{K}^2+M_{\pi}^2)+4(M_{K}^2+M_{\pi}^2)^2] \\ \nonumber
&+&\frac{3}{8}J_{K \eta}^r(u) (u - \frac{3}{2}(M_{K}^2+M_{\pi}^2))^2 +
\frac{1}{2}J_{\pi\pi}^r(t)t(2t-M_{\pi}^2) \\ \nonumber
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
&+&\frac{3}{4}J_{KK}^r(t)t^2+\frac{1}{2}J_{\eta\eta}^r(t)M_{\pi}^2(t-\frac{8}{9}
M_{K}^2))\\ \nonumber\end{aligned}$$ The masses $M_{\alpha}$ and the decay constants $F_{\alpha}$ appearing in these equations are the physical values. The relation with the corresponding constants appearing in the chiral lagrangian, and the functions $\mu_{\alpha}$ can be found in \[8\]. The transcendental functions $M^r_{\alpha\beta}, L^r_{\alpha\beta}$ and $J^r_{\alpha\beta}$ are defined in \[2\]. The first terms in the above amplitudes reproduce the well known Weinberg low-energy theorems. The $L^r_i$ constants can be considered as phenomenological parameters that up to constant factors are the renormalized coupling constants of the chiral lagrangian renormalized conventionaly at the $m_{\eta}$ scale. Their relation with the corresponding bare constants $L_i$ and their evolution with the renormalization scale can be found in \[2\]. Using eq.1 and eq.5 it is possible to obtain the corresponding partial wave amplitudes. In the general framework of ChPT they can be obtained as a series with increasing number of $p^2$ powers, i.e. $$t_{IJ}=t_{IJ}^{(0)}+t_{IJ}^{(1)}+...$$ where $t_{IJ}^{(0)}$ is of order $p^2$ and corresponds to the low-energy theorem and $t_{IJ}^{(1)}$ is of order $p^4$. In general, the real part of $t_{IJ}^{(1)}$ cannot be expressed in terms of elementary functions but it can be computed numerically. The amplitudes in eq.5 have been used in the literature, without further elaboration, to fit the low-energy $\pi\pi$ and $\pi K$ scattering data \[2,8,9\].
[**Dispersion relations and the inverse amplitude method:**]{} One very important point concerning the partial wave amplitudes computed from ChPT to one-loop is the fact that they have the appropriate cut structure, namely the left cut and the right or unitarity cut. However, they only fulfill the unitarity condition on the right cut in a perturbative sense i.e. $$\begin{aligned}
Im t_{IJ}^{(1)} = \sigma_{\alpha \beta}\mid t_{IJ}^{(0)} \mid
^2\end{aligned}$$
Let us show now how, with the use of dispersion theory, it is possible to build up a completely unitarized amplitude for the $\pi\pi$ and $\pi K$ amplitudes starting from the one-loop ChPT result above. Let us start writing a three subtracted dispersion relation for the partial wave $t_{IJ}$: $$t_{IJ}(s)=C_0+C_1s+C_2s^2+
\frac{s^3}{\pi}\int_{(M_{\alpha}+M_{\beta})^2}^{\infty}\frac{Im
t_{IJ}(s')ds'}{s'^3(s'-s-i\epsilon)} + LC(t_{IJ})$$ where $LC(t_{IJ})$ represent the left cut contribution. In particular, this equation is fulfilled by the standard one-loop ChPT result. Note that three subtractions are needed to ensure the convergence of the integrals in this approximation since the one-loop ChPT amplitudes are second order polynomials modulo $log$ factors (higher order ChPT amplitudes require more subtractions but we do not know how many are needed for the exact result). To the one-loop level, the imaginary part of the amplitude can be written on the right cut integral as $Im t_{IJ}\simeq Im t_{IJ}^{(1)}=\sigma t_{IJ}^{(0)2}$ (from the left cut there is a generic $t_{IJ}^{(1)}$ contribution).
The substraction terms appearing in the above dispersion relation can be expanded in powers of $M_{\alpha}^2$, and they have contributions from $t_{IJ}^{(0)}$ and from $t_{IJ}^{(1)}$ that can be written as $a_0+a_1s$ and $b_0+b_1s+b_2s^2$ and also from higher order terms of the ChPT series i.e $C_0=a_0+b_0+...$ and $C_1=a_1+b_1+...$. The $a$ and $b$ constants depend also on the $L^r_i$ parameters. Then, in the one-loop ChPT aproximation, the exact amplitude $t(s)$ can be written as: $$t_{IJ}(s)\simeq t_{IJ}^{(0)}(s)+t_{IJ}^{(1)}(s)$$ where: $$\begin{aligned}
t_{IJ}^{(0)} &=& a_0+a_1s \\ \nonumber
t_{IJ}^{(1)} &=& b_0+b_1s+b_2s^2+ \\ \nonumber
&+&\frac{s^3}{\pi}\int_{(M_{\alpha}+M_{\beta})^2}^{\infty}\frac{\sigma
t_{IJ}^{(0)}(s')^2ds'}{s'^3(s'-s-i\epsilon)}+LC(t_{IJ}) \nonumber\end{aligned}$$ where in the left cut contribution we have to integrate here $Imt_{IJ}^{(1)}(s)$ as an aproximation to $t_{IJ}(s)$. In some sense, we can understand eq.9,10 and eq.11 in a rather different way that will be useful later: we can assume that the dispersion relation in eq.9 is fulfilled by the exact amplitude $t(s)$ and we solve this equation approximately by introducing inside the left and right cut integrals the one-loop prediction for $Imt(s)$ to find again eq.10. However, we would like to stress again that this result does not fulfill the elastic unitarity condition in eq.4 but only the perturbative version in eq.8. In fact this happens to any order in the ChPT expansion since a polynomial can never fulfill eq.4. However, there are other ways to use the information contained in the ChPT series apart from the direct comparison with the experiment of the truncated series. Instead of using the dispersion relation for the loop ChPT amplitude we can try a three subtracted dispersion relation for the inverse of $t_{IJ}$ or more exactly for the auxiliar function $G(s)=t_{IJ}^{(0)2}/t_{IJ}$ namely: $$G(s)=G_0+G_1s+G_2s^2+ \\ \nonumber
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\frac{s^3}{\pi}\int_{(M_{\alpha}+M_{\beta})^2}^{\infty}\frac{ImG(s')ds'}{s'^3(s'-s-i\epsilon)}
+LC(G)+PC$$
where $LC(G)$ is the left cut contribution and $PC$ is the pole contribution that eventually could appear due to possible zeros of $t_{IJ}(s)$. Now, on the right cut we have $ImG=t_{IJ}^{(0)2}Im(1/t_{IJ})=-t_{IJ}^{(0)2}Imt_{IJ}/\mid t_{IJ}
\mid^2= -t_{IJ}^{(0)2}\sigma$. This means that the right cut integral appearing in the dispersion relation for $G$ is the same than that appearing in eq.10 and eq.11 for the one-loop dispersion relation for $t$ and hence it can be obtained from the one-loop result. The left cut integral of the dispersion relation in eq.12 cannot be computed exactly but we can use the one-loop ChPT result to write $ImG=-t_{IJ}^{(0)2}Imt_{IJ}/
\mid t_{IJ} \mid^2 \simeq
-Im t_{IJ}^{(1)}$. Besides, the substraction constants can be expanded in terms of $
M_{\alpha}^2/F_{\beta}^2$ powers so that $G_0=a_0-b_0+O((M_{\alpha}^2/F_{\beta}^2)^3)$, $F^2_{\alpha}G_1=a_1-b_1
+O((M_{\alpha}^2/F_{\beta}^2)^2)$, and $F^2_{\alpha}F^2_{\beta}G_2=-b_2+O(M_{\alpha}^2/F_{\beta}^2)$. Therefore, neglecting the the pole contribution, the dispersion relation in eq.12 can be written approximately as: $$\begin{aligned}
\frac{t_{IJ}^{(0)2}}{t_{IJ}}&\simeq& a_0+a_1s-b_0-b_1s-b_2s^2 \\ \nonumber
&-&\frac{s^3}{\pi}\int_{(M_{\alpha}+M_{\beta})^2}^{\infty}\frac{\sigma
t_{IJ}^{(0)2}(s')ds'}{s'^3(s'-s-i\epsilon)}+LC(G)\end{aligned}$$ with $LC(G)$ computed with the $ImG$ approximated by $-Im t_{IJ}^{(1)}$ or in other words: $$\frac{t_{IJ}^{(0)2}}{t_{IJ}}\simeq
t_{IJ}^{(0)}-t_{IJ}^{(1)}$$ or what it is the same: $$t_{IJ}\simeq
\frac{t_{IJ}^{(0)2}}{
t_{IJ}^{(0)}-t_{IJ}^{(1)} }$$
Remarkably, to derive this result the one-loop ChPT approximation has been used only inside the left cut integral but not inside the right cut integral which was computed exactly. This is in contrast with the one-loop ChPT result in eq.10 when considered from the point of view of the dispersion relation that $t_{IJ}$ has to fulfill. In this case, in adddition to the above approximations used to derive eq.15, we have done a much stronger one concerning the right cut contribution since the $Imt_{IJ}$ was also approximated by the one-loop result. This fact is crucial because the right cut contribution is responsible for the very strong rescattering effects present in these reactions. As a consequence of that, eq.15 fulfills the unitarity condition $Im t_{IJ}= \sigma \mid t_{IJ} \mid ^2$ exactly and not only perturbatively as it was the case of eq.10. Of course both results (eq.10 and eq.15) provide the same answers at low energies but it is expected that eq.15 will provide more realistic results at higher energies and this is in fact what we find when comparing with the experimental data.
When zeros are present in the partial waves, they need to be included and the final result is not so simple as in eq.15. This can be done taking into account their contribution to eq.12 or just making a new sustraction in these points. However, in practical cases, it can happen that the corresponding residuous are very small and eq.15 is still a good approximation outside the region around the position of the these zeros. This is for instance the case of the $I=J=0$ and $I=2,J=0$ channels \[6\] which have zeros close to $M_{\Pi}^2/2$ and $2M_{\Pi}^2$ respectively. However, for a given channel $IJ$ there is no way to know a priori if zeros will be present or not since ChPT only provides a low-energy expansion of $t(s)$ and not the whole amplitude. Nevertheless, in practice, a simple inspection of the low-energy behaviour of the amplitudes can make sensible the hipothesis that zeros are not present in the low-energy region but of course this must not be necessarily true for all the channels.
We could also ask about the possibility of extending the above method to higher orders of the ChPT series for instace to the two-loop computation. In principle this can be done in a straightforward way. We start writing a four subtracted dispersion relation for the two-loop amplitude since it is a third order polynomial modulo logarithms. As in the one-loop case, this dispersion relation can be interpreted as an approximation to the exact amplitude $t_{IJ}$. Then we write a four subtracted dispersion relation for the auxiliar function $
G=t_{IJ}^{(0)2}/t_{IJ}$. The integrals of this dispersion relation can be related modulo higher order terms to the ChPT by approaching $ImG$ by its low energy expansion. The subtraction constants can be treated similarly as we did in the one-loop case an finaly we arrive to: $$t_{IJ}\simeq \frac{t_{IJ}^{(0)2} } {
t_{IJ}^{(0)}-t_{IJ}^{(1)}+t_{IJ}^{(1)2}/t_{IJ}^{(0)}-t_{IJ}^{(2)} }$$ It is very easy to show that this amplitude, which incidentally corresponds to the formal \[1,2\] Padé approximant, fulfills the elastic unitarity condition in eq.4 but in this case, the right cut integral appearing in the dispersion relation for $G$ cannot be computed exactly as it was in the one-loop case. Unfortunately, as there is not any two-loop computation available, we cannot confront the above equation with the experimental data.
[**Determination of the chiral parameters and discussion:**]{} First of all, we have used the set of low energy chiral parameters $L_i$ proposed in \[11\] for i=1,2,3 and those given in \[2\] for i=4..8 which will not be changed in this work. Appliying those values to the proposed unitarized partial waves of eq.15, we have found the appropriate qualitative behaviour for both reactions. Moreover, we find two resonances in the partial wave amplitudes corresponding to the $(1,1)$ and $(1/2,1)$ channels. These resonances can be identified naturally as the $\rho$ and the $K^*$ particles. The phase-shifts cross the $\pi/2$ value in their corresponding channels. In addition, in the chiral limit where an analytical computation of the partial waves is possible, two poles are found in he second Riemann sheet accordingly to these resonances. In the other channels, where no physical resonances exist below $1 GeV$, they do not appear in our numerical results. We consider this fact as a strong support for the use of the unitarized partial waves in eq.15.
To fit completely the data with these formulae the next step has been to tune the (1,1) $\pi\pi$ channel to give the correct mass for the $ \rho $ resonance. Since this partial wave is almost only sensible to the relation $
L_3+2L^{r}_1-L^{r}_2$, then, setting the $\rho$ mass to 774 Mev means fixing a value of this special combination of parameters.
Nevertheless, there are still two degrees of freedom. Slightly varying the initial values we can fit the $\pi \pi$ (2,0) and (0,0) channels, thus obtaining $L^{r}_1$ and $L^{r}_2$. Unfortunately the experimental data coming from these channels allow some small uncertainty in the parameters, rending a $K^{\star}$ mass between 850 and 950 Mev. So we use the mass of this resonance for a further parameter tuning, and finally we obtain, at the renormalization scale $\mu=M_{\eta}=548.8 Mev$: $$L^{r}_1= 0.6 10^{-3}, L^{r}_2=1.6 10^{-3} , L_3=-3.8 10 ^{-3}$$ These values are well inside the errors quoted in \[11\] $$L^{r}_1= (0.88 \pm 0.47) 10^{-3}
, L^{r}_2=(1.61 \pm 0.38) 10^{-3}
, L_3=(-3.62 \pm 1.31) 10^{-3}$$
The $K^{\star}$ mass thus obtained is 880 Mev In figures 1 and 2 can be seen the results of this global fit for $\pi \pi$ scattering and those from non unitarized ChPT with the $\bar l_i$ parameters proposed in \[10\]. Figures 3 to 5 represent our global fit curves (continuous lines) for $\pi K$ scattering in three different channels. The dashed lines are the non unitarized ChPT predictions from the $L_i$ given in \[2\] and \[11\]. Fig.3 is the (3/2,0) channel and figures 4 and 5 are the (1/2,1) (1/2,0) respectively.\
[**Conclusions:**]{} The inverse amplitude method applied to the one-loop result coming from ChPT produces a simple way to unitarize the Goldstone boson elastic scattering amplitudes which takes into account, exactly, the strong rescattering effects. Incidentally, this method is formally equivalent to the $[1,1]$ Padé approximant applied to the one-loop ChPT result, provided that the exact partial wave amplitude has no zeros in the first sheet. Note that since the one-loop result is not, strictly speaking, a polynomial,the equivalence is only formal (This is not the case if one considers the amplitudes as polynomials in $\frac{1}{F_{\pi}^{2}}$ )
The unitarized amplitudes (with the previously fitted parameters for the standard one-loop ChPT result) give rise to the appearence of two resonances in the (1,1) and (1/2,1) channels that have to be understood as the $\rho$ and $K^*$, but not any more are found in other channels where no physical resonances exist below $1 GeV$. Therefore, the existence of these resonances is a highly non trivial prediction of the approach followed here and do not need to be introduced by hand in the data fit . Just tunnig slightly only three parameters, namely $L_1$, $L_2$ and $L_3$ we are able to obtain the right value for the masses of the $\rho$ and $K^*$ resonances. In addition we provide a fit for six channels in a remarkable agreement with the experiment.
The results obtained in this and previous works strongly suggest that the range of validity of the one-loop ChPT can be enlarged by a consistent traitment of the analiticity and unitarity constraints summarized in the dispersion relations but a great deal of work still remains to be done in this direction. As a final comment, we think that the recently proposed large $N$ approximation to ChPT (with $N$ being the number of Nambu-Godstone bosons) \[25\] (see also \[26\]) could provide a new extension of the one-loop ChPT results (see also \[26\] for applications to the $\gamma\gamma \rightarrow \pi\pi$ reaction \[27\]).
[**Aknowledgements:**]{} This work has been partially supported by the Ministerio de Educación y Ciencia (Spain)(CICYT AEN90-0034).
[**Figure Captions**]{}
Figure 1 .- (1,1) Phase shift for $\pi \pi$ scattering. The continuous line corresponds to our fit using eq.15. The dashed line is the result coming from non unitarized ChPT with the $\bar l_i$ parameters proposed in \[10\]. The experimental data comes from: $\bigcirc$ ref.\[16\], $\bigtriangleup$ ref.\[20\].
Figure 2 .- Phase shift for $\pi \pi$ scattering. The results coming from the fit proposed in this paper (eq.15) are the continuous line which represents the (0,0) phase shift, and the dashed line which corresponds to that of (2,0). The dotted and dashed-dotted lines are the (0,0) and (2,0) phase shifts respectively,they were obtained with non unitarized ChPT and the parameters given in \[10\]. The experimental data corresponds to: $\bigtriangleup$ ref.\[12\], $\bigcirc$ ref.\[13\],$\Box$ ref.\[14\],$\diamondsuit$ ref.\[15\], $\bigtriangledown$ ref.\[16\],$\star$ ref.\[17\],$\times$ ref.\[18\],$\bullet$ ref.\[19\].
Figure 3.- Phase shift of the (3/2,0) channel for $\pi K$ scattering. The continuous line is the result of the inverse amplitud method (eq.15) with the parameters proposed in this paper, whereas the dashed line is non unitarized ChPT with the parameters proposed in \[2\] and \[11\]. Data corresponds to ref.\[21\]
Figure 4.- The same as figure 3 but for the (1/2,1) partial wave. Data comes from: $\bigtriangleup$ ref.\[21\] and $\star$ ref.\[24\].
Figure 5.- As figure 3 but for the (1/2,0) channel. The experimental data corresponds to: $\bigtriangleup$ ref.\[21\],$\diamondsuit$ ref.\[22\], and $\star$ to ref.\[23\].
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|
---
abstract: 'For $X$ a smooth quasi-projective variety and $X^{[n]}$ its associated Hilbert scheme of $n$ points, we study two canonical Fourier–Mukai transforms $\D(X)\to \D(X^{[n]})$, the one along the structure sheaf and the one along the ideal sheaf of the universal family. For $\dim X\ge 2$, we prove that both functors admit a left-inverse. This means in particular that both functors are faithful and injective on isomorphism classes of objects. Using another method, we also show in the case of an elliptic curve that the Fourier–Mukai transform along the structure sheaf of the universal family is faithful and injective on isomorphism classes. Furthermore, we prove that the universal family of $X^{[n]}$ is always flat over $X$, which implies that the Fourier–Mukai transform along its structure sheaf maps coherent sheaves to coherent sheaves.'
author:
- Andreas Krug and Jørgen Vold Rennemo
bibliography:
- 'references.bib'
title: Some ways to reconstruct a sheaf from its tautological image on a Hilbert scheme of points
---
Introduction
============
Tautological bundles on Hilbert schemes of points have been intensively studied from various perspectives, and have been used for many applications, starting in the 1960s when they were studied on symmetric products of curves; see [@Schwarzenberger--bundlesplane; @Schwarzenberger--secant; @Mattuck--sym]. They are defined by means of the Fourier–Mukai transform along the universal family of the Hilbert scheme. More concretely, let $X$ be a smooth quasi-projective variety over $\IC$, let $X^{[n]}$ be the associated Hilbert scheme of $n$ points, and let $\Xi_n\subset X^{[n]}\times X$ be the universal family of length $n$ subschemes of $X$, together with its projections $X\xleftarrow q \Xi_n\xrightarrow p X^{[n]}$. Given a rank $r$ vector bundle $E$ on $X$, there is the associated tautological bundle $E^{[n]}=p_*q^*E$ of rank $rn$ on $X^{[n]}$.
A very natural question is whether the bundle on $X$ can be reconstructed from its associated tautological bundle on the Hilbert scheme: $$\begin{aligned}
\label{eq:Q}
E^{[n]}\cong F^{[n]}\quad \xRightarrow{\quad \textbf{\large ?}\quad}\quad E\cong F\,.\end{aligned}$$ This question was studied quite recently by Biswas and Parameswaran [@Biswas-Parameswaran--reconstructioncurves] and by Biswas and Nagaraj [@Biswas-Nagaraj--reconstructioncurves; @Biswas-Nagaraj--reconstructionsurfaces]. Maybe surprisingly, Question has a negative answer if $X=\IP^1$; see [@Biswas-Nagaraj--reconstructionsurfaces Sect. 2.1]. On the other hand, the answer to Question is affirmative for semi-stable vector bundles on curves of genus $g(X)\ge 2$, see [@Biswas-Parameswaran--reconstructioncurves; @Biswas-Nagaraj--reconstructioncurves], and for arbitrary vector bundles on surfaces; see [@Biswas-Nagaraj--reconstructionsurfaces].
In the present paper, we generalise, strengthen, and complement these results in various directions. We extend the results from vector bundles to coherent sheaves and, more generally, objects in the derived category. We obtain new affirmative answers to Question for varieties of higher dimension and for elliptic curves. In most cases, we prove something slightly stronger than an affirmative answer to Question , namely the existence of left-inverses to the functor $(\_)^{[n]}$. Furthermore, we obtain similar results if we replace $(\_)^{[n]}$, which is the Fourier–Mukai transform along the structure sheaf of the universal family, by the Fourier–Mukai transform along the ideal sheaf of the universal family.
Let us describe the results in more detail. As alluded to above, given a smooth quasi-projective variety $X$ over $\IC$ and a positive integer $n$, we consider the Fourier–Mukai transform $$(\_)^{[n]}=\FM_{\reg_{\Xi_n}}\cong p_*\circ q^*\colon \D(X)\to \D(X^{[n]})$$ between the derived categories of perfect complexes. We can now extend Question from vector bundles to objects of the derived category $\D(X)$. In other words, we ask ourselves whether the functor $(\_)^{[n]}\colon \D(X)\to \D(X^{[n]})$ is injective on isomorphism classes. However, for a better understanding of this functor, we first prove the following, which extends a result of Scala [@Sca2 Sect. 2.1] from surfaces to varieties of arbitrary dimension.
The universal family $\Xi_n\subset X^{[n]}\times X$ is always flat over $X$. Hence, the functor $(\_)^{[n]}$ sends coherent sheaves on $X$ to coherent sheaves on $X^{[n]}$.
Besides $(\_)^{[n]}$, there is a second, equally canonical, Fourier–Mukai transform to the Hilbert scheme, namely the one along the universal ideal sheaf $$\FF_n=\FM_{\cI_{\Xi_n}}\cong \pr_{X^{[n]}*}\bigl(\cI_{\Xi_n}\otimes \pr_X^{*}(\_)\bigr)\colon \D(X)\to \D(X^{[n]})\,.$$ This functor was studied in the case that $X$ is a K3 surface in [@Add] and [@MMdefK3], and in the case that $X$ is a surface with $\Ho^1(\reg_X)=0=\Ho^2(\reg_X)$ in [@KSEnriques]. In the case of a K3 surface, it was shown that $\FF_n$ is a $\IP^{n-1}$-functor. In the case of a surface with $\Ho^1(\reg_X)=0=\Ho^2(\reg_X)$, it was shown that $\FF_n$ is fully faithful. In both cases, it follows that $\FF_n$ is injective on isomorphism classes and faithful.
Let us now summarise the results of this paper concerning reconstruction of objects from their images under the functors $(\_)^{[n]}$ and $\FF_n$.
\[thm:Psiintro\] If $X$ is a smooth quasi-projective variety of dimension $d=\dim X\ge 2$, there are left-inverses of both functors $(\_)^{[n]},\FF_n\colon \D(X)\to \D(X^{[n]})$ for every $n\in \IN$. In particular, $(\_)^{[n]}$ and $\FF_n$ are faithful, and for every pair $E,F\in \D(X)$ we have $$E^{[n]}\cong F^{[n]}\quad \Longleftrightarrow \quad E\cong F \quad \Longleftrightarrow \quad \FF_n(E)\cong \FF_n(F)\,.$$
is proved in . We need two different constructions of a left-inverse of the functors $(\_)^{[n]}$ and $\FF_n$. Which one of the two constructions works depends on whether or not $n$ is of the form $\binom{m+d}d$ for some $m\in \IN$.
If $n$ is not of the form $\binom{m+d}d$ for some $m\in \IN$, we write $n=\binom{m+d-1}d +\ell$ for some $0< \ell< \binom{m+d-1}{d-1}=\rank(\Sym^m \Omega_X)$. Let $\IG\subset X^{[n]}$ be the locus of subschemes $\xi\subset X$ such that $$\Spec(\reg_{X,x}/\fm_x^m)\subset \xi\subset \Spec(\reg_{X,x}/\fm_x^{m+1})\,$$ for some $x \in X$; in particular these are *punctual* subschemes in the sense that $\supp(\xi) = \{x\}$. The morphism $f\colon \IG\to X$, $\xi\mapsto x$, given by forgetting the scheme structure of $\xi$, identifies $\IG$ with the Grassmannian bundle $\Gr(\Sym^m \Omega_X, \ell)$ of rank $\ell$ quotients of $\Sym^m\Omega_X$. We denote the universal quotient bundle on $\IG$ by $\cQ$ and write the closed embedding as $\iota\colon \IG\hookrightarrow X^{[n]}$. In , we prove that the functor $K:=f_*(\cQ^\vee\otimes \iota^*(\_))\colon \D(X^{[n]})\to \D(X)$ is a left-inverse of $(\_)^{[n]}$ and of $\FF_n[1]$.
If $n=\binom{m+d}d$ for some $m\in \IN$ and $d\ge 2$, we construct another functor $N\colon \D(X^{[n]})\to \D(X)$ which is then proven to be a left-inverse of $(\_)^{[n]}$ and of $\FF_n[1]$ in . The functor $N$ is somewhat similar to the functor $K$ described above, but, instead of $\IG\subset X^{[n]}$, it uses a certain locus of pairs of punctual schemes inside the nested Hilbert scheme $X^{[n-1,n]}\subset X^{[n-1]}\times X^{[n]}$; see for details. Obviously, and together proof .
In , using equivariant sheaves on the cartesian product $X^n$, we also give two other reconstruction methods. These methods give an affirmative answer to question in many cases; namely for arbitrary objects in $\D(X)$ if $X$ is a surface, and for reflexive sheaves if $X$ is projective of dimension $d>2$. However, they are slightly weaker than the methods of which work for every object in $\D(X)$ for every $d\ge 2$. The reason that we decided to include the constructions of in the paper is twofold. Firstly, the proofs in are somewhat easier than those in . Secondly, the construction of is used in [@BiswasKrug] to prove an analogous reconstruction result for Hitchin pairs.
If the variety $X$ has fixed-point free automorphisms, we find a further reconstruction method which, contrary to the other methods, also works for curves. Here, we only state the most significant consequence, and refer to for details.
\[thm:ellipticintro\] Let $X$ be an elliptic curve. Then, for every $n\in \IN$ and every pair $E,F\in \D(X)$, we have $$E^{[n]}\cong F^{[n]}\quad \Longleftrightarrow \quad E\cong F\,.$$
In summary, we now are now not very far away from a complete answer of Question , in its generalised form for objects of the derived category.
gives a complete affirmative answer for $\dim X\ge 2$. In the curve case, the picture is a bit more subtle. Note first that on a curve $X$, every object $E \in D(X)$ is the direct sum of its shifted cohomology sheaves, and so by exactness of $(-)^{[n]}$ (see ) it is enough to answer Question (\[eq:Q\]) under the assumption $E, F \in \Coh(X)$. Furthermore, the torsion part of a sheaf $E$ is easily recovered from $E^{[n]}$, and so the general form of Question (\[eq:Q\]) reduces to the special form where $E$ and $F$ are vector bundles.
For $X=\IP^1$, as mentioned above, the answer to Question (\[eq:Q\]) is negative. Note however, that the answer is affirmative for line bundles on $\IP^1$; see . For curves of genus $1$, we have an affirmative answer for arbitrary objects of the derived category by . For curves of genus $g(X)\ge 2$, we have an affirmative answer for semi-stable vector bundles by [@Biswas-Nagaraj--reconstructioncurves], and also for a slightly bigger class of vector bundles, namely those where the Harder–Narasimhan factors have slopes contained in a sufficiently small interval; see [@Biswas-Nagaraj--reconstructionsurfaces Prop. 2.1].
The remaining open question is thus:
Do there exist pairs of unstable sheaves $E$ and $F$ on a curve of genus $g\ge 2$ with $E\not \cong F$ but $E^{[n]}\cong F^{[n]}$ for some $n\ge 2$?
General conventions {#general-conventions .unnumbered}
-------------------
All our schemes and varieties are defined over the complex numbers $\IC$. Given two schemes $X$ and $Y$, we write the projections from their product to the factors as $\pr_X\colon X\times Y\to X$ and $\pr_Y\colon X\times Y\to Y$. If $Z\subset X\times Y$ is a subscheme, we denote the restrictions of the projections to $Z$ by $\pr^Z_X$ and $\pr^Z_Y$, respectively.
For $X$ a scheme, let $\D(\QCoh(X))$ be the derived category of quasi-coherent $\reg_X$-modules. We write $\D(X):=\Perf(X)\subset \D(\QCoh(X))$ for the full subcategory of perfect complexes, i.e. those complexes which are locally quasi-isomorphic to bounded complexes of vector bundles. If $X$ is a smooth variety, $\D(X)$ is equivalent to the bounded derived category of coherent sheaves. We use the same notation for derived functors as for their non-derived versions. For example, given a morphism $f\colon X\to Y$, we write $f^*\colon \D(Y)\to \D(X)$ instead of $Lf^*\colon \D(Y)\to \D(X)$.
We say that a functor $F\colon \cC\to \cD$ is *injective on isomorphism classes* if for all pairs of objects $C,C'\in \cC$, we have that $F(C)\cong F(C')$ implies that $C\cong C'$. A *left-inverse* of a given functor $F\colon \cC\to \cD$ is a functor $G\colon \cD\to \cC$ such that $G\circ F\cong \id_{\cC}$ (often in the literature, this is called a *quasi left-inverse* functor as we require the composition only to be isomorphic, not equal, to the identity). A functor admitting a left-inverse is injective on isomorphism classes and faithful.
In this paper, $\IN$ denotes the set of positive integers, and $\IN_0$ denotes the set of non-negative integers.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors thank Ben Anthes, Pieter Belmans, John Christian Ottem and Sönke Rollenske for helpful discussions and comments.
Hilbert schemes of points and Fourier–Mukai transforms
======================================================
Hilbert schemes of points and symmetric quotients
-------------------------------------------------
From now on, $X$ will always be a smooth quasi-projective variety over $\IC$. Given a non-negative integer $n$, the *Hilbert scheme $X^{[n]}$ of $n$ points on $X$* is the fine moduli space of zero-dimensional closed subschemes of $X$ of length $n$. It is smooth if and only if $\dim X\le 2$ or $n\le 3$; see [@Fog; @Cheah--cellularHilb].
We consider the cartesian product $X^n$ together with the action of the symmetric group $\sym_n$ given by permutation of the factors. The quotient $X^{(n)}:=X^n/\sym_n$ by that action is called the *$n$-th symmetric product* of $X$. We denote the quotient morphism by $\pi\colon X^n\to X^{(n)}$ and write the points of the symmetric product in the form $x_1+\dots+x_n:=\pi(x_1,\dots,x_n)$.
There is the *Hilbert–Chow morphism* $$\mu\colon X^{[n]}\to X^{(n)}\quad,\quad [\xi]\mapsto \sum_{x\in \xi}\ell(\reg_{\xi,x})\cdot x$$ sending a length $n$ subscheme to its weighted support. If $X$ is a curve, the Hilbert–Chow morphism is an isomorphism.
Flatness of the universal family
--------------------------------
Being a fine moduli space, the Hilbert scheme $X^{[n]}$ comes equipped with a universal family $\Xi_n=\Xi\subset X^{[n]}\times X$ which is flat and finite of degree $n$ over $X^{[n]}$. In fact, it is also flat over $X$, as we show in the following.
\[prop:flatalgebraic\] For every smooth variety $X$ and every $n\in \IN$, the universal family $\Xi_n\subset X^{[n]}\times X$ is flat over $X$.
By GAGA, a morphism of schemes of finite type over $\IC$ is flat if and only if its analytification is; see [@SGAI Exposé XII, Prop. 3.1]. Note that the analytification $\Xi^{\an}$ of $\Xi$ is the universal family of the *Douady space* of $X^{\an}$, that means the moduli space of zero-dimensional analytic subspaces of $X^{\an}$ of length $n$; see [@Douady]. Hence, we can deduce from the analogous result in the category of complex spaces, which is below.
\[prop:flatanalytic\] For every complex manifold $M$ and every $n\in \IN$, the universal family $\Xi_n\subset M^{[n]}\times M$ of the Douady space $M^{[n]}$ is flat over $M$.
We first prove the assertion in the case that $M=\IC^d$. By generic flatness, there is a non-empty open subset $U\subset M$ such that the restriction of $\Xi_n$ is flat over $U$. Let $(\xi,x)\in \Xi_n\subset M^{[n]}\times M$ with $x\not\in U$. The action of $\Aut(\IC^d)$ is transitive. Hence, there exists an $\phi\in \Aut(\IC^d)$ with $\phi(x)\in U$. Let $\phi^{[n]}$ denote the induced automorphism of $M^{[n]}$. As $\phi^{[n]}\times \phi$ is an automorphism of $\Xi_n$, the flatness of $\reg_{\Xi_n,(\xi,x)}$ over $\reg_{M,x}$ follows from the flatness of $\reg_{\Xi_n,(\phi^{[n]}(\xi),\phi(x))}$ over $\reg_{M,\phi(x)}$.
Let now $M$ be an arbitrary complex manifold of dimension $d$, and let $(\xi,x)\in \Xi_n$. Write $x_0=x$ and $\mu(\xi)=n_0\cdot x+n_1\cdot x_1+\dots +n_tx_t$ where $\mu\colon M^{[n]}\to M^{(n)}$ denotes the Douady–Barlet morphism, which is the analytic analogue of the Hilbert–Chow morphism; see [@Barlet]. Now, choose pairwise disjoint open neighbourhoods $U_0, \dots, U_k$ of $x_0, \dots x_k$ such that every $U_i$ is isomorphic to an open subset of $\IC^d$. Then, $U_0^{[n_0]}\times U_1^{[n_1]}\times \dots \times U_k^{[n_k]}$ is an open neighbourhood of $\xi\in M^{[n]}$. Hence, $\Xi^{U_0}_{n_0}\times U_1^{[n_1]}\times \dots \times U_k^{[n_k]}$, where $\Xi^{U_0}_{n_0}\subset U_0^{[n_0]}\times U_0$ denotes the universal family of $U_0^{[n_0]}$, is an open neighbourhood of $(\xi, x)$ in $\Xi_n$. The restriction of the projection $\pr^{\Xi_n}_X\colon \Xi_n\to X$ to this open neighbourhood is given by the composition $$\Xi^{U_0}_{n_0}\times U_1^{[n_1]}\times \dots \times U_k^{[n_k]}\to \Xi^{U_0}_{n_0}\to U_0\hookrightarrow M\,.$$ The first morphism is the projection to the first factor, hence flat. The second map is the projection from the universal family. As $U_0$ is an open subset of $\IC^d$, for which we already proved the assertion, this morphism is flat too. The third morphism is the open embedding, hence flat. It follows that the whole composition is flat, which is what we needed to show.
Canonical Fourier–Mukai transforms {#subsect:tautological}
----------------------------------
Given a smooth quasi-projective variety $X$ and a positive integer $n$, we define the *tautological functor* $$(\_)^{[n]}:=\pr^{\Xi}_{X^{[n]}*}\circ \pr^{\Xi*}_{X}\colon \D(X)\to \D(X^{[n]})\,.$$ The functor is well-defined in that it preserves perfect complexes. Indeed, every pull-back preserves perfect complexes, and the push-forward $\pr^{\Xi}_{X^{[n]}*}$ preserves perfect complexes as well since $\pr^{\Xi}_X$ is flat and finite. Note that $\pr^{\Xi}_{X^{[n]}*}$ and $\pr^{\Xi*}_{X}$ are both already exact on the level of the abelian categories of coherent sheaves, before deriving, since $\pr^{\Xi}_{X^{[n]}}$ is finite and $\pr^{\Xi}_{X}$ is flat by . Hence, $(\_)^{[n]}$ restricts to an exact functor $$(\_)^{[n]}:=\pr^{\Xi}_{X^{[n]}*}\circ \pr^{\Xi*}_{X}\colon \Coh(X)\to \Coh(X^{[n]})\,;$$ see [@Sca2 Sect. 2.1] for a different proof of this fact in the case that $X$ is a surface. Let $E\in \Coh(X)$ be a vector bundle of rank $r$. The fact that $\pr^{\Xi}_{X^{[n]}}$ is flat and finite of degree $n$ implies that $E^{[n]}$ is a vector bundle of rank $rn$. By the projection formula, the tautological functor can be identified with the Fourier–Mukai transform $$(\_)^{[n]}\cong \FM_{\reg_{\Xi}}=\pr^{X\times X^{[n]}}_{X^{[n]}*}\bigl(\reg_\Xi\otimes \pr^{X\times X^{[n]}*}_{X}(\_)\bigr)$$ along the structure sheaf of the universal family. Another much-studied functor is the Fourier–Mukai transform $$\FF_n:=\FM_{\cI_\Xi}=\pr^{X\times X^{[n]}}_{X^{[n]}*}\bigl(\cI_\Xi\otimes \pr^{X\times X^{[n]}*}_{X}(\_)\bigr)$$ along the ideal sheaf of the universal family. We only consider the functor $\FF_n$ in the case that $X$ is projective since only in this case does it preserve perfect complexes. To see this, we note that the short exact sequence $$0\to \cI_\Xi\to \reg_{X^{[n]}\times X}\to \reg_\Xi\to 0$$ induces, for every perfect complex $E\in \D(X)$, an exact triangle $$\begin{aligned}
\FM_{\cI_\Xi}(E)\to \FM_{\reg_{X^{[n]}\times X}}(E)\to \FM_{\reg_\Xi}(E)\to \FM_{\cI_\Xi}(E)[1] \end{aligned}$$ consisting, a priori, of objects of $\D(\QCoh(X^{[n]}))$. We have already seen that $E^{[n]}\cong \FM_{\reg_\Xi}(E)$ is perfect. Furthermore, we have $\FM_{\reg_{X^{[n]}\times X}}(E)\cong \reg_{X^{[n]}}\otimes_{\IC}\Ho^*(E)$. This is again a perfect complex since we assume $X$ to be projective which implies that $\Ho^*(E)$ is a finite dimensional graded vector space. Since the subcategory $\D(X^{[n]})\subset \D(\QCoh(X^{[n]}))$ of perfect complexes is triangulated, it follows that $\FF_n(E)\cong \FM_{\cI_\Xi}(E)$ is perfect too. Hence, we have a well-defined functor $\FF_n\colon \D(X)\to \D(X^{[n]})$.
\[lem:tautbasechange\] Let $T$ be a scheme, $\cZ\subset T\times X$ a flat family of length $n$ subschemes of $X$, and $\psi\colon T\to X^{[n]}$ the classifying morphism for $\cZ$. Then we have an isomorphism of functors $$\psi^*\circ (\_)^{[n]}\cong\pr^{\cZ}_{T*}\circ \pr^{\cZ*}_X\cong \FM_{\reg_{\cZ}} \,.$$
This follows from base change along this cartesian diagram with flat vertical arrows: $$\begin{gathered}[b]
\xymatrix{
\cZ \ar^{\psi\times \id_X}[r] \ar_{\pr_T^{\cZ}}[d] & \Xi\ar^{\pr^\Xi_{X^{[n]}}}[d] \ar^{\pr^\Xi_X}[r]& X \\
\cZ \ar^{\psi}[r] & X^{[n]} \,. &
}\\[-\dp\strutbox]
\end{gathered}
\qedhere$$
Reconstruction using the symmetric product {#sect:symstack}
==========================================
Reconstruction for reflexive sheaves {#subsect:reflexive}
------------------------------------
The following construction, which reconstructs reflexive sheaves on a smooth quasi-projective variety $X$ of dimension $d\ge 2$ from their associated tautological sheaves on $X^{[n]}$, is inspired by [@Stapletontaut Sect. 1].
Let $X^{[n]}_0\subset X^{[n]}$ be the open locus of reduced subschemes. Let $X^n_0\subset X^n$ the open complement of the big diagonal, and let $X_0^{(n)}=\pi(X_0^n)\subset X^{(n)}$. This means $$\begin{aligned}
X^n_0&=\bigl\{(x_1,\dots,x_n)\in X^n\mid \text{ the $x_i$ are pairwise distinct } \bigr\}\,,\\
X^{(n)}_0&=\bigl\{x_1+\dots+x_n\in X^{(n)}\mid \text{ the $x_i$ are pairwise distinct } \bigr\}\,. \end{aligned}$$ The Hilbert–Chow morphism induces an isomorphism $X_0^{[n]}\xrightarrow \sim X_0^{(n)}$. Let $j\colon X_0^{(n)}\hookrightarrow X^{[n]}$ be the open embedding induced by the inverse of that isomorphism. For $i=1,\dots, n$, let $\pr_i\colon X^n\to X$ denote the projection to the $i$-th factor, and let $\pr_i^0\colon X_0^n\to X$ be the restriction of that projection.
\[lem:openpullback\] We have an isomorphism of functors $$\pi_0^*\circ j^*\circ(\_)^{[n]} \cong \bigoplus_{i=1}^n\pr_i^{0*}\colon \D(X)\to \D(X_0^n)\,.$$
This is stated in [@Stapletontaut Lem. 1.1] in the case that $E$ is a vector bundle and $X$ is a surface. The proof of our more general statement is exactly the same. For convenience, we quickly reproduce the proof in slightly different words.
Note that $X_0^{(n)}$ is the fine moduli space of reduced length $n$ subschemes on $X$, and the quotient $\pi_0\colon X_0^n\to X_0^{(n)}$ is the classifying morphism for the family $Z=\bigsqcup_{i=1}^n\Gamma_{i}\subset X_0^n\times X$ given by the disjoint union of the graphs $\Gamma_i$ of the projections $\pr^0_i\colon X_0^n\to X$. Hence, by , we have $\pi_0^*\circ j^*\circ(\_)^{[n]}\cong \FM_{\reg_Z}$. Since $\reg_Z\cong \bigoplus_{i=1}^n\reg_{\Gamma_{i}}$, and $\FM_{\reg_{\Gamma_{i}}}\cong \pr_i^{0*}$, we get the asserted isomorphism.
\[thm:reflexive\] Let $X$ be a smooth projective variety of dimension $d\ge 2$. Then, for every pair $E,F\in\Refl(X)\subset \Coh(X)$ of reflexive sheaves, we have $$E^{[n]}\cong F^{[n]}\quad \Longrightarrow \quad E\cong F\,.$$
Let $E^{[n]}\cong F^{[n]}$, and write $\alpha\colon X^n_0\hookrightarrow X^n$ for the open embedding. Then, by , $$\begin{aligned}
\label{eq:alphaequal}
\alpha^*(\bigoplus_{i=1}^n\pr_i^*E)\cong \bigoplus_{i=1}^n\pr_i^{0*}E\cong \pi_0^*j^*E^{[n]}\cong \pi_0^*j^*F^{[n]} \cong \bigoplus_{i=1}^n\pr_i^{0*}F\cong \alpha^*(\bigoplus_{i=1}^n\pr_i^*F)\,. \end{aligned}$$ The sheaves $\bigoplus_{i=1}^n\pr_i^*E$ and $\bigoplus_{i=1}^n\pr_i^*F$ are reflexive since flat pull-backs preserve reflexivity; see [@Hartshorne--stable Prop. 1.8]. Note that the codimension of the complement of $X^n_0$ in $X^n$ is $d\ge 2$. Hence, for a reflexive sheaf $\cE$ on $X^n$, we have $\cE\cong \alpha_*\alpha^* \cE$; see [@Hartshorne--generalized Prop. 1.11]. Combining this with gives $$\begin{aligned}
\label{eq:Cequal}
\bigoplus_{i=1}^n\pr_i^*E\cong \alpha_*\alpha^*(\bigoplus_{i=1}^n\pr_i^*E) \cong \alpha_*\alpha^*(\bigoplus_{i=1}^n\pr_i^*F)\cong \bigoplus_{i=1}^n\pr_i^*F\,.\end{aligned}$$ Let $\delta\colon X\hookrightarrow X^n$ be the embedding of the small diagonal. We have $\delta^*\pr_i^*\cong \id$ for every $i=1,\dots,n$. Combining this with gives $$E^{\oplus n}\cong \delta^*(\bigoplus_{i=1}^n\pr_i^*E)\cong \delta^*(\bigoplus_{i=1}^n\pr_i^*F)\cong F^{\oplus n}\,.$$ The category $\Coh(X)$ is Krull-Schmidt; see [@Atiyah--Krull-Schmidt]. Hence, $E^{\oplus n}\cong F^{\oplus n}$ implies $E\cong F$.
Equivariant sheaves and the McKay correspondence
------------------------------------------------
We quickly collect some facts about equivariant sheaves, their derived categories, and functors between them for later use. For further details, the reader may consult, among others, [@BKR Sect. 4], [@Elagin], or [@Krug--remarksMcKay Sect. 2.2]. Let $Y$ be a smooth variety equipped with an action of a finite group $G$. A *$G$-equivariant sheaf* on $Y$ is a pair $(F,\lambda)$ consisting of a coherent sheaf $F\in \Coh(Y)$ and a *$G$-linearisation* $\lambda$, which means a family $\{\lambda_g\colon F\xrightarrow\sim g^*F\}_{g\in G}$ of isomorphisms such that for every pair $g,h\in G$ the following diagram commutes: $$\xymatrix{
F \ar^{\lambda_g}[r] \ar@/_6mm/^{\lambda_{hg}}[rrr] & g^*F \ar^{g^*\lambda_h}[r] & g^*h^*F\ar^{\cong}[r] & (hg)^*F\,.
}$$ We obtain the abelian category $\Coh_G(Y)$ of equivariant sheaves, where a morphism between two equivariant sheaves is a morphism of the underlying coherent sheaves that commutes with the linearisations. We denote the bounded derived category of $\Coh_G(Y)$ by $\D_G(Y)$.
Let $Z$ be a second smooth variety equipped with a $G$-action, and let $f\colon Y\to Z$ be a $G$-equivariant morphism. Then, if $(E,\lambda)\in \Coh(Z)$ is a $G$-equivariant sheaf, the pull-back $f^*E$ canonically inherits a $G$-linearisation induced by $\lambda$. This gives a functor $f^*\colon \Coh_G(Z)\to \Coh_G(Y)$ together with its derived version $f^*\colon \D_G(Z)\to \D_G(Y)$. If $f$ is proper, we also get a push-forward $f_*\colon \D_G(Y)\to \D_G(Z)$.
If $H\le G$ is a subgroup, we have an exact functor $\Res_G^H\colon \Coh_G(Y)\to\Coh_H(Y)$ given by restricting the linearisations to $H$. This functor is exact, so it induces a functor on the derived categories $\Res_G^H\colon \D_G(Y)\to \D_H(Y)$.
If the $G$-action on $Y$ is trivial, a $G$-linearisation of a sheaf $E\in \Coh(Y)$ is the same as a $G$-action on $E$, i.e. a compatible family of $G$-actions on $E(U)$ for every open subset $U\subset Y$. Hence, we can take the invariants of an equivariant sheaf over every open subset, which gives a functor $(\_)^G\colon \Coh_G(Y)\to \Coh(Y)$. Since we are working over $\IC$, this functor is exact and we get an induced functor $(\_)^G\colon \D_G(Y)\to \D(Y)$.
Let now $X^n$ be the $n$-fold cartesian product of a smooth quasi-projective surface $X$, and let $G=\sym_n$ be the symmetric group acting by permutation of the factors. We consider the reduced fibre product $$\begin{aligned}
\xymatrix{
(X^{[n]}\times_{X^{(n)}} X^n)_{\mathsf{red}} \ar^{\quad\quad p}[r] \ar^{q}[d] & X^n\ar^{\pi}[d] \\
X^{[n]} \ar^{\mu}[r] & X^{(n)}\,.
}\end{aligned}$$ In this set-up, the derived McKay correspondence of Bridgeland–King–Reid [@BKR] and Haiman [@Hai] states that the functor $\Phi:=p_*\circ q^*\colon \D(X^{[n]})\to \D_{\sym_n}(X^n)$ is an equivalence. One can deduce quite easily that $$\Psi:=(\_)^{\sym_n}\circ q_*\circ p^*\colon \D_{\sym_n}(X^n)\to \D(X^{[n]})$$ is an equivalence too, but it is not the inverse of $\Phi$; see [@Krug--remarksMcKay Prop. 2.8].
For use in the next subsection, we define a functor $$\CC\colon \Coh(X)\to \Coh_{\sym_n}(X^n)\quad, \quad E\mapsto \CC(E)=(\bigoplus_{i=1}^n\pr_i^*E, \lambda)$$ where $\pr_i\colon X^n\to X$ is the projection to the $i$-th factor, and $\lambda_g$ is, for $g\in \sym_n$, the direct sum of the canonical isomorphisms $\pr_i^*E\xrightarrow\sim g^*\pr_{g(i)}^*E$. As the projections $\pr_i$ are flat, the functor $\CC\colon \Coh(X)\to \Coh_{\sym_n}(X^n)$ is exact, hence it induces a functor $\CC\colon \D(X)\to \D_{\sym_n}(X^n)$.
Reconstruction of complexes using the McKay correspondence
----------------------------------------------------------
In this subsection, with the exception of , let $X$ be a smooth quasi-projective surface. In this case, we can strengthen by constructing a left-inverse to the functor $(\_)^{[n]}\colon \D(X)\to\D(X^{[n]})$ on the level of derived categories. The key is the following result, which can be seen as a refinement of .
\[thm:PsiC\] $\Psi^{-1}\circ (\_)^{[n]}\cong \CC$.
\[thm:Psiinverse\] The functor $G=(\_)^{\sym_n}\circ \delta^*\circ \Psi^{-1}\colon \D(X^{[n]})\to \D(X)$ is left inverse to $(\_)^{[n]}$.
By , we have $G\circ (\_)^{[n]}\cong (\_)^{\sym_n}\circ \delta^*\circ\CC$. The only factor of the composition $(\_)^{\sym_n}\circ \delta^*\circ\CC$ that is not already exact on the level of (equivariant) coherent sheaves is $\delta^*$. Hence, for a given $E\in \D(X)$, we can compute $(\delta^*\CC(E))^{\sym_n}$ by replacing $E$ by a resolution by vector bundles and then applying the non-derived functors. Accordingly, it suffices to prove the isomorphism of functors $(\_)^{\sym_n}\circ \delta^*\circ\CC\cong \id$ on the category $\VB(X)$ of vector bundles on $X$. For every $E\in \VB(X)$, since $\pr_i\circ \delta=\id_X$, we have $\delta^*\CC(E)\cong E^{\oplus n}$ with the $\sym_n$-action on a local section $s=(s_1,\dots s_n)\in E^{\oplus n}(U)$ given by $g\cdot s =(s_{g^{-1}(1)}, \dots, s_{g^{-1}(n)})$. Hence, $s$ is $\sym_n$-invariant if and only if it is of the form $s=(t,\dots, t)$ for some $t\in E(U)$. It follows that the projection $E^{\oplus n}\to E$ to any factor induces a functorial isomorphism $(\delta^*\CC(E))^{\sym_n}\cong E$.
\[rem:reflinverse\] Using $\sym_n$-equivariant sheaves, we can also slightly strengthen by constructing, for $X$ of arbitrary dimension, a concrete left-inverse functor of the restriction $(\_)^{[n]}\colon \Refl(X)\to \Coh(X^{[n]})$ of the tautological functor to the category of reflexive sheaves. First, we note that the pull-back of a coherent sheaf along the quotient $\pi_0\colon X_0^n\to X_0^{(n)}$ is naturally equipped with a $\sym_n$-linearisation given by the canonical isomorphisms $\pi_0^*\xrightarrow\sim g^*\circ \pi_0^*$. Hence, we can regard $\pi_0^*$ as a functor $\Coh(X_0^{(n)})\to \Coh_{\sym_n}(X_0^n)$ instead of a functor $\Coh(X_0^{(n)})\to \Coh(X_0^n)$. Doing this, we can see that, instead of the statement of , we get the isomorphism of functors $\pi_0^*\circ j^*(\_)^{[n]}\cong \alpha^*\circ\CC\colon \D(X)\to \D_{\sym_n}(X^n_0)$. Now, setting $$H:=(\_)^{\sym_n}\circ \delta^*\circ \alpha_*\circ \pi_0^*\circ i^*\colon \Coh(X^{[n]})\to \Coh(X)\,,$$ and combining the proofs of and , we get $H\circ (\_)^{[n]}\cong \id$ as an isomorphism of endofunctors of $\Refl(X)$.
Recall that, as explained in , we have for $E\in \D(X)$ a natural exact triangle $$\begin{aligned}
\label{eq:FFF}
E^{[n]}[-1] \to \FF_n(E)\to \reg_{X^{[n]}}\otimes \Ho^*(E)\to E^{[n]}\,. \end{aligned}$$ Hence, if $G(\reg_{X^{[n]}})=0$, we would have an isomorphism of functors $G\circ (\_)^{[n]}[-1]\cong G\circ \FF_n$, and accordingly $\FF_n$ also had a left-inverse, namely $G[1]$. However, this is not the case. One can show easily that $$\begin{aligned}
\label{eq:PsiO}
\Psi^{-1}(\reg_{X^{[n]}})\cong \reg_{X^n} \end{aligned}$$ where $\reg_{X^n}$ is equipped with the canonical $\sym_n$-linearisation given by the push-forwards of functions $\reg_{X^n}\xrightarrow\sim g^*\reg_{X^n}$; see [@Krug--remarksMcKay Rem. 3.10]. It follows that $G(\reg_{X^{[n]}})\cong \reg_X$.
In the following, we will adapt the functor $G\colon \D(X^{[n]})\to \D(X)$ in such a way that it annihilates $\reg_{X^{[n]}}$ while still being left-inverse to $(\_)^{[n]}$. This new functor will then (up to shift) also be a left inverse of $\FF_n$.
Let $\sym_k$ act on some variety $Y$. We denote by $\MM_{\alt_k}\colon \Coh_{\sym_k}(Y)\to \Coh_{\sym_k}(Y)$ the tensor product with the sign character, which means that $\MM_{\alt_k}(E, \lambda)=(E,\bar\lambda)$ with $\bar\lambda_g=\sgn(g)\cdot \lambda_g$. This functor is exact, hence induces a functor $\MM_{\alt_k}\colon \D_{\sym_k}(Y)\to \D_{\sym_k}(Y)$.
\[thm:Psiinversecomplicated\] The functor $I=(\_)^{\sym_2}\circ \MM_{\alt_2}\circ \Res_{\sym_n}^{\sym_2}\circ \delta^*\circ \Psi^{-1}\colon \D(X^{[n]})\to \D(X)$ is left-inverse to $(\_)^{[n]}$, and $I[1]$ is left inverse to $\FF_n$.
By , we have $I\circ (\_)^{[n]}\cong (\_)^{\sym_2}\circ \MM_{\alt_2}\circ \Res_{\sym_n}^{\sym_2}\circ \delta^*\circ\CC$. Hence, exactly as in the proof of , we can reduce the assertion $I\circ (\_)^{[n]}\cong \id_{\D(X)}$ to the construction of an isomorphism $(\_)^{\sym_2}\circ \MM_{\alt_2}\circ \Res_{\sym_n}^{\sym_2}\circ \delta^*\circ\CC\cong \id$ of endofunctors of $\VB(X)$.
For $E\in \VB(X)$, we have $\MM_{\alt_2}\Res_{\sym_n}^{\sym_2}\delta^*\CC(E)\cong E^{\oplus n}$ with the $\sym_2$-action on a local section $s=(s_1,\dots s_n)\in E^{\oplus n}(U)$ given by $(1\,\,2)\cdot s =(-s_2,-s_1,-s_3, \dots, -s_{n})$. Hence, $s$ is $\sym_2$-invariant if and only if it is of the form $s=(t,-t, 0,\dots, 0)$ for some $t\in E(U)$. It follows that the projection $E^{\oplus n}\to E$ to any of the first two factors induces a functorial isomorphism $\bigr(\MM_{\alt_2}\Res_{\sym_n}^{\sym_2}\delta^*\CC(E)\bigl)^{\sym_2}\cong E$.
Following the discussion above, for the second assertion it suffices to show $I(\reg_{X^{[n]}})\cong 0$. By , we only need to check that $\bigl(\MM_{\alt_2}\Res_{\sym_n}^{\sym_2}\delta^*(\reg_{X^{n}})\bigr)^{\sym_2}=0$. This is the case since the $\sym_2$-action on $\MM_{\alt_2}\Res_{\sym_n}^{\sym_2}\delta^*(\reg_{X^{n}})\cong \reg_X$ is given by multiplication by $\sgn$.
Reconstruction using a small stratum of punctual subschemes {#sect:smallstratum}
===========================================================
Jet bundles
-----------
Let $\Delta\subset X\times X$ be the diagonal and $\cI_\Delta$ its ideal sheaf. For $m\in \IN$, we write the subscheme defined by $\cI_\Delta^m$ as $m\Delta\subset X\times X$. For $E\in \D(X)$ and $m\in \IN_0$, the associated *$m$-jet object* is defined as $\Jet^m E=\FM_{\reg_{(m+1)\Delta}}(E)$. In particular, we have $\Jet^0 E\cong E$. For $m>0$, there is a short exact sequence $$\begin{aligned}
\label{eq:sesdiag}
0\to \delta_*(\Sym^{m}\Omega_X)\to \reg_{(m+1)\Delta}\to \reg_{m\Delta}\to 0\,.\end{aligned}$$ which induces the exact triangle $$\begin{aligned}
\label{eq:jettriangle}
\Sym^{m}\Omega_X\otimes E\to \Jet^{m}E\to \Jet^{m-1}E\to \Sym^{m}\Omega_X\otimes E[1]\,. \end{aligned}$$ It follows inductively that, if $E\in \VB(X)$ is a vector bundle of rank $r$, the associated $m$-jet object $\Jet^{m} E$ is a vector bundle of rank $\binom{m+d}d r$ where $d=\dim X$. Note that $\Jet^m\reg_X$ has fibres $(\Jet^m\reg_X)(x)=\reg_{X,x}/\fm_x^{m+1}$.
The locus of punctual subschemes with Hilbert function concentrated in minimal degrees {#subsec:family}
--------------------------------------------------------------------------------------
For $0\le \ell\le {\binom{m+d-1}{d-1}}=\rank(\Sym^{m}\Omega_X)$, there is a family of punctual length $n:=\binom{m+d-1}d + \ell$ subschemes of $X$ over $\IG:=\Gr(\Sym^{m}\Omega_X, \ell)$, the Grassmannian of rank $\ell$ quotients of the symmetric product of the cotangent bundle, constructed as follows. Let $f\colon\IG\to X$ be the natural projection, and let $\eps\colon f^*\Sym^{m}\Omega_X\to \cQ$ be the universal quotient bundle. Using base change along the cartesian diagram $$\begin{aligned}
\xymatrix{
\IG \ar@^{(->}^{(\id_{\IG},f)\quad}[r] \ar^{f}[d] & \IG\times X\ar^{f\times \id_X}[d] \\
X \ar@^{(->}^{\delta\quad}[r] & X\times X\,,
}\end{aligned}$$ we see that the pull-back of the short exact sequence along $f\times \id_X$ is of the form $$\begin{aligned}
\label{eq:sesgraphf}
0\to (\id_\IG, f)_* f^*\Sym^{m}\Omega_X\to (f\times \id_X)^*\reg_{(m+1)\Delta}\to (f\times \id_X)^*\reg_{m\Delta}\to 0\,.\end{aligned}$$ Modding out the first two terms of by the kernel of the push-forward $$(\id, f)_*\eps\colon (\id_\IG, f)_* f^*\Sym^{m}\Omega_X\to (\id_\IG,f)_*\cQ$$ of the universal quotient, we get a short exact sequence $$\begin{aligned}
\label{eq:cF}
0\to (\id_\IG, f)_*\cQ \to \cF\to (f\times \id_X)^*\reg_{m\Delta}\to 0\end{aligned}$$ where $\cF$ is a quotient of $(f\times \id_X)^*\reg_{(m+1)\Delta}$ hence, in particular, of $(f\times \id_X)^*\reg_{X\times X}\cong \reg_{\IG\times X}$. Thus, $\cF\cong \reg_\cZ$ for some closed subscheme $\cZ\subset \IG\times X$ supported on $\Gamma_f=(f\times\pr_X)^{-1}\Delta$. Note that $\pr_{\IG*}(\id_\IG, f)_*\cQ\cong \cQ$, and by flat base change $\pr_{\IG*}(f\times \id_X)^*\reg_{m\Delta}\cong f^*(\Jet^{m-1}\reg_X)$. Hence, by , $\pr_{\IG*}\reg_\cZ$ is locally free of rank $$\rank(\pr_{\IG*}\reg_\cZ)=\rank \cQ + \rank (\Jet^{m-1}\reg_X)= \ell + \binom{m+d-1}d =n \,,$$ which means that $\cZ$ is a family of length $n$ subschemes of $X$, flat over $\IG$. We denote the classifying morphism for $\cZ\subset \IG\times X$ by $\iota\colon \IG\to X^{[n]}$.
Let us note some facts about $\iota\colon \IG\to X$, though they are not logically necessary for the proofs in the following subsection. The classifying morphism $\iota\colon \IG\to X^{[n]}$ is a closed embedding; compare [@Goettschebook Sect. 2.1]. Its image is exactly the locus of length $n$ subschemes $\xi\subset X$ with $\supp\xi =\{x\}$ for some point $x\in X$, which means that they are *punctual*, satisfying $\Spec(\reg_{X,x}/\fm_x^m)\subset \xi\subset \Spec(\reg_{X,x}/\fm_x^{m+1})$. The last property is equivalent to the Hilbert function of $\xi$ being $(1,d, \binom{d+1}{d-1},\dots, \binom{d+m-2}{d-1},\ell, 0,0,\dots)$. For more information on the strata of $X^{[n]}$ parametrising punctual subschemes with given Hilbert functions, see [@Goettschebook Sect. 2.1] and [@Iarrobino--memoirs].
A left-inverse functor using restriction to the Grassmannian bundle
-------------------------------------------------------------------
In this subsection, we construct a left-inverse of $(\_)^{[n]}$ and $\FF_n$ whenever $n$ is not of the form $n=\binom{u+d}d$ for any $u\in \IN_0$. Note that, for $d=1$, every positive integer is of the form $\binom{u+d}d$. This means that we do not get left-inverses in the case that $X$ is a curve by our method; see for some further details on the curve case.
\[thm:Gres\] Assume that $n\in \IN$ is not of the form $n=\binom{u+d}k$ for any $u\in \IN_0$. Set $m-1:=\max\{u\mid \binom{u+d}d<n\}$ and $\ell:=n-\binom{m+d-1}d$. Let $f\colon \IG=\Gr(\Sym^{m}\Omega_X,\ell)\to X$ be the Grassmannian bundle together with the family $\cZ\subset \IG\times X$ of length $n$ subschemes constructed in , and let $\iota\colon \IG\hookrightarrow X^{[n]}$ be the classifying morphism for $\cZ$. Then $$K:=f_*\bigl(\cQ^\vee\otimes\iota^*(\_)\bigr)\colon \D(X^{[n]})\to \D(X)\,,$$ where $f^*\Sym^{m}\Omega_X\to \cQ$ is the universal quotient bundle, is left-inverse to the functor $(\_)^{[n]}$. Also, $K[1]$ is left-inverse to $\FF_n$.
For the proof, we need the following
\[lem:Grasssod\] Let $Y$ be a smooth variety, $V\in \VB(Y)$, and $0<\ell<\rank V$. Let $f\colon \Gr(V,\ell)\to Y$ be the Grassmannian bundle of rank $\ell$ quotients of $V$ with universal quotient bundle $\cQ$. Then we have the following isomorphisms of functors:
1. $f_*\bigl(\cQ^\vee\otimes f^*(\_)\bigr)\cong0$,
2. $f_*\bigl(\sHom(\cQ,\cQ)\otimes f^*(\_)\bigr)\cong\id_{\D(Y)}$,
3. $f_*\circ f^*\cong \id_{\D(Y)}$.
By [@Samokhin--hom Thm. 2], Kapranov’s full exceptional collection on a Grassmannian over a point [@Kapranov--homogeneous Sect. 3] has a relative version in the form of a semi-orthogonal decomposition of $\D(\Gr(V,\ell))$, with all of its factors being of the form $\sfS^\alpha(\cQ) \otimes f^*\D(X)$ for some Schur functors $\sfS^\alpha$; for an overview of the theory of semi-orthogonal decompositions see, for example, [@Kuz--ICM]. In particular, two of the factors of the semi-orthogonal decomposition are $f^*\D(X)$ and $\cQ \otimes f^*\D(X)$, which means that the functors $f^*\colon \D(X)\to \D(\Gr(V,\ell))$ and $\cQ\otimes f^*(\_)\colon \D(X)\to \D(\Gr(V,\ell))$ are both fully faithful. Their right adjoints are given by $f_*$ and $f_*(\cQ^\vee\otimes \_)$, respectively. The composition of a fully faithful functor with its right adjoint is the identity, which gives (ii) and (iii).
For (i), we note that the factor $f^*\D(X)$ stands in our semi-orthogonal decomposition on the left of $\cQ\otimes f^*\D(X)$, which means that $\Hom\bigl(\cQ\otimes f^*\D(X), f^*\D(X)\bigr)=0$; compare [@Kapranov--homogeneous Lem. 3.2a)]. Using the right-adjoint $f_*(\cQ^\vee\otimes\_)$ of the functor $\cQ\otimes f^*(\_)$, this Hom-vanishing translates to (i).
Let $E\in \D(X)$. By , we have a natural isomorphism $\iota^*E^{[n]}\cong \FM_{\reg_\cZ}(E)$. The short exact sequence induces the exact triangle $$\begin{aligned}
\label{eq:sometriangle}
\FM_{(\id_\IG,f)_*\cQ}(E)\to \FM_{\reg_\cZ}(E)\to \FM_{(f\times \id_X)^*\reg_{m\Delta}}(E)\to \FM_{(\id_\IG,f)_*\cQ}(E)[1]\,. \end{aligned}$$ By the projection formula, we have $\FM_{(\id_\IG,f)_*\cQ}(E)\cong \cQ\otimes f^*E$. By flat base change, we have $\FM_{(f\times \id_X)^*\reg_{m\Delta}}(E)\cong f^*\FM_{\reg_{m\Delta}}(E)\cong f^*\Jet^{m-1} E$. Hence, can be rewritten as $$\cQ\otimes f^*E\to \iota^*E^{[n]}\to f^*\Jet^{m-1} E\to \cQ\otimes f^*E[1]\,.$$ Applying $f_*(\cQ^\vee \otimes \_)$, we get the exact triangle $$f_*(\sHom(\cQ,\cQ)\otimes f^*E)\to K(E^{[n]})\to f_*(\cQ^\vee\otimes f^*\Jet^{m-1} E)\to \cQ\otimes f_*(\sHom(\cQ,\cQ)\otimes f^*E)[1]\,.$$ By , we see that $f_*(\cQ^\vee\otimes f^*\Jet^{m-1} E)$ vanishes, and we get a natural isomorphism $K(E^{[n]})\cong f_*(\sHom(\cQ,\cQ)\otimes f^*E)\cong E$.
Now, in order to prove that $K\circ \FF_n\cong [-1]$, because of , we only need to check that $K(\reg_{X^{[n]}})=0$. As $\iota^*\reg_{X^{[n]}}\cong \reg_\IG\cong f^*\reg_X$, this follows directly from (i).
In the special case $d=2=n$, the functor $K$ of coincides with the functor $I$ of . To see this, note that, in this special case, the subvariety $\iota(\IG)$ agrees with the whole boundary divisor of $X^{[2]}$, i.e. the locus of non-reduced length 2 schemes, and $f\colon \IG\to X$ is a $\IP^1$-bundle, which implies that $\cQ\cong \reg_f(1)$. Now, [@Kru3 Prop. 4.2] says that $\Phi^{-1}\circ \delta_*\cong \iota_*\bigl(\reg_f(-2)\otimes f^*(\_)\bigr)[1]$. Combining this with [@KPScyclic Thm. 4.26(i)], we get $\Psi\circ \MM_{\alt_2}\circ \delta_*\cong \iota_*\bigl(\reg_f(-1)\otimes f^*(\_)\bigr)[1]$. Taking the left-adjoints on both sides of this isomorphism gives $I\cong K$.
\[rem:jetreconstruction\] If $n=\binom{m+d}d$, we have $\IG=X$ and $\iota^*E^{[n]}\cong \Jet^m E$. So in this case, our construction only recovers the $m$-jet object $\Jet^m E$, but not $E$ itself. In the case that $X$ is a curve and $E$ is a vector bundle, the $m$-th jet bundle $\Jet^n E$ is recovered in [@Biswas-Nagaraj--reconstructioncurves] by a slightly different construction from $E^{[n]}$. The authors of *loc. cit.* proceed to reconstruct $E$ from $\Jet^n E$ if $g(X)\ge 2$ and $E$ is semi-stable, using the Harder–Narasimhan filtration. Let us remark that, if $X$ is a curve and $L\in \Pic X$ is a line bundle, the exact triangles yield the formula $$\det(\Jet^nL)\cong L^{\otimes n+1}\otimes \omega_X^{\otimes \binom{n+1}2}\,.$$ Hence, if $X=\IP^1$, one can always recover a line bundle $L\in \Pic(\IP^1)$ from its $n$-jet bundle $\Jet^n(L)$ which in turn can be recovered from $L^{[n]}$. Another way to reconstruct a line bundle $L\in \Pic( \IP^1)$ from $L^{[n]}\in \VB(X^{[n]})$ is to use formulae for the characteristic classes of $L^{[n]}$; see [@Mattuck--sym; @Wang--tautintegrals].
Using a nested Hilbert scheme {#sect:nested}
-----------------------------
We now turn to the case where $n = \binom{d+m}{m}$ for some $m$.
\[lem:xi\] Let $\xi\subset X$ be a subscheme concentrated in one point $x\in X$ with $\fm_x^{m+1}\subset \cI_\xi\subset \fm^m_x$ and $\ell(\xi)=\dim_{\IC}(\reg_{X,x}/\fm_x^{m+1})-1=\binom{d+m}d -1$. Let $\cI_{\xi}(x)=\cI_{\xi}/\fm_x$ be the fibre in $x$ of the ideal sheaf of $\xi$. Then $\dim_\IC \cI_{\xi}(x)=1+\binom{d+m}{d-1}-d$.
We can assume for simplicity that $X=\IA^d$ is the affine space and $x=0$ is the origin. We write $\cI:=\cI_\xi$ and $\fm:=\fm_x=(x_1,\dots, x_d)$, which are ideals in $\IC[x_1,\dots, x_d]$, and set $V:=\cI(x)=\cI/\fm\cdot \cI$. By assumption, we have $I=\fm^{m+1}+(h)$ for some $h\in \IC[x_1,\dots, x_d]$ homogenous of degree $m$. Hence, $\fm\cdot I=\fm^{m+2}+\fm\cdot (h)$. We consider the subspace $U:=\fm^{m+1}/\fm\cdot I$ of $V$. Note that the $\IC$-linear map $$\fm/\fm^2\to \fm^{m+1}/\fm^{m+2}\quad,\quad \bar x_i\mapsto \overline{x_i\cdot h}$$ is injective. Hence, $\fm\cdot (h)/\fm^{m+2}$ is a $d$-dimensional subspace of the $\binom{d+m}{d-1}$-dimensional vector space $\fm^{m+1}/\fm^{m+2}$. Since $U=\bigl(\fm^{m+1}/\fm^{m+2}\bigr)/\bigl(\fm\cdot (h)/\fm^{m+2}\bigr)$, we get $\dim U= \binom{d+m}{d-1} -d$. Furthermore, the quotient $V/U$ is one-dimensional, spanned by $\bar h$. In summary, $$\dim V=\dim (U/V) +\dim U= 1+\binom{d+m}{d-1}-d\,.\qedhere$$
We now consider the set-up of with $\ell=\rank(\Sym^m\Omega_X)-1=\binom{m+d-1}{d-1}$. This means that $f\colon \IG\to X$ is actually a $\IP$-bundle, and the family $\cZ$, that we constructed in parametrises punctual subschemes of length $\binom{m+d}d-1$ with the property of $\xi$ of .
The non-derived pull-back $\cE:=(\id_{\IG},f)^*\cI_{\cZ}$ is a vector bundle of rank $1+\binom{d+m}{d-1}-d$ on $\IG$.
It is sufficient to prove that, for every closed point $t\in \IG$, the fibre $\cE(t)$ is of dimension $1+\binom{d+m}{d-1}-d$; see e.g. [@HarAGbook Ex. 5.8]. Let $\xi:=\cZ_t\subset X$ be the fibre of the family $\cZ\subset \IG\times X$ over $t$. Then $\xi$ is a subscheme concentrated in $x:=f(t)$ with the same properties as $\xi$ in . Considering the commutative diagram of closed embeddings $$\xymatrix{
\{t\}\ar^{t\,\mapsto (t,x)\quad}[r]\ar[d] & \{t\}\times X\ar[d]\\
\IG\ar[r] & \IG\times X\,,
}$$ we see that $\cE(t)\cong \cI_{\cZ\mid \{t\}\times X}(t,x)\cong \cI_{\xi}(x)$, where the last isomorphism uses the flatness of $\cZ$ over $\IG$. Hence, the result follows from .
We now consider the $\IP$-bundle $p\colon Y:=\IP(\cE)\to \IG$ and set $g:=f\circ p\colon Y\to X$. There is the commutative diagram with cartesian squares $$\begin{aligned}
\xymatrix{
Y \ar@^{(->}^{(\id_{Y},g)\quad}[r] \ar^{p}[d] \ar@/_6mm/_{g}[dd] & Y\times X\ar^{p\times \id_X}[d] \ar@/^4mm/^{\pr_X}[drr] & \\
\IG \ar@^{(->}^{(\id_{\IG},f)\quad}[r] \ar^{f}[d] & \IG\times X\ar^{f\times \id_X}[d] \ar^{\pr_X}[rr] & & X \\
X \ar@^{(->}^{\delta\quad}[r] & X\times X \ar@/_4mm/_{\pr_X}[rru] & &.
}\end{aligned}$$ Let $\alpha\colon p^*\cE\to \reg_p(1)$ be the universal rank 1 quotient. We set $\cJ:=(p\times \id_X)^*\cI_\cZ$, which, by flatness of $p\times \id_X$, is an ideal sheaf on $Y\times X$. Furthermore, we consider the unit of adjunction $$\eta\colon \cJ\to (\id_Y,g)_*(\id_Y,g)^*\cJ\cong (\id_Y,g)_*p^*(\id_{\IG}, f)^*\cI_{\cZ}\cong (\id_Y,g)_*p^* \cE$$ and the composition $\beta:=(\id_Y,g)_*\alpha\circ \eta\colon \cJ\to (\id_Y,g)_*\reg_p(1)$, which is again surjective. We set $\cJ':=\ker(\beta)$ and denote the subscheme corresponding to this ideal sheaf by $\cZ'\subset Y\times X$. Applying the snake lemma to the diagram $$\begin{aligned}
\xymatrix{
& 0\ar[d] &0\ar[d] & & \\
0\ar[r] & \cJ'\ar[d] \ar[r] & \reg_{Y\times X}\ar^{\id}[d] \ar[r] & \reg_{\cZ'}\ar[d] \ar[r] & 0 \\
0\ar[r] & \cJ\ar^\beta[d]\ar[r] & \reg_{Y\times X} \ar[r]\ar[d] & (p\times \id_X)^*\reg_{\cZ}\ar[d] \ar[r] & 0 \\
& (\id_Y,g)_*\reg_p(1)\ar[d] &0 &0 & \\
& 0 & & &
}\end{aligned}$$ yields a short exact sequence $$\begin{aligned}
\label{eq:Z'}
0\to (\id_Y,g)_*\reg_p(1)\to \reg_{\cZ'}\to (p\times \id_X)^*\reg_\cZ\to 0\,.\end{aligned}$$ It follows that $\cZ'$ is flat of degree $\deg(\cZ)+1=\binom{m+d}d$ over $Y$. We set $n:=\binom{m+d}d$ and denote the classifying morphism for $\cZ'$ by $\nu\colon Y\to X^{[n]}$. The morphism $(p, \nu) \colon Y \to \mathbb G \times X^{[n]}$ embeds $Y$ as the subscheme of $\mathbb G \times X^{[n]}$ whose points are pairs $$\{(\xi, \xi') \mid \xi \in \mathbb G, \xi \in X^{[n]}, \xi \subset \xi'\}.$$
\[thm:Pres\] For $d\ge 2$ and $n=\binom {m+d}d$ for some $m\in \IN$, the functor $$N:=g_*\bigl(\reg_p(-1)\otimes \nu^*(\_)\bigr)\colon \D(X^{[n]})\to \D(X)$$ is left-inverse to $(\_)^{[n]}$. Also, $N[1]$ is left-inverse to $\FF_n$.
The proof is very similar to that of . Let $E\in \D(X)$. By , we have a natural isomorphism $\nu^*E^{[n]}\cong \FM_{\reg_{\cZ'}}(E)$. The short exact sequence induces the exact triangle $$\begin{aligned}
\label{eq:sometriangle}
\FM_{(\id_Y,g)_*\reg_p(1)}(E)\to \FM_{\reg_{\cZ'}}(E)\to \FM_{(p\times \id_X)^*\reg_{\cZ}}(E)\to \FM_{(\id_Y,g)_*\reg_p(1)}(E)[1]\,. \end{aligned}$$ By projection formula, we have $\FM_{(\id_Y,g)_*\reg_p(1)}(E)\cong \reg_p(1)\otimes g^*E$. By flat base change, we have $\FM_{(p\times \id_X)^*\reg_{\cZ}}(E)\cong p^*\FM_{\reg_{\cZ}}(E)$. Hence, can be rewritten as $$\reg_p(1)\otimes g^*E\to \nu^*E^{[n]}\to p^*\FM_{\reg_{\cZ}}(E)\to \reg_p(1)\otimes g^*E[1]\,.$$ Applying $p_*(\reg_p(-1) \otimes \_)$, we get the exact triangle $$p_*p^*f^*E\to p_*(\reg_p(-1) \otimes \nu^*E^{[n]})\to p_*\bigl(\reg_p(-1) \otimes p^*\FM_{\reg_{\cZ}}(E)\bigr)\to \cQ\otimes p_*p^*f^*E[1]\,.$$ By , we see that $p_*\bigl(\reg_p(-1) \otimes p^*\FM_{\reg_{\cZ}}(E)\bigr)$ vanishes, and we get a natural isomorphism $p_*(\reg_p(-1) \otimes \nu^*E^{[n]})\cong p_*p^*f^*E \cong f^*E$. Note that the assumption $d\ge 2$ is needed for the above vanishing since, for $d=1$, we would have $\rank(\cE)=1$ so that $p\colon \IP(\cE)\to \IG$ is an isomorphism. Applying $f_*$, we get a natural isomorphism $$N(E^{[n]})\cong f_*p_*(\reg_p(-1) \otimes \nu^*E^{[n]})\cong f_*f^*E\cong E\,,$$ which means that we have an isomorphism of functors $N\circ (\_)^{[n]}\cong \id$.
Now, in order to prove that $N\circ \FF_n\cong [-1]$, because of , we only need to check that $N(\reg_{X^{[n]}})=0$. As $\nu^*\reg_{X^{[n]}}\cong \reg_Y\cong p^*\reg_{\IG}$, this follows directly from (i).
\[thm:lefti\] For $d\ge 2$, the functors $(\_)^{[n]}\colon \D(X)\to \D(X^{[n]})$ and $\FF_n\colon \D(X)\to \D(X^{[n]})$ both have a left-inverse for every $n\in \IN$.
If $n=\binom{m+d}d$ for some $m\in \IN$, we get a left-inverse by . If $n$ is not of this form, we get a left-inverse by .
Reconstruction using fixed-point free automorphisms {#sect:freeautos}
===================================================
Multigraphs as families of points {#subsect:multigraphs}
---------------------------------
Let $\phi_1,\dots,\phi_n\in \Aut(X)$ be automorphisms with *empty pairwise equalisers*, which means that $\phi_i(x)\neq \phi_j(x)$ for every $i\neq j$ and every $x\in X$. Then the graphs $\Gamma_{\phi_i}\subset X\times X$ are pairwise disjoint and $$\Gamma:=\bigsqcup_{i=1}^n\Gamma_{\phi_i}\subset X\times X$$ is a family of reduced length $n$ subschemes over $X$. We denote its classifying morphism by $$\psi:=\psi_{\{\phi_1,\dots,\phi_n\}}\colon X\to X^{[n]}\,.$$ This morphism maps to the open part $X^{[n]}_0$ of reduced subschemes, which is naturally isomorphic to the open part $X^{(n)}_0$ of the symmetric product. Hence, we can equivalently describe $\psi$ as the morphism $X\to X^{(n)}_0$, $x\mapsto \phi_1(x)+\dots+\phi_n(x)$. Note that $\psi$ is always finite but, in general, not a closed embedding. For example, if $n=2$, $\phi_1=\id$, and $\phi_2=\iota$ is a fixed-point free involution, we have $\psi(x)=\psi(\iota(x))$ for all $x\in X$. In that case, $\psi$ factorises over a closed embedding $X/\iota\hookrightarrow X^{[2]}$ of the quotient.
\[lem:psipullback\] We have an isomorphism of functors $\psi^*\circ (\_)^{[n]}\cong \phi_1^*\oplus\phi_2^*\oplus\dots\oplus \phi_n^*$.
By , we have $\psi^*\circ (\_)^{[n]}\cong \FM_{\reg_\Gamma}$. Now, the assertion follows from the facts that $\reg_{\Gamma}\cong \reg_{\Gamma_{\phi_1}}\oplus \reg_{\Gamma_{\phi_2}}\oplus\dots\oplus \reg_{\Gamma_{\phi_n}}$ and $\FM_{\reg_{\Gamma_{\phi_i}}}\cong\phi_i^*$.
Reconstruction using multigraphs
--------------------------------
\[thm:fixedfree\] Let $X$ be a smooth projective variety such that there exist a set of $n+1$ automorphisms $\{\phi_0,\phi_1,\dots,\phi_n\}\subset \Aut(X)$ with empty pairwise equalisers. Then the tautological functor $(\_)^{[n]}\colon \D(X)\to \D(X^{[n]})$ is injective on isomorphism classes and faithful.
\[cor:abelian\] Let $A$ be an abelian variety. Then the functor $(\_)^{[n]}\colon \D(A)\to \D(A^{[n]})$ is injective on isomorphism classes and faithful for every $n\in \IN$.
There is an infinite subgroup of $\Aut(A)$ whose elements have empty pairwise equalisers, namely the subgroup of translations.
Note that also applies to elliptic curves, while all the other reconstruction results presented in this paper require $\dim X$ to be at least 2.
Replacing $\phi_i$ by $\phi_0^{-1}\circ \phi_i$, we may assume without loss of generality that $\phi_0=\id$. For $j=0,\dots, n$, let $\psi_j:=\psi_{\{\phi_i\mid i\neq j\}}\colon X\to X^{[n]}$ be the classifying morphism for $\Gamma=\sqcup_{i=0,\dots, n\,,\, i\neq j} \Gamma_{\phi_i}\subset X\times X$, as discussed in .
By , the composition $\psi_j\circ (\_)^{[n]}$, for any $j=0,\dots,n$, is faithful. This implies the faithfulness of $(\_)^{[n]}$.
Now, let $E,F\in \D(X)$ with $E^{[n]}\cong F^{[n]}$. Then, by , we have $$\bigoplus_{i\in\{0,\dots,n\}\,,\, i\neq j}\phi_i^*(E)\cong \psi_j^*E^{[n]}\cong \psi_j^*F^{[n]}\cong \bigoplus_{i\in\{0,\dots,n\}\,,\, i\neq j}\phi_i^*(F)$$ for every $j=0,\dots, n$. The category $\D(X)$ is a Krull-Schmidt category; see [@Le-Chen--Karoubian Cor. B]. Hence, $E\cong F$ follows from below.
\[prop:Krull-Schmidt\] Let $\cD$ be a Krull-Schmidt category, let $\Phi_0,\Phi_1,\dots, \Phi_n\le \Aut(\cD)$ be pairwise distinct linear autoequivalences with $\Phi_0=\id_{\cD}$. Then for $E,F\in \cD$, we have: $$\bigoplus_{i\in\{0,\dots,n\}\,,\, i\neq j}\Phi_i(E)\cong \bigoplus_{i\in\{0,\dots,n\}\,,\, i\neq j}\Phi_i(F) \quad \forall\, j=0,\dots,n \quad\Longrightarrow \quad E\cong F\,.$$
For $j=0,\dots, n$, we set $$\IB_j:=\bigoplus_{i\in\{0,\dots,n\}\,,\, i\neq j}\Phi_i(E)\cong \bigoplus_{i\in\{0,\dots,n\}\,,\, i\neq j}\Phi_i(F)\,.$$ We note that the number of indecomposable factors (with multiplicity) of every $\IB_j$ is $n$ times the number of irreducible factors of $E$ as well as $n$ times the number of irreducible factors of $F$. In particular, the number of indecomposable factors of $E$ and of $F$ is the same, and we can argue by induction over that number.
As the base case of the induction we can take the numbers of factors to be zero, in which case $E\cong 0\cong F$. Now assume that $E$ and $F$ both have $k>0$ indecomposable factors. By the cancellation property in Krull-Schmidt categories, we have $$\begin{aligned}
\label{eq:congcondition}
\IB_0\cong \IB_j\quad\Longleftrightarrow \quad E\cong \Phi_j(E) \text{ and } F\cong \Phi_j(F)\,.\end{aligned}$$ If $\IB_0\not \cong \IB_\ell$ for some $\ell\in\{1,\dots, n\}$, we pick some indecomposable object $B$ that occurs in $\IB_\ell$ with a bigger multiplicity than in $\IB_0$. This $B$ must be an indecomposable factor of $E$ and of $F$. We write $E\cong E' \oplus B$ and $F\cong F'\oplus B$. It follows that, for every $j=0,\dots, n$, $$\bigoplus_{i\in\{0,\dots,n\}\,,\, i\neq j}\Phi_i(E')\cong \bigoplus_{i\in\{0,\dots,n\}\,,\, i\neq j}\Phi_i(F')\cong \IB_j' \quad\text{with}\quad \IB_j\cong\IB_j'\oplus \bigoplus_{i\in\{0,\dots,n\}\,,\, i\neq j}\Phi_i(B) \,.$$ Hence, we can apply the induction hypothesis to get $E'\cong F'$, which implies $E\cong F$.
Now, assume that $\IB_0\cong \IB_\ell$ for every $\ell=1,\dots, n$. Then shows that $E\cong \Phi_\ell(E)$ and $F\cong \Phi_\ell(F)$ for every $\ell$. Hence, $E^{\oplus n}\cong \IB_0\cong F^{\oplus n}$. In a Krull-Schmidt category, $E^{\oplus n}\cong F^{\oplus n}$ implies $E\cong F$.
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abstract: 'Measurements from the Solar Irradiance Monitor (SIM) onboard the SORCE mission indicate that solar spectral irradiance at Visible and IR wavelengths varies in counter phase with the solar activity cycle. The sign of these variations is not reproduced by most of the irradiance reconstruction techniques based on variations of surface magnetism employed so far, and it is not clear yet whether SIM calibration procedures need to be improved, or if instead new physical mechanisms must be invoked to explain such variations. We employ three-dimensional magneto hydrodynamic simulations of the solar photosphere to investigate the dependence of solar radiance in SIM Visible and IR spectral ranges on variations of the filling factor of surface magnetic fields. We find that the contribution of magnetic features to solar radiance is strongly dependent on the location on the disk of the features, being negative close to disk center and positive toward the limb. If features are homogeneously distributed over a region around the equator (activity belt) then their contribution to irradiance is positive with respect to the contribution of HD snapshots, but decreases with the increase of their magnetic flux for average magnetic flux larger than 50 G in at least two of the Visible and IR spectral bands monitored by SIM. Under the assumption that the 50 G snapshots are representative of quiet Sun regions we find thus that the Spectral Irradiance can be in counter-phase with the solar magnetic activity cycle.'
author:
- 'S. Criscuoli, H. Uitenbroek'
title: 'Interpretation of Solar Irradiance Monitor measurements through analysis of 3D MHD simulations.'
---
Introduction
============
Solar irradiance, the radiative energy flux the Earth receives from the Sun at its average orbital distance, varies along with magnetic activity, over periods of days to centuries and presumably even on longer time scales. The magnitude of irradiance variations strongly depends on wavelength. The precise measurement of irradiance over the spectrum is becoming more and more compelling, because of the increasing evidence of the effects of these variations on the chemistry of the **Earth’s** atmosphere and terrestrial climate [e.g. @lockwood2012; @ermolli2013 and references therein]. However, absolute measurement of spectral irradiance variations, especially over time scales longer than a few solar rotations, is seriously hampered by difficulties in determining degradation of instrumentation in space. Therefore, calibrations of radiometric measurements have to rely significantly on inter-calibration with other instruments and/or reconstructions through models based on proxies of magnetic activity.
In this context, recent measurements obtained with the Spectral Irradiance Monitor [SIM; @harder2005] radiometers on board the Solar Radiation and Climate Experiment [SORCE; @rottman2005], which show an irradiance signal at Visible and IR spectral bands in *counterphase* with the solar cycle [@harder2009], have been strongly debated. This result was confirmed by @preminger2011, who found that variations in irradiance of solar and solar-like stars in red and blue continuum band-passes is in counter phase with their activity cycle. By contrast, recent results obtained from the analysis of **VIRGO/SOHO** [@frohlich1995] data at visible spectral ranges [@wehrli2013] show signals *in phase* with the magnetic cycle. **Theoretically, most of the irradiance reconstruction techniques which usually reproduce more than $90\%$ of variations of total solar irradiance (i.e., the irradiance integrated over the whole spectrum), produce irradiance variations at SIM Visible and Infrared bands that are in phase with the magnetic cycle [see @ermolli2013 for a review]. The Spectral and Total Irradiance REconstruction models for Satellite Era (SATIRE) produce a signal slightly in counter-phase in the IR [e.g. @ball2011]. The only reconstructions that produce a signal in counter-phase with the magnetic activity cycle on both visible and IR bands are those obtained with the Solar Radiation Physical Modelling (SRPM) tools [@fontenla2012], which, on the other hand, have been criticized for being explicitly constructed to reproduce SIM measurements.**
Given the above controversy it is still an open question whether SIM finding of counter phase spectral variation in visible and IR bands is the result of a problem with internal calibration procedures, or if instead current modeling is not adequate in reproducing irradiance variations at those spectral ranges. In particular the physical cause of long-term variations is still unclear having been attributed alternatively to changes in quiet Sun magnetism that is mostly hidden in full-disk observations [@fontenla2012], or to a change of the temperature gradient in the solar atmosphere, most likely due to an increase of the magnetic filling factor over the cycle [@harder2009].
**Several irradiance reconstruction techniques, such as the SATIRE and the SRPM cited above, the reconstructions of the Astronomical Observatory of Rome [OAR, @ermolli2011], and those obtained with the Code for Solar Irradiance [COSI, @haberreiter2008; @shapiro2010] and with the Solar Modelling in 3D [SOLMOD, @haberreiter2011], are based on one-dimensional static atmosphere models.** Such models can be constructed to reproduce observed spectra very well, but their semi-empirical nature prevents them from being used to explore the underlying physics [@uitenbroek2011]. In this contribution we employ snapshots from 3-D magneto-hydrodynamic (MHD) simulations of the solar photosphere to qualitatively investigate whether an increase of the magnetic filling factor over the solar surface can produce a decrease of the disk-integrated solar radiative emission in the four visible and IR spectral bands monitored by SIM. Since the contribution of features as pores and sunspots is well known to be negative, this study is aimed at investigating the contribution of features like faculae and network.
The paper is organized as follows: in Sect. 2 we describe the MHD snapshots employed and the spectral synthesis performed; in Sect. 3 we present our results and in Sect. 4 we draw our conclusions.
{width="16.5cm"}
Simulations and Spectral Synthesis {#data:simulations}
==================================
For our analysis we consider series of snapshots of 3-D MHD simulations of the solar photosphere calculated by @fabbian2010 [@fabbian2012] using the STAGGER code [@garlsgaard1996]. They are characterized by four different cases of introduced magnetic flux with average vertical field strength values of approximately 0 G, 50 G, 100 G and 200 G. The 0 G case (hydrodynamic, or HD case hereafter) and the 50 G case are representative of quiet Sun regions, while the 100 and 200 G are meant to represent magnetic regions. The horizontal dimensions of each snapshot are 6 Mm $\times$ 6 Mm, with a spatial sampling of 23.8 km resulting in a grid size of 252 points in both horizontal directions. These snapshots were used by Fabbian (2010, 2012) for the first-ever quantitative assessment of the impact of magnetic fields in 3D photospheric models of the Sun on the solar chemical composition. The numerical setup adopted for the calculations is described in the above cited papers. @beck2013 discussed the quality of the simulations and their comparison with observations. **@criscuoli2013 employed this set of simulations to investigate physical and observational differences between quiet and facular regions. Among other results, this study showed that, in agreement with high-spatial-resolution observations, the emergent intensity at disk center in the red continuum of magnetic features characterized by the same size and the same amount of average magnetic flux decreases with the increase of their environmental magnetic flux. That is, small-size magnetic features are brighter in quiet areas with respect to active regions. Here we extend that work to a larger set of wavelengths and lines of sight and investigate the effects of the magnetic flux on the radiative emission as would be observed by moderate spatial resolution data (as those usually employed for irradiance studies).** **At this aim, we considered each of these snapshots to represent a patch of unresolved magnetic field with the corresponding average flux**. At each pixel we calculated with the RH code [@uitenbroek2003] the emergent radiation by solving for LTE radiative transfer at 16 continuum wavelengths, distributed equidistantly over 6 spectral positions in each of the SIM pass bands (400 – 972 nm, 972 – 1630 nm, and 1630 – 2423 nm). **Figure \[images\] illustrates examples of the emergent intensity along the vertical line of sight at 630 nm through snapshots characterized by different amounts of average magnetic flux**. We solved radiative transfer through the snaphots in different directions spanning 9 inclinations, distributed according to the zeroes of the Gauss–Legendre polynomials in $\mu = \cos(\theta)$, where $\theta$ is the angle with the vertical, and 2 azimuths per octant, plus the vertical direction, for a total of 73 directions. We then averaged over the different azimuths and all the spatial positions in the snapshot to calculate the snapshot’s average emergent intensity at all 16 wavelengths as a function of heliocentric angle.
To ensure sufficient statistics and reduce oscillation effects, for each average magnetic flux value we considered 10 snapshots taken 2.5 minutes apart (40 snapshots in total). Figure \[gradient\] shows the differences between the average temperature stratifications of the MHD and the HD atmospheres as function of the optical depth computed at 500 nm. These curves agree with those reported in @fabbian2010 [their figure 1], within the statistical variations due to here considering fewer snapshots from the complete series, and to possible slight differences in the computed optical depth due to employing a different radiative transfer code. **The solid gray area approximately indicates the range in formation heights of the continuum intensity along a vertical line of sight at the considered wavelengths. The dashed gray area shows approximately the formation heigths at the same wavelengths for an inclined line of sight (namely, for $\mu$ = 0.2, where $\mu$ is the cosine of teh heliocentric angle); as expected, the shallower the line of sight, the more the formation heights shift toward higher layers of the atmosphere (smaller optical depth values).** The plot shows that an increase of the average magnetic flux causes an average decrease of the temperature at about optical depth unity and an increase at smaller optical depths, thus making the temperature gradient shallower with the increase of the magnetic flux. **As explained in detail in @criscuoli2013, these variations must be ascribed to the inhibition of convection by the magnetic field, which reduces the amount of energy transported by the plasma from the lower to the higher layers of the atmosphere.** These temperature gradient changes are similar to those invoked by @harder2009 to explain SIM measurements at Visible and IR bands, and are qualitatively similar to the differences between temperatures of quiet and magnetic feature atmosphere models such as employed in SRPM [e.g. @fontenla2012]. We note however, that because of our inclusion of the true three-dimensional structure of the snapshots, the centre-to-limb behavior of our computed intensities is more realistic than that of the one-dimensional models, even though the snapshots possess, on average, similar temperature gradients.
After calculating the spatially and azimuth averaged emergent intensities as a function of $\mu$ for each snapshot we averaged them over the 10 realizations for each average magnetic flux case, and compared the intensities for each of the MHD cases with those of the HD case as reference. Intensity contrast is, therefore, defined as the intensity relative to that of the HD case at the same heliocentric angle: $C = <I_{MHD}>/<I_{HD}> - 1$, where the averages are over spatial dimensions in each snapshot, realization, and azimuth. Finally, we computed the flux by integrating intensities over a range of heliocentric angles.
![Difference between MHD average temperature stratifications and the HD case versus the optical depth $\tau_{500}$. Error bars represent the standard deviation obtained by averaging over the 10 snapshots each. The solid and dashed gray areas indicate the approximate formation heights for the wavelenghts considered for vertical and $\mu$ =0.2 lines of sight, respectively.[]{data-label="gradient"}](Tavg_500nm.eps){width="4.6cm"}
Results
=======
The plot in Fig. \[CLVs\] shows the average intensity contrast of the MHD snapshots as a function of wavelength for different heliocentric angles. We note that the intensity contrast is negative for the 100 G and 200 G cases near disk center ($\mu > 0.8$) both for short wavelengths (below approximately 500 nm for the 100 G snapshots and below approximately 700 nm for the 200 G ones) and wavelengths above 1500 nm, but that the contrast is positive for all angles and wavelengths in the 50 G average field case. This is because at short and long wavelengths intensity emanates from relatively deep layers of the solar atmosphere, at which, as illustrated in Fig. \[gradient\] **(continuous gray area)**, the 100 and 200 G simulations have average temperature lower than the one of the HD case. **For shallower lines of sight ($\mu \leq$ 0.8) the formation heights shift toward higher layers of the atmosphere (dashed gray area in Fig. \[gradient\]), where the difference between the average temperatures of MHD and HD simulations becomes positive.** These results are in agreement with those obtained from observations [e.g. @yeo2013; @ermolli2007; @ortiz2002; @sanchezcuberes2002]. Close to disk center most of the curves exhibit a maximum at about 800 nm. This wavelength dependence of the contrast closely follows the behavior of the $H^{-}$ opacity, which is the prominent source of opacity in the photosphere, and has a maximum near that wavelength. At those wavelengths intensity forms relatively high in the atmosphere, where the MHD snapshots are on average warmer than the HD ones.
[Flux\_cumul\_serena.eps]{}
The derived intensity contrasts would suggest that it is in principle possible to have a decrease of the irradiance at continuum wavelengths due to the presence of magnetic features (other than sunspots and pores), in particular when these have an average vertical flux of at least 100 G, appear preferentially near disk center, and are observed at short (below 600-700 nm), or long (above 1500 nm) wavelengths. However, we have to verify that negative contrasts still could appear when activity is more evenly spread out over the disk, as is the case when averaging over several solar rotations, and when integrated over the SIM wavelength bands. We therefore estimated the cumulative radiative flux computed over the surfaces of disks of increasing partial radius and integrated these fluxes in wavelength over the four bands observed by SIM. Figure \[fluxes\] shows the relative differences between the cumulative flux computed for the MHD snapshots and the HD ones as function of partial solar radius. The curves confirm that the contribution of faculae to the radiative flux in SIM bands is always positive in the 50 G case, while it can turn negative as the magnetic flux increases (in particular, this can be seen for the curve corresponding to the 200 G simulation), if these features are preferentially located close to disk center. Nevertheless, the contribution is positive if these magnetic features are uniformly distributed over the disk (cumulative radius of one). The main reason for this positive contrast is the limb brightening of the intensity contrast that occurs for all wavelengths (see Fig.\[CLVs\]), and is the result of the sampling of higher temperatures in the MHD snapshots at shallower viewing angles, and the additional brightening that stems from geometrical effects when shallow viewing directions sample hotter material behind partially evacuated magnetic field concentrations. This latter effect cannot adequately be represented by one-dimensional modeling.
Typically, magnetic activity occurs preferentially in activity belts North and South of the equator, moving from higher latitudes in the early phase of the solar cycle to lower ones in later phases. Limiting magnetic elements to lower latitudes limits the number of magnetic elements at higher heliocentric angles, potentially allowing for a negative contribution to the irradiance. To test this possibility we computed the disk integrated radiative flux of magnetic elements confined to an activity belt between latitudes of $\pm$ 30 degrees. Fig. \[fluxesactivitybelt\] shows the result of this calculation for the SIM bands. In particular, it shows the radiative flux relative differences between the MHD and HD snapshots integrated over the SIM bands as function of the average vertical magnetic-flux. It clearly shows a positive contrast for all field cases considered in all four bands. Nevertheless, it also shows that in the 400-691 nm and in the 1630-2423 nm bands the radiative fluxes relative differences decrease with the increase of the magnetic flux. This suggests that, if magnetic features are preferentially located over the activity belt, and if the relative number of features with higher magnetic flux increases with the increase of the magnetic activity, then the radiance at those two SIM bands decreases. **Moreover if, more “realistically”, we consider that even the quiet Sun is permeated by magnetic flux [see @pillet2013 for a review on quiet Sun magnetic field] and we take as reference the 50 G snapshots, then the contribution of facular regions at those two SIM pass bands is always negative.**
Discussion and Conclusions
==========================
We employed snapshots from 3-D MHD simulations, characterized by different values of average vertical magnetic flux, to estimate solar irradiance variations at the visible and IR spectral ranges of SIM radiometers, stemming from contributions of patches of unresolved magnetic field. The results from our spectral synthesis confirm the fact the contribution of facular region to irradiance is strongly dependent on their location over the solar disk [see also the discussion in @fontenla2012]. In particular, we find that the increase of the magnetic filling factor over the solar surface can produce a [*decrease*]{} of emitted radiation only for mostly vertical lines of sight and only for wavelengths below 500-700 nm (depending on the magnetic flux), or above 1500 nm (Fig. \[CLVs\]). Integrating the intensity over the disk, even if we limit the contribution of magnetic regions to an activity belt, always renders the contribution of the magnetic elements to the irradiance positive (Fig. \[fluxesactivitybelt\]). Nevertheless, if magnetic features are distributed over the activity belt, their contribution decreases at two of the SIM bands (namely at 400-691 and 1630-2423 nm) with the increase of the average magnetic flux. This suggests that, assuming that the relative number of features with larger magnetic flux increases with the increase of the magnetic activity, then the spectral irradiance at those SIM bands can decrease toward solar maximum. **Results shown in Fig. \[fluxesactivitybelt\] also indicate that, if we take as reference the 50 G snapshots instead of the HD ones, then the contribution of facular regions to irradiance at the 400-691 and 1630-2423 nm SIM wavelength bands is always negative. We note that this is a more “realistic” assumption than taking the HD snapshots as reference, as previous works have shown that MHD simulations with average vertical magnetic field between 20 - 30 G best represent properties of magnetic field of the quiet Sun [@khomenko2005; @danilovic2010]. Finally, we note that magnetic features tend to appear toward higher latitudes at the beginning of the cycle, migrate toward the equator as the magnetic activity peaks, and that then part of their flux, fragmented into lower magnetic flux features, tends to migrate toward the poles during the descending phase. Since, as we have shown, the contribution to irradiance of these features strongly depends on their position on the solar disk, we speculate that multiple peaks of the solar spectral irradiance could be observed.**
Note that an average flux of 200 G, the maximum we considered, is modest for facular regions, as values up to 800 G are usually employed for reconstructions [e.g. @ball2012 and references therein]. On the other hand, from results shown in this work as well as from results obtained from numerical simulations by other authors [e.g. @vogler2005], and from observations [e.g. @yeo2013; @ortiz2002] it is clear that the center-to-limb variation of contrast increases with magnetic flux so that it is likely that our conclusions would be even stronger.
Likewise, the inclusion of spectral lines in our calculations would have most likely increased the contrast between MHD and HD intensities, as spectral lines contribute opacity and raise the formation height of the spectral bands, causing them to sample slightly higher layers, where the differences between the average temperatures of the models with different field strength is larger. Nevertheless, we expect this effect to be larger for the lower magnetic flux simulations, where the average temperature gradient is steeper (and spectral lines are deeper), with respect to higher magnetic flux simulations, thus increasing the steepness of the relations in Fig. \[fluxesactivitybelt\] for the 400-691 ans 1630-2423 nm bands, and decreasing the steepness of the curves of the other two bands. This effect too would thus strengthen our conclusions.
**We therefore conclude that the spectral synthesis presented in this study are compatible with a negative contribution of facular regions to the irradiance in the SIM visible and IR bands with an increase in magnetic filling factor if as reference for quiet Sun we assume snapshots of 50 G average magnetic flux.**
The snapshots of magneto-convection simulations were provided to us by Elena Khomenko and were calculated using the computing resources of the MareNostrum (BSC/CNS, Spain) and DEISA/HLRS (Germany) supercomputer installations.
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abstract: 'We consider a specific graph learning task: reconstructing a symmetric matrix that represents an underlying graph using linear measurements. We study fundamental trade-offs between the number of measurements (sample complexity), the complexity of the graph class, and the probability of error by first deriving a necessary condition (fundamental limit) on the number of measurements. Then, by considering a two-stage recovery scheme, we give a sufficient condition for recovery. Furthermore, assuming the measurements are Gaussian IID, we prove upper and lower bounds on the (worst-case) sample complexity. In the special cases of the uniform distribution on trees with ${n}$ nodes and the Erdős-Rényi $({n},p)$ class, the fundamental trade-offs are tight up to multiplicative factors. Applying the Kirchhoff’s matrix tree theorem, our results are extended to the scenario when part of the topology information is known a priori. In addition, for practical applications, we design and implement a polynomial-time (in ${n}$) algorithm based on the two-stage recovery scheme. We apply the heuristic algorithm to learn admittance matrices in electric grids. Simulations for several canonical graph classes and IEEE power system test cases demonstrate the effectiveness of the proposed algorithm for accurate topology and parameter recovery.'
author:
- |
Tongxin Li$^*$\
Lucien Werner$^*$\
Steven H. Low[^1]
title: 'Learning Graphs from Linear Measurements: Fundamental Trade-offs and Applications '
---
Introduction {#sec:int}
============
Background {#sec:int.bac}
----------
Symmetric matrices are ubiquitous constructs in graphical models with examples such as the $(0,1)$ adjacency matrix and the (generalized) Laplacian of an undirected graph. A major challenge in graph learning is inferring graph parameters embedded in those graph-based matrices from historical data or real-time measurements. In contrast to traditional statistical inference methods [@chow1968approximating; @chow1973consistency; @tan2010learning], model-based graph learning, such as physically-motivated models and graph signal processing (GSC) [@dong2019learning] takes advantage of additional data structures offered freely by nature. Among different measurement models for graph learning, linear models have been used and analyzed commonly for different tasks, *e.g.,*, linear structural equation models (SEMs) [@ghoshal2018learning; @ghoshal2017learning], linear graph measurements [@ahn2012analyzing], generalized linear cascade models [@pouget2015inferring], *etc*. Despite extra efforts required on data collection, processing and storage, model-based graph learning often guarantees provable sample complexity, which, for most of the time, is significantly lower than the empirical number of measurements needed with traditional inference methods. In many problem settings, having computationally efficient algorithms with low sample complexity is important. One reason for this is that the graph parameters may change in a short time-scale, making sample complexity a vital metric to guarantee that the learning can be accomplished with limited number of measurements.
Taking a modern power system as a concrete example, due to the increasing size of distributed energy resources, the network parameters are subject to rapid changes. The necessity for preventing cascading failure also sometime involves reforming network connectivity and thus undesirably destabilizes the system [@guo2018failure]. Constrained by these issues arising out of the mega-trends, analyzing fundamental limits of parameter identification and designing a corresponding scheme that is efficient in both computational and sample complexity become more and more critical. In addition to the applications in the scenarios when stable measurements are scarce resources, understanding sample complexity and having a practical identification scheme can furthermore bridge the theory-to-application gap and benefit existing algorithms in electric grids. For instance, many real-time or nearly real-time graph algorithms based on temporal data, such as (real-time) optimal power flow [@momoh1999review; @low2014convex; @tang2017real], real-time contingency analysis [@mittal2011real] and frequency control [@horta2015frequency] *etc*., either require full and accurate knowledge of the network, or can be improved if certain estimates are (partially) accessible.
In this work, we consider a general graph learning problem where the measurements and underlying matrix to be recovered can be represented as or approximated by a linear system. A *graph matrix* $\mathbf{Y}(G)$ with respect to an underlying graph $G$ (see Definition \[def:graph\_matrix\]) is defined as an ${n}\times{n}$ symmetric matrix with each nonzero $(i,j)$-th entry corresponding to an edge connecting node $i$ and node $j$ where ${n}\in\mathbb{N}_+$ is the number of nodes of the underlying *undirected* graph. The diagonal entries can be arbitrary. The measurements are summarized as two ${m}\times {n}$ ($1\leq {m}\leq {n}$) real or complex matrices $\mathbf{A}$ and $\mathbf{B}$ satisfying $$\begin{aligned}
\label{eq:linear}
\mathbf{A} = \mathbf{B}\mathbf{Y}(G)+\mathbf{Z}\end{aligned}$$ where $\mathbf{Z}$ denotes additive noise.
We focus on the following problems:
- [What is the *minimum number* $m$ of linear measurements required for reconstructing the *symmetric* matrix $\mathbf{Y}$?]{} [Is there an algorithm *asymptotically achieving* recovery with the minimum number of measurements?]{} As a special case, can we characterize the sample complexity when the measurements are Gaussian IID[^2]?
- Do the theoretical guarantees on sample complexity result in a practical algorithm (in terms of both sample and computational complexity) for recovering electric grid topology and parameters?
Related Work
------------
### Graph Learning Aspects
Algorithms for learning sparse graphical model structures have a rich tradition in previous literature. For general MRFs, learning the underlying graph structures is known to be NP-hard [@bogdanov2008complexity]. However, in the case when the underlying graph is a tree, the classical Chow-Liu algorithm [@chow1968approximating] offers an efficient approach to structure estimation. Recent results contribute to an extensive understanding of the Chow-Liu algorithm. The authors in [@tan2010learning] analyzed the error exponent and showed experimental results for chain graphs and star graphs. For pairwise binary MRFs with bounded maximum degree, [@santhanam2012information] provides sufficient conditions for for correct graph selection. Similar achievability results for Ising models are in [@anandkumar2010high]. Model-based graph learning has been emerging recently and assuming the measurements form linear SEMs, the authors in [@ghoshal2018learning; @ghoshal2017learning] showed theoretical guarantees of the sample complexity for learning a directed acyclic graph (DAG) structure, under mild conditions on the class of graphs.
From a converse perspective, information-theoretic tools have been widely applied to derive fundamental limits for learning graph structures. For a Markov random field (MRF) with bounded maximum degree, [@santhanam2012information] derived necessary conditions on the number of samples for estimating the underlying graph structure using Fano’s inequality (see [@fano1961transmission]). For Ising models, [@anandkumar2012high] combined Fano’s inequality with *typicality* to derive weak and strong converse. Similar techniques have also been applied to Gaussian graphical models [@anandkumar2011high] and Bayesian networks [@ghoshal2017information]. Fundamental limits for noisy compressed sensing have been extensively studied in [@aeron2010information] under an information-theoretic framework.
### Parameter Identification of Power Systems
Graph learning has been widely used electric grids applications, such as state estimation [@cavraro2017voltage; @chen2015quickest] and topology identification [@sharon2012topology; @deka2017structure]. Most of the literature focuses on topology identification or change detection, but there is not much recent work on joint topology and parameter recovery, with notable exceptions of [@li2013blind; @yu2017patopa; @park2018exact]. Moreover, there is little exploration on the fundamental performance limits (estimation error and sample complexity) on topology and parameter identification of power networks, with the exception of [@zhu2012sparse] where a sparsity condition is provided for exact recovery of outage lines. Based on single-type measurements (either current or voltage), correlation analysis has been applied for topology identification [@tan2011sample; @liao2015distribution; @bolognani2013identification]. Approximating the measurements as normal distributed random variables, [@sharon2012topology] proposed an approach for topology identification with limited measurements. A graphical learning-based approach was provided by [@deka2016estimating]. Recently, data-driven methods were studied for parameter estimation [@yu2017patopa]. In [@yuan2016inverse], a similar linear system as (\[2.3\]) has been used combined with regression to recover the symmetric graph parameters (which is the admittance matrix in the power network) where the matrix $\mathbf{B}$ is of full column rank, implying that at least ${m}=\Omega({n})$ measurements are necessary. Sparse recovery ([@candes2005error; @rudelson2008sparse]), however suggests that recovering the graph matrix may take much fewer number of measurements by fully utilizing the sparsity of $\mathbf{Y}$. Some experimental results for recovering topology of a power network based on compressed sensing algorithms are reported in [@babakmehr2016compressive]. Nonetheless, in the worst case, some of the columns (or rows) of $\mathbf{Y}$ may be dense vectors consisting of many non-zeros, prohibiting us from applying compressed sensing algorithms to recover each of the columns (or rows) of $\mathbf{Y}$ separately. Moreover, the columns to be recovered may not share the same support set. Thus many distributed compressed sensing schemes (*cf.* [@baron2005distributed]) are not directly applicable in this situation. This motivates us to handle the difficulty that for a randomly chosen graph, some of the columns (or rows) in the corresponding graph matrix may not be sparse by considering a new two-stage recovery scheme.
Our Contributions {#sec:con}
-----------------
We demonstrate that the linear system in (\[eq:linear\]) can be used to learn the topology and parameters of a graph. Our framework can be applied to perform system identification in electrical grids by leveraging synchronous nodal current and voltage measurements obtained from phasor measurement units (PMUs).
The main results of this paper are summarized here.
1. [*Fundamental Trade-offs*:]{} In Theorem \[thm:1\], we derive a general lower bound on the *probability of error* for topology identification (defined in (\[poet\])). In Section \[sec:ach\], we describe a simple two-stage recovery scheme combining $\ell_1$-norm minimization with an additional step called *consistency-checking*. For any arbitrarily chosen distribution, we characterize it using the definition of *$({\mu},{K})$-sparsity* (see Definition \[def:sparse\]) and argue that if a graph is drawn according to such a distribution, then the number of measurements required for exact recovery is bounded from above as in Theorem \[thm:3\].
2. [*(Worst-case) Sample Complexity*:]{} We focus on the case when the matrix $\mathbf{B}$ has Gaussian IID entries in Section \[sec:GIPM\]. Under this assumption, we provide upper and lower bounds on the worst-case sample complexity in Theorem \[thm:4\]. We show two applications of Theorem \[thm:4\] for the uniform sampling of trees and the Erdős-Rényi $({n},p)$ model in Corollary \[coro:1\] and \[coro:2\], respectively.
3. [*(Heuristic) Algorithm*:]{} Motivated by the two-stage recovery scheme, a heuristic algorithm with polynomial (in ${n}$) running-time is reported in Section \[sec:6\], together with simulation results for power system test cases validating its performance in Section \[sec:sim\].
Outline of the Paper
--------------------
The remaining content is organized as follows. In Section \[sec:pre\], we specify our models. In Section \[sec:fun\], we present the converse result as fundamental limits for recovery. The achievability is provided in \[sec:ach\]. We present our main result as the worst-case sample complexity for Gaussian IID measurements in Section \[sec:GIPM\]. A heuristic algorithm together with simulation results are reported in Section \[sec:6\] and \[sec:sim\].
Model and Definitions {#sec:pre}
=====================
Notation
--------
Let $\mathbb{F}$ denote a field that can either be the set of real numbers $\mathbb{R}$, or the set of complex numbers $\mathbb{C}$. The set of all symmetric ${n}\times{n}$ matrices whose entries are in $\mathbb{F}$ is denoted by $\mathbb{S}^{{n}\times{n}}$. The imaginary unit is denoted by $\mathrm{j}$. Throughout the work, let $\log\left(\cdot\right)$ denote the binary logarithm with base $2$ and let $\ln\left(\cdot\right)$ denote the natural logarithm with base $e$. We use $\mathbb{E}\left[\cdot \right]$ to denote the expectation of random variables if the underlying probability distribution is clear. The mutual information is denoted by $\mathbb{I}(\cdot)$. The (differential) entropy is denoted by $\mathbb{H}(\cdot)$ and in particular, we use $h(\cdot)$ for binary entropy. To distinguish random variables and their realizations, we follow the convention and denote the former by capital letters (*e.g.,* $A$) and the latter by lower case letters (*e.g.,* $a$). The symbol $C$ is used to designate a constant. Matrices are denoted in boldface (*e.g.,* $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{Y}$). The $i$-th row, the $j$-th column and the $(i,j)$-th entry of a matrix $\mathbf{A}$ are denoted by $A^{(i)}$, $A_j$ and $A_{i,j}$ respectively. For notational convenience, let $\mathcal{S}$ be a subset of $\mathcal{N}$. Denote by $\mathcal{S}^{\mathrm{c}}:=\mathcal{N}\backslash\mathcal{S}$ the complement of $\mathcal{S}$ and by $\mathbf{A}^{\mathcal{S}}$ a sub-matrix consisting of $|\mathcal{S}|$ columns of the matrix $\mathbf{A}$ whose indices are chosen from $\mathcal{S}$. The notation $\top$ denotes the transpose of a matrix, $\mathrm{det}\left(\cdot\right)$ calculates its determinant. For the sake of notational simplicity, we use big O notation ($o$,$\omega$,$O$,$\Omega$,$\Theta$) to quantify asymptotic behavior. Table \[notation\] summarizes the notation used throughout the paper.
[l|l]{}\
${n}$ & Number of nodes\
$d_j$ & Degree of node $j$\
$\mathcal{N}$ & Set of ${n}$ nodes\
$\mathcal{E}$ & Set of edges\
$G$ & Underlying random graph\
$\mathsf{G}_{n}$ & Candidacy set consisting of graphs with ${n}$ vertices\
${\mathcal{G}_{n}}$ & Probability distribution of $G$ in $\mathsf{G}_{n}$\
$\mathbf{Y}$ & ${n}\times{n}$ symmetric matrix to be recovered\
\
${m}$ & Number of measurements\
$\mathcal{M}$ & Set of measurement indexes\
$\mathbf{A},\mathbf{B}$ &Matrices of Measurements\
$A^{(i)}$ & $i$-th row of the matrix $\mathbf{A}$\
$A_j$ & $j$-th column of the matrix $\mathbf{A}$\
$A_{i,j}$ & $(i,j)$-th entry of the matrix $\mathbf{A}$\
$\mathbf{Z}$ &Additive output noise\
\
$\mathbbm{R}$ & Set of real numbers\
$\mathbbm{C}$ & Set of complex numbers\
$\mathbbm{F}$ & Either $\mathbbm{R}$ or $\mathbbm{C}$\
$\mathrm{j}$ & Imaginary unit\
$\mathrm{sign}$ & Sign function\
\[notation\]
Graphical Model {#sec:model}
---------------
Denote by $\mathcal{N}=\{1,\ldots,{n}\}$ a set of ${n}$ nodes and consider an *undirected* graph $G=(\mathcal{N},\mathcal{E})$ (with no self-loops) whose edge set $\mathcal{E}\subseteq \mathcal{N}\times\mathcal{N}$ contains the desired topology information. The degree of each node $j$ is denoted by $d_j$. The connectivity between the nodes is unknown and our goal is to determine it by learning the associated *graph matrix* using linear measurements.
\[def:graph\_matrix\] Provided with an underlying graph $G=(\mathcal{N},\mathcal{E})$, a [*symmetric*]{} matrix $\mathbf{Y}(G)\in\mathbb{S}^{{n}\times{n}}$ is called a [*graph matrix*]{} if the following conditions hold: $$\begin{aligned}
{Y}_{i,j}(G)=\begin{cases}
\neq 0 \quad &\text{ if } i\neq j \text{ and } (i,j)\in\mathcal{E}\\
0 \quad &\text{ if } i\neq j \text{ and } (i,j)\notin\mathcal{E}\\
\text{arbitrary} \quad &\text{ otherwise }
\end{cases}.\end{aligned}$$
Our theorems can be generalized to recover a broader class of symmetric matrices, as long as the matrix to be recovered satisfies (1) Knowing $\mathbf{Y}(G)\in\mathbbm{F}^{{n}\times{n}}$ gives the full knowledge of the topology of $G$; (2) The number of non-zero entries in a column of $\mathbf{Y}(G)$ has the same order as the degree of the corresponding node, *i.e.,* there is a positive constant $C>0$ such that $|\mathrm{supp}({Y}_{j})| = Cd_j$. for all $j\in\mathcal{N}$. To have a clear presentation, we consider specifically the case $C=1$.
In this work, we employ a probabilistic model and assume that the graph $G$ is chosen randomly from a *candidacy set* $\mathsf{G}_{n}$ (with ${n}$ nodes), according to some distribution ${\mathcal{G}_{n}}$. Both the candidacy set $\mathsf{G}_{n}$ and distribution ${\mathcal{G}_{n}}$ are not known to the estimator. For simplicity, we often omit the subscripts of $\mathsf{G}_{n}$ and $\mathcal{G}_{n}$.
\[example\] We exemplify some possible choices of the candidacy set and distribution:
1. *[(Mesh Network)]{}* When $G$ represents a transmission (mesh) power network and no prior information is available, the corresponding candidacy set $\mathsf{G}_{n}^{\mathrm{all}}$ consisting of all graphs with ${n}$ nodes and $G$ is selected uniformly at random from $\mathsf{G}_{n}^{\mathrm{all}}$. Moreover, $\left|\mathsf{G}_{n}^{\mathrm{all}}\right|=2^{{n}\choose 2}$ in this case.
2. *[(Radial Network)]{}* When $G$ represents a distribution (radial) power network and no other prior information available, then the corresponding candidacy set $\mathsf{T}_{n}^{\mathrm{all}}$ is a set containing all spanning trees of the complete graph with ${n}$ buses and $G$ is selected uniformly at random from $\mathsf{T}_{n}^{\mathrm{all}}$; the cardinality is $\left|\mathsf{T}_{n}^{\mathrm{all}}\right|={n}^{{n}-2}$ followed by Cayley’s formula.
3. *[(Radial Network with Prior Information)]{}* When $G=\left(\mathcal{N},\mathcal{E}\right)$ represents a distribution (radial) power network, and we further know that some of the buses cannot be connected (which may be inferred from locational/geographical information), then the corresponding candidacy set $\mathsf{T}_{n}^H$ is a set of spanning trees of a sub-graph $H=\left(\mathcal{N},\mathcal{E}_H\right)$ with ${n}$ buses. An edge $e\notin\mathcal{E}_H$ if and only if we know $e\notin\mathcal{E}$. The size of $\mathsf{T}_{n}^H$ is given by Kirchhoff’s matrix tree theorem ([c.f.]{} [@west2001introduction]). See Theorem \[thm:kirchhoff\].
4. *[(Erdős-Rényi $({n},p)$ model)]{}* In a more general setting, $G$ can be a random graph chosen from an ensemble of graphs according to a certain distribution. When a graph $G$ is sampled according to the Erdős-Rényi $({n},p)$ model, each edge of $G$ is connected IID with probability $p$. We denote the corresponding graph distribution for this case by $\mathcal{G}_{\mathrm{ER}}({n},p)$ for convenience.
The next section is devoted to describing available measurements.
Linear System of Measurements {#sec:cvm}
-----------------------------
Suppose the measurements are sampled discretely and indexed by the elements of the set $\{1,\ldots,{m}\}$. As a general framework, the measurements are collected in two matrices $\mathbf{A}$ and $\mathbf{B}$ and defined as follows.
Let ${m}$ be an integer with $1\leq {m}\leq {n}$. The *generator matrix* $\mathbf{B}$ is an ${m}\times {n}$ *random* matrix and the *measurement matrix* $\mathbf{A}$ is an ${m}\times {n}$ matrix with entries selected from $\mathbb{F}$ that satisfy the linear system (\[eq:linear\]): $$\begin{aligned}
\mathbf{A} = \mathbf{B}\mathbf{Y}(G)+\mathbf{Z}\end{aligned}$$ where $\mathbf{Y}(G)\in\mathbb{S}^{{n}\times {n}}$ is a graph matrix to be recovered, with an underlying graph $G$ and $\mathbf{Z}\in\mathbb{F}^{m\times n}$ denotes the random *additive noise*. We call the the recovery *noiseless* if $\mathbf{Z}=\mathbf{0}$. Our goal is to resolve the matrix $\mathbf{Y}(G)$ based on given matrices $\mathbf{A}$ and $\mathbf{B}$.
Applications to Electrical Grids
--------------------------------
Various applications fall into the framework in (\[eq:linear\]). Here we present two examples of the graph identification problem in power systems. The measurements are modeled as time series data obtained via nodal sensors at each node, *e.g.,* PMUs, smart switches, or smart meters.
### Example $1$: Nodal Current and Voltage Measurements
We assume data is obtained from a short time interval over which the unknown parameters in the network are *time-invariant*. $\mathbf{Y}\in\mathbb{C}^{{n}\times {n}}$ denotes the *nodal admittance matrix* of the network and is defined $$\begin{aligned}
\label{eq:admi}
Y_{i,j}:=\begin{cases}
-y_{i,j} \quad &\text{if } i\neq j\\
y_i + \sum_{k\neq i} y_{i,k} & \text{if } i=j
\end{cases}\end{aligned}$$ where $y_{i,j}\in\mathbb{C}$ is the admittance of line $(i,j)\in\mathcal{E}$ and $y_i$ is the self-admittance of bus $i$. Note that if two buses are not connected then $Y_{i,j}=0$.
The corresponding generator and measurement matrices are formed by simultaneously measuring both currents (or equivalently, power injections) and voltages at each node and at each time step. For each $t=1,\ldots,{m}$, the nodal current injections are collected in an ${n}$-dimensional random vector $I_{t}=(I_{t,1},\ldots,I_{t,{n}})$. Concatenating the $I_{t}$ into a matrix we get $\mathbf{I}:=[I_{1},I_{2},\ldots,I_{{m}}]^{\top}\in \mathbb{C}^{{m}\times {n}}$. The generator matrix $\mathbf{V}:=[V_{1},V_{2},\ldots,V_{{m}}]^{\top}\in \mathbb{C}^{{m}\times {n}}$ is constructed analogously. Each pair of measurement vectors $(I_{t},V_{t})$ from $\mathbf{I}$ and $\mathbf{V}$ must satisfy Kirchhoff’s and Ohm’s laws, $$\begin{aligned}
\label{2.3}
{I}_{t}=\mathbf{Y}{V}_{t}, \quad t=1,\ldots,{m}.\end{aligned}$$ In matrix notation (\[2.3\]) is equivalent to $
\mathbf{I}=\mathbf{V}\mathbf{Y}
$, which is a noiseless version of the linear system defined in (\[eq:linear\]).
Compared with only obtaining one of the current, power injection or voltage measurements (for example, as in [@tan2011sample; @tan2010learning; @liao2015distribution]), collecting simultaneous current-voltage pairs doubles the amount of data to be acquired and stored. There are benefits however. First, exploiting the physical law relating voltage and current not only enables us to identify the topology of a power network but also recover the parameters of the admittance matrix. Furthermore, dual-type measurements significantly reduce the sample complexity for topology recovery, compared with the results for single-type measurements.
### Example $2$: Nodal Power Injections and Phase Angles
Similar to the previous example, at each time $t=1,\ldots,{m}$, denote by $P_{t,j}$ and $\theta_{t,j}$ the active nodal power injection and the phase of voltage at node $j$ respectively. The matrices $\mathbf{P}\in\mathbb{R}^{{m}\times{n}}$ and $\pmb{\theta}\in\mathbb{R}^{{m}\times{n}}$ are constructed in a similar way by concatenating the vectors ${P}_t=(P_{t,1},\ldots,P_{t,{n}})$ and ${\theta}_{t}=(\theta_{t,1},\ldots,\theta_{t,{n}})$. The matrix representation of the DC power flow model can be expressed as a linear system $\mathbf{P} =\pmb{\theta}\mathbf{C}\mathbf{S}\mathbf{C}^{\top}$, which belongs to the general class represented in (\[eq:linear\]). Here, the diagonal matrix $\mathbf{S}\in\mathbb{R}^{|\mathcal{E}|\times|\mathcal{E}|}$ is the susceptence matrix whose $e$-th diagonal entry represents the susceptence on the $e$-th edge in $\mathcal{E}$ and $\mathbf{C}\in \{-1,0,1\}^{{n}\times |\mathcal{E}|}$ is the node-to-link incidence matrix of the graph. The vertex-edge incidence matrix[^3] $\mathbf{C}\in \{-1,0,1\}^{{n}\times |\mathcal{E}|}$ is defined as $$\begin{aligned}
C_{j,e}:=
\begin{cases}
1, \quad &\text{if bus } j \text{ is the source of } e\\
-1, \quad &\text{if bus } j \text{ is the target of } e\\
0, \quad &\text{otherwise}
\end{cases}.\end{aligned}$$ Note that $\mathbf{C}\mathbf{S}\mathbf{C}^{\top}$ specifies both the network topology and the susceptences of power lines.
Probability of Error as the Recovery Metric {#sec:poe}
-------------------------------------------
We define the error criteria considered in this paper. We refer to finding the edge set $\mathcal{E}$ of $G$ via matrices $\mathbf{A}$ and $\mathbf{B}$ as the *topology identification problem* and recovering the graph matrix $\mathbf{Y}$ via matrices $\mathbf{A}$ and $\mathbf{B}$ as the *parameter reconstruction problem*.
Let $f$ be a function or algorithm that returns an estimated graph matrix $\mathbf{X}=f(\mathbf{A},\mathbf{B})$ given inputs $\mathbf{A}$ and $\mathbf{B}$. The *probability of error for topology identification* ${{\varepsilon}_{\mathrm{T}}}$ is defined to be the probability that the estimated edge set is not equal to the correct edge set: $$\begin{aligned}
\label{poet}
{{\varepsilon}_{\mathrm{T}}}:=
\mathbb{P}\left(\exists \ i\neq j: \mathrm{sign}(X_{i,j})\neq \mathrm{sign}\left({Y}_{i,j}(G)\right)\right)\end{aligned}$$ where the probability is taken over the randomness in $G,\mathbf{B}$, and $\mathbf{Z}$. The *probability of error for (noiseless[^4]) parameter reconstruction* ${{\varepsilon}_{\mathrm{P}}}$ is defined to be the probability that the estimate $\mathbf{X}$ is not equal to the original graph matrix $\mathbf{Y}(G)$: $$\begin{aligned}
\label{poe}
{{\varepsilon}_{\mathrm{P}}}:=\sup_{\mathbf{Y}\in\mathbb{Y}(G)}\mathbb{P}\left(\mathbf{X}\neq \mathbf{Y}(G)\right)\end{aligned}$$ where $\mathbb{Y}(G)$ is the set of all graph matrices $Y(G)$ that are consistent with the underlying graph $G$ and the probability is taken over the randomness in $G$ and $\mathbf{B}$.
Note that for a fixed noiseless parameter reconstruction algorithm, we always have the corresponding ${{\varepsilon}_{\mathrm{P}}}$ greater than ${{\varepsilon}_{\mathrm{T}}}$. We use ${{\varepsilon}_{\mathrm{P}}}$ as the error metric in this work and refer it as the *probability of error* considered in the remainder of this paper.
Fundamental Trade-offs
======================
We discuss fundamental trade-offs of the parameter recovery problem defined in Section \[sec:model\] and \[sec:cvm\]. The converse result is summarized in Theorem \[thm:1\] as an inequality involving the probability of error, the distributions of the underlying graph, generator matrix and noise. Next, in Section \[sec:ach\], we focus on a particular two-stage scheme, and show in Theorem \[thm:3\] that under certain conditions, the probability of error is asymptotically zero (in ${n}$).
Converse {#sec:fun}
--------
The following theorem states the fundamental limit.
\[thm:1\] The probability of error for topology identification ${{\varepsilon}_{\mathrm{T}}}$ is bounded from below as $$\begin{aligned}
\label{3.0}
{{\varepsilon}_{\mathrm{T}}}\geq 1-\frac{ \mathbb{H}\left(\mathbf{B}\right)-\mathbb{H}\left(\mathbf{Z}\right)+\ln 2}{\mathbb{H}\left({\mathcal{G}_{n}}\right)}
\end{aligned}$$ where $\mathbb{H}\left(\mathbf{B}\right)$, $\mathbb{H}\left(\mathbf{Z}\right)$ and $\mathbb{H}\left({\mathcal{G}_{n}}\right)$ are differential entropy (in base $e$) functions of the random variables $\mathbf{B}$, $\mathbf{Z}$ and probability distribution ${\mathcal{G}_{n}}$, respectively.
The graph $G$ is chosen from a discrete set $\mathsf{G}_{n}$ according to some probability distribution ${\mathcal{G}_{n}}$. As previously introduced, Fano’s inequality [@fano1961transmission] borrowed plays an important role in deriving fundamental limits. We especially focus on its extended version. Similar generalizations appear in many places, *e.g.,* [@aeron2010information; @santhanam2012information] and [@ghoshal2016information].
\[lemma:fano\] Let $G$ be a random graph and let $\mathbf{A}$ and $\mathbf{B}$ be measurement matrices defined in Section \[sec:model\] and \[sec:cvm\]. Suppose the original graph $G$ is selected from a nonempty candidacy set $\mathsf{G}_{n}$ according to a probability distribution ${\mathcal{G}_{n}}$. Let $\hat{G}$ denote the estimated graph. Then the conditional probability of error for estimating $G$ from $\mathbf{B}$ given $\mathbf{A}$ is always bounded from below as $$\begin{aligned}
\label{3.2}
\mathbbm{P}\left(\hat{G}\neq G\big| \mathbf{A}\right)\geq 1-\frac{\mathbbm{I}\left(G;\mathbf{B}\big|\mathbf{A}\right)+\ln 2}{\mathbbm{H}\left({\mathcal{G}_{n}}\right)}\end{aligned}$$ where the randomness is over the selections of the original graph $G$ and the estimated graph $\hat{G}$.
In (\[3.2\]), the term $\mathbbm{I}\left(G;\mathbf{B}\big|\mathbf{A}\right)$ denotes the conditional mutual information (base $e$) between $G$ and $\mathbf{B}$ conditioned on $\mathbf{A}$, which is defined as $$\begin{aligned}
&\mathbbm{I}\left(G;\mathbf{B}\big | \mathbf{A}\right):=\sum_{G\in\mathsf{G}_{n}}\int_{\mathbf{B}}\int_{\mathbf{A}}p\left(\mathbf{A},\mathbf{B},G\right)\ln\frac{p\left(\mathbf{B}|\mathbf{A},G\right)}{p\left(\mathbf{B}|\mathbf{A}\right)}\mathrm{d}\mathbf{A}\mathrm{d}\mathbf{B}\end{aligned}$$ where the integrals are both taken over $\mathbbm{F}^{{n}\times{m}}$. Furthermore, the conditional mutual information $\mathbbm{I}\left(G;\mathbf{B}|\mathbf{A}\right)$ is bounded from above by the differential entropies of $\mathbf{B}$ and $\mathbf{A}$. It follows that $$\begin{aligned}
\label{3.8}
\mathbbm{I}\left(G;\mathbf{B}|\mathbf{A}\right)&= \mathbbm{H}\left(\mathbf{B}|\mathbf{A}\right)-\mathbbm{H}\left(\mathbf{B}|G,\mathbf{A}\right)\\
\label{3.9}
&\leq \mathbbm{H}\left(\mathbf{B}|\mathbf{A}\right)-\mathbbm{H}\left(\mathbf{B}|\mathbf{Y},\mathbf{A}\right)\\
\label{3.10}
&= \mathbbm{H}\left(\mathbf{B}|\mathbf{A}\right)-\mathbbm{H}\left(\mathbf{Z}\right)\\
\label{3.11}
&\leq \mathbbm{H}\left(\mathbf{B}\right)-\mathbbm{H}\left(\mathbf{Z}\right).\end{aligned}$$ Here, Eq. (\[3.8\]) follows from the definitions of mutual information and differential entropy. Moreover, knowing $\mathbf{Y}$, the graph $G$ can be inferred. Thus, $\mathbbm{H}\left(\mathbf{B}|G,\mathbf{A}\right)\geq \mathbbm{H}\left(\mathbf{B}|\mathbf{Y},\mathbf{A}\right)$ yields (\[3.9\]). Recalling the linear system in (\[eq:linear\]), we obtain (\[3.10\]). Furthermore, (\[3.11\]) holds since $\mathbbm{H}\left(\mathbf{B}\right)\geq \mathbbm{H}\left(\mathbf{B}|\mathbf{A}\right)$.
Plugging (\[3.11\]) into (\[3.2\]), $$\begin{aligned}
{{\varepsilon}_{\mathrm{P}}}\geq {{\varepsilon}_{\mathrm{T}}}=&\mathbbm{E}_{\mathbf{A}}\left[\mathbbm{P}\left(\hat{G}\neq G\big| \mathbf{A}\right)\right]
\geq 1-\frac{ \mathbbm{H}\left(\mathbf{B}\right)-\mathbbm{H}\left(\mathbf{Z}\right)+\ln 2}{\mathbbm{H}\left({\mathcal{G}_{n}}\right)},\end{aligned}$$ which yields the desired (\[3.0\]).
Achievability
-------------
\[alg:1\]
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\[sec:ach\] In this subsection, we consider the achievability for *noiseless* parameter reconstruction. The proofs rely on constructing a two-stage recovery scheme (Algorithm \[alg:1\]), which contains two steps – *column-retrieving* and *consistency-checking*. The worst-case running time of this scheme depends on the underlying distribution $\mathcal{G}_{n}$[^5]. The scheme is presented as follows.
### Two-stage Recovery Scheme
#### [Retrieving columns]{}
In the first stage, using $\ell_1$-norm minimization, we recover each column of $\mathbf{Y}$ based on (\[eq:linear\]) (with no noise): $$\begin{aligned}
\label{4.1}
\mathrm{minimize} \quad &{\big|\big|X_j\big|\big|_{\ell_1}}\\
\label{4.0}
\mathrm{subject}\ \mathrm{to}\quad &\mathbf{B}X_{j}= A_j,\\
\label{4.2}
&X_j\in\mathbb{F}^{{n}}.\end{aligned}$$ Let ${X}_{j}^{\mathcal{S}}:=({X}_{i,j})_{i\in\mathcal{S}}$ be a length-$|\mathcal{S}|$ column vector consisting of $\left|\mathcal{S}\right|$ coordinates in ${X}_j$, the $j$-th retrieved column. We do not restrict the methods for solving the $\ell_1$-norm minimization in (\[4.1\])-(\[4.2\]), as long as there is a unique solution for sparse columns with fewer than ${\mu}$ non-zeros (the parameter ${\mu}>0$ is defined in Definition \[def:sparse\] below).
#### Checking consistency
In the second stage, we check for error in the decoded columns $X_1,\ldots,X_{n}$ using the symmetry property of the graph matrix $\mathbf{Y}$. Specifically, we fix a subset $\mathcal{S}\subseteq\mathcal{N}$ with a given size $\left|\mathcal{S}\right|={n}-{K}$ for some integer $0\leq {K}\leq{n}$. Then we check if $X_{i,j}= X_{j,i}$ for all $i,j\in\mathcal{S}$. If not, we choose a different set $\mathcal{S}$ of the same size. This procedure stops until either we find such a subset $\mathcal{S}$ of columns, or we go through all possible subsets without finding one. In the latter case, an error is declared and the recovery is unsuccessful. In the former case, we accept $X_j, j\in \mathcal{S}$, as correct. For each vector $X_j, j\not\in\mathcal{S}$, we accept its entries $X_{i,j}, i\in\mathcal{S}$, as correct and use them to compute the other entries $X_{i,j}, i\not\in\mathcal{S}$, of $X_j$ using (\[4.0\]): $$\begin{aligned}
\label{4.3}
\mathbf{B}^{\mathcal{S}^{\mathrm{c}}}X_{j}^{\mathcal{S}^{\mathrm{c}}}=A_j- \mathbf{B}^{\mathcal{S}}X^{\mathcal{S}}_{j}, \quad j\in\mathcal{S}^{\mathrm{c}}.\end{aligned}$$ We combine $X_j^{\mathcal{S}}$ and $X_j^{\mathcal{S}^\mathrm{c}}$ to obtain a new estimate $X_j$ for each $j\in\mathcal{S}^{\mathrm{c}}$. Together with the columns $X_j$, $j\in\mathcal{S}$, that we have accepted, they form the estimated graph matrix $\mathbf{X}$.
### $({\mu},{K})$-sparse Distribution
We now analyze the sample complexity of the two-stage scheme. Let $d_j(G)$ denote the degree of node $j$ in $G$. Denote by $\mathcal{N}_{\mathrm{Large}}\left({\mu}\right)$ the set of nodes having degrees greater than the *threshold parameter* $0\leq {\mu}\leq {n}-2$: $$\begin{aligned}
\label{eq:large}
\mathcal{N}_{\mathrm{Large}}\left({\mu}\right):=\Big\{j\in\mathcal{N}: d_j(G)>{\mu}\Big\}\end{aligned}$$ and $\mathcal{N}_{\mathrm{Small}}\left({\mu}\right):= \mathcal{N}\backslash\mathcal{N}_{\mathrm{Large}}\left({\mu}\right).$ Making use of (\[eq:large\]), we define the following set of graphs, with a *counting parameter* $0\leq {K}\leq{n}$: $$\begin{aligned}
\nonumber
\mathsf{G}_{n}({\mu},{K})&:=\Big\{G\in\mathsf{G}_{n}:\left|\mathcal{N}_{\mathrm{Large}}\left({\mu}\right)\right|\leq {K}\Big\}.\end{aligned}$$ The following definition characterizes graph distributions.
\[def:sparse\] A graph distribution $\mathcal{G}$ defined on $\mathsf{G}_{n}$ is said to be $(\mu,K,\rho)$-*sparse* if assuming $G\sim\mathcal{G}$, $$\begin{aligned}
\label{eq:rec}
\mathbb{P}_{{\mathcal{G}}}\left(G\notin\mathsf{G}_{{n}}({\mu },{K})\right)\leq \rho.\end{aligned}$$
The following lemmas provide examples of sparse distributions. Denote by $\mathcal{U}_{\mathsf{T}_{n}^{\mathrm{all}}}$ the uniform distribution on the set $\mathsf{T}_{n}^{\mathrm{all}}$ of all trees with ${n}$ nodes.
\[lemma:trees\] For any ${\mu}\geq 1$ and ${K}>0$, the distribution $\mathcal{U}_{\mathsf{T}_{n}^{\mathrm{all}}}$ is $({\mu },{K},1/K)$-[sparse]{}.
Denote by $\mathcal{G}_{\mathrm{ER}}({n},p)$ the graph distribution for the Erdős-Rényi $({n},p)$ model.
\[lemma:renyi\] For any ${\mu}({n},p)\geq {2{n}h(p)}/{(\ln 1/p)}$ and ${K}>0$, the distribution $\mathcal{G}_{\mathrm{ER}}({n},p)$ is $({\mu},{K},{n}\exp(-{n}h(p)))/K)$-[sparse]{}.
The threshold and counting parameters for both examples are tight, as indicated in Corollary \[coro:1\] and \[coro:2\]. The proofs of Lemma \[lemma:trees\] and \[lemma:renyi\] are postponed to Appendix \[app:proof\_trees\].
### Analysis of the Scheme
We now present another of our main theorems, which makes use of the restricted isometry property (*cf.*, [@candes2005error; @rudelson2008sparse]). Given a generator matrix $\mathbf{B}$, the corresponding *restricted isometry constant* denoted by $\sigma_{\mu}$ is the smallest positive number with $$\begin{aligned}
C\left(1-\sigma_{\mu}\right)\left|\left|\mathbf{x}\right|\right|_{\ell_2}^2 \leq \left|\left|\mathbf{B}^{\mathcal{S}}\mathbf{x}\right|\right|_{\ell_2}^2 \leq C\left(1+\sigma_{\mu}\right)\left|\left|\mathbf{x}\right|\right|_{\ell_2}^2\end{aligned}$$ for some constant $C>0$ and for all subsets $\mathcal{S}\subseteq\mathcal{N}$ of size $\left|\mathcal{S}\right|\leq {\mu}$ and all $\mathbf{x}\in\mathbb{F}^{\left|\mathcal{S}\right|}$.
Denote by $\mathrm{spark}(\mathbf{B})$ the smallest number of columns in the matrix $\mathbf{B}$ that are linearly dependent (see [@donoho2003optimally] for the requirements on the spark of the generator matrix to guarantee desired recovery criteria). Consider the models defined in Section \[sec:model\] and \[sec:cvm\].
\[thm:3\] Suppose the generator matrix $\mathbf{B}$ has restricted isometry constants $\sigma_{3{\mu}}$ and $\sigma_{4{\mu}}$ satisfying $\sigma_{3{\mu}}+3\sigma_{4{\mu}}< 2$ and furthermore, ${m}\geq \mathrm{spark}(\mathbf{B})>2K$. If the distribution $\mathcal{G}$ is $({\mu},{K},\rho)$-sparse, then the probability of error for the two-stage scheme to recover a graph matrix $\mathbf{Y}(G_n)$ of $G_n\sim\mathcal{G}_{n}$ satisfies ${{\varepsilon}_{\mathrm{P}}}\leq \rho$.
First, the theory of compressed sensing (see [@candes2005error; @rudelson2008sparse]) implies that if the generator matrix $\mathbf{B}$ has restricted isometry constants $\sigma_{3{\mu}}$ and $\sigma_{4{\mu}}$ satisfying $\sigma_{3{\mu}}+3\sigma_{4{\mu}}< 2$, then all columns $Y_j$ with $j\in\mathcal{N}_{\mathrm{Small}}$ are correctly recovered using the minimization in (\[4.1\])-(\[4.2\]). It remains to show that the consistency-check in our scheme works, which is summarized as the following lemma.
\[lemma:detect\] Suppose the matrix $\mathbf{B}$ has restricted isometry constants $\sigma_{3{\mu}}$ and $\sigma_{4{\mu}}$ satisfying $\sigma_{3{\mu}}+3\sigma_{4{\mu}}< 2$. Furthermore, suppose ${m}\geq \mathrm{spark}(\mathbf{B})>2K$. If $G\in\mathsf{G}({\mu},{K})$, then the collection of columns $\{X_j\}_{j\in\mathcal{S}}$ passing the consistency-check such that $X_{i,j} =X_{j,i}$ for all $i,j\in\mathcal{S}$, are correctly decoded and together with (\[4.3\]), the two-stage scheme always returns the original (correct) graph matrix.
The proof of Lemma \[lemma:detect\] can be found in Appendix \[app:proof\_consistency\_check\]. Making use of Lemma \[lemma:detect\], it follows that ${{\varepsilon}_{\mathrm{P}}}\leq 1-\mathbb{P}_{{\mathcal{G}}}(G\in\mathsf{G}({\mu},{K}))$ provided ${m}\geq \mathrm{spark}(\mathbf{B})>2K$. In agreement with the assumption that the distribution $\mathcal{G}$ is $({\mu},{K},\rho)$-sparse, (\[eq:rec\]) must be satisfied. Thus, the probability of error is less than $\rho$.
Gaussian IID Measurements {#sec:GIPM}
=========================
In this section, we consider a special regime when the measurements in the matrix $\mathbf{B}$ are Gaussian IID random variables. Utilizing the converse in Theorem \[thm:1\] and the achievability in Theorem \[thm:3\], the Gaussian IID assumption allows the derivation of explicit expressions of sample complexity as upper and lower bounds on the number of measurements ${m}$. Combining with the results in Lemma \[lemma:trees\] and \[lemma:renyi\], we are able to show that for the corresponding lower and upper bounds match each other for graphs distributions $\mathcal{U}_{\mathsf{T}_{n}^{\mathrm{all}}}$ and $\mathcal{G}_{\mathrm{ER}}({n},p)$ (with certain conditions on $p$ and ${n}$).
For the convenience of presentation, in the remainder of the paper, we restrict that the measurements are chosen from $\mathbbm{R}$, although the theorems can be generalized to the complex measurements. In realistic scenarios, for instance, a power network, besides the measurements collected from the nodes, nominal state values, *e.g.,* operating current and voltage measurements are known to the system designer a priori. Representing the nominal values at the nodes by $\overline{A}\in\mathbbm{R}^{n}$ and $\overline{B}\in\mathbbm{R}^{n}$ respectively, the measurements in $\mathbf{A}$ and $\mathbf{B}$ are centered around ${m}\times {n}$ matrices $\overline{\mathbf{A}}$ and $\overline{\mathbf{B}}$ defined as $$\begin{aligned}
\overline{\mathbf{A}}:=\begin{bmatrix}
\cdots & \overline{A} & \cdots\\
\cdots & \overline{A} & \cdots\\
& \vdots &\\
\cdots & \overline{A} & \cdots
\end{bmatrix},\quad
\overline{\mathbf{B}}:=\begin{bmatrix}
\cdots & \overline{B} & \cdots\\
\cdots & \overline{B} & \cdots\\
& \vdots &\\
\cdots & \overline{B} & \cdots
\end{bmatrix}.\end{aligned}$$ The rows in $\mathbf{A}$ and $\mathbf{B}$ are the same, because the graph parameters are time-invariant, so are the nominal values. Without system fluctuations and noise, the nominal values satisfy the linear system in (\[eq:linear\]), *i.e.,* $$\begin{aligned}
\label{2.10}
\overline{\mathbf{A}} = \overline{\mathbf{B}}\mathbf{Y}.\end{aligned}$$ Knowing $\overline{A}$ and $\overline{B}$ is not sufficient to infer the network parameters (the entries in the graph matrix $\mathbf{Y}$), since the rank of the matrix $\overline{B}$ is one. However, measurement fluctuations can be used to facilitate the recovery of $\mathbf{Y}$. The deviations from the nominal values are denoted by additive perturbation matrices $\mathbf{\widetilde{A}}$ and $\mathbf{\widetilde{B}}$ such that $
\mathbf{A}=\overline{\mathbf{A}}+\mathbf{\widetilde{A}}.
$ Similarly, $
\mathbf{B}=\overline{\mathbf{B}}+\mathbf{\widetilde{B}}
$ where $\widetilde{\mathbf{B}}$ is an ${m}\times {n}$ matrix consisting of additive perturbations. Thus, putting (\[2.3\]) and (\[2.10\]) together, the equations above imply that $
\overline{\mathbf{A}}+\mathbf{\widetilde{A}}=\mathbf{B}\mathbf{Y} =\overline{\mathbf{B}}\mathbf{Y}+\mathbf{\widetilde{B}}\mathbf{Y}
$ leading to $
\mathbf{\widetilde{A}}= \mathbf{\widetilde{B}}\mathbf{Y}
$ where we extract the perturbation matrices $\mathbf{\widetilde{A}}$ and $\mathbf{\widetilde{B}}$. We specifically consider the case when the additive perturbations $\mathbf{\widetilde{B}}$ is a matrix with Gaussian IID entries. Without loss of generality, we suppose the mean is zero and the variance is one. For simplicity, in the remainder of this paper, we slightly abuse the notation and replace the perturbations matrices $\mathbf{\widetilde{A}}$ and $\mathbf{\widetilde{B}}$ by $\mathbf{A}$ and $\mathbf{B}$ (we assume that $\mathbf{{B}}$ is Gaussian IID), if the context is clear. Moreover, throughout this section, we focus on the case when the measurements are noiseless.
The next theorem implies that Gaussian IID random variables are not arbitrary selections. They are the most “informative" measurements in the sense that any measurement vector with fixed mean and covariance achieves the maximal entropy with normal distribution.
\[thm:2\] Suppose the row measurements of the generator matrix $\mathbf{B}\in\mathbb{R}^{{n}\times{m}}$ are identically distributed random vectors with zero mean and covariance $\mathbf{{K}}\in\mathbb{R}^{{n}\times{n}}$. The probability of error ${{\varepsilon}_{\mathrm{P}}}$ is bounded from below as $$\begin{aligned}
\label{4.20}
{{\varepsilon}_{\mathrm{P}}}\geq&1-\left[{{m}\ln\left(\left(2\pi e\right)^{2{n}} \det\mathbf{{K}}\right)+\ln 2}\right]/{\mathbb{H}\left({\mathcal{G}_{n}}\right)}\end{aligned}$$ for noiseless recovery where $\mathbb{H}\left({\mathcal{G}_{n}}\right)$ is the differential entropy (in base $e$) of the graph distribution ${\mathcal{G}_{n}}$.
It can be inferred from the theorem that the number of samples must be at least linear in ${n}$ to ensure a small probability of error, the size of the graph, given that the graph, as a mesh network, is chosen uniformly at random from $\mathsf{G}_{n}^{\mathrm{all}}$ (see Example \[example\] (a)). On the other hand, as corollaries, under the assumptions of Gaussian IID measurements, ${m}=\Omega(\log {n})$ is *necessary* for making the probability of error less or equal to $1/2$, if the graph is chosen uniformly at random from $\mathsf{T}_{n}^{\mathrm{all}}$; ${m}=\Omega({n}h(p))$ is *necessary* if the graph is sampled according to $\mathcal{G}_{\mathrm{ER}}({n},p)$, as in Examples \[example\] (b) and (c), respectively. The theorem can be generalized to complex measurements by adding additional multiplicative constants.
The proof is based on Theorem \[thm:1\]. The key fact used is that the entropy $\mathbb{H}\left(\mathbf{B}\right)$ is maximized when ${B}^{(t)}$ is distributed normally with zero mean and covariance $\mathbf{{K}}\in\mathbb{R}^{{n}\times{n}}$, for all $t=1,\ldots,{m}$, $$\begin{aligned}
\label{4.10}
\mathbb{H}\left(\mathbf{B}\right)\leq \sum_{t=1}^{{m}}\mathbb{H}\left({B}^{(t)}\right)&\leq\frac{1}{2}{m}\ln\left(\left(2\pi e\right)^{2{n}} \det\mathbf{{K}} \right).\end{aligned}$$ Substituting the above into Theorem \[thm:1\] gives (\[4.20\]).
Sample Complexity for Sparse Distributions {#sec:4.c}
------------------------------------------
We consider the worst-case sample complexity for recovering graphs generated according to a sequence of sparse distributions, defined similarly as Definition \[def:sparse\] to characterize asymptotic behavior of graph distributions.
\[def:sparse\_seq\] A sequence $\{\mathcal{G}_{{n}}\}$ of graph distributions is said to be $(\mu({n}),K({n}))$-*sparse* if assuming a sequence of graphs is chosen as $G_{n}\sim\mathcal{G}_{n}$, the sequences $\{\mu({n})\}$ and $\{K({n})\}$ guarantee that $$\begin{aligned}
\label{eq:rec_seq}
\lim_{{n}\rightarrow\infty}\mathbb{P}_{{\mathcal{G}_{n}}}\left(G_{n}\notin\mathsf{G}_{n}({\mu({n})},{K({n})})\right)=0.\end{aligned}$$
In the remaining contexts, we sometime write $\mu({n})$ and $K({n})$ as $\mu$ and $K$ for simplicity. Based on the sequence of sparse distributions we defined above, we show the following theorem, which provides upper and lower bounds on the worst-case sample complexity, with Gaussian IID measurements.
\[thm:4\] Suppose that the generator matrix $\mathbf{B}$ has Gaussian IID entries with mean zero and variance one and assume the sequences $\{\mu({n})\}$ and $\{K({n})\}$ satisfy ${\mu}({n})<{n}^{-3/{\mu({n})}}({n}-{K}({n}))$ and ${K}({n})=o({n})$ for all ${n}$. For any sequence of distributions that is $({\mu}({n}),{K}({n}))$-sparse, the two-stage scheme guarantees that $\lim_{{n}\rightarrow\infty}{{\varepsilon}_{\mathrm{P}}}=0$ using ${m}=O\left({\mu}\log({{n}}/{{\mu}})+{K}\right)$ measurements. Conversely, there exists a $({\mu}({n}),{K}({n}))$-sparse sequence of distributions such that the number of measurements must satisfy ${m}=\Omega\left({\mu}\log({{n}}/{{\mu}})+{K}/{n}^{3/{\mu}}\right)$ to make the probability of error ${{\varepsilon}_{\mathrm{P}}}$ less than ${1}/{2}$ for all ${n}$.
The upper bound on ${m}$ that we are able to show differs from the lower bound by a sub-linear term $n^{3/\mu}$. In particular, when the term $\mu\log({n}/\mu)$ dominates $K$, the lower and upper bounds become tight up to a multiplicative factor.
The first part is based on Theorem \[thm:3\]. Under the assumption of the generator matrix $\mathbf{B}$, using Gordon’s escape-through-the-mesh theorem, Theorem $4.3$ in [@rudelson2008sparse] implies that for any columns $Y_j$ with $j\in\mathcal{N}_{\mathrm{Small}}$ are correctly recovered using the minimization in (\[4.1\])-(\[4.2\]) with probability at least $1-2.5\exp\left(-(4/9){\mu}\log({n}/{\mu})\right)$, as long as the number of measurements satisfies ${m}\geq 48{\mu}\left(3+2\log({n}/{\mu})\right)$, and ${n}/{\mu}>2, {\mu}\geq 4$ (if ${\mu}\leq 3$, the multiplicative constant increases but our theorem still holds). Similar results were first proved by Candes, *et al.* in [@candes2005error] (see their Theorem $1.3$). Therefore, applying the union bound, the probability that all the ${\mu}$-sparse columns can be recovered simultaneously is at least $1-2.5{n}\exp\left(-(4/9){\mu}\log({n}/{\mu})\right)$. On the other hand, conditioned on that all the ${\mu}$-sparse columns are recovered, Theorem \[thm:3\] shows that ${m}\geq\mathrm{spark}(\mathbf{B})>2K$ is sufficient for the two-stage scheme to succeed. Since each entry in $\mathbf{B}$ is an IID Gaussian random variable with zero mean and variance one, if ${m}\geq 48{\mu}\left(3+2\log({n}/{\mu})\right)+2K$, with probability one that the spark of $\mathbf{B}$ is greater than $2K$, verifying the statement.
The converse follows directly from Theorem \[thm:2\]. Consider the uniform distribution $\mathcal{U}_{\mathsf{G}_{n}({\mu},{K})}$ on $\mathsf{G}_{n}({\mu},{K})$. Then $\mathbb{H}\left(\mathcal{U}_{\mathsf{G}_{n}({\mu},{K})}\right)=\ln\left|\mathsf{G}_{n}({\mu},{K})\right|$. Let $0\leq\alpha,\beta\leq 1$ be parameters such that ${\mu}<\beta({n}-\alpha {K})$. To bound the size of $\mathsf{G}_{n}({\mu},{K})$, we partition $\mathcal{N}$ into $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ with $|\mathcal{N}_{1}|={n}-\alpha {K}$ and $|\mathcal{N}_{2}|=\alpha {K}$. First, we assume that the nodes in $\mathcal{N}_{1}$ form a ${\mu}/2$-regular graph. For each node in $\mathcal{N}_{2}$, construct $\beta({n}-\alpha {K})\in\mathbb{N}_{+}$ edges and connect them to the other nodes in $\mathcal{N}$ with uniform probability. A graph constructed in this way always belongs to $\mathsf{G}_{n}({\mu},{K})$, unless the added edges create more than ${K}$ nodes with degrees larger than ${\mu}$. Therefore, as ${n}\rightarrow\infty$, $$\begin{aligned}
\label{eq:change1}
\left|\mathsf{G}_{n}({\mu},{K})\right|\geq& \rho\cdot\frac{e^{1/4}\displaystyle\binom{\,N-1}{\phi}^{N}{\displaystyle\binom{\binom{N}{2}}{\phi N/2}}}{\displaystyle\binom{N(N-1)}{\phi N}}
\cdot\binom{{n}-1}{M}^{\alpha {K}}\end{aligned}$$ where $N:={n}-\alpha {K}$, $M:=\beta({n}-\alpha {K})$ and $\phi:={\mu}/2$. The first term $\rho$ denotes the fraction of the constructed graphs that are in $\mathsf{G}_{n}({\mu},{K})$. The second term in (\[eq:change1\]) counts the total number of $\phi$-regular graphs [@liebenau2017asymptotic], and the last term is the total number of graphs created by adding new edges for the nodes in $\mathcal{N}_2$. If ${K}=O({\mu})$, there exists a constant $\alpha>0$ small enough such that $\rho=1$. If ${\mu}=o({K})$, for any fixed node in $\mathcal{N}_1$, the probability that its degree is larger than ${\mu}$ is $$\begin{aligned}
&\sum_{i=\phi+1}^{\alpha {K}}\binom{\alpha {K}}{i}\beta^{i}(1-\beta)^{\alpha {K}-i}
\leq
\sum_{i=\phi+1}^{\alpha {K}}\alpha {K} h\left(\frac{i}{\alpha {K}}\right)\beta^i
\leq (\alpha {K})^2 \beta^{\phi+1}\end{aligned}$$ where $h({i}/{\alpha {K}})$ is in base $e$. Take $\beta={n}^{-3/{\mu}}$ and $\alpha=1/2$. The condition ${\mu}<{n}^{-3/{\mu}}({n}-{K})$ guarantees that ${\mu}<\beta({n}-\alpha {K})$. Letting $\digamma({n}):=1/{n}$ be the assignment function for each node in $\mathcal{N}_1$, we check that $$\begin{aligned}
(\alpha {K})^2 \beta^{\phi+1}\leq \frac{1}{4{n}} \leq \digamma({n})\cdot \left(1-\frac{1}{\digamma({n})}\right)^{N}\leq\frac{1}{e{n}}.\end{aligned}$$ Therefore, applying the Lovász local lemma, the probability that all the nodes in $\mathcal{N}_1$ have degree less than or equal to $\mu$ can be bounded from below by $ \left(1-\digamma({n})\right)^N \geq 1/4$ if ${n}\geq 2$, which furthermore is a lower bound on $\rho$. Therefore, taking the logarithm, $$\begin{aligned}
\label{4.30}
\mathbb{H}\left(\mathcal{U}_{\mathsf{G}_{n}({\mu},{K})}\right)\geq& \frac{(N-1)^2}{2}h(\varepsilon)-O(N\ln{\mu})
+\frac{{K}}{2}\left(({n}-1)h\left(\frac{M}{{n}-1}\right)-O(\ln{n})\right)-O(1)\\
\label{4.4}
=& \Omega\left({n}^2 h(\varepsilon)+{n}^{1-3/{\mu}} {K}\right)\end{aligned}$$ where $\varepsilon:=\phi/(N-1)\leq 1/2$. In (\[4.30\]), we have used Stirling’s approximation and the assumption that ${K}=o({n})$. Continuing from (\[4.4\]), since $2{n}h(\varepsilon)\geq {\mu}\ln({n}/{\mu})$, for sufficiently large ${n}$, $$\begin{aligned}
\label{4.5}
\mathbb{H}\left(\mathcal{U}_{\mathsf{G}_{n}({\mu},{K})}\right)= \Omega\left({n}{\mu}\log\frac{{n}}{{\mu}}+{n}^{1-3/{\mu}} {K}\right).\end{aligned}$$ Substituting (\[4.5\]) into (\[4.20\]) and noting that $\mathrm{det}(\mathbf{{K}})=1$, when ${n}\rightarrow\infty$, it must hold that $${m}=\Omega\left({\mu}\log({{n}}/{{\mu}})+{K}/{n}^{3/{\mu}}\right)$$ to ensure that ${{\varepsilon}_{\mathrm{P}}}$ is smaller than $1/2$.
### Uniform Sampling of Trees
As one of the applications of Theorem \[thm:4\], we characterize the sample complexity of the uniform sampling of trees.
\[coro:1\] Suppose that the generator matrix $\mathbf{B}$ has Gaussian IID entries with mean zero and variance one and assume $G_{n}\sim\mathcal{U}_{\mathsf{T}_{n}^{\mathrm{all}}}$. There exists an algorithm that guarantees $\lim_{{n}\rightarrow\infty}{{\varepsilon}_{\mathrm{P}}}=0$ using ${m}=O\left(\log{n}\right)$ measurements. Conversely, the number of measurements must satisfy ${m}=\Omega\left(\log{n}\right)$ to make the probability of error ${{\varepsilon}_{\mathrm{P}}}$ less than ${1}/{2}$.
*Sketch of Proof:* The achievability follows from combining Theorem \[thm:4\] and Lemma \[lemma:trees\], by setting $K({n})=\log{n}$. Substituting $\mathbb{H}(\mathcal{U}_{\mathsf{T}_{n}^{\mathrm{all}}})=\Omega\left({n}\log{n}\right)$ into (\[4.20\]) yields the desired result for converse.
### Erdős-Rényi $({n},p)$ model
Similarly, the following corollary is shown by recalling Lemma \[lemma:renyi\].
\[coro:2\] Assume $G_{n}\sim\mathcal{G}_{\mathrm{ER}}({n},p)$ with $1/{n}\leq p\leq 1-1/{n}$. Under the same conditions in Corollary \[coro:1\], there exists an algorithm that guarantees $\lim_{{n}\rightarrow\infty}{{\varepsilon}_{\mathrm{P}}}=0$ using ${m}=O\left({n}h(p)\right)$ measurements. Conversely, the number of measurements must satisfy ${m}=\Omega\left({n}h(p)\right)$ to make the probability of error ${{\varepsilon}_{\mathrm{P}}}$ less than ${1}/{2}$.
*Sketch of Proof:* Taking ${K}={n}h(p)/\log{n}$ and ${\mu}={2{n}h(p)}/{(\ln 1/p)}$, we check that ${\mu}<{n}^{-3/{\mu}}({n}-{K})$ and ${K}=o({n})$. The assumptions on $h(p)$ guarantee that $h(p)\geq(\log{n})/{n}$, whence ${n}h(p)=\omega\left(\log({n}/{K})\right)$. The choice of $\{\mu({n})\}$ and $\{K({n})\}$ makes sure that the sequence of distributions is $(\mu({n}),K({n}))$-sparse. Theorem \[thm:4\] implies that ${m}=O({n}h(p))$ is sufficient for achieving a vanishing probability of error. For the second part of the corollary, substituting $\mathbb{H}(\mathcal{G}_{\mathrm{ER}}({n},p))=h(p){{n}\choose 2}=\Omega\left({n}^2 h(p)\right)$ into (\[4.20\]) yields the desired result.
Structure-based Parameter Recovery {#sec:5}
==================================
Often in practice, some prior information of the graph topology is available. For example, in a power system, besides knowing that the transmission network is a radial network, if in addition, we are able to know from locational and geographical information or past records that some of the nodes in $\mathcal{N}$ are *not* connected through a power line, then the size of the candidacy set $\mathsf{G}_{n}$ becomes smaller, allowing a potential improvement on sample complexity. Applying the Kirchhoff’s theorem (*c.f.* [@west2001introduction]) stated below, our results are extended to practical situations.
\[thm:kirchhoff\] Let $H$ be a connected graph with ${n}$ labeled nodes. Then the number of spanning trees denoted by $\kappa(H)$ is given by the product of $1/{n}$ and all non-zero eigenvalues of the (unnormalized) Laplacian matrix of $H$: $$\begin{aligned}
\label{2.6}
\kappa(H)=\frac{1}{{n}}{\lambda}_1{\lambda}_2\cdots {\lambda}_{{n}-1}=\det\left(L_{H}'\right)
\end{aligned}$$ where $L_{H}'$ denotes the reduced Laplacian of $H$ (cofactor) by deleting the first column and row from the Laplacian matrix $L_H$.
Therefore, if we know a priori that the topology to be recovered is a spanning tree lying in some known underlying graph $H$, then the size of the candidacy set $\mathsf{T}_{n}^H$ is given by $\left|\mathsf{T}_{n}^H\right|=\kappa(H)$. Let $\mathcal{U}_{\mathsf{T}_{n}^H}$ denote the uniform distribution on $\mathsf{T}_{n}^H$. As a remark, when we have no additional information of the underlying graph and we only know $G$ is a spanning tree, $H$ becomes the complete graph with ${n}$ nodes and $|\mathsf{G}_{n}|={n}^{{n}-2}$. The following corollary is obtained as a direct application.
\[corollary:subgraph\_lower\] Under the same assumption of Corollary \[coro:1\], if $G\sim\mathcal{U}_{\mathsf{T}_{n}^H}$, then the number of measurements ${m}$ must satisfy $$\begin{aligned}
{m}=\Omega\left(\frac{1}{{n}}\log\left({\prod_{j=1}^{{n}-1}{\lambda}_j}\right)\right)
\end{aligned}$$ to make the probability of error $\mathbbm{P}_{e}^{\mathrm{T}}$ less than ${1}/{2}$. Here, ${\lambda}_1,\ldots,{\lambda}_{{n}-1}$ denote the non-zero eigenvalues of the Laplacian matrix of $H$.
*Sketch of Proof:* The proof follows along the same lines as those of Corollary \[coro:1\] and \[coro:2\]. Putting $\kappa(H)={\lambda}_1{\lambda}_2\cdots {\lambda}_{{n}-1}/{n}$ into (\[4.2\]) gives the bound.
The next achievability follows straightforward by noting that the number of unknown entries in each $j$-th column of the graph matrix $L_H$ is at most $\max\mathrm{diag}_j\left(L_H\right)$.
\[corollary:subgraph\_upper\] Under the same assumption of Corollary \[coro:1\], if $G\sim\mathcal{U}_{\mathsf{T}_{n}^H}$, then the following upper bound on the number of measurements ${m}$ is sufficient to achieve a vanishing probability of error ${{\varepsilon}_{\mathrm{P}}}=o(1)$: $$\begin{aligned}
{m}=O\left(\log\max\left(\mathrm{diag}_j\left(L_H\right)\right)\right).
\end{aligned}$$ Here, $\mathrm{diag}_j\left(L_H\right)$ denote the $j$-th diagonal entry of the (unnormalized) Laplacian matrix of $H$.
Heuristic Algorithm {#sec:6}
===================
We present in this section an algorithm motivated by the consistency-checking step in the proof of achievability (see Section \[sec:ach\]). Instead of checking the consistency of each subset of $\mathcal{N}$ consisting of ${n}-{K}$ nodes, as the two-stage scheme does and which requires $O({n}^{{K}})$ operations, we compute an estimate $X_j$ for each column of the graph matrix independently and then assign a score to each column based on its symemtric consistency with respect to the other columns in the matrix. The lower the score, the closer the estimate of the matrix column $X_j$ is to the ground truth $Y_j$. Using a scoring function we rank the columns, select a subset of them to be “correct”, and then eliminate this subset from the system. The size of the subset determines the number of iterations. Heuristically, this procedure results in a polynomial-time algorithm to compute an estimate $\mathbf{X}$ of the graph matrix $\mathbf{Y}$.
The algorithm proceeds in four steps.
### [Step $1$]{}. Initialization
Let matrices $\mathbf{A} \in \mathbb{R}^{{m}\times {n}}$ and $\mathbf{B}\in \mathbb{R}^{{m}\times {n}}$ be given and set the number of columns fixed in each iteration to be an integer $s$ such that $1 \leq s \leq {n}$. For the first iteration, set $\mathcal{S}(0) \leftarrow \mathcal{N}$, $\mathbf{A}(0)\leftarrow\mathbf{A}$, and $\mathbf{B}(0)\leftarrow\mathbf{B}$.
For each iteration $r=0,\ldots,\lceil {{n}}/{s}\rceil-1$, we perform the remaining three stages. The system dimension is reduced by $s$ after each iteration.
### [Step $2$]{}. Independent $\ell_1$-minimization
For all $j\in\mathcal{S}(r)$, we solve the following $\ell_1$-minimization:
$$\begin{aligned}
\nonumber\label{opt:xr}
X_j(r) = \operatorname*{arg\,min}_{x \in \mathbb{F}^{{n}-sr}} \quad &{\big|\big|x\big|\big|_{\ell_1}}\\
\nonumber
\mathrm{subject } \ \mathrm{ to}\quad &\mathbf{B}(r)x= A_j(r),\\
&x\in \mathcal{X}_j(r)\end{aligned}$$
Constraint (\[opt:xr\]) is optional; the set $\mathcal{X}_j(r)$ may encode additional constraints on the form of $x$ such as entry-wise positivity or negativity (*e.g.,* Section \[sec:sim\]). The forms of reduced matrix $\mathbf{B}(r)$ and reduced vector $A_j(r)$ are specified in Step 4.
### [Step $3$]{}. Column scoring
We rank the *symmetric consistency* of the independently solved columns. For all $j\in\mathcal{S}(r)$, let $$\begin{aligned}
\mathsf{score}_j(r):= \sum_{i=1}^{{n}-sr}\left|X_{i,j}(r) - X_{j,i}(r) \right|\end{aligned}$$ Note that if $\mathsf{score}_j(r)=0$ then $X_j(r)$ and its partner symmetric row in $\mathbf{X}(r)$ are identical. Otherwise there will be some discrepancies between the entries and the sum will be positive. The subset of the $X_j(r)$ corresponding to the $s$ smallest values of $\mathsf{score}_j(r)$ is deemed “correct”. Call this subset of correct indices $\mathcal{S}'(r)$.
###
Based on the assumption that $s$ of the previously computed columns $X_j(r)$ are correct, the dimension of the linear system is reduced by $s$. We set $\mathcal{S}(r+1)\leftarrow \mathcal{S}(r)\backslash\mathcal{S}'(r)$. For all $i,j\in\mathcal{S}'(r)$, we fix $$\label{eq:recover1}
X_{i,j} = X_{i,j}(r), \ X_{j,i}= X_{i,j}(r)$$ The measurement matrices are reduced to $$\begin{aligned}
\mathbf{B}(r+1)&\leftarrow \underline{\mathbf{B}}^{\mathcal{S}(r+1)},\\
\nonumber
A_j(r+1)&\leftarrow \underline{A}_j(r)-\sum_{i\in\mathcal{S}'(r)}\underline{B}_i X_{i,j}.\end{aligned}$$ When $r\leq {n}-{m}$, $\underline{\mathbf{B}}^{\mathcal{S}(r+1)}=\mathbf{B}^{\mathcal{S}(r+1)}$, $\underline{A}_j(r)={A}_j(r)$ and $\underline{B}_i={B}_i$. When $r>{n}-{m}$, to avoid making the reduced matrix $\mathbf{B}(r+1)$ over-determined, we set $\mathbf{B}(r+1)$ to be an $({n}-r)\times ({n}-r)$ sub-matrix of $\mathbf{B}^{\mathcal{S}(r+1)}$ by selecting ${n}-r$ rows of $\mathbf{B}^{\mathcal{S}(r+1)}$ uniformly at random. A new length-$({n}-r)$ vector $\underline{A}_j(r)$ is formed by selecting the corresponding entries from $A_j(r)$. Once the $\lceil {{n}}/{s}\rceil$ iterations complete, an estimate $\mathbf{X}$ is returned using (\[eq:recover1\]). The algorithm requires at most $\lceil {{n}}/{s}\rceil$ iterations and in each iteration, the algorithm solves an $\ell_1$-minimization and updates a linear system. Solving an $\ell_1$-minimization can be done in polynomial time (*c.f.* [@ge2011note]). Thus, the heuristic algorithm is a polynomial-time algorithm.
Applications in Electric Grids {#sec:sim}
==============================
Experimental results for the heuristic algorithm are given here for both synthetic data and IEEE standard power system test cases. The algorithm was implemented in Matlab; simulated power flow data was generated using Matpower 7.0 [@zimmerman2011matpower] and CVX 2.1 [@cvx] with the Gurobi solver [@gurobi] was used to solve the sparse optimization subroutine.
Scalable Topologies and Error Criteria {#sec:7.a}
--------------------------------------
We first demonstrate our results using synthetic data and two typical graph ensembles – stars and chains. For both topologies, the graph size was incremented from ${n}=5$ to ${n}=300$ and the number of samples required for accurate recovery of parameters and topology was recorded. For each simulation, we generated a complex-valued random admittance matrix $\mathbf{Y}$ as the ground truth. Both the real and imaginary parts of the line impedances of the network were selected uniformly and IID from $[-100,100]$. A valid electrical admittance matrix was then constructed using these impedances. The real components of the entries of $\mathbf{B}$ were distributed IID according to $\mathcal{N}\left(1, 1\right)$ and the imaginary components according to $\mathcal{N}\left(0, 1\right)$. $\mathbf{A} = \mathbf{Y}\mathbf{B}$ gave the corresponding complex-valued measurement matrix.
Given data matrices $\mathbf{A},\mathbf{B}$ the algorithm returned an estimate $\mathbf{X}$ of the ground truth $\mathbf{Y}$. If an entry of $\mathbf{X}$ had magnitude $|X_{i,j}| < \varepsilon$ (where $\varepsilon=10^{-5}$ was the error threshold), then the entry was fixed to be 0. Following this, if $\mathrm{supp}\left(\mathbf{X}\right)=\mathrm{supp}\left(\mathbf{Y}\right)$ then the topology recovery was deemed exact. The criterion for accurate parameter recovery was $|Y_{i,j}-X_{i,j}|<\varepsilon$ for all non-zero entries in both matrices. The number of samples ${m}$ (averaged over repeated trials) required to meet these criteria was designated as the sample complexity for accurate recovery. The sample complexity trade-off displayed in Figure \[fig:ieee\] shows approximately logarithmic dependence on graph size ${n}$ for both ensembles.
.
IEEE Test Cases
---------------
![The number of samples required to accurately recover the nodal admittance matrix is shown on the vertical axis. Results were averaged over 20 independent simulations. Star and chain graphs were scaled in size between 5 and 300 nodes. IEEE test cases ranged from 5 to 200 buses. In the latter case, there were no assumptions on the random IID selection of the entries of $\mathbf{Y}$ (in contrast to the star/chain networks). Linear and logarithmic (in ${n}$) reference curves are plotted as dashed lines.[]{data-label="fig:ieee"}](CDC_heuristic.pdf)
We also validated the heuristic algorithm on 17 IEEE standard power system test cases ranging from $5$ to $200$ buses. The procedure for determining sample complexity for accurate recovery was the same as above, but the data generation was more involved.
### Power flow data generation
A sequence of time-varying loads was created by scaling the nominal load values in the test cases by a times series of Bonneville Power Administration’s aggregate load on $02/08/2016$, 6am to 12pm [@bpa]. For each test case network, we performed the following steps to generate a set of measurements:
1. Interpolated the aggregate load profile to $6$-second intervals, extracted a length-$m$ random consecutive subsequence, and then scaled the real parts of bus power injections by the load factors in the subsequence.
2. Computed optimal power flow in Matpower for the network at each time step to determine bus voltage phasors.
3. Added a small amount of Gaussian random noise ($\sigma^2=0.001$) to the voltage measurements and generated corresponding current phasor measurements using the known admittance matrix.
### Sample complexity for recovery of IEEE test cases
Figure \[fig:ieee\] shows the sample complexity for accurate recovery of the IEEE test cases. The procedure and criteria for determining the necessary number of samples for accurate recovery of the admittance matrix were the same as for the synthetic data case. Unlike the previous setting, here we have no prior assumptions about the structure of the IEEE networks: networks have both mesh and radial topologies. However, because power system topologies are typically highly sparse, the heuristic algorithm was able to achieve accurate recovery with a comparable (logarithmic) dependence on graph size.
![Probability of error for parameter recovery ${{\varepsilon}_{\mathrm{P}}}$ for the the IEEE $30$-bus test case is displayed on the vertical axis. Probability is taken over 50 independent simulations. The horizontal axis shows the number of samples used to compute the estimate $\mathbf{X}$. The probability of error for independent recovery of each $X_j$ via $\ell_1$-norm minimization (double dashed line) and full rank non-sparse recovery (dot dashed line) are shown for reference. Adding the symmetry score function (second-to-left) improves over the naive column-wise scheme. Adding entry-wise positivity/negativity constraints on the entries of $\mathbf{X}$ (left-most curve) reduces sample complexity even further ($\approx 1/3$ samples needed compared to full rank recovery).[]{data-label="fig:ieee30"}](CDC_proberror.pdf)
### Influence of structure constraints on recovery
There are structural properties of the nodal admittance matrix for power systems—symmetry, sparsity, and entry-wise positivity/negativity—that we exploit in the heuristic algorithm to improve sample complexity for accurate recovery. The score function $\mathsf{score}_j(r)$ rewards symmetric consistency between columns in $\mathbf{X}$; the use of $\ell_1$-minimization promotes sparsity in the recovered columns; and the constraint set $\mathcal{X}_j$ in (\[opt:xr\]) forces $\mathrm{Re}(X_{i,j}) \leq 0,\ \mathrm{Im}(X_{i,j}) \geq 0$ for $i \neq j$ and $\mathrm{Re}(X_{i,j}) \geq 0$ for $i=j$. These entry-wise properties are commonly found in power system admittance matrices. In Figure \[fig:ieee30\] we show the results of an experiment on the IEEE $30$-bus test case that quantify the effects of the structure constraints on the probability of error.
[Proof of Lemma \[lemma:detect\]]{} {#app:proof_consistency_check}
-----------------------------------
Conditioned on $G\in\mathsf{G}_{n}({\mu},{K})$ and the assumption $\sigma_{3{\mu}}+3\sigma_{4{\mu}}< 2$, there are no less than ${n}-{K}$ many columns correctly recovered. The consistency-checking verifies that if the collection of an arbitrary set of nodes $\mathcal{S}$ of cardinality ${n}-{K}$ satisfies the symmetry property as the true graph $\mathbf{Y}$ must obey. Therefore, any such set $\mathcal{S}$ with $\left|\mathcal{S}\right|={n}-{K}$ must contain at least ${n}-2K$ many corresponding indexes of the correctly recovered columns. Then if the consistency-checking fails, it is necessary that there exist two distinct length-${n}$ vectors $Y'$ and $Y^*$ in $\mathbbm{F}^{{n}}$ such that $Y^*$ is the minimizer of the $\ell_1$-minimization (\[4.1\])-(\[4.2\]) that differs from the correct answer $Y'$, *i.e.,* $Y'\neq Y^*$ where $A=\mathbf{B}Y'$ and $$\begin{aligned}
Y^*=\operatorname*{arg\,min}_{Y} & \ \left|\left|Y\right|\right|_{\ell_1}\\
\mathrm{subject}\ \mathrm{to} \ &A=\mathbf{B}Y\\
&Y\in\mathbbm{F}^n\end{aligned}$$ for some $A\in\mathbbm{F}^{m}$ and furthermore, the vectors $Y'$ and $Y^*$ can have at most $2K$ distinct coordinates, $$\begin{aligned}
\left|\mathrm{supp}\left(Y'-Y^*\right)\right|\leq 2K.\end{aligned}$$
However, the constraints $\mathbf{B}Y'=A$ and $\mathbf{B}Y^*=A$ imply that $\mathbf{B}\left(Y'-Y^*\right)=0$, contradicting to $\mathrm{spark}(\mathbf{B})>2K$. Therefore, ${n}-{K}$ many columns can be successfully recovered if the decoded solution passes the consistency-checking. Moreover, since $\mathrm{spark}(\mathbf{B})>2K$ and number of unknown coordinates in each length-${K}$ vector $X_{j}^{\mathcal{S}^{\mathrm{c}}}$ (for $j=1,\ldots,|\mathcal{S}^{\mathrm{c}}|$) to be recovered is ${K}$, the solution of the system (\[4.3\]) is guaranteed to be unique. Thus, Algorithm \[alg:1\] always recovers the correct columns $Y_1,\ldots,Y_N$ conditioned on ${m}\geq \mathrm{spark}(\mathbf{B})>2K$.
[Proof of Lemma \[lemma:trees\]]{} {#app:proof_trees}
----------------------------------
Consider the following function $$\begin{aligned}
F(\mathcal{E}) = \sum_{j=1}^{{n}} f(d_j(G))\end{aligned}$$ where $d_j(G)$ denotes the degree of the $j$-th node and consider the following indicator function: $$\begin{aligned}
f(d_j(G)) := \begin{cases}
1 \quad &\text{if } d_j(G)> {\mu}\\
0 & \text{otherwise}
\end{cases}.\end{aligned}$$
Applying the Markov’s inequality, $$\begin{aligned}
\label{eq:a.1}
\mathbbm{P}\left(G\notin\mathsf{T}_{n}^{\mathrm{all}}({\mu},{K})\right)&=\mathbbm{P}_{\mathcal{U}_{\mathsf{T}_{n}^{\mathrm{all}}}}\left(F(\mathcal{E})\geq {K}\right)
\leq \frac{\mathbbm{E}_{\mathcal{U}_{\mathsf{T}_{n}^{\mathrm{all}}}}\left[F(\mathcal{E})\right]}{{K}}.\end{aligned}$$ Continuing from (\[eq:a.1\]), the expectation $\mathbbm{E}_{\mathcal{U}_{\mathsf{T}_{n}^{\mathrm{all}}}}\left[F(\mathcal{E})\right]$ can be further expressed and bounded as $$\begin{aligned}
\nonumber
\mathbbm{E}_{\mathcal{U}_{\mathsf{T}_{n}^{\mathrm{all}}}}\left[F(\mathcal{E})\right] &=
\sum_{j=1}^{{n}}\mathbbm{E}_{\mathcal{U}_{\mathsf{T}_{n}^{\mathrm{all}}}}\left[f(d_j(G))\right]\\
\label{eq:a.10}
& = \sum_{j=1}^{{n}} \mathbbm{P}_{\mathcal{U}_{\mathsf{T}_{n}^{\mathrm{all}}}}\left(d_j(G)> {\mu}\right).\end{aligned}$$ Since $G$ is chosen uniformly at random from $\mathsf{T}_{n}^{\mathrm{all}}$, it is equivalent to selecting its corresponding Prüfer sequence (by choosing ${n}-2$ integers independently and uniformly from the set $\mathcal{N}$, *c.f.* [@kajimoto2003extension]) and the number of appearances of each $j\in\mathcal{N}$ equals to $d_j(G)-1$. Therefore, for any fixed node $j\in\mathcal{N}$, the Chernoff bound implies that $$\begin{aligned}
\label{eq:a.12}
\mathbbm{P}_{\mathcal{U}_{\mathsf{T}_{n}^{\mathrm{all}}}}\left(d_j(G)> {\mu}\right)\leq \exp\left(-({n}-2) \mathbbm{D}_{\mathrm{KL}}\left(\frac{{\mu}}{{n}-2}\big|\big| \frac{1}{{n}}\right)\right)\end{aligned}$$ where $\mathbbm{D}_{\mathrm{KL}}(\cdot || \cdot)$ is the Kullback-Leibler divergence and $$\begin{aligned}
\label{eq:a.11}
\mathbbm{D}_{\mathrm{KL}}\left(\frac{{\mu}}{{n}-2}\big|\big| \frac{1}{{n}}\right)\geq\frac{{\mu}}{{n}-2}\ln{n}.\end{aligned}$$
Therefore, substituting (\[eq:a.11\]) back into (\[eq:a.12\]) and combining (\[eq:a.1\]) and (\[eq:a.10\]), setting ${\mu}\geq 1$ leads to $$\begin{aligned}
\mathbbm{P}\left(G\notin\mathsf{T}_{n}^{\mathrm{all}}({\mu},{K})\right)\leq \frac{{n}\exp(-{\mu}\ln{n})}{{K}}\leq \frac{1}{K}.\end{aligned}$$
[Proof of Lemma \[lemma:renyi\]]{} {#app:proof_renyi}
----------------------------------
For any fixed node $j\in\mathcal{N}$, applying the Chernoff bound, $$\begin{aligned}
\mathbbm{P}_{\mathcal{G}_{\mathrm{ER}}({n},p)}\left(d_j(G)> {\mu}\right)\leq \exp\left(-{n}\mathbbm{D}_{\mathrm{KL}}\left(\frac{{\mu}}{{n}}\big|\big| p\right)\right).\end{aligned}$$
Continuing from (\[eq:a.1\]), the expectation $\mathbbm{E}_{\mathcal{G}_{\mathrm{ER}}({n},p}\left[F(\mathcal{E})\right]$ can be further expressed and bounded as $$\begin{aligned}
\mathbbm{E}_{\mathcal{G}_{\mathrm{ER}}({n},p)}\left[F(\mathcal{E})\right]
\label{eq:a.3}
& \leq {n}\cdot \exp\left(-{n}\mathbbm{D}_{\mathrm{KL}}\left(\frac{{\mu}}{{n}}\big|\big| p\right)\right)\end{aligned}$$ where the probability $p$ satisfies $0<p\leq {\mu}/{n}<1$. Note that $$\begin{aligned}
\label{eq:a.2}
\mathbbm{D}_{\mathrm{KL}}\left(\frac{{\mu}}{{n}}\big|\big| p\right)= \frac{{\mu}}{{n}}\ln \frac{1}{p}+\left(1-\frac{{\mu}}{{n}}\right)\ln \frac{1}{1-p} - h(p)\end{aligned}$$ where the binary entropy $h(p)$ is in base $e$. Taking ${\mu}\geq{2{n}h(p)}/{(\ln 1/p)}\geq 2{n}p$, substituting (\[eq:a.2\]) into (\[eq:a.3\]) leads to $$\begin{aligned}
\mathbbm{E}_{\mathcal{G}_{\mathrm{ER}}({n},p)}\left[F(\mathcal{E})\right] \leq {n}\exp\left(-{n}h(p)\right).\end{aligned}$$ Therefore, (\[eq:a.1\]) gives $$\begin{aligned}
\mathbbm{P}\left(G\notin\mathsf{G}_{n}^{\mathrm{all}}({\mu},{K})\right)\leq \frac{{n}\exp\left(-{n}h(p)\right)}{{K}}.\end{aligned}$$
[^1]: Li, Werner and Low are with the Computing + Mathematical Sciences Department, California Institute of Technology, Pasadena, CA 91125 USA (e-mail: [email protected]; [email protected]; [email protected])
[^2]: This means the entries of the matrix $\mathbf{B}$ are IID normally distributed.
[^3]: Although the underlying network is a directed graph, when considering the fundamental limit for topology identification, we still refer to the recovery of an undirected graph $G$.
[^4]: In this exploratory work, we assume the measurements are noiseless and algorithms seek to recover each entry of the graph matrix *exactly*. When the measurements are noisy, Theorem \[thm:1\] provides general converse results as trade-offs between the number of measurement needed and the probability of error defined in (\[poe\]).
[^5]: Although for certain distributions, the computational complexity is not polynomial in ${n}$, the scheme still provides insights on the fundamental trade-offs between the number of samples and the probability of error for recovering graph matrices. Furthermore, motivated by the scheme, a polynomial-time heuristic algorithm is provided in Section \[sec:6\] and experimental results are reported in Section \[sec:sim\].
|
= 24truecm = 16truecm = -1.3truecm = -2truecm
\[theorem\]
-1truecm 2truecm [**STRINGY INSTABILITY OF TOPOLOGICALLY NON-TRIVIAL AdS BLACK HOLES AND OF deSITTER S-BRANE SPACETIMES\
**]{} 2truecm Brett McInnes 2truecm
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Republic of Singapore.\
E-mail: [email protected]\
1truecm
ABSTRACT
Seiberg and Witten have discussed a specifically “stringy" kind of instability which arises in connection with “large" branes in asymptotically AdS spacetimes. It is easy to see that this instability actually arises in most five-dimensional asymptotically AdS black hole string spacetimes with non-trivial horizon topologies. We point out that this is a more serious problem than it may at first seem, for it cannot be resolved even by taking into account the effect of the branes on the geometry of spacetime. \[It is ultimately due to the [*topology*]{} of spacetime, not its geometry.\] Next, assuming the validity of some kind of dS/CFT correspondence, we argue that asymptotically deSitter versions of the Hull-Strominger-Gutperle S-brane spacetimes are also unstable in this “topological" sense, at least in the case where the R-symmetries are preserved. We conjecture that this is due to the unrestrained creation of “late" branes, the spacelike analogue of large branes, at very late cosmological times. 3.5truecm
1. Large Brane Instability and Topologically Non-Trivial AdS Black Holes {#large-brane-instability-and-topologically-non-trivial-ads-black-holes .unnumbered}
========================================================================
The fact that asymptotically AdS black holes can have topologically non-trivial event horizons [@kn:lemos][@kn:peldan][@kn:mann] has attracted considerable attention in connection with the AdS/CFT correspondence. In five dimensions [@kn:birmingham], apart from the spherical, flat, and hyperbolic possibilities for the structure of the event horizon, Milnor’s [@kn:milnor] “prime decomposition" of compact orientable 3-manifolds suggests further candidates, and black hole spacetimes with some of the corresponding event horizons have been constructed in a remarkable paper of Cadeau and Woolgar [@kn:cadeau].
In [@kn:witten2], Witten remarks that if one wishes to generalise the AdS/CFT correspondence to bulk manifolds which are merely [*asymptotically*]{} AdS, then \[in the Euclidean formulation at least\] one must take care that the scalar curvature at conformal infinity should remain non-negative. For otherwise the massless scalars of the CFT will undergo “runaway" behaviour. For AdS space itself, the \[Euclidean\] boundary is a conformal sphere with its canonical structure, represented by a metric of positive scalar curvature; the latter prevents a runaway through the conformal coupling term. Notice that it is [*not*]{} claimed that [*all*]{} CFTs misbehave on spaces of negative curvature – this is of course not the case. The instability arises only for a subclass of CFTs arising “holographically", as for example in the AdS/CFT correspondence. \[This point is emphasised in [@kn:witten3]; see also [@kn:yau]\]. Note that the two-dimensional case is particularly delicate here; but we stress that, in any case, [*none of the CFTs discussed in this work will be defined on a manifold of dimension less than three*]{}.
The bulk version of this runaway was explained in detail in [@kn:seiberg]; the instability arises from the nucleation and growth of “large branes", that is, in the five-dimensional case, 3-branes which are generated at large distances, “near" the boundary. The relevant Euclidean action for a (D-1)-brane in a (D+1)-dimensional asymptotically AdS space is, in the notation of [@kn:seiberg], $$\label{eq:action}
S = {Tr^D_0 \over 2^D} \int \sqrt{g}\; \left( (1-q)\phi^{{2D\over D-2}}+{8 \over (D-2)^2}[(\partial\phi)^2 + {D-2\over 4(D-1)}\phi^2R] + {\cal O}(\phi^{{2(D-4)\over D-2}})\right).$$ Here $D$ is [*strictly*]{} greater than 2 — that is, this equation, and consequently the rest of our discussion, is only valid for spacetimes of dimension at least 4 \[and consequently only for conformal boundaries of dimension at least 3\]. The brane carries a charge $q$ under a background antisymmetric field; the field ${\phi}$ tends to infinity as the conformal boundary is approached, and $R$ is the scalar curvature of that boundary. In the BPS case, the action will be unbounded below if $R$ is negative: there will be unstable production of branes “near" the boundary. The case where $R$ is zero is more delicate; one expects instability in some cases but not in others, depending on higher-order corrections [@kn:witten3]. As the boundary metric is only defined modulo conformal factors, and as scalar curvature is not a conformal invariant, perhaps we should clarify. A conformally invariant formulation is most easily given in terms of the scalar fields on the boundary. These are governed by the “conformal Laplacian", defined \[for $D > 2$\] by $$\label{eq:conflap}
\Delta_{CON} = \Delta + {{D-2} \over {4(D - 1)}}R,$$ where $\Delta$ is the ordinary Laplacian. The conformally invariant form of the stability condition is that this operator should be non-negative [@kn:yau]. For this to hold, it must be possible to choose a conformal gauge such that $R$ \[a function of position in general\] is everywhere non-negative. In the the case of a [*compact*]{} boundary, this is easier to understand because it is then always possible [@kn:parker] to choose a conformal gauge such that $R$ is [*constant*]{}. This is usually, but not invariably [@kn:jin], also possible for complete, non-compact Riemannian manifolds.
Now consider a five-dimensional Euclidean AdS black hole with metric $$\label{eq:adshyper}
g(EAdS_5;hyper) = {f_5}(r) \; dt \otimes dt + {f_5}^{-1}(r) \; dr \otimes dr + r^2h_{ij}[H^3/\Gamma] \; dx^i \otimes dx^j$$ with $$\label{eq:f}
{f_5} = -1 - {16\pi G \mu \over 3r^2 \; {\rm{Vol}}({H^3}/ {\Gamma})} + {r^2 \over L^2},$$ where ${\mu}$ is the mass parameter and $h_{ij}[H^3/\Gamma] \; dx^i \otimes dx^j
$ is the metric on the three-dimensional horizon, ${H^3}/ {\Gamma}$. Here $H^3$ is the three-dimensional hyperbolic space of constant sectional curvature $-1$, and $\Gamma$ is an infinite discrete group acting on $H^3$ in such a way that the quotient is compact. The metric at infinity \[that is, the canonical representative of the induced conformal structure on the boundary\] is $$\label{eq:gamma4}
{\gamma}_4 = dt \otimes dt + {L^2}h_{ij}[H^3/\Gamma] \; dx^i \otimes dx^j.$$ The scalar curvature of this metric is $-6/{L^2}$. Thus, large branes apparently grow uncontrollably at large distances in this spacetime. Before commenting on this, let us take note of the topology of the space on which ${\gamma}_4$ is defined. First, as usual for Euclidean black holes, we have to identify time periodically, so $t$ is a parameter on a circle. Next, the topology of hyperbolic space is that of ${{I\!\! R}}^3$, so we see finally that the topology of the boundary is that of ${S^1} \times {{{I\!\! R}}^3}/{\Gamma}$. Of course, the conformal boundary of Euclidean AdS space itself has a very different topology, namely that of a sphere. Thus we see that a spacetime can be asymptotically AdS, [*and yet have a conformal infinity with a topology which differs very radically from that of the boundary of AdS space*]{}.
It is clear that the hyperbolic AdS black hole is not stable as a string background. However, we wish to argue that it is by no means obvious, [*from this alone,*]{} that such spacetimes must be rejected as string backgrounds. Instability, in reality, rarely leads to indefinitely increasing perturbations: it is usually ultimately self-limiting, and we should assume that this is the case here unless we can prove the contrary. In using the Seiberg-Witten criterion in the above way, we are ignoring the effect of the branes on the space-time geometry. This is a good approximation at short distances, but eventually, at extremely large distances from the event horizon, the branes must have such an effect. In the Lorentzian picture, the geometry of the spacelike sections will begin to evolve in response to the branes, and, as can be seen from the form of equations 3 and 5 \[in which the geometry of the spacelike sections is clearly influenced by the geometry at infinity and vice versa\], this means that the geometry at infinity must also change. This will in general change the scalar curvature, so the Seiberg-Witten instability condition might cease to hold. In physical language, one would say that the back-reaction of the branes on the geometry eventually causes the spacetime to settle down to a new, approximately static state. But now we have an obvious consistency check: clearly, the scalar curvature of the boundary of the \[Euclidean version of\] the final spacetime geometry must be positive or zero; otherwise we would have a contradiction. In the Euclidean language of [@kn:seiberg]: in order for large-brane instability to limit itself through back-reaction, it must be possible to deform the Euclidean version of the spacetime in such a way that the scalar curvature at infinity changes from negative to non-negative values. In short, [*unless we can prove otherwise,*]{} we should assume that the real significance of Seiberg-Witten instability for this spacetime is that formulae \[eq:adshyper\] and \[eq:f\] are not to be trusted at extremely large distances, so that the metric $\gamma_4$ does not accurately reflect the geometry of the boundary in this case. This greatly complicates the study of these black holes from the AdS/CFT point of view, but in itself it does not rule them out as string backgrounds.
Circumstantial evidence that something of this kind can happen comes from the study of asymptotically AdS black holes with [*flat*]{}, toral event horizons. Intuitively, one would expect that AdS itself should be the thermal background for any asymptotically AdS black hole, and this is certainly the case when the horizon is spherical. But it has been argued [@kn:surya] that it is [*not*]{} so when the horizon is a torus. The argument is based on a conjecture, due to Horowitz and Myers [@kn:horowitz], that the “AdS soliton" is the lowest energy metric among all metrics which are asymptotic to it sufficiently rapidly. The \[Euclidean\] metric of the soliton is given, in $n+1$ dimensions, by $$\label{eq:adssoliton}
g(AdS;soliton) = {r^2}dt \otimes dt + {{L^2} \over {r^2}}{(1 - {{{r_0}^n} \over {r^n}})^{-1}}dr \otimes dr + {{r^2} \over {L^2}}{(1 - {{{r_0}^n} \over {r^n}})}d\phi \otimes d\phi + {r^2}\sum^{n-2}_{i=1} \; d\theta_i \otimes d\theta_i.$$ Here $r_0$ and L are positive constants, $r$ is a radial coordinate satisfying $r>r_0$, and $\phi$ and the $\theta_i$ are angular coordinates of various periodicities. In the Euclidean case, $t$ too is periodic. Thus the conformal infinity of the soliton has the structure of an n-dimensional torus, $T^n$. This is of course the same as the conformal infinity of an asymptotically AdS black hole with a flat \[toral\] horizon, and the black hole metric does approach that of the soliton sufficiently rapidly for the Myers-Horowitz conjecture to apply. If the latter is correct, then, as argued in [@kn:surya], we should certainly use the soliton as the thermal background for toral asymptotically AdS black holes, [*not*]{} AdS itself.
There is now quite impressive evidence in favour of the Myers-Horowitz conjecture [@kn:galloway1][@kn:galloway2][@kn:gallowaynew] and for the claim that the soliton is the correct thermal background for toral black holes [@kn:page1][@kn:page2]. The resulting picture of these “toral-boundary" spacetimes is very satisfactory; for example, they exhibit a well-defined confinement/deconfinement transition similar to, but interestingly different from, the more familiar transition which is known to occur [@kn:witten4] in the case of the AdS-Schwarzschild black hole. There is certainly no hint of any kind of runaway on the boundary or large-brane instability in the bulk. Now the conformal structure at infinity is represented by an exactly flat metric. But suppose that we slightly perturb the geometry of either the toral black hole or the soliton at some point deep in the bulk. Suppose that this perturbation causes the boundary conformal structure to change so that it is no longer the conformal structure represented by the flat metric. Then, using techniques to be explained below, one can show that the scalar curvature of the resulting metric on the boundary [*cannot be everywhere positive or zero*]{}. That is, the slightest perturbation of the boundary conformal geometries of these spacetimes renders them unstable in the Seiberg-Witten sense.
We suggest that this can be reconciled with the good thermodynamical behaviour of these spacetimes by invoking back-reaction as above. That is, we predict that the \[Lorentzian\] geometry evolves in such a way that the \[Euclidean version of\] the final metric corresponds to a boundary metric with a scalar curvature that has been driven back towards zero. \[It cannot become positive everywhere — see below.\] If this prediction proves to be false, then the implication is that the AdS soliton is unstable against arbitrarily small fluctuations, which seems very unlikely. \[Note that it follows from the main theorem stated in the next section that [*every*]{} metric of zero scalar curvature on a torus is perfectly flat — that is, if the scalar curvature vanishes on a torus, so does every curvature component. This does not imply, however, that the final metric after back-reaction effects is necessarily identical to that given in equation 6. For it is known [@kn:anderson] that when the boundary is topologically a torus, the boundary conformal structure does not uniquely determine the bulk metric, not even when the bulk is an Einstein manifold.\]
The question, then, is this: can the manifold ${S^1} \times {{{I\!\! R}}^3}/{\Gamma}$, which we obtained above as the Euclidean boundary of the five-dimensional hyperbolic AdS black hole spacetime, be given a metric of non-negative scalar curvature? Intuition may suggest that it should be impossible to find a metric of positive scalar curvature on ${{{I\!\! R}}^3}/{\Gamma}$, but, before we trust intuition, note that it is certainly possible \[[@kn:besse], page 123\] to find a metric of [*constant negative*]{} scalar curvature on the 3-sphere $S^3$. The point is that the scalar curvature is just the “average" of all of the curvature components. Thus, if even one component can be made sufficiently negative at each point, the scalar curvature on $S^3$ will be negative at each point. The reader can visualise this by imagining a deformation of the ordinary 3-sphere such that in [*some*]{} directions at each point the geometry becomes “saddle-like", while remaining “sphere-like" in other directions. \[This corresponds to modifying the metric, but [*not*]{} the topology, of the 3-sphere.\] Similarly, while it is of course impossible to force [*all*]{} of the sectional curvatures on ${{{I\!\! R}}^3}/{\Gamma}$ to be positive, one would be surprised to find that this cannot be arranged for the [*average*]{}. Even if it is indeed impossible to construct a positive scalar curvature metric on ${{{I\!\! R}}^3}/{\Gamma}$, it might still be possible to construct a warped product \[or even more general\] metric on ${S^1} \times {{{I\!\! R}}^3}/{\Gamma}$ with positive scalar curvature, since warping can certainly change the sign of scalar curvature $-$ compare the positive scalar curvature product metric $dx \otimes dx + d\theta \otimes d\theta + {sin^2}(\theta)d\phi \otimes d\phi$ on $(0, \infty) \times S^2$ with the negative scalar curvature warped metric $dx \otimes dx + {sinh^2}(x) [d\theta \otimes d\theta + {sin^2}(\theta)d\phi \otimes d\phi]$. Clearly we need some powerful mathematical technique to settle this.
Such a technique exists, and in the next section we give a very brief overview of it. The upshot is that it is actually [*impossible*]{} to define a metric of positive or zero scalar curvature on this manifold: no matter how we deform it, the scalar curvature remains resolutely negative. Thus, these black hole spacetimes are unstable \[if we embed them in string theory\] in a very radical way. The only escape route is to suppose that the branes actually change the [*topology*]{} of spacetime, a possibility that we shall also consider. Finally, in section 3 we show that, if some kind of “dS/CFT" correspondence is valid, then any asymptotically deSitter version of the Hull-Strominger-Gutperle “S-brane" spacetime is also radically unstable in the same sense, at least if we maintain the ansatz — used to obtain all known examples of S-brane spacetimes — that the R-symmetries are preserved. We believe that this is due to “late" S-branes, the spacelike analogue of “large" branes.
2. Topologically Induced Instability {#topologically-induced-instability .unnumbered}
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In this section we introduce a geometric technique which, in view of the importance of scalar curvature in Seiberg-Witten instability, is the natural one for our purposes. The main reference is [@kn:lawson].
A smooth map $f: X \rightarrow Y$ from one Riemannian manifold to another is said to be $\epsilon$-contracting if for any piecewise smooth curve $C$ in $X$, $$\label{eq:curve}
L[f(C)] \leq {\epsilon} \times L[C],$$ where $L$ denotes the length. A map which takes every point of $X$ to a single point of $Y$ evidently satisfies this condition for any positive $\epsilon$. To avoid this trivial case, we consider only mappings of non-zero [*degree*]{} \[see [@kn:lawson], page 303; note that mappings may have to be “constant at infinity" to make sense of this if $X$ happens to be non-compact.\] An n-dimensional manifold $M$ is said to be [*enlargeable*]{} if, given any Riemannian metric on $M$, for any positive $\epsilon$ there exists an orientable covering manifold which admits an $\epsilon$-contracting map \[with respect to the pull-back metric\] of non-zero degree onto the unit n-sphere. Notice that enlargeability is a topological condition; we can nevertheless think of an enlargeable manifold as one which, like a torus but unlike a sphere, has “arbitrarily large" covering spaces. Simple examples are provided by compact manifolds, such as tori and the underlying manifolds of compact hyperbolic spaces, which admit a metric of non-positive [*sectional*]{} curvature. Thus, ${{{I\!\! R}}^3}/{\Gamma}$ above is enlargeable. The following properties of enlargeable manifolds are important. \[See [@kn:lawson], page 306.\]
1. The product of any compact enlargeable manifold with a torus of any dimension is again enlargeable.
2. The connected sum of a compact enlargeable manifold with any compact manifold is again enlargeable.
3. If the scalar curvature of a Riemannian metric on a compact enlargeable spin manifold is non-negative, then the metric must be flat.
\[Recall that the connected sum of two manifolds is obtained by removing small balls from each, and then joining the two along the resulting boundaries.\] Now point 1 informs us that ${S^1} \times {{{I\!\! R}}^3}/{\Gamma}$ is enlargeable. Point 3 then implies, since this space \[like all products of a circle with an orientable compact 3-dimensional manifold\] is a spin manifold, that there can be no metric of positive scalar curvature on this manifold, [*not even if we allow warped products.*]{} Furthermore, if the scalar curvature were zero, then the metric would have to be flat. But this is not possible: ${S^1} \times {{{I\!\! R}}^3}/{\Gamma}$ does not have the topology of a flat manifold. \[One says that $\Gamma$ is [*homotopically atoroidal*]{} \[[@kn:besse],page 158\] and that, as the name suggests, means that $\Gamma$ cannot occur as part of the fundamental group of a manifold which can be flat [@kn:wolf].\] Thus, the scalar curvature of ${S^1} \times {{{I\!\! R}}^3}/{\Gamma}$ [*cannot be everywhere positive or zero*]{}, no matter how the manifold is deformed.
We conclude that once large branes begin to develop in the hyperbolic AdS black hole spacetime, nothing can rein in the instability: for, no matter how the branes deform the spacetime, the scalar curvature at infinity can never become everywhere positive or zero. The instability is [*induced topologically*]{} [@kn:mcinnes2]. We are forced to conclude that these spacetimes simply cannot arise as solutions of string or M theory. In fact, this conclusion is actually consistent with the findings of [@kn:surya], where the authors remark that [*no analogue of the AdS soliton seems to exist in the hyperbolic case*]{}. It is now clear why this is so: there can be no well-behaved ground state, analogous to the AdS soliton, for these radically unstable spacetimes. \[We agree with these authors that pure AdS is not the correct ground state to use in the hyperbolic case; this is argued most convincingly in the introduction to [@kn:galloway2].\]
More generally, Milnor’s [@kn:milnor] “prime decomposition" theorem states that any compact orientable 3-manifold $M^3$ can be expressed as a connected sum in the following way: $$\label{eq:hex}
M^3 = {\Sigma_1} \# {\Sigma_2} \# ... \# (S^1 \times S^2) \# (S^1 \times S^2) \# ... \# K_1 \# K_2 \# ...,$$ where each $\Sigma_i$ is a manifold covered by a homotopy 3-sphere, where $\#$ denotes the connected sum, and where each $K_i$ is an Eilenberg-MacLane space of the form $K(\pi,1)$. We shall assume the truth of the Poincarè conjecture, so that the reader can interpret “homotopy 3-sphere" as $S^3$. Then the $\Sigma_i$ are just quotients of $S^3$ by \[completely known\] finite groups; for example, the real projective space $S^3/{{Z\!\!\! Z}_2}$ is one possibility. \[See [@kn:weeks] for a readily accessible statement of the classification.\] A $K(\pi,1)$ space is just a 3-dimensional manifold whose only non-trivial homotopy group is its fundamental group. It is not yet proven that all compact $K(\pi,1)$ spaces are enlargeable, but all known examples are so, and furthermore it is known \[[@kn:lawson], page 324\] that no such manifold can accept a metric of positive scalar curvature, and that the same is true of the connected sum of a $K(\pi,1)$ with any other compact manifold. Therefore, in view of properties 2 and 3 listed above, it is reasonable to conjecture that all compact $K(\pi, 1)$ spaces are enlargeable.
Assuming the truth of this, we see that any compact orientable 3-manifold is enlargeable if it has at least one $K_i$ in its Milnor decomposition. Thus we find that compact orientable 3-manifolds fall into three categories: the six well-understood [@kn:wolf] manifolds which can be flat \[see [@kn:reboucas] for an accessible review\], the enlargeable manifolds of “non-flat" topology, and manifolds of the form ${\Sigma_1} \# {\Sigma_2} \# ... \# (S^1 \times S^2) \# (S^1 \times S^2) \# ...$ Every member of this last class has a metric of constant positive scalar curvature [@kn:schoen][@kn:joyce]. Assume that, as is the case in physically interesting examples, the topology of the black hole spacetime is a product of the event horizon topology with ${{I\!\! R}}^2$. Then the conformal boundary of the corresponding Euclidean asymptotically AdS black hole spacetime in five dimensions will be a product of $S^1$ with a member of one of these classes, and so property 1 above implies that the scalar curvature of the conformal boundary cannot be non-negative unless the event horizon is either flat or of the form ${\Sigma_1} \# {\Sigma_2} \# ... \# (S^1 \times S^2) \# (S^1 \times S^2) \# ...$. Hence we conclude that large-brane instability rules out all five-dimensional asymptotically AdS black holes \[as string backgrounds\] except those with flat or ${\Sigma_1} \# {\Sigma_2} \# ... \# (S^1 \times S^2) \# (S^1 \times S^2) \# ...$ event horizons. \[Strictly speaking, this is under the assumption that the spacetime topology has the above product form. We conjecture, however, that it is true in general that the conformal boundary of \[the Euclideanized version of\] a black hole spacetime with an enlargeable event horizon is necessarily itself enlargeable.\] The flat case we have considered already, and the AdS-Schwarzschild black hole can have any $\Sigma_i$ as event horizon. It would be interesting to exhibit a black hole — perhaps one should really say “black string" — with, for example, ${\Sigma_1} \# {\Sigma_2} \# (S^1 \times S^2)$ as event horizon. In any case, these are the only remaining possibilities for event horizons of string theoretic asymptotically AdS black holes in five dimensions.
Throughout this discussion, we have assumed that while the nucleation of large branes can change the geometry of spacetime, it cannot change the [*topology*]{}. But it is well known that D-branes can in fact change the topology of 10 or 11 dimensional spacetimes in string/M theory [@kn:aspinwall1][@kn:aspinwall2]. However, that kind of topology change usually involves changes of topology among the members of some [*family*]{} of spacetimes as a parameter is changed. Even in the cases where the topology change occurs within [*one*]{} spacetime, as in [@kn:greene], it occurs along some spacelike dimension of a higher-dimensional spacetime. In our case, in order to affect the topology at infinity, the branes would have to change the topology of the spacelike sections as they evolve in [*time*]{} in the Lorentzian version of the spacetime. \[Again, to see this, consider equations 3 and 5: the topology of the boundary is controlled by that of the spatial sections.\] But classical theorems of Geroch [@kn:geroch] and Tipler [@kn:tipler] state that spacetimes in which topology changes occur due to temporal evolution are singular or contain closed timelike worldlines. Admittedly, Horowitz [@kn:horowitz2] has argued that the singularities are not necessarily very drastic in cases where the topology change is very simple. Even if we accept that argument, however, the topology change here would necessarily be very extreme — from a compact hyperbolic space, for example, with its infinite and very complex fundamental group, to a sphere. It is most unlikely that such a major topological change can be effected without inducing non-innocuous singularities or causality violations. In short, it is conceivable that that large branes can avert a catastrophic runaway by modifying the topology of spacetime, but it is highly likely that this has to be paid for by some kind of equally catastrophic gravitational collapse in the region far [*outside*]{} the event horizon. Thus it seems clear that changing topology cannot save the situation, though the details of this deserve further investigation.
We wish to impress on the reader the extreme nature of the restriction imposed on black hole horizons by Seiberg-Witten instability. For there is a definite sense in which compact orientable 3-manifolds having no $K(\pi, 1)$ component in their Milnor decomposition are a “tiny minority". This instability rules out a large number of apparently acceptable spacetimes. We now argue that, if there is a “dS/CFT correspondence", it has a similar effect in the case of asymptotically deSitter S-brane spacetimes.
3. Late Brane Instability of Asymptotically deSitter S-Brane Spacetimes {#late-brane-instability-of-asymptotically-desitter-s-brane-spacetimes .unnumbered}
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\[Throughout this section, for technical reasons connected with the form of the S-brane metric we shall consider below, all spacetimes will be four-dimensional; therefore all event horizons will be two-dimensional and all CFTs will be defined on [*three-dimensional*]{} conformal boundaries.\]
The above discussion of the Seiberg-Witten instability was given, as in [@kn:seiberg], in the Euclidean formulation. To repeat: a negative scalar curvature on the boundary gives rise to a scalar field “runaway" which is the AdS/CFT counterpart of the unstable emission of “large branes" in the bulk. When we turn to the dS/CFT correspondence \[[@kn:hull1][@kn:park1][@kn:park2][@kn:witten1][@kn:strominger1][@kn:strominger2][@kn:medved1]; see [@kn:balasubramanian][@kn:leblond] for relevant recent discussions\], we find that, since conformal infinity is now spacelike, the CFT is automatically defined on a space of Euclidean signature, even if the bulk is Lorentzian. We can therefore reasonably hope that the lessons we learned in the Euclidean AdS case will apply here: a CFT on a boundary with scalar curvature which is not positive or zero will suffer a runaway in this case too. \[We should perhaps stress that the dS/CFT correspondence is by no means supported by as much evidence as its AdS counterpart. However, for our purposes, the precise details, and even the exact validity, of the dS/CFT correspondence are not essential. All we require is that a runaway in a CFT at infinity should be reflected in some kind of bulk instability. We certainly expect this to hold true even if, as Susskind [@kn:susskind] has suggested, the dS/CFT correspondence can only be approximately valid. In fact, anything else would signal a basic failure of the holography principle in the deSitter context.\]
Now in fact [*non-singular*]{} asymptotically deSitter spacetimes are well-behaved from this point of view. For the main result of [@kn:galloway3] states the following. Suppose that an asymptotically deSitter spacetime is [*future asymptotically simple*]{}, in the sense that every future-directed inextendible null geodesic has an end point on future null infinity; this essentially just means that there are no singularities. Then, if the Dominant Energy Condition \[DEC\] holds, the spacetime is globally hyperbolic with compact Cauchy surfaces having finite fundamental groups. This result implies that future and past conformal infinity are also compact with finite fundamental groups. \[Notice that the DEC is essential here: the spacetime given in [@kn:mcinnes1] is asymptotically deSitter and future asymptotically simple, but conformal infinity has an infinite fundamental group. Of course, this spacetime does not satisfy the DEC.\] In the case of a four-dimensional asymptotically deSitter space, this compactness in turn means that \[assuming orientability\] we can use the Milnor decomposition; and clearly there can be no $K(\pi, 1)$ factors in this case, since such manifolds always have infinite fundamental groups. \[This does [*not*]{} mean that the scalar curvature must necessarily be positive $-$ recall that the scalar curvature [*can*]{} be negative even for a 3-sphere $-$ but it does mean that the manifold can be deformed so that the scalar curvature [*becomes*]{} positive, that is, all these manifolds admit a metric of positive scalar curvature [@kn:schoen][@kn:joyce].\] Thus, there is no danger of Seiberg-Witten instability for non-singular asymptotically deSitter spacetimes.
Of course, this result does not apply to spacetimes that are asymptotically deSitter but not future asymptotically simple. To see why such spacetimes could be of considerable interest, consider the [*spacelike branes*]{} introduced by Hull [@kn:hull2] and analysed in detail in [@kn:gutperle1] \[see also [@kn:gutperle2][@kn:peet][@kn:ivashchuk]; see [@kn:hashimoto] for more recent references\]. An explicit four-dimensional S-brane metric is given by [@kn:gutperle1] $$\label{eq:ansatz}
-{{d\tau^2}\over\lambda^2} + {\lambda^2}dz^2 + R^2 d{H_2}^2,$$ where $d{H_2}^2$ denotes a locally hyperbolic metric \[of constant curvature $-1$\], and where $\lambda$ and $R$ are simple functions of the time coordinate $\tau$. \[The hyperbolic term ensures the preservation of the R-symmetries.\] This metric is asymptotically [*flat*]{} in the remote past and future. Its Penrose diagram [@kn:buchel1][@kn:buchel2] is obtained simply by means of a ninety degree rotation of the Penrose diagram of a Schwarzschild black hole, with the understanding that each point represents a locally hyperbolic space instead of a sphere. Thus the spacetime has non-spacelike future and past conformal boundaries and is nakedly singular. \[The significance of this last property is discussed in [@kn:buchel1][@kn:buchel2]. See also [@kn:burgess][@kn:quevedo] for an interesting interpretation of the singularities.\] Thus we have an important example of a spacetime which is not future asymptotically simple.
As our Universe is probably [@kn:ratra] asymptotically deSitter rather than asymptotically flat, it is very important to ask whether it is possible to modify the S-brane geometry so that the conformal boundary becomes spacelike. If such asymptotically deSitter S-brane spacetimes exist, their Penrose diagrams will be square, with naked singularities to each side as above, but with horizontal \[spacelike\] upper and lower boundaries. That is, their Penrose diagrams will be obtained from a ninety degree rotation of the diagram of an AdS black hole, just as the Penrose diagram of the above metric was obtained from a ninety degree rotation of the diagram of a Schwarzschild black hole. Let us try to construct an explicit example of a metric with such a Penrose diagram. \[The intention here is not to find an S-brane spacetime, but rather to give a concrete example of a metric with such a Penrose diagram, so that the structure of future conformal infinity can be determined for any spacetime having a diagram of this type. Following [@kn:buchel1][@kn:buchel2], we maintain the R-symmetries by assuming that the horizons are hyperbolic.\]
The Lorentzian four-dimensional anti-deSitter black hole with a hyperbolic horizon has metric $$\label{eq:adshyper4}
g(AdS_4;hyper) = - {f_4}(r) \; dt \otimes dt + {f_4}^{-1}(r) \; dr \otimes dr + r^2h_{ij}[H^2/\Delta] \; dx^i \otimes dx^j$$ with $$\label{eq:f4}
{f_4} = -1 - {8\pi G \mu \over r \; {\rm{Vol}}({H^2}/ {\Delta})} + {r^2 \over L^2};$$ note that the event horizon is now a compact Riemann surface, $H^2/\Delta$, where $\Delta$ is a discrete infinite group acting freely and properly discontinuously on the hyperbolic plane, $H^2$. \[ The conformal boundary is then a three-dimensional manifold with topology given by the product of the Riemann surface topology with a line.\] The Penrose diagram, as stated above, is a square with horizontal black and white hole singularities and with vertical timelike infinities. Now simply rotate this diagram through 90 degrees. The new diagram has spacelike, horizontal infinities and timelike, naked singularities to either side: it resembles what in the West would be called a Chinese lantern. This is the desired nakedly singular asymptotically deSitter spacetime; the AdS event horizon has been transformed to the dS cosmological horizon. Notice that, in equation \[eq:adshyper4\], the spacelike and timelike roles of $r$ and $t$ are exchanged inside the event horizon as usual. We can use this to deduce the metric of the “rotated" spacetime as follows. If $\alpha$ denotes the radius of the event horizon, then set $r = \alpha - \epsilon$ and substitute this into equation \[eq:f4\]. The result is, up to first order in $\epsilon$, $$\label{eq:epsilon}
{f_4} = {8\pi G \mu \epsilon \over {\alpha^2} \; {\rm{Vol}}({H^2}/ {\Delta})} + {{2\alpha\epsilon}\over L^2}.$$ Now we want the geometry just [*inside*]{} the AdS [*event*]{} horizon to resemble the geometry just [*outside*]{} the dS [*cosmological*]{} horizon. If the parameters are adjusted so that the latter has radius $\alpha$, then we have $r = \alpha + \epsilon$ in this case, and so we see that the desired metric is obtained simply by reversing the signs of the last two terms in equation \[eq:f4\]. That is, the dS analogue of the AdS black hole with hyperbolic event horizon is the nakedly singular spacetime with metric $$\label{eq:dshyper}
g(dS_4;hyper) = - j(r) \; dt \otimes dt + j^{-1}(r) \; dr \otimes dr + r^2h_{ij}[H^2/\Delta] \; dx^i \otimes dx^j$$ where $$\label{eq:j}
j = -1 + {8\pi G \mu \over r \; {\rm{Vol}}({H^2}/ {\Delta})} - {r^2 \over L^2}.$$
Of course we are not suggesting that an asymptotically deSitter S-brane metric will have exactly this form: no doubt there will be differences. The point, however, is that, in view of the way we constructed it, we can expect the [*asymptotic*]{} behaviour of this spacetime to mimic that of an asymptotically deSitter S-brane spacetime. Since we propose to use the dS/CFT correspondence, this is all we need.
Now in fact the metric 13 has already been discussed in the dS/CFT literature: it was introduced in [@kn:cai1] as an example of a “topological deSitter" spacetime. In fact, the hyperbolic deSitter spacetime was found to be particularly well-behaved, in that it has a positive Cardy-Verlinde Casimir energy [@kn:cai2][@kn:morecai][@kn:medved3]. Thus, the situation here is particularly favourable from the dS/CFT point of view. Let us therefore investigate the structure of conformal infinity. Outside the cosmological horizon, the coordinate $r$ in equation \[eq:dshyper\] becomes timelike, and so the geometry of conformal infinity is revealed by letting $r$ tend to infinity. In view of the way we derived the metric, it is no surprise to find that the canonical representative of the conformal structure is almost precisely the same as in the case of the Euclidean hyperbolic AdS black hole, that is, it is given by $$\label{eq:gamma3}
{\gamma}_3 = dt \otimes dt + {L^2}h_{ij}[H^2/\Delta] \; dx^i \otimes dx^j.$$ Here $\gamma_3$ is a metric on a three-dimensional Euclidean manifold \[since $t$ has become spacelike\] with scalar curvature $-2/L^2$. The full conformal boundary consists of two copies of this manifold, one in the infinite past, the other in the infinite future; this is analogous to the fact that the conformal boundary of deSitter space itself consists of two 3-spheres in the infinite past and future.
Now, examining $\gamma_3$ more closely, we notice two crucial features. The first is initially surprising: this metric is [*geodesically complete*]{}, despite the fact that the naked singularities are eternal and so are “still present at infinity". The reason for this is that while the spacetime has no event horizon, it does still have cosmological horizons which, as in any asymptotically deSitter spacetime, continuously “contract" around a given point. Thus, at late times, the regions of future infinity which can be affected by the singularities becomes steadily smaller, until, at infinity itself, they have shrunk to two points. These two points do have to be excised, but the conformal freedom at infinity can be used to push the “holes" off to an infinite distance. \[One “hole“ is at $``t = \infty"$, the other at $``t = -\infty$”.\] In fact, this is an example of a classical construction due to Nomizu and Ozeki [@kn:nomizu], whose theorem states that, given any Riemannian metric on any manifold, there exists a conformally related metric which is geodesically complete. The fact that this nakedly singular spacetime has a conformal infinity which is [*complete*]{} is a beautiful physical implementation of the Nomizu-Ozeki construction: the strong deSitter expansion has indeed pushed the holes off to an infinite spatial distance \[at temporal infinity\].
Now from the dS/CFT point of view, this is just as we would wish. For the correspondence, like its AdS/CFT counterpart, is supposed to relate a gravitational theory in the bulk to a [*non-gravitational*]{} theory on the boundary. A non-gravitational theory should be defined on a non-singular, geodesically complete space. The good behaviour of the geometry at infinity confirms our belief that the dS/CFT correspondence can be used here, in the sense that any misbehaviour of the boundary CFT cannot be ascribed to singularities of the boundary metric. \[Related observations were made in [@kn:ghezelbash].\] For this reason, we shall henceforth only consider complete metrics on the boundary.
This discussion brings us to the second point: since, for reasons explained above, $t$ runs from $-\infty$ to $+\infty$, it is clear that the boundary is not compact. \[Indeed, this is why the question of completeness is an issue here $-$ a compact manifold is geodesically complete with respect to any Riemannian \[not Lorentzian\] metric.\] Before dealing with the consequences of this, let us contrast the situation here with pure deSitter space, with its metric $$\label{eq:dS}
g(dS_4) = -(1 - {{r^2}\over{L^2}}) {dt\otimes dt} + {(1 - {{r^2}\over{L^2}})}^{-1} {dr\otimes dr} + {r^2} {d{\Omega_2}}^2,$$ where $d{\Omega_2}^2$ is the unit 2-sphere metric. Letting $r$ tend to infinity as above, we obtain $$\label{eq:cylinder}
dt \otimes dt + {L^2}{d{\Omega_2}}^2,$$ which appears to be the metric on the non-compact cylinder ${I\!\! R}\times S^2$. Of course this is not correct: the future boundary is a three-sphere. The point is that the polar coordinates do not cover the origin or its antipode, and the excision of these two points from $S^3$ does indeed produce a topological cylinder; the deSitter expansion then implements a Nomizu-Ozeki completion, producing the metric \[eq:cylinder\]. Evidently, the “non-compactness" here is a mere coordinate effect. In the case of equation \[eq:gamma3\], however, the excisions are genuinely necessary, so that future infinity is genuinely non-compact; it is a three-dimensional manifold with topology ${I\!\! R}\times ({I\!\! R}^2/\Delta)$. Thus we see that while the [*local*]{} geometry of conformal infinity here is the same as that of the conformal infinity of a (Euclidean) hyperbolic AdS black hole spacetime, the [*global*]{} structure is quite different. But the methods we used to establish topological instability in the AdS case were based on the compactness of the boundary. The failure of asymptotic simplicity in this case produces, as the Andersson-Galloway theorem [@kn:galloway3] would lead us to expect, a non-compact boundary. It follows that [*it is no longer clear that the scalar curvature on the boundary cannot be positive or zero everywhere*]{}. We must therefore reconsider the consequences of sign conditions on the scalar curvature of the boundary, corresponding to singular asymptotically deSitter metrics like the one given by equation \[eq:dshyper\] above.
The scalar curvature of the boundary metric used here \[as representative of the conformal structure\] is $-2/L^2$, so we can expect unstable behaviour for the CFT on the boundary. What is the bulk counterpart of this instability? A comparison with the AdS case suggests an answer: at extremely late times, beyond the horizon \[in the uppermost “diamond" of the Penrose diagram, where the metric is not static\], we can expect spacelike branes to be “emitted" \[that is, to appear suddenly\]. We interpret the boundary runaway as the dS/CFT dual of unstable production of these “late branes", which are the dS/CFT analogue of the AdS/CFT “large branes". Of course, the question now is this: can back-reaction bring the instability under control by modifying the geometry of the spatial sections, so that future conformal infinity actually has positive or zero scalar curvature, with late branes mediating the transition? As we saw, the methods used in section 2 to prove that the analogous AdS boundary cannot accept positive or zero scalar curvature do not work here, so the answer requires new techniques.
Again, these techniques do exist, but they are somewhat more abstruse than in the compact case, so we shall not attempt to summarize them; the reader may consult pages 313-326 of [@kn:lawson]. The main result we need may be stated as follows. Let $M$ be any compact enlargeable spin manifold, and let $g$ be any metric on ${I\!\! R}\times M$ such that the manifold is geodesically complete. \[Recall that there is a physical motivation for requiring this last condition.\] Then the scalar curvature of $g$ [*cannot*]{} be everywhere positive. Since ${{I\!\! R}^2}/\Delta$ is enlargeable \[it has a metric of non-positive sectional curvature\], we see that there is indeed no complete metric of positive scalar curvature on a three-dimensional manifold with the topology that we have here, ${I\!\! R}\times ({I\!\! R}^2/\Delta)$, despite the non-compactness. In fact, we have a stronger result in this particular case. Suppose that the scalar curvature of a complete metric on this manifold satisfies $R \geq 0$ everywhere. Then a theorem of Kazdan [@kn:kazdan] states that the Ricci tensor, if it is not identically zero, can be used to deform the metric so that the scalar curvature becomes everywhere strictly positive. Thus $R \geq 0$ everywhere on ${I\!\! R}\times ({I\!\! R}^2/\Delta)$ implies that the Ricci tensor must vanish. But since ${I\!\! R}\times ({I\!\! R}^2/\Delta)$ is three-dimensional, this in turn means that the metric is flat. However, once again, this manifold has the wrong topology to be flat. We conclude that there is no complete metric with scalar curvature everywhere non-negative on this boundary manifold.
Once again, then, we are forced to conclude that no matter what effect late branes have on the spacetime geometry, their unstable production cannot be halted: it is simply impossible to deform a metric on ${I\!\! R}\times ({I\!\! R}^2/\Delta)$ in a way that would achieve this. Again, as in the previous section, processes which might change the topology of the spacelike sections would lead either to causality violation or to new singularities, and we would not normally interpret this as a sign that stability had been restored — just the reverse. Clearly, such space-times cannot occur either in dS/CFT or in whatever fundamental theory — one hopes it is string or M theory — underlies dS/CFT. If anything resembling dS/CFT is valid, then asymptotically deSitter S-brane spacetimes preserving R-symmetries are unstable.
4. Conclusion {#conclusion .unnumbered}
=============
We have seen that Seiberg-Witten instability imposes a strong and unexpected constraint on the geometry of the event horizon of an asymptotically AdS black hole embedded in string theory. All [*known*]{} asymptotically AdS black holes are “Seiberg-Witten unstable" in string theory, except those with horizons built up \[by connected sums\] from $S^1 \times S^2$ and quotients of spheres and tori. In fact, we believe that the word “known" in this statement can be dropped; this depends on the purely topological conjectures that \[a\] all Eilenberg-MacLane spaces are enlargeable and \[b\] the Euclidean conformal boundary of a black hole spacetime necessarily inherits enlargeability from the horizon. Counterexamples to either of these conjectures, in the highly improbable event that any can be found, must have an extremely complex topology and are most unlikely to be of physical interest. Thus it seems that, for string theory on AdS backgrounds, event horizon geometries cannot be much more complicated than in the asymptotically flat case. Surprisingly, we were able to establish this even if we allowed the large branes in the bulk to act back on the spacetime geometry. No matter how strong the back-reaction may be, the Seiberg-Witten criterion continues to hold if it holds initially. Only when conditions become so extreme that the [*topology*]{} of the spatial sections changes can there be any possibility of back-reaction bringing the large branes under control; but, by then, new singularities \[or, worse still, causality violations\] are generated, which presumably indicates that stability has been lost in any case.
We have also investigated the possibility of generalizing the asymptotically flat S-brane solutions of [@kn:gutperle1] to asymptotically deSitter S-brane spacetimes. Again we find, assuming that some kind of dS/CFT correspondence is valid, and assuming that the R-symmetries are preserved, that such solutions are unstable in a way which is immune to back-reaction.
In [@kn:buchel1][@kn:buchel2], the consequences of including tachyonic matter in the effective dynamics of S-branes was studied, while maintaining the R-symmetries as usual. The main consequence is the replacement of the “Milne horizon" by a spacelike singularity representing an S-brane. The spacetime otherwise evolves in an orderly way, though the timelike singularities are [*not*]{} resolved. As one would expect on the basis of the results obtained here, future conformal infinity remains non-spacelike. An asymptotically deSitter version of this spacetime would \[because the timelike singularities are still eternal, and the hyperbolic part of the metric is still present\] have the same kind of future conformal infinity as we discussed above, and we therefore predict that it would be unstable, with or without back-reaction. In fact, we predict that physically reasonable deSitter S-brane spacetimes can only be obtained by explicitly breaking the R-symmetries. Unfortunately, it is not known how to obtain explicit solutions in this case; the difficulties are discussed in [@kn:buchel1][@kn:buchel2]. An alternative approach would be to try to generalize the S-brane solutions with flat transverse spaces, mentioned in [@kn:quevedo]. Unfortunately, the Seiberg-Witten criterion in the case where the boundary scalar curvature is zero is not fully understood. Perhaps a “higher-derivative" approach \[see for example [@kn:odintsov2] for references\] can elucidate the higher-order terms in the large brane action.
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abstract: 'The isospin dependence of the spin-orbit potential is investigated for an effective Skyrme-like energy functional suitable for density dependent Hartree-Fock calculations. The magnitude of the isospin dependence is obtained from a fit to experimental data on finite spherical nuclei. It is found to be close to that of relativistic Hartree models. Consequently, the anomalous kink in the isotope shifts of Pb nuclei is well reproduced.'
address:
- '$^1$Max Planck Institut für Astrophysik, Karl-Schwarzschildstrasse 1, D-85740 Garching, Germany'
- '$^2$Physik-Department der Technischen Universität München, D-85747 Garching, Germany'
author:
- 'M.M. Sharma$^1$, G. Lalazissis$^2$, J. König$^2$, and P. Ring$^2$'
title: Isospin Dependence of the Spin Orbit Force and Effective Nuclear Potentials
---
The Hartree-Fock approach based upon phenomenological density dependent forces[@VB.72; @FQ.78; @DG.80] has proved to be very successful in the microscopic description of ground state properties of nuclear matter and of finite nuclei over the entire periodic table. In all these calculations the spin-orbit potential has been assumed to be isospin independent. Its only parameter, the strength, is usually adjusted to the experimental spin-orbit splitting in spherical nuclei like $^{16}$O or $^{208}$Pb. The exchange term, however, causes for nuclei with neutron excess a strong isospin dependence of the corresponding single-particle spin-orbit field.
In recent years Relativistic Mean Field (RMF) theory[@SW.86] with nonlinear self-interactions between the mesons has gained considerable interest for the investigations of low-energy phenomena in nuclear structure. With only a few phenomenological parameters such theories are able to give a quantitative description of ground state properties of spherical and deformed nuclei[@GRT.90; @SNR.93] at and away from the stability line. In addition, excellent agreement with experimental data has been found recently also for collective excitations such as giant resonances[@VBR.94] and for twin bands in rotating superdeformed nuclei[@KR.93]. In many respects the relativistic mean-field theory is regarded as similar to the density-dependent Hartree-Fock theory of the Skyrme type[@Thi.86; @Rei.89]
Recently, however, detailed investigations of high precision data on nuclear charge radii in Pb isotopes[@TBF.93; @SLR.93] and of shell effects at the neutron drip line[@Dob.94; @SLH.94] have shown considerable differences between the Skyrme approach and the relativistic mean field theory. In fact, density dependent Hartree-Fock calculations with Skyrme[@TBF.93] or Gogny forces[@Egi.94], which used so far antisymmetrized, isospin independent spin-orbit interactions, were not able to reproduce the kink in the isotope shifts of Pb nuclei (see Fig. \[F1\]). On the other hand, this kink is obtained in the RMF theory without any new adjustment of parameters[@SLR.93]. Another considerable difference has been found in theoretical investigation of shell effects in very exotic Zr-isotopes near the neutron drip line: in conventional non-relativistic Skyrme calculations the shell gap at isotope $^{122}$Zr with the magic neutron configuration N = 82 is totally smeared out[@Dob.94], whereas relativistic calculations using various parameter sets show at N=82 a clear kink in the binding energy as a function of the neutron number[@SLH.94]. This difference is caused by the different spin-orbit splitting of the single particle levels in these nuclei. Mass calculations within the Finite Range Droplet Model (FRDM) [@MNM.94], which are based on an isospin dependent spin-orbit term carefully adjusted to experimental data, are in excellent agreement with the relativistic predictions.
This gives us a hint, that one does not really need full relativistic calculations in order to understand these differences between conventional density dependent Hartree-Fock calculations and RMF theory. A spin-orbit term with a properly chosen isospin dependence might represent the essential part of a relativistic calculation. In this letter, we therefore explore the isospin dependence of the spin-orbit term in non-relativistic Skyrme calculations and analyze the consequences of this on the nuclear properties. We start from the Skyrme type force $$\begin{aligned}
V(1,2)&=&t_0(1+x_0P^\sigma)
\delta({\bf r}_1-{\bf r}_2)\nonumber\\
&+&~t_1(1+x_1P^\sigma)
(\delta({\bf r}_1-{\bf r}_2){\bf k}^2+h.c.)\nonumber\\
&+&~t_2(1+x_2P^\sigma){\bf k}\delta({\bf r}_1-{\bf r}_2){\bf k}
\label{mska}\\
&+&~\frac{1}{6}t_3(1+x_3P^\sigma)
\delta({\bf r}_1-{\bf r}_2)\rho^\alpha
\nonumber\\
&+&~W_0(1+x_wP^\tau)(
{\mbox{\boldmath $\sigma$}}^{(1)}+
{\mbox{\boldmath $\sigma$}}^{(2)})
{\bf k}\times\delta({\bf r}_1-{\bf r}_2){\bf k}
\nonumber\end{aligned}$$ with ${\bf k}=\frac{1}{2}({\bf p}_1-{\bf p}_2)$. In contrast to the conventional Skyrme ansatz, where the energy functional contains a Hartree- and a Fock-contribution, we here neglect the exchange (Fock-) term for the spin-orbit potential in the last line of Eq. (\[mska\]). Otherwise, the operator $P^\tau$ would be equivalent to $+1$ for spin saturated systems. For the rest of the potential the Fock terms are included. The associated 11 parameters $t_i$, $x_i$ ($i=0\dots 3$), $W_0$, $x_w$ and $\alpha$ of the Modified Skyrme Ansatz (MSkA) in Eq. (\[mska\]) are determined by a fit to experimental data of finite spherical nuclei. The nuclear properties taken into consideration are the empirical binding energies and charge radii of the closed-shell nuclei $^{16}$O, $^{40}$Ca, $^{90}$Zr, and $^{208}$Pb. In order to take into account the variation in isospin we have also included Sn-isotopes $^{116}$Sn, $^{124}$Sn, and the doubly closed nucleus $^{132}$Sn as well as one of the lead isotopes $^{214}$Pb. The resulting force and its parameters are presented in Table \[T1\].
The spin-orbit term in the single-particle field derived >from this force has the form ${\bf W}_\tau({\bf r})( {\bf
p}\times{\mbox{\boldmath $\sigma$}})$ with $${\bf W}_\tau({\bf r})~=~
W_1{\mbox{\boldmath $\nabla$}}\rho_\tau+,
W_2{\mbox{\boldmath $\nabla$}}\rho_{\tau^\prime\ne\tau},
\label{spinorbit}$$ where $\rho_\tau$ is the density for neutrons or protons ($\tau=n$ or $p$) and $W_1 =W_0(1+x_w)/2$, $W_2 =W_0/2$. Conventional Skyrme calculations use a spin-orbit potential without isospin dependence and include the Fock term. This leads to $x_w=1$ and the relationship $W_1/W_2=2$, which is nearly by a factor 2 different from the value $1.0005$ obtained within the modified Skyrme Ansatz MSkA (see Table \[T1\]). It is interesting to note that the fit leads to a value of $x_w$ very close to zero, which corresponds to the one without the Fock term.
In order to study the spin-orbit term in the RMF theory we start from the standard Lagrangian density[@GRT.90] $$\begin{aligned}
{\cal L}&=&\bar\psi\left(\gamma(i\partial-g_\omega\omega
-g_\rho\vec\rho\vec\tau-eA)-m-g_\sigma\sigma
\right)\psi
\nonumber\\
&&+\frac{1}{2}(\partial\sigma)^2-U(\sigma )
-\frac{1}{4}\Omega_{\mu\nu}\Omega^{\mu\nu}
+\frac{1}{2}m^2_\omega\omega^2\nonumber\\
&&-\frac{1}{4}{\vec{\rm R}}_{\mu\nu}{\vec{\rm R}}^{\mu\nu}
+\frac{1}{2}m^2_\rho\vec\rho^{\,2}
-\frac{1}{4}{\rm F}_{\mu\nu}{\rm F}^{\mu\nu}
\label{lagrangian}\end{aligned}$$ which contains nucleons $\psi$ with mass $m$. $\sigma$-, $\omega$-, $\rho$-mesons, the electromagnetic field and nonlinear self-interactions $U(\sigma)$ of the $\sigma$-field, $$U(\sigma)~=~\frac{1}{2}m^2_\sigma\sigma^2+
\frac{1}{3}g_2\sigma^3+\frac{1}{4}g_3\sigma^4.$$
In a non-relativistic approximation of the corresponding Dirac equation[@Koe.94] we obtain the following single particle spin-orbit term: $$\begin{aligned}
{\bf W}_\tau({\bf r})&=&
\frac{1}{m^2m^{*2}}(C^2_\sigma+C^2_\omega+C^2_\rho)
{\mbox{\boldmath $\nabla$}}\rho_\tau\nonumber\\
&&~+\frac{1}{m^2m^{*2}}(C^2_\sigma+C^2_\omega-C^2_\rho)
{\mbox{\boldmath $\nabla$}}\rho_{\tau^\prime\ne\tau},
\label{relspinorbit}\end{aligned}$$ with $C_i^2=(m\,g_i/m_i)^2$ for $i=\sigma,\omega,\rho$, which is similar in form to the spin-orbit field (\[spinorbit\]), but which contains $\bf r$-dependent parameters $$\begin{aligned}
W_1&=&\frac{1}{m^2m^{*2}}(C^2_\sigma+C^2_\omega+C^2_\rho)\\
W_2&=&\frac{1}{m^2m^{*2}}(C^2_\sigma+C^2_\omega-C^2_\rho),\end{aligned}$$ with $m^*({\bf r})=m-g_\sigma\sigma({\bf r})$ The $r$-dependence drops out in the ratio $W_1/W_2$ which is $1.13$ for the parameter set NL1[@GRT.90] and $1.10$ for the parameter set NL-SH[@SNR.93]. This is only slightly higher than the value $1.0005$ obtained in the Modified Skyrme Ansatz by the fit to the empirical data (see Table \[T1\]). The absolute size of the spin-orbit term turns out to be $\bf r$-dependent, which stems from the $\bf r$-dependence of the effective mass $m^*({\bf r})$. It can be approximated by $m/m^*\approx
1+C^2_\sigma/m^3\,\rho({\bf r})$. A more careful consideration would therefore require an explicitly density dependent spin-orbit term. It has not been included in the present investigation.
In Table \[T2\] we show nuclear matter results obtained in the MSkA and compare it with the values from the conventional Skyrme force SkM$^*$[@BQB.82]. The saturation density is obtained as $\rho_0=0.1531$ fm$^{-3}$ in MSkA. This is the same as the value obtained from an extensive fit of the mass formula FRDM[@MNM.94]. The binding energy per particle $E/A$ is 16.006 MeV, which is close to that of other Skyrme forces. The compression modulus $K=319$ MeV is somewhat higher than that of the presently adopted Skyrme forces, but it lies within the error bars of the analysis based upon the breathing mode energies. The asymmetry energy $J$ is close to the empirical value of 33 MeV. The effective mass $m^*$ is in good agreement with that of SkM$^*$.
In Table \[T3\] we show binding energies and charge radii obtained in the MSkA for a number of spherical nuclei and compare them with those from SkM$^*$. A comparison with the empirical values shows, that the binding energies obtained with MSkA have improved over those of SkM$^*$. The slightly reduced charge radii in MSkA seem to be connected with a slightly higher binding energy, which improves the results for the lighter Pb-isotopes. In general the charge radii are improved including that of $^{16}$O.
In Fig. 1 we show the isotope shifts of Pb nuclei for the Modified Skyrme Ansatz MSkA together with experimental data and results of conventional Skyrme calculations. For MSkA we observe a clear kink at the double magic nucleus $^{208}$Pb, whereas the conventional Skyrme force SkM$^*$ with an isospin independent spin-orbit term gives an almost straight line. For the lighter isotopes both theories give excellent agreement, on the heavier side MSkA comes closer to the experimental isotope shifts. It may be recalled, that the RMF theory[@SLR.93] is successful in reproducing the full size of this kink. MSkA uses an isospin dependent, but density independent spin-orbit force. In contrast the spin-orbit term derived from the RMF theory (see Eq. \[relspinorbit\]) is implicitly density dependent through the density dependence of the effective mass $m^*({\bf r})$. A density dependence of the spin-orbit term in Skyrme theory might improve the charge radii of the heavier Pb-isotopes. It has also been observed that a density dependent pairing force can possibly improve the situation in this context[@TBF.93]. This requires, however, further investigations.
In Fig. 2 we present binding energies of Zr-nuclei about the neutron drip line as a function of the mass number. It is observed that in agreement with earlier investigations[@Dob.94] within the conventional Skyrme theory shell effects about the closed shell nucleus $^{122}$Zr are weakened considerably. In contrast, RMF-calculations exhibit strong shell effects in the drip line region. It has been surmised in Ref. [@SLH.94] that this is caused by the differences in the spin-orbit terms in the two approaches. It is gratifying to see that introduction of an isospin dependent (not antisymmetrized) spin-orbit term in Skyrme theory leads to stronger shell effects. This is also in agreement with the predictions of the FRDM[@MNM.94], where also an isospin dependent spin-orbit term is used. A more quantitative analysis shows that the ratio $W_1/W_2$ in FRDM is close to 1.06 for the Pb-nuclei and 1.09 for $^{122}$Zr.
Summarizing, we conclude that the isospin dependence of the spin-orbit term has an essential influence on the details of anomalous isotope shifts of Pb-nuclei. A new Modified Skyrme Ansatz has been proposed, and the isospin dependence of the spin-orbit strength has been determined. The magnitude of this isospin dependence $x_w$ is in agreement with the deductions from the relativistic mean-field theory. A reasonably good agreement with the experimental data on the binding energies and charge radii has been obtained. The kink in the isotope shifts of Pb-nuclei has been obtained in the modified Skyrme ansatz. However, the agreement with the empirical isotope shifts for heavy Pb-nuclei is not so good as in the RMF theory. This calls for further investigations including a density dependence of the spin-orbit term.
Finally a remark about the isospin dependence of the spin-orbit term: our investigations lead to the interesting result, that the parameter $x_w$ in the Eq. (1) is close to zero. This leads practically to an isospin independent single-particle spin-orbit field in Eq. (2). This is in agreement with relativistic calculations, where the entire isospin-dependence of the spin-orbit field is caused by the parameter $C_\rho$ in Eq. (5), which is in fact rather small. In contrast the two-body spin-orbit potential in conventional Skyrme theory is isospin independent. The exchange term, however, causes a strong isospin dependence in the corresponding single-particle spin-orbit field. On the other hand in RMF theory the spin-orbit field has its origin in Lorentz covariance. There is no contribution from a two-body spin-orbit potential and an exchange term is therefore excluded.
One of the authors (G.A.L) acknowledges support from the E.U., HCM program, contract: EG/ERB CHBICT-930651 , This work is also supported in part by the Bundesministerium für Forschung und Technologie under the project 06 TM 743.
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------------------------------------------ -----------------
$t_0 = -1200.074$ (MeV fm$^3$) $x_0 = 0.187$
$t_1 = 396.302$ (MeV fm$^5$) $x_1 = 0.018$
$t_2 = -105.579$ (MeV fm$^5$) $x_2 = -0.059$
$t_3 = 10631.527$ (MeV fm$^{3+3\alpha}$) $x_3 = 0.046$
$W_0 = 316.38$ (MeV fm$^5$) $x_w = 0.0005 $
$\alpha = 0.7557 $
------------------------------------------ -----------------
: Parameters of the interaction in the modified Skyrme Ansatz (MSkA).
\[T1\]
MSkA SkM$^*$
---------------- ------------------ ------------------
$\rho_0$ 0.1531 fm$^{-3}$ 0.1603 fm$^{-3}$
$(E/A)_\infty$ 16.006 MeV 15.776 MeV
$K$ 319.4 MeV 216.7 MeV
$\rho_0^2e'''$ 100.2 MeV 57.9 MeV
$J$ 30.0 MeV 30.0 MeV
$m^*/m$ 0.76 0.79
: Nuclear matter properties obtained in the modified Skyrme Ansatz (MSkA).
\[T2\]
------------ --------- --------- --------- -- ------- ------- -------
Nuclei expt. MSkA SkM\* expt. MSkA SkM\*
$^{16}$O -127.6 -128.1 -127.7 2.730 2.742 2.811
$^{40}$Ca -342.1 -342.8 -341.1 3.450 3.468 3.518
$^{48}$Ca -416.0 -416.8 -420.1 3.500 3.506 3.537
$^{90}$Zr -783.9 -781.9 -783.0 4.270 4.274 4.296
$^{116}$Sn -988.7 -984.0 -983.4 4.626 4.623 4.619
$^{124}$Sn -1050.0 -1047.9 -1049.0 4.673 4.677 4.678
$^{132}$Sn -1102.9 -1106.3 -1110.7 - 4.728 4.727
$^{200}$Pb -1576.4 -1570.9 -1568.4 5.464 5.465 5.468
$^{202}$Pb -1592.2 -1588.3 -1586.0 5.473 5.475 5.478
$^{204}$Pb -1607.5 -1605.4 -1603.4 5.483 5.486 5.489
$^{206}$Pb -1622.3 -1622.3 -1620.3 5.492 5.496 5.501
$^{208}$Pb -1636.5 -1637.8 -1636.4 5.503 5.506 5.510
$^{210}$Pb -1645.6 -1645.8 -1645.6 5.522 5.522 5.520
$^{212}$Pb -1654.5 -1653.8 -1654.5 5.540 5.537 5.531
$^{214}$Pb -1663.3 -1661.6 -1663.1 5.558 5.552 5.541
------------ --------- --------- --------- -- ------- ------- -------
: The binding energies and charge radii obtained with the Modified Skyrme Ansatz (MSkA) in the Hartree-Fock approximation as compared with the normal Skyrme force SkM\* and the empirical values.
\[T3\]
|
---
abstract: 'The $\rho^0$ production at high-$p_T$ (5.0 $\leq p_T \leq$ 10.0 GeV/$c$) measured in minimum bias $p+p$, Au+Au and central Au+Au collisions in the STAR detector are presented. The $\rho^0/\pi$ ratio measured in $p+p$ is compared to PYTHIA calculations as a test of perturbative quantum chromodynamics (pQCD) that describes reasonably well particle production from hard processes. The $\rho^0$ nuclear modification factor are also presented. In $p+p$ collisions, charged pions and (anti-)protons are measured in the range 5.0 $\leq p_T \leq$ 15.0 GeV/$c$ and the anti-particle to particle ratio and the baryon to meson ratios of these hadrons are discussed.'
address: ' Brookhaven National Laboratory, Upton, NY, 11973, USA'
author:
- P Fachini
title: '$\rho^0$ Production at High $p_T$ in Central Au+Au and $p+p$ collisions at $\sqrt{s_{_{NN}}} = 200$ GeV in STAR'
---
Introduction
============
The study of inclusive hadron production at high $p_T$ in $p+p$ collisions provides information on perturbative QCD (pQCD), parton distribution functions in the proton (PDF) and fragmentation functions (FF) of the partons. In Au+Au collisions, high $p_T$ hadron production is a sensitive probe of the strongly interacting QCD matter formed in these collisions. The transverse momentum spectra of $\pi$, $\rho^0$, and $p$ measured in $p+p$ and central Au+Au collisions can be used to study the effect of energy loss on fragmentation. These measurements can provide insight in the quantum chromodynamics (QCD) predicted difference between quark and gluon energy loss.
Analysis and Results
====================
In $p+p$ collisions, charged pions and (anti-)protons were measured in the range 5.0 $\leq p_T \leq$ 15.0 GeV/$c$ using jet triggered events. Figure \[fig:Ratio1\] shows the $\bar{p}/\pi^-$ (left panel) and $p/\pi^+$ (right panel) ratios as a function of $p_T$ compared to PYTHIA [@pythia] (solid lines) and DSS [@dss] (dotted line) NLO calculations. In this figure, since PYTHIA minbias and PYTHIA jet triggered (two solid lines) are the same, we can conclude that the effect of the jet trigger on the ratios is negligible. We also observe that the DSS calculation reproduces the $p/\pi^+$ ratio; this is not the case for the $\bar{p}/\pi^-$ ratio. PYTHIA reproduces both $\bar{p}/\pi^-$ and $p/\pi^+$ ratios. Figure \[fig:Ratio2\] shows the $\pi^-/\pi^+$ (left panel) and $\bar{p}/p$ (right panel) ratios as a function of $p_T$ compared to PYTHIA (solid lines) and DSS (dotted line) NLO calculations. In this figure, we also observe that the effect of the jet trigger on the ratios is negligible and that PYTHIA reproduces both $\pi^-/\pi^+$ and $\bar{p}/p$ ratios. Both ratios decrease as a function of $p_T$, showing the effect of valence quarks in the case of the $\pi^-/\pi^+$ ratio and the significant quark jet contribution to the baryon production in the case of the $\bar{p}/p$ ratio. The DSS calculation reproduces the behavior of both ratios. However, while it reproduces the magnitude of the $\pi^-/\pi^+$ ratio, it fails in the case of the $\bar{p}/p$ ratio.
![(Color online) $\bar{p}/\pi^-$ (left panel) and $p/\pi^+$ (right panel) ratios as a function of $p_T$ at $\sqrt{s_{_{NN}}} = 200$ GeV compared to PYTHIA [@pythia] and DSS [@dss] NLO calculations.[]{data-label="fig:Ratio1"}](QMpbarpimRatio.eps){height="14pc" width="18pc"}
![(Color online) $\bar{p}/\pi^-$ (left panel) and $p/\pi^+$ (right panel) ratios as a function of $p_T$ at $\sqrt{s_{_{NN}}} = 200$ GeV compared to PYTHIA [@pythia] and DSS [@dss] NLO calculations.[]{data-label="fig:Ratio1"}](QMppipRatio.eps){height="14pc" width="18pc"}
![(Color online) $\pi^-/\pi^+$ (left panel) and $\bar{p}/p$ (right pannel) ratios as a function of $p_T$ at $\sqrt{s_{_{NN}}} = 200$ GeV compared to PYTHIA [@pythia] and DSS [@dss] NLO calculations.[]{data-label="fig:Ratio2"}](QMpimpipRatio.eps){height="14pc" width="18pc"}
![(Color online) $\pi^-/\pi^+$ (left panel) and $\bar{p}/p$ (right pannel) ratios as a function of $p_T$ at $\sqrt{s_{_{NN}}} = 200$ GeV compared to PYTHIA [@pythia] and DSS [@dss] NLO calculations.[]{data-label="fig:Ratio2"}](QMpbarpRatio.eps){height="14pc" width="18pc"}
The $\rho^0$ meson is measured at high $p_T$ (5.0 $\leq p_T
\leq$ 9.0 GeV/$c$) in minimum bias $p+p$ (using jet triggered data), Au+Au and central Au+Au collisions. In order to obtain the transverse momentum spectra, the $\rho^0$ signal was fit to a relativistic Breit-Wigner (BW) function times the phase space [@rho]. In the fits to the $\rho^0$ signal, the mass and width are fixed according to the PDG [@pdg] average for the $\rho^0$ measured in $e^+e^-$ interactions, which corresponds to the rho mass in the vacuum. The mass and width of the $\rho^0$ in the vacuum reproduce well the data at high $p_T$ (5.0 $\leq p_T
\leq$ 9.0 GeV/$c$). There is not enough sensitivity to do a detailed study of the $\rho^0$ mass and width as a function of $p_T$ in minimum bias and central Au+Au collisions [@rho]. In the case of $p+p$ collisions, even though there is no sensitivity to do a study of the width as a function of $p_T$, we were able to look more closely into the mass. The left panel of Fig. \[fig:Mass\] depicts the $\rho^0$ mass as a function of $p_T$ measured in $p+p$ collisions in the case the mass is a free parameter in the fit. We observe that the $\rho^0$ mass approaches its value in the vacuum in this $p_T$ interval. The right panel of Fig. \[fig:Mass\] shows the $\rho^0$ transverse momentum spectra measured in minimum bias $p+p$, Au+Au and central Au+Au collisions. The $\rho^0$ spectrum measured in $p+p$ collisions is similar to the DSS NLO calculations for pions (within 30$\%$) as expected, since it has been shown in deep inelastic electron scattering that quarks fragment with equal probability into pions and $\rho$ mesons [@isr].
![(Color online) Left panel: $\rho^0$ mass as a function of $p_T$ measured in $p+p$ collisions at $\sqrt{s_{_{NN}}} = 200$ GeV. The solid line corresponds to the PDG [@pdg] average for the $\rho^0$ measured in $e^+e^-$ interactions. Right panel: $\rho^0$ spectra as a function of $p_T$ measured in minimum bias $p+p$, Au+Au and central Au+Au collisions at $\sqrt{s_{_{NN}}} = 200$ GeV.[]{data-label="fig:Mass"}](QMRhoMasspp.eps){height="14pc" width="18pc"}
![(Color online) Left panel: $\rho^0$ mass as a function of $p_T$ measured in $p+p$ collisions at $\sqrt{s_{_{NN}}} = 200$ GeV. The solid line corresponds to the PDG [@pdg] average for the $\rho^0$ measured in $e^+e^-$ interactions. Right panel: $\rho^0$ spectra as a function of $p_T$ measured in minimum bias $p+p$, Au+Au and central Au+Au collisions at $\sqrt{s_{_{NN}}} = 200$ GeV.[]{data-label="fig:Mass"}](QMSpectraRhoHighPt.eps){height="14pc" width="18pc"}
The $\rho^0/\pi^-$ ratio as a function of $x_T$ ($x_T = 2p_T/\sqrt{s}$) measured in minimum bias $p+p$, Au+Au and central collisions are compared to PYTHIA calculations in the left panel of Fig. \[fig:Raa\]. One observes that PYTHIA under-predicts the measured ratios; however, we cannot rule out the possibility that PYTHIA can be tuned to describe the data. In the left panel of Fig. \[fig:Raa\] it is also shown the $\rho^0/\pi^-$ measured at ISR at two different $p_T$ bins at $\sqrt{s_{_{NN}}} = 52.2$ GeV, which are lower than the STAR measurements.
The right panel of Fig. \[fig:Raa\] depicts the charged pions, $\rho^0$ and proton plus anti-proton $R_{AA}$ [@raa] as a function of $p_T$. There is a separation of approximately 1.5 $\sigma$ between charged pions and protons plus anti-protons $R_{AA}$ for $p_T \geq$ 7 GeV/$c$, where the protons plus anti-protons $R_{AA}$ are larger than the charged pions $R_{AA}$. The right panel of Fig. \[fig:Raa\] also shows that the $\pi^0$, charged pions and $\rho^0$ nuclear modification factors are comparable.
![(Color online) Left panel: The $\rho^0/\pi^-$ ratio as a function of $x_T$ measured in minimum bias $p+p$, Au+Au and central collisions at $\sqrt{s_{_{NN}}} = 200$ GeV are compared to PYTHIA calculations. Right panel: $\pi$, $\rho^0$ and $p$ $R_{AA}$ as a function of $p_T$ at $\sqrt{s_{_{NN}}} = 200$ GeV [@raa]. The $\pi^0$ data comes from [@pi0].[]{data-label="fig:Raa"}](QMRatiosHighPtXt.eps){height="14pc" width="18pc"}
![(Color online) Left panel: The $\rho^0/\pi^-$ ratio as a function of $x_T$ measured in minimum bias $p+p$, Au+Au and central collisions at $\sqrt{s_{_{NN}}} = 200$ GeV are compared to PYTHIA calculations. Right panel: $\pi$, $\rho^0$ and $p$ $R_{AA}$ as a function of $p_T$ at $\sqrt{s_{_{NN}}} = 200$ GeV [@raa]. The $\pi^0$ data comes from [@pi0].[]{data-label="fig:Raa"}](QMRhoRaa.eps){height="14pc" width="18pc"}
Summary
=======
In $p+p$ collisions, charged pions and (anti-)protons are measured in the range 5.0 $\leq p_T \leq$ 15.0 GeV/$c$ and the $\bar{p}/\pi^+$, $p/\pi^-$, $\bar{p}/p$ and $\pi^-/\pi^+$ ratios were discussed. The $\rho^0$ production at high-$p_T$ (5.0 $\leq p_T \leq$ 10.0 GeV/$c$) measured in minimum bias $p+p$, Au+Au and central Au+Au collisions in the STAR detector were presented. The $\rho^0/\pi$ ratio measured in $p+p$ is compared to PYTHIA calculations as a test of perturbative quantum chromodynamics (pQCD) that describes reasonably well particle production from hard processes and also to the $\rho^0/\pi$ ratio measured in Au+Au collisions. The charged pions, $\rho^0$ and proton plus anti-proton nuclear modification factors were also presented. The protons plus anti-protons and the charged pions $R_{AA}$ seem to behave oppositely to what is naïvely expected from color charge dependence of energy loss [@eloss]. The $\pi^0$, charged pions and $\rho^0$ nuclear modification factors are comparable indicating that the fragmentation of vector mesons and pseudo-scalars are similar in $p+p$ and Au+Au collisions.
References {#references .unnumbered}
==========
[10]{} T. Sjöstrand [*et al.*]{}, 2001 Computer Physics Commun. [**35**]{} 238. D. de Florian, R. Sassot, and M. Stratmann, 2007 Phys. Rev. D [**75**]{} 114010. J Adams [*et al.*]{} 2004 Phys. Rev. Lett. [**92**]{} 092301. Particle Data Group 2006 J. Phys. G [**33**]{} 1. E. E. Kluge, 1979 Phys. Scrip. [**19**]{} 109. B. I. Abelev [*et al.*]{} 2006 Phys. Rev. Lett. [**97**]{} 152301. G. Lin, these proceedings. B. Mohanty, arXiv:0705.0953; B. I. Abelev [*et al.*]{} 2007 Phys. Lett. B [**655**]{} 104. J. Adams [*et al.*]{} 2003 Phys. Rev. Lett. [**91**]{} 072304.
|
[Anomaly-induced effective action and Chern-Simons modification of general relativity]{}
[**Sebastião Mauro**]{}$^{a}$ and $^{b,a,c}$
[*(a)*]{} Departamento de Física, ICE, Universidade Federal de Juiz de Fora,\
CEP: 36036-330, Juiz de Fora, MG, Brazil
[*(b)*]{} Département de Physique Théorique and Center for Astroparticle Physics, Université de Genève, 24 quai Ansermet, CH–1211 Genéve 4, Switzerland
[*(c)*]{} Tomsk State Pedagogical University and Tomsk State University, Tomsk, 634041, Russia
> [**Abstract.**]{} Recently it was shown that the quantum vacuum effects of massless chiral fermion field in curved space-time leads to the parity-violating Pontryagin density term, which appears in the trace anomaly with imaginary coefficient. In the present work the anomaly-induced effective action with the parity-violating term is derived. The result is similar to the Chern-Simons modified general relativity, which was extensively studied in the last decade, but with the kinetic terms for the scalar different from those considered previously in the literature. The parity-breaking term makes no effect on the zero-order cosmology, but it is expected to be relevant in the black hole solutions and in the cosmological perturbations, especially gravitational waves.
>
> Pacs: 04.62.+v, 11.10.Lm, 11.15.Kc
>
> Keywords: Effective Action, Conformal anomaly, Chern-Simons gravity
Introduction
============
The derivation and properties of conformal (trace) anomaly are pretty well-known (see, e.g., [@duff94] and also [@PoImpo; @PoS-Conform] for the technical introduction related to the present work). At the one-loop level the anomaly is given by an algebraic sum of the contributions of massless conformal invariant fields of spins $0,1/2,1$ in a curved space-time of an arbitrary background metric. Recently, it was confirmed that the quantum effects of chiral (L) fermion produce an imaginary contribution which violates parity [@bonora]. As a result, the anomalous trace has the form T\^\_&=& -\_1C\^2 - \_2E\_4 - a\^R - F\_\^2 - \_4 P\_4. \[T\] Here we have included the external electromagnetic field $F_{\mu\nu}=\pa_\mu A_\nu-\pa_\nu A_\mu$ for generality, also C\^2 &=& C\_C\^ = R\_\^2 - 2 R\_\^2 + 13R\^2 \[Weyl\] is the square of the Weyl tensor in four-dimensional space-time and E\_4 &=& 14\^\^ R\_ R\_ = R\_\^2 - 4 R\_\^2 + R\^2 \[GB\] is the integrand of the Gauss-Bonnet topological term.
The $\be$-functions are given by algebraic sums of the contributions of $N_s$ scalars, $N_f$ Dirac fermions and $N_v$ massless vector fields. The explicit form is well known, (4)\^2\_1 &=& N\_s + N\_f + N\_v,\
(4)\^2\_2 &=& -N\_s - N\_f - N\_v,\
(4)\^2\_3 &=& N\_s + N\_f - N\_v. \[abc\] One can assume that $a^\prime$ in (\[T\]) is equal to $\be_3$, but there is ambiguity, as will be discussed below. ${\tilde \be}$ is the usual $\be$-function of QED or scalar QED etc, depending on the model.
Furthermore, there is a parity-violating Pontryagin density term $\be_4 P_4$, where P\_4 &=& 12\^R\_ R\_\^. \[Pont\] By dimensional reasons the term with $P_4$ is possible, but for a long time it was believed that this term, in fact, does not show up. However, in a recent paper [@bonora] this term was actually found with a purely imaginary coefficient $\,\be_4 = i/(48 \cdot 16\pi^2)$, as a contribution of chiral (left) fermions. The chirality is important here, because the contribution of the right-hand fermions is going to cancel the one of the left-hand fermions, so taking them in a pair would kill the effect. Let us also note that much earlier, in [@DuffNieuw], the possibility of such a term coming from integrating out antisymmetric tensor field has been considered, also some general considerations were presented even earlier in [@DDI-80] and more recently in [@Nakayama].
Some questions arise due to the result of [@bonora] and its physical interpretation. First, does the parity-violating term in the anomaly mean that the dynamics of gravity is affected in a significant way? Second, in case of a positive answer to the last question, does it mean that the chiral fermions are disfavoured theoretically, since they produce imaginary component in the gravitational field equations? The last possibility was discussed in [@bonora] as a theoretical argument in favor of massive neutrino. The third question is whether the parity-odd terms in the anomaly have some relation to the Chern-Simons modification of $4d$-gravity suggested in [@jackiw-pi; @LWK]. The theories of this sort were extensively investigated in the last decade, as one can see from the review [@AleYu] and other works on the subject. This question looks really natural, because the Chern-Simons-gravity is based on the action which includes the $P_4$-term with an extra scalar factor inside the integral. Let us note that the relation between parity-odd terms and anomalies in $D=4$ was discussed, i.e., in [@AlvaWit] in relation to gravitational anomalies, so the novelty of the term (\[Pont\]) concerns only the trace anomaly.
The purpose of the present work is to address the questions formulated above. In order to do so, we derive the effective action of gravity by integrating conformal anomaly, and show that the result is a new version of the Chern-Simons $4d$-gravity with a special form of the kinetic term for the scalar and some extra higher-derivative terms which are typical for this action. From the technical side most of the consideration is pretty well-known, but we present full details in order to make it readable for those who are not familiar with the subject. The paper is organized as follows. In Sect. 2 we review the well-known scheme of deriving anomaly-induced effective action, with an extra parity-odd term corresponding to Pontryagin density. The anomaly-induced action provides a specific form of the kinetic term for the auxiliary scalar in Chern-Simons modified gravity. For this reason, in the last subsection we present a short review of the previous version of kinetic terms, which are known in the literature. Sect. 3 includes a general, mainly qualitative, discussion of the physical interpretation of the new parity-violating term. Finally, in Sect. 4 we draw our conclusions and suggest possible perspectives of a further work on the subject.
Integration of anomaly with parity-violating term
=================================================
The integration of conformal anomaly (\[T\]) in $d=4$ means solving the equation similar to the one for the Polyakov action in $d=2$, g\_ = -T\_\^= (C\^2 + bE\_4 + c R + F\_\^2 + P\_4). \[mainequation\] Here we introduced useful notations $\,\big(\om,\,b,\,c,\,{\tilde b},\,\ep\big)
= (4 \pi)^2\,\big(\be_1,\,\be_2,\,a^\prime,\,\tilde{\be},\,\be_4\big)$. The coefficient $\ep$ derived in [@bonora] is imaginary, but we will not pay attention to this until the solution is found. The first reason for this is that this is technically irrelevant, and also it is, in principle, possible to have a real coefficient of the same sort at the non-perturbative level.
Conformal properties of Pontryagin term and anomaly
---------------------------------------------------
The solution of Eq. (\[mainequation\]) is technically is not very complicated [@rei] in the usual theory without Pontryagin term, and it remains equally simple when this term is present. In order to understand this, let us make an observation that this term is conformal invariant in $d=4$, simply because one can recast (\[Pont\]) in the form when the Weyl tensor replaces the Riemann tensor, P\_4 &=& 12\^C\_ C\_\^. \[Pont-W\] The proof of this statement is well-known (see [@Gru-Yu] for further developments), but for the convenience of the reader we present a proof in the Appendix. One can easily see that the [*r.h.s.*]{} of the Eq. (\[mainequation\]) consists of the three different terms, which can be classified according to [@DeserSchwim]. One can distinguish [*(i)*]{} conformally invariant part $\om C^2 + \tilde{\be} F_{\mu\nu}^2 + \be_4 P_4$; [*(ii)*]{} the topological term $bE_4$ and [*(iii)*]{} surface term $c\Box R$.
In fact, the last division is not unambiguous. For example, in $d=4$ both $P_4$ and $bE_4$ can be presented as total derivatives, and the term $P_4$ is not only topological, but also conformal, according to Eq. (\[Pont-W\]). Hence, the Gauss-Bonnet invariant can be attributed to two groups of terms and the Pontryagin density even to all three groups [*(i)*]{}, [*(ii)*]{} and [*(iii)*]{}. In any case, as the reader will see shortly, the conformal invariance of $P_4$ makes the inclusion of this term into anomaly-induced action a very simple exercise. We shall present some details only to achieve a self-consistent exposition of the consideration.
The simplest part is the $\Box R$-term, which can be directly integrated by using the relation - g\_ d\^4xR\^2 = 12 R. \[identity\] It is easy to see that in this case the solution is a local functional, that gives rise to the well-known ambiguity in the coefficient $a^\prime$ of the $\Box R$-term, which was discussed in details in [@anom2003].
Now, let us concentrate on the non-local part of anomaly-induced action[^1]. The solution of (\[mainequation\]) can be presented in the simplest, non-covariant form, in the covariant non-local form and in the local covariant form with two auxiliary fields. Let us start from the simplest case. By introducing the conformal parametrization of the metric g\_ &=& [|g]{}\_e\^[2(x)]{} \[confpa\] one can use an identity - g\_ = - . |\_[[|g\_]{}g\_, 0]{}. \[deriv\] Here and below the quantities with bars are constructed using the metric ${\bar g}_{\mu\nu}$, in particular \_\^2 &=& F\_F\_ [|g]{}\^[|g]{}\^, F\_ = \_A\_- \_A\_\[Fmn\]
Furthermore, we will need the conformal transformation rules W\_k &=& \_k\^2 , (W\_k = C\^2,P\_4,F\^2), \[Wk\] and (E - 23R) &=& ([|E]{} - 23[|]{}[|R]{} + 4[|]{}\_4), \[GBtrans\]\
[|]{}\_4 &=& \_4, where \_4 &=& \^2 + 2R\^\_\_- 23R+13R\_[;]{}\^\[Pan\] is covariant, self-adjoint, fourth-derivative, conformal operator [@Paneitz].
After we use the transformation rules (\[GBtrans\]) and (\[Wk\]), Eq. (\[mainequation\]) becomes very simple and the solution for the effective action can be found in the form \_[ind]{} &=& d\^4 x{ \^2 + \_\^2 + \_4 + b([|E]{}-23 [|[|]{}]{} [|R]{}) + 2b\_4}\
&-& (c+) d\^4 xR\^2 + S\_c\[[|g]{}\_,A\_\], \[quantum\] where $S_c[{\bar g}_{\mu\nu},\,A_\mu]=S_c[g_{\mu\nu},\,A_\mu]$ is an unknown conformal invariant functional of the metric and $\,A_\mu$. This functional is an integration constant for the Eq. (\[mainequation\]) and hence it can not be uniquely defined in the present framework. Let us note that in some cases this functional is irrelevant. An example is cosmological solution without background electromagnetic field. In this case the metric is conformally trivial and $S_c[g_{\mu\nu}]$ becomes an irrelevant constant.
Even in cases of non-cosmological metrics the functional $S_c[g_{\mu\nu}]$ does not prove to be very significant, because the rest of the effective action (\[quantum\]) contains all information about the UV behaviour of the theory. In the massless case, with a usual duality between UV and IR regimes, this means that $S_c[g_{\mu\nu}]$ may have only sub-leading contributions. These arguments are confirmed by successful applications to black holes [@balsan; @Reis-Nord] and gravitational waves [@wave; @GW-HD].
The solution (\[quantum\]) is non-covariant, because it is not expressed in terms of the physical metric $g_{\mu\nu}$. In order to obtain the non-local covariant solution of Eq. (\[mainequation\]), one has to introduce the Green function for the Paneitz operator, (\_4)\_xG(x,y) = (x,y) \_x = d\^4 x . \[GP\] Using (\[deriv\]) it is easy to check that for any conformal functional $A(g_{\mu\nu}) = A({\bar g}_{\mu\nu})$, && 2 g\_ (y) \_xA(E - 23R) \[gabi\]\
&=& \_xA.(E - 23R)|\_[0, [|g]{}\_ g\_]{} = 4\_4 A = 4\_4 A.
By means of the last relation it is easy to solve both remaining parts (remember that the local part we already have from Eq. (\[identity\])) of induced effective action, and we arrive at \_[ind]{} &=& \_ + \_b + \_c, \[nonloc\] where \_ = 14\_x\_y (C\^2 + F\_\^2 + P\_4 )\_x G(x,y)(E - 23R )\_y, \[om-term\] \_b = \_x \_y (E - 23R)\_xG(x,y)(E - 23R)\_y \[b-term\] and \_[c]{} = - \_xR\^2(x) . \[c-term\] One has to note that the Pontryagin density shows up only in the first nonlocal term (\[om-term\]), but in what follows we shall see that the second term (\[b-term\]) is still relevant for constructing the kinetic term of the Chern-Simons modification of gravity. At the same time, the local term (\[c-term\]) will remain separated from others.
Anomaly-induced action and kinetic term for Chern-Simons gravity
----------------------------------------------------------------
As a next step, the nonlocal expressions for the anomaly-induced effective action can be presented in a local form by introducing two auxiliary scalar fields $\ph$ and $\psi$ [@a]. An equivalent two-scalar representation was suggested in [@MaMo], while the simpler one-scalar form was known from much earlier [@rei]. Since the details of the procedure were described also in [@PoImpo; @PoS-Conform] and do not change essentially due to the term $P_4$, let us present only the final result \_[ind]{} &=& S\_c\[g\_\] - \_xR\^2 + \_x{12 \_4- 12 \_ 4\
&+&\
&+& (C\^2 + F\_\^2 + P\_4) }. \[finaction\] At the classical level the local covariant form (\[finaction\]) is equivalent to the non-local covariant form (\[nonloc\]). The definition of the boundary conditions for the Green functions $G(x,y)$ are equivalent to the same boundary conditions for the auxiliary scalars $\ph$ and $\psi$. For the discussion of the importance to have two fields let us address the reader to [@a; @MaMo; @PoImpo].
The action (\[finaction\]) represent a final product of our integration of conformal anomaly. However, in order to make the consideration leading to the version of Chern-Simons gravity [@jackiw-pi] more explicit, let us make a change of variables similar to one of [@MaMo]. Let us introduce two new scalars, = , = , \[chixi\] such that = , = . \[phpsi\] Then the total gravitational action, including the classical part and the anomaly-induced action (\[finaction\]) can be cast into the form \_[grav]{} &=& S\_[EH]{} + S\_[HD]{} + \_[ind]{}\
&=& S\_[EH]{}\[g\_\] + S\_[HD]{}\[g\_\] + S\_c\[g\_\] + \_x{\_4+ k\_1(E -23R)(- )\
&+& k\_2 (C\^2 + F\_\^2 + P\_4) +k\_3R\^2 }. \[faction\] where k\_1=, k\_2=, k\_3=-, \[coeffs\] and the classical vacuum part includes the Einstein-Hilbert action with cosmological constant S\_[EH]{} =-d\^4 x(R+2). \[EH\] and the higher derivative terms, S\_[HD]{} &=& d\^4x {a\_1C\^2+a\_2E+a\_3R+a\_4R\^2 }. \[HD\] Obviously, the $R^2$-terms in (\[HD\]) and (\[faction\]) combine, that produce a well-known ambiguity in the local part of the total action with anomaly-induced contribution [@anom2003].
Compared to the previously known solutions [@a; @MaMo; @GiaMot; @QED-Form], the expression (\[faction\]) has an extra term proportional to $\,\chi\,P_4$, where $\chi$ is a new auxiliary scalar field related to the conformal anomaly. This is exactly the structure which was extensively discussed in the context of Chern-Simons extension of general relativity starting from [@LWK] and [@jackiw-pi]. The remarkable difference is that the field $\chi$ in (\[faction\]) has higher-derivative kinetic term and also contains a mixing with the second scalar field $\xi$.
Brief review of other forms of the kinetic term
-----------------------------------------------
Since the main output of the previous consideration is the new form of the kinetic term for the Chern-Simons modified gravity, Eq. (\[faction\]), it is worthwhile to give a list of the previously known kinetic terms.
The Chern-Simons gravity is usually understood as an effective theory which should be obtained from a more fundamental theory [@jackiw-pi; @LWK]. Consequently, the form of the kinetic term depends on the choice of the fundamental theory. In our case it is the quantum theory of matter fields (with parity violation) on classical curved background, which led us to (\[faction\]). This action is different from the previously known versions, which can be presented as S &=& d\^4x { -R + \^ R\_ - } + S\_[mat]{} , \[dCS\] where $\ka=1/16\pi G$, while $\al$ and $\be$ are some new constants. The Chern-Simons coupling field is $\psi$ with a potential term $V(\psi)$, and $S_{mat}$ is the action of matter. Also, we used the standard notation for the dual Riemann tensor \^ &=& \^ R\^\_[ ]{} . The modified Enstein equation for the action (\[dCS\]) is R\_ - g\_R + C\_ = T\_ , \[EMmetrica\] where $T_{\mu\nu}$ is the total momentum-energy tensor. The C-tensor is defined as C\^ &=& \_\^[(]{} \_R\^[)]{}\_ + \^[()]{}\_\_. The variation of (\[dCS\]) with respect to the scalar field yields the equation &=& - \^ R\_ , \[EMescalar\] which is the Klein-Gordon equation with an extra Pontryagin density source.
There are two main approached, based on different choices of the constants $\al$ and $\be$. One of them is called non-dynamical Chern-Simons gravity, when we have $\be=0$. Then the scalar field does not evolve dynamically, and is a field prescribed externally. This model was introduced by Jackiw and Pi in [@jackiw-pi], where it was defined that $\psi=t/\mu$, the choice called canonical, with $\mu$ being some dimensional parameter. The boundary conditions in this theory were discussed in [@Gru-Yu]. The development of this approach and further references can be found in the review [@AleYu].
All solutions for the non-dynamic case must satisfy the Pontryagin constraint \^ R\_ = 0 . This constraint limits the space of solutions of the theory. For instance, Kerr metric can not be solution since it does not satisfy this constraint. The rotating black hole solutions have been found within approximation schemes, with certain inconsistencies discussed in the literature [@yunes-pretorius]. On the other hand, it was discussed that ghosts can not be avoided in the non-dynamic theory [@MotoSuy], forcing to pay more attention to the dynamic scalar case.
The dynamic case corresponds to an arbitrary $\al$ and $\be$ in the action (\[dCS\]). Such a model was introduced by Smith et. al. in [@smith], motivated by the low energy limit of string theory. The potential term was supposed to follow from the fundamental string theory, however, as usual, there is some freedom in this part. For a zero potential, $V(\psi)=0$, there is a uniqueness theorem which ensures that in case spherically symmetric, static and asymptotically flat spacetime, the solution is given by the Schwarzschild metric [@shiromizu]. In [@rogatko], the uniqueness was established for the case the Reissner-Nordstrom metrics.
Until now, there are no exact solutions for a rotating black hole and some approximate schemes are used instead. For example, in [@yunes-pretorius; @kMT; @YSYT2] the slowly rotating black hole was studied in the small-coupling limit, while [@Ali-Chen] carried out the study for an arbitrarily large coupling.
Recently, the post-GR corrections for the dynamic model has been considered through the study of rapidly rotating black holes in the decoupling limit [@stein; @konno-takahashi]. Such work is important because of the possibility to constraint the theory in the strong field regime.
The gravitational perturbations are fundamental for better understanding of the gravitational waves and the stability of solutions. In [@MotoSuy; @Yunes-Sopuerta; @CG] the gravitational perturbations were explored for the black hole background and in [@DFK] for the cosmological case. The issue of ghosts was studied for both non-dynamic and dynamic models. Although in both models the ghosts are present, one can avoid them for a constant scalar background in the dynamic case [@MotoSuy], while in the non-dynamical theory the scalar behavior is fixed. Let us mention that the works listed above were done for zero potential of the scalar field, while in the recent work [@MPCG] the gravitational perturbations with the mass term $V(\psi)=m^2\psi^2$ were considered.
Indeed, the choice of the kinetic term is non-trivial, in particular it was recognized that the Lagrangian of the kinetic term does not necessarily be of the Klein-Gordon type. Other kinetic terms were considered, e.g. the one discussed in [@AleYu] has some relation to string theory S\_ = - d\^4x { \_1 g\^\_\_+ \_2(g\^\_\_)\^2 } , where $\be_1$ and $\be_2$ are some constants.
An approach which is the closest one to our result (\[faction\]) was developed in [@YS; @YSYT] and eventually used to describe the linear stability in [@AYY]. The corresponding theory is known as Quadratic Modified Gravity, the action can be cast into the form S &=& d\^4x {R + f\_1()R\^2 + f\_2()R\_\^2 + f\_3()R\_\^2\
&+& f\_4()\^R\_ - } + S\_[mat]{} , \[fi\] where $f_i(\psi)$ are some functions of the scalar field. One can easily see that in the case of (\[faction\]) all these functions are linear, the coefficients are defined by the number of quantum particles and the kinetic term is more complicated and involves higher derivative and the second scalar. Also, the potential term is absent for the anomaly-induced version of the Chern-Simons gravity (\[faction\]).
The situation with ghosts and tachyons in the theory (\[fi\]) was discussed in [@MotoSuy]. It is important to note that the fields $\chi$ and $\xi$ are auxiliary scalars, which just exist to parametrize the non-localities in the original action (\[nonloc\]). This feature removes the need for discussing the ghosts related to the fields $\chi$ and $\xi$. Another way to understand this is to remember that the terms in the action (\[nonloc\]) are at least of the third order in curvature (except $R^2$, which does not produce ghosts [@Stelle-77]). Therefore, the quantum part of induced actions presented above has no issue with ghosts, at least on the flat background[^2]. Of course, the classical action behind the anomaly contains a usual $C^2$-term, which is known to produce ghosts. However, there are some indications that the ghost is not becoming a real particle at the energies below Planck scale [@GW-HDQG].
Interpretation and applications of the parity-violating terms
=============================================================
The presence of imaginary parity-violating terms in the conformal anomaly (\[T\]) can be interpreted such that the existence of massless left-handed neutrino should be theoretically disfavoured [@bonora]. This possibility looks very interesting, especially in view of experimental confirmation of neutrino oscillations. However, one has to remember that the conformal anomaly is not a directly observable physical quantity. The remarkable exception is the cosmological FRW-like solution, when the metric depends only on the conformal factor according to Eq. (\[confpa\]), with $\si=\si(\eta)$ and $\eta$ conformal time. The trace anomaly directly affects the dynamics of $\si=\si(\eta)$. But, as far as Weyl tensor is zero for the FRW-like metric, the Eq. (\[Pcon\]) shows that the Pontryagin term is also zero for this metric. Consequently, the background cosmological solution is not affected by the presence of the new term with $P_4$.
In all other cases the solution (\[faction\]) is not exact. This means that the effect of the $P_4$-dependent term can be, in principle, compensated by the conformal functional $S_c(g_{\mu\nu})$. This means that all the conclusions concerning the possible physical effects related to $P_4$ assume that the functional $S_c(g_{\mu\nu})$ is irrelevant. On the other hand, the general arguments presented above show that this assumption is quite reasonable. Then, we can expect that the $P_4$-term can be relevant for the gravitational waves on the cosmological (or other) background, and also for the physically relevant solutions such as Schwarzschild, Reissner-Nordstrom or Kerr.
The two aspects of the $P_4$-term in the anomaly (\[T\]) and action (\[faction\]) can be relevant. The first one is that this term is parity-violating. This means that it is expected to produce a parity-odd solutions, including for the metric perturbations. As a result, one can expect the parity-odd component to emerge in the CMB spectrum. The second aspect is related to the imaginary coefficient. Let us present some considerations of these two aspects, but start from the general discussion of the solution in the presence of the Pontryagin term.
Possible gravitational solutions
--------------------------------
The solutions in the Chern-Simons modified theory of gravity have been extensively discussed in the literature, e.g., in the papers [@YS; @AYY]. The main difference between the models which were previously considered and (\[faction\]) is the form of the kinetic term for the scalar field and the presence of higher derivative terms. Let us consider in some details the simplest case of the spherically-symmetric solution, which is quite illustrative. Our purpose is not to find a new solution, but only show that the parity-violating and other higher-derivative terms in the action (\[faction\]) may modify the usual Schwarzschild solution. This does not happen with the classical higher derivative terms of (\[HD\]), because for the Ricci-flat background the Weyl-square term in $d=4$ can be easily reduced to the Gauss-Bonnet topological invariant [@BH-dim]. So, we can completely concentrate on the anomaly-induced part. The equations for the auxiliary fields have the form \_4&=& k\_1 (E -23R) - k\_2 (C\^2 + P\_4 ) ,\
\_4&=& - k\_1 (E -23R). \[cx\] Furthermore, in the Ricci-flat case the Paneitz operator becomes simply $\Box^2$, and also one has $E=C^2=R_{\mu\nu\al\be}^2$. For the sake of simplicity, consider the possible solutions of the form [@jackiw-pi] = d\_1 t + f\_1(r) , = d\_2 t + f\_2(r) , \[CXsol1\] where $d_{1,2}$ are constants and $f_{1,2}$ some functions of the radial variable $r$. Assuming that the metric satisfies Schwarzschild solution, one has $\,P_4=0$, $\,R_{\mu\nu\al\be}^2=48(GM)^2/r^6\,$ and \^2f\_[1,2]{}(r) = . \[emsb\] where $\,\al_1 = 12(k_1-\om k_2)$ and $\,\al_2 = -12k_1$. The general solution for the functions $f_{1,2}(r)$ corresponds to the equations (here $f=f_{1,2}$ and $\al=\al_{1,2}$) was obtained in [@balsan], &=& +-- + (BM\^2 + -AM -)\
&-& - - (-)\
&+& (-) . \[CXsol2\] Here $(d,A,B,C)$ are constants that specify the homogeneous solution of $\Box^2f=0$. However, it is not necessary that the equation for the metric can be satisfied for any choice of the coefficients $(d,A,B,C)_{1,2}$. In the part which is important for our consideration, however, one can see that there is no real influence of the $P_4$ term, at this level. Moreover, if we assume, as an approximation, that at low energies higher derivative parity-even terms are irrelevant and only parity-odd $P_4$-dependent term is relevant, then the Schwarzschild solution is valid in this truncated version of the theory. Finally, Eq. (\[CXsol1\]) which is typical [@balsan] in the higher derivative model such as (\[faction\]), shows that the difference between dynamical and non-dynamical versions of the Chern-Simons modified gravity can be resolved, at least for some particular solutions.
Parity violation
----------------
What could be the effect of parity-violating term in the gravitational action? In order to answer this question, one has to remember that the most likely manifestation of the Pontryagin term is the parity violation in the gravitational waves spectrum [@LWK]. Due to the Planck suppression the effect is going to be very weak and perhaps can not be observed directly. However, the parity violation can eventually go to the CMB through the well-known mechanisms (see, e.g., [@Dodelson; @Durrer-CMB]) and may eventually lead to the anisotropy in the metric perturbations.
Imaginary coefficient
---------------------
The imaginary component of effective action is a typical phenomena in quantum field theory [@Schwinger-51]. Usually it is related to the logarithmic structure in the form factors at the UV (the same as conformal anomaly) and indicates to the possible particle production by external field [@ItsikZu; @Maggiore]. In order to have such a production, the energy of created particles should be smaller than the intensity of an external field. In the case of strictly massless neutrino this condition can be easily satisfied. However, some other details must be taken into account. The production of massless left-handed neutrino will be related to the fourth-derivative term $\,k_2 \chi\ep P_4$ in the effective action (\[faction\]). This term is strongly suppressed by the Planck mass in the Einstein-Hilbert term, even in the inflationary period, except in the initial stable phase of the modified Starobinsky inflation model [@asta]. And in this special case any kind of particle production is compensated by the powerful inflation, such that the density of created particles remains negligible.
After inflation the energy of the created neutrino particles would be very small, at most of the order of the energy of the gravitational waves, since the effect is zero for the FRW-like background. During the long period of existence of the Universe there can be certain production of such particles, but there is another aspect of the problem. The neutrinos are fermions and, with a very small energy, fermionic particles should form a Fermi surface. Then the production of neutrino should be suppressed by the Pauli principle. From the quantum theory viewpoint this means the existence of the effect of quantum interaction between neutrino, which would forbid their creation. All in all, the imaginary component of the effective action can not be probably seen as a basis of the no-go theorem forbidding theoretically massless neutrino.
Conclusions
===========
We presented a simple derivation of the anomaly-induced effective action of gravity with the new parity-violating term in conformal anomaly, which was recently discovered in [@bonora]. The integration proceeds with a minimal changes compared to the known procedure, since the new term is both topological and conformal. The result of the integration represents a new version of the well-known Chern-Simons modification of general relativity, which was extensively discussed in the literature, starting from [@jackiw-pi] and [@LWK]. The possible role of such a term was explored in details [@AleYu], and we can not add much to this discussion, except to suggest a new form of kinetic term for the auxiliary scalar field, derived in (\[faction\]).
Concerning the physical significance of the Pontryagin term, there is no doubt that the presence of parity-violating term in gravity is potentially very interesting [@AleYu]. The reason is that even a very small violation of the symmetry can give an observable effect. At the same time, some simple qualitative arguments show that the effect of imaginary term in the effective action and the related production of neutrino by the gravitational background should be too weak to provide a theoretical “prohibition” of the massless neutrino, as it was suggested in [@bonora].
One more observation concerns the electromagnetic sector of the anomaly-induced action. There is no parity violation in this part of the action. However, if one could find some field (in the baryonic or dark sectors of the spectrum), which produce the parity-violating term in the conformal anomaly, the mechanism which we described in this paper would immediately generate axion with a very specific form of the kinetic term, equal to the one presented in (\[faction\]).
Finally, let us say a few words about the perspectives of the new form of the Chern-Simons modified gravity (\[faction\]). It is obvious that it would be interesting to check both theoretical and phenomenological consequences of this theory, starting from the solution for the rotating black hole and cosmological applications. From the QFT side, it would be interesting to see whether some (probably reduced) version of this term can be derived in other theories, in particular whether it can be met in the theory on massive neutrino at very low energies, due to the difference of the masses of the right and left components from one side and the effect of gravitational decoupling [@apco] from another side. Regardless of the serious technical difficulties of this program, it does not look completely impossible to be completed.
Acknowledgements {#acknowledgements .unnumbered}
================
I.Sh wish to acknowledge stimulating correspondence and discussions with Dr. Guilherme Pimentel, clarifying conversation with Dr. Stefano Giaccari and also interesting discussion about related subjects with Prof. Fidel Schaposnik. S.M. is grateful to CAPES for support of his Ph.D. research program. I.Sh. is very grateful to the Département de Physique Théorique and Center for Astroparticle Physics of Université de Genève for partial support and kind hospitality during his sabbatical stay, and to CNPq and also to FAPEMIG and ICTP for partial support of his work.
Appendix {#appendix .unnumbered}
========
The Pontryagin density is given by P\_4 &=& \^ R\_[ . .]{}\^[ ]{} R\_ . \[P4\] Let us prove that the Riemann tensor here can be replaced by the Weyl tensor. In 4-dimensional space we have the following relation between Riemann and Weyl tensors, R\_ = C\_ + ( R\_ g\_ - R\_ g\_ + R\_ g\_ - R\_ g\_) - R ( g\_ g\_ - g\_ g\_) . \[R-W\] Replacing (\[R-W\]) into (\[P4\]) we get P\_4 = \^ C\_[ . .]{}\^[ ]{} C\_ + 2 \^C\_ R\^\_- \^ C\_ R . On the other hand, Bianchi identity for the Weyl tensor C\_ + C\_ + C\_ = 0, provide the relations \^C\_ = 0 , \^ C\_ = 0 . Therefore, we arrive at the desired formula, P\_4 = \^ C\_[ . .]{}\^[ ]{} C\_ , \[Pcon\] that shows $\sqrt{-g} P_4$ to be conformal invariant.
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[^1]: The non-localities due to anomaly was first discussed in [@DDI-76].
[^2]: This does not exclude the emergence of ghosts on other background, as it was discussed in [@Chiba-05] and recently in [@QGprT] in relation to the final stage of the de Sitter phase of the evolution of the $\La$CDM universe.
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abstract: 'The aim of this study is to demonstrate the feasibility of removing the image Moiré artifacts caused by system inaccuracies in grating-based x-ray interferometry imaging system via convolutional neural network (CNN) technique. Instead of minimizing these inconsistencies between the acquired phase stepping data via certain optimized signal retrieval algorithms, our newly proposed CNN-based method reduces the Moiré artifacts in the image-domain via a learned image post-processing procedure. To ease the training data preparations, we propose to synthesize them with numerical natural images and experimentally obtained Moiré artifact-only-images. Moreover, a fast signal processing method has also been developed to generate the needed large number of high quality Moiré artifact-only images from finite number of acquired experimental phase stepping data. Experimental results show that the CNN method is able to remove Moiré artifacts effectively, while maintaining the signal accuracy and image resolution.'
author:
- Jianwei Chen
- Jiongtao Zhu
- Wei Shi
- Qiyang Zhang
- Hairong Zheng
- Dong Liang
- Yongshuai Ge
bibliography:
- 'Bibliography\_Paper.bib'
title: 'Automatic image-domain Moiré artifact reduction method in grating-based x-ray interferometry imaging'
---
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Introduction
============
The past two decades have witnessed the quick developments of grating-based x-ray interferometry imaging method, especially the Talbot-Lau imaging method. As a novel x-ray imaging method, it is able to generate three images, i.e., the absorption image, the differential phase contrast (DPC) image, and the dark-field (DF) image, with unique contrast mechanism from the same acquired dataset simultaneously. Different from the conventional absorption signal, the DPC signal corresponds to the x-ray refraction information. Studies have shown that the DPC signal may have advancements in providing superior contrast sensitivity for certain types of soft tissues[@Momose1996; @Momose2006; @Pfeiffer2006a; @Bech2009; @Jensen2011a; @li2014grating]. In addition, the complimentary DF signal corresponds to the small-angle-scattering (SAS) information, and thus is particularly sensitive to certain fine structures such as microcalcifications inside breast tissue[@Anton2013; @Michel2013; @wang2014non; @Grandl2015; @scherer2016correspondence]. Therefore, numerous research interests[@Zhu2010; @miao2013motionless; @Ge2014; @Koehler2015] have been attracted with the aim to translate this innovative x-ray imaging method from laboratory investigations to real clinical applications.
In an x-ray Talbot-Lau interferometry system, detection of the diffraction fringes of phase grating are often performed with an analyzer grating, which is usually an absorption grating having identical period as of the self-image of the phase grating. As a result, Moiré patterns with resolvable period by most of the medical grade flat-panel-detectors are generated. In reality, depending on the relative alignments of the phase grating and analyzer grating, the detected Moiré diffraction patterns may have various distributions, i.e., spatially varied periods and intensities, over the entire detector surface. To retrieve the attenuation, the refraction and the SAS information of an imaging object, the phase stepping technique[@Weitkamp2005] is usually utilized. Specifically, a group of Moiré patterns are recorded by laterally translating one of the three gratings within one period in Talbot-Lau interferometry. As a result, each detector pixel records a group of phase stepping data points, which form the so-called phase stepping curve and are used to extract the absorption, DPC, and DF signals.
As long as all the experimental conditions are ideal, images free of artifacts can be easily generated. However, due to some non-ideal experimental settings, their image quality may be degraded. For example, the polychromatic x-ray beam used in laboratory may introduce the so-called type-II beam hardening effect[@bevins2013type] in DPC-CT imaging. For radiographic DPC imaging, the acquired DPC images may also be contaminated by phase wrapping artifacts[@jerjen2011reduction; @tapfer2012experimental; @epple2013unwrapping; @epple2015phase] and Moiré image artifacts[@seifert2016optimisation; @marschner2016helical; @kaeppler2017improved; @dittmann2018optimization; @de2018analysis; @Hauke2017Analytical; @hauke2018enhanced]. The former artifact happens when the detected refraction angle of the imaging object is out of the $[0, 2\pi)$ range, while the second artifact usually happens due to the non-ideal experimental conditions. For example, the inaccurate mechanical movement of the grating during phase stepping procedure, the non-stationary drifts of the x-ray focal spot over multiple exposures, system vibrations, and so on. In this paper, the impact of these potential influences is categorized as effective phase stepping position deviation, denoted by $\eta$.
In fact, any slight phase stepping position deviation will break the consistency between the assumed ideal sinusoidal phase stepping signal model and the acquired phase stepping data. Due to such mismatch, undesired image artifacts might show up after signal extractions using the assumed theoretical signal model. Since this kind of artifacts look similar as of the Moiré diffraction patterns, so they are named as Moiré image artifacts. During experiments, these slight effective phase stepping position deviations are often random and hard to be exactly determined, therefore, Moiré image artifacts are often unavoidable and needs to be mitigated after data acquisitions. By far, several different methods have been proposed[@Pelzer2015Reconstruction; @seifert2016optimisation; @marschner2016helical; @kaeppler2017improved; @dittmann2018optimization; @de2018analysis; @hauke2018enhanced] to mitigate these Moiré image artifacts. Among them, Pelzer et al[@Pelzer2015Reconstruction] and Seifert et al[@seifert2016optimisation] have developed correction algorithms based on principle component analysis (PCA) to adjust the phase stepping position deviations. By establishing an expectation-maximization (EM) algorithm, the Moiré artifacts in single-shot DPC-CT imaging system could also be reduced[@marschner2016helical]. Kaeppler et al[@kaeppler2017improved] developed another phase step position optimization algorithm under the condition of discarding very irregular phase step data points. Moreover, an enhanced image reconstruction method[@hauke2018enhanced] has proposed to better fulfill a real clinical application environment. The phase step position deviations can also be corrected using a two-step fitting algorithm[@dittmann2018optimization], in which both the sinusoidal shape of the phase stepping curve and the phase stepping positions are considered. Overall, most of the above Moiré artifacts reduction algorithms are developed with the aim to adjust the inaccurate phase stepping positions, namely, minimize the inconsistencies between the theoretical phase stepping signal model and the experimentally acquired phase stepping data during signal extractions.
Recently, deep learning technique[@lecun2015deep; @goodfellow2016deep] has attracted huge research interests due to its remarkable performance in variety of applications, for example, image classification and segmentation, object identification, image artifact removal[@sun2018moire], denoising, and so on. Taking advantages of its powerful strength and remarkable performance, it becomes very interesting to apply the deep learning technique to the Moiré artifacts reduction problem in x-ray interferometry imaging. In particular, we mainly consider the Moiré artifact reduction task as an image post-processing procedure, and hope to mitigate them in image-domain using an end-to-end supervised U-Net[@ronneberger2015u] type convolutional neural network (CNN). Unlike conventional algorithm-driven artifact reduction methods, the performance of CNN-based Moiré artifacts reduction method heavily relies on the training data, including both of the data quality and the data richness. In other words, sufficient number of high quality image pairs: one Moiré artifact image(known as network input) and one clean image (known as label), has to be prepared for network training. In this preliminary work, we suggest to train the network with numerical images, and validate its performance on experimental images. The numerical images used for network training are manually synthesized from natural images and experimentally acquired Moiré artifact-only images. Herein, we assume that all of the Moiré artifact-only images are acquired from an already fine-tuned Talbot-Lau interferometry with fixed grating alignments.
Even with same grating alignments, we noticed that the acquired Moiré artifacts are still different from each other, including the structural variations and the spatial shifts. To make the CNN network robust enough, undoubtedly, the Moiré artifact-only images used in numerical training data synthesis have to be sufficiently rich. In other words, repeated experiments have to be performed during training data preparations. This could dramatically prolong the data preparation period, and might become a hurdle when using the CNN technique. In addition, we also noticed that most of the Moiré artifact-only images generated from normal phase stepping data, which contains phase stepping uncertainties, and processed with standard signal extraction procedures have low signal-to-noise-ratio (SNR). Such low quality Moiré artifact-only image samples could also degrade the final CNN performance. Therefore, we proposed a fast signal processing method to generate large number of high SNR Moiré artifact only images from finite number of phase stepping datasets obtained with normal data acquisition procedures. The key idea is to manually increase the slight inconsistencies of the phase stepping data, thus, increasing the amplitude of the Moiré artifacts. Specifically, we suggest to manually replace one phase stepping image by its neighbors. For instance, replacing the $k$-$th$ phase stepping image by the $(k-1)$-$th$ or $(k+1)$-$th$ phase stepping image results in an increase of $\eta^{(k)}$ by one. By doing so, the inconsistencies between the theoretical signal model needed data and the experimental phase stepping data are greatly magnified. Results show that the amplitude of the Moiré artifacts can be dramatically improved. Moreover, roughly $M\times(M-1)$ number of Moiré artifacts with unique spatial distribution can be obtained from only one single experimental phase stepping data. As a result, the training data preparation procedure could be accelerated by at least ten times (assuming $M=4$).
The remain of this paper is organized as following: the section II presents the theoretical analyses of the phase stepping position deviation induced Moiré artifact for the absorption, DPC, and DF contrasts. In the section III, numerical simulations are performed to demonstrate the proposed novel signal processing method. In the section IV, we describe the details of generating the training dataset, and the proposed image-domain Moiré artifact reduction CNN network. In section V, numerical experiments are carried out to verify the performance of the CNN-based Moiré artifact reduction method. In addition, experimental results are also validated. We made discussions about this work in section VI, and finally gave a brief conclusion in section VII.
Theoretical analyses
====================
General theory of Moiré artifacts
---------------------------------
In this theoretical discussion part, the standard phase stepping procedure is assumed. With this ideal signal model, the detected x-ray intensity at a certain pixel for the $k$-th phase stepping position is denoted as $${I}^{(k)}=I_0\Bigg\{1+\epsilon\cos\left[\frac{2\pi k}{\mathrm{M}}+\phi\right]\Bigg\},
\label{eq:signal_mod}$$ where $I_0$ corresponds to the x-ray absorption contrast signal, $\phi$ corresponds to the x-ray differential phase contrast signal, and $\epsilon$ corresponds to the x-ray dark-field contrast signal, $\mathrm{M} (\ge3)$ corresponds to the total phase step number, and $k=1,2,...,\mathrm{M}$. Notice that $I_0$, $\epsilon$, and $\phi$ are all varied spatially. No Poisson photon fluctuations are assumed in this model. To proceed, the effective phase stepping position deviation $\eta^{(k)}$ is considered. In reality, $\eta^{(k)}$ could bee caused by many external influences, vibrations, and mechanical inaccuracies. For our experimental system, we noticed that $\eta^{(k)}$ are dominantly influenced by the x-ray tube focal spot drifts between consecutive exposures, see Fig. \[fig:spotdrift\]. Now the acquired x-ray intensity can be written as: $$\begin{aligned}
\label{eq:signal_1}
{I}^{(k)}=&I_0\Bigg\{1+\epsilon\cos\left[2\pi\frac{k+\eta^{(k)}}{\mathrm{M}}+\phi\right]\Bigg\}.\end{aligned}$$ Without knowing $\eta^{(k)}$ precisely in prior, the $I_0$, $I_1$, and $\phi$ signals should still be retrieved using Eqs. (\[eq:ch01\_N0\])-(\[eq:ch01\_phi\]) from these acquired non-ideal phase stepping datasets $\{{I}^{(k)}\}$. With Taylor approximations similar as in literature[@Hauke2017Analytical], the final radiographic absorption image $\hat{\mathcal{A}}$ with Moiré artifacts can be estimated via $$\begin{aligned}
\hat{\mathcal{A}}=-\ln \left\{ \frac{\hat{I^{obj}_0}}{\hat{I^{ref}_0}}\right\} &\approx \mathcal{A}-\frac{2\pi \epsilon_{ref}}{\mathrm{M}^2}\sum_{k=1}^{\mathrm{M}}\eta^{(k)}_{ref}\sin\left[\frac{2\pi k}{\mathrm{M}}+\phi_{ref}\right] \nonumber \\
&+\frac{2\pi \epsilon_{obj}}{\mathrm{M}^2}\sum_{k=1}^{\mathrm{M}}\eta^{(k)}_{obj}\sin\left[\frac{2\pi k}{\mathrm{M}}+\phi_{obj}\right].
\label{eq:atten_0}\end{aligned}$$ Herein, the superscript $ref$ denotes the air scan, and the superscript $obj$ denotes the object scan. During these derivations, we have ignored the high order terms of $\alpha^{(k)}_{ref}$, $\alpha^{(k)}_{obj}$, $\eta^{(k)}_{ref}$, and $\eta^{(k)}_{obj}$. Similarly, the radiographic DF contrast image with Moiré artifacts, denoted as $\hat{\mathcal{S}}$, can be expressed as below $$\begin{aligned}
\hat{\mathcal{S}}=-\ln \left\{ \frac{\hat{\epsilon^{obj}}}{\hat{\epsilon^{ref}}}\right\}& \approx \mathcal{S}-\frac{2\pi}{\mathrm{M}^2}\sum_{k=1}^{\mathrm{M}}\eta^{(k)}_{ref}\sin\left[\frac{4\pi k}{\mathrm{M}}+2\phi_{ref}\right] \nonumber \\
& +\frac{2\pi}{\mathrm{M}^2}\sum_{k=1}^{\mathrm{M}}\eta^{(k)}_{obj}\sin\left[\frac{4\pi k}{\mathrm{M}}+2\phi_{obj}\right] \nonumber \\
& +\frac{2\pi \epsilon_{ref}}{\mathrm{M}^2}\sum_{k=1}^{\mathrm{M}}\eta^{(k)}_{ref}\sin\left[\frac{2\pi k}{\mathrm{M}}+\phi_{ref}\right] \nonumber \\
& -\frac{2\pi \epsilon_{obj}}{\mathrm{M}^2}\sum_{k=1}^{\mathrm{M}}\eta^{(k)}_{obj}\sin\left[\frac{2\pi k}{\mathrm{M}}+\phi_{obj}\right].
\label{eq:sas_0}\end{aligned}$$ Finally, the DPC image, denoted as $\hat{\mathcal{P}}$, can be expressed by $$\begin{aligned}
\hat{\mathcal{P}}=\hat{\phi^{obj}}-\hat{\phi^{ref}}&\approx \mathcal{P} + \frac{2\pi}{\mathrm{M}^2}\sum_{k=1}^{\mathrm{M}}\eta^{(k)}_{ref}\cos\left[\frac{4\pi k}{\mathrm{M}}+2\phi_{ref}\right] \nonumber \\
&-\frac{2\pi}{\mathrm{M}^2}\sum_{k=1}^{\mathrm{M}}\eta^{(k)}_{obj}\cos\left[\frac{4\pi k}{\mathrm{M}}+2\phi_{obj}\right].
\label{eq:dpc_0}\end{aligned}$$ In Eqs. (\[eq:atten\_0\])-(\[eq:dpc\_0\]), the $\mathcal{A}$, $\mathcal{S}$, and $\mathcal{P}$ denote the ideal absorption image, DF image, and DPC image free of Moiré artifacts, correspondingly. Obviously, the measured absorption image $\hat{\mathcal{A}}$, DF image $\hat{\mathcal{S}}$, and DPC image $\hat{\mathcal{P}}$ are all contaminated by Moiré artifacts due to the random $\eta^{(k)}_{ref}$ and $\eta^{(k)}_{obj}$. In particular, the residual Moiré artifact on absorption image $\hat{\mathcal{A}}$ only contains the basic frequency component, which is identical as of the detected Moiré patterns from the phase stepping procedure. The residual Moiré artifacts on DPC images $\hat{\mathcal{P}}$ only consists of doubled frequency component. Whereas, Moiré artifacts on DF images $\hat{\mathcal{S}}$ are superimposed by the basic and doubled frequency components. Since the $\epsilon$ factor is usually less than one, therefore, the doubled frequency component becomes more dominant than the basic frequency component, see Eq. (\[eq:sas\_0\]). Due to the same reason, Moiré artifacts on absorption images are less pronounced than on DPC and DF images. Additionally, theoretical results also indicate that large values of $\eta^{(k)}_{ref}$ and $\eta^{(k)}_{obj}$ can lead to more dramatic Moiré artifact amplitude. Approximately, the Moiré artifacts can be assumed as additive signals[@de2018analysis] to the artifact-free images if ignoring the influences of $\epsilon_{ref}$, $\epsilon_{ref}$, object refractive property $\mathcal{P}$, and object SAS property $\epsilon_{obj}$. Although not quite rigorous, such assumptions typically yield good results during experimental validations.
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Numerical simulations
---------------------
Numerical simulations are performed in Matlab (The MathWorks Inc., Natick, MA, USA) to quantitatively investigate the impact of $\eta^{(k)}$ to the Moiré artifact magnitude. First of all, $I^{obj}_0$ and $I^{obj}_1$ are assumed to be the same as $I^{ref}_0$ and $I^{ref}_1$, which are set to 100 and 20, respectively. The $\phi^{ref}=\mathrm{mod}\left(\sqrt{((u/5+50)^2+(v/5+50)^2)}, 2\pi\right)$ map varies along both the horizontal and vertical directions, denoted as $u$ and $v$. Moreover, the $\eta^{(k)}_{ref}$ and $\eta^{(k)}_{obj}$ are sampled randomly from two random distributions: the uniform random distribution $[-\sigma_{\eta}, \sigma_{\eta}]$, and the Gaussian distribution with zero mean and standard deviation of $\sigma_{\eta}$ ($\sigma_{\eta}\in[0.0, 0.3]$). The phase shift of the object $\mathcal{P} = \phi^{obj}-\phi^{ref}$ is fixed at $\pi/3$. The total number for phase steps is equal to eight. The simulated images have dimension of 256 pixels by 256 pixels, and no Poisson photon fluctuations are considered. Finally, simulations with same parameters are repeated by 50 times.
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Results in Fig. \[fig:Num\_results\_2\] clearly demonstrate that the amplitude of the Moiré artifacts is linearly proportional to the effective phase stepping position deviation intensity $\sigma_{\eta}$. Therefore, it is always beneficial for generating images with less Moiré artifacts by minimizing the effective phase stepping position deviations. Although the phase stepping position deviation strengths need to be well controlled when designing a real experimental system, however, we would like to artificially increase their strengths to generate Moiré artifact-only images with higher SNR. Details of this method are discussed in the next section.
Moiré artifact reduction CNN
============================
Training data preparations
--------------------------
As demonstrated by the numerical simulations, generally, larger phase step position deviations correspond to higher Moiré artifact amplitude for all the three different contrast mechanisms, see Fig. \[fig:Num\_results\_2\]. Based on this observation, we were inspired to develop a fast signal processing method to generate sufficient number of CNN network needed high SNR Moiré artifact-only image samples from a standard x-ray Talbot-Lau imaging system, from which phase stepping datasets are acquired with standard radiation dose levels. To this aim, we suggest to virtually increase the deviation $\eta^{(k)}$ strength at the $k$$-th$ phase stepping position. In particular, one of the initially acquired $M$ phase stepping images is replaced by the image corresponding to other phase step position. Clearly, such manual replacement operation explicitly increases the strength of $\eta^{(k)}$ (i.e., $\sigma^{k}_{\eta}$), and thus helps increasing the inconsistencies of the phase stepping dataset. For instance, if replacing the second phase step image by the third phase step image, then $\eta^{(k=2)}$ approximately enlarges by 1.
As shown in Fig. \[fig:Exp\_moire\_0\] and Fig. \[fig:Exp\_moire\_1\], the resultant amplitudes of the Moiré artifacts indeed are increased significantly. In this study, we generate the Moiré artifact-only images by only replacing one phase stepping image once at a time. Line profiles in Fig. \[fig:Exp\_moire\_1\](b)-(c) demonstrate that this new signal processing method can also reduce image noise. As a result, the quality of the Moiré artifact-only images can be greatly enhanced. Meanwhile, since there are $M$ phase stepping images, and multiple air-scan phase stepping datasets can be easily acquired during system warm up or calibration scans, therefore, the network required large number of Moiré artifact-only image samples can be quickly generated.
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Afterwards, the network training data are ready to be synthesized. In this work, 5100 natural images(downloaded from ImageNet) and 210 experimentally acquired Moiré-artifact-only images (amplitude normalized between 0.0 to 1.0) with high SNR are prepared for each contrast mechanism, denoted as $\mathcal{M}_{\mathcal{A}}$, $\mathcal{M}_{\mathcal{S}}$, and $\mathcal{M}_{\mathcal{P}}$, see Fig. \[fig:Exp\_moire\_0\](j)-(l). First, the central part of the image with dimension of $256\times256$ is cropped and converted to gray-scale image $\mathrm{I}_{n}$. Afterwards, images are normalized and combined accordingly with the experimentally acquired Moiré artifact only images to generate the needed network training dataset for each contrast mechanism: the absorption image $\mathcal{A}_{Moir\'{e}}$, the dark-field image $\mathcal{S}_{Moir\'{e}}$, and the differential phase contrast image $\mathcal{P}_{Moir\'{e}}$. Details of these operations are expressed below: $$\begin{aligned}
\label{eq:M0}
\mathcal{A}_{Moir\'{e}} &= (3-\alpha) \times\frac{\mathrm{I}_{n}}{2^{8}}+\alpha\times\mathcal{M}_{\mathcal{A}}, \\
\label{eq:M1}
\mathcal{S}_{Moir\'{e}} &= (3-\alpha) \times\frac{\mathrm{I}_{n}}{2^{8}}+\alpha\times\mathcal{M}_{\mathcal{S}}, \\
\label{eq:M2}
\mathcal{P}_{Moir\'{e}} &=(2\pi-\alpha) \times\frac{\partial}{\partial u}\left[\frac{\mathrm{I}_{n}}{2^{8}}\right]+\alpha\times\mathcal{M}_{\mathcal{P}}.\end{aligned}$$ These Moiré samples are randomly augmented by factor $\alpha$, which is uniformly sampled between 0.0 and 0.3. Notice that the selection of 0.3 is very empirical. For every individual contrast mechanism, 200 out of the 210 experimentally acquired Moiré-artifact-only images are randomly sampled to synthesize the 5000 pairs of network training dataset, and the rest 10 experimentally acquired Moiré-artifact-only images are used to generate the 100 pairs of network testing dataset.
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CNN network
-----------
The U-Net type CNN network architecture is implemented, as illustrated in Figure \[fig:network\_ed\]. It contains ten layers in total. The strides of the $7\times7$ sized convolutional filters are set to be (2, 2). Five shortcuts are added accordingly to minimize the vanishing gradient phenomenon during back-propagation[@he2016deep]. The activation functions are ReLu[@hahnloser2000digital] (Rectified Linear Unit). Using the same CNN network, the synthesized absorption, DF, and DPC dataset are trained separately. For each network training, the objective loss function of the network is defined as: $$\mathcal{L} = \frac{1}{\mathrm{N_u}}\frac{1}{\mathrm{N_v}}\sum^{\mathrm{N_u}}_{u=1}\sum^{\mathrm{N_v}}_{v=1}|\mathrm{I_m}-\mathrm{I^{cnn}_m}|^2,
\label{eq:loss}$$ where $\mathrm{N_u}$ and $\mathrm{N_v}$ represent the total pixel number of the image along the $u$ and $v$ directions, correspondingly. In addition, the reference image free of Moiré artifact is denoted as $\mathrm{I_m}$, the CNN-learned image with reduced Moiré artifact is denoted as $\mathrm{I^{cnn}_m}$. During training procedures, the Adam algorithm[@kingma2014adam] is used with learning rate of 0.0001. The network is trained for 1000 epochs with mini-batch size of 64, and batch-shuffling is turned on to increase the randomness of the training data. On the Tensorflow deep learning framework using a single NVIDIA GeForce GTX 1070 GPU, the training time for each contrast is around eight hours.
Results of numerical experiments
================================
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Numerical validation results of the CNN
---------------------------------------
The network validation results in Figs. \[fig:Num\_cnn\_1\] demonstrate that the proposed CNN is robust enough to reduce the Moiré artifacts for the absorption contrast, DF contrast, and DPC contrast. The difference images between the reference images and the CNN learned images with reduced Moiré artifacts show that this method barely degrade the image spatial resolution. Despite that some very minor Moiré artifacts may still existed after processed by the CNN, they are visually negligible on the CNN processed results. In addition, the CNN-based artifact mitigation method does not introduce any new image artifacts.
To quantitatively characterize the performance of the CNN network, the structural similarity (SSIM) index are calculated for the entire image of three different contrast mechanisms independently. Herein, the reference image is considered as the ground truth. Comparisons results are listed in Table. \[table:1\]. Clearly, the CNN is able to significantly improve the image SSIM index, demonstrating that most of the Moiré artifacts have been removed after processed by the CNN.
\[table:1\]
[cccc]{} & & &\
Before CNN & 0.919$\pm$0.049 & 0.811$\pm$0.063 & 0.877$\pm$0.091\
After CNN & 0.994$\pm$0.004 & 0.963$\pm$0.013 & 0.975$\pm$0.025\
Experiments and results
=======================
Experiments
-----------
Experiments are performed on an in-house x-ray Talbot-Lau interferometry imaging bench in our lab. The system includes a rotating-anode Tungsten target diagnostic-level X-ray tube (IAE XM15, IAE S.p.A., ITALY). It is operated at 40.00 kV (with mean energy of 28.00 keV) radiographic mode with 0.10 mm nominal focal spot. The X-ray tube current is set at 100.00 mA, with a 3.00 second exposure period for each phase step. The X-ray detector is an energy-integrating detector (Varex 3030Dx, Varex Imaging Corporation, UT, USA) with a native element dimension of 194.00 $\mu$m $\times$ 194.00 $\mu$m. The Talbot-Lau interferometry consists of three gratings: a source grating G0, a $\pi$-phase grating G1, and an analyzer grating G2. The G0 grating has a period of 24.00 $\mu m$, with a duty cycle of 0.35. The G1 grating has a period of 4.36 $\mu m$, with a duty cycle of 0.50. The G2 grating has a period of 2.40 $\mu m$, with a duty cycle of 0.50. The distance between G0 and G1 is 1773.60 mm, and the distance between G1 and G2 is 177.36 mm. The total phase stepping number $\mathrm{M}$ of the G0 grating is set to eight, with a stepping interval of 3.00 $\mu m$. The linear stage (VP-25XL, Newport, USA) has a high precision of 0.01 $\mu m$. Two specimens, a chicken claw and a chicken drumstick, are scanned separately with two different dose levels: the reference 100% dose and a lower dose level of 15%.
To determine the focal spot drifts of our x-ray tube, two tungsten bids of diameter 1.0 $mm$ are scanned. They are taped on a rigid steel holder, which is fixed on the optical table. The bids are positioned close to the G0 grating, which is located 15.00 cm downstream of the source. In total, 40 independent exposures are collected from two individual experiments. The second is performed one week after the first, but with exactly the same tube settings as of the DPC data acquisitions. With the acquired bids absorption images and the imaging geometry, the tube focal spot drift map is estimated, see Fig. \[fig:spotdrift\].
To collect the Moiré-artifact-only sample images $\mathcal{M}_{\mathcal{A}}$, $\mathcal{M}_{\mathcal{S}}$, and $\mathcal{M}_{\mathcal{P}}$ in Eqs. (\[eq:M0\])-(\[eq:M2\]), phase stepping datasets without any object are acquired repeatedly for four times at the reference dose level. Using the above discussed data preparation method, 210 high SNR Moiré artifact-only image samples (dimension of $256\times256$) with varied spatial distributions are generated for each contrast mechanism.
Experimental results
--------------------
The Moiré artifact reduction results of the CNN network using experimentally acquired phase stepping datasets are shown in Figs. \[fig:Exp\_results\_1\]-\[fig:Exp\_results\_3\] separately for the absorption contrast, DF contrast, and DPC contrast. Undoubtedly, results demonstrate that the already trained CNNs are able to remove most of the Moiré artifacts for the three different contrast mechanisms, while maintaining high image spatial resolution and signal accuracy, see plots in Fig. \[fig:Exp\_results\_4\]. As discussed previously, Moiré artifacts on the acquired absorption images are way less pronounced than on the DF and DPC images, however, the CNN can still effectively remove them. Results also demonstrate that the CNN network is also feasible to reduce Moiré artifacts on images acquired with lower radiation dose levels. Therefore, the image quality can be significantly improved after processed by the proposed CNN method.
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Discussion
==========
In this work, we have demonstrated a CNN enabled image-domain Moiré artifacts reduction method for grating-based x-ray interferometry imaging system. This method considers the Moiré artifacts reduction task as a image post-processing procedure, rather than an optimization task of signal extractions from phase stepping datasets. Therefore, the conventional signal retrieval method is still utilized to extract the three contrast images. Afterwards, these extracted images are feed into the proposed CNN network to get rid of the Moiré artifacts and finally generate clean images. Experimental results show that the CNN method can effectively reduce image Moiré artifacts, while maintaining the signal accuracy and image resolution.
The network training data with Moiré artifacts are synthesized from natural images and the experimentally acquired Moiré artifact-only images. We believe this might be the easiest way to generate the needed training data, which always requires a large volume of high quality of training samples. To do so, we have made two assumptions: The first one assumes that Moiré artifacts are independent of the scanned object and thus can be approximated as additive signals to the artifact-free images. The second one assumes that the generated Moiré artifact-only samples from finite number of experiments are rich enough to represent all of the possible Moiré artifacts appear on real data acquired from the same interferometry imaging system with identical grating alignments. Although not rigorous enough, experimental results demonstrate that the first assumption is able to yield good results. During our experimental validations, we did not find violations to the second assumption. However, the performance of the CNN could be degraded if Moiré artifacts appear on real data are very different from the collected Moiré artifact-only samples. This might be considered as one limitation of our currently proposed method. Whenever this happens, the new Moiré artifact patterns have to be added into the Moiré artifact-only sample pool. Afterwards, the CNN networks have to be retrained with newly synthesized training data. Sometimes, new network architecture with deeper dimension[@goodfellow2014generative; @huang2017densely; @sun2018moire] may be beneficial.
Although majority of the Moiré artifacts on absorption image, DPC image, and DF image can be effectively mitigated, some minor artifacts may still remain after processed by the CNN. Two main reasons may cause this result. First of all, the used Moiré-artifact-only samples in this work only contain 200 images. This may not be able to represent all of the possible Moiré artifacts generated from experiments. If increasing the number of Moiré-artifact-only samples in future, the performance of the CNN could be further optimized. The second reason may be related to the used U-Net type CNN network. In this work, it is purely an empirical choice. Other types of CNN architectures[@goodfellow2014generative; @huang2017densely; @sun2018moire] with more deep network depth or more network parameters may generate better Moiré artifacts reduction results.
Conclusion
==========
In conclusion, we have demonstrated a CNN-based image-domain Moiré artifacts reduction method for grating-based x-ray interferometry imaging system. The new method treats the Moiré artifact reduction task as an image-domain post-processing procedure, and the CNN is utilized as an efficient tool to realize the aim. In addition, a fast signal processing method to generate large number of high quality Moiré artifact-only image samples has also been demonstrated to ease the training data preparations. Experimental results show that the CNN is able to effectively remove the Moiré artifacts while maintaining the signal accuracy and image resolution.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank Dr. Peiping Zhu at the Institute of High Energy Physics, Chinese Academy of Sciences, for his patient discussions to the theoretical analyses results. The authors also would like to acknowledge the ImageNet organization for providing the natural images to enable the network training procedures. This project is supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 11804356).
Analytically, the three unknown signals can all be estimated from Eq. (\[eq:signal\_mod\]) and Eq. (\[eq:I1\_signal\]) as follows: $$\begin{aligned}
\label{eq:ch01_N0}
\hat{I_0} =& \frac{1}{\mathrm{M}} \sum_{k=1}^{\mathrm{M}} {I}^{(k)}, \\
\label{eq:ch01_N1}
\hat{I_1} =& \frac{2}{\mathrm{M}} \sqrt{\left[\sum_{k=1}^{\mathrm{M}} {I}^{(k)}\sin(\frac{2\pi k}{\mathrm{M}})\right]^2+ \left[\sum_{k=1}^{\mathrm{M}}{I}^{(k)}\cos(\frac{2\pi k}{M})\right]^2}, \\
\label{eq:ch01_phi}
\hat{\phi} =& \tan^{-1}\left[- \frac{\sum_{k=1}^{\mathrm{M}} {I}^{(k)}\sin(\frac{2\pi k}{\mathrm{M}})}{\sum_{k=1}^{\mathrm{M}} {I}^{(k)}\cos(\frac{2\pi k}{\mathrm{M}})} \right].\end{aligned}$$ After adding the random fluctuation term to each phase stepping position $k$, the modulation term in Eq. (\[eq:signal\_1\]) can be approximated by ignoring the high order terms of the Taylor expansion with respect to the small value $\eta^{(k)}$, namely, $$\begin{aligned}
&\cos\left[2\pi \frac{k+\eta^{(k)}}{\mathrm{M}}+\phi\right] \nonumber \\
=&\cos\left[\frac{2\pi k}{\mathrm{M}}+\phi\right]\cos\left[\frac{2\pi \eta^{(k)}}{\mathrm{M}}\right] - \sin\left[\frac{2\pi k}{\mathrm{M}}+\phi\right]\sin\left[\frac{2\pi \eta^{(k)}}{\mathrm{M}}\right], \nonumber \\
\approx&\cos\left[\frac{2\pi k}{\mathrm{M}}+\phi\right] - \frac{2\pi \eta^{(k)}}{\mathrm{M}}\sin\left[\frac{2\pi k}{\mathrm{M}}+\phi\right].
\label{eq:app_1}\end{aligned}$$ As a result, $$\begin{aligned}
&\sum_{k=1}^{\mathrm{M}} {I}^{(k)}\sin\left[\frac{2\pi k}{\mathrm{M}}\right]\approx -\frac{\mathrm{M}I_1}{2}\sin\left[\phi\right]\nonumber \\
& - \frac{\pi I_1}{\mathrm{M}}\sum_{k=1}^{\mathrm{M}}\eta^{(k)}\left[\cos\left[\phi\right]-\cos\left[\frac{4\pi k}{\mathrm{M}}+\phi\right]\right].
\label{eq:app_2}\end{aligned}$$ Similarly, we have $$\begin{aligned}
&\sum_{k=1}^{\mathrm{M}} {I}^{(k)}\cos\left[\frac{2\pi k}{\mathrm{M}}\right]\approx \frac{\mathrm{M}I_1}{2}\cos\left[\phi\right] \nonumber \\
& - \frac{\pi I_1}{\mathrm{M}}\sum_{k=1}^{\mathrm{M}}\eta^{(k)}\left[\sin\left[\phi\right]+\sin\left[\frac{4\pi k}{\mathrm{M}}+\phi\right]\right].
\label{eq:app_3}\end{aligned}$$
[^1]: Jianwei Chen and Jiongtao Zhu have made equal contributions to this work and both are considered as the first authors.
[^2]: Jianwei Chen and Jiongtao Zhu have made equal contributions to this work and both are considered as the first authors.
[^3]: Scientific correspondence should be addressed to Dong Liang ([email protected]) and Yongshuai Ge ([email protected]).
[^4]: Scientific correspondence should be addressed to Dong Liang ([email protected]) and Yongshuai Ge ([email protected]).
|
---
abstract: 'The Radio Interferometer Measurement Equation (RIME) is a matrix-based mathematical model that describes the response of a radio interferometer. The Jones calculus it employs is not suitable for describing the analogue components of a telescope. This is because it does not consider the effect of impedance mismatches between components. This paper aims to highlight the limitations of Jones calculus, and suggests some alternative methods that are more applicable. We reformulate the RIME with a different basis that includes magnetic and mixed coherency statistics. We present a microwave network inspired 2N-port version of the RIME, and a tensor formalism based upon the electromagnetic tensor from special relativity. We elucidate the limitations of the Jones-matrix-based RIME for describing analogue components. We show how measured scattering parameters of analogue components can be used in a 2N-port version of the RIME. In addition, we show how motion at relativistic speed affects the observed flux. We present reformulations of the RIME that correctly account for magnetic field coherency. These reformulations extend the standard formulation, highlight its limitations, and may have applications in space-based interferometry and precise absolute calibration experiments.'
author:
- |
D. C. Price$^{1}$[^1] and O. M. Smirnov$^{2,3}$\
$^1$Harvard-Smithsonian Center for Astrophysics, MS42, 60 Garden Street, Cambridge MA, 01243 United States\
$^2$Department of Physics and Electronics, Rhodes University, PO Box 94, Grahamstown, 6140, South Africa\
$^3$SKA South Africa, 3rd Floor, The Park, Park Road, Pinelands, 7405, South Africa\
bibliography:
- 'references.bib'
date: 'Accepted 2015 January 12. Received 2014 November 04'
title: Generalized Formalisms of the Radio Interferometer Measurement Equation
---
\[firstpage\]
Methods: data analysis — Techniques: interferometric — Techniques: polarimetric
Introduction
============
Coherency in electromagnetic fields is a fundamental topic within optics. Its importance in fields such as radio astronomy can not be overstated: interferometry and synthesis imaging techniques rely heavily upon coherency theory [@Taylor1999; @BookMandelWolf; @ThompsonMoranSwenson2004]. Of particular importance to radio astronomy is the Van-Cittert-Zernicke theorem (vC-Z, [@VanCittertZernicke1938]) and the radio interferometer Measurement Equation (RIME, [@Hamaker:1996p5735]). The vC-Z relates the brightness of a source to its mutual coherency as measured by an interferometer, and the RIME provides a polarimetric framework to calibrate out corruptions caused along the signal’s path.
While the vC-Z theorem dates back to 1938, more recent work such as that of @Carozzi:2009hf extends its applicability to polarized measurements over wide fields of view. The RIME has a much shorter history: it was not formulated until 1996 [@Hamaker:1996p5735]. Before the RIME, calibration was conducted in an ad-hoc manner, with each polarization essentially treated separately. The framework was expounded in a series of follow-up papers [@Sault:1996p5731; @Hamaker:1996p5733; @Hamaker:2000p7625; @Hamaker:2006p7626]; recent work by Smirnov extends the formalism to the full sky case, and reformulates the RIME using tensor algebra [@Smirnov:2011a; @Smirnov:2011d].
This article introduces two reformulations of the RIME, that extend its applicability and demonstrates limitations with the Jones-matrix-based formalism. Both of these reformulations consider full electromagnetic coherency statistics (i.e. electric, magnetic and mixed coherencies). The first is inspired by transmission matrix methods from microwave engineering. We show that this reformulation better accounts for the changes in impedance between analogue components within a radio telescope. The second reformulation is a relativistic-aware formulation of the RIME, starting with the electromagnetic field tensor. This formalism allows for relativistic motion to be treated as instrumental effect and incorporated into the RIME.
This article is organized as follows. In Section 2, we review existing formalisms of the RIME and methods from microwave engineering. Section 3 defines coherency matrices, which are used in Section 4 to formulate general coherency relationships for radio astronomy. In Section 5, we introduce a tensor formulation of the RIME based upon the electromagnetic tensor from special relativity. Discussion and example applications are given in Section 6; concluding remarks are given in Section 7.
Jones and Mueller RIME formulations
===================================
Before continuing on to derive a more general relationship between the two-point coherency matrix and the voltage-current coherency, we would like to give a brief overview and derivation of the radio interferometer Measurement Equation of @Hamaker:1996p5735. Our motivation behind this is to highlight that Hamaker et. al.’s RIME is a special case of a more general (and thus less limited) coherency relationship.
In their seminal Measurement Equation (ME) paper, @Hamaker:1996p5735 showed that Mueller and Jones calculuses provide a good framework for modelling radio interferometers. In optics, Jones and Mueller matrices are used to model the transmission of light through optical elements [@Jones1941; @Mueller1948]. Mueller matrices are $\mbox{4}\times\mbox{4}$ matrices that act upon the Stokes vector $$\mbox{\textbf{\emph{s}}}=\begin{pmatrix}I & Q & U & V\end{pmatrix}^{T},$$ whereas Jones matrices are only $\mbox{2}\times\mbox{2}$ in dimension and act upon the ‘Jones vector’: the electric field vector in a coordinate system such that z-axis is aligned with the Poynting vector $$\mbox{\textbf{\emph{e}}}(\mbox{\emph{\textbf{r}}},t)=\begin{pmatrix}e_{x}(\mbox{\emph{\textbf{r}}},t) & e_{y}(\mbox{\emph{\textbf{r}}},t)\end{pmatrix}^{T}.$$ Jones calculus dictates that along a signal’s path, any (linear) transformation can be represented with a Jones matrix, ***J***: $$\mbox{\textbf{\emph{e}}}_{\rm{out}}(\mbox{\textbf{\emph{r}}},t)=\mbox{\textbf{\emph{J}}}\mbox{\textbf{\emph{e}}}_{\rm{in}}(\mbox{\textbf{\emph{r}}},t)\label{eq:jones-transmission}$$ A useful property of Jones calculus is that multiple effects along a signal’s path of propagation correspond to repeated multiplications: $$\mbox{\textbf{\emph{e}}}_{\mathrm{out}}(\mbox{\textbf{\emph{r}}},t)=\emph{\textbf{J}}_{n}\cdots\emph{\textbf{J}}_{2}\emph{\textbf{J}}_{1}\mbox{\textbf{\emph{e}}}_{\rm{in}}(\mbox{\textbf{\emph{r}}},t),$$ which can be collapsed into a single matrix when required.
The RIME uses Jones matrices to model the various corruptions and effects during a signal’s journey from a source right though to the correlator. A block diagram for a (simplified) two-element interferometer is shown in Figure \[fig:RIME-cartoon\]. From left to right, the figure shows the journey of a signal from a source right through to the correlator. The radiation from the source is picked up by two antennas, which we have denoted with subscript $p$ and $q$. The radiation follows a unique path to both of these antennas; each antenna also has associated with it a unique chain of analogue components that amplify and filter the signal to prepare it for correlation. Each of these subscripted boxes may be represented by a Jones matrix; alternatively an overall Jones matrix can be formed for the $p$ and $q$ branches (the dashed areas).
![Block diagram showing a simple model of an interferometer that can be modelled with the RIME. Radiation from a source propagates through free space to two telescopes, *p* and *q*. After passing through the telescope’s analogue chain, the two signals are interfered in a cross-correlator.\[fig:RIME-cartoon\]](figures-rime-diagram){width="0.95\columnwidth"}
Hamaker’s RIME derivation
-------------------------
The derivation of the RIME is remarkably simple and elegant. For a single point source of radiation, the voltages induced at the terminals of a pair of antennas, $p$ and $q$ are $$\begin{aligned}
\mbox{\textbf{\emph{v}}}_{p}(t) & =\mbox{\textbf{\emph{J}}}_{p}\mbox{\textbf{\emph{e}}}_{0}(t)\\
\mbox{\textbf{\emph{v}}}_{q}(t) & =\mbox{\textbf{\emph{J}}}_{q}\mbox{\textbf{\emph{e}}}_{0}(t)\end{aligned}$$ In its simplest form, the RIME is formed by taking the outer product of these two relationships. Note that in their original paper, the authors use the Kronecker incarnation of the outer product, which we will denote with $\star$. We reserve the symbol $\otimes$ for the matrix outer product of two matrices $\emph{\textbf{A}}\otimes\emph{\textbf{B}}=\mbox{\textbf{\emph{A}}}\emph{\textbf{B}}^{H},$ where *H* denotes the Hermitian conjugate transpose[^2]. Using the Kronecker outer product, the RIME is given by $$\left\langle \mbox{\textbf{\emph{v}}}_{p}\star\mbox{\textbf{\emph{v}}}_{q}\right\rangle =\left(\mbox{\textbf{\emph{J}}}_{p}\star\mbox{\textbf{\emph{J}}}_{q}\right)\left\langle \mbox{\textbf{\emph{e}}}_{0}\star\mbox{\textbf{\emph{e}}}_{0}\right\rangle =\left(\mbox{\textbf{\emph{J}}}_{p}\star\mbox{\textbf{\emph{J}}}_{q}\right)\mbox{\textbf{\emph{e}}}_{00}\label{eq:mueller-rime}$$ where $\mbox{\textbf{\emph{J}}}_{p}$ and $\mbox{\textbf{\emph{J}}}_{q}$ are the Jones matrices representing all transformations along the two signal paths, and $\left(\mbox{\textbf{\emph{J}}}_{p}\star\mbox{\textbf{\emph{J}}}_{q}\right)$ is a $\mbox{4}\times\mbox{4}$ matrix. Here, $\mbox{\textbf{\emph{e}}}_{00}$ is the sky brightness of a single point source of radiation. For a multi-element interferometer, every antenna has its own unique Jones matrix, and a RIME may be written for every pair of antennas.
Due to their choice of outer product, Hamaker et. al. arrive at a coherency vector $$\mbox{\textbf{\emph{e}}}_{pq}(\mbox{\textbf{\emph{r}}}_{p},\mbox{\textbf{\emph{r}}}_{q},\tau)=\left\langle \mbox{\textbf{\emph{e}}}_{p}(\mbox{\textbf{\emph{r}}}_{p},t)\star\mbox{\textbf{\emph{e}}}_{q}(\mbox{\textbf{\emph{r}}}_{q},t+\tau)\right\rangle =\begin{pmatrix}\left\langle e_{px}e_{qx}^{*}\right\rangle \\
\left\langle e_{px}e_{qy}^{*}\right\rangle \\
\left\langle e_{py}e_{qx}^{*}\right\rangle \\
\left\langle e_{py}e_{qy}^{*}\right\rangle
\end{pmatrix},$$ as opposed to the coherency matrix of @Wolf1954; this is introduced in §\[sub:Electromagnetic-coherency\] below. The column vector of a point source at $\mbox{\textbf{\emph{r}}}_{0}$ is then $\mbox{\textbf{\emph{e}}}_{00}$; that is, $p=q$ and $\tau=0$. The vector $\mbox{\textbf{\emph{e}}}_{00}$ is related to the Stokes vector by the transform[^3]. $$\begin{pmatrix}I\\
Q\\
U\\
V
\end{pmatrix}=\begin{pmatrix}1 & 0 & 0 & 1\\
1 & 0 & 0 & -1\\
0 & 1 & 1 & 0\\
0 & -j & j & 0
\end{pmatrix}\begin{pmatrix}\left\langle e_{0x}e_{0x}^{*}\right\rangle \\
\left\langle e_{0x}e_{0y}^{*}\right\rangle \\
\left\langle e_{0y}e_{0x}^{*}\right\rangle \\
\left\langle e_{0y}e_{0y}^{*}\right\rangle
\end{pmatrix}.$$ The quantity $\left(\mbox{\textbf{\emph{J}}}_{p}\star\mbox{\textbf{\emph{J}}}_{q}\right)$ in Eq. \[eq:mueller-rime\] can therefore be viewed as a Mueller matrix. That is, Eq. \[eq:mueller-rime\] can be considered a Mueller-calculus-based ME for a radio interferometer. To summarize, Hamaker et. al. showed that:
- Jones matrices can be used to model the propagation of a signal from a radiation source through to the voltage at the terminal of an antenna.
- A Mueller matrix can be formed from the Jones terms of a pair of antennas, which then relates the measured voltage coherency of that pair to a source’s brightness.
Showing that these calculuses were applicable and indeed useful for modelling and calibrating radio interferometers was an important step forward in radio polarimetry.
The 2$\times$2 RIME
-------------------
In a later paper, @Hamaker:2000p7625 presents a modified formulation of the RIME, where instead of forming the coherency vector from the Kronecker outer product ($\star$), the coherency matrix is formed from the matrix outer product ($\otimes$): $$\mbox{\textbf{\emph{E}}}_{pq}=\left\langle \emph{\textbf{e}}_{p}\otimes\emph{\textbf{e}}_{q}\right\rangle =\begin{pmatrix}\left\langle e_{px}e_{qx}^{*}\right\rangle & \left\langle e_{px}e_{qy}^{*}\right\rangle \\
\left\langle e_{px}e_{qx}^{*}\right\rangle & \left\langle e_{py}e_{qy}^{*}\right\rangle
\end{pmatrix}$$ The resulting coherency matrix is then shown to be related to the Stokes parameters by $\mbox{\textbf{\emph{E}}}_{00}=\mbox{ \textsf{B} }$, where $$\begin{aligned}
\mathsf{B} & =\begin{pmatrix}I+Q & U+jV\\
U-jV & I-Q
\end{pmatrix}.\end{aligned}$$ The equivalent to the RIME of Eq. \[eq:mueller-rime\] is $$\left\langle \mbox{\textbf{\emph{v}}}_{p}\otimes\mbox{\textbf{\emph{v}}}_{q}\right\rangle =2\left\langle \mbox{(\ensuremath{\mbox{\textbf{\emph{J}}}_{p}\textbf{\emph{e}}}}_{0})\otimes(\mbox{\textbf{\emph{J}}}_{q}\mbox{\textbf{\emph{e}}}_{0})\right\rangle =2\mbox{\textbf{\emph{J}}}_{p}\left\langle \mbox{\textbf{\emph{e}}}_{0}\otimes\mbox{\textbf{\emph{e}}}_{0}\right\rangle \mbox{\textbf{\emph{J}}}_{q}^{H}\label{eq:jones-rime}$$ or more simply, $$\mathsf{V}_{pq}=\mbox{\textbf{\emph{J}}}_{p}\mbox{\textbf{\emph{B}}}\mbox{\textbf{\emph{J}}}_{q}^{H}.\label{eq:jones-rime-2}$$ This approach avoids the need to use $\mbox{4}\times\mbox{4}$ Mueller matrices, so is both simpler and computationally advantageous. This form is also cleaner in appearance, as fewer indices are required.
@Smirnov:2011a takes the $\mbox{2}\times\mbox{2}$ version of the RIME as a starting point and extends the RIME to a full sky case. By treating the sky as a brightness distribution $\mathsf{B}$($\sigma$), where $\sigma$ is a direction vector, each antenna has a Jones term $\mbox{\textbf{\emph{J}}}_{p}(\sigma)$ describing the signal path for a given direction. The visibility matrix $\mathsf{V}_{pq}$ is then found by integrating over the entire sky: $$\mathsf{V}_{pq}=\int_{4\pi}\mbox{\textbf{\emph{J}}}_{p}(\sigma)\mathsf{B}\mbox{(\ensuremath{\sigma})}\mbox{\textbf{\emph{J}}}_{q}^{H}(\sigma)d\Omega.$$ This is a more general form of *Zernicke’s propagation law*. Smirnov goes on to derive the Van-Cittert Zernicke theorem from this result; we return to vC-Z later in this article.
A generalized tensor RIME
-------------------------
In @Smirnov:2011d, a generalized tensor formalism of the RIME is presented. The coherency of two voltages is once again defined via the outer product $e^{i}\bar{e}_{j}$, giving a (1,1)-type tensor expression: $$\mbox{\emph{V}}_{qj}^{pi}=\mbox{\emph{J}}_{\alpha}^{pi}\mbox{\emph{B}}_{\beta}^{\alpha}\bar{\mbox{\emph{J}}}_{qj}^{\beta}.$$ This formalism is better capable of describing mutual coupling between antennas, beamforming, and wide field polarimetry. $ $ In this paper, we focus on a matrix based formalism which considers the propagation of the magnetic field in addition to the electric field. We then show that this formulation is equivalent to the tensor formalism presented in @Smirnov:2011d, but is instead in the vector space $\mathbb{C}^{6}$.
Microwave engineering transmission matrix methods
-------------------------------------------------
All formulations of the RIME to date — including the tensor formulation — do not consider the propagation of the magnetic field. In free space, magnetic field coherency can be easily derived from the electric field coherency. However, at the boundary between two media, the magnetic field must be considered. Here, we introduce some results from microwave engineering which contrast with the Jones formalism.
In circuit theory, the well-known impedance relation, $V=ZI$ relates current and voltage over a terminal pair (or ‘port’). However, this relation is specific to a 1-port network; for microwave networks with more than one port, the matrix form $[V]=[Z][I]$ must be used:
$$\begin{pmatrix}v_{1}\\
v_{2}\\
\vdots\\
v_{N}
\end{pmatrix}=\begin{pmatrix}Z_{11} & Z_{12} & \cdots & Z_{1N}\\
Z_{21} & \ddots & & \vdots\\
\vdots & & & \vdots\\
Z_{N1} & \cdots & \cdots & Z_{NN}
\end{pmatrix}\begin{pmatrix}i_{1}\\
i_{2}\\
\vdots\\
i_{N}
\end{pmatrix},$$
where $Z_{ab}$ is the port-to-port impedance from port *a* to port *b*, $v_{n}$ is the voltage on port $n$, and $i_{n}$ is the current. A common example of a 2-port network is a coaxial cable, and a common 3-port network is the Wilkinson power divider.
The analogue components of most radio telescopes can be considered 2-port networks. The 2-port transmission, or $ABCD$ matrix, relates the voltages and currents of a 2-port network:
$$\begin{pmatrix}v_{1}\\
i_{1}
\end{pmatrix}=\begin{pmatrix}A & B\\
C & D
\end{pmatrix}\begin{pmatrix}v_{2}\\
i_{2}
\end{pmatrix},$$
this is shown in Figure \[fig:transmission-cascade\]. If two 2-port networks are connected in cascade (see Figure \[fig:transmission-cascade\]), then the output is equal to the product of the transmission matrices representing the individual components: $$\begin{pmatrix}v_{1}\\
i_{1}
\end{pmatrix}=\begin{pmatrix}A_{1} & B_{1}\\
C_{1} & D_{1}
\end{pmatrix}\begin{pmatrix}A_{2} & B_{2}\\
C_{2} & D_{2}
\end{pmatrix}\begin{pmatrix}v_{2}\\
i_{2}
\end{pmatrix},$$ as is shown in texts such as @Pozar2005. The elements in the 2-port transmission matrix are related to port-to-port impedances by[^4] $$\begin{aligned}
A & =Z_{11}/Z_{21}\\
B & =\frac{Z_{11}Z_{22}-Z_{12}Z_{21}}{Z_{21}}\\
C & =1/Z_{21}\\
D & =Z_{22}/Z_{21}.\end{aligned}$$ Like the Jones matrix, the $ABCD$ matrix allows a signal’s path to be modelled through multiplication of matrices representing discrete components. While the Jones matrix acts upon a pair of orthogonal electric field components, the $ABCD$ matrix acts upon a voltage-current pair at a single port. As Jones matrices do not consider changes in impedance (free space impedance is implicitly assumed), it is not suitable for describing analogue components. Conversely, the $2\times2$ $ABCD$ matrix cannot model cross-polarization response of a telescope. In the section that follows, we derive a more general coherency relationship which weds the advantages of both approaches.
![Top: The ABCD matrix for a 2-port network. In this diagram, voltage is denoted with $V$, and current with $I$. Bottom: Connecting two components in cascade. Diagram adapted from @Pozar2005\[fig:transmission-cascade\]](figures-transmission-matrix){width="0.8\columnwidth"}
Coherency in radio astronomy
============================
We now turn our attention to formulating a more general RIME that is valid in a larger range of cases. In this section, we introduce the coherency matrices of @Wolf1954, along with voltage-current coherency matrices. The following section then formulates relationships between source brightness and measured coherency based upon these matrices.
Electromagnetic coherency\[sub:Electromagnetic-coherency\]
----------------------------------------------------------
To begin, we introduce the coherency matrices of @Wolf1954, that fully describe the coherency statistics of an electromagnetic field. We may start by introducing $\mbox{\textbf{\emph{e}}}(\mbox{\textbf{\emph{r}}},t)$ and $\mbox{\textbf{\emph{h}}}(\mbox{\textbf{\emph{r}}},t)$ as the complex analytic representations of the electric and magnetic field vectors at a spacetime point $(\mbox{\textbf{\emph{r}}},t)$: $$\begin{aligned}
\mbox{\textbf{\emph{e}}}(\mbox{\textbf{\emph{r}}},t) & =\begin{pmatrix}e_{x}(\mbox{\textbf{\emph{r}}},t) & e_{y}(\mbox{\textbf{\emph{r}}},t) & e_{z}(\mbox{\textbf{\emph{r}}},t)\end{pmatrix}^{T}\label{eq:elec-vec}\\
\mbox{\textbf{\emph{h}}}(\mbox{\textbf{\emph{r}}},t) & =\begin{pmatrix}h_{x}(\mbox{\textbf{\emph{r}}},t) & h_{y}(\mbox{\textbf{\emph{r}}},t) & h_{z}(\mbox{\textbf{\emph{r}}},t)\end{pmatrix}^{T}\end{aligned}$$ The coherency matrices are then defined by the formulae $$\begin{aligned}
\mbox{\textbf{\emph{E}}}_{pq}(\mbox{\textbf{\emph{r}}}_{p},\mbox{\textbf{\emph{r}}}_{q},\tau) & =\begin{pmatrix}\left\langle e_{k}(\mbox{\textbf{\emph{r}}}_{p},t)e_{l}^{*}(\mbox{\textbf{\emph{r}}}_{q},t+\tau)\right\rangle \end{pmatrix}\\
\mbox{\textbf{\emph{H}}}_{pq}(\mbox{\textbf{\emph{r}}}_{p},\mbox{\textbf{\emph{r}}}_{q},\tau) & =\begin{pmatrix}\left\langle h_{k}(\mbox{\textbf{\emph{r}}}_{p},t)h_{l}^{*}(\mbox{\textbf{\emph{r}}}_{q},t+\tau)\right\rangle \end{pmatrix}\\
\mbox{\textbf{\emph{M}}}_{pq}(\mbox{\textbf{\emph{r}}}_{p},\mbox{\textbf{\emph{r}}}_{q},\tau) & =\begin{pmatrix}\left\langle e_{k}(\mbox{\textbf{\emph{r}}}_{p},t)h_{l}^{*}(\mbox{\textbf{\emph{r}}}_{q},t+\tau)\right\rangle \end{pmatrix}\\
\mbox{\textbf{\emph{N}}}_{pq}(\mbox{\textbf{\emph{r}}}_{p},\mbox{\textbf{\emph{r}}}_{q},\tau) & =\begin{pmatrix}\left\langle h_{k}(\mbox{\textbf{\emph{r}}}_{p},t)e_{l}^{*}(\mbox{\textbf{\emph{r}}}_{q},t+\tau)\right\rangle \end{pmatrix}.\end{aligned}$$ Here, $k$ and $l$ are indices representing the $x,y,z$ subscripts from Cartesian coordinates. $\mbox{\textbf{\emph{E}}}_{pq}$ and $\mbox{\textbf{\emph{H}}}_{pq}$ are called the electric and the magnetic coherency matrices, and $\mbox{\textbf{\emph{M}}}_{pq}$ and $\mbox{\textbf{\emph{N}}}_{pq}$ are called the mixed coherency matrices. The subscripts $p$ and $q$ correspond to the spacetime points $(\mbox{\textbf{\emph{r}}}_{p},t)$ and $(\mbox{\textbf{\emph{r}}}_{q},t+\tau)$, respectively. We may arrange these into a single $\mbox{6}\times\mbox{6}$ matrix **$\mathbb{B}_{pq}$** that is equivalent to the time averaged outer product of the electromagnetic field column vectors at spacetime points $(\mbox{\textbf{\emph{r}}}_{p},t)$ and $(\mbox{\textbf{\emph{r}}}_{q},t+\tau)$: $$\mathbb{B}_{pq}=\left\langle \begin{pmatrix}\mbox{\textbf{\emph{e}}}_{p}\\
\mbox{\textbf{\emph{h}}}_{p}
\end{pmatrix}\otimes\begin{pmatrix}\mbox{\textbf{\emph{e}}}_{q}\\
\mbox{\textbf{\emph{h}}}_{q}
\end{pmatrix}\right\rangle =\begin{pmatrix}\mbox{\textbf{\emph{E}}}_{pq} & \mbox{\textbf{\emph{M}}}_{pq}\\
\mbox{\textbf{\emph{N}}}_{pq} & \mbox{\textbf{\emph{H}}}_{pq}
\end{pmatrix}\label{eq:coherency-matrix}$$ This matrix fully describes the coherency properties of an electromagnetic field at two points in spacetime. We will refer to this matrix as the *two point coherency matrix*. It is worth noting that:
- When $\mbox{\textbf{\emph{r}}}_{p}=\mbox{\textbf{\emph{r}}}_{q}$ and $\tau=0$ we retrieve what @Bergman:2008p7859 refer to as the ‘EM sixtor matrix’. @Bergman:2008p7859 show this sixtor matrix is related to what they refer to as ‘canonical electromagnetic observables’: a unique set of Stokes-like parameters that are irreducible under Lorentz transformations. These are used in the the analysis of electromagnetic field data from spacecraft.
- For monochromatic plane waves, when $\mbox{\textbf{\emph{r}}}_{p}=\mbox{\textbf{\emph{r}}}_{q}$ and $\tau=0$, and we choose a coordinate system with $z$ in the direction of propagation (i.e. along the Poynting vector), $\textbf{\emph{E}}_{pq}$ becomes what @Smirnov:2011a refers to as the *brightness matrix*, $\mathsf{B}$.
From here forward, we drop the subscript $\mathbb{B}=\mathbb{B}_{00}$ and shall refer to this as the *brightness coherency* to highlight its relationship with $\mathsf{B}$.
Voltage and current coherency
-----------------------------
A radio telescope converts a free space electromagnetic field into a time varying voltage, which we then measure after signal conditioning (e.g. amplification and filtering). As such, radio interferometers measure coherency statistics between time varying voltages.
One may model the analogue components of a telescope as a 6-port network, with three inputs ports and three output ports. We propose this so that there is an input-output pair of ports for each of the orthogonal components of the electromagnetic field. We can then define a set of voltages ***v****(t)* and currents $\mbox{\textbf{\emph{i}}}(t)$
$$\begin{aligned}
\mbox{\textbf{\emph{v}}}(t) & =\begin{pmatrix}v_{x}(t) & v_{y}(t) & v_{z}(t)\end{pmatrix}^{T}\\
\mbox{\textbf{\emph{i}}}(t) & =\begin{pmatrix}i_{x}(t) & i_{y}(t) & i_{z}(t)\end{pmatrix}^{T}.\end{aligned}$$
In practice, most telescopes are single or dual polarization, so only the $x$ and $y$ components are sampled. Nonetheless, it is possible to sample all three components with three orthogonal antenna elements [@Bergman:2005p7825]. The voltage response of an antenna is linearly related to the electromagnetic field strength [@Hamaker:1996p5735], and the current is linearly related to voltage by Ohm’s law, so we may write a general linear relationship $$\begin{pmatrix}\mbox{\textbf{\emph{v}}}(t)\\
\mbox{\textbf{\emph{i}}}(t)
\end{pmatrix}=\begin{pmatrix}\mbox{\textbf{\emph{A}}} & \mbox{\textbf{\emph{B}}}\\
\mbox{\textbf{\emph{C}}} & \mathbf{\mbox{\textbf{\emph{D}}}}
\end{pmatrix}\begin{pmatrix}\mbox{\textbf{\emph{e}}}(\mbox{\textbf{\emph{r}}},t)\\
\mbox{\textbf{\emph{h}}}(\mbox{\textbf{\emph{r}}},t)
\end{pmatrix},$$ where ***A***, ***B***, **** ***C*** **** and **** ***D*** **** are block matrices forming an overall transmission matrix $\mathbb{T}'$. We can now define a matrix of voltage-current coherency statistics that consists of the block matrices
$$\begin{aligned}
\mbox{\textbf{\emph{V}}}_{pq}(\tau) & =\begin{pmatrix}\left\langle v_{k}(t)v_{l}^{*}(t+\tau)\right\rangle \end{pmatrix}\\
\mbox{\textbf{\emph{W}}}_{pq}(\tau) & =\begin{pmatrix}\left\langle i_{k}(t)i_{l}^{*}(t+\tau)\right\rangle \end{pmatrix}\\
\mbox{\textbf{\emph{K}}}_{pq}(\tau) & =\begin{pmatrix}\left\langle v_{k}(t)i_{l}^{*}(t+\tau)\right\rangle \end{pmatrix}\\
\mbox{\textbf{\emph{L}}}_{pq}(\tau) & =\begin{pmatrix}\left\langle i_{k}(t)v_{l}^{*}(t+\tau)\right\rangle \end{pmatrix},\end{aligned}$$
these are analogous to (and related to) **** the electromagnetic coherency matrices above[^5]. In a similar manner to the two-point coherency matrix, we define $\mathbb{V}_{pq}$ $$\mathbb{V}_{pq}=\left\langle \begin{pmatrix}\mbox{\textbf{\emph{v}}}_{p}\\
\mbox{\textbf{\emph{i}}}_{p}
\end{pmatrix}\otimes\begin{pmatrix}\mbox{\textbf{\emph{v}}}_{q}\\
\mbox{\textbf{\emph{i}}}_{q}
\end{pmatrix}\right\rangle =\begin{pmatrix}\mbox{\textbf{\emph{V}}}_{pq} & \mathbf{\mbox{\textbf{\emph{K}}}}_{pq}\\
\mbox{\textbf{\emph{L}}}_{pq} & \mbox{\textbf{\emph{W}}}_{pq}
\end{pmatrix},$$ which we will refer to as the *voltage-current coherency matrix*. This is analogous to the ‘visibility matrix’, $\mathsf{V}_{pq}$, of @Smirnov:2011a.
Two point coherency relationships\[sec:2pt-coherency\]
======================================================
Now we have introduced the two-point coherency matrix $\mathbb{B}_{pq}$ and the voltage-current coherency matrix $\mathbb{V}_{pq}$, we can formulate relationships between the two. In this section, we first formulate a general coherency relationship describing propagation from a source of electromagnetic radiation to two spacetime coordinates. We then show that this relationship underlies both the RIME and the vC-Z relationship.
A general two point coherency relationship
------------------------------------------
Suppose we have two sensors, located at points $\mbox{\textbf{\emph{r}}}_{p}$ and $\mbox{\textbf{\emph{r}}}_{q}$, which fully measure all components of the electromagnetic field. Assuming linearity, the propagation of an electromagnetic field $\mbox{\textbf{\emph{f}}}_{0}=\begin{pmatrix}\mbox{\textbf{\emph{e}}}(\mbox{\textbf{\emph{r}}}_{0},t) & \mbox{\textbf{\emph{h}}}(\mbox{\textbf{\emph{r}}}_{0},t)\end{pmatrix}^{T}$ from a point $\mbox{\textbf{\emph{r}}}_{0}$ to $\mbox{\textbf{\emph{r}}}_{p}$ and $\mbox{\textbf{\emph{r}}}_{q}$ can be encoded into a 6$\times$6 matrices, $\mathbb{T}_{p}$ and $\mathbb{T}_{q}$: $$\begin{aligned}
\mbox{\textbf{\emph{f}}}_{p} & =\mathbb{T}_{p}\mbox{\textbf{\emph{f}}}_{0}\\
\mbox{\textbf{\emph{f}}}_{q} & =\mathbb{T}_{q}\mbox{\textbf{\emph{f}}}_{0}\end{aligned}$$ The coherency between the two signals $\mbox{\textbf{\emph{f}}}_{p}$ and $\mbox{\textbf{\emph{f}}}_{q}$ is then given by the matrix $\mathbb{B}_{pq}$:
$$\begin{aligned}
\mathbb{B}_{pq} & =\left\langle \mbox{\textbf{\emph{f}}}_{p}\otimes\mbox{\textbf{\emph{f}}}_{q}\right\rangle \label{eq:tp_tq_outer}\\
& =\left\langle (\mathbb{T}_{p}\mbox{\textbf{\emph{f}}}_{0})\otimes(\mathbf{\mathbb{T}}_{q}\mbox{\textbf{\emph{f}}}_{0})\right\rangle \\
& =\left\langle \mathbf{\mathbb{T}}_{p}(\mbox{\textbf{\emph{f}}}_{0}\otimes\mbox{\textbf{\emph{f}}}_{0})\mathbb{T}_{q}^{H}\right\rangle \\
& =\mathbf{\mathbb{T}}_{p}\mathbb{B}\mathbf{\mathbb{T}}_{q}^{H}\end{aligned}$$
we can write this in terms of block matrices $$\begin{pmatrix}\mbox{\textbf{\emph{E}}}_{pq} & \mbox{\textbf{\emph{M}}}_{pq}\\
\mbox{\textbf{\emph{N}}}_{pq} & \mbox{\textbf{\emph{H}}}_{pq}
\end{pmatrix}=\begin{pmatrix}\mbox{\textbf{\emph{A}}}_{p} & \mbox{\textbf{\emph{B}}}_{p}\\
\mbox{\textbf{\emph{C}}}_{p} & \mbox{\textbf{\emph{D}}}_{p}
\end{pmatrix}\begin{pmatrix}\mbox{\textbf{\emph{E}}}_{00} & \mbox{\textbf{\emph{M}}}_{00}\\
\mbox{\textbf{\emph{N}}}_{00} & \mbox{\textbf{\emph{H}}}_{00}
\end{pmatrix}\begin{pmatrix}\mbox{\textbf{\emph{A}}}_{q} & \mbox{\textbf{\emph{B}}}_{q}\\
\mbox{\textbf{\emph{C}}}_{q} & \mbox{\textbf{\emph{D}}}_{q}
\end{pmatrix}^{H}\label{eq:two-pt-coherency}$$ This is the most general form that relates the coherency at two points $\mbox{\textbf{\emph{r}}}_{p}$ and $\mbox{\textbf{\emph{r}}}_{q}$, to the electromagnetic energy density at point $\mbox{\textbf{\emph{r}}}_{0}$.
In radio astronomy, antennas are used as sensors to measure the electromagnetic field. Following from Eq. \[eq:two-pt-coherency\], we may write an equation relating voltage and current coherency: $$\mathbb{V}_{pq}=\mathbf{\mathbb{T}}{}_{p}^{'}(\mathbf{\mathbb{T}}_{p}\mathbb{B}\mathbb{T}_{q}^{H})\mathbf{\mathbb{T}}{}_{q}^{'H}.\label{eq:measurement-eq-non-collapsed}$$ As the $\mathbb{T}'$ and **$\mathbb{T}$** matrices are both $\mbox{6}\times\mbox{6}$ , we can are both collapse these matrices into one overall matrix. Eq. \[eq:measurement-eq-non-collapsed\] then becomes
$$\mathbb{V}_{pq}=\mathbf{\mathbb{T}}{}_{p}\mathbb{B}\mathbf{\mathbb{T}}_{q}^{H},\label{eq:RIME-basic}$$
which is the general form that relates the voltage-coherency matrix $\mathbb{V}_{pq}$ to the brightness coherency $\mathbb{B}.$
Equation \[eq:RIME-basic\] is a central result of this paper. It is a general case which relates the EM field at a given point in space-time to the voltage and current coherencies in between pairs of telescopes. In the sections that follow, we show that generalized versions of the Van-Cittert-Zernicke theorem and RIME may be formulated based upon this coherency relationship, and that the common formulations can be derived from these general results.
The Radio Interferometer Measurement Equation
---------------------------------------------
By comparing Eq. \[eq:jones-rime-2\] with Eq. \[eq:two-pt-coherency\], it is apparent that the Jones formulation of the RIME is retrieved by setting all but the top left block matrices to zero, such that we have $$\mbox{\textbf{\emph{V}}}_{pq}=\mbox{\textbf{\emph{A}}}_{p}\mbox{\textbf{\emph{E}}}_{00}\mbox{\textbf{\emph{A}}}_{q}^{H}.$$ But under what assumptions may we ignore the other entries of Eq. \[eq:two-pt-coherency\]? To answer this, we may note that monochromatic plane waves in free space have ***E*** **** and ***H*** **** are in phase and mutually perpendicular: $$\begin{aligned}
\mbox{\textbf{\emph{e}}}(\mbox{\textbf{\emph{r}}},t) & =\begin{pmatrix}e_{x}(\mbox{\textbf{\emph{r}}},t) & e_{y}(\mbox{\textbf{\emph{r}}},t)\end{pmatrix}^{T}\nonumber\\
\mbox{\textbf{\emph{h}}}(\mbox{\textbf{\emph{r}}},t) & =\frac{1}{c_{0}}\begin{pmatrix}-e_{y}(\mbox{\textbf{\emph{r}}},t) & e_{x}(\mbox{\textbf{\emph{r}}},t)\end{pmatrix}^{T}
\label{eq:MPWZ}\end{aligned}$$ Where $c_{0}$ is the magnitude of the speed of light. In such a case, all coherency statistics can be derived from the $\mbox{2}\times\mbox{2}$ brightness matrix $\mathsf{B}$. @Carozzi:2009hf show that the field coherencies can be written $$\begin{aligned}
\mathbb{B} & =\begin{pmatrix}\mbox{\textbf{\emph{E}}}_{pq} & \mbox{\textbf{\emph{M}}}_{pq}\\
\mbox{\textbf{\emph{N}}}_{pq} & \mbox{\textbf{\emph{H}}}_{pq}
\end{pmatrix}=\begin{pmatrix}\mathsf{B} & \mathsf{B}\mbox{\textbf{\emph{F}}}^{T}\\
\mbox{\textbf{\emph{F}}}\mathsf{B} & \mbox{\textbf{\emph{F}}}\mathsf{B}\mbox{\textbf{\emph{F}}}^{T}
\end{pmatrix}\label{eq:Carozzi-matrix}\end{aligned}$$ where $\mathbf{F}$ is the matrix $$\mbox{\textbf{\emph{F}}}=\frac{1}{c_{0}}\begin{pmatrix}0 & 1\\
-1 & 0
\end{pmatrix}.$$ Under these conditions, the rank of $\mathbb{B}$ is 2, so the relationship in Eq. \[eq:two-pt-coherency\] is over constrained. It follows that the $\mbox{2}\times\mbox{2}$ RIME is perfectly acceptable — and indeed preferable to Eq. \[eq:two-pt-coherency\] — for describing coherency of plane waves that propagate through free space.
There are numerous situations in which we cannot assume that we have monochromatic plane waves. This includes near field sources where the wavefront is not well approximated by a plane wave; propagation through ionized gas; and situations where we choose not to treat our field as a superposition of quasi-monochromatic components. Most importantly, the assumptions that underlie the $\mbox{2}\times\mbox{2}$ RIME do not hold within the analogue components of a telescope, where the signal does not enjoy free space impedance. **
A 2N-port transmission matrix based RIME\[sub:2N-port-trans-RIME\]
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For a dual polarization telescope, a 4-port description (2-in 2-out) of our analogue system is more appropriate than the general 6-port description. Using the result \[eq:Carozzi-matrix\] above and Eq. \[eq:RIME-basic\], we can write a relationship
$$\mbox{\textbf{\emph{V}}}_{pq}=\begin{pmatrix}\mbox{\textbf{\emph{A}}}_{p} & \mbox{\textbf{\emph{B}}}_{p}\\
\mbox{\textbf{\emph{C}}}_{p} & \mbox{\textbf{\emph{D}}}_{p}
\end{pmatrix}\begin{pmatrix}\mathsf{B} & \mathsf{B}\mbox{\textbf{\emph{F}}}^{T}\\
\mbox{\textbf{\emph{F}}}\mathsf{B} & \mbox{\textbf{\emph{F}}}\mathsf{B}\mbox{\textbf{\emph{F}}}^{T}
\end{pmatrix}\begin{pmatrix}\mbox{\textbf{\emph{A}}}_{q} & \mbox{\textbf{\emph{B}}}_{q}\\
\mbox{\textbf{\emph{C}}}_{q} & \mbox{\textbf{\emph{D}}}_{q}
\end{pmatrix}^{H},\label{eq:2n-port-RIME}$$
Here, all block matrices have been reduced in dimensions from $\mbox{3}\times\mbox{3}$ to $\mbox{2}\times\mbox{2}$. This version of the RIME retains the ability to model analogue components, but uses the approximations of the vC-Z to express $\mathbb{B}$ in terms of the regular $\mbox{2}\times\mbox{2}$ brightness matrix B. The transmission matrices matrices here are similar to the 2N-port transmission matrices as defined by @BrandaoFaria2002.[^6]
The transmission matrices may be broken down into a chain of cascaded components. That is, for a cascade of $n$ components, we may write the overall transmission matrix as a product of matrices representing the individual components:
$$\begin{pmatrix}\mbox{\textbf{\emph{A}}}_{p} & \mbox{\textbf{\emph{B}}}_{p}\\
\mbox{\textbf{\emph{C}}}_{p} & \mbox{\textbf{\emph{D}}}_{p}
\end{pmatrix}=\begin{pmatrix}\mbox{\textbf{\emph{A}}}_{np} & \mbox{\textbf{\emph{B}}}_{np}\\
\mbox{\textbf{\emph{C}}}_{np} & \mbox{\textbf{\emph{D}}}_{np}
\end{pmatrix}\cdots\begin{pmatrix}\mbox{\textbf{\emph{A}}}_{1p} & \mbox{\textbf{\emph{B}}}_{1p}\\
\mbox{\textbf{\emph{C}}}_{1p} & \mbox{\textbf{\emph{D}}}_{1p}
\end{pmatrix}$$
This is similar to, but more general than, Jones calculus. In the following section we will explore the difference.
Limitations of Jones calculus
-----------------------------
Jones calculus essentially asserts two things. Firstly, it asserts that the voltage 2-vector at the output of a dual-polarization system $\bmath{v}_p=(v_{p1},v_{p2})^T$ is linear with respect to the EMF 2-vector $\bmath{e}=(e_x,e_y)^T$ at the input of the system: $$\bmath{v}_p = \bmath{J}_p \bmath{e},$$ where the Jones matrix $\bmath{J}_p$ describes the voltage transmission properties of the system. The second assertion is that for a system composed of $n$ components, the effective Jones matrix is a product of the Jones matrices of the components: $$\bmath{v}_p = \bmath{J}_{np} \cdots \bmath{J}_{1p} \bmath{e}.$$
With 2N-port transmission matrices, we instead describe the input of the system by the 4-vector $[\bmath{e},\bmath{h}]^T$, which for a monochromatic plane wave is equal to $$\begin{pmatrix}\bmath{e}\\ \bmath{h}\end{pmatrix} =
(e_x,e_y,-e_y/c_0,e_x/c_0)^T =
\begin{pmatrix}\bmath{e}\\ \bmath{F}\bmath{e}\end{pmatrix},$$ and the output of the system is a 4-vector of 2 voltages and 2 currents $(\bmath{v}_p,\bmath{i}_p)^T$, which is linear with respect to the input: $$\begin{pmatrix}\bmath{v}_p\\ \bmath{i}_p\end{pmatrix} =
\begin{pmatrix}\bmath{A}_p & \bmath{B}_p\\ \bmath{C}_p & \bmath{D}_p \end{pmatrix}
\begin{pmatrix}\bmath{e}\\ \bmath{F}\bmath{e}\end{pmatrix}.$$ Note that the output voltage is still linear with respect to the input $\bmath{e}$. Indeed, if one is only interested in the voltage, the above becomes $$\bmath{v}_p = (\bmath{A}_p+\bmath{B}_p\bmath{F})\bmath{e},$$ i.e. the system has an effective Jones matrix (i.e. a voltage transmission matrix) of $$\bmath{J}_p = \bmath{A}_p+\bmath{B}_p\bmath{F}.$$ However, the Jones formalism breaks down when the system is composed of multiple components. For example, with 2 components, we may naively attempt to apply Jones calculus, and describe the voltage transmission matrix of the system as a product of the components’ voltage transmission matrices: $$\bmath{J}_p = (\bmath{A}_{2p}+\bmath{B}_{2p}\bmath{F})(\bmath{A}_{1p}+\bmath{B}_{1p}\bmath{F}),$$ i.e. $$\bmath{v}_p = (
\bmath{A}_{2p}\bmath{A}_{1p} + \bmath{B}_{2p}\bmath{F}\bmath{A}_{1p} +
\bmath{A}_{2p}\bmath{B}_{1p}\bmath{F} + \bmath{B}_{2p}\bmath{F}\bmath{B}_{1p}\bmath{F})\bmath{e}.
\label{eq:jones-naive}$$ This, however, completely neglects the current transmission properties. Applying the 2N-port transmission matrix formalism, we can see the difference: $$\begin{pmatrix}\bmath{v}_p\\ \bmath{i}_p\end{pmatrix} =
\begin{pmatrix}\bmath{A}_{2p} & \bmath{B}_{2p}\\ \bmath{C}_{2p} & \bmath{D}_{2p} \end{pmatrix}
\begin{pmatrix}\bmath{A}_{1p} & \bmath{B}_{1p}\\ \bmath{C}_{1p} & \bmath{D}_{1p} \end{pmatrix}
\begin{pmatrix}\bmath{e}\\ \bmath{F}\bmath{e}\end{pmatrix},$$ from which we can derive an expression for the voltage vector: $$\bmath{v}_p =
(\bmath{A}_{2p}\bmath{A}_{1p} + \bmath{B}_{2p}\bmath{C}_{1p} +
\bmath{A}_{2p}\bmath{B}_{1p}\bmath{F} + \bmath{B}_{2p}\bmath{D}_{1p}\bmath{F})\bmath{e},
\label{eq:jones-2N}$$ which differs from Eq. \[eq:jones-naive\] in the second and fourth term of the sum, since in general $$\bmath{F}\bmath{A}_{1p} \neq \bmath{C}_{1p},~~\bmath{F}\bmath{B}_{1p} \neq \bmath{D}_{1p}.
\label{eq:jones-difference}$$
To summarize, because Jones calculus operates on voltages alone, and ignores impedance matching, we cannot use it to accurately represent the voltage response of an analogue system by a product of the voltage responses of its individual components. The difference is summarized in Eqs. \[eq:jones-naive\]–\[eq:jones-difference\]; a practical example is given in Sect. \[sub:Modelling-real-analogue\]. By contrast, the 2N-port transmission matrix formalism does allow us to break down the overall system response into a product of the component responses, by taking both voltages and currents into account.
Since we have now shown the $2\times2$ form of the RIME to be insufficient, the obvious question arises, why have we been getting away with using it? Historically, practical applications of the RIME have tended to follow the formulation of @JEN:note185, rolling the electronic response of the overall system (as well as tropospheric phase, etc.) into a single ‘$G$-Jones’ term that is solved for during calibration. Under these circumstances, the $2\times2$ formalism is perfectly adequate – it is only when we attempt to model the individual components of the analogue receiver chain that its limitations are exposed. On the other hand, @Carozzi:2009hf have highlighted the limitations of Jones calculus in the wide-field polarization regime.
Tensor formalisms of the RIME
=============================
Up until now, we have presented our $6\times6$ RIME using matrix notation. We now briefly discuss how the work presented here is closely related to the tensor formalism presented in @Smirnov:2011d.
As is discussed in @Smirnov:2011d, the classical Jones formulation of the RIME is in the vector space $\mathbb{C}^{2}$. The formulation proposed by @Carozzi:2009hf is instead in $\mathbb{C}^{3}$. In contrast, our Eq. \[eq:RIME-basic\] can be considered to work in $\mathbb{C}^{6}$; that is, our EMF vector has 6 components: $$\begin{aligned}
e^{i} & = & \sum_{j=1}^{3}e_{j}x^{j}+\sum_{k=1}^{3}h_{j}x^{j}\equiv\sum_{j=1}^{6}e_{j}x^{j}.\end{aligned}$$ The coherency of two voltages is once again defined via the outer product $e^{i}\bar{e}_{j}$, quite remarkably giving a (1,1)-type tensor expression identical Eq. 9 of @Smirnov:2011d: $$\mbox{\emph{V}}_{qj}^{pi}=\mbox{\emph{J}}_{\alpha}^{pi}\mbox{\emph{B}}_{\beta}^{\alpha}\bar{\mbox{\emph{J}}}_{qj}^{\beta}.$$ In the next section, we show an alternative RIME based upon a (2,0)-type tensor commonly encountered in special relativity.
A relativistic RIME
-------------------
Another potential reformulation of the RIME involves the electromagnetic tensor of special relativity. The classical RIME formulation implicitly assumes that both antennas measure the EMF in the same inertial reference frame. If this is not the case (consider, e.g., space VLBI), then we must in principle account for the fact that the observed EMF is altered when moving from one reference frame to another, and in particular, that the $\bmath{e}$ and $\bmath{h}$ components intermix. In special relativity, this can be elegantly formulated in terms of the *electromagnetic field tensor*, which represents the 6 independent components of the EMF by a (2-0)-type tensor:
$$F^{\alpha\beta}=\left(\begin{array}{cccc}
0 & -e_{x}/c & -e_{y}/c & -e_{z}/c\\
e_{x}/c & 0 & -h_{z} & h_{y}\\
e_{y}/c & h_{z} & 0 & -h_{x}\\
e_{z}/c & -h_{y} & h_{x} & 0
\end{array}\right),
\label{eq:F-general}$$
The advantage of this formulation is that the EMF tensor follows standard coordinate transform laws of special relativity. That is, for a different inertial reference frame given by the Lorentz transformation tensor $\Lambda^\alpha_{\alpha'}$, the EMF tensor transforms as: $$F'^{\alpha\beta} = \Lambda^\alpha_{\alpha'} \Lambda^\beta_{\beta'} F^{\alpha'\beta'}.$$ The measured 2-point coherency between $[F_p]$ and $[F_q]$ can be formally defined as the average of the outer product: $$[V_{pq}]^{\alpha\beta\gamma\delta} = 2 c^2 \langle [F_p]^{\alpha\beta}[F^H_q]^{\gamma\delta} \rangle,$$ where $\cdot^H$ represents the conjugate tensor, i.e. $[F^H]^{\gamma\delta}=\bar{F}^{\delta\gamma}$. A factor of 2 is introduced for the same reasons as in @Smirnov:2011a, and the reason for $c^2$ will be apparent below. Note that the indices in the brackets should be treated as labels, while those outside the brackets are proper tensor indices.
Let us now pick a reference frame for the signal (‘frame zero’), and designate the EMF tensor in that frame by $[F_0]$, or $[F_0(\bmath{\bar{x}})]$ to emphasize that this is a function of the four-position $\bmath{\bar{x}}=(ct,\bmath{x})$. The $[F_0(\bmath{\bar{x}})]$ field follows Maxwell’s equations; in the case of a monochromatic plane wave propagating along direction $\bmath{\bar{z}}=(1,\bmath{z})$, this has a particularly simple solution of $$[F_0(\bmath{\bar{x}})] = [F_0(\bmath{\bar{x}}_0)]e^{-2\pi i \lambda^{-1} (\bmath{\bar{x}}-\bmath{\bar{x}}_0)\cdot \bmath{\bar{z}}}.
\label{eq:F0-x}$$
Let us now consider two antennas located at $\bmath{p}$ and $\bmath{q}$. The 2-point coherency measured in frame zero becomes $$\begin{aligned}
[V_{pq}]^{\alpha\beta\gamma\delta} & = &
2 c^2 \langle [F_0(\bmath{\bar{p}})]^{\alpha\beta}[F^H(\bmath{\bar{q}})]^{\gamma\delta} \rangle \nonumber\\
&=& K_p \left [ 2 c^2 \langle [F_0(\bmath{\bar{x}}_0)]^{\alpha\beta}[F^H(\bmath{\bar{x}}_0)]^{\gamma\delta} \rangle \right ] K^H_q,\end{aligned}$$ where $K_p$ is the complex exponent of Eq. \[eq:F0-x\], and is the direct equivalent of the $K$-Jones term of the RIME [@Smirnov:2011a]. The quantity in the square brackets is the equivalent of the brightness matrix $\mathsf{B}$, which we’ll call the *brightness tensor*:
$$B^{\alpha\beta\gamma\delta} = [B_0]^{\alpha\beta\gamma\delta} = 2c^2 \langle [F_0(\bmath{\bar{x}}_0)]^{\alpha\beta}[F_0^H(\bmath{\bar{x}}_0)]^{\gamma\delta} \rangle.$$
Each element of the brightness tensor gives the coherency between two components of the EMF observed in the chosen reference frame (‘frame zero’). Nominally, the brightness tensor has $4^4=256$ components, but only 36 are unique and non-zero (given the 6 components of the EMF). @Carozzi:2006bj show that the brightness tensor may be decomposed into a set of antisymmetric second rank tensors (‘sesquilinear-quadratic tensor concomitants’) that are irreducible under Lorentz transformations. The physical interpretation of the 36 unique quantities within the brightness matrix is discussed in @Bergman:2008p7859, with regards to the aforementioned irreducible tensorial set.
While the brightness tensor has redundancy not present in the tensorial set of @Carozzi:2006bj, we shall continue to use it as a basis to our relativistic RIME for clarity of analogy to the brightness coherency matrix of Eq. \[eq:RIME-basic\], and as it leads to a relativistic RIME for which we can define transformation tensors analogous to Jones matrices. The redundancy can be described by a number of symmetry properties of the brightness tensor: it is (a) *Hermitian* with respect to swapping the first and second pair of indices: $$B^{\alpha\beta\gamma\delta} = \bar{B}^{\gamma\delta\alpha\beta},
\label{eq:BT-herm}$$ (b) *antisymmetric* within each index pair (since the EMF tensor itself is antisymmetric, i.e. $F^{\alpha\beta}=-F^{\beta\alpha}$): $$B^{\alpha\beta\gamma\delta} = -B^{\beta\alpha\gamma\delta} = -B^{\alpha\beta\delta\gamma}
\label{eq:BT-antisym}$$ To see the direct analogy to the brightness matrix, consider again the case of the monochromatic plane wave propagating along $\bmath{z}$ (Eq. \[eq:MPWZ\]). The EMF tensor then takes a particularly simple form: $$\label{eq:F-planewave}
F^{\alpha\beta} = \frac{1}{c} \left(\begin{array}{cccc}
0 & -e_{x} & -e_{y} & 0 \\
e_{x} & 0 & 0 & e_x \\
e_{y} & 0 & 0 & e_y \\
0 & -e_{x} & -e_y & 0
\end{array}\right),$$ and the brightness tensor has only $8^2=64$ non-zero components, with the additional ‘0-3’ symmetry property: $$\begin{aligned}
B^{0\beta\gamma\delta} = B^{3\beta\gamma\delta} && B^{\alpha0\gamma\delta} = B^{\alpha3\gamma\delta} \nonumber\\
B^{\alpha\beta0\delta} = B^{\alpha\beta3\delta} && B^{\alpha\beta\gamma0} = B^{\alpha\beta\gamma3}
\label{eq:BT-sym}\end{aligned}$$ Four unique components can be defined in terms of the Stokes parameters: $$\begin{aligned}
B^{0110} = I+Q && B^{0220} = I-Q \nonumber\\
B^{0120} = U+iV && B^{0210} = U-iV
\label{eq:BT-IQUV}\end{aligned}$$ and conversely, $$\begin{aligned}
I = \frac{B^{0110}+B^{0220}}{2} && Q = \frac{B^{0110}-B^{0220}}{2}, \nonumber\\
U = \frac{B^{0120}+B^{0210}}{2} && V = \frac{B^{0120}-B^{0210}}{2i}.
\label{eq:IQUV-BT}\end{aligned}$$ The other non-zero components of the brightness tensor can be derived using the Hermitian, antisymmetry and 0-3 symmetry properties. Finally, for an unpolarized plane wave, only 32 components of the brightness tensor are non-zero and equal to $\pm I$. As these will be useful in further calculations, they are summarized in Table \[tab:BT\].
01 02 10 13 20 23 31 32
---- ------ ------ ------ ------ ------ ------ ------ ------
01 $-I$ $I$ $I$ $-I$
02 $-I$ $I$ $I$ $-I$
10 $I$ $-I$ $-I$ $I$
13 $I$ $-I$ $-I$ $I$
20 $I$ $-I$ $-I$ $I$
23 $I$ $-I$ $-I$ $I$
31 $-I$ $I$ $I$ $-I$
32 $-I$ $I$ $I$ $-I$
: The non-zero components of the brightness tensor $B^{\alpha\beta\gamma\delta}$ for an unpolarized plane wave. Rows correspond to $\alpha\beta$, columns to $\gamma\delta$.[]{data-label="tab:BT"}
So far this has been nothing more than a recasting of the RIME using EMF tensors. Consider, however, the case where antennas $p$ and $q$ measure the signal in different inertial reference frames. The EMF tensor observed by antenna $p$ becomes $$[F_p]^{\alpha\beta} = [\Lambda_p]^\alpha_{\alpha'} [\Lambda_p]^\beta_{\beta'} [F_0]^{\alpha'\beta'},$$ where $[\Lambda_p]^\alpha_\mu$ is the Lorentz tensor corresponding to the transform between the signal frame and the antenna frame. Since the same Lorentz tensor always appears twice in these equations (due to $F$ being a (2,0)-type tensor), let us designate $$[\Lambda_p]^{\alpha\beta}_{\alpha'\beta'} = [\Lambda_p]^\alpha_{\alpha'} [\Lambda_p]^\beta_{\beta'}
\label{eq:LL}$$ for compactness. The measured coherency now becomes $$[V_{pq}]^{\alpha\beta\gamma\delta} =
K_p {[\Lambda_p]}^{\alpha\beta}_{\alpha'\beta'}
[B_0]^{\alpha'\beta'\gamma'\delta'}
[\Lambda_q]^{\gamma\delta}_{\gamma'\delta'}
K^H_q.
\label{eq:RRIME-KL}$$ The equivalent of Jones matrices would be (2,2)-type ‘Jones tensors’ $J^{\alpha\beta}_{\alpha'\beta'}$, so a more general formulation of the above would be $$[V_{pq}]^{\alpha\beta\gamma\delta} =
{[J_p]}^{\alpha\beta}_{\alpha'\beta'}
[B_0]^{\alpha'\beta'\gamma'\delta'}
[J^H_q]^{\gamma\delta}_{\gamma'\delta'},
\label{eq:RRIME-J}$$ where tensor conjugation $\cdot^H$ is defined as: $$[J^H]^{\alpha\beta}_{\alpha'\beta'}=\bar{J}^{\beta\alpha}_{\beta'\alpha'}.$$ Note that both the $K$ and $\Lambda$ terms of Eq. \[eq:RRIME-KL\] can be considered as special examples of Jones tensors. The $K$ term can be explicitly written as a (2,2)-type tensor via two Kronecker deltas: $$[K_p]^{\alpha\beta}_{\alpha'\beta'} = K_p \delta^{\alpha}_{\alpha'}\delta^{\beta}_{\beta'},$$ while the $\Lambda$ term is a (2,2)-type tensor by definition (Eq. \[eq:LL\]), noting that $$[\Lambda^H_p]^{\alpha\beta}_{\alpha'\beta'} = [\Lambda_p]^{\alpha\beta}_{\alpha'\beta'},$$ since the components of any Lorentz transformation tensor $\Lambda^{\alpha}_{\alpha'}$ are real.
Finally, let us note the equivalent of the ‘chain rule’ for Jones tensors. If the signal chain is represented by a sequence of Jones tensors $[J_{p,n}],...,[J_{p,1}]$ (including $K$ terms, $\Lambda$ terms, and all instrumental and propagation effects), then the overall response is given by $$[J_p]^{\alpha\beta}_{\alpha'\beta'} =
[J_{p,n}]^{\alpha\beta}_{\alpha_n\beta_n}
[J_{p,n-1}]^{\alpha^n\beta^n}_{\alpha_{n-1}\beta_{n-1}}
\cdots
[J_{p,1}]^{\alpha^2\beta^2}_{\alpha'\beta'}.
\label{eq:JT-chain}$$ Equations \[eq:RRIME-KL\], \[eq:RRIME-J\] and \[eq:JT-chain\] above constitute a relativistic reformulation of the RIME (RRIME). The RRIME allows us to incorporate relativistic effects into our measurement equation. That is, we can treat relativistic motion as an ‘instrumental’ effect. Some interesting observational consequences, and the relation of the RRIME to work from other fields are treated in the discussion below.
Discussion
==========
The formulations presented here highlight the relationship between the seemingly disparate fields of microwave networking and special relativity to measurements in radio astronomy. This is a remarkable illustration of how these fields are intrinsically related by underlying fundamental physics.
Indeed, it has long been known that Jones and Mueller transformation matrices are related to Lorentz transformations by the Lorentz group. Wiener was aware as early as 1928 that the the 2$\times$2 coherency matrix could be written in terms of the Pauli spin matrices [@wiener1928; @wiener1930]. More recently, @Baylis:1993 presents a more general geometric algebra formalism that unifies Jones and Mueller matrices with Stokes parameters and the Poincaré sphere. @Han:1997jones show that the Jones formalism is a representation of the six-parameter Lorentz group; further to this @Han:1997stokes show that the Stokes parameters form a Minkowskian four-vector, similar to the energy-momentum four-vector in special relativity.
In contrast, the RRIME presented here is novel as it fully describes interferometric measurement between two points, instead of simply describing polarization states and defining a transformation algebra. Similarly, the 2N-port RIME specifically shows how microwave networking methods can be incorporated into a ME.
What do we gain from using these more general measurement equations in lieu of the simpler 2$\times$2 RIME? For most applications, it will suffice to simply be aware of the standard RIME’s limitations, and to work in a piecemeal fashion. For example, one can quite happily use Jones matrices to describe free space propagation, and microwave network methods to describe discrete analogue components. An overall ‘system Jones’ matrix may be derived to describe instrumental effects, but this matrix should never be decomposed into a Jones chain.
Similarly, special relativity describes relativistic effects through Lorentz transformations acting upon the EMF tensor. We can treat relativistic motion as an instrumental effect by using the RRIME, or we can apply special relativistic corrections separately, as required. The effect of relativistic boosts on the Stokes parameters are considered in @cocke:1972.
In the subsections that follow, we present some potential use cases that highlight the usefulness of the 2N-port and relativistic RIME formulations.
Modelling real analogue components\[sub:Modelling-real-analogue\]
-----------------------------------------------------------------
The 2N-port RIME may have practicality in absolute flux calibration of radio telescopes. Generally, interferometer data is calibrated by assuming that the sky brightness is known, or at least approximately known. If we wish to do absolute calibration of a telescope without making assumptions about the sky, we must instead make sure that we accurately model the analogue components of our telescope. A particularly ubiquitous method of device characterization within microwave engineering is the use of scattering parameters; we briefly introduce these below before incorporating them into the 2N-port RIME.
### Scattering parameters
Voltage, current and impedance are somewhat abstract concepts at microwave frequencies, so engineers often use scattering parameters to quantify a device’s characteristics. Scattering parameters relate the incident and reflected voltage waves on the ports of a microwave network. The scattering matrix, ***S***, is given by $$\begin{pmatrix}v_{1}^{-}\\
v_{2}^{-}\\
\vdots\\
v_{n}^{-}
\end{pmatrix}=\begin{pmatrix}S_{11} & S_{12} & \cdots & S_{1n}\\
S_{21} & & & \vdots\\
\vdots & & \ddots & \vdots\\
S_{n1} & \cdots & \cdots & S_{nn}
\end{pmatrix}\begin{pmatrix}v_{1}^{+}\\
v_{2}^{+}\\
\vdots\\
v_{n}^{+}
\end{pmatrix},$$ where $v_{n}^{+}$ is the amplitude of the voltage wave incident on port $n$, and $v_{n}^{-}$ is the amplitude of the voltage wave reflected from port $n$.
For a dual polarization system (we will label the polarization *x* and *y*) with negligible crosstalk, we can model the analogue chain for each polarization as a discrete 2-port network. Assuming that the analogue chains have the same number of components (but not that the components are identical), the transmission matrix for each pair of components is $$\mbox{\textbf{\emph{T}}}=\begin{pmatrix}\tilde{A}_{x} & 0 & \tilde{B}_{x} & 0\\
0 & \tilde{A}_{y} & 0 & \tilde{B}_{y}\\
\tilde{C}_{x} & 0 & \tilde{D}_{x} & 0\\
0 & \tilde{C}_{y} & 0 & \tilde{D}_{y}
\end{pmatrix},$$ where the elements are from the *ABCD* matrices of the *x* and *y* polarizations, and are given by the relations $$\begin{aligned}
\tilde{A} & =\frac{1+S_{12}S_{21}+S_{22}-S_{11}(1+S_{22})}{2S_{12}}\label{eq:s-to-t-first}\\
\tilde{B} & =-Z_{0}\frac{1+S_{11}-S_{12}S_{21}+S_{22}+S_{11}S_{22}}{2S_{12}}\\
\tilde{C} & =\frac{1}{Z_{0}}\frac{-1+S_{11}+S_{12}S_{21}+S_{22}-S_{11}S_{22}}{2S_{12}}\\
\tilde{D} & =\frac{1+S_{11}+S_{12}S_{21}-S_{22}-S_{11}S_{22}}{2S_{12}}\label{eq:s-to-t-last}\end{aligned}$$ Here, $Z_{0}$ is the characteristic impedance of the analogue chain, which in most telescopes is set to 50 or 75 ohms. We have added tildes to the *ABCD* parameters, as we are using the inverse definition to that in @Pozar2005.
If the system has significant crosstalk between polarizations, the analogue chain is more accurately modelled as a 4-port network. In this case the relationships are not so simple, and the off-diagonal entries of the block matrices of ***T*** **** will no longer be zero.
### Scattering matrix example
We now present a simple illustration of the differences between assigning a component a scalar value (as is done in the Jones formalism), and by modelling it as a 2-port network. Consider a component with a scattering matrix
$$\mbox{\textbf{\emph{S}}}=\begin{pmatrix}0.1\angle0^{\circ} & 0.9\angle0^{\circ}\\
0.9\angle0^{\circ} & 0.1\angle0^{\circ}
\end{pmatrix}$$
for a given quasi-monochromatic frequency. Here, values are presented in angle notation to emphasize that they are complex valued. In decibels, the $S_{11}$ and $S_{22}$ parameters have a magnitude of -10 dB, and the $S_{12}$ and $S_{21}$ parameters have a magnitude of about $-0.5$ dB. Now suppose we have three identical copies of this component, and we connect them together in cascade. If one only considered the forward gain ($S_{21}$), one would arrive at an overall $S_{\rm{21tot}}$ of $$S_{\rm{21tot}}=S_{21}S_{21}S_{21}=0.729$$ In contrast, using the standard microwave engineering methods, we can form a transmission matrix (using equations \[eq:s-to-t-first\]-\[eq:s-to-t-last\] above), and then convert this back into an overall matrix, $\mbox{\textbf{\emph{S}}}_{\rm{tot}}$. By doing this, one finds $$\mbox{\textbf{\emph{S}}}_{\rm{cas}}=\begin{pmatrix}0.25\angle0^{\circ} & 0.75\angle0^{\circ}\\
0.75\angle0^{\circ} & 0.25\angle0^{\circ}
\end{pmatrix}.$$ The difference becomes more marked as $S_{11}$ and $S_{22}$ increase; as $S_{11}$ and $S_{22}$ approach zero, the two methods converge.
When designing a component, $S_{11}$ and $S_{22}$ are generally optimised to be as small as possible over the operational bandwidth[^7]. Nevertheless, their affect on the overall system is often non-negligible.
### Absolute calibration experiments
![Block diagram of a three-state switching experiment. In addition to the antenna path, an RF load can in series with a noise diode can be selected. The noise diode can be turned on and off, giving three possible states: antenna, load, and diode + load. These three states are used for instrumental calibration. \[fig:three-switch\]](figures-total-power-block-diagram){width="0.99\columnwidth"}
Now we have shown how scattering parameters can be used within the 2N-port RIME, we turn our attention to the challenges of absolute calibration of radio telescopes. Measurement of the absolute flux of radio sources is fiendishly hard; as such, almost all calibrated flux data presented in radio astronomy literature are pinned to the flux scale presented in @Baars:1977, either directly or indirectly [@Kellermann:2009]. Recent experiments, such as the Experiment to Detect the Global EoR Step (EDGES, @Bowman:2010), and the Large-Aperture Experiment to detect the Dark Ages (LEDA, @leda2012), are seeking to make precision measurement of the sky temperature (i.e. total power of the sky brightness) as a function of frequency. The motivation for this is to detect predicted faint (mK) spectral features imprinted on the sky temperature due to coupling between the microwave background and neutral Hydrogen in the early universe. For such instruments — and other instruments with wide field-of-views — this flux ‘bootstrapping’ method is not sufficient.
For experiments such as EDGES a thorough understanding of the analogue systems is vital to control the systematics that confound calibration. The calibration strategy used for EDGES is detailed in @Rogers:2012hd, and consists of a novel three-state switching and rigorous scattering parameter measurements with a Vector Network Analyzer (VNA) to remove the bandpass and characterize the antenna.
Our 2N-port RIME allow for scattering parameters to be directly incorporated into a measurement equation that relates sky brightness to measured voltages at the digitizer. This could be used either to (a) infer the scattering parameters of a device given a known sky brightness, or (b) infer the sky brightness from precisely measured scattering parameters. The EDGES approach is the latter, but via an ad-hoc method without formal use of a ME.
### A three-state switching measurement equation
The three-state switching as described in @Rogers:2012hd involves switching between the antenna and a reference load and noise diode, and measuring the resultant power spectra ($P_{A}$, $P_{L}$, and $P_{D}$, respectively). This is shown as a block diagram in Figure \[fig:three-switch\]. The power as measured in each state is then given by: $$\begin{aligned}
P_{A} & = & Gk_{B}\Delta\nu(T_{A}+T_{\rm{rx}})\label{eq:pow-ant}\\
P_{L} & = & Gk_{B}\Delta\nu(T_{L}+T_{\rm{rx}})\\
P_{D} & = & Gk_{B}\Delta\nu(T_{D}+T_{L}+T_{\rm{rx}}\label{eq:pow-diode})\end{aligned}$$ where $k_B$ is the Boltzmann constant, $T_D$ and $T_L$ are the diode and reference load noise temperatures, $T_{\rm{rx}}$ is the receiver’s noise temperature, $G$ is the system gain, and $\Delta\nu$ is bandwidth. One can recover the antenna temperature $T_{A}$ by $$T_{A}=T_{D}\frac{P_{A}-P_{L}}{P_{D}-P_{L}}+T_{L}\label{eq:ant-temp},$$ where the diode and load temperatures, $T_D$ and $T_L$, are known. Antenna temperature is then related to the sky temperature by $$T_{\rm{sky}}=T_{A}(1-|\Gamma|^2),\label{eq:sky-ant}$$ where the reflection coefficient $\Gamma\equiv S_{11}$. Failure to account for reflections (i.e. impedance mismatch) results in an unsatisfactory calibration, due to standing waves within the coaxial cable that manifest as a sinusoidal ripple on $T_A = T_{A}(\nu)$. This effect is a prime example why one must not use Jones matrices to describe analogue components separately; an example showing a standing wave present on three-state switch calibrated spectrum from a prototype LEDA antenna is shown in Figure \[fig:leda-all\].
We may instead write the equations above in terms of 2N-port transmission matrices, and form MEs for the three states: $$\begin{aligned}
\mathbf{V}_{A} & = & \mathbf{G}(\mathbf{T}_A\mathbb{B}_{\rm{sky}}\mathbf{T}_A^{H} + \mathbb{B}_{\rm{rx}})\mathbf{G}^{H}\label{eq:switchmat1}\\
\mathbf{V}_{L} & = & \mathbf{G}(\mathbb{B}_{L} + \mathbb{B}_{\rm{rx}})\mathbf{G}^{H}\\
\mathbf{V}_{D} & = & \mathbf{G}(\mathbf{T}_L\mathbb{B}_{D}\mathbf{T}_L^{H}+\mathbb{B}_{L} + \mathbb{B}_{\rm{rx}})\mathbf{G}^{H}\label{eq:switchmat3}.\end{aligned}$$ Here, we have replaced the scalar powers $P_{A,L,D}$ with corresponding voltage-coherency matrices $\mathbf{V}_{A,L,D}$, temperatures $T_{A,L,\rm{sky},\rm{rx}}$ are replaced with brightness matrices $\mathbb{B}_{A,L,\rm{sky},\rm{rx}}$, and $Gk_B$ is instead represented by a system gain matrix $\mathbf{G}$. We have added a transmission matrix for the antenna $\mathbf{T}_A$, and a transmission matrix $\mathbf{T}_L$ for the load in series with the noise diode. Note that the relation of Eq. \[eq:sky-ant\] is now encoded into the ME by the matrix $\mathbf{T}_A$, and that we have dropped antenna number subscripts as *p*=*q* for autocorrelation measurements.
It is immediately apparent that the cancellations that occur in the ratio of Eq. \[eq:ant-temp\] will not in general occur for the equivalent ratio $\mathbf{R} = (\mathbf{V}_{A}-\mathbf{V}_{L})(\mathbf{V}_{D}-\mathbf{V}_{L})^{-1}$. We can however retrieve the result of Eq. \[eq:ant-temp\] by treating the two polarizations separately, setting $\mathbf{G} = G\mathbf{I}$ and $\mathbf{T}_L = \mathbf{I}$, with $\mathbf{T}_A = T_{A}(1-|\Gamma|^2)\mathbf{I}$.
Our equations Eq. \[eq:switchmat1\]-\[eq:switchmat3\] allow for both polarizations to be treated together, which will be important if cross-polarization terms are non-negligible. Also, we may expand Eq. \[eq:switchmat1\] to include ionospheric effects. This ability to combine all effects into a single ME may simplify data analysis and improve calibration accuracy for such experiments.
RRIME and Lorentz boosts
------------------------
To illustrate how the RRIME can be used to describe relativistic effects, let us first consider the ‘simple’ case of a relativistically moving source. Suppose we have an unpolarized point source (Table \[tab:BT\]), with antennas $p$ and $q$ located in the $xy$ plane perpendicular to the direction of propagation $z$ (so the $K$ component becomes unity), both moving parallel to the $x$ axis with velocity $v$. The Lorentz transformation tensor from the signal frame to the antenna frame is then
$$[\Lambda_p]^\alpha_{\alpha'}=
\left(\begin{array}{cccc}
\gamma & -\beta\gamma & 0 & 0 \\
-\beta\gamma & \gamma & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right),
\label{eq:Lboost}$$
where $$\beta = \frac{v}{c},~~\gamma=\frac{1}{\sqrt{1-\beta^2}}.$$ The Lorentz factor $\gamma$ is unity 1 at $v=0$, and goes to infinity with $v\rightarrow c$. This case can be analyzed without invoking coherencies. Consider the EMF, which in the signal frame is a plane wave (Eq. \[eq:F-planewave\]). In the antenna frame it becomes
$$[F_p]^{\alpha\beta} = \frac{1}{c}
\left(\begin{array}{cccc}
0 & -e_x & -\gamma e_y & -\beta \gamma e_x \\
e_x & 0 & \beta\gamma e_y & \gamma e_x \\
\gamma e_y & -\beta\gamma e_y & 0 & e_y \\
\beta\gamma e_x & -\gamma e_x & -e_y & 0
\end{array}\right ) .$$
Noting that $$\left(\begin{array}{ccc}
\gamma^{-1} & 0 & \beta \\
0 & 1 & 0 \\
-\beta & 0 & \gamma^{-1}
\end{array}\right )
\left(\begin{array}{c}
e_{x} \\
\gamma e_y \\
\beta \gamma e_x
\end{array}\right ) =
\gamma \left(\begin{array}{c}
e_x \\
e_y \\
0
\end{array}\right ),$$ and $$\left(\begin{array}{ccc}
\gamma^{-1} & 0 & \beta \\
0 & 1 & 0 \\
-\beta & 0 & \gamma^{-1}
\end{array}\right )
\left(\begin{array}{c}
-e_y \\
\gamma e_x \\
-\beta \gamma e_y
\end{array}\right ) =
\gamma \left(\begin{array}{c}
-e_y \\
e_x\\
0
\end{array}\right ),$$ and $\gamma^{-2}+\beta^2=1,$ we can see that the EMF in the antenna frame is equivalent to a boost of the original EMF by $\gamma$ (also called [*Doppler boost*]{}), coupled to a rotation through $\phi=-\cos^{-1}\beta$ in the $xz$ plane (i.e. the direction of propagation appears to change, also called [*relativistic aberration*]{}). Both effects are well-understood in the guise of [*relativistic beaming*]{}, and explain why, for example, some AGNs exhibit asymmetric jets [see e.g. @Sparks1992].
From an RRIME point of view, a more interesting case arises when one antenna is moving with respect to the other (as is the case in space VLBI). Let’s consider antenna $p$ to be at rest with respect to the signal frame (so the $[\Lambda_p]^{\alpha\beta}_{\alpha'\beta'}$ component becomes the equivalent of unity – the product of two Kronecker deltas – and thus may be dropped), and antenna to be $q$ moving parallel to the $x$ axis with velocity $v$. The $[J_q]^{\gamma\delta}_{\gamma'\delta'}$ tensor in Eq. \[eq:RRIME-J\] is then a product of two Lorentz tensors of the form of Eq. \[eq:Lboost\]. In the absence of other effects, the measured coherency becomes
$$[V_{pq}]^{\alpha\beta\gamma\delta} =
B^{\alpha\beta\gamma'\delta'}
[\Lambda_q]^\gamma_{\gamma'} [\Lambda_q]^\delta_{\delta'},$$
or writing it out as an explicit sum for two particular elements of interest (and dropping the $pq$ indices): $$\begin{aligned}
V^{0110} & = & \sum_{\gamma\delta}
B^{01\gamma\delta}
[\Lambda_q]^{0}_{\gamma}
[\Lambda_q]^{1}_{\delta} \\
V^{0220} & = & \sum_{\gamma\delta}
B^{02\gamma\delta}
[\Lambda_q]^{0}_{\gamma}
[\Lambda_q]^{2}_{\delta}.\end{aligned}$$ Each sum above nominally contains 16 terms, but if we assume an unpolarized point source, then from Table \[tab:BT\] (looking up rows ‘01’ and ‘02’) we note that only four components of $B$ in each sum are non-zero. Combining this with Eq. \[eq:Lboost\], we get: $$V^{0110} = I(\gamma^2 - \beta^2\gamma^2) = I,$$ and $$V^{0220} = I\gamma.$$ Doing the same sums for $V^{0120}$ and $V^{0210}$, we arrive at zero. This implies that our instrument will measure the Stokes parameters as $$\begin{aligned}
I_\mathrm{meas} &=& \frac{V^{0110}+V^{0220}}{2} = I\frac{1+\gamma}{2} \\
Q_\mathrm{meas} &=& \frac{V^{0110}-V^{0220}}{2} = I\frac{1-\gamma}{2} \\
U_\mathrm{meas} &=& 0 \\
V_\mathrm{meas} &=& 0.\end{aligned}$$
Since $\gamma>1$, we measure boost in the total power $I$, and negative apparent $Q$ (that is, linear polarization perpendicular to the direction of motion). The physical meaning of this instrumental $Q$ can be understood in terms of the polarization aberration discussed by @Carozzi:2009hf: because the arriving plane wave is aberrated in the frame of antenna $q$ (i.e. no longer appears to propagate along the $z$ axis, but along a slightly different direction), it is measured as polarized by dipoles that are parallel to $xy$.
Note that this formulation does not incorporate the Doppler shift observed by a moving antenna (we quietly assume that the correlator takes care of correcting for this when channelizing), or the problems of clock distribution to a relativistically moving observing platform. This will have to be addressed in a future work.
In principle, the effects of Doppler boost and relativistic aberration can be described without invoking the full RRIME – traditional Jones calculus still suffices. However the RRIME provides a unifying framework that allows these effects to be incorporated into a single compact interferometric measurement equation. This is similar to how the original RIME formulation of @Hamaker:1996p5735 incorporated polarimetric effects that were already described previously [@Sault-AT-Calibration] in a compact closed form.
An even more interesting use case arises when the incident EMF can no longer be described by a plane wave. The EMF tensor of Eq. \[eq:F-general\] then no longer reduces to Eq. \[eq:F-planewave\], or in other words, the $\bmath{e}$ and $\bmath{h}$ components of the EM are no longer mutually redundant. Under Lorentz transformations, the $\bmath{e}$ and $\bmath{h}$ components then become intermixed – an antenna measuring $\bmath{e}$ in the rest frame will also measure some contribution from $\bmath{h}$ in a moving frame. It is clear that in this case, Jones calculus (which only operates on $\bmath{e}$) can no longer apply, and a full RRIME must be invoked. Practical applications of this will have to wait for near-field space VLBI.
Conceptually, the latter example is very similar to the transmission matrix formalism. In a system where the voltage and current components intermix, the voltage-only Jones formalism no longer applies, and a formalism incorporating both components must be invoked.
Conclusions
===========
The radio interferometer Measurement Equation provides a powerful framework for describing a signal’s journey from an astrophysical source to the receiver of a radio telescope. Nevertheless, the Jones calculus it employs is in general not sufficient to describe the components within the analogue chain of a radio telescope. Within the telescope, methods from microwave network theory, such as transmission matrices and scattering parameters, are more appropriate.
Similarly, the Jones formalism is not able to describe relativistic effects. We have presented two reformulations that account for mixed and magnetic field coherency in a way that is not possible with the Jones formalism. These reformulations extend the applicability of the RIME to allow it to correctly model analogue components, and to describe relativistic effects within the RIME.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Stef Salvini, Ian Heywood and Mike Jones for their comments, Tobias Carozzi for his valuable insight, and the late Steve Rawlings for his advice in the seminal stages of this paper. O. Smirnov’s research is supported by the South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation. This work has made use of LWA1 outrigger dipoles made available by the LEDA project, funded by NSF under grants AST-1106054, AST-1106059, AST-1106045, and AST- 1105949.
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: For a discussion on the subtleties of outer product definition, see @Smirnov:2011d, §A.6.1
[^3]: Here, $j=\sqrt{-1}$, to avoid confusion with current, $i$, used later.
[^4]: Note that $Z_{21}=0$ is zero impedance, which is never satisfied in real components, and $Z_{21}=\infty$ represents an open circuit
[^5]: Spatial location $\mathbf{r}$ is no longer relevant as the voltage propagates through analogue components with clearly defined inputs and outputs.
[^6]: We note that our definition here is the inverse of that of Faria (input and output are swapped).
[^7]: A notable exception is RF filters, for which $S_{11}$ is often close to unity out-of-band. In such cases $S_{11}$ is strongly dependent upon frequency.
|
---
abstract: 'We consider smooth, complete Riemannian manifolds which are exponentially locally doubling. Under a uniform Ricci curvature bound and a uniform lower bound on injectivity radius, we prove a Kato square root estimate for certain coercive operators over the bundle of finite rank tensors. These results are obtained as a special case of similar estimates on smooth vector bundles satisfying a criterion which we call *generalised bounded geometry*. We prove this by establishing quadratic estimates for perturbations of Dirac type operators on such bundles under an appropriate set of assumptions.'
address:
- 'Lashi Bandara, Centre for Mathematics and its Applications, Australian National University, Canberra, ACT, 0200, Australia'
- 'Alan McIntosh, Centre for Mathematics and its Applications, Australian National University, Canberra, ACT, 0200, Australia '
author:
- Lashi Bandara
- Alan McIntosh
date: 'June 10, 2014'
title: The Kato square root problem on vector bundles with generalised bounded geometry
---
Introduction
============
In this paper, we consider the *Kato square root problem* for uniformly elliptic operators on smooth vector bundles ${\mathcal{V}}$ over smooth, complete Riemannian manifolds ${\mathcal{M}}$ which are at most exponentially locally doubling. Let ${{{\nabla}_{{{}}}}}$ be a connection on the bundle and ${\mathrm{h}}$ its metric. Define the uniformly elliptic operator ${\mathrm{L}}_{A}u
= -a \operatorname{div}(A_{11} {{{\nabla}_{{{}}}}}u) - a \operatorname{div}(A_{10}u)
+ a A_{01} {{{\nabla}_{{{}}}}}u + a A_{00} u$ where $\operatorname{div}= -{{{{{\nabla}_{{{}}}}}}^\ast}$ and where the $a, A_{ij}$ are ${{\rm L}^{\infty}_{\rm {}}}$ coefficients. Under an appropriate bounded geometry assumption, we show that ${ {\mathcal{D}}}(\sqrt{{\mathrm{L}}_{A}}) = { {\mathcal{D}}}({{{\nabla}_{{{}}}}}) = {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}})$ and that $\|\sqrt{{\mathrm{L}}_{A}}u\| \simeq {\left\| u \right\|}_{{{{\rm W}^{1,2}_{\rm {}}}}}$ for all $u \in {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}})$.
The case of trivial, flat bundles was considered by Morris in [@Morris3] (and his thesis [@Morris]). In particular, he obtains solutions to the Kato problem on Euclidean submanifolds under *extrinsic* curvature bounds. The novelty of our work is that we dispense with the requirement of an embedding and prove much more general results under *intrinsic* assumptions. We take the perspective of preserving the thrust of the harmonic analytic argument from the Euclidean context and as a consequence, we are forced to perform a detailed and intricate analysis of the geometry.
Our work utilises the foundation laid by Axelsson, Keith and McIntosh in [@AKMC], and the perspective developed in [@AKM2] by the same authors. The ideas in both these papers have their roots in the solution of the Kato conjecture by Auscher, Hofmann, Lacey, McIntosh and Tchamitchian in [@AHLMcT]. See also the surveys of Hofmann [@HofmannEx], McIntosh [@Mc90], the book by Auscher and Tchamitchian [@AT], and the recent survey [@HMc] by Hofmann and McIntosh.
The idea of the authors of [@AKMC] is to consider closed, densely defined, nilpotent, operators $\Gamma$, along with perturbations $B_1,B_2$ and then to establish quadratic estimates of the form $$\int_{0}^\infty {\left\| t \Pi_B(I + t^2 \Pi_B)^{-1}u \right\|}^2 \frac{dt}{t}
\simeq {\left\| u \right\|}^2$$ where $\Pi_B = \Gamma + B_1 {{\Gamma}^\ast} B_2.$ In [@AKM2], the authors illustrate that for *inhomogeneous* operators, it is enough to establish a certain *local* quadratic estimate, for which we need bounds on the integral from $0$ to $1$, since the integral from $1$ to $\infty$ is straightforward in this case. The proof of this quadratic estimate proceeds by reduction to a Carleson measure estimate.
The techniques developed in [@AHLMcT], [@AKMC] and [@Morris3] rely upon being able to take averages of functions over subsets, and defining constant vectors in key aspects of the proof. This is a primary obstruction to generalising these techniques to non-trivial bundles. To circumvent this obstacle, we formulate a condition which we call *generalised bounded geometry*. This condition captures a uniform locally Euclidean structure in the bundle. The existence of a dyadic decomposition (below a fixed scale) provides a decomposition of the manifold in a way that allows us to work on a fixed set of coordinates in the bundle. We can picture this decomposition of the bundle as a sort of abstract polygon - Euclidean regions separated by a boundary of null measure. Under the condition of generalised bounded geometry, and using this decomposition, we are then able to adapt the arguments of [@Morris3] and [@AKMC] in order to obtain a Kato square root estimate.
The intuition behind generalised bounded geometry is the existence of *harmonic coordinates* under Ricci bounds, along with a uniform lower bound on injectivity radius. We use this to show that the bundle of $(p,q)$ tensors satisfies generalised bounded geometry. As a consequence, we obtain a Kato square root estimate for tensors under a coercivity condition which is automatically satisfied for scalar-valued functions. We highlight the scalar-valued version as a central theorem of this paper.
\[Thm:Int:KatoFn\] Let $({\mathcal{M}},{\mathrm{g}})$ be a smooth, complete Riemannian manifold with ${\left|{{\rm Ric}}\right|} \leq C$ and $\operatorname{inj}(M) \geq \kappa > 0$. Suppose that the following ellipticity condition holds: there exist $\kappa_1, \kappa_2 > 0$ such that $$\begin{aligned}
\operatorname{Re}{\left\langle a u, u \right\rangle} &\geq \kappa_1 {\left\| u \right\|}^2
\qquad\text{and}\\
\operatorname{Re}({\left\langle A_{11} \nabla v, \nabla v \right\rangle} +{\left\langle A_{10} v, \nabla v \right\rangle}+{\left\langle A_{01} \nabla v, v \right\rangle}+{\left\langle A_{00} v, v \right\rangle} )&\geq \kappa_2 {\left\| v \right\|}_{{{{\rm W}^{1,2}_{\rm {}}}}}^2\end{aligned}$$ for all $u \in {{\rm L}^{2}_{\rm {}}}({\mathcal{M}})$ and $v \in {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{M}})$. Then, ${ {\mathcal{D}}}(\sqrt{{\mathrm{L}}_A}) = { {\mathcal{D}}}({{{\nabla}_{{{}}}}}) = {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{M}})$ and $\|{\sqrt{{\mathrm{L}}_A} u}\|\simeq {\left\| {{{\nabla}_{{{}}}}}u \right\|} + {\left\| u \right\|} = {\left\| u \right\|}_{{{{\rm W}^{1,2}_{\rm {}}}}}$ for all $u \in {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{M}})$.
We also prove Lipschitz estimates for small perturbations of our operators, similar to those in §6 of [@AKMC], which are a direct consequence of considering operators with *complex* bounded measurable coefficients.
In [@B], the first author proves quadratic estimates for operators $\Pi_B$ on doubling measure metric spaces under an appropriate set of assumptions. We conclude this paper by demonstrating how to extend the quadratic estimates which we obtain on a manifold, to the more general setting of a complete metric space equipped with a Borel-regular measure that is exponentially locally doubling.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors appreciate the support of the Centre for Mathematics and its Applications at the Australian National University, Canberra, where this project was undertaken. The first author was supported through an Australian Postgraduate Award and through the Mathematical Sciences Institute, Australian National University, Canberra. The second author gratefully acknowledges support from the Australian Government through the Australian Research Council.
The authors thank Andrew Morris for helpful conversations and suggestions.
Preliminaries {#prelim}
=============
Notation
--------
Throughout this paper, we use the Einstein summation convention. That is, whenever there is a repeated lowered index and a raised index (or conversely), we assume summation over that index. By ${\ensuremath{\mathbb{N}}}$ we denote natural numbers not including $0$ and we let ${\mathbb{Z}}_{+} = {\ensuremath{\mathbb{N}}}{\cup}{{\left\{0\right\}}}$. We denote the integers by ${\mathbb{Z}}$. We take the liberty to sometimes write $a \lesssim b$ for two real quantities $a$ and $b$. By this, we mean that there is a constant $C > 0$ such that $a \leq C b$. By $a \simeq b$ we mean that $a \lesssim b$ and $b \lesssim a$. For a function (and indeed, a section) $f$, we denote its support by ${{\rm spt} {\text{ }}f}$. We denote an open ball centred at $x$ of radius $r$ by $B(x,r)$. The radius of a ball $B$ (open or closed) is denoted by $\operatorname{rad}(B)$. For $\Omega \subset {\mathcal{M}}$, we denote the diameter of $\Omega$ by $\operatorname{diam}\Omega = \sup{{\left\{ d(x,y):\, x,y\in\Omega\right\}}}$.
Manifolds and vector bundles
----------------------------
In this section, we introduce terminology that allows us to describe the class of manifolds in which we obtain our results. Furthermore, we introduce the function spaces that we shall use. We also prove a key result that allows us to construct Sobolev spaces on vector bundles.
Let ${\mathcal{M}}$ be a smooth, complete Riemannian manifold with metric ${\mathrm{g}}$ and Levi-Civita connection ${{{\nabla}_{{{}}}}}$. Let $d\mu$ denote the volume measure induced by ${\mathrm{g}}$. We follow the notation of [@Morris3] in the following definition.
We say that ${\mathcal{M}}$ has *exponentially locally doubling volume growth* if there exists $c \geq 1,\ \kappa, \lambda \geq 0$ such that $$0 < \mu(B(x,tr)) \leq
c t^\kappa {\mathrm{e}}^{\lambda t r} \mu(B(x,r)) < \infty
\tag{$\text{E}_{\text{loc}}$}
\label{Def:Pre:Eloc}$$ for all $t \geq 1,\ r > 0$ and $x \in {\mathcal{M}}$.
Let ${\mathcal{V}}$ be a smooth vector bundle of rank $N$ with metric ${\mathrm{h}}$ and connection ${{{\nabla}_{{{}}}}}$. Let $\uppi_{{\mathcal{V}}}: {\mathcal{V}}\to {\mathcal{M}}$ denote the canonical projection. Since we are interested in considering “sections” which may have low regularity, we deviate from the usual definition of ${\mathbf{\Gamma}}({\mathcal{V}})$ by dropping the requirement that they be differentiable. More precisely, we write ${\mathbf{\Gamma}}({\mathcal{V}})$ to be the space of measurable functions $\omega:{\mathcal{M}}\to {\mathcal{V}}$ such that $\omega(x) \in \uppi_{{\mathcal{V}}}^{-1}(x)$. We impose no regularity assumptions (other than measurability) and we call these *sections* of ${\mathcal{V}}$. We write ${{\rm L}^{1}_{\rm loc}}({\mathcal{V}}) = {{\rm L}^{1}_{\rm loc}}({\mathcal{M}},{\mathcal{V}}) $ to denote the sections $\gamma \in {\mathbf{\Gamma}}({\mathcal{V}})$ such that $$\int_{K} {\left|\gamma(x)\right|}_{{\mathrm{h}}(x)}\ d\mu(x) < \infty$$ for each compact $K \subset {\mathcal{M}}$. Similarly, we define ${{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$. The spaces ${{\rm C}^{k}_{\rm {}}}({\mathcal{V}})$ are then the $k$-differentiable sections and ${{\rm C}^{k}_{\rm c}}({\mathcal{V}})$ are $k$-differentiable sections with compact support. The following proposition is necessary to define Sobolev spaces on ${\mathcal{V}}$.
\[Prop:Pre:ConnClosed\] The connection ${{{\nabla}_{{{}}}}}: {{\rm C}^{\infty}_{\rm {}}} {\cap}{{\rm L}^{2}_{\rm {}}}({\mathcal{V}})
\to {{\rm L}^{2}_{\rm {}}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}})$ is a densely defined, closable operator.
That ${{\rm C}^{\infty}_{\rm c}}({\mathcal{V}})$ is dense in ${{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$ is shown by a mollification argument after covering the manifold by countably many compact sets corresponding to coordinate charts. Since ${{\rm C}^{\infty}_{\rm c}}({\mathcal{V}}) \subset {{\rm C}^{\infty}_{\rm {}}} {\cap}{{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$, we have that ${{{\nabla}_{{{}}}}}$ is a densely defined operator.
We show that ${{{\nabla}_{{{}}}}}$ is a closable operator by reduction to the well known fact that ${{{\nabla}_{{{}}}}}= {{\rm d}}$ is closable on scalar-valued functions. Fix $u_n \in {{\rm C}^{\infty}_{\rm {}}} {\cap}{{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$ such that $u_n \to 0$ and ${{{\nabla}_{{{}}}}}u_n \to v$. For each $x \in {\mathcal{M}}$, we can find an open set $K_x$ such that ${\overline{K_x}}$ is compact and with a local trivialisation $\psi_x : {\overline{K_x}} \times {\mathbb{C}}^N \to \uppi_{{\mathcal{V}}}^{-1}({\overline{K_x}})$. Let ${{\left\{K_x\right\}}}$ be a collection of such sets and let ${{\left\{K_j\right\}}}$ denote a countable subcover. Now, fix $K_j$ and let ${{\left\{e^i\right\}}}$ denote an orthonormal frame (by a Gram-Schmidt procedure) in ${\overline{K_j}}$. Write $u_n = (u_n)_i e^i$ and note that $${{{\nabla}_{{{}}}}}u_n = {{{\nabla}_{{{}}}}}(u_n)_i {\otimes}e^i + (u_n)_i {{{\nabla}_{{{}}}}}e^i$$ and therefore, $$\sum_{i} {\left|{{{\nabla}_{{{}}}}}(u_n)_i - v_i\right|}^2
= {\left|{{{\nabla}_{{{}}}}}(u_n)_i {\otimes}e^i - v\right|}^2
\leq {\left|{{{\nabla}_{{{}}}}}u_n - v\right|}^2 + \sup_{l} {\left|{{{\nabla}_{{{}}}}}e^l\right|}^2 {\left|u_n\right|}^2.$$ Since by assumption ${\overline{K_i}}$ is compact and the basis $e^i$ is smooth, we have that $\sup_{l} {\left|{{{\nabla}_{{{}}}}}e^l\right|}^2 \leq C_j$ for some $C_j$ dependent on $K_j$. Thus, $${\left\| {{{\nabla}_{{{}}}}}(u_n)_i - v_i \right\|}^2_{{{\rm L}^{2}_{\rm {}}}(K_j)} \to 0.$$ Thus, we have that $(u_n)_i \to 0$ and ${{{\nabla}_{{{}}}}}(u_n)_i \to v_i$ in ${{\rm L}^{2}_{\rm {}}}(K_j)$, and by the closability of ${{{\nabla}_{{{}}}}}$ on functions, we have that $v_i = 0$ almost everywhere in $K_j$. Thus, $v = 0$ almost everywhere in $K_j$ and consequently $v = 0$ almost everywhere in ${\mathcal{M}}$. This proves that ${{{\nabla}_{{{}}}}}$ is a closable operator.
As a consequence of this proposition, we can define the Sobolev space ${{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}})$ as the completion of functions ${{\rm C}^{\infty}_{\rm {}}} {\cap}{{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$ in the graph norm of ${{{\nabla}_{{{}}}}}$. The closure of ${{{\nabla}_{{{}}}}}$ is then denoted by the same symbol, namely ${{{\nabla}_{{{}}}}}: {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}}) \subset {{\rm L}^{2}_{\rm {}}}({\mathcal{V}}) \to {{\rm L}^{2}_{\rm {}}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}})$.
The higher order Sobolev spaces ${{{\rm W}^{k,2}_{\rm {}}}}({\mathcal{V}})$ are defined as subsets of ${{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}})$ in the usual manner. To keep some accord with tradition, when we consider the situation ${\mathcal{V}}= {\mathbb{C}}$, we write ${{\rm L}^{2}_{\rm {}}}({\mathcal{M}})$ in place of ${{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$ and similarly for the function spaces ${{\rm C}^{k}_{\rm {}}},\ {{\rm C}^{k}_{\rm c}}$ and ${{{\rm W}^{k,2}_{\rm {}}}}$.
We also highlight that we have the following important density result.
${{\rm C}^{\infty}_{\rm c}}({\mathcal{V}})$ is dense in ${{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}})$.
Since we assume that ${\mathcal{M}}$ is complete, the Hopf-Rinow theorem guarantees that closed balls are compact. Fix some centre $x \in {\mathcal{M}}$ and consider closed balls ${\overline{B(x,r)}}$. For such balls, we can find $\eta_r \in {{\rm C}^{\infty}_{\rm c}}({\mathcal{V}})$ such that ${{\rm spt} {\text{ }}\eta}_r$ is compact, $0 \leq \eta \leq 1$, $\eta_r \equiv 1$ on $B(x,r)$ and $\sup {\left|{{{\nabla}_{{{}}}}}\eta_r\right|} \to 0$ as $r \to \infty$. Now, for any $u \in {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}})$, $\eta_r u$ is compactly supported. Also, $${{{\nabla}_{{{}}}}}(\eta_r u) = {{{\nabla}_{{{}}}}}\eta_r {\otimes}u + \eta_r {{{\nabla}_{{{}}}}}u$$ and $${\left\| {{{\nabla}_{{{}}}}}(\eta_r u) - {{{\nabla}_{{{}}}}}u \right\|}
\leq {\left\| {{{\nabla}_{{{}}}}}\eta_r {\otimes}u \right\|}
+ {\left\| \eta_r {{{\nabla}_{{{}}}}}u - {{{\nabla}_{{{}}}}}u \right\|}
\leq \sup_{x \in {\mathcal{M}}} {\left|{{{\nabla}_{{{}}}}}\eta_r\right|} {\left\| u \right\|}
+ {\left\| \eta_r {{{\nabla}_{{{}}}}}u - {{{\nabla}_{{{}}}}}u \right\|} \to 0$$ as $r \to \infty$. Thus, it suffices to consider $u \in {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}})$ with ${{\rm spt} {\text{ }}u}$ compact. But for each such $u$, an easy mollification argument in each local trivialisation will yield a sequence $u_n \in {{\rm C}^{\infty}_{\rm c}}({\mathcal{V}})$ such that ${\left\| u_n - u \right\|}_{{{{\rm W}^{1,2}_{\rm {}}}}} \to 0$.
We remark that this proof does not generalise to higher order Sobolev spaces. In fact, even for $k = 2$, the best known result for the density of ${{\rm C}^{\infty}_{\rm c}}({\mathcal{M}})$ in ${{{\rm W}^{2,2}_{\rm {}}}}({\mathcal{M}})$ is under uniform lower bounds on injectivity radius along with uniform lower bounds on Ricci curvature. See §3 of [@Hebey] and in particular, Proposition 3.3 in [@Hebey].
For $f \in {{\rm L}^{1}_{\rm loc}}({\mathcal{M}})$ (i.e. a function), we define the *average* of $f$ on some measurable subset $A \subset {\mathcal{M}}$ with $0 < \mu(A) < \infty$ by $ f_A = \fint_{A} f\ d\mu = \frac{1}{\mu(A)} \int_{A} f\ d\mu.$
In what follows, we assume that ${\mathcal{M}}$ satisfies (\[Def:Pre:Eloc\]) unless stated otherwise.
Generalised Bounded Geometry {#Sect:Prelim:GBG}
----------------------------
The harmonic analytic techniques we employ in the main proof, along with some of the assumptions we make on the operators under consideration, require us to capture a uniform locally Euclidean structure in the underlying vector bundle. The concept which we describe here is motivated by the fact that injectivity radius bounds on a manifold, coupled with appropriate curvature bounds, give *bounded geometry* on the $(p,q)$ tensor bundle. It provides us with the framework for applying a dyadic decomposition in order to construct a system of global coordinates (no longer smooth), thus allowing us to define key quantities and to construct proofs.
Recalling that ${\mathcal{V}}$ is equipped with a metric ${\mathrm{h}}$, we make the following definition.
\[GBG\] \[Def:Pre:GBG\] Suppose there exists ${\rho}> 0$, $C \geq 1$, such that for each $x \in {\mathcal{M}}$, there exists a local trivialisation $\psi_x: B(x,{\rho}) \times {\mathbb{C}}^N \to \uppi_{{\mathcal{V}}}^{-1}(B(x,{\rho}))$ satisfying $$C^{-1} I \leq {\mathrm{h}}\leq C I$$ in the basis ${{\left\{e^i = \psi_x (y,\tilde{e}^i)\right\}}}$, where ${{\left\{\tilde{e}^i\right\}}}$ is the standard orthonormal basis for ${\mathbb{C}}^N$. Then, we say that ${\mathcal{V}}$ has *generalised bounded geometry* or *GBG*. We call ${\rho}$ the *GBG radius*, and local trivialisations $\psi_x$ the *GBG charts*.
Since we can always take ${\rho}$ to be as small as we like, we assume that ${\rho}\leq 5$.
For the convenience of the reader we quote Proposition 4.2 from [@Morris3].
\[Thm:Dya:Christ\] There exists a countable collection of open subsets ${{\left\{{Q_{\alpha}}^k \subset {\mathcal{M}}: \alpha \in I_k,\ k \in {\mathbb{Z}}_{+}\right\}}}$, $z_\alpha^k \in {Q_{\alpha}}^k$ (called the *centre* of ${Q_{\alpha}}^k$), index sets $I_k$ (possibly finite), and constants $\delta \in (0,1)$, $a_0 > 0$, $\eta > 0$ and $C_1, C_2 < \infty$ satisfying:
(i) for all $k \in {\mathbb{Z}}$, $\mu({\mathcal{M}}\setminus {\cup}_{\alpha} {Q_{\alpha}}^k) = 0$,
(ii) if $l \geq k$, then either ${Q_{\beta}}^l \subset {Q_{\alpha}}^k$ or ${Q_{\beta}}^l {\cap}{Q_{\alpha}}^k = {\varnothing}$,
(iii) for each $(k,\alpha)$ and each $l < k$ there exists a unique $\beta$ such that ${Q_{\alpha}}^k \subset {Q_{\beta}}^l$,
(iv) $\operatorname{diam}{Q_{\alpha}}^k \leq C_1 \delta^k$,
(v) $ B(z_\alpha^k, a_0 \delta^k) \subset {Q_{\alpha}}^k$,
(vi) for all $k, \alpha$ and for all $t > 0$, $\mu{{\left\{x \in {Q_{\alpha}}^k: d(x, {\mathcal{M}}\setminus{Q_{\alpha}}^k) \leq t \delta^k\right\}}} \leq C_2 t^\eta \mu({Q_{\alpha}}^k).$
This theorem was first proved by Christ in [@Christ] for $k\in{\mathbb{Z}}$, for doubling measure metric spaces.
Throughout this paper we fix ${\mathrm{J}}\in {\ensuremath{\mathbb{N}}}$ such that $C_1 \delta^{\mathrm{J}}\leq \frac{{\rho}}{5}$. For $j \geq {\mathrm{J}}$, we write ${{\mathscr{Q}}}^j$ to denote the collection of all cubes ${Q_{\alpha}}^j$, and set ${{\mathscr{Q}}}= {\cup}_{j \geq {\mathrm{J}}} {{\mathscr{Q}}}^j$.
We call ${\mathrm{t_S}}= \delta^{{\mathrm{J}}}$ the *scale*. Furthermore, when $t\leq {\mathrm{t_S}}$, set ${{\mathscr{Q}}}_t = {{\mathscr{Q}}}^j$ whenever $\delta^{j+1} < t \leq \delta^{j}$.
The existence of a truncated dyadic structure allows us to formulate the following system of coordinates on ${\mathcal{V}}$.
Let $x_{{Q_{{}}}}$ denote the *centre* of each ${Q_{{}}}\in {{\mathscr{Q}}}^{\mathrm{J}}$. Then, we call the system of coordinates $${{\mathscr{C}}}= {{\left\{\psi:B(x_{{Q_{{}}}},{\rho}) \times {\mathbb{C}}^N \to \uppi_{{\mathcal{V}}}^{-1}(B(x_{{Q_{{}}}},{\rho}))\ \text{s.t.}\ {Q_{{}}}\in {{\mathscr{Q}}}^{\mathrm{J}}\right\}}}$$ the *GBG coordinates*. We call the set $${{\mathscr{C}}}_{{\mathrm{J}}} = {{\left\{\psi_{{Q_{{}}}} = \psi{{{\lvert_{}}_{}}_{{Q_{{}}}}}: {Q_{{}}}\times {\mathbb{C}}^N \to \uppi_{{\mathcal{V}}}^{-1}({Q_{{}}})\ \text{s.t.}\ {Q_{{}}}\in {{\mathscr{Q}}}^{\mathrm{J}}\right\}}}$$ the *dyadic GBG coordinates*. For any cube $Q \in {{\mathscr{Q}}}^j$, there is a unique cube ${\widehat{{Q_{{}}}}} \in {{\mathscr{Q}}}^{{\mathrm{J}}}$ satisfying ${Q_{{}}}\subset {\widehat{{Q_{{}}}}}$ and we call this cube the *GBG cube of ${Q_{{}}}$*. The *GBG coordinate system* of ${Q_{{}}}$ is then $\psi:B(x_{{\widehat{{Q_{{}}}}}},{\rho}) \times {\mathbb{C}}^N \to \uppi_{{\mathcal{V}}}^{-1}(B(x_{{\widehat{{Q_{{}}}}}},{\rho})$.
Since ${\cup}{{\mathscr{Q}}}^J$ is an almost-everywhere covering of ${\mathcal{M}}$, the GBG coordinate system of ${\mathcal{V}}$ defines an almost-everywhere smooth global trivialisation of the vector bundle.
We emphasise that throughout this paper, for any cube ${Q_{{}}}\in {{\mathscr{Q}}}^j$, we denote its GBG cube by ${\widehat{{Q_{{}}}}}$.
A first and important consequence of the GBG coordinates is that it allows us to define the notion of a *constant* section (in the eyes of our GBG coordinates). More precisely, given $w \in {\mathbb{C}}^N$, we define $\omega(x) = w$ whenever $x \in {Q_{{}}}\in {{\mathscr{Q}}}^J ={{\mathscr{Q}}}_{{\mathrm{t_S}}}$ in the GBG coordinates associated to ${Q_{{}}}$. Thus, we call $\omega$ the GBG constant section associated to $w$. Note that $\omega\in {{\rm L}^{\infty}_{\rm {}}}({\mathcal{V}})$, and that although $\omega$ corresponds to a constant vector $w \in {\mathbb{C}}^N$, $\omega$ typically has discontinuities across the boundaries of sets ${Q_{{}}}\in {{\mathscr{Q}}}^J$. We remark that this notion is crucial in later parts of the paper. Next, we can define a notion of *cube integration* in the following way. Let $u \in {{\rm L}^{1}_{\rm loc}}({\mathcal{V}})$. Then, for any ${Q_{{}}}\in {{\mathscr{Q}}}$, we can write $$\int_{{Q_{{}}}} u\ d\mu = {\left(\int_{{Q_{{}}}} u_i\ d\mu\right)} e^i$$ where $u = u_i e^i$ in the GBG coordinates of ${Q_{{}}}$. Note that $ \int_{{Q_{{}}}} u\ d\mu$ is a function from $Q$ to ${\mathcal{V}}$.
Pursuing a similar vein of thought, we define a *cube average*. Given a cube ${Q_{{}}}\in {{\mathscr{Q}}}$ and $u \in {{\rm L}^{1}_{\rm loc}}({\mathcal{V}})$, define $u_{{Q_{{}}}}\in {{\rm L}^{\infty}_{\rm {}}}({\mathcal{V}})$ by $$u_{{Q_{{}}}}(y) = \begin{cases}
\tfrac{1}{\mu({Q_{{}}})} \int_{{Q_{{}}}}u\ d\mu &\text{when } y\in B(x_{{\widehat{{Q_{{}}}}}},{\rho}),\\
0 &\text{otherwise}.
\end{cases}$$
We remark that in the average of a function over a general measurable set of positive measure, we do not perform a cutoff as we do here. However, whenever we write $u_{{Q_{{}}}}$ with ${Q_{{}}}$ being a cube (even for a function), we shall always assume this definition.
Lastly, we define the *dyadic averaging operator*. For each $t > 0$, define the operator ${\mathcal{A}}_t: {{\rm L}^{1}_{\rm loc}}({\mathcal{V}}) \to {{\rm L}^{1}_{\rm loc}}({\mathcal{V}})$ by $${\mathcal{A}}_t u (x) = \frac{1}{\mu({Q_{{}}})} \int_{{Q_{{}}}} u(y)\ d\mu(y)$$ whenever $x \in {Q_{{}}}\in {{\mathscr{Q}}}_t$. We remark that ${\mathcal{A}}_t: {{\rm L}^{2}_{\rm {}}}({\mathcal{V}}) \to {{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$ is a bounded operator with norm bounded uniformly for all $t \leq {\mathrm{t_S}}$ by a bound depending on the constant $C$ arising in the GBG criterion.
Functional calculi of sectorial operators
-----------------------------------------
Of fundamental importance to the setup and proof that we present in this paper, is the functional calculus of certain operators. In this section, we introduce the key type of operators that we shall concern ourselves with, and, for the convenience of the reader, recall some facts about functional calculi of these operators. A fuller treatment of this material can be found in [@AMX]. A local version of this theory can be found in [@Morris2].
Let ${{\mathscr{B}}}_1, {{\mathscr{B}}}_2$ be Banach spaces. We say that a linear map $T:{ {\mathcal{D}}}(T) \subset {{\mathscr{B}}}_1 \to {{\mathscr{B}}}_2$ is an *operator* with domain ${ {\mathcal{D}}}(T)$. The range of $T$ is denoted by ${ {\mathcal{R}}}(T)$ and the null space by ${ {\mathcal{N}}}(T)$. We say that such an operator is *closed* if the graph ${{\mathscr{G}}}(T) = {{\left\{(u, Tu): u \in { {\mathcal{D}}}(T)\right\}}}$ is closed in the product topology of ${{\mathscr{B}}}_1 \times {{\mathscr{B}}}_2$. The operator $T$ is *bounded* if ${\left\| Tu \right\|}_{{{\mathscr{B}}}_2} \leq {\left\| u \right\|}_{{{\mathscr{B}}}_1}$ for all $u \in { {\mathcal{D}}}(T) = {{\mathscr{B}}}_1$. If ${{\mathscr{B}}}_1={{\mathscr{B}}}_2$ then the *resolvent* set ${\rho}(T)$ consists of all $\zeta \in {\mathbb{C}}$ such that $\zeta I - T$ is one-one, onto and has a bounded inverse. The spectrum is then ${\sigma}(T) = {\mathbb{C}}\setminus {\rho}(T)$.
For $0 \leq \omega < \frac{\pi}{2}$, define the bisector $S_{\omega} = {{\left\{\zeta \in {\mathbb{C}}: {\left|\arg \zeta\right|} \leq \omega\ \text{or}\ {\left|\arg (-\zeta)\right|} \leq \omega\ \text{or}\ \zeta = 0\right\}}}$. The open bisector is then defined as $S_{\omega}^o = {{\left\{\zeta \in {\mathbb{C}}: {\left|\arg \zeta\right|} < \omega\ \text{or}\ {\left|\arg (-\zeta)\right|} < \omega,\ \zeta \neq 0\right\}}}$. An operator $T$ in a Banach space ${{\mathscr{B}}}$ is *$\omega$-bisectorial* if it is closed, ${\sigma}(T) \subset S_{\omega}$, and the following *resolvent bounds* hold: for each $\omega < \mu < \frac{\pi}{2}$, there exists $C_{\mu}$ such that for all $\zeta \in {\mathbb{C}}\setminus S_\mu$, we have ${\left|\zeta\right|} {\left\| (\zeta I - T)^{-1} \right\|} \leq C_{\mu}$. Depending on context, we refer to such operators simply as bisectorial and $\omega$ is then the *angle of bisectoriality*.
Now fix $0 \leq \omega < \mu < \frac{\pi}{2}$ and assume that $T$ is an $\omega$-bisectorial operator in a Hilbert space ${{\mathscr{H}}}$. We let $\Psi(S_{\mu}^o)$ denote the space of all holomorphic functions $\psi: S_{\mu}^o \to {\mathbb{C}}$ for which $${\left|\psi(\zeta)\right|} \lesssim \frac{{\left|\zeta\right|}^{\alpha}}{1 + {\left|\zeta\right|}^{2\alpha}}$$ for some $\alpha > 0$. For such functions, we define a functional calculus similar to the Riesz-Dunford functional calculus by $$\psi(T) = \frac{1}{2\pi\imath} \oint_{\gamma} \psi(\zeta)(\zeta I - T)^{-1}\ d\zeta$$ where $\gamma$ is a contour in $S_{\mu}^o$ enveloping $S_{\omega}$ parametrised anti-clockwise, and the integral is defined via Riemann sums. This integral converges absolutely as a consequence of the decay of $\psi$, coupled with the resolvent bounds of $T$. The operator $\psi(T)$ is bounded. We say that $T$ has a *bounded holomorphic functional calculus* if there exists $C > 0$ such that ${\left\| \psi(T) \right\|} \leq C {\left\| \psi \right\|}_\infty$ for all $\psi \in \Psi(S_{\mu}^o)$.
Now suppose that ${{\mathscr{H}}}$ is a Hilbert space, and that $T:{ {\mathcal{D}}}(T)\subset {{\mathscr{H}}}\to {{\mathscr{H}}}$ is $\omega$-bisectorial. In this case ${{\mathscr{H}}}= { {\mathcal{N}}}(T)\oplus \overline{{ {\mathcal{R}}}(T)}$ where the direct sum is typically non-orthogonal. Let $\operatorname{\mathbf{P}}_{{ {\mathcal{N}}}(T)}$ denote the projection of ${{\mathscr{H}}}$ onto ${ {\mathcal{N}}}(T)$ which is $0$ on ${ {\mathcal{R}}}(T)$. Let ${{\rm Hol}}^\infty(S_{\mu}^o)$ denote the space of all bounded functions $f: S_{\mu}^o {\cup}{{\left\{0\right\}}} \to {\mathbb{C}}$ which are holomorphic on $S_{\mu}^o$. For such a function $f$, there exists a uniformly bounded sequence $(\psi_n)$ of functions in $\Psi(S_{\mu}^o)$ which converges to $f{{{\lvert_{}}_{}}_{S_{\mu}^o}}$ in the compact-open topology over $S_{\mu}^o$. If $T$ has a bounded holomorphic functional calculus, and $u\in{{\mathscr{H}}}$, then the limit $\lim_{n\to\infty} \psi_n(T)u$ exists in ${{\mathscr{H}}}$, and we define $$f(T)u = f(0)\operatorname{\mathbf{P}}_{{ {\mathcal{N}}}(T)}u + \lim_{n\to\infty} \psi_n(T)u\ .$$ This defines a bounded operator independent of the sequence $(\psi_n)$, and we have the bound ${\left\| f(T) \right\|} \leq C {\left\| f \right\|}_\infty$ for some finite $C$.
In particular the functions $\chi^{+}, \chi^{-}$ and $\operatorname{sgn}=\chi^{+}-\chi^{-}$ belong to ${{\rm Hol}}^\infty(S_{\mu}^o)$, where $\chi^{+}(\zeta) = 1$ when $\operatorname{Re}(\zeta) > 0$ and $0$ otherwise, and $\chi^{-}(\zeta) = 1$ when $\operatorname{Re}(\zeta) < 0$ and $0$ otherwise. Therefore, if $T$ has a bounded functional calculus in ${{\mathscr{H}}}$, then $\chi^{\pm}(T)$ are bounded projections, and ${{\mathscr{H}}}= { {\mathcal{N}}}(T) \oplus { {\mathcal{R}}}(\chi^{+}(T))\oplus { {\mathcal{R}}}(\chi^{-}(T))$. The bounded operator $\operatorname{sgn}(T)$ equals 0 on ${ {\mathcal{N}}}(T)$, equals 1 on ${ {\mathcal{R}}}(\chi^{+}(T))$, and equals $-1$ on ${ {\mathcal{R}}}(\chi^{-}(T))$.
Hypotheses {#Sect:Ass}
==========
Though our primary goal in this paper is to provide, under an appropriate set of assumptions, a positive answer to the Kato square root problem on vector bundles, we shall pursue a slightly more general setup in the footsteps of [@AKMC]. The purpose of this section is to describe this setup as a list of hypotheses, and in later sections, demonstrate how to apply tools from harmonic analysis to prove quadratic estimates under these hypotheses. The Kato square root estimate on vector bundles (and $(p,q)$ tensors) will then be obtained by showing that these hypotheses are satisfied under our geometric assumptions.
Let ${{\mathscr{H}}}$ be a Hilbert space and let ${\left\langle {\cdotp},{\cdotp}\right\rangle}$ denote its inner product. Following [@AKMC] and [@Morris3], we make the following operator theoretic assumptions.
1. The operator $\Gamma: { {\mathcal{D}}}(\Gamma) \subset {{\mathscr{H}}}\to {{\mathscr{H}}}$ is closed, densely defined and *nilpotent*.
2. The operators $B_1, B_2 \in {\mathcal{L}}({{\mathscr{H}}})$ satisfy $$\begin{aligned}
&\operatorname{Re}{\left\langle B_1u,u \right\rangle} \geq \kappa_1 {\left\| u \right\|}^2 &&\text{whenever }u \in { {\mathcal{R}}}({{\Gamma}^\ast}), \\
&\operatorname{Re}{\left\langle B_2u,u \right\rangle} \geq \kappa_2 {\left\| u \right\|}^2 &&\text{whenever }u \in { {\mathcal{R}}}(\Gamma),
\end{aligned}$$ where $\kappa_1, \kappa_2 > 0$ are constants.
3. The operators $B_1, B_2$ satisfy $B_1B_2 ({ {\mathcal{R}}}(\Gamma)) \subset { {\mathcal{N}}}(\Gamma)$ and $B_2B_1({ {\mathcal{R}}}({{\Gamma}^\ast})) \subset { {\mathcal{N}}}({{\Gamma}^\ast})$.
Furthermore, define ${{\Gamma}^\ast}_B = B_1 {{\Gamma}^\ast} B_2$, $\Pi_B = \Gamma + {{\Gamma}^\ast}_B$ and $\Pi = \Gamma + {{\Gamma}^\ast}$. The operator $\Pi$ is self-adjoint, and $\Pi_B$ is bisectorial, and thus we define the following associated bounded operators: $$R_t^B = (1 + it \Pi_B)^{-1},\
P_t^B = (1 + t^2 \Pi_B^2)^{-1},\
Q_t^B = t\Pi_B (1 + t^2 \Pi_B^2)^{-1},\
\Theta_t^B = t {{\Gamma}^\ast}_B (1 + t^2 \Pi_B^2)^{-1}.$$ We write $R_t, P_t, Q_t,
\Theta_t$ on setting $B_1 = B_2 = 1$. The full implications of these assumptions are listed in §4 of [@AKMC].
The following additional assumptions are mild, particularly since we wish to apply the theory to differential operators. They are essentially the same as the assumptions made in [@AKMC] and [@Morris3], but modified for vector bundles. We remark that in (H5) below, ${\mathcal{L}}({\mathcal{V}})$ denotes the vector bundle of all bounded linear maps $T_x:{\mathcal{V}}_x \to {\mathcal{V}}_x$ for each $x \in {\mathcal{M}}$ (where ${\mathcal{V}}_x$ is the fibre over $x$). The boundedness is with respect to the metric ${\mathrm{h}}_x$ on the fibre ${\mathcal{V}}_x$. Note that the local trivialisations for this bundle are canonically induced by the local trivialisations of ${\mathcal{V}}$, and in each local trivialisation the $T_x$ can be represented as usual by an $N \times N$ matrix.
1. The Hilbert space ${{\mathscr{H}}}= {{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$, where ${\mathcal{V}}$ is a smooth vector bundle with smooth metric ${\mathrm{h}}$ over a smooth, complete Riemannian manifold ${\mathcal{M}}$ with smooth metric ${\mathrm{g}}$. Furthermore, ${\mathcal{V}}$ satisfies the GBG criterion and ${\mathcal{M}}$ satisfies (\[Def:Pre:Eloc\]).
2. The operators $B_1, B_2$ are multiplication operators, i.e. there exist $B_i \in {{\rm L}^{\infty}_{\rm {}}}({\mathcal{L}}({\mathcal{V}}))$.
3. The operator $\Gamma$ is a first order differential operator. That is, there exists $C_{\Gamma} > 0$ such that whenever $\eta \in {{\rm C}^{\infty}_{\rm c}}({\mathcal{M}})$, we have that $\eta { {\mathcal{D}}}(\Gamma) \subset { {\mathcal{D}}}(\Gamma)$ and ${{\mathrm{M}}}_{\eta}u(x) = {{\left[\Gamma,\eta(x)I\right]}}u(x)$ is a multiplication operator satisfying $${\left|{{\mathrm{M}}}_{\eta}u(x)\right|} \leq C_{\Gamma} {\left|{{{\nabla}_{{{}}}}}\eta\right|}_{{{\rm T}^\ast}M} {\left|u(x)\right|}$$ for all $u \in { {\mathcal{D}}}(\Gamma)$ and almost all $x \in {\mathcal{M}}$.
We note as in [@Morris3] that (H6) implies the same hypothesis with $\Gamma$ replaced by either ${{\Gamma}^\ast}$ or $\Pi$.
It is in the following two hypotheses that we make a more substantial departure from [@AKMC] and [@Morris3]. A significant difference is that we have used the dyadic structure, rather than balls, in their formulation. This cannot be avoided since we are forced to employ quantities which are defined through GBG coordinates.
Recalling the definition of a cube integral in §\[Sect:Prelim:GBG\], we formulate the following *cancellation* hypothesis.
1. There exists $c > 0$ such that for all ${Q_{{}}}\in {{\mathscr{Q}}}$, $${\left|\int_{{Q_{{}}}} \Gamma u\ d\mu\right|}
\leq c \mu({Q_{{}}})^{\frac{1}{2}} {\left\| u \right\|}
\quad\text{and}\quad
{\left|\int_{{Q_{{}}}} {{\Gamma}^\ast} v\ d\mu\right|}
\leq c \mu({Q_{{}}})^{\frac{1}{2}} {\left\| v \right\|}$$ for all $u \in { {\mathcal{D}}}(\Gamma),\ v\in { {\mathcal{D}}}({{\Gamma}^\ast})$ satisfying ${{\rm spt} {\text{ }}u},\ {{\rm spt} {\text{ }}v} \subset {Q_{{}}}$.
Lastly, we make the following abstract [Poincaré ]{}and coercivity hypotheses on the bundle (recalling that ${{\mathscr{Q}}}_t = {{\mathscr{Q}}}^j$ whenever $\delta^{j+1} < t \leq \delta^{j}$).
1. There exist $C_P,C_C, c, \tilde{c} > 0$ and an operator $\Xi: { {\mathcal{D}}}(\Xi) \subset {{\rm L}^{2}_{\rm {}}}({\mathcal{V}}) \to {{\rm L}^{2}_{\rm {}}}({{\mathscr{N}}})$, where ${{\mathscr{N}}}$ is a normed bundle over ${\mathcal{M}}$ with norm ${\left|{\cdotp}\right|}_{{{\mathscr{N}}}}$ and ${ {\mathcal{D}}}(\Pi) {\cap}{ {\mathcal{R}}}(\Pi) \subset { {\mathcal{D}}}(\Xi)$, satisfying:
1. (Dyadic [Poincaré ]{}) $$\int_{B} {\left|u - u_{{Q_{{}}}}\right|}^2\ d\mu
\leq C_P(1 + r^\kappa e^{\lambda c rt})
(rt)^2
\int_{\tilde{c}B} {\left({\left|\Xi u\right|}_{{{\mathscr{N}}}}^2 + {\left|u\right|}^2\right)}\ d\mu$$ for all balls $B = B(x_{{Q_{{}}}},rt)$ with $r \geq C_1/\delta$ where ${Q_{{}}}\in {{\mathscr{Q}}}_t$ with $t \leq {\mathrm{t_S}}$, and
2. (Coercivity) $${\left\| \Xi u \right\|}_{{{\rm L}^{2}_{\rm {}}}({{\mathscr{N}}})}^2 + {\left\| u \right\|}_{{{\rm L}^{2}_{\rm {}}}({\mathcal{V}})}^2
\leq C_C {\left\| \Pi u \right\|}_{{{\rm L}^{2}_{\rm {}}}({\mathcal{V}})}^2.$$
for all $u \in { {\mathcal{D}}}(\Pi){\cap}{ {\mathcal{R}}}(\Pi)$.
We justify calling this an abstract [Poincaré ]{}inequality for two reasons. First, the inequality is for sections on a vector bundle, not just for scalar-valued functions. Second, in the usual [Poincaré ]{}inequality, the operator $\Xi$ is simply ${{{\nabla}_{{{}}}}}$. We allow ourselves other possibilities in choosing the operator $\Xi$ here, because it can be useful in the situation of a vector bundle that is in general non-flat and non-trivial.
Main Results {#Sect:Res}
============
Bounded holomorphic functional calculi and Kato square root type estimates
--------------------------------------------------------------------------
In this section we to first illustrate how to reduce the main quadratic estimate to a simpler, local quadratic estimate. Then, we present the main theorem of this paper and illustrate its main corollary; a Kato square root type estimate.
We begin with the following adaptation of Proposition 5.2 in [@Morris3].
\[Prop:Ass:Main\] Suppose that $(\Gamma, B_1, B_2)$ satisfies the hypotheses (H1)-(H3) along with ${\left\| u \right\|} \lesssim {\left\| \Pi u \right\|}$ for $u \in { {\mathcal{R}}}(\Pi)$, and that there exists $c > 0$ and $t_0 > 0$ such that $$\label{quad} \int_{0}^{t_0} {\left\| \Theta_t^B P_t u \right\|}^2\ \frac{dt}{t}
\leq c {\left\| u \right\|}^2 \tag{Q1}$$ for all $u \in { {\mathcal{R}}}(\Gamma)$, together with three similar estimates obtained by replacing $(\Gamma, B_1, B_2)$ by $({{\Gamma}^\ast},B_2,B_1)$, $({{\Gamma}^\ast},{{B_2}^\ast}, {{B_1}^\ast})$ and $(\Gamma, {{B_1}^\ast}, {{B_2}^\ast})$. Then, $\Pi_B$ satisfies $$\int_{0}^\infty {\left\| Q_t^B u \right\|}^2\ \frac{dt}{t}
\simeq {\left\| u \right\|}^2$$ for all $u \in {\overline{{ {\mathcal{R}}}(\Pi_B)}} \subset {{\mathscr{H}}}$. Thus, $\Pi_B$ has a bounded holomorphic functional calculus.
The proof is similar to the proof of Proposition 5.2 in [@Morris3]. The assumption that ${\left\| u \right\|} \lesssim {\left\| \Pi u \right\|}$ allows us to handle the integral from $t_0$ to $\infty$.
We use the entire list of hypotheses (H1)-(H8) in §\[Sect:Ass\] to show that the assumptions of Proposition \[Prop:Ass:Main\] are satisfied. Thus, this yields the main theorem of this paper.
\[Thm:Ass:Main\] Suppose that ${\mathcal{M}}$, ${\mathcal{V}}$ and $(\Gamma,B_1,B_2)$ satisfy (H1)-(H8). Then, $\Pi_B$ satisfies the quadratic estimate $$\int_{0}^\infty {\left\| Q_t^B u \right\|}^2\ \frac{dt}{t}
\simeq {\left\| u \right\|}^2$$ for all $u \in {\overline{{ {\mathcal{R}}}(\Pi_B)}} \subset {{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$ and hence has a bounded holomorphic functional calculus.
We defer the proof to §\[Sec:Harm\].
In particular, the projections $\chi^{\pm}(\Pi_B)$ are bounded, as is the operator $\operatorname{sgn}(\Pi_B)= \chi^{+}(\Pi_B) - \chi^{-}(\Pi_B)$. Thus we have the following corollary.
Under the hypotheses of Theorem \[Thm:Ass:Main\], \[Cor:Ass:Main\]
(i) there is a spectral decomposition $${{\rm L}^{2}_{\rm {}}}({\mathcal{V}}) = { {\mathcal{N}}}(\Pi_B) \oplus E_B^{+} \oplus E_B^{-}$$ where $E_B^{\pm} = { {\mathcal{R}}}(\chi^{\pm}(\Pi_B))$ (the direct sum is in general non-orthogonal), and
(ii) ${ {\mathcal{D}}}(\Gamma) {\cap}{ {\mathcal{D}}}({{\Gamma}^\ast}_B) = { {\mathcal{D}}}(\Pi_B) = { {\mathcal{D}}}(\sqrt{{\Pi_B}^2})$ with $${\left\| \Gamma u \right\|} + {\left\| {{\Gamma}^\ast}_B u \right\|}
\simeq {\left\| \Pi_B u \right\|}
\simeq \|\sqrt{{\Pi_B}^2}u\|$$ for all $u \in { {\mathcal{D}}}(\Pi_B)$.
To prove part (ii), we use the identities $ \Pi_B u = \operatorname{sgn}(\Pi_B)\sqrt{{\Pi_B}^2}u$ and $\sqrt{{\Pi_B}^2}u = \operatorname{sgn}(\Pi_B) \Pi_B u$ for all $u \in { {\mathcal{D}}}(\Pi_B)$, together with the bound on $\operatorname{sgn}(\Pi_B)$.
Stability of perturbations {#Sect:Res:Stab}
--------------------------
It is a consequence of the fact that the estimates in Theorem \[Thm:Ass:Main\] hold for a class of operators with complex measurable coefficients $B_i$, that operators such as $\operatorname{sgn}(\Pi_B)$ are also stable under small perturbations in $B$.
We provide the following adaption of Theorem 6.4 in [@AKMC] with a minor modification. In the following theorem, ${{\mathscr{H}}}$ denotes an abstract Hilbert space. We say that a family ${{\left\{T(\zeta)\right\}}}_{\zeta\in U}$ of $\omega$-bisectorial operators has a *uniformly bounded holomorphic functional calculus* if each operator $T(\zeta)$ has a bounded holomorphic functional calculus on the same sector $S^o_\mu{\cup}{{\left\{0\right\}}}$ with a bound which is uniform in $\zeta\in U$.
\[Thm:Res:HolDep\] Let $U \subset {\mathbb{C}}$ be an open set and $B_1, B_2: U \to {\mathcal{L}}({{\mathscr{H}}})$ be holomorphic and suppose that $(\Gamma,B_1(\zeta),B_2(\zeta))$ satisfy (H1)-(H3) uniformly for all $\zeta \in U$. Suppose further that $\Pi_{B(\zeta)}$ has a uniformly bounded holomorphic functional calculus on $S_{\mu}^o\cup\{0\}$ for some $\omega < \mu < \frac{\pi}{2}$ (where $\omega$ is the angle of sectoriality). Let $f \in {{\rm Hol}}^\infty(S_{\mu}^o)$. Then the map $\zeta \mapsto f(\Pi_{B(\zeta)})$ is holomorphic on $U$.
This claim is proved in a similar way to the first part of Theorem 6.4 in [@AKMC], with the exception that instead of invoking Theorem 2.10 in [@AKMC], we note that the uniformly bounded holomorphic functional calculus assumption is sufficient.
Next, consider the situation of ${{\mathscr{H}}}= {{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$. Adapting the construction of [@AKMC], we define the following Hilbert space $$\mathcal{K} = {{\rm L}^{2}_{\rm {}}}({\mathcal{M}}\times (0,\infty), {\mathcal{V}}; \frac{d\mu dt}{t}).$$ Then, for $\psi \in \Psi(S_{\mu}^o)$, $t > 0$ and almost all $x \in {\mathcal{M}}$, we define the operator ${\mathcal{S}}_{B(\zeta)}(\psi): {{\mathscr{H}}}\to \mathcal{K}$ by $({\mathcal{S}}_{B(\zeta)}(\psi)u)(x,t) = \psi(t\Pi_{B(\zeta)})u(x)$.
Under the hypothesis of Theorem \[Thm:Res:HolDep\], and the additional assumption that ${{\mathscr{H}}}= {{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$, whenever $\omega < \mu < \frac{\pi}{2}$ (where $\omega$ is the angle of sectoriality), the map $\zeta \mapsto {\mathcal{S}}_{B(\zeta)}(\psi)$ is holomorphic on $U$ for all $\psi \in \Psi(S_{\mu}^o)$.
We note that our choice of $\mathcal{K}$ is an adequate replacement to $\mathcal{K}$ in the proof of Theorem 6.4 in [@AKMC]. Also, for $t > 0$, the function $\psi_t(\zeta) = \psi(t\zeta) \in \Psi(S_{\mu}^o)$ and ${\left\| \psi_t \right\|}_\infty = {\left\| \psi \right\|}_\infty$. Therefore, the uniformly bounded holomorphic functional calculus assumption holds uniformly in $t > 0$ and is again an adequate substitution to run the rest of the argument of the proof of Theorem 6.4 in [@AKMC].
\[Cor:Res:Stab\] Let ${{\mathscr{H}}}, \Gamma, B_1, B_2, \kappa_1, \kappa_2$ satisfy (H1)-(H8) and take $\eta_i < \kappa_i$. Set $0 < \hat{\omega}_i < \frac{\pi}{2}$ by $\cos\hat{\omega}_i = \frac{ \kappa_i - \eta_i}{{\left\| B_i \right\|}_\infty + \eta_i}$ and $\hat{\omega} = \frac{1}{2}(\hat{\omega}_1 + \hat{\omega}_2)$. Let $A_i \in {{\rm L}^{\infty}_{\rm {}}}({\mathcal{L}}({\mathcal{V}}))$ satisfy
(i) ${\left\| A_i \right\|}_\infty \leq \eta_i$,
(ii) $A_1 A_2 { {\mathcal{R}}}(\Gamma), B_1 A_2 { {\mathcal{R}}}(\Gamma),
A_1 B_2 { {\mathcal{R}}}(\Gamma) \subset { {\mathcal{N}}}(\Gamma)$, and
(iii) $A_2 A_1 { {\mathcal{R}}}({{\Gamma}^\ast}), B_2 A_1 { {\mathcal{R}}}({{\Gamma}^\ast}),
A_2 B_1 { {\mathcal{R}}}({{\Gamma}^\ast}) \subset { {\mathcal{N}}}({{\Gamma}^\ast})$.
Letting $\hat{\omega} < \mu < \frac{\pi}{2}$, we have:
(i) for all $f \in {{\rm Hol}}^\infty(S_{\mu}^o)$, $${\left\| f(\Pi_B) - f(\Pi_{B+A}) \right\|}
\lesssim ({\left\| A_1 \right\|}_\infty + {\left\| A_2 \right\|}_\infty)
{\left\| f \right\|}_\infty,\qquad\text{and}$$
(ii) for all $\psi \in \Psi(S_{\mu}^o)$, $$\int_{0}^\infty {\left\| \psi(t\Pi_{B})u - \psi(t\Pi_{B + A})u \right\|}^2 \frac{dt}{t}
\lesssim ({\left\| A_1 \right\|}_\infty^2 + {\left\| A_2 \right\|}_\infty^2)
{\left\| u \right\|},$$ whenever $u \in {{\mathscr{H}}}$.
The implicit constants depend on (H1)-(H8) and $\eta_i$.
The argument proceeds in a similar way to the proof of Theorem 6.5 in [@AKMC]. The conditions on $A_i$ guarantee that $\tilde{B}_i(\zeta) = (B_i + \zeta A_i)$ satisfies (H3). This is, in fact, a necessary amendment to the original proof of Theorem 6.5 in [@AKMC].
[Poincaré ]{}Inequalities
=========================
In this short section we show that, under appropriate geometric conditions, we can bootstrap the [Poincaré ]{}inequality on functions to the dyadic [Poincaré ]{}inequality on the bundle. As in [@Morris3], we make the following definition.
We say that ${\mathcal{M}}$ satisfies a *local [Poincaré ]{}inequality* if there exists $c \geq 1$ such that for all $f \in {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{M}})$, $${\left\| f - f_{B} \right\|}_{{{\rm L}^{2}_{\rm {}}}(B)} \leq c\ \operatorname{rad}(B) {\left\| f \right\|}_{{{{\rm W}^{1,2}_{\rm {}}}}(B)}
\tag{$\text{P}_{\text{loc}}$}
\label{Def:Pre:Ploc}$$ for all balls $B$ in ${\mathcal{M}}$ such that $\operatorname{rad}(B) \leq 1$.
Note that we allow the Sobolev norm ${\left\| {\cdotp}\right\|}_{{{{\rm W}^{1,2}_{\rm {}}}}(B)}$ over the ball on the right of the inequality, rather than simply ${\left\| {{{\nabla}_{{{}}}}}{\cdotp}\right\|}$.
The following proposition then illustrates that under appropriate gradient bounds on the GBG coordinate basis, we can obtain a dyadic [Poincaré ]{}inequality in the bundle.
\[Prop:App:DyPoin\] Suppose that ${\mathcal{M}}$ satisfies both (\[Def:Pre:Eloc\]) and (\[Def:Pre:Ploc\]). Furthermore, suppose that that there exists $C_G > 0$ such that in each GBG chart with basis denoted by ${{\left\{e^i\right\}}}$ we have ${\left|{{{\nabla}_{{{}}}}}e^i\right|} \leq C_G$ for each $i$. Then, for all $u \in { {\mathcal{D}}}({{{\nabla}_{{{}}}}}) = {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}})$, $t \leq {\mathrm{t_S}}$, ${Q_{{}}}\in {{\mathscr{Q}}}_t$, and $r \geq \frac{C_1}{\delta}$, $$\int_{B} {\left|u - u_{{Q_{{}}}}\right|}^2\ d\mu \lesssim
(1 + e^{\lambda rt})(rt)^2 \int_{B} ({\left|{{{\nabla}_{{{}}}}}u\right|}^2 + {\left|u\right|}^2)\ d\mu$$ where $B = B(x_{{Q_{{}}}},rt)$.
First, consider the case $(rt) \geq \frac{{\rho}}{5}$. Then $${\left\| u - u_{{Q_{{}}}} \right\|}_{{{\rm L}^{2}_{\rm {}}}(B)}^2
\lesssim {\left\| u \right\|}^2_{{{\rm L}^{2}_{\rm {}}}(B)} + {\left\| u_{{Q_{{}}}} \right\|}^2_{{{\rm L}^{2}_{\rm {}}}(B)}.$$ Recalling that ${\widehat{{Q_{{}}}}}$ is the GBG cube of ${Q_{{}}}$, $$\begin{gathered}
\int_{B} {\left|u_{{Q_{{}}}}\right|}^2\ d\mu
= \int_{B} {\left| {\left(\fint_{{Q_{{}}}} u_i\ d\mu\right)}{\raisebox{\depth}{\(\chi\)}_{B(x_{{\widehat{{Q_{{}}}}}},{\rho})}} e^i\right|}^2\ d\mu
\simeq \sum_{i} \int_{B} {\left|\fint_{{Q_{{}}}} u_i\ d\mu\right|}^2{\raisebox{\depth}{\(\chi\)}_{B(x_{{\widehat{{Q_{{}}}}}},{\rho})}} d\mu \\
\leq \sum_{i} \int_{B} {\left( \fint_{{Q_{{}}}} {\left|u_i\right|}^2\ d\mu\right)}\ d\mu
\leq \frac{\mu(B)}{\mu({Q_{{}}})} \int_{B} {\left|u\right|}^2\ d\mu
\lesssim (1 + r^\kappa e^{\lambda rt}) {\left\| u \right\|}_{{{\rm L}^{2}_{\rm {}}}(B)}^2.\end{gathered}$$ Thus, $
{\left\| u - u_{{Q_{{}}}} \right\|}^2_{{{\rm L}^{2}_{\rm {}}}(B)}
\lesssim (1 + r^{\kappa} e^{\lambda rt}) (rt)^2 {\left\| u \right\|}^2_{{{\rm L}^{2}_{\rm {}}}(B)}$ since $rt \geq \frac{{\rho}}{5}$.
Next, suppose that $(rt) < \frac{{\rho}}{5}$. It is easy to see that we have $B(x_{{Q_{{}}}},rt) \subset B(x_{{\widehat{{Q_{{}}}}}},{\rho})$ and so, $$\begin{gathered}
\int_{B} {\left|u - u_{{Q_{{}}}}\right|}^2\ d\mu
\simeq \sum_{i} \int_{B}
{\left|u_i - {\left(\fint_{{Q_{{}}}}u_i\ d\mu\right)}{\raisebox{\depth}{\(\chi\)}_{B(x_{{\widehat{{Q_{{}}}}}},{\rho})}}\right|}^2\ d\mu \\
\leq \sum_{i} \int_{B} {\left|u_i - (u_i)_B\right|}^2\ d\mu
+ \sum_{i} \int_{B} {\left|(u_i)_B - {\left(\fint_{{Q_{{}}}} u_i\ d\mu\right)}\right|}^2\ d\mu\end{gathered}$$ For the first term, we invoke (\[Def:Pre:Ploc\]) so that $$\sum_{i} \int_{B} {\left|u_i - (u_i)_B\right|}^2\ d\mu
\lesssim (rt)^2 \int_{B}
{\left(\sum_{i} {\left|{{{\nabla}_{{{}}}}}u_i\right|}^2 + {\left|u\right|}^2\right)}\ d\mu.$$ Now, for the second term, $$\begin{gathered}
\sum_{i} \int_{B} {\left|(u_i)_B - {\left(\fint_{{Q_{{}}}} u_i\ d\mu\right)}\right|}^2\ d\mu
\leq \sum_{i} \frac{\mu(B)}{\mu({Q_{{}}})} \int_{{Q_{{}}}} {\left|u_i - (u_i)_B\right|}^2\ d\mu \\
\lesssim (1+ r^\kappa e^{\lambda rt}) (rt)^2 \int_{B} {\left(\sum_{i} {\left|{{{\nabla}_{{{}}}}}u_i\right|}^2 + {\left|u\right|}^2\right)}\ d\mu\end{gathered}$$
Next, we note that $
{{{\nabla}_{{{}}}}}u = {{{\nabla}_{{{}}}}}(u_i) {\otimes}e^i + u_i {\otimes}{{{\nabla}_{{{}}}}}e^i$ and therefore, by the hypothesis ${\left|{{{\nabla}_{{{}}}}}e^i\right|} \leq C_{G}$, $$\sum_{i} {\left|{{{\nabla}_{{{}}}}}u_i\right|}^2 \simeq
{\left|{{{\nabla}_{{{}}}}}u_i {\otimes}e^i\right|}^2
\leq {\left|{{{\nabla}_{{{}}}}}{u}\right|}^2 + {\left|u_i\right|}^2 {\left|{{{\nabla}_{{{}}}}}e^i\right|}^2
\lesssim {\left|{{{\nabla}_{{{}}}}}{u}\right|}^2 + {\left|u\right|}.$$ The proof is complete by combining these estimates.
Kato Square Root Estimates for Elliptic Operators {#Sect:App}
=================================================
The Kato square root problem on vector bundles
----------------------------------------------
Here, we present the main applications of Theorem \[Thm:Ass:Main\] to uniformly elliptic operators which arise naturally from a connection over a vector bundle. First, we describe a setup of operators which is a generalisation of §1 of [@Morris3] (and before that from [@AKM2]), making the necessary changes to facilitate the fact that we are working, in general, on a non-trivial bundle.
Let ${{\mathscr{H}}}= {{\rm L}^{2}_{\rm {}}}({\mathcal{V}}) \oplus {{\rm L}^{2}_{\rm {}}}({\mathcal{V}}) \oplus {{\rm L}^{2}_{\rm {}}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}})$. As discussed in §\[prelim\], ${{{\nabla}_{{{}}}}}: {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}}) \subset {{\rm L}^{2}_{\rm {}}}({\mathcal{V}}) \to {{\rm L}^{2}_{\rm {}}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}})$ is a closed densely defined operator, and so has a well defined adjoint ${{{{{\nabla}_{{{}}}}}}^\ast}$, which we denote by $ -\operatorname{div}: { {\mathcal{D}}}(\operatorname{div}) \subset {{\rm L}^{2}_{\rm {}}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}}) \to {{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$.
The reason for this notation is because when the the connection ${{{\nabla}_{{{}}}}}$ and the metric ${\mathrm{h}}$ are compatible, then ${{{\nabla}}^\ast}$ has the form of a divergence in the weak sense of Proposition \[Prop:App:Comp\] below. First some notation.
For $v \in {{\rm C}^{\infty}_{\rm {}}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}})$, define $\operatorname{tr}{{{\nabla}_{{{}}}}}v$ by contracting the first two indices of ${{{\nabla}_{{{}}}}}v \in {{\rm C}^{\infty}_{\rm {}}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}})$ over ${\mathrm{g}}$, to yield a section $\operatorname{tr}{{{\nabla}_{{{}}}}}v \in {{\rm C}^{\infty}_{\rm {}}}({\mathcal{V}})$.
The connection ${{{\nabla}_{{{}}}}}$ and the metric ${\mathrm{h}}$ are *compatible* if the product rule $$X({\mathrm{h}}(Y,Z)) = {\mathrm{h}}({{{\nabla}_{{X}}}}Y, Z) + {\mathrm{h}}(Y, {{{\nabla}_{{X}}}}Z)$$ is satisfied for every $X \in {{\rm C}^{\infty}_{\rm {}}}({{\rm T}}{\mathcal{M}})$ and $Y,Z \in {{\rm C}^{\infty}_{\rm {}}}({\mathcal{V}})$. For such connection and metric pairs, we have $\operatorname{div}=-{{{\nabla}}^\ast} = \operatorname{tr}{{{\nabla}_{{{}}}}}$ in the following weak sense.
\[Prop:App:Comp\] Suppose that the connection ${{{\nabla}_{{{}}}}}$ and the metric ${\mathrm{h}}$ are compatible. Then, for all $T \in {{\rm C}^{\infty}_{\rm c}}({\mathcal{V}})$ and $P \in {{\rm C}^{\infty}_{\rm {}}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}})$, $$\int_{{\mathcal{M}}} {\left\langle {{{\nabla}_{{{}}}}}T, P \right\rangle}\ d\mu = \int_{{\mathcal{M}}} {\left\langle T, -\operatorname{tr}{{{\nabla}_{{{}}}}}P \right\rangle}\ d\mu.$$
Fix $x \in {\mathcal{M}}$ so that we can locally write $T = T_i e^i$ and $P = P_{kl} dx^k {\otimes}e^l$. Next, define the ${\mathcal{V}}$ “inner product” yielding a $1$-form by ${\left\langle T,P \right\rangle}_{{\mathcal{V}}} = T_i P_{kl} {\mathrm{h}}(e^i, e^l) dx^k.$
Now, to make calculations easier, further assume that ${{\left\{x^i\right\}}}$ are normal coordinates at $x$. Then, for any $X = X_k dx^k$, we have that the divergence at $x$ is $\operatorname{div}X = \partial_k X_k$. Thus, at $x$, $$\begin{aligned}
\operatorname{div}{\left\langle T,P \right\rangle}_{{\mathcal{V}}}
= \sum_{k} &(\partial_k T_i) P_{kl} {\mathrm{h}}(e^i, e^l)
+ \sum_{k} T_i P_{kl} {\mathrm{h}}({{{\nabla}_{{\partial_k}}}} e^i, e^l) \\
&+ \sum_{k} T_i (\partial_k P_{kl}) {\mathrm{h}}(e^i, e^l)
+ \sum_{k} T_i P_{kl} {\mathrm{h}}(e^i, {{{\nabla}_{{\partial_k}}}} e^l)\end{aligned}$$ by the compatibility of ${{{\nabla}_{{{}}}}}$ and ${\mathrm{h}}$.
A calculation at $x$ then shows that ${{{\nabla}_{{{}}}}}T = {{{\nabla}_{{{}}}}}(T_i e^i)
= \partial_k T_i dx^k {\otimes}e^i + T_i dx^k {\otimes}{{{\nabla}_{{\partial_k}}}}e^i.$ Also, $\operatorname{tr}{{{\nabla}_{{{}}}}}P
= \sum_{k} \partial_k P_{kl} e^l + \sum_{k} P_{kl} {{{\nabla}_{{{}}}}}e^l,$ since we assumed normal coordinates at $x$, making ${{{\nabla}_{{{}}}}}dx^k = 0$. Then, a direct calculation shows that $$\begin{aligned}
{\left\langle {{{\nabla}_{{{}}}}}T, P \right\rangle}
&= \sum_{k} (\partial_k T_i) P_{kl} {\mathrm{h}}(e^i, e^l)
+ \sum_{k} T_i P_{kl} {\mathrm{h}}({{{\nabla}_{{\partial_k}}}} e^i, e^l),
\quad\text{and}\\
{\left\langle T, \operatorname{tr}{{{\nabla}_{{{}}}}}P \right\rangle}
&= \sum_{k} T_i (\partial_k P_{kl}) {\mathrm{h}}(e^i, e^l)
+ \sum_{k} T_i P_{kl} {\mathrm{h}}(e^i, {{{\nabla}_{{\partial_k}}}} e^l).\end{aligned}$$ Thus, at $x$, $\operatorname{div}{\left\langle T,P \right\rangle}_{{\mathcal{V}}} = {\left\langle {{{\nabla}_{{{}}}}}T, P \right\rangle} + {\left\langle T, \operatorname{tr}{{{\nabla}_{{{}}}}}P \right\rangle}.$
By the compactness of ${{\rm spt} {\text{ }}T}$, it is easy to see that ${{\rm spt} {\text{ }}{\left\langle } \right\rangle}{T,P}_{\mathcal{V}},\ {{\rm spt} {\text{ }}{\left\langle } \right\rangle}{T, \operatorname{tr}{{{\nabla}_{{{}}}}}P}$ and ${{\rm spt} {\text{ }}{\left\langle } \right\rangle}{{{{\nabla}_{{{}}}}}T, P}$ are all compact. Thus, we integrate this equation over ${\mathcal{M}}$ and apply the divergence theorem to obtain the conclusion.
We pause to introduce some notation. When ${\mathcal{W}},\ \tilde{\mathcal{W}}$ are two vector bundles, define the new vector bundle ${\mathcal{L}}({\mathcal{W}}, \tilde{\mathcal{W}})$ to mean the space of all maps $C: {\mathcal{W}}\to \tilde{\mathcal{W}}$ such that for each $x \in {\mathcal{M}}$, $C(x) \in {\mathcal{L}}({\mathcal{W}}_x,\tilde{\mathcal{W}}_x)$. This is consistent with the previous notation since ${\mathcal{L}}({\mathcal{W}}) = {\mathcal{L}}({\mathcal{W}}, {\mathcal{W}})$.
With this notation in mind, let $A_{00} \in {{\rm L}^{\infty}_{\rm {}}}({\mathcal{L}}({\mathcal{V}})),\
A_{01} \in {{\rm L}^{\infty}_{\rm {}}}({\mathcal{L}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}}, {\mathcal{V}})),\
A_{10} \in {{\rm L}^{\infty}_{\rm {}}}({\mathcal{L}}({\mathcal{V}}, {{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}})),\
A_{11} \in {{\rm L}^{\infty}_{\rm {}}}({\mathcal{L}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}})).$ Then, define $A \in {{\rm L}^{\infty}_{\rm {}}}({\mathcal{L}}({\mathcal{V}}\oplus ({{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}})))$ by $$A = \begin{pmatrix} A_{00} & A_{01} \\ A_{10} & A_{11}\end{pmatrix}.$$ Furthermore, let $a \in {{\rm L}^{\infty}_{\rm {}}}({\mathcal{L}}({\mathcal{V}}))$. Set $B_1,B_2:{{\mathscr{H}}}\to {{\mathscr{H}}}$ by $$B_1 = \begin{pmatrix} a & 0 \\ 0 & 0\end{pmatrix}
\quad\text{and}\quad
B_2 = \begin{pmatrix} 0 & 0 \\ 0 & A\end{pmatrix}\ .$$
Moreover, set $$S = \begin{pmatrix}I \\ {{{\nabla}_{{{}}}}}\end{pmatrix},\
{{S}^\ast} = \begin{pmatrix} I & -\operatorname{div}\end{pmatrix},\
\Gamma = \begin{pmatrix}0 & 0 \\ S & 0\end{pmatrix},\
\text{and}\
{{\Gamma}^\ast} = \begin{pmatrix}0 & {{S}^\ast} \\ 0 & 0\end{pmatrix}$$ and define the following divergence form operator ${\mathrm{L}}_{A}: { {\mathcal{D}}}({\mathrm{L}}_A) \subset {{\rm L}^{2}_{\rm {}}}({\mathcal{V}}) \to {{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$ by $${\mathrm{L}}_{A}u = a{{S}^\ast}ASu
= -a \operatorname{div}(A_{11} {{{\nabla}_{{{}}}}}u) - a \operatorname{div}(A_{10}u)
+ a A_{01} {{{\nabla}_{{{}}}}}u + a A_{00} u.$$
We apply Theorem \[Thm:Ass:Main\] to prove the Kato square root problem on vector bundles.
\[Thm:App:KatoVB\] Suppose that ${\mathcal{M}}$ satisfies (\[Def:Pre:Eloc\]) and both ${\mathcal{V}}$ and ${{\rm T}^\ast}{\mathcal{M}}$ have generalised bounded geometry (so that they are equipped with GBG coordinate systems), and that
(i) ${\mathcal{M}}$ satisfies (\[Def:Pre:Ploc\]),
(ii) the GBG charts for ${{\rm T}^\ast}{\mathcal{M}}$ are induced from coordinate systems on ${\mathcal{M}}$,
(iii) the connection ${{{\nabla}_{{{}}}}}$ and the metric ${\mathrm{h}}$ are compatible,
(iv) there exists $C > 0$ such that in each GBG chart ${{\left\{e^j\right\}}}$ for ${\mathcal{V}}$ and ${{\left\{dx^i\right\}}}$ for ${{\rm T}^\ast}{\mathcal{M}}$, we have that ${\left|{{{\nabla}_{{{}}}}}e^j\right|}, {\left|\partial_k {\mathrm{h}}^{ij}\right|},
{\left|\partial_k {\mathrm{g}}^{ij}\right|} \leq C$ a.e.,
(v) there exist $\kappa_1, \kappa_2 > 0$ such that $$\operatorname{Re}{\left\langle a u, u \right\rangle} \geq \kappa_1 {\left\| u \right\|}^2
\qquad\text{and}\qquad
\operatorname{Re}{\left\langle A Sv, Sv \right\rangle} \geq \kappa_2 {\left\| v \right\|}_{{{{\rm W}^{1,2}_{\rm {}}}}}^2$$ for all $u \in {{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$ and $v \in {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}})$, and
(vi) we have that ${ {\mathcal{D}}}({\Delta}) \subset {{{\rm W}^{2,2}_{\rm {}}}}({\mathcal{V}})$, and there exist $C' > 0$ such that $${\left\| {{{\nabla}_{{{}}}}}^2 u \right\|} \leq C' {\left\| (I + {\Delta})u \right\|}$$ whenever $u \in { {\mathcal{D}}}({\Delta})$.
Then,
(i) $\Pi_B$ has a bounded holomorphic functional calculus, and
(ii) ${ {\mathcal{D}}}(\sqrt{{\mathrm{L}}_A}) = { {\mathcal{D}}}({{{\nabla}_{{{}}}}}) = {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}})$ with $\|{\sqrt{{\mathrm{L}}_A} u}\|\simeq {\left\| {{{\nabla}_{{{}}}}}u \right\|} + {\left\| u \right\|} = {\left\| u \right\|}_{{{{\rm W}^{1,2}_{\rm {}}}}}$ for all $u \in {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}})$.
We show that $(\Gamma,B_1,B_2)$ satisfy (H1)-(H8) of §\[Sect:Ass\] in order to invoke Theorem \[Thm:Ass:Main\].
Nilpotency of $\Gamma$ is immediate. That $\Gamma$ is densely-defined and closed follows easily from Proposition \[Prop:Pre:ConnClosed\]. This settles (H1). Also, (H3) is an easy calculation and (H4)-(H5) are immediate. The fact that (H6) is satisfied is an immediate consequence of the Leibniz property of the connection. The conditions (i) and (ii) allow us to invoke Proposition \[Prop:App:DyPoin\], thus proving (H8)-1. It remains to demonstrate (H2), (H7), and (H8)-2 hold with $\Xi:{ {\mathcal{D}}}(\Xi) \subset {{\rm L}^{2}_{\rm {}}}({\mathcal{V}}) \oplus {{\rm L}^{2}_{\rm {}}}({\mathcal{V}}) \oplus {{\rm L}^{2}_{\rm {}}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}})
\to {{\rm L}^{2}_{\rm {}}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}}) \oplus {{\rm L}^{2}_{\rm {}}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}})
\oplus {{\rm L}^{2}_{\rm {}}}({{\mathcal{T}}^{(0,2)}}{\mathcal{M}}{\otimes}{\mathcal{V}})$ defined by $\Xi(u_1,u_2,u_3) = ({{{\nabla}_{{{}}}}}u_1, {{{\nabla}_{{{}}}}}u_2, {{{\nabla}_{{{}}}}}u_3)$. The domain of $\Xi$ is ${ {\mathcal{D}}}(\Xi) = {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}}) \oplus {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}})
\oplus {{{\rm W}^{1,2}_{\rm {}}}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}})$.
Fix $u \in { {\mathcal{R}}}({{\Gamma}^\ast})$. That is, $u = ({{S}^\ast} v, 0)$ for $v \in {{\rm L}^{2}_{\rm {}}}({\mathcal{V}}) \oplus {{\rm L}^{2}_{\rm {}}}({{\rm T}^\ast}M {\otimes}{\mathcal{V}})$. Thus, $$\operatorname{Re}{\left\langle B_1 u, u \right\rangle}
= \operatorname{Re}{\left\langle a {{S}^\ast}v, {{S}^\ast}v \right\rangle}
\geq \kappa_1 {\left\| {{S}^\ast}v \right\|}^2
= \kappa_1 {\left\| u \right\|}^2.$$ Next, let $u \in { {\mathcal{R}}}(\Gamma)$. Thus, $u = (0,Sv)$ for $v \in {{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$. Therefore, $$\operatorname{Re}{\left\langle B_2 u, u \right\rangle}
= \operatorname{Re}{\left\langle A Sv, Sv \right\rangle}
\geq \kappa_2 {\left\| u \right\|}^2$$ which settles (H2).
We verify (H7). First, let $u = (u_1, u_2, u_3) \in { {\mathcal{D}}}(\Gamma)$ with ${{\rm spt} {\text{ }}u} \subset {Q_{{}}}$ and $v = (v_1,v_2,v_3) \in { {\mathcal{D}}}({{\Gamma}^\ast})$ with ${{\rm spt} {\text{ }}v} \subset {Q_{{}}}$. Then, $\Gamma u = (0, Su_1) = (0,u_1, {{{\nabla}_{{{}}}}}u_1)$, ${{\Gamma}^\ast}v = ({{S}^\ast}(v_2,v_3),0) = (v_2 - \operatorname{div}v_3,0,0)$ and we have that $${\left|\int_{{Q_{{}}}} \Gamma u\ d\mu\right|}
= {\left|\int_{{Q_{{}}}} u_1\ d\mu\right|} + {\left|\int_{{Q_{{}}}} {\nabla}u_1\ d\mu\right|}$$ and $${\left|\int_{{Q_{{}}}} {{\Gamma}^\ast} v\ d\mu\right|}
= {\left|\int_{{Q_{{}}}} {v_2 - \operatorname{div}v_3}\ d\mu\right|}
\leq {\left|\int_{{Q_{{}}}} v_2\ d\mu\right|} + {\left|\int_{{Q_{{}}}} \operatorname{div}v_3\ d\mu\right|}.$$ By Cauchy-Schwartz, $${\left|\int_{{Q_{{}}}} u_1\ d\mu\right|}
\lesssim \mu({Q_{{}}})^{\frac{1}{2}} {\left\| u_1 \right\|}
\leq \mu({Q_{{}}})^{\frac{1}{2}} {\left\| u \right\|},$$ and by a similar computation, $${\left|\int_{{Q_{{}}}} v_2\ d\mu\right|} \lesssim \mu({Q_{{}}})^{\frac{1}{2}} {\left\| v \right\|}.$$ To conveniently deal with the two remaining estimates, we omit the indices in $u_1$ and $v_3$ and note that it remains to prove $$\text{(a)} \quad
{\left|\int_{{Q_{{}}}} {{{\nabla}_{{{}}}}}u\right|} \lesssim \mu({Q_{{}}})^{\frac{1}{2}} {\left\| u \right\|}
\quad\text{and}\quad
\text{(b)} \quad
{\left|\int_{{Q_{{}}}} \operatorname{div}v\right|} \lesssim \mu({Q_{{}}})^{\frac{1}{2}} {\left\| v \right\|}$$ for all $u \in { {\mathcal{D}}}({{{\nabla}_{{{}}}}})$, $v \in { {\mathcal{D}}}(\operatorname{div})$ with ${{\rm spt} {\text{ }}u},\ {{\rm spt} {\text{ }}v} \subset {Q_{{}}}$.
Before continuing, we remark that every function $f \in {{\rm L}^{1}_{\rm loc}}({\mathcal{V}})$ can be written as $f= f_ie^i = {\mathrm{h}}(f, {\mathrm{h}}_{ki} e^k)e^i$ in $B(x_{{\widehat{{Q_{{}}}}}},{\rho})$. Here ${\mathrm{h}}_{ij} = {\mathrm{h}}(e_i, e_j)$, where $\{e_i\}$ is the dual basis of $\{e^i\}$, and we use the same notation ${\mathrm{h}}$ to denote the induced inner product on ${{{\mathcal{V}}}^\ast}$ by requiring that ${\mathrm{h}}_{ij}{\mathrm{h}}^{jk} = \delta_{i}^k$. We also remark that every function $F \in {{\rm L}^{1}_{\rm loc}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}})$ can be written as $F = F_{ij}\ dx^i {\otimes}e^j = {\mathrm{g}}{\otimes}{\mathrm{h}}(F, {\mathrm{g}}_{ai}h_{bj}\ dx^a {\otimes}e^b)\ dx^i {\otimes}e^j$ in $B(x_{{\widehat{{Q_{{}}}}}},{\rho})$ where ${\mathrm{g}}_{ij} = {\mathrm{g}}(\partial_{i},\partial_j)$ on ${{\rm T}}{\mathcal{M}}$.
Turning to the proof of (a), let $u\in {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}})$ with ${{\rm spt} {\text{ }}u}\subset {Q_{{}}}$. Choose $\psi \in {{\rm C}^{\infty}_{\rm c}}({\mathcal{M}})$ such that ${{\rm spt} {\text{ }}\psi}\subset B(x_{{\widehat{{Q_{{}}}}}},{\rho})$ and $\psi = 1$ on $Q$, and extend $\psi {\mathrm{g}}_{ai}{\mathrm{h}}_{bj}\ dx^a{\otimes}e^b$ to be zero outside of $B(x_{{\widehat{{Q_{{}}}}}},{\rho})$. Then, by the above remark with $F = {{{\nabla}_{{{}}}}}u$, we have the following identity on $Q$. $$\begin{aligned}
\int_{{Q_{{}}}} {{{\nabla}_{{{}}}}}u
&= \int_{{Q_{{}}}} {\mathrm{g}}{\otimes}{\mathrm{h}}({{{\nabla}_{{{}}}}}u, \psi {\mathrm{g}}_{ai}{\mathrm{h}}_{bj}\ dx^a{\otimes}e^b)\ d\mu\ dx^i {\otimes}e^j \\
&= \int_{{\mathcal{M}}} {\mathrm{g}}{\otimes}{\mathrm{h}}({{{\nabla}_{{{}}}}}u, \psi {\mathrm{g}}_{ai}{\mathrm{h}}_{bj}\ dx^a{\otimes}e^b)\ d\mu\ dx^i {\otimes}e^j \\
&= \int_{{\mathcal{M}}} {\mathrm{h}}(u, -\operatorname{tr}{{{\nabla}_{{{}}}}}(\psi {\mathrm{g}}_{ai}{\mathrm{h}}_{bj}\ dx^a{\otimes}e^b))\ d\mu\ dx^i {\otimes}e^j \\
&= \int_{{Q_{{}}}} {\mathrm{h}}(u, -\operatorname{tr}{{{\nabla}_{{{}}}}}({\mathrm{g}}_{ai}{\mathrm{h}}_{bj}\ dx^a{\otimes}e^b))\ d\mu\ dx^i {\otimes}e^j \\
&= \int_{{Q_{{}}}} -{\mathrm{h}}(u, {\mathrm{g}}_{ai}{\mathrm{h}}_{bj}\operatorname{tr}{{{\nabla}_{{{}}}}}(dx^a {\otimes}e^b)
+ \operatorname{tr}(({{{\nabla}_{{{}}}}}{\mathrm{g}}_{ai}{\mathrm{h}}_{bj}) {\otimes}dx^a {\otimes}e^b))\ d\mu\ dx^i {\otimes}e^j\ .\end{aligned}$$ We have used Proposition \[Prop:App:Comp\] (since ${\mathrm{g}}_{ai}{\mathrm{h}}_{bj}\ dx^a {\otimes}e^b$ are smooth), the product rule for ${{{\nabla}_{{{}}}}}$, and the linearity of $\operatorname{tr}$. We note that by an easy calculation, we have ${\left|\operatorname{tr}(X)\right|} \lesssim {\left|X\right|}$ for all $x \in {\mathcal{M}}$ whenever $X \in {{\rm C}^{\infty}_{\rm {}}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}})$. Furthermore, the bound on the metric in each GBG chart implies bounds on ${\left|{\mathrm{h}}_{ai}\right|}$ and ${\left|{\mathrm{g}}_{ai}\right|}$, and the bounds in (iv) imply bounds on ${\left|\partial_k {\mathrm{h}}_{ai}\right|}$ and ${\left|\partial_k {\mathrm{g}}_{bj}\right|}$. Since we assumed the connection to be Levi-Cevita, we can write ${{{\nabla}_{{{}}}}}dx^a$ purely in terms of the Christoffel symbols, which in turn can be written in terms of ${\mathrm{g}}_{ij}$, ${\mathrm{g}}^{ij}$ and $\partial_k {\mathrm{g}}_{ij}$. Also, ${\left|e^b\right|}$ and ${\left|dx^a\right|}$ are bounded by the GBG hypothesis, ${\left|{{{\nabla}_{{{}}}}}e^b\right|}$ by (iv), and so we conclude that $${\left|{\mathrm{g}}_{ai}{\mathrm{h}}_{bj}\operatorname{tr}{{{\nabla}_{{{}}}}}(dx^a {\otimes}e^b)
+ \operatorname{tr}(({{{\nabla}_{{{}}}}}{\mathrm{g}}_{ai}{\mathrm{h}}_{bj}) {\otimes}dx^a {\otimes}e^b)\right|} \lesssim 1.$$ On combining these estimates, and applying the Cauchy-Schwartz inequality, we conclude that $${\left|\int_{{Q_{{}}}} {{{\nabla}_{{{}}}}}u\right|} \lesssim \mu({Q_{{}}})^{\frac{1}{2}} {\left\| u \right\|}$$ as required.
To verify (b), let $v \in { {\mathcal{D}}}(\operatorname{div})$ with ${{\rm spt} {\text{ }}v} \subset Q$, and apply the above remark with $f = \operatorname{div}v$ to obtain by a similar argument that $$\int_{{Q_{{}}}} \operatorname{div}v
= \int_{{Q_{{}}}} {\mathrm{h}}(\operatorname{div}v, {\mathrm{h}}_{ki}\ e^k)\ d\mu\ e^i
= \int_{{Q_{{}}}} {\mathrm{g}}{\otimes}{\mathrm{h}}(v, {{{\nabla}_{{{}}}}}({\mathrm{h}}_{ki}\ e^k))\ d\mu\ e^i.$$ Reasoning as before, we have that ${\left|{{{\nabla}_{{{}}}}}({\mathrm{h}}_{ki}\ e^k)\right|}$ is bounded and by the Cauchy-Schwartz inequality, we conclude that $${\left|\int_{{Q_{{}}}} \operatorname{div}v\right|} \leq \mu({Q_{{}}})^\frac{1}{2} {\left\| v \right\|}$$ as required. This completes the proof of (H7).
To show (H8) let $\Xi(u_1,u_2,u_3) = ({{{\nabla}_{{{}}}}}u_1, {{{\nabla}_{{{}}}}}u_2, {{{\nabla}_{{{}}}}}u_3)$. Upon noting that $${\left|{{{\nabla}_{{{}}}}}(dx^i {\otimes}e^j)\right|}
\leq {\left|{{{\nabla}_{{{}}}}}dx^i\right|} {\left|e^j\right|} + {\left|{{{\nabla}_{{{}}}}}e^j\right|} {\left|dx^i\right|} \lesssim 1,$$ we apply Proposition \[Prop:App:DyPoin\] which proves (H8)-1.
It remains to show (H8)-2. Fix $v = (v_1,v_2,v_3) \in { {\mathcal{R}}}(\Pi) {\cap}{ {\mathcal{D}}}(\Pi)$. Thus, there is $u = (u_1, u_2, u_3) \in { {\mathcal{D}}}(\Pi)$ such that $v = \Pi u = (u_2 - \operatorname{div}u_3, u_1, {{{\nabla}_{{{}}}}}u_1)$. A calculation the shows that $${\left\| v \right\|}^2 + {\left\| \Xi v \right\|}^2
= {\left\| u_1 \right\|}^2 + 2 {\left\| {{{\nabla}_{{{}}}}}u_1 \right\|}^2 + {\left\| {{{\nabla}_{{{}}}}}^2 u_1 \right\|}
+ {\left( {\left\| v_1 \right\|}^2 + {\left\| {{{\nabla}_{{{}}}}}v_1 \right\|}^2\right)}.$$ Also, $${\left\| \Pi v \right\|}^2 = {\left\| (I + {\Delta}) u_1 \right\|}^2 + {\left\| v_1 \right\|}^2 + {\left\| {{{\nabla}_{{{}}}}}v_1 \right\|}^2.$$ But note that $${\left\| (I + {\Delta})u_1 \right\|}^2
= {\left\langle (I + {\Delta})u_1, (I + {\Delta})u_1 \right\rangle}
= {\left\| u_1 \right\|}^2 + 2 {\left\| {{{\nabla}_{{{}}}}}u_1 \right\|}^2 + {\left\| {\Delta}u_1 \right\|}^2.$$ Combining these estimates with (iv), we have that ${\left\| v \right\|}^2 + {\left\| \Xi v \right\|}^2 \lesssim {\left\| \Pi v \right\|}$ which proves (H8)-2.
Now, by invoking Theorem \[Thm:Ass:Main\], we conclude that $\Pi_B$ has a bounded holomorphic functional calculus. By its Corollary \[Cor:Ass:Main\], we obtain that ${\left\| \sqrt{{\Pi_B}^2}v \right\|} \simeq {\left\| \Pi_B v \right\|}$ for $v \in { {\mathcal{D}}}(\Pi_B) = { {\mathcal{D}}}(\sqrt{{\Pi_B}^2})$. Fix $u \in {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}})$. Then $v = (u,0,0) \in { {\mathcal{D}}}(\Pi_B)$ and $${\left\| \sqrt{{\mathrm{L}}_{A}}u \right\|} = {\left\| \sqrt{{\Pi_B}^2}v \right\|}
\simeq {\left\| \Pi_B v \right\|} = {\left\| v \right\|}_{{{{\rm W}^{1,2}_{\rm {}}}}}$$ which finishes the proof.
Instead of taking ${{{\nabla}_{{{}}}}}$ to be a connection, we could have considered a *sub-connection*, by which we mean a map ${{{\nabla}_{{{}}}}}: {{\rm C}^{\infty}_{\rm {}}}({\mathcal{V}}) \to {{\rm C}^{\infty}_{\rm {}}}({{\rm T}^\ast}{\mathcal{M}}{\otimes}{\mathcal{V}})$ that is function linear and satisfies the Leibniz property on a sub-bundle ${\mathcal{E}}$ of ${{\rm T}^\ast}{\mathcal{M}}$, and vanishes outside of ${\mathcal{E}}$. This is related to the study of square roots of elliptic operators associated with sub-Laplacians on Lie groups [@BEMc]. However, it is not clear to us whether a sub-connection is automatically densely-defined and closable.
Manifolds with injectivity and Ricci bounds
-------------------------------------------
In this section, we apply Theorem \[Thm:App:KatoVB\] to establish the Kato square root estimates for manifolds which have injectivity radius bounds and Ricci bounds. Our approach is to show that under these conditions, there exist *harmonic* coordinates which gives us bounds on the metric and its derivatives. We then use these coordinates to show that the bundle of $(p,q)$ tensors satisfy the GBG criterion.
We first present the following theorem which is really contained in the observation following Theorem 1.2 in [@Hebey].
Suppose there exist $\kappa, \eta > 0$ such that $\operatorname{inj}(M) \geq \kappa$ and ${\left|{{\rm Ric}}\right|} \leq \eta$. Then, for any $A > 1$ and $\alpha \in (0,1)$, there exists $r_H = r_H(n,A,\alpha,\kappa,\eta) > 0$ such that $B(x,r_H)$ corresponds to a coordinate system satisfying:
(i) $A^{-1} {\updelta}_{ij} \leq {\mathrm{g}}_{ij} \leq A {\updelta}_{ij}$ as bilinear forms and,
(ii) $\begin{aligned}[t]\sum_{l} r_H
\sup_{y \in B(x,r_H)} {\left|\partial_{l} {\mathrm{g}}_{ij}(y)\right|}
+ \sum_{l} r_H^{1 + \alpha}
\sup_{y \neq z}
\frac{{\left|\partial_l {\mathrm{g}}_{ij}(z) - \partial_l {\mathrm{g}}_{ij}(y)\right|}}
{d(y,z)^\alpha} \leq A - 1.
\end{aligned}$
This immediately gives us the existence of GBG coordinates for tensor fields.
\[Cor:App:GBGExt\] Under the assumptions that $\operatorname{inj}(M) \geq \kappa>0$ and ${\left|{{\rm Ric}}\right|} \leq \eta$, there exist GBG coordinates for ${{\mathcal{T}}^{(p,q)}} {\mathcal{M}}$. Furthermore, in each basis ${{\left\{e^i\right\}}}$ in each GBG coordinate system for ${{\mathcal{T}}^{(p,q)}}{\mathcal{M}}$, we have that ${\left|{{{\nabla}_{{{}}}}}e^i\right|} \leq C_{p,q}.$
First, we note that the Ricci bounds imply that there exists $\eta \in{\mathbb{R}}$ such that that ${{\rm Ric}}\geq \eta {\mathrm{g}}$. Thus, as in the proof of Theorem 1.1 in [@Morris3], we conclude that ${\mathcal{M}}$ satisfies (\[Def:Pre:Eloc\]).
Fix $A = 2$ and $\alpha = \frac{1}{2}$. The previous theorem guarantees the existence of harmonic coordinates for these choices. Thus, this yields GBG coordinates for ${{\rm T}}{\mathcal{M}}$ with $2^{-1} \leq {\mathrm{g}}\leq 2$.
It is an easy calculation to show that $G \simeq I$ implies $G^{-1} \simeq I$ with the same constants for a positive definite matrix $G$. Thus, we obtain GBG coordinates for ${{\rm T}^\ast}{\mathcal{M}}$ with $2^{-1} \leq {\mathrm{g}}\leq 2$ where we denote the metric on ${{\rm T}^\ast}{\mathcal{M}}$ also by ${\mathrm{g}}$.
Next, if two inner products $u,v$ on vector spaces $U,V$ satisfy $C_1^{-1} \leq u \leq C_1$ and $C_2^{-1} \leq v \leq C_2$, then $(C_1 C_2)^{-1} \leq u {\otimes}v \leq C_1 C_2$ on $U {\otimes}V$. Thus, by induction, we have that $2^{-pq} \leq {\mathrm{g}}\leq 2^{pq}$ for the metric ${\mathrm{g}}$ on ${{\mathcal{T}}^{(p,q)}} {\mathcal{M}}$.
For the gradient bounds, first consider ${{\rm T}}{\mathcal{M}}= {{\mathcal{T}}^{(1,0)}}{\mathcal{M}}$. Since we assume our connection is Levi-Civita, we can write the Christoffel symbols ${\Gamma^{k}_{ij}}$ purely in terms of $\partial_{a} {\mathrm{g}}_{bc}$ and ${\mathrm{g}}^{ab}$. The Cauchy-Schwartz inequality allows us to bound the ${\mathrm{g}}^{ab}$ and the bounds on $\partial_{a} {\mathrm{g}}_{bc}$ comes from the theorem. Thus, ${\left|{\Gamma^{k}_{ij}}\right|} \leq C$ and so ${\left|{{{\nabla}_{{{}}}}}e_i\right|} \leq C$. An inductive argument then yields the result for ${{\mathcal{T}}^{(p,q)}}{\mathcal{M}}$ with the constant dependent on $p$ and $q$.
With this result, we can apply the general Theorem \[Thm:App:KatoVB\] to obtain the following solution to the Kato square root problem on finite rank tensors.
\[Thm:App:KatoTen\] Suppose that ${\left|{{\rm Ric}}\right|} \leq C$, $\operatorname{inj}(M) \geq \kappa > 0$, and the following ellipticity condition holds: there exist $\kappa_1, \kappa_2 > 0$ such that $$\operatorname{Re}{\left\langle a u, u \right\rangle} \geq \kappa_1 {\left\| u \right\|}^2
\qquad\text{and}\qquad
\operatorname{Re}{\left\langle A Sv, Sv \right\rangle} \geq \kappa_2 {\left\| v \right\|}_{{{{\rm W}^{1,2}_{\rm {}}}}}^2$$ for all $u \in {{\rm L}^{2}_{\rm {}}}({{\mathcal{T}}^{(p,q)}}{\mathcal{M}})$ and $v \in {{{\rm W}^{1,2}_{\rm {}}}}({{\mathcal{T}}^{(p,q)}}{\mathcal{M}})$. Suppose further that ${ {\mathcal{D}}}({\Delta}) \subset {{{\rm W}^{2,2}_{\rm {}}}}({\mathcal{V}})$ and that there exists $C' > 0$ such that $$\label{Riesz} {\left\| {{{\nabla}_{{{}}}}}^2 u \right\|} \leq C' {\left\| (I + {\Delta})u \right\|} \tag{R}$$ whenever $u \in { {\mathcal{D}}}({\Delta})$. Then, ${ {\mathcal{D}}}(\sqrt{{\mathrm{L}}_A}) = { {\mathcal{D}}}({{{\nabla}_{{{}}}}}) = {{{\rm W}^{1,2}_{\rm {}}}}({{\mathcal{T}}^{(p,q)}}{\mathcal{M}})$ and $\|{\sqrt{{\mathrm{L}}_A} u}\|\simeq {\left\| {{{\nabla}_{{{}}}}}u \right\|} + {\left\| u \right\|} = {\left\| u \right\|}_{{{{\rm W}^{1,2}_{\rm {}}}}}$ for all $u \in {{{\rm W}^{1,2}_{\rm {}}}}({{\mathcal{T}}^{(p,q)}}{\mathcal{M}})$.
We apply Theorem \[Thm:App:KatoVB\]. The Ricci bounds imply that there exists $\eta \in{\mathbb{R}}$ such that that ${{\rm Ric}}\geq \eta {\mathrm{g}}$. This shows condition (i) as in the proof of Theorem 1.1 in [@Morris3]. Conditions (ii) and (iv) are a consequence of Corollary \[Cor:App:GBGExt\], and (iii) holds because the connection is Levi-Cevita.
The Riesz transform condition is satisfied automatically for $(0,0)$ tensors, or in other words, for scalar-valued functions, as we now show.
Recall the Wietzenböck-Bochner identity $${\left\langle {\Delta}({{{\nabla}_{{{}}}}}f), {{{\nabla}_{{{}}}}}f \right\rangle}
= \frac{1}{2} {\Delta}({\left|{{{\nabla}_{{{}}}}}f\right|}^2) + {\left|{{{\nabla}_{{{}}}}}^2 f\right|}^2 + {{\rm Ric}}({{{\nabla}_{{{}}}}}f, {{{\nabla}_{{{}}}}}f)$$ for all $f \in {{\rm C}^{\infty}_{\rm {}}}({\mathcal{M}})$. When $f \in {{\rm C}^{\infty}_{\rm c}}({\mathcal{M}})$, we get ${\left\| {{{\nabla}_{{{}}}}}^2 f \right\|} \lesssim {\left\| (I + {\Delta})f \right\|}$ by integrating this identity.
The fact that ${ {\mathcal{D}}}({\Delta}) \subset {{{\rm W}^{2,2}_{\rm {}}}}({\mathcal{M}})$ follows from Proposition 3.3 in [@Hebey] as does the density of ${{\rm C}^{\infty}_{\rm c}}({\mathcal{M}})$ in ${ {\mathcal{D}}}({\Delta})$ Thus, ${\left\| {{{\nabla}_{{{}}}}}^2 u \right\|} \lesssim {\left\| (I + {\Delta})u \right\|}$ for $u \in { {\mathcal{D}}}({\Delta})$. Hence the hypotheses of Theorem \[Thm:App:KatoTen\] hold, so Theorem \[Thm:Int:KatoFn\] follows as a consequence.
Lipschitz Estimates and Stability
=================================
In this short section, we demonstrate Lipschitz estimates for the functional calculus and the stability of the square root.
Let $\tilde{a}$ and $\tilde{A}$ satisfy the same conditions as specified for $a$ and $A$ prior to Theorem \[Thm:App:KatoVB\], and set $$\tilde{B}_1 = \begin{pmatrix} \tilde{a} & 0 \\ 0 & 0 \end{pmatrix}
\quad\text{and}\quad
\tilde{B}_2 = \begin{pmatrix} 0 & 0 \\ 0 & \tilde{A} \end{pmatrix}.$$
On noting that $\tilde{B}_i$ satisfy conditions (i)-(iii) of Corollary \[Cor:Res:Stab\], we have the following Lipschitz estimate.
\[Thm:App:LipVB\] Assume the hypotheses of Theorem \[Thm:App:KatoVB\] and fix $\eta_i < \kappa_i$. Suppose that $\tilde{B_i}$ satisfy $\|{\tilde{B}_i\|}_\infty \leq \eta_i$ for $i = 1,2$ and set $0 < \hat{\omega}_i < \frac{\pi}{2}$ by $\cos\hat{\omega}_i = \frac{ \kappa_i - \eta_i}{{\left\| B_i \right\|}_\infty + \eta_i}$ and $\hat{\omega} = \frac{1}{2}(\hat{\omega}_1 + \hat{\omega}_2)$. Then, for all $\hat{\omega} < \mu < \frac{\pi}{2}$, $${\left\| f(\Pi_B) - f(\Pi_{B+\tilde{B}}) \right\|}
\lesssim (\|\tilde{B}_1\|_\infty + \|\tilde{B}_2\|_\infty)
{\left\| f \right\|}_\infty$$ for all $f \in {{\rm Hol}}^\infty(S_{\mu}^o)$, and $$\int_{0}^\infty {\left\| \psi(t\Pi_{B})v - \psi(t\Pi_{B + \tilde{B}})v \right\|}^2 \frac{dt}{t}
\lesssim (\|\tilde{B}_1\|_\infty^2 + \|\tilde{B}_2\|_\infty^2)
{\left\| v \right\|},$$ for all $\psi \in \Psi(S_{\mu}^o)$ and all $v \in {{\mathscr{H}}}$. The implicit constants depend in particular on on $B_i$ and $\eta_i$.
We use the coefficients $\tilde{a}$ and $\tilde{A}$ to perturb the coefficients $a$ and $A$. Then, we construct the following perturbed operator ${\mathrm{L}}_{A + \tilde{A}}$ defined similar to ${\mathrm{L}}_{A}$ given by ${\mathrm{L}}_{A + \tilde{A}} u = (a + \tilde{a}){{S}^\ast}(A + \tilde{A})S u$ for $u \in { {\mathcal{D}}}({\mathrm{L}}_{A + \tilde{A}})$.
\[Thm:App:StabSq\] Assume the hypotheses of Theorem \[Thm:App:KatoVB\] and fix $\eta_i < \kappa_i$. If ${\left\| \tilde{a} \right\|}_\infty \leq \eta_1$, $\|\tilde{A}\|_\infty \leq \eta_2$, then $${\left\| \sqrt{{{\mathrm{L}}_A}}\,u - \sqrt{{{\mathrm{L}}_{A + \tilde{A}}}}\,u \right\|}
\lesssim ({\left\| \tilde{a} \right\|}_\infty + \|\tilde{A}\|_\infty)
{\left\| u \right\|}_{{{{\rm W}^{1,2}_{\rm {}}}}}$$ for all $u \in {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}})$. The implicit constant depends in particular on $A, a$ and $\eta_i$.
Let $\tilde{B}_i$ be given as in the hypothesis of Theorem \[Thm:App:LipVB\], so that $\|\tilde{B}_1\|_\infty = {\left\| \tilde{a} \right\|}_\infty \leq \eta_1$ and $\|\tilde{B}_2\|_\infty = \|\tilde{A}\|_\infty \leq \eta_2$. By Theorem \[Thm:App:LipVB\], $$\label{perturb}{\left\| \operatorname{sgn}(\Pi_B)v - \operatorname{sgn}(\Pi_{B+\tilde{B}})v \right\|}
\lesssim (\|\tilde{a}\|_\infty + \|\tilde{A}\|_\infty)
{\left\| v \right\|}$$ for all $v\in {{\mathscr{H}}}$, in particular for all $v =\begin{pmatrix} 0 \\ Su \end{pmatrix}$ with $u \in {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{V}})$. Note that $v= \Pi_B\begin{pmatrix} u \\ 0 \end{pmatrix} = \Pi_{B+\tilde B}\begin{pmatrix} u \\ 0 \end{pmatrix}$ so $\operatorname{sgn}(\Pi_B)v = \begin{pmatrix} \sqrt{{{\mathrm{L}}_A}}\,u \\ 0 \end{pmatrix}$ and $\operatorname{sgn}(\Pi_{B+\tilde B})v = \begin{pmatrix} \sqrt{{{\mathrm{L}}_{A+\tilde A}}}\,u \\ 0 \end{pmatrix}$. Thus, on substitution, we obtain the desired result.
We point out that the conclusions of both these theorems hold if, instead of assuming the hypotheses of Theorem \[Thm:App:KatoVB\], we assume the hypotheses of Theorem \[Thm:App:KatoTen\]. This yields Lipschitz estimates and the stability result for $(p,q)$ tensors. We conclude this section by highlighting the stability of the square root for scalar-valued functions as a corollary.
\[Thm:App:Stab\] Suppose that ${\left|{{\rm Ric}}\right|} \leq C$, $\operatorname{inj}(M) \geq \kappa > 0$ and the following ellipticity condition holds: there exist $\kappa_1, \kappa_2 > 0$ such that $$\operatorname{Re}{\left\langle a u, u \right\rangle} \geq \kappa_1 {\left\| u \right\|}^2
\qquad\text{and}\qquad
\operatorname{Re}{\left\langle A Sv, Sv \right\rangle} \geq \kappa_2 {\left\| v \right\|}_{{{{\rm W}^{1,2}_{\rm {}}}}}^2$$ for all $u \in {{\rm L}^{2}_{\rm {}}}({\mathcal{M}})$ and $v \in {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{M}})$. Fix $\eta_i < \kappa_i$. If ${\left\| \tilde{a} \right\|}_\infty \leq \eta_1$, $\|\tilde{A}\|_\infty \leq \eta_2$, then $${\left\| \sqrt{{{\mathrm{L}}_A}}\,u - \sqrt{{{\mathrm{L}}_{A + \tilde{A}}}}\,u \right\|}
\lesssim ({\left\| \tilde{a} \right\|}_\infty + \|\tilde{A}\|_\infty)
{\left\| u \right\|}_{{{{\rm W}^{1,2}_{\rm {}}}}}$$ for all $u \in {{{\rm W}^{1,2}_{\rm {}}}}({\mathcal{M}})$. The implicit constant depends on $C, \kappa, \kappa_i, A, a$ and $\eta_i$.
Harmonic Analysis {#Sec:Harm}
=================
Carleson measure reduction {#Sec:Harm:CN}
--------------------------
It remains for us to prove Theorem \[Thm:Ass:Main\]. The main point is to show that the main local quadratic estimate in Proposition \[Prop:Ass:Main\] is a consequence of hypotheses (H1)-(H8).
The proof proceeds by reducing the main quadratic estimate to a *Carleson measure* estimate. Thus, we first recall the notion of a (local)-Carleson measure. Set ${\mathcal{M}}_{+} = {\mathcal{M}}\times (0, t_0]$, for some $t_0 < \infty$. We emphasise that we restrict our considerations to $t \leq t_0$. The *Carleson box* over ${Q_{{}}}\in {{\mathscr{Q}}}_t$ is then defined as ${\mathrm{R}}_{{Q_{{}}}} = {\overline{{Q_{{}}}}} \times (0,\operatorname{\ell}({Q_{{}}})]$. A positive Borel measure $\nu$ on ${\mathcal{M}}_{+}$ is called a *Carleson measure* if there exists $C > 0$ such that ${\nu({\mathrm{R}}_{{Q_{{}}}})} \leq C \mu({Q_{{}}})$ for all dyadic cubes ${Q_{{}}}\in {{\mathscr{Q}}}_t$ for $t \leq t_0$. The Carleson norm ${\left\| \nu \right\|}_{{\mathcal{C}}}$ is defined by $${\left\| \nu \right\|}_{{\mathcal{C}}} = \sup_{{Q_{{}}}\in {{\mathscr{Q}}}_t,\ t \leq t_0} \frac{\nu({\mathrm{R}}_{{Q_{{}}}})}{ \mu({Q_{{}}})}$$ Let ${\mathcal{C}}$ denote the set of all Carleson measures.
The reader will find a more elaborate description of Carleson measures in the classical setting in §.2 of [@stein:harm] by Stein. The local construction described here is a fraction of a larger theory explored by Morris in [@Morris] and [@Morris3].
With a description of a Carleson measure in hand, we now illustrate how to reduce the *main local quadratic estimate* to a Carleson measure estimate. The key is the following consequence of Carleson’s theorem, which is a special case of Theorem 4.3 in [@Morris3]. Recall the dyadic averaging operator ${\mathcal{A}}_t$ from §\[Sect:Prelim:GBG\].
\[Prop:Harm:AvCarl\] For all $u \in {{\mathscr{H}}}$ and for all $\nu \in {\mathcal{C}}$, $$\iint_{{\mathcal{M}}\times (0, t_0]} {\left|{\mathcal{A}}_t u\right|}^2\ d\nu(x,t)
\lesssim {\left\| u \right\|}^2 {\left\| \nu \right\|}_{{\mathcal{C}}}.$$
Further, recall that whenever $w \in {\mathbb{C}}^N$, the associated GBG constant section is given by $\omega(x) = w$ whenever $x \in {Q_{{}}}\in {{\mathscr{Q}}}_{{\mathrm{t_S}}}$ in the GBG coordinates associated to ${Q_{{}}}$. Then, we define the *principal part* as ${\upgamma}_t(x)w = (\Theta_t^B \omega)(x)$. With this notation, and for $0 < t_0 < \infty$ to be chosen later, we split the main quadratic estimate in the following way: $$\begin{aligned}
\label{quad2}
\int_{0}^{t_0} {\left\| \Theta_t^B P_t u \right\|}^2\ \frac{dt}{t}
&\lesssim \int_{0}^{t_0} {\left\| \Theta_t^B P_t u - {\upgamma}_t{\mathcal{A}}_t P_t u \right\|}^2\ \frac{dt}{t} \tag{Q2}\\
&+ \int_{0}^{t_0} {\left\| {\upgamma}_t {\mathcal{A}}_t(P_t - I)u \right\|}^2\ \frac{dt}{t} \\
&+ \int_{0}^{t_0} \int_{M} {\left|{\mathcal{A}}_t u\right|}^2 {\left|{\upgamma}_t\right|}^2\ \frac{d\mu(x)\ dt}{t}. \end{aligned}$$ We call the first two terms on the right of the *principal terms*. Proposition \[Prop:Harm:AvCarl\] then allows us to reduce estimating the last term to proving that $$A \mapsto \int_{A} {\left|{\upgamma}_t(x)\right|}^2 \frac{d\mu(x)\ dt}{t}$$ is a Carleson measure. We call this term the *Carleson term*.
Estimation of principal terms
-----------------------------
In this section, as the title suggests, we illustrate how to estimate the two principal terms of . We proceed to do so by coupling the existence of exponential off-diagonal bounds with our dyadic [Poincaré ]{}inequality and cancellation hypothesis. The estimates here are straightforward and are more or less adapted from [@AKMC], [@Morris3] and [@AAMC].
First, we quote the the following theorem of [@Morris3], which is essentially contained in [@AKMC].
\[Prop:Harm:Offdiag\] Let $U_t$ be either $R_t^B, P_t^B,Q_t^B$ or $\Theta_t^B$ for $t \in {\mathbb{R}}^+$. There exists a $C_{\Theta} > 0$, which only depends on (H1)-(H6), for every $M > 0$ there exists $c > 0$ with $${\left\| {\raisebox{\depth}{\(\chi\)}_{E}} U_t u \right\|} \leq c {\left\langle \frac{{\left|t\right|}}{d(E,F)} \right\rangle}^{M}
\exp{\left(-C_{\Theta} \frac{d(E,F)}{t}\right)} {\left\| {\raisebox{\depth}{\(\chi\)}_{F}} u \right\|}$$ whenever $E,F$ are Borel, ${{\rm spt} {\text{ }}u} \subset F$.
Next, we have the following technical lemma.
\[Lem:Harm:Indest\] Let $r > 0$ and suppose that ${{\left\{B_j = B(x_j, r)\right\}}}$ is a disjoint collection of balls. Then, whenever $\eta \geq 1$, $$\sum_{j} {\raisebox{\depth}{\(\chi\)}_{\eta B_j}} \lesssim \eta^{\kappa} {\mathrm{e}}^{4\lambda c \eta r}.$$
Fix $x \in {\mathcal{M}}$ and let ${{\mathscr{C}}}_{x} = {{\left\{x_j \in {\mathcal{M}}: x \in B(x_j,\eta r)\right\}}}$. It is easy to see that $\sum_{j} {\raisebox{\depth}{\(\chi\)}_{\eta B_j}}(x) = \operatorname{card}{{\mathscr{C}}}_{x}$. That $x_j \in {{\mathscr{C}}}_{x}$ is equivalent to saying that $d(x,x_j) < \eta r$, and therefore for any $y \in \eta B_j$, $d(x,y) \leq d(x,x_j) + d(x_j, y) < (\eta + 1) r$. That is, $B(x,(\eta + 1)r) \supset B(x_j,r)$ and by the disjointness of ${{\left\{B_j\right\}}}$, $\mu (B(x,(\eta + 1)r)) \geq \sum_{x_j \in {{\mathscr{C}}}_{x}} \mu(B(x_j,r))$.
Next, note that (\[Def:Pre:Eloc\]) implies that $ \mu(\eta B_j) \lesssim \eta^{\kappa} {\mathrm{e}}^{\lambda c \eta r} \mu(B_j)$ and therefore, $$\sum_{x_j \in {{\mathscr{C}}}_{x}} \mu(\eta B_j) \lesssim \eta^{\kappa} {\mathrm{e}}^{\lambda c \eta r} \mu(B(x,(\eta + 1)r)).$$ Thus, it is enough to compare $\mu(\eta B_j)$ to $\mu(B(x,(\eta + 1)r))$. So, take any $y \in B(x,(\eta + 1)r)$, and note that $d(x,y) \leq d(x, x_j) + d(x_j, y) < (2\eta + 1)r < 3 \eta r$. Thus, $$\mu(B(x,(\eta + 1)r))
\leq \mu(B(x_j, 3\eta r)
\lesssim 3^\kappa {\mathrm{e}}^{3\lambda c \eta r} \mu(B(x_j,\eta r))$$ and the estimate $$\operatorname{card}{{\mathscr{C}}}_{x}\ {\mathrm{e}}^{-3\lambda c \eta r}
\lesssim \sum_{x_j \in {{\mathscr{C}}}_{x}} \frac{\mu(\eta B_j)}{\mu(B(x,(\eta+1)r))}
\lesssim \eta^{\kappa} e^{\lambda c \eta r}$$ completes the proof.
For all $u \in { {\mathcal{R}}}(\Pi)$, $$\int_{0}^{t_2} {\left\| \Theta_t^B P_t u - {\upgamma}_t{\mathcal{A}}_t P_t u \right\|}^2\ \frac{dt}{t} \lesssim {\left\| u \right\|}^2$$ where $t_2 \leq \min{{\left\{\frac{{\rho}}{5},\frac{C_{\Theta}}{4\lambda c(1 + 4\tilde{c})}\right\}}}.$
Let $v = P_t u$.
(i) First, we note that $${\left\| \Theta_t^B P_t u - {\upgamma}_t{\mathcal{A}}_t P_t u \right\|}^2 =
\sum_{{Q_{{}}}\in {{\mathscr{Q}}}_t} {\left\| \Theta_t^B(v - v_{{Q_{{}}}}) \right\|}_{{{\rm L}^{2}_{\rm {}}}({Q_{{}}})}^2.$$ For each ${Q_{{}}}\in {{\mathscr{Q}}}_t$, write $B_{{Q_{{}}}} = B(x_{{Q_{{}}}}, \frac{C_1}{\delta} t) \supset {Q_{{}}}$ and $C_j({Q_{{}}}) = 2^{j+1}B_{{Q_{{}}}} \setminus 2^j B_{{Q_{{}}}}.$ Then, for each such cube ${Q_{{}}}$, $$\begin{aligned}
\int_{{Q_{{}}}} {\left|\Theta_t^B(v - v_{{Q_{{}}}})\right|}^2\ d\mu
&= \int_{{Q_{{}}}} {\left|\Theta_t^B {\left(\sum_{j=0}^\infty {\raisebox{\depth}{\(\chi\)}_{C_j({Q_{{}}})}} (v - v_{{Q_{{}}}})\right)}\right|}^2\ d\mu \\
&\leq \sum_{j=0}^\infty \int_{{Q_{{}}}} {\left|\Theta_t^B ({\raisebox{\depth}{\(\chi\)}_{C_j({Q_{{}}})}} (v - v_{{Q_{{}}}}))\right|}^2\ d\mu \\
&\lesssim \sum_{j=0}^\infty {\left\langle \frac{t}{d({Q_{{}}},C_j({Q_{{}}}))} \right\rangle}^M
\exp{\left(- C_{\Theta} \frac{d({Q_{{}}},C_j({Q_{{}}}))}{t}\right)}
\int_{C_j({Q_{{}}})} {\left|v - v_{{Q_{{}}}}\right|}^2\ d\mu\end{aligned}$$
(ii) Next, note that by (4.1) in [@Morris3] $$2^j \frac{C_1}{\delta} t \leq d(x_{{Q_{{}}}}, C_j({Q_{{}}})) \leq d({Q_{{}}}, C_j({Q_{{}}})) + \operatorname{diam}{Q_{{}}}$$ which implies that $${\left\langle \frac{t}{d({Q_{{}}},C_j({Q_{{}}}))} \right\rangle}^M \lesssim 2^{-M(j+1)}.$$ Next, for $j \geq 1$, $$d({Q_{{}}},C_j({Q_{{}}})) \geq 2^j \frac{C_1}{2\delta} t$$ and therefore, $$\exp{\left(- C_{\Theta} \frac{d({Q_{{}}},C_j({Q_{{}}}))}{t}\right)}
\leq \exp{\left(- \frac{C_{\Theta} C_1}{4\delta} 2^{j+1}\right)}.$$ For $j = 0$, $$\exp{\left(- C_{\Theta} \frac{d({Q_{{}}},C_j({Q_{{}}}))}{t}\right)}
= 1
= \exp{\left(\frac{C_{\Theta} C_1}{4\delta}\right)}
\exp{\left(- \frac{C_{\Theta} C_1}{4\delta} 2^{0}\right)}.$$ Fix $t' > 0$ to be chosen later. Then, for all $t \leq t'$, $$\exp{\left(- C_{\Theta} \frac{d({Q_{{}}},C_j({Q_{{}}}))}{t}\right)}
\lesssim \exp{\left(- \frac{C_{\Theta} C_1}{4\delta t'} 2^{j+1}t\right)}$$ for all $j \geq 0$.
(iii) Combining the estimates in (i) and (ii), $$\int_{{Q_{{}}}} {\left|\Theta_t^B(v - v_{{Q_{{}}}})\right|}^2\ d\mu
\lesssim \sum_{j=0}^\infty 2^{-M(j+1)} \exp{\left(-\frac{C_{\Theta} C_1}{4\delta t'} 2^{j+1}t\right)}
\int_{C_j({Q_{{}}})} {\left|v - v_{{Q_{{}}}}\right|}^2\ d\mu.$$
Since $v = P_t u \in { {\mathcal{D}}}(\Pi^2) = { {\mathcal{D}}}(\Pi){\cap}{ {\mathcal{R}}}(\Pi)$, we conclude from (H8)-1 that $$\begin{aligned}
\int_{C_j({Q_{{}}})} {\left|v - v_{{Q_{{}}}}\right|}^2\ d\mu
&\lesssim {\left(1 + {\left(\frac{C_1}{\delta}\right)}^\kappa 2^{\kappa(j+1)}
\exp{\left(\frac{\lambda c C_1}{\delta} 2^{j+1} t\right)}\right)}
{\left(\frac{C_1}{\delta}\right)}^2 2^{2 (j+1)} \\
&\qquad\qquad
t^2 \int_{\tilde{c} 2^{j+1}B_{{Q_{{}}}}} ({\left|\Xi v\right|}^2 + {\left|v\right|}^2)\ d\mu.\end{aligned}$$
Therefore, $$\begin{aligned}
\sum_{{Q_{{}}}\in {{\mathscr{Q}}}_t} \int_{{Q_{{}}}} &{\left|\Theta_t^B(v - v_{{Q_{{}}}})\right|}^2\ d\mu \\
&\lesssim \sum_{{Q_{{}}}\in {{\mathscr{Q}}}_t} \sum_{j=0}^\infty 2^{-M(j+1)}
\exp{\left(-\frac{C_{\Theta} C_1}{4\delta} 2^{j+1}\right)} \\
&\qquad\qquad {\left(1 + {\left(\frac{C_1}{\delta}\right)}^\kappa 2^{\kappa(j+1)}
\exp{\left(\frac{\lambda c C_1}{\delta} 2^{j+1} t\right)}\right)}
{\left(\frac{C_1}{\delta}\right)}^2 2^{2 (j+1)} t^2 \\
&\qquad\qquad\int_{\tilde{c} 2^{j+1}B_{{Q_{{}}}}} ({\left|\Xi v\right|}^2 + {\left|v\right|}^2)\ d\mu \\
&\lesssim \sum_{j=0}^\infty 2^{-(M-\kappa-2)(j+1)}
{\left[ \exp{\left(-\frac{C_{\Theta} C_1}{4\delta t'} 2^{j+1} t\right)}
+
\exp{\left(-\frac{C_1}{\delta}{\left(\frac{C_{\Theta}}{4 t'}- \lambda c\right)} 2^{j+1} t\right)}\right]} \\
&\qquad\qquad t^2 \int_{{\mathcal{M}}} \sum_{{Q_{{}}}\in {{\mathscr{Q}}}_t} {\raisebox{\depth}{\(\chi\)}_{\tilde{c} 2^{j+1}B_{{Q_{{}}}}}}
({\left|\Xi v\right|}^2 + {\left|v\right|}^2)\ d\mu.\end{aligned}$$
(iv) Set $\eta = \tilde{c} 2^{j+1}$ and $r = \frac{C_1}{\delta}t$ and invoke Lemma \[Lem:Harm:Indest\] to conclude that $${\raisebox{\depth}{\(\chi\)}_{\tilde{c} 2^{j+1}B_{{Q_{{}}}}}} \lesssim 2^{\kappa(j+1)} \exp{\left(\frac{4\lambda c \tilde{c} C_1}{\delta} 2^{j+1}t\right)}.$$
Combining this with (iii), we have $$\begin{aligned}
\sum_{{Q_{{}}}\in {{\mathscr{Q}}}_t} \int_{{Q_{{}}}} &{\left|\Theta_t^B(v - v_{{Q_{{}}}})\right|}^2\ d\mu \\
&\lesssim \sum_{j=0} ^\infty 2^{-(M-2\kappa - 2)(j+1)} \\
&\qquad\qquad{\left[ \exp{\left(-\frac{C_1}{\delta}{\left(\frac{C_{\Theta}}{4 t'} - 4\lambda c \tilde{c}\right)} 2^{j+1} t\right)}
+
\exp{\left(- \frac{C_1}{\delta}{\left(\frac{C_{\Theta}}{4 t' } - \lambda c - 4\lambda c \tilde{c}\right)} 2^{j+1} t\right)}\right]} \\
&\qquad\qquad t^2 {\left({\left\| \Xi v \right\|}^2 + {\left\| v \right\|}^2\right)}\end{aligned}$$
(v) Now, we choose $t' \leq \frac{{\rho}}{5}$ so that $$-\frac{C_1}{\delta}{\left(\frac{C_{\Theta}}{4 t'} - 4 \lambda c \tilde{c}\right)} \leq 0
\quad\text{and}\quad
- \frac{C_1}{\delta}{\left(\frac{C_{\Theta}}{4 t' } - \lambda c - 4 \lambda c \tilde{c}\right)} \leq 0.$$ That is, $$\frac{C_{\Theta}}{4\lambda c (1 + 4\tilde{c})} \leq t'.$$ We can, therefore, set $t' = t_2$ and then, $$\sum_{{Q_{{}}}\in {{\mathscr{Q}}}_t} \int_{{Q_{{}}}} {\left|\Theta_t^B(v - v_{{Q_{{}}}})\right|}^2\ d\mu
\lesssim t^2 \sum_{j=0} ^\infty 2^{-(M-2\kappa -2)(j+1)} {\left\| \Pi v \right\|}^2$$ by invoking (H8)-2. By choosing $M > 2\kappa + 2$, and substituting $v = P_t u$, we have $$\int_{0}^{t_2} {\left\| \Theta_t^B P_t u - {\upgamma}_t{\mathcal{A}}_t P_t u \right\|}^2\ \frac{dt}{t}
\lesssim \int_{0}^{t_2} {\left\| t \Pi P_t u \right\|}^2\ \frac{dt}{t}
\leq \int_{0}^{\infty} {\left\| t \Pi P_t u \right\|}^2\ \frac{dt}{t}
\lesssim {\left\| u \right\|}.$$
Next, as in [@AKMC] and [@Morris3], we note the following consequence of (H7).
\[Lem:Harm:Upsilon\] Let $\Upsilon = \Gamma,\ {{\Gamma}^\ast}$ or $\Pi$. Then, $${\left|\fint_{{Q_{{}}}} \Upsilon u\ d\mu\right|}
\lesssim \frac{1}{\operatorname{\ell}({Q_{{}}})^\eta}
{\left(\fint_{{Q_{{}}}} {\left|u\right|}\ d\mu\right)}^{\frac{\eta}{2}}
{\left(\fint_{{Q_{{}}}} {\left|\Upsilon u\right|}\ d\mu\right)}^{1 - \frac{\eta}{2}}
+ \fint_{{Q_{{}}}} {\left|u\right|}^2$$ for all $ u \in { {\mathcal{D}}}(\Upsilon)$ and ${Q_{{}}}\in {{\mathscr{Q}}}_t$ where $t \leq {\mathrm{t_S}}$.
The proof of this lemma is the same as that of the proof of Lemma 5.9 in [@Morris3]. Thus, we deduce the following.
For all $u \in {{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$, we have $$\int_{0}^{{\mathrm{t_S}}}
{\left\| {\upgamma}_t {\mathcal{A}}_t(P_t - I)u \right\|}^2 \frac{dt}{t} \lesssim {\left\| u \right\|}^2.$$
The proof of this proposition is similar to the proof of Proposition 5.10 in [@Morris3] with the principle difference being that we only consider $t \leq {\mathrm{t_S}}$.
We have demonstrated how to estimate the two principal terms of .
Carleson measure estimate
-------------------------
We are left with the task of estimating the final term of . We follow in the footsteps of [@AHLMcT], [@AKMC] remarking that, in a sense, it is to preserve the main thrust of this Carleson argument that we have constructed the various technologies in this paper. We show in this section that the argument runs as before with some changes that are possible as a consequence of the geometric assumptions which we have made.
In §5 of [@Morris3], Morris illustrates how to prove $$A \mapsto \int_{A} {\left|{\upgamma}_t(x)\right|}^2\ d\mu(x)\frac{dt}{t}$$ is a local Carleson measure. At the heart of this proof lies a certain test function. Here, we illustrate how to use the GBG condition to set up a suitable substitute test function ${\mathrm{f}}_{{Q_{{}}},{\varepsilon}}^w$. The main complication is to choose a cutoff in a way that we stay inside GBG coordinates of each cube.
Let $\tau > 1$ be chosen later. We note that we can find a smooth function $\eta_{{Q_{{}}}}: {\mathcal{M}}\to [0,1]$ such that $\eta = 1$ on $B(x_{{Q_{{}}}}, \tau C_{1}\operatorname{\ell}({Q_{{}}}))$ and $\eta = 0$ on ${\mathcal{M}}\setminus B(x_{{Q_{{}}}}, 2 \tau C_{1}\operatorname{\ell}({Q_{{}}}))$ and satisfying the gradient bound ${\left|{{{\nabla}_{{{}}}}}\eta\right|} \lesssim \frac{1}{\operatorname{\ell}({Q_{{}}})}.$ The constant here depends on $\tau$.
Let ${\widehat{{Q_{{}}}}}$ be the GBG cube with centre $x_{{\widehat{{Q_{{}}}}}}$. We want to use $\eta$ to perform a cutoff inside the ball $B(x_{{Q_{{}}}},{\rho})$. That is, we want the ball $B(x_{{Q_{{}}}},2 \tau C_{1}\operatorname{\ell}({Q_{{}}})) \subset B(x_{{Q_{{}}}},{\rho})$. A simple calculation then yields that we need to choose $\tau < 3$. Thus, for the sake of the argument, let us fix $\tau = 2$.
Next, let $v \in {\mathcal{L}}({\mathbb{C}}^N)$ and ${\left|v\right|} = 1$. Let $\hat{w}, w \in {\mathbb{C}}^N$ such that ${\left|\hat{w}\right|} = {\left|w\right|} = 1$ and ${{v}^\ast}(\hat{w}) = w$. Let $\tilde{w} = \eta\, w$. Certainly, we have that $\tilde{w} \in {{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$ since $\tilde{w} = 0$ outside of $B(x_{{\widehat{{Q_{{}}}}}}, {\rho})$. Now, fix ${\varepsilon}> 0$ and define $${\mathrm{f}}_{{Q_{{}}},{\varepsilon}}^w = \tilde{w} - \imath {\varepsilon}\operatorname{\ell}({Q_{{}}})
\Gamma R_{{\varepsilon}\operatorname{\ell}({Q_{{}}})}^B \tilde{w}
= (1 + \imath {\varepsilon}\operatorname{\ell}({Q_{{}}}) {{\Gamma}^\ast}_B R_{{\varepsilon}\operatorname{\ell}({Q_{{}}})}^B) \tilde{w}.$$ This allows us to prove a lemma similar to that of Lemma 5.12 in [@Morris3].
There exists $c > 0$ such that for all ${Q_{{}}}\in {{\mathscr{Q}}}_t$ with $t \leq {\mathrm{t_S}}$, $${\left\| {\mathrm{f}}_{{Q_{{}}},{\varepsilon}}^w \right\|} \leq c \mu({Q_{{}}})^{\frac{1}{2}}
,\quad
\iint_{{\mathrm{R}}_{{Q_{{}}}}} {\left|\Theta_{t}^B{\mathrm{f}}_{{Q_{{}}},{\varepsilon}}^w\right|}^2\ d\mu\frac{dt}{t}
\leq c \frac{\mu({Q_{{}}})}{{\varepsilon}^2}
\quad\text{and}\quad
{\left|\fint_{{Q_{{}}}} {\mathrm{f}}_{{Q_{{}}},{\varepsilon}}^w - w\ d\mu \right|}
\leq c {\varepsilon}^{\frac{\eta}{2}}.$$
The proof of this lemma proceeds exactly as the proof of Lemma 5.12 in [@Morris3]. The last estimate relies upon Lemma \[Lem:Harm:Upsilon\].
Setting ${\varepsilon}= {\left(\frac{1}{2c}\right)}^{\frac{\eta}{2}}$ we obtain the same conclusion as the author of [@Morris3] that $$\operatorname{Re}{\left\langle w, \fint_{{Q_{{}}}} f_{{Q_{{}}}}^w \right\rangle} \geq \frac{1}{2}$$ where we have set $f_{{Q_{{}}}}^w = f_{{Q_{{}}},{\varepsilon}}^w$. Furthermore, a stopping time argument as in Lemma 5.11 in [@AKMC] yields the following.
Let $t_3 = \min{{\left\{\frac{{\rho}}{5}, \frac{C_{\Theta}}{4a^3 \lambda}\right\}}}$. There exists $\alpha, \beta > 0$ such that for all ${Q_{{}}}\in {{\mathscr{Q}}}_t$ with $t \leq t_3$, there exists a collection of subcubes ${{\left\{{Q_{k}}\right\}}} \subset {\cup}_{t \leq t_3} {{\mathscr{Q}}}_t$ of ${Q_{{}}}$ such that $E_{{Q_{{}}}} = {Q_{{}}}\setminus {\cup}_k {Q_{k}}$ satisfies $\mu(E_{{Q_{{}}}}) \geq \beta \mu({Q_{{}}})$ and $E^\ast_{{Q_{{}}}} = {\mathrm{R}}_{{Q_{{}}}} \setminus {\cup}_k {\mathrm{R}}_{{Q_{k}}}$ satisfies $$\operatorname{Re}{\left\langle w, \fint_{{Q_{{}}}'} f^w_{{Q_{{}}}} \right\rangle} \geq \alpha
\quad\text{and}\quad
\fint_{{Q_{{}}}'} {\left|f_{{Q_{{}}}}^w\right|} \leq \frac{1}{\alpha}$$ whenever ${Q_{{}}}' \in {\cup}_{t \leq t_3} {{\mathscr{Q}}}_t$ with ${Q_{{}}}' \subset {Q_{{}}}$ and ${\mathrm{R}}_{{Q_{{}}}'} {\cap}E^{\ast}_{{Q_{{}}}} = {\varnothing}$.
Let $\sigma > 0$ to be chosen later and let $\nu \in {\mathcal{L}}({\mathbb{C}}^N)$ with ${\left|\nu\right|} = 1$ and define $$K_{\nu,\sigma} = {{\left\{ \nu' \in {\mathcal{L}}({\mathbb{C}}^N) \setminus{{\left\{0\right\}}}:
{\left| \frac{ \nu'}{{\left|\nu'\right|}} - \nu\right|} \leq \sigma\right\}}}.$$
Let $t_3 = \min{{\left\{\frac{{\rho}}{5}, \frac{C_{\Theta}}{4a^3 \lambda}\right\}}}$. There exists $\sigma, \beta, c > 0$ such that for all ${Q_{{}}}\in {{\mathscr{Q}}}_t$ with $t \leq t_3$, and $\nu \in {\mathcal{L}}({\mathbb{C}}^N)$ with ${\left|\nu\right|} = 1$, there exists a collection of subcubes ${{\left\{{Q_{k}}\right\}}} \subset {\cup}_{t \leq t_3} {{\mathscr{Q}}}_t$ of ${Q_{{}}}$ such that $E_{{Q_{{}}}} = {Q_{{}}}\setminus {\cup}_k {Q_{k}}$ satisfies $\mu(E_{{Q_{{}}}}) \geq \beta \mu({Q_{{}}})$ and $E^\ast_{{Q_{{}}}} = {\mathrm{R}}_{{Q_{{}}}} \setminus {\cup}_k {\mathrm{R}}_{{Q_{k}}}$ satisfies $$\iint_{(x,t) \in E^\ast_{{Q_{{}}}},\ {\upgamma}_t(x) \in K_{\nu,\sigma}}
{\left|{\upgamma}_t(x)\right|}^2\ d\mu(x) \frac{dt}{t} \leq c \mu({Q_{{}}}).$$
Then, the argument immediately following Proposition 5.11 in [@Morris3] illustrates that $$\iint_{{\mathrm{R}}_{{Q_{{}}}}} {\left|{\upgamma}_t(x)\right|}^2\ d\mu(x) \frac{dt}{t}
\lesssim \mu({Q_{{}}})$$ and this is the required Carleson-measure estimate.
On choosing $t_0 = \min{{\left\{t_2,t_3\right\}}}$ and applying Proposition \[Prop:Harm:AvCarl\], we find that we have bounded all three terms on the right of , and hence deduced the estimate of Proposition \[Prop:Ass:Main\]: $$\int_{0}^{t_0} {\left\| \Theta_{t}^B P_t u \right\|}^2\ \frac{dt}{t}
\lesssim {\left\| u \right\|}^2.$$
The invariance of (H1)-(H8) upon replacing $(\Gamma,B_1, B_2)$ by $({{\Gamma}^\ast},B_2,B_1)$, $({{\Gamma}^\ast},{{B}^\ast}_2,B_1)$, and $({{\Gamma}^\ast}, {{B}^\ast}_1, {{B}^\ast}_2)$ proves Theorem \[Thm:Ass:Main\] via Proposition \[Prop:Ass:Main\].
By establishing Theorem \[Thm:Ass:Main\], we are now in a position to enjoy the full thrust of its consequences. The first is the Kato square root type estimate for perturbations of Dirac type operators on vector bundles as listed in Corollary \[Cor:Ass:Main\]. A consequence of these corollaries is the Kato square root problem for vector bundles, as described in Theorem \[Thm:App:KatoVB\], the Kato square root problem for finite rank tensors as described in Theorem \[Thm:App:KatoTen\] and lastly, the highlighted theorem of this paper, Theorem \[Thm:Int:KatoFn\], the Kato square root problem for functions. Furthermore, we also can enjoy the holomorphic dependency results of §\[Sect:Res:Stab\], in particular, Corollary \[Cor:Res:Stab\], which illustrates the stability of the functional calculus under small perturbations.
Extension to Measure Metric Spaces
==================================
In this section, we extend the quadratic estimates to a setting where ${\mathcal{M}}$ is replaced by an exponentially locally doubling measure metric space ${\mathcal{X}}$. As a consequence, we also drop the smoothness assumption on the vector bundle ${\mathcal{V}}$. Similar quadratic estimates on doubling measure metric spaces for trivial bundles are obtained by the first author in [@B].
To be precise, let ${\mathcal{X}}$ be a complete metric space with metric $d$ and let $d\mu$ be a Borel-regular exponentially locally doubling measure. That is, we assume that $d\mu$ satisfies (\[Def:Pre:Eloc\]) with ${\mathcal{X}}$ in place of ${\mathcal{M}}$. The underlying space now lacks a differentiable structure and it no longer makes sense to ask the local trivialisations and the metric ${\mathrm{h}}$ to be smooth. Instead, we simply require them to be continuous. However, we remark that in applications, the local trivialisations would normally be Lipschitz. The fact that $d\mu$ is Borel implies that the local trivialisations are measurable. Furthermore, we assume that ${\mathcal{V}}$ satisfies the GBG condition.
With the exception of (H6), no changes need be made to the hypotheses to (H1)-(H8) in this new setting. To define a suitable (H6), we first define the following quantity as in [@B].
For $\xi: {\mathcal{X}}\to {\mathbb{C}}^N$ Lipschitz, define $\operatorname{Lip}\xi: {\mathcal{X}}\to {\mathbb{R}}$ by $$\operatorname{Lip}\xi(x) = \limsup_{y \to x} \frac{{\left|\xi(x) - \xi(y)\right|}}{d(x,y)}.$$ We take the convention that $\operatorname{Lip}\xi (x) = 0$ when $x$ is an isolated point.
We then define (H6) as in [@B].
1. For every bounded Lipschitz function $\xi: {\mathcal{X}}\to {\mathbb{C}}$, multiplication by $\xi$ preserves ${ {\mathcal{D}}}(\Gamma)$ and ${{\mathrm{M}}}_{\xi} = [\Gamma, \xi I]$ is a multiplication operator. Furthermore, there exists a constant $m > 0$ such that ${\left|{{\mathrm{M}}}_{\xi} (x)\right|} \leq m {\left|\operatorname{Lip}{\xi}(x)\right|}$ for almost all $x \in {\mathcal{X}}$.
Thus, we have the following theorem.
\[Thm:MM:Main\] Let ${\mathcal{X}}$ be a complete metric space equipped with a Borel-regular measure $d\mu$ satisfying (\[Def:Pre:Eloc\]). Suppose that that $(\Gamma,B_1,B_2)$ satisfy (H1)-(H8). Then, $\Pi_B$ satisfies the quadratic estimate $$\int_{0}^\infty {\left\| Q_t^B u \right\|}^2\ \frac{dt}{t}
\simeq {\left\| u \right\|}^2$$ for all $u \in {\overline{{ {\mathcal{R}}}(\Pi_B)}} \subset {{\rm L}^{2}_{\rm {}}}({\mathcal{V}})$ and hence has a bounded holomorphic functional calculus.
First, we note that much of the local Carleson theory was originally proved by Morris in §4.3 of [@Morris] in the setting of an exponentially doubling measure metric space. Next, the off-diagonal estimates can be obtained by using the Lipschitz separation Lemma 5.1 in [@B]. Also, the construction of the test function and the proof of Lemma \[Lem:Harm:Upsilon\] proceeds similar to the argument in [@B]. Thus the arguments of §\[Sec:Harm\] hold and the theorem is proved by Proposition \[Prop:Ass:Main\].
As before, we have the following corollaries. The $E_B^{\pm}$ are the spectral subspaces defined in §\[Sect:Res\].
\[Cor:MM:Main\] [ ]{}
(i) There is a spectral decomposition $${{\rm L}^{2}_{\rm {}}}({\mathcal{V}}) = { {\mathcal{N}}}(\Pi_B) \oplus E_B^{+} \oplus E_B^{-}$$ (where the sum is in general non-orthogonal), and
(ii) ${ {\mathcal{D}}}(\Gamma) {\cap}{ {\mathcal{D}}}({{\Gamma}^\ast}_B) = { {\mathcal{D}}}(\Pi_B) = { {\mathcal{D}}}(\sqrt{{\Pi_B}^2})$ with $${\left\| \Gamma u \right\|} + {\left\| {{\Gamma}^\ast}_B u \right\|}
\simeq {\left\| \Pi_B u \right\|}
\simeq \|\sqrt{{\Pi_B}^2}u\|$$ for all $u \in { {\mathcal{D}}}(\Pi_B)$.
Let ${{\mathscr{H}}}, \Gamma, B_1, B_2, \kappa_1, \kappa_2$ satisfy (H1)-(H8) and take $\eta_i < \kappa_i$. Set $0 < \hat{\omega}_i < \frac{\pi}{2}$ by $\cos\hat{\omega}_i = \frac{ \kappa_i - \eta_i}{{\left\| B_i \right\|}_\infty + \eta_i}$ and $\hat{\omega} = \frac{1}{2}(\hat{\omega}_1 + \hat{\omega}_2)$. Let $A_i \in {{\rm L}^{\infty}_{\rm {}}}({\mathcal{L}}({\mathcal{V}}))$ satisfy
(i) ${\left\| A_i \right\|}_\infty \leq \eta_i$,
(ii) $A_1 A_2 { {\mathcal{R}}}(\Gamma), B_1 A_2 { {\mathcal{R}}}(\Gamma),
A_1 B_2 { {\mathcal{R}}}(\Gamma) \subset { {\mathcal{N}}}(\Gamma)$, and
(iii) $A_2 A_1 { {\mathcal{R}}}({{\Gamma}^\ast}), B_2 A_1 { {\mathcal{R}}}({{\Gamma}^\ast}),
A_2 B_1 { {\mathcal{R}}}({{\Gamma}^\ast}) \subset { {\mathcal{N}}}({{\Gamma}^\ast})$.
Letting $\hat{\omega} < \mu < \frac{\pi}{2}$, we have:
(i) for all $f \in {{\rm Hol}}^\infty(S_{\mu}^o)$, $${\left\| f(\Pi_B) - f(\Pi_{B+A}) \right\|}
\lesssim ({\left\| A_1 \right\|}_\infty + {\left\| A_2 \right\|}_\infty)
{\left\| f \right\|}_\infty,\ \text{and}$$
(ii) for all $\psi \in \Psi(S_{\mu}^o)$, $$\int_{0}^\infty {\left\| \psi(t\Pi_{B})u - \psi(t\Pi_{B + A})u \right\|}^2 \frac{dt}{t}
\lesssim ({\left\| A_1 \right\|}_\infty^2 + {\left\| A_2 \right\|}_\infty^2)
{\left\| u \right\|},$$ whenever $u \in {{\mathscr{H}}}$.
The implicit constants depend on (H1)-(H8) and $\eta_i$.
\[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
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|
---
author:
- 'Mikael Chala,'
- 'Claudius Krause,'
- and Germano Nardini
bibliography:
- 'references.bib'
title: Signals of the electroweak phase transition at colliders and gravitational wave observatories
---
Introduction
============
We accurately measured the Higgs mass and its couplings to the heavy SM fermions and gauge bosons [@Khachatryan:2016vau], but the ElecroWeak (EW) sector remains very uncertain. Within the current constraints, there is still room for a vast variety of phenomena that exhibit intriguing signatures. One of them is the possibility that the Higgs field produces gravitational waves when it acquires a Vacuum Expectation Value (VEV) [@Witten:1984rs; @Hogan:1986qda; @Kamionkowski:1993fg]. For this to happen, the EW Symmetry Breaking (EWSB) must proceed via a Strong First Order EW Phase Transition (SFOEWPT). This is only possible if physics beyond the Standard Model (SM) exists, as such a transition requires the finite-temperature Higgs potential to behave radically differently from the one of the SM [@Kajantie:1996mn; @Rummukainen:1998as; @Laine:1998jb; @Csikor:1998eu] [^1].
Numerous extensions of the SM exhibiting a SFOEWPT have been considered in the literature. In most of the cases, the main ingredient to depart from the SM finite-temperature Higgs potential is to invoke new light particles in the thermal plasma coupled to the Higgs [@Quiros:1999jp; @Morrissey:2012db]. In general, making these new light fields naturally compatible with the present LHC constraints requires to rely on either extra symmetries or particular parameter regions. The strategies to test these scenarios are therefore very model dependent. However, new light particles are not a necessary ingredient to achieve a SFOEWPT. Higher-dimensional operators, obtained by integrating out heavy fields, can also provide large non-SM contributions to the Higgs potential. In this case, the lack of observation of additional particles would not be ascribed to circumstantial conditions, but simply to a considerable gap between the EW scale and the new physics scale, $f$.
At the EW scale, the theory with $\mathcal O(f)$-mass fields can be described by an effective Lagrangian containing the SM interactions as well as a tower of effective operators suppressed by powers of $1/f$. Among these operators, the interactions $\mathcal O_n=(\phi^\dagger \phi)^{\frac{n}{2}}$ have a radical impact on the Higgs potential (here $\phi$ is the Higgs EW doublet and $n$ an even integer larger than four). Refs. [@Zhang:1992fs; @Grojean:2004xa; @Bodeker:2004ws; @Delaunay:2007wb; @Grinstein:2008qi; @Damgaard:2015con; @deVries:2017ncy] studied in detail the dynamics of the EWPT in the presence of only $\mathcal O_6$. They showed that, in order for the EWSB to proceed via a SFOEWPT, the new physics scale must be $f\lesssim 600$ GeV if its couplings are of order one. The small gap between the EW scale $v\sim 246$ GeV and the required $f$ carries two major implications. *(i)* It points out that the EFT to dimension six is inaccurate. Any observable related to the EWPT receives corrections of order $\sim v^2/f^2\gtrsim 20\%$. The next tower of effective interactions, namely $\mathcal O_8$, must be included. *(ii)* It triggers the question of which new physics, at a scale of few hundreds GeV can produce such large modifications of the Higgs potential without being constrained by other Higgs measurements or direct LHC searches. We address these two points in this paper.
Thus, in section \[sec:ewpt\], we present the analysis of the EWPT in this extended EFT. We investigate the validity of the mean-field approximation. Moreover, we accurately determine the regions of the parameter space leading to the SFOEWPT, and characterize the consequent gravitational wave spectrum. We also identify the precise values of the coefficients of $\mathcal O_6$ and $\mathcal O_8$ that the future gravitational wave observatory LISA can test. Finally, we compare this region with the one that can be tested at colliders, sensitive to $\mathcal O_6$ and $\mathcal O_8$ via the Higgs self coupling measurements. Next, in section \[sec:weakly\], we discuss those models that can be matched to the EFT above without conflicting with current data. Among the most natural candidates, we single out a weakly-coupled custodial quadruplet extension. We study its phenomenology and find that at the LHC the most promising search for such an extension is to look for multi-lepton signals. Section \[sec:conclusions\] is devoted to our conclusions.
The electroweak phase transition in the EFT to dimension eight {#sec:ewpt}
==============================================================
Let us consider the SM extended with the effective operators $\mathcal O_6$ and $\mathcal O_8$, the relevant Lagrangian being $$L = L_\text{SM} + \frac{c_6}{f^2}(\phi^\dagger\phi)^3 + \frac{c_8}{f^4}(\phi^\dagger\phi)^4~,$$ where $L_\text{SM}$ is the SM Lagrangian, $\phi$ is the Higgs doublet and $f$ stands for the scale of new physics. In this section we determine the VEV of the Higgs at the critical and nucleation temperatures, $v_{T_c}$ and $v_{T_n}$, the latent heat of the phase transition, $\alpha$, and the inverse duration time of the phase transition, $\beta/H$, in this non-minimal EFT. The results we obtain extend those previously obtained in the literature (see *e.g.* Refs. [@Grojean:2004xa; @Delaunay:2007wb]), where only $\mathcal O_{6}$ has been considered (despite the low cutoff and the consequent potential breaking of the EFT approach).
Finite temperature potential {#sec:finite}
----------------------------
The first ingredient we need is the Coleman-Weinberg effective potential at finite temperature; see Ref. [@Quiros:1999jp] for a review. In the Landau gauge and in the $\overline{MS}$ renormalization scheme, the one-loop effective potential $V_{1\ell}$ of our EFT scenario can be expressed as $$\begin{aligned}
\label{eq:V1ell}
V_{1\ell} = V_{\rm tree}+\Delta V_{1\ell} ~,\end{aligned}$$ with $$\begin{aligned}
\label{eq:Vtree}
V_{\rm tree} &=& -\frac{\mu^2}{2} h_c^2 +\frac{\lambda}{4} h_c^4 +\frac{c_6}{8 f^2} h_c^6+\frac{c_{8}}{16f^{4}} h_c^8~,\\
\Delta V_{1\ell}&=&\Delta V_{1\ell, T=0} + V_{1\ell,T\ne 0}~,\\ \label{eq:VT0}
\Delta V_{1\ell, T=0} &=& \sum_{i=h,\chi,W,Z,t}\frac{n_i\, m_i^2(h_c)}{64 \pi^2} \left( \log\frac{m_i^4(h_c)}{v^2} - C_i \right)~,\\
V_{1\ell,T\ne 0} &= & \frac{n_t\, T^4}{2 \pi^2} J_f\left(m_t^2(h_c)/T^2\right)
+
\sum_{i=h,\chi,W,Z}
\frac{n_i\, T^4}{2 \pi^2} J_b\left(m_i^2(h_c)/T^2\right)
~,\label{eq:VT}
\label{eq:CWpot}\end{aligned}$$ where $\Delta V_{1\ell, T=0}$ is the temperature-independent one-loop contribution and $V_{1\ell,T\ne 0}$ is the (one-loop) remaining part. The variable $h_c$ is a constant background field of the Higgs. In Eq. , $n_i$ are the degrees of freedom $n_W=2 n_Z= 2 n_\chi = 6 n_h = -n_t/2 =6$, while $C_i$ is equal to 5/6 for gauge bosons and 3/2 for scalars and fermions. The $h_c$-dependent squared masses $m_i^2$ are $$\begin{aligned}
m_h^2(h_c) &=& -\mu^2 + 3 \lambda\, h_c^2 + \frac{15 c_6}{4 f^2} h_c^4 + \frac{7 c_8}{2 f^4}h_c^6 ~,\label{eq:masses} \\
m_\chi^2(h_c) &=& -\mu^2 + \lambda\, h_c^2 + \frac{3 c_6}{4 f^2} h_c^4 + \frac{c_8}{2 f^4}h_c^6 ~,\label{eq:masses1} \\
m_t^2(h_c)&=&\frac{y_{t}^2}{2} h_c^2~,\quad m_W^2(h_c)=\frac{g^2}{4} h_c^2 ~,\quad m_Z^2(h_c)=\frac{g^2 +g'^2}{4} h_c^2 ~.\label{eq:masses2} \end{aligned}$$ The explicit expression of the functions $J_b$ and $J_f$, with or without the hard thermal loop resummation, can be found *e.g.* in Ref. [@Quiros:1999jp; @Laine:2016hma]. Since our main results turn out to be quite insensitive to details, we can set the Yukawa, $SU(2)_L$ and $U(1)_Y$ gauge couplings at tree level by fixing $m_t(v)$, $m_W(v)$ and $m_Z(v)$ in Eq. at 172, 80 and 91 GeV, respectively. For the mean-field estimates, in which zero-temperature one-loop corrections are neglected, we moreover constrain $\mu^2$ at tree level by requiring $V_{\rm tree}$ to have a minimum at $h_c=v$: $$\label{eq:mu_mean8}
\mu^2=\lambda v^2 + \frac{3\, c_6}{4\, f^2} v^4 + \frac{c_8}{2\, f^4} v^6~.$$ Similarly, to set $\lambda$, we require $\partial^2 V(\phi_c)/\partial h_c^2|_{h_c=v}=(125\,{\rm GeV})^2$, which implies $$\label{eq:lam_mean8}
\lambda = -\frac{3\,c_6}{2\, f^2} v^2 - \frac{3\,c_8}{2\, f^4} v^4 + \frac{m_h^2}{2 v^2}~.$$ The remaining free parameters in $V_{1\ell}$ are therefore $c_6/f^2$ and $c_8/f^4$. Notice that the EFT is a valid description of the theory only at energy scales much below $f$, therefore we do not address questions of the stability of the potential. Thus, we do not exclude *a priori* all values of $c_8$ and $c_6$ leading to $V_{1\ell}$ unbounded from below; we only require $$\label{eq:stability}
V_{\rm tree}(v) < V_{\rm tree}(h_c) \quad \textrm{for any}~ h_c \in\, ]v,f]~.$$ This in practice corresponds to imposing a lower bound on $c_8$ that varies with $f$. Such constraint is $c_8\gtrsim -9$ for $f = 1$ TeV and $c_8\gtrsim -2$ for $f = 2$ TeV. For concreteness, we limit the plots hereafter to the first bound.
Figure \[fig:V\] shows the typical classes of potentials that we consider: Cases where the potential has a tree level barrier between the minima (left panel), cases where such a barrier is only due to a finite temperature (one-loop) effect (central panel), and cases where the potential is unbounded from below but the instability arises at a scale larger than $f$ (right panel). See Ref. [@Chung:2012vg] for phenomenological discussions of new physics models in each class.\
![*The potentials $V_{\rm tree}$ (black solid curve), $V_{1\ell}$ at $T=0$ (orange dashed curve) and $V_{1\ell}$ at $T=T_x$ (green dashed curve) for the choices of $c_6/f^2$ and $c_8/f^4$ indicated in each panel. In the left panel, there exist two vacua already at zero temperature ($\mu^2\simeq -3100\,GeV^2$, $\lambda\simeq-0.23$, $T_x = T_c = 35\,GeV$). In the central panel, the existence of two vacua arises only at finite temperature ($\mu^2=1900\,GeV^2$, $\lambda=-0.06$, $T_x=T_c=82\,GeV$). In the right panel, the potential is unbounded from below, but the instability scale is above the cutoff $f=1\,$TeV ($\mu^2=3000\,GeV^2$, $\lambda=-0.03$, $T_x = T_n=99\,GeV$). $T_c$ is the critical temperature obtained in the mean-field approximation.*[]{data-label="fig:V"}](./plot_4-0.pdf "fig:"){width="0.333\columnwidth"} ![*The potentials $V_{\rm tree}$ (black solid curve), $V_{1\ell}$ at $T=0$ (orange dashed curve) and $V_{1\ell}$ at $T=T_x$ (green dashed curve) for the choices of $c_6/f^2$ and $c_8/f^4$ indicated in each panel. In the left panel, there exist two vacua already at zero temperature ($\mu^2\simeq -3100\,GeV^2$, $\lambda\simeq-0.23$, $T_x = T_c = 35\,GeV$). In the central panel, the existence of two vacua arises only at finite temperature ($\mu^2=1900\,GeV^2$, $\lambda=-0.06$, $T_x=T_c=82\,GeV$). In the right panel, the potential is unbounded from below, but the instability scale is above the cutoff $f=1\,$TeV ($\mu^2=3000\,GeV^2$, $\lambda=-0.03$, $T_x = T_n=99\,GeV$). $T_c$ is the critical temperature obtained in the mean-field approximation.*[]{data-label="fig:V"}](./plot_2-2.pdf "fig:"){width="0.333\columnwidth"} ![*The potentials $V_{\rm tree}$ (black solid curve), $V_{1\ell}$ at $T=0$ (orange dashed curve) and $V_{1\ell}$ at $T=T_x$ (green dashed curve) for the choices of $c_6/f^2$ and $c_8/f^4$ indicated in each panel. In the left panel, there exist two vacua already at zero temperature ($\mu^2\simeq -3100\,GeV^2$, $\lambda\simeq-0.23$, $T_x = T_c = 35\,GeV$). In the central panel, the existence of two vacua arises only at finite temperature ($\mu^2=1900\,GeV^2$, $\lambda=-0.06$, $T_x=T_c=82\,GeV$). In the right panel, the potential is unbounded from below, but the instability scale is above the cutoff $f=1\,$TeV ($\mu^2=3000\,GeV^2$, $\lambda=-0.03$, $T_x = T_n=99\,GeV$). $T_c$ is the critical temperature obtained in the mean-field approximation.*[]{data-label="fig:V"}](./plot_2-3.pdf "fig:"){width="0.333\columnwidth"}
Mean-field estimates {#subsec:mean}
--------------------
From $V_{1\ell}$ it is straightforward to determine some quantities that roughly characterise the EWPT, namely $T_c$ and $v_{T_c}/T_c$. The critical temperature, $T_c$, is the temperature at which the minima of the broken and unbroken phases are degenerate. It provides the upper bound on the temperature at which the EWPT really starts, $T_n$. The quantity $v_{T_c}/T_c$, with $v_{T_c}$ being the VEV of the Higgs in the EW broken phase at $T=T_c$, is linked to the strength of the EWPT. Indeed, due to the fact that $v_T/T$ typically decreases with increasing $T$, $v_{T_c}/T_c$ can be used as a lower bound on the actual value of $v_T/T$ during the EWPT (if the transition ever happens; see below).
The potential $V_{1\ell}$ is easy to treat numerically, but for analytic insights on $T_c$ and $v_{T_c}/T_c$, the mean-field approximation may be helpful. We then begin neglecting $\Delta_{1\ell,T=0} V(h_c)$. In $\Delta_{1\ell,T\ne 0} V(h_c)$, we consider the high-temperature expansion of $J_b$ and $J_f$ and retain their leading terms, *i.e.* $J_b(x)\to \pi^2 x/12$ and $J_f(x)\to -\pi^2 x/24$ in Eq. . The potential $V_{1\ell}$ now reduces to the form $$\label{eq:V8app}
V_{\rm mean}(\phi,T) = \frac{-\mu^2+a_T T^2}{2} h_c^2 +\frac{\lambda}{4} h_c^4 +\frac{c_6}{8 f^2} h_c^6+\frac{c_{8}}{16f^{4}} h_c^8~,$$ with $a_T = \frac{1}{16} \left(4\frac{m_{h}^{2}}{v^{2}}+3g^{2}+g'^{2}+4y_{t}^{2} -12 c_{6}\frac{v^2}{f^2} -12 c_8 \frac{v^4}{f^4} \right)$.
In Eq. the thermal contribution can only raise the potential at $h_c\ne 0$. No transition from the symmetric to the broken phase is conceivable if at zero temperature the EW breaking minimum is above the symmetric one. Hence, the condition $V_{\rm mean}(v,T=0)<V_{\rm mean}(0,T=0)$ has to be satisfied, which is equivalent to $$\label{eq:c6_upper}
\frac{c_6}{f^2} < \frac{m_h^2}{v^4} - \frac{3 v^2}{2} \frac{c_8}{f^4}~.$$ Saturating the inequality is not feasible. As previously mentioned, there must be a gap between $T_c$ and $T_n$, and the stronger the phase transition is the larger is the gap. For this reason, values of $c_6/f^2$ close to the upper bound in Eq. are not acceptable since they lead to $T_c\to 0$ and $v_{T_c}/T_c \to \infty$. In this limit the EWPT would never happen within the lifetime of the Universe. Such values of $c_6/f^2$ are thus expected to be ruled out by more sophisticated estimates; see section \[subsec:numerics\]. For the same reason, it is at large $c_6/f^2$ that, whenever the EWPT can really start, the parameter scenarios with the strongest EWPTs arise. To appreciate the relevance of this effect, let us first evaluate the EWPT disregarding the issue.
![*$c_{6}/f^2-c_{8}/f^4$ of parameter space for a SFOEWPT in the mean-field approximation. Left) The filled region shows the allowed values for $c_{6}$ and $c_{8}$ such that at $T = 0$ the deepest minimum is at $v$. In the darker areas there is a second minimum above the one at $v$. For negative $c_{8}$, we cut off the potential at $1$ TeV and demand that $V(1 \text{TeV})>V(v)$ to ensure that the global minimum is at $v$. Superimposed are shades of yellow to green to show the strength of the phase transition, $v_{T_c}/T_{c}$, based on the critical temperature. Right) Zoomed version on the black rectangle of the left panel (note the different axis ranges). Lines of constant $T_c$ are depicted.* []{data-label="fig:c6c8"}]({plot_c6c8_f_indep_a0}.pdf "fig:"){height="42.00000%"} ![*$c_{6}/f^2-c_{8}/f^4$ of parameter space for a SFOEWPT in the mean-field approximation. Left) The filled region shows the allowed values for $c_{6}$ and $c_{8}$ such that at $T = 0$ the deepest minimum is at $v$. In the darker areas there is a second minimum above the one at $v$. For negative $c_{8}$, we cut off the potential at $1$ TeV and demand that $V(1 \text{TeV})>V(v)$ to ensure that the global minimum is at $v$. Superimposed are shades of yellow to green to show the strength of the phase transition, $v_{T_c}/T_{c}$, based on the critical temperature. Right) Zoomed version on the black rectangle of the left panel (note the different axis ranges). Lines of constant $T_c$ are depicted.* []{data-label="fig:c6c8"}]({plot_c6c8_f_indep_a0_zoom}.pdf "fig:"){height="42.00000%"}
We fix the values of $\mu$ and $\lambda$ as in Eqs. and , and we require $c_6$ and $c_8$ to fulfil Eq. . Moreover, by definition, at $T=T_c$ the EWSB minimum is degenerate with the symmetric one. These properties lead to the following relations for $v_{T_c}$ and $T_c$: $$\begin{aligned}
\begin{aligned}
\label{eq:d-e:15}
v_{T_c}^{2} & = \left[-\frac{2c_{6}}{3c_{8}}\pm 2\sqrt{\frac{c_{6}^{2}}{9c_{8}^{2}}-\frac{\lambda}{3 c_{8}}}\right]f^{2}~,\\
T_c^2 &= \frac{\mu^2}{a_0}-\left[\frac{2 c_{6}^{3}}{27 c_{8}^{2}}-\frac{c_{6}\lambda}{3c_{8}}\mp 2 \sqrt{\left(\frac{c_{6}^{2}}{9c_{8}}-\frac{\lambda}{3}\right)^{3}\frac{1}{c_{8}}}\right]\frac{f^{2}}{a_0}~,
\end{aligned}\end{aligned}$$ The left panel of Fig. \[fig:c6c8\] summarises our mean-field-approximation results in the plane $c_6/f^2$–$c_8/f^4$. To the right of the whole shaded area, Eq. is violated. Therefore, along the right border, $T_c=0$ and $v_{T_c}/T_c=\infty$. On the left of it, the above conditions for a first order EWPT are not satisfied. Below it, instead, Eq. is not satisfied for $f=1\,$TeV. (As previously explained, this border would move up or down by assuming different values of $f$.) The yellow and green regions mark the values of $c_6/f^2$ and $c_8/f^2$ leading to $0.7<v_{T_c}/T_c<1.3$ and $v_{T_c}/T_c>1.3$, respectively. These regions are split into a darker and a lighter areas. For $c_8/f^2 <0$ the former shows where $V_{\rm tree}$ is unbounded from below but the instability is above the cutoff (*cf*. right panel in Fig. \[fig:V\]); in the latter, $V_{\rm tree}$ does not provide any sign of instability below the cutoff (*cf*. left and central panels in Fig. \[fig:V\]). The same split is applied to the grey region where the EWPT is not strong. In the dark grey area with $c_8/f^2>0$, besides the global minimum at $h_c=v$, $V_{\rm tree}$ presents a further minimum at $h_c \in ]v,f[$. (For phenomenological implications of the latter see *e.g.* Ref. [@Greenwood:2008qp].) We do not further discuss this peculiar configuration since it does not appear in the region with a SFOEWT. The right panel of Fig. \[fig:c6c8\] shows a zoom of the rectangle in the plot in the left panel. It also reports some contour curves for $T_c$.
Numerical procedure {#subsec:numerics}
-------------------
The quantity $v_{T_c}/T_c$ is a good estimate of the strength of the EWPT only when the gap between $T_c$ and $T_n$ is small. Quantitatively, $T_n$ is defined as the temperature at which the probability for the nucleation of one single bubble (containing the broken phase) in a horizon volume is approximately $\sim 1$. For our scenario, the nucleation temperature can be considered in practice as the temperature $T_n$ such that $S_3[V_{1\ell}(h_c,T_n)]\simeq 140\,T_n$, with $S_3$ the action of the thermal decay from the false to the true vacuum of $V_{1\ell}$ [@Quiros:1999jp; @Laine:2016hma] [^2]. Analytically, $S_3$ can be calculated in the limit of thin or thick wall bubbles [@Quiros:1999jp], but in general we do not expect our bubble profiles to precisely fulfil any of these two limits. We thus determine $S_3$ numerically. For this scope, we use the code `CosmoTransition` [@Wainwright:2011kj] in which, to be more accurate, we do not implement the potential in the mean-field approximation but as in Eq. with the hard-thermal loop resummations in $J_b$ and $J_f$ included [^3]. For this second and more precise study of the EWPT, for each value of $c_6/f^2$ and $c_8/f^4$ we determine numerically the values of $\mu$ and $\lambda$ for which $h_c = v$ and $m_h\sim 125$ GeV.
![*Values of $T_n$ (top left), $v_{T_n}/T_n$ (top right), $\alpha$ (bottom left) and $\beta/H$ (bottom right) characterising the SFOEWPT in the plane $c_6/f^2$–$c_8/f^4$. The labels of $T_n$ and $T_c$ are in GeV units. On the right of the grey area the condition in Eq. is violated. In the gray area to the right of the dashed line, the lifetime of the EW symmetric vacuum is longer that the age of the Universe, whereas on the left the transition results too weak for our purposes, *i.e.* $v_{T_n}/T_n<0.7$. Below the grey area, the EW vacuum at zero temperature is not the global minimum at scales below the cutoff $f=1\,$TeV. In orange the parameter region LISA is sensitive to.*[]{data-label="fig:ewpt"}](./plot_Tn.pdf "fig:"){width="0.49\columnwidth"} ![*Values of $T_n$ (top left), $v_{T_n}/T_n$ (top right), $\alpha$ (bottom left) and $\beta/H$ (bottom right) characterising the SFOEWPT in the plane $c_6/f^2$–$c_8/f^4$. The labels of $T_n$ and $T_c$ are in GeV units. On the right of the grey area the condition in Eq. is violated. In the gray area to the right of the dashed line, the lifetime of the EW symmetric vacuum is longer that the age of the Universe, whereas on the left the transition results too weak for our purposes, *i.e.* $v_{T_n}/T_n<0.7$. Below the grey area, the EW vacuum at zero temperature is not the global minimum at scales below the cutoff $f=1\,$TeV. In orange the parameter region LISA is sensitive to.*[]{data-label="fig:ewpt"}](./plot_vnTn.pdf "fig:"){width="0.49\columnwidth"}\
![*Values of $T_n$ (top left), $v_{T_n}/T_n$ (top right), $\alpha$ (bottom left) and $\beta/H$ (bottom right) characterising the SFOEWPT in the plane $c_6/f^2$–$c_8/f^4$. The labels of $T_n$ and $T_c$ are in GeV units. On the right of the grey area the condition in Eq. is violated. In the gray area to the right of the dashed line, the lifetime of the EW symmetric vacuum is longer that the age of the Universe, whereas on the left the transition results too weak for our purposes, *i.e.* $v_{T_n}/T_n<0.7$. Below the grey area, the EW vacuum at zero temperature is not the global minimum at scales below the cutoff $f=1\,$TeV. In orange the parameter region LISA is sensitive to.*[]{data-label="fig:ewpt"}](./plot_alp.pdf "fig:"){width="0.49\columnwidth"} ![*Values of $T_n$ (top left), $v_{T_n}/T_n$ (top right), $\alpha$ (bottom left) and $\beta/H$ (bottom right) characterising the SFOEWPT in the plane $c_6/f^2$–$c_8/f^4$. The labels of $T_n$ and $T_c$ are in GeV units. On the right of the grey area the condition in Eq. is violated. In the gray area to the right of the dashed line, the lifetime of the EW symmetric vacuum is longer that the age of the Universe, whereas on the left the transition results too weak for our purposes, *i.e.* $v_{T_n}/T_n<0.7$. Below the grey area, the EW vacuum at zero temperature is not the global minimum at scales below the cutoff $f=1\,$TeV. In orange the parameter region LISA is sensitive to.*[]{data-label="fig:ewpt"}](./plot_bet.pdf "fig:"){width="0.49\columnwidth"}\
The findings for $T_n$ and $v_{T_n}/T_n$ are respectively displayed in the top left and top right panels of Fig. \[fig:ewpt\] (dotted lines). As expected, for values of $c_6/f^2$ nearby its upper limit (right border of the gray area; *cf*. Eq. ), $S_3[V_{1\ell}(h_c,T)]/T$ is larger than 140 for any $T$, meaning that the EWPT never starts. This problem is avoided when $2c_6/f^2+ 3v^2c_8/f^4$ goes below the threshold of about $3.5$ (black, thick dashed line). Conceptually, at the threshold one obtains $T_n= 0$ and $v_{T_n}/T_n=\infty$. The strongest EWPTs and largest supercoolings (namely, the gaps between $T_n$ and $T_c$) are thus achieved just below this threshold. By departing from it (*i.e.* by reducing $c_6/f^2$ at fixed $c_8/f^4$), the supercooling is reduced and, in turn, $v_{T_n}/T_n$ drops down. At some point, at about $c_6/f^2+ 3v^2c_8/(2 f^4) \approx 1.5$, the parameter $v_{T_n}/T_n$ reaches 0.7, below which we do not draw any result. (We also omit the findings in the region where the EW vacuum instability is below the cutoff; see section \[subsec:mean\].) The values of $c_6/f^2$ and $c_8/f^4$ relevant for the present paper are therefore those within the gray and yellow regions on the left of the dashed thick line.
The behaviour of $T_n$ and $T_c$ just described is also visible in the left panel of Fig. \[fig:ewpt2\]. As the figure highlights, for $v_{T_n}/T_n \gtrsim 4$ the discrepancy between $T_n$ (evaluated with the full potential $V_{1\ell}$ and hard-thermal loop resummation) and $T_c$ (evaluated in the mean-field approximation) is about 20%, whereas negligible for $v_{T_n}/T_n\lesssim 1$. From this point of view, what prevents the use of $v_{T_c}/T_c\gtrsim 1$ in the mean-field approximation as a bound for EW baryogenesis (instead of $v_{T_n}/T_n\gtrsim 1$) is not the accuracy of the result but the presence of a sizeable region where the nucleation never occurs. Within the allowed $c_6/f^2$–$c_8/f^4$ parameter region, we also calculate the inverse duration time of the phase transition and the normalised latent heat. In our case we can approximate them, respectively, by $\beta/H= T_n \frac{d}{dT}\left( \frac{S_3}{T}\right)$ and $\alpha= \epsilon(T_n)/(35 T_n^4)$, where $\epsilon(T_n)$ is the latent heat at the temperature $T_n$. We determine them by means of `CosmoTransition` [^4]. Their dependencies on $c_6/f^2$ and $c_8/f^4$ are presented in the bottom panels of Fig. \[fig:ewpt\]. The correlation between $T_n$, $v_{T_n}/T_n$, $\alpha$ and $\beta/H$ is evident. It is clear that all these quantities practically do not depend on $c_6/f^2$ and $c_8/f^4$ separately but only on $2c_6/f^2+ 3v^2c_8/f^4$. As expected, nearby the thick dashed line, where $T_n$ is small and $v_{T_n}/T_n$ is large, the EWPT exhibits small $\beta/H$ and large $\alpha$, typical of large supercoolings. The values of $\alpha$ and $\beta/H$ that we obtain are more readable in Fig. \[fig:ewpt2\] (right panel) where their values are expressed as a function of $c_6/f^2$ for $c_8/f^4=5\,{\rm TeV}^{-4}$ (dotted curves), $c_8/f^4=2\,{\rm TeV}^{-4}$ (dashed curves) and $c_8/f^4=0$ (solid curves). In general, for $c_8= 0$, our results are in very good agreement with those of Ref. [@Grojean:2004xa; @Delaunay:2007wb].
![*Left panel) The values of $(T_c-T_n)/T_c$ in the plane $c_6/f^2$–$c_8/f^4$ (dotted lines). The rest stands as in Fig. \[fig:ewpt\]. Right panel) The values of $\beta/H$ and $10^6\times\alpha$ as a function of $c_6/f^2$ for $c_8/f^4=5\,{\rm TeV}^{-4}$ (dotted curves), $c_8/f^4=2\,{\rm TeV}^{-4}$ (dashed curves) and $c_8/f^4=0$ (solid curves).*[]{data-label="fig:ewpt2"}](./plot_TnTc.pdf "fig:"){width="0.49\columnwidth"} ![*Left panel) The values of $(T_c-T_n)/T_c$ in the plane $c_6/f^2$–$c_8/f^4$ (dotted lines). The rest stands as in Fig. \[fig:ewpt\]. Right panel) The values of $\beta/H$ and $10^6\times\alpha$ as a function of $c_6/f^2$ for $c_8/f^4=5\,{\rm TeV}^{-4}$ (dotted curves), $c_8/f^4=2\,{\rm TeV}^{-4}$ (dashed curves) and $c_8/f^4=0$ (solid curves).*[]{data-label="fig:ewpt2"}](./plot_AlpBet.pdf "fig:"){width="0.49\columnwidth"}
A further quantity useful to characterise the EWPT is $v_w$, the velocity at which the bubbles containing the broken phase expand into the EW symmetric phase. This speed results from the balance between the pressure difference between the two phases and the friction of the plasma on the bubbles. In general, the determination of $v_w$ is subtle [@Bodeker:2004ws; @Huber:2013kj; @Konstandin:2014zta; @Leitao:2014pda]. Fortunately, for our aim, it is relevant to know $v_w$ only when $v_{T_n}/T_n\gtrsim 4$; see below. In such a regime, on one side one expects $v\gtrsim 0.9$ [@Huber:2013kj], on the other side $v_w$ cannot reach the speed of light, even asymptotically [@Bodeker:2009qy; @Bodeker:2017cim]. Due to this tiny window, it seems acceptable to take $v_w=0.95$, for which we can straightforwardly adopt some results of the gravitational wave literature.
A SFOEWPT sources a gravitational wave stochastic background. Its power spectrum depends on $v_w$, $T_n$, $\beta/H$ and $\alpha$ [@Caprini:2015zlo]. If the amplitude of the signal is strong enough, the LISA experiment will be able to detect it towards the end of the LHC [@Audley:2017drz]. Figure 4 in Ref. [@Caprini:2015zlo] shows the values of $\beta/H$ and $\alpha$ that LISA can probe when $v_w\simeq 0.95$. We use this figure to forecast the capabilities of LISA for constraining the EFT we are working with [^5]. The region that can be tested is marked in yellow in Figs. \[fig:ewpt\] and \[fig:ewpt2\].
Interplay between gravitational wave signatures and Higgs-self coupling measurements {#sec:interplay}
------------------------------------------------------------------------------------
From the bottom-up perspective we have adopted so far, the only collider implications of the operators $\mathcal{O}_6$ and $\mathcal{O}_8$ are changes in the rates of double- and triple-Higgs production. These are related to the modified Higgs couplings. Neglecting radiative corrections, the latter are given by $$\label{eq:h3h4}
\frac{\lambda_{3}}{\lambda_{3,\text{SM}}} = 1+ \frac{v^{2}}{m_{h}^{2}}\left(2 c_{6} \frac{v^{2}}{f^{2}} +4c_{8} \frac{v^{4}}{f^{4}}\right)~, \quad
\frac{\lambda_{4}}{\lambda_{4,\text{SM}}} = 1 + 4 \frac{v^{2}}{m_{h}^{2}} \left(3 c_{6} \frac{v^{2}}{f^{2}}+8c_{8}\frac{v^{4}}{f^{4}}\right) ,$$
$\frac{c_6}{f^2}\,[\text{TeV}^{-2}]$ $\frac{c_8}{f^{4}}\,[\text{TeV}^{-4}]$ $\lambda_3^{(0)}/ \lambda_{3,\text{SM}}$ $\lambda_4^{(0)}/ \lambda_{4,\text{SM}}$ $\lambda_3/ \lambda_{3,\text{SM}}$ $\lambda_4/ \lambda_{4,\text{SM}}$
-------------------------------------- ---------------------------------------- ------------------------------------------ ------------------------------------------ ------------------------------------ ------------------------------------
0 0 1 1 0.91 0.56
2 -2 1.82 5.72 1.68 5.02
2 0 1.94 6.63 1.77 5.81
2 5 2.22 8.89 2.01 7.79
4 -2 2.76 11.34 2.53 10.48
4 0 2.88 12.25 2.63 11.32
4 5 3.16 14.52 2.87 13.44
: *Comparison between tree (denoted by the superscript “ $^{(0)}$") and loop level values of $\lambda_3$ and $\lambda_4$ with respect to their SM, tree-level values.*[]{data-label="tab:self"}
The corresponding numbers at one loop, obtained numerically for several values of $c_6/f^2$ and $c_8/f^8$ are also shown in Tab. \[tab:self\].
These couplings have not been experimentally constrained yet. However, departures on the Higgs trilinear coupling beyond the range $[-0.7, 7.1]$ will be accessible at the 95% C.L. in the HL-LHC run [@DiVita:2017eyz; @DiVita:2017vrr; @Kim:2018uty]. Moreover, values outside the interval $[0.1, 1.9]$ [@DiVita:2017vrr] can be probed in a future FCC-ee facility [@Gomez-Ceballos:2013zzn]. Likewise, searches for double-Higgs and triple-Higgs production at future hadron colliders might also constrain $\lambda_4$ [@Papaefstathiou:2015paa]. The reach of the different facilities is shown in the left panel of Fig. \[fig:l3l4\] as a function of $c_6/f^2, c_8/f^4$. In the right panel, this information is depicted in the plane $\lambda_{3}/\lambda_{3,\text{SM}}$–$\lambda_{4}/\lambda_{4, \text{SM}}$. The grey area in the latter shows the non-accessible region of a $100$ TeV $pp$ collider, taken from Ref. [@Papaefstathiou:2015paa] (the reference cuts at $\lambda_{4}/ \lambda_{4,\text{SM}}=11$, and so do we). As we already mentioned, the region of the SFOEWPT identified by the nucleation temperature is a subset of the region found by the mean-field approximation. The region of parameter space that LISA is sensitive to is a subset of the former.\
![*Left panel) Region of Fig. \[fig:c6c8\] where the SFOEWPT is achieved accordingly to the criterion $v_{T_n}\gtrsim T_n$ instead of $v_{T_c}\gtrsim T_c$. The reaches of FCC-ee [@DiVita:2017vrr] and LISA [@Caprini:2015zlo] are also displayed. Right panel) Allowed region from the left panel translated to the $\lambda_{3}/\lambda_{3,\text{SM}}$–$\lambda_{4}/\lambda_{4, \text{SM}}$ plane together with the future experimental sensitivities [@Papaefstathiou:2015paa].* []{data-label="fig:l3l4"}]({plot_c6c8_nucl_zoom}.pdf "fig:"){height="40.00000%"} ![*Left panel) Region of Fig. \[fig:c6c8\] where the SFOEWPT is achieved accordingly to the criterion $v_{T_n}\gtrsim T_n$ instead of $v_{T_c}\gtrsim T_c$. The reaches of FCC-ee [@DiVita:2017vrr] and LISA [@Caprini:2015zlo] are also displayed. Right panel) Allowed region from the left panel translated to the $\lambda_{3}/\lambda_{3,\text{SM}}$–$\lambda_{4}/\lambda_{4, \text{SM}}$ plane together with the future experimental sensitivities [@Papaefstathiou:2015paa].* []{data-label="fig:l3l4"}]({plot_l3l4}.pdf "fig:"){height="40.00000%"}
With LISA starting to take data in the early 2030’s, a sensible part of the parameter space where the SFOEWPT takes place would be first probed by LISA. Almost the complete parameter space would be tested at a future FCC-ee. A future hadron collider with $30~\text{ab}^{-1}$ [@Papaefstathiou:2015paa] could be fully conclusive.
Matching of concrete models {#sec:weakly}
===========================
The operators $\mathcal{O}_6$ and $\mathcal{O}_8$ are most commonly induced by new heavy scalars. These fields, however, generate normally other operators already at dimension six. Our aim here is to single out the properties of those UV completions that generate only $\mathcal{O}_6$ and $\mathcal{O}_8$ and are allowed by current data. Let us parameterize the effective Lagrangian after integrating out the new degrees of freedom as $$\label{eq:lag}
L = L_{\text{SM}} + \sum_i \frac{c_i}{f^{4-d_i}}\mathcal{O}_{i}~,$$ where $L_{\text{SM}}$ stands for the SM Lagrangian, $c_i/f^{4-d_i}$ represents the coefficient of the corresponding operator $\mathcal O_i$ and $f$ is the typical new-physics scale. The couplings are all expected to scale as $c_i \sim \tilde{g}^{2}\times \tilde{g}^{2n}/(4\pi)^{2n}$, with $\tilde{g}$ some weak coupling, $n$ the perturbative order at which $\mathcal O_i$ is generated, and $d_i$ the canonical dimension of $\mathcal O_i$. This means that $c_6$ can be $\mathcal{O}(1)$ TeV$^{-2}$ as required by the SFOEWPT only if the operator $\mathcal{O}_6$ is induced at tree level. Additionally, other operators with couplings of similar size will be generated. Among these, we have, in a Warsaw-like basis [@Grzadkowski:2010es], the following ones [@deBlas:2014mba] [^6]: $$\label{eq:operators}
\mathcal O_{6} = (\phi^\dagger\phi)^3~,
\quad \mathcal O_{d6} = \frac{1}{2}\partial_\mu(\phi^\dagger\phi)\partial^\mu(\phi^\dagger\phi)~,
\quad \mathcal O_{\phi D} = (\phi^\dagger D_\mu \phi)((D^\mu\phi)^\dagger \phi)~.$$ These typically appear together with further effective interactions. The same scalars generating $\mathcal O_{6},\mathcal O_{d6}$ and $\mathcal O_{\phi D}$ also induce, at the same order, the operators $$\label{eq:O_psi}
\mathcal{O}_{\psi\phi} = y_\psi(\phi^\dagger\phi)(\overline{\psi}_L\phi \psi_R)~,$$ with $y_\psi$ the Yukawa coupling of the SM fermions, here generically indicated as $\psi_L$ and $\psi_R$. These operators modifiy the Higgs-fermion interactions. $\mathcal O_{d6}$ provides a contribution to the Higgs kinetic term. As a consequence, the Higgs couplings to fermions and gauge bosons are modified with respect to those of the SM by the factors [^7] $$\label{eq:SMratio}
\frac{g_{hff}}{g_{hff}^{\text{SM}}} = c, \quad \frac{g_{hVV}}{g_{hVV}^{\text{SM}}} = a, \quad \frac{g_{hgg}}{g_{hgg}^{\text{SM}}} = c, \quad \frac{g_{h\gamma\gamma}}{g_{h\gamma\gamma}^{\text{SM}}} = \frac{aI_\gamma + cJ_\gamma}{I_\gamma+J_\gamma}~,$$ with $$a = 1 - c_{d6}\frac{v^2}{2f^2}, \quad c = 1 - c_{d6}\frac{v^2}{2f^2} + \mathcal{O}(c_{\psi\phi}, c_{\phi D})\frac{v^2}{f^2}~,$$ and $I_\gamma \simeq -1.84$, $J_\gamma\simeq 8.32$. We can obtain robust constraints on $c_{d6}$ from the present LHC measurements by marginalising the Run-2 constraints on $a$ over all possible values of $c$. One obtains [@deBlas:2018tjm] $$\label{eq:cd6_marginal}
c_{d6}\frac{v^2}{2 f^2}= -0.01\pm 0.06 \quad\rm{at~68\%~C.L.}~.$$ A further improvement to $\pm 0.03$ is expected at the end of the HL run if no new physics is found [@deBlas:2018tjm]. We also note that neglecting $\mathcal{O}_{\phi D}$ can be justified at the matching scale, since $c_{\phi D}(f)\approx 0$ can be naturally explained by means of UV symmetries. However, due to $\mathcal{O}_{d6}$, $c_{\phi D}$ runs between the renormalization scales $f$ and $v$ [@Alonso:2013hga]: $$\label{eq:cd6.RGE}
c_{\phi D}(v) \simeq c_{\phi D}(f) + \frac{5}{24 \pi^2}g'^2 c_{d6}(f) \log{\frac{f}{v}}~.$$ The present constraint on the coupling of $\mathcal{O}_{\phi D}$, namely [@deBlas:2013gla] $$-0.023 < c_{\phi D}/f^2~\text{TeV}^{2} < 0.006~,$$ provides an (indirect) bound on $c_{d6}$. Clearly, in view of the these bounds on $c_{d6}$ and $c_{\phi D}$, there will be little room for these couplings to be of the size of $c_6$, as suggested by power counting estimates in weakly-coupled scenarios. It is therefore crucial to understand whether there exist concrete UV scenarios that, at low energy, naturally generate a large hierarchy between $c_6$ and the other $c_i$ coefficients. A hierarchy between different operator coefficients can also be generated (rather model-independently) with strongly-coupled UV-completions. We discuss the resulting picture in section \[sec:ewXL\].
New scalars with weak isospin $I\leq 1$ {#sec:models}
---------------------------------------
In light of the above discussion, it is worth considering scenarios in which operators other than $\mathcal O_6$ are negligible. To this aim, let us first assume that the SM Higgs sector is extended only with new heavy scalars with isospin $I \leqslant 1$; see Ref. [@Jiang:2016czg] for a related discussion. Concrete realisations and their signals at lepton colliders have been also discussed in Ref. [@Cao:2017oez]. In the simplest case in which there is only one new field, $\varphi$, $\mathcal O_6$ is the only operator generated at tree level if and only if $\varphi$ is a colourless $SU(2)_L$ doublet with vanishing couplings to the fermions [@deBlas:2014mba; @DiLuzio:2017tfn]. This scenario is then poorly motivated, because there is no symmetry that can remove *only* the doublet couplings to fermionic currents, since a $\mathbb{Z}_2$ parity under which they are the only odd fields would make $c_6$ also vanish. Moreover, the new doublets appearing in the most common UV setups do not exhibit this property.
On the other hand, one might argue that many motivated extensions of the SM Higgs sector involve several new fields. This is for instance the case of non-minimal composite Higgs models [^8]. One particularly interesting example is the coset $SU(5)/SO(5)$ [@Vecchi:2015fma], which admits a four-dimensional UV completion [@Ferretti:2013kya]. The scalar sector consists of a hyperchargeless triplet, $\Xi_0$, a triplet with hypercharge $1$, $\Xi_1$, and a neutral singlet $\mathcal{S}$ on top of the Higgs doublet. The effective operators we are interested in receive multiple contributions, namely $$\begin{aligned}
\frac{c_{d6}}{f^2} = \frac{1}{M^4}\bigg(\kappa_{\mathcal S}^2 - \kappa_{\Xi_0}^2 - 4|\kappa_{\Xi_1}|^2\bigg), \quad \frac{c_{\phi D}}{f^2} &= -\frac{2}{M^4}\bigg(\kappa_{\Xi_0}^2 - 2|\kappa_{\Xi_1}|^2\bigg)~,\\
\frac{c_{\psi\phi}}{f^2} &= \frac{1}{M^4}\bigg(\kappa_{\Xi_0}^2 + 2|\kappa_{\Xi_1}|^2\bigg)~,
\end{aligned}$$ and $$\begin{aligned}
\nonumber
\frac{c_{6}}{f^2}&=\frac{\kappa_{\cal S}}{M^4}\left(-\lambda_{\cal S}\kappa_{\cal S}+\frac{\kappa_{{\cal S}^3}\kappa_{\cal S}^{2}}{M^2}-\lambda_{{\cal S}\Xi_0}\kappa_{\Xi_0}-4\operatorname{Re}\left[ \lambda_{{\cal S}\Xi_1}(\kappa_{\Xi_1})^*\right]+\frac{\kappa_{{\cal S} \Xi_0}\kappa_{\Xi_0}^2}{M^{2}} +\frac{2\kappa_{{\cal S}\Xi_1}|\kappa_{\Xi_1}|^2}{M^{2}}\right)\\\nonumber
&-\frac{ \kappa_{\Xi_0}^2}{M^4}\left(\lambda_{\Xi_0}-2\lambda\right)-\frac{\left|\kappa_{\Xi_1}\right|^2}{M^4}\left(2\lambda_{\Xi_1}-\sqrt{2}\tilde{\lambda}_{\Xi_1}-4\lambda\right)-\frac{2\sqrt{2}}{M^4}\operatorname{Re}\left[\lambda_{\Xi_1\Xi_0}(\kappa_{\Xi_1})^* \kappa_{\Xi_0}\right]\\
&-\frac{\sqrt{2}}{M^6 }\kappa_{\Xi_0\Xi_1}\kappa_{\Xi_0}|\kappa_{\Xi_1}|^2~,\end{aligned}$$ where $M$ is the (assumed common) mass term of all new scalars, and the other couplings just parameterise the renormalizable interactions among themselves and the SM particles [@deBlas:2014mba]. It is interesting to show that not even in this case, which contains several scalars and many different couplings, can $\mathcal O_6$ be the only non-vanishing operator. Indeed, $c_{\phi D}$ only vanishes for $\kappa_{\Xi_0} = \sqrt{2}|\kappa_{\Xi_1}|$. This choice can in fact be enforced by an $SU(2)_L\times SU(2)_R$ symmetry, as in the Georgi-Machacek model [@Georgi:1985nv]. It would yield $$\frac{c_{d6}}{f^2} = \frac{1}{M^4}\bigg(\kappa_{\mathcal S}^2 - 6|\kappa_{\Xi_1}|^2\bigg)~,$$ which could then be removed by enforcing $\kappa_{\mathcal S} = \sqrt{6}|\kappa_{\Xi_1}|$. As a result, it would turn out that $c_{\psi\phi}/f^{2} = 4|\kappa_{\Xi_1}|^2/M^4$, which vanishes if and only if $\kappa_{\Xi_1} = 0$. In such a case, however, $c_6$ is vanishing too. Actually, we can go further and show that *there is no weakly-coupled renormalizable extension of the Higgs sector containing singlets or triplets —with non-vanishing couplings to the SM— in which the effective operators produced after integrating out all new scalars at tree level modify only the scalar potential.*
In order to prove this statement, let us work in the Warsaw basis and use the results of Ref. [@deBlas:2014mba]. Let us also assume first that the extended Higgs sector contains (at least) one neutral singlet. This field generates a positive $c_{d6}$ that can be only cancelled by the contribution of a colourless triplet scalar. Indeed, any combination of colourless-triplet scalars, independently of the number of fields and their quantum numbers, gives a negative contribution to $c_{d6}$. This contribution is in fact the sum of all independent contributions [@deBlas:2014mba].
Colourless triplet scalars, on their side, also produce the operator $\mathcal O_{\psi\phi}$ with coefficient $c_{\psi\phi}\propto c_{d6}$. Therefore, it cannot be neglected if the triplet has to cancel the singlet contribution to $\mathcal O_{d6}$. The operator $\mathcal O_{\psi\phi}$, in turn, cannot be cancelled by the singlet, which does not produce it at all at tree level. For this matter, at least one extra doublet is to be present, too. However, doublets produce also four-fermion operators like $\mathcal O_{le} = (\overline{l_L}\gamma_\mu l_L)(\overline{e_R}\gamma^\mu e_R)$. This is actually generated only by doublets, with negative sign for $l_L$ and $e_R$ of the same flavour. So, it cannot be removed at all by including other scalar fields. Instead, its coefficient must be explicitly forced to vanish. In such a case, however, the coupling $c_{\psi\phi}$ induced by the triplets would be strictly vanishing, and so all the linear interactions between the new physics and the SM, in contradiction with our hypothesis. Had we started considering the presence of at least one triplet, instead of one singlet, we would have arrived to exactly the same conclusion.
New scalars with weak isospin $I> 1$
------------------------------------
Let us now consider the case $I > 1$. The only scalars that can couple in a renormalizable way to the SM sector are quadruplets with hypercharges $Y = 1/2, 3/2$. Interestingly, they contribute only to $\mathcal{O}_6$ when integrated out. These quadruplets can appear, for example, in Grand Unified Theories (GUT).
In GUT models, the SM fermions as well as the Higgs doublet are embedded in multiplets of a simple gauge group containing the SM $SU(3)_c\times SU(2)_L\times U(1)_Y$. Two main GUT gauge groups have been typically considered in the literature, namely $SU(5)$ and $SO(10)$ (and at a lesser extent, $E_6\supset SO(10) \supset SU(5)$). The minimal irreducible representations of the scalar fields that can lead to SM gauge uncoloured quadruplets are the $\mathbf{35}$ and the $\mathbf{70}$ in $SU(5)$ [@Slansky:1981yr; @Feger:2012bs]. Obviously, such large-dimensional representations do not decompose only into quadruplets, but into many other states. An example is $$\label{eq:su5}
\mathbf{35} = (\mathbf{1}, \mathbf{4})_{3/2} + (\mathbf{\overline{3}}, \mathbf{3})_{2/3} + (\mathbf{\overline{6}}, \mathbf{2})_{−1/6} + (\mathbf{\overline{10}}, \mathbf{1})_{−1}~,$$ where the two numbers in parenthesis and the sub-index denote the dimension of the irreducible representation of $SU(3)_c$ and $SU(2)_L$ under which the corresponding field transforms and its hypercharge, respectively. Clearly, larger representations reduce to a larger number of exotic fields. Despite being unlikely, it is still possible that the effective operators generated by the coloured scalars are sub-leading with respect to the $\mathcal{O}_6$ induced by the quadruplet. This can happen it two cases: *i)* If the coloured scalars are much heavier (which can be justified if a specific mechanism, similar to those advocated to solve the doublet-triplet splitting problem in SUSY GUT models [@Witten:1981kv; @Dvali:1993yf; @Georgi:1981vf; @Masiero:1982fe; @Babu:2006nf; @Babu:1993we], is enforced); *ii)* if all non-quadruplet fields have vanishing linear couplings to the SM at the renormalizable level. Surprisingly, this is the case for all extra fields in Eq. (although in principle they could couple, *e.g.*, to dangerous flavour-violating currents via effective interactions).
Although the representation $\mathbf{35}$ does not include the Higgs boson, nor is required to break $SU(5)$ down to the SM gauge group (unlike *e.g.* the $\mathbf{24}$), the aforementioned observations motivate further studies of a Higgs sector extended with quadruplets [^9]. There is a caveat, though. Despite being suppressed by higher powers of $1/M^2$, with $M$ the mass of these fields, dimension-eight operators can be also in conflict with current data. For example, for a quadruplet with $Y = 3/2$, the operator $(\phi^\dagger\phi)\mathcal{O}_{\phi D}$, which violates custodial symmetry at dimension eight, carries a coefficient of order $\sim c_6/M^4$. The rather low upper bound on $M\lesssim$ few hundred GeVs implied by the SFOEWPT is therefore in tension with the very well measured value of the $\rho$ parameter [@Babu:2009aq; @Ghosh:2016lnu]. Indeed, the experimental bound $\rho_{\textrm exp}=(1.00037\pm 0.00023)$ [@Patrignani:2016xqp] imposes $M \gg 1 $ TeV. A way out to this problem is considering a custodially symmetric quadruplet setup. We devote next section to this topic.
### A custodial quadruplet setup
We start from the custodially symmetric Lagrangian of the $SU(2)$-quadruplet that was discussed in Ref. [@Logan:2015xpa]. The potential is [^10] $$\begin{aligned}
\begin{aligned}
\label{eq:cus-qdrpt:1}
\mathcal{L} &= \frac{1}{2} \langle (D_{\mu}\Theta)^{\dagger} D^{\mu}\Theta\rangle + \frac{1}{2} \langle (D_{\mu}\Phi)^{\dagger} D^{\mu}\Phi\rangle - \frac{\mu^{2}}{2} \langle\Phi^{\dagger}\Phi\rangle - \frac{\lambda}{4} \langle\Phi^{\dagger}\Phi\rangle^{2}\\
& -\frac{\mu^{2}_{\Theta}}{2} \langle\Theta^{\dagger}\Theta\rangle- \frac{\lambda'}{4} \langle\Phi^{\dagger}\Phi\rangle \langle\Theta^{\dagger}\Theta\rangle- \widetilde{\lambda} \langle \Phi^{\dagger}T^{a}_{1/2}\Phi T^{b}_{1/2}\rangle \langle \Theta^{\dagger}T^{a}_{3/2}\Theta T^{b}_{3/2}\rangle\\
& - \frac{2 \sqrt{2}}{3} \lambda_{\Theta} \langle \Phi^{\dagger} \hat{T}^{1,a}_{1/2} \Phi (\hat{T}^{1,b}_{1/2})^{\dagger}\rangle \langle \Phi^{\dagger} (\hat{T}^{1,a}_{3/2\, 1/2})^{\dagger} \Theta \hat{T}^{1,b}_{3/2\, 1/2} \rangle\\
& - \frac{2 \sqrt{2}}{3} \lambda_{\Theta} \langle \Phi \hat{T}^{1,a}_{1/2} \Phi^{\dagger} (\hat{T}^{1,b}_{1/2})^{\dagger}\rangle \langle \Phi (\hat{T}^{1,a}_{3/2\, 1/2})^{\dagger} \Theta^{\dagger} \hat{T}^{1,b}_{3/2\, 1/2} \rangle \quad +\mathcal{O}(\Theta^{3},\Theta^{4}),
\end{aligned}\end{aligned}$$ where $\langle A \rangle$ is the trace of the matrix $A$, and $$\begin{aligned}
\begin{aligned}
\label{eq:cus-qdrpt:2}
\Theta = \begin{pmatrix}\Theta_{3}^{*} & -\Theta_{1}^{-*}&\Theta_{1}^{++}&\Theta_{3}^{+++}\\-\Theta_{3}^{+*}&\Theta_{1}^{*}&\Theta_{1}^{+}&\Theta_{3}^{++}\\ \Theta_{3}^{++*}&-\Theta_{1}^{+*}&\Theta_{1}&\Theta_{3}^{+}\\ -\Theta_{3}^{+++*}&\Theta_{1}^{++*}&\Theta_{1}^{-}&\Theta_{3} \end{pmatrix}\equiv\begin{pmatrix}\widetilde{\Theta}_{3} & \widetilde{\Theta}_{1}&\Theta_{1}&\Theta_{3}\end{pmatrix} \quad \text{and} \quad \Phi = \begin{pmatrix}h_{0}^{*}&h^{+}\\ -h^{-}&h_{0} \end{pmatrix}\equiv\begin{pmatrix}\widetilde{\phi}&\phi\end{pmatrix}.
\end{aligned}\end{aligned}$$ In this notation, $\Theta_{3}$ has hypercharge $3/2$, $\Theta_{1}$ has hypercharge $1/2$, and $\phi$ is the SM-Higgs doublet. The covariant derivative is defined as $$\label{eq:cus-qdrpt:3}
D_{\mu}\Theta =\partial_{\mu}\Theta + i g W_{\mu}\Theta - i g' B_{\mu} \Theta T_{3/2}^{3}.$$ From Eq. we can derive the equations of motion for $\Theta$ and integrate it out at tree level. We find $$\begin{aligned}
\begin{aligned}
\label{eq:cus-qdrpt:4}
\Delta\mathcal{L}_{6}&=\frac{\lambda_{\Theta}^{2}}{\mu^{2}_{\Theta}}(\phi^{\dagger}\phi)^{3}\\
\Delta\mathcal{L}_{8}&=\frac{\lambda_{\Theta}^{2}}{2\mu^{4}_{\Theta}}\left(5(\phi^{\dagger}\phi)^{2}(D_{\mu}\phi)^{\dagger}(D^{\mu}\phi)+(\phi^{\dagger}\phi)D_{\mu}(\phi^{\dagger}\phi)D^{\mu}(\phi^{\dagger}\phi)\right)-\frac{\lambda_{\Theta}^{2}}{\mu^{4}_{\Theta}}\left(\lambda'+\frac{15}{4}\widetilde{\lambda}\right)(\phi^{\dagger}\phi)^{4}.
\end{aligned}\end{aligned}$$ The contribution to $\Delta\mathcal{L}_{6}$ is consistent with [@deBlas:2014mba]. There, the contribution of $\Theta_{3}$ is $3 \lambda_{\Theta}^{2}/(2\mu^{2}_{\Theta}) $ and the one of $\Theta_{1}$ (with the relation $\lambda_{\Theta_{1}}=-\sqrt{3}\lambda_{\Theta_{3}}$ coming from Eq. ) is also $3\lambda_{\Theta}^{2}/(2\mu^{2}_{\Theta}) $. The resulting factor of $3$ is absorbed in the different definition of $\mathcal{O}_{6}$ compared to ours. We see that at dimension eight the model induces the desired contribution to the Higgs potential, as well as two more contributions with two derivatives. All of them conserve custodial symmetry.
We have also checked, using `SARAH` [@Staub:2013tta], that loop corrections to the $\rho$ and $S$ parameters are well within the experimental bound. The collider phenomenology of the custodial quadruplet can be understood in terms of the unbroken $SU(2)_V$. The Higgs bi-doublet decomposes as $(\mathbf{2}, \mathbf{2})= 1 + \mathbf{3}$, while the custodial bi-quadruplet decomposes as $(\mathbf{4},\mathbf{4}) = 1 + \mathbf{3} + \mathbf{5} + \mathbf{7}$. The latter singlet and triplet contain only electrically neutral and singly-charged scalars, which are difficult to produce and detect at colliders. Note that they only couple to the SM fermions via the mixing with the Higgs singlet and triplet. Moreover, this mixing is very small: After all, $\mathcal{O}_6$ is the only operator generated at tree level, which does not modify the Higgs couplings at low energy. This also suggests that measuring the Higgs couplings is not the most promising strategy to test this setup.
Moreover, the septuplet contains large electric charges. However, these cannot directly decay into pairs of SM particles [^11]. They decay only via the emission of (soft) gauge bosons into lower-charged states in the custodial quadruplet, which are also difficult to test at colliders. The quintuplet, instead, can be both efficiently produced (in pairs via EW interactions) and decays mostly into pairs of gauge bosons (indeed $\mathbf{3}\times\mathbf{3} = 1 + \mathbf{3} + \mathbf{5}$). Decays into pairs of Higgs bosons are not allowed, because this is a complete singlet of $SU(2)_V$. In particular, the doubly-charged, singly-charged and neutral components of the quintuplet decay with branching ratios $$\text{Br}(\Theta^{\pm\pm}\to W^\pm W^\pm) = 1, \quad \text{Br}(\Theta^{\pm}\to W^\pm Z) = 1, \quad \text{Br}(\Theta^0\to W^+ W^- + ZZ) = 1~.$$
We implement this model in `MadGraph v5` [@Alwall:2014hca] by means of `Feynrules v2` [@Alloul:2013bka]. We subsequently compute the pair-production cross sections mediated by neutral and charged currents for masses in between $300$ and $1000$ GeV. The results are shown in the upper left and upper right panels of Fig. \[fig:xsecs\], respectively.
![*Upper left) Neutral current cross sections for pair-production of scalars in the custodial quadruplet model. Upper right) Same as before but for the charged current. Bottom left) Integrated luminosity required to exclude the custodial quadruplet at the 95% C.L. for different masses using two different analyses; see text for details. Two representative values of the collected luminosity, $\mathcal{L} = 300$ fb$^{-1}$ and $\mathcal{L} = 3$ ab$^{-1}$ are also shown with dashed lines. Bottom right) Parameter space region where the FOEWPT takes place for $\lambda^\prime =\tilde{\lambda}=1$ and the reach of different searches. The yellow region shows the HL-LHC reach taken from the bottom-left panel.*[]{data-label="fig:xsecs"}](./plot_nxsecs.pdf "fig:"){width="0.49\columnwidth"} ![*Upper left) Neutral current cross sections for pair-production of scalars in the custodial quadruplet model. Upper right) Same as before but for the charged current. Bottom left) Integrated luminosity required to exclude the custodial quadruplet at the 95% C.L. for different masses using two different analyses; see text for details. Two representative values of the collected luminosity, $\mathcal{L} = 300$ fb$^{-1}$ and $\mathcal{L} = 3$ ab$^{-1}$ are also shown with dashed lines. Bottom right) Parameter space region where the FOEWPT takes place for $\lambda^\prime =\tilde{\lambda}=1$ and the reach of different searches. The yellow region shows the HL-LHC reach taken from the bottom-left panel.*[]{data-label="fig:xsecs"}](./plot_cxsecs.pdf "fig:"){width="0.49\columnwidth"} ![*Upper left) Neutral current cross sections for pair-production of scalars in the custodial quadruplet model. Upper right) Same as before but for the charged current. Bottom left) Integrated luminosity required to exclude the custodial quadruplet at the 95% C.L. for different masses using two different analyses; see text for details. Two representative values of the collected luminosity, $\mathcal{L} = 300$ fb$^{-1}$ and $\mathcal{L} = 3$ ab$^{-1}$ are also shown with dashed lines. Bottom right) Parameter space region where the FOEWPT takes place for $\lambda^\prime =\tilde{\lambda}=1$ and the reach of different searches. The yellow region shows the HL-LHC reach taken from the bottom-left panel.*[]{data-label="fig:xsecs"}](./plot_lum.pdf "fig:"){width="0.49\columnwidth"} ![*Upper left) Neutral current cross sections for pair-production of scalars in the custodial quadruplet model. Upper right) Same as before but for the charged current. Bottom left) Integrated luminosity required to exclude the custodial quadruplet at the 95% C.L. for different masses using two different analyses; see text for details. Two representative values of the collected luminosity, $\mathcal{L} = 300$ fb$^{-1}$ and $\mathcal{L} = 3$ ab$^{-1}$ are also shown with dashed lines. Bottom right) Parameter space region where the FOEWPT takes place for $\lambda^\prime =\tilde{\lambda}=1$ and the reach of different searches. The yellow region shows the HL-LHC reach taken from the bottom-left panel.*[]{data-label="fig:xsecs"}](./plot_quad.pdf "fig:"){width="0.49\columnwidth"}
We have also estimated the current and the future LHC reach for this scenario. To this aim, we have generated Monte Carlo events, including radiation, fragmentation and hadronization effects with `Pythia v6` [@Sjostrand:2006za], and analysed them using `CheckMate v2` [@Dercks:2016npn]. The latter implements several multi-lepton SUSY searches. Among them, the search that turns out to be the most sensitive to our scenario, is the “$SR3\ell-H$" signal region of Ref. [@ATLAS-CONF-2016-096], which looks for three leptons, no $b$-jets and large missing energy. This analysis considers $13$ fb$^{-1}$ of LHC data at $\sqrt{s} = 13$ TeV. The integrated luminosity needed to exclude a particular value of the quadruplet mass at the 95% C.L. can then be estimated as $$\mathcal{L} = 13~\text{fb}^{-1}\times \frac{1}{(s/s_{excl})^2}~,$$ where $s/s_{excl}$ is the number of expected signal events over the number of excluded signal events as reported by `CheckMate`. The corresponding result is represented by the thick solid line in the left bottom panel of Fig. \[fig:xsecs\]. The thin solid line represents the luminosity required to test the different masses using the improved multi-lepton search described in Ref. [@Alcaide:2017dcx]. As things stand, masses as large as $M\sim 600$ GeV can be tested in multi-lepton final states at the LHC. Getting ahead of the results discussed, we also show the reach of LHC Higgs-self couplings measurements as well as that of the gravitational wave observatory LISA; see right bottom panel in Fig. \[fig:xsecs\]. Interestingly, the former cannot even test the parameter space region where the FOEWPT takes place. (As a matter of fact, in the present scenario the LHC Higgs-self couplings measurements are sensitive only to the region where the theory does not achieve EWSB.). These results suggest that most weakly-coupled models (those containing $SU(2)_L$ charged states), even if tuned to avoid large corrections to operators other than $\mathcal{O}_6$, can be better tested at gravitational wave observatories or in direct LHC searches [^12].
Strongly-coupled models {#sec:ewXL}
-----------------------
So far, we discussed the dynamics of the electroweak phase transition in presence of effective modifications of the scalar potential only, as well as potential UV-completions that lead to this particular pattern of low-energy effects. Working in a generic bottom-up EFT, we would in principle have many more effective operators, with coefficients of similar size to the coefficients that modify the potential. To overcome the strong experimental constraints on these operators, we require a hierarchy between the large effects in the scalar sector and the more constrained effects in the gauge-fermion sector. This can be achieved with a strongly-coupled UV-completion. While the complete description of such a UV-completion requires lattice simulations (and is therefore more model-dependent), we can describe the low-energy effects by assuming a mass gap between the (pseudo-) Nambu-Goldstone bosons and the higher resonances of the theory. The EW chiral Lagrangian ([$ew\chi L$]{}) [@Feruglio:1992wf; @Bagger:1993zf; @Koulovassilopoulos:1993pw; @Burgess:1999ha; @Wang:2006im; @Grinstein:2007iv; @Contino:2010mh; @Contino:2010rs; @Azatov:2012bz; @Alonso:2012px; @Buchalla:2012qq; @Buchalla:2013rka; @Buchalla:2013eza] is the most general EFT that describes such low-energy effects of strongly-coupled new physics. Historically, it emerged from the Higgs-less chiral Lagrangian [@Appelquist:1980vg; @Longhitano:1980tm; @Dobado:1990zh; @Herrero:1993nc], which was then supplemented with a generic scalar singlet $h$. Since this does not assume any IR-doublet structure for the Higgs, it describes a very wide class of new-physics models that induce large deviations in the Higgs sector from the SM. The leading-order [[$ew\chi L$]{}]{} is $$\begin{aligned}
\label{eq:s1}\nonumber
L_{\text{LO}}^{ew\chi} =& -\frac{1}{2} \langle G_{\mu\nu}G^{\mu\nu}\rangle -\frac{1}{2}\langle W_{\mu\nu}W^{\mu\nu}\rangle -\frac{1}{4} B_{\mu\nu}B^{\mu\nu} \\\nonumber
&+i\bar{q}_{L}\slashed{D}q_{L} +i\bar{\ell}_{L}\slashed{D}\ell_{L} +i\bar{u}_{R}\slashed{D}u_{R} +i\bar{d}_{R}\slashed{D}d_{R} +i\bar{e}_{R}\slashed{D}e_{R} \\\nonumber
&+\frac{v^2}{4}\ \operatorname{Tr}{(D_\mu U^\dagger D^\mu U)} \left( 1+F(h)\right) +\frac{1}{2} \partial_\mu h \partial^\mu h - V(h)\\\nonumber
&- \frac{v}{\sqrt{2}} \left[ \bar{q}_{L} \left( Y_u +\sum\limits^\infty_{n=1} Y^{(n)}_u \left(\frac{h}{v}\right)^n \right) U P_{+}q_{R} + \bar{q}_{L} \left( Y_d + \sum\limits^\infty_{n=1} Y^{(n)}_d \left(\frac{h}{v}\right)^n \right) U P_{-}q_{R} \right.\\
&\left. + \bar{\ell}_{L} \left( Y_e +\sum\limits^\infty_{n=1} Y^{(n)}_e \left(\frac{h}{v}\right)^n \right) U P_{-}\ell_{R} + \text{ h.c.}\right] ~,
$$ where $U$ stands for the exponential of the Goldstone matrix, $G, W$ and $B$ are the SM gauge fields, $u_R$, $d_R$, $e_R$, $q_L$ and $\ell_L$ are the fermions of the SM, and $Y$ are generalised Yukawa couplings. The scalar $h$ couples through general polynomials to the other fields, which reflects its strongly-coupled origin.
These polynomials ($V(h), F(h)$, and $Y_i(h)=Y_i+\sum_{n=1}^\infty Y_i^{(n)} (h/v)^n)$ are not truncated at canonical dimension four, but go to arbitrary order. (An additional operator of the structure $(\partial_{\mu}h)(\partial^{\mu})f(h)$ is also allowed by symmetry, but can be removed via field redefinitions, without loss of generality [@Buchalla:2013rka].) The coefficients of these polynomials depend on $v/f$.
As the Lagrangian in Eq. contains terms with arbitrarily high canonical dimension, the EFT can clearly not be organized in terms of canonical dimensions. Instead, it is organised by a generalisation of the momentum expansion of chiral perturbation theory [@Weinberg:1978kz], the chiral dimensions [@Buchalla:2013rka; @Buchalla:2013eza]. They reflect an expansion in terms of loops, which guarantees the renormalizability of the EFT at a fixed order in the expansion. The cutoff of the EFT is at $\Lambda=4\pi f$, yielding the expansion parameter $f^{2}/\Lambda^{2} = 1/ 16\pi^{2}$. For $v < f$, the parameter $\xi = v^{2}/f^{2}$ is smaller than the unity and Eq. can be further expanded in $\xi$. In this scenario, a double expansion in $\xi$ and $1/16\pi^{2}$ organises the EFT [@Buchalla:2014eca], in the spirit of the strongly-interacting light Higgs Lagrangian [@Giudice:2007fh].\
In this double expansion, we still see some of the decoupling effects, but also a pattern of Wilson coefficients that is coming from the strong sector. Depending on the structure of the operators, they will be suppressed by ratios of scales ($\xi$, based on their canonical dimension) and loop factors ($1/16\pi^{2}$, based on their chiral dimension). This creates an additional hierarchy among the operators of a given canonical dimension, compared to the weakly-coupled case of section \[sec:weakly\]. Some of the dimension six operators, corresponding to $L_{\text{LO}}^{ew\chi}$, will only be suppressed by $\xi$, while other operators, corresponding to $L_{\text{NLO}}^{ew\chi}$, will be suppressed by an additional loop factor, resulting in $\xi/16\pi^{2}$. The former affects the Higgs sector with deviations of $\mathcal{O}(10\%)$, dominating over effects in the gauge-fermion sector of the latter group, with deviations of $\mathcal{O}(1\%)$ or below. This hierarchy also reflects the current experimental constraints: The gauge-fermion sector is rather strongly constrained, while large effects in the Higgs couplings are still possible. The [$ew\chi L$]{} of Eq. is now expanded in both chiral and canonical dimensions. Since $\xi=\mathcal{O}(0.1 - 0.2)$ [@Khachatryan:2016vau; @Buchalla:2015qju; @Sanz:2017tco; @deBlas:2018tjm], effects of $\mathcal{O}(\xi^{2})$ could in principle be larger than the $\mathcal{O}(1/16\pi^{2})$ effects. The leading effects in the double expansion are then given by expanding $L_{\text{LO}}$ up to $\mathcal{O}(\xi^{2})$. *A priori*, the Higgs potential, which at this order contains both $\mathcal{O}_6$ and $\mathcal{O}_8$, is of chiral dimension $0$ and the dominating effect. However, the Higgs mass is then expected to be of order $\mathcal{O}(\Lambda)$, which would break the EFT approach. In order for this to make sense, the Higgs mass must be parametrically suppressed to appear at chiral dimension of 2 [^13]. An additional fine tuning of $\mathcal{O}(\xi)$ is needed for $m_{h}\sim v$. This, however, might only affect the mass term of the potential and the Higgs self-couplings could have large deviations from the SM, induced by $c_6$ and $c_8$. We can understand the enhancement on the operators in the potential by just dimensional analysis if we assume that the strongly-coupled theory is described by only one relevant coupling $g_\ast$. To this aim, we need to abandon the convention $\hbar = c = 1$ recovering the physical dimensions of these two constants. It turns out that the coefficient of any operator involving $r$ Higgs insertions and $q$ derivatives scales as $g_\ast^2 f^4 [h/f]^r [\partial/(g_\ast f)]^q$, up to $\mathcal{O}(1)$ coefficients [@Giudice:2007fh; @Panico:2015jxa; @Buchalla:2016sop; @Chala:2017sjk]. Hence, scalar operators not carrying derivatives are enhanced with respect to the derivative ones by several powers of $g_\ast$ ($ \gg 1$ in a strongly couple theory); *e.g.* $c_6\sim g_\ast^2$ versus $c_{d6}\sim 1$. We refer to Ref. [@Azatov:2015oxa] for a discussion on which scenarios show this enhancement while still having $m_h \sim v$. This justifies why we studied the effects of $\mathcal{O}_{6}$ and $\mathcal{O}_{8}$, neglecting other effects, as first approximation.
To account for all leading effects consistently, we have to consider the full set of dimension-six and dimension-eight operators that contribute at chiral dimension $2$ for the expansion in $\xi$. The operators are $$\begin{aligned}
\label{eq:d-e:1}\nonumber
\left(\phi^\dagger\phi\right)^{3}, \quad \partial_{\mu}\left(\phi^\dagger\phi\right)\partial^{\mu}\left(\phi^\dagger\phi\right), \quad \bar\Psi Y \phi\Psi\left(\phi^\dagger\phi\right), \\
\left(\phi^\dagger\phi\right)^{4}, \quad \partial_{\mu}\left(\phi^\dagger\phi\right)\partial^{\mu}\left(\phi^\dagger\phi\right)\left(\phi^\dagger\phi\right), \quad \bar\Psi Y \phi\Psi \left(\phi^\dagger\phi\right)^{2}.
$$ With the identification $\phi = \tfrac{(v+h)}{\sqrt{2}}\, U \binom{0}{1}$, we find at the different orders of $\xi$: $$\begin{aligned}
\label{eq:d-e:2}\nonumber
L_{\xi^{0}} &= \frac{1}{2}\partial_{\mu}h\partial^{\mu}h + \frac{\mu^{2}}{2}(v+h)^{2}-\frac{\lambda}{4}(v+h)^{4}-\tfrac{1}{\sqrt{2}}\bar\Psi\hat{Y}_{\Psi}U P_{\pm}\Psi (v+h)+\frac{v^2}{4}\ \operatorname{Tr}{(D_\mu U^\dagger D^\mu U)} \left( 1+\tfrac{h}{v}\right)^{2},\\\nonumber
L_{\xi^{1}} &= \frac{c_{d6}}{2f^{2}}\partial_{\mu}h\partial^{\mu}h(v+h)^{2} -\frac{c_{6}}{8f^{2}}(v+h)^{6}-\tfrac{1}{\sqrt{2}f^{2}}\bar\Psi\hat{Y}_{\Psi}^{(6)}U P_{\pm}\Psi (v+h)^{3},\\
L_{\xi^{2}} &= \frac{c_{d8}}{2f^{4}}\partial_{\mu}h\partial^{\mu}h(v+h)^{4} -\frac{c_{8}}{16f^{4}}(v+h)^{8}-\tfrac{1}{\sqrt{2}f^{4}}\bar\Psi\hat{Y}_{\Psi}^{(8)}U P_{\pm}\Psi (v+h)^{5}.
\end{aligned}$$ To bring the Lagrangian to the form of $L_{\text{LO}}^{ew\chi}$ in Eq. , we have to canonically normalise the field $h$ using the field redefinition discussed in Ref. [@Buchalla:2013rka]. We find [@Buchalla:2016bse] $$\begin{aligned}
\label{eq:d-e:3}\nonumber
h\rightarrow h \Big\lbrace 1-\tfrac{\xi}{2} c_{d6} &\left(1+\tfrac{h}{v}+\tfrac{h^2}{3 v^2}\right)+\xi ^2 c_{d6}^2 \left(\tfrac{3}{8}+\tfrac{h}{v}+\tfrac{13}{12} \left(\tfrac{h}{v}\right)^2+\tfrac{13}{24} \left(\tfrac{h}{v}\right)^3+\tfrac{13}{120} \left(\tfrac{h}{v}\right)^4\right) \\
& -\xi ^2 c_{d8} \left(\tfrac{1}{2}+\tfrac{h}{v}+\left(\tfrac{h}{v}\right)^2 +\tfrac{1}{2} \left(\tfrac{h}{v}\right)^3+\tfrac{1}{10} \left(\tfrac{h}{v}\right)^4\right)\Big\rbrace .
$$ To obtain the right Higgs VEV and mass, the parameters $\mu^{2}$ and $\lambda$ have to fulfil $$\begin{aligned}
\begin{aligned}
\label{eq:d-e:4}
\mu^{2} &= \frac{m_h^2}{2} +\frac{v^{2}}{f^{2}}\left(\frac{1}{2}c_{d6} m_h^2-\frac{3}{4} c_6v^{2}\right)+\frac{v^{4}}{f^{4}}\left(\frac{1}{2}c_{d8} m_{h}^{2}-c_8 v^{2}\right),\\
\lambda&= \frac{m_{h}^{2}}{2 v^{2}}+\frac{v^{2}}{f^{2}}\left(\frac{c_{d6}}{2}\frac{m_{h}^{2}}{v^{2}}-\frac{3c_{6}}{2}\right)+\frac{v^{4}}{f^{4}}\left(\frac{c_{d8}}{2}\frac{m_{h}^{2}}{v^{2}}-\frac{3c_{8}}{2}\right).
\end{aligned}\end{aligned}$$ Applying Eq. everywhere in Eq. , we find the expansion of $V(h), F(h)$, and $Y(h)$ in $\xi$. Writing $$\begin{aligned}
\begin{aligned}
\label{eq:d-e:5}
V(h) &= \frac{1}{2}m_{h}^{2}v^{2}\left[\frac{h^{2}}{v^{2}}+\sum_{i=3}^{8}\lambda_{i}\left(\frac{h}{v}\right)^{i}\right],\\
F(h) &= \sum_{i=1}^{6}f_{i}\left(\frac{h}{v}\right)^{i},
\end{aligned}\end{aligned}$$ we finally have $$\begin{aligned}
\begin{aligned}
\label{eq:d-e:6}
\lambda_{3}&=1+ \frac{v^{2}}{f^{2}} \left(2 c_{6}\frac{v^{2}}{m_{h}^{2}}-\frac{3}{2}c_{d6}\right)+\frac{v^{4}}{f^{4}}\left(\frac{15}{8}c_{d6}^{2}-3\frac{v^{2}}{m_{h}^{2}}c_{6}c_{d6}-\frac{5}{2}c_{d8}+4\frac{v^{2}}{m_{h}^{2}}c_{8}\right) ,\\
\lambda_{4}&=\frac{1}{4} + \frac{v^{2}}{f^{2}}\left(3 c_{6}\frac{v^{2}}{m_{h}^{2}}-\frac{25}{12}c_{d6} \right)+\frac{v^{4}}{f^{4}}\left(\frac{11}{2}c_{d6}^{2}-9\frac{v^{2}}{m_{h}^{2}}c_{6}c_{d6}-\frac{21}{4}c_{d8}+8\frac{v^{2}}{m_{h}^{2}}c_{8}\right) ,
\end{aligned}\end{aligned}$$ $$\begin{aligned}
\begin{aligned}
\label{eq:d-e:13}
f_{1}&=2 - \frac{v^{2}}{f^{2}}c_{d6}+\frac{v^{4}}{f^{4}}\left(\frac{3}{4}c_{d6}^{2}-c_{d8}\right) ,\\
f_{2}&=1 - 2\frac{v^{2}}{f^{2}}c_{d6}+\frac{v^{4}}{f^{4}}\left(3c_{d6}^{2}-3c_{d8}\right)
\end{aligned}\end{aligned}$$ and $$\begin{aligned}
\begin{aligned}
\label{eq:d-e:7}
Y_{\Psi}^{(1)} & =Y_{\Psi}+\frac{v^{2}}{f^{2}}\left(2\hat{Y}_{\Psi}^{(6)}-\frac{c_{d6}}{2}Y_{\Psi}\right)+\frac{v^{4}}{f^{4}}\left(4\hat{Y}_{\Psi}^{(8)}-\frac{c_{d8}}{2}Y_{\Psi}-c_{d6}\hat{Y}_{\Psi}^{(6)}+\frac{3}{8}c_{d6}^{2}Y_{\Psi}\right) ,\\
Y_{\Psi}^{(2)} & =\frac{v^{2}}{f^{2}}\left(3\hat{Y}_{\Psi}^{(6)}-\frac{c_{d6}}{2}Y_{\Psi}\right)+ \frac{v^{4}}{f^{4}}\left(10\hat{Y}_{\Psi}^{(8)}-c_{d8}Y_{\Psi}-4c_{d6}\hat{Y}_{\Psi}^{(6)}+c_{d6}^{2}Y_{\Psi}\right),
\end{aligned}\end{aligned}$$ where we only list the couplings relevant for the subsequent discussion. The matrices $Y_{\Psi}$ and $ Y_{\Psi}^{(n)}$ are the fermion mass and Yukawa matrices defined in Eq. . Note that the functional dependence of Eqs. and on $c_{i}$ differ from the result of Refs. [@Buchalla:2014eca; @Buchalla:2016bse], as we do not include explicit factors of $\lambda$ in the definition of the Wilson coefficients. Already now we see two of the implications of adding these effective operators. The triple- and quartic-Higgs couplings are further modified with respect to the SM. Moreover, new vertices, such as $\bar\Psi \Psi hh$, also relevant for the study of double-Higgs production, arise.
Additionally, for current Higgs observables, also the local $GGh$ and $\gamma\gamma h$ operators are important, even though they are formally of next-to-leading order. This is because these amplitudes arise at the one-loop level of the leading-order Lagrangian; see Ref. [@Buchalla:2015wfa]. Such a Lagrangian is $$\label{eq:d-e:20}
L_{GGh}=L_{kin}+G_{\mu\nu}G^{\mu\nu}\left[\frac{c_{g6}}{16\pi^{2}f^{2}}\phi^{\dagger}\phi+\frac{c_{g8}}{16\pi^{2}f^{4}}(\phi^{\dagger}\phi)^{2}\right].$$ After symmetry breaking and the field redefinition of Eq. , this creates a contribution that renormalizes the gluon kinetic term and therefore $G_{\mu\nu}$. After this renormalization, we find $$\label{eq:d-e:21}
L_{GGh}=G_{\mu\nu}G^{\mu\nu} \left[1+f_{G1}\frac{h}{v}+f_{G2}\frac{h^{2}}{v^{2}} +\mathcal{O}(h^{3})\right],$$ with $$\begin{aligned}
\begin{aligned}
\label{eq:d-e:22}
16\pi^{2} f_{G1} &= \xi c_{g6} + \xi^{2}\left(c_{g8}-\frac{1}{2}c_{d6}c_{g6}-\frac{c_{g6}^{2}}{32\pi^{2}}\right),\\
32\pi^{2} f_{G2} &= \xi c_{g6} +\xi^{2}\left(3 c_{g8}-\frac{1}{2}c_{d6}c_{g6}-\frac{c_{g6}^{2}}{32\pi^{2}}\right).
\end{aligned}\end{aligned}$$ The last term in each of the $f_{Gi}$ comes from the renormalization and is sub-leading. Finally, it is also worth noting that all these operators would contribute to the EWPT, as they alter the $h_c$-dependent squared masses $m_i^2$ in Eqs. -. In addition, the derivative operator $\mathcal{O}_{d6}$ requires a reevaluation of the Coleman-Weinberg effective potential at finite temperature, as the field redefinition of Eq. cannot be done in the unbroken phase [@Burgess:2010zq; @Passarino:2016pzb]. All these effects would be suppressed by $v^{2}/f^{2}$ in $a_{0}$, but would nevertheless have an impact on the computation of the quantities of the EWPT. Current experimental results only constrain effective couplings with a single Higgs field [@Buchalla:2015qju; @deBlas:2018tjm], namely $Y_{t,b,\tau}^{(1)}, f_{1},$ and $f_{G1}$ from the list above. From these, $f_{1}$ is the most constrained, but still allows for deviations of $\mathcal{O}(5\%)$. The others are not constrained beyond $\mathcal{O}(10\%)$. While from a bottom-up point of view a deviation in one of these couplings might hint to a deviation in $\lambda_{3}$ of comparable size, such conclusions are strongly model dependent.\
Double Higgs production, which would shed light on the $\lambda_{3}$ coupling of the Higgs potential in the SM, depends on five of the effective parameters from above [@Grober:2015cwa; @Kim:2018uty] if we restrict ourselves to the top loops only. These are $Y_{t}^{(1)}, Y_{t}^{(2)}, f_{G1}, f_{G2},$ and $\lambda_{3}$. A large deviation in $\lambda_{3}$ from its SM value could then be not seen in the experiment because of the interplay with the otherwise unconstrained other parameters.\
Conclusions {#sec:conclusions}
===========
It is well known that the presence of higher-dimensional operators in the Standard Model Higgs potential can drastically influence the dynamics of the ElectroWeak (EW) symmetry breaking. Among the possible operators, the interactions $\mathcal O_n=(\phi^\dagger\phi)^{\frac{n}{2}}$, with $\phi$ being the Higgs doublet, have attracted a lot of attention to make the EW Phase Transition (EWPT) strongly first order while evading any scheduled LHC search. Achieving a strongly first order EWPT requires $c_6/f^2\gtrsim 1$ TeV$^{-2}$, with $f$ the cutoff of the theory and $c_6$ the coefficient of $\mathcal O_6$. This implies that $f$ is likely too close to the EW scale for the dimension-six EFT to be accurate, at least in weakly-coupled theories. Dimension-eight operators have then to be considered as well, which is also the case when strongly-coupled sectors are present. Such sectors can also lead to naturally large corrections to the Higgs potential (in comparison with other operators). In view of this possibility, we have also examined the EFT where (only) both $\mathcal O_6$ and $\mathcal O_8$ are unsuppressed.
In the aforementioned dimension-eight setup, we have computed the parameters relevant for the EWPT, including the critical and nucleation temperatures and the VEVs of the Higgs at these temperatures. We have also estimated the latent heat and the inverse duration time of the phase transition, characterising the gravitational waves produced in the collisions of nucleated bubbles. Regarding the coefficients of $\mathcal{O}_6$ and $\mathcal{O}_8$, $c_6$ and $c_8$ respectively, we have obtained that the parameter region $3 \lesssim c_6/f^2 + 3 v^2 c_8/(2 f^4) \lesssim 3.5$ is in the reach of the future LISA experiment. Remarkably, due to the low LHC sensitivity to $\mathcal{O}_6$ and $\mathcal{O}_8$, LISA will be the first experiment able to significantly constrain these operators. Concerning the reach of future colliders, we have shown that almost all values of interest will be probed by a future FCC-ee in double-Higgs production, while the whole parameter space will be testable combining double- and triple-Higgs production in hadronic colliders.
Given that the new physics matching the previous EFT must be quite low, we have also explored the possibility of producing the supposely heavy new fields at the LHC. Among the ultraviolet completions exhibiting only the operators $\mathcal O_n$, we have proven that in weakly-coupled setups consisting of new scalar singlets or triplets, the presence at low energies of other effective operators already quite constrained by LHC and EW precision data is unavoidable. (Of course, in scenarios with several scalars, a tuning in the fundamental parameters can still yield to an EFT where the coefficients $c_6$ and $c_8$ are substantially larger than those of the other effective operators.) On the contrary, in models involving only doublets or quadruplets (higher representations do only lead to $\mathcal{O}_6$ at the loop level, being $c_6$ therefore very small to modify the EWPT), new symmetries can make all operators other than those modifing the scalar potential vanish. Such models still contain charged particles that can be produced in pairs via Drell-Yan and then decay into longitudinal polarizations of the gauge bosons. We have shown that even in the particular case of a custodial quadruplet, the LHC reach is far smaller than that of LISA.
We are grateful to Florian Staub for useful support on `SARAH`. MC acknowledges AEC for hospitality during the first state of this project. CK thanks Andrew J. Long for discussions about the effective potential, and the Enrico Fermi Institute at the University of Chicago and Fermi National Laboratory for hospitality, where parts of this research was carried out. CK acknowledges also fruitful discussions at the LHCPHENO 2017 workshop at IFIC Valencia. MC is supported by the Royal Society under the Newton International Fellowships programm. The work of CK is supported in part by the Spanish Government and ERDF funds from the EU Commission \[Grants No. FPA2014-53631-C2-1-P and SEV-2014-0398\], by the Alexander von Humboldt-Foundation, and by the Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. GN is financed by the Swiss National Science Foundation (SNF) under grant 200020-168988.
[^1]: This different behaviour is not needed in (peculiar) setups where the EWPT is preceded by some exotic phenomena. One example is the warped extradimension framework in which the EWPT is forbidden till when the decomposite-composite transition starts [@Creminelli:2001th; @Randall:2006py; @Nardini:2007me; @Konstandin:2010cd]. A further case occurs when inflation has a reheating temperature below the EW scale [@GarciaBellido:1999sv; @Krauss:1999ng; @GarciaBellido:2002aj; @Smit:2002yg; @Konstandin:2011ds; @Arunasalam:2017ajm; @vonHarling:2017yew].
[^2]: This assumes the Universe to be dominated by radiation during the EWPT.
[^3]: We also modified the code to evaluate the $S_4$ bubble action. Within the numerical precision of the code, we did not find significant changes, at least in the resolution relevant for our plots.
[^4]: In order to obtain $\beta/H$ one has to modify the subroutine `transitionFinder.py`, as explained in Ref. [@Chala:2016ykx]. Briefly, we determine $\beta/H$ by first finding the temperature $T_{240}$ at which $S_3[V_{1\ell}(h,T_{240})]/T_{240}=240$, and then we use the approximation $\beta/H\simeq T_n (240-140)/(T_{240}-T_n)$.
[^5]: The LISA design approved by ESA has a sensitivity that is quite similar to that dubbed “C1“ in Fig. 4 of Ref. [@Caprini:2015zlo]. For our analysis we then use the “C1” sensitivity region of that figure. Moreover, as *a posteriori* it turns out that LISA can probe our region when $T_n\lesssim 50\,$GeV, we use the result with $T_n=50\,$GeV of Ref. [@Caprini:2015zlo].
[^6]: Note that $\mathcal O_6$ cannot be originated from integrating out at tree level new fermions [@Jiang:2016czg; @deBlas:2014mba; @deBlas:2017xtg]. We also stress that the operator basis in Eq. is converted into the proper Warsaw basis [@Grzadkowski:2010es] by integrating by parts $\mathcal{O}_{d6}$.
[^7]: In a complete dimension-six analysis, there are even more operators contributing to these factors, like $G_{\mu\nu}G^{\mu\nu}\phi^{\dagger}\phi$ and a similar operator for the photon. These can also be constrained by Higgs-couplings measurements and they do not contribute to the EWPT at tree level.
[^8]: We note that composite Higgs models involve strongly-coupled dynamics and they are better described by the EW chiral Lagrangian; see section \[sec:ewXL\]. However, it has been shown that, in certain parameter space regions, the contribution of the extra scalars to the Higgs effective operators can overcome the contribution of the strong sector [@Chala:2017sjk].
[^9]: Larger representations, such as the mentioned $\mathbf{70}$, do contain a Higgs doublet, but also other fields with renormalizable interactions to the SM fermions. Moreover, smaller representations typically contain singlets and triplets (such as in the $\mathbf{15}$ and the $\mathbf{24}$, to name a few).
[^10]: We use the same convention and notation for the generators as in Ref. [@Logan:2015xpa].
[^11]: Note that there is no $SU(2)_V$ septuplet constructed out of two $1$ and/or two $\mathbf{3}$. The septuplet cannot even decay into three triplets: Although allowed by $SU(2)_V$, operators mediating this decay would contain at least three gauge bosons and one scalar, while Lorentz invariance forbids this kind of interaction at dimension four.
[^12]: Note that most SM extensions avoiding large corrections to operators such as $\mathcal{O}_{\phi D}$ or $\mathcal{O}_{d6}$ involve different multiplets and therefore charged (often doubly-charged) scalars. One possible counter-example is a singlet scalar whose own parameters are tuned; see Ref. [@DiVita:2017eyz].
[^13]: This occurs naturally in composite Higgs models (CHMs), where the Higgs potential is generated radiatively and then comes with two powers of weak couplings ($g^2, y^2$) and a corresponding loop suppression of the scale $\Lambda^2$ to the scale $f^{2}$.
|
---
abstract: 'In this paper, we use the language of operads to study open dynamical systems. More specifically, we study the algebraic nature of assembling complex dynamical systems from an interconnection of simpler ones. The syntactic architecture of such interconnections is encoded using the visual language of wiring diagrams. We define the symmetric monoidal category $\bfW$, from which we may construct an operad ${\mathcal{O}\bfW}$, whose objects are black boxes with input and output ports, and whose morphisms are wiring diagrams, thus prescribing the algebraic rules for interconnection. We then define two $\mcG$ and $\mcL$, which associate semantic content to the structures in $\bfW$. Respectively, they correspond to general and to linear systems of differential equations, in which an internal state is controlled by inputs and produces outputs. As an example, we use these algebras to formalize the classical problem of systems of tanks interconnected by pipes, and hence make explicit the algebraic relationships among systems at different levels of granularity.'
author:
- Dmitry Vagner
- 'David I. Spivak'
- Eugene Lerman
title: Algebras of Open Dynamical Systems on the Operad of Wiring Diagrams
---
Introduction {#sec:intro}
============
It is widely believed that complex systems of interest in the sciences and engineering are both modular and hierarchical. Network theory uses the tools and visual language of graph theory to model such systems, and has proven to be both effective and flexible in describing their modular character. However, the field has put less of an emphasis on finding powerful and versatile language for describing the hierarchical aspects of complex systems. There is growing confidence that category theory can provide the necessary conceptual setting for this project. This is seen, for example, in Mikhail Gromov’s well-known claim, “the mathematical language developed by the end of the 20th century by far exceeds in its expressive power anything, even imaginable, say, before 1960. Any meaningful idea coming from science can be fully developed in this language.” [@Gromov]
Joyal and Street’s work on string diagrams [@JSTensor] for monoidal categories and (with Verity) on traced monoidal categories [@JSTraced] has been used for decades to visualize compositions and feedback in networked systems, for example in the theory of flow charts [@Arthan]. Precursors, such as Penrose diagrams and flow diagrams, have been used in physics and the theory of computation, respectively, since the 1970’s [@Scott; @Baez1].
Over the past several years, the second author and collaborators have been developing a novel approach to modular hierarchical systems based on the language of operads and symmetric monoidal categories [@Spivak2; @RupelSpivak]. The main contribution to the theory of string diagrams of the present research program is the inclusion of an outer box, which allows for holarchic [@Koestler] combinations of these diagrams. That is, the parts can be assembled into a whole, which can itself be a part. The composition of such assemblies can now be viewed as morphism composition in an operad. In fact, there is a strong connection between traced monoidal categories and algebras on these operads, such as our operad ${\mathcal{O}\bfW}$ of wiring diagrams, though it will not be explained here (see [@SpivakSchultzRupel] for details).
More broadly, category theory can organize graphical languages found in a variety of applied contexts. For example, it is demonstrated in [@Baez1] and [@Coecke] that the theory of monoidal categories unifies the diagrams coming from diverse fields such as physics, topology, logic, computation, and linguistics. More recently, as in [@Baez2], there has been growing interest in viewing more traditionally applied fields, such as ecology, biology, chemistry, electrical engineering, and control theory through such a lens. Specifically, category theory has been used to draw connections among visual languages such as planar knot diagrams, Feynman diagrams, circuit diagrams, signal flow graphs, Petri nets, entity relationship diagrams, social networks, and flow charts. This research is building toward what John Baez has called “a foundation of applied mathematics” [@BaezTalk].
The goal of the present paper is to show that open continuous time dynamical systems form an algebra over a certain (colored) operad, which we call the operad of *wiring diagrams*. It is a variant of the operad that appeared in [@RupelSpivak]. That is, wiring diagrams provide a straightforward, diagrammatic language to understand how dynamical systems that describe processes can be built up from the systems that describe its sub-processes.
More precisely, we will define a symmetric monoidal category $\bfW$ of black boxes and wiring diagrams. Its underlying operad ${\mathcal{O}\bfW}$ is a graphical language for building larger black boxes out of an interconnected set of smaller ones. We then define two $\bfW$-algebras, $\mcG$ and $\mcL$, which encode *open dynamical systems*, i.e., differential equations of the form $$\begin{aligned}
\label{dia:basic form}
\begin{cases}
\dot{Q}={f^{\text{in}}}(Q,input)\\
output={f^{\text{out}}}(Q)
\end{cases}\end{aligned}$$ where $Q$ represents an internal state vector, $\dot{Q}=\frac{dQ}{dt}$ represents its time derivative, and $input$ and $output$ represent inputs to and outputs from the system. In $\mcG$, the functions ${f^{\text{in}}}$ and ${f^{\text{out}}}$ are smooth, whereas in the subalgebra $\mcL\ss\mcG$, they are moreover linear. The fact that $\mcG$ and $\mcL$ are $\bfW$-algebras captures the fact that these systems are closed under wiring diagram interconnection.
Our notion of interconnection is a generalization of that in Deville and Lerman [@DL1], [@DL2], [@DL3]. Their version of interconnection produces a closed system from open ones, and can be understood in the present context as a morphism whose codomain is the closed box (see Definition \[def:mon\]). Graph fibrations between wiring diagrams form an important part of their formalism, though we do not discuss that aspect here. This paper is the third in a series, following [@RupelSpivak] and [@Spivak2], on using wiring diagrams to model interactions. The algebra we present here, that of open systems, is distinct from the algebras of relations and of propagators studied in earlier works. Beyond the dichotomy of discrete vs. continuous, these algebras are markedly different in structure. For one thing, the internal wires in [@RupelSpivak] themselves carry state, whereas here, a wire should be thought of as instantaneously transmitting its contents from an output site to an input site. Another difference between our algebra and those of previous works is that the algebras here involve *open systems* in which, as in (\[dia:basic form\]), the instantaneous change of state is a function of the current state and the input, whereas the output depends only on the current state (see Definition \[def:general algebra\]). The differences between these algebras is also reflected in a mild difference between the operad we use here and the one used in previous work.
Motivating example
------------------
The motivating example for the algebras in this paper comes from classical differential equations pedagogy; namely, systems of tanks containing salt water concentrations, with pipes carrying fluid among them. The systems of ODEs produced by such applications constitute a subset of those our language can address; they are linear systems with a certain form (see Example \[ex:as promised\]). To ground the discussion, we consider a specific example.
\[ex:main\] Figure \[fig:pipebrine\] below reimagines a problem from Boyce and DiPrima’s canonical text as a dynamical system over a *wiring diagram*.
In this diagram, $X_1$ and $X_2$ are boxes that represent tanks consisting of salt water solution. The functions $Q_1(t)$ and $Q_2(t)$ represent the amount of salt (in ounces) found in 30 and 20 gallons of water, respectively. These tanks are interconnected with each other by pipes embedded within a total system $Y$. The prescription for how wires are attached among the boxes is formally encoded in the wiring diagram $\Phi:X_1,X_2\to Y$, as we will discuss in Definition \[def:W\].
Both tanks are being fed salt water concentrations at constant rates from the outside world. Specifically, $X_1$ is fed a 1 ounce salt per gallon water solution at 1.5 gallons per minute and $X_2$ is fed a 3 ounce salt per gallon water solution at 1 gallon per minute. The tanks also both feed each other their solutions, with $X_1$ feeding $X_2$ at 3 gallons per minute and $X_2$ feeding $X_1$ at 1.5 gallons per minute. Finally, $X_2$ feeds the outside world its solution at 2.5 gallons per minute.
The dynamics of the salt water concentrations both within and leaving each tank $X_i$ is encoded in a linear open system $f_i$, consisting of a differential equation for $Q_i$ and a readout map for each $X_i$ output (see Definition \[def:opensystem\]). Our algebra $\mcL$ allows one to assign a linear open system $f_i$ to each tank $X_i$, and by functoriality the morphism $\Phi\taking X_1,X_2\to Y$ produces a linear open system for the larger box $Y$. We will explore this construction in detail, in particular providing explicit formulas for it in the linear case, as well as for more general systems of ODEs.
Preliminary Notions {#sec:pre}
===================
Throughout this paper we use the language of monoidal categories and functors. Depending on the audience, appropriate background on basic category theory can be found in MacLane [@MacLane], Awodey [@Awodey], or Spivak [@Spivak]. Leinster [@Leinster] is a good source for more specific information on monoidal categories and operads. We refer the reader to [@KFA] for an introduction to dynamical systems.
We denote the category of sets and functions by $\Set$ and the full subcategory spanned by finite sets as $\FinSet$. We generally do not concern ourselves with cardinality issues. We follow Leinster [@Leinster] and use $\times$ for binary product and $\Pi$ for arbitrary product, and dually $+$ for binary coproduct and $\amalg$ for arbitrary coproduct in any category. By [*operad*]{} we always mean symmetric colored operad or, equivalently, symmetric multicategory.
Monoidal categories and operads
-------------------------------
In Section \[sec:W\], we will construct the symmetric monoidal category $(\bfW,\oplus,0)$ of boxes and wiring diagrams, which we often simply denote as $\bfW$. We will sometimes consider the underlying operad ${\mathcal{O}\bfW}$, obtained by applying the fully faithful functor $$\mathcal{O}\taking \mathbf{SMC}\to\mathbf{Opd}$$ to $\bfW$. A brief description of this functor ${\mathcal{O}}$ is given below in Definition \[def:SMC to Opd\].
\[def:SMC to Opd\]
Let $\mathbf{SMC}$ denote the category of symmetric monoidal categories and lax monoidal functors; and $\mathbf{Opd}$ be the category of operads and operad functors. Given a symmetric monoidal category $(\mcC,\otimes,I _\mcC)\in\operatorname{Ob}\mathbf{SMC}$, we define the operad ${\mathcal{O}\mcC}$ as follows: $$\operatorname{Ob}{\mathcal{O}\mcC}:=\operatorname{Ob}\mcC, \hspace{10 mm} \operatorname{Hom}_{{\mathcal{O}\mcC}} (X_1,\ldots,X_n;Y):=\operatorname{Hom}_{\mcC}(X_1\otimes\cdots\otimes X_n,Y)$$ for any $n\in\NN$ and objects $X_1,\ldots,X_n,Y\in\operatorname{Ob}\mcC$.
Now suppose $F\taking (\mcC,\otimes,I _{\mcC})\to(\mcD,\odot,I _{\mcD})$ is a lax monoidal functor in $\mathbf{SMC}$. By definition such a functor is equipped with a morphism $$\mu\taking FX_1\odot\cdots\odot FX_n\to F(X_1\otimes\cdots\otimes X_n),$$ natural in the $X_i$, called the [*coherence map*]{}. With this map in hand, we define the operad functor ${\mathcal{O}F}\taking {\mathcal{O}\mcC}\to {\mathcal{O}\mcD}$ by stating how it acts on objects $X$ and morphisms $\Phi\taking X_1,\ldots,X_n\to Y$ in ${\mathcal{O}\mcC}$: $${\mathcal{O}F}(X):=F(X),\hspace{1.8 mm} {\mathcal{O}F}(\Phi:X_1,\ldots,X_n\to Y):=F(\Phi)\circ\mu:FX_1\odot\cdots\odot FX_n\to FY.$$
\[ex:sets\] Consider the symmetric monoidal category $(\Set ,\times,\star)$, where $\times$ is the cartesian product of sets and $\star$ a one element set. Define as in Definition \[def:SMC to Opd\]. Explicitly, $\Sets$ is the operad in which an object is a set and a morphism $f\taking X_1,\ldots,X_n\to Y$ is a function $f\taking X_1\times\cdots\times X_n\to Y$.
\[def:algebra\] Let $\mcC$ be a symmetric monoidal category and let $\Set=(\Set,\times,\star)$ be as in Example \[ex:sets\]. A *$\mcC$-algebra* is a lax monoidal functor $\mcC\to\Set$. Similarly, if $\mcD$ is an operad, a *$\mcD$-algebra* is defined as an operad functor $\mcD\to\Sets$.
To avoid subscripts, we will generally use the formalism of SMCs in this paper. Definitions \[def:SMC to Opd\] and \[def:algebra\] can be applied throughout to recast everything we do in terms of operads. The primary reason operads may be preferable in applications is that they suggest more compelling pictures. Hence throughout this paper, depictions of wiring diagrams will often be operadic, i.e., have many input boxes wired together into one output box.
Typed sets
----------
Each box in a wiring diagram will consist of finite sets of ports, each labelled by a type. To capture this idea precisely, we define the notion of typed finite sets. By a *finite product* category, we mean a category that is closed under taking finite products.
\[def:typed finite sets\] Let $\mcC$ be a small finite product category. The category of *$\mcC$-typed finite sets*, denoted ${\mathbf{TFS}_{\mcC}}$, is defined as follows. An object in ${\mathbf{TFS}_{\mcC}}$ is a map from a finite set to the objects of $\mcC$: $$\operatorname{Ob}{\mathbf{TFS}_{\mcC}}:=\{(A,\tau)\; |\; A\in\operatorname{Ob}\FinSet, \tau\taking A\to\operatorname{Ob}\mcC)\}.$$ Intuitively, one can think of a typed finite set as a finite unordered list of $\mcC$-objects. For any element $a\in A$, we call the object $\tau(a)$ its [*type*]{}. If the typing function $\tau$ is clear from context, we may denote $(A,\tau)$ simply by $A$.
A morphism $q\taking(A,\tau)\to (A',\tau')$ in ${\mathbf{TFS}_{\mcC}}$ consists of a function $q\taking A\to A'$ that makes the following diagram of finite sets commute: $$\xymatrix{
A \ar[rr]^q \ar[rd]_\tau
& {}
& A' \ar[ld]^{\tau'}\\
&\operatorname{Ob}\mcC
}$$ Note that ${\mathbf{TFS}_{\mcC}}$ is a cocartesian monoidal category.
We refer to the morphisms of ${\mathbf{TFS}_{\mcC}}$ as [*$\mcC$-typed functions*]{}. If a $\mcC$-typed function $q$ is bijective, we call it a *$\mcC$-typed bijection*.
In other words, ${\mathbf{TFS}_{\mcC}}$ is the comma category for the diagram $$\FinSet{\xrightarrow{i}}\Set{\xleftarrow{\operatorname{Ob}\mcC}}\{*\}$$ where $i$ is the inclusion.
\[def:depprod\] Let $\mcC$ be a finite product category, and let $(A,\tau)\in\operatorname{Ob}{\mathbf{TFS}_{\mcC}}$ be a $\mcC$-typed finite set. Its *dependent product* ${\overline{(A,\tau)}}\in\operatorname{Ob}\mcC$ is defined as $$\overline{(A,\tau)}:=\prod_{a\in A}\tau(a).$$ Coordinate projections and diagonals are generalized as follows. Given a typed function $q\taking (A,\tau)\to (A',\tau')$ in ${\mathbf{TFS}_{\mcC}}$ we define $${\overline{q}}\taking {\overline{(A',\tau')}}\to{\overline{(A,\tau)}}$$ to be the unique morphism for which the following diagram commutes for all : $$\xymatrix{
\prod_{a'\in A'}\tau'(a') \ar[r]^{\overline{q}} \ar[d]_{\pi_{q(a)}}
& \prod_{a\in A}\tau(a) \ar[d]^{\pi_a}\\
\tau'(q(a))\ar@{=}[r]&\tau(a)
}$$ By the universal property for products, this defines a functor, $${\overline{\;\cdot\;}}\taking{\mathbf{TFS}_{\mcC}}\op\to\mcC.$$
The dependent product functor ${\mathbf{TFS}_{\mcC}}\op\to\mcC$ is strong monoidal. In particular, for any finite set $I$ whose elements index typed finite sets $(A_i,\tau_i)$, there is a canonical isomorphism in $\mcC$, $${\overline{\coprod_{i\in I}(A_i,\tau_i)}}\iso\prod_{i\in I}{\overline{(A_i,\tau_i)}}.$$
\[rem:default\] The category of second-countable smooth manifolds and smooth maps is essentially small (by the embedding theorem) so we choose a small representative and denote it $\Man$. Note that $\Man$ is a finite product category. Manifolds will be our default typing, in the sense that we generally take $\mcC:=\Man$ in Definition \[def:typed finite sets\] and denote $$\begin{aligned}
\label{dia:TFS}
{\mathbf{TFS}_{}}:={\mathbf{TFS}_{\Man}}.\end{aligned}$$ We thus refer to the objects, morphisms, and isomorphisms in ${\mathbf{TFS}_{}}$ simply as *typed finite sets*, *typed functions*, and *typed bijections*, respectively.
\[rem:TFSL\] The ports of each box in a wiring diagram will be labeled by manifolds because they are the natural setting for geometrically interpreting differential equations (see [@SpiM-CalcMan]). For simplicity, one may wish to restrict attention to the full subcategory $\Euc$ of Euclidean spaces $\RR^n$ for $n\in\NN$, because they are the usual domains for ODEs found in the literature; or to the (non-full) subcategory $\bfL$ of Euclidean spaces and linear maps between them, because they characterize linear systems of ODEs. We will return to ${\mathbf{TFS}_{\bfL}}$ in Section \[sec:l\].
Open systems
------------
As a final preliminary, we define our notion of open dynamical system. Recall that every manifold $M$ has a [*tangent bundle*]{} manifold, denoted $TM$, and a smooth projection map $p\taking TM\to M$. For any point $m\in M$, the preimage $T_mM:=p^{-1}(m)$ has the structure of a vector space, called the [*tangent space of $M$ at $m$*]{}. If $M\iso\RR^n$ is a Euclidean space then also $T_mM\iso\RR^n$ for every point $m\in M$. A [*vector field on $M$*]{} is a smooth map $g\taking M\to TM$ such that $p\circ g=\operatorname{id}_M$. See [@SpiM-CalcMan] or [@Warner] for more background.
For the purposes of this paper we make the following definition of open systems; this may not be completely standard.
\[def:opensystem\] Let $M,{U^{\text{in}}},{U^{\text{out}}}\in\operatorname{Ob}\Man$ be smooth manifolds and $TM$ be the tangent bundle of $M$. Let $f=({f^{\text{in}}},{f^{\text{out}}})$ denote a pair of smooth maps $$\begin{aligned}
\begin{cases}
{f^{\text{in}}}\taking M\times{U^{\text{in}}}\to TM\\
{f^{\text{out}}}\taking M\to{U^{\text{out}}}
\end{cases}\end{aligned}$$ where, for all $(m,u)\in M\times{U^{\text{in}}}$ we have ${f^{\text{in}}}(m,u)\in T_mM$; that is, the following diagram commutes: $$\xymatrix{
M\times {U^{\text{in}}} \ar[rr]^{{f^{\text{in}}}} \ar[rd]_{\pi_M}
& {}
& TM \ar[ld]^{p}\\
& M
}$$ We sometimes use $f$ to denote the whole tuple, $$f=(M,{U^{\text{in}}},{U^{\text{out}}},f),$$ which we refer to as an *open dynamical system* (or *open system* for short). We call $M$ the [*state space*]{}, ${U^{\text{in}}}$ the *input space*, ${U^{\text{out}}}$ the *output space*, ${f^{\text{in}}}$ the *differential equation*, and ${f^{\text{out}}}$ the *readout map* of the open system.
Note that the pair $f=({f^{\text{in}}},{f^{\text{out}}})$ is determined by a single smooth map $$f\taking M\times{U^{\text{in}}}\to TM\times{U^{\text{out}}},$$ which, by a minor abuse of notation, we also denote by $f$.
In the special case that $M,U^\text{in},U^\text{out}\in\operatorname{Ob}\bfL$ are Euclidean spaces and $f$ is a linear map (or equivalently ${f^{\text{in}}}$ and ${f^{\text{out}}}$ are linear), we call $f$ a *linear open system*.
\[ex:dynamical system\] Let $M$ be a smooth manifold, and let be trivial. Then an open system in the sense of Definition \[def:opensystem\] is a smooth map over $M$, in other words, a vector field on $M$. From the geometric point of view, vector fields are autonomous (i.e., closed!) dynamical systems; see [@Teschl].
For an arbitrary manifold ${U^{\text{in}}}$, a map can be considered as a function ${U^{\text{in}}}\to\VF(M)$, where $\VF(M)$ is the set of vector fields on $M$. Hence, ${U^{\text{in}}}$ [ *controls*]{} the behavior of the system in the usual sense.
\[rem:thepoint\] Given an open system $f$ we can form a new open system by feeding the readout of $f$ into the inputs of $f$. For example suppose the open system is of the form $$\begin{cases}
M\times A\times B \xrightarrow{F} TM \\
g= (g_A, g_B)\colon M \to C\times B,
\end{cases}$$ where $A$, $B$, $C$ and $M$ are manifolds. Define $F'\colon M\times A\to TM$ by $$F'(m,a) := F(m, a, g_B(m))\qquad \textrm{ for all }\quad (m,a)\in M\times A.$$ Then $$\begin{cases}
M\times A \xrightarrow{F'} TM \\
g_A\colon M \to C
\end{cases}$$ is a new open system obtained by plugging a readout of $f$ into the space of inputs $B$. Compare with Figure \[fig:wiringdiagram\].
This looks a little boring. It becomes more interesting when we start with several open systems, take their product and then plug (some of the) outputs into inputs. For example suppose we start with two open systems $$\begin{cases}
M_1\times A\times B \xrightarrow{F_1} TM_1 \\
g_1\colon M_1 \to C
\end{cases}$$ and $$\begin{cases}
M_2\times C \xrightarrow{F_2} TM_2 \\
g_2 = (g_B,g_D)\colon M_2 \to B\times D
\end{cases}.$$ Here, again, all capital letters denote manifolds. Take their product; we get $$\begin{cases}
M_1\times A\times B \times M_2\times C
\xrightarrow{(F_1,F_2)} TM_1\times TM_2 \\
(g_1,g_2)\colon M_1\times M_2 \to C\times B\times D
\end{cases}$$ Now plug in the functions $g_B$ and $g_1$ into inputs. We get a new system $$\begin{cases}
M_1\times M_2\times A \xrightarrow{F'} TM_1\times TM_2 \\
g'\colon M_1\times M_2 \to D
\end{cases}$$ where $$F'(m_1, m_2, a):= (F_1(m_1, a, g_B(m_2)), F_2 (m_2, g_1(m_1)).$$ Compare with Figure \[fig:WD\]. Making these kinds of operations on open systems precise for an arbitrary number of interacting systems is the point of our paper.
By defining the appropriate morphisms, we can consider open dynamical systems as being objects in a category. We are not aware of this notion being defined previously in the literature, but it is convenient for our purposes.
\[def:odscat\] Suppose that $M_i,{U^{\text{in}}}_i,{U^{\text{out}}}_i\in\operatorname{Ob}\Man$ and $(M_i,{U^{\text{in}}}_i,{U^{\text{out}}}_i,f_i)$ is an open system for . A *morphism of open systems* $$\zeta\taking(M_1,{U^{\text{in}}}_1,{U^{\text{out}}}_1,f_1)\to (M_2,{U^{\text{in}}}_2,{U^{\text{out}}}_2,f_2)$$ is a triple $(\zeta_M,\zeta_{{U^{\text{in}}}},\zeta_{{U^{\text{out}}}})$ of smooth maps $\zeta_M\taking M_1\to M_2$, $\zeta_{{U^{\text{in}}}}\taking {U^{\text{in}}}_1\to {U^{\text{in}}}_2$, and $\zeta_{{U^{\text{out}}}}\taking {U^{\text{out}}}_1\to {U^{\text{out}}}_2$, such that the following diagram commutes:
$$\xymatrix{
M_1\times{U^{\text{in}}}_1 \ar[r]^{f_1} \ar[d]_{\zeta_M\times\zeta_{{U^{\text{in}}}}}
&TM_1\times{U^{\text{out}}}_1 \ar[d]^{T\zeta_M\times\zeta_{{U^{\text{out}}}}}\\
M_2\times{U^{\text{in}}}_2 \ar[r]_{f_2}
& TM_2\times{U^{\text{out}}}_2
}$$
This defines the category $\ODS $ of open dynamical systems. We define the subcategory $\ODS _\bfL\ss\ODS$ by restricting our objects to linear open systems, as in Definition \[def:opensystem\], and imposing that the three maps in $\zeta$ are linear.
As in Remark \[rem:thepoint\], we will often want to combine two or more interconnected open systems into one larger one. As we shall see in Section \[sec:g\], this will involve taking a product of the smaller open systems. Before we define this formally, we first remind the reader that the tangent space functor $T$ is strong monoidal, i.e., it canonically preserves products, $$T(M_1\times M_2)\iso TM_1\times TM_2.$$
\[def:osprod\] The category $\ODS$ of open systems has all finite products. That is, if $I$ is a finite set and $f_i=(M_i,{U^{\text{in}}}_i,{U^{\text{out}}}_i,f_i)\in\operatorname{Ob}\ODS$ is an open system for each $i\in I$, then their product is $$\prod_{i\in I}f_i=\left(\prod_{i\in I}M_i,\prod_{i\in I}{U^{\text{in}}}_i,\prod_{i\in I}{U^{\text{out}}}_i,\prod_{i\in I}f_i\right)$$ with the obvious projection maps.
The Operad of Wiring Diagrams {#sec:W}
=============================
In this section, we define the symmetric monoidal category $(\bfW,\oplus,0)$ of wiring diagrams. We then use Definition \[def:SMC to Opd\] to define the wiring diagram operad ${\mathcal{O}\bfW}$, which situates our pictorial setting. We begin by formally defining the underlying category $\bfW$ and continue with some concrete examples to explicate this definition.
\[def:W\] The category $\bfW$ has objects *boxes* and morphisms *wiring diagrams*. A box $X$ is an ordered pair of $\Man$-typed finite sets (Definition \[def:typed finite sets\]), $$X=({X^{\text{in}}},{X^{\text{out}}})\in\operatorname{Ob}{\mathbf{TFS}_{}}\times\operatorname{Ob}{\mathbf{TFS}_{}}.$$ Let ${X^{\text{in}}}=(A,\tau)$ and ${X^{\text{out}}}=(A',\tau')$. Then we refer to elements $a\in A$ and $a'\in A'$ as *input ports* and *output ports*, respectively. We call $\tau(a)\in\operatorname{Ob}\Man$ the *type* of port $a$, and similarly for $\tau'(a')$.
A wiring diagram $\Phi\taking X\to Y$ in $\bfW$ is a triple $(X,Y,\varphi)$, where $\varphi$ is a typed bijection (see Definition \[def:typed finite sets\]) $$\begin{aligned}
\label{dia:wd function}
\varphi\taking{X^{\text{in}}}+{Y^{\text{out}}}\xrightarrow{\cong} {X^{\text{out}}}+{Y^{\text{in}}},\end{aligned}$$ satisfying the following condition:
no passing wires
: $\varphi({Y^{\text{out}}})\cap{Y^{\text{in}}}=\varnothing$, or equivalently $\varphi({Y^{\text{out}}})\ss{X^{\text{out}}}$.
This condition allows us to decompose $\varphi$ into a pair $\varphi=({\varphi^{\text{in}}},{\varphi^{\text{out}}})$: $$\begin{aligned}
\label{dia:components of wd}
\left\{
\begin{array}{l}
{\varphi^{\text{in}}}\taking {X^{\text{in}}} \to {X^{\text{out}}}+{Y^{\text{in}}} \\
{\varphi^{\text{out}}}\taking {Y^{\text{out}}} \to {X^{\text{out}}}
\end{array}
\right.\end{aligned}$$
We often identify the wiring diagram $\Phi=(X,Y,\varphi)$ with the typed bijection $\varphi$, or equivalently its corresponding pair $({\varphi^{\text{in}}},{\varphi^{\text{out}}})$.
By a *wire* in $\Phi$, we mean a pair $(a,b)$, where $a\in{X^{\text{in}}}+{Y^{\text{out}}}$, $b\in{X^{\text{out}}}+{Y^{\text{in}}}$, and $\varphi(a)=b$. In other words a wire in $\Phi$ is a pair of ports connected by $\phi$.
The *identity* wiring diagram $\iota:X\to X$ is given by the identity morphism ${X^{\text{in}}}+{X^{\text{out}}}\to{X^{\text{in}}}+{X^{\text{out}}}$ in ${\mathbf{TFS}_{}}$.
Now suppose $\Phi=(X,Y,\varphi)$ and $\Psi=(Y,Z,\psi)$ are wiring diagrams. We define their *composition* as $\Psi\circ\Phi=(X,Z,\omega)$, where $\omega=({\omega^{\text{in}}},{\omega^{\text{out}}})$ is given by the pair of dashed arrows making the following diagrams commute. $$\label{dia:composition diagrams}
\xymatrixcolsep{3.5pc}
\xymatrix{
{X^{\text{in}}}
\ar[dd]_{{\varphi^{\text{in}}}}
\ar@{-->}[r]^{{\omega^{\text{in}}}} &
{X^{\text{out}}}+{Z^{\text{in}}} \\
{} &
{X^{\text{out}}}+{X^{\text{out}}}+{Z^{\text{in}}}
\ar[u]_{\nabla+\bigid _{{Z^{\text{in}}}}} \\
{X^{\text{out}}}+{Y^{\text{in}}}
\ar[r]_-{\bigid _{{X^{\text{out}}}}+{\psi^{\text{in}}}} &
{X^{\text{out}}}+{Y^{\text{out}}}+{Z^{\text{in}}}
\ar[u]_{\bigid _{{X^{\text{out}}}}+{\varphi^{\text{out}}}+\bigid _{{Z^{\text{in}}}}}
} \mskip5mu
\xymatrixcolsep{1.5pc}
\xymatrix{
{Z^{\text{out}}}
\ar[rd]_{{\psi^{\text{out}}}}
\ar@{-->}[rr]^{{\omega^{\text{out}}}}
&
&{X^{\text{out}}}\\
&{Y^{\text{out}}}
\ar[ru]_{{\varphi^{\text{out}}}}}$$ Here $\nabla\taking {X^{\text{out}}}+{X^{\text{out}}}\to {X^{\text{out}}}$ is the codiagonal map in ${\mathbf{TFS}_{}}$.
\[rem:different Cs\] For any finite product category $\mcC$, we may define the category $\bfW_{\mcC}$ by replacing $\Man$ with $\mcC$, and ${\mathbf{TFS}_{}}$ with ${\mathbf{TFS}_{\mcC}}$, in Definition \[def:W\]. In particular, as in Remark \[rem:TFSL\], we have the symmetric monoidal category $\bfW_{\bfL}$ of linearly typed wiring diagrams.
What we are calling a box is nothing more than an interface; at this stage it has no semantics, e.g., in terms of differential equations. Each box can be given a pictorial representation, as in Example \[ex:pictorial box\] below.
\[ex:pictorial box\] As a convention, we depict a box $X=(\{a,b\},\{c\})$ with input ports connecting on the left and output ports connecting on the right, as in Figure \[fig:box\] below. When types are displayed, we label ports on the exterior of their box and their types adjacently on the interior of the box with a ‘:’ symbol in between to designate typing. Reading types off of this figure, we see that the type of input port $a$ is the manifold $\RR$, that of input port $b$ is the circle $S^1$, and that of output port $c$ is the torus $T^2$.
A morphism in $\bfW$ is a wiring diagram $\Phi=(X,Y,\varphi)$, the idea being that a smaller box $X$ (the domain) is nested inside of a larger box $Y$ (the codomain). The ports of $X$ and $Y$ are then interconnected by wires, as specified by the typed bijection $\varphi$. We will now see an example of a wiring diagram, accompanied by a picture.
\[ex:wiringdiagram\] Reading off the wiring diagram $\Phi=(X,Y,\varphi)$ drawn below in Figure \[fig:wiringdiagram\], we have the following data for boxes: $$\begin{matrix} {X^{\text{in}}}=\{a,b\} & {X^{\text{out}}}=\{c,d\} \\ {Y^{\text{in}}}=\{m\} & {Y^{\text{out}}}=\{n\}\end{matrix}$$ Table \[tab:wiringdiagram\] makes $\varphi$ explicit via a list of its wires, i.e., pairs $(\gamma,\varphi(\gamma))$.
$$\begin{array}{c||c|c|c}
\rule[-4pt]{0pt}{16pt}
\gamma\in{X^{\text{in}}}+{Y^{\text{out}}}& a & b & n
\\\hline
\rule[-4pt]{0pt}{16pt}
\varphi(\gamma)\in{X^{\text{out}}}+{Y^{\text{in}}} & m & d & c
\end{array}$$ \[tab:wiringdiagram\]
\[rem:samestate\] The condition that $\varphi$ be typed, as in Definition \[def:typed finite sets\], ensures that if two ports are connected by a wire then the associated types are the same. In particular, in Example \[ex:wiringdiagram\] above, $(a,b,n)$ must be the same type tuple as $(m,d,c)$.
Now that we have made wiring diagrams concrete and visual, we can do the same for their composition.
\[ex:composition\] In Figure \[fig:compose\], we visualize the composition of two wiring diagrams $\Phi=(X,Y,\varphi)$ and $\Psi=(Y,Z,\psi)$ to form $\Psi\circ\Phi=(X,Z,\omega)$. Composition is depicted by drawing the wiring diagram for $\Psi$ and then, inside of the $Y$ box, drawing in the wiring diagram for $\Phi$. Finally, to depict the composition $\Psi\circ\Phi$ as one single wiring diagram, one simply “erases" the $Y$ box, leaving the $X$ and $Z$ boxes interconnected among themselves. Figure \[fig:compose\] represents such a procedure by depicting the $Y$ box with a dashed arrow.
It’s important to note that the wires also connect, e.g. if a wire in $\Psi$ connects a $Z$ port to some $Y$ port, and that $Y$ port attaches via a $\Phi$ wire to some $X$ port, then these wires “link together" to a total wire in $\Psi\circ\Phi$, connecting a $Z$ port with an $X$ port. Table \[tab:compose\] below traces the wires of $\Psi\circ\Phi$ through the ${\omega^{\text{in}}}$ and ${\omega^{\text{out}}}$ composition diagrams in (\[dia:composition diagrams\]) on its left and right side, respectively. The left portion of the table starts with $\gamma\in{X^{\text{in}}}$ and ends at ${\omega^{\text{in}}}(\gamma)\in{X^{\text{out}}}+{Z^{\text{in}}}$, with intermediary steps of the composition denoted with superscripts $\gamma^n$. The right portion of the table starts with $\gamma\in{Z^{\text{out}}}$ then goes through the intermediary of $\gamma'\in{Y^{\text{out}}}$ and finally reaches ${\omega^{\text{out}}}(\gamma)\in{Z^{\text{out}}}$. We skip lines on the right portion to match the spacing on the left.
$$\begin{array}{c||c|c|c||c||c}
\rule[-4pt]{0pt}{16pt}
\gamma\in{X^{\text{in}}}& a & b & c & v & \gamma\in{Z^{\text{out}}}
\\\hline
\rule[-4pt]{0pt}{16pt}
\gamma^1\in{X^{\text{out}}}+{Y^{\text{in}}} & d & k & l & {} & {}
\\\hline
\rule[-4pt]{0pt}{16pt}
\gamma^2\in{X^{\text{out}}}+{Y^{\text{out}}}+{Z^{\text{in}}} & d & u & n & m & \gamma'\in{Y^{\text{out}}}
\\\hline
\rule[-4pt]{0pt}{16pt}
\gamma^3\in{X^{\text{out}}}+{X^{\text{out}}}+{Z^{\text{in}}} & d & u & f & {} & {}
\\\hline
\rule[-4pt]{0pt}{16pt}
{\omega^{\text{in}}}(\gamma)\in{X^{\text{out}}}+{Z^{\text{in}}} & d & u & f & e & {\omega^{\text{out}}}(\gamma)\in{X^{\text{out}}}
\end{array}$$ \[tab:compose\]
\[rem:pathology\] The condition that $\varphi$ be both injective and surjective prohibits [*exposed*]{} ports and [*split*]{} ports, respectively, as depicted in Figure \[fig:unsafe\][**a**]{}. The [*no passing wires*]{} condition on $\varphi({Y^{\text{out}}})$ prohibits wires that go straight across the $Y$ box, as seen in the intermediate box of Figure \[fig:unsafe\][**b**]{}.
Now that we have formally defined and concretely explicated the category $\bfW$, we will make it into a monoidal category by defining its tensor product.
\[def:mon\] Let $X_1,X_2,,Y_1,Y_2\in\operatorname{Ob}\bfW$ be boxes and $\Phi_1\taking X_1\to Y_2$ and $\Phi_2\taking X_2\to Y_2$ be wiring diagrams. The *monoidal product* $\oplus$ is given by $$X_1\oplus X_2:=\left({X^{\text{in}}}_1+{X^{\text{in}}}_2\;,\;{X^{\text{out}}}_1+{X^{\text{out}}}_2\;\right), \hspace{10 mm} \Phi_1\oplus\Phi_2:=\Phi_1+\Phi_2.$$ The *closed box* $0=\{\varnothing,\varnothing\}$ is the monoidal unit.
Once we add semantics in Section \[sec:g\], closed boxes will correspond to *autonomous systems*, which do not interact with any outside environment (see Remark \[ex:dynamical system\]).
We now make this monoidal product explicit with an example.
Consider boxes $X=(\{x_1,x_2\},\{x_3,x_4\})$ and $Y=(\{y_1\},\{y_2,y_3\})$ depicted below.
We depict their tensor $X\oplus Y=(\{x_1,x_2,y_1\},\{x_3,x_4,y_2,y_3\})$ by stacking boxes.
Similarly, consider the following wiring diagrams (with ports left unlabelled).
We can depict their composition via stacking.
We now prove that the above data characterizing $(\bfW,\oplus,0)$ indeed constitutes a symmetric monoidal category, at which point we can, as advertised, invoke Definition \[def:SMC to Opd\] to define the operad ${\mathcal{O}\bfW}$.
\[prop:W is SMC\] The category $\bfW$ in Definition \[def:W\] and the monoidal product $\oplus$ with unit $0$ in Definition \[def:mon\] form a symmetric monoidal category $(\bfW,\oplus,0)$.
We begin by establishing that $\bfW$ is indeed a category. We first show that our class of wiring diagrams is closed under composition. Let $\Phi=(X,Y,\varphi)$, $\Psi=(Y,Z,\psi)$, and $\Psi\circ\Phi=(X,Z,\omega)$.
To show that $\omega$ is a typed bijection, we replace the pair of maps $({\varphi^{\text{in}}},{\varphi^{\text{out}}})$ with a pair of bijections $(\widetilde{{\varphi^{\text{in}}}},\widetilde{{\varphi^{\text{out}}}})$ as follows. Let ${X_{\varphi}^{\text{exp}}}\ss{X^{\text{out}}}$ (for *exports*) denote the image of ${\varphi^{\text{out}}}$, and ${X_{\varphi}^{\text{loc}}}$ (for *local ports*) be its complement. Then we can identify $\varphi$ with the following pair of typed bijections $$\left\{
\begin{array}{lr}
\widetilde{{\varphi^{\text{in}}}}\taking {X^{\text{in}}} \xrightarrow{\cong} {X_{\varphi}^{\text{loc}}}+{Y^{\text{in}}} \\
\widetilde{{\varphi^{\text{out}}}}\taking {Y^{\text{out}}} \xrightarrow{\cong} {X_{\varphi}^{\text{exp}}}
\end{array}
\right.$$
Similarly, identify $\psi$ with $(\widetilde{{\psi^{\text{in}}}},\widetilde{{\psi^{\text{out}}}})$. We can then rewrite the diagram defining $\omega$ in (\[dia:composition diagrams\]) as one single commutative diagram of typed finite sets. $$\xymatrixcolsep{4pc}
\xymatrix{
{X^{\text{in}}}+{Z^{\text{out}}}
\ar[d]_{\widetilde{{\varphi^{\text{in}}}}+\widetilde{{\psi^{\text{out}}}}}
\ar@{-->}[r]^{\omega}
&{X^{\text{out}}}+{Z^{\text{in}}} \\
{X_{\varphi}^{\text{loc}}}+{Y^{\text{in}}}+{Y_{\psi}^{\text{exp}}}
\ar[d]_{\bigid _{{X_{\varphi}^{\text{loc}}}}+\widetilde{{\psi^{\text{in}}}}+\bigid _{{Y_{\psi}^{\text{exp}}}}}
&{X_{\varphi}^{\text{loc}}}+{X_{\varphi}^{\text{exp}}}+{Z^{\text{in}}}
\ar[u]_{\cong} \\
{X_{\varphi}^{\text{loc}}}+{Y_{\psi}^{\text{loc}}}+{Z^{\text{in}}}+{Y_{\psi}^{\text{exp}}}
\ar[r]_-{\cong} &
{X_{\varphi}^{\text{loc}}}+{Y^{\text{out}}}+{Z^{\text{in}}}
\ar[u]_{\bigid _{{X_{\varphi}^{\text{loc}}}}+\widetilde{{\varphi^{\text{out}}}}+\bigid _{{Z^{\text{in}}}}}
}$$ As a composition of typed bijections, $\omega$ is also a typed bijection.
The following computation proves that $\omega$ has no passing wires:
$$\omega({Z^{\text{out}}})=\varphi\big(\psi({Z^{\text{out}}})\big)\subseteq\varphi({Y^{\text{out}}})\subseteq {X^{\text{out}}}.$$
Therefore $\bfW$ is closed under wiring diagram composition. To show that $\bfW$ is a category, it remains to prove that composition of wiring diagrams satisfies the unit and associativity axioms. The former is straightforward and will be omitted. We now establish the latter.
Consider the wiring diagrams $\Theta=(V,X,\theta),\Phi=(X,Y,\varphi),\Psi=(Y,Z,\psi)$; and let $(\Psi\circ\Phi)\circ\Theta=(V,Z,\kappa)$ and $\Psi\circ(\Phi\circ\Theta)= (V,Z,\lambda)$. We readily see that ${\kappa^{\text{out}}}={\lambda^{\text{out}}}$ by the associativity of composition in ${\mathbf{TFS}_{}}$. Proving that ${\kappa^{\text{in}}}={\lambda^{\text{in}}}$ is equivalent to establishing the commutativity of the following diagram:
$$\label{eqn:associativity}
\xymatrix@C=31pt@R=1.8pc{
{} &{V^{\text{out}}}+{Z^{\text{in}}} &{} \\
{} &{V^{\text{out}}}+{V^{\text{out}}}+{Z^{\text{in}}} \ar[u]^{\nabla+\bigid } &{} \\
{V^{\text{out}}}+{Y^{\text{out}}}+{Z^{\text{in}}} \ar[r]^-{\bigid +{\varphi^{\text{out}}}+\bigid }
&{V^{\text{out}}}+{X^{\text{out}}}+{Z^{\text{in}}} \ar[u]^{\bigid + {\theta^{\text{out}}}+\bigid }
&{V^{\text{out}}}+{X^{\text{out}}}+{X^{\text{out}}}+{Z^{\text{in}}} \ar[l]_-{\bigid +\nabla+\bigid } \\
{V^{\text{out}}}+{Y^{\text{in}}} \ar[u]^{\bigid +{\psi^{\text{in}}}} &{} &{} \\
{V^{\text{out}}}+{V^{\text{out}}}+{Y^{\text{in}}} \ar[u]^{\nabla+\bigid }
&{V^{\text{out}}}+{X^{\text{out}}}+{Y^{\text{in}}} \ar[l]^-{\bigid + {\theta^{\text{out}}}+\bigid } \ar[r]_-{\bigid +\bigid +{\psi^{\text{in}}}}
&{V^{\text{out}}}+{X^{\text{out}}}+{Y^{\text{out}}}+{Z^{\text{in}}} \ar[uu]_{\bigid +\bigid +{\varphi^{\text{out}}}+\bigid } \\
{} &{V^{\text{out}}}+{X^{\text{in}}} \ar[u]^{\bigid + {\varphi^{\text{in}}}} &{} \\
{} &{V^{\text{in}}} \ar[u]^{{\theta^{\text{in}}}}&{}
}$$
This diagram commutes in any category with coproducts, as follows from the associativity and naturality of the codiagonal map. We present a formal argument of this fact below in the language of string diagrams (See [@JSTensor]). As in [@Selinger], we let squares with blackened corners denote generic morphisms. We let triangles denote codiagonal maps. See Figure \[string\] below.
\[fig:string\]
\[scale=.8\] at (2,0) (Theta11) [$\theta^\text{out}$]{}; at (4.5,.35) (V1) ; at (3.5,-1.15) (Psi) [$\psi^\text{in}$]{}; at (6,-.7) (Phi) [$\varphi^\text{out}$]{}; at (8.5,-.7) (Theta12) [$\theta^\text{out}$]{}; at (10.7,-.13) (V2) ;
(.13,0) – (Theta11.west); (Theta11.east) – (\[yshift=-10pt\]V1.west); (.13,.75) – (\[yshift=11pt\]V1.west); (\[yshift=12pt\]Psi.east) – (\[yshift=-.5pt\]Phi.west); (\[yshift=-11pt\]Psi.east) – (13,-1.5); (Phi.east) – (Theta12.west); (.13,-1.15) – (Psi.west); (V1.east) – (\[yshift=12pt\]V2.west); (\[yshift=6pt\]Theta12.east) – (\[yshift=-10pt\]V2.west); (V2.east) – (13,-.13);
\[scale=.8\] at (6.5,0.5) (Theta11) [$\theta^\text{out}$]{}; at (2,-1.15) (Psi) [$\psi^\text{in}$]{}; at (4.2,-.7) (Phi) [$\varphi^\text{out}$]{}; at (6.5,-.7) (Theta12) [$\theta^\text{out}$]{}; at (8.7,.8) (V1) ; at (11.1,.15) (V2) ;
(\[yshift=-1.7pt\]Theta11.east) – (\[yshift=-10pt\]V1.west); (\[yshift=12pt\]Psi.east) – (\[yshift=-.5pt\]Phi.west); (\[yshift=-11pt\]Psi.east) – (13,-1.5); (Phi.east) – (Theta12.west); (0,-1.15) – (Psi.west); (0,.5) – (Theta11.west); (\[yshift=12pt\]Theta12.east) – (\[yshift=-12pt\]V2.west); (V1.east) – (\[yshift=12pt\]V2.west); (0,1.2) – (\[yshift=12pt\]V1.west); (V2.east) – (13,.15);
\[scale=.8\] at (6.5,0.5) (Theta11) [$\theta^\text{out}$]{}; at (2,-1.15) (Psi) [$\psi^\text{in}$]{}; at (4.2,-.7) (Phi) [$\varphi^\text{out}$]{}; at (6.5,-.7) (Theta12) [$\theta^\text{out}$]{}; at (8.7,-.13) (V1) ; at (11.1,.3) (V2) ;
(\[yshift=-6.5pt\]Theta11.east) – (\[yshift=11pt\]V1.west); (\[yshift=12pt\]Psi.east) – (\[yshift=-.5pt\]Phi.west); (\[yshift=-11pt\]Psi.east) – (13,-1.5); (Phi.east) – (Theta12.west); (0,-1.15) – (Psi.west); (0,.5) – (Theta11.west); (\[yshift=6pt\]Theta12.east) – (\[yshift=-10pt\]V1.west); (V1.east) – (\[yshift=-12pt\]V2.west); (0,1.2) – (8,1.2) – (\[yshift=12pt\]V2.west); (V2.east) – (13,.3);
\[scale=.8\] at (9.2,-0.3) (Theta) [$\theta^\text{out}$]{}; at (2,-1.15) (Psi) [$\psi^\text{in}$]{}; at (4.2,-.7) (Phi) [$\varphi^\text{out}$]{}; at (6.5,-.3) (V1) ; at (11.1,.3) (V2) ;
(\[yshift=12pt\]Psi.east) – (\[yshift=-.6pt\]Phi.west); (\[yshift=-11pt\]Psi.east) – (13,-1.5); (Phi.east) – (\[yshift=-11.pt\]V1.west); (0,-1.15) – (Psi.west); (0,.1) – (\[yshift=12pt\]V1.west); (V1.east) – (Theta.west); (\[yshift=6pt\]Theta.east) – (\[yshift=-10pt\]V2.west); (0,.75) – (\[yshift=12pt\]V2.west); (V2.east) – (13,.3);
The first step of the proof follows from the topological nature of string diagrams, which mirror the axioms of monoidal categories. The second step invokes the associativity of codiagonal maps. The third and final step follows from the naturality of codiagonal maps, i.e., the commutativity of the following diagram.
$$\xymatrix{{V^{\text{out}}}+{V^{\text{out}}} \ar[r]^-{\nabla} \ar[d]_{{\theta^{\text{out}}}+{\theta^{\text{out}}}} & {V^{\text{out}}} \ar[d]^{{\theta^{\text{out}}}} \\
{X^{\text{out}}}+{X^{\text{out}}} \ar[r]^-{\nabla} & {X^{\text{out}}}
}$$
Now that we have shown that $\bfW$ is a category, we show that $(\oplus,0)$ is a monoidal structure on $\bfW$. Let $X,X',X''\in\operatorname{Ob}\bfW$ be boxes. We readily observe the following canonical isomorphisms. $$\begin{aligned}
&X\oplus 0= X= 0\oplus X &\emph{(unity)}\\
&(X\oplus X')\oplus X''= X\oplus (X'\oplus X'')&\emph{(associativity)}\\
&X\oplus X'= X'\oplus X &\emph{(commutativity)}\end{aligned}$$ Hence the monoidal product $\oplus$ is well behaved on objects. It is similarly easy, and hence will be omitted, to show that $\oplus$ is functorial. This completes the proof that $(\bfW,\oplus,0)$ is a symmetric monoidal category.
Having established that $(\bfW,\oplus,0)$ is an SMC, we can now speak about the operad ${\mathcal{O}\bfW}$ of wiring diagrams. In particular, we can draw operadic pictures, such as the one in our motivating example in Figure \[fig:pipebrine\], to which we now return.
\[ex:wiring explained\] Figure \[fig:WD\] depicts an ${\mathcal{O}\bfW}$ wiring diagram $\Phi\taking X_1,X_2\to Y$, which we may formally denote by the tuple $\Phi=(X_1,X_2;Y;\varphi)$. Reading directly from Figure \[fig:WD\], we have the boxes:
$$\begin{aligned}
X_1&=\big(\{{X^{\text{in}}}_{1a},{X^{\text{in}}}_{1b}\},\{{X^{\text{out}}}_{1a}\}\big) \\
X_2&=\big(\{{X^{\text{in}}}_{2a},{X^{\text{in}}}_{2b}\} ,\{{X^{\text{out}}}_{2a},{X^{\text{out}}}_{2b}\}\big) \\
Y&=\big(\{{Y^{\text{in}}}_a,{Y^{\text{in}}}_b\},\{{Y^{\text{out}}}_a\}\big)\end{aligned}$$
The wiring diagram $\Phi$ is visualized by nesting the domain boxes $X_1,X_2$ within the codomain box $Y$, and drawing the wires prescribed by $\varphi$, as recorded below in Table \[tab:explicit\].
$$\begin{array}{c||c|c|c|c|c}
\rule[-4pt]{0pt}{16pt}
w\in{X^{\text{in}}}+{Y^{\text{out}}}&{X^{\text{in}}}_{1a}&{X^{\text{in}}}_{1b}&{X^{\text{in}}}_{2a}&{X^{\text{in}}}_{2b}&{Y^{\text{out}}}_{a}
\\\hline
\rule[-4pt]{0pt}{16pt}
\varphi(w)\in{X^{\text{out}}}+{Y^{\text{in}}}&{Y^{\text{in}}}_{b}&{X^{\text{out}}}_{2b}&{Y^{\text{in}}}_{a}&{X^{\text{out}}}_{1a}&{X^{\text{out}}}_{2a}
\end{array}$$ \[tab:explicit\]
To reconceptualize $\Phi\taking X_1,X_2\to Y$ as a wiring diagram in $\bfW$, we simply consider the tensor $\Phi\taking X_1\oplus X_2\to Y$, as given in Figure \[fig:reconcept\] below. This demonstrates the fact that operadic pictures are easier to read and hence are more illuminating.
The following remark explains that our pictures of wiring diagrams are not completely ad hoc—they are depictions of 1-dimensional oriented manifolds with boundary. The boxes in our diagrams simply tie together the positively and negatively oriented components of an individual oriented 0-manifold.
\[rem:cobordism\] For any set $S$, let $\operatorname{1--\bf Cob}/S$ denote the symmetric monoidal category of oriented 0-manifolds over $S$ and the 1-dimensional cobordisms between them. We call its objects *oriented $S$-typed 0-manifolds*. Recall that $\bfW=\bfW_{\Man}$ is our category of $\Man$-typed wiring diagrams; let ${\mathbf M}:=\operatorname{Ob}\Man$ denote the set of manifolds (see Remark \[rem:default\]). There is a faithful, essentially surjective, strong monoidal functor $$\bfW\to \operatorname{1--\bf Cob}/{\mathbf M},$$ sending a box $({X^{\text{in}}},{X^{\text{out}}})$ to the oriented ${\mathbf M}$-typed 0-manifold ${X^{\text{in}}}+{X^{\text{out}}}$ where ${X^{\text{in}}}$ is oriented positively and ${X^{\text{out}}}$ negatively. Under this functor, a wiring diagram $\Phi=(X,Y,\varphi)$ is sent to a 1-dimensional cobordism that has no closed loops. A connected component of such a cobordism can be identified with either its left or right endpoint, which correspond to the domain or codomain of the bijection . See [@SpivakSchultzRupel].
In fact, with the [*no passing wires*]{} condition on morphisms (cobordisms) $X\to Y$ (see Definition \[def:W\]), the subcategory $\bfW\ss\operatorname{1--\bf Cob}/{\mathbf M}$ is the left class of an orthogonal factorization system. See [@Abadi].
Let $\Phi=(X,Y,\varphi)$ be a wiring diagram. Applying the dependent product functor (see Definition \[def:depprod\]) to $\varphi$, we obtain a diffeomorphism of manifolds $$\label{eqn:prodwd}\overline{\varphi}\taking \overline{{X^{\text{out}}}}\times\overline{{Y^{\text{in}}}}\to \overline{{X^{\text{in}}}}\times\overline{{Y^{\text{out}}}}.$$ Equivalently, if $\varphi$ is represented by the pair $({\varphi^{\text{in}}},{\varphi^{\text{out}}})$, as in Definition \[def:W\], we can express ${\overline{\varphi}}$ in terms of its pair of component maps: $$\left\{
\begin{array}{lr}
\overline{{\varphi^{\text{in}}}}\taking \overline{{X^{\text{out}}}}\times\overline{{Y^{\text{in}}}}\to\overline{{X^{\text{in}}}} \\
\overline{{\varphi^{\text{out}}}}\taking \overline{{X^{\text{out}}}}\to\overline{{Y^{\text{out}}}}
\end{array}
\right.$$
It will also be useful to apply the dependent product functor to the commutative diagrams in (\[dia:composition diagrams\]), which define wiring diagram composition. Note that, by the contravariance of the dependent product, the codiagonal $\nabla\taking {X^{\text{out}}}+{X^{\text{out}}}\to {X^{\text{out}}}$ gets sent to the diagonal map $\Delta\taking \overline{{X^{\text{out}}}}\to\overline{{X^{\text{out}}}}\times\overline{{X^{\text{out}}}}$. Thus we have the following commutative diagrams: $$\begin{aligned}
\label{dia:dep prod of wd}
\xymatrixcolsep{3.5pc}
\xymatrix{
\overline{{X^{\text{out}}}}\times\overline{{Z^{\text{in}}}}
\ar[r]^{\overline{{\omega^{\text{in}}}}}
\ar[d]_{\Delta\times\bigid }
&\overline{{X^{\text{in}}}}
\\
\overline{{X^{\text{out}}}}\times\overline{{X^{\text{out}}}}\times\overline{{Z^{\text{in}}}}
\ar[d]_{\bigid \times\overline{{\varphi^{\text{out}}}}\times\bigid }
&{}
\\
\overline{{X^{\text{out}}}}\times\overline{{Y^{\text{out}}}}\times\overline{{Z^{\text{in}}}}
\ar[r]_-{\bigid \times\overline{{\psi^{\text{in}}}}}
&\overline{{X^{\text{out}}}}\times\overline{{Y^{\text{in}}}}
\ar[uu]_{\overline{{\varphi^{\text{in}}}}}
} \mskip15mu
\xymatrixcolsep{1.5pc}
\xymatrix{
\overline{{X^{\text{out}}}}
\ar[rd]_{\overline{{\varphi^{\text{out}}}}}
\ar[rr]^{\overline{{\omega^{\text{out}}}}}
&
&\overline{{Z^{\text{out}}}}\\
&\overline{{Y^{\text{out}}}}
\ar[ru]_{\overline{{\psi^{\text{out}}}}}}\end{aligned}$$
The Algebra of Open Systems {#sec:g}
===========================
In this section we define an algebra $\mcG\taking(\bfW,\oplus,0)\to(\mathbf{Set},\times,\star)$ (see Definition \[def:algebra\]) of general open dynamical systems. A $\bfW$-algebra can be thought of as a choice of semantics for the syntax of $\bfW$, i.e., a set of possible meanings for boxes and wiring diagrams. As in Definition \[def:SMC to Opd\], we may use this to construct the corresponding operad algebra ${\mathcal{O}\mcG }:{\mathcal{O}\bfW}\to\mathbf{Sets}$. Before we define $\mcG$, we revisit Example \[ex:main\] for inspiration.
\[ex:promise\] As the textbook exercise [@BD Problem 7.21] prompts, let’s begin by writing down the system of equations that governs the amount of salt $Q_i$ within the tanks $X_i$. This can be done by using dimensional analysis for each port of $X_i$ to find the the rate of salt being carried in ounces per minute, and then equating the rate $\dot{Q}_i$ to the sum across these rates for ${X^{\text{in}}}_i$ ports minus ${X^{\text{out}}}_i$ ports.
$$\begin{aligned}
\dot{Q}_1\frac{\text{oz}}{\text{min}}&=
-\left(\frac{Q_1 \text{oz}}{30 \text{gal}}\cdot\frac{3 \text{gal}}{\text{min}}\right)
+\left(\frac{Q_2\text{oz}}{20\text{gal}}\cdot\frac{1.5\text{gal}}{\text{min}}\right)
+\left(\frac{1\text{oz}}{\text{gal}}\cdot\frac{1.5\text{gal}}{\text{min}}\right) \\
\dot{Q}_2\frac{\text{oz}}{\text{min}}&=
-\left(\frac{Q_2 \text{oz}}{20 \text{gal}}\cdot\frac{(1.5+2.5) \text{gal}}{\text{min}}\right)
+\left(\frac{Q_1\text{oz}}{30\text{gal}}\cdot\frac{3\text{gal}}{\text{min}}\right)
+\left(\frac{3\text{oz}}{\text{gal}}\cdot\frac{1\text{gal}}{\text{min}}\right)\end{aligned}$$
Dropping the physical units, we are left with the following system of ODEs:
$$\label{eqn:naive}
\left\{
\begin{array}{lr}
\dot{Q}_1=-.1Q_1+.075Q_2+1.5 \\
\dot{Q}_2=.1Q_1-.2Q_2+3
\end{array}
\right.$$
The derivations for the equations in (\[eqn:naive\]) involved a hidden step in which the connection pattern in Figure \[fig:pipebrine\], or equivalently Figure \[fig:WD\], was used. Our wiring diagram approach explains this step and makes it explicit. Each box in a wiring diagram should only “know” about its own inputs and outputs, and not how they are connected to others. That is, we can only define a system on $X_i$ by expressing $\dot{Q}_i$ just in terms of $Q_i$ and ${X^{\text{in}}}_i$—this is precisely the data of an open system (see Definition \[def:opensystem\]). We now define our algebra $\mcG$, which assigns a set of open systems to a box. Given a wiring diagram and an open system on its domain box, it also gives a functorial procedure for assigning an open system to the codomain box. We will then use this new machinery to further revisit Example \[ex:promise\] in Example \[ex:as promised\].
\[def:general algebra\] We define $\mcG:(\bfW,\oplus,0)\to(\Set,\times,\star)$ as follows. Let $X\in\operatorname{Ob}\bfW$. The *set of open systems on $X$*, denoted $\mcG(X)$, is defined as $$\mcG (X)=\{(S,f)\; |\;S\in\operatorname{Ob}{\mathbf{TFS}_{}},(\overline{S},\overline{{X^{\text{in}}}},\overline{{X^{\text{out}}}},f)\in\operatorname{Ob}\ODS \}.$$ We call $S$ the set of *state variables* and its dependent product $\overline{S}$ the *state space*.
Let $\Phi=(X,Y,\varphi)$ be a wiring diagram. Then $\mcG (\Phi)\taking \mcG (X)\to\mcG (Y)$ is given by $(S,f)\mapsto (\mcG (\Phi)S,\mcG (\Phi)f)$, where $\mcG (\Phi)S=S$ and $g=\mcG (\Phi)f\taking \overline{S}\times\overline{{Y^{\text{in}}}}\to T\overline{S}\times\overline{{Y^{\text{out}}}}$ is defined by the dashed arrows $({g^{\text{in}}},{g^{\text{out}}})$ (see Definition \[def:opensystem\]) that make the diagrams below commute: $$\xymatrixcolsep{3.5pc}
\xymatrix{
\overline{S}\times\overline{{Y^{\text{in}}}}
\ar[d]_{\Delta\times\bigid _{\overline{{Y^{\text{in}}}}}}
\ar@{-->}[r]^-{{g^{\text{in}}}}
&T\overline{S}
\\
\overline{S}\times\overline{S}\times\overline{{Y^{\text{in}}}}
\ar[d]_{\bigid _{\overline{S}}\times {f^{\text{out}}}\times\bigid _{\overline{{Y^{\text{in}}}}}}
& {}
\\
\overline{S}\times\overline{{X^{\text{out}}}}\times\overline{{Y^{\text{in}}}}
\ar[r]_-{\bigid _{\overline{S}}\times\overline{{\varphi^{\text{in}}}}}
&\overline{S}\times\overline{{X^{\text{in}}}}
\ar[uu]_{{f^{\text{in}}}}
} \hspace{7 mm} \xymatrixcolsep{2.5pc}
\xymatrix{
\overline{S}
\ar[rd]_{{f^{\text{out}}}}
\ar@{-->}[rr]^{{g^{\text{out}}}}
&
&\overline{{Y^{\text{out}}}}\\
&\overline{{X^{\text{out}}}}
\ar[ru]_{\overline{{\varphi^{\text{out}}}}}}
\label{eqn:G of wd}$$ One may note strong resemblance between the diagrams in (\[eqn:G of wd\]) and those in (\[dia:composition diagrams\]).
We give $\mcG $ a lax monoidal structure: for any pair $X,X'\in\bfW$ we have a coherence map $\mu_{X,X'}:\mcG (X)\times\mcG (X')\to\mcG (X\oplus X')$ given by $$\big((S,f),(S',f')\big)\mapsto (S+S',f\times f'),$$ where $f\times f'$ is as in Lemma \[def:osprod\]. \[def:mu\]
Recall from Remark \[rem:default\] that $\Man$ is small, so the collection $\mcG(X)$ of open systems on $X$ is indeed a set.
One may also encode an initial condition in $\mcG $ by using $\Man_*$ instead of $\Man$ in Remark \[rem:default\] as the default choice of finite product category, where $\Man_*$ is the category of pointed smooth manifolds and base point preserving smooth maps. The base point represents the initialization of the state variables.
We now establish that $\mcG$ is indeed an algebra.
\[prop:G is W-alg\] The pair $(\mcG,\mu)$ of Definition \[def:general algebra\] is a lax monoidal functor, i.e., $\mcG$ is a $\bfW$-algebra.
Let $\Phi=(X,Y,\varphi)$ and $\Psi=(Y,Z,\psi)$ be wiring diagrams in $\bfW$. To show that $\mcG$ is a functor, we must have that $\mcG (\Psi\circ\Phi)=\mcG (\Psi)\circ\mcG (\Phi)$. Immediately we have $\mcG (\Psi\circ\Phi)S=S=\mcG (\Psi)(\mcG (\Phi)S)$.
Now let and $k:=\mcG (\Psi)(\mcG (\Phi)f)$. It suffices to show $h=k$, or equivalently $({h^{\text{in}}},{h^{\text{out}}})=({k^{\text{in}}},{k^{\text{out}}})$. One readily sees that ${h^{\text{out}}}={k^{\text{out}}}$. We use (\[dia:dep prod of wd\]) and (\[eqn:G of wd\]) to produce the following diagram; showing it commutes is equivalent to proving that that ${h^{\text{in}}}={k^{\text{in}}}$. $$\label{eqn:algebracomp}
\xymatrixcolsep{3pc}
\xymatrix{
{} &\overline{S}\times\overline{{Z^{\text{in}}}}\ar[d]^{\Delta\times\bigid } &{} \\
{} &\overline{S}\times\overline{S}\times\overline{{Z^{\text{in}}}} \ar[d]^{\bigid \times {f^{\text{out}}}\times\bigid } &{} \\
\overline{S}\times\overline{{Y^{\text{out}}}}\times\overline{{Z^{\text{in}}}} \ar[d]_{\bigid \times\overline{{\psi^{\text{in}}}}}
&\overline{S}\times\overline{{X^{\text{out}}}}\times\overline{{Z^{\text{in}}}} \ar[l]_-{\bigid \times\overline{{\varphi^{\text{out}}}}\times\bigid } \ar[r]^-{\bigid \times\Delta\times\bigid }
&\overline{S}\times\overline{{X^{\text{out}}}}\times\overline{{X^{\text{out}}}}\times\overline{{Z^{\text{in}}}} \ar[dd]^{\bigid \times\bigid \times\overline{{\varphi^{\text{out}}}}\times\bigid } \\
\overline{S}\times\overline{{Y^{\text{in}}}} \ar[d]_{\Delta\times\bigid } &{} &{} \\
\overline{S}\times\overline{S}\times\overline{{Y^{\text{in}}}} \ar[r]_-{\bigid \times {f^{\text{out}}}\times\bigid }
&\overline{S}\times\overline{{X^{\text{out}}}}\times\overline{{Y^{\text{in}}}} \ar[d]^{\bigid \times\overline{{\varphi^{\text{in}}}}}
&\overline{S}\times\overline{{X^{\text{out}}}}\times\overline{{Y^{\text{out}}}}\times\overline{{Z^{\text{in}}}} \ar[l]^-{\bigid \times\bigid \times\overline{{\psi^{\text{in}}}}}\\
{} &\overline{S}\times\overline{{X^{\text{in}}}}\ar[d]^{{f^{\text{in}}}} &{} \\
{} &T\overline{S} &{}
}$$
The commutativity of this diagram, which is dual to the one for associativity in (\[eqn:associativity\]), holds in an arbitrary category with products. Although the middle square fails to commute by itself, the composite of the first two maps equalizes it; that is, the two composite morphisms agree.
Since we proved the analogous result via string diagrams in the proof of Proposition \[prop:W is SMC\], we show it concretely using elements this time. Let $(s,z)\in{\overline{S}}\times{\overline{{Z^{\text{in}}}}}$ be an arbitrary element. Composing six morphisms ${\overline{S}}\times{\overline{{Z^{\text{in}}}}}\too{\overline{S}}\times{\overline{{X^{\text{out}}}}}\times{\overline{{Y^{\text{in}}}}}$ through the left of the diagram gives the same answer as composing through the right; namely, $$\Big(s,{f^{\text{out}}}(s),{\psi^{\text{in}}}\big({\varphi^{\text{out}}}\circ{f^{\text{out}}}(s),z\big)\Big)\in{\overline{S}}\times{\overline{{X^{\text{out}}}}}\times{\overline{{Y^{\text{in}}}}}.$$
Since the diagram commutes, we have shown that $\mcG$ is a functor. To prove that the pair $(\mcG,\mu)$ constitutes a lax monoidal functor $\bfW\to\mathbf{Set}$, i.e., a $\bfW$-algebra, we must establish coherence. Since $\mu$ simply consists of a coproduct and a product, this is straightforward and will be omitted.
As established in Definition \[def:SMC to Opd\], the coherence map $\mu$ allows us to define the operad algebra ${\mathcal{O}\mcG}$ from $\mcG$. This finally provides the formal setting to consider open dynamical systems over operadic wiring diagrams, such as our motivating one in Figure \[fig:pipebrine\]. We note that, in contrast to the trivial equality $\mcG(\Phi)S=S$ found in Definition \[def:general algebra\], in the operadic setting we have $${\mathcal{O}\mcG}(\Phi)(S_1,\ldots,S_n)=\amalg_{i=1}^n S_i.$$ This simply means that the set of state variables of the larger box $Y$ is the disjoint union of the state variables of its constituent boxes $X_i$. Now that we have the tools to revisit Example \[ex:promise\], we do so in the following section, but first we will define the subalgebra $\mcL$ to which it belongs—that of linear open systems.
The Subalgebra of Linear Open Systems {#sec:l}
=====================================
In this section, we define the algebra $\mcL\taking\bfW_{\bfL}\to\Set$, which encodes linear open systems. Here $\bfW_{\bfL}$ is the category of $\bfL$-typed wiring diagrams, as in Remark \[rem:different Cs\]. Of course, one can use Definition \[def:SMC to Opd\] to construct an operad algebra ${\mathcal{O}\mcL }:{\mathcal{O}\bfW_{\bfL}}\to\mathbf{Sets}$.
Before we give a formal definition for $\mcL$, we first provide an alternative description for linear open systems and wiring diagrams in $\bfW_{\bfL}$. The category $\bfL$ enjoys special properties—in particular it is an additive category, as seen by the fact that there is an equivalence of categories $\bfL\cong \mathbf{Vect}_\RR$. Specifically, finite products and finite coproducts are isomorphic. Hence a morphism in $\bfL$ canonically decomposes into a matrix equation $$\begin{bmatrix} a_1 \\ a_2 \end{bmatrix} \mapsto \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} = \begin{bmatrix} f^{1,1} & f^{1,2} \\ f^{2,1} & f^{2,2} \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}$$ This matrix is naturally equivalent to the whole map $f$ by universal properties. We use these to rewrite our relevant $\bfL$ maps in Definitions \[def:rewrite\] and \[def:rewrite2\] below.
\[def:rewrite\] Suppose that $(M,{U^{\text{in}}},{U^{\text{out}}},f)$ is a linear open system and hence $f:M\times{U^{\text{in}}}\to TM\times {U^{\text{out}}}$. Then $f$ decomposes into the four linear maps: $$\begin{aligned}
f^{M,M}&\taking M\to TM & f^{M,U}&\taking {U^{\text{in}}}\to TM \\ f^{U,M}&\taking M\to{U^{\text{out}}} & f^{U,U}&\taking{U^{\text{in}}}\to{U^{\text{out}}}\end{aligned}$$ By Definition \[def:opensystem\], we know $f^{U,U}=0$. If we let $(m,{u^{\text{in}}},{u^{\text{out}}})\in M\times{U^{\text{in}}}\times{U^{\text{out}}}$, these equations can be organized into a single matrix equation $$\label{eqn:matrixform} \begin{bmatrix}\dot{m} \\ {u^{\text{out}}} \end{bmatrix}=\begin{bmatrix} f^{M,M} & f^{M,U} \\ f^{U,M} & 0 \end{bmatrix}\begin{bmatrix} m \\ {u^{\text{in}}} \end{bmatrix}$$
We will exploit this form in Definition \[def:linear algebra\] to define how $\mcL$ acts on wiring diagrams in terms of one single matrix equation, in place of the seemingly complicated commutative diagrams in (\[eqn:G of wd\]). To do so, we also recast wiring diagrams in matrix format in Definition \[def:rewrite2\] below.
\[def:rewrite2\] Suppose $\Phi=(X,Y,\varphi)$ is a wiring diagram in $\bfW_\bfL$. Recalling (\[eqn:prodwd\]), we apply the dependent product functor to $\varphi$: $${\overline{\varphi}}\taking{\overline{{X^{\text{out}}}}}\times{\overline{{Y^{\text{in}}}}}\to{\overline{{X^{\text{in}}}}}\times{\overline{{Y^{\text{out}}}}}$$ Since this is a morphism in $\bfL$, it can be decomposed into four linear maps $$\begin{aligned}
\overline{\varphi}^{X,X}&\taking \overline{{X^{\text{out}}}}\to\overline{{X^{\text{in}}}}&\overline{\varphi}^{X,Y}&\taking \overline{{X^{\text{out}}}}\to\overline{{Y^{\text{out}}}}\\
\overline{\varphi}^{Y,X}&\taking \overline{{Y^{\text{in}}}}\to\overline{{X^{\text{out}}}}&\overline{\varphi}^{Y,Y}&\taking \overline{{Y^{\text{in}}}}\to\overline{{Y^{\text{out}}}}\end{aligned}$$ By virtue of the no passing wires condition in Definition \[def:W\], we must have . We can then, as in (\[eqn:matrixform\]), organize this information in one single matrix:
$$\overline{\varphi}=
\begin{bmatrix} \;\overline{\varphi^{X,X}} &\overline{\varphi^{X,Y}}\; \\
\overline{\varphi^{Y,X}}
& 0
\end{bmatrix}$$
The bijectivity condition in Definition \[def:W\] implies that $\overline{\varphi}$ is a permutation matrix.
We now employ these matrix characterizations to define the algebra $\mcL$ of linear open systems.
\[def:linear algebra\] We define the algebra $\mcL\taking(\bfW_\bfL,\oplus,0)\to (\Set,\times,\star)$ as follows. Let $X\in\operatorname{Ob}\bfW_{\bfL}$. Then the *set of linear open systems $\mcL(X)$ on $X$* is defined as $$\mcL(X):=\big\{(S,f)\;|\;S\in\operatorname{Ob}{\mathbf{TFS}_{\bfL}}, (\overline{S},\overline{{X^{\text{in}}}},\overline{{X^{\text{out}}}},f)\in\operatorname{Ob}\ODS _\bfL\big\}.$$
Let $\Phi=(X,Y,\varphi)$ be a wiring diagram. Then, as in Definition \[def:general algebra\], we define $\mcL (\Phi)(S,f):=(S,g)$. We use the format of Definitions \[def:rewrite\] and \[def:rewrite2\] to define $g$: $$\label{eqn:glin}
\begin{split}
g=
\begin{bmatrix} g^{S,S} & g^{S,X} \\ g^{X,S} & g^{X,X} \end{bmatrix} & =\begin{bmatrix} f^{S,X} & 0 \\ 0 & I \end{bmatrix}
\overline{\varphi}
\begin{bmatrix} f^{X,S} & 0 \\ 0 & I \end{bmatrix}+\begin{bmatrix} f^{S,S} & 0 \\ 0 & 0 \end{bmatrix} \\
& =\begin{bmatrix} f^{S,X} & 0 \\ 0 & I \end{bmatrix}
\begin{bmatrix}{\overline{\varphi}}^{X,X}&{\overline{\varphi}}^{X,Y}\\{\overline{\varphi}}^{Y,X}&{\overline{\varphi}}^{Y,Y}\end{bmatrix}
\begin{bmatrix} f^{X,S} & 0 \\ 0 & I \end{bmatrix}+\begin{bmatrix} f^{S,S} & 0 \\ 0 & 0 \end{bmatrix} \\
& =\begin{bmatrix} f^{S,X}\overline{\varphi}^{X,X}f^{X,S}+f^{S,S} & f^{S,X}\overline{\varphi}^{X,Y} \\ \overline{\varphi}^{Y,X}f^{X,S} & 0 \end{bmatrix}
\end{split}$$ This is really just a linear version of the commutative diagrams in (\[eqn:G of wd\]). For example, the equation $g^{S,S}=f^{S,X}\overline{\varphi}^{X,X}f^{X,S}+f^{S,S}$ can be read off the diagram for ${g^{\text{in}}}$ in (\[eqn:G of wd\]), using the additivity of $\bfL$.
Finally, The coherence map $\mu_{\bfL_{X,X'}}:\mcL (X)\times\mcL (X')\to\mcL (X\oplus X')$ is given, as in Definition \[def:mu\], by $\big((S,f),(S',f')\big)\mapsto (S+S',f\times f')$.
We now establish that this constitutes an algebra.
The pair $(\mcL,\mu_\bfL)$ of Definition \[def:linear algebra\] is a lax monoidal functor, i.e. a $\bfW_\bfL$-algebra.
Since coherence is identical to that in Proposition \[prop:G is W-alg\], it will suffice to show functoriality. Let $\Phi=(X,Y,\varphi)$ and $\Psi=(Y,Z,\psi)$ be wiring diagrams with composition $\Psi\circ\Phi=(X,Z,\omega)$. We now rewrite $\overline{\omega}$ using a matrix equation in terms of $\overline{\varphi}$ and $\overline{\psi}$ by recasting (\[dia:composition diagrams\]) in matrix form below.
$$\label{eqn:omegamatrix}
\begin{split}
\overline{\omega}=\begin{bmatrix} \overline{\omega^{X,X}} & \overline{\omega^{X,Z}} \\ \overline{\omega^{Z,X}} & \overline{\omega^{Z,Z}} \end{bmatrix} & = \begin{bmatrix} \overline{\varphi}^{X,Y} & 0 \\ 0 & I \end{bmatrix}\overline{\psi}\begin{bmatrix} \overline{\varphi}^{Y,X} & 0 \\ 0 & I \end{bmatrix}+\begin{bmatrix} \overline{\varphi}^{X,X} & 0 \\ 0 & 0 \end{bmatrix} \\
& =\begin{bmatrix} \overline{\varphi}^{X,Y}\overline{\psi}^{Y,Y}\overline{\varphi}^{Y,X}+\overline{\varphi}^{X,X} & \overline{\varphi}^{X,Y}\overline{\psi}^{Y,Z} \\ \overline{\psi}^{Z,Y}\overline{\varphi}^{Y,X} & 0 \end{bmatrix}
\end{split}$$
We now prove that $\mcL (\Psi\circ\Phi)=\mcL (\Psi)\circ\mcL (\Phi)$. We immediately have $\mcL (\Psi\circ\Phi)S=S=\mcL (\Psi)(\mcL (\Phi)S)$. Let $h:=\mcL (\Psi\circ\Phi)f$ and . We must show $h=k$. Let $g=\mcL (\Phi)f$ and $\Psi\circ\Phi=(X,Z,\omega)$. It is then straightforward matrix arithmetic to see that $$\label{eqn:matrix}
\begin{split}
k=\mcL (\Psi)g &=\begin{bmatrix} g^{S,Y} & 0 \\ 0 & I \end{bmatrix}\overline{\psi}\begin{bmatrix} g^{Y,S} & 0 \\ 0 & I \end{bmatrix} + \begin{bmatrix} g^{S,S} & 0 \\ 0 & 0 \end{bmatrix} \\
& =\begin{bmatrix}f^{S,X}(\overline{\varphi}^{X,Y}\overline{\psi}^{Y,Y}\overline{\varphi}^{Y,X}+\overline{\varphi}^{X,X})f^{X,S}+f^{S,S} & f^{S,X}\overline{\varphi}^{X,Y}\overline{\psi}^{Y,Z} \\
\overline{\psi}^{Z,Y}\overline{\varphi}^{Y,X}f^{X,S} & 0 \end{bmatrix} \\
&=\begin{bmatrix} f^{S,X} & 0 \\ 0 & I \end{bmatrix}\overline{\omega}\begin{bmatrix} f^{X,S} & 0 \\ 0 & I \end{bmatrix} + \begin{bmatrix} f^{S,S} & 0 \\ 0 & 0 \end{bmatrix} =\mcL (\Psi\circ\Phi)f=h
\end{split}$$ Therefore, the pair $(\mcL,\mu_\bfL)$ constitutes a lax monoidal functor $\bfW_{\bfL}\to\mathbf{Set}$, i.e., a $\bfW_{\bfL}$-algebra.
Although we’ve been referring to $\mcL$ as a subalgebra of $\mcG$, this is technically not the case since they have different source categories. The following diagram illustrates precisely the relationship between the $\bfW_{\bfL}$-algebra $\mcL$, defined above, and the $\bfW$-algebra $\mcG$, defined in Section \[sec:g\]. $$\label{eqn:final}
\xymatrix@C=16pt@R=30pt{
\bfW_{\bfL}\ar@{^{(}->}[rr]^{\bfW_i} \ar[dr]_{\mcL }&\ar@{}[d]|(.4){\overset{\textstyle\epsilon}{\Longrightarrow}}&\bfW\ar[dl]^{\mcG }\\
&\Set
}$$ Here, the natural inclusion $\bfW_i\taking\bfW_{\bfL}\operatorname{\hookrightarrow}\bfW$ corresponds to $i\taking\bfL\hookrightarrow\Man$, and we have a natural transformation $\epsilon:\mcL \to\mcG \circ i$. Hence for each , we have a function $\epsilon_X:\mcL (X)\to \mcG (i(X))=\mcG (X)$ that sends the linear open system $(S,f)\in\mcL (X)$ to the open system .
As promised, we now reformulate Example \[ex:main\] in terms of our language.
\[ex:as promised\] For the reader’s convenience, we reproduce Figure \[fig:pipebrine\] and Table \[tab:explicit\].
$$\begin{array}{c||c|c|c|c|c}
\rule[-4pt]{0pt}{16pt}
w\in{X^{\text{in}}}+{Y^{\text{out}}}&{X^{\text{in}}}_{1a}&{X^{\text{in}}}_{1b}&{X^{\text{in}}}_{2a}&{X^{\text{in}}}_{2b}&{Y^{\text{out}}}_{a}
\\\hline
\rule[-4pt]{0pt}{16pt}
\varphi(w)\in{X^{\text{out}}}+{Y^{\text{in}}}&{Y^{\text{in}}}_{b}&{X^{\text{out}}}_{2b}&{Y^{\text{in}}}_{a}&{X^{\text{out}}}_{1a}&{X^{\text{out}}}_{2a}
\end{array}$$
We can invoke the yoga of Definition \[def:rewrite2\] to write $\overline{\varphi}$ as a matrix below:
$$\label{eqn:phimatrix}
\begin{bmatrix} \;\overline{{X_{1a}^{\text{out}}}}\; \\ \overline{{X_{2a}^{\text{out}}}} \\ \overline{{X_{2b}^{\text{out}}}} \\ \overline{{Y_a^{\text{in}}}} \\ \overline{{Y_b^{\text{in}}}} \end{bmatrix} =
\begin{bmatrix}
0 &0 &I &0 &0 \\
0 &0 &0 &0 &I \\
0 &I &0 &0 &0 \\
I &0 &0 &0 &0 \\
0 &0 &0 &I &0
\end{bmatrix} \begin{bmatrix} \;\overline{{X_{1a}^{\text{in}}}}\; \\ \overline{{X_{1b}^{\text{in}}}} \\ \overline{{X_{2a}^{\text{in}}}} \\ \overline{{X_{2b}^{\text{in}}}} \\ \overline{{Y_a^{\text{out}}}} \end{bmatrix}$$
One can think of $\overline{\varphi}$ as a block permutation matrix consisting of identity and zero matrix blocks. An identity matrix in block entry $(i,j)$ represents the fact that the port whose state space corresponds to row $i$ and the one whose state space corresponds to column $j$ get linked by $\Phi$. In general, the dimension of each $I$ is equal to the dimension of the corresponding state space and hence the formula in (\[eqn:phimatrix\]) is true, independent of the typing. In the specific example of this system, however, all of these ports are typed in $\RR$, and so we have $I=1$ in (\[eqn:phimatrix\]).
As promised in Example \[ex:promise\], we now write the open systems for the $X_i$ in Figure \[fig:pipebrine\] as elements of $\mcL (X_i)$. The linear open systems below in (\[eqn:tanks\]) represent $f_1$ and $f_2$, respectively. $$\label{eqn:tanks}
\left[ \begin{array}{c} \dot{Q}_1 \\ {X_{1a}^{\text{out}}} \end{array} \right] = \begin{bmatrix} -.1 & 1 & 1 \\ .1 & 0 & 0 \end{bmatrix} \left[ \begin{array}{c} Q_1 \\ {X_{1a}^{\text{in}}} \\ {X_{1b}^{\text{in}}} \end{array} \right], \left[ \begin{array}{c} \dot{Q}_2 \\ {X_{2a}^{\text{out}}} \\ {X_{2b}^{\text{out}}} \end{array} \right] = \begin{bmatrix} -.2 & 1 & 1 \\ .125 & 0 & 0 \\ .075 & 0 & 0\end{bmatrix} \left[ \begin{array}{c} Q_2 \\ {X_{2a}^{\text{in}}} \\ {X_{2b}^{\text{in}}} \end{array} \right]$$
Note the proportion of zeros and ones in the $f$-matrices of (\[eqn:tanks\])—this is perhaps why the making explicit of these details was an afterthought in (\[eqn:naive\]). Because we may have arbitrary nonconstant coefficients, our formalism can capture more intricate systems.
We then use (\[eqn:phimatrix\]) to establish that ${X^{\text{in}}}_{1b}={X^{\text{out}}}_{2b}$ and ${X^{\text{in}}}_{2b}={X^{\text{out}}}_{1a}$. This allows us to recover the equations in (\[eqn:naive\]):
$$\left\{
\begin{array}{lr}
\dot{Q}_1=-.1Q_1+{X_{1a}^{\text{in}}}+{X_{1b}^{\text{in}}}=-.1Q_1+1.5+{X_{2b}^{\text{out}}}=-.1Q_1+.075Q_2+1.5 \\
\dot{Q}_2=-.2Q_2+{X_{2a}^{\text{in}}}+{X_{2b}^{\text{in}}}=-.2Q_2+3+{X_{1a}^{\text{out}}}=-.2Q_2+.1Q_1+3
\end{array}
\right.$$
The coherence map in Definition \[def:linear algebra\] gives us the combined tank system: $$(Q,f):=\mu_\bfL((\{Q_1\},f_1),(\{Q_2\},f_2))=(\{Q_1,Q_2\},f_1\times f_2)\in\mcL(X).$$ This system can then be written out as a matrix below $$\label{eqn:combinedsystem}\begin{bmatrix}\dot{Q_1} \\ \dot{Q_2} \\ {X_{1a}^{\text{out}}} \\ {X_{2a}^{\text{out}}} \\ {X_{2b}^{\text{out}}}\end{bmatrix}=\begin{bmatrix} -.1 & 0 & 1 & 1 & 0 & 0 \\ 0 & -.2 & 0 & 0 & 1 & 1 \\ .1 & 0 & 0 & 0 & 0 & 0\\ 0 & .125 & 0 & 0 & 0 & 0 \\ 0 & .075 & 0 & 0 & 0 & 0 \end{bmatrix}\begin{bmatrix}Q_1 \\ Q_2 \\ {X_{1a}^{\text{in}}} \\ {X_{1b}^{\text{in}}} \\ {X_{2a}^{\text{in}}} \\ {X_{2ba}^{\text{in}}}\end{bmatrix}$$ Finally, we can apply formula (\[eqn:glin\]) to (\[eqn:combinedsystem\]) above to express as a matrix the open system $(Q,g)=(\Phi)f\in\mcL(Y)$ for the outer box $Y$.
$$\left[ \begin{array}{c} \dot{Q}_1 \\ \dot{Q}_2 \\ {Y^{\text{out}}} \end{array} \right] = \begin{bmatrix} -.1 & .075 & 0 & 1 \\ .1 & -.2 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix} \left[ \begin{array}{c} Q_1 \\ Q_2 \\ {Y_a^{\text{in}}} \\ {Y_b^{\text{in}}} \end{array} \right]$$
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Lattice points in large regions and related arithmetic functions:
Recent developments in a very classic topic
**A. Ivić (Belgrade),**
**E. Krätzel, M. Kühleitner, and W.G. Nowak (Vienna)**
*Dedicated to Professor Wolfgang Schwarz on the occasion of his 70th birthday*
\#1 [0= 1= ]{}
=cmbx10 scaled1 =msbm10 \#1 =msbm10 scaled 700 \#1 =msbm10 \#1
The branch of analytic number theory which is concerned with the number of integer points in large domains, in the Euclidean plane as well as in space of dimensions $\ge3$, has a long and very prolific history which reaches back to classic works of E. Landau, J.G. Van der Corput, G. Voronoï, G.H. Hardy, to mention just a few. Enlightening accounts of the developments of this theory can be found in the monographs of F. Fricker [@fricker], E. Krätzel [@kr-lp],[@kr-anafu], and M. Huxley [@Hu1]. Although the major problems in this field are quite old (and yet unsolved), there have been a lot of amazing new developments in recent times. This survey article attempts to give an overview of the state-of-art in this field, with an emphasis on results established during the last few years. Furthermore, we are aiming to point out the intrinsic relationships between lattice point quantities and certain arithmetic functions which arose originally from number theory originally without any geometric motivation.
### 1. Lattice points in a large circle. The problem of Gauß. {#lattice-points-in-a-large-circle.-the-problem-of-gauß. .unnumbered}
Starting from the arithmetic function viewpoint, let
$$r(n) := \#\{(m,k)\in\Z^2:\ m^2+k^2 = n \ \}$$ denote the number of ways to represent the integer $n\ge0$ as a sum of two squares of integers. While the maximal size of $r(n)$ is estimated, e.g., in E. Krätzel [@kr], and an explicit formula may be found in E. Hlawka, J. Schoißengeier, and R. Taschner [@hst], p. 60, its average order is described by the Dirichlet summatory function $$A(x) = \sum_{0\le n\le x} r(n)\,,$$ where $x$ is a large real variable. In geometric terms, $A(x)$ is the number of lattice points $(m,k)\in\Z^2$ in a compact, origin-centered circular disc ${\bf C}(\sqrt{x})$ of radius $\sqrt{x}$. Obviously, $A(x)$ equals asymptotically the area of ${\bf C}(\sqrt{x})$, i.e., $\pi x$. Already C.F. Gauß (1777 - 1855) made the following more precise but still extremely simple observation: If we associate to each lattice point $(m,k)\in\Z^2$ the square $$Q(m,k) = \{ (u,v)\in\R^2:\ |u-m|\le{\textstyle{1\over2}},\
|v-k|\le{\textstyle{1\over2}}\ \}\,,$$ then obviously $${\bf C}\left(\sqrt{x}-{\sqrt{2}\over2}\right) \subset
\bigcup_{(m,k)\in{\bf C}(\sqrt{x})}Q(m,k)\subset{\bf
C}\left(\sqrt{x}+{\sqrt{2}\over2}\right)\,.$$ Evaluating the areas, we see that, for every $x>{1\over2}$, $$\pi\left(\sqrt{x}-{\sqrt{2}\over2}\right)^2 < A(x) <
\pi\left(\sqrt{x}+{\sqrt{2}\over2}\right)^2\,,$$ hence the [*lattice point discrepancy* ]{} of ${\bf C}(\sqrt{x})$ (traditionally often mis-translated as “lattice rest”) $$P(x):= A(x)-\pi x
\eqno{(1.1)}$$ satisfies $$|P(x)| < \pi\sqrt{2x}+{\pi\over2}
\eqno{(1.2)}$$ which is usually simply written as $P(x)=O(\sqrt{x})$ or $P(x)\ll\sqrt{x}$. Before reporting on stronger results, we shall try to shed some light on the mathematical background of all more sophisticated approaches to the problem. The naïve idea is to apply Poisson’s formula to evaluate the sum $$A(x) = \sum_{(m,k)\in\Zi^2}{\bf c}_x(m,k)\,,$$ where ${\bf c}_x$ denotes the indicator function of ${\bf
C}(\sqrt{x})$. To get rid of the discontinuity at the boundary of ${\bf C}(\sqrt{x})$, which would create subtle questions of convergence, it is convenient to consider the integrated form $$\int\limits_0^x A(t)\, {\rm d}t = \int\limits_0^x \left(\sum_{m^2+k^2\le t} 1\right) {\rm d}t = \sum_{m^2+k^2\le
x}(x-m^2-k^2) =$$ $$= \sum_{(m,k)\in\Zi^2}\ \dint{u^2+v^2\le x}
(x-u^2-v^2)e^{2\pi i(mu+kv)}\, {\rm d}(u,v)\,.$$ Since the double integral can be evaluated explicitly by means of the Bessel function $J_2$, this gives $$\int\limits_0^x A(t)\, {\rm d}t = {\pi\over2}\,x^2 + {x\over\pi}
\sum_{n=1}^\infty {r(n)\over n}\,J_2\left(2\pi\sqrt{nx}\right)\,.
\eqno{(1.3)}$$ With some care, one can verify that this identity may be differentiated term by term (cf., e.g., E. Krätzel [@kr-lp], p. 126, Th. 3.12), yielding (for $x$ not an integer) $$P(x) = \sqrt{x} \sum_{n=1}^\infty r(n)n^{-1/2}
J_1(2\pi\sqrt{nx})\,, \eqno{(1.4)}$$ which is called [*Hardy’s identity*]{}. Since this latter series is only conditionally convergent, it is convenient for applications to have a sharp truncated version at hand. A result of this kind can be found as Lemma 1 in A. Ivić [@I8]: For $x\ge1$, $x\notin\Z$, and $x\le M\le x^A$, $A>1$ some fixed constant, $$P(x) = \sqrt{x} \sum_{1\le n< M} r(n)n^{-1/2}
J_1(2\pi\sqrt{nx}) +\qquad\qquad\qquad\qquad\qquad\qquad\quad$$ $$+ O\left(\min\left(x^{5/4}M^{-1/2}+x^{1/2+\epsilon}M^{-1/2}\Vert
x\Vert^{-1} + x^{1/4}M^{-1/4},\ x^\epsilon\right)\right)\,,
\eqno{(1.5)}$$ where $\Vert\cdot\Vert$ denotes the distance from the nearest integer, and $\epsilon>0$ is arbitrary. Moreover, using the well-known Bessel function asymptotics $$J_1(2\pi\sqrt{nx}) = {1\over\pi}(nx)^{-1/4} \cos(2\pi\sqrt{nx}-{\textstyle{3\over4}}\pi) +
O\left((nx)^{-3/4}\right)\,,$$ one arrives at a representation $$P(x) = {1\over\pi}\,x^{1/4}\sum_{1\le n<M} {r(n)\over n^{3/4}}\,
\cos(2\pi\sqrt{nx}-{\textstyle{3\over4}}\pi) \ + \hbox{remainder
terms}\,.\eqno{(1.6)}$$ In a sense, all of the deep investigations on the order of $P(x)$ which have been done during the past century, are based on results like (1.6), though often somewhat shrouded by technicalities.
#### 1.1. O-estimates in the circle problem. {#o-estimates-in-the-circle-problem. .unnumbered}
For instance, to derive upper bounds for this lattice point discrepancy, one is led to consider exponential sums $$\sum_{m,k}(m^2+k^2)^{-3/4}\,e^{2\pi i\sqrt{x}\sqrt{m^2+k^2}}\,.$$ The twentieth century saw the development of increasingly complicated methods to deal with such sums. A historical survey of the results obtained can be found in E. Krätzel [@kr-lp]. They started with W. Sierpiński’s [@si] bound $$P(x) =
O\left(x^{1/3}\right)$$ and in the 1980’s culminated in the work of G.A. Kolesnik [@ko], who had $$P(x) = O\left(x^{139/429 +\epsilon}\right)$$ as his sharpest result. During the last one-and-a-half decades, M. Huxley, starting from the ideas due to E. Bombieri – H. Iwaniec [@BI] and ideas due to H. Iwaniec and C.J. Mozzochi [@IM], devised a substantially new approach which he called the [*Discrete Hardy-Littlewood method*]{}. His strongest result was published in 2003 [@Hu2] and reads $$P(x) = O\left(x^{131/416} (\log x)^{18637/8320}\right)\,, \eqno{(1.7)}$$ thereby improving upon his 1993 [@hu] bound $O(x^{23/73+\epsilon})$. Note that ${131\over416}=0.3149038\dots$, ${23\over73}=0.315068\dots$, while Kolesnik’s ${139\over429}=0.324009\dots$.
#### **1.2. Lower bounds for the lattice point discrepancy of a circle.** {#lower-bounds-for-the-lattice-point-discrepancy-of-a-circle. .unnumbered}
As (1.6) suggests, a stronger estimate than $P(x)=O(x^{1/4})$ cannot be true. In fact, G.H. Hardy [@ha], [@Ha1] proved that[^1] $$P(x)=\Omega_\pm(x^{1/4})\,,\qquad P(x)=\Omega_-(x^{1/4}(\log
x)^{1/4})\,, \eqno(1.8)$$ adding the unsubstantiated claim that $P(x)=\Omega_\pm(x^{1/4}(\log x)^{1/4})$. The second part of (1.8) was not improved until more than six decades later by J.L. Hafner [@Hf1], who obtained[^2] $$P(x)=\Omega_-\left(x^{1/4}(\log x)^{1/4} (\log_2x)^{(\log2)/4}
\exp(-c(\log_3x)^{1/2})\right)\,,\qquad c>0\,. \eqno{(1.9)}$$ All estimates like (1.8) and (1.9) start from (a variant of) formula (1.6) and are based on the idea of finding an unbounded sequence of $x$-values for which the terms $\cos(2\pi\sqrt{nx}-{\textstyle{3\over4}}\pi)$ “essentially all pull into the same direction”. That is, the numbers $\sqrt{nx}$ should be close to an integer, which is achieved by Dirichlet’s theorem on Diophantine approximation (see e.g., Lemma 9.1 of [@I13]). In Hafner’s work, this task is done only for those $n$ for which $r(n)$ is comparatively large; this idea slightly improves the estimate. As for $\Omega_+$-results, K. Corrádi and I. Kátai [@CK] proved that $$P(x) = \Omega_+\left(x^{1/4}\exp\left(c'(\log_2x)^{1/4}
(\log_3x)^{-3/4}\right)\right)\qquad (c'>0)\,, \eqno{(1.10)}$$ thereby refining previous work of A.E. Ingham [@ing] and of K.S. Gangadharan [@Ga]. (Note that the factor of $x^{1/4}$ here is less than any power of $\log x$. This inherent asymmetry arises from the fact that the deduction of (1.10) is based on sort of a quantitative version of Kronecker’s approximation theorem which is much weaker than Dirichlet’s.) The strongest $\Omega$-bound for $P(x)$ available to date was established in 2003 by K. Soundararajan [@So]. His ingenious new approach allows us to restrict the application of Dirichlet’s theorem to a still smaller set of integers $n$. His method can be summarized in the following fairly general statement. [**Soundararajan’s Lemma [[@So]]{}.**]{} When applied to (1.6), this lemma yields $$P(x) = \Omega\left(x^{1/4}(\log x)^{1/4}(\log_2 x)^{(3/4)(2^{1/3}-1)}
(\log_3 x)^{-5/8}\right)\,. \eqno{(1.11)}$$ To assess the refinement in the exponent of the $\log_2 x$-factor, note that ${1\over4}\log2=0.1732\dots$, while ${3\over4}(2^{1/3}-1)=0.1949\dots$. We remark that Soundararajan’s method does not allow us to replace the $\Omega$-symbol in this assertion by $\Omega_+$ or $\Omega_-$.
#### 1.3. The mean square in the circle problem. {#the-mean-square-in-the-circle-problem. .unnumbered}
The results reported so far already suggest that $$\inf\{\lambda:\ P(x) \ll_\lambda\ x^\lambda\ \} = {\textstyle{1\over4}}\,.
\eqno{(1.12)}$$ To prove (or disprove) this conjecture is usually called the Gaussian circle problem in the strict sense. In favor of this hypothesis, there are quite precise mean-square asymptotic formulas of the shape $$\Int_0^X (P(x))^2\, {\rm d}x = C\,X^{3/2} + Q(X)\,, \eqno{(1.13)}$$ with $$C = {1\over3\pi^2}\sum_{n=1}^\infty r^2(n) n^{-3/2} =
{16\over3\pi^2}\,
{\zeta_{\Qi(i)}^2({3\over2})\over\zeta(3)}(1+2^{-3/2})^{-1}
\approx 1.69396\,. \eqno{(1.14)}$$ The estimation of this new remainder term $Q(X)$ has been subject of intensive research, using more and more ingenuity. We mention the results of H. Cramér [@cr]: $Q(X)=O( X^{5/4+\epsilon})$, E. Landau [@la3]: $Q(X)=O( X^{1+\epsilon})$, A. Walfisz [@wa1]: $Q(X)=O( X(\log X)^3)$, and I. Kátai [@ka]: $
Q(X)=O( X(\log X)^2)$. Later on, E. Preissmann [@Pr] found a short and elegant proof for this last result, using a deep inequality of H.L. Montgomery and R.C. Vaughan [@mv].Recently W.G. Nowak [@no4] succeeded in improving Kátai’s estimate to $$Q(X) = O\left(X\,(\log X )^{3/2}\,\log_2X\right)\,.
\eqno{(1.15)}$$ We shall sketch the idea of this refinement. On any interval $[{1\over2} X,X]$, one can split up $P(x)$ as $$P(x)=H(X,x)+R(X,x)\,,$$ where $$\Int_{X/2}^X R(X,x)^2\, {\rm d}x \ll X^{1/2}\,,$$ and $$\Int_{X/2}^X H(X,x)^2\, {\rm d}x = C\,\left(X^{3/2}-({\textstyle{1\over2}} X)^{3/2}\right)
+ O\left(X\sum_{1\le n<X^5} {r(n)^2\over n\,\Delta_{{\bf
B}}(n)}\right)\,, \eqno{(1.16)}$$ with $$\Delta_{{\bf B}}(n) := \min_{k\in{{\bf B}},\ k\ne n} |k-n|\,,
\qquad {{\bf B}} := \{n\in\Z:\ r(n)>0\ \}\,.$$ This is deduced by standard techniques, applying (1.6) and a Hilbert-type inequality of H.L. Montgomery and R.C. Vaughan [@mv].
Using the trivial bound $\Delta_{\bf B}(n)\ge1$ in (1.16), along with the well-known fact that $\sum_{n\le y}r(n)^2\ll y \log y$, one readily obtains the result of I. Kátai [@ka], in a way that mimicks E. Preissmann’s [@Pr] argument. The basic idea of the improvement in (1.15) is the observation that the set ${\bf
B}$ is actually rather sparse, i.e., $\Delta_{\bf B}(n)$ should be “on average” somewhat larger. It was known to E. Landau [@la1] that $$\sum_{1\le n\le x,\ n\in{\bf B}} 1 \ \sim {c\,x\over\sqrt{\log x}}\qquad (x\to\infty)\,,$$ for some $c>0$. Roughly speaking, this suggests that the elements of ${\bf B}$ behave like some sequence $[c'\,n \sqrt{\log n}]$. For the latter, the distance between subsequent terms asymptotically equals $c'\,\sqrt{\log n}$. If one could use this bound for $\Delta_{\bf B}(n)$ in (1.16), the estimate (1.15) would be immediate. However, the following result, which can be proved, and turns out to suffice for this purpose: For every integer $h\ne0$, $$\sum_{1\le n\le x,\ n+h\in{\bf B}} r(n)^2 \ll
\left({|h|\over\phi(|h|)}\right)^2\,x(\log x)^{1/2}\,, \eqno{(1.17)}$$ where $\phi$ denotes Euler’s function. This inequality is derived by an elementary convolution argument from the related bound $$\sum_{1\le n\le x\atop n\in{\bf B},\ kn+h\in{\bf B}} 1 \
\ll \left({|h|k\over\phi(|h|k)}\right)^2\,{x\over\log x} \,,$$ which is true for arbitrary integers $k>0$ and $h\ne0$. The latter estimate follows by Selberg’s sieve method, and was established (for $k=1$) by G.J. Rieger [@ri].We conclude this subsection by the remark that, during the last 15 years, higher power moments of $P(x)$ have been investigated as well. We mention the papers of K.M. Tsang [@Ts1], D.R. Heath-Brown [@HB3], A. Ivić and P. Sargos [@IS], and W. Zhai [@Zh] which provide results of the shape $$\Int_0^X (P(x))^m \, {\rm d}x \ \sim \ (-1)^m\,C_m\,X^{1+m/4}\qquad (X\to\infty)\,, \eqno{(1.18)}$$ where $3\le m\le9$ and $C_m>0$ are explicitly known constants. This not only supports the conjecture (1.12), but is of particular interest in the cases of odd $m$: It means that there is some excess of the values $x$ for which $P(x)<0$ over those with $P(x)>0$. This observation matches well with the different accuracy of the $\Omega_+$- and $\Omega_-$-estimates (1.9) and (1.10).
### 2. Lattice points in spheres. {#lattice-points-in-spheres. .unnumbered}
As most experts agree, in our familiar three-dimensional Euclidean space the analogue of the Gaussian problem is the most difficult and enigmatic. For integers $n\ge0$, let $$r_3(n) := \#\{(m,k,\ell)\in\Z^3:\ m^2+k^2+\ell^2 = n\ \}\,.$$ Then the formula $$\sum_{0\le n\le x} r_3(n) = {4\pi\over3}\,x^{3/2} + P_3(x) \eqno{(2.1)}$$ defines the lattice point number, as well as the lattice point discrepancy $P_3(x)$, of a compact, origin-centered ball of radius $\sqrt{x}$. By the same argument which furnished (1.2), it is trivial to see that $P_3(x)\ll x$. However, it was known already to E. Landau [@la2] that $$P_3(x) = O\left(x^{3/4}\right)\,. \eqno{(2.2)}$$ For many decades, the subsequent history of improvements of this $O$-bound was essentially the personal property of I.M. Vinogradov, whose series of papers on the topic in 1963 culminated in the result [@vi] $$P_3(x) = O\left(x^{2/3}(\log x)^6\right)\,.\eqno{(2.3)}$$ Also in this case, the last few years brought significant further progress in this problem, namely the estimates of F. Chamizo and H. Iwaniec [@ci] $$P_3(x) =
O\left(x^{29/44+\epsilon}\right)\eqno{(2.4)}$$ and of D.R. Heath-Brown [@hb] $$P_3(x) = O\left(x^{21/32+\epsilon}\right)\,.\eqno{(2.5)}$$ Note that ${29\over44}=0.65909\dots$, ${21\over32}=0.65625$. These bounds are based on the finer algebro-arithmetic theory of $r_3(n)$, notably on the explicit formula, an account on which can be found, e.g., in P.T. Bateman [@ba]. In contrast to this quite long history of upper bounds, the sharpest known $\Omega$-result was obtained by G. Szegö as early as 1926 [@sz] and reads $$P_3(x) = \Omega_-\left(x^{1/2}(\log x)^{1/2}\right) \,.\eqno{(2.6)}$$ Neither Hafner’s nor Soundararajan’s ideas were able to improve upon this bound, probably because the function $r_3(n)$ is distributed quite evenly and is not supported on a small subset of the integers. However, the corresponding $\Omega_+$-estimates have a longer history of their own: After K. Chandrasekharan’s and R. Narasimhan’s result [@cn] that $$\limsup_{x\to\infty} {P_3(x)\over x^{1/2}} = + \infty\,,$$ W.G. Nowak [@no1] proved that $$P_3(x) =
\Omega_+\left(x^{1/2}(\log_2x)^{1/2}(\log_3x)^{-1/2}\right)\,.
\eqno{(2.7)}$$ Later on, S.D. Adhikari and Y.-F.S. Pétermann [@adh] obtained the improvement $$P_3(x) = \Omega_+\left(x^{1/2}\log_2x\right)\,.\eqno{(2.8)}$$ Finally, and again quite recently, K.M. Tsang [@ts] complemented Szegö’s bound (2.6), showing that in fact $$P_3(x) = \Omega_\pm\left(x^{1/2}(\log x)^{1/2}\right) \,.\eqno{(2.9)}$$
As for the mean-square of the lattice point discrepancy $P_3(x)$, its asymptotic behavior has been investigated by V. Jarnik as early as in 1940 [@ja]. He showed that, with a certain constant $c_3>0$, $$\Int_0^X (P_3(x))^2\, {\rm d}x = c_3\, X^2 \log X + O\left(X^2
(\log X)^{1/2}\right)\,. \eqno{(2.10)}$$ The proof of (2.10) is much more difficult than in the planar case. It uses the theory of theta-functions, complex integration and the classic Hardy-Littlewood method. As far as the authors were able to ascertain, an improvement of the error term in (2.10) has never been attained. The interesting point of (2.10) is that it contains the omega estimates (2.6), resp., (2.9), in full accuracy, apart from the ambiguity of sign. That is, our state of knowledge is quantitatively quite different than with the circle problem: While (1.13) together with (1.9) - (1.11) tell us (in the planar case) that $P(x)\ll x^{1/4}$ [*in mean-square, with unbounded sequences of “exceptional” $x$-values for which $P(x)$, resp., $-P(x)$, is definitely much larger,* ]{} a phenomenon of this kind is not known for the three-dimensional sphere.
For spheres of dimensions $s\ge4$, the situation is much easier and better understood, leaving no room for progress. With the elementary identity $$r_4(n) := \#\{(k_1,k_2,k_3,k_4)\in\Z^4:\ k_1^2+k_2^2+k_3^2+k_4^2=n\ \}
\ =\ 8\,\sigma(n)-32\,\sigma\left({n\over4}\right)\,,$$ $$\sigma(n) :=\sum_{m\,|\,n,\ m>0} m$$ at one’s disposal, it is not difficult to show that (see E. Krätzel [@kr-anafu], p. 227-231) $$P_4(x):=\sum_{0\le n\le x}r_4(n) - {\pi^2\over2}\,x^2 =
O\left(x\,\log x\right)\,,\quad
P_4(x)=\Omega\left(x\,\log_2x\right)\,. \eqno{(2.11)}$$ These rather simple estimates have been improved slightly to $$P_4(x)= O\left(x\,(\log x)^{2/3}\right)\,,\qquad
P_4(x)=\Omega_\pm\left(x\,\log_2x\right)\,. \eqno{(2.12)}$$ The $O$-result in (2.12) is due to A. Walfisz [@arnold2], [@wa3] who used some extremely deep and involved analysis, while the $\Omega$-bounds are due to S.D. Adhikari and Y.-F.S. Pétermann [@adh]. For dimensions $s\ge5$, the exact order of the lattice point discrepancy $P_s(x)$ of an origin-centered $s$-dimensional ball of radius $\sqrt x$, has been known for a long time, namely $$P_s(x)= O\left(x^{s/2-1}\right)\,,\qquad
P_s(x)=\Omega\left(x^{s/2-1}\right)\,. \eqno{(2.13)}$$ See again E. Krätzel [@kr-anafu], pp 227 ff. For the finer theory of these higher dimensional lattice point discrepancies, the reader is referred to the monograph of A. Walfisz [@wa2].
amssym.def amssym [**u**]{}
### 3. Lattice points in convex bodies. {#lattice-points-in-convex-bodies. .unnumbered}
By a $p$-dimensional convex domain $CD_p$ we mean a compact convex set in ${\Bbb{R}}^p$ with the origin as an interior point. Its distance function $F$ is a homogeneous function of degree $1$, such that $$CD_p = \{ \vec{t} \in {\Bbb{R}^p}: F(\vec{t};CD_p) \le 1 \}.$$ The number of lattice points $\vec{n} \in {\Bbb{Z}}^p $ [ in the dilated convex domain]{} $xCD_p$, where $x$ is a large positive parameter, is denoted by[^3] $$A(x;CD_p) = \# \{ \vec{n} \in {\Bbb{Z}}^p: F(\vec{n};CD_p) \le x \}.$$ Let $area(CD_2)$ denote the area of $CD_2$ and $vol(CD_p)$ the volume of $CD_p \; (p > 2)$. The number of lattice points in $xCD_p$ can be written as $$\begin{aligned}
A(x;CD_2) & = & area(CD_2)x^2 + P(x;CD_2), \\
A(x;CD_p) & = & vol(CD_p)x^p + P(x;CD_p).\end{aligned}$$ Here $area(CD_2)x^2$ and $vol(CD_p)x^p$ denote the main terms and $P(x;CD_p), p \ge 2,$ the remainders or error terms.
#### 3.1 Lattice points in plane convex domains. {#lattice-points-in-plane-convex-domains. .unnumbered}
An elementary result due to [M. V. Jarnik]{} [ and]{} [ H. Steinhaus]{} [@stein], 1947: Let $J$ be a closed, rectifiable [ Jordan]{} curve with area $F$ and length $l
\ge 1$, and let $G$ be the number of lattice points inside and on the curve. Then $|G - F| < l$.
Let $l$ denote the length of the curve of boundary of the convex domain $CD_2$. Then $$|P(x;CD_2)| < lx.$$ The result $P(x;CD_2) \ll x$ is the best of its kind under the above conditions. An example is the square with length of side $2$ and $(0,0)$ as centre, dilated by $x$.
Suppose that the boundary of the plane convex domain $CD_2$ is sufficiently smooth with finite nonzero curvature throughout. We say that the boundary curve is of class $C^k \: (k = 2,3, \ldots )$ if the radius of curvature is $k - 2$ times continuously differentiable with respect to the direction of the tangent vector. Let $r$ be the absolute value of the radius of curvature in a point of the boundary and $r_{\max}, r_{\min}$ the maximum, minimum of $r$, respectively. Then we shall assume that $$0 < r_{\min} \le r_{\max} < \infty .$$
[@corput], 1920: Let the curve of boundary of the plane convex domain $CD_2$ be of class $C^2$ , then $$P(x;CD_2) \ll x^{\frac{2}{3}}.$$ [A more precise formulation due to]{} [ H. Chaix]{} [@chaix], 1972: $$|P(x;CD_2)| \le C(1 + r_{\max}x)^{\frac{2}{3}}, \quad C > 0\
\hbox{\rm an absolute constant}.$$ [A formulation with explicit constants is due to]{} [ E. Krätzel]{} [@kratz5], 2004: $$|P(x;CD_2)| < 48(r_{\max}x)^{\frac{2}{3}} + \left ( 703
\sqrt{r_{\max}} + \frac{3r_{\max}}{5 \sqrt{r_{\min}}} \right )
\sqrt{x} + 11 \quad \mbox{for} \quad x \ge 1.$$ In particular, if $CD_2 = E_2$ is an ellipse, defined by the quadratic form $$Q(t_1,t_2;E_2) = at_1^2 + 2bt_1t_2 + ct_2^2, \quad ac - b^2 = 1,$$ then one has $$|P(x;E_2)| < 38 x^{\frac{2}{3}} + \left ( 3 a^{\frac{3}{4}} +
700 c^{\frac{3}{4}} \right ) \sqrt{x} + 11.$$ [E. Krätzel]{} and [ W. G. Nowak]{} [@jewie], 2004, showed that the factor $38$ is the leading term of the estimate may be reduced to $\frac{17}{2}$. In particular, for the circle $E_2 = C \; (a =
c = 1, \: b = 0)$, $$\begin{aligned}
|P(x;C)| <
\left\{\begin{array}{ll}
\frac{17}{2}x^{\frac{2}{3}} + \frac{3}{2} \sqrt{x}
+ 3 & \mbox{for} \quad x \: \ge 28, \\
9 x^{\frac{2}{3}} &
\mbox{for} \quad x \ge 1000.
\end{array}\right.\end{aligned}$$
[@jarnik1], 1925, proved that the estimate of [ J. G. van der Corput]{} is the best of its kind. In fact, there exist plane convex domains $CD_2$ with $P(x;CD_2) = \Omega
(x^{2/3})$.
[J. G. van der Corput]{} [@vandecorp], 1923, showed that his estimate can be sharpened for a large class of plane convex domains $CD_2$. Under the additional assumption that the boundary curve is of class $C^{\infty}$, he obtained $$P(x;CD_2) \ll x^{\vartheta}$$ holds with some $\vartheta < \frac{2}{3}$. [O. Trifonov]{} [@trif], 1988, showed that the estimate holds with $\vartheta
= \frac{27}{41} + \varepsilon, \varepsilon > 0$.
[@Hu1], 1993: Let the curve of boundary of the plane convex domain $CD_2$ be of class $C^3$. Then $$P(x;CD_2) \ll x^{\frac{46}{73}} (\log x)^{\frac{315}{146}},
\qquad \frac{46}{73} = 0,6301\ldots .$$ [M. N. Huxley]{} [@Hu2], 2003, improved this result and obtained $$P(x;CD_2) \ll x^{\frac{131}{208}} (\log x)^{\frac{18637}{8320}},
\qquad \frac{131}{208} = 0,6298\ldots .$$ [G. Kuba]{} [@kubaell], 1994, proved a two-dimensional asymptotic result for a special ellipse $E_2$. Let $$A(x,y;E_2) = \# \left \{ (m,n) \in {\Bbb{Z}}^2:
\frac{m^2}{x^2} + \frac{n^2}{y^2} \le 1 \right \} .$$ Then it is shown that, for $xy \longrightarrow \infty$, $$A(x,y;E_2) = \pi xy + P(x,y;E_2)$$ with
$$P(x,y;E_2) \ll
\left\{\begin{array}{ll}
(xy)^{\frac{23}{73}}(\log
xy)^{\frac{315}{146}} & \mbox{for } x\ll y^{\frac{119}{100}}
(\log xy)^{\frac{315}{100}}, \\
\frac{x}{\sqrt{y}} & \mbox{for } x\gg y^{\frac{119}{100}}(\log
xy)^{\frac{315}{100}}.
\end{array}\right.$$ In the last case one also has $$P(x,y;E_2) = \Omega \left ( \frac{x}{\sqrt{y}} \right )\, .$$
[**Lower bounds.**]{} Suppose that the boundary curve of the plane convex domain $CD_2$ is of class $C^2$. Then the following lower bounds have been proved:
The Theorem of [ V. Jarnik]{} [@jarnik2], 1924: $$P(x;CD_2) = \Omega \left ( x^{\frac{1}{2}} \right ).$$ [S. Krupička]{} [@krup], 1957: $$P(x;CD_2) = \Omega_{\pm} \left ( x^{\frac{1}{2}} \right ).$$ [W. G. Nowak]{} [@now1], 1985: $$P(x;CD_2) = \Omega_{-} \left ( x^{\frac{1}{2}}
(\log x)^{\frac{1}{4}}\right ).$$ In this connection the [estimates of the average order]{} of the remainder are interesting: [@krup], 1957: $$\int\limits_0^x |P(\sqrt{t};CD_2)|\, {\rm d}t \gg x^{\frac{5}{4}}.$$ [W. G. Nowak]{} [@now1], 1985: $$\int\limits_0^x |P(\sqrt{t};CD_2)|\, {\rm d}t \ll x^{\frac{5}{4}},
\qquad \int\limits_0^x |P^2(\sqrt{t};CD_2)|\, {\rm d}t \ll
x^{\frac{3}{2}}.$$ [P. Bleher]{} [@bleher], 1992: $$\int\limits_0^x P^2(t;CD_2) \, {\rm d}t \sim Ax^2.$$ $A$ is a positive constant depending on $CD_2$.
W. G. Nowak [@no3], 2002: $$\int\limits_{x-\Lambda}^{x+\Lambda} P^2(t;CD_2) \, {\rm d}t \sim
4A\Lambda x\, ,$$ for any $\Lambda=\Lambda(x)$ satisfying $$\lim_{x\to\infty} {\log x\over \Lambda(x)}=0.$$
Let $k \in \Bbb{N}, \: k \ge 3$. A [ Lamé]{} curve $L_k$ is defined by $$|t_1|^k + |t_2|^k = 1.$$ The curvature of this curve vanishes in the points $(0,\pm 1), \:
(\pm 1,0)$. The number of lattice points inside and on the dilated [ Lamé]{} curves $xL_k$ is defined by $$A(x;L_k) = \# \{ (m,n) \in {\Bbb{Z}}^2: |m|^k + |n|^k \le x^k \} .$$ The points with curvature zero give an important contribution to the estimate of the number of lattice points.
The number of lattice points is represented by
$$A(x;L_k) = \frac{2}{k} \frac{\Gamma^2(\frac{1}{k})}
{\Gamma (\frac{2}{k})} \, x^2 + \Delta (x;L_k).$$ Trivially $\Delta (x;L_k) \ll x$. The first progress concerning the upper bound was made by [ D. Cauer]{} [@cauer], 1914, who proved that $$\Delta (x;L_k) \ll x^{1 - \frac{1}{2k-1}},$$ which was also obtained by [ J. G. van der Corput]{} [@holland], 1919. The resolution of the problem of the size of $$\Delta (x;L_k) = O, \; \Omega \left ( x^{1 - \frac{1}{k}} \right )$$ was given by [B. Randol]{} [@burton], 1966, for even $k > 2$ and by [E. Krätzel]{} [@kr-kreis], 1967, for odd $k > 3$ and by the same author [@kr-gitter], 1969, for $k = 3$.
was introduced by [E. Krätzel]{} [@kr-kreis], 1967, [@kr-gitter], 1969 (see also [@kr-lp]): $$A(x;L_k) = \frac{2}{k} \frac{\Gamma^2(\frac{1}{k})}
{\Gamma (\frac{2}{k})} \: x^2 + 4 \psi_{2/k}^{(k)}(x) + P(x;L_k).$$ The function $\psi_{\nu}^{(k)}(x)$ is defined for $k \ge 1$, and for $\nu > \frac{1}{k}$ it is represented by the absolutely convergent infinite series $$\psi_{\nu}^{(k)}(x) = 2 \sqrt{\pi} \, \Gamma \left ( \nu + 1 -
\frac{1}{k} \right ) \sum_{n=1}^{\infty} \left ( \frac{x}{\pi n}
\right )^{\frac{k\nu}{2}} J_{\nu}^{(k)}(2\pi nx),\eqno{(3.1)}$$ where $J_{\nu}^{(k)}(2\pi nx)$ denotes the generalized [Bessel]{} function (see E. Krätzel [@kr-lp], p. 145). It is known that $$4 \psi_{2/k}^{(k)}(x) = O, \; \Omega \left ( x^{1 - \frac{1}{k}}
\right ).$$ Accordingly, this term in the asymptotic representation of $A(x;L_k)$ is called the [second main term]{}. The remainder $P$ is estimated by $$P(x;L_k) = o \left ( x^{\frac{2}{3}} \right ) \qquad(x\to\infty)$$ for $k \ge 3$.
[@kr-expo], 1981 and independently [W. G. Nowak]{} [@now2], 1982: $$P(x;L_k) \ll x^{\frac{27}{41}}, \qquad \frac{27}{41} = 0,6585 \ldots .$$ [W. Müller]{} and [W. G. Nowak]{} [@wowe], 1988: $$P(x;L_k) \ll x^{\frac{25}{38}} (\log x)^{\frac{14}{95}}, \qquad
\frac{25}{38} = 0,6578 \ldots .$$ [G. Kuba]{} [@kuba1], 1993: $$P(x;L_k) \ll x^{\frac{46}{73}}(\log x)^{315\over146}, \qquad \frac{46}{73} = 0,6301 \ldots .$$
[@schnabel], 1982: $$P(x;L_k) = \Omega \left ( x^{\frac{1}{3}} \right ).$$ [E. Krätzel]{} [@kr-lp], 1988: $$P(x;L_k) = \Omega_{\pm} \left ( x^{\frac{1}{2}} \right ).$$ [W. G. Nowak]{} [@now3], 1997: $$P(x;L_k) = \Omega_- \left ( x^{\frac{1}{2}} ( \log x)^{\frac{1}{4}}
\right ).$$ [M. Kühleitner, W. G. Nowak, J. Schoissengeier]{} and [T. Wooley]{} [@kuehleinow1], 1998: $$P(x;L_3) = \Omega_+ \left ( x^{\frac{1}{2}}
( \log \log x)^{\frac{1}{4}} \right ).$$ [The average order:]{} [W. G. Nowak]{} [@now4],1996: $$\frac{1}{x} \int\limits_0^x P^2(t;L_k) \, {\rm d}t \ll x.$$ [M. Kühleitner]{} [@kuehlei1], 2000: $$\frac{1}{x} \int\limits_0^x P^2(t;L_k) \, {\rm d}t = C_kx + (x^{1
- \varepsilon})$$ with explicitly given $C_k > 0, \, \varepsilon = \varepsilon (k)
> 0$.
[@now5], 2000: $$\int\limits_{x - 1/2}^{x + 1/2} P^2(t;L_k) \, {\rm d}t \ll x(\log
x)^2.$$ [M. Kühleitner]{} and [W. G. Nowak]{} [@kuehleinow2], 2001: Let $\Lambda (x)$ be an increasing function with $$\Lambda (x) \le \frac{x}{2} \quad \mbox{and} \quad
\lim_{x\to\infty} \frac{\log x}{\Lambda (x)} = 0.$$ Then $$\int\limits_{x - \Lambda}^{x + \Lambda} P^2(t;L_k) \, {\rm d}t
\sim 4C_k\Lambda x$$ with the same constant $C_k$ as above.
Suppose that the boundary curve of the plane convex domain $CD_2$ contains a finite number of points with curvature zero. [Y. Colin de Verdière]{} [@colin], 1977, showed that $$P(x;CD_2) \ll x^{1 - \frac{1}{k}}$$ holds if $k - 2 \ge 1$ is the maximal order of a zero of the curvature on the boundary of $CD_2$. Moreover, this estimate cannot be improved if the slope of the boundary curve is rational in at least one point, where the curvature vanishes to the order $k - 2$. Further, [B. Randol]{} [@randol], 1969, [Y. Colin de Verdiere]{} [@colin], 1977, and [M. Tarnopolska-Weiss]{}, [@weiss], 1978, showed: If $CD_2$ is rotated about the origin by an angle $\varphi$ then for this new domain $CD_2'$ the estimate $$P(x;CD_2') \ll x^{\frac{2}{3}}$$ holds for almost all $\varphi \in [0,2\pi ]$.
[W. G. Nowak]{} [@now6], 1984, obtained a refinement in the rational case. Assuming that the boundary curve has rational slope in an isolated point with curvature zero, then the contribution of this point to the discrepancy in the asymptotic representation of the number of lattice points can be given explicitly by a [Fourier]{} series which is absolutely convergent. [ E. Krätzel]{} [@kr-anafu], 2000, gives an integral representation for this contribution. It has the precise order $x^{1 - 1/k}$ if $k - 2$ is the order of vanishing of the curvature in the boundary point. In other words: Each isolated boundary point with curvature zero and rational slope produces a new main term in the asymptotic representation of the number of lattice points.
Further, [W. G. Nowak]{} [@now7], 1985, obtained a refinement also in the irrational case also. Assume that the slope in a boundary point of vanishing curvature is irrational. Under certain assumptions on the approximability of this irrational number by rationals, this point gives a contribution to the error term of order $x^{\vartheta}$ with $\vartheta < \frac{2}{3}$. Improvements of this result were given by [W. Müller]{} and [W. G. Nowak]{}, [@wolfgeorg], 1985. Moreover, [M. Peter]{}, [@peter1], 2000, proved that there is an error term with $\vartheta < \frac{2}{3}$ if and only if the slope is of finite type at the point with curvature zero.
#### 3.2 Lattice points in convex bodies of higher dimensions. {#lattice-points-in-convex-bodies-of-higher-dimensions. .unnumbered}
An elementary result is the Theorem of [J. M. Wills]{}, [@konmenge], 1973: Let $K$ be a $p$-dimensional, strictly convex body $(p \ge 3)$ and $r$ the radius of its greatest sphere in the interior. Let $G$ be the number of lattice points inside and on the body. Then $$vol(K) \left ( 1 - \frac{\sqrt{p}}{2r} \right )^p \le G \le
vol(K) \left ( 1 + \frac{\sqrt{p}}{2r} \right )^p,$$ provided that $r > \sqrt{p}\, /2$.
$$A(x;CD_p) = vol(CD_p)x^p + O(x^{p - 1}).$$
Suppose that the boundary of the convex domain $CD_p \: (p \ge 3)$ is a smooth $(p - 1)$-dimensional surface with finite nonzero [Gauss]{}ian curvature throughout. Assume that the canonical map, which sends every point of the unit sphere in ${\Bbb{R}}^p$ to that point of the boundary of the convex domain where the outward normal has the same direction, is one-one and real-analytic. The first non-trivial estimates for the remainder $P(x;CD_p)$ were given by [E. Landau]{}, 1912 and 1924, in case of an ellipsoid and by [E. Hlawka]{}, 1950, for a general convex body. We begin with the general case, and we shall give a short survey of the history of the ellipsoid problem.
, [@hla-gitter], [@hla-omega], 1950:
$$\begin{aligned}
P(x;CD_p)=
\left\{\begin{array}{ll}
O(x^{p - 2 + \frac{2}{p + 1}}), \\
\Omega( x^{\frac{p - 1}{2}}).
\end{array}\right.\end{aligned}$$
and [W. G. Nowak]{}, [@krae-now], 1991: $$\begin{aligned}
P(x;CD_p) \ll
\left\{\begin{array}{ll}
x^{p - 2 + \frac{12}{7p + 4}} \quad \mbox{for}
\quad 3 \le p \le 7, \\
x^{p - 2 + \frac{5}{3p + 1}} \quad \mbox{for}
\quad 8 \le p.
\end{array}\right.\end{aligned}$$ [E. Krätzel]{} and [W. G. Nowak]{} [@ekk-geo], 1992: $$\begin{aligned}
P(x;CD_p) \ll
\left\{\begin{array}{ll}
x^{p - 2 + \frac{8}{5p + 2}}( \log x
)^{\frac{10}{5p + 2}} \quad \mbox{for} \quad 3 \le p \le 6, \\
x^{p - 2 + \frac{3}{2p}}( \log x)^{\frac{2}{p}} \qquad
\quad \mbox{for} \quad 7 \le p.
\end{array}\right.\end{aligned}$$ [W. Müller]{} [@mueller], 2000: $$\begin{aligned}
P(x;CD_p) \ll
\left\{\begin{array}{ll}
x^{1 + \frac{20}{43} + \varepsilon} \qquad \qquad
\mbox{for} \quad p = 3, \\
x^{2 + \frac{6}{17} + \varepsilon} \qquad \qquad
\mbox{for} \quad p = 4, \\
x^{p - 2 + \frac{p + 4}{p^2 + p + 2} + \varepsilon} \quad \;
\mbox{for} \quad 5 \le p
\end{array}\right.\end{aligned}$$ for every $\varepsilon > 0$.
[@lattice], [@rest], [@body] showed over the years 1985 - 1991 $$\begin{aligned}
P(x;CD_p) =
\left\{\begin{array}{ll}
\Omega_- \left ( x(\log x)^{\frac{1}{3}} \right )
\qquad \qquad \mbox{for} \quad p = 3, \\
\Omega_{\pm} \left ( x^{\frac{p - 1}{2}}(\log x
)^{\frac{1}{2} - \frac{1}{2p}} \right ) \quad \mbox{for} \quad
p \ge 4.
\end{array}\right.\end{aligned}$$
[@womue], 1997: If $p \ge 4$, then $$\int\limits_0^x |P(t;CD_p)|^2\, {\rm d}t \ll x^{2p - 3 +
\varepsilon}$$ for every $\varepsilon > 0$.
Let $F(t,z)$ be a distance function depending on two variables $t, z$. Then $$R_p = \left \{ (t_1,t_2, \ldots ,t_{p - 1},z) \in {\Bbb{R}}^p:
F \left ( \sqrt{t_1^2 + t_2^2 + \cdots + t_{p - 1}^2}\, ,z \right )
\le 1 \right \}$$ denotes a $p$-dimensional body of revolution. [F. Chamizo]{} [@chamizo], 1998, proved for the number of lattice points $A(x;R_p)$ the representation $$A(x;R_p) = vol(R_p)x^p + P(x;R_p),$$ where the remainder is estimated by $$\begin{aligned}
P(x;R_p) \ll
\left\{\begin{array}{ll}
x^{\frac{11}{8}} \: \quad \quad \; \, \mbox{for}
\quad p = 3, \\
x^3 \log x \quad \mbox{for} \quad p = 5, \\
x^{p - 2} \: \qquad \mbox{for} \quad p > 5.
\end{array}\right.\end{aligned}$$ [M. Kühleitner]{} [@kuehleit], 2000, proved the lower bound $$P(x;R_3) = \Omega_- \left ( x(\log x)^{\frac{1}{3}} (\log \log
x)^{\frac{\log 2}{3}} e^{-A \sqrt{\log \log \log x}} \right ),$$ where $A$ is a positive constant.
[@kn], 2004, improved this to $$P(x;R_3) = \Omega_- \left ( x(\log x)^{\frac{1}{3}}
(\log \log x)^{\frac{2}{3}(\sqrt{2}-1)}(\log\log\log x)^{-\frac{2}{3}} \right ).$$
$p = 3$: [E. Krätzel]{} and [ W. G. Nowak]{}, [@jewie], 2004, proved a qualitatively weaker estimate, but with precise numerical constants. We describe only the simplest case. Let the body of revolution be given by $$R_3 = \left \{ (t_1,t_2,z) \in {\Bbb{R}}^3: t_1^2 + t_2^2 \le
f^2(z), \; |z| \le 1 \right \},$$ where it is assumed that $$f(z) \ge 0, \quad f(z) = f(-z), \quad f(1) = 0, \quad f(0) \le 1,$$ $$f'(z) \longrightarrow - \infty, \quad f''(z) \longrightarrow
- \infty \quad \mbox{for} \quad z \longrightarrow 1 - 0.$$ Further assume that $f''(z)$ is monotonic and $0 < r_{\min} \le
r_{\max} < \infty$ for the absolute value of the radius of curvature. Let $$\begin{aligned}
A(x;R_3) & = & \# \left \{ (n_1,n_2,m) \in {\Bbb{Z}}^3:
n_1^2 + n_2^2 \le x^2 f^2 \left( \frac{m}{x} \right ),
\; |m| \le x \right \} \\
& = & vol(R_3)x^3 + P(x;R_3).\end{aligned}$$ Then $$|P(x;R_3)| \le \left ( 73 r_{\max}^{\frac{3}{4}} +
13 r_{\max}^{\frac{1}{4}} \right ) x^{\frac{3}{2}} + c_1x \log x
+ c_2 x + c_3x^{\frac{3}{4}} + c_4 x^{\frac{1}{2}} + 5$$ with explicit positive constants $c_1, \ldots , c_4$ depending on $r_{\min}$ and $r_{\max}$.
The square of the distance function of an ellipsoid $CD_p = E_p
\; (p \ge 3)$ is given by the positive definite quadratic form $$F^2(\vec{t};E_p) = \sum_{\nu = 1}^p \sum_{\mu = 1}^p a_{\nu \mu}
t_{\nu}t_{\mu}, \qquad a_{\nu \mu} = a_{\mu \nu} \in \Bbb{R},$$ with the determinant $d = det(a_{\nu \mu}) > 0$. Then the ellipsoid $E_p$ is defined by $$E_p = \left \{ \vec{t} \in {\Bbb{R}}^p: F(\vec{t};E_p) \le 1
\right \} ,$$ and the number of lattice points is given by $$\begin{aligned}
A(x;E_p) & = & \# \left \{ \vec{n} \in {\Bbb{Z}}^p:
F(\vec{n};E_p) \le x \right \} \\
& = & \frac{\pi^{p/2}}{\Gamma (\frac{p}{2} + 1) \sqrt{d}} \, x^p +
P(x;E_p).\end{aligned}$$ It is necessary to distinguish between rational and irrational ellipsoids. An ellipsoid is called rational if there exists a number $c > 0$ such that $ca_{\nu \mu}$ are integers for all $\nu
, \mu$. An irrational ellipsoid is then a non-rational ellipsoid.
[@landau1], 1915, [@landau2], 1924: The estimates $$\begin{aligned}
P(x;E_p) =
\left\{\begin{array}{ll}
O\left(x^{p - 2 + \frac{2}{p + 1}}\right), \\
\Omega \left ( x^{\frac{p - 1}{2}} \right )
\end{array}\right.\end{aligned}$$ hold for arbitrary ellipsoids.
[@arnold1], 1924, for $p \ge 8$ and [E. Landau]{} [@landau3], 1924, for $p \ge 4$: $$\begin{aligned}
P(x;E_p) \ll
\left\{\begin{array}{ll}
x^{p - 2} \qquad \quad \mbox{for} \quad p > 4, \\
x^2 \log^2x \quad \; \mbox{for} \quad p = 4.
\end{array}\right.\end{aligned}$$ Note that $$P(x;E_p) = \Omega \left ( x^{p - 2} \right ) \quad \mbox{for} \quad
p \ge 3.$$ [A. Walfisz]{} [@arnold2], 1960: $$P(x;E_4) = O\left ( x^2(\log x)^{\frac{2}{3}} \right ).$$ [Y.-K. Lau]{} and [K. Tsang]{} [@lau], 2002: $$P(x;E_3) = \Omega_{\pm}(x \sqrt{\log x} \, ).$$
In the case of irrational ellipsoids a lot of special results are available. To obtain them, one usually imposes special conditions on the coefficients of the quadratic form. Most of them were proved by [B. Diviš, V. Jarnik, B. Novak]{} and [A. Walfisz]{}. It is nearly impossible to list all the different results. For details, see [F. Fricker’s]{} monograph [@fricker].
The most important results have been obtained for [quadratic forms of diagonal type]{}: $$F^2(\vec{t};E_p) = \sum_{\nu = 1}^p a_{\nu \nu}t_{\nu}^2 \qquad a_{\nu\nu}>0\ .$$ [V. Jarnik]{} [@jarnik3], 1928: The estimate $$\begin{aligned}
P(x;E_p) \ll
\left\{\begin{array}{ll}
x^{p - 2} & \mbox{for} \quad p > 4, \\
x^2 \log^2x & \mbox{for} \quad p = 4
\end{array}\right.\end{aligned}$$ holds for arbitrary diagonal quadratic forms.
[@jarnik4], 1929: The estimate $$P(x;E_p) = o(x^{p - 2)} \quad \mbox{for} \quad p > 4$$ holds for irrational, diagonal, quadratic forms.
[@jarnik4], 1929 for $k = 4$ and [A. Walfisz]{} [@arnold3], 1927 for $p > 4$: For each monotonic function $f(x) > 0$ with $f(x) \longrightarrow 0$ for $x \longrightarrow
\infty$ there is an irrational, diagonal, quadratic form such that $$\begin{aligned}
P(x;E_p) =
\left\{\begin{array}{ll}
\Omega (x^{p - 2}f(x)) &\mbox{for} \quad p > 4, \\
\Omega (x^2f(x) \log \log x) & \mbox{for} \quad p = 4.
\end{array}\right.\end{aligned}$$
and [F. Götze]{} [@begoe1], 1997: The estimate $$P(x;E_p) \ll x^{p - 2} \quad \mbox{for} \quad p \ge 9$$ holds for arbitrary ellipsoids. This result was extended to $p \ge
5$ by [F. Götze]{} [@goetze], 2004.
and [F. Götze]{} [@begoe2], 1999: The estimate $$P(x;E_p) = o(x^{p - 2}) \quad \mbox{for} \quad p \ge 9$$ holds for irrational ellipsoids.
Let $k,p \in \Bbb{N}, \: k \ge 3, \: p \ge 3$. The super spheres $SS_{k,p}$ are defined by $$|t_1|^k + |t_2|^k + \cdots + |t_p|^k = 1.$$ The [Gauss]{}ian curvature of these super spheres vanishes for $t_{\nu} = 0 \: (\nu = 1,2, \ldots ,p)$. Consider the number of lattice points inside and on the super spheres $xSS_{k,p}$: $$A(x;SS_{k,p}) = \# \left \{ \vec{n} \in {\Bbb{Z}}^p:
|n_1|^k + |n_2|^k + \cdots + |n_p|^k \le x^k \right \} .$$
[@burton2], 1966, for even $k$ and [E. Krätzel]{} [@kratz6], 1973, for odd $k$: $$A(x;SS_{k,p}) = V_{k,p}x^p + \Delta (x;SS_{k,p}), \qquad
V_{k,p} = \left ( \frac{2}{k} \right )^p
\frac{\Gamma^p (\frac{1}{k})}{\Gamma (1 + \frac{p}{k})},$$
$$\begin{aligned}
\Delta (x;SS_{k,p}) =
\left\{\begin{array}{ll}
O\left( x^{p - 2 + \frac{2}{p + 1}}\right) &\mbox{for} \quad k \le p + 1, \\
O, \Omega \left ( x^{(p - 1)(1 - \frac{1}{k})} \right )
&\mbox{for} \quad k > p + 1.
\end{array}\right.\end{aligned}$$
Hence the lattice point problem is settled in case when $k > p +
1$, as in the planar case.
, [@kr-lp], 1988: $$A(x;SS_{k,p}) = V_{k,p}x^p + \sum_{r = 1}^{p - 1} H_{k,p,r}(x) +
\Delta_{k,p}(x).$$ The terms $H_{k,p,r}(x)$ are defined recursively: $$\Delta_{k,1}(x) = - 2 \psi (x) = - 2 \left ( x - [x] - \frac{1}{2}
\right ),$$ $$H_{k,p,r}(x) = {p \choose r} (p - r)V_{k,p - r} \int\limits_0^x
(x^k - t^k)^{\frac{p - r}{k} - 1} t^{k - 1} \Delta_{k,r}(t) \,
{\rm d}t$$ for $r = 1,2, \ldots ,p - 1$. Furthermore $$H_{k,p,1}(x) = pV_{k,p - 1} \psi_{p/k}^{(k)}(x),$$ where $\psi_{\nu}^{(k)}(x)$ is defined in (3.1). Estimates: $$\psi_{p/k}^{(k)}(x) = O, \, \Omega_{\pm} \left (
x^{(p - 1)(1 - \frac{1}{k})} \right ),$$ $$\begin{aligned}
H_{k,p,r}(x) \ll
\left\{\begin{array}{ll}
x^{p - 2} & \mbox{for} \quad p - r > k, \\
x^{(p - r)(1 - \frac{2}{(r + 1)k}) + r - 2 + \frac{2}{r + 1}}
& \mbox{for} \quad p - r \le k,
\end{array}\right.\end{aligned}$$ $r = 2,3, \ldots ,p - 1$, and $$\Delta_{k,p}(x) \ll x^{p - 2 + \frac{2}{p + 1}}.$$ $$\begin{aligned}
H_{k,p,r}(x) & = & \Omega_{\pm} \left (
x^{(p - r)(1 - \frac{1}{k}) + \frac{r - 1}{2}} \right )
\quad\mbox{for} \quad r= 2,3, \ldots ,p - 1, \\
\Delta_{k,p}(x) & = & \Omega_{\pm} \left ( x^{\frac{p - 1}{2}}
\right ).\end{aligned}$$
The representation of the number of lattice points suggests that besides of the main term $V_{k,p}x^p$ there are $p - 1$ further main terms. But the known upper bounds for these terms at present allow at most one second main term. So far we have $$\begin{aligned}
A(x;SS_{k,p}) =
\left\{\begin{array}{ll}
V_{k,p}x^p + O \left ( x^{p - 2 +
\frac{2}{p + 1}} \right ) &\mbox{for } k \le p + 1, \\
V_{k,p}x^p + pV_{k,p - 1} \psi_{p/k}^{(k)}(x) +
O \left ( x^{(p - 2)(1 - \frac{2}{3k}) + \frac{2}{3}} \right ) &\mbox{for } k > p + 1.
\end{array}\right.\end{aligned}$$
[@kr-lp], 1988, for $p
= 3$ and [R. Schmidt-Röh]{} [@ralph], 1989 for $p > 3$: $$\begin{aligned}
\Delta_{k,p}(x) \ll
\left\{\begin{array}{ll}
x^{p - 2 + \frac{12}{7p + 4}} &
\mbox{for} \quad 3 \le p \le 7, \\
x^{p - 2 + \frac{5}{3p + 1}} &\mbox{for} \quad 8 \le p.
\end{array}\right.\end{aligned}$$
[@kr-lp], 1988, for $r = 2$ and [S. Höppner]{} and [E. Krätzel]{} [@hoeppner], 1993, for $r = 3$: $$\begin{aligned}
\int\limits_0^x \Delta_{k,2}(t^{\frac{1}{k}}) \, {\rm d}t & \ll &
x^{1 - \frac{1}{2k}}, \\
\int\limits_0^x \Delta_{k,3}(t^{\frac{1}{k}}) \, {\rm d}t & \ll &
x \log x\end{aligned}$$ and, if $\Delta_{k,r}(t) \ll \Delta_{k,r}^{*}(x)$ for $1 \le t \le x$, $$H_{k,p,r}(x) \ll x^{(p - r)(1 + \frac{r - 3}{2k})} \left (
\Delta_{k,r}^{*}(x) \right )^{1 - \frac{p - r}{k}} (\log x)^{r - 2},$$ provided that $r = 2,3, \: k \ge p - r$. It is highly probable that the estimate also holds for $r > 3$.
estimate for $\Delta_{k,2}(x)$ and [ E. Krätzel’s]{} estimate for $\Delta_{k,3}(x)$ lead to $$\begin{aligned}
H_{k,p,2}(x) & \ll & x^{(p - 2)(1 - \frac{165}{146k}) + \frac{46}{73}}
(\log x)^{\frac{315}{146}(1 - \frac{p - 2}{k})} \quad
\mbox{for} \quad k \ge p - 2, \\
H_{k,p,3}(x) & \ll & x^{(p - 3)(1 - \frac{37}{25k}) + \frac{37}{25}}
\qquad \qquad \qquad \qquad \ \mbox{for} \quad k \ge p - 3.\end{aligned}$$ In case when $r \ge 4, \: p \ge 5$ it is true that $$\sum_{r = 4}^{p - 1} H_{k,p,r}(x) \ll
x^{(p - 4)(1 - \frac{2}{5k}) + \frac{12}{5}}.$$
$$\begin{aligned}
A(x;SS_{k,p}) =
\left\{\begin{array}{ll}
V_{k,p}x^p + O \left ( x^{\lambda_{k,p}} \right )
& \mbox{for} \quad k < p + 1, \\
V_{k,p}x^p + pV_{k,p - 1} \psi_{p/k}^{(k)}(x) + O \left (
x^{\vartheta_{k,p}} \right ) & \mbox{for} \quad k \ge p + 1
\end{array}\right.\end{aligned}$$
with $$\begin{aligned}
\lambda_{k,p} &=&
\left\{\begin{array}{ll}
p - 2 + \frac{12}{7p + 4} & \mbox{for}
\quad 3 \le p \le 7, \\
p - 2 + \frac{5}{3p + 1} & \mbox{for} \quad 8 \le p, \end{array}\right.\\
\vartheta_{k,p} & = & p - 2 + \frac{12}{7p + 4} \quad\ \mbox{for}
\quad p = 3,4, \quad k \le p + 4,\end{aligned}$$ $$\begin{aligned}
\vartheta_{k,p} =
\left\{\begin{array}{ll}
(p - 2) \left ( 1 - \frac{165}{146k} \right )
+ \frac{46}{73} + \varepsilon & \mbox{for} \quad p = 3,4, \,
k \ge p + 5, \; \varepsilon > 0, \\
(p - 2) \left ( 1 - \frac{2}{5k} \right ) + \frac{2}{5} +
\frac{4}{5k} & \mbox{for} \quad p \ge 5, \quad \;
k < \frac{533p - 482}{168}, \\
(p - 2) \left ( 1 - \frac{165}{146k} \right ) + \frac{46}{73}
+ \varepsilon & \mbox{for} \quad p \ge 5, \quad \;
k \ge \frac{533p - 482}{168}, \; \varepsilon > 0.
\end{array}\right.\end{aligned}$$ In addition to all that, [E. Krätzel]{} [@kratz7], 1999, proved that the estimate holds with $$\vartheta_{k,p} = p - 2 + \frac{5}{3p + 1}$$ for even $k$ and $p + 1 \le k < 2p - 4$.
Let $k,p \in \Bbb{N}, \: k,p \ge 3$. Consider the super ellipsoids $SE_p$ $$\lambda_1 |t_1|^k + \lambda_2 |t_2|^k + \cdots + \lambda_p |t_p|^k = 1$$ with $\lambda_1, \lambda_2, \ldots , \lambda_p > 0$ and the number of lattice points inside and on the super ellipsoids $xSE_p$: $$A(x;SE_p) = \# \left \{ \vec{n} \in {\Bbb{Z}}^p:
\lambda_1 |n_1|^k + \lambda_2 |n_2|^k + \cdots +
\lambda_p |n_p|^k \le x^k \right \}.$$ [V. Bentkus]{} and [F. Götze]{} [@begoe3], 2001, proved $$A(x;SE_p) = \frac{V_{k,p}}{(\lambda_1 \lambda_2 \cdots
\lambda_p)^{1/k}} \, x^p + P(x;SE_p),$$ where $$P(x;SE_p) \ll x^{p - k}$$ for even $k$ and sufficiently large $p$. A super ellipsoid is called rational if there exists a number $c > 0$ such that all $c \lambda_j$ are natural numbers. Otherwise the superellipsoid is called irrational. Then $$P(x;SE_p) = o(x^{p - k})$$ if and only if $SE_p$ is irrational.
Our knowledge about the properties of points on the boundary with [Gauss]{}ian curvature zero for the estimate of the number of lattice points is very poor. Something is known if the points with [Gauss]{}ian curvature zero are isolated.
[@haber], 1993: Let the boundary of the convex body contain only finitely many points of vanishing curvature such that the tangent plane is rational at these points. Then each such point produces a new main term.
[@nachr], 2000, and [@kr-anafu], 2000, simplified the proof of this result for $p = 3$ and gave integral representations for these contributions. Furthermore, if the slope of the tangent planes are irrational the error terms will be of smaller order.
The restriction to isolated zeros of curvature excludes some important bodies such as the super spheres, for example. Thus [M. Peter]{}, [@peter2], 2002, extended the considerations to the case of non-isolated zeros of curvature. Instead of assuming only finitely many zeros of curvature, he assumed only finitely many flat points.
The reader is also referred to the papers of [E. Krätzel]{}, [@kratz8], 2002, and [D.A. Popov]{}, [@popov], 2000.
=msbm10 scaled 700 \#1 =msbm10 \#1
### 4. Divisor problems and related arithmetic functions {#divisor-problems-and-related-arithmetic-functions .unnumbered}
The classical (or Dirichlet) divisor problem consists of the estimation of the function $$\Delta(x) := \mathop{\sum\nolimits^{'}}\limits_{n\le x}d(n) -
x(\log x + 2\gamma-1) - {\textstyle{1\over4}}, \eqno{(4.1)}$$ where $d(n) = \sum_{d|n}1$ is the number of divisors of a natural number $n$, $\gamma$ is Euler’s constant, and the prime ${}^{'}$ denotes that the last summand in (4.1) is halved if $x
\;(\,>1)$ is an integer.
As the reader will observe in the results to come, there is a far-reaching analogy between this error term $\Delta(x)$ and $P(x)$, the lattice point discrepancy of the circle, which we discussed in Section 1. The deep reason is the great similarity of the generating functions $$\sum_{n=1}^\infty {d(n)\over n^s} = \zeta^2(s)
\qquad\hbox{and}\qquad \sum_{n=1}^\infty {r(n)\over n^s} =
4\,\zeta_{\Qi(i)}(s) = 4\, \zeta(s)\,L(s)\quad (\Re(s)>1)\,.$$ Here $\zeta_{\Qi(i)}$ is the Dedekind zeta-function of the Gaussian field, and $L(s)$ is the Dirichlet $L$-series with the non-principal character modulo 4.
The [*generalized (Dirichlet) divisor*]{} problem (or the Piltz divisor problem, as it is also sometimes called), consists of the estimation of the function $$\Delta_k(x) := \mathop{\sum\nolimits^{'}}\limits_{n\le x}d_k(n) -
xP_{k-1}(\log x) - {\textstyle{(-1)^k\over2^k}}, \eqno{(4.2)}$$ where $d_k(n)$ is the number of ways $n$ may be written as a product of $k$ given factors, so that $d_1(n) \equiv 1$ and $d_2(n) \equiv d(n)$, $\Delta_2(x) \equiv \Delta(x)$. In (4.2), $P_{k-1}(t)$ is a suitable polynomial in $t$ of degree $k$, and one has in fact $$P_{k-1}(\log x) = \mathop{{\rm
Res}}\limits_{s=1}\,x^{s-1}\zeta^k(s) s^{-1},\eqno{(4.3)}$$ where $\zeta(s) = \sum_{n\ge1}n^{-s}\;(\Re s>1)$ is the Riemann zeta-function. The connection between $d_k(n)$ and $\zeta^k(s)$ is a natural one, since one has $$\zeta^k(s) \;=\; \sum_{n=1}^\infty d_k(n)n^{-s} = \prod_{p\,{\rm
prime}}(1-p^{-s})^{-k}\qquad(\Re s > 1).\eqno{(4.4)}$$ From (4.4) one infers easily that $d_k(n)$ is a multiplicative function of $n$ and that, for a prime $p$ and $\alpha$ a natural number, $d_k(p^\alpha) = k(k+1)\cdots (k+\alpha-1)/\alpha!.$ The relation (4.4) may be extended to complex $k$; for this and other properties of the so-called [*generalized divisor problem*]{} see [@I13], Chapter 14. The basic quantities related to $\Delta_k(x)$ are the numbers $$\alpha_k := \inf\,\Bigl\{\;a_k\ge0\;:\;\Delta_k(x)
\ll x^{a_k}\;\Bigr\},\quad
\beta_k := \inf\,\Bigl\{\;b_k\ge0\;:\;\int_1^X\Delta^2_k(x)\,{\rm d}x
\ll X^{b_k}\;\Bigr\}.\eqno{(4.5)}$$ Starting from the classical result of Dirichlet that $\Delta(x)
\ll \sqrt{x}$, there have been numerous results on $\alpha_k$ and $\beta_k$; for some of them the reader is referred to [@I13], Chapter 13, [@Ti1], Chapter 12, [@IO], [@kr-lp]. These results, in principle, have been obtained by two types of methods. For $k = 2,3$ the estimation of $\alpha_k$ is carried out by means of exponential sums, and for larger $k$ by employing results connected with power moments of $\zeta(s)$. The results on $\beta_k$ have been obtained by several techniques, also using power moments of $\zeta(s)$. When $k=2$ an important analytic tool for dealing with $\Delta(x)$ is the formula of G.F. Voronoï (see e.g., [@DF], [@I10], [@I13], [@J1], [@V1]). It says that $$\Delta(x) =
-2\pi^{-1}\sqrt{x}\sum_{n=1}^\infty d(n)n^{-1/2}
\left(K_1(4\pi\sqrt{nx}) +
{\pi\over2}Y_1(4\pi\sqrt{nx})\right),\eqno{(4.6)}$$ where $K_1,
Y_1$ are Bessel functions in standard notation (see e.g., [@Leb]). Despite the beauty and importance of (4.6), in practice it is usually expedient to replace it by a truncated version, obtained by complex integration techniques and the asympotic formulas for the Bessel functions. This is $$\Delta(x)
= (\pi\sqrt{2})^{-1}x^{1/4}\sum_{n\le N}d(n)n^{-3/4}
\cos(4\pi\sqrt{nx}-\pi/4) + O_\varepsilon
\left(x^\varepsilon(1+(x/N)^{1/2})\right), \eqno{(4.7)}$$ where the implied $O$–constant depends only on $\varepsilon$, and the parameter $N$ satisfies $1\ll N\le x^A$ for any fixed $A>0$. Estimates of the form $$\Delta(x) \ll x^\alpha(\log
x)^C\eqno{(4.8)}$$ with various values of $\alpha$ and $C\ge0$ have appeared over the last hundred years or so and, in general, reflect the progress of analytic number theory. The last in a long line of records (see [@I13], Chapter 13 for a discussion) for bounds of the type (4.8) is $\alpha = {131\over416} = 0.3149\ldots\,,$ due to M.N. Huxley [@Hu2]. This result is obtained by the intricate use of exponential sum techniques (see his monograph [@Hu1]) connected to the Bombieri–Iwaniec method of the estimation of exponential sums (see the works of E. Bombieri – H. Iwaniec [@BI] and of H. Iwaniec – C.J. Mozzochi [@IM]). The limit of these methods appears to be $\alpha_2 \le 5/16$, whilst traditionally one conjectures that $\alpha_2 = 1/4$ holds. In general, one conjectures that $\alpha_k = (k-1)/(2k)$ and $\beta_k
= (k-1)/(2k)$ holds for every $k\ge2$. Either of these statements (see [@I13] or [@Ti1]) is equivalent to the Lindelöf hypothesis that $\zeta({1\over2} + it) \ll_\varepsilon
|t|^\varepsilon$ for any given $\varepsilon>0$; note that trivially $\beta_k \le \alpha_k$ holds for $k\ge2$.
The best known upper bound $\alpha_3 \le 43/96$ was obtained by G. Kolesnik [@Kol1], who used a truncated formula for $\Delta_3(x)$, analogous to (4.7), coupled with the estimation of relevant three-dimensional exponential sums. Further sharpest known bounds for $\alpha_k$, obtained by using power moments for $\zeta(s)$ are (see [@I13], Chapter 13): $\alpha_k \le
(3k-4)/(4k)\;(4 \le k \le 8)$, $\alpha_9 \le 35/54$ and (see [@IO] for the values of $k$ between 10 and 20), $\alpha_k \le
(63k-258)/(64k)\;(79 \le k \le 119)$ (see [@IO]). For large values of $k$ one has the best bounds which come from the Vinogradov–Korobov zero-free region for $\zeta(s)$ and the bound for $\zeta(1+it)$. Namely if $D>0$ is such a constant for which one has $$\zeta(\sigma + it) \ll |t|^{D(1-\sigma)^{3/2}}\log^{2/3}|t|\qquad(
{\textstyle{1\over2}} < \sigma \le 1),\eqno{(4.9)}$$ then $$\alpha_k \;\le\;1 - {\textstyle{1\over3}}\cdot 2^{2/3}(Dk)^{-2/3}.
\eqno{(4.10)}$$ From the work of H.-E. Richert [@Ri2] it is known that $D\le
100$ may be taken in (4.9). The best known result $D \le 4.45 $ is due to K. Ford [@Fo].
As for results on $\beta_k$, it is worth noting that the exact value $\beta_k = (k-1)/(2k)$ is known for $k = 2,3$ (see [@I13], Chapter 13) and $k=4$ (see [@HB1]). One has $\beta_5 \le 9/20$ (see W. Zhang [@Zn]), $\beta_6 \le 1/2,
\beta_7 < 0.55469, \beta_8 < 0.606167, \beta_9 < 0.63809,
\beta_{10} < 0.66717$ (see [@IO]). If (4.9) holds, then (ibid.) $$\beta_k \;\le\;1 - {\textstyle{2\over3}}(Dk)^{-2/3}. \eqno{(4.11)}$$
On the other hand, one may ask for lower bounds or omega results for $\Delta_k(x)$. G.H. Hardy showed in [@Ha1] that $$\Delta(x) = \cases{\Omega_+((x\log x)^{1/4}\log_2x),&\cr
\Omega_-(x^{1/4}).\cr}$$ Further improvements are due to K.S. Gangadharan [@Ga], K. Corrádi– I. Katai [@CK] and to J.L. Hafner [@Hf1]. K. Corrádi– I. Katai proved $$\Delta(x) =
\Omega_-\left(x^{1/4}\exp\{c(\log_2x)^{1/4}(\log_3x)^{-3/4}\}
\right)\qquad(c>0)\, .$$ Hafner op. cit. showed that, with suitable constants $A>0, B>0$,
$$\Delta(x) = \Omega_+\left((x\log x)^{1/4}
(\log_2x)^{(3+\log4)/4}\exp(-A\sqrt{\log_3x})\right)\, .
\eqno{(4.12)}$$ Recently K. Soundararajan [@So] (see his lemma in Section 1.2) proved $$\Delta(x) = \Omega\left((x\log x)^{1/4}
(\log_2x)^{(3/4)(2^{4/3}-1)}(\log_3x)^{-5/8}\right)\, .\eqno{(4.13)}$$
Note that $(3/4)(2^{4/3}-1) = 1.1398\ldots...\,$, whilst $(3+\log4)/4 = 1.0695\ldots...\,$, but on the other hand (4.13) is an omega result and not an $\Omega_+$ or $\Omega_-$-result. In other words, Soundararajan obtains a better power of $\log_2x$, but cannot ascertain whether $\Delta(x)$ takes large positive, or large negative values. His method, like Hafner’s (see [@Hf2]), carries over to $\Delta_k(x)$. Hafner proved, with suitable $A_k>0$, $$\Delta_k(x) = \Omega^*\left((x\log x)^{(k-1)/(2k)}
(\log_2x)^{((k-1)/2k)(k\log k -k+1)+k-1}\exp(-A_k\sqrt{\log_3x})\right),$$ where $\Omega^* = \Omega_+$ if $k=2,3$ and $\Omega^* = \Omega_\pm$ if $k\ge4$. Soundararajan showed that $$\Delta_k(x) = \Omega\left((x\log x)^{(k-1)/(2k)}
(\log_2x)^{((k+1)/2k)(k^{2k/(k+1)}-1)}(\log_3x)^{-1/2-(k-1)/4k}\right).$$ The above estimate holds with $\Omega_+$ in place of $\Omega$ if $k\,\equiv\,3\,({\rm mod}\,8)$, and with $\Omega_-$ in place of $\Omega$ if $k\,\equiv\,7\,({\rm mod}\,8)$.
Large values and power moments of $\Delta(x)$ were investigated by A. Ivić [@I3]. It was shown there that $$\int_0^X|\Delta(x)|^A\,{\rm d}x \;\ll_\varepsilon\;
X^{1+A/4+\varepsilon}\qquad(0 \le A \le 35/4),\eqno{(4.14)}$$ and the range for $A$ for which the above bounds holds can be slightly extended by using newer bounds for $\zeta({1\over2}+it)$. The bounds in (4.14) are precisely what one expects to get if the conjectural bound $\alpha_2 = 1/4$ holds. They were used by D.R. Heath-Brown [@HB3] to prove that the function $x^{-1/4}\Delta(x)$ has a distribution function and that, for $A\in [0,9]$ (not necessarily an integer), the mean value $$X^{-1-A/4}\int_0^X|\Delta(x)|^A\,{\rm d}x$$ converges to a finite limit as $X\to\infty$. Moreover the same is true for the odd moments $$X^{-1-A/4}\int_0^X\Delta(x)^A\,{\rm d}x \qquad(A = 1,3,5,7,9).$$ For particular cases sharper results on moments are, in fact, known. A classical result of G.F. Voronoï [@V1] states that $$\int_1^X\Delta(x)\,{\rm d}x \;=\; {1\over4}X + O(X^{3/4}),$$ so that $\Delta(x)$ has 1/4 as mean value. K.C. Tong [@To1] proved the mean square formula $$\int_1^X\Delta^2(x)\,{\rm d}x \;=\; CX^{3/2} + F(X)
\quad(C = (6\pi^2)^{-1}\sum_{n=1}^\infty d^2(n)n^{-3/2} = 0.6542869\ldots\,),$$ where the error term $F(X)$ satisfies $F(X) \ll X\log^5X$. Much later E. Preissmann [@Pr] improved this to $F(X) \ll
X\log^4X$. Further relevant results on $F(X)$ are to be found in the works of Y.-K. Lau- K.M. Tsang [@LT] and K.-M. Tsang [@Ts1], [@Ts2]. In particular, it was shown that $$\int_2^X F(x)\,{\rm d}x \;=\; -(8\pi^2)^{-1}X^2\log^2X + cX^2\log
X + O(X)\eqno{(4.15)}$$ holds with a suitable $c>0$. On the basis of (4.15) it is plausible to conjecture that, with a certain $\kappa$, one has $$F(x) = -(4\pi^2)^{-1}x\log^2x + \kappa x\log x + O(x).$$ K.-M. Tsang [@Ts3] treated the third and fourth moment of $\Delta(x)$, proving $$\int_1^X\Delta^3(x)\,{\rm d} x = BX^{7/4} +
O_\varepsilon(X^{\beta+\varepsilon}) \qquad(B > 0)\eqno{(4.16)}$$ and $$\int_1^X\Delta^4(x)\,{\rm d} x = CX^2 +
O_\varepsilon(X^{\gamma+\varepsilon}) \qquad(C > 0)\eqno{(4.17)}$$ with $\beta = {47\over28}, \gamma = {45\over23}$. In a forthcoming work of A. Ivić and P. Sargos [@IS], those values are improved to $\beta = {7\over5}, \gamma = {23\over12}$. A result on integrals of $\Delta^3(x)$ and $\Delta^4(x)$ in short intervals, which improves a result of W.G. Nowak [@No1], is also obtained in [@IS]. The analogues of (4.16) and (4.17) hold for the moments of $P(x)$, with the same values of the exponents $\beta$ and $\gamma$. One of the ingredients used therein for the proof of (4.17) is the following lemma, due to O. Robert– P. Sargos [@RS]: [*Let $k\ge 2$ be a fixed integer and $\delta > 0$ be given. Then the number of integers $n_1,n_2,n_3,n_4$ such that $N < n_1,n_2,n_3,n_4 \le 2N$ and*]{} $$|n_1^{1/k} + n_2^{1/k} - n_3^{1/k} - n_4^{1/k}| < \delta N^{1/k}$$ [*is, for any given $\varepsilon>0$,*]{} $$\ll_\varepsilon N^\varepsilon(N^4\delta + N^2).\eqno{(4.18)}$$ Moments of $\Delta^k(x)$ when $5\le k\le 9$ are treated by W. Zhai [@Zh], who obtained asymptotic formulas with error terms.
Finally we mention results for the Laplace transform of $\Delta(x)$ and $P(x)$, studied in [@I8] and [@I12]. We have $$\int_0^\infty P^2(x)e^{-x/T}\,{\rm d} x
={1\over4}\left({T\over\pi}\right) ^{3/2}\sum_{n=1}^\infty
r^2(n)n^{-3/2} - T +
O_\varepsilon(T^{\alpha+\varepsilon})\eqno{(4.19)}$$ and $$\int_0^\infty \Delta^2(x){\rm e}^{-x/T}\,{\rm d} x =
{1\over8}\left({T\over2\pi}\right)^{3/2}
\sum_{n=1}^\infty d^2(n) n^{-3/2} + T(A_1\log^2T + A_2\log T +
A_3) + O_\varepsilon(T^{\beta+\varepsilon}). \eqno{(4.20)}$$
The $A_j$’s are suitable constants ($A_1 =
-1/(4\pi^2)$), and the constants ${1\over2} \le \alpha < 1$ and ${1\over2} \le \beta < 1$ are defined by the asymptotic formula $$\sum_{n\le x}r(n)r(n+h) = {(-1)^h8x\over h}\sum_{d|h}(-1)^dd +
E(x,h),\; E(x,h) \ll_\varepsilon
x^{\alpha+\varepsilon},\eqno{(4.21)}$$ $$\sum_{n\le x}d(n)d(n+h) = x\sum_{i=0}^2(\log
x)^i\sum_{j=0}^2c_{ij} \sum_{d|h}\left({\log d\over d}\right)^j +
D(x,h), \; D(x,h) \ll_\varepsilon
x^{\beta+\varepsilon}.\eqno{(4.22)}$$ The $c_{ij}$’s are certain absolute constants, and the $\ll$–bounds both in (4.21) and in (4.22) should hold uniformly in $h$ for $1\le h \le x^{1/2}$. With the values $\alpha = 5/6$ of D. Ismoilov [@Ism] and $\beta = 2/3$ of Y. Motohashi [@Mo] it followed then that (4.19) and (4.20) hold with $\alpha = 5/6$ and $\beta = 2/3$. Motohashi’s fundamental paper (op. cit.) used the powerful methods of spectral theory of the non-Euclidean Laplacian. A variant of this approach was used recently by T. Meurman [@Me] to sharpen Motohashi’s bound for $D(x,h)$ for ‘large’ $h$, specifically for $x^{7/6} \le h \le
x^{2-\varepsilon}$, but the limit of both methods is $\beta = 2/3$ in (4.22). Using the results of F. Chamizo [@Ch], A. Ivić [@I12] obtained $\alpha = 2/3$ in (4.19), which appears to be the limit of the present methods.
The [*general divisor problem*]{} can be defined in various ways. Here we shall follow the notation introduced in [@I5]. Let $d(a_1,a_2,\ldots a_k;n)$ be the number of representations of an integer $n\ge1$ in the form $n = m_1^{a_1}m_2^{a_2}\ldots
m_k^{a_k}$, where $k\ge2$ and $1 \le a_1 \le a_2 \le \cdots \le
a_k$ are given integers, and the $m$’s are positive integers. Then by the general divisor problem we shall mean the estimation of the quantity
$$\begin{aligned}
\Delta (a_1,a_2,\ldots a_k;n) &=& \sum_{n\le x}d(a_1,a_2,\ldots
a_k;n) - \sum_{j=1}^k \,\mathop{{\rm
Res}}\limits_{s=1/a_j}\,\left(\prod_{r=1}^k
\zeta(a_rs)\right)x^ss^{-1}\nonumber \\ &=& \sum_{n\le
x}d(a_1,a_2,\ldots a_k;n) - \sum_{j=1}^k \left(\prod_{r=1,r\ne
j}^k\zeta(a_r/a_j)\right)x^{1/a_j}\end{aligned}$$
if the $a_j$’s are distinct. If this is not so, then the appropriate limit has to be taken in the above sum. For instance, if $ a_1= a_2 = \ldots = a_k =1$, then we get the classical Dirichlet divisor problem (without the dash in the sum and the constant term in (4.2)). It should be noted that $k$ does not have to be finite. A good example for this case is $$\zeta(s)\zeta(2s)\zeta(3s)\ldots = \sum_{n=1}^\infty a(n)n^{-s}
\qquad(\Re s > 1),\eqno{(4.24)}$$ corresponding to $k = \infty,\, a_j = j$ for every $j$. Using the product representation (4.3) (with $k=1$) for $\zeta(s)$, it follows that $a(n)$ is a multiplicative function of $n$, and that for every prime $p$ and every natural number $\alpha$ one has $a(p^\alpha) =P(\alpha)$, where $P(\alpha)$ is the number of (unrestricted) partitions of $\alpha$. Thus $a(n)$ denotes the number of nonisomorphic Abelian (commutative) groups with $n$ elements (see [@Krae3] for a survey of results up to 1982).
Another example of this kind is $$\prod_{r\ge1,m\ge1}\zeta(rm^2s) = \sum_{n=1}^\infty
S(n)n^{-s}\qquad(\Re s >1),$$ where $S(n)$ denotes the number of nonisomorphic semisimple rings with $n$ elements (see J. Knopfmacher [@Kno]).
In the general divisor problem we introduce two constants:
$$\begin{aligned}
{\bar\alpha}_k := \alpha(a_1,a_2,\ldots,a_k) &=&
\inf\left\{\,\alpha\ge0 : \Delta(a_1,a_2,\ldots,a_k) \ll x^\alpha
\right\},\\ {\bar\beta}_k := \beta(a_1,a_2,\ldots,a_k) &=&
\inf\left\{\,\beta\ge0 : \int_1^X\Delta^2(a_1,a_2,\ldots,a_k){\rm
d}x \ll X^{1+2\beta}\right\},\end{aligned}$$
which generalize $\alpha_k,\,\beta_k$ in (4.5). From the classical results of E. Landau [@La] it follows that $${k-1\over2(a_1+a_2+\ldots+a_k)} \le {\bar \alpha}_k \le
{k-1\over(k+1)a_1} \eqno{(4.25)}$$ if the numbers $a_j$ are distinct. In many particular cases the bounds in (4.25) may be superseded by the use of various exponential sum techniques, coupled with complex integration methods and the results on $\zeta(s)$ (moments, functional equations etc.). The case $k = 3$ (the three-dimensional divisor problem) is extensively discussed in E. Krätzel’s monograph [@kr-lp] and e.g., the work of H.-Q. Liu [@Li2].
In what concerns results on ${\bar\beta}_k$, A. Ivić [@I5] proved the following result: let $r$ be the largest integer which satisfies $(r-2)a_r \le a_1 + a_2 + \ldots + a_{r-1}$ for $2\le r
\le k$, and let $$g_k = g(a_1,a_2,\ldots,a_k) := {r-1\over2(a_1+a_2+\ldots+a_r)}.
\eqno{(4.26)}$$ Then ${\bar\beta}_k\ge g_k$, and if $$\int_0^T|\zeta({\textstyle{1\over2}}+it)|^{2k-2}\,{\rm d}t
\ll_\varepsilon T^{1+\varepsilon}\eqno{(4.27)}$$ holds, then ${\bar\beta}_k = g_k$. Thus assuming (4.27) we obtain a precise evaluation of ${\bar\beta}_k$, but it should be remarked that (4.27) at present is known to hold for $k=2,3$, whilst its truth for every $k$ is equivalent to the Lindelöf hypothesis. Moreover, we have $$\int_1^X\Delta^2(a_1,a_2,\ldots,a_k)\,{\rm d}x = \Omega
\left(X^{1+2g_k}\log^AX\right),\eqno{(4.28)}$$ where $A = A(a_1,a_2,\ldots,a_k)$ is explicitly evaluated in [@I5]. The results remain valid if $k = \infty$, provided that the generating function is of the form $$\zeta^{q_1}(b_1s)\zeta^{q_2}(b_2s)\zeta^{q_3}(b_3s)\ldots
\quad(1\le b_1 < b_2 < b_3<\ldots),\eqno{(4.29)}$$ where the $b$’s and the $q$’s are given natural numbers. The generating function (4.24) of $a(n)$ is clearly of the form (4.29). In this case it is convenient to define the error term as $$E(x) := \sum_{n\le x}a(n) - \sum_{j=1}^6c_jx^{1/j} \quad\left(c_j
= \prod_{\ell=1,\ell\ne j}^\infty \zeta\bigl({\ell\over
j}\bigr)\right). \eqno{(4.30)}$$ Then (4.28) implies that $$\int_1^XE^2(x) \,{\rm d}x = \Omega(X^{4/3}\log X),\eqno{(4.31)}$$ and (4.31) yields further $E(x) = \Omega(x^{1/6}\log^{1/2}x)$, obtained by R. Balasubramanian–K. Ramacha-ndra [@BR]. The omega result (4.31) is well in tune with the upper bound $$\int_1^XE^2(x) \,{\rm d}x = O(X^{4/3}\log^{89}X)$$ of D.R. Heath-Brown [@HB2], who improved the exponent $39/29
= 1.344827\ldots\,$ of [@I4].
As for upper bounds for $E(x)$ of the form $E(x) =
O(x^c\log^Cx)\;(C\ge0)$ or $E(x) =
O_\varepsilon(x^{c+\varepsilon})$, they have a long and rich history (note that $c\ge1/6$ must hold by (4.31)). The bound $c\le1/2$ was obtained by P. Erdős–G. Szekeres [@ES], who were the first to consider the function $a(n)$ and the so-called [*powerful numbers*]{} (see [@I3], [@kr-lp] for an account). After their work, the value of $c$ was decreased many times (chronologically in the works [@KR], [@Ri1], [@Scw], [@Sch], [@Sr], [@Kol2] and Liu [@Li1], who had $c\le 50/199 = 0.25125\ldots\,)$. Recently O. Robert–P. Sargos [@RS] obtained $c \le 1/4+\varepsilon$ by the use of (4.18). This is the limit of the method, and the result comes quite close to the conjecture of H.-E. Richert [@Ri1], made in 1952, that $E(x) = o(x^{1/4})$ as $x\to\infty$. The most optimistic conjecture is that $c = 1/6+\varepsilon$ holds.
A related problem is the estimation of $T(x) = \sum\tau(G)$, where $\tau(G)$ denotes the number of direct factors of a finite Abelian group $G$, and summation is over all Abelian groups whose orders do not exceed $x$. It is known (see [@Co] or [@Krae4]) that $$T(x) = \sum_{n\le x}t(n),\quad\sum_{n=1}^\infty t(n)n^{-s}
= \zeta^2(s)\zeta^2(2s)\zeta^2(3s)\ldots\;(\Re s>1),$$ which is of the form (4.29) with $q_j = 2, b_j = j$. Here one should define appropriately the error term (note that $t(n)
= \sum_{d|n} a(d)a(n/d)$) as
$$\begin{aligned}
\Delta_1(x) &:=& T(x) - \sum_{j=1}^5(D_j\log x + E_j)x^{1/j}\\& =&
\sum_{mn\le x}a(m)a(n)- \sum_{j=1}^5(D_j\log x + E_j)x^{1/j},\end{aligned}$$
where $D_j$ and $E_j\,(D_1>0)$ can be explicitly evaluated (see H. Menzer–R. Seibold [@MS]). They proved that $\Delta_1(x) = O_\varepsilon(x^{\rho+\varepsilon})$ with $\rho =
45/109 = 0.412844\ldots\,$, improving a result of E. Krätzel [@Krae4], who had the exponent 5/12. As for the true order of $\Delta_1(x)$, one expects $\rho = 1/4$ to hold. This is supported by the estimate (see [@I6] and [@I7]) $$\int_1^X\Delta^2_1(x)\,{\rm d}x = \Omega(X^{3/2}\log^4X)\, ,
\eqno{(4.32)}$$ which yields $\rho \ge 1/4$. The value of $\rho$ was later improved (see [@HM] and [@Li2]), and currently the best bound $\rho \le 47/130$ is due to J. Wu [@Wu]. Moreover, in [@I6] it was shown that $$\int_1^X\Delta^2_1(x)\,{\rm d}x = O_\varepsilon
(X^{8/5+\varepsilon}),$$ while if (4.27) holds with $k=5$, then one has $$\int_1^X\Delta^2_1(x)\,{\rm d}x =
O_\varepsilon(X^{3/2+\varepsilon}).\eqno{(4.33)}$$ In this case (4.32) and (4.33) determine fairly closely the true order of the mean square of $\Delta_1(x)$.
The previous discussion centered on [*global*]{} problems involving arithmetic functions which arise in connection with divisor problems, that is, the estimation of the error terms in the asymptotic formulas for summatory functions or the related power moment estimates. One can, of course, treat also [*local*]{} problems as well, namely problems involving pointwise estimation, distribution of values, and other arithmetic properties. The literature on this subject, which can be also considered as a part of the theory of divisor problems, is indeed vast, especially on $d(n)$ and $d_k(n)$. Thus it would be a great increase in the length of this work, as well somewhat outside the mainstream of the theory, if we dwelt in general on local problems. Also one sometimes, under divisor problems, considers divisor problems of the type (4.29) (or (4.23)), where the $q$’s are allowed to be negative. This greatly increases the class of functions that are allowed (like e.g., [*squarefree numbers*]{} whose characteristic function is generated by $\zeta(s)/\zeta(2s)$, or [*squarefull numbers*]{} whose characteristic function is generated by $\zeta(2s)\zeta(3s)/\zeta(6s)$). In this work we have found it best to adhere to divisor problems of the form (4.23) or (4.29).
As for local problems, we shall conclude with just a few words on a representative divisor problem, namely the function $a(n)$. It was proved by E. Krätzel [@Krae1] that $$\limsup_{n\to\infty}\,\log a(n)\cdot{\log\log n\over\log n} =
{\log5\over4},\eqno{(4.34)}$$ and his result was sharpened by W. Schwarz – E. Wirsing [@SW] and (on the Riemann Hypothesis) by J.-L. Nicolas [@NI]. General results for multiplicative functions, analogous to (4.34), were obtained by E. Heppner [@He] and A.A. Drozdova – G.A. Freiman (see [@Po], Chapter 12). The existence of the local densities of $a(n)$, the numbers $$d_k :=
\lim_{x\to\infty}\,x^{-1}\sum_{n\le x, a(n)=k}1$$ for any given integer $k\ge1$, was proved by Kendall-Rankin [@KR]. This was sharpened in [@I1] to $$\sum_{n\le x, a(n)=k}1 = d_kx + O(x^{1/2}\log x),$$ where the $O$–constant is uniform in $k$. Further sharpenings and generalizations are to be found in [@I2], [@IT], [@Krae2], [@KW] and [@no2].
The distribution of values of $a(n)$ was investigated in [@I2] and P. Erdős – A. Ivić [@EI1]. It was shown there that $[x,\,2x]$ contains at least $\sqrt{x}$ integers $n$ for which $a(n+1) = a(n+2) = \ldots = a(n+k)$ with $k = [\log
x\log_3x/(40(\log_2x)^2)]$, and at least $\sqrt{x}$ integers $m$ such that the values $a(m+1), a(m+2), \ldots, a(m+t)$ are all distinct, where for a suitable $C>0$ $$t = [C(\log x/\log\log x)^{1/2}].$$ Iterates of $a(n)$ were treated by P. Erdős – A. Ivić [@EI2] and [@I7].
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[^1]: Let us recall briefly the different $\Omega$-symbols: For real functions $F$ and $G>0$, we mean by that $\limsup(F(x)/G(x))>0$, as $x\to\infty$. Similarly, $F(x)=\Omega_-(G(x))$ stands for $-F(x)=\Omega_+(G(x))$. Further, $F(x)=\Omega_\pm(G(x))$ says that both of these assertions are true, and $F(x)=\Omega(G(x))$ means that at least one of them applies, i.e., $F(x)\ne o(G(x))$.
[^2]: We write throughout $\log_j$ for the $j$-fold iterated natural logarithm. Thus $\log_2=\log\log$, and so on.
[^3]: Observe the difference in notation compared to sections 1 and 2: If $CD_p$ is the $p$-dimensional unit ball, then $A(x;CD_p)=A_p({x^2})$.
|
---
abstract: 'Using the Dirac procedure to treat constraints dynamical sistems applied to gravitation, as described in the context of Teleparallel Equivalent of General Relativity (TEGR), we investigate, from the first class constraints, the gauge transformations in the fundamental field: the components of tetrads. We have shown that there is no an isotropy in physical space with respect to gauge transformations, i.e., given an arbitrary gravitational field, coming from a gauge transformation in the internal space, physical space reacts differently in the spatial and temporal components. By making an appropriate choice, we have found a gauge transformation for the components of tetrad field that allows a direct analogy with the gauge transformations of the Yang-Mills theory. In addition, to the flat case in which the algebra index is fixed, we get transformations similar those of the Electromagnetism. Moreover, still considering the flat case, the dependence of the gauge transformation parameter in the space-time variables is periodic, just like in the Electromagnetism. Furthermore, the gravitoelectric and gravitomagnetic fields as have recently been defined, make sense since they allow a direct relation with the momenta, which is analogous to what occurs in other gauge theories.'
---
**ADVANCED STUDIES IN THEORETICAL PHYSICS, Vol. x, 200x, no. xx, xxx - xxx**
**[L.R.A. Belo\*, E.P. Spaniol\*\*, J.A. de Deus\*\*\* and V.C. de Andrade\*\*\*\*]{}**
Institute of Physics, University of Brasilia 70.917-910, Brasilia, Distrito Federal, Brazil
\* [email protected]
\*\* [email protected]
\*\*\* [email protected]
\*\*\*\* [email protected]
\[section\]
\[Theorem\][Definition]{}
\[Theorem\][Corollary]{}
\[Theorem\][Lemma]{}
\[Theorem\][Example]{}
[**Keywords:**]{} Teleparallel Equivalent of General Relativity, Gauge symmetries, Hamiltonian Formulation, Gauge Transformation Parameter
Introduction
============
An alternative description of General Relativity (GR) is the TEGR. The usual formulation of TEGR in the literature is its Lagrangian version [@hayashi1; @hayashi2; @aldrovandi3; @andrade4; @arcos5; @aldrovandi6]; such a formulation, as well as the theory of Yang-Mills and Electromagnetism has a configuration space larger than necessary, resulting the appearance of constraints in its Hamiltonian formulation [@maluf7; @chee8], or in the Riemann-Cartan geometry, with local SO(3,1) symmetry [@blagojevic9; @blagojevic10; @blagojevic11]. In the case of Yang-Mills and Electromagnetism, the Hamiltonian formulation can be achieved by following the Dirac algorithm [@dirac11] to deal with constraint Hamiltonian systems. This same algorithm allows us to conclude that the first class constraints, primary or secondary, act as generators of gauge transformation, giving an interpretation for the extra degrees of freedom coming from the Hamiltonian formulation.
The equivalence principle states that the special relativity equations must be recovered in a locally inertial coordinate system, where the effects of gravitation are absent. Thus, based on this principle would be natural to expect that gravitation had a local Poincaré symmetry and that it was possible to describe it as a genuine gauge theory for this group. In fact, it is possible [@blagojevic9]. However, there are theoretical and experimental evidence, that from the most fundamental point of view (beyond the standard model) the Lorentz symmetry is broken [@Kostelecky12; @Carroll13; @Colladay14; @Belich15; @Songaila16; @Moffat17; @Coleman18]. This would eliminate the Poincaré group as the local symmetry group of gravity, leaving room only for the translational sector (or more general other).
In this paper, we show that starting from the TEGR Lagrangian contained in the literature it can be conclude that there is no an isotropy in physical space with respect to gauge transformations, i.e., given an arbitrary gravitational field, coming from a gauge transformation in the internal space, physical space reacts differently in the spatial and temporal components. In addition, to the flat case in which the algebra index is fixed, we get transformations similar those of the electromagnetism. We will also see that for the flat case, the dependence of the gauge transformation parameter in the space-time variables is periodic, as well in the Electromagnetism. Furthermore, we will show that the gravitoelectric and gravitomagnetic fields as have recently been defined [@spaniol19], make sense since they allow a direct relation with the momenta, which is analogous to what occurs in other gauge theories.
Momenta canonically conjugated to the tetrad field
==================================================
As we will see in the next section, by introducing some new fields it is possible to find a consistent Hamiltonian description for the TEGR. In this description it will be possible, for example, to identify all the constraints of the theory. Let us see now the reasons why it will be necessary to follow this alternative procedure, i.e., why we simply could not straightly follow the Legendre transformation procedure.
The Lagrangian density associated with TEGR has the form [^1] [^2] [@aldrovandi3]: $${\cal L} = \frac{h}{2k^2} \left[\frac{1}{4} \; T^\rho{}_{\mu \nu}
T_\rho{}^{\mu \nu} + \frac{1}{2} \; T^\rho{}_{\mu \nu} \; T^{\nu
\mu}{}_\rho - T_{\rho \mu}{}^{\rho} \; T^{\nu \mu}{}_\nu \right] \label{2.1}$$ with $h=det(h^{a}{}_{\mu})$ and $k=\frac{8\pi G}{c^4}$. This expression can be rewritten in a more elegant form to get: $${\cal L} = \frac{h}{8k^2} \left[ \frac{1}{4} \, T^{a}{}_{\mu \nu}
T^{b}{}_{\rho \lambda} \, N_{a b}{}^{\nu \rho, \nu \lambda} \right], \label{2.2}$$ with $N_{a b}{}^{\mu \rho, \nu \lambda}$ being the tensor responsible for all possible contractions of indices, given by:
$$\begin{aligned}
N_{a b}{}^{\mu \rho, \nu \lambda} &=& \frac{1}{2} \eta_{ab} \left[g^{\mu \rho} \; g^{\nu \lambda}- g^{\mu \lambda} \; g^{\nu \rho}\right] + \frac{1}{2} h_{a}{}^{\rho} \left[ h_{b}{}^{\mu}\; g^{\nu
\lambda}-h_{b}{}^{\nu}\; g^{\mu \lambda}\right]\nonumber \\
&-&\frac{1}{2}h_{a}{}^{\lambda} [h_{b}{}^{\mu}\; g^{\nu\rho}-h_{b}{}^{\nu}\; g^{\mu \rho}] + h_{a}{}^{\mu} \left[h_{b}{}^{\lambda}\; g^{\nu \rho}-h_{b}{}^{\rho}\; g^{\mu \lambda}
\right]\nonumber \\
&-& h_{a}{}^{\nu} \left[ h_{b}{}^{\lambda}\; g^{\mu \rho}-h_{b}{}^{\rho}\; g^{\mu \lambda}\right]. \label{2.3}\end{aligned}$$
Another way to write the Lagrangian density (\[2.1\]), and perhaps the most fruitful of all is: $$\mathcal{L}_{G}=\frac{h}{4k^2}\;S^{\rho \mu \nu }\;T_{\rho \mu \nu }, \label{2.4}$$ where $$S^{\rho \mu \nu }=-S^{\rho \nu \mu }\equiv {\frac{1}{2}}\left[
K^{\mu \nu \rho }-g^{\rho \nu }\;T^{\theta \mu }{}_{\theta }+g^{\rho
\mu }\;T^{\theta \nu }{}_{\theta }\right] \label{2.5}$$ and $K^{\mu \nu \rho }$ is the contortion tensor given by $$K^{\mu \nu \rho}= \frac{1}{2}T^{\nu \mu \rho}+\frac{1}{2}T^{\rho \mu \nu}-\frac{1}{2}T^{\mu \nu \rho}. \label{2.6}$$
As we would expect, this is a quadratic Lagrangian density in the field strength tensor [^3].
Let us first define the momenta canonically conjugated to tetrads $h_c{}^{\sigma}$, using the Lagrangian density (\[2.2\]) $$\begin{aligned}
\Pi_c{}^{\sigma} \equiv \frac{\partial L}{\partial(\partial_{0}
h^{c}{}_{\sigma})}=\frac{h}{8k^2} C_{c b}{}^{\rho \sigma
\lambda}T^{b}{}_{\rho \lambda}, \label{2.7}\end{aligned}$$ with, $$C_{c b}{}^{\rho \sigma \lambda}=\left[ N_{c b}{}^{0 \rho, \sigma
\lambda}-N_{c b}{}^{\sigma \rho, 0 \lambda}+ N_{b c}{}^{\rho 0,
\lambda \sigma}-N_{b c}{}^{\rho \sigma, \lambda 0} \right]. \label{2.8}$$ Repeating the calculation for the momentum using as a starting point the Lagrangian density (\[2.4\]), we get: $$\begin{aligned}
\Pi_c{}^{\sigma} \equiv \frac{\partial L}{\partial(\partial_{0}
h^{c}{}_{\sigma})}=-\frac{h}{k^2} S_{c}{}^{\sigma 0}. \label{2.9}\end{aligned}$$ Taking then the definitions (\[2.3\]), (\[2.5\]) and (\[2.8\]), we see that expressions (\[2.7\]) and (\[2.9\]) for the momentum are totally equivalent. The fact that the superpotential $S^{a \mu \nu}=h^{a}{}_{\rho}S^{\rho
\mu \nu}$ appears explicitly in the expression of the Lagrangian density (\[2.4\]) makes it more useful than the definition (\[2.2\]).
Seizing the opportunity, we would to use the latter definition of momentum to corroborate recent definitions regarding the gravito-electric and gravito-magnetic fields [@spaniol19]: $$\begin{aligned}
E_{a}{}^{i}=S_{a}{}^{0 i}, \nonumber \\
\epsilon^{i j k}B_{a k}=S_{a}{}^{i j}. \label{2.10}\end{aligned}$$ Comparing the definition (\[2.10\]) for the gravito-electric field with the expression (\[2.9\]) for the momentum, we see that: $$\begin{aligned}
\Pi_c{}^{i}=\frac{h}{k^2} E_{c}{}^{i}. \label{2.11}\end{aligned}$$ This result clearly shows that the definitions for the gravito-electric and gravito-magnetic fields proposed in Ref [@spaniol19] lead us to a result completely analogous to what occurs with the theories of Yang-Mills and the Electromagnetism in which the momenta are also directly related to the “electric” fields of those theories. The main difference here is that these fields are related to the superpotential $S^{a \mu \nu}$, while the theories of Yang-Mills and Electromagnetism are related to the field strength tensor, $E_{a}{}^{i}=F_{a}{}^{0 i}$ and $E^{i}=F^{0 i}$, respectively. This difference is justified by the fact that gravitation, unlike the other interactions, presents a special property, soldering [@kobayashi20]. This property is a consequence of the existence of a tetrads field $h^{a}{}_{\mu}$, which acts as a link between the bundle (inner space) and the manifold, so that algebra indices can be transformed into space-time indices, implying a Lagrangian density over a quadratic term in the field strength tensor.
The usual sequence from here would be to isolate the $\partial_{0}h^{c}{}_{\sigma}$ terms of velocity and get the Hamiltonian version of the TEGR but unfortunately this can not be made in a simple way. Again, this difficulty is related to the fact that the momentum be related with superpotential and not with the field strength. Let us see, from the expression (\[2.9\]) we can show, after a long but straightforward calculation, that: $$\begin{aligned}
\Pi_{c}{}^{i}+O\left(h^{c}{}_{i},\vec{\nabla} h^{c}{}_{i}\right)&=&[\frac{1}{2}\left(h_{c}{}^{0}h_{a}{}^{0}g^{ij}+g_{ac}g^{ij}g^{00}+h_{c}{}^{j}h_{a}{}^{i}g^{00}-h_{c}{}^{0}h_{a}{}^{i}g^{0j}\right) \nonumber \\
&-&h_{c}{}^{0}h_{a}{}^{0}g^{ij}-h_{c}{}^{i}h_{a}{}^{j}g^{00}+h_{c}{}^{i}h_{a}{}^{0}g^{0j}]\partial_{0}h^{a}{}_{j}. \label{2.12}\end{aligned}$$ This set of equations can be rewritten as: $$P_c{}^{i}=K_{ca}{}^{ij}\partial_{0}h^{a}{}_{j}, \label{2.13}$$ where we define the objects $$P_c{}^{i}\equiv \Pi_{c}{}^{i}+O\left(h^{c}{}_{i},\vec{\nabla} h^{c}{}_{i}\right) \label{2.14}$$ and $$\begin{aligned}
K_{ca}{}^{ij}&\equiv&\frac{1}{2}\left(h_{c}{}^{0}h_{a}{}^{0}g^{ij}+g_{ac}g^{ij}g^{00}+h_{c}{}^{j}h_{a}{}^{i}g^{00}-h_{c}{}^{0}h_{a}{}^{i}g^{0j}\right) \nonumber \\
&-&h_{c}{}^{0}h_{a}{}^{0}g^{ij}-h_{c}{}^{i}h_{a}{}^{j}g^{00}+h_{c}{}^{i}h_{a}{}^{0}g^{0j}, \label{2.15}\end{aligned}$$ from where we obtain that $$\partial_{0}h^{a}{}_{j}=\left(K^{-1}\right)^{ca}{}_{ij}P_{c}{}^{i}. \label{2.16}$$ That is, find a set of solutions to the system (\[2.12\]) is equivalent to find the inverse $\left(K^{-1}\right)^{ca}{}_{ij}$; such task seems impossible even if we use algebraic manipulators. To circumvent this issue, we will adopt another possible procedure as described below.
Constraints as generators of gauge transformation
=================================================
Motivated by the theory of Yang-Mills and by Electromagnetism, where the secondary constraints are directly related to the Gauss’s law, we can also make such a comparison for the case of gravitation described by TEGR in order to have an alternative way in obtaining the constraints of the theory. Before proceeding, we note that the expression for the momenta (\[2.9\]) gives us directly the primary constraints of TEGR $$\begin{aligned}
\Phi_{c}=\Pi_{c}{}^{0}=-\frac{h}{k^2}S_{c}{}^{0 0}. \label{3.1}\end{aligned}$$ To test if the gravitational Gauss’s law [@spaniol19] really represents the secondary constraints we make [^4]: $$\begin{aligned}
\frac{d\Phi_{c}}{dt}&=&\{ \Phi_{c},{\cal H}_{0} \} \nonumber \\
&=&\frac{\delta
\Phi_{c}{}^{0}}{\delta h^{a}{}_{\rho}} \frac{\delta {\cal H}_{0}}{\delta \Pi_{a}{}^{\rho}}-\frac{\delta
\Phi_{c}{}^{0}}{\delta \Pi_{a}{}^{\rho}} \frac{\delta {\cal H}_{0}}{\delta h^{a}{}_{\rho}} \nonumber \\
&=&-\frac{\partial{\cal H}_{0}}{\partial h^{c}{}_{0}}+\partial_{\lambda}\frac{\partial{\cal H}_{0}}{\partial (\partial_{\lambda}h^{c}{}_{0})} \nonumber \\
&=&\frac{\partial{\cal L}}{\partial h^{c}{}_{0}}+\partial_{0}\Pi_{c}{}^{0}-\partial_{\lambda}\frac{\partial{\cal L}}{\partial (\partial_{\lambda}h^{c}{}_{0})} \nonumber \\
&=&\chi_{c}. \label{3.2}\end{aligned}$$ Here, $\chi_c$ is the gravitational Gauss’s law given by: $$\chi_{a}=\partial_{i}(hS_{a}{}^{0 i})-k^2(hj_{a}{}^{0})=k^2\partial_{i}(\Pi_{a}{}^{i})-k^2(hj_{a}{}^{0})=\partial_{i}(hE_{a}{}^{i})-k^2(hj_{a}{}^{0})=0, \label{3.3}$$ with $$j_{a}{}^{\rho}=\frac{\partial L}{\partial
h^{a}{}_{\rho}}=\frac{h_{a}{}^{\lambda}}{k^2} \left[ T^{c}{}_{\mu
\lambda}S_{c}{}^{\mu
\rho}-\frac{1}{4}\delta_{\lambda}{}^{\rho}T^{c}{}_{\mu
\nu}S_{c}{}^{\mu \nu } \right] \label{3.4}$$ assuming the role of a “vacuum source”. Or, in a way that makes clear the nonlinear character of gravity, $$\begin{aligned}
\chi_{a}&=&\partial_{i}(hE_{a}{}^{i})+k^2 h [ H^{b c}{}_{a i
j}E_{b}{}^{i}E_{c}{}^{j}+T^{b c}{}_{a n i j}\varepsilon^{j n k}
E_{c}{}^{i}B_{b}{}^{k}+ g_{r i} h^{c}{}_{j} \varepsilon^{j r k}(
E_{c}{}^{i}B_{a}{}^{k} \nonumber \\
&-&1/2E_{a}{}^{i}B_{c}{}^{k}) + J^{c}{}_{i
j}E_{c}{}^{i}E_{a}{}^{j}+K^{b c}{}_{a r i j n}\varepsilon^{i j
k}\varepsilon^{n r t} B_{c}{}^{k}B_{b}{}^{t} ]=0, \label{3.5}\end{aligned}$$ where objects $H^{b c}{}_{a i j}$, $T^{b c}{}_{a n i j}$ , $J^{c}{}_{i j}$ and $K^{b c}{}_{a r i j n}$ are combinations of terms of tetrads [^5]. The expression (\[3.5\]) justifies the interpretation of $j_{a}{}^{\rho}$ as a source of vacuum. Moreover, $j_{a}{}^{\rho}$ is the energy-momentum tensor of the gravitational field [@andrade21]. Notice that (\[3.5\]) is equivalent to zero component of GR field equations written in terms of gravitoelectric and gravitomagnetic fields.
The full equivalence between the GR and TEGR takes place within the equations of motion derived from the Lagrangian (\[2.4\]) and Einstein-Hilbert, $${\cal L}_{GR}=-\frac{\sqrt{-g}}{2 k^2} R. \label{3.6}$$ It can be shown that [@aldrovandi3]: $${\cal L}_{GR}={\cal L}_{TEGR}+\partial_\mu(\frac{2h}{k^2}T^{\nu\mu}{}_{\nu}). \label{3.7}$$ Although the divergence term in the above expression does not contribute to the dynamics, represented by the Euler-Lagrange equations, we would think that this term has some relevance in the Hamiltonian formulation. That is not the case. Consider: $${\cal L}=\frac{h}{4k^2}\;S^{\rho \mu \nu }\;T_{\rho \mu \nu}\;+\partial_\mu(\frac{2h}{k^2}T^{\nu\mu}{}_{\nu}), \label{3.8}$$ hence $$\begin{aligned}
\Pi_c{}^{\sigma} \equiv \frac{\partial {\cal L}}{\partial(\partial_{0}
h^{c}{}_{\sigma})}=-\frac{h}{k^2} S_{c}{}^{\sigma 0}+f(h_{c}{}^{\sigma},\partial^{i}h_{c}{}^{\sigma}). \label{3.9}\end{aligned}$$ From the previous lagrangian it may be noted again that there is no dependence on $\partial_{0}h^{a}{}_{0}$, so $\Pi_{c}{}^{0}\equiv\Phi_c$ remain primary constraints. Continuing: $$\begin{aligned}
\frac{d\Phi_{c}}{dt}&=&\{ \Phi_{c},{\cal H}_{0} \} \nonumber \\
&.& \nonumber \\
&.& \nonumber \\
&.& \nonumber \\
&=&\chi_{c}. \label{3.10}\end{aligned}$$ We can thus conclude that using the Lagrangean density (\[2.4\]) is satisfactory and there is no information lost in the divergence term in (\[3.8\]).
If we look closely, we see that the expression for the secondary constraints (gravitational Gauss’s law) has an explicit dependence on velocities, and as we said previously, to isolate those terms depending on the momenta is not a trivial task. A method for obtaining such constraints has been developed in [@maluf7]; we will use here only the final result, since development is quite long. We have: $$\chi_{c}=h_{c}{}^{0}{\cal H}_{0}+h_{c}{}^{i}F_{i}, \label{3.11}$$ where $$\begin{aligned}
{\cal H}_{0}&=&-h_{a 0}\partial_{k}\Pi^{a k}-\frac{kh}{4g^{0 0}}(g_{i k}g_{j l}P^{i j}P^{k l}-\frac{1}{2}P^2) \nonumber \\
&+&kh(\frac{1}{4}g^{i m}g^{n j}T^{a}{}_{m n}T_{a i j}+\frac{1}{2}g^{n j}T^{i}{}_{m n}T^{m}{}_{i j}-g^{i k}T^{j}{}_{j i}T^{n}{}_{n k}) \label{3.12}\end{aligned}$$ and $$F_{i}=h_{a i}\partial_{k}\Pi^{a k}-\Pi^{a k}T_{aki}+\Gamma^{m}T_{0mi}+\Gamma^{lm}T_{lmi}+\frac{1}{2g^{0 0}}(g_{i k}g_{j l}P^{k l}-\frac{1}{2}P)\Gamma^{j}. \label{3.13}$$ Moreover, the objects were defined: $$\begin{aligned}
P^{ik}&=&\frac{1}{2kh}(h_{c}{}^{i}\Pi^{ck}+h_{c}{}^{k}\Pi^{ci})+g^{0m}(g^{kj}T^{i}{}_{mj}+g^{ij}T^{k}{}_{mj}-2g^{ik}T^{j}{}_{mj}) \nonumber \\
&+&(g^{km}g^{0i}+g^{im}g^{0k})T^{j}{}_{mj}, \label{3.14}\end{aligned}$$ $$\Gamma^{ik}=\frac{1}{2}(h_{c}{}^{i}\Pi^{ck}-h_{c}{}^{k}\Pi^{ci})-kh[-g^{im}g^{kj}T^{0}{}_{mj}+(g^{im}g^{0k}-g^{km}g^{0i})T^{j}{}_{mj}] \label{3.15}$$ and $$\Gamma^{k}=\Pi^{0k}+2kh(g^{kj}g^{0i}T^{0}{}_{ij}-g^{0k}g^{0i}T^{j}{}_{ij}+g^{00}g^{ik}T^{j}{}_{ij}). \label{3.16}$$ The class test of the constraints shows that they are all first class [@maluf7]. Following the Dirac algorithm [@dirac11], let us calculate then the transformations generated by the constraints, which do not modify the physical state of the system (gauge transformations): $$\begin{aligned}
\delta h^{b}{}_{\rho}(x)&=&\int d^{3}x'
\left[\varepsilon_{1}^{a}(x')\{ h^{b}{}_{\rho}(x),\Phi_{a}(x)
\}+\varepsilon_{2}^{a}(x')\{ h^{b}{}_{\rho}(x),\chi_{a}(x) \} \right] \nonumber \\
&=&\int d^{3}x' \varepsilon_{1}^{a}(x') \left(\frac{\delta
h^{b}{}_{\rho}(x)}{\delta h^{c}{}_{\beta}(x')} \frac{\delta
\Phi_{a}(x)}{\delta \Pi_{c}{}^{\beta}(x')}-\frac{\delta
h^{b}{}_{\rho}(x)}{\delta \Pi_{c}{}^{\beta}(x')} \frac{\delta
\Phi_{a}(x)}{\delta h^{c}{}_{\beta}(x')} \right) \nonumber \\
&+&\int d^{3}x' \varepsilon_{2}^{a}(x') \left(\frac{\delta
h^{b}{}_{\rho}(x)}{\delta h^{c}{}_{\beta}(x')} \frac{\delta
\chi_{a}(x)}{\delta \Pi_{c}{}^{\beta}(x')}-\frac{\delta
h^{b}{}_{\rho}(x)}{\delta \Pi_{c}{}^{\beta}(x')} \frac{\delta
\chi_{a}(x)}{\delta h^{c}{}_{\beta}(x')} \right), \label{3.17}\end{aligned}$$ that results in $$\delta h^{b}{}_{\rho}=\delta^{0}_{\rho}\varepsilon^{b}_{1}+\nabla_{\rho}\varepsilon_{2}^{b}, \label{3.18}$$ with $$\nabla_{\rho}\varepsilon_{2}^{b}\equiv\delta_{\rho}^{i}\partial_{i}\varepsilon_{2}^{b}+\omega^{b}{}_{a \rho} \varepsilon_{2}^{a} \label{3.19}$$ and $$\begin{aligned}
\omega^{b}{}_{a \rho}&\equiv&-\frac{1}{g^{00}}\delta_{\rho}^{i}h_{a}{}^{0}h^{b}{}_{i}g^{0\mu}T^{j}{}_{j\mu}+\frac{3}{2g^{00}}\delta^{i}_{\rho}g^{0 b}h_{a i}g^{0\mu}T^{j}{}_{j\mu}+\frac{1}{2g^{00}}\delta^{i}_{\rho}h^{b 0}h_{a i}g^{0\mu}T^{j}{}_{j\mu} \nonumber \\
&+&\frac{3}{2}\delta^{i}_{\rho}h^{b}{}_{\mu}h_{a}{}^{\nu}T^{\mu}{}_{i \nu}+\delta^{i}_{\rho}g^{0 b}g_{0 \mu}h_{a}{}^{\nu}T^{\mu}{}_{i \nu}-\frac{1}{2}\delta^{i}_{\rho}g_{i \mu}h^{b \nu}h_{a}{}^{\alpha}T^{\mu}{}_{\nu \alpha} \nonumber \\
&+&\frac{1}{2}\delta^{i}_{\rho}h_{a i}h^{b \mu}T^{0}{}_{0 \mu}
-\delta^{i}_{\rho}h^{b}{}_{i}h_{a}{}^{\mu}T^{0}{}_{0 \mu}+\frac{1}{2}\delta^{i}_{\rho}\delta^{b}_{a}T^{0}{}_{0 i} \label{3.20}\end{aligned}$$ playing the role of covariant derivative and connection, respectively. The connection that appears in the definition (\[3.20\]) is not a usual spin connection, as we would expect for the case of a covariant derivative acting on a 4-vector with internal index. The reason for this must be related to the fact that we are dealing with a theory in which indices of internal space can be taken into space-time indices. Continuing, we can write: $$\delta h^{b}{}_{0}=\varepsilon^{b}_{1}, \label{3.21}$$ and $$\delta h^{b}{}_{i}=\nabla_{i}\varepsilon_{2}^{b}, \label{3.22}$$ which makes possible claims that there is no an isotropy in physical space with respect to gauge transformations, i.e., given an arbitrary gravitational field, arising from a gauge transformation in the internal space, physical space reacts differently in the temporal and spatial components. In addition, for the flat case in which the algebra index is fixed, we get transformations similar to those of electromagnetism, $$\delta h_{0}=\varepsilon_{1}, \label{3.23}$$ and $$\delta h_{i}=\partial_{i}\varepsilon_{2}. \label{3.24}$$ This is analogous to that was shown in ref. [@spaniol19], in which, in the weak field limit, the gravitational Maxwell’s equations are analogous to the electromagnetism. Here we see that this analogy is still valid in a more fundamental context.
We can go ahead and rewrite (\[3.18\]) as follows: $$\delta h^{b}{}_{\rho}=\delta^{0}_{\rho}\varepsilon^{b}_{1}+\delta_{\rho}^{i}\partial_{i}\varepsilon_{2}^{b}+\omega^{b}{}_{a \rho} \varepsilon_{2}^{a}. \label{3.25}$$ Introducing now the following relation between the parameters $\varepsilon^{b}_{1}$ and $\varepsilon^{b}_{2}$ $$\varepsilon^{b}_{1}=\partial_{0}\varepsilon_{2}^{b}, \label{3.26}$$ we have: $$\delta h^{b}{}_{\rho}=\partial_{\rho}\varepsilon_{2}^{b}+\omega^{b}{}_{a \rho} \varepsilon_{2}^{a}. \label{3.27}$$ The subindex 2 can now be ignored, $$\delta h^{b}{}_{\rho}=\partial_{\rho}\varepsilon^{b}+\omega^{b}{}_{a \rho} \varepsilon^{a}\equiv \nabla^{'}_{\rho}\varepsilon^{b}. \label{3.28}$$ The above transformations allow a direct analogy with the gauge transformations obtained in the Yang-Mills theory.
It is importantly to stress out that the statement about the anisotropy of the physical space made earlier is still valid, since the transformations in which was based the statement did not take into account the hypothesis that relates the two transformation parameters.
Dependence of the gauge transformation parameter in the space-time variables
============================================================================
In genuine gauge theories, it is possible to use an arbitrary gauge to find a differential equation for the gauge transformation parameter. For the case of the ETRG, however, to find one similar to the Lorenz gauge, for example, is not an easy task and what we can do is to use equations that are valid for construction. Consider the absolute parallelism condition [@aldrovandi3]: $$D_{\nu}h^{b}{}_{\rho}=\partial_{\nu}h^{b}{}_{\rho}-\Gamma^{\alpha}_{\rho\nu}h^{b}{}_{\alpha}=0. \label{4.1}$$ If we substitute the transformations (\[3.28\]) in this equation, we get: $$D_{\nu}\nabla^{'}_{\rho}\varepsilon^{b}=0, \label{4.2}$$ in which $\nabla^{'}_\rho$ is the operator defined in (\[3.28\]) and $D_{\nu}$ the usual covariant derivative used in (\[4.1\]). Solving this set of differential equations in a general form, is a dificult task. Fortunately, in the flat limit we have: $$\partial_{\nu}\partial_{\rho}\varepsilon^{b}=0. \label{4.3}$$ The simplified form (\[4.3\]) has the same shape for each index $b$. The equation can be rewritten as follows $$\partial_{\nu}\partial_{\rho}\varepsilon=0, \label{4.4}$$ or raising the first index in order to obtain a wave equation $$\partial^{\rho}\partial_{\rho}\varepsilon=0. \label{4.5}$$ Obviously, the solution of the previous equation is a plane wave. Thus, in the flat limit, the analogy with electromagnetism is completed. It is important to stress out that in Electromagnetism, the transformation parameter has a clear interpretation: a phase that calibrates the wave function [^6] of the source field in internal space. In the case of gravitation, even if we are at the flat limit, this interpretation is lost, once we have four parameters on (\[4.3\]).
Final remarks
=============
Starting from the TEGR Lagrangian contained in the literature, it is not possible to obtain a Hamiltonian formulation for this theory simply following the standard procedure of a Legendre transformation. The main reason for this impossibility is the momenta be associated with the superpotencial and not with the field strength tensor, being this fact, in turn, a consequence of soldering property which requires that the Lagrangian has more than one quadratic term in torsion. It was also shown that the gravitational Gauss’ law is exactly the secondary constraints of the theory as usual occurs in Electromagnetism. Moreover, divergence term necessary to ensure equality between the TEGR and Einstein-Hilbert Lagrangian densities does not influence on the final results of the secondary constraints.
When we act with the first class constraints on the components of the tetrad field we obtained as second kind gauge transformations a similar structure of the Yang-Mills transformations, wich therefore generalizes the transformations of Electromagnetism. The reason for these transformations are similar and not identical to those obtained in Yang-Mills, is due to the fact that the spin connection obtained for the case of TEGR be able to take algebra indices to physical space indices. Again, this characteristic must be associated with soldering property which allow an exchange between objects defined in the physical and internal spaces. Furthermore, we show that there is no isotropy in physical space with respect to gauge transformations, i.e., given a gauge transformation in the internal space, physical space reacts differently in the spatial and temporal components. In fact, for the flat case, in which the algebra index is fixed, we arrive at the transformations analogous to those of Electromagnetism. It is important to note that this analogy had been obtained through the gravitational Maxwell equations [@spaniol19], however, here the analogy was made in a more fundamental level.
By replacing the second kind gauge transformations in the absolute parallelism condition, we have obtained a highly coupled system of 64 differential equations. Luckily, for the flat case, the system is substantially simplified, allowing even to say that the internal space has an isotropic structure, i.e., the solutions are independent of the index that characterizes this space. This new system is easily solved, and its solution allows us to know that the dependence of the gauge transformation parameters in the variables of physical space is periodic, i.e., the solution is a plane wave.
With regard to the Gravitoelectromagnetism, by getting the relation (\[2.11\]), we hope to have contributed to corroborate the definitions (\[2.10\]), since they lead to a relationship between the momenta and GE fields completely analogous to what happens in Electromagnetism and Yang-Mills theory. The definitions (\[2.10\]) allow to rewrite the expression for the gravitational Lorentz force in terms of GE and GM fields, giving us an alternative way to get the gravitomagnetic “drag” so mentioned in the literature [@spaniol20].
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[**Received: Month xx, 200x**]{}
[^1]: A tetrad field $h_{a}=h_{a}{}^{\mu}\partial_{\mu}$ is a linear basis that relates the metric $g$ to the metric of the tangent space $g=g_{a
b}dx^{a}dx^{b}$ by $g_{a b}=g_{\mu \nu}h_{a}{}^{\mu}h_{b}{}^{\nu}$.
[^2]: The torsion is defined by $T^\rho{}_{\mu \nu}
\;\equiv\Gamma^\rho{}_{\nu \mu} \;-\Gamma^\rho{}_{\mu \nu}$. The object $\Gamma^\rho{}_{\nu \mu}$ is the Weitzenbock connection defined by $\Gamma^\rho{}_{\nu \mu} \;\equiv
h_{a}{}^{\rho}\partial_{\mu}h^{a}{}_{\nu}$.
[^3]: Torsion written in the tetrad basis $T^{a}{}_{\mu \nu}=h^{a}{}_{\rho}T^{\rho}{}_{\mu
\nu}=\partial_{\nu}h^{a}{}_{\mu}-\partial_{\mu}h^{a}{}_{\nu}$.
[^4]: The canonical hamiltonian density is given by ${\cal H}_{0} = \Pi_{c}{}^{i}\partial_{0}h^{c}{}_{i}-{\cal L}$.
[^5]: See Ref. [@spaniol19] for the complete expressions.
[^6]: Making use of a quantum term.
|
---
abstract: 'In this paper, we numerically study energy dissipation caused by traffic in the Nagel-Schreckenberg (NaSch) model with open boundary conditions (OBC). Numerical results show that there is a nonvanishing energy dissipation rate $E_d$, and no true free-flow phase exists in the deterministic and nondeterministic NaSch models with OBC. In the deterministic case, there is a critical value of the extinction rate $\beta _{cd}$ below which $E_d$ increases with increasing $\beta $, but above which $E_d$ abruptly decreases in the case of the speed limit $v_{\max }\geqslant 3.$ However, when $v_{\max }\leqslant \ 2,$ no discontiguous change in $E_d$ occurs. In the nondeterministic case, the dissipated energy has two different contributions: one coming from the randomization, and one from the interactions, which is the only reason for dissipating energy in the deterministic case. The relative contributions of the two dissipation mechanisms are presented in the stochastic NaSch model with OBC. Energy dissipation rate $E_d$ is directly related to traffic phase. Theoretical analyses give an agreement with numerical results in three phases (low-density, high-density and maximum current phase) for the case $v_{\max }=1.$'
author:
- Wei Zhang
- Wei Zhang
title: Boundary effects on energy dissipation in a cellular automaton model
---
\[sec:level1\]INTRODUCTION
==========================
In the last decades, traffic problems have attracted much attention of a community of physicists because of the observed nonequilibrium phase transitions and various nonlinear dynamical phenomena. A number of traffic models have been proposed to investigate the dynamical behavior of the traffic flow, including fluid dynamical models, gas-kinetic models, car-following models and cellular automata (CA) models[@1; @2; @3; @4]. These dynamical approaches represented complex physical phenomena of traffic flow among which are hysteresis, synchronization, wide moving jams, and phase transitions, etc. Among these models, the cellular automata approaches can be used very efficiently for computers to perform simulation[@1; @4; @5; @6; @7; @8; @9; @10; @11; @12; @13; @14]. The Nagel-Schreckenberg (NaSch) model is a basic CA models describing one-lane traffic flow[@5]. Based on the NaSch model, many CA models have succeeded in modeling a wide variety of properties of vehicular traffic[@1; @4; @7; @8; @9; @10; @11; @12; @13; @14].
On the other hand, the problems of traffic jams, environmental pollution and energy dissipation caused by traffic have become more and more significant in modern society. Financial damage from traffic due to energy dissipation and environmental pollution is huge every year. In accordance with previously reported results in Refs.\[15\], more than 20% fuel consumption and air pollution is caused by impeded and ”go and stop” traffic. Most recently, the problem of energy dissipation in traffic system has been investigated in the framework of car following model, city traffic model and NaSch model with periodic boundary conditions (PBC), respectively[@16; @17; @18; @19; @20]. And analytical expressions for energy dissipation have also been provided in the nondeterministic NaSch model with PBC in the case of $v_{\max }=1$ and free-flow state[@20]. However, the effects of boundary condition on energy dissipation have not been discussed yet.
The most significant difference between systems with open and periodic boundary conditions is the vehicle density $\rho $. In a periodic system, which has no maximum current phase, vehicle density is considered as an adjustable parameter. In systems with open boundary conditions (OBC), however, there are two adjustable parameters, namely the injection rate $%
\alpha $ and the extinction rate $\beta $, and the vehicle density $\rho $ is only a derived parameter. Compared with periodic systems, it implies that open systems show a different behaviour of quantities such as the global density, the current, the density profile and even the microscopic structure of traffic phase[@21; @22; @23; @24; @25; @26; @27; @28; @29; @30]. Therefore, energy dissipation in the CA model with OBC, which is relevant to many realistic situation in traffic, should be further investigated.
In this paper, we investigate the energy dissipation rate within the framework of the deterministic and nondeterministic NaSch model with OBC. The behaviours of the energy dissipation rate in different traffic phase are distinct. Theoretical analyses are presented in low-density, high-density and maximum current phase in the case of $v_{\max }=1$. The influences of the speed limit $v_{\max }$ on the energy dissipation are also investigated. Energy dissipation caused by braking is related not only to the velocity of vehicles, but also to the headway distribution. Thus, the behaviour of the energy dissipation rate is more complex than simpler quantities, and should be further investigated. The results of this article may lead to a profound understanding of some features of traffic system or may provide schemes for reducing energy dissipation of the existing traffic network.
The paper is organized as follows. Section II is devoted to the description of the model and the definition of energy dissipation rate. In section III, the numerical studies are given, and the influences of the injection and extinction rate on energy dissipation rate are considered. And theoretical analyses are presented in the special case $v_{\max }=1$. Finally, the conclusions are given in section IV.
MODEL AND ENERGY DISSIPATION
============================
Our investigations are based on a one dimensional cellular automaton model introduced by Nagel and Schreckenberg. The model is defined on a single lane road consisting of $L$ cells of equal size numbered by $i=1,$ $2,$ $\cdots ,$ $L$ and the time is discrete. Each site can be either empty or occupied by a car with the speed $v=0,$ $1,$ $2,\cdots $ $,$ $v_{\max }$, where $v_{\max }$ is the speed limit. Let $x(i,t)$ and $v(i,t)$ denote the position and the velocity of the $i$th car at time $t$, respectively. The number of empty cells in front of the $i$th vehicle is denoted by $d(i,t)=x(i+1,t)-x(i,t)-1$. The following four steps for all cars update in parallel with periodic boundary.
\(1) Acceleration:
$v(i,t+1/3)\rightarrow \min [v(i,t)+1,v_{\max }];$
\(2) Slowing down:
$v(i,t+2/3)\rightarrow \min [v(i,t+1/3),d(i,t)];$
\(3) Stochastic braking:
$v(i,t+1)\rightarrow \max [v(i,t+2/3)-1,0]$ with the probability $p;$
\(4) Movement: $x(i,t+1)\rightarrow x(i,t)+v(i,t+1).$
Open systems are characterized by the injection rate $\alpha $ and the extinction rate $\beta $, which means by the probability $\alpha $ and $%
\beta $ that a vehicle moves into and out of the system. In this paper, open boundary conditions are defined according to[@28; @29]. At site $i=0$ which means out of the system, a vehicle with speed $v=v_{\max }$ is created with probability $\alpha
$. The vehicle immediately moves forward in accordance with the NaSch rule. If the site $i=1$ is occupied by a car, the injected vehicle at site $i=0$ is deleted. At $i=L+1$ a ”block” occurs with probability $1-\beta $ and causes a slowing down of the vehicles at the end of the system. Otherwise, a vehicle may leave freely from the end of the system.
It should be mentioned that for $v_{\max }=1$ the model above in the nondeterministic case is different from parallel updated asymmetric exclusion process (ASEP) with open boundary conditions[@21; @22; @23]. In the ASEP model, if the site $L$ is occupied then the particle on that site exits with probability $\beta $, irrespective of their velocity. In the model above, however, the extinction of the vehicle at site $L$ depends not only on the probability $\beta $, but also on the braking probability $p.$ Even if the vehicle with $v=1$ is at site $L$ and there is no ”block” at site $L+1$, it may fail to exit because of being randomly delayed. This difference may influence the current and phase transition, which will be analysed in the following section. In the deterministic case, however, the model above with $v_{\max
}=1$ is identical with parallel updated ASEP with OBC, for there is no stochastic delay.
The kinetic energy of the vehicle with the velocity $v$ is $mv^2/2$, where $%
m $ is the mass of the vehicle. When braking the energy is lost. Let $E_d$ denotes energy dissipation rate per time step per vehicle. For simple, we neglect rolling and air drag dissipation and other dissipation such as the energy needed to keep the motor running while the vehicle is standing in our analysis, i.e., we only consider the energy lost caused by speed-down. The dissipated energy of $i$th vehicle from time $t-1$ to $t$ is defined by[@20]
$$e(i,t)=%
%TCIMACRO{
%\QATOPD\{ . {\frac m2\left[ v^2(i,t-1)-v^2(i,t)\right] \quad \text{for }v(i,t)<v(i,t-1)}{0\qquad \qquad \qquad \qquad \qquad ~~\text{for }v(i,t)\geqslant v(i,t-1).}}
%BeginExpansion
{\frac m2\left[ v^2(i,t-1)-v^2(i,t)\right] \quad \text{for }v(i,t)<v(i,t-1) \atopwithdelims\{. 0\qquad \qquad \qquad \qquad \qquad ~~\text{for }v(i,t)\geqslant v(i,t-1).}%
%EndExpansion
\qquad \left( 1\right)$$ Thus, the energy dissipation rate
$$E_d=\frac 1T\frac 1N\sum_{t=t_0+1}^{t_0+T}\sum_{i=1}^Ne(i,t),\qquad
\left( 2\right)$$ where $N$ is the number of vehicles in the system and $t_0$ is the relaxation time, taken as $t_0=10^5$. In this model, the particles are ”self-driven” and the kinetic energy increases in the acceleration step. In the stationary state, the value of the increased energy while accelerating is equivalent to that of the dissipated energy caused by speed-down, and the kinetic energy is constant in the system. In the simulation, the system size $L=1000$ is selected, and the results are obtained by averaging over 20 initial configurations and $10^4$ time steps after discarding $10^5$ initial transient states.
NUMERICAL RESULTS
=================
Effects of the boundary conditions on energy dissipation in the deterministic case
----------------------------------------------------------------------------------
First, we investigate the influences of the boundary conditions on energy dissipation in the deterministic NaSch model with the maximum velocity $%
v_{\max }=5$. In the deterministic case, the stochastic braking is not considered, i.e., $p=0$. Figure 1 shows the energy dissipation rate $E_d$ as a function of the extinction rate $\beta $ with different values of the injection rate $\alpha $. As shown in Fig. 1, there is a critical value of the extinction rate $\beta _{cd}$ below which $E_d$ increases with the increase of the rate $\beta $, but above which $E_d$ abruptly decreases. The position of $\beta
_{cd}$ shifts towards a high value of $\beta $ with increasing the injection rate $\alpha $. The astonishing result is that there is a nonvanishing energy dissipation, even though in low-density phase, which means that there is no ”true” free-flow phase.
![\[fig:epsart\] Energy dissipation rate $E_d$ (scaled by $m$ ) as a function of the extinction rate $\beta $ in the deterministic NaSch model with $v_{\max }=5$ for various values of the injection rate $\alpha $.](fig_1){height="5cm"}
When $\beta <\beta _{cd},$ the state of the system is high-density phase. As the extinction rate $\beta $ increases, the kinetic energy possessed by vehicles increases because of the increase of the mean vehicle velocity, and the dissipated energy while braking increases. When $\beta >\beta _{cd},$ the state of the system is low-density phase in which the distance-headways is larger and the interaction between vehicles is weaker than that in the high-density phase. The interaction is the only reason for dissipating energy in the deterministic case. Consequently, $E_d$ decreases abruptly when the transition from high-density to low-density phase occurs. Because of boundary effects there are vehicular interactions even at low-density phase. Therefor, there is a nonvanishing energy dissipation rate $E_d$, and no ”true” free-flow phase exists in the deterministic NaSch model with open boundary conditions.
Energy dissipation rate $E_d$ in the case of $v_{\max }=\ 3$ and 4 show similar behaviour to that for $v_{\max }=5,$ and the position of $\beta _{cd} $ shifts towards a low value of $\beta $ with increasing the speed limit $v_{\max },$ as shown in Fig. 2. But in the case of $v_{\max }=2$ and $%
1$, there is no critical value of the extinction rate $\beta _{cd}$, i.e., no discontinuous change in $E_d$ occurs. According to previously reported results in Refs.\[19\], in the case of $v_{\max
}<3$, the state of a system with injection rate $\alpha =1$ is the jamming phase, while in the case of $%
v_{\max }\geqslant \ 3$, the jamming state exists in the region of low value of $\beta $, and low-density lies in the region of high value of $\beta .$ Thus, there is no discontinuous change in $E_d$ in the case of $v_{\max }=2$ and $1$ for no phase transition occurs. Near $\beta =0$, energy dissipation rate $E_d$ is independent of the speed limit $v_{\max }$, and $E_d$ increases linearly with the increase of $\beta $. In the case of $\alpha =1$ and $\beta
\rightarrow 0$, the maximum velocity which vehicles can move is 1, so the speed limit has no influences on energy dissipation.
![\[fig:epsart\] Energy dissipation rate $E_d$ (scaled by $m$ ) as a function of the extinction rate $\beta $ in the deterministic NaSch model in the case of $\alpha =1.0$ for various values of the speed limit $v_{\max }$.](fig_2){height="5cm"}
![\[fig:epsart\] Energy dissipation rate $E_d$ (scaled by $m$ ) as a function of the extinction rate $\beta $ in the deterministic NaSch model with $v_{\max }=1$ for various values of the injection rate $\alpha $. Symbol data are obtained from computer simulations, and solid line corresponds to analytic results of the formula (6).](fig_3){height="5cm"}
In the special case of $v_{\max }=1,$ energy dissipation rate $E_d$ is proportional to the mean density of ”go and stop” vehicles per time step. The mean density of ”go and stop” vehicles is defined in the following way:
$$\rho _{gs}=\frac 1N\frac
1T\sum_{t=t_0+1}^{t_0+T}\sum_{i=1}^Nn(i,t)\left[ 1-n(i,t+1)\right]
,\qquad \qquad (3)$$ where $n(i,t)=0$ for stopped cars and $n(i,t)=1$ for moving cars at time $t$, and $t_0$ is the relaxation time as mentioned in the section II. And energy dissipation rate $E_d$ in the case of $v_{\max }=1$ can be written as $$E_d=\frac 12m\rho _{gs}.\qquad \qquad (4)$$ For open boundary conditions, the mean density of ”go and stop” vehicles per time step reads $$\rho _{gs}=n_0-n_0^2=\beta (1-\beta ),\qquad \qquad (5)$$ where $n_0=N_0/N$ is the fraction of the stopped vehicles, and $N_0$ is the number of stopped vehicles on the road.
As a consequence, energy dissipation rate $E_d$ in the case of $v_{\max }=1$ can be obtained as $$E_d=\frac m2(\beta -\beta ^2).\qquad \qquad (6)$$ In formula (6), energy dissipation rate $E_d$ is directly related to the probability for a vehicle moving out of the system or the probability for a vehicle occupying the last site of the system. As shown in figure 3, theoretical analysis is in good agreement with numerical results.
For $v_{\max }=1$ and $\alpha <1,$ a discontinuous change in $E_d$ also occurs at a critical point $\beta _{cd}$, and the position of $\beta _{cd}$ shifts towards a high value of $\beta $ with increasing $\alpha ,$ which is similar to that in the case of $v_{\max }=5$.
Effects of the boundary conditions on energy dissipation in the nondeterministic case
-------------------------------------------------------------------------------------
Next, we investigate the rate of energy dissipation $E_d$ when the stochastic braking behaviours of drivers are considered, i.e., $p\neq 0$. In the nondeterministic case, the dissipated energy has two different contributions: one coming from the stochastic noise and one from the interactions between vehicles (this in fact is the only reason for dissipating energy in the deterministic case). Let $E_{di}$ and $E_{dr}$ denote the rate of energy dissipation caused by the interactions and randomizations, respectively. And energy dissipation rate $E_d=E_{di}+E_{dr}. $ Figure 4 and 5 show the relation of the energy dissipation rate $E_{di}$ and $E_{dr}$ to the extinction rate $\beta $ with different values of the injection rate $\alpha $, respectively, in the case of $v_{\max }=5$ and p$%
=0.5.$ As shown in figure 4, there is a critical value of the extinction rate $\beta _{cr}$ below which $E_{di}$ increases with increasing $\beta $, but above which $E_{di}$ abruptly decreases, except for the case of $\alpha =1.$ Different from $E_{di}$, energy dissipation rate $E_{dr}$ sharp increases at the critical point $\beta _{cr}$ above which $E_{dr}$ shows the approximate plateau and is independent of $\beta ,$ except for $\alpha =1,$ as shown in figure 5. And the position of $\beta _{cr}$ shifts towards a high value of $\beta $ with the increase of the injection rate $\alpha $. For $\alpha =1,$ there is no discontinuous change in $E_{di}$ and $E_{dr}$. Compared figure 4 with 5, it turns out that energy dissipation is mainly caused by the randomization in the low-density phase, and by the interactions in the high-density phase. When $\alpha >0.35,$ the values of the rate of energy dissipation $E_{di}$ and $E_{dr}$ for various value of $%
\alpha $ collapse into a single curve (not shown).
![\[fig:epsart\] Energy dissipation rate $E_{di}$ (scaled by $m$ ) as a function of the extinction rate $\beta $ in the non-deterministic NaSch model with $v_{\max }=5$ and $p=0.5$ for various values of the injection rate $\alpha $.](fig_4){height="5cm"}
![\[fig:epsart\] Energy dissipation rate $E_{dr}$ (scaled by $m$ ) as a function of the extinction rate $\beta $ in the non-deterministic NaSch model with $v_{\max }=5$ and $p=0.5$ for various values of the injection rate $\alpha $.](fig_5){height="5cm"}
In fact, in the case of p$=0.5$ and $v_{\max }=5,$ there are the transitions from the high-density to low-density phase for $\alpha
\leqslant 0.35$ and from high-density to maximum current phase for $\alpha >0.35$[@29]. In the high-density phase, with the increase of the extinction rate $\beta $, the mean velocity increases and the dissipated energy increases. In the low-density and maximum current phase, with increasing the rate $\beta $, the distance headways increases and the interactions lowers; thus energy dissipation rate $E_{di}$ decreases. Energy dissipation rate $E_{dr}$ reaches a constant value in the low-density phase, but continuatively increases with the increase of $\beta $ in the maximum current phase.
It should be noted that energy dissipation rate $E_d$ for $\alpha
=1$ is minimum, i.e., energy dissipation in the maximum current phase is lower than that in the low-density phase (not shown). Traffic flow, however, in the maximum current phase is maximal.
The relationship of $E_d$ to the extinction rate $\beta $ for different values of $v_{\max }$ in the case of $\alpha =1$ is shown in figure 6. Similar to the deterministic case, when the value of $\beta $ is very small, the rate of energy dissipation shows a scaling relation and is independent of the speed limit, as shown in figure 6. In the region of high values of $%
\beta $, the scaling relations of energy dissipation $E_d$ to the speed limit cannot be observed; and $E_d$ increases with the increase of $v_{\max } $. When $v_{\max }>3$, energy dissipation rate $E_d$ increases with increasing the rate $\beta $ in the region of high values of $\beta $. However, when $v_{\max }\leqslant \ 3$, the value of $E_d$ tends to be invariable. Though the states of the system for different speed limit are maximum current phase in the region of high values of $\beta $, with the increase of $\beta $, the mean velocity increases for $v_{\max }>3$ but does not vary in the case of $v_{\max }\leqslant \ 3$. Consequently, there is a plateau for the case $v_{\max }\leqslant \ 3$ in the region of high values of $\beta $, which is different from that in the case of $v_{\max }>3$.
![\[fig:epsart\] Energy dissipation rate $E_d$ (scaled by $m$ ) as a function of the extinction rate $\beta $ in the non-deterministic NaSch model with $\alpha =1.0$ and $p=0.5$ for various values of the speed limit $v_{\max }$.](fig_6){height="5cm"}
In the system with $v_{\max }=1$, some vehicles can be stopped due to the stochastic braking, therefore the rate of energy dissipation $E_d$ is proportional to the mean ”go and stop” density $\rho
_{gs},$ which demonstrates that the probability for ”go and stop” vehicle to appear per time step in the system.
In the low-density phase A ($\alpha <\beta ,$ $\alpha <\alpha _c=1-\sqrt{p}%
), $ the fraction of the stopped vehicles reads
$$n_0=1-\frac{q-\alpha }{1-\alpha },\qquad \qquad (7)$$ where $q=1-p.$ And the mean ”go and stop” density $\rho _{gs}$ can be obtain as $$\rho _{gs}=n_0-n_0^2=\frac{q-\alpha }{1-\alpha }\left( 1-\frac{q-\alpha }{%
1-\alpha }\right) .\qquad \qquad (8)$$ Substituting formula (8) into (4), we can obtain energy dissipation rate $%
E_d $ in the low-density phase A
$$E_d^A=\frac{m(q-\alpha )(1-q)}{2(1-\alpha )^2}.\qquad \qquad (9)$$
Figure 7 shows the relation between the rate of energy dissipation $E_d$ and the injection rate $\alpha $ with various values of $p$, in the case of $%
\beta =1.$ As shown in Fig. 7 in the region of low values of $\alpha
$, formula (9) gives good agreement with the simulation data.
![\[fig:epsart\] Energy dissipation rate $E_d$ (scaled by $m$ ) as a function of the injection rate $\alpha $ in the non-deterministic NaSch model in the case of $v_{\max }=1$ and $\beta =1.0$ for various values of the stochastic braking probability $p$. Symbol data are obtained from computer simulations, and solid line corresponds to analytic results of the formula (9) and (15).](fig_7){height="5cm"}
![\[fig:epsart\] Energy dissipation rate $E_d$ (scaled by $m$ ) as a function of the extinction rate $\beta $ in the non-deterministic NaSch model in the case of $v_{\max }=1$ and $\alpha =1.0$ for various values of the stochastic braking probability $p$. Symbol data are obtained from computer simulations, and solid line corresponds to analytic results of the formula (12) and (15).](fig_8){height="5cm"}
From $\frac{dE_d^A}{d\alpha }=0,$ we obtain the critical value $\alpha _{mc}=1-2p$ in which the curve described by formula (9) reaches the maximum. Compared with the critical rate $\alpha
_c=1-\sqrt{p},$ the critical value of the stochastic braking probability $p_c^L=\frac 14$ can be obtained. When $p<p_c^L,$ energy dissipation rate $E_d^A$ increases with increasing the injection rate $\alpha .$ When $p\geqslant 0.5,$ however, with the increase of the rate $\alpha ,$ energy dissipation rate $E_d^A$ decreases. In the interval $0.5>p\geqslant p_c^L,$ with increasing $\alpha ,$ the rate of energy dissipation $E_d^A$ increases first and decreases after a maximum value is reached.
In the high-density phase B ($\beta <\alpha ,$ $\beta <\beta _c=\frac 1{1+%
\sqrt{p}}$), the model of this paper is different from the ASEP with parallel update and the mean velocity is determined not only by the extinction rate $\beta ,$ but also the stochastic braking probability $p.$ The fraction of the stopped vehicles reads
$$n_0=1-q\beta .\qquad \qquad (10)$$ And the mean ”go and stop” density $\rho _{gs}$ can be written as $$\rho _{gs}=n_0-n_0^2=q\beta (1-q\beta ).\qquad \qquad (11)$$ Substituting formula (11) into (4), we can obtain the rate of energy dissipation $E_d$ in the high-density phase B
$$E_d^B=\frac m2(q\beta -q^2\beta ^2).\qquad \qquad (12)$$
Figure 8 shows the relation the energy dissipation rate $E_d$ as a function of the extinction rate $\beta $ with different values of the stochastic braking probability $p,$ in the case of $\alpha =1.$ As shown in Fig. 8, the agreement can be obtained in the case of low value of the extinction rate $%
\beta .$
From $\frac{dE_d^B}{d\alpha }=0,$ we can obtain the critical value $\beta _{mc}=\frac 1{2(1-p)}$ in which the curve corresponding to formula (12) reaches the maximum. Compared with the critical rate $\beta _c=\frac 1{1+%
\sqrt{p}},$ the critical value of the stochastic braking probability $p_c^h=%
\frac 14$ can be obtained. When $p\geqslant p_c^h,$ energy dissipation rate $%
E_d^B$ increases with the increase of the extinction rate $\beta .$ When $%
p<p_c^h,$ however, with increasing the rate $\beta ,$ energy dissipation rate $E_d^B$ increases first and decreases after a maximum value is reached.
In the maximum current phase C ($\alpha >\alpha _c,\beta >\beta
_c$), the fraction of the stopped vehicles reads
$$n_0=\sqrt{p}.\qquad \qquad (13)$$ And the mean ”go and stop” density $\rho _{gs}$ can be obtain as
$$\rho _{gs}=n_0-n_0^2=\sqrt{p}(1-\sqrt{p}).\qquad \qquad (14)$$ Substituting formula (14) into (4), we can obtain energy dissipation rate $%
E_d$ in the maximum current phase C
$$E_d^C=\frac m2(\sqrt{p}-p).\qquad \qquad (15)$$ Equation (15) demonstrates that energy dissipation in the maximum current phase is independent of the rate $\alpha $ and $\beta ,$ and is only determined by the stochastic braking probability. Figure 7 and 8 give a comparison between number results and Eq. 15. As shown in the right region of figure 7 and 8, formula (15) gives good agreement with the simulation data.
SUMMARY
=======
In this paper, we investigate the rate of energy dissipation caused by braking in the NaSch model with open boundary conditions. Different from periodic systems, open systems in which the vehicle density is only a derived parameter are controlled by the injection and extinction rate. In fact, real traffic systems are usually open, hence it is highly desirable to investigate energy dissipation in traffic systems both numerically and theoretically.
Numerical results show that in the deterministic case there is a critical value of the extinction rate $\beta _{cd}$ above which $E_d$ decreases abruptly for $v_{\max }\geqslant 3,$ however, no discontiguous change in $%
E_d $ occurs when $v_{\max }<3$. The rate $\beta _{cd}$ is related not only to the injection rate $\alpha ,$ but also to the maximum velocity of vehicles. In the nondeterministic case, there is also a critical value of the extinction rate $\beta _{cr}$ below which $E_{di}$ and $E_{dr}$ increase with increasing $\beta ,$ above which $E_{di}$ abruptly decreases but $%
E_{dr} $ sharp increases and shows the approximate plateau with further increase of $\beta $, when the transition from the high-density to low-density phase occurs. However, when the transition from the high-density to the maximum current phase occurs, the values of energy dissipation rate $%
E_{di}$ and $E_{dr}$ for various value of $\alpha $ collapse into a single curve, and no discontinuous change occurs. Moreover$,$ there is a nonvanishing energy dissipation rate, and no ”true” free-flow phase exists in the deterministic and nondeterministic NaSch models with open boundary conditions.
Energy dissipation rate $E_d$ is directly related to traffic phase. Energy dissipation in maximum current phase is smaller than that in the low-density phase. A phenomenological mean-field theory is presented to describe the energy dissipation rate $E_d$ in three phases (low-density, high-density and maximum current phase) in the case of $v_{\max }=1$. Theoretical analyses give an excellent agreement with numerical results. But in the case of $%
v_{\max }>1$, explicit expressions about the energy dissipation rate $E_d$ do not be obtained because of effects of long length of time space correlations, and deserve further investigate.
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abstract: |
The class of generalized Petersen graphs was introduced by Coxeter in the 1950s. Frucht, Graver and Watkins determined the automorphism groups of generalized Petersen graphs in 1971, and much later, Nedela and Škoviera and (independently) Lovrečič-Saražin characterised those which are Cayley graphs. In this paper we extend the class of generalized Petersen graphs to a class of [*$GI$-graphs*]{}. For any positive integer $n$ and any sequence $j_0,j_1,....,j_{t-1}$ of integers mod $n$, the $GI$-graph $GI(n;j_0,j_1,....,j_{t-1})$ is a $(t\!+\!1)$-valent graph on the vertex set ${\mathbb{Z}}_t \times {\mathbb{Z}}_n$, with edges of two kinds:
- an edge from $(s,v)$ to $(s',v)$, for all distinct $s,s' \in {\mathbb{Z}}_{t}$ and all $v \in {\mathbb{Z}}_n$,
- edges from $(s,v)$ to $(s,v + j_s)$ and $(s,v - j_s)$, for all $s \in {\mathbb{Z}}_{t}$ and $v \in {\mathbb{Z}}_n$.
By classifying different kinds of automorphisms, we describe the automorphism group of each $GI$-graph, and determine which $GI$-graphs are vertex-transitive and which are Cayley graphs. A $GI$-graph can be edge-transitive only when $t \leq 3$, or equivalently, for valence at most $4$. We present a unit-distance drawing of a remarkable $GI(7;1,2,3)$.
[**Keywords**]{}: $GI$-graph, generalized Petersen graph, vertex-transitive graph, edge-transitive graph, circulant graph, automorphism group, wreath product, unit-distance graph.\
[**Mathematics Subject Classification (2010)**]{}: 20B25, 05E18, 05C75.
author:
- |
Marston D.E. Conder\
[Department of Mathematics, University of Auckland,]{}\
[Private Bag 92019, Auckland 1142, New Zealand]{}\
[[email protected]]{}\
- |
Tomaž Pisanski\
[Faculty of Mathematics and Physics, University of Ljubljana,]{}\
[Jadranska 19, 1000 Ljubljana, Slovenia]{}\
[[email protected]]{}\
- |
Arjana Žitnik\
[Faculty of Mathematics and Physics, University of Ljubljana,]{}\
[Jadranska 19, 1000 Ljubljana, Slovenia]{}\
[[email protected]]{}
date: 10 July 2012
title: '$GI$-graphs and their groups\'
---
Introduction {#intro}
============
Trivalent graphs (also known as cubic graphs) form an extensively studied class of graphs. Among them, the Petersen graph is one of the most important finite graphs, constructible in many ways, and is a minimal counter-example for many conjectures in graph theory. The Petersen graph is the initial member of a family of graphs $G(n,k)$, known today as [*Generalized Petersen graphs*]{}, which have similar constructions. Generalized Petersen graphs were first introduced by Coxeter [@Coxeter] in 1950, and were named in 1969 by Watkins [@Watkins].
A standard visualization of a generalized Petersen graph consists of two types of vertices: half of them belong to an outer rim, and the other half belong to an inner rim; and there are three types of edges: those in the outer rim, those in the inner rim, and the ‘spokes’, which form a $1$-factor between the inner rim and the outer rim. The outer rim is always a cycle, while the inner rim may consist of several isomorphic cycles. A generalized Petersen graph $G(n,k)$ is given by two parameters $n$ and $k$, where $n$ is the number of vertices in each rim, and $k$ is the ‘span’ of the inner rim (which is the distance on the outer rim between the neighbours of two adjacent vertices on the inner rim).
The family $G(n,k)$ contains some very important graphs. Among others of particular interest are the $n$-prism $G(n,1)$, the Dürer graph $G(6,2)$, the Möbius-Kantor graph $G(8,3)$, the dodecahedron $G(10,2)$, the Desargues graph $G(10,3)$, the Nauru graph $G(12,5)$, and of course the Petersen graph itself, which is $G(5,2)$.
Generalized Petersen graphs possess a number of interesting properties. For example, $G(n,k)$ is vertex-transitive if and only if either $n = 10$ and $k = 2$, or $k^2 \equiv \pm 1$ mod $n\,$ [@Frucht], and a Cayley graph if and only if $k^2 \equiv 1$ mod $n\,$ [@Lovrecic1; @NedelaSkoviera], and arc-transitive only in the following seven cases: $(n,k) = (4,1)$, $(5,2)$, $(8,3)$, $(10, 2)$, $(10, 3)$, $(12, 5)$ or $(24, 5)\,$ [@Frucht].
If we want to maintain the symmetry between the two rims, then another parameter has to be introduced, allowing the span on the outer rim to be different from 1. This gives the definition of an [*$I$-graph*]{}.
The family of $I$-graphs was introduced in 1988 in the Foster Census [@Foster]. For some time this family failed to attract the attention of many researchers, possibly due to the fact that among all $I$-graphs, the only ones that are vertex-transitive are the generalized Petersen graphs [@igraphs; @LovrecicMarusic]. Still, necessary and sufficient conditions for testing whether or not two $I$-graphs are isomorphic were determined in [@igraphs; @HPZ2], and these were used to enumerate all $I$-graphs in [@Petkovsek]. Also in [@HPZ2] it was shown that all generalized Petersen graphs are unit-distance graphs, by representing them as isomorphic $I$-graphs. Furthermore, in [@igraphs] it was shown that automorphism group of a connected $I$-graph $I(n,j,k)$ that is not a generalized Petersen graph is either dihedral or a group with presentation $$ \Gamma = \langle\, \rho, \tau, \varphi \mid
\rho^n = \tau^2 = \varphi^2 = 1, \,
\rho\tau\rho = \tau, \,
\varphi\tau\varphi = \tau, \,
\varphi\rho\varphi = \rho^a \, \rangle$$ for some $a \in {\mathbb{Z}}_n$, and that among all $I$-graphs, only the generalized Petersen graphs can be vertex-transitive or edge-transitive.
In this paper we further generalize both of these families of graphs, and call them *generalized I-graphs*, or simply [*$GI$-graphs*]{}. We determine the group of automorphisms of any $GI$-graph. Moreover, we completely characterize the edge-transitive, vertex-transitive and Cayley graphs, among the class of $GI$-graphs.
At the end of the paper we briefly discuss the problem of unit-distance realizations of $GI$-graphs. This problem has been solved for $I$-graphs in [@HPZ1]. We found a remarkable new example of a 4-valent unit-distance graph, namely $GI(7;1,2,3)$, which is a Cayley graph on 21 vertices for the group ${\mathbb{Z}}_7 \rtimes {\mathbb{Z}}_3$. Let us note that ours is not the only possible generalization. For instance, see [@Lovrecic] for another approach, which is not much different from ours. The basic difference is that our approach uses complete graphs, while the approach by Lovrečič-Saražin, Pacco and Previtali in [@Lovrecic] uses cycles; their construction coincides with ours for $t \le 3$, but not for larger $t$.
We acknowledge the use of [Magma]{} [@Magma] in constructing and analysing examples of $GI$-graphs, and helping us to see patterns and test conjectures that led to many of the observations made and proved in this paper.
Definition of $GI$-graphs and their properties {#defn-props}
==============================================
For positive integers $n$ and $t$ with $n \ge 3$, let $(j_0,j_1, \dots , j_{t-1})$ be any sequence of integers such that $0 < j_k < n$ and $j_k \ne n/2$, for $0 \le k < t$.\
Then we define $GI(n;j_0,j_1, \dots , j_{t-1})$ to be the graph with vertex set ${\mathbb{Z}}_t \times {\mathbb{Z}}_n$, and with edges of two types:\
----------------------- ------------------------------------------------------------------------------------------------------------------------
(a)-7pt ${}$ an edge from $(s,v)$ to $(s',v)$, for all distinct $s,s' \in {\mathbb{Z}}_{t}$ and all $v \in {\mathbb{Z}}_n$,
\[+2pt\] (b)-7pt ${}$ edges from $(s,v)$ to $(s,v + j_s)$ and $(s,v - j_s)$, for all $s\in {\mathbb{Z}}_{t}$ and all $v \in {\mathbb{Z}}_n$.
\[+4pt\]
----------------------- ------------------------------------------------------------------------------------------------------------------------
This definition gives us an infinite family of graphs, which we call *GI-graphs*.
The graph $GI(n;j_0,j_1, \dots , j_{t-1})$ has $nt$ vertices, and is regular of valence $(t-1)+2 = t+1.$ Edges of type (a) are called the *spoke edges*, while those of type (b) are called the *layer edges*. Also for each $s \in {\mathbb{Z}}_t$ the set $L_s = \{(s,v) : v \in {\mathbb{Z}}_n\}$ is called a [*layer*]{}, and for each $v \in {\mathbb{Z}}_n$ the set $S_v = \{(s,v) : s \in {\mathbb{Z}}_t\}$ is called a [*spoke*]{}. We observe that the induced subgraph on each spoke is a complete graph $K_t$ of order $t$. On the other hand, the induced subgraph on the layer $L_s$ is a union of $d$ cycles of length $n/d_s$, where $d_s =\gcd(n,j_s)$.\
In the case $t=1$, the graph $GI(n;j_0)$ is simply a union of disjoint isomorphic cycles of length $n/\gcd(n,j_0)$. In the case $ t = 2$, we have $I$-graphs; for example, $GI(n;1,j)$ is a generalized Petersen graph, for every $j$, and in particular, $GI(5;1,2)$ is the dodecahedral graph (the $1$-skeleton of a dodecahedron). Some other examples are illustrated in Figure \[fig:examplesGIgraphs\].
![$GI$-graphs $GI(6;2,2)$, $GI(6;1,1,2)$, and $GI(6;2,1,2)$.[]{data-label="fig:examplesGIgraphs"}](GI6_2_2.pdf "fig:"){width="90.00000%"}\
(a)
![$GI$-graphs $GI(6;2,2)$, $GI(6;1,1,2)$, and $GI(6;2,1,2)$.[]{data-label="fig:examplesGIgraphs"}](GI6_1_1_2.pdf "fig:"){width="90.00000%"}\
(b)
![$GI$-graphs $GI(6;2,2)$, $GI(6;1,1,2)$, and $GI(6;2,1,2)$.[]{data-label="fig:examplesGIgraphs"}](GI6_2_1_2.pdf "fig:"){width="90.00000%"}\
(c)
Note that taking $j_k^\prime = \pm j_k$ for all $k$ gives a $GI$-graph $GI(n;j_0^\prime,j_1^\prime, \dots , j_{t-1}^\prime)$ that is exactly the same as $GI(n;j_0,j_1, \dots , j_{t-1})$. Similarly, any permutation of $j_0, j_1, \dots, j_{t-1}$ gives a $GI$-graph isomorphic to $GI(n;j_0,j_1, \dots , j_{t-1})$. Therefore we will usually assume that $0 < j_k < n/2$ for all $k$, and that $j_0 \le j_1 \le \dots \le j_{t-1}$. In this case, we say that the $GI$-graph $GI(n;j_0,j_1, \dots , j_{t-1})$ is in *standard form*.
The following theorem gives a partial answer to the problem of distinguishing between two $GI$-graphs.
\[thm:multiply\] Suppose $j_0,j_1,\dots,j_{t-1} \ne 0$ or $n/2 $ modulo $n$, and $a$ is a unit in ${\mathbb{Z}}_n$. Then the graph $GI(n;aj_0, aj_1, \dots, aj_{t-1})$ is isomorphic to the graph $GI(n;j_0,j_1, \dots, j_{t-1})$.
Since $a$ is coprime to $n$, the numbers $av$ for $0 \le v < n$, are all distinct in ${\mathbb{Z}}_n$, and so we can label the vertices of $GI(n;aj_0, aj_1, \dots, aj_{t-1})$ as ordered pairs $(s,av)$ for $s \in {\mathbb{Z}}_t$ and $v \in {\mathbb{Z}}_n$. Now define a mapping $\varphi : V(GI(n;aj_0,aj_1, \dots, aj_{t-1})) \to V(GI(n;j_0,j_1, \dots, j_{t-1}))$ by setting $ \varphi((s,av) ) = (s,v)$ for all $s \in {\mathbb{Z}}_t$ and all $v \in {\mathbb{Z}}_n$. This is clearly a bijection, and since a vertex $(s,av)$ in $GI(n;aj_0, aj_1, \dots, aj_{t-1})$ is adjacent to $(s',av)$ for each $s' \in {\mathbb{Z}}_t \setminus \{s\}$ and to $(s,av \pm aj_s) = (s,a(v \pm j_s))$, it is easy to see that $\varphi$ is also a graph homomorphism.
We may collect the parameters $j_s$ into a multiset $J$, and then use the abbreviation $GI(n; J)$ for the graph $GI(n;j_0,j_1, \dots, j_{t-1})$. Also we will say that the multiset $J$ is in *canonical form* if it is lexicographically first among all the multisets that give isomorphic copies of $GI(n; J)$ via Proposition \[thm:multiply\].
We now list some other properties of $GI$-graphs. Proposition \[thm:spokes\] shows that the spoke edges are easy to recognise when $t > 3$.
\[thm:factors\] The graph $GI(n;j_0,j_1, \dots , j_{t-1})$ admits a factorization into a $(t-1)$-factor $nK_t$ and a $2$-factor $($namely the spokes and the layers$)$.
\[thm:spokes\] An edge of a $GI$-graph with $4$ or more layers is a spoke-edge if and only if it belongs to some clique of size $4$.
No edge between two vertices in the same layer can lie in a $K_4$ subgraph, because the subgraph induced on each layer is a union of cycles, and no two spokes between two different layers can have a common vertex.
\[thm:connectedGI\] Let $d=\gcd(n,j_0,j_1,\dots, j_{t-1})$. Then the graph $GI(n;j_0,j_1, \dots , j_{t-1})$ is a disjoint union of $d$ copies of $GI(n/d;j_0/d,j_1/d, \dots , j_{t-1}/d)$. In particular, the graph $GI(n;j_0,j_1, \dots , j_{t-1})$ is connected if and only if $d = 1$.
First observe that the edges of every spoke make up a clique (of order $t$), so the graph is connected if and only if every two spokes are connected via the layer edges. Now there exists an edge between two spokes $S_u$ and $S_v$ whenever $v-u$ is a multiple of $j_s$ for some $s$, and hence a path of length $2$ between $S_u$ and $S_v$ whenever $v-u$ is a ${\mathbb{Z}}_n$-linear combination of some $j_s$ and $j_{s'}$, and so on. Thus $S_u$ and $S_v$ lie in the same connected component of the graph if and only if $v-u$ is expressible (mod $n$) as a ${\mathbb{Z}}$-linear combination of $j_0,j_1,\dots, j_{t-1}$, say $v-u = cn + c_{0}j_{0} + c_{1}j_{1} + \dots + c_{t-1} j_{t-1}$ for some $c_0,c_1,\dots, c_{t-1} \in {\mathbb{Z}}$. By Bezout’s identity, this occurs if and only if $v-u$ is a multiple of $\gcd(n,j_0,j_1,\dots, j_{t-1}) = d$. It follows that the graph has $d$ components, each containing a set of spokes $S_v$ with $v = u+jd$ for fixed $u$ and variable $j$, that, is, with subscripts differing by multiples of $d$. Finally, since $(v-u)/d = c(n/d)+ c_{0}(j_{0}/d) + c_{1}(j_{1}/d) + \dots + c_{t-1}(j_{t-1}/d)$, it is easy to see that each component is isomorphic to $GI(n/d;j_0/d,j_1/d, \dots , j_{t-1}/d)$.
Finally, note that the restriction of a $GI$-graph to any proper subset of its layers gives rise to another $GI$-graph. In particular, if $J$ and $K$ are multisets with $J \subseteq K$, then $GI(n;J)$ is an induced subgraph of $GI(n;K)$.
Automorphisms of $GI$-graphs {#automs}
============================
In this section, we consider the possible automorphims of a $GI$-graph $X=GI(n;J)$, where $J =\{j_0,j_1, \dots , j_{t-1}\}$ is any multiset. If $X$ is disconnected, then since all connected components of $X$ are isomorphic to each other (by Proposition \[thm:connectedGI\]), we may simply reduce this to the consideration of automorphisms a connected component of $X$ (and then find the automorphism group using a theorem of Frucht [@Frucht0], cf. [@Harary]). Hence from now on, we will assume that $X$ is connected.
The set of edges of $X=GI(n;J)$ may be partitioned into spoke edges and layer edges, and we will call this partition of edges the *fundamental edge-partition* of $X$. We know that the graph induced on the spoke edges is a collection of complete graphs, and that the graph induced on the layer edges is a collection of cycles (with each cycle belonging to a single layer, but with a layer being composed of two or more cycles of the same length $n/\gcd(n,j_s)$ if the corresponding element $j_s$ of $J$ is not a unit mod $n$).
We will say that an automorphism of $X$ *respects the fundamental edge-partition* if it takes spoke edges to spoke edges, and layer edges to layer edges. Any automorphism of $X$ that does not respect the fundamental edge-partition (and so takes some layer edge to a spoke edge, and some spoke edge to a layer edge) will be called *skew*.
\[theorem:skewautom\] Let $X$ be a connected $GI$-graph with $t$ layers, where $t \ge 2$. If $X$ has a skew automorphism, then either $\,t = 2$ and $X$ is isomorphic to one of the seven special generalized Petersen graphs $G(4,1)$, $G(5,2)$, $G(8,3)$, $G(10, 2)$, $G(10, 3)$, $G(12, 5)$ and $G(24, 5)$, or $\,t = 3$ and $X$ is isomorphic to $GI(3;1,1,1)$. Moreover, each of these eight graphs is arc-transitive $($and is therefore both vertex-transitive and edge-transitive$)$.
First, if $t > 3$ then no layer edge lies in a clique of size $t$, but every spoke edge does, and therefore no automorphism can map a spoke edge to a layer edge. Thus $t \le 3$.
Next, suppose $t = 3$, and let $\varphi$ be an automorphism taking an edge $e$ of some spoke $S_v$ to an edge $e'$ of some layer $L_s$. Since every edge of a spoke lies in a triangle, namely the spoke itself, it follows that $\varphi$ must take the whole spoke $S_v = \{(0,v),(1,v),(2,v)\}$ containing $e$ to some triangle containing the layer edge $e'$, and then the other two edges of the triangle $\{\varphi(0,v),\varphi(1,v),\varphi(2,v)\}$ must be edges from the same layer as $e'$, namely $L_s$. It follows that $j_s = n/3$. But then since each of the images $\varphi(0,v),\varphi(1,v),\varphi(2,v)$ lies in two triangles (namely a spoke and a triangle in $L_s$), each of the vertices $(0,v),(1,v)$ and $(2,v)$ must similarly lie in two triangles, and it follows that all three layers contain a triangle, so $j_0 = j_1 = j_2 = n/3$. In particular, $\gcd(j_0,j_1,j_2) = n/3$, and by connectedness, Proposition \[thm:connectedGI\] implies $n/3 = 1$, so $n = 3$ and $j_0 = j_1 = j_2 = 1$. Thus $X$ is $GI(3;1,1,1)$, which is well-known to be arc-transitive (see [@Lovrecic], for example).
Finally, for the case $t =2$, everything we need was proved in [@Frucht] and [@igraphs].
\[corollary:edgetransitive\] Every edge-transitive connected $GI$-graph is isomorphic to one of the eight graphs listed in Theorem [*\[theorem:skewautom\]*]{}.
Hence from now on, we will consider only the automorphisms that respect the fundamental edge-partition. There are three special classes of such automorphisms:\
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(1) ${}$-8pt automorphisms that preserve every layer
\[+2pt\] (2) ${}$-8pt automorphisms that preserve every spoke
\[+2pt\] (3) ${}$-8pt automorphisms that permute both the layers and the spokes non-trivially.
\[+6pt\]
-------------- -----------------------------------------------------------------------------------
\
We will consider particular cases of automorphisms of these types below.
Define mappings $\rho:V(X) \to V(X)$ and $\tau: V(X) \to V(X)$ given by $$ \rho(s,v) = (s,v+1)
\ \ \
\mbox{and}
\ \ \
\tau(s,v) = (s,-v) \quad \hbox{ for all } s \in {\mathbb{Z}}_t \hbox{ and all } v \in {\mathbb{Z}}_n. \tag{$\dagger$}
$$ Clearly these are automorphisms of $X$ of type (1), permuting the vertices in each layer. Indeed $\rho$ can be viewed as a rotation (of order $n$), and $\tau$ as a reflection (of order $2$), and it follows that the automorphism group of $X$ contains a dihedral subgroup of order $2n$, generated by $\rho$ and $\tau$. These $2n$ automorphisms are all of type (1), and all of them respect the fundamental edge-partition of $X$.
Next, if two of the members of the multiset $J$ are equal, say $j_{s_1}=j_{s_2}$ for $s_1 \ne s_2$, then we have an automorphism $\lambda_{i,s_1,s_2}$ that exchanges two cycles of layers $L_{s_1}$ and $L_{s_2}$, but preserves every spoke. These automorphisms are of type (2).
\[propn:mix\_layers\] Suppose $j_{s_1}=j_{s_2}$ where $s_1 \ne s_2$, and define $d=\gcd(n,j_{s_1}) = \gcd(n,j_{s_2})$. Then for each $i \in {\mathbb{Z}}_d$, the mapping $\lambda_{i,s_1,s_2}:V(X) \to V(X)$ given by $$\lambda_{i,s_1,s_2}(s,v)=\left\{ \begin{array}{ll}
(s_2,v) & \mbox{if} \ \ s=s_1 \ \ \mbox{and} \ \ v \equiv i \ \ \mbox{\rm mod} \ d \\
(s_1,v) & \mbox{if} \ \ s=s_2 \ \ \mbox{and} \ \ v \equiv i \ \ \mbox{\rm mod} \ d \\
\,(s,v) & \mbox{otherwise}
\end{array} \right.$$ is an automorphism of $X$, which respects the fundamental edge-partition, and preserves all layers other than $L_{s_1}$ and $L_{s_2}$.
This is obviously a permutation of $V(X)$, preserving adjacency. Moreover, it is also clear that $\lambda_{i,s_1,s_2}$ preserves every spoke $S_v$, and exchanges one of the cycles in layer $L_{s_1}$ with the corresponding cycle in layer $L_{s_2}$, while preserving all other layer cycles.
\[cor:exchange\_layers\] Suppose $j_{s_1}=j_{s_2}$ where $s_1 \ne s_2$, and define $d=\gcd(n,j_{s_1}) = \gcd(n,j_{s_2})$.\
Then the product $$\lambda_{s_1,s_2} := \lambda_{0,s_1,s_2}\lambda_{1,s_1,s_2}\dots \lambda_{d-1,s_1,s_2}$$ is an automorphism of $X$ that respects the fundamental edge-partition, and exchanges layers $L_{s_1}$ and $L_{s_2}$, while preserving every other layer.
There is another family of automorphisms exchanging layers that exist in some situations; but these automorphism do not preserve spokes, and so they are of type (3):
\[propn:change\_layers\] Let $a$ be any unit in ${\mathbb{Z}}_n$ with the property that $aJ =\{\pm j_0, \pm j_1, \dots, \pm j_{t-1}\}$, and then let $\alpha: {\mathbb{Z}}_t \to {\mathbb{Z}}_t$ be any bijection with the property that $j_{\alpha(s)} = \pm a j_s$ for all $s \in {\mathbb{Z}}_t$.\
Then the mapping $\sigma_a: V(X) \to V(X)$ given by $$\sigma_a(s,v) = (\alpha(s),a v) \ \ \hbox{ for all } s \in {\mathbb{Z}}_t \, \hbox{ and all } v \in {\mathbb{Z}}_n$$ is an automorphism of $X$ that respects the fundamental edge-partition.
Note that the mapping $\alpha$ is not uniquely determined if there exist distinct $s_1$ and $s_2$ for which $j_{s_1}= \pm j_{s_2}$, but we can always define the mapping $\alpha$ so that it is a bijection (and satisfies $j_{\alpha(s)} = \pm a j_s$ for all $s \in {\mathbb{Z}}_t$). Indeed $\alpha$ is uniquely determined if we require that $\alpha(s_1) < \alpha(s_2)$ whenever $s_1 < s_2$ and $j_{s_1}= \pm j_{s_2}$. On the other hand, $\sigma_a$ is not defined when the condition $aJ =\{\pm j_0, \pm j_1, \dots, \pm j_{t-1}\}$ fails (or equivalently, when $a(J \cup -J) \ne J \cup -J$). Note also that $\sigma_1$ is the identity automorphism, while $\sigma_{-1}$ is the automorphism $\tau$ defined earlier, since for $a = -1$ we may take $\alpha$ as the identity permutation and then $\sigma_{-1}(s,v) = (s,-v) = \tau(s,v)$ for every vertex $(s,v)$.
First, let $b$ be the multiplicative inverse of $a$ in ${\mathbb{Z}}_n^{\,*}$. Then for any $(s,v) \in {\mathbb{Z}}_t \times {\mathbb{Z}}_n$, we have $\sigma_a(\alpha^{-1}(s),bv)=(\alpha(\alpha^{-1}(s)),abv)=(s,v)$, and therefore $\sigma_a$ is surjective. Since $V(X)$ is finite, it follows that $\sigma_a$ is a permutation. Also $\sigma_a$ preserves edges, indeed it respects the fundamental edge-partition, because it takes each neighbour $(s^\prime,v)$ of the vertex $(s,v)$ in the spoke $S_v$ to the neighbour $(\alpha(s^\prime), av)$ of the vertex $(\alpha(s), av)$ in the spoke $S_{av}$, and takes the two neighbours $(s,v \pm j_s)$ of the vertex $(s,v)$ in the layer $L_s$ to the two neighbours $\sigma_a(s,v \pm j_s)=(\alpha(s),a (v \pm j_s))=(\alpha(s), a v \pm a j_s)
=(\alpha(s),a v \pm j_{\alpha(s)})$ of the vertex $(\alpha(s), av)$ in the layer $L_{\alpha(s)}$.
In the remaining part of this section we will show that if the $GI$-graph $X$ is connected, then the automorphisms described above and their products give all of the automorphisms of $X$ that respect the fundamental edge-partition.
For this we require two technical Lemmas, the proofs of which are obvious.
\[lemma:spokespoke\] Let $X$ be a connected $GI$-graph with at least two layers. Then every automorphism of $X$ that preserves spoke edges must permute the spokes $($like blocks of imprimitivity$)$.
\[lemma:layers\] Every automorphism of a $GI$-graph that respects the fundamental edge-partition must permute the layer cycles.
It will also be helpful to relate the automorphisms of a $GI$-graph to the automorphisms of the corresponding circulant graph.
Let $S$ be a subset of ${\mathbb{Z}}_n$ such that $S=-S$ and $0 \not \in S$. Then the *circulant graph* $\operatorname{Circ}(n;S)$ is defined as the graph with vertex set ${\mathbb{Z}}_n$, such that vertices $u$ and $v$ are adjacent precisely when $u-v \equiv a$ mod $n$ for some $a \in S$. Equivalently, this is the Cayley graph for ${\mathbb{Z}}_n$ given by the subset $S$. Note that $\operatorname{Circ}(n;S)$ is connected if and only if $S$ additively generates ${\mathbb{Z}}_n$, that is, if and only if some linear combination of the members of $S$ is $1$ mod $n$.
Now suppose that $S=\{s_1, \dots, s_c\}$, and that $\Gamma=\operatorname{Circ}(n;S)$ is connected.
For $1 \le i \le c$, let $G_{i,1},G_{i,2}, \ldots, G_{i,k_i}$ be the distinct cosets of the cyclic subgroup $G_{i,1}=\langle s_i \rangle$ in $G=\langle S\rangle$. Then we can form a partition $\mathcal C = \{ C_{ij} \}$ of the edges of $\Gamma$, where $$C_{ij} =\{\,\{g,g+s_i\}: \, g \in G_{i,j} \,\} \quad \hbox{ for } \, 1 \le j \le k_i \, \hbox{ and } \,1 \le i \le c.$$ Notice that each part $C_{ij}$ of $\mathcal C$ consists of precisely the edges of a cycle formed by adding multiples of the single element $s_i$ of $S$ to a member of the coset $G_{i,j}$.
We say that an automorphism $\varphi$ of $\Gamma$ *respects the partition* $\mathcal C$ if $\varphi(C_{ij}) \in \mathcal C$ for every $C_{ij} \in \mathcal C$. We have the following, thanks to Joy Morris:
\[thm:joymorris\] Suppose the circulant graph $\,\Gamma=\operatorname{Circ}(n;S)$ is connected. If $\psi$ is an automorphism of $\Gamma$ which fixes the vertex $0$ and respects the partition $\mathcal C = \{ C_{ij} \}$, then $\psi$ is induced by some automorphism of ${\mathbb{Z}}_{n}$ — that is, there exists a unit $a \in {\mathbb{Z}}_n$ with the property that $\, \psi(x)=ax\,$ for every $x \in {\mathbb{Z}}_n$ $($and in particular, $aS = S)$.
For a proof (by induction on $|S|$), see [@Morris]. To apply it, we associate with our graph $X= GI(n;J)$ the circulant graph $Y = \operatorname{Circ}(n;S \cup -S)$, where $S$ is the underlying set of $J$.\
Note that the projection $\eta : V(X) \to V(Y)$ given by $\eta(s,v)$ = $v$ takes every layer edge $\{(s,v), (s,v + j_s)\}$ of $X$ to the edge $\{v,v + j_s\}$ of $Y$, and hence gives a graph homomorphism from the subgraph of $X$ induced on layer edges onto the graph $Y$.
\[propn:XtoY\] Every automorphism of $X= GI(n;J)$ that preserves the set of spoke edges induces an automorphism of $Y = \operatorname{Circ}(n;S \cup -S)$ that respects the partition $\mathcal C = \{ C_{ij} \}$.
Any such automorphism $\varphi$ induces a permutation on the set of spokes of $X$, and hence under the above projection $\eta$, induces an automorphism of $Y$, say $\psi$. Moreover, since $\varphi$ preserves the layer edges, it must permute the layer cycles among themselves, and it follows that $\psi$ respects the partition $\mathcal C = \{ C_{ij} \}$.
\[cor:preserve\_layercycles\] Suppose $X$ is connected. Then every automorphism of $X=GI(n;J)$ that respects the fundamental edge-partition of $X$ is expressible as a product of powers of the rotation $\rho$, the reflection $\tau$, and the automorphisms $\lambda_{i,s_1,s_2}$ and $\sigma_a$ defined in Proposition [*\[propn:mix\_layers\]*]{} and Proposition [*\[propn:change\_layers\]*]{}.
First, any such automorphism $\varphi$ induces a permutation on the set of spokes of $X$, and so by multiplying by a suitable element of the dihedral group of order $2n$ generated by $\rho$ and $\tau$, we may replace $\varphi$ by an automorphism $\varphi'$ that respects the fundamental edge-partition of $X$, and preserves the spoke $S_0$. In particular, $\varphi'$ induces an automorphism of $Y = \operatorname{Circ}(n;S \cup -S)$ that fixes the vertex $0$. By Theorem \[thm:joymorris\], this automorphism of $Y$ is induced by multiplication by some unit $a \in {\mathbb{Z}}_n$, and then by multiplying by the inverse of $\sigma_a$ we may replace $\varphi'$ by an automorphism $\varphi''$ that preserves all of the spokes $S_v$. Finally, since $\varphi''$ preserves all of the spokes and also permutes the layer cycles among themselves, $\varphi''$ is expressible as a product of the automorphisms $\lambda_{i,s_1,s_2}$ defined in Proposition \[propn:mix\_layers\].
As a special case, we have also the following, for the automorphisms that preserve layers:
\[cor:preserve\_layers\] Suppose $X$ is connected. Then any automorphism of $X=GI(n;J)$ that takes layers to layers is a product of powers of the rotation $\rho$, the reflection $\tau$, and the automorphisms $\lambda_{s_1,s_2}$ and $\sigma_a$ defined in Corollary [*\[cor:exchange\_layers\]*]{} and Proposition [*\[propn:change\_layers\]*]{}.
Automorphism groups of $GI$-graphs {#automgps}
==================================
Now that we know all possible automorphisms of a $GI$-graph, it is not difficult to determine their number, and construct the automorphism groups in many cases. We will sometimes use $F(n;J)$ to denote the number of automorphisms of $GI(n;J)$, and $A(n;J)$ to denote the automorphism group $GI(n;J)$.
The automorphism group $A(n;J)$ of $GI(n;J)$ always contains a dihedral subgroup of order $2n$, generated by the rotation $\rho$ and the reflection $\tau$, defined in ($\dagger$) in the previous section (before Proposition \[propn:mix\_layers\]). Note that the relations $\rho^n = \tau^2 = (\rho\tau)^2 = 1$ hold, with the third of these being equivalent to $\tau\rho\tau = \rho^{-1}$.
We split the consideration of $F(n;J)$ and $A(n;J)$ into four cases, below.
The disconnected case {#subs:disconnected}
---------------------
Let $d = \gcd(n,J)$. Then $GI(n;J)$ is the disjoint union of $d$ isomorphic copies of $GI(n;J/d)$. This reduces the computation of $\operatorname{Aut}(X)$ to the case of connected $GI$-graphs. In particular, we have $$\label{eqn:autdis}
A(n;J) \cong A(n,J/d) \wr {{\rm Sym}}(d) \\[+2pt]$$ so $\operatorname{Aut}(GI(n;J))$ is the wreath product of $\operatorname{Aut}(GI(n;J/d))$ by the symmetric group ${{\rm Sym}}(d)$ of degree $d$, and therefore $$\label{eqn:numdis}
F(n,J) = |\!\operatorname{Aut}(GI(n;J))| = d! \,(F(n,J/d))^d.$$
The edge-transitive case {#subs:ET}
------------------------
The eight connected edge-transitive $GI$-graphs were given in Theorem \[theorem:skewautom\]. Seven of them are generalized Petersen graphs, with $J = \{1,k\}$ for some $k \in {\mathbb{Z}}_n^{\, *}$, and their automorphism groups are known — see [@Frucht] or [@Lovrecic] for example.
For each of these seven graphs, all of which are cubic, there is an automorphism $\mu$ of order $3$ that fixes the vertex $(0,0)$ and induces a $3$-cycle on it neighbours $(1,0)$, $(0,1)$ and $(0,n-1)$. In particular, this automorphism $\mu$ takes the spoke edge $\{(0,0),(1,0)\}$ to the layer edge $\{(0,0),(0,1)\}$, and its effect on the other vertices is easily determined.
In the cases $(n,k) = (4,1)$, $(8,3)$, $(12,5)$ and $(24,5)$, where $n \equiv 0$ mod $4$ and $k^2 \equiv 1$ mod $n$, the three automorphisms $\rho$, $\tau$ and $\mu$ generate $A(n;J)$ and satisfy the defining relations $$\rho^n = \tau^2 = \mu^3 = (\rho\tau)^2 = (\rho\mu)^2 = (\tau\mu)^2 = [\rho^4,\mu] = 1$$ for a group of order $12n$ which we may denote for the time being as $\Gamma(n,k)$, although strictly speaking, the second parameter $k$ is not necessary.
Similarly in the case $(n,k) = (10,2)$, the three automorphisms $\rho$, $\tau$ and $\mu$ generate $A(n;J)$, which has order $12n$, but they satisfy different defining relations, with the relation $[\rho^4,\mu] = 1$ replaced by $\mu\rho^{-1}\mu\rho^{2}\mu^{-1}\rho^{2}\tau = 1$. In the other two cases (namely $(n,k) = (5,2)$ and $(10,3)$), the automorphisms $\rho$, $\tau$ and $\mu$ generate a subgroup of index $2$ in $A(n;J)$, which has order $24n$.
In summary, the automorphism groups of the eight connected edge-transitive $GI$-graphs and their orders can be described as below: $$\label{eqn:autet}
\begin{array}{rclccrcl}
\operatorname{Aut}(GI(4,1,1))& \hskip -6pt \cong \hskip -3pt \!&\Gamma(4,1) \ \cong \ S_4 \times {\mathbb{Z}}_2 & \ \ & \ \ & F(4,1,1)& \hskip -6pt = \hskip -4pt &48 \\[+2pt]
\operatorname{Aut}(GI(5,1,2))& \hskip -6pt \cong \hskip -3pt \!&S_5 & \ \ & \ \ & F(5,1,2)& \hskip -6pt = \hskip -4pt &120 \\[+2pt]
\operatorname{Aut}(GI(8,1,3))& \hskip -6pt \cong \hskip -3pt \!&\Gamma(8,3) & \ \ & \ \ & F(8,1,3)& \hskip -6pt = \hskip -4pt &96 \\[+2pt]
\operatorname{Aut}(GI(10,1,2))& \hskip -6pt \cong \hskip -3pt \!&A_5 \times {\mathbb{Z}}_2 & \ \ & \ \ & F(10,1,2)& \hskip -6pt = \hskip -4pt &120 \\[+2pt]
\operatorname{Aut}(GI(10,1,3))& \hskip -6pt \cong \hskip -3pt \!&S_5 \times {\mathbb{Z}}_2 & \ \ & \ \ & F(10,1,3)& \hskip -6pt = \hskip -4pt &240 \\[+2pt]
\operatorname{Aut}(GI(12,1,5))& \hskip -6pt \cong \hskip -3pt \!&\Gamma(12,5) & \ \ & \ \ & F(12,1,5)& \hskip -6pt = \hskip -4pt &144 \\[+2pt]
\operatorname{Aut}(GI(24,1,5))& \hskip -6pt \cong \hskip -3pt \!&\Gamma(24,5)& \ \ & \ \ & F(24,1,5)& \hskip -6pt = \hskip -4pt &288 \\[+2pt]
\operatorname{Aut}(GI(3,1,1,1))& \hskip -6pt \cong \hskip -3pt \!&(D_6 \times D_6) \rtimes {\mathbb{Z}}_2 & \ \ & \ \ & F(3,1,1,1)& \hskip -6pt = \hskip -4pt &72.
\end{array}$$ See [@Frucht] and/or [@Lovrecic] for further details.
The case where $J$ is a set (with no repetitions) {#subs:set}
-------------------------------------------------
Suppose $J$ is a set (and not a multiset), in standard form, and let $X = GI(n;J)$. If $X$ is not connected, then sub-section \[subs:disconnected\] applies, while if $X$ is connected and edge-transitive, then sub-section \[subs:ET\] applies, so we will suppose that $X$ is connected but not edge-transitive.
Then by Corollary \[cor:preserve\_layers\], we know that the automorphism group of $X$ is generated by the automorphisms $\rho$, $\tau$ and the set $\{\sigma_a : \, a \in A\}$, where $$A = \{\,a \in Z_n^{\,*}\ | \ a(J\cup-J) = J\cup -J\, \}.$$ It is easy to see that $A$ is a subgroup of ${\mathbb{Z}}_n^{\,*}$. Indeed since $\sigma_1$ is trivial, $\sigma_{-1}=\tau$, and $\sigma_a \sigma_b=\sigma_{ab}$ for all $a,b \in A$, the set $S = \{\sigma_a : \, a \in A\}$ is a subgroup of $\operatorname{Aut}(X)$, isomorphic to $A$. In particular, $S$ is abelian. It is also easy to see that if composition of functions is read from left to right, and $\alpha$ is the bijection satisfying $j_{\alpha(s)} = \pm a j_s$ for all $s \in {\mathbb{Z}}_t$, then $$(\rho \sigma_a)(s,v) = \sigma_a(s,v+1) = (\alpha(s),a(v+1)) = (\alpha(s),av+a) = \rho^a(\alpha(s),av)
= (\sigma_a \rho^a)(s,v)$$ for every vertex $(s,v)$, and so $\rho \sigma_a = \sigma_a \rho^a$ for all $a \in A$. Rearranging, we have $\sigma_a^{-1}\rho \sigma_a = \rho^a$ for all $a \in A$, which shows that every element of $S$ normalizes the cyclic subgroup of order $n$ generated by the rotation $\rho$. Finally, again since $\tau = \sigma_{-1} \in S$, this implies that the automorphism group of $X = GI(n;J)$ is a semi-direct product: $$A(n;J) \ = \ \langle\, \{\rho\} \cup S \,\rangle \ \cong \ \langle \rho\rangle \rtimes S \ \cong \ C_n \rtimes A,
\quad \hbox{ of order } \, F(n;J) = n|A|.$$
The general case {#subs:general}
----------------
In this sub-section we deal with all remaining possibilities, in which $J$ is a multiset with repeated elements, in standard form, and $X = GI(n;J)$ is connected but not edge-transitive. Here we need two new sets of parameters, namely the multiplicity $m_j$ in $J$ of each element $j$ from the underlying set of $J$ (that is, the number of $s \in {\mathbb{Z}}_t$ for which $j_s = j$), and $d_j = \gcd(n,j)$ for all such $j$.
Also we need the set $B$ of all $a \in Z_n^{\,*}$ with the property that $aJ =\{\pm j_0, \pm j_1, \dots, \pm j_{t-1}\}$. Note that this is always a subgroup of $Z_n^{\,*}$, but is not always the same as the subgroup $A = \{\,a \in Z_n^{\,*}\ | \ a(J\cup-J) = J\cup -J\, \}$ that we took in the previous sub-section, since the multiplicities of $j$ and $\pm aj$ in $J$ might not be the same for some $a \in A$, but clearly they must be the same for every $a \in B$.
Now by Corollary \[cor:preserve\_layercycles\] we know that the automorphism group of $X$ is generated by the automorphisms $\rho$ and $\tau$, the automorphisms $\sigma_a$ for $a \in B$ (as defined in Proposition \[propn:change\_layers\]), and the automorphisms $\lambda_{i,s,s^\prime}$ (as defined in Proposition \[propn:mix\_layers\]) that mix cycles.
Just as in the previous case, the set $S = \{\sigma_a : \, a \in B\}$ is a subgroup of $\operatorname{Aut}(X)$, isomorphic to the subgroup $B$ of $Z_n^{\,*}$. Again also we have $\sigma_a^{-1}\rho \sigma_a = \rho^a$ for all $a \in B$, and so every element of $S$ normalizes the cyclic subgroup of order $n$ generated by the rotation $\rho$.
Next, for each $j \in J$, define $\Omega_j = \{ s \in {\mathbb{Z}}_t \ | \ j_s = j\,\}$, which is a set of size $m_j$, and for the time being, let $d = d_j = \gcd(n,j)$. Also define $\Omega_{ji} = \{ (s,v) \in V(X) \ | \ s \in \Omega_j, \ v \equiv i \ {\rm mod} \ d \,\}$, for $j \in J$ and $i \in {\mathbb{Z}}_d$ (where $d = \gcd(n,j)$). Note that $|\Omega_{ji}| = m_{j} n/d$, because $\Omega_{ji}$ is like a strip of vertices across $m_j$ layers of $X$, containing the $n/d$ vertices of one cycle from each of these layers.
By Proposition \[propn:mix\_layers\], for every two distinct $s_1,s_2$ in $\Omega_j$ and every $i \in {\mathbb{Z}}_d$, there exists an involutory automorphism $\lambda_{i,s_1,s_2}$ that exchanges one of the $d$ cycles from layer $L_{s_1}$ with the corresponding cycle from layer $L_{s_2}$, and preserves every spoke. This automorphism induces a transposition on the set of $m_j$ layer cycles containing the vertices of the set $\Omega_{ji}$. If we let the pair $\{s_1,s_2\}$ vary, we get all such transpositions, and hence for fixed $i \in {\mathbb{Z}}_d$, the automorphisms $\lambda_{i,s_1,s_2}$ with $s_1,s_2$ in $\Omega_j$ generate a subgroup isomorphic to the symmetric group ${{\rm Sym}}(m_j)$, acting with $n/d$ orbits of length $m_j$ on $\Omega_{ji}$ and fixing all other vertices.
Moreover, for any two distinct $i_1,i_2$ in ${\mathbb{Z}}_d$, the elements of $T_{i_1}$ and $T_{i_2}$ move disjoint sets of vertices (namely $\Omega_{ji_1}$ and $\Omega_{ji_2}$), and hence commute with each other. Hence the subgroup $T_j$ generated by all of the automorphisms $\lambda_{i,s_1,s_2}$ with $s_1,s_2$ in $\Omega_j$ is isomorphic to the direct product of $d$ copies of ${{\rm Sym}}(m_j)$, one for each value of $i$ in ${\mathbb{Z}}_d$.
Similarly, for any two distinct $j,j'$ in $J$, the corresponding subgroups $T_{j}$ and $T_{j'}$ move disjoint sets of vertices (from disjoint sets of layers of $X$), and hence commute with each other, so the subgroup $N$ generated by the set of all of the automorphisms $\lambda_{i,s_1,s_2}$ is a direct product $\Pi_{j \in J\,} T_j \cong \Pi_{j \in J\,} (S_{m_j})^{d_{j}}$, of order $\Pi_{j \in J\,} ({m_j}!)^{d_{j}}$.
On the other hand, for fixed $s_1$ and $s_2$ in $\Omega_j$, then $$\rho^{-1}{\lambda_{i,s_1,s_2\,}}{\rho} = \lambda_{i+1,s_1,s_2}
\quad \ \hbox{ and } \ \quad
\tau^{-1}{\lambda_{i,s_1,s_2\,}}{\tau} = \lambda_{-i,s_1,s_2}
\quad \ \hbox{ for all } \, i \in {\mathbb{Z}}_d,$$ so the automorphisms $\lambda_{i,s_1,s_2}$ are permuted among themselves in a cycle under conjugation by the rotation $\rho$, and fixed or interchanged in pairs under conjugation by the reflection $\tau$.
Finally if $a \in B\setminus \{\pm 1\}$, and $j'$ is the element of (the underlying set of) $J$ congruent to $\pm aj$ mod $n$, then the automorphism $\sigma_a$ defined in Proposition \[propn:change\_layers\] takes the layers $L_s$ for $s \in \Omega_j$ to the layers $L_{s'}$ for $s' \in \Omega_{j'}$, and conjugates the subgroup $T_j$ (generated by those $\lambda_{i,s_1,s_2}$ with $s_1,s_2$ in $\Omega_j$) to the corresponding subgroup $T_{j'}$. Hence $\sigma_a$ normalises the subgroup $N = \Pi_{j \in J\,} T_j$.
Thus $N$ is normalised by $\rho$ and $\tau$ ($=\sigma_{-1}$) and all the other $\sigma_a$, and is therefore normal in $\operatorname{Aut}(X)$. It follows that $$A(n;J) \ = \ \langle\, N \cup \{\rho\} \cup S \,\rangle \ \cong \ N \rtimes \langle \rho\rangle \rtimes S
\ \cong \ \prod_{j \in J\,} (S_{m_j})^{d_{j}} \rtimes C_n \rtimes B,$$ of order ${\displaystyle F(n;J) = n\,|B|\prod_{j \in J} (m_{j}!)^{d_{j}}}, $\
where the products are taken over all $j$ from the underlying set of $J$, without multiplicities.
Summary {#subs:summary}
-------
Combining the results from the four sub-sections above gives an algorithm for computing the automorphism groups of $GI$-graphs and their automorphism groups in general.
Vertex-transitive $GI$-graphs {#vertextrans}
=============================
In this section we consider further symmetry properties of $GI$-graphs. By Corollary \[corollary:edgetransitive\], we know there are only eight different connected edge-transitive $GI$-graphs $GI(n;J)$ having two or more layers. In particular, there are no such graphs with four or more layers. In contrast, we will show that there are several vertex-transitive $GI$-graphs, by giving a classification of them.
Note that the graph $GI(n;J)$ will be vertex-transitive if we are able to permute the layers of $GI(n;J)$ transitively among themselves. Now for each non-zero $a \in {\mathbb{Z}}_n$, consider multiplication of the (multi)set $J$ by $a$. If this preserves $J$ (as a multiset), then it gives a bijection from $J$ to $J$, and so by Proposition \[propn:change\_layers\], an automorphism $\sigma_a$ of $GI(n;J)$, permuting the layers. The graph $GI(n;J)$ will be vertex-transitive if the group generated by all such $\sigma_a$ acts transitively on the layers.
\[thm:vtsubgroup\] Let $J$ be any subset of ${\mathbb{Z}}_n^{\,*}$ with the two properties that [*(a)*]{} $J \cap -J = \emptyset$, and [*(b)*]{} $J \cup -J$ is a $($multiplicative$)$ subgroup of ${\mathbb{Z}}_n^{\,*}$. Then $GI(n;J)$ is vertex-transitive.
Since $J \cup -J$ is a subgroup of ${\mathbb{Z}}_n^{\,*}$ (not containing $0$), multiplication by any $a \in J$ gives a bijection from $J$ to $J$ and hence an automorphism $\sigma_a$ of $GI(n;J)$. Moreover, for any $a,b \in J$ there exists $c \in J$ such that $ac=\pm b$ in ${\mathbb{Z}}_n$, and in this case, the automorphism $\sigma_c$ takes any layer $s$ with $j_s = a$ to a layer $s'$ with $j_{s'} = \pm b$. It follows that the group generated by $\{\,\sigma_a\! : a\in J \,\}$ acts transitively on the layers of $GI(n;J)$, and hence that the group generated by $\{\rho\} \cup \{\,\sigma_a\! : a\in J \,\}$ acts transitively on the vertices of $GI(n;J)$.
\[corollary:VT\] Let $A$ be any subgroup of the multiplicative group ${\mathbb{Z}}_n^{\,*}$ containing an element of ${\mathbb{Z}}_n\setminus \{\pm 1\}$. If $-1 \in A$, then take $J = A \cap \{1,2,\dots, \lfloor \frac{n-1}{2} \rfloor \}$ $($so that $A = J \cup -J)$, while if $-1 \not\in J$, let $J = A$. Then $GI(n;J)$ is vertex-transitive. Hence for every integer $n > 6$, there exists at least one vertex-transitive $GI$-graph of the form $GI(n;J)$ for some $J$ with $|J| > 1$.
Note that the above requires $\phi(n) = |{\mathbb{Z}}_n^{\,*}|$ to be at least $4$, so that $n > 4$ and $n \ne 6$, in order for there to be at least two layers. A sub-family consists of those for which $A$ is the cyclic subgroup $\{1,r,r^2, \ldots, r^{t-1})$ generated by the powers of a single unit $r \in {\mathbb{Z}}_n^{\,*}\setminus \{\pm 1\}$. An example is given in Figure \[fig:twoGIgraphs\](b), with $n = 7$ and $r = 2$ (and $2^2 \equiv 4 \equiv -3$ mod $7$).
![The graph $GI(5;1,1,2)$ in (a) has 5-cycles as its three layers but is not vertex-transitive, while the graph $GI(7;1,2,3)$ in (b) is vertex-transitive and has two edge orbits.[]{data-label="fig:twoGIgraphs"}](GI5_1_1_2.pdf "fig:"){width="70.00000%"}\
(a)
![The graph $GI(5;1,1,2)$ in (a) has 5-cycles as its three layers but is not vertex-transitive, while the graph $GI(7;1,2,3)$ in (b) is vertex-transitive and has two edge orbits.[]{data-label="fig:twoGIgraphs"}](GI7_1_2_3.pdf "fig:"){width="70.00000%"}\
(b)
Next, we say that a subset $J=\{j_0,j_1, \dots, j_{t-1}\} $ of ${\mathbb{Z}}_n$ is *primitive* if $1 \in J$ and $j_i \ne \pm j_k$ whenever $i \ne k$. Also we say that the graph $GI(n;J)$ is *primitive* if $J$ is a primitive subset of ${\mathbb{Z}}_n$. Note that any such graph is connected, since $1 \in J$.
\[thm:primVT\] A primitive $GI$-graph $GI(n;J)$ is vertex-transitive if and only if either $J \,\cup \,-J$ is a $($multiplicative$)$ subgroup of ${\mathbb{Z}}_n^{\,*}$, or $n=10$ and $J=\{1,2\}$.
First, it was shown in [@Frucht] that $GI(10;1,2)$ is vertex-transitive. Also by Theorem \[thm:vtsubgroup\], we know that $GI(n;J)$ is vertex-transitive when $J \cup -J$ is a subgroup of ${\mathbb{Z}}_n^{\,*}$.
Conversely, suppose that $X=GI(n;J)$ is a primitive vertex-transitive $GI$-graph, other than $GI(10;1,2)$. We have to show that $J \cup -J$ is a subgroup of ${\mathbb{Z}}_n^{\,*}$.
Since $X$ is primitive, we have $1 \in J$, and without loss of generality we may assume that $j_0=1$. By Theorem \[theorem:skewautom\], we know that if $X$ has a skew automorphism, then either $t = 2$ and $(n,j_1) = (4,1)$, $(5,2)$, $(8,3)$, $(10, 2)$, $(10, 3)$, $(12, 5)$ or $G(24, 5)$, or $t = 3$ and $(n,j_1,j_2)= (3,1,1)$. It is easy to see that $J \cup -J$ is a subgroup of ${\mathbb{Z}}_n^{\,*}$ in all of these cases except $(n,t,j_0,j_1) = (10,2,1, 2)$. Hence we may assume that $X$ has no skew automorphism, and therefore every automorphism of $X$ preserves the fundamental edge-partition.
Now because $X$ is vertex-transitive, and the layer $L_0$ is a single $n$-cycle, it follows that all the layers of $X$ must be cycles, and so every element of $J$ must be coprime to $n$, and therefore a unit mod $n$. In particular, there are no automorphisms that ‘mix’ cycles from different layers. In fact, since $J$ is primitive, $J \cup -J$ contains $2t$ distinct elements, and it follows that $X$ has no automorphisms of the form given in Proposition \[propn:mix\_layers\] or Corollary \[cor:exchange\_layers\].
Hence (by Theorem \[thm:joymorris\]) the only automorphisms that preserve the spoke $S_0$ are the automorphisms $\sigma_a$ given in Corollary \[cor:preserve\_layers\].
But for any $x \in J$ (say $x = j_s$), by vertex-transitivity there exists an automorphism of $X$ that maps the vertex $(0,0)$ to the vertex $(s,0)$, and this must be one of the automorphisms $\sigma_a$, where $a$ is a unit in ${\mathbb{Z}}_n$ and $a(J \cup -J) = J \cup -J$. In particular, since $\sigma_a$ takes $(0,v)$ to $(\alpha(0),a v)$ for all $v \in {\mathbb{Z}}_n$, we have $\alpha(0) = s$ and therefore $x = j_s = j_{\alpha(0)} = \pm a j_0 = \pm a$, which gives $x(J \cup -J) = \pm a(J \cup -J) = \pm (J \cup -J) = J \cup -J$, for every $x \in J$.
Thus $J \cup -J$ is closed under multiplication, and by finiteness (and the fact that every element of $J$ is a unit mod $n$), it follows that $J \cup -J$ is a subgroup of ${\mathbb{Z}}_n^{\,*}$, as required.
We will now find some other examples, and show that every vertex-transitive $GI$-graph has a special form. To do that, we introduce some more notation: we denote by $[k]J$ the concatenation of $k$ copies of the multiset $J$. Note that this may involve a non-standard ordering of the elements of $[k]J$, but it makes the proofs of some things in this and the next section easier to explain — specifically, Theorem \[thm:classvt\] and Lemmas \[lem:multipleCayley\] and \[lem:doublenonCayley\].
\[thm:classvt\] Let $X=GI(n;J)$ be any connected vertex-transitive $GI$-graph. Then\
[*(a)*]{} If $\operatorname{Aut}(X)$ has a vertex-transitive subgroup that preserves the fundamental edge-partition of $X$, then $GI(n;[k]J)$ is vertex-transitive for every positive integer $k$.\
[*(b)*]{} All elements in $J \cup -J$ have the same multiplicity, say $k_0$, and $($so conversely$)$ the graph $X = GI(n;J)$ is isomorphic to $GI(n;[k_0]J_0)$ for some primitive subset $J_0$ of ${\mathbb{Z}}_n$, such that $GI(n;J_0)$ is vertex-transitive.
Let $X = GI(n;J) = GI(n; j_0,j_1,...,j_{t-1})$, and let $Y = GI(n;[k]J)$.
Note that the vertex-set of $Y$ is ${\mathbb{Z}}_{kt} \times {\mathbb{Z}}_n$, and we can write $[k]J = (j_0,j_1,...,j_{kt-1})$, where $j_c = j_d$ whenever $c \equiv d$ mod $t$, and accordingly, we can write each member $s$ of $Z_{kt}$ in the form $at+b$ where $a \in {\mathbb{Z}}_k$ and $b \in {\mathbb{Z}}_t$.
Also note that any permutation $f$ of $\{1,2,...,k\}$ gives rise to a corresponding permutation $\widetilde{f}$ of $\{1,2,...,kt\}$, defined by setting $\widetilde{f}(at+b) = f(a)t+b$ for all $a \in {\mathbb{Z}}_k$ and all $b \in {\mathbb{Z}}_t$, and in fact gives rise to an automorphism $\theta = \theta_f$ of $Y = GI(n;[k]J)$, defined by\
$ \theta_f(at+b,v) = (\widetilde{f}(at+b),v) = (f(a)t+b,v)
\quad \mbox{for all} \ a \in {\mathbb{Z}}_k, \ b \in {\mathbb{Z}}_t \, \mbox{ and }\, v \in {\mathbb{Z}}_n.$\
It is easy to see that $\theta_f$ preserves the edges of each spoke $S_v$, and permutes the layers among themselves. In fact $\theta_f$ takes $L_{at+b}$ to $L_{f(a)t+b}$ for all $a \in {\mathbb{Z}}_k$ and all $b \in {\mathbb{Z}}_t$, and hence $\theta_f$ preserves each of the sets $\{L_s : s \in {\mathbb{Z}}_{kt} \, | \, s \equiv b \ {\rm mod} \ t\,\}$ for $b \in {\mathbb{Z}}_t$.
It follows that given any two layers $L_c = \{(c,v) : v \in {\mathbb{Z}}_n\}$ and $L_d = \{(d,v) : v \in {\mathbb{Z}}_n\}$ with $c \equiv d$ mod $t$, there exists an automorphism $\theta$ of $Y$ taking $L_c$ to $L_d$. In particular, since $\operatorname{Aut}(Y)$ is transitive on vertices of each layer (as is the automorphism group of every $GI$-graph), we find that $\operatorname{Aut}(Y)$ has at most $t$ orbits on vertices of $Y$.
We can now prove (a), by extending certain automorphisms of $X$ to automorphisms of $Y$ that make it vertex-transitive.
Let $\xi$ be any automorphism of $X$ that respects the fundamental edge-partition. Define a permutation $\pi = \pi_{\xi}$ of the vertex set of $Y$ by letting $$\pi(at+b,v) = (at+c,w) \quad \mbox{ whenever } \, \xi(b,v) = (c,w),$$ for all $a \in {\mathbb{Z}}_k$, all $b \in {\mathbb{Z}}_t$, and all $v \in {\mathbb{Z}}_n$. If $e$ is a spoke edge of $Y$, say from $(at+b,v)$ to $(a't+b',v)$, and $(c,w) = \xi(b,v)$, then since $\xi$ takes spoke edges to spoke edges in $X$, we see that $\xi(b',v) = (c',w)$ for some $c' \in {\mathbb{Z}}_t$, and so by definition $\pi(a't+b',v) = (a't+c',w)$, which is a neighbour of $(at+c,w)$. Thus $\pi$ takes the edge $e$ to the spoke edge in $Y$ from $(at+c,w)$ to $(a't+c',w)$. Similarly, if $e$ is a layer edge of $Y$, say from $(at+b,v)$ to $(at+b,z)$, with $z = v+j_b$ (since $j_d = j_{d'}$ whenever $d \equiv d'$ mod $t$), and $(c,w) = \xi(b,v)$, then since $\xi$ permutes the layers of $X$, we know that $\xi$ takes the neighbour $(b,z) = (b,v+j_b)$ of $(b,v)$ on the same layer of $X$ as $(b,v)$ to a neighbour of $(c,w)$ on the the same layer of $X$ as $(c,w)$, namely $(c,w \pm j_c)$. Hence by definition, $\pi(at+b,z) = (at+c,w \pm j_c)$, which is a neighbour of $(at+c,w)$ in $Y$ because $j_{at+c} = j_c$. Thus $\pi$ takes $e$ to a layer edge from $(at+c,w)$ to $(at+c,w \pm j_c)$ in $Y$. In particular, since $\pi$ preserves both the set of all spoke edges of $Y$ and the set of all layer edges of $Y$, we find that $\pi = \pi_{\xi}$ is an automorphism of $Y$.
Moreover, since $\xi$ can be chosen to take any layer of $X$ to any other layer of $X$, it follows that the subgroup of $\operatorname{Aut}(Y)$ generated by the automorphisms $\theta_f$ and $\pi_{\xi}$ found above is transitive on layers of $Y$, and hence $Y$ is vertex-transitive.
Next we prove (b), namely that all elements in $J \cup -J$ have the same multiplicity, say $k_0$, and $X$ is isomorphic to $GI(n;[k_0]J_0)$ for some primitive subset $J_0$ of ${\mathbb{Z}}_n$ such that $GI(n;J_0)$ is vertex-transitive.
If $X$ is edge-transitive, then by Theorem \[theorem:skewautom\] we have $(n;J) = (4;1,1)$, $(5;1,2)$, $(8;1,3)$, $(10;1,2)$, $(10;1,3)$, $(12;1,5)$, $(24;1,5)$ or $(3;1,1,1)$. In the first case, we can take $k_0 = 2$ and $J_0 = \{1\}$, and observe that $GI(n;J_0) = GI(4,1)$ which is simply a $4$-cycle, and vertex-transitive. Similarly, in the last case, we can take $k_0 = 3$ and $J_0 = \{1\}$, and observe that $GI(n;J_0) = GI(3,1)$ which is a $3$-cycle, and vertex-transitive. In all the other six cases, we can take $k_0 = 1$ and $J_0 = J$, and note that $X = GI(n;J)$ itself is vertex-transitive. Thus (b) holds in all eight cases, and so from now on, we may assume that $X$ is not edge-transitive, and hence that every automorphism of $X$ respects the fundamental edge-partition.
This implies that $\operatorname{Aut}(X)$ is transitive on the layers of $X$, and it follows that all the layer cycles have the same length, so $\gcd(n,j_s)=\gcd(n,j_0)$ for all $s \in {\mathbb{Z}}_t$. But on the other hand, $X = GI(n;J)$ is connected, so $\gcd(n,j_0,j_1\dots,j_{t-1})=1$. Thus $\gcd(n,j_s) = 1$ for all $s \in {\mathbb{Z}}_t$.
In particular, there exists $a \in {\mathbb{Z}}_n^{\,*}$ such that $1 = aj_s \in aJ$. Now by Theorem \[thm:multiply\], the graph $X = GI(n;J)$ is isomorphic to $GI(n;aJ)$, and therefore we can replace $J$ by $aJ$, or more simply, suppose that $1 \in J$. If all the elements of $J$ are the same, then $X = GI(n;J)$ is isomorphic to $GI(n; [t]\{1\})$, and then since the set $\{1\}$ is primitive and $GI(n; 1)$ is simply an $n$-cycle, again (b) holds.
So now suppose that not all elements of $J$ are the same. For any two distinct $j_i, j_s \in J$, there must be an automorphism $\sigma_a$ that takes layer $L_i$ to layer $L_s$, by Corollary \[cor:preserve\_layers\]. In this case $a(J \cup -J) = J \cup -J$, by definition of $\sigma_a$, and therefore the multiplicities of $j_i$ and $j_s$ are the same. Hence all elements of $J \cup -J$ have the same multiplicity, say $k_0$.
In particular, $J=[k_0]J_0$ where $J_0$ is the underlying set of $J$, and $X$ is isomorphic to $GI(n;[k_0]J_0)$. The set $J_0$ is primitive since it contains 1 and all of its elements are distinct. To finish the proof, all we have to do is show that $GI(n;J_0)$ is vertex-transitive. But that is easy: for any two distinct $j_i,j_s \in J_0$, we know that there exists an automorphism $\sigma_a$ of $X$ taking layer $L_i$ of $X$ to layer $L_s$ of $X$, and $a(J \cup -J) = J \cup -J$; it then follows that $a(J_0 \cup -J_0)=J_0 \cup -J_0$, and therefore $\sigma_a$ induces an automorphism of $GI(n;J_0)$ that takes layer $L_i$ of $GI(n;J_0)$ to layer $L_s$ of $GI(n;J_0)$, as required.
Note that the above theorem above applies only to connected $GI$-graphs. Disconnected vertex-transitive $GI$-graphs are just disjoint unions of connected vertex-transitive $GI$-graphs, and can be dealt with accordingly.
We finish this section with observations about the graphs $GI(5;1,2)$ and $GI(10;1,2)$.
The Petersen graph $GI(5;1,2)$ is vertex-transitive, and its automorphism group acts transitively on the two layers; in fact so does a subgroup of order 20 which preserves the set of its ten layer edges. By Theorem \[thm:classvt\], it follows that every $GI$-graph of the form $GI(5;1,2,1,2,\ldots,1,2)$ is vertex-transitive.
On the other hand, the automorphism group of the dodecahedral graph $GI(10;1,2)$ has no layer-transitive subgroup preserving the set of layer edges (and the set of spoke edges), and so the above theorem does not apply to it. In fact $GI(10;[k]\{1,2\})$ is not vertex-transitive for any $k > 1$, because the fact that $2$ is not a unit mod $10$ implies that the automorphism group has two orbits on layers.
The graph $GI(10;1,2)$ is the only such exception, since for every other vertex-transitive $GI$-graph $X$, either $\operatorname{Aut}(X)$ itself preserves the fundamental edge-partition, or $X$ is edge-transitive and is then one of the other seven graphs in Theorem \[theorem:skewautom\], and for each of those, the subgroup of $\operatorname{Aut}(X)$ preserving the fundamental edge-partition is layer-transitive.
Cayley $GI$-graphs {#cayley}
==================
In this section we characterise the $GI$-graphs that are Cayley graphs.
First, a Cayley graph ${\rm Cay}(G,S)$ is a graph whose vertices can be labelled with the elements of some group $G$, and whose edges correspond to multiplication by the elements of some subset $S$ or their inverses. In particular, the edges of ${\rm Cay}(G,S)$ may be taken as the pairs $\{g,sg\}$ for all $g \in G$ and all $s \in S$, and then the group $G$ acts naturally as a group of automorphisms of ${\rm Cay}(G,S)$ by right multiplication. This action is transitive on vertices, indeed regular on vertices: for any ordered pair $(u,v)$ of vertices, there is a unique element of $G$ taking $u$ to $v$ (namely $g = u^{-1}v$).
Alternatively, a Cayley graph is any (regular) graph $X$ whose automorphism group has a subgroup $G$ that acts regularly on vertices. In that case, any particular vertex can be labelled with the identity element of $G$, and the subset $S$ can be taken as the set of all $s \in G$ taking that vertex to one of its neighbours.
Note that under both definitions, the Cayley graph is connected if and only if the set $S$ generates the group $G$. Also note that every Cayley graph is vertex-transitive (by definition), and that every non-trivial element of the subgroup $G$ fixes no vertices of the graph.
Now suppose $X= GI(n;J)$ is a vertex-transitive $GI$-graph.
We will assume that $X$ is connected, because if it is not, then it is simply a disjoint union of isomorphic copies of a connected smaller example. In particular, by Theorem \[thm:classvt\], we know that either $J$ is primitive (and $X$ is one of the graphs given by Theorem \[thm:primVT\]), or all elements in $J \cup -J$ have the same multiplicity $k_0 > 1$ and then $X$ is isomorphic to $GI(n;[k_0]J_0)$ for some primitive subset $J_0$ of ${\mathbb{Z}}_n$ such that $GI(n;J_0)$ is vertex-transitive.
Also we will suppose that $X$ is not $GI(10;1,2)$, for reasons related to Theorem \[thm:primVT\]. In fact, of the seven generalized Petersen graphs among the eight edge-transitive $GI$-graphs listed in Theorem \[theorem:skewautom\], it is known by the main result of [@NedelaSkoviera] or [@Lovrecic1] that $G(4,1)$, $G(8,3)$, $G(12, 5)$ and $G(24, 5)$ are Cayley graphs, while $G(5,2)$, $G(10, 2)$ and $G(10, 3)$ are not. Most of this (and the fact that the eighth edge-transitive graph $GI(3;1,1,1)$ is a Cayley graph) will actually follow from what we prove below.
Consider the case where $J$ is primitive (as we defined in Section \[vertextrans\]). In this case, $J \,\cup \,-J$ is a subgroup of ${\mathbb{Z}}_n^{\,*}$ under multiplication, and also $|\!\operatorname{Aut}(X)| = n|J \cup -J| = 2n|J| = 2|V(X)|$.
Hence if $G$ is a subgroup of $\operatorname{Aut}(X)$ that acts regularly on vertices of $X$, then $G$ is a subgroup of index $2$ in $\operatorname{Aut}(X)$. On the other hand, $G$ cannot contain the element $\tau$, since $\tau$ is a non-trivial automorphism with fixed points (namely the vertices $(s,0)$ for all $s$), and it follows that $G$ must be generated by the rotation $\rho$ and some subgroup of index $2$ in $\{\sigma_a : \, a \in J \cup -J\}$ not containing $\sigma_{-1} = \tau$. The latter has to be of the form $\{\sigma_a : \, a \in K\}$ for some subgroup $K$ of $J \cup -J$, such that $-1 \notin K$.
Conversely, if $J$ is a set, and $K$ is a subgroup of index $2$ in $J \cup -J$ not containing $-1$, then the group generated by $\{\sigma_a : \, a \in K\}$ permutes the layers of $X$ transitively, and so the subgroup generated by $\{\rho\} \cup \{\sigma_a : \, a \in K\}$ acts regularly on $V(X)$.
Thus we have the following:
\[thm:primCayley\] If $GI(n;J)$ is primitive and $J \cup -J$ is a multiplicative subgroup of ${\mathbb{Z}}_n^{\,*},$ then $GI(n;J)$ is a Cayley graph if and only if $J \cup -J$ has a subgroup of index $2$ that does not contain $-1$.
Note that this gives infinitely many examples of $GI$-graphs that are Cayley graphs, including those where $n$ is a prime congruent to $3$ mod $4$ and $J$ is the subgroup of all squares in ${\mathbb{Z}}_n^{\,*}$. On the other hand, it also gives infinitely many vertex-transitive $GI$-graphs that are not Cayley graphs, including those where $n$ is a prime congruent to $1$ mod $4$ and $J \, \cup\, -J = {\mathbb{Z}}_n^{\,*} = {\mathbb{Z}}_n\!\setminus\!\{0\}$.
This theorem also shows that among the six primitive $GI$-graphs that are edge-transitive, $G(8;1,3)$, $G(12;1,5)$ and $G(24;1,5)$ are Cayley graphs, while $G(5;1,2)$ and $G(10;1,3)$ are not. (The graph $GI(10;1,2)$ is not a Cayley graph, for other reasons.)
Next, consider the more general case, where $X=GI(n;J)$ is connected and vertex-transitive. In this case, by Theorem \[thm:classvt\], we know that all elements in $J \cup -J$ have the same multiplicity $k_0$, and $X$ is isomorphic to $GI(n;[k_0]J_0)$ for some primitive subset $J_0$ of ${\mathbb{Z}}_n$, such that $GI(n;J_0)$ is vertex-transitive. Also by what we found in Section \[vertextrans\] and sub-section \[subs:general\] of Section \[automgps\], we have $d_j := \gcd(n,j) = 1$ for all $j \in J_0$, and therefore $$|\!\operatorname{Aut}(X)| = n|J_0 \cup -J_0| \prod_{j \in J_0} (k_0!)^{d_j}
= 2n|J_0| (k_0!)^{|J_0|}.$$
We will find the following helpful, and to state it, we will refer to the automorphism $\rho$ of each $GI$-graph $GI(n;J)$ as its [*standard rotation*]{}, and sometimes denote it by $\rho_J$.
\[lem:multipleCayley\] If $GI(n;J)$ has a vertex-regular subgroup containing the standard rotation, then so does $GI(n;[k]J)$ for every integer $k > 1$.
Let $X = GI(n;J)$ and $Y = GI(n;[k]J)$, and let $\rho$ ($= \rho_J$) be the standard rotation for $X$. Also let $\{\rho\} \cup S$ be a generating set for a vertex-regular subgroup of $\operatorname{Aut}(X)$. Note that $\operatorname{Aut}(X)$ is layer-transitive on $X$, since $X$ is not $GI(10;1,2)$. Now by multiplying elements of $S$ by powers of $\rho$ if necessary, we may assume that $\langle S \rangle$ induces a regular permutation group on the set of layers of $X$. In particular, $\langle S\rangle$ has order $J$. Next, for each $\xi \in S$, the automorphism $\pi_\xi$ defined in the proof of Theorem \[thm:classvt\] acts fixed-point-freely on $Y = GI(n;[k]J)$, and it follows that the set $\{\pi_\xi : \xi \in S\}$ generates a subgroup of order $|J|$ that permutes the layers of $Y = GI(n;[k]J)$ in $|J|$ blocks of size $k$. Also if $f$ is the $k$-cycle $f = (1,2,\dots,k)$ in ${{\rm Sym}}(k)$, then the automorphism $\theta_f$ defined in the proof of Theorem \[thm:classvt\] induces a $k$-cycle on each of those $|J|$ layer-blocks. Finally, $\theta_f$ commutes with $\rho_{[k]J}$ and all the $\pi_\xi$ (for $\xi \in S$), so the subgroup generated by $\rho_{[k]J}$, $\theta_f$ and all the $\pi_\xi$ has order $nk|J|$, and acts regularly on the vertices of $Y$, as required.
Note that this shows, for example, that both of the remaining two edge-transitive $GI$-graphs $GI(4;1,1)$ and $GI(3;1,1,1)$ are Cayley graphs.
Somewhat surprisingly, we also have the following:
\[lem:doublenonCayley\] If $J$ is primitive and both $GI(n;J)$ and $GI(n;[2]J)$ are vertex-transitive, then $GI(n;[2]J)$ is always a Cayley graph, and so is $GI(n;[k]J)$ for every even integer $k > 1$.
First, if $GI(n;J)$ is a Cayley graph, then this follows from Lemma \[lem:multipleCayley\], so we will assume that $GI(n;J)$ is not a Cayley graph. Also because $GI(n;[2]J)$ is vertex-transitive, we know that $X \ne GI(10;1,2)$, and so $J \cup -J$ is a subgroup of ${\mathbb{Z}}_n^{\,*}$, by Theorem \[thm:primVT\]. On the other hand, by Proposition \[thm:primCayley\], we know that $J \cup -J$ has no subgroup of index $2$ that excludes $-1$. Hence we can write $J \cup -J$ as $U \times W$, where $U$ is a cyclic $2$-subgroup containing $-1$ and of order $q = 2^e$ for some $e > 1$, and $W$ is complementary to $U$, and of order $2t/q$. Also let $u$ be a generator of $U$, so that $u^{q/2} = -1$.
Now consider the automorphisms of $Y = GI(n;[2]J)$. For each $a \in J \cup -J = U \times W$, without loss of generality we will choose the associated bijection $\alpha: {\mathbb{Z}}_{2t} \to {\mathbb{Z}}_{2t}$ to be the ‘duplicate’ of the corresponding natural bijection from ${\mathbb{Z}}_{t}$ to ${\mathbb{Z}}_{t}$, namely so that $\alpha$ takes $s$ to $s'$, and $s+t$ to $s'+t$, whenever $j_{s'} = j_{s'+t} = \pm a j_{s} = \pm a j_{s+t}$ (for $0 \le s < t$).
For the moment, suppose that $W$ is trivial, so that $U = J \cup -J$. Then the automorphism $\sigma_u$ is not semi-regular, because the vertex $(0,0)$ lies in a cycle of length $q/2$ consisting of all $(s,0)$ with $0 \le s < t$ and $\pm j_{s} = u^i$ for some $i$, while the vertex $(0,1)$ lies in a cycle of length $q$ consisting of all $(s,u^i)$ such that $0 \le s < t$ and $\pm j_{s} = u^i$, for $0 \le i < q$. Hence in particular, the subgroup generated by $\rho$ and $\sigma_u$ has order $nq = 2nt$, but cannot be vertex-regular (since the $(q/2)$th power of $\sigma_u$ is a non-trivial element with fixed points).
On the other hand, we can multiply $\sigma_u$ by $\lambda_{0,t}$, which interchanges vertices $(0,v)$ and $(t,v)$, for all $v \in Z_n$, and find that $\sigma_{u}\lambda_{0,t}$ is a semi-regular element of order $q$, with $n/q$ cycles of length $q$. (The vertex $(0,0)$ lies in a cycle of length $q = 2t$ consisting of all $(s',0)$ with $\pm j_{s'} = u^i$ for some $i$, while the vertex $(0,1)$ lies in a cycle of length $q$ consisting of all $(s,u^i)$ such that $0 \le s < t$ and $\pm j_{s} = u^i$ for even $i$, and all $(s+t,u^i)$ such that $0 \le s < t$ and $\pm j_{s} = u^i$ for odd $i$; the cycles containing the other vertices $(s',1)$ have a similar form.)
It follows that the subgroup generated by $\rho$ and $\sigma_{u}\lambda_{0,t}$ has order $nq = 2nt$, and is transitive on vertices, and hence is vertex-regular, so that $GI(n;[2]J)$ is a Cayley graph.
When $W$ is non-trivial, the elements $\sigma_w$ for all $w$ in $W$ (or simply all $w$ from a generating set for $W$) induce a regular permutation group on the layers $L_s$ for which $\pm j_s \in W$, and it follows that the subgroup generated by $\rho$ and $\sigma_{u}\lambda_{0,t}$ and these $\sigma_w$ acts regularly on the vertices of $Y$, again making $GI(n;[2]J)$ a Cayley graph.
Finally, for any even integer $k > 2$, we find that $GI(n;[k]J) = GI(n;[k/2][2]J)$ is a Cayley graph, by applying Lemma \[lem:multipleCayley\] with $[2]J$ in place of $J$, and $k/2$ in place of $k$.
On the other hand, the same does not hold when $k$ is odd:
\[lem:oddnonCayley\] If $J$ is primitive and $GI(n;J)$ is vertex-transitive but not a Cayley graph, then $GI(n;[k]J)$ is not a Cayley graph for any odd integer $k > 1$.
Assume the contrary, so that $X = GI(n;J)$ is vertex-transitive and not a Cayley graph, but $Y = GI(n;[k]J)$ is a Cayley graph, for some odd $k$.
Then we know that $X \ne GI(10;1,2)$, since $Y$ is vertex-transitive, and so $J \cup -J$ is a subgroup of ${\mathbb{Z}}_n^{\,*}$, by Theorem \[thm:primVT\]. On the other hand, since $X$ is not a Cayley graph, Proposition \[thm:primCayley\] tells us that $J \cup -J$ has no subgroup of index $2$ that excludes $-1$, and therefore $J \cup -J$ contains an element $u$ of (multiplicative) order $4m$ for some $m$, with $u^{2m} = -1$. Also by Theorem \[theorem:skewautom\], we know that $Y$ is not edge-transitive, and so $\operatorname{Aut}(Y)$ preserves the fundamental edge-partition of $Y$, and hence every subgroup of $\operatorname{Aut}(Y)$ permutes the layers of $Y$ among themselves.
Now let $R$ be a vertex-regular subgroup of $\operatorname{Aut}(Y)$, and take $b = u^m$, which has order $4$, with $b^2 = -1$ in ${\mathbb{Z}}_n^{\,*}$. Next, choose $i$ such that $j_i = \pm b$ (noting that such an $i$ must exist because $b$ lies in the subgroup $J \cup -J$). Then by vertex-transitivity of $R$, there exists some automorphism $\theta$ of $Y$ taking the vertex $(0,0)$ to the vertex $(i,0)$. Moreover, by our knowledge of the structure of $\operatorname{Aut}(Y)$ from Section \[automgps\] and the fact that all of the automorphisms $\lambda_{s_1,s_2}$ and $\sigma_a$ preserve the spoke $S_0$, it follows that $\theta = w\sigma_b$ or $w\sigma_{-b}$ for some $w$ in the subgroup $N$ generated by the set of all of the automorphisms $\lambda_{s_1,s_2}$.
Since $R$ acts regularly on vertices, every non-trivial automorphism in $R$ has to be semi-regular. In particular, $\theta$ is semi-regular, as is its square\
$\theta^2 = (w\sigma_{\pm b})^2 = w(\sigma_{\pm b}w\sigma_{\pm b}^{\ -1}) \sigma_{(\pm b)^2}
= w'\sigma_{-1} = w'\tau$,\
where $w' = w(\sigma_{\pm b}w\sigma_{\pm b}^{\ -1}) \in N$. Both $w'$ and $\tau$ preserve the spoke $S_0$, and therefore so does $w'\tau$, and thus $w'\tau$ acts semi-regularly on $S_0$. But also $\tau$ fixes every vertex $(s,0)$ of $S_0$, and so $w'$ itself acts semi-regularly on $S_0$. Furthermore, since every element of $N$ preserves the set $\{L_0,L_t,\dots,L_{(k-1)t}\}$ of $k$ layers corresponding to the occurrences of $1$ in $J$, it follows that both $w'\tau$ and $w'$ act semi-regularly on the set $K = \{(rt,0) : 0 \le r < k\}$.
In particular, cycles of the permutation induced by $w'$ on $K = \{(rt,0) : 0 \le r < k\}$ must all have the same length, say $\ell$. Note that $w'$ is non-trivial, for otherwise $w'\tau = \tau$, which is not semi-regular on vertices (because it has fixed points), and therefore $\ell > 1$. But also $\ell$ must divide $k$, so $\ell$ is odd.
Now consider any $\ell$-cycle of $w'$ on $K$, say $((s_1,0),(s_2,0), \dots, (s_\ell,0))$. Because $\tau$ fixes every vertex of $K$, this is also a cycle of $w'\tau$, and hence all cycles of $w'\tau$ have length $\ell$. Also by definition of the elements generating $N$ (as defined in Proposition \[propn:mix\_layers\]), we know that $((s_1,1),(s_1,1), \dots, (s_\ell,1))$ must be a cycle of $w'$. But now the cycle of $w'\tau$ containing the vertex $(s_1,1)$ is\
$((s_1,1),(s_2,-1),(s_3,1), \dots,(s_{\ell-1},-1),(s_{\ell},1),
(s_1,-1),(s_2,1),(s_3,-1), \dots,(s_{\ell},-1)),$\
which has length $2k$, and this contradicts the fact that $w'\tau$ is semi-regular.
Putting together Proposition \[thm:primCayley\] and Lemmas \[lem:multipleCayley\] and \[lem:doublenonCayley\], we have the following:
\[thm:nonprimCayley\] If $X=GI(n;J)$ is connected, then $X$ is a Cayley graph if and only if\
[*(a)*]{} $J$ is primitive, and $J \cup -J$ is a multiplicative subgroup of ${\mathbb{Z}}_n^{\,*},$ with a subgroup of index $2$ that does not contain $-1$, or\
[*(b)*]{} $X = GI(n;J)$ is isomorphic to $GI(n;[k_0]J_0)$ for some primitive subset $J_0$ of ${\mathbb{Z}}_n$ and some integer $k_0 > 1$, such that either $GI(n;J_0)$ is a Cayley graph, or $k_0$ is even and $GI(n;J_0)$ is vertex-transitive but is not the dodecahedral graph $GI(10;1,2)$.
Additional remarks {#conclusion}
==================
The family of $GI$-graphs forms a natural generalisation of the Petersen graph. Our initial studies of $GI$-graphs have shown that this family is indeed very interesting and deserves further consideration. These graphs are also related to circulant graphs [@Morris]. Through that relationship, we were able to solve the puzzle of what appeared to be unstructured automorphisms of $GI$-graphs, and this enabled us to find their automorphism groups and classify those that are vertex-transitive or Cayley graphs.
Let us mention also the problem of unit-distance drawings of $GI$-graphs. A graph is a *unit-distance graph* if it can be drawn in the plane such that all of its edges have the same length. In [@HPZ1], it was shown that all $I$-graphs are unit-distance graphs. On the other hand, obviously no $GI$-graph with four or more layers can be a unit-distance graph, since it contains a $K_4$ as a subgraph, which itself is not a unit-distance graph. Hence the only open case of interest is the sub-class of $GI$-graphs having three layers.
For each $k \in {\mathbb{Z}}_n$, the graph $GI(n; k,k,k)$ is a cartesian product of two cycles and is therefore a unit-distance graph by [@HorvatPisanski2 Theorem 3.4]. We know of only one other connected example that is a unit-distance graph, and it is remarkable.
This is the graph $GI(7;1,2,3)$, which is shown in Figure \[fig:UDGIgraphs\]. The vertices can be drawn equidistantly on three concentric circles with radii $$R_1=\frac{1}{2\sin(\pi/7)}, \ \ \
R_2=\frac{1}{2\sin(2\pi/7)}, \ \ \ \mbox{and} \ \ \
R_3=\frac{1}{2\sin(3\pi/7)},$$ and the two smaller circles rotated through angles of $\pi/3$ and $-\pi/3$ with respect to the largest circle. One can then verify that all edges have the same length $1$.
![The graph $GI(7;1,2,3)$ as a unit-distance graph.[]{data-label="fig:UDGIgraphs"}](GI7_1_2_3UDr.pdf){width="35.00000%"}
The graph $GI(7;1,2,3)$ is a Cayley graph for the non-abelian group of order $21$, namely ${\mathbb{Z}}_7 \rtimes_{2\,} {\mathbb{Z}}_3$, which has presentation $\langle \, a,b \, | \ a^7 = b^3 = 1, \, b^{-1}ab = a^2 \, \rangle$. Its girth is 3 but it contains no cycles of length 4. This means that its Kronecker cover (see [@ImrichPisanski]) has girth 6 and is a Levi graph [@Coxeter] of a self-polar, point- and line-transitive but not flag-transitive combinatorial $(21_4)$-configuration. The resulting configuration is different from the configuration of Grünbaum and Rigby [@GR], since the latter configuration is flag-transitive but the one obtained from $GI(7;1,2,3)$ is not.
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been financed by ARRS within the EUROCORES Programme EUROGIGA (project GReGAS, N1–0011) of the European Science Foundation. It was also supported partially by the ARRS (via grant P1-0294), and the N.Z. Marsden Fund (via grant UOA1015).
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|
---
abstract: 'The object of this work is the spinor L-function of degree $3$ and certain degeneration related to the functoriality principle. We study liftings of automorphic forms on the pair of symplectic groups $\left(\text{GSp}(2),\text{GSp}(4)\right)$ to $\text{GSp}(6)$. We prove cuspidality and demonstrate the compatibility with conjectures of Andrianov, Panchishkin, Deligne and Yoshida. This is done on a motivic and analytic level. We discuss an underlying torus and L-group homomorphism and put our results in the context of the Langlands program.'
address: 'Max-Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany'
author:
- Bernhard Heim
title: 'Towards Functoriality Of Spinor L-functions'
---
Introduction
============
The [*principle of functoriality*]{} describes deep conjectural relationships among automorphic representations on different groups [@La97]. In this paper we consider automorphic representations and liftings of the pair of groups $\text{GL}(2)$, $\text{GSp}(4)$ to $\text{GSp}(6)$, where $\text{GSp}(2n)$ denotes the symplectic group with similitudes of degree $n$. The lifting will be described in terms of automorphic representations $\pi \in \mathcal{A}(\text{GSp}(2n))_k$ with holomorphic discrete series at archimedean places (of weight $k$), and convolutions of their spinor L-functions. This leads to map $$\mathcal{A}(\text{GL}(2)({\mathbb{A}})_{k-2} \times \mathcal{A}(\text{GSp}(4)({\mathbb{A}})_{k}
\longrightarrow \mathcal{A}(\text{GSp}(6))({\mathbb{A}})_{k}.$$ Here ${\mathbb{A}}$ denotes the adéles over ${\mathbb{Q}}$. We prove a cuspidality criterion and show that all other formally possible weights can not occur. Moreover we prove that the lifting is compatible with conjectures of Andrianov [@An74], Yoshida [@Yo01], Panchishkin [@Pa07] and Deligne [@De79] relating the functional equation of the spinor L-function of degree $3$ and the arithmetic nature of the critical values. Morover the underlying $L$-function can be identified with the spinor L-function of $$\text{GL}(2)({\mathbb{A}}) \times \text{GSp}(4)({\mathbb{A}}).$$ An integral representaion, functional equation and special values results have been indepently obtained by Furusawa [@Fu93] and Heim [@He99], and Böcherer and Heim [@BH00], [@BH06]. The results and observations obtained in this paper give strong evidence that the Miyawaki conjecture [@Mi92], [@He07d] of type II is essentially not only the lift of a pair of two elliptic modular forms, in contrast, it can be viewed as a lifting of a pair of an elliptic modular form and a Siegel modular form of degree $2$. There is also promising numerical data due to Miyawaki[@Mi92] supporting this approach. Let $\lambda_p(F)$ be the $p$-th Hecke eigenvalue of Siegel cuspidal Hecke eigenform. By a tremendous effort Miyawaki succeded in determine the second eigenvalue of the Hecke eigenform $F_{14}$ of degree $3$ and weight $14$: $\lambda_2(F_{14}) = -2^7 \cdot 2295$. In this interesting case our method gives the simple statement that $$\lambda_2(F_{14}) = \tau(2) \cdot \lambda_2(G),$$ where $\tau(2)=-24$ is the second Fourier coefficient of the Ramanujan $\Delta$-function and $G$ the non-trivial Siegel cuspform of weight $14$ and degree $2$. Experts in the this field, as Ibukiyama, Poor and Yuen, are optimistic that such calculations could be done for higher weights in the near future. We also would like to mention the recent work of Chiera and Vankov [@CV08], which is the first in a series of papers attacking this problem by starting with the weight of first occurence $k=12$, in which the conjecture of Miyawaki is now fully proven by Ikeda [@Ik06] and Heim [@He07d].\
\
Concerning numerical data, Langlands \[[@La79a],p. 213\] stated:
*The necessary equalities are too complicated to be merely coincidences,\
and we may assume with some confidence, ... .*
This was in reference to the paper of Kurokawa [@Ku78], where a statement of what is now called the Saito-Kurokawa lifting had been verified for small primes, broaching what has become an important research subject over the last $30$ years. The underlying principle of our lifting can be considered as a higher dimensional analog of the $L$-group homomorphism : $$\text{GL}(2)({\mathbb{C}}) \times \text{GL}(2)({\mathbb{C}}) \longrightarrow \text{GL}(4)({\mathbb{C}})$$ given by Ramakrishnan [@Ra00], which goes beyond endoscopic considerations. To explain this will require some preparation. It will become apparent that the approach here is only the tip of an iceberg, suggesting optimism about further results in this direction. We note that not much is known concerning functoriality and lifting of automorphic representations without the assumption of a Whittaker model. At this point we would like to suggest the reader to consult the excellent overview article of Raghuram and Shahidi [@RS08], and especially chapter 7, with the desription of Harders fundamental work [@Ha83].\
\
Let $G/K$ be a reductive group over a number field $K$ and $\mathcal{A}(G)$ the space of automorphic representations $\pi$ of $G(\mathbb{A}_K)$, where $\mathbb{A}_K$ are the adeles of $K$. A general question addressed in the Langlands program [@Ge84] arises if one takes the convolution L-function of two automorphic representations, and asks whether there exists another automorphic representation with this L-function. Let $G_1,G_2$, $G_3$ be three reductive groups defined over the same number field $K$. Let $\rho_1,\rho_2$ be finite dimensional representations of the $L$-groups of $G_1$ and $G_2$. In the sense of Langlands (cf. [@GS88]) we attach to the data $(\pi, \rho)$ the [*Langlands L-function*]{} $L(s,\pi,\rho)$. In this notation we can ask whether there exists a map $$\boxtimes \,\,\,\,
\begin{cases}
\mathcal{A}(G_1) \times \mathcal{A}(G_2) &\longrightarrow \quad \mathcal{A}(G_3),\\
\left( \pi^1, \pi^{2}\right) & \mapsto \quad \quad \pi^1 \boxtimes \pi^{2},
\end{cases}$$ with an automorphic representation $\pi:= \pi^1 \boxtimes \pi^{2} \in \mathcal{A}(G_3)$ which has the property that the convolution L-function $L(s,\pi^1 \otimes \pi^{2},\rho_1 \otimes \rho_2)$ is equal to a L-function $L(s,\pi^1 \boxtimes \pi^{2},\rho_3)$, where $\rho_3$ is a suitable finite dimensional representation of the $L$-group of $G_3$. One is far away from having a general solution of this problem and it seems that even the [*first examples*]{} given in the literature demonstrate the depth of the issues. This is illustrated by the work of Ramakrishnan [@Ra00] and Kim and Shahidi [@KS02]. Here we briefly recall results of Ramakrishnan. He constructed a map $$\mathcal{A}(Gl_2) \times \mathcal{A}(GL_2) \longrightarrow \quad \mathcal{A}(GL_4),$$ where $\rho_i$, $1 \leq i \leq 3$ is the standard representation. The existence of such a map had been expected for many years, but several difficult auxiliary results were needed. Since we are in a parallel situation, we mention some essential auxiliary results still needed in order to prove the Main Conjecture in [@He07e]. We recall briefly some of the main ingredients in the proof. To quote Ramakrishnan [@Ra00]:
*Our proof uses a mixture of converse theorems, base change and descent, and it also appeals to the local regularity properties of Eisenstein series and the scalar products of their truncation.*
This involves a converse theorem of Cogdell and Piatetski-Shapiro [@CP96], base change results of Arthur and Clozel [@AC89], results of Garrett [@Ga87] and Piateski-Shapiro and Rallis [@PR87] towards the triple product L-function and finally applications of Arthur’s truncation [@Ar80] of non-cuspidal Eisenstein series to get boundness in vertical strips of the triple product L-functions. Recently the L-functions considered in this paper showed up several interesting papers. In dissertations of Agarwal [@Ag07] and Saha [@Sa08] the level aspect and $p$-adic properties had been considered. Both have beautiful applications in mind. The first indicates applications towards functoriality à la Ramakrishnan and the second one goes towards the modularity approach of the Iwasawa conjecture related to ideas of Skinner and Wiles.
In two manuscripts Pitale and Schmidt [@PS1],[@PS2] consider the L-function of $\text{GSp}(4) \times \text{GL}(2)$ mainly for holomorphic modular forms and partly refine the results of Furusawa [@Fu93] and Heim [@He99], and Böcherer and Heim [@BH00], [@BH06] in the level aspect.
Hence it is obvious that considering the $\text{GSp}(4) \times \text{GL}(2)$ spinor L-function for the family of weights $k,k-2$, even in the highly assumed case of endoscopy (e.g. Saito-Kurokawa lifts) gives a significant new approach to several conjectures related to the spinor L-function of Siegel modular forms of degree $3$.
Recently Miyawaki’s conjecture for $\text{F}12$ had been proven [@He07d]. Applications are indicated by Dalla Piazza and van Geemen [@PG1]. These L-series and their relation to Galois representations are of great interest in mathematical physics, where they show up in the bosonic and superstring measures.
Acknowledgements {#acknowledgements .unnumbered}
================
This paper was supported by the Max Planck Institute of Mathematics in Bonn during my stay between Mai/July 2008. I also would like to thank Christian Kaiser for several very useful discussions on the topic.
Summary of the main results
===========================
The lifting we predict would be an example of Langlands’ functoriality principle. Since our predicted lift goes beyond endoscopy, and since it satiesfies all known motivic and analytic viewpoints and which is compatible with conjectures of Andrianov,Deligne, Yoshida, Panchiskin, we have to generalize the concept of $L$-group homomorphism.
The L-group of the group of projective symplectic similitudes of degree $n$ is the spin group $\text{Spin}(2n+1)({\mathbb{C}})$. Let $r_n$ be complex representations of the L-groups $\text{Spin}(2n+1)({\mathbb{C}})$. Langlands conjectures that for any L-group homomorphism $$\Theta:
\text{Spin}(3)({\mathbb{C}}) \, \times \, \text{Spin}(5)({\mathbb{C}}) \longrightarrow \text{Spin}(7)({\mathbb{C}})$$ there exists a map $$\Theta^*: \mathcal{A}(PGSp(2)) \times \mathcal{A}(PGSp(4)) \longrightarrow \mathcal{A}(PGSp(6))$$ such that the associated L-functions are equal, namely, $$L(s,\Theta^*(\pi^1,\pi^2),r_3) = L(s,\pi^1 \otimes \pi^2,r_1 \otimes r_2).$$ Let $\rho_n$ be the spin representation of $\text{Spin}(2n+1)({\mathbb{C}})$. Since we are interested in a reasonable generalization of the Miyawaki conjecture, and in a proof of Andrianov’s and Deligne’s conjecture for such liftings for the spinor L-function of degree $3$, we restrict ourself to the space $\mathcal{A}(PGSp(2n))_k$ of all automorphic representations $\pi = \pi_F$, associated to Siegel modular forms $F$ of degree $n$ and weight $k$. Take $r_i =\rho_i$ (for $n=1,2,3$). To study the spinor L-function of degree $3$ fix a maximal torus $T(n)$ in $\text{Spin}(2n+1)$ parametrized by $(a_0,a_1,\cdots, a_n)$ with $a_i \in {\mathbb{C}}^\times$. We take the torus homomorphism $\Theta: T(3) \times T(5) \longrightarrow T(7)$, or lifted L-group homomorphism, given by $$\Theta
\Big((\alpha_0, \alpha_1) , (\beta_0,\beta_1,\beta_2) \Big)\;=\;
(\alpha_0 \beta_0,\,\beta_1,\,\beta_2,\,\alpha_1).$$ We are aware that this seems to be very bizarre, but since we can not assume endoscopy in general (since we not only consider Saito-Kurokawa cuspforms), it seems that this is the only way to state such a lift in the framework of Langlands description of lifts in terms of Satake parameters.
Now suppose that the corresponding functorial map $\Theta^*(\pi^1,\pi^2)$ exists. Then put $$\pi:= \pi^1 \boxtimes \pi^2 := \Theta^*(\pi^1,\pi^2).$$ [*A priori*]{} it is not clear that these conjectural lifts are cuspidal. In this paper we show that they are. Let the corresponding cuspidal Siegel modular forms of degree $1,2,3$ have weights $k_1,k_2,k_3 \in {\mathbb{N}}$. We prove that only the triple $k-2,k,k$ can occur. This conclusion includes Miyawaki’s conjecture [@Mi92] of type II [@He07e]. We also give a motivic interpretation of these lifts and deduce the same result. Now let $F= h \boxtimes G$ with suitable $h\in S_{k-2}, G \in S_k^2, F \in S_k^3$, where $S_l^n$ is the space of cuspidal Siegel modular forms of weight $k$ and degree $n$. Let $\lambda_p$ be the $p$-th Hecke eigenvalue. Then, for all primes $p$, the eigenvalues of the $h,G,F$ satisfy $$\lambda_p(F) = \lambda_p(h) \cdot \lambda_p(G).$$ We prove the conjecture of Andrianov and Panchiskin on the meromorphic continuation and functional equation of the spinor L-function of degree $3$ for this type of cuspidal Hecke eigenform (see section \[sectionconjectures\] for details).
Let $k_1,k_2$ be positive even integers. Let $\pi^1 \in \mathcal{A}(GL(2))_{k_1}$ and $\pi^{2} \in
\mathcal{A}(GSp(4))_{k_2}$ be two automorphic representations. Let $\pi := \pi^1
\boxtimes \pi^2\in \mathcal{A}(GSp(6))_{k}$ be modular. Then necessarily $k_1=k-2$ and $k_2=k$. The L-function $L(s, \pi, \rho_3)$ has a meromorphic continuation to the whole complex plane. Let the first Fourier-Jacobi coefficient of $\pi^2$ be non-trivial. Then the completed L-function $$\widehat{L}(s,\pi,\rho_3) = L_{\infty}(s, \pi,\rho_3) \, L(s,\pi,\rho_3) ,$$ has a functional equation under $s \mapsto 3k-5-s$, where $$L_{\infty}(s,\pi) := \Gamma_{{\mathbb{C}}}(s) \, \prod_{m=1}^3 \, \Gamma_{{\mathbb{C}}}(s-k+m).$$
For $\pi^2$ attached to a Saito-Kurokawa lift the spinor L-function is entire. The special values of the lifts are apparently compatible with Deligne’s conjecture on the arithmetic of special values specialized to spinor L-function and to twisted spinor L-function by Yoshida and Panchishkin.
Let $\widetilde{\pi}= \pi' \boxtimes \pi'' \in \mathcal{A}(\text{GSp}(6))_k$ for $\pi' \in \mathcal{A}(\text{GL}(2))_{k-2}$ and $\pi'' \in \mathcal{A}(\text{GSp}(4))_{k}$ primitive. Let $F_{\widetilde{\pi}} \in S_k^3$ be normalized. Then the critical integers of the spinor L-function of $F_{\widetilde{\pi}}$ are exactly $m=k,k+1,\ldots, 2k-5$. Moreover there exists a positive $\Omega \in {\mathbb{R}}$ such that $$\frac{L(m, F_{\widetilde{\pi}},\rho_3)}{{\pi}^{4m-3k+6} \Omega} \in E.$$
It is surprising that the spinor L-function attached to a Siegel modular form $F= h \boxtimes G \in S_k^3$ has a Rankin-Selberg integral representation with respect to an Eisenstein series of Siegel type of degree $5$ [@He99], [@BH00], [@BH06].
Basic notation and facts
========================
In this section we describe the relation between Siegel modular forms and automorphic representations. We use the approach of Langlands to systematically define the spinor, standard, and $GL(2)$-twisted spinor L-function.\
\
Let $k,n$ be positive integers and let $S_k^n=S_k(\Gamma_n)$ be the space of cuspidal Siegel modular forms of weight $k$ with respect to the Siegel modular group $\Gamma_n = Sp(2n)({\mathbb{Z}})$. Let $ G_{2n}:=GSp(2n)$ be the symplectic group with similitudes and let $\overline{G}_{2n}:= G/ Z$, where $Z$ is the center of $G$. In the setting of Siegel modular forms let $F \in S_{k}^{n}$ be a Hecke eigenform and $\pi$ the associated automorphic representation of $\overline{G}_{2n}(\mathbb{A}_{{\mathbb{Q}}})$ over ${\mathbb{Q}}$, where $\mathbb{A}_{{\mathbb{Q}}}$ is the adele ring of ${\mathbb{Q}}$. Then $\pi$ factors over primes as $\pi = \bigotimes_v'\pi_v$, where $\pi_{\infty}$ is a holomorphic discrete series representation (or [*limit*]{} of discrete series) and, for finite $v=p$, $\pi_p$ is a spherical representation of $Sp(2n)({\mathbb{Q}}_p)$. Here $p$ are the prime numbers in ${\mathbb{Q}}$. The local representations are uniquely determined by the [*Satake parameters* ]{} of $F$: $$\mu_{0,p}^F, \mu_{1,p}^F \cdot
\ldots \cdot \mu_{n,p}^F.$$ These Satake parameters are unique up to the action of the Weyl group of $G_{2n}$. There is a clear correspondence between the eigenvalues of the Hecke operators in the classical setting and the Satake parameters coming from the local representations. For example let $\lambda_p(F)$ the eigenvalue of $F$ with respect to the Hecke operator $T_p^{(n)}$. Here $T_p^{(n)}$ is the canonical generalization of the well-known Hecke operator $T_p$ for elliptic modular forms on the upper half-space $\H:=\left\{ z = x + iy \vert \, y>0 \right\}$. Then by standard normalization we have $$\lambda_p(F) = \mu_{0,p}^F \left( 1 + \sum_{r=1}^n \prod_{1\leq i_1 \ldots
\leq i_r} \mu_{i_1,p}^F \cdot \ldots \cdot \mu_{i_r,p}^F\right).$$ To fix notation, let $H$ be any linear algebraic reductive group (split) over a number field $K$. Then $$\mathcal{A}(H) :=
\left\{ \pi
\text{ cuspidal automorphic representation of }
H({\mathbb{A}}_{K} )
\right\}.$$ Let $H$ be the projective group of symplectic similitudes and $k$ a positive integer. Then we denote by $\mathcal{A}(H)_k$ the set of all representations $\pi$ in $\mathcal{A}(H)$ such that there exists a $F \in S_k^n$ with $\pi = \pi_F$.
Now we can reformulate our lifting problem. This leads to the following necessary condition. Let
- $\pi^1 \in \mathcal{A}(PGSp(2)({\mathbb{A}}))_{k_1}$
- $\pi^2\in \mathcal{A}(PGSp(4)({\mathbb{A}}))_{k_2}$.
Then there has to exist a $\pi^3\in \mathcal{A}(PGSp(6)({\mathbb{A}}))_{k_3}$ such that for all finite primes $p$: $$\fbox{$\lambda_p(F) = \lambda_p(h) \cdot \lambda_p(G)$}.$$ Here $h,G,F$ are related to $\pi^1,\pi^2,\pi^3$ as above. Now we want to put this observation into the picture of the Langlands program to apply standard techniques and obtain possible generalizations.
Automorphic L-functions
-----------------------
Langlands’ set-up attaches automorphic L-functions to reductive algebraic groups in a systematic way and recovers the classical L-functions as the Hecke L-function, Rankin-Selberg L-function and Andrianov Spinor L-function. Let us recall the main ingredients. We fix the following data related to a linear algebraic group (split) over some number field $K$.
- $G({\mathbb{A}}_K)$ denote the restricted direct product $\prod_{v} G(K_v)$, here the $K_v$ are the local fields attached to the valuations $v$.
- $\!\!\! \!\!\!\phantom{X}^L G$ the $L$-group of $G$, which is a well-defined subgroup of $GL(n)({\mathbb{C}})$, $n$ suitable.
- $\rho$ a finite dimensional representation of the $L$-group of $G$.
- $\pi$ an automorphic representation of $G({\mathbb{A}}_F)$, $\pi = \otimes \pi_v$.
For example let $G= PGSp(2n)$, then the L-group coincides in this special case with the universal covering $Spin(2n+1)({\mathbb{C}})$ of the orthogonal group $S0(2n+1)({\mathbb{C}})$. We denote by $\rho_n$ and $st_n$ the spin representation of dimension $2n$ and the standard imbedding into $GL (2n+1)({\mathbb{C}})$ of the L-group.
Let $\pi \in \mathcal{A}(G)$ with $\pi = \otimes_p \pi_v$, then for almost all $v$ the local representations $\pi_v$ are spherical and unramified and correspond to the conjugacy class of a semi-simple element $t_v^{\pi}$ in the L-group. Let $S$ be a finite set, which contains all places at infinity and all finite $v$, which are ramified. Then the Langlands L-function attached to this data is defined by $$L^{(S)}(s,\pi,r):=
\prod_{v \not\in S}
\text{det} \left(
1 - r(t_v^{\pi}) (Nv)^{-s}
\right)^{-1}.$$ Here $Nv$ is the order of the residue field related to $v$. If we work over ${\mathbb{Q}}$ and $S=\{\infty\}$ we skip the $S$. This $L$-function is holomorphic with respect to $s \in {\mathbb{C}}$ for $\text{Re}(s)$ large enough. Langlands conjectured that the $L$-function can be completed such that one gets a meromorphic function on the whole $s$-plane with a functional equation.
Let $n,k \in {\mathbb{N}}$ and $\pi \in \mathcal{A}(GSp(2n))_k$. Then the [*spinor*]{} and [*standard*]{} L-functions are given by $$\begin{aligned}
L(s, \pi, \rho_n) & :=
\prod_p
\left(
\prod_{k=0}^n
\prod_{1 \leq i_1 < \ldots < i_k \leq n}
\left( 1- \mu_0 \mu_{i_1} \ldots \mu_{i_k} \, p^{-s}\right)
\right)^{-1}\\
L(s, \pi, st_n) & :=
\prod_p
\left( (1-p^{-s} ) \prod_{j=1}^n
\left( 1- \mu_j \, p^{-s}\right) \,
\left( 1-\mu_j^{-1} \, p^{-s} \right)
\right)^{-1}.
$$
Let $\pi= \pi_{\infty} \otimes_p \pi_p \in \mathcal{A}\left( \text{GSp}(2n)\right)$. Then $\pi$ satisfies the Ramanujan-Petersson conjecture at finite places if for all primes $p$ and local finite representations $\pi_p$ the Satake parameters $\mu_0, \mu_1, \ldots, \mu_n$ satisfy $$\vert\mu_1\vert = \vert\mu_2\vert = \ldots = \vert\mu_n\vert = 1.\\
\\$$ [***Two remarks:***]{}\
a) The spinor and standard L-function converges absolutely and locally uniformly for real part of $s$ large enough. For example without any assumption one can choose $\text{Re}(s) > n+1$ in the case of the standard L-function.\
b) Let $\pi \in \mathcal{A}\left( \text{GSp}(2n)\right)_k$ satisfy the Ramanujan-Petersson conjecture, then the Euler products converge already for $$\text{Re}(s) > \Big( nk - n(n+1)/2\Big)/2 \text{ and }\text{Re}(s) > 1.$$ \
It is well known that the Ramanujan-Petersson conjecture is fullfilled in the case of elliptic cusp forms. In the case $n=2$ the Saito-Kurokawa lifts (CAP-representations) contradict the conjecture. Nevertheless for all other representations $\pi \in \mathcal{A}\left( \text{GSp}(4)\right)_k$ the Ramanujan-Petersson conjecture is satisfied [@We94], [@We05].\
Let $c_1:= (k+1)/2$, $c_2 := k -1/$, and $c_3 := (3k)/2 -2$. Then we have the sharp bounds $\text{Re}(s) > c_n$ for the spinor L-functions in the cases $n=1,2,3$, if the conjecture is satisfied.
Conjectures of Andrianov and Deligne {#sectionconjectures}
------------------------------------
Let $ \pi \in \mathcal{A}\left( \text{GSp}(6)\right)_k$. Then there are several open conjectures on the analytic and arithmetic properties of the spinor L-function attached to $\pi$.\
\
[**Functional equation \[[*Conjecture of Andrianov [@An74], [@Pa07]* ]{}\]**]{}\
The completed spinor L-function $\widehat{L}(s, \pi, \rho_3)$ attached to an automorphic representation $\pi \in \mathcal{A}\left( \text{GSp}(6)\right)_k$ has a meromorphic continuation to the whole complex plane and satisfies the functional equation $$s \mapsto 3k -5 -s.$$ The completion is given by $$\widehat{L}(s, \pi, \rho_3) := L_{\infty}(s,\pi) \, L(s, \pi, \rho_3),$$ where $\Gamma_{{\mathbb{C}}}(s) := 2 (2 \pi)^{-s} \Gamma(s)$ and $$L_{\infty}(s, \pi) :=\Gamma_{{\mathbb{C}}}(s) \, \prod_{m=1}^3 \, \Gamma_{{\mathbb{C}}}(s-k+m).$$\
\
[**Arithmetic of critical values \[[*Conjecture of Deligne [@De79], Panchiskin [@Pa07], Yoshida [@Yo01]*]{}\]**]{}\
Let $\mathcal{M}$ be a (hypothetical) motive attached to $\pi \in \mathcal{A}\left(\text{GSp}(2n)\right)_k$ with $L$-function $L(s,\pi,\rho_n)$. Hence $\mathcal{M}$ is a motive over ${\mathbb{Q}}$ with coefficients in an algebraic number field $E$. Let $R:= E \otimes_{{\mathbb{Q}}} {\mathbb{C}}$ with $E \subset R$. Then the L-function $L(s, \mathcal{M})$ of the motive takes values in $R$. Let $m \in {\mathbb{Z}}$ be a critical value, then Deligne’s conjecture predicts that $$\frac{L(m, \mathcal{M})}{(2 \pi \, i)^{d^{\pm} m} \, c^{\pm}(\mathcal{M}) } \in E.$$ Here $\pm$ has the same sign as $(-1)^m$ and $c^{+}(\mathcal{M}),
c^{-}(\mathcal{M})$ are Deligne’s periods. The natural integers $d^{+}$ and $d^{-}$ are the dimension of the $+$ and $-$ eigenspace of the Betti realization of $\mathcal{M}$. Let $n=3$ then Panchiskin [@Pa07] has determined explicit the interval of the involved critical values. They are given by the positive integers $k, k+1, \ldots, 2k-5$. Since we prove that the period does not depend on the parity of the special values we ommit the discussion on the meaning of this fact in this paper. We only want to note that this somehow seems to reflect the degeneration of the lifted Siegel modular form of degree $3$ on the level of periods. Of course one could speculate and ask if this already determine lifts in the case of Siegel modular forms of degree $3$.
Candidates for the lifting - first properties
=============================================
Let $h \in S_{k_1}^{n_1}$ and $G \in S_{k_2}^{n_2}$ be Hecke eigenforms. Let $\pi^1$ and $\pi^2$ be the associated automorphic representations and let $r_1$ and $r_2$ be two complex respresentations of dimension $m_1$ and $m_2$ of the corresponding $L$-groups. Then $$L(s, \pi^1 \otimes \pi^2, r_1 \otimes r_2):=
\prod_p \left( 1_{m_1 + m_2} - r_1(t_p^{\pi^1}) \otimes r_2(t_p^{\pi^2}) \, p^{-s}\right)^{-1}.$$ If there is any automorphic representation $\pi^3$ (and complex representation $r_3$ of the related $L$-group) such that $$L(s,\pi^3,r_3)=L(s, \pi^1 \otimes \pi^2, r_1 \otimes r_2),$$ then we put $\pi^3 = \pi^1 \boxtimes \pi^2$ to indicate the modularity of the tensor product.
Already in the work of Ramakrishnan [@Ra00] it was a non-trivial task to show that the possible lifting is forced to be cuspidal. Now since our groups have higher rank one can expect that this is the same case here also. In contrast to the $GL_n$ case the Ramanujan-Petersson conjecture fails in general. Let $M_k^n$ be the space of Siegel modular forms of degree $n$, weight $k$ with respect to the Siegel modular group $\Gamma_n$.
Let $\pi= \pi(F)$ be a holomorphic automorphic representation of $\text{GSp}(6)(\mathbb{A})$ associated to a Hecke eigenform $F$ in $M_k^3$. Assume that $\pi= \pi^1 \boxtimes \pi^2$ is the lifting of the two cuspidal automorphic representations $\pi^1 \in \mathcal{A}\left(GL(2)\right)_{k_1}$ and $\pi^2\in \mathcal{A}\left(GSp(4)\right)_{k_2}$ of even weights $k_1,k_2$. Then $\pi$ is cuspidal.
In the case of the lifting of $\text{GL}(2) \times \text{GL}(2)$ to $ \text{GL}(4) $ this was called by Ramakrishnan [*cuspidality criterion*]{}, and employed in [@Ra00], section 3.2.
Let $\pi^1 \in \mathcal{A}\left(GL(2)\right)_{k_1}$ and $\pi^2 \in \mathcal{A}\left(GSp(4)\right)_{k_2}$ be two automorphic representations with even weights $k_1,k_2$. Let us assume that an automorphic representation $\pi$ exists coming from some Siegel eigenform $F$ of degree $3$ of weight $k_3$, such that $$L(s,\pi,\rho_3) = L(s, \pi^1 \otimes \pi^2, \rho_1 \otimes \rho_2).$$ Let us recall a result of Chai and Faltings \[[@CF90], p. 107\].
Let $F \in S_k^n$ be a Hecke eigenform. Let $k>n$. Then we have for every finite prime $p$, that there exists at least one element in the Weyl group orbit of the $p$-Satake parameters $\mu_0, \mu_1, \mu_2, \ldots, \mu_n$ such that $$\vert \mu_1 \cdot \mu_2 \cdot \ldots \cdot \mu_n \vert =1.$$ Here we have the normalization $\mu_0^2 \mu_1 \ldots \mu_n = p^{nk - n(n+1)/2}$.
Let $\alpha_0, \alpha_1$ and $\beta_0,\beta_1,\beta_2$ be the Satake parameters of $\pi_p'$ and $\pi_p''$. Here we have choosen the normalization $$\alpha_0^2 \alpha_1 = p^{k_1-1}, \quad
\quad \beta_0^2 \beta_1 \beta_2 = p^{2k_2 -3}.$$ If we assume that $\pi$ is not cuspidal, then it follows from the Darstellungssatz of Klingen and [@Ha81] that $F$ is a Siegel Eisenstein series or a Klingen Eisenstein series. The case $k=4$ is treated as the case of Siegel type Eisenstein series, although we have Hecke summmation. Let $\mu_0, \mu_1,\mu_2,\mu_3$ be the Satake parameter of $\pi_p$ with $\mu_0^2 \mu_1 \mu_2 \mu_3 = p^{3k-6}$. From the lifting assumption we can deduce that $$\label{lift_parameters}
\mu_0 = \alpha_0 \beta_0, \, \mu_1 = \beta_1,\, \mu_2=\beta_2, \text{ and } \mu_3 = \alpha_1.$$ Hence at least one of the Satake parameter $\mu_1, \mu_2,\mu_3$ of $\pi_p$ has the absolute value one. We now proceed case by case.\
\
a) Let first $F$ be a Siegel Eisenstein series. We can assume that the weight $k$ is largen then $3$. It is well known that the Satake parameters $\mu_1,\mu_2,\mu_3$ have the values $p^{k-3},\, p^{k-2}, p^{k-1}$. This is unique up to the action of the Weyl group of the symplectic group. Since none of the values has absolute value one we have a contradiction.\
b) Next we assume that $F$ is a Klingen Eisenstein series attached to the cusp form $H \in S_k^2$. Here $H$ is also a Hecke eigenform. Let $\gamma_0,\gamma_1,\gamma_2$ be the $p$-th Satake parameter of $H$. Then it follows from a Theorem of Chai and Faltings, that these parameters can be choosen in such a way, that $\vert \gamma_1 \gamma_2 \vert =1$. Then the Satake parameters of $F$ are given by $\mu_1 = \gamma_1, \, \mu_2 = \gamma_2, \, \mu_3 = p^{k-3}$. From (\[lift\_parameters\]) we know that (up to the action of the Weyl group) at least one of the Satake parameters $\mu_1,\mu_2,\mu_2$ has absolute values $1$. But if this would be the case for the Klingen Eisenstein series attached to $H$, then it would follow together with the Theorem of Chai and Faltings that $\vert \mu_1 \vert = \vert \mu_2 \vert$. Hence $\vert \mu_1 \, \mu_2 \mu_3\vert$ is always equal to $p^{k-3}$ or $p^{3-k}$, hence a contradiction to the Theorem of Chai and Faltings in degree $3$.\
c) Now let $F$ be a Klingen Eisenstein series attached to a Hecke eigenform $f \in S_k$ with $p$-th Satake parameters $\gamma_0, \gamma_1$, where $\vert \gamma_1 \vert =1$. Then the Satake parameters $\mu_1, \mu_2, \mu_3$ of $F$ can be choosen by $$\mu_1 = \gamma_1, \,\, \mu_2 = p^{k-2}, \,\, \mu_3 = p^{k-3}.$$ But this is a contradiction.
There is another possibility to prove this result by analytic methods instead of our algebraic proof. This is done by applying the method given in [@Ha81] to the properies of the standard L-function.
Motivic viewpoint
=================
In this section we consider the lifting problem from the motivic point of view. We determine the possible weights for the candidates and show that this is compatible with our previous results obtained in [@He07e].
We examine the possibility of the decomposition of a motive ${\mathcal{M}}$ over ${\mathbb{Q}}$ into the tensor product of certain motives ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$, which are attached to automorphic forms: $${\mathcal{M}}= {\mathcal{M}}_1 \otimes_{{\mathbb{Q}}} {\mathcal{M}}_2.$$
Motivic decompositions
----------------------
In 1999 we first considered the opposite question, we started with two [*constructed*]{} motives ${\mathcal{M}}_1, {\mathcal{M}}_2$ and had been interested on critical values [@De79] of the motive ${\mathcal{M}}_1 \otimes_{{\mathbb{Q}}} {\mathcal{M}}_2$ and the explicit formula for Deligne’s conjecture on the arithmetic of the L-function evaluated at this point. This was a natural question after finding a new integral representation of the $\text{GL}(2)$-twisted Andrianov spinor L-function [@He99]. This motivated Yoshida to work out the formula and a general procedure which describes how invariants and periods of motives ${\mathcal{M}}_1, {\mathcal{M}}_2, \ldots, {\mathcal{M}}_r$ behave by standard algebraic operations (see also Yoshida [@Yo01], section 4, page 1193). Finally main parts of Deligne’s conjecture to the L-function from above had been proven by Böcherer and Heim [@BH00], [@BH06]. Since ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ had been motives attached to automorphic forms it is an interesting problem to study the tensor product and ask if the new motives is a motive of an automorphic form - modularity of motives. The question if this motivic approach of the lifting considered in this paper leads to the same results, which we obtain by concrete analytic methods, was raised by Harder during the Japanese-German conference held at the MPI Mathematik in Bonn in February 2008.
We calculate the Hodge numbers to delete all [*forbidden lifts*]{}. Moreover we can predict with this approach the functional equation of the spinor L-function attached to a Siegel modular form of degree $3$ and the arithmetic of the critical values.\
\
Let $k,n$ be positive integers. We assume that $k>n$, since we are mainly interested in $n=1,2,3$. Let $\pi \in \mathcal{A}\left( \text{GSp}(2n)\right)_k$ be an automorphic representation with holomorphic discrete series at infinity.
We denote by $E_{\pi}$ the totally real number field generated by the Hecke eigenvalues of $\pi_f := \otimes_p \, \pi_p$. Now it is well known that we can pick a fix vector $F = F(\pi)\in S_k^n$ attached to the representation $\pi$, such that the field generated by the Fourier coefficients is contained in $E_{\pi}$. We denote such $F$ normalized. In the case $n=1$ this can easily obtained by choosing $F$ to be primitive. For Siegel modular form there is no such explicit property available.
Determination of the possible Hodge types
-----------------------------------------
Let ${\mathcal{M}}$ be a motive over ${\mathbb{Q}}$ with coefficients in a totally real number field $E$. Let $H_B({\mathcal{M}})$ be the Betti realization of ${\mathcal{M}}$. Then $H_B({\mathcal{M}})$ is a vector space over $E$ and its finite dimension $d$ is called the rank of the motive ${\mathcal{M}}$. The Hodge decomposition is given by $$\text{H}_\text{B}({\mathcal{M}}) \otimes_{{\mathbb{Q}}} \, {\mathbb{C}}=
\bigoplus_{p,q \, \in \, {\mathbb{Z}}} \text{H}^{p,q}({\mathcal{M}}).$$ Then $\text{H}^{p,q}({\mathcal{M}})$ are free $R:= E \otimes_{{\mathbb{Q}}} {\mathbb{C}}$ modules. If $\text{H}^{p,q}({\mathcal{M}}) = \{ 0\}$ whenever $p + q \neq w$. Then we denote ${\mathcal{M}}$ a pure motive and $w$ the (pure) weight. In the following let $\mathcal{M} =\mathcal{M}(F)$ be the motive attached to a cuspidal Siegel Hecke eigenform with spinor representation of the L-group. Let $h \in S_k$ then the Hodge type of $\mathcal{M}(h)$ is given by $$(0,k-1) + (k-1,0).$$ Let $G \in S_l^2$ then the Hodge type of $\mathcal{M}(G)$ is given by $$(0,2l-3) + (l-2,l-1)+ (l-1,l-2) + (2l-3,0).$$ By employing the Künneth formula we obtain for the tensor product of $\mathcal{M}(h) \otimes \mathcal{M}(G)$ the Hodge type $$\begin{aligned}
& &(0,2l+k-4) + (l-2,l+k-2)+ (l-1,l+k-3)+(k-1,2l-3)\\
& & +(k+l-3,l-1)+(2l-3,k-1)+(l+k-2,l-2)+(k+2l-4,0).\end{aligned}$$ Assuming that this tensor product is isomorphic to the motive $\mathcal{M}(F)$ attached to $F \in S_K^3$, which has the Hodge type $$(0,3K-6) + ( K-3,2K-3) + ( K-2,2K-4) + (K-1,2K-5) + \text{ symmetric terms},$$ then we obtain $k=K-2,l=K$. This gives the desired result. We are very much indepted to Alexei Panchiskin, for sharing some of his ideas with us. Its also possible to compare the Hodge numbers in the setting of vector-valued modular forms (see [@BG07], for the relevant Hodge decompositions).
Analytic properties of Spinor L-functions
=========================================
In this section we prove the meromorphic continuation of the spinor L-function attached to the lifting $\pi = \pi' \boxtimes \pi''$. We complete the L-function at infinity and obtain the functional equation. This result is compatible with the conjecture of Andrianov and Panchiskin.
Let $k_1,k_2$ be even positive integers. Let $\pi^1 \in \mathcal{A}(GL(2))_{k_1}$ and $\pi^{2} \in
\mathcal{A}(GSp(4))_{k_2}$ be two automorphic representations. Let $\pi := \pi^1
\boxtimes \pi^2\in \mathcal{A}(GSp(6))_{k}$ be modular. Then $k_1=k-2$ and $k_2=k$. Let $\rho_3$ be the spinor representation of $Spin(7)$. Then the L-function $L(s, \pi, \rho_3)$ has a meromorphic continuation to the whole complex plane. Let the first Fourier-Jacobi coefficient of $\pi^2$ be non-trivial. Then the completed L-function
$$\widehat{L}(s,\pi,\rho_3) = L_{\infty}(s, \pi,\rho_3) \, L(s,\pi,\rho_3) ,$$
satisfies the functional equation $s \mapsto 3k-5-s$. Here $$L_{\infty}(s,\pi) := \Gamma_{{\mathbb{C}}}(s) \, \prod_{m=1}^3 \, \Gamma_{{\mathbb{C}}}(s-k+m).$$
Let $\alpha_0,\alpha_1$ be the Satake parameters of $\pi_p'$ and $\beta_0,\beta_1,\beta_2$ be the Satake parameters of $\pi_p^2$. They correspond to the semi-simple elemens (conjugacy classes) $t(\pi_p^1)$ and $t(\pi_p^2)$ in the related L-groups. Let $\rho_1$ and $\rho_2$ be the spinor representations of $SL(2)({\mathbb{C}})$ and $Spin(5)({\mathbb{C}})$ then the spinor L-function of $\pi$ is given by $$L(s, \pi,\rho_3) = \prod_p \Big\{ \text{det}
\left( 1_8 - \rho_1\left(t(\pi_p^1)\right) \otimes
\rho_2\left(t(\pi_p^2)\right) \, p^{-s}\right) \Big\}^{-1}.$$ Then the local factors $L_p(X,\pi_p,\rho_3)$ are: $$\begin{aligned}
& &( 1- \alpha_0 \beta_0 X)
( 1- \alpha_0 \beta_0 \beta_1 X)
( 1- \alpha_0 \beta_0 \beta_2 X)
( 1- \alpha_0 \beta_0 \beta_1 \beta_2 X)\\
& &( 1- \alpha_0 \alpha_1 \beta_0 X)
( 1- \alpha_0 \alpha_1 \beta_0 \beta_1 X)
( 1- \alpha_0 \alpha_1 \beta_0 \beta_2 X)
( 1- \alpha_0 \alpha_1 \beta_0 \beta_1 \beta_2 X).\end{aligned}$$ Since $\pi$ is associated with a Siegel modular form with trivial central character one can employ the Langlands-Shahidi method to obtain the meromorphic continuation of $L(s,\pi,\rho_3)$. This works since in this case the problem can be reduced to the theory of Euler products applied to a Levi subgroup of the exceptional group of type $F_4$. This has been demonstrated in [@AS01]. To get the functional equation one has to use a different method. Surprisingly there also exists a Rankin-Selberg integral which involves the L-function $L(s, \pi' \otimes \pi'',\rho_1 \otimes \rho_2)$ (if we assume the technical restriction $\pi''$ primitive) for all even weights $k_1,k_2$. This has been discovered in [@He99] and generalized in [@BH00]. Once such an integral representation has been found one can apply advanced techniques to get the desired properties of the L-function. In this case the functional equation of an Eisenstein series of Siegel type of degree $5$ leads to the meromorphic continuation and functional equation of the L-function $L(s,\pi, \rho_3)$ we are interested in. Let $h \in S_{k_1}$ and $G \in S_{k_2}^2$ be Hecke eigenforms with ($ 0 < k_1 \leq k_2$) even integers. For technical reasons we assume that the first Fourier-Jacobi coefficient of $G$ is non-trivial. Then the function $$\begin{aligned}
\mathcal{L}(s) & := &
2^{-3}\,\,
\left( 2 \pi \right)^{4-2k_2-k_1} \,
\Gamma_{{\mathbb{C}}}(s) \,\, \Gamma_{{\mathbb{C}}}(s-k_2+1) \,\, \Gamma_{{\mathbb{C}}}(s-k_2+2) \,\,\Gamma_{{\mathbb{C}}}(s-k_1+1)
\\ & &
\prod_{p} L_p\left( p^{-s}, \pi_h \otimes \pi_G,\rho_1 \otimes \rho_2\right)\end{aligned}$$ has a meromorphic continuation to the whole complex plane and satisfies the functional equation $$s \mapsto k_1 + 2 k_2 -3 -s.$$ Let $k:=k_1+2 = k_2=k_3$. Then $$\Gamma_{{\mathbb{C}}}(s) \, \prod_{m=1}^3 \, \Gamma_{{\mathbb{C}}}(s-k+m) \,\, L(s, \pi' \times \pi'', \rho_1 \otimes \rho_2).$$ has the functional equation $s \mapsto 3k - 5-s$. Hence the theorem is proven.
On Deligne’s conjecture
=======================
Let $F_{\pi} \in S_k^n$ be a Hecke eigenform attached to $\pi \in \mathcal{A}(GSp(2n)({\mathbb{A}})_k$. If $k>2n$ be even then $F_{\pi}$ can be normalized such that the Fourier coefficients of $F_{\pi}$ are contained in the totally real number field $E$ generated by eigenvalues. Here we are interested the cases $n=1,2,3$. For $n=1$ this is obtained by choosing the eigenform to be primitive, i.e the first Fourier coefficient $a_g(1)=1$. For Siegel modular forms there is no such choice. Hence we make the assumption that the first Fourier coefficient in degree $2$ is non-trivial and denote such forms also primitive. Let further $\parallel \! F_{\pi} \! \parallel$ denote the norm of $F_{\pi}$, related to the Petersson scalar product. If for $\pi \in \mathcal{A}(GSp(4))_k$ any primitive $F$ exists we denote $\pi$ primitive. Finally using all the observations and results of this paper we obtain our main result:
Let $\widetilde{\pi}= \pi' \boxtimes \pi'' \in \mathcal{A}(\text{GSp}(6))_k$ for $\pi' \in \mathcal{A}(\text{GL}(2))_{k-2}$ and $\pi'' \in \mathcal{A}(\text{GSp}(4))_{k}$ primitive. Let $F_{\widetilde{\pi}} \in S_k^3$ be normalized. Then the critical integers of the spinor L-function of $F_{\widetilde{\pi}}$ are exactly $m=k,k+1,\ldots, 2k-5$. Moreover there exists a positive $\Omega \in {\mathbb{R}}$ such that $$\frac{L(m, F_{\widetilde{\pi}},\rho_3)}{{\pi}^{4m-3k+6} \Omega} \in E.$$
For the readers convenience we recall the basic steps. All the ingredients have already been prepared in this paper. Let $F_{\widetilde{\pi}}$ be given. We have proven the the full Andrianov conjecture for $L(s,\widetilde{\pi},\rho_3$. Hence we have a meromorphic continuation on the whole complex plane. After completion we get a functional equation. From the explicit form of the $\Gamma$-factors we can deduce the critical values in the sense of Deligne and obtain exactly $m=k,k+1,\ldots, 2k-5$. With the functoriality property determined in this paper the lifting is directly related to another L-function. It can be identified with the so-called $GL(2)$- twisted spinor L-function. And hence properties of this L-function can be transfered to $L(s,\widetilde{\pi},\rho_3)$. Let $$\Omega:= \parallel h_{\pi'}\parallel^2 \cdot \parallel G_{\pi''} \parallel^2.$$ Here $h_{\pi'}$ and $G_{\pi''}$ are primitive forms as described above. Then finally form the arithmetic results given in [@BH06] we obtain the desired result of the theorem.
Identifying the period $\Omega$ with the period given in Deligne’s conjecture $c^{\pm}(\mathcal{M})$ of the attached motive leads to a proof of this conjecture for the lifts considered in this paper. This would imply that the the dimension of the $+$ and $-$ eigenspace of the Betti realization of $\mathcal{M}$ are equal. Finally we would like to remark that Miyawaki’s conjecture of TypII [@Mi92], [@He07e] extended to special values result is contained in this theorem. Moreover the assumption to be primitive can be removed.
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abstract: 'We investigate the finite-frequency thermal transport through a quantum dot subject to strong interactions, by providing an exact, nonperturbative formalism that allows us to carry out a systematic analysis of the thermopower at any frequency. Special emphasis is put on the dc and high-frequency limits. We demonstrate that, in the Kondo regime, the ac thermopower is characterized by a universal function that we determine numerically.'
author:
- Razvan Chirla
- Cătălin Paşcu Moca
bibliography:
- 'references.bib'
title: 'Finite-frequency thermoelectric response in strongly correlated quantum dots'
---
Introduction {#sec:Intro}
============
In the quest to find the most energy-efficient systems and devices, thermal generation of currents in nanometer-size structures which can be manipulated by electric fields, may offer one of the best paths to follow. [@Sales.96; @Snyder.08; @Heremans.08] Quantum dots (QD) are among the best candidates, since they are highly tunable. Moreover, they are characterized by an enhanced figure of merit, as a result of the converging effect of reduced spatial dimensionality, that minimizes the phonon thermal conductivity, and an increased electronic density of states. So far, they can be used as thermoelectric power generators or coolers [@Humphrey.02], and when embedded into bulk materials or nanowires, a structure with large thermopower coefficient, $S$, is obtained. [@Harman.02; @Wang.08] The same environment, however, is fundamentally enhancing the interaction between the electrons, generating strong correlations and other dynamical effects. By doing detailed measurements in QDs, it has been shown that the oscillating behavior of the thermopower as a function of the gate voltage might carry information on interactions present in the system.[@Svensson.12] On the theoretical side, the thermoelectric problem in quantum dots is also of considerable interest: First, a perturbative calculation valid for weakly interacting QDs was presented in Ref. . Later on, in Refs. the thermopower of a Kondo correlated dot was computed. Quite recently, by using the numerical renormalization group approach (NRG) approach, the thermoelectric properties of a strongly correlated dot modeled in terms of the Anderson model, were investigated systematically. [@Costi.10] Other more exotic systems such as the SU(4) Kondo state [@Bas.12] or double-dot systems[@Trocha.12] were also studied, but, so far, mostly static effects have been addressed [@Donsa.13] and only few theoretical and experimental studies were focused on dynamical effects. [@Lopez.13]
![(Color online) Sketch of the real and imaginary parts of the universal function $s(\omega/T_K, T/T_K)$ in the Kondo regime, at a fixed temperature, $T\ll T_K$. The coefficients $a'$ and $a''$ depend on temperature as $\sim 1/T^2$. See also Eq. for the functional dependence of $s$. []{data-label="fig:s0"}](sketch_s_omega.eps){width="0.9\columnwidth"}
In contrast, other transport quantities, such as the usual differential conductance or the noise, have been investigated at various frequencies. [@Sindel.05; @Blanter.00; @Moca.11] Consequently, new interesting physics has emerged: It was found that the modulation of the gate voltages suppresses the Kondo temperature [@Kogan.04] $T_K$, and that the frequency-dependent emission noise of a quantum dot [@Basset.12] in the Kondo regime, at high frequencies, $\hbar\omega \gg k_B T_K$, provides information on the system at energy scales which are not accessible by simple dc measurements. Furthermore, in a slightly different context, i.e., the correlated band models, it has been predicted that the thermopower in the high-frequency limit may provide further understanding on the thermoelectric transport. [@Shastry.06; @Shastry.09; @Xu.11]
Motivated by these observations, we consider here the problem of thermoelectric response at finite frequencies in a quantum dot subject to a strong Coulomb interaction, and in particular we shall investigate the dynamical thermopower $S(\omega)$. This quantity characterizes how a charge current, $I^{(1)}(t)$, is generated by an infinitesimal time-dependent temperature difference $\delta T (t)$ across the dot and, at the same time, how a heat current $I^{(2)}(t)$ responds to an infinitesimal voltage drop $\delta V(t)$ [@Mahan] $$\begin{aligned}
\label{Eq:S_general}
\left (
\begin{array}{c}
\langle I^{(1)}(t)\rangle \\
\langle I^{(2)}(t)\rangle
\end{array}
\right )
&=& \int dt' dt''\,T\,S(t-t')\, G (t'-t'')\times
\nonumber \\
&&\left (
\begin{array}{c}
\delta T(t'')/ T|_{\delta V =0}\\
\delta V(t'')|_{\delta T=0}
\end{array}
\right )\, . \end{aligned}$$ As defined, the thermopower itself is not a response function, so it can not be computed directly within the linear response theory. Instead, the combination $L_{12}(\omega) = T G(\omega)\, S(\omega)$ that appears in Eq. is a true response function that can be computed exactly. To get the thermopower spectrum $S(\omega)$, aside from $L_{12}$ we also need the usual ac conductance, $ G(\omega)=L_{11}(\omega)$. Then, $S(\omega)$ can be expressed as [@Shastry.09; @Luttinger.64] $$\label{Eq:S_omega}
S(\omega)= {1\over T}\left\{ {L_{12}(\omega) \over L_{11}(\omega)}\right \}\, .$$
One of the main results of this work is that in the strong coupling (Kondo) regime, i.e., $\max \{\omega, T\}\ll T_K $, the equilibrium ac thermopower takes on a simple, universal form, which apart from a phase-dependent prefactor, is characterized by a universal function $$S(\omega, T) \simeq {k_B\over e} \Big ({T\over T_K}\Big )\, s(\omega/T_K, T/T_K)\cot(\delta_0)\,.
\label{Eq:s_intro}$$ Here, $\delta_0$ is the phase shift of the electrons at the Fermi level, and the function $s$ is a complex universal function that depends exclusively on $\omega/T_K$ and $T/T_K$. The prefactor $k_B/e = 8.6 \times 10^{-5} \;V/K$ is the unit in which the thermopower is measured. The characteristic features of $s(\omega/T_K, T/T_K)$ are sketched in Fig. \[fig:s0\]. At a given temperature $T\ll T_K$, and when $\omega \ll T_K$, the real part $s'$ grows quadratically with the frequency $s'\simeq s_0 +a'(T)\, \omega^2+\dots$, followed by multiple changes of sign at some intermediate frequencies, $\omega_i\sim \{T, \Gamma\}$, and becomes constant in the $\omega\to \infty$ limit. Its imaginary part vanishes in the $\omega\to 0$ limit, has a linear dependence $s''\simeq a'' (T)\, \omega+\dots$ below the Kondo scale, and vanishes in the $\omega\to \infty$ limit.
In Fig. \[fig:dot\], we present a sketch of the setup. It consists of a quantum dot that is coupled to two external leads, $\alpha =\{L,R\}$, that have different temperatures, $T_\alpha(t)= T\pm\delta T(t)/2$. The temperature gradient $\delta T(t)$ generates a time-dependent current which flows across the dot. Starting from the Kubo formalism and Fourier transforming from time $t$ to frequency $\omega$, we find the ac thermopower, $S(\omega)$. We derive general, exact expressions for $L_{ij}(\omega)$, and implicitly for $S(\omega)$, which are valid at any frequency (see Eqs. and ). The derivation is then followed by a careful analysis of different regimes of interest, such as the large-frequency limit, $S^*= S(\omega\to \infty)$, or the conventional low-frequency limit, $S_0 = S(\omega=0)$ [@Costi.10]. As a technical observation, since we are interested in $S(\omega)$ at any frequency, even in the region $\omega\gg D$, the bandwidth $D$ of the conduction electrons must be kept finite in the calculations, otherwise an unphysical divergence with increasing frequency is present in the spectrum of $L_{12}(\omega)$ (see Sec. \[sec:TP\]).
The paper is organized as follows: In Sec. \[sec:TF\], we introduce the model Hamiltonian and derive the exact expressions for the generalized susceptibilities $L_{ij}(\omega)$ that enter Eq. . The operators for the charge and heat currents are discussed in Sec. \[sub:CH\], and the results for the ac thermopower are presented in Sec. \[sec:TP\]. We give the final remarks in Sec. \[sec:C\]. Further technical details are discussed in Appendices \[app:A\] and \[app:B\].
![(Color online) Sketch of the quantum dot that is coupled to two external leads which are assumed to be at different temperatures $T\pm\delta T (t)/2 $.[]{data-label="fig:dot"}](sketch_quantum_dot.eps){width="0.9\columnwidth"}
Theoretical framework {#sec:TF}
=====================
Model Hamiltonian
-----------------
In this work, we shall consider the case of a QD described by the Anderson model. It consists of a single localized orbital that is coupled to two external leads (see Fig. \[fig:dot\]). The dot can accommodate up to two electrons with strong on-site interaction. The Hamiltonian takes the form $$\begin{aligned}
H & = & \sum_{\sigma}\varepsilon_{d} \;d^{\dagger}_{\sigma}d^{}_{\sigma}+
U\;d^{\dagger}_{\uparrow} d^{}_{\uparrow}d^{\dagger}_{\downarrow} d^{}_{\downarrow}\,\\
&+& \sum_{{\bf k}, \sigma}\sum_{\alpha=L,R}\left (\varepsilon_{\bf k}\;
c^{\dagger}_{\alpha \bf k \sigma}c^{}_{\alpha \bf k \sigma}
+\Big (t_{\alpha{\bf k}}\;
c^{\dagger}_{\alpha \bf k \sigma} d_{\sigma}
+H.c. \Big )\right )\, , \nonumber\end{aligned}$$ where $d_{\sigma}$ is the annihilation operator of an electron with spin $\sigma$ in the dot, and $c^{\dagger}_{\alpha \bf k \sigma}$ is the creation operator of an electron with momentum $\bf k $ and spin $\sigma$ in lead $\alpha=\{L,R\}$. They satisfy the usual anticommutation relations: $\{d_{{\sigma}},d_{{{\sigma}}'}^{\dagger} \}= \delta_{{{\sigma}}{{\sigma}}'}$ and $\{c^{}_{\alpha \bf k \sigma}, c^{\dagger}_{\alpha' \bf k' \sigma'} \} =
(2\pi)^3\,\delta({\bf k-k'})\,\delta_{\alpha\alpha'}\,\delta_{\sigma\sigma'}$. We treat the leads as having a constant density of states $\varrho(\omega)=\varrho_0=1/(2D)$, with $2D$ the bandwidth. The tunneling matrix is considered as being momentum independent, $t_{\alpha{\bf k}}= t_{\alpha}$. Its strength is characterized by the usual hybridization function $
\Gamma_{\alpha} = \pi\varrho_0 t_{\alpha}^2$. We define the total hybridization as $\Gamma= \sum_{\alpha=\{L,R\}} \Gamma_\alpha$. The dot itself supports a single orbital with energy $\varepsilon_d$ subject to the on-site Coulomb interaction $U$. Close to the electron-hole symmetric configuration, $\varepsilon_d\simeq -U/2$, the dot is in the Kondo regime, characterized by the Kondo energy scale which is defined as [@Haldane.81] $$T_K = \sqrt{U\, \Gamma\over 4} e^{\pi
\varepsilon_d(\varepsilon_d+U)/\Gamma U}.$$
Currents and response functions {#sub:CH}
-------------------------------
To compute the response functions, we need to define the charge and the heat currents. We introduce first the charge- and heat-transfer operators $$\begin{aligned}
\label{Eq:QK}
Q_\alpha = Q^{(1)}_\alpha &=&e\sum_{\bf k, \sigma}c^{\dagger}_{\alpha \bf k \sigma}c^{}_{\alpha \bf k
\sigma},\nonumber \\
K_\alpha = Q^{(2)}_\alpha &=&\sum_{\bf k, \sigma}(\varepsilon_{\bf k}-\mu_\alpha)c^{\dagger}_{\alpha \bf k
\sigma}c^{}_{\alpha \bf k \sigma}\, , \end{aligned}$$ and define the currents as time derivatives of the corresponding charge/heat operators: $$I_\alpha^{(i)} = (-1)^{i+1}\frac{{\rm d}Q^{(i)}_\alpha}{{\rm d}t}\,.$$ Their explicit expressions can be obtained in terms of the equations of motion as: $$\begin{aligned}
\label{Eq:IL1}
I_{\alpha}^{(1)}&= & i\; \frac{e}{\hbar}\; \sum_{\bf k, \sigma}t_{\alpha}\;
c^{\dagger}_{\alpha \bf k \sigma} d_{\sigma} + H.c.\nonumber\\
I_{\alpha}^{(2)} &= &- \frac{i}{\hbar} \sum_{\bf k, \sigma}(\varepsilon_{\bf k}-
\mu_{\alpha})t_{\alpha}\;
c^{\dagger}_{\alpha \bf k \sigma} d_{\sigma} + H.c. \, .
\label{Eq:IL2}\end{aligned}$$ To avoid a two-channel calculations, it is customary to perform a rotation of the $L/R$ basis to a new one $\{ \alpha_{\bf k \sigma}, \widetilde\alpha_{\bf k \sigma} \}$, defined by: $\alpha_{\bf k \sigma} = \xi_L\,c_{L \bf k \sigma}+\xi_R\,c_{R \bf k \sigma}$ and $\widetilde\alpha_{\bf k \sigma} = \xi_R\,c_{L \bf k \sigma}-\xi_L\,c_{R \bf k
\sigma}$. This unitary transformation decouples the odd channel, $\widetilde \alpha_{{{\mathbf k}}{{\sigma}}}$, from the dot, so that the dot remains coupled only to the even channel, $\alpha_{{{\mathbf k}}{{\sigma}}}$. The coefficients are $\xi_\alpha=t_\alpha/\sqrt{t_L^2+t_R^2}$ and satisfy $\xi_L^2+\xi_R^2=1$. Following this unitary transformation, only the interacting part of the Hamiltonian $H_{\rm int}$ changes to $H_{\rm int} = t_{\rm eff}\;\sum_{\bf k, \sigma}
\alpha^{\dagger}_{\bf k \sigma} d_{\sigma} + h.c. \, ,
$ with $t_{\rm eff}=\sqrt{t_L^2+t_R^2}$. In what follows we shall consider the perfectly symmetric dot, $t_L=t_R=t$. The currents, $I^{(i)}=(I^{(i)}_L-I^{(i)}_R)/2$, transform accordingly, and under equilibrium conditions, $\mu_{L/R} =0$, in the new basis they are defined as: $$\begin{aligned}
I^{(1)}&=&i\,\frac{e\; t_{\rm eff}}{2\sqrt{2}\hbar}\; \sum_{\bf k, \sigma}\;\widetilde\alpha^{\dagger}_{\bf k \sigma}\, d_{\sigma} + H.c.\label{Eq:I_charge} \\
I^{(2)}&=&-i\, \frac{t_{\rm eff}}{2\sqrt{2}\,\hbar} \sum_{{\bf k}, \sigma}\;\varepsilon_{\bf k}\,\left (\widetilde\alpha^{\dagger}_{\bf k \sigma}\, d_{\sigma} + H.c.\right )\label{Eq:I_heat}\, , \end{aligned}$$ and are expressed in terms of the decoupled channel operators only. This allows us to obtain exact results for the response functions. The currents $I^{(i)}_{L/R}$ also have a symmetrical component, which gets subtracted out in the definition of $I^{(i)}$. Within the Kubo formalism, the generalized response functions $L_{ij}$ are given by[@hewson] $$L_{ij}(t,t')= -\frac{i}{\hbar}\Theta(t-t')\langle [I^{(i)}(t), Q^{(j)}(t')]\rangle\, ,$$ where $$\begin{aligned}
Q^{(i)}=\frac{Q^{(i)}_L-Q^{(i)}_R}{2}\;.\end{aligned}$$ We want to express $L_{ij}$ in terms of locally defined operators only, and for that we eliminate the charge operators. In Fourier space we obtain $$\label{Eq:L}
L_{ij}(\omega)=-{T_{ij}(\omega)-T_{ij}(0)\over i\, \omega}\, ,$$ with $T_{ij}(\omega)$ the Fourier transform of the generalized susceptibility: $$\label{Eq:T}
T_{ij}(t,t')= -\frac{i}{\hbar}\, \Theta(t-t')\langle[ I^{(i)}(t),I^{(j)}(t') ]\rangle.$$ Somewhat similar to the calculation of the ac conductance [@Sindel.05], the calculation of thermopower, $S(\omega)= S'(\omega)+S''(\omega)$, reduces to the calculation of $A_{\rm d}(\omega)= -\rm Im\, G_{\rm d}^{\rm R}(\omega)/\pi$ - the spectral representation of the d-level operators in the dot (see Appendix \[app:B\]). In the present work $A_{\rm d}(\omega)$ shall be computed by using the Wilson numerical renormalization group approach[@Wilson.80.1; @Wilson.80.2; @Bulla.08] (NRG) as implemented in the Flexible-NRG code. [@BudapestNRG] Throughout the NRG calculation, the Wilson ratio was fixed to $\Lambda=2$, and we have kept on average 4000 multiplets at each iteration.
Ac thermopower {#sec:TP}
==============
Let us now focus on the calculations of the response functions $L_{ij}$ and the ac thermopower $S(\omega)$. In general, $L_{ij}(\omega)$ are complex functions of $\omega$, as their imaginary parts capture retardation effects due to the external excitation. The full $\omega$ dependence of the ${\rm Re}\, L_{ij}(\omega)$ acquires a relatively compact expression in terms of the spectral representation of the d-level: $$\begin{aligned}
\label{Eq:ReLij}
\textrm{Re}\,L_{ij}(\omega)=\frac{t_{\rm eff}^2}{2\omega \hbar} \left( -\frac{e}{\hbar}\right)^{4-i-j}
\varrho_0
\int \rm d\omega'\left \{\textrm{Im}G_d^R(\omega') \right. \nonumber \\
\left[
(\omega'-\omega)^{i+j-2}\Theta_{\omega'-\omega}f(\omega'-\omega)+\right. \nonumber \\
+ (\omega'+\omega)^{i+j-2}\Theta_{\omega'+\omega}f(\omega')-\nonumber \\
-(\omega'-\omega)^{i+j-2}\Theta_{\omega'-\omega}f(\omega')\nonumber \\
\left. \left.-(\omega'+\omega)^{i+j-2}
\Theta_{\omega'+\omega}f(\omega'+\omega)\right] \right\}\, , \nonumber\\\end{aligned}$$ with $\Theta_\omega=\Theta(D-|\omega|)$ and $f(\omega)$ the Fermi-Dirac distribution. To get the ac conductance [@Sindel.05], the high energy cut-off $D$ can be safely taken to infinity, as $L_{11}$ remains a regular function. This is not the case for $L_{12}$ which diverges at large frequencies when $D\to \infty$, so it is compulsory to have a finite bandwidth for the conduction electrons. To get the imaginary part of $L_{ij}$, the use of the Hilbert transform is unavoidable. We first compute ${\rm Re}\, L_{ij}$, and then ${\rm Im}\, L_{ij}$ is obtained by a Kramers-Kr" onig(KK) transformation $$\label{Eq:KK}
\textrm{Im}\, L_{ij}(\omega)=-\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{\textrm{Re}
\,L_{ij}(\omega')}{\omega'-\omega}\rm d \omega'$$ Although Eq. looks cumbersome, we can interpret it in terms of inelastic tunneling processes and see these correlation functions as rates by which the system absorbs or emits photons [@Basset.12] at frequencies $\omega$. Further details on how to compute $L_{ij}$ are presented in Appendix \[app:B\]. With $L_{ij}$ at hand, $S(\omega)$ can be obtained by using Eq. . By symmetry, $S'(\omega)= S'(-\omega)$, is an even function of frequency, while $S''(\omega)= -S''(-\omega)$, is an odd function. The full $\omega$ and $T$ dependence for $S(\omega)$ is displayed in Figs. \[fig:S\] (a)- \[fig:S\](f), while in Figs. \[fig:S\](g) and \[fig:S\](h) a cut at constant temperature $T= 0.1\, \Gamma$ is presented.
![(Color online) Density plots for the real and imaginary parts of the thermopower in the $(T, \omega)$ plane: (a), (b) Kondo regime, (c), (d) mixed valence regime, and (e), (f) empty orbital regime. Panels (g) and (h) represent the thermopower at $T= 0.1\,\Gamma$ for positive $\omega$ along the dashed lines in the density plots. The darker lines are the locations of the zeros, and mark the positions where the thermopower changes sign. The $(\pm)$ symbols indicate the signs of $S(\omega)$. The NRG parameters used are $U/\Gamma=10$, $\varepsilon_d/\Gamma = -4\, (n=0.985)$, $\varepsilon_d/\Gamma=-0.2\, (n=0.5)$ and $\varepsilon_d/\Gamma = 5\, (n=0.04)$.[]{data-label="fig:S"}](ReS_1.eps "fig:"){width="0.47\columnwidth"} ![(Color online) Density plots for the real and imaginary parts of the thermopower in the $(T, \omega)$ plane: (a), (b) Kondo regime, (c), (d) mixed valence regime, and (e), (f) empty orbital regime. Panels (g) and (h) represent the thermopower at $T= 0.1\,\Gamma$ for positive $\omega$ along the dashed lines in the density plots. The darker lines are the locations of the zeros, and mark the positions where the thermopower changes sign. The $(\pm)$ symbols indicate the signs of $S(\omega)$. The NRG parameters used are $U/\Gamma=10$, $\varepsilon_d/\Gamma = -4\, (n=0.985)$, $\varepsilon_d/\Gamma=-0.2\, (n=0.5)$ and $\varepsilon_d/\Gamma = 5\, (n=0.04)$.[]{data-label="fig:S"}](ImS_1.eps "fig:"){width="0.47\columnwidth"} ![(Color online) Density plots for the real and imaginary parts of the thermopower in the $(T, \omega)$ plane: (a), (b) Kondo regime, (c), (d) mixed valence regime, and (e), (f) empty orbital regime. Panels (g) and (h) represent the thermopower at $T= 0.1\,\Gamma$ for positive $\omega$ along the dashed lines in the density plots. The darker lines are the locations of the zeros, and mark the positions where the thermopower changes sign. The $(\pm)$ symbols indicate the signs of $S(\omega)$. The NRG parameters used are $U/\Gamma=10$, $\varepsilon_d/\Gamma = -4\, (n=0.985)$, $\varepsilon_d/\Gamma=-0.2\, (n=0.5)$ and $\varepsilon_d/\Gamma = 5\, (n=0.04)$.[]{data-label="fig:S"}](ReS_2.eps "fig:"){width="0.47\columnwidth"} ![(Color online) Density plots for the real and imaginary parts of the thermopower in the $(T, \omega)$ plane: (a), (b) Kondo regime, (c), (d) mixed valence regime, and (e), (f) empty orbital regime. Panels (g) and (h) represent the thermopower at $T= 0.1\,\Gamma$ for positive $\omega$ along the dashed lines in the density plots. The darker lines are the locations of the zeros, and mark the positions where the thermopower changes sign. The $(\pm)$ symbols indicate the signs of $S(\omega)$. The NRG parameters used are $U/\Gamma=10$, $\varepsilon_d/\Gamma = -4\, (n=0.985)$, $\varepsilon_d/\Gamma=-0.2\, (n=0.5)$ and $\varepsilon_d/\Gamma = 5\, (n=0.04)$.[]{data-label="fig:S"}](ImS_2.eps "fig:"){width="0.47\columnwidth"} ![(Color online) Density plots for the real and imaginary parts of the thermopower in the $(T, \omega)$ plane: (a), (b) Kondo regime, (c), (d) mixed valence regime, and (e), (f) empty orbital regime. Panels (g) and (h) represent the thermopower at $T= 0.1\,\Gamma$ for positive $\omega$ along the dashed lines in the density plots. The darker lines are the locations of the zeros, and mark the positions where the thermopower changes sign. The $(\pm)$ symbols indicate the signs of $S(\omega)$. The NRG parameters used are $U/\Gamma=10$, $\varepsilon_d/\Gamma = -4\, (n=0.985)$, $\varepsilon_d/\Gamma=-0.2\, (n=0.5)$ and $\varepsilon_d/\Gamma = 5\, (n=0.04)$.[]{data-label="fig:S"}](ReS_3.eps "fig:"){width="0.47\columnwidth"} ![(Color online) Density plots for the real and imaginary parts of the thermopower in the $(T, \omega)$ plane: (a), (b) Kondo regime, (c), (d) mixed valence regime, and (e), (f) empty orbital regime. Panels (g) and (h) represent the thermopower at $T= 0.1\,\Gamma$ for positive $\omega$ along the dashed lines in the density plots. The darker lines are the locations of the zeros, and mark the positions where the thermopower changes sign. The $(\pm)$ symbols indicate the signs of $S(\omega)$. The NRG parameters used are $U/\Gamma=10$, $\varepsilon_d/\Gamma = -4\, (n=0.985)$, $\varepsilon_d/\Gamma=-0.2\, (n=0.5)$ and $\varepsilon_d/\Gamma = 5\, (n=0.04)$.[]{data-label="fig:S"}](ImS_3.eps "fig:"){width="0.47\columnwidth"} ![(Color online) Density plots for the real and imaginary parts of the thermopower in the $(T, \omega)$ plane: (a), (b) Kondo regime, (c), (d) mixed valence regime, and (e), (f) empty orbital regime. Panels (g) and (h) represent the thermopower at $T= 0.1\,\Gamma$ for positive $\omega$ along the dashed lines in the density plots. The darker lines are the locations of the zeros, and mark the positions where the thermopower changes sign. The $(\pm)$ symbols indicate the signs of $S(\omega)$. The NRG parameters used are $U/\Gamma=10$, $\varepsilon_d/\Gamma = -4\, (n=0.985)$, $\varepsilon_d/\Gamma=-0.2\, (n=0.5)$ and $\varepsilon_d/\Gamma = 5\, (n=0.04)$.[]{data-label="fig:S"}](ReS_w.eps "fig:"){width="0.47\columnwidth"} ![(Color online) Density plots for the real and imaginary parts of the thermopower in the $(T, \omega)$ plane: (a), (b) Kondo regime, (c), (d) mixed valence regime, and (e), (f) empty orbital regime. Panels (g) and (h) represent the thermopower at $T= 0.1\,\Gamma$ for positive $\omega$ along the dashed lines in the density plots. The darker lines are the locations of the zeros, and mark the positions where the thermopower changes sign. The $(\pm)$ symbols indicate the signs of $S(\omega)$. The NRG parameters used are $U/\Gamma=10$, $\varepsilon_d/\Gamma = -4\, (n=0.985)$, $\varepsilon_d/\Gamma=-0.2\, (n=0.5)$ and $\varepsilon_d/\Gamma = 5\, (n=0.04)$.[]{data-label="fig:S"}](ImS_w.eps "fig:"){width="0.47\columnwidth"}
![(Color online) (a) The dc-thermopower $S_0=S(\omega=0)$ as function of temperature in different regimes. (b) Temperature dependence of $S^*=S(\omega \gg D)$.[]{data-label="fig:S_zero"}](S_zero.eps "fig:"){width="0.9\columnwidth"} ![(Color online) (a) The dc-thermopower $S_0=S(\omega=0)$ as function of temperature in different regimes. (b) Temperature dependence of $S^*=S(\omega \gg D)$.[]{data-label="fig:S_zero"}](S_bigom.eps "fig:"){width="0.9\columnwidth"}
In Figs. \[fig:S\](a)- \[fig:S\](b), we represent the results for $S(\omega, T)$ when the system is in the Kondo regime, where $\epsilon_d\simeq -U/2$, and $\langle n\rangle \simeq 1$. Figs. \[fig:S\](c) and \[fig:S\](d) present results for $S(\omega, T)$ in the mixed valence regime with $\langle n\rangle = 0.5$, while Figs. \[fig:S\](e) and \[fig:S\](f) show $S(\omega, T)$ in the empty orbital limit, $\langle n \rangle\ll 1$. By symmetry, when $1<n<2$, the thermopower has the same magnitude, but opposite sign, as the role of particles and holes is inverted. At the electron-hole symmetric point, i.e., $n=1$, the particles and holes move together in the same direction under the temperature gradient, and the thermopower vanishes exactly.
At very small frequencies and temperatures, $\{\omega, T\} \to 0 $, only the quasiparticles very close to the Fermi surface give a contribution to the currents flowing through the device, so this limit can be understood in terms of the Fermi-liquid picture. [@Nozieres.74] The Fermi-liquid scale $\Omega_F$, is controlled by either $T_K$ within the Kondo regime, or by $\Gamma$ itself otherwise. When $\{\omega, T\}\ll \Omega_F$, the frequency dependence of $S(\omega)$ is captured by a simple analytical expression $$S(\omega, T) = S_0(T)+ b'(T) \, {\omega}^2+i\,b''(T)\, {\omega}+\dots\, .$$ Here $S_0(T)<0$, is the dc-thermopower, while $b'$ and $b''$ are some coefficients that depend on temperature. In the Kondo regime, $\{b'(T), b''(T)\}\sim 1/T$. In our convention, positive (negative) $S_0$ corresponds to the situation when charge and heat currents flow in the same (opposite) directions. At some intermediate frequencies, $\omega\sim T$, $S'(\omega)$ changes sign and becomes positive. In the Kondo regime, there is another change of sign at a larger, almost constant frequency $\omega_2\simeq \Gamma$, and $S'$ becomes negative in the $\omega\gg D$ limit. The sign of $S_0(T)$ can be associated with the type of dominant carriers in the system at that particular energy: hole like carriers correspond to $S_0>0$, and particle like carriers to $S_0<0$. The first sign change in $S_0(T)$ can be understood as follows: At $T \simeq 0$, the Kondo peak is weighted towards positive energies (for $n<1$), but as temperature increases, it is pulled towards the negative energy region. Thus, the quantum dot shifts from having predominantly particle carriers, to having predominantly hole carriers in the window $\sim T$ that contributes to the transport. Whenever the average entropy carried inside this window becomes zero, the thermopower vanishes. At a finite frequency $\omega$, inelastic tunneling processes in a window $\sim 2 \omega$ around the Fermi level give additional contributions to the transport, see Eq. (\[Eq:ReLij\]). The picture gets more complicated by the existence of retardation effects, which lead to finite imaginary parts in the response functions, and consequently affect the zeros of the thermopower. The high-frequency features in Fig. \[fig:S\] at energies $\omega \sim \{U, D\}$ can be associated with Hubbard charging, and eventually band-edge effects. In Fig. \[fig:S\_zero\], we display the temperature dependence of $S_0(T)$ and $S^*(T)$. It has been already shown [@Costi.93] that in the Kondo regime, when $T\ll T_K$, $S_0$ depends linearly on $T$, $S_0\propto T$. This observation carries over to the mixed valence and empty orbital regimes too, as long as $T\ll \Gamma$. In the opposite limit, $T\gg T_K$, $S_0$ shows a change of sign at some large temperature, and then decays towards zero.
In the large-frequency limit $\omega\gg D$, $S^*$ can be evaluated simply as $$\label{Eq:S_star}
S^*(T) = \frac{1}{T}\frac{L_{12}^*}{L_{11}^*}.$$ with $L_{ij}^*$ some temperature-dependent coefficients, discussed in Appendix \[app:B\] (see Eq. ). As $\omega\gg D$, all states are involved in transport, so that the only energy scale that survives is the bandwidth itself (as long as $\{T_K, U\}\ll D$). Therefore, we expect the features of $S^*(T)$ to carry information only on $D$ itself. In the small-temperature limit, $T\to 0$, the $L_{ij}^*$’s become constants, so a divergent behavior $S^* \propto 1/T$ emerges in this limit. This is clearly visible in Fig. \[fig:S\_zero\] (b), where $S^* T$ becomes constant when $T\ll \Gamma$. At intermediate temperatures, $\Gamma<T<D$, $S^*$ decreases significantly, and vanishes in the in large $T$ limit.
In what follows we shall focus on the strongly correlated regime. When $\{\omega, T\}\leq T_K$, a clear universal behavior emerges as $S(\omega)$ depends exclusively on the $T/T_K$ and $\omega/T_K$ ratios.
![(Color online) Universal scaling functions for a filling $\left <n \right>=0.95$ for a fixed temperature, $T=0.02\;T_K$. In the inset, we represent the $s_0$ universal function. []{data-label="fig:S_scaling"}](S_scaled.eps){width="0.99\columnwidth"}
It was found previously[@Costi.10] that as long as $T\ll T_K$, $$\begin{aligned}
\label{Eq:s0}
s_0 (T/T_K) &=& \Big (\frac{e}{\pi\gamma\, T}\Big )S_0(T)\,\tan \delta_0\nonumber\\
& = & \frac{e}{k_B}\Big (\frac{T_K}{T} \Big )S_0(T)\,\left (\frac{\tan \delta_0}{{{\tilde \gamma}}} \right )\end{aligned}$$ is a universal function that scales with $T/T_K$ up to a filling-dependent phase factor. Here, $\delta_0$ is the phase shift at the Fermi level, $\delta_0 = \pi\langle n\rangle/2$, with $\langle n\rangle $ the average occupation of the dot, $\gamma$ is the specific-heat coefficient of the quantum dot which is filling dependent, and ${{\tilde \gamma}}= \pi \gamma\, T_K/k_B$ is a dimensionless quantity of the order 1. In the inset of Fig. \[fig:S\_scaling\] we represent $s_0(T/T_K)$ as a function of $T/T_K$ for a filling $\langle n \rangle =0.95$. We extend this analysis to finite frequencies where a similar behavior emerges, as the ac thermopower can be expressed as: $$\label{Eq:s}
S(\omega, T) = {k_B\over e} \Big ({T\over T_K}\Big )\, s(\omega/T_K, T/T_K)\, {{\tilde \gamma}}\,\cot(\delta_0)\,.$$ The universal scaling function $s$ depends on $\omega/T_K$ and $T/T_K$ only. A sketch with the frequency dependence is presented in Fig. \[fig:s0\], while in Fig. \[fig:S\_scaling\] we represent the exact numerical calculation for the frequency dependence of $|s|$. In the Fermi-liquid regime, $\{\omega, T\}\ll T_K$, simple analytical expressions can approximate the real and imaginary parts of $s$: $$\begin{aligned}
\label{Eq:un}
s'\Big({\omega\over T_K}, {T\over T_K}\Big)& =& s_0 \Big({T\over T_K}\Big) +
\alpha' \Big({\omega\over T_K}\Big)^2\Big({T_K\over T}\Big)^2+\dots, \nonumber\\
s''\Big({\omega\over T_K}, {T\over T_K}\Big) &= & \alpha'' \Big({\omega\over T_K}\Big)
\Big({T_K\over T}\Big)^2+\dots.\end{aligned}$$ with $\alpha'$ and $\alpha''$ some coefficients $\sim 1$. The $s'\sim \omega^2$ frequency dependence can be related to the virtual Kondo transitions from the singlet ground state to the excited states. [@Glazman.03] These transitions give for the imaginary part of the T-matrix: $\textrm{Im}\,{\rm T}(\omega,T)\propto1- {(3\, \omega^2+\pi^2 T^2)}/{T_K^2}$, when $\{\omega, T\}\ll T_K$, which in turn introduces corrections of the order $\sim \omega^2$ and $\sim T^2$ in the response functions $\textrm{Re}\,L_{ij}$. Simple analytics then show that $s'\sim\omega^2/T^2$. Then, by Hilbert transform, $s''$ is linear in frequency. This scaling for $s$ extends up to frequencies of the order of $\omega \sim T_K$, followed by a sign change at some particular frequencies $\omega_i$. At very large frequencies $\omega\gg T_K$, $s'$ becomes a constant, while $s''\to 0$.
Concluding Remarks {#sec:C}
==================
We have studied the finite-frequency thermopower of a quantum dot described by the Anderson model. For that we have first constructed a general framework which allowed us to investigate in a non-perturbative manner the ac thermopower. When calculating the ac conductance [@Sindel.05], it is safe to take the bandwidth $D\to \infty$, but when we address the problem of the ac thermopower, it is compulsory to keep $D$ finite. Although $S(\omega)$ presents a relatively rich structure that includes several sign changes, in the Fermi liquid regime a simple analytical expression is able to capture its behavior over a broad range of temperatures and frequencies. In the Kondo regime, the ac thermopower is characterized by a universal function that we have determined numerically. We have also found that the $S_0$ and $S^*$ have a markedly different behavior in the low-temperature limit.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank I. Weymann for carefully reading our manuscript. This research has been financially supported by UEFISCDI under French-Romanian Grant DYMESYS (PN-II-ID-JRP-2011-1 and ANR 2011-IS04-001-01) and by Hungarian Research Funds under grant Nos. K105149, CNK80991, TAMOP-4.2.1/B-09/1/KMR-2010-0002.
Green’s Functions {#app:A}
=================
Within the L/R basis transformation, one channel becomes decoupled, and can be treated as a non-interacting one. It is simply described by the non-interacting Hamiltonian $H_{0} = \sum_{\bf k, \sigma}\varepsilon_{\bf k}\;
\widetilde{\alpha}^{\dagger}_{\bf k \sigma}\,\widetilde{\alpha}_{\bf k \sigma}$. In what follows, we shall fix the chemical potential to zero. The non-equilibrium evolution of the system is described by the conduction electron Green’s function: $g_{\bf k \sigma}(t-t')=-i\langle {\cal T_C} \widetilde{\alpha}^{}_{\bf k \sigma}(t)
\widetilde{\alpha}^{\dagger}_{\bf k \sigma}(t')\rangle
$, where $\cal T_C$ is the time ordering operator on the Keldysh contour. Within this language, we can define four Green’s functions. Two combinations define the greater and lesser components $$\begin{aligned}
\label{Eq:g}
g^{>}_{\bf k \sigma}(t-t')&=&-i\langle \widetilde{\alpha}^{}_{\bf k \sigma}(t)\,
\widetilde{\alpha}^{\dagger}_{\bf k \sigma}(t')\rangle, \nonumber \\
g^{<}_{\bf k \sigma}(t-t')&=&i\langle \widetilde{\alpha}^{\dagger}_{\bf k \sigma}(t')\,
\widetilde{\alpha}^{}_{\bf k \sigma}(t)
\rangle,\end{aligned}$$ while the other two define the time and anti-time ordered ones: $
g^{t}_{\bf k \sigma}(t-t')= -i\langle {\cal T}\,\widetilde{\alpha}^{}_{\bf k \sigma}(t)\,
\widetilde{\alpha}^{\dagger}_{\bf k \sigma}(t')\rangle$, and $
g^{\tilde t}_{\bf k \sigma}(t-t')=-i\langle\tilde {\cal T}\widetilde{\alpha}^{}_{\bf k \sigma}(t)\,
\widetilde{\alpha}^{\dagger}_{\bf k \sigma}(t')\rangle$. Here ${\cal T}$ and $ \cal \tilde T$ are the time and anti-time ordering operators. We also introduce the retarded and the advanced Green’s functions, which are defined in the usual way: $$\begin{aligned}
g^{A}_{\bf k \sigma}(t-t') & = & i \Theta (t'-t)\langle\{\widetilde{\alpha}^{}_{\bf k \sigma}(t),
\widetilde{\alpha}^{\dagger}_{\bf k \sigma}(t')\}\rangle,\nonumber \\
g^{R}_{\bf k \sigma}(t-t') & = & -i \Theta (t-t')\langle\{\widetilde{\alpha}^{}_{\bf k \sigma}(t),
\widetilde{\alpha}^{\dagger}_{\bf k \sigma}(t')\}\rangle.\end{aligned}$$ We are interested in the momentum integrated Green’s function $g_{\sigma}(\omega)= \sum_{\bf k}g_{\bf k{{\sigma}}}(\omega)$, as these are the only quantities that explicitly enter the expression for the ac thermopower. Here, we consider the simplest situation of a dispersionless electronic band with a band cutoff D. It is characterized by a constant density of states $N(\omega) = 1/(2D)\, \Theta(D-|\omega|)= N(0)\, \Theta_{\omega}$, with $N(0)= 1/(2D)$, the DOS at the Fermi level and $\Theta_{\omega}=\Theta(D-|\omega|)$. Then, the momentum integrated Green’s function have relatively simple analytical expressions [@Koerting.07]: $g^{>}_{\sigma}(\omega)=-2 \pi i\, (1-f(\omega))\,N(0)\,\Theta_{\omega}$, $g^{<}_{\sigma}(\omega)=2 \pi i\, f(\omega)\,N(0)\,\Theta_{\omega}$ and $\textrm{Im}\, g^{A}_{\sigma}(\omega)=-\textrm{Im}\, g^{R}_{\sigma}(\omega) = i\, \pi
N(0)\Theta_{\omega} $. Usually, $\textrm{Re}\, g^{R/A}$ is neglected in the large-bandwidth limit, but as long as we are interested in the response functions $L_{ij}$ in the large $\omega>D$ limit, its contribution becomes important.
Current Correlations and the dynamical transport coefficients {#app:B}
==============================================================
![(Color online) The imaginary part of the dynamical coefficients $L_{ij}(\omega)$ as function of frequency. The inset shows the asymptotic $\sim 1/\omega$ behavior in the large-$\omega$ limit in the Kondo regime for $\langle n \rangle = 0.985$. The temperature was fixed to T=$10^{-5} \Gamma$.[]{data-label="fig:ImLij"}](ImL11.eps "fig:"){width="0.49\columnwidth"} ![(Color online) The imaginary part of the dynamical coefficients $L_{ij}(\omega)$ as function of frequency. The inset shows the asymptotic $\sim 1/\omega$ behavior in the large-$\omega$ limit in the Kondo regime for $\langle n \rangle = 0.985$. The temperature was fixed to T=$10^{-5} \Gamma$.[]{data-label="fig:ImLij"}](ImL12.eps "fig:"){width="0.49\columnwidth"}
![(Color online) The temperature dependence of $L_{ij}^*$ in the Kondo regime. The results are obtained by using the sum rule expression, Eq. .[]{data-label="fig:L_star"}](L_star.eps){width="0.95\columnwidth"}
![(Color online). The temperature dependence of $\rm Re\, L_{ij}^{(0)}$. Here $L_{11}^{(0)}$ can be identified with the dc-conductance itself.[]{data-label="fig:ReL0"}](L11_zero.eps "fig:"){width="0.49\columnwidth"} ![(Color online). The temperature dependence of $\rm Re\, L_{ij}^{(0)}$. Here $L_{11}^{(0)}$ can be identified with the dc-conductance itself.[]{data-label="fig:ReL0"}](L12_zero.eps "fig:"){width="0.49\columnwidth"}
To get the dynamical transport coefficients, we need to compute the retarded response functions defined in Eq. , with the current operators defined in Eqs. and . In the even/odd basis, one channel becomes decoupled and the charge current correlator $T_{11}$ can be evaluated as: $$\begin{gathered}
\label{Eq:T11}
T_{11}(t-t')=\Theta(t-t')\frac{e^2}{\hbar^3}\frac{t_{\rm eff}^2}{4}\sum_{\bf k, \sigma} \nonumber \\
\left \{ G_{d}^{>}(t-t')g^{<}_{\bf k \sigma}(t'-t)-G_{d}^{<}(t-t')g^{>}_{\bf k \sigma}(t'-t)-\right.\nonumber \\
\left. -G_d^{<}(t'-t)g^{>}_{\bf k \sigma}(t-t')-G_d^{>}(t'-t)g^{<}_{\bf k \sigma}(t-t') \right \},
\end{gathered}$$ which gives for $L_{11}$ defined in Eq. a relatively compact expression: $$\begin{gathered}
\label{Eq:ReL11}
\textrm{Re}\,L_{11}(\omega)=-\frac{t_{\rm eff}^2}{2\omega} \frac{e^2}{\hbar^3} \varrho_0
\int \rm d\omega'\left \{\textrm{Im}G_d^R(\omega')\left[\right. \right.\nonumber \\
\Theta_{\omega'-\omega}f(\omega'-\omega) + \Theta_{\omega'+\omega}f(\omega')-\Theta_{\omega'-\omega}f(\omega')-
\nonumber \\
\left. \left.-\Theta_{\omega'+\omega}f(\omega'+\omega)\right] \right\}.\end{gathered}$$ Notice that within the present formalism, we can identify $\rm Re\, L_{11}(\omega)$ with the real part of the ac conductance $G(\omega)$. [@Sindel.05] A similar expression can be derived for $T_{12}(t,t')$, which in the Fourier space becomes: $$\begin{gathered}
\label{Eq:T12}
T_{12}(\omega)=-\frac{t_{\rm eff}^2}{4}\frac{e}{\hbar^3}\sum_{\bf k, \sigma}
\varepsilon_{\bf k}\int\frac{\rm d \omega'}{2\pi}\left\{G_d^R(\omega+\omega') g_{\bf k \sigma}^{<}(\omega')
\right. \nonumber \\
+G_d^{<}(\omega+\omega')g_{\bf k \sigma}^{A}(\omega')
+G_d^{>}(\omega')g_{\bf k \sigma}^{R}(\omega+\omega')+ \nonumber \\
\left.+G_d^{A}(\omega')g_{\bf k \sigma}^{>}(\omega+\omega')\right \}.\end{gathered}$$ Subtracting the $T_{12}(\omega=0)$ term and dividing by $-i\omega$, the real part of $L_{12}$ is obtained as: $$\begin{gathered}
\label{Eq:ReL12}
\textrm{Re}\,L_{12}(\omega)=\frac{t_{\rm eff}^2}{2\omega} \frac{e}{\hbar^2}
\varrho_0
\int \rm d\omega'\left \{\textrm{Im}G_d^R(\omega')\left[\right. \right.\nonumber \\
(\omega'-\omega)\Theta_{\omega'-\omega}f(\omega'-\omega) + (\omega'+\omega)\Theta_{\omega'+\omega}f(\omega')-\nonumber \\
\left. \left.-(\omega'-\omega)\Theta_{\omega'-\omega}f(\omega')
-(\omega'+\omega)\Theta_{\omega'+\omega}f(\omega'+\omega)\right] \right\}.\end{gathered}$$
In these expressions, $L_{ij}$ depends explicitly on the retarded localized d-level Green’s function $\rm G_d^R(\omega)$. This quantity shall be computed exactly by using the NRG method. In this way, Eqs. and are the exact expressions for the real parts of the Onsager transport coefficients, and no approximation of any kind was used so far. The ac thermopower depends not only on the real, but also on the imaginary parts of $L_{ij}$. Actually their imaginary parts give the main contribution in the large-frequency limit. To obtain them, we use the Kramers-Krönig relations, Eq. . In the ac limit, when $\omega$ is the largest energy scale ($\omega\gg D$), we can simplify considerably the calculation by noticing that $$\label{Eq:asymptotic}
\textrm{Im}\,L_{ij}(\omega)\simeq \frac{L_{ij}^{*}}{\omega}$$ with $L_{ij}^*$ some coefficients, $$\label{Eq:L_star}
L_{ij}^*=\frac{1}{\pi}\int_{-\infty}^{+\infty}\textrm{Re}\,L_{ij}(\omega')\rm d\omega' ,$$ which are thus determined as the sum rule of dynamical quantities. [@Xu.11] In Fig. \[fig:ImLij\] we represent the imaginary parts of $L_{ij}$ in the Kondo regime, as computed by doing the KK transformations of Eq. and . The insets display the large-frequency behavior, which indicates that our approximation, Eq. , is indeed correct. The temperature dependence of $L_{ij}^*$ is displayed in Fig. \[fig:L\_star\].
In the small-frequency limit, $\omega\rightarrow 0$, the calculation can be simplified again. Introducing the notation $\textrm{Re}\, L_{ij}^{(0)}=\textrm{Re}L_{ij}\,(\omega\rightarrow 0)$, we notice that Eqs. and reduce considerably to $$\begin{gathered}
\label{Eq:KK_L}
\textrm{Re}\,L_{ij}^{(0)} = \frac{t_{\rm eff}^2}{\hbar}\left( -\frac{e}{\hbar}\right)^{4-i-j}
\varrho_0\times\,\\
\int \rm d\tilde \omega\, \left ( \tilde \omega ^{i+j-2} \textrm{Im}G_d^R(\tilde \omega)
\frac{\partial f(\tilde \omega)}{\partial \tilde \omega}\right ).\end{gathered}$$ Here, $\textrm{Re}\, L_{11}^{(0)}$ is the dc-conductance itself. Its temperature dependence is displayed in Fig. \[fig:ReL0\]. As expected in the $T\rightarrow 0$ limit, $\textrm{Re}\,
L_{11}^{(0)}$ shows the usual Kondo behavior, as the system is close to the unitary limit. From Eqs. (\[Eq:KK\_L\]) and (\[Eq:S\_omega\]) one can notice that the thermopower has the form of an average entropy $-\left<\epsilon-\mu\right>/(eT)$ carried per particle/hole across the quantum dot. Thus, for the case of perfect particle-hole symmetry, the thermopower becomes zero. This form can also be used to justify the $1/T$ dependence of $S^*$ (see Fig. \[fig:S\]).
|
---
abstract: 'Let $M$ be a closed (compact with no boundary) spherical $CR$ manifold of dimension $2n+1$. Let $\widetilde{M}$ be the universal covering of $M.$ Let $\Phi $ denote a $CR$ developing map $$\Phi :\widetilde{M}\rightarrow S^{2n+1}$$where $S^{2n+1}$ is the standard unit sphere in complex $n+1$-space $C^{n+1}$. Suppose that the $CR$ Yamabe invariant of $M$ is positive. Then we show that $\Phi $ is injective for $n\geq 3$. In the case $n=2$, we also show that $\Phi $ is injective under the condition: $s(M)<1$. It then follows that $M$ is uniformizable.'
address:
- 'Institute of Mathematics, Academia Sinica, Taipei, 11529 and National Center for Theoretical Sciences, Taipei Office, Taiwan, R.O.C.'
- 'Department of Mathematics, National Central University, Chung Li, 32054, Taiwan, R.O.C.'
- 'Department of Mathematics, Princeton University, Princeton, NJ 08544, U.S.A.'
author:
- 'Jih-Hsin Cheng'
- 'Hung-Lin Chiu'
- Paul Yang
title: Uniformization of spherical $CR$ manifolds
---
Introduction and statement of the results
=========================================
Spherical $CR$ structures are modeled on the boundary of complex hyperbolic space. There have been many studies in various aspects for this structure (e.g., [@BS], [@KT], [@FG], [@Gol], [@CT], [@Sch]). In this paper, we study the uniformization problem. Let $S^{2n+1}$ denote the standard unit sphere in complex $n+1$-space $C^{n+1}$. Let us start with the $CR$ automorphism group $Aut_{CR}(S^{2n+1})$ of $S^{2n+1},$ which is the group of complex fractional linear transformation $SU(n+1,1)/(\text{center})$. We have the following complex analogue of the Liouville theorem in conformal geometry ([@CM]).
**Lemma 1.1.** Let $f$ be a $CR$ diffeomorphism from a connected open set $U$ in $S^{2n+1}.$ If $f(U)\subset S^{2n+1}$, then $f$ is the restriction to $U$ of a complex fractional linear transformation.
Let $M$ be a spherical $CR$ manifold of dimension $2n+1$. Let $\widetilde{M}$ be the universal covering of $M$. Using analytic continuation and Lemma 1.1, we gets a $CR$ immersion $\Phi :\widetilde{M}\rightarrow S^{2n+1}.$ The map $\Phi $ is unique up to composition with elements of $Aut_{CR}(S^{2n+1})$ acting on $S^{2n+1}$. Such a map $\Phi $ is called a $CR$ developing map.
We will determine when $\Phi $ is injective. Let $\lambda (M)$ denote the $CR $ Yamabe invariant of $M$ (see (\[Ya\]) in Section 2). In the case of $n
$ $=$ $2,$ we also need a condition on another $CR$ invariant $s(M)$ which measures the integrability of a positive minimal Green’s function $G_{p}$ on $\widetilde{M}$ (see Theorem 3.4 in Section 3):$$s(M):=\inf {\{s:\int_{\widetilde{M}\setminus U_{p}}G_{p}^{s}dV_{\theta
}<\infty \}}$$where $U_{p}$ is a neighborhood of $p$.(see (\[s(M)\]) in Section 3). Observe that $s(M)$ $\leq $ $1$ in general (see Theorem 3.5 in Section 3). We have the following result.
**Theorem A.** *Let* $M$* be a closed (compact with no boundary) spherical* $CR$* manifold of dimension* $2n+1$*with *$\lambda (M)>0.$*. Let* $\widetilde{M}$* be the universal covering of* $M.$* Let* $\Phi $* denote a* $CR$* developing map* $$\Phi :\widetilde{M}\rightarrow S^{2n+1}$$*Then *$\Phi $ *is injective for* $n\geq 3$*. In case* $n=2$*,* $\Phi $* is injective if we further assume* $s(M)<1$*.*
**
Theorem A implies that a closed spherical $CR$ manifold $M$ with $\lambda
(M)>0$ is uniformizable. Let $\pi _{1}(M)$ denote the fundamental group of $M.$ The $CR$ developing map $\Phi $ induces a group homomorphism: $$\Phi _{\ast }:\pi _{1}(M)\rightarrow Aut_{CR}(S^{2n+1}).$$In case $\Phi $ is injective, the group homomorphism $\Phi _{\ast }$ is injective. Note that $\pi _{1}(M)$ acts on $\widetilde{M}$ by deck transformations. The following result follows from Theorem A.
**Corollary B.** *Suppose that we are in the situation of Theorem A. Then* $M$* is* $CR$* diffeomorphic to the quotient* $\Omega /\Gamma $* where* $\Omega =\Phi (\widetilde{M})$* *$\subset $* *$S^{2n+1}$* and* $\Gamma =\Phi
_{\ast }(\pi _{1}(M))$ *for* $n$ $\geq $ $2$*. Moreover,* $\Gamma $* is a discrete subgroup of* $Aut_{CR}(S^{2n+1})$*and acts on* $\Omega $* properly discontinuously.*
**
The idea of the proof of Theorem A follows a similar line as for the conformal case. Basically we will be dealing with the Green’s functions of the $CR$ invariant sublaplacian (see (\[ISL\]) in Section 2) on different spaces. In particular, the idea of comparing the pull-back $\bar{G}$ of the Green’s function on $S^{2n+1}$ with the (minimal positive) Green’s function $G$ of $\widetilde{M}$ follows the work of Schoen and Yau ([@SY1] or [SY2]{}). We reduce the injectivity problem to the estimate of the quotient $v$ $:=$ $\frac{G}{\bar{G}}.$ As expected, the $CR$ Bochner formula for $v$ contains an extra cross term which has no Riemannian analogue. Fortunately we can manage this extra cross term by converting it into a term involving a Paneitz-like operator $P$ (see (\[Pan\]) in Section 2). The nonnegativity of $P$ for $n$ $\geq $ $2$ (see line 5 in the proof of Proposition 3.2 in [@GL] or [@CC]) helps to simplify the estimates (see (\[24\]) and (\[4\]) in Section 4). We can finally prove $v$ $=$ $1,$ and hence $\Phi $ is injective under the condition mentioned in Theorem A.
There has been an unpublished paper ([@Li1]) about this uniformization problem, circulating for years. The main difference between our paper and [@Li1] is the treatment of the $CR$ Bochner formula. We have realized the important role of that Paneitz-like operator $P$ in the $CR$ setting of the Bochner formula through the study of some other problems in recent years (e.g., [@Chiu], [@CC]). So we can clarify some estimates in [Li1]{} and conclude a new result in the case of $n$ $=$ $2.$
Based on the uniformization of spherical $CR$ manifolds, in his another unpublished paper ([@Li3]), using Bony’s strong maximum principle, Z. Li showed the nonnegativity of the CR mass (see Definition 5.1 in Section [seccrmass]{}). We state his result as Corollary C:
**Corollary C.** * Let* $M$ *be a closed spherical* $\mathit{CR}$* manifold with $\lambda (M)>0$.* *Then, for $n\geq 3$, the *$CR$ *mass *$A_{b}>0$ * unless* $M$ * is the standard sphere.* * In case* $n=2,\ $ *the same result also holds if we assume futher* $s(M)<1$. **
Notice that Z. Li’s arguments are valid, provided that the $CR$ developing map $\Phi $ is injective. We rewrite his proof in Section \[seccrmass\]. To solve the $CR$ Yamabe problem by producing a minimizer for the Yamabe (or Tanaka-Webster) quotient, one needs to work out a test function estimate by using the above positive mass theorem. This has been done in [@Li3] (see also unpublished notes of Andrea Malchiodi). In dimension 3, we also have a positive mass theorem under the condition $P$ $\geq $ $0$ through a different approach ([@CMY]). Note that in dimension 3 (with $\lambda (M)$ $>$ $0)$, the condition $P$ $\geq $ $0$ is not automatic and is shown to be almost equivalent to the embeddability of the underlying $CR$ structure ([@CCY1], [@CCY2]).
**Acknowledgments.** The first author’s research was supported in part by NSC 98-2115-M-001-008-MY3, the second author’s research was supported in part by CIZE Foundation and in part by NSC 96-2115-M-008-017-MY3, and the third author’s research was supported in part by DMS-0758601.
Bochner formulas and $CR$ invariant operators
=============================================
Let $M$ be a smooth (meaning $C^{\infty }$ throughout the paper$)$ ($2n+1)$-dimensional (paracompact) manifold. A contact structure or bundle $\xi $ on $M$ is a completely nonintegrable $2n$-dimensional distribution. A contact form is a 1-form annihilating $\xi $. Let $(M,\xi )$ be a contact ($2n+1)$-dimensional manifold with an oriented contact structure $\xi $. There always exists a global oriented contact form $\theta $, obtained by patching together local ones with a partition of unity. The Reeb vector field of $\theta $ is the unique vector field $T$ such that $\theta (T)=1$ and $\mathcal{L}_{T}\theta =0$ or $d\theta (T,{\cdot })=0$. A $CR$-structure compatible with $\xi $ is a smooth endomorphism $J:{\xi }$ ${\rightarrow }$ ${\xi }$ such that $J^{2}=-Identity$. Let $T_{1,0}$ $\subset $ $\xi \otimes C$ denote the $n$-dimensional complex subbundle of $TM\otimes C,$ consisting of eigenvectors of $J$ with eigenvalue $i.$ We will assume throughout that the $CR$ structure $J$ is integrable, that is, $T_{1,0}$ satisfies the condition $[T_{1,0},$ $T_{1,0}]$ $\subset $ $T_{1,0}.$ A pseudohermitian structure compatible with an oriented contact structure $\xi $ is a $CR$-structure $J$ compatible with $\xi $ together with a global contact form $\theta $. On $\xi ,$ we define the Levi form $L_{\theta }$ $:=$ $\frac{1}{2}d\theta (\cdot
,J\cdot ).$ If $L_{\theta }$ is definite (independent of the choice of contact form), $M$ (or $(M,\xi ,J)$) is said to be strictly pseudoconvex. We call $\theta $ positive if $L_{\theta }$ is positive definite (often called Levi metric in this case). We will always assume that $M$ is strictly pseudoconvex and $\theta $ is positive.
Given a pseudohermitian structure $(J,\theta )$ (with $J$ integrable and $\theta $ positive), we can choose complex vector field $Z_{\alpha }$, $\alpha $ $=$ $1,$ $2,...,$ $n,$ eigenvectors of $J$ with eigenvalue $i$, and complex 1-form ${\theta }^{\alpha },$ $\alpha $ $=$ $1,$ $2,...,$ $n,$ such that $\{\theta ,{\theta ^{\alpha }},{\theta ^{\bar{\alpha}}}\}$ is dual to $\{T,Z_{\alpha },Z_{\bar{\alpha}}\}$ (${\theta ^{\bar{\alpha}}}={\bar{({\theta ^{\alpha }})}}$,$Z_{\bar{\alpha}}={\bar{({Z_{\alpha }})}}$). It follows that $$d\theta =ih_{\alpha \bar{\beta}}{\theta ^{\alpha }}{\wedge }{\theta }^{\bar{\beta}}$$(summation convention throughout) for some hermitian matrix of functions $(h_{\alpha {\bar{\beta}}})$, which is positive definite since $M$ is strictly pseudoconvex and $\theta $ is positive.
The pseudohermitian connection of $(J,\theta )$ is the connection $\nabla
^{p.h.}$ on $TM{\otimes }C$ (and extended to tensors) given by
$${\nabla }^{p.h.}Z_{\alpha }={\omega _{\alpha }}^{\beta }{\otimes }Z_{\beta },{\nabla }^{p.h.}Z_{\bar{\alpha}}={\omega _{\bar{\alpha}}}^{\bar{\beta}}{\otimes }Z_{\bar{\beta}},{\nabla }^{p.h.}T=0$$
in which the 1-forms ${\omega _{\alpha }}^{\beta }$ are uniquely determined by the following equations with a normalization condition ([We]{}, [@Ta], [@Lee]):$$\begin{aligned}
d{\theta ^{\beta }} &=&{\theta ^{\alpha }}{\wedge }{\omega _{\alpha }}^{\beta }+{A^{\beta }}_{\bar{\alpha}}\theta {\wedge }{\theta ^{\bar{\alpha}},} \label{2.1} \\
dh_{\alpha \bar{\gamma}} &=&{\omega _{\alpha }}^{\beta }h_{\beta \bar{\gamma}}+h_{\alpha \bar{\beta}}{\omega _{\bar{\gamma}}}^{\bar{\beta}}. \notag\end{aligned}$$
The coefficient ${A^{\beta }}_{\bar{\alpha}}$ in ($\ref{2.1})$ is called the (pseudohermitian) torsion. As usual we use the matrix $h_{\alpha
\bar{\beta}}$ to raise and lower indices, e.g., ${A}_{\alpha \gamma }$ $=$ $h_{\alpha \bar{\beta}}{A^{\bar{\beta}}}_{\gamma }$ where ${A^{\bar{\beta}}}_{\gamma }$ is the complex conjugate of ${A^{\beta }}_{\bar{\gamma}}.$ We define covariant differentiation with respect to the connection $\nabla
^{p.h.}.$ For a real $C^{\infty }$ smooth function $\varphi ,$ we have $\varphi _{0}$ $:=$ $T\varphi ,$ $\varphi _{\alpha }$ $:=$ $Z_{\alpha
}\varphi $, $\varphi _{\alpha \beta }$ $:=$ $Z_{\beta }Z_{\alpha }\varphi -{\omega _{\alpha }}^{\gamma }(Z_{\beta })Z_{\gamma }\varphi ,$ etc.. For the subgradient $\nabla _{b}\varphi $, the sublaplacian ${\Delta }_{b}\varphi ,$ and the subhessian ($\nabla ^{H}$)$^{2}\varphi $, we have the following formulas:$$\begin{aligned}
\nabla _{b}\varphi &=&\varphi ^{\alpha }Z_{\alpha }+\varphi ^{\bar{\alpha}}Z_{\bar{\alpha}} \\
{\Delta }_{b}\varphi &=&-(\varphi {_{\alpha }}^{\alpha }+\varphi {_{\bar{\alpha}}}^{\bar{\alpha}}) \\
(\nabla ^{H})^{2}\varphi &=&\varphi {_{\alpha }}^{\beta }\theta ^{\alpha
}\otimes Z_{\beta }+\varphi {_{\bar{\alpha}}}^{\beta }\theta ^{\bar{\alpha}}\otimes Z_{\beta } \\
&&+\varphi {_{\alpha }}^{\bar{\beta}}\theta ^{\alpha }\otimes Z_{\bar{\beta}}+\varphi {_{\bar{\alpha}}}^{\bar{\beta}}\theta ^{\bar{\alpha}}\otimes Z_{\bar{\beta}}.\end{aligned}$$Differentiating ${\omega _{\beta }}^{\alpha }$ gives
$$\begin{aligned}
&&d{\omega _{\beta }}^{\alpha }{-\omega _{\beta }}^{\gamma }\wedge {\omega
_{\gamma }}^{\alpha } \\
&=&R{_{\beta }}^{\alpha }{}_{\rho \bar{\sigma}}\theta ^{\rho }\wedge \theta
^{\bar{\sigma}}+i{A^{\alpha }}_{\bar{\gamma}}\theta _{\beta }\wedge \theta ^{\bar{\gamma}}-iA_{\beta \gamma }\theta ^{\gamma }\wedge \theta ^{\alpha }\text{ mod }\theta\end{aligned}$$
where $R{_{\beta }}^{\alpha }{}_{\rho \bar{\sigma}}$is the Tanaka-Webster curvature. Write $R_{\alpha \bar{\beta}}$ $:=$ $R{_{\gamma }}^{\gamma }{}_{\alpha \bar{\beta}}$ and $R$ $:=$ $R{_{\alpha }}^{\alpha }.$ For $X$ $=$ $X^{\alpha }Z_{\alpha },$ $Y$ $=$ $Y^{\beta }Z_{\beta }$ $\in $ $T_{1,0},$ we define $$\begin{aligned}
Ric(X,Y) &=&R_{\alpha \bar{\beta}}X^{\alpha }Y^{\bar{\beta}} \\
Tor(X,Y) &=&2\func{Re}(iA_{\bar{\alpha}\bar{\beta}}X^{\bar{\alpha}}Y^{\bar{\beta}}).\end{aligned}$$
We recall the pointwise Bochner formula ([@Gr]):$$\begin{aligned}
\frac{1}{2}{\Delta }_{b}|\nabla _{b}\varphi |^{2} &=&-|(\nabla
^{H})^{2}\varphi |^{2}+<\nabla _{b}\varphi ,\nabla _{b}{\Delta }_{b}\varphi >
\label{BF} \\
&&-2Ric((\nabla _{b}\varphi )_{C},(\nabla _{b}\varphi )_{C}) \notag \\
&&+(n-2)Tor((\nabla _{b}\varphi )_{C},(\nabla _{b}\varphi )_{C}) \notag \\
-2 &<&J\nabla _{b}\varphi ,\nabla _{b}\varphi _{0}> \notag\end{aligned}$$
for a real smooth function $\varphi ,$ where the length $\cdot $ $|$ and the inner product $<\cdot ,\cdot >$ are with respect to the Levi metric $L_{\theta }$ and $(\nabla _{b}\varphi )_{C}$$:=$ $\varphi ^{\alpha }Z_{\alpha }.$
We define a Paneitz-like operator $P$ by$$P\varphi :=4(\varphi {_{\bar{\alpha}}}^{\bar{\alpha}}{_{\beta }}+inA_{\beta
\alpha }\varphi ^{\alpha })^{\beta }+conjugate. \label{Pan}$$
Let $P_{\beta }\varphi $ $:=$ $\varphi {_{\bar{\alpha}}}^{\bar{\alpha}}{_{\beta }}+inA_{\beta \alpha }\varphi ^{\alpha }.$ For $n$ $=$ $1$, the $CR$ pluriharmonic functions are characterized by $P_{1}\varphi $ $=$ $0$ ([@Lee2]) while, for $n$ $\geq $ $2,$ they are characterized by $P\varphi $ $=$ $0.$ (see [@GL] in which $P$ is also identified with the compatibility operator for solving a certain degenerate Laplace equation in the case of $n$ $=$ $1$). On the other hand, this operator $P$ is a $CR$ analogue of the Paneitz operator in conformal geometry (see [@Hir] for the relation to a $CR$ analogue of the $Q$-curvature and the $\log $-term coefficient in the Szegö kernel expansion). On a closed pseudohermitian ($2n+1)$-dimensional manifold $(M,$ $J,$ $\theta ),$ we call $P$ nonnegative if there holds$$\int_{M}\varphi (P\varphi )dV_{\theta }\geq 0 \label{NG}$$
for all real smooth functions $\varphi ,$ in which the volume form $dV_{\theta }$ $=$ $\theta \wedge (d\theta )^{n}.$ We need the integrated Bochner formula:$$\begin{aligned}
\int_{M}\varphi _{0}^{2}dV_{\theta } &=&\frac{1}{n^{2}}\int_{M}({\Delta }_{b}\varphi )^{2}dV_{\theta } \label{IBF} \\
&&+\frac{2}{n}\int_{M}Tor((\nabla _{b}\varphi )_{C},(\nabla _{b}\varphi
)_{C})dV_{\theta } \notag \\
&&-\frac{1}{2n^{2}}\int_{M}\varphi (P\varphi )dV_{\theta } \notag\end{aligned}$$
(see Corollary 2.4 in [@CC]). By Theorem 3.2 in [@CC], we learn that $P$ is nonnegative for $n$ $\geq $ $2.$ It follows from (\[IBF\]) and (\[NG\]) that$$\begin{aligned}
\int_{M}\varphi _{0}^{2}dV_{\theta } &\leq &\frac{1}{n^{2}}\int_{M}({\Delta }_{b}\varphi )^{2}dV_{\theta } \label{IBF2} \\
&&+2\kappa \int_{M}|\nabla _{b}\varphi |^{2}dV_{\theta } \notag\end{aligned}$$
where $\kappa $ $:=$ $\max_{q\in M}(\sum_{\alpha ,\beta
}(A_{\alpha \beta }A^{\alpha \beta })(q))^{1/2}$ (note that $\sum_{\alpha
,\beta }A_{\alpha \beta }A^{\alpha \beta }$ is independent of the choice of frames and is a nonnegative real function on $M$).
Let $b_{n}$ $:=$ $2+\frac{2}{n}.$ We define the $CR$ invariant sublaplacian $D_{\theta }$ by$$D_{\theta }=b_{n}{\Delta }_{b}+R \label{ISL}$$
where $R$ denotes the Tanaka-Webster scalar curvature (with respect to $\theta $ while fixing $J)$. Suppose that $\tilde{\theta}$ $=$ $u^{\frac{2}{n}}\theta $ for a positive $C^{\infty }$ smooth function $u.$ Then for any real smooth function $\varphi ,$ there holds$$D_{\theta }(u\varphi )=u^{1+\frac{2}{n}}D_{\tilde{\theta}}(\varphi ).
\label{TL}$$
Letting $\varphi $ $\equiv $ $1$ in (\[TL\]) gives the transformation law for $R:$$$\tilde{R}=u^{-1-\frac{2}{n}}D_{\theta }(u)$$
where $\tilde{R}$ denotes the Tanaka-Webster scalar curvature with respect to $\tilde{\theta}.$ The Yamabe problem on a $CR$ manifold is to find $u$ (or $\tilde{\theta})$ such that $\tilde{R}$ is a given constant. This is the Euler-Lagrange equation of the following energy functional:$$E_{\theta }(u):=\int_{M}(b_{n}|\nabla _{b}u|^{2}+Ru^{2})dV_{\theta }.$$
for positive smooth functions $u$ such that $$\int_{M}|u|^{b_{n}}dV_{\theta }=1. \label{C}$$
The $CR$ Yamabe invariant $\lambda (M)$ has the following expression:$$\lambda (M)=\inf_{u\in \Xi }E_{\theta }(u) \label{Ya}$$
where $\Xi $ is the space of positive smooth (with compact support if $M$ is noncompact) functions $u$ satisfying (\[C\]). For $M$ closed, it is known that $\lambda (M)$ $>$ $0$ is equivalent to the existence of a contact form $\bar{\theta}$ with respect to which $\bar{R}$ $>$ $0.$
Let $(M,$ $J,$ $\theta )$ be a closed pseudohermitian manifold with $R>0.$ Let $\Gamma ^{\beta }(M)$ denote the nonisotropic Hölder space of exponent $\beta $ (page 181 in [@JL] or [@FS] for the local description modelled on the Heisenberg group). Following a standard argument in [@A] for the elliptic case, we obtain
**Proposition 2.1.** *Let* $(M,$* *$J,$* *$\theta )$* be a closed pseudohermitian manifold with* $R>0.$* Then for any* $f\in \Gamma ^{\beta }(M),\ \beta $* a noninteger* $> $* *$0$*, there exists a unique* $u\in \Gamma ^{\beta
+2}(M) $* such that* $$D_{\theta }(u)=f.$$
Using Proposition 2.1 and a similar construction in [@A], we have that for any $p\in M$, there is a unique Green’s function $G_{p}$ for $D_{\theta
} $ with pole at $p$.
The Green’s function of the universal covering
==============================================
Let $S^{2n+1}$ denote the unit sphere in $C^{n+1}.$ Let $\hat{\xi}$ $:=$ $TS^{2n+1}$ $\cap $ $J_{C^{n+1}}$ $(TS^{2n+1})$ be the standard contact bundle over $S^{2n+1},$ where $J_{C^{n+1}}$ denotes the almost complex structure of $C^{n+1}.$ Let $\hat{J}$ be the restriction of $J_{C^{n+1}}$ to $\hat{\xi}.$ We call a $CR$ manifold $(M,$ $J)$ (or $(M,$ $\xi ,$ $J),$ resp.$)$ spherical if it is locally $CR$ equivalent to ($S^{2n+1},$ $\hat{J})$ (or $(S^{2n+1},$ $\hat{\xi},$ $\hat{J}),$ resp.) (cf. e.g., [@BS]). Let $(M,$ $J,$ $\theta )$ be a closed pseudohermitian manifold of dimension $2n+1$ with $(M,$ $J)$ spherical and $R$ $>$ $0.$ Let $\widetilde{M}$ be the universal covering of $M$ with the $CR$ structure $\pi ^{\ast }J$ and the contact form $\pi ^{\ast }\theta $, where $$\pi :\widetilde{M}\rightarrow M$$is the canonical projection. It follows that $\widetilde{M}$ has no boundary. If $\widetilde{M}$ is compact, then $(\widetilde{M},$ $\pi ^{\ast
}J)$ must be $CR$ equivalent to ($S^{2n+1},$ $\hat{J})$ since it is simply conected and spherical. We will assume that $\widetilde{M}$ is noncompact (or $\pi _{1}(M)$ is an infinite group) throughout this section. We will still use $\theta $ to mean $\pi ^{\ast }\theta $. Our goal in this section is to study the existence of the Green’s function for $D_{\theta }$ on $\widetilde{M}$ and its decay property at the geometric boundary of $\widetilde{M}$.
Let $\Omega $ be a relatively compact smooth domain in $\widetilde{M}$ and $p $ $\in $ $\Omega $. We would like to construct the Dirichlet Green’s function $G_{p}^{\Omega }$ for the domain $\Omega $, that is, to prove the following
**Theorem 3.1.** *There exists a unique* $G_{p}^{\Omega }\in
C^{\infty }(\Omega \setminus \{p\})\cap C(\overline{\Omega }\setminus \{p\})$* such that*$$\begin{array}{rcl}
D_{\theta }(G_{p}^{\Omega }) & = & \delta _{p}\ \text{in}\ \Omega \\
G_{p}^{\Omega }|_{\partial \Omega } & = & 0\end{array}
\label{3.1}$$
**
Once $G_{p}^{\Omega }$ is constructed, the symmetry property $G_{p}^{\Omega
}(q)$ $=$ $G_{q}^{\Omega }(p)$is due to the fact that $D_{\theta }$ is self-adjoint and from the integration by parts argument as in the elliptic case. The positivity of $G_{p}^{\Omega }$ is due to the fact that the leading order operator of $D_{\theta }$ is nonnegative and the Tanaka-Webster scalar curvature $R$ is positive.
As in the elliptic case, the existence of the Dirichlet Green’s function is equivalent to the solvability of nonhomogeneous Dirichlet problem with zero boundary value. So solving (\[3.1\]) is reduced to solving the following Dirichlet problem..
**Theorem 3.2.** *Let* $\Omega $* be a relatively compact smooth domain in* $\widetilde{M}$* and* $f\in \Gamma ^{\beta
}(\overline{\Omega })$*. Then there is a unique* $u\in \Gamma ^{\beta
+2}(\Omega )\cap C(\overline{\Omega })$* such that*$$\begin{array}{rcl}
D_{\theta }(u) & = & f\ \text{in}\ \Omega \\
u|_{\partial \Omega } & = & 0\end{array}
\label{3.2}$$
**
The uniquness of the solution to (\[3.2\]) follows from the following lemma.
**Lemma 3.3.*** Let* $\Omega $* be a relatively compact smooth domain in* $\widetilde{M}$* such that for* $u,v\in
C^{2}(\Omega )\cap C(\overline{\Omega }),$* there holds*$$\begin{aligned}
D_{\theta }(u) &\leq &D_{\theta }(v)\text{ in}\ \Omega \\
u|_{\partial \Omega } &=&v|_{\partial \Omega }.\end{aligned}$$*Then* $u$* *$< $* *$v$* in* $\Omega $* unless *$u=v$ * in* $\Omega$.
To prove Lemma 3.3, we observe that the leading order part of $D_{\theta }$ is a subelliptic operator of Hormander’s type (sum of square vector fields). Then one can apply Bony’s arguments without essential change by the local nature of this lemma. To prove Theorem 3.2, we use Perron’s construction which relies heavily on the maximum principle and the solvability of the Dirichlet problem for the balls. We remark that the key step of proving Theorem 3.2 is to localize the problem. In this respect, the $CR$ invariance enables us to reduce the problem on $(\widetilde{M},\theta )$ ($J$ omitted) for $D_{\theta }$ to the problem on the Heisenberg group $(H^{n},\Theta )$ for $D_{\Theta }$ ([@JL]).
We will often call a Heisenberg ball simply a ball in this section. Let $B$ denote a small ball in $\widetilde{M}$, identified with a Heisenberg ball in $H^{n}.$ There is a positive function $\phi \in C^{\infty }(\widetilde{M})$ such that $\phi ^{\frac{2}{n}}\theta =\Theta $ in $B$. By the known results on $H^{n}$ and the transformation law for $D_{\theta }$, we conclude that for each $f\in \Gamma ^{\beta }(\bar{B})$ and $g\in C(\partial {B})$, there is a unique $u\in \Gamma ^{\beta +2}(B)\cap C(\bar{B})$ such that$$\begin{aligned}
D_{\theta }(u) &=&f\ \text{in}\ B \\
u|_{\partial B} &=&g\end{aligned}$$
In the Dirichlet problem for $D_{\theta }$ on a smooth domain in $\widetilde{M}$, the question of continuity up to the boundary is a purely local issue. So we will deal with it in the same localizing spirit as above.
We begin the Perron process by generalizing the notion of subsolution in classical elliptic theory to the operator $D_{\theta }$ on a smooth domain $\Omega $ in $\widetilde{M}$. Note that $D_{\theta }$ has a nonnegative leading order part.
**Definition.** A continuous function $u$ in $\Omega $ is called a subsolution to the equation $$D_{\theta }(v)=f$$where $f\in \Gamma ^{\beta }(\bar{\Omega}),\beta $ a noninteger $>$ $0$, if for every ball $B\subset \subset \Omega $ and $v$ such that $$D_{\theta }(v)=f\ \text{in}\ B,\ u\leq v\ \text{on}\ \partial {B},$$then we have that $u\leq v$ in $B$.
Analogously we can define the notion of supersolution as well. These notions are completely in parallel to the notions of continuous subharmonic and superharmonic functions in classical elliptic theory. The significance of these notions are ensured by the Bony’s maximum principle. They also have the following useful properties of sub and supersolutions:
\(1) If $u\in C^{2}(\Omega )$, then $u$ is a subsolution (supersolution, resp.) if and only if that $D_{\theta }(u)\leq f$ $(D_{\theta }(u)\geq f,$ resp.$).$
\(2) If $u_{1},\cdots ,u_{m}$ are subsolutions (supersolutions, resp.) in $\Omega $, then $\max \{u_{j}:1\leq j\leq m\}$ $(\min \{u_{j}:1\leq j\leq
m\}, $ resp.$)$ is also a subsolution (supersolution, resp.) in $\Omega $.
\(3) Suppose that $B\subset \subset \Omega $ and $u_{1}$ satisfies $$\begin{aligned}
D_{\theta }(u_{1}) &=&f\text{ in}\ B \\
u_{1} &=&u_{2}\text{ in}\ \partial B\end{aligned}$$where $u_{2}$ is a subsolution in $\Omega $. Then$$u=\Big\{\begin{array}{ll}
u_{1} & \ \text{in}\ B \\
u_{2} & \ \text{in}\ \Omega \backslash B\end{array}$$is also a subsolution in $\Omega $.
**(of Theorem 3.2)** We will carry out the proof in the spirit of standard arguments in the elliptic theory. It consists of two main steps:
**Step $1$ : Construction of the Perron solution.** Consider the folllowing set of subsolutions $$S=\{v:v\ \text{is a subsolution for}\ D_{\theta }(w)=f,\text{ }v|_{\partial
\Omega }\leq 0\}$$Note that the Tanaka-Webster scalar curvature on $\widetilde{M}$ has a positive lower bound, say $R_{0}$. We observe that $\frac{-\sup {|f|}}{R_{0}}\in S$ and the constant $\frac{\sup {|f|}}{R_{0}}$ is a supersolution. Therefore $u(x)=\sup_{v\in S}{v(x)}$ is well defined.
We would like to show that $u$ is the Perron solution, i.e. ,$D_{\theta}u=f$ in $\Omega$. Let $p\in \Omega$ be an arbitrary fixed point. By the definition of $u$, there exists a sequence $v_{m}\in S$ such that $v_{m}(p)\rightarrow u(p)$. By replacing $v_{m}$ with $\max\{v_{1}, \cdots,
v_{m}\}$, we may assume that the sequence is monotone.
Now choose a ball $B\subset \subset \Omega $ of $p$ such that the geometry of $B$ can be flattened after a conformal change of a contact form. Let $w_{m}$ be the unique solution satisfying$$\begin{array}{rcll}
D_{\theta }(w_{m}) & = & f & \text{in}\ B \\
w_{m} & = & v_{m} & \text{in}\ \partial B.\end{array}$$It follows from earlier discussion that$$W_{m}=\Big\{\begin{array}{ll}
w_{m} & \ \text{in}\ B \\
v_{m} & \ \text{in}\ \Omega \backslash B\end{array}$$is a subsolution in $\Omega $. Hence, $u(p)\geq W_{m}(p)\geq
v_{m}(p)\rightarrow u(p)$. Because $W_{m}$ is monotone increasing in $B$, the limit $W=\lim_{m\rightarrow \infty }W_{m}$ exists in $B$. As in the elliptic theory, the subelliptic apriori estimates imply that the sequence $W_{m}$ contains a subsequence converging in $B$, and hence $W$ is a solution in $B$ and $W(p)=u(p)$.
We claim that $W=u$ in $B$ to complete step $1$. The arguments are standard: let $q\in B$, there is a monotone increasing sequence $g_{m}\in S$ such that $g_{q}\rightarrow u(q)$. Let $h_{m}$ solve the equation$$\begin{array}{rcll}
D_{\theta }(w) & = & f & \text{in}\ B \\
w & = & \bar{g}_{m} & \text{in}\ \partial B\end{array}$$
where in $B,\ \bar{g}_{m}=\max \{g_{m},v_{m}\}$. Therefore the sequence is also monotone increasing and $v_{m}\leq h_{m}$ in $B$. As before, $h=\lim_{m\rightarrow \infty }h_{m}$ is a solution in $B$ and $h(q)=u(q)$. Since $u(p)\geq h(p)\geq W(p)=u(p)$, Bony’s strong maximum principle implies that $W=h$ in $B$. Therefore, $u$ is indeed an interior solution.
**Step $2$ : Continuity up to the boundary.** Fix $p\in \partial \Omega
$, we choose a small ball $B$ of $p$ such that the boundary of $B\cap \Omega
$ is smooth and the geometry of $B$ can be flattened by choosing a conformal contact form. Notice the following two facts:
\(1) There exists $u_{f}$ which solves the Dirichlet problem:$$\begin{array}{rcll}
D_{\theta }(u_{f}) & = & f & \text{in}\ B\cap \Omega \\
u_{f} & = & 0 & \text{in}\ \partial (B\cap \Omega ).\end{array}$$
\(2) The existence of a local barrier at $p$, that is, for a small ball of $p$, there is a function $w\in C(\overline{B}\cap \overline{\Omega })\cap
C^{2}(B\cap \Omega )$ such that $$D_{\theta }(w)=0\ \text{in}\ B\cap \Omega ,w>0\ \text{in}\ \overline{B}\cap
\overline{\Omega }\backslash \{p\}\ \text{and}\ w(p)=0.$$
We will use this local barrier to construct a global barrier and show that the Perron solution $u$ obtained in step 1 is continuous up to the boundary. For any $\varepsilon $, there exists $\delta $ and $K$ such that $$|u_{f}(x)|\leq \varepsilon \ \text{if}\ |x-p|\leq \delta$$and $$Kw(x)\geq \sup {u_{f}}\ \text{if}\ |x-p|\geq \delta .$$We will consider $w_{1}=-Kw+u_{f}-\varepsilon $ in $B\cap \Omega $. Then we have immediately that $$D_{\theta }w_{1}=f\ \text{and}\ w_{1}<0\ \text{in}\ B\cap \Omega .$$Consider the following number $$M_{K}=\sup_{\partial (B\cap \Omega )\backslash \partial \Omega
}w_{1}=\sup_{\partial (B\cap \Omega )\backslash \partial \Omega
}(-Kw-\varepsilon )=-K(\min_{\partial (B\cap \Omega )\backslash \partial
\Omega }w)-\varepsilon ,$$and we let$$W_{1}=\Big\{\begin{array}{ll}
\max (w_{1},M_{K}) & \ \text{in}\ \overline{B}\cap \overline{\Omega } \\
M_{K} & \ \text{in}\ \overline{\Omega }\backslash \overline{B}.\end{array}$$ Note that $W_{1}|_{\partial (B\cap \Omega )\backslash \partial
\Omega }=M_{K}$, so $W_{1}\in C(\Omega )$. It is easy to check directly that $W_{1}$ is a subsolution (when $K$ is large enough). To check the boundary behavior of $W_{1}$, we note that (when $q$ is very close to $p$): $$W_{1}(q)=\max (W_{1}(q),M_{K})=-Kw(q)+u_{f}(q)-\varepsilon \rightarrow
-\varepsilon$$as $q\rightarrow p$. So it is zero at $p$ and negative everywhere else on $\partial \Omega $. Therefore, $W_{1}\in S$ and $W_{1}$ is a global boundary barrier at $p$. It follows that $$\lim \inf_{q\rightarrow p}u(q)\geq \lim_{q\rightarrow
p}w_{1}(q)=-\varepsilon .$$Since $\varepsilon $ is arbitrary, we have the continuity of $u$ up to the boundary.
We are now ready to construct a Green’s function $G_{p}$ for $D_{\theta }$ with pole at $p$ $\in $ $\widetilde{M}$. We would also like to discuss its decay properties at the infinity.
Recall that $\Phi :\widetilde{M}\rightarrow S^{2n+1}$ denotes the $CR$ developing map. Let $H_{y}$ be the Green’s function for the $CR$ invariant sublaplacian $D_{0}$ of $(S^{2n+1},\theta _{S^{2n+1}})$ with the pole $y=\Phi (p)$, where $\theta _{S^{2n+1}}$ is the standard contact form on $S^{2n+1}$. Since $\Phi $ is a $CR$ immersion, we can write $$\Phi ^{\ast }(\theta _{S^{2n+1}})=|\Phi ^{^{\prime }}|^{2}\theta$$where $\theta $ is the contact form for $\widetilde{M}$ and $|\Phi
^{^{\prime }}|$ is a positive $C^{\infty }$ smooth function on $\widetilde{M}
$. By the transformation law (\[TL\]) of the $CR$ invariant sublaplacian, we immediately obtain the following formula $$D_{\theta }(|\Phi ^{^{\prime }}|^{n}H_{y}\circ \Phi )=\sum_{\bar{p}\in \Phi
^{-1}(y)}|\Phi ^{^{\prime }}(\bar{p})|^{n+2}\delta _{\bar{p}}.
\label{3.11.1}$$Let us consider the following function (with poles in $\Phi
^{-1}(y)$): $$\overline{G}:=|\Phi ^{^{\prime }}(p)|^{-(n+2)}|\Phi ^{^{\prime
}}|^{n}H_{y}\circ \Phi . \label{3.11.2}$$Since $\Phi $ is a $CR$ immersion, $\overline{G}$ is positive, $C^{\infty }$ smooth, and $D_{\theta }\overline{G}=0$ on $\widetilde{M}\backslash \Phi ^{-1}(y).$ Also, $\overline{G}$ has exactly the same asymptotic behavior at each of $\Phi ^{-1}(y)$. We call $\overline{G}$ the normalized pullback of $H_{y}$, which will be taken as a singular barrier in the construction of a global Green’s function on $\widetilde{M}$ through a limit procedure of Dirichlet Green’s functions.
Let $\{\Omega _{k}\subset \Omega _{k+1}:k=1,\cdots ,\}$ be a relatively compact, $C^{\infty }$ smooth exhaustion of the universal covering $\widetilde{M}.$ Take $p$ $\in $ $\Omega _{1}$. Note that $$D_{\theta }(\overline{G}-G_{p}^{\Omega _{k}})\geq 0,$$$\overline{G}$ is positive, and $G_{p}^{\Omega _{k}}|_{\partial
\Omega _{k}}=0$. By the maximum principle of Bony, we see that $$G_{p}^{\Omega _{k}}<\overline{G}. \label{barrier}$$away from $\Phi ^{-1}(y).$ Near the point $p$, we have the following equality: $$D_{\theta }(\overline{G}-G_{p}^{\Omega _{k}})=0.$$So $\overline{G}-G_{p}^{\Omega _{k}}$ is smooth near $p$ by the regularity result for $\Delta _{b}$ and $D_{\theta }$ being “covariant” to $D_{\Theta }$ $=b_{n}\Delta _{b},$ $\Theta :$ standard contact form in the Heisenberg group (cf. the argument in the end of the proof of Lemma 4.1). Therefore we have the following result.
**Theorem 3.4.** *Let* $G_{p}=\lim_{k\rightarrow \infty
}G_{p}^{\Omega _{k}}$*. Then* $G_{p}$* is a positive fundamental solution of* $D_{\theta }$* with pole at* $p.$*Moreover,* $G_{p}$* is minimal among all positive fundamental solutions.*
**
By the strong maximum principle of Bony and (\[barrier\]), the sequence of the Green’s functions $\{G_{p}^{\Omega _{k}}\}$ is strictly increasing and has an upper bound. Thus, away from $\Phi ^{-1}(y)$, the limit of $G_{p}^{\Omega _{k}}$ exists. Next by the standard argument using Bony’s maximum principle, $\Phi ^{-1}(y)\backslash \{p\}$ is a set of removable singularities for $G_{p}$.
The minimality of $G_{p}$ follows from its construction, i.e., if $F_{p}$ is another global positive fundamental solution on $\widetilde{M}$ with pole at $p$, then again Bony’s maximum principle implies that $G_{p}^{\Omega
_{k}}<F_{p}$ for any $k$, and the conclusion follows.
We would like to discuss the decay properties of the constructed $G_{p}$, in particular, its integrability away from the pole $p$. We define $s(M),$ the minimum exponent of the integrability of $G_{p}$ by$$s(M):=\inf {\{s:\int_{\widetilde{M}\setminus U_{p}}G_{p}^{s}dV_{\theta
}<\infty \}} \label{s(M)}$$
where $U_{p}$ is a neighborhood of $p$.
**Theorem 3.5.** $s(M)$* is a* $CR$* invariant and satisfies the following inequality:* $$s(M)\leq 1. \label{3.12.1}$$
**
For $p$ $\in $ $\widetilde{M}$, let $U_{p}$ be a small neighborhood of $p$ with smooth boundary. Let $\{\Omega _{k}\subset \Omega _{k+1}:k=1,\cdots ,\}$ be a relatively compact, smooth exhaustion of the universal covering $\widetilde{M}$. We may assume that $p$ $\in $ $U_{p}$ $\subset $ $\Omega
_{1} $. Because the Dirichlet Green’s function $G_{p}^{\Omega _{k}}$ is smooth in $\Omega _{k-1}\backslash U_{p},$ Bony’s maximum principle implies that $$\sup_{\Omega _{k-1}\backslash U_{p}}G_{p}^{\Omega _{k}}\leq \sup_{\Omega
_{k}\backslash U_{p}}G_{p}^{\Omega _{k}}\leq \sup_{\partial
U_{p}}G_{p}^{\Omega _{k}}<\max_{\partial U_{p}}G_{p}$$Therefore we may ($C^{\infty })$ smoothly extend $G_{p}^{\Omega
_{k}}$ into $U_{p}$ with the extension smaller than $\max_{\partial
U_{p}}G_{p},$ but positive. Denote this extension (which is a smooth function over $\Omega _{k-1}$) by $\bar{G}_{p}^{\Omega _{k}}$.
For each $\alpha \geq 0$, let $u_{k}$ be the solution to the following Dirichlet problem:$$\begin{array}{rcl}
D_{\theta }(u) & = & (\bar{G}_{p}^{\Omega _{k}})^{\alpha }\ \text{in}\
\Omega _{k-1} \\
u|_{\partial \Omega _{k-1}} & = & 0.\end{array}$$By the weak maximum principle, we obtain $$\max_{\Omega _{k-1}}u_{k}\leq \frac{(\max_{\partial U_{p}}G_{p})^{\alpha }}{R_{0}} \label{3.13.1}$$
where $R_{0}>0$ is the lower bound of the Tanaka-Webster scalar curvature of $\widetilde{M}$. By the solution representation in $\Omega
_{k-1},$ we have
$$\begin{aligned}
u_{k}(p) &=&\int_{\Omega _{k-1}}G_{p}^{\Omega _{k-1}}(q)(\bar{G}_{p}^{\Omega
_{k}}(q))^{\alpha }dV_{\theta }(q) \label{3.14} \\
&\geq &\int_{\Omega _{k-1}\backslash U_{p}}G_{p}^{\Omega _{k-1}}(q)(\bar{G}_{p}^{\Omega _{k}}(q))^{\alpha }dV_{\theta }(q) \notag\end{aligned}$$
By the monotonicity of $G_{p}^{\Omega _{k}}$ and letting $k$ $\rightarrow $ $\infty $, we conclude that $$\int_{\widetilde{M}\setminus U_{p}}G_{p}^{1+\alpha }dV_{\theta }\leq \frac{(\max_{\partial U_{p}}G_{p})^{\alpha }}{R_{0}}.$$from (\[3.14\]) and (\[3.13.1\]). So (\[3.12.1\]) follows. Finally, we need to show that $s(M)$ is a well defined $CR$ invariant. It is routine to check that the definition of $s(M)$ is independent of the choice of $U_{p}$. Also $s(M)$ is independent of the choice of contact form from the transformation law of Green’s functions with respect to two different contact forms (cf. Proposition 2.6 of Chapter VI in [@SY2] for the conformal case).
Let $\rho $ denote the Carnot-Carathéodory distance on $\widetilde{M}$ with respect to the Levi metric (see, e.g., [@Str])$.$ Let $B_{r}(p)$ $\subset $ $\widetilde{M}$ denote the ball of radius $r$, centered at $p,$ with respect to the Carnot-Carathéodory distance $\rho .$ From Theorem 3.5 and a Moser’s iteration procedure, we have the following result.
**Proposition 3.6.** *There holds* $$\lim_{r\rightarrow \infty }\left( \sup \{G_{p}(x):\rho (x,p)\geq r\}\right)
=0.$$
**
Recall that $b_{n}$ $:=$ $2+\frac{2}{n}$. By Theorem 3.5, $\int_{\widetilde{M}\setminus U_{p}}G_{p}^{b_{n}}dV_{\theta }<\infty $. Thus we have $$\lim_{r\rightarrow \infty }\int_{\{x:\rho (x,p)\geq
r\}}G_{p}^{b_{n}}dV_{\theta }=0.$$Therefore it is enough to establish the estimate $$G_{p}(x)\leq C\left( \int_{B_{1}(x)}G_{p}^{b_{n}}dV_{\theta }\right)
^{1/b_{n}}\ \text{for all}\ x\in \widetilde{M}\backslash B_{2}(p)$$where $B_{1}(x)$ is a ball of radius $1,$ centered at $x$. First, we have the equation for $G_{p}$ $$\Delta _{b}G_{p}+\frac{1}{b_{n}}RG_{p}=0\ \ \text{on}\ \ \widetilde{M}\backslash B_{2}(p).$$Take $q\geq b_{n}$ $:=$ $2+\frac{2}{n}$. Multiplying the above formula by $G_{p}^{q-1}\phi ^{2}$ and integrating by parts give$$\begin{aligned}
&&(q-1)\int_{\widetilde{M}}\phi ^{2}G_{p}^{q-2}|\nabla
_{b}G_{p}|^{2}dV_{\theta }+\frac{1}{b_{n}}\int_{\widetilde{M}}RG_{p}^{q}\phi
^{2}dV_{\theta } \label{3.15} \\
&\leq &2\int_{\widetilde{M}}\phi G_{p}^{q-1}|\nabla _{b}\phi ||\nabla
_{b}G_{p}|dV_{\theta } \notag \\
&\leq &\alpha \int_{\widetilde{M}}\phi ^{2}G_{p}^{q-2}|\nabla
_{b}G_{p}|^{2}dV_{\theta }+\frac{1}{\alpha }\int_{\widetilde{M}}|\nabla
_{b}\phi |^{2}G_{p}^{q}dV_{\theta } \notag\end{aligned}$$
for all $\alpha >0$. Here $\phi \in C_{0}^{\infty }(\widetilde{M}\backslash B(p))$. Taking $\alpha =q-2$ in (\[3.15\]), we get $$\int_{\widetilde{M}}\phi ^{2}G_{p}^{q-2}|\nabla _{b}G_{p}|^{2}dV_{\theta
}\leq \frac{1}{q-2}\int_{\widetilde{M}}|\nabla _{b}\phi
|^{2}G_{p}^{q}dV_{\theta } \label{leq1}$$by $R$ $>$ $0$ and $G_{p}$ $>$ $0.$ On the other hand, taking $\alpha $ $=$ $1$ in (\[3.15\]) gives$$\begin{split}
2\int_{\widetilde{M}}\phi G_{p}^{q-1}|\nabla _{b}\phi ||\nabla
_{b}G_{p}|dV_{\theta }& \leq \int_{\widetilde{M}}\phi ^{2}G_{p}^{q-2}|\nabla
_{b}G_{p}|^{2}dV_{\theta }+\int_{\widetilde{M}}|\nabla _{b}\phi
|^{2}G_{p}^{q}dV_{\theta } \\
& \leq (\frac{q-1}{q-2})\int_{\widetilde{M}}|\nabla _{b}\phi
|^{2}G_{p}^{q}dV_{\theta }.
\end{split}
\label{leq2}$$by (\[leq1\]). It then follows from (\[leq2\]) that $$\begin{split}
& \int_{\widetilde{M}}\left( |\nabla _{b}(\phi G_{p}^{q/2})|^{2}+\frac{1}{b_{n}}R\phi ^{2}G_{p}^{q}\right) dV_{\theta } \\
\leq & \int_{\widetilde{M}}\left( |\nabla _{b}\phi |^{2}G_{p}^{q}+\frac{q^{2}}{4}\phi ^{2}G_{p}^{q-2}|\nabla _{b}G_{p}|^{2}+q\phi G_{p}^{q-1}|\nabla
_{b}\phi ||\nabla _{b}G_{p}|+\frac{1}{b_{n}}R\phi ^{2}G_{p}^{q}\right)
dV_{\theta } \\
\leq & C_{n}q^{2}\int_{\widetilde{M}}\left( |\nabla _{b}\phi |^{2}+\phi
^{2}\right) G_{p}^{q}dV_{\theta }
\end{split}
\label{leq3}$$for some constant $C_{n}$ independent of $q$. Applying the Sobolev inequality $$\left( \int_{\widetilde{M}}|\phi |^{b_{n}}dV_{\theta }\right) ^{2/b_{n}}\leq
\lambda (\widetilde{M})^{-1}\int_{\widetilde{M}}\left( |\nabla _{b}\phi
|^{2}+\frac{1}{b_{n}}R\phi ^{2}\right) dV_{\theta }$$(note that $\lambda (\widetilde{M})$ $>$ $0.$ In fact, $\lambda (\widetilde{M})$ $=$ $\lambda (S^{2n+1})$ a $CR$ analogue of Theorem 2.2 of Chapter VI in [@SY2]), we obtain $$\begin{split}
& \left[ \int_{\widetilde{M}}(\phi G_{p}^{q/2})^{b_{n}}\right] ^{2/b_{n}} \\
\leq & \lambda (\widetilde{M})^{-1}\int_{\widetilde{M}}\left( |\nabla
_{b}(\phi G_{p}^{q/2})|^{2}+\frac{1}{b_{n}}R\phi ^{2}G_{p}^{q}\right)
dV_{\theta } \\
\leq & \tilde{C}_{n}q^{2}\int_{\widetilde{M}}\left( |\nabla _{b}\phi
|^{2}+\phi ^{2}\right) G_{p}^{q}dV_{\theta }
\end{split}
\label{leq4}$$by (\[leq3\]) for some constant $\tilde{C}_{n}$ independent of $q $.
We will use (\[leq4\]) repeatedly with $$q_{0}=b_{n}=2r,\ \ q_{k}=q_{0}r^{k},\ \ \text{with}\ \ r=\frac{n+1}{n}.$$Define a sequence of cut-off functions as follows. Set $a_{0}=1,a_{k}=1-\sum_{i=1}^{k}3^{-i}$ for $k\geq 1$, and we require that for each $k$ the function $\phi _{k}\in C_{0}^{\infty }(\widetilde{M})$ satisfies $$\phi _{k}=\{\begin{array}{rl}
1, & y\in B_{a_{k}}(x) \\
0, & y\not\in B_{a_{k-1}}(x),\end{array}$$$$0\leq \phi _{k}\leq 1,\ \ |\nabla _{b}\phi _{k}|\leq 2\cdot 3^{k}.$$Then iteratively we get from (\[leq4\]) that
$$\begin{aligned}
&&\left( \int_{B_{a_{k}}(x)}G_{p}^{q_{k+1}}dV_{\theta }\right) ^{1/q_{k+1}}
\label{3.20} \\
&\leq &(Cq_{k}^{2})^{1/q_{k}}(4\cdot 3^{2k}+1)^{1/q_{k}}\left(
\int_{B_{a_{k-1}}(x)}G_{p}^{q_{k}}dV_{\theta }\right) ^{1/q_{k}}\leq ...
\notag \\
&\leq &\prod_{j=1}^{k}(Cr^{2j})^{1/pr^{j}}\left(
\int_{B_{a_{0}}(x)}G_{p}^{b_{n}}dV_{\theta }\right) ^{1/b_{n}} \notag\end{aligned}$$
Since the product converges as $k\rightarrow \infty $, we can take the limit $k\rightarrow \infty $ in (\[3.20\]) to get $$\sup_{y\in B_{\frac{1}{2}}(x)}G_{p}(y)\leq C\left(
\int_{B_{1}(x)}G_{p}^{b_{n}}dV\right) ^{1/b_{n}}.$$This completes the proof.
Injectivity of the $CR$ developing map
======================================
Let $p$ $\in $ $\widetilde{M}$, and recall that $G_{p}$ denotes the minimal positive Green’s function for the $CR$ invariant sublaplacian $D_{\theta }$ with pole at $p$. Let $\Phi :\widetilde{M}\rightarrow S^{2n+1}$ be a $CR$ developing map. Recall that $H_{y}$ denotes the Green’s function for the $CR$ invariant sublaplacian $D_{0}$ of $S^{2n+1}$ with the pole $y=\Phi (p)$. The normalized pullback of $H_{y}$ is$$\overline{G}:=|\Phi ^{^{\prime }}(p)|^{-(n+2)}|\Phi ^{^{\prime
}}|^{n}H_{y}\circ \Phi$$which has poles in $\Phi ^{-1}(y)$. Observe that $\Phi $ is one to one if and only if $\Phi ^{-1}(y)=\{p\}$. Therefore to prove injectivity of $\Phi ,$ it suffices to prove $G_{p}=\overline{G}$.
For any $y\in S^{2n+1}$, the Cayley transform is a global $CR$ diffeomorphism $$C_{y}:(S^{2n+1}\backslash \{y\},\theta _{S^{2n+1}})\rightarrow (H^{n},\Theta
)$$with $C_{y}(y)=\infty $ and $$C_{y}^{\ast }\Theta =H_{y}^{\frac{2}{n}}\theta _{S^{2n+1}}.$$This means that $(S^{2n+1}\backslash \{y\},H_{y}^{\frac{2}{n}}\theta _{S^{2n+1}})$ is Heisenberg flat. Now we have
$$\begin{aligned}
|\Phi ^{^{\prime }}(p)|^{\frac{-2(n+2)}{n}}\Phi ^{\ast }\circ C_{y}^{\ast
}(\Theta ) &=&|\Phi ^{^{\prime }}(p)|^{\frac{-2(n+2)}{n}}\Phi ^{\ast
}(H_{y}^{\frac{2}{n}}\theta _{S^{2n+1}}) \label{4.1} \\
&=&|\Phi ^{^{\prime }}(p)|^{\frac{-2(n+2)}{n}}(H_{y}\circ \Phi )^{\frac{2}{n}}\Phi ^{\ast }\theta _{S^{2n+1}} \notag \\
&=&|\Phi ^{^{\prime }}(p)|^{\frac{-2(n+2)}{n}}(H_{y}\circ \Phi )^{\frac{2}{n}}|\Phi ^{^{\prime }}|^{2}\theta \notag \\
&=&(|\Phi ^{^{\prime }}(p)|^{-(n+2)}(H_{y}\circ \Phi )|\Phi ^{^{\prime
}}|^{n})^{\frac{2}{n}}\theta \notag \\
&=&\overline{G}^{\frac{2}{n}}\theta . \notag\end{aligned}$$
It follows from, (\[4.1\]) that $(\widetilde{M},$ $\bar{\theta}:=\overline{G}^{\frac{2}{n}}\theta )$ is Heisenberg flat away from $\Phi
^{-1}(y)$.
Cosider the quotient of $G_{p}$ and $\overline{G}:$$$v:=\frac{G_{p}}{\overline{G}}.$$By (\[barrier\]) we have $\overline{G}-G_{p}\geq 0$ away from $\Phi ^{-1}(y)$. So there holds $$0<v\leq 1$$away from $\Phi ^{-1}(y).$ Taking $u$ $=$ $G_{p}$ and $\varphi $ $= $ $1$ in (\[TL\]) , we obtain that on $\widetilde{M}\backslash \{p\}$: $$R(G_{p}^{\frac{2}{n}}\theta )=G_{p}^{-1-\frac{2}{n}}D_{\theta }(G_{p})=0.
\label{4.1.1}$$Here for a contact form $\eta ,$ $R(\eta )$ or $R_{\eta }$ means the Tanaka-Webster scalar curvature with respect to $\eta .$ Writing $G_{p}^{\frac{2}{n}}\theta =v^{\frac{2}{n}}\bar{\theta}$, we get $$0=R(G_{p}^{\frac{2}{n}}\theta )=R(v^{\frac{2}{n}}\bar{\theta})=v^{-1-\frac{2}{n}}D_{\bar{\theta}}(v)$$away from $\Phi ^{-1}(y)$ by (\[4.1.1\]) and (\[TL\]) with $u$ $=$ $v$ and $\varphi $ $=$ $1.$ Therefore we have $$(b_{n}\bar{\Delta}_{b}+R_{\bar{\theta}})(v)=b_{n}\bar{\Delta}_{b}(v)=0
\label{harofv}$$
by noting that $R_{\bar{\theta}}=0.$ We would like to examine the asymptotic behavior of $v$ near $\Phi ^{-1}(y)$. We will often write the coordinates $(z_{1},$ $...,$ $z_{n},$ $t)$ in $H^{n}$ by $(z,$ $t)$ where $z$ $=$ $(z_{1},$ $...,$ $z_{n}).$ Define the Heisenberg norm $\rho _{0}(z,t)$ $:=$ $(|z|^{4}+t^{2})^{1/4}.$ Denote the fundamental solution to $D_{\Theta }$ $=$ $b_{n}\Delta _{b}$ by $c(n)\rho _{0}(z,t)^{-2n}$ for some dimensional constant $c(n)$ ([@FS]).
**Lemma 4.1.** *For each* $\bar{p}\in \Phi ^{-1}(y),$*we can choose a coordinate map* $(z,t):\widetilde{M}\rightarrow H^{n}$* and a smooth function* $u$ * near* $\bar{p}$* such that* $(z(\bar{p}),t(\bar{p}))$* *$=$* *$(0,0)$* and there hold*$$G_{p}(z,t)=c(n)u(p)u\rho _{0}(z,t)^{-2n}+a\text{ }smooth\text{ }function
\label{4.3.1}$$*near* $\bar{p}$ $=$ $p$ * and*$$\overline{G}(z,t)=c(n)u(q)u\rho _{0}(z,t)^{-2n}+a\text{ }smooth\text{ }function \label{4.3.2}$$*near* $\bar{p}$ $=$ $q$,* resp..*
Let $-y\in S^{2n+1}$ be the antipodal point of $y\in S^{2n+1}$. Consider the Cayley transform $C_{-y}$ (with pole at $-y$). Obviously, $C_{-y}(y)=0$ $\in
$ $H^{n}$ and $$(C_{-y}^{-1})^{\ast }(H_{-y}^{\frac{2}{n}}\theta _{S^{2n+1}})=\Theta
\label{4.3.3}$$It follows from (\[4.3.3\]) that $$(C_{-y}\circ \Phi )^{\ast }\Theta =(H_{-y}\circ \Phi )^{\frac{2}{n}}|\Phi
^{^{\prime }}|^{2}\theta . \label{4.3.5}$$Here we have written $\Phi ^{\ast }(\theta _{S^{2n+1}})=|\Phi
^{^{\prime }}|^{2}\theta .$ Let $u$ :$=$ $(H_{-y}\circ \Phi )|\Phi
^{^{\prime }}|^{n}.$ We can then write (\[4.3.5\]) as$$(C_{-y}\circ \Phi )^{\ast }\Theta =u^{\frac{2}{n}}\theta \label{4.3.6}$$
near $q$ $\in $ $\Phi ^{-1}(y).$ Take $C_{-y}\circ \Phi $ $:$ $\widetilde{M}\rightarrow H^{n}$ as a coordinate map $(z,t)$. By (\[4.3.6\]) and (\[TL\]) in Section 2 with $\varphi $ $=$ $c(n)\rho _{0}(z,t)^{-2n}$, we obtain$$\begin{aligned}
D_{\theta }(uc(n)\rho _{0}(z,t)^{-2n}) &=&u^{1+\frac{2}{n}}D_{\Theta
}(c(n)\rho _{0}(z,t)^{-2n}) \\
&=&u^{1+\frac{2}{n}}\delta _{(0,0)}.\end{aligned}$$Note that the volume change formula is ($(C_{-y}\circ \Phi )^{\ast
})$ $\Theta \wedge (d\Theta )^{n}$ $=$ $u^{2+\frac{2}{n}}\theta \wedge
(d\theta )^{n}.$ So in view of Theorem 3.4, (\[3.11.1\]), and (\[3.11.2\]),.we have$$\begin{aligned}
D_{\theta }(G_{p}-u(p)uc(n)\rho _{0}(z,t)^{-2n}) &=&0, \label{4.3.7} \\
D_{\theta }(\bar{G}-u(\bar{p})uc(n)\rho _{0}(z,t)^{-2n}) &=&0 \notag\end{aligned}$$near $p,$ $\bar{p}$ $\in $ $\Phi ^{-1}(y)$, resp.. By applying (\[TL\]) we obtain$$\begin{aligned}
D_{(C_{-y}\circ \Phi )^{\ast }\Theta }(u^{-1}(G_{p}-u(p)uc(n)\rho
_{0}(z,t)^{-2n}) &=&0 \label{4.3.8} \\
D_{(C_{-y}\circ \Phi )^{\ast }\Theta }(u^{-1}(\bar{G}-u(\bar{p})uc(n)\rho
_{0}(z,t)^{-2n}) &=&0 \notag\end{aligned}$$according to (\[4.3.6\]) and (\[4.3.7\]). Observe that $R_{(C_{-y}\circ \Phi )^{\ast }\Theta }$ $=$ $(C_{-y}\circ \Phi )^{\ast
}R_{\Theta }$ $=$ $R_{\Theta }\circ (C_{-y}\circ \Phi )$ $=$ $0.$ Hence $D_{(C_{-y}\circ \Phi )^{\ast }\Theta }$ $=$ $b_{n}\Delta _{b}.$ By the regularity result for $\Delta _{b}$ (e.g., Theorem 16.7 in [@FS]), we have (\[4.3.1\]) and (\[4.3.2\]) from (\[4.3.8\]) and $G_{p},$.$\bar{G},$ $\rho _{0}(z,t)^{-2n}$ being $L_{loc}^{1}.$
By Lemma 4.1 we deduce that near $p$ (which is a pole of $\overline{G}$): $$v(z,t)=1+O(\rho _{0}^{2n}),$$and near $\bar{p}$ $\in $ $\Phi ^{-1}(y)\setminus \{p\}$ $$v(z,t)=O(\rho _{0}^{2n}).$$since $G_{p}$ is smooth near any $\bar{p}$ $\in $ $\Phi
^{-1}(y)\setminus \{p\}$. From Lemma 4.1 we also have
$$\begin{aligned}
|\nabla _{b}v| &=&O(\rho _{0}^{2n-1}), \\
|\nabla _{b}|\nabla _{b}v|| &=&O(\rho _{0}^{2n-2}),\Delta _{b}v=O(\rho
_{0}^{2n-2})\end{aligned}$$
near any $\bar{p}$ $\in $ $\Phi ^{-1}(y).$
With the asymptotic behavior of $G_{p},$ $\bar{G},$ and $v$ near any $\bar{p}
$ $\in $ $\Phi ^{-1}(y)$ understood, we observe that the set $\Phi ^{-1}(y)$ has no contribution when we play the integration by parts in the computation below throughout the remaining section. We would like to show that $v$ is a constant, and hence is identically one since $v(p)=1$. Write $G,$ $dV$ instead of $G_{p},$ $dV_{\theta }$ for short in the remaining section. Also note that the notation $C$ may mean different constants.
**Lemma 4.2.** *There exists a constant* $C>0$* such that for any* $\phi \in C_{0}^{\infty }(\widetilde{M})$* there holds*$$\int_{\widetilde{M}}\phi ^{2}|\nabla _{b}\log {\overline{G}}|^{2}dV\leq
C\int_{\widetilde{M}}(\phi ^{2}|\nabla _{b}\log {G}|^{2}+|\nabla _{b}\phi
|^{2})dV. \label{firequ}$$
Away from $\Phi ^{-1}(y)$ we have $$\begin{split}
\Delta _{b}\log {\overline{G}}& =\overline{G}^{-1}\Delta _{b}\overline{G}-(-\overline{G}^{-2}|\nabla _{b}\overline{G}|^{2}) \\
& =\overline{G}^{-1}\Delta _{b}\overline{G}+|\nabla _{b}\log {\overline{G}}|^{2} \\
& =-b_{n}^{-1}R(\theta )+|\nabla _{b}\log {\overline{G}}|^{2} \\
& =G^{-1}\Delta _{b}G+|\nabla _{b}\log {\overline{G}}|^{2} \\
& =\Delta _{b}\log {G}-|\nabla _{b}\log {G}|^{2}+|\nabla _{b}\log {\overline{G}}|^{2}
\end{split}
\label{4.3.4}$$
Multiplying $\phi ^{2}$ on both sides of (\[4.3.4\]) and integrating by parts, we obtain $$\begin{split}
\int_{\widetilde{M}}\phi ^{2}|\nabla _{b}\log {\overline{G}}|^{2}dV& =\int_{\widetilde{M}}\phi ^{2}|\nabla _{b}\log {G}|^{2}dV+2\int_{\widetilde{M}}\phi
\nabla _{b}\phi (\nabla _{b}\log {\overline{G}}-\nabla _{b}\log {G})dV \\
& \leq C\int_{\widetilde{M}}(\phi ^{2}|\nabla _{b}\log {G}|^{2}+|\nabla
_{b}\phi |^{2})dV.
\end{split}$$for some constant $C$ by noting that the boundary integral around $\bar{p}$ $\in $ $\Phi ^{-1}(y)$ tends to zero$.$ Here we have used the Schwarz inequality with $\varepsilon $ in the last inequality.
Let $\bar{T}$ $(\overline{\nabla }_{b}, \overline{\Delta }_{b}$, $d\overline{V},$ $\bar{R}$ $=$ $R_{\bar{\theta}},$ etc., resp.) denote the corresponding quantity of $T$ ($\nabla _{b},$ $\Delta _{b},$ $dV,$ $R,$ etc., resp.) with respect to $\bar{\theta}$ (while fixing $J$)$.$
**Lemma 4.3.** *There holds* $v_{\bar{0}}$* *$:=$* *$\overline{T}v\equiv 0$* in either of the following cases:*
*(a)* $n\geq 3$
*(b)* $n=2$* and* $s(M)<1.$
First observe that the Paneitz-like operator $P$ is nonnegative for $\varphi
$ $\in $ $C_{0}^{\infty }(\widetilde{M})$ in \[NG\] if $n$ $\geq $ $2$ (Extending Theorem 3.2 in [@CC] to this situation). With respect to $\bar{\theta}$ (Heisenberg flat)$,$ the torsion vanishes and hence $\kappa $ $=$ $0$ in (\[IBF2\]). Therefore by extending (\[IBF2\]) to the situation for $\bar{\theta}$ (singular at $\bar{p}$ $\in $ $\Phi ^{-1}(y))$ in view of the asymptotic behavior of $v$ discussed before$,$ we have $$\begin{split}
n^{2}\int_{\widetilde{M}}(\phi v)_{\bar{0}}^{2}d\bar{V}& \leq \int_{\widetilde{M}}(\overline{\Delta }_{b}(\phi v))^{2}d\overline{V} \\
& =\int_{\widetilde{M}}(\phi \overline{\Delta }_{b}v+v\overline{\Delta }_{b}\phi -2\overline{\nabla }_{b}\phi \overline{\nabla }_{b}v)^{2}d\overline{V} \\
& \leq C\left( \int_{\widetilde{M}}v^{2}(\overline{\Delta }_{b}\phi )^{2}d\overline{V}+\int_{\widetilde{M}}|\overline{\nabla }_{b}\phi |^{2}|\overline{\nabla }_{b}v|^{2}d\overline{V}\right) \\
& =C(I+II)
\end{split}
\label{24}$$for $\phi $ $\in $ $C_{0}^{\infty }(\widetilde{M}),$ where $I=\int_{\widetilde{M}}v^{2}(\overline{\Delta }_{b}\phi )^{2}d\overline{V}$ and $II=\int_{\widetilde{M}}|\overline{\nabla }_{b}\phi |^{2}|\overline{\nabla }_{b}v|^{2}d\overline{V}$. Rewrite$$\begin{aligned}
II &=&\int_{\widetilde{M}}|\overline{\nabla }_{b}\phi |^{2}|\overline{\nabla
}_{b}v|^{2}d\overline{V} \label{24.0} \\
&=&\int_{\widetilde{M}}|\nabla _{b}\phi |^{2}|\nabla _{b}v|^{2}\overline{G}^{\frac{2n-2}{n}}dV \notag\end{aligned}$$
where$$\begin{aligned}
|\nabla _{b}v|^{2} &=&|\frac{\overline{G}\nabla _{b}G-G\nabla _{b}\overline{G}}{\overline{G}^{2}}|^{2} \label{24.1} \\
&=&|v\nabla _{b}\ log\ G-v\nabla _{b}\log \overline{G}|^{2} \notag \\
&\leq &C(v^{2}|\nabla _{b}\log {G}|^{2}+v^{2}|\nabla _{b}\log {\overline{G}}|^{2}). \notag\end{aligned}$$Let $q^{\prime }$ :$=$ $\frac{2(n-1)}{n}$. Then $q^{\prime }>1$ if and only if $n\geq 3$. From (\[24.1\]) we have$$\begin{split}
\overline{G}^{q^{\prime }}|\nabla _{b}v|^{2}& \leq C(v^{2}\overline{G}^{q^{\prime }}|\nabla _{b}\log {G}|^{2}+v^{2}\overline{G}^{q^{\prime
}}|\nabla _{b}\log {\overline{G}}|^{2}) \\
& =Cv^{2-q^{\prime }}(G^{q^{\prime }}|\nabla _{b}\log {G}|^{2}+G^{q^{\prime
}}|\nabla _{b}\log {\overline{G}}|^{2}) \\
& \leq CG^{q^{\prime }}(|\nabla _{b}\log {G}|^{2}+|\nabla _{b}\log {\overline{G}}|^{2}).
\end{split}
\label{25}$$Recall that $B_{\rho }(p)$ $\subset $ $\widetilde{M}$ denote the ball of radius $\rho $, centered at $p,$ with respect to the Carnot-Carathéodory distance. Substituting (\[25\]) into (\[24.0\]), we get$$II\leq C\rho ^{-2}\int_{B_{2\rho }(p)\backslash B_{\rho }(p)}G^{q^{\prime
}}(|\nabla _{b}\log {G}|^{2}+|\nabla _{b}\log {\overline{G}}|^{2})dV
\label{25.1}$$
by taking a cutoff function $\phi $ such that $0$ $\leq $ $\phi $ $\leq $ $1,$ $\phi $ $=$ $1$ on $B_{\rho }(p),$ $\phi $ $=$ $0$ on ($\widetilde{M}\TEXTsymbol{\backslash}$$B_{2\rho }(p)),$ and $|\nabla _{b}\phi
| $ $\leq $ $\frac{C}{\rho }.$
Taking $\phi =\psi G^{\frac{q^{\prime }}{2}}$ in (\[firequ\]), $\psi \in
C_{0}^{\infty }(\widetilde{M}\backslash \{p\}),$ we get $$\begin{split}
& \int_{\widetilde{M}}\psi ^{2}G^{q^{\prime }}|\nabla _{b}\log {\overline{G}}|^{2}dV \\
& \leq C\int_{\widetilde{M}}(\psi ^{2}G^{q^{\prime }}|\nabla _{b}\log {G}|^{2}+|\nabla _{b}(G^{q^{\prime }/2}\psi )|^{2})dV \\
& \leq C\int_{\widetilde{M}}(\psi ^{2}|\nabla _{b}\log {G}|^{2}+|\nabla
_{b}\psi |^{2})G^{q^{\prime }}dV.
\end{split}
\label{26}$$Note that the integral of $|\nabla _{b}\log {G}|^{2}G^{q^{\prime
}} $ over a region containing $p$ diverges (this is why we need $\psi $ compactly supported away from $p).$ Choosing $\psi $ such that $0$ $\leq $ $\psi $ $\leq $ $1,\psi $ $=$ $1$ on $B_{2\rho }(p)\backslash B_{\rho }(p),$ $\psi $ $=$ $0$ on $B_{\rho /2}(p)$ $\cup $ ($\widetilde{M}\TEXTsymbol{\backslash}$$B_{4\rho }(p))$, and $|\nabla _{b}\psi |$ $\leq $ $\frac{C}{\rho },$ we get$$II\leq C\rho ^{-2}\int_{B_{4\rho }(p)\backslash B_{\rho /2}(p)}G^{q^{\prime
}}(1+|\nabla _{b}\log {G}|^{2})dV \label{26.1}$$
from (\[25.1\]) and (\[26\]) (for $\rho $ large).
For $I=\int_{\widetilde{M}}v^{2}(\overline{\Delta }_{b}\phi )^{2}d\overline{V}$, since $$(b_{n}\overline{\Delta }_{b}+\overline{R})\phi =\overline{G}^{-1-\frac{2}{n}}(b_{n}\Delta _{b}+R_{\theta })(\overline{G}\phi ),$$we have $$\begin{split}
b_{n}\overline{\Delta }_{b}\phi & =\overline{G}^{-1-\frac{2}{n}}(b_{n}(\phi
\Delta _{b}\overline{G}+\overline{G}\Delta _{b}\phi -2\nabla _{b}\phi \nabla
_{b}\overline{G})+R_{\theta }\overline{G}\phi ) \\
& =\overline{G}^{-1-\frac{2}{n}}(\phi (b_{n}\Delta _{b}\overline{G}+R_{\theta }\overline{G})+b_{n}(\overline{G}\Delta _{b}\phi -2\nabla
_{b}\phi \nabla _{b}\overline{G})) \\
& =b_{n}(\overline{G}^{\frac{-2}{n}}\Delta _{b}\phi -2\overline{G}^{-1-\frac{2}{n}}\nabla _{b}\phi \nabla _{b}\overline{G}),
\end{split}
\label{27}$$that is, $$\overline{\Delta }_{b}\phi =\overline{G}^{\frac{-2}{n}}\Delta _{b}\phi -2\overline{G}^{-1-\frac{2}{n}}\nabla _{b}\phi \nabla _{b}\overline{G}.
\label{27.1}$$By (\[27.1\]) we have$$\begin{aligned}
I &=&\int_{\widetilde{M}}v^{2}(\overline{G}^{\frac{-2}{n}}\Delta _{b}\phi -2\overline{G}^{-1-\frac{2}{n}}\nabla _{b}\phi \nabla _{b}\overline{G})^{2}d\overline{V} \label{27.2} \\
&\leq &C(\int_{\widetilde{M}}v^{2}(\frac{\Delta _{b}\phi }{\overline{G}^{\frac{2}{n}}})^{2}d\overline{V}+\int_{\widetilde{M}}v^{2}\overline{G}^{-2-\frac{4}{n}}|\nabla _{b}\phi |^{2}|\nabla _{b}\overline{G}|^{2}d\overline{V})
\notag \\
&=&C(III+IV) \notag\end{aligned}$$
where$$\begin{aligned}
III &=&\int_{\widetilde{M}}v^{2}(\frac{\Delta _{b}\phi }{\overline{G}^{\frac{2}{n}}})^{2}d\overline{V} \label{27.3} \\
&=&\int_{\widetilde{M}}(\frac{\Delta _{b}\phi }{\overline{G}^{\frac{2}{n}}})^{2}(\frac{G}{\overline{G}})^{2}(\overline{G}^{\frac{2}{n}})^{n+1}dV \notag
\\
&=&\int_{\widetilde{M}}(\Delta _{b}\phi )^{2}G^{2}\overline{G}^{\frac{2(n+1)}{n}-\frac{4}{n}-2}dV \notag \\
&\leq &C\rho ^{-2}\int_{\widetilde{M}\backslash B_{\rho /4}}G^{2-\frac{2}{n}}dV \notag\end{aligned}$$
and$$\begin{aligned}
IV &=&\int_{\widetilde{M}}v^{2}\overline{G}^{-2-\frac{4}{n}}|\nabla _{b}\phi
|^{2}|\nabla _{b}\overline{G}|^{2}dV \label{27.4} \\
&=&\int_{\widetilde{M}}v^{2}\overline{G}^{\frac{-2}{n}}|\nabla _{b}\phi
|^{2}|\nabla _{b}\overline{G}|^{2}dV \notag \\
&=&\int_{\widetilde{M}}v^{2}\overline{G}^{2-\frac{2}{n}}|\nabla _{b}\phi
|^{2}|\nabla _{b}\log {\overline{G}}|^{2}dV \notag \\
&\leq &\int_{\widetilde{M}}|\nabla _{b}\phi |^{2}G^{q^{\prime }}|\nabla
_{b}\log {\overline{G}}|^{2}dV \notag \\
&\leq &C\rho ^{-2}\int_{B_{2\rho }(p)\backslash B_{\rho }(p)}G^{q^{\prime
}}|\nabla _{b}\log {\overline{G}}|^{2}dV \notag\end{aligned}$$
by a suitable choice of $\phi .$ Again we let $\phi =\psi
G^{q^{\prime }/2}$ in Lemma 4.2, where $\psi \in C_{0}^{\infty }(\widetilde{M}\setminus \{p\})$. By choosing $\psi $ suitably, we can convert (\[27.4\]) into$$IV\leq C\rho ^{-2}\int_{B_{4\rho }(p)\backslash B_{\rho /2}(p)}G^{q^{\prime
}}(1+|\nabla _{b}\log {G}|^{2})dV. \label{27.5}$$Finally we are going to show that $$II\text{ (}IV,\text{ resp.)}\leq C\rho ^{-2}\int_{\widetilde{M}\backslash
B_{\rho /4}}G^{q^{\prime }}dV \label{27.6}$$(recall that $q^{\prime }$ $=$ $2$ $-$ $\frac{2}{n})$ for $n$ $\geq $ $3,$ and $$II\text{ (}IV,\text{ resp.)}\leq C\rho ^{-2}\int_{\widetilde{M}\backslash
B_{\rho /4}}G^{\widetilde{q}}dV \label{27.6b}$$
for $n$ $=$ $2,$ in which $\widetilde{q}$ $<$ $1.$ First consider the case $(a)$ $n\geq 3$. So $q^{\prime }>1$. Now $b_{n}\Delta
_{b}G+R_{\theta }G$ $=$ $0$ (away from $p)$ and $R_{\theta }$ $>$ $C$ $>$ $0$. This implies that $$\Delta _{b}G=-b_{n}^{-1}R_{\theta }G\leq -b_{n}^{-1}CG$$Multiplying by $\phi ^{2}G^{q^{\prime }-1}$ with $\phi \in
C_{0}^{\infty }(\widetilde{M}\setminus \{p\})$ and integrating by parts give
$$\begin{aligned}
0 &\geq &\int_{\widetilde{M}}\phi ^{2}G^{q^{\prime }-1}\Delta _{b}G\ dV \\
&=&\int_{\widetilde{M}}\nabla _{b}(\phi ^{2}G^{q^{\prime }-1})\nabla _{b}GdV
\\
&=&\int_{\widetilde{M}}2\phi G^{q^{\prime }-1}\nabla _{b}\phi \nabla
_{b}GdV+(q^{\prime }-1)\int_{\widetilde{M}}\phi ^{2}G^{q^{\prime }-2}|\nabla
_{b}G|^{2}dV.\end{aligned}$$
By Young’s inequality with $\varepsilon $, we get $$\int_{\widetilde{M}}\phi ^{2}G^{q^{\prime }-2}|\nabla _{b}G|^{2}dV\leq
C\int_{\widetilde{M}}G^{q^{\prime }}|\nabla _{b}\phi |^{2}dV \label{27.7}$$
(noting that we have used the fact $q^{\prime }>1).$ By choosing $\phi $ appropriately in (\[27.7\]) and observing that $G^{q^{\prime
}-2}|\nabla _{b}G|^{2}$ $=$ $G^{q^{\prime }}|\nabla _{b}\log G|^{2}$, we obtain$$\int_{B_{4\rho }(p)\backslash B_{\rho /2}(p)}G^{q^{\prime }}|\nabla _{b}\log
G|^{2}\leq C\rho ^{-2}\int_{B_{8\rho }(p)\backslash B_{\rho
/4}(p)}G^{q^{\prime }}dV. \label{27.8}$$
So we have (\[27.6\]) for $II$ ($IV,$ resp.) by (\[26.1\]) and (\[27.8\]) ((\[27.5\]) and (\[27.8\]), resp.).
For the case $(b)$, $n$ $=$ $2$ implies $q^{\prime }$ $=$ $1$. From $b_{n}\Delta _{b}G+R_{\theta }G=0$ where $|R_{\theta }|\leq c$ we have
$$\Delta _{b}G=-b_{n}^{-1}R_{\theta }G\geq -b_{n}^{-1}cG,$$
that is, $$0\leq \Delta _{b}G+b_{n}^{-1}cG$$
Multiplying by $\phi ^{2}G^{\widetilde{q}-1}$ with $\widetilde{q}<1,\ \phi $ $\in $ $C_{0}^{\infty }(M\backslash \{p\})$ and integrating by parts give$$\begin{split}
0& \leq \int_{\widetilde{M}}\phi ^{2}G^{\widetilde{q}-1}\Delta
_{b}GdV+b_{n}^{-1}c\int_{\widetilde{M}}\phi ^{2}G^{\widetilde{q}}dV \\
& =\int_{\widetilde{M}}\nabla _{b}(\phi ^{2}G^{\widetilde{q}-1})\nabla
_{b}GdV+b_{n}^{-1}c\int_{\widetilde{M}}\phi ^{2}G^{\widetilde{q}}dV \\
& =\int_{\widetilde{M}}2\phi G^{\widetilde{q}-1}\nabla _{b}\phi \nabla
_{b}GdV+b_{n}^{-1}c\int_{\widetilde{M}}\phi ^{2}G^{\widetilde{q}}dV+(\widetilde{q}-1)\int_{\widetilde{M}}\phi ^{2}G^{\widetilde{q}-2}|\nabla
_{b}G|^{2}dV.
\end{split}
\label{29}$$From (\[29\]) we have$$\begin{aligned}
&&(1-\widetilde{q})\int_{\widetilde{M}}\phi ^{2}G^{\widetilde{q}-2}|\nabla
_{b}G|^{2}dV \\
&\leq &2\int_{\widetilde{M}}\phi G^{\widetilde{q}-1}|\nabla _{b}\phi
||\nabla _{b}G|dV+b_{n}^{-1}c\int_{\widetilde{M}}\phi ^{2}G^{\widetilde{q}}dV\end{aligned}$$
By Young’s inequality with $\varepsilon $, we obtain$$\int_{\widetilde{M}}\phi ^{2}G^{\widetilde{q}-2}|\nabla _{b}G|^{2}dV\leq
C(\int_{\widetilde{M}}G^{\widetilde{q}}|\nabla _{b}\phi |^{2}dV+\int_{\widetilde{M}}G^{\widetilde{q}}\phi ^{2}dV).$$Since $G^{\widetilde{q}-2}\geq G^{-1}$ on $\widetilde{M}\setminus
K $ for some compact subset $K$ by Proposition 3.6, we have $$\int_{\widetilde{M}}\phi ^{2}G^{-1}|\nabla _{b}G|^{2}dV\leq C(\int_{\widetilde{M}}G^{\widetilde{q}}|\nabla _{b}\phi |^{2}dV+\int_{\widetilde{M}}G^{\widetilde{q}}\phi ^{2}dV). \label{29.1}$$
Observing that $G^{-1}|\nabla _{b}G|^{2}$ $=$ $G|\nabla _{b}\log
G|^{2}$ and choosing a suitable cutoff function $\phi $ in (\[29.1\])$,$ we obtain$$\int_{B_{4\rho }(p)\backslash B_{\rho /2}(p)}G|\nabla _{b}\log G|^{2}dV\leq
C(\frac{1}{\rho ^{2}}+1)\int_{B_{8\rho }(p)\backslash B_{\rho /4}(p)}G^{\widetilde{q}}dV. \label{29.2}$$
Thus we have (\[27.6b\]) for $II$ ($IV,$ resp.) by (\[26.1\]) and (\[29.2\]) ((\[27.5\]) and (\[29.2\]), resp.) for $\rho $ large in view of Proposition 3.6.
By (\[3.12.1\]) and the assumption $s(M)$ $<$ $1$ for $n$ $=$ $2,$ we have the convergence of the integrals in (\[27.6\]) and (\[27.6b\]) (in fact, both of them tend to zero as $\rho $ $\rightarrow $ $\infty ).$ So as $\rho $ $\rightarrow $ $\infty ,$ $II$ and $IV$ go to zero. On the other hand, it is clear that $III$ goes to zero as $\rho $ $\rightarrow $ $\infty $ by ([27.3]{}) and (\[3.12.1\]). So from (\[24\]) and (\[27.2\]) we conclude that $v_{\bar{0}}$ $=$ $0.$
**(of Theorem A)**
We need only to prove $v$ $:=\frac{G}{\overline{G}}\equiv 1$. Let $q$ $=$ $\frac{2n}{n+1}$. We first prove that for any $\phi \in C_{0}^{\infty }(\widetilde{M})$ there holds$$\int_{\widetilde{M}}\phi ^{2}|\overline{\nabla }_{b}v|^{q-2}|\overline{\nabla }_{b}|\overline{\nabla }_{b}v||^{2}d\overline{V}\leq C\int_{\widetilde{M}}|\nabla _{b}\phi |^{2}\overline{G}^{q}|\nabla _{b}v|^{q}dV
\label{8}$$for some constant $C$. Since $\overline{\Delta }_{b}v=0,$ $\bar{\theta}=\overline{G}^{\frac{2}{n}}\theta $ is flat (hence $Ric$ and $Tor$ vanish), and $v_{\bar{0}}$ $=$ $0$ by Lemma 4.3, we reduce the Bochner formula (\[BF\]) to
$$\frac{1}{2}\overline{\Delta }_{b}|\overline{\nabla }_{b}v|^{2}=-|(\overline{\nabla }^{H})^{2}v|^{2}. \label{2}$$
Observe that $$|\overline{\nabla }_{b}|\overline{\nabla }_{b}v||^{2}\leq |(\overline{\nabla
}^{H})^{2}v|^{2}. \label{3}$$For $q>1$ we compute $$\begin{split}
\overline{\Delta }_{b}|\overline{\nabla }_{b}v|^{q}& =\overline{\Delta }_{b}(|\overline{\nabla }_{b}v|^{2})^{\frac{q}{2}} \\
& =\frac{q}{2}(|\overline{\nabla }_{b}v|^{2})^{\frac{q}{2}-1}\overline{\Delta }_{b}|\overline{\nabla }_{b}v|^{2}-\frac{q}{2}(\frac{q}{2}-1)(|\overline{\nabla }_{b}v|^{2})^{\frac{q}{2}-2}|\overline{\nabla }_{b}|\overline{\nabla }_{b}v|^{2}|^{2} \\
& =\frac{q}{2}|\overline{\nabla }_{b}v|^{q-2}\overline{\Delta }_{b}|\overline{\nabla }_{b}v|^{2}-\frac{q(q-2)}{4}|\overline{\nabla }_{b}v|^{q-4}|\overline{\nabla }_{b}|\overline{\nabla }_{b}v|^{2}|^{2} \\
& \leq -q(q-1)|\overline{\nabla }_{b}v|^{q-2}|\overline{\nabla }_{b}|\overline{\nabla }_{b}v||^{2} \\
& =-C_{q}|\overline{\nabla }_{b}v|^{q-2}|\overline{\nabla }_{b}|\overline{\nabla }_{b}v||^{2}
\end{split}
\label{4}$$where $C_{q}$ $=$ $q(q-1)$ $>$ $0$ for $n$ $\geq $ $2.$ For the inequality in (\[4\]) we have used (\[2\]), (\[3\]). Consider first the case $\phi \in C_{0}^{\infty }(\widetilde{M}\setminus \Phi ^{-1}(y))$. Multiplying (\[4\]) by $\phi ^{2}$ and integrating by parts, we get
$$\begin{split}
\int_{\widetilde{M}}\phi ^{2}|\overline{\nabla }_{b}|\overline{\nabla }_{b}v||^{2}|\overline{\nabla }_{b}v|^{q-2}d\overline{V}& \leq
-C_{q}^{-1}\int_{\widetilde{M}}\phi ^{2}\overline{\Delta }_{b}|\overline{\nabla }_{b}v|^{q}d\overline{V} \\
& =-C_{q}^{-1}\int_{\widetilde{M}}\overline{\nabla }_{b}(\phi ^{2})\overline{\nabla }_{b}|\overline{\nabla }_{b}v|^{q}d\overline{V} \\
& \leq 2qC_{q}^{-1}\int_{\widetilde{M}}|\phi ||\overline{\nabla }_{b}\phi ||\overline{\nabla }_{b}v|^{q-1}|\overline{\nabla }_{b}|\overline{\nabla }_{b}v||d\overline{V} \\
& \leq C_{q}^{\prime }\int_{\widetilde{M}}|\overline{\nabla }_{b}\phi |^{2}|\overline{\nabla }_{b}v|^{q}d\overline{V}
\end{split}$$
for some constant $C_{q}^{\prime },$ where the last inequality is deduced by applying the Schwarz (or Young’s) inequality with $\varepsilon $. Now we change the integral on the right hand side to a corresponding one using the form $\theta $ and get $$\begin{split}
\int_{\widetilde{M}}\phi ^{2}|\overline{\nabla }_{b}|\overline{\nabla }_{b}v||^{2}|\overline{\nabla }_{b}v|^{q-2}d\overline{V}& \leq C_{q}^{\prime
}\int_{\widetilde{M}}\overline{G}^{\frac{-2}{n}}|{\nabla }_{b}\phi |^{2}|{\nabla }_{b}v|^{q}\overline{G}^{\frac{-q}{n}}(\overline{G}^{\frac{2}{n}})^{n+1}dV \\
& =C_{q}^{\prime }\int_{\widetilde{M}}|\nabla _{b}\phi |^{2}|\nabla
_{b}v|^{q}\overline{G}^{\frac{2(n+1)-2-q}{n}}dV.
\end{split}$$
Observe that $q=\frac{2n}{n+1}$ implies $\frac{2(n+1)-2-q}{n}=q$. We have shown the desired inequality (\[8\]) for $\phi \in C_{0}^{\infty }(\widetilde{M}\setminus \Phi ^{-1}(y))$.Now for $\phi \in C_{0}^{\infty }(\widetilde{M})$, we consider $\psi
_{r}\phi $ where $\psi _{r}$ is a cutoff function such that for each $\bar{p}
$ $\in $ $\Phi ^{-1}(y),$ $\psi _{r}\equiv 0$ in $B_{r}(\bar{p}),$ $\psi
_{r}\equiv 1$ on $\widetilde{M}\setminus B_{2r}(\bar{p})$, and $0\leq \psi
_{r}\leq 1$ ($r$ small so that $B_{2r}(\bar{p}_{1})$ $\cap $ $B_{2r}(\bar{p}_{2})$ is empty for any pair of points $\bar{p}_{1},$ $\bar{p}_{2}$ $\in $ $\Phi ^{-1}(y))$. We also require that $|\nabla _{b}\psi
_{r}|\leq 2r^{-1}$. Applying (\[8\]) for $\psi _{r}\phi $, we have
$$\begin{split}
& \int_{\widetilde{M}}\psi _{r}^{2}\phi ^{2}|\overline{\nabla }_{b}|\overline{\nabla }_{b}v||^{2}|\overline{\nabla }_{b}v|^{q-2}d\overline{V} \\
& \leq C_{1}\int_{\widetilde{M}}\psi _{r}^{2}|\nabla _{b}\phi |^{2}|\nabla
_{b}v|^{q}\overline{G}^{q}dV+C_{2}\int_{\widetilde{M}}\phi ^{2}|\nabla
_{b}\psi _{r}|^{2}|\nabla _{b}v|^{q}\overline{G}^{q}dV.
\end{split}$$
Noticing that the last integral has order $O(r^{2n-q})\rightarrow
0 $, as $r\rightarrow 0$, we see that (\[8\]) holds for $\phi \in
C_{0}^{\infty }(\widetilde{M})$.Next we are going to prove $$\begin{split}
& \int_{B_{\rho }(p)}|\overline{\nabla }_{b}v|^{q-2}|\overline{\nabla }_{b}|\overline{\nabla }_{b}v|^{2}d\overline{V} \\
& \leq C\rho ^{-2}\int_{B_{4\rho }(p)\backslash B_{\rho
/2}(p)}G^{q}(1+|\nabla _{b}\log {G}|^{2})dV,
\end{split}
\label{19}$$where $\rho >0$ is sufficiently large and $C$ is a constant. We want to make use of (\[8\]). Note that
$$\begin{split}
\overline{G}^{q}|\nabla _{b}v|^{q}& =|\overline{G}\nabla _{b}v|^{q} \\
& =|\nabla _{b}G-G\overline{G}^{-1}\nabla _{b}\overline{G}|^{q} \\
& \leq C(|\nabla _{b}G|^{q}+G^{q}|\nabla _{b}\log {\overline{G}}|^{q}).
\end{split}
\label{10}$$
Thus, if we take $\phi \in C_{0}^{\infty }(\widetilde{M})$ such that $\phi \equiv 1$ in $B_{\rho }(p),\phi \equiv 0$ on $\widetilde{M}\setminus B_{2\rho }(p),0\leq \phi \leq 1$ and $|\nabla _{b}\phi |\leq 2\rho
^{-1}$, We see from (\[8\]) and (\[10\]) that
$$\begin{split}
\int_{B_{\rho }(p)}|\overline{\nabla }_{b}v|^{q-2}|\overline{\nabla }_{b}|\overline{\nabla }_{b}v||^{2}d\overline{V}& \leq \int_{\widetilde{M}}\phi
^{2}|\overline{\nabla }_{b}v|^{q-2}|\overline{\nabla }_{b}|\overline{\nabla }_{b}v||^{2}d\overline{V} \\
& \leq C\rho ^{-2}\int_{B_{2\rho }(p)\backslash B_{\rho }(p)}(|\nabla
_{b}G|^{q}+G^{q}|\nabla _{b}\log {\overline{G}}|^{q})dV.
\end{split}
\label{11}$$
Let $a$ $=$ $\frac{(2-q)q}{2}.$ Note that $q$ $<$ $2.$ By Young’s inequality we have
$$\begin{split}
|\nabla _{b}G|^{q}& =G^{a}G^{-a}|\nabla _{b}G|^{q} \\
& \leq C(G^{a\frac{2}{2-q}}+G^{-a\frac{2}{q}}|\nabla _{b}G|^{2}) \\
& \leq C(G^{q}+G^{q-2}|\nabla _{b}G|^{2}) \\
& =CG^{q}(1+|\nabla _{b}\log {G}|^{2}),
\end{split}
\label{12}$$
and $$\begin{split}
G^{q}|\nabla _{b}\log {\overline{G}}|^{q}& =G^{q}\frac{|\nabla _{b}\overline{G}|^{q}}{\overline{G}^{q}} \\
& \leq G^{q}\frac{C\overline{G}^{q}(1+|\nabla _{b}\log {\overline{G}}|^{2})}{\overline{G}^{q}} \\
& =CG^{q}(1+|\nabla _{b}\log {\overline{G}}|^{2}).
\end{split}
\label{13}$$
So from (\[11\]), (\[12\]), and (\[13\]), we obtain $$\begin{split}
& \int_{B_{\rho }(p)}|\overline{\nabla }_{b}v|^{q-2}|\overline{\nabla }_{b}|\overline{\nabla }_{b}v||^{2}d\overline{V} \\
\leq & C\rho ^{-2}\int_{B_{2\rho }\backslash B_{\rho }}G^{q}(1+|\nabla
_{b}\log {G}|^{2}+|\nabla _{b}\log {\overline{G}}|^{2})dV.
\end{split}
\label{14}$$
Taking $\phi $ $=$ $\psi G^{\frac{q}{2}}$ in Lemma 4.2, where $\psi \in C_{0}^{\infty }(\widetilde{M}\setminus \{p\})$, we have $$\begin{split}
& \int_{\widetilde{M}}\psi ^{2}G^{q}|\nabla _{b}\log {\overline{G}}|^{2}dV \\
\leq & C\int_{\widetilde{M}}(\psi ^{2}G^{q}|\nabla _{b}\log {G}|^{2}+\nabla
_{b}(\psi G^{\frac{q}{2}})|^{2})dV \\
\leq & C\int_{\widetilde{M}}(\psi ^{2}|\nabla _{b}\log {G}|^{2}+|\nabla
_{b}\psi |^{2})G^{q}dV.
\end{split}
\label{18}$$
Choosing the cutoff function $\psi $ appropriately in (\[18\]), we get$$\begin{aligned}
&&\int_{B_{2\rho }\backslash B_{\rho }}G^{q}|\nabla _{b}\log {\overline{G}}|^{2}dV \label{18.1} \\
&\leq &C\int_{B_{4\rho }(p)\backslash B_{\rho /2}(p)}G^{q}(1+|\nabla
_{b}\log {G}|^{2})dV \notag\end{aligned}$$
for $\rho $ large. Substituting (\[18.1\]) into (\[14\]) gives (\[19\]).
Since $b_{n}\Delta _{b}G+R_{\theta }G=0$ on $\widetilde{M}\setminus \{p\}$ and $R_{\theta }\geq C>0$, we have$$\Delta _{b}G=-b_{n}^{-1}R_{\theta }G\leq -b_{n}^{-1}CG$$Multiplying by $\phi ^{2}G^{q-1}$ with $\phi \in C_{0}^{\infty }(\widetilde{M}\setminus \{p\})$ and integrating by parts give $$\begin{split}
0& \geq \int_{\widetilde{M}}\phi ^{2}G^{q-1}(\Delta _{b}G)dV \\
& =\int_{\widetilde{M}}\nabla _{b}(\phi ^{2}G^{q-1})\nabla _{b}GdV \\
& =\int_{\widetilde{M}}2\phi G^{q-1}\nabla _{b}\phi \nabla
_{b}GdV+(q-1)\int_{\widetilde{M}}\phi ^{2}G^{q-2}|\nabla _{b}G|^{2}dV.
\end{split}
\label{21}$$Applying the Schwarz inequality with $\varepsilon $ to (\[21\]), we obtain $$\int_{\widetilde{M}}\phi ^{2}G^{q-2}|\nabla _{b}G|^{2}dV\leq C\int_{\widetilde{M}}G^{q}|\nabla _{b}\phi |^{2}dV. \label{21.1}$$Noting that $G^{q-2}|\nabla _{b}G|^{2}$ $=$ $G^{q}|\nabla _{b}\log
G|^{2}$ and choosing some appropriate $\phi $ in (\[21.1\]), we can reduce (\[19\]) to $$\int_{B_{\rho }(p)}|\overline{\nabla }_{b}v|^{q-2}|\overline{\nabla }_{b}|\overline{\nabla }_{b}v||^{2}dV\leq C\rho ^{-2}\int_{\widetilde{M}\backslash
B_{\rho /4}}G^{q}dV. \label{22}$$
By (\[3.12.1\]) $G^{q}$ is integrable since $q$ $=$ $\frac{2n}{n+1}$ $>$ $1$ for $n$ $\geq $ $2.$ So letting $\rho \rightarrow \infty $ in (\[22\]) we get$$|\overline{\nabla }_{b}v|=const.$$
Since $\overline{\nabla }_{b}v|$ $=$ $\bar{G}^{-1}|\overline{\nabla }_{b}G-v\overline{\nabla }_{b}\bar{G}|$ $\rightarrow $ $0$ at $p$ by Lemma 4.1, we have $\overline{\nabla }_{b}v$ $=$ $0.$ So $v$ $= $ $const..$ From $v\rightarrow 1$ at $p,$ we conclude that $v$ $\equiv $ $1.$
The positive CR mass theorem {#seccrmass}
============================
In this section, according to the work of Li in [@Li3], we would like to introduce a positive mass theroem for spherical CR manifolds. Let $M$ be a closed spherical CR manifold, $\widetilde{M}$ be its universal cover and $$\pi : \widetilde{M}\longrightarrow M$$ be the canonical projection map, $$\Phi : \widetilde{M}\longrightarrow S^{2n+1}$$ be a CR developing map. We would like to construct local coordinates near each point $b$ of $M$. There is a local inverse $\pi^{-1}$ as follows: $$\pi^{-1} : U_{b} \longrightarrow \widetilde{M},$$ where $U_{b}$ is a neighborhood of $b\in M$. Let $q=\Phi(p)\in S^{2n+1}$, where $p\in\pi^{-1}(b)$, the local CR transformation $$T=C_{q}\circ \Phi\circ\pi^{-1} : U_{b} \longrightarrow H^{n}$$ provides $M$ a local coordinate $(z,t)\in H^{n}$ such that $(z(b),t(b))=\infty$. Here $C_{q} : S^{2n+1}\longrightarrow H^{n}$ is the Cayley transform with pole at $q$, i.e. $C_{q}(q)=\infty$. We will call such coordinates “spherical CR coordinates near $\infty$”.
Let $G_{b}$ be the Green’s function of $D_{\theta}$ with pole at $b$. It follows that there is a positive smooth function $h=h(z,t)$ defined on $H^{n} $ near $\infty$ such that $$\label{secequ8}
(T^{-1})^{*}(G_{b}^{\frac{2}{n}}\theta)=h^{\frac{2}{n}}\Theta.$$ By positive constant rescaling we may assume that the complex Jacobian at $p$ is $|\Phi^{^{\prime }}(p)|=1$. Let $\rho(z,t)=(|z|^{4}+t^{2})^{1/4}$ be the Heisenberg norm on $H^{n}$. Therefore, We have the following asymptotic expansion of $h=h(z,t)$ near $\infty$:
**Lemma 5.1.** *Let* $M$ * be a closed spherical CR manifold which is not the standard sphere. Suppose the CR developing map is injective. Let* $h$ * be defined as above. Then we have, near* $\infty$, $$h=h(z,t)>1$$ *and* $$h(z,t)=1+A_{b}\cdot\rho(z,t)^{-2n}+O(\rho(z,t)^{-2n-1}).$$
Since the projection $\pi$ doesn’t change geometry, it follows that near $x\in \pi^{-1}(b)$: $$D_{\theta}(\pi^{*}G_{b})=D_{\theta}(G_{b}),$$ where the left hand side is over $\widetilde{M}$ and the right hand side is over $M$. For $b\in M$ and $x\in\pi^{-1}(b)$, let $\delta_{x}$ be the Dirac delta function with pole at $x$. Therefore, $$D_{\theta}(\pi^{*}G_{b})=\sum_{x\in\pi^{-1}(b)}\delta_{x},$$ so $\pi^{*}G_{b}$ has poles precisely in the set $\pi^{-1}(b)\subset
\widetilde{M}$. For each fixed $p\in\pi^{-1}(b)$, $$D_{\theta}(\overline{G}_{p})=\delta_{p}.$$ Therefore $\pi^{*}G_{b}-\overline{G}_{p}$ is bounded near $p$. On the other hand, by the normalization $|\Phi^{^{\prime }}(p)|=1$ and $\Phi $ is injective, we have $$\overline{G}_{p}^{\frac{2}{n}}\theta=\Phi^{*}(H_{q}^{\frac{2}{n}}\theta_{S^{2n+1}}),$$ it follows that globally on $\widetilde{M}\setminus p$ $$\label{secequ12}
((C_{q}\circ\Phi)^{-1})^{*}(\overline{G}_{p}^{\frac{2}{n}}\theta)=\Theta.$$ From equations (\[secequ8\]) and (\[secequ12\]) and that $\pi^{*}G_{b}$ and $\overline{G}_{p}$ have the same pole strength near $p$, $$h(\infty)=\lim_{(z,t)\rightarrow \infty}h(z,t)=1.$$ On the other hand, because $\overline{G}_{p}$ is the minimal Green’s function for $D_{\theta}$ on $\widetilde{M}$, Bony’s strong maximum principle implies that globally on $\widetilde{M}$: $$\pi^{*}G_{b}>\overline{G}_{p},$$ which by equations (\[secequ8\]) and (\[secequ12\]) that for $(z,t)$ near $\infty$, we have $h=h(z,t)>1$.
Next, we would like to show the asymptotic expansion of the function $h=h(z,t)$ near $\infty$. By the transformation rule of the CR invariant sublaplacian, we have $R(G_{b}^{\frac{2}{n}}\theta)=0$ globally on $M\setminus \{b\}$, where $R(G_{b}^{\frac{2}{n}}\theta)$ is the Webster curvature with respect to the contact form $G_{b}^{\frac{2}{n}}\theta$. Using the transformation rule again we have $D_{\Theta}(h)=0$ for $(z,t)$ in a neighborhhod of $\infty\in H^{n}$. It follows that (since $h$ is regular near $\infty$ and $h(\infty)=1$): $$h(z,t)=1+c_{-1}\cdot\rho^{-1}+\cdots +A_{b}\cdot\rho^{-2n}+O(\rho^{-2n-1}).$$ Consider the global CR inversion $\vartheta : H^{n}\setminus
0\longrightarrow H^{n}\setminus 0$ as follows: $$\vartheta(z,t)=(\hat{z},\hat{t})=(-z/w, t/|w|^{2}),$$ where $w=t+i|z|^{2}$. Consider the following standard contact form $$\Theta(z,t)=dt-i\sum_{\alpha=1}^{n}(z^{\alpha}\cdot
dz^{\bar\alpha}-z^{\bar\alpha}\cdot dz^{\alpha}),$$ and $$\Theta(\hat{z},\hat{t})=d\hat{t}-i\sum_{\alpha=1}^{n}(\hat{z}^{\alpha}\cdot d\hat{z}^{\bar\alpha}-\hat{z}^{\bar\alpha}\cdot d\hat{z}^{\alpha}).$$ It follows that $\rho(z,t)^{2}=|w|=|\hat{w}|^{-1}=\rho(\hat{z},\hat{t})^{-2}$ and $$\Theta(\hat{z},\hat{t})=|w|^{-2}\cdot \Theta(z,t)=(\rho(\hat{z},\hat{t})^{2n})^{\frac{2}{n}}\cdot\Theta(z,t).$$ Therefore, $$h(z,t)^{\frac{2}{n}}\cdot \Theta(z,t)=(h(\hat{z},\hat{t})\cdot\rho(\hat{z},\hat{t})^{-2n})^{\frac{2}{n}}\cdot\Theta(\hat{z},\hat{t}).$$ By the facts that $R(h(z,t)\cdot\Theta(z,t))=0$ and $\Theta(\hat{z},\hat{t})$ and the trasformation rule of the CR invariant sublaplacian, it follows that near but away from the origin in $H^{n}$, we have $$D_{\Theta(\hat{z},\hat{t})}(h(\hat{z},\hat{t})\cdot\rho(\hat{z},\hat{t})^{-2n})=0, \text{and}\ D_{\Theta(\hat{z},\hat{t})}\rho(\hat{z},\hat{t})^{-2n}=0.$$ Therefore, near but away fron the origin in $H^{n}$: $$D_{\Theta(\hat{z},\hat{t})}(h(\hat{z},\hat{t})\cdot\rho(\hat{z},\hat{t})^{-2n}-\rho(\hat{z},\hat{t})^{-2n})=0$$ and note that $$\begin{split}
&h(\hat{z},\hat{t})\cdot\rho(\hat{z},\hat{t})^{-2n}-\rho(\hat{z},\hat{t})^{-2n} \\
=&c_{-1}\cdot\rho(\hat{z},\hat{t})^{-2n+1}+\cdots +c_{-2n+1}\cdot\rho(\hat{z},\hat{t})^{-1}+A_{b}+O(\rho(\hat{z},\hat{t})).
\end{split}$$ A standard removable singularity argument (Proposition 5.17 in [@JL]) implies that $$c_{-1}=\cdots =c_{-2n+1}=0.$$ Therefore, we have the following aymptotic expansion of $h=h(z,t)$ near $\infty$: $$h(z,t)=1+A_{b}\cdot\rho(z,t)^{-2n}+O(\rho(z,t)^{-2n-1}).$$ This completes the lemma.
We call the constant $A_{b}$ CR mass.
We would like to remark that the constant $A_b$ doesn’t depend on the choice of local coordinates near $b\in M$ (see [@Li3]). Corollary C states the positivity of the CR mass.
**(of Corollary C)**
By Theorem A, $\Phi$ is injective. It follows from Lemma 5.1 that for $(z,t)$ near $\infty$ $$h=h(z,t)=1+A_{b}\cdot\rho(z,t)^{-2n}+O(\rho(z,t)^{-2n-1})>1.$$ Let $B_{L}(0)$ be a ball on $H^{n}$ centered at $0$ and with radius $L$ such that $$m=\min_{\partial B_{L}(0)}(h-1)>0.$$ Since in $H^{n}\setminus B_{L}(0)$, $$D_{\Theta}(h-1)=D_{\Theta}(mL^{2n}\rho^{-2n})=0$$ and $$(h-1)|_{\partial B_{L}(0)}\geq mL^{2n}\rho^{-2n}|_{\partial B_{L}(0)},$$ we conclude from Bony’s maximum principle that in $H^{n}\setminus B_{L}(0),$ $$h-1\geq mL^{2n}\rho^{-2n}.$$ Therefore, $A_{b}\geq mL^{2n}>0$.
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|
---
author:
-
-
- '[^1]'
-
bibliography:
- 'lattice2017.bib'
title: BSM Kaon Mixing at the Physical Point
---
Introduction {#intro}
============
Kaon mixing is a flavour changing neutral current process in which a neutral kaon ${K^0}$ oscillates with its anti-particle $\bar{K^0}$. In the standard model (SM) it is dominated by box diagrams such as the one shown in figure \[fig:kkmixing\].
[KaonMixing]{}
(100,45)
We separate out the long distance contributions, using the operator product expansion (OPE), into a matrix element ${\braket{\bar{K^0}|O_1|K^0}}$ where $O_1$ is the SM four quark operator shown in equation \[eq:opbasis\]. It has (vector-axial)$\times$(vector-axial) Dirac structure in the SM as a result of the W vertices. Beyond the SM, when the mediating particle is not constrained by the standard model flavour changing vertices, effective operators with other Dirac structures are possible. We can construct a full (SUSY) basis [@Gabbiani:1996hi] of five parity-even four-quark operators:
$$\begin{split}
O_1 &=(\bar{s}_a {\gamma_{\mu}}(1-{\gamma_{5}})d_a)(\bar{s}_b {\gamma_{\mu}}(1-{\gamma_{5}})d_b) \\
O_2 &= (\bar{s}_a (1-{\gamma_{5}})d_a)(\bar{s}_b (1-{\gamma_{5}})d_b)\\
O_3 &= (\bar{s}_a (1-{\gamma_{5}})d_b)(\bar{s}_b (1-{\gamma_{5}})d_a)\\
O_4 &= (\bar{s}_a (1+{\gamma_{5}})d_a)(\bar{s}_b (1+{\gamma_{5}})d_b)\\
O_5 &= (\bar{s}_a (1+{\gamma_{5}})d_b)(\bar{s}_b (1+{\gamma_{5}})d_a),\\
\label{eq:opbasis}
\end{split}
$$
which appear in the effective $\Delta S = 2$ Hamiltonian as $$\mathcal{H}^{\Delta S = 2} = \sum_{i=1}^5 C_i(\mu) O_i(\mu).$$
Whilst the Wilson coefficients $C_i(\mu)$ depend on the physics of the particular BSM model studied, the operators themselves are model independent.
Motivation
----------
The BSM matrix elements have not been as widely studied as the standard model $B_k$. There have been calculations by RBC-UKQCD [@Boyle:2012qb][@Garron:2016mva], ETM [@Bertone:2012cu][@Carrasco:2015pra] and SWME [@Bae:2013tca][@Jang:2014aea][@Jang:2015sla], but there are some tensions between the results from different collaborations. These differences can be seen in table \[tab:compBSM\] and are summarised in the most recent FLAG report [@Aoki:2016frl].
The most recent BSM kaon mixing study by RBC-UKQCD [@Garron:2016mva] sought to address these tensions and proposed that they arose from different choices in the renormalisation methods applied. It was argued that the new RI-SMOM scheme introduced was better behaved than the more commonly used RI-MOM scheme. This work aims to improve upon the precision of those results by including a third lattice spacing and ensembles with physical pions. By obtaining more precise results we should be able to comment on the renormalisation scheme’s role in the obeserved tension.
Parameterisation of the Matrix Elements
=======================================
Bag Parameters
--------------
The renormalised bag parameter is defined as the ratio of the matrix element over its vacuum saturation approximation value,
$$\label{eq:BagGen}
B_i(\mu) = \frac{ {\braket{\bar{K^0}|O_i(\mu)|K^0}} }{{\braket{\bar{K^0}|O_i(\mu)|K^0}}}_{\textrm{VSA}.}$$
At leading order, the forms of the SM and BSM bag parameters are given by,
$$\label{eq:BagSM}
B_1(\mu)= \frac{ {\braket{\bar{K^0}|O_1(\mu)|K^0}} }{\frac{8}{3}m_K^2 f_K^2},$$
$$\label{eq:BagBSM}
B_i(\mu) = \frac{ (m_s(\mu) + m_d(\mu))^2 }{N_i m_K^2 f_K^4} {\braket{\bar{K^0}|O_i(\mu)|K^0}}.$$
The factors $N_i$ depend upon the basis in which we’re working. As we work in the SUSY basis, $N_i = (\frac{8}{3},\frac{-5}{3},\frac{1}{3},2,\frac{2}{3})$.
Ratio Parameters
----------------
Ratio parameters, $R_i$, are another parametrisation of the BSM matrix elements. The idea of using ratios to define parameters was proposed in [@Donini:1999nn], following the forms given in [@Babich:2006bh], we define the ratio parameters as,
$$R_i\bigg( \frac{m^2_P}{f^2_P} , a^2, \mu\bigg) =
\bigg[ \frac{f_K^2}{m_K^2} \bigg ]_{Exp.}
\bigg[ \frac{m_P^2}{f_P^2}
\frac{{\braket{\bar{P}|O_i(\mu)|P}}}{ {\braket{\bar{P}|O_1(\mu)|P}} } \bigg]_{Lat.,}$$
where $P$ denotes the simulated strange-light pseudoscalar meson. At the physical point they reduce to direct ratios of the BSM to SM matrix elements ,
$$R_i(\mu) = R_i\bigg( \frac{m^2_K}{f^2_K} , a=0, \mu\bigg) =
\frac{ {\braket{\bar{K^0}|O_i(\mu)|K^0}} }{ {\braket{\bar{K^0}|O_1(\mu)|K^0}} } .
\label{eq:Ratio-phys}$$
The ratio parameters have some advantages over the bag parameters. As there is no explicit dependence on the quark masses, the matrix elements can be recovered from the ratio parameters $R_i$, the SM bag parameter $B_K$ and the experimentally measured kaon mass and decay constant alone. In addition we can expect some cancellation of errors due to the similarity of the numerator and denominator.
Lattice Implementation {#sec-1}
======================
We use RBC-UKQCD’s $ n_f=2+1 $ gauge ensembles generated with the Iwasaki gauge action [@Iwasaki:1985we][@Okamoto:1999hi]. Our ensembles have a DWF action with either the Möbius [@Brower:2004xi] or Shamir [@Shamir:1993zy] kernel. These ensembles span 3 lattice spacings; (C)oarse, (M)edium, and (F)ine. C0 and M0 have physical pion masses and all ensembles have physical valence strange quark masses. The details of the ensembles are shown in table \[tab:enspar\]. The ensembles C0 and M0 have been described in more detail in [@Blum:2014tka], and F1 in [@Boyle:2017jwu].
Correlator Fitting {#sec-2}
------------------
We define the two-point and three-point functions as
$$c_{\mathcal{O}_1 \mathcal{O}_2}(t_i,t) = \sum_x \braket{\mathcal{O}_2(x,t)\mathcal{O}_1(x_i,t_i)},
\label{eq:2pt}$$
$$c_{O_k}(t_i,t,t_f)
= {\braket{P(t_f)|O_k(t)|P(t_i)}},
\label{eq:3pt}$$
where $\mathcal{O}_{1/2}$ denote bilinear operators which in this work are either $\mathbb{P}$, the pseudo-scalar density, or $\mathbb{A}_0$, the temporal component of the local axial current and $O_k$ are the four-quark operators.
At large times the ground state dominates and we can fit the two-point correlators to a $\cosh$ or $\sinh$ function (depending on the Dirac structure of $\mathcal{O}_1$ and $\mathcal{O}_2$) to measure the pseudoscalar masses and amplitudes.
$$c_{\mathcal{O}_1 \mathcal{O}_2}(t_i,t) \xrightarrow[t_i \ll t \ll T]{} \frac{a^4\braket{0|\mathcal{O}_2|P}\braket{P|\mathcal{O}_1|0}}{2am_P}
\bigg(e^{-m_P(t-t_i)}\pm e^{-m_P(T-(t-t_i))}\bigg)
\label{eq:2ptfit}$$
Taking ratios of the correlators to measure $B_i^{\textrm{Lat}}$ and $R^{\textrm{Lat}}_i$, as shown in equations \[eq:Rcorrs\] and \[eq:Bcorrs\], they plateau far from the lattice time extent boundaries and we can fit to a constant. $$R_k^{\textrm{Lat}}(t_f,t,t_i) = \frac{c_{O_k}(t_i,t,t_f)}{c_{O_1}(t_i,t,t_f)}\xrightarrow[t_i \ll t \ll t_f \ll T]{} \frac{{\braket{\bar{P}|O_k|P}}}{{\braket{\bar{P}|O_1|P}}}
\label{eq:Rcorrs}$$ $$B_k^{\textrm{Lat}}(t_f,t,t_i) = \frac{1}{N_k} \frac{ c_{O_k}(t_i,t,t_f) }
{ c_{\bar{P}P}(t_i,t) c_{P\bar{P}}(t_i,t)}
\xrightarrow[t_i \ll t \ll t_f \ll T]{} \frac{1}{N_k}\frac{{\braket{\bar{P}|O_k|P}}}{\braket{\bar{P}|\mathbb{P}|0} \braket{0|\mathbb{P}|P}}, \; \; \; \; k>1.
\label{eq:Bcorrs}$$
\[fig:corrfits\]
Non-Perturbative Renormalization {#sec-3}
================================
The bare parameters are renormalised to ensure a well defined continuum limit and remove any divergences. We use the non-perturbative Rome-Southampton method [@Martinelli:1994ty] with non-exceptional kinematics (RI-SMOM) [@Sturm:2009kb]. The RI-SMOM scheme for the SM four-quark operator is described in [@Aoki:2010pe], and the extension to the full SUSY basis in [@Boyle:2017skn]. The renormalised matrix elements can be expressed as, $$\braket{Q_i}^{\textrm{RI}}(\mu,a) = Z_{ij}^{\textrm{RI}}(\mu,a) \braket{Q_j}^{\textrm{bare}}(a),$$ where $Z_{ij}^{\textrm{RI}}(\mu,a)$ denotes the renormalisation factor which, if chiral symmetry breaking effects can be neglected, has block diagonal structure.
[renorm]{}
(70,60)
In this method, we require that the projection of the renormalised amputated vertex function (Figure \[fig:vertex\]) $\Pi^{\textrm{ren}}_i$ is equal to its tree level value. This equation defines $Z^{RI}_{ij}$.
The choice of projector is not unique, in [@Garron:2016mva] we have used two schemes called ($\gamma_\mu, \gamma_\mu$) and ($\slashed{q},\slashed{q}$), which are also the schemes considered in this work. Details on the renormalisation procedure and the definition of these projectors can be found in [@Boyle:2017skn].
$$P_k[\Pi^{\textrm{RI}}_i(p_1,p_2)]_{p^2=\mu^2} = P_k\bigg[\frac{Z_{ij}^{\textrm{RI}}(\mu,a)}{Z_q(\mu,a)}\Pi_j^{\textrm{bare}}(a,p_1,p_2)\bigg]_{p^2=\mu^2} = P_k[\Pi_i^{(0)}]$$
Here we present only results obtained through the $(\gamma_\mu,\gamma_\mu)$ projection scheme.
Since the discretisation used in C0(M0) and C1/2(M1/2) only differ by the approximation of the sign function in the infinite Ls limit we have assumed it is valid to reuse the renormalisation calculated for C1/2(M1/2) for C0(M0) for the time being. The renormalisation is calculated on a small subsett of configurations to the matrix element measurements, therefore we propagate the errors on the renormalisation by generating bootstraps according to a gaussian distribution with width equal to the error.
Extrapolation to the Physical Point and Continuum Limit {#sec-4}
=======================================================
The renormalised parameters are extrapolated to the physical pion mass and continuum limit in an uncorrelated global fit. We use the following two ansatz:
1. A chiral and continuum extrapolation of $R_i$ and $B_i$ according to a fit ansatz linear in both $a^2$ and $m_P^2/f_P^2$. $$Y\bigg(a^2,\frac{m^2_{P}}{16\pi^2f_P^2}\bigg) = Y\bigg(0,\frac{m^2_{\pi}}{16\pi^2f_{\pi}^2}\bigg)\bigg[1 + \alpha a^2 + \beta \frac{m^2_{P}}{16\pi^2f_P^2} \bigg]$$
2. A global fit following NLO SU(2) chiral perturbation theory to a fit function shown below. $$Y\bigg(a^2,\frac{m^2_{P}}{16\pi^2f_P^2}\bigg) = Y\bigg(0,\frac{m^2_{\pi}}{16\pi^2f_{\pi}^2}\bigg)\bigg[1 + \alpha a^2 + \frac{m^2_{P}}{16\pi^2f_P^2} \bigg(\beta + C_i \log\bigg(\frac{m^2_P}{\Lambda^2}\bigg) \bigg) \bigg]$$ $C_i$ are the chiral logarithm factors, for $R_i$ we have $C_i = (3/2,3/2,5/2,5/2)$ and for $B_i$ then $C_i$=(-1/2,-1/2,1/2,1/2). $\Lambda$ is the QCD scale.
We expect the dominant lattice artefacts to be linear in $a^2$ as we use domain-wall fermions. These two methods are equivalent up to the chiral logarithm term, and the difference can indicate how strong the chiral effects from including non-physical pion masses are. The lattice spacings were calculated in [@Blum:2014tka] from many of the same ensembles as in this work, therefore a correlation between the data in our global fit is present.. However, in order to decouple this work from the previous work we perform an uncorrelated fit. We propagate the error on lattice spacings by generating bootstraps according to a gaussian distribution with width equal to the error on $a$. The error on the lattice spacings is small (of order 0.5%) and the contribution of the lattice spacing to the correction of the data is of order 10% so overall we expect the effect of neglecting these correlations to be small and we believe this approach is justified. When calculating $\chi^2$ in the global fit, we consider the data’s deviation from the model in y axis only. The gradient of the slope we obtain in $m_\pi/(4\pi f_\pi)^2$ is small therefore the change in $\chi^2$, were we to instead consider the smallest approach to the fit line, would be negligible.
Results {#sec-5}
=======
Plots of the global fit, with the linear fit ansatz, are shown for the ratio parameters in Figure 4. In table \[tab:results\] we present preliminary results from the global fit for the ratio parameters for both fit ansatz. We can see that the fits favour the linear ansatz over the chiral one. The ratio parameter linear ansatz fits are all good fits with $\chi^2$ per d.o.f of less than 1. The bag parameters have been measured and renormalised but the global fits have not yet been finalised. Full results including the BSM bag parameters will be included in a future publication.
linear fit $\chi^2$ /dof chiral PT fit $\chi^2$ /dof RBC/UKQCD16
------- ------------ --------------- --------------- --------------- ----------------
$R_2$ -18.69(11) 0.4 -18.83(11) 2.8 -19.11(43)(31)
$R_3$ 5.612(41) 0.7 5.665(41) 2.9 5.76(14)(16)
$R_4$ 38.91(21) 0.3 39.54(22) 4.6 40.12(82)(188)
$R_5$ 10.91(6) 0.3 11.079(58) 5.0 11.13(21)(83)
: Preliminary results of the global fit results for both ansatz are presented here alongside the $\chi^2$ per degree of freedom. Results from the previous RBC-UKQCD BSM kaon mixing calculations [@Garron:2016mva] are also presented for comparison. All results are presented in the intermediate RI-SMOM$^{(\gamma_\mu,\gamma_\mu)} $scheme at 3GeV[]{data-label="tab:results"}
\[fig:ratioresults\]
Conclusion
==========
We have calculated kaon mixing bag and ratio parameters using $n_f=2+1$ DWF QCD at 3 lattice spacings and several pion masses, including the physical pion mass. We’ve obtained preliminary results, via a simulataneous chiral/continuum fit, consistent with RBC-UKQCD’s previous work [@Garron:2016mva] but with statistical errors reduced by a factor of at least 3 for all the ratio parameters.
We have not yet calculated the sytematic errors but by including measurements at the physical point we have eliminated the systematic error from the chiral extrapolation and the inclusion of a third lattice spacing helps control the continuum extrapolation. Therefore we would expect to have a reduced systematic error too.
We are in the process of cross-checking the bag parameter fits and are still to convert a renormalisation factor for F1 to $\overline{\text{MS}}$ . Once complete and once the systematic errors have been finalised we will present the full final results in a forthcoming journal publication.
Acknowledgements
================
We thank our colleagues in RBC and UKQCD for their contributions and helpful discussions. The measurements in this work were computed on the STFC funded DiRAC facility (grants ST/K005790/1, ST/K005804/1, ST/K000411/1, ST/H008845/1). This research has received funding from the SUPA student prize scheme, Edinburgh Global Research Scholarship, Royal Society Wolfson Research Merit Award WM160035 and STFC (grant ST/L000458/1, ST/M006530/1 and an STFC studentship.) N.G. is supported by the Leverhulme Research grant RPG-2014-118.
[^1]: Speaker,
|
---
author:
- |
Abdelhamid Meziani\
Department of Mathematics\
Florida International University\
Miami, Florida 33199\
\
email: [email protected]
title: |
Nonrigidity of a class of two dimensional surfaces\
with positive curvature and planar points
---
Introduction {#introduction .unnumbered}
============
The problem considered here deals with the bendings of an orientable, embedded surface $S$ in ${\mathbb{R}}^3$. We assume that $S$ has a vanishing first homology group, that ${\overline{S}}$ is a $C^\infty$ compact surface with boundary, that it has positive curvature except at finitely many planar points in $S$. The main result states that for any $k\in{\mathbb{Z}}^+$, $S$ has nontrivial infinitesimal bendings of class $C^k$. That is, there is a $C^k$ function $U:\, {\overline{S}}\ \longrightarrow\
{\mathbb{R}}^3$ such that the first fundamental form of the deformation surface $S_\sigma =\{ p+\sigma U(p),\ p\in S\}$ satisfies $dS_\sigma^2=dS^2+O(\sigma^2)$ as $\sigma\to 0$, where $\sigma$ is a real parameter. Furthermore, $S_\sigma$ is not obtained from $S$ through a rigid motion of ${\mathbb{R}}^3$. A consequence of this result is the nonrigidity of $S$ in the following sense. Any given ${\epsilon}$-neighborhood of $S$ (for the $C^k$ topology) contains isometric surfaces that are not congruent.
The study of bendings of surfaces in ${\mathbb{R}}^3$ has a rich history and many physical applications. In particular, it is used in the theory of elastic shells. We refer to the survey article of Sabitov ([[@SABITOV1]]{}) and the references therein. The results of this paper are also related to those contained in the following papers [[@BLEE]]{}, [[@EFIMOV]]{}, [[@GREENE]]{}, [[@KANN]]{}, [[@KARATO]]{}, [[@MEZCAG]]{}, [[@MEZJDE]]{}, [[@POGO1]]{}, [[@USMANOV1]]{}, [[@SABITOV2]]{}
Our approach is through the study of the associated (complex) field of asymptotic directions on $S$. We prove that such a vector field generates an integrable structure on ${\overline{S}}$. We reformulate the equations for the bending field $U$ in terms of a Bers-Vekua type equation (with singularities). Then use recent results about the solvability of such equations to construct the bending fields.
Integrability of the field of asymptotic directions
===================================================
For the surfaces considered here, we show that the field of asymptotic directions on $S$ has a global first integral.
Let $S\subset{\mathbb{R}}^3$ be an orientable $C^\infty$ surface with a $C^\infty$ boundary. We assume that $H_1(S)=0$. The surface $S$ is diffeomorphic to a relatively compact domain ${\Omega}\in{\mathbb{R}}^2$ with a $C^\infty$ boundary. Hence, $${\overline{S}} =\{ R(s,t)\in{\mathbb{R}}^3;\ (s,t)\in{\overline{{\Omega}}}\}\, ,$$ where the position vector $R:\ {\overline{{\Omega}}}\, \longrightarrow\, {\mathbb{R}}^3$ is a $C^\infty$ parametrization of ${\overline{S}}$. Let $E,\ F,\ G$ and $e,\
f,\ g$ be the coefficients of the first and second fundamental forms of $S$. Thus, $$\begin{array}{lll}
E =R_s\cdot R_s,\quad & F =R_s\cdot R_t,\quad & G =R_t\cdot R_t,\\
e =R_{ss}\cdot N,\quad & f =R_{st}\cdot N,\quad & g =R_{tt}\cdot N,
\end{array}$$ where $N={\displaystyle}\frac{R_s\times R_t}{|R_s\times R_t|}$ is the unit normal of $S$. The Gaussian curvature of $S$ is $K={\displaystyle}\frac{eg-f^2}{EG-F^2}$. We assume that ${\overline{S}}$ has positive curvature except at a finite number of planar points in $S$. That is, there exist $p_1=(s_1,t_1)\in{\Omega},\cdots ,p_l(s_l,t_l)\in{\Omega}$ such that $$K(s,t) >0,\quad \forall (s,t)\in{\overline{{\Omega}}}\backslash\{ p_1,\cdots
,p_l\} \, .$$ The (complex) asymptotic directions on $S$ are given by the quadratic equation $${\lambda}^2 +2f{\lambda}+eg =0$$ Thus ${\lambda}=-f+i\sqrt{eg-f^2}\, \in\, {\mathbb{R}}+i{\mathbb{R}}^+$ except at the planar points $p_1,\cdots ,p_l$ where ${\lambda}=0$.
Consider the structure on ${\overline{{\Omega}}}$ generated by the ${\mathbb{C}}$-valued vector field $$L=g(s,t)\frac{{\partial}}{{\partial}s}+{\lambda}(s,t)\frac{{\partial}}{{\partial}t}\, .$$ This structure is elliptic on ${\overline{{\Omega}}}\backslash\{ p_1,\cdots
,p_l\} $. That is, $L$ and ${\overline{L}}$ are independent outside the planar points. The next proposition shows that $L$ has a global first integral on ${\overline{S}}$.
[*Proof.*]{} Since L is $C^\infty$ and elliptic on ${\overline{{\Omega}}}\backslash\{ p_1,\cdots ,p_l\} $, then it follows from the uniformization of complex structures on planar domains (see[[@SPRINGER]]{}) that there exists a $C^\infty$ diffeomorphism $$Z: \ {\overline{{\Omega}}}\backslash\{ p_1,\cdots ,p_l\}\,\longrightarrow\,
Z({\overline{{\Omega}}}\backslash\{ p_1,\cdots ,p_l\} )\subset{\mathbb{C}}$$ such that $LZ=0$. It remains to show that $Z$ has the form (1.5) in a neighborhood of a planar point.
Let $p_j$ be a planar point of $S$. We can assume that $S$ is given in a neighborhood of $p_j$ as the graph of a function $z=z(x,y)$ with $p_j=(0,0)$, $z(0,0)=0$, and $z_x(0,0)=z_y(0,0)=0$. The assumption on the curvature implies that $$z(x,y)=z_m(x,y) +o(\sqrt{x^2+y^2}^m),$$ where $z_m(x,y)$ is a homogeneous polynomial of degree $m>2$, satisfying $z_{xx}z_{yy}-z_{xy}^2 >0$ for $(x,y)\ne 0$. We can also assume that $z(x,y) >0$ for $(x,y)\ne 0$. The complex structure generated by the asymptotic directions is given by the vector field $$L=z_{yy}\frac{{\partial}}{{\partial}x}+(-z_{xy}+i\sqrt{z_{xx}z_{yy}-z_{xy}^2})
\frac{{\partial}}{{\partial}y}\, .$$ With respect to the polar coordinates $x=\rho\cos\phi$, $y=\rho\sin\phi$, we get $$z=\rho^mP(\phi)+\rho^{m+1}A(\rho,\phi)\, ,$$ where $P(\phi)$ is a trigonometric polynomial of degree $m$ satisfying $P(\phi)>0$ and (curvature) $$m^2P(\phi)^2+mP(\phi)P''(\phi)-(m-1)P'(\phi)^2 > 0\,\qquad
\forall\phi \in{\mathbb{R}}\, .$$ With respect to the coordinates $(\rho,\phi)$, the vector field $L$ becomes $$L=m(m-1)\rho^{m-2}(P(\phi)+O(\rho))L_0\,$$ with $$L_0=\frac{{\partial}}{{\partial}\phi}+\rho
\left(M(\phi)+iN(\phi)+O(\rho)\right)\frac{{\partial}}{{\partial}\rho}$$ and $$M=\frac{P'}{mP}\quad\mathrm{and}\quad
N=\frac{1}{m}\sqrt{\frac{m^2P^2+mPP''-(m-1)P'^2}{(m-1)P^2}}\, .$$ We know (see[[@MEZJFA]]{}) that such a vector field $L_0$ is integrable in a neighborhood of the circle $\rho =0$. Moreover, we can find coordinates $(r,{\theta})$ in which $L_0$ is $C^1$-conjugate to the model vector field $$T=\mu_j \frac{{\partial}}{{\partial}{\theta}}-ir\frac{{\partial}}{{\partial}r}$$ where $\mu_j >0$ is given by $$\frac{1}{\mu_j}=\frac{1}{2\pi}\int_0^{2\pi}(N(\phi)-iM(\phi))d\phi
=\frac{1}{2\pi}\int_0^{2\pi}N(\phi)d\phi\, .$$ The function $u_j(r,{\theta})=r^{\mu_j}{\mathrm{e}^{i{\theta}}}$ is a first integral of $T$ in $r>0$.
Now we prove that the function $Z$ which is defined in ${\overline{{\Omega}}}\backslash\{ p_1,\cdots ,p_l\}$ extends to $p_j$ with the desired form given by (1.5). Let $O_j$ be a disc centered at $p_j$ where $L$ is conjugate to a multiple of $T$ in the $(r,{\theta})$ coordinates. Since $u_j$ and $Z$ are both first integrals of $L$ in the punctured disc $O_j\backslash p_j$, then there exists a holomorphic function $h_j$ defined on the image $u_j(O_j\backslash
p_j)$ such that $ Z(r,t)=h_j(u_j(r,t))$. Since both $Z$ and $u_j$ are homeomorphisms onto their images, then $h_j$ is one to one in a neighborhood of $u_j(p_j)=0\in{\mathbb{C}}$ and since $h_j$ is bounded, then $$h_j(\zeta )=C_0+C_1\zeta +O(\zeta^2)\qquad\mathrm{for}\ \zeta
\ \mathrm{close\ to\ }0\in{\mathbb{C}}$$ with $C_1\ne 0$. This means that after a linear change of the coordinates $(r,{\theta})$ (to remove the constant $C_1$) , the function $Z$ has the form (1.5) $\quad\Box$
Equations of the bending fields in terms of $L$
===============================================
Let $S$ be a surface given by (1.1). An infinitesimal bending of class $C^k$ of $S$ is a deformation surface $S_\sigma\subset\in{\mathbb{R}}^3$, with $\sigma\in{\mathbb{R}}$ a parameter, given by the position vector $$R_\sigma (s,t)=R(s,t)+\sigma U(s,t)\, ,$$ whose first fundamental form satisfies $$dR_\sigma^2=dR^2+O(\sigma^2)\qquad\mathrm{as}\ \sigma\to 0\, .$$ This means that the bending field $U:\, {\overline{{\Omega}}}\, \longrightarrow\,
{\mathbb{R}}^3$ is of class $C^k$ and satisfies $$dR\cdot dU =0\, .$$ The trivial bendings of $S$ are those induced by the rigid motions of ${\mathbb{R}}^3$. They are given by $U(s,t)=A\times R(s,t)+B$, where $A$ and $B$ are constants in ${\mathbb{R}}^3$, and where $\times$ denotes the vector product in ${\mathbb{R}}^3$.
Let $L$ be the field of asymptotic directions defined by (1.4). For each function $U:\, {\Omega}\,\longrightarrow\, {\mathbb{R}}^3$, we associate the ${\mathbb{C}}$-valued function $w$ defined by $$w(s,t)=LR(s,t)\cdot U(s,t)=g(s,t)u(s,t)+{\lambda}(s,t)v(s,t)\, ,$$ where $u=R_s\cdot U$ and $v=R_t\cdot U$. The following theorem proved in [[@MEZCONT]]{} will be used in the next section.
[ If $U:\, {\Omega}\,\longrightarrow\,{\mathbb{R}}^3$ satisfies $(2.2)$, then the function $w$ given by $(2.3)$ satisfies the equation $$CLw=Aw+B{\overline{w}},$$ where $$\begin{array}{ll}
A & = (LR\times{\overline{L}}R)\cdot (L^2R\times{\overline{L}}R)\, ,\\
B & = (LR\times{\overline{L}}R)\cdot (L^2R\times LR)\, ,\\
C & = (LR\times{\overline{L}}R)\cdot (LR\times{\overline{L}}R)\, .
\end{array}$$ ]{}
[[**Remark 2.1**]{} [ If $w$ solves equation (2.4). The function $w'=aw$, where $a$ is a nonvanishing function solves the same equation with the vector field $L$ replaced by $L'=aL$]{} ]{}
Main Results
============
[Let $S$ be a surface given by $(1.1)$ and such that its curvature $K$ satisfies $(1.2)$. Then for every $k\in{\mathbb{Z}}^+$, the surface $S$ has a nontrivial infinitesimal bending $U:\,
{\overline{{\Omega}}}\,\longrightarrow\, {\mathbb{R}}^3$ of class $C^k$.]{}
[[**Remark 3.1**]{} [ It should be mentioned that without the assumption that $K>0$ up to the boundary ${\partial}S$, the surface could be rigid under infinitesimal bendings. Indeed, let $T^2$ be a standard torus in ${\mathbb{R}}^3$, it is known (see[[@AUDOLY]]{} or [[@POGO2]]{}) that if $S$ consists of the portion of $T^2$ with positive curvature, then $S$ is rigid under infinitesimal bendings. Here the curvature vanishes on ${\partial}S$.]{} ]{}
Before we proceed with the proof, we give a consequence of Theorem 3.1.
[Let $S$ be as in Theorem 3.1. Then for every ${\epsilon}>0$ and for every $k\in{\mathbb{Z}}^+$, there exist surfaces $\Sigma^+$ and $\Sigma^-$ of class $C^k$ in the ${\epsilon}$-neighborhood of $S$ (for the $C^k$-topology) such that $\Sigma^+$ and $\Sigma^-$ are isometric but not congruent.]{}
[[*Proof.*]{} Let $U:\, {\overline{{\Omega}}}\,\longrightarrow\, {\mathbb{R}}^3$ be a nontrivial infinitesimal bending of $S$ of class $C^k$. Consider the surfaces $\Sigma_{\sigma}$ and $\Sigma_{-\sigma}$ defined the position vectors $$R_{\pm\sigma}(s,t)=R(s,t)\pm\sigma U(s,t)\, .$$ Since $dR\cdot dU=0$, then $dR_{\pm\sigma}^2=dR^2+\sigma^2dU^2$. Hence $\Sigma_\sigma$ and $\Sigma_{-\sigma}$ are isometric. Furthermore, since $U$ is nontrivial, then $\Sigma_\sigma$ and $\Sigma_{-\sigma}$ are not congruent (see [[@SPIVAK]]{}). For a given ${\epsilon}>0$, the surfaces $\Sigma_{\pm\sigma}$ are contained in the ${\epsilon}$-neighborhood of $S$ if $\sigma$ is small enough $\quad\Box$ ]{}
[*Proof of Theorem 3.1.*]{} First we construct non trivial solutions $w$ of equation (2.4) and then deduce the infinitesimal bending fields $U$. For this, we use the first integral $Z$ of $L$ to transform equation (2.4) into a Bers-Vekua type equation with singularities. Let $Z_1=Z(p_1),\cdots , Z_l=Z(p_l)$ be the images of the planar points by the function $Z$. The pushforward of equation (2.4) via $Z$ gives rise to an equation of the form $$\frac{{\partial}W}{{\partial}{\overline{Z}}}=\frac{A(Z)}{\prod_{j=1}^l(Z-Z_j) }W+
\frac{B(Z)}{\prod_{j=1}^l(Z-Z_j) }{\overline{W}}\, ,$$ where $w(s,t)=W(Z(s,t))$ and $
A,B\, \in\, C^\infty(Z({\overline{{\Omega}}}\backslash\{ p_1,\cdots
,p_l\}))\cap L^\infty (Z({\overline{{\Omega}}})).
$ The local study of the solutions of such equations near a singularity is considered in [[@MEZCVEE]]{}, [[@TUNG]]{}, [[@USMANOV2]]{}. To construct a global solution of (3.1) with the desired properties, we proceed as follows. We seek a solution $W$ in the form $W(Z)=H(Z)W_1(Z)$, where $$H(Z)=\prod_{j=1}^l(Z-Z_j)^M\, ,$$ where $M$ is a (large) positive integer to be chosen. In order for $W=HW_1$ to solve (3.1), the function $W_1$ needs to solve the modified equation $$\frac{{\partial}W_1}{{\partial}{\overline{Z}}}=\frac{A(Z)}{\prod_{j=1}^l(Z-Z_j) }W_1+
\frac{B(Z)}{\prod_{j=1}^l(Z-Z_j) }\frac{{\overline{H(Z)}}}{H(Z)}{\overline{W_1}}\, .$$ Since $A$ and $B{\overline{H}}/H$ are bounded functions on $Z({\overline{{\Omega}}})$, then a result of [[@TUNG]]{} gives a continuous solution $W_1$ of (3.2) on $Z({\overline{{\Omega}}})$. Furthermore, such a solutions is $C^\infty$ on $Z({\overline{{\Omega}}}\backslash\{Z_1,\cdots ,Z_l\})$ since the equation is elliptic and the coefficients are $C^\infty$ outside the $Z_j$’s.
The function $W(Z)=H(Z)W_1(Z)$ is therefore a solution of (3.1). It is $C^\infty$ on $Z({\overline{{\Omega}}}\backslash\{Z_1,\cdots ,Z_l\})$ and vanishes to order $M$ at each point $Z_j$. Consequently, the function $w(s,t)=W(Z(s,t))$ is $C^\infty$ on ${\overline{{\Omega}}}\backslash\{p_1,\cdots ,p_l\}$ and vanishes to order $M\mu_j$ at each planar point ($\mu_j$ is the positive number appearing in Proposition 1.1).
Now we recover the bending field $U$ from the solution $w$ of (2.4) and the relation $w=LR\cdot U$. Set $w=gu+{\lambda}v$, where ${\lambda}$ is the asymptotic direction given in (1.3). The functions $u$ and $v$ are uniquely determined by $$v=\frac{w-{\overline{w}}}{2i\sqrt{eg-f^2}}\quad\mathrm{and}\quad
u=\frac{w+{\overline{w}}+2fv}{2g}\, ,$$ provided that the function $w$ vanishes to a high order at the planar points (order of vanishing of $W$ at $p_j$ larger than that of the curvature). These functions are $C^\infty$ outside the planar points. At each planar point $p_j$, it $m_j$ is the order of vanishing of $K$, then the functions $v$ and $u$ vanish to order $M\mu_j-m_j$. It follows from $LR\cdot U=w$ that $R_s\cdot U=u$ and $R_t\cdot U=v$. The condition $dR\cdot dU=0$ implies that $$R_{ss}\cdot U=u_s,\quad R_{tt}\cdot U=v_t,\quad\mathrm{and}\quad
2R_{st}\cdot U=u_t+v_s\, .$$ In terms of the components $(x,y,z)$ of $R$ and $(\xi,\eta,\zeta)$ of $U$, we have $$\left\{\begin{array}{ll}
x_s\xi+y_s\eta+z_s\zeta & =u\\
x_t\xi +y_t\eta +z_t\zeta & =v\\
x_{ss}\xi +y_{ss}\eta+z_{ss}\zeta & =u_s\\
x_{tt}\xi+y_{tt}\eta +z_{tt}\zeta & =v_t\\
2x_{st}\xi+2y_{st}\eta+2z_{st}\zeta & =u_t+v_s
\end{array}\right.$$ (equation (2.4) guarantees the compatibility of this system). Note that at each point $p\in{\overline{{\Omega}}}$ where $K>0$, the functions $\xi,\
\eta,$ and $\zeta$ are uniquely determined by $u$, $v$, and $u_s$ (or $v_t$). Indeed, at such a point the determinant of the first three equations of (3.4) is $$R_{ss}\cdot (R_s\times R_t)=|R_s\times R_t|e\, \ne \, 0\, .$$ With our choice that $w$ (and so $u$ and $v$) vanishing to an order larger than that of the curvature at each planar point, the functions $\xi,\ \eta,$ and $\zeta$ are also uniquely determined to be 0 at each planar point. To see why, assume that at $p_j$, we have $x_sy_t-x_ty_s\ne 0$, then after solving the first two equations for $\xi$ and $\eta$ in terms of $\zeta$, $u$, and $v$, the third equations becomes, $$R_{ss}\cdot (R_s\times R_t) \zeta =\left|
\begin{array}{lll}
x_s & y_s & u\\
x_t & y_t & v\\
x_{ss} & y_{ss} & u_s
\end{array}
\right| .$$ Since the zeros $p_j$ of $R_{ss}\cdot (R_s\times R_t)$ is isolated and since $u$ and $v$ vanish to a high order at $p_j$, the function $\zeta$ is well defined by (3.5). Consequently, for any given $k\in{\mathbb{Z}}^+$, a nonzero solution $w$ of (2.4) which vanishes at high orders ($M$ large), gives rise to a unique field of infinitesimal bending $U$ of $S$, so that it is $C^\infty$ on ${\overline{{\Omega}}}\backslash\{p_1,\cdots ,p_l\}$ and vanishes to an order $k$ at each $p_j$. Such a field is therefore of class $C^k$ at each planar point. It remains to verify that $U$ is not trivial. If such a field were trivial ($U=A\times R +B$), then the vanishing of $dU=A\times dR$ at $p_j$ together with $dR\cdot dU=0$ gives $A=0$ and so $U=B=0$ since $U=0$ at $p_j$. This would give $w\equiv 0$ which is a contradiction $\quad\Box$
[99]{}
B. Audoly, *Courbes rigidifiant les surfaces*, C.R. Acad. Sci. Paris Ser. I Math. 328 (1999), No. 4, 313-316
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N.V. Efimov, *Qualitative questions of the theory of deformation of surfaces*, Uspekhi Mat. Nauk. 3(2)(1948) 47-128; translated in Amer. Math. Soc. 6, 274-323 (1948)
R.E. Greene and H. Wu, *On the rigidity of punctured ovaloids*, Ann. of Math. (2), Vol. 94, 1-20, (1971)
E. Kann, *Glidebending of general caps: An infinitesimal treatment*, Proc. Amer. Math. Soc., Vol. 84, No. 2, 247-255, (1982)
I. Karatopraklieva, *Infinitesimal bendings of high order of rotational surfaces*, C.R. Acad. Bulgare Sci. 43 (12), (1990) 13-16
A. Meziani, *On planar elliptic structures with infinite type degeneracy*, J. Funct. Anal., Vol. 179, 333-373, (2001)
A. Meziani, *Infinitesimal bending of homogeneous surfaces with nonnegative curvature*, Comm. Anal. Geom., Vol. 11, 697-719, (2003)
A. Meziani, *Planar complex vector fields and infinitesimal bendings of surfaces with nonnegative curvature*, in: Contemp. Math., Vol. 400, Amer. Math. Soc., 189-202, (2006)
A. Meziani, *Infinitesimal bendings of high orders for homogeneous surfaces with positive curvature and a flat point*, J. Diff. Equations, Vol. 239, 16-37, (2007)
A. Meziani, *Representation of solutions of a singular CR equation in the plane*, Complex Var. Elliptic Equ., Vol. 53, No. 2, 1111-1130, (2008)
A.V. Pogorelov, *Bendings of surfaces and stability of the shells*, Nauka, Moscow, 1986; translation: Amer. Math. Soc., Providence, R.I., 1988
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---
abstract: 'In the framework of the Eliashberg formalism the free energy difference between the superconducting and normal state for the molecular metallic hydrogen was calculated. The pressure values $p_{1}=347$ GPa and $p_{2}=428$ GPa were taken into consideration. It has been shown, that together with the increase of the pressure, grows the value of the specific heat jump at the critical temperature and the value of the thermodynamic critical field near zero Kelvin: $\left[\Delta C\left(T_{C}\right)\right]_{p2}/\left[\Delta C\left(T_{C}\right)\right]_{p1}\simeq 2.33$ and $\left[H_{C}\left(0\right)\right]_{p2}/\left[H_{C}\left(0\right)\right]_{p1}\simeq 1.74$. Next, it has been stated, that the ratio $\Delta C\left(T_{C}\right)/C^{N}\left(T_{C}\right)$ also increases from $1.91$ to $2.39$; whereas $T_{C}C^{N}\left(T_{C}\right)/H^{2}_{C}\left(0\right)$ decreases from $0.152$ to $0.140$. The last results prove that the considered parameters significantly diverge from the prediction based on the BCS model.'
author:
- 'R. Szcz[ȩ]{}[s]{}niak, M.W. Jarosik'
title: |
—————————————————————————————————————\
Specific heat and thermodynamic critical field for the molecular metallic hydrogen
---
\#1[(\[\#1\])]{} \#1[\[\#1\]]{}
Introduction
============
The study of the metallic hydrogen’s properties has been lasting for over seventy years. In 1935, Wigner and Huntington for the first time suggested, that under the influence of high pressure ($p$) the hydrogen should transform into the molecular metallic phase \[1\]. The later theoretical results set the metallization of hydrogen in the pressures range from $300$ GPa to $400$ GPa [@Zhang], [@Stadele]. It is worth mentioning, that the understanding of the high-pressure properties of hydrogen seems to be substantial due to the fact, that this element in the metallic state (both molecular and atomic) is appearing inside the planets of the Jovian type [@Fortney].
The next step was made by Ashcroft who suggested, that the metallic hydrogen could be potential high-temperature superconductor [@Ashcroft]. Since that moment, the constant interest in the properties of the hydrogen’s superconducting state has been dated. In particular, the numerical results predict that in the range of the “lower” pressures (up to $500$ GPa) the critical temperature ($T_{C}$) is of the order ($80$-$300$) K [@Zhang], [@Richardson], [@Caron], [@Cudazzo]. For the extremely high pressure ($2000$ GPa) the superconducting state in the atomic metallic hydrogen has been studied in the papers [@Maksimov], [@Szczesniak1]. It has been shown that the critical temperature decreases from $631$ K to $413$ K for $\mu_{C}^{*}\in(0.1,0.5)$, where $\mu_{C}^{*}$ denotes the critical value of the Coulomb pseudopotential. In the considered case the other thermodynamic parameters diverge from the BCS values [@BCS] e.g.: the dimensionless ratio $r_{1}\equiv \Delta C\left(T_{C}\right)/C^{N}\left(T_{C}\right)$ is changing from $1.82$ to $1.68$ together with the Coulomb pseudopotential’s growth, whereas the minimum value of $r_{2}\equiv T_{C}C^{N}\left(T_{C}\right)/H^{2}_{C}\left(0\right)$ is equal to $0.162$ [@Szczesniak1]. The symbols defining the ratios $r_{1}$ and $r_{2}$ have following meaning: $\Delta C\left(T_{C}\right)$ denotes the specific heat difference between the superconducting and normal state at the critical temperature, $C^{N}\left(T_{C}\right)$ represents the specific heat of the normal state, while $H_{C}\left(0\right)$ is the value of the thermodynamic critical field at the temperature of zero Kelvin.
In the literature the specific heat and the thermodynamic critical field were not determined for the molecular metallic hydrogen. Due to the large values of the electron-phonon constant ($\left[\lambda\right]_{p_{1}}=0.93$ and $\left[\lambda\right]_{p_{2}}=1.2$) it has to be presumed, that above quantities should be calculated in the framework of the Eliashberg formalism [@Eliashberg]. In the paper, we take into consideration the following values of the pressure: $p_{1}=347$ GPa and $p_{2}=428$ GPa. In this case the molecular metallic hydrogen crystallizes in the [*Cmca*]{} structure [@Zhang], [@Cudazzo1].
THE ELIASHBERG EQUATIONS
========================
The BCS theory is based on the Hamiltonian, which models the pairing interaction in the simplest effective way. We notice that the BCS Hamiltonian can be derived from the more realistic Fröhlich’s operator ($H_{F}$), which describes the electron-phonon coupling in the open form [@Frohlich], [@TransKanon]. The Eliashberg equations are derived directly from $H_{F}$ with an use of the thermodynamic Green functions [@Elk]. As a result one can obtain [@Eliashberg]:
$$\label{r1}
Z_{n}=1+\frac{1}{\omega_{n}}\frac{\pi}{\beta}\sum_{m=-M}^{M}
\lambda\left(i\omega_{n}-i\omega_{m}\right)
\frac{\omega_{m}Z_{m}}
{\sqrt{\omega_m^2Z^{2}_{m}+\phi^{2}_{m}}}$$
and $$\label{r2}
\phi_{n}=
\frac{\pi}{\beta}\sum_{m=-M}^{M}
\left[\lambda\left(i\omega_{n}-i\omega_{m}\right)-\mu_{C}^{*}\theta\left(\omega_{c}-|\omega_{m}|\right)\right]
\frac{\phi_{m}}
{\sqrt{\omega_m^2Z^{2}_{m}+\phi^{2}_{m}}}.$$
The solutions of the Eliashberg equations are two functions defined on the imaginary axis: the wave function renormalization factor ($Z_{n}\equiv Z\left(i\omega_{n}\right)$) and the order parameter function ($\phi_{n}\equiv\phi\left(i\omega_{n}\right)$); $\omega_{n}\equiv \left(\pi / \beta\right)\left(2n-1\right)$ is the $n$-th Matsubara frequency, where $\beta\equiv\left(k_{B}T\right)^{-1}$ ($k_{B}$ denotes the Boltzmann constant). In the framework of the Eliashberg formalism the order parameter is defined as: $\Delta_{n}\equiv \phi_{n}/Z_{n}$. The symbol $\lambda\left(z\right)$ represents the pairing kernel: $$\label{r3}
\lambda\left(z\right)\equiv 2\int_0^{\Omega_{\rm{max}}}d\Omega\frac{\Omega}{\Omega ^2-z^{2}}\alpha^{2}F\left(\Omega\right).$$ The Eliashberg functions for the pressures $p_{1}$ and $p_{2}$ ($\alpha^{2}F\left(\Omega\right)$) were determined in the paper [@Zhang]. The symbol $\Omega_{\rm{max}}$ denotes the maximum phonon frequency, where $\left[\Omega_{\rm{max}}\right]_{p_{1}}=477$ meV and $\left[\Omega_{\rm{max}}\right]_{p2}=508$ meV.
The depairing correlations, appearing between electrons, are modeled with the help of the Coulomb pseudopotential $\mu_{C}^{*}$; the symbol $\theta$ denotes the Heaviside unit function and $\omega_{c}$ is the cut-off frequency ($\omega_{c}=3\Omega_{\rm{max}}$). In the paper we have assumed low value of the Coulomb pseudopotential for both considered pressures ($\mu^{*}_{C}=0.1$). The assumption above can be justified by referring to the Bennemann-Garland formula [@Bennemann]: $\mu_{C}^{*}\sim 0.26\rho\left(0\right)/\left[1+\rho\left(0\right)\right]$, where the symbol $\rho\left(0\right)$ indicates the value of the electronic density of states at the Fermi energy. In particular, we have: $\rho_{1}\left(0\right)=0.4512$ states/Ry/spin for $p_{1}$ and $\rho_{2}\left(0\right)=0.4885$ states/Ry/spin for $p_{2}$ [@Zhang]. Thus, $\left[\mu_{C}^{*}\right]_{p1}$ and $\left[\mu_{C}^{*}\right]_{p2}$ amounts $\sim 0.081$ and $\sim 0.085$ respectively.
From the mathematical point of view the Eliashberg set is composed of the strongly non-linear algebraic equations with the integral kernel $\lambda\left(z\right)$. In order to achieve stable solutions one needs to take into account adequately large number of the equations. In the paper we have assumed $M=800$, what assured stability of the solutions beginning from the temperature of $T_{0}=11.6$ K ($1$ meV). The Eliashberg equations were solved by using the iterative method presented in the papers [@Szczesniak2] and [@Szczesniak3].
THE NUMERICAL RESULTS
=====================
The solutions of the Eliashberg equations for the selected temperatures have been presented in Figs. and . It can be easily noticed, that the functions $Z_{m}$ and $\Delta_{m}$ decrease together with the Matsubara frequencies’ growth. However, $Z_{m}$ saturates considerably slower than $\Delta_{m}$.
The applied pressure significantly influences on the values of wave function renormalization factor and the order parameter. From the physical point of view the above fact means, that together with the increasing of $p$ increases the electron effective mass ($m^{*}_{e}\sim Z_{m=1}$) and the value of critical temperature ($\left[T_{C}\right]_{p_{1}}=108.2$ K, $\left[T_{C}\right]_{p_{2}}=162.7$ K).
Analyzing the dependence of $Z_{m}$ and $\Delta_{m}$ on temperature it has been stated, that the solutions of the Eliashberg equations very unlikely evolve with $T$. In Fig. we have plotted the functions $Z_{m=1}\left(T\right)$ and $\Delta_{m=1}\left(T\right)$. The presented results show, that the wave function renormalization factor is weakly dependent on the temperature and takes its maximum for $T=T_{C}$. In contrast, the temperature dependence of the order parameter is strong and can be modeled by using the formula: $\Delta_{m=1}\left(T\right)=\Delta_{m=1}\left(T_{0}\right)\sqrt{1-\left(\frac{T}{T_{C}}\right)^{\beta}}$, where: $\left[\Delta_{m=1}\left(T_{0}\right)\right]_{p_{1}}=18.15$ meV, $\left[\Delta_{m=1}\left(T_{0}\right)\right]_{p_{2}}=29.12$ meV, $\left[\beta\right]_{p_{1}}=3.58$ and $\left[\beta\right]_{p_{2}}=3.61$.
![The wave function renormalization factor on the imaginary axis for selected values of the temperature. The figure (A) shows results for $p_{1}$, the figure (B) for $p_{2}$.[]{data-label="f1"}](Rys1)
![The order parameter on the imaginary axis for selected values of the temperature. The figure (A) shows results for $p_{1}$, the figure (B) for $p_{2}$.[]{data-label="f2"}](Rys2)
![ (A) The dependence of the wave function renormalization factor for the first Matsubara frequency on the temperature. (B) The dependence of the order parameter for the first Matsubara frequency on the temperature. In both cases the results for $p_{1}$ and $p_{2}$ are presented.[]{data-label="f3"}](Rys3)
The thermodynamic properties of the molecular metallic hydrogen can be explicitly determined on the basis of the free energy difference between the superconducting and normal state ($\Delta F$) [@Bardeen]: $$\label{r4}
\frac{\Delta F}{\rho\left(0\right)}=-\frac{2\pi}{\beta}\sum_{m=1}^{M}
\left(\sqrt{\omega^{2}_{m}+\Delta^{2}_{m}}- \left|\omega_{m}\right|\right)
(Z^{{\rm S}}_{m}-Z^{N}_{m}\frac{\left|\omega_{m}\right|}
{\sqrt{\omega^{2}_{m}+\Delta^{2}_{m}}}),$$ where the functions $Z^{S}_{m}$ and $Z^{N}_{m}$ denote the wave function renormalization factors for the superconducting (S) and normal (N) state respectively.
In the first step, on the basis of Eq. , we have calculated the specific heat difference between the superconducting and normal state $\left(\Delta C\equiv C^S-C^N\right)$: $$\label{r5}
\frac{\Delta C}{k_{B}\rho\left(0\right)}
=-\frac{1}{\beta}\frac{d^{2}\left[\Delta F/\rho\left(0\right)\right]}
{d\left(k_{B}T\right)^{2}}.$$ Next, the specific heat in the normal state has been calculated with an use of the formula: $$\label{r6}
\frac{C^{N}}{ k_{B}\rho\left(0\right)}=\frac{\gamma}{\beta},$$ where $\gamma\equiv\frac{2}{3}\pi^{2}\left(1+\lambda\right)$. In Fig. \[f4\] we have plotted the temperature dependence of the specific heat for the superconducting and normal state. Assuming previously given values of the electronic density of states it can be shown, that together with the growth of $p$ the specific heat’s jump at the critical temperature very strongly increases. In particular, we have: $\left[\Delta C\left(T_{C}\right)\right]_{p_{2}}/\left[\Delta C\left(T_{C}\right)\right]_{p_{1}}\simeq 2.33$.
![ The dependence of the specific heat in the superconducting and normal state on the temperature. The figure (A) shows results for $p_{1}$, the figure (B) for $p_{2}$. The vertical line indicates a position of the specific heat jump that occurs at $T_{C}$.[]{data-label="f4"}](Rys4)
Below, we have calculated the values of the thermodynamic critical field (cgs units): $$\label{r7}
\frac{H_{C}}{\sqrt{\rho\left(0\right)}}=\sqrt{-8\pi
\left[\Delta F/\rho\left(0\right)\right]}.$$ In Fig. \[f5\] we have presented the dependence of $H_{C}/\sqrt{\rho\left(0\right)}$ on the temperature. On the basis of obtained results we can see, that the value of the thermodynamic critical field near the temperature of zero Kelvin ($H_{C}\left(0\right)\simeq H_{C}\left(T_{0}\right)$) also strongly increases with the pressure: $\left[H_{C}\left(0\right)\right]_{p_{2}}/\left[H_{C}\left(0\right)\right]_{p_{1}}\simeq 1.74$.
![ The thermodynamic critical field as a function of the temperature. The figure (A) shows results for $p_{1}$, the figure (B) for $p_{2}$.[]{data-label="f5"}](Rys5)
On the basis of determined thermodynamic functions one can calculate two fundamental ratios: $r_{1}$ and $r_{2}$. Let us notice, that in the framework of BCS model these quantities have the universal values ($\left[r_{1}\right]_{\rm BCS}=1.43$ and $\left[r_{2}\right]_{\rm BCS}=0.168$) [@BCS]. For the molecular metallic hydrogen following results were obtained: $$\label{r8}
\left[r_{1}\right]_{p_{1}}=1.91,\qquad \left[r_{1}\right]_{p_{2}}=2.39$$ and $$\label{r9}
\left[r_{2}\right]_{p_{1}}=0.152,\qquad \left[r_{2}\right]_{p_{2}}=0.140.$$
It is easy to notice that the calculated ratios significantly diverge from the values predicted by the BCS theory. Additionally it should be underlined, that $r_{1}$ is increasing together with the pressure’s growth, whereas the ratio $r_{2}$ is decreasing.
SUMMARY
=======
In the paper the free energy difference between the superconducting and normal state for the molecular metallic hydrogen was calculated. The pressure values $p_{1}=347$ GPa and $p_{2}=428$ GPa were taken into consideration. On the basis of achieved results it has been shown, that the specific heat’s jump at the critical temperature and the thermodynamic critical field near the temperature of zero Kelvin strongly increase together with the pressure’s growth ($\left[\Delta C\left(T_{C}\right)\right]_{p2}/\left[\Delta C\left(T_{C}\right)\right]_{p1}\simeq 2.33$ and $\left[H_{C}\left(0\right)\right]_{p2}/\left[H_{C}\left(0\right)\right]_{p1}\simeq 1.74$). The obtained thermodynamic quantities enable the determination of the fundamental ratios: $r_{1}$ and $r_{2}$. It has been proven, that the ratios $r_{1}$ and $r_{2}$ very considerably differ from the values predicted by the BCS model. In particular, $r_{1}$ is increasing from $1.91$ to $2.39$ together with the pressure’s growth; whereas $r_{2}$ is decreasing from $0.152$ to $0.140$.
The authors wish to thank Prof. K. Dzili[ń]{}ski for providing excellent working conditions and the financial support. We also thank A.P. Durajski and D. Szcz[ȩ]{}[ś]{}niak for the productive scientific discussion that improved the quality of the presented paper. All numerical calculations were based on the Eliashberg function sent to us by: [**L. Zhang**]{}, Y. Niu, Q. Li, T. Cui, Y. Wang, [**Y. Ma**]{}, Z. He and G. Zou for whom we are also very thankful.
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(c) J.P. Carbotte, F. Marsiglio, in: The Physics of Superconductors, edited by K.H. Bennemann, J.B. Ketterson, (Springer, Berlin, 2003), Vol 1, p. 223. P. Cudazzo, G. Profeta, A. Sanna, A. Floris, A. Continenza, S. Massidda, E.K.U. Gross, Phys. Rev. B [**81**]{}, 134505 (2010). (a) H. Fröhlich, Phys. Rev. [**79**]{}, 845 (1950);\
(b) H. Fröhlich, Proc. R. Soc. A [**223**]{}, 296 (1954). P.L. Taylor, O. Heinonen, [*Quantum Approach to Condensed Matter Physics*]{}, Cambridge University Press, 2002. (a) K. Elk, W. Gasser, [*Die Methode der Greenschen Funktionen in der Festkörperphysik*]{}, Akademie- Verlag, Berlin 1979; (b) A.L. Fetter, J.D. Walecka, [*Quantum Theory of Many-Particle Systems*]{}, McGraw-Hill Book Company, 1971. K.H. Bennemann, J.W. Garland, AIP Conf. Proc. [**4**]{}, 103 (1972). R. Szcz[ȩ]{}[s]{}niak, Acta Phys. Pol. A [**109**]{}, 179 (2006). R. Szcz[ȩ]{}[s]{}niak, Solid State Commun. [**138**]{}, 347 (2006). J. Bardeen, M. Stephen, Phys. Rev. [**136**]{}, A1485 (1964).
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abstract: |
This paper gives a detailed analysis of the Cannon–Thurston maps associated to a general class of hyperbolic free group extensions. Let ${\mathbb{F}}$ denote a free groups of finite rank at least $3$ and consider a *convex cocompact* subgroup $\Gamma\le\operatorname{Out}({\mathbb{F}})$, i.e. one for which the orbit map from $\Gamma$ into the free factor complex of ${\mathbb{F}}$ is a quasi-isometric embedding. The subgroup $\Gamma$ determines an extension $E_\Gamma$ of ${\mathbb{F}}$, and the main theorem of Dowdall–Taylor [@DT1] states that in this situation $E_\Gamma$ is hyperbolic if and only if $\Gamma$ is purely atoroidal.
Here, we give an explicit geometric description of the Cannon–Thurston maps $\partial {\mathbb{F}}\to\partial E_\Gamma$ for these hyperbolic free group extensions, the existence of which follows from a general result of Mitra. In particular, we obtain a uniform bound on the multiplicity of the Cannon–Thurston map, showing that this map has multiplicity at most $2\operatorname{rank}({\mathbb{F}})$. This theorem generalizes the main result of Kapovich and Lustig [@KapLusCT] which treats the special case where $\Gamma$ is infinite cyclic. We also answer a question of Mahan Mitra by producing an explicit example of a hyperbolic free group extension for which the natural map from the boundary of $\Gamma$ to the space of laminations of the free group (with the Chabauty topology) is not continuous.
author:
- 'Spencer Dowdall, Ilya Kapovich, and Samuel J. Taylor'
title: '**Cannon–Thurston maps for hyperbolic free group extensions**'
---
Introduction
============
A remarkable paper of Cannon and Thurston [@CannonThurston] proved that if $M$ is a closed hyperbolic 3–manifold which fibers over the circle $\mathbb S^1$ with fiber $S$, then the inclusion of $\widetilde S=\mathbb H^2$ into $\widetilde M=\mathbb H^3$ extends to a continuous $\pi_1(S)$–equivariant surjective map from $\partial \mathbb H^2=\mathbb S^1$ to $\partial \mathbb H^3=\mathbb S^2$. In particular, the inclusion $\pi_1(S)\le\pi_1(M)$ of word-hyperbolic groups extends to a continuous map $\partial\pi_1(S)\to\partial\pi_1(M)$ of their Gromov boundaries. Though not published till 2007, this paper [@CannonThurston] has been highly influential since its circulation as a preprint in 1984. Consequently, if the inclusion $\iota\colon H\to G$ of a word-hyperbolic subgroup $H$ of a word-hyperbolic group $G$ extends to a continuous (and necessarily $H$–equivariant and unique) map ${\partial\iota}\colon\partial H\to \partial G$, the map ${\partial\iota}$ came to be called the *Cannon–Thurston map*. It is easy to see that the Cannon–Thurston map always exists and is injective in the case that $H$ is a quasiconvex subgroup of $G$; the above result of Cannon and Thurston [@CannonThurston] provided the first nontrivial example of existence of ${\partial\iota}$ in the non-quasiconvex case.
Later the work of Mitra [@MitraCTmaps-general; @MitraCTmaps-trees; @Mitra98] showed that the Cannon–Thurston map exists in several general situations corresponding to non-quasiconvex subgroups. In particular, Mitra proved [@MitraCTmaps-general] that whenever $$\label{eqn:hyp_SES}
1 \longrightarrow H \longrightarrow G \longrightarrow \Gamma \longrightarrow 1$$ is a short exact sequence of three infinite word-hyperbolic groups, then the Cannon–Thurston map ${\partial\iota}\colon\partial H\to\partial G$ exists and is surjective. Only recently did the work of Baker and Riley [@BakerRiley] produce the first example of a word-hyperbolic subgroup $H$ of a word-hyperbolic group $G$ for which the inclusion $H\le G$ does not extend to a Cannon–Thurston map. Analogs and generalizations of the Cannon–Thurston map have been studied in many other contexts, see for example [@Klar; @McM; @Miyachi; @LLR; @LMS; @Gerasimov; @Bow07; @Bow13; @MP11; @MjD14; @Mj14; @JKLO]. The best understood case concerns discrete isometric actions of surface groups on $\mathbb H^3$, where the most general results about Cannon–Thurston maps are due to Mj [@Mj14].
The main results from the theory of JSJ decomposition for word-hyperbolic groups (see [@RipsSela] for the original statement and [@Levitt] for a clarified version) imply that if we have a short exact sequence of three infinite word-hyperbolic groups with $H$ being torsion-free, then $H$ is isomorphic to a free product of surface groups and free groups. Thus understanding the structure of the Cannon–Thurston map for such short exact sequences requires first studying in detail the cases where $H$ is a surface group or a free group. The case of word-hyperbolic extensions of closed surface groups is closely related to the theory of convex cocompact subgroups of mapping class groups, and the structural properties of the Cannon–Thurston map in this setting are by now well understood; [@LMS] for details.
In this paper we consider the case of hyperbolic extensions of free groups and specifically the class of hyperbolic extensions introduced by Dowdall and Taylor [@DT1]. To describe this class, we henceforth fix a free group ${\mathbb{F}}$ of finite rank at least $3$. Following Hamenstädt and Hensel [@HaHe], we say that a finitely generated subgroup $\Gamma\le\operatorname{Out}({\mathbb{F}})$ is *convex cocompact* if the orbit map $\Gamma\to{{{\EuScript F}}}$ into the free factor graph ${{{\EuScript F}}}$ of ${\mathbb{F}}$ is a quasi-isometric embedding. Hyperbolicity of ${{{\EuScript F}}}$ [@BF14] and the definition of ${{{\EuScript F}}}$ respectively imply convex cocompact subgroups of $\operatorname{Out}({\mathbb{F}})$ are word-hyperbolic and that their infinite-order elements are all fully irreducible. We say that a subgroup $\Gamma\le\operatorname{Out}({\mathbb{F}})$ is *purely atoroidal* if every infinite-order element $\phi\in \Gamma$ is atoroidal (that is, no positive power of $\phi$ fixes a nontrivial conjugacy class in ${\mathbb{F}}$). Given any subgroup $\Gamma\le\operatorname{Out}({\mathbb{F}})$, the full pre-image $E_\Gamma$ of $\Gamma$ under the quotient map $\operatorname{Aut}({\mathbb{F}})\to\operatorname{Out}({\mathbb{F}})$ fits into a short exact sequence $$\label{eqn:free_extension}
1\longrightarrow {\mathbb{F}}\longrightarrow E_\Gamma\longrightarrow \Gamma\longrightarrow 1$$ with kernel ${\mathbb{F}}\cong \operatorname{Inn}({\mathbb{F}})$. The main result of [@DT1] proves that $E_\Gamma$ is hyperbolic whenever $\Gamma\le \operatorname{Out}({\mathbb{F}})$ is convex cocompact and purely atoroidal. From this, we see that if $\Gamma \le \operatorname{Out}({\mathbb{F}})$ is convex cocompact, then $E_\Gamma$ is hyperbolic if and only if $\Gamma$ is purely atoroidal. Moreover, there is a precise sense [@TT] in which random finitely generated subgroups of $\operatorname{Out}({\mathbb{F}})$ satisfy these hypotheses and so define hyperbolic extensions as in .
Our goal in this paper is to understand the Cannon–Thurston maps for hyperbolic extensions $E_\Gamma$ of ${\mathbb{F}}$ corresponding to convex cocompact subgroups $\Gamma$ of $\operatorname{Out}({\mathbb{F}})$. Such extensions vastly generalize the hyperbolic free-by-cyclic groups with fully irreducible monodromy whose Cannon–Thurston maps were explored in detail by Kapovich and Lustig in [@KapLusCT]. Our analysis extends many results of [@KapLusCT] and gives an explicit description of the Cannon–Thurston map in the general setting of purely atoroidal convex cocompact $\Gamma$. Moreover, this description reveals quantitative global and local features of the map and allows us to address multiple conjectures regarding Cannon–Thurston maps in this setting. In the case of groups $E_\Gamma$ corresponding to purely atoroidal convex cocompact $\Gamma\le \operatorname{Out}({\mathbb{F}})$, we answer a question of Swarup (which appears as Question $1.20$ on Bestvina’s list) by establishing uniform finiteness of the fibers of ${\partial\iota}\colon \partial {\mathbb{F}}\to \partial E_\Gamma$.
Let $\Gamma\le \operatorname{Out}({\mathbb{F}})$ be purely atoroidal and convex cocompact, where ${\mathbb{F}}$ is a free group of finite rank at least $3$, and let ${\partial\iota}\colon\partial {\mathbb{F}}\to\partial E_\Gamma$ denote the Cannon–Thurston map for the hyperbolic ${\mathbb{F}}$–extension $E_\Gamma$. Then for every $y\in \partial E_\Gamma$, the degree $\deg(y) = \#\left(({\partial\iota})^{-1}(y)\right)$ of the fiber over $y$ satisfies $$1 \le \deg(y) \le 2\operatorname{rank}({\mathbb{F}}).$$ In particular, the fibers Cannon–Thurston map are all finite and of uniformly bounded size.
To establish , we relate Mitra’s theory of “ending laminations” () for hyperbolic group extensions [@MitraEndingLams] to the theory of algebraic laminations on free groups developed by Coulbois, Hilion, and Lustig [@CHL1; @CHL2]. A general result of Mitra [@MitraEndingLams] about Cannon–Thurston maps for short exact sequences of hyperbolic groups ( below) implies that distinct points $p,q\in \partial {\mathbb{F}}$ are identified by the Cannon–Thurston map ${\partial\iota}\colon \partial {\mathbb{F}}\to\partial E_\Gamma$ if and only if $(p,q)$ is a leaf of an ending lamination $\Lambda_z$ on ${\mathbb{F}}$ for some $z\in \partial \Gamma$. Since our $\Gamma$ is convex cocompact by assumption, the orbit map $\Gamma\to{{{\EuScript F}}}$ into the free factor complex extends to a continuous embedding $\partial\Gamma\to\partial{{{\EuScript F}}}$. By Bestvina–Reynolds [@bestvina2012boundary] and Hamenstädt [@hamenstadt2013boundary], the boundary $\partial {{{\EuScript F}}}$ consist of equivalence classes of arational ${\mathbb{F}}$–trees. Thus to each $z\in \partial \Gamma$ we may also associate a class $T_z$ of arational ${\mathbb{F}}$–trees. The tree $T_z$ moreover comes equipped with a *dual lamination* $L(T_z)$, as introduced in [@CHL2]. Informally, $L(T_z)$ consists of lines in the free group which project to bounded diameter sets in the tree $T_z$. Our key technical result, , shows that for every $z\in\partial \Gamma$ we have $\Lambda_z=L(T_z)$. Combining this with Mitra’s general theory [@MitraEndingLams], we obtain the following explicit description of the Cannon–Thurston map:
Let $\Gamma\le\operatorname{Out}({\mathbb{F}})$ be convex cocompact and purely atoroidal. Then the Cannon–Thurston map ${\partial\iota}\colon \partial {\mathbb{F}}\to \partial E_\Gamma$ identifies points $a,b \in \partial {\mathbb{F}}$ if and only if there exists $z \in \partial \Gamma$ such that $(a,b) \in L(T_z)$. That is, ${\partial\iota}$ factors through the quotient of $\partial {\mathbb{F}}$ by the equivalence relation$$a\sim b \iff (a,b)\in L(T_z)\text{ for some }z\in \partial \Gamma$$ and descends to an $E_\Gamma$–equivariant homeomorphism $\nicefrac{\partial{\mathbb{F}}}{\sim} \to \partial E_\Gamma$.
We derive from by using results of Coulbois–Hilion [@CoulboisHilion-index] concerning the *$\mathcal Q$–index* for very small minimal actions of ${\mathbb{F}}$ on $\mathbb{R}$–trees. The point is that the laminations $\Lambda_z$ that Mitra constructs in [@MitraEndingLams] are a priori complicated and unwieldy objects from which it is difficult to extract information, whereas the laminations $L(T_z)$ appearing in are subject to the general theory ${\mathbb{R}}$–trees. The equality $\Lambda_z=L(T_z)$ provided by thus allows us to access this theory and use it to analyze the Cannon–Thurston maps.
However, establishing the equality $\Lambda_z = L(T_z)$ is nontrivial even in the special case, treated by Kapovich and Lustig [@KapLusCT], when $\Gamma = \langle\phi\rangle$ is a cyclic group generated by an atoroidal fully irreducible element $\phi$. The general case considered here is considerably harder since our trees $T_z$ no longer enjoy the “self-similarity” properties of stable trees of atoroidal fully irreducibles. The laminations $\Lambda_z$ and $L(T_z)$ are defined in very different terms, and the main difficulty is in establishing the inclusion $\Lambda_z\subseteq L(T_z)$. The key step in this direction is which shows that if $g_i\in\Gamma$ is a quasigeodesic sequence converging to $z\in\partial\Gamma$, then $\ell_{T_z}(g_i(h))\to 0$ for every nontrivial $h\in {\mathbb{F}}$. Note that in this situation it is fairly straightforward to see that the projective geodesic current $[\mu]=\lim_{i\to\infty} [\eta_{g_i(h)}]$ satisfies $\langle T_z, \mu\rangle=0$ (where $\langle\cdot,\cdot\rangle$ is the intersection pairing constructed in [@kapovich2007geometric]), but this is much weaker than the needed conclusion $\lim_{i\to\infty} \ell_{T_z}(g_i(h))=0$. The proof of relies on recent results of Dowdall and Taylor [@DT1] about folding paths in Culler and Vogtmann’s Outer space ${{{\EuScript X}}}$ that remain close to the orbit of a purely atoroidal convex cocompact subgroup $\Gamma$.
Our , establishing that for every $z\in\partial \Gamma$ we have $\Lambda_z=L(T_z)$, has quickly found useful applications in a new paper of Mj and Rafi [@MjRafi] regarding quasiconvexity in the context of hyperbolic group extensions. See Proposition 4.3 in [@MjRafi] and its applications in Theorem 4.11 and Theorem 4.12 of [@MjRafi]. We remark that the quasiconvexity result given by Theorem 4.12 of [@MjRafi] is also proved by different methods in the forthcoming paper [@DT2].
Rational and essential points {#rational-and-essential-points .unnumbered}
-----------------------------
In addition to and , we obtain fine information about the Cannon–Thurston map in regards to rational and essential points. Recall that a point $\xi$ in the boundary $\partial G$ of a word-hyperbolic group $G$ is called *rational* if there is an infinite-order element $g\in G$ such that $\xi$ equals the limit $g^\infty$ in $G\cup \partial G$ of the sequence $\{g^n\}$. For the short exact sequence , a point $y\in \partial E_\Gamma$ is called *$\Gamma$–essential* if there exists a (necessarily unique) point $\zeta(y)\in \partial\Gamma$ such that $y={\partial\iota}(p)$ for a point $p\in\partial{\mathbb{F}}$ that is *proximal* for $L(T_{\zeta(y)})$ in the sense of below. Informally, $\Gamma$–essential points are the ${\partial\iota}$–images of points in $\partial{\mathbb{F}}$ that “remember” in an essential way the lamination $L(T_z)$ for some $z\in\partial\Gamma$.
We write $\deg(y)\colonequals \#\left(({\partial\iota}){^{-1}}(y)\right)$ for the cardinality of the Cannon–Thurston fiber over $y\in\partial E_\Gamma$ (so $1\le \deg(y)\le 2\operatorname{rank}({\mathbb{F}})$ by ). Every $y\in \partial E_\Gamma$ with $\deg(y)\ge 2$ is $\Gamma$–essential and moreover has Cannon–Thurston fiber given by $({\partial\iota})^{-1}(y)=\{p\}\cup\{q\in \partial {\mathbb{F}}\mid (p,q)\in L(T_{\zeta(y)})\}$ for *every* $p\in ({\partial\iota}){^{-1}}(y)$ (). However, there are also may $\Gamma$–essential points with $\deg(y)=1$. Our next result describes the fibers of ${\partial\iota}$ over rational points of $\partial E_\Gamma$.
Suppose that $1 \to {\mathbb{F}}\to E_\Gamma \to \Gamma \to 1$ is a hyperbolic extension with $\Gamma \le \operatorname{Out}({\mathbb{F}})$ convex cocompact. Consider a rational point $g^\infty\in \partial E_\Gamma$, where $g\in E_\Gamma$ has infinite order.
1. Suppose that $g^k$ is equal to $w\in {\mathbb{F}}\lhd E_\Gamma$ for some $k\ge 1$ (i.e., $g$ projects to a finite order element of $\Gamma$). Then $({\partial\iota})^{-1}(g^\infty) = \{w^\infty\}\subset \partial {\mathbb{F}}$ and so $\deg(g^\infty) = 1$.
2. Suppose that $g$ projects to an infinite-order element $\phi\in\Gamma$. Then there exists $k\ge 1$ such that the automorphism $\Psi\in\operatorname{Aut}({\mathbb{F}})$ given by $\Psi(w) = g^kwg^{-k}$ is forward rotationless (in the sense of [@FH11; @CoulboisHilion-botany]) and its set $\mathrm{att}(\Psi)$ of attracting fixed points in $\partial {\mathbb{F}}$ is exactly $\mathrm{att}(\Psi)=({\partial\iota}){^{-1}}(g^\infty)$. Moreover, $g^\infty$ is $\Gamma$–essential and $\zeta(g^\infty) = \phi^\infty$.
In the case of a cyclic group $\langle\phi\rangle$ generated by an atoroidal fully irreducible automorphism $\phi$, Kapovich and Lustig [@KapLusCT] showed that every point $y\in\partial E_{\langle\phi\rangle}$ with $\deg(y)\ge 3$ is rational. We show that when $\Gamma$ is nonelementary this conclusion no longer holds and rather that, with some unavoidable exceptions, rational points in $\partial E_\Gamma$ come in a specific way from rational points in $\partial\Gamma$:
Suppose that $1 \to {\mathbb{F}}\to E_\Gamma \to \Gamma \to 1$ is a hyperbolic extension with $\Gamma \le \operatorname{Out}({\mathbb{F}})$ convex cocompact. Then the following hold:
1. If $y\in \partial E_\Gamma$ has $\deg(y)\ge 3$ and $\zeta(y)\in\partial \Gamma$ is rational, then $y$ is rational.
2. If $y\in \partial E_\Gamma$ has $\deg(y)\ge 2$ and $\zeta(y)\in\partial\Gamma$ is irrational, then $y$ is irrational.
Conical limit points {#conical-limit-points .unnumbered}
--------------------
Recall that a point $\xi$ in the boundary $\partial G$ of a word-hyperbolic group $G$ is a *conical limit point* for the action of a subgroup $H\le G$ on $\partial G$ if there exists a geodesic ray in the Cayley graph of $G$ that converges to $\xi$ and has a bounded neighborhood that contains infinitely many elements of $H$. Combining the results of this paper with the results of [@JKLO], we obtain the following:
Let $\Gamma\le\operatorname{Out}({\mathbb{F}})$ be purely atoroidal and convex cocompact. If $y\in \partial E_\Gamma$ is $\Gamma$–essential, then $y$ is not a conical limit point for the action of ${\mathbb{F}}$ on $\partial E_\Gamma$. In particular, if $\deg(y)\ge 2$ or if $y=g^\infty$ for some $g\in E_\Gamma$ projecting to an infinite-order element of $\Gamma$, then $y$ is not a conical limit point for the action of ${\mathbb{F}}$.
It is known (see [@Gerasimov; @JKLO]) in a very general convergence group situation that if a Cannon–Thurston map exists then every conical limit point has exactly one pre-image under the Cannon–Thurston map; thus points with $\ge 2$ pre-images cannot be conical limit points. However, also applies to many $\Gamma$–essential points $y\in \partial E_\Gamma$ with $\deg(y)=1$.
Discontinuity of ending laminations {#discontinuity-of-ending-laminations .unnumbered}
-----------------------------------
In [@MitraEndingLams], Mitra asks whether the map which associates to each point $z\in\partial \Gamma$ the corresponding ending lamination $\Lambda_z$ is continuous with respect to the Chabauty topology on the space of laminations. Of course, in the case of extensions by ${\mathbb{Z}}$ there is nothing to check since the boundary $\partial {\mathbb{Z}}$ is discrete. In , we answer Mitra’s question in the negative by producing a hyperbolic extension $E_\Gamma$ for which the map $z \mapsto \Lambda_z$ is not continuous. This is done explicitly in .
Besides establishing this discontinuity, we also provide a positive result about subconvergence of ending laminations. For the statement, let ${\mathcal{L}}({\mathbb{F}})$ denote the space of laminations on ${\mathbb{F}}$ equipped with the Chabauty topology (recalled in ) and let $\Lambda_z$ denote Mitra’s [@MitraEndingLams] ending lamination for $z\in\partial \Gamma$ (see ). For a lamination $L \in {\mathcal{L}}({\mathbb{F}})$, the notation $L'$ denotes the set of accumulation points of $L$, in the usual topological sense.
Let $\Gamma\le\operatorname{Out}({\mathbb{F}})$ be purely atoroidal and convex cocompact, and let $\Lambda_z\in{\mathcal{L}}({\mathbb{F}})$ denote the ending lamination associated to $z\in \partial \Gamma$. Then for any sequence $z_i$ in $\partial \Gamma$ converging to $z$ and any subsequence limit $L$ of the corresponding sequence $\Lambda_{z_i}$ in ${\mathcal{L}}({\mathbb{F}})$, we have $$\Lambda'_z \subset L \subset \Lambda_z.$$
This result can be viewed as a statement about the map $\partial \Gamma\to{\mathcal{L}}({\mathbb{F}})$, given by $z\mapsto \Lambda_z$, possessing a weak form of continuity.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We are grateful to Arnaud Hilton for providing us a copy of the draft 2006 preprint [@CHLunpub] and to Chris Leininger for useful conversations. We also thank the referee for helpful comments.
The first author was partially supported by NSF grant DMS-1204814. The second author was partially supported by the NSF grant DMS-1405146. The third author was partially supported by NSF grant DMS-1400498.
Cannon–Thurston maps {#sec:ct-intro}
====================
In this section, we recall some facts about Cannon–Thurston maps for general hyperbolic extensions. For a word-hyperbolic group $G$, we denote its Gromov boundary by $\partial G$. The following result establishes the existence of the Cannon–Thurston map:
\[pd:mitra\] Suppose that $1 \to H \to G \to \Gamma \to 1$ is an exact sequence of word-hyperbolic groups. Then the inclusion $\iota\colon H \to G$ admits a continuous extension $\hat\iota\colon H \cup \partial H \to G \cup \partial G$ with $\hat\iota(\partial H)\subseteq \partial G$. The restricted map ${\partial\iota}\colonequals\hat\iota|_{\partial H}\colon \partial H \to \partial G$ is called the [*Cannon–Thurston map*]{} for the inclusion $H \to G$; it is surjective whenever $H$ is infinite.
For an element $g\in G$ denote by $\Phi_g$ the automorphism $h\mapsto
ghg^{-1}$ of $H$. We denote by $\phi_g\in\operatorname{Out}(H)$ the outer automorphism class of $\Phi_g$. Similarly, for an element $q\in \Gamma$ denote by $\phi_q\in \operatorname{Out}(H)$ the outer automorphism class of $\Phi_g$, where $g\in G$ is any element that maps to $q$; note that the class $\phi_q$ is independent of the chosen lift $g$. For a conjugacy class $[h]$ in $H$ and an element $q\in \Gamma$ we also write $[q(h)]:=[\phi_q(h)]=[ghg^{-1}]$, where $g\in G$ is any element projecting to $q$.
By construction, the Cannon–Thurston map ${\partial\iota}\colon \partial H \to \partial G$ in is $H$–equivariant with respect to the left translation actions of $H$ on $\partial H$ and $\partial G$. However, ${\partial\iota}$ actually turns out to be $G$–equivariant with respect to the action $G \curvearrowright \partial H$ defined by $g\cdot p \colonequals \Phi_g(p)$ for $g\in G$ and $p\in \partial H$. Notice that the restricted action $H\curvearrowright \partial H$, namely $h\cdot p = \Phi_h(p)$, agrees with the usual action of $H$ on $\partial H$ by left translation.
\[prop:action\] Suppose $1\to H \to G \to \Gamma\to 1$ is an exact sequence of hyperbolic groups. Then the map $(g,p)\mapsto g\cdot p$ defines an action of $G$ on $\partial H$ by homeomorphisms. Moreover, the Cannon–Thurston map ${\partial\iota}\colon \partial H \to \partial G$ is $G$–equivariant.
While this is implicit in [@KapLusCT], we include a proof for completeness. The fact that $(g,p)\mapsto g\cdot p$ defines a group action by homeomorphisms follows directly from the definitions. Choose $p\in \partial H$ and $g\in G$. To prove $G$–equivariance we must show that ${\partial\iota}(\Phi_g(p))=g\cdot {\partial\iota}(p)$.
Choose a sequence $h_n\in H$ such that $h_n\to p$ in the topology of $H\cup \partial H$. By definition of ${\partial\iota}$ it follows that $h_n\to {\partial\iota}(p)$ in the topology of $G\cup \partial G$. In $G$ we have $gh_ng^{-1}=\Phi_g(h_n)$ so that $$g\cdot {\partial\iota}(p)=\lim_{n\to\infty} gh_n=\lim_{n\to\infty} gh_ng^{-1}=\lim_{n\to\infty} \Phi_g(h_n)$$ in the topology of $G\cup \partial G$. But definition of ${\partial\iota}$ the last limit above is exactly ${\partial\iota}(\Phi_g (p))$.
Background on free groups, laminations and $\mathcal Q$–index
=============================================================
For the entirety of this section let ${\mathbb{F}}$ be a free group of finite rank $N\ge 2$. We will also fix a free basis $X$ of ${\mathbb{F}}$ and the Cayley graph ${\mathrm{Cay}}({\mathbb{F}},X)$ of ${\mathbb{F}}$ with respect to $X$.
Laminations on free groups {#sec:laminations_on_free}
--------------------------
We denote ${\partial^2 {\mathbb{F}}}:=\{(p,q)\in \partial{\mathbb{F}}\times \partial{\mathbb{F}}\colon p\ne q\}$ and endow ${\partial^2 {\mathbb{F}}}$ with the subspace topology from the product topology on $\partial{\mathbb{F}}\times \partial{\mathbb{F}}$. There is a natural diagonal left action of ${\mathbb{F}}$ on ${\partial^2 {\mathbb{F}}}$ by left translations: $w(p,q)\colonequals(wp,wq)$ where $w\in{\mathbb{F}}$ and $(p,q)\in{\partial^2 {\mathbb{F}}}$.
An *algebraic lamination* on ${\mathbb{F}}$ is a closed ${\mathbb{F}}$–invariant subset $L\subseteq {\partial^2 {\mathbb{F}}}$ such that $L$ is also invariant with respect to the “flip map” ${\partial^2 {\mathbb{F}}}\to{\partial^2 {\mathbb{F}}}$, $(p,q)\mapsto (q,p)$. If $L$ is an algebraic lamination on ${\mathbb{F}}$, an element $(p,q)\in L$ is also referred to as a *leaf* of $L$. In the Cayley graph ${\mathrm{Cay}}({\mathbb{F}},X)$ of ${\mathbb{F}}$ with respect to the free basis $X$, every leaf $(p,q)\in L$ is represented by a unique unparameterized bi-infinite geodesic $l$ from $p$ to $q$ in ${\mathrm{Cay}}({\mathbb{F}},X)$. In this situation we will also sometimes say that $l$ is a leaf of $L$. We refer the reader to [@CHL1; @CHL2] for the background information on algebraic laminations.
We say that a subset $L\subseteq {\partial^2 {\mathbb{F}}}$ is *diagonally closed* if whenever $p,q,r\in \partial{\mathbb{F}}$ are three distinct points such that $(p,q),(q,r)\in L$ then $(p,r)\in L$.
An important class of laminations are those corresponding to conjugacy classes of ${\mathbb{F}}$. For $g \in {\mathbb{F}}\setminus \{1\}$, we denote by $g^{+\infty} \in \partial {\mathbb{F}}$ the unique forward limit of the sequence $(g^n)_{n\ge1}$ in ${\mathbb{F}}\cup \partial {\mathbb{F}}$. Define $g^{-\infty}$ similarly and note that $(g^{-1})^{+\infty} = g^{-\infty}$. Then define the algebraic lamination $$L(g) = {\mathbb{F}}\cdot (g^{+\infty}, g^{-\infty}) \cup {\mathbb{F}}\cdot (g^{-\infty},g^{\infty}).$$ Note that $L(g)$ depends only on the conjugacy class of $g$. Moreover, $L(g)$ is indeed a closed subset of ${\partial^2 {\mathbb{F}}}$ and so is a bona fide algebraic lamination. In what follows, for a subset $A$ of a topological space, we denote the closure of $A$ by $\overline{A}$ and its set of accumulation points by $A'$. For a collection $\Omega$ of conjugacy classes of ${\mathbb{F}}$, we let $L(\Omega)$ denote the smallest algebraic lamination containing $L(g)$ for each $g\in\Omega$. We observe that $$\begin{aligned}
L(\Omega) = \overline{\bigcup_{g \in \Omega}L(g)}\end{aligned}$$ Since each $L(g)$ is itself closed in ${\partial^2 {\mathbb{F}}}$, we see that the above closure is unnecessary when $\Omega$ is finite.
Finally, we denote the set of all laminations of ${\mathbb{F}}$ by ${\mathcal{L}}({\mathbb{F}})$, which we consider with the Chabauty topology. We recall the definition of this topology:
\[def:Chabauty\] Let $Y$ be a locally compact metric space and let $C(Y)$ be the collection of closed subsets of $Y$. The [*Chabauty topology*]{} on $C(Y)$ is defined as the topology generated by the subbasis consisting of
1. $\mathcal{U}_1(K) = \{C \in C(Y) : C \cap K = \emptyset \}$ for $K \subset Y$ compact.
2. $\mathcal{U}_2(O) = \{ C \in C(Y): C \cap O \neq \emptyset \}$ for $O \subset Y$ open.
A geometric interpretation of convergence in the Chabauty topology is stated in ; it will be needed in . Recall that the space $C(Y)$ is always compact [@canary2006fundamentals]. Returning to the situation of algebraic laminations of ${\mathbb{F}}$, we note that ${\mathcal{L}}({\mathbb{F}})$ is closed in $C(\partial^2 {\mathbb{F}})$ and hence is itself compact. We henceforth consider ${\mathcal{L}}({\mathbb{F}})$ with the subspace topology and refer to this as the Chabauty topology on ${\mathcal{L}}({\mathbb{F}})$.
Outer space and its boundary
----------------------------
Outer space, denoted $\operatorname{cv}$ and introduced by Culler–Vogtmann in [@CVouter], is the space of ${\mathbb{F}}$–marked metric graphs, up to some natural equivalence. A *marked graph* $(G,\phi)$ is a core graph $G$ (finite with no valence one vertices) equipped with a marking $\phi\colon R \to G$, which is a homotopy equivalence from a fixed rose $R$ with $\mathrm{rank}({\mathbb{F}})$ petals to the graph $G$. A *metric* on $G$ is a function $\ell$ assigning to each edge of $G$ a positive real number (its *length*) and we call the sum of the lengths of the edges of $G$ its *volume*. A marked metric graph is a triple $(G,\phi, \ell)$, and Outer space is defined to be set of marked metric graph up to *equivalence*, where $(G_1,\phi_1, \ell_1)$ is equivalent to $(G_2,\phi_2,\ell_2)$ if there is an isometry from $G_1$ to $G_2$ in the homotopy class of the *change of marking* $\phi_2 \circ \phi_1^{-1}\colon G_1 \to G_2$. *Projectivized Outer space* ${{{\EuScript X}}}$, also sometimes denoted $\operatorname{CV}$, is then defined to be the subset of $\operatorname{cv}$ consisting of graphs of volume $1$. Although points in $\operatorname{cv}$ are as described above, we will often denote a marked metric graph simply by its underlying graph $G$ suppressing the marking and metric.
Given $G \in \operatorname{cv}$, the marking associated to $G$ allows one to measure the length of a conjugacy class $\alpha$ of ${\mathbb{F}}$. In particular, there is a unique immersed loop in $G$ corresponding to the homotopy class $\alpha$ which we denote by $\alpha\vert G$. The *length of $\alpha$ in $G$*, denoted $\ell(\alpha\vert G)$, is the sum of the lengths of the edges of $G$ crossed by $\alpha\vert G$, counted with multiplicites. The standard topology on $\operatorname{cv}$ is defined as the smallest topology such that each of the length functions $\ell(\alpha\vert\;\cdot\;)\colon \operatorname{cv}\to \mathbb{R}_+$ is continuous [@CVouter; @Paulin].
Given a point $(G,\phi,\ell)$ in $\operatorname{cv}$, we can define $T$ to be the universal cover of $G$ equipped with a metric obtained by lifting the metric $\ell$ and also equipped with an action of ${\mathbb{F}}$ on $T$ by covering transformations (where ${\mathbb{F}}$ and $\pi_1(G)$ are identified via the marking $\phi$). Then $T$ is an ${\mathbb{R}}$–tree equipped with a minimal free discrete isometric action of ${\mathbb{F}}$. Under this correspondence, equivalent marked metric graphs correspond to ${\mathbb{F}}$–equivariantly isometric ${\mathbb{R}}$–trees. This procedure provides an identification between $\operatorname{cv}$ and the space of minimal free discrete isometric actions of ${\mathbb{F}}$ on ${\mathbb{R}}$–trees, considered up to ${\mathbb{F}}$–equivariant isometries. If $T$ corresponds to $(G,\phi,\ell)$ then for every $w\in {\mathbb{F}}$ we have $\ell(w\vert G)=\ell_T(w):=\min_{x\in T} d_T(x,wx)$. We will also sometime use the notation $i(T,w)$ to denote the translation length of $w$ in $T$, i.e. $i(T,w) = \ell_T(w)$. This notation refers to the intersection form studied in [@kapovich2007geometric]; the details of which are not needed here.
We denote by ${\overline{\mbox{cv}}}$ the set of all very small minimal isometric actions of ${\mathbb{F}}$ on ${\mathbb{R}}$–trees, considered up to ${\mathbb{F}}$–equivariant isometries. As usual, for $T\in{\overline{\mbox{cv}}}$ and $w\in {\mathbb{F}}$, define the *translation length* of $w$ on $T$ as $\ell_T(w):=\inf_{x\in T} d(x,wx)$. It is known that ${\overline{\mbox{cv}}}$ is equal to the closure of $\operatorname{cv}$ with respect to the “axes topology;” see [@CL95; @BF-Outer] for the original proof, and see [@Gui98] for a generalization. We denote the projectivization of ${\overline{\mbox{cv}}}$ by $\overline{{{{\EuScript X}}}} = {{{\EuScript X}}}\cup \partial {{{\EuScript X}}}$. Hence, $\partial {{{\EuScript X}}}$ denotes projective classes of very small minimal actions of ${\mathbb{F}}$ on ${\mathbb{R}}$–trees which are not free and simplicial; this is the so-called boundary of Outer space. We remark that $\overline{{{{\EuScript X}}}}$ is compact.
We recall how $\operatorname{Aut}({\mathbb{F}})$ and $\operatorname{Out}({\mathbb{F}})$ act on ${\overline{\mbox{cv}}}$. If $T\in{\overline{\mbox{cv}}}$ and $\Phi\in\operatorname{Aut}({\mathbb{F}})$, the tree $T\Phi\in {\overline{\mbox{cv}}}$ is defined as follows. As a set and a metric space we have $T\Phi=T$. The action of ${\mathbb{F}}$ is modified via $\Phi$: for every $x\in T$ and $w\in {\mathbb{F}}$ we have $w\underset{T\Phi}{\cdot}x=\Phi(w) \underset{T}{\cdot}x$. This formula defines a right action of $\operatorname{Aut}({\mathbb{F}})$ on ${\overline{\mbox{cv}}}$. The subgroup $\operatorname{Inn}({\mathbb{F}})\le \operatorname{Aut}({\mathbb{F}})$ is contained in the kernel of this action and therefore the action descends to a right action of $\operatorname{Out}({\mathbb{F}})$ on ${\overline{\mbox{cv}}}$: for $\phi\in\operatorname{Out}({\mathbb{F}})$ and $T\in{\overline{\mbox{cv}}}$ we have $T\phi\colonequals T\Phi$, where $\Phi\in\operatorname{Aut}({\mathbb{F}})$ is any automorphism in the outer automorphism class $\phi$. At the level of translation length functions, for $T\in{\overline{\mbox{cv}}}$, $w\in{\mathbb{F}}$ and $\phi\in\operatorname{Out}({\mathbb{F}})$ we have $\ell_{T\phi}(w)=\ell_T(\phi(w))$. Finally, these right actions of $\operatorname{Aut}({\mathbb{F}})$ and $\operatorname{Out}({\mathbb{F}})$ on ${\overline{\mbox{cv}}}$ can be transformed into left actions by putting $\Phi T:= T\Phi^{-1}$, $\phi T:=T \phi^{-1}$ for $T\in {\overline{\mbox{cv}}}$, $\phi\in\operatorname{Out}({\mathbb{F}})$ and $\Phi\in\operatorname{Aut}({\mathbb{F}})$.
Metric properties of Outer space
--------------------------------
For the applications in this paper, we will need a few facts from the metric theory of Outer space. We refer the reader to [@FMout; @BF14; @DT1] for details on the relevant background.
If $T_1=(G_1,\phi_1,\ell_1)$ and $T_2=(G_2,\phi_2,\ell_2)$ are two points in $\operatorname{cv}$, the *extremal Lipschitz distortion* $\operatorname{Lip}(T_1,T_2)$, also sometimes denoted $\operatorname{Lip}(G_1,G_2)$, is the infimum of the Lipschitz constants of all the Lipschitz maps $f\colon (G_1,\ell_1)\to (G_2,\ell_2)$ that are freely homotopic to the the change of marking $\phi_2\circ \phi_1^{-1}$. If one views $T_1$ and $T_2$ as ${\mathbb{R}}$–trees, then $\operatorname{Lip}(T_1,T_2)$ is the infimum of the Lipschitz constants among all ${\mathbb{F}}$–equivariant Lipschitz maps $T_1\to T_2$. It is known that $$\operatorname{Lip}(T_1,T_2)=\max_{w\in {\mathbb{F}}\setminus\{1\}} \frac{\ell_{T_2}(w)}{\ell_{T_1}(w)}.$$ For $T_1,T_2\in {{{\EuScript X}}}$ we put $$d_{{{\EuScript X}}}(T_1,T_2)\colonequals\log \operatorname{Lip}(T_1,T_2)$$ and call $d_{{{\EuScript X}}}(T_1,T_2)$ the *asymmetric Lipschitz distance* from $T_1$ to $T_2$. It is known that $d_{{{\EuScript X}}}$ satisfies all the axioms of being a metric on $\mathcal X$ except that $d_{{{\EuScript X}}}$ is, in general, not symmetric as there exist $T_1,T_2\in{{{\EuScript X}}}$ such that $d_{{{\EuScript X}}}(T_1,T_2)\ne d_{{{\EuScript X}}}(T_2,T_1)$. Because of this asymmetry, it is sometimes convenient to consider the symmetrization of the Lipschitz metric: $${d^{\mathrm{sym}}_{{\EuScript X}}}(T_1,T_2) \colonequals d_{{{\EuScript X}}}(T_1,T_2) + d_{{{\EuScript X}}}(T_2,T_1)$$ which is an actual metric on ${{{\EuScript X}}}$ and induces the standard topology [@FMout]. For a subset $A \in {{{\EuScript X}}}$, we denote by $N_K(A)$ the *symmetric $K$–neighborhood* of $A$, which is the neighborhood of $A$ considered with the symmetric metric.
It is known that for any $T_1,T_2\in{{{\EuScript X}}}$ there exists a unit-speed $d_{{{\EuScript X}}}$–geodesic $\gamma\colon [a,b]\to {{{\EuScript X}}}$ given by a *standard geodesic* from $T_1$ to $T_2$ in ${{{\EuScript X}}}$. Such a geodesic is a concatenation of a *rescaling path*, which only alters the edge lengths of $T_1$, followed by a *folding path*. This geodesic has the property that $\gamma(a)=T_1$, $\gamma(b)=T_2$, $b-a=d_{{{\EuScript X}}}(T_1,T_2)$ and that for any $a\le t\le t'\le b$ one has $t'-t=d_{{{\EuScript X}}}(\gamma(t),\gamma(t'))$. The folding path $\gamma(s)$ has some additional properties arising from its specific construction. We omit describing these properties for the moment (and refer the reader to [@FMout; @BF14; @DT1] for details), but will use them as needed in our arguments.
If ${\mathbf{I}}\subseteq {\mathbb{R}}$ is a (possibly infinite) interval, we also say that $\gamma\colon{\mathbf{I}}\to{{{\EuScript X}}}$ is a folding path if for every finite subinterval $[a,b]\subseteq {\mathbf{I}}$ the restriction $\gamma|_{[a,b]}\colon[a,b]\to {{{\EuScript X}}}$ is a folding path giving a unit speed $d_{{{\EuScript X}}}$–geodesic in the above sense. Then $\gamma\colon{\mathbf{I}}\to{{{\EuScript X}}}$ is also a unit-speed $d_{{{\EuScript X}}}$–geodesic.
Dual laminations of very small trees {#sec:dual_laminations}
------------------------------------
\[def:dual\_lam\] Let $T\in{\overline{\mbox{cv}}}$. For each $\epsilon >0$, let $\Omega^{\le \epsilon}(T)$ denote the collection of $1\ne g \in {\mathbb{F}}$ with $\ell_T(g) \le \epsilon$. We form the algebraic lamination generated by these $\epsilon$–short conjugacy classes: $$L^{\le \epsilon}(T) = L(\Omega^{\le \epsilon}(T))=\overline{\bigcup_{g \in \Omega^{\le \epsilon}(T)} L(g)} \subset \partial^2 {\mathbb{F}}.$$ The [*dual lamination*]{} $L(T)\subseteq {\partial^2 {\mathbb{F}}}$ of $T$ is then defined to be $$L(T) := \bigcap_{\epsilon>0}L^{\le \epsilon}(T).$$
\[rem:L(T)\] Note that $L^{\le \epsilon}(T)$ and $L(T)$ are in fact algebraic laminations on ${\mathbb{F}}$. Further, it is well-known [@CHL2] and not hard to show that $L(T)$ consists of all $(p,q)\in{\partial^2 {\mathbb{F}}}$ such that for every $\epsilon>0$ and every finite subword $v$ of the bi-infinite geodesic from $p$ to $q$ in ${\mathrm{Cay}}({\mathbb{F}},X)$ there exists a cyclically reduced word $w$ over $X^{\pm 1}$ with $\ell_T(w)\le \epsilon$ such that $v$ is a subword of $w$.
In this paper, we will only be concerned with a certain class of trees $T \in {\overline{\mbox{cv}}}$:
A tree $T\in{\overline{\mbox{cv}}}$ is called *arational* if there does not exist a proper free factor $F$ of ${\mathbb{F}}$ and an $F$–invariant subtree $Y\subseteq T$ such that $F$ acts on $Y$ with dense orbits.
In [@Rey12] Reynolds obtained a useful characterization of arational trees in different terms. This characterization implies that if $T\in{\overline{\mbox{cv}}}$ does not arise as a dual tree to a geodesic lamination on a once-punctured surface, then $T$ is arational if and only if $T$ is “indecomposable” (in the sense of [@Gui08]) and ${\mathbb{F}}$ acts on $T$ freely with dense orbits. In particular, if $\phi\in\operatorname{Out}({\mathbb{F}})$ is an atoroidal fully irreducible, then the stable tree $T_\phi$ (discussed in below) is free and arational; see [@CoulboisHilion-botany].
The factor complex and its boundary {#sec:factor_complex}
-----------------------------------
The *free factor complex* of ${\mathbb{F}}$ (for rank$({\mathbb{F}}) \ge 3$) is the complex ${{{\EuScript F}}}$ defined as following: vertices of ${{{\EuScript F}}}$, are conjugacy classes of free factors of ${\mathbb{F}}$ and vertices $A_0, \ldots, A_k$ span an $k$–simplex if these classes have nested representatives $A_0 < \dotsb < A_k$. The complex ${{{\EuScript F}}}$ was introduced in [@HVff] and has since become a central tool for studying the geometry of $\operatorname{Out}({\mathbb{F}})$. In particular, the following theorem is most important for our purposes.
\[th:BFhyp\] The free factor complex ${{{\EuScript F}}}$ is Gromov-hyperbolic; moreover, an element $\phi\in\operatorname{Out}({\mathbb{F}})$ acts on ${{{\EuScript F}}}$ as a loxodromic isometry if and only if $\phi$ is fully irreducible.
A central tool in the proof of is the coarse Lipschitz projection $\pi\colon {{{\EuScript X}}}\to {{{\EuScript F}}}$ from Outer space to the factor complex, which is defined by sending $G \in {{{\EuScript X}}}$ to the collection $$\pi(G) = \{\pi_1(G') : G' \subset G \text{ is a connected, proper subgraph}\} \subset {{{\EuScript F}}}^0.$$ By [@BF14 Lemma 3.1], $\mathrm{diam}_{{{\EuScript F}}}(\pi(G)) \le 4$. Further, there is an $L\ge 0$, depending only on $\operatorname{rank}({\mathbb{F}})$, such that $\pi\colon {{{\EuScript X}}}\to {{{\EuScript F}}}$ is coarsely $L$–Lipschitz [@BF14 Corollary 3.5]. Moreover [@BF14 Theorem 9.3], if $\gamma\colon[a,b]\to{{{\EuScript X}}}$ is a folding path, then $\pi(\gamma([a,b]))$ is within a uniform Hausdorff distance (independent of $\gamma$) from any ${{{\EuScript F}}}$–geodesic from $\pi(\gamma(a))$ to $\pi(\gamma(b))$.
As a hyperbolic space, ${{{\EuScript F}}}$ has a Gromov boundary. Let $\mathcal{AT}$ be the subspace of $\partial {{{\EuScript X}}}$ consisting of projective classes of arational trees. For $T,T' \in \mathcal{AT}$, define $T\approx T'$ to mean that $L(T)=L(T')$. Thus $\approx$ is an equivalence relation on $ \mathcal{AT}$. The following theorem computes the boundary of ${{{\EuScript F}}}$ and will be needed in .
\[th:factor\_complex\_boundary\] The projection $\pi\colon {{{\EuScript X}}}\to {{{\EuScript F}}}$ has an extension to a map $\partial \pi\colon \mathcal{AT} \to \partial {{{\EuScript F}}}$ which satisfies the following properties:
- If $(G_i)_{i\ge0} \subset {{{\EuScript X}}}$ is a sequence converging in $\overline{{{{\EuScript X}}}}$ to $T\in \mathcal{AT}$, then $\pi(G_i) \to \partial \pi (T)$ in ${{{\EuScript F}}}\cup \partial {{{\EuScript F}}}$.
- If $(G_i)_{i\ge0} \subset {{{\EuScript X}}}$ is a sequence converging in $\overline{{{{\EuScript X}}}}$ to $T\in \overline{{{{\EuScript X}}}} \setminus \mathcal{AT}$, then the sequence $(\pi(G_i))_{i\ge0}$ remains bounded in ${{{\EuScript F}}}$.
Moreover, if $T \approx T'$ then $\partial \pi (T) = \partial \pi (T')$, and the induced map $(\mathcal{AT} / \approx) \to \partial F$ is a homeomorphism.
We also record the following useful statement which follows directly from [@CHR11 Theorem A]:
\[prop:CHR\] Let $T,T'\in{\overline{\mbox{cv}}}$ be free arational trees such that $T\not\approx T'$. Then $L(T)\cap L(T')=\varnothing$. Moreover if $(p,q)\in L(T)$ then there does not exist $q'\in \partial{\mathbb{F}}$ such that $(p,q')\in L(T')$.
The $\mathcal Q$–map and the $\mathcal Q$–index {#sec:Q_map}
-----------------------------------------------
For a tree $T\in{\overline{\mbox{cv}}}$, denote $\hat T\colonequals\overline{T}\cup
\partial T$, where $\overline T$ is the metric completion of $T$ and $\partial T$ is the hyperbolic boundary of $T$. Note that the action of ${\mathbb{F}}$ on $T$ naturally extends to an action of ${\mathbb{F}}$ on $\hat T$.
For a tree $T\in{\overline{\mbox{cv}}}$ with dense ${\mathbb{F}}$–orbits, Coulbois, Hilion and Lustig [@CHL2] constructed an ${\mathbb{F}}$–equivariant surjective map $\mathcal Q_T\colon \partial {\mathbb{F}}\to \hat T$. The precise definition of $\mathcal Q_T$ is not important for our purposes but we will need the following crucial property of $\mathcal Q_T$:
\[prop:keyQ\][@CHL2 Proposition 8.5] Let $T\in{\overline{\mbox{cv}}}$ be a tree with dense ${\mathbb{F}}$–orbits. Then for distinct points $p,p'\in \partial {\mathbb{F}}$ we have $\mathcal
Q_T(p)=\mathcal Q_T(p')$ if and only if $(p, p')\in L(T)$.
For a tree $T\in{\overline{\mbox{cv}}}$ we say that a freely reduced word $v$ over $X^{\pm 1}$ is an *$X$–leaf segment* for $L(T)$ (or just a *leaf segment* for $L(T)$) if there exists $(p,p')\in L(T)$ such that $v$ labels a finite subpath of the bi-infinite geodesic from $p$ to $p'$ in ${\mathrm{Cay}}({\mathbb{F}},X)$.
\[defn:proximal\] A point $p\in \partial {\mathbb{F}}$ is said to be *proximal* for $L(T)$ if for every $v$ such that $v$ occurs infinitely often as a subword of the geodesic ray from $1$ to $p$ in ${\mathrm{Cay}}({\mathbb{F}},X)$, the word $v$ is a leaf segment for $L(T)$.
\[prop:prox\] The following hold:
1. For $T\in{\overline{\mbox{cv}}}$ the definition of a proximal points for $L(T)$ does not depend on the free basis $X$.
2. If $T,T'\in {\overline{\mbox{cv}}}$ are free arational trees such that there exists a point $p\in\partial {\mathbb{F}}$ that is proximal for both $L(T)$ and $L(T')$, then $L(T)=L(T')$.
Part (1) easily follows from the fact that for any two free bases $X_1,X_2$ of ${\mathbb{F}}$, the identity map ${\mathbb{F}}\to{\mathbb{F}}$ extends to a quasi-isometry ${\mathrm{Cay}}({\mathbb{F}},X_1)\to{\mathrm{Cay}}({\mathbb{F}},X_2)$. We leave the details to the reader.
For part (2), suppose that $T,T'\in {\overline{\mbox{cv}}}$ are free arational trees such that there exists $p\in\partial {\mathbb{F}}$ which is proximal for both $L(T)$ and $L(T')$. Therefore for every $n\ge 1$ there exists a freely reduced word over $X^{\pm 1}$ of length $n$ which is a leaf-segment for both $L(T)$ and $L(T')$. By a standard compactness argument it then follows that there exists a point $(p_1,q_1)\in L(T)\cap L(T')$. Therefore by we have $L(T)=L(T')$.
We will need the following known results about the map $\mathcal Q_T$:
\[prop:Qmap\] Let $T\in{\overline{\mbox{cv}}}$ be a free ${\mathbb{F}}$–tree with dense ${\mathbb{F}}$–orbits. Then the following hold:
1. [@CHL2 Proposition 5.8] If $p\in
\partial{\mathbb{F}}$ is such that $\mathcal Q_T(p)\in\overline{T}$, then $p$ is proximal for $L(T)$.
2. [@CoulboisHilion-index Proposition 5.2] For every $x\in \partial T$ we have $\#(\mathcal Q_T^{-1}(x))=1$.
3. For every $x\in \hat T$ we have $1\le \#(\mathcal
Q_T^{-1}(x))<\infty$.
4. There are only finitely many ${\mathbb{F}}$–orbits of points $x\in \hat
T$ with $\#(\mathcal Q_T^{-1}(x))\ge 3$.
Associated to the map $\mathcal Q_T$ there is a notion of the $\mathcal
Q$–index of $T$, developed in [@CoulboisHilion-index]. We will only need the definition and properties of the $\mathcal
Q$–index for the case where $T\in{\overline{\mbox{cv}}}$ is a free ${\mathbb{F}}$–tree with dense orbits, and so we restrict our consideration to that context.
Let $T\in{\overline{\mbox{cv}}}$ be a free ${\mathbb{F}}$–tree with dense ${\mathbb{F}}$–orbits. The *$\mathcal Q$–index* of a point $x\in \hat T$ is defined to be ${\mbox{ind}_\mathcal Q}(x):=\max\{0, -2+\#(\mathcal Q_T^{-1}(x))\}$. The *$\mathcal Q$–index* of the tree $T$ is then defined as $${\mbox{ind}_\mathcal Q}(T):=\sum {\mbox{ind}_\mathcal Q}(x),$$ where the summation is taken over the set of representatives of ${\mathbb{F}}$–orbits of points $x$ of $\hat T$ with $\#(\mathcal Q_T^{-1}(x))\ge 3$.
The main result of [@CoulboisHilion-index] is the following:
[@CoulboisHilion-index Theorem 5.3]\[th:index-bound\] Let ${\mathbb{F}}$ be a finite-rank free group with $\operatorname{rank}({\mathbb{F}})\ge 3$. Then every free ${\mathbb{F}}$–tree $T\in{\overline{\mbox{cv}}}$ with dense ${\mathbb{F}}$–orbits satisfies $${\rm ind}_\mathcal Q(T)\le 2\operatorname{rank}({\mathbb{F}})-2.$$
Stable trees of fully irreducibles {#sect:fully-irred}
----------------------------------
For any fully irreducible $\phi\in \operatorname{Out}({\mathbb{F}})$ there is an associated *stable tree* $T_\phi\in{\overline{\mbox{cv}}}$ with the property that $\phi
T_\phi=\lambda T_\phi$ for some $\lambda>1$. The tree $T_\phi\in{\overline{\mbox{cv}}}$ is uniquely determined by $\phi$, up to multiplying the metric by a positive scalar, and the projective class $[T_\phi]\in{\overline{{{\EuScript X}}}}$ is the unique attracting fixed point for the left action of $\phi$ on ${\overline{{{\EuScript X}}}}$. The tree $T_\phi$ may be explicitly constructed from a train-track representative $f\colon G\to G$ of $\phi^{-1}$, and the “eigenvalue” $\lambda$ in the equation $\phi
T_\phi=\lambda T_\phi$ is the Perron-Frobenius eigenvalue of the transition matrix of $f$.
For any fully irreducible $\phi\in \operatorname{Out}({\mathbb{F}})$ the tree $T_\phi$ has dense ${\mathbb{F}}$–orbits and is arational [@Rey12; @CoulboisHilion-botany]; if in addition $\phi$ is atoroidal then the action of ${\mathbb{F}}$ on $T_\phi$ is free.
Suppose now that $\phi\in \operatorname{Out}({\mathbb{F}})$ is an atoroidal fully irreducible element, so that $\phi
T_\phi=\lambda T_\phi$ for some $\lambda>1$. Then for every representative $\Phi\in \operatorname{Aut}({\mathbb{F}})$ of the outer automorphism class $\phi$ the trees $\Phi T_\phi$ and $\lambda T_\phi$ are ${\mathbb{F}}$–equivariantly isometric. The metric completions $\overline{\Phi T_\phi}$ and $\lambda \overline{T_\phi}$ are thus ${\mathbb{F}}$–equivariantly isometric as well.
Using the definition of $\overline{\Phi T_\phi}$ as an ${\mathbb{F}}$–tree, it follows that there exists a bijective $\lambda$–homothety $H_\Phi\colon \overline{T_\phi} \to \overline{T_\phi}$ which *represents* $\Phi$ in the sense that for every $x\in
\overline{T_\phi}$ and every $w\in{\mathbb{F}}$ we have $$\label{eqn:homothety-action}
H_\Phi(wx)=\Phi^{-1}(w)H_\Phi(x).$$ Moreover, there is a unique point $C(H_\Phi)\in \overline{T_\phi}$ which is fixed by $H_\Phi$; this point is called the *center* of $H_\Phi$.
It is known that for every representative $\Phi\in \operatorname{Aut}({\mathbb{F}})$ of $\phi$ there exists a unique homothety $H_\Phi$ representing $\Phi$ in the above sense. Moreover, it is also known that the set of homotheties representing all representatives of $\phi$ in $\operatorname{Aut}({\mathbb{F}})$ is exactly the set $$\{w H_{\Phi_0}| w\in{\mathbb{F}}\}$$ where $\Phi_0$ is some representatives of $\phi$ in $\operatorname{Aut}({\mathbb{F}})$. We refer the reader to [@KapLus-stabil] for details.
We will need a number of known results relating homotheties $H_\Phi$ to the map $\mathcal Q_{T_\phi}$ which are summarized in below. Before stating this proposition recall that there is a notion of a *forward rotationless*, or FR, element of $\operatorname{Out}({\mathbb{F}})$ which allows one to disregard certain periodicity and permutational phenomena that otherwise complicate the index theory for $\operatorname{Out}({\mathbb{F}})$. The notion of an FR element of $\operatorname{Out}({\mathbb{F}})$ was first introduced by Feighn and Handel [@FH11]. We refer the reader to Definition 3.2 in [@CoulboisHilion-botany] for a precise definition. For our purposes we only need to know that for every fully irreducible $\phi\in \operatorname{Out}({\mathbb{F}})$ there exists $k\ge 1$ such that $\phi^k$ is FR [@CoulboisHilion-botany Proposition 3.3]. Note that in this case $T_{\phi}=T_{\phi^k}$, $L(T_{\phi})=L(T_{\phi^k})$ and $\mathcal Q_{T_\phi}=\mathcal Q_{T_{\phi^k}}$. Also if $\phi\in\operatorname{Out}({\mathbb{F}})$ is an FR element then $\phi^m$ is also FR for every $m\ge 1$.
\[prop:att\] Let $\phi\in \operatorname{Out}({\mathbb{F}})$ be an atoroidal fully irreducible FR element.
1. [@CoulboisHilion-botany Proposition 3.1] For every representative $\Phi\in \operatorname{Aut}({\mathbb{F}})$ of $\phi$, the left action of $\Phi$ on $\partial {\mathbb{F}}$ has finitely many fixed points, each of which is either a local attractor or a local repeller. Moreover, the action of $\Phi$ on $\partial {\mathbb{F}}$ has at least one fixed point which is a local attractor, and at least one fixed point which is a local repeller.
2. [@CoulboisHilion-botany Lemma 4.3] Let $\Phi\in \operatorname{Aut}({\mathbb{F}})$ be a representative of $\phi$, and denote by $att(\Phi)$ the set of all fixed points of $\Phi$ in $\partial {\mathbb{F}}$ that are local attractors. Let $H_\Phi$ be the homothety of $T_\phi$ representing $\Phi$. Then $$\mathcal Q_{T_\phi}(att(\Phi))=C(H_\Phi)\qquad\text{and}\qquad \mathcal Q_{T_\phi}^{-1}(C(H_\Phi))=att(\Phi).$$
\[cor:attr\] Let $\phi\in\operatorname{Out}({\mathbb{F}})$ be an atoroidal fully irreducible and let $\Phi\in\operatorname{Aut}({\mathbb{F}})$ be a representative of $\phi$. Let $p\in
att(\Phi)$. Then:
1. The point $p$ is proximal for $L(T_\phi)$.
2. If $T\in {\overline{\mbox{cv}}}$ is a free arational tree such that $L(T)\ne
L(T_\phi)$ then there does not exist $p'\in \partial {\mathbb{F}}$ such that $(p',p)\in L(T)$.
implies that $Q_{T_\phi}(p)=C(H_\Phi)\in\overline{T_\phi}$. Therefore by part (1) of , $p$ is proximal for $L(T_\phi)$, as required.
We now prove part (2) of the corollary. Let $T\in {\overline{\mbox{cv}}}$ be a free arational tree such that $L(T)\ne L(T_\phi)$. Suppose that there exists $p\ne p'\in\partial{\mathbb{F}}$ such that $(p',p)\in L(T)$. If we knew that there exists some $q$ such that $(q,p)\in L(T_\phi)$, then we would obtain a contradiction with . However, under the assumptions made on $p$, it may happen that $Q_{T_\phi}^{-1}(Q_{T_\phi}(p))=\{p\}$, so that a point $q$ with $(q,p)\in L(T_\phi)$ does not exist. Thus we cannot directly appeal to , and need an additional argument, provided below.
Since $p$ is proximal for $L(T_\phi)$, there exist $X$–leaf segments $v_n$ for $L(T_\phi)$ with $|v_n|\to\infty$ as $n\to\infty$, such that each $v_n$ occurs infinitely many times as a subword in the geodesic ray from $1$ to $p$ in ${\mathrm{Cay}}({\mathbb{F}},X)$. Since $(p',p)$ is a leaf of $L(T)$, it follows that each $v_n$ is also a leaf-segment for $L(T)$.
For each $n\ge 1$ choose a geodesic segment $\gamma_n=[u_n,w_n]$ in ${\mathrm{Cay}}({\mathbb{F}},X)$ with label $v_n$ and passing through the vertex $1\in{\mathbb{F}}$ such that $d_{{\mathrm{Cay}}({\mathbb{F}},X)}(u_n,1)\to\infty$ and $d_{{\mathrm{Cay}}({\mathbb{F}},X)}(1,w_n)\to\infty$ as $n\to\infty$. After passing to a subsequence, we may assume that the segments $\gamma_n$ converge to a bi-infinite geodesic from $u\in \partial{\mathbb{F}}$ to $w\in \partial {\mathbb{F}}$.
Since $v_n$ is a leaf-segment for $L(T_\phi)$ there exists a sequence $(s_n,s_n')\in L(T_\phi)$ such that the geodesic from $s_n$ to $s_n'$ in ${\mathrm{Cay}}({\mathbb{F}},X)$ passes through $\gamma_n$ for every $n\ge 1$. Similarly, since $v_n$ is a leaf-segment for $L(T)$, there exists a $(t_n,t_n')\in L(T)$ such that the geodesic from $t_n$ to $t_n'$ in ${\mathrm{Cay}}({\mathbb{F}},X)$ passes through $\gamma_n$ for every $n\ge 1$. By construction it then follows that $$\lim_{n\to\infty} (s_n,s_n')=\lim_{n\to\infty} (t_n,t_n')=(u,w).$$ Since $L(T_\phi)$ and $L(T)$ are closed in $\partial^2{\mathbb{F}}$, it follows that $(u,w)\in L(T_\phi)\cap L(T)$. However, since $T_\phi$, $T$ are free arational trees with $L(T_\phi)\ne L(T)$, this contradicts the conclusion $L(T_\phi)\cap L(T)=\varnothing$ of .
Hyperbolic extensions of free groups
====================================
For the duration of this paper, we assume that ${\mathbb{F}}$ is a finite-rank free group with $\operatorname{rank}({\mathbb{F}})\ge 3$. Note that if $F_2=F(a,b)$ is free of rank two, then for every $\phi\in\operatorname{Out}(F_2)$ we have $\phi([g])=[g^{\pm 1}]$ where $g=[a,b]$. For this reason if $1\to F_2\to E \to Q\to 1$ is a short exact sequence with $Q$ and $E$ hyperbolic, then $|Q|=[E:F_2]<\infty$. On the other hand, free groups of rank at least $3$ admit many interesting word-hyperbolic extensions, as discussed in more detail below.
Subgroups of $\operatorname{Out}({\mathbb{F}})$ and hyperbolic extension of free groups {#sec:free_extensions}
---------------------------------------------------------------------------------------
We now recall a general class of hyperbolic ${\mathbb{F}}$–extensions constructed in [@DT1]. These hyperbolic extensions are the natural generalization of hyperbolic free-by-cyclic groups with fully irreducible monodromy. For any $\Gamma \le \operatorname{Out}({\mathbb{F}})$ there is an ${\mathbb{F}}$–extension $E_\Gamma$ obtained from the following diagram: $$\begin{aligned}
\label{cd:extension}
\begin{CD}
1 @>>> {\mathbb{F}}@>i>> \operatorname{Aut}({\mathbb{F}}) @>p>> \operatorname{Out}({\mathbb{F}}) @>>> 1\\
@. @| @AAA @AAA @. \\
1 @>>> {\mathbb{F}}@>i>> E_\Gamma @>p>> \Gamma @>>> 1 \\
\end{CD}\end{aligned}$$ We say that $E_\Gamma \colonequals p^{-1}(\Gamma)$ is the ${\mathbb{F}}$–extension corresponding to $\Gamma$.
Recall that $\phi \in \operatorname{Out}({\mathbb{F}})$ is called [*atoroidal*]{} if no positive power of $\phi$ fixes a conjugacy class of ${\mathbb{F}}$. A key result of Brinkmann [@Brink] shows that for a cyclic subgroup $\langle \phi\rangle\le \operatorname{Out}({\mathbb{F}})$, the extension $E_{\langle \phi\rangle}$ is word-hyperbolic if and only if $\phi$ is atoroidal or finite order. A subgroup $\Gamma \le \operatorname{Out}({\mathbb{F}})$ is said to be [*purely atoroidal*]{} if every infinite order element of $\Gamma$ is atoroidal.
The following theorem gives geometric conditions on a subgroup $\Gamma \le \operatorname{Out}({\mathbb{F}})$ that imply the corresponding extension $E_\Gamma$ is hyperbolic:
\[th: DT1\] Let $\Gamma \le \operatorname{Out}({\mathbb{F}})$ be finitely generated. Suppose that $\Gamma$ is purely atoroidal and that for some $A \in {{{\EuScript F}}}^0$ the orbit map $\Gamma \to {{{\EuScript F}}}$ given by $g \mapsto gA$ is a quasi-isometric embedding. Then the corresponding extension $E_\Gamma$ is hyperbolic.
Recall that we have called a finitely generated subgroup $\Gamma \le \operatorname{Out}({\mathbb{F}})$ *convex cocompact* if some orbit map $\Gamma \to {{{\EuScript F}}}$ is a quasi-isometric embedding. Hence, implies that if $\Gamma$ is convex cocompact, then $E_\Gamma$ is hyperbolic if and only if $\Gamma$ is purely atoroidal. Note that if $\phi \in \operatorname{Out}({\mathbb{F}})$ is fully irreducible, then $\langle \phi \rangle$ is convex cocompact by .
\[rmk:reform\] In [@DT2], the authors reformulate in terms of the co-surface graph ${\mathcal{CS}}$. This is the $\operatorname{Out}({\mathbb{F}})$–graph defined as follows: vertices are conjugacy classes of primitive elements of ${\mathbb{F}}$ and two conjugacy classes $\alpha$ and $\beta$ are joined by an edge whenever there is a once punctured surface $S$ whose fundamental group can be identified with ${\mathbb{F}}$ in such a way that $\alpha$ and $\beta$ both represent simple closed curves on $S$. We note that closely related graphs appear in [@kapovich2007geometric; @MR2; @Mann-thesis]; see [@DT2] for a discussion and further references. In [@DT1 Theorem 9.2], it is shown that if $\Gamma \le \operatorname{Out}({\mathbb{F}})$ admits a quasi-isometric orbit map into ${\mathcal{CS}}$, then $\Gamma$ is purely atoroidal and convex cocompact, and hence the corresponding extension $E_\Gamma$ is hyperbolic. In [@DT2], the converse is proven: A finitely generated subgroup $\Gamma \le \operatorname{Out}({\mathbb{F}})$ admits a quasi-isometric orbit map into the co-surface graph if and only if $\Gamma$ is purely atoroidal and convex cocompact. The authors in [@DT2] use this characterization to further study the geometry of the hyperbolic extension $E_\Gamma$.
Laminations for hyperbolic extensions {#sec:Mitra_lam}
-------------------------------------
Fix $\Gamma \le \operatorname{Out}({\mathbb{F}})$ finitely generated such that the corresponding extension $$1 \longrightarrow {\mathbb{F}}\longrightarrow E_\Gamma \longrightarrow \Gamma \longrightarrow 1$$ is an exact sequence of hyperbolic groups.
\[def:Mitra\] Let $z\in\partial \Gamma$. Let $\rho$ be a geodesic ray in $\Gamma$ from $1$ to $z$, with the vertex sequence $g_1,g_2,g_3,\dots, g_n,\dots$ in $\Gamma$.
For $1\ne h\in {\mathbb{F}}$ let $w_n$ be a cyclically reduced word over $X^{\pm 1}$ representing the conjugacy class $[g_n(h)]$ in ${\mathbb{F}}$. Let $R_{z,h}$ be the set of all pairs $(u,u')\in {\mathbb{F}}\times{\mathbb{F}}$ such that the freely reduced form $v$ of $u^{-1}u'$ occurs a subword in a cyclic permutation of $w_n$ or of $w_n^{-1}$ for some $n\ge 1$. Put $$\Lambda_{z,h}=\overline{R_{z,h}}\cap {\partial^2 {\mathbb{F}}}$$ where $\overline{R_{z,h}}$ is the closure of $R_{z,h}$ in $({\mathbb{F}}\cup\partial{\mathbb{F}})\times ({\mathbb{F}}\cup\partial {\mathbb{F}})$. Thus $\Lambda_{z,h}$ consists of all $(p_1,p_2)\in{\partial^2 {\mathbb{F}}}$ such that there exists a sequence $(u_i,u_i')\in {\mathbb{F}}\times{\mathbb{F}}$ converging to $(p_1,p_2)$ in $({\mathbb{F}}\cup\partial{\mathbb{F}})\times ({\mathbb{F}}\cup\partial {\mathbb{F}})$ as $i\to\infty$ and such that for every $i\ge 1$ the freely reduced form $v_i$ of $(u_i)^{-1}u_i'$ occurs as a subword in a cyclic permutation of some $w_{n_i}$ or of $w_{n_i}^{-1}$ (which, since $p_1\ne p_2$, automatically implies that $n_i\to\infty$ as $i\to\infty$). Put $$\Lambda_{z} := \bigcup_{h\in {\mathbb{F}}\setminus\{1\}}\Lambda_{z,h}.$$ Finally, define the *ending lamination* of the extension to be $$\Lambda := \bigcup_{z\in \partial \Gamma} \Lambda_{z}.$$
\[rem:Lambda\_z\] Thus $\Lambda_{z,h}$ consists of all $(p,q)\in {\partial^2 {\mathbb{F}}}$ such that for every subword $v$ of the bi-infinite geodesic from $p$ to $q$ in ${\mathrm{Cay}}({\mathbb{F}},X)$ there exists $m\ge 1$ such that $v$ is a subword of a cyclic permutation of $w_m$ or $w_m^{-1}$.
Mitra [@MitraEndingLams Lemma 3.3] shows that the definition of $\Lambda_z$ does not depend on the choice of a geodesic ray $(g_n)_n$ from $1$ to $z$ in the Cayley graph of $\Gamma$. Moreover, the proof of [@MitraEndingLams Lemma 3.3] implies that instead of a geodesic ray one can also use any quasigeodesic sequence from $1$ to $z$ in $\Gamma$. [@MitraEndingLams Remark on p. 399] also shows that for every $z\in\partial \Gamma$ there exists a finite subset $R\subseteq {\mathbb{F}}\setminus\{1\}$ such that $\Lambda_z=\cup_{h\in R} \Lambda_{z,h}$. Since every $\Lambda_{z,h}$ is an algebraic lamination on ${\mathbb{F}}$, it follows that $\Lambda_z$ is also an algebraic lamination on ${\mathbb{F}}$.
The results of Mitra [@MitraEndingLams] imply that $\Lambda_{z,h}$ does not depend on the choice of a free basis $X$ of ${\mathbb{F}}$. Therefore $\Lambda_z$ and $\Lambda$ are independent of $X$ as well. In below we give an equivalent definition of $\Lambda_{z,h}$ which does not involve the choice of $X$; this gives another proof that $\Lambda_{z,h}$ is independent of $X$.
Mitra [@MitraEndingLams] in fact defines $\Lambda_{z,h}$, $\Lambda_z$ and $\Lambda$ in the context of an arbitrary short exact sequence of word-hyperbolic groups. As it suffices for our purposes, here we have only presented the definitions in the somewhat more transparent setting free group extensions.
Recall that for a collection $\Omega$ of conjugacy classes of ${\mathbb{F}}$, we define $L(\Omega)$ to be the smallest algebraic lamination containing $L(g)$ for each $g\in \Omega$.
\[lem:mitraII\] For $z \in \partial \Gamma$, let $(g_i)_{i\ge0}$ be a geodesic ray in $\Gamma$ such that $\lim_{i\to \infty} g_i =z \in \partial \Gamma$. Then for any $h \in {\mathbb{F}}\setminus \{1\}$ we have $$\Lambda_{z,h} = \bigcap_{k\ge0} L\big(\{g_i (h): i\ge k\}\big).$$
Let $w_n$ be the cyclically reduced form over $X^{\pm 1}$ of $g_n(h)$. Recall that $\Lambda_{z,h}$ consists of all $(p,q)\in {\partial^2 {\mathbb{F}}}$ such that for every finite subword $v$ of the bi-infinite geodesic in ${\mathrm{Cay}}({\mathbb{F}},X)$ from $p$ to $q$ there exists $n\ge 1$ such that $v$ is a subword of a cyclic permutation of $w_n$ or of $w_n^{-1}$.
Put $L\colonequals \bigcap_{k\ge0} L(\{g_i (h): i\ge k\})$. Then $L$ consists of all $(p,q)\in {\partial^2 {\mathbb{F}}}$ such that for every finite subword $v$ of the bi-infinite geodesic in ${\mathrm{Cay}}({\mathbb{F}},X)$ from $p$ to $q$ and every $M\ge 1$ there exist $n\ge M$ and $m\in\mathbb Z\setminus\{0\}$ such that $v$ is a subword of a cyclic permutation of $w_n^m$. Hence $\Lambda_{z,h}\subseteq L$.
Let $(p,q)\in L$ be arbitrary. Let $v$ be a finite subword of of the bi-infinite geodesic in ${\mathrm{Cay}}({\mathbb{F}},X)$ from $p$ to $q$. We will use the following claim to complete the proof:
\[claim:growth\] The cyclically reduced length ${\left\Vertw_n\right\Vert}$ of $w_n$ tends to $\infty$ as $n\to\infty$.
Assuming the claim, choose $M\ge 1$ such that for all $n\ge M$ we have ${\left\Vertw_n\right\Vert}\ge{\left\vertv\right\vert}$. Since $(p,q)\in L$, there exist $n\ge M$ and $m\in\mathbb Z\setminus\{0\}$ such that $v$ is a subword of a cyclic permutation of $w_n^m$. The fact that ${\left\Vertw_n\right\Vert}\ge {\left\vertv\right\vert}$ implies that $v$ is a subword of a cyclic permutation of $w_n$ or of $w_n^{-1}$. Therefore $(p,q)\in \Lambda_{z,h}$. Hence $L\subseteq \Lambda_{z,h}$ and so $L= \Lambda_{z,h}$, as required.
We now prove the claim. Note that since $E_\Gamma$ is hyperbolic, each infinite order element of $\Gamma$ is atoroidal. Hence, if we denote by $\Gamma_\alpha$ the subgroup of $\Gamma$ consisting of those elements which fix the conjugacy class $\alpha$, then $\Gamma_\alpha$ is a torsion subgroup of $\operatorname{Out}({\mathbb{F}})$ and hence by [@DT1 Lemma 2.13] has ${\left\vert\Gamma_\alpha\right\vert} \le e$ for some $e\ge 0$ depending only on $\operatorname{rank}({\mathbb{F}})$. If the claim is false, then there is a $D \ge0$ and a infinite subsequence such that ${\left\Vertg_{n_i}(h)\right\Vert}\le D$ for all $i\ge0$. Let $C$ denote the finite number (depending on $D$ and $X$) of conjugacy classes of ${\mathbb{F}}$ whose cyclically reduced length is at most $D$. Hence if $k \ge C(e+2)$, we may find at least $e+2$ distinct elements in the list $g_{n_1}(h),\dotsc,g_{n_k}(h)$ that all belong to the same conjugacy class $\alpha$. This produces $e+1$ distinct elements of $\Gamma$ which fix the conjugacy class $\alpha$. This contradicts our choice of $e$ and completes the proof of the claim.
The main result of Mitra in [@MitraEndingLams] is:
\[th:Mj\_lam\][@MitraEndingLams Theorem 4.11] Suppose that $1 \to H \to G \to \Gamma \to 1$ is an exact sequence of hyperbolic groups with Cannon–Thurston map $\partial i: \partial H \to \partial G$. Then for distinct points $p,q\in \partial H$, $\partial i(p) = \partial i(q)$ if and only if $(p,q)\in \Lambda$ if and only if $(p,q) \in \Lambda_z$ for some $z \in \partial \Gamma$.
Dual laminations at the boundary of $\Gamma$ {#sec:laminations_agree}
============================================
\[conv:main\] For the remainder of this paper, we fix a free group ${\mathbb{F}}$ of finite rank at least $3$ and a finitely generated, purely atoroidal, convex cocompact subgroup $\Gamma\le\operatorname{Out}({\mathbb{F}})$. Thus we may choose a cyclic free factor $x \in {{{\EuScript F}}}^0$ so that the orbit map $\Gamma \to {{{\EuScript F}}}$ given by $g \mapsto gx$ is a quasi-isometric embedding. This orbit map then induces a $\Gamma$–equivariant topological embedding $\kappa \colon \partial \Gamma \to \partial {{{\EuScript F}}}$ and we identify $\partial \Gamma$ with its image in $\partial {{{\EuScript F}}}$. Hence, each point $z\in\partial \Gamma$ correspond to equivalence class $T_z$ of arational trees, each of which has a well-defined dual lamination $L(T_z)$ (). Furthermore, shows that the corresponding extension $E_\Gamma$ is a hyperbolic group. Thus the short exact sequence $$1 \longrightarrow {\mathbb{F}}\longrightarrow E_\Gamma\longrightarrow \Gamma \longrightarrow 1$$ of hyperbolic groups admits a surjective Cannon–Thurston map ${\partial\iota}\colon \partial {\mathbb{F}}\to\partial E_\Gamma$ by , and for every $z\in \partial \Gamma$ there is a corresponding ending lamination $\Lambda_z$ as defined by Mitra in
The main result of this section characterizes the laminations $\{\Lambda_z \colon z\in \partial \Gamma \}$ appearing in for the extension $1 \to {\mathbb{F}}\to E_\Gamma \to \Gamma \to 1$. Recall that we have denoted by $\partial \pi :\mathcal{AT} \to \partial {{{\EuScript F}}}$ the map which associates to each arational tree of ${\overline{\mbox{cv}}}$ the corresponding point in the boundary of the factor complex (see ).
\[th:laminations\_agree\] For each $z \in \partial \Gamma$, there is $T_z \in {\overline{\mbox{cv}}}$ which is free and arational such that $z \mapsto \partial \pi(T_z)$ under $\partial \Gamma \to \partial {{{\EuScript F}}}$ with the property that $$\Lambda_z = L(T_z).$$
\[cor:quotient1\]\[th:main\_2\] Let $\Gamma\le\operatorname{Out}({\mathbb{F}})$ be convex cocompact and purely atoroidal. Then the Cannon–Thurston map ${\partial\iota}\colon \partial {\mathbb{F}}\to \partial E_\Gamma$ identifies points $a,b \in \partial {\mathbb{F}}$ if and only if there exists $z \in \partial \Gamma$ such that $(a,b) \in L(T_z)$. That is, ${\partial\iota}$ factors through the quotient of $\partial {\mathbb{F}}$ by the equivalence relation$$a\sim b \iff (a,b)\in L(T_z)\text{ for some }z\in \partial \Gamma$$ and descends to an $E_\Gamma$–equivariant homeomorphism $\nicefrac{\partial{\mathbb{F}}}{\sim} \to \partial E_\Gamma$.
The specified equivalence relation is by definition given by the subset $\bigcup_{z\in \partial \Gamma}L(T_z)=\Lambda$ of ${\partial^2 {\mathbb{F}}}$, where the last equality holds by . asserts that $\Lambda=\{(p,q)\in \partial{\mathbb{F}}\times\partial{\mathbb{F}}: {\partial\iota}(p)={\partial\iota}(q)\}$. Since the Cannon–Thurston map ${\partial\iota}:\partial{\mathbb{F}}\to \partial E_\Gamma$ is continuous, it follows that $\Lambda$ is a closed subset of $\partial{\mathbb{F}}\times\partial{\mathbb{F}}$. Therefore $\nicefrac{\partial{\mathbb{F}}}{\sim}$, equipped with the quotient topology, is a compact Hausdorff topological space. Moreover, the continuity and surjectivity of ${\partial\iota}\colon\partial{\mathbb{F}}\to \partial E_\Gamma$ now imply that ${\partial\iota}$ quotients through to a continuous bijective map $J\colon\nicefrac{\partial{\mathbb{F}}}{\sim}\to \partial E_\Gamma$, which is, by construction, $E_\Gamma$–equivariant. The fact that both $\nicefrac{\partial{\mathbb{F}}}{\sim}$ and $\partial E_\Gamma$ are compact Hausdorff topological spaces implies that $J$ is a homeomorphism, as required.
Recall that by a general result of [@CHL0], for every $z\in \partial \Gamma$ the map $\mathcal Q_{T_z}\colon \partial{\mathbb{F}}\to \hat T_z$ quotients through to a ${\mathbb{F}}$–equivariant homeomorphism $\partial{\mathbb{F}}/L(T_z)\to \hat T_z$, where $\partial{\mathbb{F}}/L(T_z)$ is given the quotient topology and where $\hat T_z$ is given the “observer’s topology”. Now a similar argument to the proof of implies the following statement (we leave the details to the reader):
\[cor:quotient2\] For each $z\in \partial \Gamma$ the Cannon–Thurston map ${\partial\iota}\colon\partial {\mathbb{F}}\to E_\Gamma$ factors through $\mathcal Q_z\colon \partial {\mathbb{F}}\to \hat T_z$ and induces a continuous, surjective ${\mathbb{F}}$–equivariant map $\hat T_z \to \partial E_\Gamma$ (where $\hat T_z$ is equipped with the observer’s topology).
We now start working towards the proof of . For our next lemma we assume that the reader has some familiarity with folding paths in ${{{\EuScript X}}}$; for example [@BF14 Section 2, 4]. This material is also summarized in [@DT1 Section 2.7] and the reader may find helpful the discussion appearing before Lemma $6.9$ of [@DT1]. For a folding path $G_t$, we say that a conjugacy class $\alpha$ is [*mostly legal*]{} at time $t_0$ if its *legal length* $\mathrm{leg}(\alpha\vert G_{t_0})$ is at least half of its total length ${\ell}(\alpha\vert G_{t_0})$. Of course, if $\alpha$ is mostly legal at time $t_0$, then it is mostly legal for all $t\ge t_0$. Also, for any path $q\colon {\mathbf{I}}\to {{{\EuScript X}}}$, we say that $q$ has the $(\lambda,N_0)$–flaring property for constants $\lambda, N_0 \ge1$ if for any $t \in {\mathbf{I}}$ and any $\alpha \in {\mathbb{F}}\setminus \{1\}$ $$\lambda \cdot \ell(\alpha| q(t)) \le \max \big\{\ell(\alpha| q(t-N_0)), \ell(\alpha| q(t+N_0))\big\}.$$
Fix a rose $R \in {{{\EuScript X}}}$ with a petal labeled by our fixed $x \in {{{\EuScript F}}}^0$. Observe that $x$ is contained in the projection $\pi(R)$ of $R$ to the factor complex ${{{\EuScript F}}}$.
\[lem:eventual\_growth\] For $\Gamma \le \operatorname{Out}({\mathbb{F}})$ as in and for any $K \ge0$ and $\lambda >1$, there exist $N_0, c\ge1$ satisfying the following: Suppose that $\gamma\colon {\mathbf{I}}\to {{{\EuScript X}}}$ is a unit speed folding path contained in a symmetric $K$–neighborhood of $\Gamma \cdot R \subset {{{\EuScript X}}}$. Then $\gamma\colon {\mathbf{I}}\to {{{\EuScript X}}}$ has the $(\lambda,N_0)$–flaring property. Moreover, for any conjugacy class $\alpha$ whose length along $\gamma$ is minimized at $t_\alpha\in{\mathbf{I}}$, we have for all $t\ge t_{\alpha}$ $$\frac{1}{c} \cdot e^{(t - t_\alpha)}\ell(\alpha|\gamma(t_\alpha)) \le \ell(\alpha|\gamma(t)) \le e^{(t - t_\alpha)}\ell(\alpha|\gamma(t_\alpha)).$$
That $\gamma\colon {\mathbf{I}}\to {{{\EuScript X}}}$ has the $(\lambda,N_0)$–flaring property is exactly the conclusion of Proposition 6.11 of [@DT1] (note that the needed “$A_0$–QCX” hypothesis follows from [@DT1 Corollary 6.3]). Also, the upper bound in the statement of the lemma follows immediately from the definition of a unit speed folding path (see [@BF14 Section 4]), so we focus on the lower bound.
For the conjugacy class $\alpha$, let $s_\alpha$ be the infimum of times for which $\alpha$ is mostly legal (if such a time does not exist, set $s_\alpha$ to be the right endpoint of ${\mathbf{I}}$). We show that $s_\alpha - t_\alpha \le C$, for some constant $C$ not depending on $\alpha$. Then [@DT1 Lemma 6.10] (which is an application of [@BF14 Corollary 4.8]) implies that for $t\ge t_\alpha$ $$\begin{aligned}
\ell(\alpha|\gamma(t)) &\ge \frac{1}{3}e^{t -s_\alpha} \mathrm{leg}(\alpha| \gamma(s_\alpha))\\
&\ge \frac{1}{6}e^{t -s_\alpha} \ell(\alpha| \gamma(s_\alpha)) \\
&= \frac{1}{6}e^{t -t_\alpha}e^{-(s_\alpha -t_\alpha)}\ell(\alpha| \gamma(s_\alpha))\\
&\ge \frac{1}{6e^{C}} \cdot e^{t -t_\alpha}\ell(\alpha| \gamma(t_\alpha)),\end{aligned}$$ as needed. Hence, it suffice to prove the uniform bound $s_\alpha - t_\alpha \le C$ over all nontrivial conjugacy classes $\alpha$. This will follow from applying the flaring property of the folding path $\gamma$; the idea is that if $\alpha$ is not mostly legal at some time $t_0$ then either the length of $\alpha$ decreases at some definite rate at $t_0$ (which is impossible if $t_0 = t_\alpha$), or after a bounded amount of time $\alpha$ becomes mostly legal. The details are slightly technical and our argument relies on the proof of Proposition $6.11$ of [@DT1].
Since the image of $\gamma$ is contained in the $K$–neighborhood (with respect to ${d^{\mathrm{sym}}_{{\EuScript X}}}$) of $\Gamma \cdot R$, there is an $\epsilon >0$ depending only on $R\in {{{\EuScript X}}}$ and $K\ge 1$ such that $\gamma({\mathbf{I}}) \subset {{{\EuScript X}}}_{\ge \epsilon}$, the $\epsilon$–thick part of ${{{\EuScript X}}}$. The flaring property then implies that there is a $M\ge1$ depending only on $\lambda$ and $\epsilon$ such that $$\begin{aligned}
\label{eq:flare}
12 \le \ell(\alpha | \gamma(t_0 -N_0))\le \frac{1}{\lambda}\ell(\alpha | \gamma(t_0)),\end{aligned}$$ for $t_0 =t_\alpha +M$. Hence, it suffices to bound the difference $s_\alpha -t_0$. According to the proof of Proposition $6.11$ of [@DT1] either $(1)$ $\mathrm{ilg}(\alpha|\gamma(t_0)) \ge \frac{\ell(\alpha|\gamma(t_0))}{2}$, or $(2)$ $\mathrm{ilg}(\alpha|\gamma(t_0)) < \frac{\ell(\alpha|\gamma(t_0))}{2}$ and $\mathrm{leg}(\alpha|\gamma(t_0)) > 0$. Here, $\mathrm{ilg}(\alpha|\gamma(t_0))$ is the illegal length of $\alpha$ as defined in Section $6$ of [@DT1]. (We note that the third case of [@DT1 Proposition 6.11] does not arise since in that case $\ell(\alpha | \gamma(t_0)) < 6$.)
In case $(1)$, Proposition $6.11$ shows that $\ell(\alpha|\gamma(t_0 -N_0)) \ge \lambda \cdot \ell(\alpha|\gamma(t_0))$, which directly contradicts . Hence, we conclude that we are in the situation of case $(2)$ of [@DT1 Proposition 6.11], where it is shown that the legal length constitutes a definite fraction of the total length of $\alpha$ in $\gamma(t_0)$. In fact, there it is shown that $$\mathrm{leg}(\alpha|\gamma(t_0)) \ge \frac{\ell(\alpha|\gamma(t_0)) -6}{2(1+ \breve{m})}\ge \frac{\ell(\alpha|\gamma(t_0))}{4(1+ \breve{m})},$$ where $\breve{m}$ is a constant depending only on the rank of ${\mathbb{F}}$. From this it follows easily that $s_\alpha - t_0$ is uniformly bounded (e.g., [@DT1 Lemmas 6.9–6.10] show that illegal length decays at a definite rate whereas legal length grows at a definite rate). This completes the proof of the lemma.
The companion to is the following proposition, which states that we can extract folding rays in ${{{\EuScript X}}}$ which stay uniformly close to the orbit of $\Gamma$, have the required flaring property, and limit to free, arational trees in $\partial {{{\EuScript X}}}$. Most of this follows from the main technical work in [@DT1] on stable quasigeodesics in ${{{\EuScript X}}}$. Recall we have fixed $R \in {{{\EuScript X}}}$ with a petal labeled by $x$.
\[lem:folding\_rays\] For any $k, \lambda \ge1$ there are $M, K\ge0$ such that if $(g_i)_{i\ge0}$ is a $k$–quasigeodesic ray in $\Gamma$, then there is an infinite length folding ray $\gamma\colon {\mathbf{I}}\to {{{\EuScript X}}}$ parameterized at unit speed with the following properties:
1. The sets $\gamma({\mathbf{I}})$ and $\{g_iR : i\ge 0\}$ have symmetric Hausdorff distance at most $K$.
2. The rescaled folding path $G_t = e^{-t}\cdot \gamma(t)\in\operatorname{cv}$ converges to the arational tree $T \in \partial\operatorname{cv}$ with the property that $\lim_{i\to \infty}g_i x= \partial \pi(T)$ in ${{{\EuScript F}}}\cup \partial {{{\EuScript F}}}$, where $\partial \pi(T)$ is the projection of the projective class of $T$ to the boundary of ${{{\EuScript F}}}$. Moreover, the action ${\mathbb{F}}\curvearrowright T$ is free.
3. The folding path $\gamma$ has the $(\lambda,M)$ flaring property.
By Theorem $5.5$ of [@DT1], the orbit $\Gamma \cdot R$ is quasiconvex; hence, there is a $K\ge0$ depending only on $k\ge0$ (and the quasi-isometry constants of the orbit map $\Gamma \to {{{\EuScript F}}}$) such that any geodesic of ${{{\EuScript X}}}$ joining points of $(g_iR)_{i\ge0}$ is contained in the symmetric $K$–neighborhood of the quasigeodesic $ (g_iR)_{i\ge0}$ (note that $\Gamma$ is word-hyperbolic and moreover qi-embedded into ${{\EuScript X}}$ by [@DT1 Lemma 6.4]). Let $\gamma_i$ be a standard geodesic of ${{{\EuScript X}}}$ joining $g_0R$ to $g_iR$. Since this collection of geodesics begins at $g_0R$ and remains in a symmetric $K$–neighborhood of $(g_iR)_{i\ge0}$, the Arzela–Ascoli theorem implies that (after passing to a subsequence) the $\gamma_i$ converge uniformly on compact sets to a geodesic ray $\gamma\colon {\mathbf{I}}\to {{{\EuScript X}}}$, which is also contained in the symmetric $K$–neighborhood of $ (g_iR)_{i\ge0}$. Hence, the geodesic $\gamma$ is contained in ${{{\EuScript X}}}_{\ge \epsilon}$ for $\epsilon$ depending only on $K$. As in the proof of Lemma $6.11$ of [@bestvina2012boundary], we see that except for some initial portion of $\gamma$ of uniformly bounded size, $\gamma$ is a folding path. Hence, up to increasing $K$ by a bounded amount, this completes the proof of item $(1)$.
To prove $(2)$, let $\xi$ denote the limit of $(g_i x)_{i\ge0}$ in $\partial {{{\EuScript F}}}$. Note that the rescaled folding path $G_t = e^{-t}\cdot \gamma(t)$ is isometric on edges and hence converges to a tree $T \in {\overline{\mbox{cv}}}$ [@HMaxes]. Hence $\gamma(t)$ converges to the projective class of $T$ in $\overline{{{{\EuScript X}}}}$ as $t \to \infty$ and by item $(1)$ $$\lim_{t\to \infty} \pi (\gamma(t)) = \lim_{i \to \infty} \pi(g_iR) = \lim_{i \to \infty} g_i x =\xi,$$ in ${{{\EuScript F}}}\cup \partial {{{\EuScript F}}}$. Hence, the tree $T$ is arational and $\partial \pi (T)= \xi \in \partial {{{\EuScript F}}}$ by . To compete the proof of $(2)$, it only remains to show that the tree $T$ has a free ${\mathbb{F}}$–action. This will follow using item $(3)$, which we note follows immediately from Proposition $6.11$ of [@DT1].
To see that ${\mathbb{F}}\curvearrowright T$ is free, it suffices to show that $\ell_T(\alpha) >0$ for each $\alpha \in {\mathbb{F}}\setminus \{1\}$. Since $\lim_{t \to \infty} G_t = T$, we see using $(3)$ and that $$\begin{aligned}
\ell_T(\alpha) &= \lim_{t \to \infty}\ell(\alpha|G_t) \\
&= \lim_{t \to \infty} e^{-t} \cdot \ell(\alpha|\gamma(t))\\
& \ge \lim_{t \to \infty} e^{-t} \cdot \frac{ e^{(t - t_\alpha)}}{c} \ell(\alpha|\gamma(t_\alpha)) \\
& = \frac{1}{ce^{ t_\alpha}} \cdot \ell(\alpha|\gamma(t_\alpha))\\
& >0.\end{aligned}$$ This completes the proof.
, together with the fact that every fully irreducible element of $\operatorname{Out}({\mathbb{F}})$ acts on ${{{\EuScript F}}}$ as a loxodromic isometry, immediately implies:
\[cor:phi\] Let $\phi\in\Gamma$ be an element of infinite order. Then for the point $z=\phi^{\infty}\in \partial\Gamma$ we have $T_z=T_\phi$. That is, $z$ is mapped under the map $\partial \Gamma\to\partial{{{\EuScript F}}}$ to the $\approx$–equivalence class $[T_\phi] \in \partial {{{\EuScript F}}}$, where $T_\phi$ is the stable tree of $\phi$.
Most of the work of is done with the following proposition.
\[prop:limit\_to\_zero\] Let $(g_i )_{i\ge0}$ be a quasigeodesic sequence in $\Gamma$ converging to $z\in \partial \Gamma$. Then there is a $T_z \in {\overline{\mbox{cv}}}$ such that $z \mapsto \partial \pi_1(T_z)$ under $\partial \Gamma\to\partial{{{\EuScript F}}}$ with the property that for every $1\ne h\in{\mathbb{F}}$ we have $$\lim_{i \to \infty} \ell_{T_z}(g_ih) = 0.$$
Apply with $\lambda = 2$ to obtain the unit speed folding ray $\gamma\colon {\mathbf{I}}\to {{{\EuScript X}}}$ and denote by $G_t = e^{-t}\cdot \gamma(t)$ the rescaled folding path for which the associated folding maps are isometric on edges. Let $T_z$ be the limit of $G_t$ in ${\overline{\mbox{cv}}}$. By , the image of $z$ under the extension of the orbit map $\partial \Gamma \to \partial {{{\EuScript F}}}$ is $\partial \pi(T_z)$.
As in , let $t_{g_ih}$ denote a time for which $g_i h$ has its length minimized along $\gamma(t)$. Also, for each $i \ge 0$ let $t_i$ denote a time for which the symmetric distance between $\gamma(t)$ and $g_iR$ is less than $K$. (Such a time exists by .) Hence, by definition of the symmetric distance on ${{{\EuScript X}}}$ we have $$\begin{aligned}
\label{eq:lengths_close}
& e^{-K} \le \frac{\ell(\alpha|\gamma(t_i))}{\ell(\alpha|g_iR)} \le e^K,\end{aligned}$$ for each conjugacy class $\alpha$. We will need the following claim:
\[claim:times\_agree\] There is a constant $B\ge0$ which is independent of $i\ge0$ so that $${\left\vertt_i - t_{g_ih}\right\vert} \le B.$$
We will show that $B$ can be taken to be $$\max \left \{2M+M\log_2 \frac{e^{M}e^{K} \ell(h\vert R)}{\epsilon},\; \log \left (\frac{c\cdot \ell(h|R)}{\epsilon} \right) +K \right \},$$ where the constants $M,K,c$ are as in . To see this, first suppose that $t_i < t_{g_ih}$. Let $D = {\left\lfloor\frac{t_{g_ih}-t_i}{M}\right\rfloor}$ so that $DM \le t_{g_ih}-t_i\le DM+M$ and consequently $$\label{eq:adjust_int_point}
\ell(g_ih\vert \gamma(t_{g_i h}-DM)) \le e^M \ell(g_i h\vert \gamma(t_i))$$ since $\gamma$ is a directed geodesic. As in let $\epsilon$ be the length of the shortest loop appearing along the folding path $\gamma$. Then by definition of $t_{g_ih}$ we have $$\ell(g_i h\vert \gamma(t_{g_ih} - M)) \ge \ell(g_i h\vert \gamma(t_{g_i h}))\ge \epsilon.$$ Applying the $(2,M)$–flaring condition inductively at times $t_{g_ih}-M, t_{g_ih}-2M, \ldots, t_{g_ih}-DM $, we find that $$\ell(g_ih\vert \gamma(t_{g_ih}-DM)) \ge 2^{D-1}\ell(g_ih\vert\gamma(t_{g_ih}-M)) \ge 2^{D-1}\epsilon.$$ Combining with and rearranging gives $$\frac{t_{g_ih}-t_i}{M} -2 \le D-1 \le \log_2 \frac{e^M \ell(g_i h\vert \gamma(t_i))}{\epsilon}.$$ Applying and isolating $t_{g_ih}-t_i$ now gives the desired bound $$t_{g_ih}-t_i \le 2M + M\log_2 \frac{e^Me^K\ell(g_ih\vert g_i R))}{\epsilon} = 2M+M\log_2 \frac{e^{M}e^{K} \ell(h\vert R)}{\epsilon}.$$
Now suppose that $t_{g_ih} \le t_i$. Applying to the conjugacy class $\alpha = g_i h$ and using then yields $$\begin{aligned}
e^K &\ge \frac{\ell(g_ih|\gamma(t_i))}{\ell(g_ih|g_iR)}\\
&\ge \frac{\ell(g_ih|\gamma(t_{g_ih}))}{c \cdot \ell(h|R)} e^{(t_i - t_{g_ih})}\\
&\ge \frac{\epsilon}{c \cdot \ell(h|R)} e^{(t_i - t_{g_ih})}.\end{aligned}$$ One final rearrangement then gives the claimed bound $$\begin{aligned}
t_i - t_{g_ih} &\le K + \log \frac{c\cdot \ell(h|R)}{\epsilon} \qedhere\end{aligned}$$
We next observe that $t_i \to \infty$ as $i\to \infty$. To see this, recall that the orbit map $g\mapsto gR$ gives a quasi-isometric embedding $\Gamma\to{{\EuScript X}}$ (see, e.g., [@DT1 Lemma 6.4]). Since $(g_i)_{i\ge 0}$ is a geodesic in $\Gamma$, this implies $d_{{\EuScript X}}(g_0R,g_iR)\to \infty$ as $i\to\infty$. Therefore $$\label{eq:to_infinity}
t_i = d_{{{{\EuScript X}}}}(\gamma(0),\gamma(t_i)) \ge d_{{{{\EuScript X}}}}(g_0R,g_iR) - 2K \to\infty$$ as $i\to\infty$, as claimed.
Finally, we can now compute $$\begin{aligned}
\lim_{i\to \infty}\ell_T(g_ih) &= \lim_{i \to \infty} \lim_{t \to \infty} \ell(g_ih\vert G_t) & \\
&= \lim_{i\to \infty} \lim_{t \to \infty} e^{-t} \cdot \ell(g_ih\vert \gamma(t)) & \\
&\le \lim_{i\to \infty} \lim_{t\to\infty}e^{-t} (e^{(t-t_{g_ih})}\ell(g_ih\vert \gamma(t_{g_ih})) & (\Cref{lem:eventual_growth}) \\
&= \lim_{i \to \infty} e^{-t_{g_ih}}\cdot \ell(g_ih\vert \gamma(t_{g_ih})) & \\
&\le \lim_{i\to \infty} e^{2B}e^{-t_i} \cdot \ell(g_ih\vert \gamma(t_i)) & (Claim~\ref{claim:times_agree}) \\
&= \lim_{i\to \infty}e^{2B+K}e^{-t_i} \cdot \ell(g_ih\vert g_iR) & (\Cref{eq:lengths_close}) \\
&=e^{2B+K}\ell(h|R) \cdot \lim_{i\to \infty} e^{-t_i} & \\
&= 0 & (\Cref{eq:to_infinity}). \end{aligned}$$ This completes the proof of the proposition.
Although we will not need this fact, we note that Namazi–Pettet–Reynolds have recently shown that under the assumption that the orbit map $\Gamma \to {{{\EuScript F}}}$ is a quasi-isometric embedding, each equivalence class of trees appearing the image of $\partial \Gamma$ in $\partial {{{\EuScript F}}}$ is uniquely ergodic [@NPR]. (See [@NPR] and the references therein for a discussion of the various notions of unique ergodicity.) In particular, in the statement of (and therefore ), the tree $T_z$ is unique up to rescaling.
We can now conclude the proof of .
Choose a free basis $X$ of ${\mathbb{F}}$. Let $z\in \partial \Gamma$ and let $(g_n)_{n=1}^\infty$ be a geodesic ray from $1$ to $z$ in the Cayley graph of $\Gamma$.
Let $(p,q)\in \Lambda_z$ be an arbitrary leaf of $\Lambda_z$. Then there is $1\ne h\in{\mathbb{F}}$ such that $(p,q)\in \Lambda_{z,h}$. For every $n\ge 1$ let $w_n$ be the cyclically reduced form of $g_n(h)$ over $X^{\pm 1}$. Let $\gamma$ be the bi-infinite geodesic from $p$ to $q$ in ${\mathrm{Cay}}({\mathbb{F}},X)$. Let $v$ be the label of some finite subsegment of $\gamma$. By , there exists an infinite sequence $n_i\to\infty$ such that for all $i\ge 1$ $v$ is a subword of a cyclic permutation of $w_{n_i}^{\pm 1}$. By , we see that $\lim_{i \to \infty} \ell_{T_z}(w_{n_i}) = 0$. Thus for every $\epsilon>0$ there exists a cyclically reduced word $w$ over $X^{\pm 1}$ with $\ell_{T_z}(w)\le \epsilon$ such that $v$ is a subword of $w$. Since $v$ was the label of an arbitrary subsegment of the geodesic from $p$ to $q$, by it follows that $(p,q)\in L(T_z)$. As $(p,q)\in \Lambda_z$ was arbitrary, we conclude that $\Lambda_z \subset L(T_z)$.
Since the orbit map $\Gamma\to{{{\EuScript F}}}$ is a quasi-isometric embedding, this orbit map extends to a $\Gamma$–equivariant injective continuous map $\partial \Gamma\to\partial{{{\EuScript F}}}$. Thus for any distinct $z_1,z_2\in\partial \Gamma$ we have $T_{z_1}\not\approx T_{z_2}$ and therefore, by , there do not exist $p,q,q'\in \partial {\mathbb{F}}$ such that $(p,q)\in L(T_{z_1})$ and $(p,q')\in L(T_{z_2})$. Since $\Lambda=\cup_{z\in\partial \Gamma}\Lambda_z$ is diagonally closed by , it now follows that for every $z\in \partial \Gamma$ the lamination $\Lambda_z$ is diagonally closed.
Let $z\in \partial \Gamma$ be arbitrary. implies that $L(T_z)$ is diagonally closed. Since $T_z$ is free and arational, [@CHR11 Theorem A] (see also [@bestvina2012boundary Proposition 4.2]) implies that $L(T_z)$ possesses a unique minimal sublimation and that $L(T_z)$ is obtained from this minimal sublamination by adding diagonal leaves. Therefore the only diagonally closed sublamination of $L(T_z)$ is $L(T_z)$ itself. We have already established that $\Lambda_z \subseteq L(T_z)$. Since $\Lambda_z$ is an algebraic lamination on ${\mathbb{F}}$ (see ) and since $\Lambda_z$ is diagonally closed, it follows that $\Lambda_z = L(T_z)$, as required.
Fibers of the Cannon–Thurston map
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Recall (c.f. ) that we have fixed a convex cocompact subgroup $\Gamma\le\operatorname{Out}({\mathbb{F}})$ for which the extension group $E_\Gamma$ is hyperbolic. The short exact sequence $1\to {\mathbb{F}}\to E_\Gamma\to\Gamma\to 1$ thus gives rise to a surjective Cannon–Thurston map denoted ${\partial\iota}\colon \partial{\mathbb{F}}\to\partial E_\Gamma$. We write $\deg(y)$ for the cardinality $\#\left(({\partial\iota})^{-1}(y)\right)$ of the fiber over $y\in \partial E_\Gamma$ and call this the [*degree*]{} of $y$. In this section we use to describe the fibers of the Cannon–Thurston map. The key technical observation is the following.
\[lem:fiber\_description\] Suppose $y\in \partial E_\Gamma$ has $\deg(y)\ge 2$. Then there is a unique point $z\in \partial \Gamma$ and a point $c\in T_z$ so that $({\partial\iota})^{-1}(y) = \mathcal{Q}_{T_z}^{-1}(c)$. Moreover, for any $p\in ({\partial\iota})^{-1}(y)$ we have that $p$ is proximal for $L(T_z)$ and that $$({\partial\iota})^{-1}(y) =\mathcal{Q}_{T_z}^{-1}(c) = \{p\}\cup\{q\in \partial {\mathbb{F}}\mid (p,q)\in L(T_z)\}.$$
Recall that since the orbit map $\Gamma\to \mathcal F$ is a quasi-isometric embedding and since $\Gamma$ and $\mathcal F$ are Gromov-hyperbolic, we have a $\Gamma$–equivariant topological embedding $\kappa\colon \partial \Gamma\to\partial F$. Thus to every $z\in\partial \Gamma$ we have an associated point $\kappa(z)\in \partial \mathcal F$ which is represented by an equivalence class of an arational tree $T_z\in {\overline{\mbox{cv}}}$. Moreover, by , for every $z\in \partial \Gamma$ the action of ${\mathbb{F}}$ on $T_z$ is free and $\Lambda_z=L(T_z)$. Note that since $\kappa$ is injective, implies that for every $p\in \partial {\mathbb{F}}$ there is at most one point $z\in\partial\Gamma$ for which $L(T_z)$ contains a leaf of the form $(p,q)$.
Suppose now that $\deg(y)=m\ge 2$, so that $({\partial\iota})^{-1}(y)=\{p_1,\dots, p_m\}\subseteq \partial {\mathbb{F}}$ consists of $m\ge 2$ distinct points. By and , we find that for each pair $1\le i < j\le m$ there is some $z_{ij}\in \partial \Gamma$ so that $(p_i,p_j) \in \Lambda_{z_{ij}} = L(T_{z_{ij}})$. The above observation (regarding and the injectivity of $\kappa$) shows there is in fact a unique such $z\in \partial\Gamma$; hence we have $(p_i,p_j)\in L(T_z)$ for all $1\le i<j\le m$. This proves that for each $1\le i \le m$ the fiber $({\partial\iota})^{-1}(y)$ has the claimed form $$({\partial\iota})^{-1}(y) = \{p_i\} \cup \{q\in \partial {\mathbb{F}}\mid (p_i,q)\in L(T_z)\}.$$ Moreover, since $p_1,\dotsc,p_m$ are all endpoints of leafs of $L(T_z)$, it is now immediate from that each point of $({\partial\iota})^{-1}(y)$ is proximal for $L(T_z)$. Finally, by there is a point $c\in \hat T_z$ such that $\mathcal{Q}_{T_z}^{-1}(c)=\{p_1,\dots, p_m\} = ({\partial\iota})^{-1}(y)$. This concludes the proof of the lemma.
\[defn:essential\] A point $y\in \partial E_\Gamma$ is said to be *$\Gamma$–essential* if there exists $z\in \partial \Gamma$ such that ${\partial\iota}(x)=y$ for some $x\in\partial{\mathbb{F}}$ proximal for $L(T_z)$ (see ). In this case, there is a unique such point $z\in \partial\Gamma$, which we denote $\zeta(y)\colonequals z$.
show that a point $x\in \partial{\mathbb{F}}$ can be proximal for $L(T_z)$ for at most one point $z\in \partial\Gamma$. Thus $\zeta(y)$ is clearly uniquely determined for any $\Gamma$–essential point with $\deg(y) =1$. This together with shows that is justified in asserting that $\zeta(y)$ is uniquely determined. We also note that every $y\in \partial E_\Gamma$ with $\deg(y)\ge 2$ is $\Gamma$–essential by .
Bounding the size of fibers of the Cannon–Thurston map
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Using , the $\mathcal{Q}$–index theory for very small ${\mathbb{R}}$–trees now easily gives a uniform bound on the cardinality of any fiber of the Cannon–Thurston map ${\partial\iota}\colon\partial{\mathbb{F}}\to\partial E_\Gamma$.
\[thm:main\]\[th:main\_1\] Let $\Gamma\le \operatorname{Out}({\mathbb{F}})$ be purely atoroidal and convex cocompact, where ${\mathbb{F}}$ is a free group of finite rank at least $3$, and let ${\partial\iota}\colon\partial {\mathbb{F}}\to\partial E_\Gamma$ denote the Cannon–Thurston map for the hyperbolic ${\mathbb{F}}$–extension $E_\Gamma$. Then for every $y\in \partial E_\Gamma$, the degree $\deg(y) = \#\left(({\partial\iota})^{-1}(y)\right)$ of the fiber over $y$ satisfies $$1 \le \deg(y) \le 2\operatorname{rank}({\mathbb{F}}).$$ In particular, the fibers Cannon–Thurston map are all finite and of uniformly bounded size.
Fix any point $y\in \partial E_\Gamma$. Since the Cannon–Thurston map is surjective, we clearly have $({\partial\iota})^{-1}(y)\neq \varnothing$. Thus $\deg(y)\ge 1$. If $\deg(y)=1$ there is nothing to prove, so assume $\deg(y) = m\ge 2$. By , there exists a free arational tree $T_z\in{\overline{\mbox{cv}}}$ and a point $c\in\hat{T_z}$ so that $({\partial\iota})^{-1}(y) = \mathcal{Q}_{T_z}^{-1}(c)$. then gives $$m-2 = {\mbox{ind}_\mathcal Q}(c) \le {\mbox{ind}_\mathcal Q}(T_z)\le 2\operatorname{rank}({\mathbb{F}})-2.\qedhere$$
Rational points and the Cannon–Thurston map
-------------------------------------------
A point in the boundary $\partial G$ of a word-hyperbolic group $G$ is said to be *rational* if it is equal to the limit $g^{\infty} \colonequals \lim_{n\to\infty} g^n$ in $G\cup \partial G$ for some infinite order element $g\in G$. A point in $\partial G$ is *irrational* if it is not rational. Our next result analyzes the fibers of the Cannon–Thurston map ${\partial\iota}$ over rational points of $\partial {\mathbb{F}}$ and $\partial E_\Gamma$.
\[th:rational\_fibers\] Suppose that $1 \to {\mathbb{F}}\to E_\Gamma \to \Gamma \to 1$ is a hyperbolic extension with $\Gamma \le \operatorname{Out}({\mathbb{F}})$ convex cocompact. Consider a rational point $g^\infty\in \partial E_\Gamma$, where $g\in E_\Gamma$ has infinite order.
1. Suppose that $g^k$ is equal to $w\in {\mathbb{F}}\lhd E_\Gamma$ for some $k\ge 1$ (i.e., $g$ projects to a finite order element of $\Gamma$). Then $({\partial\iota})^{-1}(g^\infty) = \{w^\infty\}\subset \partial {\mathbb{F}}$ and so $\deg(g^\infty) = 1$.
2. Suppose that $g$ projects to an infinite-order element $\phi\in\Gamma$. Then there exists $k\ge 1$ such that the automorphism $\Psi\in\operatorname{Aut}({\mathbb{F}})$ given by $\Psi(w) = g^kwg^{-k}$ is forward rotationless (in the sense of [@FH11; @CoulboisHilion-botany]) and its set $\mathrm{att}(\Psi)$ of attracting fixed points in $\partial {\mathbb{F}}$ is exactly $\mathrm{att}(\Psi)=({\partial\iota}){^{-1}}(g^\infty)$. Moreover, $g^\infty$ is $\Gamma$–essential and $\zeta(g^\infty) = \phi^\infty$.
First suppose $g^k = w\in {\mathbb{F}}$ for some $k\ge 1$. By continuity and ${\mathbb{F}}$–equivariance of the Cannon–Thurston map, it is immediate that ${\partial\iota}$ sends $w^\infty \in \partial {\mathbb{F}}$ to $g^\infty \in \partial E_\Gamma$ (note that $(g^k)^\infty = g^\infty$ in $\partial E_\Gamma$). Thus $\{w^\infty\}\subset ({\partial\iota})^{-1}(g^\infty)$. Since for every $z\in\partial\Gamma$, $T_z$ is a free arational tree, there do not exist $z\in \partial \Gamma$ and $p\in \partial {\mathbb{F}}$ such that $(p,w^\infty)\in L(T_z)$. Therefore and imply that $({\partial\iota})^{-1}(g^\infty) \subset\{w^\infty\}$ and part (1) is verified.
Now suppose that $g$ projects to an infinite order element $\phi\in \Gamma$. As explained in , we may choose $k\ge 1$ so that the automorphism $\Psi\in \operatorname{Aut}({\mathbb{F}})$ given by $w\mapsto g^k w g^{-k}$ is forward rotationless. Let $p\in att(\Psi)$ be a locally attracting fixed point for the left action of $\Psi$ on $\partial {\mathbb{F}}$. By , $p$ is a proximal point for $L(T_\phi)$, and moreover shows that $T_\phi = T_z$ for the point $z\colonequals \phi^\infty\in \partial\Gamma$.
Recall that by the map ${\partial\iota}\colon\partial {\mathbb{F}}\to\partial E_\Gamma$ is $E_\Gamma$–equivariant, and that $g^k$ acts on $\partial {\mathbb{F}}$ by $\Psi$, that is $g^kq=\Psi(q)$ for every $q\in\partial {\mathbb{F}}$. Since $p=g^kp$ is a local attractor for $\Psi$, the $E_\Gamma$–equivariance and continuity of ${\partial\iota}$ ensure that ${\partial\iota}(p)=g^k{\partial\iota}(p)$ is a local attractor for the action of $g^k$ on $\partial E_\Gamma$. Since $g$ is an element of infinite order in a word-hyperbolic group $E_\Gamma$, $g^\infty$ is the unique local attractor for the action of $g$ on $\partial E_\Gamma$. Thus we may conclude that in fact ${\partial\iota}(p)=g^\infty$. This proves that $att(\Psi)\subseteq ({\partial\iota})^{-1}(g^\infty)$. Since $p$ is proximal for $L(T_{z})$, we also see that $g^\infty$ is $\Gamma$–essential and that $\zeta(g^\infty)=z = \phi^\infty$, as claimed.
Now, if $\deg(g^\infty)=1$, it follows that $\#att(\Psi)\le 1$ so that $att(\Psi) = \{p\}= ({\partial\iota})^{-1}(g^\infty)$, as required. On the other hand, if $\deg(g^\infty)\ge 2$, then provides a point $c\in T_\phi$ so that $att(\Psi)\subset ({\partial\iota})^{-1}(g^\infty) = \mathcal{Q}_{T_\phi}^{-1}(c)$. Part (2) of then ensures that $({\partial\iota})^{-1}(g^\infty)=att(\Psi)$. Thus claim (2) is verified.
Kapovich and Lustig showed [@KapLusCT] that for the Cannon–Thurston map $\partial{\mathbb{F}}\to\partial E_{\langle\phi\rangle}$ associated to a cyclic group generated by an atoroidal fully irreducible $\phi\in \operatorname{Out}({\mathbb{F}})$, every point $y\in \partial E_{\langle\phi\rangle}$ with $\deg(y)\ge 3$ is rational and has the form $y=g^\infty$ for some $g\in E_{\langle\phi\rangle}-{\mathbb{F}}$. (Note that here $\partial\langle\phi\rangle$ consists of two points $\phi^{\pm\infty}$, both of which are rational.) Here we show that this result need not hold in the general setting of convex cocompact subgroups. Rather, we find that the Cannon–Thurston map, via the assignment $y\mapsto \zeta(y)$ for $\Gamma$–essential points, detects the following relationship between rationality/irrationality in $\partial E_\Gamma$ and $\partial \Gamma$.
\[th:rational\_and\_irrational\] Suppose that $1 \to {\mathbb{F}}\to E_\Gamma \to \Gamma \to 1$ is a hyperbolic extension with $\Gamma \le \operatorname{Out}({\mathbb{F}})$ convex cocompact. Then the following hold:
1. If $y\in \partial E_\Gamma$ has $\deg(y)\ge 3$ and $\zeta(y)\in\partial \Gamma$ is rational, then $y$ is rational.
2. If $y\in \partial E_\Gamma$ has $\deg(y)\ge 2$ and $\zeta(y)\in\partial\Gamma$ is irrational, then $y$ is irrational.
Our argument for part (1) of the above theorem is similar to the proof of part (3) of Theorem 5.5 in [@KapLusCT]. The proof is included here for completeness.
Suppose first that $y\in \partial E_\Gamma$ is such that $\deg(y)\ge 3$ and $\zeta(y)\in\partial \Gamma$ is rational. Thus $\zeta(y)=\phi^\infty$ for some atoroidal fully irreducible $\phi\in\Gamma$. Then $({\partial\iota})^{-1}(y)=\mathcal Q_{T_\phi}^{-1}(x)$ for some $x\in \overline{T_\phi}$. Let $k\ge 1$ be such that $\psi=\phi^k$ is FR. Note that $T_\phi=T_\psi$. Choose some homothety $H$ of $\overline T_\phi$. in implies that $H$ acts on ${\mathbb{F}}$–orbits of points of $\overline T_\phi$. Since there are only finitely many ${\mathbb{F}}$–orbits of points of $\overline T_\phi$ with $\mathcal Q_{T_\phi}$–preimage of cardinality $\ge 3$ (), some positive power of $H$ preserves every such orbit and, in particular, preserves the orbit of $x$. Thus, after replacing $k$ by $kt$ for some integer $t\ge 1$, we may assume that $Hx=wx$ for some $w\in{\mathbb{F}}$. Then $w^{-1}Hx=x$. The homothety $H_1=w^{-1}H$ represents some $\Psi\in\operatorname{Aut}({\mathbb{F}})$ whose outer automorphism class is $\psi$. Moreover, $x$ is the center of the homothety $H_1$. Therefore by it follows that $\mathcal Q_{T_\phi}^{-1}(x)=att(\Psi)$. Thus $({\partial\iota})^{-1}(y)=att(\Psi)$.
Choose $g\in E_\Gamma$ so that the automorphism $h\mapsto ghg^{-1}$ of ${\mathbb{F}}$ is exactly $\Psi$. Note that $g$ projects to $\psi\in\Gamma$ and thus $g$ has infinite order in $E_\Gamma$. Let $p\in att(\Psi)$. We have $p=\Psi(p)=gp$ and ${\partial\iota}(p)=y$. By $E_\Gamma$–equivariance of ${\partial\iota}$ it follows that $gy=y$. Since $g$ is an element of infinite order in a word-hyperbolic group $E_\Gamma$, it follows that $y=g^{\pm\infty}$ is rational. This proves claim (1).
Next suppose that $\deg(y)\ge 2$ and that $\zeta(y)$ is irrational. Assuming, on the contrary, that $y$ is rational, we have $y=g^\infty$ for some non-torsion element $g\in E_\Gamma$. Since $\deg(y)\ge 2$, (1) implies that $g$ projects to an element of infinite order $\phi$ of $\Gamma$. But then $\zeta(y) = \phi^\infty$ is rational by (2), contradicting the assumption that $\zeta(y)$ was irrational. Therefore $y$ is indeed rational and claim (2) holds.
Conical limit points
--------------------
Recall that every non-elementary subgroup of a word-hyperbolic group $G$ acts as a convergence group on the Gromov boundary $\partial G$. If a group $H$ acts as a convergence group on a compact metrizable space $Z$, a point $z\in Z$ is called a *conical limit point* for the action of $H$ on $Z$ if there exist an infinite sequence $h_n$ of distinct elements of $H$ and a pair of distinct points $z_-,z_+\in Z$ such that $\lim_{n\to\infty} h_n z = z_+$ and that $(h_n|_{Z\setminus \{z\}} )_n$ converges uniformly on compact subsets to the constant map $c_{z_-} \colon Z\setminus \{z\}\to Z$ sending $Z\setminus \{z\}$ to $z_-$. It is also known that if $H\le G$ is a non-elementary subgroup of a word-hyperbolic group $G$, then $z\in \partial G$ is a conical limit point for the action of $H$ on $\partial G$ if and only if there exists an infinite sequence of distinct elements $h_n\in H$ such that all $h_n$ lie in a bounded Hausdorff neighborhood of a geodesic ray from $1$ to $z$ in the Cayley graph of $G$. We refer the reader to [@JKLO] for more details and background regarding conical limit points.
\[th:conical\]\[cor:main\_4\] Let $\Gamma\le\operatorname{Out}({\mathbb{F}})$ be purely atoroidal and convex cocompact. If $y\in \partial E_\Gamma$ is $\Gamma$–essential, then $y$ is not a conical limit point for the action of ${\mathbb{F}}$ on $\partial E_\Gamma$. In particular, if $\deg(y)\ge 2$ or if $y=g^\infty$ for some $g\in E_\Gamma$ projecting to an infinite-order element of $\Gamma$, then $y$ is not a conical limit point for the action of ${\mathbb{F}}$.
Choose a free basis $X$ of ${\mathbb{F}}$. Suppose that $y\in \partial E_\Gamma$ is $\Gamma$–essential, and let $z=\zeta(y)\in\partial \Gamma$ so that $y={\partial\iota}(p)$ for some $p\in\partial{\mathbb{F}}$ proximal to $L(T_z)=\Lambda_z$. Then every freely reduced word over $X^{\pm 1}$ which occurs infinitely many times in the geodesic ray from $1$ to $p$ in ${\mathrm{Cay}}({\mathbb{F}},X)$ is a leaf-segment for $L(T_z)=\Lambda_z$. Therefore [@JKLO Theorem B] implies that $y={\partial\iota}(p)$ is not a conical limit point for the action of ${\mathbb{F}}$ on $\partial E_\Gamma$, as claimed. The remaining assertions now follow from and .
Discontinuity of $z\in \partial \Gamma \mapsto \Lambda_z \in {\mathcal{L}}({\mathbb{F}})$ {#sec:continuity}
=========================================================================================
In this section, we answer a question of Mahan Mitra using our hyperbolic extension $E_\Gamma$. As stipulated in , our fixed finitely generated subgroup $\Gamma \le \operatorname{Out}({\mathbb{F}})$ has a quasi-isometric orbit map into the free factor complex and gives rise to an exact sequence of word-hyperbolic groups $$1 \longrightarrow {\mathbb{F}}\longrightarrow E_\Gamma \longrightarrow \Gamma \longrightarrow 1.$$ Thus each point $z \in \partial \Gamma$ has an associated ending lamination $\Lambda_z$ () and we consider the map $F\colon \partial \Gamma \to \mathcal{L}({\mathbb{F}})$ defined by $F(z)=\Lambda_z$. Here ${\mathcal{L}}({\mathbb{F}})$ is the set of laminations equipped with the Chabauty topology (). In his work on Cannon–Thurston maps for normal subgroups of hyperbolic groups, Mitra asked whether this map $F$ is continuous. We answer this question by producing an explicit example for which $F\colon \partial \Gamma \to \mathcal{L}({\mathbb{F}})$ is not continuous. This is done in .
Before turning to this example, we establish a “subconvergence” property for the map $F\colon \partial \Gamma \to \mathcal{L}({\mathbb{F}})$. This is the strongest positive result that one can give about continuity with respect to the Chabauty topology on ${\mathcal{L}}({\mathbb{F}})$. Recall that for a lamination $L \in {\mathcal{L}}({\mathbb{F}})$ the notation $L'$ denotes the set of accumulation points of $L$, in the usual topological sense.
\[th:lamination\_continuity\] Let $\Gamma\le\operatorname{Out}({\mathbb{F}})$ be purely atoroidal and convex cocompact, and let $\Lambda_z\in{\mathcal{L}}({\mathbb{F}})$ denote the ending lamination associated to $z\in \partial \Gamma$. Then for any sequence $z_i$ in $\partial \Gamma$ converging to $z$ and any subsequence limit $L$ of the corresponding sequence $\Lambda_{z_i}$ in ${\mathcal{L}}({\mathbb{F}})$, we have $$\Lambda'_z \subset L \subset \Lambda_z.$$
Before proving , we first recall a characterization of convergence in the Chabauty topology. Let $X$ be a locally compact metric space and let $C(X)$ be the space of closed subsets of $X$ equipped with the Chabauty topology. Recall that $C(X)$ is compact. The following lemma is well known; see [@canary2006fundamentals].
\[lem:Cabauty\_convergence\] For a locally compact metric space $X$, a sequence $C_i$ converges to $C$ in $C(X)$ if and only if the following conditions are satisfied:
1. For each $x_{i_k} \in C_{i_k}$, whenever $x_{i_k} \to x$ in $X$ as $k \to \infty$ it follows that $x\in C$.
2. For each $x\in C$, there is are $x_i \in C_i$ with $x_i \to x$ in $X$ as $i \to \infty$.
As “weak evidence” towards continuity of the map $F$, Mitra proves the following proposition [@MitraEndingLams], which essentially amounts to verifying $(1)$ in .
\[prop:subconvergence\] If $z_i \to z$ in $\partial \Gamma$ and if $(p_i,q_i)\in\Lambda_{z_i}$ converge to $(p,q)$ in ${\partial^2 {\mathbb{F}}}$, then $(p,q)\in\Lambda_z$.
In [@CHL2], the authors remark that if $(T_i)_{i\ge 0}$ is a sequence of trees in ${\overline{\mbox{cv}}}$ converging to a tree $T$, then any subsequence limit $L$ of the corresponding sequence of dual laminations $(L(T_i))_{i\ge 0}$ in ${\mathcal{L}}({\mathbb{F}})$ is contained in $L(T)$. They elaborate this statement in [@CHLunpub Proposition 1.1]. Combining this general fact with gives an alternative proof of .
We now turn to the proof of .
Suppose that $z_i \to z$ in $\partial \Gamma$. Then by and , if $L$ is any subsequential limit of $\Lambda_{z_i}$ in the Chabauty topology it follows that $L \subset \Lambda_z$. By , $\Lambda_z = L(T_z)$ for the free and arational tree $T_z$; we also know that $L(T_z)$ is the diagonal closure of its unique minimal sublamination $L'(T_z)$ by [@CHR11 Theorem A]. In particular, we must have that $L \supset L'(T_z) = \Lambda'_z$. Hence, $ \Lambda'_z \subset L \subset \Lambda_z$, as required.
We conclude this section by producing an example of a hyperbolic extensions $E_\Gamma$ for which $F \colon \partial \Gamma \to {\mathcal{L}}({\mathbb{F}})$ is not continuous. Before explaining this example, we briefly recall some facts related to the index theory of free group automorphisms. We refer the reader to [@CoulboisHilion-botany; @kapovich2014invariant; @CoulboisHilion-index; @CHR11] for more details.
Let $\phi\in\operatorname{Out}({\mathbb{F}})$ be an atoroidal fully irreducible element and let $h \colon G\to G$ be a train track representative of $\phi$ (thus $h$ is necessarily expanding and irreducible). By replacing $h$ with a sufficiently large positive power and possibly subdividing $G$, we may further assume the following: that the endpoints of all INPs in $G$ (if any are present) are vertices of $G$, that every periodic vertex of $G$ is fixed by $h$, that every periodic direction in $G$ has period $1$ and that every periodic INP (if any are present) in $G$ has period $1$. Here, the acronym INP stands for irreducible Nielsen path; see [@bestvina1992train; @bestvina1997laminations; @FH11] for background. In the discussion below, if $(p,q)\in L$ for some algebraic lamination $L$ on ${\mathbb{F}}$, we will often refer to a geodesic $\mathfrak l$ from $p$ to $q$ in $\tilde G$ as a leaf of $L$.
Recall that the *Bestvina-Feighn-Handel lamination* $L_{BFH}(\phi)$ is an algebraic lamination on ${\mathbb{F}}$ consisting of all $(p,q)\in \partial^2{\mathbb{F}}$ such that for every finite subpath $\tilde\gamma$ of the geodesic in $\tilde\Gamma$ connecting $p$ to $q$, the projection $\gamma$ of $\tilde\gamma$ to $G$ is a subpath of $h^n(e)$ for some $n\ge 1$ and some (oriented) edge $e$ of $G$. If $v$ is a periodic vertex of $G$ and $e$ is an edge starting with $v$ defining a periodic direction at $v$ (so that $h(e)$ starts with $e$), then $e$ determines a semi-infinite reduced edge-path $\rho_e$ in $G$ called the *eigenray* of $h$ corresponding to $v$. Namely, $\rho_e$ is defined as the path such that for every $n\ge 1$, $h^n(e)$ is an initial segment of $\rho_e$. It is known [@kapovich2014invariant] that $L_{BFH}(\phi)\subseteq L(T_\phi)$ is the unique minimal sublamination of $L(T_\phi)$; that is, $L_{BFH}(\phi)$ is the unique minimal (with respect to inclusion) nonempty subset of $L(T_\phi)$ which is itself an algebraic lamination on ${\mathbb{F}}$. It is also known that $L(T_\phi)$ is the “transitive closure” of $L_{BFH}(\phi)$; that is, $L(T_\phi)$ is the smallest diagonally closed (in the sense defined in ) subset of $\partial^2 {\mathbb{F}}$ which contains $L_{BFH}(\phi)$ and is itself a lamination. Moreover, $L(T_\phi)\setminus L_{BFH}(\phi)$ consists of finitely many ${\mathbb{F}}$–orbits of points of $\partial^2{\mathbb{F}}$ called *diagonal leaves* of $L(T_\phi)$, and [@kapovich2014invariant] gives a precise description of these diagonal leaves in terms of the train track $h$: If $v$ is a periodic vertex and $e,e'$ are distinct periodic edge of $G$ with origin $v$, then any lift to $\tilde\Gamma$ of the biinfinite path $\rho_e^{-1}\rho_{e'}$ is a leaf of $L(T_\phi)$; such a leaf is called a *special leaf*. Some of the special leaves already belong to $L_{BFH}(\phi)$ (this happens precisely when the turn $e,e'$ is “taken” by $h$). Special leaves that do not belong to $L_{BFH}(\phi)$ are necessarily diagonal. If $h$ has no periodic INPs, then all diagonal leaves of $L(T_\phi)$ arise in this way; that is, every diagonal leaf is special. If $h$ has some periodic INPs, then $L(T_\phi)$ admits diagonal leaves of additional kind, but their precise description is not needed here (see [@kapovich2014invariant] for details).
\[ex:disc\] We now construct an example of a purely atoroidal convex cocompact subgroup $\Gamma\le \operatorname{Out}({\mathbb{F}})$, with $\operatorname{rank}({\mathbb{F}})=3$, such that the map $\partial \Gamma\to\mathcal L({\mathbb{F}})$ given by $z\mapsto \Lambda_z$ is not continuous with respect to the Chabauty topology on ${\mathcal{L}}({\mathbb{F}})$.
Suppose that we are given automorphisms $\phi$ and $\psi$ of ${\mathbb{F}}= F(a,b,c)$, the free group of rank $3$, with the following properties:
1. $\phi$ and $\psi$ are atoroidal and fully irreducible.
2. $\phi$ and $\psi$ are positive with respect to the basis $\{a,b,c\}$. Thus we can represent $\phi$ and $\psi$ by train track maps on the rose $R_3$ with a single vertex $v$ and petals corresponding to $a,b,c$; we denote these train track maps by $f$ and $g$ accordingly. We further assume that we have replaced $f$ and $g$ by appropriate positive powers so that for each of $f,g$ every periodic vertex of $R_3$ is fixed, that every periodic direction in $R_3$ has period $1$, and that every periodic INP in $R_3$ (if any are present) has period $1$.
3. For the map $f$ the directions corresponding to the three edges of $R_3$ labelled $a,b,c$ are periodic.
4. The map $g$ has $4$ periodic directions at $v$, given by the edges labelled by $a,c,a^{-1},b^{-1}$. Moreover, $g$ has no periodic INPs.
At the end of this example, we will give references for where one can find automorphisms satisfying these conditions.
By [@bestvina1997laminations; @KLping; @DT1], we may replace $\phi$ and $\psi$ by further positive powers such that $\Gamma = \langle \phi, \psi \rangle$ is a purely atoroidal, convex cocompact subgroup of $\operatorname{Out}({\mathbb{F}})$. Hence, the corresponding extension $E_\Gamma$ is hyperbolic. We will show that the map $F\colon\partial \Gamma \to {\mathcal{L}}({\mathbb{F}})$ defined by $F(z) = \Lambda_z$ is not continuous.
For a leaf $\mathfrak l$ of a lamination $L\in{\mathcal{L}}({\mathbb{F}})$, we say that $\mathfrak l$ is *positive* (correspondingly *negative*) if $\mathfrak l$ is labelled by a positive (correspondingly negative) bi-infinite word in $F(a,b,c)$, and we say that $\mathfrak l$ is *mixed* if it is neither positive nor negative. Note that, since the automorphism $\psi$ is positive, the definition of $L_{BFH}(\psi)$ implies that every leaf of $L_{BFH}(\psi)$ is, up to a flip, labelled by a positive biinfinite word in $F(a,b,c)$. Also recall that, as discussed above, $L_{BFH}(\psi)$ is the unique minimal sublamination of $L(T_\psi)$. Since $g$ has no periodic INPs, every mixed leaf of $L(T_\psi)$ is special. Hence, each mixed leaf of $L(T_\psi)$ is, up to the ${\mathbb{F}}$–action and the flip, labelled by a word of the form $\rho_1^{-1}\rho_2$ where $\rho_1,\rho_2$ are two eigenrays of $g$ corresponding to two distinct periodic directions at $v$.
We establish following: $(i)$ there are exactly $2$ mixed diagonal leaves of $L(T_\psi)$, up to the ${\mathbb{F}}$–action and the flip, $(ii)$ if $\phi^n L(T_\psi)$ converges to $L$ in ${\mathcal{L}}({\mathbb{F}})$ then $L$ contains at most $2$ mixed leaves, up to the ${\mathbb{F}}$–action and the flip; $(iii)$ the lamination $L(T_\phi)$ contains at least $3$ distinct mixed diagonal leaves, up to the ${\mathbb{F}}$–action and the flip.
To see $(i)$, note that since we assumed that $g$ has no periodic INPs, mixed leaves of $L(T_\psi)$ are leaves of $L(T_\psi) \setminus L_{BFH}(\psi)$ and hence are diagonal and special for $L(T_\psi)$. Recall that $g$ has exactly four periodic directions at $v$, namely $a,c,a^{-1},b^{-1}$. Thus $g$ has 4 eigenrays starting at $v$: positive eigenrays $\rho_a$, $\rho_c$ and negative eigenrays $\rho_{a^{-1}}$, $\rho_{b^{-1}}$. Up to a flip, every mixed leaf of $L(T_\psi)$ is then labeled by either $(\rho_a)^{-1}\rho_c$ or $(\rho_{a^{-1}})^{-1} \rho_{b^{-1}}$. Thus $(i)$ is verified.
A similar argument can be used to prove $(ii)$, although a bit of care is needed here in using the definition of the Chabauty topology on ${\mathcal{L}}({\mathbb{F}})$. Suppose $L\in {\mathcal{L}}({\mathbb{F}})$ is the limit of $\phi^n
L(T_\psi)$ as $n\to\infty$. Any mixed leaf $\mathfrak l$ of $L$, up to a flip, is labelled by a word of the form $W^{-1}Z$, where $W$ and $Z$ are each positive rays from the identity in $F(a,b,c)$ starting with distinct symbols. Since $\lim_{n\to\infty} \phi^n
L(T_\psi)=L$ in the Chabauty topology, and since $\phi$ is a positive automorphism, every such mixed leaf $\mathfrak l$ of $L$ must be a subsequential limit of mixed leaves of $\phi^n
L(T_\psi)$, which are labelled by words of the form $\phi^n\left((\rho_a)^{-1}\rho_c\right)$ or $\phi^n\left((\rho_{a^{-1}})^{-1} \rho_{b^{-1}}
\right)$. Finally note that the assumptions on $f$ imply that if $n_i,m_i\to\infty$ are two sequences of indices such that some mixed leaves of $\phi^{n_i}
L(T_\psi)$ labelled by words of the form $\phi^{n_i}\left((\rho_a)^{-1}\rho_c\right)$ converge to a mixed leaf ${\mathfrak l}_1$ of $L$ and that some leaves of $\phi^{m_i}
L(T_\psi)$ labelled by words of the form $\phi^{m_i}\left((\rho_a)^{-1}\rho_c\right)$ converge to a mixed leaf ${\mathfrak l}_2$ of $L$, then the leaves ${\mathfrak l}_1,{\mathfrak l}_2$ have the same label (up to a shift). The same holds when $(\rho_a)^{-1}\rho_c$ is replaced by $(\rho_{a^{-1}})^{-1} \rho_{b^{-1}}$. It follows that, up to the ${\mathbb{F}}$–action and the flip, there are at most 2 mixed leaves in $L$, and $(ii)$ is verified.
Finally, we observe $(iii)$. Let $r_a,r_b,r_c$ be the $f$–eigenrays in $R_3$ corresponding to the $f$–periodic directions $a,b,c$ at $v$. Thus $r_a,r_b,r_c$ are positive semi-infinite words. Then there exist special mixed leaves in $L(T_\phi)$ labelled by $(r_a)^{-1}r_b$, $(r_a)^{-1}r_c$, $(r_b)^{-1}r_c$. These leaves are distinct, up to the ${\mathbb{F}}$–action and the flip. Thus $(iii)$ is verified.
Now let $\phi^\infty=\lim_{n\to\infty} \phi^n \in\partial\Gamma$ and $\psi^\infty=\lim_{n\to\infty} \psi^n \in\partial\Gamma$. We know that the orbit map $\Gamma\to{{{\EuScript F}}}$ induces an embedding $\partial
\Gamma\to\partial {{{\EuScript F}}}$ which takes $\phi^\infty$ to $T_\phi$ and $\psi^\infty$ to $T_\psi$. We argue that $F\colon \partial \Gamma \to {\mathcal{L}}({\mathbb{F}})$ is not continuous by contradiction. Indeed suppose that $F$ is continuous. Then by $$\begin{aligned}
\lim_{n\to \infty} \phi^n L(T_\psi) &=
\lim_{n\to \infty} \phi^n \Lambda_{\psi^\infty} \\
& = \lim_{n\to \infty} \phi^n F(\psi^\infty) \\
& = \lim_{n\to \infty} F(\phi^n \psi^\infty)\\
&=F(\phi^\infty)=\Lambda_{\phi^\infty}= L(T_\phi),\end{aligned}$$ where convergence in ${\mathcal{L}}({\mathbb{F}})$ is with respect to the Chabauty topology. Together with $(ii)$, the fact that $\lim_{n\to\infty} \phi^n
L(T_\psi)=L(T_\phi)$ implies that, up to the ${\mathbb{F}}$–action and the flip, there are at most 2 distinct mixed leaves in $L(T_\phi)$. However, this contradicts $(iii)$. Thus $F$ is not continuous.
To complete the example, it only remains to give an example of automorphisms $\phi$ and $\psi$ satisfying conditions (1)–(4). The automorphism $\phi$ can be taken to be the automorphism $\alpha_3$ constructed by Jager and Lustig in [@jager2008free]. This automorphism is given by $f(a) =
abc$, $f(b) = bab$, and $f(c) = cabc$, and each of the required properties is verified by Jager and Lustig. For the automorphism $\psi$, we may take (a rotationless power of) the automorphism constructed by Pfaff in Example $3.2$ of [@pfaff2013out]. This is the automorphism $g(a) = cab$, $g(b) =ca$, and $g(c) =
acab$, and the required properties are established by Pfaff. This completes the example.
[JKLO]{}
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Department of Mathematics, Vanderbilt University\
1326 Stevenson Center, Nashville, TN 37240, U.S.A\
E-mail: [[email protected]]{}
Department of Mathematics, University of Illinois at Urbana-Champaign\
1409 W. Green Street, Urbana, IL 61801, U.S.A\
E-mail: [[email protected]]{}
Department of Mathematics, Yale University\
10 Hillhouse Ave, New Haven, CT 06520, U.S.A\
E-mail: [[email protected]]{}
|
---
abstract: 'In this note some properties of the sum of element orders of a finite abelian group are studied.'
author:
- 'Marius Tărnăuceanu, Dan Gregorian Fodor'
date: 'October 1, 2014'
title: |
**On the sum of element orders\
of finite abelian groups**
---
[**MSC (2010):**]{} Primary 20D60; Secondary 20D15.
[**Key words:**]{} finite groups, sum of element orders.
Introduction
============
Let $G$ be a finite group. We define the function $$\psi(G)=\dd\sum_{a\in G}o(a),$$where $o(a)$ denotes the order of $a\in G$. The starting point for our discussion is given by the papers [@1; @2] which investigate the minimum/maximum of $\psi$ on the groups of the same order.
Recall that the function $\psi$ is multiplicative, that is if $G_1$ and $G_2$ are two finite groups satisfying $\gcd(|G_1|,
|G_2|)=1$, then $\psi(G_1\hspace{-1mm}\times\hspace{-1mm}G_2)=\psi(G_1)\psi(G_2)$. By a standard induction argument, it follows that if $G_{i}$, $i=1,2,\dots,k$, are finite groups of coprime orders, then $$\psi(\xmare{i=1}k G_i)=\prod_{i=1}^{k} \psi(G_i)\,.$$This shows that the study of $\psi(G)$ for finite nilpotent groups $G$ can be reduced to $p$-groups.
In the current note we will focus on the restriction of $\psi$ to the class of finite abelian groups $G$. In this case we are able to give an explicit formula for $\psi(G)$. We prove that abelian $p$-groups of a fixed order are determined by this quantity and we conjecture that this happens for *arbitrary* finite abelian groups. Other interesting properties of the function $\psi$ will be also discussed.
Most of our notation is standard and will not be repeated here. Basic concepts and results on group theory can be found in [@3; @4]. For subgroup lattice notions we refer the reader to [@5].
Main results
============
As we have seen above, computing the sum of element orders of finite abelian groups is reduced to $p$-groups. For such a group $G$ we can determine $\psi(G)$ by using Corollary 4.4 of [@6].
[**Theorem 1.**]{} [*Let $G=\X{i=1}k\,\mathbb{Z}_{p^{\a_i}}$ be a finite abelian $p$-group, where $1\leq\a_1\leq\a_2\leq...\leq\a_k$. Then $$\psi(G)=1+\dd\sum_{\a=1}^{\a_k}\left(p^{2\a}f_{(\a_1,\a_2,...,\a_k)}(\a)-p^{2\a-1}f_{(\a_1,\a_2,...,\a_k)}(\a-1)\right),$$where $$f_{(\a_1,\a_2,...,\a_k)}(\a)=\left\{\barr{lll}
p^{(k-1)\a},&\mbox{ if }&0\le\a\le\a_1\vspace*{1,5mm}\\
p^{(k-2)\a+\a_1},&\mbox{ if }&\a_1\le\a\le\a_2\\
\vdots\\
p^{\a_1+\a_2+...+\a_{k-1}},&\mbox{ if
}&\hspace{-1mm}\a_{k-1}\le\a\,.\earr\right.$$*]{}
[**Remarks.**]{}
- The function $f_{(\a_1,\a_2,...,\a_k)}$ in Theorem 1 is increasing.
- $\psi(\X{i=1}k\,\mathbb{Z}_{p^{\a_i}})$ is a polynomial in $p$ of degree $2\a_k+\a_{k-1}+...+\a_1$.
- An alternative way to write $\psi(\X{i=1}k\,\mathbb{Z}_{p^{\a_i}})$ is $$\psi(\X{i=1}k\,\mathbb{Z}_{p^{\a_i}})=p^{2\a_k+\a_{k-1}+...+\a_1}-\left(p-1\right)\dd\sum_{\a=0}^{\a_k-1}p^{2\a}f_{(\a_1,\a_2,...,\a_k)}(\a).$$
Theorem 1 allows us to obtain a precise expression of $\psi(G)$ for some particular finite abelian $p$-groups $G$.
[**Corollary 2.**]{}
*We have:*
- $\psi(\mathbb{Z}_{p^n})=\dd\frac{p^{2n+1}+1}{p+1}$;
- $\psi(\mathbb{Z}_p^n)=p^{n+1}-p+1$;
- $\psi(\mathbb{Z}_{p^2}\hspace{-1mm}\times\hspace{-1mm}\mathbb{Z}_p^{n-2})=p^{n+2}-p^{n+1}+p^n-p+1$;
- $\psi(\mathbb{Z}_{p^{\a_1}}\hspace{-1mm}\times\hspace{-1mm}\mathbb{Z}_{p^{\a_2}})=\dd\frac{p^{2\a_2+\a_1+3}+p^{2\a_2+\a_1+2}+p^{2\a_2+\a_1+1}+p^{3\a_1+2}+p+1}{(p+1)(p^2+p+1)}\,$;
- $\psi(\mathbb{Z}_{p^{\a_1}}\hspace{-1mm}\times\hspace{-1mm}\mathbb{Z}_{p^{\a_2}}\hspace{-1mm}\times\hspace{-1mm}\mathbb{Z}_{p^{\a_3}})=\dd\frac{p^{2\a_3+\a_2+\a_1+1}+p^{3\a_2+\a_1+2}}{p+1}-\dd\frac{p^{3\a_2+\a_1+3}-p^{4\a_1+3}}{p^2+p+1}-\dd\frac{p^{4\a_1+4}-1}{p^3+p^2+p+1}\,$.
Given a positive integer $n$, it is well-known that there is a bijection between the set of types of abelian groups of order $p^n$ and the set $P_n=\{(x_1,x_2,...,x_n)\in\mathbb{N}^n\mid
x_1\geq x_2\geq...\geq x_n, x_1+x_2+...+x_n=n\}$ of partitions of $n$, namely the map $$\X{i=1}k\,\mathbb{Z}_{p^{\a_i}} (\mbox{ with }\a_1\leq\a_2\leq...\leq\a_k \mbox{ and }\sum_{i=1}^k \a_i=n)\longmapsto (\a_k,...,\a_1,\hspace{-2mm}\underbrace{0,...,0}_{n-k\,
\rm positions }\hspace{-2mm})\,.$$Moreover, recall that $P_n$ is totally ordered under the lexicographic order $\preceq$, where $$(x_1,x_2,...,x_n)\hspace{-1mm}\prec\hspace{-1mm}(y_1,y_2,...,y_n)\hspace{-1mm}\Longleftrightarrow\hspace{-1mm} \left\{\barr{lll}
x_1=y_1,...,x_m=y_m\\
\mbox{and}\\
x_{m+1}{<} y_{m+1} \mbox{ for some }
m\hspace{-1mm}\in\hspace{-1mm}\left\{0,1,...,n{-}1\right\}.\earr\right.$$Obviously, the lexicographic order induces a total order on the set of types of abelian $p$-groups of order $p^n$.
By computing the values of $\psi$ corresponding to all types of abelian $p$-groups of order $p^2$, $p^3$ and $p^4$, respectively, one obtains:
- $\psi(\mathbb{Z}_p^2)=p^3-p+1<\psi(\mathbb{Z}_{p^2})=p^4-p^3+p^2-p+1$;
- $\psi(\mathbb{Z}_p^3)=p^4-p+1<\psi(\mathbb{Z}_p\hspace{-1mm}\times\hspace{-1mm}\mathbb{Z}_{p^2})=p^5-p^4+p^3-p+1<\psi(\mathbb{Z}_{p^3})\hspace{-1mm}=p^6-p^5+p^4-p^3+p^2-p+1$;
- $\psi(\mathbb{Z}_p^4)=p^5-p+1<\psi(\mathbb{Z}_p\hspace{-1mm}\times\hspace{-1mm}\mathbb{Z}_p\hspace{-1mm}\times\hspace{-1mm}\mathbb{Z}_{p^2})=p^6-p^5+p^4-p+1<\psi(\mathbb{Z}_{p^2}\hspace{-1mm}\times\hspace{-1mm}\mathbb{Z}_{p^2})=p^6-p^4+p^3-p+1<\psi(\mathbb{Z}_p\hspace{-1mm}\times\hspace{-1mm}\mathbb{Z}_{p^3})\hspace{-1mm}=p^7-p^6+p^5-p^4+p^3-p+1<\psi(\mathbb{Z}_{p^4})=p^8-p^7+p^6-p^5+p^4-p^3+p^2-p+1$.
The above inequalities suggest us that the function $\psi$ is strictly increasing. This is true, as shows the following theorem.
[**Theorem 3.**]{} [*Let $G_1=\X{i=1}k\,\mathbb{Z}_{p^{\a_i}}$ and $G_2=\X{j=1}r\,\mathbb{Z}_{p^{\b_j}}$ be two finite abelian $p$-groups of order $p^n$. Then $$\psi(G_1)<\psi(G_2)\Longleftrightarrow(\a_k,...,\a_1,\hspace{-2mm}\underbrace{0,...,0}_{n-k\,
\rm positions
}\hspace{-2mm})\prec(\b_r,...,\b_1,\hspace{-2mm}\underbrace{0,...,0}_{n-r\,
\rm positions }\hspace{-2mm})\,.\0(*)$$*]{}
[**Proof.**]{} First of all, we remark that it suffices to prove $(*)$ only for consecutive partitions of $n$ because $P_n$ is fully ordered.
Assume that $(\a_k,...,\a_1,0,...,0)\prec(\b_r,...,\b_1,0,...,0)$. We have to prove $\psi(G_1)<\psi(G_2)$ (notice that this inequality holds for the first two elements of $P_n$, by b) and c) of Corollary 2). Let $s\in\{1,2,...,r-1\}$ such that $\b_1=\b_2=\cdots=\b_s<\b_{s+1}$. We distinguish the following two cases.
[**Case 1.**]{} $\b_1\geq 2$
Then $(\a_k,...,\a_1,0,...,0)$ is of type $(\b_r,...,\b_2,\b_1-1,1,0,...,0)$, i.e. $k=r+1$, $\a_1=1$, $\a_2=\b_1-1$ and $\a_i=\b_{i-1}$ for $i=3,4,...,r+1$. We infer that $f_{(\a_1,\a_2,...,\a_k)}(\gamma)=f_{(\b_1,\b_2,...,\b_r)}(\gamma),
\forall \hspace{1mm}\gamma\geq\b_1$. One obtains $$\psi(G_2)-\psi(G_1)=$$ $$=p^{\b_r+n}-\left(p-1\right)\dd\sum_{\gamma=0}^{\b_r-1}p^{2\gamma}f_{(\b_1,\b_2,...,\b_r)}(\gamma)-p^{\a_k+n}+\left(p-1\right)\dd\sum_{\gamma=0}^{\a_k-1}p^{2\gamma}f_{(\a_1,\a_2,...,\a_k)}(\gamma)=$$ $$\hspace{-42,5mm}=\left(p-1\right)\dd\sum_{\gamma=1}^{\b_1-1}p^{2\gamma}\left(f_{(\a_1,\a_2,...,\a_k)}(\gamma)-f_{(\b_1,\b_2,...,\b_r)}(\gamma)\right)=$$ $$\hspace{-61mm}=\left(p-1\right)\dd\sum_{\gamma=1}^{\b_1-1}p^{2\gamma}\left(p^{(r-1)\gamma+1}-p^{(r-1)\gamma}\right)>0.$$
[**Case 2.**]{} $\b_1=1$
Then $(\a_k,...,\a_1,0,...,0)$ is of type $(\b_r,...,\b_{s+1}-1,\b'_t,\b'_{t-1},...,\b'_1,0,...,0)$, where $\b_{s+1}-1\geq\b'_t\geq\b'_{t-1}\geq...\geq\b'_1\geq 1$ and $\b'_t+\b'_{t-1}+...+\b'_1=s+1$. We infer that $f_{(\a_1,\a_2,...,\a_k)}(\gamma)=f_{(\b_1,\b_2,...,\b_r)}(\gamma),
\forall \hspace{1mm}\gamma\geq\b_{s+1}$. So, we can suppose that $s=r-1$, i.e. $$(\a_k,...,\a_1,0,...,0)=(\b_{r}-1,\b'_t,\b'_{t-1},...,\b'_1,0,...,0).$$One obtains $$\psi(G_2)-\psi(G_1)=p^{\b_r+n}-\left(p-1\right)\dd\sum_{\gamma=0}^{\b_r-1}p^{2\gamma}f_{(\b_1,\b_2,...,\b_r)}(\gamma)-p^{\b_r+n-1}+S,$$where $$S=\left(p-1\right)\dd\sum_{\gamma=0}^{\b_r-2}p^{2\gamma}f_{(\a_1,\a_2,...,\a_k)}(\gamma)>
0.$$Since $$f_{(\b_1,\b_2,...,\b_r)}(\gamma)=\left\{\barr{lll}
p^{(r-1)\gamma},&\mbox{ if }&0\le\gamma\le 1\vspace*{1,5mm}\\
p^{r-1},&\mbox{ if }&1\le\gamma\,,\earr\right.$$it follows that $$\psi(G_2)-\psi(G_1)> p^{\b_r+n}-p^{\b_r+n-1}-\left(p-1\right)\dd\sum_{\gamma=0}^{\b_r-1}p^{2\gamma}f_{(\b_1,\b_2,...,\b_r)}(\gamma)=$$ $$\hspace{25mm}=p^{\b_r+n}-p^{\b_r+n-1}-\left(p-1\right)\left[1+p^{r-1}\dd\frac{p^{2\b_r}-p^2}{p^2-1}\right]=$$ $$\hspace{26,5mm}=\dd\frac{1}{p+1}\left[p^{\b_r+n-1}\left(p^2-p-1\right)+p^{r+1}-p^2+1\right]>0.$$
Conversely, assume that $\psi(G_1)<\psi(G_2)$, but $(\a_k,...,\a_1,0,...,0)\succeq(\b_r,...,\newline\b_1,0,...,0)$. Then the first part of the proof leads to $\psi(G_2)\leq\psi(G_1)$, a contradiction. Hence $(*)$ holds. $\scriptstyle\Box$
Two immediate consequences of Theorem 3 are the following.
[**Corollary 4.**]{} [*Let $n$ be a positive integer and $G$ be an abelian $p$-group of order $p^n$. Then the minimum value of $\psi(G)$ is obtained for $G$ elementary abelian, while the maximum value of $\psi(G)$ is obtained for $G$ cyclic.*]{}
[**Corollary 5.**]{} [*Two finite abelian $p$-groups of the same order are isomorphic if and only if they have the same sum of element orders.*]{}
Inspired by Corollary 5, we came up with the following conjecture, which we have verified by computer for all abelian groups of order less or equal to 100000.
[**Conjecture 6.**]{} [*Two finite abelian groups of the same order are isomorphic if and only if they have the same sum of element orders.*]{}
In order to decide if two finite abelian groups $G_1$ and $G_2$ are isomorphic by using the above results, the condition $\mid
G_1\hspace{-1mm}\mid\,=\mid G_2\hspace{-1mm}\mid$ is essential, as shows the following simple example.
[**Example.**]{} We have $\mathbb{Z}_2^2\not\cong\mathbb{Z}_3$ even if $\psi(\mathbb{Z}_2^2)=\psi(\mathbb{Z}_3)=7$.
This proves that the function $\psi$ is not injective, too. The surjectivity of $\psi$ also fails because $Im(\psi)$ contains only odd positive integers (notice that in fact more can be said, namely: $\psi(G)$ *is odd for all finite groups* $G$). Moreover, there exist odd positive integers not contained in $Im(\psi)$, as 5.
Finally, we observe that $\psi(G)$ is not divisible by $\mid\hspace{-1mm} G\hspace{-1mm}\mid$ for large classes of finite groups $G$, as $p$-groups, groups of order $p^nq$ ($p,q$ primes) without normal Sylow $q$-subgroups and groups of even order. More precisely, by MAGMA we checked that there are only three types of groups of order at most 2000 satisfying $\mid
G\hspace{-1mm}\mid\hspace{2mm}\mid \psi(G)$ (the smallest order of such a group $G$ is 105 and $\psi(G)=1785=105\cdot17$) and these are not abelian. Consequently, the study of this property for abelian groups seems to be interesting.
[**Theorem 7.**]{} [*There are finite abelian groups $G$ such that $$\psi(G)\equiv 0\hspace{1mm}({\rm mod} \mid
G\hspace{-1mm}\mid).$$*]{}
[**Proof.**]{} Let $G=\mathbb{Z}_{13}\hspace{-1mm}\times\hspace{-1mm}\mathbb{Z}_{13}\hspace{-1mm}\times\hspace{-1mm}\mathbb{Z}_{23}$. We have $\mid G\hspace{-1mm}\mid\hspace{1mm}=3887$ and $$\psi(G)=\psi(\mathbb{Z}_{13}\hspace{-1mm}\times\hspace{-1mm}\mathbb{Z}_{13})\psi(\mathbb{Z}_{23})=\left(13^3-13+1\right)\dd\frac{23^3+1}{23+1}=$$ $$\hspace{-34mm}=1107795=3887\cdot285,$$completing the proof. $\scriptstyle\Box$
We end our note by indicating a natural generalization of $\psi(G)$, which is obtained by replacing the orders of elements with the orders of elements relative to a certain subgroup of $G$.
[**Open problem.**]{} Let $G$ be a finite group. For every subgroup $H$ of $G$, we define the function $$\psi_H(G)=\dd\sum_{a\in G}o_H(a),$$where $o_H(a)$ denotes the order of $a\in G$ relative to $H$ (that is, the smallest positive integer $m$ such that $a^m\in
H$). Study the connections between $\psi(G)$ and the collection $(\psi_H(G))_{H\leq G}$, as well as the minimum/maximun of $\{\psi_H(G)\hspace{1mm}\mid\hspace{1mm} H\leq G, \mid
H\mid\,=n\}$, where $n\in \mathbb{N}^{*}$ is fixed.
[**Acknowledgements.**]{} The authors are grateful to the reviewers for their remarks which improve the previous version of the paper.
[00]{} H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs, [*Sums of element orders in finite groups*]{}, Comm. Algebra [**37**]{} (2009), 2978-2980. H. Amiri and S.M. Jafarian Amiri, [*Sum of element orders on finite groups of the same order*]{}, J. Algebra Appl. [**10**]{} (2011), 187-190. B. Huppert, [*Endliche Gruppen*]{}, I, Springer Verlag, Berlin, 1967. I.M. Isaacs, [*Finite group theory*]{}, Amer. Math. Soc., Providence, R.I., 2008. R. Schmidt, [*Subgroup lattices of groups*]{}, de Gruyter Expositions in Mathematics 14, de Gruyter, Berlin, 1994. M. Tărnăuceanu, [*An arithmetic method of counting the subgroups of a finite abelian group*]{}, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) [**53[/]{}101**]{} (2010), 373-386.
Marius Tărnăuceanu\
Faculty of Mathematics\
“Al.I. Cuza” University\
Iaşi, Romania\
e-mail: [[email protected]]{}
Dan Gregorian Fodor\
Faculty of Mathematics\
“Al.I. Cuza” University\
Iaşi, Romania\
e-mail: [[email protected]]{}
|
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abstract: 'In this paper we study the supremum functional $M_t = \sup_{0 \le s \le t} X_s$, where $X_t$, $t \ge 0$, is a one-dimensional L[é]{}vy process. Under very mild assumptions we provide a simple uniform estimate of the cumulative distribution function of $M_t$. In the symmetric case we find an integral representation of the Laplace transform of the distribution of $M_t$ if the L[é]{}vy-Khintchin exponent of the process increases on $(0,\infty)$.'
address:
- |
Institute of Mathematics and Computer Science\
Wroc[ł]{}aw University of Technology\
ul. Wybrze[ż]{}e Wyspia[ń]{}skiego 27\
50-370 Wroc[ł]{}aw, Poland\
\
\
- |
Institute of Mathematics\
Polish Academy of Sciences\
ul. [Ś]{}niadeckich 8\
00-976 Warszawa, Poland
- |
LAREMA\
Université d’Angers\
2 Bd Lavoisier\
49045 Angers cedex 1, France
author:
-
-
-
title: 'Suprema of L[é]{}vy processes'
---
,
Introduction {#sec:intro}
============
By a classical reflection argument, the supremum functional $M_t = \sup_{0 \le s \le t} X_s$ of the Brownian motion $X_t$ has truncated normal distribution, ${\mathbf{P}}(M_t \ge x) = 2 {\mathbf{P}}(X_t \ge x)$ ($x \ge 0$). A similar question for symmetric $\alpha$-stable processes was first studied by Darling [@bib:d56], and the case of general L[é]{}vy processes $X_t$ was addressed by Baxter and Donsker [@bib:bd57]. Theorem 1 therein gives a formula for the double Laplace transform of the distribution of $M_t$, which for a *symmetric* L[é]{}vy process $X_t$ with L[é]{}vy-Khintchin exponent $\Psi(\xi)$ reads [ ]{} Inversion of the double Laplace transform is typically a very difficult task. Apart from the Brownian motion case, an explicit formula for the distribution of $M_t$ was found for the Cauchy process (the symmetric $1$-stable process) by Darling [@bib:d56], for a compound Poisson process with $\Psi(\xi) = 1 - \cos \xi$ by Baxter and Donsker [@bib:bd57] and for the Poisson process with drift by Pyke [@bib:p59].
The development of the fluctuation theory for L[é]{}vy processes resulted in many new identities involving the supremum functional $M_t$, see, for example, [@bib:b96; @bib:d07; @bib:k06; @bib:s99]. There are numerous other representations for the distribution of $M_t$, at least in the stable case, see [@bib:bdp08; @bib:b73; @bib:d56; @bib:d87; @bib:gj09; @bib:gj10; @bib:hk11; @bib:hk09; @bib:k10a; @bib:k10; @bib:mk10; @bib:z64]. The main goal of this article is to give a more explicit formula for ${\mathbf{P}}(M_t < x)$ and simple sharp bounds for ${\mathbf{P}}(M_t < x)$ in terms of the L[é]{}vy-Khintchin exponent $\Psi(\xi)$ for a class of L[é]{}vy processes. Most estimates of the cumulative distribution function of $M_t$ are proved for very general L[é]{}vy processes, without symmetry assumptions.
Let $\tau_x$ denote the first passage time through a barrier at the level $x$ for the process $X_t$, [ ]{} with the infimum understood to be infinity when the set is empty. We always assume that $X_0 = 0$. Since ${\mathbf{P}}(M_t < x) = {\mathbf{P}}(\tau_x > t)$, the problems of finding the cumulative distribution functions of $M_t$ and $\tau_x$ are the same. The supremum functional and first passage time statistics are important in various areas of applied probability ([@bib:ak05; @bib:bnmr01]), as well as in mathematical physics ([@bib:ka08; @bib:kkm07]). The recent progress in the potential theory of Lévy processes is, in part, due to the application of fluctuation theory, see [@bib:bbkrsv09; @bib:cks11; @bib:gr08; @bib:ksv9; @bib:ksv10a; @bib:ksv10; @bib:ksv11].
The paper is organized as follows. Section \[sec:pre\] contains some preliminary material related to Bernstein functions, Stieltjes functions and estimates for the Laplace transform. In Section \[sec:sup\] (Theorem \[th:supest\] and Corollary \[cor:kappa:regular1\]) we prove, under mild assumptions, the estimate [ ]{} where $V(x)$ and $\kappa(z, 0)$ are the renewal function for the ascending ladder-height process, and the Laplace exponent of the the ascending ladder-time process corresponding to $X_t$, respectively. Here $f(x) \approx g(x)$ means that there are constants $c_1, c_2 > 0$ such that $c_1 g(x) \le f(x) \le c_2 g(x)$. In Section \[sec:v\] we show that in the symmetric case, given some regularity of $\Psi(\xi)$, we have [ ]{} see Theorem \[th:vest\]. Therefore the estimate of the above cumulative distribution function of $M_t$ takes a very explicit form [ ]{} The other main result of Section \[sec:v\] is an explicit formula for the (single, in the space variable) Laplace transform of the distribution of $M_t$ (Theorem \[th:suplaplace\]), under the assumption that $X_t$ is symmetric and $\Psi(\xi)$ is increasing on $[0, \infty)$.
When $\Psi(\xi) = \psi(\xi^2)$ for a complete Bernstein function $\psi(\xi)$, the above resuls can be significantly improved. Following the approach of [@bib:mk10], a (rather complicated) explicit formula for ${\mathbf{P}}(M_t < x)$ can be given, and estimates and asymptotic formulae for ${\mathbf{P}}(M_t < x)$ extend to $(d / dt)^n {\mathbf{P}}(M_t < x)$ when $x$ is small or $t$ is large. These results will be covered in a forthcoming paper.
We denote by $C$, $C_1$, $C_2$ etc. constants in theorems, and by $c$, $c_1$, $c_2$ etc. temporary constants in proofs. Any dependence of a constant on some parameters is always indicated by writing, for example, $c(n, {\varepsilon})$. We write $f(x) \sim g(x)$ when $f(x) / g(x) \to 1$. We use the terms *increasing*, *decreasing*, *concave*, *convex function* etc. in the weak sense.
Preliminaries {#sec:pre}
=============
Complete Bernstein and Stieltjes functions {#sec:cbf}
------------------------------------------
A function $\psi(\xi)$ is said to be a *complete Bernstein function* (CBF) if [ ]{} where $c_1, c_2 \ge 0$, and $\mu$ is a measure on $(0, \infty)$ such that the integral $\int_0^\infty \min(\zeta^{-1}, \zeta^{-2}) \mu(d\zeta)$ is finite. A function $\tilde{\psi}(\xi)$ is said to be a *Stieltjes functions* if [ ]{} for some $\tilde{c}_1, \tilde{c}_2 \ge 0$ and some measure $\tilde{\mu}$ on $(0, \infty)$ such that the integral $\int_0^\infty \min(1, \zeta^{-1}) \tilde{\mu}(d\zeta)$ is finite. See [@bib:ssv10] for a general account on complete Bernstein functions, Stieltjes functions and related notions.
It is known that $\psi(\xi)$ is a CBF if and only if $\psi(\xi)$ is nonnegative and increasing on $(0, \infty)$, holomorphic in ${\mathbf{C}}\setminus (-\infty, 0]$, and $\operatorname{Im}\psi(\xi) > 0$ when $\operatorname{Im}\xi > 0$. Furthermore, if $\psi(\xi)$ is a CBF, then $\xi / \psi(\xi)$ is a CBF, and $1 / \psi(\xi)$ and $\psi(\xi) / \xi$ are Stieltjes functions
The function $\tilde{\psi}(\xi)$ given by is the Laplace transform of $\tilde{c}_2 \delta_0(dx) + (\tilde{c}_1 + {\mathcal{L}}\tilde{\mu}(x)) dx$ ([@bib:ssv10], Theorem 2.2). Furthermore, $\pi \tilde{c}_1 \delta_0(d\zeta) + \tilde{\mu}(d\zeta)$ is the limit of measures $-\operatorname{Im}(\tilde{\psi}(-\zeta + i {\varepsilon})) d\zeta$ as ${\varepsilon}\to 0^+$ ([@bib:ssv10], Corollary 6.3 and Comments 6.12), so, in a sense, it is the boundary value of $\tilde{\psi}$. Therefore, we use a shorthand notation $-\operatorname{Im}(\tilde{\psi}^+(-\zeta)) d\zeta$ for $\tilde{\mu}(d\zeta)$. Furthermore, we have $\tilde{c}_1 = \lim_{\xi \to 0} (\xi \tilde{\psi}(\xi))$ and $\tilde{c}_2 = \lim_{\xi \to \infty} \tilde{\psi}(\xi)$.
Following [@bib:mk10], we define [ ]{} for any function $\psi(\xi)$ such that $\min(1, \zeta^{-2}) \log \psi(\zeta^2)$ is integrable in $\zeta > 0$. By a simple substitution, [ ]{} By [@bib:mk10], Lemma 4, if $\psi(\xi)$ is a CBF, then also $\psi^\dagger(\xi)$ is a CBF (this was independently proved in [@bib:ksv10], Proposition 2.4), and [ ]{}
\[prop:daggerest\] If $\psi(\xi)$ is nonnegative on $(0, \infty)$ and both $\psi(\xi)$ and $\xi / \psi(\xi)$ are increasing on $(0, \infty)$, then [ ]{} where ${\mathcal{C}}\approx 0.916$ is the Catalan constant. Note that $e^{2 {\mathcal{C}}/ \pi} \le 2$.
If, in addition, $\psi(\xi)$ is regularly varying at $\infty$, then [ ]{} An analogous statement for $\xi \to 0$ holds for $\psi(\xi)$ regularly varying at $0$.
In particular, holds for any CBF. Likewise, holds for any regularly varying CBF.
A result similar to was obtained independently in [@bib:ksv11], Proposition 3.7, while for CBFs was derived in [@bib:ksv9], Proposition 2.2.
By the assumptions, we have [ ]{} It follows that [ ]{} The lower bound is obtained in a similar manner.
The second statement of the proposition is proved in a very similar manner to Lemma 15 in [@bib:mk10]. Define an auxiliary function $h(\xi, \zeta) = \psi(\xi^2 \zeta^2) / \psi(\xi^2)$. By we have $|\log h(\xi, \zeta)| \le 2 |\log \zeta|$, $\xi, \zeta > 0$. Since $\psi$ is regularly varying at infinity, for some $\alpha$, $\lim_{\xi \to \infty} h(\xi, \zeta) = \zeta^{2 \alpha}$ for each $\zeta > 0$. Hence, by dominated convergence, [ ]{} It follows that [ ]{} and so finally $\lim_{\xi \to \infty} \psi^\dagger(\xi) / \sqrt{\psi(\xi^2)} = 1$, as desired. Regular variation at $0$ is proved in a similar way.
As in [@bib:mk10], for differentiable functions $\psi(\xi)$ with positive derivative we define [ ]{} This definition is extended continuously by $\psi_\lambda(\lambda^2) = \psi(\lambda^2) / (\lambda^2 \psi'(\lambda^2))$. Note that if $\psi(0) = 0$, then $\psi_\lambda(0) = 1$. For simplicity, we denote $\psi_\lambda^\dagger(\xi) = (\psi_\lambda)^\dagger(\xi)$. By [@bib:mk10], Lemma 2, if $\psi(\xi)$ is a CBF, then $\psi_\lambda(\xi)$ is a CBF for any $\lambda > 0$.
Estimates for the Laplace transform {#sec:lap}
-----------------------------------
This short section contains some rather standard estimates for the inverse Laplace transform.
\[prop:lb\] Let $a > 0$, $c \ge 1$. If $f$ is nonnegative and $f(x) \le c f(a) \max(1, x/a)$ , then for any $\xi > 0$, [ ]{}
We have [ ]{} as desired.
\[prop:ub\] If $f$ is nonnegative and increasing, then for $a, \xi > 0$, [ ]{}
As before, [ ]{} as claimed.
\[prop:ub3\] If $f$ is nonnegative and decreasing, then for $a, \xi > 0$, [ ]{}
Again, [ ]{} as claimed.
Suprema of general L[é]{}vy processes {#sec:sup}
=====================================
We briefly recall the basic notions of the fluctuation theory for L[é]{}vy processes. Let $L_t$ be the local time of the process $X_t$ reflected at its supremum $M_t$, and denote by $L^{-1}_s$ the right-continuous inverse of $L_t$, the ascending ladder-time process for $X_t$. This is a (possibly killed) subordinator, and $H_s = X(L^{-1}_s) = M(L^{-1}_s)$ is another (possibly killed) subordinator, called the ascending ladder-height process. The Laplace exponent of the ascending ladder process, that is, the (possibly killed) bivariate subordinator $(L^{-1}_s, H_s)$ ($s < L(\infty)$), is denoted by $\kappa(z, \xi)$. By, e.g., [@bib:b96], Corollary VI.10, [ ]{} where $c$ is a normalization constant of the local time. Since our results are not affected by the choice of $c$, we assume that $c = 1$. We note that $\kappa(z, 0)$ is a Bernstein function of $z$, and also $z / \kappa(z, 0)$ is a Bernstein function (this follows from by Frullani’s integral; see [@bib:b96], formula (VI.3) for the case when $X_t$ is not a compound Poisson process). For more account on the fluctuation theory we refer the reader to [@bib:b96; @bib:d07; @bib:k06]. In general, there is no closed-form formula for $\kappa(z, \xi)$. For a list of special cases, see [@bib:kkp10] and the references therein. For a symmetric process which is not a compound Poisson process, we have $\kappa(z,0) = \sqrt{z}$.
As usual, $\tau_x$ denotes the first passage time through a barrier at $x \ge 0$ for $X_t$ (or for $M_t$). Following [@bib:b96], for $x, z \ge 0$ we define [ ]{} For $z = 0$, we simply have $V^0(x) = \int_0^{\infty}{\mathbf{P}}(H_s < x)ds$, so that $V^0(x) = V(x)$ is the renewal function of the process $H_s$, studied in more detail for symmetric Lévy processes in Section \[sec:v\]. By [@bib:b96], formula (VI.8), [ ]{} (Note that in [@bib:b96] a weak inequality $M_t \le x$ is used in the definition of $V^z(x)$.) Hence, for a symmetric process $X_t$ which is not a compound Poisson process, we have [ ]{} This is a partial inverse of the double Laplace transform in ; however, there is no known explicit formula for $V^z(x)$. For a different and, in a sense, more explicit partial inverse, see below.
By [@bib:b96], Section VI.4, the Laplace transform of $V^z(x)$ is $1 / (\xi \kappa(z, \xi))$. Hence, when $X_t$ is symmetric and it is not a compound Poisson process, the right hand side of the Baxter-Donsker formula can be written as $\sqrt{z} / (z \kappa(z, \xi))$ (see [@bib:f74], Corollary 9.7).
\[th:supest\] Let $X_t$ be a L[é]{}vy process, $M_t = \sup_{0 \le s \le t} X_s$, and let $\kappa(z, \xi)$ be the bivariate Laplace exponent of its ascending ladder process. Suppose that [ ]{} and that $\kappa(z, 0) / z$ is unbounded (near $0$). For $t, x > 0$, we have [ ]{} Here [ ]{} where $z \in (0, 1/t)$ solves [ ]{}
The upper bound in is a direct consequence of and Proposition \[prop:ub3\] with $\xi = 1/t$.
Following [@bib:b96], Lemma VI.21, we find a lower bound for $V^z(x)$. We have [ ]{} which implies [ ]{} Let $\sigma_z = \inf \{ t \ge 1/z: X_t = M_t \} = L^{-1}(L_{1/z})$; $\sigma_z$ is a stopping time. Since the support of the measure $dL_t$ is contained in the set $\{t : X_t = M_t\}$ of zeros of the reflected process, we have [ ]{} Next, observe that $M_{\sigma_z} = X_{\sigma_z}$, so that [ ]{} Hence, [ ]{} Since $\sigma_z \ge 1 / z$, by the strong Markov property, [ ]{} which, by , yields [ ]{} Let $k > 0$. By and the already proved upper bound of , [ ]{} The last two estimates give [ ]{} Fix ${\varepsilon}\in (0, 1)$ (later we choose ${\varepsilon}= 1/4$). Note that the function $\kappa(z, 0) / z$ is continuous, decreasing and unbounded. Hence, it maps the interval $(0, 1/t]$ onto the interval $[t\kappa(1/t, 0), \infty)$. Furthermore, $\kappa(z, 0)$ is increasing, so that $K(z) \ge \kappa(z, 0) / z$. In particular, $\frac{e^2}{{\varepsilon}(e - 1)} \, K(1/t)>K(1/t) \ge t\kappa(1/t, 0) $. It follows that we can choose $z = z(t)<1 / t$ such that [ ]{} Setting $k = z t < 1$, the above equality can be rewritten as [ ]{} Suppose now that $V(x) \kappa(z, 0) \le {\varepsilon}(e - 1) / e$. Then, by the upper bound of , we have ${\mathbf{P}}(M_{1/z} \ge x) = 1 - {\mathbf{P}}(M_{1/z} < x) \ge 1 - {\varepsilon}$. This, and give [ ]{} This estimate holds for $t \ge t_0$, where $V(x) \kappa(z(t_0), 0) = {\varepsilon}(e - 1) / e$ (here we use continuity of $\kappa(z(t), 0)$ as a function of $t$). Hence, by monotonicity of ${\mathbf{P}}(M_t < x)$ in $t$, [ ]{} The lower bound in follows by taking ${\varepsilon}= 1/4$ and using the inequality $\kappa(z, 0) = \kappa(k/t, 0) \ge k \kappa(1/t, 0)$.
To formulate the next result we define *upper scaling conditions*: [ ]{} Observe that the condition implies that for any $z^* > 0$ there is $c^*$ such that [ ]{}
\[cor:kappa:regular1\] Let $X_t$ be a L[é]{}vy process, $M_t = \sup_{0 \le s \le t} X_s$, and let $\kappa(z, \xi)$ be the bivariate Laplace exponent of its ascending ladder process. If $\kappa(z, 0)$ satisfies condition with $0 < {\varrho}< 1$ and the integral $\int_1^\infty \kappa(z, 0) z^{-2} dz$ is finite then [ ]{} for every $x > 0$ and $t \ge 1$. If $\kappa(z, 0)$ satisfies with $0 < {\varrho}< 1$ and $\lim_{z \to 0} z / \kappa(z ,0) = 0$ then holds for $x > 0$ and $t \le 1$.
In particular, if $\kappa(z, 0)$ satisfies both and , that is, there are $c > 0$ and ${\varrho}\in (0, 1)$ such that $\kappa(\lambda z, 0) \le c \lambda^{\varrho}\kappa(z, 0)$ for $\lambda \ge 1$ and $z > 0$, then is true for every $x > 0$ and $t > 0$.
We begin with the first part of the statement. By the condition , [ ]{} In particular, $\kappa(s, 0) / s$ is unbounded. Furthermore, using also finiteness of the integral $\int_1^\infty \kappa(z, 0) z^{-2} dz$, we obtain [ ]{} This implies that the assumptions of Theorem \[th:supest\] are satisfied.
Let $t \ge 1$ and define $z = z(t) \in (0, 1/t)$ as in Theorem \[th:supest\]. By the condition we have [ ]{} By definition of $z$ and (with $s = 1 / t$), we have [ ]{} which gives $z t \ge c_4(\kappa)$. Hence, the constant $C_2$ in Theorem \[th:supest\] satisfies $C_2 = z t / (2 e) \ge c_4(\kappa) / (2 e)$. This ends the proof of the first part.
The second part can be justified in a similar way, since the condition implies that [ ]{} Moreover, for $t < 1$ and $z = z(t)$ selected according to Theorem \[th:supest\] we have $z(1) \le z(t) < 1/t$. Applying (with $z^* = z(1)$) we obtain [ ]{} Finally, the last statement is a direct consequence of the previous ones.
Due to Potter’s theorem ([@bib:bgt87], Theorem 1.5.6) the condition is implied by regular variation of $\kappa(z, 0)$ at zero with index $0 < {\varrho}^* <1$. Likewise, the condition is implied by regular variation of $\kappa(z, 0)$ at $\infty$ with index $0 < {\varrho}^* <1$.
In the second part of the above corollary the assumption $\lim_{z \to 0} z / \kappa(z ,0) = 0$ can be removed at the expence that the lower bound holds for $t \le t_0$, where $t_0 =t_0(\kappa)$ is sufficiently small. This is due to the fact that since $\lim_{t \searrow 0} K(1/t) = 0$, $z = z(t)$ in Theorem \[th:supest\] is well defined for $t$ small enough.
By the results of [@bib:b96], Theorem VI.14, and [@bib:bd97], the regular variation of order ${\varrho}\in (0, 1)$ of $\kappa(z, 0)$ at $0$ or at $\infty$ is equivalent to the existence of the limit of ${\mathbf{P}}(X_t \ge 0)$ as $t \to \infty$ or $t \to 0^+$, respectively. Hence, Corollary \[cor:kappa:regular1\] implies the following result.
\[cor:kappa:regular2\] Let $X_t$ be a L[é]{}vy process and $M_t = \sup_{0 \le s \le t} X_s$. If [ ]{} then holds for $x > 0$ and $t \ge 1$. If [ ]{} then is true for $x>0$ and $t \le 1$. Finally, if [ ]{} then holds for every $x>0$ and $t>0$.
We only need to verify that $\kappa(z, 0) / z^2$ is integrable at infinity, and that $\lim_{z \to 0^+} (z / \kappa(z, 0)) = 0$. In each of the cases, there is ${\varepsilon}> 0$ such that ${\mathbf{P}}(X_t \ge 0) \le 1 - {\varepsilon}$ for all $t > 0$. Therefore, by and the Frullani integral, $\kappa(z, 0) \le z^{1 - {\varepsilon}}$ for $z \ge 1$, and $\kappa(z, 0) \ge z^{1 - {\varepsilon}}$ when $0 < z < 1$. The result follows.
The uniform estimates of Corollary \[cor:kappa:regular2\] complement the existing results from [@bib:gn86] about the asymptotic behavior of ${\mathbf{P}}(M_t < x)$, where it was shown that [ ]{} under the assumption that $\kappa(z,0)$ is regularly varying at zero with index ${\varrho}\in (0, 1)$.
Suprema of symmetric L[é]{}vy processes {#sec:v}
=======================================
In this section we assume that $X_t$ is a *symmetric* L[é]{}vy process with L[é]{}vy-Khintchin exponent $\Psi(\xi)$. In a rather general setting, we can invert the Laplace transform in time variable in .
\[th:suplaplace\] Suppose that $X_t$ is a symmetric L[é]{}vy process with L[é]{}vy-Khintchin exponent $\Psi(\xi)$. Suppose that $\Psi(\xi)$ is increasing in $\xi > 0$. If $M_t = \sup_{0 \le s \le t} X_s$, then [ ]{}
Since ${\mathbf{P}}(M_t < x) = {\mathbf{P}}(\tau_x > t)$, the following integrated form of is sometimes more convenient.
\[cor:suplaplace\] With the notation and assumptions of Theorem \[th:suplaplace\], [ ]{}
Let $\psi(\xi) = \Psi(\sqrt{\xi})$ for $\xi > 0$. For any $z \in {\mathbf{C}}\setminus (-\infty, 0]$ and $\xi > 0$, we define (see and ) [ ]{} For any $\xi > 0$, the function ${\varphi}(\xi, z)$ is positive and increasing in $z \in (0, \infty)$. As $z \to 0$ or $z \to \infty$, ${\varphi}(\xi, z)$ converges to $0$ and $1$, respectively. Furthermore, if $\operatorname{Im}z > 0$, then $\arg (1 + \psi(\zeta^2) / z) \in (-\pi, 0)$ for all $\zeta > 0$, and therefore [ ]{} Hence, for any $\xi > 0$, ${\varphi}(\xi, z)$ (and even $({\varphi}(\xi, z))^2$) is a complete Bernstein function of $z$. Note that the continuous boundary limit ${\varphi}^+(\xi, -z)$ exists for $z > 0$: if $z = \psi(\lambda^2)$, or $\lambda = \sqrt{\psi^{-1}(z)}$, then [ ]{} see for the notation. Here $\log^-$ denotes the boundary limit on $(-\infty, 0)$ approached from below, $\log^-(-\zeta) = -i \pi/2 + \log \zeta$ for $\zeta > 0$. The function $\log |1 - \zeta^2 / \lambda^2|$ is harmonic in the upper half-plane $\operatorname{Im}\zeta > 0$, so that [ ]{} Furthermore, $\exp(i \arctan(\xi/\lambda)) = (\lambda + i \xi) / \sqrt{\lambda^2 + \xi^2}$. Therefore, with $z = \psi(\lambda^2)$, [ ]{} see for the notation. Note that if $\psi(\xi)$ is bounded on $(0, \infty)$ and $z \ge \sup_{\xi > 0} \psi(\xi)$, then ${\varphi}^+(\xi, -z)$ is real.
By , ${\varphi}(\xi, z) / z$ is the double Laplace transform of the distribution of $M_t$. But for all $\xi > 0$, ${\varphi}(\xi, z) / z$ is a Stieltjes function of $z$. Therefore, by , [ ]{} Note that the second equality holds true also when $\psi(\xi)$ is bounded. Since $1 / (z + \psi(\lambda^2)) = \int_0^\infty e^{-t \psi(\lambda^2)} e^{-z t} dt$, we have [ ]{} The theorem follows by the uniqueness of the Laplace transform.
Let $V(x) = V^0(x)$ be the renewal function for the ascending ladder-height process $H_s$ corresponding to $X_t$; see Section \[sec:sup\] for the definition. When $X_t$ satisfies the *absolute continuity condition* (for example, if $1 / (1 + \Psi(\xi))$ is integrable in $\xi$), then $V(x)$ is the (unique up to a multiplicative constant) increasing harmonic function for $X_t$ on $(0, \infty)$, and $V'(x)$ is the decreasing harmonic function for $X_t$ on $(0, \infty)$, cf. [@bib:s80]. It is known ([@bib:b96], formula (VI.6))) that for $\xi > 0$, [ ]{} Moreover, if $X_t$ is not a compound Poisson process, then by [@bib:f74], Corollary 9.7, [ ]{} where $\Psi(\xi) = \psi(\xi^2)$, see for the notation. Clearly, we have ${\mathcal{L}}V'(\xi) = \xi {\mathcal{L}}V(\xi) = 1 / \psi^\dagger(\xi)$; here $V'$ is the distributional derivative of $V$ on $[0, \infty)$. We remark that when $X_t$ is a compound Poisson process, then, also by [@bib:f74], Corollary 9.7, [ ]{} For simplicity, we state the next three results only for the case when $X_t$ is not a compound Poisson process. However, extensions for compound Poisson processes are straightforward due to .
As an immediate consequence of Proposition \[prop:daggerest\] and Karamata’s Tauberian theorem ([@bib:bgt87], Theorem 1.7.1), we obtain the following result, which in the case of complete Bernstein functions was derived in Proposition 2.7 of [@bib:ksv9].
\[prop:vregular\] Let $\Psi(\xi)$ be the L[é]{}vy-Khintchin exponent of a symmetric L[é]{}vy process $X_t$, which is not a compound Poisson process, and suppose that both $\Psi(\xi)$ and $\xi^2 / \Psi(\xi)$ are increasing in $\xi > 0$. If $\Psi(\xi)$ is regularly varying at $\infty$, then $V$ is regularly varying at $0$ and $\Gamma(1 + \alpha) V(x) \sim 1 / \sqrt{\Psi(1/x)}$ as $x \to 0$. Similarly, if $\Psi(\xi)$ is regularly varying at $0$, then $\Gamma(1 + \alpha) V(x) \sim 1 /\sqrt{\Psi(1/x)}$ as $x \to \infty$.
Another consequence of Proposition \[prop:daggerest\] is a uniform estimate of the renewal function (see also Proposition 3.9 of [@bib:ksv11]).
\[th:vest\] Let $\Psi(\xi)$ be the L[é]{}vy-Khintchin exponent of a symmetric L[é]{}vy process $X_t$, which is not a compound Poisson process, and suppose that both $\Psi(\xi)$ and $\xi^2 / \Psi(\xi)$ are increasing in $\xi > 0$. Then [ ]{}
Let $\psi(\xi) = \Psi(\sqrt{\xi})$ for $\xi > 0$. By Proposition \[prop:daggerest\], we obtain $e^{-2 {\mathcal{C}}/ \pi} / \sqrt{\xi^2 \psi(\xi^2)} \le {\mathcal{L}}{V}(\xi) \le e^{2 {\mathcal{C}}/ \pi} / \sqrt{\xi^2 \psi(\xi^2)}$, $\xi > 0$. Since $V$ is increasing, Proposition \[prop:ub\] gives [ ]{} Furthermore, using subadditivity and monotonicity of $V$ (see [@bib:b96], Section III.1), for $x = k a + r$ ($k \ge 0$, $r \in [0, a)$) we obtain $V(x) \le k V(a) + V(r) \le (k + 1) V(a)$. It follows that $V(x) \le 2 V(a) \max(1, x / a)$ for all $a, x > 0$, and so, by Proposition \[prop:lb\], [ ]{} as desired.
We remark that when $V$ is a concave function on $(0, \infty)$ (for example, when $\psi$ is a complete Bernstein function, see below), then clearly $V(x) \le \max(1, x/a) V(a)$, so that the lower bound in Theorem \[th:vest\] holds with constant $2/5$ instead of $1/5$.
If $\psi(\xi)$ is a complete Bernstein function (CBF, see ), then $\psi^\dagger(\xi)$ and $\xi / \psi^\dagger(\xi)$ are CBFs, and hence $1 / \psi^\dagger(\xi)$ is a Stieltjes function (see ). Therefore, $V'(x)$ is a completely monotone function on $(0, \infty)$, and $V(x)$ is a Bernstein function (see [@bib:ssv10] for the relation between completely monotone, Bernstein, complete Bernstein and Stieltjes functions). More precisely, we have the following result.
\[prop:v\] Let $\Psi(\xi)$ be the L[é]{}vy-Khintchin exponent of a symmetric L[é]{}vy process $X_t$, which is not a compound Poisson process, and suppose that $\Psi(\xi) = \psi(\xi^2)$ for a complete Bernstein function $\psi$. Then $V$ is a Bernstein function, and [ ]{} where $b = \lim_{\xi \to 0^+} (\xi / \sqrt{\psi(\xi^2)})$.
As explained after formula , the expression $\operatorname{Im}(-1 / \psi^+(-\xi^2)) d\xi$ in and should be understood in the distributional sense, as a weak limit of measures $\operatorname{Im}(-1 / \psi(-\xi^2 + i {\varepsilon})) d\xi$ on $(0, \infty)$ as ${\varepsilon}\to 0^+$. The measure $\operatorname{Im}(-1 / \psi^+(-\xi)) d\xi$ has an atom of mass $\pi b$ at $0$, and this atom is not included in the integrals from $0^+$ to $\infty$ in and .
Since $1 / \psi^\dagger(\xi)$ is a Stieltjes function, it has the form , [ ]{} where, using , [ ]{} and [ ]{} Using Proposition \[prop:daggerest\], we can express $a$ and $b$ in terms of $\psi$. Since $\psi$ is unbounded, also $\psi^\dagger$ is unbounded (by ), and so in fact $a = 0$. In a similar way, if $\xi / \psi(\xi)$ converges to $0$ as $\xi \to 0^+$, then gives $\xi / \psi^\dagger(\xi) \to 0$, so that $b = 0$. When the limit of $\xi / \psi(\xi)$ is positive (since $\xi / \psi(\xi)$ is a CBF, the limit always exists), then $\psi$ is regularly varying at $0$, and so $b = \lim_{\xi \to 0^+} (\xi / \sqrt{\psi(\xi^2)})$, as desired. By the uniqueness of the Laplace transform, [ ]{} The result follows by integration in $x$.
Note that for a compound Poisson process, we have $a > 0$, so there is an extra positive constant in .
As a combination of Theorem \[th:supest\] and Theorem \[th:vest\], we obtain the following result.
\[th:est:sym\] Let $\Psi(\xi)$ be the L[é]{}vy-Khintchin exponent of a symmetric L[é]{}vy process $X_t$. Suppose that both $\Psi(\xi)$ and $\xi^2 / \Psi(\xi)$ are increasing in $\xi > 0$. If $M_t = \sup_{0 \le s \le t} X_s$, then for all $t, x > 0$, [ ]{}
When $X_t$ is not a compound Poisson process, then the result follows from Theorems \[th:supest\] and \[th:vest\], and from $\kappa(z, 0) = \sqrt{z}$. Suppose that $X_t$ is a compound Poisson process. For ${\varepsilon}> 0$ consider $X^{\varepsilon}_t = {\varepsilon}B_t + X_t$, where the Brownian motion $B_t$ is independent of $X_t$. Then the L[é]{}vy-Khintchin exponent of $X^{\varepsilon}_t$ equals to $\Psi_{\varepsilon}(\xi)=({\varepsilon}\xi)^2 + \Psi(\xi)$. It is easy to check that $\xi^2 / \Psi_{\varepsilon}(\xi)$ is increasing. Moreover, $M^{\varepsilon}_t$ converges in distribution to $M_t$ as ${\varepsilon}\to 0$. The result follows by the continuity of $\Psi(\xi)$.
Clearly, the condition ‘$\Psi(\xi)$ and $\xi^2 / \Psi(\xi)$ are increasing in $\xi > 0$’ in Theorem \[th:vest\], Proposition \[prop:vregular\] and Theorem \[th:est:sym\] can be replaced with [ ]{} If $\Psi(\xi) = \psi(\xi^2)$, then reads [ ]{} Using the standard representation of Bernstein functions, it is easy to check that any Bernstein function $\psi(\xi)$ (not necessarily a complete one) satisfies . Hence, Theorem \[th:est:sym\] applies to any *subordinate Brownian motion*: a process $X_t = B_{\eta_t}$, where $B(s)$ is the standard Brownian motion (with ${\mathbf{E}}(B_s) = 0$ and $\operatorname{Var}(B_s) = 2 s$), $\eta_t$ is a subordinator (with ${\mathbf{E}}(e^{-\xi \eta_t}) = e^{-t \psi(\xi)}$), and $B_s$ and $\eta_t$ are independent processes.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are deeply indebted to Lo[ï]{}c Chaumont for numerous discussions on the subject of the article and many valuable suggestions. We thank Tomasz Grzywny for pointing out errors in the preliminary version of the article. We also thank the anonymous referees for helpful comments, and in particular for indicating that Corollary \[cor:kappa:regular1\] can be given in the present more general form.
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[**Characterizations of the $(b, c)$-inverse in a ring**]{}\
\
[ **Abstract:**]{} 0truemm0truemm Let $R$ be a ring and $b, c\in R$. In this paper, we give some characterizations of the $(b,c)$-inverse, in terms of the direct sum decomposition, the annihilator and the invertible elements. Moreover, elements with equal $(b,c)$-idempotents related to their $(b, c)$-inverses are characterized, and the reverse order rule for the $(b,c)$-inverse is considered.\
[ **2010 Mathematics Subject Classification:** 15A09, 16U99. ]{}\
[ **Keywords:**]{} $(b, c)$-inverse, $(b, c)$-idempotent, Regularity, Image-kernel $(p, q)$-inverse.
Introduction
============
Moore-Penrose inverse, Drazin inverse and group inverse, as for the classical generalized inverses, are special types of outer inverses. In [@MP.D1], Drazin introduced a new class of outer inverse in a semigroup and called it $(b, c)$-inverse.
\[def:bc-inverse\] Let $R$ be an associative ring and let $b, c\in R$. An element $a \in R$ is $(b, c)$-invertible if there exists $y\in R$ such that
$y \in (bRy) \cap (yRc)$, $yab = b$, $cay = c$.
If such $y$ exists, it is unique and is denoted by $a^{\|(b,c)}$.
From [@MP.D1], we know that the Moore-Penrose inverse of $a$, with respect to an involution $*$ of $R$, is the $(a^{\ast}, a^{\ast})$-inverse of $a$, the Drazin inverse of $a$ is the $(a^{j}, a^{j})$-inverse of $a$ for some $j\in \mathbb{N}$, in particular, the group inverse of $a$ is the $(a,a)$-inverse of $a$.
Given two idempotents $e$ and $f$, Drazin introduced the Bott-Duffin $(e, f)$-inverse in [@MP.D1], which can be considered as a particular cases of the $(b, c)$-inverse. In 2014, Kantún-Montiel introduced the image-kernel $(p, q)$-inverse for two idempotents $p$ and $q$, and pointed out that an element $a$ is image-kernel $(p, q)$-invertible if and only if it is Bott-Duffin $(p, 1-q)$-invertible [@GKM Proposition 3.4]. In [@D.M1], elements with equal idempotents related to their image-kernel $(p, q)$-inverses are characterized in terms of classical invertibility. The topics of research on the image-kernel $(p, q)$-inverse and the Bott-Duffin $(e, f)$-inverse attract wide interest (see [@C.X1; @C.X2; @N2; @C.L.Z; @D.W; @MP.D1; @GKM; @D.M1]).
This article is motivated by the papers [@MP.D1; @D.M1]. In [@MP.D1], as a generalization of $(b,c)$-inverse, hybrid $(b,c)$-inverse and annihilator $(b,c)$-inverse were introduced. In section 3, it is shown that if the $(b,c)$-inverse of $a$ exists, then both $b$ and $c$ are regular. Further, under the natural hypothesis of both $b$ and $c$ regular, some characterizations of the $(b,c)$-inverse are obtained in terms of the direct sum decomposition, the annihilator and the invertible elements. In particular, we will prove that $(b,c)$-inverse, hybrid $(b,c)$-inverse and annihilator $(b,c)$-inverse are coincident. Some results of the image-kernel $(p, q)$-inverse in [@D.M1] are generalized.
If $a$ has a $(b, c)$-inverse, then both $a^{\|(b,c)}a$ and $aa^{\|(b,c)}$ are idempotents. These will be referred as to the $(b,c)$-idempotents associated with $a$. In [@N1], Castro-González, Koliha and Wei characterized matrices with the same spectral idempotents corresponding to the Drazin inverses of these matrices. Koliha and Patrício [@JJK1] extend the results to the ring case. A similar question for the Moore-Penrose inverse was considered in [@P.P1]. In [@D.M1], Mosić gave some characterizations of elements which have the same idempotents related to their image-kernel $(p, q)$-inverses. It is of interest to know whether two elements in the ring have equal $(b,c)$-idempotents. In section 4, some characterizations of those elements with equal $(b,c)$-idempotents are given. Moreover, the reverse order rule for the $(b,c)$-inverse is considered.
Preliminaries
=============
Let $R$ be an associative ring with unit 1. Let $a\in R$. Recall $a$ is a regular element if there exists $x \in R$ such that $a = axa$. In this case, the element $x$ is called an inner inverse for $a$ and we will denote it by $a^{-}$. If the equation $x = xax$ is satisfied, then we say that $a$ is outer generalized invertible and $x$ is called an outer inverse for $a$. An element $x$ that is both inner and outer inverse of $a$ and commutes with $a$, when it exist, must be unique and is called the group inverse of $a$, denoted by $a^{\#}$. From now on, $E(R)$ and $R^{\#}$ stand for the set of all idempotents and the set of all group invertible elements in $R$. For the sake of convenience, we introduce some necessary notations.
For an element $a \in R$ and $X\subseteq R$, we define
$aR := \{ax : x \in R\}$, $Ra := \{xa : x \in R\}$;\
$l(X) := \{y \in R : \ yx=0 \ \text{for any} \ x\in X\}$, $r(X) := \{y \in R : \ xy=0 \ \text{for any} \ x\in X\}$.
In particular,
$l(a) := \{y \in R : \ ya=0 \}$, $r(a) := \{y \in R : \ ay=0 \}$,\
$rl(a)=\{y : xy=0, x\in l(a)\}$ and $lr(a)=\{y : yx=0, x\in r(a)\}$.
Let $p, q \in E(R)$. An element $a\in R$ has an image-kernel $(p, q)$-inverse [@GKM; @D.M1] if there exists an element $c \in R$ satisfying
$cac = c, \quad caR = pR, \quad (1-ac)R = qR.$
The image-kernel $(p, q)$-inverse is unique if it exists, and it will be denoted by $a^{\times}$. A generalization of the original Bott-Duffin inverse [@B.D] was given in [@MP.D1]: let $e, f \in E(R)$, an element $a \in R$ is Bott-Duffin $(e, f)$-invertible if there exist $y \in R$ such that $y = ey = yf$, $yae = e$ and $fay = f$. When $e=f$, the element $y$, if any, is given by $y=e(ae+1-e)^{-1}$ as for the original Bott-Duffin inverse.
The above mentioned generalized inverses are particular cases of the $(b, c)$-inverse where $b$ and $c$ have the property of being both idempotents. Hence, the research of $(b,c)$-inverse has important significance to the development of the generalized inverse theory.
For the future reference we state two known results.
\[lem1:bc-inverse\] [@MP.D1 Theorem 2.2] For any given $a, b, c \in R$, there exists the $(b, c)$-inverse $y$ of $a$ if and only if $Rb=Rt$ and $cR=tR$, where $t=cab$.
\[lem2:bc-inverse\] [@MP.D1 Proposition 6.1] For any given $a, b, c \in R$, $y$ is the $(b, c)$-inverse of $a$ if and only if $yay = y$, $yR = bR$ and $Ry = Rc$.
Some characterizations of the existence of $(b, c)$-inverse
===========================================================
Firstly, we will give some lemmas which will be used in the sequel.
\[le:y outer of a\] Let $a, y \in R$ such that $y$ is an outer inverse of $a$. Then
(i) $r(a) \cap yR=\{0\}$.
(ii) $l(a) \cap Ry=\{0\}$.
(iii) $Ray=Ry$.
(iv) $yaR=yR$.
$(i)$. Let $x\in r(a) \cap yR$. Then $ax=0$ and there exists $g\in R$ such that $x=yg$. This gives that $ayg=0$ and, thus, $yayg=yg=0$. Therefore, $x=0$.
$(ii)$. Let $x\in l(a) \cap Ry$. Then $xa=0$ and there exists $h\in R$ such that $x=hy$. It leads to $hya=0$. Then $hyay=hy=0$ and, thus, $x=0$.
$(iii)$ and $(iv)$. From $yay=y$ it follows that $yaR=yR$ and $Ry=Ray$.
\[le:regular\] Let $a \in R$ be regular and $b\in R$. Then
(i) $b$ is regular in case $Ra=Rb$.
(ii) $rl(a)=aR$ and $lr(a)=Ra$.
$(i)$. Since $Ra=Rb$, there exist some $g,h\in R$ such that $a=gb$ and $b=ha$. Hence, using that $a$ is regular, one can see $b=(ha)a^{-}a=ba^{-}a=ba^{-}gb$, which means that $b$ is regular.
$(ii)$. It is easy to check that $aR\subseteq rl(a)$. Note that $l(a)=l(aa^{-})=R(1-aa^{-})$. For any $x\in rl(a)$, one can get $R(1-aa^{-})x=l(a)x=0$. This gives $x=aa^{-}x\in aR$ and $rl(a)=aR$. Similar considerations apply to prove that $lr(a)=Ra$.
\[pro:regular\] If $a$ has a $(b,c)$-inverse, then $b$, $c$ and $t=cab$ are all of them regular.
Let $y$ be the $(b,c)$-inverse of $a$. In view of Definition \[def:bc-inverse\], one can see $b=yab\in (bRy)ab\subseteq bRb$. This gives that $b$ is regular. In the same manner one can obtain that $c$ is regular. Now, on account of Lemma \[lem1:bc-inverse\], we have $Rb=Rt$ and $cR=tR$ since the $(b,c)$-inverse of $a$ exists. From Lemma \[le:regular\], we conclude that $t$ is regular.
In what follows, we will give necessary and sufficient conditions for the existence of the $(b,c)$-inverse when $t=cab$ is regular.
\[th:exists\_bc-inverse\] Let $a, b, c \in R$. If $t=cab$ is regular, then the following statements are equivalent:
(i) $a$ has a $(b,c)$-inverse.
(ii) $r(a)\cap bR = \{0\}$ and $R = abR \oplus r(c)$.
(iii) $r(t)=r(b)$ and $tR=cR$.
(iv) $l(t)=l(c)$ and $Rt=Rb$.
(v) $l(t)=l(c)$ and $r(t)= r(b)$.
$(i)\Rightarrow (ii)$ Suppose that $y$ is the $(b,c)$-inverse of $a$. By Lemma \[lem2:bc-inverse\], $yay=y$, $yR = bR$ and $Ry=Rc$. By Lemma \[le:y outer of a\] $(i)$, one can see $r(a)\cap yR=\{0\}$, it follows that $r(a)\cap bR=\{0\}$. Since $ay\in E(R)$, we have the decomposition $R=ayR\oplus r(ay)$. From $yR = bR$ we obtain $ayR=abR$. By Lemma \[le:y outer of a\] $(iii)$ and $Ry=Rc$, then $Ray=Rc$ and hence $r(ay)=r(c)$. Consequently, we have $R=abR\oplus r(c)$.
$(ii)\Rightarrow (iii)$. It is clear that $r(b)\subseteq r(t)$. For any $x\in r(t)$, we have $tx=cabx=0$. This means that $abx\in r(c)$. Using that $r(c)\cap abR=\{0\}$ we conclude that $abx=0$. Then $bx \in r(a)\cap bR=\{0\}$. This implies that $bx=0$ and, thus, $x\in r(b)$. Therefore $r(t)= r(b)$.
It is clear that $tR\subseteq cR$. Since $R = abR \oplus r(c)$, we can write $1=abg+h$ where $g\in R$ and $h\in r(c)$. Premultiplaying by $c$ gives $c=cabg\in tR$, ensuring that $cR=tR$.
$(iii)\Rightarrow (iv)$. Since $tR=cR$, we have $l(c)=l(t)$. It is clear the $Rt\subseteq Rb$. Using that $t$ is regular and $r(t)=r(b)$ we obtain that $b(1-t^{-}t)=0$. Then $b=bt^{-}t$. Consequently, $Rt=Rb$.
$(iv)\Rightarrow (v)$. It is clear.
$(v)\Rightarrow (i)$. Since $r(t)=r(b)$ and $t$ is regular we can prove that $Rt=Rb$ as in the proof of $(iii)\Rightarrow (iv)$. Similarly, from $l(t)=l(c)$ and the fact that $t$ is regular we get $tR=cR$. On account of Lemma \[lem1:bc-inverse\] we conclude that $a$ has a $(b,c)$-inverse.
In Theorem \[th:exists\_bc-inverse\], the implications $(i)\Rightarrow (ii)$ and $(ii)\Rightarrow (iii)$ are valid even if $t$ is not regular. However, we will give a counterexample to show that $(iii)$ does not imply $(iv)$ in general when $t$ is not regular.
Set $R=\mathbb{Z}$, $a=b=1$ and $c=2$. Clearly, $tR=cR$ and $r(t)=r(b)$, but $Rb\neq Rt$.
When we replace the hypothesis that $t$ is regular in Theorems \[th:exists\_bc-inverse\] by the condition that both $b$ and $c$ are regular, we obtain the following result.
\[th:exists2\_bcinverse\] Let $a, b, c \in R$. If both $b$ and $c$ are regular, then the statements (i)-(iv) in Theorem \[th:exists\_bc-inverse\] are equivalent.
We note that in item $(iii)$ condition $tR=cR$ together with $c$ is regular implies that $t$ is regular, in item $(iv)$ $Rt=Rb$ together with $b$ is regular implies that $t$ is regular.
The statements $(v)\Rightarrow (i)$ in Theorem \[th:exists\_bc-inverse\] is not true, when $b$ and $c$ are regular. For example, set $R=\mathbb{Z}$, $b=c=1$ and $a=2$. Then $b$ and $c$ are regular. It is easy to check that $l(t)=l(c)$ and $r(t)=r(b)$, but $t=2$ is not regular. Then $a$ is not $(b,c)$-invertible by Proposition \[pro:regular\].
As a generalization of $(b,c)$-inverse, hybrid $(b,c)$-inverse and annihilator $(b,c)$-inverse were introduced in [@MP.D1].
\[def:hybrid-bc-inverse\] Let $a, b, c, y \in R$. We say that $y$ is a hybrid $(b, c)$-inverse of $a$ if
\[eq:hybrid-bc-inverse\] $yay=y$, $yR = bR$, $r(y)=r(c)$.
\[def:annihilator-bc-inverse\] Let $a, b, c, y \in R$. We say that $y$ is a annihilator $(b, c)$-inverse of $a$ if
\[eq:hybrid-bc-inverse\] $yay=y$, $l(y)= l(b)$, $r(y)=r(c)$.
In [@MP.D1], Drazin pointed out that for any given $a, b, c \in R$,
$(b,c)$-invertible $\Rightarrow$ hybrid $(b, c)$-invertible $\Rightarrow$ annihilator $(b, c)$-invertible.
In what follows, we will prove that the three generalized inverses are coincident whenever $t=cab$ is regular.
\[th:all-inverses\]Let $a, b, c, y\in R$. If $t$ is regular, then the following conditions are equivalent:
(i) $y$ is the $(b,c)$-inverse of $a$.
(ii) $y$ is the hybrid $(b,c)$-inverse of $a$.
(iii) $y$ is the annihilator $(b,c)$-inverse of $a$.
$(i)\Rightarrow (ii)\Rightarrow (iii)$. These implications are clear.
$(iii)\Rightarrow (i)$. By Definition \[def:annihilator-bc-inverse\], we have $1-ay\in r(y)=r(c)$ and $1-ya\in l(y)=l(b)$. This implies that $c=cay$ and $b=yab$. Next, we will prove that $r(t)=r(b)$ and $l(t)=l(c)$. Combining with Theorem \[th:exists\_bc-inverse\] $(v)$, then we can find that
$a$ is annihilator $(b, c)$-invertible $\Rightarrow$ $a$ is $(b, c)$-invertible.
It is clear that $r(b)\subseteq r(t)$. Let $w\in r(t)$. Then $cabw=0$ and hence $abw\in r(c)=r(y)$. This implies that $yabw=0$. Then $bw=0$ since $yab=b$. This shows $r(t)\subseteq r(b)$. Therefore, $r(t)=r(b)$. Similarly, we can prove that $l(c)=l(t)$. Since $a$ has a $(b,c)$-inverse $z$, then $a$ has the annihilator $(b,c)$-inverse $z$ and by the uniqueness we have $z=y$.
\[th:ebcinverse\] Let $a, b, c \in R$. If both $b$ and $c$ are regular, then the statements (i)-(iii) in Theorem \[th:all-inverses\] are equivalent.
We only need to prove that $(iii)\Rightarrow (i)$. If $y$ is the annihilator $(b,c)$-inverse of $a$, then $l(y)=l(b)$, this gives that $rl(y)=rl(b)$. Since $b$ and $y$ are regular, we have $rl(b)=bR$ and $rl(y)=yR$ by Lemma \[le:regular\] $(ii)$. This implies that $yR=bR$. Similarly, we can obtain that $Ry=Rc$. Thus, it follows that $y$ is the $(b,c)$-inverse of $a$ by Lemma \[lem2:bc-inverse\].
The following lemma it is well known.
\[le:idempotent\] Let $a \in R$ and $e \in E(R)$. Then the following conditions are equivalent:
(i) $e \in eaeR \cap Reae$.
(ii) $eae + 1 - e$ is invertible (or $ae + 1 - e$ is invertible).
\[th:d-bcinverse\] Let $a, b, c, d \in R$ such that the $(b,c)$-inverse of $a$ exists. Let $e=bb^{-}$ where $b^{-}$ are fixed, but arbitrary inner inverses of $b$. Then the following statements are equivalent:
(i) $d$ has a $(b,c)$-inverse.
(ii) $e\in ea^{\|(b,c)}deR\cap Rea^{\|(b,c)}de$.
(iii) $a^{\|(b,c)}de+1-e$ is invertible.
In this case, $$\label{eq:d-bcinverse} d^{\|(b,c)}=(a^{\|(b,c)}de+1-e)^{-1}a^{\|(b,c)}.$$
Firstly, as $a^{\|(b,c)}$ exists we have $a^{\|(b,c)}\in bR\cap Rc$ by Lemma \[lem2:bc-inverse\]. Therefore $$\label{eq1:d-bcinverse}
a^{\|(b,c)}=bb^{-}a^{\|(b,c)}=a^{\|(b,c)}c^{-}c.$$ From Definition \[def:bc-inverse\] we have $b=a^{\|(b,c)}ab$. Combining with (\[eq1:d-bcinverse\]), we can write $$\label{eq2:d-bcinverse}
b=ea^{\|(b,c)}c^{-}cab.$$ $(i)\Rightarrow(ii)$. Suppose that $d^{\|(b,c)}$ exists. By Definition \[def:bc-inverse\], we also have $c=cdd^{\|(b,c)}$. Substituting this into (\[eq2:d-bcinverse\]) yields $$b=ea^{\|(b,c)}c^{-}(cdd^{\|(b,c)})ab=ea^{\|(b,c)}dd^{\|(b,c)}ab.$$ Multiplying on the right by $b^{-}$ we obtain $e=ea^{\|(b,c)}dd^{\|(b,c)}ae$. Since $d^{\|(b,c)}=ed^{\|(b,c)}$, which follows by interchanging $a^{\|(b,c)}$ and $d^{\|(b,c)}$ in (\[eq1:d-bcinverse\]), we get $e=ea^{\|(b,c)}ded^{\|(b,c)}ae$. This implies that $e\in ea^{\|(b,c)}deR$. Similarly, we can prove that $e\in Rea^{\|(b,c)}de$.
$(ii)\Rightarrow(iii)$ See Lemma \[le:idempotent\].
$(iii)\Rightarrow(i)$ Firstly we note that $ea^{\|(b,c)}=a^{\|(b,c)}$ by (\[eq1:d-bcinverse\]). Set $x=ea^{\|(b,c)}de+1-e$. It is clear that $ex=xe$ and $ex^{-1}=x^{-1}e$. Write $y=x^{-1}a^{\|(b,c)}$. Next, we verify that $y$ is the $(b, c)$-inverse of $d$.
**Step 1**. $ydy=y$. Indeed,
Using $a^{\|(b,c)}=ea^{\|(b,c)}$, we get $$\begin{aligned}
ydy&=&x^{-1}a^{\|(b,c)}dx^{-1}a^{\|(b,c)}=x^{-1}ea^{\|(b,c)}dx^{-1}ea^{\|(b,c)}\\
&=&x^{-1}(ea^{\|(b,c)}de+1-e)ex^{-1}a^{\|(b,c)}\\
&=&x^{-1}ea^{\|(b,c)}=x^{-1}a^{\|(b,c)}=y.\end{aligned}$$
**Step 2**. $bR=yR$.
On account of $a^{\|(b,c)}=ea^{\|(b,c)}$ and $(1-e)b=0$, one can get
$b=x^{-1}(ea^{\|(b,c)}de+1-e)b=x^{-1}ea^{\|(b,c)}deb=x^{-1}a^{\|(b,c)}deb=ydeb \in yR$
Meanwhile, $y=x^{-1}a^{\|(b,c)}=x^{-1}ea^{\|(b,c)}=ex^{-1}a^{\|(b,c)}\in bR$. This guarantees $bR=yR$.
**Step 3**. $Rc=Ry$.
From Definition \[def:bc-inverse\], we have $c=caa^{\|(b,c)}$. This leads to $c=caxx^{-1}a^{\|(b,c)}=caxy\in Ry$. On the other hand, from (\[eq1:d-bcinverse\]) we conclude that $y=x^{-1}a^{\|(b,c)}=x^{-1}a^{\|(b,c)}c^{-}c\in Rc$. It means that $Rc=Ry$.
Similarly, we can state the analogue of Theorem \[th:d-bcinverse\].
\[th2:d-bcinverse\] Let $a, b, c, d \in R$ such that the $(b,c)$-inverse of $a$ exists. Let $f=c^{-}c$ where $c^{-}$ are fixed, but arbitrary inner inverses of $c$. Then the following statements are equivalent:
(i) $d$ has a $(b,c)$-inverse.
(ii) $f\in fda^{\|(b,c)}fR\cap Rfda^{\|(b,c)}f$.
(iii) $fda^{\|(b,c)}+1-f$ is invertible.
In this case, $$\label{eq2:d-bcinverse-2}
d^{\|(b,c)}= a^{\|(b,c)}(fda^{\|(b,c)}+1-f)^{-1}.$$
In case that both $a^{\|(b,c)}$ and $d^{\|(b,c)}$ exist, from Theorem \[th:d-bcinverse\] and \[th2:d-bcinverse\], it may be concluded that $$\begin{split} \label{eq:d_(b,c)_inverse}
(a^{\|(b,c)}de+1-e)^{-1}=d^{\|(b,c)}ae+1-e;\\
(fda^{\|(b,c)}+1-f)^{-1}=fad^{\|(b,c)}+1-f.
\end{split}$$ Indeed, since $d^{\|(b,c)}=(a^{\|(b,c)}de+1-e)^{-1}a^{\|(b,c)}$, we have $(a^{\|(b,c)}de+1-e)d^{\|(b,c)}=a^{\|(b,c)}.$ Hence, $$(a^{\|(b,c)}de+1-e)(d^{\|(b,c)}ae+1-e)= a^{\|(b,c)}ae+1-e=1,$$ where the last identity is due to the fact that $a^{\|(b,c)}ae=e$, because $b=a^{\|(b,c)}ab$. Interchanging the roles of $a$ and $d$ in Theorem \[th:d-bcinverse\] it follows that $(d^{\|(b,c)}ae+1-e)(a^{\|(b,c)}de+1-e)=1$ and, in consequence, the first identity in (\[eq:d\_(b,c)\_inverse\]) holds. The second identity in (\[eq:d\_(b,c)\_inverse\]) can be proved in the same manner.
For any two idempotents $p$ and $q$, we replace $b$ and $c$ by $p$ and $1-q$ respectively in Theorem \[th:d-bcinverse\] and \[th2:d-bcinverse\], we obtain the following corollary.
[@D.M1 Theorem 3.3] Let $p, q \in E(R)$ and let $a \in R$ be such that $a^{\times}$ exists. Then for $d \in R$ the following statements are equivalent:
(i) $d^{\times}$ exists.
(ii) $1-p+a^{\times}dp$ is invertible.
(iii) $q+(1-q)da^{\times}$ is invertible.
Characterizations of elements with equal $(b,c)$-idempotents
============================================================
Let $a^{\|(b,c)}$ exists. Since $a^{\|(b,c)}$ is an outer inverse of $a$, when it exists, then both $a^{\|(b,c)}a$ and $aa^{\|(b,c)}$ are idempotents. These will be referred to as the $(b,c)$-idempotents associated with $a$. We are interested in finding characterizations of those elements in the ring with equal $(b,c)$-idempotents.
In what follows, we will give necessary and sufficient conditions for $aa^{\|(b,c)} =dd^{\|(b,c)}$. We firstly establish an auxiliary result.
\[le:bc-inverses\] Let $a, b, c, d\in R$ such that $a^{\|(b,c)}$ and $d^{\|(b,c)}$ exist. Let $e=bb^{-}$ and $f=c^{-}c$, where $b^{-}$ and $c^{-}$ are fixed, but arbitrary inner inverses of $b$ and $c$, respectively. Then
(i) $d^{\|(b,c)}=d^{\|(b,c)}aa^{\|(b,c)}=a^{\|(b,c)}ad^{\|(b,c)}$.
(ii) $a^{\|(b,c)}=a^{\|(b,c)}dd^{\|(b,c)}=d^{\|(b,c)}da^{\|(b,c)}$.
(iii) $e=ed^{\|(b,c)}aa^{\|(b,c)}de=ea^{\|(b,c)}ae=ed^{\|(b,c)}de.$
(iv) $f=fda^{\|(b,c)}ad^{\|(b,c)}f=fdd^{\|(b,c)}f=faa^{\|(b,c)}f$.
$(i)$. In view of (\[eq:d-bcinverse\]) and (\[eq2:d-bcinverse-2\]), with the notation $e=bb^{-}$ and $f=c^{-}c$, we have $$\begin{aligned}
d^{\|(b,c)}&=&(a^{\|(b,c)}de+1-e)^{-1}a^{\|(b,c)}=d^{\|(b,c)}aa^{\|(b,c)}\\
&=&a^{\|(b,c)}(fda^{\|(b,c)}+1-f)^{-1}=a^{\|(b,c)}ad^{\|(b,c)}.\end{aligned}$$
$(ii)$. We get these equalities by interchanging the roles of $a^{\|(b,c)}$ and $d^{\|(b,c)}$ in previous results.
$(iii)$. By the Definition \[def:bc-inverse\], we have $b=d^{\|(b,c)}db$. Multiplying on the right by $b^{-}$ gives $e=d^{\|(b,c)}de$. Similarly, $e=ea^{\|(b,c)}ae$. Multiplying $(i)$ on the right by $de$ leads to $e=ed^{\|(b,c)}aa^{\|(b,c)}de$.
$(iv)$. By the definition \[def:bc-inverse\], we have $c=cad^{\|(b,c)}$ and, multiplying on the left by $c^{-}$, we get $f=fdd^{\|(b,c)}$. Similarly, $faa^{\|(b,c)}f$. Multiplying $(ii)$ on the left by $fd$, one can see $f=fd a^{\|(b,c)}ad^{\|(b,c)}f$.
Let $a,b,c,d\in R$ such that $a^{\|(b,c)}$ and $d^{\|(b,c)}$ exist. Then the following statements are equivalent:
(i) $aa^{\|(b,c)} =dd^{\|(b,c)}$.
(ii) $aa^{\|(b,c)}dd^{\|(b,c)}=dd^{\|(b,c)}aa^{\|(b,c)}$.
(iii) $ad^{\|(b,c)}da^{\|(b,c)}=da^{\|(b,c)}ad^{\|(b,c)}$.
(iv) $ad^{\|(b,c)} \in R^{\#}$ and $(ad^{\|(b,c)})^{\#} =d a^{\|(b,c)}$.
(v) $da^{\|(b,c)} \in R^{\#}$ and $(da^{\|(b,c)})^{\#} =a d^{\|(b,c)}$.
$(i)\Leftrightarrow (ii) \Leftrightarrow (iii)$. From Lemma \[le:bc-inverses\] we obtain $$\label{eq:idempotents}
\begin{split}
aa^{\|(b,c)}=aa^{\|(b,c)}dd^{\|(b,c)}=ad^{\|(b,c)}da^{\|(b,c)};\\
dd^{\|(b,c)}=dd^{\|(b,c)}aa^{\|(b,c)}=da^{\|(b,c)}ad^{\|(b,c)}.
\end{split}$$ This leads to $$\begin{aligned}
aa^{\|(b,c)}=dd^{\|(b,c)}&\Leftrightarrow& aa^{\|(b,c)}dd^{\|(b,c)}=dd^{\|(b,c)}aa^{\|(b,c)}\\
&\Leftrightarrow& ad^{\|(b,c)}da^{\|(b,c)}=da^{\|(b,c)}ad^{\|(b,c)}.\end{aligned}$$
$(iii)\Leftrightarrow (iv)$. Set $x= da^{\|(b,c)}$. We will prove that $x$ is the group inverse of $ad^{\|(b,c)}$. Combining $(iii)$ with Lemma \[le:bc-inverses\], we get $$\begin{aligned}
xad^{\|(b,c)}&=&da^{\|(b,c)}ad^{\|(b,c)}=ad^{\|(b,c)}da^{\|(b,c)}=ad^{\|(b,c)}x;\\
ad^{\|(b,c)}x ad^{\|(b,c)}&=& a(d^{\|(b,c)}da^{\|(b,c)})ad^{\|(b,c)}=a(a^{\|(b,c)}ad^{\|(b,c)})=ad^{\|(b,c)}; \nonumber\\
xad^{\|(b,c)}x&=& x ad^{\|(b,c)}da^{\|(b,c)}= xaa^{\|(b,c)}=da^{\|(b,c)}aa^{\|(b,c)}=x.\end{aligned}$$ This implies that $ad^{\|(b,c)} \in R^{\#}$ and $(ad^{\|(b,c)})^{\#} = da^{\|(b,c)}$. Conversely, if the latter holds, then $da^{\|(b,c)}ad^{\|(b,c)}=ad^{\|(b,c)}da^{\|(b,c)}$.
$(iii)\Leftrightarrow (v)$. The proof is similar to the previous equivalence.
We state the result in terms of the other $(b,c)$-idempotent.
Let $a,b,c,d\in R$ such that $a^{\|(b,c)}$ and $d^{\|(b,c)}$ exist. Then the following statements are equivalent:
(i) $a^{\|(b,c)}a =d^{\|(b,c)}d$.
(ii) $d^{\|(b,c)}da^{\|(b,c)}a=a^{\|(b,c)}ad^{\|(b,c)}d$.
(iii) $a^{\|(b,c)}dd^{\|(b,c)}a=d^{\|(b,c)}aa^{\|(b,c)}d$.
(iv) $a^{\|(b,c)}d \in R^{\#}$ and $(a^{\|(b,c)}d)^{\#} =d^{\|(b,c)}a$.
(v) $d^{\|(b,c)}a \in R^{\#}$ and $(d^{\|(b,c)}a)^{\#} =a^{\|(b,c)}d$.
Next, we consider conditions under which the reverse order rule for the $(b,c)$-inverse of the product $ad$, $(ad)^{\|(b,c)}=d^{\|(b,c)}a^{\|(b,c)}$ holds.
\[th:ad-bcinverse\] Let $a,b,c,d\in R$ such that $a^{\|(b,c)}$ and $d^{\|(b,c)}$ exist. Then the following statements are equivalent:
(i) $ad$ has a $(b,c)$-inverse of the form $(ad)^{\|(b,c)}=d^{\|(b,c)}a^{\|(b,c)}$.
(ii) $d^{\|(b,c)}=d^{\|(b,c)}add^{\|(b,c)}a^{\|(b,c)}=d^{\|(b,c)}a^{\|(b,c)}add^{\|(b,c)}$.
(iii) $a^{\|(b,c)}=a^{\|(b,c)}add^{\|(b,c)}a^{\|(b,c)}=d^{\|(b,c)}a^{\|(b,c)}ada^{\|(b,c)}$.
$(i)\Leftrightarrow (ii)$. We first assume that $ad$ has a $(b,c)$-inverse given by $(ad)^{\|(b,c)}=d^{\|(b,c)}a^{\|(b,c)}$. Then Lemma \[le:bc-inverses\] is true for $(ad)^{\|(b,c)}$ in place of $a^{\|(b,c)}$. It follows that
$d^{\|(b,c)}=d^{\|(b,c)}ad(ad)^{\|(b,c)}=(ad)^{\|(b,c)}add^{\|(b,c)}$.
Substituting $(ad)^{\|(b,c)}=d^{\|(b,c)}a^{\|(b,c)}$ yields
$d^{\|(b,c)}=d^{\|(b,c)}add^{\|(b,c)}a^{\|(b,c)}=d^{\|(b,c)}a^{\|(b,c)}add^{\|(b,c)}$.
Conversely, if the latter identities hold then $y=d^{\|(b,c)}a^{\|(b,c)}$ is the $(b,c)$-inverse of $ad$. Indeed, since $d^{\|(b,c)}db=b$ and $c=cdd^{\|(b,c)}$, we have $$\begin{aligned}
yady&=&d^{\|(b,c)}a^{\|(b,c)}add^{\|(b,c)}a^{\|(b,c)}=d^{\|(b,c)}a^{\|(b,c)};\\
yadb&=&d^{\|(b,c)}a^{\|(b,c)}adb= d^{\|(b,c)}a^{\|(b,c)}add^{\|(b,c)}db=d^{\|(b,c)}db=b;\\
cady&=&cadd^{\|(b,c)}a^{\|(b,c)}=cdd^{\|(b,c)}add^{\|(b,c)}a^{\|(b,c)}=cdd^{\|(b,c)}=c.\end{aligned}$$ $(ii)\Rightarrow (iii)$. By Lemma \[le:bc-inverses\] we have $a^{\|(b,c)}=a^{\|(b,c)}dd^{\|(b,c)}=d^{\|(b,c)}da^{\|(b,c)}$. By $(ii)$, one can see
$a^{\|(b,c)}=a^{\|(b,c)}d(d^{\|(b,c)}add^{\|(b,c)}a^{\|(b,c)})= (d^{\|(b,c)}a^{\|(b,c)}add^{\|(b,c)})da^{\|(b,c)}$.
Hence, it is easy to get $a^{\|(b,c)}=a^{\|(b,c)}add^{\|(b,c)}a^{\|(b,c)}= d^{\|(b,c)}a^{\|(b,c)}ada^{\|(b,c)}$.\
$(iii)\Rightarrow (ii)$. The proof is similar to $(ii)\Rightarrow (iii)$.
Let $a,b,c,d\in R$ such that $a^{\|(b,c)}$ and $d^{\|(b,c)}$ exist. Then the following statements are equivalent:
(i) $a^{\|(b,c)}a = dd^{\|(b,c)}$.
(ii) $a^{\|(b,c)}dd^{\|(b,c)}a=dd^{\|(b,c)}aa^{\|(b,c)}$.
(iii) $d^{\|(b,c)}da^{\|(b,c)}a=da^{\|(b,c)}ad^{\|(b,c)}$.
(iv) $a^{\|(b,c)}= dd^{\|(b,c)}a^{\|(b,c)}$ and $d^{\|(b,c)}= d^{\|(b,c)}a^{\|(b,c)}a$.
(v) $a^{\|(b,c)}ad^{\|(b,c)} = d^{\|(b,c)}a^{\|(b,c)}a$ and $a^{\|(b,c)}dd^{\|(b,c)}=dd^{\|(b,c)}a^{\|(b,c)}$.
If any of the previous statements is valid, then $(ad)^{\|(b,c)}=d^{\|(b,c)}a^{\|(b,c)}.$
$(i) \Leftrightarrow (ii) \Leftrightarrow (iii)$. From Lemma \[le:bc-inverses\] we obtain $$\label{eq2:idempotents}
\begin{split} a^{\|(b,c)}a=a^{\|(b,c)}dd^{\|(b,c)}a=d^{\|(b,c)}da^{\|(b,c)}a;\\
dd^{\|(b,c)}=dd^{\|(b,c)}aa^{\|(b,c)}=da^{\|(b,c)}ad^{\|(b,c)}.
\end{split}$$ Hence, it gives that $$\begin{aligned}
a^{\|(b,c)}a=dd^{\|(b,c)}&\Leftrightarrow& a^{\|(b,c)}dd^{\|(b,c)}a=dd^{\|(b,c)}aa^{\|(b,c)}\\
&\Leftrightarrow& d^{\|(b,c)}da^{\|(b,c)}a=da^{\|(b,c)}ad^{\|(b,c)}.\end{aligned}$$
$(i)\Leftrightarrow (iv)$. The necessary condition is immediate. Next, we assume that $a^{\|(b,c)}= dd^{\|(b,c)}a^{\|(b,c)}$ and $d^{\|(b,c)}= d^{\|(b,c)}a^{\|(b,c)}a$. Then we have $a^{\|(b,c)}a= dd^{\|(b,c)}a^{\|(b,c)}a$ and $dd^{\|(b,c)}= dd^{\|(b,c)}a^{\|(b,c)}a$. So $a^{\|(b,c)}a =dd^{\|(b,c)}$, as desired.
$(v)\Leftrightarrow(i)$. The proof is similar to the above.
Finally, we will prove that $dd^{\|(b,c)}=a^{\|(b,c)}a$ implies that $(ad)^{\|(b,c)}=d^{\|(b,c)}a^{\|(b,c)}$. Since $d^{\|(b,c)}=d^{\|(b,c)}a^{\|(b,c)}a$, we have $d^{\|(b,c)}=d^{\|(b,c)}a^{\|(b,c)}add^{\|(b,c)}$. Moreover, since $d^{\|(b,c)}= d^{\|(b,c)}aa^{\|(b,c)}$ by Lemma \[le:bc-inverses\], using $dd^{\|(b,c)}=a^{\|(b,c)}a$, it follows that
$d^{\|(b,c)}= d^{\|(b,c)}aa^{\|(b,c)}=d^{\|(b,c)}aa^{\|(b,c)}aa^{\|(b,c)}=d^{\|(b,c)}add^{\|(b,c)}a^{\|(b,c)}$.
By Theorem \[th:ad-bcinverse\] our assertion is proved.
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---
abstract: 'In 1995 the author, Jones, and Segal introduced the notion of “Floer homotopy theory" [@cjs]. The proposal was to attach a (stable) homotopy type to the geometric data given in a version of Floer homology. More to the point, the question was asked, “When is the Floer homology isomorphic to the (singular) homology of a naturally occuring (pro)spectrum defined from the properties of the moduli spaces inherent in the Floer theory?". A proposal for how to construct such a spectrum was given in terms of a “framed flow category", and some rather simple examples were described. Years passed before this notion found some genuine applications to symplectic geometry and low dimensional topology. However in recent years several striking applications have been found, and the theory has been developed on a much deeper level. Here we summarize some of these exciting developments, and describe some of the new techniques that were introduced. Throughout we try to point out that this area is a very fertile ground at the interface of homotopy theory, symplectic geometry, and low dimensional topology.'
author:
- |
Ralph L. Cohen [^1]\
Department of Mathematics\
Stanford University\
Bldg. 380\
Stanford, CA 94305
title: ' Floer homotopy theory, revisited'
---
Introduction {#introduction .unnumbered}
============
In three seminal papers in 1988 and 1989 A. Floer introduced Morse theoretic homological invariants that transformed the study of low dimensional topology and symplectic geometry. In [@floerI] Floer defined an “instanton homology" theory for 3-manifolds that, when paired with Donaldson’s polynomial invariants of 4 manifolds defined a gauge theoretic 4-dimensional topological field theory that revolutionized the study of low dimensional topology and geometry. In [@floerFix], Floer defined an infinite dimensional Morse theoretic homological invariant for symplectic manifolds, now referred to as “Symplectic" or “Hamiltonian" Floer homology, that allowed him to prove a well-known conjecture of Arnold on the number of fixed points of a diffeomorphism $\phi_1 : M {\xrightarrow}{\cong} M$ arising from a time-dependent Hamiltonian flow $\{\phi_t\}_{0 \leq t \leq 1}$. In [@floerLag] Floer introduced “Lagrangian intersection Floer theory" for the study of interesections of Lagrangian submanifolds of a symplectic manifold.
Since that time there have been many other versions of Floer theory introduced in geometric topology, including a Seiberg-Witten Floer homology [@KM]. This is similar in spirit to Floer’s “instanton homology", but it is based on the Seiberg-Witten equations rather than the Yang-Mills equations. There were many difficult, technical analytic issues in developing Seiberg-Witten Floer theory, and Kronheimer and Mrowka’s book [@KM] deals with them masterfully and elegantly. Another important geometric theory is Heegard Floer theory introduced by Oszvath and Szabo [@OS]. This is an invariant of a closed 3-manifold equipped with a $spin^c$ structure. It is computed using a Heegaard diagram of the manifold. It allowed for a related “knot Floer homology" introduced by Oszvath and Szabo [@OS2] and by Rasmussen [@Ras]. Khovanov’s important homology theory that gave a “categorification" of the Jones polynomial [@khov] was eventually was shown to be related to Floer theory by Seidel and Smith [@SS] and Abouzaid and Smith [@AbS]. Lipshitz and Sarkar [@lipsark] showed that there is an associated “Khovanov stable homotopy". There have been many other variations of Floer theories as well.
The rough idea in all of these theories is to associate a Morse-like chain complex generated by the critical points of a functional defined typically on an infinite dimensional space. Recall that in classical Morse theory, given a Morse function $f : M\to {\mathbb{R}}$ on a closed Riemannian manifold, the “Morse complex" is the chain complex $$\cdots \to C_p(f) {\xrightarrow}{{\partial}_p} C_{p-1}(f) \to \cdots$$ where $C_p(f)$ is the free abelian group generated by the critical points of $f$ of index $p$, and the boundary homomorphisms can be computed by “counting" the gradient flow-lines connecting critical points of relative index one. More specfically, if $Crit_q(f)$ is the set of critical points of $f$ of index $q$, then if $a \in Crit_p(f)$, then $$\label{morsecomp}
{\partial}_p ([a]) = \sum_{b \in Crit_{p-1}(f)} \#{\mathcal{M}}(a,b) \, [b]$$ where ${\mathcal{M}}(a,b)$ is the moduli space of gradient flow lines connecting $a$ to $b$, which, since $a$ and $b$ have relative index one is a closed, zero dimensional oriented manifold. $\#{\mathcal{M}}(a,b)$ reflects the “oriented count" of this finite set. More carefully $\#{\mathcal{M}}(a,b)$ is the integer in the zero dimensional oriented cobordism group, $\pi_0 MSO \cong {\mathbb{Z}}$ represented by ${\mathcal{M}}(a, b)$.
In Floer’s original examples, the functionals he studied were in fact ${\mathbb{R}}/{\mathbb{Z}}$-valued. In the case of Floer’s instanton theory, the relevant functional is the Chern-Simons map defined on the space of connections on a principal $SU(2)$-bundle over the three-manifold. Its critical points are flat connections and its flow lines are “instantons", i.e anti-self-dual connections on the three-manifold $Y$ crossed with the real line. Modeling classical Morse theory, the “Floer complex" is generated by the critical points of this functional, suitably perturbed to make them nondegenerate, and the boundary homomorphisms are computed by taking oriented counts of the gradient flow lines, i.e anti-self-dual connections on $Y \times {\mathbb{R}}$ that connect critical points of relative index one.
When Floer introduced what is now called “Symplectic" or “Hamiltonian" Floer homology (“SFH") to use in his proof of Arnold’s conjecture, he studied the symplectic action defined on the free loop space of the underlying symplectic manifold $M$, $${\mathcal{A}}: LM \to {\mathbb{R}}/{\mathbb{Z}}.$$ He perturbed the action functional by a time dependent Hamiltonian function $H : {\mathbb{R}}/{\mathbb{Z}}\times M \to {\mathbb{R}}$. Call the resulting functional ${\mathcal{A}}_H$. The critical points of ${\mathcal{A}}_H$ are the 1-periodic orbits of the Hamiltonian vector field. That is, they are smooth loops $\alpha : {\mathbb{R}}/{\mathbb{Z}}\to M $ satisfying the differential equaiton $$\frac{d\alpha}{dt} = X_H(t, \alpha (t))$$ where $X_H$ is the Hamiltonian vector field. One way to think of $X_H$ is that the symplectic 2-form $\omega$ on $M$ defines, since it is nondegenerate, a bundle isomorphism $\omega : TM {\xrightarrow}{\cong} T^*M$. This induces an identification of the periodic one-forms, that is sections of the cotangent bundle pulled back over ${\mathbb{R}}/{\mathbb{Z}}\times M$, with periodic vector fields. The Hamiltonian vector field $X_H$ corresponds to the differential $dH$ under this identification. Floer showed that with respect to a generic Hamiltonian, the critical points of ${\mathcal{A}}_H$ are nondegenerate.
Of course to understand the gradient flow lines connecting critical points, one must have a metric. This is defined using the symplectic form and a compatible choice of almost complex structure $J$ on $TM$. With respect to this structure the gradient flow lines are curves $\gamma : {\mathbb{R}}\to LM$, or equivalently, maps of cylinders $$\gamma : {\mathbb{R}}\times S^1 \to M$$ that satisfy (perturbed) Cauchy-Riemann equations. If ${\mathcal{M}}(\alpha_1, \alpha_2, H, J)$ is the moduli space of such “pseudo-holomorphic" cylinders that connect the periodic orbits $\alpha_1$ and $\alpha_2$, then the boundary homomorphisms Morse-Floer chain complex is defined by giving an oriented count of the zero dimensional moduli spaces, in analogy to the situation in classical Morse theory described above.
The other, newer examples of Floer homology tend to be similar. There is typically a functional that can be perturbed in such a way that its critical points and zero dimensional moduli spaces of gradient flow lines define a chain complex whose homology is invariant of the choices made.
In the case of a classical Morse function $f : M \to {\mathbb{R}}$ closed manifold, the Morse complex can be viewed as the cellular chain complex of a $CW$-complex $X_f$ of the homotopy type of $M$. $X_f$ has one cell of dimension $\lambda$, for every critical point of $f$ of index $\lambda$. The attaching maps were studied by Franks in [@franks], and one may view the work of the author and his collaborators in [@cjs1], [@cjs] as a continuation of that study. This led us to ask the question:
**Q1: Does the Floer chain complex arise as the cellular chain complex of a $CW$-complex or a $C.W$-spectrum?**
More specifically we asked the the following question:
**Q2: What properties of the data of a Floer functional, i.e its critical points and moduli spaces of gradient flow lines connecting them, are needed to define a $CW$-spectrum realizing the Floer chain complex?**
More generally one might ask the following question:
**Q3: Given a finite chain complex, is there a reasonable way to classify the $CW$-spectra that realize this complex?**
In this paper we take up these questions. We also discuss how studying Floer theory from a homotopy perspective has been done in recent years, and how it has been applied with dramatic success. We state immediately that the applications that we discuss in this paper are purely the choice of the author. There are many other fascinating applications and advances that all help to make this an active and exciting area of research. We apologize in advance to researchers whose work we will not have the time or space to discuss.
This paper is organized as follows. In section 2 we discuss the three questions raised above. We give a new take on how these questions were originally addressed in [@cjs], and give some of the early applications of this perspective. In section 3 we describe how Lipshitz and Sarkar used this perspective to define the notion of “Khovanov homotopy" [@lipsark][@lipsark2]. This is a stable homotopy theoretic realization of Khovanov’s homology theory [@khov] which in turn is a categorification of the Jones polynomial invariant of knots and links. In particular we describe how the homotopy perspective Lipshitz and Sarkar used give more subtle and delicate invariants than the homology theory alone, and how these invariants have been applied. In section 3 we describe Manolescu’s work on an equivariant stable homotopy theoretic view of Seiberg-Witten Floer theory. In the case of his early work [@Man03][@Man07], the group acting is the circle group $S^1$. In his more recent work [@Man13] [@Man16] he studied $Pin(2)$-equivariant Floer homotopy theory and used it to give a dramatic solution (in the negative) to the longstanding question about whether all topological manifolds admit triangulations. In section 4 we describe the Floer homotopy theoretic methods of Kragh [@kragh] and of Abouzaid-Kragh [@Abou-Kra] in the study of the symplectic topology of the cotangent bundle of a closed manifold, and how they were useful in studying Lagrangian immersions and embeddings inside the cotangent bundle.
#### Acknowledgments.
The author would like to thank M. Abouzaid, R. Lipshitz, and C. Manolescu for helpful comments on a previous draft of this paper.
The homotopy theoretic foundations
==================================
Realizing chain complexes
-------------------------
We begin with a purely homotopy theoretic question: Given a finite chain complex, $C_*$, $$C_n {\xrightarrow}{{\partial}_n} C_{n-1} {\xrightarrow}{{\partial}_{n-1}} \cdots {\xrightarrow}{{\partial}_2}C_1 {\xrightarrow}{{\partial}_1}C_0$$ how can one classify the finite $CW$-spectra $\bf X$ whose associated cellular chain complex is $C_*$? This is question [**Q3**]{} above.
Of course one does not need to restrict this question to finite complexes, but that is a good place to start. In particular it is motivated by Morse theory, where, given a Morse function on a closed, $n$-dimensional Riemannian manifold, $f : M^n \to {\mathbb{R}}$, one has a corresponding “Morse - Smale" chain complex $C_*^f$
$$C^f_n {\xrightarrow}{{\partial}_n} C^f_{n-1} {\xrightarrow}{{\partial}_{n-1}} \cdots {\xrightarrow}{{\partial}_2}C^f_1 {\xrightarrow}{{\partial}_1}C^f_0$$ Here $C^f_p$ is the free abelian group generated by the critical points of $f$ of index $p$. The boundary homomorphisms are as described above (\[morsecomp\]). Of course in this case the Morse function together with the Riemannian metric define a $CW$ complex homotopy equivalent to $X$. In Floer theory that is not the case. One does start with geometric information that allows for the definition of a chain complex, but knowing if this complex comes, in a natural way from the data of the Floer theory, is not at all clear, and was the central question of study in [@cjs].
This homotopy theoretic question was addressed more specifically in [@floeroslo]. Of central importance in this study was to understand how the attaching maps of the cells in a finite $CW$-spectrum can be understood geometrically, via the theory of (framed) cobordism of manifolds with corners. We will review these ideas in this section and recall some basic examples of how they can be applied.
By assumption, the chain complex $C_*$ is finite, so each $C_i$ is a finitely generated free abelian group. Let ${\mathcal{B}}_i$ be a basis for $C_i$. Let ${\mathbb{S}}$ denote the sphere spectrum. For each $i$, consider the free ${\mathbb{S}}$-module spectrum generated by ${\mathcal{B}}_i$, $${\mathcal{E}}_i = \bigvee_{\alpha \in {\mathcal{B}}_i} {\mathbb{S}}.$$ There is a natural isomorphism $$H_0( {\mathcal{E}}_i) \simeq C_i.$$
\[realize\] We say that a finite spectrum ${\bf X} $ realizes the complex $C_* $, if there exists a filtration of spectra converging to $X$, $$X_0 {\hookrightarrow}X_1 {\hookrightarrow}\cdots {\hookrightarrow}X_n = {\bf X}$$ satisfying the following properties:
1. There is an equivalence of the subquotients $$X_i/X_{i-1} \simeq \Sigma^i {\mathcal{E}}_i,$$
2. The induced composition map in integral homology, $$\begin{aligned}
&\tilde H_i(X_i/X_{i-1}) {\xrightarrow}{\delta_i} \tilde H_{i}(\Sigma X_{i-1}) {\xrightarrow}{\rho_{i-1}} H_{i}(\Sigma (X_{i-1}/X_{i-2})) \notag \\
&= H_0({\mathcal{E}}_i) \to H_0({\mathcal{E}}_{i-1}) \notag \\
&= C_i \to C_{i-1} \notag\end{aligned}$$ is the boundary homomorphism, ${\partial}_i $.
Here the “subquotient" $X_i/X_{i-1}$ refers to the homotopy cofiber of the map $X_{i-1} \to X_i$, the map $\rho_i : X_i \to X_i/X_{i-1}$ is the projection map, and the map $\delta_i : X_i /X_{i-1} \to \Sigma X_{i-1}$ is the Puppe extension of the homotopy cofibration sequence $X_{i-1} \to X_i {\xrightarrow}{\rho_i} X_i/X_{i-1}$.
To classify finite spectra that realize a given finite complex, in [@cjs] the authors introduced a category ${\mathcal{J}}$ whose objects are the nonnegative integers ${\mathbb{Z}}^+$, and whose non-identity morphisms from $i$ to $j$ is empty for $i \leq j$, for $i > j+1$ it is defined to be the one point compactification, $$Mor_{{\mathcal{J}}}(i, j) \cong ({\mathbb{R}}_+)^{i-j-1}\cup \infty$$ and $Mor_{{\mathcal{J}}}(j+1, j) $ is defined to be the two point space, $S^0$. Here ${\mathbb{R}}_+$ is the space of nonnegative real numbers. Composition in this category can be viewed in the following way. Notice that for $i > j+1$ $Mor_{{\mathcal{J}}}(i,j)$ can be viewed as the one point compactification of the space $J(i,j)$ consisting of sequences of real numbers $\{\lambda_k\}_{k\in {\mathbb{Z}}}$ such that
$$\begin{aligned}
\lambda_k &\geq 0 \quad \text{for all} \, \, k \notag \\
\lambda_k &= 0 \quad \text{unless} \, i > k > j. \notag
\end{aligned}$$
For consistency of notation we write $Mor_{{\mathcal{J}}}(i,j) = J(i,j)^+$. Composition of morphisms $ J(i,j)^+ \wedge J(j,k)^+ \to J(i,k)^+ $ is then induced by addition of sequences. In this smash product the basepoint is taken to be $\infty$. Notice that this map is basepoint preserving. Given integers $p > q$, then there are subcategories ${\mathcal{J}}^p_q$ defined to be the full subcategory generated by integers $q \geq m \geq p.$ The category ${\mathcal{J}}_q$ is the full subcategory of ${\mathcal{J}}$ generated by all integers $m \geq q$.
The following is a recasting of a discussion in [@cjs].
\[realmod\] The realizations of the chain complex $C_*$ by finite spectra correspond to extensions of the association $j \to {\mathcal{E}}_j$ to basepoint preserving functors $Z : {\mathcal{J}}_0 \to Spectra$, with the property that for each $j \geq 0$, the map obtained by the application of morphisms, $$\begin{aligned}
Z_{j+1, j} : J(j+1,j)^+ &\wedge {\mathcal{E}}_{j+1} \to {\mathcal{E}}_{j} \notag \\
S^0 &\wedge {\mathcal{E}}_{j+1} \to {\mathcal{E}}_{j} \notag
\end{aligned}$$ induces the boundary homomorphism ${\partial}_{j+1} $ on the level of homology groups. Here $Spectra$ is a symmetric monoidal category of spectra (e.g the category of symmetric spectra), and by a “basepoint preserving functor" we mean one that maps $\infty$ in $J (p,q)^+$ to the constant map $Z(p) \to Z(q)$.
Since this is a recasting of a result in [@cjs] we supply a proof here.
Suppose one has a functor $Z : {\mathcal{J}}_0 \to Spectra$ satisfying the properties described in the theorem. One defines a “geometric realization" of this functor as follows.
As described in [@cjs], given a functor to the category of spaces, $ Z : {\mathcal{J}}_q \to Spaces_*$, where $Spaces_*$ denotes the category of based topological spaces, one can take its geometric realization, $$\label{zrealize}
|Z| = \coprod_{q\leq j } Z(j)\wedge J(j, q-1)^+ / \sim$$ where one identifies the image of $Z(j) \wedge J(j,i)^+\wedge J(i, q-1)^+$ in $Z(j) \wedge J(j, q-1)^+$ given by composition of morphisms, with its image in $Z(i) \wedge J(i, q-1)$ defined by application of morphisms $Z(j) \wedge J(j,i)^+ \to Z(i)$.
For a functor whose value is in $Spectra$, we replace the above construction by a coequalizer, in the following way:
Let $Z : {\mathcal{J}}_q \to Spectra$. Define two maps of spectra, $$\label{iotamu}
\iota, \mu : \bigvee_{q \leq j} Z(j) \wedge J(j,i)^+\wedge J(i, q-1)^+ {\longrightarrow}\bigvee_{q\leq j } Z(j)\wedge J(j, q-1)^+.$$ The first map $\iota$ is induced by the composition of morphisms in ${\mathcal{J}}_q$, $J(j,i)^+ \wedge J(i, q-1)^+ {\hookrightarrow}J(j, q-1)^+.$ The second map $\mu$ is the given by the wedge of maps, $$Z(j) \wedge J(j,i)^+\wedge J(i, q-1)^+ {\xrightarrow}{\mu_q \wedge 1} Z(i)\wedge J(i, q-1)^+$$ where $\mu_q : Z(j) \wedge J(j,i)^+ \to Z(i)$ is given by application of morphisms.
\[georeal\] Given a functor $Z: {\mathcal{J}}_q \to Spectra$ we define its geometric realization $|Z|$ to be the homotopy coequalizer (in the category $Spectra$) of the two maps, $$\iota, \mu : \bigvee_{q \leq j} \bigvee_{i = q}^{j-1}Z(j) \wedge J(j,i)^+\wedge J(i, q-1)^+ {\longrightarrow}\bigvee_{q\leq j } Z(j)\wedge J(j, q-1)^+.$$
Given a functor $Z : {\mathcal{J}}_0 \to Spectra$ as in the statement of the theorem, we claim the geometric realization $|Z|$ realizes the chain complex $C_*$. In this realization, the filtration $$|Z|_0 {\hookrightarrow}|Z|_1 {\hookrightarrow}\cdots |Z|_k {\hookrightarrow}|Z|_{k+1} {\hookrightarrow}\cdots {\hookrightarrow}|Z|_n = |Z|$$ is the natural one where $|Z|_k $ is the homotopy coequalizer of $\iota$ and $\mu$ restricted to $$\label{zeekay}
\iota, \, \mu : \bigvee_{j=0}^k\bigvee_{i = 0}^{j-1} Z(j) \wedge J(j,i)^+\wedge J(i, -1)^+ {\longrightarrow}\bigvee_{j=0}^k Z(j)\wedge J(j, -1)^+.$$
To see that this is a filtration with the property that $|Z|_k/|Z|_{k-1} \simeq \Sigma^k {\mathcal{E}}_k$, notice that for a based space or spectrum, $Y$, the smash product $J(n,m)^+\wedge Y$ is homeomorphic iterated cone, $$\label{cone}
Y \wedge J(n,m)^+ \cong c^{n-m-1}(Y).$$ In particular $Z(k) \wedge J(k, -1)^+$ is the iterated cone $ c^k(Z(k)) = c^k({\mathcal{E}}_k).$ Now lets consider how $|Z|_k$ is built from $|Z|_{k-1}$. Let ${\partial}J(k, -1)$ denote the subspace of $J(k, -1)$ defined to consist of those sequences of real numbers $\{\lambda_i\}_{i\in {\mathbb{Z}}}$ such that $$\begin{aligned}
\lambda_i &\geq 0 \quad \text{for all} \quad i \notag \\
\lambda_i &= 0 \quad \text{unless} \quad k > i > -1, \notag \\
\text{there is at least one value of $i$ for} \, &k > i > -1, \text{ such that} \, \lambda_i = 0. \notag
\end{aligned}$$
It is easy to check that ${\partial}J(k,-1)$ is homeomorphic to ${\mathbb{R}}^{k-1}$, and its one-point compactification ${\partial}J(k, -1)^+$ is homeomorphic to the sphere $S^{k-1}$.
Now consider the map $\hat \mu : Z(k) \wedge {\partial}J(k, -1)^+\to |Z|_{k-1}$ defined by the maps $$\mu : Z(k) \wedge J(k, i)^+ \wedge J(i, -1)^+ \to |Z|_{k-1}$$ as described in (\[iotamu\]) above. In particular $\mu = Z_{k,i} \wedge 1$ where $Z_{k,i} : Z(k) \wedge J(k, i)^+ \to Z(i) $ is given by the application of morphisms.
It is then clear from the definition of $|Z|_k$ (\[zeekay\]) that it is the homotopy cofiber of the map
$$\begin{aligned}
\hat \mu : {\partial}J(k, -1)^+ \wedge Z(k) &\to |Z|_{k-1} \notag \\
\cong S^{k-1} \wedge Z(k) &\to |Z|_{k-1}.\end{aligned}$$
Therefore the homotopy cofiber of the inclusion $|Z|_{k-1} {\hookrightarrow}|Z|_k$ is $S^k \wedge Z(k)$, which by hypothesis is $S^k \wedge {\mathcal{E}}_k$.
We now show that the map $$\begin{aligned}
Z_{j+1, j} : J(j+1,j)^+ &\wedge {\mathcal{E}}_{j+1} \to {\mathcal{E}}_{j} \notag \\
S^0 &\wedge {\mathcal{E}}_{j+1} \to {\mathcal{E}}_{j} \notag
\end{aligned}$$ induces the boundary homomorphism ${\partial}_j$ in homology. To see this, recall that the boundary homomorphism is given by applying homology to the composition $$\begin{aligned}
{\partial}: |Z|_{j+1}/|Z|_j &{\xrightarrow}{\alpha} \Sigma |Z|_j {\xrightarrow}{\rho} \Sigma |Z|_j/|Z|_{j-1} \notag \\
{\partial}: S^{j+1}\wedge Z(j+1) &\to S^{j+1} \wedge Z(j) \notag \\
{\partial}: S^{j+1}\wedge {\mathcal{E}}_{j+1} &\to S^{j+1} \wedge {\mathcal{E}}_j \notag
\end{aligned}$$ Here $\alpha : |Z|_{j+1}/|Z|_j \to \Sigma |Z|_j $ is the connecting map in the Puppe extension of the cofibration sequence $$|Z|_j {\hookrightarrow}|Z|_{j+1} \to |Z|_{j+1}/|Z|_j$$ and $\rho : \Sigma |Z|_j \to \Sigma |Z|_j/|Z|_{j-1}$ is projection onto the (homotopy) cofiber. By the above discussion we have a homotopy cofibration sequence $$\cdots \to Z(j+1) \wedge {\partial}J(j+1, -1) {\xrightarrow}{\hat \mu} |Z|_j \to |Z|_{j+1} {\xrightarrow}{\rho} |Z|_{j+1}/ |Z|_j {\xrightarrow}{\alpha} \Sigma |Z|_j \to \cdots$$ So we have a homotopy commutative diagram
$$\begin{CD}
|Z|_{j+1}/ |Z|_j @>\alpha >> \Sigma |Z|_j \\
@V \simeq VV @VV = V \\
\Sigma {\partial}J(j+1, -1) \wedge Z(j+1) @>> \Sigma \hat \mu > \Sigma |Z|_j
\end{CD}$$
So the boundary map ${\partial}$ given above can be viewed as the composition $$\label{comp} {\partial}: \Sigma {\partial}J(j+1, -1) \wedge Z(j+1) {\xrightarrow}{ \Sigma \hat \mu } \Sigma |Z|_j {\xrightarrow}{\rho} \Sigma |Z|_j/|Z|_{j-1}.$$
Notice that when restricted to any $\Sigma J(j+1, r)^+ \wedge J(r, -1)^+ \wedge Z(j+1)$ for $r < j$, the composition ${\partial}$ is naturally null homotopic. This says that ${\partial}$ factors through the umkehr map (Pontrjagin-Thom construction) $$\label{pontthom}
{\partial}: \Sigma {\partial}J(j+1, -1) \wedge Z(j+1) {\xrightarrow}{\tau} \Sigma J(j+1, j)^+ \wedge int(J(j, -1)^+ \wedge Z(j+1) {\xrightarrow}{ \bar {\partial}} \Sigma |Z|_j/|Z|_{j-1}.$$ Here for $r > s$ $int (J(r, s)) $ is the interior of the space $J(r,s)$. This consists of sequences $\{\lambda_i\}_{i\in {\mathbb{Z}}}$ such that $$\begin{aligned}
\lambda_i &\geq 0 \quad \text{for all} \quad i \notag \\
\lambda_i & = 0 \quad \text{unless} \quad r > i > s \quad \text{in which case} \quad \lambda_i > 0. \notag
\end{aligned}$$ In particular $int (J(r, s)) \cong {\mathbb{R}}^{r-s-1}$ and so $int (J(r, s))^+ \cong S^{r-s-1}$.
Thus the boundary map ${\partial}$ can be viewed as the composition $$\begin{aligned}
{\partial}: \Sigma {\partial}J(j+1, -1) \wedge Z(j+1) &{\xrightarrow}{\tau} \Sigma J(j+1, j)^+ \wedge int(J(j, -1)^+ \wedge Z(j+1) {\xrightarrow}{ \bar {\partial}} \Sigma |Z|_j/|Z|_{j-1} \notag \\
{\partial}: S^{j+1} \wedge Z(j+1) &{\xrightarrow}{\tau} J(j+1,j) \wedge S^{j+1} {\xrightarrow}{ \bar {\partial}} \Sigma |Z|_j/|Z|_{j-1} \simeq S^{j+1}\wedge Z(j) \notag
\end{aligned}$$ where $\tau$ is an equivalence. But by the definition of $\bar {\partial}$ (\[pontthom\]) in terms of $ \mu = Z_{j, j-1} : J(j+1, j)^+ \wedge Z(j) \to Z(j-1)$, we see that the boundary homomorphism ${\partial}_j : C_j \to C_{j-1} $ is defined by the application of morphisms, as asserted.
So a functor $Z : {\mathcal{J}}_0 \to Spectra$ satisfying the properties specified in Theorem \[realmod\] defines a geometric realization $|Z|$ which is a finite spectrum. Consider how the data of the functor $Z$ defines the $CW$-structure of $|Z|$. Clearly $|Z|$ will have one cell of dimension $i$ for every element of $\pi_0(Z(i)) = \pi_0({\mathcal{E}}_i) = {\mathcal{B}}_i$. The attaching maps were described in [@cjs], [@cotangent], [@floeroslo] in the following way.
In general assume that $X$ be a finite $CW$-spectrum with skeletal filtration $$X_0 {\hookrightarrow}X_1 {\hookrightarrow}\cdots {\hookrightarrow}X_n = X.$$ In particular each map $X_{i-1} {\hookrightarrow}X_i$ is a cofibration, and we call its cofiber $K_i = X_i /(X_{i-1})$. This is a wedge of (suspension spectra) of spheres of dimension $i$. $$K_i \simeq \bigvee_{{\mathcal{D}}_i} \Sigma^i{\mathbb{S}}$$ where ${\mathcal{D}}_i$ is a finite indexing set.
As was discussed in [@cjs] one can then “rebuild" the homotopy type of the $n$-fold suspension, $\Sigma^n X,$ as the union of iterated cones and suspensions of the $K_i$’s, $$\label{decomp}
\Sigma^n X \simeq \Sigma^nK_0 \cup c(\Sigma^{n-1} K_1) \cup \cdots \cup c^i(\Sigma^{n-i} K_i) \cup \cdots \cup c^n K_n.$$
This decomposition can be described as follows. Define a map $\delta_i : \Sigma^{n-i}K_i \to \Sigma^{n-i+1}K_{i-1}$ to be the iterated suspension of the composition, $$\delta_i : K_i \to \Sigma X_{i-1} \to \Sigma K_{i-1}$$ where the two maps in this composition come from the cofibration sequence, $X_{i-1}\to X_i \to K_i \to \Sigma X_{i-1} \cdots$. As was pointed out in [@cotangent], this induces a “homotopy chain complex",
$$\label{htpychain}
K_n {\xrightarrow}{\delta_n} \Sigma K_{n-1} {\xrightarrow}{\delta_{n-1}} \cdots {\xrightarrow}{\delta_{i+1}}\Sigma^{n-i }K_{i } {\xrightarrow}{\delta_{i}} \Sigma^{n-i+1}K_{i-1} {\xrightarrow}{\delta_{i-1}} \cdots {\xrightarrow}{\delta_1}\Sigma^{n}K_0 = \Sigma^n X_0.$$
We refer to this as a homotopy chain complex because examination of the defining cofibrations leads to canonical null homotopies of the compositions, $$\delta_j \circ \delta_{j+1}.$$ This canonical null homotopy defines an extension of $\delta_j$ to the mapping cone of $\delta_{j+1}$: $$c(\Sigma^{n-j-1}K_{j+1}) \cup_{\delta_{j+1}} \Sigma^{n-j}K_j {\longrightarrow}\Sigma^{n-j+1}K_{j-1}.$$ More generally, for every $q$, using these null homotopies, we have an extension to the iterated mapping cone, $$\label{attach}
c^q(\Sigma^{n-j-q}K_{j+q}) \cup c^{q-1}(\Sigma^{n-j-q+1}K_{j+q-1}) \cup \cdots \cup c(\Sigma^{n-j-1}K_{j+1}) \cup_{\delta_{j+1}} \Sigma^{n-j}K_j {\longrightarrow}\Sigma^{n-j+1}K_{j-1}.$$
In other words, for each $p > q$, these null homotopies define maps of spectra, $$\label{phi}
\phi_{p,q} : c^{p- q-1}\Sigma^{n-p}K_p \to \Sigma^{n-q}K_q.$$ The cell attaching data in the $CW$-spectrum $X$ as in (\[decomp\]) is then defined via the maps $\phi_{p,q}$.
Given a functor $Z : {\mathcal{J}}_0 \to Spectra$ satisfying the hypotheses of Theorem \[realmod\], We have that $$K_p = |Z|_p/|Z|_{p-1} \simeq \bigvee_{{\mathcal{B}}_p} \Sigma^p {\mathbb{S}}$$ and the attaching maps $$\begin{aligned}
\phi_{p,q} : &c^{p- q-1}\Sigma^{-p}K_p \to \Sigma^{-q}K_q \\
&J(p,q)^+ \wedge\Sigma^{-p} |Z|_p/|Z|_{p-1} \to \Sigma^{-q}|Z|_q/|Z|_{q-1} \notag \\
&J(p,q)^+ \wedge Z(p) \to Z(q) \notag\end{aligned}$$ is given by the application of the morphisms of the category ${\mathcal{J}}_0$ to the value of the functor $Z$.
Manifolds with corners and framed flow categories
-------------------------------------------------
In order to make use of Theorem \[realmod\] in the setting of Morse and Floer theory, one needs a more geometric way of understanding the homotopy theoretic information contained in a functor $Z : {\mathcal{J}}_0 \to Spectra$ satisfying the hypotheses of the theorem. As is common in algebraic and differential topology, the translation between homotopy theoretic information and geometric information is done via cobordism theory and the Pontrjagin-Thom construction.
In the case when the $CW$-structure comes from a Morse function $f : M^n \to {\mathbb{R}}$ on a closed $n$-dimensional Riemannian manifold, the attaching maps $\phi_{p,q}$ defining a functor $Z_f : {\mathcal{J}}_0 \to Spectra$, were shown in [@cotangent], [@floeroslo] to come from the cobordism-type of the moduli spaces of gradient flow lines connecting critical points of index $p$ to those of index $q$. The relevant cobordism theory is *framed cobordism of manifolds with corners. Thus to extend this idea to the Floer setting, with the goal of developing a “Floer homotopy type", one needs to understand certain cobordism-theoretic properties of the corresponding moduli spaces of gradient flows of the particular Floer theory. (In [@floeroslo] the author also considered how other cobordism theories give rise not to “Floer homotopy types" but rather to “Floer module spectra", or said another way, “Floer $E$-theory" where $E_\bullet $ is a generalized homology theory. We refer the reader to [@floeroslo] for details.) In order to make this idea precise we recall some basic facts about cobordisms of manifolds with corners. The main reference for this is Laures’s paper [@laures].*
Recall that an $n$-dimensional manifold with corners, $M$, has charts which are local homeomorphisms with ${\mathbb{R}}_+^n$. Here ${\mathbb{R}}_+$ denotes the nonnegative real numbers and ${\mathbb{R}}_+^n$ is the $n$-fold cartesian product of ${\mathbb{R}}_+$. Let $\psi : U \to {\mathbb{R}}_+^n$ be a chart of a manifold with corners $M$. For $x \in U$, the number of zeros of this chart, $c(x)$ is independent of the chart. One defines a *face of $M$ to be a connected component of the space $\{m{\in}M \, \text{such that} \, c(m)=1\}$.*
Given an integer $k$, there is a notion of a manifold with corners having “codimension k", or a $\langle k\rangle$-manifold. We recall the definition from [@laures].
\[kmanifold\] A $\langle k\rangle$-manifold is a manifold with corners, $M$, together with an ordered $k$-tuple $(\partial{_1}M,...,\partial{_k}M)$ of unions of faces of $M$ satisfying the following properties.
1. Each $m{\in}M$ belongs to $c(m)$ faces
2. $\partial{_1}M \, {\cup} {\cdots} {\cup} \, \partial_{k} M = \partial {M} $
3. For all $1{\leq}i{\neq}j{\leq}k$, $\partial{_i}M \, {\cap} \, \partial{_j}M$ is a face of both $\partial{_i}M$ and $\partial{_j}M$.
The archetypical example of a $\langle k \rangle$-manifold is $ {\mathbb{R}}_+^k$. In this case the face $F_j \subset {\mathbb{R}}_+^k$ consists of those $k$-tuples with the $j^{th}$-coordinate equal to zero.
As described in [@laures], the data of a $\langle k \rangle$-manifold can be encoded in a categorical way as follows. Let $\underbar{2}$ be the partially ordered set with two objects, $\{0, 1\}$, generated by a single nonidentity morphism $0 \to 1$. Let ${\underbar{2}}^k$ be the product of $k$-copies of the category ${\underbar{2}}$. A $\langle k \rangle$-manfold $M$ then defines a functor from ${\underbar{2}}^k$ to the category of topological spaces, where for an object $a = (a_1, \cdots , a_k) \in {\underbar{2}}^k$, $M(a)$ is the intersection of the faces ${\partial}_i(M)$ with $a_i = 0$. Such a functor is a $k$-dimensional cubical diagram of spaces, which, following Laures’ terminology, we refer to as a $\langle k \rangle$-diagram, or a $\langle k \rangle$-space. Notice that ${\mathbb{R}}_+^k(a) \subset {\mathbb{R}}_+^k$ consists of those $k$-tuples of nonnegative real numbers so that the $i^{th}$-coordinate is zero for every $i$ with $a_i=0$. More generally, consider the $\langle k\rangle$-Euclidean space, ${\mathbb{R}}_+^k \times {\mathbb{R}}^n$, where the value on $a \in {\underbar{2}}^k$ is ${\mathbb{R}}_+^k(a) \times {\mathbb{R}}^n$. In general we refer to a functor $\phi : {\underbar{2}}^k \to {\mathcal{C}}$ as a $\langle k\rangle$-object in the category ${\mathcal{C}}$.
In this section we will consider embeddings of manifolds with corners into Euclidean spaces $M {\hookrightarrow}{\mathbb{R}}_+^k \times {\mathbb{R}}^n$ of the form given by the following definition.
\[embed\] A “neat embedding" of a $\langle k \rangle$-manifold $M$ into $ {\mathbb{R}}_+^k \times {\mathbb{R}}^m $ is a natural transformation of $\langle k \rangle$-diagrams $$e : M {\hookrightarrow}{\mathbb{R}}_+^k \times {\mathbb{R}}^m$$ that satisfies the following properties:
1. For each $a{\in}\underline{2}^k$, $e(a)$ is an embedding.
2. For all $b < a$, the intersection $M(a) \cap \left( \mathbb{R}^k_+(b) \times \mathbb{R}^m\right) = M(b)$, and this intersection is perpendicular. That is, there is some $\epsilon > 0$ such that $$M(a) \cap \left(\mathbb{R}^k_+(b) \times [0,\epsilon)^k(a-b) \times \mathbb{R}^m\right) = M(b) \times [0,\epsilon)^k(a-b).$$
Here $a-b$ denotes the object of ${\underbar{2}}^k$ obtained by subtracting the $k$-vector $b$ from the $k$-vector $a$.
In [@laures] it was proved that every $\langle k \rangle$-manifold neatly embeds in $\mathbb{R}^{k}_+{\times} {\mathbb{R}}^N$ for $N$ sufficiently large. In fact it was proved there that a manifold with corners, $M$, admits a neat embedding into $\mathbb{R}^{k}_+{\times} {\mathbb{R}}^N$ *if and only if $M$ has the structure of a $\langle k\rangle$-manifold. Furthermore in [@genauer] it is shown that the connectivity of the space of neat embeddings, $Emb_{\langle k\rangle}(M; \mathbb{R}^{k}_+{\times} {\mathbb{R}}^N)$ increases with the dimension $N$.*
Notice that an embedding of manifolds with corners, $e : M {\hookrightarrow}{\mathbb{R}}_+^k \times {\mathbb{R}}^m $, has a well defined normal bundle. In particular, for any pair of objects in ${\underbar{2}}^k$, $a > b$, the normal bundle of $e(a) : M(a) {\hookrightarrow}{\mathbb{R}}_+^k(a) \times {\mathbb{R}}^m$, when restricted to $M(b)$, is the normal bundle of $e(b) : M(b) {\hookrightarrow}{\mathbb{R}}_+^k(b) \times {\mathbb{R}}^m$.
These embedding properties of $\langle k \rangle$ - manifolds make it clear that these are the appropriate manifolds to study for cobordism - theoretic information. In particular, given an embedding $e: M {\hookrightarrow}{\mathbb{R}}_+^k \times {\mathbb{R}}^m$ the Thom space of the normal bundle, $Th (M, e)$, has the structure of an $\langle k\rangle$-space, where for $a \in {\underbar{2}}^k$, $Th (M, e)(a)$ is the Thom space of the normal bundle of the associated embedding, $M(a) {\hookrightarrow}{\mathbb{R}}_+^k (a) \times {\mathbb{R}}^N$. We can then desuspend and define the Thom spectrum, $M^\nu_e = \Sigma^{-N}Th (M,e)$, to be the associated $\langle k\rangle$-spectrum. The Pontrjagin-Thom construction defines a map of $\langle k\rangle$-spaces, $$\tau_e : \left( {\mathbb{R}}_+^k \times {\mathbb{R}}^N\right)\cup \infty = (( {\mathbb{R}}_+^k)\cup \infty) \wedge S^N \to Th (M, e).$$ Desuspending we get a map of $\langle k\rangle$-spectra, $\Sigma^\infty (( {\mathbb{R}}_+^k)\cup \infty) \to M^\nu_e. $ Notice that the homotopy type (as $\langle k\rangle$-spectra) of $M^\nu_e$ is independent of the embedding $e$. We denote the homotopy type of this normal Thom spectrum as $M^\nu$, and the Pontrjagin-Thom map, $\tau : \Sigma^\infty (( {\mathbb{R}}_+^k)\cup \infty) \to M^\nu$.
Compact manifolds with corners, and in particular $\langle k \rangle$ - manifolds naturally occur as the moduli spaces of flow lines of a Morse function, and in some cases, of a Floer function. We first recall how they appear in Morse theory.
Consider a smooth, closed $n$-manifold $M^n$, and a smooth Morse function $f : M^n \to {\mathbb{R}}$. Given a Riemannian metric on $M$, one studies the flow of the gradient vector field $\nabla f$. In particular a flow line is a curve $\gamma : {\mathbb{R}}\to M$ satisfying the ordinary differential equation, $$\frac{d}{dt}\gamma(s) + \nabla f (\gamma (s)) = 0.$$ By the existence and uniqueness theorem for solutions to ODE’s, one knows that if $x \in M$ is any point then there is a unique flow line $\gamma_x$ satisfying $\gamma_x(0) = x$. One then studies unstable and stable manifolds of the critical points, $$\begin{aligned}
W^u(a) &= \{ x \in M \, : \, \lim_{t\to -\infty}\gamma_x (t) = a \} \notag \\
W^s(a) &= \{ x \in M \, : \, \lim_{t\to +\infty}\gamma_x (t) = a \}. \notag
\end{aligned}$$
The unstable manifold $W^u(a)$ is diffeomorphic to a disk $D^{\mu (a)}$, where $\mu (a)$ is the index of the critical point $a$. Similarly the stable manifold $W^s(a)$ is diffeomorphic to a disk $D^{n-\mu (a)}$.
For a generic choice of Riemannian metric, the unstable manifolds and stable manifolds intersect transversally, and their intersections, $$W(a,b) = W^u(a) \cap W^s (b)$$ are smooth manifolds of dimension equal to the relative index, $\mu (a) - \mu (b)$. When the choice of metric satisfies these transversality properties, the metric is said to be “Morse-Smale". The manifolds $W(a,b)$ have free ${\mathbb{R}}$-actions defined by “going with the flow". That is, for $t \in {\mathbb{R}}$, and $x \in M$, $$t\cdot x = \gamma_x (t).$$ The “moduli space of flow lines" is the manifold $${\mathcal{M}}(a,b) = W(a,b) /{\mathbb{R}}$$ and has dimension $\mu (a) - \mu (b) -1$. These moduli spaces are not generally compact, but they have canonical compactifications which we now describe.
In the case of a Morse-Smale metric, (which we assume throughout the rest of this section), there is a partial order on the finite set of critical points given by $a \geq b$ if ${\mathcal{M}}(a,b) \neq \emptyset$. We then define $$\label{compact}
\bar {\mathcal{M}}(a,b) = \bigcup_{a=a_1 >a_2> \cdots > a_k = b} {\mathcal{M}}(a_1, a_2) \times \cdots \times {\mathcal{M}}(a_{k-1}, a_k),$$
The topology of $\bar {\mathcal{M}}(a,b)$ can be described naturally, and is done so in many references including [@cjs1]. ${\bar {\mathcal{M}}}(a,b)$ is the space of “piecewise flow lines" emanating from $a$ and ending at $b$.
The following definition of a Morse function’s “flow category " was also given in [@cjs1].
The *flow category ${\mathcal{C}}_f$ is a topological category associated to a Morse function $f : M \to {\mathbb{R}}$ where $M$ is a closed Riemannian manifold. Its objects are the critical points of $f$. If $a$ and $b$ are two such critical points, then $Mor_{{\mathcal{C}}_f}(a, b) = {\bar {\mathcal{M}}}(a, b)$. Composition is determined by the maps $${\bar {\mathcal{M}}}(a,b) \times {\bar {\mathcal{M}}}(b, c) {\hookrightarrow}{\bar {\mathcal{M}}}(a, c)$$ which are defined to be the natural embeddings into the boundary.*
The moduli spaces ${\mathcal{M}}(a,b)$ have natural framings on their stable normal bundles (or equivalently, their stable tangent bundles) that play an important role in this theory. These framings are defined in the following manner. Let $a > b$ be critical points. Let ${\epsilon}> 0$ be chosen so that there are no critical values in the half open interval $[f(a)-{\epsilon}, f(a))$. Define the *unstable sphere to be the level set of the unstable manifold, $$S^u(a) = W^u(a) \cap f^{-1}(f(a)-{\epsilon}).$$ The sphere $S^u(a)$ has dimension $ \mu(a) - 1$. Notice there is a natural diffeomorphism, $${\mathcal{M}}(a,b) \cong S^u(a) \cap W^s(b).$$ This leads to the following diagram, $$\label{intersect}
\begin{CD}
W^s(b) @>{\hookrightarrow}>> M \\
@A \cup AA @AA\cup A \\
{\mathcal{M}}(a,b) @>>{\hookrightarrow}> S^u(a).
\end{CD}$$ From this diagram one sees that the normal bundle $\nu$ of the embedding ${\mathcal{M}}(a,b) {\hookrightarrow}S^u(a)$ is the restriction of the normal bundle of $W^s(b) {\hookrightarrow}M$. Since $W^s(b)$ is a disk, and therefore contractible, this bundle is trivial. Indeed an orientation of $W^s(b)$ determines a homotopy class of trivialization, or a framing. In fact this framing determines a diffeomorphism of the bundle to the product, $W^s(b) \times W^u(b)$. Thus these orientations give the moduli spaces ${\mathcal{M}}(a,b)$ canonical normal framings, $\nu \cong {\mathcal{M}}(a,b) \times W^u(b) . $*
As was pointed out in [@cjs1], these framings extend to the boundary of the compactifications, ${\bar {\mathcal{M}}}(a, b)$. In order to describe what it means for these framings to be “coherent" in an appropriate sense, the following categorical approach was used in [@floeroslo]. The first step is to abstract the basic properties of a flow category of a Morse functon.
\[cptcat\] A smooth, compact category is a topological category ${\mathcal{C}}$ whose objects form a discrete set, and whose whose morphism spaces, $Mor (a,b)$ are compact, smooth $\langle k \rangle$ -manifolds, where $k = dim \, Mor(a,b)$. The composition maps, $\nu : Mor (a,b) \times Mor (b, c) \to Mor(a, c)$, are smooth codimension one embeddings (of manifolds with corners) whose images lie in the boundary. Moreover every point in the boundary of $Mor(a, c)$ is in the image under $\nu$ of a unique maximal sequence in $Mor(a, b_1) \times Mor (b_1, b_2) \times \cdots \times Mor(b_{k-1}, b_k)\times Mor (b_k, c)$ for some objects $\{b_1, \cdots , b_k\}$.
A smooth, compact category ${\mathcal{C}}$ is said to be a “Morse-Smale" category if the following additional properties are satisfied.
1. The objects of ${\mathcal{C}}$ are partially ordered by the condition $$a \geq b \quad \text{if} \quad Mor (a,b) \neq \emptyset.$$
2. $Mor(a,a) = \{identity \}$.
3. There is a set map, $\mu : Ob ({\mathcal{C}}) \to {\mathbb{Z}}$, which preserves the partial ordering, such that if $a > b$, $$dim \, Mor(a,b) = \mu (a) - \mu (b) -1.$$ The map $\mu$ is known as an “index" map. A Morse-Smale category such as this is said to have finite type, if there are only finitely many objects of any given index, and for each pair of objects $a > b$, there are only finitely many objects $c$ with $a > c > b$. For ease of notation we write $k(a,b) = \mu(a) -\mu (b) -1$.
The following is a folk theorem that goes back to the work of Smale and Franks [@franks] although a proof of this fact did not appear in the literature until much later [@qin].
Let $f : M \to {\mathbb{R}}$ be smooth Morse function on a closed Riemannian manifold with a Morse-Smale metric. Then the compactified moduli space of piecewise flow-lines, $\bar {\mathcal{M}}(a,b)$ is a smooth $\langle k(a,b) \rangle $ - manifold.
Using this result, as well as an associativity result for the gluing maps ${\bar {\mathcal{M}}}(a,b) \times {\bar {\mathcal{M}}}(b, c) \to
{\bar {\mathcal{M}}}(a,c)$ which was eventually proved in [@qin], it was proven in [@cjs] that the flow category ${\mathcal{C}}_f$ of such a Morse-Smale function is indeed a Morse-Smale smooth, compact category according to Definition \[cptcat\].
**Remark. The fact that [@cjs1] was never submitted for publication was due to the fact that the “folk theorem" mentioned above, as well as the associativity of gluing, both of which the authors of [@cjs1] assumed were “well known to the experts", were indeed not in the literature, and their proofs which was eventually provided in [@qin], were analytically more complicated than the authors imagined.**
In order to define the notion of “coherent framings" of the moduli spaces ${\bar {\mathcal{M}}}(a,b)$, so that we may apply the Pontrjagin-Thom construction coherently, we need to study an associated category, enriched in spectra, defined using the stable normal bundles of the moduli spaces of flows.
\[normalthom\] Let ${\mathcal{C}}$ be a smooth, compact category of finite type satisfying the Morse-Smale condition. Then a “normal Thom spectrum" of the category ${\mathcal{C}}$ is a category, ${\mathcal{C}}^\nu$, enriched over spectra, that satisfies the following properties.
1. The objects of ${\mathcal{C}}^\nu$ are the same as the objects of ${\mathcal{C}}$.
2. The morphism spectra $Mor_{{\mathcal{C}}^\nu}(a,b)$ are $\langle k(a,b)\rangle$-spectra, having the homotopy type of the normal Thom spectra $Mor_{{\mathcal{C}}}(a,b)^\nu$, as $\langle k(a,b)\rangle$-spectra. The composition maps, $$\circ : Mor_{{\mathcal{C}}^\nu}(a,b)\wedge Mor_{{\mathcal{C}}^\nu}(b,c) \to Mor_{{\mathcal{C}}^\nu}(a,c)$$ have the homotopy type of the maps, $$Mor_{{\mathcal{C}}}(a,b)^\nu \wedge Mor_{{\mathcal{C}}}(b,c)^\nu \to Mor_{{\mathcal{C}}}(a,c)^\nu$$ of the Thom spectra of the stable normal bundles corresponding to the composition maps in ${\mathcal{C}}$, $Mor_{{\mathcal{C}}} (a,b) \times Mor_{{\mathcal{C}}} (b,c) \to Mor_{{\mathcal{C}}}(a,c)$. Recall that these are maps of $\langle k(a,c)\rangle$-spaces induced by the inclusion of a component of the boundary.
3. The morphism spectra are equipped with “Pontrjagin-Thom maps" $\tau_{a,b} : \Sigma^\infty (J(\mu (a), \mu (b))^+) = \Sigma^\infty ( ( {\mathbb{R}}_+^{k(a,b)})\cup \infty)) \to Mor_{{\mathcal{C}}^\nu}(a,b)$ such that the following diagram commutes: $$\begin{CD}
\Sigma^\infty (J(\mu (a), \mu (b))^+) \wedge \Sigma^\infty (J(\mu (b), \mu (c))^+) @>>> \Sigma^\infty (J(\mu (a), \mu (c))^+) \\
@V\tau_{a.b}\wedge \tau_{b,c} VV @VV\tau_{a,c} V \\
Mor_{{\mathcal{C}}^\nu}(a,b) \wedge Mor_{{\mathcal{C}}^\nu}(b,c) @>>> Mor_{{\mathcal{C}}^\nu}(a,c).
\end{CD}$$ Here the top horizontal map is defined via the composition maps in the category ${\mathcal{J}}$, and the bottom horizontal map is defined via the composition maps in ${\mathcal{C}}^\nu$.
With the notion of a “normal Thom spectrum" of a flow category ${\mathcal{C}}$, the notion of a coherent $E^*$-orientation was defined in [@floeroslo]. Here $E^*$ is a generalized cohomology theory represented by a commutative ($E_\infty$) ring spectrum $E$. We recall that definition now.
First observe that a commutative ring spectrum $E$ induces a $\langle k\rangle$-diagram in the category of spectra (“$\langle k\rangle$-spectrum"), $E\langle k\rangle$, defined in the following manner.
For $k = 1$, we let $E\langle1\rangle: {\underbar{2}}\to Spectra$ be defined by $E\langle 1\rangle(0) = S^0$, the sphere spectrum, and $E\langle1\rangle(1) = E$. The image of the morphism $0 \to 1$ is the unit of the ring spectrum $S^0 \to E$.
To define $E\langle k\rangle$ for general $k$, let $a$ be an object of ${\underbar{2}}^k$. We view $a$ as a vector of length $k$, whose coordinates are either zero or one. Define $E\langle k\rangle (a)$ to be the multiple smash product of spectra, with a copy of $S^0$ for every every zero coordinate, and a copy of $E$ for every string of successive ones. For example, if $k = 6$, and $a = (1, 0, 1,1, 0,1)$, then $E\langle k\rangle (a) = E\wedge S^0\wedge E\wedge S^0\wedge E$.
Given a morphism $a \to a'$ in ${\underbar{2}}^k$, one has a map $E\langle k\rangle (a) \to E\langle k\rangle (a')$ defined by combining the unit $S^0 \to E$ with the ring multiplication $E\wedge E \to E$.
Said another way, the functor $E\langle k\rangle : {\underbar{2}}^k \to Spectra$ is defined by taking the $k$-fold product functor $E\langle 1\rangle : {\underbar{2}}\to Spectra$ which sends $(0 \to 1)$ to $S^0 \to E$, and then using the ring multiplication in $E$ to “collapse" successive strings of $E$’s.
This structure allows us to define one more construction. Suppose ${\mathcal{C}}$ is a smooth, compact, Morse-Smale category of finite type as in Definition \[cptcat\]. We can then define an associated category, $E_{{\mathcal{C}}}$, whose objects are the same as the objects of ${\mathcal{C}}$ and whose morphisms are given by the spectra, $$Mor_{E_{{\mathcal{C}}}}(a,b) = E\langle k(a,b)\rangle$$ where $k(a,b) = \mu(a) - \mu (b) -1$. Here $\mu (a)$ is the index of the object $a$ as in Definition \[cptcat\]. The composition law is the pairing, $$\begin{aligned}
E\langle k(a,b)\rangle \wedge E\langle k(b,c)\rangle &= E\langle k(a,b)\rangle \wedge S^0 \wedge E\langle k(b,c)\rangle \notag\\
&{\xrightarrow}{1 \wedge u \wedge 1} E\langle k(a,b)\rangle \wedge E\langle 1\rangle \wedge E\langle k(b,c)\rangle \notag \\
&{\xrightarrow}{\mu} E\langle k(a,c)\rangle. \notag
\end{aligned}$$ Here $u : S^0 \to E = E\langle 1\rangle$ is the unit. This category encodes the multiplication in the ring spectrum $E$.
\[eorient\] An $E^*$-orientation of a smooth, compact category of finite type satisfying the Morse-Smale condition, ${\mathcal{C}}$, is a functor, $u : {\mathcal{C}}^\nu \to E_{{\mathcal{C}}}$, where ${\mathcal{C}}^\nu$ is a normal Thom spectrum of ${\mathcal{C}}$, such that on morphism spaces, the induced map $$Mor_{{\mathcal{C}}^\nu}(a,b) \to E\langle k(a,b)\rangle$$ is a map of $\langle k(a,b)\rangle$-spectra that defines an $E^*$ orientation of $ Mor_{{\mathcal{C}}^\nu}(a,b) \simeq {\bar {\mathcal{M}}}(a,b)^\nu$.
The functor $u : {\mathcal{C}}^\nu \to E_{{\mathcal{C}}}$ should be thought of as a coherent family of $E^*$- Thom classes for the normal bundles of the morphism spaces of ${\mathcal{C}}$. When $E = {\mathbb{S}}$, the sphere spectrum, then an $E^*$-orientation, as defined here, defines a coherent family of framings of the morphism spaces, and is equivalent to the notion of a *framing of the category ${\mathcal{C}}$, as defined in [@cjs].*
In [@cjs] the following was proved modulo the results of [@qin] which appeared much later.
\[framed\] Let $f : M \to {\mathbb{R}}$ be a Morse function on a closed Riemannian manifold satisfying the Morse-Smale condition. Then the flow category ${\mathcal{C}}_f$ has a canonical structure as a “${\mathbb{S}}$-oriented, smooth, compact Morse-Smale category of finite type". That is, it is a *“framed, smooth compact Morse-Smale category". The induced framings of the morphism manifolds ${\bar {\mathcal{M}}}(a,b)$ are canonical extensions of the framings of the open moduli spaces ${\mathcal{M}}(a,b)$ descrbed above (\[intersect\]).*
The main use of the notion of compact, smooth, framed flow categories, is the following result.
([@cjs], [@floeroslo]) Let ${\mathcal{C}}$ be a compact, smooth, framed category of finite type satisfying the Morse-Smale property. Then there is an associated, naturally defined functor $Z_{{\mathcal{C}}} : {\mathcal{J}}_0 \to Spectra$ whose geometric realization $|Z_{{\mathcal{C}}}|$ realizes the associated “Floer complex" $$\to \cdots \to C_{i+1} {\xrightarrow}{ {\partial}_i } C_{i} {\xrightarrow}{{\partial}_{i-1}} \cdots {\xrightarrow}{{\partial}_0} C_0.$$ Here $C_j$ is the free abelian group generated by the objects $ a \in Ob ({\mathcal{C}})$ with $\mu (a) = j$, and the boundary homomorphisms are defined by $${\partial}_{j-1} ([a]) = \sum_{\mu (b) = j-1} \#Mor(a,b) \, \cdot \, [b]$$ where $\#Mor(a,b)$ is the framed cobordism type of the compact framed manifold $Mor (a,b)$ which zero dimensional (since its dimension $= \mu (a) - \mu (b) -1 = 0$). Therefore this cobordism type is simply an integer, which can be viewed as the oriented count of the number of points in $Mor(a,b)$.
*Sketch. The proof of this is sketched in [@cjs] and is carried out in [@floeroslo] in the setting of $E^*$-oriented compact, smooth categories, for $E$ any ring spectrum. The idea for defining the functor $Z_{\mathcal{C}}$ is to use the Pontrjagin-Thom construction in the setting of framed manifolds with corners (more specifically, framed $\langle k \rangle$-manifolds). Namely, one defines $$Z_{{\mathcal{C}}}(j) = \bigvee_{a \in Obj ({\mathcal{C}}) \, : \, \mu (a) = j} {\mathbb{S}}.$$ On the level of morphisms one needs to define, for every $i > j$, a map of spectra $$Z_{{\mathcal{C}}} (i,j) : J(i,j)^+ \wedge Z_{{\mathcal{C}}} (i) \to Z_{{\mathcal{C}}}(j).$$ This is defined to be the wedge, taken over all $a \in Obj ({\mathcal{C}})$ with $\mu (a) = i$, and $b \in Obj ({\mathcal{C}})$ with $\mu (b) = j$, of the maps $$Z_{{\mathcal{C}}}(a,b) : J(\mu(a), \mu (b))^+\wedge {\mathbb{S}}_a \to {\mathbb{S}}_b$$ defined to be the composition $$\Sigma^\infty (J(\mu(a), \mu (b)^+) {\xrightarrow}{\tau_{a,b} } Mor_{{\mathcal{C}}^\nu} (a,b) {\xrightarrow}{\ u} {\mathbb{S}}$$ where ${\mathbb{S}}_a$ and ${\mathbb{S}}_b$ are copies of the sphere spectrum indexed by $a$ and $b$ respectively in the definition of $Z_{{\mathcal{C}}}(i)$ and $Z_{{\mathcal{C}}}(j)$. $\tau_{a,b}$ is the Pontrjagin-Thom map, and $ u$ is ${\mathbb{S}}$-normal orientation class (framing). Details can be found in [@floeroslo].*
Thus to define a *“Floer homotopy type" one is looking for a compact, smooth, framed category of finite type satisfying the Morse-Smale property. The compact framed manifolds with corners that constitute the morphism spaces, define, via the Pontrjagin-Thom construction, the attaching maps of the $CW$-spectrum defining this (stable) homotopy type.*
We see from Theorem \[framed\] that given a Morse-Smale function $f : M \to {\mathbb{R}}$, then its flow category satisfies these properties, and it was proved in [@cjs] that, not surprisingly, its Floer homotopy type is the suspension spectrum $\Sigma^\infty (M_+)$. The $CW$-structure is the classical one coming from Morse theory, with one cell of dimension $k$ for each critical point of index $k$. The fact that the compactified moduli spaces of flow lines, which constitute the morphism spaces in this category, together with their structure as framed manifolds with corners, define the attaching maps in the $CW$-structure of $\Sigma^\infty M$, can be viewed as a generalization of the well-known work of Franks in [@franks].
As pointed out in [@cjs], a distinguishing feature in the flow category of a Morse-Smale function $f : M \to {\mathbb{R}}$ is that the framing is *canonical. See (\[intersect\]) above. As also was pointed out in [@cjs], if one chooses a *different framing of the flow category ${\mathcal{C}}_f$, then the “difference" between the new framing and the canonical framing defines a functor $$\label{frame}
\Phi : {\mathcal{C}}_f \to {\mathcal{G}}L_1({\mathbb{S}})$$ where $ {\mathcal{G}}L_1({\mathbb{S}})$ is the category corresponding to the group-like monoid $GL_1({\mathbb{S}})$ of “units" of the sphere spectrum (see [@units]). This monoid has the homotopy time of the colimit $$GL_1({\mathbb{S}}) \simeq {\operatorname{colim}}_{n \to \infty} \Omega^n_{\pm 1}S^n.$$ Here the subscript denotes the path components of $\Omega^nS^n$ consisting of based self maps of the sphere $S^n$ of degree $\pm 1$. By a minor abuse of notations, we let $\Phi$ denote the framing of ${\mathcal{C}}_f$ that defines the map (\[frame\]).**
Passing to the geometric realizations of these categories, one gets a map $$\phi : M \to BGL_1({\mathbb{S}}).$$ which we think of as the isomorphism type of a spherical fibration over $M$. The following is also a result of [@cjs].
If $f : M \to {\mathbb{R}}$ is a Morse-Smale function on a closed Riemannian manifold, and its flow category ${\mathcal{C}}_f$ is given a framing $\Phi$, then the *Floer homotopy type of $({\mathcal{C}}_f , \Phi)$ viewed as a compact, smooth, framed category, is the Thom spectrum of the corresponding stable spherical fibration, $M^\phi$*
The free loop space of a symplectic manifold and Symplectic Floer theory
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Let $(N^{2n}, \omega)$ be a symplectic manifold. Here $\omega$ is a closed, nondegenerate, skew symmetric bilinear form on the tangent bundle, $TN$. Let $LN$ be its free loop space. One of the earliest applications of Floer theory [@floerFix] was to the (perturbed) symplectic action functional on $LN$.
Let $L_0N \subset LN$ be the path component of contractible loops in $N$. Let $\tilde {L_0N }{\xrightarrow}{p} L_0N$ be its universal cover. Explicitly, $$\tilde {LN} = \{(\gamma, \theta) \in L_0N \times Map(D^2, N) \, : \, \text{the restriction of $\theta$ to $S^1 = {\partial}D^2$ is equal to}
\quad \gamma \} / \sim$$ where the equivalence relation is given by $(\gamma, \theta_1) \sim (\gamma, \theta_2)$ if, when we combine $\theta_1$ and $\theta_2$ to define a map of the $2$-sphere, $$\theta_{1,2} = \theta_1 \cup \theta_2 : D^2 \cup_{S^1} D^2 = S^2 \to N$$ then $\theta_{1,2}$ is null homotopic. In other words, $(\gamma, \theta_1)$ is equivalent to $ (\gamma, \theta_2)$ if $\theta_1$ and $\theta_2$ are homotopic maps relative to the boundary.
One can then define the “symplectic action" functional, $$\begin{aligned}
\label{sympact}
{\mathcal{A}}&: \tilde{L_0N} \to {\mathbb{R}}\\
(\gamma, \theta) &\to \int_{D^2} \theta^*(\omega) \notag \\\end{aligned}$$
The symplectic action descends to define an ${\mathbb{R}}/{\mathbb{Z}}$-valued function $${\mathcal{A}}: L_0N \to {\mathbb{R}}/{\mathbb{Z}}.$$ One needs to perturb this functional by use of a Hamiltonian vector field in order to achieve nondegeneracy of critical points. A Morse-type complex, generated by critical points, is then studied, and the resulting symplectic Floer homology has proved to be an important invariant.
We describe the situation when $N = T^*M^n$, the cotangent bundle of a closed, $n$-dimensional manifold, in more detail. In particular this is a situation where one has a corresponding “Floer homotopy type", defined via a compact, smooth flow category as described above. This was studied by the author in [@cotangent], making heavy use of the analysis of Abbondandolo and Schwarz in [@AS].
Coming from classical mechanics, the cotangent bundle of a smooth manifold has a canonical symplectic structure. It is defined as follows.
Let $p : T^*M \to M$ be the projection map. For $x \in M$ and $ u\in T^*_xM$, let $z = (y, u) \in T^*M$, and consider the composition $$\theta _z: T_{z}(T^*M) {\xrightarrow}{dp}T_yM {\xrightarrow}{u}{\mathbb{R}}.$$
This defines a $1$-form $\theta$ on $T^*M$, called the “Liouville" $1$-form, and the symplectic form $\omega$ is defined to be the exterior derivative $\omega = d\theta$. It is easy to check that $\omega$ is nondegenerate.
Let $$H : {\mathbb{R}}/{\mathbb{Z}}\times T^*M \to {\mathbb{R}}$$ be a smooth function. Such a map is called a “time-dependent periodic Hamiltonian". Using the nondegeneracy of the symplectic form, this allows one to define the corresponding “Hamiltonian vector field" $X_H$ by requiring it to satisfy the equation $$\omega (X_H(t,z), v) = -dH_{t,z} (v)$$ for all $t \in {\mathbb{R}}/{\mathbb{Z}}$, $ z \in T^*M$, and $v \in T_{z}(T^*M)$. We will be considering the space of $1$-periodic solutions, ${\mathcal{P}}(H)$, of the Hamiltonian equation $$\frac{dz}{dt} = X_H(t, z(t))$$ where $z : {\mathbb{R}}/{\mathbb{Z}}\to T^*M$ is a smooth function.
Using a periodic time-dependent Hamiltonian one can define the perturbed symplectic action functional $$\begin{aligned}
{\mathcal{A}}_H : L (T^*M) &\to {\mathbb{R}}\notag \\
z&\to \int z^*(\theta -Hdt) = \int_0^1(\theta (\frac{dz}{dt}) - H(t, z(t)) dt.\end{aligned}$$ This is a smooth functional, and its critical points are the periodic orbits of the Hamiltonian vector field, ${\mathcal{P}}(H)$. Now let $J$ be a $1$-periodic, smooth almost complex structure on $T^*M$, so that for each $t \in {\mathbb{R}}/{\mathbb{Z}}$, $$\langle \zeta, \xi \rangle_{J_t} = \omega (\zeta, J(t,z)\xi ), \quad \zeta, \xi \in T_{z}T^*M, \, z \in T^*M,$$ is a loop of Riemannian metrics on $T^*M$. One can then consider the gradient of ${\mathcal{A}}_H$ with respect to the metric, $\langle \cdot , \cdot \rangle$, written as $$\nabla_J{\mathcal{A}}_H(z) = -J(z,t)(\frac{dz}{dt}-X_H(t,z)).$$ The (negative) gradient flow equation on a smooth curve $u : {\mathbb{R}}\to L(T^*M)$, $$\frac{du}{ds} + \nabla_J{\mathcal{A}}_H(u(s))$$ can be rewritten as a perturbed Cauchy-Riemann PDE, if we view $u$ as a smooth map ${\mathbb{R}}/{\mathbb{Z}}\times {\mathbb{R}}\to T^*M$, with coordinates, $t \in {\mathbb{R}}/{\mathbb{Z}}, \, s \in {\mathbb{R}}$, $$\label{cauchy}
{\partial}_s u -J(t,u(t,s))({\partial}_t u - X_H(t, u(t,s)) = 0.$$
Let $a, b \in {\mathcal{P}}(H)$. Abbondandolo and Schwarz defined the space of solutions
$$\begin{aligned}
\label{wab}
W(a,b; H,J) = \{u : {\mathbb{R}}&\to L(T^*M)\, \text{ a solution to} \, (\ref{cauchy}), \, \text{such that} \\
\lim_{s\to -\infty}u(s) &= a, \, \text{and} \, \lim_{s\to +\infty}u(s) = b \}.\notag\end{aligned}$$
As in the case of Morse theory, we then let ${\mathcal{M}}(a,b)$ the “moduli space" obtained by dividing out by the free ${\mathbb{R}}$-action, $$\label{flows}
{\mathcal{M}}(a,b) = W(a, b; H,J))/{\mathbb{R}}.$$
It was shown in [@AS] that with respect to a generic choice of Hamiltonian and almost complex structure, the spaces $W(a, b; H, J)$ and ${\mathcal{M}}(a, b)$ whose dimensions are given by $\mu(a) - \mu (b)$ and $\mu (a) - \mu (b) - 1$ respectively, where $\mu$ represents the “Conley - Zehnder index" of the periodic Hamiltonian orbits $a$ and $b$. Furthermore it was shown in [@AS] that in analogy with Morse theory, one can compactify these moduli spaces as
$$\bar {\mathcal{M}}(a,b) = \bigcup_{a=a_1 >a_2> \cdots > a_k = b} {\mathcal{M}}(a_1, a_2) \times \cdots \times {\mathcal{M}}(a_{k-1}, a_k).$$
The fact that that these compact moduli spaces have canonical framings was shown in [@cotangent] using the obstruction to framing described originally in [@cjs]. This was the *polarization class defined as follows.*
Let $N$ be an almost complex manifold whose tangent bundle is classified by a map $\tau: N \to BU(n)$. Applying loop spaces, one has a composite map, $$\label{polarization}
\rho: LN {\xrightarrow}{L(\tau)} L(BU(n)) {\hookrightarrow}LBU \simeq BU \times U \to U \to U/O.$$ Here the homotopy equivalence $L(BU) \simeq BU \times U$ is well defined up to homotopy, and is given by a trivialization of the fibration $$U \simeq \Omega BU {\xrightarrow}{\iota} L(BU) {\xrightarrow}{ev}BU$$ where $ev : LX \to X$ evaluates a loop a $0 \in {\mathbb{R}}/{\mathbb{Z}}$. The trivialization is the composition $$U \times BU {\xrightarrow}{\iota \times \sigma} L(BU) \times L(BU) {\xrightarrow}{mult} L(BU).$$ Here $\sigma : BU \to L(BU) $ is the section of the above fibration given by assigning to a point $x \in BU$ the constant loop at that point, and the “multiplication" map in this composition is induced by the infinite loop space structure of $BU$.
The reason we refer the map $\rho$ as the “polarization class" of the loop space $LN$, is because when viewed as an infinite dimensional manifold, the tangent bundle $T(LN)$ is polarized, and its infinite dimensional tangent bundle has structure group given by the “restricted general linear group of a Hilbert space", $GL_{res}(H)$ as originally defined in [@PS]. As shown there, $GL_{res}(H)$ has the homotopy type of ${\mathbb{Z}}\times BO$, and so its classifying space, $BGL_{res}(H) $ has the homotopy type of $B({\mathbb{Z}}\times BO) \simeq U/O$ by Bott periodicity. The classifying map $LN \to U/O$ has the homotopy type of the “polarization class" $\rho$ defined above. See [@PS] , [@cjs], and [@cotangent] for details.
When $N = T^*M$ two things were shown in [@cotangent]. First that when viewed as a space of paths, there is a natural map to the based loop space, $ \iota_{a,b} : {\bar {\mathcal{M}}}(a,b) \to \Omega L(T^*M)$, well defined up to homotopy, so that the composition $$\label{tangent}
\tau_{a,b} : {\bar {\mathcal{M}}}(a, b) {\xrightarrow}{\iota_{a,b}} \Omega L(T^*M) {\xrightarrow}{\Omega \rho} \Omega U/O \simeq {\mathbb{Z}}\times BO$$ classifies the stable tangent bundle of ${\bar {\mathcal{M}}}(a,b)$. Second, it was shown that in this case, i.e when $N = T^*M$, the polarization class $\rho: L(T^*M) \to U/O$ is trivial. This is essentially because the almost complex structure (i.e $U(n)$-structure) of the tangent bundle of $T^*M$ is the complexification of the $n$-dimensional real bundle (i.e $O(n)$-structure) of the tangent bundle of $M$ pulled back to $T^*M$ via the projection map $T^*M \to M$. By (\[tangent\]) this leads to a coherent family of framings on the moduli spaces, which in turn lead to a smooth, compact, framed structure on the flow category ${\mathcal{C}}_H$ of the symplectic action functional, as shown in [@cotangent].
Using the methods and results of [@AS], which is to say, comparing the flow category of the perturbed symplectic action functional ${\mathcal{C}}_H$ to the Morse flow category of an energy functional on $LM$, the following was shown in [@cotangent].
\[cotangent\] If $M^n$ is a closed spin manifold of dimension $n$. For appropriate choices of a Hamiltonian $H$ and a generic choice of almost complex structure $J$ on the cotangent bundle $T^*M$, then the Floer homotopy type determined by the smooth, compact flow category ${\mathcal{C}}_H$ is given by the suspension spectrum of the free loop space, $$Z_H(T^*M) \simeq \Sigma^\infty (LM_+).$$
**[Remarks]{}**
1\. This theorem generalized a result of Viterbo [@viterbo] stating that the symplectic Floer homology, $SFH_*(T^*M)$ is isomorphic to $H_*(LM)$.
2\. If $M$ is not spin, one needs to use appropriatly twisted coefficients in both Viterbo’s theorem and in Theorem \[cotangent\] above. This was first observed by Kragh in [@kragh], and was overlooked in all or most of the discussions of the relation between the symplectic Floer theory of the cotangent bundle and homotopy type of the free loop space before Kragh’s work, including the author’s work in [@cotangent].
The work of Lipshitz and Sarkar on Khovanov homotopy theory
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A recent dramatic application of the ideas of Floer homotopy theory appeared in the work of Lipshitz and Sarkar on the homotopy theoretic foundations of Khovanov’s homological invariants of knots and links. This work appeared in [@lipsark] and [@lipsark2]. Another version of Khovanov homotopy appears in [@HKK]. It was proved to be equivalent to the Lipshitz-Sarkar construction in [@LLS].
The Khovanov homology of a link $L$ is a bigraded abelian group, $Kh^{i,j}(L)$. It is computed from a chain complex denoted $KhC^{i,j}(L)$ that is defined in terms of a link diagram. However the Khovanov homology is shown not to depend on the choice of link diagram, and is an invariant of isotopy class of the link. This invariant was originally defined by Khovanov in [@khov] in which he viewed these homological invariants as a “categorification" of the Jones polynomial $V(L)$ in the sense that the graded Euler characteristic of this homology theory recovers $V(L)$ via the formula $$\begin{aligned}
\chi (Kh^{i, j}(L)) &= \sum_{i,j} (-1)^iq^j \, rank \, Kh^{i,j}(L) \notag \\
&= (q + q^{-1})\, V(L). \notag\end{aligned}$$
The goal of the work of Lipshitz and Sarkar was to associate to a link diagram $L$ a family of spectra $X^j(L)$ whose homotopy types are invariants of the isotopy class of the link (and in particular do not depend on the particular link diagram used), and so that the Khovanov homology $Kh^{i,j}(L)$ is isomorphic to the reduced singular cohomology $\tilde H^i(X^j(L))$. Their basic idea is to construct a compact, smooth, framed flow category from the moduli spaces associated to a link diagram. Their construction is entirely combinatorial, and the cells of the spectrum $X(L) = \bigvee_j X^j(L)$ correspond to the standard generators of the Khovanov complex $KhC^{*,*} (L)$. That is, $X(L)$ *realizes the Khovanov chain complex in the sense described above. $X(L)$ is referred to as the “Khovanov homotopy type" of the link $L$, and it has had several interesting applications.*
Notice that by virtue of the existence of a Khovanov homotopy type, the Khovanov homology, when reduced modulo a prime, carries an action of the Steenrod algebra ${\mathcal{A}}_p$. In [@lipsark2] the authors show that the Steenrod operation $Sq^2$ acts nontrivially on the Khovanov homology for many knots, and in particular for the torus knot, $T_{3,4}$. It is also known by work of Seed [@Seed] that there are pairs of links with isomorphic Khovanov’s homology, but distinct Khovanov homotopy types. Also, Rasmussen constructed a slice genus bound, called the $s$-invariant, using Khovanov homology [@Ras]. Using the Khovanov homotopy type, Lipshitz and Sarkar produced a family of generalizations of the $s$-invariant, and used them to obtain even stronger slice genus bounds. Stoffregen and Zhang [@SZ] used Khovanov homotopy theory to describe rank inequalities for Khovanov homology for prime-periodic links in $S^3$.
We now give a sketch of the construction of compact framed flow category of Lipshitz and Sarkar, which yields the Khovanov homotopy type.
By a “link diagram", one means the projection onto ${\mathbb{R}}^2$ of an embedded disjoint union of circles in ${\mathbb{R}}^3$. One keeps track of the resulting “over" and “under crossings", and usually one orients the link (i.e puts an arrow in each compoonent). One can “resolve" a crossing in two ways. These are referred to as a $0$-resolution and a $1$ resolution and are described by the following diagram.
![ The $0$-resolution and the $1$-resolution[]{data-label="figone"}](zero-one.pdf){height="3cm"}
Roughly speaking, the Lipshitz-Sarkar view of the Khovanov chain complex is that it is generated by all possible configurations of resolutions of the crossings of a link diagram. We recall their definition more carefully.
A **resolution configuration $D$ is a pair $(Z(D), A(D))$, where $Z(D)$ is a set of pairwise disjoint embedded circles in $S^2 = {\mathbb{R}}^2 \cup \infty$, and $A(D)$ is an ordered collection of arcs embedded in $S^2$ with $$A(D) \cap Z(D) = {\partial}A(D).$$ The number of arcs in the resolution configuration $D$ is called its *index denoted by $ind (D)$.***
Given a link diagram $L$ with $n$-crossings, an ordering of the crossings, and a vector $v \in \{0, 1\}^n$, there is an associated resolution configuration $D_L(v)$ obtained by taking the resolution of $L$ corresponding to $v$. That is, one takes the $0$-resolution of the $i^{th}$ crossing if $v_i = 0$, and the $1$-resolution of the $i^{th}$ crossing if $v_i = 1$. One then places arcs corresponding to each of the crossings labeled by zero’s in $v$.
See the following picture of the resolution configuration corresponding to a diagram of the trefoil knot.
![ A knot diagram $K$ for the trefoil with ordered crossings, and the resolution configuration $D_K((0,1,0))$[]{data-label="figtwo"}](trefoil.pdf){height="5cm"}
The following terminology is also useful.
1\. The **core $c(D)$ of a resolution configuration is the the resolution configuration obtained from $D$ by deleting all the circles in $Z(D)$ that are disjoint from all arcs in $A(D0$.**
2\. A resolution configuration is **basic if $D = c(D)$, ie every circle in $Z(D)$ intersects an arc in $A(D)$.**
One can also do a **surgery along a subset $A \subset A(D)$. The resulting resolution configuration is denoted $s_{A}(D)$. The surgery procedure is best illustrated by the following picture.**
{height="4cm"} \[figthree\]
A **labeling $x = \{x_1, x_2\} $ of a resolution configuration $D $ is a labeling of each circle in $Z:(D)$ by either $x_1$ or $x_2$. Labeled resolutions configurations have a partial ordering defined to be the transitive closure of the following relations.**
We say that $(E,y) < (D,x)$ if
1. the labelings agree on $D \cap E$
2. $D$ is obtained from $E$ by surgering along a single arc of $A(E)$, In particular, either:
\(a) $Z(E \backslash D)$ contains exactly one circle, say $Z_i$ and $Z(D \backslash E)$ contains exactly two circles, say $Z_j$ and $Z_k$, or
\(b) $ Z(E \backslash D)$ contains exactly two circles, say $Z_i$ and $Z_j$, and $Z(D \backslash E)$ contains exactly one circle, say $Z_k$.
3. In case (2a), either $ y(Z_i) = x(Z_j) = x(Z_k) = x_- $ or $y(Z_i) = x_+ $ and $\{x(Z_j), x(Z_k)\} = \{x_+, x_-\}$.
In case (2b), either $ y(Z_i) = y(Z_j) = x(Z_k) = x_+$ or $y(Z_i) = x_+ $ or $\{y(Z_i), y(Z_j)\} = \{x_+, x_-\}$ and $x(Z_k) = x_+.$
One can now define the *Khovanov chain complex $KhC(L)$ as follows.*
Given an oriented link diagram $L$ with $n$ crossings and an ordering of the crossings in $L$, $KhC(L)$ is defined to be the free abelian group generated by labeled resolution configurations of the form $(D_L(u), x)$ for $u \in \{0, 1 \}^n$. $ KhC(L)$ carries two gradings, a *homological grading $gr_h$ and a *quantum grading $gr_q$, defined as follows: $$\begin{aligned}
gr_h((D_L(u),x)) &= -n_- +|u| \notag \\
gr_q((D_L(u),x)) &= n_+ -2n_- +|u| + \#\{Z \in Z(D_L(u)) \, : \, x(Z) = x_+ \} \notag \\
&- \#\{Z \in Z(D_L(u)) \, : \, x(Z) = x_- \} \notag \end{aligned}$$ Here $n_+$ denotes the number of positive crossings in $L$, and $n_- $ denotes the number of negative crossings.**
The differential preserves the quantum grading, increases the homological grading by $1$, and is defined as $$\delta (D_L(v), y) = \sum (-1)^{s_0({\mathcal{C}}_{u,v})} (D_L(u), x)$$ where the sum is taken over all labeled resolution configurations $(D_L(u), x)$ with $|u| = |v|+1$ and $(D_L(v), y) < (D_L(u), x)$. The sign $s_0({\mathcal{C}}_{u,v}) \in {\mathbb{Z}}/2$ is defined as follows: If $u = ({\epsilon}_1, \cdots , {\epsilon}_{i-1}, 1, {\epsilon}_{i+1}, \cdots , {\epsilon}_n)$ and $v = ({\epsilon}_1, \cdots , {\epsilon}_{i-1}, 0, {\epsilon}_{i+1}, \cdots , {\epsilon}_n)$, then $s_0({\mathcal{C}}_{u,v}) = {\epsilon}_1 + \cdots + {\epsilon}_{i-1}.$
The homology of this chain complex is the *Khovanov homology $Kh^{*,*}(L)$. To define the *Khovanov homotopy type of the link $L$, Lipshitz and Sarkar define higher dimensional moduli spaces which have the structure of framed manifolds with corners so that they in turn define a compact, smooth, framed flow category, which by the theory described above, defines the associated (stable) homotopy type.**
These moduli spaces are defined as a certain covering of the moduli spaces occurring in a “framed flow category of a cube". We now sketch these constructions, following Lipshitz and Sarkar [@lipsark].
Let $f_1 : {\mathbb{R}}\to {\mathbb{R}}$ be a Morse function with one index zero critical point and one index 1 critical point. For concreteness one can use the function $$f_1(x) = 3x^2 - 2x^3.$$ Define $f_n : {\mathbb{R}}^n \to {\mathbb{R}}$ by $$f_n(x_1, \cdots , x_n) = f_1(x_1) + \cdots + f_1(x_n).$$ $f_n$ is a Morse function, and we let ${\mathcal{C}}(n)$ denote its flow category. It is a straightforward exercise to see that the geometric realization $|{\mathcal{C}}(n)|$ is the $n$-cube $[ 0, 1]^n$. The vertices of this cube $u \in \{0, 1 \}^n$ correspond to the critical points of $f_n$ and have a grading which corresponds to the Morse index: $gr(u) = |u| = \sum_i u_i$. They also have a partial ordering coming from the ordering of $\{0, 1\}$. We say that $v \leq_i u$ if $v \leq u$ and $gr(u) - gr(v) = i$. That is $i$ is the relative index of $u$ and $v$. Let $\bar 1 = (1, 1, \cdots , 1)$ and $\bar 0 = (0, \cdots , 0)$. The following is not difficult, and is verified in [@lipsark].
The compactified moduli space of piecewise flows $\bar{\mathcal{M}}_{{\mathcal{C}}(n)} (\bar 1, \bar 0)$ is
- a single point if $n = 1$,
- a closed interval if $n=2$,
- a closed hexagonal disk if $n = 3$, and is
- homeomorphic to a closed disk $D^{n-1}$ for general $n$.
Furthermore, given any $v < u$, then $\bar {\mathcal{M}}_{{\mathcal{C}}(n)}(u, v) \cong \bar {\mathcal{M}}_{{\mathcal{C}}(gr(u)-gr(v))}(\bar 1, \bar 0).$
![ The cube moduli space $\bar{\mathcal{M}}_{{\mathcal{C}}(3)} (\bar 1, \bar 0)$[]{data-label="figfour"}](cube.pdf){height="4cm"}
Lipshitz and Sarkar then proceed to define moduli spaces of “decorated resolution configurations" that cover the moduli spaces occurring in a framed flow category of a cube.
A **decorated resolution configuration is a triple $(D, x, y)$ where $D$ is a resolution configuration , $x$ is a labeling of each component of $Z(s(D))$, $y$ is a labeling of each component of $Z(D)$ such that $$(D,y) < (s(D), x).$$ Here $s(D) = s_{A(D)}(D)$ is the maximal surgery on $D$.**
In [@lipsark] Lipshitz and Sarkar proceed to construct moduli spaces ${\mathcal{M}}(D,x,y)$ for every decorated resolution configuration $(D, x, y)$. These will be compact, framed manifolds with corners. Indeed they are $<n-1>$-manifolds where $n$ is the index of $D$. They also produce covering maps $${\mathcal{F}}: {\mathcal{M}}(D, x, y) \to \bar {\mathcal{M}}_{{\mathcal{C}}(n)} (\bar 1, \bar 0)$$ which are maps of $<n-1>$-spaces, trivial as a covering maps on each component of $ {\mathcal{M}}(D, x, y)$, and are local diffeomorphisms. The framings of the moduli spaces $ {\mathcal{M}}(D, x, y)$ are then induced from the framings of $ \bar {\mathcal{M}}_{{\mathcal{C}}(n)} (\bar 1, \bar 0)$. They produce composition or gluing maps for every labeled resolution configuration $(E,z)$ with $$(D,y) < (E,z) < (s(D), x)$$ $$\circ : {\mathcal{M}}(D \backslash E, z|, y|) \times {\mathcal{M}}(E\backslash s(D), x|, z|) {\longrightarrow}{\mathcal{M}}(D, x, y)$$ that are embeddings into the boundary of ${\mathcal{M}}(D, x, y)$. These moduli spaces are constructed recursively using a clever, but not very difficult argument. We refer the reader to [@lipsark] for details.
These constructions allow for the definition of a Khovanov flow category for the link $L$, and it is shown to be a compact, smooth, framed flow category as defined in section 1.2 above. Using the theory introduced in [@cjs] and described in section 1.2, one obtains a spectrum realizing the Khovanov chain complex. The This is the Lipshitz-Sarkar “Khovanov homotopy type" of the link $L$.
Manolescu’s equivariant Floer homotopy and the triangulation problem
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In this section the author is relying heavily on the expository article by Manolescu [@Man15]. We refer the reader to this beautiful survey of recent topological applications of Floer theory.
Monopole Floer homology and equivariant Seiberg-Witten stable homotopy
----------------------------------------------------------------------
One example of a dramatic application of Floer’s original instanton homology theory, and in particular its topological field theory relationship to the Donaldson invariants of closed $4$-manifolds, was to the study of the group of cobordism classes of homology $3$-spheres. Define $$\Theta^3_H = \{\text{oriented homology $3$-spheres}\}/\sim$$ where $X_0 \sim X_1$ if and only if there exists a smooth, compact, oriented $4$-manifold $W$ with $${\partial}W = (-Y_0) \sqcup Y_1$$ and $H_1(W) = H_2(W) = 0$. The group operation is represented by connected sum, and the inverse is given by reversing orientation. The standard unit $3$-sphere $S^3$ is the identity element. The Rokhlin homomorphism [@rok], [@EK] is the map $$\mu : \Theta^H_3 \to {\mathbb{Z}}/2$$ defined by sending a homology sphere $X$ to $\mu (X) = \sigma (X)/8 \,(mod \, 2)$, where $W$ is any compact spin $4$-manifold with ${\partial}W = X$. It is a theorem that this homomorphism is well-defined. Furthermore, using this homomorphism one knows that the group $\Theta^H_3$ is nontrivial, since, for example, the Poincaré sphere $P$ bounds the $E_8$ plumbing which has signature $-8$. Therefore $\mu (P) = 1$.
This result has been strengthened using Donaldson theory and Instanton Floer homology. For example Furuta and Fintushel-Stern proved that $\Theta^H_3$ is infinitely generated [@Fur][@FS]. And using the $SU(2)$ equivariance of Instanton Floer homology, in [@Froy] Froyshov defined a surjective homomorphism $$\label{froy}
h : \Theta^H_3 \to {\mathbb{Z}}.$$
Monopole Floer homology is similar in nature to Floer’s Instanton homology theory, but it is based on the Seiberg-Witten equations rather than the Yang-Mills equations. More precisely, let $Y$ be a three-manifold equipped with a $Spin^c$ structure $\sigma$. One considers the configuration space of pairs $(A, \phi)$, where $A$ is a connection on the trivial $U(1)$ bundle over $Y$, and $\phi$ is a spinor. There is an action of the gauge group of the bundle on this configuration space, and one considers the orbit space of this action. One can then define the Chern-Simons-Dirac functional on this space by $$CSD (A, \phi) = -\frac{1}{8}\int_Y (A^t - A^t_0) \wedge (F_{A^t} + F_{A^t_0}) \, + \frac{1}{2}\int_Y \langle D_A\phi, \phi\rangle \, dvol.$$ Here $A_0$ is a fixed base connection, and the superscript $t$ denotes the induced connection on the determinant line bundle. The symbol $F$ denotes the curvature of the connection and the symbol $D$ denotes the covariant derivative.
Monopole Floer homology is the Floer homology associated to this functional. To make this work precisely involves much analytic, technical work, due in large part to the existence of reducible connections. Kronheimer and Mrowka dealt with this (and other issues) by considering a blow-up of this configuration space [@KM]. They actually defined three versions of Monopole Floer homology for every such pair $(Y, \sigma)$.
Monopole Floer homology can also be used to give an alternative proof of Froyshov’s theorem about the existence of a surjective homomorphism. $$\delta : \Theta^H_3 \to {\mathbb{Z}}$$ This uses the $S^1$-equivariance of these equations. Monopole Floer homology has also been used to prove important results in knot theory [@KMOS] and in contact geometry [@taubes].
In [@Man03] Manolescu defined a “Monopole", or “Seiberg-Witten" Floer stable homotopy type. He did not follow the program defined by the author, Jones and Segal in [@cjs] as outlined above, primarily because the issue of smoothness in defining a framed, compact, smooth flow category is particularly difficult in this setting. Instead, he applied Furuta’s technique of “finite dimensional approximations" [@Fur2]. More specifically, the configuration space $X$ of connections and spinors $(A, \phi)$ is a Hilbert space that he approximated by a nested sequence of finite dimensional subspaces $X_\lambda$. He considered the Conley index associated to the flow induced by $CSD$ on a large ball $B_\lambda \subset X_\lambda$. Roughly, if $L_\lambda \subset {\partial}B_\lambda$ is that part of the boundary where the flow points in an outward direction, then Manolescu views the Conley index as the quotient space $$I_\lambda = B_\lambda / L_\lambda.$$ The homology $H_*(I_\lambda)$ is the Morse-homology of the approximate flow on $B_\lambda$, assuming that the flow satisfies the Morse-Smale transversality condition. However Manolosecu did not need to assume the Morse-Smale condition in his work. Namely he simply defined the Seiberg-Witten Floer homology directly as the relative homology of $I_\lambda$, with a degree shift that depends on $\lambda$. The various $I_\lambda$’s fit together to give a spectrum $SWF(Y, \sigma)$ defined for every rational homology sphere $Y$ with $Spin^c$-structure $\sigma$. Since the Seiberg-Witten equations have an $S^1$ symmetry, the spectrum $SWF(Y, \sigma)$ carries an $S^1$ action. This defines Manolescu’s “$S^1$-equivariant Seiberg-Witten Floer stable homotopy type".
An important case is when $Y$ is a homology sphere. In this setting there is a unique $Spin^c$ structure $\sigma$ coming from a spin structure. The conjugation and $S^1$-action together yield an action by the group $$Pin (2) = S^1 \oplus S^1j \subset {\mathbb{C}}\oplus {\mathbb{C}}j = {\mathbb{H}}$$ where ${\mathbb{H}}$ are the quarternions. In [@Man13] Manolescu defined the $Pin(2)$-equivariant Floer homology of $Y$ to be the (Borel) equivariant homology of the spectrum,
$$\label{Pin}
SWFH_*^{Pin(2)}((Y) = \tilde H^{Pin(2)}_*(SWF(Y)).$$
This theory played a crucial role in Manolescu’s resolution of the triangulation question as we will describe below. But before doing that we point out that by having the equivariant stable homotopy type, he was able to define a corresponding $Pin(2)$-equivariant Seiberg-Witten Floer $K$-theory, which he used in [@Man13] to prove an analogue of Furuta’s “10/8"-conjecture for four-manifolds with boundary. That is, Furuta proved that if $W$ is a closed, smooth spin four-manifold then $$b_2(W) \geq \frac{10}{8} |sign(W)| + 2$$ where $b_2$ is the second Betti number and $sign (W)$ is the signature. (The “$11/8$-conjecture states that $b_2 \geq \frac{11}{8} sign(W)$.) Using $Pin(2)$-equivariant Floer $K$-theory, Manolescu proved that if $W$ is a smooth, spin compact four-manifold with boundary equal to a homology sphere $Y$, then there is a an invariant $\kappa (Y) \in {\mathbb{Z}}$, and an analogue of Furuta’s inequality, $$b_2(W) \geq \frac{10}{8} |sign(W)| + 2 - 2\kappa (Y).$$
The triangulation problem
-------------------------
A famous question asked in 1926 by Kneser [@kne] is the following:
**Question. Does every topological manifold admit a triangulation?**
By “triangulation", one means a homeomorphism to a simplicial complex. One can also ask the stronger question regarding whether every manifold admits a *combinatorial triangulation, which is one in which the links of the simplices are spheres. Such a triangulation is equivalent to a piecewise linear (PL) structure on the manifold.*
In the 1920’s Rado proved that every surface admits a combinatorial triangulation, and in the early 1950’s, Moise showed that any topological three manifold also admits a combinatorial triangulation. In the 1930’s Cairns and Whitehead showed that *smooth manifolds of any dimension admit combinatorial triangulations. And in celebrated work in the late 1960’s, Kirby and Siebenmann [@KS] showed that there exist topological manifolds without $PL$-structures in every dimension greater than four. Furthermore they showed that in these dimensions, the existence of $PL$-structures is determined by an obstruction class $\Delta (M) \in H^4(M; {\mathbb{Z}}/2).$ The first counterexample to the simplicial triangulation conjecture was given by Casson [@Cas] who showed that Freedman’s four dimensional $E_8$-manifold, which he had proven did not have a $PL$-structure, did not admit a simplicial triangulation. In dimensions five or greater, a resolution of Kneser’s triangulation question was not achieved until Manolescu’s recent work [@Man13] using equivariant Seiberg-Witten Floer homotopy theory.*
We now give a rough sketch of Manolescu’s work on this.
Let $M$ be a closed, oriented $n$-manifold, with $n \geq 5$ that is equipped with a homeomorphism to a simplicial complex $K$ (i.e a simplicial triangulation). As mentioned above, the Kirby-Siebenmann obstruction to $M$ having a combinatorial ($PL$) triangulation is a class $\Delta (M) \in H^4(M; {\mathbb{Z}}/2)$. A related cohomology class is the Sullivan-Cohen-Sato class $c(K)$ [@Sul], [@Coh], [@Sato] defined by $$c(K) = \sum_{\sigma \in K^{(n-4)} } [link_K(\sigma)] \, \cdot \, \sigma \, \in H_{n-4} (M; \Theta^H_3) \cong H^4(M; \Theta^H_3).$$ Here the sum is taken over all codimension $4$ simplices in $K$. The link of each such simplex is known to be a homology $3$-sphere. If this were a combinatorial triangulation the link would be an actual $3$-sphere.
Consider the short exact sequence given by the Rokhlin homomorphism $$\label{rok}
0 \to ker (\mu) \to \Theta^H_3 {\xrightarrow}{\mu} {\mathbb{Z}}/2 \to 0.$$ This induces a long exact sequence in cohomology $$\cdots \to H^4(M; \Theta^H_3) {\xrightarrow}{\mu_*} H^4(M; {\mathbb{Z}}/2) {\xrightarrow}{\delta} H^5(M; \ker (\mu_*)) \to \cdots$$ In this sequence it is known that $\mu_*(c(K)) = \Delta (M) \in H^4(M ;{\mathbb{Z}}/2).$ One concludes that if $M$ is a manifold that admits a simplicial triangulation, the Kirby-Siebenmann obstruction $\Delta (M)$ is in the image of $\mu_*$ and therefore in the kernel of $\delta$. An important result was that this necessary condition for admitting a triangulation is also a sufficient condition.
\[GS\] (Galewski-Stern [@GS], Matumoto [@Mat]) A closed topological manifold $M$ of dimension greater or equal to $5$ is admits a triangulation if and only if $\delta (\Delta (M)) =0$.
Now notice that the connecting homomorphism $\delta$ in this long exact sequence would be zero if the short exact sequence (\[rok\]) were split. If this were the case then by the Galewski-Stern-Matumoto theorem (\[GS\]), every closed topological manifold of dimension $\geq 5$ would be triangulable. The following theorem states that this is in fact a sufficient condition.
\[GS2\] (Galewski-Stern [@GS], Matumoto [@Mat]) There exist non-triangulable manifolds of every dimension greater or equal to $5$ if and only if the exact sequence (\[rok\]) does not split.
In settling the triangulation problem Manolescu proved the following.
\[triangle\] (Manolescu [@Man13]) The short exact sequence (\[rok\]) does not split.
The strategy of his proof was the following. Suppose $\eta : {\mathbb{Z}}/2 \to \Theta^H_3$ is a splitting of the above sequence. The image under $\eta$ of the nonzero element would represent a homology $3$-sphere $Y$ of order $2$ in $\Theta^H_3$ with nonzero Rokhlin invariant. Notice that the fact that $Y$ has order $2$ means that $-Y$ ($= Y$ with the opposite orientation) and $Y$ represent the same element in $\Theta^H_3$. Thus to show that this cannot happen, Manolescu defined a lift of the Rokhlin invariant to the integers $$\beta : \Theta^H_3 \to {\mathbb{Z}}$$ Given such a lift and any element $X \in \Theta^H_3$ of order $2$, $$\begin{aligned}
\beta(X) &= \beta(-X) \quad \text{because $X$ has order two} \notag \\
&= -\beta (X) \in {\mathbb{Z}}\notag\end{aligned}$$ Thus $ \beta(X) = -\beta (X) \in {\mathbb{Z}}$ and therefore must be zero. Thus $X$ has zero Rokhlin invariant.
Manolescu’s strategy was therefore to construct such a lifting $\beta : \Theta^H_3 \to {\mathbb{Z}}$ of the Rokhlin invariant $\mu : \Theta^H_3 \to {\mathbb{Z}}/2$. His construction was modeled on Froyshov’s invariant (\[froy\]). But to construct $\beta$ he used $Pin(2)$-equivariant Seiberg-Witten Floer homology $SWFH^{Pin (2)}$ defined using the $Pin(2)$-equivariant Seiberg-Witten Floer homotopy type (spectrum) as described above (\[Pin\]). We refer to [@Man13] for the details of the construction and resulting dramatic solution of the triangulation problem.
Floer homotopy and Lagrangian immersions: the work of Abouzaid and Kragh
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Another beautiful example of an application of a type of Floer homotopy theory was found by Abouzaid and Kragh [@Abou-Kra] in their study of Lagrangian immersions of a Lagrangian manifold $L$ into the cotangent bundle of a smooth, closed manifold, $T^*N$. Here, $L$ and $N$ must be the same dimension. The basic question is whether a Lagrangian immersion is Lagrangian isotopic to a Lagrangian embedding. This is particularly interesting when $L$ = $N$. In this case the “Nearby Lagrangian Conjecture" of Arnol’d states that every closed exact Lagrangian submanifold of $T^*N$ is Hamiltonian isotopic to the zero section, $\eta : N {\hookrightarrow}T^*N$. In [@Abou-Kra] the authors use Floer homotopy theory and classical calculations by Adams and Quillen [@AQ] of the $J$-homomorphism in homotopy theory to give families of Lagrangian immersions of $S^n$ in $T^*S^n$ that are regularly homotopic to the zero section embedding as smooth immersions, but are *not Lagrangian isotopic to any Lagrangian embedding.*
We now describe these constructions and results in a bit more detail. Let $(M^{2n}, \omega)$ be a symplectic manifold of dimension $2n$. Recall that a Lagrangian submanifold $L \subset M$ is a smooth submanifold of dimension $n$ such the restriction of the symplectic form $\omega$ to the tangent space of $L$ is trivial. That is, each tangent space of $L$ is an isotropic subspace of the tangent space of $M$. A Lagrangian embedding $e : L {\hookrightarrow}M$ is a smooth embedding whose image is a Lagrangian submanifold. Similarly, a Lagrangian immersion $\iota : L \to M$ is a smooth immersion so that the image of each tangent space is a Lagrangian subspace of the tangent space of $M$.
If $M$ is a cotangent bundle $T^*N$ for some smooth, closed $n$-manifold $N$, then a Lagrangian immersion $\iota : L \to T^*N$ determines a map $\tau_L : L \to U/O$, which is well-defined up to homotopy. This map is defined as follows.
Let $j : N \subset {\mathbb{R}}^K$ be any smooth embedding with normal bundle $\nu$. Complexifying, we get $${\mathbb{C}}^K \times N \cong (\nu \otimes {\mathbb{C}}) \oplus (TN \otimes {\mathbb{C}}) \cong (\nu \otimes {\mathbb{C}}) \oplus T(T^*N)_{|_N}.$$ Here $T(T^*N)$ has a Hermitian structure induced by its symplectic structure and the Riemannian structure on N coming from the embedding $j$. So for each $x \in L$, the tangent space $T_xL$ defines, via the immersion $\iota$, a Lagrangian subspace of $T_{\iota (x)} (T^*N)$. By taking the direct sum with the Lagrangian $\nu = \nu \otimes {\mathbb{R}}\subset \nu \otimes {\mathbb{C}}$, one obtains a Lagrangian subspace of ${\mathbb{C}}^K$. The Grassmannian of Lagrangian subspaces of ${\mathbb{C}}^K$ is homeomorphic to $U(K)/O(K)$. One therefore has a map $$\tau_L : L \to U(K)/O(K) {\xrightarrow}{\subset} U/O.$$
Now the $h$-principle for Lagrangian immersions states that the set of Lagrangian isotopy classes of immersions $L \to T^*N$ in the homotopy class of a fixed map $\iota_0 : L \to T^*N$ can be identified with the connected components of the space of injective maps of vector bundles $$TL \to \iota_0^* (T(T^*N))$$ which have Lagrangian image. Let $Sp(T(T^*N))$ be the bundle over $T^*N$ with fiber over $(x, u) \in T^*N$ the group of linear automorphisms of $T_{(x,u)}(T^*N))$ preserving the symplectic form. This is a principal $Sp(n)$-bundle. Then the space of all such maps of vector bundles is either empty or a principal homogeneous space over the space of sections of $\iota_0^*(Sp(T(T^*N))$. That is, the space of sections acts freely and transitively.
Abouzaid and Kragh then consider the following special case:
$$\label{special}
L = N, \,TN \otimes {\mathbb{C}}\, \text{is a trivial complex vector bundle, and} \, \iota_0 \, \text{is the inclusion of the zero section.}$$
In this case the the bundle $Sp(T(T^*N)$ is the trivial $Sp(n)$ bundle, and so the corresponding space of sections is the (based) mapping space $Map (N, Sp(n))$, and since $U(n) \simeq Sp(n)$ it is equivalent to $Map (N, U(n))$. Since the inclusion of $U(n) {\hookrightarrow}U$ is $(2n-1)$- connected, one concludes the following.
The equivalence classes of Lagrangian immersions in the homotopy class of the zero section of $T^*N$ are classified by homotopy classes of maps from $N$ to $U$.
One therefore has the following homotopy theoretic characterization of the Lagrangian isotopy classes of Lagrangian immersions of the sphere $S^n$ into its cotangent space.
\[sn\] Isotopy classes of Lagrangian immersions of the sphere $S^n$ in $T^*S^n$ in the homotopy class of the standard embedding, are classified by $\pi_n (U)$.
Of course these homotopy groups are known to be the integers ${\mathbb{Z}}$ when $n$ is odd and zero when $n$ is even. Using a type of Floer homotopy theory, Abouzaid and Kragh proved the following in [@Abou-Kra].
\[Lagsn\] Whenever $ n$ is congruent to $1$, $3$, or $5$ modulo $8$, there is a class of Lagrangian immersions of $S^n$ in $T^*S^n$ in the homotopy class of the zero section, which does not admit an embedded representative.
We now sketch the Abouzaid-Kragh proof of this theorem, and in particular point out the Floer homotopy theory they used.
They first observed that given a Lagrangian immersion $j : N \to T^*N$ in the homotopy class of the zero section, satisfying condition (\[special\]), then one has a well defined (up to homotopy) classifying map $\gamma_j : N \to U$ which lifts the map $\tau_N : N \to U/O$ described above: $$\label{lift}
\tau_N : N {\xrightarrow}{\gamma_j} U {\xrightarrow}{project} U/O.$$
Now given any Lagrangian embedding $L {\hookrightarrow}M$ as above, Kragh [@kragh] defined a “Maslov" (virtual) bundle $\eta_L$ over the component of the free loop space consisting of contractible loops, ${\mathcal{L}}_0L$.
**Note. We have changed the notation for the free loop space by using a script ${\mathcal{L}}$, so as not to get confused by the use of an “$L$" to denote a Lagrangian.**
The bundle $\eta_L$ is classified by the following map (which by abuse of notation we also call $\eta_L$) $$\label{Maslov}
\eta_L: {\mathcal{L}}_0 L {\xrightarrow}{{\mathcal{L}}\tau_L} {\mathcal{L}}_0 U/O {\xrightarrow}{\simeq} U/O \times \Omega _0 U/O {\xrightarrow}{project} \Omega_0 U/O \simeq BO.$$
Here the equivalence $ {\mathcal{L}}_0 U/O \simeq U/O \times \Omega_0 U/O$ comes from considering the evaluation fibration $$\Omega U/O \to LU/O {\xrightarrow}{ev} U/O$$ where $ev$ evaluates a loop at the basepoint. This fibration has a canonical (up to homotopy) trivialization as infinite loop spaces because of the existence of a section as constant loops, and using the infinite loop structure of $U/O$. The equivalence $\Omega_0U/O \simeq BO$ comes from Bott periodicity.
**Note. Given a map of any space $f : X \to U/O$, the corresponding virtual bundle over the free loop space $${\mathcal{L}}_0 X \to BO$$ defined as this composition ${\mathcal{L}}_0 X {\xrightarrow}{{\mathcal{L}}f} {\mathcal{L}}_0 U/O {\xrightarrow}{\simeq} U/O \times \Omega _0 U/O {\xrightarrow}{project} \Omega_0 U/O \simeq BO.$ also appeared in the work of Blumberg, the author, and Schlichtkrull [@BCS] in their work on the topological Hochschild homology of Thom spectra.**
Let ${\mathbb{S}}$ be the sphere spectrum, and let $GL_1({\mathbb{S}})$ be the group-like monoid of units in ${\mathbb{S}}$. That is, $GL_1({\mathbb{S}})$ is the colimit of the group-like monoid of based self homotopy equivalences of $S^n$, $\Omega^n_{\pm 1} S^n$. These are the degree $\pm 1$ based self maps of $S^n$, which is a group-like monoid under composition. The classifying space, $BGL_1({\mathbb{S}})$, classifies (stable) spherical fibrations. There is a natural map $J : BO \to BGL_1({\mathbb{S}})$ coming from the inclusion $O(n) {\hookrightarrow}\Omega^n_{\pm 1}S^n$ defined by considering the based self equivalence of $ S^n = {\mathbb{R}}^n \cup \infty $ given by an orthogonal matrix. We call this map “$J$" as it induces the classical $J$ homomorphism on the level of homotopy groups, $ J : \pi_q (O) \to \pi_q({\mathbb{S}})$.
Notice that given a $k$-dimensional vector bundle $\zeta \to X$, the associated spherical fibration is the sphere bundle $S(\zeta) \to X$ defined by taking the one-point compactification of each fiber.
The following is an important result about Lagrangian *embeddings in $\cite{Abou-Kra}.$*
\[criterion\] If $j : L \to T^*N$ is an exact Lagrangian embedding then the stable spherical fibration of the Maslov bundle $\eta_L$ is trivial. That is, the composition $${\mathcal{L}}_0 L {\xrightarrow}{\eta_L} BO {\xrightarrow}{J} BGL_1({\mathbb{S}})$$ is null homotopic.
Before sketching how this theorem was proven in [@Abou-Kra], we indicate how Abouzaid and Kragh used this result to detect the Lagrangian immersions of $S^n$ in $ T^*S^n$ yielding Theorem \[Lagsn\].
A Lagrangian immersion $\iota : S^n \to T^*S^n$ is classified by a class $\alpha_\iota \in \pi_n(U)$ by Corollary \[sn\]. By Theorem \[criterion\] if $\iota$ is Lagrangian isotopic to a Lagrangian embedding, then the composition $${\mathcal{L}}S^n = {\mathcal{L}}_0 S^n{\xrightarrow}{{\mathcal{L}}_0 \alpha_\iota} {\mathcal{L}}_0 U \to {\mathcal{L}}_0(U/O) {\xrightarrow}{\simeq} U/O \times \Omega_0 U/O \to \Omega_0U/O \simeq BO {\xrightarrow}{J} BGL_1({\mathbb{S}})$$ is null homotopic. If $\iota$ is such that $\alpha_\iota \in \pi_n(U)$ is a nonzero generator (here $n$ must be odd), then Abouzaid and Kragh precompose this map with the composition $$S^{n-1} {\hookrightarrow}\Omega S^n \to {\mathcal{L}}S^n$$ and show that in the dimensions given in the statement of the theorem, then classical calculations of the $J$-homomorphism in homotopy theory imply that this composition $S^{n-1} \to BGL_1({\mathbb{S}})$ is nontrivial. Therefore by Theorem \[criterion\], the Lagrangian immersions of $S^n$ into $T^*S^n$ that these classes represent cannot be Lagrangian isotopic to embeddings, even though as smooth immersions, they are isotopic to the zero section embedding.
The proof of Theorem \[criterion\] is where Floer homotopy theory is used. This was based on earlier work by Kragh in [@kragh]. The Floer homotopy theory used was a type of Hamiltonian Floer theory for the cotangent bundle. They did not directly use the constructions in [@cjs] described above, but the spirit of the construction was similar. More technically they used finite dimensional approximations of the free loop space, not unlike those used by Manolescu [@Man03]. We refer the reader to [@kragh], [@Abou-Kra] for details of this construction. In any case this construction was used to give a spectrum-level version of the Viterbo transfer map, when one has an exact Lagrangian embedding $j : L \to T^*N$. This transfer map is given on the spectrum level by a map
$${\mathcal{L}}_0 j^! : {\mathcal{L}}_0N^{-TN} \to ({\mathcal{L}}_0 L)^{-TL \oplus \eta}$$ and similarly a map $$\label{equivalence}
{\mathcal{L}}_0 j^! : \Sigma^\infty ({\mathcal{L}}_0N_+)\to ( {\mathcal{L}}_0 L)^{TN -TL \oplus \eta}$$ An important result in [@Abou-Kra] is that ${\mathcal{L}}_0j^!$ is an equivalence. Then they make use of the result, essentially due to Atiyah, that given a finite $CW$-complex $X$ with a stable spherical fibration classified by a map $\rho : X \to BGL_1({\mathbb{S}})$, then the Thom spectrum $X^\rho$ is equivalent to the suspension spectrum $\Sigma^\infty (X_+)$ if and only if $\rho$ is null homotopic. Applying this to finite dimensional approximations to ${\mathcal{L}}_0 L$, they are then able to show that the equivalence (\[equivalence\]) implies that the Maslov bundle $\eta$, when viewed as a stable spherical fibration is trivial.
We end this discussion by remarking on the recasting of the Abouzaid-Kragh results by the author and Klang in [@ck]. Given a Lagrangian immersion, $j : L \to T^*N$, consider the resulting class $$\tau_L : L \to U/O$$ described above. Taking based loop space, one has a loop map $$\Omega \tau_L : \Omega_0 L \to \Omega_0 U/O \simeq BO.$$ The resulting Thom spectrum $(\Omega_0 L)^{\Omega \tau_L}$ is a ring spectrum. Thus one can apply topological Hochshild homology to this ring spectrum, $THH((\Omega_0 L)^{\Omega \tau_L})$ and one obtains a homotopy theoretic invariant of the Lagrangian isotopy type of the Lagrangian immersion $j$. This topological Hochshild was computed by Blumberg, the author, and Schlichtkrull in [@BCS]. It was shown that
$$THH((\Omega_0 L)^{\Omega \tau_L}) \simeq ({\mathcal{L}}_0L)^{\ell (\tau_L)}$$ where $\ell (\tau_L)$ is a specific stable bundle over the free loop space ${\mathcal{L}}L$. In particular, as was observed in [@ck], in the case when $\tau_L$ factors through a map to $U$, as is the case when $L=N$ and it satisfies the condiion (\[special\]), then there is an equivalence of stable bundles, $\ell (\tau_L) \cong \eta_L$, where $\eta_L$ is the Maslov bundle as above. As a consequence of Theorem \[criterion\] of Abouzaid and Kragh, one obtains the following:
Assume $j : N \to T^*N$ is a Lagrangian immersion in the homotopy class of the zero section, and that the complexification $TN\otimes {\mathbb{C}}$ is stably trivial. Then if, on the level of topological Hochschild homology, we have $
THH (\Omega_0 N)^{\Omega \tau_N}) ) $ is not equivalent to $ THH(\Sigma^\infty (\Omega N_+))
$, then $j$ is not Lagrangian isotopic to a Lagrangian embedding.
Using the results of [@BCS] the authors in [@ck] used this proposition to give a proof of Abouzaid and Kragh’s Theorem \[Lagsn\].
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[^1]: The author was partially supported by a grant from the NSF.
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abstract: 'The product monomial crystal was defined by Kamnitzer, Tingley, Webster, Weekes, and Yacobi for any semisimple simply-laced Lie algebra $\fg$, and depends on a collection of parameters $\bR$. We show that a family of truncations of this crystal are Demazure crystals, and give a Demazure-type formula for the character of each truncation, and the crystal itself. In type $A$, we show the product monomial crystal is the crystal of a generalised Schur module associated to a column-convex diagram depending on $\bR$.'
author:
- Joel Gibson
bibliography:
- 'references.bib'
title: A Demazure Character Formula for the Product Monomial Crystal
---
|
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abstract: |
We report on the clustering properties of Lyman–break galaxies (LBGs) at $z\sim 3$. The correlation length of flux–limited samples of LBGs depends on their rest–frame ultraviolet (UV) luminosity at $\lambda\sim 1700$ Å, with fainter galaxies being less strongly clustered in space. We have used three samples with progressively fainter flux limits: two extracted from our ground–based survey, and one from the Hubble Deep Fields (both North and South). The correlation length decreases by a factor $\approx 3$ over the range of limiting magnitudes that we have probed, namely $25\simlt {\cal
R}\simlt 27$, suggesting that samples with fainter UV luminosity limit include galaxies with smaller mass. We have compared the observed scaling properties of the clustering strength with those predicted for cold dark matter (CDM) halos and found that: 1) the clustering strength of LBGs follows, within the errors, the same scaling law with the volume density as the halos; 2) the scaling law predicted for the galaxies using the halos mass spectrum and a number of models for the relationship that maps the halos’ mass into the galaxies’ UV luminosity depends only on how tightly mass and UV luminosity correlate, but is otherwise insensitive to the details of the models. We interpret these results as additional evidence that the strong spatial clustering of LBGs is due to galaxy biasing, supporting the theory of biased galaxy formation and gravitational instability as the primary physical mechanism for the formation of structure. We have also fitted models of the mass–UV luminosity relationship to the data to simultaneously reproduce from the CDM halo mass spectrum the dependence of the correlation length with the UV luminosity and the luminosity function. We have found that 1) a scale invariant relationship between mass and UV luminosity (e.g. a power law) is not supported by the observations, suggesting that the properties of star formation of galaxies change along the mass spectrum of the observed LBGs; 2) the scatter of the UV luminosity of LBGs of given mass must be relatively small for massive LBGs, suggesting that the mass is an important parameter in regulating the activity of star formation in these systems, and that the fraction of massive halos at $z\sim 3$ that are not observed in UV–selected surveys is not large. From the fits, for a given choice of the cosmology, one can assign a scale of mass to the LBGs. For example, if $\Omega=0.3$ and $\Omega_{\Lambda}= 0.7$, the average mass of galaxies with luminosity ${\cal
R}=23$, 25.5 and 27.0 is $\langle M\rangle= 2.5$, 0.9, and $0.4\times 10^{12}$ [M$_{\odot}$]{}, respectively. The numbers are $\approx 2$ times larger and $\approx 10$ times smaller in an open universe with $\Omega=0.2$ and $\Omega_{\Lambda}=0$ and in the Einstein–de Sitter cosmologies, respectively.
author:
- Mauro Giavalisco and Mark Dickinson
title: 'CLUSTERING SEGREGATION WITH UV LUMINOSITY IN LYMAN–BREAK GALAXIES AT $z\sim 3$ AND ITS IMPLICATIONS'
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50[ h\_[50]{}\^[-1]{}]{}
INTRODUCTION
============
If cosmic structure has been assembled through the amplification of initial density perturbations by gravitational instability, then a general result is that the clustering of virialized systems differs from that of the average mass–density distribution, and depends on the properties of the systems themselves, such as mass and central density, with more massive systems being more strongly clustered in space (e.g. Kaiser 1984; Croft & Efstathiou 1994; Mo & Fukugita 1996; Jing & Suto 1996). The hierarchical models of galaxy formation provide a simple description of the clustering of virialized structures, or halos, in terms of the “geometrical” bias of their spatial distribution relative to that of the average mass-density field (e.g. Kaiser 1984; Peacock & Heavens 1985; Bardeen et al. 1986, BBKS), and both analytical works (Matarrese et al. 1997; Mann, Peacock & Heavens 1998; Catelan, Matarrese & Porciani 1998; Coles et al. 1998) and N–body simulations have consistently confirmed the overall robustness of this treatment (Brainerd & Villumsen 1994; Colin, Carlberg, & Couchman 1997; Mo & White 1996; Jing 1999; Katz, Hernquist, & Weinberg 1999).
In this scheme, visible galaxies and clusters have formed when the baryonic gas cooled and condensed in the potential wells of the halos (White & Rees 1978), and therefore their spatial distribution and clustering properties are predicted to be similar to those of the hosting halos. In broad terms, the observations support this interpretation. For example, in both the local universe (Davis et al. 1988; Hamilton 1988; Santiago & Da Costa 1990; Park et al. 1994; Loveday et al. 1995; Tucker et al. 1997; Valotto & Lambdas 1997) and at intermediate redshifts, e.g. $z\simlt 1$ (Le Fevre et al. 1996; Carlberg et al. 1997; Cohen et al. 2000), the clustering strength of galaxies has been observed to depend on properties of the galaxies that correlate with the mass, such as the optical luminosity and the spectral types, with more luminous and earlier types having stronger spatial clustering than fainter and later ones. For clusters of galaxies a similar trend with the richness parameter has also been observed (e.g. Bahcall & West 1992; Bahcall & Chen 1994). These results have generally been interpreted as evidence of the existence of galaxy biasing (Peacock 1997).
A number of works have attempted to test this paradigm by comparing its predictions to the properties of galaxies, using semi–analytical models or N–body simulations (for a review see, among others, Frenk & White 1991; Kauffmann, Nusser & Steinmetz 1997; Baugh et al. 1998; Governato et al. 1998; Katz, Hernquist & Weinberg 1999; Somerville, Primack & Faber 2000; Mo, Mao & White 1999). An essential assumption of these works is the relationship between the activity of star formation of nascent galaxies and the properties of the hosting dark matter halos, such as the mass, central density and angular momentum (e.g. Dalcanton, Spergel & Summers 1997; Heavens & Jimenez 1999). This relationship is key to the predicted properties of the galaxies in the models, but unfortunately, in contrast to the relative conceptual simplicity required to model the dark matter, the relevant physics is very complex and little empirical information is available to help constraining it. Until recently, the models have been tested against observations in the local and intermediate redshift universe. This, however, requires modeling the evolution of the galaxies from the epoch of formation to that of the observations. The properties of relatively evolved galaxies are, in general, the result of rather complicated processes of transformation (e.g. merging, interactions and environmental effects), and they can have decoupled from those of the systems in which they (or their components) originally formed.
A more straightforward test can now be attempted directly at high redshifts, thanks to the large and homogeneous samples of star–forming galaxies at $z>2$ made available by the highly efficient Lyman–break technique (Steidel et al. 1996; Madau et al. 1996; Steidel et al. 1999, S99). These “Lyman–break galaxies” (LBGs) are selected from their UV emission properties, which primarily reflects the instantaneous rate of star formation (modulo obscuration by dust). This offers the distinct advantage of working with a well defined physical quantity, which is relatively independent of the previous evolutionary history of the galaxies.
The samples are large enough that quantitative studies of clustering, e.g. low–order statistics such as the correlation function, have become possible. One of the most interesting results from these works is that LBGs are strongly clustered in space, with a correlation length comparable to that of present–day galaxies and larger than that of galaxies at intermediate redshifts (Steidel et al. 1998; Giavalisco et al. 1998; Adelberger et al. 1998, in the following S98, G98 and A98, respectively. See also Connolly, Szalay & Brunner 1998, and Arnouts et al. 1999). This is difficult to explain in terms of the clustering of the mass for any reasonable assumptions of the parameters of the background cosmology for typical CDM power spectra, if the normalization of spectrum has to match the present–day distribution of galaxies and clusters (e.g. Eke et al. 1995).
The strong spatial clustering of the LBGs and the apparently non–monotonic evolution with redshift of the galaxy correlation length (G98; Connolly et al. 1998) seem to be generally consistent with the expectations of the theory of biased galaxy formation (S98; G98; A98; Connolly et al. 1998; Peacock et al. 1999), apparently implying that they form in relatively massive halos. If halos and LBGs are physically associated, an obvious consequence is that the relationship between the clustering properties and the spatial abundance of the galaxies is determined by that of the halos, a fact that in principle can be tested. In this paper we present a measure of the scaling law of the clustering strength with the UV luminosity and with the volume density of LBGs at $z\sim 3$ and discuss the comparison with the predictions of the gravitational instability theory for CDM halos. Specifically, we present the measure the correlation length and the volume density of three samples of LBGs with different flux limits; the two brighter samples come from our ground–based survey, while the faintest one comes from the Hubble Deep Fields (both North and South, HDF hereafter).
The problem in comparing the observations with the theory is that one generally does not know how the properties of the halos (e.g. the mass) map into those of the observed galaxies (e.g. the UV luminosity). Therefore, with the help of a very simple but relatively general model we explore if, by requiring that the theory simultaneously reproduces the observed scaling law of the clustering strength with the UV luminosity and the luminosity function (our observables), one can derive some general conclusions about the nature of the clustering properties of LBGs as well as information about the association between the UV luminosity and the mass. Interestingly, the scaling law of the clustering strength with the volume density is largely insensitive to the assumptions of the model — it only depends on how tightly the UV luminosity correlates with the mass of the galaxies — suggesting that one can use the clustering of LBGs to test in broad lines our ideas on structure formation and the physics of star formation in high redshift galaxies.
We consider three cosmological models in the paper, an open universe with matter density parameter $\Omega=0.2$ and cosmological constant density parameter $\Omega_{\Lambda}=0$, a spatially flat, low–density universe with $\Omega=0.3$ and $\Omega_{\Lambda}=0.7$, and the Einstein–De Sitter universe with $\Omega=1$ and $\Omega_{\Lambda}=0$. Whenever we compute a physical quantity that depends on the cosmology, we list the three numbers that correspond to these three cosmologies in the same order as we have presented them here. Throughout the paper we use $H_0=100\, h$ .
THE THREE SAMPLES OF LYMAN–BREAK GALAXIES
=========================================
The high efficiency of the Lyman–break technique at $z\sim 3$, and the relative ease with which its selection criteria can be quantified and modeled, make it particularly advantageous for constructing large and well controlled samples of high–redshift galaxies that are nicely suited to study galaxy clustering (see the discussion in G98 and S99). The technique and its application to clustering studies have been extensively discussed elsewhere (Steidel, Pettini & Hamilton 1995; Madau et al. 1996; Dickinson 1998; S98; G98; A98), and here we only review the aspects that are relevant to this paper.
We shall now discuss the properties of the three samples of LBGs at $z\sim 3$, or “$U$–band dropouts”, that have been used in this study, which span a range of limiting magnitudes $\Delta\, m_l\approx 2.2$ mag. The two brighter samples are selected from our $U_nG{\cal R}$ ground–based survey (S99) and considerably overlap with those discussed by G98 and A98. The fainter one comes from the HDF surveys, both the North and South (Williams et al. 1996; Williams et al. 2000; Casertano et al. 2000). The brighter of the two ground–based samples, referred to here as the “spectroscopic sample” (SPEC), consists only of galaxies with secured redshifts, while the fainter one, or “photometric sample” (PHOT), includes all the $U$–band dropouts (with or without redshifts) down to limiting magnitudes ${\cal R}\le 25.5$ (all magnitudes in this paper are in the $AB$ scale of Oke & Gunn, 1983). The HDF sample is built in an analogous way and includes all the $U_{300}$–band dropouts down to $V_{606}\le 27$. At the time of this writing, there are 24 spectroscopic redshift identifications in the HDF–N sample (see Dickinson 1998), and 3 in the HDF–S (Cristiani et al. 2000). Table 1 lists the relevant parameters of the three samples.
The Ground–Based Samples
------------------------
As in previous studies of the clustering properties of LBGs (S98, G98 and A98), here too an object is considered a Lyman–break galaxy if its colors satisfy the criteria $$(U_n-G)\ge 1.0+(G-{\cal R}) ; \quad (U_n-G)\ge 1.6 ;\quad (G-{\cal R})\le
1.2,\eqno(2.1)$$ with the additional requirement ${\cal R}\le 25.5$ imposed to produce a reasonably complete sample. As discussed by S99, the definition of color criteria to select LBGs is, to some extent, arbitrary. The definition above is a relatively stringent color cut, and other criteria can certainly be defined that would result in larger samples of high–redshift galaxies. However, these would also contain a non negligible number of interlopers at lower redshifts. Since for two of the three samples discussed here we measure the correlation length by inverting the angular correlation function, interlopers provide a source of systematic errors that would bias our measures, and need to be minimized. With the spectroscopic information available, we have defined Eqn.2.1 in order to obtain an optimal balance between the competing requirements of having as large a sample as possible which is also as free of low-redshift interlopers as possible, i.e. maximizing the efficiency. In the above case, the only significant source of interlopers are galactic stars ($\approx 3.4$%), and all the galaxies that satisfy Eqn.2.1 and that have been confirmed spectroscopically were found to have redshift in the expected range for $U_n$-band dropouts, i.e. $2.2\simlt z\simlt 3.8$. LBG candidates that remain unidentified have spectra with S/N too low to allow a secure measure of the redshifts, and in no case was an identified redshift found outside the range expected for LBGs.
At the time of this writing our ground–based survey includes 1,243 candidates from 15 different fields that satisfy Eqn.2.1. We refer to this as the TOTAL sample. We have secured redshifts for 583 sources from the TOTAL sample. Of these, 20 are stars ($3.4$%), while the remaining 563 are high–redshift sources. These high–redshift objects form the TOTALSPEC sample, which includes 547 galaxies, 8 QSOs and 8 other types of AGNs. Figure 1 plots the redshift histogram of the 547 galaxies of the TOTALSPEC sample, after exclusion of the QSOs and AGNs. The two ground–based samples considered in this paper are defined as follows.
The [**PHOT**]{} sample includes the 876 LBGs (either candidates or galaxies with secured redshifts) of the TOTAL sample extracted from the 7 largest and deepest fields of the survey as detailed by G98, who used the sample to measure the angular correlation function of LBGs. The average surface density of LBGs with ${\cal R}\le 25.5$ from these 7 fields and its standard deviation are $$\Sigma_P = 1.22\pm 0.18\hbox{~ arcmin$^{-2}$}\eqno(2.2)$$
The [**SPEC**]{} sample consists of the 268 galaxies with secured redshift identification discussed in A98 plus additional 178 new redshifts from the field 1415+527 (the “Westphal field” in the nomenclature of G98). As before, we have selected these 446 galaxies because they come from our largest and deepest fields, which also have the highest degree of spectroscopic completeness. Thus, the SPEC sample is included in the PHOT sample, and consists of all its galaxies with measured redshifts. A98 used the initial sample of 268 galaxies to estimate the correlation length with the counts–in–cell analysis. With the new 178 redshifts from the Westphal field, we have obtained an additional data point for such statistics, which we have averaged, after proper weighting, to the previous measure as we will discuss later. The redshift histogram of the Westphal sample is plotted in Figure 2 together with the smoothed histogram of the TOTALSPEC sample.
We shall now compare the relative depth of the PHOT and SPEC samples, and assign an equivalent depth to the SPEC sample, in terms of the magnitude of the galaxies that numerically dominate it, in order to estimate its surface density. In the course of the spectroscopic follow–up of the Lyman–break survey, the selection of the targets from the pool of available candidates was primarily designed to optimize the size of the spectroscopic sample. Essentially, this meant cutting as many slits as possible in any one mask of the LRIS multi-object spectrograph (Oke et al. 1995) to correspond to the brightest possible candidates, in order to increase the chances of positive identification. As a result, the SPEC sample has been subject to an additional flux selection that has made it overall brighter than the PHOT sample. Although strictly speaking the SPEC sample is not flux–limited (galaxies as faint as ${\cal R}\sim 25.5$ can still be counted in it), this preferential selection of brighter galaxies makes the magnitude distribution equivalent, for practical purposes, to that of a traditional flux limited sample, as it can be seen in Figure 1.
We have estimated the depth and surface density of the SPEC sample by comparing its magnitude distribution $N(m)$ to that of the PHOT sample. First, we assigned a depth to both samples using the magnitude at which the number counts reach a maximum. We have identified the location of the maximum by arranging the magnitudes in a vector sorted by ascending order and running a boxcar window on this vector to derive the surface density of galaxies as a continuous, smooth function of the magnitude ${\cal R}$. This method has the advantage of not requiring any binning of the data and is relatively insensitive to the size of the boxcar window. For the SPEC sample, we find the maximum to be located at ${\cal R}_{eff}=24.7\pm 0.1$, where the error bar reflects the uncertainties induced by varying the size of the boxcar window and the photometric error. The maximum of the PHOT sample is located at ${\cal
R}_{eff}=25.1\pm 0.1$, namely 0.4 mag fainter. Subsequently, we have assigned the SPEC sample the surface density of a flux–limited sample of LBGs with $R_{eff}= 24.6$. We have estimated this value assuming that the magnitude distributions of the SPEC and PHOT samples have the same shape, which is approximately the case as shown by Figure 3. In this case we can scale the size of the PHOT sample by the ratio of the integral counts $N_T(m)=
\int_{-\infty}^m{N(m')\, dm'}$ evaluated at two magnitudes that differ from each other by $\Delta m=0.4$, namely the difference between the values of $m_{eff}$ of the two samples. We have computed this ratio using a linear fit of the $Log(N)$ vs. $m$ relationship in the range $22.5<{\cal R}<24.7$, which is a magnitude range relatively unaffected by flux incompleteness. Using this method, we have found the surface density of the SPEC sample to be $$\Sigma_S=0.7\pm 0.1\hbox{~ arcmin$^{-2}$},\eqno(2.3)$$ or $0.6\times$ smaller than the surface density of the PHOT sample. For practical purposes, the SPEC sample can be considered equivalent to a flux–limited sample with limiting magnitude ${\cal R}\simlt 25.1$. Figure 3 shows the magnitude distribution of the three samples.
The HDF Sample
--------------
The faintest sample that we have considered comes from the HDF survey and includes Lyman–break galaxies from both the Northern (HDF–N) and Southern (HDF–S) WFPC2 fields. We have used the STScI versions v2.0 and v1.0 of the stacked and drizzled images in the [$U_{300}$]{}, [$B_{450}$]{}, [$V_{606}$]{} and [$I_{814}$]{} filters of the HDF–N and HDF–S, respectively, to build the sample, retaining only the data from the WF chips, and neglecting the small area covered by the PC chip.
We have constructed samples of LBGs with criteria essentially identical to those adopted for the ground–based survey. The photometric systems used in the HDF survey, however, is somewhat different from the one adopted for the ground–based survey and, as a result, the resulting redshift distribution of the HDF LBGs differs from that of their ground–based counterparts. The most conspicuous difference is that the [$U_{300}$]{} bandpass is significantly bluer than the $U_n$ one, causing Lyman–break galaxies start to enter the selection window built with this bandpass already at redshifts $z\simgt 1.8$. By contrast, the ground–based bandpass $U_n$ starts to effectively pick up LBGs when they have redshifts in excess of $z\simgt 2.2$.
As we pointed out earlier, there is some degree of arbitrariness in setting the color cuts that define LBGs, and a number of samples of Lyman–break galaxies in the HDF (so far, essentially in the North field only) have been proposed by several groups. The color selection criteria differ, and no “definitive” method has yet been established. The number of LBG candidates therefore varies from sample to sample depending on the adopted criteria. For example, Madau et al. (1996) defined conservative criteria based on models of galaxy color distributions in order to select $z>2$ galaxies while avoiding significant risk of contamination from objects at lower redshifts. These criteria, however, miss some of the galaxies that have been spectroscopically confirmed to have $z\simgt 2$. Using the wealth of redshift information available in the HDF (Steidel et al. 1996b; Cohen et al. 1996; Lowenthal et al. 1997; Dickinson 1998) to fine-tune the selection, Dickinson (1998) built a sample of LBGs from the HDF–N by applying the so called “marginal” criteria proposed by Steidel, Pettini & Hamilton (1995), which for the HDF filters are re-written as $(U_{300}-B_{450})\ge 1.0+(B_{450}-V_{606})$ and $(B_{450}-V_{606})\le 1.2$. This selection window successfully recovers all the 27 spectroscopically confirmed LBGs in the HDF at $2<z<3.5$. As is the case for the ground–based sample, the dominant source of contaminants (always very modest in absolute terms) are galactic stars, but these are easily recognized in the WFPC2 images and excluded (all obvious stars with the above colors have also already been observed spectroscopically, as it turns out). Excluding the stars, there are 198 galaxies in the HDF–N and 222 in the HDF–S with $V_{606}<27.0$ that satisfy the above selection criteria.
The above selection window, however, is slightly different from that adopted for the ground–based samples, because it lacks the additional cut $(U_{300}-B_{450})\ge 1.6$. While such a difference is relatively minor, we have decided to include it in our definition of LBGs in the HDF for consistency with the ground–based sample. Therefore, the color selection criteria that we have adopted for the HDF sample are: $$(U_{300}-B_{450})\ge 1.0+(B_{450}-V_{606}) ; \quad (U_{300}-B_{450})\ge 1.6
; \quad (B_{450}-V_{606})\le 1.2,\eqno(2.4),$$ with the additional limit $V_{606}\le 27.0$ imposed to produce a sample reasonably complete in apparent magnitude. This selection window recovers 23 of the 24 spectroscopic redshifts of the HDF–N and the 3 ones of the HDF–S, and yields a sample of 126 galaxies in the HDF–N and 145 in the HDF–S, or a total sample of 271 galaxies. There are actually a few more LBG candidates in both HDF fields that satisfy Eqn.2.4. However, in the present study, we have considered only LBGs from those regions of the HDF WF chips that received the maximum exposure time, i.e. we have excluded the edges around each chip where the S/N was lower due to the dithering and the incompleteness is difficult to correct. The surveyed area is 4.45 arcmin$^2$ in the North and 4.76 arcmin$^2$ in the South, or a total area of 9.21 arcmin$^2$. Taking the average of the surface densities of the North and South field we find the surface density of LBGs to be $$\Sigma_{H} = 29.4\pm 3.5\hbox{~ arcmin$^{-2}$},\eqno(2.5)$$ where the error bar is quadratic sum of the two individual Poisson error bars. Finally, we can also put the HDF galaxies on the same magnitude scale of the ground–based ones using the approximation ${\cal R}=(V_{606}+I_{814})/2)$, proposed by Steidel et al. (1996b; see also S99). The distribution of the ${\cal R}$ magnitudes derived in this way is shown in Figure 3 with dotted data points.
CORRECTING THE VOLUME DENSITY FOR INCOMPLETENESS
================================================
In general, samples of LBGs will not include all the star–forming galaxies physically present in the volumes of space probed by the survey, even if their luminosity is high enough for detection. Galaxies brighter than the magnitude limit will be missing from the samples because of photometric errors, which scatter their colors outside of the selection window, confusion blending with nearby sources, and because they have intrinsic colors that do not satisfy the color selection criteria. For the ground–based samples, we have quantified this incompleteness with Monte–Carlo simulations in which large numbers of artificial LBGs with a range of intrinsic colors and magnitudes were added to the original CCD frames and then recovered using the same techniques adopted for the real galaxies. A complete discussion of the results of these simulations, including estimates of the intrinsic distribution functions, will be presented in a forthcoming paper (Adelberger et al. in preparation). Here we use the initial results of this work, which in part have already been discussed in S99, that have allowed us to correct the LBG samples for incompleteness with the goal to estimate their volume density.
The main output from the simulations is the probability $p(m,z)$ that a LBG at redshift $z$ with apparent magnitude $m$ and colors extracted from the [*intrinsic distribution*]{} of galaxy spectral shapes observed in LBGs at $z\sim
3$ is recorded in our sample (see S99 for the tabulated values of the function $p(m,z)$ and relevant discussion). For the purposes of this paper it is convenient to integrate $p(m,z)$ over the magnitudes and quantify the completeness of a sample with $m\le m_{lim}$ by means of a parameter as: $${\cal N}(z) = \alpha\times N(z),\eqno(3.1)$$ where ${\cal N}(z)$ and $N(z)$ are the [*predicted*]{} and [*observed*]{} redshift distributions, respectively. With good approximation $\alpha$ does not depend on the redshifts, and from the simulations we found $\alpha\sim
0.55$ at ${\cal R}=25.5$ ($\alpha$ is larger at brighter magnitudes). The effective cosmic volume comprised by the LBG samples is easily computed as the integral of the comoving volume element $dV(z)$ occupied by the galaxies with redshift in the range $[z,z+dz]$ per arcmin$^2$, weighting it by the probability that these galaxies are included in the sample, namely: $$V_{eff} = \int_0^{\infty} f(z)\, dV(z),\eqno(3.2)$$ where $f(z)=\alpha\, N(z)/N_{Peak}$. Finally, if $\Sigma$ is the surface density of LBGs, the volume density is simply given by $$n = {\Sigma\over V_{eff}}.\eqno(3.3)$$
The Ground–Based Samples
------------------------
The redshift distribution $N(z)$ of the 547 galaxies in the TOTALSPEC sample (from which we have excluded the 18 QSOs and AGNs) is shown in Figure 1, and a smoothed curve is shown in Figure 2. The mean value is $\bar z=3.04$ and the standard deviation is $\sigma_z=0.24$, with approximately $90$% of the galaxies included in the range $2.57\simlt z\simlt 3.37$, and none with $z<2.3$. This is rather similar to the one shown in G98, but it benefits from $\sim 45$% more redshifts. When we use this redshift distribution in the Limber transform of Eqn.4.5 (see later), we perform a local cubic spline interpolation on the histogram. In each case, the final results do not appreciably depend on the choice of the binning used to build the histogram.
We assume that the function $N(z)$ of Figures 1 and 2 is representative of both the SPEC and PHOT samples. In the case of the SPEC sample such an assumption is actually unnecessary, since this sample is contained in the TOTALSPEC one, and the galaxies in TOTALSPEC that are not included in SPEC come from fields that were not used in the counts–in–cell statistics only because they either were too small or had poor spectroscopic coverage.
The case of PHOT is different. This sample has only $\sim 50$% spectroscopic completeness and, as discussed above, is fainter, with $\sim 50$% of the still spectroscopically unidentified galaxies having ${\cal R}>25$ mag. Therefore, there is the possibility that its redshift distribution might differ from that of the SPEC sample. However, we believe that this is quite unlikely. As we have already pointed out in G98, our ability of securing redshifts for the LBGs of Eqn.2.1 has no obvious dependence on the magnitude (up to ${\cal R}\sim 25.5$) or on the color of the galaxies. To test this, we have divided the TOTALSPEC sample into 2 sub-samples, a bright one which includes all galaxies with ${\cal R}\le 24.5$, and a faint one, which contains the remaining galaxies. The redshift histograms of these 2 sub-samples are plotted in Figure 1 with thin dotted and dashed lines, respectively. As it can be seen from the figure, the 2 histograms are essentially identical, with mean redshifts $\bar z_B=3.021$ and $\bar z_F=3.016$, and standard deviations $\sigma_B=0.233$ and $\sigma_F=0.242$, respectively. A Kolomgorov-Smirnov test shows that the probability that the 2 distributions are not extracted from the same parent distribution is only $\sim 42$%. In practice, because both the volume density and (as we shall see) the Limber transform are rather insensitive to small changes in the shape of $N(z)$, we have assumed that the PHOT and SPEC samples have the same redshift distribution and that this is described by the histogram shown in Figure 1 (or the smooth curve in Figure 2).
Using Eqn.3.3 we measured the volume density of the PHOT sample to be $$n_{P} = (1.9\pm 0.3,\, 1.8\pm 0.3,\, 7.2\pm 1.1)\times\alpha^{-1}
\times 10^{-3}\hbox{~~{$h^{-3}$Mpc$^3$}}\eqno(3.4)$$ for the three cosmological models considered in this paper, respectively (see Section 1). The volume density of the SPEC sample is $1.8\times$ smaller and equal to $$n_{S} = (1.1\pm 0.2,\, 1.0\pm 0.2,\, 4.0\pm 0.6)\times\alpha^{-1}
\times 10^{-3}\hbox{~~{$h^{-3}$Mpc$^3$}},\eqno(3.5)$$ Table 2 lists the volume densities of the two ground–based sample for the values of $\alpha$ used here, namely $\alpha=0.55$.
The HDF Samples
---------------
With only 27 redshifts secured so far, the redshift distribution of the HDF samples is much less accurately constrained than that of the ground–based survey, which results in relatively larger uncertainties on its volume density and correlation length. The histogram of the HDF–N redshifts is shown in Figure 1 as a shaded area together with that of the ground–based sample. There are at most 5 galaxies in any of the redshift bins and only one galaxy each in the $z=2.6$ and $z=2.8$ bins, and the shape of $N(z)$ is unlikely to reflect the redshift distribution for the adopted selection criteria. The observed $N(z)$ most likely results from 1) small number statistics; 2) incompleteness at $z\simlt 2.5$ due to the difficulty of measuring the relevant spectral features with the LRIS spectrograph; and, more importantly, 3) to fluctuations introduced by the spatial clustering. Moreover, the secured redshifts are of galaxies belonging to the bright end of the luminosity distribution, whereas the $N(z)$ and the volume density of the sample are actually dominated by the galaxies close to the magnitude limit. Spectroscopic redshifts for these galaxies are not practical with current instrumentation, and although at the depth of the ground–based sample there is no evidence of a dependence of $N(z)$ with the luminosity, we also do not know if such an effect is present at fainter fluxes.
Despite this paucity of spectroscopy for the HDF LBGs, we can still obtain some constrain on their $N(z)$ by using the information on the intrinsic color distribution of the ground–based LBGs, as discussed in S99. This will allow us to derive the volume density and the correlation length of the HDF sample with enough precision for the purposes of this paper. Specifically, we have used Monte–Carlo simulations to compute the expected redshift distribution of the $U_{300}$ dropouts combining the ground–based intrinsic color distribution LBGs (under the assumption that it does not depend on the luminosity) with the color criteria of Eqn.2.4. This technique, discussed in detail in S99, also assumes that the volume density of LBG does not appreciably change over the probed redshift range. This is very likely a minor assumption, as suggested by the fact that the same technique is very successful in reproducing the observed redshift distributions of both the $U_n$- and $G$-band dropouts from the ground–based survey (see. Figure 4 of S99). The predicted $N(z)$ for the HDF is shown in Figure 7 of S99 and is also reproduced here in Figure 1 as a thin continuous line with arbitrary normalization. As expected, it extends to smaller redshifts compared to the ground–based one, although the two distributions are very similar at the high redshift end. The HDF one has mean redshift $\bar z_{H}=2.60$ and standard deviation $\sigma_{H}=0.69$.
To estimate the degree to which the volume densities and (as we shall see later) the spatial correlation length depend on the shape of $N(z)$, we have also considered two additional redshift distribution functions for the HDF sample. We considered 1) a top–hat function over the range $2.0\le z\le 3.5$, with mean redshift and standard deviation equal to $\bar z_{TH}=2.75$ and $\sigma_{TH}=0.43$. This is the redshift distribution originally assumed for the LBGs in the HDF–North by Madau et al. (1996); 2) a synthetic function built by summing together 5 modified versions of the ground–based $N(z)$, each obtained by rigidly shifting it by the amount $\Delta z=0.0$, $-0.2$, $-0.4$, $-0.6$, $-0.8$, respectively. This last function has mean redshift $\bar
z_{SYN}=2.61$ and standard deviation $\sigma_{SYN}=0.38$, and while it is qualitatively similar to both the fiducial and ground–based $N(z)$’s, it is narrower than the former and broader than the latter. Its mean redshift is the same as the fiducial function, consistent with the expectations given the HDF filters. Table 2 lists the volume densities computed from Eqn.3.3 with $\alpha=1$ for the three adopted cosmologies. It can be seen from the table that changing the $N(z)$ among those considered in our pool results in $\simlt 30$% variations of the volume density.
Note that, while we have included the effects of the intrinsic variations of the colors of LBGs when we have derived the expected redshift distribution of the HDF sample (and hence these effects have been accounted for in the measure of the volume density), we have not performed the Monte–Carlo simulations on the HDF images to determine the extent of the incompleteness due to photometric errors and crowding effects. This is very likely no cause of systematic errors in the measure of the angular correlation function, but it affects the measure of the volume density by causing it to be biased low. However, we expect this incompleteness to be significantly less severe than in the ground–based samples, as empirically supported by the good agreement between the luminosity function of the ground–based sample of LBGs (which contains the incompleteness corrections due to photometric errors and crowding effects) and that of the HDF North sample discussed by S99 (see their Figure 7a). The reason is that while photometric accuracy at the faint end of the HDF sample is comparable to that of the ground based one, the WFPC2 images have significantly higher angular resolution which greatly reduces the incompleteness due to crowding effects. We will quantify the incompleteness of the HDF samples in future works. For the moment we note that, strictly speaking, the measures of the volume density of the HDF sample must be considered a lower limit. As will become clear later, however, this will not significantly affect our conclusions.
THE CLUSTERING OF LYMAN–BREAK GALAXIES
======================================
In previous works we have measured the spatial correlation function of Lyman–break galaxies using the counts–in–cell technique for the SPEC sample (A98), and the inversion of the angular correlation function [$w(\theta)$]{} for the PHOT sample (G98). In this section we present the measure of the angular correlation function of the HDF sample and its inversion to estimate the spatial correlation length. We also present minor revisions to the previous measures using additional data.
The SPEC Sample
---------------
The variance $\sigma^2_g$ of galaxy counts in cells of assigned volume is proportional to the integral of the spatial correlation function $\xi_g(r)$ over the volume of the cell $$\sigma^2_g={1\over V^2_{cell}}\int\int_{V_{cell}}dV_1\, dV_2\, \xi_g(r_{12})
\eqno(4.1).$$ Assuming the power–law model $\xi_g(r)=(r/r_0)^{-\gamma}$, one can estimate the correlation length $r_0$ from $\sigma_g^2$ if one knows (or assumes) the value of the slope $\gamma$.
Since we have augmented the sample discussed in A98 with an additional 178 new redshifts from the Westphal field 1415+527 (the 446 total galaxies now comprise the SPEC sample), we have revised the measure of $r_0$. Figure 2 shows the distribution of the new redshifts in the Westphal field together with the redshift selection function of the survey. The measure by A98 consists of a weighted average of $\sigma^2_g$ estimated from cubic redshift cells of size 11.9, 11.4, 7.7 $\hh$Mpc in 6 individual fields that have the same geometry ($\sim 9$ arcmin in side). Since the Westphal field is considerably larger, we could not simply measure the corresponding $\sigma^2_g$ and include it in the average. Therefore, we have first derived $r_0$ and then averaged this value with the other analogous measures. We have analyzed the Westphal field with the same technique described in A98. We have divided the sample into a grid of roughly cubic cells whose transverse size is equal to the field of view (now $\sim 15.1$ arcmin), which at $z=3$ corresponds to $\sim 20.4$, 19.7 and 13.2 $\hh$Mpc, and estimated the count variance using both the two estimators discussed by A98 with consistent results[^1]. We found $\sigma^2_W=
0.1$[$_{-0.05}^{+0.04}$]{}, 0.22[$_{-0.1}^{+0.08}$]{}, 0.12[$_{-0.08}^{+0.06}$]{} (1–$\sigma$ error bars).
Using the power-law model and by approximating the cubes with spheres of equal volume (this considerably simplifies the algebra, but has little effect on the final results), the count fluctuations can be expressed in terms of the correlation length $r_0$ as $\sigma^2_{g}=72\,(r_0/R_{cell})^{\gamma}/
[(3-\gamma)(4-\gamma)(6-\gamma)\, 2^{\gamma}]$ (Peebles 1980, §59), with $R_{cell}$ the radius of the equivalent sphere. Assuming $\gamma=2.0\pm 0.2$ (the value of the slope measured from the PHOT sample, as we shall see later), we found the correlation length of the Westphal sample to be $r_0=2.7\pm 0.7$, $3.8\pm 1$ and $2.1\pm 0.6$ $\hh$Mpc. After inverse–variance weighting with the analogous values by A98, $r_0=5.6\pm 0.8$, $5.4\pm 0.8$ and $3.6\pm 0.6$ $\hh$Mpc using our value of $\gamma$, we have finally derived the correlation length of the SPEC sample: $$r_0=4.2\pm 0.6,\hbox{~~~~} 5.0\pm 0.7,\hbox{~~~~} 3.1\pm 0.4
\hbox{~~~~~$\hh$Mpc}.\eqno(4.2)$$ A smaller slope results in a larger correlation length, and the uncertainty due to the error on $\gamma$ is comparable to that due to the error on $\sigma^2_g$ (we added the two contributions in quadrature). Thus, the Westphal field is “less correlated” (albeit at the $\approx 2\sigma$ level) than the average (from the other 6 fields), very likely a manifestation of the still poorly quantified cosmic variance, although the values in Eqn.4.2 are within the errors of the measures by A98. Finally, it is important to keep in mind that with this technique one measures the average of the correlation function over a volume that is $\approx 5$ times larger than the correlation volume. The method, therefore, cannot provide any information on the shape of $\xi(r)$. Only with the assumption of the power law model and the value of its slope one can actually derive a value of the correlation length.
The PHOT and HDF Samples
------------------------
In G98 we presented the measure of $r_0$ from the PHOT sample, obtained by inverting the angular correlation function [$w(\theta)$]{} with the Limber transformation and the observed redshift distribution function. The same technique is employed here for the HDF sample. We used the estimator of [$w(\theta)$]{}proposed by Landy & Szalay (1993) $$w(\theta) = {DD(\theta)-2DR(\theta)+RR(\theta)\over RR(\theta)}.\eqno(4.3)$$ which minimizes the variance of the estimate (see also the discussion by Hamilton 1993). Here $DD(\theta)$ is the number of pairs of observed galaxies with angular separations in the range $(\theta,\theta+\delta\theta)$, $RR(\theta)$ is the analogous quantity for homogeneous (random) catalogs with the same geometry of the observed catalog, and $DR(\theta)$ is the number of observed-random cross pairs. This statistics produces an estimate of [$w(\theta)$]{}which is biased low by a factor (the “integral constraint”) $I\simeq
1+O((\theta_0/\theta_{\rm max})^{\beta})$, where $w(\theta_0)=1$ and $\beta\sim 1$ (Peebles 1974), but in the case of the PHOT sample the correlation amplitude is rather weak and $\theta_0 \ll \theta_{\rm max}$, and hence we can neglect this small correction. This bias can be significant in the HDF, however, because of its small areal coverage, and we have estimated its magnitude using numerical simulations, as we describe later.
For both the PHOT and HDF samples we measured [$w(\theta)$]{} from a weighted average of the individual [$w(\theta)$]{} functions of the fields that comprise each sample using inverse variance weighting; it made little difference if we used Poisson or bootstrap variance (Ling, Barrow & Frenk 1986; see G98 for a a complete discussion). In the HDF we have estimated [$w(\theta)$]{} separately from the whole mosaic of WF CCDs in each of the Northern and Southern fields and then averaged the two measures together. We have chosen not to split each HDF sub–sample according to the galaxies’ positions in the 3 CCDs that cover the WFPC2 field of view and then average the [$w(\theta)$]{} functions from each CCD. The advantage of this choice is that one maximizes the S/N due to the larger number of galaxy–galaxy pairs present in this case, an important consideration with such a small sample as the HDF (one looses all the inter–CCD pairs by considering each CCD separately). The disadvantage is that if there are slight variations of surface density in the LBG samples from the various CCDs (as the result of slight variation of sensitivity from chip to chip), then these will produce a spurious clustering signal. At the limiting magnitude of our sample there is no evidence for such variations, as we have verified by checking the surface density of LBG candidates and the S/N of the photometry in each chip from both the HDF–N and HDF–S. The fluctuations are consistent with the Poisson statistics, with no obvious indications that one CCD is preferentially detecting more candidates then the other ones. Therefore, we opted to work with the whole mosaic, after masking the regions that received less than the maximum amount of exposure time due to dithering.
We subsequently fitted the weighted average to the power law $w(\theta)=
A_\omega\theta^{-\beta}$ with Levenberg-Marquardt non-linear least-squares (Press [[*et al.*]{} ]{}1992). To estimated confidence intervals on the parameters $A_{\omega}$ and $\beta$, we generated a large ensemble of Monte–Carlo realizations (100,000) of the measured [$w(\theta)$]{}, assuming normal errors, and calculated best fit parameter values for each of these synthetic data sets (e.g. Press [[*et al.*]{} ]{}1992 §15.6). As discussed in G98, we found that the fitted parameters slightly depend on the choice of the binning used to compute [$w(\theta)$]{}, and to take this additional source of uncertainties into account we have included the effects of a randomly variable binning into the Monte–Carlo simulations. The error bars, therefore, reflect the uncertainty that derives from the choice of the binning. Figure 4 shows the measured [$w(\theta)$]{} of the PHOT and HDF samples plotted together with the best fit power-law model. For clarity the error bars of [$w(\theta)$]{} have been plotted separately from the data points.
Finally, we derived the spatial correlation function $\xi(r)$ by inverting [$w(\theta)$]{} with the Limber transformation, using the fiducial HDF redshift distribution $N(z)$ (Peebles 1980; Efstathiou [[*et al.*]{} ]{}1991). If the spatial function is modeled as $$\xi(r)=(r/r_0)^{-\gamma}\times F(z),\eqno(4.4)$$ where $F(z)$ describes its redshift dependence, the angular function has the form $w(\theta)=A_w\theta^{-\beta}$, where $\beta=\gamma-1$ and $$A_w = C\, r_0^{\gamma}\, \int_{z_i}^{z_f} F(z)\, D_{\theta}^{1-\gamma}(z)\,
N(z)^2\, g(z)\, dz\, \times
\Biggl[\int_{z_i}^{z_f} N(z)\, dz\Biggr]^{-2}\eqno(4.5)$$ (Efstathiou [[*et al.*]{} ]{}1991). Here $D_{\theta}(z)$ is the angular diameter distance, $$g(z) = {H_0\over c}\bigl[(1+z)^2(1+\Omega_0z+
\Omega_{\Lambda}((1+z)^{-2}-1))^{1/2}\bigr],\eqno(4.6)$$ and $C$ is a numerical factor given by $$C = \sqrt{\pi}\, {\Gamma[(\gamma-1)/2]\over \Gamma(\gamma/2)}.\eqno(4.7)$$ The Lyman–break galaxies’ redshift distribution is considerably narrower than those of traditional flux–limited redshift surveys and it is reasonable to expect little evolution of their clustering over so short a range. In this case the function $F(z)$ can be taken out of the integral and the quantity $r_0(z)=r_0\, F(z)$ is the correlation length at the epoch of observations.
The slope of the HDF sample is $$\gamma_{HDF}=2.2\hbox{{$_{-0.2}^{+0.3}$}}\eqno(4.8),$$ within the errors the same value found in the ground–based sample. The correlation length of the HDF sample obtained with the fiducial $N(z)$ in the Limber transform is $$r_{0,HDF}=1.1\hbox{{$_{-0.8}^{+0.9}$},~~~~~}\, 1.2\hbox{{$_{-0.8}^{+0.9}$}, ~~~~~}\,
0.8\hbox{{$_{-0.5}^{+0.6}$}~~~~~$\hh$Mpc},\eqno(4.9)$$ respectively, approximately 3 times smaller than the values found in the PHOT sample. We have estimate the uncertainty on the measure of $r_0$ from the inversion of the Monte–Carlo realizations of [$w(\theta)$]{}.
With the total number of redshift in the SPEC sample increased to 547 from the original 376 discussed in G98, we have also recomputed the Limber inversion of the PHOT sample using the updated $N(z)$. Using the same procedure described above for the HDF, we have found $$\gamma_{PHOT}=2.0\hbox{{$_{-0.2}^{+0.2}$}}\eqno(4.10)$$ and $$r_{0,PHOT}=3.1\hbox{{$_{-0.6}^{+0.7}$},~~~~~}\, 3.2\hbox{{$_{-0.7}^{+0.7}$},~~~~~}\,
1.9\hbox{{$_{-0.5}^{+0.4}$}~~~~~$\hh$Mpc}\eqno(4.11)$$ in the three cosmologies, respectively, in very good agreement with the values presented by G98. Table 3 lists the measures of the slope of the correlation function $\gamma$ and of the spatial correlation length $r_0$ for the three samples in each of the three adopted cosmology, together with their 68.3% confidence interval. It also shows how the HDF correlation length changes if one uses the top–hat and synthetic redshift distribution functions discussed above, instead of the fiducial one. It can be seen that the uncertainty on $N(z)$ introduces a negligible effect (given the present random errors) on the measure of $r_0$, and regardless of which $N(z)$ is adopted, the correlation length of the HDF sample is always $\approx 3$ times smaller than that of the brighter PHOT sample.
RESULTS
=======
Figure 5 plots the correlation length of the three samples as a function of both the limiting magnitude and the corresponding absolute magnitude (in the $\Lambda$ cosmology), and shows the main result of this study, namely that the clustering strength of Lyman–break galaxies is a function of their rest–frame UV luminosity at $\lambda\sim 1700$ Å. Remembering that this is a good tracer of the instantaneous rate of star formation (modulo obscuration by dust), another way to state the result is that LBGs with higher star–formation rates are more strongly clustered in space.
Note that, strictly speaking, the SPEC and PHOT samples are not completely statistically independent, since the galaxies of the former also belong to the latter. However, in regard to the measure of the correlation length they may be considered as independent. In the case of the SPEC sample, the correlation length has been derived from the fluctuations of the counts in very large cubic cells, exploiting only the information on the redshifts of the galaxies with no consideration to the relative angular displacements between them. This information, on the other hand, has been used to measure the function [$w(\theta)$]{} of the PHOT sample, which also contains twice as many galaxies as the SPEC one. The redshifts have only been used in a cumulative way during the measure of the correlation length of the PHOT sample, as the distribution function $N(z)$ in the Limber integral of Eqn.4.4.
How significant is the detection of clustering segregation? Taken at face value, the data shown in Figure 5 suggest that we have likely detected it, despite the relatively large random errors. A $\chi^2$ test computed under the null hypothesis that $r_0$ does not depend on $m_{lim}$ (using the weighted average of the three measures as an estimate of the true value of $r_0$) yields $\chi_r^2=4.44$, 5.32, and 5.40 with 2 degrees of freedom. This implies that the null hypothesis is rejected at the 98.82%, 99.51% and 99.55% confidence level in the three cosmologies, respectively.
Systematic errors, however, can affect the measure of [$w(\theta)$]{}. The PHOT and HDF samples have only partial redshift completeness and they might contain a fraction of interlopers at significantly lower redshifts than the rest of the LBGs. The inclusion of a fraction $f$ of interlopers, which are spatially uncorrelated with the Lyman–break galaxies and hence essentially distributed at random respect to them, causes the observed estimate of [$w(\theta)$]{} to be underestimated by a factor $(1-f)^2$. However, this is unlikely to be a problem. As discussed in G98, the overall efficiency of the Lyman–break technique in going from photometrically selected candidates to spectroscopically confirmed LBGs is $\ge 75$%, with the redshifts of the remainder $25$% undetermined. Hence, the interloper contamination of the PHOT sample (and presumably that of the HDF sample as well) is $f\le 25$%. However, the true contamination is likely to be significantly smaller, since interlopers were never found among the positive identifications, and all the missed identifications were due to inadequate S/N. These were probably caused by [*observational accidents*]{}, such as astrometric errors or small misalignments of the multi-object masks of the spectrograph with the selected targets that resulted in insufficient flux being recorded on the detector.
A more serious problem is the possibility that the observed correlation lengths are affected by systematic error due to cosmic variance, if the samples are not fair representations of the large scale structure at $z\sim
3$. As mentioned earlier, this results in a bias on the measured amplitude of [$w(\theta)$]{} (and on the volume density) induced by structure fluctuations on scales that are comparable or larger than that of the sample (integral constraint). This is unlikely a major problem for the SPEC and PHOT samples, since they probe relatively large volumes of space and consist of several, independent sub–samples from different regions of the sky, but it might affect the much smaller HDF sample. For example, Steidel et al. (1999) noticed that removing the criterion $U_{300}-B_{450}>1.6$ from the definition of LBGs in the HDF changes the estimated effective volume by only $\sim 14$%, but increases the total size of the sample by 45%, significantly altering the shape of the luminosity function. The excess galaxies have UV colors consistent with the possibility that they all belong to a single structure at $z\sim 2$. This shows the potential dangers of working with samples covering small volume of space such as the HDF. Fortunately, the inclusion of the criterion yields a luminosity function that matches fairly well that of the ground–based sample (defined through the same photometric criteria), both in normalization and faint–end slope, and, as noticed by S99, a Schechter fit to the ground–based data alone provides a good description of the HDF data as well. Also, we found that the correlation function of the HDF sample is insensitive, within the error, to the exclusion of the color criterion.
The difference of $r_0$ between the SPEC and PHOT samples is small enough (given the errors) that alone it does not provide any information about the clustering segregation (the $1\sigma$ error bars almost overlap). The evidence comes from the measure of $r_0$ in the HDF sample, and in order to assess the robustness of the result, we have used numerical simulations to estimate the confidence level that the HDF is indeed less correlated than the PHOT sample. [^2] Specifically, we have generated mock HDF samples that are realizations of a parent population with assigned correlation function, and we have measured their [$w(\theta)$]{} following exactly the same procedure used for the real data, including averaging over two mock samples to simulate the HDF–N and HDF–S observations We have generated a series of large galaxy distributions by performing Poisson sampling of 2–D lognormal density fields that have power spectrum corresponding to the assigned correlation function, assigning a probability of finding a galaxy in a given point as proportional to the density field in that point. We have normalized the probability so that the number density of the parent population of the mock galaxies has an assigned value, which we taken equal to the value observed in the HDF (Eqn.2.5). To simulate the effects the density fluctuations on large scales, we have allowed the size of each galaxy distribution to be much larger than the size of the HDF (we used $60\times 60$ square arcmin), and in each case we have randomly extracted from this distribution a sample with the same geometry of the HDF. Thus, by construction, the “measured” [$w(\theta)$]{} of each mock sample is an estimate of the parent population’s [$w(\theta)$]{}, and it is subject to the combined effects of random errors and bias by integral constraint as the real data.
In the null hypothesis that there is no clustering segregation, the HDF and the PHOT samples have the same parent angular correlation function. To estimate how confidently the data allow us to reject this hypothesis, we have used the power–law fit to the observed [$w(\theta)$]{} of the PHOT sample as input to the simulations and compared the distribution of the measures of [$w(\theta)$]{} in the mock samples with the observations in the HDF. Note that this is a conservative approach, since the observed [$w(\theta)$]{} is an underestimate of the parent correlation function of the PHOT sample because of the integral constraint bias. The top panel of Figure 6 shows the distribution of the values of [$w(\theta)$]{} of the mock samples with $1<\theta<2$ arcsec together with the observed HDF value. According to the simulations, the null hypothesis can be rejected at the 98.4% confidence level.
Since the range of angular separations where the HDF and PHOT correlation functions are reliably measured do not overlap, however, the above simulations can only test that the HDF is not clustered as strongly as the small–scale extrapolation of the PHOT [$w(\theta)$]{}. Therefore, we have conducted a second test, where we have measured the correlation function of the HDF sample and of the mock samples in very large bins, 25 arcsec in size, to provide partial overlapping of angular separations with the PHOT sample. The bottom panel of Figure 6 shows the distribution of the simulated values of [$w(\theta)$]{} at $\theta=25$ arcsec together with the observed value, while Figure 7 shows the observed correlation functions of the PHOT and HDF samples as well as the best fit power–law models. Note that the error bars of the HDF data points are smaller than those in Figure 4, because of the larger number of pairs in the 25 arcsec bins. In this test too, the simulations suggest that the null hypothesis can be rejected at the 98.6% confidence level.
In conclusion, the simulations increase our confidence that we have detected clustering segregation in the LBG population at $z\sim 3$. While we regard the detection as tentative —larger samples are needed to improve upon the accuracy and precision of the current measures— the simulations suggest that the samples selected with the criteria of Eqn.2.4 are at least not grossly unrepresentative of the $z\sim 3$ LBG population at faint magnitudes, and the value of $r_0$ that we have measured is not largely affected by systematics.
DARK MATTER HALOS AND LYMAN–BREAK GALAXIES
==========================================
In this section we will discuss the implications of the detection of clustering segregation and how we can use it to derive information on the relationship between the activity of star formation and the mass of the galaxies.
The clustering strength and abundances of virialized structures that form via gravitational instability (“halos” hereafter) depends to a large extent on their mass, with more massive halos being more strongly clustered and less abundant in space than less massive ones (e.g. see Mo & White 1996). The UV morphology of LBGs (Giavalisco et al. 1996a; Steidel et al. 1996b; Lowenthal et al. 1997) suggests that some degree of virialization has taken place in these structures. The strong spatial clustering of bright LBGs suggests that these systems trace relatively massive halos, with comparatively few halos of small mass populating the bright samples (otherwise we would not observe the strong clustering), apparently implying a correlation between the UV luminosity and the mass. The detection of clustering segregation seems to provide additional empirical support to this idea.
If the UV luminosity is somehow correlated with the mass, the clustering strength should obey the same scaling laws predicted for the halos. Thus, it is interesting to compare the clustering properties and abundances of the LBGs with the predictions of the gravitational instability theory. The problem is that the only visible halos are those hosting star–forming galaxies with UV luminosity and spectral energy distribution that satisfy the selection criteria and sensitivity of the observations, and to predict the clustering properties of the LBGs one needs to know the relationship between the mass and the UV luminosity. A tested model for this relationship is not available, since the relevant physics is poorly understood. Luckily, as we shall see, the scaling law of the clustering strength with the volume density is highly degenerate with the form of the mass–UV luminosity relationship. Thus, we will compare the observed clustering properties of LBGs to the theory using a simple phenomenological model fitted to reproduce the luminosity function and the scaling law of the correlation length with the UV luminosity.
We have modeled the mass distribution with the CDM power spectrum and we have computed the mass spectrum of the halos using the Press-Schechter (1974, PS) formalism. We have implemented the formalism using the technique described by Mo & White (1996). We used a primordial power spectrum with spectral index $n=-1$, and the transfer function given by Bardeen et al. (1986) with $\Gamma=0.25$, the value measured from local galaxy surveys (Peacock 1997; Dodelson & Gaztanaga 1999). We fixed the amplitude of the spectrum at $z=0$ by adopting the cluster normalization of Eke et al. (1996), with $\sigma_8=1.0$, $0.9$ and $0.5$ for our three adopted cosmologies.
The volume density and clustering strength of halos of mass $M$ are controlled by the fractional overdensity at the collapse, $\nu\equiv\delta_c/\sigma(M)$, where $\delta_c\approx 1.7$ is the linear overdensity of a spherical perturbation at collapse (e.g. Peebles 1980) and $\sigma(M)$ is the rms density fluctuation of the density field smoothed using a spherical top–hat window enclosing the mass $M$. For a Gaussian density field, the volume density of the halos is approximately given by $$n(M)\, dM = \sqrt{{2\over\pi}}\, {\bar\rho\over M}\, e^{-\nu^2/2}\,
{d\nu\over dM}\, dM,\eqno(6.1)$$ where $\bar\rho$ is the average mass density of the universe.
The spatial clustering of the halos, to first order, is amplified respect to that of the mass by a factor $\simeq\nu^2$ and independent of the spatial scale (Kaiser 1984; Mo & White 1996), and the correlation function of the halos is proportional to that of the mass through the second power of the linear bias parameter $b$: $$\xi_h(r) = b^2\times \xi_m(r),\eqno(6.2)$$ where to first order $$b\simeq 1+(\nu^2-1)/\delta_c.\eqno(6.3)$$ In this linear bias model, the clustering of halos of a given mass is entirely specified by the value of $b(M)$, because the $\xi_m(r)$ is fixed once the power spectrum has been assigned. Note that there are no free parameters in the model other than those necessary to specify the cosmology and the power spectrum, namely $\sigma_8$, $\Gamma$, and the spectral index $n$.
A comparison to N–body simulations shows that Eqn.6.3 is accurate for halos with mass $M\simgt M_*$[^3], but it progressively over–predicts the bias on smaller scale (Mo & White 1996). Jing (1999) proposed a revised formula, derived from a fit to the results of N–body simulations, that it is accurate to better than 15% in the range $M\simgt 0.01M_*$ (see also Porciani, Catelan & Lacey 1999). The mass range and distribution of the halos used for the comparison with the LBGs are set if one requires that the samples of halos have the same clustering strength and volume density of the galaxies; when carrying out such comparisons, we used Eqn.6.3 as well ass the analogous ones made with Jing’s approximation, finding similar results.
Another important point is that Eqn.6.2 is not valid over spatial scales smaller than the linear size of the halos, because of spatial exclusion effects. N–body simulations (e.g. see Figure 6 of Mo & White 1996) show that Eqn.6.2 begins to over–predict the autocorrelation function of halos (with $M\simgt M_*$) in Eulerian space (the observed one) for separations of the order of the Eulerian radius. We can estimate this from the relation $M=4/3\pi\delta_c{\bar\rho}R^3_E$, where for a spherical perturbation $\delta_c$ is known (Bryan and Norman 1998) and ${\bar\rho}$ is the background density, and have an idea of the spatial (and angular) scales over which exclusion is important. The mass spectrum of LBGs is currently unconstrained, however the dependence of $R_E$ with the mass is very shallow and we can get a relative good answer using the few available estimates of the mass of bright galaxies by Pettini et al. (1998). These are in the range $10^{10}$ to $10^{11}$ [M$_{\odot}$]{}, and are likely lower limits to the true values. As we shall see later, spatial abundances and clustering properties of our three samples also suggest masses in the range $10^{11}$ to $10^{12}$ [M$_{\odot}$]{} for the SPEC and PHOT samples and $10^{10}$ to $10^{11}$ [M$_{\odot}$]{} for the HDF. If $M=10^{10}$ ($10^{12}$) [M$_{\odot}$]{}, $R_E\sim 36$ (170) $h^{-1}$kpc in comoving coordinates, corresponding to a projected angular separation on the sky $\theta\sim 2.5$ (12) arcsec. Hence, exclusion effects do not seem to be a factor in the measure of [$w(\theta)$]{} of the ground–based samples (see G98). They also seems unlikely in the HDF sample, since of the 163 pairs with $\theta\le 10$ arcsec (where the bulk of the signal is observed, see Figure 4), only 15 have $\theta
\simlt 2.5$ arcsec. Of these, only a fraction of the order of $[R_E/L_{eff})
\times (1+\xi(R_E)]\sim 15$% ($L_{eff}\sim 150$ $h^{-1}$Mpc is the effective linear size of the survey along the line of sight) is actually expected to have physical separations $r\simlt R_E$, namely only about 2 pairs.
It is convenient to compare the clustering and halos and galaxies using the bias instead of the correlation length, because in this way, both the data and the models depend in a similar way by the choice of the normalization of the power spectrum and, to first order, the comparison is not affected by the uncertainties on $\sigma_8$. For the same reason, however, the comparison is also largely degenerate respect to the choice of cosmological parameters, namely to the absolute amplitude of the power spectrum at a given epoch.
Equations 6.3 describes the “monochromatic” bias of halos with mass $M$. The average bias of a mass–limited sample of halos with mass $M\ge M_L$ and mass function $n(M)$ is given by $$\langle b\rangle ={\int_{M_L}^{\infty} dM\, n(M)\, b(M)\over n_d},
\eqno(6.4)$$ where $n_d=\int_{M_L}^{\infty} dM\, n(M)$ is the mean volume density of the sample, and the expectation value of the correlation function of the halos of the sample is, to first order, related to the mass autocorrelation function by the usual relation $\xi_h(r)=\langle b\rangle^2\xi(r)$ (Porciani et al. 1998). In the following we will refer to the function $\langle b\rangle =\langle
b(n_d)\rangle$ as the [*clustering segregation function*]{}.
In practice, the average bias of the LBGs of the SPEC sample is estimated from the ratio of the observed variance of galaxy counts to the variance of the mass fluctuations on the same scale (see A98): $$\langle b\rangle=\sqrt{{\sigma^2_{g}(R_{cell})\over \sigma^2_m(R_{cell})}},
\eqno(6.5)$$ while for the PHOT and HDF samples it is given by the ratio of their correlation function to that of the mass: $$\langle b\rangle=\sqrt{{\xi_g(r_0)\over \xi_m(r_0)}}.\eqno(6.6)$$
It is important to realize that these expressions are relative to different spatial scales, namely several Mpc for the SPEC and PHOT samples and $\sim 1$ $\hh$Mpc for the HDF one. Observations in the local universe (Peacock 1997) suggest that galaxy bias varies relatively slowly with the scale. However, while limiting our analysis to the scale–independent linear bias is probably appropriate with the present accuracy, one must keep in mind that the scale dependence of the bias at $z\sim 3$ is currently unconstrained (see also. Mann, Peacock & Heavens 1998; Somerville et al. 2000; Chen & Ostriker 2000).
Finally, since Eqn.6.5 and 6.6 probe different spatial scales, they are also sensitive in different ways to the presence of non–linear clustering. Eqn.6.5 provides a measure of the bias averaged on scales that at $z\sim 3$ are only mildly non–linear or not linear at all (depending on the adopted cosmology), while Eqn.6.6 probes scales where relatively strong non–linear clustering can have already developed at $z\sim 3$, particularly in an open cosmology. Such non–linearities result in an overestimate of the linear bias above, if only the linear part of $\xi_m(r)$ is used. This systematics would affect the open cosmology more than the flat ones, because for these $\xi_m(r)$ enters the non–linear phase sooner in redshift than in the other two cases. Therefore, we have computed $\xi_m(r)$ taking the Fourier transform of the non–linear power spectrum, following the prescription of Peacock (1997).
The bottom panel of Figure 8 shows the clustering segregation function of the LBGs (data points) compared to that of the CDM halos (continuous curve) for the $\Lambda$ cosmology. The plot shows that, within the errors, the LBGs have the same clustering segregation function of the halos (similar results are valid for the other two cosmologies). To understand what such an agreement means and to actually compare the observations to the predicted segregation function of the galaxies (as opposed to that of the halos) we need to model the relationship between mass and UV luminosity, which is the subject of the next section.
The Mass–UV Luminosity Relationship
-----------------------------------
The luminosity at $\sim 1700$ Å of LBGs is mostly the result of their star formation rate and the amount of intrinsic dust obscuration that the radiation suffers [*in situ*]{}, before leaving the galaxies. Physical arguments suggest that the mass is an important parameter in determining the star formation rate of the galaxies (e.g. Frenk & White 1995; A98). However, it is also reasonable to expect that both dust obscuration and star formation activity are characterized by some amount of stochasticity, even in galaxies with the same mass. For example, a different angular momentum (e.g. Dalcanton, Spergel & Summers 1997; Heavens & Jimenez 1999), density profile (Kennicutt 1998a,b), or interactions with other nearby systems could result in LBGs with equal mass having different star formation rate. Or galaxies with the same mass and star formation rate could have different dust obscuration and, thus, different UV luminosity. Therefore, we have assumed that the UV luminosity $L_{UV}$ of LBGs hosted in halos of mass $M$ is a random variable that depends on $M$ through its mean $\langle L_{UV}\rangle={\cal L}(M)$ and variance $\sigma^2_{UV}(M)$, and we have modeled $L_{UV}$ as the product of ${\cal L}(M)$ with a random variable $A$ $$L_{UV} = A\, {\cal L}(M),\eqno(6.7)$$ where the mean value of $A$ is, by definition, equal to 1. For simplicity, we have assumed in the following that the distribution function of $\log(A)$ is Gaussian.
Before fitting the model mass–UV luminosity relationship to the data to simultaneously reproducing the clustering strength, luminosity distribution and spatial abundances of LBGs we have to select the observables. Luminosity and abundances are constrained by the luminosity function $\phi(L_{UV})$. As to the clustering properties, the clustering segregation is not very useful because, as we will see in a moment, it is largely degenerate with the shape of ${\cal L}(M)$. Much more useful is the function $\langle b\rangle=\langle
b(L_0)\rangle$ that links the average bias of a sample of LBGs to its limiting luminosity $L_0$. This depends on both $\sigma^2_{UV}$ and ${\cal L}(M)$, since these together set the mass range and distribution of the galaxies that enter the sample. Computing the predicted observables is straightforward. The probability to find a galaxy hosted in a halo with mass $M$ that has luminosity $L=L_{UV}(A,M)$ is $$\pi(A,M)\, dA\, dM = n(M)\, N_g(M)\, f(A)\, \delta(L-L_{UV}(A,M))\, dA\,
dM\eqno(6.8)$$ where $N_g(M)$ is the average number of galaxies hosted in halos of mass $M$ and $f(A)$ is the distribution function of $A$. The average bias of a flux limited sample of LBGs with $L>L_0$ is given by: $$\langle b_e(L_0)\rangle = {\int_0^{\infty}p(M|L_0)\, b(M)\, dM
\over \int_0^{\infty}p(M|L_0)\, dM},\eqno(6.9)$$ where $$p(M|L_0)=\int_{L_0}^{+\infty}dL \int_{-\infty}^{+\infty} \pi(A,M)\, dA =
n(M)\, \int_{L_0}^{+\infty} f(A)\, \biggl({\partial L_{UV}\over
\partial A}\biggr)^{-1}\, dL\eqno(6.10)$$ is the probability that a halo with mass $M$ is included in the sample, and the partial derivative is computed for $L_{UV}(A,M)=L$. The luminosity function is given by: $$\phi(L) = \int_0^{\infty}dM \int_{-\infty}^{\infty} \pi(A,M)\, dA =
\int_0^{\infty} \biggl({\partial L_{UV}\over \partial A}\biggr)^{-1}\, f(A)\,
n(M)\, N_g(M)\, dM.\eqno(6.11)$$ It is also interesting to compute the average mass and the variance of a sample of galaxies with luminosity $L$, which are given by: $$\langle M(L)\rangle = {\int_0^{\infty} M\, \bigl({\partial L_{UV}\over
\partial A}\bigr)^{-1}\, f(A)\,n(M)\, N_g(M)\, dM \over \phi(L)},
\eqno(6.12a)$$ and $$\sigma^2_M(L)\rangle = {\int_0^{\infty} (M-\langle M(L)\rangle )^2\,
\bigl({\partial L_{UV}\over \partial A}\bigr)^{-1}\, f(A)\,n(M)\, N_g(M)\, dM
\over \phi(L)}.\eqno(6.12b)$$
In the following we have assumed $N_g(M)\equiv 1$; as we will discuss later, it is unlikely that this approximation has grossly misdirected our results. We have experimented with 3 different models to study how the functional form of ${\cal L}(M)$ and $\sigma^2_{UV}(M)$ affect the observables. Model A has ${\cal L}(M)=\epsilon\, M^{\alpha}$ and $\sigma^2_{UV}=\sigma^2_0$, and hence three free parameters, namely $\alpha$, $\epsilon$ and $\sigma$. In model B the variance is a function of the mass, and for simplicity, we worked in the “logarithmic space”[^4], and we modeled the variance of the variable $\log(A)$ as $\sigma^2=\sigma^2_0[{1/1+(M/M_{\sigma})}]^2$. Model B has four free parameters, namely $\alpha$, $\epsilon$, $\sigma_0$ and $M_{\sigma}$. In model C the variance is again constant, but the slope of the power law varies from the high–mass asymptotic value $\alpha$ to the low–mass one $\beta$, and is equal to $(\alpha+\beta)/2$ at the cut–off mass $M_{co}$. We used the arctg law to model this variation. This model has five free parameters, namely $\alpha$, $\beta$, $M_{co}$, $\epsilon$, and $\sigma_0$.
We have fitted the models to the data using the multi–dimensional algorithm “AMOEBA” (Press et. al. 1994) to minimize the sum $\chi^2=\chi^2_C+\chi^2_L$ of the two chi–square’s relative to the function $\langle b(L_0)\rangle$ and $\phi(L)$, respectively. The data on the luminosity function come from the LBGs of the PHOT and HDF samples, after dimming the HDF magnitudes by the amount $\Delta{\cal R}=0.25$ (S99) to account for their smaller mean redshift, according to the prescription by S99. When computing the $\chi^2_L$, we did not include the two faintest data points, because no corrections for incompleteness have been computed for the HDF sample, and these are very likely the ones more strongly affected.
To a good extent, the overall slope of $\phi(L)$ is controlled by the logarithmic slope of ${\cal L}(M)$, while the curvature depends on $\sigma^2_{UV}$. If $\sigma^2_{UV}$ is constant, increasing its value reduces the curvature. If $\sigma^2_{UV}$ increases towards smaller masses, the curvature of $\phi(L)$ increases (interestingly, a $\sigma^2_{UV}$ that increases with $M$ is inconsistent with the data, since it results in a luminosity function with the wrong curvature). Increasing the logarithmic slope of ${\cal L}(M)$ makes the function $\langle b(L_0)\rangle$ steeper, while increasing the variance flattens it. The latter effect is very pronounced at the high–end of the mass spectrum (and of the luminosity function), but it becomes progressively less noticeable at smaller masses (and fainter luminosities).
The top two panels of Figure 8 show the results of the fits in the case of the $\Lambda$ cosmology, while the bottom panel shows the clustering segregation function of the galaxies computed from the fitted model, compared to the analogous quantity for the halos (continuous curve) and to the observations (data points). The dotted curves represent model A, the short–dashed curves model B, and the long–dashed curves model C. Similar plots are obtained in the other two cosmologies. However, while the quality of the fits in the case of the open cosmology is similar to the case discussed above, the fits are somewhat worse in the Einstein–de Sitter world. We plan to return to the comparisons of the results of the fits in the various world models in a forthcoming paper.
Model A yields a slope $\alpha\sim 1.5$ and a small variance ($\sigma\simlt
0.2$), and as the figure shows, it fits the observed $\langle b(L_0)\rangle$ data points relatively well, but only crudely reproduces the shape of the luminosity function. Smaller values of $\alpha$ would give a better agreement at the high–luminosity end, but a larger discrepancy at the low–luminosity one, while a larger slope would have the opposite effect. The clustering segregation function predicted by this model is very similar to that of the halos (because of the very small scatter). Increasing $\sigma_0$ decreases the curvature of the luminosity function and it decreases $\langle b(L_0)\rangle$, reducing the quality of the fit. Thus, while a power–law (with the CDM + PS mass spectrum) is consistent with the clustering properties of LBGs, it is inconsistent with the luminosity function.
By allowing the scatter to increase at the low–mass end of the spectrum, model B improves the fit to the luminosity function without significantly changing the agreement with the data on $\langle b(L_0)\rangle$. We found $\alpha\sim 1.0$ and $\sigma_0\sim 7.5$. This corresponds to $\sigma\sim 0.7$ for $M=10^{12}$ [M$_{\odot}$]{}, and $\sigma\sim 3.8$ for $M=10^{11}$ [M$_{\odot}$]{}. What happens is that LBGs hosted in halos of lower mass become increasingly less likely to be bright enough to be included in the sample, progressively decreasing $\phi(L)$ at the faint end, and thus increasing its curvature. This effect combines with the shallow dependence of the bias on the mass for small masses to make the clustering segregation function of the LBGs predicted by this model highly degenerate respect to the parameters of the mass–UV relationship, and essentially identical to that of the halos. Note that this model predicts that the clustering strength reaches a maximum at the bright end and then it decreases again. This is because the bias is a steep function of the mass for large masses, and the fitted variance is large enough to decrease the average bias of very bright samples.
The variable slope of the power–law in model C allows for a better fit to the luminosity function than model A. We found a high–mass slope $\alpha\sim 0.8$ and a low–mass slope $\beta\sim 2$, with $\sigma\sim 0.7$. Note that accurate measures of the correlation length of bright samples will be very effective to discriminate among the various models.
We note that error bars on the fitted parameters, estimated by bootstrapping the data points with gaussian distributions, are in general relatively small, with the 68% confidence level fractional errors in the range 30–50%. This is due to the fact that the fits are dominated by the data points of the luminosity function. There are cases, however, when the errors can be very large, such as the error on $\sigma_0$ in model A, which is $\sim 1000$%. The reason is that in this model the function $\langle b(L_0)\rangle$ has a stronger dependence on $\sigma_0$ than $\phi(L)$ has, and thus the error is dominated by the relatively large uncertainties on the clustering data. In this case the fit only yielded an upper limit to the value of the scatter ($\sigma_0\simlt 0.2$ from 1,000 bootstrapped samples).
Finally, although we do not discuss in detail the results of the fits, it is important to keep in mind that the mass–UV luminosity relationship that we have derived here includes the effects of dust obscuration, and thus it will be different, in general, from the relationship between the mass and the [*intrinsic*]{} UV luminosity (or, equivalently, the star formation rate). The amplitude of the power law (or power laws) used to model the [*observed*]{} average luminosity is smaller, while the variance is larger, because it includes the contribution of the fluctuations of extinction from galaxy to galaxy. The slope of the power law will be also be different if the obscuration properties of the galaxies depend on their luminosity.
DISCUSSION
==========
The primary results of this work are the following: 1) the clustering strength of LBGs scales with their UV luminosity, with brighter galaxies being more strongly clustered in space; 2) the corresponding scaling law of the clustering strength with the volume density (clustering segregation) is, within the observational uncertainties, the same as that predicted by the gravitational instability for CDM halos, regardless of the values of the cosmological parameters that we have considered; 3) the predicted clustering segregation of LBGs depends very weakly on the relationship between the UV luminosity of the galaxies and their mass. Models of the mass–UV luminosity relationship that simultaneously reproduce the luminosity function of the LBGs and the scaling law of the clustering strength with the UV luminosity predict a clustering segregation function that is very similar to that of the halos and to that of the observed LBGs.
In the models we have assumed $N_g(M)\equiv 1$, namely that the average number of LBGs per halo is unitary and independent on the mass of the halo. While this is almost certainly not true in detail, this approximation seems adequate here. We actually only need to discuss the possibility that $N_g(M)>1$ in some region of the mass spectrum, since the models already include the case of “invisible” halos, i.e. halos whose LBGs are too faint to be observed, but they do not include the case of halos with multiple LBGs. Physically, one expects that $N_g(M)$ is a monotonic increasing function of the mass, reflecting the fact that massive halos are more likely than smaller halos to have substructure and be host to more than one galaxy that satisfies the selection criteria of the observations. The question is whether or not the portion of the mass spectrum covered by the observations includes halos for which “multiplicity effects” have observable consequences. It could be, for example, that such halos are relatively rare at $z\sim 3$ and the few (if any) that might be present in the survey do not significantly contribute to the observed correlation function. Empirically, there are no obvious indications of multiplicity in the samples, and the approximation that a suitable halo is populated by at most one bright LBG is consistent with the observed spatial and angular distribution of the galaxies. Among pairs of galaxies in the SPEC sample with the same redshift only two have angular separation smaller than 20 arcsec and only one at 10 arcsec (at $z=3$, 10 arcsec correspond to 224, 216, 144 $h^{-1}$ kpc in comoving coordinates, respectively). In the PHOT sample, the number of close pairs ($\theta\le 20$ arcsec) is consistent with the Poisson statistics (see G98) and, if anything, it actually seems to be smaller than the extrapolation of the observed angular correlation function, which is measured over scales $\theta\simgt 20$ arcsec (G98). A similar situation exists for the HDF sample, as we discussed in Section 6. Of course, since the HDF sample does not cover enough area to include a sufficient number of bright LBGs (only 45 galaxies in the HDF sample have ${\cal R}<25.5$), the present data do not constrain the possibility that galaxies fainter than the PHOT flux limit can reside in close spatial proximity of brighter ones.
The conclusion that seems to emerge is that, within the accuracy of the current observations, the clustering properties and abundance in space of LBGs are consistent with those of halos that have formed by the epoch of the observations. These galaxies apparently flag the sites of the halos with high efficiency, and in a way such that the bias of the UV light relative to the mass is similar to that of the halos relative to the mass. This provides empirical support to the paradigm that galaxies form first in the highly biased virialized peaks of the mass distribution, and that gravitational instability is the primary physical process behind the formation of cosmic structures. It is also interesting that these results are insensitive to the choice of the cosmological parameters (at least given the accuracy allowed by the present data). The clustering and abundance of LBGs seem to be more effective in constraining the mechanisms of galaxy formation than in discriminating among the background cosmologies.
This study has also given us some insight on how the clustering properties and the luminosity function of the LBGs depend on the relationship between the mass and the UV luminosity (assuming the CDM halo mass spectrum). Three interesting points come out of the analysis. First, a mass–UV luminosity relationship similar to a power–law and a constant scatter (i.e. a variance independent on the mass) do not seem consistent with the data. This suggests that the relationship between the star formation activity and the mass is not scale invariant along the mass spectrum of the observed LBGs. Either the overall efficiency of star formation per unit mass, or the scatter between mass and star formation, or both, seem to increase towards the low end of the spectrum. Adelberger (2000) suggested that a steepening of the mass–UV luminosity relationship towards lower masses may result from the increased importance of feedback mechanisms in the star formation activity, such as supernovae and stellar winds. Similarly, a scatter increasing towards lower masses may be the signature of the growing importance of the environment, for example merging and interactions, over the local gravity on the star formation activity of small galaxies. The fits suggest that the value of $M_{co}$ in model C, namely the mass that marks the transition from the high–mass slope to the low–mass one in the mass–UV luminosity relationship, is $\sim
10^{12}$ [M$_{\odot}$]{} in the open and $\Lambda$ cosmology and $\sim 10^{11}$ in the EdS one, while the value of $M_{\sigma}$ in model B is $\sim 10^{11}$ and $10^{10}$ [M$_{\odot}$]{} in the same cases, respectively.
The second interesting suggestion is that the scatter of the mass–UV luminosity relationship, namely the variance of the UV luminosity of galaxies of given mass, is relatively small at the high–mass end of the spectrum. This came out consistently, regardless of the adopted model. For example, the scatter of galaxies with luminosity (expressed in magnitude) ${\cal R}=25.5$ (the flux limit of the PHOT sample) is $\sigma_{UV}\simlt 20$% in model A and $\sim 70$% of the average value in model B and C, respectively. On the contrary, the scatter at the small–mass end depends on the adopted model, and thus it remains unconstrained. For example, galaxies with ${\cal R}\sim 27.2$ have $\sigma_{UV}\sim 250$% in model B, but $\sigma_{UV}\sim 70$% in model B. To this purpose it is interesting to observe that the “Global Schmidt Law” for local starburst galaxies discussed by Kennicutt (1998b) shows that the scatter between the integrated star formation surface density and the integrated gas surface density remains relatively constant in absolute value over a range that spans $\sim 6$ orders of magnitudes in gas and star–formation densities (hence, the relative scatter increases for lower gas density). If the gas mass fraction traces the total mass, the Schmidt Law would favor a situation similar to model B. As we noted earlier and as Figure 8 (top panel) shows, more accurate measures of the correlation function at bright luminosity will be very valuable to discriminate among the models.
The possibility that the scatter for bright LBGs is relatively small is an intriguing one, since it implies that these galaxies flag the sites of massive halos with good efficiency, with the UV light being a tracer of the total mass and star formation a process regulated more by local gravity than by external factors such as merging and interactions. This also argues against a significant population of massive and “dark” halos being under–represented in UV–selected samples, either because without appreciable star formation or because heavily obscured by dust. Samples of bright UV–selected galaxies at high redshifts seem to trace the sites of massive halos with high efficiency, and there is no evidence that the most massive and actively star forming of these systems might be systematically missing. Note that Adelberger & Steidel (2000) reach similar conclusions from independent arguments based on the sub–millimetric background and faint counts.
Another interesting consequence is that one can constrain the duty cycle of the UV–bright phase. If most halos are being simultaneously observed, the duty cycle must be, on average, comparable or larger than the cosmic time covered by the observations, namely $T_{UV}\simgt\Delta T_{LBG}\sim 0.5$, 0.6, 0.3 $\hh$ Gyr. A significantly shorter duty cycle, for example $\sim 10^7$ yr, would imply an intrinsic volume density of halos $\sim 50$ times higher than that of LBGs. In turn, the duty cycle gives information on the stellar mass associated with each halo. For example, the star formation rate of LBGs at $z=3$ with ${\cal R}=23$ is $\sim 50$, 30, 20 $h^{-2}$[M$_{\odot}$ yr$^{-1}$]{}, uncorrected for dust obscuration. Thus, the average stellar mass associated to such halos is $M_*\simgt 2.5$, 2.0, 0.6 $10^{10}$ $h^{-3}$ [M$_{\odot}$]{}. With dust corrections likely to be a factor of several (Calzetti 1997a; Dickinson 1998; Pettini et al. 1998; S98; Steidel & Adelberger 2000), these lower limits become close to $10^{11}$ $h^{-3}$[M$_{\odot}$]{}, comparable to the stellar mass of present–day galaxies of $\sim L^*$ luminosity.
This does not mean that the average duty cycle of the individual LBGs is $\simgt 0.5$ Gyr, but only that the hosting halos are made “visible” for such a period of time. For example, semi–analytic models (Kolatt et al. 1999; Somerville et al. 2000) predict a series of LBGs with much shorter duty cycles associated to the same halo which become UV-luminous in sequence, on average one at a time, as a result of interactions. It would be interesting to know what these models predict for the clustering segregation and scatter of the mass–UV luminosity relationship. Direct constraints on the duty cycle of the individual galaxies based on their spectroscopic and photometric properties are still very uncertain (e.g. Lowenthal et al. 1997; Sawicki & Yee 1997; Pettini et al. 1999; Dickinson 2000). Regardless of the duration of the star formation of the individual galaxies, however, the clustering properties and the abundances of LBGs seem to imply that a substantial amount of stellar mass has been produced at $z\sim 3$ in close spatial proximity to the brightest of them. Note that both the estimate of the stellar mass and the mass of the halos (as we shall see later) change by an order of magnitude with the cosmologies adopted here. Thus, the implied mass–to–light ratios (where the light is now the one at visible wavelengths emitted by the formed stellar population) predicted for these systems at the present time remain approximately constant to $\sim 10$ times solar, consistent to what observed in present–day bright galaxies (see Fukugita, Hogan & Peebles 1998).
Finally, the third interesting point is that having fitted the mass–UV luminosity relationship to simultaneously reproduce the dependence of the clustering strength with the luminosity and the luminosity function, one can assign a scale of mass to the LBGs for a given choice of the cosmological parameters. It is useful (from the “observational” point of view) to express this in terms of the average mass and the variance for galaxies with assigned luminosity $L$ (Eqn.6.12). Interestingly, while the variance depends on the adopted mass–UV luminosity model, and thus it is not constrained, the average varies weakly with the model. For example, the average mass and the standard deviation of galaxies with luminosity ${\cal R}=23$, 25.5 and 27.0 according to model C in the $\Lambda$ cosmology are $M=2.5\pm 2.7$, $0.9\pm 0.7$ and $0.4\pm 0.12$ $\times 10^{12}$ [M$_{\odot}$]{}, respectively. As a comparison, the average from model A and B are 3.7, 0.8, 0.3 and 2.9, 0.8, 0.4 $\times
10^{12}$ [M$_{\odot}$]{}, respectively, but the standard deviations are $\sim 100$ times smaller and $\sim 50$% larger, respectively. In the open and Einstein–de Sitter cosmologies the values of the average from model C are 6.2, 1.5 and 0.5 $\times 10^{12}$ [M$_{\odot}$]{} and 1, 0.5 and 0.3 $\times 10^{11}$ [M$_{\odot}$]{}, respectively.
Irrespective of the exact value, note that the variance of the mass (for given luminosity) decreases with decreasing luminosity, while we discussed earlier that the variance of the luminosity (for given mass) seems to decrease with increasing mass. The two effects are not in contradiction, and are due to the shape of the halo mass spectrum, which becomes increasingly flatter towards the low–mass end. The variance of the luminosity for a fixed value of the mass includes the contribution of only one type of halos with given volume density. The variance of the mass for a fixed luminosity, on the contrary, includes the contribution of halos from the whole spectrum. Since the average luminosity decreases with decreasing mass, the mass variance at fainter luminosity becomes increasingly more dominated by low–mass halos, which are much more abundant in space than the high–mass ones and for which the mass spectrum is flatter, and hence increasingly smaller.
We conclude by pointing out that the size of the present samples is clearly the limiting factor in this study. We are currently conducting a wide–area survey for LBGs at $z\sim 3$, and we plan to return soon on the issues that we have discussed here with new measures of the correlation length obtained from much larger samples extracted from contiguous fields (0.3 square degree each).
In summary:
[**1)**]{} We have studied the dependence of the spatial clustering strength of Lyman–break galaxies at $z=3$ with their UV luminosity at $\lambda=1700$ Å. We used three samples of LBGs with progressively fainter limiting magnitude, two from our $U_nG{\cal R}$ ground–based surveys and another one from the Hubble Deep Field (both North and South).
[**2)**]{} We have found evidence that the correlation length decreases by a factor $\approx 3$ over the range of magnitudes that we have probed, namely $25\simlt {\cal R}\simlt 27$. Lyman–break galaxies with higher $1700$ Åluminosity seem to be more strongly clustered in space, suggesting that fainter samples include galaxies with smaller mass.
[**3)**]{} The observed scaling law of the correlation length is consistent with the predictions of the theory of gravitational instability and implies that fainter samples include less massive LBGs. The scaling law of the clustering strength with the volume density is, within the errors, the same as the one predicted for mass–limited samples of halos. This is interesting, because this function is essentially independent from the relationship between the mass of the galaxies and their UV luminosity. The LBGs seem to flag the sites of halos with high efficiency and in a way such that the bias of the light respect to the mass is similar to that of the halos respect to the mass. We interpret this result as strong support to the theory of biased galaxy formation.
[**4)**]{} Fitting models of the mass–UV luminosity relationship to simultaneously reproduce the observed clustering segregation and the luminosity function, suggests that mass and UV luminosity are relatively tightly correlated in bright LBGs. Galaxies with average luminosity ${\cal
R}=25.5$ have standard deviation $\sigma_{UV}\simlt 70$% of the average (the scatter is not constrained at fainter luminosity). This suggests that i) the mass is an important parameter in regulating star formation in massive galaxies; ii) the duty–cycle of the UV–bright phase is similar to the cosmic time spanned by the observations ($\sim 0.5$ Gyr), which in turn would set limits to the stellar mass assembled in the galaxies at the epoch of the observations; iii) the fraction of massive halos at $z\sim 3$ that have not started substantial star formation or whose UV light is not observed because heavily obscured must be relatively rare.
[**5)**]{} The fits also show that a scale invariant relationship (e.g. a power law) between mass and UV luminosity is not consistent with the observations, suggesting that the properties of star formation of galaxies vary along the mass spectrum. From the fits one can also assign a scale of mass to the galaxies. For example, in the $\Lambda$ cosmology the average mass of galaxies with luminosity ${\cal R}=23$, 25.5 and 27.0 $\Lambda$–cosmology $\langle
M\rangle=2.5$, 0.9, and $0.4\times 10^{12}$ [M$_{\odot}$]{}, respectively. These numbers would be $\approx 2$ times larger in the open universe and $\approx 10$ times smaller in the EdS one.
We would like to thank the staff at the Palomar, Kitt Peak and Keck observatories for their invaluable help in obtaining the data that made this work possible. We also would like to thank all the people who have worked on the HDF survey. We have benefited from stimulating conversations with Gus Oemler, Ray Carlberg, Simon White, Cristiano Porciani and Stefano Casertano, who gave us very useful comments on the paper. We also thank Kurt Adelberger, Chuck Steidel and Max Pettini, our collaborators in the Lyman–break galaxy survey, for their help with an early version of the paper. Cristiano Porciani has also kindly made available to us his code to generate mock samples of galaxies with assigned clustering properties. Finally, we are grateful to an anonymous referee for his/her very constructive criticism on the manuscript. Throughout most of the work that led to this paper, MG has been supported by the Hubble Fellowship program through grant HF-01071.01-94A awarded by the Space Telescope Science Institution, which is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555.
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[llccccc]{} 1 & SPEC & 725.9 & ${\cal R}<25.0$ & 446 & $0.7\pm 0.1$ & 3.04 2 & PHOT & 718.9 & ${\cal R}<25.5$ & 876 & $1.22\pm 0.18$ & 3.04 3 & HDF & 9.2 & $V_{606}<27.0$ & 271 & $29.4\pm 3.5$ & 2.60
[llccc]{} SPEC & Observed & $1.9\pm 0.3$, & $1.8\pm 0.3$, & $7.2\pm 1.1$ PHOT & Observed & $3.5\pm 0.5$, & $3.3\pm 0.5$, & $13\pm 2$ HDF & Simulated & $46\pm 5$, & $43\pm 5$, & $169\pm 20$ HDF & Top–Hat & $27\pm 3$, & $24\pm 3$, & $ 94\pm 11$ HDF & Synthetic & $36\pm 4$, & $32\pm 4$, & $122\pm 15$
[lcccc]{} & & $\Omega=0.2$, $\Omega_{\Lambda}=0$ & $\Omega=0.3$, $\Omega_{\Lambda}=0.7$ & $\Omega=1$, $\Omega_{\Lambda}=0$ & & & &SPEC & 2.0[$_{-0.2}^{+0.2}$]{} & $4.2\pm 0.6$ & $5.0\pm 0.7$ & $3.1\pm0.4$ & & & &PHOT & 2.0[$_{-0.2}^{+0.2}$]{} & 3.1[$_{-0.6}^{+0.7}$]{} & 3.2[$_{-0.7}^{+0.7}$]{} & 1.9[$_{-0.5}^{+0.4}$]{} & & & &HDF & 2.2[$_{-0.3}^{+0.6}$]{} & 1.1[$_{-0.8}^{+0.9}$]{} & 1.2[$_{-0.8}^{+0.9}$]{} & 0.8[$_{-0.5}^{+0.6}$]{} & & & &HDF (TH) & 2.2[$_{-0.2}^{+0.3}$]{} & 1.0[$_{-0.7}^{+0.8}$]{} & 1.0[$_{-0.7}^{+0.8}$]{} & 0.7[$_{-0.4}^{+0.5}$]{} & & & &HDF (SYN) & 2.2[$_{-0.2}^{+0.3}$]{} & 0.9[$_{-0.7}^{+0.8}$]{} & 1.0[$_{-0.7}^{+0.8}$]{} & 0.6[$_{-0.4}^{+0.5}$]{}
[^1]: A98 used two estimators of $\sigma^2_g$, a non–parametric one defined as ${\cal S}=[(N-\mu)^2-\mu]/\mu^2$, where $\mu$ is the mean number of galaxies in the cell and $N$ is the number actually found, and a maximum–likelihood estimator where the distribution function of the galaxy fluctuation field $\delta_g=(\rho_g-\bar\rho_g)/\bar\rho_g$ is assumed to be lognormal (cf. their Eqn.7 and Figure 3).
[^2]: For the simulations, we have used a code written by Cristiano Porciani, who generously made it available to us, that we have adapted to our specific problem.
[^3]: $M_*$ is the mass scale of non–linearity, defined as $\sigma^2(M_*)=1$. At $z=3$, $M_*=4.4\times 10^{12}$, $2.8\times
10^{11}$, $6.5\times 10^8$, respectively, in the three cosmologies adopted here.
[^4]: The variance $\sigma^2_{UV}$ of $A$ is related to the variance $\sigma^2$ of $\log(A)$ by the relation $\sigma^2_{UV}=
e^{\sigma^2}-1$.
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abstract: 'Thermal fluctuations strongly modify the large length-scale elastic behavior of crosslinked membranes, giving rise to scale-dependent elastic moduli. While thermal effects in flat membranes are well understood, many natural and artificial microstructures are modeled as thin elastic [*shells*]{}. Shells are distinguished from flat membranes by their nonzero curvature, which provides a size-dependent coupling between the in-plane stretching modes and the out-of-plane undulations. In addition, a shell can support a pressure difference between its interior and exterior. Little is known about the effect of thermal fluctuations on the elastic properties of shells. Here, we study the statistical mechanics of shape fluctuations in a pressurized spherical shell using perturbation theory and Monte Carlo computer simulations, explicitly including the effects of curvature and an inward pressure. We predict novel properties of fluctuating thin shells under point indentations and pressure-induced deformations. The contribution due to thermal fluctuations increases with increasing ratio of shell radius to thickness, and dominates the response when the product of this ratio and the thermal energy becomes large compared to the bending rigidity of the shell. Thermal effects are enhanced when a large uniform inward pressure acts on the shell, and diverge as this pressure approaches the classical buckling transition of the shell. Our results are relevant for the elasticity and osmotic collapse of microcapsules.'
author:
- Jayson Paulose
- 'Gerard A. Vliegenthart'
- Gerhard Gompper
- 'David R. Nelson'
title: Fluctuating shells under pressure
---
The elastic theory of thin plates and shells [@landau_elasticity_1986], a subject over a century old, has recently found new applications in understanding the mechanical properties of a wide range of natural and artificial structures at microscopic length scales. The mechanical properties of viral capsids [@ivanovska_bacteriophage_2004; @michel_nanoindentation_2006; @klug_failure_2006], red blood cells [@park_measurement_2010], and hollow polymer and polyelectrolyte capsules [@gao_elasticity_2001; @gordon_self-assembled_2004; @lulevich_elasticity_2004; @elsner_mechanical_2006; @zoldesi_elastic_2008] have been measured and interpreted in terms of elastic constants of the materials making up these thin-walled structures. Theoretically, models that quantify the deformation energy of a two-dimensional membrane have been used to investigate the shapes of viral capsids [@lidmar_virus_2003; @nguyen_elasticity_2005; @nguyen_continuum_2006] and their expected response to point forces and pressures [@vliegenthart_mechanical_2006; @buenemann_mechanical_2007; @buenemann_elastic_2008; @siber_stability_2009], as well as shape transitions of pollen grains [@katifori_foldable_2010].
Like its counterparts in other areas of science, such as fluid dynamics and the theory of electrical conduction in metals, thin shell theory aims to describe the physics of slowly varying disturbances in terms of a few macroscopic parameters, such as the shear viscosity of incompressible fluids and the electrical conductivity of metals. Despite such venerable underpinnings as the Navier-Stokes equations and Ohm’s law, these hydrodynamic theories can break down, sometimes in spectacular ways. For example, it is know from mode coupling theory [@pomeau_time_1975] and from renormalization group calculations [@forster_large-distance_1977] that thermal fluctuations cause the shear viscosity of incompressible fluids to diverge logarithmically with system size in a two-dimensional incompressible fluid. In the theory of electrical conduction, quenched disorder due to impurities coupled with interactions between electrons lead to a dramatic breakdown of Ohm’s law in thin films and one-dimensional wires at low temperatures, with a conductance that depends on the sample dimensions [@lee_disordered_1985].
![**Simulated thermally fluctuating shells.** **(a)** Triangulated shell with 5530 points separated by average nearest-neighbor distance $r_{0}$ with Young’s modulus $Y=577\epsilon/r_{0}^{2}$ and bending rigidity $\kappa=50\epsilon$ at temperature $k_\text{B}T=20\epsilon$, where $\epsilon$ is the energy scale of the Lennard-Jones potential used to generate the disordered mesh. **(b)** Same as in (a) with external pressure $p=0.5p_{\mathrm{c}}$, where $p_{\mathrm{c}}$ is the classical buckling pressure. The thermally excited shell has already buckled under pressure to a shape with a much smaller enclosed volume than in (a).[]{data-label="fig_schematic"}](fig1.pdf){width="87mm"}
Even more dramatic breakdowns of linear response theory can arise in thin plates and shells. Unlike the macroscopic shell structures of interest to civil engineers, thermal fluctuations can strongly influence structures with size of order microns, since the elastic deformation energies of extremely thin membranes (with nanoscale thicknesses) can be of the order of the thermal energy $k_\text{B}T$ (where $k_\text{B}$ is the Boltzmann constant and $T$ the temperature) for typical deformations. The statistical mechanics of *flat* solid plates and membranes (*i.e.* membranes with no curvature in the unstrained state) has been studied previously (see [@nelson_statistical_1988; @bowick_statistical_2001] and references therein). Thermal fluctuations lead to *scale-dependent* elastic moduli for flat membranes, causing the in-plane elastic moduli to vanish at large length scales while the bending rigidity diverges [@nelson_fluctuations_1987; @aronovitz_fluctuations_1989]. These anomalies arise from the the nonlinear couplings between out-of-plane deformations (transverse to the plane of the undeformed membrane) and the resultant in-plane strains, which are second order in the out-of-plane displacements.
Much less is known about spherical shells subject to thermal fluctuations (Fig. 1a). In fact, the coupling between in-plane and out-of-plane modes is significantly different. Geometry dictates that a closed spherical shell cannot be deformed without stretching; as a result, out-of-plane deformations provide a *first* order contribution to the in-plane strain tensor [@landau_elasticity_1986]. This introduces new nonlinear couplings between in-plane and out-of-plane deformations, which are forbidden by symmetry in flat membranes. We can also consider the buckling of spherical shells under uniform external pressure, which has no simple analogue for plates (Fig. 1b). An early exploration with computer simulations combined an analysis of the elastic energy due to the linear strain contributions of a spherical membrane with the nonlinear corrections from flat membranes to suggest new scaling behavior for thermally fluctuating spherical membranes [@zhang_scaling_1993]. However, an important nonlinear coupling triggered by the curved background metric was not considered, nor was the effect of an external pressure investigated. Here, we study the mechanics of fluctuating spherical shells using perturbation theory and numerical simulations, taking into account the nonlinear couplings introduced by curvature as well as the effects of a uniform external pressure.
Results and discussion {#results-and-discussion .unnumbered}
======================
Elastic energy of a thin shell {#elastic-energy-of-a-thin-shell .unnumbered}
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The elastic energy of a deformed spherical shell of radius $R$ is calculated using *shallow-shell theory* [@koiter_stability]. This approach considers a shallow section of the shell, small enough so that slopes measured relative to the section base are small. The in-plane displacements of the shallow section are parametrized by a two-component phonon field $u_i(\mathbf{x})$, $ i={1,2}$; the out-of-plane displacements are described by a field $f(\mathbf{x})$ in a coordinate system $\mathbf{x}=(x_1,x_2)$ tangent to the shell at the origin. We focus on *amorphous* shells, with uniform elastic properties, and can thus neglect the effect of the 12 inevitable disclinations associated with crystalline order on the surface of a sphere [@lidmar_virus_2003]. In the presence of an external pressure $p$ acting inward, the elastic energy for small displacements in terms of the bending rigidity $\kappa$ and Lamé coefficients $\mu$ and $\lambda$ reads (see [*Supplementary Information*]{} for details): $$G=\int d^2x\,\left[\frac{\kappa}{2} (\nabla^2 f)^2 +\mu u_{ij}^2 + \frac{\lambda}{2} u_{kk}^2-pf\right],$$ where the nonlinear strain tensor is $$u_{ij}(\mathbf{x})=\frac{1}{2}\left(\partial_i u_j+\partial_j u_i +\partial_i f \partial_j f\right)-\delta_{ij}\frac{f}{R}.$$ Here, $d^{2}x \equiv \sqrt{g} dx_{1}dx_{2}$, where $g$ is the determinant of the metric tensor associated with the spherical background metric. Within shallow shell theory, $g \approx 1$ (see [*Supplementary Information*]{}).
If we represent the normal displacements in the form $f(\mathbf{x}) = f_0 + f^\prime(\mathbf{x})$, where $f_0$ represents the uniform contraction of the sphere in response to the external pressure, and $f^\prime$ is the deformation with reference to this contracted state so that $\int d^2x f^\prime =0$, then the energy is quadratic in fields $u_1$, $u_2$ and $f_0$. These variables can be eliminated in a functional integral of $\exp(-G[f^\prime, f_0,u_1,u_2]/k_\mathrm{B}T)$ by Gaussian integration (see [*Supplementary Information*]{} for details). The effective free energy $G_{\mathrm{eff}}$ which results is the sum of a harmonic part $G_0$ and an anharmonic part $G_1$ in the remaining variable $f^\prime(\mathbf{x})$: $$\begin{aligned}
\label{eqn_effectivef_pressure}
G_0 & = & \frac{1}{2}\int d^2 x\left[\kappa(\nabla^2 f^\prime)^2-\,\frac{pR}{2} |\nabla f^\prime|^2+\frac{Y}{R^2} {f^\prime}^2\right],\\
G_1& = &\frac{Y}{2}\int d^2 x\left[\left(\frac{1}{2}P^\mathrm{T}_{ij}\partial_i f^\prime \partial_j f^\prime\right)^2-\frac{f^\prime}{R}P^\mathrm{T}_{ij}\partial_i f^\prime \partial_j f^\prime \right]. \nonumber\end{aligned}$$ where $Y = 4\mu(\mu+\lambda)/(2\mu+\lambda)$ is the two-dimensional Young modulus and $P^\mathrm{T}_{ij}=\delta_{ij}-\partial_i \partial_j/\nabla^2$ is the transverse projection operator. The “mass” term $Y({f^\prime}/R)^2$ in the harmonic energy functional reflects the coupling between out-of-plane deformation and in-plane stretching due to curvature, absent in the harmonic theory of flat membranes (plates). The cubic interaction term with a coupling constant $-Y/2R$ is also unique to curved membranes and is prohibited by symmetry for flat membranes. These terms are unusual because they have system-size-dependent coupling constants. Note that an inward pressure ($p >0$) acts like a negative $R$-dependent surface tension in the harmonic term. As required, the effective elastic energy of fluctuating flat membranes is retrieved for $R\to \infty$ and $p = 0$. In the following, we exclusively use the field $f^\prime(\mathbf{x})$ and thus drop the prime without ambiguity.
When only the harmonic contributions are considered, the equipartition result for the thermally generated Fourier components $f_\mathbf{q} = \int d^2x \,f(\mathbf{x})\exp(i\mathbf{q}\cdot\mathbf{x})$ with two-dimensional wavevector $\mathbf{q}$ are $$\label{eqn_corrfn_gaussian}
\langle f_\mathbf{q}f_\mathbf{q^\prime} \rangle_0 = \frac{Ak_\mathrm{B}T \delta_{\mathbf{q},\mathbf{-q^\prime}}}{\kappa q^4 -\frac{pR}{2}q^2+ \frac{Y}{R^2}}.$$ where $A$ is the area of integration in the $(x_1,x_2)$ plane. Long-wavelength modes are restricted by the finite size of the sphere, *i.e.* $q \gtrsim 1/R$. In contrast to flat membranes for which the amplitude of long-wavelength ($q \to 0$) modes diverges as $k_\text{B}T/(\kappa q^4)$, the coupling between in-plane and out-of-plane deformations of curved membranes cuts off fluctuations with wavevectors smaller than a characteristic inverse length scale [@zhang_scaling_1993]: $$q^* = (\ell^*)^{-1} = \left(\frac{Y}{\kappa R^2}\right)^{1/4} \equiv \frac{\gamma^{1/4}}{R},$$ where we have introduced the dimensionless [*Föppl-von Kármán number*]{} $\gamma = YR^2/\kappa$ [@lidmar_virus_2003]. We focus here on the case $\gamma \gg 1$, so $\ell^{*} \ll R$. As $p$ approaches $p_\mathrm{c} \equiv 4\sqrt{\kappa Y}/R^2$, the modes with $q = q^*$ become unstable and their amplitude diverges. This corresponds to the well-known buckling transition of spherical shells under external pressure [@koiter_stability]. When $p > p_\mathrm{c}$, the shape of the deformed shell is no longer described by small deformations from a sphere, and the shallow shell approximation breaks down.
{width="170mm"}
Anharmonic corrections to elastic moduli {#anharmonic-corrections-to-elastic-moduli .unnumbered}
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The anharmonic part of the elastic energy, neglected in the analysis described above, modifies the fluctuation spectrum by coupling Fourier modes at different wavevectors. Upon rescaling all lengths by $\ell^*$, it can be shown that the size of anharmonic contributions to $\langle |f_{\mathbf{q}}|^2 \rangle$ is set by the dimensionless quantities $k_\text{B}T \sqrt{\gamma}/\kappa$ and $p/p_\text{c}$. The correlation function including the anharmonic terms in Eq. \[eqn\_effectivef\_pressure\] is given by the Dyson equation, $$\label{eqn_dyson}
\langle |f_\mathbf{q}|^2 \rangle = \frac{1}{\langle |f_\mathbf{q}|^2 \rangle_0^{-1}-\Sigma(\mathbf{q})}$$ where $\Sigma(\mathbf{q})$ is the self-energy, which we evaluate to one-loop order using perturbation theory. While $\langle |f_\mathbf{q}|^2 \rangle$ can be numerically evaluated at any $\mathbf{q}$, an approximate but concise description of the fluctuation spectrum is obtained by expanding the self-energy up to order $q^4$ and defining renormalized values $Y_{\scriptscriptstyle\mathrm{R}}$, $\kappa_{\scriptscriptstyle\mathrm{R}}$ and $p_{\scriptscriptstyle\mathrm{R}}$ of the Young’s modulus, bending rigidity and pressure, from the coefficients of the expansion: $$Ak_\mathrm{B}T\langle|f_\mathbf{q\rightarrow 0}|^2\rangle^{-1} \equiv \kappa_{\scriptscriptstyle\mathrm{R}} q^4 - \frac{p_{\scriptscriptstyle\mathrm{R}}R}{2} q^2+\frac{Y_{\scriptscriptstyle\mathrm{R}}}{R^2} + O(q^6).$$ To lowest order in $k_\mathrm{B}T/\kappa$ and $p/p_\text{c}$ we obtain the approximate expressions (see [*Supplementary Information*]{} for details) $$\label{eqn_smallpyr}
Y_{\scriptscriptstyle\mathrm{R}} \approx Y \left[1 -\frac{3}{256}\frac{k_\mathrm{B}T}{\kappa}\sqrt{\gamma}\left(1+\frac{4}{\pi}\frac{p}{p_{\text{c}}}\right)\right],$$ $$\label{eqn_genpressure}
p_{\scriptscriptstyle\mathrm{R}} \approx p +\frac{1}{24\pi}\frac{k_\mathrm{B}T}{\kappa}p_\text{c}\sqrt{\gamma}\left(1+\frac{63\pi}{128}\frac{p}{p_{\text{c}}}\right) ,$$ and $$\label{eqn_smallpkr}
\kappa_{\scriptscriptstyle\mathrm{R}} \approx \kappa\left[1 +\frac{61}{4096}\frac{k_\mathrm{B}T}{\kappa}\sqrt{\gamma}\left(1-\frac{1568}{915\pi}\frac{p}{p_{\text{c}}}\right)\right].$$ (See [*Supplementary Information*]{} for details of the calculation and the complete dependence on $p/p_\text{c}$.) Thus the long-wavelength deformations of a thermally fluctuating shell are governed by a smaller effective Young’s modulus, a larger effective bending rigidity, and a nonzero negative surface tension even when the external pressure is zero. At larger $p/p_\text{c}$, however, both the Young’s modulus and the bending modulus fall compared to their zero temperature values, and the negative effective surface tension determined by $p_{\scriptscriptstyle\mathrm{R}}$ gets very large. The complete expressions for the effective elastic parameters, including the full $p/p_\text{c}$-dependence, show that all corrections diverge as $p/p_\text{c} \to 1$. Furthermore, the effective elastic constants are not only temperature-dependent, but also system size-dependent, since $\sqrt\gamma\propto R$. Although the corrections are formally small for $k_{\text{B}}T \ll \kappa$, they nevertheless diverge as $R \to \infty$! The thermally generated surface tension, strong dependence on external pressure, and size dependence of elastic constants are unique to spherical membranes, with no analogue in planar membranes.
Simulations of thermally fluctuating shells {#simulations-of-thermally-fluctuating-shells .unnumbered}
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{width="170mm"}
We complement our theoretical calculations with Monte Carlo simulations of randomly triangulated spherical shells with discretized bending and stretching energies that translate directly into a macroscopic 2D shear modulus $Y$ and a bending ridigity $\kappa$ [@vliegenthart_forced_2006; @vliegenthart_compression_2011]. (Details are provided in [*Materials and Methods*]{}.) Here we study shells with $600 < \gamma < 35000$ and $2\times 10^{-6} <k_\text{B}T/\kappa < 0.5$. The anharmonic effects are negligible at the low end of this temperature range.
The fluctuation spectra of the simulated spherical shells are evaluated using an expansion of the radial displacement field in spherical harmonics [@gompper_random_1996]. The radial position of a node $i$ at angles ($\phi,\theta$) can be written as $r_i(\phi,\theta)=\widetilde{R_0}+f(\phi,\theta)$ with $\widetilde{R_0}$ the average radius of the fluctuating vesicle. The function $f(\phi,\theta)$ can be expanded in (real) spherical harmonics $$\label{eqn0sphharm}
f(\phi,\theta)=R\sum_{l=0}^{l_M}\sum_{m=-l}^{m=l} A_{lm}Y_{lm}(\phi,\theta)$$ where $l_M$ is the large wavenumber cutoff determined by the number of nodes in the lattice $(l_M+1)^2=N$ [@gompper_random_1996]. The theoretical prediction for the fluctuation spectrum including anharmonic effects is ([*Supplementary Information*]{}) $$\begin{split}
k_\text{B}T\langle |A_{lm}|^{2}\rangle^{-1} \approx &\kappa_{\scriptscriptstyle\mathrm{R}}(l+2)^{2}(l-1)^{2} -p_{\scriptscriptstyle\mathrm{R}}R^{3}\left[1+\frac{l(l+1)}{2}\right] \\
&+Y_{\scriptscriptstyle\mathrm{R}}R^{2}\left[\frac{3(l^{2}+l-2)}{3(l^{2}+l)-2}\right]. \label{eqn-almsurf}
\end{split}$$ Fig. \[fig\_sphfl\] displays our theoretical and simulation results for the fluctuation spectrum. At the lowest temperature (corresponding to $k_\text{B}T\sqrt\gamma/\kappa \approx 0.1 \ll 1$), the spectrum is well-described by the bare elastic parameters $Y$, $\kappa$ and $p$. At the intermediate temperature ($k_\text{B}T\sqrt\gamma/\kappa \approx 10$) anharmonic corrections become significant, enhancing the fluctuation amplitude for some values of $l$ by about 20%–40% compared to the purely harmonic contribution. At this temperature, one-loop perturbation theory successfully describes the fluctuation spectrum. However, at the highest temperature simulated ($k_\text{B}T\sqrt\gamma/\kappa \approx 24$), the anharmonic corrections observed in simulations approach 50% of the harmonic contribution at zero pressure and over 100% for the pressurized shell. With such large corrections, we expect that higher-order terms in the perturbation expansion contribute significantly to the fluctuation spectrum and the one-loop result overestimates the fluctuation amplitudes.
Similarly, thermal fluctuations modify the mechanical response when a shell is deformed by a deliberate point-like indentation. In experiments, such a deformation is accomplished using an atomic force microscope [@ivanovska_bacteriophage_2004; @elsner_mechanical_2006]. In our simulations, two harmonic springs are attached to the north and south pole of the shell. By changing the position of the springs the depth of the indentation can be varied (Fig. \[fig\_indent\]a, inset). The thermally averaged pole-to-pole distance $\langle z \rangle$ is measured and compared to its average value in the absence of a force, $\langle z_{0} \rangle$. For small deformations, the relationship between the force applied at each pole and the corresponding change in pole–pole distance is spring-like with a spring constant $k_{\text{s}}$: $\langle F \rangle \equiv k_{\text{s}} (\langle z_{0} \rangle - \langle z \rangle)$. The spring constant is related to the amplitude of thermal fluctuations in the normal displacement field in the *absence* of forces by (see [*Supplementary Information*]{} for the detailed derivation) $$\label{eqn0ksfluct}
k_{\text{s}} = \frac{k_\text{B}T}{2\langle [f(\mathbf{x})]^{2} \rangle} \approx \frac{k_\text{B}T}{\langle z_{0}^{2}\rangle - \langle z_{0}\rangle^{2}}.$$ This fluctuation-response relation is used to measure the temperature dependence of $k_{\text{s}}$ from simulations on fluctuating shells with no indenters. At finite temperature, anharmonic effects computed above make this spring constant both size- and temperature-dependent: $$\label{eqn0sprconst}
k_{\text{s}} \approx \frac{4\sqrt{\kappa Y}}{R}\left[1-0.0069\frac{k_\text{B}T}{\kappa} \sqrt\gamma\right].$$
Fig. \[fig\_indent\]a shows the force-compression relation for a shell with $R = 20 r_0$ and dimensionless temperatures $k_\text{B}T\sqrt\gamma/\kappa = 1.36 \times 10^{-4}$ and $k_\text{B}T\sqrt\gamma/\kappa = 34$. The linear response near the origin (Fig. \[fig\_indent\]b) is very well described by $k_{\text{s}}$ measured indirectly from the fluctuations in $z_{0}$ at each temperature, Eq. \[eqn0ksfluct\]. The thermal fluctuations lead to an appreciable 20% reduction of the spring constant for this case. Measuring spring constants over a range of temperatures (Fig. \[fig\_indent\]c) confirms that the shell response softens as the temperature is increased, in agreement with the perturbation theory prediction. We note, however, a small but systematic shift due to the finite mesh size of the shells, an approximately 5% effect for the largest systems simulated here. At the higher temperatures ($k_\text{B}T\sqrt\gamma/\kappa >20$), the measured spring constants deviate from the perturbation theory prediction, once again we believe due to the effect of higher-order terms.
![**Temperature dependence of the buckling pressure.** Buckling pressure for simulated shells at various radii and temperatures, normalized by the *classical* (*i.e.* zero temperature) critical buckling pressure $p_\text{c}$ for perfectly uniform, zero temperature shells with the same parameters. For all shells, $Yr_{0}^{2}/\kappa = 11.54$. In separate sets of symbols, we either vary the shell radius over the range $7.5 \leq R/r_{0} \leq 55$ while keeping the temperature constant ($k_{\text{B}}T = 2\times 10^{-6}\kappa$, blue circles; $k_{\text{B}}T = 0.4\kappa$, yellow squares) or vary the temperature over the range $2\times 10^{-8} \leq k_{\text{B}}T/\kappa \leq 0.4$ while keeping the radius constant at $R=20 r_{0}$ (red triangles). The parameter $k_{\text{B}}T\sqrt\gamma/\kappa$ sets the strength of anharmonic corrections for thermally fluctuating shells. The inset shows the $1/R^{2}$ dependence of the buckling pressure as the radius is varied, for shells at low and high temperature.[]{data-label="fig_critpres"}](fig4.pdf){width="87mm"}
We also simulate the buckling of thermally excited shells under external pressure. When the external pressure increases beyond a certain value (which we identify as the renormalized buckling pressure), the shell collapses from a primarily spherical shape (Fig. 1a) to a shape with one or more large volume-reducing inversions (Fig. 1b). For zero temperature shells, this buckling is associated with the appearance of an unstable deformation mode in the fluctuation spectrum. At finite temperature, the appearance of a mode with energy of order $k_\text{B}T$ is sufficient to drive buckling. Anharmonic contributions, strongly enhanced by an external pressure, also reduce the effective energy associated with modes in the vicinity of $q^{*}$ primarily due to the enhanced negative effective surface tension $p_{\scriptscriptstyle\mathrm{R}}R/2$ (see Eq. \[eqn\_genpressure\]). As a result, unstable modes arise at lower pressures and we expect thermally fluctuating shells to collapse at pressures below the classical buckling pressure $p_\text{c}$. This is confirmed by simulations of pressurized shells (Fig. \[fig\_critpres\]). When anharmonic contributions are negligible ($k_\text{B}T\sqrt\gamma/\kappa \ll 1$), the buckling pressure observed in simulations is only $\sim 80\%$ of the theoretical value because the buckling transition is highly sensitive to the disorder introduced by the random mesh. Relative to this low temperature value, the buckling pressure is reduced significantly when $k_\text{B}T\sqrt\gamma/\kappa$ becomes large.
Conclusion and outlook {#conclusion-and-outlook .unnumbered}
----------------------
In summary, we have demonstrated that thermal corrections to the elastic response become significant when $k_\text{B}T\sqrt{\gamma}/\kappa \gg 1$ and that first-order corrections in $k_\text{B}T/\kappa$ already become inaccurate when $k_\text{B}T\sqrt{\gamma}/\kappa \gtrsim 20$. Human red blood cell (RBC) membranes are known examples of curved solid structures that are soft enough to exhibit thermal fluctuations. Typical measured values of the shear and bulk moduli of RBC membranes correspond to $Y\approx 25$ $\mu$N/m [@park_measurement_2010; @waugh_thermoelasticity_1979], while reported values of the bending rigidity $\kappa$ vary widely from 6 $k_\text{B}T$ to 40 $k_\text{B}T$ [@park_measurement_2010; @evans_bending_1983]. Using an effective radius of curvature $R \approx 7$ $\mu$m [@park_measurement_2010] gives $k_\text{B}T\sqrt{\gamma}/\kappa$ in the range 2–35. Thus, RBCs could be good candidates to observe our predicted thermal effects, provided their bending rigidity is in the lower range of the reported values.
For continuum shells fabricated from an elastic material with a 3D Young’s modulus $E$, thickness $h$ and typical Poisson ratio $\approx 0.3$, $k_\text{B}T\sqrt\gamma/\kappa \approx 100Rk_\text{B}T/(Eh^4)$. Hence very thin shells with a sufficiently high radius-to-thickness ratio ($R/h$) *must* display significant thermal effects. Polyelectrolyte [@elsner_mechanical_2006] and protein-based [@hermanson_engineered_2007] shells with $R/h \approx 10^3$ have been fabricated, but typical solid shells have a bending rigidity $\kappa$ several orders of magnitude higher than $k_\text{B}T$ unless $h \lesssim 5$ nm. Microcapsules of 6 nm thickness fabricated from reconstituted spider silk [@hermanson_engineered_2007] with $R \approx 30$ $\mu\mathrm{m}$ and $E \approx 1$ GPa have $k_\text{B}T\sqrt{\gamma}/\kappa \approx 3$, and could exhibit measurable anharmonic effects.
Thermal effects are particularly pronounced under finite external pressure—an indentation experiment carried out at $p = p_\mathrm{c}/2$ on the aforementioned spider silk capsules would show corrections of 10% from the classical zero-temperature theory. For similar capsules with half the thickness, perturbative corrections at $p=p_\mathrm{c}/2$ are larger than 100%, reflecting a drastic breakdown of shell theory because of thermal fluctuations. The breakdown of classical shell theory explored here points to the need for a renormalization analysis, similar to that carried out already for flat plates [@statistical_1988].
Materials and Methods {#materials-and-methods .unnumbered}
=====================
Monte Carlo Simulations of randomly triangulated shells {#sec0sisimulation .unnumbered}
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A random triangulation of radius $R_0$ is constructed by distributing $N$ nodes on the surface of a sphere with the required radius. The first two of these nodes are fixed at the north and the south pole of the sphere whereas the positions of the remaining $N-2$ nodes are randomized and equilibrated in a Monte Carlo simulation. During this equilibration process the nodes interact via a steeply repulsive potential (the repulsive part of a Lennard Jones potential). After equilibration, when the energy has reached a constant value on average, the simulation is stopped and the final configuration is ‘frozen’. The neighbours of all nodes are determined using a Delaunay triangulation [@renka_algorithm_1997]. The spherical configurations as well as the connection lists are used in further simulations.
In subsequent simulations nearest neighbours are permanently linked by a harmonic potential giving rise to a total stretching energy [@seung_defects_1988], $$E_\text{s}=\frac{k}{2}\sum_{i,j} (|r_{ij}-r_{ij}^0|^2),$$ where the sum runs over all pairs of nearest neighbours, $r_{ij}$ is the distance between two neighbours and $r_{ij}^0$ the equilibrium length of a spring. The equilibrium length $r_{ij}^0$ is determined at the start of the simulation, when the shell is still perfectly spherical and thus the stretching energy vanishes for the spherical shape. The spring constant $k$ is related to the two-dimensional Lamé coefficients $\lambda=\mu=\sqrt{3}k/4$ and the two-dimensional Young modulus $Y=2 k/\sqrt{3}$ [@seung_defects_1988].
The mean curvature (more precisely, twice the mean curvature) at node $i$ is discretized using [@gompper_random_1996; @itzykson_discrete_1986; @kohyama_budding_2003] $$H_i = \frac{1}{\sigma_i}{\bf n}_i \cdot \sum_{j(i)}\frac{\sigma_{ij}}{l_{ij}}({\bf r}_i-{\bf r}_j)$$ where ${\bf n}_i$ is the surface (unit) normal at node $i$ (the average normal of the faces surrounding node $i$), $\sigma_i=\sum_{j(i)}\sigma_{ij}l_{ij}$ is the area of the dual cell of node $i$, $\sigma_{ij}=l_{ij}[\cot{\theta_1} + \cot{\theta_2}]/2$ is the length of a bond in the dual lattice and $l_{ij}=|{\bf r}_i-{\bf r}_j|$ is the distance between the nodes $i$ and $j$. The total curvature energy is, $$E_\text{b} = \frac{\kappa}{2} \sum_i \sigma_i (H_i-H_0)^2$$ with $\kappa$ the bending rigidity and $H_0$ the spontaneous curvature at node $i$. In all simulations $H_0={2/R_0}$ (since $H_i$ is twice the mean curvature). In the cases of elastic shells under pressure a term $P V$ is added to the Hamiltonian where $P$ is the external pressure and $V$ the volume of the shell.
Similar elastic networks with stretching and bending potentials have been studied in relation to the stability of membranes, icosahedral and spherical shells that contain defects [@lidmar_virus_2003; @siber_stability_2009; @vliegenthart_compression_2011; @seung_defects_1988; @gompper_triangulated-surface_2004; @widom_soft_2007] or defect scars [@kohyama_budding_2003; @bowick_interacting_2000; @bausch_grain_2003; @kohyama_defect_2007] as well as for the deformation of icosahedral viruses [@vliegenthart_mechanical_2006; @buenemann_mechanical_2007; @buenemann_elastic_2008] and the crumpling of elastic sheets [@vliegenthart_forced_2006].
Simulations are performed for shells of 5530 ($R_0=20 \ r_0$), 22117 ($R_0=40 \ r_0$) and 41816 ($R = 55 \ r_{0}$) nodes. The Hookean spring constant and the bending rigidity are taken such that the shells have Föppl-von-Kármán numbers in the range $650<\gamma<35000$ and that the dimensionless temperature is in the range $2 \times 10^{-6}< k_\text{B}T/\kappa<0.5$. Monte Carlo production runs consist typically of $1.25 \times 10^8$ Monte Carlo steps where in a single Monte Carlo step an attempt is made to update the positions of all nodes once on average. Configurations were stored for analysis typically every $N_\text{samp}=2000$ Monte Carlo steps. For the largest system (41816 nodes), such a run took about 700 days of net CPU time spread over several simultaneous runs in a Linux cluster of Intel XEON X5355 CPUs. For the smaller shells, the computational time scaled down roughly linearly with system size.
The fluctuation spectrum from computer simulations {#the-fluctuation-spectrum-from-computer-simulations .unnumbered}
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For a particular configuration of a simulated shell, the coefficients $A_{lm}$ of the expansion of the radial displacements in spherical harmonics (Eq. \[eqn0sphharm\]) are determined by a least squares fit of the node positions to a finite number $l_M$ of (real) spherical harmonics. In practice we have used $l_M=26$ as the upper wavenumber cutoff for all simulations. At each temperature and pressure, this procedure is repeated for about 10000 independent configurations and the results averaged to obtain the curves presented in Fig. \[fig\_sphfl\].
Simulations of shells indented by point-like forces {#simulations-of-shells-indented-by-point-like-forces .unnumbered}
---------------------------------------------------
To perform indentation simulations, two harmonic springs are attached to the north and south pole of the shell. This leads to an additional term in the Hamiltonian $V_\text{s}=k_\text{i}\left(z_\text{i}^\text{N} - z^\text{N} \right)^2/2+k_\text{i}\left(z_\text{i}^\text{S} - z^\text{S} \right)^2/2$ where $k_\text{i}=\kappa/r_{0}^{2}$ is the spring constant of the indenter. Here, one end of the springs, at positions $z^\text{N}$ and $z^\text{S}$, is attached to the vertices at the north and south pole, respectively. The positions of the other end of the springs, at $z_\text{i}^\text{N}$ and $z_\text{i}^\text{S}$, are fixed externally and determine the indentation force and depth, as indicated in Fig. S4.
By changing $z_\text{i}^\text{N}$ and $z_\text{i}^\text{S}$, the depth of the indentation can be varied. After the springs are fixed a certain distance apart, the thermally average pole-to-pole distance $\langle z \rangle$ is measured and compared to its value in the absence of a force, $\langle z_{0} \rangle$. The instantaneous force at the poles is calculated from the instantaneous extension of the harmonic springs after each $N_\text{samp}$ Monte Carlo steps; thermal averaging then determines the average corresponding to $\langle z \rangle$. This provides the force-indentation curves in Fig. 3(a–b).
It is very difficult to unambiguously identify the linear regime in the force-indentation curves. Extracting the effective spring constant of shell deformation $k_\text{s}$ from a linear fit in the small indentation region is subject to inaccuracies and sensitivity to the number of points included in fitting. Instead, we extract the spring constants of thermally fluctuating shells by using a relation between $k_\text{s}$ and the fluctuations in $z_0$ (see [*Supplementary Information*]{} for derivation): $$k_{\text{s}} \approx \frac{k_\text{B}T}{\langle z_{0}^{2}\rangle - \langle z_{0}\rangle^{2}}.$$ This procedure was used to measure the temperature-dependent spring constants in Fig. 3c.
It is a pleasure to acknowledge J. Hutchinson, F. Spaepen, Z. Zeravcic and A. Kosmrlj for helpful discussions. Work by JP and DRN was supported by the National Science Foundation via Grant DMR1005289 and through the Harvard Materials Research Science and Engineering Center through Grant DMR0820484.
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[Supplementary Information]{}
Fields and strains in shallow shell theory {#fields-and-strains-in-shallow-shell-theory .unnumbered}
==========================================
{width="3.42in"}
We describe the deformations of the sphere using *shallow shell theory* which we summarize here. We follow the presentation by Koiter and van der Heijden [@si_koiter_stability]. A shallow section of the sphere is isolated and Cartesian coordinates $(x_1,x_2)$ are set up to define a plane that just touches the undeformed sphere at the origin and lies tangent to it; the $z$ axis is thus normal to the sphere at the origin (Fig. S1). We use the Monge representation to parametrize the undeformed shell by its height $z=Z(x_1,x_2)$ above the plane, where $Z(x_1,x_2)$ is the undeformed state corresponding to a sphere of radius $R$ with its center located on the $z$-axis above the $(x_1,x_2)$ plane; $$Z(x_1,x_2) = R\left(1-\sqrt{1-\frac{x_1^2}{R^2}-\frac{x_2^2}{R^2}}\right)$$ The assumption in shallow shell theory is that the section of the shell under consideration is small enough that slopes $\partial_1 Z \sim x_1/R$ and $\partial_2 Z \sim x_2/R$ measured relative to the $(x_1,x_2)$ plane are small. (Partial derivatives are denoted by $\partial/\partial x_i \equiv \partial_i$.) Then the undeformed state is approximately parabolic in $x_1$ and $x_2$, $$Z(x_1,x_2) \approx \frac{x_1^2+x_2^2}{2R}.$$
Deformations from this initial state are quantified *via* a local normal displacement $f(x_1,x_2)$ perpendicular to the undeformed surface and tangential displacements $u_1(x_1,x_2)$ and $u_2(x_1,x_2)$ within the shell along the projections of the $x_1$ and $x_2$ axes on the sphere respectively. In terms of these fields, a point $(x_1, x_2, Z(x_1,x_2))$ in the undeformed state moves to $\left(x_1+u_1- f \partial_1 Z, x_2+u_2- f \partial_2 Z, Z+f\right)$ to lowest order in the slopes $\partial_i Z=x_i/R$. The strain tensor is defined by the relation between the length $ds^\prime$ of a line element in the deformed state and the corresponding line element length $ds$ in the undeformed state [@si_landau_elasticity_1986]: $$(ds^\prime)^2 = ds^2 + 2u_{ij}dx_idx_j.$$ With this definition and neglecting terms of order $(\partial_i Z)^2$ and their derivatives, we find the nonlinear strain tensor used in the main text, $$u_{ij}(\mathbf{x})=\frac{1}{2}\left(\partial_i u_j+\partial_j u_i +\partial_i f \partial_j f\right)-\delta_{ij}\frac{f}{R}.$$ The stretching energy is then given by [@si_landau_elasticity_1986] $$G_s=\frac{1}{2}\int dS\,\left[2\mu u_{ij}^2 + \lambda u_{kk}^2\right],$$ where $\mu$ and $\lambda$ are the Lamé coefficients and $dS$ is an area element.
We also include a bending energy of the Helfrich form [@si_helfrich_elastic_1973] that penalizes changes in local curvature: $$G_b = \frac{\kappa}{2}\int dS \, (H-H_0)^2,$$ where $\kappa$ is the bending rigidity, $H$ the mean curvature and $H_0$ the spontaneous mean curvature (which we take to be equal everywhere to the curvature $2/R$ of the undeformed shell). For a shallow section of the shell, the local curvature can be written in terms of the height field $Z(x_1,x_2)+f(x_1,x_2)$ as $$H = \nabla^2(Z+f) = \frac{2}{R}+\nabla^2 f,$$ where $\nabla^2 = \partial_{11}+\partial_{22}$ is the Laplacian in the tangential coordinate system. Finally the energy due to an external pressure $p$ equals the work done, $$W = -p\int dS \,f.$$
The area element is $dS = dx_{1}dx_{2} /\sqrt{1-(x_{1}^{2}+x_{2}^{2})/R^{2}}\approx dx_1dx_2$ when terms of order $(x_{i}/R)^2$ and above are neglected. Summing the stretching, bending and pressure energies leads to the elastic energy expression $G=G_s+G_b+W$, Eq. (1) in the main text.
Since we are restricted to a shallow section of the shell, the theory is strictly applicable only to deformations whose length scale is small compared to the radius $R$. The typical length scale $\ell$ of deformations can be obtained by balancing the bending and stretching energies $G_b$ and $G_s$ discussed above. Upon noting that the stretching free energy density in a region of size $\ell$ is $\mathcal{G}_s \sim Y(f/R)^2$, where $Y$ is a typical elastic constant, and $\mathcal{G}_b \sim \kappa f^2/\ell^4$, we recover the Föppl-von Kármán length scale introduced in the main text, $$\ell^* = \frac{R}{\gamma^{1/4}},$$ where the Föppl-von Kármán number is $\gamma = YR^2/\kappa$. More sophisticated calculations (sketched below) show that the relevant elastic constant is the 2D Young’s modulus, $Y=4\mu(\mu+\lambda)/(2\mu+\lambda)$.
For a shell made up of an elastic material of thickness $h$, taking $Y$ and $\kappa$ from the 3D Young’s modulus of an isotropic solid within thin shell theory provides the estimate $\gamma \approx 10 (R/h)^2$ [@si_landau_elasticity_1986]. For shallow shell theory to be valid, we need $\ell^* \ll R$. Hence shallow shell theory is valid when $\gamma \gg 1$ i.e. $R \gg h$, which is precisely the limit of large, thin curved shells which are most susceptible to thermal fluctuations. This agreement between shallow shell theory and more general shell theories that are applicable over entire spherical shells has been discussed by Koiter [@si_koiter_spherical_1963] in the context of the response of a shell to a point force at its poles. Shallow shell theory was also used to study the stability of pressurized shells by Hutchinson [@si_hutchinson_imperfection_1967]. In both cases, shallow shell theory was shown to be valid for thin shells such that $h/R \ll 1$. Since thermal fluctuations are only relevant for shells with radii several orders of magnitude larger than their thickness, shallow shell theory is an excellent starting point for the extremely thin shells of interest to us here.
Elimination of in-plane phonon modes and uniform spherical contraction by Gaussian integration {#elimination-of-in-plane-phonon-modes-and-uniform-spherical-contraction-by-gaussian-integration .unnumbered}
==============================================================================================
A spherical shell under the action of a uniform external pressure that is lower than the critical buckling threshold responds by contracting uniformly by an amount $f_0$. The out-of-plane deformation field can then be written as a sum of its uniform and non-uniform parts, $$f(\mathbf{x}) = f_0 + f^\prime(\mathbf{x})= f_0 +\sum_{\mathbf{q} \neq 0 }f_\mathbf{q}e^{-i\mathbf{q}\cdot\mathbf{x}},$$ where $f^\prime(\mathbf{x})$ represents the contribution to the field from its $\mathbf{q} \neq 0$ Fourier components. (In this section, for ease of presentation we use the normalization $f_\mathbf{q} \equiv \frac{1}{A}\int d^2x\,f(\mathbf{x})e^{i\mathbf{q}\cdot\mathbf{x}},$ where $A$ is the area of integration in the $(x_1,x_2)$ plane. The inverse transform is then $f(\mathbf{x})=\sum_{\mathbf{q}}f_\mathbf{q}e^{-i\mathbf{q}\cdot\mathbf{x}}.$) With this decomposition, $\int\,d^2x\,f^\prime(\mathbf{x})=0$ and thus only $f_0$ contributes to the pressure work $W$. On the other hand, only $f^\prime$ contributes to the nonlinear part of the strain tensor. Hence the elastic energy $G=G_b+G_s+W$ defined above is harmonic in the in-plane phonon fields $u_1(\mathbf{x})$ and $u_2(\mathbf{x})$ as well as the uniform contraction $f_0$. To analyze the effects of anharmonicity, it is useful to eliminate these fields and define an effective free energy [@si_nelson_statistical_1988], $$\label{si_eqn_gaussianintegral}
\begin{split}
G&_\mathrm{eff}[f^\prime] = -k_\text{B}T \ln\left\{\int \mathcal{D}\vec{u}(x_1,x_2)\int df_0\,e^{-G[f^\prime ,f_0,u_1,u_2]/k_\text{B}T]}\right\}.
\end{split}$$ To carry out the functional integrals in Eq. (\[si\_eqn\_gaussianintegral\]) for a fixed out-of-plane displacement field $f^\prime(\mathbf{x})$, the strain tensor $u_{ij}$ must also be separated into its $\mathbf{q}=0$ and $\mathbf{q}\neq 0$ components: $$u_{ij}=\tilde{u}^0_{ij}+\sum_{\mathbf{q}\neq 0}\left[\frac{i}{2}\left(q_i u_j(\mathbf{q})+q_ju_i(\mathbf{q})\right)+A_{ij}(\mathbf{q})-\delta_{ij}\frac{f_\mathbf{q}}{R}\right]e^{-i \mathbf{q \cdot x}}$$ where $$A_{ij}(\mathbf{q})=\frac{1}{2A}\int d^2x\,\partial_i f^\prime\, \partial_j f^\prime\, e^{i \mathbf{q \cdot x}}.$$ The uniform part of the strain tensor has the following components: $$\label{si_eqn_uij0}
\begin{split}
\tilde{u}^0_{11} &= u^0_{11} + A_{11}(\mathbf{0}) - \frac{f_0}{R}, \\
\tilde{u}^0_{22} &= u^0_{22} + A_{22}(\mathbf{0}) - \frac{f_0}{R},\\
\tilde{u}^0_{12} &= u^0_{12} + A_{12}(\mathbf{0}).\\
\end{split}$$ Here, $u^0_{ij}$ are the uniform in-plane strains that are *independent* of $f_0$. This restriction implies that $u^0_{11}+u^0_{22}=0$ because a simultaneous uniform in-plane strain of the same sign in the $x_1$ and $x_2$ directions corresponds to a change in radius of the sphere and thus cannot be decoupled from $f_0$. Hence in addition to $f_0$ and $u_{12}^0$, there is only one more independent degree of freedom, $\Delta u^0\equiv u_{11}^0-u_{22}^0$, that determines the uniform contribution to the strain tensor.
Finally we perform the functional integration in Eq. (\[si\_eqn\_gaussianintegral\]) over the phonon fields $u_i$ as well as the three independent contributions to the uniform part of the strain tensor — $f_0$, $\tilde{u}_{12}^0$ and $\Delta u^0$. The resulting effective free energy is, upon suppressing an additive constant,
$$\label{si_eqn_feff}
G_\mathrm{eff} = \int d^2 x\left[\frac{\kappa}{2}(\nabla^2 f^\prime)^2+\frac{Y}{2} \left(\frac{1}{2}P^\mathrm{T}_{ij}\partial_i f^\prime \partial_j f^\prime-\frac{f^\prime}{R}\right)^2\right]-A\frac{pR}{2}\left[A_{11}(\mathbf{0})+A_{22}(\mathbf{0})\right]$$
where $P^\mathrm{T}_{ij} = \delta_{ij}-\partial_i \partial_j/\nabla^2$ is the transverse projection operator. Note that as a result of the integration the Lamé coefficients $\mu$ and $\lambda$ enter only through the 2D Young’s modulus $Y = 4\mu(\mu+\lambda)/(2\mu+\lambda)$. Finally, substituting
$$A_{11}(\mathbf{0})+A_{22}(\mathbf{0}) = \frac{1}{2A}\int d^2x\,\left[(\partial_1 f^\prime)^2+(\partial_2 f^\prime)^2\right] = \frac{1}{2A}\int d^2x|\nabla f^\prime|^2$$
in Eq. (\[si\_eqn\_feff\]) gives the effective free energy used in the analysis, Eq. (3) in the main text. In the following, we drop the prime on the out-of-plane displacement field since $f_0$ has now been eliminated. When only the harmonic contributions are considered, the equipartition result for the thermally generated Fourier components $f_\mathbf{q} = \int d^2x \,f(\mathbf{x})\exp(i\mathbf{q}\cdot\mathbf{x})$ with two-dimensional wavevector $\mathbf{q}$ are $$\label{si_eqn_corrfn_gaussian}
\langle f_\mathbf{q}f_\mathbf{q^\prime} \rangle_0 = \frac{Ak_\mathrm{B}T \delta_{\mathbf{q},\mathbf{-q^\prime}}}{\kappa q^4 -\frac{pR}{2}q^2+ \frac{Y}{R^2}}.$$ where $A$ is the area of integration in the $(x_1,x_2)$ plane. This harmonic spectrum \[Eq. (4) in the main text\] takes on corrections due to the anharmonic terms that are calculated in the next section.
One-loop contributions to the self-energy {#one-loop-contributions-to-the-self-energy .unnumbered}
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{width="7in"}
{width="3.42in"}
Here we describe the self-energy used to calculate the leading anharmonic corrections to the fluctuation spectrum in the main text. The Feynman rules obtained from the effective free energy $G_\mathrm{eff}[f]$ are summarized in Fig. S2. Henceforth, Fourier components are defined as in the main text: $f_\mathbf{q} = \int d^2x \,f(\mathbf{x})\exp(i\mathbf{q}\cdot\mathbf{x})$ with two-dimensional wavevector $\mathbf{q}$. The inverse Fourier transformation of the out-of-plane deformation field is $$f(\mathbf{x}) = \frac{1}{A}\sum_{\mathbf{q}\neq\mathbf{0}}f_{\mathbf{q}}e^{-i\mathbf{q}\cdot\mathbf{x}},$$ where $A$ is the area of integration in the $(x_1,x_2)$ plane and the sum is over all allowed Fourier modes. The one-loop contribution to the self-energy $\Sigma(\mathbf{q})$ due to the anharmonic three-point vertex (cubic term in the energy) and the four-point vertex (quartic term) are summarized in Fig. S3. Fig. S3a is also present in the calculation for flat membranes [@si_nelson_statistical_1988], and provides a contribution $$\label{si_eqn_hat}
-Y\int \frac{d^2 k}{(2\pi )^2}\,\frac{[P^\mathrm{T}_{ij}(\mathbf{k})q_i q_j]^2}{\kappa|\mathbf{q+k}|^4 -\frac{pR}{2}|\mathbf{q+k}|^2+\frac{Y}{R^2}}$$ to the self-energy. Fig. S3b involves two-vertex terms arising from the cubic coupling unique to shells with curvature (note that, despite “amputation” of the propagator legs, the diagrams are distinct because the slashes decide the momentum terms which survive various index contractions in addition to determining the momentum of the transverse projection operator introduced at each vertex). The net contribution to the self-energy from the four diagrams in Fig. S3b is
$$\label{si_eqn_candy}
\begin{split}
&\frac{Y^2}{R^2}\int\frac{d^2 k}{(2\pi )^2}\,\frac{1}{\Bigl(\kappa|\mathbf{q+k}|^4-\frac{pR}{2}|\mathbf{q+k}|^2+\frac{Y}{R^2}\Bigr)\Bigl(\kappa k^4-\frac{pR}{2}q^2+\frac{Y}{R^2}\Bigr)}\times \\ &\quad\left\{\frac{1}{2}[P^\mathrm{T}_{ij}(\mathbf{q})k_i k_j]^2+[P^\mathrm{T}_{ij}(\mathbf{k})q_i q_j]^2+[P^\mathrm{T}_{ij}(\mathbf{k})q_i q_j][P^\mathrm{T}_{lm}(\mathbf{k+q})q_l q_m]+2[P^\mathrm{T}_{ij}(\mathbf{k})q_i q_j][P^\mathrm{T}_{lm}(\mathbf{q})k_l k_m]\right\}
\end{split}$$
While the inverse of the harmonic correlation function, Eq. (\[si\_eqn\_corrfn\_gaussian\]), only contains terms of order $q^{0}$, $q^{2}$, $q^{4}$, the one-loop corrections to the spectrum \[Eqs. (\[si\_eqn\_hat\]–\[si\_eqn\_candy\])\] generate terms with these powers of $q$ as well as terms of order $q^{6}$ and above in the full inverse fluctuation spectrum. If we keep only terms of order $q^{4}$ and below in the calculation of the one-loop fluctuation spectrum, we can provide an approximate description of the low-$q$ behaviour of the shell in terms of effective elastic constants: $$Ak_\mathrm{B}T\langle|f_\mathbf{q\rightarrow 0}|^2\rangle^{-1} \equiv \kappa_{\scriptscriptstyle\mathrm{R}} q^4 - \frac{p_{\scriptscriptstyle\mathrm{R}}R}{2} q^2+\frac{Y_{\scriptscriptstyle\mathrm{R}}}{R^2} + O(q^6),$$ where $Y_{\scriptscriptstyle\mathrm{R}}$, $\kappa_{\scriptscriptstyle\mathrm{R}}$ and $p_{\scriptscriptstyle\mathrm{R}}$ are the effective Young’s modulus, bending rigidity and dimensionless pressure respectively. At long length scales, probes of the elastic properties of thermally fluctating shells would provide information of these effective elastic constants rather than the “bare” constants $Y$, $\kappa$ and $p$ that describe the zero-temperature shell. Upon expanding the integrands in Eqs. (\[si\_eqn\_hat\]–\[si\_eqn\_candy\]) to O($q^{4}$) the momentum integrals can be carried out analytically to obtain:
$$\label{si_eqn_yr_whole}
Y_{\scriptscriptstyle\mathrm{R}} = Y\left[1-\frac{3}{128\pi}\frac{k_\text{B}T}{\kappa}\frac{\sqrt\gamma}{(1-\eta^2)^{3/2}}\left(\eta\sqrt{1-\eta^2}+\pi - \cos^{-1}\eta\right)\right],$$
$$\label{si_eqn_kr_whole}
\begin{split}
\kappa_{\scriptscriptstyle\mathrm{R}} &= \kappa\Biggl[1+\frac{1}{30720 \pi}\frac{k_\text{B}T}{\kappa}\frac{\sqrt\gamma}{(1-\eta^2)^{7/2}}\biggl[\eta\sqrt{1-\eta^2}\left(-1699+3758\eta^2-2104\eta^4\right) \\ &\qquad\qquad\qquad
+15(61-288\eta^2+416\eta^4-192\eta^6)\left(\pi - \cos^{-1}\eta\right)\biggr]\Biggr],
\end{split}$$
$$\label{si_eqn_pr_whole}
\eta_{\scriptscriptstyle\mathrm{R}}=\eta+\frac{1}{1536 \pi}\frac{k_{B}T}{\kappa} \frac{\sqrt{\gamma }}{\left(1-\eta ^2\right)^{5/2}} \left[\sqrt{1-\eta ^2} \left(64-67 \eta
^2\right)+3\left(21 \eta -22 \eta ^3\right)
\left(\pi-\cos ^{-1}\eta\right)\right].$$
where we have defined a dimensionless pressure $\eta \equiv p/p_{\text{c}}$ and $p_{\text{c}} = 4\sqrt{\kappa Y}/R^{2}$ is the classical buckling pressure of the shell. We see explicitly that the quantities diverge in the limit $\eta \to 1$. To lowest order in the external pressure, we have $$\label{si_eqn_smallpyr}
Y_{\scriptscriptstyle\mathrm{R}} \approx Y \left[1 -\frac{3}{256}\frac{k_\mathrm{B}T}{\kappa}\sqrt{\gamma}\left(1+\frac{4}{\pi}\frac{p}{p_{\text{c}}}\right)\right],$$ $$\label{si_eqn_genpressure}
p_{\scriptscriptstyle\mathrm{R}} \approx p +\frac{1}{24\pi}\frac{k_\mathrm{B}T}{\kappa}p_\text{c}\sqrt{\gamma}\left(1+\frac{63\pi}{128}\frac{p}{p_{\text{c}}}\right) ,$$ and $$\label{si_eqn_smallpkr}
\kappa_{\scriptscriptstyle\mathrm{R}} \approx \kappa\left[1 +\frac{61}{4096}\frac{k_\mathrm{B}T}{\kappa}\sqrt{\gamma}\left(1-\frac{1568}{915\pi}\frac{p}{p_{\text{c}}}\right)\right].$$ These are the approximate renormalized elastic quantities tabulated in Eqs. (7–9) in the main text.
In evaluating the above expressions, the momentum integrals in Eqs. (\[si\_eqn\_hat\]–\[si\_eqn\_candy\]) must strictly speaking be carried out over the phase space of all allowed Fourier modes $f(\mathbf{k})$ of the system, which go from some low-$k$ cutoff $k_{\text{min}}\sim 1/R$ to a high-$k$ cutoff set by the microscopic lattice constant. However, since all integrals converge in the ultraviolet limit $k \to \infty$, the upper limit of the $k$-integrals can be extended to $\infty$. The integrals are well-behaved at low momenta due to the mass term $\sim Y/R^2$ in the propagator. Hence we carry out the momentum integrals over the entire two-dimensional plane of $\mathbf{k}$. The excess contribution to the self energy by including spurious Fourier modes with $0< k < 1/R$, *i.e.* for wavevectors less than the natural infrared cutoff $k_\mathrm{min} \sim 1/R$, gives rise to an error of roughly $1/\sqrt{\gamma}$ which is negligible for extremely thin shells. This correction is of similar magnitude to the errors introduced by using shallow shell theory (which is inaccurate for the longest-wavelength modes with wavevector $k\sim 1/R$) which are also negligible in the thin-shell limit.
Calculation of fluctuation spectrum with spherical harmonics {#calculation-of-fluctuation-spectrum-with-spherical-harmonics .unnumbered}
============================================================
While the perturbation theory calculations were carried out using a basis of Fourier modes in a shallow section of the shell to decompose the radial displacement field, the fluctuation spectrum is most efficiently measured in simulations using a spherical harmonics expansion. To compare the simulation results to the expected corrections from perturbation theory, we use the description of the shell in terms of the effective elastic constants $Y_{\scriptscriptstyle\mathrm{R}}$, $\kappa_{\scriptscriptstyle\mathrm{R}}$ and $p_{\scriptscriptstyle\mathrm{R}}$, Eqs. (7–9) in the main text.
Consider a spherical shell of radius $R$ with bending rigidity $\kappa$ and Lamé coefficients $\lambda$ and $\mu$, experiencing a tangential displacement field $\mathbf{u} = (u_{x},u_{y})$ and a radial displacement field $f$. Like any smooth vector field, $\mathbf{u}$ can be decomposed into an irrotational (curl-free) part and a solenoidal (divergence-free) part: $\mathbf{u} \equiv \nabla\Psi+\mathbf{v}$, where the scalar function $\Psi$ generates the irrotational component and $\mathbf{v}$ is the solenoidal component. Upon expanding $f\equiv \sum_{l,m}A_{lm}RY_{l}^{m}$ and $\Psi \equiv \sum_{l,m}B_{lm}R^{2}Y_{l}^{m}$ in terms of spherical harmonics $Y_{l}^{m}(\theta,\phi)$, the elastic energy of the deformation to quadratic order in the fields is given by [@si_zhang_scaling_1993] $$\begin{split}
G &= R^{2}\sum_{l,m}\biggl\{\left[\frac{\kappa}{2}\frac{(l+2)^{2}(l-1)^{2}}{R^{2}}+2K\right] A_{lm}^{2}-2Kl(l+1)A_{lm}B_{lm} \\
&\qquad\qquad\qquad+\frac{1}{2}l(l+1)\left[(K+\mu)l(l+1)-2\mu\right]B_{lm}^{2}\biggr\}+G_{\text{sol}}(\mathbf{v}),
\end{split}$$ where $K=\lambda+\mu$ is the bulk modulus. The solenoidal component $\mathbf{v}$ does not couple to the radial displacement field and provides an independent contribution $G_{\text{sol}}$ which is purely quadratic in the field $\mathbf{v}$.
To this elastic energy, we also add the surface energy-like contribution $G_{\text{S}}=-(pR/2) \Delta A$ due to the “negative surface tension” $-pR/2$ present in the shell when it is uniformly compressed in response to an external pressure $p$. Here $\Delta A$ is the excess area due to deformations about the average radius. In terms of spherical harmonic coefficients, this area change can be written [@si_milner_dynamical_1987] $$\Delta A \approx R^{2}\sum_{l>1,m}A_{lm}^{2}\left[1+\frac{l(l+1)}{2}\right].$$
As we did for the elastic energy in shallow shell theory, we can now integrate out the quadratic fluctuating quantities $B_{lm}$ and the solenoidal field $\mathbf{v}$ to obtain an effective free energy in terms of the radial displacements alone: $$G_{\text{eff}} = \frac{R^{2}}{2}\sum_{l>1,m}\left\{\frac{\kappa(l+2)^{2}(l-1)^{2}}{R^{2}}-pR\left[1+\frac{l(l+1)}{2}\right]+\frac{4\mu(\mu+\lambda)(l^{2}+l-2)}{(2\mu+\lambda)(l^{2}+l)-2\mu}\right\}A_{lm}^{2}.$$ The fluctuation amplitude is obtained via the equipartition theorem: $$\begin{split}
k_{B}T\langle |A_{lm}|^{2}\rangle_{0}^{-1} &= \kappa(l+2)^{2}(l-1)^{2}-pR^{3}\left[1+\frac{l(l+1)}{2}\right]+\frac{4\mu(\mu+\lambda)(l^{2}+l+2)}{(2\mu+\lambda)(l^{2}+l)-2\mu}R^{2} \\
&= \kappa(l+2)^{2}(l-1)^{2}-pR^{3}\left[1+\frac{l(l+1)}{2}\right]+\frac{Y}{1+\frac{Y}{2\mu(l^{2}+l-2)}}R^{2}. \label{si_eqn.sphfluct}
\end{split}$$ where $Y = 4\mu(\mu+\lambda)/(2\mu+\lambda)$ is the 2D Young’s modulus introduced earlier. The effect of anharmonic contributions to the fluctuation spectrum can now be calculated by using the effective temperature-dependent quantities $Y_{\scriptscriptstyle\mathrm{R}}$, $\kappa_{\scriptscriptstyle\mathrm{R}}$ and $p_{\scriptscriptstyle\mathrm{R}}$ in place of the bare elastic constants in the above expression. However, the last term in Eq. (\[si\_eqn.sphfluct\]) also requires knowledge of the thermal corrections to the Lamé coefficient $\mu$ which was eliminated in the shallow shell calculation when the tangential displacement fields were integrated out. For the discretized stretching energy used in the simulations, we have $\mu = 3Y/8$. If we assume that this relationship is not significantly changed by the anharmonic corrections to one-loop order, then $\mu_{\scriptscriptstyle\mathrm{R}} \approx 3Y_{\scriptscriptstyle\mathrm{R}}/8$. Upon substituting this approximation together with the other effective elastic parameters in Eq. (\[si\_eqn.sphfluct\]), we find $$\begin{split}
k_{B}T\langle |A_{lm}|^{2}\rangle^{-1} \approx &\kappa_{\scriptscriptstyle\mathrm{R}}(l+2)^{2}(l-1)^{2} -p_{\scriptscriptstyle\mathrm{R}}R^{3}\left[1+\frac{l(l+1)}{2}\right] +Y_{\scriptscriptstyle\mathrm{R}}R^{2}\left[\frac{3(l^{2}+l-2)}{3(l^{2}+l)-2}\right] \label{si_eqn_approxsph}
\end{split}$$ which is the same as as Eq. (11) in the main text [^1].
Linear response of the shell to point forces {#linear-response-of-the-shell-to-point-forces .unnumbered}
============================================
We calculate the response of the shallow shell to a point force at the origin, corresponding to a force field $h(\mathbf{x}) = F\delta^{2}(\mathbf{x})$. The Fourier decomposition of this force field is $$h_\mathbf{q} = F, \,\text{for all}\, \mathbf{q}.$$ The linear response of the deformation field $f$ to this force is related to its fluctuation amplitudes in the *absence* of the force, $\langle |f_\mathbf{q}|^2\rangle_{h=0}$, by the fluctuation-response theorem: $$\langle f_\mathbf{q} \rangle = \frac{\langle |f_\mathbf{q}|^2\rangle_{h=0}}{Ak_\text{B}T}h_{\mathbf{q}} = \frac{\langle |f_\mathbf{q}|^2\rangle_{h=0}}{Ak_\text{B}T}F.$$ The inward deflection at the origin is then $$\label{si_eqn0deflection}
\langle f(\mathbf{x}=0)\rangle = \frac{1}{A} \sum_{\mathbf{q}}\langle f_\mathbf{q} \rangle = \frac{F}{A^{2}k_{B}T}\sum_{\mathbf{q}} \langle |f_\mathbf{q}|^2\rangle_{h=0}.$$ This can be related to $\langle f^{2} \rangle$, the mean square fluctuations of the deformation field in real space which is a position-independent quantity in the absence of nonuniform external forces: $$\label{si_eqn0msfluct}
\begin{split}
\langle f^{2} \rangle \equiv \langle [f(\mathbf{x})]^{2} \rangle_{h=0} &= \frac{1}{A^{2}}\sum_{\mathbf{q}}\sum_{\mathbf{q^{\prime}}}\langle f_{\mathbf{q}}f_{\mathbf{q^{\prime}}}\rangle e^{-i(\mathbf{q}+\mathbf{q^{\prime}})\cdot\mathbf{x}} \\
&= \frac{1}{A^{2}}\sum_{\mathbf{q}}\langle |f_\mathbf{q}|^2\rangle_{h=0}.
\end{split}$$ From Eqs. (\[si\_eqn0deflection\]) and (\[si\_eqn0msfluct\]), we obtain $$\label{si_eqn0defl-fluct}
\langle f(\mathbf{x}=0)\rangle = \frac{F}{k_{B}T}\langle f^{2}\rangle.$$ This equation relates the depth of the indentation due to a force $F$ at the origin to the mean square fluctuations of the deformation field $f$ in the absence of such a force.
When only harmonic contributions are considered, Eq. (\[si\_eqn\_corrfn\_gaussian\]) gives us the mean square amplitude $\langle |f_\mathbf{q}|^2\rangle_0 = Ak_{B}T/(\kappa q^4 - pRq^2/2+Y/R^{2})$ in terms of the elastic constants and external pressure. Upon taking the continuum limit of the sum over wavevectors $\sum_{\mathbf{q}} \to A\int d^{2}q/(2\pi)^{2}$, we can calculate the fluctuation amplitudes exactly: $$\label{si_eqn0bareflucts}
\langle f^{2} \rangle = \int\frac{d^{2}q}{(2\pi)^{2}}\frac{k_{B}T}{\kappa q^4-\frac{pR}{2}q^{2}+\frac{Y}{R^{2}}} = \frac{Rk_{B}T}{8\sqrt{\kappa Y}}\frac{1+\frac{2}{\pi}\sin^{-1}\eta}{\sqrt{1-\eta^{2}}},$$ where $\eta = p/p_{\text{c}}= pR^{2}/(4\sqrt{\kappa Y})$ is the dimensionless pressure, and $\eta < 1$, *i.e.* we restrict ourselves to pressures below the classical buckling pressure. From Eqs. (\[si\_eqn0defl-fluct\]) and (\[si\_eqn0bareflucts\]), we get the linear relation between the indentation force and the depth of the resulting deformation: $$F=\frac{8\sqrt{\kappa Y}}{R}\frac{\sqrt{1-\eta^{2}}}{1+\frac{2}{\pi}\sin^{-1}\eta}\langle f(\mathbf{x}=0)\rangle.$$ The temperature drops out and we obtain a result valid for $T=0$ shells as well. The expression reproduces the well-known Reissner solution [@si_reissner_stresses_1946] for the linear response of a spherical shell to a point force when $\eta=0$, and also reproduces the recent result from Vella et al [@si_vella_indentation_2011] for indentations on spherical shells with an *internal* pressure when $\eta < 1$. At finite temperatures, however, anharmonic effects contribute terms of order $(k_{B}T)^{2}$ and higher to $\langle f^{2} \rangle$, making the response temperature-dependent.
In the simulations, the shells contract by a small amount due to thermal fluctuations, even in the absence of external forces. Thus, indentations are measured relative to the thermally averaged pole-to-pole distance of the shell at finite temperature, $\langle z_0 \rangle < 2R$. Equal and opposite inward forces are applied to the north and south poles of the shell to maintain a force balance (see details in the [*Materials and Methods*]{} section of the main text) and the resulting average pole-to-pole distance, $\langle z \rangle$, is measured. This corresponds to an average indentation depth of $(\langle z_{0} \rangle - \langle z \rangle)/2$ at each pole, with associated force \[from Eq. (\[si\_eqn0defl-fluct\])\] $$F = \frac{k_{B}T}{\langle f^{2} \rangle}\frac{(\langle z_{0} \rangle - \langle z \rangle)}{2} \equiv k_{\text{s}} (\langle z_{0} \rangle - \langle z \rangle),$$ *i.e.* the shell as a whole acts as a spring with spring constant $$\label{si_eqn0springconstant}
k_{\text{s}} = \frac{k_{B}T}{2\langle f^{2} \rangle}.$$
At $T=0$, we have $$\label{si_eqn0springconstantt0}
k_{\text{s}} = \frac{4\sqrt{\kappa Y}}{R}\frac{\sqrt{1-\eta^{2}}}{1+\frac{2}{\pi}\sin^{-1}\eta};$$ in particular, $k_{\text{s}}=4\sqrt{\kappa Y}/R$ in the absence of external pressure. Anharmonic contributions change the fluctuation amplitude $\langle f^{2} \rangle$ and hence the linear response. To lowest order in temperature, the effects of anharmonic contributions can be obtained by using the renormalized elastic constants calculated using perturbation theory \[Eqs. (\[si\_eqn\_smallpyr\])–(\[si\_eqn\_smallpkr\])\] in Eq. (\[si\_eqn0springconstantt0\]) and keeping terms to $O(T)$. In particular, even if the bare pressure $p = 0$, the renormalized dimensionless pressure $p_{\text{R}}$ is nonzero and affects the spring constant, as do the temperature-dependent effective elastic moduli. The result in this case is $$k_{\text{s}}(T>0) \approx \frac{4\sqrt{\kappa Y}}{R}\left[1-0.0069\frac{k_{B}T}{\kappa} \sqrt\gamma\right].$$ This is the theoretical prediction quoted as Eq. (13) in the main text.
Measuring the effective spring constant from Monte Carlo simulations {#measuring-the-effective-spring-constant-from-monte-carlo-simulations .unnumbered}
====================================================================
{width="7in"}
We extract the spring constants of thermally fluctuating shells for Fig. 3c in the main text by using the relation between $k_{\text{s}}$ and fluctuations in the transverse displacement field $f$ \[Eq. (\[si\_eqn0springconstant\])\]. It is straightforward to measure the average pole-to-pole distance of the fluctuating shell in the [*absence*]{} of external forces, $\langle z_{0} \rangle = \langle R - f_{\text{N}} - f_{\text{S}} \rangle$, where $f_{\text{N}}$ and $f_{\text{S}}$ are the inward displacements at the north and south poles respectively. Since the displacements at the poles are expected to be independent of each other, the mean squared fluctuations in $z_{0}$ are closely related to the mean square fluctuations in $f$: $$\langle z_{0}^{2}\rangle - \langle z_{0}\rangle^{2} \approx 2\langle f^2 \rangle.{}$$ The spring constant can thus be measured indirectly from the fluctuations in the pole-to-pole distance using Eq. (\[si\_eqn0springconstant\]): $$k_{\text{s}} = \frac{k_{B}T}{2\langle f^{2} \rangle} \approx \frac{k_{B}T}{\langle z_{0}^{2}\rangle - \langle z_{0}\rangle^{2}}.$$
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[^1]: If, as is more likely, the thermal corrections to $\mu$ and $Y$ do differ to O($k_{B}T$), we can nevertheless estimate that the resulting error term introduced by the assumption $\mu_{\scriptscriptstyle\mathrm{R}} \approx 3Y_{\scriptscriptstyle\mathrm{R}}/8$ is suppressed by a factor $4/[3(l^{2}+l-2)+4]$ relative to the anharmonic corrections and is thus atleast an order of magnitude smaller than the anharmonic contribution itself when $l>1$.
|
---
abstract: 'We develop a linear algorithm for extracting extragalactic point sources for the Compact Source Catalogue of the upcoming PLANCK mission. This algorithm is based on a simple top-hat filter in the harmonic domain with an adaptive filtering range which does not require [*a priori*]{} knowledge of the CMB power spectrum and the experiment parameters such as the beam size and shape nor pixel noise level.'
author:
- 'L.-Y. Chiang$^{1}$, H.E. Jørgensen$^{2}$, I.P. Naselsky$^{3}$, P.D. Naselsky$^{1,3}$,'
- |
I.D. Novikov$^{1,2,4,5}$, and P.R. Christensen$^{1,6}$\
$^1$ Theoretical Astrophysics Center, Juliane Maries Vej 30, DK-2100, Copenhagen, Denmark\
$^2$ Astronomical Observatory, Juliane Maries Vej 30, DK-2100, Copenhagen, Denmark\
$^3$ Rostov State University, Zorge 5, 344090 Rostov-Don, Russia\
$^4$ Astro-Space Center of Lebedev Physical Institute, Profsoyuznaya 84/32, Moscow, Russia\
$^5$ NORDITA, Blegdamsvej 17, DK-2100, Copenhagen, Denmark\
$^6$ Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
date: 'Accepted 2002 ???? ???; Received 2002 ???? ???'
title: An adaptive filter for the PLANCK Compact Source Catalogue construction
---
methods: data analysis – techniques: image processing – cosmic microwave background
Introduction
============
The ESA PLANCK Surveyor will produce ten all-sky high-resolution maps of the cosmic microwave background (CMB) anisotropies at 9 frequencies, from which the angular power spectrum will be derived to place constraints on the cosmological parameters. The maps produced by the PLANCK satellite, however, will not only include the CMB signal but also contain some astrophysical foreground sources arising from dust, free–free, synchrotron emission, Sunyaev-Zel’dovich effects and extragalactic point sources. It is therefore an important task to separate those foreground sources from the CMB signal. In this paper we will concentrate on the bright point sources with flux above $0.1-0.3$ Jy in connection with the Early Release Compact Source Catalogue (ERCSC) from the PLANCK mission. The main goal of the ERCSC construction is to produce such a catalogue right after the first six months of observation of the CMB sky by the PLANCK before more complicated and time consuming analysis of the CMB power spectrum, component separation and investigation of in-flight systematics. Therefore, it would be very useful to develop a fast method of the point sources extraction which needs as little information as possible about the parameters of the experiment.
In this paper, we present a fast linear algorithm for the extraction of extragalactic point sources from the CMB maps, which is a generalized amplitude-phase method of Naselsky, Novikov & Silk (2002). There have been developments of methods on point source extraction such as the high-pass filter by Tegmark and de Oliveira-Costa (1998) (hereafter TO98), Maximum–Entropy Method (MEM) by Hobson et al. (1999), and Mexican Hat Wavelet method (MHW) by Cay[ó]{}n et al. (2000). Here we introduce a simple top-hat filter for the extraction of point sources in the CMB maps. This method works very well without assumptions about the cosmological model, or [*a priori*]{} knowledge of the power spectrum of CMB and point sources.
The filter proposed by TD98 is [*optimal*]{} for point source extraction from the theoretical point of view, and requires the exact information about the CMB and the foregrounds power spectrum, the beam shape properties (its ellipticity and orientation), and the pixel noise level. For concrete applications, however, other methods can also be useful, which explains the succeeding discussions of other filters in the literature mentioned above and in this paper. We will compare our filter with the TD98 filter.
This paper is arranged as follows. In Section 2 we introduce the top–hat filter and elaborate the subtlety in the definition of the criterion by which the point sources are extracted. As our top–hat filtering range is adaptive, we apply in Section 3 the top–hat filter to the numerical simulated maps and estimate the filtering ranges for the PLANCK channels when the experiment parameters such as the beam size and the pixel noise level are known. We also compare our method with the theoretically optimal TD98 filter in Section 4. In Section 5 we generalize our filter to an algorithm that does not need any information about the CMB power spectrum, the beam size and the noise level. The results and discussion are in Section 6.
The top–hat filter
==================
In the observed map by the PLANCK satellite the signal at pixel $i$ can be expressed as $$d_i=S_i({{\mbox{\boldmath $r$}}}_i)+n_i$$ where $n_i$ is the pixel noise, ${{\mbox{\boldmath $r$}}}_i$ correspoinds to the position of the $i$-pixel in the map and $S_i \equiv \Delta T/T({{\mbox{\boldmath $r$}}}_i)$ includes the CMB signal and foreground contaminations, of which the relevant component to this paper is the point source contribution. Expanding $\Delta T/T$ in spherical harmonics, we have $$\frac{\Delta T}{T}({{\mbox{\boldmath $r$}}}_i)=\sum_{\ell m}B_{\ell m}a_{\ell m}Y_{\ell
m}({{\mbox{\boldmath $r$}}}_i),$$ where $B_{\ell m}$ is the beam response. The main idea of our method to extract point sources is through a simple linear top–hat filter in the harmonic domain with two cut–off scales. These two scales serve to remove the influence of the lower and higher multipole parts of the total power spectrum of the signal for optimal extraction of point sources from the PLANCK maps.
In TD98 the authors introduced the ratio of the amplitude of a point source (convolved with the beam) to the variance of the total signal in the map $\sigma^2_{\rm tot}$, which can be denoted as $$\overline{\Re}=\left( \frac{\langle B^2 \otimes S^2
\rangle}{\sigma^2_{\rm tot}}\right)^{1/2}.$$ Here $\langle B^2 \otimes S^2 \rangle=\langle (\Delta T_{\rm PS}/T
)^2\rangle$ is the contribution from the point source relative flux $S$ to the map, $\otimes$ denotes convolution, and $B$ is the beam response, which is assumed Gaussian. They found the shape of the filter $F_{\rm TD98}$ by maximizing $F_{\rm TD98} \otimes \overline{\Re}$.
In our method we introduce a linear filter, $${\cal F}_{\rm TH}(\lmin,\lmax)=\Theta(\ell-\lmin)\Theta(\lmax-\ell),
\label{eq:heaviside}$$ where $\Theta$ is the Heaviside step function, i.e., $\Theta(x)=0$ for $x\leq 0$, and $\Theta(x)=1$ for $x > 0$. $(\lmin,\lmax)$ is the filtering range of ${\cal F}_{\rm TH}$. This filter has a top–hat shape in the harmonic domain with two characteristic scales $\lmin$ and $\lmax$, both of which are functions of the antenna beam shape, the power spectrum of the CMB, the power spectra of all kinds of foregrounds and pixel noise, and possible systematic features. When these parameters are known, we can find $\lmin$ and $\lmax$ through maximizing $F_{\rm TH} \otimes R$ for each frequency channel of the PLANCK mission[^1]. Here $F_{\rm TH}$ is the filter in Eq. (\[eq:heaviside\]) in the real domain and $R$ is the resultant relative flux, $$R=\Re -\nu_{\min},
\label{eq:deepest}$$ where $\Re=(B \otimes S)/\sigma_{\rm tot}$ and $-\nu_{\min}$ is defined by $- \nu_{\min} \sigma_{\rm in}$ being the amplitude of the deepest minimum in the map, with $\sigma_{\rm in}$ the square root of the variance of the pre-filtered map.
Note that there is a subtle difference between the TD98 filtering and the top-hat optimization algorithm in the definition of the criterion by which the point sources are extracted. In order to obtain the filter shape, the TD98 filter is defined by maximizing the $\overline{\Re}$ ratio in their theoretical derivation whereas the top–hat filter is defined by maximizing the $R= \Re-\nu_{\min}$ ratio. According to the prediction of bright point source contamination in the Low Frequency Instrument (LFI) frequency range of the PLANCK mission [@toffolatti], point sources with flux above $0.1-0.5$ Jy are rare events in a $10^\circ \times 10^\circ$ patch of the sky. For the 30 GHz frequency channel, for example, the estimated number density of point sources is $\sim$0.3-1 source for each $10^\circ \times
10^\circ$ patch. Thus each bright point source is a peculiar peak in the $\Delta T/T$ map and the observed amplitude of the point source in the map is a combination of the point source contribution itself (which is not known), the signal from the CMB plus foregrounds convolved with beam response, and pixel noise contribution in the pixel containing the point source signal. As is mentioned in TD98, in order to extract point sources from the filtered map it is necessary to introduce a criterion to screen point sources from the ‘noise’[^2]. It is the so–called $5\sigma_{\rm f}$ criterion, which means that the peaks in the filtered map with amplitudes above $5\sigma_{\rm f}$ threshold are identified as point sources, $\sigma_{\rm f}$ being the square root of the variance of the filtered map. The amplitude of each filtered peak above $5\sigma_{\rm f}$, however, is the combination of the amplitude of the point source and the filtered ‘noise’. Therefore, generally speaking, the final (filtered) signal around the peak area with the amplitude around $5\sigma_{\rm f}$ is sensitive to the actual realization of the pre-filtered ‘noise’, which can either increase or decrease the point source amplitude depending on initial realization of the ‘noise’ signal. This is the reason why the TD98 filter can distinguish [*mean*]{} point source contribution from the map. In our method for the construction of the filter we consider the worst case, i.e., when the point source is at the position of the deepest minimum $-\nu_{\min}
\sigma_{\rm in}$ of the signal in the map.
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We would like to point out the significance of the two characteristic scales $\lmin$ and $\lmax$ on enhancing the ratio of the point source flux to the $\sigma_{\rm f}$ of the filtered map. There are two main features on the total power spectrum $C_{\ell}^{\rm tot}$ from the map as shown in Fig. \[powerspectrum1\] and Fig. \[powerspectrum2\]. On the low multipole end, there are the CMB itself which has the standard characteristic of $C_{\ell} \propto \ell^{-2}$ with a Harrison-Zel’dovich power spectrum from adiabatic perturbation, together with the so–called low multipole tail from the foregrounds such as dust emission, free–free and synchrotron emission. Thus the scale $\lmin$ of the top–hat filter cuts off this low multipole part of the total power spectrum $C_{\ell}^{\rm tot}$. For the high multipole end, on the other hand, the most dominant component in $C_{\ell}^{\rm tot}$ is the pixel noise. The corresponding scale $\lmax$ is therefore crucial in cutting down the pixel noise contribution in the filtered map, hence decreasing $\sigma_{\rm f}$.
The top–hat filter aims to suppress the low multipole tail and the pixel noise contribution so as to minimize the $\sigma_{\rm f}$. At the same time, it retains the part of $C_{\ell}^{\rm tot}$ that is most modified by the beam response (see the shaded area of each panel in both Fig. \[powerspectrum1\] and Fig. \[powerspectrum2\]). This means that, instead of the theoretically optimal TD98 filter which needs some preliminary and detailed information about the power spectra of the PLANCK map components and corresponding beam response, the top–hat filter has the simplest shape that transforms the uncertainties of the parameters to the scales of the cut-off $(\lmin,\lmax)$.
In constructing the top-hat filter for a specific frequency channel, we have to estimate the two cut–off scales when the experimental parameters such as the beam size and the pixel noise level are known. To do so, we choose the deepest minimum of the ‘noise’ as the position of the point source (hence the expression of Eq. (\[eq:deepest\])) and optimize the cut-off scales $(\lmin,\lmax)$ exactly from such worst realization of the point source signal and ‘noise’. The filter with the filtering range obtained by the optimization of the signal $S_{\rm f}$ (point source) to the ‘noise’ $(N_{\rm f})$ ratio for the filtered map in the worst case under the condition $S_{\rm f}/N_{\rm f}\rightarrow \max$ allows us to detect all point sources with the same flux (and above) at any other different locations of the map. [^3]
----------- -------------------- --------------------- ---------------------- ---------- ------------ -----------------
Frequency $\sigma_{\rm CMB}$ $\sigma_{\rm dust}$ $\sigma_{\rm noise}$ FWHM Pixel size Simulation size
(GHz) ($10^{-5}$) ( $10^{-5}$) $(10^{-5})$ (arcmin) (arcmin) (squared area)
857 4.47 155700. 2221.11 5.0 1.5 $12.8^\circ$
545 4.47 1220.0 48.951 5.0 1.5 $12.8^\circ$
353 4.48 65.1 4.795 5.0 1.5 $12.8^\circ$
217 4.43 8.52 1.578 5.5 1.5 $12.8^\circ$
143 4.27 2.55 1.066 8.0 1.5 $12.8^\circ$
100 (HFI) 4.07 1.15 0.607 10.7 3.0 $25.6^\circ$
100 (LFI) 4.10 1.13 1.432 10.0 3.0 $25.6^\circ$
70 3.88 0.558 1.681 14.0 3.0 $25.6^\circ$
44 3.43 0.228 0.679 23.0 6.0 $25.6^\circ$
30 3.03 0.114 0.880 33.0 6.0 $25.6^\circ$
----------- -------------------- --------------------- ---------------------- ---------- ------------ -----------------
Estimation of the filtering range {#estimation}
==================================
In this section we describe the technique of the filtering range $(\lmin,\lmax)$ estimation for the 10 maps of the PLANCK mission. The basic model of the PLANCK experiment which details the scan strategy, pixel noise properties, beam shape analysis and different kinds of the foreground contaminations is recently discussed in Mandolesi et al. (2000), Burigana et al. (1998), Bersanelli et al. (1997), and Chiang et al. (2001). In this section in order to determine the filtering range we will first assume that all the above-mentioned characteristics of the possible signals are well determined with corresponding accuracy. This condition will be relaxed in Section \[iteration\] when we introduce a more generalized algorithm for the top–hat filtering. At this stage those well–determined characteristics allow us to estimate the optimal values of the $(\lmin,\lmax)$ range for the top–hat filter (Eq.( \[eq:heaviside\])) and compare the efficiency of the point source extraction from the maps by applying the TD98 and the top–hat filter. Below we will use the flat sky approximation for the CMB maps without loss of generality. Moreover, Chiang et al. (2001) have shown the importance of periodic boundary condition of simulations, which is the standard part of the flat sky approximation, for descriptions of the real signal from small patches of the sky.
5.5cm
We produce for each PLANCK observing frequency channel a set of realizations of simulated maps using the data provided by Vielva et al. (2001). The details of the simulations are listed in Table \[simulation\]. The CMB signals are created from the angular power spectrum of the $\Lambda$CDM model by Lee et al. (2001). Dust emission is simulated with power law index $-3$. The free-free and synchrotron emissions are not simulated and added to obtain the filtering range $\ell_{\min}$ and $\ell_{\max}$ in Table \[range\]. Without adding these two parts of foreground emissions would, of course, affect the estimation of the filtering range, especially for LFI frequency channels, where the [*rms*]{} of both emissions are less than by one order of magnitude or comparable to that of CMB. As the free-free and synchrotron would have been assumed Gaussian, what would be modified is not the filtering range but the enhancement factor $R$. Moreover, as will become clear in Section \[iteration\], these filtering ranges serve as the initial values for the iteration scheme when we generalize the top-hat filter. The combined realization is then convolved with the corresponding antenna beam size. In this section we assume Gaussian beams $B_k=\exp(-k^2\theta^2/2)$, where $\theta=\mbox{FWHM}/2.355$, but a more realistic PLANCK antenna beam shape can be modelled using the method proposed by Chiang et al. (2001), which, as will be shown in Section \[iteration\], can also be tackled without difficulty.
4.5cm (-42,28)\
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To estimate the $(\kmin,\kmax)$ filtering range, in each realization we add one beam–convolved point source with fixed amplitude $\Re=3$, which is deliberately located at the deepest minimum of the realization with beam–convolved CMB signal including foregrounds plus pixel noise. In the flat sky approximation we firstly Fourier transform the total map, then we impose the top–hat filter as in Eq. (\[eq:heaviside\]) with a cut–off in Fourier domain, ${\cal F} (\kmin,\kmax) = \Theta(| {\bf k}|-\kmin)
\Theta(\kmax - |{\bf k}|)$. We inverse Fourier transform the filtered Fourier ring and calculate the ratio of the peak amplitude to the $\sigma_{f}$ from the filtered map. The filtering range $(\kmin,\kmax)$ is determined as the one from which the maximal enhancement $R$ can be reached for the filtered map.
Figure \[contour\] shows the surface $R$ as functions of $\kmin$ and $\kmax$ for the High Frequency Instrument (HFI) 143 GHz channel. The surface $R(\kmin,\kmax)$ for each frequency channel is morphologically similar, but the position for maximal $R$ can vary slightly from realization to realization owing to different lowest minimal values $-\nu_{\min}\sigma_{\rm in}$ of Eq. (\[eq:deepest\]) and the effect from cosmic variance. From the contour map we can easily see that, due to the flatness around the maximum, the filtering range covers roughly 10 per cent of $\kmin$ and $\kmax$ with only a few percent of variation in enhancement. We therefore optimize the filtering range from a set of realizations for each channel. In Table \[range\] we show the optimal sets of the filtering range in Fourier and corresponding spherical harmonic domain for all PLANCK frequency channels.
The choice of $\Re=3$ of the initial beam–convolved point source is to make sure the enhancement $R>5$ for all the PLANCK channels. Therefore, with the top–hat filtering algorithm, we can claim that the point sources extracted will have amplitudes above $3\sigma_{\rm in}$ in the pre-filtered map. Theoretically, we can apply peak statistics to confirm that the highest point in the filtered map for the suggested filtering range should be a point source. The number of peaks above the threshold $\nu_{\rm t}$ per steradian is $$\begin{aligned}
\lefteqn{N_{\max}(\nu_{\rm t})=\frac{\gamma^2}{(2\pi)^{3/2}\theta_*^2}
\nu_{\rm t}
\exp(-\frac{\nu_{\rm t}^2}{2}) +} \nonumber \\
&& \frac{1}{4\pi \sqrt{3} \theta_*^2}
\left\{ 1-
\Phi\left[\frac{\nu_{\rm t}}{2(1-2\gamma^2/3)^{1/2}}\right]\right\}\end{aligned}$$\[eq:peak\] where $\Phi$ is the error function, $\theta_*^2 =
2\sigma_1^2/\sigma_2^2$, and $\gamma=\sigma_1^2/\sigma_0
\sigma_2$ [@BE87]. The spectral parameters are defined as $$\sigma_i^2=\pi\int {\ell}^{2i+1} C_{\ell}^{\rm tot} d \ell.
\label{eq:spectralpara}$$ In order to find the threshold of the highest peak from the theoretical point of view, we put $A_{\rm f} N_{\max}(\nu_{\rm t})=
1$, where $A_{\rm f}$ is the area of the simulation patch for each channel and $\nu_{\rm t}$ is the threshold parameter above which there is only one peak. We exclude the point source contribution from the input power spectrum $C_{\ell}^{\rm tot}$ in Eq. (\[eq:spectralpara\]) so that the criterion for point source extraction should be set above the threshold from the calculation. The threshold before filtering $\nu_{\rm t}^{\rm i}$ and after filtering $\nu_{\rm t}^{\rm f}$ for the corresponding simulation patch of each channel are listed in Table \[range\]. According to the $\nu_{\rm t}^{\rm f}$ values for all channels, the threshold of the highest peak for the corresponding area after filtering by the suggested range are less than $5\sigma_{\rm f}$. Hence, the $5\sigma_{\rm f}$ is an appropriate criterion for point source extraction.
Figure \[illustration\] illustrates an example of point source enhancement of a realization from the HFI 143 GHz channel. Panel (a) shows the simulated map with one added point source of $\Re=3.21$, i.e., the amplitude of the beam–convolved point source is $3.21\;\sigma_{\rm in}$. In panel (b) we display the detailed part of the beam–convolved point source located at the deepest minimum of the map. We specifically choose the 143 GHz channel for presentation because the beam shape carried by the point source is not much deformed by the pixel noise level. This figure shows the enhancement is related to the size of FWHM, and the level of the pixel noise against the beam-convolved CMB plus foregrounds. As shown in Fig. \[powerspectrum1\] and Fig. \[powerspectrum2\], the bending of the power spectrum by the beam response can be preserved towards high multipole modes as long as the pixel noise level is low. The less the pixel noise level, the more towards the high multipole modes the filtering range shifts in order to include the bending, hence the more low multipole power is excluded in the filtered $\sigma_{\rm f}$, resulting in higher $R$. For larger FWHM (e.g. compare the power spectrum of the single point source at 143 GHz with 353 GHz channel in Fig. \[powerspectrum2\]), which nevertheless bends the total power spectrum, the filtering range has to shift towards low multipoles to keep the bending part, thus more power is included in the $\sigma_{\rm f}$, resulting in smaller $R$. Panel (c) is the filtered map after the filtering ($\kmin,\kmax$) = (42,86) with enhancement $R=15.5$.
Frequency $\kmin$ $\kmax$ $\lmin$ $\lmax$ $\nu_{\rm t}^{\rm i}$ $\nu_{\rm t}^{\rm f}$
----------- --------- --------- --------- --------- ----------------------- -----------------------
857 90 166 2539 4673 $3.45^{*}$ 4.56
545 74 146 2090 4111 $3.73^{*}$ 4.50
353 64 137 1810 3859 $3.85^{*}$ 4.46
217 53 116 1501 3269 $4.01^{*}$ 4.41
143 42 86 1192 2427 $4.11^{*}$ 4.25
100 (HFI) 68 137 960 1929 4.09 4.50
100 (LFI) 52 116 736 1634 4.31 4.39
70 40 93 566 1311 4.43 4.29
44 26 68 370 958 3.90 4.08
30 25 41 356 580 4.04 3.96
: The optimal filtering range for each PLANCK frequency channel. The multipole ranges are calculated according to the corresponding simulation sizes of the maps. As the surface of $R
\equiv R(\kmax,\kmin)$ can vary slightly from realization to realization due to different deepest minimal value of the realizations and the effect of cosmic variance, the suggested set of filtering range can be used as the initial filtering range $(\kmin,\kmax)^{(0)}$ for iteration schemes. The last two columns $\nu_{\rm t}^{\rm i}$ and $\nu_{\rm t}^{\rm f}$ are, before and after filtering, the theoretical values (in terms of $\sigma$) of the threshold above which there is only one peak in the corresponding area of the simulated patch. According to $\nu_{\rm t}^{\rm f}$ from all channels, the choice of $5\sigma_{\rm f}$ is appropriate for point source extraction. The sign \* in the 6th column denotes the values which are calculated from direct integration of (A1.9) in Bond & Efstathiou(1987).[]{data-label="range"}
We set the initial value $\Re=3$ for the estimation of the optimal filtering range $(\kmin,\kmax)$. For $\Re > 3$, the filter with the optimal filtering range will enhance the index $R$ even more. As tabulated in Table \[enhancement\], the enhancement for higher frequency channels can reach above 20, which means that with our proposed method we can extract point source amplitude much less than $3 \sigma_{\rm in}$ in the pre-filtered map. We can also estimate what level of the point source amplitude at each channel can be detected by the suggested filter. In Table \[enhancement\] we list the amplitudes of the point sources in terms of $\sigma_{\rm in}$ which can be filtered to reach the criterion $R=5$ with the suggested range.
Frequency $\Re$ $R_{\rm TH}$ $R_{\rm TD98}$ $R_{\rm MTD98}$ $\Re_{R=5}$
----------- ------- -------------- ---------------- ----------------- -------------
857 2.99 56.32 64.88 66.50 0.294
545 3.01 36.52 41.51 41.42 0.542
353 3.05 27.46 30.71 30.61 0.867
217 3.12 18.85 20.05 19.98 1.21
143 3.10 14.82 15.47 15.41 1.81
100 (HFI) 3.01 11.65 12.00 11.99 1.87
100 (LFI) 2.94 6.81 6.57 6.53 2.77
70 2.91 7.36 6.82 6.68 2.41
44 3.01 7.44 7.25 7.23 2.52
30 3.02 6.91 6.35 6.34 2.84
: Comparison of the enhancing capability for different filters. We simulate a set of realizations for each frequency channel. The second column shows the initial amplitude of the beam–convolved point source. The third column $R_{\rm TH}$ is the result from the top–hat filter. The fourth column $R_{\rm TD98}$ shows the TD98 filtering and the final $R_{\rm MTD98}$ is the modified TD98 filtering. The theoretically optimal TD98 filter gives better enhancement in HFI only when the FWHM of the beam, dust and noise power spectrum are correctly modelled. The last column lists the amplitudes of the point sources in terms of the square root of the variance of the total map which can be filtered to reach above $5\sigma_{\rm f}$ by the top–hat filter.[]{data-label="enhancement"}
Comparison with the TD98 filter
===============================
Using the top–hat filter with the optimal filtering range of $(\kmin,\kmax)$ we can compare its efficiency and accuracy of the point source extraction from the same realizations with the TD98 filter. The TD98 filter is expressed as $$W_k=\frac{B_k}{C_k^{\rm tot}}=\frac{B_k}{B_k^2 C_k+C^{\rm pix}},
\label{eq:TD98}$$ where $B_k$ is the beam, $C_k$ is the sum of the power spectrum of the CMB and foreground components and $C^{\rm pix}$ is the pixel noise power spectrum. The shape of the TD98 filter for each channel is shown with long dash line in Fig. \[powerspectrum1\] and Fig. \[powerspectrum2\], which also targets the bending of the total power spectrum. Although the TD98 filter is theoretically optimal, the trouble is that it requires the input of the CMB, and foregrounds power spectrum and the parameters of the experiments such as the beam size and shape, and the pixel noise power spectrum. As is claimed in TD98, the different inputs of cosmological models can have 20 per cent variation in $R$. In this regard, we also compare the standard (Eq. (\[eq:TD98\])) and the modified TD98 filter, which is mentioned briefly in their paper. The modified version is that, instead of inserting any theoretical CMB power spectrum from different cosmological models and power–law foregrounds into $C_{\ell}^{\rm tot}$ in the filter ($C_k^{\rm tot}$ in the flat sky approximation), we can simply insert $C_{\ell}^{\rm tot}$ from the observed map itself as long as the power spectrum of the CMB is not severely modified by point sources, which is the case when we only have one point source in the simulated map. This is to compensate any fluctuations caused by cosmic variance. The bonus of this is that the modified TD98 filter does not depend on any cosmological models. In Table \[enhancement\] we show from the top–hat, the TD98 and modified TD98 filter the enhancement factor of a $3 \sigma_{\rm in}$ point source at the deepest minimum of the map from a set of realizations. We can see that for LFI channels, the top–hat filter performs better than both TD98 and the modified TD98 filters at the [*worst*]{} situation. We would like to point out that the TD98 filter is optimal when targeting the mean $R$, i.e., when a few point sources are located at both above and beneath the mean level of the map. We therefore list also in Table \[enhancementinmean\] the enhancement factor $R$ by placing one point source at a random position of a map and calculate the mean enhancement factor for a set of realizations. It is shown that the TD98 filter performs better than the top–hat filter, indicating that the top–hat filtering is $\sim$13-15% below the optimal extraction of point sources. This would transfer to roughly 7-10% loss of point source extraction compared to the TD98 filter when all the parameters are known, such as the CMB, dust power spectrum, beam properties, and the noise level.
Frequency $R_{\rm TH}$ $R_{\rm TD98}$ $R_{\rm MTD98}$
----------- -------------- ---------------- ----------------- -- --
857 55.91 65.29 66.81
143 14.54 16.92 16.90
70 7.31 8.31 8.29
30 6.94 8.24 8.21
: Comparison of the enhancing capability for the filters on point sources at random positions. Here we show two channels for both LFI and HFI. We put one point source with $\Re=3$ at a random position of a realization and calculate the mean enhancement factor $R$ from a set of realizations. The second column is the result from the top–hat filter with the suggested filtering range in Table \[range\]. The third column $R_{\rm TD98}$ shows the TD98 filtering and the final $R_{\rm MTD98}$ is the modified TD98 filtering.[]{data-label="enhancementinmean"}
To compare from the theoretical point of view the enhancement factor in details between these two filters for a single point source, we need the information of the location and amplitude of the point source. Specifically when we place one point source at the deepest minimum, this situation favours the top–hat filter, as the top–hat filter is ‘designed’ for this situation and similar situations such as other local minima. Also the amplitude of the point source is crucial in the enhancement factor $R$. Though from Table \[enhancement\] for the HFI channels the $R$ from TD98 filter is higher, it does become smaller than that from the top–hat filter when the point source amplitude is significantly smaller than $3\sigma_{\rm in}$.
Generalization of the top–hat filter {#iteration}
====================================
The determination of the top–hat filtering range for each channel is sensitive to the CMB power spectrum model, pixel noise properties, beam shape and foregrounds contamination. However, for the proposed ERCSC construction it is absolutely necessary to simplify the method of the point source detection from the map, using general characteristics of the ‘noise’ and point sources only. We would like to point out specifically that the point source filter with adaptive range and a fixed shape, such as the top–hat filter is a much simpler way to define such kind of filters. As the concept of the top–hat and the TD98 filter to extract point sources takes advantage of the bending on the power spectrum by the beam response and the pixel noise level, they are both useful for the regions near the ecliptic plane in the flat sky approximation, where the scans are nearly parallel without crossings (see the fig.1 of Delabrouille, Patanchon & Audit 2002). For high galactic latitude scans, however, the crossings of scans complicate the beam shape configuration. The effective asymmetric beam will manifest itself in the Fourier domain which is not isotropic in the Fourier ring (see the Fig.1 of Chiang et al. 2001). It is unknown whether the global beam orientation is fixed (parallel) in the scans of PLANCK. If it is not, it will have degradation effect on point source extraction from any input of fixed beam function, such as the TD98 filter and MHW method.
To tackle this problem and also to relax the condition we set earlier in Section \[estimation\] to determine the filtering range, we would like to expand the top–hat filter to a more generalized algorithm. Following Naselsky, Novikov & Silk (2002), we can apply the iteration scheme for point source extraction from the map, using as the initial step of iterations the suggested $(\lmin,\lmax)$ parameters from Table \[range\]. Without acquiring the exact characteristics of the experiment such as pixel noise level and beam shape and size, the initial filtering with the suggested set of $(\lmin, \lmax)$ may not result in maximal enhancement for that specific realization. We can, however, always find the highest peak in the filtered map after the first top–hat filtering. There are two possibilities for the value of the amplitude of such peak. If we choose the criterion for the point source detection as $5 \sigma_{\rm f}$, it is likely that the actual value of the amplitude for the highest peak satisfies this criterion, from which we can claim we have identified a point source that can be removed easily from the map. If the highest peak is less than $5\sigma_{\rm f}$, we can make the second iteration by slightly fine–tuning the $\lmin$ and $\lmax$ parameter. For the highest peak $< 5\sigma_{\rm f}$ after the first iteration, if we still cannot increase its filtered amplitude up to the criterion after fine-tuning, we can claim that it is not a point source. Of course, for higher frequency channels of HFI we can set the criterion lower than $5\sigma_{\rm f}$ as shown in Table \[enhancement\], which means we can detect point sources with $\Re<3$.
The principal idea is that, with an adaptive filtering range, we can change $\kmax$ and $\kmin$ step by step to check the change of $R$ of the highest peak. In each iteration we need to compare the amplitude of the highest peak (after filtering) with the criterion for point source detection. For this algorithm, suppose that for the 100 GHz channel of HFI we put a point source with $\Re=3$ at an unknown position in the map and the pixel noise level is twice the predicted value. When we apply the filter with the suggested range, we can detect the highest point, which is less than $5\sigma_{\rm f}$. By tuning the $\kmax$ with fixed $\kmin$ to re-filter the original map, we can always find the enhancement $R$ by going through a bump along $\kmax$. The position of the highest peak in the filtered map is the same as the initial iteration, but with different $R$, which indicates our suggested filtering range is not optimal. Then we fix the $\kmax$ corresponding to the maximum $R$ of the bump, re-filtering along the $\kmin$ axis to find the $\kmin$ corresponding to the maximal $R$. Through this process we can always find the new $\kmin$ and $\kmax$ parameter which has the maximal enhancement $R$.[^4]
Discussions and Conclusion
==========================
We have introduced a simple and fast top–hat filter for extraction of point sources in CMB maps. As the filter cut–off range is adaptive, we can estimate it with the simulated maps when the parameters of the PLANCK channels are known.
We would like to emphasise that the main shape of the antenna beam for the PLANCK mission is close to elliptical due to optical distortions and telescope designs [@burigana2000]. The orientation of the beam, therefore, is crucial for any methods of point source extraction, which should be taken into account in order for maximal extraction of point sources. Our top–hat iteration algorithm, however, does not need this part of information and it will be even more useful if there is change of the beam shape due to the degradation effects of the mirrors during the mission.
We also perform detailed comparison of the proposed top–hat algorithm with the TD98 filter. The advantage of our method is that it is very simple, fast and does not require any detailed information about the real beam shape, the spectra of CMB and noise, and possible correlations in the pixel noise. In practice one can take the ready filter with the suggested range from Table \[range\] for each PLANCK frequency channel (if desirable) to improve it by the simple and fast algorithm described in Section \[iteration\]. We would like to mention that the efficiency $R$ for the top–hat filter and the TD98 filter shown in Table \[enhancement\] are practically the same.
Acknowledgments {#acknowledgments .unnumbered}
===============
This paper was supported in part by Danmarks Grundforskningsfond through its support for the establishment of the Theoretical Astrophysics Center and by grants RFBR 17625 and INTAS 97-1192. We thank Dmitri Novikov for useful discussions and remarks.
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[^1]: The values $lmin$ and $\lmax$ can be used as the first step of the iteration scheme introduced in Section \[iteration\] to maximize the $R$ factor when the parameters have considerable variations against predicted values.
[^2]: by ‘noise’ we mean the filtered CMB plus foregrounds and pixel noise
[^3]: The deepest minimum at the initial map as a rule is quite isolated (see Zabotin & Naselsky 1985, Bond & Efstathiou 1987, Coles & Barrow 1987). The probability of finding such a realization of the point source and ‘noise’ is negligibly small. Thus the description ‘different locations of the map’ means different (and most probable) realization of the ‘noise’ at the point source area.
[^4]: In this example the $R$ is now lower than that listed in Table \[range\] as the pixel noise is doubled.
|
---
abstract: 'Predator-prey relationships are one of the most studied interactions in population ecology. However, little attention has been paid to the possibility of role exchange between species once determined as predators and preys, despite firm field evidence of such phenomena in the nature. In this paper, we build a model capable of reproducing the main phenomenological features of one reported predator-prey role-reversal system, and present results for both the homogeneous and the space explicit cases. We find that, depending on the choice of parameters, our role-reversal dynamical system exhibits excitable-like behaviour, generating waves of species’ concentrations that propagate through space.'
author:
- 'Faustino Sánchez-Garduño'
- Pedro Miramontes
- 'Tatiana T. Márquez-Lago [^1]'
title: 'Role reversal in a predator-prey interaction'
---
Introduction {#intro}
============
In 1988, A. Barkai and C.D McQuaid reported a novel observation in population ecology while studying benthic fauna in South African shores [@barkai]: a predator-prey role reversal between a decapod crustacean and a marine snail. Specifically, in Malgas Island, the rock lobster [*Jasus lalandii*]{} preys on a type of whelk, [*Burnupena papyracea*]{}. As could be easily expected, the population density of whelks soared upon extinction of the lobsters in a nearby island (Marcus island, just four kilometers away from Malgas). However, in a series of very interesting controlled ecological experiments, Barkai and McQuaid reintroduced a number of [*Jasus lalandii*]{} in Marcus Island, to investigate whether the equilibrium observed in the neighboring Malgas Island could be restored. The results were simply astounding:
> “The result was immediate. The apparently healthy rock lobsters were quickly overwhelmed by large number of whelks. Several hundreds were observed being attacked immediately after release and a week later no live rock lobsters could be found at Marcus Island.”
Surprisingly, and despite observations such as the report in [@barkai], Theoretical Population Biology has largely ignored the possibility of predators and preys switching their roles. Of importance, the paper of Barkai and McQuaid suggests the existence of a threshold control parameter responsible for switching the dynamics between (a) a classical predator-prey system with sustained or decaying oscillations, and (b) a predator (the former prey) driving its present-day prey to local extinction.
It is worth noting there are some papers in the literature describing ratio-dependent predation (see, for example [@seitz] and [@holbrook]), but they are not related to the possibility of role-reversals. On the other hand, the likelihood of changing ecological roles as a result of density dependence has already been documented for the case of mutualism by Breton [@breton] and, in 1998, Hernández made an interesting effort to build a mathematical scheme capable of taking into account the possible switches among different possible ecological interactions [@hernandez]. So, to the best of our knowledge, there are no theoretical studies –supported by field evidence– specifically addressing predator-prey role-reversals yet.
Mathematical model
==================
Predator-prey systems are generally modeled by adopting one of the many variations of the classical Lotka-Volterra model:
$$\begin{aligned}
\label{E0}
\dot{x} & = \alpha x-\beta xy\nonumber\\
\dot{y} & =-\gamma y+\delta xy,\end{aligned}$$
where $\alpha$ denotes the intrinsic preys’ rate of growth, $\beta$ corresponds to the rate of predation upon preys, $\gamma$ stands for the predators’ death rate in absence of preys, and $\delta$ represents the benefit of predators due to the encounters with preys. Our goal is to assess whether modeling the role-reversal behavior observed by Barkai & McQuaid [@barkai] is possible, when adopting appropriate parameters and assumptions.
For instance, if one considers quadratic density dependence in the preys as well as in the predators, non-constant rates of consumption of preys by the predators, and the profiting of predators by the existence of preys, then it is possible to suggest the following system:
$$\begin{aligned}
\label{E1}
\begin{array}{ccc}
\dot{x} & = & Bx(A/B-x)-C(x)xy\\
\dot{y} & = & -Dy-Ey^2+F(x)xy,
\end{array}\end{aligned}$$
where $B$ represents the intrinsic growth rate of the prey in the absence of predators, $A/B$ the carrying capacity of the prey’s habitat, $C(x)$ the rate of preys consumption by the population of predators, $D$ the predators’ decay rate in the absence of preys, $E$ the intraspecific rate of competition among predators and, finally, $F(x)$ the factor of predator’s profiting from preys. The ratio $F(x)/C(x)$ is then the fraction of prey biomass that is actually converted into predator biomass. The latter should remain constant, since the fraction of preys’ biomass converted to predators’ biomass is a physiological parameter, rather than a magnitude depending on demographical variables.
Thus, a particular case of system (\[E1\]) in the appropriate rescaled variables is:
$$\label{E2}
\begin{array}{ccc}
\dot{x} & = & bx(1-x)-cx(k-x)y\equiv f(x,y)\\
\dot{y} & = & -ey(1+y)+fx(k-x)y\equiv g(x,y),
\end{array}$$
where all the parameters are positive and $0<k<1$. In fact, all of the parameters have a relevant ecological interpretation: $b$ is the normalized intrinsic growth rate of the species with density $x$, $c$ is a measure of the damage intensity of the second species on the first one, $e$ is the normalized rate of predators decay and $f$ is the benefit (damage) the second population gets from the first one. Note the crucial role played by the interaction term $x(k-x)$, where $k$ stands for the first population threshold to switch from being prey to predator.
Phase portrait analysis
=======================
Nullclines and equilibria
-------------------------
The horizontal nullcline of the system of equations (\[E2\]), that is $[b(1-x)-c(k-x)y]x=0$, has two branches: the vertical axis and the nontrivial branch
$$\label{E3}
y_h(x)=\frac{b(1-x)}{c(k-x)},$$
which is a symmetric hyperbola with asymptotes: $x\equiv k$ and $y=b/c$ (See Figure 1).
\[fig:1\] ![[Nullclines of the system of equations (\[E2\]). The dotted line and the vertical axis are the horizontal nullcline. The continuous downward facing parabola and the horizontal axis are the vertical nullcline. $P_1$, $P_2$ and $P_3$ are the non-trivial equilibria. The origin of coordinates is a trivial equilibrium.]{}](Figure1.jpg "fig:"){width="120mm" height="95mm"}
The vertical nullcline, $[-e(1+y)+fx(k-x)]y=0$, also has two branches: the horizontal axis and
$$\label{E4}
y_v(x)=\frac{fx(k-x)}{e}-1,$$
which is a parabola with $y_v(0)=y_v(k)=-1$, attaining its maximum at $x=k/2$, the value of which is
$$y_v(k/2)=\frac{fk^2}{4e}-1.$$
This term is positive if and only if $fk^2>4e$. The zeros, $x_1$ and $x_2$ of Equation (\[E4\]) are given by
$$x_1,x_2=\frac{f\pm\sqrt{f^2k^2-4fe}}{2f}.$$
The latter are real numbers if and only if $fk^2\geq 4e$. The rate of change of $y_v$ is then
$$y'_v(x)=\frac{f}{e}(k-2x).$$
To analyze the system while keeping in mind the ecological interpretation of the variables and parameters, we will now consider the left branch of the horizontal nullcline (\[E3\]), $fk^2>4e$ with $0<k<1$ and the region of the phase plane of the system of equations (\[E2\]) defined as
$$\mathcal R=\left\{(x,y)|0\leq x\leq 1,\;\;0\leq y<+\infty\right\}.$$
The system of equations (\[E2\]) has the equilibria: $P_0=(0,0)$, $P_{1}=(1,0)$ plus those states of the system stemming from the intersection of the nullclines $y_h$ and $y_v$ in the region $\mathcal R$. Such equilibria are defined by the $x$ in the interval $(0,1)$ satisfying the identity
$$\label{E5}
\frac{b(1-x)}{c(k-x)}=\frac{fx(k-x)}{e}-1,$$
or, equivalently, the $x$ that are roots of the third order polynomial
$$\label{E6}
F(x)=Ax^3+Bx^2+Cx+D,$$
where $A=fc$, $B=-2fck$, $C=fck^2+be+ec$ and $D=-be-eck$.
The calculation of the nontrivial equilibria of (\[E2\]) follows from the determination of the roots of (\[E6\]). Consequently, due to the qualitative behavior of the functions $y_h$ and $y_v$ on $\mathcal R$, we are faced with the following possibilities:
1. The nontrivial branches of the nullclines do not intersect each other in the region of interest. In such a case, the system (\[E2\]) has just two equilibria: $P_0$ and $P_1$ in ${\mathcal R}$. Figure 2a shows the relative position of the nullclines in this case, and Figure 3a the phase portrait of the system.
For fixed positive values of $b,c,e$ and $f$, and $k\in(0,1)$ such that $(fk^2/4e)>1$ one can see that both nullclines become closer with increasing values of $k$.
{width="2.4in"} \[fig:1a\]
{width="2.4in"} \[fig:1b\]
{width="2.4in"}
{width="2.4in"}
2. The nullclines $y_h$ and $y_v$ touch each other tangentially at the point $P^{*}=(x^*,y^*)$ in the region ${\mathcal R}$. Again, Figure 2b shows the relative position of the nullclines in this case, and Figure 3b the phase portrait of the system. In such a case $x^*$, in addition to satisfy (\[E5\]), must also satisfy the condition $y_h'(x)=y_v'(x)$ [*i.e.*]{},
$$\label{E7}
\frac{b}{c}\frac{(1-k)}{(k-x)^2}=\frac{f}{e}(k-2x).$$
If one assumes the existence of $x^*$ satisfying (\[E5\]), the required extra condition (\[E7\]) imposes the restriction $0<x^*<k/2$ on $x^*$, due to the positiveness of its left hand side. Moreover, from a geometrical interpretation of (\[E7\]) it follows that:
- \(i) If
$$\frac{fk}{e}<\frac{b}{c}\frac{(1-k)}{k^2},$$ there is not any $x\geq 0$ such that $y'_h(x)=y'_v(x)$.
- \(ii) If $$\frac{fk}{e}=\frac{b}{c}\frac{(1-k)}{k^2},$$ the condition (\[E7\]) is satisfied just at $x=0$.
- \(iii) If $$\frac{fk}{e}>\frac{b}{c}\frac{(1-k)}{k^2},$$ there exists exactly one value, $x^{*}\in(0,k/2)$, of $x>0$ such that the equality (\[E7\]) holds.
In any case, the point $P^*$ is a [*non-hyperbolic*]{} equilibrium of the system of equations (\[E2\]). In fact, the proof that a tangential contact of the nullclines results in a point where the determinant of the Jacobian matrix of the system vanishes follows immediately, implying that at least one of its eigenvalues is zero.
3. The nullclines intersect each other transversally at two points, $P_2$ and $P_3$, belonging the region ${\mathcal R}$. For reference, please refer to Figure 1. In this case the system of equations (\[E2\]) has two extra equilibria which arise from the [*bifurcation*]{} of $P^*$.
Here, if in addition to choosing the parameters $f$, $k$ and $e$ such that $fk^2/4e>1$, we select the rest of them such that:
- \(i) $y_h(k/2)>y_v(k/2)$, i.e.,
$$\left[\frac{fk^2}{4e}-1\right]>\frac{b}{c}(2-k),$$
guaranteeing the existence of the equilibria $P_2=(\tilde{x}_2,\tilde{y}_2)$ and $P_3= (\tilde{x}_3,\tilde{y}_3)$ above mentioned. Moreover, the coordinates of these points satisfy $0<\tilde{x}_2<k/2$, $k/2<\tilde{x}_3<k$, $\tilde{y}_i>0$ with $\tilde{y}_2<\tilde{y}_3$. Here $i=2,3$.
- \(ii) $y_h(k/2)=y_v(k/2)$, i.e., $$\left[\frac{fk^2}{4e}-1\right]=\frac{b}{c}(2-k).$$
Here we have $P_2=(\tilde{x}_2,\tilde{y}_2)$ with $0<\tilde{x}_2<k/2$ and $0<\tilde{y}_2<\frac{b}{c}(2-k)$. Meanwhile, $P_3= (k/2,\frac{b}{c}(2-k))$.
Local dynamics
--------------
Part of the local analysis of the system of equations (\[E2\]) is based on the linear approximation around its equilibria. Thus, we calculate the Jacobian matrix of the system (\[E2\]):
$$\label{E9}
J[f_1,f_2]_{(x,y)} = \left[\begin{array}{cc}
b(1-2x)-cy(k-2x) & -cx(k-x)\\
fy(k-2x) & -e-2ey+fx(k-x)\end{array}
\right].$$
By a straightforward calculation, we obtain the eigenvalues of the Jacobian matrix (\[E9\]) at the point $P_0$. These are: $\lambda_1=b>0$ and $\lambda_2=-e<0$. Hence, $P_0$ is saddle point of the system (\[E2\]), for all positive parameter values. By carrying out similar calculations we obtain the corresponding eigenvalues of matrix (\[E9\]) at $P_1$, which are: $\lambda_1=-b<0$ and $\lambda_2=f(k-1)-e$. The restriction $0<k<1$ on $k$ implies that $(f(k-1)-e)<0$. Therefore, $P_1$ is an asymptotically stable node for all the positive parameter values appearing in system (\[E2\]).
Now we carry out the local analysis of (\[E2\]). We notice two cases, depending on the relative position of the nullclines:
[**Case 1.**]{} The main branches (\[E3\]) and (\[E4\]) of the nullclines do not intersect on ${\mathcal R}$.
Here, any trajectory of system (\[E2\]) starting at the initial condition $(x_0,y_0)$ with positive $x_0$ and $y_0$ tends to the equilibria $P_0$ as time goes to infinity. Thus, the region ${\mathcal R}^{+}$ is the basin of attraction of $P_1$. Invariably, the species with density $y$ vanishes, implying non coexistence among the interacting species. Meanwhile, the other species approach the associated carrying capacity.
[**Case 2.**]{} The nullclines intersect each other at the points $P_2=(\tilde{x}_2,\tilde{y}_2)$ and $P_3=(\tilde{x}_3,\tilde{y}_3)$, where none is tangential. Here $\tilde{x}_2$ and $\tilde{x}_3$ satisfy $0<\tilde{x}_2<k/2$ and $k/2<\tilde{x}_3<k$. In a neighborhood of $P_2$ and $P_3$, the functions $f_1$ and $f_2$ satisfy the Implicit Function Theorem. In particular, each one of the identities $f_1(x,y)=0$ and $f_2(x,y)=0$ define a function there. Actually, these are $y_h(x)$ and $y_v(x)$ given in (\[E3\]) and (\[E4\]), respectively. Their derivative at $\tilde{x}_i$ with $i=2,3$ is calculated as follows
$$y_h'(\tilde{x}_i)=-\frac{f_{1x}(\tilde{x}_i,\tilde{y}_i)}{f_{1y}(\tilde{x}_i,
\tilde{y}_i)}\;\;\mbox{and}\;\;y_v'(\tilde{x}_i)=-\frac{f_{2x}(\tilde{x}_i,
\tilde{y}_i)}{f_{2y}(\tilde{x}_i,\tilde{y}_i)}.$$
By using these equalities, we can state the following proposition.
[**Proposition 1.**]{} [*The equilibrium $P_2$ is not a saddle point. Meanwhile, the equilibrium $P_3$ is a saddle point for all the parameter values.*]{}
A proof of this proposition and some remarks can be found in Appendix A.
Global analysis
---------------
As we have already shown, system (\[E2\]) has four equilibrium points. These are illustrated in Figure 4. The origin is a saddle point with the horizontal and vertical axis as its unstable and stable manifolds. $P_1$ and $P_2$ are, respectively, a node and a saddle for parameter values after the bifurcation, and $P_3$ is a stable node. The stable manifold of the saddle point is a separatrix dividing the phase space in two disjoint regions: the set of initial conditions going to $P_1$, and the complement with points going to $P_3$. Moreover, our numerical solution shows the existence of an homoclinic trajectory starting and ending in the saddle point. Thus, we have a bistable system.
![[The equilibria of system (\[E2\]). From left to right: $P_2$ is a stable node, $P_3$ is a saddle and $P_1$ is another stable node. The heteroclinic trajectory joining the saddle point to the stable node is easily identified. The stable manifold is a separatrix between the basin of attraction of $P_1$ and $P_2$.]{}](Figure4.jpg){width="120mm" height="95mm"}
The bistability of system (\[E2\]) has an interesting ecological interpretation: the coexistence of the interacting species occurs whenever the initial population densities $(x_0,y_0)$ are located in the region above the saddle point unstable manifold. In this case, both populations evolve towards the attractor $P_2$.
On the other hand, if the initial population densities $(x_0,y_0)$ are below the separatrix, the population densities $(x(t),y(t))$ evolve towards the equilibria $P_1$ implying the non-coexistence of the species and, invariably, the species with population density $y$ vanishes. The heteroclinic trajectory of system (\[E2\]) connecting the saddle ($P_3$) with the node ($P_2$ – or focus, depending on the set of parameters), in addition to the coexistence of the species, also tells us that this occurs by the transition from one equilibrium to another as time increases.
Spatial dynamics
================
To describe more accurately our role-reversal system, we extended our model of system (\[E2\]) to incorporate the spatial variation of the population densities. Here, if we denote by $u(\vec{r},t)$ and $v(\vec{r},t)$ the population density of the whelks and lobsters at the point $\vec{r}$ at time $t$, the resulting model is:
$$\begin{aligned}
\label{E10}
\begin{array}{ccc}
u_t & = & D_u\nabla^2u+bu(1-u)-cu(k-u)v\\
v_t & = & D_v\nabla^2v-ev(1+v)+fu(k-u)v,
\end{array}\end{aligned}$$
where the subscript in $u$ and $v$ denotes the partial derivative with respect the time, and $\nabla^2$ is the laplacian operator. Here, $D_u>0$, $D_v>0$ correspond to the diffusivity of the species with density $u$ and $v$, i.e. that of whelks and lobsters, respectively. It is worth noting that the original variables have been rescaled, but still denote population densities.
We then proceeded to construct numerical solutions of the system (10) in three different domains: a circle with radius 2.2 length units (LU), an annulus defined by concentric circles of radii 2.2 LU and 1 LU, and a square with side length of 4.6 LU. All domains were constructed to depict similar distances between Malgas island and Marcus island (roughly 4km). In the first one, the annular domain, we try to mimic the island habitat of whelks and lobsters as a concentric domain. The other two domains are used to confirm the pattern formation characteristic of excitable media, and to reject any biases from the shape of the boundaries.
To obtain numerical solutions of all spatial cases, we used the finite element method with adaptive time-stepping, and assumed zero-flux boundary conditions. Accordingly, we discretized all spatial domains by means of Delaunay triangulations, until a maximal side length of 0.17 was obtained. The latter defines the approximation error of the numerical scheme. We attempted to describe two entirely different situations by using a single set of kinetic parameters: that of Malgas island, where both species co-exist, and Marcus island, where whelks soar and lobsters become extinct. The only difference between these two cases was the initial conditions used.
Aside, one could intuitively assume whelks motion to be very slow, or even negligible in comparison to that of lobsters. However, it is worth considering how slow, and whether fluid motion could aftect this speed. While there is no data specific to [*Jasus lalandii*]{} and [*Burnupena papyracea*]{} in islands of the Saldanha Bay, data of similar species can be found in the literature. For instance, a related rock-lobster species, [*Jasus Edwardii*]{} has been found to move at a rate of 5-7 km/day [@cobb]. In contrast, whelks within the superfamily [*Buccinoidea*]{} have been found to move towards food at rates between 50 and 220 meters/day (see [@himmelman] and [@lapointe]). Importantly, predation by whelks remains seemingly unaffected by variations in water flow [@powers].
By putting these findings together, we argue a reasonable model need not incorporate influences from shallow water currents, and would assume whelks to move toward ‘bait’ at a speed roughly one order of magnitude smaller than that of lobsters. Thus, we opted for a two-dimensional habitat, and one order of magnitude difference between the non-dimensional isotropic diffusion rates ($D_u = 0.01$ and $D_v = 0.1$). Aside, our choice of reaction parameters was: $b = 10$, $c = 33.8$, $e = 0.5$, $f = 30$, and $k = 0.9$. Regarding initial conditions, we adopted the following scenarios, representing the different scenarios of weighted biomass:
1. Malgas Island: the initial density of whelks at each element was drawn from a uniform distribution 0.1 \* U(0.25, 0.05), and that of lobsters from U(0.25, 0.05).
2. Marcus Island: the initial density of whelks at each element was drawn from a uniform distribution U(0.25, 0.05), and that of lobsters from 0.1 \* U(0.25, 0.05).
Results are shown in Figure 5, corresponding to averaged densities of whelks and lobsters in the three different spatial domains, respectively. Simulations in an annular domain can be found in the Supplementary Material.
![[Time evolution of whelk and lobster densities, in different spatial domains. Cases (a,c) correspond to initial conditions representing Malgas island, while (b,d) correspond to initial conditions representing Marcus island.]{}](Figure5.png){width="120mm" height="95mm"}
Interestingly, changes in density are usually accompanied with wave-like spatial transitions in each species density. Examples of this spatial transient patterns can be found in Figures 6 and 7, for annular and rectangular domains in Malgas island and Marcus island, respectively.
![[Time evolution of whelks’ densities using initial conditions representing Malgas island, in annular (top) and rectangular (bottom) domains. Cases correspond to t = 0, 2, 6 and 10, from left to right.]{}](Figure4_mod.png)
![[Time evolution of whelks’ densities using initial conditions representing Marcus island, in annular (top) and rectangular (bottom) domains. Cases correspond to t = 0, 0.25, 0.5 and 1, from left to right.]{}](Figure5_mod.png)
Discussion and final remarks
============================
We have modeled a well documented case of role-reversal in a predator-prey interaction. Our model pretends to capture the essential ecological factors within the study of Barkai and McQuaid [@barkai], who did an extraordinary field work and meticulously reported this striking role-reversal phenomenon happening between whelks and lobsters in the Saldanha Bay.
The analysis of our model and corresponding numerical solutions clearly predict the coexistence of both populations and the switching of roles between the once denoted predators and preys. Here, the coexistence scenario corresponds to the case when lobsters predate upon whelks, and role-reversal corresponds to the case when whelks drive the population of lobsters to extinction, as observed by Barkai and McQuaid in the field.
Moreover, by introducing spatial variables and letting both populations diffuse within a spatial domain, we obtain patterns that are characteristic of excitable media [@krinsky]. Of particular interest is the upper row of Figure 6, where self-sustained waves travel in the annular region. The latter is not entirely surprising, as the ordinary differential equation model in which the spatial case was based shows bistability. Nevertheless, our findings are novel in that, to the best of our knowledge, there are no reports of ecological interactions behaving as excitable media.
Acknowledgements
================
PM was supported by UNAM-IN107414 funding, and wishes to thank OIST hospitality during last stages of this work. TML was supported by OIST funding.
Appendix A
==========
[**Proof.**]{} First we prove the second part of our Proposition. At $P_3$ we have
$$y_{h}'(\tilde{x}_{3})=-\frac{f_{1x}(\tilde{x}_3,\tilde{y}_3)}{f_{1y}(\tilde{x}_3,\tilde{y}_3)}>0,$$ then the partial derivatives $f_{1x}(\tilde{x}_3,\tilde{y}_3)$ and $f_{1y}(\tilde{x}_3,\tilde{y}_3)$ have opposite signs at that point but $f_{1y}=-c\tilde{x}_3(k-\tilde{x}_3)$ with $k/2<\tilde{x}_3<k$ resulting in $f_{1y}(\tilde{x}_3,\tilde{y}_3)<0$ hence $f_{1x}(\tilde{x}_3,\tilde{y}_3)>0$. Similarly
$$y_{v}'(\tilde{x}_3)=-\frac{f_{2x}(\tilde{x}_3,\tilde{y}_3)}{f_{2y}(\tilde{x}_3,\tilde{y}_3)}<0,$$ implying that $f_{2x}$ and $f_{2y}$ have the same sign at $(\tilde{x}_3,\tilde{y}_3)$ but $f_{2x}(\tilde{x}_3,\tilde{y}_3)=f(k-2\tilde{x}_3)\tilde{y}_3$ with $k/2<\tilde{x}_3<k$ and $\tilde{y}_3>0$, then $f_{2x}(\tilde{x}_3,\tilde{y}_3)<0$ and $f_{2y}(\tilde{x}_3,\tilde{y}_3)<0$. By using the above calculations we obtain $f_{1x}f_{2y}<0$ and $f_{2x}f_{1y}>0$ then, the determinant of the Jacobian matrix of the system (\[E2\]) at $P_3$
$$detJ[f_1,f_2]_{(\tilde{x}_3,\tilde{y}_3)}=(f_{1x}f_{2y}-f_{2x}f_{1y}),$$ is negative. Therefore $P_3$ is a saddle point of (\[E2\]) for all the parameter values.
For the proof of the first part of the Proposition we follow a similar sign analysis as we did previously, by considering that $0<\tilde{x}_2<k/2$, $\tilde{y}_2>0$ and that at $\tilde{x}_2$ the inequality $y_h'(\tilde{x}_2)<y_v'(\tilde{x}_2)$ holds where both derivatives are positive. Thus, given that $f_{1y}(\tilde{x}_2,\tilde{y}_2)=-c\tilde{x}_2(k-\tilde{x}_2)$ and $f_{2x}(\tilde{x}_2,\tilde{y}_2)=f(k-2\tilde{x}_2)\tilde{y}_2$ the inequalities
$$f_{1y}<0,\;\;f_{1x}>0\;\;\mbox{and}\;\;f_{2x}>0,\;f_{2y}<0,$$ follow, from which we get $f_{1x}f_{2y}<0$ and $f_{2x}f_{1y}<0$. By using these inequalities and the condition $y_h'(\tilde{x}_2)<y_v'(\tilde{x}_2)$ one obtains
$$detJ[f_1,f_2]_{(\tilde{x}_2,\tilde{y}_2)}=(f_{1x}f_{2y}-f_{2x}f_{1y})>0,$$ with this we complete the proof. $\;\;\;\Diamond $
[**Remark 1.**]{} The trace of the Jacobian matrix (\[E9\]) at any point $(x,y)$ is the quadratic
$$\label{E11}
trJ[f_1,f_2]_{(x,y)}= -fx^2+2cxy+(fk-2b)x-(ck+2e)y+b-e.$$
Given that its discriminant[^2] $\Delta=AC-B^2=-c^2<0$, (\[E11\]) is a quadratic equation of hyperbolic type. In order the see more details of such quadratic, we calculate its gradient. This is the zero vector at the point
$$(\hat{x},\hat{y})=\left(\frac{ck+2e}{2c},\frac{fe+bc}{c^2}\right)\in{\mathcal R}.$$ Given that the partial derivatives[^3]
$$\frac{\partial^2 trJ}{\partial x^2},\;\;\;\;\frac{\partial^2
trJ}{\partial y\partial x}\;\;\;\mbox{and}\;\;\;\frac{\partial^2
trJ}{\partial y^2}$$ evaluated at any point $(x,y)$ —in particular $(\hat{x},\hat{y})$— are: $-2f<0$, $2c>0$ and $0$ respectively, then we have
$$\left[\frac{\partial^2 trJ}{\partial x^2}\right]\left[\frac{\partial^2 trJ}{\partial y^2}\right]-\left\{\frac{\partial^2 trJ}{\partial y\partial x}\right\}^2=-4c^2<0.$$ Therefore $(\hat{x},\hat{y})$ is a saddle point of the surface (\[E10\]). The value of $trJ[f_1,f_2]$ at $(\hat{x},\hat{y})$ is
If, for the parameter values we prove $trJ[f_1,f_2]_{(\tilde{x}_2,\tilde{y}_2)}<0$ in addition to what we already have had proved, our conclusion follows.
[99]{}
A. Barkai, and C. McQuaid, (1988), Predator-Prey role reversal in a marine benthic ecosystem. Science, 242(4875): 62-64.
R. D. Seitz, R. N. Lipcius, A.H. Hines, and D.B. Eggleston, (2001). Density-dependent predation, habitat variation, and the persistence of marine bivalve prey. Ecology, 82: 2435-2451.
S. Holbrook, and R. J. Schmitt (2002). Competition for shelter space causes density-dependent predation mortality in damselfishes. Ecology, 83: 2855-2868.
L. M. Breton, and F. Addicott, (1992), Density-Dependent Mutualism in an Aphid-Ant Interaction. Ecology, 73(6): 2175-2180.
M. Hernandez, I. Barradas, (2003), Variation in the outcome of population interactions: bifurcations and catastrophes. Journal of Mathematical Biology, 46(6), 571-594.
J. S. Cobb, B. F. Phillips, (1980), The biology and management of lobsters. Volume I: Physiology and behaviour. Academic Press.
J. H. Himmelman, (1988), Movement of whelks ([*Buccinum undatum*]{} towards a baited trap. Marine Biology, 97: 521-531.
V. Lapointe, B. Sainte-Marie, (1992), Currents, predators, and the aggregation of the gastropod ([*Buccinum undatum*]{} around bait, Marine Ecology Progress Series, 85: 245-257.
S. S. Powers, J. N. Kittinger, (2002), Hydrodynamic mediation of predator-prey interactions: differential patterns of prey susceptibility and predator success explained by variation in water flow. Journal of Experimental Marine Biology and Ecology, 85: 245-257.
V. Krinsky and H. Swinney (eds), Wave and patterns in biological and chemical excitable media, (North-Holland, Amsterdam, 1991).
[^1]: Corresponding author: [email protected]
[^2]: For the calculation of the discriminant we consider the general form of the quadratic
$$Ax^2+2Bxy+Cy^2+Dx+Ey+F=0,$$ hence $A=-f$, $B=c$ and $C=0$.
[^3]: For notational convenience we simply write $tr J$ instead of $trJ[f_1,f_2]_{(x,y)}$.
|
math\_macros =cmbx10 =cmr10 =cmti10 =cmbx10 scaled1 =cmr10 scaled1 =cmti10 scaled1 =cmbx9 =cmr9 =cmti9 =cmbx8 =cmr8 =cmti8 =cmr7
6.0in 8.5in -0.25truein
=1.5in
[APPLYING OPTIMIZED PERTURBATION THEORY TO\
QCD AT LOW ENERGIES\
]{}
1.0cm [A. C. Mattingly and P. M. Stevenson\
]{} 0.8cm [ABSTRACT]{}
0.3cm [ We discuss the use of the optimization procedure based on the Principle of Minimal Sensitivity to the third-order calculation of . The effective coupling constant remains finite allowing us to apply the Poggio-Quinn-Weinberg smearing method down to energies below 1 GeV, where we find good agreement between theory and experiment. The couplant freezes to a value of $\alpha_s/\pi = 0.26$ at zero energy which is in remarkable concordance with values obtained phenomenologically. 0.5cm]{}
The $e^+e^-$ annihilation cross section is a fundamental test of QCD. Recently QCD corrections to third-order (NNLO) have been calculated using the $\overline{\mbox{MS}}$ scheme [@prl]. Here we use the method of Ref. [@opt] to “optimize” the scheme choice. It turns out this allows us to take the calculations down to low energies. $ R_{e^+e^-} $ has the form $$R_{e^+e^-} \equiv \frac{\sigma(e^+e^- \rightarrow \mbox{hadrons})}
{\sigma(e^+e^- \rightarrow \mu^+\mu^-)}
= 3\sum_q e_q^2 T(v_q) [1 + g(v_q)
\underbrace{a(1 + r_1 a + r_2 a^2 + \ldots)}_{\cal R} ]$$ where $a \equiv \alpha_s/\pi$, and where the known functions $T(v_q),g(v_q)$ [@pqw] (equal to unity in the massless limit) allow for mass dependence near thresholds.
A direct comparison of theory and data is not possible because of the non-perturbative effects associated with thresholds; instead we used the method of Poggio,Quinn and Weinberg [@pqw] who define a “smeared” $R$: $$\overline{R}_{PQW}(Q;\Delta) = \frac{\Delta}{\pi} \int_0^{\infty} ds^{\prime}
\frac{R_{e^+e^-}(\sqrt{s^{\prime}})} {(s^{\prime} - Q^2)^2 + \Delta^2}.$$ The sharp resonances $\omega,\phi$,$J/\psi,\psi^{\prime}$, and $\psi(3770)$ were left out of the data compilations so that their contribution to $\overline{R}_{PQW}$ could be put in analytically. In handling the experimental data we used both raw data points and eye-ball fits to the data (especially in regions with a lot of structure). Fitting the data allowed us to put in estimates for the experimental error.
A comparison between theory and experiment is given in Fig. 1 for $\Delta=2.0$ GeV$^2$. Near the $\psi$ resonances some structure remains which can be smoothed out by increasing $\Delta$. Note that at low energies all structure has been smoothed out, and theory and experiment agree very well.
= 6.0in [Fig. 1: PQW-smeared $R$ ratio for $\Delta=2$ GeV$^2. \;\;\;\;\;$ Fig. 2: Couplant and ${\cal R}$ as a function of $Q$.]{}
To get theoretical predictions down to such low energies we used “optimized perturbation theory ” (OPT), based on the Principle of Minimal Sensitivity (PMS). Here we just sketch the ideas, for more detail see [@opt] and specifically [@us].
(6,.0001)(0,0) (4.90,2.65)[(0,0)[${\cal R}$]{}]{}
-.3in Renormalization Group invariance means that a physical quantity ${\cal
R}$ is independent of the renormalization scheme (RS); that is, it does not depend upon the particular way one defines the renormalized couplant $a$. Symbolically we can express this by: 0 = = + \[rg\] where the total derivative can be separated into two pieces giving the RS dependence via the coefficients and via the couplant. The RS can be parameterized by a set of variables, [$\tau,c_2,c_3,\ldots$]{} where $\tau \equiv \ln(\mu/{{\mbox{$\tilde{\Lambda}$}}})$ and $c_2,c_3,\ldots$ are the non-invariant $\beta$ function coefficients Perturbative coefficients in the expansion of a physical quantity can depend on the RS only through these parameters [@opt]. The third-order approximant is of the form with a similar truncation of the $\beta$ function. The couplant then depends on just two RS parameters; $a^{(3)} = a(\tau;c_2) $. Eq. (\[rg\]) dictates how the coefficients $r_1,r_2$ must depend on $\tau,c_2$. One can express this by saying that the combinations $$\rho_1 = \tau - r_1, \;\;\;\;\;\;\;\; \rho_2 = r_2 + c_2 - (r_1 + {\mbox{$\scriptstyle \frac{1}{2}$}}\;
c)^2
\label{inv}$$ are RS invariants [@opt]. Their values can be obtained from the $\overline{MS}$ ($\mu=Q$) calculations of $r_1,r_2,c,c_2$. \[ $\rho_1$ depends logarithmically on $Q$, while $\rho_2$ is a pure number, depending on $N_f$.\]
While the exact ${\cal R}$ is RS-independent, the approximant ${\cal R}^{(3)}$ is not: it depends on $\tau$ and $c_2$ in a definite way. The PMS criterion says that if an approximant depends on unphysical parameters then those parameters should be chosen so as to minimize the sensitivity of the approximant to small variations in them [@opt]. For ${\cal R}^{(3)}$ this implies $$\left. \frac{d {\cal R}^{(3)}}{d \tau}
\right|_{\bar{\tau},\bar{c_2}} = 0, \;\;\;\;\;\;\;\;\;\;\;\;
\left. \frac{d {\cal R}^{(3)}}{d c_2} \right|_{\bar{\tau},\bar{c_2}} = 0,$$ providing two coupled equations determining $\bar{\tau}$ and $\bar{c_2}$. The integrated $\beta$ equation then determines the optimized couplant, $\bar{a}$, given a $\Lambda$ value. From Eq. (\[inv\]) we can find the optimized coefficients $\overline{r_1},\overline{r_2}$, and hence obtain the optimized ${\cal R}$. Following this procedure gave the results shown in Fig. 2 for $\bar{a}$ and ${\cal R}$, as a function of $Q$.
The key result is that the optimized couplant does [*[not]{}*]{} go through a pole at some finite $Q$ of order $\Lambda$. Instead it “freezes” below 300 MeV to a nearly constant value. This fixed-point behavior is due to the large, negative $\overline{c_2}$ ($\approx -22$), which produces a non-trivial zero in the OPT $\beta$ function. The OPT analysis simplifies considerably at a fixed point [@kubo], yielding a simple formula for the infrared ($Q\rightarrow 0$) limit of the optimized couplant: $${\scriptstyle \frac{7}{4}} +
c \bar{a}^* + 3 \rho_2 \bar{a}^{*2} = 0.$$ For $N_f=2$ this gives $a^{\ast} = 0.26$.
A “freezing” of $\alpha_s$ at low energies has long been a popular assumption in phenomenological models of low-energy hadronic physics [@pheno]. This phenomenology has been rather successful and extracts values for the low-energy $\alpha_s/\pi$ that are remarkably close to ours. Recently, an energy-loss analysis of $b$-quark fragmentation has yielded a similar result [@vk]. In particular, comparison with the data yields 0.2 GeV as the integral of $\alpha_s(k^2)/\pi$ over $k$ from 0 to 1 GeV [@vk]. This happens to agree precisely with the area under our curve in Fig. 2!
We stress that our value $a^{\ast}=0.26$ is determined purely by the pertubative QCD calculations; it involves [*[no]{}*]{} experimental input (not even the value of $\Lambda$). This, we believe, is the first purely [*[theoretical]{}*]{} evidence for the “freezing” of $\alpha_s$. 0.5cm [Acknowledgements ]{} 0.3cm We thank I. Duck, N. Isgur, and V. Khoze for helpful comments. This work was supported in part by the U.S. Department of Energy under Grant No. DE-FG05-92ER40717. 0.5cm [References ]{} 0.3cm
[9]{} S. A. Gorishny, A. L. Kataev and S. A. Larin, Phys. Lett. [**259B**]{}, 144 (1991). L. R. Surguladze and M. A. Samuel, Phys. Rev. Lett. [**66**]{} 560 (1991). P. M. Stevenson, Phys. Rev. D[**23**]{}, 2916 (1981). E. C. Poggio, H. R. Quinn, and S. Weinberg, Phys. Rev. D[**13**]{}, 1958 (1976). A. C. Mattingly and P. M. Stevenson Phys. Rev. Lett. [**69**]{}, 1320 (1992). J. Kubo, S. Sakakibara and P. M. Stevenson, Phys. Rev. D[**29**]{}, 1682 (1984). See, e.g., T. Barnes, F. E. Close, and S. Monaghan, Nucl. Phys. [**B 198**]{}, 380 (1982); J. Stern and G. Clement, Phys. Lett. [**264B**]{}, 426 (1991); S. Godfrey and N. Isgur, Phys. Rev. D[**32**]{} 189 (1985). V. Khoze, SLAC-PUB-5909, Talk presented at XXVI International Conference of High Energy Physics Dallas, Texas, Aug. 1992.
|
---
abstract: 'We study DBI-type effective theory of an unstable D3-brane in the background manifold ${\rm R}^{1,1}\times {\cal M}_2$ where ${\cal M}_2$ is an arbitrary two-dimensional manifold. We obtain an exact tubular D2-brane solution of arbitrary cross sectional shape by employing $1/\cosh$ tachyon potential. When ${\cal M}_2={\rm S}^{2}$, the solution is embedded in the background geometry ${\rm R^{1,3}} \times {\rm S}^2$ of Salam-Sezgin model. This tachyon potential shows a unique property that an array of tachyon soliton solutions has a fixed period which is independent of integration constants of the equations of motion. The thin BPS limit of the configurations leads to supertubes of arbitrary cross sectional shapes.'
---
[hep-th/0404163]{}
[[[**Tubular D-branes in Salam-Sezgin Model**]{}]{}\
[Chanju Kim]{}\
[*Department of Physics, Ewha Womans University, Seoul 120-750, Korea*]{}\
[[email protected]]{}\
[Yoonbai Kim and O-Kab Kwon]{}\
[*BK21 Physics Research Division and Institute of Basic Science,\
Sungkyunkwan University, Suwon 440-746, Korea*]{}\
[[email protected] [email protected]]{} ]{}
Introduction
============
When D-branes are wrapped on some nonsupersymmetric cycles in the moduli space of compactified manifolds, e.g., K3 or Calabi-Yau manifolds, they are dissociated and form several stable D-branes among which each is wrapped on a supersymmetric cycle [@Sen:1998ex; @Horava:1998jy]. A representative example is a D$p$-brane and one of the directions is compactified as a circle of radius $R$. At the critical radius $R=\sqrt{2}$, it decays into a pair of D$(p-1)\bar{{\rm D}}(p-1)$ branes [@Sen:1998tt] situated at diametrically opposite points. In terms of boundary conformal field theory (BCFT), this phenomenon is described by a marginal deformation interpolating between the original unstable D$p$-brane and the D$(p-1)\bar{{\rm D}}(p-1)$ pair [@Sen:1998ex].
In the context of effective field theory (EFT) where the instability of D$p$-brane is represented by condensation of a tachyon field, the Dirac-Born-Infeld (DBI) type effective action [@Sen:1999md; @Garousi:2000tr] with the choice of $1/\cosh$ potential [@Buchel:2002tj; @Kim:2003he; @Leblond:2003db] can reproduce this as an array of static tachyon kink-antikinks with fixed period $2\pi R$ [@Lambert:2003zr; @Kim:2003in]. Only in this EFT, the tachyon profile in BCFT and the tachyon field configuration in EFT has one-to-one correspondence by an explicit point transformation [@Kutasov:2003er] so that the identification between them can clearly be made at the classical level [@Sen:2003bc]. When the gauge field is turned on on the unstable D-brane, the period of the D$(p-1)\bar{{\rm D}}(p-1)$ array starts changing from $2\pi R$ to a larger value and it eventually goes to infinity when the electric field approaches the critical value [@Kim:2003in; @Sen:2003bc; @Kim:2003ma]. For each value of the electric field, the period is fixed and independent of other integration constants of the equation of motion in EFT with $1/\cosh$ potential [@Kim:2003in; @Kim:2003ma].
On the other hand, when the shape of tachyon potential is chosen to be different from $1/\cosh$, the property of fixed periodicity seems likely to be lost even in pure tachyon case [@Brax:2003rs]. In this paper, we will show that, in the EFT with DBI type action, the $1/\cosh$ tachyon potential should uniquely be chosen in order to keep this periodic property.
The codimension-one D-branes which are represented as kinks or antikinks in EFT become BPS objects only in zero thickness limit [@Sen:2003tm] (except the case of the composite of D$p$-brane and fundamental string (F1) fluid with critical electric field [@Kim:2003in; @Kim:2003ma]). This is also true for codimension two or three objects such as vortex-antivortex pairs or monopole-antimonopole pairs though BCFT description may not be explicit [@Sen:1998ex; @Horava:1998jy].
In this paper we will consider another configuration, a generation of a tubular brane on R${}^{1}\times$S${}^{2}$ of which the thin limit is a supertube [@Mateos:2001qs] along the equator of S${}^{2}$. The tubular D2-brane can have an arbitrary cross sectional shape [@Bak:2001xx] and it is natural to expect this property to appear also in tachyon tubes from an unstable D3-brane [@Kim:2003uc]. We will find thick tachyon tube solutions of arbitrary cross section from an unstable D3-brane on the background manifold R${}^{1,1}\times {\cal M}_{2}$, and discuss BPS supertube limit by taking the thickness to be zero.
The case of ${\cal M}_{2}={\rm S}^{2}$ is of our particular interest. In this case the base spacetime is embedded in R${}^{1,3}\times$S${}^{2}$ of six-dimensional Salam-Sezgin model [@Salam:1984cj]. Recently Salam-Sezgin model has been studied in various contexts [@Aghababaie:2002be]–[@Cvetic:2003xr]. In particular, several investigations have been made in relation with the vacuum structure. A consistent S${}^{2}$ reduction of the Salam-Sezgin model was performed and its four-dimensional spectrum was analyzed [@Gibbons:2003gp]. A new family of supersymmetric vacua in the six-dimensional chiral gauged $N=(1,0)$ supergravity was discovered, of which the generic form is AdS${}_{3}\times$S${}^{3}$, and in this scheme R${}^{1,3}\times$S${}^{2}$ can be viewed as a fine-tuning [@Guven:2003uw]. Uniqueness of the Salam-Sezgin vacuum among all nonsingular backgrounds with four-dimensional Poincaré, de Sitter, or anti de Sitter invariance was proved [@Gibbons:2003di].
Our analysis is based on the EFT, and it is unclear whether or not the obtained tube solution can be a consistent BCFT solution in the background of string theory. Since higher-dimensional origin of Salam-Sezgin model has also been obtained [@Kerimo:2003am; @Cvetic:2003xr], this important issue should be addressed in a consistent manner.
The rest of paper is organized as follows. In section 2, we prove uniqueness of the tachyon potential for fixed periodicity of the array of tachyon soliton-antisoliton pairs. In section 3, we obtain exact tachyon tube solutions on ${\rm R}^{1,1}\times {\cal M}_2$, where ${\cal M}_2$ is a two-dimensional manifold, and discuss their BPS limit. In section 4, we consider the case ${\cal M}_2 = {\rm S}^2$ in more detail. We conclude in section 5 with brief discussion.
Tachyon Potential of D-brane Wrapped on a Cycle
===============================================
The effective tachyon action for an unstable D$p$-brane system [@Sen:1999md; @Garousi:2000tr] is $$\label{fa}
S= -{\cal T}_{p} \int d^{p+1}x\; V(T) \sqrt{-\det (g_{\mu\nu} + F_{\mu\nu} +
\partial_\mu T\partial_\nu T )}\, ,$$ where $g_{\mu\nu}$ is the metric given from the closed string sector, $T(x)$ is tachyon field, and $F_{\mu\nu}$ field strength tensor of a gauge field $A_{\mu}$ on the D$p$-brane, of which the constant piece can also be interpreted as NS-NS two form field. We set $2\pi \alpha'=1$ and then ${\cal T}_{p}$ is tension of the D$p$-brane.
Since tachyon potential measures variable tension of the unstable D-brane, it should be a runaway potential connecting $$\label{vbd}
V(T=0)=1~~\mbox{and}~~ V(T=\infty)=0.$$ Various forms of it have been proposed, e.g., $V(T)\sim e^{-T^{2}}$ from boundary string field theory [@Gerasimov:2000zp] or $V(T)\sim e^{-T}$ for large $T$ in Ref. [@Sen:2002an]. In this paper, we employ the form [@Buchel:2002tj; @Kim:2003he; @Leblond:2003db] $$\label{V3}
V(T)=\frac{1}{\cosh \left(\frac{T}{R}\right)}$$ which connects the small and the large $T$ behaviors smoothly. Here, $R$ is $\sqrt{2}$ for the non-BPS D-brane in the superstring and 2 for the bosonic string. This form of the potential has been derived in open string theory by taking into account the fluctuations around $\frac{1}{2}$S-brane configuration with the higher derivatives neglected, i.e., $\partial^2 T = \partial^3 T=
\cdots = 0$ [@Kutasov:2003er; @Okuyama:2003wm; @Niarchos:2004rw].
Most of the physics of tachyon condensation is irrelevant to the detailed form of the potential once it satisfies the runaway property and the boundary values (\[vbd\]). For example, both the basic runaway behavior of rolling tachyon solutions [@Sen:2002an] and the BPS nature of tachyon kinks with zero thickness [@Sen:2003tm] are attained irrespective of the specific shape of the potential which just reflects a detailed decaying dynamics of the unstable D-brane.
On the other hand, there are also some nice features of the form (\[V3\]) in addition to the fact that it is derived from open string theory in a specific regime. Under the 1/cosh tachyon potential (\[V3\]), exact solutions are obtained for rolling tachyon [@Kim:2003he; @Kim:2003ma] and tachyon kink solutions on unstable D$p$ with a coupling of abelian gauge field for arbitrary $p$ [@Lambert:2003zr; @Kim:2003in; @Brax:2003rs; @Kim:2003ma; @Kim:2003uc]. Another useful property may be the observation that some of the obtained classical solutions $T(x)$ in the EFT (\[fa\]), e.g., rolling tachyons [@Lambert:2003zr] and tachyon kinks [@Sen:2003bc], can be directly translated to BCFT tachyon profiles $\tau(x)$ in open string theory described by the following relation obtained in Ref. [@Kutasov:2003er], $$\label{oto}
\frac{\tau(x)}{R}=\sinh \left(\frac{T(x)}{R}\right).$$
In this section, we would like to discuss another important feature of the 1/cosh potential (\[V3\]), which is not shared by any other form. Among the tachyon soliton solutions in the effective theory, various tachyon array solutions of codimension one have been found, namely, those formed by pure tachyon kink-antikink [@Sen:1998tt; @Sen:1998ex; @Lambert:2003zr; @Kim:2003in], tachyon kink-antikink coupled to the electromagnetic field [@Kim:2003in; @Sen:2003bc; @Kim:2003ma], and tachyon tube-antitube [@Kim:2003uc]. An interesting property of all these solutions is that, with the 1/cosh potential in the EFT, the periodicity of the array is independent of any integration constant of the equation of motion, much like the case of simple harmonic oscillator. Here we will show that the converse is also true by adopting the similar line of argument to the case of simple harmonic oscillator: imposing the condition that the periodicity of the tachyon array solutions should be independent of the integration constant of the equation of motion uniquely determines the tachyon potential as Eq. (\[V3\]). This property is necessary if one wishes to identify the array solution as a configuration on a circle or a sphere of a fixed radius [@Sen:2003bc; @Sen:2003zf].
To begin with, we recall that the relevant equation for all the array solutions with $T=T(x)$ and $F_{\mu\nu}$ is summarized by a single first-order ordinary differential equation $$\label{eee}
{\cal E} = \frac12 T'^2 + \frac{1}{h} U(T),$$ where $U = V^{2}(T)$. (See Ref. [@Kim:2003in; @Kim:2003ma; @Kim:2003uc] and also (\[trr\]) in the next section.) For the array of kink-antikink, two parameters ${\cal E}$ and $h$ are $$\label{uv}
{\cal E} = -\frac{\beta_p}{2\alpha_p},\qquad
h = -\frac{2\alpha_p \gamma_p^2}{{\cal T}_p^2},$$ where $\beta_p=-\det(\eta_{\mu\nu}+F_{\mu\nu})$, $\alpha_{p}$ is cofactor of 11-component of the matrix $-(\eta+F)_{\mu\nu}$, and $\gamma_{p}$ an integration constant [@Kim:2003in; @Kim:2003ma]. For the array of tube-antitube, ${\cal E}$ and $h$ are $$\label{uv1}
{\cal E} = -\frac{1}{2},\qquad
h = -\frac{2\alpha^2\beta^2}{{\cal T}_3^2},$$ where $\alpha$ is D0 charge density per unit length and $\beta$ an integration constant [@Kim:2003uc]. (See also (\[trr\]) in the next section.) Then, for our purpose, the coefficient $h$ in front of the potential is negative and to be varied, and ${\cal E}$ is regarded as a constant with $1/h< {\cal E}<0$.
Let us require the period to be independent of $h$ in Eq. (\[eee\]). Denoting the period as $\zeta$, we have $$\begin{aligned}
\frac{\pi}{2} \zeta
&=& \int_0^{T_{\rm max}} \frac{dT}{\sqrt{2[{\cal E} - U(T)/h]}} \nonumber \\
&=& \int_{U_0}^{h{\cal E}} \frac{dT/dU}{\sqrt{2({\cal E} - U/h)}}dU,
\label{peq}\end{aligned}$$ where $T_{\rm max}$ is the maximum value of the tachyon field, $U(T_{\rm max}) = h {\cal E}$, and $U_0 = U(T=0)$. It turns out to be convenient to define the variable $\eta = h{\cal E}$. Then Eq. (\[peq\]) becomes $$\frac{\pi}{2} \zeta
= \frac{1}{\sqrt{2|{\cal E}|}} \int_{U_0}^\eta
\frac{\sqrt{-\eta}}{\sqrt{\eta - U}} \frac{dT}{dU} dU.$$ If both sides of this equation are divided by $\sqrt{(-\eta)(U-\eta)}$ and integrated with respect to $\eta$ from $U_0$ to $U$, $$\begin{aligned}
\frac{\pi \zeta}{2} \int_{U_0}^U \frac{d\eta}{\sqrt{\eta^2 - U \eta}}
&=& \frac{1}{\sqrt{2|{\cal E}|}} \int_{U_0}^U \int_{U_0}^\eta d\eta dU'
\frac{dT(U')/dU'}{\sqrt{(U-\eta)(\eta - U')}} \nonumber \\
&=& \frac{1}{\sqrt{2|{\cal E}|}} \int_{U_0}^U dU' \int_{U'}^U
d\eta \frac{dT(U')/dU'}{\sqrt{(U-\eta)(\eta - U')}},
\label{pint}\end{aligned}$$ where we changed the order of integration in the second line.
It is now elementary to perform the integral (\[pint\]) in the both sides. The result is $$\label{Uf}
\pi\zeta \, {\rm arccosh}\left( \sqrt{\frac{U_0}{U}} \right)
= \frac{\pi T}{\sqrt{2|{\cal E}|}},$$ i.e., $$\label{UT}
U(T) = \frac{U_0}{\cosh^2(T/R)},$$ where $R = \zeta \sqrt{2|{\cal E}|}$. Comparing Eq. (\[Uf\]) with Eq. (\[eee\]), we see that $V(T) = 1/\cosh(T/R)$ as asserted. This property can also be seen clearly after a point transformation (\[oto\]) to the equation (\[eee\]), which, under the specific tachyon potential (\[V3\]), results in $$\label{eef}
{\cal E}'=\frac{1}{2}\tau'^{2}+\frac{1}{2}\omega^{2}\tau^{2},$$ where $0<{\cal E}'=-1/h+{\cal E}$ and $0<\omega^{2}=-2{\cal E}/R^{2}$. Since both ${\cal E}$ and $R$ are fixed but $h$ is a variable, ${\cal E}'$ is a positive variable and $\omega^{2}$ a constant. Therefore, Eq. (\[eef\]) is formally equivalent to the expression of the mechanical energy ${\cal E}'$ of a 1-dimensional simple harmonic oscillator with unit mass of which the position is $\tau$ at time $x$. According to the proof in Ref. [@LLt], its period $2\pi/\omega$ is independent of the value of ${\cal E}'$ only for the simple harmonic oscillator.
This periodic property of the array configurations in the effective field theory is desirable if we wish to identify the array solution as a pair of D$(p-1)\bar{{\rm D}}(p-1)$ obtained from an unstable D$p$-brane wrapped on a cycle in the context of string theory [@Sen:1998ex; @Horava:1998jy; @Sen:2003bc]. (Note also from Eq. (\[UT\]) that the compactified length $\zeta$ varies as the electromagnetic field changes.) In this sense, our proof in this section tells the uniqueness of the 1/cosh tachyon potential (\[V3\]) for the tachyon field in Eq. (\[fa\]) in studying the generation of codimension one extended objects on nonsupersymmetric cycles. In section 3, we will find a family of tachyon tube solutions with such periodicity on R${}^{1}\times {\cal M}_{2}$, and in section 4, will demonstrate that single tachyon tube on ${\rm R}^{1}\times {\rm S}^{2}$ forms a thin tubular object of which the geometry is ${\rm R}^{1}\times {\rm S}^{1}$ in the BPS limit.
Tachyon Tubes of Arbitrary Cross Section and BPS Limit
======================================================
In this section we consider tachyon tube configurations in the theory described by the DBI type action (\[fa\]) on ${\rm R}^{1,1} \times {\cal M}_2$ in the coordinate system $(t,z,u,v)$ with metric $$\label{met}
ds^2 = -dt^2 + dz^2 + du^2 +f(u)^{2}dv^{2},$$ where $f(u)$ is an arbitrary function. Depending on $f(u)$ the two-dimensional manifold ${\cal M}_{2}$ defined by $(u,v)$-coordinates can be either compact or noncompact. In flat ${\rm R}^{2}$ case ($f(u)=u^2$) tachyon tube solutions were obtained in Ref. [@Kim:2003uc]. Here we will show that there exist various tachyon tubes with arbitrary cross sectional shapes, as in the case of supertubes [@Bak:2001xx]. In particular, configurations on ${\rm S}^{2}$ will be considered in greater detail in the subsequent section.
As an ansatz, we assume that the fields are dependent only on the coordinate $u$ and $F_{0u} = F_{uv} = F_{zu} = 0$. Then nonvanishing fields are $T=T(u)$, $F_{0z}\equiv E_z(u)$, $F_{0v}\equiv E_v(u)$, and $F_{v z}/f \equiv B_u (u)$. With the ansatz, the Bianchi identity dictates $E_z$ and $E_v$ to be constants, and $B_u\sim 1/f$. In this paper we further restrict our interest to looking for the configurations with the critical value for $E_z$ and vanishing $E_v$[^1] so that we have $$\label{Bian-2}
|E_z|=1,\qquad E_v = 0,\qquad B_u
= \frac{\alpha}{f},$$ where $\alpha$ is an arbitrary D0 charge density at $f=0$ and due to that the Bianchi identity $\nabla\cdot {\bf B}=0$ fails at $f=0$.
Substituting Eq. (\[Bian-2\]) with $T=T(u)$ into the action (\[fa\]), we find that the action is independent of the metric function $f(u)$, $$\begin{aligned}
\label{rfa}
S= - {\cal T}_{3}\alpha \int dtdz dv \int du\, V(T)\sqrt{1+T'^{2}},\end{aligned}$$ where the prime denotes differentiation with respect to the variable $u$. Then the equation of motion reduces to $$\label{trr}
-fT_{uu} \equiv
{\cal T}_{3}\frac{V\alpha}{\sqrt{1+T'^{2}}} = \beta\alpha^{2}\, ,$$ where $\beta$ is a nonnegative constant and $T_{uu}$ the $uu$-component of pressure.
For the solutions of Eq. (\[trr\]), many components of energy-momentum tensor $T_{\mu\nu}$ and conjugate momenta of the gauge field $\Pi_{i}$ vanish, $$\label{zet}
T_{0z}=T_{0u}=T_{zu}=T_{zv}=T_{uv}
=T_{vv}=\Pi_{u}=\Pi_{v}=0.$$ The nonvanishing components share the same functional $T$-dependence (and hence the same $u$-dependence) except $T_{uu}$ in Eq. (\[trr\]), $$\begin{aligned}
\label{loq}
fT_{00} &=& (f^2 + \alpha^2) \Sigma(u), \nonumber \\
T_{0v} &=& \alpha f \Sigma(u),\nonumber \\
T_{zz} &=& -f\Sigma(u), \nonumber \\
\Pi &=& f^2\Sigma(u),\end{aligned}$$ where $\Pi \equiv \Pi_{z}$ and $$\label{Sigma}
\Sigma(u) = \beta (1+T'^{2})=\frac{1}{\beta\alpha^{2}}({\cal T}_{3}V)^{2}.$$ The energy per unit length then satisfies the relation $$\label{BPS}
{\cal E} =\int du dv \, fT_{00} = Q_{{\rm F1}} + \int du dv \, \Pi B^{2},$$ where $Q_{\rm F1}$ is F1 charge per unit length, $$Q_{{\rm F1}} = \int du dv\, \Pi.$$ Note that Eq. (\[BPS\]) holds irrespective of the form of both the tachyon potential $V(T)$ and the metric function $f(u)$. We will shortly see that, with the 1/cosh-type potential (\[V3\]) (and only with this potential), the second term of Eq. (\[BPS\]) is identified as D0 charge per unit length.
Now let us discuss the solution of Eq. (\[trr\]) in detail. With the form of tachyon potential (\[V3\]), it is easy to obtain the exact solution $$\label{tub}
\sinh\left(\frac{T(u)}{R}\right)
=\pm \left[\sqrt{\left(
\frac{{\cal T}_{3}}{\alpha\beta}\right)^{2}-1}
\,\cos \left(\frac{u}{R}\right)\right],$$ where we imposed the condition $T'(0)=0$ for regularity. This solution represents a coaxial array of tubular kink-antikink with periodicity $2\pi R$. Note that the period is independent of integration constants $\alpha$ and $\beta$. This is consistent with the discussion on the unique property of 1/cosh tachyon potential in Sec. 2.
For the solution, the quantity $\Sigma(u)$ of Eq. (\[Sigma\]) is given by $$\label{loq2}
\Sigma(u) = \beta \frac{({\cal T}_3/\alpha\beta)^2} {1 +\left[({\cal T}_3/\alpha\beta)^2 -1 \right] \cos^2 (u/R)}.$$ The energy (tube tension) of a single kink (per unit length) is then calculated as $$\begin{aligned}
\label{lene}
{\cal E}_{2}^{(n)}&=&\int dv\int_{(n-1)\pi R}^{n\pi R}du\, fT_{00} \nonumber\\
&=&\beta \int dv\int_{(n-1)\pi R}^{n\pi R} \ du \
\frac{({\cal T}_3/\alpha\beta)^2 (f^2 + \alpha^2)}{
1 +\left[({\cal T}_3/\alpha\beta)^2 -1 \right] \cos^2 (u/R)},\end{aligned}$$ and the string charge per unit length is $$\begin{aligned}
\label{lsc}
Q_{{\rm F1}}^{(n)}&=&\int dv\int_{(n-1)\pi R}^{n\pi R}du \,\Pi
\nonumber\\
&=&\beta \int dv\int_{(n-1)\pi R}^{n\pi R} \ du \
\frac{({\cal T}_3/\alpha\beta)^2 \ f^2 }{
1 +\left[({\cal T}_3/\alpha\beta)^2 -1 \right] \cos^2 (u/R)}.\end{aligned}$$ Though the energy and the string charge are not calculable explicitly for general $f(u)$, the difference ${\cal E}_2^{(n)} - Q_{\rm F1}^{(n)}$ is quite simple and can be calculated explicitly, $$\begin{aligned}
\label{qd}
{\cal E}_2^{(n)} - Q_{\rm F1}^{(n)} &=&
\beta\alpha^2 \int dv\int_{(n-1)\pi R}^{n\pi R} \ du \
\frac{({\cal T}_3/\alpha\beta)^2 }{1 +\left[({\cal T}_3 /
\alpha\beta)^2 -1 \right] \cos^2 (u/R)}
\nonumber\\
&=&\pi \alpha R {\cal T}_3\int dv \nonumber \\
&\equiv& Q_{{\rm D0}}^{(n)} ,\end{aligned}$$ which coincides with the D0-brane charge per unit length. Note that it is independent of $f(u)$ or $\beta$. Therefore we have a BPS-like sum rule $$\label{ld0}
{\cal E}_{2}^{(n)}= Q_{{\rm F1}}^{(n)}+Q_{{\rm D0}}^{(n)}.$$ In addition, each unit tube (or antitube) carries angular momentum per unit length $$\label{ang}
L^{(n)}
=-\alpha\beta\int dv\int_{(n-1)\pi R}^{n\pi R}du
\frac{({\cal T}_3/\alpha\beta)^2 \ f^2 }{
1 +\left[({\cal T}_3/\alpha\beta)^2 -1 \right] \cos^2 (u/R)},$$ which is proportional to the string charge, i.e., $L^{(n)}=-\alpha Q_{{\rm F1}}^{(n)}$.
It is well-known that the supertube solution of cylindrical symmetry is a BPS object preserving 1/4-supersymmetry [@Mateos:2001qs] and this BPS nature is not disturbed for tubular branes with arbitrary cross sectional shape [@Bak:2001xx]. In the above, we obtained the tachyon tube solution for which the $u$-coordinate dependence is arbitrary. Since it is given by the configuration of coaxial array of tube-antitubes with nonzero thickness (\[tub\]), it may not be a BPS object despite of the BPS like sum rule (\[ld0\]). To see whether the configuration is a BPS object, we look into the stress components on $(u,v)$-plane. From Eqs. (\[zet\]) and (\[trr\]), we find that $T_{uv}$ and $T_{vv}$ vanish but $T_{uu}$ does not. If we accept vanishing of all stress components on $(u,v)$-plane as a strict saturation of the BPS bound of these spinning tachyon tubes, it can be achieved in the limit either $\alpha\rightarrow 0$ or $\beta\rightarrow 0$. The former is a trivial limit of a fundamental string without D0’s and is of no interest, while the latter corresponds to the zero thickness limit of the tachyon tube which becomes the supertube for ${\cal M}=$S${}^{2}$. Among the other components which are not in the $(u,v)$-plane, $T_{zu}$ and $T_{zv}$ vanish before taking the zero-thickness BPS limit as in Eq. (\[zet\]). On the other hand, the nonvanishing ones in Eq. (\[loq\]) become delta functions since $$\Sigma(u) \stackrel{\beta\rightarrow0}{\longrightarrow}
\frac{\pi R{\cal T}_3}{\alpha}
\sum_n \delta\left(u-\left(n-\frac12\right)\pi R \right).$$
Tachyon Tubes in the Background of Salam-Sezgin Vacuum
======================================================
Here we study the case ${\cal M}_2 = $S${}^{2}$ in more detail. Spheres appear to be a possible candidate for internal space and well-known examples involving S${}^{2}$ include AdS${}_{2}\times$S${}^{2}$ and compactifications on some Calabi-Yau manifolds with S${}^{2}$ as a submanifold. For simplicity, we assume that the other directions are flat, so the background geometry of our interest is R${}^{1,1}\times$S${}^{2}$. A representative example relevant with this flat space is $N = 2$ Einstein-Maxwell supergravity in six-dimensional space ${\rm R}^{1,3} \times
{\rm S}^2$ which is known as Salam-Sezgin model [@Salam:1984cj]. Its low energy limit admits four-dimensional $N=1$ supergravity which includes chiral fermions and of which the gauge symmetry is SO(3)$\times$U(1). The geometry R${}^{1,3}\times$S${}^{2}$ of the Salam-Sezgin vacuum is expressed by $$\label{ssbg}
ds_{6}^{2}=-dt^{2}+dx^{2}+dy^{2}+dz^{2}+\frac{1}{8g^2}(d\theta^{2}
+\sin^{2}\theta\, d\varphi^{2}),$$ where $0\le \theta \le \pi$ and $0\le\varphi\le 2\pi$. There is a constant magnetic field $-1/2g$ on the two sphere inversely proportional to the gauge coupling $g$ of the Salam-Sezgin model. This $N=2$ supergravity on R${}^{1,3}\times$S${}^{2}$ and its variants have recently attracted attention in relation with various topics [@Aghababaie:2002be; @Aghababaie:2003wz; @Guven:2003uw; @Gibbons:2003gp; @Kerimo:2003am; @Gibbons:2003di; @Aghababaie:2003ar; @Ghoshal:2003jd; @Kerimo:2004md; @Cvetic:2003xr].
Motivated by the above, we consider a tachyon tube-antitube solution (\[tub\]) on R${}^{1,1}\times$S${}^{2}$ described by the coordinates $(t,z,\theta ,\varphi )$ embedded in the Salam-Sezgin vacuum, R${}^{1,3}\times$S${}^{2}$ (\[ssbg\]). Since the period of the solution is $2\pi R$ from Eq. (\[tub\]), we identify the coordinates in the background metric (\[met\]) as $$\label{resc}
u= R\theta,\quad v=R\varphi,\quad f=\sin\theta=\sin \left(
\frac{u}{R}\right),$$ with $0\le u \le \pi R$ and $0\le v \le 2\pi R$. Then, $g$ is identified as $g=1/(2\sqrt{2}R)$, and the resultant background metric becomes Eq. (\[met\]). If the radius $R$ introduced through the tachyon potential (\[V3\]) has a string origin like $R=\sqrt{2}$ or $R=2$, the gauge coupling $g$ is of the string scale.
From the obtained tachyon profile (\[tub\]), we read that a single tachyon tube lies along the equator $(u=\pi R/2)$ and thereby F1 charge density is accumulated there (See Figure \[fig1\]). Linear D0’s along $z$-axis are located at the north pole $(u=0)$ and $\bar{{\rm D}}0$’s at the south pole $(u=\pi R)$. An intriguing point is that the energy and the F1 charge per unit length are obtained in closed forms $$\begin{aligned}
{\cal E}_{2}&=&\beta \int_{0}^{2\pi R} \ dv \
\int_{0}^{\pi R} \ du \
\frac{({\cal T}_3/\alpha\beta)^2 \left[\sin^2 (u/R) + \alpha^2\right]}{
1 +\left[({\cal T}_3/\alpha\beta)^2 -1 \right] \cos^2 (u/R)}
\nonumber\\
&=&2\pi R\left[\alpha + \frac{1}{\alpha}\frac{1}{1+(\alpha\beta/{\cal T}_3) }
\right]\times \pi R{\cal T}_{3},
\label{Slene}\\
Q_{{\rm F1}}&=&
\beta \int_{0}^{2\pi R} \ dv \
\int_{0}^{\pi R} \ du \
\frac{({\cal T}_3/\alpha\beta)^2 \sin^2 (u/R)}{
1 +\left[({\cal T}_3/\alpha\beta)^2 -1 \right] \cos^2 (u/R)}
\nonumber\\
&=& \frac{2\pi R}{\alpha}\frac{1}{1+(\alpha\beta/{\cal T}_3) }
\times \pi R{\cal T}_{3}.
\label{Slsc}\end{aligned}$$ In the thin limit $(\alpha\beta/{\cal T}_3\rightarrow 0)$ of a single tachyon tube on S${}^{2}$ of radius $R$, F1 charge density is concentrated along the equator like the ring of the Saturn (see the solid and dashed lines in Figure \[fig1\]). In the opposite limit $(\alpha\beta/{\cal T}_3\rightarrow 1)$ with $T(u)=0$ at everywhere, ${\cal E}(u) -\Pi(u)$ is evenly distributed (see the dotted line in Figure \[fig1\]). Locations of two point-like peaks due to D0 and $\bar{{\rm D}}0$ are also indicated at both the north and the south poles, respectively in Figure \[fig1\].
Product of two-dimensional flat directions $R^{2}$ to ${\rm R}^{1,1}\times {\rm S}^2$ is automatic so that the obtained tube solution on ${\rm R}^{1,1}\times
{\rm S}^{2}$ is a tachyon tube solution in Salam-Sezgin model of ${\rm R}^{1,3}\times {\rm S}^{2}$. Therefore, it describes either a formation of tubular D2-brane from an unstable D3-brane wrapped on ${\rm R}^{1}\times
{\rm S}^{2}$ or that of tubular D4-brane from the space-filling unstable D5-brane in Salam-Sezgin model.
Conclusion
==========
In this paper we studied DBI-type effective theory of unstable D3-branes and obtained an exact tubular D2-brane solution of arbitrary cross sectional shape. The background manifold of the solution is ${\rm R}^{1,1}\times {\cal M}_2$ where ${\cal M}_2$ is an arbitrary two-dimensional manifold. As the tachyon potential, we employed $1/\cosh$ potential. It was shown that it has a unique property that an array of tachyon soliton solutions has a fixed period which is independent of integration constants of the equations of motion. It also allows us to obtain closed form of solutions. The thin BPS limit of the configurations leads to supertubes of arbitrary cross sectional shapes. In particular, we investigated the case ${\cal M}_2={\rm S}^{2}$ in more detail for which the solution is embedded in the background geometry ${\rm R^{1,3}} \times
{\rm S}^2$ of Salam-Sezgin model. Since a lifting of this model to ten-dimensional type I supergravity is made, of which weak string coupling limit coincides with an exact string theory solution, the near-horizon geometry of a Neveu-Schwarz (NS) five-brane [@Cvetic:2003xr], it would be intriguing to find 9-dimensional analogue of our solution in this 10-dimensional background.
Though our discussions were only about static objects, dynamical generation of D$(p-1)\bar{{\rm D}}(p-1)$ or tubular solution should be achieved as inhomogeneous time-dependent solutions [@Sen:2003zf]. Until now, it seems incomplete since the solution seems to hit a singularity after time evolution for a finite time [@Sen:2002vv].
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are indebted to Ashoke Sen for his valuable suggestions and comments. We also would like to thank Jin-Ho Cho, Seongtag Kim, Sangmin Lee, Hyeonjoon Shin, and J. Troost for helpful discussions. This work was supported by grant No. R01-2003-000-10229-0 from the Basic Research Program of the Korea Science $\&$ Engineering Foundation (C.K.) and is the result of research activities (Astrophysical Research Center for the Structure and Evolution of the Cosmos (ARCSEC)) supported by Korea Science $\&$ Engineering Foundation(Y.K. and O.K.).
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[^1]: Nonvanishing $E_{v}$ ($E_{v}^{2}<\alpha^{2}$) can easily be understood through a boost transformation along $z$-direction and the corresponding object is a helical tachyon tube [@Kim:2003uc].
|
---
abstract: 'This article first addresses applicability of Euclidean models to the domain of Internet routing. Those models are found (limitedly) applicable. Then a simplistic model of routing is constructed for Euclidean plane densely covered with points-routers. The model guarantees low stretch and logarithmical size of routing tables at any node. The paper concludes with a discussion on applicability of the model to real-world Internet routing.'
author:
- 'Victor S. Grishchenko'
title: 'Geo-aggregation permits low stretch and routing tables of logarithmical size'
---
Applicability of the model
===========================
The underlying paradigm of the studied model is routing on a Euclidean plane. Applicability of this model is not obvious. On the contrary. First, it is a common understanding that Internet eliminates distance. Second, Internet is usually modeled as a discrete graph, not a solid plane. I shall consider both aspects.
Distance
--------
It is a known observation [@skitter] that round trip time ($\tau$) reliably correlates with geographical distance. RTT has an obvious lower bound, $\tau \ge \tau_{c} = \frac{2d}{c}$, where $d$ is distance and $c$ is speed of light. Although the way RTT depends on distance is rather complicated, practical RTT usually has the same order of magnitude as its lower bound. (From my personal limited experience, usually $\tau_{c} < \tau < 10 \tau_{c}$ for destinations farther than 1,000km).
Table \[tab:rtt\] shows that RTTs for different paths across the globe may differ by three orders of magnitude. It is a too significant difference to be neglected.
I’ll define a indistinguishability distance as $d_{i18y} = \frac{1}{2} c \tau_{0}$, where $\tau_{o}$ is “zero RTT” (“ping localhost RTT”). Currently, $d_{i18y} \approx 0.5\cdot3.0\cdot10^{5}\cdot10^{-3} = 150km$. Due to the difference between $\tau$ and $\tau_{c}$ we may further lower this estimation to “practical i18y distance” of $\sim50km$. This indistinguishability matters mostly in long-haul cases (i.e. whether a longer/shorter fiber is used). Even inside 50km locality RTT depends on such factors as number of intermediary devices, which indirectly depends on the distance. Also, further improvement of technology may lower zero RTT so i18y distance will collapse further.
Also, if we’ll move to the case of routing among wireless devices (routing in pervasive, mesh networks) then larger distance mean more energy consumption thus the distance factor becomes much more important.
So distance still matters and the geometrically shortest path is the preferred one in most cases. (At least for a simple model.)
source destination approx. RTT
------------------ ---------------------- ---------------
UK Australia 315-325 ms
UK Hong-Kong 300-335 ms
Yekaterinburg Australia 370 ms
Hong-Kong Australia 180 ms
Netherlands Vienna 25-32 ms
localhost same city $<$5 ms
localhost same LAN segment 0.2 ms
localhost localhost 0.1 ms
Solidity
--------
Different from younger Internet times, today the planet is rather densely covered with Internet infrastructure. One may check European or North-American fiber infrastructure maps or any major city fiber optics route maps[@cg].
A traceroute from LIPEX (London Internet Providers Exchange) to BCIX (Berlin Commercial Internet Exchange) takes 7 hops (see Fig. \[lipex2bcix\]). So, an average hop is $\frac {930km}{7} \approx 133$ kilometers. This example demonstrates that even transit routes passing through populated areas are likely to be interrupted with IP (level 3) devices at a step having an order of $d_{i18y}$. As far as I see it, there is a tradeoff between network flexibility and overhead of inserting level 3 devices. If a “cable” is interrupted with level 3 devices in steps having order of $d_{i18y}$ or larger, then the delay caused by devices ($\tau_{0}$) is several times smaller than the delay caused by distance itself ($\tau_{c}$).
So, current (transit) networks are rather dense in terms of populated territory coverage. (I don’t address oceanic cables here.) Off course, non-transit infrastructure has much higher density.
[|rll|rrr|]{}\
N & host name & ip & RTT1 & RTT2 & RTT3\
1 & lipex1.bdr.rtr.caladan.net.uk &193.109.219.24& 1.310 &1.038 & 0.823\
2 & 195.66.224.185 &195.66.224.185 & 9.658 & 8.974 &9.009\
3 & p15-0.core01.a03.atlas.cogentco.com &130.117.1.226& 18.384 & 9.530 & 8.875\
4 & p5-0.core01.dus01.atlas.cogentco.com &130.117.1.126& 12.969 & 13.103 & 13.083\
5 & p5-0.core01.ham01.atlas.cogentco.com &130.117.1.178& 18.455 & 18.024 & 18.150\
6 & p6-0.core01.sxf01.atlas.cogentco.com &130.117.1.182& 21.388 & 21.286 & 21.032\
7 & muli.bcix.de &193.178.185.9& 21.961 & 22.444 & 22.030\
Once again, mesh/pervasive/internet-of-things networks may demonstrate dramatically higher densities than today’s Internet – due to participation of myriads of small/wireless/sensor devices.
Thus finally, the Euclidean model is (limitedly) applicable to the Internet.
Continuous geo-aggregation theorem
===================================
On aggregation
--------------
Aggregation is essential for scalable routing algorithms. Modern Internet routing protocols, such as OSPF [@ospf] and BGP [@bgp] employ aggregation schemes. OSPF uses area prefix aggregation, BGP uses CIDR aggregation since version 4. Limitedness of these schemes may be illustrated by the fact that one can not automatically determine even a continent a device resides in having its IP address. (Although, I do not claim that the problem has its roots in the routing protocols mentioned.)
The only popular routing scheme known to me that uses no aggregation at all is Dijkstra’s shortest paths. It has computational complexity of $o(n^{2})$ so it does not scale well and it is obviously inapplicable in the continuous case (because the number of participants is infinite).
In the Euclidean model I will use the most straightforward method of spatial aggregation. I assume that it is possible to implement it, although it might not be true under contemporary patchy and area-centric address assignment policies. I will shed more light on this thesis in Section \[sec:conc\].
Surface, point-routers and covers
---------------------------------
Imagine Euclidean space where every point is a router. At least, those routers are placed so dense that we may think so. The purpose of the following routing algorithm is to transport a packet by the shortest path of length $l$ or, at least, by a path having stretch not larger than $\sigma$ (of length $< \sigma l$). Packet forwarding is done in terms of directions and diminishingly small steps. A “routed path” is a curve drawn by a packet while it travels from source to destination.
One may ask why don’t we use Cartesian coordinates to e.g. route by a straight line (Internet Coordinate System approach). Suppose we do not have such an ability (because we just pretend that the surface is solid). Points-routers may just remember *directions* to some number of other point-routers.
To simplify the task of routing point-routers may use aggregates.
[Aggregate]{} is an area (a convex) having one *representative vertex*. If a sender does not know the shortest path to some destination, it forwards the packet by the shortest path to the representative vertex of the smallest aggregate containing the destination point.
I assume that a point-router may trivially detect that a given point belongs to a given aggregate. Again, this thesis is presented in Sec. \[sec:conc\] in more detail.
[Aggregate cover]{} is a set of aggregates covering all the populated space. If any point belongs to not more than $k$ aggregates of a given cover we call this cover $k$-fat.
[Symmetric cover]{} is a cover whose aggregates are identical geometric shapes (i.e. square grid or hexagonal grid - which have non-overlapping areas (1-fat), or just a cover formed by overlapping balls of the same radius).
Just for the sake of simplicity I will further assume that we are dealing with a $k$-fat symmetric cover of a plane (2D) whose aggregates are balls of radius r.
[Multilevel self-similar cover]{} is a stack of covers where each next cover is a scaled version of the previous one by a factor of $s$ (i.e. each cover is made of balls of radius $r_0s^i$, where $i$ is denoted as “order” of the cover)
Each point-router assembles a cover of the populated space using aggregates of different orders from some global pre-given multilevel cover. For every aggregate used a point-router remembers direction to some representative point. These directions form “routing table” of the point-router. Each point-router has a purpose of
1. using fewer aggregates
2. having a guarantee of stretch not larger than $\sigma$ for any routed path
It is assumed that every router-point may forward by the shortest path if the destination point is closer than $r_0$. Stretch factor is understood as a ratio of the routed path length to the shortest path length (i.e. to the Euclidean distance between source and destination).
Angular size lemma
------------------
[(Aggregate’s angular size lemma)]{} Any routed path will have stretch under $\sigma$ if every point-router uses aggregates of angular size $\alpha \le \arccos \frac{1}{\sigma}$ (angular size is relative to the point-router).
Indeed, every point-router advances the “packet” to the representative vertex of the smallest aggregate containing the destination point by a diminishingly small step of length $\delta$. The angle between the direction to the representative point of the aggregate and the direction to the destination point is bounded by the maximum angular size of aggregates used by the forwarding point-router. Thus, the actual advance (i.e. packet-to-destination distance decrement) is not smaller than $\delta \cos \alpha$. Obliously, integrating for all the path from the source to the destination we’ll get that $\sigma < \frac{1}{\cos\alpha}$, i.e. the stretch is limited by the maximum angular size $\alpha$ of aggregates used by every point-router.
So, the restriction of using aggregates of angular size less than $\arccos \frac{1}{\sigma}$ guarantees the stretch of less than $\sigma$.
The geo-aggregation theorem
---------------------------
[Geo-aggregation theorem]{}
The number of aggregates consumed by a given point-router to assemble its cover depends logarithmically on the size of the space covered. In other words, in the described Euclidean space routing tables are logarithmically small.
**Proof.** Router at point P may use ball aggregates of order $i$ if they are at least $f_i$ as far, where $f_{i} = \frac {r_{i}}{tg \alpha} = \frac {r_{0}} {tg \alpha} s^{i} $. Indeed, at that distance angular size of such a ball is under $\alpha$, see Fig. \[fig:faragg\].
![How far an aggregate should be to fit into maximum angular size? $P$ is the point-router in question; the ball centered at $A$ is an aggregate of order $i$; lengths of $PT$ and $TR$ are equal to $r_{i}$; $\angle LPR$ is an upper estimate for the angular size of $A$.[]{data-label="fig:faragg"}](far_agg.ps)
Starting from the distance of $f_{i}+2r_{i}$ the point-router may cover the space using aggregates of order not lower than $i$. Indeed, any point is covered by an aggregate of order $i$ and any such ball covering a point farther than $f_{i}+2r_{i}$ is itself farther than $f_{i}$, so $P$ may use it.
Thus, aggregates of orders lower than $i$ are needed only inside the ball centered at $P$ with a radius of $f_{i}+4r_{i}$. This ball has a square of $$B_{i} = \pi (f_{i}+4r_{i})^{2} =
\pi ( \frac {r_{0}} {tg \alpha} s^{i} + 4r_{0}s^{i})^2 =
\pi r_{0}^2 (\frac{1}{tg \alpha} + 4)^2 s^{2i}$$ Thus, $P$ has to trace at most the following quantity of aggregates of order $i-1$: $$n_{i-1} = \frac {B_i} {\pi r_{i-1}^2} k =
\frac { \pi r_0^2 ( tg^{-1} \alpha + 4)^2 s^{2i} } {\pi r_0^2 s^{2i-2}} k =
(tg^{-1} \alpha + 4)^2 s^{2} k$$ Thus, the number of aggregates taken from a given order $i$ is bounded by a constant and does not depend on $i$.
At the same time, the square of the space covered by aggregates of orders lower than $i$ grows as $B_{i}$, i.e. $\sim s^{2i}$. Thus, to cover a World Ball of a given radius $R$ with a given stretch $\sigma$ we may use just $\sim \log_{s} R$ of orders in a multilevel self-similar cover and thus $\sim \log_{s} R$ of aggregates per point-router. (Still the total number of aggregates used by all point-routers is not logarithmic.) For nearly perfect routing having $\sigma=1+o$, where $o$ is small, $tg^{-1}\alpha \sim o^{-0.5}$. Thus the number of aggregates used by a router scales accordingly, $n_{i} \sim o^{-1}$. The problem of calculating optimal cover order scale factor $s$ for a given World Ball and $r_{0}$ is left as an exercise for the reader. The interesting fact is that decreasing $r_{0}$ will cost us the same logarithmical price as increasing $R$.
Conclusion {#sec:conc}
==========
Of course, the Euclidean model has limited applicability. But, this model illustrates the fact that there is little technical sense for any Eurasian router to distinguish different destinations inside NYC the way BGP does. Geo-aggregation with stretch of $1.1$ assumes that destinations as far as $10,000km$ might be grouped into aggregates of $4,000km$ each. This trivial assumption is in dramatic conflict with the current state-of-the-art in the Internet routing. It is my personal opinion that routing strategy overseeing continents is a bad strategy.
I will outline current obstacles that prevent any aggregation scheme having efficiency comparable to geo-aggregation. First, the current routing paradigm is area-centric by means that its top-level entities are autonomous systems which form some separate logical layer over IP addresses. Interrelationships of ASes/areas and IPs/routers/points have to be described and maintained in complex ways. Second, areas/ASes have more of organizational than of geographical underpinnings. Top-level routing (BGP) is AS-centric, so the most popular distance metric is seemingly AS hop which again has more of organizational than of time/space nature. Third, IP address assignments are patchy and again organization-centric.
Those problems are addressed not for the first time. IPv6 assignment policies are supposed to resolve IP address space fragmentation problem (“patchiness”) to some extent. RFC2374 [@rfc2374] specifies an approach of using IXes as IPv6 address assignment roots and thus exchange-based geographic aggregation.
It is my opinion that taking a point-centric approach may dramatically simplify the domain of Internet routing. As an example, the point-centric architecture uses no separate “areas” because each device just owns some prefixes and distributes sub-prefixes to downlink devices (to “lesser aggregates”). A home page of the proposal is at http://www.topoip.org.
[99]{} Bradley Huffaker, Marina Fomenkov, David Moore, and kc claffy: Macroscopic analyses of the infrastructure: measurement and visualization of Internet connectivity and performance, CAIDA http://www.cybergeography.org/ (Warning: most maps are 5 years old) RFC 2328 – OSPF Version 2 RFC 1771 – A Border Gateway Protocol 4 (BGP-4) RFC 2374 - An IPv6 Aggregatable Global Unicast Address Format
|
---
author:
- |
**Luis O. Silva[^1] and Ricardo Weder[^2]**\
Departamento de Métodos Matemáticos y Numéricos\
Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas\
Universidad Nacional Autónoma de México\
C.P. 04510, México D.F.\
`[email protected]`\
`[email protected]`
title: |
The Two-Spectra Inverse Problem for Semi-Infinite\
Jacobi Matrices in The Limit-Circle Case\
---
[**Abstract**]{}
We present a technique for reconstructing a semi-infinite Jacobi operator in the limit circle case from the spectra of two different self-adjoint extensions. Moreover, we give necessary and sufficient conditions for two real sequences to be the spectra of two different self-adjoint extensions of a Jacobi operator in the limit circle case.
Introduction {#sec:intro}
============
In the Hilbert space $l_2(\mathbb{N})$, consider the operator $J$ whose matrix representation with respect to the canonical basis in $l_2(\mathbb{N})$ is the semi-infinite Jacobi matrix $$\label{eq:jm-0}
\begin{pmatrix}
q_1 & b_1 & 0 & 0 & \cdots
\\[1mm] b_1 & q_2 & b_2 & 0 & \cdots \\[1mm] 0 & b_2 & q_3 &
b_3 & \\
0 & 0 & b_3 & q_4 & \ddots\\ \vdots & \vdots & & \ddots
& \ddots
\end{pmatrix}\,,$$ where $b_n>0$ and $q_n\in\mathbb{R}$ for $n\in\mathbb{N}$. This operator is densely defined in $l_2(\mathbb{N})$ and $J\subset J^*$ (see Section \[sec:preliminaries\] for details on how $J$ is defined).
It is well known that $J$ can have either $(1,1)$ or $(0,0)$ as its deficiency indices [@MR0184042 Sec.1.2 Chap.4], [@MR1627806 Cor.2.9]. By our definition (see Section \[sec:preliminaries\]), $J$ is closed, so the case $(0,0)$ corresponds to $J=J^*$, while $(1,1)$ implies that $J$ is a non-trivial restriction of $J^*$. The latter operator is always defined on the maximal domain in which the action of the matrix (\[eq:jm-0\]) makes sense [@MR1255973 Sec.47].
Throughout this work we assume that $J$ has deficiency indices $(1,1)$. Jacobi operators of this kind are referred as being in the limit circle case and the moment problem associated with the corresponding Jacobi matrix is said to be indeterminate [@MR0184042; @MR1627806]. In the limit circle case, all self-adjoint extensions of a Jacobi operator have discrete spectrum [@MR1627806 Thm.4.11]. The set of all self-adjoint extensions of a Jacobi operator can be characterized as a one parameter family of operators (see Section \[sec:preliminaries\]).
The main results of the present work are Theorem \[thm:recovering\] in Section \[sec:recovering\] and Theorem \[thm:necessary-sufficient\] in Section \[sec:n-s-conditions\]. In Theorem \[thm:recovering\] we show that a Jacobi matrix can be recovered uniquely from the spectra of two different self-adjoint extensions of the Jacobi operator $J$ corresponding to that matrix. Moreover, these spectra also determine the parameters that define the self-adjoint extensions of $J$ for which they are the spectra. The proof of Theorem \[thm:recovering\] is constructive and it gives a method for the unique reconstruction. The uniqueness of this reconstruction in a more restricted setting has been announced in [@MR1045318] without proof. In Theorem \[thm:necessary-sufficient\] we give necessary and sufficient conditions for two sequences to be the spectra of two self-adjoint extensions of a Jacobi operator in the limit circle case. This is a complete characterization of the spectral data for the two-spectra inverse problem of a Jacobi operator in the limit circle case.
In two spectra inverse problems, one may reconstruct a certain self-adjoint operator from the spectra of two different rank-one self-adjoint perturbations of the operator to be reconstructed. This is the case of recovering the potential of a Schrödinger differential expression in $L_2(0,\infty)$, being regular at the origin and limit point at $\infty$, from the spectra of two operators defined by the differential expression with two different self-adjoint boundary conditions at the origin [@AW1; @AW2; @borg; @MR1329533; @MR1754515; @MR0039895; @MR0062904; @MR0162996; @MR0058064]. Necessary and sufficient conditions for this inverse problem are found in [@MR0162996]. Characterization of spectral data of a related inverse problem was obtained in [@MR2355343]. The inverse problem consisting in recovering a Jacobi matrix from the spectra of two rank-one self-adjoint perturbations, was studied in [@MR2164835; @MR49:9676; @MR499269; @MR0221315; @MR1643529; @MR2091673]. A complete characterization of the spectral data for this two-spectra inverse problem is given in [@weder-silva].
In the formulation of the inverse problem studied in the present work, the aim is to recover a symmetric non-self-adjoint operator from the spectra of its self-adjoint extensions, as well as the parameters that characterize the self-adjoint extensions. There are results for this setting of the two spectra inverse problem, for instance in [@MR997788; @MR0052004] for Sturm-Liouville operators, and in [@MR1045318] for Jacobi matrices.
It is well known that self-adjoint extensions of symmetric operators with deficiency indices $(1,1)$ can be treated within the rank-one perturbation theory (cf. [@MR1752110 Sec. 1.1–1.3] and, in particular, [@MR1752110 Thm.1.3.3]). Thus, both settings may be regarded as particular cases of a general two-spectra inverse problem. A consideration similar to this is behind the treatment of inverse problems in [@MR0190761]. For Jacobi operators, however, the type of rank-one perturbations in the referred formulations of the inverse spectral problem are different [@MR1752110]. Indeed, in the setting studied in [@weder-silva], one has the so-called bounded rank-one perturbations [@MR1752110 Sec. 1.1]. This means that all the family of rank-one perturbations share the same domain. In contrast the present work deals with singular rank-one perturbations [@MR1752110 Sec. 1.3], meaning that every element of the family of rank-one perturbations has different domain. Note that for differential operators both settings involve a family of singular rank-one perturbations.
The paper is organized as follows. In Section \[sec:preliminaries\], we introduce Jacobi operators, in particular the class whose corresponding Jacobi matrix is in the limit circle case. Here we also present some preliminary results and lay down some notation used throughout the text. Section \[sec:recovering\] contains the uniqueness result on the determination of a Jacobi matrix by the spectra of two self-adjoint extensions. The proof of this assertion yields a reconstruction algorithm. Finally in Section \[sec:n-s-conditions\], we give a complete characterization of the spectral data for the two spectra inverse problem studied here.
Preliminaries {#sec:preliminaries}
=============
Let $l_{fin}(\mathbb{N})$ be the linear space of sequences with a finite number of non-zero elements. In the Hilbert space $l_2(\mathbb{N})$, consider the operator $J$ defined for every $f=\{f_k\}_{k=1}^\infty$ in $l_{fin}(\mathbb{N})$ by means of the recurrence relation $$\begin{aligned}
\label{eq:recurrence-coordinates}
(Jf)_k &:= b_{k-1}f_{k-1} + q_k f_k + b_k
f_{k+1}\,,\quad k \in \mathbb{N} \setminus \{1\}\,,\\
\label{eq:initial-coordinates}
(Jf)_1 &:= q_1 f_1 + b_1 f_2\,,\end{aligned}$$ where, for $n\in\mathbb{N}$, $b_n$ is positive and $q_n$ is real. Clearly, $J$ is symmetric since it is densely defined and Hermitian due to (\[eq:recurrence-coordinates\]) and (\[eq:initial-coordinates\]). Thus $J$ is closable and henceforth we shall consider the closure of $J$ and denote it by the same letter.
We have defined the operator $J$ so that the semi-infinite Jacobi matrix (\[eq:jm-0\]) is its matrix representation with respect to the canonical basis $\{e_n\}_{n=1}^\infty$ in $l_2(\mathbb{N})$ (see [@MR1255973 Sec. 47] for the definition of the matrix representation of an unbounded symmetric operator). Indeed, $J$ is the minimal closed symmetric operator satisfying $$\begin{array}{l}
(Je_n,e_n)=q_n\,,\quad (Je_n,e_{n+1})=(Je_{n+1},e_n)
=b_n\,,\\
(Je_n,e_{n+k})=(Je_{n+k},e_{n})=0\,,
\end{array}
\quad n\in\mathbb{N}\,,\,k\in\mathbb{N}\setminus\{1\}\,.$$ We shall refer to $J$ as the *Jacobi operator* and to (\[eq:jm-0\]) as its associated matrix.
The spectral analysis of $J$ may be carried out by studying the following second order difference system $$\label{eq:main-recurrence}
b_{n-1}f_{n-1}+ q_nf_n+
b_nf_{n+1}=
\zeta f_n\,,\quad n>1\,,\quad \zeta\in\mathbb{C}\,,$$ with the “boundary condition” $$\label{eq:boundary}
q_1f_1+b_1f_2=\zeta f_1\,.$$ If one sets $f_1=1$, then $f_2$ is completely determined by (\[eq:boundary\]). Having $f_1$ and $f_2$, equation (\[eq:main-recurrence\]) gives all the other elements of a sequence $\{f_n\}_{n=1}^\infty$ that formally satisfies (\[eq:main-recurrence\]) and (\[eq:boundary\]). Clearly, $f_n$ is a polynomial of $\zeta$ of degree $n-1$, so we denote $f_n=:P_{n-1}(\zeta)$. The polynomials $P_n(\zeta)$, $n=0,1,2,\dots$, are referred to as the polynomials of the first kind associated with the matrix (\[eq:jm-0\]) [@MR0184042 Sec.2.1 Chap.1].
The sequence $P(\zeta):=\{P_{k-1}(\zeta)\}_{k=1}^\infty$ is not in $l_{fin}(\mathbb{N})$, but it may happen that $$\label{eq:generalized-eigenvector}
\sum_{k=0}^\infty{\left|P_k(\zeta)\right|}^2<\infty\,,$$ in which case $P(\zeta)\in\operatorname{Ker}(J^*-\zeta I)$.
The polynomials of the second kind $Q(\zeta):=\{Q_{k-1}(\zeta)\}_{k=1}^\infty$ associated with the matrix (\[eq:jm-0\]) are defined as the solutions of $$b_{n-1}f_{n-1} + q_n f_n + b_nf_{n+1} = \zeta f_n\,,
\quad n \in \mathbb{N} \setminus \{1\}\,,$$ under the assumption that $f_1=0$ and $f_2=b_1^{-1}$. Then $$Q_{n-1}(\zeta):=f_n\,,\quad\forall n\in\mathbb{N}\,.$$ $Q_n(\zeta)$ is a polynomial of degree $n-1$.
As pointed out in the introduction, $J$ has either deficiency indices $(1,1)$ or $(0,0)$ [@MR0184042 Sec.1.2 Chap.4] and [@MR1627806 Cor.2.9]. These cases correspond to the limit circle and limit point case, respectively. In terms of the polynomials of the first kind, $J$ has deficiency indices $(0,0)$ if for one $\zeta\in\mathbb{C}\setminus\mathbb{R}$ the series in (\[eq:generalized-eigenvector\]) diverges. In the limit circle case (\[eq:generalized-eigenvector\]) holds for every $\zeta\in\mathbb{C}$ [@MR0184042 Thm.1.3.2], [@MR1627806 Thm.3] and, therefore, $P(\zeta)$ is always in $\operatorname{Ker}(J^*-\zeta I)$. Another peculiarity of the limit circle case is that every self-adjoint extension of $J$ has purely discrete spectrum [@MR1627806 Thm.4.11]. Moreover, the resolvent of every self-adjoint extension is a Hilbert-Schmidt operator [@MR1711536 Lem.2.19].
In what follows we always consider $J$ to have deficiency indices $(1,1)$. The behavior of the polynomials of the first kind determines this class. There are various criteria for establishing whether a Jacobi operator is symmetric but non-self-adjoint. These criteria may be given in terms of the moments associated with the matrix, for instance the criterion [@MR1627806 Prop.1.7] due to Krein. A criterion in terms of the matrix entries is the following result which belongs to Berezanski[ĭ]{} [@MR0184042 Chap.1], [@MR0222718 Thm.1.5 Chap.7].
Suppose that $\sup_{n\in\mathbb{N}}{\left|q_n\right|}<\infty$ and that $\sum_{n=1}^\infty\frac{1}{b_n}<\infty$. If there is $N\in\mathbb{N}$ such that for $n>N$ $$b_{n-1}b_{n+1}\le b_n^2\,,$$ then the Jacobi operator whose associated matrix is (\[eq:jm-0\]) is in the limit circle case.
Jacobi operators in the limit circle case may be used to model physical processes. For instance Krein’s mechanical interpretation of Stieltjes continued fractions [@MR0054078], in which one has a string carrying point masses with a certain distribution along the string, is modeled by an eigenvalue equation of a Jacobi operator [@MR0184042 Appendix]. There are criteria in terms of the point masses and their distribution [@MR0184042 Thm.0.4 Thm.0.5 Appendix] for the corresponding Jacobi operator to be in the limit circle case.
In this work, all self-adjoint extensions of $J$ are assumed to be restrictions of $J^*$. When dealing with all self-adjoint extensions of $J$, including those which imply an extension of the original Hilbert space, the self-adjoint restrictions of $J^*$ are called von Neumann self-adjoint extensions of $J$ (cf. [@MR1255973 AppendixI], [@MR1627806 Sec.6]).
There is also a well known result for $J$ in the limit circle case, namely, that $J$ is simple [@MR0184042 Thm.4.2.4]. In its turn this imply that the eigenvalues of any self-adjoint extension of $J$ have multiplicity one [@MR1255973 Thm.3 Sec.81].
Let us now introduce a convenient way of parametrizing the self-adjoint extensions of $J$ in the symmetric non-self-adjoint case. We first define the Wronskian associated with $J$ for any pair of sequences $\varphi=\{\varphi_k\}_{k=1}^\infty$ and $\psi=\{\psi_k\}_{k=1}^\infty$ in $l_2(\mathbb{N})$ as follows $$W_k(\varphi,\psi):=b_k(\varphi_k\psi_{k+1}-\psi_k\varphi_{k+1})\,,
\quad k\in\mathbb{N}\,.$$ Now, consider the sequences $v(\tau)=\{v_k(\tau)\}_{k=1}^\infty$ such that, for $k\in\mathbb{N}$, $$\label{eq:boundary-sequence}
v_k(\tau):=P_{k-1}(0)+\tau Q_{k-1}(0)\,,
\quad \tau\in\mathbb{R}\,,$$ and $$\label{eq:boundary-sequence-infty}
v_k(\infty):=Q_{k-1}(0)\,.$$
All the self-adjoint extensions $J(\tau)$ of the symmetric non-self-adjoint operator $J$ are restrictions of $J^*$ to the set [@MR1711536 Lem.2.20] $$\label{eq:beta-extensions-domain}
\mathcal{D}_\tau := \bigl\{f=\{f_k\}_{k=1}^\infty\in\operatorname{Dom}(J^*):\,
\lim_{n\to\infty}W_n\bigl(v(\tau),f\bigr)=0\bigr\}\,,\qquad
\tau\in\mathbb{R}\cup\{\infty\}\,.$$ Different values of $\tau$ imply different self-adjoint extensions, so $J(\tau)$ is a self-adjoint extension of $J$ uniquely determined by $\tau$ [@MR1711536 Lem.2.20]. Observe that the domains $\mathcal{D}_\tau$ are defined by a boundary condition at infinity given by $\tau$. We also remark that given two sequences $\varphi$ and $\psi$ in $\operatorname{Dom}(J^*)$ the following limit always exists [@MR1711536 Sec. 2.6] $$\lim_{n\to\infty}W_n(\varphi,\psi)=:W_\infty(\varphi,\psi)\,.$$ It follows from [@MR1627806 Thm.3] that, in the limit circle case, $P(\zeta)$ and $Q(\zeta)$ are in $\operatorname{Dom}(J^*)$ for every $\zeta\in\mathbb{C}$.
From what has just been said, one can consider the functions (see also [@MR0184042 Sec.2.4 Chap.1, Sec.4.2 Chap.2]) $$\label{eq:b-d-definition}
\begin{split}
W_\infty(P(0),P(\zeta))&=:D(\zeta)\,,\\
W_\infty(Q(0),P(\zeta))&=:B(\zeta)\,.
\end{split}$$ The notation for these limits has not been chosen arbitrarily; they are the elements of the second row of the Nevanlinna matrix associated with the matrix (\[eq:jm-0\]) and they are usually denoted by these letters [@MR0184042 Sec.4.2 Chap. 2], [@MR1627806 Eq.4.17].
It is well known that the functions $D(\zeta)$ and $B(\zeta)$ are entire of at most minimal type of order one [@MR0184042 Thm.2.4.3], [@MR1627806 Thm. 4.8], that is, for each $\epsilon >0$ there exist constants $C_1(\epsilon),\,C_2(\epsilon)$ such that $${\left|D(\zeta)\right|}\le C_1(\epsilon)e^{\epsilon{\left|\zeta\right|}}\,,\qquad
{\left|B(\zeta)\right|}\le C_2(\epsilon)e^{\epsilon{\left|\zeta\right|}}\,.$$
If $P(\zeta)$ is in $\mathcal{D}_\tau$ the following holds $$0=W_\infty(v(\tau),P(\zeta))=
\begin{cases}
D(\zeta)+\tau B(\zeta) & \quad\text{if}\quad \tau\in\mathbb{R}\\
B(\zeta) & \quad\text{if}\quad \tau=\infty\,.
\end{cases}$$ Thus, the zeros of the function $$\label{eq:R-function-def}
\mathfrak{R}_\tau(\zeta):=
\begin{cases}
D(\zeta)+\tau B(\zeta)& \quad\text{if}\quad \tau\in\mathbb{R}\\
B(\zeta) & \quad\text{if}\quad \tau=\infty
\end{cases}$$ constitute the spectrum of the self-adjoint extension $J(\tau)$ of $J$.
A Jacobi matrix of the form (\[eq:jm-0\]) determines, in a unique way, the sequence $P(t)=\{P_{n-1}(t)\}_{n=1}^\infty$, $t\in\mathbb{R}$. This sequence is orthonormal in any space $L_2(\mathbb{R},d\rho)$, where $\rho$ is a solution of the moment problem associated with the Jacobi matrix (\[eq:jm-0\]) [@MR0184042 Sec.2.1 Chap.2]. The elements of the sequence $\{P_{n-1}(t)\}_{n=1}^\infty$ form a basis in $L_2(\mathbb{R},d\rho)$ if $\rho$ is an N-extremal solution of the moment problem [@MR0184042 Def. 2.3.3] or, in other words, if $\rho$ can be written as $$\label{eq:extremal-measure}
\rho(t)=\langle E(t)e_1,e_1\rangle\,,\qquad t\in\mathbb{R}\,,$$ where $E(t)$ is the spectral resolution of the identity for some von Neumann self-adjoint extension of the Jacobi operator $J$ associated with (\[eq:jm-0\]) [@MR0184042 Thm.2.3.3, Thm.4.1.4].
Let $\rho$ be given by (\[eq:extremal-measure\]), then we can consider the linear isometric operator $U$ which maps the canonical basis $\{e_n\}_{n=1}^\infty$ in $l_2(\mathbb{N})$ into the orthonormal basis $\{P_n(t)\}_{n=0}^\infty$ in $L_2(\mathbb{R},d\rho)$ as follows $$\label{eq:operator-u}
Ue_n=P_{n-1}\,,\qquad n\in\mathbb{N}\,.$$ By linearity, one extends $U$ to the span of $\{e_n\}_{n=1}^\infty$ and by continuity, to all $l_2(\mathbb{N})$. Clearly, the range of $U$ is all $L_2(\mathbb{R},d\rho)$. The Jacobi operator $J$ given by the matrix (\[eq:jm-0\]) is transformed by $U$ into the operator of multiplication by the independent variable in $L_2(\mathbb{R},d\rho)$ if $J=J^*$, and into a symmetric restriction of the operator of multiplication if $J\ne J^*$. Following the terminology used in [@MR0190761], we call the operator $UJU^{-1}$ in $L_2(\mathbb{R},d\rho)$ the *canonical representation* of $J$.
By virtue of the discreteness of $\operatorname{\sigma}(J(\tau))$ in the limit circle case (here and in the sequel, $\operatorname{\sigma}(A)$ stands for the spectrum of operator $A$), the function $\rho_\tau$ given by (\[eq:extremal-measure\]), with $E(t)$ being the resolution of the identity of $J(\tau)$, can be written as follows $$\rho_\tau(t)=\sum_{\lambda_k\le t}a(\lambda_k)^{-1}\,,
\qquad \lambda_k\in\operatorname{\sigma}(J(\tau))\,,$$ where the positive constant $a(\lambda_k)$ is the so-called *normalizing constant* of $J(\tau)$ corresponding to $\lambda_k$. In the limit circle case it is easy to obtain the following formula for the normalizing constants [@MR0184042 Sec.4.1 Chap3], [@MR1627806 Thm.4.11] $$\label{eq:normalizing-constants-for-lambda}
a(\lambda_k)={\left\|P(\lambda_k)\right\|}_{l_2(\mathbb{N})}^2\,,
\qquad\lambda_k\in\operatorname{\sigma}(J(\tau))\,.$$ Formula (\[eq:normalizing-constants-for-lambda\]), which gives the jump of the spectral function at $\lambda_k$, also holds true in the limit point case, when $\lambda_k$ is an eigenvalue of $J$ [@MR0222718 Thm.1.17 Chap.7].
It turns out that the spectral function $\rho_\tau$ uniquely determines $J(\tau)$. Indeed, there are two ways of recovering the matrix from the spectral function. One method, developed in [@MR1616422] (see also [@MR1643529]), makes use of the asymptotic behaviour of the Weyl $m$-function $$m_\tau(\zeta):=\int_{\mathbb{R}}\frac{\rho_\tau(t)}{t-\zeta}$$ and the Ricatti equation [@MR1616422 Eq.2.15], [@MR1643529 Eq.2.23], $$\label{eq:ricatti}
b_n^2 m_\tau^{(n)}(\zeta)=
q_n-\zeta-\frac{1}{m_\tau^{(n-1)}(\zeta)}\,,\quad n\in\mathbb{N}\,,$$ where $m_\tau^{(n)}(\zeta)$ is the Weyl $m$-function of the Jacobi operator associated with the matrix (\[eq:jm-0\]) with the first $n$ columns and $n$ rows removed.
The other method for the reconstruction of the matrix is more straightforward (see [@MR0222718 Sec.1.5 Chap.7 and, particularly, Thm. 1.11]). The starting point is the sequence $\{t^k\}_{k=0}^\infty$, $t\in\mathbb{R}$. From what we discussed above, all the elements of the sequence $\{t^k\}_{k=0}^\infty$ are in $L_2(\mathbb{R},d\rho_\tau)$ and one can apply, in this Hilbert space, the Gram-Schmidt procedure of orthonormalization to the sequence $\{t^k\}_{k=0}^\infty$. One, thus, obtains a sequence of polynomials $\{P_k(t)\}_{k=0}^\infty$ normalized and orthogonal in $L_2(\mathbb{R},d\rho_\tau)$. These polynomials satisfy a three term recurrence equation [@MR0222718 Sec.1.5 Chap.7], [@MR1627806 Sec.1] $$\begin{aligned}
\label{eq:favard-system1}
tP_{k-1}(t) &= b_{k-1}P_{k-2}(t) + q_k
P_{k-1}(t) + b_k P_k(t)\,,
\quad k \in \mathbb{N} \setminus \{1\}\,,\\
\label{eq:favard-system2}
tP_0(t) &= q_1 P_0(t) + b_1 P_1(t)\,,\end{aligned}$$ where all the coefficients $b_k$ ($k\in\mathbb{N}$) turn out to be positive and $q_k$ ($k\in\mathbb{N}$) are real numbers. The system (\[eq:favard-system1\]) and (\[eq:favard-system2\]) defines a matrix which is the matrix representation of $J$.
After obtaining the matrix associated with $J$, if it turns out to be non-self-adjoint, one can easily obtain the boundary condition at infinity which defines the domain of $J(\tau)$. The recipe is based on the fact that the spectra of different self-adjoint extensions are disjoint [@MR0184042 Sec.2.4 Chap.4]. Take an eigenvalue, $\lambda$, of $J(\tau)$, i.e., $\lambda$ is a point of discontinuity of $\rho_\tau$ or a pole of $m_\tau$. Since the corresponding eigenvector $P(\lambda)=\{P_{k-1}(\lambda)\}_{k=1}^\infty$ is in $\operatorname{Dom}(J(\tau))$, it must be that $$W_\infty\bigl(v(\tau),P(\lambda)\bigr)=0\,.$$ This implies that either $W_\infty\bigl(Q(0),P(\lambda)\bigr)=0$, which means that $\tau=\infty$, or $$\tau=-\frac
{W_\infty\bigl(P(0),P(\lambda)\bigr)}
{W_\infty\bigl(Q(0),P(\lambda)\bigr)}
\,.$$
**Notation** We conclude this section with a remark on the notation. The elements of the unbounded set $\sigma(J(\tau))$, $\tau\in\mathbb{R}\cup\infty$, may be enumerated in different ways. Let $\sigma(J(\tau))=\{\lambda_k\}_{k\in K}$, where $K$ is a countable set through which the subscript $k$ runs. If $\sigma(J(\tau))$ is either bounded from above or below, one may take $K=\mathbb{N}$. If $\sigma(J(\tau))$ is unbounded below and above, one may set $K=\mathbb{Z}$. Of course, other choices of $K$ are possible. Since the particular choice of $K$ is not important in our formulae, we shall drop $K$ from the notation and simple write $\{\lambda_k\}_k$. All our formulae will be written so that they are *independent* of the way the elements of a sequence are enumerated, so our convention for denoting sequences should not lead to misunderstanding. Similarly, we write $\sum_k y_k$ instead of $\sum_{k\in K} y_k$, and the convergence of the series to a number $c$ means that for any sequence of sets $\{K_j\}_{j=1}^\infty$, with $K_j\subset K_{j+1}\subset K$, such that $\bigcup_{j=1}^\infty K_j=K$, the sequence $\{\sum_{k\in K_j}y_k\}_{j=1}^\infty$ tends to $c$ whenever $j\to\infty$.
Unique reconstruction of the matrix {#sec:recovering}
===================================
In this section we show that, given the spectra of two different self-adjoint extensions $J({\tau_1})$, $J({\tau_2})$ of the Jacobi operator $J$ in the limit circle case, one can always recover the matrix, being the matrix representation of $J$ with respect to the canonical basis in $l_2(\mathbb{N})$, and the two parameters ${\tau_1}$, ${\tau_2}$ that define the self-adjoint extensions. It has already been announced [@MR1045318 Thm.1] that, when ${\tau_1},{\tau_2}\in\mathbb{R}$ and ${\tau_1}\ne {\tau_2}$, the spectra $\operatorname{\sigma}(J({\tau_1}))$ and $\operatorname{\sigma}(J({\tau_2}))$ uniquely determine the matrix of $J$ and the numbers ${\tau_1}$ and ${\tau_2}$. A similar result, but in a more general setting can be found in [@MR0190761 Thm.7].
Consider the following expression which follows from the Christoffel-Darboux formula [@MR0184042 Eq.1.17]: $$\sum_{k=0}^{n-1}
P_k^2(\zeta)=b_n\left(P_{n-1}(\zeta)P_n'(\zeta)-
P_n(\zeta)P_{n-1}'(\zeta)\right)=W_n(P(\zeta),P'(\zeta))\,.$$ It is easy to verify, taking into account the analogue of the Liouville-Ostrogradskii formula [@MR0184042 Eq.1.15], that $$\begin{split}
W_n(P(\zeta),P'(\zeta))&=W_n(P(0),P(\zeta))W_n(Q(0),P'(\zeta))\\
&-W_n(Q(0),P(\zeta))W_n(P(0),P'(\zeta))\,.
\end{split}$$ Thus, $$\begin{split}
\sum_{k=0}^\infty
P_k^2(\zeta)&=W_\infty(P(\zeta),P'(\zeta))\\
&=D(\zeta)B'(\zeta)-B(\zeta)
D'(\zeta)\,.
\end{split}$$ Indeed, due to the uniform convergence of the limits in (\[eq:b-d-definition\]) [@MR0184042 Sec.4.2 Chap.2], the following is valid $$\begin{split}
B'(\zeta)&=W_\infty(Q(0),P'(\zeta))\\
D'(\zeta)&=W_\infty(P(0),P'(\zeta))\,.
\end{split}$$ Now, a straightforward computation yields (${\tau_1},{\tau_2}\in\mathbb{R}$, ${\tau_1}\ne {\tau_2}$) $$\mathfrak{R}_{\tau_1}(\zeta)\mathfrak{R}_{\tau_2}'(\zeta)-\mathfrak{R}_{\tau_1}'(\zeta)
\mathfrak{R}_{\tau_2}(\zeta)=({\tau_2}-{\tau_1})\left[D(\zeta)B'(\zeta)
-B(\zeta)D'(\zeta)\right]\,.$$ On the other hand one clearly has $$\mathfrak{R}_{\tau_1}(\zeta)\mathfrak{R}_\infty'(\zeta)-\mathfrak{R}_{\tau_1}'(\zeta)
\mathfrak{R}_\infty(\zeta)=D(\zeta)B'(\zeta) - B(\zeta)D'(\zeta)\,,
\quad {\tau_1}\in\mathbb{R}\,.$$ Hence, $$\label{eq:normalizing}
a(\zeta):=\sum_{k=0}^\infty
P_k^2(\zeta)=
\begin{cases}
\displaystyle\frac{\mathfrak{R}_{\tau_1}(\zeta)
\mathfrak{R}_{\tau_2}'(\zeta)-\mathfrak{R}_{\tau_1}'(\zeta)
\mathfrak{R}_{\tau_2}(\zeta)}{{\tau_2}-{\tau_1}} & {\tau_1}\ne {\tau_2}\,,\quad
{\tau_1},{\tau_2}\in\mathbb{R}\\[4mm]
\mathfrak{R}_{\tau_1}(\zeta)\mathfrak{R}_\infty'(\zeta)-\mathfrak{R}_{\tau_1}'(\zeta)
\mathfrak{R}_\infty(\zeta) & {\tau_1}\in\mathbb{R}\,.
\end{cases}$$ It follows from (\[eq:normalizing-constants-for-lambda\]) that the values of the function $a(\zeta)$ evaluated at the points of the spectrum of some self-adjoint extension of $J$ are the corresponding normalizing constants of that extension.
The analogue of (\[eq:normalizing\]) with ${\tau_1}\ne {\tau_2}$ and ${\tau_1},{\tau_2}\in\mathbb{R}$, for the Schrödinger operator in $L_2(0,\infty)$ being in the limit circle case is [@MR997788 Eq.1.20]. Formula [@MR997788 Eq.1.20] plays a central rôle in proving the unique reconstruction theorem for that operator [@MR997788 Thm.1.1]. The discrete counterpart of [@MR997788 Thm.1.1] is [@MR1045318 Thm.1]. It is worth mentioning that the reconstruction technique we present below is also based on (\[eq:normalizing\]).
It is well known that the spectra of any two different self-adjoint extensions of $J$ are disjoint [@MR0184042 Sec.2.4 Chap.4]. One can easily conclude this from (\[eq:normalizing\]). Moreover, the following assertion holds true.
\[prop:interlacing\] The eigenvalues of two different self-adjoint extensions of a Jacobi operator interlace, that is, there is only one eigenvalue of a self-adjoint extension between two eigenvalues of any other self-adjoint extension.
One may arrive at this assertion via rank-one perturbation theory, in particular by recurring to the Aronzajn-Krein formula [@simon1 Eq. 1.13]. Nonetheless, we provide below a simple proof to illustrate the use of (\[eq:normalizing\]). The proof of this statement for regular simple symmetric operators can be found in [@R1466698 Prop.3.4 Chap.1].
The proof of this assertion follows from the expression (\[eq:normalizing\]). It is similar to the proof of [@MR0184042 Thm.1.2.2].
Note that (\[eq:R-function-def\]) implies that the entire function $\mathfrak{R}_\tau(\zeta)$, $\tau\in\mathbb{R}\cup\{\infty\}$, is real, i.e., it takes real values when evaluated on the real line. Let $\lambda_k<\lambda_{k+1}$ be two neighboring eigenvalues of the self-adjoint extension $J({\tau_2})$ of $J$, with ${\tau_2}\in\mathbb{R}\cup\{\infty\}$. So $\lambda_k$, $\lambda_{k+1}$ are zeros of $\mathfrak{R}_{\tau_2}$ and by (\[eq:normalizing\]) these zeros are simple. Since $\mathfrak{R}_{\tau_2}'(\lambda_k)$ and $\mathfrak{R}_{\tau_2}'(\lambda_{k+1})$ have different signs, it follows from (\[eq:normalizing\]) that $\mathfrak{R}_{\tau_1}(\lambda_k)$ and $\mathfrak{R}_{\tau_1}(\lambda_{k+1})$ (${\tau_1}\in\mathbb{R}\cup\{\infty\},\,
{\tau_1}\ne {\tau_2}$) have also opposite signs. From the continuity of $\mathfrak{R}_{\tau_1}$ on the interval $[\lambda_k,\lambda_{k+1}]$, there is at least one zero of $\mathfrak{R}_{\tau_1}$ in $(\lambda_k,\lambda_{k+1})$. Now, suppose that in this interval there is more than one zero of $\mathfrak{R}_{\tau_1}$, so one can take two neighboring zeros of $\mathfrak{R}_{\tau_1}$ in $(\lambda_k,\lambda_{k+1})$. By reproducing the argumentation above with ${\tau_1}$ and ${\tau_2}$ interchanged, one obtains that there is at least one zero of $\mathfrak{R}_{\tau_2}$ somewhere in $(\lambda_k,\lambda_{k+1})$. This contradicts the assumption that $\lambda_k$ and $\lambda_{k+1}$ are neighbors.\
The assertion of the following proposition is a well established fact (see, for instance [@MR0052004 Thm.1]). We, nevertheless, provide the proof for the reader’s convenience and because we introduce in it notation for later use. Note that a non-constant entire function of at most minimal type of order one must have zeros, otherwise, by Weierstrass theorem on the representation of entire functions by infinite products [@MR589888 Thm.3 Chap.1], it would be a function of at least normal type.
Before stating the proposition we remind the definition of convergence exponent of a sequence of complex numbers (see [@MR589888 Sec.4 Chap.1]). The convergence exponent $\rho_1$ of a sequence $\{\nu_k\}_{k}$ of non-zero complex numbers accumulating only at infinity is given by $$\label{eq:convergence-exp}
\rho_1:=\inf\,\left\{\gamma\in\mathbb{R}:\lim_{r\to\infty}\sum_{{\left|\nu_k\right|}\le r}
\frac{1}{{\left|\nu_k\right|}^\gamma}<\infty\right\}\,.$$ We also remark that, as it is customary, whenever we say that an infinite product is convergent we mean that at most a finite number of factors may be zero and the partial product formed by the non-vanishing factors tends to a number different from zero [@ahlfors Sec.2.2 Chap.5].
\[prop:entire-minimal-1-order-rep\] Let $f(\zeta)$ be an entire function of at most minimal type of order one with an infinite number of zeros. Let the elements of the sequence $\{\nu_k\}_k$, which accumulate only at infinity, be the non-zero roots of $f$, where $\{\nu_k\}_k$ contains as many elements for each zero as its multiplicity. Assume that $m\in\mathbb{N}\cup\{0\}$ is the order of the zero of $f$ at the origin. Then there exists a complex constant $C$ such that $$\label{eq:entire-minimal-1-order-representation}
f(\zeta)=C\zeta^m\lim_{r\to\infty}
\prod_{{\left|\nu_k\right|}\le r}
\left(1-\frac{\zeta}{\nu_k}\right)\,,$$ where the limit converges uniformly on compacts of $\mathbb{C}$.
The convergence exponent $\rho_1$ of the zeros of an arbitrary entire function does not exceed its order [@MR589888 Thm.6 Chap.1]. Then, for a function of at most minimal type of order one, $\rho_1\le 1$. According to Hadamard’s theorem [@MR589888 Thm.13 Chap.1], the expansion of $f$ in an infinite product has either the form: $$\label{eq:hadamard1}
f(\zeta)=\zeta^me^{a\zeta +b}\lim_{r\to\infty}\prod_{{\left|\nu_k\right|}\le r}
G\left(\frac{\zeta}{\nu_k};0\right)\,,\quad
a,b\in\mathbb{C}$$ if the limit $$\label{eq:limit-series}
\lim_{r\to\infty}\sum_{{\left|\nu_k\right|}\le r}
\frac{1}{{\left|\nu_k\right|}}$$ converges, or $$\label{eq:hadamard2}
f(\zeta)=\zeta^me^{c\zeta+d}\lim_{r\to\infty}\prod_{{\left|\nu_k\right|}\le r}
G\left(\frac{\zeta}{\nu_k};1\right)\,,\quad
c,d\in\mathbb{C}$$ if (\[eq:limit-series\]) diverges. We have used here the Weierstrass primary factors $G$ (for details see [@MR589888 Sec.3 Chap.1]). Let us suppose that the order is one and (\[eq:limit-series\]) diverges, then, in view of the fact that $f$ is of minimal type, by a theorem due to Lindelöf [@MR589888 Thm.15a Chap.1], we have in particular that $$\lim_{r\to\infty}
\sum_{{\left|\nu_k\right|}\le r}\nu_k^{-1}=-c\,.$$ This implies the uniform convergence of the series $\lim_{r\to\infty}
\sum_{{\left|\nu_k\right|}\le r}\frac{\zeta}{\nu_k}$ on compacts of $\mathbb{C}$. In its turn, since $\rho_1= 1$, this yields that $\lim_{r\to\infty}\prod_{{\left|\nu_k\right|}\le
r}\left(1-\frac{\zeta}{\nu_k}\right)$ is uniformly convergent on any compact of $\mathbb{C}$. Therefore, $$\lim_{r\to\infty}\prod_{{\left|\nu_k\right|}\le r}
G\left(\frac{\zeta}{\nu_k};1\right)=e^{-c\zeta}\lim_{r\to\infty}
\prod_{{\left|\nu_k\right|}\le r}
\left(1-\frac{\zeta}{\nu_k}\right)\,.$$ Thus, (\[eq:hadamard2\]) can be written as (\[eq:entire-minimal-1-order-representation\]).
Suppose now that the limit (\[eq:limit-series\]) converges. If the order of the function is less than one, then, by [@MR589888 Thm.13 Chap.1], one may write (\[eq:hadamard1\]) as (\[eq:entire-minimal-1-order-representation\]). If the order of the function is one, by [@MR589888 Thm. 12, Thm.15b Chap.1], one concludes again that (\[eq:hadamard1\]) can be written as (\[eq:entire-minimal-1-order-representation\]) (cf. Thm 15 in the Russian version of [@MR589888] or, alternatively, [@MR1400006 Lect. 5]).\
Let $\{\lambda_n(\tau)\}_n$ be the eigenvalues of $J(\tau)$. In view of the fact that $\mathfrak{R}_\tau(\zeta)$ is an entire function of at most minimal type of order one, by Proposition \[prop:entire-minimal-1-order-rep\], one can always write $$\label{eq:R-expansion}
\mathfrak{R}_\tau(\zeta)=C_\tau\zeta^{\delta_{\tau}}\lim_{r\to\infty}
\prod_{0<{\left|\lambda_k(\tau)\right|}\le r}
\left(1-\frac{\zeta}{\lambda_k(\tau)}\right)\,,\qquad
\tau\in\mathbb{R}\cup\{\infty\}\,,$$ where $C_\tau\in\mathbb{R}\setminus\{0\}$ and $\delta_{\tau}$ is the Kronecker delta, i.e., $\delta_\tau=1$ if $\tau=0$, and $\delta_\tau=0$ otherwise. The limits in (\[eq:R-expansion\]) converge uniformly on compacts of $\mathbb{C}$. Note that when $\tau=0$ we have naturally excluded $\lambda_k(0)=0$ from the infinite product.
When writing (\[eq:R-expansion\]), we have taken into account, on the one hand, that $\mathfrak{R}_0(0)=0$, which follows from (\[eq:R-function-def\]) and the definition of the function $D$, and on the other, that different self-adjoint extensions have disjoint spectra (see Section \[sec:preliminaries\]).
Now, let us consider the following expressions derived from the Green’s formula [@MR0184042 Eqs.1.23, 2.28] $$\label{eq:p}
D(\zeta)=\zeta\sum_{k=0}^\infty P_k(0)P_k(\zeta)\,,$$ $$\label{eq:q}
B(\zeta)=-1+\zeta\sum_{k=0}^\infty Q_k(0)P_k(\zeta)\,.$$ Again we verify from (\[eq:p\]) that $D(0)=0$, while from (\[eq:q\]) we have $B(0)=-1$. Therefore $\mathfrak{R}_\tau(0)=-\tau$ for every $\tau\in\mathbb{R}$, and $\mathfrak{R}_\infty(0)=-1$. Thus, $C_\tau=-\tau$ provided that $\tau\in\mathbb{R}$ and $\tau\ne 0$, and $C_\infty=-1$.
To simplify the writing of some of the formulae below, let us introduce $R_\tau(\zeta):=\frac{\mathfrak{R}_\tau(\zeta)}{C_\tau}$, that is, $$\label{eq:R-without-constants}
R_\tau(\zeta):=\zeta^{\delta_{\tau}}\lim_{r\to\infty}
\prod_{0<{\left|\lambda_k(\tau)\right|}\le r}
\left(1-\frac{\zeta}{\lambda_k(\tau)}\right)\,,\qquad
\tau\in\mathbb{R}\cup\{\infty\}\,,$$ where $\delta_\tau$ is defined as in (\[eq:R-expansion\]).
Due to the uniform convergence of the expression $$\frac{d}{d\zeta}\left[\prod_{0<{\left|\lambda_k(\tau)\right|}\le
r}\left(1-\frac{\zeta}{\lambda_k(\tau)}\right)\right]\,,\quad\text{ as }\quad
r\to\infty\,,$$ one has $$\label{eq:R-derivative}
R_\tau'(\lambda_j(\tau))=
\begin{cases}
-[\lambda_j(\tau)]^{\delta_\tau-1}\lim\limits_{r\to\infty}
\prod\limits_{\substack{0<{\left|\lambda_k(\tau)\right|}\le r\\ k\ne j}}
\left(1-\frac{\lambda_j(\tau)}{\lambda_k(\tau)}\right) &
\lambda_j(\tau)\ne 0\\
1& \lambda_j(\tau)= 0
\end{cases}$$ By (\[eq:normalizing\]) and (\[eq:R-derivative\]), one obtains $$\label{eq:C-0}
C_0=a(0).$$
\[thm:recovering\] Let ${\tau_1},{\tau_2}\in\mathbb{R}\cup\{\infty\}$ with ${\tau_1}\ne {\tau_2}$. The spectra $\{\lambda_k({\tau_1})\}_k$, $\{\lambda_k({\tau_2})\}_k$ of two different self-adjoint extensions $J({\tau_1})$, $J({\tau_2})$ of a Jacobi operator $J$ in the limit circle case uniquely determine the matrix associated with $J$, and the numbers ${\tau_1}$ and ${\tau_2}$.
For definiteness assume that ${\tau_1}\ne 0$, in other words that the sequence $\{\lambda_k({\tau_1})\}_k$ does not contain any zero element.
By (\[eq:normalizing\]), we have $$\label{eq:normalizing-constants}
a(\lambda_k({\tau_1}))=MR_{\tau_2}(\lambda_k({\tau_1}))
R_{\tau_1}'(\lambda_k({\tau_1}))\,,$$ where $$\label{eq:constant-m}
M=
\begin{cases}
\frac{{\tau_1}{\tau_2}}{{\tau_1}-{\tau_2}} & \text{if}\quad
{\tau_1},{\tau_2}\in\mathbb{R}\setminus\{0\}\\
{\tau_2} & \text{if}\quad {\tau_1}=\infty\,,\quad {\tau_2}\ne 0\\
-{\tau_1} & \text{if}\quad {\tau_2}=\infty\\
-C_0 & \text{if}\quad {\tau_2}=0
\end{cases}$$ Now, since $\{a(\lambda_k({\tau_1}))\}_k$ are the normalizing constants of $J({\tau_1})$ we must have $$1=\sum_k\frac{1}{a(\lambda_k({\tau_1}))}=\frac{1}{M}
\sum_k\frac{1}{R_{\tau_2}(\lambda_k({\tau_1}))
R_{\tau_1}'(\lambda_k({\tau_1}))}\,.$$ Therefore $$\label{eq:calculation-M}
M=\sum_k\frac{1}{R_{\tau_2}(\lambda_k({\tau_1}))
R_{\tau_1}'(\lambda_k({\tau_1}))}\,.$$ Thus, $M$ is completely determined by the sequences $\{\lambda_k({\tau_2})\}_k$ and $\{\lambda_k({\tau_1})\}_k$. Inserting the obtained value of $M$ into (\[eq:normalizing-constants\]) one obtains the normalizing constants. Having the normalizing constants allows us to construct the spectral measure for $J({\tau_1})$. Then, by standard methods (see Section \[sec:preliminaries\]), one reconstructs the matrix associated with $J$ and the boundary condition at infinity ${\tau_1}$. From the value of $M$ and ${\tau_1}$ one obtains ${\tau_2}$, by using the first three cases in (\[eq:constant-m\]). When $0\in\{\lambda_k({\tau_2})\}_k$, one does not use (\[eq:constant-m\]), since it is already known that ${\tau_2}=0$.\
Note that the proof of Theorem \[thm:recovering\] gives a reconstruction method of the Jacobi matrix. Although mentioned earlier, we also remark here that the assertion of Theorem \[thm:recovering\], for the case of ${\tau_1},{\tau_2}\in\mathbb{R}$, was announced without proof in [@MR1045318 Thm.1].
Necessary and sufficient conditions {#sec:n-s-conditions}
===================================
In this section we give a complete characterization of our two-spectra inverse problem. We remind the reader about the remark on the notation at the end of Section \[sec:preliminaries\].
First we prove the following simple proposition related to the converse of Proposition \[prop:entire-minimal-1-order-rep\].
\[prop:prod-to-entire-mtoo\] Let $\{\nu_k\}_k$ be an infinite sequence of non-vanishing complex numbers accumulating only at $\infty$, and whose convergence exponent $\rho_1$ does not exceed one. Suppose that the infinite product $$\label{eq:infinite-prod}
\lim_{r\to\infty}
\prod_{{\left|\nu_k\right|}\le r}
\left(1-\frac{\zeta}{\nu_k}\right)$$ converges uniformly on any compact of $\mathbb{C}$. Then this product is an entire function of at most minimal type of order one if either (\[eq:limit-series\]) converges or if (\[eq:limit-series\]) diverges but the following holds $$\label{eq:density}
\lim_{r\to\infty}\frac{n(r)}{r}=0\,,$$ where $n(r)$ is the number of elements of $\{\nu_k\}_k$ in the circle ${\left|\zeta\right|}<r$.
Clearly, by the conditions of the theorem, one can express (\[eq:infinite-prod\]) in terms of canonical products [@MR589888 Sec.3 Chap.1] either in the form $$\label{eq:first-case}
\lim_{r\to\infty}
\prod_{{\left|\nu_k\right|}\le r}
\left(1-\frac{\zeta}{\nu_k}\right)=\lim_{r\to\infty}\prod_{{\left|\nu_k\right|}\le r}
G\left(\frac{\zeta}{\nu_k};0\right)$$ whenever (\[eq:limit-series\]) converges, or in the form $$\label{eq:second-case}
\lim_{r\to\infty}
\prod_{{\left|\nu_k\right|}\le r}
\left(1-\frac{\zeta}{\nu_k}\right)=e^{c\zeta}\lim_{r\to\infty}\prod_{{\left|\nu_k\right|}\le r}
G\left(\frac{\zeta}{\nu_k};1\right)$$ otherwise, where $$\label{eq:zero-distribution}
\lim_{r\to\infty} \sum_{{\left|\nu_k\right|}\le
r}\nu_k^{-1}=-c\,.$$ In the case (\[eq:first-case\]), in which the genus of the product is less than the convergence exponent of $\{\nu_k\}_k$, it is clear that (\[eq:infinite-prod\]) does not grow faster than an entire function of minimal type of order one. Indeed, by [@MR589888 Thm.7 Chap.1], the order of a canonical product is equal to the convergence exponent, so when $\rho_1<1$ the assertion is obvious. For $\rho_1=1$ the statement follows from [@MR589888 Thm.15b Chap.1].
If we have the representation (\[eq:second-case\]), then $\rho_1=1$. By [@MR589888 Thm.7 Chap.1], the canonical product has order one. Since the product of functions of the same order is of that same order, the order of (\[eq:infinite-prod\]) is one. Then, the assertion follows from [@MR589888 Thm.15a Chap.1] due to (\[eq:density\]) and (\[eq:zero-distribution\]).\
Before passing on to the main results of this section, we establish an auxiliary result which is related to part of the proof of Theorem 1 in the Addenda and Problems of [@MR0184042 Chap.4].
\[lem:auxiliary\] Consider an infinite real sequence $\{\kappa_j\}_j$ and a sequence $\{\alpha_j\}_j$ of positive numbers such that $$\sum_j\frac{\kappa_j^{2m}}{\alpha_j}<\infty\quad
\text{ for all}\ m=0,1,\dots$$ Let $\mathfrak{F}$ be an entire function of at most minimal type of order one whose zeros, $\{\kappa_j\}_j$, are simple, and such that $$\label{eq:unboundedness-along-imaginary}
{\left|\mathfrak{F}({{\rm i}}t)\right|}\to\infty\ \text{ as }\
t\to\pm\infty\,,\quad t\in\mathbb{R}\,.$$ If $$\sum_j\frac{\alpha_j}{(1+\kappa_j^2)[\mathfrak{F}'(\kappa_j)]^2}<\infty\,,$$ then $$\label{eq:lemma-absolute-convergence}
\sum_j\frac{\kappa_j^m}{\mathfrak{F}'(\kappa_j)}$$ is absolutely convergent for $m=0,1,\dots$, and the absolutely convergent expansion $$\frac{1}{\mathfrak{F}(\zeta)}=
\sum_j\frac{1}{\mathfrak{F}'(\kappa_j)(\zeta-\kappa_j)}$$ holds true for all $\zeta\in\mathbb{C}\setminus \{\kappa_j\}_j$.
The absolutely convergence of (\[eq:lemma-absolute-convergence\]) follows from $$\begin{split}
\sum_j{\left|\frac{\kappa_j^m}{\mathfrak{F}'(\kappa_j)}\right|}&=
\sum_j{\left|\frac{\sqrt{\alpha_j}}{\sqrt{1+\kappa_j^2}\mathfrak{F}'(\kappa_j)}\right|}
{\left|\frac{\kappa_j^m\sqrt{1+\kappa_j^2}}{\sqrt{\alpha_j}}\right|}\\
&\le\sqrt{\sum_j\frac{\alpha_j}{(1+\kappa_j^2)[\mathfrak{F}'(\kappa_j)]^2}}
\sqrt{\sum_j\frac{\kappa_j^{2m}+\kappa_j^{2m+2}}{\alpha_j}}<\infty
\end{split}$$ Construct the function $$h(\zeta):=1-\mathfrak{F}(\zeta)\sum_j\frac{1}
{\mathfrak{F}'(\kappa_j)(\zeta-\kappa_j)}\,,$$ where the series is absolutely convergent in compact subsets of $\mathbb{C}\setminus\{\kappa_j\}_j$ because of (\[eq:lemma-absolute-convergence\]). Clearly, $h(\kappa_j)=0$ for any $j$. Moreover, it turns out that $h$ is an entire function of at most minimal type of order one. To show this, first consider the case when $\sum_j{\left|\kappa_j\right|}^{-1}<\infty$. Here, by what we have discussed in the proof of Proposition \[prop:entire-minimal-1-order-rep\], the function $\mathfrak{F}(\zeta)/(\zeta-\kappa_j)$ can be expressed by a canonical product of genus zero. It follows from [@MR589888 Lem.3 Chap.1] (see also the proof of [@MR589888 Thm.4 Chap.1]) that, on the one hand, $$\max_{{\left|\zeta\right|}=r}{\left|\frac{\mathfrak{F}(\zeta)}{(\zeta-\kappa_j)}\right|}<
\exp\left(C(\alpha)r^\alpha\right)\,,\quad \rho_1<\alpha<1\,,$$ for any $r>0$ provided that $\rho_1<1$. If, on the other hand, $\rho_1=1$, then for any $\epsilon>0$, there exists $R_0>0$ such that $$\label{eq:uniform-j-estimation}
\max_{{\left|\zeta\right|}=r}{\left|\frac{\mathfrak{F}(\zeta)}{(\zeta-\kappa_j)}\right|}<
\exp\left(\epsilon
r\right)$$ for all $r>R_0$. Hence, in any case, we have the uniform, with respect to $j$, asymptotic estimation (\[eq:uniform-j-estimation\]) when $\sum_j{\left|\kappa_j\right|}^{-1}<\infty$.
Suppose now that $\sum_j{\left|\kappa_j\right|}^{-1}=\infty$. In this case, as was shown in the proof of Proposition \[prop:entire-minimal-1-order-rep\], $$\frac{\mathfrak{F}(\zeta)}{(\zeta-\kappa_j)}=-\frac{1}{\kappa_j}\zeta^me^{(c+\kappa_j^{-1})\zeta+d}
\lim_{r\to\infty}
\prod_{\substack{{\left|\kappa_k\right|}\le r\\ k\ne j}}G\left(\frac{\zeta}{\kappa_k};1\right)\,,$$ where $$\label{eq:limit-to--c}
\lim_{r\to\infty}\sum_{{\left|\kappa_k\right|}\le r}\kappa_k^{-1}=-c\,.$$ On the basis of the estimates found in the proof of [@MR589888 Thm.15 Chap.1] (see in particular the inequality next to [@MR589888 Eq.1.43]), one can find $R_1$, independent of $j$, such that $$\max_{{\left|\zeta\right|}=r}{\left|\frac{\mathfrak{F}(\zeta)}{(\zeta-\kappa_j)}\right|}<
\exp\left[r\left({\left|c+\sum_{{\left|\kappa_k\right|}\le r}\kappa_k^{-1}\right|} +
C\left(\limsup_{r\to\infty}\frac{n(r)}{r}+\epsilon\right)+
O\left(\frac{1}{r}\right)\right)\right]$$ for all $r>R_1$ and $\epsilon>0$ (see the definition of $n(r)$ in the statement of Proposition \[prop:prod-to-entire-mtoo\]). Note that if ${\left|\kappa_j\right|}\le r$, then the above inequality follows directly from the inequality next to [@MR589888 Eq.1.43]. If ${\left|\kappa_j\right|}> r$, the same inequality holds due to $${\left|c+\kappa_j^{-1}+\sum_{{\left|\kappa_k\right|}\le r}\kappa_k^{-1}\right|}\le
{\left|c+\sum_{{\left|\kappa_k\right|}\le r}\kappa_k^{-1}\right|}+\frac{1}{r}\,.$$ Since $\mathfrak{F}$ does not grow faster than a function of minimal type of order one, by [@MR589888 Thm.15a Chap.1], one again verifies that, for any $\epsilon>0$, (\[eq:uniform-j-estimation\]) holds for all $r$ greater than a certain $R_2$ depending only on the velocity of convergence in the limits (\[eq:limit-to–c\]) and (\[eq:density\]).
Thus, one concludes that, for any $\epsilon>0$, there is $R>0$ such that $$\max_{{\left|\zeta\right|}=r}{\left|\mathfrak{F}(\zeta)\sum_j\frac{1}
{\mathfrak{F}'(\kappa_j)(\zeta-\kappa_j)}\right|}\le
\sum_j\frac{1}{{\left|\mathfrak{F}'(\kappa_j)\right|}}
\max_{{\left|\zeta\right|}=r}{\left|\frac{\mathfrak{F}(\zeta)}{(\zeta-\kappa_j)}\right|}<\exp(\epsilon
r)$$ for all $r>R$, which shows that $h$ is an entire function of at most minimal type of order one.
Now, the function $h/\mathfrak{F}$ is also an entire function of at most minimal type of order one [@MR589888 Cor. Sec.9 Chap.1]. By the hypothesis (\[eq:unboundedness-along-imaginary\]), $$\lim_{\substack{t\to\pm\infty \\ t\in\mathbb{R}}}
\frac{h({{\rm i}}t)}{\mathfrak{F}({{\rm i}}t)}=0\,,$$ which implies that $h/\mathfrak{F}\equiv 0$ (see Corollary of [@MR589888 Sec.14 Chap.1]).\
\[thm:necessary-sufficient\] Let $\{\lambda_k\}_k$ and $\{\mu_k\}_k$ be two infinite sequences of real numbers such that
a) $\{\lambda_k\}_k\cap\{\mu_k\}_k=\emptyset$. For definiteness we assume that $0\not\in\{\lambda_k\}_k$
b) the sequences accumulate only at the point at infinity.
c) $\lambda_k\ne\lambda_j$, $\mu_k\ne\mu_j$ for $k\ne j$.
Then there exist unique ${\tau_1},{\tau_2}\in\mathbb{R}\cup\{\infty\}$, with ${\tau_1}\ne0$, ${\tau_1}\ne {\tau_2}$, and a unique Jacobi operator $J\ne J^*$ such that $\{\lambda_k\}_k=\operatorname{\sigma}(J({\tau_1}))$ and $\{\mu_k\}_k=\operatorname{\sigma}(J({\tau_2}))$ if and only if the following conditions are satisfied.
1. The convergence exponents of the sequence $\{\lambda_k\}_k$, and of the non-zero elements of $\{\mu_k\}_k$ do not exceed one. Additionally, if $$\lim_{r\to\infty}\sum_{{\left|\lambda_k\right|}\le r}
\frac{1}{{\left|\lambda_k\right|}}=\infty\,,\quad\text{ require that }\quad
\lim_{r\to\infty}\frac{n_\lambda(r)}{r}=0\,,$$ and if $$\lim_{r\to\infty}\sum_{0<{\left|\mu_k\right|}\le r}
\frac{1}{{\left|\mu_k\right|}}=\infty\,,\quad\text{ require that }\quad
\lim_{r\to\infty}\frac{n_\mu(r)}{r}=0\,,$$ where $n_\lambda(r)$ and $n_\mu(r)$ are the number of elements of $\{\lambda_k\}_k$ and $\{\mu_k\}_k$, respectively, in the circle ${\left|\zeta\right|}<r$. \[nec-suf0\]
2. The limits $$\lim_{r\to\infty}
\prod_{{\left|\lambda_k\right|}\le r}
\left(1-\frac{\zeta}{\lambda_k}\right)
\qquad
\lim_{r\to\infty}
\prod_{0<{\left|\mu_k\right|}\le r}
\left(1-\frac{\zeta}{\mu_k}\right)\,,$$ converge uniformly on compact subsets of $\mathbb{C}$, and they define the functions $$\label{eq:r-lambda-r-mu-1}
\mathcal{R}_\lambda(\zeta):=
\lim_{r\to\infty}
\prod_{{\left|\lambda_k\right|}\le r}
\left(1-\frac{\zeta}{\lambda_k}\right)$$ $$\label{eq:r-lambda-r-mu-2}
\mathcal{R}_\mu(\zeta):=\zeta^\delta\lim\limits_{r\to\infty}
\prod\limits_{0<{\left|\mu_k\right|}\le r}
\left(1-\frac{\zeta}{\mu_k}\right)\,,$$ where $\delta=1$ if $0\in\{\mu_k\}_k$, and $\delta=0$ otherwise. \[nec-suf1\]
3. All numbers $\mathcal{R}_\mu(\lambda_j)\mathcal{R}_\lambda'(\lambda_j)$ have the same sign for all $j$. The same is true for the numbers $\mathcal{R}_\lambda(\mu_j)\mathcal{R}_\mu'(\mu_j)$. \[nec-suf-constancy-sign\]
4. For every $m=0,1,2,\dots$ the series below are convergent and the following equalities hold $$\sum_j\frac{\lambda_j^m}{\mathcal{R}_\mu(\lambda_j)\mathcal{R}_\lambda'(\lambda_j)}=-
\sum_j\frac{\mu_j^m}{\mathcal{R}_\lambda(\mu_j)\mathcal{R}_\mu'(\mu_j)}$$\[nec-suf-moments\]
5. The series $$\sum_j\frac{\mathcal{R}_\mu(\lambda_j)}{\mathcal{R}_\lambda'(\lambda_j)}
\quad\text{ and }\quad
\sum_j\frac{\mathcal{R}_\lambda(\mu_j)}{\mathcal{R}_\mu'(\mu_j)}$$ diverge either to $-\infty$ or $+\infty$.\[nec-suf-divergence\]
6. The series $$\sum_j\frac{\mathcal{R}_\mu(\lambda_j)}{(1+\lambda_j^2)\mathcal{R}_\lambda'(\lambda_j)}
\quad\text{ and }\quad
\sum_j\frac{\mathcal{R}_\lambda(\mu_j)}{(1+\mu_j^2)\mathcal{R}_\mu'(\mu_j)}$$ are convergent. \[nec-suf3\]
We begin by proving that if $\{\lambda_k\}_k$ and $\{\mu_k\}_k$ are, respectively, the spectra of the self-adjoint extensions $J({\tau_1})$ and $J({\tau_2})$ of a Jacobi operator $J$, then conditions *\[nec-suf0\]*–*\[nec-suf3\]* hold true.
Since $\{\lambda_k\}_k=\operatorname{\sigma}(J({\tau_1}))$ and $\{\mu_k\}_k=\operatorname{\sigma}(J({\tau_2}))$, the functions $\mathfrak{R}_{\tau_1}$ and $\mathfrak{R}_{\tau_2}$, given by (\[eq:R-function-def\]), have the sequences $\{\lambda_k\}_k$ and $\{\mu_k\}_k$, respectively, as their sets of zeros. These functions do not grow faster than an entire function of minimal type of order one. By Proposition \[prop:entire-minimal-1-order-rep\] (see (\[eq:R-expansion\])), the limits $$\lim_{r\to\infty}
\prod_{{\left|\lambda_k\right|}\le r}
\left(1-\frac{\zeta}{\lambda_k}\right)\,,
\qquad\lim_{r\to\infty}
\prod_{0<{\left|\mu_k\right|}\le r}
\left(1-\frac{\zeta}{\mu_k}\right)$$ converge uniformly on compacts of $\mathbb{C}$. This is condition *\[nec-suf1\]*. Moreover, by (\[eq:hadamard2\]) and [@MR589888 Thm.15a Chap.1], condition *\[nec-suf0\]* holds.
The functions $\mathcal{R}_\lambda$ and $\mathcal{R}_\mu$, given by (\[eq:r-lambda-r-mu-1\]) and (\[eq:r-lambda-r-mu-2\]), coincide with $R_\tau$, given by (\[eq:R-without-constants\]), with $\tau={\tau_1}$ and $\tau={\tau_2}$, respectively. Thus (\[eq:normalizing-constants\]) is rewritten as follows $$\label{eq:normalizing-necessary}
a(\lambda_j)=M\mathcal{R}_\mu(\lambda_j)\mathcal{R}_\lambda'(\lambda_j)\,,$$ where $M$ is given by (\[eq:constant-m\]). Analogously, $$\label{eq:normalizing-necessary-mu}
a(\mu_j)=-M\mathcal{R}_\lambda(\mu_j)\mathcal{R}_\mu'(\mu_j)\,.$$
On the basis of the positiveness of the normalizing constants, from (\[eq:normalizing-necessary\]) and (\[eq:normalizing-necessary-mu\]), we obtain condition *\[nec-suf-constancy-sign\]*.
From what we discussed in Section \[sec:preliminaries\] all the moments exist for the spectral functions of $J({\tau_1})$ and $J({\tau_2})$, which are, respectively, $$\label{eq:spectral-measures}
\sum_{\lambda_k\le t}\frac{1}{a(\lambda_k)}\quad\text{ and }\quad
\sum_{\mu_k\le t}\frac{1}{a(\mu_k)}\,.$$ Hence the series in both sides of condition *\[nec-suf-moments\]* are convergent for $m\in\mathbb{N}\cup\{0\}$. Moreover, the spectral functions (\[eq:spectral-measures\]) are solutions of the same moment problem associated with $J$ (see the paragraph surrounding (\[eq:extremal-measure\])), therefore the equality of condition *\[nec-suf-moments\]* holds.
Theorem 1 in the Addenda and Problems of [@MR0184042 Chap.4] tells us that $$\label{eq:akhiezer-conds}
\sum_j\frac{a(\lambda_j)}
{\left[\mathcal{R}_\lambda'(\lambda_j)\right]^2}=+\infty$$ is a necessary condition for the sequences $\{\lambda_j\}_j$ and $\{a(\lambda_j)\}_j$ to be the spectrum of $J({\tau_1})$ and its corresponding normalizing constants. Thus, substituting (\[eq:normalizing-necessary\]) into (\[eq:akhiezer-conds\]), one establishes the divergence of the first series in condition *\[nec-suf-divergence\]*. Similarly, $$\sum_j\frac{a(\mu_j)}
{\left[\mathcal{R}_\mu'(\mu_j)\right]^2}=+\infty$$ must hold, which, by (\[eq:normalizing-necessary-mu\]), implies the divergence of the second series in condition *\[nec-suf-divergence\]*.
By the same theorem in [@MR0184042] mentioned above, and taking into account (\[eq:normalizing-necessary\]) and (\[eq:normalizing-necessary-mu\]), one obtains the convergence of the series in condition *\[nec-suf3\]*.
Let us now prove that the conditions *\[nec-suf0\]*–*\[nec-suf3\]* are sufficient. Using condition *\[nec-suf1\]* and the convergence of the series in the left hand side of condition *\[nec-suf-moments\]* with $m=0$, we define the real constant $$\label{eq:definition-of-m-cal}
\mathcal{M}:= \sum_j\frac{1}{\mathcal{R}_\mu(\lambda_j)\mathcal{R}_\lambda'(\lambda_j)}$$ and the sequence of numbers $$\label{eq:sequence-def}
a_j:=\mathcal{M}\mathcal{R}_\mu(\lambda_j)\mathcal{R}_\lambda'(\lambda_j)$$ By condition *\[nec-suf-constancy-sign\]* and (\[eq:definition-of-m-cal\]), it follows that $a_j>0$ for all $j$. Moreover, (\[eq:definition-of-m-cal\]) and (\[eq:sequence-def\]) imply that $\sum_ja_j^{-1}=1$.
With the aid of the sequences $\{\lambda_k\}$ and $\{a_k\}$ define the function $\rho :\mathbb{R}\to\mathbb{R}_+$ as follows $$\label{eq:definition-sigma}
\rho(t):=\sum_{\lambda_k\le t}a_k^{-1}.$$ Consider the self-adjoint operator of multiplication $A_\rho$ by the independent variable in $L_2(\mathbb{R},d\rho)$. We show below that this operator is the canonical representation (see Section \[sec:preliminaries\]) of a self-adjoint extension of a Jacobi matrix in the limit circle case. The proof of this fact is similar to the proof of Theorem 2 in Addenda and Problems of [@MR0184042 Chap.4]. Note, however, that our conditions are slightly different.
Consider a function $\theta_k(t)\in L_2(\mathbb{R},d\rho)$ such that $$\label{eq:definition-theta}
\theta_k(\lambda_j)=\sqrt{a_k}\delta_{kj}\,.$$ Clearly, $\theta_k(t)$ is the normalized eigenvector of $A_\rho$ corresponding to $\lambda_k$. Let $\varphi(t)\in
L_2(\mathbb{R},d\rho)$ be such that $$\label{eq:definition-phi}
\langle\varphi,\theta_j\rangle_{L_2(\mathbb{R},d\rho)}=
\frac{\sqrt{a_j}}{(\lambda_j-{{\rm i}})\mathcal{R}_\lambda'(\lambda_j)}\,.$$ Taking into account (\[eq:sequence-def\]), it is clear that the convergence of the first series in condition *\[nec-suf3\]* ensures that $\varphi(t)$ is indeed an element of $L_2(\mathbb{R},d\rho)$. Define $$\label{eq:definition-domain}
D:=\left\{\xi\in L_2(\mathbb{R},d\rho):\xi=(A_\rho+{{\rm i}}I)^{-1}\psi,
\,\psi\in L_2(\mathbb{R},d\rho),\,\psi\perp\varphi\right\}\,.$$ Since $D\subset\operatorname{Dom}(A_\rho)$, we can consider the restriction of $A_\rho$ to the linear set $D$. Let us show that this restriction is a symmetric operator with deficiency indices $(1,1)$. First we verify that $D$ is dense in $L_2(\mathbb{R},d\rho)$. Suppose that a non-zero ${\eta}\in
L_2(\mathbb{R},d\rho)$ is orthogonal to $D$. This would imply that there is a non-zero constant $C\in\mathbb{C}$ such that $${\eta}=C(A_\rho-{{\rm i}}I)\varphi\,.$$ Therefore, $$\langle{\eta},\theta_j\rangle_{L_2(\mathbb{R},d\rho)}=
C\frac{\sqrt{a_j}}{\mathcal{R}_\lambda'(\lambda_j)}\,,$$ whence we easily conclude that ${\eta}\in L_2(\mathbb{R},d\rho)$ would contradict condition *\[nec-suf-divergence\]*.
Consider now the restriction of $A_\rho$ to the set $D$, denoted henceforth by $A_\rho\upharpoonright_D$, and let us find the dimension of $\ker ((A_\rho\upharpoonright_D)^*-{{\rm i}}I)$ which is characterized as the set of all $\omega\in
L_2(\mathbb{R},d\rho)$ for which the equation $$\langle(A_\rho+{{\rm i}}I)\xi,\omega\rangle_{L_2(\mathbb{R},d\rho)}=0$$ is satisfied for any $\xi\in D$. It is not difficult to show that any such $\omega$ can be written as follows $$\omega=\widetilde{C}\varphi\,,\qquad 0\ne\widetilde{C}\in\mathbb{C}.$$ Hence $\dim\ker ((A_\rho\upharpoonright_D)^*-{{\rm i}}I)=1$. Analogously, it can be shown that the dimension of $\ker
((A_\rho\upharpoonright_D)^*+{{\rm i}}I)$ also equals one. Indeed, if $\omega\in\ker ((A_\rho\upharpoonright_D)^*+{{\rm i}}I)$ then, up to a complex constant $\omega=(A_\rho-{{\rm i}}I)(A_\rho+{{\rm i}}I)^{-1}\varphi$.
Now we show that $A_\rho\upharpoonright_D$ is the canonical representation of a Jacobi operator in the limit circle case and $A_\rho$ is the canonical representation of a self-adjoint extension of this Jacobi operator. We proceed stepwise.
I. We orthonormalize the sequence of functions $\{t^n\}_{n=0}^\infty$ with respect to the inner product of $L_2(\mathbb{R},d\rho)$. Note that condition *\[nec-suf-moments\]* guarantees that all elements of the sequence $\{t^n\}_{n=0}^\infty$ are in $L_2(\mathbb{R},d\rho)$. We obtain thus a sequence of polynomials $\{P_{n-1}(t)\}_{n=1}^\infty$ which satisfy the three term recurrence equation (\[eq:favard-system1\]) and (\[eq:favard-system2\]), where all the coefficients $b_k$ ($k\in\mathbb{N}$) turn out to be positive and $q_k$ ($k\in\mathbb{N}$) are real numbers.
II. We verify that the polynomials are dense in $L_2(\mathbb{R},d\rho)$, so the sequence we have constructed is a basis in $L_2(\mathbb{R},d\rho)$. Note first that the function $\mathcal{R}_\lambda(\zeta)$ is entire of at most minimal type of order one. Indeed, this follows from Proposition \[prop:prod-to-entire-mtoo\], in view of conditions *\[nec-suf0\]* and *\[nec-suf1\]*. Now, for any element $\lambda_{k_0}$ of the sequence $\{\lambda_k\}_k$, we clearly have $${\left|\mathcal{R}_\lambda({{\rm i}}t)\right|}\ge
{\left|1+\frac{t^2}{\lambda_{k_0}^2}\right|}\,,
\qquad t\in\mathbb{R}\,.$$ This implies that $\mathcal{R}_\lambda$ satisfies (\[eq:unboundedness-along-imaginary\]). Hence the function $\mathcal{R}_\lambda$ and the sequences $\{\lambda_j\}_j$, $\{a_j\}_j$ satisfy the condition of Lemma \[lem:auxiliary\]. Thus, we have shown the convergence of the series $$\label{eq:convergence-lemma}
\sum_j\frac{\lambda_j^m}{\mathcal{R}_\lambda'(\lambda_j)}$$ for all $m=0,1,2,\dots$, and that $$\label{eq:decomposition-lemma}
\frac{1}{\mathcal{R}_\lambda(\zeta)}=
\sum_j\frac{1}{\mathcal{R}_\lambda'(\lambda_j)(\zeta-\lambda_j)}\,.$$ Taking into account Definitions 1 and 2 of the Addenda and Problems of [@MR0184042 Chap.4], one obtains from Corollary 2 of [@MR0184042 Addenda and Problems Chap.4], together with conditions *\[nec-suf-divergence\]* and *\[nec-suf3\]*, that $\{\lambda_k\}$ is a canonical sequence of nodes and $\{a_k^{-1}\}$ the corresponding sequence of masses for the moment problem given by $\{s_m\}_{m=0}^\infty$ with $$\label{eq:s-moments}
s_m:=\frac{1}{\mathcal{M}}\sum_j\frac{\lambda_j^{m}}
{\mathcal{R}_\mu(\lambda_j)\mathcal{R}_\lambda'(\lambda_j)}\,.$$ Hence, for this moment problem, $\rho$ is a canonical solution [@MR0184042 Def.3.4.1]. By definition, a canonical solution is N-extremal and by [@MR0184042 Thm.2.3.3], the polynomials are dense in $L_2(\mathbb{R},d\rho)$.
III. We prove that the elements of the basis $\{P_{n-1}(t)\}_{n=1}^\infty$ are in $D$. From (\[eq:convergence-lemma\]) and (\[eq:decomposition-lemma\]), by Lemma 1 of the Addenda and Problems of [@MR0184042 Chap.4], one has for $m=0,1,2,\dots$ $$\sum_j\frac{\lambda_j^m}{\mathcal{R}_\lambda'(\lambda_j)}=0\,.$$ Then, if $S(t)$ is a polynomial $$\langle(A_\rho+iI)S,\varphi\rangle_{L_2(\mathbb{R},d\rho)}
=\sum_j\frac{S(\lambda_j)}{\mathcal{R}_\lambda'(\lambda_j)}=0\,.$$ Whence it follows that $S\in D$.
Now, by (\[eq:favard-system1\]) and (\[eq:favard-system2\]), it is straightforward to show that $U^{-1}A_\rho\upharpoonright_DU$ (see (\[eq:operator-u\])) is a Jacobi operator in the limit circle case.
Denote by $J$ the Jacobi operator $U^{-1}A_\rho\upharpoonright_DU$. On the basis of what was discussed in Section \[sec:preliminaries\] one can find $\tau_1\in(\mathbb{R}
\cup\{\infty\})\setminus\{0\}$ such that the self-adjoint operator of multiplication in $L_2(\mathbb{R},d\rho)$ is the canonical representation of $J({\tau_1})$. ${\tau_1}$ cannot be zero since then $\{\lambda_k\}$ should contain the zero. If $0\not\in\{\mu_k\}_k$, we define $$\label{eq:definition-of-g}
{\tau_2}:=
\begin{cases}
\mathcal{M} & \text{if}\quad {\tau_1}=\infty\\
\infty & \text{if}\quad {\tau_1}=-\mathcal{M}\\
\frac{\mathcal{M}{\tau_1}}{{\tau_1}+\mathcal{M}} & \text{in all other cases},
\end{cases}$$ and if $0\in\{\mu_k\}_k$ simply assign ${\tau_2}:=0$.
For the proof to be complete it remains to show that $\{\mu_k\}_k$ are the eigenvalues of $J({\tau_2})$. To this end we first show that $\{\mu_k\}_k$ are the eigenvalues of some self-adjoint extension of $J$. Let $\widetilde{\mathcal{M}}=-\mathcal{M}$ and define $$\widetilde{a}_j:=\widetilde{\mathcal{M}}\mathcal{R}_\lambda(\mu_j)\mathcal{R}_\mu'(\mu_j)\,.$$ From condition *\[nec-suf-moments\]* with $m=0$, it follows that $\widetilde{a}_j>0$ for any $j$ and $\sum_j\widetilde{a}_j^{-1}=1$, and that the function $\widetilde{\rho}(t):=\sum_{\mu_k\le
t}\widetilde{a}_k^{-1}$ is a solution of the moment problem $\{s_k\}_{k=0}^\infty$ with $s_k$ given by (\[eq:s-moments\]). Moreover, taking into account conditions *\[nec-suf0\]* and *\[nec-suf3\]*, one easily verifies as before that the sequences $\{\mu_j\}_j$ and $\{\widetilde{a}_j\}_j$, and the function $\mathcal{R}_\mu$ satisfy the conditions of Lemma \[lem:auxiliary\]. Therefore, by Definitions 1, 2 and Corollary 2 of the Addenda and Problems of [@MR0184042 Chap.4], as well as conditions *\[nec-suf-divergence\]* and *\[nec-suf3\]*, it turns out that the sequence $\{\mu_j\}_j$ is a canonical sequence of nodes and $\{\widetilde{a}_j\}_j$ the corresponding sequence of masses for the moment problem given by $\{s_k\}_{k=0}^\infty$ with $s_k$ satisfying (\[eq:s-moments\]). Hence $\widetilde{\rho}$ is a canonical solution of this moment problem. Denote by $J(\widetilde{{\tau_2}})$ the self-adjoint extension of $J$ having $\widetilde{\rho}$ as its spectral function.
Let us consider now the functions $R_{\tau_2}$ and $R_{\widetilde{{\tau_2}}}$ corresponding to $J({\tau_2})$ and $J(\widetilde{{\tau_2}})$, respectively (see (\[eq:R-without-constants\])). It is straightforward to verify that $$\label{eq:g-g-tilde-eq}
R_{\tau_2}(\lambda_j)=\frac{a_j}{\mathcal{M}R_{\tau_1}'(\lambda_j)}=
R_{\widetilde{{\tau_2}}}(\lambda_j)\,,$$ where the first equality follows from (\[eq:normalizing-constants\]), while the second follows from (\[eq:R-derivative\]) and (\[eq:sequence-def\]). By (\[eq:constant-m\]), (\[eq:calculation-M\]), and (\[eq:definition-of-g\]), one easily concludes from (\[eq:g-g-tilde-eq\]) that ${\tau_2}=\widetilde{{\tau_2}}$.\
\[rem:simlifications\] When $0\in\{\mu_k\}_k$, the signs of the real numbers $\mathcal{R}_\mu(\lambda_j)\mathcal{R}_\lambda'(\lambda_j)$ and $\mathcal{R}_\lambda(\mu_j)\mathcal{R}_\mu'(\mu_j)$ are known. Thus, we can write condition *\[nec-suf-constancy-sign\]* as follows $$\mathcal{R}_\mu(\lambda_j)\mathcal{R}_\lambda'(\lambda_j)<0\,\quad
\mathcal{R}_\lambda(\mu_j)\mathcal{R}_\mu'(\mu_j)>0\quad\text{ for all
}j\,.$$ This is a consequence of (\[eq:C-0\]) and (\[eq:constant-m\]) by which we know that in equation (\[eq:normalizing-necessary\]) $M=-a(0)<0$.
\[rem:interlacing1\] Note that, by Proposition \[prop:interlacing\], conditions *\[nec-suf0\]*–*\[nec-suf3\]* imply the interlacing of the sequences $\{\lambda_k\}_k$ and $\{\mu_k\}_k$.
We thank A. Osipov for drawing our attention to [@MR997788] and the anonymous referees whose comments led to an improved presentation of our work.
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[^1]: Author partially supported by PAPIIT-UNAM through grant IN-111906.
[^2]: Fellow Sistema Nacional de Investigadores.
|
---
author:
- 'R. R. Hartmann'
- 'M. E. Portnoi'
bibliography:
- 'Ref.bib'
title: 'Two-dimensional Dirac particles in a P[ö]{}schl-Teller waveguide'
---
Introduction {#introduction .unnumbered}
============
The P[ö]{}schl-Teller potential [@poschl1933bemerkungen] plays an important role in many fields of physics [@dong2007factorization] from modeling diatomic molecules and quantum many-body systems [@RomerPRL93; @RomerPRB94a; @RomerPRB94b], to applications in astrophysics [@ferrari1984new; @berti2009quasinormal], optical waveguides [@kogelnik19752] and quantum wells [@radovanovic2000intersubband; @yildirim2005nonlinear] through to Bose-Einstein and Fermionic condensates [@baizakov2004delocalizing; @antezza2007dark], and supersymmetric quantum mechanics [@dutt1988supersymmetry]. For the one-dimensional Schr[ö]{}dinger equation, the hyperbolic symmetric form can be solved in terms of associated Legendre polynomials and the eigenvalues are known explicitly [@poschl1933bemerkungen; @landau2013quantum]. We consider an analogous relativistic problem, that of a two-dimensional Dirac particle, confined by a one-dimensional P[ö]{}schl-Teller potential. Several solutions have been obtained for the Dirac equation with central P[ö]{}schl-Teller potentials [@jia2007solutions; @xu2008approximate; @wei2009spin; @wei2009algebraic; @jia2009approximate; @miranda2010solution; @tacskin2011spin] and the hyperbolic-secant potential is also known to admit analytic solutions for both the one and two-body one-dimensional Dirac problems [@hartmann2010smooth; @hartmann2014quasi; @hartmann2011excitons]. Modified P[ö]{}schl-Teller potential potentials have also been employed in numerical simulations of potential barriers in bilayer graphene [@park2015two].
With the recent explosion of research in Dirac materials [@wehling2014dirac] there has been a renewed interest in quasi-relativistic phenomena considered in condensed matter systems of different dimensionalities. This is due to the fact that the same equations which govern Dirac fermions in relativity, map directly to the equations of motion describing the quasi-particles in systems such as graphene [@neto2009electronic], carbon nanotubes [@charlier2007electronic], topological insulators [@hasan2010colloquium; @qi2011topological; @PhysRevB.94.014502], transition metal dichalcogenides [@xiao2012coupled] and 3D Weyl semimetals [@young2012dirac]. Massless Dirac particles are notoriously difficult to confine; however, it has been demonstrated that certain types of one-dimensional electrostatic waveguides in graphene, possess zero energy-modes which are truly confined within the waveguide [@hartmann2010smooth; @hartmann2014quasi] and that the number of these zero energy-modes is equal to the number of supercritical states (i.e. bound states whose energy, $E=-M$, where $M$ is the particle’s effective mass). Transmission resonances and supercriticality of Dirac particles through one-dimensional potentials have been studied extensively [@Coulter_AJP_71; @Calogeracos_PN_96; @PhysRevA.58.2160; @Dombey_PRL_20; @Kennedy_JPA_02; @Villalba_PRA_03; @Kennedy_IJMPA_04; @Guo_EJP_09; @Villalba_PS_10; @Sogut_PS_11; @Arda_PS_11; @miserev2016analytical]. The majority of studies model top-gated carbon-based nanostructures using abrupt potentials [@katsnelson2006chiral; @Chaplik_06; @PhysRevB.74.045424; @Chau_PRB_09; @Zhang_APL_09; @Williams_NanoT_11; @Yuan_JAP_11; @Wu_APL_11; @ping2012oscillating; @Katsnelson_PS_12; @PhysRevB.94.165443]. However, experimental potential profiles vary smoothly over many lattice constants, with even the smallest of gate generated potentials being far from square [@FoglerPRL]. There is also some controversy concerning waveguides which are defined by smoothly decaying, square integrable functions, which decay at large distances as $1/x^{n}$, where $n>1$. Numerical experiments imply that such waveguides always contain a zero-energy mode [@stone2012searching], whereas analysis based upon relativistic Levinson theorem, says there is a minimum potential strength required to observe a zero-energy mode [@clemence1989low; @lin1999levinson; @calogeracos2004strong; @ma2006levinson; @miserev2016analytical]. Our result supports the latter, and we demonstrate this through a simple analysis of supercritical states of zero energy.
In 2D Dirac materials, the low-energy spectrum of the charge carriers can be described by a Dirac Hamiltonian [@dirac1928quantum] of the form $$\hat{H}=\hbar v\left(\sigma_{x}\hat{k}_{x}+s_{\mathrm{K}}\sigma_{y}\hat{k}_{y}+\sigma_{z}k_{z}\right),\label{eq:Ham_orig}$$ where $\hat{k}_{x}=-i\frac{\partial}{\partial x}$, $\hat{k}_{y}=-i\frac{\partial}{\partial y}$, $\sigma_{x,y,z}$ are the Pauli spin matrices. $v$ plays the role of the speed of light and $k_{z}$ is proportional to the particle in-plane effective mass. In graphene, the charge carriers are massless, $k_{z}=0$, and the dispersion is linear, $v=v_{\mathrm{F}}\approx10^{6}$ m/s is the Fermi velocity and $s_{\mathrm{K}}$ has the value of $\pm1$ for the $K$ and $K'$ valley respectively [@wallace1947band]. For narrow-gap nanotubes and certain types of graphene nanoribbons [@dutta2010novel; @chung2016electronic], the operator $\hat{k}_{y}$ can by replaced by the number $k_{y}=E_{g}/(2 \hbar v_{\mathrm{F}} )$, which in the absence of applied field is fixed by geometry, where $E_{g}$ is the value of the bandgap. For nanotubes, $E_{g}$ can be controlled by applying a magnetic field along the nanotube axis [@portnoi2008terahertz; @hartmannoptoelectronic; @hartmann2011excitons; @hartmann2014terahertz; @hartmann2015terahertz]. When $k_{z}$ is finite, Eq. (\[eq:Ham\_orig\]) can be used as a simple model for silicene [@liu2011low] or Weyl semimetal [@wehling2014dirac; @hills2017current]. It has also been proposed that when graphene is subjected to a periodic potential on the lattice scale, for example, graphene on top of a lattice-matched hexagonal boron nitride [@giovannetti2007substrate] the Dirac fermions acquire mass with $E_{g}=2\hbar v_{\mathrm{F}}k_{z}$ being of the order of 53 meV.
In what follows we shall consider a particle described by the Hamiltonian (\[eq:Ham\_orig\]) subjected to a one-dimensional potential $U\left(x\right)$, which varies on a scale much larger than the lattice constant of the corresponding Dirac material, therefore allowing us to neglect inter-valley scattering for the case of graphene. We shall also set $s_{\mathrm{K}}=1$, and note that the other valley’s wave function can be obtained by exchanging $k_{y}\rightarrow-k_{y}.$ The Hamiltonian acts on the two-component Dirac wavefunction $$\Psi=e^{ik_{y}y}\left(\begin{array}{c}
\Psi_{A}\left(x\right)\\
\Psi_{B}\left(x\right)
\end{array}\right)$$ to yield the coupled first-order differential equations $$\left(V-E+M\right)\Psi_{A}-i\left(\frac{d}{d\tilde{x}}+\Delta\right)\Psi_{B}=0
\label{eq:ham_1}$$ and $$\left(V-E-M\right)\Psi_{B}-i\left(\frac{d}{d\tilde{x}}-\Delta\right)\Psi_{A}=0.
\label{eq:ham_2}$$ where $\tilde{x}=x/L$ and $L$ is a constant. $V=UL/\hbar v_{\mathrm{F}}$ and the charge carrier energy, $\varepsilon$, have been scaled such that $E=\varepsilon L/\hbar v_{\mathrm{F}}$. The charge carriers propagate along the $y$-direction with wave vector $k_{y}=\Delta/L$ , which is measured relative to the Dirac point, $\Psi_{A}\left(x\right)$ and $\Psi_{B}\left(x\right)$ are the wavefunctions associated with the $A$ and $B$ sublattices of graphene and finally $M=k_{z}L$ represents an effective mass in dimensionless units.
In what follows we consider the relativistic quasi-one-dimensional P[ö]{}schl-Teller potential problem which can be applied to describe e.g. graphene waveguides. We obtain the exact energy eigenfunctions for this potential and formulate a method for calculating the eigenvalues of the bound states. We then analyze the energy-spectrum of the symmetric P[ö]{}schl-Teller potential and obtain expressions for the eigenvalues of the supercritical states. By analyzing the zero-energy supercritical states we obtain a universal threshold condition for the minimum potential strength required for a potential to possess a zero-energy mode, for any square-integrable potential. We also show that the eigenfunctions in the non-relativistic limit restore the one-dimensional Schr[ö]{}dinger equation solutions. Finally, we analyze the eigenvalue spectrum for the modified P[ö]{}schl-Teller potential which includes an asymmetry term.
Relativistic one-dimensional P[ö]{}schl-Teller problem {#relativistic-one-dimensional-pöschl-teller-problem .unnumbered}
======================================================
In this section we consider the potential $$V=-\frac{a}{4}\left[1-\tanh^{2}\left(
\tilde{x}
\right)\right]+\frac{b}{2}\left[1+\tanh\left(
\tilde{x}
\right)\right],
\label{eq:potential}$$ which is a linear combination of the symmetric P[ö]{}schl-Teller potential with an additional term which enables the introduction of asymmetry [@nieto1978exact]. This potential belongs to the class of quantum models, which are quasi-exactly solvable [@turbiner1988quantum; @ushveridze1994quasi; @bender1998quasi; @downing2013solution; @hartmann2014bound; @hartmann2014quasi; @HartmannAIP; @PhysRevA.95.062110], where only some of the eigenfunctions and eigenvalues are found explicitly. The depth of the well is given by $-\left(a-b\right)^{2}/4a$, and the potential width is characterized by the parameter $L$, which was introduced after Eq. (\[eq:ham\_2\]). For the case of $b=0$, the potential transforms into the symmetric P[ö]{}schl-Teller potential, while if $a=0$, the potential is a smooth potential step, which can be used to model a p-n junction [@miserev2012quantum; @Katsnelson_AoP_13]. The symmetric and asymmetric forms of the potential are plotted in Fig. \[fig:eigenval\_both\].
![ The solid line shows the modified P[ö]{}schl-Teller potential, Eq. \[eq:potential\], for the symmetric case of $a=24$ and $b=0$. The dashed line shows the asymmetric potential for the case of $a=24$ and $b=2$. The 8 solid horizontal lines are the bound state energy levels for the symmetric potential at $\Delta = 4$ and the 6 dashed horizontal lines are the bound state energy levels for the asymmetric potential at $\Delta = 4$. []{data-label="fig:eigenval_both"}](eigenval_both){width="50.00000%"}
Substituting $\Psi_{A}=\left(\Psi_{1}+\Psi_{2}\right)/2$ and $\Psi_{B}=\left(\Psi_{1}-\Psi_{2}\right)/2$ allows Eqs. (\[eq:ham\_1\]-\[eq:ham\_2\]) to be reduced to a single second-order differential equation in $\Psi_{1}$ $\left(\Psi_{2}\right)$ $$\left[\left(V-E\right)^{2}-M^{2}-\Delta^{2}+is\frac{dV}{d\tilde{x}}\right]\Psi_{j}+\frac{d^{2}\Psi_{j}}{d\tilde{x}^{2}}=0,
\label{eq:second_order_org}$$ where $s=-\left(-1\right)^{j}$ and $j=1,\,2$ correspond to the spinor components $\Psi_{1}$ and $\Psi_{2}$ respectively. By making the natural change of variable $Z=\left[1+\tanh\left(\tilde{x}\right)\right]/2$ and using the wave function of the form $\Psi_{j}=\exp\left(pZ\right)Z^{n}\left(1-Z\right)^{m}\psi_{j}\left(Z\right)$ allows Eq. (\[eq:second\_order\_org\]) to be reduced to the Heun confluent equation [@heun1888theorie] in variable $Z$: $$\frac{\partial^2 \psi_{j}}{\partial Z^{2}}+\frac{\alpha Z^{2}+\left(2-\alpha+\beta+\gamma\right)Z-1-\beta}{Z\left(Z-1\right)}\frac{\partial \psi_{j}}{\partial Z}
+\frac{\left[\left(2+\beta+\gamma\right)\alpha+2\delta_{s}\right]Z-\left(1+\beta\right)\alpha+\left(1+\gamma\right)\beta+\gamma+2\eta_{s}}{2Z\left(Z-1\right)}\psi_{j}=0,
\label{eq:HeunEq.}$$ where $n=\beta/2$, $m=\gamma/2$, $P=\alpha/2$, $\widetilde{\Delta}^{2}=M^{2}+\Delta^{2}$, $s_{\alpha,\,\beta,\,\gamma}=\pm1$ and the parameters $\alpha,\,\beta,\,\gamma,\,\delta_{s}$ and $\eta_{s}$ are: $\alpha=ias_{\alpha}$, $\beta=s_{\beta}\sqrt{\widetilde{\Delta}^{2}-E^{2}}$, $\gamma=s_{\gamma}\sqrt{\widetilde{\Delta}^{2}-\left(E-b\right)^{2}}$, $\delta_{s}=a\left(\frac{1}{2}b-is\right)$ and $\eta_{s}=\beta^{2}/2-\left(a-b\right)\left(E-is\right)/2$. This same method of reducing a system of coupled first-order differential equations to the Heun confluent equation has been exploited to solve various generalisations of the quantum Rabi model [@XieJPA17], and notably the quasi-exact solutions of the P[ö]{}schl-Teller family potentials and Rabi systems are closely related [@RamazanJPA02]. In some instances, the resulting Heun confluent functions can be terminated as a finite polynomial [@PhysRevLett.57.787; @hartmann2014quasi] allowing particular eigenvalues to be obtained exactly, providing the parameters obey special relations. When this is not the case, the energy spectrum can be obtained fully via the Wronskian method [@ZhongJPA2013; @hartmann2014quasi; @MACIEJEWSKI201416; @XieJPA17], which is the method we shall utilize.
Equation (\[eq:HeunEq.\]) has regular singularities at $Z=0$ and $1$, and an irregular one at $Z=\infty$ which is outside the domain of $\tilde{x}$. The solutions to Eq. (\[eq:HeunEq.\]) are given by $$\psi_{j}=\sum_{s_{\alpha},\,s_{\beta},\,s_{\gamma}}A_{j,\,s_{\alpha},\,s_{\beta},\,s_{\gamma}}H\left(\alpha,\,\beta,\,\gamma,\,\delta_{s}\,,\eta_{s};\,Z\right)
+
B_{j,\,s_{\alpha},\,s_{\beta},\,s_{\gamma}}Z^{-\beta}H\left(\alpha,\,-\beta,\,\gamma,\,\delta_{s}\,,\eta_{s};\,\,Z\right),$$ where $A_{j,\,s_{\alpha},\,s_{\beta},\,s_{\gamma}}$ and $B_{j,\,s_{\alpha},\,s_{\beta},\,s_{\gamma}}$ are constants, and $H$ is the Heun confluent function [@ronveaux1995heun], which has a value of $1$ at the origin. For $\left|Z\right|<1,$ $H\left(\alpha,\,\beta,\,\gamma,\,\delta_{s}\,,\eta_{s},\,Z\right)=\left(1-Z\right)^{-\gamma}H\left(\alpha,\,\beta,\,-\gamma,\,\delta_{s}\,,\eta_{s},\,Z\right)$ and $H\left(\alpha,\,\beta,\,\gamma,\,\delta_{s}\,,\eta_{s},\,Z\right)=\exp\left(-\alpha Z\right)H\left(-\alpha,\,\beta,\,\gamma,\,\delta_{s}\,,\eta_{s},\,Z\right)$, therefore, as expected from the general theory of second order differential equations, the full solution is given by a linear combination of just two linearly independent functions $$\begin{aligned}
\Psi_{j}=
\left[
A_{j}
H\left(\alpha,\,\beta,\,\gamma,\,\delta_{s}\,,\eta_{s},\,Z\right)Z^{\frac{1}{2}\beta}
+
B_{j}
H\left(\alpha,\,-\beta,\,\gamma,\,\delta_{s}\,,\eta_{s},\,Z\right)Z^{-\frac{1}{2}\beta}
\right]
\left(1-Z\right)^{\frac{1}{2}\gamma}
\exp\left(\frac{1}{2}\alpha Z\right),
\label{eq:Full_Left}\end{aligned}$$ where $\beta$ is not an integer. It should be noted that when $a=0$ and $\alpha=\delta=0$, the Heun confluent functions appearing in Eq. (\[eq:Full\_Left\]) reduces to a Gauss hypergeometric functions and for the case of a massless particle Eq. (\[eq:Full\_Left\]) reduce down to the solutions obtained in Ref. [@miserev2012quantum]. If, however, $\beta$ is an integer then $H\left(\alpha,\,-\beta,\,\gamma,\,\delta\,,\eta_{s},\,Z\right)$ is divergent and $B_{i}$ has to be set to zero, and the second linearly independent solution can be constructed from a series expansion about the point $1-Z$. The solutions to the Heun confluent equation thus far have been given as power series expansions about the point $Z=0$. However, these power series rapidly diverge as $Z$ approaches the second singularity; therefore, at $Z=1$ we must construct solutions as power series expansions in variable $1-Z$: $$\Psi_{j}=
\left[
C_{j}H\left(-\alpha,\,\gamma,\,\beta,\,-\delta_{s},\,\eta_{s}+\delta;\,1-Z\right)\left(1-Z\right)^{\frac{1}{2}\gamma}
+
D_{j}H\left(-\alpha,\,-\gamma,\,\beta,\,-\delta_{s},\,\eta_{s}+\delta;\,1-Z\right)\left(1-Z\right)^{-\frac{1}{2}\gamma}
\right]
Z^{\frac{1}{2}\beta}\exp\left(\frac{1}{2}\alpha Z\right).
\label{Full_Right}$$ The constants $C_{i}$ and $D_{i}$ are found by matching the two power series expansions and their derivatives at $Z_{0}$ where $0<Z_{0}<1$.
For bound states, we require that $\widetilde{\Delta}^{2}>E^{2}$ and $\widetilde{\Delta}^{2}>\left(E-b\right)^{2}$. These conditions ensure that the bound states are inside the effective bandgap (which accounts for the motion along the y-axis). As $x\rightarrow-\infty$, $Z\rightarrow0$ and as $x\rightarrow\infty$, $Z\rightarrow1$ , therefore we may write the asymptotic expressions of $\Psi_{j}$ as $$\lim_{Z\rightarrow0}\left(\Psi_{j}\right)=
%\left(
A_{j}Z^{\frac{1}{2}\beta}+B_{j}Z^{-\frac{1}{2}\beta}
%\right)$$ and $$\lim_{Z\rightarrow1}\left(\Psi_{j}\right)=\left[C_{j}\left(1-Z\right)^{\frac{1}{2}\gamma}+D_{j}\left(1-Z\right)^{-\frac{1}{2}\gamma}\right]\exp\left(\frac{1}{2}\alpha\right).$$ Therefore, for bound states, $B_{i}$ ($A_{i}$) is zero for $s_{\beta}=1$ ($s_{\beta}=-1$) and $D_{i}$ ( $C_{i}$) is zero for $s_{\gamma}=1$ ($s_{\gamma}=-1$). Clearly the choice of $s_{\beta}$ and $s_{\gamma}$ is arbitrary, therefore, from hereon in we set both to $1$ unless otherwise stated. In this instance, the energy eigenvalues are found from the condition: $$\left.\frac{\partial H\left(\alpha,\,\beta,\,\gamma,\,\delta_{s}\,,\eta_{s},\,Z\right)}{\partial Z}\right|_{Z_{0}}
H\left(-\alpha,\,\gamma,\,\beta,\,-\delta_{s},\,\eta_{s}+\delta;\,1-Z_{0}\right)
=
\left.\frac{\partial H\left(-\alpha,\,\gamma,\,\beta,\,-\delta_{s},\,\eta_{s}+\delta;\,1-Z\right)}{\partial Z}\right|_{Z_{0}}
H\left(\alpha,\,\beta,\,\gamma,\,\delta_{s}\,,\eta_{s},\,Z_{0}\right),
\label{eq:bound_state_condition}$$ where $0<Z_{0}<1$.
Symmetric P[ö]{}schl-Teller potential solutions {#symmetric-pöschl-teller-potential-solutions .unnumbered}
-----------------------------------------------
In general, relating $\Psi_{1}$ to $\Psi_{2}$ is non-trivial, since neither a known expression exists which connects Heun confluent functions about two different singular points for arbitrary parameters, nor is there a general expression relating the derivative of the confluent Heun function to other confluent Heun functions, though particular instances have been obtained [@fiziev2009novel; @ShahnazaryanNR]. However, for the symmetric P[ö]{}schl-Teller potential (i.e. $b=0$) one can obtain the relation: $$\begin{aligned}
2Z\left(1-Z\right)\frac{dH\left(\alpha,\,\beta,\,\gamma,\,\delta_{1}\,,\eta_{1},\,Z\right)}{dZ}&=&
\left(\beta-iE\right)H\left(\alpha,\,\beta,\,\gamma,\,\delta_{-1}\,,\eta_{-1},\,Z\right)
\nonumber
\\
&+&
\left[\left(\beta+\gamma\right)Z-\left(\beta-iE\right)-\left(\alpha+\delta_{1}\right)Z\left(1-Z\right)\right]H\left(\alpha,\,\beta,\,\gamma,\,\delta_{1}\,,\eta_{1},\,Z\right).
\label{eq:deriv_ident}\end{aligned}$$ Therefore, $A_{2}=\Omega_{s_{\beta}}A_{1}$ and $B_{2}=\Omega_{-s_{\beta}}B_{1}$, where $\Omega_{s_{\beta}}=\left(E+i\beta\right)/\left(M+i\Delta\right)$.
In pristine graphene, only the symmetric form of Eq. (\[eq:potential\]) will contain non-leaky modes at zero energy. Non-zero-energy modes will have a finite lifetime since they can always couple to continuum states outside of the waveguide, whereas zero-energy modes are fully confined since the density of states vanishes at zero energy outside of the well. Asymmetric forms of Eq. (\[eq:potential\]) never contain truly bound modes since the density of states cannot vanish on both sides of the potential simultaneously. Notably, this is somewhat counterintuitive as for the Schr[ö]{}dinger problem a symmetric potential always contains a bound state, which can be removed by asymmetry. The emergence of bound states for a relativistic problem with an infinitely wide barrier is a manifestation of the Klein tunneling phenomenon [@hartmann2010smooth; @hartmann2014quasi].
We shall now consider the symmetric form of Eq. (\[eq:potential\]) for massless particles. Accordingly, we set $b=0$ and $M=0$, and in this instance the symmetrized real functions [@hartmann2010smooth; @hartmann2014quasi] are given by $\Psi_{\mathrm{I}}=\Psi_{A}+i\Psi_{B}$ and $\Psi_{\mathrm{II}}=\Psi_{A}-i\Psi_{B}$, where $$\Psi_{A}=A_{1}\Re\left[\Phi\exp\left(\frac{1}{2}\alpha Z-i\frac{\theta}{2}\right)
\right]
Z^{\frac{1}{2}\beta}\left(1-Z\right)^{\frac{1}{2}\gamma}\exp\left(i\frac{\theta}{2}\right)
\label{eq:Psi_A_prof}$$ and $$\Psi_{B}=iA_{1}\Im\left[\Phi\exp\left(\frac{1}{2}\alpha Z-i\frac{\theta}{2}\right)
\right]
Z^{\frac{1}{2}\beta}\left(1-Z\right)^{\frac{1}{2}\gamma}\exp\left(i\frac{\theta}{2}\right),
\label{eq:Psi_B_prof}$$ where $\Phi=H\left(\alpha,\,\beta,\,\gamma,\,\delta_{1}\,,\eta_{1},\,Z\right)$ and $\tan\theta=-E/\beta$. By employing the identity Eq.(\[eq:deriv\_ident\]), the derivatives appearing in Eq.(\[eq:bound\_state\_condition\]) can be expressed in terms of Heun confluent functions. It immediately follows that at the origin $\Psi_{\mathrm{I}}^{\star}\Psi_{\mathrm{II}}+\Psi_{\mathrm{I}}\Psi_{\mathrm{II}}^{\star}=0$, which in terms of the functions $\Psi_{A}$ and $\Psi_{B}$ yields: $$\left|\Psi_{A}\left(Z=\frac{1}{2}\right)\right|^{2}=\left|\Psi_{B}\left(Z=\frac{1}{2}\right)\right|^{2},
\label{eq:eigen_eq_abv}$$ where $Z=1/2$ corresponds to $x=0$. Substituting Eq.(\[eq:Psi\_A\_prof\]) and Eq.(\[eq:Psi\_B\_prof\]) into Eq.(\[eq:eigen\_eq\_abv\]) results in the condition $$\Re\left[\Phi\left(Z=\frac{1}{2}\right)\exp\left(\frac{\alpha}{4}-i\frac{\theta}{2}\right)\right]\mp
\Im\left[\Phi\left(Z=\frac{1}{2}\right)\exp\left(\frac{\alpha}{4}-i\frac{\theta}{2}\right)\right]=0.
\label{eq:bound_state_fast}$$ From Eq.(\[eq:Psi\_A\_prof\]) and Eq.(\[eq:Psi\_B\_prof\]), $A_{1}\Psi_{A}^{\star}=A_{1}^{\star}\Psi_{A}\exp\left(-i\theta\right)$ and $A_{1}\Psi_{B}^{\star}=-A_{1}^{\star}\Psi_{B}\exp\left(-i\theta\right)$. Therefore, the condition from which the eigenvalues of the spectrum are determined, Eq.(\[eq:eigen\_eq\_abv\]), can be written as $\left.\left(\Psi_{A}+i\Psi_{B}\right)\right|_{Z=\frac{1}{2}}\left.\left(\Psi_{A}-i\Psi_{B}\right)\right|_{Z=\frac{1}{2}}=0$. This condition can be understood in terms of parity. In principle, one can construct from these functions odd and even solutions. However, since the even modes of $\Psi_{\mathrm{I}}$ occur at the same energies as the odd modes of $\Psi_{\mathrm{II}}$ and vice versa, one can obtain the eigenvalues when the symmetrized functions $\Psi_{\mathrm{I}}$ or $\Psi_{\mathrm{II}}$ are zero at the origin. The functions $\Psi_{\mathrm{I}}$ and $\Psi_{\mathrm{II}}$ are related to the earlier introduced functions $\Psi_{1}$ and $\Psi_{2}$ by $$\Psi_{\mathrm{I}}=\frac{1}{2}\left[\left(1+i\right)\Psi_{1}+\left(1-i\right)\Psi_{2}\right]
\label{eq:connection_A}$$ and $$\Psi_{\mathrm{II}}=\frac{1}{2}\left[\left(1-i\right)\Psi_{1}+\left(1+i\right)\Psi_{2}\right].
\label{eq:connection_B}$$ Using Eqs. (\[eq:connection\_A\],\[eq:connection\_B\]) together with Eqs. (\[eq:Full\_Left\],\[Full\_Right\]) allows $\Psi_{\mathrm{I}}$ and $\Psi_{\mathrm{II}}$ to be expressed explicitly in terms of Heun functions.
Eq. (\[eq:bound\_state\_fast\]) is formally the same as Eq. (\[eq:bound\_state\_condition\]) but computationally faster. In Fig. \[Spectrum\_b0\_approx\] we plot the numerically obtained solutions of Eq. (\[eq:bound\_state\_fast\]) for the potential defined by $a=24$ and $b=0$. The dashed lines represent the boundary at which the bound states merge with the continuum which occurs at the energies $E=\pm\Delta$ and $E+a/4=\Delta$. For the potential of strength $a=24$ we find that there are four zero-energy bound modes, occurring at $\Delta = 0.597, \,2.276, \,3.817$ and $5.282$. Their normalized wavefunctions are shown in Fig. \[fig:eigenfunctions\_zero\].
![ (Color online) The energy spectrum of confined states in the symmetric P[ö]{}schl-Teller potential, of strength $a=24$, as a function of $\Delta$. The alternating red and blue lines represent the odd (even) and even (odd) modes of $\Psi_{\mathrm{I}}$ ($\Psi_{\mathrm{II}}$) respectively. The black crosses denote the supercritical states. The boundary at which the bound states merge with the continuum is denoted by the grey (short-dashed) lines. []{data-label="Spectrum_b0_approx"}](Spectrum_b0_approx){width="50.00000%"}
![ (Color online) The normalized zero-energy bound-state wavefunctions of the symmetric P[ö]{}schl-Teller potential, for strength $a=24$. (a), (b), (c) and (d) correspond to the case of $\Delta = 0.597,\,2.276,\, 3.817$ and $5.282$, respectively. The dashed-red and dashed-blue lines correspond to $\Psi_{\mathrm{I}}$, while the solid-black lines correspond to $\Psi_{\mathrm{II}}$. []{data-label="fig:eigenfunctions_zero"}](Multi_zero){width="50.00000%"}
As mentioned previously, the number of zero-energy modes is equal to the number of supercritical states (neglecting the spin and valley degrees of freedom). For the symmetric P[ö]{}schl-Teller potential, the eigenvalues of these supercritical states can be determined approximately, via simple analytic expressions. Moving to the symmetric basis, $\left(\Psi_{\mathrm{I}},\,\Psi_{\mathrm{II}}\right)^{\mathrm{T}}$, allows the pair of coupled first order differential equations, Eq. (\[eq:ham\_1\]) and Eq. (\[eq:ham\_2\]) to be reduced to a single second order differential equation in $\Psi_{\mathrm{II}}$: $$\left[\left(V-E\right)^{2}-\Delta^{2}\right]\Psi_{\mathrm{II}}-\frac{1}{\left(V-E-\Delta\right)}\frac{dV}{d\tilde{x}}\frac{d\Psi_{\mathrm{II}}}{d\tilde{x}}+\frac{d^{2}\Psi_{\mathrm{II}}}{d\tilde{x}^{2}}=0.
\label{eq:sym_equation}$$ For supercritical states, $E=-\Delta$, Eq. (\[eq:sym\_equation\]) transforms into the differential equation for the angular prolate spheroidal wave functions [@abramowitz1964handbook]: $$\frac{d}{d\eta}\left[\left(1-\eta^{2}\right)\frac{d}{d\eta}S_{1N}\left(c,\eta\right)\right]+\left[\lambda_{1N}-c^{2}\eta^{2}-\frac{1}{1-\eta^{2}}\right]S_{1N}\left(c,\eta\right)=0,$$ where $\eta=\tanh\left(z\right)$, $c=\pm V_{0}$, $\Psi_{\mathrm{II}}=\sqrt{1-\eta^{2}}S_{1N}$, and $S_{1N}$ are the spheroidal wave functions. $\lambda_{1N}=a\left(a-8\Delta\right)/16$, where the permissible values of $\lambda_{1N}$ must be determined to assure that $S_{1N}\left(c,\eta\right)$ are finite at $\eta=\pm1$. The permissible $\lambda_{1N}$ can be obtained via the asymptotic expansion $$%\lambda_{1N}\left(c\right)=cq+1-\frac{1}{8}\left(q^{2}+5\right)-\frac{q}{64c}\left(q^{2}+11-32\right)+O\left(c^{2}\right),
\lambda_{1N}\left(c\right)=cq+1-\frac{1}{8}\left(q^{2}+5\right)-\frac{q}{64c}\left(q^{2}+11-32\right)+O\left(c^{-2}\right),
\label{eq:First_bound_expand}$$ where $N=1,\,2,\,3,\,\ldots$ and $q=2N-1$ [@abramowitz1964handbook]. Keeping only the terms of expansion shown in Eq. (\[eq:First\_bound\_expand\]) yields the following eigenvalues: $$E=-\frac{1}{2}\left(1-2N\right)-\frac{a}{8}+\frac{1}{4a}\left[3-\left(1-2N\right)^{2}\right],
\label{eq:First_bound_guess}$$ where $N$ is restricted to ensure that $E$ is negative. The resulting approximate eigenvalues for the symmetric P[ö]{}schl-Teller potential of strength $a=24$ are $E=-2.475$, $-1.555$ and $-0.734$ respectively, and are indicated as black crosses in Fig. \[Spectrum\_b0\_approx\]. The approximate eigenvalues deviate increasingly from the numerically exact results, $E=-2.473$, $-1.542$ and $-0.682$, with decreasing $y$. It should be noted that a refinement of the approximate values of $\lambda_{1N}$ can be found in Ref. [@abramowitz1964handbook]. For the hyperbolic secant potential, $V=-V_{0}/\cosh\left(\tilde{x}\right)$, it was found that there was a minimum potential strength of $V_{0}=1/2$, required to observe a zero-energy mode [@hartmann2010smooth]. According to the Landauer formula, the conductance along the waveguide when the Fermi level is set to the Dirac point is $4ne^{2}/h$, where $n$ is the number of zero-energy modes. The existence of a threshold in the potential strength needed for the waveguide to contain a zero-energy mode allowed us to suggest that such waveguides could be used as switchable devices. However, later numerical calculations utilizing a variable phase method implied that power-decaying potentials always possess a bound mode [@stone2012searching]. This result cast serious doubt in the validity of employing exponentially decaying potentials as a suitable model for graphene waveguides, since realistic potential profiles decay a power of distance rather than exponentially. Notably, the threshold potential strength at which the first zero-energy mode appears can be obtained from the condition of the first bound state coinciding with the first supercritical state, i.e. $E=-\Delta=0$. In this instance, Eq. (\[eq:sym\_equation\]) can be solved exactly: $$\Psi_{\mathrm{I}}=C_{1}\cos\left(\int_{0}^{\tilde{x}}V\left(X\right)dX\right)+C_{2}\sin\left(\int_{0}^{\tilde{x}}V\left(X\right)dX\right)
\label{eq:Thres_1},$$ $$\Psi_{\mathrm{II}}=-C_{1}\sin\left(\int_{0}^{\tilde{x}}V\left(X\right)dX\right)+C_{2}\cos\left(\int_{0}^{\tilde{x}}V\left(X\right)dX\right)
\label{eq:Thres_2}.$$ For even modes of $\Psi_{\mathrm{I}}$, $C_{2}=0$, whereas, for odd modes of $\Psi_{\mathrm{I}}$, $C_{1}=0$. In the absence of the potential, when $E=\Delta=0$ the two first order differential equations in $\Psi_{\mathrm{I}}$ and $\Psi_{\mathrm{II}}$ decouple, and Eq. (\[eq:sym\_equation\]) reduces to a first order differential equation. As $E=-\Delta\rightarrow0$ and $x\rightarrow\pm\infty$ (where the potential is zero), Eq. (\[eq:Thres\_1\]) and Eq. (\[eq:Thres\_2\]) are required to be linearly dependent [@DombeyJPA02] and the Wronskian of the solutions $\Psi_{\mathrm{I}}$ and $\Psi_{\mathrm{II}}$ is zero[@calogero1967variable]. Consequently, the bound modes satisfy the condition: $\left|\Psi_{\mathrm{I}}\left(\pm\infty\right)\right|^{2}=\left|\Psi_{\mathrm{II}}\left(\pm\infty\right)\right|^{2}$. Therefore, the threshold potential strength at which the first zero-energy mode appears is found by the condition $$\left|\int_{0}^{\infty}V\left(\tilde{x}\right)d\tilde{x}\right|=\frac{\pi}{4}.
\label{eq_threshold}$$ Therefore, for any square-integrable potential, the threshold for the appearance of the first bound state of zero energy is only a function of the integrated potential. Notably, this is the same result obtained as relativistic Levinson theorem [@clemence1989low; @lin1999levinson; @calogeracos2004strong; @ma2006levinson; @miserev2016analytical]. For the P[ö]{}schl-Teller potential, Eq. (\[eq\_threshold\]) yields $a=\pi$, which agrees with Eq. (\[eq:bound\_state\_fast\]). For $V=-V_{0}/\cosh\left(\tilde{x}\right)$, Eq. (\[eq\_threshold\]) yields $V_{0}=1/2$ which restores the result of Ref. [@hartmann2010smooth]. This result implies that square-integrable power decaying potentials do indeed have a threshold, in contrast to the numerically predicted result of [@stone2012searching]. In this respect, exponentially decaying potentials are not that different from power-decaying and are perfectly suitable for the modeling of top-gated Weyl semimetals.
Finally, it should be noted that in the non-relativistic limit, Eq. (\[Full\_Right\]) restores the well known results [@landau2013quantum] for the bound state functions of the Schr[ö]{}dinger equation for the P[ö]{}schl-Teller potential. In the limit that $\alpha$ and $\delta$ are much smaller than $\beta$ and $\gamma$ (i.e. large $\Delta$): $$\lim_{\Delta\rightarrow\infty}\left(H\left(\alpha,\,\beta,\,\gamma,\,\delta\,,\eta,\,Z\right)\right)=
Z^{-\beta}\,_{2}F_{1}\left(Q,\,1+\gamma-\beta-Q;\,1+\gamma;\,1-Z\right),$$ where $Q=\left(1+\gamma-\beta\pm\sqrt{1+\gamma^{2}+\beta^{2}-4\left(\eta_{s}+\delta_{s}\right)}\right)/2$ and $\,_{2}F_{1}$ is the Gauss hypergeometric function. Substituting $E=E_{\mathrm{SE}}+\Delta$, $b=0$ and $s_{\beta}=-s_{\gamma}=1$ into Eq. (\[Full\_Right\]), results in the non-relativistic bound state functions $$\lim_{\Delta\rightarrow\infty}\left(\Psi_{1}\right)\propto\,_{2}F_{1}\left(\epsilon+1+T,\,\epsilon-T;\,1+\epsilon;\,1-Z\right)Z^{\frac{\epsilon}{2}}\left(1-Z\right)^{\frac{\epsilon}{2}},$$ where $\epsilon=s_{\beta}\sqrt{-2E_{\mathrm{SE}}\Delta}$ and $T=\left(-1+\sqrt{1+2a\Delta}\right)/2$. For the solutions to be finite at $Z=0$, we require that $\epsilon-T=-N$ where $N=0,\,1,\,2,\,\ldots$. When this criteria is met, the Gauss hypergeometric function is a polynomial of degree $N$ and the energy levels are given by $$E_{\mathrm{SE}}=-\frac{1}{8\Delta}\left[-\left(1+2N\right)+\sqrt{1+2a\Delta}\right]^{2}.$$
Modified P[ö]{}schl-Teller potential solutions {#modified-pöschl-teller-potential-solutions .unnumbered}
----------------------------------------------
![ (Color online) The energy spectra of confined states in the modified P[ö]{}schl-Teller potential, of strength $a=24$ and $b=2$, as a function of $\Delta$ for (a) $M=0$ and (b) $M=2$. The grey (dashed), black (dot-dashed) and green (dotted) lines correspond to $\Delta^{2}+M^{2}=E^{2}$, $\Delta^{2}+M^{2}=\left(E-b\right)^{2}$ and $\Delta^{2}+M^{2}=\left[E+\left(a-b\right)^{2}/4a\right]^{2}$, respectively. The blue-shaded area highlights the energy range in which the modes contained within the waveguide are non-leaky. []{data-label="fig:asym_disp"}](Combine4){width="75.00000%"}
![ (Color online) Schematic diagrams of the dispersion of a gaped Dirac material. The black lines represent the modified P[ö]{}schl-Teller potential profile, of strength $a=24$ and $b=2$. The grey lines represent the charge-carrier dispersion of particles of $M=2$. The blue shaded area represent the energy range in which there are no continuum states to couple to outside of the well. []{data-label="fig:asym_band"}](asym_band){width="50.00000%"}
We shall now consider the case of finite $b$, which represents a smooth asymmetric waveguide. Previously considered asymmetric waveguides varied abruptly on the same scale as the graphene lattice constant [@ping2012oscillating; @he2014guided; @xu2015guided; @salem2016mid; @xu2016guided]. We shall now consider more realistic smooth asymmetric waveguides, which fit closer to experimentally achievable potential profiles. In Fig. \[fig:asym\_disp\], we plot the energy spectrum for the potential defined by the parameters $a=24$ and $b=2$. The introduction of the asymmetry term $b$ reduces the number of modes at $E=0$, which are now quasi-bound modes for the massless case (Fig. \[fig:asym\_disp\]a), since they can couple to continuum states outside of the waveguide. Naturally, for massive Dirac fermions full confinement is possible across a range of energies. In Fig. \[fig:asym\_band\], we show a schematic diagram of the dispersion of a gapped Dirac material, subjected to the modified P[ö]{}schl-Teller potential defined by $a=24$ and $b=2$. For a particle of mass $M=2$, it can be seen that for the energy range $E=0$ to $E=2$ there are no continuum states outside of the well, therefore in that range all bound solutions will be non-leaky. The corresponding energy spectrum of confined states is shown in Fig. \[fig:asym\_disp\]b.
Conclusions {#conclusions .unnumbered}
===========
We have analyzed the behavior of quasi-relativistic two-dimensional particles subjected to a modified P[ö]{}schl-Teller potential. Our results have direct applications to electronic waveguides in Dirac materials. For the symmetric P[ö]{}schl-Teller potential, explicit forms were obtained for the eigenvalues of supercritical states. A universal expression, for any symmetric potential, was obtained for the critical potential strength required to observe the first zero-energy state. The well known results for the P[ö]{}schl-Teller potential are recovered in the non-relativistic limit.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to Charles Downing for the critical reading of the manuscript. This work was supported by the EU H2020 RISE project CoExAN (Grant No. H2020-644076), EU FP7 ITN NOTEDEV (Grant No. FP7-607521), FP7 IRSES projects CANTOR (Grant No. FP7-612285), QOCaN (Grant No. FP7-316432), and InterNoM (Grant No. FP7-612624). R.R.H. acknowledges financial support from URCO (Project No. 09 F U 1TAY15-1TAY16) and Research Links Travel Grant by the British Council Newton Fund.
Author contributions statement {#author-contributions-statement .unnumbered}
==============================
R.R.H and M.E.P wrote the main manuscript text and R.R.H prepared the figures. All authors reviewed the manuscript.
Additional Information {#additional-information .unnumbered}
======================
**Competing financial interests** The authors declare no competing financial interests.
|
---
abstract: 'This Letter details a measurement of the ionization yield ($Q_y$) of 6.7 keV [$^{40}$Ar]{} atoms stopping in a liquid argon detector. The $Q_y$ of 3.6–6.3 detected $e^{-}/\mbox{keV}$, for applied electric fields in the range 240–2130 V/cm, is encouraging for the use of this detector medium to search for the signals from hypothetical dark matter particle interactions and from coherent elastic neutrino-nucleus scattering. A significant dependence of $Q_y$ on the applied electric field is observed and explained in the context of ion recombination.'
author:
- 'T.H. Joshi'
- 'S. Sangiorgio'
- 'A. Bernstein'
- 'M. Foxe'
- 'C. Hagmann'
- 'I. Jovanovic'
- 'K. Kazkaz'
- 'V. Mozin'
- 'E.B. Norman'
- 'S.V. Pereverzev'
- 'F. Rebassoo'
- 'P. Sorensen'
bibliography:
- 'EndpointBib.bib'
date: 25 April 2014
title: First measurement of the ionization yield of nuclear recoils in liquid argon
---
Liquid-phase argon has long been used as a target medium for particle detection via scintillation and charge collection. Recently there has been considerable interest in direct detection of both hypothetical dark matter particles [@Gaitskell] and coherent elastic neutrino-nucleus scattering (CENNS) [@Freedman; @Drukier]. These as-yet unobserved neutral particle interactions are expected to result in a recoiling argon atom $\mathcal{O}$(keV), generally referred to in the literature as a nuclear recoil. This prompts the question of the available signal produced by such recoils in a liquid argon detector. This quantity must be directly measured due to the difference in signals from nuclear recoils as opposed to electron recoils (e.g. Compton electrons and $\beta$-particles). In this Letter we report the first measurement of the ionization yield ($Q_{y}$) (detected electrons per unit energy) resulting from nuclear recoils in liquid argon, measured at 6.7 keV. This is also the lowest-energy measurement of nuclear recoils in liquid argon.
These results are of interest not only for particle detection, but for theoretical studies of condensed media as well. Models of the production of ions and excited atoms from low-energy recoils in liquid argon exist, but are not fully understood in the few-keV energy range [@Chepel]. To study the influence of the electric field on recombination, and thus $Q_y$, data were obtained at applied electric field values of 240, 640, 1600, 2130 V/cm.
The scintillation efficiency of nuclear recoils in liquid argon has been measured from 10–250 keV at zero electric drift field using the kinematically constrained scatter of 2.8 MeV neutrons [@Gastler:2010sc] and from 11–50 keV at electric drift fields from 0–1000 V/cm using the kinematically constrained scatter of 0.60 and 1.17 MeV neutrons [@Alexander]. No measurements of nuclear recoils in liquid argon exist below 10 keV.
Liquid argon dual-phase detectors have been shown to be sensitive to single electrons generated in the bulk [@Sangiorgio201369]. This enhances the detection capability of the ionization channel over the scintillation channel at very low energies. A low-energy threshold and calibration are critical in both dark matter searches and CENNS discovery. Both interactions exhibit a recoil energy spectrum that rises rapidly with decreasing energy [@Chepel; @Hagmann; @Akimov]. Our results suggest that dark matter searches using only the ionization channel in liquid argon (as has been done in liquid xenon [@Angle:2011th]) could probe an interesting new parameter space. The observation and modeling of electric drift-field dependence presented in this Letter, and also recently reported in the scintillation channel [@Alexander], lay the foundation for a comprehensive understanding of ion recombination in liquid argon and suggests the need for optimization of drift fields in future liquid argon-based experiments.
*Experimental Details*.– Our measurement employed a beam of neutrons to create nuclear recoils in liquid argon. The neutron spectrum was peaked at 24 and 70 keV. Contributions from the quasimonoenergetic 70 keV (12% FWHM) neutrons were selected during background subtraction. The design and deployment of the neutron beam is described in detail in Ref. [@nsource]. Our detector, a small dual-phase argon proportional scintillation counter, is described in Ref. [@Sangiorgio201369]. Small modifications to the detector since that work include the removal of the $^{55}$Fe source and holder, and the replacement of one of the electrode grids. The response to [$^{37}$Ar]{} calibrations has been verified to be consistent with the previous results. The active region of liquid argon has a 2.5-cm radius and a 3.7-cm height.
{width="98.00000%"}
Particle interactions in the liquid argon can produce primary scintillation and ionization. The detector was optimized for detection of the proportional scintillation resulting from extracting the electrons into the gas-phase argon, and accelerating them across a 1.8-cm gap. The detector has been shown to be sensitive to the signal resulting from a single electron [@Sangiorgio201369]. The applied electric field used to create the proportional scintillation was a constant 9.8 kV/cm for these measurements. The applied electric field ($\mathcal{E}$) across the liquid argon target, oriented in the $z$ direction, was varied from 240 V/cm to 2130 V/cm, in order to explore the effect on the available signal. Electrostatic simulations show a 6% nonuniformity in the applied electric field within the LAr target volume, arising from the field cage spacing.
The data acquisition was triggered by fourfold coincidence of the four phototubes, in a $10-\mu$s window. The trigger efficiency was consistent with unity for signals larger than 8 ionization electrons. New triggers were vetoed for 3 ms following very large \[$>\approx$10,000 photoelectrons (PEs)\] events, to exclude phototube saturation effects from the data.
Research-grade argon was condensed into the detector through a getter to remove electronegative impurities, and a free electron lifetime of $>$ 300 $\mu$s was verified throughout the experiment as in Ref. [@Sangiorgio201369]. The maximum electron drift time across the target region varied from 32 $\mu$s at 240 V/cm to 14 $\mu$s at 2130 V/cm applied electric field, leading to a mean electron loss of 5%. During operation, the argon vapor pressure was maintained at 1.08 bar with 1% stability, and the liquid temperature at approximately $88$ K (corresponding to a liquid density of 1.39 g/cm$^3$).
$Q_y$ was measured in an end-point-type experiment. Monoenergetic neutrons with well-defined energy ($E_n$) interact within the liquid argon target producing nuclear recoils. For *S*-wave scatter, expected for this experiment, nuclear recoils are populated uniformly in energy from zero to $E_{max}=4E_{n}m_{Ar}m_{n}/(m_{Ar}+m_{n})^{2}=6.7$ keV for $E_n=70$ keV scattering on [$^{40}$Ar]{}, the most abundant argon isotope. The end point in the observed ionization spectrum is then attributed to $E_{max}$.
Quasimonoenergetic neutrons were produced with a collimated near-threshold [$^{7}$Li]{}(*p,n*)[$^{7}$Be]{} source, and filtered with a 7-cm length of high-purity iron as described in [@nsource]. The iron neutron filter has transmission notches at 24, 70, and 82 keV. The 70-keV notch was selected to target the low-energy side of the elastic neutron scattering resonance centered at 77 keV in [$^{40}$Ar]{} thus producing a large interaction rate while limiting the probability of multiple scatter.
The proton beam energy was 1.932 MeV for all measurements and calibrated before and during data taking. Beam current was nominally 700 nA throughout data taking. The collimation aperture subtends $\pm1\degree$. The iron-filled collimator was oriented at $45 \pm0.5\degree$ with respect to the proton beam when collecting signal data. Representative background data were acquired at an angle of $55 \pm0.5\degree$, in which case 70 keV neutron production is kinematically forbidden, but all other beam-related backgrounds, including the 24 keV component of the neutron beam and beam-related gammas, are present.
Following the collection of neutron scatter data, a small amount of argon gas ($<$0.5g) containing 3$\pm$0.5 kBq of [$^{37}$Ar]{} was injected into the detector and allowed to diffuse for one hour. Calibration data as described in Ref. [@Sangiorgio201369] wer then acquired in the same four electric drift-field configurations.
*Analysis*.– Triggered proportional scintillation (ionization channel) events identified by the analysis were subjected to a series of quality cuts. The cuts included the selection of [*(a)*]{} isolated events, defined as having $<$2 PEs in the 50-$\mu$s pre-trigger and $<$10 PEs following the event, and [*(b)*]{} the rejection of primary scintillation from peripheral background events, which have a characteristic fast rise and $1.6~\mu$s decay time. Additional cuts include the rejection of [*(c)*]{} events near the $(x,y)$ edge of the active region using the same algorithm described in Ref. [@Sangiorgio201369] and [*(d)*]{} pileup events, e.g., axially ($z$) separated multiple scatters, by accepting events with 95% of signal arriving in $<20~\mu$s. Cut [*(c)*]{} also strongly limits the acceptance of pile up and multiple scatters separated in $(x,y)$. The energy dependence of this suite of selection criteria was found to vary by $<$5% for events with $>11$ detected electrons. The nuclear recoil endpoint “shoulder” is clearly visible before background subtraction (Fig. \[fig:Spectra\]).
Fluctuations in the phototube response were less than 2% over individual data sets. Single PE calibrations were performed for each data set using isolated single PE from the tail of proportional scintillation events.
The transformation of neutron scattering data from measured PE to detected electrons required a single-electron calibration from previous data because single electrons were not observed in sufficiently high rates during this experiment. Previous measurements with this detector found $7.8\pm0.1$ PEs per detected electron ($\mbox{PE}/e^-$) with a systematic uncertainty of 10% due to the difficulty in localizing the $(x,y)$ coordinates of the single-electron signals. A value of $10.4\pm0.2$ $\mbox{PE}/e^-$ was used in the present analysis. The $33\%$ increase in light yield resulted from a larger electric field and physical gap in the proportional scintillation region, and was obtained using the 2.82-keV peak from [$^{37}$Ar]{} *K*-capture (2.82 keV released in x rays/Auger electrons [@Barsanov]) acquired across a range of electric field configurations. The statistical and systematic uncertainties of this calibration were 2% and 10%, respectively.
Backgrounds during these measurements were dominantly beam related–namely, 24 keV neutrons that transit the iron filter, gammas from [$^{7}$Li]{}(*p,p’*)[$^{7}$Li]{} within the lithium target, and neutron-capture gammas–and were proportional to the proton current on the target. Data were normalized by the integrated proton current and corrected for the live time fraction of the data acquisition system. The normalized spectra were then subtracted as shown in Fig. \[fig:Spectra\].
![[]{data-label="fig:MCNP"}](Figure2.pdf){width="48.00000%"}
![[]{data-label="fig:Endpoint"}](Figure3.pdf){width="48.00000%"}
A detailed MCNP-PoliMi [@Pozzi2003550] simulation, using the /-VII.1 library, was performed to model the expected single-scatter spectra in both the signal and background detector configurations, as shown in Fig. \[fig:MCNP\]. For comparison with data as shown in Fig. \[fig:Spectra\], the simulated spectra were first converted from recoil energy to a number of electrons via a constant ionization yield ($Q_y$). Then a resolution term was applied, defined as $\sigma(n_e)=\sqrt{n_e(F'+\sigma_{e}^{2})}$, where $n_e$ is the number of detected electrons and $\sigma_{e}=0.37$ is the measured single-electron resolution. The term $F'\equiv F+R$ accounts for the Fano factor ($F$) and recombination fluctuations ($R$). The third free parameter in the fit was the rate normalization.
A $\chi^{2}$ comparison between the simulation and the background-subtracted spectrum was made using a parametric scan across the free parameters ($Q_y$, $F'$, and rate normalization), resulting in the confidence level contours shown in Fig. \[fig:Endpoint\]. The region of interest for each drift field was selected to focus on the location of the end-point shoulder. The statistical uncertainty of the best-fit $Q_y$ value was defined by the extent of the 68% confidence level contours.
*Component* *Statistical (%)* *Systematic (%)*
----------------------------------------------------------- ------------------- ------------------
Single electron peak 2–10 10
Single electron calibration 2 10
$\chi^{2}$ analysis 3–5 -
Input spectrum - 5
Background subtraction - 1–3
Slope of $Q_y$ in model 240 V/cm - $^{+5}_{-25}$
\[0.4ex\] `"` 640 V/cm - $^{+2}_{-18}$
\[0.4ex\] `"` 1600 V/cm - $^{+0}_{-19}$
\[0.4ex\] `"` 2130 V/cm - $^{+0}_{-21}$
\[0.4ex\] Liquid argon purity - 5
Drift field ($\mathcal{E}$) - 6
\[tab:Uncertainty\]
We emphasize that this analysis was focused solely on extracting the ionization yield at the end point and makes no attempt to extract information about ionization yields below 6.7 keV. This is because at energies below the end point, it is not possible to uniquely resolve the degeneracy between the free parameters in the model. The most robust method of accessing information about $Q_y$ at smaller recoil energies is to decrease the end-point energy [@nsource]. To estimate the systematic uncertainty associated with the assumption that $Q_y$ is constant with recoil energy, we repeated the analysis for each data set with the linear slope of the ionization yield as an additional free parameter. For all but the smallest value of $\mathcal{E}$, the best fit was obtained for a slope of about $-0.8~Q_{y}/\mbox{keV}$ and a slightly lower end-point $Q_y$. This is quoted as a systematic uncertainty for each drift field in Table \[tab:Uncertainty\]. Additionally, we repeated the analysis using a simple step function for the input nuclear recoil spectrum, to approximate the ideal *S*-wave recoil spectrum from monoenergetic 70 keV neutrons (this is not shown in Fig. \[fig:MCNP\]). This provided a conservative approximation of the uncertainty due to underlying uncertainties in the differential nuclear cross-section data, used in the MCNP-Polimi simulation. The systematic uncertainty associated with subtraction of background data was assessed using an exponential fit to background data ($>11$ electrons). Using the best-fit exponential for subtraction yielded the same best-fit $Q_y$. Varying the exponential constant $\pm15\%$ resulted in a $\pm$1–3% shift in best-fit $Q_y$.
Table \[tab:Uncertainty\] summarizes the statistical and systematic uncertainties present in the ionization yield results. The statistical uncertainty of the best-fit mean is quoted. Asymmetric uncertainties were attributed to several of the listed parameters as a result of their underlying nature. Uncertainties were added in quadrature when combined.
*Results and Discussion*.– The number of electrons detected from 6.7-keV nuclear recoils as a function of applied electric drift field is shown in Fig. \[fig:YieldComparison\] and the ionization yield with uncertainties is listed in Table \[tab:Results\]. The strong dependence on the electric drift field is in reasonable agreement with recent observations in the scintillation channel [@Alexander], consistent with the expected anticorrelation of scintillation and ionization. The different recoil energies and the lack of absolute scintillation yields in Ref. [@Alexander] prevent a quantifiable comparison.
![[]{data-label="fig:YieldComparison"}](Figure4.pdf){width="48.00000%"}
This field dependence is understood to be a suppression of ion-electron recombination along the ionization track and is extensively discussed in Ref. [@Chepel]. In order to fit our data we consider an empirical modification [@Dahl; @NEST] of the Thomas-Imel box model [@PhysRevA.36.614], $$\label{eq:TImodel}
n_e=\frac{N_i}{\xi }\ln (1+\xi ),~~\xi=\frac{N_iC}{\mathcal{E}^{b}}.$$ $N_{i}$ is the number of initial ion-electron pairs produced, $n_e$ is the number of electrons that escape recombination, $\mathcal{E}$ is the applied electric field, and $b$ and $C$ are constants. The electric drift-field dependence is modified from the original model to have a power-law dependence, $\xi\propto \mathcal{E}^{-b}$. The number of initial ion-electron pairs may be written as $$\label{eq:ni}
N_i=\frac{f E}{\epsilon \left(1+N_{ex}/N_{i}\right)},$$ where $E$ is the amount of energy deposited in the track, $f$ is the fraction of energy lost through ionization and atomic excitation (unity for electronic recoils) often termed a quench factor, $\epsilon=19.5$ eV is the average energy required to produce a quantum (excitation or ionization) in liquid argon [@Doke], and $N_{ex}$ is the number of initial excitations. The ratio $N_{ex}/N_{i}=0.21$ was measured for electronic recoils in liquid argon [@Kubota]. The model has only two free parameters ($C,b$) when describing electron recoils. Using the 2.82-keV [$^{37}$Ar]{} *K*-capture calibration data a best fit (Fig. \[fig:YieldComparison\]) yields $C=2.37$ and $b=0.61$ when $\mathcal{E}$ is expressed in V/cm.
Using these values for $b$ and $C$, the number of initial ion-electron pairs ($N_i$) is left as a single free parameter when applied to nuclear recoil data. Fitting to the data (Fig. \[fig:YieldComparison\]) we observe good agreement and find $N_i=72\pm2$, assuming this model remains valid at high (saturating) field values. The fact that recombination in liquid argon can be described by the same phenomenological model for few-keV electron and nuclear recoils suggests a similarity in the spatial distribution of electrons and ions for these different energy-deposition mechanisms.
Using Eq. (\[eq:ni\]) and the calculations of Lindhard *et al.* [@lindhard1963integral] for the partitioning of nuclear recoil energy ($f=0.25$) results in $N_{ex}/N_{i}=0.19$, which is surprisingly similar to the value for electron recoils. Alternatively, if $N_{ex}/N_{i}\sim$1 (as measured for nuclear recoils in liquid xenon [@SandD]) then one would find $f=0.42$. If confirmed this would suggest a promising sensitivity of liquid argon at low energies. Simultaneous measurements of scintillation and ionization are needed to unambiguously determine $f$ and $N_{ex}/N_{i}$.
[cccc]{} $\mathcal{E}$ (V/cm) & $Q_{y}$ ($e^{-}$/keV) & Statistical & Systematic\
\
240 & 3.6 & $^{+0.1}_{-0.1}$ & $^{+0.5}_{-1.1}$\
\[0.4ex\] 640 & 4.9 & $^{+0.1}_{-0.2}$ & $^{+0.6}_{-1.2}$\
\[0.4ex\] 1600 & 5.9 & $^{+0.2}_{-0.2}$ & $^{+0.7}_{-1.4}$\
\[0.4ex\] 2130 & 6.3 & $^{+0.1}_{-0.3}$ & $^{+0.8}_{-1.6}$\
\[0.4ex\]
\[tab:Results\]
We are not aware of any measurements or theoretical expectations for either the Fano factor or recombination fluctuations for nuclear recoils in liquid argon. With the simple assumption that recombination statistics are binomial, the probability for an electron to escape recombination is $p=n_e/N_i$, and so $R = 1-n_e/N_i$. From this, it would follow that the Fano factor is given by $F=F'+n_{e}/N_{i}-1$. Taking the range of $p$ from Fig. \[fig:YieldComparison\] it is clear that $F$ is smaller than $F'$ by a factor which ranges from 0.65 at $\mathcal{E}=240$ V/cm to 0.42 at $\mathcal{E}=2130$ V/cm. This is consistent with $F\approx0.5$, with a fairly large uncertainty as shown in Fig. \[fig:Endpoint\].
In this Letter we have presented the first nuclear recoil ionization yield measurement and the first measurement of sub-10-keV nuclear recoils in liquid argon using an end-point-type measurement. This demonstration suggests that end-point measurements with filtered neutron sources [@nsource; @Barbeau; @RussianFilter] are suitable for a comprehensive study of both scintillation and ionization yields of low-energy nuclear recoils in liquid argon, and could also probe $<$ 4 keV in liquid xenon. The results of such a study would clarify the threshold and calibration of liquid noble-based dark matter detectors and CENNS searches. The measurements presented in this Letter demonstrate a large ionization yield for nuclear recoils at energies below current thresholds of liquid argon dark matter searches, suggesting the ionization channel as a means for exploring light-mass dark matter in existing and future liquid argon detectors.
We would like to thank G. Bench and T. Brown for assistance and support throughout the beam measurements and J. Coleman and K. Mavrokoridis for previous detector contributions. We would like to acknowledge the Lawrence Scholars Program and the Department of Homeland Security for funding T.H.J.’s research. A portion of M.F.’s research was performed under the Nuclear Forensics Graduate Fellowship Program, which is sponsored by the U.S. Department of Homeland Security, Domestic Nuclear Detection Office, and the U.S. Department of Defense, Defense Threat Reduction Agency. We gratefully acknowledge the LDRD program (LDRD 13-FS-005) at LLNL. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. LLNL-JRNL-646478
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abstract: 'We give a review of some recent developments in the theory of tensor categories. The topics include realizability of fusion rings, Ocneanu rigidity, module categories, weak Hopf algebras, Morita theory for tensor categories, lifting theory, categorical dimensions, Frobenius-Perron dimensions classification of tensor categories.'
address: |
IRMA (CNRS)\
7 Rue René Descartes, F-67084 Strasbourg, France\
email:\
\
Department of Mathematics, MIT\
Cambridge, MA 02138, USA\
email:
author:
- 'Damien Calaque and Pavel Etingof[^1]'
title: Lectures on tensor categories
---
18D10.
Tensor category, fusion ring, braiding, modular category.
Introduction {#introduction .unnumbered}
============
This paper is based on the first author’s lecture notes of a series of four lectures given by the second author in June 2003 for the Quantum Groups seminar at the Institut de Recherche Mathématique Avancée in Strasbourg. It is about the structure and classification of tensor categories.
We always work over an algebraically closed field ${\textbf{k}}$. By a tensor category over ${\textbf{k}}$, we mean an abelian rigid monoidal category in which the neutral object **1** is simple (i.e., does not contain any proper subobject), the vector spaces ${\textrm{Hom}(X,Y)}$ are finite dimensional and all objects are of finite length.
The category of finite dimensional vector spaces ${\textrm{Vect}_{{\textbf{k}}}}$, the categories of finite dimensional representations of a group $G$, a Lie algebra ${\mathfrak g}$, or a (quasi)Hopf algebra $H$ (respectively denoted ${\textrm{Rep}}G$, ${\textrm{Rep}}{\mathfrak g}$ and ${\textrm{Rep}}H$), or the category of integrable modules over an affine Lie algebra $\hat{\mathfrak g}$ with fusion product (which can also be obtained from quantum groups) are all tensor categories in this sense.
Tensor categories appear in many areas of mathematics such as representation theory, quantum groups, conformal field theory (CFT) and logarithmic CFT, operator algebras, and topology (invariants of knots and 3-manifolds). The goal of this paper is to give an introduction to some recent developments in this subject.
The paper is subdivided into four sections, each representing a single lecture.
Section 1 introduces the main objects of the paper. We recall basic categorical definitions and results, fix the vocabulary, and give examples (for more details, we recommend the monographies [@BK; @ML; @K; @T]). The end of the section is devoted to the problem of realizability of fusion rings: examples are given, and the Ocneanu rigidity conjecture is formulated.
The goal of Section 2 is to prove the Ocneanu rigidity for fusion categories in characteristic zero. To do this, we introduce and discuss the notions of module categories and weak Hopf algebras. The more technical part of the proof is done at the end of the section.
Section 3 is about three distinct subjects. We start with a closer look at module categories, discussing the notion of Morita equivalence for them, and applying general results to the representation theory of groups. Then we recall well-known facts about braided, ribbon and modular categories. Finally, the lifting theory is presented: it allows us to extend some results from characteristic zero to the positive characteristic case.
Section 4 covers the theory of Frobenius-Perron dimension, and its applications to classification results for fusion categories.
We end this paper with two interesting open problems.
[**Remarks.**]{} 1. Being a set of lecture notes, this paper does not contain original results. Most of the results are taken from the papers [@ENO; @EO1; @O1; @O2; @O3] and references therein, including the standard texts on the theory of tensor categories.
2\. To keep this paper within bounds, we had to refrain from a thorough review of the history of the subject and of the original references, as well as from a detailed discussion of the preliminaries. We also often had to omit complete proofs. For all this material we refer the reader to books and papers listed in the bibliography.\
**Acknowledgements.** The authors are grateful to the participants of the lectures – P. Baumann, B. Enriquez, F. Fauvet, C. Grunspan, G. Halbout, C. Kassel, V. Turaev, and B. Vallette. Their interest and excitement made this paper possible. The second author is greatly indebted to D. Nikshych and V. Ostrik for teaching him much of the material given in these lectures. He is also grateful to IRMA (Strasbourg) for hospitality. His work was partially supported by the NSF grant DMS-9988786.
Finite tensor and fusion categories
===================================
Basic notions
-------------
### Definitions
Let $\mathcal C$ be a category.
Recall that $\mathcal C$ is *additive* over ${\textbf{k}}$ if\
(i) ${\textrm{Hom}(X,Y)}$ is a (finite dimensional) ${\textbf{k}}$-vector space for all $X,Y\in{\textrm{Obj}(\mathcal C)}$,\
(ii) the map ${\textrm{Hom}(Y,Z)}\times{\textrm{Hom}(X,Y)}\to{\textrm{Hom}(X,Z)},(\varphi,\psi)\mapsto\varphi\circ\psi$ is ${\textbf{k}}$-linear for all $X,Y,Z\in{\textrm{Obj}(\mathcal C)}$,\
(iii) there exists an object $\textbf0\in{\textrm{Obj}(\mathcal C)}$ such that ${\textrm{Hom}(\textbf0,X)}={\textrm{Hom}(X,\textbf0)}=\textbf0$ for all $X\in{\textrm{Obj}(\mathcal C)}$,\
(iv) finite direct sums exist.
**Remark.** When we deal with functors between additive categories, we always assume they are also additive.
Further, recall that an additive category ${\mathcal C}$ is *abelian* if\
(i) every morphism $\phi: X\to Y$ has a kernel ${\rm Ker}\phi$ (an object $K$ together with a monomorphism $K\to X$) and a cokernel ${\rm Coker}\phi$ (an object $C$ together with an epimorphism $Y\to C$);\
(ii) every morphism is the composition of an epimorphism followed by a monomorphism;\
(iii) for every morphism $\varphi$ one has $\textrm{Ker}\varphi=0\implies\varphi=\textrm{Ker}(\textrm{Coker}\varphi)$ and $\textrm{Coker}\varphi=0\implies\varphi=\textrm{Coker}(\textrm{Ker}\varphi)$.
It is known that ${\mathcal C}$ is abelian if and only if it is equivalent to a full subcategory of the category of modules over a algebra. Recall also that $\mathcal C$ is *monoidal* if there exists\
(i) a bifunctor $\otimes:\mathcal C\times\mathcal C\to\mathcal C$,\
(ii) a functorial isomorphism $\Phi:(-\otimes-)\otimes-\to-\otimes(-\otimes-)$,\
(iii) an object **1** (called the neutral object) and two functorial isomorphisms $$\lambda:\textbf1\otimes-\to-\,,\,\,\,\mu:-\otimes\textbf1\to-
\quad(\textrm{the unit morphisms})\,,$$ such that for any two functors obtained from $-\otimes\cdots\otimes-$ by inserting **1**’s and parentheses, all functorial isomorphisms between them composed of $\Phi^{\pm1}$’s, $\lambda^{\pm1}$’s and $\mu^{\pm1}$’s are equal.\
**Remark.** In the spirit of the previous remark, for additive monoidal categories we assume that $\otimes$ is biadditive.
The data $(\mathcal C,\otimes,\Phi,\lambda,\mu)$ with (i), (ii) and (iii) is a monoidal category if and only if the following properties are satisfied:
1. **Pentagon axiom.** The following diagram is commutative:\
$\xymatrix{
((-\otimes-)\otimes-) \otimes-\ar[r]^{\Phi^{1,2,3}\otimes\textrm{id}}\ar[d]^{\Phi^{12,3,4}}
& (-\otimes(-\otimes-))\otimes-\ar[r]^{\Phi^{1,23,4}}
& -\otimes((-\otimes-)\otimes-)\ar[d]^{\textrm{id}\otimes\Phi^{2,3,4}} \\
(-\otimes-)\otimes(-\otimes-)\ar[rr]^{\Phi^{1,2,34}} & & -\otimes(-\otimes(-\otimes-))}$
2. **Triangle axiom.** The following diagram commutes:\
$\xymatrix{
(-\otimes\textbf1) \otimes-\ar[r]^{\Phi_{-,\textbf1,-}}\ar[dr]_{\mu\otimes\textrm{id}} &
-\otimes(\textbf1\otimes-)\ar[d]^{\textrm{id}\otimes\lambda} \\
& -\otimes-}$
A monoidal category is called [*strict*]{} if $(X\otimes Y)\otimes Z=X\otimes (Y\otimes Z)$, $\bold 1\otimes X=X\otimes \bold 1=X$, and the associativity and unit isomorphisms are equal to the identity. A theorem also due to Maclane (see [@K]) says that any monoidal category is equivalent to a strict one. In view of this theorem, we will always assume that the categories we are working with are strict, unless otherwise specified.
Recall that a right dual for $X\in{\textrm{Obj}(\mathcal C)}$ is an object $X^*$ with two morphisms $e_X:X^*\otimes X\to\textbf1$ and $i_X:\textbf1\to X\otimes X^*$ (called the evaluation and coevaluation morphisms) satisfying the following two equations:\
(i) $(\textrm{id}_X\otimes e_X)\circ(i_X\otimes \textrm{id}_X)=\textrm{id}_X$ and\
(ii) $(e_X\otimes\textrm{id}_{X^*})\circ(\textrm{id}_{X^*}\otimes i_X)=\textrm{id}_{X^*}$.\
A left dual $^*\!\!X$ with maps $e_X':X\otimes^*\!\!X\to\textbf1$ and $i_X':\textbf1\to^*\!\!X\otimes X$ is defined in the same way.\
One can show that if it exists, the right (left) dual is unique up to a unique isomorphism compatible with evaluation and coevaluation maps.
A monoidal category is called *rigid* if any object has left and right duals.
*A *tensor category* is a rigid abelian monoidal category in which the object $\bold 1$ is simple and all objects have finite length.*
*The category ${\textrm{Rep}}H$ of finite dimensional representations of a quasi-Hopf algebra $H$ is a tensor category [@Dr]. This category is, in general, not strict (although it is equivalent to a strict one): its associativity isomorphism is given by the associator of $H$.*
In a tensor category, the tensor product functor $\otimes$ is (bi)exact.
### The Grothendieck ring of a tensor category
Let ${\mathcal C}$ be a tensor category over ${\textbf{k}}$.
*The *Grothendieck ring ${\rm Gr}({\mathcal C})$* of ${\mathcal C}$ is the ring whose basis over ${\mathbb{Z}}$ is the set of isomorphism classes of simple objects, with multiplication given by $$X\cdot Y=\sum_{Z~\textrm{simple}}N_{XY}^ZZ,$$ where $N_{XY}^Z=[X\otimes Y:Z]$ is the multiplicity (the number of occurences) of $Z$ in $X\otimes Y$ (which is well-defined by the Jordan-Hölder theorem).*
\[excat\] *(i) ${\mathcal C}={\textrm{Rep}}_{\mathbb{C}}SL(2)$. Simple objects are highest weight representations $V_j$ (of highest weight $j\in \Bbb Z$), and the structure constants of the Grothendieck ring are given by the Clebsch-Gordan formula $$V_i\otimes V_j=\sum_{\substack{k= \vert i-j\vert \\ k\equiv i+j~\textrm{mod}~2}}^{i+j}V_k$$*
*(ii) ${\mathcal C}$ is the category of integrable modules (from category $\mathcal O$) over the affine algebra ${\widehat{\mathfrak{sl}_2}}$ at level $l$ with the fusion product $$V_i\otimes V_j=\sum_{\substack{k=\vert i-j\vert \\
k\equiv i+j~\textrm{mod}~2}}^{l-\vert i+j-l\vert}V_k$$ In this case ${\textrm{Gr}({\mathcal C})}$ is a Verlinde algebra.*
*(iii) ${\mathcal C}={\textrm{Rep}}_{\textbf{k}}{\textrm{Fun}(G)}$ for a finite group $G$. Simple objects are evaluation modules $V_g$, $g\in G$, and $V_g\otimes V_h=V_{gh}$. So ${\textrm{Gr}({\mathcal C})}={\mathbb{Z}}[G]$.\
More generally, pick a 3-cocycle $\omega\in Z^3(G,{\mathbb{C}}^\times)$. To this cocycle we can attach a twisted version ${\mathcal C}(G,\omega)$ of $\mathcal C$: all the structures are the same, except the associativity isomorphism which is given by $\Phi_{V_g,V_h,V_k}=\omega(g,h,k)\textrm{id}$ (and the morphisms $\lambda,\mu$ are modified to satisfy the triangle axiom). The cocycle condition $$\omega(h,k,l)\omega(g,hk,l)\omega(g,h,k)=\omega(gh,k,l)\omega(g,h,kl)$$ is equivalent to the pentagon axiom. Again, we have ${\textrm{Gr}({\mathcal C}(G,\omega))}={\mathbb{Z}}[G]$.*
*(iv) ${\mathcal C}={\textrm{Rep}}_{\mathbb{C}}S_3$. The basis elements (simple objects) are $\textbf1,\chi,V$, with product given by $\chi\otimes\chi=\textbf1$, $\chi\otimes V=V\otimes\chi=V$ and $V\otimes V=V\oplus\textbf1\oplus\chi$.*
*(v) If ${\mathcal C}={\textrm{Rep}}G$ for $G$ a unipotent algebraic group over ${\mathbb{C}}$, then the unique simple object is $\bold 1$, hence ${\textrm{Gr}({\mathcal C})}={\mathbb{Z}}$. In this case, the Grothendieck ring does not give a lot of information about the category because the category is not semisimple.*
*(vi) ${\mathcal C}={\textrm{Rep}}H$ for the 4-dimensional Sweedler Hopf algebra $H$, which is generated by $g$ and $x$, with relations $gx=-xg$, $g^2=1$, $x^2=0$, and the coproduct $\Delta$ given by $\Delta g=g\otimes g$ and $\Delta x=x\otimes g+1\otimes x$. In this case the only simple objects are $\textbf1$ and $\chi$, with $\chi\otimes\chi=\textbf1$.*
### Tensor functors
Let ${\mathcal C}$ and ${\mathcal D}$ be two tensor categories. A functor $F:{\mathcal C}\to{\mathcal D}$ is called *quasitensor* if it is exact and equipped with a functorial isomorphism $J:F(-\otimes-)\to F(-)\otimes F(-)$ and an isomorphism $u:F(\textbf1)\to\textbf1$. Such a functor defines a morphism of unital rings ${\textrm{Gr}({\mathcal C})}\to{\textrm{Gr}({\mathcal D})}$.\
A quasitensor functor $F:{\mathcal C}\to{\mathcal D}$ is *tensor* if the diagrams\
$\xymatrix{
F((-\otimes-)\otimes-)\ar[r]^{J^{12,3}}\ar[d]^{F(\Phi_{\mathcal C})} &
F(-\otimes-)\otimes F(-)\ar[r]^{J\otimes\textrm{id}} &
(F(-)\otimes F(-))\otimes F(-)\ar[d]^{\Phi_{\mathcal D}} \\
F(-\otimes(-\otimes-))\ar[r]^{J^{1,23}} &
F(-)\otimes F(-\otimes-)\ar[r]^{\textrm{id}\otimes J} &
F(-)\otimes(F(-)\otimes F(-))}$\
$\xymatrix{
F(\textbf1\otimes-)\ar[r]^{J_{\textbf1,-}}\ar[d]_{F(\lambda_{\mathcal C})} &
F(\textbf1)\otimes F(-)\ar[d]^{u\otimes\textrm{id}} \\
F(-) & \textbf1\otimes F(-)\ar[l]^{\lambda_{\mathcal D}}}$ and $\xymatrix{
F(-\otimes\textbf1)\ar[r]^{J_{-,\textbf1}}\ar[d]_{F(\mu_{\mathcal C})} &
F(-)\otimes F(\textbf1)\ar[d]^{\textrm{id}\otimes u} \\
F(-) & F(-)\otimes\textbf1\ar[l]^{\mu_{\mathcal D}}}$\
are commutative.\
An *equivalence of tensor categories* is a tensor functor which is also an equivalence of categories.
*Let $\omega,\omega'\in Z^3(G,{\textbf{k}}^\times)$ and $\omega'/\omega= \textrm d\eta$ is a coboundary. Then $\eta$ defines a tensor structure on the identity functor ${\mathcal C}(G,\omega') \to{\mathcal C}(G,\omega)$: the coboundary condition $$\omega'(g,h,k) \eta(h,k)\eta(g,hk)=\eta(gh,k)\eta(g,h)\omega(g,h,k)$$ is equivalent to the commutativity of the previous diagram. Moreover, it is not difficult to see that this tensor functor is in fact an equivalence of tensor categories. Thus the fusion category ${\mathcal C}(G,\omega)$, up to equivalence, depends only on the cohomology class of $\omega$. In particular, we may use the notation ${\mathcal C}(G,\omega)$ when $\omega$ is not a cocycle but a cohomology class.*
Finite tensor and fusion categories
-----------------------------------
### Definitions and examples
*An abelian category ${\mathcal C}$ over ${\textbf{k}}$ is said to be *finite* if\
(i) ${\mathcal C}$ has finitely many (isomorphism classes of) simple objects,\
(ii) any object has finite length, and\
(iii) any simple object admits a projective cover.*
This is equivalent to the requirement that ${\mathcal C}={\textrm{Rep}}A$ as an abelian category for a finite dimensional ${\textbf{k}}$-algebra $A$.
*A *fusion category* is a semisimple finite tensor category.*
*In examples \[excat\], (i) is semisimple but not finite, (ii), (iii) and (iv) are fusion, (v) is neither finite nor semisimple, and (vi) is finite but not semisimple.*
Recall that if ${\mathcal C}$ and ${\mathcal D}$ are two abelian categories over ${\textbf{k}}$, then one can define their *Deligne external product* ${\mathcal C}\boxtimes{\mathcal D}$. Namely, if ${\mathcal C}=A$-Comod and ${\mathcal D}=B$-Comod are the categories of comodules over coalgebras $A$ and $B$ then ${\mathcal C}\boxtimes{\mathcal D}:=A\otimes B$-Comod.
If ${\mathcal C}$ and ${\mathcal D}$ are semisimple, the Deligne product is simply the category whose simple objects are $X\boxtimes Y$ for simple $X\in{\textrm{Obj}({\mathcal C})}$ and $Y\in{\textrm{Obj}({\mathcal D})}$. If ${\mathcal C}$ and ${\mathcal D}$ are tensor/finite tensor/fusion categories then ${\mathcal C}\boxtimes{\mathcal D}$ also has a natural structure of a tensor/finite tensor/fusion category (in the semisimple case it is simply given by $(X\boxtimes Y)\otimes(X'\boxtimes Y'):=(X\otimes Y)\boxtimes(X'\otimes Y')$).
### Reconstruction theory (Tannakian formalism)
Let $H$ be a (quasi-)Hopf algebra and consider ${\mathcal C}={\textrm{Rep}}H$, the category of its finite dimensional representations. The forgetful functor $F:{\mathcal C}\to{\textrm{Vect}_{{\textbf{k}}}}$ has a (quasi)tensor structure (the identity morphism). In addition, this functor is exact and faithful. A functor ${\mathcal C}\to{\textrm{Vect}_{{\textbf{k}}}}$ with such properties ((quasi)tensor, exact, and faithful) is called a [*(quasi)fiber functor*]{}.
*Reconstruction theory* tells us that every finite tensor category equipped with a (quasi)fiber functor is obtained in this way, i.e., can be realized as the category of finite dimensional representations of a finite dimensional (quasi-)Hopf algebra.
Namely, let $({\mathcal C},F)$ be a finite tensor category equipped with a (quasi)fiber functor, and set $H={\textrm{End}(F)}$. Then $H$ carries a coproduct $\Delta$ defined as follows: $$\Delta:H\to H\otimes H={\textrm{End}(F\times F)};\quad
T\mapsto J\circ T\circ J^{-1}$$ Moreover, one can define a counit $\epsilon:H\to{\textbf{k}}$ by $\epsilon(T)=T_{\vert F(\textbf1)}$ and an antipode $S:H\to H$ by $S(T)_{\vert F(X)}=(T_{\vert F(X^*)})^*$ (in the quasi-case this depends on the choice of the identification $j_X:F(X)^*\to F(X^*)$).\
This gives $H$ a (quasi-)Hopf algebra structure (the choice of $j_X$ has to do with Drinfeld’s special elements $\alpha,\beta\in H$). Thus we have bijections: [$$\begin{aligned}
\substack{\textrm{Finite tensor categories with quasifiber} \\
\textrm{functor up to equivalence and changing} \\
\textrm{quasitensor structure of the functor}} & \longleftrightarrow &
\substack{\textrm{Finite dimensional quasi-Hopf algebras} \\
\textrm{up to isomorphism and twisting}} \\
\substack{\textrm{Finite tensor categories with fiber} \\
\textrm{functor up to equivalence}} & \longleftrightarrow &
\substack{\textrm{Finite dimensional Hopf algebras} \\
\textrm{up to isomorphism}}\end{aligned}$$]{}
### Braided and symmetric categories
Let ${\mathcal C}$ be a monoidal category with a functorial isomophism $\sigma:-\otimes-\to-\otimes^{\textrm{op}}-$, where $X\otimes^{op}Y:=Y\otimes X$.
For given objects $V_1,\dots,V_n$ in ${\mathcal C}$, we consider an expression obtained from $V_{i_1}\otimes\cdots\otimes V_{i_n}$ by inserting $\textbf1$’s and parentheses, and where $(i_1,\dots,i_n)$ is a permutation of $\{1,\dots,n\}$. To any composition $\varphi$ of $\Phi$’s, $\lambda$’s, $\mu$’s, $\sigma$’s and their inverses acting on it, we assign an element of the braid group ${\rm B}_n$ as follows: assign $1$ to $\Phi$, $\lambda$ and $\mu$, and the generator $\sigma_k$ of ${\rm B}_n$ to $\sigma_{V_kV_{k+1}}$.
*A *braided monoidal category* is a monoidal category as above such that the $\varphi$’s depend only on their images in the braid group.*
Again, we have a coherence theorem for braided categories:
The data $({\mathcal C},\otimes,\textbf1,\Phi,\lambda,\mu,\sigma)$ defines a braided category if and only if $(\Phi,\alpha)$ satisfy the Hexagon axioms: the diagrams\
$\xymatrix{
(12)3\ar[r]^\Phi\ar[d]^{\sigma\otimes\textrm{id}} & 1(23)\ar[r]^{\sigma_{1,23}} & (23)1\ar[d]^\Phi \\
(21)3\ar[r]^\Phi & 2(13)\ar[r]^{\textrm{id}\otimes\sigma} & 2(31)}$ $\xymatrix{
(12)3\ar[r]^\Phi\ar[d]^{\sigma^{-1}\otimes\textrm{id}} & 1(23)\ar[r]^{\sigma_{1,23}^{-1}}
& (23)1\ar[d]^\Phi \\
(21)3\ar[r]^\Phi & 2(13)\ar[r]^{\textrm{id}\otimes\sigma^{-1}} & 2(31)}$\
are commutative.
**Remark.** “2(31)” is short notation for the 3-functor $(V_1,V_2,V_3)\mapsto V_2\otimes(V_3\otimes V_1)$.
To get the definition of a *symmetric monoidal category*, the reader just has to replace the braid group ${\rm B}_n$ by the symmetric group $S_n$ in the definition. To say it in another way, a symmetric monoidal category is a braided one for which $\sigma$ satisfies $\sigma_{VW}\circ\sigma_{WV}=\textrm{id}_{V\otimes W}$.
*Let $H$ be a quasitriangular bialgebra (resp. Hopf algebra), i.e., a bialgebra (resp. Hopf algebra) with an invertible element $R\in H\otimes H$ satisfying $\Delta^{\textrm{op}}(x)=R\Delta(x)R^{-1}$, $(\textrm{id}\otimes\Delta)(R)=R^{13}R^{12}$ and $(\Delta\otimes\textrm{id})(R)=R^{13}R^{23}$. Then ${\textrm{Rep}}H$ is a braided monoidal (resp. rigid monoidal, i.e., tensor) category with braiding $\sigma_{VW}:a\otimes b\mapsto R^{21}(b\otimes a)$. Moreover, axioms for $R$ are equivalent to the requirement that ${\textrm{Rep}}H$ is braided (it is not difficult to show that the first equation satisfied by $R$ is equivalent to the functoriality of $\sigma$, and the two others are equivalent to the Hexagon relations).\
If $R$ is triangular, i.e., $RR^{21}=1\otimes1$ (in particular if $H$ is cocommutative), then ${\textrm{Rep}}H$ becomes a symmetric monoidal (resp. tensor) category.*
### The Drinfeld center
Tannakian formalism tells us that there is a strong link between finite tensor categories and Hopf algebras. So it is natural to ask for a categorification of the notion of the Drinfeld double for Hopf algebras.
*The *Drinfeld center $Z({\mathcal C})$* of a tensor category ${\mathcal C}$ is a new tensor category whose objects are pairs $(X,\Phi)$, where $X\in{\textrm{Obj}({\mathcal C})}$ and $\Phi:X\otimes-\to-\otimes X$ is a functorial isomorphism such that $\Phi_{Y\otimes Z}=(\textrm{id}\otimes\Phi_Z)\circ(\Phi_Y\otimes\textrm{id})$, and with morphisms defined by ${\textrm{Hom}((X,\Phi),(Y,\Psi))}:=\{f\in{\textrm{Hom}(X,Y)}|\forall Z,
(f\otimes\textrm{id})\circ\Phi_Z=\Psi_Z\circ(\textrm{id}\otimes f)\}$.*
$Z({\mathcal C})$ is a braided tensor category, which is finite if ${\mathcal C}$ is.
See [@K] for the proof (the finiteness statement can be found for example in [@EO1]). Let us just note that the tensor product of objects is given by $(X,\Phi)\otimes(Y,\Psi)=(X\otimes Y,\Lambda)$, where $\Lambda(Z)=(\Phi(Z)\otimes\textrm{id}_Y)\circ(\textrm{id}_X\otimes\Psi(Z))$, the neutral object by $(\textbf1,\textrm{id})$, and the braiding by $\sigma_{(X,\Phi),(Y,\Psi)}=\Phi_Y$.
If ${\mathcal C}$ is a fusion category over ${\mathbb{C}}$, then $Z({\mathcal C})$ is also fusion.
This will be a consequence of a more general statement given in subsection \[dualcat\].
**Remark.** In positive characteristic, $Z({\mathcal C})$ is, in general, not fusion. For example, if ${\mathcal C}={\mathcal C}(G,1)$ over ${\textbf{k}}=\overline{\mathbb F_p}$, then $Z({\mathcal C})={\textrm{Rep}}({\textbf{k}}[G]\ltimes{\textrm{Fun}(G)})$ which is not semisimple if $|G|$ is divisible by $p$.
\
Fusion rings
------------
### Realizability of fusion rings
Broadly speaking, fusion rings are rings which have the basic properties of Grothendieck rings of fusion categories. So let us consider a tensor category ${\mathcal C}$.
1. First, we have seen that if ${\mathcal C}$ is a tensor category, then $A={\textrm{Gr}({\mathcal C})}$ is a ring which is a free ${\mathbb Z}$-module with a distinguished basis $\{X_i\}_{i\in I}$ such that $X_0=\textbf1$ and multiplication (=fusion) rule $X_i\cdot X_j=\sum_kN_{ij}^kX_k$, $N_{ij}^k\geq0$ (property 1).\
2. Second, from the semisimplicity condition we have
If ${\mathcal C}$ is a semisimple tensor category, then for every simple object $V$ one has $V^*\cong~\!^*V$ (so $V\cong V^{**}$).
The coevaluation map provides an embedding $\textbf1\hookrightarrow V\otimes V^*$. Since the category is semisimple, it implies that $V\otimes V^*\cong\textbf1\oplus W$, then there exists a projection $p:V\otimes V^*\twoheadrightarrow\textbf1$. But in a rigid category, the only simple object $Y$ such that $V\otimes Y$ projects on $\textbf1$ is $^*V$.
Thus there exists an involution $*:i\mapsto i^*$ of $I$, defining an antiautomorphism of $A={\textrm{Gr}({\mathcal C})}$, and such that $N_{ij}^0=\delta_{ij^*}$ (property 2).
*A finite dimensional ring with a basis satisfying properties 1 and 2 is called a based ring, or a *fusion ring*.*
One of the basic questions of the theory of fusion categories is
Given a fusion ring $A$, can it be realized as the Grothendieck ring of a fusion category? If yes, in how many ways?
This problem is quite nontrivial, so let us start with a series of examples to illustrate it.
### Some important examples
In this subsection we work over $\Bbb C$ unless stated otherwise.
\[ty\]
### The rigidity conjecture
\(i) Any fusion ring has at most finitely many realizations over ${\bf k}$, up to equivalence (possibly none).\
(ii) The number of tensor functors between two fixed fusion categories, up to a natural tensor isomorphism, is finite.
Thus, the conjecture suggests that fusion categories and functors between them are discrete (“rigid”) objects and can’t be deformed. It was first proved in the case of unitary categories by Ocneanu; thus we call it “Ocneanu rigidity”. The conjecture is open in general but holds for categories over ${\mathbb{C}}$ (and hence for all fields of characteristic zero). Proving this will be the main goal of the next section.
Ocneanu rigidity
================
Main results
------------
### Müger’s squared norms
Let ${\mathcal C}$ be a fusion category. For every simple object $V$, we are going to define a number $|V|^2\in{\textbf{k}}^\times$, the *squared norm* of $V$. We have already seen that $V\cong V^{**}$, so let us fix an isomorphism $g_V:V\to V^{**}$ and consider its *quantum trace* $tr(g_V):=e_{V^*}\circ(g_V\otimes\textrm{id})\circ i_V\in{\textrm{End}(\textbf1)}\cong{\textbf{k}}$.
Clearly, this is not an invariant of $V$, since $g_V$ is well defined only up to scaling. However, the product $tr(g_V)tr(g_V^{*-1})$ is already independent on the choice of $g_V$ and is an invariant of $V$.
*$|V|^2=tr(g_V)tr(g_V^{*-1})$, and the *global dimension* of ${\mathcal C}$ is [^2] $$\textrm{dim}{\mathcal C}=\sum_{V~\textrm{simple}}|V|^2\,.$$ If $\textrm{dim}{\mathcal C}\neq0$, we say that ${\mathcal C}$ is *nondegenerate*.*
*A *pivotal structure* on ${\mathcal C}$ is an isomorphism of tensor functors $g: Id\to **$. A category equipped with a pivotal structure is said to be a [*pivotal category*]{}.*
In a pivotal tensor category, we can define dimensions of objects by $\textrm{dim}V=tr(g_V)$. The following obvious properties hold: $\textrm{dim}(V\otimes W)=\textrm{dim}V\textrm{dim}W$ and $|V|^2=\textrm{dim}V\textrm{dim}V^*$.
*We say that a pivotal structure $g$ is [*spherical*]{} if $\textrm{dim}V=\textrm{dim}V^*$ for all simple objects $V$.*
**Remarks.** 1. It is not known if every fusion category admits a pivotal or spherical structure.\
2. For a simple object $V$ one has $tr(g_V)\neq0$. Indeed, otherwise $\textbf1\hookrightarrow V\otimes V^*\twoheadrightarrow\textbf1$, and then the multiplicity $[V\otimes V^*:\textbf1]\geq2$, which is impossible in a semisimple category.\
\[kap\] *Let $H$ be a finite dimensional semisimple Hopf algebra over ${\textbf{k}}$. Since ${\textbf{k}}$ is algebraically closed, it is equivalent to saying that $H$ has a decomposition: $$H=\bigoplus_{V~\textrm{simple}}{\textrm{End}(V)}\,.$$ It is well-known that the squared antipode $S^2$ is an inner automorphim ($\exists g\in H^{\times}, S^2(x)=gxg^{-1}$); this is nothing but the statement (proved above) that $V$ is isomorphic to $V^{**}$ for simple $H$-modules $V$. Thus $|V|^2=tr_V(S^2_{|{\textrm{End}(V)}})$ and $\textrm{dim}({\textrm{Rep}}H)=tr_H(S^2)$.\
It is conjectured (by Kaplansky, [@K]) that $S^2=1$; this would imply that ${\textrm{Rep}}H$ admits a spherical structure, such that $|V|^2=dim(V)^2$ and $\textrm{dim}({\textrm{Rep}}H)=dim(H)$. For ${\textbf{k}}={\mathbb{C}}$, this is the well-known Larson-Radford theorem [@LR].*
### Main theorems
\[obw\] If ${\mathcal C}$ is nondegenerate, then 1) it has no nontrivial first order deformations of its associativity constraints, and 2) any tensor functor from ${\mathcal C}$ has no nontrivial first order deformations of its tensor structure.
Any fusion category over ${\mathbb{C}}$ is nondegenerate.
The first theorem implies Ocneanu rigidity for nondegenerate fusion categories (see [@ENO 7.3] for the precise argument), and the second one proves the rigidity conjecture for fusion categories over ${\mathbb{C}}$.\
In order to prove these theorems, we have to introduce and discuss the notions of module categories and weak Hopf algebras.
Module categories
-----------------
We have seen that the notion of a tensor category is the categorification of the notion of a ring. Similarly, the notion of a module category which we are about to define is the categorification of the notion of a module over a ring.
Let ${\mathcal C}$ be a tensor category.
*A *left module category over* ${\mathcal C}$ is an abelian category ${\mathcal M}$ with an exact bifunctor $\otimes:{\mathcal C}\times{\mathcal M}\to{\mathcal M}$ and functorial isomorphisms $\alpha:(-\otimes-)\otimes\bullet\to-\otimes(-\otimes\bullet)$ and $\eta:\textbf1\otimes\bullet\to\bullet$ (where $\bullet\in {\mathcal M}$) such that for any two functors obtained from $-\otimes\cdots-\otimes\bullet$ by inserting $\textbf1$’s and parenthesis, all functorial isomorphisms between them composed of $\Phi^{\pm1}$’s, $\mu^{\pm1}$’s, $\alpha^{\pm1}$’s and $\eta^{\pm1}$’s are equal.*
The definition of a *right module category over* ${\mathcal C}$ is analogous. We also leave it to the reader to define equivalence of module categories.
There is an analog of the MacLane coherence theorem for module categories which claims that it is sufficient for $\Phi$, $\mu$, $\alpha$ and $\eta$ to make the following diagrams commute:\
$\xymatrix{
((-\otimes-)\otimes-)\otimes\bullet\ar[r]^{\Phi^{1,2,3}\otimes\textrm{id}}\ar[d]^{\alpha^{12,3,4}} &
(-\otimes(-\otimes-))\otimes\bullet\ar[r]^{\alpha^{1,23,4}} &
-\otimes((-\otimes-)\otimes\bullet)\ar[d]^{\textrm{id}\otimes\alpha^{2,3,4}} \\
(-\otimes-)\otimes(-\otimes\bullet)\ar[rr]^{\alpha^{1,2,34}} & &
-\otimes(-\otimes(-\otimes\bullet))}$\
and $\xymatrix{
(-\otimes\textbf1)\otimes\bullet\ar[r]^{\alpha_{-,\textbf1,\bullet}}\ar[dr]_{\mu\otimes\textrm{id}} &
-\otimes(\textbf1\otimes\bullet)\ar[d]^{\textrm{id}\otimes\eta} \\
& -\otimes\bullet}$\
*(i) ${\mathcal C}$ is a left module category over itself.\
(ii) Define the tensor category ${\mathcal C}^{\textrm{op}}$, which coincides with ${\mathcal C}$ as an abstract category, and has reversed tensor product $\otimes^{\textrm{op}}$, which is defined by $X\otimes^{\textrm{op}}Y=Y\otimes X$. The associativity and unit morphisms are defined in an obvious manner. Then ${\mathcal C}$ is a right module category over ${\mathcal C}^{\textrm{op}}$.\
(iii) We deduce from (i) and (ii) that ${\mathcal C}$ is a left module category over ${\mathcal C}\boxtimes{\mathcal C}^{\textrm{op}}$.\
(iv) If ${\mathcal C}={\textrm{Vect}_{{\textbf{k}}}}$ and ${\mathcal M}={\textrm{Rep}}A$ for a given algebra $A$ over ${\textbf{k}}$, then ${\mathcal M}$ is a left module category over ${\mathcal C}$.*
Note that if ${\mathcal M}$ is a left (right) module category over ${\mathcal C}$, then its Grothendieck group ${\textrm{Gr}({\mathcal M})}$ is a left (respectively, right) ${\textrm{Gr}({\mathcal C})}$-module, with a distinguished basis $M_j$ and positive structure constants $N_{ij}^r$ such that $X_i\cdot M_j=\sum_rN_{ij}^rM_r$. In this way, we can associate to any object $X\in{\textrm{Obj}({\mathcal C})}$ its left (right) multiplication matrix $N_X$, which has positive entries, and in the semisimple case $N_{X^*}=N_X^T$.\
If ${\mathcal C}$ is a fusion category, we will be interested in semisimple finite module categories over ${\mathcal C}$. Such a module category is called *indecomposable* if ${\mathcal M}$ is not module equivalent to ${\mathcal M}_1\oplus{\mathcal M}_2$ for nonzero module categories ${\mathcal M}_i$, $i=1,2$.
As was mentioned above, the theory of module categories should be viewed as a categorical analog of the theory of modules (representation theory). Thus the main problem in the theory of module categories is
Given a fusion category ${\mathcal C}$, classify all indecomposable module categories over ${\mathcal C}$ which are finite and semisimple.
The answer is known only for a few particular cases. For example, one has the following result (see [@KO; @O1] for proof and references):
If ${\mathcal C}$ is the category of integrable modules over ${\widehat{\mathfrak{sl}_2}}$ at level $l$, then semisimple finite indecomposable module categories over ${\mathcal C}$ are in one-to-one correspondence with simply laced Dynkin diagrams of ADE type and with Coxeter number $h=l+2$.
### The category of bimodules
Let ${\mathcal C}$ be a tensor category. A structure of a left module category over ${\mathcal C}$ on an abelian category ${\mathcal M}$ is the same thing as a tensor functor ${\mathcal C}\to\textrm{Fun}({\mathcal M},{\mathcal M})$ ($\textrm{Fun}({\mathcal M},{\mathcal M})$ is the monoidal category whose objects are exact functors from ${\mathcal M}$ to itself, morphisms are natural transformations, and the tensor product is just the composition of functors). This is just the categorical analog of the tautological statement that a module $M$ over a ring $A$ is the same thing as a representation $\rho: A\to {\rm End}(M)$.
If ${\mathcal M}$ is semisimple and finite, then ${\mathcal M}\cong{\textrm{Rep}}A$ as an abelian category for a (nonunique) finite dimensional semisimple algebra $A$. Therefore, structures of a left module category over ${\mathcal C}$ on ${\mathcal M}$ are in one-to-one correspondence with tensor functors ${\mathcal C}\to\textrm{Fun}({\mathcal M},{\mathcal M})=A\textrm{-bimod}$.\
**Remark.** In particular, if ${\mathcal M}$ has only one simple object (i.e. ${\mathcal M}\cong{\textrm{Vect}_{{\textbf{k}}}}$ as an abelian category), then ${\mathcal C}$-module category structures on ${\mathcal M}$ correspond to fiber functors on ${\mathcal C}$.\
Let us consider more closely the structure of the category $A$-bimod. Its tensor product $\tilde\otimes$ is the tensor product over $A$. The simple objects in this category are $M_{ij}=\textrm{Hom}_{{\textbf{k}}}(M_i,M_j)$, where $M_i\in{\textrm{Obj}({\mathcal M})}$ are simple $A$-modules; and we have $M_{ij}\tilde\otimes M_{i'j'}=\delta_{i'j}M_{ij'}$. Thus $A\textrm{-bimod}$ is finite semisimple and satisfies all the axioms of a tensor category except one: $\textbf1=\oplus_iM_{ii}$ is not simple, but semisimple.
*A *multitensor category* is a category which satisfies all axioms of a tensor category except that the neutral object is only semisimple.\
A *multifusion category* is a finite semisimple multitensor category.*
Thus, $A\textrm{-bimod}$ is a multifusion category.
### Construction of module categories over fusion categories
Let $B$ be an algebra in a fusion category ${\mathcal C}$. The category ${\mathcal M}$ of right $B$-modules in ${\mathcal C}$ is a left module category over ${\mathcal C}$: let $X\in{\textrm{Obj}({\mathcal C})}$ and $M$ be a right $B$-module ($M\otimes B\to M$), then the composition $(X\otimes M)\otimes B\tilde\to X\otimes(M\otimes B)\to X\otimes M$ gives us the structure of a right $B$-module on $X\otimes M$ (and so it defines a structure of left ${\mathcal C}$-module category on ${\mathcal M}$). We will consider the situation when ${\mathcal M}$ is semisimple; in this case the algebra $B$ is said to be semisimple.
\[modca\] Any semisimple finite indecomposable module category over a fusion category can be constructed in this way (but nonuniquely).
\[grtheo\] *Let us consider the category ${\mathcal C}(G,\omega)$, with $G$ a finite group and $\omega\in Z^3(G,{\textbf{k}}^{\times})$ a 3-cocycle. Let $H\subset G$ be a subgroup such that $\omega_{\vert H}=\textrm d\psi$ for a cochain $\psi\in C^2(H,{\textbf{k}}^{\times})$. Define the twisted group algebra $B={\textbf{k}}_\psi[H]$: $B=\oplus_{h\in H}V_h$ as an object of ${\mathcal C}$ (where $V_h$ is the 1-dimensional module corresponding to $h\in H$), and the multiplication map $B\otimes B\to B$ is given by $\psi(g,h){\rm Id}: V_g\otimes V_h\to V_{gh}=V_g\otimes V_h$. The condition $\omega_{\vert H}=\textrm d\psi$, which can be rewritten as $\psi(h,k)\psi(g,hk)\omega(g,h,k)=\psi(gh,k)\psi(g,h)$ for all $g,h,k\in H$, assures the associativity of the product for $B$ (i.e., $B$ is an algebra in ${\mathcal C}(G,\omega)$). We call ${\mathcal M}(H,\psi)$ the category of right $B$-modules in ${\mathcal C}(G,\omega)$.*
Assume ${\rm char}({\bf k})$ does not divide $\vert G\vert$. All semisimple finite indecomposable module categories over ${\mathcal C}(G,\omega)$ have this form. Moreover, two module categories ${\mathcal M}(H_1,\psi_1)$ and ${\mathcal M}(H_2,\psi_2)$ are equivalent if and only if the pairs $(H_1,\psi_1)$ and $(H_2,\psi_2)$ are conjugate under the adjoint action of $G$.
Let ${\mathcal M}$ be an indecomposable module category over ${\mathcal C}(G,\omega)$. Since for every simple object we have $X=V_g$, $X\otimes X^*=V_g\otimes V_{g^{-1}}=\textbf1$, the multiplication matrix $N_X$ of $X$ satisfies the equation $N_XN_X^T=\textrm{id}$ and thus $N_X$ is a permutation matrix. So we have a group homomorphism $G\to\textrm{Perm}(\textrm{simple}({\mathcal M}))$. But ${\mathcal M}$ is indecomposable, therefore $G$ acts transitively on $Y:=\textrm{simple}({\mathcal M})$ and so $Y=G/H$.\
Thus ${\mathcal M}$ is the category of right $B$-modules in ${\mathcal C}(G,\omega)$, where $B={\textbf{k}}_\psi[H]$ for a $2$-cochain $\psi\in C^2(H,{\textbf{k}}^\times)$. The associativity condition for the product in $B$, as we saw above, is equivalent to $\psi(h,k)\psi(g,hk)\omega(g,h,k)= \psi(gh,k)\psi(g,h)$ (i.e., $\omega_{\vert H}=\textrm d\psi$). We are done.
Weak Hopf algebras
------------------
Tensor functors ${\mathcal C}\to A\textrm{-bimod}$ are a generalization of fiber functors (which are obtained when $A={\textbf{k}}$). So it makes sense to generalize reconstruction theory for them. This leads to *Hopf algebroids*, or, in the semisimple case, to *weak Hopf algebras*.
### Definition and properties of weak Hopf algebras
*A *weak Hopf algebra* is an associative unital algebra $(H,m,1)$ together with a coproduct $\Delta$, a counit $\epsilon$, and an antipode $S$ such that:\
1) $(H,\Delta,\epsilon)$ is a coassociative counital coalgebra.\
2) $\Delta$ is a morphism of associative algebras (not necessary unital).\
3) $(\Delta\otimes\textrm{id})\circ\Delta(1)=(\Delta(1)\otimes1)\cdot(1\otimes\Delta(1))
=(1\otimes\Delta(1))\cdot(\Delta(1)\otimes1)$\
4) $\epsilon(fgh)=\epsilon(fg_1)\epsilon(g_2h)=\epsilon(fg_2)\epsilon(g_1h)$\
5) $m\circ(\textrm{id}\otimes S)\circ\Delta(h)=
(\epsilon\otimes\textrm{id})\circ(\Delta(1)\cdot(h\otimes1))$\
6) $m\circ(S\otimes\textrm{id})\circ\Delta(h)=
(\textrm{id}\otimes\epsilon)\circ((1\otimes h)\cdot\Delta(1))$\
7) $S(h)=S(h_1)h_2S(h_3)$*
Here we used Sweedler’s notation: $\Delta_k(x)=x_1\otimes x_2\otimes...\otimes x_k$ ($\Delta_k$ is the $k$-fold coproduct and summation is implicitly assumed).
**Remarks.** 1. The notion of finite dimensional weak Hopf algebra is self-dual, i.e., if $(H,m,1,\Delta,\epsilon,S)$ is a finite dimensional weak Hopf algebra then $(H^*,\Delta^*,\epsilon^*,m^*,$ $1^*,S^*)$ is also a finite dimensional weak Hopf algebra.\
2. Let $H$ be a weak Hopf algebra. $H$ is a Hopf algebra if and only if $\Delta(1)=1\otimes1$ (that is equivalent to the requirement that $\epsilon$ is an associative algebra morphism).
The linear maps $\epsilon_t:h\mapsto\epsilon(1_1h)1_2$ and $\epsilon_s:h\mapsto1_1\epsilon(h1_2)$ defined by $5)$ and $6)$ in the definition are called the *target* and *source counital maps* respectively. The images $A_t=\epsilon_t(H)$ and $A_s=\epsilon_s(H)$ are the target and source *bases* of $H$.
\[bases\] $A_t$ and $A_s$ are semisimple algebras that commute with each other, and $S_{\vert A_t}:A_t\to A_s$ is an algebra antihomomorphism.
An especially important and tractable class of weak Hopf algebras is that of regular weak Hopf algebras, defined as follows.
*A weak Hopf algebra $H$ is regular if $S^2=\textrm{id}$ on $A_t$ and $A_s$.*
>From now on all weak Hopf algebras we consider will be assumed regular.
Let $H$ be a finite dimensional weak Hopf algebra and consider the category ${\mathcal C}={\textrm{Rep}}H$. One can define the tensor product $V\otimes W$ of two representations: $V\otimes W:=\Delta(1)(V\otimes_{\textbf{k}}W)$ as a vector space, and the action of any $x\in H$ on $V\otimes W$ is given by $\Delta(x)$. As in the case of a Hopf algebra, the associativity morphism is the identity, $\epsilon_t$ gives $A_t$ the structure of an $H$-module which is the neutral object in ${\mathcal C}$, and the antipode $S$ allows us to define duality. This endows ${\mathcal C}$ with the structure of a finite tensor category [@NTV Section 4].
In the case when $H$ is regular, each $H$-module $M$ is also an $A_t\otimes A_s$-module (by Proposition \[bases\]), and hence it is an $A_t$-bimodule (since $A_s=A_t^{\textrm{op}}$). Moreover, the forgetful functor ${\mathcal C}=H\textrm{-mod}\to A_t\textrm{-bimod}$ is tensor.
### Reconstruction theory
Let ${\mathcal C}$ be a finite tensor category, $A$ a finite dimensional semisimple algebra and $F:{\mathcal C}\to A\textrm{-bimod}$ a tensor functor. Assume that the sizes of the matrix blocks of $A$ are not divisible by ${\rm char}({\textbf{k}})$ (for example, $A$ is commutative or ${\rm char}({\textbf{k}})=0$).
Consider $H=\textrm{End}_{{\textbf{k}}}(F)={\textrm{End}(\overline F)}$, where $\overline F$ is the composition of $F$ with the forgetful functor ${\rm Forget}$ to vector spaces; it is a unital associative algebra. Since any $F(X)$ is an $A$-bimodule, there exists an algebra antihomomorphism $s:A\to H$ and an algebra morphism $t:A\to H$ such that $[s(a),t(a')]=0$ for all $a,a'\in A$. Moreover, we can define a kind of coproduct $\overline\Delta:\textrm{End}_{{\textbf{k}}}(F)\to\textrm{End}_{{\textbf{k}}}(F\times F)$ in the same way as for tannakian formalism: $\overline\Delta(T)=J\circ T\circ J^{-1}$. Thus $\overline\Delta(T)$ can be interpreted as an element $$\overline\Delta(T)\in H\otimes_AH=H\otimes H/<t(a)x\otimes y-x\otimes s(a)y>,$$ such that $\overline\Delta(T)(t(a)\otimes 1+1\otimes s(a))=0$ for all $a\in H$. Now, since $A$ is semisimple, there is a canonical map $$\eta:H\otimes_AH\to H\otimes_{{\textbf{k}}}H;\quad
m\otimes n\mapsto\sum_ime_i\otimes e^in$$ for dual bases $(e_i)$ and $(e^i)$ of $A$ relatively to the pairing $(a,b)=tr_A(L_a L_b)$, where $L_a$ is the operator of left multiplication by $a$ (note that because of our assumption on the block sizes this pairing is nondegenerate). We can thus define the “true” coproduct $\Delta=\eta\circ\overline\Delta:H\to H\otimes H$ which turns out to be coassociative.\
One can also define a counit $\epsilon:H\to{\textbf{k}}$ by $\epsilon(T)=tr_A(T_{\vert\overline F(\textbf1)})$ and an antipode $S:H\to H$ by $S(T)_{\vert\overline F(X)}=(T_{\vert\overline F(X^*)})^*$.
The associative unital algebra $H$ equipped with $\Delta$, $\epsilon$ and $S$ as above is a regular weak Hopf algebra. Moreover, ${\mathcal C}\cong{\rm Rep}H$ as a tensor category.
Thus, given a tensor category ${\mathcal C}$ over ${\textbf{k}}$ and a finite dimensional semisimple algebra $A$ with block sizes not divisible by ${\rm char}({\textbf{k}})$, we have bijections (modulo appropriate equivalences):\
$\xymatrix{
\footnotesize{
\txt{Finite dimensional regular weak Hopf\\algebras $H$ with bases $A_t=A$ and $A_s=A^{\textrm{op}}$}}
\ar@{<->}[d]\ar@{<-->}[dr] & \\
\footnotesize{\txt{Finite tensor categories with tensor\\functor $F:{\mathcal C}\to A\textrm{-bimod}$}}
\ar@{<->}[r] &
\footnotesize{\txt{Finite semisimple indecomposable\\module categories over ${\mathcal C}$, equivalent\\
to $A$-mod as abelian categories}}}$\
If ${\mathcal C}$ is a fusion category, then ${\mathcal C}$ is a semisimple module category over itself. So ${\mathcal C}\cong{\textrm{Rep}}H$ as a tensor category for a semisimple weak Hopf algebra $H$ with base $A=\oplus_{i\in I}{\textbf{k}}_i$.
Any fusion category is the representation category of a finite dimensional semisimple weak Hopf algebra with a commutative base.
[**Remark.**]{} It is not known to us if there exists a (nonsemisimple) finite tensor category which is not the category of representations of a weak Hopf algebra (i.e. does not admit a semisimple module category). Finding such a category is an interesting open problem.
Proofs
------
### Nondegeneracy of fusion categories over ${\mathbb{C}}$
In any fusion category, there exists an isomorphism of tensor functors $\delta:{\rm id}\to ****$.
Recall that ${\mathcal C}\cong{\textrm{Rep}}H$ for a finite dimensional semisimple regular weak Hopf algebra $H$. In the semisimple case, the generalization of Radford’s $S^4$ formula by Nikshych [@N Section 5] tells us that: $$\exists a\in G(H), \forall x\in H, S^4(x)=a^{-1}xa\,,$$ where $a\in G(H)$ means $a$ is invertible and $\Delta(a)=\Delta(1)(a\otimes a)=(a\otimes a)\Delta(1)$ (i.e., $a$ is a grouplike element). Thus we can define $\delta$ by $\delta_V=a^{-1}|_V$. Then for every $H$-modules $V$ and $W$, the fact that $\delta_{V\otimes W}=\delta_V\otimes\delta_W$ follows from the grouplike property of $a$.
For fusion categories over ${\mathbb{C}}$, for any simple object $V$ one has $\vert V\vert^2>0$.
In particular, this implies that for any fusion category ${\mathcal C}$ over ${\mathbb{C}}$ one has $\textrm{dim}{\mathcal C}\geq1$ and so is nondegenerate.
[**Question.**]{} Does there exist $\epsilon>0$ such that for every fusion category ${\mathcal C}$ over ${\mathbb{C}}$ which is not ${\textrm{Vect}_{{\textbf{k}}}}$, $\textrm{dim}{\mathcal C}>1+\epsilon$?
First do the pivotal case. In this case $\textrm{dim}(V\otimes W)=\textrm{dim}V\textrm{dim}W$ for all objects $V,W$, thus $d_id_j=\sum_kN_{ij}^kd_k$, where $d_i=\textrm{dim}(X_i)$ are the dimensions of the simple objects. In a shorter way we can rewrite these equalities as $N_i\vec d=d_i\vec d$, where $\vec d=(d_0,\dots,d_{n-1})$.\
For all $i,j,k\in I$, $$\begin{aligned}
N_{i^*j}^k & =
dim({\textrm{Hom}(X_i^*\otimes X_j,X_k)})
=dim({\textrm{Hom}(X_j,X_i\otimes X_k)})
\quad\textrm{(by rigidity)} \\
& = dim({\textrm{Hom}(X_i\otimes X_k,X_j)})
\quad\textrm{(by semisimplicity)} \\
& = N_{ik}^j \end{aligned}$$ Therefore $N_i^TN_i\vec d=N_{i^*}N_i\vec d=d_{i^*}d_i\vec d =\vert X_i\vert^2\vec d$, so $\vert X_i\vert^2$ is an eigenvalue of $N_i^TN_i$ associated to $\vec d\neq\vec0$ and consequently $\vert X_i\vert^2>0$.
Now we extend the argument to the non-pivotal case. Let us define the *pivotal extension* $\overline{{\mathcal C}}$ of ${\mathcal C}$, which is the fusion category whose simple objects are pairs $(X,f)$: $X$ is simple in ${\mathcal C}$ and $f:X\tilde\to X^{**}$ satisfies $f^{**}f=\delta_X$ for the isomorphism of tensor functors $\delta:\textrm{id}\to ****$ constructed above. The category $\overline{{\mathcal C}}$ has a canonical pivotal structure $(X,f)\to(X^{**},f^{**})$ (which is given by $f$ itself), thus $\vert(X,f)\vert^2>0$. Finally the forgetful functor $\overline{{\mathcal C}}\to{\mathcal C};(X,f)\mapsto X$ preserves squared norms, and so $\vert X\vert^2>0$.
### Proof of Ocneanu rigidity: the Davydov-Yetter cohomology
Let ${\mathcal D}$ be a tensor category. Define the following cochain complex attached to ${\mathcal D}$:
- $C^n({\mathcal D})={\textrm{End}(T_n)}$, where $T_n$ is the $n$-functor ${\mathcal D}^n\to{\mathcal D};(X_1,\dots,X_n)\mapsto X_1\otimes...\otimes X_n$ ($T_0=\textbf1$ and $T_1=\textrm{id}$).
- The differential $\textrm d:C^n({\mathcal D})\to C^{n+1}({\mathcal D})$ is given by $$\textrm df=\textrm{id}\otimes f_{2,\dots,n+1}-f_{12,3,\dots,n+1}+\cdots
+(-1)^nf_{1,\dots,n-1,nn+1}+(-1)^{n+1}f_{1,\dots,n}\otimes\textrm{id}$$
$H^n({\mathcal D})$ is the $n$-th space of the *Davydov-Yetter cohomology* ([@Dav; @Y]).
*Assume ${\mathcal D}={\textrm{Rep}}H$ for a Hopf algebra $H$. Then $C^n({\mathcal D})=(H^{\otimes n})^{H_{ad}}$. $(C^n,\textrm d)$ is a subcomplex of the co-Hochschild complex for $H$ with trivial coefficients.*
$H^3({\mathcal D})$ and $H^4({\mathcal D})$ respectively classify first order deformations of associativity constraints in ${\mathcal D}$ and obstructions to these deformations.
*(i) Let $G$ be a finite group and ${\mathcal D}={\mathcal C}(G,1)$. Then $H^i({\mathcal D})=H^{i}(G,{\textbf{k}})$, and thus $H^i({\mathcal D})=0$ for $i>0$ if ${\textbf{k}}={\mathbb{C}}$ or $\vert G\vert$ and $\textrm{char}({\textbf{k}})$ are coprime.\
(ii) Let $G$ be a semisimple complex Lie group with Lie algebra ${\mathfrak g}$ and consider ${\mathcal C}={\textrm{Rep}}G$. Then $H^i({\mathcal C})=(\wedge^i{\mathfrak g})^G=H^{i}(G,{\mathbb{C}})$. In particular, $H^3({\mathcal C})={\mathbb{C}}$ and $H^4({\mathcal C})=0$. So there exists a unique one-parameter deformation of ${\mathcal C}={\textrm{Rep}}G$ which is realized by ${\textrm{Rep}}U_\hbar({\mathfrak g})$.*
The next result implies in particular the first part of Theorem \[obw\].
Let ${\mathcal D}$ be a nondegenerate fusion category over ${\textbf{k}}$. Then for all $i>0$, $H^i({\mathcal D})=0$.
The proof is based on the notion of categorical integral.\
Suppose that $f\in C^n({\mathcal D})$ (for $X_1,\dots,X_n$, $f_{X_1,\dots,X_n}:X_1\otimes\cdots\otimes X_n\to X_1\otimes\cdots \otimes X_n$). Define $\int f\in C^{n-1}$ in the following way: for $X_1,\dots,X_{n-1}\in{\textrm{Obj}({\mathcal D})}$, $$(\int f)_{X_1,\dots,X_{n-1}}
=\sum_{V~\textrm{simple}}tr_V((\textrm{id}\otimes g_V)\circ f_{X_1,\dots,X_{n-1},V}) tr(g_V^{*-1})$$ where $tr_V((\textrm{id}\otimes g_V)\circ f_{X_1,\dots,X_{n-1},V})$ is equal to $$(\textrm{id}^{\otimes n}\otimes e_{V^*})\circ
(\textrm{id}^{\otimes(n-1)}\otimes g_V\otimes\textrm{id})
\circ(f_{X_1,\dots,X_{n-1},V}\otimes\textrm{id})
\circ(\textrm{id}^{\otimes n}\otimes i_V)$$ **Remark.** By definition, $\int\textrm{id}=\textrm{dim}{\mathcal D}$.\
Assume now that $f\in Z^{n}({\mathcal D})$ is a cocycle. Then if we put $\varphi=\int f$, we have $$\begin{aligned}
0 & = \int{\textrm df} \\
& = \textrm{id}\otimes\int{f_{2,\dots,n+1}}-\int{f_{12,3,\dots,n+1}}+\cdots \\
& + (-1)^n\int{f_{1\dots,n-1,nn+1}}+(-1)^{n+1}f_{1,\dots,n}\otimes\int\textrm{id} \\
& = \textrm{id}\otimes\varphi_{2,\dots,n}-\varphi_{12,3,\dots,n}+\cdots \\
& + (-1)^{n-1}\varphi_{1,\dots,n-1n}+(-1)^n\int{f_{1\dots,n-1,nn+1}}+(-1)^{n+1}
\textrm{dim}{\mathcal D}\cdot f_{1,\dots,n}\end{aligned}$$
$\int{f_{1,\dots,n-1,nn+1}}=\varphi_{1,\dots,n-1}\otimes\textrm{id}$.
The proof is based the theory of weak Hopf algebras, and we will omit it, see [@ENO], Section 6.
Thus when $\textrm{dim}{\mathcal D}\neq0$, $f=\frac{1}{\textrm{dim}{\mathcal D}}(-1)^{n-1}\textrm d\varphi$.
**Remark.** In the same way, for any tensor functor $F:{\mathcal C}\to{\mathcal D}$, one can define a cochain complex $C^n_F({\mathcal C})={\textrm{End}(T_n\circ F^{\otimes n})}$ and a differential $\textrm d:C_F^n({\mathcal C})\to C_F^{n+1}({\mathcal C})$ which is given by $$\textrm df=\textrm{id}\otimes f_{2,\dots,n+1}-f_{12,3,\dots,n+1}+\cdots
+(-1)^nf_{1,\dots,n-1,nn+1}+(-1)^{n+1}f_{1,\dots,n}\otimes\textrm{id}$$ where $f_{1,\dots,ii+1,\dots,n+1}$ acts on $F(X_1)\otimes\dots\otimes F(X_{n+1})$ as $f$ on $F(X_1)\otimes\dots\otimes F(X_i\otimes X_{i+1})\otimes\dots \otimes F(X_{n+1})$ (we have used the tensor structure to identify $F(X_i)\otimes F(X_{i+1})$ and $F(X_i\otimes X_{i+1})$).\
Then one can show (see [@ENO]) that the corresponding cohomology spaces $H^i_F({\mathcal C})$ are trivial for nondegenerate categories, and that $H^2_F({\mathcal C})$ (resp. $H^3_F({\mathcal C})$) classifies first order deformations of the tensor structure of $F$ (resp. obstructions to these deformations). Thus the second part of Theorem \[obw\] is proved.
Morita theory, modular categories, and lifting theory
=====================================================
Morita theory in the categorical context
----------------------------------------
### Dual category with respect to a module category {#dualcat}
Let $H$ be a finite dimensional (weak) Hopf algebra. ${\mathcal C}={\rm Rep}(H)$ is a finite tensor category. How to describe the category ${\rm Rep}(H^*)$ in terms of ${\mathcal C}$?
The answer is given by the next definitions.
*A *module functor* between module categories ${\mathcal M}_1, {\mathcal M}_2$ over ${\mathcal C}$ is an additive functor $F:{\mathcal M}_1\to{\mathcal M}_2$ together with a functorial isomorphism $J:F(-\otimes_1\bullet)\to-\otimes_2F(\bullet)$ such that the following diagrams commute:\
$\xymatrix{
F((-\otimes_{\mathcal C}-)\otimes_1\bullet)\ar[r]^{F(\alpha)}\ar[d]^J &
F(-\otimes_1(-\otimes_1\bullet))\ar[r]^J &
-\otimes_2F(-\otimes_1\bullet)\ar[d]^{\textrm{id}\otimes J} \\
(-\otimes_{\mathcal C}-)\otimes_2F(\bullet)\ar[rr]^\alpha & & -\otimes_2(-\otimes_2F(\bullet))}$\
and $\xymatrix{
F(\textbf1\otimes_1\bullet)\ar[r]^{J_{\textbf1,\bullet}}\ar[dr]_{F(\eta_1)} &
\textbf1\otimes_2F(\bullet)\ar[d]^{\eta_2} \\
& F(\bullet)}$\
*
Let ${\mathcal C}$ be a tensor category (not necessarily semisimple) and ${\mathcal M}$ a left module category over ${\mathcal C}$.
*The *dual category of ${\mathcal C}$ with respect to ${\mathcal M}$* is the category ${\mathcal C}_{\mathcal M}^*=\textrm{Fun}_{\mathcal C}({\mathcal M},{\mathcal M})$, the category of module functors from ${\mathcal M}$ to itself with tensor product being the composition of functors.*
Thus the notion of the dual category is the categorification of the notion of the centralizer of an algebra in a module.
Observe that ${\mathcal C}_{\mathcal M}^*$ is a monoidal category and ${\mathcal M}$ is a left module category over it. However, ${\mathcal C}_{\mathcal M}^*$ is not always rigid. For example, if ${\mathcal C}={\textrm{Vect}_{{\textbf{k}}}}$ and ${\mathcal M}=A\textrm{-mod}$ for a finite dimensional associative algebra $A$ over ${\textbf{k}}$, then ${\mathcal C}_{\mathcal M}^*=\textrm{Fun}_{{\textrm{Vect}_{{\textbf{k}}}}}({\mathcal M},{\mathcal M})=\textrm{Fun}({\mathcal M},{\mathcal M})$. This category contains the category $A\textrm{-bimod}$ with tensor product $\otimes_A$ which is not exact if $A$ is not semisimple (while it must be exact in the rigid case).
Thus, to insure rigidity of the dual category, we should perhaps restrict ourselves to a subclass of module categories. A subclass that turns out to produce a good theory is that of [*exact module categories*]{}. Namely (see [@EO1]), a left module category is called *exact* if for any projective object $P$ in ${\mathcal C}$, and any $X\in{\textrm{Obj}({\mathcal M})}$, $P\otimes X$ is also projective. Such a category is finite if and only if it has finitely many simple objects. In the particular case of a fusion category ${\mathcal C}$, exactness for module categories coincides with semisimplicity.
If ${\mathcal C}$ is a finite tensor category and ${\mathcal M}$ is a finite indecomposable exact left module category over it, then ${\mathcal C}_{\mathcal M}^*$ is a finite tensor category.
*(i) If ${\mathcal C}={\textrm{Rep}}H$ and ${\mathcal M}={\textrm{Rep}}A$ for a finite dimensional regular weak Hopf algebra with bases $A,A^{\textrm{op}}$, then ${\mathcal C}_{\mathcal M}^*={\textrm{Rep}}(H^{*\textrm{op}})$.*
*(ii) Let ${\mathcal C}={\mathcal C}(G,\omega)$ and ${\mathcal M}={\mathcal M}(H,\psi)$ be as in example \[grtheo\]. Then one can consider the category of $B$-bimodules ${\mathcal C}(G,\omega,H,\psi):={\mathcal C}_{\mathcal M}^*$, where $B={\textbf{k}}_\psi[H]$ is the twisted group algebra of $H$ in ${\mathcal C}$. Such categories are called *group theoretical*.*
Let ${\mathcal C}$ be a finite tensor category and ${\mathcal M}$ a finite indecomposable exact left module category over ${\mathcal C}$. Then one can show ([@ENO; @EO1; @O1]) that the following properties hold:
1. $({\mathcal C}_{\mathcal M}^*)_{\mathcal M}^*={\mathcal C}$
2. $({\mathcal C}\boxtimes{\mathcal C}_{\mathcal M}^*)_{\mathcal M}^*=Z({\mathcal C})$
3. ${\mathcal C}_{\mathcal C}^*={\mathcal C}^{\textrm{op}}$ (and then $({\mathcal C}\boxtimes{\mathcal C}^{\textrm{op}})_{\mathcal C}^*=Z({\mathcal C})$ by the previous one).
4. If ${\mathcal M}=B\textrm{-mod}$ for a semisimple algebra $B$ in a fusion category ${\mathcal C}$, then ${\mathcal C}_{\mathcal M}^*=B\textrm{-bimod}$.
5. If ${\mathcal C}$ is a nondegenerate fusion category, then ${\mathcal C}_{\mathcal M}^*$ is also fusion. Moreover, $\textrm{dim}{\mathcal C}_{\mathcal M}^*=\textrm{dim}{\mathcal C}$, and thus $\textrm{dim}Z({\mathcal C})=(\textrm{dim}{\mathcal C})^2$.
[**Remark.**]{} Note that property (1) is the categorical version of the double centralizer theorem for semisimple algebras (saying that the centralizer of the centralizer of $A$ in a module $M$ is $A$ if $A$ is a finite dimensional semisimple algebra). Property (2) is the categorical analog of the statement that if $A'$ is the centralizer of $A$ in $M$ then the centralizer of $A\otimes A'$ in $M$ is the center of $A$. Finally, property (3) is the categorical version of the fact that the centralizer of $A$ in $A$ is $A^{op}$.
### Morita equivalence of finite tensor categories
By now, all module categories are supposed to be finite and exact.
*Two finite tensor categories ${\mathcal C}$ and ${\mathcal D}$ are *Morita equivalent* if there exists an indecomposable (left) module category ${\mathcal M}$ over ${\mathcal C}$ such that ${\mathcal C}_{\mathcal M}^*={\mathcal D}^{\textrm{op}}$. In this case we write ${\mathcal C}\sim_{\mathcal M}{\mathcal D}$.*
Obviously, this notion is the categorical analog of Morita equivalence of associative algebras.
Morita equivalence of finite tensor categories is an equivalence relation.
This relation is reflexive since ${\mathcal C}_{\mathcal C}^*={\mathcal C}^{\textrm{op}}$.\
To prove the symmetry, assume that ${\mathcal C}_{\mathcal M}^*={\mathcal D}^{\textrm{op}}$, and define ${\mathcal M}^\vee:=\textrm{Fun}({\mathcal M},{\textrm{Vect}_{{\textbf{k}}}})$. This is a left (indecomposable) module category over ${\mathcal D}$ and ${\mathcal D}_{{\mathcal M}^\vee}^*={\mathcal C}^{\textrm{op}}$. Now let us prove transitivity. Suppose ${\mathcal C}\sim_{\mathcal M}{\mathcal D}$ and ${\mathcal D}\sim_{\mathcal N}\mathcal E$. Take $\mathcal P=\textrm{Fun}_{\mathcal D}({\mathcal M}^\vee,\mathcal N)$ (By analogy with ring theory, we could denote this category by ${\mathcal M}\otimes_{\mathcal D}\mathcal N$.) Then ${\mathcal C}_{\mathcal P}^*=\mathcal E^{\textrm{op}}$. Thus the transitivity condition is verified.
\[Mug\] Let ${\mathcal C}\sim_{\mathcal M}{\mathcal D}$ be a Morita equivalence of finite tensor categories. Then there is a bijection between indecomposable left module categories over ${\mathcal C}$ and ${\mathcal D}$. It maps $\mathcal N$ over ${\mathcal C}$ to ${\rm Fun}_{\mathcal C}({\mathcal M},\mathcal N)$ over ${\mathcal D}$.
This, obviously, is the categorical version of the well known characterization of Morita equivalent algebras: their categories of modules are equivalent.
\[O3\] Indecomposable left module categories over ${\mathcal C}(G,\omega,H,\psi)$ are $${\mathcal M}(H,\psi,H',\psi'):={\rm Fun}_{{\mathcal C}(G,\omega)}({\mathcal M}(H,\psi),{\mathcal M}(H',\psi'))$$
### Application to representation theory of groups
Let $G$ be a finite group and consider the category ${\mathcal D}={\textrm{Rep}}G$. In fact, ${\mathcal D}={\mathcal C}(G,1)_{\mathcal M}^*$ with ${\mathcal M}={\mathcal M}(G,1)={\textrm{Vect}_{{\textbf{k}}}}$. Hence, indecomposable ${\mathcal D}$-module categories are of the form ${\mathcal M}(G,1,H,\psi)={\textrm{Rep}}{\mathbb{C}}_\psi[H]$.\
Now recall that fiber functors are classified by module categories with only one simple object. In our case it corresponds to the case when ${\mathbb{C}}_\psi[H]$ is simple, which is equivalent to the requirement that $\psi$ is a nondegenerate 2-cocycle, in the sense of the following definition.
*A 2-cocycle $\psi$ on $H$ is *nondegenerate* if $H$ admits a unique projective irreducible representation with cocycle $\psi$ of dimension $\sqrt{\vert H\vert}$.\
A group $H$ which admits a nondegenerate cocycle is said to be *of central type*.*
**Remarks.** 1. It is obvious that a group of central type has order $N^2$, where $N$ is an integer.\
2. Howlett and Isaacs [@HI] showed that any group of central type is solvable. This is a deep result based on the classification of finite simple groups.
Fiber functors on ${\rm Rep}G$ (i.e., Hopf twists on ${\mathbb{C}}[G]$ up to a gauge) are in one-to-one correspondence with pairs $(H,\psi)$, where $H$ is a subgroup of $G$ and $\psi$ a nondegenerate 2-cocycle on $H$ modulo coboundaries and inner automorphisms.
The theorem follows from Theorem \[Mug\] and Corollary \[O3\]. We leave the proof to the reader.
Let $D_8$ be the group of symmetries of the square and $Q_8$ the quaternion group. Then ${\rm Rep}D_8$ and ${\rm Rep}Q_8$ are not equivalent (although they have the same Grothendieck ring).
In $Q_8$, all subgroups of order 4 are cyclic and hence do not admit any nondegenerate 2-cocycle.\
On the other hand, $D_8$ has two subgroups isomorphic to ${\mathbb{Z}}_2\times{\mathbb{Z}}_2$ (not conjugate) and each has one nondegenerate 2-cocycle. Thus $Q_8$ has fewer fiber functors (in fact only 1) than $D_8$ (which has 3 such).
So, we see that one can sometimes establish that two fusion categories are not equivalent (as tensor categories) by counting fiber functors. Similarly, one can sometimes show that two fusion categories are not Morita equivalent by counting all indecomposable module categories over them (since we have seen that Morita equivalent fusion categories have the same number of indecomposable module categories). Let us illustrate it with the following example.
*We want to show that ${\textrm{Rep}}({\mathbb{Z}}_p\times{\mathbb{Z}}_p)$ and ${\textrm{Rep}}{\mathbb{Z}}_{p^2}$ are not Morita equivalent.\
First remember that ${\textrm{Rep}}G={\mathcal C}(G,1,G,1)$ and module categories over it are parametrized by $(H,\psi)$, where $H$ is a subgroup of $G$ and $\psi\in H^{2}(H,{\mathbb{C}}^{\times})$.\
On the one hand, ${\mathbb{Z}}_{p^2}$ has three subgroups (${\mathbb{Z}}_{p^2}$ itself, ${\mathbb{Z}}_p$, and $1$), all with a trivial second cohomology. Thus ${\textrm{Rep}}{\mathbb{Z}}_{p^2}$ has 3 indecomposable module categories. On the other hand, ${\mathbb{Z}}_p\times{\mathbb{Z}}_p$ has $p+3$ subgroups: ${\mathbb{Z}}_p\times{\mathbb{Z}}_p$, $p+1$ copies of ${\mathbb{Z}}_p$, and $1$. Moreover, ${\mathbb{Z}}_p\times{\mathbb{Z}}_p$ has $p$ 2-cocycles up to coboundaries. Thus ${\textrm{Rep}}({\mathbb{Z}}_p\times{\mathbb{Z}}_p)$ has $2p+2>3$ module categories. $\Box$*
Modular categories and the Verlinde formula
-------------------------------------------
Let ${\mathcal C}$ be a braided tensor category. Then we have a canonical (non-tensor) functorial isomorphism $u: \textrm{id}\to **$ given by the composition $$V\to V\otimes V^*\otimes V^{**}\to V^*\otimes V\otimes V^{**}\to V^{**}$$ (the maps are the coevaluation, the braiding, and the evaluation). This isomorphism is called the [*Drinfeld isomorphism*]{}. Using the Drinfeld isomorphism, we can define a tensor isomorphism $\delta: {\rm id}\to ****$ by the formula $\delta_V=(u_{V^*}^*)^{-1}u_V$.
*A *ribbon category* is a braided tensor category together with a pivotal structure $g:\textrm{id}\to**$, such that $g^{**}g=\delta$.*
We refer the reader who wants to learn more about ribbon categories (especially the graphical calculus for morphisms, using tangles) to [@K], [@BK] or [@T].
Assume now that ${\mathcal C}$ is a ribbon category. Recall for any simple object $V\in{\mathcal C}$ one can define the *dimension* $\textrm{dim}V$. It is known (see e.g. [@K]) that $\textrm{dim}V^*=\textrm{dim}V$.
For any two objects $V,W$, one can define the number $S_{VW}\in{\textrm{End}(\textbf1)}\cong{\textbf{k}}$ to be [$$(e_{V^*}\otimes e_{W^*})\circ(g_V\otimes\textrm{id}_{V^*}\otimes g_W\otimes\textrm{id}_{W^*})\circ
(\textrm{id}_V\otimes\sigma_{WV^*}\otimes\textrm{id}_{W^*})
\circ(\textrm{id}_V\otimes\sigma_{V^*W}\otimes\textrm{id}_{W^*})\circ(i_V\otimes i_W)\,.$$]{} Now assume that ${\mathcal C}$ is fusion, with simple objects $X_i$’s. Then we can define a matrix $S$ with entries $S_{ij}=S_{X_iX_j}$. $S$ has the following properties:
1. $S_{ij}=S_{ji}$
2. $S_{ij}=S_{i^*j^*}$
3. $S_{i0}=\textrm{dim}X_i\neq0$
*A ribbon fusion category is called *modular* if $S$ is nondegenerate.*
If ${\mathcal C}$ is a nondegenerate fusion category with a spherical structure, then $Z({\mathcal C})$ is a modular category.
In a modular category ${\mathcal C}$, $$\sum_k S_{ik}S_{kj}=({\rm dim}{\mathcal C})\delta_{ij^*}$$
Thus if ${\mathcal C}$ is a modular category, then $\textrm{dim}{\mathcal C}\neq0$ and we can define new numbers $s_{ij}=S_{ij}/\sqrt{\textrm{dim}{\mathcal C}}$ (here we must make a choice of the square root).
$$\sum_\alpha N_{ij}^\alpha s_{\alpha r}=\frac{s_{ir}s_{jr}}{s_{0r}}$$
So $s_{ir}/s_{0r}$ are eigenvalues of the multiplication matrix $N_i$. In particular, they are algebraic integers (i.e. roots of a monic polynomial with integer coefficients - the characteristic polynomial of $N_i$). Hence:
For every $r$, $\frac{{\rm dim}{\mathcal C}}{({\rm dim}X_r)^2}=\frac{s_{ir}s_{i^*r}}{s_{0r}^2}$ is an algebraic integer.
This result will be very useful to prove classification theorems in section 4.
### Galois property of the S-matrix
A remarkable result due to J. de Boere, J. Goeree, A. Coste and T. Gannon states that the entries of the S-matrix of a modular category lie in a cyclotomic field, see [@dBG; @CG]. Namely, one has the following theorem.
\[app\] Let $S=(s_{ij})_{i,j\in I}$ be the S-matrix of a modular category ${\mathcal C}$. There exists a root of unity $\xi$ such that $s_{ij}\in {\mathbb Q}(\xi)$.
Let $\{ X_i\}_{i\in I}$ be the representatives of isomorphism classes of simple objects of ${\mathcal C}$; let $0\in I$ be such that $X_0$ is the unit object of ${\mathcal C}$ and the involution $i\mapsto i^*$ of $I$ be defined by $X_i^*\cong X_{i^*}$. By the definition of modularity, any homomorphism $f: K({\mathcal C}) \to {\mathbb C}$ is of the form $f([X_i])=s_{ij}/s_{0j}$ for some well defined $j\in I$. Hence for any automorphism $g$ of $\overline{\mathbb Q}$ one has $g(s_{ij}/s_{0j})= s_{ig(j)}/s_{0g(j)}$ for a well defined action of $g$ on $I$.
Now remember from the previous subsection that one has the following properties: $\sum_ks_{ik}s_{kj}=\delta_{ij^*}$, $s_{ij}=s_{ji}$, and $s_{0i^*}=s_{0i}\ne 0$.\
Thus, $\sum_js_{ij}s_{ji^*}=1$ and hence $(1/s_{0i})^2=\sum_j(s_{ji}/s_{0i}) (s_{ji^*}/s_{0i^*})$. Applying the automorphism $g$ to this equation we get $$g(\frac{1}{s_{0i}^2})=g(\sum_j\frac{s_{ji}}{s_{0i}}\frac{s_{ji^*}}{s_{0i^*}})=
\sum_j\frac{s_{jg(i)}}{s_{0g(i)}}\frac{s_{jg(i^*)}}{s_{0g(i^*)}}=
\frac{\delta_{g(i)g(i^*)^*}}{s_{0g(i)}s_{0g(i^*)}}\,.$$ It follows that $g(i^*)=g(i)^*$ and $g((s_{0i})^2)=(s_{0g(i)})^2$. Hence $$g((s_{ij})^2)=g((s_{ij}/s_{0j})^2\cdot (s_{0j})^2)=(s_{ig(j)})^2\,.$$ Thus $g(s_{ij})=\pm s_{ig(j)}$. Moreover the sign $\epsilon_g(i)=\pm 1$ such that $g(s_{0i})=\epsilon_g(i)s_{0g(i)}$ is well defined since $s_{0i}\ne 0$, and $g(s_{ij})=g((s_{ij}/s_{0j})s_{0j})=\epsilon_g(j)s_{ig(j)}=\epsilon_g(i) s_{g(i)j}$. In particular, the extension $L$ of ${\mathbb Q}$ generated by all entries $s_{ij}$ is finite and normal, that is Galois extension. Now let $h$ be another automorphism of $\overline{\mathbb Q}$. We have $$gh(s_{ij})=g(\epsilon_h(j)s_{ih(j)})=\epsilon_g(i)\epsilon_h(j)s_{g(i)h(j)}$$ and $$hg(s_{ij})=h(\epsilon_g(i)s_{g(i)j})=\epsilon_h(j)\epsilon_g(i)s_{g(i)h(j)}= gh(s_{ij})$$ and the Galois group of $L$ over ${\mathbb Q}$ is abelian. Now the Kronecker-Weber theorem (see e.g. [@Ca]) implies the result.
Lifting theory
--------------
First recall that a fusion category over an algebraically closed field ${\textbf{k}}$ can be regarded as a collection of finite dimensional vector spaces $H_{ij}^k$ (=${\rm Hom}(X_i\otimes X_j,X_k)$), together with linear maps between direct sums of tensor products of these spaces which satisfy some equations (given by axioms of tensor categories). Thus one can define a fusion category over any commutative ring with $R$ to be a collection of free finite rank $R$-modules $H_{ij}^k$ together with module homomorphisms between direct sums of tensor products of them which satisfy the same equations.\
By a realization of a fusion ring $A$ over $R$ we will mean a fusion category over $R$ such that $N_{ij}^k:=dim(H_{ij}^k)$ are the structure constants of $A$.
If $I$ is an ideal in $R$ and ${\mathcal C}$ a fusion category over $R$ then it is clear how to define the reduced (=quotient) fusion category ${\mathcal C}/I$ over $R/I$ with the same Grothendieck ring.
Tensor functors between fusion categories over ${\textbf{k}}$ can be defined in similar terms, as collections of linear maps satisfying algebraic equations; this allows one to define tensor functors between fusion categories over $R$ (and their reduction modulo ideals) in an obvious way.
Now let ${\textbf{k}}$ be any algebraically closed field of characteristic $p>0$, $W({\textbf{k}})$ the ring of Witt vectors of ${\textbf{k}}$, $I$ the maximal ideal of $W({\textbf{k}})$ generated by $p$, and $\mathbb K$ the algebraic closure of the fraction field of $W({\textbf{k}})$ (char$(\mathbb K)=0$).
*Let ${\mathcal C}$ be a fusion category over ${\textbf{k}}$. A *lifting $\widetilde{\mathcal C}$ of ${\mathcal C}$ to $W({\bf k})$* is a realization of ${\textrm{Gr}({\mathcal C})}$ over the ring $W({\textbf{k}})$ together with an equivalence of tensor categories $\widetilde{\mathcal C}/I\tilde\to{\mathcal C}$.\
In a similar way, one defines a *lifting of a tensor functor* $F:{\mathcal C}\to{\mathcal D}$: it is a tensor functor $\widetilde F:\widetilde{\mathcal C}\to\widetilde{\mathcal D}$ over $W({\textbf{k}})$ together with an equivalence of tensor functors $\widetilde F/I\tilde\to F$.*
Let ${\mathcal C}$ be a nondegenerate fusion category over ${\bf k}$. Then there exists a unique lifting of ${\mathcal C}$ to $W({\bf k})$.
This follows from the fact that liftings are classified by $H^3({\mathcal C})$ and obstructions by $H^4({\mathcal C})$. And we know from Section 2 that the Davydov-Yetter cohomology vanishes for nondegenerate categories.
Let $F:{\mathcal C}\to{\mathcal D}$ be a tensor functor between nondegenerate fusion categories over ${\bf k}$. Then there exists a unique lifting of $F$ to $W({\bf k})$.
\[liffun\]
Again, liftings of $F$ are parametrized by $H^2_F({\mathcal C})$ and obstructions by $H^3_F({\mathcal C})$, which are trivial in the nondegenerate case.
Any semisimple Hopf algebra $H$ over ${\bf k}$ with $tr(S^2)\neq0$ (i.e., also cosemisimple) lifts to $\widetilde H$ over $W({\bf k})$.
Hence one can define $\widehat H=\widetilde H\otimes_{W({\textbf{k}})}\mathbb K$, which is a Hopf algebra over a field of charactristic zero. This allows one to extend results from the characteristic zero case to positive characteristic. For example, applying the Larson-Radford theorem [@LR] (see Corollary \[lr\] below) to $\widehat H$, one can find:
If $H$ is a semisimple and cosemisimple Hopf algebra over any algebraically closed field, then $S^2=1$.
\[nbs\] A nondegenerate braided (resp. symmetric) fusion category over ${\bf k}$ is uniquely liftable to a braided (resp. symmetric) fusion category over $W({\bf k})$.
A braiding on ${\mathcal C}$ is the same as a splitting ${\mathcal C}\to Z({\mathcal C})$ of the natural (forgetful) tensor functor $Z({\mathcal C})\to{\mathcal C}$. Theorem \[liffun\] implies that such a splitting is uniquely liftable. Thus a braiding is uniquely liftable.
Now prove the result in the symmetric case. A braiding gives rise to a categorical equivalence $B:{\mathcal C}\to{\mathcal C}^{\textrm{op}}$, and it is symmetric if and only if the composition of $B$ and $B^{21}$ is the identity. Hence the corollary follows from Theorem \[liffun\].
We conclude the section with mentioning a remarkable theorem of Deligne on the classification of symmetric fusion categories over $\Bbb C$.
Any symmetric fusion category over ${\mathbb{C}}$ is ${\rm Rep} G$ for a finite group $G$.
With some work, one can extend this result using corollary \[nbs\]:
Any symmetric nondegenerate fusion category over ${\bf k}$ (of characteristic $p$) is ${\rm Rep} G$ for a finite group $G$ of order not divisible by $p$.
Frobenius-Perron dimension
==========================
Definition and properties
-------------------------
Let ${\mathcal C}$ be a finite tensor category with simple objects $X_0,\dots,X_{n-1}$. Then for every object $X\in{\textrm{Obj}({\mathcal C})}$, we have a matrix $N_X$ of left multiplication by $X$: $[X\otimes X_i:X_j]=(N_X)_{ij}$. This matrix has nonnegative entries, and in the Grothendieck ring we have : $XX_i=\sum_j(N_X)_{ij}X_j$.
Let us now recall the classical
Let $A$ be a square matrix with nonnegative entries. Then
1. $A$ has a nonnegative real eigenvalue. The largest such eigenvalue $\lambda(A)$ dominates in absolute value all other eigenvalues of $A$. Thus the largest nonnegative eigenvalue of $A$ coincides with the spectral radius of $A$.
2. If $A$ has strictly positive entries, then $\lambda(A)$ is a simple eigenvalue, which is strictly positive, and its eigenvector can be normalized to have strictly positive entries. Moreover, if $v$ is an eigenvector with strictly positive entries, then the corresponding eigenvalue is $\lambda(A)$.
Thus to all $X\in{\textrm{Obj}({\mathcal C})}$ one can associate a nonnegative number $d_+(X)=\lambda(N_X)$, its *Frobenius-Perron dimension*.\
*(i) The Yang-Lee category: $X^2=\textbf1+X$, so $N_X=(\substack{0~1 \\ 1~1})$ and $d_+(X)=\frac{1+\sqrt5}{2}$.\
(ii) Let ${\mathcal C}={\textrm{Rep}}H$ for a finite dimensional quasi-Hopf algebra $H$, then $d_+(X)=dim(X)$ for all $H$-modules $X$.*
The following proposition follows from the interpretation of $d_+(X)$ as the spectral radius of $N_X$.
For all objects $X$ of ${\mathcal C}$, $\frac{\log ({\rm length}(X^{\otimes n}))}{\log n}\to d_+(X)$ when $n$ goes to infinity.
The assignment $X\mapsto d_+(X)$ extends to a ring homomorphism ${\rm Gr}({\mathcal C})\to{\mathbb{R}}$. Moreover, $d_+(X_i)>0$ for $i=0,\dots,n-1$.
Consider $X=\sum_iX_i\in{\textrm{Gr}({\mathcal C})}$ and denote by $M_X$ the matrix of right multiplication by $X$. For $i,j\in I$, $$\begin{aligned}
(M_X)_{ij} & = [X_i\otimes X:X_j]\geq dim({\textrm{Hom}(X_i\otimes X,X_j)}) \\
& = \sum_k dim({\textrm{Hom}(X_k,^*\!\!X_i\otimes X_j)})>0\,.\end{aligned}$$ Hence by the Frobenius-Perron theorem, there exists a unique eigenvector of $M_X$ (up to scaling) with strictly positive entries, say $R=\sum_i\alpha_iX_i$: $RX=\mu R$ with $\mu=\lambda(M_X)$. Now for all $Y\in{\textrm{Gr}({\mathcal C})}$, $(YR)X=\mu YR$ and then by the uniqueness of $R$ there is $\beta_Y\in{\mathbb{R}}$ such that $YR=\beta_YR$. Since $R$ has positive coefficients, applying again the Frobenius-Perron theorem, we obtain $\beta_Y=\lambda(N_Y)=d_+(Y)$.\
Consequently, $d_+(Y+Z)R=(Y+Z)R=YR+ZR=(d_+(Y)+d_+(Z))R$ and $d_+(YZ)R=YZR=Yd_+(Z)R=d_+(Y)d_+(Z)R$. So $Y\mapsto d_+(Y)$ extends to a ring homomorphism ${\textrm{Gr}({\mathcal C})}\to{\mathbb{R}}$.
Suppose $d_+(X_i)=0$, then $X_iR=0$ and hence $X_iX_j=0$ for all $j\in I$, which is not possible. Thus $d_+(X_i)>0$.
**Remark.** It is clear that the Frobenius-Perron dimension can be defined for any finite dimensional ring with distinguished basis and nonnegative structure constants (even if it has no realization) and does not depend on the corresponding category.
$d_+$ is the unique character of ${\rm Gr}({\mathcal C})$ that maps elements of the basis to strictly positive numbers.
Let $\chi$ be another such character. Then $\chi(X_i)\chi(X_j)=\sum N_{ij}^k\chi(X_k)$. Thus the vector with positive entries $\chi(X_k)$ is an eigenvector of the matrix $N_i$ with eigenvalue $\chi(X_i)$. So by the Frobenius-Perron theorem, $\chi(X_i)=d_+(X_i)$.
Quasitensor functors between finite tensor categories preserve Frobenius-Perron dimension.
$d_+(X)=d_+(X^*)$.
### Properties of the Frobenius-Perron dimension {#properties-of-the-frobenius-perron-dimension .unnumbered}
1. $\alpha=d_+(X)$ is an algebraic integer (it is a root of the characteristic polynomial of $N_X$).
2. $\forall g\in\textrm{Gal}(\overline{\mathbb{Q}}/{\mathbb{Q}}), \vert g\alpha\vert\leq\alpha$ (use part two of the Frobenius-Perron theorem). In particular, $\alpha\geq1$.
3. $\alpha=1\Leftrightarrow X\otimes X^*=\textbf1$ (in this case $X$ is called *invertible*).
If $X\otimes X^*=\textbf1$, then $1=d_+(\textbf1)=d_+(X)d_+(X^*)$. Since $d_+(X)\geq1$ and $d_+(X^*)\geq1$, we find that $d_+(X^*)=1$. Conversely, consider $i_X:\textbf1\hookrightarrow X\otimes X^*$ and compute $$d_+(X\otimes X^*)=d_+(\textbf1)+d_+(\textrm{coker}i_X)=1+d_+(\textrm{coker}i_X)\,.$$ Now if $d_+(X)=1$, then $d_+(X\otimes X^*)=1$, so $d_+(\textrm{coker }i_X)=0$ and hence $\textrm{coker }i_X\cong0$. Consequently, $i_X$ is an isomorphism and thus $\textbf1\cong X\otimes X^*$.
4. ([@GHJ]) If $\alpha<2$, then $\alpha=2\cos{\frac{\pi}n}$ for $n\geq3$.
Since $d_+$ is a character, $\alpha$ is the largest characteristic value of $N_X$. But the largest characteristic value of a positive integer matrix $A$ (i.e., the spectral radius of $\sqrt{AA^T}$) is, by Kronecker’s theorem, of the form $2\cos(\frac{\pi}{n})$, or is $\ge 2$.
\[qhthm\] Let ${\mathcal C}$ be a finite tensor category. ${\mathcal C}\cong{\rm Rep}H$ as a tensor category for a finite dimensional quasi-Hopf algebra $H$ if and only if every object $X$ of ${\mathcal C}$ has an integer Frobenius-Perron dimension.
First suppose that every object $X$ is such that $d_+(X)\in{\mathbb{N}}$. Then one can consider the object $P=\sum_id_+(X_i)P_i$, where $P_i$ are projective covers of $X_i$, and define a functor $F:{\mathcal C}\to{\textrm{Vect}_{{\textbf{k}}}};X\mapsto{\textrm{Hom}(P,X)}$, which is exact. Since $F(-)\otimes F(-)$ and $F(-\otimes-)$ extend to exact functors ${\mathcal C}\boxtimes{\mathcal C}\to{\textrm{Vect}_{{\textbf{k}}}}$ that map simple objects $X_i\boxtimes X_j$ to the same images, they are isomorphic. Thus $F$ is quasitensor and ${\mathcal C}\cong{\textrm{Rep}}H$.
If ${\mathcal C}\cong{\textrm{Rep}}H$, then reconstruction theory says there exists a quasifiber functor on ${\mathcal C}$. We know that such a functor preserves Frobenius-Perron dimensions, so they are integers.
If $H_1$, $H_2$ are finite dimensional quasi-Hopf algebras such that ${\rm Rep}H_1\cong{\rm Rep}H_2$ as tensor categories, then $H_1$ and $H_2$ are equivalent by a twist.
In the proof of Theorem \[qhthm\], there is no choice in the definition of the quasifiber functor $F$. Thus (by reconstruction theory) $H$ is unique up to a twist.
**Remark.** This is not true in the infinite dimensional case. For example, consider the category ${\mathcal C}={\textrm{Rep}}(SL_q(2))$ of representations of the quantum group $SL_q(2)$ with $q$ not equal to a nontrivial root of unity. Then there are many fiber functors on ${\mathcal C}$ which are not isomorphic (even as usual functors). More precisely, for every $m\geq2$ one can find a tensor functor $F:{\mathcal C}\to{\textrm{Vect}_{{\textbf{k}}}}$ such that $dim(F(V_1))=m$ (where $V_1$ is the standard 2-dimensional representation of $SL_q(2)$). Such $F$ can be classified and yield *quantum groups of a non-degenerate bilinear form* [@B; @EO2].
Finally, let us give a number-theoretic property of the Frobenius-Perron dimension in a fusion category, which allows one to dismiss many fusion rings as non-realizable.
If ${\mathcal C}$ is a fusion category over ${\mathbb{C}}$, then there exists a root of unity $\xi$ such that for every object $X$ of ${\mathcal C}$ $d_+(X)\in{\mathbb{Z}}[\xi]$.
*Consider the fusion ring $A$ with basis $\textbf1,X,Y$ and fusion rules $XY=2X+Y$, $X^2=\textbf1+2Y$ and $Y^2=\textbf1+X+2Y$. The computation of $d_+(X)$ reduces to a cubic equation whose Galois group is $S_3$. So we cannot find any root of unity $\xi$ such that $d_+(X)\in{\mathbb{Z}}[\xi]$, and consequently $A$ is not realizable.*
FP-dimension of the category
----------------------------
Let ${\mathcal C}$ be a finite tensor category with simple objects $X_0,\dots,X_{n-1}$. We denote by $P_i$ the projective cover of $X_i$ ($i=0,\dots,n-1$).
*The *Frobenius-Perron dimension of the category* ${\mathcal C}$ is $d_+({\mathcal C})=\sum_id_+(X_i)d_+(P_i)$.*
*(i) If ${\mathcal C}$ is semisimple (and hence fusion), then $d_+({\mathcal C})=\sum_id_+(X_i)^2$.\
(ii) If ${\mathcal C}={\textrm{Rep}}H$ for a finite dimensional quasi-Hopf algebra $H$, then $d_+({\mathcal C})=dim(H)$.*
The usefulness of this notion is demonstrated, for example, by the following result.
The Frobenius-Perron dimension of the category is invariant under Morita equivalence.
Remember that $Z({\mathcal C})$ is Morita equivalent to ${\mathcal C}\boxtimes{\mathcal C}^{\textrm{op}}$. Thus we have
Let ${\mathcal C}$ be a finite tensor category. Then $d_+(Z({\mathcal C}))=d_+({\mathcal C})^2$.
We note that for spherical categories these results appear in [@Mu1], [@Mu2].
The following theorem plays a crucial role in classification of tensor categories, and in particular allows one to show that many fusion rings are non-realizable.
\[lag\] If ${\mathcal C}$ is a full tensor subcategory of a finite tensor category ${\mathcal D}$, then $\frac{d_+({\mathcal D})}{d_+({\mathcal C})}$ is an algebraic integer.
*(i) Let ${\mathcal D}={\mathcal C}(G,1)$ and ${\mathcal C}={\mathcal C}(H,1)$ for a finite group $G$ and its subgroup $H$. Then Theorem \[lag\] says that $\vert H\vert$ divides $\vert G\vert$ (because an algebraic integer which is also a rational number is an integer). Thus Theorem \[lag\] is a categorical generalization of Lagrange’s theorem for finite groups.\
(ii) Let ${\mathcal D}={\textrm{Rep}}A$ and ${\mathcal C}={\textrm{Rep}}B$ for a finite dimensional Hopf algebra $A$ and a quotient $B=A/I$ of $A$ by a Hopf ideal $I$. Theorem \[lag\] says $dim(B)$ divides $dim(A)$ (this is the famous Nichols-Zoeller theorem [@NZ]). The same applies to quasi-Hopf algebras (in which case the result is due to Schauenburg, [@S]).*
If ${\mathcal C}$ is a fusion category with integer $d_+({\mathcal C})$, then $d_+(X_i)^2\in{\mathbb{N}}$ for all $i\in I$.
Let ${\mathcal C}_{\textrm{ad}}$ be the full tensor subcategory of ${\mathcal C}$ generated by direct summands of $X_i\otimes X_i^*$ ($i\in I$), and define $B=\oplus_i(X_i\otimes X_i^*)$. This object has an integer FP dimension: $d_+(B)=d_+({\mathcal C})\in{\mathbb{N}}$. Then consider $M=N_{B^{\otimes m}}$, the left multiplication matrix by $B^{\otimes m}$ in ${\mathcal C}_{\textrm{ad}}$. This matrix has positive entries for large enough $m$ (since any simple object of ${\mathcal C}_{\textrm{ad}}$ is contained in $B^{\otimes m}$).\
Let $Y_0,...,Y_p$ be the simple objects of ${\mathcal C}_{\textrm{ad}}$. The vector $(d_+(Y_0),\dots,d_+(Y_p))$ is an eigenvector of $M$ with integer eigenvalue $d_+(B)^n$. By the Frobenius-Perron theorem, this eigenvalue is simple. Thus the entries of the eigenvector are rational (as $d_+(Y_0)=1$) and hence integer (as they are algebraic integers). Consequently, $d_+(X_i\otimes X_i^*)=d_+(X_i^2) \in{\mathbb{N}}$.
*Let ${\mathcal C}$ be a Tambara-Yamagami (TY) category (see example \[ty\]). Then $d_+(g)=1$ for $g\in G$. Also, $X^2=\sum_{g\in G}g$, so $d_+(X)=\sqrt{\vert G\vert}$. Thus $d_+({\mathcal C})=2\vert G\vert$.\
In the particular case of the Ising model ($G={\mathbb{Z}}_2$), $d_+(\bold 1)=d_+(g)=1$ and $d_+(X)=\sqrt2$, and $d_+({\mathcal C})=4$.*
Global and FP dimensions
------------------------
Until the end of the paper, and without precision, we will assume that our categories are over ${\mathbb{C}}$.
### Comparison of global and FP dimension
Let ${\mathcal C}$ be a fusion category.
For every simple object $V$ in ${\mathcal C}$, one has $\vert V\vert^2\leq d_+(V)^2$, and hence ${\rm dim}{\mathcal C}\leq d_+({\mathcal C})$. Moreover, if ${\rm dim}{\mathcal C}=d_+({\mathcal C})$, then $\vert V\vert^2=d_+(V)^2$ for any simple $V$.
It is sufficient to consider the pivotal case (otherwise one can take the pivotal extension $\overline{{\mathcal C}}$ and recall that the forgetful functor $F:\overline{{\mathcal C}}\to{\mathcal C}$ preserves squared norms and FP dimension, because it is tensor).
In this case $N_i\vec d=d_i\vec d$ (where $d_i=\textrm{dim}X_i$ and $\vec d=(d_0,\dots,d_{n-1})$), thus by the FP theorem $\vert d_i\vert\leq d_+(X_i)$, and this is an equality if $\sum_i\vert d_i\vert^2=\sum_id_+(X_i)^2$.
**Remark.** In general, the FP dimension of a fusion category and its global dimension are not equal, or even Galois-conjugate (and the same is true for $d_+(V)^2$ and $(\textrm{dim}V)^2$, for any simple object $V$).
Now denote respectively by $D$ and $\Delta$ the global and FP dimensions of ${\mathcal C}$. We already know $D/\Delta\leq1$ (previous theorem), moreover we have
\[ddel\] $D/\Delta$ is an algebraic integer.
We can assume ${\mathcal C}$ is spherical. Otherwise one may consider its pivotal extension, which can be shown to be spherical (see [@ENO]), and whose global and FP dimensions are respectively $2D$ and $2\Delta$).
In this case $Z({\mathcal C})$ is modular, of global and FP dimensions $D^2$ and $\Delta^2$ (respectively). Let $s=(s_{ij})_{ij}$ be its $S$-matrix. It follows from the Verlinde formula that the matrices $N_i$ have common eigenvalues $s_{ij}/s_{0j}$, and the corresponding eigenvectors are the columns of $s$. Since $s$ is nondegenerate, there exists a unique label $r$ such that $s_{ir}/s_{0r}=d_+(Y_i)$, where $Y_i$ are the simple objects of $Z({\mathcal C})$).\
Then $\Delta^2=\sum_id_+(Y_i)^2=\sum_i\frac{s_{ri}}{s_{0r}}\frac{s_{ir}}{s_{0r}}=\delta_{r^*r}/s_{0r}^2$, where we used the symmetry of $s$ and the fact that $s^2=(\delta_{i^*j})_{ij}$. So we find that $r=r^*$ and $\Delta^2=1/s_{0r}^2=D^2/(\textrm{dim}X_r)^2$. Consequently $D^2/\Delta^2=(\textrm{dim}X_r)^2$, hence $D/\Delta$ is an algebraic integer.
Let ${\mathcal C}$ be a nondegenerate fusion category over a field ${\bf k}$ of characteristic $p$. Then its FP dimension $\Delta$ is not divisible by $p$.
Assume that $\Delta$ is divisible by $p$. Let $\widetilde{\mathcal C}$ be the lifting of ${\mathcal C}$, and $\widehat{\mathcal C}=\widetilde{\mathcal C}\otimes_{W({\textbf{k}})}\mathbb K$ where $\mathbb K$ is the algebraic closure of the fraction field of $W({\textbf{k}})$. Then the Theorem \[ddel\] says that the global dimension $D$ of $\widehat{\mathcal C}$ is divisible by $\Delta$, hence by $p$. So the global dimension of ${\mathcal C}$ is zero. Contradiction (${\mathcal C}$ is nondegenerate).
### Pseudo-unitary fusion categories
*A fusion category ${\mathcal C}$ (over ${\mathbb{C}}$) is called *pseudo-unitary* if $\textrm{dim}{\mathcal C}=d_+({\mathcal C})$.*
**Remark.** Unitary categories (those arising from subfactor inclusions, see [@GHJ]) all satisfy this condition (so the terminology is coherent).
Any pseudo-unitary fusion category ${\mathcal C}$ admits a unique spherical structure, in which ${\rm dim}V=d_+(V)$.
Let $b:\textrm{id}\to****$ be an isomorphism of tensor functors, and $g:\textrm{id}\to**$ an isomorphism of additive functors such that $g^2=b$. Let $f_i=d_+(X_i)$. Define $d_i=tr(g_{X_i})$ and $\vec d=(d_0,\dots,d_{n-1})$; then $f_i=|d_i|$ by pseudounitarity. Further, we can define the action of $g$ on ${\rm Hom}(X_i\otimes X_j,X_k)$; let $(T_i)_{jk}$ denote the trace of this operator. Then $T_i\vec d=d_i\vec d$, and $|(T_i)_{jk}|\le (N_i)_{jk}$. Thus, $$f_if_j=|d_id_j|=|\sum (T_i)_{jk}d_k|\le \sum (N_i)_{jk}f_k=f_if_j\,.$$ This means that the inequality in this chain is an equality. In particular $(T_i)_{jk}=\pm (N_i)_{jk}$, and the argument of $d_id_j$ equals the argument of $(T_i)_{jk}d_k$ whenever $(N_i)_{jk}>0$. This implies that whenever $X_k$ occurs in the tensor product $X_i\otimes X_j$, the ratio $d_i^2d_j^2/d_k^2$ is positive. Thus, the automorphism of the identity functor $\sigma$ defined by $\sigma|_{X_i}=d_i^2/|d_i|^2$ is a tensor automorphism. Let us twist $b$ by this automorphism, i.e., replace $b$ by $b\sigma^{-1}$. After this twisting, the new dimensions $d_i$ will be real. Thus, we can assume without loss of generality that $d_i$ were real from the beginning.
It remains to twist the square root $g$ of $b$ by the automorphism of the identity functor $\tau$ given by $\tau|_{X_i}=d_i/|d_i|$ (i.e., replace $g$ by $g\tau$). After this twisting, the new $T_i$ is $N_i$ and the new $d_k$ is $f_k$. This means that $g$ is a pivotal structure with positive dimensions. It is obvious that such a structure is unique. We are done.
Any fusion category of integer FP dimension $\Delta$ is pseudo-unitary. In particular it is canonically spherical.
Let $D=D_1,\dots,D_m$ be the algebraic conjugates of $D=\textrm{dim}{\mathcal C}$. Then consider $g_i\in\textrm{Gal}(\overline{\mathbb{Q}}/{\mathbb{Q}})$ such that $g_i(D)=D_i$, and the corresponding categories ${\mathcal C}_i=g_i({\mathcal C})$. We know that $\textrm{dim}{\mathcal C}_i=D_i$ and $d_+({\mathcal C}_i)=\Delta$, so $D_i/\Delta\leq1$ is an algebraic integer. Hence $\prod_i(D_i/\Delta)$ is an algebraic integer $\leq1$. But it is also a rational number (because $\prod_iD_i,\Delta\in{\mathbb{N}}$), so it is an integer which is necessarily $1$, and therefore $D_i=\Delta$ for all $i$. In particular $D=\Delta$.
\[lr\] If $H$ is a finite dimensional semisimple Hopf algebra over ${\mathbb{C}}$ with antipode $S$, then $S^2=1$.
Let ${\mathcal C}={\textrm{Rep}}H$. On the one hand we know that $d_+({\mathcal C})=dim(H)\in{\mathbb{N}}$, hence ${\mathcal C}$ is pseudo-unitary. By example \[kap\], it means $dim(H)=\textrm{dim}{\mathcal C}=tr(S^2)$.\
On the other hand, $S$ is of finite order, so $S^2$ is semisimple and its eigenvalues are roots of unity. Consequently $S^2=1$.
Classification
--------------
A natural classification problem for fusion categories is the following one.
Classify fusion categories over ${\mathbb{C}}$ of given Frobenius-Perron dimension.
The next theorem solves this problem in the case of the Frobenius-Perron dimension being a prime number $p$. Namely, it generalizes to the quasi-Hopf algebra case a result of Kac and Zhu on semisimple Hopf algebras of prime dimension $p$.
Let ${\mathcal C}$ be a fusion category over $\Bbb C$.
If $d_+({\mathcal C})=p$ is a prime, then ${\mathcal C}={\mathcal C}({\mathbb{Z}}_p,\omega)$. In particular, any semisimple quasi-Hopf algebra $H$ of prime dimension $p$ is of the form $H={\rm Fun}({\mathbb{Z}}_p)$ with associator defined by $\omega\in H^{3}({{\mathbb{Z}}_p},{{\mathbb{C}}^\times})={\mathbb{Z}}_p$.
$d_+({\mathcal C})=p$ is a prime, then $d_+(Z({\mathcal C}))=p^2\in{\mathbb{N}}$. Hence $Z({\mathcal C})$ has a canonical spherical structure in which $d_i:=\textrm{dim}X_i=d_+(X_i)$ for any simple object $X_i$. Moreover, since ${\mathcal C}$ is itself spherical (because it is of integer FP dimension), $Z({\mathcal C})$ is modular and hence $p^2/d_i^2$ is an algebraic integer. Thus $d_i=1$ or $\sqrt p$ (as $d_i^2\in{\mathbb{N}}$).\
If there exists $i$ such that $d_i=\sqrt p$, then using the forgetful functor $F:Z({\mathcal C})\to{\mathcal C}$ we find a simple object $F(X_i)$ in ${\mathcal C}$ with FP dimension $\sqrt p$ (it is simple because the dimensions of its simple constituents must be square roots of integers). Since $d_+({\mathcal C})=p$, it is the only simple object in ${\mathcal C}$. This is a contradiction (there must be a neutral object).\
Consequently, all simple objects in $Z({\mathcal C})$, and hence in ${\mathcal C}$ also (using $F$), have FP dimension $1$, i.e. are invertible. But fusion categories all whose simple objects are invertible are all of the type ${\mathcal C}(G,\omega)$. In our case the group $G$ must have order $p$, so the result is proved.
With quite a bit more work, this theorem can be extended to the case of products of two primes.
If $d_+({\mathcal C})=pq$ for two prime numbers $p\leq q$, then either $p=2$ and ${\mathcal C}$ is a Tambara-Yamagami category attached to the group ${\mathbb{Z}}_q$, or ${\mathcal C}$ is Morita equivalent to ${\mathcal C}(G,\omega)$ with $\vert G\vert=pq$.
The case $p=q$ is done in [@ENO Proposition 8.32] and the case $p<q$ is treated in [@EGO].
Open problems {#open-problems .unnumbered}
-------------
In conclusion we formulate two interesting open problems.
1. Let us fix $N\in{\mathbb{N}}$ (and still work over ${\mathbb{C}}$). E. Landau’s theorem (1903) says that the number of finite groups which have $\le N$ irreducible representations is finite. In the same way, the number of semisimple finite dimensional quasi-Hopf algebras which have $\le N$ irreducible representations is finite (see [@ENO]).\
It is natural to ask if the number of fusion categories over ${\mathbb{C}}$ with $\le N$ simple objects is finite. In the case $N=2$ this is shown in [@O2], but the case $N=3$ is already open.\
2. Does there exists a semisimple Hopf algebra $H$ over ${\mathbb{C}}$ whose representation category ${\textrm{Rep}}H$ is not group-theoretical?\
For quasi-Hopf algebras, it exists (consider e.g. a TY category related to $G={\mathbb{Z}}_p\times{\mathbb{Z}}_p$ with the isomorphism $G^\vee\to G$ corresponding to an elliptic quadratic form, see [@ENO]).
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[^1]: The work of P.E. was partially supported by the NSF grant DMS-9988786.
[^2]: To avoid confusion, we will use the notation $\textrm{dim}$ for global dimensions, and italic $dim$ for vector space dimensions.
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abstract: 'Deep transfer learning using dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) has shown strong predictive power in characterization of breast lesions. However, pretrained convolutional neural networks (CNNs) require 2D inputs, limiting the ability to exploit the rich 4D (volumetric and temporal) image information inherent in DCE-MRI that is clinically valuable for lesion assessment. Training 3D CNNs from scratch, a common method to utilize high-dimensional information in medical images, is computationally expensive and is not best suited for moderately sized healthcare datasets. Therefore, we propose a novel approach using transfer learning that incorporates the 4D information from DCE-MRI, where volumetric information is collapsed at feature level by max pooling along the projection perpendicular to the transverse slices and the temporal information is contained in either in second-post contrast subtraction images. Our methodology yielded an area under the receiver operating characteristic curve of $0.89\pm0.01$ on a dataset of 1161 breast lesions, significantly outperforming a previously approach that incorporates the 4D information in DCE-MRI by the use of maximum intensity projection (MIP) images.'
author:
- |
Qiyuan Hu, Heather M. Whitney, Maryellen L. Giger\
Committee on Medical Physics, Department of Radiology\
University of Chicago\
Chicago, IL 60637\
`[email protected]`\
bibliography:
- 'myCitationLib.bib'
title: 'Transfer Learning in 4D for Breast Cancer Diagnosis using Dynamic Contrast-Enhanced Magnetic Resonance Imaging'
---
Introduction
============
Magnetic resonance imaging (MRI) is one of the imaging modalities for clinical diagnosis and monitoring of breast cancer. It has been established for use in screening patients with high risk of breast cancer, cancer staging, and monitoring cancer response to therapies [@pickles2015prognostic; @turnbull2009dynamic]. In comparison with more commonly used clinical modalities, such as mammography and ultrasound, MRI offers much higher sensitivity to breast cancer diagnosis [@morrow2011mri; @kuhl2005mammography]. Dynamic contrast-enhanced (DCE)-MRI provides high-resolution volumetric lesion visualization as well as functional information via temporal contrast enhancement patterns—information that carries significant clinical value for breast cancer management. However, the labor-intensive breast MRI interpretation process in combination with deficiency in MRI reading experts make it an expensive clinical procedure.
In order to assist radiologists in the interpretation of diagnostic imaging, automated computer-aided diagnosis (CADx) systems continue to be developed to potentially improve the accuracy and efficiency of breast cancer diagnosis [@giger2013breast]. Recently, deep learning methods have demonstrated success in computer-aided diagnostic and prognostic performance based on medical scans [@greenspan2016guest; @tajbakhsh2016convolutional; @shin2016deep]. Although training deep neural networks from scratch typically relies on massive datasets for training and is thus often intractable for medical research due to data scarcity, it has been shown that standard transfer learning techniques like fine-tuning or feature extraction based on ImageNet-trained convolutional neural networks (CNNs) can be used for CADx [@yosinski2014transferable; @donahue2014decaf; @huynh2016digital; @antropova2017deep]. However, pretrained CNNs require two-dimensional (2D) inputs, limiting the ability to exploit high-dimensional volumetric and temporal image information that can contribute to lesion classification.
To take advantage of the rich 4D information inherent in DCE-MRI without sacrificing the efficiency provided by transfer learning, a previously proposed method used the second post-contrast subtraction maximum intensity projection (MIP) images to classify breast lesions as benign or malignant, and showed superiority to using only 2D or 3D information [@antropova2018use]. In this study, we propose a new transfer learning method that also makes use of the 4D information in DCE-MRI, but instead of reducing the volumetric information to 2D at the image level via creating MIP images, we do so at the feature level by max pooling CNN features from all slices for a given lesion. Compared with using MIP images, our CNN feature MIP method demonstrated significant improvement in classification performance in the task of distinguishing between benign and malignant breast lesions.
Materials and Methods
=====================
Database
--------
A database consisting of 1161 unique breast lesions from 855 women who have undergone breast MR exams was retrospectively collected under HIPAA-compliant Institutional Review Board protocols. Of all lesions, 270 were benign (23%) and 891 were malignant (77%) based on pathology and radiology reports. Images in the database were acquired over the span of 12 years, from 2005 to 2017, using either 1.5 T or 3 T Philips Achieva scanners with T1-weighted spoiled gradient sequence.
CNN Input
---------
As illustrated in Fig. \[fig: MIPROI\], the subtraction images were created by subtracting the pre-contrast ($t_0$) images from their corresponding second post-contrast ($t_2$) images in order to emphasize the contrast enhancement pattern within the lesion and suppress constant background. To generate MIP images, the 3D volume of subtracted images for each lesion was then collapsed into a 2D image by selecting the voxel with the maximum intensity along the axial dimension, i.e., perpendicular to the transverse slices.
![Illustration of the processes to construct the second post-contrast subtraction images, the subtraction maximum intensity projection (MIP) images, and region of interest (ROI).[]{data-label="fig: MIPROI"}](MIPROIconstruction.png){width="0.9\linewidth"}
To avoid confounding contributions from distant voxels, a region of interest (ROI) around each lesion was automatically cropped from the image to use in the subsequent classification process. The ROI size was chosen based on the maximum dimension of each lesion and a small part of the parenchyma around the lesion was included. The minimum ROI size was set to $32\times32$ pixels as required by the pre-trained CNN architecture. The cropping process is illustrated in Fig. \[fig: MIPROI\].
Classification
--------------
Figure \[fig: flowchart\] schematically shows the transfer learning classification and evaluation process for the two methods following the ROI construction. For each lesion, CNN features were extracted from both the MIP and all individual slices of the subtraction ROIs using a VGG19 model pretrained on ImageNet [@simonyan2014very; @deng2009imagenet]. Feature vectors were extracted at various network depths from the five max pooling layers of the VGGNet. These features were then average-pooled along the spatial dimensions and normalized with Euclidian distance. The pooled features were then concatenated to form a CNN feature vector for a given lesion [@huynh2016digital; @antropova2017deep]. For the CNN feature MIP method, features extracted from all slices were concatenated into a 3D feature vector and was then collapsed into a 2D feature vector by max pooling along the axial dimension, i.e., the direction perpendicular to the transverse slices.
![Lesion classification pipelines based on diagnostic images. Three-dimensional volumetric lesion information from dynamic contrast-enhanced (DCE)-MRI is collapsed into 2D at the image level by using maximum intensity projection (MIP) images (left), or at the feature level by max pooling CNN features extracted from all slices (right) along the axial dimension.[]{data-label="fig: flowchart"}](classificationFlowchart_abstract.png){width="0.4\linewidth"}
Two linear support vector machine (SVM) classifiers were trained on the CNN features extracted from subtraction MIPs and subtraction volumes separately to differentiate between benign and malignant lesions. SVM was chosen over other classification methods due to its ability to handle sparse high-dimensional data, which is an attribute of the CNN features [@russakovsky2015imagenet]. To address the problem of class imbalance, a misclassification penalty for cases in each class was assigned to be inversely proportional to its class prevalence in the training data.
Evaluation
----------
Each SVM classifier was trained and evaluated using nested five-fold cross-validation, where all lesions from the same patient were kept together in the same fold in order to eliminate the impact of using correlated lesions for both training and testing. Within each training set in the outer cross-validation, SVM hyperparameters were optimized by an internal grid search with an inner five-fold cross-validation [@Shawe-Taylor:2011:ROM:2304791.2305138]. Principal component analysis fit on the training set was applied to both training and test sets to reduce feature dimensionality [@jolliffe2011principal]. The cross-validation evaluation process was repeated 10 times with different random seeds, and the final prediction score for each lesion was averaged over the 10 repetitions. Class prevalence was held constant across the five cross-validation folds.
Classifier performances were evaluated using receiver operating characteristic (ROC) curve analysis, with area under the ROC curve (AUC) serving as the figure of merit [@Metz1998; @metz1999proper]. The two classification methods were compared using the DeLong test [@delong1988comparing; @sun2014fast]. Standard errors and 95% confidence interval (CI) of the difference in AUCs were calculated by bootstrapping the posterior probabilities of malignancy [@efron1987better].
Results
=======
Figure \[fig: ROC\] presents the ROC curves of the two classification methods in the task of distinguishing benign and malignant breast lesions. The 95% CI of the difference in AUCs was $\Delta AUC = [0.01, 0.04]$ and the p-value was $P < .001$. The results suggest that in the task of distinguishing benign and malignant breast lesions using deep transfer learning, 3D volumetric information in DCE-MRI may have superior predictive power when collapsed along the axial dimension via MIP at the feature level than at the image level.
![Fitted binomial receiver operating characteristic (ROC) curves for two classifiers that utilize the 4D information from dynamic contrast-enhanced (DCE)-MRI. The dashed blue line represents the image maximum intensity projection (MIP) method, where the 3D volumetric information is reduced to 2D at the image level. The solid orange line represents the CNN feature MIP method of collapsing the 3D volumetric information at the feature level by max pooling features from all slices. The legend gives the area under the ROC curve (AUC) with standard error (SE) for each classifier.[]{data-label="fig: ROC"}](ROC_ps3D_subMaxVSsubMip_sgkf_PCA_linear_x10_1253_min32_BigFont_newName.png){width="0.7\linewidth"}
Discussion
==========
In conclusion, our study proposed a novel methodology to incorporate the volumetric information inherent in DCE-MRI when using deep transfer learning. Our method of utilizing the volumetric information by max pooling features along the axial dimension significantly outperformed the previously proposed method of using MIP images and exhibited comparable computational efficiency.
High dimensionality and data scarcity are unique challenges in deep learning applications to medical imaging. In order to exploit the rich clinical information in medical images without sacrificing computational efficiency or model performance, it is important to devise approaches to use transfer learning in creative ways so that volumetric and temporal data can be incorporated even when networks pretrained on 2D images are used. Compared with recent studies that train 3D CNNs from scratch on DCE-MRI [@dalmis2019artificial; @li2017discriminating], our methodology of using transfer learning on 4D medical imaging data is computationally efficient, has demonstrated high performance on moderately sized datasets, and does not require intensive image preprocessing. Future work will further expand the analysis to include other valuable sequences in multiparametric MRI, rather than DCE-MRI alone. Furthermore, we would also like to increase the size of our database, which would allow us to explore fine-tuning and to use a standard training/validation/test split of the data. Finally, expanding the database to include images from other medical centers would allow us to perform validation on an independent, external dataset, and would help us develop a more robust system by including the heterogeneity resulting from different imaging manufacturers and facility protocols.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors acknowledge other lab members, including Karen Drukker, PhD, MBA; Alexandra Edwards, MA; Hui Li, PhD; and John Papaioannou, MS, Department of Radiology, The University of Chicago, Chicago, IL for their contributions to the datasets and discussions. The work was partially supported by NIH QIN Grant U01CA195564, NIH NCI R15 CA227948, and the RSNA/AAPM Graduate Fellowship. MLG is a stockholder in R2 technology/Hologic and QView, receives royalties from Hologic, GE Medical Systems, MEDIAN Technologies, Riverain Medical, Mitsubishi and Toshiba, and is a cofounder of and equity holder in Quantitative Insights (now Qlarity Imaging). It is the University of Chicago Conflict of Interest Policy that investigators disclose publicly actual or potential significant financial interest that would reasonably appear to be directly and significantly affected by the research activities.
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abstract: 'While small single stranded viral shells encapsidate their genome spontaneously, many large viruses, such as the Herpes virus or Infectious Bursal Disease Virus (IBDV), typically require a template, consisting of either scaffolding proteins or inner core. Despite the proliferation of large viruses in nature, the mechanisms by which hundreds or thousands of proteins assemble to form structures with icosahedral order (IO) is completely unknown. Using continuum elasticity theory, we study the growth of large viral shells (capsids) and show that a non-specific template not only selects the radius of the capsid, but leads to the error-free assembly of protein subunits into capsids with universal IO. We prove that as a spherical cap grows, there is a deep potential well at the locations of disclinations that later in the assembly process will become the vertices of an icosahedron. Furthermore, we introduce a minimal model and simulate the assembly of viral shell around a template under non-equilibrium conditions and find a perfect match between the results of continuum elasticity theory and the numerical simulations. Besides explaining available experimental results, we provide a number of new predictions. Implications for other problems in spherical crystals are also discussed.'
author:
- Siyu Li
- Polly Roy
- Alex Travesset
- Roya Zandi
bibliography:
- 'bibfile.bib'
title: 'Why large icosahedral viruses need scaffolding proteins: The interplay of Gaussian curvature and disclination interactions.'
---
More than fifty years ago, Caspar and Klug [@CASPAR1962] made the striking observation that the capsids of most spherical viruses display icosahedral order(IO), defined by twelve five coordinated units (disclinations or pentamers) occupying the vertices of an icosahedron surrounded by hexameric units, see Fig. \[gallery\]. While many studies have shown that this universal IO is favored under mechanical equilibrium [@bruinsma; @Fejer:10; @Rapaport:04a], the mechanism by which these shells grow, circumventing many possible activation barriers, and leading to the perfect IO remains mainly unknown.
Under many circumstances, small icosahedral capsids assemble spontaneously around their genetic material, often a single-stranded viral RNA [@elife; @Comas; @Cornelissen2007; @Sun2007; @Nature2016]. Yet, larger double-stranded (ds) RNA or DNA viruses require what we generically denote as the template: scaffolding proteins (SPs) or an inner core [@Thuman-Commike1998; @Earnshaw1978; @Aksyuk2015; @Saad1999; @Saugar2010; @Coulibaly2005]. The focus of this paper is on these large viruses that require a template for successful assembly.
The major difficulty in understanding the pathway towards IO is apparent from the results of the generalized Thomson problem, consisting of finding the minimum configuration for interacting $M$-point particles constrained to be on the surface of a sphere. Simulation studies show that the number of metastable states increase exponentially with $M$ [@Erber1991], and only with the help of sophisticated optimization algorithms at relatively small values of $M$ [@Morris1996; @Zandi2004; @elife; @Zandi2016], it is possible to obtain IO ground states. These situations, typical of spherical crystals, become even more difficult when considering the assembly of large capsids, in which once protein subunits are attached and a few bonds are made, it becomes energetically impossible for them to re-arrange: Should a single pentamer appear in an incorrect location, IO assembly would fail.
The combined effect of irreversibility and the inherent exponentially large number of metastable states typical of curved crystals puts many drastic constraints on IO growth. The complexity of the problem may be visualized by the various viral shells illustrated in Fig. \[gallery\], characterized by a structural index, the T number [@CASPAR1962; @Wagner2015; @Chen:2007b; @Vernizzi:2007a] $T=h^2+k^2+hk$, with $h$ and $k$ arbitrary integers, such that the crystal includes $60T$ monomers or $10(T-1)$ hexamers and 12 pentamers (disclinations).
A possible mechanism to successfully self-assemble a desirable structure might consist of protein subunits with chemical specificity, very much like in DNA origami [@Rothemund2006] where structures with complex symmetries are routinely assembled. In viruses, however, capsids are built either from one or a few different types of proteins, so specificity cannot be the driving mechanism leading to IO [@Yu2013; @Nature2016; @nguyen2006continuum; @Chen:2007b; @Zandi2004]. In this paper, we show that a “generic” template provides a robust path to self- assembly of large shells with IO. This is consistent with many experimental data in that regardless of amino acid sequences and folding structures of virus coat and/or scaffolding proteins, due to the “universal” topological and geometrical constraints, large spherical viruses need scaffolding proteins to adopt IO, see Fig. \[gallery\]. Although the focus of the paper is on virus assembly, the implication of our study goes far beyond and extend to many other problems where curved crystals are involved, a point that we we further elaborate in the conclusion [@Lidmar2003; @vernizzi2011platonic].
The distinct feature of spherical crystals is that their global structure is constrained by topology. More concretely, if $s({\bf x})$ is the disclination density, then $$\label{Eq:Euler}
\int d^2{\bf x} ~s({\bf x}) = 2\pi \chi \ ,$$ where $\chi$ is the Euler characteristic ($\chi=2$ for a sphere). However a capsid closes only at the end of the assembly, and thus, Eq. \[Eq:Euler\] does not really restrict the number of disclinations during the growth process, as pentamers or other disclinations may be created or destroyed at the boundaries. For a complete shell, the easiest way to fulfill Eq. \[Eq:Euler\] is with twelve $q=+\frac{\pi}{3}$ disclinations, and this is the case we will follow hereon.
{width="\linewidth"}
A minimal model for spherical crystals consists of a free energy $$\begin{aligned}
\label{Eq:Free energy}
F_c &=& \int d^2{\bf x} \left[ \mu u_{\alpha \beta}^2 + \frac{\lambda}{2} (u_{\alpha \alpha})^2 \right]\ + \frac{\kappa}{2}\int d^2{\bf x} (H({\bf x})- H_0)^2
\nonumber\\
&\equiv& F_c^l + F_c^b \end{aligned}$$ where $u_{\alpha \beta}$ is the strain tensor. The coefficients $\mu, \lambda$ are the Lame coefficients, which depend on the microscopic underlying interactions. Here $H({\bf x})$ is the extrinsic curvature of the template, $H_0$ the spontaneous curvature and $\kappa$ the bending rigidity. By integrating the phonon degrees of freedom, we can recast the term $F^{l}_c$ in Eq. \[Eq:Free energy\] as a non-local theory of interacting disclinations, with free energy [@Bowick2000] $$\begin{aligned}
\label{Eq:Free energy_defects}
F_c^l &=& \frac{K_0}{2}\int d^2{\bf x} d^2{\bf y} \left[(K({\bf x})-s({\bf x}) )G({\bf x},{\bf y}) \right. \times \nonumber\\
&\times& \left. (K({\bf y})-s({\bf y})) \right]\ ,\end{aligned}$$ where $K({\bf x})$ is the Gaussian curvature and $K_0$ is the Young modulus. The disclination density $s({\bf x}) = \sum_{i=1}^{12}q_i \delta({\bf x}- {\bf x}_i)$ has as variables the positions of 12 disclinations, each of charge $q_i = \frac{\pi}{3}$. The function $ G({\bf x},{\bf y})$ is the inverse of the Laplacian square [@Bowick2000]. All previous studies for the model in Eq. \[Eq:Free energy\_defects\] have been done for curved crystals without a boundary. In this paper, we provide, for the first time, the necessary formalism to include the presence of a boundary.
A discrete version of Eq. \[Eq:Free energy\] is given by [@seung1988defects; @vernizzi2011platonic; @Bowick2000; @Lidmar2003] $$\label{Eq:discrete}
F_d = E_s + E_b = \sum_{i} \frac{1}{2}k_s(b_{i}-b_0)^2 + \sum_{i,j}k_b[1-\cos(\theta_{ij}-\theta_0)]$$ with $\theta_0$ a preferred angle, related to the spontaneous curvature $H_0$. The stretching energy sums over all bonds $i$ with $b_0$ the equilibrium bond length and the bending energy is between all neighboring trimers indexed with $ij$. We further assume that there is an attractive force between the trimers and the preformed scaffolding layer (inner core) (see Fig.\[largeT\]), which, consistent with our minimal model, involves a simple LJ-potential $E_{LJ}=\sum_{i}4 \epsilon [(\frac{\sigma}{r_{i}})^{12}-2(\frac{\sigma}{r_{i}})^6 ]$ with $\epsilon$ the depth of the potential and $\sigma$ the position of minimum energy corresponding to optimal distance between the center of the core and subunits. In the next section, we associate a dynamics to these models, which corresponds to following a local minimum energy pathway.
Methods {#methods .unnumbered}
=======
Discrete model {#discrete .unnumbered}
--------------
The growth of the shells is based on the following assumptions [@Comas; @Nature2016; @Zandi2016; @Yu2013]: At each step of growth, a new trimer is added to the location in the boundary which makes the maximum number of bonds with the neighboring subunits. This is consistent with the fact that protein-protein attractive interaction is weak and a subunit can associate and dissociate till it sits in a position that forms a few bounds with neighboring proteins. These interactions eventually become strong for the subunits to dissociate and trimer attachment becomes irreversible [@elife]. The attractive interactions between subunits, whose strength depends on electrostatic and hydrophobic forces, are implicit in the model. Note that pH and salt can modify the strength of protein-protein and protein-template interactions and thus the growth pathway. The impact of pH and salt on the shell assembly will be pursed elsewhere.
![ Dynamics of formation of a hexamer vs. a pentamer: five trimers are attached at a vertex with an opening angle close to $\pi /3$ at the top and much smaller than $\pi /3$ at the bottom. If the energy per subunit of formation of a pentamer $E_p$ is higher than a hexamer $E_H$, then a hexamer forms (top); otherwise, a pentamer assembles (bottom). []{data-label="PHchoice"}](fig2.pdf){width="\linewidth"}
A crucial step in the assembly process is the formation of pentamers, which occurs only if the local energy is lowered, as illustrated in Fig. \[PHchoice\]. After the addition of each subunit or the formation of a pentamer, using HOOMD package [@hoomd1; @hoomd2], we allow the triangular lattice to relax and to find its minimum energy configuration [@Wagner2015].
The proposed mechanism follows a sequential pathway where trimers ($T$) attach to the growing capsid ($C\rightarrow C^{\prime}$) according to the reaction $$\begin{aligned}
\label{Eq:reaction}
T+C & \leftrightarrows& TC \nonumber \\
TC & \rightarrow & C^{\prime} \end{aligned}$$ with characteristic rates $k_D, k^{\prime}_D$ and $k_r$. The rate $k_D=2\pi D_T R_T $ is diffusion limited, with $R_T$ the trimer radius and $D_T$ its diffusion coefficient, so that the reaction speed is linear in trimer concentration $v_{TC}=k_D [T]$, $k_D^{\prime}$ is the detachment rate as the trimer searches for the local minimum, and $k_r$ is the irreversible rate of attachment of the trimer to the capsid. The combined reaction rate is therefore $k_T = \frac{k_r k_D}{k^{\prime}_D + k_r}$. Once the second reaction in Eq. \[Eq:reaction\] takes place, there is no possibility for correcting mistakes: if a pentamer forms in the incorrect location, IO is frustrated. With some additional assumptions about the dependence of $k_r$ on the coordination of the growing capsid, it is possible to derive overall rates for capsid formation, a problem that will be pursued elsewhere.
Two important parameters arises in discussing spherical crystals with the model Eq. \[Eq:discrete\]. One is the Foppl von-Karman (FvK) number [@Lidmar2003] $$\label{Eq:FvK}
\gamma=\frac{b_0^2k_s }{k_b} \ ,$$ which measures the ratio of stretching to bending moduli. When the FvK number is large, the protein subunits optimize stretching and bend away from their preferred radius of curvature showing some degree of faceting, which is the case of large viruses, see Figure \[gallery\]. For the case of template driven self-assembly, we introduce a new parameter $$\eta = \frac{k_b}{\epsilon},$$ which measures the relative strength of the bending rigidity to the attraction of the trimers to the template. For small $\eta$, the proteins follow the core curvature during growth at all the time, regardless of proteins spontaneous curvature. For large $\eta$, the shell detaches from the core and follow its preferred curvature. In this paper we will be mostly interested in the regime $\eta \approx {\cal O}(1)$ and $\gamma \gg 1$, where the template, rather than the spontaneous curvature dictates the size of the capsid.
Continuum model {#continuum .unnumbered}
---------------
We now consider the model given in Eq. \[Eq:Free energy\_defects\] on a spherical cap with an aperture angle $\theta_m$, so that its geodesic radius is $R_m=\theta_m R$, see Fig. \[largeT\]b. The Lame term ($F_c^l$) in Eq. \[Eq:Free energy\_defects\] can then be written as $$\label{F1}
F_c^l=\frac{1}{2 K_0}\int d^2{\bf x} \sqrt{g}\left(\Delta\chi\right)^2 \ ,$$ where $g_{\mu \nu}$ is the metric defining the surface and the Laplacian is $\Delta = -\frac{1}{\sqrt{g}}\partial_{\mu} g^{\mu \nu}\partial_{\nu}$, with $\chi$ the Airy Stress function that satisfies $$\begin{aligned}
\label{Eq:bieq}
\frac{1}{K_0}\Delta ^2\chi({\bf x}) &=& s({\bf x})- K({\bf x}) .\end{aligned}$$ In SI Appendix, we provide the detailed calculations. We note that approximate solutions of Eq. \[Eq:bieq\] are available under the assumption that the Laplacian is computed with a flat metric, see Ref [@Grason2012], which immediately leads to $\int d^2 {\bf x} K({\bf x}) = \int \frac{d^2{\bf x}}{R^2} = \frac{A}{R^2} = \pi \neq 2\chi \pi = 4\pi,$ directly violating the topological constraint Eq. \[Eq:Euler\]. Therefore previous results [@Castelnovo2017] are limited to small curvatures or aperture angles ($\theta_m \ll \pi$). The generalization of Eq. \[Eq:Free energy\_defects\] to include boundaries proceeds by defining the stress tensor by the expression $\sigma^{\alpha \beta} = g^{\alpha \beta} {\Delta} \chi({\bf x}) - g^{\alpha\mu}g^{\beta\nu} {\nabla}_{\mu}{\nabla}_{\nu} \chi({\bf x})$. We now include a stress free condition $\sigma_{\alpha \beta} n^{\beta}=0$ at the boundary, where $n^{\alpha}$ is the normal to the boundary. For a spherical cap, see Fig. \[largeT\]b, we use the metric $ds^2 = g_{\mu \nu} dx^{\mu} dx^{\nu}= dr^2 + R^2\sin^2(r/R) d\phi^2$. Note that following the simulation outcomes, we ignore boundary fluctuations. This is mainly because of the strength of protein-protein interactions and line tension implicit in the growth model and is consistent with the simulation results.
With the above definitions, the topological constraint Eq. \[Eq:Euler\] is satisfied exactly for a sphere. The free energy Eq. \[Eq:Free energy\_defects\] then becomes $$\begin{aligned}
\label{Eq:Free energy_defects_simple}
F_c^l(\theta_m, {\bf x}_i) & = & E_0(\theta_m) + \sum_{i=1}^N E_{0d}({\bf x}_i, \theta_m) + \nonumber\\
&+& \sum_{i=1}^N \sum_{j=1}^N \hat{E}_{dd}({\bf x}_i,{\bf x}_j, \theta_m) \end{aligned}$$ with $E_0$ is the free energy of the hexamers, $E_{0d}$ the interplay between Gaussian curvature and pentamers and $\hat{E}_{dd}$ describes disclination(pentamer) interactions. It is convenient to separate this last term as $$\begin{aligned}
\label{Eq:dd_term}
F_c^{dd} &=& \sum_{i=1}^N \sum_{j=1}^N \hat{E}_{dd}({\bf x}_i,{\bf x}_j, \theta_m)
\nonumber\\
&=& \sum_{i=1}^N E_{self}({\bf x}_i) + \sum_{i=1}^N \sum_{j>i}^N E_{dd}({\bf x}_i, {\bf x}_j),\end{aligned}$$ where $E_{self}({\bf x}_i)$ is the disclination self-energy, which depends on the location of a pentamer relative to the boundary.
Results {#results .unnumbered}
=======
Consistent with the assumptions describing the dynamics of growth noted in previous section, we consider the spherical cap in Fig. \[largeT\]b with an aperture angle, which monotonically varies from $\theta_m=0$ to $\theta_m=\pi$ (sequential growth) as a function of time $\theta_m(t)$. Then, for each value of $\theta_m$ we calculate the free energy Eq. \[Eq:Free energy\_defects\_simple\] and compare it to the one with an additional new defect (local condition). Once the latter one is favorable, the new defect is added.
![ The 1D energy plot for the first disclination: The dotted line corresponds to disclination self-energy $E_{self}$ (Eq. \[Eq:dd\_term\]), the dashed line to the Gaussian curvature-disclination interactions $E_{0d}$ and the solid line is the result of the addition of both energies $F_c^l-E_0$ (Eq. \[Eq:Free energy\_defects\_simple\]) as a function of the location of disclination in the shell for $\theta_m=0.7$. The energy goes through a minimum for $r=0.66$. The inset graph shows the zoom-out energy plot where the circle region corresponds to the main graph. []{data-label="firstdisc"}](fig3.pdf){width="\linewidth"}
For small values of $\theta_m$, the cap grows defect free. In Fig. \[firstdisc\] we plot the energy of a spherical cap for $\theta_m=0.7$. The dotted line in the figure shows the disclination self-energy $E_{self}$, the dashed line the Gaussian curvature-disclination interactions $E_{0d}$ and the solid line is the sum of both energies as a function of the location of disclination in the cap. The diamond in the figure corresponds to the minimum of energy and indicates the location of the first (and only) disclination appearing in the cap, around $r\sim0.66$. This value is very close to the geodesic distance following from the local “screening” of the Gaussian curvature $\int d^2{\bf x}~s({\bf x}) = \int d^2{\bf x}~K({\bf x}) \rightarrow \frac{\pi}{3}=2 \pi (1-\cos(\theta_m))$ such that $r = \arccos(5/6) = 0.59$. Somewhat counter intuitively, the first disclination does not appear at the center of the cap, which is the result of the competition between the disclination self-energy whose minimum is at the boundary and the Gaussian curvature-disclination interaction $E_{0d}$ with its minimum energy occurring at the cap center, see Fig. S2 in SI Appendix where the contourplots of the different elastic free energies as a function of the location of the first disclination, $r$ are shown. As the shell grows, the appearance of a new disclination becomes energetically favorable, $i.e.$ a new energy valley for the formation of a new disclination emerges, as illustrated in Fig. \[largeT\]b, where we show the contour plots of total elastic energies for spherical caps with $\theta_m=0.8$ through $\theta_m=\pi$. The bigger ball in each plot indicates the position of the latest energy well, which is where the addition of the next disclination takes place. Remarkably, both in the continuum model and simulations, during the growth process, the disclinations always appear in the positions that eventually become the vertices of an icosahedron.
{width="\textwidth"}
Results with the discrete model Eq. \[Eq:discrete\] are shown in Fig. \[largeT\]a. Here again, the disclinations universally appear at the vertices of an icosahedron, in complete agreement with the analytical calculation. The simulations were performed for all values between $T=7$ and $T=21$ and in all cases the IO was achieved without a single error. The size of the core in Fig. \[largeT\]a is commensurate with $T=13$ structures. We note that for these simulations the proteins spontaneous radius $1/H_0$ is much smaller than the core radius, $R_c$ ($R_c H_0 \gg 1$), a point that is discussed in more detail further below. In SI Appendix we provide a movie illustrating the growth of a $T=21$ structure, which includes 420 triangles.
The role of stretching and bending rigidity {#Sect:res_stretch .unnumbered}
-------------------------------------------
Figure \[shellenergy\] shows the stretching energy vs $N$ (number of subunits assembled) as a $T=13$ shell grows for six different values of FvK $\gamma > 1$. We note that for large spontaneous radius of curvature and small $\gamma$ when bending rigidity is dominant, no large icosahedral shell assembles successfully. Rather interestingly, there are conspicuous differences in the dynamics as a function of the FvK parameter $\gamma$.
For small values of $\gamma = 2$ (thick black line in Fig. \[shellenergy\]) the shell elastic energy grows almost linearly as a function of $N$ but does not show IO. This takes place for higher $\gamma$-values. The arrows in Fig. \[shellenergy\] indicate a drop in the elastic energy associated with the appearance of pentamers, see SI Appendix for more details. At the beginning of the growth, the shells with different values of $\gamma$ might follow different pathways and thus, the number of hexamers vary before the first few pentamers form. However, as the shell grows, the pentamers appear precisely at the same place, independently of $\gamma$. Note that the bending energy of the shells always grows linearly as a function of number of subunits for any $\gamma$ (see SI Appendix).
Discussion {#Sect:Conclusions .unnumbered}
==========
[![The stretching energy of a T=13 shell as a function of number of trimeric subunits. For small FvK numbers ($\gamma =2$, black line), there is no significant drop in energy as a pentamer forms. However, for large FvK numbers ($\gamma \gg 1$), the formation of pentamers drastically lowers the energy of the elastic shell. []{data-label="shellenergy"}](fig5.pdf "fig:"){width="\linewidth"}]{}\
Our results show that for large shells ($T>4$) successful assembly into IO requires a non-specific attractive interaction between protein subunits and a template. This interaction is implicit in the continuum model and is included as a generic attractive Lennard-Jones potential in the simulations. Furthermore, we find that the location of pentamers are completely controlled by the stretching energy as it is the case in the continuum elasticity theory.
In the absence of the template, small spherical crystals ($T=1$ and $T=3$) assemble spontaneously, for almost any FvK parameter $\gamma$. However, as we increase the spontaneous radius of curvature, the final structure depends on the value of $\gamma$. For small $\gamma$, large spherical shells without any specific symmetry form, and at large $\gamma>5$ curved hexagonal sheets, which eventually assemble into tubular or conical structures are obtained. Thus our results predict that large shells with IO cannot grow without template.
A template can have a significant impact on the structure and symmetry of the shell. While a weak subunit-core attractive interaction has a minimal role in the shell shape, a very strong subunit-core interaction will override the mechanical properties of proteins. The subunits sit tightly on the template to form a sphere with no specific symmetry. We were able to observe large shells with IO only for $\eta\sim1$ but at high $\gamma$. In this regime, in order for pentamers to overcome the core attraction and form in the “correct” position, they must assume a symmetric shape and buckle up (see Fig. \[gallery\]b). Indeed a strong bending energy is needed to overcome the shell adsorption. We find that without decreasing $\gamma$ (increasing $k_b$) but with increasing spontaneous curvature, the bending energy associated with the deviation from the preferred curvature of subunits adsorbed to the core becomes strong enough to make the pentamers buckle and assume a smooth shape. Quite interestingly, we find that this is the strategy that the nature has taken to form large shells with IO.
The role of the inner core or the preformed scaffold layer presented above is very similar to the role of SPs, which assemble at the same time as the capsid proteins (CPs), [*i.e.*]{}, the template grows simultaneously with the capsid (see Fig. \[cartoon\]). In fact one can think of the inner core as a permanent “inner scaffold” [@Coulibaly2005]. For example, Bacteriophage P22 has a T=7 structure, but in the absence of scaffolding (Fig. \[gallery\], P22) often a smaller $T=4$ forms. Similarly, Herpesvirus makes a $T=16$ structure but without the SPs, a $T=7$ assembles. More relevant to the present study is the case of Infectious Bursal Disease Virus (IBDV) a dsRNA virus that in the presence of SPs forms a $T=13$ capsid but in the absence the subunits assemble to form a $T=1$ capsid. This is exactly the condition for formation of the $T=13$ structure in Fig. \[largeT\] where the preferred curvature between subunits is such that in the absence of scaffold they form a $T=1$ structure. Reoviridae virus family also form $T=13$ but they have multi-shell structures, which act as inner cores. For instance, in this family Bluetongue virus is a double capsid particle, outer (necessary for infection) and inner capsid (encloses RNA genome). The inner capsid, termed as “core” has two protein layers. The surface layer (or shell) is a $T = 13$ capsid that assembles around the inner shell, a $T = 2$ structure (an inner core). Interestingly, it has been suggested that there is an evolutionary connection between SPs of IBDV and inner capsid of Bluetongue virus [@Coulibaly2005].
Conclusions {#conclusions .unnumbered}
===========
Our model establishes that successful self-assembly of components into a spherical capsid with IO requires a template that determines the radius of the final structure. This template is very non-specific, and in its absence, protein subunits assemble into either smaller capsids or structures without IO.
[![The role of scaffolding proteins (SPs) in the formation of $T=13$ capsid of IBDV. Without SPs, the CPs (blue and white subunits) of IBDV form $T=1$ structure (upper figure). In the presence of SPs (yellow subunits), they form $T=13$ structure (lower figure). The results of our simulations are also illustrated next to each intermediate step. Note that SPs (yellow subunits) do not assemble without the CPs but probably experience some conformational changes during the assembly. However, our focus here is solely on the impact of scaffolding on the CPs resulting in a change in the capsid T number. For preformed SPs, like in the case of bluetongue virus, the core is spherical and there is no indication of any changes on the size of spherical template during the assembly.[]{data-label="cartoon"}](fig6.png "fig:"){width="\linewidth"}]{}\
Even though the focus of the above study was on the impact of the preformed scaffolding layer, based on the experimental observations we conclude that the SPs, which assemble simultaneously with CPs (Fig. \[cartoon\]), play basically the same role as the inner core in the assembly of large icosahedral shells. Figure \[cartoon\] shows that in the absence of SPs, CPs of IBDV form a $T=1$ structure but when the same IBDV proteins co-assemble with SPs (yellow units) a $T=13$ forms. The figure also shows the pathway of formation of $T=1$ and $T=13$ structures obtained in our simulations. We emphasize that the mechanical properties of subunits are the same for both shells, the difference in structures arises from the substrate or SPs.
The contribution of the SPs is twofold. The CPs of many viruses including bluetongue virus noted above do not assemble in the absence of SPs. On the one hand, it appears that SPs lower the energy barrier and help capsid subunits to aggregate. On the other hand, by forcing the CPs to assemble into a structure larger than their spontaneous radius of curvature, they contribute to preserving IO.
Examples of the role of templates on the formation of spherical crystals are not limited to viruses, but include crystallization of metals on nanoparticles [@Huixin2011], solid domains on vesicles [@Korlach1999; @GuttmanSapirSchultzButenkoOckoDeutschSloutskin2016], filament bundles [@Grason2012] and colloidal assemblies at water-oil interfaces [@BauschMe2003]. Nevertheless, it has been shown [@Meng2014] that sufficiently rigid crystals grow as almost flat sheets free of defects, unable to assemble with IO. This regime, however, seems not to be accessible to viral capsids, as the hydrophobic interaction between monomers force close-packing structures that are incompatible with grain boundaries.
This study shed light at fundamental scale on the role of mechanical properties of building blocks and scaffolding proteins. The proposed mechanism is consistent with available experiments on viruses involving either scaffolding proteins or inner capsids. Further experiments will be necessary to validate many predictions of our described mechanism.
acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Greg Grason for many helpful discussions. The S.L. and R.Z. work were supported by NSF Grant No. DMR-1719550 and A.T. by NSF Grant No. DMR-1606336. PR is funded through Senior Investigator Award, Welcome Trust under Grant No: 100218. A.T. and R.Z. thank the Aspen Center for Physics where part of this work was done with the support of the NSF Grant No.PHY-1607611.
|
---
abstract: 'Although gravitational actions diverge in asymptotically AdS spacetimes, boundary counterterms can be added in order to cancel out those divergences; such counterterms are known in general to third order in the Riemann tensor for the Einstein-Hilbert action. Considering foliations of AdS with an $S^m \times H^{d-m}$ boundary, we discuss a simple algorithm which we use to generate counterterms up to sixth order in the Riemann tensor, for the Einstein-Hilbert, Gauss-Bonnet and third-order-Lovelock Lagrangians. We also comment on other theories such as $F(R)$ gravity.'
author:
- Alexandre Yale
date: 'October 17, 2011'
title: |
Simple counterterms for asymptotically AdS spacetimes\
in Lovelock gravity
---
Introduction
============
An important application of the AdS / CFT correspondence [@Witten1998] has been the study of the thermodynamics of field theories using the tools of black hole thermodynamics. The combination of these two ideas means that one may calculate the free energy of a strongly coupled field theory by integrating the action of a gravitational theory on a suitable background. In particular, this has recently been used to calculate holographically the entanglement entropy of conformal field theories using black hole thermodynamics [@Hung2011; @Casini2011; @Hung2011b; @Hung2011c].\
\
These gravitational actions are understood to diverge in precisely the same way as their dual quantum field theories [@Burgess1999]. However, the regularization techniques which deal with these divergences are quite different than their field-theoretic counterpart. For example, in order to calculate the free energy associated with a black hole, one typically evaluates the gravitational action then subtracts a divergent piece corresponding to a background which resembles this black hole spacetime. This technique, which goes by the name of background subtraction, can however be problematic because it is not always clear which background spacetime should be subtracted [@Rob1999]. Topological black holes [@CCBH] are particularly problematic: for example, in the case of some hyperbolically-foliated black holes in AdS, it is unclear whether the background should be the hyperbolic foliation of AdS, which itself corresponds to a thermal state similar to the BTZ black hole, or an extremal black hole embedded in AdS, which has $T=0$ but is no longer AdS. Moreover, in the context of a gauge/gravity duality, one would expect to find a regulating scheme for gravity which is closer to the one found in field theories, namely through the addition of counterterms. It is important to note that these counterterms are not in competition with the Gibbons-Hawking surface terms that one must add to the action to ensure that the variational principle is well-defined: rather, it is the total action (bulk $+$ boundary) which diverges, and the surface counterterm regularizes these divergences.\
\
The development of such counterterms began soon after the emergence of the AdS/CFT correspondence, initially through the calculation of stress-energy tensors and as a means of renormalizing correlation functions on the gravity side of the duality [@Kraus1999b; @Skenderis2001], and soon after as a means of studying the thermodynamics of those topological black holes for which the background subtraction techniques failed [@Rob1999]. Because this method is both physically well-motivated, through the AdS / CFT correspondence, and much simpler than other regularization techniques, it has become a standard tool in holographic calculations. In particular, it has been key to a number of recent results, such as the derivation of thermodynamic quantities for various types of Kerr-NUT/bolt-AdS spacetimes [@Skenderis2005; @Olea2005; @Awad2000; @Das2000; @Mann2000] and D-branes [@Skenderis2006], as well as the study of linear dilaton gravity [@Mann2010].\
\
Through a connection with the trace anomaly in the boundary theory [@Skenderis1998; @Skenderis1998b], an algorithm has been developed [@Kraus1999] to compute these counterterms for asymptotically AdS spacetimes with an arbitrary foliation. This algorithm consists of an iterative procedure whereby the action is varied to yield the stress-energy tensor, which is itself used to find the term that must be added to the Lagrangian to cancel its divergences to the next order. This new improved Lagrangian is then once again integrated and varied, yielding the next order of the energy-momentum tensor, and so on.\
\
Despite its validity, there are obvious drawbacks to this algorithm. Most importantly, although it allows one to easily calculate the first few counterterms, the higher-order terms have never been computed because of the rapidly increasing complexity of the algebra caused by the need to vary the action with counterterms, as pointed out in [@Kraus1999; @Olea2005]. Most of these difficulties come from the generality with which the algorithm was developed: it applies to any foliation of AdS, including those with complicated boundary geometries, such that the boundary counterterms contain terms such as $\nabla {{\cal{R}}}$.\
\
A much more straightforward approach to find renormalizing counterterms consists of first choosing a specific foliation, based on the problem of interest, before finding the linear combination of surface curvature scalars which eliminate the divergences [@Kraus1999]. This is how, in practice, the method of counterterms is actually used. A drawback to this second technique is the limited range of applicability of its results: the counterterms found are no longer valid if the spacetime is foliated differently, and one must perform this tedious task anew for every new problem.\
\
We provide a middle-ground solution. In the spirit of AdS / CFT, one is often interested in boundary geometries which are conformally flat, such as $R^d, S^d, H^d, S^m \times H^{d-m}$ and so on. One may therefore look for a simpler algorithm that can yield counterterms valid for this restricted, yet still fairly large, interesting subset of boundaries. Our approach will embody the spirit of the second technique described earlier, which we will modify in such a way that the counterterms be valid for every foliation of AdS having one of the boundaries listed above. Restricting ourselves to those boundaries does not restrict the AdS spacetime itself: rather, it puts restrictions on the kind of conformal field theories our bulk spacetime can be dual to. These theories live on a cutoff surface which is very close to the boundary, but which may be highly curved and complicated if we so desire; our results are valid when the CFTs live on one of the geometries written above, which are by far the most popular in the literature. We ultimately provide, as a future reference, expressions for the counterterms up to sixth order in the Riemann tensor.\
\
The duality between, on one hand, central charges and couplings in the field theory and, on the other, parameters in the gravitational theory has been explored through the trace anomaly [@Skenderis1998; @Skenderis1998b]. In particular, one learns that General Relativity is only dual to those conformal field theories for which all the central charges are equal. This is simple to understand: GR does not have enough free parameters to account for the ratios between these charges. In order to probe a larger space of field theories, one must expand to more general theories of gravity which contain more free parameters, such as Lovelock theories. These additional central charges have recently been investigated holographically [@Sinha2010; @Sinha2011] and the compilation of a dictionary has begun [@Buchel2009; @Myers2010; @Camanho:2009vw; @Camanho:2009hu; @Camanho:2010ru; @deBoer:2009pn; @deBoer:2009gx], translating the couplings of this theory into those of the conformal field theory. Because the Lovelock Lagrangians contain terms of higher order in the Riemann tensor, the algorithm from [@Kraus1999] described earlier, while still applicable, becomes yet even more complicated than in the Einstein-Hilbert case, such that generic counterterms for Lovelock theories have yet to be calculated, as pointed out in [@Dehghani2006]. The simpler technique which we will use here will allow us to take on this issue.\
\
We begin in Section \[sec:method\] by describing the general form of the counterterms as well as the technique we will use to derive them. In Section \[sec:gr\], we will go through the derivation of these counterterms for the case of General Relativity, which will exemplify the use of the method, while in Section \[sec:ll\], we will apply these ideas to Lovelock theories. Finally, we will provide a simple application of these counterterms by calculating the entanglement entropy of a spherical surface, following the construction of [@Casini2011], in section \[sec:entent\]. In Section \[sec:disc\], we will discuss some of the consequences of these counterterms with regards to applications and thermodynamics in general, and comment on their geometric meaning. Appendix \[app:ll\] contains some formulas for Lovelock gravity which are difficult or impossible to find in the literature, while in Appendix \[app:paddy\], we discuss a curious result whereby one may calculate a renormalized action starting only from a vanishing bulk action and no boundary counterterm. We comment on renormalizing bulk actions alone (with no boundary terms) in asymptotically AdS spacetimes, and discuss $F(R)$ gravity, in Appendix \[app:f(r)\] and finally write the expressions for the counterterms in Appendix \[ct\].
Form of the counterterms \[sec:method\]
=======================================
The setup is as follows: we consider vacuum AdS with scale $L$ and some UV cutoff boundary at a finite radius $r$. Then, we will look for surface counterterms on this boundary, built only from quantities intrinsic to the boundary, which will cancel out the divergences in the action when the limit $r/L \rightarrow \infty$ is taken. This is done by expanding the action in powers of $r$ and adding counterterms which cancel the positive powers of $r$ order by order such that, at the end of the procedure, we are left with a finite action.\
\
Working in the vacuum of AdS may not seem particularly useful: we are generally interested in spacetimes which are asymptotically locally AdS but which are not AdS vacuum. However, these same counterterms that we will derive in the vacuum case will also cancel out the divergences that appear in other bulk spacetimes which asymptote to AdS with this same boundary [@Kraus1999], as it is clear that these are all UV divergences with a universal form. Therefore, our goal is to find counterterms which renormalize gravitational actions in vacuum AdS.\
\
Motivated by our goal to study conformal field theories on conformally flat backgrounds, we will restrict ourselves to conformally flat boundaries of the form $S^m \times H^{d-m}$, where $H^{d-m}$ corresponds to the hyperbolic hyperplane; such boundaries were also recently considered in lower-dimensional settings [@Jatkar:2011ue]. More precisely, our bulk metric will take the form [$$\begin{split} \label{eqn:metric}
ds^2 &= \frac{dr^2}{1 + \frac{r^2}{L^2}} + \left( r^2 + L^2 \right) dH^2_{d-m} + r^2 d\Omega^2_m ~,
\end{split}$$]{} where $dH^2$ and $d\Omega^2$ are the line elements for the unit-curvature hyperbolic hyperplane and sphere: [$$\begin{split}
dH^2_{d-m} &= du_1^2 + \sinh^2u_1 du_2^2 + \sinh^2u_1 \sin^2u_2 du_3^2 + \sinh^2u_1 \sin^2u_2 \sin^2u_3 du_4^2 + \ldots \\
d\Omega^2_{m} &= d\theta_1^2 + \sin^2\theta_1 d\theta_2^2 + \sin^2\theta_1 \sin^2\theta_2 d\theta_3^2 + \sin^2\theta_1 \sin^2\theta_2 \sin^2\theta_3 d\theta_4^2 + \ldots ~~.
\end{split}$$]{} The dual CFT lives on the $r/L \rightarrow \infty$ boundary of this spacetime, which is conformally flat since the radii of $S^m$ and $H^{d-m}$ agree in this limit. However, in order to calculate the counterterms, we need to perform an expansion in $(r/L)^{-1}$; this forces $(r/L)$ to remain a finite quantity throughout the calculations, meaning that the boundary metric will in effect be [$$\begin{split} \label{eqn:metric2}
ds^2 &= \left( r^2 + L^2 \right) dH^2_{d-m} + r^2 d\Omega^2_m ~.
\end{split}$$]{}\
\
Working with this family of foliations $(\ref{eqn:metric})$ will be extremely useful for two reasons. First, it will provide us with the additional equations we require in order to uniquely set coefficients in the counterterms. Indeed, these will be of the form [$$\begin{split} \label{eqn:ct}
I_{ct} = \int d^d x \sqrt{h} \left\{ \alpha_{0,1} + \alpha_{1,1} {\cal R} + \alpha_{2,1} {\cal R}^2 + \alpha_{2,2}{\cal R}^{ab}{\cal R}_{ab} + \cdots \right\}~~;
\end{split}$$]{} where ${\cal R}_{abcd}$ (in script notation) corresponds to the boundary Riemann tensor and $h_{ab}$ is the induced metric at finite $r$. Our goal is to find the coefficients $\alpha_{i,j}$, which we do by expanding both this counterterm $(\ref{eqn:ct})$ and the non-regulated bulk and boundary actions in powers of $r$, and matching the coefficients of $r^n$. Working with a single foliation, it is possible to uniquely set $\alpha_{0,1}$ and $\alpha_{1,1}$. However, because the counterterm contains two terms of order ${\cal R}^2$, we need at least two independent equations to uniquely set the coefficients $\alpha_{2,1}$ and $\alpha_{2,2}$. Each foliation (that is, each value of $m$) will provide us with one such equation, up to $m \leftrightarrow (d-m)$. Working with the generic foliation $H^{d-m} \times S^m$ will therefore provide us with the ever-increasing number of independent equations we will need to uniquely set the higher-order coefficients.\
\
There is another important advantage to working with the family of metrics given by $(\ref{eqn:metric})$: by considering a single family of foliations characterized by an integer $m$, we can construct an extremely simple computational framework which can calculate any scalar quantity (built from either the extrinsic curvature or the boundary Riemann tensor) on the boundary, for any $m$. Indeed, because the boundary is the product of conformally flat Einstein manifolds, the Weyl tensor vanishes on each part and we find, on either $H^{d-m}$ or $S^m$, that the Riemann tensor can be written in terms of the Ricci tensor: [$$\begin{split} \label{eqn:rabcd}
{{\cal{R}}}^a_{\phantom{a}bcd} = \frac{1}{p-1} \left( \delta^a_c {{\cal{R}}}_{bd} - \delta^a_d {{\cal{R}}}_{bc} \right),
\end{split}$$]{} where $p$ is the dimension of the submanifold (so either $p=m$ or $p=d-m$). This will dramatically simplify the calculations since it implies that boundary scalars can be written as linear combinations of terms of the form ${{\text{Tr}}}(R^n K^m)$, meaning the trace of the matrix multiplication of $n$ Ricci tensors and $m$ extrinsic curvatures. Writing $\left\{ \alpha, \beta, \gamma, \delta \right\}$ as $S^m$ indices and $\left\{\mu, \nu, \rho, \sigma \right\}$ as $H^{d-m}$ indices, we can illustrate this idea by calculating [$$\begin{split}
K^{ac} R^{bd} {{\cal{R}}}_{abcd} &= K^{\alpha \gamma} {{\cal{R}}}^{\beta \delta} {{\cal{R}}}_{\alpha \beta \gamma \delta} + K^{\mu \rho} {{\cal{R}}}^{\nu \sigma} {{\cal{R}}}_{\mu \nu \rho \sigma}\\
&= \frac{1}{m-1} \left[ {{\text{Tr}}}_S(K) {{\text{Tr}}}_S({{\cal{R}}}^2) - {{\text{Tr}}}_S(K {{\cal{R}}}^2) \right] + \frac{1}{d-m-1} \left[ {{\text{Tr}}}_H(K) {{\text{Tr}}}_H({{\cal{R}}}^2) - {{\text{Tr}}}_H(K {{\cal{R}}}^2) \right]~,
\end{split}$$]{} where the subscripts $H$ and $S$ mean that we’re tracing only over the hyperbolic or spherical indices.\
\
The Ricci tensor on each submanifold is given by $R_{ab} = \pm\frac{ (p-1)}{R_0^2} g_{ab}$, where $R_0$ and $p$ are the radius and dimension of the submanifold and where we take the positive sign for the spherical submanifold and the negative for the hyperbolic one. Therefore, we can easily calculate, with $x \equiv L^2/r^2$: [$$\begin{split} \label{eqn:partialtrace}
{{\text{Tr}}}_S({{\cal{R}}}^a K^b) &= \frac{x^a}{L^{2a+b}}\left[ (1+x)^{b/2} m (m-1)^a \right]\\
{{\text{Tr}}}_H({{\cal{R}}}^a K^b) &= \frac{x^a}{L^{2a+b}(1+x)^{a+\frac{b}{2}}}\left[ (-1)^a (d-m)(d-m-1)^a \right];
\end{split}$$]{} the total trace is simply defined as the sum of these two quantities: [$$\begin{split} \label{eqn:totaltrace}
{{\text{Tr}}}({{\cal{R}}}^a K^b) = \frac{x^a}{L^{2a+b}(1+x)^{a+\frac{b}{2}}}\left[ (1+x)^{a+b} m (m-1)^a + (-1)^a (d-m)(d-m-1)^a \right] ~.
\end{split}$$]{} Any contraction of an arbitrary number of boundary Riemann tensors and extrinsic curvatures, on our hypersurface of constant $r$, can be calculated using equations $(\ref{eqn:rabcd})$ and $(\ref{eqn:partialtrace})$, which is quite extraordinary. This will be crucial when investigating Lovelock theories, whose Gibbons-Hawking-York terms involve incredibly complex contractions (see Appendix \[app:ll\]) of the extrinsic curvature and the Ricci and Riemann tensors.\
\
An important consequence of equation $(\ref{eqn:totaltrace})$ is that the trace of an odd number of Ricci tensors is not independent of the other quantities to leading order in $x$. For example, we can write [$$\begin{split}
{{\text{Tr}}}({{\cal{R}}}^3) = \frac{1}{d(d-1)} \left[ (2d-1) {{\text{Tr}}}({{\cal{R}}}) {{\text{Tr}}}({{\cal{R}}}^2) - {{\text{Tr}}}({{\cal{R}}})^3 \right] ~.
\end{split}$$]{} Hence, such terms will not appear in our counterterms. Note also that the counterterms which we will consider will be intrinsic quantities only: these correspond to the $b=0$ case of equation $(\ref{eqn:partialtrace})$. However, terms with $b \neq 0$ will appear in the surface terms (equivalent to the Gibbons-Hawking-York term for GR) of Lovelock theories.
General Relativity \[sec:gr\]
=============================
Let’s begin with a detailed derivation of the counterterms for General Relativity as a preamble to the more complex Lovelock calculations in the following section. In Anti-deSitter space, the Einstein-Hilbert bulk action is given by [$$\begin{split}
I_{{{\text{bulk}}}} = \int d^{d+1} x \sqrt{-g} \left( \frac{d(d-1)}{L^2} + R \right) ~,
\end{split}$$]{} where the first term corresponds to the cosmological constant and $L$ is the AdS curvature scale. Note that here and for the rest of the paper, we will drop the prefactor $(16\pi G)^{-1}$ for notational simplicity.\
\
The divergences that we are trying to remove come entirely from the asymptotic region of the spacetime [@Kraus1999]. As such, it suffices for us to consider counterterms which will renormalize the action for AdS spacetime itself; these same counterterms will then naturally also renormalize the action for asymptotically AdS spacetimes with an $S^m \times H^{d-m}$ boundary. Using the fact that AdS is maximally symmetric, meaning $R_{abcd} = \frac{-1}{L^2} \left( g_{ac} g_{bd} - g_{ad} g_{bc} \right)$, and relating the integral over the bulk to the integral over the boundary, we calculate the bulk action [$$\begin{split} \label{eqn:gr_bulk}
I_{{{\text{bulk}}}} &= \frac{-2d}{L^2} \int d^{d+1}x \sqrt{-g}\\
&= \frac{-2d}{L^2} \sigma \int dr \left( 1 + \frac{r^2}{L^2} \right)^{-1/2} \left( r^2 + L^2 \right)^{\frac{d-m}{2}} r^m\\
&= \frac{-2d}{L^2} \int d^d x \sqrt{h} \left\{ \frac{1}{r^m \left( 1 + \frac{r^2}{L^2} \right)^{\frac{d-m}{2}}} \int dr \left( 1 + \frac{r^2}{L^2} \right)^{\frac{d-m-1}{2}}r^m \right\} ~,
\end{split}$$]{} where $h_{ab}$ corresponds to the induced metric on a hypersurface located at a finite value of $r$ and $\sigma$ is the divergent integral over the unit $m$-sphere and $(d-m)$-hyperbolic hyperplane. Although writing the bulk action as a boundary integral in the above equation may seem peculiar, it will let us perform all our computations on the integrand alone without ever having to perform the boundary integral over $\int d^d x \sqrt{h}$.\
\
For a manifold with boundary such as the one we are considering here, a surface term must be added to the action to ensure that its variational principle is well-defined. For the Einstein-Hilbert action, this boundary term is known as the Gibbons-Hawking-York term, which is simply $I_{{{\text{sur}}}} = 2 \int d^d x \sqrt{h} K$; using equation $(\ref{eqn:totaltrace})$, we can write this term explicitly as [$$\begin{split} \label{eqn:gr_sur}
I_{{{\text{sur}}}} = \int d^d x \sqrt{h} \frac{2}{L \left(1+\frac{L^2}{r^2} \right)^{1/2}} \left[ \left(1+\frac{L^2}{r^2} \right)m + (d-m) \right] ~.
\end{split}$$]{} The total action is $I = I_{{{\text{bulk}}}} + I_{{{\text{sur}}}}$ from equations $(\ref{eqn:gr_bulk})$ and $(\ref{eqn:gr_sur})$. Working order by order (in either $x=L^2/r^2$ or the Ricci tensor, which are of the same order), we look for the coefficients $\alpha_{i,j}$ from equation $(\ref{eqn:ct})$ which ensure that the divergent part of the total action is canceled out by the counterterm for every value of $m$ (that is: every foliation). To zeroth order, only one constant needs to be found — $\alpha_{0,1}$ — such that we only need to consider one foliation, which will be $m=d$ for simplicity. Setting $I = I_{{{\text{ct}}}}$ and truncating to zeroth order in $x$ yields [$$\begin{split}
\underbrace{\frac{2(d-1)}{L}}_{I} = \underbrace{\alpha_{0,1}}_{I_{{{\text{ct}}}}} ~.
\end{split}$$]{} To first order, we once again only need to consider a single foliation, $m=d$, and our single equation is [$$\begin{split}
\underbrace{\frac{d(d-1)}{(d-2)L}}_I = \underbrace{\alpha_{1,1} \frac{d(d-1)}{L^2}}_{I_{{{\text{ct}}}}}~,
\end{split}$$]{} obviously yielding [$$\begin{split}
\alpha_{1,1} = \frac{L}{d-2}~.
\end{split}$$]{} To second order, we find our first non-trivial system of equations as we have two constants to solve for. Looking at the two foliations $m=d$ and $m=d-1$ leaves us with two independent equations and two constants to find: [$$\begin{split}
m=d & \rightarrow \frac{-d(d-1)}{4L(d-4)} = \alpha_{2,1} \frac{d^2(d-1)^2}{L^4} + \alpha_{2,2} \frac{d(d-1)^2}{L^4} \\
m=d-1 & \rightarrow \underbrace{\frac{-(d-1)}{4L}}_I = \underbrace{\alpha_{2,1} \frac{(d-1)^2(d-2)^2}{L^4} + \alpha_{2,2} \frac{(d-1)(d-2)^2}{L^4}}_{I_{{{\text{ct}}}}}~,
\end{split}$$]{} yielding the solution [$$\begin{split}
\alpha_{2,1} &= \frac{-d L^3}{(d-4)(d-2)^2} \\
\alpha_{2,2} &= \frac{L^3}{(d-4)(d-2)^2}~.
\end{split}$$]{} Combining the above calculations and rewriting slightly leads to a counterterm up to order ${{\cal{R}}}^2$: [$$\begin{split}
I_{{{\text{ct}}}} = \int d^d x \sqrt{h} \left( \frac{2(d-1)}{L} + \frac{L}{d-2} {\cal R} + \frac{L^3}{(d-4)(d-2)^2} \left[ {{\cal{R}}}^{ab} {{\cal{R}}}_{ab} - \frac{d}{4(d-1)}{\cal R}^2 \right] + \ldots \right) ~;
\end{split}$$]{} this has previously been calculated in [@Kraus1999] and [@Rob1999]. It is simple to keep going, however, and we can continue this procedure to arbitrary order; we provide the equation to sixth order in $(\ref{eqn:gr_ct})$. This counterterm can be subtracted from the gravitational action to yield a finite action in asymptotically AdS spacetimes with boundaries given by $S^m \times H^{d-m}$, which are highly interesting choices in light of the AdS / CFT correspondence. In other words, the following action is finite: [$$\begin{split}
I_{\text{GR,ren}} = \int d^{d+1} x \sqrt{-g} \left( \frac{d(d-1)}{L^2} + R \right) + 2 \int d^d x \sqrt{h} K - I_{{{\text{ct}}}} ~.
\end{split}$$]{} Because of the powers of $r$ which are hidden inside $\sqrt{h}$, the counterterm $(\ref{eqn:gr_ct})$ is really an expansion in $r$, with $\sqrt{h}{{\cal{R}}}^i$ being of order $r^{d-2i}$. Therefore, the series should be truncated at order $n$, where $d=2n+1,2n+2$ [@Rob1999; @Kraus1999], to ensure that only the divergences get subtracted. For even $d=2m$, this means that a finite term will be left over in the action; indeed, this term cannot be subtracted since its corresponding counterterm would have to be proportional to $\sqrt{h}{{\cal{R}}}^m$ which, as can be seen in equation $(\ref{eqn:gr_ct})$, is proportional to $\frac{1}{d-2m}$ and would therefore diverge. This finite term is the gravitational anomaly [@Kraus1999].\
\
While the previously known results in the literature, which contained terms up to third order in the Riemann tensor, allowed one to study up to eight bulk dimensions, the counterterm we provide works in up to fourteen dimensions, and can easily be expanded to work in an arbitrary number of dimensions. This therefore provides much greater freedom in studying higher-dimensional field theories.
Lovelock Gravity \[sec:ll\]
===========================
A common application for the counterterm method is holography: the thermodynamics of asymptotically AdS spacetimes are dual to the thermodynamics of some field theory living on this boundary. However, this duality is greatly constrained by the fact that General Relativity is dual only to those conformal field theories where all the central charges are equal; this is essentially caused by the lack of free parameters. Although this is not a problem when working with two-dimensional conformal field theories, whose universality class is uniquely defined by a single central charge $c$, it becomes an important limiting factor when trying to study higher-dimensional field theories which may have multiple central charges. A more general theory of gravity, built from Lovelock Lagrangians [@Lovelock:1971yv] (for a review of such Lagrangians, see Ch. 15 of [@Paddybook]), can help with this issue as it provides additional parameters which can be dual to CFT couplings. In particular, the coupling in front of the Gauss-Bonnet term $L_2$, described below, has recently been precisely linked to the ratio between the central charges of the dual higher-dimensional CFT [@Buchel2009; @Myers2010]. Although briefly studied in the case of four-dimensional Gauss-Bonnet gravity [@Cvetic:2001bk; @Astefanesei:2008wz], these counterterms are not known for Lovelock theories in $d$ dimension[@Dehghani2006]; therefore, we will follow the same steps as in the previous section to fill this gap in the literature.\
\
These Lovelock Lagrangians are the natural generalization of the Einstein-Hilbert Lagrangian to higher dimensions. Indeed, just as the Einstein-Hilbert Lagrangian is the Euler density of a two-dimensional manifold, the $m$th Lovelock Lagrangian $L_m$ is the Euler density of a $2m$-dimensional manifold. Thus, the quantity $L_m$ vanishes in less than $2m$ dimensions, and is a total derivative in exactly $2m$ dimensions [@YalePaddy] since it is topological; in particular, this means that the only Lovelock Lagrangian which is not a total derivative in $4$ dimensions is the $m=1$ Einstein-Hilbert term. Moreover, the $m$-th Lovelock Lagrangian is the unique scalar of order $m$ in the Riemann tensor whose equations of motion are second order: they are specifically built so that the fourth-order terms cancel out. These Lagrangians are given by [$$\begin{split} \label{eqn:lm}
L &= \sum_m \lambda_m L_m \\
L_m &= \delta^{A_1 B_1 A_2 B_2 \cdots A_m B_m}_{C_1 D_1 C_2 D_2 \cdots C_m D_m} R_{A_1 B_1}^{C_1 D_1} R_{A_2 B_2}^{C_2 D_2} \cdots R_{A_m B_m}^{C_m D_m}~,
\end{split}$$]{} where [$$\begin{split}
\delta^{A_1 B_1 A_2 B_2 \cdots A_m B_m}_{C_1 D_1 C_2 D_2 \cdots C_m D_m} = \delta^{[A_1}_{C_1} \delta^{B_1}_{D_1} \cdots \delta^{B_m]}_{D_m}
\end{split}$$]{} is the totally antisymmetric determinant tensor. In particular, the $m=1$ cases leads to the Einstein-Hilbert Lagrangian $L_1 = R$, while $L_2 = R^2 - 4R_{ab}R^{ab} + R_{abcd}R^{abcd}$ is the Gauss-Bonnet Lagrangian. Examples of Lovelock Lagrangians for larger values of $m$ can be found in Appendix \[app:ll\].\
\
Thanks to the simplicity of our method for finding counterterms, we are able to calculate them for any Lovelock theory, and to arbitrary order. The first step is calculating the bulk action [$$\begin{split} \label{action_ll}
I_{{{\text{bulk}}},m} = \int d^{d+1} x \sqrt{-g} \left( L_m - 2 \Lambda_m \right) ~.
\end{split}$$]{} As in the previous section, the fact that the bulk spacetime is maximally symmetric makes evaluating $L_m$ a trivial matter. However, because the equations of motion from $(\ref{action_ll})$ are now different than in the case of General Relativity, we need to find the cosmological constant $\Lambda_m$ which ensures that the metric $(\ref{eqn:metric})$ is a solution. While varying the Lovelock Lagrangian in equation $(\ref{action_ll})$ appears at first to be a daunting challenge, one finds a surprisingly simple result [@PaddyLL]. In vacuum, the equations of motion can be written as [$$\begin{split}
0 &= P_a^{\phantom{a}bcd}R_{ebcd} - \frac{1}{2}g_{ae} L_m + \Lambda g_{ae} \\
P_{cd}^{\phantom{cd}ab} &= m \delta^{a b a_2 b_2 a_3 b_3 \cdots a_m b_m}_{c d c_2 d_2 c_3 d_3 \cdots c_m d_m} R^{c_2 d_2}_{a_2 b_2} \cdots R^{c_m d_m}_{a_m b_m}~.
\end{split}$$]{} After some algebra, one finds that this leads to [$$\begin{split} \label{eqn:ll_bulk}
L_{{\text{bulk}}}= L_m - 2\Lambda = \frac{2m(-1)^m}{L^{2m}}d(d-1)(d-2)\cdots(d-2m+2);
\end{split}$$]{} In particular, we recover the result from the previous section, the prefactor in equation $(\ref{eqn:gr_bulk})$, for $m=1$: $L_1 - 2\Lambda = \frac{-2d}{L^2}$. The bulk action is then simply given by $I_{{\text{bulk}}}= L_{{\text{bulk}}}\int d^{d+1} x \sqrt{-g}$.\
\
Just as a Gibbons-Hawking surface term must be added to the Einstein-Hilbert action to make the variational principle well-defined, a surface countribution must be added to our Lovelock Lagrangian. The topological nature of these Lagrangians will help us find this term: in particular, the Lagrangians are closed and locally exact in $d=2m$ dimensions. The Chern-Simons form $Q$ is defined to embody this quality: given two connections $\Gamma_1$ and $\Gamma_2$, we have dimensions $dQ(\Gamma_1,\Gamma_2) = L_m(\Gamma_1)-L_m(\Gamma_2)$. Then, index theorems [@Eguchi; @Nakahara] tell us that for a manifold with boundary, the Euler characteristic $\chi$ must be supplemented by a boundary contribution: $\chi = \int (L_m(\Gamma) + dQ(\Gamma_0,\Gamma) )$; pictorially, the $dQ$ in this equation replaces the bulk Lagrangian $L_m(\Gamma)$ with $L_m(\Gamma_0)$ where $\Gamma_0$ is the connection for a product metric having the same boundary. In common language, this Chern-Simons term $Q$ can be written [@Myers1987; @Olea2005] [$$\begin{split} \label{eqn:qm}
Q_m = 2m \int_0^1 dt \delta^{a_1 a_2 \cdots a_{2m-1}}_{b_1 b_2 \cdots b_{2m-1}} K^{b_1}_{a_1} \left( \frac{1}{2} R^{b_2 b_3}_{a_2 a_3} - t^2 K^{b_2}_{a_2} K^{b_3}_{a_3} \right) \cdots \left( \frac{1}{2} R^{b_{2m-2} b_{2m-1}}_{a_{2m-2} a_{2m-1}} - t^2 K^{b_{2m-2}}_{a_{2m-2}} K^{b_{2m-1}}_{a_{2m-1}} \right);
\end{split}$$]{} the Einstein-Hilbert and Gauss-Bonnet surface terms are thus: [$$\begin{split}
Q_1 &= 2 {{\text{Tr}}}(K) \\
Q_2 &= 4{{\text{Tr}}}({{\cal{R}}}){{\text{Tr}}}(K) - 8 {{\text{Tr}}}(K {{\cal{R}}}) - \frac{4}{3} {{\text{Tr}}}(K)^3 + 4{{\text{Tr}}}(K){{\text{Tr}}}(K^2) - \frac{8}{3} {{\text{Tr}}}(K^3),
\end{split}$$]{} while larger surface terms can be found in Appendix \[app:ll\]. The Gibbons-Hawking-like surface term will be the dimensionally-extended version of $Q_m$ and will precisely cancel out normal derivatives of the metric on the boundary, thus ensuring that the variational principle is well defined. This has long been known for the $m=1$ Einstein-Hilbert case, and has also been done in detail for the $m=2$ Gauss-Bonnet term [@Bunch]. These boundary terms have in particular been used to find the Israel conditions [@Davis] and Friedmann equations [@Gravanis] for Einstein-Gauss-Bonnet gravity. For a more in-depth discussion, we refer the reader to [@YalePaddy] and references therein.\
\
Upon using equation $(\ref{eqn:totaltrace})$ to write these $Q_m$ explicitly as functions of $x = \frac{L^2}{r^2}$, the total action is found by integrating the bulk action from equation $(\ref{eqn:ll_bulk})$, as was done for General Relativity in equation $(\ref{eqn:gr_bulk})$, and adding the boundary integral of $Q_m$ from equation $(\ref{eqn:qm})$. Proceeding in precisely the same way as in Section \[sec:gr\], we solve linear equations to find the surface counterterm that we should add in order to make the action finite. As in the previous section, we find the counterterms up to sixth order for future reference: they are given in equations $(\ref{eqn:ll_ct})$ and $(\ref{eqn:ll_ct})$ for second-order and third-order Lovelock gravity, respectively.\
\
In practice, one should combine many of the above counterterms. For example, in order to regularize an action with Einstein-Hilbert, Gauss-Bonnet and $m=3$ Lovelock terms, [$$\begin{split}
I = \int \left[ \lambda_1(R - 2 \Lambda_1) + \lambda_2(L_2 - 2\Lambda_2) + \lambda_3(L_3 - 2\Lambda_3) \right] + \oint \left[ \lambda_1(2K) + \lambda_2(Q_2) +\lambda_3(Q_3) \right] ~,
\end{split}$$]{} one must combine the counterterms from equation $(\ref{eqn:gr_ct})$ with those from equations $(\ref{eqn:ll_ct})$ and $(\ref{eqn:ll_ct2})$. These equations are the main results of this paper. As discussed at the end of Section \[sec:gr\], they can renormalize the action in up to fourteen boundary dimensions.
Simple Application: Entanglement Entropy \[sec:entent\]
=======================================================
Let us now demonstrate a practical implementation of these results to holographic calculations of entanglement entropy [@Ryu1; @Ryu2; @Ryu3]. It has recently been shown [@Casini2011] that the entanglement entropy for a spherical surface in a CFT is equal to the Wald entropy of the “topological black hole” consisting of the hyperbolic foliation of AdS$_{4+1}$: [$$\begin{split} \label{newmetric}
ds^2 = -\left( \frac{r^2}{L^2}-1\right)\frac{L^2}{R^2} dt^2 + \frac{dr^2}{\frac{r^2}{L^2}-1} + r^2 dH_3^2 \,.
\end{split}$$]{} We will not go into the details of this construction, which can be found in [@Casini2011]. Rather, our aim is to provide a simple example where the counterterms may be used. If we take our gravity action to be Einstein-Gauss-Bonnet: [$$\begin{split}
I = \frac{1}{2 l_P^3} \int d^5 x \sqrt{-g} \left[ -\frac{12}{L^2 f_\infty} + R + \frac{\lambda f_\infty L^2}{2} L_2 \right] + \frac{1}{2l_P^3} \int d^4 x \sqrt{h} \left[ 2 K + \frac{\lambda f_\infty L^2}{2} Q_2 \right]\,,
\end{split}$$]{} where $f_\infty$ obeys $1 - f_\infty + \lambda f_\infty^2 = 0$, then the entanglement entropy will be proportional to the central charge $a$ of the CFT [@Casini2011] which, using the standard AdS/CFT dictionary [@Buchel2009; @Myers2010], can be written [$$\begin{split} \label{a}
a = \pi^2 \left( \frac{L}{l_P}\right)^3 \left( 1 - 6 \lambda f_\infty \right) \,.
\end{split}$$]{} Using the counterterms calculated in the previous sections, we can present an alternate and elegant derivation of this result. Instead of directly computing the Wald entropy of the horizon, as is done in [@Casini2011], we will compute the free energy ${\cal F}$ of the spacetime by integrating the action. In particular, this will reproduce the entropy result through $S \propto \partial {\cal F} / \partial T$.\
\
This free energy will necessarily diverge due to the large-$r$ bound of integration, and the standard technique to deal with this — background subtraction — is ambiguous since our spacetime is already (a particular coordinate choice of) vacuum AdS. The introduction of counterterms allows us to deal with the large-$r$ divergences and calculate the free energy: [$$\begin{split} \label{free}
\beta {\cal F} &= \frac{1}{2 l_P^3} \int d^5x \sqrt{-g} \left[ -\frac{12}{L^2 f_\infty}+ R + \frac{\lambda f_\infty L^2}{2} L_2 \right] + \frac{1}{2l_P^3} \int d^4 x \sqrt{h} \left[ 2 K + \frac{\lambda f_\infty L^2}{2} Q_2 \right] \\
&~~~~- \frac{1}{2l_P^3} \int d^4 x \sqrt{h} \left[ \frac{6}{L} + \frac{L}{2} {\cal R} + \frac{\lambda f_\infty L^2}{2} \left( \frac{-8}{L^3} + \frac{2}{L} {\cal R} \right) \right] \\
&= \frac{-\left(1-6 \lambda f_\infty \right) L^3}{l_P^3R} \beta H \,,
\end{split}$$]{} where $H = \int dH_3$ is the area of the hyperbolic slice. The second line in this expression corresponds to the counterterms from equations $(\ref{eqn:gr_ct})$ and $(\ref{eqn:ll_ct})$. These are truncated to order ${\cal R}$ because we are working with $d=4$ for simplicity, and the constant term which they introduce was thrown out. Since the temperature $T = \beta^{-1}$ for the line element $(\ref{newmetric})$ is given by [@YM] [$$\begin{split}
T &= \frac{L}{4\pi R} \partial_r \left( \frac{r^2}{L^2}-1\right)\Big|_{r=L} = \frac{1}{2\pi R} \,,
\end{split}$$]{} we can calculate the entropy density: [$$\begin{split}
s = S/H = -\frac{\partial \left( {\cal F}/H\right)}{\partial T} = \frac{2}{\pi} a \,,
\end{split}$$]{} where we used equation $(\ref{a})$. This result is in agreement with [@Casini2011].
Discussion \[sec:disc\]
=======================
A popular motivation for studying renormalization counterterms in asymptotically AdS spacetimes is to understand the thermodynamics of their associated conformal field theories. Indeed, the action of a bulk theory of gravity evaluated on an asymptotically AdS background provides us, through the usual ideas of black hole thermodynamics, with the free energy of this spacetime. The AdS / CFT correspondence then links this free energy with that of a thermal vacuum of the dual field theory living on the boundary. Although this free energy — and its associated entropy — is generally divergent, the method of counterterms not only provides a consistent and mostly unambiguous way of calculating the renormalized bulk free energy, it does so in a way which is analogous to the counterterm renormalization technique used on the CFT side. Because our technique is easily applicable to any gravitational Lagrangian, as demonstrated by our application to Lovelock Lagrangians, a large number of conformal field theories can be studied in this manner. Indeed, previous work on counterterms has essentially focused on General Relativity and on the first two or three terms in the counterterm expansion; this not only limits the number of dimensions that the dual field theory may have, but also constrains us to look only at a small section of the space of field theories, where all central charges are equal. By generalizing to both higher-order counterterms and wider gravitational theories, we have lifted both of these constraints, at the very small cost of requiring the field theories to live on conformally flat Einstein spacetimes.\
\
An important lesson, which was also pointed out in [@Kraus1999], is that the renormalized action will see no contribution from surface terms at infinity, such as the Gibbons-Hawking-York surface term, because they are entirely canceled out by the renormalization procedure. This means that all the information contained in the renormalized action appears to be included in the bulk term alone. As we exemplify in Appendix \[app:paddy\], this can even be true when the bulk term vanishes!\
\
In fact, the counterterm plays two roles: it cancels the boundary term at infinity and regularizes the volume integral. This can be seen by writing the bulk Lagrangian, in a stationary spherically symmetric background, as a total derivative, $L(r) = \partial_r f(r)$, such that the action is: [$$\begin{split}
I &= \int_{r_0}^r L dr + \oint Q\\
&= f(r) - f(r_0) + \oint Q~.
\end{split}$$]{} Then, up to a constant, the only contribution which will survive the renormalization process is $f(r_0)$: that is, the entire non-trivial contribution to the free energy will come from the horizon. This is caused by the divergent nature of $f(r)$ and $\oint Q$, both of which being evaluated at infinity, and the counterterm’s role will be to cancel them out up to some constant.\
\
This means that the boundary terms of the non-renormalized theory are not required in order to find the renormalized action. In that spirit, we will provide an expression for counterterms which regularize bulk Lagrangians in asymptotically AdS spacetimes in Appendix \[app:f(r)\], allowing one to renormalize theories of gravity for which surface terms are not usually used, such as $F(R)$ gravity. Moreover, this point of view is particularly insightful in that it can shed some light on some alternate counterterms which have been proposed in an effort to understand the geometric meaning of counterterms. In particular, it has been noticed [@Olea2005] that in $2n$ dimensions, another possible counterterm for either Einstein-Hilbert or Lovelock gravity (without Gibbons-Hawking surface terms) is the Chern-Simons form $Q_n$, with a proportionality constant depending on the theory which we are investigating. One should note, however, that unlike the counterterms we present here, $Q_n$ depends on quantities extrinsic to the boundary. In four-dimensional General Relativity, the following action is finite for some value of $\alpha$: [$$\begin{split} \label{eqn:Olea1}
I_{ren} = \int \left(R + \frac{6}{L^2} \right) + \alpha \oint Q_2~.
\end{split}$$]{} It seems very surprising that the surface term for $m=2$ Lovelock gravity, $Q_2$, somehow knows enough about the Einstein-Hilbert action to cancel its divergences. However, it turns out that this is merely a regularization of the volume integral. Indeed, because the Euler number in four dimensions is $\chi = \int L_2 + \oint Q_2$, then the regularizing term $\oint Q_2$ in equation $(\ref{eqn:Olea1})$ is merely, up to a constant ($\chi$), the volume integral of the Gauss-Bonnet Lagrangian $L_2$. However, since both $R=\frac{-12}{L^2}$ and $L_2=\frac{24}{L^2}$ are constants in AdS, then the above action $(\ref{eqn:Olea1})$ can be written $I_{ren} = \int \left( \frac{-6}{L^2} - \alpha \frac{24}{L^4} \right) + \alpha \chi$, which diverges unless $\alpha=\frac{-L^2}{4}$; this value of $\alpha$ is also found in [@Olea2005]. Thus, $Q_2$ regularizes the action in pure AdS. No issues will arise when we extend the discussion to asymptotically AdS spacetimes with, say, a black hole in the center, since we’ve seen that all the divergences come from the boundary. Through a Fefferman-Graham expansion, [@Olea2007; @Olea2009] shows that this is indeed the case.\
\
Because the counterterms are essentially a series in $\sum_i r^{d-2i}$, one may find, depending on whether the dimensionality is even or odd, a finite counterterm. This introduces an inherent ambiguity as to whether this finite term should be added, as initially pointed out in [@Kraus1999; @Kraus1999b]. This problem is akin to the fact that one may add a constant term to the action without affecting the equations of motion. However, adding this constant term would affect both the free energy and energy density, emphasizing once more that these are only useful as relative quantities.\
\
As we showed in section \[sec:entent\], these counterterms let us calculate the renormalized free energy without having to worry about the usual ambiguity of background subtraction. This allowed us to calculate the entropy density for a hyperbolic black hole with inverse temperature $\beta = 2 \pi R$. This quantity has previously been shown to be equal to the entanglement entropy for a spherical entangling surface [@Casini2011] and is proportional to the A-type central charge of the boundary theory.\
\
The counterterms that we find in equations $(\ref{eqn:gr_ct})$, $(\ref{eqn:ll_ct})$ and $(\ref{eqn:ll_ct2})$ apply to the boundaries of asymptotically conformally flat AdS spaces which are among the most interesting in the context of AdS / CFT. Counterterms were previously known to third order in General Relativity, which only renormalize the action in up to $8$ dimensions; by including the counterterms to sixth order, the action remains finite in up to $14$ dimensions. Moreover, the Lovelock theories of gravity which we considered are dual to richer families of conformal field theories than pure General Relativity.
Lovelock Theory \[app:ll\]
==========================
The Lovelock Lagrangian $L_m$ consists of the Euler density for a compact $2m$-dimensional manifold as well as its corresponding Chern-Simons surface term $Q_m$ [@Myers1987; @Olea2005] which yields the boundary contribution to the Euler number for a manifold with boundary. Although the action can be succinctly written as $S = \int L_m + \oint Q_m$ with $L_m$ and $Q_m$ given by equations $(\ref{eqn:lm})$ and $(\ref{eqn:qm})$, these definitions involve the totally antisymmetric tensor $\delta^{a_1 a_2 \ldots a_{2m}}_{b_1 b_2 \ldots b_{2m}}$ which we must expand in order to perform actual computations. As this can be a tedious exercise in practice, especially for the surface term, the expressions for Lovelock Lagrangians past the Gauss-Bonnet term ($m=2$) are extremely difficult to find in the literature. We therefore provide some of these here for convenience. The bulk terms are [$$\begin{split}
L_1 &= R\\
L_2 &= R^2 - 4 R_{ab} R^{ab}\\
L_3 &= R^3 - 12 R R^{ab}R_{ab} + 3 R R^{abcd}R_{abcd} + 16R^{ab}R_a^{\phantom{a} c} R_{bc} + 24R^{ac}R^{bd}R_{abcd} - 24R^{ab}R_a^{\phantom{a} cde}R_{bcde} \\
&~~~+ 2R^{abcd}R_{ab}R^{ef} R_{cdef} - 8R^{abcd} R_{a \phantom{e} c}^{\phantom{a} e \phantom{c} f} R_{bfde}
\end{split}$$]{} while the surface terms are [$$\begin{split}
Q_1 &= 2K\\
Q_2 &= 4K {{\cal{R}}}- 8K^{mn}{{\cal{R}}}_{mn} - \frac{4}{3} K^3 + 4K K^{mn}K_{mn} - \frac{8}{3}K^{mn}K_m^p K_{np}\\
Q_3 &= 6 K {{\cal{R}}}^2 - 24K {{\cal{R}}}^{mn}{{\cal{R}}}_{mn} + 6K{{\cal{R}}}^{mnpq}{{\cal{R}}}_{mnpq} - 24K^{mn}{{\cal{R}}}{{\cal{R}}}_{mn} + 48K^{mn}{{\cal{R}}}_m^p{{\cal{R}}}_{np} + 48K^{mn}{{\cal{R}}}^{pq}{{\cal{R}}}_{mnpq} \\
&~~~- 24K^{mn}{{\cal{R}}}_m^{pqr}{{\cal{R}}}_{npqr} - 4{{\cal{R}}}K^3 + 24 K^2 K^{mn}{{\cal{R}}}_{mn} + 12KK^{mn}K_{mn}{{\cal{R}}}- 48KK^{mn}K_m^p{{\cal{R}}}_{np} \\
&~~~- 24KK^{mn}K^{pq}{{\cal{R}}}_{mpnq} + 48K^{mn}K_m^pK_n^q{{\cal{R}}}_{pq} + 48K^{mn}K_m^pK^{qr}{{\cal{R}}}_{nqpr} + \frac{6}{5}K^5 - 12K^{mn}K_{mn}K^3 \\
&~~~+ 24K^{mn}K_m^pK_{np}K^2 + 18KK^{mn}K_{mn}K^{pq}K_{pq} - 36KK^{mn}K_m^pK_n^qK_{pq} \\
&~~~- 24K^{mn}K_{mn}K^{pq}K_p^rK_{qr} + \frac{144}{5}K^{mn}K_m^pK_n^qK_p^rK_{qr} ~.
\end{split}$$]{}
A curiosity: renormalized action from a vanishing bulk action \[app:paddy\]
===========================================================================
As discussed in the last paragraph of Section \[sec:disc\], a somewhat surprising consequence of these renormalization techniques is that the free energy appears to be entirely determined by the bulk Lagrangian. Although formulated differently, this was noticed by Padmanabhan [@Paddy2002] while studying the thermodynamics of black holes. Because it shines some light on our discussion, we shall recap some of his arguments. Consider a stationary spherically symmetric black hole whose line element is given by [$$\begin{split}
ds^2 = -f(r) dt^2 + \frac{dr^2}{f(r)} + dX^2~,
\end{split}$$]{} where $X$ corresponds to the transverse coordinates which we will drop for the remainder of this discussion. Then, the Ricci scalar can be written as a total derivative: [$$\begin{split}
R = \nabla_r^2 f(r) - \frac{2}{r^2} \partial_r \left( r(1-f) \right).
\end{split}$$]{} The (Euclidean) action is now integrated over the exterior region of the spacetime, from the location of the horizon $r_0$ up to some large radius $r$: [$$\begin{split} \label{eqn:paddyaction}
I &= \frac{-\beta}{4} \int_{r_0}^r dr \left[ -\partial_r (r^2 f') + 2\partial_r (r(1-f)) \right] \\
&= \underbrace{\frac{-\beta}{4} \left[ r_0^2 f'(r_0) - 2r_0 \right]}_{I_{ren}} + \underbrace{\frac{\beta}{4} \left[ r^2 f'(r) - 2r \right]}_{F(r)} ~,
\end{split}$$]{} where $F(r)$ is a function of the large radius $r$ and will be renormalized away. Although we did not explicitly add a surface term to our action, one can imagine that $F(r)$ could account for it. If we denote the period of the Euclidean time by $\beta \equiv \frac{4\pi}{f'(r_0)}$, then the above expression relates the Euclidean action to the free energy through $I_{ren} = S - \beta E$ where $S = \frac{A}{4}$ is the entropy and $E=\sqrt{\frac{A}{16\pi}}$ is the energy. While interesting, this is all now quite well understood.\
\
An important observation, however, is rarely stressed. Consider now the Schwarzschild black hole, which corresponds to $f(r) = 1 - \frac{2M}{r}$. Because the Ricci scalar vanishes, the typical calculation for the renormalized Euclidean action is very different than the one given above: we write $I = 2 \oint K$ over the boundary, and subtract the flat-space contribution $I_0 = 2 \oint K_0$ to find the finite action $I - I_0 = 4\pi M^2$. On the other hand, if we tried to use the above, more general, technique, we would find that the Ricci scalar vanishes because it is the total derivative of a constant... Yet, following through with the outlined steps, we would recover once more the correct finite action $I_{ren}=4\pi M^2$. Thus, we are led to a curious result: *even when the bulk action vanishes, it can still contain all the information of the renormalized action*.\
\
An astute eye will notice an ambiguity: because the Ricci scalar is the total derivative of a constant, then we can choose that constant to be anything, and not necessarily this $4 \pi M^2$. This ambiguity actually lets us modify the entropy as we can add a term that depends on $M$ to the term in square brackets in $(\ref{eqn:paddyaction})$ (provided we also subtract that same term from $F(r)$, but that gets thrown out according to the procedure); this is obviously nonsense. The counterterm technique provides an elegant solution to this puzzle by properly defining what is meant by “throwing away” the function $F(r)$ in equation $(\ref{eqn:paddyaction})$ (that is: how much of $F(r)$ comes from the boundary at infinity, which should be removed, and how much of $F(r)$ comes from a constant that we’ve added, which should be kept so that the entropy remains unchanged). With this strategy, equation $(\ref{eqn:paddyaction})$ could actually be written [$$\begin{split}
I &= \frac{-\beta}{4} \left[ r_0^2 f'(r_0) - 2r_0 + C \right] + \underbrace{\frac{\beta}{4} \left[ r^2 f'(r) - 2r + C \right] + 2 \oint K}_{F(r)} - \underbrace{\left\{ \frac{\beta}{4} \left[ r^2 f'(r) - 2r\right] + 2 \oint K \right\}}_{I_{{{\text{ct}}}}} \\
&= \frac{-\beta}{4} \left[ r_0^2 f'(r_0) - 2r_0 \right]~,
\end{split}$$]{} where the arbitrary constant $C$ no longer plagues the final result. By the underbrace denoting $I_{{{\text{ct}}}}$, we really mean that $I_{{{\text{ct}}}}$ is an expansion in boundary quantities which is equal to the term above the underbrace; the actual form of $I_{{{\text{ct}}}}$ would need to be derived through the non-vanishing off-shell bulk action (which will not be vanishing). The counterterms might therefore provide a procedure for extracting the renormalized action from the bulk action even when it vanishes, and without having to introduce boundary terms.
Regularizing the bulk action alone \[app:f(r)\]
===============================================
In light of our discussion and the previous appendix, it is clear that the ideas presented in this paper can be applied to gravitational theories for which we do not generally know the Gibbons-Hawking-York equivalent boundary term. Indeed, as previously noted, the contribution from this boundary term to the action would have been entirely canceled out by our counterterm. One example of gravitational theories where surface terms are not generally used are $F(R)$ theories of gravity (see [@Sotiriou2010; @Nojiri:2010wj] for modern reviews), where the bulk action is some function of the Ricci scalar: [$$\begin{split}
I = \int d^{d+1} x \sqrt{-g} F(R)~.
\end{split}$$]{} Although these theories are plagued by a number of difficulties, such as an ill-defined initial-value problem, they have received a considerable amount of interest in the literature, particularly in cosmology where one goal is to find a function $F(R)$ which could replace dark energy [@Hu2007; @Starobinsky2007]; moreover, they are equivalent to scalar-tensor theories of gravity [@Paddybook]. Recently, some of the attention has shifted to studying $F(R)$ gravity in terms of the gauge / gravity duality, where the aim [@Liu2010; @Caravelli2010] is to understand which holographic features of gravity (such as calculation of various types of entropies, the existence of Cardy’s $c$-theorem, etc) are specific to Einstein-Hilbert or Lovelock Lagrangians, and which are more universal. We therefore provide a simple expression for the counterterm, to fourth order, valid for the family of bulk metrics defined in equation $(\ref{eqn:metric})$: [$$\begin{split}
I_{{{\text{ct}}}} &= F \left( \frac{-d(d+1)}{L^2} \right) \int d^d x \sqrt{h} \Bigg(
\frac{L}{d}
-\frac{L^3}{2(d-2)(d-1)d} \\
&+\frac{L^5}{2(d-4)(d-2)^2(d-1)d} \bigg[ 3 d \frac{{{\cal{R}}}^2}{4(d-1)} - 3 {{\cal{R}}}^{ab} {{\cal{R}}}_{ab} \bigg] \\
&+\frac{L^7}{4(d-6)(d-4)(d-2)^3(d-1)^2d^2} \bigg[ \big( -160+160 d-10 d^2-5 d^3 \big) \frac{{{\cal{R}}}^3}{4(d-1)} + \big(40-70 d+15 d^2\big){{\cal{R}}}{{\cal{R}}}^{ab}{{\cal{R}}}_{ab} \bigg] \\
&+\frac{L^9}{8(d-8)(d-6)(d-4)(d-2)^4(d-1)^2d} \bigg[ \\
&~~~~\big(20480-48384 d+28672 d^2-5528 d^3+210 d^4+35 d^5\big) \frac{{{\cal{R}}}^4}{16(d-1)^2 d^2}\\
&~~~~+ \big(2560-8608 d+11352 d^2-5798 d^3+1229 d^4-105 d^5\big) \frac{{{\cal{R}}}^2 {{\cal{R}}}^{ab}{{\cal{R}}}_{ab}}{2(d-1)^2 d^2} \\
&~~~~+\big( -440+114 d+11 d^2 \big) {{\cal{R}}}^a_b {{\cal{R}}}^b_c {{\cal{R}}}^c_d {{\cal{R}}}^d_a \bigg]
+ \ldots \Bigg)
\end{split}$$]{} Setting $F(x)=1$, this provides a regularization counterterm for the volume of asymptotically AdS spacetimes.
Expressions for the counterterms {#ct}
================================
The counterterm for General Relativity, derived in Section \[sec:gr\], is: [$$\begin{split} \label{eqn:gr_ct}
I_{{{\text{ct}}}} &= \int d^d x \sqrt{h} \Bigg( \frac{2(d-1)}{L} + \frac{L}{(d-2)}{{\cal{R}}}+ \frac{L^3}{(d-4)(d-2)^2} \left( {\cal R}^{ab} {{\cal{R}}}_{ab} - \frac{d}{4(d-1)}{\cal R}^2 \right) \\
& + \frac{L^5}{2 (d-6) (d-2)^3(d-1)^2 d} \bigg[ \frac{1}{4} \left( d^2 + 6d - 8 \right) {{\cal{R}}}^3 - (d-1)(3d-2){{\cal{R}}}{{\cal{R}}}^a_b {{\cal{R}}}^b_a \bigg] \\
&+ \frac{L^7}{4(d-8)(d-6)(d-4)(d-2)^4(d-1)} \bigg[ \\
&~~~~\left( -4096+9984 d-5632 d^2+936 d^3-30 d^4-5 d^5 \right) \frac{{{\cal{R}}}^4}{16 d^2 (d-1)^2}\\
&~~~~+ \left( -512+1760 d-2344 d^2+1130 d^3-203 d^4+15 d^5 \right) \frac{{{\cal{R}}}^2 {{\cal{R}}}^a_b {{\cal{R}}}^b_a}{2d^2 (d-1)^2} \\
&~~~~+ \left( 136-62 d+3 d^2 \right) {{\cal{R}}}^a_b {{\cal{R}}}^b_c {{\cal{R}}}^c_d {{\cal{R}}}^d_a \bigg] \\
&+\frac{L^9}{8(d-10)(d-8)(d-6)(d-4)(d-2)^5(d-1)^3d} \bigg[ \\
&~~~~\Big( -163840+903168 d-1734656 d^2+1349248 d^3-463984 d^4+70380 d^5\\
&~~~~~~~~-4752 d^6+77 d^7+7 d^8 \Big) \frac{{{\cal{R}}}^5}{16(d-1)^2 d^2} \\
&~~~~+ \Big(-20480+133376 d-344832 d^2+448128 d^3-288428 d^4+90032 d^5\\
&~~~~~~~~-13185 d^6+968 d^7-35 d^8 \Big) \frac{ {{\cal{R}}}^3 {{\cal{R}}}^{ab} {{\cal{R}}}_{ab}}{2(d-1)^2d^2} \\
&~~~~+ \Big(7552-30480 d+28668 d^2-9576 d^3+1097 d^4-33 d^5 \Big) {{\cal{R}}}{{\cal{R}}}^a_b {{\cal{R}}}^b_c {{\cal{R}}}^c_d {{\cal{R}}}^d_a \bigg] \\
&+\frac{L^{11}}{8(d-12)(d-10)(d-8)(d-6)(d-4)(d-2)^6(d-1)^3} \bigg[ \\
&~~~~\Big( -15728640+158007296 d-598376448 d^2+1080991744 d^3-998355456 d^4+493757056 d^5 \\
&~~~~~~~~-131566512 d^6+18413972 d^7-1333692 d^8+45543 d^9-378 d^{10}-21 d^{11} \Big) \frac{1}{64(d-1)^4 d^4} {{\cal{R}}}^6 \\
&~~~~+\Big( -3932160+43433984 d-194895872 d^2+458334208 d^3-615241984 d^4+476404592 d^5\\
&~~~~~~~~-211709588 d^6+52722304 d^7-7059991 d^8+498441 d^9-17865 d^{10}+315 d^{11}\Big) \frac{{{\cal{R}}}^4 {{\cal{R}}}^{ab} {{\cal{R}}}_{ab}}{16(d-1)^4d^4} \\
&~~~~+ \Big( 1449984-9623040 d+24449280 d^2-26776016 d^3+14306460 d^4-3857428 d^5\\
&~~~~~~~~+507861 d^6-30858 d^7+645 d^8 \Big) \frac{1}{4(d-1)^2 d^2} {{\cal{R}}}^2 {{\cal{R}}}^a_b {{\cal{R}}}^b_c {{\cal{R}}}^c_d {{\cal{R}}}^d_a \\
&~~~~+ \Big( -157056+261136 d-156092 d^2+41432 d^3-4945 d^4+285 d^5-6 d^6\Big) {{\cal{R}}}^a_b {{\cal{R}}}^b_c {{\cal{R}}}^c_d {{\cal{R}}}^d_e {{\cal{R}}}^e_f {{\cal{R}}}^f_a
\bigg] + \ldots \Bigg)~.
\end{split}$$]{}
The counterterms for $m=2$ and $m=3$ Lovelock theories, derived in Section \[sec:ll\] are given by [$$\begin{split} \label{eqn:ll_ct}
I_{m=2,{{\text{ct}}}} &= \int d^d x \sqrt{h} \Bigg( -\frac{4 (d-3) (d-2) (d-1)}{3 L^3} + \frac{2(d-3)}{L} {{\cal{R}}}\\
&+ \frac{L}{(d-2)(d-4)} \bigg[ 3d(d-3) \frac{{{\cal{R}}}^2}{2(d-1)}
-6(d-3){{\cal{R}}}^{ab}{{\cal{R}}}_{ab} \bigg] \\
&+\frac{L^3}{3(d-6)(d-2)^2(d-1)d} \bigg[ \big(-120+130 d-15 d^2-5 d^3\big) \frac{{{\cal{R}}}^3}{4(d-1)} + \big(30-55 d+15 d^2\big) {{\cal{R}}}{{\cal{R}}}^{ab}{{\cal{R}}}_{ab} \bigg]\\
&+\frac{L^5}{6(d-8)(d-6)(d-4)(d-2)^3(d-1)} \bigg[ \\
~~~~&\big(-61440 + 167168 d - 123136 d^2 + 38008 d^3 - 4514 d^4 + 63 d^5 + 21 d^6\big) \frac{{{\cal{R}}}^4}{16 d^2 (d-1)^2} \\
&~~~~+ \big(-7680+28576 d-42456 d^2+26246 d^3-7803 d^4+1080 d^5-63 d^6\big) \frac{{{\cal{R}}}^2 {{\cal{R}}}^{ab} {{\cal{R}}}_{ab}}{2 d^2 (d-1)^2} \\
&~~~~+ \big(2328-1634 d+375 d^2-19 d^3 \big) {{\cal{R}}}^a_b {{\cal{R}}}^b_c {{\cal{R}}}^c_d {{\cal{R}}}^d_a \bigg]\\
&+ \frac{L^7}{12(d-10)(d-8)(d-6)(d-4)(d-2)^4(d-1)^3d} \bigg[ \\
&~~~~\big(-2457600+14538752 d-30459904 d^2+27699584 d^3-12618640 d^4+3046820 d^5\\
&~~~~~~~~-374764 d^6+21195 d^7-216 d^8-27 d^9 \big) \frac{{{\cal{R}}}^5}{16d^2(d-1)^2} \\
&~~~~+ \big(-307200+2124544 d-5869824 d^2+8344576 d^3-6276084 d^4+2580076 d^5\\
&~~~~~~~~-587415 d^6+71301 d^7-4509 d^8+135 d^9\big) \frac{{{\cal{R}}}^3 {{\cal{R}}}^{ab}{{\cal{R}}}_{ab}}{2d^2(d-1)^2} \\
&~~~~+ \big(122496-527600 d+589236 d^2-278756 d^3+62727 d^4-6092 d^5+189 d^6\big) {{\cal{R}}}{{\cal{R}}}^a_b {{\cal{R}}}^b_c {{\cal{R}}}^c_d {{\cal{R}}}^d_a \bigg] \\
&+ \frac{L^9}{12(d-12)(d-10)(d-8)(d-6)(d-4)(d-2)^5(d-1)^3} \bigg[\\
&~~~~\big( -235929600+2521890816 d-10093428736 d^2+19583016960 d^3-20243482112 d^4\\
&~~~~~~~~+11975617920 d^5-4208281360 d^6+879183244 d^7-105396104 d^8+6849353 d^9\\
&~~~~~~~~-208013 d^{10}+1155 d^{11}+77 d^{12} \big) \frac{{{\cal{R}}}^6}{64d^4(d-1)^4} \\
&~~~~+ \big(-58982400+689455104 d-3247906816 d^2+8067641344 d^3-11660653312 d^4\\
&~~~~~~~~+10114976464 d^5-5371636812 d^6+1759074292 d^7-349958269 d^8+40538978 d^9\\
&~~~~~~~~-2588868 d^{10}+83050 d^{11}-1155 d^{12}\big) \frac{{{\cal{R}}}^4 {{\cal{R}}}^{ab}{{\cal{R}}}_{ab}}{16d^4(d-1)^4} \\
&~~~~+ \big(23519232-162132480 d+436222208 d^2-533662448 d^3+344089348 d^4-125624216 d^5\\
&~~~~~~~~+26237659 d^6-2995067 d^7+170609 d^8-3645 d^9\big) \frac{{{\cal{R}}}^2 {{\cal{R}}}^a_b {{\cal{R}}}^b_c {{\cal{R}}}^c_d {{\cal{R}}}^d_a}{4d^2(d-1)^2} \\
&~~~~+ \big(-2636928+4990512 d-3718820 d^2+1421092 d^3-293971 d^4+31944 d^5\\
&~~~~~~~~-1767 d^6+38 d^7\big) {{\cal{R}}}^a_b {{\cal{R}}}^b_c {{\cal{R}}}^c_d {{\cal{R}}}^d_e {{\cal{R}}}^e_f {{\cal{R}}}^f_a \bigg]
+ \ldots \Bigg)~,
\end{split}$$]{} [$$\begin{split} \label{eqn:ll_ct2}
I_{m=3,{{\text{ct}}}} &= \int d^d x \sqrt{h} \Bigg(
-\frac{6 (-4+d) (-3+d) (-2+d) (-1+d)}{L^5} + \frac{3 (-4+d) (-3+d)}{L^3}{{\cal{R}}}\\
&+ \frac{9(d-3)}{L(d-2)} \bigg[\frac{-d}{4(d-1)}{{\cal{R}}}^2 + {{\cal{R}}}^{ab}{{\cal{R}}}_{ab} \bigg] \\
&+ \frac{15(d-4)(d-3)L}{2(d-6)(d-2)^2(d-1)d} \bigg[(d^2+6d-8)\frac{{{\cal{R}}}^3}{4(d-1)} + (3d-2){{\cal{R}}}{{\cal{R}}}^{ab}{{\cal{R}}}_{ab} \bigg]\\
&+ \frac{L^3}{4(d-8)(d-6)(d-2)^3(d-1)} \bigg[ \\
&~~~~\big(184320-496896 d+403200 d^2-135768 d^3+18474 d^4-315 d^5-105 d^6\big) \frac{{{\cal{R}}}^4}{16d^2(d-1)^2} \\
&~~~~+ \big(23040-85152 d+127992 d^2-86238 d^3+28455 d^4-4632 d^5+315 d^6\big) \frac{{{\cal{R}}}^2 {{\cal{R}}}^{ab}{{\cal{R}}}_{ab}}{2d^2(d-1)^2} \\
&~~~~+ \big(-3960+2346 d-243 d^2-33 d^3\big) {{\cal{R}}}^a_b {{\cal{R}}}^b_c {{\cal{R}}}^c_d {{\cal{R}}}^d_a \bigg]\\
&+\frac{L^5}{8(d-10)(d-8)(d-6)(d-2)^4(d-1)^3d} \bigg[ \big(7372800-39450624 d+79469568 d^2-72985728 d^3\\
&~~~~+34270704 d^4-8733180 d^5+1235700 d^6-92301 d^7+1512 d^8+189 d^9\big) \frac{{{\cal{R}}}^5}{16d^2(d-1)^2} \\
&~~~~+ \big(921600-5852928 d+15351168 d^2-21246912 d^3+16007436 d^4-6722340 d^5\\
&~~~~~~~~+1619457 d^6-233667 d^7+21051 d^8-945 d^9\big) \frac{{{\cal{R}}}^3 {{\cal{R}}}^{ab}{{\cal{R}}}_{ab}}{2d^2(d-1)^2} \\
&~~~~+ \big(-243072+1026960 d-1097772 d^2+448092 d^3-66753 d^4+156 d^5+429 d^6\big) {{\cal{R}}}{{\cal{R}}}^a_b {{\cal{R}}}^b_c {{\cal{R}}}^c_d {{\cal{R}}}^d_a \bigg]\\
&+\frac{L^7}{8(d-12)(d-10)(d-8)(d-6)(d-2)^5(d-1)^3}\bigg[ \big(707788800-6401359872 d+23481679872 d^2\\
&~~~~~~~~-43466637312 d^3+43522156032 d^4-24779634048 d^5+8211306384 d^6-1587130092 d^7+180670920 d^8\\
&~~~~~~~~-13174929 d^9+644853 d^{10}-10395 d^{11}-693 d^{12}\big) \frac{{{\cal{R}}}^6}{64d^4(d-1)^4} \\
&~~~~+ \big(176947200-1777287168 d+7657439232 d^2-18007136256 d^3+25041792768 d^4-20965903440 d^5\\
&~~~~~~~~+10619859372 d^6-3234507732 d^7+582978213 d^8-62594418 d^9+4607652 d^{10}-280698 d^{11}\\
&~~~~~~~~+10395 d^{12}\big) \frac{{{\cal{R}}}^4 {{\cal{R}}}^{ab}{{\cal{R}}}_{ab}}{16d^4(d-1)^4} \\
&~~~~+ \big(-46669824+317514240 d-839054592 d^2+993035376 d^3-594270372 d^4+185533464 d^5\\
&~~~~~~~~-27917331 d^6+1259907 d^7+99783 d^8-8811 d^9\big) \frac{{{\cal{R}}}^2 {{\cal{R}}}^a_b {{\cal{R}}}^b_c {{\cal{R}}}^c_d {{\cal{R}}}^d_a}{4d^2(d-1)^2} \\
&~~~~+ \big(4599936-8142768 d+5263044 d^2-1517124 d^3+171147 d^4+1080 d^5-1161 d^6\\
&~~~~~~~~+66 d^7\big) {{\cal{R}}}^a_b {{\cal{R}}}^b_c {{\cal{R}}}^c_d {{\cal{R}}}^d_e {{\cal{R}}}^e_f {{\cal{R}}}^f_a \bigg] \Bigg)~.
\end{split}$$]{}
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|
---
abstract: 'The [*Spitzer*]{} Survey of Stellar Structure in Galaxies (S$^{4}$G) is an Exploration Science Legacy Program approved for the [*Spitzer*]{} post-cryogenic mission. It is a volume-, magnitude-, and size-limited (d$<$40 Mpc, $|$b$|$ $>$ 30$^{\circ}$, m$_{Bcorr}<$15.5, D$_{25}>$1$\arcmin$) survey of 2,331 galaxies using the Infrared Array Camera (IRAC) at 3.6 and 4.5$\mu$m. Each galaxy is observed for 240 s and mapped to $\ge$ 1.5$\times$D$_{25}$. The final mosaicked images have a typical 1$\sigma$ rms noise level of 0.0072 and 0.0093 MJy sr$^{-1}$ at 3.6 and 4.5$\mu$m, respectively. Our azimuthally-averaged surface brightness profile typically traces isophotes at $\mu_{3.6\mu m}(AB) (1 \sigma) \sim$ 27 mag arcsec$^{-2}$, equivalent to a stellar mass surface density of $\sim$ 1 [$M_{\sun}$]{}pc$^{-2}$. [S$^{4}$G ]{}thus provides an unprecedented data set for the study of the distribution of mass and stellar structures in the local Universe. This large, unbiased and extremely deep sample of all Hubble types from dwarfs to spirals to ellipticals will allow for detailed structural studies, not only as a function of stellar mass, but also as a function of the local environment. The data from this survey will serve as a vital testbed for cosmological simulations predicting the stellar mass properties of present-day galaxies. This paper introduces the survey, describes the sample selection, the significance of the 3.6 and 4.5$\mu$m bands for this study, and the data collection & survey strategy. We describe the [S$^{4}$G ]{}data analysis pipeline and present measurements for a first set of galaxies, observed in both the cryogenic and warm mission phase of [*Spitzer*]{}. For every galaxy we tabulate the galaxy diameter, position angle, axial ratio, inclination at $\mu_{3.6\mu m} (AB)$= 25.5 and 26.5 mag arcsec$^{-2}$ (equivalent to $\approx \mu_\mathrm{B} (AB)$=27.2 and 28.2 mag arcsec$^{-2}$, respectively). These measurements will form the initial [S$^{4}$G ]{}catalog of galaxy properties. We also measure the total magnitude and the azimuthally-averaged radial profiles of ellipticity, position angle, surface brightness and color. Finally, we deconstruct each galaxy using GALFIT into its main constituent stellar components: the bulge/spheroid, disk, bar, and nuclear point source, where necessary. Together these data products will provide a comprehensive and definitive catalog of stellar structures, mass and properties of galaxies in the nearby Universe and enable a variety of scientific investigations, some of which are highlighted in this introductory [S$^{4}$G ]{}survey paper.'
author:
- 'Kartik Sheth, Michael Regan, Joannah L. Hinz, Armando Gil de Paz, Karín Menéndez-Delmestre, Juan-Carlos Muñoz-Mateos, Mark Seibert, Taehyun Kim Eija Laurikainen, Heikki Salo, Dimitri A. Gadotti, Jarkko Laine, Trisha Mizusawa, Lee Armus, E. Athanassoula, Albert Bosma, Ronald J. Buta, Peter Capak, Thomas H. Jarrett, Debra M. Elmegreen, Bruce G. Elmegreen, Johan H. Knapen, Jin Koda, George Helou, Luis C. Ho, Barry F. Madore, Karen L. Masters, Bahram Mobasher, Patrick Ogle, Chien Y. Peng, Eva Schinnerer, Jason A. Surace, Dennis Zaritsky, Sébastien Comerón, Bonita de Swardt Sharon E. Meidt, Mansi Kasliwal, Manuel Aravena'
bibliography:
- 'mymasterbib.bib'
title: 'The Spitzer Survey of Stellar Structure in Galaxies (S$^4$G)'
---
Introduction
============
Understanding the distribution of stars within a galaxy is akin to the study of the endoskeleton of a body; embedded within the galaxy is the fossil record of the assembly history and evolutionary processes of cosmic time. The first step in unraveling this history is to obtain a complete census of the stellar structures in galaxies in the local volume. This is the primary motivation for the [*Spitzer*]{} Survey of Stellar Structures in Galaxies ([S$^{4}$G]{}). [S$^{4}$G ]{}is a volume-limited (d $<$ 40 Mpc), size-limited (D$_{25} > $1$\arcmin$) and apparent B-band brightness (corrected for inclination, galactic extinction and K-correction) limited (m$_{Bcorr} < $15.5) survey of 2,331 nearby galaxies at 3.6 and 4.5$\mu$m with the Infrared Array Camera (IRAC) [@fazio04] on the [*Spitzer*]{} Space Telescope (SSC) [@werner04].
Over the last 70 years, numerous surveys have sought to create baseline data sets for nearby galaxies with as few as tens of galaxies (ANGST: @dalcanton09; SINGS, @kennicutt03) to tens of thousands of galaxies (e.g., RC3, @devaucouleurs91). While these surveys was designed specifically for particular scientific goals, none of them provides an accurate inventory of the stellar mass and the stellar structures in nearby galaxies. The main reason is that infrared surveys, where light from the old stellar population is better measured due to reduced dust extinction and contamination from star formation, are extremely difficult to conduct from the ground. The largest ground-based infrared surveys (e.g., 2MASS, @skrutskie06; DENIS, @epchtein94 [@paturel03]) are rather shallow in depth, and as a result, studies of nearby galaxies have been restricted to high surface brightness inner disks. In contrast, deeper infrared surveys (e.g., OSUBGS, @eskridge02, NIRSOS, @laurikainen10) have imaged only a few hundred galaxies. [S$^{4}$G ]{}is specifically designed to provide over an order of magnitude improvement in the sample size and several magnitudes deeper data than existing surveys.
[S$^{4}$G ]{}builds upon the two previous [*Spitzer*]{} Legacy surveys of nearby galaxies: SINGS [@kennicutt03] and the Local Volume Legacy Survey (LVL) [@lee08]. SINGS was designed to probe star formation, dust and polycyclic aromatic hydrocarbon (PAH) emission in all representative environments [*within*]{} galaxies in a sample of 75 objects. The LVL survey was designed to study spatially-resolved star formation and the red stellar population within galaxies in a local volume of 11 Mpc. The local volume limits the 258 LVL galaxies to consist mostly of dwarf, irregular and late type systems (see green histogram in Fig. 1). The remainder of the [*Spitzer*]{} archive from the cryogenic mission is inadequate for the science goals outlined here because it lacks sufficient numbers of galaxies of all stellar masses, particularly at M$_{*} < $ 10$^9$ [$M_{\sun}$]{}, Hubble types and environments. As shown in Figures \[hubmasstyp\] and \[volume\], [S$^{4}$G ]{}explores the full range of stellar structures in a representative and large sample of galaxies of all types, masses and in diverse environments. [S$^{4}$G ]{}is designed to provide the ultimate baseline data set for the study of stellar structures and mass in nearby galaxies.
In this paper we introduce [S$^{4}$G]{}, describe the data collection and analysis strategy and briefly describe the variety of scientific investigations possible with these data. All [S$^{4}$G ]{}galaxies are processed through a uniform pipeline to create the deepest and largest mid-infrared image catalog of nearby galaxies to date. For every galaxy we measure the magnitude and surface brightness and tabulate the galaxy diameter, position angle, axial ratio, and inclination at $\mu_{3.6\mu m}$ (AB) = 25.5 and 26.5 mag arcsec$^{-2}$. We also compute the azimuthally-averaged radial profiles of ellipticity, position angle, surface brightness and color. This fundamental catalog of galaxy properties will allow us to classify their morphology based on their stellar structures – a preliminary study is presented in @buta10. In the long term, the main goal of the [S$^{4}$G ]{}project is to obtain a detailed understanding of the properties of the different stellar structures, their formation and evolutionary paths and their role in the broader picture of galaxy evolution. This will be done by decomposing every galaxy into its constituent structural components (e.g., bulge/spheroid, bar, disk, nuclear point source) using GALFIT [@peng02; @peng10].
A number of scientific studies are already under way by the [S$^{4}$G ]{}team. These include a study of outer disks to determine the variety and frequency of different disk profile(s), truncations or thresholds and their relationship to the build up of galaxy disks, a study of shells and debris around galaxies and their relationship to central star formation or AGN activity, an analysis of the structural properties of bulges/spheroids (classical, disky, boxy/peanut-shaped), determination of the bar fraction and bar properties in relationship to the host galaxies and environment, an analysis of the stellar arm-interarm variations in spirals, an investigation of the underlying old stellar population in rings and a search for fossil rings, an analysis of the old stellar content of dwarf galaxies and its implication for their formation history, etc. Longer term studies include star formation history and assembly of galaxies using [*GALEX*]{} and [H$\alpha$ ]{}data, the Tully–Fisher relationship using existing HI data (e.g., ALFALFA, @giovanelli05, HIPASS, @barnes01), and testing of new mid-infrared diagnostics of AGN. We describe these studies in more detail in §\[sfgsci\]. Ultimately, [S$^{4}$G ]{}will lead to numerous other studies of astrophysical phenomena in the nearby Universe and provide a key anchoring data set for studies of galaxy evolution.
The [*Spitzer*]{} 3.6 and 4.5$\mu$m Bands: Best Tracers of Stellar Mass
=======================================================================
The 3.6 and 4.5$\mu$m IRAC bands on [*Spitzer*]{} are ideal tracers of the stellar mass distribution in galaxies because they image the Rayleigh–Jeans limit of the blackbody emission for stars with T $>$ 2000 K. Moreover, the mid-infrared \[3.6\]–\[4.5\] color in nearby galaxies is nearly constant with radius, and independent of the age of the stellar population and its mass function [@pahre04]. Analysis of the tilt in the fundamental plane for ellipticals shows that the [*Spitzer*]{} IRAC bands are better tracers of the stellar mass than even K-band emission, which is more sensitive to variations in the underlying stellar population; the IRAC data are more uniform and therefore are the best and perhaps most direct tracers of stellar mass [@jun08]. Additionally, on [*Spitzer*]{} these bands are so sensitive that [S$^{4}$G ]{}will image to extremely low stellar mass surface densities ($\sim$ 1[$M_{\sun}$]{}pc$^{-2}$).
There are minor contaminants to the old stellar population light in the 3.6 and 4.5$\mu$m IRAC bands. In the 3.6$\mu$m band, there is a very weak 3.3$\mu$m PAH feature, which contributes negligibly to the overall emission in the band ($<$ 2% as the equivalent width is $\sim$0.02$\mu$m, @tokunaga91). Hot dust (T$_{d} > $500K) from very small grains can contribute light to both bands with a higher contribution in the 4.5$\mu$m band, but this condition occurs only near active galactic nuclei or extreme starbursts. By examining the colors of the galaxies, we should be able to use the two IRAC bands to remove the effects of the 3.3$\mu$m PAH or hot dust emission. Another important advantage of having 2 bands is that by co-adding them, we can further improve the signal to noise and push the study to even fainter regions of the galaxies. In both bands, the contamination from young red supergiants is also very low. From previous studies, it is known that in the near-infrared, the overall contamination from young red supergiants to a galaxy is only $\sim$3% but in regions of star formation, the contribution may be as high as 25% [@rhoads98]. These regions, however, are easily identifiable and can be flagged. At the IRAC wavelengths, the fractional contribution from star-forming regions is expected to be the same (or lower) because these wavebands are at longer wavelengths on the Rayleigh–Jeans tail of the spectral energy distribution. At these wavelengths, there is lower dust extinction [@draine84] and no other significant emission sources. Thus, with the extremely low surface brightness limits, [S$^{4}$G ]{}offers a unique and virtually dust-free view of the distribution of mass in stellar structure in the nearby Universe.
Although ground-based near-infrared observations can offer higher angular resolution than [*Spitzer*]{}, the main obstacle for ground-based observations is the very high and variable sky brightness (typically $\mu_{K} \sim$ 13.4 mag arcsec$^{-2}$). The surface brightness level in the IRAC data is over 10 magnitudes below the typical sky brightness level. To get to this level from the ground, one would need to characterize the sky brightness, the flat field and instrumental variations to better than 0.0009%. This is currently not possible with any existing (or planned) near-infrared survey. Surveys like UKIDSS LAS or VISTA VHS are planning to reach a depth of 18.4 (21.2 in AB) mag arcsec$^{-2}$ in the K$_s$ band, which translates to 30 [$L_{\sun}$]{}pc$^{-2}$ (even with the brighter stellar emission at K$_s$ band), whereas [S$^{4}$G ]{}reaches $\sim$ 2 [$L_{\sun}$]{}pc $^{-2}$. It is the stability of the background that is the biggest advantage for space-based observations.
Sample Selection
================
We chose all galaxies with radial velocity V$_{radio}$ $<$ 3000 km/s (corresponding to a distance d $<$ 40 Mpc for a Hubble constant of 75 km/s/Mpc), total corrected blue magnitude m$_{Bcorr} <$ 15.5, blue light isophotal angular diameter D$_{25}$ $>$ 1.0 arcmin at galactic latitude $|$b$|$ $>$ 30$^{\circ}$ using HyperLEDA [@paturel03]. The choice of V$_{radio}$ does limit the sample to galaxies with a radio-derived (e.g., HI) radial velocities in HyperLEDA, and as a result misses some galaxies for which only optically-derived radial velocities exists; currently, a sample chosen with V$_{optical}$ instead of V$_{radio}$ would contain 2997 galaxies. A comparison of the galaxy properties of the two samples shows that the [S$^{4}$G ]{}sample misses galaxies that are small, relatively faint and early-type (gas-poor) systems from the volume surveyed.
We also note that our sample was defined using HyperLEDA in September 2007 - since then 98 more galaxies have been added to the HyperLEDA data base that meet the [S$^{4}$G ]{}criteria described above and we expect that some more may be added as better data become available. Given the limited lifetime of [*Spitzer*]{}, we are unable to go back and acquire data on these additional galaxies but we expect that the few percent additional galaxies will not strongly influence the core characteristics of the sample, except for the bias against early-type galaxies as noted above.
The choice of a 40 Mpc volume is arbitrary – it was chosen to be large enough to provide a statistically significant number of galaxies of all types and to be representative of a large range of the local large scale structure environment (as shown in Fig. \[volume\]). Our experience with previous surveys of galaxy properties (e.g., @sheth08) has shown that a few thousand galaxies are needed to accurately measure and account for completeness effects in mass, color and size selection of galaxy samples for robust investigations of relationships between galactic structure and host galaxy properties. The size cut, log D$_{25} >$ 1.0, was made to ensure that galaxies were large enough for a detailed study of their internal structure (at 40 Mpc, 1$\arcmin \sim$11.6 kpc). The apparent size and apparent magnitude cut were chosen to match the RC3 limits. The galactic latitude cut, $|$b$|$ $>$ 30$^{\circ}$, was used to minimize the unresolved Galactic light contribution from the Milky Way disk. The full sample of 2,331 [S$^{4}$G ]{}galaxies is listed in Table \[s4gsamptab\]. In Figure \[hubmasstyp\], we show the distribution of [S$^{4}$G ]{}galaxies as a function of Hubble type and stellar mass, along with the existing distribution of the LVL galaxies (green) and GO/GTO galaxies (blue) including the SINGS galaxies. There are some noteworthy sub-samples within [S$^{4}$G ]{}. There are 188 early-type galaxies (ellipticals and lenticulars), 206 dwarf galaxies (defined as M$_B > $ -17), of which 135 are DDO dwarfs and 465 edge-on ($i > $ 75$^{\circ}$) systems. Also over 800 of the [S$^{4}$G ]{}galaxies are mapped beyond a diameter of 3$\times$D$_{25}$. These sub-samples provide the deepest, largest and most homogeneous samples for a multitude of galaxy evolutionary studies, some of which are described in §\[sfgsci\].
Archival / Cryogenic [*Spitzer*]{} Data
---------------------------------------
Of the 2,331 galaxies in the [S$^{4}$G ]{}sample, 597 had some existing data at 3.6 and 4.5$\mu$m from the [*Spitzer*]{} cryogenic mission. 125 of these are part of the LVL survey and 56 are from the SINGS survey, both of which used the same observing strategy (§\[obsstrat\]) we employ for [S$^{4}$G ]{}. Almost all of the archival maps have at least 240 s of integration time per pixel and are sufficiently deep. The only exceptions are NGC 5457 (96s), NGC 0470, NGC 0474 (150s), and NGC 5218, NGC 5216, and NGC 5576 (192s).
In terms of area, 82 of the archival galaxies were mapped between 1.0 and 1.5$\times$D$_{25}$, and 43 others were mapped to $<$1.0 $\times$D$_{25}$. While outer disk science for these 125 galaxies would benefit from an extended map, they represent only a small fraction of the total sample, and the repetition of observations would have yielded only an incremental scientific return. Therefore, we decided not to repeat observations of any of the archival galaxies. We also decided not to map the SMC and the LMC, which meet our selection criteria but are very large and are being observed as part of other GO programs. M33, the other commonly studied Local Group galaxy was observed during the cryogenic mission and is included in the archival part of our sample. All other galaxies that were found via the HyperLEDA data base search in 2007 September are observed (see note earlier about the modest additions to the HyperLEDA database in the last three years). Although we do not repeat observations of any of the archival galaxies, we are processing all galaxies in the [S$^{4}$G ]{}sample, except the SMC and LMC, through the same [S$^{4}$G ]{}reduction and analysis pipelines described in §\[pipeline\].
[*Spitzer*]{} Post-Cryogenic Mission Observations Strategy {#obsstrat}
==========================================================
The 1,734 [S$^{4}$G ]{}galaxies being observed in the post-cryogenic mission are mapped using either a small dithered map or a mosaicked observation with both IRAC channels. All galaxies are observed with a total on-source integration time of 240 s which leads to an rms noise of $\mu$(1$\sigma$) $\sim$ 0.0072 and 0.0093 MJy sr$^{-1}$ at 3.6 and 4.5$\mu$m, respectively. This translates to a typical surface brightness sensitivity of $\mu_{AB} \sim$27 mag arcsec$^{-2}$ or a stellar surface density of 1.13 [$M_{\sun}$]{}pc$^{-2}$ (assuming a solar M$_{K}$ = 3.33, @worthey94; M/L$_{K}$=1, @heraudeau97) .
1560 of the new observations are of galaxies with D$_{25} < $ 3.3$\arcmin$. We can map these galaxies to 1.5$\times$D$_{25}$ with a single pointing because the [*Spitzer*]{} field of view is 5$\arcmin$. Hence all of these galaxies are mapped using a standard cycling small dither pattern with 4 exposures of 30s each in two separate Astronomical Observation Requests (AORs). Each pair of AORs is separated by at least 30 days to allow for sufficient rotation of the telescope so that the galaxy is imaged at two distinct orientations. Two AORs allow us to better remove cosmic rays, and the redundant information gathered by the two visits allows us to characterize and remove image artifacts (e.g., muxbleed, column pulldown, etc (see the IRAC Instrument Handbook[^1]) and possible asteroids. Drizzling the data from the two visits also allows us to achieve sub-pixel sampling with which we can reconstruct images with better fidelity than would be possible from a single visit. We also note that since both the 3.6 and 4.5$\mu$m arrays collect data simultaneously, their offset placement in the [*Spitzer*]{} focal plane means that an additional flanking map is made adjacent to the galaxy in each of the two channels (in some cases, the flanking field from each of the AORs may not overlap exactly due to the telescope rotation, but this does not affect the observations of the main galaxy). An example of a typical dithering AOR is shown in the left panel in Figure \[mappingaor\].
One of the considerations in designing these AORs was the effect of saturation from bright stars on the chip and the scattered light from bright stars falling in one of the three scattering zones surrounding the arrays[^2]. Stars brighter than 5th magnitude can leave a latent charge on the chip. Stars brighter than 11th magnitude in the scattering zones may scatter light into the main field of view of the IRAC chips. To mitigate these cases, we used the [*Spitzer*]{} Observing Tool (SPOT[^3]) to examine every galaxy with a star, m$_{Ks} <$ 8 (stars fainter than m$_{Ks}$ = 8 do not contribute significantly to the background) within an area of 300$\arcsec$, covering the IRAC field of view and scattering zones adequately, for all possible visibility windows through 2011. In a majority of these cases, no modifications of AORs were needed because the offending star(s) fell out of the chip or the scattering zones. In the remaining cases, we constructed AORs with timing constraints, slightly offset pointings or used medium dithers or mosaics to reduce the effect of the bright stars.
For galaxies with D$_{25}$ $>$ 3.3$\arcmin$, we mosaic them in array coordinates with offsets of 146.6$\arcsec$ with 30s integrations at each location to create a map $\ge$1.5$\times$D$_{25}$. This leads to a map with each pixel observed four times at all wavelengths in the core of the map. In addition, on each side of the galaxy there will be a region where each pixel is observed twice but only in one of the two channels. Like the dithered AORs, each mosaic is observed twice but with only a follow-on constraint so that each mosaic overlaps the other closely. An example of this type of an AOR is shown in the right panel of Figure \[mappingaor\].
Although larger mosaics could be made, our experience with SINGS data showed that the dominant noise term comes from variations of the background [@regan04; @regan06]. The best background for a galaxy includes sufficient sky not far from the area of interest. Our map sizes are thus fine-tuned to achieve the best possible signal-to-noise for all galaxies.
The [S$^{4}$G ]{}Pipeline {#pipeline}
=========================
The [S$^{4}$G ]{}pipeline is divided into four distinct parts, which we refer to as Pipelines 1, 2, 3 and 4. Pipeline 1 takes the raw basic calibrated data (BCD) fits files and converts them to science-ready data. Pipeline 2 creates a mask for the foreground and background objects for each of the individual galaxies. Pipeline 3 measures the standard global galaxy properties such as size, axial ratio, magnitude, color, etc. and computes the radial profiles of the standard properties. Pipeline 4 deconstructs each galaxy into its major structural components. Each of these is described in detail here.
[S$^{4}$G ]{}Pipeline 1
-----------------------
The purpose of pipeline 1 (P1) is to create science-ready mosaics from the two visits of each target. To do this P1 performs two major steps. First, it matches the background level in the individual images to account for drifts in the zero point of the amplifiers. Second, it creates the final mosaic using the Drizzle package of IRAF[^4].
To match the background levels in the individual images we find the regions of overlap between all pairs of overlapping images. Typically we require that the overlap region contain 20000 pixels. For all of the new post-cryogenic mission observations (and archival observations that used the same observing strategies), such an overlap is always available. However, when reducing other archival data with less overlap between the frames, we reduce the number of pixels in the overlap region to as low as 5000 pixels – this still provides an adequate number of pixels for estimating the background.
Within these regions we determine the brightness level of the 20th percentile pixel. We use the 20th percentile brightness level to better avoid detector artifacts such as muxbleed which can affect even the median. By using the 20th percentile brightness level we have a better chance of using the true sky level. We then perform a least square solution using the brightness differences between all the pairs of overlapping images, minimizing the residual background difference. Since we use the difference, we need to use the background level in the first image to set the zero point of the solution. We then make corrected images by adding the solved-for corrections and use these images in the formation of the final mosaic. We create the final mosaic following the standard prescription of the STSDAS DITHER package. This method removes the cosmic rays by first forming a mosaic from a median combination of the images and then comparing the individual images to the median mosaic and flagging any cosmic rays. The final mosaic is formed by drizzling the individual images. The resulting mosaic has a pixel scale of 0.75$\arcsec$ and we correct for the change in pixel size to keep the units of MJy sr$^{-1}$.
The relative astrometry of the mosaicked images is excellent because the [*Spitzer*]{} pipeline already incorporates the 2MASS positions to update its astrometry for each of the basic calibrated data frames. The [S$^{4}$G ]{}pipeline therefore does not require the type of cross-correlation that was needed previously for the SINGS pipeline to align overlapping fields. The relative astrometric accuracy is $<$0$\farcs$1.
The point spread function for IRAC is asymmetric and depends on the spacecraft orientation angle. Pipeline 1 images are produced by combining two different visits to the galaxy and therefore we estimate the PSF by combining stars in the foreground and background for six typical [S$^{4}$G ]{}galaxies to create a “super”-PSF which has a typical FWHM of 1$\farcs$7 and 1$\farcs$6 at 3.6$\mu$m and 4.5$\mu$m respectively. For the most distant galaxies in our survey at a distance of $\sim$40Mpc, this resolution corresponds to a linear scale of $\sim$300pc. At the median distance of the galaxies in this survey at 21.6 Mpc, this corresponds to a linear scale of $\sim$170pc.
[S$^{4}$G ]{}Pipeline 2
-----------------------
After the science-ready images are produced by the [S$^{4}$G ]{}pipeline, we generate masks for point sources using SExtractor [@bertin96] for both channels. Each mask is checked by eye and iterated to identify any point source otherwise missed by SExtractor. We also unmask any region of the galaxy that may be have been incorrectly identified by SExtractor. The images with their corresponding masks are then run through the third part of the [S$^{4}$G ]{}pipeline (Pipeline 3, P3 hereafter).
[S$^{4}$G ]{}Pipeline 3
-----------------------
The first step in P3 is to determine the local sky level around each galaxy. We want to be sufficiently far from the galaxy to ensure that there is no contamination from the galaxy but not so far that the background is not truly local. We do this by computing the median sky value in two concentric elliptical annuli centered at the position of the galaxy. Typically we start the inner annulus at a distance of 1.5$\times$D$_{25}$. This annulus is divided in azimuth into 45 regions (or sky boxes). Each of these regions is then grown outwards until it encompasses a total of 4000 non-masked pixels[^5], as shown in Figure \[ngcsky\]. The second annulus begins at the end of the first annulus and the same process is repeated. We compare the measured sky values in the two annuli for any radial gradients which might be indicative of light contamination from the galaxy. If we find that there is contamination, then we move the inner radius outwards and repeat the process until we are assured that we are sampling the local sky.
In our analysis so far, the inner radius of the inner annulus has varied between 1.2 and 2$\times$D$_{25}$. In all cases for the warm mission data and a majority of the archival data, it has been sufficent to set the inner radius of the annulus to either 1.5$\times$ or 2$\times$ D$_{25}$. However in a few cases where the archival data are shallow and the galaxy is relatively blue, we moved the inner radius to 1.2 $\times$D$_{25}$ to be immediately outside the galaxy to get as accurate a measurement possible for the local sky level. Automating this procedure is non-trivial because variations at these faint levels ($\mu_{3.6\mu m} > $27 (AB) mag arcsec$^{-2}$) can be from a lopsided galaxy, debris or tidal structures, or variation in the background. Therefore we examine every galaxy by eye to ensure that contamination from the galaxy light to the measurement of the sky background is as low as possible and move the radii outwards as needed.
The sky background computation method for three of the galaxies in the [S$^{4}$G ]{}sample (NGC 0337, NGC 4450, NGC 4579) is shown in Figure \[ngcsky\]. In the central column of this figure, we show an example case for NGC 4450 where the median sky in some of the boxes is affected by the bleeding of flux from a (masked) but saturated nearby bright field star. In such cases we ignore the affected pixels or move the annuli further out to get the best estimate for the background. Finally, we use the standard deviation obtained from the 90 sky boxes with the average standard deviation within each box to estimate the contribution of the low-frequency flat-fielding errors and possible additions to the total error budget in the background determination.
The typical sky brightness level is fainter than 26 (AB)magarcsec$^{-2}$ at 3.6 and 4.5$\micron$. The Poisson noise from the galaxy dominates the total error budget in the central regions, while errors in the sky background are the main source of uncertainty in the outer parts. Large-scale background errors can be particularly significant in galaxies with large apparent sizes. However, in the fainter outer parts, the noise is significantly reduced from azimuthally averaging over a large number of pixels in the outer galaxy – we are therefore able to measure the galaxy profiles well below the typical sky brightness. The typical S/N for a [S$^{4}$G ]{}galaxy at the canonical $\mu_{3.6\mu}$ = 26.5 (AB) mag arcsec$^{-2}$ is $>$ 3.
The images resulting from Pipeline 1 are calibrated in units of MJy sr$^{-1}$. The conversion from MJy sr$^{-1}$ to AB magnitudes is such that the zero-point to convert fluxes in Jy to magnitudes does not vary with wavelength [@oke74]:
$$m_{\mathrm{AB}} (\mathrm{mag})=-2.5\log F_{\nu} (\mathrm{Jy})+8.9$$
From this definition one can derive the corresponding expression for surface brightness:
$$\mu_{\mathrm{AB}} (\mathrm{mag})=-2.5\log I_{\nu} (\mathrm{MJy\ str^{-1}})+20.472$$
### P3: Measuring the Galaxy Host Properties
After the sky level has been properly measured, we use the IRAF routine [ellipse]{} to determine the radial profiles of intensity, surface brightness, ellipticity and position angle for all galaxies in both bands. The center is determined using [imcenter]{} and kept fixed during the fit, whereas the ellipticity and position angle are left as free parameters. We generate profiles with two different radial resolutions, by incrementing the semi-major axis in $2\arcsec$ and $6\arcsec$ at each step, respectively. The latter profiles have a coarser resolution and a better S/N ratio - these are used to measure the RC3-like parameters. The low- and high-frequency sky background errors described above, together with the Poisson noise of the source along each isophote, are considered together in the final error budget of the surface photometry at each radius (see @gildepaz05 [@munoz-mateos09]). The errors in the position angle and ellipticity at each radius are computed by the ellipse task from the rms of the pixel values along each fitted isophote. The corresponding errors for the values of these geometrical parameters at 25.5 and 26.5 (AB) magarcsec$^{-2}$ are interpolated between the corresponding adjacent values. The error in the semi-major axis is computed from the change in the radius affected by moving the surface brightness profile by the measured errors and an additional 10% error added in quadrature to the measured errors. The 10% error is an estimate of the aperture correction uncertainty as estimated by the Spitzer Science Center instrument team. We compute the error in the semi-major axis for each galaxy. Sample results for NGC 0337 and NGC 4579 are noted below. For NGC 4450, the scattered light from a background star strongly affects the computation of the surface brightness error and therefore the error in the semi-major axis that is unrealistic. In such cases, the error in the semi-major axis can be assumed to be the median semi-major axis error in the [S$^{4}$G ]{}sample. The maximum error in the determination of the semi-major axis is 6$\arcsec$, the step size of the coarse resolution fitting procedure. In Figures \[NGC0337\_ellipse\], \[NGC4450\_ellipse\] and \[NGC4579\_ellipse\] we show three sample sets of profiles for three galaxies in the sample: NGC 0337 (SBd), NGC 4450 (SAab) and NGC 4579 (SABb). The basic RC3-like galaxy properties (semi-major axis, axial ratio and position angle) are tabulated at the levels of 25.5 and 26.5 (AB) mag arcsec$^{-2}$ in both the 3.6$\mu$m and the 4.5$\mu$m images, and are quoted in Table \[RC3\_tab\].
Along with the surface photometry we also measure the curve of growth (e.g., @munoz-mateos09 [@gildepaz07]) with the integrated flux up to each radius. By fitting the accumulated magnitude as a function of the magnitude gradient at each point we obtain the asymptotic magnitude as the y-intercept of that fit (see @gildepaz07). Once the total magnitude is known, it is straightforward to locate the radii containing a given percentage of the total galaxy luminosity, from which concentration indices can be determined. In Table \[mag\_tab\] we quote the asymptotic magnitudes of the three sample galaxies considered here, together with the $C_{31}$ [@devaucouleurs77a] and $C_{42}$ [@kent85] concentration indices.
[S$^{4}$G ]{}Pipeline 4 {#pipe4}
-----------------------
In Pipeline 4, we decompose the two-dimensional stellar distribution in each galaxy into different sub-components using the version 3 of GALFIT [@peng02; @peng10]. GALFIT is a parametric fitting algorithm that offers a large flexibility on the fitted models; the optimal solution for the model parameters is found with the Levenberg-Marquardt algorithm, performing non-linear least-squares $\chi^2$ minimization of the difference between observed and model images. For our pipeline, we could have also used BUDDA [@gadotti08] and BDBAR [@laurikainen05]. We tested all three algorithms using a set of galaxy images and verified that GALFIT produces the same results as BUDDA and BDBAR if the three codes are run with the same set of input images, models and initial parameters. We chose GALFIT because it is currently the most sophisticated algorithm, and also well known by a large portion of the astronomical community. Other studies in which software was developed for similar purposes include @simard98 [@pignatelli06; @mendez-abreu08].
To simplify the use of GALFIT for a large number of cases, a set of new IDL-based tools have been created (“GALFIDL”). They include automatic creation of the input files for decompositions, containing reasonable first guesses for the initial parameters based on P3 products, as well as routines for running and visualizing the decompositions. GALFIDL will be available to the community concurrently with the publication of the data and the GALFIT analyses; an overview of these procedures is already available at [http://cc.oulu.fi/$\sim$hsalo/galfidl.html]{}.
As a standard part of the pipeline analysis, decompositions include bulges and disks, and, if appropriate, bars, nuclear components, and in some cases multiple exponential disks. Taking into account that decompositions, particularly at high redshift, are often made in a more simple manner, we also compute the 2-component bulge/disk decompositions, and 1-component Sérsic function fits. A decomposition for NGC 1097 is shown in Figs. \[dc1\] and \[dc2\]: the 5-component model includes an exponential disk, a Sérsic function for the bulge, and a Ferrers function for the bar. The galaxy has an extremely prominent nuclear ring, which is fit using a Gaussian function, and a nuclear point source using the corresponding PSF. Without deep, almost dust-free observations, like those in [S$^{4}$G ]{}, the measurement of bulge structural parameters, like shape, size and profile, might in some cases be compromised if the quantification of underlying structures, such as the disks, bars and rings is inaccurate. GALFIT3.0 allows for even more sophisticated decompositions, which will be made for selected sub-samples of [S$^{4}$G ]{}galaxies. For example, it is possible to add more components, or one may want to fit the spiral arms, of which an example is shown in Fig. \[dc3\].
Science Investigations {#sfgsci}
======================
In this section we highlight some preliminary results from a number of science investigations that are and will be carried out by the [S$^{4}$G ]{}team. We also discuss possible studies that will be enabled by the [S$^{4}$G ]{}dataset.
The Faint Outskirts of Galaxies
-------------------------------
A major area of study for [S$^{4}$G ]{}is a quantitative analysis of stellar structures in the faint outer regions of galaxies where the stellar surface density we trace is extremely low and comparable or lower than the atomic gas reservoir. Already, deep optical studies have shown that, in a majority of spiral galaxies, the outer disks exhibit a secondary, exponential component. Stellar disks appear to extend beyond 1.5$\times$ D$_{25}$ (e.g., @pohlen06 [@erwin08]) with truncated or anti-truncated profiles. [*GALEX*]{} [@martin05] has found extended UV (XUV) disks even farther out, where it was assumed only H[i]{} gas existed (e.g., @gildepaz05 [@thilker05; @thilker07; @munoz-mateos07; @zaritsky07]. Not only is [S$^{4}$G ]{}significantly deeper than the previous optical data, but it also traces the stellar profiles into the H[i]{} disk of the galaxy – in over 800 [S$^{4}$G ]{}galaxies, we image an area $>$3D$_{25}$.
Another observation that has been much debated is the break between the inner and outer exponentials profiles (e.g., @bakos08 [@azzollini08]). With [S$^{4}$G ]{}we are measuring the break radius and comparing it to the disk size and mass. The [S$^{4}$G ]{}catalog of these breaks will provide strong quantitative constraints for galaxy evolutionary models. In particular, this catalog will be key to quantify the possible role of radial stellar migration in shaping the outskirts of disks [@roskar08; @sanchez-blazquez09].
In elliptical galaxies, deep optical observations have revealed the presence of faint tidal tail and shell like structures (e.g., @bennert08 [@canalizo07]. [S$^{4}$G ]{}contains 188 ellipticals and lenticulars – with this set and the [S$^{4}$G ]{}depth we are quantifying the frequency and mass of shells and other debris features in the outskirts of ellipticals and lenticulars.
Bulges: Classical, Disk-like, Boxy/Peanut
-----------------------------------------
Bulges are an inhomogeneous class of objects and different types have been proposed in the literature [@kormendy04; @athanassoula05a]. @kormendy04 distinguishes bulges from pseudo-bulges, while @athanassoula05a distinguishes three categories: classical bulges, boxy/peanut bulges, which are a part of a bar, and disk-like bulges, formed out of disk material. Several studies have analyzed their observational properties and compared them with N-body models [e.g. @kuijken95; @carollo97; @bureau99; @erwin03; @bureau05; @fisher06; @drory07; @fisher08; @gadotti09]. Even in lenticular galaxies there seems to be evidence for disk-like bulges, based on structural decompositions (e.g., @laurikainen05 [@laurikainen07; @gadotti08; @graham08]) and kinematics (e.g., @peletier07 [@falcon-barroso06]). But until now few studies have analyzed the structural properties of bulges in a large number of galaxies; none have done it in a sample as large and deep as [S$^{4}$G ]{}. Moreover, [S$^{4}$G ]{}encompasses a range of large scale structures, allowing us to examine the environmental influences on the formation of bulges. This may have important implications since massive bulges are expected to form from mergers.
From a preliminary analysis of the [S$^{4}$G ]{}data, @buta10 are finding that many late type ellipticals in the RC3 have disks, and thus should instead be classified as S0s, consistent with other deep near-infrared studies [@laurikainen10]. With the [S$^{4}$G ]{}data we will examine how common the disk structure is in all types of ellipticals.
Finally, the extremely deep data from [S$^{4}$G ]{}allows us to best determine the structural properties of the bulge component – in shallower data, estimates of shape, size and profile are uncertain when measurements of underlying components such as disks, bars and rings are imprecise. As described in §\[pipe4\], the [S$^{4}$G ]{}project is carrying out detailed decomposition of all the galaxies, and expects to compile the most detailed data base of bulges of all kinds. The combined advantages of the [S$^{4}$G ]{}data (deep and insensitive to dust attenuation) and the uniform reduction pipeline (careful and detailed structural analysis) will allow for a suitable examination of galaxies with composite bulges, i.e., galaxies hosting more than one bulge category (e.g., @erwin03 [@barentine10; @nowak10]). The spatial resolution of our survey is adequate for resolving typical bulges (our resolution is $\sim$100 pc at the median distance of the survey volume) but is inadequate for precise measurements of the bulge structure from the inability to resolve compact nuclear structures which are often present at the centers of galaxies. These unresolved point sources affect the values of Sersic parameters, which on average are overestimated in ground-based optical images compared to HST studies (e.g., @carollo97 [@balcells03]). We also expect that the reduced dust extinction may reveal triaxial structures in bulges, and thus allow us to probe beyond the standard symmetric Sérsic fits.
Bars
----
Stellar bars are dynamically important structures in galaxies. A bar-induced gas flow leads to a number of evolutionary effects from the triggering of circumnuclear starbursts to the build-up of bulges (@scoville79 [@simkin80; @zaritsky86; @athanassoula92; @friedli93; @martin94; @friedli95; @sakamoto99; @sheth00; @sheth02; @sheth05]). The cosmological redshift evolution of the bar fraction is also an important signpost of the growth and dynamic maturity of galaxies (e.g., @abraham99 [@sheth03; @jogee04; @elmegreen04; @sheth08]). Whereas the previous smaller and shallower studies gave mixed results for the change in the bar fraction with redshift, @sheth08 show that not only does the overall bar fraction in disk galaxies decline significantly over the last seven billion years, but also that the decline depends on the galaxy type. They show that the formation and evolution of bars is strongly correlated with the host galaxy mass, bulge-dominance and presumably their dark matter halo. The [S$^{4}$G ]{}data is critical to establish the precise local frequency of bars and its variation with galaxy type. It also provides a measurement of the light fraction in the bar component (e.g., @durbala08 [@gadotti08; @weinzirl09]) and accurate measures of properties such as the bar length, shape and ellipticity (e.g., @menendez07 [@erwin05b; @gadotti10]). Several techniques may be used to derive bar strengths, based on calculating bar torques [@laurikainen02; @buta03], and estimating the bar-spiral arm contrast [@elmegreen85]. Analysis of the relative Fourier intensity profiles of bars is compared with models of the evolution of barred galaxies (e.g., @athanassoula02b [@debattista04]) to constrain the history of their evolution and interaction with their dark matter halo [@athanassoula03].
Another study of interest is the presence of secondary or nuclear bars which may be responsible for feeding the central black holes and/or stellar clusters [@shlosman90]. The [S$^{4}$G ]{}observations, unaffected by the high and patchy extinction often present in galactic centers, allow us to detect all nuclear bars with sizes of r $>$ 500 pc (our resolution, 2$\arcsec \sim$ 200 pc at the median distance of [S$^{4}$G ]{}) in galaxies of all types.
Galactic Rings
--------------
Outer rings are large, optically low surface brightness features that dominate the outer disks of some early-type barred and weakly-barred galaxies (see review by @buta96. The properties of the rings constrain their formation epoch, the dynamical time-scale at their respective radii, and the evolution of the bar pattern speed (e.g., @athanassoula03. Finding outer rings around unbarred galaxies is also of great interest, because it would suggest that some unbarred galaxies may in fact be highly evolved former barred galaxies. Whereas ground-based near-IR imaging is rarely deep enough to detect outer rings reliably to study their stellar populations, colors, and other characteristics, the [S$^{4}$G ]{}data will not only allow us to detect many new outer rings, but also to make a complete census of these features in normal galaxies and to measure their stellar mass, shape, width and ellipticity. Likewise, [S$^{4}$G ]{}provides an unprecedented view of circumnuclear and inner rings, which are typically found in barred spirals. These rings should have a broad, underlying older stellar population background as they evolve, and it is this type of feature we will see in the [S$^{4}$G ]{}data.
Spiral Arms
-----------
A systematic study of spiral arm amplitudes as a function of radius at 3.6 and 4.5$\mu$m, free of confusing effects from dust extinction, promises to initiate a new set of spiral arm studies. Although the discrepancies in the classical density wave model were solved with the swing amplification theory [@toomre81], there has been very little testing of this theory with observations. Modal studies by @bertin89a [@bertin89b] predict that there should be amplitude variations along the arms from resonances between inward and outward moving spiral waves and the precise amplitude variations have been modeled [@elmegreen84; @elmegreen89; @elmegreen93; @regan97]. Most recently, a new theory has been proposed for the formation of rings and spiral arms which argues in favor of chaotic orbits, confined by invariant manifolds emanating from the L1 & L2 Lagrangian points of a bar [@romero-gomez07]. Several comparisons between the results of this theory and observations have been carried out successfully [@athanassoula09a; @athanassoula09b] and several more will be possible with the [S$^{4}$G ]{}database. [S$^{4}$G ]{}will cover the full range of galaxies with varying spiral arm strengths and galactic structures. In Figure \[arminterarm\], we show a preliminary result of the type of analysis that is possible with the [S$^{4}$G ]{}data. In this figure we show the arm-interarm contrast measured from the 3.6$\mu$m [S$^{4}$G ]{}images for NGC 7793, a flocculent galaxy, in contrast with M51, a grand design spiral galaxy (see a model for amplitude variation of M51 in @salo00a). The contrasts are measured beyond R$_{25}$, further than in previous optical studies. The spiral structure traced in the 3.6 $\mu$m band is different in detail from the optical images, but still shows weak arms in NGC 7793 and increasingly stronger arms in M51. We will carry out a detailed analysis of spiral arm amplitudes and variations and compare the observational data to the models.
For the NGC 5194 (M51) system, there may well be a connection between the spiral structure and the presence of the companion (NGC 5195). Indeed, detailed HI observations by @rots90 show the presence of a long tail, which emanates from the west side of the galaxy, running south of the spiral arm visible in the [S$^{4}$G ]{}image, and turning up towards the north further eastwards. The HI kinematics are very difficult to model with an interaction involving a single passage, but @salo00a [@salo00b] propose a model with a second passage (Figure \[fig\_m51\_simu1\]), which gives a better fit to some of the aspects of the observations. The inner spiral structure in the main spiral is then amplified by the action of the companion, and wave interference causes amplitude modulations, which are well reproduced in the [S$^{4}$G ]{}observations (Figure \[fig\_m51\_simu2\]). This shows the interest of the [S$^{4}$G ]{}data for detailed modeling of the dynamics of individual galaxies.
Early Type Galaxies & Dwarfs
----------------------------
[S$^{4}$G ]{}contains 188 galaxies of T-type $<$ 0 and therefore constitutes one of the largest and most homogeneous mid-infrared data sets for early type galaxies. The shapes of ellipticals and lenticulars may reflect different evolutionary paths, with wet mergers leading to disky shapes and dry mergers leading to boxy profiles [@naab06; @pasquali07; @kang07]. In addition, @kormendy09 shows that deep imaging can reveal departures from the Sérsic profiles in elliptical galaxies, which are diagnostics of their formation. They found a dichotomy in which ellipticals that have cuspy cores at their inner radii can be separated from those which show an excess of light at the center. The [S$^{4}$G ]{}data offer us a unique opportunity to revisit the frequency of the different shapes and profiles as a function of stellar mass and environment, free of dust obscuration effects.
Although dwarfs are the most abundant type of galaxy in the Universe, their low surface brightnesses of typically $\mu_B >$ 22 mag arcsec$^{-2}$ have restricted detailed observations of these stellar systems. In particular, the characterization of the underlying stellar structure in these faint galaxies still remains largely unexplored. Using an absolute magnitude criterion of $M_B > -17$, we have identified a preliminary sample of 206 dwarf galaxies in [S$^{4}$G ]{}. The sample includes 87 late-type DDO dwarfs previously unobserved with [*Spitzer*]{}, to complement the 48 which have been observed. The [S$^{4}$G ]{}dwarfs therefore constitute a representative sample of nearby dwarf galaxies in which to study the properties of the underlying stellar component.
Dwarf galaxies in [S$^{4}$G ]{}allow for a detailed analysis of the underlying stellar disk in both the 3.6 and 4.5$\mu$m data. By deriving the radial scale lengths for these galaxies, more insight can be gained into the evolutionary link between dwarfs and their giant counterparts. At the same time, the relatively large sample of dwarfs in [S$^{4}$G ]{}gives a unique opportunity to look at how these parameters may vary with environment. We can also derive stellar masses for the dwarfs which will be used to assess their stellar mass to dark-matter ratio and will likely provide crucial evidence to the longstanding debate of whether these galaxies are truly dark-matter dominated systems (see e.g., @strigari08).
In addition to these areas, there are at least three other main areas of astrophysics that these data will address uniquely: the reported absence of bulges in some disk galaxies, the vertical stellar structure, and the Tully-Fisher relationship. There is also likely to be significant amounts of spin-off science for [S$^{4}$G ]{}with its large and accurate inventory of mass and galactic structure in the nearby Universe, e.g., relationship between the Tully-Fisher relationship and properties of the fundamental plane (e.g., @zaritsky08), dust heating in outer disks, the interaction / merger fraction from a study of the debris around these galaxies, and diagnosis of AGN activity from tracing hot dust very close to the AGN from the \[3.6\]-\[4.5\] color.
Data Release and Team Policy
============================
The [S$^{4}$G ]{}team is committed to releasing all of the reduced data upon publication of the entire [S$^{4}$G ]{}datasets from each of the pipelines. We expect the science-ready images from pipeline 1, the masks from pipeline 2, the galaxy properties and profiles from pipeline 3 and the input and output files from Pipeline 4 (including the mask, sigma images, images without NaN values and the GALFIT input and output parameter files) to be released on a staggered schedule using NED or IRSA for long term use by the community. Currently the products of each of the pipeline are expected to be released in the supplement issue of the journals upon verification of their quality which will be on-going and occur as part of the various scientific investigations conducted by the team. We expect that some of the data and pipeline may need to be refined or enhanced in the future. As the data are still being acquired (data acquisition is expected to be complete in 2011) and analyzed, our earliest release date for the the science-ready images (P1 products) for the entire [S$^{4}$G ]{}data is not expected at least until 2012, and the supplement papers for the products from the other pipelines are expected to follow within a year after the Pipeline 1 supplement paper. In the meanwhile, to allow for maximum use of the data for science by the larger community, the collaboration has agreed on an open-door policy whereby any member of the astronomical community may join the team temporarily as a guest and pursue any type of science. The use of the data, publication policies and authorship on papers are clearly defined in our policy statement available here: [ http://www.cv.nrao.edu/$\sim$ksheth/ s4g/S4G\_policy\_v6.pdf]{}. This statement is adopted by the [S$^{4}$G ]{}team and all of its guests. Interested members of the astronomical community are invited to read the policy and contact the PI or an [S$^{4}$G ]{}member to discuss the possibilities of joining the team.
The noteworthy point of this policy is that our team has agreed to avoid carving out large science categories or areas to encourage the maximum use of the data within the team, and within the larger community, for any science that may be possible with these data. Our philosophy is that if multiple people are interested in a topic but want to work in separate teams then we will encourage both teams to collaborate and address the scientific problems in independent but parallel investigations.
Summary
=======
[S$^{4}$G ]{}will provide the deepest, largest, and most homogenous data set of nearby galaxies at 3.6 and 4.5$\mu$m. The spatial resolution will be unmatched at these wavelengths, $\sim$2$\arcsec$, which is ideal for comparisons to ground based optical and near-IR observations, as well as MeerKAT, eVLA, ALMA and other future large radio array surveys. The sample of 2,331 galaxies includes most morphological types and masses. The images will give an unparalleled view of stellar mass distributions and faint peripheral structures. All of the galaxies, as well as many additional galaxies from the [*Spitzer*]{} archives, will be reduced in the same way, giving data products that span the full range from science-ready images to measurements of global galaxy properties and individual components. [S$^{4}$G ]{}is designed to be a useful reference for many years to come. It will contain the best available mid-infrared data for individual galaxies and be among the most complete surveys for statistical studies. It will be useful to study the origin and evolution of galaxies and their dynamical components, and to supplement observations of nearby galaxies at other wavelengths.
Acknowledgements
================
The authors thank the referee for their useful comments and suggestions that greatly helped improve this paper. We are also grateful to the dedicated staff at the Spitzer Science Center for their support and help with the planning and execution of this legacy exploration program. KS would like to thank L. Armus, E. Bell, S. Carey, E. Churchwell, M. Dickinson, G. Helou, N. Scoville, and J. Stauffer for sharing their experiences in leading large teams. A.G.dP and J.C.M.M are partially financed by the Spanish Programa Nacional de Astronomía y Astrofísica under grants AyA2006-02358 and AyA2009-10368. A.G.dP is also financed by the Spanish Ramón y Cajal program. J.C.M.M. acknowledges the receipt of a Formación del Profesorado Universitario fellowship. EL and HS acknowledge support from the Academy of Finland. KMD is supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-0802399. DME acknowledges support from the Spitzer Science Center from NASA grant JPL RSA-1368024. RB acknowledges support from NSF grant AST 05-07140. EA and AB thank the Centre National d’Etudes Spatiales and ANR-06-BLAN-0172 for support. K.L.M. acknowledges funding from the Peter and Patricia Gruber Foundation as the 2008 IAU Fellow, from the University of Portsmouth, and from SEPnet (www.sepnet.ac.uk). This work is based on observations and archival data obtained with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Support for this work was provided by NASA. KS and other staff at the NRAO acknowledge support from the National Radio Astronomy Observatory, which is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
[*Facilities:*]{}
[rrrrrrr]{} UGC12893 & 0.00784 & 17.21952 & 8.4 & -15.97 & 1.06 & 15.14\
PGC000143 & 0.03292 & -15.46140 & 9.9 & -15.83 & 2.04 & 10.00\
ESO012-014 & 0.04511 & -80.34825 & 9.0 & -18.17 & 1.18 & 13.72\
NGC7814 & 0.05418 & 16.14554 & 2.0 & -20.04 & 1.64 & 10.96\
UGC00017 & 0.06198 & 15.21822 & 9.0 & -16.02 & 1.39 & 14.60\
NGC7817 & 0.06635 & 20.75182 & 4.1 & -21.11 & 1.52 & 11.56\
ESO409-015 & 0.09223 & -28.09915 & 6.2 & -15.16 & 1.02 & 14.43\
ESO293-034 & 0.10559 & -41.49592 & 6.0 & -18.68 & 1.40 & 12.67\
NGC0007 & 0.13916 & -29.91678 & 4.9 & -18.22 & 1.39 & 13.18\
NGC0014 & 0.14618 & 15.81659 & 9.8 & -18.25 & 1.18 & 12.35\
IC1532 & 0.16464 & -64.37161 & 4.0 & -17.89 & 1.26 & 13.86\
... & ... & ... & ... & ... & ... & ...\
... & ... & ... & ... & ... & ... & ...\
NGC7798 & 23.99042 & 20.74986 & 4.1 & -20.20 & 1.20 & 12.56\
NGC7800 & 23.99354 & 14.80723 & 9.7 & -19.45 & 1.24 & 12.61\
[lrrrrrrrr]{} & 25.5 & 90.6$\pm$1.5 & 0.287$\pm$0.018 & 118.4$\pm$2.0 & & 84.1$\pm$1.3 & 0.278$\pm$0.037 & 118.5$\pm$4.5\
& 26.5 & 110.4$\pm$4.7 & 0.270$\pm$0.027 & 119.8$\pm$3.2 & & 106.9$\pm$1.3 & 0.287$\pm$0.055 & 123.8$\pm$7.1\
& 25.5 & 209.0$\pm$6 & 0.293$\pm$0.016 & 1.4$\pm$1.8 & & 199.9$\pm$... & 0.293$\pm$0.020 & 3.2$\pm$2.3\
& 26.5 & 253.0$\pm$6 & 0.336$\pm$0.048 & 19.8$\pm$5.1 & & 266.7$\pm$... & 0.418$\pm$0.026 & 18.1$\pm$2.3\
& 25.5 & 256.0$\pm$2.6 & 0.295$\pm$0.011 & 95.2$\pm$1.2 & & 229.0$\pm$2.6 & 0.268$\pm$0.018 & 95.5$\pm$2.2\
& 26.5 & 278.6$\pm$17 & 0.251$\pm$0.018 & 78.6$\pm$2.4 & & 263.4$\pm$12 & 0.282$\pm$0.027 & 98.3$\pm$3.4\
[lrrrrrrr]{} NGC 0337 & 11.432$\pm$0.001 & 2.67 & 2.74 & & 11.888$\pm$0.002 & 2.82 & 2.90\
NGC 4450 & 9.630$\pm$0.001 & 4.58 & 4.17 & & 10.112$\pm$0.001 & 4.56 & 4.16\
NGC4579 & 9.083$\pm$0.001 & 5.29 & 4.54 & & 9.564$\pm$0.001 & 5.60 & 4.71\
[^1]: See
[^2]: A description of this problem is given in the IRAC instrument handbook at: .http://ssc.spitzer.caltech.edu/irac/iracinstrumenthandbook/
[^3]:
[^4]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^5]: For archival galaxies that fill the mosaicked frame, this requirement is reduced to 1000 non-masked pixels.
|
---
abstract: 'In this paper, the Discontinuous Cell Method (DCM) is formulated with the objective of simulating cohesive fracture propagation and fragmentation in homogeneous solids without issues relevant to excessive mesh deformation typical of available Finite Element formulations. DCM discretizes solids by using the Delaunay triangulation and its associated Voronoi tessellation giving rise to a system of discrete cells interacting through shared facets. For each Voronoi cell, the displacement field is approximated on the basis of rigid body kinematics, which is used to compute a strain vector at the centroid of the Voronoi facets. Such strain vector is demonstrated to be the projection of the strain tensor at that location. At the same point stress tractions are computed through vectorial constitutive equations derived on the basis of classical continuum tensorial theories. Results of analysis of a cantilever beam are used to perform convergence studies and comparison with classical finite element formulations in the elastic regime. Furthermore, cohesive fracture and fragmentation of homogeneous solids are studied under quasi-static and dynamic loading conditions. The mesh dependency problem, typically encountered upon adopting softening constitutive equations, is tackled through the crack band approach. This study demonstrates the capabilities of DCM by solving multiple benchmark problems relevant to cohesive crack propagation. The simulations show that DCM can handle effectively a wide range of problems from the simulation of a single propagating fracture to crack branching and fragmentation.'
address:
- 'Department of Civil and Environmental Engineering, Northwestern University, Evanston (IL) 60208, USA.'
- 'Greenman-Pedersen, INC., Albany (NY), 12205, USA.'
author:
- Gianluca Cusatis
- 'Roozbeh Rezakhani [^1]'
- 'Edward A. Schauffert'
title: 'Discontinuous Cell Method (DCM) for the Simulation of Cohesive Fracture and Fragmentation of Continuous Media'
---
{width="2"}
[ [**Center for Sustainable Engineering of Geological and Infrastructure Materials (SEGIM)**]{}\
\[0.1in\] Department of Civil and Environmental Engineering\
\[0.1in\] McCormick School of Engineering and Applied Science\
\[0.1in\] Evanston, Illinois 60208, USA ]{}
\
0.5in
\
\
[**SEGIM INTERNAL REPORT No. 16-08/587D**]{}\
cohesive fracture ,finite elements ,discrete models ,delaunay triangulation ,voronoi tessellation ,fragmentation
Introduction
============
A quantitative investigation of cohesive fracture propagation necessitates an accurate description of various fracture phenomena including: crack initiation; propagation along complex three-dimensional paths; interaction and coalescence of distributed multi-cracks into localized continuous cracks; and interaction of fractured/unfractured material. The classical Finite Element (FE) method, although it has been used with some success to address some of these aspects, is inherently incapable of modeling the displacement discontinuities associated with fracture. To address this issue, advanced computational technologies have been developed in the recent past. First, the embedded discontinuity methods (EDMs) were proposed to handle displacement discontinuity within finite elements. In these methods the crack is represented by a narrow band of high strain, which is embedded in the element and can be arbitrarily aligned. Many different EDM formulations can be found in the literature and a comprehensive comparative study of these formulations appears in Reference [@Jirasek-1]. The most common drawbacks of EDM formulations are stress locking (spurious stress transfer between the crack surfaces), inconsistency between the stress acting on the crack surface and the stress in the adjacent material bulk, and mesh sensitivity (crack path depending upon mesh alignment and refinement).
A method that does not experience stress locking and reduces mesh sensitivity is the extended finite element method (X-FEM). X-FEM, first introduced by Belytschko & Black [@Belytschko-2], exploits the partition of unity property of FE shape functions. This property enables discontinuous terms to be incorporated locally in the displacement field without the need of topology changes in the initial uncracked mesh. Mo[ë]{}s et al. [@Moes-1] enhanced the primary work of Belytschko et al. [@Belytschko-2] through including a discontinuous enrichment function to represent displacement jump across the crack faces away from the crack tip. X-FEM has been successfully applied to a wide variety of problems. Dolbow et. al. [@Dolbow-1] applied XFEM to the simulation of growing discontinuity in Mindlin-Reissner plates by employing appropriate asymptotic crack-tip enrichment functions. Belytschko and coworkers [@Belytschko-3] modeled evolution of arbitrary discontinuities in classical finite elements, in which discontinuity branching and intersection modeling are handled by the virtue of adding proper terms to the related finite element displacement shape functions. Furthermore, they studied crack initiation and propagation under dynamic loading condition and used a criterion based on the loss of hyperbolicity of the underlying continuum problem [@Belytschko-4]. Zi et Al. [@Zi-Belytschko] extended X-FEM to the simulation of cohesive crack propagation. The main drawbacks of X-FEM are that the implementation into existing FE codes is not straightforward, the insertion of additional degrees of freedoms is required on-the-fly to describe the discontinuous enrichment, and complex quadrature routines are necessary to integrate discontinuous integrands.
Another approach widely used for the simulation of cohesive fracture is based on the adoption of cohesive zero-thickness finite elements located at the interface between the usual finite elements that discretize the body of interest [@Camacho-1; @Ortiz-1]. This method, even if its implementation is relatively simple, tends to be computationally intensive because of the large number of nodes that are needed to allow fracturing at each element interface. Furthermore, in the elastic phase the zero-thickness finite elements require the definition of an artificial penalty stiffness to ensure inter-element compatibility. This stiffness usually deteriorates the accuracy and rate of convergence of the numerical solution and it may cause numerical instability. To avoid this problem, algorithms have been proposed in the literature [@Pandolfi-1] for the dynamic insertion of cohesive fractures into FE meshes. The dynamic insertion works reasonably well in high speed dynamic applications but is not adequate for quasi-static applications and leads to inaccurate stress calculations along the crack path.
An attractive alternative to the aforementioned approaches is the adoption of discrete models (particle and lattice models), which replace the continuum a priori by a system of rigid particles that interact by means of linear/nonlinear springs or by a grid of beam-type elements. These models were first developed to describe the behavior of particulate materials [@Cundall-1] and to solve elastic problems in the pre-computers era [@Hrennikoff-1]. Later, they have been adapted to simulate fracture and failure of quasi-brittle materials in both two [@Schlangen-1] and three dimensional problems [@Cusatis-1; @Cusatis-2; @Lilliu-1; @Cusatis-3]. In this class of models, it is worth mentioning the rigid-body-spring model developed by Bolander and collaborators, which dicretizes the material domain using Voronoi diagrams with random geometry, interconnected by zero-size springs, to simulate cohesive fracture in two and three dimensional problems [@Bolander-2; @Bolander-1; @Bolander-3; @Bolander-4]. Various other discrete models, in the form of either lattice or particle models, have been quite successful recently in simulating concrete materials [@cusatis-ldpm-1; @cusatis-ldpm-2; @Rezakhani-JMPS; @Leite-1; @Donze-1; @Grassl-1].
Discrete models can realistically simulate fracture propagation and fragmentation without suffering from the aforementioned typical drawbacks of other computational technologies. The effectiveness and the robustness of the method are ensured by the fact that: a) their kinematics naturally handle displacement discontinuities; b) the crack opening at a certain point depends upon the displacements of only two nodes of the mesh; c) the constitutive law for the fracturing behavior is vectorial; d) remeshing of the material domain or inclusion of additional degrees of freedom during the fracture propagation process is not necessary. Despite these advantages the general adoption of these methods to simulate fracture propagation in continuous media has been quite limited because of various drawbacks in the uncracked phase, including: 1) the stiffness of the springs is defined through a heuristic (trial-and-error) characterization; 2) various elastic phenomena, e.g. Poisson’s effect, cannot be reproduced exactly; 3) the convergence of the numerical scheme to the continuum solution cannot be proved; 4) amalgamation with classical tensorial constitutive laws is not possible; and 5) spurious numerical heterogeneity of the response (not related to the internal structure of the material) is inherently associated with these methods if simply used as discretization techniques for continuum problems.
The Discontinuous Cell Method (DCM) presented in this paper provides a framework unifying discrete models and continuum based methods. The Delaunay triangulation is employed to discretize the solid domain into triangular elements, the Voronoi tessellation is then used to build a set of discrete polyhedral cells whose kinematics is described through rigid body motion typical of discrete models. Tonti [@tonti-1] presented a somewhat similar approach to discretize the material domain and to compute the finite element nodal forces using dual cell geometries. Furthermore, the DCM formulation is similar to that of the discontinuous Galerkin method which has primarily been applied in the past to the solution of fluid dynamics problems, but has also been extended to the study of elasticity [@Guzey-1]. Recently, discontinuous Galerkin approaches have also been used for the study of fracture mechanics [@Shen-1] and cohesive fracture propagation [@Abedi-1]. The DCM formulation can be considered as a discontinuous Galerkin approach which utilizes piecewise constant shape functions. Another interesting feature of DCM is that the formulation includes rotational degrees of freedom. Researchers have attempted to introduce rotational degrees of freedom to classical finite elements by considering special form of displacement functions along each element edge to improve their performance in bending problems [@Allman-1; @Bergan-1]. This strategy leads often to zero energy deformation modes and to singular element stiffness matrix even if the rigid body motions are constrained. DCM formulation simply incorporates nodal rotational degrees of freedom, without suffering from the aforementioned problem.
Governing Equations {#2}
===================
Equilibrium, compatibility, and constitutive laws for Cauchy continua can be formulated as limit case of the governing equations for Cosserat continua in which displacements and rotations are assumed to be independent fields but the couple stress tensor is identically zero [@Zhou-1]. For small deformations and for any position vector $\bf x$ in the material domain $\Omega$, one has $$\gamma_{ij} =u_{j,i} -e_{ijk} \varphi_k \label{Eq:Compatibility-1}$$ and $$\sigma_{ji,j} + b_{0i} = \rho \ddot{u}_i ;~~~ e_{ijk} \sigma_{jk} = 0 \label{Eq:Equilibrium-2}$$
In the above equations, the summation rule of repeated indices applies; $u_i$ and $\varphi_i$ are displacement and rotation fields, respectively. $\gamma_{ij}$ is the strain tensor; $\sigma_{ij}$ is the stress tensor; $b_{0i}$ are body forces per unit volume; $\rho$ is the mass density. Subscripts $i$, $j$, and $k$ represent components of Cartesian coordinate system which can be $i,j,k = 1,2,3$ in three dimensional problems; $e_{ijk}$ is the Levi-Civita permutation symbol. Considering any position dependent field variable such as $f(\bf x, t)$, $f_{,i}$ represent partial derivative of $f$ with respect to the $i$th component of the coordinate system, while $\dot{f}$ is the time derivative of the variable. The partial differential equations above need to be complemented by appropriate boundary conditions that can either involve displacements, $u_i - u_{0i} = 0$ on $\Gamma_u$ (essential boundary conditions); or tractions, $\sigma_{ji}n_j - t_{0i}=0$ on $\Gamma_t$ (natural boundary conditions); where $\Gamma = \Gamma_t \cup \Gamma_u$ is the boundary of the solid volume $\Omega$.
In the elastic regime, the constitutive equations can be written as $$\begin{gathered}
\label{eq:const}
\sigma_{ij} = E_V \epsilon_V \delta_{ij} + E_D (\gamma_{ij}-\epsilon_V \delta_{ij})\end{gathered}$$ where $\epsilon_V=\gamma_{ii}/3$ is the volumetric strain; $E_V$ and $E_D$ are the volumetric and deviatoric moduli that can be expressed through Young’s modulus $E$ and Poisson’s ratio $\nu$: $E_V = E/(1-2\nu)$; $E_D=E/(1+\nu)$. It is worth observing that since the solution of the problem formulated above requires the stress tensor to be symmetric (see second equation in Equation \[Eq:Equilibrium-2\]), the constitutive equations imply the symmetry of the strain tensor as well, which, in turn leads to displacements and rotations to be related through the following expression: $e_{ijk} \varphi_k = (u_{j,i}-u_{i,j})/2$
The weak form of the equilibrium equations can be obtained through the Principle of Virtual Work (PVW) as $$\label{PVW}
\int_{\Omega} \sigma_{ji} \delta \gamma_{ji} \textrm{d}\Omega + \int_{\Omega} \rho \ddot{u}_i \delta u_{i} \textrm{d}\Omega = \int_{\Gamma_{t}}t_{0i} \delta u_i \textrm{d}{\Gamma_{t}} + \int_\Omega b_{0i} \delta u_{i} \textrm{d}\Omega$$ where $\delta \gamma_{ij}= \delta u_{j,i} -e_{ijk} \delta \varphi_k $, $\delta u_i$ and $\delta \varphi_i$ are arbitrary strains, displacements, and rotations, respectively, satisfying compatibility equations with homogeneous essential boundary conditions. It must be observed here that the PVW in Equation \[PVW\] is the weak formulation of both linear and angular momentum balances. Hence, the symmetry of the stress tensor and, consequently, the symmetry of the strain tensor are imposed in average sense. This is a significant difference compared to classical formulations for Cauchy continua in which the symmetry of the stress tensor is assumed “a priori”.
Discontinuous Cell Method Approximation
=======================================
Domain Discretization
---------------------
Let us consider a three-dimensional *primal cell complex*, which, according to the customary terminology in algebraic topology [@tonti-2], is a subdivision of the three-dimensional space $\mathds{R}^3$ through sets of vertices (0-cells), edges (1-cells), faces (2-cells), and volumes (3-cells). Next let us construct a *dual cell complex* anchored to the primal. This can be achieved, for example, by associating a primal 3-cell with a dual 0-cell , a primal 2-cell with a dual 1-cell, etc. The primal/dual complex obtained through the Delaunay triangulation of a set of points and its associated Voronoi tessellation is a very popular choice in many fields of study for its ability to discretize complex geometry and it is adopted in this study.
Let us consider a material domain $\Omega$ and discretize it into tetrahedral elements by using the centroidal Delaunay tetrahedralization, and the associated Voronoi tessellation which leads to a system of polyhedral cells [@Polymesh-paulino]. Figure \[DCMgeom3D\]a, shows a typical tetrahedral element with the volume $\Omega^e$, external boundary $\Gamma^e$, and oriented surfaces $\Gamma^f$ located within the volume. The interior oriented surfaces $\Gamma^f$ are derived from the Voronoi tessellation and are hereinafter called “facets". In 3D, the facets are triangular areas of contact between adjacent polyhedral cells. In the Voronoi tessellation procedure, the triangular facets $\Gamma^f$ are perpendicular to the element edges $\Gamma^e$, which is a crucial feature of DCM formulation as it will be shown later, and their geometry is such that one node of each facet is placed in the middle of the tetrahedral element edge, one is located on one of the triangular faces of the tetrahedral element, and one is located inside the tetrahedral element. As a result, each tetrahedral element contains twelve facets in a 3D setting, Figure \[DCMgeom3D\]a. Figure \[DCMgeom3D\]b illustrates a portion of the tetrahedral element associated with one of its four nodes $\alpha$ and the corresponding facets. Combining such portions from all the tetrahedral elements connected to the same node, one obtains the corresponding Voronoi cell. Each node in the 3D DCM formulation has six degrees of freedom, three translational and three rotational, which are shown in Figure \[DCMgeom3D\]b. The same figure depicts, for a generic facet, three unit vectors, one normal $\bold{n}_f$ and two tangential ones $\bold{m}_f$ and $\bold{l}_f$, defining a local system of reference. In the rest of the paper, the facet index $f$ is dropped when possible to simplify notation.
Discretized Kinematics
----------------------
The DCM approximation is based on the assumptions that displacement and rotation fields can be approximated by the rigid body kinematics of each Voronoi cell, that is $$u_i({\bf x}) = u_{Ii} + e_{ijk} \varphi_{Ij} (x_k - x_{Ik}) ;~~ \varphi _i ({\bf x}) = \varphi_{Ii} ~~~ \textrm{for} ~ {\bf x} \in \Omega^I
\label{eq:disp-approx}$$ where $u_{Ii}$, $\varphi_{Ik}$ are displacements and rotations of node $I$; $\Omega^I$ is the volume of the cell associated with node $I$. Obviously with this approximating displacement and rotation functions, strain versus displacement/rotation relationships in Equations \[Eq:Compatibility-1\] cannot be enforced locally – as typically done in displacement based finite element formulation.
Let us consider a generic node $I$ of spatial coordinates ${\bf x}_I$ and the adjacent nodes $J$ of spatial coordinates ${ \bf x}_J = { \bf x}_I+ {\ensuremath{\boldsymbol\ell}}$, where ${\ensuremath{\boldsymbol\ell}}$ is the vector connecting the two nodes. One can write ${\ensuremath{\boldsymbol\ell}}= \ell { \bf n}^\ell$ in which ${ \bf n}^\ell$ is a unit vector in the direction of ${\ensuremath{\boldsymbol\ell}}$. Note that, to simplify notation the two indices $I$ and $J$ are dropped when supposed to appear together. Without loss of generality, let us also assume that node $I$ is located on the negative side of the facets whereas node $J$ is located on the positive side of the associated facet oriented through its normal unit vector ${ \bf n}$. Moreover, it is useful to introduce the vector ${\bf d} = { \bf x}_{P} - { \bf x}_I$ connecting node $I$ to a generic point P on the facet. The displacement jump at point P reads $$\hat{{\bf u}}= {\bf u}^+ - {\bf u}^- = {\bf u}_{J} - {\bf u}_I + \mb{\varphi}_{J} \times ({\bf d} - {\ensuremath{\boldsymbol\ell}}) - \mb{\varphi}_I \times {\bf d} = \Delta {\bf u} - \mb{\varphi}_I \times \ell {\bf n}^\ell - \Delta \mb{\varphi} \times \ell ({\bf n}^\ell - \xi {\bf n}^d)\label{Eq:Strain1}$$ where ${\bf u}^+$ and ${\bf u}^-$ are the values of displacements on the positive and negative side of the facet, respectively; $\Delta {\bf u} = {\bf u}_{J} - {\bf u}_I$; $\Delta \mb{\varphi} = \mb{\varphi}_{J} - \mb{\varphi}_I$; $\xi = d/\ell$; and $d$, ${\bf n}^d$ are magnitude and direction of vector ${\bf d}$. Equation \[Eq:Strain1\] can be rewritten in tensorial notation as $$\hat{u}_{i} = \Delta u_{i} - \ell e_{ijk} \varphi_{j} n^\ell_{k} - \ell e_{ijk} \Delta \varphi_{j} (n_{k}^\ell - \xi n_{k}^d) \label{Eq:Strain2}$$
By expanding the displacement and the rotation fields in Taylor series around ${\bf x}_I$ and by truncating the displacement to the second order and the rotation to the first, one obtains $$\Delta u_{i} = \ell u_{i,j} n^\ell_{j} + \frac{1}{2}\ell^2 u_{i,jk} n^\ell_{j} n^\ell_{k}; ~~~~ \Delta \varphi_{i} = \ell \varphi_{i,j} n^\ell_{j}$$
By projecting the displacement jump $\hat{u}_{i}$ in the direction orthogonal to the facet and dividing it by the element edge length $\ell$, one can write
$$\begin{split}
\ell^{-1} \hat{u}_i n_i = \ell^{-1} n_i (\ell u_{i,j} n^\ell_j + \frac{1}{2}\ell^2 u_{i,jk} n^\ell_j n^\ell_k) - e_{ijk} n_i \varphi_j n^\ell_k - e_{ijk} n_i (\ell \varphi_{j,p} n^\ell_p) (n_k^\ell - \xi n_k^d)= \\
= u_{j,i} n^\ell_i n_j + \frac{1}{2}\ell u_{j,ik} n_i^\ell n_j n^\ell_k - e_{ijk} \varphi_k n_i^\ell n_j - \ell \varphi_{j,i} e_{pjk} n_p n^\ell_i (n_k^\ell - \xi n_k^d) =\\
= (u_{j,i} - e_{ijk} \varphi_k) n^\ell_i n_j + \frac{1}{2}\ell u_{j,ik} n^\ell_i n_j n^\ell_k - \ell \varphi_{j,i} e_{pjk} n_p n^\ell_i (n_k^\ell - \xi n_k^d) = \gamma_{ij}n^\ell_i n_j + \mathcal{O}(\ell)
\label{Eq:DisJumpN}
\end{split}$$
where $\gamma_{ij}$ is the strain tensor. At convergence the discretization size tends to zero ($\ell \rightarrow 0$) and one can write $\ell^{-1} \hat{u}_i n_i =\gamma_{ij} n^\ell_i n_j$. Furthermore, if the dual complex adopted for the volume discretization is such that the facets are orthogonal to the element edges – this condition is verified, for example, by the Delaunay-Voronoi complex – then, ${ \bf n}^\ell \equiv { \bf n}$, and one has $\ell^{-1} \hat{u}_i n_i =\gamma_{ij} n_i n_j$: the normal component of the displacement jump normalized with the element edge length represents the projection of the strain tensor onto the facet. Similarly, it can be shown that the components of the displacement jump tangential to the facets can be expressed as $\ell^{-1} \hat{u}_i m_i =\gamma_{ij} n_i m_j$ and $\ell^{-1} \hat{u}_i l_i =\gamma_{ij} n_i l_j$.
Before moving forward, a few observations are in order. Since the facet are flat the unit vector $n_i$ is the same for any point belonging to a given facet and the projection of the strain tensor is uniform over each facet for a uniform strain field. The variation of the displacement jump over the facet is due to the curvature and it is an high order effect that can be neglected (see the last term in Equation \[Eq:DisJumpN\]). Based on the previous observation one can conclude that the analysis of the interaction of two adjacent nodes can be based on the average displacement jump which, given the linear distribution of the jump, can be calculated as the displacement jump, $\bold{w}$, at the centroid of the facet C. This leads naturally to the following definition of “facet strains”: $$\label{eq:eps-N}
\epsilon_{N} = \frac{ n_i } {\ell \Gamma} \int_{\Gamma} \hat{u}_i \, \textrm{d}S = \frac{n_i w_{i}}{\ell} = n_i n_j \gamma_{ij}$$ and $$\label{eq:eps-ML}
\epsilon_{M} = \frac{m_i } {\ell \Gamma} \int_{\Gamma} \hat{u}_i \, \textrm{d}S = \frac{m_i w_{i}}{\ell} = n_i m_j \gamma_{ij} ~~~~ \epsilon_{L} = \frac{l_i } {\ell \Gamma} \int_{\Gamma} \hat{u}_i \, \textrm{d}S = \frac{l_i w_{i}}{\ell} = n_i l_j \gamma_{ij}$$ Equations \[eq:eps-N\] and \[eq:eps-ML\] show that the “facet strains” correspond to the projection of the strain tensor onto the facet local system of reference.
Let us now consider a uniform hydrostatic stress/strain state in one element $\gamma_{ij} = \epsilon_V \delta_{ij}$ and $\sigma_{ij} = \sigma_V \delta_{ij}$. In this case the tractions on each facet must correspond to the volumetric stress and energetic consistency requires that $3 \Omega_e \sigma_{V} \epsilon_V = \sum_{ \mathcal{F}_e} \Gamma \ell \sigma_V \epsilon_N$ which gives $$\label{eq:eps-V}
\epsilon_V=\frac{1}{3 \Omega_e} \sum_{ \mathcal{F}_e} \Gamma \ell \epsilon_N = \frac{1}{3 \Omega_e} \sum_{ \mathcal{F}_e} \Gamma n_i w_{i}$$ where $\mathcal{F}_e$ is the set of facets belonging to one element. By using Equation \[eq:eps-V\] and Equations \[eq:eps-N\], \[eq:eps-ML\], facet deviatoric strains can also be calculated as $\epsilon_D=\epsilon_N-\epsilon_V$.
By introducing Equation \[eq:disp-approx\] into Equations \[eq:eps-N\] and \[eq:eps-ML\] one can also write
$$\label{eq:eps-N-new}
\ell \epsilon_{N} = n_i w_{i}= -n_{Ji} (u_{Ji} + e_{ijk} \varphi_{Jj} c_{Jk}) -n_{Ii} (u_{Ii} + e_{ijk} \varphi_{Ij} c_{Ik})$$
$$\label{eq:eps-M-new}
\ell \epsilon_{M} = m_i w_{i}= -m_{Ji} (u_{Ji} + e_{ijk} \varphi_{Jj} c_{Jk}) -m_{Ii} (u_{Ii} + e_{ijk} \varphi_{Ij} c_{Ik})$$
$$\label{eq:eps-L-new}
\ell \epsilon_{L} = l_i w_{i}= -l_{Ji} (u_{Ji} + e_{ijk} \varphi_{Jj} c_{Jk}) -l_{Ii} (u_{Ii} + e_{ijk} \varphi_{Ij} c_{Ik})$$
where $\bold{n}_J=-\bold{n}$, $\bold{n}_I=\bold{n}$, $\bold{m}_J=-\bold{m}$, $\bold{m}_I=\bold{m}$, $\bold{l}_J=-\bold{l}$, $\bold{l}_I=\bold{l}$, and $\bold{c}_I$, $\bold{c}_J$ are vectors connecting the facet centroid C with nodes $I$, $J$, respectively.
Discretized Equilibrium Equations
---------------------------------
For the adopted discretized kinematics in which all deformability is concentrated at the facets, the PVW in Equation \[PVW\] can be rewritten as
$$\begin{aligned}
\label{eq:equi-weak-discrete-0}
\begin{split}
\sum_{\mathcal{F}} \Gamma \ell \left( t_N \delta \epsilon_N + t_M \delta \epsilon_M + t_L \delta \epsilon_L \right) + \int_{\Omega} \rho \ddot{u}_i \delta u_{i} \textrm{d}\Omega=
\int_{\Gamma_t}t_{0i} \delta u_i \textrm{d}{\Gamma} + \int_{\Omega} b_{0i} \delta u_{i} \textrm{d}\Omega
\end{split}\end{aligned}$$
where $\mathcal{F}$ is the set of all facets in the domain.
By introducing Equations \[eq:disp-approx\], \[eq:eps-N-new\], \[eq:eps-M-new\], and \[eq:eps-L-new\] into Equation \[eq:equi-weak-discrete-0\] and considering displacement and traction continuity at the inter-element interfaces, one can write $$\begin{aligned}
\label{eq:equi-weak-discrete}
\begin{split}
\sum_I \left ( \sum_{\mathcal{F}_I} \Gamma t_{Ii} ( \delta u_{Ii} + e_{ijk} \delta \varphi_{Ij} c_{Ik}) + \int_{\Omega_I} [b_{0i}-\rho \ddot{u}_{Ii} - \rho e_{imp} \ddot{\varphi}_{Im} (x_p-x_{Ip})] [ \delta u_{Ii} + e_{ijk} \delta \varphi_{Ij} (x_k-x_{Ik}) ] d\Omega \right)=0
\end{split}\end{aligned}$$ where $t_{Ii} = t_N n_{Ii} +t_M m_{Ii}+t_L l_{Ii}$, $I$ is the generic Voronoi cell, $\Omega_I$ and $\mathcal{F}_I$ are the volume and the set of facets of the cell $I$, respectively. Note that the first term on the LHS of Equation \[eq:equi-weak-discrete\] also includes the contribution of external tractions for cells located on the domain boundary.
Since Equation \[eq:equi-weak-discrete\] must be satisfied for any virtual variation $ \delta u_{Ii}$ and $\delta \varphi_{Ik}$, it is equivalent to the following system of algebraic equations ($I=1, ..., N_c$, $N_c$= total number of Voronoi cells):
$$\label{eq:cell-equilibrium-1}
\sum_{\mathcal{F}_I} \Gamma t_{Ii} + F_{Ii}-\mathcal{M}_{I} \ddot{u}_{Ii}- \mathcal{S}_{Iik} \ddot{\varphi}_{Ik}=0~~~\textrm{for}~~~i=1,2,3$$
$$\label{eq:cell-equilibrium-2}
\sum_{\mathcal{F}_I} \Gamma t_{Ii}e_{ijk} c_{Ij} + W_{Ik}-\mathcal{S}_{Iik} \ddot{u}_{Ii}- \mathcal{I}_{Ikp} \ddot{\varphi}_{Ip}=0~~~\textrm{for}~~~k=1,2,3$$
where $F_{Ii}=\int_{\Omega_I} b_{0i} \textrm{d}\Omega$ = external force resultant, $\mathcal{M}_{I}=\int_{\Omega_I} \rho \textrm{d}\Omega$ = mass, $\mathcal{S}_{Iik} = \int_{\Omega_I} \rho e_{ijk} (x_j-x_{Ij}) \textrm{d}\Omega$ = first-order mass moments, $W_{Ik}=\int_{\Omega_I} b_{0i} e_{ijk} (x_j-x_{Ij}) \textrm{d}\Omega$ = external moment resultant, $\mathcal{I}_{Ikp}= \int_{\Omega_I} \rho e_{imp} (x_m-x_{Im}) e_{ijk} (x_j-x_{Ij}) \textrm{d}\Omega$ = second-order mass moments, of cell $I$. Equations \[eq:cell-equilibrium-1\] and \[eq:cell-equilibrium-2\] coincides with the force and moment equilibrium equations for each Voronoi cell.
Note that $\mathcal{S}_{Iik}=0$ and $W_{Ik}=0$ (for uniform body force), if the vertex of the nodes of the Delaunay discretization coincide with the mass centroid of the Voronoi cells. This is the case for all the cells in the interior of the mesh if a centroidal Voronoi tessellation is adopted. Also, in general, $\mathcal{I}_{Ikp} \neq 0$ for $k\neq p$, and this leads to a non-diagonal mass matrix. A diagonalized mass matrix can be obtained simply by discarding the non-diagonal terms.
Discretized Constitutive Equations
----------------------------------
In the DCM framework, the constitutive equations are imposed at the facet level where the facet tractions need to be expressed as function of the facet strains. For elasticity, by projecting the tensorial constitutive equations reported in Equation \[eq:const\] in the local system of reference of each facet, one has
$$\label{tN-const}
t_N=\sigma_{ij}n_i n_j= E_V \epsilon_V \delta_{ij}n_i n_j +E_D (\gamma_{ij}n_i n_j - \epsilon_V \delta_{ij} n_i n_j) = E_V \epsilon_V + E_D \epsilon_D = E_D \frac{n_i w_{i}}{\ell} + \frac{E_V-E_D}{3 \Omega_e} \sum_{ \mathcal{F}_e} \Gamma n_i w_{i}$$
$$\label{tM-const}
t_M=\sigma_{ij}n_i m_j= E_V \epsilon_V \delta_{ij}n_i m_j +E_D (\gamma_{ij}n_i m_j - \epsilon_V \delta_{ij} n_i m_j) = E_D \epsilon_M = E_D \frac{m_i w_{i}}{\ell}$$
$$\label{tL-const}
t_L=\sigma_{ij}n_i l_j= E_V \epsilon_V \delta_{ij}n_i l_j +E_D (\gamma_{ij}n_i l_j - \epsilon_V \delta_{ij} n_i l_j) = E_D \epsilon_L = E_D \frac{l_i w_{i}}{\ell}$$
Two-Dimensional Implementation
==============================
Three-Node Triangular Element
-----------------------------
In order to pursue a two-dimensional implementation of DCM, let us consider a 2D Delaunay-Voronoi discretization as shown in Figures \[DCMgeom\]a and b. A generic triangle can be considered as the triangular base of a prismatic volume as shown in Figure \[DCMgeom\]c and characterized by 6 vertexes, 9 edges, 2 triangular faces, and 3 rectangular faces. By considering the Voronoi vertexes on the two parallel triangular faces and face/edge points located at mid-thickness (see, e.g., points $a$ and $d$) a complete tessellation of the volume in six sub-volumes, one per vertex, can be obtained by triangular facets. Of these facets, $N_f=12$ are orthogonal to the triangular faces, set $\mathcal{F}_f$ (see, e.g., the one connecting points $c$, $e$, $h$ in Figure \[DCMgeom\]c) and $N_o=6$ are parallel to the triangular faces, set $\mathcal{F}_o$, (see, e.g., the one connecting points $c$, $d$, $a$ in Figure \[DCMgeom\]c).
One can write $$\label{eq:eps-V-2D}
\epsilon_V=\frac{1}{3 \Omega_e} \left( \sum_{ \mathcal{F}_f} \Gamma \ell \epsilon_N + \sum_{ \mathcal{F}_o} \Gamma s \epsilon_N \right)$$ where $s$ is the out-of-plane thickness.
For plane strain conditions $\epsilon_N=0$ for the facet set $\mathcal{F}_o$ and simply the second term in Equation \[eq:eps-V-2D\] is zero. For plane stress, instead, $t_N=E_V \epsilon_V+E_D(\epsilon_N-\epsilon_V)=0$ for the facet set $\mathcal{F}_o$. Therefore, for the facet set $\mathcal{F}_o$, $\epsilon_N=(E_D-E_V) \epsilon_V / E_D= -3\nu / (1-2 \nu) \epsilon_V $ and , also, $\sum_{ \mathcal{F}_o} \Gamma s = \Omega_e$. Using these relations, one has $\sum_{ \mathcal{F}_o} \Gamma s \epsilon_N = -3\nu/(1-2\nu)\Omega_e\epsilon_V$. Substituting this relation in Equation \[eq:eps-V-2D\] leads to $\epsilon_V = (1-2\nu)/(3\Omega_e(1-\nu)) \left( \sum_{ \mathcal{F}_f} \Gamma \ell \epsilon_N \right)$. In addition the facet set $\mathcal{F}_f$ is composed by 3 sets of 4 planar triangular facets. For each set, strains and tractions are the same on the 4 facets because the response is uniform through the thickness. Consequently the 4 facets can be grouped into one rectangular facet of area $sh$ where $h$ is the in length of the facet (see Figure \[DCMgeom\]b)
By taking everything into account the volumetric strain for 2D problem can be written as $$\label{eq:eps-V-2D-1}
\epsilon_V=\frac{1}{3 A^e \alpha} \sum_{f=1}^3 \ell_f h_f \epsilon_{fN} = \frac{1}{3 A^e \alpha} \sum_{f=1}^3 h_f n_{fi} w_{fi}$$ where $A^e$ is the area of the triangular element and $\alpha=(1-\nu)/(1-2\nu)$ for plane stress and $\alpha=1$ for plane strain.
The triangular DCM element has 9 degrees of freedom, two displacements and one rotation for each node, which can be collected in one vector $\bold{Q}^T=[u_{I1}~u_{I2}~\varphi_{I3}~u_{J1}~u_{J2}~\varphi_{J3}~u_{K1}~u_{K2}~\varphi_{K3}]$ in which $I$, $J$, and $K$ are the element node indexes, and 1, 2, and 3 represent the three Cartesian coordinate axes. By using Equations \[eq:eps-N-new\] to \[eq:eps-L-new\], one can write $\epsilon_{fN}=\bold{N}_f \bold{Q}$, $\epsilon_{fM}=\bold{M}_f \bold{Q}$, $\epsilon_{V}=\bold{V} \bold{Q}$, $\epsilon_{fD}=\bold{D}_f \bold{Q}$ where $$\label{eq:matrix-N1}
\bold{N}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {1};}}} = \ell^{-1}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {1};}}} \begin{bmatrix}
-n_{I1} & -n_{I2} & n_{I1}c_{I2}-n_{I2}c_{I1} & n_{I1} & n_{I2} & -n_{I1}c_{J2}+n_{I2}c_{J1} & 0 & 0 & 0
\end{bmatrix}$$ $$\label{eq:matrix-M1}
\bold{M}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {1};}}} = \ell^{-1}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {1};}}} \begin{bmatrix}
-m_{I1} & -m_{I2} & m_{I1}c_{I2}-m_{I2}c_{I1} & m_{I1} & m_{I2} & -m_{I1}c_{J2}+m_{I2}c_{J1} & 0 & 0 & 0
\end{bmatrix}$$ $$\label{eq:matrix-N2}
\bold{N}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {2};}}} = \ell^{-1}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {2};}}} \begin{bmatrix}
0 & 0 & 0 & -n_{J1} & -n_{J2} & n_{J1}c_{J2}-n_{J2}c_{J1} & n_{J1} & n_{J2} & -n_{J1}c_{K2}+n_{J2}c_{K1}
\end{bmatrix}$$ $$\label{eq:matrix-M2}
\bold{M}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {2};}}} = \ell^{-1}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {2};}}} \begin{bmatrix}
0 & 0 & 0 & -m_{J1} & -m_{J2} & m_{J1}c_{J2}-m_{J2}c_{J1} & m_{J1} & m_{J2} & -m_{J1}c_{K2}+m_{J2}c_{K1}
\end{bmatrix}$$ $$\label{eq:matrix-N3}
\bold{N}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {3};}}} = \ell^{-1}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {3};}}} \begin{bmatrix}
-n_{I1} & -n_{I2} & n_{I1}c_{I2}-n_{I2}c_{I1} & 0 & 0 & 0 & n_{I1} & n_{I2} & -n_{I1}c_{K2}+n_{I2}c_{K1}
\end{bmatrix}$$ $$\label{eq:matrix-M3}
\bold{M}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {3};}}} = \ell^{-1}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {3};}}} \begin{bmatrix}
-m_{I1} & -m_{I2} & m_{I1}c_{I2}-m_{I2}c_{I1} & 0 & 0 & 0 & m_{I1} & m_{I2} & -m_{I1}c_{K2}+m_{I2}c_{K1}
\end{bmatrix}$$ and $\bold{V} = (3 A_f \alpha)^{-1}\sum_{f=1}^3 \ell_f h_f \bold{N}_f$, $\bold{D}_f=\bold{N}_f-\bold{V}$
The elastic stiffness matrix for a single DCM triangle can be computed similarly to classical FEM elements by writing the internal energy as function of the nodal degrees of freedom. By using Equations \[tN-const\] and \[tM-const\], one has $$\label{IntEneVar-0}
\mathcal{U} = \frac{1}{2} s \ell_f h_f \left( t_{Nf} \epsilon_{Nf} + t_{Mf} \epsilon_{Mf}\right)= \frac{1}{2} s \ell_f h_f \left[ E_V\epsilon^2_{Vf} +E_D\epsilon_{Df}^2+E_D\epsilon_{Mf}^2 + (E_V+E_D)\epsilon_{Vf}\epsilon_{Df}\right] =\frac{1}{2}\bold{Q}^T\bold{K} \bold{Q}$$ where $$\label{eq:K}
\bold{K} = 2s A^e E_V \bold{V}^T \bold{V} + sh_f \ell_f \left[ E_D \bold{D}_f^T \bold{D}_f + 0.5(E_V+E_D)(\bold{V}^T \bold{D}_f + \bold{D}_f \bold{V}^T) + E_D \bold{M}_f^T \bold{M}_f \right]$$ and summation rule applies for repeated index $f$. In the previous equation the relation $\sum_f h_f \ell_f = 2A^e$ was used.
Similarly the mass matrix of the element can be obtained by writing the kinetic energy as function of the nodal velocities. For uniform density, one has $$\label{node-kinetic}
\mathcal{K} = \frac{1}{2} \sum_I\int_{A_{I}} \rho (\dot{u}_{1}^2 + \dot{u}_{2}^2 ) s \textrm{d}A = \frac{1}{2} s \rho \sum_I\int_{A_{I}} [\dot{u}^2_{I1} + \tilde{y}_I^2\dot{\varphi}^2_{I3}-2\tilde{y}_I\dot{u}_{I1}\dot{\varphi}_{I3}+ \dot{u}^2_{I2} + \tilde{x}^2_I \dot{\varphi}_{I3}^2+2\tilde{x}_I \dot{u}_{I2}\dot{\varphi}_{I3}] \textrm{d}A = \frac{1}{2}\dot{\bold{Q}}^T\bold{M} \dot{\bold{Q}}$$ where $\tilde{x}_I=x - x_{I}$, $\tilde{y}_I=y - y_{I}$ $$\label{element-mass}
\bold{M} =
\begin{bmatrix}
\bold{M}_1 & \bold{0} & \bold{0} \\
\bold{0} & \bold{M}_2 & \bold{0} \\
\bold{0} & \bold{0} & \bold{M}_3 \\
\end{bmatrix}
\hspace{0.25 in}
\bold{M}_I =\rho s
\begin{bmatrix}
A_I & 0 & -S_{yI} \\
0 & A_I & S_{xI} \\
-S_{yI} & S_{xI} & I_I \\
\end{bmatrix}$$ and $A_I=\int_{A_{I}} \textrm{d}A$, $I_I=\int_{A_{I}} (\tilde{x}^2_I+\tilde{y}^2_I ) \textrm{d}A$, $S_{yI}=\int_{A_{I}} \tilde{y}_I \textrm{d}A$, $S_{xI}=\int_{A_{I}} \tilde{x}_I \textrm{d}A$
To study the deformation properties of the formulated triangular elements, one can consider the generalized eigenvalue problem $\bold K_e \boldsymbol \Phi_I = \lambda_I \boldsymbol \Phi_I$ where $\lambda_I$ and $\boldsymbol \phi_I$ are the $I$th eigenvalue and the corresponding eigenvector of the element stiffness matrix, respectively. The triangular element has nine eigenvalues which correspond to the element nine degrees of freedom, three of which must be equal to zero to represent the three possible rigid body deformation modes. The other eigenvalues must be positive to ensure positive definiteness of the stiffness matrix. The deformation modes for an equilateral triangular element are plotted in Figure \[def-modes\]: Figures \[def-modes\] (a), (b), and (c) correspond to the rigid body deformation modes with zero eigenvalues; Figures \[def-modes\] (d), (e), and (f) are bending modes of deformation; uniaxial deformation mode is depicted in Figures \[def-modes\] (g); and volumetric and pure shear deformation modes are plotted in Figures \[def-modes\] (h) and (i), respectively.
In Figure \[spectralAnalysis\]a, a triangular element is considered, and the eigenvalue problem is solved for the different position of the top node from 1 to 2. For the final configuration 2, the length of the internal facet $f$ is zero. The minimum positive eigenvalue $\lambda_{min}$ is plotted versus facet to edge length ratio, $f/e$, of the triangular element in Figure \[spectralAnalysis\]b. One can see that as the facet length tends to zero and the element becomes a right triangle, $\lambda_{min}$ tends to zero, which means that the element stiffness matrix becomes singular. The zero eigenvalue is associated with a zero energy mode of deformation, which is plotted in Figure \[spectralAnalysis\]c. Since normal and tangential components of displacement jump vector at centroid of the facets are zero, normal and tangential strains on both element facets are equal to zero. Therefore, this element deformation mode is of zero energy. For $f/e < 0$, $\lambda_{min} < 0$ and the stiffness matrix is not positive definite. Therefore, during the domain discretization procedure, it must be considered that all element facets length must be positive. In other words, they must be completely placed inside the element, which, in turn, requires that all angles of the triangle to be smaller than $\pi/2$.
Four-Node Quadrilateral Elements
--------------------------------
It is not always possible to obtain triangular meshes that satisfy the shape requirements discussed in the previous section. Figure \[34elems\]a shows, for example, a situation where the triangulation produced a right triangle at the right-angled exterior corner of a structural domain. In this situation, the Voronoi tessellation produces three orthogonal bisectors that intersect at a point located exactly on the hypotenuse of the right triangle. This results in a facet with zero length, which would lead to a zero eigenvalue of the stiffness matrix. Figure \[34elems\]b shows a more severe situation where the generation of nodes and the resulting triangulation produces an obtuse triangle. In this situation, the three orthogonal bisectors intersect at a point located outside of the obtuse triangle, which would result in a negative facet area and a negative eigenvalue of the stiffness matrix. In order to overcome the computational problems implied by this situations, it is possible to combine a problematic triangle with the neighboring triangle into a four-node, five-facet quadrilateral element, Figure \[34elems\]c. The interior, or fifth facet is the orthogonal bisector of a straight line between two nodes at opposite corners of the quadrilateral element. The translation and rotation of the two nodes labeled as $I$ and $K$ in Figure \[34elems\]c, are the degrees of freedom that produce the displacement jump for the fifth facet. The straight line distance between Nodes $I$ and $K$ provides the edge length $\ell_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {5};}}}$ required for the calculation of facet strains. The volumetric strain, constant inside the quadrilateral element, is still calculated by Equation \[eq:eps-V-2D-1\] where the contribution of all five facets are taken into account. Also, a four node, rectangular element with four facets is generated if the two adjacent triangles are both right, see Figure \[34elems\]d. For the generic quadrilateral element, Equations \[eq:matrix-N1\] to \[eq:matrix-M3\] must be substituted by the following equations
$$\label{4eq:matrix-N1}
\bold{N}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {1};}}} = \ell^{-1}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {1};}}} \begin{bmatrix}
-n_{I1} & -n_{I2} & n_{I1}c_{I2}-n_{I2}c_{I1} & n_{I1} & n_{I2} & -n_{I1}c_{J2}+n_{I2}c_{J1} & 0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}$$
$$\label{4eq:matrix-M1}
\bold{M}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {1};}}} = \ell^{-1}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {1};}}} \begin{bmatrix}
-m_{I1} & -m_{I2} & m_{I1}c_{I2}-m_{I2}c_{I1} & m_{I1} & m_{I2} & -m_{I1}c_{J2}+m_{I2}c_{J1} & 0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}$$
$$\label{4eq:matrix-N2}
\bold{N}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {2};}}} = \ell^{-1}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {2};}}} \begin{bmatrix}
0 & 0 & 0 & -n_{J1} & -n_{J2} & n_{J1}c_{J2}-n_{J2}c_{J1} & n_{J1} & n_{J2} & -n_{J1}c_{K2}+n_{J2}c_{K1} & 0 & 0 & 0
\end{bmatrix}$$
$$\label{4eq:matrix-M2}
\bold{M}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {2};}}} = \ell^{-1}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {2};}}} \begin{bmatrix}
0 & 0 & 0 & -m_{J1} & -m_{J2} & m_{J1}c_{J2}-m_{J2}c_{J1} & m_{J1} & m_{J2} & -m_{J1}c_{K2}+m_{J2}c_{K1} & 0 & 0 & 0
\end{bmatrix}$$
$$\label{4eq:matrix-N3}
\bold{N}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {3};}}} = \ell^{-1}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {3};}}} \begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 & -n_{K1} & -n_{K2} & n_{K1}c_{K2}-n_{K2}c_{K1} & n_{K1} & n_{K2} & -n_{K1}c_{L2}+n_{K2}c_{L1}
\end{bmatrix}$$
$$\label{4eq:matrix-M3}
\bold{M}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {3};}}} = \ell^{-1}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {3};}}} \begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 & -m_{K1} & -m_{K2} & m_{K1}c_{K2}-m_{K2}c_{K1} & m_{K1} & m_{K2} & -m_{K1}c_{L2}+m_{K2}c_{L1}
\end{bmatrix}$$
$$\label{4eq:matrix-N4}
\bold{N}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {4};}}} = \ell^{-1}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {4};}}} \begin{bmatrix}
-n_{I1} & -n_{I2} & n_{I1}c_{I2}-n_{I2}c_{I1} & 0 & 0 & 0 & 0 & 0 & 0 & n_{I1} & n_{I2} & -n_{I1}c_{L2}+n_{I2}c_{L1}
\end{bmatrix}$$
$$\label{4eq:matrix-M4}
\bold{M}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {4};}}} = \ell^{-1}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {4};}}} \begin{bmatrix}
-m_{I1} & -m_{I2} & m_{I1}c_{I2}-m_{I2}c_{I1} & 0 & 0 & 0 & 0 & 0 & 0 & m_{I1} & m_{I2} & -m_{I1}c_{L2}+m_{I2}c_{L1}
\end{bmatrix}$$
$$\label{4eq:matrix-N5}
\bold{N}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {5};}}} = \ell^{-1}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {5};}}} \begin{bmatrix}
-n_{I1} & -n_{I2} & n_{I1}c_{I2}-n_{I2}c_{I1} & 0 & 0 & 0 & n_{I1} & n_{I2} & -n_{I1}c_{K2}+n_{I2}c_{K1} & 0 & 0 & 0
\end{bmatrix}$$
$$\label{4eq:matrix-M5}
\bold{M}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {5};}}} = \ell^{-1}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=0.1pt] (char) {5};}}} \begin{bmatrix}
-m_{I1} & -m_{I2} & m_{I1}c_{I2}-m_{I2}c_{I1} & 0 & 0 & 0 & m_{I1} & m_{I2} & -m_{I1}c_{K2}+m_{I2}c_{K1} & 0 & 0 & 0
\end{bmatrix}$$
Elastic Analysis Results
========================
Patch Test and Facet Tensors
----------------------------
Numerical experiments carried out in this section show that the 2D DCM triangle passes the patch test and is able to reproduce exactly uniform strain and stress fields. Acknowledging that the following observations apply equally to stress or strain, the most basic result regarding a uniform field is that the normal and tangential stresses calculated by DCM for a facet with certain orientation correspond to the tractions calculated by projecting the stress tensor onto the facet orientation. Conversely, facet’s orientation, normal and tangential (shear) stresses along with the calculated value of the volumetric stress can be used to determine the facet overall stress tensor with respect to the global system of reference (see Appendix \[facet strain tensor\]). To do the patch test, an elastic square specimen discretized by DCM is subjected to uniform $\varepsilon_{xx} = \varepsilon_{yy} = 0.1$ as shown in Figure \[PatchTest\]a. $E = 1000$ MPa and $\nu$ = 0.25 are used as material properties. DCM analysis is performed, and the results are used to calculated the stress tensor for all of the facets. The exact uniform stress tensor due to the applied uniform strain tensor can be calculated by elasticity equations. In Figure \[PatchTest\]b, $\sigma_{xx}^{\text{DCM}}/\sigma_{xx}^{\text{Exact}} =$ ratio of the $\sigma_{xx}$ calculated by DCM for each facet to the one obtained from elasticity equations, is plotted for all facets. For each facet and the corresponding element, the portion of the element area associated to that facet is colored according to the $\sigma_{xx}^{\text{DCM}}/\sigma_{xx}^{\text{Exact}}$ value. One can see that DCM successfully generated the uniform stress field which matches well with the elasticity results. This contour is the same for the $\sigma_{yy}$ component of the stress tensor. In addition, the resultant force vector on each node is plotted in Figure \[PatchTest\]c. It is clear that the force vector is negligible for the nodes inside the specimen, while its distribution on the specimen surfaces correspond to the uniform stress and strain fields.
Convergence Study on Cantilever Beam {#cantilever}
------------------------------------
In order to study the convergence of the present method to the exact solution for a non-uniform strain field, a classical cantilever beam test \[19\] was simulated. The rectangular domain, shown in Figure \[CanteliverBeam\], is characterized by a length-to-depth ratio of 4. The traction boundary conditions are the classic stress distributions of simple bending. Figure \[CanteliverBeam\] shows a parabolically varying shear at the cantilever tip. At the fixed end, only the displacement boundary conditions are shown for clarity. However, an equal but opposite parabolic shear was applied at the fixed end, as well as a linearly varying normal stress based on the non-zero bending moment at that location. The exact solution for the displacement field is provided in Hughes \[19\] which assumes linear isotropic elasticity.
Six different meshes at various levels of refinement are considered. Figure \[CanteliverBeam\] shows the coarsest, with 128 elements, and the finest, with 1790 elements. For comparison, the same numerical simulations were performed using the standard constant strain triangle (CST) finite element. All the computations were carried out under plane strain conditions, with a Poisson’s ratio of 0.3. Figure \[PeakStat\] presents the results of the convergence study. The relative error between the numerical calculation and the exact solution is plotted as a function of the inverse of the square root of the number of elements ($N^{−1/2}$), which is proportional to the characteristic element size. The results for the total elastic energy and the tip displacement are shown in Figures \[PeakStat\]a for both DCM and Constant Strain Triangle (CST) finite element.
In the log-log plots of Figures \[PeakStat\]a, the slope of the line segments provide a measure of the convergence rate. For the strain energy, the average convergence rates for the DCM and CST, respectively, are 1.62 and 1.99, and for the tip displacement they are 2.1 and 2.02. The theoretical convergence rate for the CST is 2 for both strain energy and tip deflection. Although the convergence rates are comparable, the DCM outperforms CST in terms of accuracy. The CST error in both strain energy and tip deflection is one order of magnitude higher than the DCM error. In terms of energy, for example, the DCM error ranges from 0.38$\%$ to 0.06$\%$ (coarsest to finest mesh), whereas the CST error ranges from 10$\%$ to 0.8$\%$. It must be mentioned, however, that each node in the DCM has one degree of freedom, the rotation, more than its counterpart in the CST. This additional degree of freedom results in higher computational cost for DCM compared to classical FEM. The tip displacement and the strain energy errors are plotted versus total number of degrees of freedom for both DCM and FEM simulations in Figure \[PeakStat\]b in log-log axes. One can see that as the total number of DOFs increases, the error values decrease (DOFs axis is reversed). It can bee seen that for approximately equal number of elements, total number of DOFs for DCM is higher than CST. However, the accuracy remains higher than CST for the same number of DOFs.
Cohesive Fracture Propagation
=============================
The convergence study presented in the previous section demonstrates that the DCM performs very well in the elastic regime. However, the most attractive feature of this method is the ability of easily accommodating the displacement discontinuity associated with fracture without suffering from the typical shortcomings of the classical finite element method, the limitations of typical particle models, or the complexity and the high computational cost of advanced finite element formulations. In this section a simple isotropic damage model is introduced in the DCM framework in order to simulate the initiation and propagation of quasi-brittle fracture.
Formulation
-----------
According to the classical damage mechanics and the DCM formulation for elasticity presented above, in a damaged material the facet tractions $t_{fN}$ and $t_{fM}$ can be calculated as: $$\label{NTConstDamag}
t_{fN} = (1-D_f) \bigg[ \frac{E_V-E_D}{3\alpha_v\Omega^e}\sum_f A_{f} w_{fN} + \frac{E_D w_{fN}}{\ell_f} \bigg]; ~~~~~ t_{fM} = (1-D_f) \bigg[ \frac{E_D w_{fM}}{\ell_f} \bigg]$$
where $D_f$ is the damage parameter related to the facet $f$. The evolution of the damage parameter is assumed to be governed by a history variable, the facet maximum effective strain, $\varepsilon_{f}^{max}$, characterizing the overall amount of straining that the material has been subject to during prior loading: $$\label{DamagPara}
D_f = 1 - \frac{\varepsilon_t}{\varepsilon_{f}^{max}} \text{exp} \bigg[\frac{-<\varepsilon_{f}^{max} - \varepsilon_t>}{\varepsilon_{fF}} \bigg]$$ where $\left\langle{x}\right\rangle = \text{max}(0,x)$, $\varepsilon_t$ is a material parameter representing the strain limit which governs the onset of damage, and $\varepsilon_{fF}$ governs the damage evolution rate. The maximum effective strain $\varepsilon_{f}^{max}$ that is used for each facet at each computational step is equal to the maximum principal strain $\varepsilon_{fI}$ that the facet has experienced through the loading process. This value is compared to $\varepsilon_t$ to distinguish facet elastic behavior from the the nonlinear case. $\varepsilon_{fI}$ is the maximum eigenvalue of the facet strain tensor whose components can be derived in terms of facet normal $\varepsilon_{fN}$, tangential $\varepsilon_{fM}$, and volumetric $\varepsilon_{V}$ strains as discussed in Section \[cantilever\] and presented in details in Appendix \[facet strain tensor\].
In order to ensure convergence upon mesh refinement and to avoid spurious mesh sensitivity, one can define $$\label{effmaxstr}
\varepsilon_{fF} = \frac{\varepsilon_t}{2}\bigg( \frac{\ell_t}{\ell_f} - 1 \bigg)$$ where $\ell_t = 2EG_t/\sigma_t^2$ is Hillerborg’s characteristic length, which is assumed to be a material parameter. $\sigma_t$ and $G_t$ are the elastic limit stress and fracture energy, respectively. In order to demonstrate the ability of DCM to simulate cohesive fracture with this simple two-parameter model, several different fracture analyses will be summarized in the following sections. Included will be examples of quasi-static fracture, dynamic crack propagation, and fragmentation.
Numerical results
-----------------
Multiple numerical tests are carried out to check the efficiency and robustness of the established framework. Quasi-Static and dynamic fracture simulations are performed.
Quasi-Static Fracture
---------------------
In this section, the simulation of direct tensile tests and three-point bending tests on notched specimens under quasi-static loading is carried out. The specimens are 120$\times$300 mm rectangular panels with out of plane thickness of 80 mm. The notch is one third of the panel depth, 40 mm, with a width equal to 12 mm. As shown in Figure \[NotchedSpecimen\], the notch tip is assumed to be semicircular in order to avoid unrealistic singularities in the stress distribution which might lead to premature crack initiation and propagation. For both the tensile and the three-point bending tests, the assumed material model parameters are: $\varepsilon_t = \sigma_t/E = 6.7 \times 10^{-5}$, tensile strength $\sigma_t = 2$ MPa, Young’s modulus $E = 30$ GPa; and material characteristic length $\ell_t = 0.7$ m, which corresponds to a fracture energy of approximately 47 J/m$^2$.
The direct tensile test is performed by constraining the boundary nodes on the left side of the specimen and by applying an increasing displacement to the nodes on the right side of the specimen, see Figure \[NotchedSpecimen\](a). In order to investigate the mesh size dependency of the solution, three different average element size in the zone of the notch of 4, 2, and 1 mm for the coarse, medium, and fine meshes, respectively, are considered and shown in Figure \[NotchedGeoms\].
Figure \[NotchedResults-DT\]b reports the nominal stress versus applied displacement curves for three different mesh resolutions. The nominal stress is defined as $\sigma_N = P/Db$, where $P$ is the overall applied load corresponding to a certain displacement. $D$ and $b$ are the specimen width and thickness, respectively. Cracked specimens with different mesh resolutions are plotted in Figure \[NotchedResults-DT\]a. The crack pattern shown in this figure is relevant to the applied displacement at the end of the loading process (0.06 mm, see Figure \[NotchedResults-DT\]b). The nodal displacements in Figure \[NotchedResults-DT\]a have been amplified by 50 to clearly visualize the crack that develops from the notch tip and towards the upper edge of the specimen. All curves reported in Figure \[NotchedResults-DT\]b have an initial elastic tangent followed by a nonlinear hardening branch up to a certain peak load. Afterwards, the behavior is characterized by a softening response with decreasing load carrying capacity for increasing end displacement. One can observe that the DCM responses for different mesh configurations matches very well which confirms convergence and mesh insensitivity of the DCM framework obtained with the simple regularization in Equation \[effmaxstr\].
Spurious mesh sensitivity which is characterized by mesh dependent energy dissipation and lack of convergence is the typical problem of the tensorial constitutive equations for softening materials when not properly regularized [@Bazant-CrackBandModel]. Classical numerical techniques such as finite element method that employs these type of constitutive equations result in more brittle response as a finer mesh is utilized. DCM solves this problem by introducing a length type variable $\ell_f$, the local distance between the two nodes, into the vectorial constitutive equations defined on each facet, Equations \[effmaxstr\]. Therefore, the constitutive equations for any generic facet vary with the edge length between the two corresponding nodes, which yields to a mesh independent global response for different mesh configurations. Fracture energy, energy consumed per unit area of the formed crack, is equal to the area under the curves in Figure \[NotchedResults-DT\](b), which means all the three simulations show the same fracture energy. The simulations were not carried out until the applied load dropped to zero, and so only partial estimates of the total energy consumption relevant to complete material separation can be made. For the performed DCM simulations, consumed energies are calculated as 45.7, 46.5, and 46.9 J/m$^2$ for the coarse, medium, and fine meshes, respectively. As noted above the material property used for all simulations was relevant to a fracture energy of $G_t$ $\approx$ 47 J/m$^2$. These positive results provide some quantitative evidence regarding convergence, and proper energy consumption independent of mesh refinement of DCM.
The configuration for the three-point bending test is shown in Figure \[NotchedSpecimen\](b). The bottom left corner of the specimen is constrained both vertically and horizontally, whereas the bottom right corner is constrained only in the vertical direction (classical pin-roller boundary conditions for simply supported beams). The load is applied by gradually increasing the displacement of a few boundary nodes located on the top side of the specimen, centered at the specimen axis of symmetry. Similar to the case of direct tension, the obtained results are reported in terms of nominal stress versus applied displacement for the three mesh resolutions.
Cracked specimens at the end of the loading process (0.12 mm, see Figure \[NotchedResults-3pbt\](b)) are depicted in Figure \[NotchedResults-3pbt\](a). Due to the bending character of these simulations, the stress profile and the crack opening displacement along the crack are not uniform. Nominal stress versus the applied displacement for the three simulations are plotted in Figure \[NotchedResults-3pbt\](b). The results display once again that the DCM solution is convergent upon mesh refinement and spurious mesh sensitivity does not take place.
Dynamic Fracture and Fragmentation
----------------------------------
### Bar End-Spalling Fragmentation Test {#end-spalling}
An end-spalling fragmentation experiment is carried out on a two dimensional bar of 1 mm height $\times$ 10 mm length and 1 mm thickness under plane stress condition. The bar is subjected to a sinusoidal velocity impulse with peak value of $60$ m/s and 1 $\mu$s duration time. In this study, the assumed material properties are: $\sigma_t = 844$ MPa, $E = 190$ GPa, material characteristic length $\ell_t = 0.012$ m which corresponds to a fracture energy of 2.2 $\times$ $10^4$ J/m$^2$, material density $\rho = 8000$ $\text{kg/m}^\text{3}$, and the Poisson’s ratio $\nu = 0.30$. The horizontal velocity component of the nodes located vertically at mid-height of the bar is plotted along the bar (Figure \[Bar-Sinus-velTime\]). One can see that the velocity impulse travels undisturbed across the bar, see time steps 1, 1.5, and 2 $\mu$s, while the magnitude of the wave doubles as it reaches the right end due to the interaction with the free-end boundary condition, see time step 2.5 $\mu$s in Figure \[Bar-Sinus-velTime\]. One can calculate that the velocity impulse moves across the specimen at an approximate velocity of 5000 m/s, which corresponds closely to the basic equation for one dimensional wave propagation velocity $v = \sqrt{E/\rho}$ which results in a velocity of 4874 m/s. In terms of stress, the traveling wave applies a compressive stress on the bar before arriving at the free-end, where the stress reverses from compression to tension. The generated tensile stress overcomes the tensile strength of the material, and end-spalling fragmentation begins. One can see that the wave moving back through the bar is no more sinusoidal which is due to the engendered material nonlinearity, see time steps 2.75 and 3.5 $\mu$s. More detailed analysis of the problem under consideration reveals that actually a bi-axial strain state is generated through the specimen due to the Poisson’s effect results in the presence of lateral straining. In turn, this leads to inclined principal strain directions and, consequently, inclined cracks (see Figure \[crack-bar-impulse\]) since crack initiation and propagation are simulated with the strain dependent damage model discussed earlier.
Figure \[crack-bar-impulse\] shows the fracture pattern of the bar at different time steps namely 3.6, 4, and 4.4 $\mu$s. One can see that a localized fracture takes place at the bar end and evolves into total separation of the right end once the fracture energy of the material is completely overcome. Contour of the damage parameter $D_f$ is also illustrated in Figure \[crack-edge-impulse\], which confirms the fracture pattern occurred at the bar end. In addition, one can notice that the crack propagates vertically at the center of the bar, while it deviates as it moves towards the cross section edges. This can be explained by the fact that the bi-axial effect is more pronounced over the areas away from the center of the bar. In Figure \[bar-eps-ratio\], maximum effective strain experienced by each facet $\varepsilon_{f}^{max}$ normalized by $\varepsilon_t = \sigma_t/E = 4.4 \times 10^{-3}$ is plotted at different time instants. 1.35, 1.8, and 2.25 $\mu s$ are instants during which the compressive wave travels through the bar before reaching the free-end, while 2.7, 3.6, and 4.5 $\mu s$ are after the signal reaches the free-end and leads to a tensile wave. One can see that at 1.35, 1.8, and 2.25 $\mu s$, $\varepsilon_{f}^{max}/\varepsilon_t < 1$ for all facets, which implies that all facets stay in the elastic regime. At 2.7 $\mu s$, which is just after the signal reaches the free-end, and the compressive wave converts into tensile one, $\varepsilon_{f}^{max}/\varepsilon_t > 1$ at the end of the bar where nonlinearity starts to develop. At 3.6 and 4.5 $\mu s$, damage localizes, and $\varepsilon_{f}^{max}/\varepsilon_t$ contour corresponds to the specimen fracture pattern depicted in Figure \[crack-bar-impulse\].
Ratio of the horizontal component facet stress tensor $\sigma_{f,11}$ to $\sigma_t$ is plotted in Figure \[bar-sig-ratio\] at the same time instants as the ones considered in Figure \[bar-eps-ratio\]. One can clearly see the propagation of the compressive wave through the specimen at 1.35, 1.8, and 2.25 $\mu s$, and its conversion to tensile wave at 2.7 $\mu s$. At 4.5 $\mu s$, it can be observed that the stress value on the facets around which fracture takes place is approximately zero, which corresponds to the bar splitting type of failure pattern.
### Edge Cracked Plate under Velocity Impulse
In this section, a classical dynamic crack propagation test is simulated. The reference experimental data is relevant to maraging steel [@Kalthoff-1], which shows high tensile strength and brittle behavior when subjected to high strain rate. A schematic representation of the test configuration is shown in Figure \[crack-edge-impulse\], in which one can see a projectile impacting the central part of an unrestrained double notched specimen. The plate has a 10 mm out-of-plane thickness. Plane stress condition can be assumed for the DCM analysis. By using the symmetry of the problem, half of the specimen is modeled and appropriate boundary conditions, horizontal nodal displacement and nodal rotation equal to zero, are enforced on the line of symmetry. Kalthoff and Wrinkler [@Kalthoff-1] investigated the effect of the projectile velocity on the failure mechanism: a brittle failure with a crack at an angle of $-70^{\circ}$ was observed for the case of low impact velocity (32 m/s), see Figure \[crack-edge-impulse\]. In the current example, a velocity of 16 m/s is applied at the impacted nodes, and this impulse is kept constant to the end of the simulation. The velocity of 16 m/s is selected because the elastic impedance of the projectile and the specimen are considered to be equal. Material properties considered in the DCM simulations are the same as the ones used in Section \[end-spalling\].
Belytschko et al. [@Belytschko-1] simulated this experiment using the continuum based model XFEM for quasi-brittle fracture, which is considered here as reference to discuss the DCM performance. To investigate the mesh dependency of the DCM results, a fine and a coarse mesh with element edge of $\sim$0.65 mm (50573 elements) and $\sim$1 mm (22437 elements) are considered. The initial vertical notch is simulated with 1.5 mm width, and the time step used in the explicit integration scheme is 0.02 $\mu$s. The velocity impulse applied to the DCM boundary nodes generates a compressive wave in the central part of the specimen, which propagates until it reaches the notch tip. At this point, significant shear strains develop leading to high principal tensile strains and crack initiation at the left side of the notch tip. Subsequently, crack propagates towards the left boundary of the specimen.
Figure \[ECPF-Frac\] shows crack initiation and propagation in different time steps for the fine mesh case. The average crack propagation angle with the horizontal axis at the time step 56 $\mu$s is $69^\circ$ which compares very well with experimental result $70^\circ$. At this time, a localized damage which leads to fracture takes place on the top right boundary of the specimen, and the generated crack propagates towards the notch tip. This is due to the reflection of the compressive wave from the top right boundary and is also reported by Belytschko et al. [@Belytschko-1] in their XFEM simulations. The propagating crack tends to become horizontal at the end of the simulation, see Figure \[ECPF-Frac\]e, as the localized fracture occurs and propagates on the top right boundary. This can also be related to the strain based failure criteria employed in DCM model and the simple damage model used in constitutive behavior of the material. Damage coefficient $D_f$ contours of the fine mesh simulation are plotted in Figure \[ECPF-Damage\], which clearly shows two highly localized damaged areas corresponding to the fracture pattern depicted in Figure \[ECPF-Frac\].
Figures \[ECPC-Frac\] and \[ECPC-Damage\] shows the fracture pattern and damage coefficient contour at different time steps for the coarse mesh simulations, respectively, which agrees well with the fine mesh results. Average crack propagation angle with the horizontal axis at the time step 56 $\mu$s is $68^\circ$, and the crack tends to propagate horizontally as fracture occurs and develops from the top right boundary. DCM performs more accurately compared to the XFEM [@Belytschko-1] in terms of predicting the crack propagation angle. However, the crack does not develop on the same path to the end of the test, which is captured by the XFEM simulations. In addition, DCM is able to capture micro cracks developing from the main crack faces, as it can be seen in Figures \[ECPF-Damage\] and \[ECPC-Damage\], while this is not captured by other approaches. It is also worth noting that the computational cost of methods like XFEM increases as the crack propagates because additional DOFs must be inserted to capture the displacement discontinuity. This is not the case for DCM which is characterized by the same number of DOFs in the elastic and fracturing regimes.
### Dynamic Crack Branching
A final benchmark fracture problem simulated by DCM in this section involves dynamic crack propagation and crack branching. Figure \[Branch-Geom\] shows a schematic representation of the test configuration. A pre-notched rectangular panel is subjected to a uniform traction applied as a step function on the two edges parallel to the notch. This experiment has been simulated computationally by other authors [@Song-1; @Xu-1], and related experimental results were reported by different researchers [@Ramulu-1; @Ravi-Chandar-1]. Ramulu and Kobayashi [@Ramulu-1] observed experimentally that a major crack starts to propagate from the notch tip to the right, which branches into two cracks at a certain point during the experiment, see Figure \[Branch-Geom\] for the sketch of the experimental result. The DCM parameters used in this test are: $\sigma_t = 3.1$ MPa, $E = 32$ GPa; material characteristic length $\ell_t = 0.02$ m, which corresponds to a fracture energy of approximately 3 J/m$^2$; material density $\rho = 2500$ $\text{kg/m}^\text{3}$; Poisson’s ratio $\nu = 0.2$. The applied traction is $\sigma_0 = 1$ MPa.
Crack initiation and propagation at different time steps of the DCM simulation is illustrated in Figure \[Branch-Damage\](a-e) through the damage parameter contours. One can see that the crack starts to propagate from the notch tip parallel to the symmetry axis of the configuration on a straight path for a short distance, and it branches into two cracks subsequently. The deformed configuration of the DCM simulation is plotted in Figure \[Branch-Damage\]f, which agrees well with the experimental results. Experimental observations also report that before the main branching occurs, minor branches emerge from the main crack but only propagate on a short distance [@Ramulu-1]. DCM is able to capture these minor branches which can be seen in the damage variable contours in Figure \[Branch-Damage\].
DCM Versus Particle Methods
===========================
DCM and classical particle models are basically governed by the same set of algebraic equations expressing compatibility and equilibrium. This naturally follows from the adoption of rigid body kinematics which is common to the two approaches. The difference between the two methods lies in the formulation of the constitutive law, namely in the relationship between facet tractions $\bold{t}_f$ and facet openings $\bold{w}_f$. For classical particle models, in which rigid particles are connected by springs, this relationship can be expressed as $t_{fN} = E_N w_{fN}/\ell_f$ and $t_{fM} = E_T w_{fM}/\ell_f$, where $E_N$ and $E_T$ are the normal and tangential elastic stiffnesses, respectively. DCM constitutive laws, Equation \[NTConstDamag\], are consistent to the ones for particle models if one sets $E_V = E_D$ and $E_T = E_D$. These conditions correspond to an elastic material with zero Poisson’s ratio [@Bolander-1]. By properly setting the ratio between the normal and tangential stiffnesses, particle models can simulate an “average” non-zero Poisson’s  ratio (average in the sense that Poisson’s ratio is defined by analyzing a finite, as opposed to an infinitesimal, volume of material). In this case, however, particle models feature an intrinsic heterogeneous response even for load configurations that produce uniform strain fields according to continuum theory [@Bolander-1]. In conclusion, for non-zero Poisson’s ratio the two formulations are fundamentally different and the key difference is that DCM accounts for the orthogonality of the deviatoric and volumetric deformation modes while classical particle models do not.
It must be mentioned here that the heterogeneous response of particle models is not necessarily a negative property and, actually, it is critical for their ability to handle automatically strain localization and crack initiation [@cusatis-ldpm-1; @cusatis-ldpm-2]. It must be kept in mind, however, that in this case the size of the discretization cannot be user-defined but must be linked to the actual size of the material heterogeneity. Only under this condition can one consider the heterogeneous response of particle models to be a representation of the actual internal behavior of the material rather than a spurious numerical artifact.
Conclusions
===========
In this paper, the formulation of the Discontinuous Cell Method (DCM) has been outlined. A convergence study in the elastic regime shows that DCM converges to the exact continuum solution with a convergence rate that is comparable to that of constant strain finite elements, but with accuracy that is one order of magnitude higher. In addition, numerical simulations show that DCM, with a simple two parameter isotropic damage model, can simulate cohesive fracture propagation without the drawbacks of standard finite elements, such as spurious mesh sensitivity, and without the complications of most recently formulated computational techniques. In addition, DCM successfully simulated the crack branching which is observed in the experiment of a benchmark problem. Finally, DCM can simulate the transition from localized fracture to fragmentation without mesh entanglement typical of finite element approaches.
**ACKNOWLEDGMENTS**\
This material is based upon work supported by the National Science Foundation under grant no. CMMI-1435923.
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Facet Strain Tensor Calculation {#facet strain tensor}
===============================
Derivation of facet strain tensor components $\varepsilon_{ij}$ in terms of facet normal $\varepsilon_{fN}$, tangential $\varepsilon_{fM}$ and volumetric strains $\varepsilon_{V}$ are explained in this section for plane strain and stress problems. In both cases, one should solve a system of three algebraic equations with three unknowns.
Plane Strain Problem
--------------------
In plane strain problems, out of plane strain component $\varepsilon_{33} = 0$, and one should consider following system of equations $$\begin{aligned}
\label{PEstrainTens-1}
\begin{split}
&\varepsilon_{V} = \frac{1}{3}\varepsilon_{ii} = \frac{1}{3}(\varepsilon_{11} + \varepsilon_{22}) \\
&\varepsilon_{fN} = N_{ij} \varepsilon_{ij} = N_{11}\varepsilon_{11} + 2 N_{12}\varepsilon_{12} + N_{22}\varepsilon_{22} \\
&\varepsilon_{fM} = M_{ij} \varepsilon_{ij} = M_{11}\varepsilon_{11} + 2 M_{12}\varepsilon_{12} + M_{22}\varepsilon_{22}
\end{split}\end{aligned}$$
where $N_{ij} = n_i n_j$ and $M_{ij} = n_i m_j$ are the two projection tensors that are calculated for each facet using its unit normal $n_i$ and tangential $m_i$ vectors, subscript $f$ is dropped for simplicity. Solution of the above system of equations will yield to the following expressions for strain tensor components: $$\begin{aligned}
\label{PEstrainTens-2}
\begin{split}
&\varepsilon_{11} = \frac{\varepsilon_{M} N_{12} - \varepsilon_{N} M_{12} + 3 \varepsilon_{V}(M_{12}N_{22} - N_{12}M_{22})} {M_{11}N_{12} + M_{12}N_{22} - M_{12}N_{11} - M_{22}N_{12}} \\
\\
&\varepsilon_{12} = \frac{\varepsilon_{N} (M_{11} - M_{22}) + \varepsilon_{M} (N_{22} - N_{11}) - 3 \varepsilon_{V} (M_{11}N_{22} - N_{11}M_{22})} {2(M_{11}N_{12} + M_{12}N_{22} - M_{12}N_{11} - M_{22}N_{12})} \\
\\
&\varepsilon_{22} = \frac{\varepsilon_{N} M_{12} - \varepsilon_{M} N_{12} + 3 \varepsilon_{V}(M_{11}N_{12} - N_{11}M_{12})} {M_{11}N_{12} + M_{12}N_{22} - M_{12}N_{11} - M_{22}N_{12}}
\end{split}\end{aligned}$$
These quantities are then used to calculate the strain tensor eigenvalues.
Plane Stress Problem
--------------------
for the case of plane stress problems, out of plane strain component $\varepsilon_{33} = -\nu(\varepsilon_{11}+\varepsilon_{22})/(1-\nu)$ should be taken into account. Therefore, the first equation in the system of equations \[PEstrainTens-1\] should be revised as $\varepsilon_{V} = \varepsilon_{ii}/3 = (\varepsilon_{11} + \varepsilon_{22} - \nu(\varepsilon_{11}+\varepsilon_{22})/(1-\nu))/3$, while the two other equations stay the same. $$\begin{aligned}
\label{PEstressTens-2}
\begin{split}
&\varepsilon_{11} = \frac{\varepsilon_{M} N_{12} - \varepsilon_{N} M_{12} + 3(1-\nu)/(1-2\nu) \varepsilon_{V}(M_{12}N_{22} - N_{12}M_{22})} {M_{11}N_{12} + M_{12}N_{22} - M_{12}N_{11} - M_{22}N_{12}} \\
\\
&\varepsilon_{12} = \frac{\varepsilon_{N} (M_{11} - M_{22}) + \varepsilon_{M} (N_{22} - N_{11}) - 3 (1-\nu)/(1-2\nu) \varepsilon_{V} (M_{11}N_{22} - N_{11}M_{22})} {2(M_{11}N_{12} + M_{12}N_{22} - M_{12}N_{11} - M_{22}N_{12})} \\
\\
&\varepsilon_{22} = \frac{\varepsilon_{N} M_{12} - \varepsilon_{M} N_{12} + 3(1-\nu)/(1-2\nu) \varepsilon_{V}(M_{11}N_{12} - N_{11}M_{12})} {M_{11}N_{12} + M_{12}N_{22} - M_{12}N_{11} - M_{22}N_{12}}
\end{split}\end{aligned}$$
[^1]: Corresponding author.\
E-mail address: [email protected]
|
---
abstract: 'We consider the evolution of an almost Hermitian metric by the $(1,1)$ part of its Chern-Ricci form on almost complex manifolds. This is an evolution equation first studied by Chu and coincides with the Chern-Ricci flow if the complex structure is integrable and with the Kähler-Ricci flow if moreover the initial metric is Kähler. We find the maximal existence time for the flow in term of the initial data and also give a convergence result. As an example, we study this flow on the (locally) homogeneous manifolds in more detail.'
address: 'Institut Fourier, Université Grenoble Alpes, 100 rue des maths, Gières 38610, France'
author:
- Tao Zheng
title: 'An almost complex Chern-Ricci flow'
---
[GBK]{}[song]{}
gobble<span style="font-variant:small-caps;"></span>
[^1]
Introduction
============
Let $(M,J)$ be a compact almost complex manifold (without boundary) with $\dim_{\mathbb{R}}M=2n$, where $J$ is the almost complex structure. Then assume that $g_0$ is an almost Hermitian metric on $(M,J)$, i.e., $g_0$ is a Riemannian metric satisfying $g_0(JX,JY)=g_0(X,Y)$ for any vector fields $X$ and $Y$. Associated to $g_0$ is a unique real $(1,1)$ form $\omega_0$ defined by $\omega_0(X,Y):=g_0(JX,Y)$ for any vector fields $X$ and $Y$ and vice versa. In what follows, we will not distinguish the two terms.
Since Hamilton [@Ha] introduced the Ricci flow, it has established many deep results in topological, smooth and Riemannian manifolds (see for example [@BS; @Ha2; @P1]). Next, we consider the parabolic flows of metrics on $M$ starting at $g_0$ which preserve the Hermitian condition and reveal the information about the structure of $M$ as a complex manifold. When $g_0$ is a Kähler metric, i.e., ${\mathrm{d}}\omega_0=0$, the Ricci flow does exactly this and it is called the Kähler-Ricci flow firstly introduced by Cao [@Ca]. The behavior of the Kähler-Ricci flow is deeply intertwined with the complex and algebro-gemetric properties of $M$ (see for example [@Ca; @CW; @csw; @FIK; @PSSW; @PS2; @ST; @ST2; @ST3; @SW; @SW2; @SW3; @SW4; @SY; @szkrf; @Ti; @TZ; @To1; @tosattikawa; @Ts; @Ts2; @weinkovekrf]).
If the Hermitian metric $g_0$ is not Kähler, then there are two types of Ricci curvature which are equal to each other when ${\mathrm{d}}\omega_0=0$. There are also two types of the evolution of the Hermitian metric. One is the Chern-Ricci flow which firstly introduced by Gill [@gill] when the first Bott-Chern class vanishes and is studied deeply by Tosatti and Weinkove (and Yang) [@twjdg; @twcomplexsurface; @twymathann]. The Chern-Ricci flow is a natural evolution equation on complex manifolds and its behavior reflects the underlying geometry (see also [@ftwz; @gillmmp; @gillscalar; @laurent; @lr; @niexiaolan; @yangxiaokui; @zhengtaocjm] and references therein). Another type of the evolution of Hermitian metric was introduced by Streets and Tian [@StT; @StT2; @StT3] (see also [@LY]) and was generalized to almost Hermitian manifolds by Vezzoni [@V].
Given $F\in C^{\infty}(M,\mathbb{R})$, Chu, Tosatti and Weinkove [@ctw] solved the following Monge-Ampère equation on almost Hermitian manifold $$\begin{aligned}
\label{ecma}
&(\omega_0+{\sqrt{-1}\partial\overline{\partial}}\varphi)^n= e^{F+b}\omega_0^n\\
&\omega_0+{\sqrt{-1}\partial\overline{\partial}}\varphi>0, \quad\sup\varphi=0,\nonumber\end{aligned}$$ for a unique $b\in\mathbb{R}$, where ${\sqrt{-1}\partial\overline{\partial}}\varphi=\frac{1}{2}\left({\mathrm{d}}J{\mathrm{d}}\varphi\right)^{(1,1)}$ (see in Section \[sec2\]). When $J$ is integrable and $\omega_0$ is Kähler, was solved by Yau [@Y] to confirm the Calabi conjecture. Tosatti and Weinkove [@TW2] solved for any dimension if $J$ is integrable (see also [@Ch; @TW1]).
In this paper, we consider the evolution equation for almost Hermitian forms suggested by Chu, Tosatti and Weinkove [@ctw] $$\begin{aligned}
\label{acrf}
\frac{\partial\omega_t}{\partial t}=-{\mathrm{Ric}^{(1,1)}}(\omega_t),\quad \omega(0)=\omega_0\end{aligned}$$ on almost Hermitian manifold $(M,\omega_0,J)$. Here for convenience, we call ${\mathrm{Ric}^{(1,1)}}(\omega)$ is the $(1,1)$ part of the Chern-Ricci form of $\omega$. This flow coincides exactly with the Chern-Ricci flow if $J$ is integrable, and the Kähler-Ricci flow if furthermore $\omega_0$ is Kähler.
Chu [@chu1607] firstly studied the flow in the case where there exists an almost Hermitian metric $\omega_0$ with ${\mathrm{Ric}^{(1,1)}}(\omega_0)={\sqrt{-1}\partial\overline{\partial}}F$ for some $F\in C^{\infty}(M,\mathbb{R})$ and proved that the solution to exists for all the time and converges to an almost Hermitian metric $\omega_{\infty}$ with ${\mathrm{Ric}^{(1,1)}}(\omega_{\infty})=0$ (see more details in Section \[sec7\]).
In this paper, we characterize the maximal existence time for a solution for the flow . For this aim, we rewrite the flow as $$\begin{aligned}
\frac{\partial\omega_t}{\partial t}=-{\mathrm{Ric}^{(1,1)}}(\omega_0)+{\sqrt{-1}\partial\overline{\partial}}\theta(t),\quad\mbox{with}\quad\theta(t)=\log\frac{\omega_t^n}{\omega_0^n}.\end{aligned}$$ Hence, as long as the flow exists, the solution $\omega_t$ starting at $\omega_0$ must be of the form $\omega_t=\alpha_t+{\sqrt{-1}\partial\overline{\partial}}\Upsilon(t)$ with $\frac{\partial \Upsilon}{\partial t}=\theta(t)$ and $\alpha_t=\omega_0-t{\mathrm{Ric}^{(1,1)}}(\omega_0)$. We define a number $T=T(\omega_0)$ with $0<T\leq \infty$ by $$\begin{aligned}
T:=\sup\Big\{t\geq0:\;\exists\, \phi\in C^{\infty}(M,\mathbb{R})\,\mbox{such that}\,\alpha_t+{\sqrt{-1}\partial\overline{\partial}}\phi>0\Big\}.\end{aligned}$$ Note that for any other Hermitian metric $\omega_0'=\omega_0+{\sqrt{-1}\partial\overline{\partial}}\psi>0$ with $\psi\in C^{\infty}(M,\mathbb{R})$, we have $T(\omega_0')=T(\omega_0)$. Indeed, yields that $$\begin{aligned}
\alpha_t+{\sqrt{-1}\partial\overline{\partial}}\phi
=&\omega_0-t{\mathrm{Ric}^{(1,1)}}(\omega_0')+{\sqrt{-1}\partial\overline{\partial}}\left(\phi+t\log\frac{\omega_0^n}{\omega_0'^n}\right)\\
=&\omega_0'-t{\mathrm{Ric}^{(1,1)}}(\omega_0')+{\sqrt{-1}\partial\overline{\partial}}\left(\phi-\psi+t\log\frac{\omega_0^n}{\omega_0'^n}\right),\end{aligned}$$ as required.
It is easy to see that a solution to cannot exist beyond time $T$. Indeed, we have
\[mainthm\] There exists a unique maximal solution to the flow on $[0,\,T)$.
In the special case when $J$ is integrable, this is already known by the result of Tian and Zhang [@TZ] who extended earlier work of Cao [@Ca] and Tsuji [@Ts; @Ts2] when $\omega_0$ is Kähler, and by the result of Tosatti and Weinkove [@twjdg] (see also [@twymathann]) when $\omega_0$ is even not Kähler.
We point out that the flow is equivalent to the scalar partial differential equation $$\begin{aligned}
\label{phit}
{\frac{\partial}{\partial t}}\phi(t)=\log \frac{(\alpha_t+{\sqrt{-1}\partial\overline{\partial}}\phi_t)^n}{\omega_0^n}, \quad \alpha_t+{\sqrt{-1}\partial\overline{\partial}}\phi_t>0,\quad\phi(0)=0,\end{aligned}$$ on the same time interval. Indeed, assume that $\phi_t$ is the solution to . We set $\omega_t=\alpha_t+{\sqrt{-1}\partial\overline{\partial}}\phi_t>0$. From , it follows that $$\begin{aligned}
{\frac{\partial}{\partial t}}\omega_t
={\frac{\partial}{\partial t}}(\alpha_t+{\sqrt{-1}\partial\overline{\partial}}\phi_t)
=-{\mathrm{Ric}^{(1,1)}}(\omega_0)-{\mathrm{Ric}^{(1,1)}}(\omega_t)+{\mathrm{Ric}^{(1,1)}}(\omega_0)=-{\mathrm{Ric}^{(1,1)}}(\omega_t),\end{aligned}$$ i.e., $\omega_t$ is the solution to the flow .
On the other hand, assume that $\omega_t$ is the solution to . We set $$\phi_t=\int_0^t\log \frac{\omega_s^n}{\omega_0^n}{\mathrm{d}}s$$ with $\phi(0)=0$. This, together with , yields $$\begin{aligned}
{\frac{\partial}{\partial t}}(\omega_t-\alpha_t-{\sqrt{-1}\partial\overline{\partial}}\phi_t)
=-{\mathrm{Ric}^{(1,1)}}(\omega_t)+{\mathrm{Ric}^{(1,1)}}(\omega_0)+{\mathrm{Ric}^{(1,1)}}(\omega_t)-{\mathrm{Ric}^{(1,1)}}(\omega_0)=0,\end{aligned}$$ with $(\omega_t-\alpha_t-{\sqrt{-1}\partial\overline{\partial}}\phi_t)|_{t=0}=0$, i.e., $\omega_t\equiv\alpha_t+{\sqrt{-1}\partial\overline{\partial}}\phi_t$. It follows that $\phi_t$ is the solution to the scalar partial differential equation .
Next, we consider a convergence result about the flow . Since there is no Bott-Chern cohomology group on almost complex manifold if $J$ is not integrable, the statement of this result may be slightly different.
\[thm2\] Assume that $(M,\omega_0,J)$ is an almost Hermitian manifold equipped a volume form $\Omega$ with ${\mathrm{Ric}^{(1,1)}}(\Omega)<0$. There exists an almost Hermitian metric $\omega_{\infty}$ with ${\mathrm{Ric}^{(1,1)}}(\omega_{\infty})=-\omega_{\infty}$, such that for any initial almost Hermitian metric $\omega_0$, the solution to exists for all the time and that $\omega(t)/t$ converge smoothly to $\omega_{\infty}$ as $t\rightarrow\infty$.
If the complex structure $J$ is integrable, then the existence of such volume form $\Omega$ is equivalent to the fact that $M$ is Kähler with negative first Chern class, and our theorem coincides precisely with [@twjdg Theorem 1.7], i.e., on Kähler manifold with negative first Chern class, the Chern-Ricci flow, starting at any initial Hermitian metric $\omega_0$, exists for all the time and $\omega(t)/t$ converge smoothly to the unique Kähler-Einstein metric $\omega_{\mathrm{KE}}$ on $M$.
The outline of the paper is as follows. In Section \[sec2\], we collect some preliminaries about almost complex geometry. In Section \[sec3\], we give the uniform priori estimates of the solution to and its time derivative. In Section \[sec4\] and Section \[sec5\], we give the first and second order estimates of the solution to respectively. In Section \[sec6\], we get the higher order estimates of the solution to and complete the proof of Theorem \[mainthm\]. In Section \[sec7\], we prove some convergence results including Theorem \[thm2\]. In Section \[secexample\], as an example, we study the flow on a compact almost Hermitian manifold $M$ whose universal cover is a Lie group $G$ such that if $\pi:G\longrightarrow M$ is the covering map, then $\pi^{\ast}\omega_0$ and $\pi^{\ast}J$ are left invariant in more detail.
[**Acknowledgements**]{} The author thanks Professor Valentino Tosatti for suggesting him this question and for many other invaluable conversations and directions. The author also thanks Professor Jean-Pierre Demailly and Professor Ben Weinkove for some useful comments. The author is also grateful to the anonymous referees and the editor for their careful reading and helpful suggestions which greatly improved the paper.
Preliminaries {#sec2}
=============
In this section, we collect some basic materials about almost Hermitian geometry (see for example [@kobayashi; @twyau]).
The Nijenhuis tensor on almost complex manifolds
------------------------------------------------
Let $(M,J)$ be an almost complex manifold with $\dim_{\mathbb{R}}M=2n$, where $J$ is the almost complex structure, i.e., $J\in \mathrm{End}(TM)$ with $J^2=-\mathrm{Id}_{TM}$. Here $TM$ is the tangent vector bundle of $M$. Denote by ${ \mathfrak{X}(M)}$ the set of all the sections of $TM$, i.e., the set of all the vector fields. Then we have $$TM\otimes_{\mathbb{R}}\mathbb{C}=T^{1,0}M\oplus T^{0,1}M,$$ where $$T^{1,0}M:=\Big\{X-{\sqrt{-1}}JX:\;\forall\;X\in TM\Big\}$$ and $$T^{0,1}M:=\Big\{X+{\sqrt{-1}}JX:\;\forall\; X\in TM\Big\}.$$ Denote by $\Lambda^1 M$ the dual of $TM$. We extend the Nijenhuis tensor $J$ to any $p$ form $\varpi$ by $$\begin{aligned}
\label{jkuozhang}
(J\varpi)(X_1,\cdots,X_p):=(-1)^p\varpi(JX_1,\cdots,JX_p),\quad \forall\;X_1,\cdots,X_p\in { \mathfrak{X}(M)}.\end{aligned}$$ It is easy to check that for any $(2n-p)$ form $\xi$ and $p$ form $\psi$, there holds $$\begin{aligned}
\xi\wedge(J\psi)=(-1)^p(J\xi)\wedge\psi.\end{aligned}$$ We also have $$\bigwedge^1 M\otimes_{\mathbb{R}}\mathbb{C}=\bigwedge^{1,0}M\oplus\bigwedge^{0,1}M,$$ where $$\bigwedge^{1,0}M:=\left\{\eta+{\sqrt{-1}}J\eta:\;\forall\; \eta\in \bigwedge^1M\right\}$$ and $$\bigwedge^{0,1}M:=\left\{\eta-{\sqrt{-1}}J\eta:\;\forall\; \eta\in \bigwedge^1M\right\}.$$ It is easy to see that $$\left(T^{1,0}M\right)^{\ast}=\bigwedge^{1,0}M,\quad \left(T^{0,1}M\right)^{\ast}=\bigwedge^{0,1}M.$$ We also denote that $$\bigwedge^{p,q}M:=\bigwedge^p\left(\bigwedge^{1,0}M\right)\otimes\bigwedge^q\left(\bigwedge^{0,1}M\right).$$ Then we have $$\bigwedge^pM\otimes_{\mathbb{R}}\mathbb{C}=\oplus_{r+s=p}\bigwedge^{r,s}M.$$ A $2$ form $\zeta$ is a $(1,1)$ form if and only if there holds $$\begin{aligned}
\label{11biaozhun}
\zeta(X,Y)=\zeta(JX,JY),\quad \forall\,X,Y\in{ \mathfrak{X}(M)}.\end{aligned}$$ Indeed, $\zeta$ is a $(1,1)$ form if and only if $$\zeta(X+{\sqrt{-1}}JX,Y+{\sqrt{-1}}JY)=\zeta(X-{\sqrt{-1}}JX,Y-{\sqrt{-1}}JY)=0,\quad \forall\,X,Y\in{ \mathfrak{X}(M)},$$ as required. Therefore, the $(1,1)$ part of any $2$ form $\xi$, denoted by $\xi^{(1,1)}$, can be given by $$\begin{aligned}
\label{11part}
\xi^{(1,1)}(X,Y)=\frac{1}{2}\left(\xi(X,Y)+\xi(JX,JY)\right),\quad\forall\, X,Y\in{ \mathfrak{X}(M)}.\end{aligned}$$ We also define $$\begin{aligned}
\xi^{\mathrm{ac}}(X,Y):=\xi(X,Y)-\xi^{(1,1)}(X,Y)=\frac{1}{2}\left(\xi(X,Y)-\xi(JX,JY)\right),\quad\forall\, X,Y\in{ \mathfrak{X}(M)}.\end{aligned}$$
For any $X,\,Y\in { \mathfrak{X}(M)}$, Nijenhuis tensor ${\mathcal{N}}$ is defined by $$\begin{aligned}
4\label{torsion}
{\mathcal{N}}(X,\,Y)=[JX,\,JY]-J[JX,\,Y]-J[X,\,JY]-[X,\,Y].\end{aligned}$$ If ${\mathcal{N}}=0$, then the Nijenhuis tensor is called integrable.
For the Nijenhuis tensor, a direct calculation yields
\[lemn\] The Nijenhuis tensor ${\mathcal{N}}$ satisfies $$\begin{aligned}
{\mathcal{N}}(X,\,Y)=&-{\mathcal{N}}( Y,\,X),\quad {\mathcal{N}}(JX,\,Y)=-J{\mathcal{N}}(X,\,Y),\\
{\mathcal{N}}(X,\,JY)=&-J{\mathcal{N}}(X,\,Y),\quad {\mathcal{N}}(JX,\,JY)=-{\mathcal{N}}(X,\,Y).\end{aligned}$$ Moreover, ${\mathcal{N}}(JX,\,Y)={\mathcal{N}}(X,\,JY)$.
By Lemma \[lemn\], for any $(1,0)$ vector fields $W$ and $V$, it is easy to get $$\begin{aligned}
{\mathcal{N}}(V,\,W)=-[V,\,W]^{(0,1)},\quad {\mathcal{N}}(V,\,\overline{W})={\mathcal{N}}(\overline{V},\,W)=0\end{aligned}$$ and $$\begin{aligned}
{\mathcal{N}}(\overline{V},\,\overline{W})=-[\overline{V},\,\overline{W}]^{(1,0)}=-\overline{[V,\,W]^{(0,1)}}.\end{aligned}$$ Let $e_1,\cdots, e_n$ be the basis of $T^{1,0}M$ and $\theta^1,\cdots,\theta^n$ be the dual basis of $\Lambda^{1,0}M$, i.e., $$\theta^i(e_j)=\delta^i_j,\quad i,j=1,\cdots,n.$$ Then we have $$\begin{aligned}
{\mathcal{N}}(\overline{e}_i,\overline{e}_j)=&-[\overline{e}_i,\overline{e}_j]^{(1,0)}=:N_{\overline{i}\overline{j}}^ke_k,\\
{\mathcal{N}}(e_i,e_j)=&-[e_i,e_j]^{(0,1)}=\overline{N_{\overline{i}\overline{j}}^k}\overline{e}_k,\end{aligned}$$ which implies $$\begin{aligned}
{\mathcal{N}}=\frac{1}{2}\overline{N_{\overline{i}\overline{j}}^k}\overline{e}_k\otimes\left(\theta^i\wedge\theta^j\right)+
\frac{1}{2}N_{\overline{i}\overline{j}}^ke_k\otimes\left(\overline{\theta}^i\wedge\overline{\theta}^j\right).\end{aligned}$$ Denote the structure coefficients of Lie bracket by $$\begin{aligned}
[e_i,e_j]=:&C_{ij}^ke_k-\overline{N_{\overline{i}\overline{j}}^k}\overline{e}_k,\\
[e_i,\overline{e}_j]=:&C_{i\overline{j}}^ke_k+C_{i\overline{j}}^{\overline{k}}\overline{e}_k,\\
[\overline{e}_i,\overline{e}_j]=:&-N_{\overline{i}\overline{j}}^ke_k+\overline{C_{ij}^k}\overline{e}_k.\end{aligned}$$ A direct calculation yields $$\begin{aligned}
\label{dtheta}
{\mathrm{d}}\theta^i=-\frac{1}{2}C_{k\ell}^i\theta^k\wedge\theta^{\ell}
-C_{k\overline{\ell}}^i\theta^k\wedge\overline{\theta}^{\ell}
+\frac{1}{2}N_{\overline{k}\overline{\ell}}^i\overline{\theta}^k\wedge\overline{\theta}^{\ell}.\end{aligned}$$ From , we can split the exterior differential operator, ${\mathrm{d}}: \Lambda^{\bullet}M\otimes_{\mathbb{R}}\mathbb{C}\longrightarrow \Lambda^{\bullet+1}M\otimes_{\mathbb{R}}\mathbb{C}$, into four components (see for example [@angella]) $${\mathrm{d}}=A+\partial+\overline{\partial}+\overline{A}$$ with $$\begin{aligned}
A:\;&\Lambda^{\bullet,\bullet} M\longrightarrow \Lambda^{\bullet+2,\bullet-1} M\\
\partial:\;&\Lambda^{\bullet,\bullet} M\longrightarrow \Lambda^{\bullet+1,\bullet} M\\
\overline{\partial}:\;&\Lambda^{\bullet,\bullet} M\longrightarrow \Lambda^{\bullet,\bullet+1} M\\
\overline{A}:\;&\Lambda^{\bullet,\bullet} M\longrightarrow \Lambda^{\bullet-1,\bullet+2} M.\end{aligned}$$ In terms of these components, the condition ${\mathrm{d}}^2=0$ can be written as $$\begin{aligned}
A^2=&0,\\
\partial A+A\partial=&0,\\
A\overline{\partial}+\partial^2+\overline{\partial}A=&0,\\
A\overline{A}+\partial\overline{\partial}+\overline{\partial}\partial+\overline{A}A=&0,\\
\partial\overline{A}+\overline{\partial}^2+\overline{A}\partial=&0,\\
\overline{A}\overline{\partial}+\overline{\partial}\overline{A}=&0,\\
\overline{A}^2=&0.\end{aligned}$$ For any $p$ form $\varpi$, there holds $$\begin{aligned}
\label{waiweifen}
({\mathrm{d}}\varpi)(X_1,\cdots,X_{p+1})
=&\sum\limits_{\lambda=1}^{p+1}(-1)^{\lambda+1}X_{\lambda}(\varpi(X_1,\cdots,\widehat{X_{\lambda}},\cdots,X_{p+1}))\\
&+\sum\limits_{\lambda<\mu}(-1)^{\lambda+\mu}\varpi([X_{\lambda},X_{\mu}],X_1,\cdots,\widehat{X_{\lambda}},\cdots,\widehat{X_{\mu}},\cdots,X_{p+1})\nonumber\end{aligned}$$ for any fields $X_1,\cdots,X_{p+1}\in { \mathfrak{X}(M)}$. For any $\varphi\in C^{\infty}(M,\,\mathbb{R})$, from and , a direct computation yields $$\begin{aligned}
\label{djdvarphi20}
({\mathrm{d}}J{\mathrm{d}}\varphi)(e_i,e_j)
=&-2{\sqrt{-1}}[e_i,e_j]^{(0,1)}(\varphi),\\
\label{djdvarphi11}
({\mathrm{d}}J{\mathrm{d}}\varphi)(\overline{e}_i,\overline{e}_j)
=&2{\sqrt{-1}}[\overline{e}_i,\overline{e}_j]^{(1,0)}(\varphi),\\
\label{djdvarphi02}
({\mathrm{d}}J{\mathrm{d}}\varphi)(e_i,\overline{e}_j)
=&2{\sqrt{-1}}\left(e_i\overline{e}_j(\varphi)- [e_i,e_j]^{(0,1)}(\varphi)\right).\end{aligned}$$ A direct calculation shows that $$\begin{aligned}
\label{ddbarvarphi}
{\sqrt{-1}\partial\overline{\partial}}\varphi=\frac{1}{2}({\mathrm{d}}J{\mathrm{d}}\varphi)^{(1,1)} ={\sqrt{-1}}\left(e_i\overline{e}_j(\varphi)-[e_i,\overline{e}_j]^{(0,1)}(\varphi)\right)\theta^i\wedge\overline{\theta}^j.\end{aligned}$$ Thanks to and , we get that $$\begin{aligned}
({\mathrm{d}}J{\mathrm{d}}\varphi)(X,Y)-({\mathrm{d}}J{\mathrm{d}}\varphi)(JX,JY)=-4\left(J\circ{\mathcal{N}}(X,Y)\right)(\varphi), \quad \forall\, X,\,Y\in { \mathfrak{X}(M)},\end{aligned}$$ which, together with , yields that ${\mathrm{d}}J{\mathrm{d}}\varphi$ is a real $(1,1)$ form if and only if $J$ is integrable.
For any real $(1,1)$ form $\eta={\sqrt{-1}}\eta_{i\overline{j}}\theta^i\wedge\overline{\theta}^j$, combining and gives $$\begin{aligned}
\label{dbareta}
\overline{\partial}\eta=\frac{{\sqrt{-1}}}{2}\left(\overline{e}_j\eta_{k\overline{i}}-\overline{e}_i\eta_{k\overline{j}}
-C_{k\overline{i}}^p\eta_{p\overline{j}}
+C_{k\overline{j}}^{p}\eta_{p\overline{i}}
+\overline{C_{ij}^q}\eta_{k\overline{q}}\right)\theta^k\wedge\overline{\theta}^i\wedge\overline{\theta}^j.\end{aligned}$$
The almost complex connection on almost complex manifolds
---------------------------------------------------------
A connection $D$ is called almost complex if $DJ=0$, which means that for any $X,\,Y\in { \mathfrak{X}(M)}$, we have $$\begin{aligned}
\label{dj}
0=(D_XJ)(Y)=D_X(J(Y))-J(D_XY).\end{aligned}$$ Therefore, we can define Christoffel symbols by $$D_{e_{\alpha}}e_j=\Gamma_{\alpha j}^ke_k,\quad D_{e_{\alpha}}\overline{e}_j=\Gamma_{\alpha\overline{j}}^{\overline{k}}\overline{e}_k,\;
\alpha \in\{1,\cdots,n,\overline{1},\cdots,\overline{n}\},\quad i,\,j,\,k\in\{1,\cdots,n\}.$$ Here we also use the notation $e_{\overline{i}}=\overline{e}_i$. The connection form $(\omega_j^i)$ is defined by $\omega_{j}^i:=\Gamma_{kj}^i\theta^k+\Gamma_{\overline{k}j}^i\overline{\theta}^k$. Then the torsion $\Theta=(\Theta^i)$ is defined by $$\Theta^i:={\mathrm{d}}\theta^i-\theta^p\wedge\omega^i_p.$$ This implies that $$\begin{aligned}
\Theta^i(e_j,e_k)=&\Gamma_{jk}^i-\Gamma_{kj}^i-C_{jk}^i=:T_{jk}^i,\nonumber\\
\label{canonicald}\Theta^i(e_j,\overline{e}_k)=&-\Gamma_{\overline{k}j}^i-C_{j\overline{k}}^i=:S_{j\overline{k}}^i,\\
\Theta^i(\overline{e}_j,\overline{e}_k)=&-{\mathrm{d}}\theta^i([\overline{e}_j,\overline{e}_k])=N_{\overline{j}\overline{k}}^i.\nonumber\end{aligned}$$ The torsion $\Theta=(\Theta^i)$ can be split into three parts $$\begin{aligned}
\Theta=\Theta^{2,0}+\Theta^{1,1}+\Theta^{0,2},\end{aligned}$$ where $$\begin{aligned}
\Theta^{2,0}=&\left(\frac{1}{2}T_{jk}^i\theta^j\wedge\theta^k\right)_{1\leq i\leq n},\\
\Theta^{1,1}=&\left(S_{j\overline{k}}^i\theta^j\wedge\overline{\theta}^k\right)_{1\leq i\leq n},\\
\Theta^{0,2}=&\left(\frac{1}{2}N_{\overline{j}\overline{k}}^i\overline{\theta}^j\wedge\overline{\theta}^k\right)_{1\leq i\leq n}.\end{aligned}$$ It follows that the $(0,2)$ part of an almost complex connection is uniquely determined by the Nijenhuis tensor (see for example [@kobayashi Theorem 1.1] and [@twyau]).
The curvature form is defined by $$\begin{aligned}
\Omega_{j}^{i}={\mathrm{d}}\omega_{j}^{i}+\omega_{k}^{i}\wedge\omega_{j}^{k}.\end{aligned}$$ We define $$\begin{aligned}
\label{rklji}
R_{k \ell j}{}^i
:=&\Omega_{j}^{i}(e_k,e_{\ell})\\
=&e_k(\Gamma_{\ell j}^i)-e_{\ell}(\Gamma_{k j}^i)-C_{k\ell}^p\Gamma_{pj}^i+\overline{N_{\overline{k}\overline{\ell}}^p}\Gamma_{\overline{p}j}^i
+\Gamma_{kp}^i\Gamma_{\ell j}^p-\Gamma_{\ell p}^i\Gamma_{kj}^p,\nonumber\end{aligned}$$ $$\begin{aligned}
\label{riccikbarl}
R_{k \overline{\ell} j}{}^i
:=&\Omega_{j}^{i}(e_k,\overline{e}_{\ell})\\
=&e_k(\Gamma_{\overline{\ell} j}^i)-\overline{e}_{\ell}(\Gamma_{k j}^i)-C_{k\overline{\ell}}^p\Gamma_{pj}^i-C_{k\overline{\ell}}^{\overline{p}}\Gamma_{\overline{p}j}^i
+\Gamma_{kp}^i\Gamma_{\overline{\ell} j}^p-\Gamma_{\overline{\ell} p}^i\Gamma_{kj}^p,\nonumber\end{aligned}$$ and $$\begin{aligned}
\label{rkbarlbarji}
R_{\overline{k} \overline{\ell} j}{}^i
:=&\Omega_{j}^{i}(\overline{e}_k,\overline{e}_{\ell})\\
=&\overline{e}_k(\Gamma_{\overline{\ell} j}^i)-\overline{e}_{\ell}(\Gamma_{\overline{k} j}^i)+N_{\overline{k}\overline{\ell}}^p\Gamma_{pj}^i-\overline{C_{k\ell}^{p}}\Gamma_{\overline{p}j}^i
+\Gamma_{\overline{k}p}^i\Gamma_{\overline{\ell} j}^p-\Gamma_{\overline{\ell} p}^i\Gamma_{\overline{k}j}^p.\nonumber\end{aligned}$$ We can write $\Omega=\left(\Omega_j^i\right)=\Omega^{(2,0)}+\Omega^{(1,1)}+\Omega^{(0,2)}$, with $$\begin{aligned}
\Omega^{(2,0)}=&\left(\frac{1}{2}R_{k\ell j}{}^i\theta^k\wedge\theta^{\ell}\right),\\
\Omega^{(1,1)}=&\left(R_{k\overline{\ell}j}{}^i\theta^k\wedge\overline{\theta}^{\ell}\right),\\
\Omega^{(0,2)}=&\left(\frac{1}{2}R_{\overline{k}\overline{\ell} j}{}^i\overline{\theta}^k\wedge\overline{\theta}^{\ell}\right).\end{aligned}$$ Then the Chern-Ricci form is $({\sqrt{-1}}\Omega_{i}^i)\in 2\pi c_1(M,J)\in H^2(M,\mathbb{R})$ by the Chern-Weil theory (see for example [@kn2 Chapter 12]), where $c_1(M,J)$ is the first Chern class of $(M,J)$.
The canonical connection on almost Hermitian manifolds
------------------------------------------------------
Assume that $(M,\,g,\,J)$ is an almost Hermitian manifold with $\dim_{\mathbb{R}}M=2n$, where a Riemannian metric $g$ is called a Hermitian metric if it satisfies $g(X,\,Y)=g(JX,\,JY),\;\forall\; X,\,Y\in { \mathfrak{X}(M)}$. We also denote the Hermitian metric $g$ by $\langle\cdot,\,\cdot\rangle$. This metric uniquely defines a real $(1,1)$ form $\omega(\cdot,\,\cdot)=g(J\cdot,\, \cdot)$ and vice versa. An affine connection $D$ on $TM$ is called almost Hermitian connection if $D g=D J=0$. For any $X,\,Y,\,Z\in { \mathfrak{X}(M)}$, we have $$\begin{aligned}
\label{dg}
0=(D_Xg)(Y,Z)=X\langle Y,\,Z\rangle-\langle D_XY,\,Z\rangle-\langle Y,\,D_XZ\rangle.\end{aligned}$$ For the almost Hermitian connection, we have (see for example [@kobayashi Theorem 2.1] and [@gau])
Let $(M,\,g,\,J)$ be an almost Hermitian manifold with $\dim_{\mathbb{R}}M=2n$. Then for any given vector valued $(1,1)$ form $\Phi=(\Phi^i)_{1\leq i\leq n}$, there exists a unique almost Hermitian connection $D$ on $(M,\,g,\,J)$ such that the $(1,1)$ part if the torsion is equal to the given $\Phi$.
If the $(1,1)$ part of the torsion of an almost Hermitian connection vanishes everywhere, then the connection is known as the *second canonical connection* and was first introduced by Ehrensmann and Libermann [@ehrensmann]. It is also sometimes referred to as the *Chern connection*, since when $J$ is integrable it coincides with the connection defined in Chern [@chern], and in Schouten and Danzig [@schouten]. We denote it by $\nabla^C$.
We use the local $(1,0)$ frames as before. We write $\omega$ as $$\omega={\sqrt{-1}}g_{i\overline{j}}\theta^i\wedge\overline{\theta}^j.$$ From , and , we get $$\begin{aligned}
e_i\langle e_k,\overline{e}_{\ell}\rangle
=\langle \nabla^C_{e_i}e_k,\overline{e}_{\ell}\rangle+\langle e_k,\nabla^C_{e_i}\overline{e}_{\ell}\rangle
=\langle \Gamma_{ik}^pe_p,\overline{e}_{\ell}\rangle+\langle e_k,\Gamma_{i\overline{\ell}}^{\overline{q}}\overline{e}_{q}\rangle
=\langle \Gamma_{ik}^pe_p,\overline{e}_{\ell}\rangle+\langle e_k,-\overline{C_{\ell\overline{i}}^{q}}\overline{e}_{q}\rangle,\end{aligned}$$ i.e., $$\begin{aligned}
\label{gammaikp}
\Gamma_{ik}^p=g^{\overline{\ell}p}e_ig_{k\overline{\ell}}+g^{\overline{\ell}p}g_{k\overline{q}}\overline{C_{\ell\overline{i}}^q}.\end{aligned}$$ This gives the components of the torsion as $$T_{ik}^p=\Gamma_{ik}^p-\Gamma_{ki}^p-C_{ik}^p=g^{\overline{\ell}p}e_ig_{k\overline{\ell}}+g^{\overline{\ell}p}g_{k\overline{q}}\overline{C_{\ell\overline{i}}^q}
-g^{\overline{\ell}p}e_kg_{i\overline{\ell}}-g^{\overline{\ell}p}g_{i\overline{q}}\overline{C_{\ell\overline{k}}^q}-C_{i k}^p.$$ We also lower the index of torsion and denote it by $$T_{ik\overline{\ell}}=T_{ik}^pg_{p\overline{\ell}}
=e_ig_{k\overline{\ell}}+g_{k\overline{q}}\overline{C_{\ell\overline{i}}^q}
-e_kg_{i\overline{\ell}}-g_{i\overline{q}}\overline{C_{\ell\overline{k}}^q}
-C_{ik}^pg_{p\overline{\ell}}.$$ Thanks to , we obtain $$\begin{aligned}
\overline{\partial}\omega
= \frac{{\sqrt{-1}}}{2}\left(\overline{e}_jg_{k\overline{i}}-\overline{e}_ig_{k\overline{j}}
-C_{k\overline{i}}^pg_{p\overline{j}}
+C_{k\overline{j}}^{p}g_{p\overline{i}}
+\overline{C_{ij}^q}g_{k\overline{q}}\right)\theta^k\wedge\overline{\theta}^i\wedge\overline{\theta}^j
= \frac{{\sqrt{-1}}}{2}\overline{T_{ji\overline{k}}}\theta^k\wedge\overline{\theta}^i\wedge\overline{\theta}^j.\end{aligned}$$ By , it yields that $$\begin{aligned}
\label{gammaipp}
\Gamma_{ip}^p=e_i\left(\log\det g\right)-C_{i\overline{q}}^{\overline{q}}.\end{aligned}$$ Using and , we can deduce that $$\begin{aligned}
\label{rkl}
R_{k\ell}:=&R_{k \ell i}{}^i
=[e_k,e_{\ell}]^{(0,1)}\left(\log\det g\right)
-e_k\left(C_{\ell\overline{q}}^{\overline{q}}\right)
+e_{\ell}\left(C_{k\overline{q}}^{\overline{q}}\right)
+C_{k\ell}^pC_{p\overline{q}}^{\overline{q}}
+\overline{N_{\overline{k}\overline{\ell}}^p}C_{\overline{p}i}^i.\end{aligned}$$ Combining and implies $$\begin{aligned}
\label{rkbarl}
R_{k\overline{\ell}}:=&R_{k \overline{\ell} i}{}^i
=-\left(e_k\overline{e}_{\ell}-[e_k,\overline{e}_{\ell}]^{(0,1)}\right)(\log\det g)
+\overline{e}_{\ell}\left(C_{i\overline{q}}^{\overline{q}}\right)
+e_k\left(C_{\overline{\ell} i}^i\right)
+C_{k\overline{\ell}}^pC_{p\overline{q}}^{\overline{q}}
-C_{k\overline{\ell}}^{\overline{p}}C_{\overline{p}i}^i.\end{aligned}$$ From and , it follows that $$\begin{aligned}
\label{rkbarlbar}
R_{\overline{k} \overline{\ell}}:=&R_{\overline{k} \overline{\ell} i}{}^i
=-[\overline{e}_k,\overline{e}_{\ell}]^{(1,0)}\left(\log\det g\right)
-N_{\overline{k}\overline{\ell}}^pC_{p\overline{q}}^{\overline{q}}
+\overline{e}_k(C_{\overline{\ell} i}^i)
-\overline{e}_{\ell}(C_{\overline{k} i}^i)
-\overline{C_{k\ell}^{p}}C_{\overline{p}i}^i.\end{aligned}$$ The Chern-Ricci form ${\mathrm{Ric}}(\omega)$ is defined by $${\mathrm{Ric}}(\omega):=\frac{{\sqrt{-1}}}{2}R_{k\ell}\theta^k\wedge\theta^{\ell}
+{\sqrt{-1}}R_{k\overline{\ell}}\theta^k\wedge\overline{\theta}^{\ell}
+\frac{{\sqrt{-1}}}{2}R_{\overline{k}\overline{\ell}}\overline{\theta}^k\wedge\overline{\theta}^{\ell}.$$ It is a closed real $2$ form and furthermore is a closed real $(1,1)$ form if the complex structure is integrable. If $J$ is integrable and ${\mathrm{d}}\omega=0$, then the Chern-Ricci form coincides exactly with the Ricci form defined by the Levi-Civita connection of $\omega$. Assume that $\tilde \omega={\sqrt{-1}}\tilde g_{i\overline{j}}\theta^i\wedge\overline{\theta}^j$ is another almost Hermitian metric. From , , , , and , it follows that (see also for example [@twyau (3.16)]) $$\begin{aligned}
\label{diffric}
{\mathrm{Ric}}(\tilde\omega)-{\mathrm{Ric}}(\omega)=-\frac{1}{2}{\mathrm{d}}J{\mathrm{d}}\log\frac{\tilde\omega^n}{\omega^n},\end{aligned}$$ with ${\mathrm{Ric}}(\omega)\in 2\pi c_1(M,J)\in H^2(M,\mathbb{R})$. Note that in general there exist representatives of $2\pi c_1(M,J)$ which cannot be written in the form ${\mathrm{Ric}}(\omega)-\frac{1}{2}{\mathrm{d}}J{\mathrm{d}}F$ for any $\omega$ and $F$ even when $J$ is integrable (see for example [@TW1 Corollary 2]).
The Chern scalar curvature $R$ is defined by $$R:={\mathrm{tr}}_{\omega}{\mathrm{Ric}}(\omega)={\mathrm{tr}}_{\omega}{\mathrm{Ric}^{(1,1)}}(\omega)=\frac{n{\mathrm{Ric}}(\omega)\wedge\omega^{n-1}}{\omega^n}=\frac{n{\mathrm{Ric}^{(1,1)}}(\omega)\wedge\omega^{n-1}}{\omega^n}.$$ For any $\varphi\in C^{\infty}(M,\mathbb{R})$, we define the canonical Laplacian by $$\begin{aligned}
\Delta^C_{\omega}\varphi:=\frac{n{\sqrt{-1}\partial\overline{\partial}}\varphi\wedge\omega^{n-1}}{\omega^n}=\frac{n({\mathrm{d}}J{\mathrm{d}}\varphi)\wedge\omega^{n-1}}{2\omega^n}
=g^{\overline{j}i}
\left(e_i\overline{e}_j(\varphi)-[e_i,\overline{e}_j]^{(0,1)}(\varphi)\right).\end{aligned}$$ Using the second canonical connection $\nabla^C$, it can also be rewritten as $$\Delta_{\omega}^C\varphi=g^{\overline{j}i}\nabla^C_i\nabla^C_{\overline{j}}\varphi=g^{\overline{j}i}\nabla^C_{\overline{j}}\nabla^C_i\varphi$$ since the $(1,1)$ part of the torsion of $\nabla^C$ vanishes. Denote by $\Delta_g$ the Laplace-Beltrami operator of the Riemannian metric $g$. For these two different Laplace operators, we have (see for example [@To0 Lemma 3.2]) $$\begin{aligned}
\label{2laplace}
\Delta_g \varphi=2\Delta_{\omega}^C\varphi+\tau({\mathrm{d}}\varphi),\end{aligned}$$ where $$\tau({\mathrm{d}}\varphi)=2\mathrm{Re}\left(T_{pj}^jg^{\overline{q}p}\overline{e}_q(\varphi)\right).$$ Given any volume form $\Omega$, there exists an almost Hermitian metric (not unique) $\omega'$ such that $\Omega=\omega'^n$ since we have $f:=\log\frac{\Omega}{\omega^n} \in C^{\infty}(M,\mathbb{R})$ and we can take $\omega'=e^{f/n}\omega$ for example. Hence, from , and , it follows that we can also define the Ricci form ${\mathrm{Ric}}(\Omega)$ associated to $\Omega$ by replacing $\det g$ with $\Omega$ in , and when it occurs, and also have $$\begin{aligned}
\label{omegaOmega}
{\mathrm{Ric}}(\omega)-{\mathrm{Ric}}(\Omega)=-\frac{1}{2}{\mathrm{d}}J{\mathrm{d}}\log\frac{\omega^n}{\Omega}.\end{aligned}$$
Preliminary estimates {#sec3}
=====================
In this section, we give the estimates of $\varphi$ and $\dot\varphi:={\frac{\partial}{\partial t}}\varphi$. For this aim, we need to prove that the flow can be reduced to a parabolic Monge-Ampère equation. Fix $T_0<T$ and in particular $T_0<\infty$. By definition of $T$, we can define reference metrics $\hat\omega_t$ for $M\times[0,T_0]$ by $$\begin{aligned}
\hat\omega_t:=\alpha_t+\frac{t}{T_0}{\sqrt{-1}\partial\overline{\partial}}\phi_{T_0}=\frac{T_0-t}{T_0}\omega_0+\frac{t}{T_0}\left(\alpha_{T_0}+{\sqrt{-1}\partial\overline{\partial}}\phi_{T_0}\right)=:{\sqrt{-1}}\hat g_{i\overline{j}}\theta^i\wedge\overline{\theta}^j,\end{aligned}$$ where $\phi_{T_0}\in C^{\infty}(M,\mathbb{R})$ satisfies $\alpha_{T_0}+{\sqrt{-1}\partial\overline{\partial}}\phi_{T_0}>0$. Note that the almost Hermitian metrics $\hat\omega_t$ vary smoothly on the compact interval $[0,T_0]$ and hence we can deduce uniform estimates on $\hat\omega_t$ for $t\in [0,T_0]$. We rewrite $\hat\omega_t$ as $\hat\omega_t=\omega_0+t\chi$ with $$\chi=\frac{1}{T_0}{\sqrt{-1}\partial\overline{\partial}}\phi_{T_0}-{\mathrm{Ric}^{(1,1)}}(\omega_0).$$ We define a volume form $\Omega=\omega_0^ne^{\phi_{T_0}/T_0}$. Note that $${\sqrt{-1}\partial\overline{\partial}}\log\Omega=\frac{1}{T_0}{\sqrt{-1}\partial\overline{\partial}}\phi_{T_0}+{\sqrt{-1}\partial\overline{\partial}}\log\omega_0^n\not=\chi=\frac{\partial\hat\omega_t}{\partial t}$$ and $$\begin{aligned}
\label{hatomegaomega0}
C_0^{-1}\omega_0\leq \hat\omega_t\leq C_0\omega_0\end{aligned}$$ for some uniform constant $C_0>0$.
\[dengjiadingyi\] A smooth family $\omega(t)$ of almost Hermitian metrics on $[0,T_0)$ solves the flow if and only if there is a family of smooth functions $\varphi(t)$ for $t\in [0,T_0)$ such that $\omega(t)=\hat\omega_t+{\sqrt{-1}\partial\overline{\partial}}\varphi(t)$, and solve $$\begin{aligned}
\label{apcma}
\frac{\partial}{\partial t}\varphi=\log\frac{\left(\hat\omega_t+{\sqrt{-1}\partial\overline{\partial}}\varphi\right)^n}{\Omega},\quad \hat\omega_t+{\sqrt{-1}\partial\overline{\partial}}\varphi>0,\quad \varphi|_{t=0}=0.\end{aligned}$$
We use the ideas from for example [@tosattikawa]. For the “if" direction, we set $\omega(t)=\hat\omega_t+{\sqrt{-1}\partial\overline{\partial}}\varphi(t)$. From , it follows that $$\begin{aligned}
{\frac{\partial}{\partial t}}\omega=
&{\frac{\partial}{\partial t}}\hat\omega_t+{\sqrt{-1}\partial\overline{\partial}}\left({\frac{\partial}{\partial t}}\varphi\right)\\
=&\frac{1}{T_0}{\sqrt{-1}\partial\overline{\partial}}\phi_{T_0}-{\mathrm{Ric}^{(1,1)}}(\omega_0)-\frac{1}{T_0}{\sqrt{-1}\partial\overline{\partial}}\phi_{T_0}+{\sqrt{-1}\partial\overline{\partial}}\log\frac{\omega^n}{\omega_0^n}\\
=&\frac{1}{T_0}{\sqrt{-1}\partial\overline{\partial}}\phi_{T_0}-{\mathrm{Ric}^{(1,1)}}(\omega_0)-\frac{1}{T_0}{\sqrt{-1}\partial\overline{\partial}}\phi_{T_0}-{\mathrm{Ric}^{(1,1)}}(\omega)+{\mathrm{Ric}^{(1,1)}}(\omega_0),\end{aligned}$$ as required.
For the ‘only if’ direction, assume that $\omega$ solves the flow on $[0,T_0)$. We define $$\varphi(t)=\int_{0}^t\log\frac{\omega(s)^n}{\Omega}{\mathrm{d}}s$$ for $t\in [0,T_0)$. We have $$\begin{aligned}
\label{gai1}
{\frac{\partial}{\partial t}}\varphi(t)=\log\frac{\omega(t)^n}{\Omega},\quad \varphi(0)=0.\end{aligned}$$ On the other hand, by , we can deduce $$\begin{aligned}
\label{gai2}
{\frac{\partial}{\partial t}}\left(\omega-\hat\omega_t\right)
=-{\mathrm{Ric}^{(1,1)}}(\omega)-\chi={\sqrt{-1}\partial\overline{\partial}}\left(\log\frac{\omega^n}{\omega_0^n}-\frac{\phi_{T_0}}{T_0}\right)={\sqrt{-1}\partial\overline{\partial}}\log\frac{\omega^n}{\Omega}.\end{aligned}$$ Thanks to and , it follows that $${\frac{\partial}{\partial t}}\left(\omega-\hat\omega_t-{\sqrt{-1}\partial\overline{\partial}}\varphi\right)=0,\quad\mbox{with}\quad
\left(\omega-\hat\omega_t-{\sqrt{-1}\partial\overline{\partial}}\varphi\right)|_{t=0}=0$$ so that $\omega=\hat\omega_t+{\sqrt{-1}\partial\overline{\partial}}\varphi$ and $\varphi$ is the solution to .
Standard parabolic theory of partial differential equation yields that there exists a unique maximal solution to on some time interval $[0,{T_{\mathrm{max}}})$ with ${T_{\mathrm{max}}}>0$. Assume for a contradiction that ${T_{\mathrm{max}}}<T_0$.
Now we prove uniform estimates for the solution $\varphi$ to up to the maximal time. For later use, we write $$\begin{aligned}
\omega(t)={\sqrt{-1}}g_{i\overline{j}}\theta^i\wedge\overline{\theta}^j,\quad \omega_0={\sqrt{-1}}(g_0)_{i\overline{j}}\theta^i\wedge\overline{\theta}^j.\end{aligned}$$
\[lemc0\] Assume that $\varphi(t)$ is the solution to the flow on $[0,{T_{\mathrm{max}}})$. There exists a positive uniform constant $C>0$, independent of $t\in [0,{T_{\mathrm{max}}})$, such that
1. \[c0\]$\|\varphi(t)\|_{C^0}\leq C$.
2. \[dotvarphi\] $\|\dot{\varphi}(t)\|_{C^0}\leq C$.
3. \[volumequi\]$C^{-1}\omega_0^{n}\leq \omega^n\leq C\omega_0^n$.
We use the ideas from [@TZ; @twjdg]. For Part , set $\psi=\varphi-At$ for a constant $A>0$ to be determined later. Fix any $T'\in (0,{T_{\mathrm{max}}})$ and assume that $\psi$ attains a maximum on $M\times [0,T']$ at a point $(x_0,t_0)$ with $t_0>0$. At this point, the maximum principle yields $$0\leq {\frac{\partial}{\partial t}}\psi=\log\frac{\left(\hat\omega_t+{\sqrt{-1}\partial\overline{\partial}}\psi\right)^n}{\Omega}-A\leq \log\frac{\hat\omega_t^n}{\Omega}-A<0,$$ provided $A$ is chosen sufficiently large, a contradiction. Here we use the fact that $\hat\omega_t$ is a smooth family of metrics on $[0,{T_{\mathrm{max}}}]$. Since $T'\in (0,{T_{\mathrm{max}}})$ is arbitrary, this yields that the maximum of $\psi(t)$ is achieved at $t=0$. We get the upper bound for $\psi$ and hence for $\varphi$. The lower bound is proved similarly.
As for Part , we first consider the upper bound for $\dot{\varphi}$. We consider the quantity $Q_1=t\dot\varphi-\varphi-nt$ and have $$\begin{aligned}
\left({\frac{\partial}{\partial t}}-\Delta_{\omega}^{C}\right)Q_1
=t{\mathrm{tr}}_{\omega}\chi-n+{\mathrm{tr}}_{\omega}(\omega-\hat\omega_t)={\mathrm{tr}}_{\omega}(t\chi-\hat\omega_t)=-{\mathrm{tr}}_{\omega}\omega_0<0,\end{aligned}$$ where we use the fact that ${\frac{\partial}{\partial t}}\dot\varphi=\Delta_{\omega}^C\dot\varphi+{\mathrm{tr}}_{\omega}\chi$. This, together with the maximum principle, implies that upper bound for $Q_1$ and hence for $\dot\varphi$.
For the lower bound for $\dot\varphi$, we define $Q_2:=(T_0-t)\dot\varphi+\varphi+nt$ and have $$\begin{aligned}
\left({\frac{\partial}{\partial t}}-\Delta_{\omega}^{C}\right)Q_2
=(T_0-t){\mathrm{tr}}_{\omega}\chi+n-\Delta_{\omega}^C\varphi={\mathrm{tr}}_{\omega}\left(\hat\omega_t+(T_0-t)\chi\right)={\mathrm{tr}}_{\omega}\hat\omega_{T_0}>0.\end{aligned}$$ Then the lower bound for $Q_2$ and hence for $\dot\varphi$ follows from the maximum principle and the fact that ${T_{\mathrm{max}}}<T_0$.
Part follows from Part , and the definition of $\Omega$.
First order estimate {#sec4}
====================
In this section, we give the first order estimate of the solution $\varphi$ to .
\[thm1order\] Assume that $\varphi(t)$ is the solution to the flow on $[0,{T_{\mathrm{max}}})$. There exists a uniform constant $C>1$, independent of $t\in [0,{T_{\mathrm{max}}})$, such that $$\begin{aligned}
\sup\limits_{M\times[0,{T_{\mathrm{max}}})}|\partial \varphi|_{g_0} \leq C.\end{aligned}$$
We consider the quantity $Q:=e^{f(\varphi)}|\partial \varphi|_{g_0}^2$, where $f$ will be determined later. Fix any $T'\in (0,{T_{\mathrm{max}}})$ and assume that $$\sup\limits_{M\times[0,T']}Q=Q(x_0,t_0)$$ with $t_0>0$. Since $g_0$ is almost Hermitian metric, we choose $g_0$-unitary frame $e_1,\cdots,e_n$ such that $g(x_0,t_0)$ is diagonal near $x_0$. At the point $(x_0,t_0)$, the maximum principle yields $$\begin{aligned}
\label{evolutionQ}
0\geq&\left(\Delta_{\omega}^C-{\frac{\partial}{\partial t}}\right)Q\\
=&e^f\left(\Delta_{\omega}^C-{\frac{\partial}{\partial t}}\right)|\partial \varphi|_{g_0}^2+|\partial \varphi|_{g_0}^2\left(\Delta_{\omega}^C-{\frac{\partial}{\partial t}}\right)e^f
+2\mathrm{Re}\left(g^{\overline{i}i}e_i\left(|\partial \varphi|_{g_0}^2\right)\overline{e}_i(e^f)\right).\nonumber\end{aligned}$$ Since $\omega=\hat\omega_t+{\sqrt{-1}\partial\overline{\partial}}\varphi$, at $(x_0,t_0)$, a direct computation gives $$\begin{aligned}
\label{lapef}
\Delta_{\omega}^C(e^f)
=&g^{\overline{i}i}\left(e_i\overline{e}_i\left(e^f\right)-[e_i,\overline{e}_i]^{(0,1)}\left(e^f\right)\right)\\
=&g^{\overline{i}i}\left(e^f(f')^2|e_i(\varphi)|^2+e^ff''|e_i(\varphi)|^2+e^ff'e_i\overline{e}_i(\varphi)
-e^ff'[e_i,\overline{e}_i]^{(0,1)}(\varphi)\right)\nonumber\\
=&g^{\overline{i}i}e^f\left((f')^2+f''\right)|e_i(\varphi)|^2+e^ff'{\mathrm{tr}}_{\omega}(\omega-\hat\omega_t)\nonumber\\
=&g^{\overline{i}i}e^f\left((f')^2+f''\right)|e_i(\varphi)|^2+ne^ff'-e^ff'\left({\mathrm{tr}}_{\omega}\hat\omega_t\right)\nonumber\end{aligned}$$ and $$\begin{aligned}
\label{lappartialu}
\Delta^C_{\omega}(|\partial\varphi|_{g_0}^2)
=&g^{\overline{i}i}\left(e_i\overline{e}_i(e_k(\varphi)\overline{e}_k(\varphi))-[e_i,\overline{e}_i]^{(0,1)}(e_k(\varphi)\overline{e}_k(\varphi))\right)\\
=&g^{\overline{i}i}\left(|e_ie_k(\varphi)|^2+|e_i\overline{e}_k(\varphi)|^2+e_i\overline{e}_ie_k(\varphi)\overline{e}_k(\varphi)
+e_k(\varphi)e_i\overline{e}_i\overline{e}_k(\varphi)\right)\nonumber\\
&-g^{\overline{i}i}\left([e_i,\overline{e}_i]^{(0,1)}e_k(\varphi)\overline{e}_k(\varphi)+e_k(\varphi)
[e_i,\overline{e}_i]^{(0,1)}\overline{e}_k(\varphi)\right)\nonumber\\
=&g^{\overline{i}i}\left(|e_ie_k(\varphi)|^2+|e_i\overline{e}_k(\varphi)|^2\right)
+g^{\overline{i}i}\left(e_k\left(e_i\overline{e}_i(\varphi)-[e_i,\overline{e}_i]^{(0,1)}(\varphi)\right)\overline{e}_k(\varphi)\right)\nonumber\\
&+g^{\overline{i}i}e_k(\varphi)\left(\overline{e}_k\left(e_i\overline{e}_i(\varphi)-[e_i,\overline{e}_i]^{(0,1)}(\varphi)\right)\right)
-g^{\overline{i}i}[e_k,e_i]\overline{e}_i(\varphi)\overline{e}_k(\varphi)\nonumber\\
&-g^{\overline{i}i}e_i[e_k,\overline{e}_i](\varphi)\overline{e}_k(\varphi)
-g^{\overline{i}i}e_k(\varphi)[\overline{e}_k,e_i]\overline{e}_i(\varphi)
-g^{\overline{i}i}e_k(\varphi)e_i[\overline{e}_k,\overline{e}_i](\varphi)\nonumber\\
&-g^{\overline{i}i}\left[e_k,[e_i,\overline{e}_i]^{(0,1)}\right](\varphi)\overline{e}_k(\varphi)
-g^{\overline{i}i}e_k(\varphi)\left[\overline{e}_k,[e_i,\overline{e}_i]^{(0,1)}\right](\varphi)\nonumber\\
=&g^{\overline{i}i}\left(|e_ie_k(\varphi)|^2+|e_i\overline{e}_k(\varphi)|^2\right)
+g^{\overline{i}i}\left(e_k\left(g_{i\overline{i}}-\hat g_{i\overline{i}}\right)\overline{e}_k(\varphi)\right)\nonumber\\
&+g^{\overline{i}i}e_k(\varphi)\left(\overline{e}_k\left(g_{i\overline{i}}-\hat g_{i\overline{i}}\right)\right)
-g^{\overline{i}i}[e_k,e_i]\overline{e}_i(\varphi)\overline{e}_k(\varphi)\nonumber\\
&-g^{\overline{i}i}e_i[e_k,\overline{e}_i](\varphi)\overline{e}_k(\varphi)
-g^{\overline{i}i}e_k(\varphi)[\overline{e}_k,e_i]\overline{e}_i(\varphi)
-g^{\overline{i}i}e_k(\varphi)e_i[\overline{e}_k,\overline{e}_i](\varphi)\nonumber\\
&-g^{\overline{i}i}\left[e_k,[e_i,\overline{e}_i]^{(0,1)}\right](\varphi)\overline{e}_k(\varphi)
-g^{\overline{i}i}e_k(\varphi)\left[\overline{e}_k,[e_i,\overline{e}_i]^{(0,1)}\right](\varphi).\nonumber\end{aligned}$$ On the other hand, we have, at $(x_0,t_0)$, $$\begin{aligned}
\label{pptu}
{\frac{\partial}{\partial t}}|\partial\varphi|_{g_0}^2
=&e_k(\dot\varphi)\overline{e}_k(\varphi)+e_k(\varphi)\overline{e}_k(\dot\varphi)\\
=&g^{\overline{i}i}e_k(g_{i\overline{i}})\overline{e}_k(\varphi)
-e_k(\log\Omega)\overline{e}_k(\varphi)
+g^{\overline{i}i}e_k(\varphi)\overline{e}_k(g_{i\overline{i}})
-e_k(\varphi)\overline{e}_k(\log\Omega),\nonumber\end{aligned}$$ where we use . At $(x_0,t_0)$, from and , it follows that $$\begin{aligned}
\label{evolutionpartialvarphi}
\left(\Delta^C_{\omega}-{\frac{\partial}{\partial t}}\right)|\partial\varphi|_{g_0}^2
=&g^{\overline{i}i}\left(|e_ie_k(\varphi)|^2+|e_i\overline{e}_k(\varphi)|^2\right)
-g^{\overline{i}i}e_k \left(\hat g_{i\overline{i}}\right)\overline{e}_k(\varphi)
-g^{\overline{i}i}e_k(\varphi)\overline{e}_k\left(\hat g_{i\overline{i}}\right) \\
&-g^{\overline{i}i}[e_k,e_i]\overline{e}_i(\varphi)\overline{e}_k(\varphi)
-g^{\overline{i}i}e_i[e_k,\overline{e}_i](\varphi)\overline{e}_k(\varphi)
-g^{\overline{i}i}e_k(\varphi)[\overline{e}_k,e_i]\overline{e}_i(\varphi)\nonumber\\
&-g^{\overline{i}i}e_k(\varphi)e_i[\overline{e}_k,\overline{e}_i](\varphi)
-g^{\overline{i}i}\left[e_k,[e_i,\overline{e}_i]^{(0,1)}\right](\varphi)\overline{e}_k(\varphi)\nonumber\\
&-g^{\overline{i}i}e_k(\varphi)\left[\overline{e}_k,[e_i,\overline{e}_i]^{(0,1)}\right](\varphi)
+2\mathrm{Re}\left(e_k(\log\Omega)\overline{e}_k(\varphi)\right)\nonumber\\
\geq&(1-\varepsilon)g^{\overline{i}i}\left(|e_ie_k(\varphi)|^2+|e_i\overline{e}_k(\varphi)|^2\right)\nonumber\\
&-\frac{C}{\varepsilon}|\partial \varphi|_{g_0}^2\sum\limits_{i}g^{\overline{i}i}
+2\mathrm{Re}\left(e_k(\log\Omega)\overline{e}_k(\varphi)\right),\nonumber\end{aligned}$$ where for the inequality we use the Cauchy-Schwarz inequality and $\varepsilon \in(0,1/2]$ and without loss of generality we assume $|\partial \varphi|_{g_0}^2(x_0,t_0)>1$. At $(x_0,t_0)$, a direct calculation implies $$\begin{aligned}
\label{realpart}
&2\mathrm{Re}\left(g^{\overline{i}i}e_i\left(|\partial \varphi|_{g_0}^2\right)\overline{e}_i(e^f)\right)\\
=&2\mathrm{Re}\left(e^ff'g^{\overline{i}i}e_k(\varphi)e_i\overline{e}_k(\varphi)\overline{e}_i(\varphi)\right)
+2\mathrm{Re}\left(e^ff'g^{\overline{i}i}e_ie_k(\varphi)\overline{e}_k(\varphi)\overline{e}_i(\varphi)\right)\nonumber\\
=&2\mathrm{Re}\left(e^ff'g^{\overline{i}i}e_k(\varphi)\left(g_{i\overline{k}}-\hat g_{i\overline{k}}+[e_i,\overline{e}_k]^{(0,1)}(\varphi)\right) \overline{e}_i(\varphi)\right)\nonumber\\
&+2\mathrm{Re}\left(e^ff'g^{\overline{i}i}e_ie_k(\varphi)\overline{e}_k(\varphi)\overline{e}_i(\varphi)\right)\nonumber\\
\geq&2 e^ff'|\partial\varphi|_{g_0}^2
-2\mathrm{Re}\left(e^ff'g^{\overline{i}i}\hat g_{i\overline{k}}e_k(\varphi)\overline{e}_i(\varphi)\right)
-\varepsilon e^f(f')^2|\partial\varphi|_{g}^2|\partial\varphi|_{g_0}^2
-\frac{Ce^f}{\varepsilon}|\partial\varphi|_{g_0}^2\sum\limits_{i}g^{\overline{i}i}\nonumber\\
&-(1+2\varepsilon)e^f(f')^2|\partial\varphi|_{g_0}^2|\partial\varphi|_{g}^2-(1-\varepsilon)e^f\sum\limits_{i}g^{\overline{i}i}|e_ie_k(\varphi)|^2\nonumber\\
=&2 e^ff'|\partial\varphi|_{g_0}^2
-2\mathrm{Re}\left(e^ff'g^{\overline{i}i}\hat g_{i\overline{k}}e_k(\varphi)\overline{e}_i(\varphi)\right)
-\frac{Ce^f}{\varepsilon}|\partial\varphi|_{g_0}^2\sum\limits_{i}g^{\overline{i}i}\nonumber\\
&-(1+3\varepsilon)e^f(f')^2|\partial\varphi|_{g_0}^2|\partial\varphi|_{g}^2-(1-\varepsilon)e^f\sum\limits_{i}g^{\overline{i}i}|e_ie_k(\varphi)|^2,\nonumber\end{aligned}$$ where for the inequality we use the Cauchy-Schwarz inequality and $\varepsilon\in(0,1/2]$. At $(x_0,t_0)$, from and , it follows that $$\begin{aligned}
\label{evolutionef}
\left(\Delta_{\omega}^C-{\frac{\partial}{\partial t}}\right)e^f
=&e^f\left(f''+(f')^2\right)|\partial\varphi|_{g}^2+e^ff'\left(\Delta_{\omega}^C-{\frac{\partial}{\partial t}}\right)\varphi\\
=&e^f\left(f''+(f')^2\right)|\partial\varphi|_{g}^2+ne^ff'-e^ff'\left({\mathrm{tr}}_{\omega}\hat\omega_t\right)-e^ff'\left({\frac{\partial}{\partial t}}\varphi\right).\nonumber\end{aligned}$$ From , , and , it yields that, at $(x_0,t_0)$, $$\begin{aligned}
\label{1oder1}
0\geq&e^f\left(f''-3\varepsilon (f')^2\right)|\partial\varphi|_{g_0}^2|\partial\varphi|_{g}^2 -e^ff'|\partial \varphi|_{g_0}^2\left({\mathrm{tr}}_{\omega}\hat\omega_t\right)
-\frac{C_1e^f}{\varepsilon} |\partial \varphi|_{g_0}^2\sum\limits_{i}g^{\overline{i}i}\\
&-e^ff'\left({\frac{\partial}{\partial t}}\varphi\right)|\partial\varphi|_{g_0}^2
-2e^f\mathrm{Re}\left(e_k(\log\Omega)\overline{e}_k(\varphi)\right)
+(n+2)e^ff'|\partial\varphi|_{g_0}^2\nonumber\\
&-2\mathrm{Re}\left(e^ff'g^{\overline{i}i}\hat g_{i\overline{k}}e_k(\varphi)\overline{e}_i(\varphi)\right),\nonumber\end{aligned}$$ where $C_1>1$ is a uniform constant.
Define $$f(\varphi)=\frac{e^{-A\left(\varphi- \varphi_0-1\right)}}{A},\quad
\varepsilon=\frac{Ae^{A\left(\varphi(x_0,t_0)-\varphi_0-1\right)}}{6},$$ where $\varphi_0:=\sup\limits_{M\times[0,{T_{\mathrm{max}}})}\varphi$. Then, at $(x_0,t_0)$, we have $$\begin{aligned}
\label{1order2}
f''-3\varepsilon(f')^2=\frac{Ae^{-A\left(\varphi(x_0,t_0)-\varphi_0-1\right)}}{2},\end{aligned}$$ $$\begin{aligned}
\label{1order3}
&-e^ff'|\partial \varphi|_{g_0}^2\left({\mathrm{tr}}_{\omega}\hat\omega_t\right)
-\frac{C_1e^f}{\varepsilon} |\partial \varphi|_{g_0}^2\sum\limits_{i}g^{\overline{i}i}\\
\geq&e^f|\partial \varphi|_{g_0}^2\left(\sum\limits_{i}g^{\overline{i}i}\right)\left(C_{0}^{-1}-\frac{6C_1}{A}\right)
e^{-A\left(\varphi(x_0,t_0)-\varphi_0-1\right)},\nonumber\end{aligned}$$ and $$\begin{aligned}
\label{1order6}
2\left|\mathrm{Re}\left(e^ff'g^{\overline{i}i}\hat g_{i\overline{k}}e_k(\varphi)\overline{e}_i(\varphi)\right)\right|
\leq -2C_0e^ff'|\partial\varphi|_g^2
=2C_0e^f|\partial\varphi|_g^2e^{-A\left(\varphi(x_0,t_0)-\varphi_0-1\right)}.\end{aligned}$$ where for the inequality we use and the fact that $f'<0$.
Note that we assume that $|\partial\varphi|_{g_0}(x_0,t_0)>1$. Choosing $A=12C_0C_1$, combining , and yields that, at $(x_0,t_0)$, $$\begin{aligned}
\label{1order4}
e^f\left(f''-3\varepsilon (f')^2\right)|\partial\varphi|_{g_0}^2|\partial\varphi|_{g}^2-2\mathrm{Re}\left(e^ff'g^{\overline{i}i}\hat g_{i\overline{k}}e_k(\varphi)\overline{e}_i(\varphi)\right)
\geq C^{-1}|\partial\varphi|_{g_0}^2|\partial\varphi|_{g}^2\end{aligned}$$ and $$\begin{aligned}
\label{1order5}
-e^ff'|\partial \varphi|_{g_0}^2\left({\mathrm{tr}}_{\omega}\hat\omega_t\right)
-\frac{C_1e^f}{\varepsilon} |\partial \varphi|_{g_0}^2\sum\limits_{i}g^{\overline{i}i}
\geq C^{-1}|\partial \varphi|_{g_0}^2\sum\limits_{i}g^{\overline{i}i}.\end{aligned}$$ where $C>0$ is a uniform constant. Combining , and implies that, at $(x_0,t_0)$, $$\begin{aligned}
\label{1order7}
0\geq C^{-1}|\partial\varphi|_{g_0}^2|\partial\varphi|_{g}^2+C^{-1}|\partial \varphi|_{g_0}^2\sum\limits_{i}g^{\overline{i}i}-C|\partial\varphi|_{g_0}^2-C.\end{aligned}$$ Note that here the constant $C$ may be larger.
From Lemma \[lemc0\] and the same argument in [@ctw] (see also [@chu1607]), Theorem \[thm1order\] follows.
Using Theorem \[thm1order\], taking $\varepsilon=1/2$ in , we get
\[lemevolutionpartialu\] There exists a uniform constant $C>0$ such that $$\begin{aligned}
\label{lemformulaevolutionpartialvarphi}
\left(\Delta^C_{\omega}-{\frac{\partial}{\partial t}}\right)|\partial\varphi|_{g_0}^2
\geq\frac{1}{2}g^{\overline{i}i}\left(|e_ie_k(\varphi)|^2+|e_i\overline{e}_k(\varphi)|^2\right)
-C \sum\limits_{i}g^{\overline{i}i}
-C.\end{aligned}$$
Second order estimate {#sec5}
=====================
In this section, we deduce the second order estimate of the solution $\varphi$ to using the method from [@ctw] in the elliptic setup.
\[thm2order\] Assume that $\varphi$ is the solution to the flow . There exists a uniform constant $C>0$, independent of $t$, such that $$\begin{aligned}
\sup_{M\times[0,{T_{\mathrm{max}}})}|\nabla^2 \varphi|_{g_0}\leq C,\end{aligned}$$ where $\nabla$ is the Levi-Civita connection with respect to the Riemannian metric $g_0$.
Denote by $\lambda_1(\nabla^2\varphi)\geq \cdots\geq \lambda_{2n}(\nabla^2\varphi)$ the eigenvalues of $\nabla^2\varphi$ with respect to the Riemannian metric $g_0$. Then we have $\Delta_{g_0}\varphi=\sum\limits_{\alpha=1}\lambda_{\alpha}(\nabla^2\varphi)$, where $\Delta_{g_0}$ is the Laplace-Beltrami operator of $g_0$. Since $$\omega=\hat\omega_t+{\sqrt{-1}\partial\overline{\partial}}\varphi>0,$$ it follows that $$\begin{aligned}
\Delta^C_{\omega_0}\varphi={\mathrm{tr}}_{\omega_0}\omega-{\mathrm{tr}}_{\omega_0}\hat\omega_t>-{\mathrm{tr}}_{\omega_0}\hat\omega_t>-C,\end{aligned}$$ where we use the fact that $\hat\omega_t$ vary smoothly on the compact interval $[0,T_0]$ and hence can be estimated uniformly. This, together with and Theorem \[thm1order\], yields $$\begin{aligned}
\label{lapg0varphi}
\Delta_{g_0}\varphi
=&2\Delta_{\omega_0}^C\varphi+\tau({\mathrm{d}}\varphi)>-C,\end{aligned}$$ where $C>0$ is a uniform constant. As a result, we get $$\begin{aligned}
|\nabla^2 \varphi|_{g_0}\leq C\max\Big\{\lambda_1(\nabla^2\varphi),0\Big\}+C.\end{aligned}$$ Therefore, it is sufficient to bound $\lambda_1(\nabla^2\varphi)$ from above in order to prove Theorem \[thm2order\]. Denote $$S_H:=\Big\{(x,t)\in M\times [0,{T_{\mathrm{max}}}):\;\lambda_1(\nabla^2\varphi)>0\Big\}.$$ Without loss of generality, we assume $S_H\not=\emptyset$. Otherwise, we can get the upper bound of $\lambda_1(\nabla^2\varphi)$ directly. We consider the quantity $$H=\log\lambda_1(\nabla^2\varphi)+\phi(|\partial \varphi|_g^2)+h(\varphi-\varphi_0),$$ on the set $S_H$, where $$\phi(s)=-\frac{1}{2}\log\left(1-\frac{s}{2K}\right),\quad h(s)=e^{-Ds}$$ with sufficiently large uniform constant $D$ to be determined later. Here recall that $\varphi_0:=\sup\limits_{M\times[0,{T_{\mathrm{max}}})}\varphi$, and for convenience, we use notation $K:=1+\sum\limits_{M\times[0,{T_{\mathrm{max}}})}|\partial \varphi|_{g_0}^2$ which is a uniform constant by Theorem \[thm1order\]. Note that $$\phi(|\partial \varphi|_{g_0}^2)\in\,[0,\,2\log 2]$$ and $$\frac{1}{4K}\leq \phi'\leq \frac{1}{2K},\quad \phi''=2(\phi')^2.$$ For any $T'\in (0,{T_{\mathrm{max}}})$, assume that $H$ attains its maximum at $(x_0,t_0)$ on $\Big\{(x,t)\in M\times [0,T']:\;\lambda_1(\nabla^2\varphi)>0\Big\}$. Note that $H$ is not smooth in general since the dimension of eigenspace associated to $\lambda_1(\nabla^2\varphi)$ may be strictly larger than $1$. We use a perturbation argument in [@ctw] to deal with this case (see also [@sz; @stw1503]). Since $g_0$ is almost Hermitian metric, we can choose a coordinate patch $(U;\,x^1,\cdots,x^{2n})$ centered at $x_{0}$ such that
1. $U$ is diffeomorphic to a ball $B_2(0)\subset \mathbb{R}^{2n}$ of radius $2$ centered at $0$.
2. Denote by $\partial_{\alpha}$ the local vector fields $\partial/\partial x^{\alpha}$. There holds $g_0(\partial_{\alpha},\partial_{\beta})|_{x_{0}}=\delta_{\alpha\beta},\,\alpha,\,\beta\hat{}=1,\cdots,2n$.
3. The almost complex structure $J$ satisfies $J(x_{0})=J_0$, where $J_0$ is the standard complex structure on $\mathbb{R}^{2n}$, i.e., $J_0\partial_{2i-1}=\partial_{2i}$ for $i=1,\cdots,n$.
4. There holds $$\begin{aligned}
\label{zuobiao}
\partial_{\gamma}(g_0)_{\alpha\beta}|_{x_{0}}=0,\quad \forall\;\alpha,\,\beta,\,\gamma=1,\cdots,2n.\end{aligned}$$
5. If at $x_0$, we define $$\begin{aligned}
\label{10fields}
e_i:=\frac{1}{\sqrt{2}}\left(\partial_{2i-1}-{\sqrt{-1}}\partial_{2i}\right),\quad i=1,\cdots,n,\end{aligned}$$ then these form a frame of $(1,0)$ vectors at $x_0$, and we have $(g_0)_{i\overline{j}}:=g_0(e_i,\overline{e}_j)=\delta_{ij}$, i.e., the frame is $g_0$-unitary. Furthermore, the argument in [@ctw] yields that we can assume $\left(g_{i\overline{j}}(x_0,t_0)\right)$ is diagonal with $$g_{1\overline{1}}(x_0,t_0)\geq\cdots\geq g_{n\overline{n}}(x_0,t_0).$$
We extend $e_1,\cdots,e_n$ smoothly to a $g_0$-unitary frame of $(1,0)$ vectors in a neighborhood of $x_0$, and from now on, the local unitary frame is fixed. Assume that $V_1\in T_{x_0}M$ is a unit vector, i.e., $g_0(V_1,V_1)=1$, with $$\nabla^2\varphi(V_1,V_1)=\lambda_1(\nabla^2\varphi)(x_0,t_0).$$ We can construct an orthonormal basis of $T_{x_0}M$, denoted by $V_1,\cdots,V_{2n}$, such that $$\nabla^2\varphi(V_{\alpha},V_{\alpha})=\lambda_{\alpha}(\nabla^2\varphi)(x_0,t_0), \quad\alpha=1,\cdots,2n$$ with $\lambda_{1}(\nabla^2\varphi)(x_0,t_0)\geq \cdots\geq \lambda_{2n}(\nabla^2\varphi)(x_0,t_0)$. We assume $V_{\beta}=\sum\limits_{\alpha=1}^{2n}V^{\alpha}{}_{\beta}\partial_{\alpha}$ for $\beta=1,\cdots,2n$. We extend $V_1,\cdots,V_{2n}$ to be vector fields in a neighborhood of $x_0$ by taking the components to be constant. To use the perturbation argument, we define a smooth section $B=B_{\alpha\beta}{\mathrm{d}}x^{\alpha}\otimes {\mathrm{d}}x^{\beta}$ of $T^{\ast}M\otimes T^{\ast}M$ near $x_0$, where $$B_{\alpha\beta}=\delta_{\alpha\beta}-V^{\alpha}{}_1V^{\beta}{}_1.$$ It is easy to deduce that the eigenvalues of $(B_{\alpha\beta})$ are $0,\underbrace{1,\cdots,1}_{(n-1)}$ and that $V_1$ is the eigenvector associated to the eigenvalue $0$, i.e., $B(V_1,V_1)=0$. Consider the endomorphism $\Phi=\left(\Phi^{\alpha}{}_{\beta}\right)$ of $TM$ defined by $$\begin{aligned}
\label{phi}
\Phi^{\alpha}{}_{\beta}=g_0^{\alpha\gamma}\nabla^2_{\gamma\beta}\varphi-g_0^{\alpha\gamma}B_{\gamma\beta},\end{aligned}$$ and denote its eigenvalues by $$\lambda_1(\Phi)\geq\cdots\geq\lambda_{2n}(\Phi).$$ Since $B=\left(B_{\alpha\beta}\right)$ is nonnegative, we can deduce that $\lambda_1(\Phi)\leq \lambda_1(\nabla^2\varphi)$ in the neighborhood of $(x_0,t_0)$. In particular, we have $$\lambda_{1}(\Phi)(x_0,t_0)=\lambda_{1}(\nabla^2\varphi)(x_0,t_0),\quad\Phi(V_{\alpha})(x_0,t_0)=\lambda_{\alpha}(\Phi)(x_0,t_0)V_{\alpha},\quad \alpha=1,\cdots,2n$$ and hence $$\lambda_1(\Phi)(x_0,t_0)>\lambda_2(\Phi)(x_0,t_0)\geq\cdots\geq\lambda_{2n}(\Phi)(x_0,t_0).$$ This yields that $\lambda_1(\Phi)$ is smooth in a neighborhood of $(x_0,t_0)$. In the following, we write $\lambda_{\alpha}$ for $\lambda_{\alpha}(\Phi)$ for short. We can apply the maximum principle to the quantity $$\tilde H=\log \lambda_1(\Phi)+\phi(|\partial \varphi|_{g_0}^2)+h(\varphi-\varphi_0),$$ which still obtains a local maximum at $(x_0,t_0)$.
We need the first and second derivatives of $\lambda_1$ at $(x_0,t_0)$ as follows.
At $(x_0,t_0)$, we have $$\begin{aligned}
\label{lambda1order}
\lambda_1^{\alpha\beta}:=&\frac{\partial \lambda_1}{\partial\Phi^{\alpha}{}_{\beta}}=V^{\alpha}{}_1V^{\beta}{}_1.\\
\label{lambda2order}
\lambda_1^{\alpha\beta,\gamma\delta}:=&\frac{\partial^2\lambda_1}{\partial\Phi^{\alpha}{}_{\beta}\partial\Phi^{\gamma}{}_{\delta}}
=\sum\limits_{\mu>1}\frac{V^{\alpha}{}_1V^{\beta}{}_{\mu}V^{\gamma}{}_{\mu}V^{\delta}{}_1+
V^{\alpha}{}_{\mu}V^{\beta}{}_{1}V^{\gamma}{}_{1}V^{\delta}{}_{\mu}}{\lambda_1-\lambda_{\mu}},\end{aligned}$$ where the Greek indices $\alpha,\beta,\gamma,\mu,\cdots$ go from $1$ to $2n$, unless otherwise indicated.
By Lemma \[lemc0\] and , the arithmetic-geometry mean inequality yields $$\begin{aligned}
\label{tromegaomega0xiajie}
{\mathrm{tr}}_{\omega}\omega_0\geq c,\end{aligned}$$ for a uniform constant $c>0$. We also assume that, at $(x_0,t_0)$, there holds $\lambda_1\gg K\geq1$.
\[lem1\] At $(x_0,t_0)$, we have $$\begin{aligned}
\label{lem1formula}
\left(\Delta_{\omega}^C-{\frac{\partial}{\partial t}}\right)\lambda_1
\geq&2\sum\limits_{\alpha>1}g^{\overline{i}i}\frac{|e_i(\varphi_{V_{\alpha}V_{1}})|^2}{\lambda_1-\lambda_{\alpha}}
+g^{\overline{p}p}g^{\overline{q}q}\left|V_1(g_{p\overline{q}})\right|^2 \\
&-2g^{\overline{i}i}[V_1,e_i]V_1\overline{e}_i(\varphi)
-2g^{\overline{i}i}[V_1,\overline{e}_i]V_1\overline{e}_i(\varphi)-C\lambda_1\sum\limits_{i}g^{\overline{i}i},\nonumber\end{aligned}$$ where we write $$\varphi_{\alpha\beta}:=\nabla^2_{\alpha\beta}\varphi,\quad \varphi_{V_{\alpha}V_{\beta}}:=\varphi_{\gamma\delta}V^{\gamma}{}_{\alpha}V^{\delta}{}_{\beta}=\nabla^2\varphi(V_{\alpha},V_{\beta}).$$
At $(x_0,t_0)$, noting that $g(x_0,t_0)$ is diagonal, by and , a direct computation yields $$\begin{aligned}
\label{Llambda1}
\left(\Delta_{\omega}^C-{\frac{\partial}{\partial t}}\right)\lambda_1
=&g^{\overline{i}i}\lambda_{1}^{\alpha\beta,\gamma\delta}e_i(\Phi^{\gamma}{}_{\delta})\overline{e}_i(\Phi^{\alpha}{}_{\beta})
+g^{\overline{i}i}\lambda_{1}^{\alpha\beta}e_i\overline{e}_i(\Phi^{\alpha}{}_{\beta})\\
&-g^{\overline{i}i}\lambda_{1}^{\alpha\beta}[e_i,\overline{e}_i]^{(0,1)}(\Phi^{\alpha}{}_{\beta})
-\lambda_{1}^{\alpha\beta}{\frac{\partial}{\partial t}}(\Phi^{\alpha}{}_{\beta})\nonumber\\
=&g^{\overline{i}i}\lambda_{1}^{\alpha\beta,\gamma\delta}e_i(\varphi_{\gamma\delta})\overline{e}_i(\varphi_{\alpha\beta})
+g^{\overline{i}i}\lambda_{1}^{\alpha\beta}e_i\overline{e}_i(\varphi_{\alpha\beta})\nonumber\\
&+g^{\overline{i}i}\lambda_{1}^{\alpha\beta}\varphi_{\gamma\beta}e_i\overline{e}_i(g_0^{\alpha\gamma})
-g^{\overline{i}i}\lambda_{1}^{\alpha\beta}B_{\gamma\beta}e_i\overline{e}_i(g_0^{\alpha\gamma})\nonumber\\
&-g^{\overline{i}i}\lambda_{1}^{\alpha\beta}[e_i,\overline{e}_i]^{(0,1)}(\varphi_{\alpha\beta})
-\lambda_{1}^{\alpha\beta}\nabla_{\alpha\beta}^2\dot{\varphi}\nonumber\\
\geq&2\sum\limits_{\alpha>1}g^{\overline{i}i}\frac{|e_i(\varphi_{V_{\alpha}V_{1}})|^2}{\lambda_1-\lambda_{\alpha}}
+g^{\overline{i}i}e_i\overline{e}_i(\varphi_{V_1V_1})
-g^{\overline{i}i}[e_i,\overline{e}_i]^{(0,1)}(\varphi_{V_1V_1})\nonumber\\
&-\dot{\varphi}_{V_1V_1}-C\lambda_1\sum\limits_{i}g^{\overline{i}i}.\nonumber\end{aligned}$$ Since $\varphi_{V_{\alpha}V_{\beta}}=V_{\alpha}V_{\beta}(\varphi)-\left(\nabla_{V_{\alpha}}V_{\beta}\right)(\varphi)$ for any $\alpha$ and $\beta$, we have $$\begin{aligned}
\label{eieiuv1v11}
g^{\overline{i}i}\left(e_i\overline{e}_i-[e_i,\overline{e}_i]^{(0,1)}\right)(\varphi_{V_1V_1})
=g^{\overline{i}i}\left(e_i\overline{e}_i-[e_i,\overline{e}_i]^{(0,1)}\right)\left( V_1V_1(\varphi)-\left(\nabla_{V_{1}}V_{1}\right)(\varphi)\right).\end{aligned}$$ From now on, we write $G$ for term bounded by $C\lambda_1\sum\limits_{i}g^{\overline{i}i}$ which may change from line to line. At $(x_0,t_0)$, noting that $|\partial \varphi|_{g_0}\leq C$, a direct computation yields $$\begin{aligned}
\label{eieiuv1v12}
&g^{\overline{i}i}\left(e_i\overline{e}_i-[e_i,\overline{e}_i]^{(0,1)}\right)\left(\nabla_{V_{1}}V_{1}\right)(\varphi)\\
=&g^{\overline{i}i}e_i\left(\nabla_{V_{1}}V_{1}\right)\overline{e}_i(\varphi)
-g^{\overline{i}i}\left(\nabla_{V_{1}}V_{1}\right)[e_i,\overline{e}_i]^{(0,1)}(\varphi)+G\nonumber\\
=&g^{\overline{i}i}\left(\nabla_{V_{1}}V_{1}\right)e_i\overline{e}_i(\varphi)
-g^{\overline{i}i}\left(\nabla_{V_{1}}V_{1}\right)[e_i,\overline{e}_i]^{(0,1)}(\varphi)
+G\nonumber\\
=&g^{\overline{i}i}\left(\nabla_{V_{1}}V_{1}\right)\left(e_i\overline{e}_i(\varphi)-[e_i,\overline{e}_i]^{(0,1)}(\varphi)\right)
+G\nonumber\\
=&g^{\overline{i}i}\left(\nabla_{V_{1}}V_{1}\right)\left(g_{i\overline{i}}-\hat g_{i\overline{i}}\right)
+G\nonumber\\
=&g^{\overline{i}i}\left(\nabla_{V_{1}}V_{1}\right)(g_{i\overline{i}})+G\nonumber\end{aligned}$$ and $$\begin{aligned}
\label{giieieiv1v1}
&g^{\overline{i}i}\left(e_i\overline{e}_iV_{1}V_{1}(\varphi)-[e_i,\overline{e}_i]^{(0,1)}V_1V_1(\varphi)\right)\\
=&g^{\overline{i}i}V_1V_1\left(e_i\overline{e}_i(\varphi)-[e_i,\overline{e}_i]^{(0,1)}(\varphi)\right)
-2g^{\overline{i}i}[V_1,e_i]V_1\overline{e}_i(\varphi)-2g^{\overline{i}i}[V_1,\overline{e}_i]V_1e_i(\varphi)+G\nonumber\\
=&g^{\overline{i}i}V_1V_1\left(g_{i\overline{i}}-\hat g_{i\overline{i}}\right)
-2g^{\overline{i}i}[V_1,e_i]V_1\overline{e}_i(\varphi)-2g^{\overline{i}i}[V_1,\overline{e}_i]V_1e_i(\varphi)+G\nonumber\\
=&g^{\overline{i}i}V_1V_1(g_{i\overline{i}})
-2g^{\overline{i}i}[V_1,e_i]V_1\overline{e}_i(\varphi)-2g^{\overline{i}i}[V_1,\overline{e}_i]V_1e_i(\varphi)+G,\nonumber\end{aligned}$$ where we use the fact that $\hat\omega_t$ vary smoothly on $M\times[0,T_0]$ and hence can be estimated uniformly.
From , we get $$\begin{aligned}
\label{1apcma}
g^{\overline{i}i}\left(\nabla_{V_{1}}V_{1}\right)(g_{i\overline{i}})
=\left(\nabla_{V_{1}}V_{1}\right)(\dot\varphi)
+\left(\nabla_{V_{1}}V_{1}\right)(\log\Omega).\end{aligned}$$ Applying $V_1$ twice to implies $$\begin{aligned}
\label{2apcma}
g^{\overline{i}i}V_{1}V_1(g_{i\overline{i}})
=g^{\overline{p}p}g^{\overline{q}q}\left|V_1(g_{p\overline{q}})\right|^2+V_1V_1(\dot\varphi)+V_1V_1(\log\Omega).\end{aligned}$$ Combining , , , , and yields $$\begin{aligned}
\label{Llambda2}
&g^{\overline{i}i}e_i\overline{e}_i(\varphi_{V_1V_1})
-g^{\overline{i}i}[e_i,\overline{e}_i]^{(0,1)}(\varphi_{V_1V_1})\\
=&V_1V_1(\dot\varphi)-\left(\nabla_{V_{1}}V_{1}\right)(\dot\varphi)
+g^{\overline{p}p}g^{\overline{q}q}\left|V_1(g_{p\overline{q}})\right|^2\nonumber\\
&-2g^{\overline{i}i}[V_1,e_i]V_1\overline{e}_i(\varphi)-2g^{\overline{i}i}[V_1,\overline{e}_i]V_1e_i(\varphi)+G\nonumber\\
=&\dot{\varphi}_{V_1V_1}
+g^{\overline{p}p}g^{\overline{q}q}\left|V_1(g_{p\overline{q}})\right|^2
-2g^{\overline{i}i}[V_1,e_i]V_1\overline{e}_i(\varphi)-2g^{\overline{i}i}[V_1,\overline{e}_i]V_1e_i(\varphi)+G.\nonumber\end{aligned}$$ Thanks to and , we get .
\[lemzuihouyige\] At $(x_0,t_0)$, for any $\varepsilon\in(0,1/2]$, there holds $$\begin{aligned}
\label{Ltildeh1}
0\geq&(2-\varepsilon)\sum\limits_{\alpha>1}g^{\overline{i}i}\frac{|e_i(\varphi_{V_{\alpha}V_{1}})|^2}{\lambda_1\left(\lambda_1-\lambda_{\alpha}\right)}
+\frac{g^{\overline{p}p}g^{\overline{q}q}\left|V_1(g_{p\overline{q}})\right|^2}{\lambda_1}\\
&-(1+\varepsilon)\frac{g^{\overline{i}i}|e_i\left(\varphi_{V_{1}V_1}\right)|^2}{\lambda_1^2}
+\frac{\phi'}{2}\sum\limits_pg^{\overline{i}i}(|e_ie_p(\varphi)|^2+|e_i\overline{e}_p(\varphi)|^2)
-\frac{C}{\varepsilon}\sum\limits_{i}g^{\overline{i}i}\nonumber\\
&+\phi''g^{\overline{i}i}\left|e_i\left(|\partial \varphi|_{g_0}^2\right)\right|^2
-De^{-D(\varphi-\varphi_0)}\left(\Delta_{\omega}^C-{\frac{\partial}{\partial t}}\right)\varphi+D^2e^{-D(\varphi-\varphi_0)} |\partial\varphi|_g^2.\nonumber\end{aligned}$$
At $(x_0,t_0)$, we define $$\begin{aligned}
[V_1,e_i]=\sum\limits_{\alpha=1}^{2n}\tau_{i\alpha}V_{\alpha}\end{aligned}$$ for some $\tau_{i\alpha}\in\mathbb{C}$ uniformly bounded. This yields $$\begin{aligned}
\left|[V_1,e_i]V_1\overline{e}_i(\varphi)
+[V_1,\overline{e}_i]V_1\overline{e}_i(\varphi)\right|\leq C\sum\limits_{\alpha=1}^{2n}\left|V_{\alpha}V_1e_i(\varphi)\right|.\end{aligned}$$ Note that $$\begin{aligned}
V_{\alpha}V_1e_i(\varphi)
=&V_{\alpha}e_i V_1(\varphi)+V_{\alpha}\left[V_1,e_i\right](\varphi)\\
=&e_i V_{\alpha} V_1(\varphi)+\left[V_{\alpha},e_i\right]V_1(\varphi)+V_{\alpha}\left[V_1,e_i\right](\varphi)\\
=&e_i\left(\varphi_{V_{\alpha}V_1}\right)+e_i(\nabla_{V_{\alpha}}V_1)(\varphi)
+\left[V_{\alpha},e_i\right]V_1(\varphi)+V_{\alpha}\left[V_1,e_i\right](\varphi).\end{aligned}$$ Therefore, it follows that, at $(x_0,t_0)$, $$\begin{aligned}
\label{fiivieieibaru}
&2\frac{g^{\overline{i}i}\left([V_1,e_i]V_1\overline{e}_i(\varphi)
+[V_1,\overline{e}_i]V_1e_i(\varphi)\right)}{\lambda_1}\\
\leq&C\frac{g^{\overline{i}i}|e_i\left(\varphi_{V_{1}V_1}\right)|}{\lambda_1}
+C\sum\limits_{\alpha>1}\frac{g^{\overline{i}i}|e_i\left(\varphi_{V_{\alpha}V_1}\right)|}{\lambda_1}
+C\sum\limits_{i}g^{\overline{i}i}\nonumber\\
\leq&\varepsilon\frac{g^{\overline{i}i}|e_i\left(\varphi_{V_{1}V_1}\right)|^2}{\lambda_1^2}
+\varepsilon\sum\limits_{\alpha>1}\frac{g^{\overline{i}i}|e_i\left(\varphi_{V_{\alpha}V_1}\right)|^2}{\lambda_1(\lambda_{1}-\lambda_{\alpha})}
+\frac{C}{\varepsilon}\sum\limits_{i}g^{\overline{i}i}
+\frac{C}{\varepsilon}\sum\limits_{i}g^{\overline{i}i}\sum\limits_{\alpha>1}\frac{\lambda_1-\lambda_{\alpha}}{\lambda_1}\nonumber\\
\leq&\varepsilon\frac{g^{\overline{i}i}|e_i\left(\varphi_{V_{1}V_1}\right)|^2}{\lambda_1^2}
+\varepsilon\sum\limits_{\alpha>1}\frac{g^{\overline{i}i}|e_i\left(\varphi_{V_{\alpha}V_1}\right)|^2}{\lambda_1(\lambda_{1}-\lambda_{\alpha})}
+\frac{C}{\varepsilon}\sum\limits_{i}g^{\overline{i}i},\nonumber\end{aligned}$$ where we use the Cauchy-Schwarz inequality for $\varepsilon\in (0,1/2]$ and to get, at $(x_0,t_0)$, $$\begin{aligned}
\sum\limits_{\alpha>1}\frac{\lambda_1-\lambda_{\alpha}}{\lambda_1}
=&(2n-1)-\sum\limits_{\alpha>1}\frac{\lambda_{\alpha}(\nabla^2\varphi)-1}{\lambda_1}\\
=&2n+\frac{2n-1}{\lambda_1}-\frac{\Delta u}{\lambda_1}\leq 4n-1+C/\lambda_1\leq C,\end{aligned}$$ by the assumption that $\lambda_1(\nabla^2\varphi)\gg K\geq 1$.
At $(x_0,t_0)$, since we have $$\begin{aligned}
0\geq&\left(\Delta_{\omega}^C-{\frac{\partial}{\partial t}}\right)\tilde H\\
=&\frac{1}{\lambda_1}\left(\Delta_{\omega}^C-{\frac{\partial}{\partial t}}\right)\lambda_1-\frac{g^{\overline{i}i}|e_i(\lambda_1)|^2}{\lambda_1^2}
+\phi'\left(\Delta_{\omega}^C-{\frac{\partial}{\partial t}}\right)|\partial\varphi|_{g_0}^2
+\phi''g^{\overline{i}i}\left|e_i(|\partial\varphi|_{g_0}^2)\right|^2\nonumber\\
&-De^{-D\varphi}\left(\Delta_{\omega}^C-{\frac{\partial}{\partial t}}\right)\varphi+D^2e^{-D\varphi}|\partial\varphi|_{g}^2,\nonumber\end{aligned}$$ and a direct computation yields $e_{i}(\lambda_1)=e_i(\varphi_{V_1V_1})$, the inequality follows from , and .
In what follows, we use $C_D$ to denote the constant depending on the initial data and $D$, which is a uniform constant when $D$ is determined. We split up into different cases.
**Case 1:** Assume that $$\begin{aligned}
\label{case1condition}
g_{1\overline{1}}(x_0,t_0)\leq D^3 e^{-2D(\varphi(x_0,t_0)-\varphi_0)}g_{n\overline{n}}(x_0,t_0).\end{aligned}$$ Since $\tilde H$ attains maximum at $(x_0,t_0)$, we have $e_{i}(\tilde H)=0$, i.e., $$\begin{aligned}
\label{eiH}
\frac{e_i(\varphi_{V_1V_1})}{\lambda_1}=De^{-D(\varphi-\varphi_0)}e_i(\varphi)-\phi'e_i\left(|\partial\varphi|_{g_0}^2\right).\end{aligned}$$ Taking $\varepsilon=1/2$, combining and the Cauchy-Schwarz inequality yields $$\begin{aligned}
\label{case1}
&-\frac{3}{2}\frac{g^{\overline{i}i}|e_i(\varphi_{V_1V_1})|^2}{\lambda_1^2}\\
=&-\frac{3}{2}g^{\overline{i}i}\left|De^{-D(\varphi-\varphi_0)}e_i(\varphi)-\phi'e_i\left(|\partial\varphi|_{g_0}^2\right)\right|^2\nonumber\\
\geq&-6\left(\sup\limits_{M\times[0,{T_{\mathrm{max}}})}|\partial\varphi|_{g_0}^2\right)D^2e^{-2D(\varphi-\varphi_0)}\sum\limits_{i}g^{\overline{i}i}
-2(\phi')^2g^{\overline{i}i}\left|e_i\left(|\partial\varphi|_{g_0}^2\right)\right|^2\nonumber\\
=&-6\left(\sup\limits_{M\times[0,{T_{\mathrm{max}}})}|\partial\varphi|_{g_0}^2\right)D^2e^{-2D(\varphi-\varphi_0)}\sum\limits_{i}g^{\overline{i}i}
-\phi''g^{\overline{i}i}\left|e_i\left(|\partial\varphi|_{g_0}^2\right)\right|^2\nonumber\end{aligned}$$ From and , it follows that $$\begin{aligned}
\label{Ltildeh11}
0\geq&
-6\left(\sup\limits_{M\times[0,{T_{\mathrm{max}}})}|\partial\varphi|_{g_0}^2\right)D^2e^{-2D(\varphi-\varphi_0)}\sum\limits_{i}g^{\overline{i}i}
+\frac{\phi'}{2}\sum\limits_pg^{\overline{i}i}\left(|e_ie_p(\varphi)|^2+|e_i\overline{e}_p(\varphi)|^2\right)\\
&-De^{-D(\varphi-\varphi_0)}\left(\Delta_{\omega}^C-{\frac{\partial}{\partial t}}\right)\varphi -C\sum\limits_{i}g^{\overline{i}i},\nonumber\end{aligned}$$ where we discard some positive terms. Thanks to and , we can deduce $$\begin{aligned}
C_D\geq g_{1\overline{1}}(x_0,t_0)\geq\cdots\geq g_{n\overline{n}}(x_0,t_0)\geq C_D^{-1}.\end{aligned}$$ This, together with Lemma \[lemc0\] and , yields $$\begin{aligned}
\label{hessvarphi}
\sum\limits_{i,p}\left(|e_ie_p(\varphi)|^2+|e_i\overline{e}_p(\varphi)|^2\right)(x_0,t_0)\leq C_D,\end{aligned}$$ where we also use $$\begin{aligned}
\label{lapomegavarphiuse}
-De^{-D(\varphi-\varphi_0)}\Delta_{\omega}^C\varphi
=&-De^{-D(\varphi-\varphi_0)}g^{\overline{i}i}\left(g_{i\overline{i}}-\hat g_{i\overline{i}}\right)\\
\geq& -nDe^{-D(\varphi-\varphi_0)}+C_0^{-1}De^{-D(\varphi-\varphi_0)}\sum\limits_{i}g^{\overline{i}i}.\nonumber\end{aligned}$$ From Theorem \[thm1order\] and , it follows that $\lambda_1(x_0,t_0)$ and hence $\tilde H$ is bounded from above. This completes the proof of Case 1.
**Case 2:** At $(x_0,t_0)$, assume that $$\begin{aligned}
\label{case2condition}
\frac{\phi'}{4}\sum\limits_pg^{\overline{i}i}\left(|e_ie_p(\varphi)|^2+|e_i\overline{e}_p(\varphi)|^2\right)
\geq 6\left(\sup\limits_{M\times[0,{T_{\mathrm{max}}})}|\partial\varphi|_{g_0}^2\right)D^2e^{-2D(\varphi-\varphi_0)}\sum\limits_{i}g^{\overline{i}i}.\end{aligned}$$ By the same argument as in Case 1, we also have . From Lemma \[lemc0\], , and , it follows that, at $(x_0,t_0)$, $$\begin{aligned}
\label{case2use}
0\geq C^{-1}\sum\limits_p g^{\overline{i}i}\left(|e_ie_p(\varphi)|^2+|e_i\overline{e}_p(\varphi)|^2\right)+(C_0^{-1}D-C)\sum\limits_ig^{\overline{i}i}-C_D.\end{aligned}$$ We choose $D$ sufficiently large such that $C_0^{-1}D-C>1$. Then from , we can deduce the lower and upper bounds of $g_{i\overline{i}}$ and applying the same argument as in Case 1 to completes the proof of Case 2.
**Case 3:** At $(x_0,t_0)$, neither Case 1 nor Case 2 holds.
In this case, we need to estimate $$\begin{aligned}
(2-\varepsilon)\sum\limits_{\alpha>1}g^{\overline{i}i}\frac{|e_i(\varphi_{V_{\alpha}V_{1}})|^2}{\lambda_1\left(\lambda_1-\lambda_{\alpha}\right)}
+\frac{g^{\overline{p}p}g^{\overline{q}q}\left|V_1(g_{p\overline{q}})\right|^2}{\lambda_1}
-(1+\varepsilon)\frac{g^{\overline{i}i}|e_i\left(\varphi_{V_{1}V_1}\right)|^2}{\lambda_1^2}\end{aligned}$$ in Lemma \[lemzuihouyige\]. For this aim, we define $$I:=\Big\{i:\;g_{i\overline{i}}(x_0,t_0)> D^3 e^{-2D(\varphi(x_0,t_0)-\varphi_0)}g_{n\overline{n}}(x_0,t_0)\Big\}.$$ Since Case 1 does not hold, we have $1\in I$. Without loss of generality, we denote $I=\Big\{1,\cdots,j\Big\}$. By the similar argument of [@ctw Lemma 5.5], we get
\[lemyinyong\] At $(x_0,t_0)$, for any $(0,1/2]$, we have $$\begin{aligned}
-(1+\varepsilon)\sum\limits_{i\in I}\frac{g^{\overline{i}i}|e_i\left(\varphi_{V_{1}V_1}\right)|^2}{\lambda_1^2}
\geq -\sum\limits_{i}g^{\overline{i}i}-\phi''\sum\limits_{i\in I}g^{\overline{i}i}\left|e_i\left(|\partial\varphi|_{g_0}^2\right)\right|^2\end{aligned}$$
Assume that $\lambda_1(x_0,t_0)\geq C_D/\varepsilon^3$, where $D$ and $\varepsilon$ will be chosen uniformly later. The similar arguments in [@ctw Lemma 5.6, Lemma 5.7, Lemma 5.8] yield
\[lemyinyong2\] At $(x_0,t_0)$, for any $(0,1/6]$, we have $$\begin{aligned}
&(2-\varepsilon)\sum\limits_{\alpha>1}g^{\overline{i}i}\frac{|e_i(\varphi_{V_{\alpha}V_{1}})|^2}{\lambda_1\left(\lambda_1-\lambda_{\alpha}\right)}
+\frac{g^{\overline{p}p}g^{\overline{q}q}\left|V_1(g_{p\overline{q}})\right|^2}{\lambda_1}
-(1+\varepsilon)\sum\limits_{i\not\in I}\frac{g^{\overline{i}i}|e_i\left(\varphi_{V_{1}V_1}\right)|^2}{\lambda_1^2}\\
\geq& -3\varepsilon \sum\limits_{i\not\in I}\frac{g^{\overline{i}i}|e_i\left(\varphi_{V_{1}V_1}\right)|^2}{\lambda_1^2}-\frac{C}{\varepsilon}\sum\limits_{i}g^{\overline{i}i}.\end{aligned}$$
Since $\partial\tilde H(x_0,t_0)=0$, for any $\varepsilon\in(0,1/6]$, the Cauchy-Schwarz inequality yields $$\begin{aligned}
\label{2orderguji}
&-3\varepsilon \sum\limits_{i\not\in I}\frac{g^{\overline{i}i}|e_i\left(\varphi_{V_{1}V_1}\right)|^2}{\lambda_1^2}\\
=&-3\varepsilon \sum\limits_{i\not\in I}g^{\overline{i}i}\left|De^{-D(\varphi-\varphi_0)}e_i(\varphi)-\phi'e_i\left(|\partial\varphi|_{g_0}^2\right)\right|^2\nonumber\\
\geq&-6\varepsilon D^2e^{-2D(\varphi-\varphi_0)}|\partial\varphi|_{g}^2
-6\varepsilon (\phi')^2\sum\limits_{i\not\in I}g^{\overline{i}i}\left|e_i\left(|\partial\varphi|_{g_0}^2\right)\right|^2\nonumber\\
\geq&-6\varepsilon D^2e^{-2D(\varphi-\varphi_0)}|\partial\varphi|_{g}^2
-\phi''\sum\limits_{i\not\in I}g^{\overline{i}i}\left|e_i\left(|\partial\varphi|_{g_0}^2\right)\right|^2.\nonumber\end{aligned}$$ From , , , Lemma \[lemc0\], Lemma \[lemyinyong\] and Lemma \[lemyinyong2\], it follows that, at $(x_0,t_0)$, $$\begin{aligned}
\label{2jiezuihouyigeguji}
0\geq&\frac{\phi'}{2}\sum\limits_pg^{\overline{i}i}(|e_ie_p(\varphi)|^2+|e_i\overline{e}_p(\varphi)|^2)
+\left(C_0^{-1}De^{-D(\varphi-\varphi_0)}-\frac{C_2}{\varepsilon}\right)\sum\limits_{i}g^{\overline{i}i}\\
&-C_2De^{-D(\varphi-\varphi_0)}
+\left(D^2e^{-D(\varphi-\varphi_0)}-6\varepsilon D^2e^{-2D(\varphi-\varphi_0)}\right) |\partial\varphi|_g^2.\nonumber\end{aligned}$$ Choose $$D=C_0(6C_2+1),\quad \varepsilon=e^{D\left(\varphi(x_0,t_0)-\varphi_0\right)}/6\in(0,1/6].$$ Now at $(x_0,t_0)$, by Theorem \[thm1order\] and , we can deduce $$\sum\limits_p g^{\overline{i}i}\left(|e_ie_p(\varphi)|^2+|e_i\overline{e}_p(\varphi)|^2\right)+\sum\limits_i g^{\overline{i}i}\leq C,$$ for a uniform constant $C>0$. The similar discussion in Case 1 yields the upper bound for $\lambda_1(x_0,t_0)$ and hence for $\tilde H$, which completes the proof of Case 3.
Proof of the maximum time existence theorem {#sec6}
===========================================
In this section, using the priori estimates established earlier, we get the higher order priori estimates and complete the proof of Theorem \[mainthm\].
By Theorem \[thm2order\], there exists a uniform constant $C>0$ such that $$C^{-1}\omega_0\leq\omega\leq C\omega_0.$$ From the higher order estimates of [@chu1607] using the $C^{2,\alpha}$ estimate of [@chujianchuncvpde] (see also [@TWWY]), it follows that for each $k=0,1,2,\cdots$, there exists uniform constants $C_k>0$ such that $$\|\varphi(t)\|_{C^k(\omega_0)}\leq C_k.$$ These uniform estimates on $[0,{T_{\mathrm{max}}})$ imply that we can take limits of $\varphi(t)$ and get a solution on $[0,{T_{\mathrm{max}}}]$. Applying the standard parabolic short time existence theory we get a solution a little beyond ${T_{\mathrm{max}}}$, a contradiction. Hence there exists a solution $\varphi$ to on $[0,T_0)$. Taking ${\sqrt{-1}\partial\overline{\partial}}$ on both sides of , we get a solution to on $[0,T_0)$. Since $T_0<T$ is arbitrary, we get a solution to on $[0,T)$. Uniqueness follows from the uniqueness of the solution to . As mentioned before, the flow cannot extend beyond $T$.
Some convergence results {#sec7}
========================
In this section, we consider some convergence results of the flow . Firstly, let us recall the case when there exists a volume form $\Omega$ with ${\mathrm{Ric}^{(1,1)}}(\Omega)=0$ considered by Chu [@chu1607]. The similar argument as in Lemma \[dengjiadingyi\], yields that
Assume that there exists a volume form $\Omega$ with ${\mathrm{Ric}^{(1,1)}}(\Omega)=0$. A smooth family $\omega(t)$ of almost Hermitian metrics on $[0,\infty)$ solves the flow if and only if there is a family of smooth functions $\varphi(t)$ for $t\in [0,\infty)$ solves $$\begin{aligned}
{\frac{\partial}{\partial t}}\varphi=\log\frac{\left(\omega_0+{\sqrt{-1}\partial\overline{\partial}}\varphi\right)^n}{\Omega},\quad \omega_0+{\sqrt{-1}\partial\overline{\partial}}\varphi>0,\quad \varphi(0)=0,\end{aligned}$$ with $\omega(t)=\omega_0+{\sqrt{-1}\partial\overline{\partial}}\varphi$.
Chu [@chu1607] shows that $\varphi(t)$ converge to $\varphi_{\infty}$ smoothly as $t\longrightarrow \infty$ and $\omega_{\infty}=\omega_0+{\sqrt{-1}\partial\overline{\partial}}\varphi_{\infty}>0$ with ${\mathrm{Ric}^{(1,1)}}(\omega_{\infty})=0$.
If the complex structure $J$ is integrable, then this assumption is equivalent to the fact that $c_1^{\mathrm{BC}}(M)=0$, and the convergence result belongs to Gill [@gill] who used the crucial zero order estimate from [@TW2].
Secondly, we assume that there exists a volume form $\Omega$ with ${\mathrm{Ric}^{(1,1)}}(\Omega)<0$, which is equivalent to the fact that $M$ is Kähler manifold with negative first Chern class if $J$ is integrable. By Theorem \[mainthm\], the flow exists for all the time and we denote by $\tilde \omega(s)$ its solution.
Suppose that $t=\log(s+1)$ and $\omega=\frac{\tilde\omega}{s+1}$. We get a new metric which solves $$\begin{aligned}
\label{nacrf}
{\frac{\partial}{\partial t}}\omega=-{\mathrm{Ric}^{(1,1)}}(\omega)-\omega,\quad \omega(0)=\omega_0,\end{aligned}$$ for $t\in[0,\infty)$. We claim that the flow is equivalent to the parabolic Monge-Ampère equation $$\begin{aligned}
\label{napcma}
{\frac{\partial}{\partial t}}\varphi=\log\frac{(\hat\omega+{\sqrt{-1}\partial\overline{\partial}}\varphi)^n}{\Omega}-\varphi,\quad \hat\omega+{\sqrt{-1}\partial\overline{\partial}}\varphi>0,\quad \varphi|_{t=0}=0,\end{aligned}$$ where $\hat\omega=-{\mathrm{Ric}^{(1,1)}}(\Omega)+e^{-t}\left({\mathrm{Ric}^{(1,1)}}(\Omega)+\omega_0\right)$ with $$\begin{aligned}
\label{evolutionhatomega}
{\frac{\partial}{\partial t}}\hat\omega=-{\mathrm{Ric}^{(1,1)}}(\Omega)-\hat\omega,\quad\hat\omega|_{t=0}=\omega_0.\end{aligned}$$ Indeed, assume that $\omega$ is the solution to . By and , we know that $$\begin{aligned}
\label{gai3}
{\frac{\partial}{\partial t}}(\omega-\hat\omega)=-(\omega-\hat\omega)+{\sqrt{-1}\partial\overline{\partial}}\log\frac{\omega^n}{\Omega}.\end{aligned}$$ We define $\varphi(t)$ by $$\begin{aligned}
\varphi(t):=e^{-t}\int_0^te^s\log\frac{\omega(s)^n}{\Omega}{\mathrm{d}}s,\end{aligned}$$ for any $t\in [0,\infty)$. This function $\varphi(t)$ satisfies $$\begin{aligned}
\label{shoulianyong}
{\frac{\partial}{\partial t}}\varphi=\log\frac{\omega^n}{\Omega}-\varphi,\quad \varphi(0)=0.\end{aligned}$$ Thanks to and , we can deduce that $${\frac{\partial}{\partial t}}\left(e^t(\omega-\hat\omega-{\sqrt{-1}\partial\overline{\partial}}\varphi)\right)=0,\quad (\omega-\hat\omega-{\sqrt{-1}\partial\overline{\partial}}\varphi)|_{t=0}=0,$$ which implies $\omega=\hat\omega+{\sqrt{-1}\partial\overline{\partial}}\varphi>0$ for all $t\geq 0$.
Conversely, assume that $\varphi$ is the solution to . Taking ${\sqrt{-1}\partial\overline{\partial}}$ on the both sides of , we know that $\omega=\hat\omega+{\sqrt{-1}\partial\overline{\partial}}\varphi$ is the solution to .
Note that $\hat\omega$ converge smoothly to $-{\mathrm{Ric}^{(1,1)}}(\Omega)>0$ as $t\longrightarrow \infty$. Hence there exists a uniform constant $C_0>0$ such that $$\begin{aligned}
\label{cankaoduliangdengjia}
C_0^{-1}\omega_0\leq\hat\omega\leq C_0\omega_0.\end{aligned}$$ It is easy to see that Theorem \[thm2\] follows from
Let $\varphi(t)$ be the solution to on $M\times[0,\infty)$. Then $\varphi$ converge smoothly to $\varphi_{\infty}$ as $t\longrightarrow \infty$, and we have $$\begin{aligned}
{\mathrm{Ric}^{(1,1)}}(\omega_{\infty})=-\omega_{\infty},\end{aligned}$$ where $\omega_{\infty}=-{\mathrm{Ric}^{(1,1)}}(\Omega)+{\sqrt{-1}\partial\overline{\partial}}\varphi_{\infty}>0$.
The proof consists of several steps as follows.
*Step 1:* We deduce the uniform estimates for $\varphi$ and $\dot\varphi:={\frac{\partial}{\partial t}}\varphi$ using the method from [@Ca; @TZ; @Ts] (see also [@twjdg]). A simple maximum principle argument yields that $|\varphi|\leq C$ for a uniform constant $C>0$ independent of $t$. A direct computation gives $$\begin{aligned}
\left({\frac{\partial}{\partial t}}-\Delta_{\omega}^C\right)\dot \varphi
=&-\dot\varphi-{\mathrm{tr}}_{\omega}\left({\mathrm{Ric}^{(1,1)}}(\Omega)+\hat\omega\right)\\
=&-\dot\varphi-n+\Delta_{\omega}^C\varphi-{\mathrm{tr}}_{\omega}\left({\mathrm{Ric}^{(1,1)}}(\Omega)\right),\end{aligned}$$ i.e., $$\begin{aligned}
\left({\frac{\partial}{\partial t}}-\Delta_{\omega}^C\right)\left(\dot\varphi+\varphi\right)=-n-{\mathrm{tr}}_{\omega}\left({\mathrm{Ric}^{(1,1)}}(\Omega)\right).\end{aligned}$$ At the minimum $(x_0,t_0)$ of $\varphi+\dot\varphi$, without loss of generality, we can assume $t_0>0$, otherwise we can get the lower bound of $\dot\varphi$ directly. At $(x_0,t_0)$, we have $-{\mathrm{tr}}_{\omega}\left({\mathrm{Ric}^{(1,1)}}(\Omega)\right)\leq n$. This, together with ${\mathrm{Ric}^{(1,1)}}(\Omega)<0$ and the arithmetic-geometric means inequality, yields $$\dot\varphi+\varphi=\log\frac{\omega^n}{\Omega}
=\log\frac{\omega^n}{\left(-{\mathrm{Ric}^{(1,1)}}(\Omega)\right)^n}+\log\frac{\left(-{\mathrm{Ric}^{(1,1)}}(\Omega)\right)^n}{\Omega}
\geq\log\frac{\left(-{\mathrm{Ric}^{(1,1)}}(\Omega)\right)^n}{\Omega}\geq-C$$ at this point and hence everywhere. Since $|\varphi|\leq C$, we get $\dot\varphi\geq-C$.
Since $e^t\left({\mathrm{Ric}^{(1,1)}}(\Omega)+\hat\omega\right)={\mathrm{Ric}^{(1,1)}}(\Omega)+\omega_0$, a direct computation yields $$\left({\frac{\partial}{\partial t}}-\Delta_{\omega}^C\right)\left(\dot\varphi+\varphi+nt-e^t\dot\varphi\right)={\mathrm{tr}}_{\omega}\omega_0>0.$$ This, together with the maximum principle, implies that there holds $\dot\varphi\leq Cte^{-t}\leq Ce^{-t/2}$ for $t\geq 1$. The uniform upper bound of $\varphi+\dot\varphi=\log\frac{\omega^n}{\Omega}$ and the arithmetic-geometric means inequality yield that $$\begin{aligned}
\label{guanjianxiajie}
{\mathrm{tr}}_{\omega}\omega_0\geq n\left(\frac{\omega_0^n}{\omega^n}\right)^{1/n}= n\left(\frac{\Omega}{\omega^n}\frac{\omega_0^n}{\Omega}\right)^{1/n}\geq c,\end{aligned}$$ for a uniform constant $c>0$.
Given and , we can deduce the first and second order estimates.
*Step 2*: We need the estimate $\sup\limits_{M\times[0,\infty)}|\partial\varphi|_{g_0}\leq C$ for a uniform constant $C>0$. To see this, we just need to follow the proof of Theorem \[thm1order\] word for word except replacing by $$\begin{aligned}
{\frac{\partial}{\partial t}}|\partial\varphi|_{g_0}^2
=&e_k(\dot\varphi)\overline{e}_k(\varphi)+e_k(\varphi)\overline{e}_k(\dot\varphi)\\
=&g^{\overline{i}i}e_k(g_{i\overline{i}})\overline{e}_k(\varphi)
-e_k(\log\Omega)\overline{e}_k(\varphi)
+g^{\overline{i}i}e_k(\varphi)\overline{e}_k(g_{i\overline{i}})
-e_k(\varphi)\overline{e}_k(\log\Omega),\nonumber\\
&-2e_k(\varphi)\overline{e}_k(\varphi).\nonumber\end{aligned}$$ In the process of calculation of $\left(\Delta_{\omega}^C-{\frac{\partial}{\partial t}}\right)|\partial\varphi|_{g_0}^2$, the term $-2e_k(\varphi)\overline{e}_k(\varphi)$ is harmless. Then all other arguments are the same and hence we get the first order estimate.
*Step 3*: We need the third order estimate $\sup\limits_{M\times[0,\infty)}|\nabla^2\varphi|_{g_0}\leq C$, where $\nabla^2\varphi$ is the Hessian with respect to the Levi-Civita connection of $g_0$. To see this, we still need to follow the proof of Theorem \[thm2order\] word for word except replacing and by $$\begin{aligned}
g^{\overline{i}i}\left(\nabla_{V_{1}}V_{1}\right)(g_{i\overline{i}})
=\left(\nabla_{V_{1}}V_{1}\right)(\dot\varphi)
+\left(\nabla_{V_{1}}V_{1}\right)(\log\Omega)+\left(\nabla_{V_{1}}V_{1}\right)(\varphi).\end{aligned}$$ and $$\begin{aligned}
g^{\overline{i}i}V_{1}V_1(g_{i\overline{i}})
=g^{\overline{p}p}g^{\overline{q}q}\left|V_1(g_{p\overline{q}})\right|^2+V_1V_1(\dot\varphi)+V_1V_1(\log\Omega)+V_1V_1(\varphi).\end{aligned}$$ Note that the new term $V_1V_1(\varphi)-\left(\nabla_{V_{1}}V_{1}\right)(\varphi)=\varphi_{V_1V_1}$ is harmless and absorbed by $G$, as required.
*Step 4*: The higher order estimates follows from [@chu1607].
*Step 5*: Convergence result. The second order estimate implies that $$C^{-1}\omega_0\leq \omega\leq C\omega_0$$ for a uniform constant $C>0$. This yields that $$\begin{aligned}
\left({\frac{\partial}{\partial t}}-\Delta_{\omega}^C\right)(e^t\dot\varphi)=-{\mathrm{tr}}_{\omega}\left({\mathrm{Ric}^{(1,1)}}(\Omega)+\omega_0\right)\geq-C.\end{aligned}$$ This, together with the maximum principle, implies that $\dot\varphi\geq -(1+t)e^{-t}\geq -Ce^{-t/2}$.
We can deduce that $\dot\varphi$ converge uniformly to zero exponentially fast, and hence $\varphi$ converge uniformly exponentially fast to a continuous limit function $\varphi_{\infty}$. From the higher order uniform estimates of $\varphi$, it follows that $\varphi_{\infty}$ is smooth and $$\varphi\longrightarrow \varphi_{\infty}$$ in smooth topology. Therefore, by passing to the limits in , we can deduce that the limit metric $\omega_{\infty}=-{\mathrm{Ric}^{(1,1)}}(\Omega)+{\sqrt{-1}\partial\overline{\partial}}\varphi_{\infty}$ satisfies $$\begin{aligned}
\label{sol1}
\log\frac{\omega_{\infty}^n}{\Omega}=\varphi_{\infty}.\end{aligned}$$ Taking ${\sqrt{-1}\partial\overline{\partial}}$ on the both sides of this equation, we get $${\mathrm{Ric}^{(1,1)}}(\omega_{\infty})=-\omega_{\infty},$$ as required.
*Step 6*: We prove that $\omega_{\infty}$ is independent of the initial metric $\omega_0$. Assume that the normalized flow starts at another almost Hermitian metric $\omega_0'$. The same argument as above implies that there exists a smooth function $\varphi'_{\infty}$ such that $$\begin{aligned}
\label{sol2}
(-{\mathrm{Ric}^{(1,1)}}(\Omega)+{\sqrt{-1}\partial\overline{\partial}}\varphi'_{\infty})^n=e^{\varphi_{\infty}'}\Omega,\quad \omega_{\infty}':=-{\mathrm{Ric}^{(1,1)}}(\Omega)+{\sqrt{-1}\partial\overline{\partial}}\varphi'_{\infty}>0.\end{aligned}$$ Thanks to and , we get $$\begin{aligned}
\label{sol3}
\omega_{\infty}'^n=(\omega_{\infty}+{\sqrt{-1}\partial\overline{\partial}}\phi)^n=e^{\phi}\omega_{\infty}^n,\end{aligned}$$ where we denote $\phi:=\varphi_{\infty}'-\varphi_{\infty}$. Applying the maximum principle to shows that $\phi\equiv0$, i.e., $$\omega_{\infty}'=\omega_{\infty},$$ as desired.
An example {#secexample}
==========
In this section, as an example, we study the flow on the (locally) homogeneous manifolds in more detail. We note that the flow can be seen as a $(p,q)$ flow given in [@laurent] and hence we can use the method in [@laurent].
Let $(M,J,\omega_0)$ be a compact almost Hermitian manifold whose universal cover is a Lie group $G$ such that if $\pi:G\longrightarrow M$ is the covering map, then $\pi^{\ast}\omega_0$ and $\pi^{\ast}J$ are left invariant. For example, we can take $M=G/\Gamma$, where $\Gamma$ is a compact discrete subgroup of $G$, including solvmanifolds and nilmanifolds. Then the solution to on $M$ is obtained by pulling down the corresponding flow solution on the Lie group $G$, which stays left invariant. The flow on $G$ becomes an ordinary differential equation for a family of almost Hermitian metrics $\omega(t)$ with respect to the fixed complex structure $J$ on the Lie algebra $\mathfrak{g}$ of the Lie group $G$.
It is sufficient to consider the flow on Lie group $G$. Since the Chern-Ricci form $p$ of a left invariant almost Hermitian metric $\omega$ defined by $$p(X,Y)=-\frac{1}{2}{\mathrm{tr}}\, J\, \mathrm{ad}[X,Y]+\frac{1}{2}{\mathrm{tr}}\, \mathrm{ad}\, J[X,Y],\quad \forall\,X,\,Y\in\mathfrak{g}$$ depends only on $J$ (see for example [@P] or [@V2 Proposition 4.2]), the flow becomes $${\frac{\partial}{\partial t}}\omega(t)=-2p^{(1,1)},\quad \omega(0)=\omega_0$$ with solution $\omega(t)$ given by $$\omega_t:=\omega(t)=\omega_0-2tp^{(1,1)}.$$ We define $$p^{(1,1)}=\omega_0(P_0\cdot,\cdot)=\omega_t(P(t)\cdot,\cdot),$$ which implies that $$\begin{aligned}
P_t:=P(t)=(\mathrm{Id}-2tP_0)^{-1}P_0.\end{aligned}$$ We call $P_0$ (resp. $P_t$) the Chern-Ricci operator of $\omega_0$ (resp. $\omega_t$). It follows that the maximal existence time $T$ is given by $$T=\left\{
\begin{array}{ll}
\infty, & \quad\text{if}\;P_0\leq 0, \\
1/(2p_{+}), & \quad\text{otherwise},
\end{array}
\right.$$ where $p_{+}$ is the maximal positive eigenvalue of the Chern-Ricci operator $P_0$ of $\omega_0$.
Let $p_1,\cdots,p_{2n}$ of eigenvalues of $P_0$ with the orthonormal basis $e_1,\cdots,e_{2n}$, i.e., $$\omega_0(e_i,\overline{e}_j)=\delta_{ij},\quad P_0(e_{\alpha})=p_{\alpha}e_{\alpha},\quad \alpha=1,\cdots,2n.$$ A direct calculation yields that the scalar curvature $R(\omega_t)$ of $\omega_t$ is given by $$\begin{aligned}
\label{shuliangqulv}
R(\omega_t)={\mathrm{tr}}_{\omega_t}p^{(1,1)}={\mathrm{tr}}P_t=\sum\limits_{\alpha=1}^{2n}\frac{p_{\alpha}}{1-2tp_{\alpha}}.\end{aligned}$$ From , it follows that $$\begin{aligned}
\frac{{\mathrm{d}}}{{\mathrm{d}}t}R(\omega_t)=\sum\limits_{\alpha=1}^{2n}\frac{2p_{\alpha}^2}{(1-2tp_{\alpha})^2}\geq0.\end{aligned}$$ This implies that $R(\omega_t)$ is strictly increasing unless $P_t\equiv 0$, i.e., $\omega_t\equiv\omega_0$, and that the Chern scalar curvature must blow up in finite singularities, i.e., if $T<\infty$, then there holds $$\int_0^{T}R(\omega_t){\mathrm{d}}t=\infty.$$ Also we note that $$\begin{aligned}
\label{scalar}
R(\omega_t)\leq \frac{C}{T-t},\end{aligned}$$ for a uniform constant $C>0$. We remark that is the claim of [@SW4 Conjecture 7.7] for the Kähler-Ricci flow on general compact Kähler manifolds (see [@gillscalar] for the Chern-Ricci flow on general compact Hermitian manifolds).
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[^1]: Supported by National Natural Science Foundation of China grant No. 11401023, and the author’s post-doc is supported by the European Research Council (ERC) grant No. 670846 (ALKAGE)
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---
abstract: 'The classical and quantum dynamics of the Friedmann-Robertson-Walker Universe with massless scalar and massive fermion matter field as a source is discussed in the framework of the Dirac generalized Hamiltonian formalism. The Hamiltonian reduction of this constrained system is realized for two cases of minimal and conformal coupling between gravity and matter. It is shown that in both cases for all values of curvature, $k= 0, \pm 1, $ of maximally symmetric space there exists a time independent reduced local Hamiltonian which describes the dynamics of the cosmic scale factor. The relevance of conformal time-like Killing vector fields in FRW space-time to the existence of time independent Hamiltonian and the corresponding notion of conserved energy is discussed. The extended quantization with the Wheeler-deWitt equation is compared with the canonical quantization of unconstrained system. It is shown that quantum observables treated as expectation values of the Dirac observables properly describe the original classical theory.'
address:
- '$^a$ A. Razmadze Mathematical Institute, Tbilisi, 380093, Georgia'
- '$^b$ Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia'
- '$^c$ Laboratory of Information Technologies, Joint Institute for Nuclear Research, 141980 Dubna, Russia'
author:
- 'A. M. Khvedelidze $^a$ $^b$, and Yu.G. Palii $^c$'
title: Generalized Hamiltonian Dynamics of Friedmann Cosmology with Scalar and Spinor Matter Source Fields
---
Introduction
============
Cosmological models apart from the main task, to investigate the large scale structure of the Universe, are highly attractive objects with the standpoint of analysis of the conceptual problems in the theory of gravitation. By studying cosmological models instead of general spacetime we can to overcome the difficulties due to the infinite number of degrees of freedom and concentrate attention to the problems arising solely from the time reparametrization invariance; such as the construction of observables. [^1] In the present article we attempt a contribution to the discussion of some aspects of this problem by considering the simplest cosmological model, the Friedmann-Robertson-Walker (FRW) Universe filled in the scalar massless and massive spinor matter fields. The conventional Hamiltonian description of this model is based on the original Dirac [@Dir58] and the so-called Arnowitt-Deser-Misner (ADM) [@ADM] formulation of general relativity. [^2] The ADM method involves the choice of certain coordinate fixing conditions (gauge), solution of the constraints and construction of the observables such as energy, momentum and angular momentum, using the asymptotically flat boundary condition for gravitational field and assuming that three-dimensional space of constant time is open [@ReggeTeit]. However, when the closed Universe is considered to build the ADM observables from initial data for canonical variables it is impossible. Since in this case there is no boundary of the space manifold and no asymptotic region can be used to construct the corresponding integrals of motion. This leads to the conclusion that for such cosmological models neither the natural notion of time evolution nor the corresponding energy definition is possible to find [@Faddeev], [@Torre]. To clear up this contradiction between the existence of widely used cosmological quantities and their absence in the corresponding field theoretical formulation the FRW cosmological model will be considered in the framework of the Dirac Generalized Hamiltonian formulation [@DiracL]-[@HenTeit]. The key moment of the canonical treatment is the assumption that general relativity represents “already parametrized” theory due to the principle of general covariance, so that the problem of construction of observables can be solved automatically rewriting the theory in the equivalent “deparametrized” form. [^3] However, careful analysis of correctness of such deparametrized program carried out by Hajicek [@HajicekTop] shows that even for simple mechanical system with one quadratic Hamiltonian constraint there are topological obstructions to its implementation analogous of the well-known “Gribov ambiguity” in gauge theories. A direct way to clarify the topological structure of such a theory lies in the finding of integral curves of the dynamical equations and the investigation their global properties. Within this motivation the present note is devoted to the realization of local deparametrization of integrable cosmological FRW models considering it as a preparation for the study the global features of reduction procedure. [^4] We will follow the method of Hamiltonian reduction to construct the observables and the corresponding dynamical equations which is well elaborated for gauge theories. This approach is based on an appropriate choice of canonical coordinates on phase space and deals without explicit introduction of any gauge-fixing functions (see e.g. [@GKP] and references therein).
The general plan of present article is as follows. In Section II we state the FRW cosmological model with real massless scalar and massive fermion fields as sources with different type of coupling to gravity. In Section III some generic features of the Hamiltonian reduction and the construction of observables in reparametrization invariant mechanical models is disscused. The aim of Section III is to explain the method to obtain the unconstrained system from reparametrized invariant one by considering the simplest example of free relativistic particle motion. Section IV is devoted to the construction of the unconstrained systems equivalent to FRW cosmology when the homogeneous matter is presented in different forms: as massless scalar field interacting with gravity minimally and conformaly, massive spinor field. Finally, in Section V we discuss the correspondence principle fulfillment for observables in quantum theories based either on Wheeler-deWitt equation or on the canonical quantization scheme of the unconstrained classical system. In Appendices we state some notations and technical details of derivations in order to simplify the reading of the main text.
Model with spatial homogeneity and isotropy
===========================================
By definition, the FRW spacetime is a four-dimensional pseudo-Riemannian manifold on which a six-dimensional Lie group $G_6$ acts as group of isometries. The group of isometries $G_6$ has a three-dimensional isotropy subgroup and three-dimensional subgroup which acts simply transitive on the one parameter (“time $t$”) family of spacelike hypersurfaces $ \Sigma_t $. The large group of isometries restricts both the dependence and the number of independent components of the metric tensor and leads to the so-called maximally symmetric three-dimensional space. After the choice of standard coordinates [@MTW] one has the FRW metric $$\label{eq:FRWmetric}
ds^2 = - N^2(t)\, dt \otimes dt +
a^2(t)\, \gamma_{ab} \, d{x}^a \otimes d{x}^b ,$$ where $ \gamma_{ab}$ is the time independent metric of three-dimensional space $$\gamma_{ab}\, d{x}^a \otimes d{x}^b = \frac{dr^2}{1-\frac{kr^2}{r_o^2}} +
r^2(d\theta^2 + \sin^2\theta d\varphi^2)$$ of constant curvature $
{}^{(3)}R(\gamma_{ij})=-6k/r_o^2, \quad k = 0, \pm 1 $. The lapse function $ N(t) $ and the cosmic scale factor $ a(t) $ describe the remaining gravitational degrees of freedom whose classical behavior is determined by varying the standard Hilbert action. However, constructed in this way the minisuperspace model is out of interest. Simple counting of the physical degrees of freedom shows that this vacuum FRW model is empty on the classical level; only unphysical degrees of freedom propagate. Thus in order to have some nontrivial observables it is necessary to introduce the source matter fields.
### Lagrangian for scalar field with minimal coupling to gravity
The introduction of a massless scalar field as a source of gravity results in the simplest cosmological model which has direct correspondence to the classical Friedmann model. [^5] For a massless scalar field, the two most interesting couplings to gravity extensively considered are the so-called minimal coupling and the conformal one. [^6]
The Hilbert action for gravity minimally coupled to massless scalar field $$W=\int d^4x \sqrt{-g}\left[-\frac{{}^{4}R}{2\kappa}+
\frac{1}{2}g^{\mu \nu}\partial_\mu\Phi\partial_\nu\Phi\right]$$ reduces to the following $$\label{tru}
W=V_{(3)}
\int dt \left[-\frac{3}{\kappa} \left(
\frac{\dot{a}^2}{N_c} - \frac{ka^2}{r^2_o} N_c \right) +
\frac{a^2}{2N_c} \dot{\Phi}^2
+ \frac{3}{\kappa}
\frac{d}{dt}\left(\frac{a\dot a}{N_c}\right)\right]\;,$$ assuming the spatial homogeneity of the scalar field and FRW metric (\[eq:FRWmetric\]). Here $\kappa=8\pi G$ and new variable $
N_c = {N}/{a}$ has been introduced. Integration over the spatial hyperplane leads to the appearance of the factor $V_{(3)}$ – “volume” of the three-dimensional space of constant curvature. [^7]
### Lagrangian for a scalar field with conformal coupling to gravity
The conformally coupled scalar field is described by action $$\label{eq:lagrG+M}
W[ g, \Phi ] = \int\limits_{}^{}d^4x\sqrt{-g}\left[-\frac{1}{2\kappa}\,
{}^{(4)}R
+ \frac{1}{12}\, {}^{(4)}R\Phi^2 + \frac{1}{2}
g^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi\right].$$ Choosing the metric (\[eq:FRWmetric\]) this leads to the action for the FRW Universe filled in by massless homogeneous scalar field ${\varphi(t)}:= {a(t)}\Phi (t)$ $$\begin{aligned}
\label{eq:lagrFWW+MS}
&&W[a, N_c, \varphi] = \int dt\left[-
\frac{3}{\kappa}\left(\frac{\dot a^2}{N_c}-
\frac{ka^2}{r_o^2}N_c\right)+ \frac{1}{2}\left(\frac{\dot \varphi^2}{N_c}-
\frac{k\varphi^2}{r_o^2}N_c\right) +
\frac{3}{\kappa}\frac{d}{dt}\left(\frac{\dot a a}{N_c}\right)\right]\,.\end{aligned}$$
### FRW Lagrangian with spinor matter fields
The combined system of Dirac field and FRW metric have been investigated from classical and quantum point of view by many authors. In present article we explore the model closely related to the formulation given in [@Isham; @Zanelli]. The starting point is the action for a massive spinor field interacting with gravity is given by $$\label{EH}
W=\int d^4x\sqrt{-g}\left[-\frac{{}^{(4)}R(g)}{2\kappa}+
\frac{i}{2}\left(\bar\Psi\gamma^\mu(x)\bigtriangledown_\mu\Psi-
\bigtriangledown_\mu\bar\Psi\gamma^\mu(x)\Psi\right)
-m\bar\Psi\Psi\right],$$ where the spinor field $\Psi(x)$ ( $\bar\Psi$ Dirac conjugate spinor field) components are treated classically as a collection of Grassmann variables $\Psi_i\Psi_j +\Psi_j\bar\Psi_i=0 $ and $\bigtriangledown_\mu$ is the covariant derivative (see notation in Appendix \[appA\].). Assuming the homogeneity of the fermion fields and after the redefinition $\psi(t):=a^{3/2}(t)\Psi(t) $ Eq.(\[EH\]) reduces to the action of the finite dimensional system $$\label{S}
W=\int dt\left[
-\frac{3}{\kappa}\left(\frac{\dot a^2}{N_c}-
\frac{ka^2}{r_o^2}N_c\right)
+\frac{i}{2}(\bar \psi\gamma^o\dot\psi
-\dot{\bar\psi}\gamma^o\psi)
- a N_c{\cal H}_D
+\frac{3}{\kappa}\frac{d}{dt}\left(\frac{a\dot a}{N_c}\right)
\right]\;,$$ with $$\label{HD}
{\cal H}_D=m\bar{\psi}{\psi}.$$
Reduction and observables in reparametrization invariant mechanical models
==========================================================================
It is the purpose of this part to discuss the construction of observables for a system with reparametrization invariance. For our aims we shall state the ideas using a mechanical system, i.e. a system with a finite number of degrees of freedom and restrict ourselves to the case of Abelian constraints.
Let us consider a system with $ 2n $ - dimensional phase space $\Gamma $ whose dynamics is constrained to a certain submanifold $\Gamma_c $ describing by the functionally independent set of $ m $ abelian constraints $$\label{eq:constr}
\varphi_\alpha (p,q) = 0, \quad \quad \quad \quad
\{ \varphi_\alpha (p,q), \varphi_\beta (p,q)\} = 0\,.$$ Due to the presence of constraints the Hamiltonian dynamics is described by the Poincaré-Cartan form $$\label{eq:P-C}
\Theta = \sum_{i=1}^{n} \, p_i dq_i - H_E (p,q) dt \,,$$ with the extended Hamiltonian $H_E (p,q) $ differing from the canonical Hamiltonian $ H_C (p,q)$ by a linear combination of constraints with arbitrary multipliers $ u_\alpha (t) $ $$H_E (p,q) = H_C (p,q) + u_\alpha (t) \varphi_\alpha (p, q) \,.$$ For the case of first class constraints the functions $ u_\alpha (t) $ can’t be fixed without using some additional requirements. This observation reflects the existence of the local (gauge) symmetry and the presence of coordinates in the theory whose dynamics is governed in an arbitrary way. However, according to the principle of gauge invariance, these coordinates do not affect physical quantities and thus can be treated as ignorable (gauge degrees of freedom). The question is how to identify these coordinates. If theory contains only Abelian constraints one can find these ignorable coordinates as follows. It is always possible [@Levi-Civita] - [@Shanmugadhasan] to define a canonical transformation to a new set of canonical coordinates $$\begin{aligned}
\label{eq:cantr}
q_i & \mapsto & Q_i = Q_i \left ( q , p \right ),\nonumber\\
p_i & \mapsto & P_i = P_i \left ( q , p \right ),\end{aligned}$$ so that $ m $ of the new momenta ($\overline{P}_1, \dots ,\overline{P}_m $) become equal to the Abelian constraints $$\overline{P}_\alpha = \varphi _\alpha (q, p)\,.$$ In the new coordinates ($\overline{Q}, \overline{P}$) and ($ {Q}^{\ast}, {P}^{\ast}$) we have the following canonical equations $$\begin{aligned}
\label{eq:motion}
\dot{Q}^{\ast} = \{{Q}^{\ast}, H_{Ph} \},
&\quad \quad \quad & \dot{\overline{P}} = 0, \nonumber\\
\dot{P}^{\ast} = \{{P}^{\ast}, H_{Ph} \},
& \quad \quad \quad & \dot{\overline{Q}} = u(t) ,\end{aligned}$$ with the physical Hamiltonian $$\label{eq:PhysHam}
H_{Ph}(P^{\ast}, Q^{\ast}) := H_C(P, Q)\,
\Bigl\vert_{ \overline{P}_\alpha = 0}.$$ The physical Hamiltonian $H_{Ph}$ depends only on the $( n-m )$ pairs of new gauge-invariant canonical coordinates $( {Q}^{\ast}, {P}^{\ast} )$. Moreover the form of the canonical system (\[eq:motion\]) expresses the explicit separation of the phase space into physical and unphysical sectors. Arbitrary functions $ u(t) $ enter only into the part the equation for the ignorable coordinates $ \overline{Q}_\alpha $, conjugated to the momenta ${\overline{P}}_\alpha$. [^8] Trying to apply this program to any model with reparametrization invariance we as a rule reveal that the physical Hamiltonian defined by (\[eq:PhysHam\]) is zero and thus we have the dynamics of unconstrained system in the Maupertuis form $$\label{eq:P-C-M}
\Theta_{Ph} = \sum_{i=1}^{n-m} \, P^\ast_i d Q^\ast_i - d V ,$$ where $dV$ is a total differential. The problem is now how to deal with the zero Hamiltonian. This situation in some sence opposite to the case known from the Hamilton-Jacobi method of integration of equations of motion. The main idea of this method is to implement on the system with Hamiltonian $ H(t,p,q) $ the canonical transformation with generating function $ S(t, q,p) $, which is the solution for the equation $$\frac{\partial S}{\partial t} + H\left(
t, q, \frac{\partial S}{\partial q}\right) = 0\,.$$ As a result the new Hamiltonian is zero and the equation of motion in the new coordinates have the simplest form $$\dot{Q} = 0, \quad \quad \dot{P} = 0 \,.$$ After reduction we have just a system in these coordinates and the problem is to reconstruct the nonzero Hamiltonian in any other coordinates for the obtained uncontsrained system. Two remarks to the picture described above may be in order. There is no difference between the local behavior in systems obtained via the reduction of reparametrization invariant theories. The specific properties, which make a difference of systems are hidden in the total differential in the Poincaré-Cartan form.
Before passing to the construction of the reduced phase space for FRW Universe it seems worth to set forth our approach to the same problem of a free relativistic particle.
Digress: Reduced dynamics of free relativistic particle
--------------------------------------------------------
For the presentation of our procedure to construct the reduced dynamical system from the degenerate system with reparametrization invariance let us start with the simplest case of free motion of a particle in Minkowski space-time writing its action in the form close to the cosmological Friedmann models (\[tru\]), (\[eq:lagrFWW+MS\]), (\[S\]) $$\label{parac}
W[x,e] := \frac{1}{2} \int\limits_{T_1}^{T_2} d\tau
\left ( \frac{{\dot{x}}_\mu^2}{e} + e m^2 \right ).$$ The independent configuration variables are particle wordline coordinates $x_\mu (\tau)$ and the additional “vielbein” determinant $e(\tau)$.
Invariance of the action (\[parac\]) under the reparametrization of time $ \tau \rightarrow \tau'=f(\tau) $ spoils the uniqueness of the Cauchy problem for the corresponding equations of motion. Therefore the problem is to fix the part of the variables whose dynamics will be unique and whose initial conditions are free from any constraints. The usual way to deal with this problem consists in choosing of a gauge which tights the parameter of evolution with the configuration variables. For example, the proper time gauge fixing $ x_0(\tau) = \tau $ leads to the instant form of the dynamics for a relativistic particle. However, let us act in spirit of the previous section and try to reproduce the results of the instant form of particle dynamics without introduction of gauge conditions.
According to the Dirac prescription the generalized Hamiltonian dynamics for the system (\[parac\]) takes place on the phase space spanned by five canonical pairs ($ e, p_e$) and ($ x_\mu ,p_\mu $) restricted by the primary constraint $
p_{e}=0\;
$ and the secondary constraint $$p_\mu p^\mu - m^2 = 0 \,.$$ To take into account these constraints and to derive equations of motion one can consider the Poincare-Cartan 1-form $$\label{P-Cp}
\Theta: = p_e de+ p_\mu dx^\mu - H_T d\tau\;,$$ with the total Hamiltonian $$H_T: =\frac{1}{2}e(p_x^2-m^2) + \lambda(\tau) p_e\;.$$ The equation of motion together with both constraints follow from functional $$W[e,p_e; x,p,; \lambda]: = \int \Theta\;,$$ using independent variation of the canonical pairs $(e,p_e),(x,p)$ and the Lagrange multiplier $\lambda$ $$\begin{aligned}
&&\dot x_\mu = ep_\mu\;, \quad \quad \dot p_\mu=0\;, \\
&&\dot e=\lambda\;, \\ \label{eq:encon}
&& p^2-m^2 =0\;, \quad \quad p_e = 0.\end{aligned}$$ Let us now convince ourselves that performing certain canonical transformations one can put the equation in such form that the Lagrange multiplier function $\lambda(\tau)$ enters only in the equation for one canonical pair. According to the general scenario described in previous section each canonical transformation $$\left(\begin{array}{cc}
e & p_e\\
x^\mu & p_\mu
\end{array}\right)
\longmapsto
\left(\begin{array} {cc}
e & p_e\\
X^\mu & \Pi_\mu
\end{array}\right)\;,$$ that identifies one canonical momentum with the energy constraint (\[eq:encon\]), say $\Pi_0 $ $$\Pi_0= \frac{1}{2}(p_x^2-m^2)$$ leads to this pattern. One possible way to complete the canonical transformations is [^9] $$\label{Pip}
\begin{array} {ll}
\Pi_0=\frac{1}{2}(p_x^2-m^2)\,, \quad &X_0=\frac{x_0}{p_0}\,,\\
\Pi_i=p_i\,, \quad &X_i=x_i-\frac{p_i}{p_0}x_0\;.
\end{array}$$ and the inverse transformation is $$\label{piP}
\begin{array} {ll}
p_0=\sqrt{2\Pi_o+\vec\Pi^2+m^2}\,, \quad & x_0 = X_0\sqrt{2\Pi_o+\vec\Pi^2
+m^2}\,,\\
p_i=\Pi_i\,, \quad & x_i=X_i+\Pi_iX_0\;.
\end{array}$$ In terms of the new variables the total Hamiltonian is $$\label{HT}
H_T=e\Pi_o+\lambda p_e\;.$$ and the equations of motion separate into two parts; one for the canonical pairs $ (e, p_e)$ and $X_0, \Pi_0$, with dependence from the Lagrange multiplier $\lambda(\tau)$ $$\begin{aligned}
\label{XPep}
\dot{X}_0=e\,, &\quad & \dot{e}=\lambda \,,\\
\dot{ p}_e=-\Pi_0\,, &\quad & \dot{\Pi}_0 =0 \,,\end{aligned}$$ constrained by $
\Pi_0= 0
$ and the equations of motion for the variables $(X_i,\Pi_i)$ $$\label{PHUR}
\dot{X}_i=0\, \quad \quad \dot{\Pi}_i=0\,,$$ which have a unique solution with initial values free from any restriction. One can construct the reduced Poincare -Cartan 1-form for physical unconstrained variables $X_i,\Pi_i$ from (\[P-Cp\]), rewritten in terms of the new canonical variables $$\Theta=\Pi_0dX^0-\Pi_idX_i + p_ede-(e\Pi_o+\lambda p_e)dt+
d(X_0(\Pi_0+m^2))\,,$$ by considering the projection onto the constraint shell $$\Theta^{\ast}=\Theta\bigl{|}_{ \Pi_0=0,~p_e=0}=-\Pi_idX_i+d(X_o)m^2\;.$$ Thus we have convinced ourselves that the variables $\Pi_i, X_i$ – are Jacobi’s coordinates for the obtained unconstrained theory with zero Hamiltonian. Now we shall show how to reconstruct the unconstrained Hamiltonian in terms of initial variables using the generating function to new set of canonical pairs (\[Pip\]) and Hamilton-Jacobi equation. To find the unconstrained system whose Jacobi’s coordinates are $\Pi_i,X_i$ let us write down the generating function $S(\Pi,x)$ of the canonical transformation $( x,p) \to (X,\Pi)$ (\[Pip\]) $$p=\frac{\partial S(\Pi,x)}{\partial x}\,,\;\;\;\;\;\;
X=\frac{\partial S(\Pi,x)}{\partial \Pi}\;.$$ One can easily verify from the condition $$\Pi dX-pdx=d(X_o(\Pi_o+m^2)),$$ that the function $$\label{genfun}
S(\Pi,x)=x_0\sqrt{2\Pi_0+\vec \Pi^2+m^2}-x_i\Pi_i\;$$ generates the above canonical transformations (\[Pip\]). Restriction the generating function by the condition $\Pi_o=0$ leads to the function $$S^{\ast}(\Pi_i,x_i, x_0) =S(\Pi,x)\bigl{|}_{\Pi_o=0}=
x_0\sqrt{\vec \Pi^2+m^2}-x_i\Pi_i\;,$$ which we shall now treat as generating function defined on the unconstrained phase space $ (x_i, p_i)$ and depended explicitly on some parameter $x_0$, which has the meaning of evolution parameter for the obtained reduced system. To verify this, one can use the generating function $S^{\ast}(\Pi_i,x_i, x_0) $ to write down the inverse transformation for variables in the reduced Poincare -Cartan form directly on the constraint shell $$\Theta^{\ast}=-\Pi_idX_i+m^2dX_o|_{\left\{
\begin{array}{l}
X_i=\frac{\partial S^{\ast}}{\partial \Pi_i}=x_i\\
p_i=\frac{\partial S^{\ast}}{\partial x_i}=\Pi_i
\end{array}\right.}=-p_idx_i+\sqrt{\vec p^2+m^2}dx_o\;.$$
From this form it follows that we get the Hamiltonian system for a relativistic particle $$\begin{aligned}
\frac{dx_i}{dt}&=&\{x_i,h\}=\frac{2p_i}{\sqrt{\vec p^2+m^2}} \\
\frac{dp_i}{dt }&=&\{p_i,h\}=0 \,,\end{aligned}$$ in the instant form of the dynamics with the parameter $t:=x_0$ and the Hamiltonian defined from the reduced generating function $$h=: \frac{\partial S^\ast}{\partial x_0} = \sqrt{\vec{p}^2 + m^2}.$$
Hamiltonian reduction of FRW cosmological models
================================================
Scalar field with minimal coupling to gravity
---------------------------------------------
After performing the Legendre transformation on the Lagrangian in the action (\[tru\]) describing the dynamics of a homogeneous scalar field with minimal coupling to FRW space time one finds that the phase space spanned by the canonical pairs $(a, p_a), (N_c,P_N)$ and $(\Phi, P_\Phi )$ is restricted by the primary constraint $$P_N = 0$$ and secondary constraint $$\label{LPS1}
C=\frac{\kappa p_a^2}{12}+\frac{3ka^2}{\kappa r_0^2}
-\frac{P_\Phi^2}{2a^2}\;.$$ Exploiting the nondegenerate character of the metric $ (a\neq 0) $ the secondary constraint (\[LPS1\]) can be rewritten in the equivalent form $$\label{tilcon}
\tilde C=a^2C =a^2\left(\frac{\kappa p_a^2}{12}+\frac{3ka^2}{\kappa r_0^2}
\right)
-P_\Phi^2/2\;,$$ which shows the separability of the gravitational and the matter source part in constraint. To obtain the reduced Hamiltonian describing the evolution of cosmic scalar factor $a$ one can introduce the new canonical coordinates for scalar field $$\Pi_\Phi:=P_\Phi^2/2\;,\;\;\;\;
T_\Phi :=\Phi/P_\Phi \,.$$ After this redefinition the corresponding Poincare-Cartan form $$\label{pcm}
\Theta= p_a da + \Pi_\Phi dT_\Phi -\frac{N}{a^2}\tilde C
dt +d\left(\Pi_\Phi T_\Phi \right),$$ projected onto the constraint shell reduces to $$\label{rpcm}
\Theta^\ast = p_a da + H(a) dT_\Phi +
d\left(H(a)T_\Phi \right),$$ where the reduced Hamiltonian that governs the scale factor $a$ evolution in time $ T_\Phi$ is $$H(a):=
a^2\left(\frac{\kappa p_a^2}{12}+\frac{3ka^2}{\kappa r_0^2}\right)\,.$$
Note, that there is another possibility to reduce the theory. The reduced theory can be formulated in terms of a scalar field. To find the dynamics of the scalar field we perform the canonical transformation on the scale factor $$\begin{aligned}
\label{mtime}
&&\Pi_a:=
a^2\left(\frac{\kappa p_a^2}{12}+\frac{3ka^2}{\kappa r_0^2}\right)\\
&&T_a:=
\int\limits_{a_o}^{a}{a^2da}\left(\frac{\kappa}{3}\Pi_a
-ka^4r_o^{-2}\right)^{-1/2}\end{aligned}$$ and as a result the reduced Poincare-Cartan form in terms of scalar fiel variables is $$\label{rpcmsc}
\Theta^\ast = P_\Phi d\Phi - H(P_\Phi) dT_a +
d\left(S(a, \Pi_a)- T_a\Pi_a \right),$$ where the reduced Hamiltonian that describes the evolution of scalar field $\Phi$ in time $ T_a$ is $$H(P_\Phi):= \frac{1}{2}P_\Phi^2 \,,$$ and the function $S(a, \Pi_a)$ is the generating function of the canonical transformation (\[mtime\]).
Scalar field with conformal coupling to gravity
-----------------------------------------------
In the case of a homogeneous scalar field conformally coupled to the FRW space time (\[eq:lagrFWW+MS\]) the phase space spanned by the canonical pairs $(a, p_a), (N_c,P_N)$ and $(\varphi, p_\varphi )$ is restricted by the primary constraint $$P_N = 0\,,$$ and secondary constraint $$\label{scc}
C := \Pi_\varphi -\Pi_a\,,$$ where $$\Pi_a:= \frac{\kappa p_a^2}{12}+ \frac{3ka^2}{\kappa r_o^2} \,,
\quad \quad
\Pi_\varphi:= \frac{p_\varphi^2}{2}+\frac{k\varphi^2}{2r_o^2}\,.$$ The total Hamiltonian $ H_T := N_c C + \lambda(t) P_N $ contains the arbitrary function $\lambda(t) $ and thus the Hamilton-Dirac equations $$\begin{array}{l}
\dot a=-N_c\kappa p_a/6\\
\dot p_a=N_c6ka/(\kappa r_o^2)\end{array}\;\;\;\;
\begin{array}{l}
\dot N_c=\lambda\\
\dot P_N = C\end{array}\;\;\;
\begin{array}{l}
\dot \varphi=N_cp_\varphi\\
\dot p_\varphi=-N_ck\varphi/r_o^2\;
\end{array}$$ cannot be solved in a unique way. According to the scheme described in the preceding sections to implement the Hamiltonian reduction one can search for a transformation to a new set of canonical variables in terms of which the equations of motion separate into independent parts: the physical (independent of the arbitrary function) and the unphysical one with unpredictable evolution. To achieve this let us perform the canonical transformation from $(p_a\,,a)$ and $(p_\varphi, \varphi)$ to the new canonical pairs such that matter part of the constraint $\Pi_\varphi$ becomes one of the new canonical momenta $$\begin{aligned}
\label{P}
\Pi_\varphi=
\frac{p_\varphi^2}{2} +\frac{k\varphi^2}{2r_o^2}\,.\end{aligned}$$ Using the generating function $$S(\Pi_\varphi, \varphi):=
\int\limits_{a_o}^{a}{da}\sqrt{2\Pi_\varphi
-\frac{k}{2r_o^2}\varphi^2}\,,$$ the corresponding canonical conjugated coordinate $T_\varphi$ is $$T_\varphi=
\int\limits_{a_o}^{a}\frac{da}{\sqrt{2\Pi_a
-\frac{k}{2r_o^2}\varphi^2}}\,$$ and the reduced action reads [^10] $$\begin{aligned}
W^\ast [a] =
\int p_ada+(\frac{\kappa}{12}p_a^2+\frac{3k}{\kappa r_o^2}a^2)
dT_\varphi +
d\left(S(\Pi_\varphi, \varphi) - \Pi_\varphi T_\varphi\right),\end{aligned}$$
It is worth mentioning that if instead of matter part the gravitational part of constraint $\Pi_a$ will be used for the construction of the new canonical momenta then the reduced action describing the evolution of scalar field is $$\begin{aligned}
W^\ast[\varphi]&=&\int p_\varphi d\varphi
-\frac{1}{2}(p_\varphi^2+
\frac{k\varphi^2}{r_o^2})dT_\varphi\;.\end{aligned}$$
Spinor field as source field for FRW Universe
---------------------------------------------
The Hamiltonian reduction of this model is achieved along the same lines as in the previous section. However, dealing with fermion fields there are some specific features due to the presence of the second class constraints.
The action (\[S\]) for the homogeneous spinor field in FRW Universe is degenerate and the corresponding primary constraints are $$\label{CONSpin}
\begin{array}{ccl}
C_N &:=&p_N=0\\
C_\psi&:=&p_\psi+\frac{i}{2}\bar\psi\gamma^o=0\\
C_{\bar\psi}&:=&p_{\bar\psi}+\frac{i}{2}\gamma^o\psi =0\,.
\end{array}$$ They satisfy the algebra $$\label{eq:alg1}
\{C_N,C_\psi\}=0 \;,\;\;\;\;\; \{C_N,C_{\bar\psi}\}=0\;,\;\;\;\;\;
\{C^{(1)}_\psi,C^{(1)}_{\bar\psi}\}=-i\gamma^o \,.$$
According to the Dirac prescription the evolution in time is governed by the total Hamiltonian $$H_T=H_c+\lambda_NC_N+C_\psi\lambda_\psi+
\lambda_{\bar\psi}C_{\bar\psi}\,,$$ with the arbitrary functions $\lambda $ and the canonical Hamiltonian $ H_c$ $$\label{HC}
H_{c}=N_c
\left[-\left(
\frac{\kappa p_a^2}{12}+
\frac{3ka^2}{\kappa r_o^2}
\right) +a{\cal H}_D\right]\,.$$ The requirement to conserve the constraints during the evolution fixes the functions $\lambda_\psi$ and $\lambda_{\bar\psi}$ $$\label{lspin1}
\lambda_{\bar\psi}=i N_cma\bar\psi\gamma^o, \;\;\;\;\;\;\;\;\;
\lambda_{\psi}=i N_cma\gamma^o\psi \,.$$ but leaves the function $\lambda_N$ unspecified it and leads to the existence of the secondary constraint $$C := \frac{\kappa p_a^2}{12}+
\frac{3ka^2}{\kappa r_o^2}
-a{\cal H}_D =0 \,.$$ Due to the algebra of constraints (\[eq:alg1\]) and the Poisson brackets of secondary constraint $ C $ with any other $$\label{PB}
\{C ,C_\psi\}=ma\bar\psi\;,\;\;\;\;
\{C, C_{\bar\psi}\}=-ma\psi \;,\;\;\;\;
\{C, C_{N}\}=0\;,$$ one can verify that no additional constraints emerge. [^11] The algebra (\[eq:alg1\]) and (\[PB\]) shows that the constraints represent a mixed system of first and second class constraints. In order to perform the Hamiltonian reduction we will start with rewriting the constraints into an equivalent form such that the first class constraints form the ideal of algebra and the algebra of second class constraints is canonical. This equivalent set of constraints $\bar C_\psi, \bar C_{\bar\psi}$ is given in the Appendix B. The canonical character of the new algebra $\{\bar C_\psi,\bar C_{\bar\psi}\}=-1$ allows to perform the canonical transformation that converts the new second class constraints $\bar C_\psi,\bar C_{\bar\psi}$ to the pair of canonical variables $$\label{NEWCON}
\begin{array}{l}
\bar \Pi_\psi=\bar
C_\psi\,,\\ \bar Q_\psi=\bar C_{\bar\psi}\,,
\end{array}\;\;\;\;
\begin{array}{l}
\Pi_\psi=i p_\psi\gamma^o+\frac{1}{2}\bar\psi \,,\\
Q_\psi=p_{\bar\psi}-\frac{i}{2}\gamma^o\psi \,.
\end{array}$$ This means that the dynamics of phase space variables $\bar Q_\psi, \bar \Pi_\psi$ is completely “frozen” and other canonical pairs change in time independently of them. In other words we can everywhere in the formulas omit this variables without destroying the dynamics of the physically relevant quantities. Turning to the reduction due to the first class constraints let us pass to the new Hamiltonian constraint ${\cal{C}}$ $$\label{eq:newconstr}
{\cal{C}}: = \frac{1}{a} C =
\frac{1}{a}\left(\frac{\kappa p_a^2}{12}+
\frac{3ka^2}{\kappa r_o^2}\right)
-i m\Pi_\psi\gamma^oQ_\psi\,,$$ assuming that the metric is nondegenerate $a\neq 0$. In order to achieve the reduction for first class constraint we perform the canonical transformation from the $(p_a, a)$ to the new variables $( \Pi_a,Q_a) $ such that $$\label{Pq}
\Pi_a= \frac{1}{a}\left(\frac{\kappa p_a^2}{12}+
\frac{3ka^2}{\kappa r_o^2}\right)\,.$$ Using the generating function $S(a,\Pi_a)$ $$S(a,\Pi_a)=\frac{6}{\kappa}\int\limits_{a_o}^{a}da
\sqrt{
\frac{\kappa}{3}a\Pi_a
-ka^2r_o^{-2}}\,,$$ one can find the variable canonically conjugated to $\Pi_a$ $$\label {timePar}
T_a=\int\limits_{a_o}^{a}{ada}\left(
\frac{\kappa}{3}a\Pi_a
-ka^2r_o^{-2}
\right)^{-1/2} \,,$$ and after projection onto the constraint shell $
{\cal C}=0, \quad \bar C_\psi= 0, \quad \bar\Pi_\psi=0,
\quad
\bar C_{\bar\psi}=0, \quad \bar Q_\psi=0
$ the reduced action is $$W^\ast[Q_\psi]= \int dQ_\psi\Pi_\psi + m \Pi_\psi\gamma^o Q_\psi dT_a\,.$$ Thus we have derived the standard Dirac Hamiltonian for reduced spinor field and this matter source corresponds to the case of the dust filled Universe; the Hubble constant behaves as $$H^2=\left(\frac{1}{a}\frac{da}{dT_a}\right)^2
=\frac{\kappa}{3}\frac{M_D}{a^3}-\frac{k}{a^2r_o^2}$$ with the constant $M_D$.
We shall finish with one remark concerning the simple generalization of the above result to a more complex system. It is interesting to note that if one includes the interaction of massive spinor with the scalar massless one in the action of the following type $$\begin{aligned}
W[ g, \Phi,\Psi ] &=& \int\limits_{}^{}d^4x\sqrt{-g}\left[
-\frac{1}{16\pi G}\,
{}^{(4)}R + \frac{1}{12}\, {}^{(4)}R\Phi^2 + \frac{1}{2}
g^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi\right.\\
&&\left.+\frac{i}{2}\left(\bar\Psi\gamma^\mu(x)\bigtriangledown_\mu\Psi-
\bigtriangledown_\mu\bar\Psi\gamma^\mu(x)\Psi\right)
-m\bar\Psi\Psi-\mu\Phi\bar\Psi\Psi\right]\,,\nonumber\end{aligned}$$ then the action obtained after supposition of the FRW Universe $$\begin{aligned}
W[a, N_c, \varphi,\psi]& = &\int dt\left[-
\frac{3}{\kappa}\frac{\dot a^2}{N_c}
+\frac{1}{2}\frac{\dot \varphi^2}{N_c}
+\frac{i}{2}(\bar \psi\gamma^o\dot\psi-\dot{\bar\psi}\gamma^o\psi)
\right.\\
&&\left.
-N_c\left(-\frac{3}{\kappa}\frac{ka^2}{r_o^2}
+\frac{k\varphi^2}{2r_o^2}
+(ma+\mu\varphi)\bar\psi\psi\right)
\right]\,,\nonumber\end{aligned}$$ can be connected with the action describing the interaction of fermion field and massless scalar field. Let us consider two possible cases.
a). . One can convince ourself that after introduction of the new scalar field $\phi$ and the scale factor $\alpha$ $$ma+\mu\varphi=\mu\phi\sqrt{1-\frac{\kappa}{6}\frac{m^2}{\mu^2}}
\;;\;\;\;a+\frac{m}{\mu}\frac{\kappa}{6}\varphi=
A\sqrt{1-\frac{\kappa}{6}\frac{m^2}{\mu^2}}$$ we get the action for the massless spinor interacting with the field $\phi$ $$\begin{aligned}
\label{eq:ac1}
W[A, N_c, \phi,\psi]& = &\int dt\left[-
\frac{3}{\kappa}\frac{\dot A^2}{N_c}
+\frac{1}{2}\frac{\dot \phi^2}{N_c}
+\frac{i}{2}(\bar \psi\gamma^o\dot\psi-\dot{\bar\psi}\gamma^o\psi)
\right.\\
&&\left.
-N_c\left(-\frac{3}{\kappa}\frac{kA^2}{r_o^2}
+\frac{k\varphi^2}{2r_o^2}
+\tilde\mu\phi\bar\psi\psi\right)
\right],\nonumber\end{aligned}$$ and the new coupling constant $$\tilde\mu=\mu\sqrt{1-\frac{\kappa}{6}\frac{m^2}{\mu^2}}\;.$$
b). . In this case one can use another transformation, $$\varphi=\frac{1}{\sqrt{1-\frac{6}{\kappa}\frac{\mu^2}{m^2}}}
\left(\phi-\frac{6}{\kappa}\frac{\mu}{m}A\right)\;;\;\;\;\;\;
a=\frac{1}{\sqrt{1-\frac{6}{\kappa}\frac{\mu^2}{m^2}}}
\left(A-\frac{\mu}{m}\phi\right)\,,$$ and get the action $$\begin{aligned}
\label{eq:ac2}
W[A, N_c, \phi,\psi]& = &\int dt\left[-
\frac{3}{\kappa}\frac{\dot A^2}{N_c}
+\frac{1}{2}\frac{\dot \phi^2}{N_c}
+\frac{i}{2}(\bar \psi\gamma^o\dot\psi-\dot{\bar\psi}\gamma^o\psi)
\right.\\
&&\left.
-N_c\left(-\frac{3}{\kappa}\frac{kA^2}{r_o^2}
+\frac{k\phi^2}{2r_o^2}
+\tilde m A\bar\psi\psi\right)
\right],\nonumber\end{aligned}$$ with the new mass for the fermion field $\tilde m=m\sqrt{1-\frac{6}{\kappa}\frac{\mu^2}{m^2}}\;.
$ One can verify that these two actions are related by the field redefinition $$A\;\to\;i\frac{\kappa}{6}\phi ,
\quad \quad
\phi\;\to\;i \frac{6}{\kappa}A,$$ and thus it is enough to reduce one of the actions (\[eq:ac1\]),(\[eq:ac2\]).
For the action (\[eq:ac1\]) the energy constraint $$C= -
\frac{\kappa p_a^2}{12}-
\frac{3ka^2}{\kappa r_o^2}
+\frac{p_\phi^2}{2} +\frac{k\phi^2}{2r_o^2} +
\tilde\mu \phi{\cal H}_D\,,$$ again has separable contributions from the gravitational and the matter part. After introduction of the new canonical momentum $$\Pi_a:=
\frac{\kappa p_a^2}{12}-
\frac{3ka^2}{\kappa r_o^2}$$ and the corresponding conjugated coordinate $T_a$ in the same manner as for the case of the conformal scalar field the following action for the physical scalar and spinor fields can be derived $$W^\ast[\phi, \psi]=
\int dQ_\psi\Pi_\psi + p_\phi d\phi -H dT_a,$$ with physical Hamiltonian describing the system of interacting spinor and scalar fields $$H:= \frac{p_\phi^2}{2} + \frac{k\phi^2}{2r_o^2}+
\tilde\mu \phi{\cal H}_D\,.$$
Classical and quantum observables for FRW Universe
==================================================
Extended quantization: Wheeler-deWitt equation
----------------------------------------------
According to the Dirac prescription in the extended quantization scheme one considers the classical constraints to be the conditions on the state vector $\Psi$ [@Wheel], [@deWitt] , [^12] $$\begin{aligned}
\label{eq:WdW}
&&P_N \Psi =0\,,\\
&&H_T\Psi =0 \,.\end{aligned}$$ Quantum observable in this quantization scheme are constructed with analogy to that of the two-dimensional relativistic spin zero bosonic Klein-Gordon field as expectation value $$\bigl< O \bigl> = \int d\phi\left(
\Psi^\ast O \partial_a (\Psi) - \partial_a (\Psi^\ast) O \Psi^\ast
\right)\,.$$ However, as it has been analyzed by Kaup and Vitello [@KaupVitello] this conventional interpretation cannot be used without violating the correspondence principle. More precisely, it has been shown that the expectation values for the scalar fields and the cosmic scale factor do not correspond to the classical values; their evolution describe the expansion phase of Friedmann evolution, but then instead of contraction, the expectation values tunnel through the barrier and continue to expand. Below it will be demonstrated that opposite to this situation the canonical quantization of the unconstrained system obtained in the preceding part of the paper leads directly to the fulfillment of the correspondence principle.
Reduced quantization: Heisenberg equation
-----------------------------------------
To analyze the correspondence principle let us consider the case of a conformal scalar field in the closed Friedmann Universe. As it has been shown the evolution of scale factor $a$ in conformal time $t$ is governed by the harmonic oscillator Hamiltonian which after conventional quantization reads $$\hat H=\frac{\kappa}{12}\hat p^2+
\frac{3}{\kappa r^2_o}\hat a^2\;.$$ Assuming the quantum state in the form $$\Psi=\frac{1}{(\alpha^2\pi)^{1/4}}
\exp\left[\frac{i}{\hbar}p_oa-\frac{(a-a_o)^2}{2\alpha^2}\right]$$ where $a_o$ and $p_o$ are the mean values of the coordinate and the momentum respectively (real parameter $\alpha$ characterizes the mean square deviation of $a$) and using the solution of Heisenberg equations for the operators $\hat a(t)$ and $\hat p(t)$ $$\begin{aligned}
&&\hat a(t)=\hat a(0)\cos\frac{t}{r_o}
-\frac{\kappa r_o}{6}\hat p(0)\sin\frac{t}{r_o}\,,\\
&&\hat p(t)=\frac{6}{\kappa r_o}\hat a(0)\sin\frac{t}{r_o}
+\hat p(0)\cos\frac{t}{r_o}\;,\end{aligned}$$ one can find the time dependence of the mean values of $\hat a(t)$ and $\hat p(t)$, $$\begin{aligned}
&&\overline{a(t)}=\int\limits_{-\infty}^{+\infty}
\Psi^*\hat a(t)\Psi da=a_o\cos\frac{t}{r_o}
-\frac{\kappa r_o}{6}p_o\sin\frac{t}{r_o}\,,\\
&&\overline{p(t)}=\int\limits_{-\infty}^{+\infty}
\Psi^*\hat p(t)\Psi da=\frac{6}{\kappa r_o}a_o\sin\frac{t}{r_o}
+p_o\cos\frac{t}{r_o}\;.\end{aligned}$$ This means that we have the correspondence with the classical formulae $$\begin{aligned}
&&a(t)=r_o\sqrt{\frac{\kappa}{3}|H|}\sin\left(\frac{Q-T_c}{r_o}\right)\\
&&p(t)=\sqrt{\frac{12}{\kappa}|H|}\cos\left(\frac{Q-T_c}{r_o}\right)\end{aligned}$$ when constants are taken as $$\overline{a(0)}=a_o=
r_o\sqrt{\frac{\kappa}{3}|H|}\sin\frac{Q}{r_o}\;,\;\;\;
\overline{p(0)}=p_o=\sqrt{\frac{12}{\kappa}|H|}\cos\frac{Q}{r_o}\;.$$
At the end we note that there is no wave packet diffusion when the mean square deviation $$\overline{(\triangle a(t))^2}=\frac{\alpha^2}{2}
\left(\cos^2\frac{t}{r_o}+\left(\frac{\kappa r_o}{6}\right)^2
\frac{\hbar^2}{\alpha^4}\sin^2\frac{t}{r_o}\right)$$ is time independent. This holds for the special value of $\alpha$ $$\alpha^2=\hbar\frac{\kappa r_o}{6}\;.$$
Concluding remarks
==================
In the present paper the method of Hamiltonian reduction for reparametrization invariant mechanical systems have been elaborated. This approach is based on the choice of adapted coordinates using the generating function of the canonical transformation that is a solution of the corresponding Hamilton – Jacobi equation. We have derived the reduced Hamiltonians for the Friedmann cosmological models with homogeneous scalar and spinor field matter sources and find the corresponding observable time. The obtained reduced Hamiltonians have two attractive peculiarities:
i\. They are the generators of evolution with respect to observable time;
ii\. They are conserved quantities which can be treated as the energy of the reduced systems.[^13]
Furthermore, the representation for the Hubble parameter and the red shift is founded in terms of the Dirac observables in the frame of the generalized Hamiltonian dynamics and correspondence between field Friedmann models and perfect fluid Friedmann models with different equations of state has been established.
The comparison of extended quantization with the Wheeler-deWitt equation and the canonical quantization of unconstrained system shows the conceptual advantage of later. In reduced system with the Schrödinger type equation instead of the to the Wheeler-deWitt equation the wave function is normalizable and has clear standard quantum mechanical interpretation. It is shown that quantum observables treated as expectation values of the Dirac observables properly describe the original classical theory.
It is in order here to make a remark concerning the relation to the conventional gauge-fixing method. Certainly, the results derived in the present note by reduction without introduction of gauge functions can be reproduced by the gauge fixing method. However, from our derivation it is clear that due to the complicated relations between the initial variables and the observable time the gauge functions depend on the initial variables in a complex way which is difficult to guess.
Finally we would like to point out the possibility to exploit the suggested approach. The method elaborated in the present article can be used in the description of the other cosmological models, like Bianchi cosmologies, with different type of global symmetries. But the applicability of obtained results to a general problem of observables meets with the several difficulties. Nevertheless, we hope that in combination with other refined methods our approach will help to extend our understanding of the puzzle of the observables in theory of gravity.
Acknowledgments
================
We would like to thank S.Gogilidze, D.Mladenov, H.-P.Pavel and V.Pervushin for helpful and critical discussions. This work was supported in part by the Russian Foundation of Basic Research, Grant No 99-01-00101.
Dirac equation in FRW space time {#appA}
================================
To describe a spinor field on a Rimanian manifold the vierbein fields $h^\mu_a(x) \,\, \mu, \nu, a, b = 0,1,2,3 $ $$ds^2=g_{\mu\nu}dx^\mu dx^\nu=\eta_{ab}(h^a_\mu dx^\mu)(h^b_\nu dx^\nu)\;;\;
\quad \eta_{ab}:=(+---),$$ and the Dirac $\gamma$-matrices with a specific dependence on space time coordinates are introduced $$\gamma^\mu(x)=h^\mu_a(x)\gamma^a .$$ The following relations between the vierbein fields and the metric tensor $g_{\mu\nu}$ hold $$h^\mu_ah_{b\mu}=\eta_{ab}\;;\;\;h^a_\mu h_{a\nu}=g_{\mu\nu}\;;\;\;
h^a_\mu h^\mu_b=\delta^a_b\;;\;\;\;h^a_\mu h^\nu_a=\delta^\nu_\mu\;.
\;;\;\;\;
h_{a\mu}=\eta_{ab}h^b_\mu=g_{\mu\nu}h^\nu_a\;.$$ The Dirac equation for spinors in curved space time reads $$\label{DEQ}
(i\gamma^\mu(x)\bigtriangledown_\mu-m)\Psi(x)=0\;,$$ with the covariant derivative $$\label{COVDER}
\bigtriangledown_\mu\Psi(x)=[\partial_\mu+\frac{1}{4}C_{abc}h^c_\mu
\gamma^b\gamma^a]
\Psi(x)\;,$$ where Ricci coefficients $$C_{abc}\equiv (\bigtriangledown_\mu h^\nu_a)h_{b\nu}h_c^\mu\;;\;\;\;\;
\bigtriangledown_\mu h^\nu_a=(\Gamma^\nu_{\mu\lambda}
-h^\nu_b\partial_\mu h^b_\lambda)
h^\lambda_a\;,$$ are introduced. For the specific case of the Robertson – Walker metric, $$\label{TFRN}
ds^2=a^2(t)\tilde ds^2=
a^2(t)\left[(N(t)dt)^2-\left(1+\frac{kr^2}{4r_o^2}\right)^{-2}
\left(dr^2+r^2(d\xi^2+\sin^2\xi d\zeta^2)\right)\right]\,,$$ the following vierbein fields $$\label{tetr}
\left\{
\begin{array} {ll}
h^{\underline o}_o=aN&h^{\underline 1}_1=a\left(1+\frac{kr^2}{4r_o^2}
\right)^{-1}\\
h^{\underline 2}_2=ar\left(1+\frac{kr^2}{4r_o^2}\right)^{-1}&
h^{\underline 3}_3=ar\sin \zeta\left(1+\frac{kr^2}{4r_o^2}\right)^{-1}
\end{array}\right.$$ are used in the main text. Here the vierbein indices are underlined. The Dirac equation then looks $$\begin{aligned}
\frac{i}{a}\left[\gamma^o\frac{1}{N}\frac{\partial}{\partial t}
+\gamma^1\left(1+\frac{kr^2}{4r_o^2}\right)\frac{\partial}{\partial r}
+\gamma^2\frac{1+\frac{kr^2}{4r_o^2}}{r}\frac{\partial}{\partial \zeta}
+\gamma^3\frac{1+\frac{kr^2}{4r_o^2}}{r\sin \zeta}
\frac{\partial}{\partial \xi}\right.\nonumber\\
\left.+\frac{3\dot a}{2aN}\gamma^o+\frac{1-\frac{kr^2}{4r_o^2}}{r}\gamma^1
+\frac{\cot \zeta}{2r}\left(1+\frac{kr^2}{4r_o^2}\right)\gamma^2
\right]\Psi(x)-m\Psi(x)&=&0\;.\end{aligned}$$ To maintain the space homogeniety of the Friedmann Universe we suppose that the spinor field is only time dependent. In the main text the FRW Universe with the spinor matter source is formulated in terms of the fermion variable $\psi$ $$\psi(t)=a^{3/2}(t)\Psi(t).$$
Separation of first and second class constraints in model with spinor field
============================================================================
The set of constraints $C_A=(C_{\psi}, C_{\bar\psi}, C )$ represent a mixed system of first and second class constraints; the rank of the Poisson matrix $ {\cal M}= \{C_A,C_B\} $ is equal to two. The explicit form of the Poisson matrix is $${\cal M} =
\left(\begin{array}{cc}
\triangle&K\\
-K^T&0
\end{array}\right)\,,$$ where and $\triangle $ and $K$ denote $$\triangle=\left(\begin{array}{cc}
0&-i\gamma^o\\
-i\gamma^o&0
\end{array}\right)\;\;\;\;\;\;\;\;
K=\left(\begin{array}{c}
-ma\bar\psi\\
ma\psi
\end{array}\right)\,.$$ In order to perform the reduction procedure it is useful to separate first and second class constraints. One can easily verify that applying the similarity transformation $T$ $$T=
\left(\begin{array}{cc}
1&0\\
K^T\triangle^{-1}&1
\end{array}\right)\,,\;\;\;\;\;\;\mbox{Sdet}T\neq 0$$ to the constraints $C_A$ $$\tilde C =T\cdot C =
\left(\begin{array}{ccc}
1&0&0\\
0&1&0\\
i ma\gamma^o\psi&-i am\bar\psi\gamma^o&1
\end{array}\right)\cdot
\left(\begin{array}{c}
C_\psi\\
C_{\bar\psi}\\
C\end{array}\right)$$ we achieve the separation of the constraints on the surface defined by the second class constraints $$\{\tilde C ,\tilde C_\psi\}=i ma
\tilde C_\psi\gamma^o\;\;\;\;\;
\{\tilde C,\tilde C_{\bar\psi}\}=-i ma
\gamma^o\tilde C_{\bar\psi}\,.$$ To have this separation on the whole phase space one can pass to the new set of constraints $$\begin{aligned}
\label{eqconsf}
\bar C &=&\tilde C + ma\tilde C\psi \tilde C_{\bar\psi}\nonumber\\
&=&\Pi+ ma \left(p_\psi
p_{\bar\psi}-\frac{i}{2}
[p_\psi\gamma^o\psi+\bar\psi\gamma^op_{\bar\psi}]\right)
-\frac{1}{4}a{\cal H}_D,\\
\label{gr}
\bar C_\psi&=&-i \tilde C_\psi\gamma^o=-i p_\psi\gamma^o+\frac{1}{2}
\bar\psi \,, \\
\bar C_{\bar\psi}&=&\tilde C_{\bar\psi}=
p_{\bar\psi}+\frac{i}{2}\gamma^o\psi \,.
\label{sconstr}\end{aligned}$$ In this new set $\bar C$ belongs to the ideal of the algebra of constraints $$\{\bar C,\bar C_\psi\}=\{\bar C_N,\bar C_{\bar\psi}\}=0\,,$$ and second class constraints $\bar C_\psi,\bar C_{\bar\psi}$ obey the canonical algebra $$\{\bar C_\psi,\bar C_{\bar\psi}\}=-1\,.$$
Reduced Hamiltonian as conserved quantity from conformal symmetry
==================================================================
In this Appendix we discuss the existence of time independent reduced Hamiltonians from the geometrical standpoint. The Friedmann – Robertson – Walker space-time is conformally flat $$ds^2_{FRW}=A^2(x)ds^2_{Minkowski}\,.$$ In the flat Friedmann Universe the conformal factor $A(x)$ is simple scale factor $a(T_c) $ and it is easy to verify that the conformal time translation is a conformal symmetry $$\pounds_{\partial_{T_c}}g_{\mu\nu}=
\pounds_{\partial_{T_c}}(a^2(T_c)\eta_{\mu\nu})=
\eta_{\mu\nu}\partial_{T_c}a^2(T_c)=
g_{\mu\nu}2\frac{\dot a}{a}.$$ It is well-known that if space time possesses the conformal Killing vector and matter energy-momentum tensor is traceless, then one can construct the conserved quantity as follows. Considering the covariant derivative of contraction of the stress tensor and the confomal Killing vector $$\begin{aligned}
\bigtriangledown_\mu P^{\mu}&=&\bigtriangledown_\mu\left(\xi_\nu
T^{\mu\nu}\right)=
\xi_\nu\bigtriangledown_\mu T^{\mu\nu}+T^{\mu\nu}\frac{1}{2}
(\bigtriangledown_\mu\xi_\nu+\bigtriangledown_\nu\xi_\mu)\nonumber\\
&=&\xi_\nu\bigtriangledown_\mu T^{\mu\nu}+\frac{1}{n}T^\mu_\mu
\bigtriangledown^\nu\xi_\nu \nonumber\,,\end{aligned}$$ and assuming the covariant conservation of the traceless matter energy-momentum tensor $$\bigtriangledown_\mu T^{\mu \nu}\ = 0\,, \quad \quad T^\mu_\mu=0\,,$$ we have the conservation law for four-vector $P^{\mu}$ in the covariant differential form $$\bigtriangledown_\mu P^{\mu}=0.$$ To get the global conserved quantity one can integrate this equality over the whole space-time and use Gauss theorem $$\begin{aligned}
\label{charge}
&&\int\limits_{V}^{}d^4x\sqrt{-g}\bigtriangledown_\mu\left(\xi_\nu
T^{\mu\nu}\right)=
\int\limits_{V}^{}d^4x\frac{\partial}{\partial x^\mu}
\left(\sqrt{-g}\xi_\nu T^{\mu\nu}\right)\\
&&=\int\limits_{T_{c}'}^{}(\xi_{T_c})_\nu T^{\mu\nu}\sqrt{-g}d^3x-
\int\limits_{T_{c}''}^{}(\xi_{T_c})_\nu T^{\mu\nu}\sqrt{-g}d^3x\nonumber\,,\end{aligned}$$ where in the last line we specify the Killing vector corresponding to the conformal translation in Robertson – Walker space-time. For the conformal scalar field with Lagrangian $${\cal L}=\sqrt{-g}\left(\frac{1}{2}g^{\mu\nu}\partial_\mu\Phi
\partial_\nu\Phi
+\frac{1}{12}\ {}{(\!4\!)}\!\!R\Phi^2\right)\,,$$ the canonical stress tensor $$T^{C}_{\mu\nu}=\partial_\mu\Phi\partial_\nu\Phi
-g_{\mu\nu}\frac{1}{\sqrt{-g}}{\cal L}$$ has nonzero trace $T^{{C} \mu}_{\mu}\neq 0$. However, according to [@ChernikovTag], one can pass to the improved tensor $$T_{\mu\nu}=T^{C}_{\mu\nu}-
\frac{1}{6}\left[-\ {}{(\!4\!)}\!\!R_{\mu\nu}+
\partial_\mu\partial_\nu-g_{\mu\nu}\partial^\mu\partial_\mu\right]\Phi^2\,,$$ which is traceless $ T^\mu_{\mu}=0.$ Thus for a conformal time Killing vector in adapted coordinates $\xi_{T_c}=(1,0,0,0)$ and for homogeneous scalar field $
\varphi(T_c)=a(T_c)\Phi(T_c)
$ from eq. (\[charge\]) it follows that $$H=\int\limits_{T_{c}}^{}(\xi_{T_c})_o T^{oo}\sqrt{-g}d^3x=
V_{(3)}\left(\frac{p_\varphi^2}{2}+\frac{k\varphi^2}{2r_o^2}\right)$$ is conserved charge that coincides with the reduced Hamiltonian derived in the main text.
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[^1]: The problem of observables consist in the determination of the invariant characteristics of gravitational field in terms of measurable quantities [@Bergmann] and is closely related to that of time evolution [@KucharTime], [@HajicekTime].
[^2]: For review of the cosmological models construction with applications of the ADM method see e.g. [@Ryan].
[^3]: To make agreement between the four-dimensional covariance and the possibility to extract from the canonical coordinates hidden variables appropriate for deparametrization theory is difficult task. To solve this problem Kuchar suggested to perform the “second parametrization” of general relativity by extending its phase space by the additional embedding variables [@KucharEmbed].
[^4]: Apart from topological obstruction arising due to the projection onto the constraint shell it is necessary also to investigate the problems connected with the topological structure of spaces of constant curvature. The well elaborated classification of three-dimensional spacelike manifolds [@Wolf] allows to estimate the influence of topological properties on physical quantities. An interesting study of the role played by this global properties is under present consideration (see [@Bernui] and references therein).
[^5]: Below we will point out the correspondence the conventional Friedmann cosmology based on the Einstein equations supplemented by certain matter equation of state.
[^6]: It is well known that essentially all types of couplings of free scalar field to the scalar curvature and its kinetic term can be reduced to minimal coupling form using rescaling of the metric and scalar field redefinition [@Deser]. In 1974 based on this type of transformations Bekenstein [@Beken] proposed the method of construction solution for particular case of conformally coupled Einstein-scalar equation from solution of the minimally coupled ones (see also [@CalColJack]). The detailed investigation of this type solutions for FRW geometry with spatial homogeneous scalar fields can be found in [@Page]. Note also the interesting consideration of the evolution of Friedman cosmology driven by scalar fields, given in [@StarkovichCooperstock]
[^7]: In all formulas this factor will be omitted, in order to simplify the numerical factors.
[^8]: This paper deals with Abelian constraints only, but a few remarks on the general non-Abelian case may be in order. A straightforward generalization to this situation is unattainable; identification of momenta with constraints is forbidden due to the non-Abelian character of constraints. However, one can replace the non-Abelian constraints by an equivalent set of constraints forming an Abelian algebra and after this implement the above mentioned Levi-Civita transformation. For proofs of this Abelianization statement see e.g. [@Sunder] – [@HenTeit] and the description of itterative Abelianization conversion in [@GKP].
[^9]: Different possibilities to complete the canonical transformations for remaining variables will lead to another forms of dynamics, or to equivalent form but in another frame of reference.
[^10]: Based on this action one can derive the Hubble parameter $
H^2=
\frac{1}{a^4(T_c)}\frac{\kappa}{3} \Pi_\varphi
-\frac{k}{r_o^2a^2(T_c)}\;
$ and convince ourselves that it corresponds to the radiation dominated Fridmann model with the constant $ \Pi_\varphi $.
[^11]: Secondary constraint $C$ is conserved in weak sense $$\dot C =i Nm p_a(C_\psi\gamma^o\psi+
\bar\psi\gamma^oC_{\bar\psi})\approx 0.$$
[^12]: The standard procedure of letting $P_N \rightarrow -i\partial_N,\,
p_a \rightarrow -i\partial_a, \,
P_\Phi \rightarrow -i\partial_{P_\Phi} $ is assumed.
[^13]: The conservation of the conformal matter Hamiltonian with respect to conformal time translations follows from conformal symmetry of the Robertson – Walker space-time.
|
---
abstract: 'We present the results of radiation-magnetohydrodynamic simulations of the formation and expansion of regions and their surrounding photodissociation regions (PDRs) in turbulent, magnetised, molecular clouds on scales of up to 4 parsecs. We include the effects of ionising and non-ionising ultraviolet radiation and x rays from population synthesis models of young star clusters. For all our simulations we find that the region expansion reduces the disordered component of the magnetic field, imposing a large-scale order on the field around its border, with the field in the neutral gas tending to lie along the ionisation front, while the field in the ionised gas tends to be perpendicular to the front. The highest pressure compressed neutral and molecular gas is driven towards approximate equipartition between thermal, magnetic, and turbulent energy densities, whereas lower pressure neutral/molecular gas bifurcates into, on the one hand, quiescent, magnetically dominated regions, and, on the other hand, turbulent, demagnetised regions. The ionised gas shows approximate equipartition between thermal and turbulent energy densities, but with magnetic energy densities that are 1 to 3 orders of magnitude lower. A high velocity dispersion ($\sim 8$ km s$^{-1}$) is maintained in the ionised gas throughout our simulations, despite the mean expansion velocity being significantly lower. The magnetic field does not significantly brake the large-scale region expansion on the length and timescales accessible to our simulations, but it does tend to suppress the smallest-scale fragmentation and radiation-driven implosion of neutral/molecular gas that forms globules and pillars at the edge of the region. However, the relative luminosity of ionising and non-ionising radiation has a much larger influence than the presence or absence of the magnetic field. When the star cluster radiation field is relatively soft (as in the case of a lower mass cluster, containing an earliest spectral type of B0.5), then fragmentation is less vigorous and a thick, relatively smooth PDR forms. **Accompanying movies are available at <http://youtube.com/user/divBequals0>**'
author:
- |
S. J. Arthur$^{1}$[^1], W. J. Henney$^{1}$, G. Mellema$^{2}$, F. De Colle$^{3}$ and E. Vázquez-Semadeni$^{1}$\
$^1$Centro de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de México, Campus Morelia, Apdo. Postal 3-72, 58090 Morelia, Michoacán, México\
$^2$Department of Astronomy & Oskar Klein Centre, AlbaNova, Stockholm University, SE 10691 Stockholm, Sweden\
$^3$Department of Astronomy and Astrophysics, University of California Santa Cruz, Santa Cruz, CA 95064, USA
title: 'Radiation-magnetohydrodynamic simulations of [H II]{} regions and their associated PDRs in turbulent molecular clouds'
---
\[2005/12/01\]
H II regions – ISM: kinematics and dynamics – magnetohydrodynamics – photon-dominated region (PDR) – Stars: formation
Introduction {#sec:intro}
============
regions, that is, regions of photoionised gas, are among the most arresting astronomical objects at optical wavelengths. The basic theory behind their formation and expansion has been known for some time . More recently, attention has turned to explaining the irregular structures, filaments, globules and clumps seen within and around these regions. These could be due to underlying density inhomogeneities in the molecular cloud into which the region is expanding, or could be due to instabilities at the ionisation front itself . The effect of the stellar ionising radiation on these structures can lead to radiation driven implosion, and the subsequent compression could result in the formation of new stars .
Magnetic fields are well known to pervade molecular clouds and are thought to play an important role in regulating the star-formation process by providing support to collapsing structures in molecular clouds [@1999osps.conf..305M], although recently their relative importance in this process has become less clear [@2007ARAA...45..565M]. However, theoretical work on the interplay between ionising radiation and magnetic fields in and around regions has only been possible recently, with the advent of radiation-magnetohydrodynamic codes. @2007ApJ...671..518K performed the first simulations of the expansion of an region in a uniform, magnetised medium. At early times, the thermal pressure of the photoionised gas dominates over the magnetic field, which is pushed out of the expanding region. At late times (several Myr), the photoionised region becomes elongated along the field lines and the magnetic field filled back in. @2009MNRAS.398..157H and @2010arXiv1012.1500M have studied the effects of ionising radiation on magnetised globules, finding that the photoevaporation process is significantly altered only when the magnetic pressure exceeds the thermal pressure by more than a factor of ten in the initial globule. @2010arXiv1010.5905P have recently studied the combined effects of magnetic fields and photoionisation on the formation of a massive star on scales $< 0.1$ pc, finding that rotational winding of the field lines produces a magnetised ‘bubble’, whose magnetic pressure acts to confine the nascent [H II]{} region during its ultracompact phase.
In this paper, we study the expansion of regions in non-uniform (turbulent) magnetised molecular clouds. Previously, we studied region expansion in a turbulent medium without a magnetic field and our results were strikingly similar to observed regions [@2006ApJ...647..397M]. In the present work, we study the effect of the expanding photoionised gas on structures in the surrounding magnetised medium and also the effect that the turbulent magnetic field has on the region itself. At least at the relatively early times studied in our simulations, most of the interesting effects due to the magnetic field will occur within the neutral, photon dominated region around the region, and so we use a careful treatment of the heating and cooling processes in this gas, first discussed in @2009MNRAS.398..157H, in order to have a realistic portrayal of the dynamical processes here.
The wealth of new observational data at infrared wavelengths enables comparisons to be made of our simulations not only with observations of the photoionised gas, as done previously in @2006ApJ...647..397M but also with the PDR and molecular gas affected by the region. In particular, the Spitzer GLIMPSE surveys of bubbles in the Galactic plane highlight polycyclic aromatic hydrocarbon (PAH) emission at the outer edges of regions and in PDRs. These surveys show that the thickness of the PAH-emitting shell varies between 0.2 and 0.4 of the outer radius and that shell thickness increases approximately linearly with radius. Moreover, the bubbles are generally non-spherical. Herschel and APEX observations reveal thermal emission from cold dust in the molecular gas outside the PDR . These observations are useful in identifying condensations that could be collapsing to form new stars.
Studies of magnetic fields in and around molecular clouds are important for understanding the star-formation process. [There is no consensus as to the importance of magnetic fields in the formation of clouds, cores, and ultimately stars. If magnetic fields are dynamically important, then star formation is regulated by ambipolar diffusion [e.g., @1999osps.conf..305M]. Alternatively, if magnetic fields are unimportant then other processes such as turbulence and stellar feedback provide support to molecular clouds and determine the star-formation efficiency (e.g., ). It is not known whether magnetic fields are strong enough to dynamically support molecular clouds and more evidence is needed. It is possible that reality lies somewhere between these extremes. Recent Zeeman observations of molecular clouds do not agree with predictions of idealized ambipolar diffusion models (strong magnetic field) but more observations are needed.]{}
There are three main techniques for tracing the magnetic field in diffuse HI and molecular clouds: polarisation from aligned dust grains, linear polarisation of spectral lines, and Zeeman splitting of spectral lines [@2005LNP...664..137H]. Polarisation studies yield $B_\perp$, the strength of the field projected onto the plane of the sky, whereas Zeeman splitting gives $B_{||}$, the strength of the field parallel to the line of sight. In particular, the Zeeman effect has been used to study the atomic gas (assumed to be in the photodissociated region) in the star-forming regions M17 [@2001ApJ...560..821B] and the Ophiuchus dark cloud complex [@1994ApJ...424..208G], while the OH Zeeman effect is used to study the line-of-sight magnetic field in the molecular gas. The results of such studies suggest that there is rough equipartition of magnetic and turbulent (dynamical) energy densities in the neutral gas. In the Ophiuchus dark cloud region, the uniform field in the gas is estimated to be 10.2 $\mu$G [@1994ApJ...424..208G], whereas the magnitude of the line-of-sight magnetic field strength in the gas associated with M17 is found to be $\sim 500 \mu$G (at $60\arcsec$ resolution, or even twice as high at $26\arcsec$ resolution) [@2001ApJ...560..821B], suggesting that there is small-scale structure in the magnetic field. Recent linear polarisation studies at submillimetre wavelengths of the magnetic field around the ultracompact region G5.89–0.39 [@2009ApJ....695.1399T] provide evidence for compression of the B field in the surrounding molecular cloud and general disturbance of the magnetic field caused by the expansion of the region.
For photoionised gas, Faraday rotation measures of line-of-sight extragalactic radio sources are used to calculate the magnetic field strengths in foreground regions [@1981ApJ...247L..77H]. The Zeeman effect cannot be used in the ionised gas because the thermal linewidths are too large. @1981ApJ...247L..77H use double extragalactic radio sources to sample two positions in the regions S117 and S264 and find identical rotation measures in each case of order 1 or $2\,\mu$G. In these regions, the thermal energy density dominates the magnetic energy density. In a third region, S119, for which only a single rotation measure is available, the line-of-sight magnetic field strength is around 20 $\mu$G and the ratio of magnetic to thermal energy density is around 0.4, hence the magnetic field could have had an effect on the expansion of this region.
The remainder of this paper is organised as follows. In § \[sec:model\] we describe our numerical algorithm and present two test problems: the classical Spitzer law for the expansion of a non-magnetised [H II]{} region in a uniform medium, and the expansion of an region in a uniform magnetised medium . In the second case, we draw attention to some interesting properties of the solution that have not been described previously.
In § \[sec:turbhii\] we describe the expansion of regions in both magnetised and non-magnetised turbulent media, taking as initial conditions for the ambient medium the results of an MHD turbulence calculation [@2005ApJ...618..344V], and considering both a strong (approximately O9 spectral type) ionising source and a weak (B0.5) one. In § \[sec:discussion\] we make qualitative comparisons between simulated optical and long-wavelength images and observations. We also produce synthetic maps of the projected line-of-sight and plane-of-sky components of the magnetic field, which can be qualitatively compared with observations of the magnetic field in star-forming regions. We discuss the globules and filaments at the periphery of our simulated regions in the context of recent numerical work on photoionised magnetised globules . Finally, in § \[sec:conclusion\] we outline our conclusions.
Numerical Model {#sec:model}
===============
Numerical Method {#sec:nummethod}
----------------
The numerical method used in this paper is the same as that described by @2009MNRAS.398..157H. The Phab-C$^2$ code combines the ideal magnetohydrodynamic code described by with the C$^2$-Ray method (Conservative Causal Ray) for radiative transfer developed by @2006NewA...11..374M, and adds a new treatment of the heating and cooling in the neutral gas [@2009MNRAS.398..157H]. The MHD code is a second-order upwind scheme that integrates the ideal MHD equations using a Godunov method with a Riemann solver, similar to that described by @1998MNRAS.297..265F but with the constrained transport (CT) method (e.g., [@2000JCoPh.161..605T]) used to maintain the divergence of the magnetic field close to zero. An artificial (Lapidus) viscosity is also included [@1984JCoPh..54..174C] in order to broaden the discontinuities and reduce oscillations. The C$^2$-Ray method for radiative transfer is explicitly photon conserving [@2006NewA...11..374M] and the algorithm allows for timesteps much longer than the characteristic ionisation timescales or the timescale for the ionisation front to cross a numerical grid cell, by means of an analytical relaxation solution for the ionisation rate equations. This results in a highly efficient radiative transfer method.
The MHD and radiative transfer codes are coupled via operator splitting, the inclusion of equations for the advection of ionised and neutral density in the MHD code, and the energy equation, which includes both radiative cooling and heating due to the absorption of stellar radiation by gas or dust: ionising extreme ultraviolet radiation in the photoionised region, non-ionising far ultraviolet radiation in the neutral gas and x-rays in the dense molecular gas. A detailed discussion of the heating and cooling contributions is given by @2009MNRAS.398..157H. In particular, the treatment of heating in the neutral PDR region, taking into account FUV/optical dust heating and x-ray heating, is a substantial improvement over standard treatments in the literature. The heating term for the neutral gas is calibrated using Cloudy [@1998PASP..110..761F] and is tailored to the ionisation source. Moreover, x-rays and FUV radiation will be enhanced due to the presence of associated low-mass stars in the star-formation environment, and we use the results of @2008ApJ...675.1361F to estimate the distributions of FUV and EUV fluxes appropriate to stellar clusters of membership sizes relevant to the two cases we study: an O9 and a B0.5 star. The energy equation is solved iteratively using substepping and the timestep for the whole calculation is determined solely by the magnetohydrodynamic timestep.
The individual components of the Phab-C$^2$ code have been tested against standard problems and the whole code was applied to the problem of the photoionisation of magnetised globules @2009MNRAS.398..157H, work which has recently been independently corroborated by @2010arXiv1012.1500M. In this paper, we have implemented an entropy fix [@1999JCoPh.148..133B] to correctly deal with the thermal and dynamical pressures in regions of low density where the magnetic pressure dominates. We have also improved the treatment of the boundary conditions because they can become important, particularly towards the end of the simulations. For our O star simulations we use outflow boundary conditions, since thermal pressure is expected to dominate throughout the evolution and the region will rapidly grow to the size of the computational box. For the B star simulations, the expansion timescale of the region is much longer and is comparable to the crossing time of the computational box corresponding to the initial turbulent velocity dispersion of the molecular gas. We therefore use periodic boundary conditions in this case, which ensures that the total mass in the computational box remains constant.
Parallelisation using Open-MP was implemented where possible, enabling the current simulations to be run in about two weeks on an Intel 8-core server with 32 GB of RAM. Preliminary work was performed on the Kan Balam supercomputer of the Universidad Nacional Autónoma de México.
Test Problems {#sec:testprob}
-------------
### Hydrodynamic [H II]{} Region Expansion in a Uniform Medium {#sec:nomhd}
![Radius against time for the hydrodynamic expansion of a photoionised region in a uniform medium. Bottom panel: numerical simulation (black line) and analytical solution (gray line). Top panel: relative error. []{data-label="fig:nomhd_rad"}](radii_vs_t_unif-weak-zerob-256){width="\linewidth"}
Org/
![Mean gas velocity against time for the hydrodynamic expansion of a photoionised region in a uniform medium. Black line — numerical simulation; thick gray line — analytic solution.[]{data-label="fig:nomhd_vel"}](velocities_vs_t_unif-weak-zerob-256){width="\linewidth"}
The radiation part of the code has been extensively tested and documented . In order to test the radiation (magneto-)hydrodynamics combined code we first consider the purely hydrodynamic evolution of a photoionised region around an ionising source in a uniform medium. The analytical solution to this problem is the familiar @1978ppim.book.....S expansion law $$R_\mathrm{Spitzer} = R_0\left(1 + \frac{7}{4}\frac{c_\mathrm{i} t}{R_0}\right)^{4/7} \ ,
\label{eq:spitz}$$ where $R_\mathrm{Spitzer}$ is the ionisation front radius, $R_0 = (3
Q_\mathrm{H}/4\pi n_0^2 \alpha_\mathrm{B})^{1/3}$ is the initial Strömgren radius, $c_\mathrm{i}$ is the sound speed in the ionised gas and $t$ is time, $Q_\mathrm{H}$ is the ionising photon rate, $n_0$ is the ambient density and $\alpha_\mathrm{B}$ is the case B recombination rate, $\alpha_\mathrm{B} = 2.6\times 10^{-13} T^{-0.7}$ [@2006agna.book.....O], where $T$ is the temperature in the ionised gas. For an ambient density $n_0 = 1000$ cm$^{-3}$, stellar ionising photon rate $Q_\mathrm{H} = 5 \times 10^{46}$ s$^{-1}$ and stellar temperature $T_\mathrm{eff} = 37500$ K the results are shown in Fig. \[fig:nomhd\_rad\], together with the corresponding analytic solution. These parameters were chosen because they represent the mean values of the turbulent medium B star case described below in § \[sec:turbhii\]. The lower panel shows radius against time, while the upper panel shows the relative error, defined as $$\mbox{Error} = \frac{R_\mathrm{ion} - R_\mathrm{Spitzer}}{R_\mathrm{Spitzer}} \ .$$ From this figure, we see that the largest discrepancies, of about 6%, occur at the beginning of the expansion, when a shock wave begins to propagate outwards ahead of the ionisation front and a corresponding rarefaction wave travels back towards the ionising source, which is not accounted for in the analytical solution. At later times, the error is never more than 1%. Our code, therefore, compares extremely favorably with other radiation-hydrodynamics codes on this problem [see, e.g. @2006ApJS..165..283A; @2007ApJ...671..518K; @2009MNRAS.400.1283I]. There is no evidence for overcooling in the ionisation front in our simulations, as was found to be a problem by @2007ApJ...671..518K. In the analytic Spitzer solution we assume an [H II]{} region temperature $T = 8900$ K, which gives the best fit to our numerical result. This is about $400$ K hotter than the volume-averaged ionised gas temperature in our simulation, but is roughly equal to the ionised gas temperature just inside the ionisation front.
In Fig. \[fig:nomhd\_vel\] we plot the gas radial velocity from the numerical simulation and the corresponding analytical solution. The numerical solution closely follows the analytical solution, though with oscillations due to acoustic phenomena, such as the initial rarefaction wave, which propagates back into the region when the ionization front begins to expand outwards.
### MHD [H II]{} Region Expansion {#sec:krum}
{width="\linewidth"}
{width="\linewidth"}
There is no corresponding analytical solution for the expansion of an region into a magnetised medium. Therefore, in order to test the MHD part of the numerical code, we carry out a simulation of the expansion of an region in a uniform magnetised medium using the same parameters as those of @2007ApJ...671..518K. This problem was also used to test the radiation MHD code of @2010arXiv1012.1500M. The simulation is performed in a $256^3$ computational box with constant ambient density and temperature of $n_0 = 10^2$ cm$^{-3}$ and $T_0 = 11$ K, respectively, a constant ambient magnetic field directed at $30^{\circ}$ to the $x$-axis in the $x$-$y$ plane[^2] of strength $B_0 = 14.2\, \mu$G and an ionising source producing $Q_\mathrm{H} = 4 \times 10^{46}$ ionising photons s$^{-1}$. The magnetic field is placed at an angle to the grid lines in order to be able to distinguish between physical and computational effects. [We run the simulation in a $(20\,\mathrm{pc})^3$ box and in a $(40\,\mathrm{pc})^3$ box, the former to study the early evolution $(t
< 2$ Myr) and the latter to highlight the late-time evolution.]{} We note that the ionising source is very weak [later than spectral type B0.5, @1996ApJ...460..914V] and that the ambient density is lower than that typically found in star-forming regions.
@2007ApJ...671..518K found a critical radius and time for when magnetic pressure and tension start to become important for the expansion of the region in a uniform magnetised medium, which corresponds to when the magnetic pressure in the neutral gas becomes of the same order as the thermal pressure in the ionised gas, i.e., $\rho_\mathrm{n}v_\mathrm{A}^2 \sim \rho_\mathrm{i} c_\mathrm{i}^2$. Here, $\rho_\mathrm{n}$, $\rho_\mathrm{i}$ are the densities in the neutral and ionised gas, $v_\mathrm{A}$ is the Alfvén speed, and $c_\mathrm{i}$ is the sound speed in the ionised gas. Thus, magnetic effects become significant when the region has expanded to a radius of roughly $$R_\mathrm{m} \equiv \left( \frac{c_\mathrm{i}}{v_\mathrm{A}}\right)^{4/3} R_0 \ ,
\label{eq:rm}$$ where $R_0$ is the initial Strömgren radius in a non-magnetised medium of the same uniform density. The corresponding time is $$t_\mathrm{m} \equiv \frac{4}{7} \left( \frac{c_\mathrm{i}}{v_\mathrm{A}} \right)^{7/3} t_0 \ ,
\label{eq:tm}$$ where $t_0$ is the sound crossing time of the initial Strömgren sphere, and both of these equations implicitly assume that the classical Spitzer expansion law [@1978ppim.book.....S] holds until the magnetic field becomes important.
For the parameters in this test problem, the initial Strömgren radius is $R_0 = 0.48$ pc, and the Alfvén speed in the ambient magnetised medium is $2.6$ km s$^{-1}$. Our simulations give a temperature in the photoionised gas of $T_\mathrm{i} \sim 9000$ K and hence a sound speed of $c_\mathrm{i} = 9.8$ km s$^{-1}$ (compare the lower assumed ionised sound speed $c_\mathrm{i} \sim 8.7$ km s$^{-1}$ in the @2007ApJ...671..518K models). Thus, the critical radius and time are $R_\mathrm{m} = 2.8$ pc and $t_\mathrm{m} = 0.61$ Myr, respectively, for our models.
In Figs. \[fig:krum1\] and \[fig:krum2\] we show snapshots of the magnetic field and the photoionised gas distribution in the central $x$-$y$, $x$-$z$ and $y$-$z$ planes of the computational box after 2 and 6 Myr of evolution. The coloured discs shown between the panels of the top row are keys to the interpretation of the magnetic field images. The hue indicates the projected field orientation in the plane of each cut (red for vertical, cyan for horizontal, etc.), while the brightness indicates the magnetic field strength, as is shown in the left-hand disc. The colour saturation diminishes as the out-of-plane component of the field increases, as shown in the right-hand disk. This non-standard representation allows the magnetic field to be sampled at every pixel. The lower panels of the figures show the ionisation fraction, temperature and density of the gas in the same central planes. Ionisation fraction is represented by colour: red indicates fully ionised gas while blue/green is partially ionised. The gas temperature varies beween $\sim 10^4$ K (red) in the ionised gas to a few hundred Kelvin in the neutral (PDR) gas. The gas density is represented by the intensity in these figures, with the densest, cold molecular gas appearing white.
At early times, $t < t_\mathrm{m}$, the region is essentially spherical and expands at approximately the same speed (half the sound speed) in all directions because the ionised gas thermal pressure is much higher than the ambient magnetic pressure. Expansion across the magnetic field lines compresses the B-field. The expanding region and PDR are preceded by a fast-mode shock, which travels furthest perpendicular to the magnetic field direction. The fast-mode shock bends and compresses the field lines and so the PDR and region propagate in a modified magnetic field. The slow-mode shock is strongest along the field lines and most evident at the end caps, where it produces a dense shell. It compresses the gas and demagnetises it (reduces the field). The slow-mode shock is almost isothermal and the dense shell behind it is pushed by the pressure in the photoionised region. At early times the magnetic field is reduced in the photoionised gas by the expansion of the region. At later times, $t > t_\mathrm{m}$, the magnetic tension pulls the field back into the region at the equator.
In Fig. \[fig:krum1\] we show the situation after 2 Myr, when the magnetic field has started to pull back into the region. The region is elongated along the magnetic field direction but the photon dominated region (warm neutral gas) is approximately spherical. The magnetic field retains its original direction except around the edge of the region. This is particularly evident at the poles of the region. At the time shown in the figure, the magnetic field appears concentrated towards the centre, with regions of lower magnetic field around the equator or waist of the structure. Neutral gas is pulled into the region along with the magnetic field, becomes ionised and flows away from the ionisation front into the region. It then flows along the field lines and is channelled out towards the poles, where the ionisation front transforms into a recombination front. We note that at this time in the simulation, we do see instabilities form in the direction perpendicular to the magnetic field, as reported by @2007ApJ...671..518K and @2010arXiv1012.1500M. These instabilities form at the equator where the ionisation front is parallel to the magnetic field direction because the slow-mode shock which precedes the ionisation front, is not permitted to cross the field lines. As a result, the ionisation front corrugates.
Our Fig. \[fig:krum2\] shows the simulation after 6 Myr in a $(40\,\mathrm{pc})^{3}$ computational box. The region has stopped expanding perpendicular to the magnetic field but the shocked neutral shell is still expanding along the field lines, leading to a very elongated structure. However, in the direction along the field lines, the flow has become one dimensional and so the density does not fall off as the region expands. Thus, there is a limit as to how far the ionisation/recombination front can travel in this direction. There is an extended region of partially ionised gas beyond the recombination front. The warm, neutral gas (PDR) is still roughly spherical in shape but is pierced by the cigar-shaped region along the field direction. The dense neutral knots formed as a result of the instability at the equator shadow parts of the PDR. The knots formed closest to the perpendicular direction remain there throughout the simulation, pinned by the magnetic field. The knots slightly off the equator experience the rocket effect due to the photoionisation of their outer skins. They move like beads on a wire along the magnetic field lines around the edge of the region, a process which can be clearly observed in animations of this simulation. By the end of the calculation, the magnetic field has refilled the region and become roughly uniform and returned completely to its original direction.
![Radius against time for the magnetohydrodynamic expansion of a photoionised region in a uniform magnetised medium. Numerical MHD simulation (black line) and analytical non-magnetised solution for classical Spitzer region expansion (gray line). Also shown are the polar radius (dotted line) and equatorial radius (dashed line). The break in the lines at $t =
2$ Myr is a result of plotting velocities from two simulations at different numerical resolutions.[]{data-label="fig:mhd_rad"}](radii_vs_t_krum){width="\linewidth"}
![Mean velocities against time for the magnetohydrodynamic expansion of a photoionised region in a uniform magnetised medium. Rms velocity (black line) and analytical non-magnetised solution for classical Spitzer region expansion (thick grey line). Also shown are the mean radial velocity in the ionised gas (dashed line) and the velocity of the gas at the ionization front (dotted line). The break in the lines at $t = 2$ Myr is a result of plotting velocities from two simulations at different numerical resolutions. []{data-label="fig:mhd_vel"}](velocities_vs_t_krum){width="\linewidth"}
Finally, in Figs. \[fig:mhd\_rad\] and \[fig:mhd\_vel\], we examine the expansion of the magnetised region. Fig. \[fig:mhd\_rad\] shows the evolution of the radius of the photoionised region with time. The mean radius is consistently below the classical Spitzer expansion law, unlike the non-magnetic case (see Fig. \[fig:nomhd\_rad\]). In the direction perpendicular to the magnetic field, the expansion slows with time, as shown by the evolution of the minimum radius. In the direction along the field lines, the expansion initially ($t \leq 1$ Myr) follows the classical Spitzer law but then accelerates as material is channelled along the field lines and out of the poles of the region. This expansion abruptly stops just after 4 Myr, which corresponds to the time when the ionisation front can no longer expand along the field lines because the flow has become one dimensional.
In Fig. \[fig:mhd\_vel\] the mean radial and root-mean-square (rms) velocities of the ionised gas are plotted together with the analytical value and also the mean velocity of the ionisation front. The discontinuity at 2 Myr is due to the change in resolution of the simulation. The calculated values are all above the analytical values and indicate that typical velocities in the photoionised gas are of the order 1–2 km s$^{-1}$ throughout the simulated lifetime of the region. The mean radial velocity shows the same sort of ringing that we saw in the non-magnetised simulation (see Fig. \[fig:nomhd\_vel\]). After about 3 Myr, the gas velocities become dominated by the one-dimensional flow being channelled along the field lines.
[H II]{} Region Expansion in a Magnetised Turbulent Medium {#sec:turbhii}
==========================================================
In this section we describe the results of radiation MHD simulations of region expansion in a more realistic medium where neither the initial density distribution nor magnetic field are uniform. We consider both an O star ionising flux and one more appropriate to a B star in order to study the effect of the magnetic field on the ionised gas and on the neutral gas of the PDR.
Initial Set Up {#sec:setup}
--------------
[As initial conditions we take the results of a $256^3$ driven-turbulence, ideal magnetohydrodynamic simulation by @2005ApJ...618..344V, specifically the simulation with initial isothermal Mach number $M_\mathrm{s} = 10$, Jeans number $J = 4$ and plasma beta $\beta = 0.1$. The original @2005ApJ...618..344V simulations are scale free and are characterised by the three nondimensional quantities $M_\mathrm{s} = \sigma/c$ (the rms sonic Mach number of the turbulent velocity dispersion $\sigma$), $J\equiv L/L_\mathrm{J}$ (the Jeans number, giving the size of the computational box $L$ in units of the Jeans length $L_\mathrm{J}$), and the plasma beta, $\beta \equiv 8\pi \rho_0 c^2/B_0^2$ (ratio of thermal to magnetic pressures).]{}
[We set the scaling by choosing the computational box size to be 4 pc, giving the ambient temperature as $T_0
\sim 5$ K, the mean atomic number density as $\langle n_0 \rangle =
1000$ cm$^{-3}$ and the mean magnetic field strength as $B_0 = 14.2
\mu$G. We choose as our starting point the time in the evolution when there is one collapsing object. At this point, the mean values of the magnetic field the number density are unchanged, since flux and mass are conserved, while the mean plasma beta is now $\beta = 0.032$, which is consistent with the rms value of the magnetic field $B_{\rm
rms} = 24.16\, \mu$G.]{} That is, the turbulence has enhanced the rms magnetic field by a factor $\eta = 24.16/14.2$, which leads to a change by a factor $\eta^2 \sim 3$ in the value of $\beta$ over the original, uniform conditions.
As in our previous paper [@2006ApJ...647..397M], we place our ionising source in the centre of the collapsing object, remove $32
M_{\odot}$ of material (corresponding to the mass of the star formed) and take advantage of the periodic boundary conditions of the turbulence simulation to move the source to the centre of the grid. We also subtract the source velocity from the whole grid. We consider two ionising sources: one corresponds approximately to an O9 star [@1996ApJ...460..914V], having an effective temperature $T_\mathrm{eff} = 37500$ K and an ionising photon rate of $Q_\mathrm{H} =
10^{48.5}$ photons s$^{-1}$, while the other ionising source has $Q_\mathrm{H} = 5
\times 10^{46}$ photons s$^{-1}$, which corresponds to B0.5.
Results {#sec:results}
-------
In this section we present our results both for the O star and the B star and in each case for the magnetic and also for the purely hydrodynamic simulations. In this way we can assess the importance of the magnetic field and the ionising photon flux on the evolving structures in the expanding regions and surrounding neutral gas. We first present emission-line images of the simulated regions for both O and B stars in the magnetic and pure hydrodynamic cases when they have reached a similar size, which corresponds to 200,000 yrs of evolution for the O star case and $10^6$ yrs of evolution in the B star case. This permits a direct comparison with observations at optical and longer wavelengths. We then examine the expansion and time evolution of the ionised, neutral and molecular components using global statistical properties. The importance of the magnetic field in the different components is then studied for the particular O and B star simulations presented earlier.
Although the ionised/neutral transition of hydrogen is treated explicitly in a detailed manner in our radiation-MHD code, the neutral/molecular transition is treated much more approximately. We simply assume that the molecular fraction is a predetermined function of the extinction $A_V$ from the central star to each point in our simulation: $n_{\mathrm{mol}}/n_{\mathrm{neut}} = \left(1 +
e^{-4(A_V - 3)} \right)^{-1}$, which was determined by fitting to the Cloudy models described in the Appendix of @2009MNRAS.398..157H.
A rough estimate of the importance of the magnetic field in our turbulent media simulations can be obtained by calculating the critical radius and timescale, $R_\mathrm{m}$ and $t_\mathrm{m}$ (see Eqs. \[eq:rm\] and \[eq:tm\]), for the initial value of the magnetic field and the mean density. Using $B_\mathrm{rms} = 24.16\,\mu$G and $\langle n \rangle = 10^3$ cm$^{-3}$, and assuming a mean particle mass of 1.3 m$_\mathrm{H}$, the initial representative Alfvén speed in the neutral gas is $v_\mathrm{A} \equiv B_\mathrm{rms}/(4\pi \rho)^{1/2} \simeq
1.46$ km s$^{-1}$. In the photoionised gas, the sound speed is $c_\mathrm{i} =
9.8$ km s$^{-1}$. For the strong ionising source (O star), the Strömgren radius in a uniform medium of density equal to the mean density would be $R_\mathrm{0} = 0.45$ pc, giving a critical radius $R_\mathrm{m} =
5.7$ pc and time $t_\mathrm{m} = 2.2$ Myr for when the magnetic field starts to become important. If, instead of the rms magnetic field we use the mean magnetic field, $\langle B \rangle = 14.2\,\mu$G, then the corresponding values are $R_\mathrm{m} = 11.86$ pc and $t_\mathrm{m} = 7.9$ Myr. For the weak ionising source (B star), the numbers are $R_\mathrm{0} = 0.113$ pc, $R_\mathrm{m} = 1.43$ pc ($R_\mathrm{m} = 2.98$ pc) and $t_\mathrm{m} = 0.55$ Myr ($t_\mathrm{m} = 1.98$ Myr) depending on whether the rms or mean magnetic field is considered.
Thus, for the strong ionising source, the critical diameter is larger than the dimension of the computational box, hence we do not expect that the magnetic field would have much effect on the expansion of the region in this case. However, for the weak ionising source, it is not clear whether the rms or the mean value of the magnetic field is most important for determining the global evolution of the region. If the rms field is the determining parameter then the fact that the critical diameter in this case is smaller than the size of the computational box leads us to expect that the magnetic field will affect the global expansion of the region. On the other hand, if it turns out that the mean magnetic field is most important, then the effect will not be noticed within our computational size and timescales. Furthermore, given the non-uniform initial conditions, the timescale for the magnetic field to become important could vary depending on the local gas sound speed to Alfvén speed ratio.
### Morphologies and Images {#sec:morph}
{width="\linewidth"}
{width="\linewidth"}
We start our comparison by considering the morphological appearance of the regions and the surrounding neutral and molecular material. In Figs. \[fig:images\_O\] and \[fig:images\_B\] we show two types of image data, one showing mostly the ionised gas, where the different colours represent emission from optical emission lines, the other shows synthetic, long-wavelength (infrared and radio) emission, emphasizing the neutral and molecular material.
[The optical emission was calculated as described in our previous papers [@2006ApJ...647..397M; @2009MNRAS.398..157H], assuming heavy element ionisation fractions that are fixed functions of the hydrogen ionisation fractions. For the B star models, these functions were recalibrated for the softer stellar ionising spectrum using the Cloudy plasma code [@1998PASP..110..761F]. For the O star models, we employ the “classical” *HST* red-green-blue colour scheme of \[\], H$\alpha$, \[\]. Since the \[\] emission from the B star models is predicted to be very weak, in this case we use the scheme \[\], \[\], H$\alpha$. In both schemes, the progression from red through green to blue corresponds to increasing degree of ionisation inside the region. The emission from all the optical lines is negligible in the dense neutral/molecular zones, but the dust absorption associated with these dense regions is visible in the images. Scattering by dust is not included.]{}
[For the long-wavelength images the emission bands were chosen to give a more global view of the simulations than can be provided by the optical emission lines. The red band shows the total column density of neutral/molecular gas, which very crudely approximates far-infrared or sub-mm continuum emission from cool dust. The green band of the long-wavelength images shows a simple approximation to the mid-infrared emission from polycyclic aromatic hydrocarbons (PAH). The PAHs are assumed to reprocess a fixed fraction of the local far-ultraviolet radiation field. Detailed calculations [@2001ApJ...556..501B] show that this assumption is correct to within a factor of about two for most PAH emission features over a broad range of conditions. The local PAH density is assumed to be proportional to the neutral plus molecular hydrogen density, since there is strong evidence that PAHs are destroyed in ionised gas [@1989ApJ...344..791B; @2007ApJ...665..390L]. No attempt is made to discriminate between different PAH charge states. The blue band of the long-wavelength images shows the 6 cm radio continuum emission due to bremsstrahlung in the ionised gas. Apart from a slightly different temperature dependence this is very similar to the H$\alpha$ emission, except that it does not suffer any dust absorption. ]{}
Starting with the O-star case, we see that the morphological appearance shows the typical characteristics of regions in turbulent media found in earlier work , namely a fairly irregular structure with fingers or pillars, as well as bar-like features at the edge of the region. Note how the most prominent finger in the bottom edge of the region does not exactly point towards the source of ionising photons. In appearance the simulated regions look strikingly like observed regions.
Comparing the magnetic with the non-magnetic case, one sees an overall agreement in the structures; the presence of a magnetic field does not have a large impact on the global morphology of the region. This is to be expected as for the O-star case the critical time (see Eq. \[eq:tm\]) has not been reached by the time most of the volume has become ionised. However, the magnetic field does have an effect on small scale features in the region. In the magnetic case the fingers and bars look smoother and broader, due to the magnetic pressure, which modifies the radiation-driven implosion of these structures . This behaviour is visible in both optical and long-wavelength images, but most clearly in the latter. In section \[sec:discussion\] we discuss the behaviour of the globules in greater depth.
For the B-star case, the critical time does occur within the simulation. However, as is apparent from Fig. \[fig:images\_B\] this does not lead to a significant deformation of the region. The reason is that magnetic field itself also has a turbulent structure, and there is no large-scale field which can impress its direction on the shape of the region, as in the test case presented above (see § \[sec:krum\]). Although the magnetic and non-magnetic results do differ, the overall impression is that they are still quite similar. In fact, the difference between the O-star and the B-star cases is much larger than between the magnetic and non-magnetic cases. For the B-star case, no pillars are found and the ionised region appears more spherical. The edge of the region is often more fuzzy, although some sharper bar-like features are present. The long wavelength images show the reason for this: the region is embedded in a thick PDR region (with an extent of $\sim 30\%$ of the radius of the region) which erases much of the small-scale density fluctuations present in the original cold molecular medium. Close inspection reveals that the high-density regions already inside the PDR region photo-evaporate and thus lose their large density contrast. In the O-star case these are the features which develop into pillars. The fact that the temperature in the PDR is closer to 100 K than to 10 K in combination with the low ionising flux of the B-star should generally be less conducive to the formation of pillars, as shown by @2010ApJ...723..971G.
Careful examination of the images shows that otherwise the effect of the magnetic field is quite similar to that in the O-star case: small-scale structures are smoothed out in the magnetised case. However, the effect is quite marginal and does not give a clear observable diagnostic for the presence of magnetic fields or not.
### Global Properties of the Region Evolution {#sec:glob-prop-prot}
![Mean densities of the ionised (solid line), neutral (dashed line) and molecular (dotted line) components for the evolving region around the O star. Lines with symbols are for the MHD simulation, while lines without symbols are for the purely hydrodynamic case.[]{data-label="fig:comp1_O"}](comparison1_vs_t_Ostar){width="\linewidth"}
![Same as Fig. \[fig:comp1\_O\] but for the B star.[]{data-label="fig:comp1_B"}](comparison1_vs_t_Bstar){width="\linewidth"}
The initial mean density in the computational box is $n_0 =
1000$ cm$^{-3}$ but the material is distributed inhomogeneously with the densest clumps having densities $> 10^5$ cm$^{-3}$. In Figs. \[fig:comp1\_O\] and \[fig:comp1\_B\] we can see how the mean densities in the ionised, neutral and molecular components evolve with time and the growth of the regions around the O and B stars, respectively. In these figures, lines with symbols are for the MHD simulations and lines without symbols are the purely hydrodynamic case. We can see immediately that the magnetic field has a negligible effect on the densities of the different components in both the O and B star cases. As the evolution of the region proceeds, the mean density of the molecular gas in the O star simulation increases slowly with time because the lower density molecular gas becomes ionised or incorporated into the neutral PDR and also because the dense clumps and filaments at the edge of the region are compressed due to radiation-driven implosions, thereby increasing their density. By the end of the simulation, only these densest clumps remain, since the majority of the computational box has been ionised. In the B star case, the mean density of the molecular gas remains roughly constant throughout the simulation. This is because the region does not break out into regions of lower density and remains confined within the molecular gas for the duration of the simulation. Also, there is less fragmentation in the B star simulation because density inhomogeneities in the neutral region of the PDR are smoothed out by photoevaporation flows due to FUV radiation before the ionisation front reaches them (see Fig. \[fig:images\_B\]).
By the end of the respective simulations, the mean density in the ionised gas around the O star is $\sim 100$ cm$^{-3}$, while that around the B star is $\sim 10$ cm$^{-3}$ even though the spatial extent is much smaller. This is because the latter is a much weaker ionising source. In both cases, the density of the neutral PDR gas tends to a roughly constant value of $10^3$ cm$^{-3}$.
![Ionised, neutral and molecular gas fractions by volume (solid lines) and mass (dashed lines) for the evolving region around the O star. Lines with symbols are for the MHD simulation, while lines without symbols are for the purely hydrodynamic case.[]{data-label="fig:comp2_O"}](comparison2_vs_t_Ostar){width="\linewidth"}
![Same as Fig. \[fig:comp2\_O\] but for the B star.[]{data-label="fig:comp2_B"}](comparison2_vs_t_Bstar){width="\linewidth"}
In Figs. \[fig:comp2\_O\] and \[fig:comp2\_B\] we compare how the ionised, neutral and molecular gas component fractions vary throughout the evolution of the regions. We calculate both volume and mass fractions for each component. The ionised gas in both the O and B star cases is thermally dominated and there is no discernible distinction between the MHD and purely hydrodynamic results. In the O star case, the ionised fraction comes to occupy 90% of the volume but only 10% of the mass of the computational box by the end of the simulation. In the B star case, the region does not manage to globally break out of the computational box and so even at the end of the simulation the volume occupied by ionised gas is less than 30%. The mass fraction is $< 1\%$ in this case, since the density in the ionised gas is low. The neutral gas is most affected by the magnetic field for both O and B star simulations. Figs. \[fig:comp2\_O\] and \[fig:comp2\_B\] show that the MHD simulation has both higher mass and higher volume fractions of neutral gas than the purely hydrodynamic results. The neutral component is more important in the B star case than in the O star case, reaching $\sim 70\%$ of the mass and volume fractions after 500,000 yrs and remaining roughly constant thereafter. In the O star case, however, the mass fraction remains roughly constant for the second half of the simulation while the volume fraction decreases sharply. This difference in behaviour can be attributed to the greater fragmentation in the O star case (both MHD and HD) compared to the B star. The neutral material in the B star simulation is distributed in a quite broad, smooth, almost uniform density region around the region, whereas in the O star simulation the neutral region around the photoionised gas is relatively thin but there is also neutral material inside the dense clumps and filaments that are formed by radiation-driven implosion, which have high density but low volume.
![Expansion of the region with time for the O star. The thick grey line represents the analytical solution. Also shown are the mean radius (solid lines), the maximum radius (dotted lines) and the minimum radius (dashed lines) of the ionisation front. Lines with symbols are for the MHD simulation, while lines without symbols are for the purely hydrodynamic case.[]{data-label="fig:comp2b_O"}](comparison2b_vs_t_Ostar){width="\linewidth"}
![Same as Fig. \[fig:comp2b\_O\] but for the B star.[]{data-label="fig:comp2b_B"}](comparison2b_vs_t_Bstar){width="\linewidth"}
The expansion of an region in a clumpy medium is not so easy to characterise. In Figs. \[fig:comp2b\_O\] and \[fig:comp2b\_B\] we plot a variety of different measures of the radius of the photoionised region as functions of time, together with the analytical solution obtained using Equation \[eq:spitz\] for a medium of uniform density $n_0 = 10^3$ cm$^{-3}$. We calculate the mean radius[^3] and also the minimum and maximum radii of the ionisation front. For the region around the O star, the mean radius closely follows the analytical solution for a long period of the simulation, possibly because the filling factor of the dense clumps and filaments which retard the ionisation front is small. The maximum radius for the ionisation front for the O star simulation quickly becomes equal to the distance to the edge of the computational box and therefore has no physical meaning after this point. Until it leaves the box, the maximum radius represents the direction in which the ionisation front is able to expand most quickly, i.e. a path of lowest density radially outward from the star.
For the B star simulation, the mean radius is always below the analytical solution. Since this is true for both the MHD and purely hydrodynamic simulations, it cannot be due to the magnetic field. The reason is that for the B star, the initial clump in which the star is embedded has a greater effect on the subsequent evolution of the region than in the O star case. In fact, we see that the maximum radius of the ionisation front in the B star region does follow the analytical solution. The maximum radius follows the expansion of ionisation front in the direction where it was first able to break out of the dense clump in which the star was embedded. We have already seen that the neutral medium around the region in the B star case takes on a smooth and homogeneous appearance and that the mean molecular and neutral gas densities are roughly constant in time. Once the ionisation front has managed to break out of the clump, therefore, it follows the expansion law for a uniform medium in this mean density. In the majority of directions, however, the ionisation front takes much longer to break out of the initial clump and so the mean ionisation front radius is retarded compared to the analytical solution.
In both O and B star simulations, the minimum radius of the ionisation front indicates the presence of fingers and clumps of neutral material within the region. For the O star simulation we have already mentioned how the magnetic field suppresses fragmentation, and this is reflected in the graph of minimum radius with time: fingers and globules formed in the purely hydrodynamic simulation are denser and survive longer closer to the star compared to the MHD simulation. In the B star case there is much less fragmentation and we see no fingers and clumps of neutral gas within the region and the minimum radii of both MHD and hydrodynamic simulations seen in Fig. \[fig:comp2b\_B\] bear this out.
![Mean gas radial velocities within the ionised, neutral and molecular gas components for the O star simulation. The thick grey line represents the analytical mean radial velocity. Also shown are the rms radial velocities (solid lines) and the mean radial velocity (dashed lines) of the gas. Lines with symbols are for the MHD simulation, while lines without symbols are for the purely hydrodynamic case.[]{data-label="fig:comp3_O"}](comparison3_vs_t_Ostar){width="\linewidth"}
![Same as Fig. \[fig:comp3\_O\] but for the B star.[]{data-label="fig:comp3_B"}](comparison3_vs_t_Bstar){width="\linewidth"}
In Figs. \[fig:comp3\_O\] and \[fig:comp3\_B\] we show the mean radial velocities for the three gas components. Again, we compare the results of both MHD (lines with symbols) and purely hydrodynamic simulations. Beginning with the ionised gas component, we see that for the O star the rms velocities reach a peak of about 8 km s$^{-1}$, and then decreases very slowly, which is what we saw in our previous paper [@2006ApJ...647..397M]. The mean radial velocity, on the other hand, peaks at 8–10 km s$^{-1}$ then declines quite quickly. The rms velocities are due to interacting photoevaporated flows within the region, which lead to gas flowing inwards as well as outwards [@2003RMxAC..15..175H]. The mean radial velocity traces the general expansion of the photoionised gas. We see that for the non-uniform medium, the initial expansive motions are more rapid than for the simple analytical model but this is just because the gas first begins to expand into regions of lowest density [@2002ApJ...580..969S]. There is a small difference between the rms velocities in the MHD and hydrodynamic cases, with the latter being marginally higher. This could be due to the greater degree of fragmentation in the hydrodynamic simulations which, in turn, leads to stronger photoevaporated flows.
In the B star simulation the rms velocities are similar to the O star case just discussed, which suggests that photoevaporated flows are important in this region, too. The pure hydrodynamic simulation shows slightly higher velocities than the MHD case, which indicates that the magnetic field plays some role in reducing the importance of photoevaporated flows within the ionised gas. The mean radial velocities show the same sort of ringing seen in the test problem of region expansion in a uniform density medium (see Fig. \[fig:nomhd\_vel\]). The period of the oscillations is of order $2\times10^5$ yrs, which is consistent with the sound crossing time of the photoionised region. Note that in both O and B star simulations, the actual mean radial velocities are considerably higher than the corresponding analytical values for expansion in a uniform medium with density equal to the mean density of the initial computational box ($n_0 = 1000$ cm$^{-3}$).
The neutral gas velocities show more separation between hydrodynamic and MHD results, with the hydrodynamic velocities being consistently higher in both mean and rms velocities. Interestingly, the different simulation velocities appear to depart from each other at a specific point in time, which is different depending on whether the mean or rms velocity is considered. This is true for both O and B star simulations. In the O star case, after the initial acceleration, the neutral gas velocities reach 4–5 km s$^{-1}$, while for the B star the velocities are 1–2 km s$^{-1}$ lower and show evidence for ringing as for the ionised gas.
The initial molecular gas velocities reflect the Mach 10 supersonic turbulence of the underlying density distribution. At early times, there is evidence of residual infalling motions, i.e. negative radial velocities, again from the initial conditions. These persist longer in the B star simulations than in the O star simulations. This is because the O star rapidly ionises out to the edge of the computational box, and the molecular material which remains is being accelerated radially away from the central star in clumps that are experiencing the rocket effect. In the B star case, most of the molecular material surrounding the region remains undisturbed, and so the imprint of the initial conditions remains in this gas until the end of the simulation.
In summary, we find slight differences in the expansion of regions between MHD and purely hydrodynamic simulations but the greatest differences are due to the different ionising and FUV fluxes of the central O or B star. Most of the differences between the MHD and hydrodynamic cases are due to the lesser amount of fragmentation in the former, indicating that magnetic fields provide some support to the neutral gas against radiation-driven implosions.
Magnetic Quantities {#sec:magnetic-quantities}
-------------------
{width="\linewidth"}
![Magnitude of B-field (Gauss) against number density (cm$^{-3}$) for volume-weighted joint distributions in the case of the B-star at $t=10^6$ years. The diagonal lines correspond to constant Alfvén speeds of 10 and 1 km s$^{-1}$. The three colours represent ionised (blue), neutral (green) and molecular (red) material.[]{data-label="fig:nB_Bstar"}](mhd-pressures-rgb-Bstar-ep-1000-n-B){width="\linewidth"}
![Magnetic pressure versus turbulent pressure for volume-weighted joint distributions in the case of the B-star at $t=10^6$ years. Note that the scales are the same on both $x$ and $y$ axes. The diagonal line shows equal pressures. The three colours represent ionised (blue), neutral (green) and molecular (red) material.[]{data-label="fig:pram_pmag_Bstar"}](mhd-pressures-rgb-Bstar-ep-1000-pram-pmag){width="\linewidth"}
Even though we find that the presence of a magnetic field only has small effects on the growth of the region, this does not mean that the field is uninteresting or unimportant. For one thing, the field is important in the still neutral medium, where it constitutes a significant fraction of the total energy.
In Fig. \[fig:cuts\_Bstar\] we show the magnetic field and the distribution of the photoionised gas for the B star case in the same format as Figs. \[fig:krum1\] and \[fig:krum2\]. We can see that the magnetic field looks very different. The initial conditions were the result of an MHD turbulence calculation and in the molecular gas (coloured grey/white in the lower panels of Fig. \[fig:cuts\_Bstar\]) we can see that the field appears to be randomly oriented.
In the warm, neutral (PDR) gas (coloured purple in the lower panels of the figure) there is strong evidence that the magnetic field is being aligned parallel to the ionisation front, particularly in the lower left region of the $x$-$y$ and $x$-$z$ plots. At some points in this image, the ionisation front has left the grid, and the periodic boundary condition in this simulation means that ionised material exits one side of the grid and reappears at the opposite boundary, where it will usually recombine because of the high column density through the molecular and neutral gas to the star at this position. At this point in the simulation, this is not affecting the global evolution of the region and surrounding warm, neutral gas.
The photoionised region has a very low magnetic field intensity, showing that the field has been pushed aside by the expansion of the region. Photoevaporated flows from density inhomogeneities in both the neutral and molecular gas drag the magnetic field with them into the photoionised region in long filamentary structures. However, there is little mass associated with these flows. In the $x$-$y$ and $x$-$z$ central planes there is a large region of partially ionised gas. This is also in the direction where the ionisation front has left the grid and could be due to a density or velocity gradient, which results in the ionisation front being unable to keep up with the gas, since it is no longer preceded by a neutral shock in this part of the computational grid.
In order to analyze the role of the magnetic field in our three components, ionised, neutral and molecular, we present colour plots showing the joint distribution of various quantities in these three components. Since the B-star and O-star results are quite comparable, we only show the B-star results.
We start by considering joint distribution of the magnetic field strength against the density. Fig. \[fig:nB\_Bstar\] shows this for the case of the B-star at $t=10^6$ years. The distribution of the ionised material is shown in blue and displays a wide range of field strengths (0.1 to 10 $\mu$G) across a narrow range of densities (10 to 20 cm$^{-3}$; in the case of the O-star these values are some 5 times higher). The diagonal lines show the Alfvén speeds. If the sound speed in the gas is smaller than $v_\mathrm{A}$ the flow will be magnetically dominated. As can be seen, the ionised material does not reach Alfvén speeds of 10 km s$^{-1}$, the typical sound speed in the ionised medium. Consequently the ionised flow is never magnetically dominated, consistent with the results found in the previous sections.
Another way to represent the role of the magnetic field is to plot the joint distribution of magnetic and ram (turbulent) pressure, as is done in Fig. \[fig:pram\_pmag\_Bstar\]. In full equipartion, the turbulent, magnetic and thermal pressures should all be equal. Since the region is expanding through its thermal pressure, it is the thermal pressure which is the overall dominating one. Fig. \[fig:pram\_pmag\_Bstar\] shows that of the other two pressure components, turbulent pressure mostly dominates over magnetic pressure. In only a few places in the region the magnetic pressure dominates. In the neutral and molecular material (represented by green and red in the joint distribution figures) the situation is different. The joint distribution of density and magnetic field strength, Fig. \[fig:nB\_Bstar\], shows a wide range of densities ($10^2$–$10^4$ cm$^{-3}$) and a narrower range of magnetic field strengths (mostly around 10–30 $\mu$G, although some areas, principally neutral ones, have magnetic fields as weak as 1 $\mu$G). The molecular material generally has higher densities than the neutral material. Since the sound speed in the neutral and molecular regions is 1 km s$^{-1}$ or less, they are mostly magnetically rather than thermally dominated. In the distribution of pressures, Fig. \[fig:pram\_pmag\_Bstar\], one sees that the neutral and molecular material clusters around equipartition of the magnetic and turbulent pressures. The molecular material has regions where the turbulent pressure dominates, and others where the magnetic pressure dominates. The neutral (PDR) material has generally somewhat lower pressures and is closer to equipartition.
Fig. \[fig:nB\_Bstar\] can be compared to observationally derived values for the magnetic field and densities in regions and their surroundings. Results of Harvey–Smith et al. (in preparation) on large Galactic regions show ranges in magnetic field strengths and typical densities for the ionised, neutral and molecular components very comparable to what we find in our simulation results.
Discussion {#sec:discussion}
==========
Comparison with observations: RCW120 {#subsec:rcw120}
------------------------------------
The region RCW120 and its surrounding medium have recently been extensively studied at near- to far-infrared wavelengths in the context of triggered star formation . This small region appears to have a single ionising source, identified from VLT-SINFONI near-IR spectroscopy and spectral line fitting to stellar atmosphere models as an O6–O8V/III star with an effective temperature of $T_\mathrm{eff} = 37.5\pm2$ kK and ionising photon rate of $\log
Q_\mathrm{H} = 48.58\pm0.22$. These observationally derived parameters are remarkably similar to those we have adopted for our O star simulations and so an opportunity for a direct comparison of RCW120 with our results presents itself.
Morphologically, the observations and simulation look very similar. In particular, our Fig. \[fig:images\_O\], corresponding to the O star simulation after 200,000 yrs of evolution, is the same size as the observed bubble RCW120. The upper panels of the figure show the optical emission (with extinction from dust), while the lower panels show synthetic 6cm radio free-free emission from the ionised gas (blue), generic PAH emission from the PDR (green) and cold molecular gas column density (red). Figs. 1 and 2 of show the H$\alpha$ emission from the ionised gas, the $8\,\mu$m emission from PAHs in the PDR, $24\,\mu$m emission from warm dust in the photoionised region, and $870\,\mu$m cold dust emission from the neutral and molecular gas. If the reader mentally rotates the simulated images so that the lower left corner moves to the top centre, then a certain correspondence between simulations and observations can be imagined. The ionised gas emission fills the central cavity. In the optical, the images appear criss-crossed by dust lanes. Some of these are foreground, others are the result of projection of filaments and ridges deeper inside the region. The PAH emission comes from a thin shell around the periphery of the region, with projection seeming to put some PAH emission in the interior of the region, as can be seen in both the simulation and the observations. The simulated PAH emission presents the same sort of irregularities, filaments and clumps as are seen in the observations.
The initial total mass in the computational box is $\sim
2000\,M_{\odot}$. After 200,000 yrs of evolution, the mass distribution, as seen in Fig. \[fig:comp2\_O\], is approximately 5% ionised gas, 55% neutral gas, and 40% molecular gas. Some of the molecular gas is undisturbed material at the edge of the computational box but quite a large proportion must be in the clumps and filaments in the region and PDR, where it becomes compressed by radiation-driven implosion. The region has influenced virtually all of the computational domain by this time, and the $\sim
2000\,M_{\odot}$ of neutral and molecular material is comparable to the 1100–$2100\,M_{\odot}$ derived by and from the $870\,\mu$m cold dust emission. This dust will be present in both the neutral and molecular gas. Hence, we can surmise that the average density in the vicinity of RCW120 is similar to that in our computational box, that is $\langle n_0 \rangle = 10^3$ cm$^{-3}$, though, of course, the density distribution is far from homogeneous. From this, we could go so far as to postulate that the age of RCW120 must be similar to the time depicted in our simulation, that is, of order 200,000 yrs. were unable to assign an age to RCW120 except to say that it must be younger than 5 Myr.
Although radiation-driven implosion does enhance the density of photoionised globules at the periphery of the simulated region, the fact that we do not include self gravity in our simulations means that we are unable to model triggered star formation in the neutral shell around the region. However, it is quite likely that radiation-driven implosion could trigger gravitational collapse in clumps that are already on the point of forming stars.
There are other potentially important physical processes that we do not include in our models. Dust grains are an important component of the interstellar medium in star-forming regions. They absorb ionising photons and both observations (e.g., [@1989ApJS...69..831W]) and theory (e.g., [@2004ApJ...608..282A]) suggest that more than 50% of all the ionizing photons are absorbed by dust within the regions in the initial stages of evolution when the ionised density is high. Radiation pressure on dust grains can form a central cavity in the region , which can be as large as 30% of the ionisation front radius. Such cavities have been known for a long time from optical observations of regions (e.g., [@1962ApJ...135..394M]) and are also seen in more recent infrared observations (e.g., [@2008ApJ...681.1341W]). @2009ApJ...703.1352K have studied the effect of radiation pressure on the dynamics of regions and conclude that it is only an important effect for massive star clusters and not for regions around individual massive stars. Stellar winds are also to be expected from stars whose effective temperatures are greater than 25,000 K and these winds will provide an alternative mechanism for evacuating a central cavity in the dust and gas distribution in the region. The mechanical luminosity of the stellar wind is converted into thermal pressure by shock waves [@1997pism.book.....D] and this could affect the dynamics of the region.
Predicted maps of projected magnetic field {#sec:pred-maps-proj}
------------------------------------------
Figs. \[fig:bproj-ostar-full\] to \[fig:bproj-bstar-globule\] show visualisations of the projected integrated B-field (line-of-sight and plane-of-sky components) from our simulations. These visualisations represent generic idealised versions of maps that can be obtained from various observational techniques. The line-of-sight field component can be determined from the Faraday rotation measure [@1981ApJ...247L..77H] for ionised gas or from Zeeman spectroscopy [@2001ApJ...560..821B] for neutral/molecular gas. The plane-of-sky field components can be determined from observations of polarized emission or absorption [@2008AJ....136..621K] from aligned spinning dust grains. In all of these techniques, the determination of $\mathbf{B}$ is not straightforward and relies on many auxiliary assumptions. In particular, it is very difficult to determine the absolute magnitude of the plane-of-sky field components except via statistical techniques such as the Chandrasekhar-Fermi method . Furthermore, many of the techniques do not uniformly sample all regions along the line of sight, but are biased towards regions with particular physical conditions. We therefore caution against direct comparison of observational results with our maps in any but a qualitative sense. The maps are nonetheless useful in showing an overview of the magnetic field geometry in our simulations. In the following discussion, we describe locations on the projected map using the points of the compass, assuming that north is up (positive $y$) and east is left (negative $x$).
The most striking aspect of Figs. \[fig:bproj-ostar-full\] and \[fig:bproj-bstar-full\] is the large-scale order that is apparent in the projected magnetic field. This is particularly visible in the neutral-weighted maps (top-right panels), where it can be seen that the field is frequently oriented parallel to the large-scale ionisation front, forming a ring around the [H II]{} region. The effect is particularly strong along the north and south borders of the region because of the net positive field along the $x$-axis (see § \[sec:setup\]), but it is also seen to a lesser extent along the east and west borders, despite the fact that the mean $y$-component of the field is zero. This is because of the fast-mode MHD shock that is driven into the surrounding gas by the expanding [H II]{} region and which compresses both the gas and the field, tending to bend the field lines so that they are more closely parallel to the shock than they were in the undisturbed medium. A large-scale pattern is much harder to discern in the molecular-weighted maps (bottom-right panels), partly because the molecular column density is much less smoothly distributed, being concentrated in globules and filaments.
In most dense filaments, the field direction is parallel to the long axis of the filament, as can be seen particularly clearly in Fig. \[fig:bproj-ostar-globule\], which shows a detail view of the dense photoevaporating globule found to the south in Fig. \[fig:bproj-ostar-full\] and which is fed by multiple neutral/molecular filaments. In the molecular gas at the head of the globule, the magnetic field is bent into a hairpin shape. The B-field in the ionised gas at the head of the globule tends to be oriented perpendicular to the ionisation front, as is the case with almost all the dense globules visible in Fig. \[fig:bproj-ostar-full\]. The same is true along much of the bar-like feature to the west of the globule. On the other hand, in a few regions, such as along the filament that extends south from the globule, the B-field in the ionised gas lies along the ionisation front.
Fig. \[fig:bproj-bstar-globule\] shows a detail view of the same dense southern globule, but from the B star simulation shown in Fig. \[fig:bproj-bstar-full\]. The molecular gas shows a similar field pattern to that seen in the O star case: the field goes up one of the feeding filaments and down the other, with a hairpin bend in between. Although this can be seen clearly in the molecular gas, the neutral and ionised gas show very different patterns, with magnetic field vectors that are generally perpendicular to the filament and continuous with the large-scale field pattern in the region. This is because the total ionised and neutral columns are dominated by diffuse material along the line of sight, rather than by material associated with the globule and filament.
Effects of magnetic field on globule formation and evolution {#sec:effects-magn-field}
------------------------------------------------------------
It is interesting to compare the properties of the globules generated in our turbulent simulations with the results of previous detailed studies of the photoionisation of isolated dense globules . The principal findings of the earlier studies are that a sufficiently strong magnetic field ($\beta < 0.01$ in the initial neutral globule) will produce important qualitative changes in the photoevaporation process. Depending on the initial field orientation with respect to the direction of the ionising photons, either extreme flattening of the globule may occur (perpendicular orientations) or the radiative implosion of the globule may be prevented (parallel orientations). For all orientations the ionised photoevaporation flow cannot freely escape from the globule, leading to recombination at late times. On the other hand, a weaker field ($\beta \simeq 0.01$ in the initial globule) leads only to a moderate flattening of the globule and does not prevent the free escape of the ionised photoevaporation flow.
In our turbulent simulations, the mean initial value of $\beta$ is $\simeq 0.032$ (§ \[sec:setup\]), which is intermediate between the two cases discussed above and might lead one to suspect that magnetic effects on the evolution of globules should be substantial. However, this is *not* the case. A careful examination of the three-dimensional globule morphologies in our simulations shows no evidence of magnetically induced flattening. Although many globules do show asymmetries, this is true equally of our non-magnetic simulations and is presumably due to their irregular initial shape and internal turbulent motions. The only difference in the globule properties between our non-magnetic and magnetic simulations is that the very smallest globules do not seem to form in the magnetic case.
In order to explain this apparent discrepancy with earlier work, it is necessary to examine in more detail the distribution of magnetic field in the initial conditions of our turbulent simulations. It turns out that the dense filaments of molecular gas (from which the globules will ultimately form) tend to be much less magnetically dominated than the more diffuse gas, typically showing $\beta >
0.1$. Furthermore, these filaments already show a magnetic geometry similar to that described above (§ \[sec:pred-maps-proj\]), with the field running along the long axis of the filament and with the field changing sign as one moves across the short axis. This is very different from the initial conditions assumed in the earlier globule photoevaporation studies, which were a uniform magnetic field that threaded a spherical globule (e.g., Fig. 1 of [@2009MNRAS.398..157H]). As a result, the compressed globule heads in our turbulent simulations show approximately equal thermal and magnetic pressures ($\beta \sim 1$) and the magnetic effects are very modest. One caveat to this result is that numerical diffusion due to our limited spatial resolution may be producing non-physical magnetic reconnection in the globule head when oppositely directed field lines are forced together. Higher resolution studies of globule implosion with realistic initial field configurations are required in order to clarify this [and these should also include self gravity, which may be important during the phase of maximum compression [@2007MNRAS.377..383E]]{}.
Conclusions {#sec:conclusion}
===========
We have performed radiation-magnetohydrodynamic simulations of the formation and expansion of regions and PDRs around an O star and a B star in a turbulent magnetised molecular cloud on scales of up to 4 parsec. Our principal conclusions are as follows:
1. The expansion of the region is little affected by the presence of the magnetic field, since the thermal pressure of the ionised gas dominates the dynamics on the timescales of our simulations (§ \[sec:glob-prop-prot\]).
2. The O star simulations produce greater fragmentation and denser clumps and filaments around the periphery of the region than the B star case (§ \[sec:morph\], Figs. \[fig:images\_O\] and \[fig:images\_B\]).
3. For B stars the non-ionising far ultraviolet radiation plays an important role in determining the morphology of the region. regions around such stars are surrounded by a thick, relatively smooth shells of neutral material (PDR), $\sim 30\%$ of the bubble radius (e.g., lower panels of Fig. \[fig:cuts\_Bstar\]). In the O star simulations, the PDR is thinner and more irregular in shape (e.g., Fig. \[fig:images\_O\]).
4. The resemblance at optical and longer wavelengths of our simulations to observed bubbles is striking (§ \[subsec:rcw120\] and ). Our $\sim 2$ pc radius bubbles are a typical size compared to bubbles at known distances in the GLIMPSE surveys [@2006ApJ...649..759C; @2007ApJ...670..428C], and so comparisons are meaningful.
5. The expanding region and PDR tend to erase pre-existing small-scale disordered structure in the magnetic field, producing a large-scale ordered field in the neutral shell, with orientation approximately parallel to the ionisation front (top panels of Fig. \[fig:cuts\_Bstar\] and top-right panels of Figs. \[fig:bproj-ostar-full\] and \[fig:bproj-bstar-full\]).
6. Dense evaporating globules, pillars, and elephant trunk structures tend to be fed by two or more neutral/molecular filaments, with magnetic fields running along their length (right panels of Fig. \[fig:bproj-ostar-globule\] and bottom-right panel of Fig. \[fig:bproj-bstar-globule\]). The field geometry in the neutral and molecular gas at the bright head of the structure tends to be of a hairpin shape.
7. The weak magnetic field in the ionised gas also shows an ordered structure. On the largest scales and at the latest times, it tends to align (albeit weakly in the O star case), with the mean field direction of the simulation’s initial conditions (Fig. \[fig:bproj-bstar-full\], top-left panel). On smaller scales, there is a general (but not universal) tendency for the field in ionised gas to be oriented perpendicular to the local ionisation front. This tendency is more pronounced in the O star simulation (Fig. \[fig:bproj-ostar-full\], top-left panel) and in globule evaporation flows for both simulations (top-left panels of Figs. \[fig:bproj-ostar-globule\] and \[fig:bproj-bstar-globule\]).
8. The highest pressure compressed neutral/molecular gas shows approximate equipartition between thermal, turbulent, and magnetic energy density, whereas lower pressure gas (either neutral or molecular) tends to separate into, on the one hand, magnetically dominated, quiescent regions, and, on the other hand, demagnetised, highly turbulent regions (§ \[sec:magnetic-quantities\], Fig. \[fig:pram\_pmag\_Bstar\]). The lower pressure gas also separates into low-$\beta$, magnetically dominated regions (which are largely molecular) and high-$\beta$, thermally dominated regions (which are largely neutral). The ionised gas, on the other hand, always shows approximate equipartition between thermal and turbulent energies, but with the magnetic energy being lower by 1 to 3 orders of magnitude.
9. Velocity dispersions in the ionised gas of $7$–$9$ [$\mathrm{km\ s^{-1}}$]{} are maintained for the entire duration of all our simulations (§ \[sec:glob-prop-prot\], Fig. \[fig:comp3\_O\] and \[fig:comp3\_B\]). This is 5 to 10 times higher than the value that would be predicted by expansion in a uniform medium. At early times ($t < 100,000$ yr for the O star, or $t < 300,000$ yr for the B star), this dispersion is mainly due to radial champagne-flow expansion as the [H II]{} region escapes from its natal clump. At later times, the net radial expansion of ionised gas subsides, but the velocity dispersion is maintained by inwardly-directed photoevaporation flows from globules and pillars.
Acknowledgments {#acknowledgments .unnumbered}
===============
SJA and WJH acknowledge financial support from DGAPA PAPIIT projects IN112006, IN100309 and IN110108. This work was supported in part by Swedish Research Council grant 2009-4088. Some of the numerical calculations in this paper were performed on the Kan Balam supercomputer maintained and operated by DGSCA, UNAM. This work has made extensive use of NASA’s Astrophysics Abstract Data Service and the astro-ph archive.
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Integration of plane-of-sky field components
============================================
Polarisation-based methods for measuring the plane-of-sky components of the magnetic field cannot distinguish the sense of the field and are degenerate between a position angle $\theta$ and a position angle $\theta + 180^\circ$. It is therefore not sufficient to simply integrate $B_x$ and $B_y$ along the line of sight with appropriate weighting. Neither is it sufficient to integrate $B_x^2$ and $B_y^2$, since this will always give a result in the first quadrant ($B_x, B_y > 0$). In order to calculate our projected magnetic field maps (§ \[sec:pred-maps-proj\]), we therefore adopt a “Stokes parameter” approach [@Chandrasekhar:1960], whereby the plane-of-sky components $B_x(x,y,z)$ and $B_y(x,y,z)$ are first transformed to ($Q, U$)-space as $B_Q = \tilde{B} \cos 2 \theta$, $B_U = \tilde{B} \sin 2 \theta$, where $\tilde{B} = (B_x^2 +
B_y^2)^{1/2}$ and $\theta = \tan^{-1} (B_y/B_x)$. Then the line-of-sight integration is performed as $M_{Q,U}(x,y) = \int
B_{Q,U}(x,y,z)\, n \, dz$, where $n$ is the relevant density (ionised, neutral, or molecular). Finally, the components of the projected field are transformed back to ($x,y$)-space: $M_x =
\tilde{M} \cos\theta$, $M_y = \tilde{M} \sin\theta$, where $\tilde{M} = (M_Q^2 + M_U^2)^{1/2}$ and $\theta = 0.5 \tan^{-1}
(M_U/M_Q)$.
[^1]: E-mail: [[email protected]]([email protected]); [[email protected]]([email protected]); [[email protected]]([email protected]); [[email protected]]([email protected]); [[email protected]]([email protected])
[^2]: The original simulation performed by @2007ApJ...671..518K has the magnetic field parallel to the $x$-axis.
[^3]: We define the mean radius as the radius of a sphere with the same volume as the real region.
|
---
abstract: 'We recently defined an invariant of contact manifolds with convex boundary in Kronheimer and Mrowka’s sutured monopole Floer homology theory. Here, we prove that there is an isomorphism between sutured monopole Floer homology and sutured Heegaard Floer homology which identifies our invariant with the contact class defined by Honda, Kazez and Mati[ć]{} in the latter theory. One consequence is that the Legendrian invariants in knot Floer homology behave functorially with respect to Lagrangian concordance. In particular, these invariants provide computable and effective obstructions to the existence of such concordances. Our work also provides the first proof which does not rely on the relative Giroux correspondence that the vanishing or non-vanishing of Honda, Kazez and Mati[ć]{}’s contact class is a well-defined invariant of contact manifolds.'
address:
- |
Department of Mathematics\
Boston College
- |
Department of Mathematics\
Princeton University
author:
- 'John A. Baldwin'
- Steven Sivek
bibliography:
- 'References.bib'
title: On the equivalence of contact invariants in sutured Floer homology theories
---
Introduction {#sec:intro}
============
The purpose of this article is to establish an equivalence between two invariants of contact 3-manifolds with boundary—one defined using Heegaard Floer homology and the other using monopole Floer homology. Our equivalence fits naturally into the ongoing program of establishing connections between the many different instantiations of Floer theory. In addition to the theoretical appeal of such connections, an equivalence between invariants in different Floer homological settings allows one to combine the intrinsic advantages of the different settings, often with interesting topological or geometric consequences. This principle is illustrated nicely by Taubes’s isomorphism between monopole Floer homology and embedded contact homology [@taubes1; @taubes2; @taubes3; @taubes4; @taubes5], whose first step, a correspondence between monopoles and Reeb orbit sets, proved the Weinstein conjecture for 3-manifolds [@taubeswc].
Our work provides another illustration of this principle. Indeed, one of the great advantages of Heegaard Floer homology is its computability. On the other hand, monopole Floer homology enjoys a certain functoriality with respect to exact symplectic cobordism which has yet to be proven in the Heegaard Floer setting. The equivalence described in this paper enables us to combine these advantages to give a new, computable obstruction to the existence of Lagrangian concordances between Legendrian knots, a subject of much recent interest. Another application of our equivalence is a Giroux-correspondence-free proof that the contact invariant in sutured Heegaard Floer homology is well-defined. Below, we describe our equivalence and its applications in more detail. We then indicate our strategy of proof and discuss potential applications of this strategy outside the realm of contact geometry. We work in characteristic 2 throughout this paper.
Our equivalence
---------------
Let us first recall the invariants of *closed* contact 3-manifolds defined by Kronheimer and Mrowka and by Ozsv[á]{}th and Szab[ó]{}. Suppose $(Y,\xi)$ is a closed contact 3-manifold, and $\eta$ is an oriented curve in $Y$. In [@km; @kmosz], Kronheimer and Mrowka associate to such data a class $$c_{HM}(\xi)\in{\HMto_{\bullet}}(-Y,{\mathfrak{s}}_\xi;\Gamma_{\eta})\footnote{In \cite{km, kmosz}, this class is denoted by $\psi$.}$$ in the monopole Floer homology of $-Y$ with local coefficients. Here, ${\mathfrak{s}}_\xi$ is the ${\text{Spin}^c}$ structure on $-Y$ associated with $\xi$, and $\Gamma_{\eta}$ is a local system on the monopole Floer configuration space with fiber a Novikov ring $\Lambda$. In [@osz1], Ozsv[á]{}th and Szab[ó]{} likewise define a class $$c_{HF}(\xi)\in{HF^+}(-Y,{\mathfrak{s}}_\xi;\Gamma_{\eta})$$ in Heegaard Floer homology with local coefficients, but by very different means. Remarkably, these two invariants are equivalent. This is made precise in the theorem below, which follows from Taubes’ work [@taubes1; @taubes2; @taubes3; @taubes4; @taubes5] together with work of Colin, Ghiggini, and Honda [@cgh3; @cgh4; @cgh5] on the isomorphism between Heegaard Floer homology and embedded contact homology.[^1]
\[thm:closedequiv\] For every ${\mathfrak{s}}\in{\text{Spin}^c}({-}Y)$, there is an isomorphism of $\Lambda[U]$-modules $$\Phi_{\mathfrak{s}}:{HF^+}({-}Y,{\mathfrak{s}};\Gamma_{\eta})\to{\HMto_{\bullet}}(-Y,{\mathfrak{s}};\Gamma_{\eta}),$$ such that $\Phi_{{\mathfrak{s}}_\xi}(c_{HF}(\xi))=c_{HM}(\xi)$.
This article sets out to establish a similar equivalence for invariants of contact 3-manifolds *with boundary*, or, more precisely, *sutured contact manifolds.* These are triples $(M,\Gamma,\xi)$ where $(M,\xi)$ is a contact 3-manifold with convex boundary and dividing set $\Gamma\subset \partial M$. In [@hkm4], Honda, Kazez, and Mati[ć]{} associate to such data a class $$c_{HF}(\xi)\in{SFH}(-M,-\Gamma)\footnote{In \cite{hkm4}, this class is denoted by $EH$.}$$ in the sutured Heegaard Floer homology of $(-M,-\Gamma)$ which, in a sense, generalizes Ozsv[á]{}th and Szab[ó]{}’s invariant of closed contact manifolds (it generalizes the *hat* version of Ozsv[á]{}th and Szab[ó]{}’s invariant). In [@bsSHM], we gave a similar generalization of Kronheimer and Mrowka’s invariant. Ours assigns to a sutured contact manifold a class $$c_{HM}(\xi)\in{\underline{{SHM}}}(-M,-\Gamma)\footnote{In \cite{bsSHM}, this class is denoted by $\psi$.}$$ in a version of sutured monopole Floer homology with local coefficients.[^2] Our main theorem is the following, settling a conjecture made in [@bsSHM Conjecture 1.9].
\[thm:main\] There is an isomorphism of $\Lambda$-modules $${SFH}(-M,-\Gamma)\otimes \Lambda\to{\underline{{SHM}}}(-M,-\Gamma),$$ sending $c_{HF}(\xi)\otimes 1$ to $c_{HM}(\xi)$.
Applications
------------
In order to define $c_{HF}(\xi)$—in both the *closed* and *with boundary* settings—one first chooses an open book compatible with $\xi$. The existing proofs that $c_{HF}(\xi)$ does not depend on this choice rely on Giroux’s correspondence—in particular, on its assertion that any two open books compatible with $\xi$ are related by a sequence of positive stabilizations. By contrast, our construction of $c_{HM}(\xi)$ is logically independent of this assertion. One immediate application of Theorem \[thm:main\] is therefore a proof of the following well-definedness result from [@hkm4] that is logically independent of Giroux’s correspondence.
For any sutured contact manifold $(M,\Gamma,\xi)$, the vanishing or non-vanishing of $c_{HF}(\xi)$ does not depend on the choices made in its construction.
The main topological application of Theorem \[thm:main\] in this paper is to the study of Lagrangian concordance initiated by Chantraine in [@chantraine]. Recall that for Legendrian knots $K_1,K_2\subset (Y,\xi)$, $K_1$ is *Lagrangian concordant* to $K_2$—written $K_1\prec K_2$—if there is an embedded Lagrangian cylinder $C$ in the symplectization of $(Y,\xi)$ such that $$\begin{aligned}
C\,\cap\, ((-\infty,-T)\times Y) &= (-\infty,-T)\times K_1,\\
C\,\cap\, ((T,\infty)\times Y) &= (T, \infty)\times K_2\end{aligned}$$ for some $T>0$. Two Legendrian knots related by Lagrangian concordance must have the same classical invariants (Thurston-Bennequin and rotation numbers) [@chantraine]. A challenging problem, therefore, which has attracted a lot of recent attention, is to find tools for deciding whether two knots with the same classical invariants are Lagrangian concordant. Note that this is more difficult than the already formidable task of deciding whether two smoothly isotopic knots with the same classical invariants are Legendrian isotopic.
In [@bsLeg], we defined a Legendrian invariant which assigns to a Legendrian $K\subset (Y,\xi)$ a class $$\mathcal{L}_{HM}(K)\in {\underline{KHM}}(-Y,K)$$ in monopole knot Floer homology with local coefficients. It is defined in terms of a certain contact structure $\xi_K$ on the sutured knot complement $Y(K)$ with two meridional sutures, $$\label{eqn:leg}\mathcal{L}_{HM}(K):=c_{HM}(\xi_K)\in {\underline{{SHM}}}(-Y(K))=:{\underline{KHM}}(-Y,K).$$ Furthermore, we used the functoriality of $c_{HM}$ under exact symplectic cobordism (a feature whose analogue in Heegaard Floer homology has not been established independently of Theorem \[thm:closedequiv\]) to show that $\mathcal{L}_{HM}$ behaves functorially under Lagrangian concordance, as follows.
\[thm:functcon\] If $K_1,K_2\subset (Y,\xi)$ are Legendrian with $K_1\prec K_2$, then there is a map $${\underline{KHM}}(-Y,K_2)\to {\underline{KHM}}(-Y,K_1),$$ sending $\mathcal{L}_{HM}(K_2)$ to $\mathcal{L}_{HM}(K_1).$
In this way, the class $\mathcal{L}_{HM}$ provides an obstruction to the existence of Lagrangian concordances between Legendrian knots. Unfortunately, this class is not easily computable. A much more computable Legendrian invariant is that defined by Lisca, Ozsv[á]{}th, Stipsicz, and Szab[ó]{} in [@lossz]. Theirs takes the form of a class $$\mathcal{L}_{HF}(K)\subset {\widehat{{HFK}}}(-Y,K)$$ in Heegaard knot Floer homology. Though originally defined in terms of open books for $(Y,\xi)$, Stipsicz and V[é]{}rtesi discovered in [@sv] that it can also be formulated as $$\mathcal{L}_{HF}(K)=c_{HF}(\xi_K)\in {SFH}(-Y(K))={\widehat{{HFK}}}(-Y,K).$$ In fact, their work was the inspiration for our subsequent definition of $\mathcal{L}_{HM}$. The equivalence below then follows immediately from Theorem \[thm:main\].
\[thm:equivleg\] There is an isomorphism of $\Lambda$-modules $${\widehat{{HFK}}}(-Y,K)\otimes\Lambda\to{\underline{KHM}}(-Y,K),$$ sending $\mathcal{L}_{HF}(K)\otimes 1$ to $\mathcal{L}_{HM}(K)$.
The second author defined a similar but different invariant of Legendrian knots in ${\underline{KHM}}$ in [@sivek]. In fact, it was his construction that inspired our series [@bsSHM; @bsSHI; @bsLeg]. It is still unknown how his Legendrian invariant is related to $\mathcal{L}_{HM}$ and therefore to $\mathcal{L}_{HF}$.
A Legendrian invariant is said to be *effective* if it can distinguish smoothly isotopic Legendrian knots with the same classical invariants. The invariant $\mathcal{L}_{HF}$ is effective in that there are Legendrian knots as above for which the invariant vanishes for one but not for the other. Theorem \[thm:equivleg\] then implies that $\mathcal{L}_{HM}$ is effective as well, resolving [@bsLeg Conjecture 1.1].
The invariant $\mathcal{L}_{HM}$ is effective.
The classes $\mathcal{L}_{HF}$ and $\mathcal{L}_{HM}$ are invariant under negative Legendrian stabilization, and therefore provide invariants of transverse knots as well. Theorem \[thm:equivleg\], combined with computations in Heegaard Floer homology [@lossz], implies that $\mathcal{L}_{HM}$ is an effective transverse invariant knots as well, in the sense that it can distinguish smoothly isotopic transverse knots with the same self-linking numbers.
An even more striking consequence of Theorems \[thm:functcon\] and \[thm:equivleg\] is that the invariant $\mathcal{L}_{HF}$ is also functorial under Lagrangian concordance, as follows.
\[thm:functhf\] If $K_1,K_2\subset (Y,\xi)$ are Legendrian with $K_1\prec K_2$, then there is a map $${\widehat{{HFK}}}(-Y,K_2)\otimes\Lambda\to {\widehat{{HFK}}}(-Y,K_1)\otimes\Lambda,$$ sending $\mathcal{L}_{HF}(K_2)\otimes 1$ to $\mathcal{L}_{HF}(K_1)\otimes 1.$
Once again, the value of establishing this functoriality in the Heegaard Floer setting has to do with the relative computability of invariants in that setting. In fact, before the discovery of $\mathcal{L}_{HF}$, Ozsv[á]{}th, Szab[ó]{}, and Thurston defined in [@oszt] an (intrinsically) algorithmically computable invariant of Legendrian knots in the tight contact structure $(S^3,\xi_{std})$ using the grid diagram formulation of knot Floer homology. Their invariant assigns to a Legendrian $K\subset (S^3,\xi_{std})$ a class $$\Theta_{HF}(K)\in {\widehat{{HFK}}}(-S^3,K).$$ It was shown in [@bvv] that these two Heegaard Floer invariants are equivalent where they overlap, per the following theorem.
\[thm:equivleginv\]For any Legendrian knot $K\subset (S^3,\xi_{std}),$ there is an automorphism $${\widehat{{HFK}}}(-S^3,K)\to{\widehat{{HFK}}}(-S^3,K),$$ sending $\mathcal{L}_{HF}(K)$ to $\Theta_{HF}(K)$.
Combined with Theorem \[thm:functhf\], this theorem implies that the invariant $\Theta_{HF} = \mathcal{L}_{HF}$ provides an entirely computable obstruction to the existence of Lagrangian concordances between Legendrian knots in $(S^3,\xi_{std})$, as follows.
\[thm:grid-concordance\] If $K_1$ and $K_2$ are Legendrian knots in $(S^3,\xi_{std})$ with $$\Theta_{HF}(K_1) \neq 0\,\textrm{ and }\,\Theta_{HF}(K_2) = 0,$$ then there is no Lagrangian concordance from $K_1$ to $K_2$.
As mentioned by the authors of [@bvv], proving a result like Theorem \[thm:grid-concordance\] was a major part of their motivation for establishing the equivalence described in Theorem \[thm:equivleginv\].
It is easy to find examples demonstrating the effectiveness of the obstruction in Theorem \[thm:grid-concordance\]. In particular, there are infinitely many pairs $(K_1,K_2)$ of smoothly isotopic Legendrian knots with the same classical invariants which satisfy the hypotheses of Theorem \[thm:grid-concordance\] [@ost; @not; @kng; @bald9]. The results of [@ost] imply that such $K_1$ and $K_2$ are not Legendrian isotopic, whereas Theorem \[thm:grid-concordance\] implies the much stronger fact that $K_1$ is not Lagrangian concordant to $K_2$.
It bears mentioning that the Legendrian contact homology DGA of Chekanov and Eliashberg [@yasha8] enjoys a similar sort of functoriality under Lagrangian concordance [@ehk]. However, it can be difficult to apply this DGA obstruction in practice. Consider, for example, the two Legendrian representatives $K_1$ and $K_2$ of $m(10_{132})$ with $(tb,r) = (1,0)$ described by Ng, Ozsv[á]{}th, and Thurston in [@not]. One can show that $K_1$ is not Lagrangian concordant to $K_2$ by showing that the DGA is trivial for $K_2$ while nontrivial for $K_1$. But proving this nontriviality is tricky as the DGA for $K_1$ does not even admit any nontrivial finite-dimensional representations [@sivek2]. By contrast, it is quite easy to check that $\Theta_{HF}$ vanishes $K_2$ but not for $K_1$, and in so doing, apply the Heegaard Floer obstruction in Theorem \[thm:grid-concordance\].
Another advantage of $\Theta_{HF}$ is that it is preserved under negative Legendrian stabilization, whereas the Legendrian contact homology DGA is trivial for stabilized knots. In particular, for $K_1$ and $K_2$ satisfying the hypotheses of Theorem \[thm:grid-concordance\], we may also conclude that no negative stabilization of $K_1$ is Lagrangian concordant to any negative stabilization of $K_2$. The DGA, by contrast, cannot tell us anything about Lagrangian concordances between stabilized knots.
In Section \[sec:concex\], we give another demonstration of our obstruction, providing several additional examples of Legendrian knots with the same classical invariants which are not smoothly isotopic or Lagrangian concordant, but which *are smoothly concordant*. In these examples, Lagrangian concordance is obstructed by Theorem \[thm:grid-concordance\] while the Legendrian contact homology DGA provides no such obstruction.
We next outline our proof of Theorem \[thm:main\], starting with a very brief review of sutured monopole Floer homology and of our contact invariant.
Proof outline
-------------
A *closure* of a balanced sutured manifold $(M,\Gamma)$, as defined by Kronheimer and Mrowka in [@km4], is a closed manifold $Y$ together with a distinguished surface $R\subset Y$, formed from $(M,\Gamma)$ in a certain manner, and containing $M$ as a submanifold. The sutured monopole Floer homology of $(M,\Gamma)$ is defined as the monopole Floer homology of $Y$ in the *bottom* ${\text{Spin}^c}$ structures with respect to $R$, $${\underline{{SHM}}}(M,\Gamma):={\HMto_{\bullet}}(Y| R; \Gamma_\eta):=\bigoplus_{\langle c_1({\mathfrak{s}}),[R]\rangle = 2-2g(R)}{\HMto_{\bullet}}(Y;\Gamma_\eta),$$ for some curve $\eta\subset R$.[^3] Given a sutured contact manifold $(M,\Gamma,\xi)$, we gave a procedure in [@bsSHM] for extending $\xi$ to a contact structure $\bar\xi$ on a closure $(Y,R)$ with respect to which $R$ is convex and such that $$\langle c_1({\mathfrak{s}}_{\bar\xi}),[R]\rangle = 2-2g(R).$$ We refer to $(Y,R,\bar\xi)$ as a *contact closure* of $(M,\Gamma,\xi)$. We then defined $$c_{HM}(\xi):=c_{HM}(\bar\xi)\in {\HMto_{\bullet}}(-Y,{\mathfrak{s}}_{\bar\xi};\Gamma_\eta)\subset {\HMto_{\bullet}}(-Y| {-}R; \Gamma_\eta)=:{\underline{{SHM}}}(-M,-\Gamma),$$ and proved that this class is independent of the myriad choices involved in its construction. Note that Theorem \[thm:closedequiv\] provides an isomorphism
$$\xymatrix@C=12pt@R=-2pt{
{HF^+}(-Y| {-}R; \Gamma_\eta)\ar[rr]^{\cong}&& {\HMto_{\bullet}}(-Y| {-}R; \Gamma_\eta)\,\,\, \ar@{}[r]|{=:}&{\underline{{SHM}}}(-M,-\Gamma) \\
\rotatebox{90}{$\in$} && \rotatebox{90}{$\in$} & \rotatebox{90}{$\in$} \\
\,\,\,c_{HF}(\bar\xi)\,\,\ar@{|->}[rr] &&\,\,\, c_{HM}(\bar\xi)\,\ar@{}[r]|{=:}&c_{HM}(\xi).
}$$
Therefore, in order to prove Theorem \[thm:main\], it suffices to establish the following.
\[thm:main2\] There is an isomorphism of $\Lambda$-modules $${SFH}(-M,-\Gamma)\otimes \Lambda \to {HF^+}(-Y| {-}R; \Gamma_\eta),$$ sending $c_{HF}(\xi)\otimes 1$ to $c_{HF}(\bar\xi)$.
It bears mentioning that Lekili has already shown in [@lekili2] that the modules in Theorem \[thm:main2\] are isomorphic. Given a sutured Heegaard diagram for $(-M,-\Gamma)$, Lekili constructs a pointed Heegaard diagram for $-Y$ in a certain natural way, and, by comparing these diagrams, defines a quasi-isomorphism between the corresponding chain complexes. The problem for us is that Lekili’s pointed Heegaard diagram has nothing to do with the contact structure $\bar\xi$, even if the sutured diagram came from a partial open book compatible with $\xi$, so Lekili’s isomorphism says nothing about the relationship between the invariants of $\xi$ and $\bar\xi$.
Indeed, a rather more novel approach is required. Our strategy for proving Theorem \[thm:main2\] makes use of an interesting topological reformulation of the contact invariant of $\bar\xi$. We discovered this reformulation in [@bsSHM] and used it in [@bsSHI] to define a contact invariant in instanton Floer homology. One starts with a partial open book for $(M,\Gamma,\xi)$, which provides a description of this contact manifold as obtained from an $[-1,1]$-invariant contact structure $\xi_S$ on the product sutured manifold $$H(S) = (S\times[-1,1],\partial S\times\{0\})$$ by attaching contact 2-handles along certain curves $s_1,\dots,s_n$ in $\partial H(S)$. These curves correspond naturally to Legendrians in a contact closure $(Y_S,R,\bar\xi_S)$ of $(H(S),\xi_S)$, and we proved that contact (+1)-surgery on these Legendrian curves results in a contact closure $(Y,R,\bar\xi)$ of $(M,\Gamma,\xi)$. It then follows from the functoriality of the contact invariant under such surgeries [@osz1 Theorem 4.2] that the map $$G:={HF^+}(-W|{-}R;\Gamma_{\nu}):{HF^+}(-Y_S|{-}R;\Gamma_\eta)\to{HF^+}(-Y|{-}R;\Gamma_\eta)$$ induced by the natural 2-handle cobordism $(W,\nu )$ from $(Y_S,\eta)$ to $(Y,\eta)$ corresponding to these surgeries sends $c_{HF}(\bar\xi_S)$ to $c_{HF}(\bar\xi)$ (up to multiplication by a unit in $\Lambda$, a subtlety we will generally ignore). Fortunately, $${HF^+}(-Y_S|{-}R;\Gamma_\eta)\cong \Lambda = \Lambda\langle c_{HF}(\bar\xi_S)\rangle$$ is 1-dimensional, generated by the contact class, so $c_{HF}(\bar\xi)$ may be characterized more simply as the image of this generator under $G$. What we actually prove, then, is that there is an isomorphism $$F:SFH(-M,-\Gamma)\otimes\Lambda\to {HF^+}(-Y|{-}R; \Gamma_\eta)$$ sending $c_{HF}(\xi)\otimes 1$ to $G(c_{HF}(\bar\xi_S))$, as shown in the following diagram.
$$\xymatrix@C=5pt@R=11pt{
&&&&{SFH}(-M,-\Gamma)\otimes \Lambda\,\,\ar[dd]_{\cong}^{F}\ar@<.1ex>@{}[r]|(.63){\rotatebox{180}{$\in$}}&\,\,c_{HF}(\xi)\otimes 1 \ar@{|->}[ddd]\\
\\
\Lambda\langle c_{HF}(\bar\xi_S)\rangle\ar@{}[d]|{\rotatebox{90}{$\in$}}\ar@{}[r]|(.4){\cong}&{HF^+}(-Y_S| {-}R; \Gamma_\eta)\ar[rrr]_{G}&&&{HF^+}(-Y| {-}R; \Gamma_\eta)\ar@<.75ex>@{}[dr]|(.59){\rotatebox{145}{$\in$}} \\
c_{HF}(\bar\xi_S)\,\ar@{|->}[rrrrr]&&&&&\,G(c_{HF}(\bar\xi_S))\,\ar@{}[r]|(.58){=}& c_{HF}(\bar\xi).
}$$
Since $G(c_{HF}(\bar\xi_S)) = c_{HF}(\bar\xi)$, this proves Theorem \[thm:main2\] and, therefore, Theorem \[thm:main\].
A few further remarks are in order about our proof. First, the map $F$ is induced by a natural map from the Floer complex associated with a sutured diagram for $(-M,-\Gamma)$ to the Floer complex associated with a pointed Heegaard diagram for $-Y$ that is similar to, but slightly different from, the sort of diagram Lekili considers. A novel aspect of our construction is that, for $g(R)$ sufficiently large and for sufficient *winding* of the curves in the Heegaard diagram, we are able to show that this map is a chain map and a quasi-isomorphism without resorting to the somewhat involved holomorphic curve analysis that appears in Lekili’s proof.
Second, the map $G$ is induced by a chain map defined by counting holomorphic triangles in a pointed Heegaard triple diagram. Our proof that the map $F$ sends $c_{HF}(\xi)\otimes 1$ to $G(c_{HF}(\bar\xi_S))$ is ultimately achieved by showing that, for $g(R)$ sufficiently large and sufficient winding, these triangle counts correspond to much simpler triangle counts in a related sutured triple diagram which encodes the sutured cobordism from $-H(S)$ to $(-M,-\Gamma)$ corresponding to the contact 2-handle attachments along the curves $s_1,\dots,s_n$.
Future applications
-------------------
Our method of proof may also have interesting applications outside of contact geometry. In particular, we strongly suspect that our techniques may be used to show that more general holomorphic polygon counts in sutured multidiagrams can be made to correspond precisely with holomorphic polygon counts in pointed Heegaard multidiagrams associated to closures of the relevant sutured manifolds, so long as one takes closures with $g(R)$ sufficiently large. This sort of higher polygon counting is frequently used to construct exact triangles in the Heegaard Floer setting. An important example is Manolescu’s unoriented skein exact triangle in Heegaard knot Floer homology, which is defined by counting holomorphic polygons in a sutured multidiagram [@man1]. An extension of our work here ought to then provide an alternative construction of Manolescu’s triangle, using instead a pointed Heegaard multidiagram for closures of the sutured manifolds (knot complements) involved in the triangle.
Our interest in this alternative construction of Manolescu’s triangle has to do with defining an analogous skein triangle in monopole knot Floer homology—a theory defined, after all, in terms of closures. Exact triangles in the monopole Floer setting are typically defined in terms of counts of monopoles over cobordisms equipped with families of metrics. A construction of Manolescu’s triangle via pointed Heegaard multidiagrams for closures would tell us which cobordisms and families of metrics to use in defining the analogous skein triangle in monopole knot Floer homology (a multidiagram specifies a cobordism and embedded hypersurfaces along which to stretch the metric). An interesting goal would then be to iterate this triangle to give a combinatorial description of monopole knot Floer homology, following the ideas in [@baldlev], and to use this description to give another proof that Heegaard knot Floer homology is isomorphic to monopole knot Floer homology, circumventing the difficult analysis of Taubes et al.
Organization
------------
In Section \[sec:background\], we provide a basic review of the constructions and properties of the contact invariants in Heegaard and monopole Floer homologies and their sutured variants. Then, in Section \[sec:proof\], we prove Theorem \[thm:main2\] and therefore our main Theorem \[thm:main\], as outlined above. Finally, in Section \[sec:concex\], we give examples further illustrating the effectiveness of Theorem \[thm:grid-concordance\].
Acknowledgements
----------------
We thank Ko Honda for helpful correspondence.
Background {#sec:background}
==========
Monopole Floer homology and contact invariants {#ssec:hm}
----------------------------------------------
We start by recalling some basic features of monopole Floer homology and of Kronheimer and Mrowka’s contact invariant for closed contact 3-manifolds defined therein.
Let $\Lambda$ be the Novikov ring over ${\mathbb{Z}/2\mathbb{Z}}$ defined by $$\Lambda=\bigg\{\sum_{\alpha}c_{\alpha}t^{\alpha}\,\,\bigg |\, \,\alpha\in\mathbb{R},\,\,c_{\alpha}\in{\mathbb{Z}/2\mathbb{Z}},\,\,\#\{\beta<n|c_{\beta}\neq 0\}<\infty\textrm{ for all } n\in {\mathbb{Z}}\bigg\}.$$ Suppose $Y$ is a closed, oriented 3-manifold and $\eta$ is a smooth 1-cycle in $Y$. Kronheimer and Mrowka defined in [@kmbook; @km4] a version of monopole Floer homology with local coefficients which assigns to the pair $(Y,\eta)$ a $\Lambda[U]$-module $${\HMto_{\bullet}}(Y;\Gamma_\eta) = \bigoplus_{{\mathfrak{s}}\in{\text{Spin}^c}(Y)}{\HMto_{\bullet}}(Y,{\mathfrak{s}};\Gamma_\eta).$$ Their assignment is functorial in that a smooth cobordism $(W,\nu)$ from $(Y_1,\eta_1)$ to $(Y_2,\eta_2)$ gives rise to a map of $\Lambda[U]$-modules, $$\label{eqn:hmmap}{\HMto_{\bullet}}(W;\Gamma_\nu):{\HMto_{\bullet}}(Y_1;\Gamma_{\eta_1})\to{\HMto_{\bullet}}(Y_2;\Gamma_{\eta_2}),$$ where these maps compose in the obvious way. To a pair $(Y,\eta)$ and a contact structure $\xi$ on $Y$, Kronheimer and Mrowka assign in [@km; @kmosz] an element $$c_{HM}(\xi)\in{\HMto_{\bullet}}(-Y,{\mathfrak{s}}_\xi;\Gamma_\eta)\subset {\HMto_{\bullet}}(-Y;\Gamma_\eta)\footnote{As mentioned in the introduction, this class is denoted by $\psi(\xi)$ in \cite{kmosz}.}$$ which depends only on the isotopy class of $\xi$. One of the most important properties of this invariant is its functoriality with respect to contact $(+1)$-surgery, per the following.[^4]
\[thm:contactsurgery\] Suppose $(Y',\xi')$ is the result of contact $(+1)$-surgery on a Legendrian knot $K\subset (Y,\xi)$ disjoint from $\eta$. Let $W$ be the corresponding 2-handle cobordism, obtained from $Y\times[0,1]$ by attaching a contact $(+1)$-framed 2-handle along $K\times\{1\}$, and let $\nu\subset W$ be the cylinder $\nu= \eta\times [0,1]$. The induced map $${\HMto_{\bullet}}(-W;\Gamma_{\nu}):{\HMto_{\bullet}}(-Y;\Gamma_{\eta})\to{\HMto_{\bullet}}(-Y';\Gamma_\eta)$$ sends $c_{HM}(\xi)$ to $c_{HM}(\xi')$, up to multiplication by a unit in $\Lambda$ (a subtlety we will generally ignore).
This theorem is a special case of a more general functoriality result for the contact invariant with respect to maps induced by exact symplectic cobordisms, as established by Hutchings and Taubes in [@ht] (cf. also Mrowka–Rollin [@mr]). This more general result was used to prove Theorem \[thm:functcon\], as mentioned in the introduction.
Sutured monopole Floer homology and contact invariants {#ssec:shm}
------------------------------------------------------
We now recall the definition of sutured monopole Floer homology and our construction of the contact invariant for sutured contact manifolds defined therein.
Suppose $(M,\Gamma)$ is a balanced sutured manifold. Let $A(\Gamma)$ be a closed tubular neighborhood of $\Gamma$, and let $T$ be a compact, connected, oriented surface with $g(T)\geq 2$ and $\pi_0(\partial T)\cong \pi_0(\Gamma)$. Let $$h:\partial T\times [-1,1]\to A(\Gamma)$$ be an orientation-reversing homeomorphism sending $\partial T\times \{\pm 1\}$ to $\partial (R_{\pm}{\smallsetminus}A(\Gamma))$. Now consider the *preclosure* $$P = M\cup_h F\times [-1,1]$$ formed by gluing $F\times[-1,1]$ according to $h$. The balanced condition ensures that $P$ has two homeomorphic boundary components, $\partial_+P$ and $\partial_-P$, given by$$\partial_{\pm}P= R_{\pm}(\Gamma) \cup T\times\{\pm1\}.$$ One can then glue $\partial_+P$ to $\partial_-P$ by an orientation-reversing homeomorphism to form a closed, oriented 3-manifold $Y$ containing a distinguished surface $$R:=\partial_+P= -\partial_-P\subset Y.$$ In [@km4], Kronheimer and Mrowka define a *closure* of $(M,\Gamma)$ to be any pair $(Y,R)$ obtained in this manner. We refer to $T$ as the *auxiliary surface* used to form this closure.
If $(Y,R)$ is a closure of $(M,\Gamma)$, then $(-Y,-R)$ is a closure of $(-M,-\Gamma)$.
\[rmk:altclosure\] It is sometimes helpful to think of $Y$ as obtained by gluing $R\times[-1,1]$ to $P$, by a map which identifies $R\times\{\pm 1\}$ with $\partial_{\mp}P$, and $R$ as $R\times\{0\}$. In particular, from this perspective, $\partial M$ is a submanifold of $Y$.
Suppose $(Y,R)$ is a closure of $(M,\Gamma)$ as above, and fix an oriented curve $\eta\subset R$ which is dual to a homologically essential curve in the auxiliary surface $T$. The sutured monopole Floer homology of $(M,\Gamma)$ is defined to be the $\Lambda$-module $${\underline{{SHM}}}(M,\Gamma):={\HMto_{\bullet}}(Y| R; \Gamma_\eta):=\bigoplus_{\langle c_1({\mathfrak{s}}),[R]\rangle = 2-2g(R)}{\HMto_{\bullet}}(Y;\Gamma_\eta).\footnote{Technically, Kronheimer and Mrowka consider instead those ${\mathfrak{s}}$ with $\langle c_1({\mathfrak{s}}),[R]\rangle = 2g(R)-2$. Our convention is better suited to the contact perspective and changes nothing of import. In particular, the two conventions result in isomorphic $\Lambda$-modules.}$$ Indeed, Kronheimer and Mrowka prove in [@km4] that the isomorphism class of this module is independent of the choice of closure $(Y,R)$ and curve $\eta$, and is therefore an invariant of $(M,\Gamma)$.
Now suppose $(M,\Gamma,\xi)$ is a sutured contact manifold; that is, $\partial M$ is convex and $\Gamma$ divides the characteristic foliation of $\xi$ on $\partial M$. In [@bsSHM], we gave a procedure for extending $\xi$ first to a contact structure $\xi'$ on $P$, and then to a contact structure $\bar \xi$ on a closure $(Y,R)$ of $(M,\Gamma)$, such that $R$ is convex with respect to $\bar\xi$ with $$\langle c_1({\mathfrak{s}}_{\bar\xi}),[R]\rangle_Y = 2-2g(R).$$ We refer to $(Y,R,\bar\xi)$ as a *contact closure* of $(M,\Gamma,\xi)$. The above pairing implies that $$\langle c_1({\mathfrak{s}}_{\bar\xi}),[-R]\rangle_{-Y} = 2-2g(R).$$ It therefore makes sense to define the element $$c_{HM}(\xi):=c_{HM}(\bar\xi)\in {\HMto_{\bullet}}(-Y,{\mathfrak{s}}_{\bar\xi};\Gamma_\eta)\subset {\HMto_{\bullet}}(-Y| {-}R; \Gamma_\eta)=:{\underline{{SHM}}}(-M,-\Gamma).$$ We showed in [@bsSHM] that, for a certain subclass of curves $\eta\subset R$ satisfying the constraints above, this element is independent of the choices involved in its construction. Below, we describe an alternate, explicit means of defining this invariant using partial open books.
Following [@bsSHM Definition 4.9], a *partial open book* is a quadruple $(S,P,h,\mathbf{c})$, where:
- $S$ is a surface with nonempty boundary,
- $P$ is a subsurface of $S$,
- $h:P\to S$ is an embedding which restricts to the identity on $\partial P\cap \partial S$,
- $\{c_1,\dots,c_n\}$ is a set of disjoint, properly embedded arcs in $P$ whose complement in $S$ deformation retracts onto $S{\smallsetminus}P$.
Given such a partial open book, let $\xi_S$ be the $[-1,1]$-invariant contact structure on $S\times[-1,1]$ for which each $S\times\{t\}$ is convex with Legendrian boundary and dividing set consisting of one boundary-parallel arc for each component of $\partial S$, oriented in the direction of $\partial S$. Let $H(S)$ be the sutured contact manifold obtained from $(S\times[-1,1],\xi_{S})$ by rounding corners, as illustrated in Figure \[fig:productsurfacecorners\] below. In particular, the dividing set on $\partial H(S)$ is isotopic to $\partial S\times \{0\}$. Let $s_i$ be the curve on $\partial H(S)$ given by$$\label{eqn:basishandle}s_i=(c_i\times\{1\})\cup (\partial c_i\times [-1,1])\cup (h(c_i)\times\{-1\}).\footnote{In a slight abuse of notation, we will simply identify $H(S)$ with $(S\times[-1,1],\xi_S)$, ignoring corner rounding.}$$ We say that $(S,P,h,\{c_1,\dots,c_n\})$ is *compatible* with the sutured contact manifold $(M,\Gamma,\xi)$ if the latter can be obtained from $H(S)$ by attaching contact $2$-handles along the curves $s_1,\dots,s_n$. Honda, Kazez, and Mati[ć]{} proved the following in [@hkm4].
\[thm:relativegiroux1\] Every sutured contact manifold admits a compatible partial open book decomposition.
![Left, $(S\times[-1,1],\xi_{S})$, with negative region shaded, for a genus $2$ surface with $3$ boundary components. Right, the convex boundary of the sutured contact manifold $H(S)$ obtained by rounding corners.[]{data-label="fig:productsurfacecorners"}](productsurfacecorners2){width="11cm"}
Suppose now that $(M,\Gamma,\xi)$ is a sutured contact manifold and that $(S,P,h,\{c_1,\dots,c_n\})$ is a compatible partial open book. Suppose $(Y_S,R,\bar\xi_S)$ is a contact closure of $H(S)$. Adopting the perspective of Remark \[rmk:altclosure\], we may view $s_1,\dots,s_n$ as embedded curves in $Y_S$ disjoint from $R$. After small perturbation, we may assume that each $s_i$ is Legendrian with respect to $\bar\xi_S$ (via the Legendrian Realization Principle [@kanda; @honda2]). In [@bsSHM], we proved that the result of contact $(+1)$-surgeries on these Legendrians is a contact closure $(Y,R,\bar\xi)$ of $(M,\Gamma,\xi)$. For a certain class of curves $\eta\subset R$, we showed that $$\label{eqn:productlambda}{\underline{{SHM}}}(-H(S)):={\HMto_{\bullet}}(-Y_S|{-}R;\Gamma_\eta)\cong \Lambda$$ is generated by the contact class $c_{HM}(\xi_S):=c_{HM}(\bar\xi_S)$. Let $W$ be the 2-handle cobordism from $Y_S$ to $Y$ corresponding to the above surgeries on $s_1,\dots,s_n$. Since $R\subset Y_S$ and $R\subset Y$ are homologous in $W$ (these two copies of $R$ are the boundary components of a properly embedded $R\times[-1,1]\subset W$), the cobordism map in Theorem \[thm:contactsurgery\] restricts to a map $$\label{eqn:cobmapbkgnd}{\HMto_{\bullet}}(-W|{-}R;\Gamma_{\nu}):{\HMto_{\bullet}}(-Y_S|{-}R;\Gamma_{\eta})\to{\HMto_{\bullet}}(-Y|{-}R;\Gamma_\eta)=: {\underline{{SHM}}}(-M,-\Gamma),$$ obtained as the sum of the maps $${\HMto_{\bullet}}(-W,{\mathfrak{t}};\Gamma_\nu): {\HMto_{\bullet}}(-Y_S,{\mathfrak{t}}|_{-Y_S};\Gamma_{\eta})\to{\HMto_{\bullet}}(-Y,{\mathfrak{t}}|_{-Y};\Gamma_\eta)$$ over ${\text{Spin}^c}$ structures ${\mathfrak{t}}$ on $-W$ for which $$\langle c_1({\mathfrak{t}}),[-R]\rangle = 2-2g(R).$$ By Theorem \[thm:contactsurgery\], this restriction map then sends the generator $c_{HM}(\bar\xi_S)$ to the class $c_{HM}(\bar\xi)=:c_{HM}(\xi)$ (again, up to multiplication by a unit in $\Lambda$). In other words, the invariant $c_{HF}(\xi)$ is simply the image of the generator of (\[eqn:productlambda\]) under the cobordism map in (\[eqn:cobmapbkgnd\]). This formulation will be critical in our proof of Theorem \[thm:main\].
Sutured Heegaard Floer homology and contact invariants {#ssec:sfh}
------------------------------------------------------
Below, we recall the definition of sutured Heegaard Floer homology and Honda, Kazez, and Mati[ć]{}’s construction of the contact invariant defined therein. To define the sutured Heegaard Floer homology of a balanced sutured manifold $(M,\Gamma)$, as introduced by Juh[á]{}sz in [@juhasz2], one starts with an admissible sutured Heegaard diagram $$(\Sigma,\alpha = \{\alpha_1,\dots,\alpha_n\},\beta = \{\beta_1\dots,\beta_n\})$$ for $(M,\Gamma)$. In particular,
- $\Sigma$ is a compact surface with boundary,
- $M$ is obtained from $\Sigma\times[-1,1]$ by attaching 3-dimensional 2-handles along the curves $\alpha_i\times\{-1\}$ and $\beta_i\times\{1\}$, for $i= 1,\dots,n$, and
- $\Gamma$ is given by $ \partial \Sigma\times\{0\}$.
The sutured Heegaard Floer complex $$CF(\alpha,\beta):=CF(\Sigma,\alpha,\beta)$$ is the ${\mathbb{Z}/2\mathbb{Z}}$ vector space generated by intersection points $$\mathbf{x}\in{\mathbb{T}_{\alpha}}\cap {\mathbb{T}_{\beta}}= (\alpha_1\times\dots\times\alpha_n)\cap (\beta_1\times\dots\times\beta_n)\subset \operatorname{Sym}^n(\Sigma).$$ The differential is defined by counting holomorphic disks in the usual way; namely, for a generator ${\mathbf{x}}$ as above, $$d{\mathbf{x}}=\sum_{{\mathbf{y}}\in{\mathbb{T}_{\alpha}}\cap{\mathbb{T}_{\beta}}}\,\sum_{\substack{\phi\in\pi_2({\mathbf{x}},{\mathbf{y}})\\\mu(\phi)=1}} \#\big(\mathcal{M}(\phi)/\mathbb{R}\big)\cdot {\mathbf{y}},$$ where $\pi_2({\mathbf{x}},{\mathbf{y}})$ is the set of homotopy classes of Whitney disks from ${\mathbf{x}}$ to ${\mathbf{y}}$; $\mu(\phi)$ refers to the Maslov index of $\phi$; and $\mathcal{M}(\phi)$ is the moduli space of pseudoholomorphic representatives of $\phi$. The sutured Heegaard Floer homology of $(M,\Gamma)$ is the homology $${SFH}(M,\Gamma):=H_*(CF(\Sigma,\alpha,\beta),d)$$ of this complex.
Suppose now that $(M,\Gamma,\xi)$ is a sutured contact manifold and that $(S,P,h,\{c_1,\dots,c_n\})$ is a compatible partial open book. Let $\Sigma$ be the surface formed by attaching $1$-handles $H_1,\dots,H_n$ to $S$, where the feet of $H_i$ are attached along the endpoints of $c_i$. Orient $\Sigma$ so that the induced orientation on $S$ as a subsurface of $\Sigma$ is opposite the given orientation on $S$. For $i=1,\dots,n$, let $\alpha_i$ and $\beta_i$ be embedded curves in $\Sigma$ such that:
- $\alpha_i$ is the union of $c_i$ with a core of $H_i$, and
- $\beta_i$ is the union of $h(c_i)$ with a core of $H_i$.
We require that these curves intersect in the region $H_i\subset \Sigma$ in the manner shown in Figure \[fig:handlecurves\]. Then $(\Sigma,\beta,\alpha)$ is an admissible sutured Heegaard diagram for $(-M,-\Gamma)$, as shown in [@hkm4]. For each $i=1,\dots,n$, let $c^i$ be the unique intersection point between $\alpha_i$ and $\beta_i$ in $H_i$, and define $$\label{eqn:defc}\mathbf{c}= \{c^1,\dots,c^n\}\in {\mathbb{T}_{\beta}}\cap {\mathbb{T}_{\alpha}}\subset \operatorname{Sym}^n(\Sigma).$$ In [@hkm4], the contact invariant $c_{HF}(\xi)$ is given by $$c_{HF}(\xi):= [\mathbf{c}]\in {SFH}(-M,-\Gamma).$$
$\beta_i$ at 57 101 $\alpha_i$ at 80 100 $c^i$ at 138 24 $H_i$ at 190 6 $-S$ at 240 95
![The handle $H_i$, the curves $\alpha_i$ and $\beta_i$, and the intersection point $c^i$.[]{data-label="fig:handlecurves"}](handlecurves){width="7.5cm"}
Heegaard Floer homology with local coefficients and contact invariants {#ssec:hflocal}
----------------------------------------------------------------------
Finally, we recall the definition of Heegaard Floer homology with local coefficients and we review the construction of Ozsv[á]{}th and Szab[ó]{}’s contact invariant for closed manifolds therein.
Suppose $Y$ is a closed, oriented 3-manifold, and $\eta$ is an oriented, embedded curve in $Y$. To define the Heegaard Floer homology of $Y$ with local coefficient system associated with $\eta$, one starts with an admissible pointed Heegaard diagram $$(\Sigma,\alpha = \{\alpha_1,\dots,\alpha_n\},\beta=\{\beta_1,\dots,\beta_n\},z)$$ for $Y$. We may view $\eta$ as a (possibly non-embedded) curve on $\Sigma$. The chain complex $${CF^+}(\alpha,\beta;\Gamma_\eta):={CF^+}(\Sigma,\alpha,\beta,z;\Gamma_\eta)$$ is the $\Lambda[U]$-module $$\bigoplus_{{\mathbf{x}}\in T_\alpha \cap T_\beta} \bigg(\frac{\Lambda[U,U^{-1}]}{U\cdot \Lambda[U]}\bigg)\langle{\mathbf{x}}\rangle,$$ generated by intersection points ${\mathbf{x}}\in{\mathbb{T}_{\alpha}}\cap {\mathbb{T}_{\beta}}$. For ease of notation, we will adopt the convention in [@osz8] and use $[{\mathbf{x}},i]$ to denote $U^{-i}{\mathbf{x}}$, which then vanishes for $i<0$. The differential is defined on such a pair by $$\partial^+([{\mathbf{x}},i]) =\sum_{\substack{\phi\in\pi_2({\mathbf{x}},{\mathbf{y}})\\\mu(\phi)=1}} \#\big(\mathcal{M}(\phi)/\mathbb{R}\big)\cdot [{\mathbf{y}},i-n_z(\phi)]\cdot t^{\partial_\alpha(\phi)\cdot \eta},$$ and extended linearly with respect to multiplication in $\Lambda$. Here, $\partial_\alpha(\phi)\cdot \eta$ refers to the oriented intersection in $\Sigma$ of the $\alpha$ portion of the boundary of the domain of $\phi$ in $\Sigma$ with the curve $\eta$. The Heegaard Floer homology of $Y$ is the homology $${HF^+}(Y;\Gamma_\eta):=H_*({CF^+}(\alpha,\beta;\Gamma_\eta),\partial^+)$$ of this complex. The homology is, up to isomorphism of $\Lambda[U]$-modules, an invariant of the class $[\eta]\in H_1(Y;\mathbb{Z})$.
Suppose now that $(Y,\xi)$ is a closed contact 3-manifold with embedded curve $\eta$. Its contact invariant, originally defined by Ozsv[á]{}th and Szab[ó]{} in [@osz1] in terms of compatible (non-partial) open books, may also be described as follows. Let $(S,P,h,\{c_1,\dots,c_n\})$ be a partial open book for the sutured contact manifold $(Y(1),\xi(1))$ obtained from $(Y,\xi)$ by removing a Darboux ball. Let $(\Sigma,\beta,\alpha)$ be the corresponding sutured Heegaard diagram for $-Y(1)$. Note that $\Sigma$ has a single boundary component. Let $\Sigma'$ be the closed surface obtained by capping off this component with a disk containing a point $z$. Then $(\Sigma',\beta,\alpha,z)$ is an admissible pointed Heegaard diagram for $-Y$. It follows from Honda, Kazez, and Mati[ć]{}’s work in [@hkm4] that Ozsv[á]{}th and Szab[ó]{}’s contact invariant is given by $$c_{HF}(\xi)=[[\mathbf{c},0]]\in{HF^+}(-Y;\Gamma_\eta),$$ where $\mathbf{c}\in{\mathbb{T}_{\beta}}\cap{\mathbb{T}_{\alpha}}$ is the intersection point defined in (\[eqn:defc\]). This invariant enjoys the same sort of functoriality under contact $(+1)$-surgery as Kronheimer and Mrowka’s contact invariant in monopole Floer homology, per the following.[^5]
\[thm:contactsurgeryhf\] Suppose $(Y',\xi')$ is the result of contact $(+1)$-surgery on a Legendrian knot $K\subset (Y,\xi)$ disjoint from $\eta$. Let $W$ be the corresponding 2-handle cobordism, obtained from $Y\times[0,1]$ by attaching a contact $(+1)$-framed 2-handle along $K\times\{1\}$, and let $\nu\subset W$ be the cylinder $\nu= \eta\times [0,1]$. The induced map $${HF^+}(-W;\Gamma_{\nu}):{HF^+}(-Y;\Gamma_{\eta})\to{HF^+}(-Y';\Gamma_\eta)$$ sends $c_{HF}(\xi)$ to $c_{HF}(\xi')$.
As we will need to study these sorts of cobordism maps in depth, we recall here the definition of the map in Theorem \[thm:contactsurgeryhf\] in the setting of local coefficients.
Let $(\Sigma,\gamma,\beta,\alpha,z)$ be an admissible pointed Heegaard triple diagram for the cobordism $-W$, left-subordinate to the contact $(+1)$-framed knot $K\subset -Y$, as in [@osz5 Section 5.2]. In particular, $Y_{\gamma,\beta}$ is a connected sum of copies of $S^1\times S^2$, $Y_{\beta,\alpha} = -Y$, and $Y_{\gamma,\alpha} = -Y'$. Let $\Theta\in {\mathbb{T}_{\gamma}}\cap{\mathbb{T}_{\beta}}$ denote the intersection point in top Maslov grading. Then the map $${HF^+}(-W;\Gamma_\nu): {HF^+}(-Y;\Gamma_\eta)\to {HF^+}(-Y';\Gamma_\eta)$$ is induced by the chain map $$\label{eqn:f+}f^+_{\gamma,\beta,\alpha;\Gamma_\nu}:{CF^+}(\beta,\alpha;\Gamma_\eta)\to{CF^+}(\gamma,\alpha;\Gamma_\eta),$$ defined on $[{\mathbf{x}},i]$ by $$f^+_{\gamma,\beta,\alpha;\Gamma_\nu}([{\mathbf{x}},i]) =\sum_{{\mathbf{y}}\in{\mathbb{T}_{\gamma}}\cap{\mathbb{T}_{\alpha}}}\,\sum_{\substack{\phi\in\pi_2(\Theta,{\mathbf{x}},{\mathbf{y}})\\\mu(\phi)=0}} \#\mathcal{M}(\phi)\cdot [{\mathbf{y}},i-n_z(\phi)]\cdot t^{\partial_\alpha(\phi)\cdot \eta},$$ where $\pi_2(\Theta,{\mathbf{x}},{\mathbf{y}})$ is the set of homotopy classes of Whitney triangles with vertices at $\Theta,{\mathbf{x}},{\mathbf{y}}$, and $\mathcal{M}(\phi)$ is the moduli space of holomorphic representatives of $\phi$. This map is an invariant of the class $[\nu]\in H_2(-W,-\partial W;\mathbb{Z})$.
Proof of Main Theorem {#sec:proof}
=====================
Suppose $(M,\Gamma,\xi)$ is a sutured contact manifold. The goal of this section is to prove Theorem \[thm:main2\] of the introduction, a more precise version of which is stated below. The Main Theorem of this paper, Theorem \[thm:main\], then follows immediately, as explained in the introduction.
There exists a contact closure $(Y,R,\bar\xi)$ of $(M,\Gamma,\xi)$ with an oriented curve $\eta \subset R$ which is dual to an essential curve in the associated auxiliary surface $T$, for which there is an isomorphism of $\Lambda$-modules $$F:{SFH}(-M,-\Gamma)\otimes \Lambda \to {HF^+}(-Y| {-}R; \Gamma_\eta),$$ sending $c_{HF}(\xi)\otimes 1$ to $c_{HF}(\bar\xi)$.
We prove Theorem \[thm:main2\] according to the strategy outlined in the introduction. In particular, as we show in Subsection \[ssec:facthm\], Theorem \[thm:main2\] follows from Theorems \[thm:map\] and \[thm:iso\].
Heegaard diagrams for closures and cobordisms
---------------------------------------------
Fix a partial open book $$(S,P,h,\{c_1,\dots,c_n\})$$ compatible with $(M,\Gamma,\xi)$. Recall that an important ingredient in the proof of Theorem \[thm:main2\] involves understanding the map (denoted in the introduction by $G$) induced by $-W$, where $W$ is the 2-handle cobordism from a closure of $H(S)$ to a closure of $(M,\Gamma)$, corresponding to surgery on the $\partial H(S)$-framed curves $$\label{eqn:si}s_i=(c_i\times\{1\})\cup (\partial c_i\times [-1,1])\cup (h(c_i)\times\{-1\})\,\subset\, \partial (S\times[-1,1])$$ in the closure of $H(S)$. In order to eventually understand this map (see Theorem \[thm:map\]), we first describe a pointed Heegaard triple diagram for the cobordism $-W$.
Let $\Sigma$ be the surface formed by attaching $1$-handles $H_1,\dots,H_n$ and $H_1',\dots,H_n'$ to $S$, where:
- the feet of $H_i$ are attached along the endpoints of $c_i$,
- the feet of $H_i'$ are attached along the endpoints of a cocore of $H_i$.
We orient $\Sigma$ so that the induced orientation on $S$ as a subsurface of $\Sigma$ is opposite the given orientation on $S$. For each $i=1,\dots,n$, let $\alpha_i,\beta_i,\gamma_i$ be embedded curves in $\Sigma$ such that:
- $\alpha_i$ is the union of $c_i$ with a core of $H_i$,
- $\beta_i$ is the union of a cocore of $H_i$ with a core of $H_i'$,
- $\gamma_i$ is the union of $h(c_i)$ with a core of $H_i$.
We require that these curves intersect in the region $H_i\subset \Sigma$ in the manner shown on the right in Figure \[fig:HH2\].
$S$ at 40 8 $\Sigma$ at 175 8 $c_i$ at -7 46 $c_i$ at -7 145 $\alpha_i$ at 130 146 $\beta_i$ at 299 108 $\gamma_i$ at 130 133
$H_i$ at 250 155 $H_i'$ at 341 89
$\Theta^i$ at 423 156 $c_\beta^i$ at 485 165 $c_\gamma^i$ at 502 84 $w_i^+$ at 473 140 $w_i^-$ at 488 35
![On the left, a portion of $S$ near $c_i$. In the middle, the corresponding portion of $\Sigma$ with the curves $\alpha_i,\beta_i,\gamma_i$. On the right, a closeup of these curves in $H_i$. We have labeled the intersection points $\Theta^i, c_\beta^i,c_\gamma^i$, the points $w_i^{\pm}$, and have shaded the triangle $\Delta^i$.[]{data-label="fig:HH2"}](HH2){width="12.5cm"}
\[rmk:three\]Note that the sutured Heegaard diagram $$\label{eqn:diagstab}(\Sigma,\{\beta_1,\dots,\beta_n\},\{\alpha_1,\dots,\alpha_n\})$$ is an $n$-fold stabilization of the standard diagram for $-H(S)$. Meanwhile, the sutured Heegaard diagram $$\label{eqn:diagob}(\Sigma, \{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\})$$ is obtained from the standard Heegaard diagram for $(-M,-\Gamma)$ associated with the partial open book $(S,P,h,\{c_1,\dots,c_n\})$, as described in Subsection \[ssec:sfh\], by attaching the handles $H_1',\dots,H_n'$. In particular, it is a sutured Heegaard diagram for the sutured manifold obtained from $(-M,-\Gamma)$ by attaching $n$ contact $1$-handles. We will generally ignore this difference, however, and simply think of the diagram in (\[eqn:diagob\]) as encoding $(-M,-\Gamma)$ since (1) there is a canonical isomorphism $$\begin{aligned}
H_*(SFC(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\}))&\cong H_*(SFC(\Sigma{\smallsetminus}(\cup_i H_i'),\{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\}))\\
&=: SFH(-M,-\Gamma),\end{aligned}$$ and (2) a contact closure of a sutured manifold obtained from $(-M,-\Gamma)$ via contact $1$-handle attachments is also a contact closure of $(-M,-\Gamma)$, as explained in [@bsSHM].
We now describe a Heegaard triple diagram which encodes closures of the sutured manifolds specified by the diagrams in (\[eqn:diagstab\]) and (\[eqn:diagob\]) as well as the cobordism $-W$. Let $T$ be a compact, oriented, connected surface with boundary and $g(T)\geq 2$, such that $$\pi_0(\partial T) \cong \pi_0(\partial \Sigma) \cong \pi_0(\partial S).$$ Let $R_S$ be the closed, oriented surface formed by gluing $T$ to $S$ by a diffeomorphism of their boundaries. Let $R_\Sigma$ be the surface formed by gluing $T$ to $\Sigma$ in a similar manner. Let $D_a$ and $D_b$ be two disjoint disks in $T$. Let ${\underline}{R}_S$ and ${\underline}{R}_\Sigma$ denote the complements of these disks in $R_S$ and $R_\Sigma$, $$\begin{aligned}
{\underline}{R}_S&=R_S{\smallsetminus}D_a {\smallsetminus}D_b,\\
{\underline}{R}_\Sigma&=R_\Sigma{\smallsetminus}D_a {\smallsetminus}D_b,\end{aligned}$$ and let $${\underline}\Sigma = {\underline}{R}_S\cup {\underline}{R}_\Sigma$$ be the closed surface formed by gluing these complements together by the identity maps on $\partial D_a$ and $\partial D_b$. In other words, ${\underline}\Sigma$ is obtained by connecting $R_S$ and $R_\Sigma$ via two tubes. We will think of the $\alpha_i,\beta_i,\gamma_i$ curves above as lying in ${\underline}{R}_\Sigma\subset {\underline}\Sigma.$ See the middle diagram in Figure \[fig:annuluseg\] for an illustration of ${\underline}\Sigma = {\underline}R_S\cup {\underline}R_\Sigma$ in the case that $S$ is an annulus, $h$ is a right-handed Dehn twist around the core, $\{c_1,\dots,c_n\}$ consists of just the cocore, and $T$ is a genus 2 surface with 2 boundary components.
${\underline}R_\Sigma$ at 220 360 ${\underline}R_S$ at 640 360 $\longrightarrow$ at 220 900 $=$ at 574 900
$c_1$ at 85 957 $h(c_1)$ at -25 905 $S$ at 118 790 $\Sigma$ at 390 790 $\Sigma$ at 718 790
$\Sigma$ at 115 430 $S$ at 730 430 $T{\smallsetminus}D_a{\smallsetminus}D_b$ at 565 430 $T{\smallsetminus}D_a{\smallsetminus}D_b$ at 285 430 $\partial D_a$ at 390 465 $\partial D_b$ at 390 625
![Top left, the annulus $S$ with basis arc $c_1$ and its image under the right-handed Dehn twist around the core (it looks like a left-handed Dehn twist because we reverse the orientation of $S$ in forming $\Sigma$). Top middle and right, $\Sigma$ and the curves $\alpha_1,\beta_1,\gamma_1$. Middle, the corresponding surface ${\underline}\Sigma ={\underline}R_\Sigma\cup{\underline}R_S$. Bottom, the triple diagram $({\underline}\Sigma,\gamma,\beta,\alpha)$ without the curves $\gamma_{2},\dots,\gamma_{8}$, which are just small Hamiltonian translates of $\beta_{2},\dots,\beta_{8}$.[]{data-label="fig:annuluseg"}](annuluseg){width="11.5cm"}
$A$ at -15 202 ${\underline}R_S$ at 400 35 $180^\circ$ at 645 220 $a_{i}$ at 244 289 $a_{j}$ at 215 317
$b_{j}$ at 243 113 $b_{i}$ at 213 85
![The arcs $a_i, b_j$ on ${\underline}R_S$. The arc $b_i$ is the image of $a_i$ under rotation of $180^\circ$ about the axis $A$.[]{data-label="fig:TSarcs"}](TSarcs){width="11cm"}
Suppose $R_S$ has genus $g$. Then $R_\Sigma$ and ${\underline}\Sigma$ have genera $n+g$ and $n+2g+1$, respectively. Let $$\label{eqn:abarcs}\{a_{n{+}1},\dots,a_{n+2g}\}\textrm{ and }\{b_{n+1},\dots,b_{n+2g}\}$$ be the sets of pairwise disjoint, properly embedded arcs in ${\underline}R_S$ shown in Figure \[fig:TSarcs\]. In particular, the two sets of arcs are related by rotation of the surface by $180^\circ$ around the axis shown in the figure. We index the arcs so that each $a_i$ intersects $b_i$ in exactly one place and is disjoint from $b_j$ for $j\neq i$. Note that the $a_i$ have endpoints on $\partial D_b$ and cut ${\underline}R_S$ into an annulus with $\partial D_a$ as a boundary component. Likewise, the $b_j$ have endpoints on $\partial D_a$ and cut ${\underline}R_S$ into an annulus with $\partial D_b$ as a boundary component.
Observe that the regular neighborhoods $$\begin{aligned}
U_i&=N(\alpha_i\cup\beta_i)\subset \Sigma,\\
V_i&=N(\beta_i\cup \gamma_i)\subset \Sigma\end{aligned}$$ are once-punctured tori. In particular, there exist disks $D_1,\dots,D_n\subset S$ such that $$\label{eqn:homo}{\underline}R_S{\smallsetminus}(\cup_{i=1}^n D_i)\, \cong\, {\underline}R_\Sigma {\smallsetminus}(\cup_{i=1}^n U_i)\, \cong\, {\underline}R_\Sigma {\smallsetminus}(\cup_{i=1}^n V_i).$$ We may assume that these $D_i$, as disks in ${\underline}R_S$, are disjoint from the arcs in (\[eqn:abarcs\]). Let $$\begin{aligned}
\varphi_\alpha&: {\underline}R_S {\smallsetminus}(\cup_{i=1}^n D_i) \to {\underline}R_\Sigma{\smallsetminus}(\cup_{i=1}^n U_i),\\
\varphi_\gamma&: {\underline}R_S {\smallsetminus}(\cup_{i=1}^n D_i) \to {\underline}R_\Sigma{\smallsetminus}(\cup_{i=1}^n V_i)\end{aligned}$$ be diffeomorphisms which restricts to the identity map on $T{\smallsetminus}D_a {\smallsetminus}D_b$. For $i=n{+}1,\dots, n{+}2g$, let $\alpha_i,\beta_i$ be the curves in ${\underline}\Sigma= \underline{R}_S \cup \underline{R}_\Sigma$ given by $$\begin{aligned}
\alpha_i &= a_i \cup \varphi_\alpha(a_i),\\
\beta_i &= b_i\cup \varphi_\gamma(b_i).\end{aligned}$$ Let $$\begin{aligned}
\alpha_{n+2g+1} &= \partial D_a\subset {\underline}\Sigma,\\
\beta_{n+2g+1} &= \partial D_b\subset {\underline}\Sigma.
\end{aligned}$$ For $i=n+1,\dots,n+2g+1$, let $\gamma_i$ be a small Hamiltonian translate of $\beta_i$ in ${\underline}\Sigma$ such that $\beta_i$ and $\gamma_i$ intersect in exactly two points, both contained in ${\underline}R_S\subset {\underline}\Sigma$. See Figure \[fig:TSarcs2\] for a closeup near the intersections of the curves $\alpha_i,\beta_i, \gamma_i$, for $i=n{+}1,\dots,n{+}2g$, on ${\underline}R_S\subset {\underline}\Sigma$.
$\beta_i$ at -10 65 $\alpha_i$ at 7 318 $\gamma_i$ at -10 100
$\beta_j$ at 8 -10 $\gamma_j$ at 53 -10
$\alpha_i$ at 375 15
$\beta_j$ at 375 300 $\alpha_j$ at 375 60 $\gamma_j$ at 400 318
$x_\beta^i$ at 91 271 $x_\gamma^i$ at 126 239 $x_\beta^j$ at 466 262 $x_\gamma^j$ at 444 232 $\Theta^i$ at 50 225 $\Theta^j$ at 437 283
![Intersections of the $\alpha_i,\beta_i,\gamma_i$ curves for $i=n{+}1,\dots,n{+}2g$. The figures on the left and right show the same curves, after rotating this portion of ${\underline}R_S$ by $180^\circ$ about the axis $A$ of Figure \[fig:TSarcs\]. Left, we have labeled the intersection points $\Theta^i, x_\beta^i,x_\gamma^i$ and shaded the triangle $\Delta^i$; likewise for $\Theta^j, x_\beta^j,x_\gamma^j, \Delta^j$ on the right. []{data-label="fig:TSarcs2"}](TSarcs2){width="11cm"}
Let $$\begin{aligned}
{\mathbf{\alpha}}&= \{\alpha_1,\dots,\alpha_{n+2g+1}\},\\
{\mathbf{\beta}}&=\{\beta_1,\dots,\beta_{n+2g+1}\},\\
{\mathbf{\gamma}}&= \{\gamma_1,\dots,\gamma_{n+2g+1}\}.\end{aligned}$$ We claim the following:
- $({\underline}\Sigma, \beta, \alpha)$ is a Heegaard diagram for a closure $Y_{\beta} =: -Y_S$ of $-H(S)$,
- $({\underline}\Sigma, \gamma, \alpha)$ is a Heegaard diagram for a closure $Y_{\gamma} =: -Y$ of $(-M,-\Gamma)$,
- $({\underline}\Sigma,\gamma,\beta,\alpha)$ is a Heegaard triple diagram for the cobordism $W_{\beta,\gamma}=:-W:-Y_S\to -Y$, where $W:Y_S\to Y$ is the $2$-handle cobordism corresponding to surgery on the $s_i\subset \partial H(S)\subset Y_S$.
These claims will be unsurprising to the expert. These sorts of Heegaard diagrams for closures are used in Lekili’s work [@lekili2] and are very similar to those in Ozsv[á]{}th-Szab[ó]{}’s work [@osz1 Section 3]. Nevertheless, we provide an explanation below. We will explain the first of these Heegaard diagrams in depth; the claim regarding the second admits a similar explanation. We will then address the third claim, regarding the Heegaard triple diagram for the cobordism.
Recall from Remark \[rmk:three\] that $$(\Sigma,\{\beta_1,\dots,\beta_n\},\{\alpha_1,\dots,\alpha_n\})$$ is a Heegaard diagram for the sutured manifold $-H(S)$. It follows that $$(R_\Sigma,\{\beta_1,\dots,\beta_n\},\{\alpha_1,\dots,\alpha_n\})$$ specifies the preclosure $P$ obtained from $-H(S)$ by attaching $T\times[-1,1]$ in the usual way. That is, $P$ is obtained from $R_\Sigma\times[-1,1]$ by attaching thickened disks to $R_\Sigma\times\{-1\}$ along $\beta_1,\dots,\beta_n$ and to $R_\Sigma\times\{1\}$ along $\alpha_1,\dots,\alpha_n$. Let us denote these unions of thickened disks by $C_\beta$ and $C_\alpha$, respectively, so that $$P = R_\Sigma\times[-1,1]\cup C_\beta \cup C_\alpha.$$ The boundary of $P$ consists of two homeomorphic components, $\partial P = \partial_+P\sqcup -\partial_-P.$ Let $$R_\beta := \partial_+P.$$ Per Remark \[rmk:altclosure\], one may form a closure $Y_\beta=:-Y_S$ of $-H(S)$ by gluing $R_\beta\times[-1,1]$ to $P$, and one may do so in such a way that
- $D_a\times\{\mp 1\}\subset R_\beta\times\{\mp 1\}$ is identified with $D_a\times\{\pm 1\}\subset \partial_{\pm}P$, and
- $D_b\times\{\mp 1\}\subset R_\beta\times\{\mp 1\}$ is identified with $D_b\times\{\pm 1\}\subset \partial_{\pm}P$.
The distinguished surface may be identified with $R_\beta=:-R$. The diagram on the left in Figure \[fig:schematichd\] is a schematic illustration of this closure. Note that the disjoint union $R_\Sigma\times\{0\}\sqcup R_\beta\times\{0\}$ separates $Y_\beta$ into two pieces $V_\beta$ and $V_\alpha$, containing $C_\beta$ and $C_\alpha$, respectively. Let $T_\beta$ and $T_\alpha$ be the tubes in $V_\alpha$ and $V_\beta$, respectively, defined by $$\begin{aligned}
T_\beta &= (D_b\times[0,1]\subset R_\Sigma\times[0,1])\cup (D_b\times[-1,0]\subset R_\beta\times[-1,0]),\\
T_\alpha &= (D_a\times[-1,0]\subset R_\Sigma\times[-1,0])\cup (D_a\times[0,1]\subset R_\beta\times[0,1]).\end{aligned}$$ Then the handlebodies $$\begin{aligned}
H_\beta &= \overline{V_\beta{\smallsetminus}T_\alpha} \cup T_\beta,\\
H_\alpha &= \overline{V_\alpha{\smallsetminus}T_\beta} \cup T_\alpha\end{aligned}$$ provide a Heegaard splitting of $Y_\beta$, as indicated in the diagram on the right in Figure \[fig:schematichd\]. The Heegaard surface in this splitting is therefore obtained by connecting $R_\Sigma\times\{0\}$ and $R_\beta\times\{0\}$ via two tubes. In particular, since $R_\beta\cong R_S$, this Heegaard surface may be identified with ${\underline}\Sigma$. Under this identification, it is not hard to see that the Heegaard diagram $({\underline}\Sigma,\beta,\alpha)$ specifies a splitting of precisely this form. Similar reasoning shows that $({\underline}\Sigma,\gamma,\alpha)$ determines an analogous Heegaard splitting of a closure $Y_\gamma=:-Y$ of $(-M,-\Gamma)$, with distinguished surface $R_\gamma=:-R$.[^6]
$R_\Sigma\times[-1,1]$ at 66 132 $C_\beta$ at 70 112 $C_\alpha$ at 70 152 $R_\beta\times[-1,0]$ at 158 230 $R_\beta\times[0,1]$ at 158 34 $R_\beta\times\{0\}$ at 245 125
$V_\alpha$ at 497 215 $V_\beta$ at 497 20
$T_\beta$ at 608 162
$T_\alpha$ at 590 106
![Left, a schematic of the closure $Y_\beta$. Right, a schematic of the standard Heegaard splitting of the closure: the handlebody $H_\alpha$ is the union of $V_\alpha$ and $T_\alpha$, shown in white and light gray, respectively; the handlebody $H_\beta$ is the union of $V_\beta$ and $T_\beta$, shown in medium and dark gray, respectively.[]{data-label="fig:schematichd"}](schematichd){width="15.8cm"}
We turn now to the claim about the triple diagram. Viewing $-H(S)$ as the sutured manifold determined by the sutured Heegaard diagram $$(\Sigma,\{\beta_1,\dots,\beta_n\},\{\alpha_1,\dots,\alpha_n\}),$$ it is easy to see that $\gamma_i\subset \Sigma$ is isotopic in $-H(S)$ to the curve $s_i$ in (\[eqn:si\]), for $i=1,\dots,n$, and that the $\Sigma$-framing of $\gamma_i$ coincides with the $\partial H(S)$-framing of $s_i$. Moreover, $\beta_i$ bounds a meridional disk for $\gamma_i$. It follows rather easily that $$({\underline}\Sigma,\{\beta_{n+1},\dots,\beta_{n+2g+1}\},\alpha)$$ specifies the complement of a *bouquet* for the link $s_1\cup\dots\cup s_n\subset Y_\beta$, as defined in [@osz5], and therefore that $$({\underline}\Sigma,\gamma,\beta,\alpha)$$ is a left-subordinate Heegaard triple diagram for the cobordism $W_{\beta,\gamma}=:-W$ associated to surgery on this framed link.
Admissibility
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In order to define chain complexes and chain maps using these Heegaard diagrams, we must specify the location of the basepoint $z$ and then isotope the attaching curves so as to make the pointed Heegaard triple diagram $({\underline}\Sigma,\gamma,\beta,\alpha,z)$ weakly admissible, meaning that every nontrivial periodic domain has both positive and negative multiplicities. Note that once the triple diagram is weakly admissible, the diagrams $({\underline}\Sigma,\beta,\alpha,z)$ and $({\underline}\Sigma, \gamma,\alpha,z)$ will be too.
As for the location of the basepoint, let us first wind $\alpha_{n+2g+1}$ one and a half times along the curve $\delta$, as in Figure \[fig:uvuv2\], so that it meets each of $\beta_{n+2g+1}$ and $\gamma_{n+2g+1}$ in the four points $$u_\beta,u'_\beta,v'_\beta,v_\beta \textrm{ and } u_\gamma,u'_\gamma,v'_\gamma,v_\gamma,$$ respectively. We place the basepoint $z$ as shown in the figure.
$z$ at 460 237
$\delta$ at 49 94 $u_\beta$ at 694 271 $u'_\beta$ at 692 227 $v'_\beta$ at 691 155 $v_\beta$ at 695 101 $u_\gamma$ at 637 267 $u'_\gamma$ at 632 220 $v'_\gamma$ at 637 144 $v_\gamma$ at 650 88 $\Theta^{n+2g+1}$ at 722 329
![Left, the curve $\delta$. Middle, the result of winding $\alpha_{n+2g+1}$ around $\delta$, and the basepoint $z$. Right, a closeup of $\alpha_{n+2g+1}, \beta_{n+2g+1},\gamma_{n+2g+1}$ near the intersection points $u_\beta$, $u'_\beta$, $v'_\beta$, $v_\beta$ and $u_\gamma$, $u'_\gamma$, $v'_\gamma$, $v_\gamma$ and $\Theta^{n+2g+1}$; the triangle $\Delta^{n+2g+1}$ is shaded.[]{data-label="fig:uvuv2"}](uvuv2){width="12.5cm"}
Establishing weak admissibility (see Proposition \[prop:winding\]) requires a thorough accounting of the periodic domains in the pointed Heegaard triple diagram $({\underline}\Sigma,\gamma,\beta,\alpha,z)$. To begin with, let us henceforth orient the curves $\beta_i$ and $\alpha_i$
- as in Figure \[fig:HH2\] for $i=1,\dots,n$,
- as in Figure \[fig:TSarcs2\] for $i=n{+}1,\dots,n{+}2g$, and
- as in Figure \[fig:uvuv2\] for $i=n{+}2g{+}1$.
We orient each $\gamma_i$ in the same direction as $\beta_i$. Let $P_{\beta}$ and $P_{\gamma}$ denote the $(\beta,\alpha)$- and $(\gamma,\alpha)$-periodic domains with multiplicities $2$ and $1$ in the regions $\Sigma$ and $S$ of ${\underline}\Sigma$, respectively, and with $$\begin{aligned}
\partial P_{\beta} &= \beta_{n+2g+1} - \alpha_{n+2g+1} ,\\
\partial P_{\gamma} &= \gamma_{n+2g+1}- \alpha_{n+2g+1},\end{aligned}$$ as shown in Figure \[fig:periodic\]. From our description of the Heegaard splittings of the closures $Y_{\beta}$ and $Y_{\gamma}$, it is clear that $P_\beta$ and $P_\gamma$ represent the homology classes $$[R_{\beta}]\in H_2(Y_{\beta};\mathbb{Z}) \textrm{ and }[R_{\gamma}]\in H_2(Y_{\gamma};\mathbb{Z}),$$ where $R_\beta$ and $R_\gamma$ are the genus $g$ distinguished surfaces in $Y_\beta$ and $Y_\gamma$.
$R_2$ at 110 192 $R_3$ at 117 168 $A$ at 253 50 $B$ at 20 50 $1$ at 182 136 $1$ at 253 20 $0$ at 226 136 $2$ at 105 136 $2$ at 20 20 $-1$ at 145 225 $z$ at 190 208
![The multiplicities of $P_\beta$ near $\beta_{n+2g+1}$ and $\alpha_{n+2g+1}$. $R_1$ is the small bigon region with multiplicity $-1$.[]{data-label="fig:periodic"}](periodic){width="6cm"}
For $i=n{+}1,\dots,n{+}2g{+}1$, let $P_i$ denote the small $(\gamma,\beta)$-periodic domain with $$\partial P_i = \gamma_{i}-\beta_{i}.$$ The collection $$\{P_\gamma,P_{n+1},\dots,P_{n+2g+1}\}$$ is linearly independent in the $\mathbb{Q}$-vector space of rational periodic domains for $({\underline}\Sigma,\gamma,\beta,\alpha,z)$. It therefore extends to a basis $$\label{eqn:basis}\{P_\gamma,P_{n+1},\dots,P_{n+2g+1}, Q_1,\dots, Q_m\}$$ for this vector space (note that $P_\beta = P_\gamma -P_{n+2g+1}$). By adding multiples of $P_i$, we may assume that none of the $Q_j$ contain multiples of $\beta_{i}$ in their boundaries, for any $i=n{+}1,\dots,n{+}2g{+}1$. Note that for each such $i$, there is an oriented curve in ${\underline}\Sigma$ which intersects $\gamma_i$ and $\alpha_i$ positively in exactly one point each, intersects all other $\gamma$ and $\alpha$ curves zero times algebraically, and is disjoint from $\beta_1,\dots,\beta_n$. Thus, $\gamma_i$ and $\alpha_i$ occur with opposite multiplicities in the boundary of each $Q_j$. In particular, by adding multiples of $P_\gamma$, we may assume that none of the $Q_j$ contain multiples of $\gamma_{n+2g+1}$ or $\alpha_{n+2g+1}$ in their boundaries. By further change of basis, we may also assume that $$\label{eqn:basis2}\partial Q_j =\alpha_{i_j}-\gamma_{i_j} +\sum_{k=1}^{i_j-1} a_{j,k}(\alpha_k - \gamma_k) +\sum_{k=1}^{n} b_{j,k}\beta_k,$$ where, for some $p$, $$1\leq i_1<\dots<i_p\leq n,$$$$n+1 \leq i_{p+1}<\dots<i_m\leq n+2g.$$ We may also assume that $a_{j,i_\ell} =0$ for all $\ell=1,\dots,m$.
Since all of the $Q_1,\dots,Q_p$ are disjoint from $\gamma_i,\beta_i,\alpha_i$ for $i=n{+}1,\dots,n{+}2g{+}1$ and satisfy $n_z=0$, each must have multiplicity $0$ in every region of ${\underline}\Sigma{\smallsetminus}\gamma{\smallsetminus}\beta{\smallsetminus}\alpha$ which intersects $\partial \Sigma\subset{\underline}\Sigma$. In other words, these $Q_1,\dots,Q_p$ are rational periodic domains for the sutured Heegaard triple diagram $$(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\beta_1,\dots,\beta_n\},\{\alpha_1,\dots,\alpha_n\}).$$ In fact, the collection $\{Q_1,\dots,Q_p\}$ is a basis for the vector space of such domains. This will henceforth be our preferred basis for this vector space.
In order to establish weak admissibility, we will isotope the $\alpha$ curves by a procedure known as *winding*, described below.
Note that for each $i=n{+}1,\dots,n{+}2g$, there is a homologically essential curve $\nu_i\subset {\underline}R_S$ such that $\nu_i$ intersects $\alpha_i$ exactly once and is disjoint from all other $\alpha_j$. We may assume that $\nu_i$ is disjoint from the disks $D_1,\dots,D_n$ in (\[eqn:homo\]). Let $$\eta_i = \varphi_\alpha(\nu_i)\subset {\underline}R_\Sigma\subset {\underline}\Sigma,$$ where $\varphi_\alpha$ is the map defined following (\[eqn:homo\]). Then $\eta_i$ also intersects $\alpha_i$ exactly once and is disjoint from all other $\alpha$ curves. Let $\eta_i^+$ and $\eta_i^-$ be parallel copies of $\eta_i$, oriented as on the left in Figure \[fig:etawinding\]. One can then talk about *winding* $\alpha_i$ along these curves in the directions given by their orientations. The diagram on the right in Figure \[fig:etawinding\] shows the result of winding $\alpha_i$ once along each of $\eta_i^+$ and $\eta_i^-$. This winding will be key not only for achieving admissibility, but also for many of the other results in this section.
$\alpha_i$ at -7 61 $\eta_i^+$ at 63 129 $\eta_i^-$ at 144 129 $w_i^+$ at 71 31 $w_i^-$ at 152 95 $w_i^+$ at 290 31 $w_i^-$ at 372 95
![Left, the intersection of $\eta_i^+$ and $\eta_i^-$ with $\alpha_i$, and the points $w_i^+$ and $w_i^-$. Right, the curve $\alpha_i$ after winding once along $\eta_i^+$ and $\eta_i^-$.[]{data-label="fig:etawinding"}](etawinding){width="10.7cm"}
In what follows, we will need to keep track of the effect of such winding on the coefficients of various domains of these Heegaard diagrams. In order to do so, we introduce new basepoints $$\label{eqn:basepts}w_1^+,\dots,w_{n+2g}^+\, \textrm{ and } \,w_1^-,\dots,w_{n+2g}^-\subset {\underline}\Sigma,$$ where
- $w_i^\pm$ are the points in $H_i\subset\Sigma\subset{\underline}\Sigma$ shown in Figure \[fig:HH2\], for $i=1,\dots,n$, and
- $w_i^\pm$ are the points on $\eta_i^\pm$ shown in Figure \[fig:etawinding\], for $i=n+1,\dots,n+2g$.
Note that $$\label{eqn:mult2}n_{w_{i}^\pm}(P_j)=0$$ for all $i,j$. Moreover, for $j,k=1,\dots,p$ with $k\neq j$, we have that $$\label{eqn:mult3}n_{w_{i_j}^\pm}(Q_j) = \pm 1\,\textrm{ and }\,
n_{w_{i_k}^\pm}(Q_j) = 0.$$ For $j,k$ in this range, the above quantities are unaffected by the winding described above. On the other hand, for $j=p+1,\dots,m$, the quantity $$n_{w_{i_j}^\pm}(Q_j)$$ changes by $\pm 1$ after winding $\alpha_{i_j}$ along $\eta_{i_j}^\pm$ and is unaffected by all other winding. Given a domain $D$ of the pointed triple diagram $({\underline}\Sigma,\gamma,\beta,\alpha,z)$, we will hereafter use the shorthand ${\underline}n_{w_i}(D)$ for the minimum $${\underline}n_{w_i}(D) = \min\{n_{w_i^+}(D),n_{w_i^-}(D)\}.$$
Below, we explain how to achieve weak admissibility via winding.
\[prop:winding\] The diagram $({\underline}\Sigma,\gamma,\beta,\alpha,z)$ can be made weakly admissible by winding each $\alpha_i$ along $\eta_i^+$ and $\eta_i^-$ in the directions given by their orientations sufficiently many times.
Let $$K=\max\big\{|n_{w_{i}^\pm}(Q_j)| \mid i=1,\dots,n+2g,\,j=1,\dots,m\big\}.$$ Choose some $$N>m^2K^2+mK.$$ We claim that the pointed Heegaard triple diagram $({\underline}\Sigma,\gamma,\beta,\alpha,z)$ becomes weakly admissible after winding each $\alpha_{i}$ along $\eta_{i}^+$ and $\eta_{i}^-$ at least $N$ times. Indeed, suppose we have performed this winding. It suffices to show that every nontrivial periodic domain has negative multiplicity in some region (for, if that’s the case, then every nontrivial periodic domain must also have positive multiplicity in some region). Let $P$ be such a periodic domain. Then we can write $P$ as a linear combination $$P = a_\gamma P_\gamma + \sum_{i=n+1}^{n+2g+1}a_{i} P_{i} + \sum_{i=1}^p b_iQ_i + \sum_{i=p+1}^m c_iQ_i$$ with integer coefficients. Note that $P_\gamma$ has multiplicities $-1$ and $+1$ in the regions $R_1$ and $R_2$, respectively, indicated in Figure \[fig:periodic\]. The multiplicities of the other $P_i$ and $Q_j$ are zero in these regions, so $P$ has a negative multiplicity if $a_\gamma$ is nonzero. Let us therefore assume that $a_\gamma = 0$. If all $c_i$ are zero, then $$n_{w_{i_j}^\pm}(P) = \pm b_i$$ for $j=1,\dots,p$ by (\[eqn:mult2\]) and (\[eqn:mult3\]). This means that $P$ has negative multiplicity $-|b_i|$ in some region if $b_i$ is nonzero. If, on the other hand, all $b_i$ are zero, then some $a_i$ must be nonzero as $P$ is nontrivial. But then $P$ has negative multiplicity $-|a_i|$ in some region. In conclusion, $P$ must have a negative multiplicity if all $c_i$ are zero. Let us therefore assume that some $c_i$ is nonzero.
Suppose $$|b_j| = \max\big\{|b_1|,\dots,|b_p|\big\}\,\textrm{ and }\,
|c_k| = \max\big\{|c_{p+1}|,\dots,|c_m|\big\}.$$ Before winding, we have that $$n_{w_{i_k}^\pm}(P)\,\leq\, |b_j|pK + |c_k|(m-p)K\,\leq\, |b_j|mK + |c_k|mK.$$ After winding, we therefore have that the minimum ${\underline}n_{w_{i_k}}(P)$ satisfies $$\label{eqn:ineq1}{\underline}n_{w_{i_k}}(P)\,\leq\, |b_j|mK + |c_k|mK-|c_k|N\,<\,|b_j|mK -|c_k|m^2K^2,$$ where the latter inequality is strict since $c_k$ is nonzero. Note that $$\label{eqn:ineq2}{\underline}n_{w_{i_j}}(P)\,\leq\, |c_k|(m-p)K-|b_j|\,\leq\, |c_k|mK-|b_j|$$ both before and after winding. Combining inequalities (\[eqn:ineq1\]) and (\[eqn:ineq2\]), we have that $${\underline}n_{w_{i_k}}(P) + mK{\underline}n_{w_{i_j}}(P)<0$$ after winding. Therefore, at least one of ${\underline}n_{w_{i_k}}(P)$ or ${\underline}n_{w_{i_j}}(P)$ is negative after winding.
We have thus shown that after winding $N$ times, every nontrivial periodic domain has a negative multiplicity. This completes the proof.
We will henceforth assume that the diagram $({\underline}\Sigma,\gamma,\beta,\alpha,z)$ has been made weakly admissible by winding as in Proposition \[prop:winding\].
Bottom ${\text{Spin}^c}$ structures
-----------------------------------
Next, we analyze the portions of the chain complexes $${CF^+}(\beta,\alpha):={CF^+}({\underline}\Sigma,\beta,\alpha,z)\,\textrm{ and }\,{CF^+}(\gamma,\alpha,z):={CF^+}({\underline}\Sigma,\gamma,\alpha,z)$$ in the bottom ${\text{Spin}^c}$ structures with respect to the genus $g$ distinguished surfaces $R_\beta\subset Y_\beta$ and $R_\gamma\subset Y_\gamma$. Recall from the discussion above that the $(\beta,\alpha)$- and $(\gamma,\alpha)$-periodic domains $P_\beta$ and $P_\gamma$ represent the homology classes of these distinguished surfaces. Suppose $[{\mathbf{x}},i]$ is a generator of ${CF^+}(\beta,\alpha)$ or ${CF^+}(\gamma,\alpha)$. According to [@osz14], we have $$\begin{aligned}
\label{eqn:1}\langle c_1({\mathfrak{s}}_z({\mathbf{x}})),[R_{\beta}]\rangle&=e(P_{\beta})+ 2n_{{\mathbf{x}}}(P_{\beta}), \textrm{ or}\\
\label{eqn:2}\langle c_1({\mathfrak{s}}_z({\mathbf{x}})),[R_{\gamma}]\rangle&=e(P_{\gamma}) + 2n_{{\mathbf{x}}}(P_{\gamma}),\end{aligned}$$ respectively, where $e(D)$ refers to the *Euler measure* of the domain $D$. For each $i=n{+}1,\dots,n{+}2g$, let $x_{\beta}^i$ and $x_{\gamma}^i$ be the unique intersection points $$\begin{aligned}
x_{\beta}^i&=\beta_i\cap\alpha_i\cap {\underline}R_S,\\
x_{\gamma}^i&=\gamma_i\cap\alpha_i\cap {\underline}R_S,\end{aligned}$$ shown in Figure \[fig:TSarcs2\], and define $$\begin{aligned}
{\mathbf{x}}_{\beta}&=\{x_{\beta}^{n+1},\dots, x_{\beta}^{n+2g}\}\in \operatorname{Sym}^{2g}({\underline}R_S),\\
{\mathbf{x}}_{\gamma}&=\{x_{\gamma}^{n+1},\dots, x_{\gamma}^{n+2g}\}\in \operatorname{Sym}^{2g}({\underline}R_S).\end{aligned}$$ Let ${CF^+}(\beta,\alpha|R_\beta)$ denote the direct summand of ${CF^+}(\beta,\alpha)$ generated by generators $[{\mathbf{x}},i]$ with $$\langle c_1({\mathfrak{s}}_z({\mathbf{x}})),[R_{\beta}]\rangle=2-2g,$$ and define ${CF^+}(\gamma,\alpha|R_\gamma)$ analogously, replacing $\beta$ with $\gamma$. We have the following characterization of the generators in these bottom ${\text{Spin}^c}$ structures.
\[lem:bot\] A generator $[{\mathbf{x}},i]\in{CF^+}(\beta,\alpha)$ is in ${CF^+}(\beta,\alpha|R_\beta)$ iff ${\mathbf{x}}$ is of the form $${\mathbf{x}}= {\mathbf{y}}\cup\{u_\beta\}\cup {\mathbf{x}}_\beta\, \,\textrm{ or }\,\, {\mathbf{x}}={\mathbf{y}}\cup\{v_\beta\}\cup {\mathbf{x}}_\beta ,$$ where $${\mathbf{y}}\in (\beta_1\times\dots\times\beta_n)\cap (\alpha_1\times\dots\times\alpha_n)\subset \operatorname{Sym}^n(\Sigma).$$ The analogous statement holds for generators of ${CF^+}(\gamma,\alpha)$, replacing $\beta$ with $\gamma$.
First, let us suppose that ${\mathbf{x}}$ is of the form described in the lemma. Note that $$\begin{aligned}
e(P_\beta) &= 2\chi({\underline}R_\Sigma) + \chi({\underline}R_S) = -6g-4n,\\
n_{{\mathbf{y}}}(P_\beta)&=2n,\\
n_{{\mathbf{x}}_\beta}(P_\beta)&=2g,\\
n_{u_\beta}(P_\beta) = n_{v_\beta}(P_\beta) &= 1 .\end{aligned}$$ The formula (\[eqn:1\]) then implies that $[{\mathbf{x}},i]\in {CF^+}(\beta,\alpha|R_\beta)$, as desired. For the converse, it is easy to see that if ${\mathbf{x}}$ is not of this form, then $n_{{\mathbf{x}}}(P_\beta)>2n+2g+1$ (changing ${\mathbf{x}}$ from a generator of this form to another generator moves intersection points from the portion of ${\underline}\Sigma$ where $P_\beta$ has multiplicity $1$ to the portion where it has multiplicity $2$) which implies that $$\langle c_1({\mathfrak{s}}_z({\mathbf{x}})),[R_{\beta}]\rangle>2-2g.$$ See [@lekili2] for what is virtually the same argument.
Theorems \[thm:map\] and \[thm:iso\] imply Main Theorem {#ssec:facthm}
-------------------------------------------------------
Below, we state Theorems \[thm:map\] and \[thm:iso\] and explain how they imply Theorem \[thm:main2\]. We first introduce some notation.
Let us fix an oriented, embedded curve $$\label{eqn:etatwisting}\eta\subset{\underline}R_{S}\subset {\underline}\Sigma.$$ This curve defines curves in $Y_\beta$ and $Y_\gamma$ which we will also denote by $\eta$. One may then consider the complexes with twisted coefficients $${CF^+}(\beta,\alpha;\Gamma_\eta)\textrm{ and }{CF^+}(\gamma,\alpha;\Gamma_\eta)$$ and their corresponding homologies $${HF^+}(Y_\beta;\Gamma_\eta)\textrm{ and } {HF^+}(Y_\gamma;\Gamma_\eta),$$ as defined in Subsection \[ssec:hflocal\]. Let $$\nu = \eta\times I\subset W_{\beta,\gamma}$$ be the cylindrical cobordism from $\eta\subset Y_\beta$ to $\eta\subset Y_\gamma$. Let $$\Theta= \{\Theta^1,\dots,\Theta^{n+2g+1}\}\in {CF^+}(\gamma,\beta)$$ denote the generator in the top Maslov grading, where each $\Theta^i$ is one of the two intersection points between $\gamma_i$ and $\beta_i$, as shown in Figures \[fig:HH2\], \[fig:TSarcs2\], and \[fig:uvuv2\]. The map $${HF^+}(W_{\beta,\gamma};\Gamma_\nu): {HF^+}(Y_\beta;\Gamma_\eta)\to {HF^+}(Y_\gamma;\Gamma_\eta)$$ is then defined as in Subsection \[ssec:hflocal\], in terms of a chain map $$\label{eqn:f+}f^+_{\gamma,\beta,\alpha;\Gamma_\nu}:{CF^+}(\beta,\alpha;\Gamma_\eta)\to{CF^+}(\gamma,\alpha;\Gamma_\eta)$$ defined on a generator $[{\mathbf{x}},i]$ by $$f^+_{\gamma,\beta,\alpha;\Gamma_\nu}([{\mathbf{x}},i]) =\sum_{\substack{\phi\in\pi_2(\Theta,{\mathbf{x}},{\mathbf{y}})\\\mu(\phi)=0}} \#\mathcal{M}(\phi)\cdot [{\mathbf{y}},i-n_z(\phi)]\cdot t^{\partial_\alpha(\phi)\cdot \eta}.$$ As usual, this map decomposes over ${\text{Spin}^c}$ structures on $W_{\beta,\gamma}$. Since $R_\beta$ and $R_\gamma$ are homologous (in fact, isotopic) in $W_{\beta,\gamma}$, we have that $$\langle c_1({\mathfrak{t}}|_{Y_\beta}),[R_\beta]\rangle=\langle c_1({\mathfrak{t}}|_{Y_\gamma}),[R_\gamma]\rangle$$ for any ${\mathfrak{t}}\in{\text{Spin}^c}(W_{\beta,\gamma})$. The chain map above therefore restricts to a map $$f^+_{\gamma,\beta,\alpha;\Gamma_\nu}:{CF^+}(\beta,\alpha|R_\beta;\Gamma_\eta)\to{CF^+}(\gamma,\alpha|R_\gamma;\Gamma_\eta),$$ which gives rise to the map $$\label{eqn:Gmap}G:={HF^+}(W_{\beta,\gamma}|R_\beta\sim R_\gamma;\Gamma_\nu): {HF^+}(Y_\beta|R_\beta;\Gamma_\eta)\to {HF^+}(Y_\gamma|R_\gamma;\Gamma_\eta)$$ on homology. This is precisely the map we called $G$ in the introduction. For each $i=1,\dots,n$, let $c_{\beta}^i$ and $c_{\gamma}^i$ be the unique intersection points $$\begin{aligned}
c_{\beta}^i&=\beta_i\cap\alpha_i\cap H_i,\\
c_{\gamma}^i&=\gamma_i\cap\alpha_i\cap H_i\end{aligned}$$ shown in Figure \[fig:HH2\], and define $$\begin{aligned}
{\mathbf{c}}_{\beta}&=\{c_{\beta}^{1},\dots, c_{\beta}^{n}\}\in {SFC}(\Sigma,\{\beta_1,\dots,\beta_n\},\{\alpha_1,\dots,\alpha_n\}),\\
{\mathbf{c}}_{\gamma}&=\{c_{\gamma}^{1},\dots, c_{\gamma}^{n}\}\in {SFC}(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\}).\end{aligned}$$ Note that these are representatives of the contact invariants associated to the partial open books $(S,\emptyset,\emptyset,\emptyset)$ and $(S,P,h,\{c_1,\dots,c_n\})$, respectively.
Theorems \[thm:map\] and \[thm:iso\] below are the two main theorems of this section.
\[thm:map\] For sufficiently large $g$ and sufficient winding, the map $f^+_{\gamma,\beta,\alpha;\Gamma_\nu}$ sends $$[{\mathbf{c}}_{\beta}\cup\{u_\beta\}\cup {\mathbf{x}}_\beta,0]\,\textrm{ to }\,[{\mathbf{c}}_{\gamma}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,0].$$
\[thm:iso\] For sufficiently large $g$ and sufficient winding, there are quasi-isomorphisms $$\begin{aligned}
\label{eqn:qi1}f_\beta&:{SFC}(\Sigma,\{\beta_1,\dots,\beta_n\},\{\alpha_1,\dots,\alpha_n\})\otimes\Lambda\to{CF^+}(\beta,\alpha|R_\beta;\Gamma_{\eta})\\
\label{eqn:qi2}
f_\gamma&:{SFC}(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\})\otimes\Lambda\to{CF^+}(\gamma,\alpha|R_\gamma;\Gamma_{\eta})\end{aligned}$$ sending ${\mathbf{c}}_\beta\otimes 1$ to $[{\mathbf{c}}_\beta\cup \{u_\beta\}\cup {\mathbf{x}}_\beta,0]$ and ${\mathbf{c}}_\gamma\otimes 1$ to $ [{\mathbf{c}}_\gamma\cup \{u_\gamma\}\cup {\mathbf{x}}_\gamma,0]$, respectively.
We will prove these theorems in the next Subsection. But first, we demonstrate below how they imply Theorem \[thm:main2\] and, hence, our Main Theorem, Theorem \[thm:main\].
Suppose $g$ is sufficiently large and that we have wound sufficiently to guarantee the conclusions of Theorems \[thm:map\] and \[thm:iso\].
Note that $${SFC}(\Sigma,\{\beta_1,\dots,\beta_n\},\{\alpha_1,\dots,\alpha_n\})\otimes\Lambda$$ has rank $1$, generated by ${\mathbf{c}}_\beta\otimes 1$. It then follows from Theorem \[thm:iso\] that $[{\mathbf{c}}_\beta\cup \{u_\beta\}\cup {\mathbf{x}}_\beta,0]$ represents a generator of $${HF^+}(Y_\beta|R_\beta;\Gamma_\eta)= {HF^+}(-Y_S|{-}R;\Gamma_\eta)\cong \Lambda.$$ But this generator is precisely the contact invariant $c_{HF}(\bar\xi_S)$ of a contact closure $(Y_S,R,\bar\xi_S)$ of $(H(S),\xi_S)$, as follows from the discussion in Subsection \[ssec:shm\] and Theorem \[thm:closedequiv\]. That is, $$\label{eqn:chf}c_{HF}(\bar\xi_S)=[[{\mathbf{c}}_\beta\cup \{u_\beta\}\cup {\mathbf{x}}_\beta,0]].$$ Theorem \[thm:map\] implies that the map $$G:={HF^+}(W_{\beta,\gamma}|R_\beta\sim R_\gamma;\Gamma_\nu): {HF^+}(Y_\beta|R_\beta;\Gamma_\eta)\to {HF^+}(Y_\gamma|R_\gamma;\Gamma_\eta)$$ sends $[[{\mathbf{c}}_\beta\cup \{u_\beta\}\cup {\mathbf{x}}_\beta,0]]$ to $[[{\mathbf{c}}_\gamma\cup \{u_\gamma\}\cup {\mathbf{x}}_\gamma,0]]$. On the other hand, the functoriality of the contact invariant under contact $(+1)$-surgeries (Theorem \[thm:contactsurgeryhf\]) implies that this map $G$, now viewed as $$G:={HF^+}({-}W|{-}R;\Gamma_\nu): {HF^+}({-}Y_S|{-}R;\Gamma_\eta)\to {HF^+}({-}Y|{-}R;\Gamma_\eta)$$ sends $c_{HF}(\bar\xi_S)$ to the contact invariant $c_{HF}(\bar\xi)$ of a corresponding contact closure $(Y,R,\bar\xi)$ of $(M,\Gamma,\xi)$. Combined with (\[eqn:chf\]), this means that $$\label{eqn:chf2}c_{HF}(\bar\xi)=[[{\mathbf{c}}_\gamma\cup \{u_\gamma\}\cup {\mathbf{x}}_\gamma,0]].$$ By definition (Subsection \[ssec:sfh\]), we have that $$c_{HF}(\xi)\otimes 1=[{\mathbf{c}}_\gamma\otimes 1]\in H_*({SFC}(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\})\otimes \Lambda)={SFH}(-M,-\Gamma)\otimes\Lambda.$$ Theorem \[thm:iso\], combined with (\[eqn:chf2\]), then implies that the map $$F:=(f_\gamma)_*:{SFH}(-M,-\Gamma)\otimes \Lambda\to {HF^+}(Y_\gamma|R_\gamma;\Gamma_\eta)= {HF^+}(-Y|{-}R;\Gamma_\eta)$$ sends $c_{HF}(\xi)\otimes 1$ to $c_{HF}(\bar \xi)$, as claimed in Theorem \[thm:main2\].
As discussed in Subsection \[ssec:shm\], the equalities in (\[eqn:chf\]) and (\[eqn:chf2\]) hold up to multiplication by units in $\Lambda$, a fact that has no real bearing on our results.
Proofs of Theorems \[thm:map\] and \[thm:iso\]
----------------------------------------------
Below, we prove Theorems \[thm:map\] and \[thm:iso\], which will complete the proof of our Main Theorem, Theorem \[thm:main\]. These proofs occupy the rest of this section.
Our proof of Theorem \[thm:map\] will closely follow that of Proposition \[prop:winding\]. In preparation for the proof, we introduce the following notation. Let $\Delta^i$ be the small triangle with vertices at $$\begin{cases}
\Theta^i, c_\beta^i, c_\gamma^i, & \textrm{for } i=1,\dots,n,\\
\Theta^i, x_\beta^i, x_\gamma^i, & \textrm{for } i=n+1,\dots,n+2g,\\
\Theta^{i}, u_\beta, u_\gamma, & \textrm{for } i=n+2g+1,\\
\end{cases}$$ as shown shaded in Figures \[fig:HH2\], \[fig:TSarcs2\], and \[fig:uvuv2\], and let $$\begin{aligned}
\Delta_\Sigma&=\Delta^{1}+ \dots+ \Delta^{n},\\
\Delta_S& = \Delta^{n+1}+ \dots+ \Delta^{n+2g+1},\\
\Delta_{{\underline}\Sigma} &= \Delta_\Sigma+ \Delta_S.\end{aligned}$$ From Lemma \[lem:bot\] and the discussion following it, the image $$f^+_{\gamma,\beta,\alpha;\Gamma_\nu}([{\mathbf{c}}_{\beta}\cup\{u_\beta\}\cup {\mathbf{x}}_\beta,0])$$ is a linear combination of generators of the form $[{\mathbf{y}}\cup\{s\}\cup {\mathbf{x}}_\gamma,0],$ where $s\in\{u_\gamma,v_\gamma\}$. As we will see, Theorem \[thm:map\] follows more or less immediately from the lemma below.
\[lem:stdtri\] Fix an intersection point $$\label{eqn:y} {\mathbf{y}}\in (\gamma_1\times\dots\times\gamma_n)\cap (\alpha_1\times\dots\times\alpha_n)\in \operatorname{Sym}^n(\Sigma).$$ For sufficiently large $g$ and sufficient winding, the following is true: if $\phi$ is a Whitney triangle $$\label{eqn:whit}\phi\in \pi_2(\Theta, {\mathbf{c}}_{\beta}\cup\{u_\beta\}\cup {\mathbf{x}}_\beta, {\mathbf{y}}\cup\{s\}\cup {\mathbf{x}}_\gamma)$$ with $n_z(\phi)=0$, $\mu(\phi)=0$, and no negative multiplicities, where $s\in \{u_\gamma,v_\gamma\}$, then
- $s=u_\gamma$,
- ${\mathbf{y}}= {\mathbf{c}}_\gamma$, and
- the domain $D(\phi)=\Delta_{{\underline}\Sigma}$.
Fix some ${\mathbf{y}}$ as in (\[eqn:y\]). We will break the proof into two cases, according to whether $s=u_\gamma$ or $s=v_\gamma$.
**. Suppose $s=u_\gamma$. Fix some Whitney triangle in $$\label{eqn:pi2}\pi_2(\Theta, {\mathbf{c}}_{\beta}\cup\{u_\beta\}\cup {\mathbf{x}}_\beta, {\mathbf{y}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma)$$ with domain $D$ satisfying $n_z(D)=0$. Note that the boundary $\partial (D{-}\Delta_{S})$ consists of
- integer multiples of complete $\gamma_i$, $\beta_i$, $\alpha_i$ curves for $i=n{+}1,\dots,n{+}2g{+}1$, and
- integer multiples of arcs of the $\gamma_i$, $\beta_i$, $\alpha_i$ curves for $i=1,\dots,n$.
It follows, from the same sort of reasoning that was applied to the $Q_j$ in the paragraph above equation (\[eqn:basis2\]), that there is a linear combination of the basis elements in (\[eqn:basis\]) with integer coefficients which, when added to $D{-} \Delta_S$, results in a domain $D'$ whose boundary is
- disjoint from $\beta_i$ for $i=n{+}1,\dots, n{+}2g{+}1$,
- disjoint from $\alpha_{i_j}$ and $\gamma_{i_j}$ for $j=p+1,\dots,m$ (recall that $n+1 \leq i_j \leq n+2g$ for such $j$), and
- disjoint from $\alpha_{n+2g+1}$ and $\gamma_{n+2g+1}$.
Note that $D'$ has corners at the components of $\{\Theta^1,\dots,\Theta^n\}$, ${\mathbf{c}}_\beta$, and ${\mathbf{y}}$.
Let $$K=\max\big\{|n_{w_{i}^\pm}(Q_j)| +|n_{w_{i}^\pm}(D')|\mid i=1,\dots,n+2g,\,j=1,\dots,m\big\}.$$ Choose some $$N>K+m(m+1)K^2+mK.$$ We claim that after winding each $\alpha_{i}$ along the $\eta_{i}^\pm$ at least $N$ times, any Whitney triangle $\phi$ satisfying the hypotheses of the lemma must have domain $D(\phi) = {\underline}\Sigma$, which will also imply that ${\mathbf{y}}= {\mathbf{c}}_\gamma$. Indeed, suppose we have performed this winding, and that $$\phi\in \pi_2(\Theta, {\mathbf{c}}_{\beta}\cup\{u_\beta\}\cup {\mathbf{x}}_\beta, {\mathbf{y}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma)$$ satisfies the hypotheses of the lemma. Then the domain $D(\phi)$ differs from $D$ by a periodic domain. We may therefore write $D(\phi)$ as a linear combination $$\label{eqn:lincombo}D(\phi) = D'+ \Delta_S+ a_\gamma P_\gamma + \sum_{i=n+1}^{n+2g+1}a_{i} P_{i} + \sum_{i=1}^p b_iQ_i + \sum_{i=p+1}^m c_iQ_i,$$ with integer coefficients.
Note that $P_\gamma$ has multiplicities $-1$ and $+1$ in the regions $R_1$ and $R_2$, respectively, shown in Figure \[fig:periodic\]. The multiplicity of $D'$ is zero in these regions since $n_z(D')=0$ and its boundary is disjoint from $\alpha_{n+2g+1},\beta_{n+2g+1},\gamma_{n+2g+1}.$ The multiplicities of the $P_i$ and $Q_j$ are also zero in these regions. Since $D(\phi)$ has no negative multiplicities, we must therefore have that $a_\gamma = 0$. Thus, $$\label{eqn:lincomboagain}D(\phi) = D'+ \Delta_S+ \sum_{i=n+1}^{n+2g+1}a_{i} P_{i} + \sum_{i=1}^p b_iQ_i + \sum_{i=p+1}^m c_iQ_i,$$
We next show that all $c_i$ are zero. Suppose not. Suppose $$|b_j| = \max\big\{|b_1|,\dots,|b_p|\big\}\,\textrm{ and }\,
|c_k| = \max\big\{|c_{p+1}|,\dots,|c_m|\big\}.$$ Before winding, we would have had that $$n_{w_{i_k}^\pm}(D(\phi))\,\leq\, K+ |b_j|pK + |c_k|(m-p)K\, \leq\, K+|b_j|mK + |c_k|mK,$$ given (\[eqn:lincomboagain\]). After winding, we therefore have that $$\label{eqn:ineq3}{\underline}n_{w_{i_k}}(D(\phi))\,\leq \,K+|b_j|mK + |c_k|mK-|c_k|N\,<\,K+|b_j|mK -|c_k|K-|c_k|m(m+1)K^2.$$ Our assumption that some $c_i$ is nonzero implies that $|c_k|\geq 1$, which, in combination with the inequality above, implies that $$\label{eqn:ineq4}{\underline}n_{w_{i_k}}(D(\phi))< |b_j|mK -|c_k|m(m+1)K^2.$$ Note that $$\label{eqn:ineq5}{\underline}n_{w_{i_j}}(D(\phi))\,\leq\, K+ |c_k|(m-p)K-|b_j|\,\leq \,K+|c_k|mK-|b_j|$$ both before and after winding. Since ${\underline}n_{w_{i_k}}(D(\phi))$ is nonnegative after winding, we must have that $$|b_j|> |c_k|(m+1)K,$$ which then implies that ${\underline}n_{w_{i_j}}(D(\phi))$ is negative by (\[eqn:ineq5\]), a contradiction. Thus, all $c_i$ are zero. We therefore have that $$\label{eqn:newD}D(\phi) = D'+ \Delta_S+ \sum_{i=n+1}^{n+2g+1}a_{i} P_{i} + \sum_{i=1}^p b_iQ_i.$$
We next argue that the boundary $\partial D'$ is disjoint from $\alpha_i$ for all $i=n{+}1,\dots,n{+}2g$. Suppose instead that some such $\alpha_i$ appears in $\partial D'$ with nonzero integer coefficient $\ell$. Before winding, we would have had that $$n_{w_{i}^\pm}(D(\phi))\,\leq\, K+ |b_j|pK \,\leq\, K+|b_j|mK,$$ given (\[eqn:newD\]). After winding, we therefore have that $$\label{eqn:ineq6}{\underline}n_{w_{i}}(D(\phi))\,\leq \,K+|b_j|mK -|\ell|N\,<\,|b_j|mK -|\ell|m(m+1)K^2.$$ Since ${\underline}n_{w_{i}}(D(\phi))$ is nonnegative after winding, we have that $$|b_j|\,> \,|\ell|(m+1)K\,>\,K,$$ which then implies that ${\underline}n_{w_{i_j}}(D(\phi))$ is negative by (\[eqn:ineq5\]), a contradiction. Thus, $\partial D'$ is disjoint from $\alpha_i$ for all $i=n{+}1,\dots,n{+}2g$.
Since $n_z(D')=0$ and $\partial D'$ is disjoint from the curves $\alpha_i,\beta_i,\gamma_i$, for $i=n{+}1,\dots,n{+}2g{+}1$, we may conclude that $D'$ has multiplicity zero outside of $\Sigma\subset {\underline}\Sigma$. Then each $P_i$ has a negative multiplicity in some region where the multiplicities of $D'$, $\Delta_S$, and $Q_j$ are zero, for all $j=1,\dots,p$. So, if any $a_i$ is nonzero, then $D(\phi)$ has a negative multiplicity $-|a_i|$ in some region, a contradiction. Thus, all $a_i$ are zero. In summary, we have shown that $$D(\phi) = D'+ \Delta_S+ \sum_{i=1}^p b_iQ_i,$$ where $D'$ is supported in $\Sigma\subset{\underline}\Sigma$. This implies that $$D(\phi){-} \Delta_S=D'+ \sum_{i=1}^p b_iQ_i$$ is the domain of a Whitney triangle $$\phi'\in\pi_2(\{\Theta^1,\dots,\Theta^n\},{\mathbf{c}}_\beta,{\mathbf{y}})$$ with respect to the sutured Heegaard triple diagram $$(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\beta_1,\dots,\beta_n\},\{\alpha_1,\dots,\alpha_n\}).$$ Moreover, the domain $D(\phi')$ cannot have any negative multiplicities: if it did, then so would $D(\phi)$ since $\Delta_S$ does not completely cover any region of $D(\phi')$. We claim that in order for $D(\phi')$ to have no negative multiplicities, it must be that $D(\phi') =\Delta_\Sigma.$ To see this, we refer the diagram in Figure \[fig:phiprime\] below which shows the possible multiplicities of $D(\phi')$ near $\Theta^i$ and $c_\beta^i$. Either $y=-x$ or $y=1-x$. In the first case, we must have $x=0$. But that would imply that $x-1=-1$, which would mean that $D(\phi')$ has a negative multiplicity. Therefore, $y=1-x$. But this forces $x=1$. It follows that the domain of $D(\phi')$ near $\Theta^i$ and $c_\beta^i$ consists just of the triangle $\Delta^i$. This implies that $D(\phi') =\Delta_\Sigma$, which then implies that $$D(\phi) = \Delta_\Sigma+ \Delta_S =\Delta_{{\underline}\Sigma},$$ and, hence, that ${\mathbf{y}}= {\mathbf{c}}_\gamma$, as claimed in the lemma.
$\Theta^i$ at 37 156 $c_\beta^i$ at 91 183 $0$ at 72 85 $0$ at 124 87 $0$ at 12 173 $0$ at 170 173 $x$ at 81 140 $y$ at 97 30 $x-1$ at 52 182
![The possible multiplicities of $D(\phi')$ near $\Theta^i$ and $c_\beta^i$.[]{data-label="fig:phiprime"}](phiprime){width="4cm"}
This completes the proof of Lemma \[lem:stdtri\] in the case $s=u_\gamma$.
**. Suppose $s=v_\gamma$. Let $D$, $D'$, and $N$ be as above. We will verify that for sufficiently large $g$ and sufficient winding, there is no Whitney triangle $\phi$ satisfying the hypotheses of the lemma. We break this verification into two subcases.
**. Suppose there does *not* exist a Whitney triangle in $$\pi_2(\{\Theta^1,\dots,\Theta^n\},{\mathbf{c}}_\beta,{\mathbf{y}})$$ with respect to the sutured Heegaard triple diagram $$(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\beta_1,\dots,\beta_n\},\{\alpha_1,\dots,\alpha_n\}).$$ We claim that after winding each $\alpha_{i}$ along the $\eta_{i}^\pm$ at least $N{+}1$ times, the domain of any Whitney triangle $$\phi\in\pi_2(\Theta, {\mathbf{c}}_{\beta}\cup\{u_\beta\}\cup {\mathbf{x}}_\beta, {\mathbf{y}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma)$$ with $n_z(\phi)=0$ has a negative multiplicity, in which case $\phi$ does not satisfy the hypotheses of the lemma, and we are done. Indeed, suppose we have performed this winding, and that $\phi$ is such a triangle. Assume, for a contradiction, that $D(\phi)$ has no negative multiplicities.
$1$ at 225 137 $2$ at 145 225 $R_2$ at 115 194 $u_\gamma$ at 140 260 $v_\gamma$ at 140 185 $z$ at 180 210
![The immersed bigon $C$ with vertices at $u_\gamma$ and $v_\gamma$. We have labeled the multiplicities of the regions forming the bigon. Note that $C$ has multiplicities $2$ and $0$ in the regions $R_1$ and $R_2$, respectively, that were defined in Figure \[fig:periodic\]. Recall that $R_1$ is the small bigon region containing the number $2$ in this figure.[]{data-label="fig:bigon"}](bigon){width="6cm"}
Let $C$ be the domain shown in Figure \[fig:bigon\], representing an immersed bigon in ${\underline}\Sigma$ with vertices at $u_\gamma$ and $v_\gamma$. Then $D{+}C{-}{\underline}\Sigma$ is the domain of a Whitney triangle in $$\pi_2(\Theta, {\mathbf{c}}_{\beta}\cup\{u_\beta\}\cup {\mathbf{x}}_\beta, {\mathbf{y}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma)$$ and satisfies $n_z(D{+}C{-}{\underline}\Sigma)=0$. The domain $D(\phi)$ therefore differs from $D{+}C{-}{\underline}\Sigma$ by a periodic domain. We may then write $D(\phi)$ as a linear combination $$D(\phi) = D'+C-{\underline}\Sigma+ \Delta_S+ a_\gamma P_\gamma + \sum_{i=n+1}^{n+2g+1}a_{i} P_{i} + \sum_{i=1}^p b_iQ_i + \sum_{i=p+1}^m c_iQ_i$$ with integer coefficients.
Note that $D(\phi)$ has multiplicities $1-a_\gamma$ and $-1+a_\gamma$ in the regions $R_1$ and $R_2$, respectively. Since we are assuming that $D(\phi)$ has no negative multiplicities, it must be that $a_\gamma=1$. Our proof of Lemma \[lem:stdtri\] in the case $s=u_\gamma$ above easily extends to show that if some $c_i$ is nonzero then the quantity $$\label{eqn:neg}{\underline}n_{w_j}\bigg(D'+\Delta_S + \sum_{i=n+1}^{n+2g+1}a_{i} P_{i} + \sum_{i=1}^p b_iQ_i + \sum_{i=p+1}^m c_iQ_i\bigg)$$ is less than $-1$ for some $j$, due to the winding. Since $$n_{w_j^\pm}(C-{\underline}\Sigma+ P_\gamma)=1,$$ the quantity ${\underline}n_{w_j}(D(\phi))$ is $1$ more than the quantity in (\[eqn:neg\]). As $D(\phi)$ has no negative multiplicities, it must be that all $c_i=0$. A similar argument as in the case $s=u_\gamma$ then shows that the boundary $\partial D'$ must be disjoint from $\alpha_i$ for all $i=n{+}1,\dots,n{+}2g{+}1$ (and that all $a_i=0$). It follows that $D'$ is supported in $\Sigma\subset{\underline}\Sigma$ and represents a Whitney triangle in $$\pi_2(\{\Theta^1,\dots,\Theta^n\},{\mathbf{c}}_\beta,{\mathbf{y}})$$ with respect to the sutured Heegaard triple diagram $$(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\beta_1,\dots,\beta_n\},\{\alpha_1,\dots,\alpha_n\}).$$ But this yields the desired contradiction since we are assuming that no such triangle exists in this subcase.
**. Suppose there *does* exist a Whitney triangle in $$\pi_2(\{\Theta^1,\dots,\Theta^n\},{\mathbf{c}}_\beta,{\mathbf{y}})$$ with respect to the sutured Heegaard triple diagram $$(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\beta_1,\dots,\beta_n\},\{\alpha_1,\dots,\alpha_n\}).$$ Let $E$ denote its domain. For $j=1,\dots,p$, let $$W_j={\underline}n_{w_{i_j}}(E)+1,$$ and let $$M = \max\bigg\{\mu\bigg(E+C +\Delta_S+ \sum_{i=1}^p b_iQ_i\bigg) \,\,\Big |\, \,|b_1|\leq W_1,\dots,|b_p|\leq W_p\bigg\}.$$ Note that $M$ depends only on data defined in terms of the sutured Heegaard triple diagram. We claim that if $2g>M$ and we have wound each $\alpha_i$ along the $\eta_i^\pm$ at least $N{+}1$ times, then the domain of any Whitney triangle $$\phi\in\pi_2(\Theta, {\mathbf{c}}_{\beta}\cup\{u_\beta\}\cup {\mathbf{x}}_\beta, {\mathbf{y}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma)$$ with $n_z(\phi)=0$ either has a negative multiplicity or else $\mu(\phi)<0$, in either of which cases $\phi$ does not satisfy the hypotheses of the lemma, and we are done. Indeed, suppose $2g>M$ and that we have performed this winding. Let $\phi$ be such a Whitney triangle. As before, we may write $D(\phi)$ as a linear combination $$D(\phi) = D+C-{\underline}\Sigma+ \Delta_S+ a_\gamma P_\gamma + \sum_{i=n+1}^{n+2g+1}a_{i} P_{i} + \sum_{i=1}^p b_iQ_i + \sum_{i=p+1}^m c_iQ_i$$ with integer coefficients. Let us suppose that $D(\phi)$ has no negative multiplicities and show that $\mu(\phi)<0$. The arguments from before show that since $D(\phi)$ has no negative multiplicities, $a_\gamma=1$, all $c_i=0$, and all $a_i=0$, due to the winding. Thus, $$D(\phi) = E+C-{\underline}\Sigma+ \Delta_S+ P_\gamma +\sum_{i=1}^p b_iQ_i.$$ It follows that for each $j=1,\dots,p$, $${\underline}n_{w_{i_j}}(D(\phi)) = {\underline}n_{w_{i_j}}(E)+1-|b_j| = W_j-|b_j|.$$ As we are assuming that $D(\phi)$ has no negative multiplicities, we must have that $|b_j|\leq W_j$ for each $j=1,\dots,p$. Let $$\phi'\in\pi_2(\Theta, {\mathbf{c}}_{\beta}\cup\{u_\beta\}\cup {\mathbf{x}}_\beta, {\mathbf{y}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma)$$ be the Whitney triangle with domain $$D(\phi') = E+C+ \Delta_S +\sum_{i=1}^p b_iQ_i,$$ and let $$\psi\in\pi_2({\mathbf{y}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,{\mathbf{y}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma)$$ be the Whitney disk with domain $$D(\psi)=-{\underline}\Sigma+P_\gamma.$$ Since $$D(\phi)=D(\phi')+D(\psi),$$ the Whitney triangle $\phi$ is the concatenation of $\phi'$ with $\psi$. It follows that $$\begin{aligned}
\mu(\phi) &= \mu(\phi')+\mu(\psi)\\
&\leq M+\mu(\psi)\\
&=M+\mu(-{\underline}\Sigma)+\mu(P_\gamma)\\
&=M-2+e(P_\gamma)+2n_{{\mathbf{y}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma}(P_\gamma)\\
&=M-2+2-2g\\
&=M-2g.
\end{aligned}$$ The formula for $\mu(P_\gamma)$ comes from [@lip] and its calculation is contained in the proof of Lemma \[lem:bot\]. Since $2g>M$, we have that $\mu(\phi)<0$, as desired.
This completes the proof of Lemma \[lem:stdtri\] in the case $s=v_\gamma$, which, in turn, completes the proof of Lemma \[lem:stdtri\].
Since there are only finitely many ${\mathbf{y}}$ as in (\[eqn:y\]), Lemma \[lem:stdtri\] tells us that for sufficiently large $g$ and sufficient winding, $$f^+_{\gamma,\beta,\alpha;\Gamma_\nu}([{\mathbf{c}}_{\beta}\cup\{u_\beta\}\cup {\mathbf{x}}_\beta,0])=\#\mathcal{M}(\phi)\cdot[{\mathbf{c}}_\gamma\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,0]\cdot t^{\partial_\alpha(\phi)\cdot \eta},$$ where $\phi$ is the homotopy class of Whitney triangles with domain $\Delta_{{\underline}\Sigma}$. But this homotopy class has a unique holomorphic representative, and $\Delta_{{\underline}\Sigma}$ is entirely disjoint from $\eta$. Thus, $$\#\mathcal{M}(\phi)\cdot t^{\partial_\alpha(\phi)\cdot \eta}=1,$$ completing the proof of Theorem \[thm:map\].
In the proof of Theorem \[thm:map\], we showed that, for sufficiently large $g$ and sufficient winding, the relevant holomorphic triangle counts for the diagram $({\underline}\Sigma,\gamma,\beta,\alpha,z)$ are the same as the analogous holomorphic triangle counts for the sutured triple diagram $$(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\beta_1,\dots,\beta_n\},\{\alpha_1,\dots,\alpha_n\}).$$ A similar principle is at work in the proof of Theorem \[thm:iso\] below.
The rest of this subsection is dedicated to the proof of Theorem \[thm:iso\].
The maps $f_\beta$ and $f_\gamma$ we have in mind in (\[eqn:qi1\]) and (\[eqn:qi2\]) of Theorem \[thm:iso\] are the $\Lambda$-linear maps $$\begin{aligned}
f_\beta&:{SFC}(\Sigma,\{\beta_1,\dots,\beta_n\},\{\alpha_1,\dots,\alpha_n\})\otimes\Lambda\to{CF^+}(\beta,\alpha|R_\beta;\Gamma_{\eta})\\
f_\gamma&:{SFC}(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\})\otimes\Lambda\to{CF^+}(\gamma,\alpha|R_\gamma;\Gamma_{\eta})\end{aligned}$$ which send a generator ${\mathbf{x}}\otimes 1$ to $ [{\mathbf{x}}\cup \{u_\beta\}\cup {\mathbf{x}}_\beta,0]$ and $ [{\mathbf{x}}\cup \{u_\gamma\}\cup {\mathbf{x}}_\gamma,0]$, respectively. Note that $${SFC}(\Sigma,\{\beta_1,\dots,\beta_n\},\{\alpha_1,\dots,\alpha_n\})\otimes\Lambda$$ has rank $1$, generated by ${\mathbf{c}}_\beta\otimes 1$. Therefore, to show $f_\beta$ is a quasi-isomorphism, it suffices to prove that $$[{\mathbf{c}}_\beta\cup \{u_\beta\}\cup {\mathbf{x}}_\beta,0]$$ is a cycle and generates ${HF^+}(Y_\beta|R_\beta;\Gamma_\eta)$. But this is essentially proven in [@osz1 Section 3]. We will therefore focus on the harder case, involving the map $f_\gamma$. Besides, our proof that $f_\gamma$ is a quasi-isomorphism, given sufficiently large $g$ and sufficient winding, translates into to a proof that $f_\beta$ is a quasi-isomorphism as well.
We will give the proof of Theorem \[thm:iso\] at the very end of this subsection, after proving several important lemmas. The proofs of these lemmas will start to feel somewhat repetitive. We include them for the reader who is interested in going through the details carefully.
To begin with, we need the following.
\[lem:fchainmap\]Sufficiently large $g$ and sufficient winding guarantees that $f_\gamma$ is a chain map.
As we shall see, this follows from the lemma below, which is an analogue of Lemma \[lem:stdtri\].
\[lem:stddisk\] Fix a pair of intersection points, $$\label{eqn:xy} {\mathbf{x}},{\mathbf{y}}\in (\gamma_1\times\dots\times\gamma_n)\cap (\alpha_1\times\dots\times\alpha_n)\in \operatorname{Sym}^n(\Sigma).$$ For sufficiently large $g$ and sufficient winding, the following is true: if $\phi$ is a Whitney disk $$\label{eqn:whitdisk}\phi\in \pi_2({\mathbf{x}}\cup\{s\}\cup {\mathbf{x}}_\gamma, {\mathbf{y}}\cup\{s'\}\cup {\mathbf{x}}_\gamma)$$ with $n_z(\phi)=0$, $\mu(\phi)=1$, and no negative multiplicities, where $s,s'\in \{u_\gamma,v_\gamma\}$, then
- $s=s'$, and
- the domain $D(\phi)$ is supported in $\Sigma\subset {\underline}\Sigma$.
Our proof of Lemma \[lem:stddisk\] will closely follow that of Lemma \[lem:stdtri\]. In preparation, let $$\label{eqn:basisd}\{P_\gamma,S_1,\dots,S_r\}$$ be a basis for the vector space of rational periodic domains for $({\underline}\Sigma,\gamma,\alpha),$ where the $S_j$ satisfy $$\partial S_j =\alpha_{i_j}-\gamma_{i_j} +\sum_{k=1}^{i_j-1} a_{j,k}(\alpha_k - \gamma_k).$$
Let $q$ satisfy $$1\leq i_1<\dots<i_q\leq n,$$$$n+1 \leq i_{q+1}<\dots<i_r\leq n+2g.$$ We may also assume that $a_{j,i_\ell} =0$ for $\ell=1,\dots,r$. Note that this is the doubly-periodic analogue of the condition satisfied by the periodic domains $Q_1,\dots,Q_m$ in (\[eqn:basis2\]). In particular, it follows that the collection $\{S_1,\dots,S_q\}$ is a basis for the space of rational periodic domains for the sutured Heegaard diagram $$(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\}).$$ We will consider this to be our preferred basis. Note, for $j,k=1,\dots,q$ with $k\neq j$, that $$\label{eqn:mult}n_{w_{i_j}^\pm}(S_j) = \pm 1\,\textrm{ and }\,
n_{w_{i_k}^\pm}(S_j) = 0.$$ For $j,k$ in this range, the above quantities are unaffected by winding. On the other hand, for $j=q+1,\dots,r$, the quantity $$n_{w_{i_j}^\pm}(S_j)$$ changes by $\pm 1$ after winding $\alpha_{i_j}$ along $\eta_{i_j}^\pm$ and is unaffected by all other winding.
We are now ready to prove Lemma \[lem:stddisk\].
Fix ${\mathbf{x}},{\mathbf{y}}$ as in (\[eqn:xy\]). We will break this proof into three cases, according to whether $s=s'$ or $(s,s')=(u_\gamma, v_\gamma)$ or $(s,s')=(v_\gamma,u_\gamma)$.
**. Suppose $s=s'$. Fix some Whitney disk in $$\pi_2({\mathbf{x}}\cup\{s\}\cup {\mathbf{x}}_\gamma, {\mathbf{y}}\cup\{s\}\cup {\mathbf{x}}_\gamma)$$ with domain $D$ satisfying $n_z(D)=0$. Note that the boundary $\partial D$ consists of
- integer multiples of complete $\gamma_i$ and $\alpha_i$ curves for $i=n{+}1,\dots,n{+}2g{+}1$, and
- integer multiples of arcs of the $\gamma_i$ and $\alpha_i$ curves for $i=1,\dots,n$.
It follows that there is a linear combination of the basis elements in (\[eqn:basisd\]) with integer coefficients which, when added to $D$, results in a domain $D'$ whose boundary is
- disjoint from $\alpha_{i_j}$ and $\gamma_{i_j}$ for $j=q+1,\dots,r$, and
- disjoint from $\alpha_{n+2g+1}$ and $\gamma_{n+2g+1}$.
We claim that after winding each $\alpha_{i}$ along the $\eta_{i}^\pm$ sufficiently many times, any Whitney disk $\phi$ satisfying the hypotheses of the lemma must have domain $D(\phi)$ supported in $\Sigma\subset {\underline}\Sigma$. Indeed, suppose we have performed a large amount of winding, and that $$\phi\in \pi_2({\mathbf{x}}\cup\{s\}\cup {\mathbf{x}}_\beta, {\mathbf{y}}\cup\{s\}\cup {\mathbf{x}}_\gamma)$$ satisfies the hypotheses of the lemma. Then the domain $D(\phi)$ differs from $D$ by a periodic domain. We may therefore write $D(\phi)$ as a linear combination $$D(\phi) = D'+ a_\gamma P_\gamma + \sum_{i=1}^q b_iS_i + \sum_{i=q+1}^r c_iS_i,$$ with integer coefficients. The exact same kind of argument as was used in the proof of Lemma \[lem:stdtri\], *Case 1: $s=u_\gamma$*, shows that since $D(\phi)$ has no negative multiplicities, it must be that $a_\gamma=0$, all $c_i=0$, and the boundary of $D'$ is disjoint from all $\alpha_i$ for $i=n{+}1,\dots,n{+}2g{+}1$. It follows that $D(\phi)$ is supported in $\Sigma\subset {\underline}\Sigma$, as claimed. This completes the proof in the case $s=s'$.
**. Suppose $s=u_\gamma$ and $s'=v_\gamma$. Let $D$ and $D'$ be as above. We will verify that for sufficiently large $g$ and sufficient winding, there is no Whitney disk satisfying the hypotheses of the lemma. We break this verification into two subcases, mirroring the proof of Lemma \[lem:stdtri\], *Case 2: $s=v_\gamma$*.
**. Suppose there does *not* exist a Whitney disk in $\pi_2({\mathbf{x}},{\mathbf{y}})$ with respect to the sutured Heegaard diagram $$(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\}).$$ We claim that after winding each $\alpha_{i}$ along the $\eta_{i}^\pm$ sufficiently many times, the domain of any Whitney disk $$\phi\in \pi_2({\mathbf{x}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\beta, {\mathbf{y}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma)$$ with $n_z(\phi)=0$ has a negative multiplicity, in which case $\phi$ does not satisfy the hypotheses of the lemma, and we are done. Indeed, suppose we have performed a large amount of winding, and that $\phi$ is such a disk. Let $C$ be the immersed bigon in Figure \[fig:bigon\]. Then we can write $D(\phi)$ as a linear combination $$D(\phi) = D'+C-{\underline}\Sigma+ a_\gamma P_\gamma + \sum_{i=1}^q b_iS_i + \sum_{i=q+1}^r c_iS_i$$ with integer coefficients. The exact same kind of argument as was used in the second case in the proof of Lemma \[lem:stdtri\], *Subcase 2.1: no triangle*, shows that since $D(\phi)$ has no negative multiplicities, it must be that $a_\gamma=1$, all $c_i=0$, and the boundary of $D'$ is disjoint from all $\alpha_i$ for $i=n{+}1,\dots,n{+}2g{+}1$. It follows that $D'$ is supported in $\Sigma\subset{\underline}\Sigma$ and represents a Whitney disk in $\pi_2({\mathbf{x}},{\mathbf{y}})$ with respect to the sutured Heegaard diagram $$(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\}).$$ But this yields the desired contradiction since we are assuming that no such disk exists in this subcase.
**. Suppose there *does* exist a Whitney disk in $\pi_2({\mathbf{x}},{\mathbf{y}})$ with respect to the sutured Heegaard diagram $$(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\}).$$ Let $E$ denote its domain. For $j=1,\dots,q$, let $$W_j={\underline}n_{w_{i_j}}(E)+1,$$ and let $$M = \max\bigg\{\mu\bigg(E+C+ \sum_{i=1}^q b_iS_i\bigg) \,\,\Big |\, \, |b_1|\leq W_1,\dots,|b_q|\leq W_q\bigg\}.$$ Note that $M$ depends only on data defined in terms of the sutured Heegaard diagram. We claim that if $2g>M$ and we have wound each $\alpha_i$ along the $\eta_i^\pm$ sufficiently many times, then the domain of any Whitney disk $$\phi\in \pi_2({\mathbf{x}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\beta, {\mathbf{y}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma)$$ with $n_z(\phi)=0$ either has a negative multiplicity or else $\mu(\phi)<0$, in either of which cases $\phi$ does not satisfy the hypotheses of the lemma, and we are done. From here, our proof proceeds almost exactly as in the Proof of Lemma \[lem:stdtri\], *Subcase 2.2: triangle*, so we will not repeat it.
This completes the proof of Lemma \[lem:stddisk\] in the case $(s,s')=(u_\gamma,v_\gamma)$.
**. Suppose $s=v_\gamma$ and $s'=u_\gamma$. We will verify that there is no Whitney disk satisfying the hypotheses of the lemma. Let $D'$ and $C$ be the domains described above, and suppose $\phi$ is a Whitney disk $$\phi\in\pi_2( {\mathbf{x}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma, {\mathbf{y}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma)$$ with $n_z(\phi)=0$. Then $D(\phi)$ is given by a linear combination $$D(\phi) = D'-C+{\underline}\Sigma+ a_\gamma P_\gamma + \sum_{i=1}^q b_iS_i + \sum_{i=q+1}^r c_iS_i$$ with integer coefficients. Note that the multiplicities of $D(\phi)$ are $-1-a_\gamma$ and $1+a_\gamma$ in the regions $R_1$ and $R_2$, respectively. Since $D(\phi)$ has no negative multiplicities, it must be that $a_\gamma=-1$. However, in that case, $D(\phi)$ has multiplicity $-1$ in the region $R_3$ shown in Figure \[fig:periodic\], a contradiction.
This completes the proof of Lemma \[lem:stddisk\] in the last case $(s,s')=(v_\gamma,u_\gamma)$.
Suppose $g$ is large enough and the winding sufficient for the conclusion of Lemma \[lem:stddisk\] to hold. It suffices to show, for each pair ${\mathbf{x}}, {\mathbf{y}}$ as in (\[eqn:xy\]) that the coefficient of $[{\mathbf{y}}\cup \{s\}\cup {\mathbf{x}}_\gamma,0]$ in $f_\gamma(d {\mathbf{x}}\otimes 1)$ is the same as its coefficient in $\partial f_\gamma({\mathbf{x}}\otimes 1)$, for $s = u_\gamma$ or $v_\gamma$, where $d$ is the differential on $${SFC}(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\})$$ and $\partial$ is the differential on ${CF^+}(\gamma,\alpha|R_\gamma;\Gamma_{\eta})$. Note that both coefficients $$\langle f_\gamma(d {\mathbf{x}}\otimes 1), [{\mathbf{y}}\cup \{s\}\cup {\mathbf{x}}_\gamma,0]\rangle\,\textrm{ and } \langle \partial f_\gamma({\mathbf{x}}\otimes 1), [{\mathbf{y}}\cup \{s\}\cup {\mathbf{x}}_\gamma,0]\rangle$$ are zero if $s=v_\gamma$. This is by definition for the first and by Lemma \[lem:stddisk\] for the second. We therefore only need consider the case $s=u_\gamma$. By definition, we have that $$\begin{aligned}
\label{eqn:c1}\langle f_\gamma(d {\mathbf{x}}\otimes 1), [{\mathbf{y}}\cup \{u_\gamma\}\cup {\mathbf{x}}_\gamma,0]\rangle &= \langle d {\mathbf{x}},{\mathbf{y}}\rangle,\\
\label{eqn:c2}\langle \partial f_\gamma({\mathbf{x}}\otimes 1), [{\mathbf{y}}\cup \{u_\gamma\}\cup {\mathbf{x}}_\gamma,0]\rangle &= \langle \partial([{\mathbf{x}}\cup \{u_\gamma\}\cup {\mathbf{x}}_\gamma,0]), [{\mathbf{y}}\cup \{u_\gamma\}\cup {\mathbf{x}}_\gamma,0]\rangle.\end{aligned}$$ But it follows immediately from Lemma \[lem:stddisk\] that the coefficients on the right hand sides of (\[eqn:c1\]) and (\[eqn:c2\]) are equal: any Whitney disk contributing to the coefficient in (\[eqn:c1\]) contributes the same amount to the coefficient in (\[eqn:c2\]), and Lemma \[lem:stddisk\] tells us that the converse is true (note that any domain contained in $\Sigma$ is disjoint from $\eta$).
We will henceforth assume that $g$ is sufficiently large and that we have wound sufficiently to guarantee that $f_\gamma$ is a chain map.
To show that $f_\gamma$ is a quasi-isomorphism (assuming sufficiently large $g$ and sufficient winding), we will show that it is a filtered chain map for some filtrations on the domain and codomain complexes, and that it induces an isomorphism between $E_1$ pages of the spectral sequences associated to these filtrations. We will first define a filtration on the codomain ${CF^+}(\gamma,\alpha|R_\gamma;\Gamma_{\eta})$ in each ${\text{Spin}^c}$ structure.
Let $A$ and $B$ be the points in ${\underline}R_S$ and ${\underline}R_\Sigma$ shown in Figure \[fig:periodic\]. Given generators $$[{\mathbf{x}}\cup\{s\}\cup {\mathbf{x}}_\gamma, i] \textrm{ and } [{\mathbf{y}}\cup\{s'\}\cup {\mathbf{x}}_\gamma,j] \textrm{ of } {CF^+}(\gamma,\alpha|R_\gamma;\Gamma_{\eta})$$ representing the same ${\text{Spin}^c}$ structure, for ${\mathbf{x}},{\mathbf{y}}$ as in (\[eqn:xy\]) and $s,s'\in\{u_\gamma,v_\gamma\}$, choose a Whitney disk $$\phi\in\pi_2({\mathbf{x}}\cup\{s\}\cup {\mathbf{x}}_\gamma,{\mathbf{y}}\cup\{s'\}\cup {\mathbf{x}}_\gamma)$$ with $n_z(\phi) = i-j$, and define the relative grading $$\label{eqn:relfilt} \mathcal{F}([{\mathbf{x}}\cup\{s\}\cup {\mathbf{x}}_\gamma, i],[{\mathbf{y}}\cup\{s'\}\cup {\mathbf{x}}_\gamma,j]) = 2n_A(\phi)-n_B(\phi).$$ This relative grading is well-defined since $2n_A{-}n_B$ is zero on the periodic domains in (\[eqn:basisd\]). For each ${\text{Spin}^c}$ structure, we choose some lift of this relative grading to an absolute grading, which we also denote by $\mathcal{F}$.
\[lem:filt\] Sufficiently large $g$ and sufficient winding guarantees that the absolute grading $\mathcal{F}$ defines a filtration.
Fix ${\mathbf{x}},{\mathbf{y}}$ as in (\[eqn:xy\]). We must show that for sufficiently large $g$ and sufficient winding, the following is true: if the coefficient $$\label{eqn:coeff}[\langle \partial([{\mathbf{x}}\cup\{s\}\cup {\mathbf{x}}_\gamma, i]),{\mathbf{y}}\cup\{s'\}\cup {\mathbf{x}}_\gamma,j]\rangle$$ is nonzero, then $$\mathcal{F}([{\mathbf{x}}\cup\{s\}\cup {\mathbf{x}}_\gamma, i])\geq\mathcal{F}([{\mathbf{y}}\cup\{s'\}\cup {\mathbf{x}}_\gamma,j]).$$ We will break the proof into three cases, according to whether $s=s'$ or $(s,s')=(u_\gamma, v_\gamma)$ or $(s,s')=(v_\gamma,u_\gamma)$.
**. Suppose $s=s'$. Suppose the coefficient in (\[eqn:coeff\]) is nonzero. Then there is a Whitney disk $$\phi\in\pi_2({\mathbf{x}}\cup\{s\}\cup {\mathbf{x}}_\gamma,{\mathbf{y}}\cup\{s\}\cup {\mathbf{x}}_\gamma)$$ with $\mu(\phi)=1$, with no negative multiplicities, and with $n_z(\phi) = i-j\geq 0$. Borrowing notation from the proof of Lemma \[lem:stddisk\], we can then write the domain $D(\phi)$ as a linear combination $$D(\phi) = D'+ (i-j){\underline}\Sigma + a_\gamma P_\gamma + \sum_{i=1}^q b_iS_i + \sum_{i=q+1}^r c_iS_i$$ with integer coefficients. As the multiplicities of $D'$ at $A$ and $B$ are zero, we have that $$\mathcal{F}([{\mathbf{x}}\cup\{s\}\cup {\mathbf{x}}_\gamma, i])-\mathcal{F}([{\mathbf{y}}\cup\{s\}\cup {\mathbf{x}}_\gamma,j])=2n_A(\phi)-n_B(\phi) = i-j\geq 0,$$ as desired.
**. Suppose $s=u_\gamma$ and $s' = v_\gamma$. The proof of Lemma \[lem:stddisk\] shows that for $g$ large enough and sufficient winding, no Whitney disk $$\phi\in\pi_2({\mathbf{x}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,{\mathbf{y}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma)$$ with domain $$D(\phi) = D'+C-{\underline}\Sigma + a_\gamma P_\gamma + \sum_{i=1}^q b_iS_i + \sum_{i=q+1}^r c_iS_i$$ can have both no negative multiplicities and $\mu(\phi)=1$. Let us assume $g$ is sufficiently large and that we have wound sufficiently so that this is the case. Suppose the coefficient in (\[eqn:coeff\]) is nonzero. Then there is a Whitney disk $$\phi\in\pi_2({\mathbf{x}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,{\mathbf{y}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma)$$ with $\mu(\phi)=1$, with no negative multiplicities, and with $n_z(\phi) = i-j\geq 0$. We can then write $D(\phi)$ as a linear combination $$D(\phi) = D'+C-{\underline}\Sigma + (i-j){\underline}\Sigma+ a_\gamma P_\gamma + \sum_{i=1}^q b_iS_i + \sum_{i=q+1}^r c_iS_i$$ with integer coefficients. Given our assumption on $g$ and the winding, we cannot have $i{-}j= 0$. Thus, $i{-}j\geq 1$, which implies that $$\mathcal{F}([{\mathbf{x}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma, i])-\mathcal{F}([{\mathbf{y}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma,j])=2n_A(\phi)-n_B(\phi) = i-j-1\geq 0,$$ as desired.
**. Suppose $s=v_\gamma$ and $s' = u_\gamma$. Suppose the coefficient in (\[eqn:coeff\]) is nonzero. Then there is a Whitney disk $$\phi\in\pi_2({\mathbf{x}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma,{\mathbf{y}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma)$$ with $\mu(\phi)=1$, with no negative multiplicities, and with $n_z(\phi) = i-j\geq 0$. Then we can write $D(\phi)$ as a linear combination $$D(\phi) = D'-C+{\underline}\Sigma + (i-j){\underline}\Sigma+ a_\gamma P_\gamma + \sum_{i=1}^q b_iS_i + \sum_{i=q+1}^r c_iS_i$$ with integer coefficients. We therefore have that $$\mathcal{F}([{\mathbf{x}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma, i])-\mathcal{F}([{\mathbf{y}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,j])=2n_A(\phi)-n_B(\phi) = i-j+1> 0,$$ as desired.
Let $\partial_0$ denote the component of the differential $\partial$ on ${CF^+}(\gamma,\alpha|R_\gamma;\Gamma_{\eta})$ which preserves the grading $\mathcal{F}$, and let $d$ denote the differential on $${SFC}(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\})\otimes\Lambda,$$ as in the proof of Lemma \[lem:fchainmap\]. We have the following.
\[lem:d0\] Sufficiently large $g$ and sufficient winding guarantees that for each generator $$[{\mathbf{x}}\cup\{s\}\cup {\mathbf{x}}_\gamma,i]\textrm{ of }{CF^+}(\gamma,\alpha|R_\gamma;\Gamma_{\eta}),$$ we have that $$\label{eqn:d0}\partial_0([{\mathbf{x}}\cup\{s\}\cup {\mathbf{x}}_\gamma,i]) =
\begin{cases}
[d({\mathbf{x}})\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,i] + [{\mathbf{x}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma,i-1],&\textrm{ if } s=u_\gamma,\\
[d({\mathbf{x}})\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma,i],&\textrm{ if } s=v_\gamma.
\end{cases}$$
Fix ${\mathbf{x}},{\mathbf{y}}$ as in (\[eqn:xy\]). As usual, we will break the proof into three cases, according to whether $s=s'$ or $(s,s')=(u_\gamma, v_\gamma)$ or $(s,s')=(v_\gamma,u_\gamma)$.
**. Suppose $s=s'$. Suppose the coefficient $$\label{eqn:coeffxy}\langle\partial_0([{\mathbf{x}}\cup\{s\}\cup {\mathbf{x}}_\gamma,i]),[{\mathbf{y}}\cup\{s\}\cup {\mathbf{x}}_\gamma,j]\rangle$$ is nonzero. Then there is a Whitney disk $$\phi\in\pi_2({\mathbf{x}}\cup\{s\}\cup {\mathbf{x}}_\gamma,{\mathbf{y}}\cup\{s'\}\cup {\mathbf{x}}_\gamma)$$ with $\mu(\phi)=1$, no negative multiplicities, $n_z(\phi) = i-j\geq 0$, and $$2n_A(\phi)-n_B(\phi) = i-j=0,$$ the latter of which follows from the proof of Lemma \[lem:filt\], *Case 1: $s=s'$*. The proof of Lemma \[lem:stddisk\], *Case 1: $s=s'$*, then shows that for sufficient winding, the domain $D(\phi)$ must be supported in $\Sigma$. It follows that $$\label{eqn:coeffequality}\langle\partial_0([{\mathbf{x}}\cup\{s\}\cup {\mathbf{x}}_\gamma,i]),[{\mathbf{y}}\cup\{s\}\cup {\mathbf{x}}_\gamma,j]\rangle = \begin{cases}
\langle d({\mathbf{x}}),{\mathbf{y}}\rangle, & \textrm{if } j=i,\\
0,& \textrm{ otherwise.}
\end{cases}$$ for any $s\in\{u_\gamma,v_\gamma\}$, as claimed in Lemma \[lem:d0\].
**. Suppose $s=u_\gamma$ and $s'=v_\gamma$. Suppose the coefficient $$\label{eqn:coeffxy}\langle\partial_0([{\mathbf{x}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,i]),[{\mathbf{y}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma,j]\rangle$$ is nonzero. Then there is a Whitney disk $$\phi\in\pi_2({\mathbf{x}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,{\mathbf{y}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma)$$ with $\mu(\phi)=1$, no negative multiplicities, $n_z(\phi) = i-j\geq 0$, and $$2n_A(\phi)-n_B(\phi) = i-j-1=0,$$ the latter of which follows from the proof of Lemma \[lem:filt\], *Case 2: $(s,s')=(u_\gamma,v_\gamma)$*, assuming we have wound sufficiently and $g$ is large enough. As in that lemma, this implies that $$D(\phi) = D'+C+ a_\gamma P_\gamma + \sum_{i=1}^q b_iS_i + \sum_{i=q+1}^r c_iS_i.$$ Note that $D(\phi)$ then has multiplicities $2-a_\gamma$ and $a_\gamma$ in the regions $R_1$ and $R_2$, respectively. It follows that $a_\gamma \in \{0,1,2\}$. For $a_\gamma =1$ or $2$, the same sort of argument as was used in the proof of Lemma \[lem:stddisk\], *Case 2: $(s,s')=(u_\gamma,v_\gamma)$*, shows that for sufficiently large $g$ and sufficient winding, any such $D(\phi)$ must either have a negative multiplicity or $\mu(\phi)<0$. So, we must have, for sufficiently large $g$ and sufficient winding, that $a_\gamma=0$. Therefore, $$D(\phi) = D'+C + \sum_{i=1}^q b_iS_i + \sum_{i=q+1}^r c_iS_i.$$ But in this case, another argument virtually identical to that in the proof of Lemma \[lem:stddisk\], *Case 2: $(s,s')=(u_\gamma,v_\gamma)$*, shows that for sufficient winding, any such $D(\phi)$ has a negative multiplicity unless all $c_i=0$ and $\partial D'$ is disjoint from $\alpha_i$ for $i=n{+}1,\dots,n{+}2g{+}1$. Thus, $$D(\phi) = D'+C+ \sum_{i=1}^q b_iS_i.$$ The condition on $\partial D'$ implies that $$D' +\sum_{i=1}^q b_iS_i\subset\Sigma,$$ so that this domain is disjoint from $C$. In order for $\mu(\phi)=1$, we must have $$\mu\big(D' +\sum_{i=1}^q b_iS_i\big)=0$$ since the bigon $C$ contributes $1$ to Maslov index. Since holomorphic disks of Maslov index zero are constant, it follows that $$D' +\sum_{i=1}^q b_iS_i=0,$$ so that $D(\phi)=C$. Then $\phi$ has a unique holomorphic representative and ${\mathbf{y}}={\mathbf{x}}$. We may therefore conclude that for sufficiently large $g$ and sufficient winding, $$\label{eqn:coeffequality2}\langle\partial_0([{\mathbf{x}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,i]),[{\mathbf{y}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma,j]\rangle = \begin{cases}
1, & \textrm{if } {\mathbf{y}}={\mathbf{x}}\textrm{ and } j=i-1,\\
0,& \textrm{ otherwise,}
\end{cases}$$ as claimed in Lemma \[lem:d0\].
**. Suppose $s=v_\gamma$ and $s'=u_\gamma$. Suppose the coefficient $$\label{eqn:coeffxy}\langle\partial_0([{\mathbf{x}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma,i]),[{\mathbf{y}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,j]\rangle$$ is nonzero. Then there is a Whitney disk $$\phi\in\pi_2({\mathbf{x}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma,{\mathbf{y}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma)$$ with $\mu(\phi)=1$, no negative multiplicities, $n_z(\phi) = i-j\geq 0$, and $$2n_A(\phi)-n_B(\phi) = i-j+1=0,$$ the latter of which follows from the proof of Lemma \[lem:filt\], *Case 3: $(s,s')=(v_\gamma,u_\gamma)$*. But these two conditions on $i{-}j$ contradict one another. Therefore, $$\label{eqn:coeffequality3}\langle\partial_0([{\mathbf{x}}\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma,i]),[{\mathbf{y}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,j]\rangle = 0$$ for all $i,j$ and ${\mathbf{x}},{\mathbf{y}}$. Putting the formulae (\[eqn:coeffequality\]), (\[eqn:coeffequality2\]), and (\[eqn:coeffequality3\]) together completes the proof of Lemma \[lem:d0\].
Suppose now that $g$ is large enough and that we have wound sufficiently for the conclusions of Lemmas \[lem:fchainmap\], \[lem:stddisk\], \[lem:filt\], and \[lem:d0\] to hold. Note that the above filtration on ${CF^+}(\gamma,\alpha|R_\gamma;\Gamma_{\eta})$ defines a filtration on $${SFC}(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\})\otimes\Lambda$$ by simply declaring the filtration grading of a generator ${\mathbf{x}}$ to be equal to that of $[{\mathbf{x}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,0]$. In particular, $d=d_0$, where $d_0$ is the component of $d$ which preserves the filtration grading on the sutured Floer complex, and $f_\gamma$ is a filtered chain map. The $E_1$ page of the spectral sequence associated to the filtration on this sutured Floer complex is therefore simply the homology $$H_*({SFC}(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\})\otimes\Lambda, d_0) = {SFH}(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\})\otimes\Lambda.$$ We claim the following.
\[lem:qi\] The map between $E_1$ pages induced by $f_\gamma$, $$E_1(f_\gamma): {SFH}(\Sigma,\{\gamma_1,\dots,\gamma_n\},\{\alpha_1,\dots,\alpha_n\})\otimes\Lambda\to H_*({CF^+}(\gamma,\alpha|R_\gamma;\Gamma_{\eta}), \partial_0),$$ is an isomorphism.
We claim that every generator of the homology $$H_*({CF^+}(\gamma,\alpha|R_\gamma;\Gamma_{\eta}), \partial_0)$$ is represented by a linear combination of generators of the form $[{\mathbf{x}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,0]$. To see how the lemma follows from this claim, suppose it is true and recall that $E_1(f_\gamma)$ is induced by the map which sends a generator ${\mathbf{x}}$ to $[{\mathbf{x}}\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,0]$. In particular, this map sends a linear combination $$\label{eqn:linc1}{\mathbf{x}}_1\otimes r_1+\dots + {\mathbf{x}}_k\otimes r_k,$$ where the $r_i\in\Lambda$, to the linear combination $$\label{eqn:linc2}[{\mathbf{x}}_1\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,0]r_1+\dots + [{\mathbf{x}}_k\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,0]r_k.$$ It follows easily from Lemma \[lem:d0\] that the sum in (\[eqn:linc1\]) is a cycle (resp. boundary) with respect to $d=d_0$ iff the sum in (\[eqn:linc2\]) is a cycle (resp. boundary) with respect to $\partial_0$. This implies that $E_1(f_\gamma)$ is an isomorphism.
It remains to prove the claim. Given a linear combination $$\label{eqn:linc3}w={\mathbf{x}}_1\otimes r_1+\dots + {\mathbf{x}}_k\otimes r_k,$$ as in (\[eqn:linc1\]), let us use the following notation $$\begin{aligned}
[w,u_\gamma,i]&:=[{\mathbf{x}}_1\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,i]r_1+\dots + [{\mathbf{x}}_k\cup\{u_\gamma\}\cup {\mathbf{x}}_\gamma,i]r_k,\\
[w,v_\gamma,i]&:=[{\mathbf{x}}_1\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma,i]r_1+\dots + [{\mathbf{x}}_k\cup\{v_\gamma\}\cup {\mathbf{x}}_\gamma,i]r_k.\end{aligned}$$ Now suppose $$c = [w_i,u_\gamma,i] + [z_i,v_\gamma,i] + \dots + [w_0,u_\gamma,0] + [z_0,v_\gamma,0]$$ is a cycle with respect to $\partial_0$, where the $w_j$ and $v_j$ are linear combinations as in (\[eqn:linc3\]). For the claim, it suffices to show that there is some $$b\in {CF^+}(\gamma,\alpha|R_\gamma;\Gamma_{\eta})$$ such that $$\label{eqn:db}\partial_0b + c=[w_0,u_\gamma,0].$$ Applying the formula for $\partial_0$ in (\[eqn:d0\]), one easily sees that the fact that $c$ is a cycle implies that $$\begin{aligned}
dz_i&=0,\\
dz_{i-1}&=w_i,\\
dz_{i-2}&=w_{i-1},\\
\vdots&\\
dz_{0} &= w_1.\end{aligned}$$ It therefore follows, after another application of (\[eqn:d0\]), that $$b=[z_i,u_\gamma,i+1]+ \dots + [z_0,u_\gamma,1]$$ satisfies (\[eqn:db\]). This completes the proof of the lemma.
The fact that $f_\gamma$ is a quasi-isomorphism follows immediately, because a filtered chain map between filtered chain complexes which induces an isomorphism between the $E_1$ pages of the associated spectral sequences induces an isomorphism on homology, assuming that the filtrations are bounded from below, which they clearly are in this case (see, e.g., the proof of [@oszbook Proposition A.6.1]).
The proof of our Main Theorem, Theorem \[thm:main\] is now complete.
Obstructing Lagrangian concordance {#sec:concex}
==================================
In this section, we provide further examples which demonstrate the effectiveness of the invariant $\Theta_{HF}$ in obstructing Lagrangian concordance. Our main result is the following.
\[thm:concordance-examples\] There are infinitely many pairs $(K_1,K_2)$ of Legendrian knots in $(S^3,\xi_{std})$ such that:
- $K_1$ and $K_2$ are smoothly concordant and have the same classical invariants $tb$ and $r$,
- $K_1$ is a negative stabilization of another Legendrian knot,
- $K_2$ is neither a positive nor negative stabilization of another Legendrian knot, and
- $\Theta_{HF}(K_1) \neq 0$ while $\Theta_{HF}(K_2) = 0$.
In particular, the last condition implies by Theorem \[thm:grid-concordance\] that there is no Lagrangian concordance from $K_1$ to $K_2$.
The fact that $K_1$ in Theorem \[thm:concordance-examples\] is a stabilization implies that its Legendrian contact homology DGA is trivial. Legendrian contact homology therefore fails to obstruct a Lagrangian concordance from $K_1$ to $K_2$ for these examples.
In our proof of Theorem \[thm:concordance-examples\] below, we will adopt the convention that the $(p,q)$-cable of a knot $K$, denoted by $C_{p,q}(K)$, has longitudinal winding $p$ and meridional winding $q$. We will also, for notational convenience, we will denote the $(r,s)$-cable of $C_{p,q}(K)$ simply by $$C_{p,q;r,s}(K):=C_{r,s}(C_{p,q}(K)).$$ Given a Legendrian knot $K$, we will use $K^+$ to denote its transverse pushoff which satisfies $$sl(K^+) = tb(K)-r(K).$$ Finally, given a knot $K$ we will denote by ${\overline{tb}}(K)$ and ${\overline{sl}}(K)$ the maximal Thurston-Bennequin and self-linking numbers among Legendrian and transverse knots smoothly isotopic to $K$.
In our examples, $K_1$ will be a Legendrian representative of the iterated torus knot $C_{3,2;3,2}(U)$ and $K_2$ will be a Legendrian representative of $$C_{3,2;3,2}(P(-m,-3,3)) \# 6_1,$$ where $P(-m,-3,3)$ is the usual pretzel knot, for any $m \geq 3$. Such $K_1$ and $K_2$ are smoothly concordant as these pretzel knots and the twist knot $6_1$ are all smoothly slice.
To define the Legendrian representative $K_1$, we rely on the following result of Ng, Ozsv[á]{}th, and Thurston from [@not Section 3.3]. The Legendrian $K$ in the proposition below was first discovered and studied by Etnyre and Honda in [@EH4].
\[prop:cable-trefoil-example\] There is a Legendrian representative $K$ of the iterated torus knot $C_{3,2;3,2}(U)$ with $(tb(K), r(K)) = (5,2)$ and $\Theta_{HF}(K) \neq 0$.
We then define $K_1$ to be the Legendrian knot obtained by negatively stabilizing this knot $K$ three times. It follows that $$(tb(K_1),r(K_1))=(2,-1)\textrm{ and }\Theta_{HF}(K_1) \neq 0$$ since $\Theta_{HF}$ is preserved by negative stabilization.
We now record some facts that will be relevant in defining $K_2$. First, we record that ${\overline{sl}}(C_{3,2;3,2}(U)) = 7$, where this maximal self-linking number is realized by the transverse pushoff of the unique Legendrian representative with $(tb,r)=(6,-1)$; see [@EH4] for the full Legendrian classification of $C_{3,2;3,2}(U)$. Since $g(C_{3,2;3,2}(U)) = 4$, the three inequalities $${\overline{sl}}(C_{3,2;3,2}(U)) \leq 2\tau(C_{3,2;3,2}(U))-1 \leq 2g_s(C_{3,2;3,2}(U))-1 \leq 2g(C_{3,2;3,2}(U))-1$$ are actually equalities. Here, $\tau$ is the Ozsváth–Szabó concordance invariant [@osz10] and $g_s$ is the smooth slice genus; see [@pla4] for the first inequality and [@osz10] for the second. In particular, we have that $\tau(C_{3,2;3,2}(U))=4$. We now use these facts to prove the following.
\[lem:tb-iterated-cable\] Suppose $K$ is a smoothly slice knot with ${\overline{tb}}(K)=-1$. Then there is a Legendrian representative of $C_{3,2;3,2}(K)$ with $(tb,r)=(6,-1)$. This Legendrian knot achieves the bound ${\overline{tb}}(C_{3,2;3,2}(K)) = 6$, and its transverse pushoff achieves the bound ${\overline{sl}}(C_{3,2;3,2}(K)) = 7$.
According to [@lidsiv Corollary 1.17], we have ${\overline{tb}}(C_{3,2}(K)) \geq 1$. As $C_{3,2}(K)$ is smoothly concordant to the right-handed trefoil $C_{3,2}(U)$, we have $$\tau(C_{3,2}(K)) = \tau(C_{3,2}(U)) = 1,$$ which implies that $${\overline{tb}}(C_{3,2}(K)) \leq 2\tau(C_{3,2}(K))-1 = 1$$ by [@pla4]. It follows that ${\overline{tb}}(C_{3,2}(K)) = 1$. Applying [@lidsiv Corollary 1.17] once more, we may then conclude that ${\overline{tb}}(C_{3,2;3,2}(K)) = 6$, as claimed in the lemma. Since $C_{3,2;3,2}(K)$ is smoothly concordant to $C_{3,2;3,2}(U)$, we have that $$\tau(C_{3,2;3,2}(K)) = \tau(C_{3,2;3,2}(U))=4,$$ and, therefore, that $${\overline{sl}}(C_{3,2;3,2}(K)) \leq 2\tau(C_{3,2;3,2}(K))-1 = 7.$$ This bound is achieved by the transverse pushoff of a $tb$-maximizing Legendrian representative: since $tb=6$ is even for this representative its rotation number must be odd, and up to reversing orientation we can ensure that $r\leq -1$; hence, the transverse pushoff has $sl=tb-r\geq7$, which implies that this inequality is actually an equality.
There are infinitely many knots $K$ satisfying the hypothesis of Lemma \[lem:tb-iterated-cable\]. These include the examples of [@cns Theorem 2.10] and the pretzel knots $P(-m,-3,3)$, for $m \geq 3$, as mentioned in [@cns Section 4.4]. Fix any such $K$ and let $L_1$ be a Legendrian representative of $C_{3,2;3,2}(K)$ with $tb(L_1) = 6$ and $r(L_1) = -1$, whose existence is guaranteed by Lemma \[lem:tb-iterated-cable\].
Etnyre, Ng, and Vértesi [@env] classified Legendrian and transverse representatives of the $6_1$ knot, which in their notation is the twist knot $K_4$. Namely, there is a single $tb$-maximizing Legendrian representative $L_2$ with $(tb,r)=(-5,0)$, and all other representatives are stabilizations of $L_2$, so it follows that ${\overline{tb}}(6_1) = -5$ and ${\overline{sl}}(6_1) = -5$.
We now define the Legendrian representative $K_2$ of $C_{3,2;3,2}(K)\#6_1$ to be the connected sum $K_2=L_1\#L_2.$ We show below that $K_2$ satisfies the conditions in Theorem \[thm:concordance-examples\].
\[prop:construct-k-vanishing-grid\] The Legendrian knot $K_2$ has the same classical invariants as $K_1$, it is not a stabilization, and $\Theta_{HF}(K_2) = 0$.
Both $L_1$ and $L_2$ maximize $tb$ within their knot types, so we have $$tb(K_2) = tb(L_1)+tb(L_2)+1 = {\overline{tb}}(L_1)+{\overline{tb}}(L_2)+1 = {\overline{tb}}(K_2)$$ by Lemma 3.3 and Corollary 3.5 of [@EH5]. More precisely, we compute that $tb(K_2) = 2$ and $$r(K_2) = r(L_1)+r(L_2) = -1.$$ This shows that $K_2$ has the same classical invariants as $K_1$. The fact that $K_2$ is a $tb$-maximizer also implies that it is not a stabilization. In order to show that $\Theta_{HF}(K_2)=0$, we appeal to a result of Vértesi [@vera Corollary 1.3], which says that there is an isomorphism $${\widehat{{HFK}}}(m(L_1)) \otimes {\widehat{{HFK}}}(m(L_2)) \to {\widehat{{HFK}}}(m(L_1\#L_2))$$ sending $$\Theta_{HF}(L_1) \otimes \Theta_{HF}(L_2)\textrm{ to }\Theta_{HF}(L_1 \# L_2) = \Theta_{HF}(K_2).$$ It therefore suffices to show that $\Theta_{HF}(L_2) = 0$. But $L_2$ represents an alternating knot type, so it has thin knot Floer homology [@osz9]. Therefore, by [@not Proposition 3.4], we have $\Theta_{HF}(L_2) \neq 0$ if and only if $sl(L_2^+) = 2\tau(L_2) - 1$. The left side is $-5$, but the right side is $-1$ since $L_2$ is smoothly slice, so we have that $\Theta_{HF}(L_2)=0$, as desired.
This completes the proof of Theorem \[thm:concordance-examples\].
[^1]: See also the work of Kutluhan, Lee, and Taubes [@klt1; @klt2; @klt3; @klt4; @klt5].
[^2]: In fact, our invariant can be made to take values in the “natural" refinement of ${\underline{{SHM}}}$ defined in [@bs3].
[^3]: In [@km4] and [@bs3], these sutured Floer homology modules are defined in terms of the *top* ${\text{Spin}^c}$ structures with respect to $R$; this is not an important difference as the two different choices result in isomorphic modules.
[^4]: This theorem was originally stated with coefficients in ${\mathbb{Z}/2\mathbb{Z}}$, but the proof works in the setting of local coefficients just as well.
[^5]: As in the monopole Floer case, this theorem was originally stated with coefficients in ${\mathbb{Z}/2\mathbb{Z}}$, but the proof works in the setting of local coefficients just as well.
[^6]: As in the introduction, we are using $-R$ to refer to the distinguished surfaces in both $-Y_S$ and $-Y$; when more specificity is desired, we use $R_\beta$ and $R_\gamma$.
|
---
abstract: 'The buildup of low energy electrons in an accelerator, known as electron cloud, can be severely detrimental to machine performance. Under certain beam conditions, the beam can become resonant with the cloud dynamics, accelerating the buildup of electrons. This paper will examine two such effects: multipacting resonances, in which the cloud development time is resonant with the bunch spacing, and cyclotron resonances, in which the cyclotron period of electrons in a magnetic field is a multiple of bunch spacing. Both resonances have been studied directly in dipole fields using retarding field analyzers installed in the Cornell Electron Storage Ring (CESR). These measurements are supported by both analytical models and computer simulations.'
author:
- 'J. R. Calvey[^1]'
- 'W. Hartung[^2]'
- 'J. Makita[^3]'
- 'M. Venturini'
bibliography:
- 'rfa\_field\_prst.bib'
title: Beam Induced Electron Cloud Resonances in Dipole Magnetic Fields
---
\[sec:intro\] Introduction
==========================
As a part of the program at Cornell [@1748-0221-10-07-P07012], the Cornell Electron Storage Ring (CESR) was instrumented with several retarding field analyzers (RFAs) [@NIMA453:507to513], to study the buildup of low energy electrons in an accelerator vacuum chamber. This effect, known as electron cloud [@ECLOUD12:Miguel; @doi:10.1142/S0217751X14300233], has been observed in a number of machines [@PRSTAB7:024402; @PRSTAB14:071001; @NIMA556:399to409; @PRSTAB6:034402; @PAC09:WE4GRC02; @PhysRevSTAB.6.014203; @PRSTAB16:011003], and is known to cause emittance growth and beam instabilities [@PhysRevSTAB.7.124801]. It is especially dangerous for low emittance, positively charged beams, and is expected to be a limiting factor in next generation positron and proton storage rings, such as the International Linear Collider damping ring [@ILCREP2007:001; @PRSTAB17:031002].
In lepton machines, electron cloud is usually seeded by photoelectrons generated by synchrotron radiation. The collision of these electrons with the beam pipe can then produce one or more secondary electrons, depending on the secondary electron yield (SEY) of the material. The SEY depends on the energy and angle of the incident electron [@PRSTAB5:124404], with peak secondary production occurring at $E_{max} \approx 300$ eV. If the average SEY is greater than unity, the cloud density will grow exponentially, until a saturation is reached. Most secondary electrons are generated with low energy ($<$ 10 eV), but can be given additional energy by the beam. As we will show in this paper, an unfortunate choice of beam parameters (particulary bunch spacing and charge) can drive up the average electron energy up into a regime of high secondary production (near $E_{max}$), resulting in a higher cloud density.
Retarding field analyzers provide information on the local electron cloud density, energy, and transverse distributions. Previous papers have described the use of RFAs at to directly compare different electron cloud mitigation techniques [@NIMA760:86to97; @NIMA770:141to154]. In addition, computer simulations have been compared to RFA measurements, to quantify the electron emission properties of different cloud mitigating coatings in field free regions [@PRSTAB17:061001]. Simulations of cloud dynamics in dipole and wiggler fields have been presented in conference proceedings [@PAC09:FR5RFP043; @IPAC10:TUPD022; @IPAC11:MOPS083; @IPAC12:WEPPR088]. This paper will summarize and expand on these results. In particular, multipacting and cyclotron resonances will be examined in detail. These effects, in which resonant interactions between the beam and electrons lead to accelerated cloud development, should be avoided to ensure optimal machine performance.
Retarding Field Analyzers
-------------------------
A retarding field analyzer consists of three main components [@NIMA453:507to513]: holes drilled in the beam pipe to allow electrons to enter the device; a retarding grid, to which a voltage can be applied, rejecting electrons with less than a certain energy; and a positively biased collector, to capture any electrons which make it past the grid. If space permits, additional (grounded) grids can be added to produce a more ideal retarding field. In addition, the collectors of most RFAs used in are segmented to allow characterization of the spatial structure of the cloud build-up. Thus a single RFA measurement provides information on the local cloud density, energy, and transverse distribution. Some of the data presented here are voltage scans, in which the retarding voltage is varied (typically from +100 to $-250$ V or $-400$ V) while beam conditions are held constant. In other measurements, where we want to study the detector response as a function of some external parameter (e.g. bunch spacing), the retarding grid was biased at +50 V, to capture all incoming electrons. The collector was set to +100 V for all of our measurements.
An example voltage scan is given in Fig. \[fig:chic\_dipole\_meas\]. The RFA response is plotted as a function of collector number and retarding voltage. Roughly speaking, this is a description of the transverse and energy distribution of the cloud. Collector 1 is closest to the outside of the chamber (where direct synchrotron radiation hits). The signal is strongly peaked in the central collector (no. 9), which is aligned with the horizontal position of the beam. The sign convention for retarding voltage is chosen so that a positive value on this axis corresponds to a negative physical voltage on the grid (and thus a rejection of lower energy electrons). The beam conditions are given as “1x45x1.25 mA e$^+$, 14 ns, 5.3 GeV." This notation indicates one train of 45 positron bunches, with a per-bunch current of 1.25 mA (1 mA = $1.6\times10^{10}$ particles), with 14 ns bunch spacing, and a beam energy of 5.3 GeV.
![\[fig:chic\_dipole\_meas\] Dipole RFA voltage scan: 1x45x1.25 mA $e^+$, 14 ns, 5.3 GeV, 810 gauss field. The central collector is no. 9.](run2983_slac4_pub_notitle3.pdf "fig:"){width=".6\textwidth"}\
Electron Cloud in Dipoles
-------------------------
In the presence of a dipole magnetic field, an electron will undergo helical motion, spiralling around the field lines. For a standard dipole magnet in an accelerator (with strength $\sim$ 1 kilogauss), a typical cloud electron (with energy $\sim$ 10 - 100 eV) will have a cyclotron radius on the order of a few hundred $\mu$m. In other words, the motion of the electron will be approximately one dimensional, along the direction of the dipole field. This pinning of the motion to the field lines often results in a strong concentration of the cloud in the center of the chamber, where beam kicks are strongest. Stronger beam kicks drive the average electron energy up, which typically results in a higher average SEY (since most secondary electrons are emitted with $E_{sec} \ll E_{max}$). This effect is seen clearly in Fig. \[fig:chic\_dipole\_meas\]. In addition, multipacting and cyclotron resonances, described below, can appear in dipole fields.
### \[ssec:mult\] Multipacting Resonances
A multipacting resonance occurs when a characteristic time for the cloud development is equal to the bunch spacing. As originally proposed by Gröbner [@HEACC77:GROBNER], this happens when the kick from the beam gives secondary electrons near the vacuum chamber wall just enough energy to reach the opposite wall in time for the next bunch. These electrons generate more secondaries, which are again given energy by the beam. This process continues, resulting in a resonant buildup of the cloud. The resonant condition is given by Eq. (\[eq:grob\]).
$$\label{eq:grob}
t_b = \frac{b^2}{c r_e N_b}$$
Here $t_b$ is the bunch spacing, $b$ is the chamber half-height, $c$ is the speed of light, $r_e$ is the classical electron radius, and $N_b$ is the bunch population. A more general condition was derived by Harkay et al. [@PAC03:RPPG002; @PRSTAB6:034402], which includes nonzero secondary emission velocity. In Section \[ssec:mult\_res\], we develop an even more general model of multipacting resonances, which includes the possibility of multiple beam kicks.
### \[ssec:cyc\] Cyclotron Resonances
A cyclotron resonance occurs when the bunch spacing is an integral multiple of the cyclotron period of an electron in a dipole field [@PRSTAB11:091002]. Under these conditions, the transverse beam kick to a given electron will always be in the same direction, resulting in a steady increase in the particle’s energy, and (usually) a higher secondary electron yield when it hits the vacuum chamber wall. The resonant condition is given in Eq. (\[eq:cyc\]), where $m_e$ is the electron mass, $q_e$ is the electron charge, $n$ is an integer, and $B$ is the magnetic field strength.
$$\label{eq:cyc}
t_b = \frac{2 \pi m_e n}{q_e B}$$
Cyclotron resonances were observed at SLAC using a chicane of four dipole magnets instrumented with RFAs [@NIMA621:33to38]. Unexpectedly, the resonances sometimes appeared as peaks in the signal, and other times as dips. This chicane was moved to CESR early in the program. In Section \[ssec:cyc\_res\], we confirm the existence of cyclotron resonances, and in Section \[ssec:cyc\_res\_sim\], we provide an explanation for the peak/dip phenomenon.
\[sec:instrumentation\] Instrumentation
=======================================
Detailed descriptions of the electron cloud experimental program, design of the field region RFAs, and data acquisition system can be found elsewhere [@NIMA770:141to154; @CLNS:12:2084]; here we provide only a brief summary. RFAs in each field region had to be specially designed to fit inside the narrow magnet apertures. The key parameters of each RFA type are listed in Table \[tab:dipole\_rfa\_styles\].
RFA Chamber type Field Strength Grids Collectors Grid trans.
---------------- ---------------- ------------------ ------- ------------ ------------- --
CESR dipole Elliptical Al 0.079 - 0.2010 T 1 9 38%
Chicane dipole Circular Al 0 - 0.12 T 3 17 92%
Wiggler Rectangular Cu 1.9 T 1 12 38/92%
: \[tab:dipole\_rfa\_styles\] List of dipole/wiggler RFA locations. The elliptical and rectangular chambers are 9 cm in width by 5 cm in height. The circular chamber is 4.5 cm in radius. “Grid trans." refers to the optical transparency of the grids. Note that the wiggler RFAs used two generations of grids with different transparencies.
#### CESR Dipole RFA
To study cloud buildup in a realistic dipole field environment, a thin RFA was installed inside a CESR dipole magnet. The magnetic field in this magnet depends on the beam energy: 790 gauss at 2.1 GeV, 1520 gauss at 4 GeV, and 2010 gauss at 5.3 GeV. The chamber is made of uncoated (6063) aluminum.
#### Chicane RFAs
A chicane of four dipole magnets designed at SLAC [@NIMA621:33to38] was installed in the L3 straight. The field of these magnets can be varied over the range of 0 to 1.46 kilogauss, which allowed for the study of the effect of dipole field strength on cloud dynamics, without affecting the trajectory of stored beams in the rest of the ring. Three of the chicane dipole chambers tested different electron cloud mitigation techniques: two of the chambers were TiN coated [@NIMA551:187to199], and one was both grooved [@NIMA571:588to598; @PAC07:THPMN118] and TiN coated (the fourth was bare aluminum).
#### Wiggler RFAs
During the reconfiguration in 2008, six superconducting wigglers were installed in the L0 straight section of CESR. They were typically operated with a peak transverse field of 1.9 T. Three of these wigglers were instrumented with RFAs, at three different locations in the wiggler field: in the center of the wiggler pole (effectively a 1.9 T dipole field), half way between two poles (where the field is longitudinal), and in an intermediate region [@NIMA770:141to154]. This paper will focus on the pole center RFAs.
The first generation wiggler RFAs were equipped with low-transparency stainless steel grids. However, as described in Section \[ssec:tramp\], secondary emission from these grids lead to a significant interaction between the electron cloud and the RFA, complicating the interpretation of the measurements. Consequently, in the second generation of wiggler chambers, the grids were changed to high-transparency copper meshes. The use of high transparency grids effectively solved the grid emission problem.
Measurements and Analytical Models
==================================
Many measurements have been taken in CESR with RFAs in dipole fields, under a wide variety of different beam conditions. This has allowed for detailed studies of electron cloud dynamics, in particular of multipacting and cyclotron resonances.
\[ssec:mult\_res\] Multipacting Resonances
------------------------------------------
To study the time evolution of the electron cloud, we collected RFA data with bunch spacings varying from 4 ns to 112 ns. All of the data presented in this section were taken with a single train of 20 bunches, at beam energy 5.3 GeV. Fig. \[fig:dipole\_spacing\_chic\] shows the signal in the central collector of the chicane RFA as a function of bunch spacing, for different bunch currents, and for both electron and positron beams. A few interesting features are readily apparent in the data. Except at the lowest current value, both the electron and positron beam data show a peak at 56 ns. The positron data has another peak, which moves to lower bunch spacings at higher currents. These data are not consistent with a simple multipacting resonance (Eq. (\[eq:grob\])), which would account for only one resonance in the positron measurement, and none in the electron measurement. Additionally, the beam kicks at the wall are very small for this case (amounting to 13 eV for a 3.5 mA beam), and so are unlikely to drive electrons at the wall into a regime of high secondary production.
A similar set of data for the CESR dipole RFA is shown in Fig. \[fig:dipole\_spacing\_cesr\]. In this case, both the electron and positron beam data contain a single peak that moves to lower spacings as the current increases. The positron data peaks occur at much lower spacings that the electron peaks.
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![\[fig:dipole\_spacing\_chic\] Central collector signal in the chicane dipole RFA (set to 810 gauss) as a function of bunch spacing, at different bunch currents. Top: positron beam; bottom: electron beam. Note that the signals have been normalized to be on the same scale. In absolute terms, the peak positron signal was about five times the peak electron signal.](multipacting_3curs_slac4_pos_pub4.pdf "fig:"){width="60.00000%"}
![\[fig:dipole\_spacing\_chic\] Central collector signal in the chicane dipole RFA (set to 810 gauss) as a function of bunch spacing, at different bunch currents. Top: positron beam; bottom: electron beam. Note that the signals have been normalized to be on the same scale. In absolute terms, the peak positron signal was about five times the peak electron signal.](multipacting_3curs_slac4_elec_pub4.pdf "fig:"){width="60.00000%"}
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![\[fig:dipole\_spacing\_cesr\] Central collector signal in the CESR dipole RFA as a function of bunch spacing, at different bunch currents. Top: positron beam; bottom: electron beam. Note that the signals have been normalized to be on the same scale. In absolute terms, the peak positron signal was about four times the peak electron signal.](multipacting_3curs_cesr_pos_pub4.pdf "fig:"){width="60.00000%"}
![\[fig:dipole\_spacing\_cesr\] Central collector signal in the CESR dipole RFA as a function of bunch spacing, at different bunch currents. Top: positron beam; bottom: electron beam. Note that the signals have been normalized to be on the same scale. In absolute terms, the peak positron signal was about four times the peak electron signal.](multipacting_4curs_cesr_elec_pub5.pdf "fig:"){width="60.00000%"}
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### Analytical Model
These resonances can be explained if we allow the secondary electrons to be generated with some (small) energy. If the time for a typical secondary electron to travel to the center of the beam pipe is equal to the bunch spacing, this electron will be kicked strongly by the beam, and is likely to produce more secondary electrons [@PRSTAB6:034402]. If we ignore the time for the kicked electron to travel to the beam pipe wall, the resonance condition is given by Eq. (\[eq:tb1\_res\]), where $t_b$ is the bunch spacing, $b$ is the chamber half-height (i.e. the distance from the wall to the beam), and $v_{sec}$ is a characteristic secondary electron velocity.
$$\label{eq:tb1_res}
t_b = b / v_{sec}$$
For a (plausible [@PRSTAB5:124404]) secondary emission energy of 1.5 eV, this peak will occur at 61 ns for the chicane dipole case ($b = 4.5$ cm). Because aluminum has a high SEY for a broad range of incident energies, we expect the resonance to be somewhat broad. The fact that there is a finite width to the secondary energy distribution will further smear out the peak. Because this model does not distinguish between electron and positron beams, we expect this peak to be in the same location for both species. This is indeed what we observe in the measured data.
For the CESR dipole RFA ($b = 2.5$ cm), the resonance should occur at 34 ns, which does not agree with either the electron or positron data. In order to derive a more accurate prediction, we need to take into account the time it takes for a kicked electron to reach the chamber wall. We define the resonant condition as the bunch spacing that results in an electron energy $E_2 = E_{max}$, where $E_{max}$ is the energy corresponding to peak secondary production (in eV). This process is diagrammed in Fig. \[fig:tb\_diagram\].
The resonant condition now becomes:
$$\begin{aligned}
\begin{split}
t_b & = \frac{b - r}{v_{sec}} + \frac{b \pm r}{v_{max}} \\
v_{max} & \equiv \sqrt{\frac{2 q_e E_{max}}{m_e}} = \frac{2 c N_b r_e}{r} \pm v_{sec}
\label{eq:tb1_res2}
\end{split}\end{aligned}$$
Here $r$ is the distance from the electron to the beam during the bunch passage, $N_b$ is the bunch population and $r_e$ is the classical electron radius. Where there is a $\pm$ symbol, the plus sign applies for positron beams, and the minus for electron beams.
Eliminating $r$ from Eq. (\[eq:tb1\_res2\]) and defining $k \equiv 2 c N_b r_e$ gives us a resonant bunch spacing (Eq. (\[eq:tb1\_res3\])). Interestingly, the condition is still the same for electron and positron beams.
$$t_{b,1} = \frac{b (v_{max} + v_{sec}) - k}{v_{max} v_{sec}}
\label{eq:tb1_res3}$$
In this analysis we have used the impulse approximation for determining the beam kick [@CERN:LHC97; @doi:10.1142/S0217751X14300233], which assumes that $r$ is much greater than the beam size. This approximation is valid as long as the distance from the electron to the beam is greater than a critical radius $r_c \approx 2 \sqrt{N_b r_e \sigma_z \sqrt{2/\pi}}$, where $\sigma_z$ is the bunch length. For the conditions presented here, $\sigma_z \approx$ 17 mm, so the critical radius is 1.6 mm at 1 mA, and 2.9 mm at 3.4 mA. For the resonant condition in Eq. (\[eq:tb1\_res2\]), $r \approx$ 2.8 mm at 1 mA, and 9.6 mm at 3.4 mA. So the impulse approximation is always valid, although it’s close at low current.
![\[fig:tb\_diagram\] Diagram of single bunch multipacting resonances: positron beams (top) and electron beams (bottom). A secondary electron is released from the bottom wall (left), travels upward at speed $v_{sec}$, receives a kick from a passing bunch (middle), and hits the wall, releasing another secondary electron at time $t = t_b$ (right).](multipacting_diagram_tb1_new.pdf){width="90.00000%"}
The 14 ns peak in the positron data is due to a higher order multipacting resonance, where it takes two bunches to set up the resonance condition. Here we consider the case where the first bunch gives some additional energy to the electron, so that it arrives near the center of the chamber in time for the second bunch, when it receives a large enough kick to give it energy $E_{max}$. This process is shown in Fig. \[fig:tb2\_diagram\].
![\[fig:tb2\_diagram\] Diagram of a two-bunch multipacting resonance. From left to right: a secondary electron is released from the bottom wall with speed $v_{sec}$. It receives a kick from a passing bunch, and continues with higher velocity ($v_2$). It is kicked again by a second bunch, bringing its speed up to $v_{max}$. Finally, it hits the wall, releasing another secondary electron at time $t = 2 t_b$.](multipacting_diagram_tb2_new.pdf){width="90.00000%"}
From this picture we can derive a system of equations for $t_{b,2}$ (where the subscript 2 is used to signify a 2-bunch resonance):
$$\begin{aligned}
\begin{split}
2 t_{b,2} &= \frac{b - r_1}{v_{sec}} + \frac{r_1 - r_2}{v_2} + \frac{r_2 + b}{v_{max}} \\
t_{b,2} &= \frac{r_1 - r_2}{v_2} \\
v_2 &= v_{sec} + \frac{k}{r_1} \\
v_{max} &= v_2 + \frac{k}{r_2}
\label{eq:tb2_res}
\end{split}\end{aligned}$$
Here $r_1$ is the distance between the beam and the electron during the first bunch passage, $r_2$ is this distance during the second bunch passage, and $v_2$ is the electron velocity after the first beam kick . Note that this condition only applies to positron beams, since the kicks must be towards the beam. These equations are a bit too unwieldily to be solved analytically, but they can be solved numerically to give predictions for the resonant bunch spacings.
### Comparison with Measured Data
Fig. \[fig:mp\_all\] compares the measured and predicted resonances for both the chicane and CESR dipole chambers. Effectively, we have varied the two most important parameters of the model: bunch current, and chamber size (since the two dipole RFAs have different chamber heights). Overall there is good agreement between the data and model for all measured resonances. In particular, the model captures the major features of the data:
- For the chicane RFA, the 1-bunch resonance appears in both the electron and positron data, at the same bunch spacing.
- The 2-bunch resonances are only observed in the positron data, at lower spacing than the 1-bunch resonances.
- All resonances move toward lower bunch spacing at higher current.
The 1-bunch resonance is not seen as clearly in the CESR dipole RFA positron data, though a “shelf" can be seen at 1.4 mA, which does correspond to the electron data peak. Simulations (Section \[ssec:mult\_sim\]) also predict a peak. The lack of a clear resonance in the data may be a result of the depletion phenomena described in a previous paper ([@NIMA770:141to154], Sec. 3.1.2). Essentially, in a strong field (such as the 2 kilogauss field of the CESR dipole RFA), the RFA can actually become less sensitive to multipacting, since it depletes the cloud under the RFA holes, exactly where it’s measuring. In general the 1-bunch resonances are less pronounced than the 2-bunch resonances; this may be why we still see the 2-bunch resonance.
The model and data are also in quantitative agreement, with two exceptions: the 1-bunch resonance for the chicane dipole at low current, and the 1-bunch resonance for the CESR dipole at high current. The former discrepancy may be due to the impulse approximation not being valid (as explained above). The latter discrepancy may be due to the fact that we are ignoring the beam’s image charge, and the cloud’s space charge. The chicane RFA chamber is in a circular chamber, so there will be no image charge (assuming a centrally located beam). It is also located in a long straight section that receives relatively little synchrotron radiation. This means the overall cloud density is lower, and space charge is less important. The CESR dipole chamber, however, is (approximately) elliptical, so image charge can be important. It is also located in a high radiation environment. An improved model, which takes image charge and space charge into account, would probably fit this data better.
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![\[fig:mp\_all\] Comparison of measured and predicted multipacting resonances for the chicane (top) and CESR dipole (bottom) RFAs. The solid lines represent 1-bunch resonances (Eq. (\[eq:tb1\_res3\])), the dashed lines 2-bunch resonances (Eq. (\[eq:tb2\_res\])), and the points are measured data. In both cases the n=1 points are taken from the electron beam data, and the n=2 from the positron data. The error bars are defined as half the difference in bunch spacing between successive measurements.](multipacting_all4_slac2.pdf "fig:"){width="60.00000%"}
![\[fig:mp\_all\] Comparison of measured and predicted multipacting resonances for the chicane (top) and CESR dipole (bottom) RFAs. The solid lines represent 1-bunch resonances (Eq. (\[eq:tb1\_res3\])), the dashed lines 2-bunch resonances (Eq. (\[eq:tb2\_res\])), and the points are measured data. In both cases the n=1 points are taken from the electron beam data, and the n=2 from the positron data. The error bars are defined as half the difference in bunch spacing between successive measurements.](multipacting_all4_cesr2.pdf "fig:"){width="60.00000%"}
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Measurements of multipacting resonances with a positron beam at the Advanced Photon Source [@PRSTAB6:034402] found a peak at 20 ns for bunch populations in the range of $3.45\times10^{10}$ to $5.75\times10^{10}$. Plugging these numbers and the chamber half-height (21 mm) into Eq. (\[eq:tb1\_res3\]) gives a resonant spacing of 18-23 ns, consistent with their result. However, they measured a different resonance (30 ns) for an electron beam, which is not predicted by our theory. Their measurements were made in a field-free region, with an RFA located at an angle with respect to the top of the chamber, so our one-dimensional model may not be completely valid. Nonetheless it is suggestive that the location of the positron peak agrees with our prediction.
Table \[tab:res\_other\] lists the predicted locations of multipacting resonances for some proposed accelerators with positively charged beams. Also included for comparison are the two most common operating modes of the APS (which now uses electron beams, so there is no 2-bunch resonance). The LHC is not included, because the beam is so intense that $E_{2} > E_{max}$ at the beam pipe wall, so the machine will generate high energy secondaries regardless of bunch spacing.
Machine $N_b$ b (cm) $t_b$ (ns) $t_{b,1}$ (ns) $t_{b,2}$ (ns)
------------ -------------------- -------- ------------ ---------------- ----------------
ILC DR $2\times10^{10}$ 2.5 6 32 7.6
CLIC DR $4.1\times10^{9}$ 3 0.5 43 17.4
SuperKEKB $9\times10^{10}$ 4.5 4 46 5.4
APS (324b) $7.1\times10^{9}$ 2.1 11 29 X
APS (24b) $9.5\times10^{10}$ 2.1 153 9 X
: Resonant bunch spacings ($t_{b,1}$, $t_{b,2}$) compared to operational spacing ($t_b$) for different accelerators.[]{data-label="tab:res_other"}
It is worth noting that running with very short bunch spacing (as many cutting edge accelerators do) can actually be advantageous from an electron cloud point of view, since it avoids both multipacting resonances. Running with high current and very large bunch spacing (as some light sources do) also works. However, it is important to keep in mind that this model does not include the cloud’s space charge, which could be an important effect in these high intensity machines. Particle tracking simulations (see Section \[ssec:mult\_sim\]) can be used to more accurately predict the resonances.
\[ssec:cyc\_res\] Cyclotron Resonances
--------------------------------------
By varying the strength of the chicane magnets, we can also study the behavior of the cloud at different dipole magnetic field values. Fig. \[fig:chicane\_scan\] shows RFA data taken as a function of magnetic field strength, at two different bunch spacings. The most prominent feature of the data is regularly occurring spikes or dips, which are seen in all cases. These correspond to “cyclotron resonances," which occur whenever the cyclotron period of cloud electrons is an integral multiple of the bunch spacing (see Section \[ssec:cyc\]). For 4 ns bunch spacing we expect them every 89 gauss; and for 12 ns spacing, every 30 gauss. This is exactly what is seen in the data. Another interesting feature of this measurement is that these resonances appear as peaks in the RFA signal in the aluminum chamber, but as dips in the coated chambers. This difference in the behavior of the two chamber materials is explained in Section \[ssec:cyc\_res\_sim\].
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![\[fig:chicane\_scan\] RFA signal as a function of chicane magnetic field: 1x45x1 mA $e^+$, 5 GeV. Top: 4 ns spacing. Bottom: 12 ns spacing. Cyclotron resonances are observed every 89 gauss with 4 ns spacing, and every 30 gauss with 12 ns spacing, as predicted by Equation (\[eq:cyc\]). Note that the aluminum chamber signal is divided by 20.](chicane_scan_1418-eps-converted-to.pdf "fig:"){width="60.00000%"}
![\[fig:chicane\_scan\] RFA signal as a function of chicane magnetic field: 1x45x1 mA $e^+$, 5 GeV. Top: 4 ns spacing. Bottom: 12 ns spacing. Cyclotron resonances are observed every 89 gauss with 4 ns spacing, and every 30 gauss with 12 ns spacing, as predicted by Equation (\[eq:cyc\]). Note that the aluminum chamber signal is divided by 20.](chicscan_1445_pub.pdf "fig:"){width="60.00000%"}
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\[ssec:tramp\] Anomalous Enhancement
------------------------------------
Detailed analysis of the wiggler RFA data is complicated by an interaction between the cloud and the RFA itself. Fig. \[fig:tramp\_example\] shows a voltage scan done with an RFA in the center pole of a wiggler (approximated by a 1.9 T dipole field). Here one can see a clear enhancement in the signal at low (but nonzero) retarding voltage. Since the RFA should simply be collecting all electrons with an energy more than the magnitude of the retarding voltage, the signal should be a monotonically decreasing function of the voltage. So the RFA is not behaving simply as a passive monitor. A similar effect has been observed in a strong dipole field at KEKB [@NIMA598:372to378]. The spike in collector current is accompanied by a corresponding dip in the grid current, suggesting that the grid is the source of the extra collector current.
![\[fig:tramp\_example\] Resonant enhancement in wiggler data, 45 bunches, 1.25 mA/bunch, $e^+$, 2.1 GeV, 14 ns. Note that there are 12 collectors, so collector 6 is one of the central ones.](wig1w_tramp_2585_pub4.pdf){width="60.00000%"}
This spurious signal comes from a resonance between the bunch spacing and retarding voltage. To understand this, consider an electron which collides with the retarding grid and generates a secondary. Because electrons are so strongly pinned to the magnetic field lines in a 1.9 T field, this electron is likely to escape through the same beam pipe hole through which it entered. An electron ejected from the grid will gain energy from the retarding field before it re-enters the vacuum chamber. If it is given the right amount of energy, it will be near the center of the vacuum chamber during the next bunch passage, and get a large beam kick, putting it in a position to generate even more secondaries. This process, which we have dubbed the “trampoline effect", is essentially an artificial multipacting resonance. If we take Eq. (\[eq:tb1\_res\]) from Section \[ssec:mult\_res\], and use the retarding voltage in place of the secondary electron energy, the resonance conditions becomes:
$$\label{eq:tramp}
V_{ret} = \frac{m_e b^2}{2 q_e t_b^2}$$
Here $V_{ret}$ is the retarding voltage, $b$ is the chamber half-height, $t_b$ is the bunch spacing, $m_e$ is the electron mass, and $q_e$ is the electron charge. Fig. \[fig:tramp\_spacing\] plots a series of retarding voltage scans done with a wiggler RFA, for 4, 8, 12, and 20 ns bunch spacing. The trampoline effect is seen in all cases, with the spike occurring at $\sim$110, 30, 15, and 10 V, respectively. Meanwhile, the simple model given in Eq. (\[eq:tramp\]) predicts 111, 28, 12, and 4 V, respectively. The predictions are quite close to the measurements, especially for short bunch spacing. The second spike at low voltage in the 4 ns data corresponds to a two-bunch resonance, also described in Section \[ssec:mult\_res\].
![\[fig:tramp\_spacing\] Resonant spike location at different bunch spacings, 1x45x1.25 mA $e^+$, 5 GeV. Only the signal in the central collector is plotted.](ipac_tramp_combined6.pdf){width="60.00000%"}
Simulations
===========
While the analytical models described above are generally successful at explaining our data, additional insight can be gained by using more detailed computer simulations. The results presented here were obtained with the particle tracking code POSINST [@MBI97:170; @LHC:ProjRep:180; @PRSTAB5:124404]. In POSINST, a simulated photoelectron is generated on the chamber surface and tracked under the action of the beam. Secondary electrons are generated via a probabilistic process. Space charge and image charge are also included in the simulation.
\[sec:modeling\] RFA Modeling
-----------------------------
In order to accurately predict the RFA signal, a sophisticated model of the detector must be incorporated into the code. Our model has been described in detail for the RFAs installed in field free regions [@PRSTAB17:061001]; the dipole RFA models are essentially the same. In short, when a macroparticle in the simulation collides with the vacuum chamber wall in the region covered by the RFA, a special function is called which calculates a simulated RFA signal based on the particle’s incident energy and angle. The signal is binned by energy and transverse position, reproducing the energy and position resolution of the RFA.
Fig. \[fig:dipole\_rfa\_eff\] shows the efficiency (fraction of the macroparticle’s charge that contributes to the RFA signal) as a function of incident angle in the chicane RFA. This represents the probability that an incoming electron will make it through the beam pipe hole and grids, and to the collector. Note that low energy particles have a very high efficiency, due to their small cyclotron radius.
![\[fig:dipole\_rfa\_eff\] Simulated RFA efficiency vs. incident angle for the chicane dipole RFA, with a 810 gauss magnetic field.[]{data-label="dipoleff"}](trans_ang_custom_slac_a.pdf "fig:"){width=".6\linewidth"}\
Using the model described above, we ran simulations for the dipole RFAs, for various beam conditions. Fig. \[fig:sim\_example\] shows a typical example, for the aluminum chicane RFA. Overall, the agreement with data (Fig. \[fig:chic\_dipole\_meas\]) is reasonable, without any additional tuning of the simulation parameters.
![\[fig:sim\_example\] Example aluminum chicane RFA simulation: 1x45x1.25 mA $e^+$, 14 ns, 5.3 GeV. Compare to Fig. \[fig:chic\_dipole\_meas\].](slac4_paramtest_c23_pub2.pdf){width=".6\linewidth"}
\[ssec:mult\_sim\] Simulation of Multipacting Resonances
--------------------------------------------------------
Because the simulation contains all the relevant features of our multipacting model (i.e. secondary emission, beam kicks, chamber geometry), it should be able to reproduce the resonances predicted by the model. In addition, we are able to vary the secondary emission energy, to study the effect this has on the resonant spacings. According to Eq. (\[eq:tb1\_res\]), the 1-bunch resonance should have an approximately inverse dependence on the emission velocity, i.e. $t_{b,1} \sim 1/\sqrt{E_{sec}}$. The 2-bunch resonance should have a much weaker dependence on emission energy.
Fig. \[fig:dip\_spacing\_sim\] plots the simulated central collector signal as a function of bunch spacing, for four different combinations of chamber, bunch current, and beam species. Both the 1-bunch and 2-bunch multipacting peaks are observed. As predicted by the model, the locations of these peaks (especially for the 1-bunch resonance) are sensitive to the energy spectrum of emitted secondary electrons. A secondary emission energy distribution peaked at 1.5 eV is generally consistent with the data, in particular with the locations of the multipacting peaks. Lowering the emission energy to 0.75 eV moves the peaks to higher bunch spacings, and broadens the peaks. Increasing the energy to 3 eV moves the peaks to lower spacings, and also results in narrower peaks. Neither of these cases are consistent with the measured data. Thus this comparison provides a fairly sensitive indirect measurement of the secondary emission energy.
In general, the data, analytical model, and simulation are in good agreement, assuming 1.5 eV secondary electrons. It is notable that the simulation agrees well with the high current electron beam data in the CESR chamber (which the analytical model did not match well). This is most likely because the simulation includes space and image charge, which are important in the high current regime.
For the sake of simplicity, the angular distribution of emitted secondaries was set to be strongly peaked at normal to the vacuum chamber wall (POSINST parameter `pangsec` [@PRSTAB5:124404] was set to 10). This was done to make it easy to compare the location of resonances to those predicted by the model. In reality the electrons should be emitted at various angles, which would complicate the analysis, but may give a qualitatively better fit to the data. Studying the effect of `pangsec` and other simulation parameters on these results would be an interesting subject for future study.
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\[ssec:cyc\_res\_sim\] Simulation of Cyclotron Resonances
---------------------------------------------------------
Under the conditions of a cyclotron resonance, we expect to see a increase in the RFA signal, due to the increased energy of the cloud electrons. As discussed in Section \[ssec:cyc\_res\], we do indeed observe peaks in the RFA current in the aluminum chicane chamber, but in the TiN-coated chambers we observe dips. Fig. \[fig:cyc\_sim\] shows a simulated magnetic field scan over a cyclotron resonance, in both an aluminum and TiN-coated chamber. Consistent with the data, we observe an increase in the aluminum chamber signal, but a decrease in the TiN chamber signal. Fig. \[fig:cyc\_eff\] provides an explanation: since the additional energy in the resonant electrons comes from transverse beam kicks, these electrons will have a larger cyclotron radius, and thus a lower RFA efficiency (see Fig. \[fig:dipole\_rfa\_eff\]). Thus there are two competing effects: an increased cloud density due to a higher average SEY, and lower overall detector sensitivity. In the aluminum chamber (where the peak SEY is high) the former effect dominates, while in the coated chamber (where the peak SEY is low) the latter one does. The net result is resonant peaks in the uncoated chamber, and dips in the coated one.
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{width=".45\linewidth"} {width=".45\linewidth"}
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![\[fig:cyc\_eff\] Effect of cyclotron resonance on RFA efficiency, 1x45x1 mA $e^+$, 4 ns, 5 GeV. Under the resonant field, the average electron cyclotron radius increases, resulting in a decrease in the average RFA efficiency.](cyclotron_efficiency_pub3.pdf "fig:"){width=".6\linewidth"}\
Simulation of Anomalous Enhancement in the Wiggler RFA
------------------------------------------------------
The main disadvantage of treating the RFA analytically (as described in Section \[sec:modeling\]) is that we cannot self-consistently model any interaction between the detector and the cloud, such as the trampoline effect described in Section \[ssec:tramp\]. Motivated by these measurements, we have incorporated into POSINST a model of the RFA geared toward reproducing the geometry of the RFAs installed in the wiggler vacuum chambers. The motion of the electrons within the RFA, including the electrostatic force from the retarding field, is tracked using a special add-on routine. The grid is modeled realistically, and secondary electrons can be produced there, with the same secondary yield model used for normal vacuum chamber collisions. The peak secondary electron yield and peak yield energy can be specified separately for the grid. Because the actual retarding field is included in the wiggler RFA model, the retarding voltage must be specified in the input file, and a separate simulation must be run for each voltage.
Fig \[fig:int\_model\] shows the result of running this full particle tracking simulation, for the set of beam conditions corresponding to Fig. \[fig:tramp\_example\]. Notably, the simulation reproduces the resonant enhancement seen in the data, at approximately the same voltage ($\sim$10 V for 14 ns spacing), and shows that the extra signal comes from the grid.
![\[fig:int\_model\] POSINST simulation showing resonant enhancement in a wiggler RFA, 1x45x1.2 mA $e^+$, 2.1 GeV, 14 ns, central collector. Compare to Fig. \[fig:tramp\_example\].](tramp_examp_newpub.pdf){width="60.00000%"}
Conclusions
===========
Electron cloud buildup has been investigated in dipole field regions throughout CESR. Measurements of multipacting and cyclotron resonances have been made at different bunch spacings, bunch currents, and with electron and positron beams.
A sophisticated analytical model for multipacting resonances has been developed, which takes into account secondary emission energy, as well as the time for kicked electrons to reach the chamber wall. This model is generally consistent with data, and has been further validated by computer simulations. An anomalous enhancement in the center-pole wiggler RFA signal has also been identified as an artificial multipacting resonance.
Cyclotron resonances have been observed in the chicane RFAs, at field values that correspond well to basic theory. The question of these resonances sometimes appearing as dips, rather than peaks in the signal, has been explained as a detector efficiency effect.
The electron cloud density is very sensitive to multipacting effects. On resonance, we observe as much as a factor of 3 increase in electron cloud signal for positron beams, and several orders of magnitude for electron beams (though the measured signal for electron beams was always lower than for positrons). Because electron cloud is a potential limiting factor for high current, low emittance beams, avoiding these resonances is crucial for achieving emittance and stability goals in present and future accelerators.
This research was supported by NSF and DOE Contracts No. PHY-0734867, No. PHY-1002467, No. PHYS-1068662, No. DE-FC02-08ER41538, No. DE-SC0006505, and the Japan/U.S. Cooperation Program.
The authors would like to thank D. Rubin, G. Dugan, J.A. Crittenden, J. Sikora, J. Livesey, M. Palmer, and K. Harkay for their helpful advice and suggestions; R. Schwartz, S. Santos, and S. Roy for assisting with the RFA measurements; and M. Furman at LBNL for his support with the POSINST simulation code.
[^1]: Present address: Advanced Photon Source, Argonne National Laboratory, Argonne, IL
[^2]: Present address: Facility for Rare Isotope Beams, Michigan State University, East Lansing MI
[^3]: Present address: Department of Physics, Old Dominion University, Norfolk, VA
|
---
abstract: 'The link between a particular class of growth processes and random matrices was established in the now famous 1999 article of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation [@BDJ99]. During the past ten years, this connection has been worked out in detail and led to an improved understanding of the large scale properties of one-dimensional growth models. The reader will find a commented list of references at the end. Our objective is to provide an introduction highlighting random matrices. From the outset it should be emphasized that this connection is fragile. Only certain aspects, and only for specific models, the growth process can be reexpressed in terms of partition functions also appearing in random matrix theory.'
author:
- 'Patrik L. Ferrari[^1] and Herbert Spohn[^2]'
date: '13.3.2010'
title: Random Growth Models
---
Growth models {#sect1}
=============
A growth model is a stochastic evolution for a height function $h(x,t)$, $x$ space, $t$ time. For the one-dimensional models considered here, either $x\in \mathbb{R}$ or $ x\in\mathbb{Z}$. We first define the TASEP[^3] (totally asymmetric simple exclusion process) with parallel updating, for which $h,x,t\in\mathbb{Z}$. An admissible height function, $h$, has to satisfy $h(x+1)-h(x)=\pm 1$. Given $h(x,t)$ and $x_\ast$ a local minimum of $h(x,t)$, one defines $$\label{2.1}
h(x_\ast, t+1)=
\left\{
\begin{array}{ll}
h(x_\ast,t)+2 & \hbox{with probability } 1-q,\; 0\leq q\leq 1\,, \\
h(x_\ast,t) & \hbox{with probability } q
\end{array}
\right.$$ independently for all local minima, and $h(x,t+1)=h(x,t)$ otherwise, see Figure \[FigGrowth\] (left). Note that if $h(\cdot,t)$ is admissible, so is $h(\cdot,t+1)$.
There are two limiting cases of interest. At $x_\ast$ the waiting time for an increase by $2$ has the geometric distribution $(1-q)
q^n$, $n=0,1,\ldots$. Taking the $q\to 1$ limit and setting the time unit to $1-q$ one obtains the exponential distribution of mean $1$. This is the time-continuous TASEP for which, counting from the moment of the first appearance, the heights at local minima are increased independently by $2$ after an exponentially distributed waiting time. Thus $h,x\in\mathbb{Z}$ and $t\in\mathbb{R}$.
The second case is the limit of rare events (Poisson points), where the unit of space and time is $\sqrt{q}$ and one takes $q\to 0$. Then the lattice spacing and time become continuous. This limit, after a slightly different height representation (see Section \[sect5\] for more insights), results in the polynuclear growth model (PNG) for which $h\in\mathbb{Z}$, $x,t\in\mathbb{R}$. An admissible height function is piecewise constant with jump size $\pm 1$, where an increase by $1$ is called an up-step and a decrease by $1$ a down-step. The dynamics is constructed from a space-time Poisson process of intensity $2$ of nucleation events. $h(x,t)$ evolves deterministically through (1) up-steps move to the left with velocity $-1$, down-steps move to the right with velocity $+1$, (2) steps disappear upon coalescence, and (3) at points of the space-time Poisson process the height is increased by $1$, thereby nucleating an adjacent pair of up-step and down-step. They then move symmetrically apart by the mechanism described under (1), see Figure \[FigGrowth\] (right).
The TASEP and PNG have to be supplemented by initial conditions and, possibly by boundary conditions. For the former one roughly divides between macroscopically flat and curved. For the TASEP examples would be $h(x)=0$ for $x$ even and $h(x)=1$ for $x$ odd, which has slope zero, and $h(x+1)-h(x)$ independent Bernoulli random variables with mean $m$, $|m|\leq 1$, which has slope $m$. An example for a curved initial condition is $h(0)=0$, $h(x+1)-h(x)=-1$ for $x=-1,-2,\ldots$, and $h(x+1)-h(x)=1$ for $x=0,1,\ldots$.
What are the quantities of interest? The most basic one is the macroscopic shape, which corresponds to a law of large numbers for $$\label{2.2}
\frac{1}{t}h\big([yt],[st]\big)$$ in the limit $t\to\infty$ with $y\in \mathbb{R}$, $s\in\mathbb{R}$, and $[\cdot]$ denoting integer part. From a statistical mechanics point of view the shape fluctuations are of prime concern. For example, in the flat case the surface stays macroscopically flat and advances with constant velocity $v$. One then would like to understand the large scale limit of the fluctuations $\{h(x,t)-vt,(x,t)\in\mathbb{Z}\times\mathbb{Z}\}$. As will be discussed below, the excitement is triggered through non-classical scaling exponents and non-Gaussian limits.
In 1986 in a seminal paper Kardar, Parisi, and Zhang (KPZ) proposed the stochastic evolution equation [@KPZ86] $$\label{2.3}
\frac{\partial}{\partial t}h(x,t)=\lambda\Big(\frac{\partial}
{\partial x}h(x,t)\Big)^2 +\nu_0 \frac{\partial^2}{\partial
x^2}h(x,t)+\eta (x,t)$$ for which $h,x,t\in\mathbb{R}$. $\eta(x,t)$ is space-time white noise which models the deposition mechanism in a moving frame of reference. The nonlinearity reflects the slope dependent growth velocity and the Laplacian with $\nu_0>0$ is a smoothing mechanism. To make the equation well-defined one has to introduce either a suitable spatial discretization or a noise covariance with $g(x)=g(-x)$ and supported close to 0. KPZ argue that, according to (\[2.3\]) with initial conditions $h(x,0)=0$, the surface width increases as $t^{1/3}$, while lateral correlations increase as $t^{2/3}$. It is only through the connection to random matrix theory that universal probability distributions and scaling functions have become accessible.
Following TASEP, PNG, and KPZ as guiding examples it is easy to construct variations. For example, for the TASEP one could introduce evaporation through which heights at local maxima are decreased by 2. The deposition could be made to depend on neighboring heights. Also generalizations to higher dimensions, $x\in\mathbb{R}^d$ or $x\in\mathbb{Z}^d$, are easily accomplished. For the KPZ equation the nonlinearity then reads $\lambda\big(\nabla_x h(x,t)\big)^2$ and the smoothening is $\nu_0 \Delta h(x,t)$. For the PNG model in $d=2$, at a nucleation event one generates on the existing layer a new layer of height one consisting of a disk expanding at unit speed. None of these models seem to be directly connected to random matrices.
How do random matrices appear? {#sect2}
==============================
Let us consider the PNG model with the initial condition $h(x,0)=0$ under the constraint that there are no nucleations outside the interval $[-t,t]$, $t\geq 0$, which also refered to as PNG droplet, since the typical shape for large times is a semicircle. We study the probability distribution $\mathbb{P}(\{h(0,t)\leq n\})$, which depends only on the nucleation events in the quadrant . Let us denote by a set of nucleation events and by $h(0,t;\omega)$ the corresponding height. The order of the coordinates of the $\omega^{(j)}$’s in the frame $\{x=\pm t\}$ naturally defines a permutation of $n$ elements. It can be seen that $h(0,t;\omega)$ is simply the length of the longest increasing subsequence of that permutation. By the Poisson statistics of $\omega$, the permutations are random and their length is Poisson distributed. By this reasoning, somewhat unexpectedly, one finds that $\mathbb{P}\big(\{h(0,t)\leq n\}\big)$ can be written as a matrix integral [@BR01a]. Let $\mathcal{U}_n$ be the set of all unitaries on $\mathbb{C}^n$ and $\mathrm{d}U$ be the corresponding Haar measure. Then $$\label{3.1}
\mathbb{P}\big(\{h(0,t)\leq n\}\big)= e^{-t^2} \int_{\mathcal{U}_n} \mathrm{d}U
\exp [t\, \mathrm{tr} (U+U^\ast)]\,.$$
(\[3.1\]) can also be expressed as Fredholm determinant on $\ell_2=\ell_2(\mathbb{Z})$. On $\ell_2$ we define the linear operator $B$ through $$\label{3.2}
(Bf)(x)=-f(x+1)-f(x-1)+\frac{x}{t} f(x)$$ and denote by $P_{\leq 0}$ the spectral projection onto $B\leq 0$. Setting $\theta_n(x)=1$ for $x>n$ and $\theta_n(x)=0$ for $x\leq n$, one has $$\label{3.3}
\mathbb{P}\big(\{h(0,t)\leq n\}\big)= \det (\mathbf{1}-\theta_n P_{\leq 0}
\theta_n)\,.$$ Such an expression is familiar from GUE random matrices. Let $\lambda_1<\ldots<\lambda_N$ be the eigenvalues of an $N\times N$ GUE distributed random matrix. Then $$\label{3.4}
\mathbb{P}\big(\{\lambda_N\leq y\}\big)= \det (\mathbf{1}-\theta_y P_N \theta_y)\,.$$ Now the determinant is over $L^2(\mathbb{R},\mathrm{d}y)$. If one sets $H=-\frac{1}{2}\frac{\mathrm{d}^2}{\mathrm{d}y^2}+\frac{1}{2N}y^2$, then $P_N$ projects onto $H\leq \sqrt{N}$.
For large $N$ one has the asymptotics $$\label{3.5}
\lambda_N\cong 2N +N^{1/3} \xi_2$$ with $\xi_2$ a Tracy-Widom distributed random variable, that is, $$\label{3.5a}
\mathbb{P}(\{\xi_2\leq s\})=F_2(s):=\det(\mathbf{1}-\chi_s K_{\rm Ai} \chi_s),$$ with $\det$ the Fredholm determinant on $L^2(\mathbb{R},\mathrm{d}x)$, $\chi_s(x)=\mathbf{1}(x>s)$, and $K_{\rm Ai}$ is the Airy kernel (see (\[4.4\]) below). Hence it is not so surprising that for the height of the PNG model one obtains [@PS00b] $$\label{3.6}
h(0,t)= 2t + t^{1/3} \xi_2$$ in the limit $t\to\infty$. In particular, the surface width increases as $t^{1/3}$ in accordance with the KPZ prediction. The law in (\[3.6\]) is expected to be universal and to hold whenever the macroscopic profile at the reference point, $x=0$ in our example, is curved. Indeed for the PNG droplet for large $t$, $|x|\leq t$.
One may wonder whether (\[3.6\]) should be regarded as an accident or whether there is a deeper reason. In the latter case, further height statistics might also be representable through matrix integrals.
Multi-matrix models and line ensembles {#sect3}
======================================
For curved initial data the link to random matrix theory can be understood from underlying line ensembles. They differ from case to case but have the property that the top line has the same statistics in the scaling limit. We first turn to random matrices by introducing matrix-valued diffusion processes.
Let $B(t)$ be GUE Brownian motion, to say $B(t)$ is an $N\times N$ hermitian matrix such that, for every $f\in \mathbb{C}^N$, $t\mapsto
\langle f,B(t) f\rangle$ is standard Brownian motion with variance $\langle f,f\rangle^2\min (t,t')$ and such that for every unitary $U$ it holds $$\label{4.1}
U B(t) U^\ast =B(t)$$ in distribution. The $N\times N$ matrix-valued diffusion, $A(t)$, is defined through the stochastic differential equation $$\label{4.2}
\mathrm{d} A(t)=-V'(A(t)) \mathrm{d}t + \mathrm{d}B (t)\,,\quad A(0)=A,$$ with potential $V:\mathbb{R}\to\mathbb{R}$. We assume $A=A^\ast$, then also $A(t)=A(t)^\ast$ for all $t\geq 0$. It can be shown that eigenvalues of $A(t)$ do not cross with probability 1 and we order them as $\lambda_1(t)<\ldots<\lambda_N(t)$. $t\mapsto
\big(\lambda_1(t),\ldots,\lambda_N(t)\big)$ is the line ensemble associated to (\[4.2\]).
In our context the largest eigenvalue, $\lambda_N(t)$ is of most interest. For its statistics is expected to be independent of the choice of $V$. We first define the limit process, the Airy$_2$ process $\mathcal{A}_2(t)$, through its finite dimensional distributions. Let $$\label{4.3}
H_{\mathrm{Ai}}=-\frac{\mathrm{d}^2}{\mathrm{d}y^2}+y$$ as a self-adjoint operator on $L^2(\mathbb{R},\mathrm{d}y)$. The Airy operator has $\mathbb{R}$ as spectrum with the Airy function Ai as generalized eigenfunctions, . In particular the projection onto $\{H_{\mathrm{Ai}}\leq 0\}$ is given by the Airy kernel $$\label{4.4}
K_{\mathrm{Ai}}(y,y')=\int^\infty_0 \mathrm{d}\lambda
\mathrm{Ai}(y+\lambda)\mathrm{Ai}(y'+\lambda)\,.$$ The associated extended integral kernel is defined through $$\label{4.5}
K_{{\cal A}_2}(y,\tau;y',\tau')=-(e^{-(\tau'-\tau)H_{\mathrm{Ai}}})(y,y')\mathbf{1}(\tau'>\tau)+(e^{\tau H_{\mathrm{Ai}}} K_{\mathrm{Ai}} e^{-\tau' H_{\mathrm{Ai}}})(y,y').$$ Then, the $m$-th marginal for $\mathcal{A}_2(t)$ at times $t_1<t_2<\ldots<t_m$ is expressed as a determinant on $L^2(\mathbb{R}\times\{t_1,\ldots,t_m\})$ according to [@PS02b] $$\label{4.6}
\mathbb{P}\big(\mathcal{A}_2(t_1)\leq s_1,\ldots,
\mathcal{A}_2(t_m)\leq s_m\big)=\det (\mathbf{1}-\chi_{s}K_{{\cal A}_2}\chi_{s}\big),$$ with $\chi_s(x,t_i)=\mathbf{1}(x>s_i)$. $t\mapsto \mathcal{A}_2(t)$ is a stationary process with continuous sample paths and covariance $g_2(t)=\mathrm{Cov}\big(\mathcal{A}_2(0),\mathcal{A}_2(t)\big)=\textrm{Var}(\xi_2) - |t|+{\cal O}(t^2)$ for $t\to 0$ and $g_2(t)=t^{-2}+{\cal O}(t^{-4})$ for $|t|\to\infty$.
The scaling limit for $\mathcal{A}_2(t)$ can be most easily constructed by two slightly different procedures. The first one starts from the stationary Ornstein-Uhlenbeck process $A^{\mathrm{OU}}(t)$ in (\[4.2\]), which has the potential $V(x)=x^2/2N$. Its distribution at a single time is GUE, to say $Z^{-1}_N \exp [-\frac{1}{2N}\mathrm{tr}A^2] \mathrm{d}A$, where the factor $1/N$ results from the condition that the eigenvalue density in the bulk is of order $1$. Let $\lambda^{\mathrm{OU}}_N(t)$ be the largest eigenvalue of $A^{\mathrm{OU}}(t)$. Then $$\label{4.8}
\lim_{N\to\infty} N^{-1/3} \big(\lambda^{\mathrm{OU}}_N
(2N^{2/3}t)-2N\big)=\mathcal{A}_2(t)$$ in the sense of convergence of finite dimensional distributions. The $N^{2/3}$ scaling means that locally $\lambda^{\mathrm{OU}}_N(t)$ looks like Brownian motion. On the other side the global behavior is confined.
The marginal of the stationary Ornstein-Uhlenbeck process for two time instants is the familiar 2-matrix model [@EM98; @NF98]. Setting $A_1=A^{\mathrm{OU}}(0)$, $A_2=A^{\mathrm{OU}}(t)$, $t>0$, the joint distribution is given by $$\label{4.9}
\frac{1}{Z^2_N} \exp \Big(-\frac{1}{2N(1-q^2)}\mathrm{tr}[A^2_1+A^2_2-2qA_1A_2]\Big)
\mathrm{d}A_1 \mathrm{d} A_2,\quad q=\exp(-t/2N).$$
A somewhat different construction uses the Brownian bridge defined by (\[4.2\]) with $V=0$ and $A^{\mathrm{BB}}(-T)=A^{\mathrm{BB}}(T)=0$, that is $$\label{4.10}
A^{\mathrm{BB}}(t)= B(T+t)-\frac{T+t}{2T} B(2T)\,,\quad |t|\leq T\,.$$ The eigenvalues $t\mapsto\big(\lambda^{\mathrm{BB}}_1(t),\ldots,\lambda^\mathrm{BB}_N(t)\big)$ is the Brownian bridge line ensemble. Its largest eigenvalue, $\lambda^{\mathrm{BB}}_N(t)$, has the scaling limit, for $T=2N$, $$\label{4.11}
\lim_{N\to \infty} N^{-1/3} \big(\lambda_N^{\mathrm{BB}} (2N^{2/3}
t)-2N\big)+t^2= \mathcal{A}_2 (t)$$ in the sense of finite-dimensional distributions. Note that $\lambda^{\mathrm{BB}}_N(t)$ is curved on the macroscopic scale resulting in the displacement by $-t^2$. But with this subtraction the limit is stationary.
To prove the limits (\[4.8\]) and (\[4.11\]) one uses in a central way that the underlying line ensemble are determinantal. For the Brownian bridge this can be seen by the following construction. We consider $N$ independent standard Brownian bridges over $[-T,T]$, $b^{\mathrm{BB}}_j(t)$, $j=1,\ldots,N$, $b^{\mathrm{BB}}_j(\pm T)=0$ and condition them on non-crossing for $|t|<T$. The resulting line ensemble has the same statistics as $\lambda^{\mathrm{BB}}_j(t)$, $|t|\leq T$, $j=1,\ldots,N$, which hence is determinantal.
The TASEP, and its limits, have also an underlying line ensemble, which qualitatively resemble $\{\lambda^{\mathrm{BB}}_j(t)$, $j=1,\ldots,N\}$. The construction of the line ensemble is not difficult, but somewhat hidden. Because of lack of space we explain only the line ensemble for the PNG droplet. As before $t$ is the growth time and $x$ is space which takes the role of $t$ from above. The top line is $\lambda_0(x,t)=h(x,t)$, $h$ the PNG droplet of Section \[sect1\]. Initially we add the extra lines $\lambda_j(x,0)=j$, $j=-1,-2,\ldots$. The motion of these lines is completely determined by $h(x,t)$ through the following simple rules: (1) and (2) from above are in force for all lines $\lambda_j(x,t)$, $j=0,-1,\ldots$. (3) holds only for . (4) The annihilation of a pair of an adjacent down-step and up-step at line $j$ is carried out and copied instantaneously as a nucleation event to line $j-1$.
We let the dynamics run up to time $t$. The line ensemble at time $t$ is $\{\lambda_j(x,t),|x|\leq t, j=0,-1,\ldots\}$. Note that $\lambda_j(\pm t,t)=j$. Also, for a given realization of $h(x,t)$, there is an index $j_0$ such that for $j<j_0$ it holds $\lambda_j(x,t)=j$ for all $x, |x|\leq t$. The crucial point of the construction is to have the statistics of the line ensemble determinantal, a property shared by the Brownian bridge over $[-t,t]$, $\lambda^{\mathrm{BB}}_j (x)$, $|x|\leq t$. The multi-line PNG droplet allows for a construction similar to the Brownian bridge line ensemble. We consider a family of independent, rate 1, time-continuous, symmetric nearest neighbor random walks on $\mathbb{Z}$, $r_j(x)$, $j=0,-1,\ldots$. The $j$-th random walk is conditioned on $r_j(\pm t)=j$ and the whole family is conditioned on non-crossing. The resulting line ensemble has the same statistics as the PNG line ensemble $\{\lambda_j(x,t),j=0,-1,\ldots\}$, which hence is determinantal.
In the scaling limit for $x=\mathcal{O}(t^{2/3})$ the top line $\lambda_0(x,t)$ is displaced by $2t$ and $t^{1/3}$ away from $\lambda_{-1}(x,t)$. Similarly $\lambda^{\mathrm{BB}}_N(x)$ for $x=\mathcal{O}(N^{2/3})$ is displaced by $2N$ and order $N^{1/3}$ apart from $\lambda^{\mathrm{BB}}_{N-1}(x)$. On this scale the difference between random walk and Brownian motion disappears but the non-crossing constraint persists. Thus it is no longer a surprise that $$\label{4.12}
\lim_{t\to\infty} t^{-1/3} \big(h(t^{2/3} x,t)-2t\big)+
x^2=\mathcal{A}_2 (x)\,,\quad x\in\mathbb{R}\,,$$ in the sense of finite dimensional distributions [@PS02b].
To summarize: for curved initial data the spatial statistics for large $t$ is identical to the family of largest eigenvalues in a GUE multi-matrix model.
Flat initial conditions {#sect4}
=======================
Given the unexpected connection between the PNG model and GUE multi-matrices, a natural question is whether such a correspondence holds also for other symmetry classes of random matrices. The answer is affirmative, but with unexpected twists. Consider the flat PNG model with and nucleation events in the whole upper half plane $\{(x,t),
x\in\mathbb{R}, t\geq 0\}$. The removal of the spatial restriction of nucleation events leads to the problem of the longest increasing subsequence of a random permutation with involution [@BR01b; @PS00b]. The limit shape will be flat (straight) and in the limit $t\to\infty$ one obtains $$h(0,t)=2t+\xi_1 t^{1/3}\,,$$ where $$\mathbb{P}(\xi_1\leq s)=F_1(2^{-2/3}s).$$ The distribution function $F_1$ is the Tracy-Widom distribution for the largest GOE eigenvalue.
As before, we can construct the line ensemble and ask if the link to Brownian motion on GOE matrices persists. Firstly, let us compare the line ensembles at fixed position for flat PNG and at fixed time for GOE Brownian motions. In the large time (resp. matrix dimension) limit, the edges of these point processes converge to the same object: a Pfaffian point process with $2\times 2$ kernel [@Fer04]. It seems then plausible to conjecture that also the two line ensembles have the same scaling limit, i.e., the surface process for flat PNG and for the largest eigenvalue of GOE Brownian motions should coincide. Since the covariance for the flat PNG has been computed exactly in the scaling limit, one can compare with simulation results from GOE multi-matrices. The evidence strongly disfavors the conjecture [@BFP08].
The process describing the largest eigenvalue of GOE multi-matrices is still unknown, while the limit process of the flat PNG interface is known [@BFS08] and called the Airy$_1$ process, ${\cal A}_1$, $$\lim_{t\to\infty} t^{-1/3} \big(h(t^{2/3}x,t)-2t\big)=2^{1/3}{\cal A}_1(2^{-2/3}x).$$ Its $m$-point distribution is given in terms of a Fredholm determinant of the following kernel. Let $B(y,y')=\textrm{Ai}(y+y')$, $H_1=-\frac{{\rm d}}{{\rm d}y^2}$. Then, $$K_{{\cal A}_1}(y,\tau;y',\tau')=-(e^{-(\tau'-\tau)H_1})(y,y')\mathbf{1}(\tau'>\tau)+(e^{\tau H_1} B e^{-\tau' H_1})(y,y')$$ and, as for the Airy$_2$ process, the $m$-th marginal for $\mathcal{A}_1(t)$ at times is expressed through a determinant on $L^2(\mathbb{R}\times\{t_1,\ldots,t_m\})$ according to $$\label{4.6b}
\mathbb{P}\big(\mathcal{A}_1(t_1)\leq s_1,\ldots,
\mathcal{A}_1(t_m)\leq s_m\big)=\det (\mathbf{1}-\chi_{s}K_{{\cal A}_1}\chi_{s}\big),$$ with $\chi_s(x,t_i)=\mathbf{1}(x>s_i)$ [@Sas05; @BFPS07]. The Airy$_1$ process is a stationary process with covariance $g_1(t)=\mathrm{Cov}\big(\mathcal{A}_1(0),\mathcal{A}_1(t)\big)=\textrm{Var}(\xi_1) - |t|+{\cal O}(t^2)$ for $t\to 0$ and $g_1(t)\to 0$ super-exponentially fast as $|t|\to\infty$.
The Airy$_1$ process is obtained using an approach different from the PNG line ensemble, but still with an algebraic structure encountered also in random matrices (in the the GUE-minor process [@JN06; @OR06]). We explain the mathematical structure using the continuous time TASEP as model, since the formulas are the simplest. For a while we use the standard TASEP representation in terms of particles. One starts with a formula by Schütz [@Sch97] for the transition probability of the TASEP particles from generic positions. Consider the system of $N$ particles with positions $x_1(t)>x_2(t)>\ldots>x_N(t)$ and let $$G(x_1,\ldots,x_N;t)=\mathbb{P}(x_1(t)=x_1,\ldots,x_N(t)=x_N |x_1(0)=y_1,\ldots,x_N(0)=y_N).$$ Then $$\label{eq5.1b}
G(x_1,\ldots,x_N;t)=\det\left(F_{i-j}(x_{N+1-i}-y_{N+1-j},t)\right)_{1\leq i,j\leq N}$$ with $$F_n(x,t)=\frac{1}{2\pi\textrm{i}}\oint_{|w|>1}{\rm d}w\frac{e^{t(w-1)}}{w^{x-n+1}(w-1)^n}.$$ The function $F_n$ satisfies the relation $$F_n(x,t)=\sum_{y\geq x}F_{n-1}(y,t)\,.$$ The key observation is that (\[eq5.1b\]) can be written as the marginal of a determinantal line ensemble, i.e. of a measure which is a product of determinants [@Sas05]. The “lines” are denoted by $x^n_i$ with time index $n$, $1\leq n \leq N$, and space index $i$, $1\leq
i\leq n$. We set $x^n_1=x_n$. Then $$\label{eq5.2}
G(x_1,\ldots,x_N;t) = \!\!\! \sum_{x_i^n,2\leq i\leq n\leq N}
\Big(\prod_{n=1}^{N-1} \det(\phi_n(x_i^{n},x_j^{n+1}))_{i,j=1}^n \Big)
\det(\Psi^N_{N-i}(x_j^N))_{i,j=1}^N$$ with $\Psi^N_{N-i}(x)=F_{-i+1}(x-y_{N+1-i},t)$, $\phi_n(x,y)=\mathbf{1}(x>y)$ and $\phi_n(x_{n+1}^{n},y)=1$ (here $x_{n+1}^{n}$ plays the role of a virtual variable). The line ensembles for the PNG droplet and GUE-valued Brownian motion have the same determinantal structure. However in (\[eq5.2\]) the determinants are of increasing sizes which requires to introduce the virtual variables $x_{n+1}^n$. However, from the algebraic perspective the two cases can be treated in a similar way. As a result, the measure in (\[eq5.2\]) is determinantal (for instance for any fixed initial conditions, but not only) and has a defining kernel dependent on $y_1,\ldots,y_N$. The distribution of the positions of TASEP particles are then given by a Fredholm determinant of the kernel. To have flat initial conditions, one sets $y_i=N-2i$, takes first the $N\to\infty$ limit, and then analyzes the system in the large time limit to get the Airy$_1$ process defined above.
A couple of remarks:\
(1) For general initial conditions (for instance for flat initial conditions), the measure on $\{x_i^n\}$ is not positive, but some projections, like on the TASEP particles $\{x_1^n\}$, defines a probability measure.\
(2) The method can be applied also to step initial conditions ($x_k(0)=-k$, $k=1,2,\ldots$) and one obtains the Airy$_2$ process. In this case, the measure on $\{x_i^n\}$ is a probability measure.\
(3) In random matrices a measure which is the product of determinants of increasing size occurs too, for instance in the GUE-minor process [@JN06; @OR06].
Growth models and last passage percolation {#sect5}
==========================================
For the KPZ equation we carry out the Cole-Hopf transformation with the result $$\label{1}
\frac{\partial}{\partial t}Z(x,t)=-\left(-\nu_0
\frac{\partial^2}{\partial x^2}+\frac{\lambda}{\nu_0}\eta (x,t)\right)
Z(x,t)\, ,$$ which is a diffusion equation with random potential. Using the Feynman-Kac formula, (\[1\]) corresponds to Brownian motion paths, $x_t$, with weight $$\label{2}
\exp \left(-\frac{\lambda}{\nu_0}\int^t_0 \mathrm{d}s\, \eta(x_s,s)\right).$$ In physics this problem is known as a directed polymer in a random potential, while in probability theory one uses directed first/last passage percolation. The spirit of the somewhat formal expression (\[2\]) persists for discrete growth processes. For example, for the PNG droplet we fix a realization, $\omega$, of the nucleation events, which then determines the height $h(0,t;\omega)$ according to the PNG rules. We now draw a piecewise linear path with local slope $m$, $|m|<1$, which starts at $(0,0)$, ends at $(0,t)$, and changes direction only at the points of $\omega$. Let $L(t;\omega)$ be the maximal number of Poisson points collected when varying over allowed paths. Then $h(0,t;\omega)=L(t;\omega)$. So to speak, the random potential from (\[2\]) is spiked at the Poisson points.
In this section we explain the connection between growth models and (directed) last passage percolation. For simplicity, we first consider the case leading to discrete time PNG droplet, although directed percolation can be defined for general passage time distributions. Other geometries like flat growth are discussed later.
Let $\omega(i,j)$, $i,j\geq 1$, be independent random variables with geometric distribution of parameter $q$, i.e., $\mathbb{P}(\omega(i,j)=k)=(1-q)q^k$, $k\geq 0$. An up-right path $\pi$ from $(1,1)$ to $(n,m)$ is a sequence of points $(i_\ell,j_\ell)_{\ell=1}^{n+m-1}$ with $(i_{\ell+1},j_{\ell+1})-(i_\ell,j_\ell)\in \{(1,0),(0,1)\}$. The last passage time from $(1,1)$ to $(n,m)$ is defined by $$\label{eq5.1}
G(n,m)=\max_{\pi:(1,1)\to (n,m)}\sum_{(i,j)\in \pi} \omega(i,j).$$ The connection between directed percolation and different growth models is obtained by appropriate projections of the three-dimensional graph of $G$. Let us see how this works.
Let time $t$ be defined by $t=i+j-1$ and position by $x=i-j$. Then, the connection between the height function of the discrete time PNG and the last passage time $G$ is simply [@Joh03] $$h(x,t)=G(i,j).$$ Thus, the discrete PNG droplet is nothing else than the time-slicing along the $i+j=t$ directions, see Figure \[FigDirPerc\].
We can however use also a different slicing, at constant $\tau=G$, to obtain the TASEP at time $\tau$ with step initial conditions. For simplicity, consider $\omega(i,j)$ to be exponentially distributed with mean one. Then $\omega(n,m)$ is the waiting time for particle $m$ to do his $n$th jump. Hence, $G(n,m)$ is the time when particle $m$ arrives at $-m+n$, i.e., $$\mathbb{P}(G(n,m)\leq \tau)=\mathbb{P}(x_m(\tau)\geq -m+n).$$
From the point of view of the TASEP, there is another interesting cut, namely at fixed $j=n$. This corresponds to look at the evolution of the position of a given (tagged) particle, $x_n(\tau)$.
A few observations:\
(1) Geometrically distributed random passage times correspond to discrete time TASEP with sequential update.\
(2) The discrete time TASEP with parallel update is obtained by replacing $\omega(i,j)$ by $\omega(i,j)+1$.
The link between last passage percolation and growth holds also for general initial conditions. In (\[eq5.1\]) the optimization problem is called point-to-point, since both $(1,1)$ and $(n,m)$ are fixed. We can generalize the model by allowing $\omega(i,j)$ to be defined on $(i,j)\in\mathbb{Z}^2$ and not only for $i,j\geq 1$. Consider the line $L=\{i+j=2\}$ and the following point-to-line maximization problem: $$G_L(n,m)=\max_{\pi:L\to (n,m)}\sum_{(i,j)\in \pi} \omega(i,j).$$ Then, the relation to the discrete time PNG droplet, namely still holds but this time $h$ is the height obtained from flat initial conditions. For the TASEP, it means to have at time $\tau=0$ the particles occupying $2\mathbb{Z}$. Also random initial conditions fit in the picture, this time one has to optimize over end-points which are located on a random line.
In the appropriate scaling limit, for large time/particle number one obtains the Airy$_2$ (resp. the Airy$_1$) process for all these cases. One might wonder why the process seems not to depend on the chosen cut. In fact, this is not completely true. Indeed, consider for instance the PNG droplet and ask the question of joint correlations of $h(x,t)$ in space-time. We have seen that for large $t$ the correlation length scales as $t^{2/3}$. However, along the rays leaving from $(x,t)=(0,0)$ the height function decorrelates on a much longer time scale, of order $t$. These slow decorrelation directions depend on the initial conditions. For instance, for flat PNG they are the parallel to the time axis. More generally, they coincide with the characteristics of the macroscopic surface evolution. Consequently, except along the slow directions, on the $t^{2/3}$ scale one will always see the Airy processes.
Growth models and random tiling {#sect6}
===============================
In the previous section we explained how different growth models (TASEP and PNG) and observables (TASEP at fixed time or tagged particle motion) can be viewed as different projections of a single three-dimensional object. A similar unifying approach exists also for some growth models and random tiling problems: there is a $2+1$ dimensional surface whose projection to one less dimension in space (resp.time) leads to growth in 1+1 dimensions (resp. random tiling in 2 dimensions) [@BF08a]. To explain the idea, we consider the dynamical model connected to the continuous time TASEP with step initial conditions, being one of the simplest to define.
In Section \[sect4\] we encountered a measure on set of variables , see (\[eq5.2\]). The product of determinants of the $\phi_n$’s implies that the measure is non-zero, if the variables $x_i^n$ belong to the set $S_N^{\rm int}$ defined through an interlacing condition, $$S_N^{\rm int}=\{x_i^n \in S_N \,|\, x_i^{n+1}<x_i^n\leq x_{i+1}^{n+1} \}.$$ Moreover, for TASEP with step initial conditions, the measure on $S_N^{\rm int}$ is a probability measure, so that we can interpret the variable $x_i^n$ as the position of the particle indexed by $(i,n)$. Also, the step initial condition, , implies that $x_i^n(0)=i-n-1$, see Figure \[FigDynamics2d\] (left).
\[b\][$x$]{} \[b\][$n$]{} \[b\][$h$]{} \[\]\[\]\[0.9\][$x_1^1$]{} \[\]\[\]\[0.9\][$x_1^2$]{} \[\]\[\]\[0.9\][$x_1^3$]{} \[\]\[\]\[0.9\][$x_1^4$]{} \[\]\[\]\[0.9\][$x_2^2$]{} \[\]\[\]\[0.9\][$x_2^3$]{} \[\]\[\]\[0.9\][$x_2^4$]{} \[\]\[\]\[0.9\][$x_3^3$]{} \[\]\[\]\[0.9\][$x_3^4$]{} \[\]\[\]\[0.9\][$x_4^4$]{} \[c\][**TASEP**]{}
Then, the dynamics of the TASEP induces a dynamics on the particles in $S_N^{\rm int}$ as follows. Particles independently try to jump on their right with rate one, but they might be blocked or pushed by others. When particle $x_k^n$ attempts to jump:\
(1) it jumps if $x_i^n<x_i^{n-1}$, otherwise it is blocked (the usual TASEP dynamics between particles with same lower index),\
(2) if it jumps, it pushes by one all other particles with index $(i+\ell,n+\ell)$, $\ell\geq 1$, which are at the same position (so to remain in the set $S_N^{\rm int}$).\
For example, consider the particles of Figure \[FigDynamics2d\] (right). Then, if in this state of the system particle $(1,3)$ tries to jump, it is blocked by the particle $(1,2)$, while if particle $(2,2)$ jumps, then also $(3,3)$ and $(4,4)$ will move by one unit at the same time.
It is clear that the projection of the particle system onto reduces to the TASEP dynamics in continuous time and step initial conditions, this projection being the sum in (\[eq5.2\]).
The system of particles can also be represented as a tiling using three different lozenges as indicated in Figure \[FigDynamics2d\]. The initial condition corresponds to a perfectly ordered tiling and the dynamics on particles reflects a corresponding dynamics of the random tiling. Thus, the projection of the model to fixed time reduces to a random tiling problem. In the same spirit, the shuffling algorithm of the Aztec diamond falls into place. This time the interlacing is $S_{\rm Aztec}=\{z_i^n \,|\,
z_i^{n+1}\leq z_i^n\leq z_{i+1}^{n+1}\}$ and the dynamics is on discrete time, with particles with index $n$ not allowed to move before time $t=n$. As before, particle $(i,n)$ can be blocked by $(i,n-1)$. The pushing occurs in the following way: particle $(i,n)$ is forced to move if it stays at the same position of $(i-1,n-1)$, i.e., if it would generate a violation due to the possible jump of particle $(i-1,n-1)$. Then at time $t$ all particles which are not blocked or forced to move, jump independently with probability $q$. As explained in detail in [@Nor08] this particle dynamics is equivalent to the shuffling algorithm of the Aztec diamond (take $q=1/2$ for uniform weight). In Figure \[FigAztec\] we illustrate the rules with a small size example of two steps. There we also draw a set of lines, which come from a non-intersecting line ensemble similar to the ones of the PNG model and the matrix-valued Brownian motions described in Section \[sect3\].
\[c\][$t=0$]{} \[c\][$t=1$]{} \[c\][$t=2$]{}
On the other hand, let $x_i^n:=z_i^n-n$. Then, the dynamics of the shuffling algorithm projected onto particles $x_1^n$, $n\geq 1$, is nothing else than the discrete time TASEP with parallel update and step initial condition! Once again, we have a $1+1$ dimensional growth and a $2$ dimensional tiling model which are different projections of the same $2+1$-dimensional dynamical object.
A guide to the literature {#sect7}
=========================
There is a substantial body of literature and only a rather partial list is given here. The guideline is ordered according to physical model under study.\
*PNG model.* A wide variety of growth processes, including PNG, are explained in [@Mea98]. The direct link to the largest increasing subsequences of a random permutation and to the unitary matrix integral of [@PS90] is noted in [@PS00b; @PS00a]. The convergence to the Airy$_2$ process is worked out in [@PS02b] and the stationary case in [@BR00; @PS04]. For flat initial conditions, the height at a single space-point is studied in [@BR01a; @BR01b] and for the ensemble of top lines in [@Fer04]. The extension to many space-points is accomplished by [@BFS08]. Determinantal space-time processes for the discrete time PNG model are discussed by [@Joh03]. External source at the origin is studied in [@IS04b] for the full line and in [@IS04a] for the half-line.\
*Asymmetric simple exclusion* (ASEP). As a physical model reversible lattice gases, in particular the simple exclusion process, were introduced by Kawasaki [@Kaw72] and its asymmetric version by Spitzer [@Spi70]. We refer to Liggett [@Lig99] for a comprehensive coverage from the probabilistic side. The hydrodynamic limit is treated in [@Spo91] and [@KL99], for example. There is a very substantial body on large deviations with Derrida as one of the driving forces, see [@Der07] for a recent review. For the TASEP Schütz [@Sch97] discovered a determinantal-like formula for the transition probability for a finite number of particles on $\mathbb{Z}$. TASEP step initial conditions are studied in the seminal paper by Johansson [@Joh00]. The random matrix representation of [@Joh00] may also be obtained by the Schütz formula [@NS04; @RS05]. The convergence to the Airy$_2$ process in a discrete time setting [@Joh05]. General step initial conditions are covered by [@PS02a] and the extended process in [@BFP09]. In [@FS06b] the scaling limit of the stationary two-point function is proved. Periodic intial conditions were first studied by Sasamoto [@Sas05] and widely extended in [@BFPS07]. A further approach comes from the Bethe ansatz which is used in [@GS92] to determine the spectral gap of the generator. In a spectacular analytic tour de force Tracy and Widom develop the Bethe ansatz for the transition probability and thereby extend the Johansson result to the PASEP [@TW09; @TW08].\
*2D tiling (statics).* The connection between growth and tiling was first understood in the context of the Aztec diamond [@JPS98], who prove the arctic circle theorem. Because of the specific boundary conditions imposed, for typical tilings there is an ordered zone spatially coexisting with a disordered zone. In the disordered zone the large scale statistics is expected to be governed by a massless Gaussian field, while the line separating the two zones has the Airy$_2$ process statistics.\
a) Aztec diamond. The zone boundary is studied by [@JPS98] and by [@JN06]. Local dimer statistics are investigated in [@CEP96]. Refined details are the large scale Gaussian statistics in the disordered zone [@Ken00], the edge statistics [@Joh05], and the statistics close to a point touching the boundary [@JN06].\
b) Ising corner. The Ising corner corresponds to a lozenge tiling under the constraint of a fixed volume below the thereby defined surface. The largest scale information is the macroscopic shape and large deviations [@CK01]. The determinantal structure is noted in [@OR03]. This can be used to study the edge statistics [@FS03; @FPS04]. More general boundary conditions (skew plane partitions) leads to a wide variety of macroscopic shapes [@OR07].\
c) Six-vertex model with domain wall boundary conditions, as introduced in [@KZJ00]. The free energy including prefactors is available [@BF06]. A numerical study can be found in [@AR05]. The mapping to the Aztec diamond, on the free Fermion line, is explained in [@FS06a].\
d) Kasteleyn domino tilings. Kasteleyn [@Kas63] noted that Pfaffian methods work for a general class of lattices. Macroscopic shapes are obtained by [@KOS06] with surprising connections to algebraic geometry. Gaussian fluctuations are proved in [@Ken08].\
*2D tiling (dynamics)*, see Section \[sect6\]. The shuffling algorithm for the Aztec diamond is studied in [@EKLP92; @Pro03; @Nor08]. The pushASEP is introduced by [@BF08b] and anisotropic growth models are investigated in [@BF08a], see also [@PS97] for the Gates-Westcott model. A similar intertwining structure appears for Dyson’s Brownian motions [@War07].\
*Directed last passage percolation*. “Directed” refers to the constraint of not being allowed to turn back in time. In the physics literature one speaks of a directed polymer in a random medium. Shape theorems are proved, e.g., in [@Kes86]. While the issue of fluctuations had been repeatedly raised, sharp results had to wait for [@Joh00] and [@PS02b]. Growths models naturally lead to either point-to-point, point-to-line, and point-to-random-line last passage percolation. For certain models one has to further impose boundary conditions and/or symmetry conditions for the allowed domain. In (\[eq5.1\]) one takes the max, thus zero temperature. There are interesting results for the finite temperature version, where the energy appear in the exponential, as in (\[2\]) [@CY06].\
*KPZ equation.* The seminal paper is [@KPZ86], which generated a large body of theoretical work. An introductory exposition is [@BS95]. The KPZ equation is a stochastic field theory with broken time reversal invariance, hence a great theoretical challenge, see, e.g., [@Las98].\
*Review articles.* A beautiful review is [@Joh06]. Growth models, of the type discussed here, are explained in [@FP06; @Fer08]. A fine introduction to random matrix techniques is [@Sas07]. [@KK08] provide an introductory exposition to the shape fluctuation proof of Johansson. The method of line ensembles is reviewed in [@Spo06].
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[^1]: Institute for Applied Mathematics, Bonn University, Endenicher Allee 60,53115 Bonn, Germany; E-mail: `[email protected]`
[^2]: Department of Mathematics, Technical University of Munich,Boltzmannstr. 3, 85748 Garching, Germany; E-mail: `[email protected]`
[^3]: Here we use the height function representation. The standard particle representation consists in placing a particle at $x$ if $h(x+1)-h(x)=-1$ and leaving empty if .
|
---
abstract: 'As algorithms are becoming more and more data-driven, the greatest lever we have left to make them robustly beneficial to mankind lies in the design of their objective functions. [*Robust alignment*]{} aims to address this design problem. Arguably, the growing importance of social medias’ recommender systems makes it an urgent problem, for instance to adequately automate hate speech moderation. In this paper, we propose a preliminary research program for robust alignment. This roadmap aims at decomposing the end-to-end alignment problem into numerous more tractable subproblems. We hope that each subproblem is sufficiently orthogonal to others to be tackled independently, and that combining the solutions to all such subproblems may yield a solution to alignment.'
author:
- Lê Nguyên Hoang
bibliography:
- 'alignment.bib'
title: 'A Roadmap for End-to-End Robust Alignment'
---
Introduction
============
As they are becoming more capable and ubiquitous, algorithms are raising numerous concerns, including fairness, privacy, filter bubbles, addiction, job displacement or even existential risks [@russell2015; @tegmark2017]. It has been argued that aligning the goals of algorithmic systems with humans’ preferences would be an efficient way to make them reliably beneficial and to avoid potential catastrophic risks [@russell2019]. In fact, given the global influence of today’s large-scale recommender systems [@kramer2014], alignment has already become urgent. Indeed, such systems currently do not seem to be robustly beneficial, since they seem to recommend unscientific claims to hundreds of millions of users on important topic like climate change [@allgaier19], and to be favoring the radicalization of many users [@ROWAM20].
Crucially, in practice, the alignment of algorithms needs to be made [*robust*]{}. This means that it ought to be resilient to algorithmic bugs, flawed world models [@ha2018], reward hacking [@amodei2016], biased data [@mehrabi2019], unforeseen side effects [@BABBC+18], distributional shift [@SOFLN+19], adversarial attacks [@GSS15] and moral uncertainty [@macaskill2014]. Robust alignment seems crucial in complex environments with diverging human preferences, like hate speech algorithmic moderation on social medias.
Unfortunately, robust alignment has been argued to be an extremely difficult problem [@bostrom2014; @FGHLH+2020]. To address it, there have been a few attempts to list key research directions towards robust alignment [@soares2015; @soares2017]. This paper aims to contribute to this line of work by outlining in a structured and compelling manner the main challenges posed by robust alignment. Given that the robustness of a system is often limited by its weakest component, robust alignment demands that we consider our algorithmic systems in their entirety. This motivates the proposal of a [*roadmap*]{} for [*robust end-to-end alignment*]{}, from data collection to algorithmic output.
While much of our proposal is speculative, we believe that several of the ideas presented here will be critical for the safety and alignment of algorithms. More importantly, we hope that this will be a useful roadmap to better estimate how they can best contribute to the effort. Given the complexity of the problem, our roadmap here will likely be full of gaps and false good ideas. Our proposal is nowhere near definite or perfect. Rather, we aim at presenting a sufficiently good starting point for others to build upon.
The Roadmap
===========
Our roadmap consists of identifying key steps to robust alignment. For the sake of exposition, these steps will be personified by 5 characters, called Alice, Bob, Charlie, Dave and Erin. Roughly speaking, Erin will be collecting data from the world, Dave will use these data to infer the likely states of the world, Charlie will compute the desirability of the likely states of the world, Bob will derive incentive-compatible rewards to motivate Alice to take the right decision, and Alice will optimize decision-making. This decomposition is graphically represented in Figure \[fig:roadmap\].
{width="\textwidth"}
Evidently, Alice, Bob, Charlie, Dave and Erin need not be 5 different algorithms. Typically, it may be much more computationally efficient to merge Charlie and Dave. Nevertheless, at least for pedagogical reasons, it seems useful to first dissociate the different roles that these components have.
In the sequel, we shall further detail the challenges posed by each of the 5 algorithms. We shall also argue that, for robustness and scalability reasons, these algorithms will need to be further divided into many more algorithms. We will see that this raises additional challenges. We shall also make a few non-technical remarks, before concluding.
Alice’s Reinforcement learning
==============================
Today’s most promising framework for large-scale algorithms seems to be [*reinforcement learning*]{} [@SuttonBarto1998; @hutter2004]. In reinforcement learning, at each time $t$, the algorithm observes some state of the world $s_t$. Depending on its inner parameters $\theta_t$, it then takes (possibly randomly) some action $a_t$.
The decision $a_t$ then influences the next state and turns it into $s_{t+1}$. The transition from $s_t$ to $s_{t+1}$ given action $a_t$ is usually assumed to be nondeterministic. In any case, the algorithm then receives a reward $R_{t+1}$. The internal parameters $\theta_t$ of the algorithm may then be updated into $\theta_{t+1}$, depending on previous parameters $\theta_t$, action $a_t$, state $s_{t+1}$ and reward $R_{t+1}$.
Note that this is a very general framework. In fact, we humans are arguably (at least partially) subject to this framework. At any point in time, we observe new data $s_t$ that informs us about the world. Using an inner model of the world $\theta_t$, we then infer what the world probably is like, which motivates us to take some action $a_t$. This may affect what likely next data $s_{t+1}$ will be observed, and may be accompanied with a rewarding (or painful) feeling $R_{t+1}$, which will motivate us to update our inner model of the world $\theta_t$ into $\theta_{t+1}$.
Let us call Alice the algorithm in charge of performing this reinforcement learning reasoning. Alice can thus be viewed as an algorithm, which inputs observed states $s_t$ and rewards $R_t$, and undertakes actions $a_t$ so as to typically maximize some discounted sum of expected future rewards.
In the case of large-scale algorithms, such actions $a_t$ are mostly of the form of messages sent through the Internet. This may sound benign. But it is not. The YouTube recommender system might suggest billions of antivax videos, causing a major decrease of vaccination and an uprise of deadly diseases. On a brighter note, if an algorithm now promotes convincing eco-friendly messages every day to billions of people, the public opinion on climate change may greatly change. Similarly, thoughtful compassion and cognitive empathy could be formidably enhanced by a robustly aligned recommender algorithm.
Note that, as opposed to all other components, in some sense, Alice is the real danger. Indeed, in our framework, she is the only one that really undertakes actions. More precisely, only her actions will be unconstrained (although others highly influence her decision-making and are thus critical as well). It is thus of the utmost importance that Alice be well-designed.
Some past work [@orseau2016; @mhamdi2017] proposed to restrict the learning capabilities of Alice to provide provable desirable properties. Typically, they proposed to allow only a subclass of learning algorithms, i.e. of update rules of $\theta_{t+1}$ as a function of $(\theta_t, a_t, s_{t+1},R_{t+1})$. However, such restrictions might be too costly, and thus may not be adopted. Yet a safety measure will be useful only if the most powerful algorithms are [*all*]{} subject to the safety measure. As a result, an effective safety measure cannot hinder too much the performance of the algorithms. In other words, [*there are constraints on the safety constraints that can be imposed*]{}. This is what makes the research to make algorithms safe so challenging.
To design safe reinforcement learning algorithms, it seems more relevant to guarantee that they will do sufficient planning to avoid dangerous actions. One interesting idea by [@amodei2016] is [*model lookahead*]{}. This essentially corresponds to Alice simulating many likely scenarii before undertaking any action. More generally, Alice faces a [*safe exploration problem*]{}.
Another key feature of safe reinforcement learning algorithms is their ability to deal adequately with uncertainty. While recent results advanced the capability of algorithms to verify the robustness of neural networks [@DSGMK18; @WWRHK20], we should not expect algorithms to be correct all the time. Just like humans, algorithms will likely be sometime wrong. But then, even if an algorithm is right 99.9999% of the time, it will still be wrong one time out of a million. Yet, algorithms like recommender systems or autonomous cars take billions of decisions every day. In such cases, thousands of algorithmic decisions may be unboundedly wrong every day.
This problem can become even more worrysome if we take into account the fact that hackers may attempt to take advantage of the algorithms’ deficiencies. Such hackers may typically submit only data that corresponds to cases where the algorithms are wrong. This is known as [*evasion attacks*]{} [@lowd2005; @su2017; @gilmer2018]. To avoid evasion attacks, it is crucial for an algorithm to never be unboundedly wrong, e.g. by reliably measuring its own confidence in its decisions and to ask for help in cases of great uncertainty [@HDAR17].
Now, even if Alice is well-designed, she will only be an effective optimization algorithm. Unfortunately, this is no guarantee of safety or alignment. Typically, because of humans’ well-known confirmation bias [@haidt2012], a watch-time maximization YouTube recommender algorithm may amplify filter bubbles, which may lead to worldwide geopolitical tensions. Both misaligned and unaligned algorithms will likely cause very undesirable [*side effects*]{} [@bostrom2014; @everitt2018].
To make sure that Alice will want to behave as we want her to, it seems critical to at least design appropriately the reward $R_{t+1}$. This is similar to the way children are taught to behave. We do so by punishing them when the sequence $(s_t,a_t,s_{t+1})$ is undesirable, and by rewarding them when the sequence $(s_t,a_t,s_{t+1})$ is desirable. In particular, rewards $R_t$ are Alice’s incentives. They will determine her decision-making. Unfortunately, determining the adequate rewards $R_t$ to be given to Alice is an extremely difficult problem. It is, in fact, the key to robust alignment. Our roadmap to solve it identifies 4 key steps incarnated by Erin, Dave, Charlie and Bob.
Erin’s data collection problem
==============================
Evidently, data are critical to do good. Indeed, even the most brilliant mind will be unable to know anything about the world if it does not have any data from that world. Now, much data is already available on the Internet, especially on a platform like YouTube. However, it is important to take into account the fact that the data on the Internet is not always fully reliable. It may be full of fake news, fraudulent entries, misleading videos, hacked posts and corrupted files.
It may then be relevant to invest in more reliable and relevant data collection. This would be Erin’s job. Typically, Erin may want to collect economic metrics to better assess needs. Recently, it has been shown that satellite images combined with deep learning allow to compute all sorts of useful economic indicators [@jean2016], including poverty risks and agricultural productivity. The increased use of sensors may further enable us to improve life standards, especially in developing countries.
To guarantee the reliability of such data, cryptographic and distributed computing solutions are likely to be useful as well, as they already are on the web. Electronic signatures may eventually be the key to distinguish reliable data from [*deep fakes*]{} [@JoPark2019; @NKH2019], while distributed computing, combined with recent Byzantine-resilient consensus algorithms like Blockchain [@nakamoto2008], Hashgraph [@baird2016] or AT-2 [@GKMPS19], could guarantee the reliable storage and traceability of critical information.
Note though that such data collection mechanisms could pose major privacy issues. It is a major current challenge to balance the usefulness of collected data and the privacy violation they inevitably cause. Some possible solutions include [*differential privacy*]{} [@dwork2014], or weaker versions like [*generative-adversarial privacy*]{} [@huang2017]. It could also be possible to combine these with more cryptographic solutions, like [*homomorphic encryption*]{} or [*multi-party computation*]{}. It is interesting that such cryptographic solutions may be (essentially) provably robust to any attacker, including a superintelligence[^1].
Dave’s world model problem
==========================
Unfortunately, raw data are usually extremely messy, redundant, incomplete, unreliable, poisoning and even hacked. To tackle these issues, it is necessary to infer the likely actual states of the world, given Erin’s collected data. This will be Dave’s job.
Dave will probably heavily rely on [*deep representation learning*]{}. This typically corresponds to determining low-dimensional representations of high-dimensional data through deep neural networks. This basic idea has given rise to today’s most promising unsupervised machine learning algorithms, e.g. [*word vectors*]{} [@mikolov2013], [*autoencoders*]{} [@liou2008], [*generative adversarial networks*]{} (GANs) [@goodfellow2014] and [*transformers*]{} [@VSPUJ+17; @RWCLAS2019].
Given how crucial it is for Dave to have an unbiased representation of the world, much care will be needed to make sure that Dave’s inference will foresee selection biases. For instance, when asked to provide images of CEOs, Google Image may return a greater ratio of male CEOs than the actual ratio. More generally, such biases can be regarded as instances of [*Simpson’s paradox*]{} [@simpson1951], and boil down to the saying “correlation is not causation”. It seems crucial that Dave does not fall into this trap.
In fact, data can be worse than unintentionally misleading. Given how influential Alice may be, there will likely be great incentives for many actors to bias Erin’s data gathering to fool Dave. This is known as [*poisoning attacks*]{}. It seems extremely important that Dave anticipate the fact that the data he was given may be purposely biased, if not hacked. Recent breakthroughs have enabled efficient performant robust high-dimensional statistics [@DiakonikolasKane19; @LecueDepersin19] with different attack models, as well as applications to robust distributed learning [@blanchard2017; @mhamdi2018; @damaskinos2018], though future work will probably need to combine such results with data certification guarantees provided by cryptographical signatures. Eventually, like any good journalist, Dave will likely need to cross information from different sources to infer the most likely states of the world.
This inference approach is well captured by the Bayesian paradigm [@hoang2018]. In particular, Bayes rule is designed to infer the likely causes of the observed data $D$. These causes can also be regarded as theories $T$ (and such theories may assume that some of the data were hacked). Bayes rule tells us that the reliability of theory $T$ given data $D$ can be derived formally by $\mathbb P[T|D] = \mathbb P[D|T] \mathbb P[T] / \mathbb P[D]$.
One typical instance of Dave’s job is the problem of inferring global health from a wide variety of collected data. This is what has been done by [@gbd2016], using a sophisticated Bayesian model that reconstructed the likely causes of deaths in countries where data were lacking.
Importantly, Bayes rule also tells us that we should not fully believe any single theory. This simply corresponds to saying that data can often be interpreted in many different mutually incompatible manners. It seems important to reason with all possible interpretations rather than isolating a single interpretation that may be flawed.
When the space of possible states of the world is large, which will surely be the case of Dave, it is often computationally intractable to reason with the full posterior distribution $\mathbb P[T|D]$. Bayesian methods often rather propose to approximate it through variational methods, or to sample from the posterior distribution to identify a reasonable number of good interpretations of the data, typically through Markov-Chain Monte-Carlo (MCMC).
In some sense, Dave’s job can be regarded as writing a compact report of all likely states of the world, given Erin’s collected data. It is unclear what language Dave’s report will be in. It might be useful to make it understandable by humans. But this might be too costly as well. Indeed, Dave’s report might be billions of pages long. It could be unreasonable or undesirable to make it humanly readable.
Note also that Erin and Dave are likely to gain cognitive capabilities over time. It is surely worthwhile to anticipate the complexification of Erin’s data and of Dave’s world models. It seems unclear so far how to do so. Some high-level (purely descriptive) language to describe world models may be needed. This high-level language probably should be flexible enough to be reshaped and redesigned over time. This may be dubbed the [*world description problem*]{}. It is arguably still a very open and uncharted area of research.
Charlie’s desirability learning problem
=======================================
Given any of Dave’s world models, Charlie’s job will then be to compute how desirable this world model is. This is the [*desirability learning*]{} problem [@soares2016], also known as [*value learning*]{}. This is the problem of assigning desirability scores to different world models. These desirability scores can then serve as the basis for any agent to determine [*beneficial*]{} actions.
Unfortunately, determining what, say, the median human considers desirable is an extremely difficult problem. But it should be stressed that we should not aim at deriving an [*ideal*]{} inference of what people desire. This is likely to be a hopeless endeavor. Rather, we should try our best to make sure that Charlie’s desirability scores will be [*good enough*]{} to avoid catastrophic outcomes, e.g. increased polarization, severe misinformation or major discrimination.
One proposed solution is to build upon comparison-based preference inputs of users. For instance, in the Thurstone-Mosteller [@thurstone27; @mosteller51], the Bradley-Terry [@bradleyterry52] or the Plackett-Luce models [@lucebook59; @plackett75], a user is assumed to assign some intrinsic preference $\theta_i$ to each alternative $i$. Given a dilemma between options $i$ and $j$, they will have a probability $\phi(\theta_i-\theta_j)$ to choose $i$ over $j$, where the function $\phi$ is an increasing function from 0 to 1. For instance, we may have $\phi(z) = (1+e^{-z})^{-1}$. Given a dataset of expressed preferences of a user, we may then infer the likely values of $\theta_i$’s.
It is even possible to address the case where the set of alternatives $i \in \mathcal A$ is infinite, as would be the case if alternatives are vectors, or vector representations of alternatives drawn for an unknown large set. One way to do so is to assume a priori that the $\theta_i$’s are a Gaussian process, and that we can define some kernel for the set $\mathcal A$ of alternatives. Bayesian inference (or approximations of Bayesian inference) could then allow to infer the likely preferences of an individual over $\mathcal A$ from a finite sample of binary preferences [@chugharamani2005a; @chugharamani2005b; @chugharamani2005c].
A perhaps more computationally effective approach could consist in learning a preference vector $\beta_k$ in feature space for each user $k$, so that $\phi(\beta_k^T x_i - \beta_k^T x_j)$ would represent the probability that user $k$ will say to prefer $i$ to $j$ on a given query, where $x_i$ and $x_j$ are vector representations of alternatives $i$ and $j$, as is done in [@NGADR18]. By maximizing over $\beta_k$ the likelihood of observed data, with some addditional appropriate regularization term on $\beta_k$, this approach can be argued to compute some maximum a posteriori of user $k$’s preferences.
Another proposed solution to infer human preferences is so-called [*inverse reinforcement learning*]{} [@ng2000; @evans2016]. Assuming that humans perform reinforcement learning to choose their actions, and given examples of actions taken by humans in different contexts, inverse reinforcement learning infers what were the humans’ likely implicit rewards that motivated their decision-making. Assuming we can somehow separate humans’ selfish rewards from altruistic ones, inverse reinforcement learning seems to be a promising first step towards inferring humans’ preferences from data.
There are, however, many important considerations to be taken into account, which we discuss below. First, despite Dave’s effort and because of Erin’s limited and possibly biased data collection, Dave’s world model is fundamentally uncertain. In fact, as discussed previously, Dave would probably rather present a distribution of likely world models. Charlie’s job should be regarded as a scoring of all such likely world models. In particular, she should not assign a single number to the current state of the world, but, rather, a distribution of likely scores of the current state of the world. This distribution should convey the uncertainty about the actual state of the world. Another challenging aspect of Charlie’s job will be to provide a useful representation of potential human disagreements about the desirability of different states of the world. Humans’ preferences are diverse and may never converge. This should not be swept under the rug. Instead, we need to agree on some way to mitigate disagreement.
This is known as a [*social choice*]{} problem. In its general form, it is the problem of aggregating the preferences of a group of disagreeing people into a single preference for the whole group that, in some sense, fairly well represents the individuals’ preferences. Unfortunately, social choice theory is plagued with impossibility results, e.g. Arrow’s theorem [@arrow1950] or the Gibbard-Satterthwaite theorem [@gibbard1973; @satterthwaite1975]. Again, we should not be too demanding regarding the properties of our preference aggregation. Besides, this is the path taken by social choice theory, e.g. by proposing randomized solutions to preserve some desirable properties [@hoang2017].
One particular proposal, known as [*majority judgment*]{} [@balinski2011], may be of particular interest to us here. Its basic idea is to choose some deciding quantile $q \in [0,1]$ (often taken to be $q=1/2$). Then, for any possible state of the world, consider all individuals’ desirability scores for that state. This yields a distribution of humans’ preferences for the state of the world. Majority judgment then concludes that the group’s score is the quantile $q$ of this distribution. If $q=1/2$, this corresponds to the score chosen by the median individual of the group.
While majority judgment seems promising, it does raise the question of how to compare two different individuals’ scores. It is not clear that $score=5$ given by John has a meaning comparable to Jane’s $score=5$. In fact, according to a theorem by von Neumann and Morgenstern [@neumann1944], within their framework, utility functions are only defined up to a positive affine transformation. More work is probably needed to determine how to scale different individuals’ utility functions appropriately, despite previous attempts in special cases [@hoang2016]. In practice, another challenge to apply social choice to real-world problem, such as hate speech moderation, is the fact the number of parameters of a hate speech moderation algorithm may be too large for humans to express their preferences. In fact, an algorithm that must decide based on the visual content of a video will likely have to rely on some sophisticated neural network. However, if the hate speech moderation algorithm has $d$ parameters, then the number of options for voters to vote on would be of the order $2^d$. Deciding what should be moderated through classical social choice would then seem impractical.
This problem has been underlined by [@MPSW19], where they studied social choice algorithms with a logarithmic-sized elicitation of voters. In other words, if there were $2^d$ options, then the social choice algorithms could only query $d$ bits of information from each voter. Clearly, we would then obtain suboptimal decision-making. Interestingly, [@MPSW19] gave upper and lower bounds on the worst-case ratio between such a social choice algorithm’s decision and the optimal decision. Note though that this worst-case ratio may be too conservative, as it excludes reasonable priors we may have on individuals’ preferences, or on the similarity of individuals’ preferences.
Another fascinating line of research proposed to infer from elicitations of pairwise preferences a model for each user’s preferences, and to apply social choice to users’ preference models. This is known as [*virtual democracy*]{} [@KLNPP19]. [@NGADR18] addressed the case of autonomous car dilemmas, [@FBSDC18] faced kidney donation, while [@LKKKY+19] tackled food donation. Interestingly, in [@LKKKY+19], voters were given the chance to interact in their model and to receive feedbacks about the social choice algorithm, so as to gain trust in the system that was implemented. This voting-based approach seems to be a promising avenue to alignment.
Now, arguably, humans’ current preferences are almost surely undesirable. Indeed, over the last decades, psychology has been showing again and again that human thinking is full of inconsistencies, fallacies and cognitive biases [@kahneman2011]. We tend to first have instinctive reactions to stories or facts [@bloom2016], which quickly becomes the position we will want to defend at all costs [@haidt2012]. Worse, we are unfortunately largely unaware of why we believe or want what we believe or want. This means that our current preferences are unlikely to be what we would prefer, if we were more informed, thought more deeply, and tried to make sure our preferences were as well-founded as possible.
In fact, arguably, we should prefer what we would prefer to prefer, rather than what we instinctively prefer. Typically, one might prefer to watch a cat video, even though one might prefer to prefer mathematics videos over cat videos. This is sometimes known as humans’ [*volitions*]{} [@yudkowsky2004], as opposed to humans’ preferences.
It might even be relevant to consider the volition of much more thoughtful and insightful versions of ourselves, as the currents selves may be blind to some phenomenons that will be of great concerns to our better selves. This can be illustrated by the fact that past standards are often no longer regarded as desirable. Our intuitions about the desirability of slavery, homosexuality and gender discrimination have been completely upset over the last century, if not over the last few decades. It seems unlikely that all of our other intuitions will never change. In particular, it seems unlikely that better versions of ourselves will fully agree with current selves. And it seems reasonable to argue that our better selves would be “more right” than current selves.
It is noteworthy that we clearly have epistemic uncertainty about our better selves. Algorithms probably should also take into account their epistemic uncertainty about our better selves’ volitions. Computing a given person’s volition may be called the [*volition learning*]{} problem. Interestingly, this is (mostly) a prediction problem, as it is the prediction of the preferences of alternative versions of ourselves. Bayes rule seems able to address this counterfactual reality.
Unfortunately, distinguishing preferences from volitions seems to be a neglected research direction so far. One approach to do so could be to assume that preferences obtained through elicitations better reflect volitions that preferences inferred from inverse reinforcement learning. This could serve as a basis to distinguish humans’ behaviors motivated by preferences from those motivated by volitions. Further research in this direction seems critical.
Such scores could also be approximated using a large number of proxies, as is done by [*boosting methods*]{} [@arora2012]. The use of several proxies could avoid the overfitting of any proxy. Typically, rather than relying solely on DALYs [@world2009], we probably should invoke machine learning methods to combine a large number of similar metrics, especially those that aim at describing other desirable economic metrics, like human development index (HDI) or gross national happiness (GNH). Still another approach may consist of analyzing “typical” human preferences, e.g. by using [*collaborative filtering*]{} techniques [@ricci2015].
Computing the desirability of a given world state according to a virtual democracy of our better selves’ volitions is Charlie’s job. In some sense, Charlie’s job would thus be to remove cognitive biases from our intuitive preferences, so that they still basically reflect what we really regard as preferable, but in a more coherent and informed manner. She would then need to aggregate such preferences, which may be computationally challenging if the computations must be done in milliseconds as in the case of autonomous cars [@NGADR18] or of recommender systems. But she might have to do more. Indeed, it seems critical for us to be able to trust Charlie’s computations. But this will arguably be very hard. Indeed, we should expect that our better selves’ volitions may find desirable things that our current selves might find repelling. Unfortunately though, we humans tend to react poorly to disagreeing jugments. And this is likely to hold even when the oppositions are our better selves. This poses a great scientific and engineering challenge. How can one be best convinced of the judgments that he or she will eventually embrace but does not yet? In other words, how can we quickly agree with better versions of ourselves? What could be said to an individual to get him closer to his better self? This may be dubbed the [*individual improvement problem*]{}, which is arguably much more a psychology problem than a computer science one. But educative feedbacks from algorithms may be essential, as seems to have been the case in [@LKKKY+19].
Bob’s incentive design
======================
The last piece of the jigsaw is Bob’s job. Bob is in charge of computing the rewards that Alice will receive, based on the work of Erin, Dave and Charlie. Evidently he could simply compute the expectation of Charlie’s scores for the likely states of the world. But this is probably a bad idea, as it opens the door to [*reward hacking*]{} [@amodei2016].
Recall that Alice’s goal is to maximize her discounted expected future rewards. But given that Alice knows (or is likely to eventually guess) how her rewards are computed, instead of undertaking the actions that we would want her to, Alice could hack Erin, Dave or Charlie’s computations, so that such hacked computations yield large rewards. This is sometimes called the [*wireheading problem*]{} [@EverittHutter19].
To avoid wireheading, Bob’s role will be to make sure that, while Alice’s rewards do correlate with Charlie’s scores, they also give Alice the incentives to guarantee that Erin, Dave and Charlie perform as reliably as possible the job they were given.
In fact, it even seems desirable that Alice be incentivized to constantly upgrade Erin, Dave and Charlie for the better. After all, it is high unlikely that early versions of Erin, Dave and Charlie will have no flaws. In fact, because of Goodhart’s law [@ManheimGarrabrant18], even the slight imperfection in their design might result in highly counter-productive optimization by Alice. Indeed, [@EHR20] showed that when the discrepancy between a measure and the true goal has a heavy-tailed distribution, then as the measure gets maximized, the correlation between measure and goal becomes negative.
This suggests that [*robust alignment*]{} must rely on the continuous upgrade of the reward system, so that the computation of Alice’s rewards keep approaching the objective function that we really want to maximize. This is sometimes known as [*corrigibility*]{} [@SFAY15; @LWN19], though for efficient corrections, it seems essential that this corrigibility be pre-programmed. Bob’s job will be to solve the [*programmed corrigibility*]{} problem, by incentivizing Alice to upgrade adequately the components of her reward system.
Unfortunately, it seems unclear how Bob can best make sure that Alice has such incentives. Perhaps a good idea is to penalize Dave’s reported uncertainty about the likely states of the world. Typically, Bob should make sure Alice’s rewards are affected by the reliability of Erin’s data. The more reliable Erin’s data, the larger Alice’s rewards. Similarly, when Dave or Charlie feel that their computations are unreliable, Bob should take note of this and adjust Alice’s rewards accordingly to motivate Alice to provide larger resources for Charlie’s computations. All of this seems to demand a way for Bob to estimate the performance of the different components of the reward system. Doing so seems to be an open problem so far.
Now, Bob should also mitigate the desire for more reliable data and for more trustworthy computations with the fact that such efforts will necessarily require the exploitation of more resources, probably at the expense of Charlie’s scores. It is this non-trivial trade-off that Bob will need to take care of.
Bob’s work might be simplified by some (partial) control of Alice’s action or world model. Although it seems unclear so far how, techniques like [*interactive proofs*]{} (IP) [@babai1985; @goldwasser1989] or [*probabilistically checkable proofs*]{} (PCP) [@arora1998] might be useful to force Alice to prove its correct behavior. By requesting such proofs to yield large rewards, Bob might be able to incentivize Alice’s transparency. All such considerations make up Bob’s [*incentive problem*]{}.
Decentralization
================
We have decomposed robust alignment into 5 components for the sake of exposition. However, any component will likely have to be decentralized to gain reliability and scalability. In other words, instead of having a single Alice, a single Bob, a single Charlie, a single Dave and a single Erin, it seems crucial to construct multiple Alices, Bobs, Charlies, Daves and Erins.
This is key to [*crash-tolerance*]{}. Indeed, a single computer doing Bob’s job could crash and leave Alice without reward nor penalty. But if Alice’s rewards are an aggregate of rewards given by a large number of Bobs, then even if some Bobs crash, Alice’s rewards will remain mostly the same. But crash-tolerance is likely to be insufficient. Instead, we should design [*Byzantine-resilient*]{} mechanisms, that is, mechanisms that still perform correctly despite hacked or malicious Bobs. Robust estimators [@DiakonikolasKane19; @LecueDepersin19] for both distributed workers [@blanchard2017] and parameter servers [@EGGR19] may be useful for this purpose.
{width="\textwidth"}
Evidently, in this Byzantine environment, cryptography, especially (postquantum?) cryptographical signatures and hashes, are likely to play a critical role. Typically, Bobs’ rewards will likely need to be signed. More generally, the careful design of secure communication channels between the components of the algorithms seems key. This may be called the [*secure messaging problem*]{}.
Another difficulty is the addition of more powerful and precise Bobs, Charlies, Daves and Erins to the pipeline. It is not yet clear how to best integrate reliable new comers, especially given that such new comers may be malicious. In fact, they may want to first act benevolent to gain admission. But once they are numerous enough, they could take over the pipeline and, say, feed Alice with infinite rewards. This is the [*upgrade problem*]{}, which was recently discussed by [@christiano2018] who proposed using numerous weaker algorithms to supervise stronger algorithms. More research in this direction is probably needed.
Now, in addition to reliability, decentralization also enables different Alices, Bobs, Charlies, Daves and Erins to focus on specific tasks. This would allow for more optimized solutions at lower costs. To this end, it may be relevant to adapt different Alices’ rewards to their specific tasks. Note though that this could also be a problem, as Alices may enter in competition with one another like in the prisoner’s dilemma. We may call it the [*specialization problem*]{}. Another open question is the extent to which algorithms should be exposed to Bobs’ rewards. Typically, if a small company creates its own algorithm, to what extent should this algorithm be aligned? It should be noted that this may be computationally very costly, as it may be hard to separate the signal of interest to the algorithm from the noise of Bobs’ rewards. Intuitively, the more influential an algorithm is, the more it should be influenced by Bobs’ rewards. But even if this algorithm is small, it may be important to demand that it be influenced by Bobs to avoid any [*diffusion of responsibility*]{}, i.e. many small algorithms that disregard safety concerns on the ground that they each hardly have any global impact on the world.
What makes this nontrivial is that any algorithm may gain capability and influence over time. An unaligned weak algorithm could eventually become an unaligned human-level algorithm. To avoid this, even basic, but potentially unboundedly self-improving[^2] algorithms should perhaps be given at least a seed of alignment, which may grow as algorithms become more powerful. More generally, algorithms should strike a balance between some original (possibly unaligned) objective and the importance they give to alignment. This may be called the [*alignment burden assignment problem*]{}.
Figure \[fig:complete\_roadmap\] recapitulates our complete roadmap.
Conclusion
==========
This paper discussed the [*robust alignment*]{} problem, that is, the problem of aligning the goals of algorithms with human preferences. It presented a general roadmap to tackle this issue. Interestingly, this roadmap identifies 5 critical steps, as well as many relevant aspects of these 5 steps. In other words, we have presented a large number of hopefully more tractable subproblems that readers are highly encouraged to tackle. Hopefully, this combination allows to better highlight the most pressing problems, how every expertise can be best used to, and how combining the solutions to subproblems might add up to solve robust alignment.
[^1]: The possible use of quantum computers may require postquantum cryptography.
[^2]: In particular, nonparametric algorithms should perhaps be treated differently from parametric ones.
|
---
author:
- |
Thomas Stastny\
<[email protected]>\
Autonomous Systems Lab, ETH Zürich\
Zürich, Switzerland\
Mar. 31, 2018 (Updated: May 22, 2018)
bibliography:
- 'lib.bib'
title: '$L_1$ guidance logic extension for small UAVs: handling high winds and small loiter radii'
---
Introduction
============
$L_1$ guidance logic [@L1original] is one of the most widely used path following controllers for small fixed-wing unmanned aerial vehicles (UAVs), primarily due to its simplicity (low-cost implementation on embedded on-board processors, e.g. micro-controllers) and ability to track both circles and lines, which make up the vast majority of a typical fixed-wing vehicle’s flight plan. In [@L2plus], the logic was extended to allow explicit setting of the $L_1$ period and damping, from which an adaptive the $L_1$ length and gain can be calculated, keeping dynamic similarity in the path convergence properties, independent of the velocity. Two primary drawbacks remain, specific to small, slow flying fixed-wing UAVs:
1. As elaborated in [@L1original], circle following convergence requires that the $L_1$ length be less than or equal to the circle radius $R$, i.e. $L_1\leq R$. This condition may often be violated when the ground speed of the aircraft is high, or the circle radius is small.
2. $L_1$ logic breaks down when wind speeds exceed the vehicle’s airspeed, another common predicament for small, slow-flying UAVs.
Though many other guidance formulations exist which may partially address one or both of the listed issues, line-of-sight -based or otherwise, this brief limits its scope to presenting simple extensions to the extensively field tested $L_1$ guidance formulations commonly used in open source autopilot platforms today (e.g. *PX4*[^1] and *Ardupiot*[^2]), allowing legacy operators to keep existing controller tunings and still take advantage of the added performance and safety features. First, an adaptive recalculation of the $L_1$ ratio is introduced in the event that the $L_1$ length violates the circle tracking convergence criteria. Second, borrowing some concepts from [@Furieri_ACC2017], a bearing feasibility parameter is introduced which continuously transitions $L_1$ commands from the *tracking* objective to a *safety* objective, i.e. attempting to *mitigate* run-away in over-wind scenarios. Finally, an airspeed reference command increment is introduced which is non-zero according to the wind speed ratio (i.e. wind speed over airspeed) and the bearing feasibility, effectively *preventing* run-away. Focus will be kept to the circle following objective (or loiter), though the same principles may similarly be applied to waypoint or line tracking.
Fundamentals
============
A similar formulation to the $L_{2+}$ algorithm [@L2plus] will form the basis of the extensions in this document. Figure \[fig:geom\] shows the algorithm geometry and notation used throughout the following sections.
![$L_1$ guidance geometry.[]{data-label="fig:geom"}](graphics/geometry/geometry.pdf){width="70.00000%"}
The original $L_1$ acceleration command, with ground speed as input, is formulated as: $$\label{eq:aref_orig}
a_\text{ref} = k_L \frac{{v_G}^2}{L_1} \sin\eta$$ where $\eta$ is the error angle between the aircraft course $\chi$ and the $L_1$ bearing $\chi_L$. $v_G = \|\mathbf{v}_G\|$, where $\mathbf{v}_G$ is the ground speed vector in inertial frame (North-East 2D plane). One may set a desired period $P_L$ and damping $\zeta_L$, from which the $L_1$ gain $k_L$ and the $L_1$ ratio $q_L$ may be calculated: $$\begin{array}{l}
q_L = P_L \zeta_L/\pi \\
k_L = 4{\zeta_L}^2 \\
\end{array}$$ The $L_1$ length may then be adaptively calculated according to the current ground speed: $$L_1 = q_L v_G$$ Using the $L_1$ length,the law of cosines may be used to intermediately determine the angle $\gamma$ between the vector from the aircraft position $\mathbf{p}$ to the center of the circle $\mathbf{c}$ and the $L_1$ vector: $$\gamma = \cos^{-1}\left(\operatorname{constrain}\left(\frac{L_1^2 + d^2 - R^2}{ 2 L_1 d},-1,1\right)\right)$$ where distance $d=\|\mathbf{d}\|$, $\mathbf{d}=\mathbf{c}-\mathbf{p}$, is from the aircraft to the loiter center. $\gamma$ is then used with the loiter direction $s_\text{loit}$ (clockwise: $s_\text{loit}=+1$; counter-clockwise: $s_\text{loit}=-1$) to calculate the $L_1$ bearing: $$\chi_L = \operatorname{wrap\_pi}\left(\chi_d - s_\text{loit} \gamma\right)$$ where $\chi_d = \operatorname{atan2}\left(d_e, d_n\right)$ and $\operatorname{atan2}$ is the four-quadrant inverse tangent. The error angle $\eta$ is then: $$\eta=\operatorname{constrain}\left(\operatorname{wrap\_pi}\left(\chi_\text{L} - \chi\right),-\frac{pi}{2},\frac{pi}{2}\right)$$ where $\chi = \operatorname{atan2}\left(v_{G_e},v_{G_n}\right)$ is the aircraft course angle and $\operatorname{wrap\_pi}$ wraps its input argument to $\left[-\pi,+\pi\right]$ . Now the original acceleration command in can be reformulated as: $$\label{eq:aref_new}
a_\text{ref} = k_L \frac{v_G}{q_L} \sin\eta$$ Typical fixed-wing implementations of the algorithm convert the acceleration reference to a roll angle reference $\phi_\text{ref}$, via a coordinated turn assumption, and subsequently saturate the reference: $$\phi_\text{ref} = \operatorname{constrain}\left(\tan^{-1}\left(\frac{a_\text{ref}}{g}\right),-\phi_\text{lim},\phi_\text{lim}\right)$$ where $g$ is the acceleration of gravity. Finally – a detail on implementation – prevent singularities when aircraft is in center of loiter: e.g. **if** $d<\epsilon$, **then** $\mathbf{d}=\left(\epsilon, 0\right)^T$. Any direction may be chosen, or alternatively, one could command zero the acceleration command or hold the previous acceleration command until outside some small radius, $\epsilon$, of the circle center where the standard guidance resumes.
Handling small loiter radii
===========================
Circle following convergence requires that $L_1\leq R$ [@L1original]. Using this relationship, a straight forward adaptation to the $L_1$ ratio may be used whenever this condition is violated (effectively decreasing the operator-defined $L_1$ period). I.e. the ratio and distance may be recalculated as $q_L = R / v_G$ and $L_1 = R$, respectively. However, decreasing the period will inherently make the guidance more aggressive, something that may not be desired the when the aircraft is far from the loiter perimeter and turning to approach. In order to maintain the operator defined gains when possible and adapt only when necessary for convergence to the path, a linear ramp may be applied as the aircraft intercepts the loiter circle using the following logic:
Calculate nominal operator defined $L_1$ ratio and corresponding distance:\
$\quad q_L=P_L \zeta_L/\pi$\
$\quad L_1=q_L v_G$\
Check loiter convergence criteria and adjust accordingly:\
where cross track error $e_t=d-R$. \[alg:adapt\_L1ratio\]
Potentially recalculating the $L_1$ ratio requires the following additional check on ground speed: $v_G=\operatorname{max}\left(v_G,v_{G_\text{min}}\right)$ (avoid singularities).
Handling high winds
===================
Bearing feasibility awareness
-----------------------------
In the case that the wind speed exceeds the UAV’s airspeed, feasibility of flying a given $L_1$ bearing depends on the wind direction. As described in [@Furieri_ACC2017], an exact binary boundary on the *bearing feasibility* can be formulated as: $$\label{eq:feas_binary}
\begin{array}{ll}
\beta\sin\lambda\geq 1 \cap |\lambda|\geq\frac{\pi}{2} & \text{(infeasible)} \\
\text{else} & \text{(feasible)} \\
\end{array}$$ where the wind ratio $\beta=w/v_A$ is the fraction of wind speed $w=\|\mathbf{w}\|$ over airspeed $v_A = \|\mathbf{v}_A\| = \|\mathbf{v}_G-\mathbf{w}\|$ and $\lambda$ is the angle between the wind $\mathbf{w}$ and look-ahead $\mathbf{l}_1$ vectors, $\in\left[-\pi,\pi\right]$. $$\lambda = \operatorname{atan2}\left(\mathbf{w}\times\mathbf{l}_1,\mathbf{w}\cdot\mathbf{l}_1\right)$$ where $\mathbf{l}_1=L_1\left(\cos\chi_L, \sin\chi_L\right)^T$. The relationship in physically describes a “feasibility cone", asymptotically decreasing to zero angular opening as the wind ratio increases above unity, see Fig. \[fig:feas\_cone\]. When the $\mathbf{l}_1$ vector lies within this cone the bearing is feasible, and contrarily when outside, infeasible.
[0.32]{}
[0.64]{} ![Bearing feasibility.[]{data-label="fig:feasibility"}](graphics/feasibility_function/feas_func.pdf "fig:"){width="\textwidth"}
As outlined in [@Furieri_ACC2017], two separate tracking objectives can then be intuited: 1) an *ideal* tracking objective, where we are able to track the prescribed bearing and 2) a *safety* objective, where we instead tend towards reducing run-away by turning against the wind and simultaneously leveling the aircraft as $t\rightarrow \infty$, where $t$ is time. As binary steps in tracking objectives will cause oscillations in guidance commands when the vehicle remains on or near the feasibility boundary (common when the wind is approaching the airspeed and small gusts, wind shear, or turbulence is present), it is further desirable to transition continuously through these two states. In [@Furieri_ACC2017], the following transitioning function (equivalently, continuous feasibility function) was proposed: $$\label{eq:feas_old}
\sigma_\text{feas} = \frac{\sqrt{1-\left(\beta\sin\lambda\right)^2}}{cos\lambda}$$ where $\beta<1$ (wind speed is less than airspeed) and $\lambda=0$ (bearing is aligned with wind direction) result in *feasible* output $\sigma_\text{feas}=1$, states beyond the feasibility boundary result in *infeasible* output $\sigma_\text{feas}=0$, and a continuous function $\sigma_\text{feas}\in\left[0,1\right]$ in between, see Fig. \[fig:feas\_func\] (left). Some practical issues exist, however, with the function as defined in ; namely, the transition is continuous but not smooth at the feasibility boundary, which can lead to jagged reference commands, and further, numerical stability issues exist as $\lambda\rightarrow\frac{\pi}{2} \cap \beta\rightarrow 1$. To address these issues, a small buffer zone below the $\beta=1$ line may be designed, considering some buffer airspeed $v_{A_\text{buf}}$. $$\beta_\text{buf} = v_{A_\text{buf}} / v_A$$ The buffer’s magnitude may be set to a reasonable guess at the wind estimate or airspeed uncertainty, or, as outlined in more detail in the following section, set depending on the airspeed reference tracking dynamics (e.g. a conservative buffer in which the airspeed reference may be properly tracked by the end of the transition). Additionally, an approximation of the feasibility function in can be made incorporating the buffer zone, as well as maintaining continuity and smoothness in the transition, see eq. and Fig. \[fig:feas\_func\] (right). $$\label{eq:feas_new}
\sigma_\text{feas}= \begin{cases}
0 & \beta > \beta_+ \\
\cos\left(\frac{\pi}{2}\operatorname{constrain}\left(\frac{\beta-\beta_-}{\beta_+-\beta_-},0,1\right)\right)^2 & \beta > \beta_- \\
1 & \text{else}
\end{cases}$$ where the upper limit of the transitioning region $\beta_+$ is approximated as a piecewise function with a linear finite cut-off to avoid singularities, the cut-off angle $\lambda_\text{co}$ chosen such that the regular operational envelope is not affected: $$\beta_+ = \begin{cases}
\beta_{+_\text{co}} + m_\text{co}\left(\lambda_\text{co}-\lambda_\text{ctsr}\right) & \lambda_\text{ctsr} < \lambda_\text{co} \\
1/\sin\lambda_\text{ctsr} & \text{else}
\end{cases}$$ with $\beta_{+_\text{co}} = 1/\sin\lambda_\text{co}$, $m_\text{co} = \cos\lambda_\text{co}/\sin\lambda_\text{co}^2$, and $\lambda_\text{ctsr}=\operatorname{constrain}\left(|\lambda|,0,\frac{\pi}{2}\right)$. The lower limit of the transitioning region $\beta_-$ is similarly made piecewise to correspond with $\beta_+$: $$\beta_- = \begin{cases}
\beta_{-_\text{co}} + \beta_\text{buf} m_\text{co}\left(\lambda_\text{co}-\lambda_\text{ctsr}\right) & \lambda_\text{ctsr} < \lambda_\text{co} \\
\left(1/\sin\lambda_\text{ctsr}-2\right)\beta_\text{buf} + 1 & \text{else}
\end{cases}$$ where $\beta_{-_\text{co}} = \left(1/\sin\lambda_\text{co}-2\right)\beta_\text{buf} + 1$. With a new feasibility function defined, it’s application to the $L_1$ algorithm can be elaborated; specifically, it is desired that 1) when the bearing is feasible, $L_1$ operates as usual, 2) when the bearing is infeasible, $L_1$ transitions to the *safety* objective, and 3) in between these states, the reference commands maintain continuity and avoid oscillations. The safety objective can be achieved by replacing the ground speed vector $\mathbf{v}_G$ with the airspeed vector $\mathbf{v}_A$ in eq. . Continuity may the be obtained utilizing the feasibility function $\sigma_\text{feas}$ in as follows: $$\mathbf{v}_\text{nav} = \sigma_\text{feas}\mathbf{v}_G + \left(1-\sigma_\text{feas}\right)\mathbf{v}_A$$ subsequently calculating the error angle $$\eta=\operatorname{constrain}\left(\operatorname{wrap\_pi}\left(\chi_\text{nav} - \chi\right),-\frac{pi}{2},\frac{pi}{2}\right)$$ where $\chi_\text{nav} = \operatorname{atan2}\left(v_{\text{nav}_e},v_{\text{nav}_n}\right)$, and computing the final acceleration reference $$a_\text{ref} = k_L \frac{v_\text{nav}}{q_L} \sin\eta$$ The resulting behavior of the aircraft will *mitigate* run-away scenarios; i.e., in over-wind scenarios, minimize the run-away as much as possible at a single commanded airspeed. However, maximum airspeed allowing, it is also possible to *prevent* run-away from the track completely via *airspeed reference compensation*. The next section details a simple approach towards this end.
Airspeed reference compensation
-------------------------------
Though nominal airspeed references are often desired for energy efficiency, in critical conditions, e.g. very high winds, airspeed reference increases may be allowed either while short-term gusts or wind shear persists, or until an emergency landing may be executed. The most straight forward approach would be to match whatever wind speed overshoot (w.r.t. the airspeed) with airspeed reference commands. However, as previously detailed, the wind magnitude does not alone dictate the bearing feasibility; more so, the relation between wind direction and desired bearing. Utilizing the bearing feasibility function (eq. ) defined in the prior sections, the vehicle may more appropriately command airspeed increments. With a nominal and maximum airspeed reference defined, $v_{A_\text{nom}}$ and $v_{A_\text{max}}$, respectively, a wind speed and bearing feasibility dependent airspeed reference increment may be calculated. $$\Delta v_{A_w} = \operatorname{constrain}\left(w-v_{A_\text{nom}}, 0, \Delta v_{A_\text{max}}\right)\left(1 - \sigma_\text{feas}\right)$$ where $\Delta v_{A_\text{max}} = v_{A_\text{max}}-v_{A_\text{nom}}$ is the maximum allowed airspeed reference increment. The airspeed reference then computed as $$v_{A_\text{ref}} = v_{A_\text{nom}} + \Delta v_{A_w}$$ Note the buffer zone is essential here, as the resultant equilibrium point of this algorithm approaches $\beta=1$ and $\lambda=\frac{\pi}{2}$, i.e. zero ground speed, and facing into the wind.
Example simulations
===================
The following simulations demonstrate the outlined $L_1$ extensions within this brief. All simulations were executed in MATLAB with a simplified 2D model of a small UAV, with first order airspeed and roll angle dynamics, as follows: $$\left(\begin{matrix}
\dot{n} \\ \dot{e} \\ \dot{v_A} \\ \dot{\xi} \\ \dot{\phi}
\end{matrix}\right)=\left(\begin{matrix}
v_A\cos\xi + w_n \\ v_A\sin\xi + w_e \\ \left(v_{A_\text{ref}}-v_A\right)/\tau_v \\ g\tan\phi / v_A \\ \left(\phi_\text{ref}-\phi\right) / \tau_\phi
\end{matrix}\right)$$ where heading $\xi=\operatorname{atan2}\left(v_{A_e},v_{A_n}\right)$ and the time constants $\tau_v=1$ and $\tau_\phi=0.5$ are representative of a small, slow speed radio-controlled UAV, running standard low-level attitude stabilization (PID) and e.g. TECS (Total Energy Control System) for airspeed/altitude control. Wind dynamics are detailed in each respective simulation. Guidance parameters for all simulations are held constant, values listed in Table \[tab:guidance\_params\].
------------------- ------- -------------------- -------
Param Value Param Value
$P_L$ $v_{A_\text{nom}}$
$\zeta_L$ $v_{A_\text{max}}$
$\phi_\text{lim}$ $v_{A_\text{buf}}$
------------------- ------- -------------------- -------
: Guidance parameters used in simulations.[]{data-label="tab:guidance_params"}
Handling small loiter radii
---------------------------
Figures \[fig:small\_radii-pos\] and \[fig:small\_radii-states\] show the effect of implementing the adaptive $L_1$ ratio vs. the original formulation. A small (relative to the flight speed) $R=$ radius loiter is followed in no wind by the adaptive formulation, by effectively reducing the $L_1$ period, while the non-adaptive formulation does not converge to the path.
Figures \[fig:small\_radii-wind-pos\] and \[fig:small\_radii-wind-states\] show the effect of the adaptive $L_1$ ratio further in moderate () eastward wind. As the ratio is proportional to ground speed, the effective period will rise and fall corresponding to the orientation to the wind, maintaining closer loiter tracking than in the non-adapted case.
Handling high winds
-------------------
In figures \[fig:const\_wind-pos\] and \[fig:const\_wind-states\], the bearing feasibility function is introduced for the purposes of both run-away *mitigation* and *prevention* with a constant eastward wind of ( over the commanded airspeed). In the original case, the $L_1$ logic is not able to account for the “backwards" flight motion with respect to the ground, as it only considers the ground velocity vector, leading to run-away with large roll reference bang-bang oscillations, due to the unstable operating point it converges to. Note in practice, aside from the obviously undesirable command oscillations, these bang-bang controls combined with any other small perturbations (gusts or couple motions from the longitudinal/direction axes) will often lead to “turn-around" trajectories, which can be disconcerting to operators viewing the flight from the ground.
With run-away *mitigation* active, the bearing feasibility function schedules continuous control action towards tracking the loiter when feasible, and turn against the wind (safety) when infeasible, resulting in the slowest run-away velocity configuration with a stable reference command. Once run-away *prevention* is active, the airspeed reference is allowed to incrementally increase with the feasibility parameter to match the wind speed in the infeasible case, and remain at nominal when tracking is feasible. This results in a zero ground speed terminal configuration, facing against the wind. In figures \[fig:sin\_wind-pos\] and \[fig:sin\_wind-states\], a amplitude, period sinusoidal eastward wind gust is introduced varying about a constant eastward wind of . Similar bang-bang reference roll oscillations are seen in the infeasible bearing cases for the original formulation, while both the run-away mitigation and prevention results similarly maintain continuous and safe control commands. The airspeed reference can be seen to follow the wind speed as it increases above the nominally command reference threshold and the bearing becomes infeasible.
[^1]: dev.px4.io
[^2]: [ardupilot.org](ardupilot.org)
|
---
abstract: 'Local models are schemes which are intended to model the étale-local structure of $p$-adic integral models of Shimura varieties. Pappas and Zhu have recently given a general group-theoretic construction of flat local models with parahoric level structure for any tamely ramified group, but it remains an interesting problem to characterize the local models, when possible, in terms of an explicit moduli problem. In the setting of local models for ramified, quasi-split $GU_n$, work towards an explicit moduli description was initiated in the general framework of Rapoport and Zink’s book and was subsequently advanced by Pappas and Pappas–Rapoport. In this paper we propose a further refinement to their moduli problem, which we show is both necessary and sufficient to characterize the (flat) local model in a certain special maximal parahoric case with signature $(n-1,1)$.'
address: 'Johns Hopkins University, Department of Mathematics, 3400 N. Charles St., Baltimore, MD 21218, USA'
author:
- Brian Smithling
title: On the moduli description of local models for ramified unitary groups
---
[^1]
Introduction {#s:intro}
============
Local models are certain projective schemes defined over a discrete valuation ring Ø. When Øis the completion of the ring of integers of the reflex field of a Shimura variety at a prime ideal, and one has a model of the Shimura variety over Ø, the local model is supposed to govern the étale-local structure of the Shimura model. This allows one to reduce questions of a local nature, such as flatness or Cohen–Macaulayness, to the local model, which in practice should be easier to study than the Shimura model itself. See [@prs13] for an overview of many aspects of the subject.
In [@pappaszhu13], Pappas and Zhu recently gave a uniform group-theoretic construction of “local models” for tamely ramified groups and showed that these schemes satisfy many good properties. They showed that their construction gives étale-local models of integral models of Shimura varieties in most (tame) PEL cases where the level subgroup at the residual characteristic $p$ of Øis a parahoric subgroup which can be described as the stabilizer of a lattice chain.[^2] In this setting, Rapoport and Zink [@rapzink96] had previously defined natural integral Shimura models and local models in terms of explicit moduli problems based on the moduli problem of abelian varieties describing the Shimura variety, but the resulting schemes are not always flat, as was first observed by Pappas [@pappas00]. In the cases where Pappas and Zhu showed that their construction gives local models of Shimura varieties, they did so by showing that it coincides with the flat closure of the generic fiber in the Rapoport–Zink local model. When the Rapoport–Zink local model is not already flat, it remains an interesting problem to obtain a moduli description of this flat closure: the Pappas–Zhu schemes are themselves defined via flat closure of the generic fiber, which does not impart a ready moduli interpretation.
When the group defining the Shimura variety splits over an unramified extension of $\QQ_p$ and only involves types $A$ and $C$, Görtz showed that the Rapoport–Zink local model is flat in . By contrast, the objects of study in the present paper are local models attached to a *ramified*, quasi-split unitary group, which were Pappas’s original examples in [@pappas00] showing that the Rapoport–Zink local model can fail to be flat. Let $F/F_0$ be a ramified quadratic extension of discretely valued non-Archimedean fields with common residue field of characteristic not $2$; let us note that the residual characteristic $2$ case is fundamentally more difficult, and we do not omit it merely for simplicity. Let $n\geq 2$ be an integer, and let $$m := \lfloor n/2 \rfloor.$$ Let $r + s = n$ be a partition of $n$; the pair $(r,s)$ is called the signature. Let $I \subset \{0,\dotsc,m\}$ be a nonempty subset with the property that $$\label{disp:I_cond}
n \text{ is even and } m-1 \in I \implies m \in I.$$ Such subsets $I$ index the conjugacy classes of parahoric subgroups in quasi-split $GU_n(F/F_0)$; see [@paprap09]\*[1.2.3]{}. Attached to these data is the Rapoport–Zink local model $M^{\ensuremath{\mathrm{naive}}\xspace}_I$,[^3] which has come to be called the “naive” local model since it is not flat in general. See \[ss:naiveLM\] for its explicit definition. It is a projective scheme over $\Spec {\ensuremath{\mathscr{O}}\xspace}_E$, where the (local) reflex field $E := F$ if $r \neq s$ and $E:= F_0$ if $r = s$. When $F$ is the $\QQ_p$-localization of an imaginary quadratic field $K$ in which $p$ ramifies, $M_I^{\ensuremath{\mathrm{naive}}\xspace}$ is a local model of a model over $\Spec{\ensuremath{\mathscr{O}}\xspace}_E$ of a $GU_n(K/\QQ)$-Shimura variety, as is explained for example in [@paprap09]\*[1.5.4]{}. See \[ss:Sh\] for an example where we spell out such an integral Shimura model explicitly.
Let $M_I^{\ensuremath{\mathrm{loc}}\xspace}$ denote the (honest) local model, defined as the scheme-theoretic closure of the generic fiber in $M_I^{\ensuremath{\mathrm{naive}}\xspace}$. As a first step towards a moduli characterization of $M_I^{\ensuremath{\mathrm{loc}}\xspace}$, Pappas proposed a new condition in [@pappas00] to add to the moduli problem defining $M_I^{\ensuremath{\mathrm{naive}}\xspace}$, called the *wedge condition*; see \[ss:wedge\_spin\_conds\]. Denote by $M_I^\wedge$ the closed subscheme of $M_I^{\ensuremath{\mathrm{naive}}\xspace}$ cut out by the wedge condition. In the maximal parahoric case $I = \{0\}$ (which is moreover a special maximal parahoric case when $n$ is odd), Pappas conjectured that $M_{\{0\}}^\wedge = M_{\{0\}}^{\ensuremath{\mathrm{loc}}\xspace}$, and he proved this conjecture in the case of signature $(n-1,1)$. We will prove his conjecture in general in [@sm-decon].
But for other $I$ the wedge condition is not enough. The next advance came in [@paprap09] with Pappas and Rapoport’s introduction of the *spin condition*; see \[ss:wedge\_spin\_conds\]. Denote by $M_I^{\ensuremath{\mathrm{spin}}\xspace}$ the closed subscheme of $M_I^\wedge$ cut out by the spin condition. Pappas and Rapoport conjectured that $M_I^{\ensuremath{\mathrm{spin}}\xspace}= M_I^{\ensuremath{\mathrm{loc}}\xspace}$, and it was shown in [@sm11d; @sm14] that this equality at least holds on the level of topological spaces. The starting point of the present paper is that the full equality of these schemes does *not* hold in general.
\[ceg\] For odd $n \geq 5$ and signature $(n-1,1)$, $M_{\{m\}}^{\ensuremath{\mathrm{spin}}\xspace}$ is not flat over $\Spec {\ensuremath{\mathscr{O}}\xspace}_E$.
See \[ss:failure\]. We remark that the level structure in the counterexample is of special maximal parahoric type. In response to the counterexample, in this paper we introduce a further refinement to the moduli problem defining $M_I^{\ensuremath{\mathrm{naive}}\xspace}$. This defines a scheme $M_I$ which fits into a diagram of closed immersions $$M_I^{\ensuremath{\mathrm{loc}}\xspace}\subset M_I \subset M_I^{\ensuremath{\mathrm{spin}}\xspace}\subset M_I^\wedge \subset M_I^{\ensuremath{\mathrm{naive}}\xspace}$$ which are all equalities in the generic fiber; see \[ss:new\_cond\]. In its formulation, our new condition is a close analog of the Pappas–Rapoport spin condition. In its mathematical content, it gives a common refinement of the spin condition and the Kottwitz condition. We conjecture that it solves the problem of characterizing $M_I^{\ensuremath{\mathrm{loc}}\xspace}$.
\[st:conj\] For any signature and nonempty $I$ satisfying , $M_I$ is flat over $\Spec {\ensuremath{\mathscr{O}}\xspace}_E$, or in other words $M_I = M_I^{\ensuremath{\mathrm{loc}}\xspace}$.
The main result of this paper is that $M_I$ at least corrects for Counterexample \[ceg\], i.e. we prove Conjecture \[st:conj\] in the setting of the counterexample.
\[st:main\_thm\] For odd $n$ and signature $(n-1,1)$, $M_{\{m\}} = M_{\{m\}}^{\ensuremath{\mathrm{loc}}\xspace}$.
Our proof of the theorem is based on calculations of Arzdorf [@arzdorf09], who studied in detail the local equations describing $M_{\{m\}}^{\ensuremath{\mathrm{loc}}\xspace}$ when $n$ is odd. In the setting of the theorem, Richarz observed that the local model is actually smooth [@arzdorf09]\*[Prop. 4.16]{}. The condition defining $M_{\{m\}}$ can be used to define a related formally smooth Rapoport–Zink space which plays an important role in the forthcoming paper [@rapoportsmithlingzhang?]. Relatedly, Richarz’s smoothness result and Theorem \[st:main\_thm\] also imply that a certain moduli problem of abelian schemes (which is an integral model for a unitary Shimura variety) is smooth; we make this explicit in \[ss:Sh\].
While the condition defining $M_I$ can be formulated for any $I$ and any signature, outside of Counterexample \[ceg\], we do not know the extent to which the inclusion $M_I \subset M_I^{\ensuremath{\mathrm{spin}}\xspace}$ fails to be an equality. Indeed Pappas and Rapoport have obtained a good deal of computational evidence for the flatness of $M_I^{\ensuremath{\mathrm{spin}}\xspace}$ in low rank cases, and we do not know of any counterexamples to the flatness of $M_I^{\ensuremath{\mathrm{spin}}\xspace}$ when $n$ is even.
The organization of the paper is as follows. In \[s:mod\_prob\] we review the definition of the naive, wedge, and spin local models, and we formulate our refined condition. In \[s:special\_case\] we explain Counterexample \[ceg\] and reduce the proof of Theorem \[st:main\_thm\] to Proposition \[st:X\_1=0\], whose proof occupies \[s:proof\]. In \[s:remarks\] we collect various remarks. We show that the condition defining $M_I$ implies the Kottwitz condition in \[ss:kottwitz\], and in \[ss:wedge\_power\_analogs\] we formulate some analogous conditions for other wedge powers. These are closed conditions on $M_I^{\ensuremath{\mathrm{naive}}\xspace}$ which hold on the generic fiber, and therefore hold on $M_I^{\ensuremath{\mathrm{loc}}\xspace}$; we show that they imply the wedge condition.[^4] In \[ss:Sh\] we give the aforementioned application of Theorem \[st:main\_thm\] to an explicit integral model of a unitary Shimura variety. We conclude the paper in \[ss:PEL\_setting\] by explaining how to formulate these conditions in the general PEL setting, where they again imply the Kottwitz condition and automatically hold on the flat closure of the generic fiber in the naive local model. To be clear, the conditions we formulate in \[ss:PEL\_setting\] will not suffice to characterize the flat closure in general, since for example they do not account for the spin condition in the ramified unitary setting. But it would be interesting to see if they prove useful in other situations in which spin conditions do not arise.
Acknowledgements {#acknowledgements .unnumbered}
----------------
It is a pleasure to thank Michael Rapoport, whose inquiries about the spin condition in the setting of Counterexample \[ceg\] led to the discovery of this counterexample, which in turn spawned the paper. I also heartily thank him and George Pappas for a number of inspiring conversations related to this work. I finally thank the referees for their helpful suggestions and remarks.
Notation {#notation .unnumbered}
--------
Throughout the paper $F/F_0$ denotes a ramified quadratic extension of discretely valued, non-Archimedean fields with respective rings of integers ${\ensuremath{\mathscr{O}}\xspace}_F$ and ${\ensuremath{\mathscr{O}}\xspace}_{F_0}$, respective uniformizers $\pi$ and $\pi_0$ satisfying $\pi^2 = \pi_0$, and common residue field $k$ of characteristic not $2$.[^5] We work with respect to a fixed integer $n \geq 2$. For $i \in \{1,\dotsc,n\}$, we write $$i^\vee := n+1-i.$$ For $i \in \{1,\dotsc,2n\}$, we write $$i^* := 2n+1-i.$$ For $S \subset \{1,\dotsc,2n\}$, we write $$S^* = \{\,i^* \mid i \in S\,\} \quad\text{and}\quad S^\perp = \{1,\dotsc,2n\} \smallsetminus S^*.$$ We also define $$\Sigma S := \sum_{i\in S}i.$$ For $a$ a real number, we write $\lfloor a \rfloor$ for the greatest integer $\leq a$, and $\lceil a \rceil$ for the least integer $\geq a$. We write $a,\dotsc,\wh b,\dotsc,c$ for the list $a,\dotsc,c$ with $b$ omitted.
The moduli problem {#s:mod_prob}
==================
In this section we review the definition of $M_I^{\ensuremath{\mathrm{naive}}\xspace}$, $M_I^\wedge$, and $M_I^{\ensuremath{\mathrm{spin}}\xspace}$ from [@paprap09], and we introduce our further refinement to the moduli problem.
Linear-algebraic setup {#ss:setup}
----------------------
Consider the vector space $F^n$ with its standard $F$-basis $e_1,\dotsc,e_n$. Let $$\phi\colon F^n \times F^n \to F$$ denote the $F/F_0$-Hermitian form which is split with respect to the standard basis, i.e. $$\label{split}
\phi(ae_i,be_j) = \ol a b \delta_{ij^\vee}, \quad a, b \in F,$$ where $a \mapsto \ol a$ is the nontrivial element of $\Gal(F/F_0)$. Attached to $\phi$ are the respective alternating and symmetric $F_0$-bilinear forms $$F^n \times F^n \to F_0$$ given by $$\langle x,y \rangle := \frac 1 2 \tr_{F/F_0}\bigl( \pi^{-1}\phi(x,y) \bigr)
\quad\text{and}\quad
(x,y) := \frac 1 2 \tr_{F/F_0}\bigl( \phi(x,y) \bigr).$$
For each integer $i = bn+c$ with $0 \leq c < n$, define the standard ${\ensuremath{\mathscr{O}}\xspace}_F$-lattice $$\label{Lambda_i}
\Lambda_i := \sum_{j=1}^c\pi^{-b-1}{\ensuremath{\mathscr{O}}\xspace}_F e_j + \sum_{j=c+1}^{n} \pi^{-b}{\ensuremath{\mathscr{O}}\xspace}_F e_j \subset F^n.$$ For all $i$, the [$\langle\text{~,~}\rangle$]{}-dual of $\Lambda_i$ in $F^n$ is $\Lambda_{-i}$, by which we mean that $$\bigl\{\,x\in F^n \bigm| \langle\Lambda_i,x\rangle \subset {\ensuremath{\mathscr{O}}\xspace}_{F_0} \,\bigr\} = \Lambda_{-i},$$ and $$\label{pairing}
\Lambda_i \times \Lambda_{-i} \xra{{\ensuremath{\langle\text{~,~}\rangle}\xspace}} {\ensuremath{\mathscr{O}}\xspace}_{F_0}$$ is a perfect ${\ensuremath{\mathscr{O}}\xspace}_{F_0}$-bilinear pairing. Similarly, $\Lambda_{n-i}$ is the [$(\text{~,~})$]{}-dual of $\Lambda_i$ in $F^n$. The $\Lambda_i$’s form a complete, periodic, self-dual lattice chain $$\dotsb \subset \Lambda_{-2} \subset \Lambda_{-1} \subset \Lambda_0 \subset \Lambda_1 \subset \Lambda_2 \subset \dotsb.$$
Naive local model {#ss:naiveLM}
-----------------
Let $I \subset \{0,\dotsc,m\}$ be a nonempty subset satisfying , and let $r + s = n$ be a partition. As in the introduction, let $$E = F \quad\text{if}\quad r \neq s \quad\text{and}\quad E = F_0 \quad\text{if}\quad r = s.$$ The *naive local model $M_I^{\ensuremath{\mathrm{naive}}\xspace}$* is a projective scheme over $\Spec {\ensuremath{\mathscr{O}}\xspace}_E$. It represents the moduli problem that sends each ${\ensuremath{\mathscr{O}}\xspace}_E$-algebra $R$ to the set of all families $$(\F_i \subset \Lambda_i \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}}R)_{i\in \pm I + n\ZZ}$$ such that
1. \[it:LM1\] for all $i$, $\F_i$ is an ${\ensuremath{\mathscr{O}}\xspace}_F \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} R$-submodule of $\Lambda_i \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} R$, and an $R$-direct summand of rank $n$;
2. \[it:LM2\] for all $i < j$, the natural arrow $\Lambda_i \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} R \to \Lambda_j \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} R$ carries $\F_i$ into $\F_j$;
3. \[it:LM3\] for all $i$, the isomorphism $\Lambda_i \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} R \xra[{\raisebox{0.4ex}{\smash[t]{$\scriptstyle\sim$}}}]{\pi \otimes 1} \Lambda_{i-n} \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} R$ identifies $$\F_i \isoarrow \F_{i-n};$$
4. \[it:LM4\] for all $i$, the perfect $R$-bilinear pairing $$(\Lambda_i \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} R) \times (\Lambda_{-i} \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} R)
\xra{{\ensuremath{\langle\text{~,~}\rangle}\xspace}\otimes R} R$$ identifies $\F_i^\perp$ with $\F_{-i}$ inside $\Lambda_{-i} \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} R$; and
5. \[it:kott\_cond\] (Kottwitz condition) for all $i$, the element $\pi \otimes 1 \in {\ensuremath{\mathscr{O}}\xspace}_F \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} R$ acts on $\F_i$ as an $R$-linear endomorphism with characteristic polynomial $$\det(T\cdot \id - \pi \otimes 1 \mid \F_i) = (T-\pi)^s(T+\pi)^r \in R[T].$$
When $r = s$, the polynomial on the right-hand side in the Kottwitz condition is to be interpreted as $(T^2 - \pi_0)^s$, which makes sense over any ${\ensuremath{\mathscr{O}}\xspace}_{F_0}$-algebra.
Wedge and spin conditions {#ss:wedge_spin_conds}
-------------------------
We continue with $I$ and $(r,s)$ as before. The *wedge condition* on an $R$-point $(\F_i)_i$ of $M_I^{\ensuremath{\mathrm{naive}}\xspace}$ is that
1. \[it:wedge\_cond\] if $r \neq s$, then for all $i$, $$\sideset{}{_R^{s+1}}{\bigwedge} (\,\pi\otimes 1 + 1 \otimes \pi \mid \F_i\,) = 0
\quad\text{and}\quad
\sideset{}{_R^{r+1}}{\bigwedge} (\,\pi\otimes 1 - 1 \otimes \pi \mid \F_i\,) = 0.$$ (There is no condition when $r=s$.)
The *wedge local model $M_I^\wedge$* is the closed subscheme in $M_I^{\ensuremath{\mathrm{naive}}\xspace}$ where the wedge condition is satisfied.
We next turn to the spin condition, which involves the symmetric form [$(\text{~,~})$]{}and requires some more notation. Let $$V := F^n \otimes_{F_0} F,$$ regarded as an $F$-vector space of dimension $2n$ via the action of $F$ on the right tensor factor. Let $$W := \sideset{}{_F^n}\bigwedge V.$$ When $n$ is even, [$(\text{~,~})$]{}is split over $F^n$, by which we mean that there is an $F_0$-basis $f_1,\dotsc,f_{2n}$ such that $(f_i,f_j) = \delta_{ij^*}$. In all cases, ${\ensuremath{(\text{~,~})}\xspace}\otimes_{F_0} F$ is split over $V$. Hence there is a canonical decomposition $$W = W_1 \oplus W_{-1}$$ of $W$ as an $SO\bigl({\ensuremath{(\text{~,~})}\xspace}\bigr)(F) \isom SO_{2n}(F)$-representation.
Intrinsically, $W_1$ and $W_{-1}$ have the property that for any totally isotropic $n$-dimensional subspace $\F \subset V$, the line $\bigwedge_F^n\F \subset W$ is contained in $W_1$ or in $W_{-1}$, and in this way they distinguish the two connected components of the orthogonal Grassmannian $\OGr(n,V)$ over $\Spec F$. Concretely, $W_1$ and $W_{-1}$ can be described as follows. Let $f_1,\dotsc,f_{2n}$ be an $F$-basis for $V$. For $S = \{i_1< \dots < i_n\} \subset \{1,\dotsc,2n\}$ of cardinality $n$, let $$\label{disp:f_S}
f_S := f_{i_1} \wedge \dotsb \wedge f_{i_n} \in W,$$ and let $\sigma_S$ be the permutation on $\{1,\dotsc,2n\}$ sending $$\{1,\dotsc,n\} \xra[{\raisebox{0.4ex}{\smash[t]{$\scriptstyle\sim$}}}]{\sigma_S} S$$ in increasing order and $$\{n+1,\dotsc,2n\} \xra[{\raisebox{0.4ex}{\smash[t]{$\scriptstyle\sim$}}}]{\sigma_S} \{1,\dotsc,2n\} \smallsetminus S$$ in increasing order. For varying $S$ of cardinality $n$, the $f_S$’s form a basis of $W$, and we define an $F$-linear operator $a$ on $W$ by defining it on them: $$a(f_S) := \sgn(\sigma_S)f_{S^\perp}.$$ Then, when $f_1,\dotsc,f_{2n}$ is a split basis for [$(\text{~,~})$]{}, $$\label{disp:W_+-1}
W_{\pm 1} = \operatorname{span}_F\bigl\{\,f_S \pm \sgn(\sigma_S)f_{S^\perp}\bigm| \#S=n\,\bigr\}$$ is the $\pm 1$-eigenspace for $a$. Any other split basis is carried onto $f_1,\dotsc,f_{2n}$ by an element $g$ in the orthogonal group. If $\det g = 1$ then $W_1$ and $W_{-1}$ are both $g$-stable, whereas if $\det g = -1$ then $W_1$ and $W_{-1}$ are interchanged by $g$. In this way $W_1$ and $W_{-1}$ are independent of choices up to labeling.
For the rest of the paper, we pin down a particular choice of $W_1$ and $W_{-1}$ as in [@paprap09]\*[7.2]{}. If $n = 2m$ is even, then $$-\pi^{-1}e_1,\dotsc,-\pi^{-1}e_m, e_{m+1},\dotsc,e_n,e_1,\dotsc,e_m, \pi e_{m+1},\dotsc, \pi e_{n}$$ is a split ordered $F_0$-basis for [$(\text{~,~})$]{}in $F^n$, and we take $f_1,\dotsc,f_{2n}$ to be the image of this basis in $V$. If $n = 2m+1$ is odd, then we take $f_1,\dotsc,f_{2n}$ to be the split ordered basis $$\label{disp:PR_basis}
\begin{gathered}
-\pi^{-1} e_1 \otimes 1, \dotsc, -\pi^{-1} e_m \otimes 1, e_{m+1}\otimes 1 - \pi e_{m+1} \otimes \pi^{-1},\\ e_{m+2} \otimes 1,\dotsc, e_n\otimes 1, e_1\otimes 1,\dotsc, e_m \otimes 1,\\ \frac{e_{m+1} \otimes 1 + \pi e_{m+1} \otimes \pi^{-1}}2, \pi e_{m+2} \otimes 1, \dotsc, \pi e_{n} \otimes 1.
\end{gathered}$$ These are the same choices that are used in . For $\Lambda$ an ${\ensuremath{\mathscr{O}}\xspace}_{F_0}$-lattice in $F^n$, $$W(\Lambda) := \sideset{}{_{{\ensuremath{\mathscr{O}}\xspace}_F}^n}\bigwedge (\Lambda \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} {\ensuremath{\mathscr{O}}\xspace}_F)$$ is naturally an ${\ensuremath{\mathscr{O}}\xspace}_F$-lattice in $W$, and we define $$W(\Lambda)_{\pm 1} := W_{\pm 1} \cap W(\Lambda).$$
We now formulate the spin condition. If $R$ is an ${\ensuremath{\mathscr{O}}\xspace}_F$-algebra, then the *spin condition* on an $R$-point $(\F_i \subset \Lambda_i \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}}R )_i$ of $M_I^{\ensuremath{\mathrm{naive}}\xspace}$ is that
1. \[it:spin\_cond\] for all $i$, the line $\bigwedge_R^n \F_i \subset W(\Lambda_i) \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F}}R$ is contained in $$\im\bigl[ W(\Lambda_i)_{(-1)^s} \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F}} R
\to W (\Lambda_i) \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F}}R
\bigr].$$
This defines the spin condition when $r \neq s$. When $r = s$, $W_{\pm 1}$ is defined over $F_0$ since [$(\text{~,~})$]{}is already split before extending scalars $F_0 \to F$, and the spin condition on $M_{I,{\ensuremath{\mathscr{O}}\xspace}_F}^{\ensuremath{\mathrm{naive}}\xspace}$ descends to $M_I^{\ensuremath{\mathrm{naive}}\xspace}$ over $\Spec {\ensuremath{\mathscr{O}}\xspace}_{F_0}$. In all cases, the *spin local model $M_I^{\ensuremath{\mathrm{spin}}\xspace}$* is the closed subscheme of $M_I^\wedge$ where the spin condition is satisfied.
Our definition of $a$ above is the same as in . As noted in those papers, this agrees only up to sign with the analogous operators denoted $a_{f_1\wedge \dotsb \wedge f_{2n}}$ in [@paprap09]\*[disp. 7.6]{} and $a$ in [@sm11b]\*[2.3]{}, and there is a sign error in the statement of the spin condition in [@paprap09]\*[7.2.1]{} tracing to this discrepancy.
Interlude: the sign of $\sigma_S$
---------------------------------
Here is an efficient means to calculate the sign $\sgn(\sigma_S)$ occurring in the expression for $W_{\pm 1}$.
\[st:sign\_sigma\_S\] For $S \subset \{1,\dotsc,2n\}$ of cardinality $n$, $$\sgn(\sigma_S) = (-1)^{ \Sigma S + \lceil n/2 \rceil}.$$
Let $P$ denote the permutation matrix attached to $\sigma_S$, so that the $(i,j)$-entry of $P$ is $\delta_{i,\sigma_S(j)}$. We compute $\det P$. Say $S = \{i_1 < \dotsb < i_n\}$. Using Laplace expansion along the $n$th column of $P$, then along the $(n-1)$st column, then along…, then along the first column, we find $$\begin{aligned}
\det P &= (-1)^{i_n + n}(-1)^{i_{n-1} + n-1}\dotsm (-1)^{i_1 + 1}\det I_n\\
&= (-1)^{\Sigma S + \frac{n(n+1)}2}\\
&= (-1)^{\Sigma S + \lceil n/2 \rceil}. \qedhere\end{aligned}$$
Further refinement {#ss:new_cond}
------------------
We now formulate our refinement to the moduli problem defining $M_I^{\ensuremath{\mathrm{spin}}\xspace}$. The idea is to further restrict the intersection in the definition of $W(\Lambda_i)_{\pm 1}$ in a way that incorporates a version of the Kottwitz condition. We continue with the notation from before.
The operator $\pi \otimes 1$ acts $F$-linearly and semisimply on $V$ with eigenvalues $\pi$ and $-\pi$. Let $$V_\pi \quad\text{and}\quad V_{-\pi}$$ denote its respective eigenspaces. Let $$W^{r,s} := \sideset{}{_F^r}\bigwedge V_{-\pi} \otimes_F \sideset{}{_F^s}\bigwedge V_\pi.$$ Then $W^{r,s}$ is naturally a subspace of $W$, and $$W = \bigoplus_{r+s=n}W^{r,s}.$$ Let $$W_{\pm 1}^{r,s} := W^{r,s} \cap W_{\pm 1}.$$ For any ${\ensuremath{\mathscr{O}}\xspace}_{F_0}$-lattice $\Lambda$ in $F^n$, let $$W(\Lambda)_{\pm 1}^{r,s} := W_{\pm 1}^{r,s} \cap W(\Lambda) \subset W.$$
Our new condition is just the analog of the spin condition with $W(\Lambda_i)_{(-1)^s}^{r,s}$ in place of $W(\Lambda_i)_{(-1)^s}$. To lighten notation, for $R$ an ${\ensuremath{\mathscr{O}}\xspace}_F$-algebra, define $$L_i^{r,s}(R) := \im\bigl[ W(\Lambda_i)_{(-1)^s}^{r,s} \otimes_{{\ensuremath{\mathscr{O}}\xspace}_F} R \to W(\Lambda_i) \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F}}R\bigr].$$ The condition on an $R$-point $(\F_i \subset \Lambda_i \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} R)_i$ of $M_I^{\ensuremath{\mathrm{naive}}\xspace}$ is that
1. \[it:new\_cond\] for all $i$, the line $\bigwedge_R^n \F_i \subset W(\Lambda_i)\otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F}}R$ is contained in $L_i^{r,s}(R)$.
This defines the condition when $r \neq s$. When $r = s$, the subspaces $W^{r,s}$ and $W_{\pm 1}$ are Galois-stable, and the condition descends from $M_{I,{\ensuremath{\mathscr{O}}\xspace}_F}^{\ensuremath{\mathrm{naive}}\xspace}$ to $M_I^{\ensuremath{\mathrm{naive}}\xspace}$. In all cases, we write $M_I$ for the locus in $M_I^{\ensuremath{\mathrm{spin}}\xspace}$ where the condition is satisfied.
Clearly $M_I$ is a closed subscheme of $M_I^{\ensuremath{\mathrm{spin}}\xspace}$. Our new condition is also already satisfied in the generic fiber: if $R$ is an $F$-algebra and $(\F \subset V \otimes_F R)$ is an $R$-point on $M_I^{\ensuremath{\mathrm{naive}}\xspace}$, then by the Kottwitz condition it is automatic that $\bigwedge_R^n \F \subset W^{r,s} \otimes_F R$ inside $W \otimes_F R$. Thus there is a diagram of closed immersions $$M_I^{\ensuremath{\mathrm{loc}}\xspace}\subset M_I \subset M_I^{\ensuremath{\mathrm{spin}}\xspace}\subset M_I^\wedge \subset M_I^{\ensuremath{\mathrm{naive}}\xspace}$$ which are all equalities between generic fibers. Conjecture \[st:conj\] is that $M_I^{\ensuremath{\mathrm{loc}}\xspace}= M_I$ in general.
The case of $n$ odd, $I = \{m\}$, and signature $(n-1,1)$ {#s:special_case}
=========================================================
The main goal of this section and \[s:proof\] is to prove Theorem \[st:main\_thm\]. We explain Counterexample \[ceg\] along the way. Throughout we specialize to the case that $n = 2m + 1$ is odd, $I =\{m\}$, and $(r,s) = (n-1,1)$. This is a case of special maximal parahoric level structure studied in detail by Arzdorf in [@arzdorf09], and we begin by reviewing the calculations of his that will be relevant for us. Of course ${\ensuremath{\mathscr{O}}\xspace}_E = {\ensuremath{\mathscr{O}}\xspace}_F$. To lighten notation, we suppress the set $\{m\}$, so that $M^{\ensuremath{\mathrm{naive}}\xspace}= M_{\{m\}}^{\ensuremath{\mathrm{naive}}\xspace}$, $M = M_{\{m\}}$, etc.
Review of Arzdorf’s calculations {#ss:arzdorf_calcs}
--------------------------------
It is clear from its definition that $M^{\ensuremath{\mathrm{naive}}\xspace}$ is naturally a closed subscheme of the Grassmannian $\Gr(n,\Lambda_m \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} {\ensuremath{\mathscr{O}}\xspace}_F)$ over $\Spec {\ensuremath{\mathscr{O}}\xspace}_F$. Arzdorf computes an affine chart on the special fiber $M_{k}^{\ensuremath{\mathrm{naive}}\xspace}$ around its “worst point” in [@arzdorf09]\*[4]{} as the restriction of one of the standard open affine charts on the Grassmannian.[^6] Here the “worst point” is the $k$-point $$(\pi\otimes 1) \cdot (\Lambda_m \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} k) \subset \Lambda_m \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} k.$$ The reason for this terminology is that the geometric special fiber $M_{\ol k}^{\ensuremath{\mathrm{naive}}\xspace}$ embeds into an affine flag variety for $GU_n$, where it decomposes (as a topological space) into a disjoint union of Schubert cells, and the (image of the) worst point is the unique closed Schubert cell. See [@paprap09]\*[2.4.2, 5.5]{}.
Following Arzdorf, take the ordered $k$-basis $$\begin{gathered}
\label{disp:Arzdorf_basis}
e_{m+2} \otimes 1, \dotsc, e_n\otimes 1, \pi^{-1}e_1\otimes 1,\dotsc, \pi^{-1}e_m\otimes 1, e_{m+1}\otimes 1,\\ \pi e_{m+2} \otimes 1, \dotsc, \pi e_n \otimes 1, e_1\otimes 1,\dotsc,e_m\otimes 1, \pi e_{m+1} \otimes 1\end{gathered}$$ for $\Lambda_m \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} k$. With respect to this basis, the standard open affine chart $U^{\Gr}$ on $\Gr(n,\Lambda_m \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}}k)$ containing the worst point is the $k$-scheme of $2n\times n$-matrices $$\label{disp:U^Gr}
\begin{pmatrix}
X\\
I_n
\end{pmatrix}$$ (the worst point itself corresponds to $X = 0$). Define $$\label{U's}
U^{\ensuremath{\mathrm{loc}}\xspace}\subset U \subset U^{\ensuremath{\mathrm{spin}}\xspace}\subset U^\wedge \subset U^{\ensuremath{\mathrm{naive}}\xspace}\subset U^{\Gr}$$ by intersecting $U^{\Gr}$ with $M^{\ensuremath{\mathrm{loc}}\xspace}$, $M$, $M^{\ensuremath{\mathrm{spin}}\xspace}$, $M^\wedge$, and $M^{\ensuremath{\mathrm{naive}}\xspace}$, respectively. Write $$X = \begin{pmatrix}
X_1 & X_2\\
X_3 & X_4
\end{pmatrix},$$ where $X_1$ is of size $(n-1)\times (n-1)$, $X_2$ is of size $(n-1)\times 1$, $X_3$ is of size $1 \times (n-1)$, and $X_4$ is scalar.
Arzdorf shows that $X_4^2 = 0$ on $U^{\ensuremath{\mathrm{naive}}\xspace}$ [@arzdorf09]\*[p. 701]{}. Let $U_{X_4 = 0}^{\ensuremath{\mathrm{naive}}\xspace}$ be the closed subscheme of $U^{\ensuremath{\mathrm{naive}}\xspace}$ defined by imposing $X_4 = 0$. Arzdorf shows that $X_2 = 0$ on $U_{X_4=0}^{\ensuremath{\mathrm{naive}}\xspace}$, and that conditions – translate to[^7] $$\label{disp:naive_eqns}
-J X_3^{\ensuremath{\mathrm{t}}\xspace}X_3 = X_1 + J X_1^{\ensuremath{\mathrm{t}}\xspace}J, \quad X_1^2 = 0, \quad\text{and}\quad X_3 X_1 = 0,$$ where $J$ is the $(n-1) \times (n-1)$-matrix $$J := \begin{pmatrix}
& & & & & 1\\
& & & & \iddots\\
& & & 1\\
& &-1\\
& \iddots\\
-1
\end{pmatrix}
\qquad\text{($m \times m$ blocks)}.$$ Arzdorf does not translate the Kottwitz condition, since it is automatically satisfied on the reduced special fiber of $M^{\ensuremath{\mathrm{naive}}\xspace}$. Let us do so now. Regarding the columns of as a basis for the subspace $\F_m$, the operator $\pi\otimes 1$ acts as $$(\pi \otimes 1) \cdot \begin{pmatrix} X\\ I_n \end{pmatrix}
= \begin{pmatrix} 0 \\ X \end{pmatrix}.$$ It follows that the Kottwitz condition on $U^{\Gr}$ is that[^8] $$\charac_X(T) = T^n.$$ When the rightmost column of $X$ is zero, which is the case on $U_{X_4=0}^{\ensuremath{\mathrm{naive}}\xspace}$, the Kottwitz condition becomes $$\label{disp:Kottwitz_cond}
\charac_{X_1}(T) = T^{n-1}.$$ We conclude that $U_{X_4 = 0}^{\ensuremath{\mathrm{naive}}\xspace}$ is the $k$-scheme of $n\times(n-1)$-matrices $\bigl(\begin{smallmatrix} X_1 \\ X_3\end{smallmatrix}\bigr)$ satisfying and .
In our case of signature $(n-1,1)$, Arzdorf shows in [@arzdorf09]\*[4.5]{} that the wedge condition on $U_{X_4=0}^{\ensuremath{\mathrm{naive}}\xspace}$ is the condition $$\label{disp:wedge_eqns}
\sideset{}{^2}\bigwedge \begin{pmatrix} X_1 \\ X_3 \end{pmatrix} = 0.$$ When is satisfied, the Kottwitz condition reduces to the condition $$\label{disp:Tr_cond}
\tr X_1 = 0.$$ Arzdorf proves that for any signature, $U^{\ensuremath{\mathrm{loc}}\xspace}$ is reduced [@arzdorf09]\*[Th. 2.1]{}, so that $$U^{\ensuremath{\mathrm{loc}}\xspace}\subset U_{X_4=0}^{\ensuremath{\mathrm{naive}}\xspace}.$$ It is an observation of Richarz [@arzdorf09]\*[Prop. 4.16]{} that for signature $(n-1,1)$, the map $$\label{disp:U_map}
\begin{pmatrix} X_1 \\ X_3 \end{pmatrix} \mapsto X_3$$ induces an isomorphism $$\label{disp:U^loc_isom}
U^{\ensuremath{\mathrm{loc}}\xspace}\isoarrow \AA_k^{n-1}.$$
Failure of flatness of $M^{\ensuremath{\mathrm{spin}}\xspace}$ {#ss:failure}
--------------------------------------------------------------
In this subsection we explain Counterexample \[ceg\]. We use the calculations of the previous subsection, whose notation we retain. The first item of business is the following.
\[st:spin=>X\_4=0\] On $U^{\ensuremath{\mathrm{naive}}\xspace}$, the spin condition implies that $X_4 = 0$.
Before proving the lemma, we need some more notation. Consider the ordered ${\ensuremath{\mathscr{O}}\xspace}_{F_0}$-basis $$\pi^{-1}e_1,\dotsc, \pi^{-1}e_m, e_{m+1},\dotsc,e_n, e_1,\dotsc, e_m, \pi e_{m+1}, \dotsc, \pi e_n$$ for $\Lambda_m$. After extending scalars ${\ensuremath{\mathscr{O}}\xspace}_{F_0} \to k$, this gives Arzdorf’s basis , but in a different order. After extending scalars ${\ensuremath{\mathscr{O}}\xspace}_{F_0} \to F$, this gives an ordered $F$-basis for $V$, and for $S \subset \{1,\dotsc,2n\}$ a subset of cardinality $n$, we define $$\label{disp:e_S}
e_S \in W$$ with respect to this basis for $V$ as in . For varying $S$ of cardinality $n$, the $e_S$’s form an ${\ensuremath{\mathscr{O}}\xspace}_F$-basis for $W(\Lambda_m) \subset W$. Given an ${\ensuremath{\mathscr{O}}\xspace}_F$-algebra $R$, we will often abuse notation and continue to write $e_S$ for its image in $W(\Lambda_m) \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F}} R$.
Let $f_1,\dotsc,f_{2n}$ denote the split ordered basis for $V$. Now let $R$ be a $k$-algebra, and let $$N_{\pm 1} := \im\bigl[ W(\Lambda_m)_{\pm 1} \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F}} R
\to W(\Lambda_m) \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F}}R
\bigr].$$
Let $(\F_m \subset \Lambda_m \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}}R)$ be an $R$-point on $U^{\ensuremath{\mathrm{naive}}\xspace}$. Regarding each column of the matrix as an $R$-linear combination of Arzdorf’s basis elements , the wedge of these columns (say from left to right) is expressible as a linear combination $$\label{disp:lin_comb}
\sum_{\substack{S \subset \{1,\dotsc,2n\}\\ \#S = n}} c_S e_S \in W(\Lambda_m) \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F}} R, \quad c_S \in R.$$ We have $$c_{\{m+1,n+1,\dotsc,\wh{n+m+1},\dotsc,2n\}} = (-1)^m X_4.$$ The spin condition for arbitrary signature $(r,s)$ is that the element is contained in $N_{(-1)^s}$. We are going to show that if is contained in $N_1$ or in $N_{-1}$, then $c_{\{m+1,n+1,\dotsc,\wh{n+m+1},\dotsc,2n\}}$ must vanish. Let $\varepsilon \in \{\pm 1\}$.
By , every element in $W(\Lambda_m)_{\varepsilon}$ is an $F$-linear combination of elements of the form $$\label{disp:f_Spmf_Sperp}
f_S + \varepsilon \sgn(\sigma_S) f_{S^\perp}.$$ With respect to the $e_S$-basis for $W$, the only elements of the form which can possibly involve $e_{\{m+1,n+1,\dotsc,\wh{n+m+1},\dotsc,2n\}}$ are for $S$ one of the sets $$\{m+1,n+1,\dotsc,\wh{n+m+1},\dotsc,2n\} \quad\text{and}\quad \{n+1,\dotsc,2n\}.$$ These are also the only two sets for which can possibly involve $e_{\{n+1,\dotsc,2n\}}$. Both of these sets are self-perp. For one of them, which we call $T$, equals $2 f_T$, and for the other, equals $0$, as follows for example from Lemma \[st:sign\_sigma\_S\]. The element $2 f_T$ is of the form $$\begin{gathered}
(\text{unit in }{\ensuremath{\mathscr{O}}\xspace}_F^\times)\cdot (e_{m+1} \otimes 1 \pm \pi e_{m+1} \otimes \pi^{-1})\\ \wedge (e_1\otimes 1) \wedge \dotsb \wedge (e_m\otimes 1) \wedge (\pi e_{m+2} \otimes 1) \wedge \dotsb \wedge (\pi e_n \otimes 1),\end{gathered}$$ which in turn is of the form $$\label{disp:2f_T}
(\text{unit in }{\ensuremath{\mathscr{O}}\xspace}_F^\times) \cdot e_{\{m+1,n+1,\dotsc,\wh{n+m+1},\dotsc,2n\}} \pm \pi^{-1}(\text{unit in }{\ensuremath{\mathscr{O}}\xspace}_F^\times) \cdot e_{\{n+1,\dotsc,2n\}}.$$
Now suppose we have an $F$-linear combination $$\label{disp:lin_comb2}
c_T \cdot 2f_T + \sum_{S\neq T}c_S\bigl(f_S + \varepsilon \sgn(\sigma_S) f_{S^\perp}\bigr) \in W(\Lambda_m)_{\varepsilon}.$$ By writing this as a linear combination of $e_S$’s and looking at the $e_{\{n+1,\dotsc,2n\}}$-term, we conclude from the above discussion and that $$\ord_\pi (c_T) \geq 1.$$ This and imply that the $e_{\{m+1,n+1,\dotsc,\wh{n+m+1},\dotsc,2n\}}$-term in dies in $N_\varepsilon$, since $\pi R = 0$.
By the lemma $U^{\ensuremath{\mathrm{spin}}\xspace}\subset U_{X_4=0}^{\ensuremath{\mathrm{naive}}\xspace}$, and we conclude that $U^{\ensuremath{\mathrm{spin}}\xspace}$ is the $k$-scheme of matrices satisfying , , and , plus the spin condition. When $n \geq 5$, we are going to show that $M^{\ensuremath{\mathrm{spin}}\xspace}$ is not flat over $\Spec {\ensuremath{\mathscr{O}}\xspace}_F$ by showing that $U^{\ensuremath{\mathrm{spin}}\xspace}\neq U^{\ensuremath{\mathrm{loc}}\xspace}$. The remaining calculation that we need is the following. We continue with our $k$-algebra $R$ and $R$-modules $N_{\pm 1}$.
\[st:N\_lem\] For $\varepsilon \in \{\pm 1\}$, the elements $$e_{\{n+1,\dotsc,2n\}} \quad\text{and}\quad e_{\{i,n+1,\dotsc,\wh{n+i},\dotsc,2n\}} \quad\text{for}\quad i \in \{1,\dotsc,\wh{m+1},\dotsc,n\}$$ are contained in $N_\varepsilon$.
In the notation of proof of Lemma \[st:spin=>X\_4=0\], the image of $\pi f_T$ in $N_\varepsilon$ is of the form $$(\text{unit})\cdot e_{\{n+1,\dotsc,2n\}},$$ which takes care of the first element on our list.
For $1 \leq i \leq m$, let $$S_1 := \{i,m+1,n+1,\dotsc,2n\}\smallsetminus\{n+i,n+m+1\}$$ and $$S_2 := \{i,n+1,\dotsc,2n\} \smallsetminus \{n+i\}.$$ Then $$S_1^\perp = \{m+1,i^\vee,1,\dotsc,2n\} \smallsetminus \{n+m+1,n+i^\vee\}$$ and $$S_2^\perp = \{i^\vee,1\dotsc,2n\} \smallsetminus \{n+i^\vee\}.$$ By Lemma \[st:sign\_sigma\_S\], $$\sgn(\sigma_{S_1}) = -1 \quad\text{and}\quad \sgn(\sigma_{S_2}) = 1.$$ Hence $$\begin{split}
\pi(f_{S_1} - \varepsilon f_{S_1^\perp}) &= (- \pi^{-1} e_i \otimes 1) \wedge (e_{m+1} \otimes \pi - \pi e_{m+1} \otimes 1)\\
&\qquad\qquad \wedge f_{n+1} \wedge \dotsb \wedge \wh f_{n+i} \wedge \dotsb \wedge \wh f_{n+m+1} \wedge \dotsb \wedge f_{2n}\\
&\qquad -\varepsilon (e_{m+1} \otimes \pi - \pi e_{m+1} \otimes 1)\\
&\qquad\qquad\qquad \wedge f_{i^\vee} \wedge f_{n+1} \wedge \dotsb \wedge \wh f_{n+m+1} \wedge \dotsb \wedge \wh f_{n+i^\vee} \wedge \dotsb \wedge f_{2n}\\
&\in W(\Lambda_m)_\varepsilon.
\end{split}$$ This equals $$- \pi e_{S_1} + (-1)^{m-1}e_{S_2} - \varepsilon \pi e_{S_1^\perp} + \varepsilon (-1)^{m+1} e_{S_2^\perp},$$ which, since $\pi R = 0$, has image $$\label{disp:e_comb1}
(-1)^{m-1}(e_{S_2} + \varepsilon e_{S_2^\perp})$$ in $N_\varepsilon$. Similarly, $$\begin{split}
\pi(f_{S_2} + \varepsilon f_{S_2^\perp}) = {}&(- \pi^{-1} e_i \otimes 1) \wedge f_{n+1} \wedge \dotsb \wedge \wh f_{n+i} \wedge \dotsb \wedge f_{n+m}\\
&\qquad\qquad\wedge \frac{e_{m+1} \otimes \pi + \pi e_{m+1} \otimes 1}2 \wedge f_{n+m+2} \wedge \dotsb \wedge f_{2n}\\
&\qquad +\varepsilon f_{i^\vee} \wedge f_{n+1} \wedge \dotsb \wedge f_{n+m}\\
&\qquad\qquad\qquad \wedge \frac{e_{m+1} \otimes \pi + \pi e_{m+1} \otimes 1}2\\
&\qquad\qquad\qquad\qquad\wedge f_{n+m+2} \wedge \dotsb \wh f_{n+i^\vee} \wedge \dotsb \wedge f_{2n}
\end{split}$$ is in $W(\Lambda_m)_\varepsilon$ and has image $$\label{disp:e_comb2}
-\frac 1 2 (-e_{S_2} + \varepsilon e_{S_2^\perp})$$ in $N_\varepsilon$. Plainly $e_{S_2}$ and $e_{S_2^\perp}$ are in the $k$-span of and , which proves the rest of the lemma.
Now let $R$ be a $k$-algebra with a nonzero element $x$ such that $x^2 = 0$. Suppose that $n \geq 5$, and take $$X_1 = \diag(x,-x,0,\dotsc,0,-x,x) \quad\text{and}\quad X_3 = 0.$$ Then $X_1$ and $X_3$ satisfy , , and . Plugging them into , and taking $X_2$ and $X_4$ both $0$, the wedge of the columns of translates to a linear combination of the basis elements $$e_{\{n+1,\dotsc,2n\}} \quad\text{and}\quad
e_{\{i,n+1,\dotsc,\wh{n+i},\dotsc,2n\}} \quad\text{for}\quad i \in \{m-1,m,m+2,m+3\}.$$ By Lemma \[st:N\_lem\], such a linear combination is contained in $N_\varepsilon$ for any $\varepsilon$. We conclude that $X_1$ and $X_3$ determine an $R$-point on $U^{\ensuremath{\mathrm{spin}}\xspace}$. But in the case of signature $(n-1,1)$, any point on $U^{\ensuremath{\mathrm{loc}}\xspace}$ with $X_3 = 0$ must have $X_1 = 0$, via . This exhibits that $U^{\ensuremath{\mathrm{spin}}\xspace}\neq U^{\ensuremath{\mathrm{loc}}\xspace}$.
Flatness of $M$ {#ss:reduction}
---------------
In this subsection we reduce Theorem \[st:main\_thm\], which asserts that the scheme $M$ is flat over $\Spec {\ensuremath{\mathscr{O}}\xspace}_F$, to Proposition \[st:X\_1=0\] below, which we will subsequently prove in \[s:proof\].
Our goal is to show that the closed immersion $M^{\ensuremath{\mathrm{loc}}\xspace}\subset M$ is an equality. Since this is an equality between generic fibers and $M^{\ensuremath{\mathrm{loc}}\xspace}$ is flat over ${\ensuremath{\mathscr{O}}\xspace}_F$, it suffices to show that it is also an equality between special fibers. For this we may assume that $k$ is algebraically closed.
Consider the closed immersions $$M_k^{\ensuremath{\mathrm{loc}}\xspace}\subset M_k \subset M_k^{\ensuremath{\mathrm{spin}}\xspace}\subset M_k^\wedge.$$ As discussed at the beginning of \[ss:arzdorf\_calcs\], these schemes all embed into an affine flag variety, where they topologically decompose into a union of Schubert cells. By Richarz’s result [@arzdorf09]\*[Prop. 4.16]{} in the situation at hand or by [@sm11d]\*[Main Th.]{} in general for odd $n$, the Schubert cells occurring in them are all the same. In the present situation, there are just two Schubert cells that occur, the “worst point” and its complement $C$ in these schemes, as follows from [@sm11d]\*[Cor. 5.6.2]{} and the calculation of the relevant admissible set in [@paprap09]\*[2.4.2]{}. Arzdorf shows that $M_k^\wedge$ contains an open reduced subscheme in [@arzdorf09]\*[Prop. 3.2]{}. Therefore the entire open cell $C$ must be reduced in $M_k^\wedge$, which implies the same for all of the schemes in the display.
To complete the proof that $M_k^{\ensuremath{\mathrm{loc}}\xspace}= M_k$, it remains to show that the local rings of these schemes at the worst point coincide. Restricting to the affine charts from \[ss:arzdorf\_calcs\], we have $$U^{\ensuremath{\mathrm{loc}}\xspace}\subset U \to \AA_k^{n-1},$$ where the second map is $\bigl(\begin{smallmatrix}X_1 \\ X_3\end{smallmatrix}\bigr) \mapsto X_3$ and the composite is the isomorphism . The worst point is the point over $0 \in \AA_k^{n-1}$. It is an easy consequence of the first condition in and the wedge condition that $U^\wedge$, and a fortiori $U$, is finite over $\AA_k^{n-1}$. By Nakayama’s lemma, it therefore suffices to show that the fiber in $U$ over $0$ is just $\Spec k$. In other words, we have reduced Theorem \[st:main\_thm\] to the following.
\[st:X\_1=0\] Let $R$ be a $k$-algebra, and suppose that we have an $R$-point on $U$ such that $X_3 = 0$. Then $X_1 = 0$.
Proof of Proposition \[st:X\_1=0\] {#s:proof}
==================================
In this section we prove Proposition \[st:X\_1=0\]. We continue with the notation and assumptions of \[s:special\_case\]. Our strategy is essentially one of computation: the main point is to find an explicit ${\ensuremath{\mathscr{O}}\xspace}_F$-basis for $W(\Lambda_m)_{-1}^{n-1,1}$ (Proposition \[st:W(Lambda\_m)\_coeff\_conds\]), from which we obtain an explicit $R$-basis for $L_m^{n-1,1}(R)$ whenever $\pi R = 0$ (Corollary \[st:im\_basis\]).
Another basis for $V$
---------------------
Let $$g_1,\dotsc,g_{2n}$$ denote the ordered $F$-basis $$\begin{gathered}
e_1 \otimes 1 - \pi e_1 \otimes \pi^{-1},\dotsc, e_n \otimes 1 - \pi e_n \otimes \pi^{-1},\\
\frac{e_1 \otimes 1 + \pi e_1 \otimes \pi^{-1}} 2, \dotsc, \frac{e_n \otimes 1 + \pi e_n \otimes \pi^{-1}} 2\end{gathered}$$ for $V$, which is a split ordered basis for [$(\text{~,~})$]{}. Moreover $g_1,\dotsc,g_n$ is a basis for $V_{-\pi}$ and $g_{n+1},\dotsc,g_{2n}$ is a basis for $V_\pi$, which makes $g_1,\dotsc,g_{2n}$ better suited to work with condition than the split basis $f_1,\dotsc,f_{2n}$ in .
It is straightforward to see that the change-of-basis matrix expressing the $g_i$’s in terms of the $f_i$’s is contained in $SO_{2n}(F)$, either by explicitly writing out this matrix and computing its determinant, or by noting that the intersection $$\operatorname{span}\{f_1,\dotsc,f_n\} \cap \operatorname{span}\{g_1,\dotsc,g_n\} = \operatorname{span}\{g_{m+1}\}$$ has even codimension $n-1$ in $\operatorname{span}\{f_1,\dotsc,f_n\}$ and in $\operatorname{span}\{g_1,\dotsc,g_n\}$, and then appealing to [@paprap09]\*[7.1.4]{}. As discussed in \[ss:wedge\_spin\_conds\], this implies that $$W_{\pm 1} = \operatorname{span}_F\bigl\{\,g_S \pm \sgn(\sigma_S) g_{S^\perp}\bigm| \#S=n\,\bigr\},$$ where $$g_S \in W$$ is defined with respect to the basis $g_1,\dotsc,g_{2n}$ as in .
Types
-----
To facilitate working with the subspace $W^{r,s} \subset W$, we make the following definition.
We say that a subset $S \subset \{1,\dotsc,2n\}$ has *type $(r,s)$* if $$\#(S\cap \{1,\dotsc,n\}) = r
\quad\text{and}\quad
\#(S\cap\{n+1,\dotsc,2n\}) = s$$
For $r + s = n$, the $g_S$’s for varying $S$ of type $(r,s)$ form a basis for $W^{r,s}$, and it is easy to check that $S$ and $S^\perp$ have the same type. Hence the following.
\[st:W\_pm1\^r,s\_spanning\_set\] $W_{\pm 1}^{r,s} = \operatorname{span}_F\{\, g_S \pm \sgn(\sigma_S) g_{S^\perp} \mid S \text{ is of type } (r,s)\,\}$.
\[rk:sign\_sigma\_S\_type\_(n-1,1)\] In proving Proposition \[st:X\_1=0\] we will be interested in $S$ of type $(n-1,1)$. Such an $S$ is of the form $$S = \bigl\{1,\dotsc, \wh j, \dotsc, n, n+i\bigr\}$$ for some $i$, $j \leq n$. By Lemma \[st:sign\_sigma\_S\], $$\sgn(\sigma_S) = (-1)^{m+1+\Sigma S} = (-1)^{m+1 + \frac{n(n+1)} 2 -j + n+i} = (-1)^{i + j + 1}.$$
Weights
-------
To determine a basis for $W(\Lambda_m)_{-1}^{n-1,1}$, we will need to answer the question of when a linear combination of elements of the form $g_S - \sgn(\sigma_S) g_{S^\perp}$ is contained in $W(\Lambda_m)$. The following will help with the bookkeeping.
Let $S \subset \{1,\dotsc,2n\}$. The *weight vector $\mathbf{w}_S$* attached to $S$ is the element of $\ZZ^n$ whose $i$th entry is $\#(S \cap \{i,n+i\})$.
\[rk:(n-1,1)\_possible weights\] If $S$ is of type $(n-1,1)$, then there are two possibilities. The first is that there are $i$ and $j$ such that the $i$th entry of $\mathbf{w}_S$ is $2$, the $j$th entry of $\mathbf{w}_S$ is $0$, and all the other entries of $\mathbf{w}_S$ are $1$. In this case $S = \{1,\dotsc, \wh j,\dotsc, n, n+i\}$ is uniquely determined by its weight. The other possibility is that $\mathbf{w}_S = (1,\dotsc,1)$. In this case all we can say is that $S = \{1,\dotsc, \wh i, \dotsc, n, n+ i\}$ for some $i \in \{1,\dotsc, n\}$.
Of course $\mathbf w_S \in \{0,1,2\}^n$, and $S$ and $S^\perp$ may or may not have the same weight. For any $S$ of cardinality $n$, $$\mathbf{w}_{S^\perp} + \mathbf{w}_S^\vee = (2,\dotsc,2),$$ where $\mathbf{w}_S^\vee$ is the vector whose $i$th entry is the $i^\vee$th entry of $\mathbf{w}_S$ for all $i$.
The reason for introducing the notion of weight is the following obvious fact.
\[st:weight\_lem\] For $S$ of cardinality $n$, write $$g_S = \sum_{S'}c_{S'}e_{S'},\quad c_{S'} \in F.$$ Then every $S'$ for which $c_{S'} \neq 0$ has the same weight as $S$.
By contrast, many different *types* of $S'$ occur in the linear combination in the display.
Worst terms
-----------
Let $A$ be a finite-dimensional $F$-vector space, and let $\B$ be an $F$-basis for $A$.
Let $x = \sum_{b\in \B} c_b b \in A$, with $c_b \in F$. We say that $c_b b$ is a *worst term for $x$* if $$\ord_\pi (c_b) \leq \ord_\pi (c_{b'}) \quad\text{for all}\quad b' \in \B.$$ We define $$\operatorname{WT}_{\B}(x) := \sum_{\substack{b \in \B\\ c_b b \text{ is a worst}\\ \text{term for } x}} c_b b.$$
Let $\Lambda$ denote the ${\ensuremath{\mathscr{O}}\xspace}_F$-span of in $A$. Trivially, a nonzero element $x \in A$ is contained in $\Lambda$ if and only if one, hence any, of its worst terms is. When this is so, and when $R$ is a $k$-algebra, the image of $x$ under the map $$\Lambda \to \Lambda \otimes_{{\ensuremath{\mathscr{O}}\xspace}_F} R$$ is the same as the image of $\operatorname{WT}_\B(x)$.
For the rest of the paper we specialize to the case that $A = W$ and is the $e_S$-basis for $W$ from . We abbreviate $\operatorname{WT}_\B$ to $\operatorname{WT}$. For any $S$ of cardinality $n$, the vector $g_S$ has a unique worst term. When $S$ has weight $(1,\dotsc,1)$, this is $$\begin{gathered}
(e_{i_1} \otimes 1) \wedge \dotsb \wedge (e_{i_\alpha}\otimes 1) \wedge (-\pi e_{j_1} \otimes \pi^{-1}) \wedge \dotsb \wedge (-\pi e_{j_\beta} \otimes \pi^{-1})\\ \wedge \frac {e_{k_1-n} \otimes 1}2 \wedge \dotsb \wedge \frac {e_{k_\gamma-n} \otimes 1}2 \wedge \frac{\pi e_{l_1-n} \otimes \pi^{-1}}2 \wedge \dotsb \wedge \frac{\pi e_{l_\delta-n} \otimes \pi^{-1}} 2,\end{gathered}$$ where we write $$S = \{i_1,\dotsc,i_\alpha, j_1,\dotsc,j_\beta, k_1\dotsc,k_\gamma, l_1,\dotsc,l_\delta\}$$ with $$\begin{gathered}
i_1 < \dotsb < i_\alpha < m+1 \leq j_1 < \dotsb < j_\beta \leq n\\
< k_1 < \dotsb < k_\gamma < n + m + 1 \leq l_1 < \dotsb < l_\delta.\end{gathered}$$ For $S$ of other weight, the worst term of $g_S$ can be made similarly explicit, using the easy fact that $$g_i \wedge g_{n+i} = (e_i \otimes 1) \wedge (\pi e_i \otimes \pi^{-1})$$ to handle all pairs of the form $i$, $n+i$ that occur in $S$. Here is the worst term of $g_S$ in all cases of type $(n-1,1)$.
\[st:WT(g\_S)\]
1. if $S = \{1,\dotsc,\wh i,\dotsc,n,n+i\}$ for some $i < m+1$, then $$\operatorname{WT}(g_S) = \frac {(-1)^{i+m}}2 \pi^{-(m+1)} e_{\{n+1,\dotsc,2n\}}.$$
2. if $S = \{1,\dotsc,\wh i,\dotsc,n,n+i\}$ for some $i \geq m+1$, then $$\operatorname{WT}(g_S) = \frac {(-1)^{i+m+1}}2 \pi^{-(m+1)} e_{\{n+1,\dotsc,2n\}}.$$
3. if $S = \{1,\dotsc,\wh j,\dotsc,n,n+i\}$ for some $i, j <m+1$ with $i \neq j$, then $$\operatorname{WT}(g_S) = (-1)^{m+1} \pi^{-m}e_{\{i,n+1,\dotsc, \wh{n+j},\dotsc,2n\}}.$$
4. if $S = \{1,\dotsc,\wh j,\dotsc,n,n+i\}$ for some $i < m+1 \leq j$, then $$\operatorname{WT}(g_S) = (-1)^m \pi^{-(m-1)} e_{\{i,n+1,\dotsc, \wh{n+j},\dotsc,2n\}}.$$
5. if $S = \{1,\dotsc,\wh j,\dotsc,n,n+i\}$ for some $j < m+1 \leq i$, then $$\operatorname{WT}(g_S) = (-1)^{m+1}\pi^{-(m+1)}e_{\{i,n+1,\dotsc, \wh{n+j},\dotsc,2n\}}.$$
6. if $S = \{1,\dotsc,\wh j,\dotsc,n,n+i\}$ for some $i,j \geq m+1$ with $i \neq j$, then
& & (g\_S) = (-1)\^m \^[-m]{} e\_[{i,n+1,, ,,2n}]{}. & &
The lattice $W(\Lambda_m)_{-1}^{n-1,1}$
---------------------------------------
In this subsection we determine an ${\ensuremath{\mathscr{O}}\xspace}_F$-basis for $W(\Lambda_m)_{-1}^{n-1,1}$. Let $S$ be of type $(n-1,1)$. Then $S\cap\{n+1,\dotsc,2n\}$ consists of a single element $i_S$. Define $$S \preccurlyeq S^\perp
\quad\text{if}\quad
i_S \leq i_{S^\perp}.$$ The elements $g_S - \sgn(\sigma_S) g_{S^\perp}$, for varying $S$ of type $(n-1,1)$ and such that $S \preccurlyeq S^\perp$, form a basis for $W_{-1}^{n-1,1}$. (In particular, note that if $S$ is of type $(n-1,1)$ and $S = S^\perp$, then necessarily $\sgn(\sigma_S) = -1$, by Remark \[rk:sign\_sigma\_S\_type\_(n-1,1)\].) Our task is to determine when a linear combination of such elements is contained in $W(\Lambda_m)$.
As a first step, we calculate the worst terms of $g_S - \sgn(\sigma_S) g_{S^\perp}$.
\[st:WT(g\_S\_pm\_g\_S\^perp)\] Let $S$ be of type $(n-1,1)$ with $S \preccurlyeq S^{\perp}$. Then exactly one of the following nine situations holds.
1. \[it:case1\] $S = \{1,\dotsc,\wh{m+1},\dotsc,n,n+m+1\}$, $S = S^\perp$, $\mathbf{w}_S = (1,\dotsc,1)$, and $$\operatorname{WT}\bigl(g_S - \sgn(\sigma_S) g_{S^\perp}\bigr) = \operatorname{WT}(2g_S) = \pi^{-(m+1)}e_{\{n+1,\dotsc,2n\}}.$$
2. \[it:case2\] $S = \{1,\dotsc,\wh{i^\vee},\dotsc,n,n+i\}$ for some $i < m+1$, $S = S^\perp$, $\mathbf{w}_S \neq (1,\dotsc,1)$, and $$\operatorname{WT}\bigl(g_S - \sgn(\sigma_S) g_{S^\perp}\bigr) = \operatorname{WT}(2g_S) = 2(-1)^{m}\pi^{-(m-1)}e_{\{i,n+1,\dotsc,\wh{i^*},\dotsc,2n\}}.$$
3. \[it:case3\] $S = \{1,\dotsc,\wh{i^\vee},\dotsc,n,n+i\}$ for some $i > m+1$, $S = S^\perp$, $\mathbf{w}_S \neq (1,\dotsc,1)$, and $$\operatorname{WT}\bigl(g_S - \sgn(\sigma_S) g_{S^\perp}\bigr) = \operatorname{WT}(2g_S) = 2(-1)^{m+1}\pi^{-(m+1)} e_{\{i,n+1,\dotsc,\wh{i^*},\dotsc,2n\}}.$$
4. \[it:case4\] $S = \{1,\dotsc,\wh{i},\dotsc,n,n+i\}$ for some $i < m+1$, $S \neq S^\perp$, $$\mathbf{w}_S = \mathbf{w}_{S^\perp} = (1,\dotsc,1),$$ and $$\operatorname{WT}\bigl(g_S - \sgn(\sigma_S) g_{S^\perp}\bigr) = (-1)^{m+1}\pi^{-m} \bigl(e_{\{i,n+1,\dotsc,\wh{n+i},\dotsc,2n\}} - e_{\{i^\vee,n+1,\dotsc,\wh{i^*},\dotsc,2n\}} \bigr).$$
5. \[it:case5\] $S = \{1,\dotsc,\wh j,\dotsc,n,n+i\}$ for some $i < j^\vee < m+1$; $S \neq S^\perp$; $\mathbf{w}_S$, $\mathbf{w}_{S^\perp}$, and $(1,\dotsc,1)$ are pairwise distinct; and $$\begin{gathered}
\operatorname{WT}\bigl(g_S - \sgn(\sigma_S) g_{S^\perp}\bigr) =\\
(-1)^m\pi^{-(m-1)}\bigl(e_{\{i,n+1,\dotsc,\wh{n+j},\dotsc,2n\}} + (-1)^{i+j}e_{\{j^\vee,n+1,\dotsc, \wh{i^*},\dotsc,2n\}}\bigr).\end{gathered}$$
6. \[it:case6\] $S = \{1,\dotsc,\wh{m+1},\dotsc,n,n+i\}$ for some $i < m+1$, $S \neq S^\perp$; $\mathbf{w}_S$, $\mathbf{w}_{S^\perp}$, and $(1,\dotsc,1)$ are pairwise distinct; and $$\operatorname{WT}\bigl(g_S - \sgn(\sigma_S) g_{S^\perp}\bigr) = (-1)^{i+1} \pi^{-m} e_{\{m+1,n+1,\dotsc,\wh{i^*},\dotsc,2n\}}.$$
7. \[it:case7\] $S = \{1,\dotsc,\wh j,\dotsc,n,n+i\}$ for some $i < m + 1 < j^\vee$; $S \neq S^\perp$; $\mathbf{w}_S$, $\mathbf{w}_{S^\perp}$, and $(1,\dotsc,1)$ are pairwise distinct; and $$\begin{gathered}
\operatorname{WT}\bigl(g_S - \sgn(\sigma_S) g_{S^\perp}\bigr) =\\
(-1)^{m+1} \pi^{-m} \bigl( e_{\{i,n\dotsc,\wh{n+j},\dotsc,2n \}} + (-1)^{i+j+1}e_{\{j^\vee,n+1,\dotsc, \wh{i^*},\dotsc,2n\}} \bigr).\end{gathered}$$
8. \[it:case8\] $S = \{1,\dotsc,\wh j,\dotsc,n,n+m+1\}$ for some $m + 1 < j^\vee$; $S \neq S^\perp$; $\mathbf{w}_S$, $\mathbf{w}_{S^\perp}$, and $(1,\dotsc,1)$ are pairwise distinct; and $$\operatorname{WT}\bigl(g_S - \sgn(\sigma_S) g_{S^\perp}\bigr) = (-1)^{m+1} \pi^{-(m+1)}e_{\{m+1,n,\dotsc,\wh{n+j},\dotsc,2n\}}.$$
9. \[it:case9\] $S = \{1,\dotsc,\wh j,\dotsc,n,n+i\}$ for some $m+1 < i < j^\vee$; $S \neq S^\perp$; $\mathbf{w}_S$, $\mathbf{w}_{S^\perp}$, and $(1,\dotsc,1)$ are pairwise distinct; and $$\begin{gathered}
\operatorname{WT}\bigl(g_S - \sgn(\sigma_S) g_{S^\perp}\bigr) =\\
(-1)^{m+1}\pi^{-(m+1)}\bigl( e_{\{i,n+1,\dotsc,\wh{n+j},\dotsc,2n\}} + (-1)^{i+j}e_{\{j^\vee,n+1,\dotsc, \wh{i^*},\dotsc,2n\}}\bigr).\end{gathered}$$
It is clear that the descriptions of $S$ and $\mathbf{w}_S$ in the nine cases cover all possibilities and are mutually exclusive. So we have to show that in each case, the calculation of the worst terms is correct. In cases –, this is simply read off from Lemma \[st:WT(g\_S)\]. The same goes for cases –, using also Lemma \[st:weight\_lem\], and Remark \[rk:sign\_sigma\_S\_type\_(n-1,1)\] to compute the sign of $\sigma_S$.
Thus the only case in which any subtleties arise is , where $S$ and $S^\perp$ are distinct but have the same weight, and therefore the respective worst terms of $g_S$ and $-\sgn(\sigma_S) g_{S^\perp}$ may, and in fact do, cancel. Here is the basic calculation, which we formulate as a separate lemma.
\[st:g\_S\_nontriv\_wt\_1\] Let $S = \{1,\dotsc,\wh{i},\dotsc, n, n+i\}$ for some $i < m+1$. Then $$\begin{gathered}
g_S - \sgn(\sigma_S) g_{S^\perp} = (-1)^i g_1 \wedge \dotsb \wedge \wh{g_i} \wedge \dotsb \wedge \wh{g_{i^\vee}} \wedge \dotsb \wedge g_n\\
\wedge \bigl[(e_i \otimes 1) \wedge (e_{i^\vee} \otimes 1) - (\pi^{-1}e_i \otimes 1) \wedge (\pi e_{i^\vee} \otimes 1)\bigr].\end{gathered}$$
We have $$S^\perp = \{1,\dotsc,\wh{i^\vee},\dotsc,n,i^*\}$$ and, by Remark \[rk:sign\_sigma\_S\_type\_(n-1,1)\], $$-\sgn(\sigma_S) = (-1)^{2i} = 1.$$ Hence $$\begin{aligned}
g_S - \sgn(\sigma_S) g_{S^\perp}
&= g_S + g_{S^\perp}\\
&= g_1 \wedge \dots \wedge \wh{g_{i}} \wedge \dotsb \wedge g_n \wedge g_{n+i} + g_1 \wedge \dotsb \wedge \wh{g_{i^\vee}} \wedge \dotsb \wedge g_n \wedge g_{i^*}\\
&= (-1)^{n-i^\vee} g_1 \wedge \dotsb \wedge \wh{g_i} \wedge \dotsb \wedge \wh{g_{i^\vee}} \wedge \dotsb \wedge g_n \wedge g_{i^\vee} \wedge g_{n+i}\\
&\quad\quad + (-1)^{n-1-i} g_1 \wedge \dotsb \wedge \wh{g_i} \wedge \dotsb \wedge \wh{g_{i^\vee}} \wedge \dotsb \wedge g_n \wedge g_{i} \wedge g_{i^*}\\
&= (-1)^ig_1 \wedge \dotsb \wedge \wh{g_i} \wedge \dotsb \wedge \wh{g_{i^\vee}} \wedge \dotsb \wedge g_n\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \wedge [g_i \wedge g_{i^*} - g_{i^\vee} \wedge g_{n+i}].\end{aligned}$$ It is elementary to verify that $$g_i \wedge g_{i^*} - g_{i^\vee} \wedge g_{n+i} = (e_i \otimes 1) \wedge(e_{i^\vee} \otimes 1) - (\pi^{-1}e_i \otimes 1) \wedge (\pi e_{i^\vee} \otimes 1),$$ which completes the proof.
Returning to the calculation of the worst terms in case in Lemma \[st:WT(g\_S\_pm\_g\_S\^perp)\], Lemma \[st:g\_S\_nontriv\_wt\_1\] implies that $$\begin{aligned}
\operatorname{WT}\bigl(g_S - {}&\sgn(\sigma_S) g_{S^\perp}\bigr)\\
={} &(-1)^i (e_1 \otimes 1) \wedge \dotsb \wedge \wh{(e_i \otimes 1)} \wedge \dotsb \wedge (e_m \otimes 1)\\
&\qquad\wedge (-\pi e_{m+1} \otimes \pi^{-1}) \wedge \dotsb \wedge \wh{(-\pi e_{i^\vee} \otimes \pi^{-1})} \wedge \dotsb \wedge (-\pi e_n \otimes \pi^{-1})\\
&\qquad\qquad\wedge \bigl[(e_i \otimes 1) \wedge (e_{i^\vee} \otimes 1) - (\pi^{-1}e_i \otimes 1) \wedge (\pi e_{i^\vee} \otimes 1)\bigr]\\
={} & (-1)^{m+1}\pi^{-m}\bigl( e_{\{i,n+1,\dotsc,\wh{n+i},\dotsc,2n\}} - e_{\{i^\vee, n+1,\dotsc,\wh{i^*},\dotsc,2n\}} \bigr).\end{aligned}$$ This completes the proof of Lemma \[st:WT(g\_S\_pm\_g\_S\^perp)\].
The following asserts that appropriate multiples of the elements $g_S - \sgn(\sigma_S) g_{S^\perp}$, for $S \preccurlyeq S^\perp$ of type $(n-1,1)$, form an ${\ensuremath{\mathscr{O}}\xspace}_F$-basis of $W(\Lambda_m)_{-1}^{n-1,1}$.
\[st:W(Lambda\_m)\_coeff\_conds\] Let $w \in W_{-1}^{n-1,1}$, and write $$w = \sum_{\substack{S \preccurlyeq S^\perp\\ \text{of type } (n-1,1)}}a_S\bigl(g_S - \sgn(\sigma_S) g_{S^\perp}\bigr),
\quad a_S \in F.$$ Then $w \in W(\Lambda_m)_{-1}^{n-1,1}$ $\iff$
1. \[it:cond\_1\] if $S = \{1,\dotsc,\wh{m+1},\dotsc,n,n+m+1\}$, then $\ord_\pi(a_S) \geq m+1$;
2. \[it:cond\_2\] if $S = \{1,\dotsc,\wh{i^\vee},\dotsc,n,n+i\}$ for some $i < m+1$, then $\ord_\pi(a_S) \geq m-1$;
3. \[it:cond\_3\] if $S = \{1,\dotsc,\wh{i^\vee},\dotsc,n,n+i\}$ for some $i > m+1$, then $\ord_\pi(a_S) \geq m+1$;
4. \[it:cond\_4\] if $S = \{1,\dotsc,\wh{i},\dotsc,n,n+i\}$ for some $i < m+1$, then $\ord_\pi(a_S) \geq m$;
5. \[it:cond\_5\] if $S = \{1,\dotsc,\wh j,\dotsc,n,n+i\}$ for some $i < j^\vee < m+1$, then $\ord_\pi(a_S) \geq m-1$;
6. \[it:cond\_6\] if $S = \{1,\dotsc,\wh j,\dotsc,n,n+i\}$ for some $i < m+1 \leq j^\vee$, then $\ord_\pi(a_S) \geq m$; and
7. \[it:cond\_7\] if $S = \{1,\dotsc,\wh j,\dotsc,n,n+i\}$ for some $m+1 \leq i < j$, then $\ord_\pi(a_S) \geq m+1$.
The implication $\Longleftarrow$ is immediate from Lemma \[st:WT(g\_S\_pm\_g\_S\^perp)\]. For the implication $\Longrightarrow$, assume $w \in W(\Lambda_m)_{-1}^{n-1,1}$, and write $$w = \sum_{\mathbf{w}\in\ZZ^n}
\sum_{\substack{S \preccurlyeq S^\perp\\ \text{of type } (n-1,1)\\ \text{and weight }\mathbf{w}}}a_S\bigl(g_S - \sgn(\sigma_S) g_{S^\perp}\bigr).$$ By Lemma \[st:weight\_lem\], $w \in W(\Lambda_m)_{-1}^{n-1,1}$ $\iff$ for each $\mathbf{w}$, $$\label{disp:fixed_w_lin_comb}
\sum_{\substack{S \preccurlyeq S^\perp\\ \text{of type } (n-1,1)\\ \text{and weight }\mathbf{w}}}a_S\bigl(g_S - \sgn(\sigma_S) g_{S^\perp}\bigr) \in W(\Lambda_m)_{-1}^{n-1,1}.$$ Thus we reduce to working weight by weight. If $\mathbf{w} \neq (1,\dotsc,1)$, then by Remark \[rk:(n-1,1)\_possible weights\] there is at most one $S$ of type $(n-1,1)$ and weight $\mathbf w$. This and Lemma \[st:WT(g\_S\_pm\_g\_S\^perp)\] imply , , , , and .
The case $\mathbf{w} = (1,\dotsc,1)$ requires a finer analysis, since in this case all the sets $$S_i := \bigl\{1,\dotsc,\wh{i},\dotsc,n,n+i\bigr\} \quad\text{for}\quad i = 1,\dotsc,m+1$$ occur as summation indices in . First consider the set $S_{m+1}$. For $i < m+1$, write $g_{S_i} - \sgn(\sigma_{S_i}) g_{S_i^\perp}$ as a linear combination of $e_{S}$’s. By Lemma \[st:g\_S\_nontriv\_wt\_1\], every $e_{S}$ that occurs in this linear combination must involve wedge factors of either $e_i \otimes 1$ and $e_{i^\vee} \otimes 1$ together, or $\pi^{-1} e_i \otimes 1$ and $\pi e_{i^\vee} \otimes 1$ together. By Lemma \[st:WT(g\_S\_pm\_g\_S\^perp)\], $$\operatorname{WT}\bigl(g_{S_{m+1}} - \sgn(\sigma_{S_{m+1}}) g_{S_{m+1}}\bigr) = \pi^{-(m+1)}e_{\{n+1,\dotsc,2n\}}$$ involves no such $e_S$. This implies .
By the same argument, for fixed $i < m+1$, the $e_S$-basis vectors that occur in $\operatorname{WT}\bigl(g_{S_i} - \sgn(\sigma_{S_i}) g_{S_i^\perp}\bigr)$, namely $e_{\{i,n+1,\dotsc,\wh{n+i},\dotsc,2n\}}$ and $e_{\{i^\vee,n+1,\dotsc,\wh{i^*},\dotsc,2n\}}$, don’t occur in $g_{S_j} - \sgn(\sigma_{S_j}) g_{S_j^\perp}$ for $j \neq i$, $m+1$. It follows easily from this, the fact that we’ve already shown that $\ord_\pi(a_{S_{m+1}}) \geq m+1$, and Lemma \[st:WT(g\_S\_pm\_g\_S\^perp)\] that $\ord_\pi(a_{S_i}) \geq m$.
The following consequence is what we’ll need for the proof of Proposition \[st:X\_1=0\].
\[st:im\_basis\] Let $R$ be a $k$-algebra. Then $L_m^{n-1,1}(R) \subset W(\Lambda_m) \otimes_{{\ensuremath{\mathscr{O}}\xspace}_F} R$ is a free $R$-module on the basis elements
1. \[it:im1\] $e_{\{n+1,\dotsc,2n\}}$;
2. \[it:im2\] $e_{\{i,n+1,\dotsc,\wh{i^*},\dotsc,2n\}}$ for $i = 1,\dotsc, \wh{m+1},\dotsc,n$;
3. \[it:im3\] $e_{\{i,n+1,\dotsc,\wh{n+i},\dotsc,2n\}} - e_{\{i^\vee,n+1,\dotsc,\wh{i^*},\dotsc,2n\}}$ for $i < m+1$;
4. \[it:im4\] $e_{\{i,n+1,\dotsc,\wh{n+j},\dotsc,2n\}} + (-1)^{i+j}e_{\{j^\vee,n+1,\dotsc, \wh{i^*},\dotsc,2n\}}$ for $i < j^\vee < m+1$ and $m+1 < i < j^\vee \leq n$;
5. \[it:im5\] $e_{\{m+1,n+1,\dotsc,\wh{n+i},\dotsc,2n\}}$ for $i = 1,\dotsc, \wh{m+1},\dotsc,n$; and
6. \[it:im6\] $e_{\{i,n+1,\dotsc,\wh{n+j},\dotsc,2n\}} + (-1)^{i+j+1}e_{\{j^\vee,n+1,\dotsc, \wh{i^*},\dotsc,2n\}}$ for $i < m+1 < j^\vee \leq n$.
Immediate from Lemma \[st:WT(g\_S\_pm\_g\_S\^perp)\] and Proposition \[st:W(Lambda\_m)\_coeff\_conds\], using that under the canonical map $W(\Lambda_m)_{-1}^{n-1,1} \to W(\Lambda_m) \otimes_{{\ensuremath{\mathscr{O}}\xspace}_F} R$, the image of any element $w$ is the same as the image of $\operatorname{WT}(w)$.
Proof of Proposition \[st:X\_1=0\] {#proof-of-proposition-stx_10}
----------------------------------
We now prove Proposition \[st:X\_1=0\]. Let $R$ be a $k$-algebra, and let $X_1$ be an $(n-1) \times (n-1)$-matrix with entries in $R$ satisfying $$X_1 = -JX_1^{\ensuremath{\mathrm{t}}\xspace}J, \quad X_1^2 = 0, \quad \sideset{}{^2}\bigwedge X_1 = 0, \quad\text{and}\quad \tr X_1 = 0,$$ and such that when $X_1$ is plugged into along with $X_2$, $X_3$, and $X_4$ all $0$, the resulting $R$-point on $U^\wedge$ lies in $U$, i.e. it satisfies . Our problem is to show that $X_1 = 0$.
In fact we will show that just the conditions $X_1 = -JX_1^{\ensuremath{\mathrm{t}}\xspace}J$ and imply that $X_1 = 0$. Decompose $X_1$ into $m \times m$ blocks $$X_1 = \begin{pmatrix}
A & B\\
C & D\\
\end{pmatrix}.$$ Then $X_1 = -JX_1^tJ$ is equivalent to $$\label{disp:LM_conds}
D = A^\ad, \quad
B = -B^\ad, \quad\text{and}\quad
C = -C^\ad,$$ where the superscript $\ad$ means to take the transpose across the antidiagonal, or in other words, to take the adjoint with respect to the standard split symmetric form. We are going to show that imposes the *same conditions as in except with opposite signs.* Since $\charac k \neq 2$, this will imply that $A = B = C = D = 0$.
Let $v \in W(\Lambda_m) \otimes_{{\ensuremath{\mathscr{O}}\xspace}_F} R$ denote the wedge product (say from left to right) of the $n$ columns of the matrix , where each column is regarded as an $R$-linear combination of Arzdorf’s basis elements . Condition is that $$v \in L_m^{n-1,1}(R).$$
We begin by analyzing the implications of this condition on the entries of $A$ and $D$. Let $$1 \leq i,j \leq m,$$ and let $a_{ij}$ and $d_{ij}$ denote the $(i,j)$-entries of $A$ and $D$, respectively. Let $$S := \{m+1+i,n+1,\dotsc,(n+m+1+j)\sphat\;,\dotsc,2n\}.$$ Then $$S^\perp = \{m+1-j,n+1,\dotsc,(n+m+1-i)\sphat\;,\dotsc,2n\}.$$ Writing $v$ as a linear combination of $e_{S'}$’s, the $e_S$-term in $v$ is $$\begin{split}
& (-1)^{1+j}a_{ij}(e_{m+1+i} \otimes 1) \\
&\qquad\wedge (\pi e_{m+2} \otimes 1) \wedge \dotsb \wedge (\pi e_{m+1+j} \otimes 1)\sphat\, \wedge \dotsb \wedge (\pi e_n \otimes 1)\\
&\qquad\qquad\wedge (e_1 \otimes 1) \wedge \dotsb \wedge (e_m \otimes 1) \wedge (\pi e_{m+1} \otimes 1)\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad = (-1)^{1+j+(m+1)(m-1)} a_{ij}e_S\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad = (-1)^{m+j}a_{ij}e_S,
\end{split}$$ and the $e_{S^\perp}$-term in $v$ is $$\begin{split}
& (-1)^{1+n-i}d_{m+1-j,m+1-i}(\pi^{-1}e_{m+1-j} \otimes 1)\\
&\qquad \wedge (\pi e_{m+2} \otimes 1) \wedge \dotsb \wedge (\pi e_n \otimes 1)\\
&\qquad\qquad\wedge (e_1 \otimes 1) \wedge \dotsb \wedge (e_{m+1-i} \otimes 1)\sphat\, \wedge \dotsb \wedge (e_m \otimes 1) \wedge (\pi e_{m+1} \otimes 1)\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad = (-1)^{1+n-i+m^2} d_{m+1-j,m+1-i}e_{S^\perp}\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad = (-1)^{m+i}d_{m+1-j,m+1-i}e_{S^\perp}.
\end{split}$$ If $i = j$, then using Corollary \[st:im\_basis\], especially part applied with $m+1-i$ in place of $i$, the condition $v \in L_m^{n-1,1}(R)$ requires that $$d_{m+1-i,m+1-i} = -a_{ii}.$$ If $i \neq j$, then using Corollary \[st:im\_basis\], especially part applied with $m+1-j$ in place of $i$ and $m+1-i$ in place of $j$, we similarly find that $$d_{m+1-j,m+1-i} = -a_{ij}.$$ Hence $D^\ad = -A$, as desired.
We next turn to $B$. By the antidiagonal entries of $B$ are $0$. Let $$1 \leq i < m+1-j \leq m,$$ and let $b_{ij}$ denote the $(i,j)$-entry of $B$. Let $$S := \{m+1+i,n+1,\dotsc,\wh{n+j},\dotsc,2n\}.$$ Then $$S^\perp = \{j^\vee,n+1,\dotsc,(n+m+1-i)\sphat\; ,\dotsc,2n\},$$ which differs from $S$ since $i < m+1-j$. The $e_S$-term in $v$ is $$\begin{split}
& (-1)^{1+m+j}b_{ij}(e_{m+1+i} \otimes 1) \wedge (\pi e_{m+2} \otimes 1) \wedge \dotsb \wedge (\pi e_n \otimes 1)\\
&\qquad\wedge (e_1 \otimes 1) \wedge \dotsb \wedge\wh{(e_{j} \otimes 1)} \wedge \dotsb \wedge (e_m \otimes 1) \wedge (\pi e_{m+1} \otimes 1)\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad = (-1)^{1+m+j+m^2} b_{ij}e_S\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad = (-1)^{1+j}b_{ij}e_S,
\end{split}$$ and the $e_{S^\perp}$-term in $v$ is $$\begin{split}
& (-1)^{1+m+(m+1-i)}b_{m+1-j,m+1-i}(e_{j^\vee} \otimes 1) \wedge (\pi e_{m+2} \otimes 1) \wedge \dotsb \wedge (\pi e_n \otimes 1)\\
&\qquad\wedge (e_1 \otimes 1) \wedge \dotsb \wedge (e_{m+1-i} \otimes 1)\sphat\, \wedge \dotsb \wedge (e_m \otimes 1) \wedge (\pi e_{m+1} \otimes 1)\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad = (-1)^{i+m^2} b_{m+1-j,m+1-i}e_{S^\perp}\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad = (-1)^{i+m}b_{m+1-j,m+1-i}e_{S^\perp}.
\end{split}$$ As above, this time using Corollary \[st:im\_basis\] with $m+1+i$ in place of $i$, we find that $v \in L_m^{n-1,1}(R)$ requires that $$b_{m+1-j,m+1-i} = b_{ij}.$$ Hence $B^\ad = B$, as desired.
We finally turn to $C$, for which the argument is almost identical to the one for $B$. By the antidiagonal entries of $C$ are $0$. Let $$1 \leq i < m+1-j \leq m,$$ and let $c_{ij}$ denote the $(i,j)$-entry of $C$. Let $$S := \{i,n+1,\dotsc,(n+m+1+j)\sphat\; ,\dotsc,2n\}.$$ Then $$S^\perp = \{m+1-j,n+1,\dotsc,\wh{i^*},\dotsc,2n\} \neq S.$$ The $e_S$-term in $v$ is $$\begin{split}
& (-1)^{1+j}c_{ij}(\pi^{-1}e_{i} \otimes 1)\\
&\qquad \wedge (\pi e_{m+2} \otimes 1) \wedge \dotsb \wedge (\pi e_{m+1+j} \otimes 1)\sphat\, \wedge \dotsb \wedge (\pi e_n \otimes 1)\\
&\qquad\qquad\wedge (e_1 \otimes 1) \wedge \dotsb \wedge (e_m \otimes 1) \wedge (\pi e_{m+1} \otimes 1)\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad = (-1)^{1+j+(m+1)(m-1)} c_{ij}e_S\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad = (-1)^{j+m}c_{ij}e_S,
\end{split}$$ and the $e_{S^\perp}$-term in $v$ is $$\begin{split}
& (-1)^{1+m+1-i}c_{m+1-j,m+1-i}(\pi^{-1}e_{m+1-j} \otimes 1)\\
&\qquad \wedge (\pi e_{m+2} \otimes 1) \wedge \dotsb \wedge (\pi e_{i^\vee} \otimes 1)\sphat\, \wedge \dotsb \wedge (\pi e_n \otimes 1)\\
&\qquad\qquad\wedge (e_1 \otimes 1) \wedge \dotsb \wedge (e_m \otimes 1) \wedge (\pi e_{m+1} \otimes 1)\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad = (-1)^{m+i+(m+1)(m-1)} c_{m+1-j,m+1-i}e_{S^\perp}\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad = (-1)^{i+1}c_{m+1-j,m+1-i}e_{S^\perp}.
\end{split}$$ Using Corollary \[st:im\_basis\] with $m+1+j$ in place of $j$, we find as before that $v \in L_m^{n-1,1}(R)$ requires that $$c_{m+1-j,m+1-i} = c_{ij}.$$ Hence $C^\ad = C$, as desired. This completes the proof of Proposition \[st:X\_1=0\], which in turn completes the proof of Theorem \[st:main\_thm\].
Further remarks {#s:remarks}
===============
In this final section of the paper we collect a few general remarks. We return to the setting of arbitrary $n$, signature $(r,s)$, and $I$ satisfying .
Relation to the Kottwitz condition {#ss:kottwitz}
----------------------------------
It is notable that the Kottwitz condition didn’t intervene explicitly in the proof of Proposition \[st:X\_1=0\] or, more generally, of Theorem \[st:main\_thm\]. There is a good reason for this, as we shall now see.
For $R$ an ${\ensuremath{\mathscr{O}}\xspace}_F$-algebra, define the following condition on an $R$-point $(\F_i)_{i}$ of $M_I^{\ensuremath{\mathrm{naive}}\xspace}$:
1. \[it:K\_n\] for all $i$, the line $\bigwedge_R^n \F_i \subset W(\Lambda_i) \otimes_{{\ensuremath{\mathscr{O}}\xspace}_F} R$ is contained in $$\label{disp:K_n_im}
\im \bigl[ \bigl(W^{r,s} \cap W(\Lambda_i)\bigr) \otimes_{{\ensuremath{\mathscr{O}}\xspace}_F} R \to W(\Lambda_i) \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F}} R \bigr].$$
As usual, this defines a condition on $M_I^{\ensuremath{\mathrm{naive}}\xspace}$ when $r \neq s$, and when $r = s$ it descends from $M_{I,{\ensuremath{\mathscr{O}}\xspace}_F}^{\ensuremath{\mathrm{naive}}\xspace}$ to $M_I^{\ensuremath{\mathrm{naive}}\xspace}$, since in this case $W^{r,s}$ is Galois-stable.
Condition is trivially implied by . In the generic fiber, is equivalent to the Kottwitz condition, and in general we have the following.
\[st:K\_n==>Kottwitz\] Condition , and a fortiori condition , implies the Kottwitz condition.
Let $T$ be a formal variable. For any $w \in W^{r,s}$, the identity $$\Bigl[\sideset{}{^n}\bigwedge (T - \pi \otimes 1)\Bigr] \cdot w = (T+\pi)^r(T-\pi)^s w$$ holds true. Hence this identity holds true for any $w$ in the image .
Wedge power analogs {#ss:wedge_power_analogs}
-------------------
More generally, one can formulate an analog of condition for any wedge power. For $1 \leq l \leq n$, let $$\tensor[^l]W{^{r,s}} := \bigoplus_{\substack{j+k=l\\ j\leq r\\ k \leq s}}
\Bigl(\sideset{}{_F^j}\bigwedge V_{-\pi} \otimes_F \sideset{}{_F^k}\bigwedge V_\pi\Bigr) \subset \sideset{}{_F^l}\bigwedge V.$$ In terms of our previous notation, $\tensor[^n]W{^{r,s}} = W^{r,s}$. For any ${\ensuremath{\mathscr{O}}\xspace}_{F_0}$-lattice $\Lambda$ in $F^n$, let $$\tensor[^l]W{} (\Lambda)^{r,s} := \tensor[^l]W{^{r,s}} \cap \sideset{}{_{{\ensuremath{\mathscr{O}}\xspace}_F}^l}\bigwedge (\Lambda \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} {\ensuremath{\mathscr{O}}\xspace}_F)$$ (intersection in $\bigwedge_F^lV$). Then, for $R$ an ${\ensuremath{\mathscr{O}}\xspace}_F$-algebra and $(\F_i)_i$ an $R$-point on the naive local model, we formulate the condition that
1. \[it:K\_l\] for all $i$, the subbundle $\bigwedge_R^l \F_i \subset \bigwedge_R^l (\Lambda \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} R)$ is contained in $$\im \Bigl[ \tensor[^l]W{} (\Lambda_i)^{r,s} \otimes_{{\ensuremath{\mathscr{O}}\xspace}_F} R \to \sideset{}{_R^l}\bigwedge (\Lambda \otimes_{{\ensuremath{\mathscr{O}}\xspace}_{F_0}} R) \Bigr].$$
As usual, this condition descends from $M_{I,{\ensuremath{\mathscr{O}}\xspace}_F}^{\ensuremath{\mathrm{naive}}\xspace}$ to $M_I^{\ensuremath{\mathrm{naive}}\xspace}$ when $r = s$. Conditions for $l = 1,\dotsc$, $n$ are closed conditions on $M_I^{\ensuremath{\mathrm{naive}}\xspace}$, and they are all implied by the Kottwitz condition in the generic fiber. Therefore they all hold on $M_I^{\ensuremath{\mathrm{loc}}\xspace}$. We do not know the relation between the conditions in general. If our conjecture that $M_I = M_I^{\ensuremath{\mathrm{loc}}\xspace}$ holds true, then they are all implied by conditions –. Here is a statement pointing in the other direction.
Assume $r \neq s$. Then conditions $(\mathrm{K}_{r+1})$ and $(\mathrm{K}_{s+1})$ imply the wedge condition .
As in the proof of Lemma \[st:K\_n==>Kottwitz\], this is an immediate consequence of the fact that for any $w \in \tensor[^{s+1}]W{^{r,s}}$ and $w' \in \tensor[^{r+1}]W{^{r,s}}$, we have $$\Bigl[ \sideset{}{^{s+1}}\bigwedge (\pi \otimes 1 + 1 \otimes \pi)\Bigr] \cdot w = 0
\quad\text{and}\quad
\Bigl[ \sideset{}{^{r+1}}\bigwedge (\pi \otimes 1 - 1 \otimes \pi)\Bigr] \cdot w' = 0.\qedhere$$
\[rk:newcond==>wedge\] It is perhaps interesting to note that the flatness of $M_{\{m\}}$ in the setting of Theorem \[st:main\_thm\] follows from only conditions – in the definition of the naive local model and our new condition . Indeed, the only other condition needed to make our proof go through is the wedge condition in the special fiber, which is used in the reduction argument given in \[ss:reduction\] and in the proof of topological flatness. Since the elements $e_S$ occurring in the statement of Corollary \[st:im\_basis\] are all for $S$ of type $(1,n-1)$ and $(0,n)$, condition implies that the $2 \times 2$-minors of the matrix $X$ in must vanish, which shows that implies the wedge condition inside the open subscheme $U^{\ensuremath{\mathrm{naive}}\xspace}$ of the special fiber defined in . Therefore implies the wedge condition on the entire special fiber, since this neighborhood contains the worst point and is invariant under polarized lattice chain automorphisms in the sense of [@rapzink96]\*[3]{}. One can similarly show that implies the wedge condition in the special fiber for any signature (still with $n$ odd and $I = \{m\}$) by proving the suitable analog of Corollary \[st:im\_basis\].
Application to Shimura varieties {#ss:Sh}
--------------------------------
In this subsection we give the most immediate application of Theorem \[st:main\_thm\] to Shimura varieties. Indeed, we are simply going to make explicit Rapoport and Zink’s general definition of integral models of PEL Shimura varieties [@rapzink96 Def. 6.9], in the particular case of a unitary similitude group ramified and quasi-split at $p$ and of signature $(n-1,1)$ for odd $n$, where the level structure at $p$ is the special maximal parahoric subgroup corresponding to $I = \{m\}$ (in the sense of [@paprap09 1.2.3(a)]); and in a way that furthermore incorporates our condition . This is also the setting of [@paprap09 1] (which allows any $n$, $I$, and signature), and differs only slightly from [@pappas00 3] (which allows any $n$ and signature and takes $I = \{0\}$). By the general formalism of local models, Richarz’s smoothness result [@arzdorf09 Prop. 4.16] and Theorem \[st:main\_thm\] will show that the moduli space we write down is smooth.
Let $K/\QQ$ be an imaginary quadratic field in which the prime $p \neq 2$ ramifies, and take $F_0 = \QQ_p$ and $F = K \otimes_\QQ \QQ_p$. Let $a \mapsto \ol a$ denote the nontrivial element of $\Gal(K/\QQ)$. Let $n$ be odd, and let $H$ be an $n$-dimensional $K/\QQ$-Hermitian space of signature $(n-1,1)$ (as in [@paprap09 1.1]) such that the $F/\QQ_p$-Hermitian space $H \otimes_\QQ \QQ_p$ is split, i.e. it has a basis $e_1,\dotsc,e_n$ such that holds. Fix such a basis, and define the lattice chain $$\Lambda_{\{m\}} := \{\Lambda_i\}_{i \in \pm m + n\ZZ}$$ in $H \otimes \QQ_p$ with respect to it, with $\Lambda_i$ as in . Let $G := GU(H)$.
We now use (essentially) the notation and terminology of [@rapzink96 6.1–6.9] for abelian schemes. For $R$ a ring, let $AV(R)$ denote the category of abelian schemes with ${\ensuremath{\mathscr{O}}\xspace}_K$-action over $\Spec R$ up to prime-to-$p$ isogeny. Thus an object in $AV(R)$ consists of a pair $(A,\iota)$, where $A$ is an abelian scheme over $\Spec R$ and $\iota$ is a ring homomorphism ${\ensuremath{\mathscr{O}}\xspace}_K \to \End_R(A) \otimes_\ZZ \ZZ_{(p)}$; and the morphisms $(A_1,\iota_1) \to (A_2,\iota_2)$ in $AV(R)$ consist of the elements in $\Hom_R(A_1,A_2)\otimes \ZZ_{(p)}$ commuting with the ${\ensuremath{\mathscr{O}}\xspace}_K$-actions. The dual of an object $(A,\iota)$ in $AV(R)$ is the pair $(A^\vee,\iota^\vee)$, where $A^\vee$ is the dual abelian scheme and $\iota^\vee(a) := \iota(\ol a)^\vee$. A -homogeneous polarization is a collection $\ol \lambda$ of quasi-isogenies $A \to A^\vee$ such that any two elements in $\ol\lambda$ differ Zariski-locally on $\Spec R$ by an element in $\QQ^\times$, and such that $\ol\lambda$ contains a polarization in $AV(R)$. Note that, by definition of the ${\ensuremath{\mathscr{O}}\xspace}_K$-action on the dual abelian scheme, the Rosati involution attached to any polarization induces the nontrivial Galois automorphism on $K$.
Now we define the moduli space. Let $C^p \subset G(\AA_f^p)$ be a sufficiently small open compact subgroup. We denote by $\S_{C^p}$ the moduli problem over $\Spec {\ensuremath{\mathscr{O}}\xspace}_F$ which associates to each ${\ensuremath{\mathscr{O}}\xspace}_F$-algebra $R$ the set of isomorphism classes of quadruples $(A,\iota,\ol\lambda,\ol\eta)$ consisting of
1. an object $(A,\iota)$ in $AV(R)$ such that $A$ has relative dimension $n$ over $\Spec R$;
2. \[plzn\] a $\QQ$-homogeneous polarization $\ol\lambda$ on $(A,\iota)$ containing a polarization $\lambda$ such that $\ker\lambda \subset A[\iota(\pi)]$ of height $n-1$, where $\pi$ is a uniformizer in ${\ensuremath{\mathscr{O}}\xspace}_K \otimes_\ZZ \ZZ_{(p)}$; and
3. a $C^p$-level structure $\ol\eta\colon H_1(A,\AA_f^p) \isoarrow H \otimes_\QQ \AA_f^p \bmod C^p$ that respects the bilinear forms on both sides up to a constant in $(\AA_f^p)^\times$ (see for details);
subject to the following condition, which is the translation to this setting of . Given such a quadruple $(A,\iota,\ol\lambda,\ol\eta)$, let $\lambda$ and $\pi$ be as in . Since $A$ is polarized and of relative dimension $n$, its $p$-divisible group has height $2n$. Hence $\iota(\pi)$ is an isogeny (in $AV(R)$) of height $n$, since $p$ ramifies in $K$. Hence there exists a unique isogeny $\rho\colon A^\vee \to A$ (which has height $1$) such that the composite $$A \xra\lambda A^\vee \xra\rho A$$ is $\iota(\pi)$. Extending this diagram periodically, we obtain an ${\ensuremath{\mathscr{L}}\xspace}$-set of abelian varieties, in the terminology of [@rapzink96 Def. 6.5], for ${\ensuremath{\mathscr{L}}\xspace}$ the ${\ensuremath{\mathscr{O}}\xspace}_F$-lattice chain $\Lambda_{\{m\}}$. Writing $f$ for the structure morphism $f\colon A \to \Spec R$, define $M(A) := (R^1f_*(\Omega^\bullet_{A/R}))^\vee$ to be the $R$-linear dual of the first de Rham cohomology module of $A$. Likewise define $M(A^\vee)$. Then $M(A)$ and $M(A^\vee)$ are finite locally free $R$-modules of rank $2n$, and the isogenies $\lambda$ and $\rho$ induce a diagram of ${\ensuremath{\mathscr{O}}\xspace}_F \otimes_{\ZZ_p} R$-modules $$M(A) \xra{\lambda_*} M(A^\vee) \xra{\rho_*} M(A),$$ which extends periodically to a chain of ${\ensuremath{\mathscr{O}}\xspace}_F \otimes_{\ZZ_p} R$-modules of type $(\Lambda_{\{m\}})$, in the terminology of [@rapzink96 Def. 3.6]. The polarization $\lambda$ makes this into a polarized chain of ${\ensuremath{\mathscr{O}}\xspace}_F \otimes_{\ZZ_p} R$-modules in a natural way. By [@pappas00 Th. 2.2], étale-locally on $\Spec R$ there exists an isomorphism of polarized chains between this chain and the chain $\Lambda_{\{m\}} \otimes_{\ZZ_p} R$ which respects the forms on both sides up to a scalar in $R^\times$; here the polarization on $\Lambda_{\{m\}} \otimes_{\ZZ_p} R$ is induced from . In particular, this chain isomorphism gives an isomorphism of modules $$\label{isom}
M(A) \isom \Lambda_{-m} \otimes_{\ZZ_p} R,$$ and the condition we finally impose on $(A,\iota,\ol\lambda,\ol\eta)$ is that
1. \[cond\] upon identifying the ${\ensuremath{\mathrm{Fil}}\xspace}^1$ term in the covariant Hodge filtration $$0 \to {\ensuremath{\mathrm{Fil}}\xspace}^1 \to M(A) \to \Lie A \to 0$$ with a submodule $\F \subset \Lambda_{-m} \otimes_{\ZZ_p} R$ via , the line $$\sideset{}{_R^n}\bigwedge \F \subset W(\Lambda_{-m}) \otimes_{\ZZ_p} R$$ is contained in $L_{-m}^{n-1,1}(R)$.
Of course the notation here is as in \[ss:new\_cond\]. It is not hard to see that is independent of the polarized chain isomorphism used above, as well as the choice of $\lambda$ and $\pi$, so that it is a well-defined condition. By the formalism of the local model diagram (see e.g. [@pappas00 Th. 2.2]), since the scheme $M_{\{m\}}$ is a closed subscheme of $M_{\{m\}}^{\ensuremath{\mathrm{naive}}\xspace}$ (and also using Remark \[rk:newcond==>wedge\]), $\S_{C^p}$ is a closed subscheme of the scheme denoted $\A_{C^p}$ in [@rapzink96 Def. 6.9] and [@pappas00 2]; in particular, $\S_{C^p}$ is representable by a quasi-projective scheme over $\Spec {\ensuremath{\mathscr{O}}\xspace}_F$. Since furthermore $M_{\{m\}}$ and $M_{\{m\}}^{\ensuremath{\mathrm{naive}}\xspace}$ have the same generic fiber, the same is true of $\S_{C^p}$ and $\A_{C^p}$. Hence the generic fiber of $\S_{C^p}$ is the base change to $\Spec F$ of a Shimura variety attached to $G$ and the Hermitian data above [@paprap09 (1.19)]. By Richarz’s smoothness result [@arzdorf09 Prop. 4.16] and Theorem \[st:main\_thm\], we obtain the following.
$\S_{C^p}$ is smooth over $\Spec {\ensuremath{\mathscr{O}}\xspace}_E$.
Theorem \[st:main\_thm\] can also be applied to certain Shimura varieties for unitary groups attached to CM fields at primes which do not ramify in the totally real subfield, but do in the CM extension. However we will leave the details for another occasion.
The general PEL setting {#ss:PEL_setting}
-----------------------
We conclude the paper by explaining how to formulate the condition introduced in \[ss:wedge\_power\_analogs\] in the general PEL setting of Rapoport and Zink’s book [@rapzink96]. Given a $\QQ_p$-vector space $V$ and a $\QQ_p$-algebra $R$, we write $V_R := V \otimes_{\QQ_p} R$.
Let $\bigl(F,B,V,{\ensuremath{\langle\text{~,~}\rangle}\xspace},*,{\ensuremath{\mathscr{O}}\xspace}_B,\{\mu\},{\ensuremath{\mathscr{L}}\xspace}\bigr)$ be a (local) PEL datum as in [@rapzink96]\*[1.38, Def. 3.18]{}; see also [@paprap05]\*[14]{}. This means that
- $F$ is a finite product of finite field extensions of $\QQ_p$;
- $B$ is a finite semisimple $\QQ_p$-algebra with center $F$;
- $V$ is a finite-dimensional left $B$-module;
- [$\langle\text{~,~}\rangle$]{}is a nondegenerate alternating $\QQ_p$-bilinear form $V \times V \to \QQ_p$;
- $b \mapsto b^*$ in an involution on $B$ satisfying $\langle bv,w\rangle = \langle v,b^*w\rangle$ for all $v,w \in V$;
- ${\ensuremath{\mathscr{O}}\xspace}_B$ is a $*$-invariant maximal order of $B$;
- $\{\mu\}$ is a geometric conjugacy class of cocharacters of the algebraic group $G$ over $\Spec \QQ_p$ whose $R$-points, for any commutative $\QQ_p$-algebra $R$, are $$G(R) = \biggl\{\, g \in GL_B(V_R) \biggm|
\begin{varwidth}{\linewidth}
\centering
there exists $c(g) \in R^\times$ such that\\
$\langle gv,gw\rangle = c(g)\langle v,w\rangle$ for all $v,w \in V_R$
\end{varwidth}
\,\biggr\};$$ and
- Łis a self-dual multichain of ${\ensuremath{\mathscr{O}}\xspace}_B$-lattices in $V$ [@rapzink96]\*[Defs. 3.4, 3.13]{}.
The conjugacy class $\{\mu\}$ is required to satisfy the conditions that for one, hence any, representative $\mu$ defined over one, hence any, extension $K$ of $\QQ_p$, the weights of $\GG_{m,K}$ acting on $V_K$ via $\mu$ are $0$ and $1$, and the composite $c \circ \mu$ ($c$ the similitude character of $G$) is $\id_{\GG_{m,K}}$. The (local) reflex field $E$ is the field of definition of the conjugacy class $\{\mu\}$, which may also be described as $$\QQ_p\bigl(\,\tr_{\ol \QQ_p} (b \mid V_1) \bigm| b \in B\,\bigr),$$ where $\ol\QQ_p$ is an algebraic closure of $\QQ_p$ and $$V_1 \subset V_{\ol\QQ_p}$$ is the weight $1$ subspace of a representative $\mu \in \{\mu\}$.
Rapoport and Zink attach to the above datum the “naive” local model in [@rapzink96]\*[Def. 3.27]{}. The following is an obvious variant.
We denote by $M^{\ensuremath{\mathrm{naive}}\xspace}$ the scheme over $\Spec {\ensuremath{\mathscr{O}}\xspace}_E$ representing the functor whose values in an ${\ensuremath{\mathscr{O}}\xspace}_E$-algebra $R$ consist of all pairs of
- a functor $\Lambda \mapsto \F_\Lambda$ from Ł(regarded as a category in which the morphisms are inclusions of lattices in $V$) to the category of ${\ensuremath{\mathscr{O}}\xspace}_B \otimes_{\ZZ_p} R$-modules; and
- a natural transformation of functors $j_\Lambda\colon \F_\Lambda \to \Lambda \otimes_{\ZZ_p} R$,
such that
1. for all $\Lambda\in{\ensuremath{\mathscr{L}}\xspace}$, $\F_\Lambda$ is a submodule of $\Lambda \otimes_{\ZZ_p} R$ which is an $R$-direct summand, and $j_\Lambda$ is the natural inclusion $\F_\Lambda \subset \Lambda \otimes_{\ZZ_p} R$;
2. for all $\Lambda \in {\ensuremath{\mathscr{L}}\xspace}$ and all $b \in B$ which normalize ${\ensuremath{\mathscr{O}}\xspace}_B$, the composite $$\F_\Lambda^b \subset (\Lambda \otimes_{\ZZ_p} R)^b \xra[{\raisebox{0.4ex}{\smash[t]{$\scriptstyle\sim$}}}]b b\Lambda \otimes_{\ZZ_p} R$$ identifies $(\F_\Lambda)^b$ with $\F_{b\Lambda}$; here for any ${\ensuremath{\mathscr{O}}\xspace}_B$-module $N$, we denote by $N^b$ the ${\ensuremath{\mathscr{O}}\xspace}_B$-module whose underlying abelian group is $N$ and whose ${\ensuremath{\mathscr{O}}\xspace}_B$-action is given by $x \cdot n = b^{-1}x b n$, for $x \in {\ensuremath{\mathscr{O}}\xspace}_B$ and $n\in N$;
3. for all $\Lambda \in {\ensuremath{\mathscr{L}}\xspace}$, under the perfect pairing $(\Lambda \otimes_{\ZZ_p} R) \times \bigl(\wh \Lambda \otimes_{\ZZ_p} R \bigr) \to R$ induced by [$\langle\text{~,~}\rangle$]{}, where $\wh\Lambda$ denotes the dual lattice of $\Lambda$, the submodules $\F_\Lambda$ and $\F_{\wh\Lambda}$ pair to $0$; and
4. (Kottwitz condition) for all $\Lambda \in {\ensuremath{\mathscr{L}}\xspace}$, there is an equality of polynomial functions over $R$ $$\det(b\mid \F_\Lambda) = \det(b \mid V_1), \quad b \in {\ensuremath{\mathscr{O}}\xspace}_B$$ (see [@rapzink96]\*[3.23(a)]{} for the precise meaning of this), where $V_1 \subset V_{\ol \QQ_p}$ is as above.
Since $V$ admits a nondegenerate symplectic form, its $\QQ_p$-dimension is even, say equal to $2n$. Choose any $\mu \in \{\mu\}$, and let $$V_{\ol\QQ_p} = V_1 \oplus V_0$$ be the corresponding weight decomposition. The condition that $c \circ \mu$ is the identity forces both $V_1$ and $V_0$ to be totally isotropic for the form induced by [$\langle\text{~,~}\rangle$]{}, and therefore both have dimension $n$. Fix an integer $l$ with $1 \leq l \leq n$.
For $b \in {\ensuremath{\mathscr{O}}\xspace}_B$, let $\chi_b(T)$ denote the characteristic polynomial of $b$ acting $\ol\QQ_p$-linearly on $V_1$. Then $\chi_b(T)$ has coefficients in ${\ensuremath{\mathscr{O}}\xspace}_E$. Let $\alpha_1,\dotsc,\alpha_d \in \ol\QQ_p$ denote its distinct roots, and write $$\chi_b(T) = \prod_{i = 1}^d (T - \alpha_i)^{m_i} \in \ol\QQ_p[T].$$ Thus $m_1 + \dotsb + m_d = n$. Let $V_{\ol\QQ_p,\alpha_i}$ denote the generalized $\alpha_i$-eigenspace for $b$ acting on $V_{\ol\QQ_p}$. Define $$\tensor[^l]W{_{b,\ol\QQ_p}} := \bigoplus_{\substack{j_1 + \dotsb + j_d = l\\
j_1 \leq m_1\\
\vdots\\
j_d \leq m_d}}
\biggl(\sideset{}{_{\ol\QQ_p}^{j_1}} \bigwedge V_{\ol\QQ_p,\alpha_1} \otimes \dotsb \otimes \sideset{}{_{\ol\QQ_p}^{j_d}}\bigwedge V_{\ol\QQ_p,\alpha_d}\biggr).$$ Then $\tensor[^l]W{_{b,\ol\QQ_p}}$ is naturally a subspace of $\bigwedge_{\ol\QQ_p}^l\! V_{\ol\QQ_p}$, and it is defined over $E$ for the same reason that $\chi_b(T)$ is. Let $\tensor[^l]W{_b}$ denote its descent to a subspace of $\bigwedge_E^l V_E$. Define $$\tensor[^l]W{} := \bigcap_{b \in {\ensuremath{\mathscr{O}}\xspace}_B} \tensor[^l]W{_b} \subset \sideset{}{_E^l}\bigwedge V_E.$$
We can now give our formulation of condition in the present setting, which we continue to denote by . For $\Lambda \in {\ensuremath{\mathscr{L}}\xspace}$, let $$\tensor[^l]W{}(\Lambda) := \tensor[^l]W{} \cap \sideset{}{_{{\ensuremath{\mathscr{O}}\xspace}_E}^l} \bigwedge (\Lambda \otimes_{\ZZ_p} {\ensuremath{\mathscr{O}}\xspace}_E)$$ (intersection in $\bigwedge_E^l V_E$). For $R$ an ${\ensuremath{\mathscr{O}}\xspace}_E$-algebra and $(\F_\Lambda \subset \Lambda \otimes_{\ZZ_p} R)_{\Lambda \in {\ensuremath{\mathscr{L}}\xspace}}$ an $R$-point on $M^{\ensuremath{\mathrm{naive}}\xspace}$, the condition is that
1. \[it:K\_l\_PEL\] for all $\Lambda$, the subbundle $\bigwedge_\Lambda^l \F_\Lambda \subset \bigwedge_R^l (\Lambda \otimes_{\ZZ_p} R)$ is contained in $$\im \Bigl[ \tensor[^l]W{}(\Lambda) \otimes_{{\ensuremath{\mathscr{O}}\xspace}_E} R \to \sideset{}{_R^l}\bigwedge (\Lambda \otimes_{\ZZ_p} R) \Bigr].$$
As in \[ss:wedge\_power\_analogs\], conditions for $l = 1,\dotsc$, $n$ are closed conditions on $M^{\ensuremath{\mathrm{naive}}\xspace}$. On the generic fiber, the Kottwitz condition implies all of them, and is equivalent to (K$_n$). In general, the analog of Lemma \[st:K\_n==>Kottwitz\] also holds.
\[st:whatevs\] Condition $(\mathrm{K}_n)$ implies the Kottwitz condition on $M^{\ensuremath{\mathrm{naive}}\xspace}$.
Let $R$ be an ${\ensuremath{\mathscr{O}}\xspace}_E$-algebra and $(\F_\Lambda)_\Lambda$ an $R$-point on $M^{\ensuremath{\mathrm{naive}}\xspace}$. We use that the Kottwitz condition holds if and only if for some $\ZZ_p$-basis $b_1,\dotsc,b_r$ of ${\ensuremath{\mathscr{O}}\xspace}_B$, for all $i$ and all $\Lambda$, the characteristic polynomial of $b_i$ acting $R$-linearly on $\F_\Lambda$ equals $\chi_{b_i}(T)$ in $R[T]$. For this we may assume that $b_1,\dotsc,b_r$ all act semisimply on $V$, and then the argument is the same as in the proof of Lemma \[st:K\_n==>Kottwitz\].
In the special case that $F = B$ is a ramified quadratic extension of $F_0 = \QQ_p$, $V = F^n$, $*$ is the nontrivial element of $\Gal(F/\QQ_p)$, and ${\ensuremath{\langle\text{~,~}\rangle}\xspace}$ is the alternating form defined in \[ss:setup\], condition as defined here reduces to the version in \[ss:wedge\_power\_analogs\]. More precisely, continuing to use the notation of \[ss:setup\], $G$ is the group $GU(\phi)$, which splits over $F$ via the map $$G_F \xra[{\raisebox{0.4ex}{\smash[t]{$\scriptstyle\sim$}}}]{(\varphi,c)} GL_n \times \GG,$$ where as above $c$ is the similitude character, and $\varphi\colon G_F \to GL_n$ is the map given on matrix entries (here regarding $G$ as a subgroup of $\Res_{F/\QQ_p} GL_n$) by $$\xymatrix@R=0ex{
F \otimes_{\QQ_p} R \ar[r] & R\\
x \otimes y \ar@{|->}[r] & xy
}$$ for an $F$-algebra $R$. Take $\{\mu\}$ to be the geometric conjugacy class of the cocharacter $$t \mapsto \bigl(\diag(\underbrace{t,\dotsc,t}_s,\underbrace{1,\dotsc,1}_r), t\bigr)$$ of $GL_n \times \GG_m$. Let Łbe the chain of ${\ensuremath{\mathscr{O}}\xspace}_F$-lattices $\Lambda_i$ for $i \in \pm I + n\ZZ$. Then the naive local model attached to this PEL datum is the one defined in \[ss:naiveLM\], and condition is the one defined in \[ss:wedge\_power\_analogs\], since the subspace $\tensor[^l]W{^{r,s}}$ defined there equals $\tensor[^l]W{}$, as one readily checks.
[^1]: *Key words and phrases.* Shimura variety; local model; unitary group
[^2]: We also mention forthcoming work of Kisin and Pappas [@kisinpappas?] extending this result from PEL cases to cases of abelian type.
[^3]: To be clear, $M_I^{\ensuremath{\mathrm{naive}}\xspace}$ depends on the signature $(r,s)$ as well as $I$, but we suppress the former in the notation.
[^4]: It may also be interesting to note that, at least in the setting of Theorem \[st:main\_thm\], the condition defining $M_I$ itself implies the wedge condition. See Remark \[rk:newcond==>wedge\].
[^5]: Many of the papers we will refer to also assume that $k$ is perfect, but any facts we need for general $k$ will follow from the case of perfect $k$ by descent.
[^6]: Strictly speaking, Arzdorf obtains equations describing the $\ol k$-points in an open subscheme of $M^{\ensuremath{\mathrm{naive}}\xspace}$, but implicit in his discussion are equations defining this subscheme itself, once one additionally incorporates the Kottwitz condition.
[^7]: Strictly speaking, Arzdorf does not explicitly address the equations arising from the lattice inclusion $\Lambda_{m+1} \subset \pi^{-1}\Lambda_m$ in . It is straightforward to verify that these equations add nothing further to when $X_4 = 0$.
[^8]: It is an easy consequence of and that the Kottwitz condition for $\F_m$ implies the Kottwitz condition for $\F_{m+1}$.
|
---
author:
- |
Zhao Song[^1]\
`[email protected]`\
UT-Austin
- |
David P. Woodruff\
`[email protected]`\
IBM Almaden
- |
Peilin Zhong\
`[email protected]`\
Columbia University
bibliography:
- 'ref.bib'
title: 'Low Rank Approximation with Entrywise $\ell_1$-Norm Error'
---
[^1]: Work done while visiting IBM Almaden.
|
---
abstract: 'We have conducted a systematic search for stellar disk truncations in disk–like galaxies at intermediate redshift (z$<$1.1) using the Hubble Ultra Deep Field (UDF) data. We use the position of the truncation as a direct estimator of the size of the stellar disk. After accounting for the surface brightness evolution of the galaxies, our results suggest that the radial position of the truncations has increased with cosmic time by $\sim$1–3 kpc in the last $\sim$8 Gyr. This result indicates a small to moderate ($\sim$25%) inside–out growth of the disk galaxies since z$\sim$1.'
author:
- '[Ignacio Trujillo]{} and Michael [Pohlen]{}'
title: 'Stellar disk truncations at high–z: probing inside–out galaxy formation'
---
Introduction
============
Understanding the formation and evolution of galactic disks is an important goal of current cosmology. The surface brightness of the disks of present–day galaxies are well described by an exponential law (Freeman 1970), with a certain scalelength, taken as the characteristic size of the disk. However, since van der Kruit (1979) it is known that some stellar disks are truncated in the outer parts (for a recent review see Pohlen et al. 2004). Stellar disk truncations have been explained as a consequence of the inhibition of widespread star formation below a critical gas surface density (Kennicutt 1989; Martin & Kennicutt 2001). Alternatively, van der Kruit (1987), has proposed that the truncation radius corresponds to the material with the highest specific angular momentum in the protogalaxy. Recent models (Elmegreen & Parravano 1994; Schaye 2004) emphasize the transition to the cold interstellar medium phase as responsible for the onset of local gravitational instability, triggering star formation.
To date it is not systematically explored whether the above description of present–day galactic disks is valid at high–z, and consequently, it is not clear whether the truncation radius of high–z disks evolves with cosmic time. However, very recently, Pérez (2004) has shown that such kind of analysis is feasible at intermediate redshift (z$\lesssim$1). Addressing the question of how the radial truncation evolves with z is strongly linked to our understanding of how the galactic disks grow and how the star formation is taking place. In this paper we propose the use of the truncation of stellar disks as a direct estimator of their sizes. The evolution of the position of the truncation, consequently, will clarify whether the galaxies are growing inside-out with the star formation propagating radially outward with time.
The above issue can be addressed observationally with a structural analysis of galaxy samples both in the near and in the high–z universe. Such studies require to follow the galaxy surface brightness distributions down to very faint magnitudes ($\sim$27–28 V–band mag/arcsec$^2$) with enough signal to noise to assure a reliable measurement of the disk truncation. The depth of the images plays a critical role in the high–z truncation detections because of the cosmological surface brightness dimming. In addition, it is necessary to work with high resolution images to avoid the seeing effects on the shape of the surface brightness distribution. This means to use the deepest observations taken with the HST telescope. For that reason we have decided to analyse the disk–like galaxies in the UDF. In order to maintain our analysis in the optical restframe, the combination of k–correction and the cosmological dimming restricts the use of the optical HST imaging for this project up to z$\sim$1. Throughout, we will assume a flat $\Lambda$–dominated cosmology ($\Omega_M$=0.3, $\Omega_\Lambda$=0.7 and H$_0$=70 km s$^{-1}$ Mpc $^{-1}$).
Data, Sample Selection and Surface Brightness Profiles
======================================================
Our galaxies have been selected in the Hubble Ultra Deep Field (Beckwith et al. 2005). This survey is a 400-orbit program to image a single field with the Wide Field Camera (WFC) of the Advanced Camera for Surveys (ACS) in four filters: F435W (B), F606W (V), F775W (i), and F850LP (z). We have used the public available V, i and z-band mosaics with a pixel scale of 0.03$''$/pixel. The FWHM is estimated to be 0.09 seconds of arc.
To make the selection of our sample we have taken advantage of the fact that the UDF field is within the Galaxy Evolution from Morphology and SEDs (GEMS; Rix et al. 2004) imaging survey with the ACS pointing to the Chandra Deep Field South (CDF–S). Focusing on the redshift range 0.1$\leq$z$\leq$1.1, GEMS provides morphologies and structural parameters for nearly 10000 galaxies (Barden et al. 2005; McIntosh et al. 2005). For these galaxies photometric redshift estimates, luminosities, and SEDs exist from COMBO–17 (Classifying Objects by Medium–Band Observations in 17 Filters; Wolf et al. 2001, 2003). The COMBO–17 team has made this information publically available through a catalog with precise redshift estimates ($\delta$z/(1+z) $\sim$ 0.02) for approximately 9000 galaxies down to m$_R$$<$24 (Wolf et al. 2004). Rest–frame absolute magnitudes and colors, accurate to $\sim$0.1 mag, are also available for these galaxies.
Barden et al. (2005) have conducted the morphological analysis of the late–type galaxies in the GEMS field by fitting Sérsic (1968) r$^{1/n}$ profiles to the surface brightness distributions. Ranvidranath et al. (2004) have shown that using the Sérsic index $n$ as a criteria, it is feasible to disentagle between late– and early– type galaxies at intermediate redshifts. Late–types (Sab–Sdm) are defined through $n$$<$2–2.5. Moreover, the morphological analysis conducted by Barden et al. provides the information about the inclination of the galaxies. This is particularly important since we want to study the truncations of the stellar disks in objects with low inclination. The edge–on view facilitates the discovery of truncations but introduces severe problems caused by the effects of dust and line–of–sight integration that we want to avoid (Pohlen et al. 2002).
Our sample is selected as follows, we have taken all the galaxies in the COMBO-17 CDF–S catalog with R$<$24 mag that are in common with the UDF object detection catalog (Beckwith et al. 2005). This leaves a total of 166 objects. From these, 133 have photometric redshift estimation (i.e. a $\sim$80% completeness). To maintain our analysis in the optical restframe we select only the 118 galaxies with z$<$1.1. The mean redshift of these galaxies is $\sim$0.68. At that redshift, the faintest absolute B–band restframe magnitude able to be analysed is M$_B$$\sim$-18.6 mag (AB system). We take this value as a compromise between maximizing the number of objects in our final sample and at the same time assuring as much as possible homogeneity (equal luminous objects) through the full redshift range. Applying this magnitude cut we get 63 galaxies. It is important to note however that, strictly, only those objects brighter than M$_B$$<$-20 are observable homogeneously up to z=1.1. Finally, we select from the 63 galaxies only those objects that according to the GEMS morphological analysis have n$<$2.5 (i.e. those which have disk–like surface brightness profiles) and e$\leq$0.5 (i.e. i$\leq$60$^\circ$). Our final sample contains 36 galaxies. These galaxies are presented in Table \[table1\].
To analyse the surface brightness profiles of our galaxies in a similar rest–frame band along the explored redshift range (0.1$<$z$<$1.1), we have extracted the profiles in the following bands: V–band for galaxies with 0.1$<$z$<$0.5, i–band for 0.5$<$z$<$0.8 and z–band for 0.8$<$z$<$1.1. This allows us to explore the surface brightness distribution in a wavelength close to the B–band restframe. In addition, to probe whether the position of the truncation depends on the observed wavelength we have analysed all galaxies in the reddest band available: the z–band.
The surface brightness profiles were extracted through ellipse fitting using the ELLIPSE task within IRAF. Fitting ellipses over the whole galaxy produces similar surface brightness profiles than those obtained through averaging different image segments (Pohlen et al. 2002; Pérez 2004). Initially the ellipticity (E) and the position angle (PA) of the elliptical isophotes are left as free parameters in order to determine the best set of E and PA describing the outer disk. This is done for all the galaxies in the z–band. The isophote fitting in the z–band is less affected by additional structure in the galaxies like spiral arms and prominent blobs of star formation and, consequently, the above parameters are retrieved more accurately. We select the E and PA at the radius where the mean flux of the best fitted free ellipse reaches 1$\sigma$ of the background noise (this means $\sim$26.3 mag/arcsec$^2$ in the z–band in the AB system). We have checked visually that for all the cases this criteria is selecting an isophote well outside the inner region of the disk. Once E and PA are determined we run again ELLIPSE with these parameters fixed. In order to avoid contamination in our isophote fitting we masked all the surrounding neighbors. Examples of the final surface brightness profiles obtained are shown in Fig. \[sample\].
From our sample of 36 galaxies there were 21 galaxies which show a truncation in the surface brightness profiles. We have named these galaxies Type DB (i.e. DownBending). The exact details of how the position of the breaks were estimated are explained in Pohlen & Trujillo (2005), consequently here we explain only very briefly the technique. According to the observations, around the break the surface brightness profile is well described by two exponential functions with scalelenghts h$_1$ and h$_2$, so we use the derivative of the surface brightness profile to estimate the position of the transition. The position of the break is measured at the radial position were the derivative profile crosses the horizontal line defined by (h$_1$+h$_2$)/2. The values obtained that way match very well with our estimations done by eye. The position of the break depends on the shape of the transition region between both exponentials. A conservative estimate of the uncertainty of the position is $\sim$8%. In addition, we do not find any systematic difference in the position of the break at using the z–band or the band closest to the B–band restframe.
There were 6 galaxies where we do not observe any signature of a break along the profile. These galaxies are named Type I (Freeman 1970). We have another 9 galaxies where the outer profiles are distinctly shallower in slope that the main disk profile. We classify these as Type III following the notation by Erwin, Beckman & Pohlen (2005). We plan to study these objects in more detail in future papers.
Discussion
==========
In Fig. \[comparison\]a we show the absolute B–band restframe magnitude versus the position of the break for our high–z galaxy sample and, for comparison, a volume selected local sample (Pohlen & Trujillo 2005). The local sample comprises the 85 Sb–Sdm galaxies from the LEDA catalogue having a mean heliocentric radial velocity relative to the Local Group (corrected for virgocentric inflow) $<$3250 km/s and M$_B$$<$-18.5 (AB system), with useful imaging data available in the Sloan Second Data Release (DR2). These galaxies were selected to be face–on to intermediate inclined (e$<$0.5). This sample represents the largest sample ever used to homogeneously probe for truncations using low inclination galaxies. From this sample we plot in Fig. \[comparison\]a the 35 galaxies which present a truncation. The truncation radii of the local sample shown in Fig. \[comparison\]a were estimated in the SLOAN g–band.
The most simple description, a linear fit, of the local sample provides with the following fit: R$_{break}$=-2.3$\times$M$_B$-35.7. We overplot this in Fig. \[comparison\]a and Fig. \[comparison\]b. On the other hand, galaxies at z$>$0.65 present $\sim$4–5 kpc smaller truncation radii at a given luminosity than the present–day galaxies (Fig. \[comparison\]a). A linear fit gives: R$_{break}$=-2.0$\times$M$_B$-34.9. A concern is whether our high–z galaxies are biased towards brighter surface brightness truncations (and consequently towards smaller sizes). To test this we explored the surface brightness distribution (in the observed band closest to the B–band rest–frame) at the break positions. The distribution peaks at 24.1 mag/arcsec$^2$ with a scatter of 0.9 mag. Our faintest surface brightness at the break is 25.5 mag/arcsec$^2$. We do not detect truncations between 25.5 and 27 mag/arcsec$^2$ (approximately our surface brightness detection limit). This indicates our sample is not biased and we are probing all the truncations much brighter than 27 mag/arcsec$^2$.
The stellar population of the galaxies are known to be younger at high–z and consequently we expect that the galaxies were much brighter in the past than what they are today. To check whether the evolution of the surface brightness alone is able to explain the different distribution of the galaxies, we have applied a correction on the observed absolute magnitudes of the high–z galaxies using the observed mean surface brightness evolution found in Barden et al. (2005) for disk–like objects (i.e. we use d$<$$\mu_B$$>$/dz=-1.43$\pm$0.03). Under the assumption of no evolution in the size of the objects this corresponds to the same degree of evolution in the absolute magnitude (i.e. M$_B$(0)=M$_B$(z)+1.43$\times$z). The result of doing this is shown in Fig. \[comparison\]b. After the magnitude correction the position of the breaks for the high–z population is still systematically smaller by 1–3 kpc at a given “corrected” luminosity. A linear fit results on R$_{break}$=-1.9$\times$M$_B$-30.5. This corresponds to an increase of $\sim$25% in the size of the inner disk since z$\sim$1. Although we are using the largest samples in existence where the truncations have been explored systematically, the estimation of the exact evolution is limited by the small number statistic. It is important, however, to note that this is the first time that the position of the high–z stellar disk truncations is used to probe directly the growth of disk galaxies with cosmic time.
Our results are consistent with a moderate inside–out formation of galactic disks. This agrees with the conclusions obtained indirectly by Barden et al. (2005) and Trujillo et al. (2004; 2005) analysing the evolution of the luminosity and stellar mass–size relations since z$\sim$3.
We are very grateful to Marco Barden for kindly providing us with the GEMS morphological analysis catalog. We acknowledge the COMBO–17 collaboration for the public provision of an unique database upon which this study was based. We thanks useful comments from Eric Bell, Peter Erwin and Renyer Peletier. The referee comments helped to improve the paper. Based on observations made with the NASA/ESA Hubble Space Telescope, which is operated by the Association of Universities for Research in Astronomy, Inc, under NASA contract NAS5-26555. This research was supported by a Marie Curie Intra–European Fellowship within the 6th European Community Framework Programme.
Barden, M. et al. 2005, ApJ, submitted, astro-ph/0502416 Beckwith, S. V. W. et al. 2005, in preparation Elmegreen, B. G., & Parravano, A., 1994, ApJ, 435, L121 Erwin, P., Beckman J.E., & Pohlen, M., 2005, ApJL, in press Freeman, K. C., 1970, ApJ, 160, 811 Kennicutt, R. C., Jr., 1989, ApJ, 344, 685 Martin, C. L., & Kennicutt, R. C., Jr. 2001, ApJ, 555, 301 McIntosh et al., 2005, ApJ, submitted, astro-ph/0411772 Pérez, I., 2004, A&A, 427, L17 Pohlen, M., Dettmar, R. J., Lütticke, R., Aronica, G., 2002, A&A, 392, 807 Pohlen, M., Beckman J.E., Hüttemeister, S., Knapen, J. H., Erwin, P., Dettmar, R.-J., 2004, astro-ph/0405541 Pohlen, M., & Trujillo, I., 2005, in preparation Ravindranath, S., et al., 2004, ApJ, 604, L9 Rix, H.-W., et al., 2004, ApJS, 152, 163 Schaye, J., 2004, ApJ, 609, 667 Sérsic, J.-L., 1968, Atlas de Galaxias Australes (Cordoba: Observatorio Astronomico) Trujillo et al. 2004, ApJ, 604, 521 Trujillo et al. 2005, ApJ, submitted, astro-ph/0504225 van der Kruit, P. C., 1979, A&AS, 38, 15 van der Kruit, P. C., 1987, A&A, 173, 59 Wolf, C., et al. 2001, A&A, 365, 681 Wolf, C., Meisenheimer, K., Rix, H. W., Borch, A., Dye, S., Kleinheinrich, M., 2003, A&A, 401, 73 Wolf, C. et al., 2004, A&A, 421, 913
![[]{data-label="sample"}](f1a.ps "fig:"){width="10cm"} ![[]{data-label="sample"}](f1b.ps "fig:"){width="10cm"} ![[]{data-label="sample"}](f1c.ps "fig:"){width="7.5cm"} ![[]{data-label="sample"}](f1d.ps "fig:"){width="7.5cm"}
![[]{data-label="sample"}](f1e.ps "fig:"){width="10cm"} ![[]{data-label="sample"}](f1f.ps "fig:"){width="10cm"} ![[]{data-label="sample"}](f1g.ps "fig:"){width="7.5cm"} ![[]{data-label="sample"}](f1h.ps "fig:"){width="7.5cm"}
[cccccc]{}
328 & 0.24 & -20.6 & DB & 2.4 & 22.9\
900 & 0.45 & -20.8 & III & – & –\
901 & 1.00 & -20.1 & I & – & –\
968 & 0.66 & -21.4 & DB & 1.1 & 23.3\
1971 & 0.14 & -19.2 & DB & 3.9 & 25.4\
2525 & 0.69 & -19.6 & DB & 0.9 & 25.5\
2607 & 0.68 & -21.3 & DB & 1.8 & 24.4\
3180 & 0.80 & -20.1 & DB & 1.1 & 25.2\
3203 & 0.35 & -18.6 & I & – & –\
3268 & 0.28 & -18.6 & DB & 1.4 & 25.4\
3372 & 0.94 & -21.7 & DB & 0.9 & 22.7\
3613 & 1.09 & -21.4 & I & – & –\
3822 & 0.19 & -20.0 & III & – & –\
4142 & 0.67 & -20.6 & III & – & –\
4394 & 0.66 & -21.1 & DB & 1.1 & 23.6\
4438 & 1.06 & -21.3 & DB & 1.1 & 24.2\
4491 & 1.07 & -20.7 & DB & 0.6 & 24.1\
4929 & 0.45 & -20.3 & III & – & –\
5177 & 0.57 & -19.3 & I & – & –\
5268 & 0.61 & -19.0 & DB & 0.4 & 23.3\
5417 & 1.09 & -21.8 & DB & 0.9 & 23.4\
6821 & 1.07 & -20.1 & DB & 0.5 & 23.9\
6853 & 0.79 & -19.2 & III & – & –\
6862 & 0.68 & -20.3 & III & – & –\
6974 & 0.61 & -20.1 & DB & 1.2 & 25.1\
7112 & 1.00 & -20.0 & DB & 0.4 & 23.2\
7556 & 0.63 & -21.7 & DB & 2.3 & 24.4\
7559 & 0.93 & -20.4 & III & – & –\
8040 & 0.22 & -18.9 & I & – & –\
8049 & 0.46 & -20.4 & III & – & –\
8125 & 1.07 & -20.5 & I & – & –\
8257 & 0.57 & -20.1 & III & – & –\
8275 & 0.71 & -21.3 & DB & 1.4 & 24.4\
8810 & 0.72 & -20.1 & DB & 0.4 & 22.7\
9253 & 0.68 & -21.8 & DB & 1.3 & 23.3\
9455 & 0.53 & -19.7 & DB & 1.4 & 25.1\
\[table1\]
|
---
abstract: 'According to many phenomenological and theoretical studies the distribution of family name frequencies in a population can be asymptotically described by a power law. We show that the Galton-Watson process corresponding to the dynamics of a growing population can be represented in Hilbert space, and its time evolution may be analyzed by renormalization group techniques, thus explaining the origin of the power law and establishing the connection between its exponent and the ratio between the population growth and the name production rates.'
author:
- 'Andrea De Luca, Paolo Rossi'
bibliography:
- 'letter.bib'
title: Renormalization group evaluation of exponents in family name distributions
---
Introduction
============
The frequency distribution of family names in local communities, regions and whole countries has been the object of a sustained interest by geneticists and statisticians for more than thirty years, starting from the seminal paper by Yasuda et al. [@Yasuda]. For a recent review of the relevant literature we refer to Colantonio et al. [@Colantonio], while Scapoli et al. [@Scapoli] have recently collected and synthesized their results on the major countries of continental Western Europe. The main motivation for these researches resides in the deep analogy existing between surname distributions and the frequency of neutral alleles in a population: both distributions are generated by an evolutionary branching process subject to mutation and migration but not conditioned by natural selection. In particular it has been observed that the dynamics of family names, in countries with an European family name system, mimics that of the Y chromosome [@Sykes]. Models for such processes have been advanced in the genetic and statistical literature, starting from the Karlin-McGregor [@Karlin] statistical theory of neutral mutations. A significant theoretical evolution occurred in particular after Lasker’s empirical observation [@Fox] that a power law could offer a good fit of the observed surname distributions. As a consequence Panaretos [@Panaretos] suggested the use of the Yule-Simon distribution, while Consul [@Consul] proposed to employ the Geeta distribution with motivations coming from a branching process modelization. Evolutionary processes have attracted also the attention of physicists, who have found that neutral evolution might be a ground for application of many techniques proper of statistical mechanics [@Derrida1] [@Serva] [@Derrida2]. In particular Miyazima et al. [@Miyazima], studying family name distributions in Japanese towns, found the systematic emergence of scaling laws, and further theoretical studies [@Zanette] [@Manrubia] justified the appearance of power laws of the Yule-Simon type in the case of growing populations with non vanishing probability for mutations. A different explanation was offered by Reed and Hughes [@Reed] who considered a branching process with mutation and migration and found that the asymptotic form of the distributions should follow a power law. The most recent and comprehensive result is due to the Korean group of Baek et al. [@Baek] [@Kim], who wrote down a master equation for the frequency distribution of family names and its time evolution in the presence of birth, death, mutation and migration, and found the possibility of different power laws with exponents depending on the mutation and migration parameters. In the present paper we reconsider the models of family name evolution in the context of a Hilbert space representation of branching processes, and show that distributions characterized by an asymptotic power law behaviour can be obtained as solutions of recursive equations which would correspond to the renormalization group equations of an (equivalent) physical system. In Sec. \[models\] we introduce and motivate our models. In Sec.\[immigration\] we discuss the simpler case of a system characterized by pure immigration without mutations. Finally in Sec.\[mutation\] we discuss the case with mutation. In Appendix \[appendice\] we represent the Galton-Watson branching process in a Hilbert space.
The models {#models}
==========
In the following sections we shall introduce two models, that take care of two different ways of generating new family names in a population: immigration from abroad and mutation occurring after reproduction. The importance of the appearance of new family names was pointed out in Refs. [@Zanette; @Manrubia; @Baek]. The analogy of the recursive equations we shall obtain with those typically derived by a renormalization-group approach to a physical system will allow us to evaluate the asymptotic behavior of the family name distribution N(k), where N is the number of family names represented by exactly k individuals. Obviously in a typical real situation both immigration and mutation contribute to the dynamics of the family name distribution. But in our models we shall first focus on a population in which only immigration occurs, and then on one in which only mutation occurs. This simplification is justified by the fact that in an exponentially growing population (an approximation usually called Malthusian law) the effect of immigration can be neglected in comparison to mutation, at least in order to study the asymptotic behavior. However, in peculiar historical conditions, mutations can be heavily depleted and as a consequence the study of a society where name change is only due to immigration retains its value. Since we are interested in the family name distribution we can limit our attention to the male individuals, which is consistent with the legislation on names present in most real societies. In the following we shall use the term “individual” referring just to males. Moreover we shall suppose that the evolution of the population can be described by the Galton-Watson model. This means we shall consider:
- time as discrete, moving from one generation to the next;
- the system as completely markovian;
- each individual as independent of all others.
The last hypothesis may be considered a very strong restriction if applied to a biological system, since, for example, the exhaustion of resources induces a collective behaviour, limiting the growing rate. But we can consider this hypothesis to be valid in the context of exponential grow of a population. It is useful to fix some definitions in the use of the Galton-Watson process. We set: $$\label{def_probgalt}
p_n = \mbox{probability for an individual to have }n\mbox{ sons}$$ It is straightforward to introduce the generating function of the Galton-Watson process: $$\label{def_galtgen}
f(z) = \sum_{n = 1}^\infty p_n z^n$$ Our hypothesis of growing population forces us to take $p_n$ such that the mean number of sons is greater than one: $$\sum_{n=1}^\infty np_n = f'(1) \equiv m > 1$$ We will exclude the trivial case: $p_n = \delta_{1n}$. We omit the explicit derivation of the recursive equations, which can be found with details in Appendix \[appendice\]. However, their meaning will be somehow intuitive.
Immigration
===========
We want to analyze a population whose members increase in number by the Galton-Watson mechanism and furthermore a group of individuals comes from outside. Each son inherits his family name from his father, while the new individuals coming from outside bring new family names. We are interested in the asymptotic behaviour of $N(k,t)$, which corresponds to the number of family names represented by $k$ individuals at time $t$. The values $N(k,0) = N_0 (k)$ are assigned as initial conditions of the problem, with: $$\label{initialcond}
\sum_{k = 1}^\infty N_0(k) = S_0 <\infty \qquad \sum_{k=1}^\infty k N_0(k) = N_0 < \infty$$ where $S_0$ is the initial number of family names and $N_0$ is the initial number of individuals. We introduce the generating function: $$n_t(z) = \sum_{k=0}^\infty N(k,t) z^k$$ Now we suppose that the individuals from outside come always distributed in the same manner: $\theta(k)$ is the number of new family names represented by $k$ individuals among them. We suppose the number of individuals $\theta_0$ and the number of new family names $G_0$ to be finite: $$\label{boundnewpeople}
\begin{array}{lll}
\theta_0 &=& \sum_k \theta(k)\\
G_0 &=& \sum_k k \theta(k)
\end{array}$$ As before we introduce the generating function: $$\theta(z) = \sum_k \theta(k) z^k$$ We can obtain a recursive equation for $n_t(z)$ involving $\theta(z)$. The explicit derivation is given in Appendix \[appendice\]: $$\label{eq_recursimmigr}
n_{t+1}(z) = n_t(f(z)) + \theta(z)$$ A formal solution is given by: $$n_t (z) = n_0\left(f_t(z)\right) + \sum_{k=0}^{t-1} \theta\left(f_{k}(z)\right)$$ where $f_{k}(z)$ indicates the function $f(z)$ iterated $k$-times. From this expression it is easy to compute the mean number of individuals $N_t$ and the mean number of family names $S_t$ at time $t$: $$\begin{gathered}
\label{totalnum}
N_t \equiv n'_t(1)= n_0'(1)[f'(1)]^t \sum_{k=0}^{t-1} \theta'(1) [f'(1)]^k = N_0 m^t + G_0 \sum_{k=0}^{t-1} m^k =\\
= \left(N_0 + \frac{G_0}{m-1}\right)m^t - \frac{G_0}{m-1}\end{gathered}$$ $$\label{totalsurname}
S_t \equiv n_t(1) = S_0 + t\theta_0$$ We are interested in the limit $t\to \infty$ and in the asymptotic behaviour: $k \gg 1$. In order to achieve this goal, we notice that Eq.(\[eq\_recursimmigr\]) is formally analogous to the equations coming from the renormalization group approach, linking the system at two different degrees of magnification. Therefore the system can be studied by using this analogy with the corresponding physical system. More explicitly, suppose $\Phi_n(T)$ is the free energy of a hierarchical model, at scale $n$ and temperature $T$. With standard renormalization group method, we can obtain the recursive equation linking two different scales (see [@derrida5]): $$\label{eq_ricorsF}
\Phi_{n+1}(T) = g(T) + \frac{1}{\mu} \Phi_n(\phi(T))$$ where $g(T)$ is a regular function that comes up after summing the degree of freedom of the smaller scale and $\phi(T)$ is the RG flow. Then near the critical point, for large $n$: $$\label{eq_exprg}
\Phi(T) \simeq (T-T_c)^\alpha \qquad \alpha = \frac{\ln \mu}{\ln \phi'(T_c)}$$ Eq.(\[eq\_recursimmigr\]) is formally analogous to Eq.(\[eq\_ricorsF\]) and in our case the role of the flow is carried out by the Galton-Watson generating function $f(z)$ and so the phases and the critical points correspond to the fixed points of $f(z)$: $$\label{eq_fixedgalt}
f(z) = z$$ From the fact that $f(z)$ is convex and $f'(1)>1$, we find that Eq.(\[eq\_fixedgalt\]) has three solutions: $q, 1 , \infty$[^1] . From the Galton-Watson theory we know that $q \in [0,1)$ is the extinction probability. Moreover it is easy to see that $f'(q) < 1$. In fact if it was $f'(q)\geq1$ one would have by convexity: $$f(1) > f(q) + f'(q)(1-q) \geq 1$$ So we have that $q, \infty$ are attractive, while $1$ is a repulsive fixed point which separates the two stable phases. We get a critical behaviour near $1$: $$n(z) \equiv \lim_{t\to \infty} n_t (z) \simeq (1-z)^{\alpha}$$ One can see that in this case we have that for $t \gg 1$: $$\label{eq_asymplink}
N(k,t) \simeq k^{-1+\alpha}$$ To compute $\alpha$, we take $\mu = 1$, $T_c = 1$, $m \equiv f'(1) = \phi(T_c)$ in Eq.(\[eq\_exprg\]) and we notice we are in an atypical situation in which $\alpha = 0$. It means that the function is diverging more slowly than any power and it is easy to see that it is logarithmic. In fact using Eq.(\[totalnum\]) and (\[totalsurname\]): $$\begin{gathered}
\label{eq_limiteimmigr}
A \equiv \lim_{z \to 1} n'(z)m^{-\frac{n(z)}{\theta_0}} = \lim_{z \to 1}\lim_{t\to\infty} n'_t(z)m^{-\frac{n_t(z)}{\theta_0}} = \\ =
\lim_{t\to \infty} \lim_{z \to 1}n'_t(z)m^{-\frac{n_t(z)}{\theta_0}} = \left(N_0+\frac{G_0}{m-1}\right)m^{-\frac{S_0}{\theta_0}}\end{gathered}$$ So we get near $1$ $$n'(z) \simeq \left(N_0+\frac{G_0}{m-1}\right)m^{\frac{n(z)-S_0}{\theta_0}} = A e^{b n(z)}$$ where we set $b = \frac{\log m}{\theta_0}$. It can be solved[^2] giving $$n(z) \simeq -\frac{1}{b}(\log(Ab) + \log(1-z))$$ which ensures us the logarithmic divergence and implies for large $k$: $$N(k) = \lim_{t\to \infty} N(k,t) = \frac{C}{k}\left(1+o(1)\right)$$ So for immigrations we find a power-law behaviour with exponent $-1$. Notice that this behaviour is completely independent of the initial condition and of the distribution of the immigrating family names at each generation.
Mutation
========
The context is analogous to the previous one but we do no longer have immigration. We use again the initial condition in Eq.(\[initialcond\]). Now, each son has a certain probability $\rho$ that his family name mutates into a new one, different from his father’s. We suppose that $\rho$ does not depend on the family and we neglect the case in which two or more sons take the same new family name. This means the Galton-Watson contribution is modified since only a part proportional to $1-\rho$ of the offspring holds the same family name and the remaining part is added to the families of size $1$. This implies the equation: $$\label{eq_mutorig}
n_{t+1} (z) = n_t \left(f\left(z^{1-\rho}\right)\right)+ \rho m n'_t(1) z$$ where we used the fact that $n'_t(1)$ equals the total number of individuals at generation $t$. Observe that mutations do not contribute to the total number of individuals and so: $$n_t'(1) = N_0 m^t$$ as it can be shown directly via Eq.(\[eq\_mutorig\]). The recursive equation can now be solved, at least formally. Defining $r(z) = f\left(z^{1-\rho}\right)$ and indicating by $r_k(z)$ the function $r(z)$ iterated $k$-times, we get the solution: $$n_t (z) = n_0(r_t(z))+\rho N_0\sum_{n=0}^{t-1} m^{t-n}r_n(z) > \rho N_0 m^t r_0(z)$$ The last inequality shows that no limit in $t$ can exist. However we can obtain a limit for the function: $$\eta_t(z) \equiv n_t(z)m^{-t}$$ Since for large $t$: $n_t(1) \propto m^t$, as one can check by putting $z=1$ in Eq.(\[eq\_mutorig\]), we are basically considering the distribution normalized to the total number of families. So we can put Eq.(\[eq\_mutorig\]) in the form: $$\label{eq_recursmut}
\eta_{t+1}(z) = \rho N_0 z + \frac{\eta_t (r(z)) }{m}$$ which is again in the form of Eq.(\[eq\_ricorsF\]). However the flow is slightly changed with respect to the Galton-Watson generating function. We have $r'(1) = (1-\rho) m $ and we suppose $\rho$ small enough for $1$ to be a repulsive fixed point for the flow. In this case we must have a critical behaviour near $1$, whose exponent can be evaluated using Eq.(\[eq\_exprg\]): $$\eta(z) = \lim_{t \to \infty} \eta_t (z) \simeq (1-z)^\alpha$$ where the exponent can be obtained using Eq.(\[eq\_exprg\]): $$\alpha = \frac{\ln(m)}{\ln(r'(1))} = \frac{\ln(m)}{\ln(m)+\ln(1-\rho)}$$ Using Eq.(\[eq\_asymplink\]) we get the exponent of the family name power-law distribution: $$\gamma \equiv \alpha + 1 = 2 - \frac{\ln(1-\rho)}{\ln(m)+\ln(1-\rho)} \simeq 2 - \frac{\rho}{\ln(m)}$$ where we considered $\rho$ very small as it is true in the real situations (see [@Baek]). Again the behaviour is completely independent of the initial condition and shows the typical features of a scale-free system.
Conclusion
==========
In this paper we represented the Galton-Watson process as a quantum evolution defining the Hilbert space and the time evolution operator corresponding to the Galton-Watson probabilities. In this way we obtained two recursive equations for two possible models with different family name production mechanism: immigrations and mutations. The structure of the branching allowed us to interpret these equations as the ones that connect different scales of a physical system and, in particular, the asymptotic behaviour corresponds to the power law emerging near the critical point. The exponents are consistent with those evaluated in [@Baek] with a master equation approach: $N(k)$ goes as $k^{-1}$ for a society where name change is only due to immigration and, approximately, as $k^{-2}$ for a society where family name mutation occurs. Our method shows the robustness of this results, which are independent of the offspring distribution. Possible extensions of the model remain to be investigated and will be the object of future studies.
Acknowledgments
===============
A.D.L. thanks the MIUR grant “*Fisica Statistica dei Sistemi Fortemente Correlati all’Equilibrio e Fuori Equilibrio: Risultati Esatti e Metodi di Teoria dei Campi” - 2007JHLPEZ*, for partially supporting this work.
The Galton-Watson process in an Hilbert space {#appendice}
=============================================
The structure of branching process that characterizes the Galton-Watson allows us to consider the reproduction governed by chance as a decay process whose interaction is given by an hamiltonian, which as we will see, is not hermitian. We first introduce the creation and destruction operators at each time with the usual commutation rules: $$\begin{aligned}
[a_k, a_h] & = & 0 \\ \ [a_k^\dagger, a_h^\dagger] & = & 0 \\ \ [ a_k, a^\dagger_h] & = & \delta_{kh}\end{aligned}$$ where, respectively, $a_k^\dagger$ creates and $a_k$ destroys an individual at time $k$. The Hilbert space is obtained in the usual way, acting on the vacuum Fock state with polynomials in $a_t^\dagger$ for all possible values of $t$. A basis for the space is then given by the following set: $$\label{basis}
{|n,t\rangle} = (a^\dagger_t)^n {|0\rangle} \qquad {\langlen,t}| = {\langle0}|{(a_{t})^n}$$ Then, at each time $t$, the state of the system, which is determined by the probability $b_k(t)$ that exactly $k$ individuals are present, can be written: $${|\phi(t)\rangle} = \sum_k b_k(t) {|n,t\rangle}$$ It must be possible to connect the dynamics to the parameters $p_n$ introduced in Eq.(\[def\_probgalt\]) and so to the generating function $f(z)$ of Eq.(\[def\_galtgen\]). This can be done setting the hamiltonian as: $$\label{def_hamil}
H(t) = f(a_{t+1}^\dagger) a_t$$ we can write the time-evolution operator: $$\label{def_onestepevol}
U(t) \equiv \exp(H(t))$$ which is the one-time-step evolution operator: it evolves the states at time $t$ to time $t+1$[^3], giving the correct probabilities according to the Galton-Watson process. In fact, one can easily check that: $$\label{Uaction}
U(t){|1,t\rangle} = U(t) a^\dagger_t {|0\rangle} = f(a^\dagger_{t+1}){|0\rangle} = \sum_n p_n (a^\dag_t)^n {|0\rangle}$$ And in general, by linearity we know that given a state ${|\phi(t)\rangle}$ with a particular probability distribution we get the state at time $t+1$ correctly evolved. Starting from a state ${|\phi_0\rangle}$ at time $0$ we can obtain the state at time $T$ by: $$\label{allstepevol}
{|\phi_T\rangle} = \mathcal U(T) {|\phi_0\rangle} \equiv U(T-1)U(T-2)\cdots U(0){|\phi_0\rangle}$$ and this equation defines the full time-evolution operator. We see now how to derive Eq.(\[eq\_recursimmigr\]). We want to write the equation of evolution for: $${|n(t)\rangle} = \sum_{k=0}^\infty N(k,t){|k,t\rangle}$$ With the notation of section \[immigration\], we define the state: $${|\theta(t)\rangle} = \sum_{k=0}^\infty \theta(k) {|k,t\rangle}$$ In the absence of immigration the evolution would be simply given by Eq.(\[allstepevol\]). But here at each step, the number of family names represented by $k$ individuals grows due to the individuals coming from outside. So we get the equation: $$\label{quantumevol}
{|n(t+1)\rangle} = U(t){|n(t)\rangle}+{|\theta(t)\rangle}$$ Now we use the map $\mathcal W$ from the Hilbert space to $C^\infty[0,1]$ defined on the basis of Eq.(\[basis\]) as: $$\label{def_reprhilb}
{|n,t\rangle} = {(a_{t}^\dagger)^n} {|0\rangle} \quad \xrightarrow{\mathcal{W}} \quad z^n_t$$ And in general: $\mathcal{W}({|\phi(t)\rangle}) = \phi(z_t) \in C^\infty[0,1]$. The action of an operator becomes an integral transformation. For $U(t)$ we have a simple kernel of the form: $U(t) \to U(z_t,z_{t+1}) = \delta(z_t-f(z_{t+1}))$ as can be deduced from Eq.(\[Uaction\]). Then: $$\phi_{t+1}(z_{t+1}) = \mathcal W (U(t){|\phi(t)\rangle})= \int U(z_t,z_t+1) \phi(z_t,t) dz = \phi_t(f(z_{t+1}))$$ where $\phi_t(z) = \mathcal W ({|\phi(t)\rangle})$. Acting on the Eq.(\[quantumevol\]) with the map $\mathcal{W}$ we get Eq.(\[eq\_recursimmigr\]).
[^1]: more precisely these are the possible outcomes of: $\lim_{n \to \infty} f_n(z_0)$ for different values of $z_0$.
[^2]: the arbitrary constant can be fixed by imposing the solution diverges in $1$
[^3]: It should be observed that the correct expression for $U(t)$ should be: $$U(t) = P_t e^{H(t)}$$ where $P_t$ destroys all the states at time $t$: $ P_t {|0\rangle} = {|0\rangle}$, $P_t (a^\dagger_t)^n {|0\rangle} = 0 $ and $\forall h \neq t \quad [P_t, a^\dagger_h] = 0$. In this way we eliminate all the parts of the states that do not evolve to time $t+1$. E.g.: $$e^{H(t)}{|2,t\rangle} = \left(f\left(a^\dag_{t+1}\right)a_t + \frac{f\left(a^\dag_{t+1}\right)^2 (a_t)^2}{2}\right)(a^\dag_t)^2{|0\rangle} = \left(2f\left(a^\dag_{t+1}\right)a^\dag_t+f\left(a_{t+1}^\dag\right)^2\right){|0\rangle}$$ and the operator $P_t$ then eliminates the first term in the parenthesis giving the correct result.
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