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abstract: 'In this paper we study the learnability of deep random networks from both theoretical and practical points of view. On the theoretical front, we show that the learnability of random deep networks with sign activation drops exponentially with its depth. On the practical front, we find that the learnability drops sharply with depth even with the state-of-the-art training methods, suggesting that our stylized theoretical results are closer to reality.'
author:
- |
Abhimanyu Das Sreenivas Gollapudi Ravi Kumar Rina Panigrahy\
Google\
Mountain View, CA
bibliography:
- 'learnability.bib'
title: On the Learnability of Deep Random Networks
---
intro relatedwork prelim repulsion learnability relu kindep conc
Appendix {#appendix .unnumbered}
========
exp proof
|
---
abstract: 'The $X=M$ conjecture of Hatayama et al. asserts the equality between the one-dimensional configuration sum $X$ expressed as the generating function of crystal paths with energy statistics and the fermionic formula $M$ for all affine Kac–Moody algebra. In this paper we prove the $X=M$ conjecture for tensor products of Kirillov–Reshetikhin crystals $B^{1,s}$ associated to symmetric powers for all nonexceptional affine algebras.'
address:
- |
Department of Mathematics\
University of California\
One Shields Ave.\
Davis, CA 95616-8633 U.S.A.
- |
Department of Mathematics\
Virginia Tech\
Blacksburg, VA 24061-0123 USA
author:
- Anne Schilling
- Mark Shimozono
title: $X=M$ for symmetric powers
---
Introduction
============
In two extraordinary papers, Hatayama et al. [@HKOTT:2001; @HKOTY:1999] recently conjectured the equality between the one-dimensional configuration sum $X$ and the fermionic formula $M$ for all affine Kac–Moody algebras. The one-dimensional configuration sum $X$ originates from the corner-transfer-matrix method [@Baxter:1982] used to solve exactly solvable lattice models in statistical mechanics. It is the generating function of highest weight crystal paths graded by the energy statistic. The fermionic formula $M$ comes from the Bethe Ansatz [@Bethe:1931] and exhibits the quasiparticle structure of the underlying model. In combinatorial terms, it can be written as the generating function of rigged configurations.
The one-dimensional configuration sum depends on the underlying tensor product of crystals. In [@HKOTT:2001; @HKOTY:1999], the $X=M$ conjecture was formulated for tensor products of Kirillov–Reshetikhin (KR) crystals $B^{r,s}$. Kirillov–Reshetikhin crystals are crystals for finite-dimensional irreducible modules over quantum affine algebras. The irreducible finite-dimensional $U'_q({\mathfrak{g}})$-modules were classified by Chari and Pressley [@CP:1995; @CP:1998] in terms of Drinfeld polynomials. The Kirillov–Reshetikhin modules $W^{r,s}$, labeled by a Dynkin node $r$ of the underlying classical algebra and a positive integer $s$, form a special class of these finite-dimensional modules. They naturally correspond to the weight $s\Lambda_r$, where $\Lambda_r$ is the $r$-th fundamental weight of ${\mathfrak{g}}$. It was conjectured in [@HKOTT:2001; @HKOTY:1999], that there exists a crystal $B^{r,s}$ for each $W^{r,s}$. In general, the existence of $B^{r,s}$ is still an open question. For type $A_n^{(1)}$ the crystal $B^{r,s}$ is known to exist [@KKMMNN:1992] and its combinatorial structure has been studied [@Sh:2002]. The crystals $B^{1,s}$ for nonexceptional types, which are relevant for this paper, are also known to exist and their combinatorics has been worked out [@KKM:1994; @KKMMNN:1992].
The purpose of this paper is to establish the $X=M$ conjecture for tensor products of KR crystals of the form $B^{1,s}$ for nonexceptional affine algebras. This extends [@OSS:2002a], where $X=M$ is proved for tensor powers of $B^{1,1}$, and [@KR:1988; @KSS:2002], where $X=M$ is proved for type $A_n^{(1)}$.
Our method to prove $X=M$ for symmetric powers combines various previous results and techniques. $X=M$ is first proved for ${\mathfrak{g}}$ such that ${\overline{{\mathfrak{g}}}}$ is simply-laced (see Corollary \[cor:X=M AD\]). This is accomplished by exhibiting a grade-preserving bijection from $U'_q({\overline{{\mathfrak{g}}}})$-highest weight vectors (paths) to rigged configurations (RCs). This was already proved for the root system $A_n^{(1)}$ [@KSS:2002]. For type $D_n^{(1)}$ we exhibit such a path-RC bijection. The proof essentially reduces to the previously known $s=1$ case [@OSS:2002a] using the “splitting" maps $B^{1,s}\rightarrow B^{1,s-1}\otimes B^{1,1}$ which are $U_q({\overline{{\mathfrak{g}}}})$-equivariant grade-preserving embeddings.
To prove that the bijection preserves the grading, we consider an involution denoted $*$ on crystal graphs that combines contragredient duality with the action of the longest element $w_0$ of the Weyl group of ${\overline{{\mathfrak{g}}}}$. This duality on the crystal graph, corresponds under the path-RC bijection to the involution on RCs given by complementing the quantum numbers with respect to the vacancy numbers.
We then reduce to the case that ${\overline{{\mathfrak{g}}}}$ is simply-laced. This is achieved using the embedding of an affine algebra ${\mathfrak{g}}$ into one (call it ${\mathfrak{g}}_Y$) whose canonical simple Lie subalgebra is simply-laced. On the $X$ side we use the virtual crystal construction developed in [@OSS:2003a; @OSS:2003b]. It is shown in [@OSS:2003b] that the KR $U'_q({\mathfrak{g}})$-crystals $B^{1,s}$ embed into tensor products of KR $U'_q({\mathfrak{g}}_Y)$-crystals such that the grading is respected. One may define the $VX$ (“virtual X") formula in terms of the image of this embedding and show that $X=VX$ (see section \[ss:VX=V\]). This is proved for tensor products of crystals $B^{1,s}$ in [@OSS:2003b]. On the $M$ side, it is observed in [@OSS:2003b] that the RCs giving the fermionic formula $M$ for type ${\mathfrak{g}}$, embed into the set of RCs giving a fermionic formula for type ${\mathfrak{g}}_Y$. Let us denote by $VM$ (“virtual $M$") the generating function over the image of this embedding of fermionic formulas. It is shown in [@OSS:2003b] that $M=VM$. It then suffices to prove $VX=VM$. That is, one must show that the path-to-RC bijection that has already been established for the simply-laced cases, restricts to a bijection between the subsets of objects in the formulas $VX$ and $VM$. This is shown in Theorem \[th:virtualbij\] and as a corollary proves $X=M$ for nonsimply-laced algebras, as stated in Corollary \[cor:X=VX=VM=M\].
In section \[s:X\] we review the crystal theory, the definition of the one-dimensional configuration sum $X$, contragredient duality and the $*$ involution. Virtual crystals are reviewed in section \[s:virtual\]. Right and left splitting of crystals are discussed in sections \[s:right\] and \[s:left\], respectively. Rigged configurations and the analogs of the splitting maps are subject of section \[sec:RC\]. The fermionic formulas $M$ and their virtual counterparts $VM$ are stated in section \[s:fermionic\]. The $X=M$ conjecture for types $A_n^{(1)}$ and $D_n^{(1)}$ is proven in section \[sec:bij\] by establishing a statistics preserving bijection. Finally, in section \[sec:virtualbij\] the equality $X=VX=VM=M$ is established for nonsimply-laced types.
Formula $X$ {#s:X}
===========
Affine algebras
---------------
Let ${\mathfrak{g}}\supset {\mathfrak{g}}' \supset {\overline{{\mathfrak{g}}}}$ be a nonexceptional affine Kac-Moody algebra, its derived subalgebra and canonical simple Lie subalgebra [@Kac:1985]. Denote the corresponding quantized universal enveloping algebras by $U_q({\mathfrak{g}})\supset
U_q'({\mathfrak{g}})\supset U_q({\overline{{\mathfrak{g}}}})$ [@Ka:95]. Let $I={\bar{I}}\cup \{0\}$ (resp. ${\bar{I}}$) be the vertex set of the Dynkin diagram of ${\mathfrak{g}}$ (resp. ${\overline{{\mathfrak{g}}}}$). For $i\in I$, let $\alpha_i$, $h_i$, ${\Lambda}_i$ be the simple roots, simple coroots, and fundamental weights of ${\mathfrak{g}}$. Let $\{{\overline{\Lambda}}_i\mid i\in {\bar{I}}\}$ be the fundamental weights of ${\overline{{\mathfrak{g}}}}$. Let $(a_0,a_1,\dotsc,a_n)$ be the smallest tuple of positive integers giving a dependency relation on the columns of the Cartan matrix of ${\mathfrak{g}}$. Write $a_i^\vee$ for the corresponding integers for the Langlands dual Lie algebra, the one whose Cartan matrix is the transpose of that of ${\mathfrak{g}}$. Let $c=\sum_{i\in I} a_i^\vee h_i$ be the canonical central element and $\delta=\sum_{i\in I} a_i \alpha_i$ the generator of null roots. Let $Q,Q^\vee,P$ be the root, coroot, and weight lattices of ${\mathfrak{g}}$. Let ${\langle \cdot\,,\,\cdot\rangle}:Q^\vee\otimes P \rightarrow
{\mathbb{Z}}$ be the pairing such that ${\langle h_i\,,\,{\Lambda}_j\rangle}=\delta_{ij}$. Let $P\rightarrow {P'}\rightarrow{\overline{P}}$ be the natural surjections of weight lattices of ${\mathfrak{g}}\supset {\mathfrak{g}}'\supset {\overline{{\mathfrak{g}}}}$. Let ${\overline{P}}^+\subset {\overline{P}}$ be the dominant weights for ${\overline{{\mathfrak{g}}}}$. Let $W$ and ${\overline{W}}$ be the Weyl groups of ${\mathfrak{g}}$ and ${\overline{{\mathfrak{g}}}}$ respectively.
Crystal graphs
--------------
Let $M$ be a finite-dimensional $U'_q({\mathfrak{g}})$-module. Such modules are not highest weight modules (except for the zero module) and therefore need not have a crystal base. Suppose $M$ has a crystal base $B$. This is a special basis of $M$; it possesses the structure of a colored directed graph called the **crystal graph**. By abuse of notation the vertex set of the crystal graph is also denoted $B$. The edges of the crystal graph are colored by the set $I$. It has the following properties (that of a regular ${\overline{P}}$-weighted $I$-crystal):
1. Fix an $i\in I$. If all edges are removed except those colored $i$, the connected components are finite directed linear paths called the **$i$-strings** of $B$. Given $b\in B$, define $f_i(b)$ (resp. $e_i(b)$) to be the vertex following (resp. preceding) $b$ in its $i$-string; if there is no such vertex, declare the result to be the special symbol $\emptyset$. Define ${\varphi}_i(b)$ (resp. ${\varepsilon}_i(b)$) to be the number of arrows from $b$ to the end (resp. beginning) of its $i$-string.
2. There is a function ${\mathrm{wt}}:B\rightarrow {\overline{P}}$ such that $$\begin{split}
{\mathrm{wt}}(f_i(b))&={\mathrm{wt}}(b)-\alpha_i \\
{\varphi}_i(b)-{\varepsilon}_i(b) &= {\langle h_i\,,\,{\mathrm{wt}}(b)\rangle}.
\end{split}$$
A **morphism** $g:B\rightarrow B'$ of ${\overline{P}}$-weighted $I$-crystals is a map $g:B\cup\{\emptyset\}\rightarrow
B'\cup\{\emptyset\}$ such that $g(\emptyset)=\emptyset$ and for any $b\in B$ and $i\in I$, $g(f_i(b))=f_i(g(b))$ and $g(e_i(b))=e_i(g(b))$. An isomorphism of crystals is a morphism of crystals which is a bijection whose inverse bijection is also a morphism of crystals.
If $B_i$ is the crystal base of the $U'_q({\mathfrak{g}})$-module $M_i$ for $i=1,2$ then the **tensor product** $M_2 \otimes M_1$ is a $U'_q({\mathfrak{g}})$-module with crystal base denoted $B_2\otimes B_1$. Its vertex set is just the cartesian product $B_2 \times B_1$. Its edges are given in terms of those of $B_1$ and $B_2$ as follows.
We use the opposite of Kashiwara’s tensor product convention.
One has ${\mathrm{wt}}(b_2\otimes b_1)={\mathrm{wt}}(b_2)+{\mathrm{wt}}(b_1)$ and $$\begin{split}
f_i(b_2\otimes b_1) &= \begin{cases}
f_i(b_2)\otimes b_1 &\text{if ${\varepsilon}_i(b_2)\ge{\varphi}_i(b_1)$} \\
b_2 \otimes f_i(b_1) &\text{otherwise,}
\end{cases} \\
e_i(b_2\otimes b_1) &= \begin{cases}
e_i(b_2)\otimes b_1 &\text{if ${\varepsilon}_i(b_2)>{\varphi}_i(b_1)$} \\
b_2 \otimes e_i(b_1) &\text{otherwise,}
\end{cases}
\end{split}$$ where the result is declared to be $\emptyset$ if either of its tensor factors are.
The tensor product construction is associative up to isomorphism.
Define ${\varphi},{\varepsilon}:B\rightarrow{P'}$ by $${\varphi}(b) = \sum_{i\in I} {\varphi}_i(b) {\Lambda}_i \qquad\qquad {\varepsilon}(b) = \sum_{i\in I} {\varepsilon}_i(b) {\Lambda}_i.$$
Every irreducible integrable finite-dimensional $U_q({\overline{{\mathfrak{g}}}})$-module is a highest weight module with some highest weight ${\lambda}\in{\overline{P}}^+$; denote its crystal graph by $B({\lambda})$. It is a ${\overline{P}}$-weighted ${\bar{I}}$-crystal with a unique classical highest weight vector.
A **classical component** of the crystal graph $B$ of a $U'_q({\mathfrak{g}})$-module is a connected component of the graph obtained by removing all $0$-arrows from $B$. The vertex $b\in B$ is a **classical highest weight vector** if ${\varepsilon}_i(b)=0$ for all $i\in {\bar{I}}$. Each classical component of a $U'_q({\mathfrak{g}})$-module has a unique classical highest weight vector.
Finite crystals {#ss:fin}
---------------
Let ${\mathcal{C}^{\mathrm{fin}}}$ be the category of **finite** crystals as defined in [@HKKOT:2000]. Every $B\in {\mathcal{C}^{\mathrm{fin}}}$ has the following properties.
1. $B$ is the crystal base of an irreducible $U'_q({\mathfrak{g}})$-module and is therefore connected.
2. There is a weight ${\lambda}\in {\overline{P}}^+$ such that there is a unique $u(B)\in B$ with ${\mathrm{wt}}(u(B))={\lambda}$ and for all $b\in B$, ${\mathrm{wt}}(b)$ is in the convex hull of ${\overline{W}}{\lambda}$.
${\mathcal{C}^{\mathrm{fin}}}$ is a tensor category [@HKKOT:2000]. If $B,B'\in {\mathcal{C}^{\mathrm{fin}}}$ then $B \otimes B'\in {\mathcal{C}^{\mathrm{fin}}}$ is connected and $u(B\otimes
B')=u(B)\otimes u(B')$. Due to the existence of the universal $R$-matrix for $U'_q({\overline{{\mathfrak{g}}}})$ it follows from [@KMN:1992] that:
1. There is a unique $U'_q({\mathfrak{g}})$-crystal isomorphism $R_{B,B'}:B \otimes B'\rightarrow B'\otimes B$ called the **combinatorial $R$-matrix**.
2. There is a unique function (the **local coenergy**) $H=H_{B,B'}:B \otimes
B'\rightarrow {\mathbb{Z}}_{\ge0}$ that is constant on classical components, zero on $u(B\otimes B')$, and is such that if $R_{B,B'}(b \otimes
b')=c'\otimes c$ then $$\label{eq:H}
H(e_0(b\otimes b')) = H(b\otimes b') +
\begin{cases}
1 &
\begin{aligned}
\text{if } e_0(b\otimes b')&=e_0(b)\otimes b' \text{ and } \\
e_0(c'\otimes c)&=e_0(c')\otimes c
\end{aligned} \\
-1 &
\begin{aligned}
\text{if } e_0(b\otimes b')&=b \otimes e_0(b') \text{ and } \\
e_0(c'\otimes c)&=c'\otimes e_0(c)
\end{aligned} \\
0 &
\text{otherwise.}
\end{cases}$$
The combinatorial $R$-matrices satisfy $$\begin{split}
R_{B,B} &= 1_{B\otimes B} \\
R_{B_1,B_2} \circ R_{B_2,B_1} &= 1_{B_2\otimes B_1}
\end{split}$$ and the Yang Baxter equation, the equality of isomorphisms $B_3\otimes B_2 \otimes B_1\rightarrow B_1 \otimes B_2 \otimes
B_3$ given by $$\label{eq:RYB}
\begin{split}
&(1_{B_1} \otimes R_{B_3,B_2})\circ (R_{B_3,B_1}\otimes 1_{B_2})
\circ (1_{B_3} \otimes R_{B_2,B_1}) \\
=\, &(R_{B_2,B_1} \otimes 1_{B_3}) \circ (1_{B_2} \otimes
R_{B_3,B_1}) \circ (R_{B_3,B_2} \otimes 1_{B_1}).
\end{split}$$ We shall abuse notation and write $R_j$ (resp. $H_j$) to denote the application of an appropriate combinatorial $R$-matrix (resp. local coenergy function) on the $(j+1)$-th and $j$-th tensor factors *from the right*. Then reads $R_1 R_2 R_1=R_2 R_1 R_2$. One has the following identities on a three-fold tensor product: $$\begin{split}
H_2 + H_1 R_2 &= H_2 R_1 + H_1 R_2 R_1 \\
H_1 + H_2 R_1 &= H_1 R_2 + H_2 R_1 R_2.
\end{split}$$
\[pp:Hbraided\] [@OSS:2003a] Let $B=B_L\otimes\dotsm\otimes B_1$ and $B'=B_M'\otimes\dotsm \otimes B_1'$.
1. $R_{B,B'}$ is equal to any composition of $R$-matrices of the form $R_{B_i,B_j'}$ which shuffle the $B_i$ to the right, past the $B_j'$.
2. For $b\otimes b'\in B\otimes B'$, the value of $H_{B,B'}$ is the sum of the values $H_{B_i \otimes B_j'}$ evaluated at the pairs of elements in $B_i
\otimes B_j'$ that must be switched by an $R$-matrix $R_{B_i
,B_j'}$ in the computation of $R_{B,B'}(b\otimes b')$.
Categories ${\mathcal{C}}$ and ${{\mathcal{C}}^A}$ of KR crystals {#ss:tensorcats}
-----------------------------------------------------------------
We work with two categories of crystals. Let ${\mathfrak{g}}$ be of nonexceptional affine type. The KR modules $W^{(1)}_s$ and their crystal bases $B^s:=B^{1,s}$ were constructed in [@KKM:1994]. See also [@OSS:2003b] for an explicit description of $B^s$. Let ${\mathcal{C}}$ be the category of tensor products of KR crystals of the form $B^s$. One has that ${\mathcal{C}}\subset {\mathcal{C}^{\mathrm{fin}}}$.
The crystal $B^s$ has the $U_q({\overline{{\mathfrak{g}}}})$-decomposition $$\label{eq:ClDecomp}
B^s \cong \begin{cases}
B(s{\overline{\Lambda}}_1) & \text{for $A_n^{(1)}$, $B_n^{(1)}$, $D_n^{(1)}$, $A_{2n-1}^{(2)}$} \\
\displaystyle{\bigoplus_{r=0}^s B((s-r){\overline{\Lambda}}_1)}
&\text{for $A_{2n}^{(2)},D_{n+1}^{(2)}$} \\
\displaystyle{\bigoplus_{r=0}^{\lfloor \frac{s}{2}\rfloor}
B((s-2r){\overline{\Lambda}}_1)} & \text{for $C_n^{(1)}$, $A_{2n}^{(2)\dagger}$.}
\end{cases}$$ In particular $u(B^s)$ is the unique vector of weight $s{\overline{\Lambda}}_1$ in $B^s$.
Let ${{\mathcal{C}}^A}$ be the category of all tensor products of KR crystals $B^{r,s}$ in type $A_n^{(1)}$. Here $B^{r,s} \cong B(s{\overline{\Lambda}}_r)$. So $u(B^{r,s})$ is the unique vector in $B^{r,s}$ of weight $s{\overline{\Lambda}}_r$. $B^{r,s}$ consists of the semistandard Young tableaux of shape given by an $r\times s$ rectangle, with entries in the set $\{1,2,\dotsc,n+1\}$ [@KN:1994]. The structure of $B^{r,s}$ as an affine crystal was given explicitly in [@Sh:2002].
We fix some notation for $B\in {\mathcal{C}}$ or $B\in{{\mathcal{C}}^A}$. Let ${\mathcal{H}}={\bar{I}}\times
{\mathbb{Z}}_{>0}$ where recall that ${\bar{I}}=\{1,2,\dotsc,n\}$ is the set of Dynkin nodes for ${\overline{{\mathfrak{g}}}}$. The multiplicity array of $B$ is the array $L=(L_i^{(a)}\mid(a,i)\in{\mathcal{H}})$ such that $L_i^{(a)}$ is the number of times $B^{a,i}$ occurs as a tensor factor in $B$ for all $(a,i)\in {\mathcal{H}}$. Up to reordering of tensor factors $B=\bigotimes_{(a,i)\in{\mathcal{H}}} (B^{a,i})^{\otimes L_i^{(a)}}$.
Intrinsic coenergy {#ss:D}
------------------
For $B\in{\mathcal{C}^{\mathrm{fin}}}$, say that $D:B\rightarrow {\mathbb{Z}}$ is an **intrinsic coenergy** function for $B$ if $D(u(B))=0$, $D$ is constant on $U_q({\overline{{\mathfrak{g}}}})$-components, and $$D(e_0(b)) - D(b) \le 1\qquad\text{for all $b\in B$.}$$ A **graded crystal** is a pair $(B,D)$ where $B\in {\mathcal{C}^{\mathrm{fin}}}$ and $D$ is an intrinsic coenergy function on $B$.
We shall give each $B\in{\mathcal{C}}$ a particular graded crystal structure.
For $B \in {\mathcal{C}^{\mathrm{fin}}}$ define $${\mathrm{level}}(B) = \min \{ {\langle c\,,\,{\varphi}(b)\rangle} \mid b\in B \}.$$ One may verify that there is a unique element $b^{\natural}\in B^s$ such that $${\varphi}(b^{\natural})={\mathrm{level}}(B^s){\Lambda}_0.$$ Define the intrinsic coenergy function $D_{B^s}:B^s\rightarrow{\mathbb{Z}}$ by $$D_{B^s}(b) = H_{B^s,B^s}(b \otimes b^{\natural})-H_{B^s,B^s}(u(B^s)\otimes b^{\natural}).$$
\[ex:DKR\] $D_{B^s}$ has value $r$ on the $r$-th summand in .
\[pp:tensor\] [@OSS:2003a] Graded crystals form a tensor category as follows. If $(B_j,D_j)$ is a graded crystal for $1\le j\le L$, then their tensor product $B=B_L\otimes\dotsm\otimes B_1$ is a graded crystal with $$\label{eq:DNY}
D_B = \sum_{1\le i<j\le L} H_i R_{i+1} R_{i+2}\dotsm R_{j-1}
+ \sum_{j=1}^L D_{B_j} R_1 R_2 \dotsm R_{j-1}$$ where $D_{B_j}$ acts on the rightmost tensor factor.
$X$ formula
-----------
Let $(B,D)$ be a graded crystal. For ${\lambda}\in{\overline{P}}^+$ let $P(B,{\lambda})$ be the set of classical highest weight vectors in $B$ of weight ${\lambda}$. Define the one-dimensional sum $$\label{eq:X}
X_{B,{\lambda}}(q) = \sum_{b\in P(B,{\lambda})} q^{D_B(b)/a_0}.$$ Recall that $a_0=1$ unless ${\mathfrak{g}}=A_{2n}^{(2)}$ in which case $a_0=2$.
Contragredient duality {#ss:dual}
----------------------
Given a $U'_q({\mathfrak{g}})$-module $M$ with crystal base $B$, the contragredient dual module $M^\vee$ has a crystal base $B^\vee=\{b^\vee\mid b\in B \}$ such that $$\begin{split}
{\mathrm{wt}}(b^\vee) &= - {\mathrm{wt}}(b) \\
f_i(b^\vee) &= e_i(b)^\vee
\end{split}$$ for $i\in I$ and $b\in B$ such that $e_i(b)\not=\emptyset$.
\[pp:dualtensor\] $$(B_2\otimes B_1)^\vee \cong B_1^\vee \otimes B_2^\vee.$$
\[ex:dualA\] Assume type $A_n^{(1)}$. We have $$\label{eq:dualKR}
B^{r,s\vee} \cong B^{n+1-r,s}.$$ The composite map $$B^{r,s} \overset{\vee}{\longrightarrow} B^{r,s\vee} \cong
B^{n+1-r,s}$$ is given explicitly as follows. Let $b\in B^{r,s}$. Replace each column of $b$, viewed as a subset of $\{1,2,\dotsc,n+1\}$ of size $r$, by the column of size $n+1-r$ given by its complement. Then reverse the order of the columns. For $n=5$, $r=2$, and $s=3$, a tableau $b\in B^{r,s}$ and its image in $B^{n+1-r,s}$ are given below: $$\young(112,346)\mapsto \young(122,334,455,566).$$
\[ex:dualArow\] By definition $B^{1,1\vee}$ is defined by replacing each element of $b\in B^{1,1}$ by an element $b^\vee$ and reversing arrows. $B^{1,s\vee}$ can be realized by the weakly increasing words of length $s$ in the alphabet $\{(n+1)^\vee<\dotsm<2^\vee<1^\vee\}$. The arrow-reversing map from $B^s$ to $B^{s\vee}$ is given by taking a word of length $s$, replacing each symbol $i$ with $i^\vee$, and reversing.
Dynkin automorphisms {#ss:dynkin}
--------------------
Let $\sigma$ be an automorphism of the Dynkin diagram of ${\mathfrak{g}}$. This induces isometries $\sigma:P\rightarrow P$ and $\sigma:{\overline{P}}\rightarrow{\overline{P}}$ given by $\sigma({\Lambda}_i)={\Lambda}_{\sigma(i)}$ for $i\in I$, $\sigma(\delta)=\delta$, and $\sigma({\overline{\Lambda}}_i)={\overline{\Lambda}}_{\sigma(i)}$ for $i\in {\bar{I}}$.
If $M$ is a $U'_q({\mathfrak{g}})$-module with crystal base $B$, then by carrying out the construction of $M$ but with $i$ replaced everywhere by $\sigma(i)$, there is a $U'_q({\mathfrak{g}})$-module $M^\sigma$ with crystal base $B^\sigma$ and a bijection $\sigma:B\rightarrow B^\sigma$ such that $$\begin{split}
{\mathrm{wt}}(\sigma(b)) &= \sigma({\mathrm{wt}}(b)) \\
\sigma(e_i(b)) &= e_{\sigma(i)}(b) \\
\sigma(f_i(b)) &= f_{\sigma(i)}(b)
\end{split}$$ for all $b\in B$ and $i\in I$.
In particular, if the appropriate KR modules have been constructed then $$(B^{r,s})^\sigma=B^{\sigma(r),s}.$$
The Dynkin involution ${\tau}$ {#ssec:sigma}
------------------------------
We fix a canonical Dynkin automorphism ${\tau}$ of the affine Dynkin diagram in the following manner. There is a length-preserving involution on ${\overline{W}}$ given by conjugation by the longest element $w_0\in{\overline{W}}$. Restricting this involution to elements of length one, one obtains an involution ${\tau}$ on the set of simple reflections $\{s_i\mid i\in {\bar{I}}\}$ of ${\overline{W}}$. For simplicity of notation this can be written as an involution on the index set ${\bar{I}}$. This gives an automorphism of the Dynkin diagram of ${\tau}$. Call the resulting Dynkin automorphism ${\tau}$.
Explicitly, ${\tau}$ is the identity except when ${\overline{{\mathfrak{g}}}}=A_{n-1}$ where ${\tau}$ exchanges $i$ and $n-i$, and ${\overline{{\mathfrak{g}}}}=D_n$ with $n$ odd, where ${\tau}$ exchanges $n-1$ and $n$ and fixes all other Dynkin nodes.
${\tau}$ may be extended to the Dynkin diagram of ${\mathfrak{g}}$ by fixing the $0$ node. It satisfies $w_0 s_i w_0 = s_{{\tau}(i)}$ for all $i\in I$.
The automorphism ${\tau}$ induces the following action on the weight lattice $P$: $${\tau}({\Lambda}_i) = {\Lambda}_{{\tau}(i)}\qquad\text{for $i\in I$.} \\$$ One may show that this is equivalent to $${\tau}({\Lambda}) = - w_0 \,{\Lambda}\qquad\text{for ${\Lambda}\in P$.}$$ In particular $$\label{eq:sigalpha}
{\tau}(\alpha_i) = \alpha_{{\tau}(i)} = - w_0\, \alpha_i.$$
The $*$ involution
------------------
Let $M$ be a $U'_q({\mathfrak{g}})$-module with crystal base $B$. With ${\tau}$ as above, define the module $$M^* = M^{{\tau}\vee}.$$ It has crystal base $B^*$, with elements $b^*$ for $b\in B$ such that $$\label{eq:wtstar}
{\mathrm{wt}}(b^*)= w_0 {\mathrm{wt}}(b)$$ and $$\label{eq:defstar}
\begin{split}
e_i(b^*) &= f_{{\tau}(i)}(b)^* \\
f_i(b^*) &= e_{{\tau}(i)}(b)^*
\end{split}$$ for all $i\in I$.
\[rem:starcomp\] By for $i\in{\bar{I}}$ it follows that the map $*$ sends classical components of $B$ to classical components of $B^*$, which by must have the same classical highest weight.
\[pp:star\] $(B_1\otimes B_2)^*\cong B_2^*\otimes B_1^*$ with $(b_1\otimes b_2)^* \mapsto b_2^* \otimes b_1^*$.
\[conj:crystalstar\] Let $B\in
{\mathcal{C}^{\mathrm{fin}}}$. Then there is a unique involution $*:B\rightarrow B$ such that and hold.
Uniqueness follows from the connectedness of $B$ and the fact that $u(B)$ is the unique vector in $B$ of its weight.
\[rem:Rstar\] The crystals satisfying Conjecture \[conj:crystalstar\] form a tensor category. Given involutions $*$ on $B_1$ and $B_2$ satisfying Conjecture \[conj:crystalstar\], define $*$ on $B_1
\otimes B_2$ by $(b_1\otimes b_2)^*= R(b_2^* \otimes b_1^*)$.
\[rem:starclass\] For ${\lambda}\in{\overline{P}}^+$ define the involution $*$ on $B({\lambda})$ to be the unique map that sends the highest weight vector $u_{\lambda}$ to the lowest weight vector (the unique vector of weight $w_0({\lambda})$) and satisfies for $i\in {\bar{I}}$. By it follows that ${\mathrm{wt}}(b^*)=w_0{\mathrm{wt}}(b)$ for all $b\in B({\lambda})$.
Explicitly, the involution $*$ on the $U_q({\overline{{\mathfrak{g}}}})$-crystal $B({\overline{\Lambda}}_1)$ is given by $$\begin{aligned}[2]
& & i&\leftrightarrow \bar{i} \\
& & {\circ}&\leftrightarrow{\circ}\end{aligned}$$ except for $$\begin{aligned}[2]
\text{${\overline{{\mathfrak{g}}}}=A_{n-1}$:}& &i&\leftrightarrow n+1-i \\
\text{${\overline{{\mathfrak{g}}}}=D_n$, $n$ odd:} & & n&\leftrightarrow n \qquad
\bar{n}\leftrightarrow\bar{n}.
\end{aligned}$$ Here we use that the crystal of $B({\overline{\Lambda}}_1)$ has underlying set [@KN:1994] $$\begin{aligned}
&\{1<2<\cdots<n\}
&& \text{for $A_{n-1}$}\\
&\{1<2<\cdots<n<{\circ}<\bar{n}<\cdots<\bar{2}<\bar{1} \}
&& \text{for $B_n$} \\
&\{1<2<\cdots<n<\bar{n}<\cdots<\bar{2}<\bar{1} \}
&& \text{for $C_n$} \\
&\{1<2<\cdots<\begin{matrix}{n} \\ {\bar{n}} \end{matrix} <\cdots<\bar{2}<\bar{1} \}
&& \text{for $D_n$.}\end{aligned}$$
Explicit formula for $*$
------------------------
We wish to determine the map $*$ of Conjecture \[conj:crystalstar\] explicitly for $B^s\in{\mathcal{C}}$ and $B^{r,s}\in
{{\mathcal{C}}^A}$. The map $*:B^s\rightarrow B^s$ must stabilize classical components by Remark \[rem:starcomp\] and the multiplicity-freeness of $B^s$ as a classical crystal. On each classical component $B(s'{\overline{\Lambda}}_1)$ of $B^s$, $*$ is uniquely defined by Remark \[rem:starclass\]. Using the $U_q({\overline{{\mathfrak{g}}}})$-embedding $B(s'{\overline{\Lambda}}_1)\rightarrow B({\overline{\Lambda}}_1)^{\otimes s'}$ and Proposition \[pp:star\], we have $(b_1b_2\dotsm b_{s'})^*=b_{s'}^* \dotsm
b_2^* b_1^*$. For $B^{r,s}\in {{\mathcal{C}}^A}$ and $b\in B^{r,s}$, $b^*$ is the tableau obtained by replacing every entry $c$ of $b$ by $c^*$ and then rotating by 180 degrees. The resulting tableau is sometimes called the **antitableau** of $b$.
For type $D_5^{(1)}$ we have $$\young(113{\bar{5}})^{\; *}=\young({\bar{5}}{\bar{3}}{\bar{1}}{\bar{1}})$$ For type $A_4^{(1)}$ $${\young(11,23)}^{\; *}=\young(34,55)\,.$$
\[pp:srstar\] With $*$ defined as above, Conjecture \[conj:crystalstar\] holds for $B^s\in {\mathcal{C}}$ and $B^{r,s}\in {{\mathcal{C}}^A}$.
\[rem:star\] From now on the notation $*$ will only be used in the following way. Let $B=B_L\otimes\dotsm\otimes B_1$ be a tensor product of factors $B_j=B^{s_j}$. Since $*$ may be regarded as an involution on $B^s$, by Proposition \[pp:star\] we may write $B^*=B_1^*\otimes\dotsm\otimes B_L^*=B_1\otimes\dotsm\otimes B_L$ for the **reversed** tensor product. Then $*:B\rightarrow B^*$ is defined by $(b_L\otimes\dotsm\otimes b_1)^*\mapsto
b_1^*\otimes\dotsm\otimes b_L^*$.
\[pp:R\*\] Let $R_j$ be the $R$-matrix acting at the $j^{\mathrm{th}}$ and $(j+1)^{\mathrm{st}}$ tensor positions from the right. On an $L$-fold tensor product of crystals of the form $B^s$, $$\label{eq:R*}
R_j \circ * = * \circ R_{L-j}$$ for $1\le j\le L-1$.
One may reduce to the case $L=2$. Since $B_2\otimes B_1$ is connected, $R$ is an isomorphism, and since holds, it suffices to check on $u(B_2\otimes B_1)$. But this holds by weight considerations.
Virtual crystals {#s:virtual}
================
We review the virtual crystal construction [@OSS:2003a; @OSS:2003b]. This allows one to reduce the study of affine crystal graphs to those of simply-laced type.
Embeddings of affine algebras {#ss:VX}
-----------------------------
Any affine algebra ${\mathfrak{g}}$ of type $X$ can be embedded into a simply-laced affine algebra ${\mathfrak{g}}_Y$ of type $Y$ [@HKOTT:2001]. For ${\mathfrak{g}}$ nonexceptional the embeddings are listed below. The notation $A_{2n}^{(2)}$ and $A_{2n}^{(2)\dagger}$ is used for two different vertex labelings of the same Dynkin diagram, in which $\alpha_0$ is respectively the extra short and extra long root. $$\label{eq:embed}
\begin{split}
C_n^{(1)},A_{2n}^{(2)},A_{2n}^{(2)\dagger},D_{n+1}^{(2)} &\hookrightarrow
A_{2n-1}^{(1)} \\
B_n^{(1)},A_{2n-1}^{(2)} &\hookrightarrow D_{n+1}^{(1)}.
\end{split}$$
Folding automorphism
--------------------
Let ${\sigma}$ be the following automorphism of the Dynkin diagram of $Y$. For $A_{2n-1}^{(1)}$, ${\sigma}(i)=2n-i$ (mod $2n$). For type $D_{n+1}^{(1)}$, ${\sigma}$ exchanges the nodes $n$ and $n+1$ and fixes all others.
Let $I^X$ and $I^Y$ be the vertex sets of the diagrams $X$ and $Y$ respectively, $I^Y/{\sigma}$ the set of orbits of the action of ${\sigma}$ on $I^Y$, and ${\iota}:I^X\rightarrow I^Y/{\sigma}$ a bijection which preserves edges and sends $0$ to $0$.
If $Y=A_{2n-1}^{(1)}$, then ${\iota}(0)=\{0\}$, ${\iota}(i)=\{i,2n-i\}$ for $0<i<n$ and ${\iota}(n)=\{n\}$.
If $Y=D_{n+1}^{(1)}$, then ${\iota}(i)=i$ for $i<n$ and ${\iota}(n)=\{n,n+1\}$.
Embedding of weight lattices
----------------------------
For $i\in I^X$ define ${\gamma}_i$ as follows.
1. Let $Y=D_{n+1}^{(1)}$.
1. Suppose the arrow points towards the component of $0$. Then ${\gamma}_i=1$ for all $i\in I^X$.
2. Suppose the arrow points away from the component of $0$. Then ${\gamma}_i$ is the order of ${\sigma}$ for $i$ in the component of $0$ and is $1$ otherwise.
2. Let $Y=A_{2n-1}^{(1)}$. Then ${\gamma}_i=1$ for $1\le i\le n-1$. For $i\in\{0,n\}$, ${\gamma}_i=2$ (which is the order of ${\sigma}$) if the arrow incident to $i$ points away from it and is $1$ otherwise.
For $X=B_n^{(1)}$ and $Y=D_{n+1}^{(1)}$ we have ${\gamma}_i=2$ if $0\le i \le n-1$ and ${\gamma}_n=1$. For $X=A_{2n-1}^{(2)}$ and $Y=D_{n+1}^{(1)}$ we have ${\gamma}_i=1$ for all $i$.
The embedding ${\Psi}:P^X\to P^Y$ of weight lattices is defined by $${\Psi}({\Lambda}^X_i) = {\gamma}_i \sum_{j\in {\iota}(i)} {\Lambda}^Y_j.$$ As a consequence we have $$\label{eq:Vdelta}
\begin{split}
{\Psi}(\alpha^X_i) &= {\gamma}_i \sum_{j\in {\iota}(i)} \alpha^Y_j \\
{\Psi}(\delta^X) &= a_0^X {\gamma}_0 \,\delta^Y.
\end{split}$$
Virtual crystals {#virtual-crystals}
----------------
Fix an embedding ${\mathfrak{g}}_X\hookrightarrow {\mathfrak{g}}_Y$ in .
Let ${\widehat{V}}$ be a $Y$-crystal. For $i\in I^X$ define the virtual crystal operators ${\hat{e}}_i,{\hat{f}}_i$ on ${\widehat{V}}$, as the composites of $Y$-crystal operators $e_j,f_j$ given by $${\hat{e}}_i = \prod_{j\in {\iota}(i)} e_j^{{\gamma}_i}\qquad\qquad {\hat{f}}_i =
\prod_{j\in {\iota}(i)} f_j^{{\gamma}_i} .$$
A **virtual crystal** (aligned in the sense of [@OSS:2003a; @OSS:2003b]) is an injection ${\Psi}:B\rightarrow {\widehat{V}}$ from an $X$-crystal $B$ to a $Y$-crystal ${\widehat{V}}$ such that:
1. For all $b\in B$, $i\in I^X$, and $j\in {\iota}(i)\subset
I^Y$, ${\varphi}_j({\Psi}(b))={\gamma}_i {\varphi}_i(b)$ and ${\varepsilon}_j({\Psi}(b))={\gamma}_i {\varepsilon}_i(b)$.
2. ${\Psi}\circ e_i = {\hat{e}}_i$ and ${\Psi}\circ f_i = {\hat{f}}_i$ for all $i\in
I^X$.
A virtual crystal realizes the $X$-crystal $B$ as the subset of the $Y$-crystal ${\widehat{V}}$ given by its image under ${\Psi}$, equipped with the virtual Kashiwara operators ${\hat{e}}_i$ and ${\hat{f}}_i$.
A morphism $g$ of virtual crystals ${\Psi}:B\rightarrow{\widehat{V}}$ and ${\Psi}':B'\rightarrow {\widehat{V}}'$ consists of a morphism $g_X:B\rightarrow B'$ of $X$-crystals and a morphism $g_Y:{\widehat{V}}\rightarrow {\widehat{V}}'$ of $Y$-crystals, such that the diagram commutes: $$\begin{CD}
B @>{{\Psi}}>> {\widehat{V}}\\
@V{g_X}VV @VV{g_Y}V \\
B' @>>{{\Psi}'}> {\widehat{V}}'
\end{CD}$$ An isomorphism $g$ of virtual crystals is a morphism $(g_X,g_Y)$ such that $g_X$ (resp. $g_Y$) is an isomorphism of $X$- (resp. $Y$-) crystals.
Tensor product of virtual crystals
----------------------------------
Let ${\Psi}:B\rightarrow {\widehat{V}}$ and ${\Psi}':B'\rightarrow {\widehat{V}}'$ be virtual crystals. It is straightforward to verify that ${\Psi}\otimes {\Psi}':B \otimes B' \rightarrow {\widehat{V}}\otimes {\widehat{V}}'$ is a virtual crystal. Virtual crystals form a tensor category [@OSS:2003a].
Virtual $B^s$
-------------
We recall from [@OSS:2003b] the virtual crystal construction of $B^s=B^{1,s}$ for ${\mathfrak{g}}$ of nonexceptional affine type. Let ${\widehat{V}}^s$ be given by $${\widehat{V}}^s = \begin{cases}
B_Y^{s\vee} \otimes B_Y^s & \text{if ${\mathfrak{g}}_Y=A_{2n-1}^{(1)}$} \\
B_Y^s &\text{if ${\mathfrak{g}}_Y=D_{n+1}^{(1)}$ and ${\mathfrak{g}}=A_{2n-1}^{(2)}$} \\
B_Y^{2s} & \text{if ${\mathfrak{g}}_Y=D_{n+1}^{(1)}$ and ${\mathfrak{g}}=B_n^{(1)}$.}
\end{cases}$$
\[th:V\] [@OSS:2003b] There is a unique virtual crystal ${\Psi}:B^s\rightarrow {\widehat{V}}^s$ such that ${\Psi}(u(B^s)) = u({\widehat{V}}^s)$.
Let $X=B_3^{(1)}$ and $Y=D_4^{(1)}$. Then ${\widehat{V}}^s=B_Y^{2s}$. Let $b=\young(1\circ{\bar{2}})\in B_X^3$. Then ${\Psi}(b)=\young(113{\bar{3}}{\bar{2}}{\bar{2}})$ and $f_3(b)=\young(1{\bar{3}}{\bar{2}})$. Furthermore $${\hat{f}}_3({\Psi}(b))=f_3\circ
f_4({\Psi}(b))=\young(11{\bar{3}}{\bar{3}}{\bar{2}}{\bar{2}}).$$
Virtual R-matrix
----------------
\[pp:VR\] [@OSS:2003b] Let ${\hat{R}}:{\widehat{V}}^t\otimes{\widehat{V}}^s\rightarrow {\widehat{V}}^s\otimes{\widehat{V}}^t$ be the composition of combinatorial $R$-matrices of type $Y$. Then the diagram commutes: $$\begin{CD}
B^t \otimes B^s @>{{\Psi}}>> {\widehat{V}}^t \otimes {\widehat{V}}^s \\
@V{R}VV @VV{{\hat{R}}}V \\
B^s \otimes B^t @>>{{\Psi}}> {\widehat{V}}^s \otimes {\widehat{V}}^t.
\end{CD}$$ That is, the pair $(R,{\hat{R}})$ is an isomorphism of virtual crystals ${\Psi}:B^t\otimes B^s\rightarrow {\widehat{V}}^t\otimes {\widehat{V}}^s$ and ${\Psi}:B^s\otimes B^t\rightarrow {\widehat{V}}^s \otimes {\widehat{V}}^t$.
Virtual local coenergy
----------------------
\[pp:VH\] [@OSS:2003b] Let ${\Psi}:B\rightarrow{\widehat{V}}$ and ${\Psi}':B'\rightarrow {\widehat{V}}'$ be virtual crystals where $B,B'\in{\mathcal{C}}$ both of type $X$. Then $$H^X_{B,B'} = \frac{1}{{\gamma}_0}\cdot H^Y_{{\widehat{V}},{\widehat{V}}'} \circ ({\Psi}\otimes
{\Psi}').$$
Virtual graded crystal
----------------------
\[pp:VD\] [@OSS:2003b] Let $B\in {\mathcal{C}}$ be a crystal of type $X$ and ${\Psi}:B\rightarrow {\widehat{V}}$ the corresponding virtual crystal. Then $$D^X = \frac{1}{{\gamma}_0}\cdot D^Y \circ \Psi.$$
Virtual $X$ formula {#ss:VX=V}
-------------------
Let $B\in {\mathcal{C}}$ be a crystal of type $X$. Let ${\Psi}:B\rightarrow{\widehat{V}}$ be the corresponding virtual crystal. For ${\lambda}\in{\overline{P}}^+$ let ${P^v}(B,{\lambda})$ be image under ${\Psi}$ of the set $P(B,{\lambda})$. Define the virtual X formula by $$VX_{B,{\lambda}}(q) = \sum_{b\in {P^v}(B,{\lambda})} q^{D({\Psi}(b))/{\gamma}_0}.$$
\[th:X=VX\] ($X=VX$) [@OSS:2003b] For ${\mathfrak{g}}$ of nonexceptional affine type $X$ and $B\in{\mathcal{C}}$ a crystal of type $X$, one has $X_{B,{\lambda}}(q)=VX_{B,{\lambda}}(q)$.
Virtual crystals and $*$-duality
--------------------------------
We believe that the following is true for any virtual crystal, namely, that up to $R$-matrices, ${\Psi}$ takes the $*$ involution of type $X$ to the $*$ involution of type $Y$.
\[pp:\*emb\] Let $B^s\in{\mathcal{C}}$ and let ${\Psi}:B^s\rightarrow{\widehat{V}}^s$ be a virtual crystal. Then the following diagram commutes, where $\iota$ is either a composition of $R$-matrices or the identity: $$\begin{CD}
B^s @>{{\Psi}}>> {\widehat{V}}^s \\
@V{*}VV @VV{\iota\circ *}V \\
B^s @>>{{\Psi}}> {\widehat{V}}^s
\end{CD}$$
Note that for the non-simply-laced types $X$ the Dynkin involution $\tau_X$ is the identity. The virtual raising and lowering operators are invariant under $\tau_Y$. It is therefore sufficient to check the above commutation on $v\in P(B^s)$, where $P(B^s)$ is the set of classical highest weight vectors in $B^s$. But $B^s$ and ${\widehat{V}}^s$ are multiplicity-free as a classical crystals and $*$ stabilizes classical components and modifies the weight of a crystal element by applying $w_0$. The following are equivalent:
1. ${\Psi}(v)^*\in {\widehat{V}}^s$ is a classical lowest weight vector and ${\mathrm{wt}}({\Psi}(v)^*)=w_0^Y({\Psi}({\lambda}))$.
2. ${\Psi}(v)\in P({\widehat{V}}^s)$ and ${\mathrm{wt}}({\Psi}(v))={\Psi}({\lambda})$.
3. $v\in P(B^s)$ and ${\mathrm{wt}}(v)={\lambda}$.
4. $v^*\in B^s$ is a classical lowest weight vector and ${\mathrm{wt}}(v^*)=w_0^X{\lambda}$.
5. ${\Psi}(v^*)$ is a classical lowest weight vector and ${\mathrm{wt}}({\Psi}(v^*))={\Psi}(w_0^X{\lambda})$.
One may verify that $w_0^Y({\Psi}({\lambda}))={\Psi}(w_0^X({\lambda}))$ using linearity, to reduce to the case ${\lambda}={\overline{\Lambda}}_i^X$ for $i\in{\bar{I}}$. It follows that ${\Psi}(v)^*$ and ${\Psi}(v^*)$ are classical lowest weight vectors in ${\widehat{V}}^s$ of the same weight. But then they must be equal.
Right splitting {#s:right}
===============
Let ${\mathfrak{g}}$ be of nonexceptional affine type. We define a family of $U_q({\overline{{\mathfrak{g}}}})$-crystal embeddings which is well-behaved with respect to intrinsic coenergy. They are denoted ${\mathrm{rs}}:={\mathrm{rs}}_{r;a,b}$ which stands for “right-split", because when $b=0$, the map splits off the rightmost column of an element in $B^{r,s}$.
\[conj:split\] Let $a-2\ge b\ge0$. Suppose ${\mathcal{C}}'$ is a set of KR crystals whose modules have been constructed, which contains $B^{r,s}$ for a particular $r\in{\bar{I}}$ and all $s\in{\mathbb{Z}}_{>0}$. Then there is an injective $U_q({\overline{{\mathfrak{g}}}})$-crystal morphism $${\mathrm{rs}}_{r;a,b}: B^{r,a} \otimes B^{r,b} \rightarrow
B^{r,a-1}\otimes B^{r,b+1}$$ such that for any crystal $B$ which is the tensor product of crystals in ${\mathcal{C}}'$, the map $$\label{eq:1rsmap}
1_B \otimes {\mathrm{rs}}_{r;a,b}: B \otimes B^{r,a} \otimes B^{r,b}
\rightarrow B \otimes B^{r,a-1} \otimes B^{r,b+1}$$ is an injective $U_q({\overline{{\mathfrak{g}}}})$-crystal morphism which preserves intrinsic coenergy.
\[th:Asplit\] Conjecture \[conj:split\] holds for ${\mathfrak{g}}=A_n^{(1)}$ for all $r\in{\bar{I}}$ and the set ${\mathcal{C}}'$ of all KR crystals.
This follows from [@Sh:2001a; @Sh:2001b; @Sh:2002].
\[th:splitrow\] Conjecture \[conj:split\] holds for any nonexceptional affine algebra ${\mathfrak{g}}$ for $r=1$ and ${\mathcal{C}}'$ the set of KR crystals of the form $B^{1,s}$.
The proof of Theorem \[th:splitrow\] occupies the remainder of this section.
Explicit definition of splitting
--------------------------------
This paper only requires the case $b=0$ for the map ${\mathrm{rs}}$. Except for type $A_n^{(1)}$ only the case $r=1$ is needed. For $s\ge2$ define the map ${\mathrm{rs}}:={\mathrm{rs}}_{1;s,0}$ as follows. For types $A_n^{(1)}$, $D_n^{(1)}$, $B_n^{(1)}$, and $A_{2n-1}^{(2)}$, define ${\mathrm{rs}}:B^s\rightarrow B^{s-1} \otimes B^1$ by ${\mathrm{rs}}(wx)=w\otimes x$ for $x\in B^1$ and $w\in B^{s-1}$ such that $wx\in B^s$. For the other types, in addition to the above rules we have ${\mathrm{rs}}(x)={{\varnothing}}\otimes x$ for $x\in B({\overline{\Lambda}}_1)\subseteq B^s$, and ${\mathrm{rs}}({{\varnothing}})=\bar{1}\otimes 1$. For $B^{r,s}\in {{\mathcal{C}}^A}$ and $b\in B^{r,s}$, let ${\mathrm{rs}}(b)=b_2\otimes
b_1$, where $b_1$ is the rightmost column of the rectangular tableau $b$ and $b_2$ is the rest of $b$.
\[rem:rs\] Suppose $s\ge 2$. Here $r=1$ for ${\mathcal{C}}$. For $B^{r,s}\in {\mathcal{C}}$ (or ${{\mathcal{C}}^A}$) we write ${\mathrm{rs}}$ for the map $1_B \otimes {\mathrm{rs}}$ on $B \otimes B^{r,s}$ and write ${\mathrm{rs}}(B\otimes
B^{r,s}):=B\otimes B^{r,s-1}\otimes B^{r,1}$.
Simply-laced ${\mathfrak{g}}$
-----------------------------
${\mathfrak{g}}=A_n^{(1)}$ is covered by Theorem \[th:Asplit\]. The other simply-laced nonexceptional family is ${\mathfrak{g}}=D_n^{(1)}$.
It is straightforward to check directly using the explicit description of $B^s$ in [@OSS:2003b] that ${\mathrm{rs}}$ is an injective $U_q({\overline{{\mathfrak{g}}}})$-crystal morphism. Let $B$ be the tensor product of crystals in ${\mathcal{C}}'$. To check that $1_B\otimes {\mathrm{rs}}$ preserves intrinsic coenergy, by it suffices to check this property for $B$ the trivial crystal and for $B=B^t$. Since $1_B
\otimes {\mathrm{rs}}$ is a $U_q({\overline{{\mathfrak{g}}}})$-crystal morphism, it is sufficient to prove that intrinsic coenergy is preserved for classical highest weight vectors. Suppose $B$ is trivial. By $B^s$ has a single classical highest weight vector, namely, $u(B^s)=1^s$. By Example \[ex:DKR\] $D_{B^s}=0$. On the other hand ${\mathrm{rs}}(1^s)=1^{s-1} \otimes 1 = u(B^{s-1} \otimes B^1)$ so its intrinsic energy is also zero. For $B=B^t$ we require the following Lemma, which is easily verified directly.
\[lem:D2P\] For ${\mathfrak{g}}=D_n^{(1)}$ and $s,t\ge 1$, $P(B^t\otimes B^s)$ consists of the elements $$v_{p,q}^{t,s} = 1^{t-p-q}\,2^p\,\bar{1}^q \otimes 1^s$$ where $p+q\le\min(s,t)$. In particular $B^t \otimes B^s$ is multiplicity-free as a $U_q(D_n)$-crystal.
Recall that $D_{B^t}$ and $D_{B^s}$ are identically zero by Example \[ex:DKR\]. By and explicit calculation, $$\label{eq:DD2}
D_{B^t\otimes B^s}(v_{p,q}^{t,s}) = H_{B^t,B^s}(v_{p,q}^{t,s})=p+2q.$$ Since $R$ is a $U_q(D_n^{(1)})$-crystal isomorphism, Lemma \[lem:D2P\] implies that $$R_{B^t,B^s}(v_{p,q}^{t,s}) = v_{p,q}^{s,t}.$$ We compute $H_{B^t,B^{s-1} \otimes B^1}$ on ${\mathrm{rs}}(v_{p,q}^{t,s})=(1^{t-p-q}2^p\bar{1}^q\otimes 1^{s-1} \otimes
1)$ using Proposition \[pp:Hbraided\]. We have $$R_{B^t,B^{s-1}}(1^{t-p-q}2^p\bar{1}^q \otimes 1^{s-1})=
\begin{cases}
1^{s-1-p-q}2^p\bar{1}^q \otimes 1^t
& \text{if $p+q<s$}\\
\overline{1}^{q-1} \otimes 1^{t-1} \bar{1}
& \text{if $p+q=s$, $q=s$}\\
2^{p-1} \overline{1}^q \otimes 1^{t-1}2
& \text{if $p+q=s$, $q<s$.}
\end{cases}$$ By we have $$H_{B^t,B^{s-1}}(1^{t-p-q}2^p\bar{1}^q \otimes 1^{s-1})=
\begin{cases}
p+2q & \text{if $p+q<s$}\\
p+2q-2 & \text{if $p+q=s$, $q=s$}\\
p+2q-1 & \text{if $p+q=s$, $q<s$.}
\end{cases}$$ By we have $H(1^t\otimes 1)=0$, $H(1^{t-1}\bar{1}\otimes 1)=2$, and $H(1^{t-1}2\otimes 1)=1$. It follows in any case that $1_{B^t}\otimes {\mathrm{rs}}$ preserves intrinsic coenergy.
Non-simply-laced ${\mathfrak{g}}$
---------------------------------
Suppose ${\mathfrak{g}}$ is not simply-laced. Let ${\mathfrak{g}}\hookrightarrow {\mathfrak{g}}_Y$ be as in . It is not hard to show that ${\mathrm{rs}}_X$ is a $U_q({\overline{{\mathfrak{g}}}})$-crystal injection. To show that the map preserves intrinsic coenergy (and thereby complete the proof of Theorem \[th:splitrow\]), by Proposition \[pp:VD\] the following result suffices.
\[pp:vrs\] There is an injective $U_q({\mathfrak{g}}_Y)$-crystal map ${\widehat{{\mathrm{rs}}}}:{\widehat{V}}^s\rightarrow {\widehat{V}}^{s-1} \otimes {\widehat{V}}^1$ such that:
1. \[it:vrsdiag\] The following diagram commutes: $$\label{eq:vrsdiag}
\begin{CD}
B^s_X @>{\Psi}>> {\widehat{V}}^s\\
@V{{\mathrm{rs}}_X}VV @VV{{\widehat{{\mathrm{rs}}}}}V \\
B_X^{s-1} \otimes B_X^1 @>>{\Psi\otimes\Psi}> {\widehat{V}}^{s-1} \otimes
{\widehat{V}}^1
\end{CD}$$
2. \[it:vD\] For any $B\in{\mathcal{C}}$, let ${\Psi}:B\rightarrow{\widehat{V}}$ be its virtual crystal embedding. Then $1_{{\widehat{V}}} \otimes {\widehat{{\mathrm{rs}}}}$ preserves intrinsic coenergy.
3. \[it:vres\] If $v\in {\widehat{V}}^s$ and ${\widehat{{\mathrm{rs}}}}(v)\in{\mathrm{Im}}(\Psi\otimes\Psi)$ then $v\in{\mathrm{Im}}(\Psi)$.
Suppose first that $Y=A_{2n-1}^{(1)}$. Then ${\widehat{V}}^s =
B_Y^{s\vee} \otimes B_Y^s$. Define the map ${\widehat{{\mathrm{rs}}}}:{\widehat{V}}^s\rightarrow
{\widehat{V}}^{s-1}\otimes {\widehat{V}}^1$ by the composition $$\label{eq:vrsAdef}
\begin{split}
&B_Y^{s\vee} \otimes B_Y^s \stackrel{1\otimes {\mathrm{rs}}_Y}{\longrightarrow}
B_Y^{s\vee} \otimes B_Y^{s-1} \otimes B_Y^1\stackrel{R}{\longrightarrow}
B_Y^{s-1} \otimes B_Y^1 \otimes B_Y^{s\vee}\\
\stackrel{1\otimes 1\otimes {\mathrm{rs}}_Y^\vee}{\longrightarrow}&
B_Y^{s-1} \otimes B_Y^1 \otimes B_Y^{s-1\vee} \otimes B_Y^{1\vee}
\stackrel{R}{\longrightarrow}
B_Y^{s-1\vee}\otimes B_Y^{s-1} \otimes B_Y^{1\vee} \otimes B_Y^1.
\end{split}$$ Here ${\mathrm{rs}}^\vee_Y(wx)=w\otimes x$ where $wx\in B_Y^{s\vee}$, $w\in
B_Y^{s-1\vee}$ and $x\in B_Y^{1\vee}$. Note that is a composition of combinatorial $R$-matrices and ${\mathrm{rs}}$ maps for type $A$. Point \[it:vD\] holds by Theorem \[th:Asplit\].
One need only verify \[it:vrsdiag\] on classical highest weight vectors, by the definition of $\Psi$ and the fact that ${\mathrm{rs}}_Y$ and ${\mathrm{rs}}^\vee_Y$ (resp. ${\mathrm{rs}}_X$) are morphisms of $U_q(\overline{Y})$- (resp. $U_q(\overline{X})$-) crystals.
Let $N=2n$. The classical highest weight vectors in $B_X^s$ have the form $1^{s-p}$ for $0\le p\le s$; if ${\mathfrak{g}}$ is $C_n^{(1)}$ or $A_{2n}^{(2)\dagger}$ then $p$ must also be even. For $p<s$ the element $1^{s-p}\in B((s-p){\overline{\Lambda}}_1)\subset B_X^s$ is sent to the following elements under the maps in : $$\begin{matrix}
1^{s-p} & 1^{s-1-p} \otimes 1 \\ N^{\vee(s-p)} 1^{\vee p} \otimes
1^s & N^{\vee s-p-1} 1^{\vee p} \otimes 1^{s-1} \otimes N^\vee
\otimes 1
\end{matrix}$$ where the intermediate results under the maps in are given by $N^{\vee(s-p)} 1^{\vee p} \otimes
1^{s-1} \otimes 1$, $1^{s-p-1}N^p \otimes 1 \otimes N^{\vee s}$, $1^{s-p-1}N^p \otimes 1 \otimes N^{\vee(s-1)} \otimes N^\vee$, and $N^{\vee s-p-1} 1^{\vee p} \otimes 1^{s-1} \otimes N^\vee \otimes
1$.
Under the maps in , the element ${\varnothing}$ is sent to $$\begin{matrix}
{\varnothing}& \overline{1} \otimes 1 \\
N^s \otimes N^{\vee s} & 1^{\vee s-1} \otimes 1^{s-2}N \otimes
N^\vee \otimes 1
\end{matrix}$$ with intermediate values in given by $N^s
\otimes N^{\vee s-1} \otimes N^\vee$, $1^{\vee s-1} \otimes 1^\vee
\otimes 1^s$, $1^{\vee s-1} \otimes 1^\vee \otimes 1^{s-1} \otimes
1$, and $1^{\vee s-1} \otimes 1^{s-2}N \otimes N^\vee \otimes 1$.
Since these are all the possible classical highest weight vectors, point \[it:vrsdiag\] follows.
For point \[it:vres\], let $v\in {\widehat{V}}^s$ and ${\widehat{{\mathrm{rs}}}}(v)\in{\mathrm{Im}}({\Psi}\otimes{\Psi})$. Without loss of generality we may assume that $v\in P({\widehat{V}}^s)$ since ${\widehat{{\mathrm{rs}}}}$ is a $U_q(\overline{Y})$-morphism. Now $v$ must have the form $v_{s,p}:=N^{\vee(s-p)} 1^{\vee p} \otimes 1^s$ for $0\le p\le s$. By computations similar to those above, ${\widehat{{\mathrm{rs}}}}(v)=v_{s-1,p} \otimes
{\Psi}(1)$ if $p<s$ and ${\widehat{{\mathrm{rs}}}}(v)=1^{\vee(s-1)}\otimes 1^{s-2}N\otimes
{\Psi}(1)$ if $p=s$. But ${\widehat{{\mathrm{rs}}}}(v)\in{\mathrm{Im}}({\Psi}\otimes{\Psi})$ means that $v_{s-1,p}\in {\mathrm{Im}}({\Psi})$ if $p<s$ and $1^{\vee(s-1)}\otimes
1^{s-2}N\in{\mathrm{Im}}({\Psi})$ if $p=s$. The parity condition for this to occur implies the parity condition that guarantees that $v_{s,p}\in{\mathrm{Im}}({\Psi})$.
Suppose next that $Y=D_{n+1}^{(1)}$ and $X=A_{2n-1}^{(2)}$. Then ${\widehat{V}}^s=B_Y^s$. Define ${\widehat{{\mathrm{rs}}}}={\mathrm{rs}}_Y:B_Y^s\rightarrow B_Y^{s-1}\otimes
B_Y^1$. Point \[it:vD\] follows by the simply-laced $D_n^{(1)}$ case. Point \[it:vres\] is trivial. For point \[it:vrsdiag\] it is enough to consider elements of $P(B^s)=\{1^s\}$. Under the maps in , $1^s$ goes to $$\begin{matrix}
1^s & 1^{s-1} \otimes 1 \\
1^s & 1^{s-1} \otimes 1
\end{matrix}$$ and commutes.
Suppose that $Y=D_{n+1}^{(1)}$ and $X=B_n^{(1)}$. Then ${\widehat{V}}^s=B_Y^{2s}$. Define ${\widehat{{\mathrm{rs}}}}:B_Y^{2s} \rightarrow B_Y^{2s-2}
\otimes B_Y^2$ by $wv\mapsto w\otimes v$ where $wv\in B_Y^{2s}$, $w\in B_Y^{2s-2}$, and $v\in B_Y^2$. This map is clearly injective and $U_q(\overline{Y})$-equivariant. Point \[it:vres\] is obvious. For point \[it:vrsdiag\] it is enough to consider the unique element $1^s\in P(B_X^s)$. Under $1^s$ goes to $$\begin{matrix}
1^s & 1^{s-1} \otimes 1 \\
1^{2s} & 1^{2s-2} \otimes 1^2
\end{matrix}$$ so that commutes. For point \[it:vD\] define ${\widehat{{\mathrm{rs}}}}':B_Y^{2s}\rightarrow B_Y^{2s-2} \otimes B_Y^1 \otimes B_Y^1$ by the composite map $$\begin{split}
B_Y^{2s} &\stackrel{{\mathrm{rs}}_Y}{\longrightarrow} B_Y^{2s-1}\otimes B_Y^1
\stackrel{R}{\longrightarrow} B_Y^1\otimes B_Y^{2s-1}\\
&\stackrel{1\otimes {\mathrm{rs}}_Y}{\longrightarrow} B_Y^1\otimes B_Y^{2s-2}\otimes B_Y^1
\stackrel{R}{\longrightarrow} B_Y^{2s-2} \otimes B_Y^1 \otimes B_Y^1.
\end{split}$$ Since ${\widehat{{\mathrm{rs}}}}'$ is the composition of ${\mathrm{rs}}_Y$ maps and $R$-matrices, it preserves intrinsic coenergy by the simply-laced case. It suffices to show that $$\xymatrix{
B_Y^{2s} \ar[dd]^{{\widehat{{\mathrm{rs}}}}} \ar[dr]^{{\widehat{{\mathrm{rs}}}}'} & \\
{} & {B_Y^{2s-2} \otimes B_Y^1 \otimes B_Y^1} \\
{B_Y^{2s-2} \otimes B_Y^2} \ar[ur]_{1\otimes {\mathrm{rs}}_Y} &
}$$ commutes since ${\widehat{{\mathrm{rs}}}}'$ and $1\otimes {\mathrm{rs}}_Y$ both preserve intrinsic coenergy. It suffices to check this for the lone classical highest weight vector $1^{2s}\in P(B_Y^{2s})$. Clearly ${\widehat{{\mathrm{rs}}}}'(1^{2s})=1^{2s-2}\otimes 1\otimes1$, while ${\widehat{{\mathrm{rs}}}}(1^{2s})=1^{2s-2}\otimes 1^2$ and this is sent by $1\otimes
{\mathrm{rs}}_Y$ to $1^{2s-2}\otimes 1\otimes 1$, as desired.
Left splitting and duality {#s:left}
==========================
We define dual analogues of the intrinsic coenergy $D$ and right splitting.
Tail coenergy
-------------
For $B^s\in {\mathcal{C}}$ define ${\overleftarrow{D}}_{B^s}=D_{B^s}$. For $B^{r,s}\in {{\mathcal{C}}^A}$, define ${\overleftarrow{D}}_{B^{r,s}}=D_{B^{r,s}}=0$. If $B_1,B_2,\dotsc,B_L\in {\mathcal{C}}$ (or ${{\mathcal{C}}^A}$) and $B=B_L\otimes\dotsm\otimes B_1$ are such that ${\overleftarrow{D}}_{B_j}:B_j\rightarrow{\mathbb{Z}}_{\ge0}$ are given, then define $$\label{eq:tailDNY}
{\overleftarrow{D}}_B = \sum_{1\le i<j\le L} H_{j-1} R_{j-2} \dotsm R_{i+1} R_i
+ \sum_{j=1}^L {\overleftarrow{D}}_{B_j} R_{L-1} R_{L-2} \dotsm R_j$$ with ${\overleftarrow{D}}_{B_j}$ acting on the leftmost tensor position. This is a different associative tensor product on graded crystals than the one given in subsection \[ss:D\].
Recall the notation $B^*$ of Remark \[rem:star\].
\[pp:tail\] Let $B\in {\mathcal{C}}$ (or $B\in {{\mathcal{C}}^A}$) and $b\in B$. Then ${\overleftarrow{D}}_B(b)=D_{B^*}(b^*)$.
For $B$ a single KR crystal, the result follows from the fact that the involution $*$ on $B$ stabilizes classical components. By Proposition \[pp:R\*\] and comparing with it suffices to show that $$\label{eq:H*}
H_{B_1,B_2}(b^*)=H_{B_2,B_1}(b)$$ for $B_1,B_2$ KR crystals. Since $B_2\otimes B_1$ is connected, the proof may proceed by induction on the number of steps (either of the form $e_i$ or $f_i$) in $B_2\otimes B_1$ from $u(B_2\otimes B_1)$ to $b$. Suppose first that $b=u(B_2\otimes B_1)$. By the definition of $u(B)$ in subsection \[ss:fin\], $B_2 \otimes B_1$ (and therefore $B_1\otimes B_2)$ contain a unique classical component isomorphic to $B({\lambda})$ where ${\lambda}={\mathrm{wt}}(b)$. And $B({\lambda})$ contains a unique vector of the extremal weight $w_0{\lambda}$. Since ${\mathrm{wt}}(b^*)=w_0{\mathrm{wt}}(b)$ it follows that $b^*$ and $u(B_1\otimes B_2)$ are in the same classical component, so that $H_{B_1,B_2}(b^*)=H_{B_1,B_2}(b)=0$ by the definition of $H$.
Now suppose $b=f_i(c)$ where $c$ is closer to $u(B_2\otimes B_1)$ than $b$ is. If $i\not=0$ then we are done since both sides of do not change under passing from $c$ to $b$, by the definition of $H$ and . So assume $i=0$. By $b^*=e_0(c^*)$. But then one may conclude the validity of for $b$ from that of $c$ using rules for the Kashiwara operators on the tensor product and .
Define ${\overleftarrow{X}}$ just like the one-dimensional sum $X$ but use ${\overleftarrow{D}}_B$ instead of $D_B$. Proposition \[pp:tail\] has this corollary.
${\overleftarrow{X}}(B,{\lambda})=X(B,{\lambda})$.
Left splitting
--------------
Whenever the right splitting map ${\mathrm{rs}}:B^{r,s}\rightarrow B^{r,s-1} \otimes B^{r,1}$ is defined, we may define the left-splitting map ${\mathrm{ls}}:B^{r,s}\rightarrow
B^{r,1}\otimes B^{r,s-1}$ by the commutation of the diagram $$\label{eq:splitstar}
\begin{CD}
B^{r,s} @>{{\mathrm{ls}}}>> B^{r,1} \otimes B^{r,s-1} \\
@V{*}VV @VV{*}V \\
B^{r,s} @>>{{\mathrm{rs}}}> B^{r,s-1} \otimes B^{r,1}.
\end{CD}$$ In particular, it is defined for $B^s\in {\mathcal{C}}$ and $B^{r,s}\in{{\mathcal{C}}^A}$.
\[cor:lsplitdef\] Here $r=1$ for the category ${\mathcal{C}}$. ${\mathrm{ls}}$ is a $U_q({\overline{{\mathfrak{g}}}})$-crystal embedding such that, for any $B^{r,s}\in {\mathcal{C}}$ (or ${{\mathcal{C}}^A}$) and for any $B\in {\mathcal{C}}$ (or ${{\mathcal{C}}^A}$), the map $${\mathrm{ls}}\otimes 1_B: B^{r,s} \otimes B \rightarrow B^{r,1} \otimes
B^{r,s-1}\otimes B$$ is injective and preserves ${\overleftarrow{D}}$.
${\mathrm{ls}}$ is a $U_q({\overline{{\mathfrak{g}}}})$-crystal embedding since ${\mathrm{rs}}$ is, by Theorem \[th:splitrow\], the definition of $*$ and . For the preservation of ${\overleftarrow{D}}$, let $b_1\otimes
b_2\in B^{r,s}\otimes B$. We have $${\overleftarrow{D}}({\mathrm{ls}}(b_1) \otimes b_2) =
D(b_2^* \otimes {\mathrm{ls}}(b_1)^*) =
D(b_2^* \otimes {\mathrm{rs}}(b_1^*)) =
D(b_2^* \otimes b_1^*) =
{\overleftarrow{D}}(b_1 \otimes b_2)$$ by Proposition \[pp:tail\] and .
\[rem:splitnote\] Suppose $s\ge 2$. Here $r=1$ for ${\mathcal{C}}$ as usual. For $B^{r,s}\in {\mathcal{C}}$ (or ${{\mathcal{C}}^A}$) we write ${\mathrm{ls}}$ for the map ${\mathrm{ls}}\otimes 1_B$ on $B^{r,s}\otimes B$. Also we write ${\mathrm{ls}}(B^{r,s}\otimes B) := B^{r,1} \otimes B^{r,s-1}\otimes B$.
Explicit left-splitting
-----------------------
\[lem:lsplitrow\] For $B^s\in {\mathcal{C}}$ the map ${\mathrm{ls}}:B^s\rightarrow B^1 \otimes B^{s-1}$ is given explicitly by ${\mathrm{ls}}(xw)=x\otimes w$ for $x\in B^1$ and $w\in B^{s-1}$ such that $xw\in B^s$, ${\mathrm{ls}}(x)=x\otimes {{\varnothing}}$ for $x\in B({\overline{\Lambda}}_1)\subseteq
B^s$, ${\mathrm{ls}}({{\varnothing}})=\bar{1}\otimes 1$. For $B^{r,s}\in {{\mathcal{C}}^A}$ and $b\in
B^{r,s}$, ${\mathrm{ls}}(b)=b_2 \otimes b_1$ where $b_2$ is the leftmost column in the $r\times s$ semistandard tableau $b$ and $b_1$ is the rest of $b$.
Box-splitting {#ss:bs}
-------------
Let $B^{r,1}\in {{\mathcal{C}}^A}$ with $r\ge2$. There is a $U_q({\overline{{\mathfrak{g}}}})$-crystal embedding ${\mathrm{lb}}:B^{r,1}\rightarrow B^{1,1} \otimes B^{r-1,1}$ given by $b\mapsto b_2 \otimes b_1$ where $b_2$ is the bottommost entry in the column tableau $b$ of height $r$, and $b_1$ is the remainder of $b$. There is a $U_q({\overline{{\mathfrak{g}}}})$-crystal embedding ${\mathrm{rb}}:B^{r,1}\rightarrow B^{r-1,1}\otimes B^{1,1}$ given by $b\mapsto
b_2 \otimes b_1$ where $b_1$ is the topmost entry in the column $b$ and $b_2$ is the rest of $b$.
The map ${\mathrm{lb}}$ is only used to define the path-RC bijection for $B\in{{\mathcal{C}}^A}$ in section \[sec:bij\].
In general the morphism ${\mathrm{rb}}$ does not preserve intrinsic coenergy, but another grading called intrinsic energy. It was proved in [@KSS:2002] that the path-RC bijection preserves the grading for ${{\mathcal{C}}^A}$ using a different method, namely, the rank-level duality for type $A^{(1)}$.
Projections and commutations
----------------------------
Define the (“left-hat") map ${\mathrm{lh}}:B_2\otimes B_1\rightarrow B_1$ by $b_2\otimes b_1\mapsto b_1$. It just removes the left tensor factor. Define the “right-hat" map ${\mathrm{rh}}:B_2\otimes B_1\rightarrow B_2$ by $b_2\otimes b_1\mapsto b_2$.
It is immediate that the following diagram commutes: $$\label{eq:hat*}
\begin{CD}
B_2 \otimes B_1 @>{{\mathrm{lh}}}>> B_1 \\
@V{*}VV @VV{*}V \\
B_1 \otimes B_2 @>>{{\mathrm{rh}}}> B_1
\end{CD}$$
Let $P(B)$ be the set of classical highest weight vectors in $B$, or equivalently, the set of classical components of $B$.
\[lem:hwvhat\] The maps ${\mathrm{lh}}:B_2\otimes B_1\rightarrow B_1$ and ${\mathrm{rh}}:B_2\otimes
B_1\rightarrow B_2$ induce maps ${\mathrm{lh}}:P(B_2\otimes B_1)\rightarrow
P(B_1)$ and ${\mathrm{rh}}:P(B_2\otimes B_1) \rightarrow P(B_2)$.
If $b_2\otimes b_1$ is a classical highest weight vector of $B_2\otimes B_1$ then by the definitions, $b_1$ is a classical highest weight vector of $B_1$. Thus ${\mathrm{lh}}$ is well-defined on components.
For ${\mathrm{rh}}$ we work with classical components. By the map $*$ takes classical components to classical components. But then ${\mathrm{rh}}$ is well-defined on components since ${\mathrm{lh}}$ is, by .
Let $b=\young(3) \otimes \young(2{\bar{2}}) \otimes
\young(12) \otimes \young(1)\in P(B^1\otimes B^2\otimes B^2 \otimes
B^1)$ of type $D_4^{(1)}$. Then ${\mathrm{lh}}(b)=\young(2{\bar{2}}) \otimes
\young(12) \otimes \young(1)$ and ${\mathrm{rh}}(b)=\young(3) \otimes
\young(2{\bar{2}}) \otimes \young(12)$. The induced map on highest weight vectors yields ${\mathrm{rh}}(b)=\young(3)\otimes\young(22)\otimes\young(11)\;$
One has the commutation of induced maps on classical highest weight vectors: $$\begin{CD}
P(B_2 \otimes B_1) @>{{\mathrm{lh}}}>> P(B_1) \\
@V{*}VV @VV{*}V \\
P(B_1 \otimes B_2) @>>{{\mathrm{rh}}}> P(B_1)
\end{CD}$$
\[rem:hatnote\] From now on, unless explicitly indicated otherwise, we only consider the map ${\mathrm{lh}}$ (resp. ${\mathrm{rh}}$) on tensor products whose left (resp. right) factor is $B^1$. In these cases, we use the notation ${\mathrm{lh}}(B^1\otimes B)=B$ and ${\mathrm{rh}}(B\otimes B^1)=B$.
For ${\lambda}\in{\overline{P}}^+$ let $${\lambda^-}= \{ \mu\in {\overline{P}}^+ \mid \text{$B({\lambda})$ occurs in $B^1 \otimes B(\mu)$ } \}$$ where $B^1$ is regarded as a $U_q({\overline{{\mathfrak{g}}}})$-crystal by restriction.
By Lemma \[lem:hwvhat\] there are well-defined bijections $$\begin{aligned}
\label{eq:lhbij}
{\mathrm{lh}}:P(B,{\lambda})&\rightarrow \bigcup_{\mu\in {\lambda^-}} P({\mathrm{lh}}(B),\mu) \\
\label{eq:rhbij}
{\mathrm{rh}}:P(B,{\lambda})&\rightarrow \bigcup_{\mu\in {\lambda^-}} P({\mathrm{rh}}(B),\mu)\end{aligned}$$ except in the case ${\mathfrak{g}}=D_{n+1}^{(2)}$. Note that $B^1$ has at most one vector of each weight except when ${\mathfrak{g}}=D_{n+1}^{(2)}$, which has two vectors $0$ and ${{\varnothing}}$ of weight $0$. If $\mu={\lambda}$, then there can be elements $b\in P({\mathrm{lh}}(B),{\lambda})$ such that both $0\otimes b$ and ${{\varnothing}}\otimes b$ are in $P(B,{\lambda})$. If so, then the right hand side of must be modified to include two copies of $b$, one coming from ${{\varnothing}}\otimes b$ and the other from $0\otimes b$. There is no analogous problem for ${\mathrm{rh}}$ since $0\not\in P(B^1)$.
\[pp:cryscomm\] Let $r=r'=1$ for ${\mathcal{C}}$.
1. \[pt:lhrh\] $[{\mathrm{lh}},{\mathrm{rh}}]=0$ on $B^1 \otimes B \otimes B^1$.
2. \[pt:lhrs\] $[{\mathrm{lh}},{\mathrm{rs}}]=0$ on $B^1 \otimes B \otimes
B^{r,s}$ for $s\ge2$.
3. \[pt:rhls\] $[{\mathrm{rh}},{\mathrm{ls}}]=0$ on $B^{r,s} \otimes B \otimes
B^1$ for $s\ge2$.
4. \[pt:lsrs\] $[{\mathrm{ls}},{\mathrm{rs}}]=0$ on $B^{r,s} \otimes B \otimes
B^{r',s'}$ for $s,s'\ge2$.
5. \[pt:hat\*\] $*\circ {\mathrm{lh}}= {\mathrm{rh}}\circ *$ on $B^1 \otimes B$.
Moreover, these commutations also hold for the induced maps on sets of classical highest weight vectors.
The operators on the entire crystals commute more or less by definition. We now prove that these identities hold for the induced maps between sets of classical highest weight vectors.
The proof is again trivial except for cases involving ${\mathrm{rh}}$. Point \[pt:lhrh\] follows from Lemma \[lem:hwvhat\]. Point \[pt:rhls\] follows from Lemma \[lem:hwvhat\] and the $U_q({\overline{{\mathfrak{g}}}})$-equivariance of ${\mathrm{ls}}$ given in Corollary \[cor:lsplitdef\]. Finally, point \[pt:hat\*\] follows from Lemma \[lem:hwvhat\] and the fact that the map $*$ respects classical raising and lowering operators.
Right hat and classical highest weight vectors
----------------------------------------------
We need to know precisely how the highest weights change when passing from an element of $P(B^1\otimes B \otimes B^1)$ to $P(B)$ via either ${\mathrm{rh}}\circ {\mathrm{lh}}$ or ${\mathrm{lh}}\circ {\mathrm{rh}}$. In this section we assume type $D_n$. The answer is given by van Leeuwen [@vL:1998]. We translate his answer into the language of partitions.
Let $P$ be the set of dominant weights that can occur in a tensor product of crystals of the form $B({\overline{\Lambda}}_1)$. A dominant weight $\sum_{i=1}^n a_i {\overline{\Lambda}}_i$ is in $P$ if and only if $a_{n-1}$ and $a_n$ have the same parity. We put a graph structure on $P$ by declaring that weights ${\lambda}$ and $\mu$ are adjacent if there is an element $x\in B({\overline{\Lambda}}_1)$ such that ${\lambda}-\mu={\mathrm{wt}}(x)$.
We realize $P$ as a subset of ${\mathbb{Z}}^n$ by letting ${\overline{\Lambda}}_i=(1^i,0^{n-i})$ for $1\le i\le n-2$, ${\overline{\Lambda}}_{n-1}=\frac{1}{2}(1^n)$ and ${\overline{\Lambda}}_n=\frac{1}{2}(1^{n-1},-1)$. As such $P$ is given by the tuples ${\lambda}=({\lambda}_1,{\lambda}_2,\dotsc,{\lambda}_n)\in{\mathbb{Z}}^n$ with ${\lambda}_1\ge {\lambda}_2
\ge\dotsm \ge {\lambda}_{n-1} \ge | {\lambda}_n |$.
We modify this notation slightly in order to use partitions. Let $Y$ be the lattice of partitions ${\lambda}=({\lambda}_1\ge{\lambda}_2\ge\dotsm\ge{\lambda}_n)\in{\mathbb{Z}}^n_{\ge0}$ with at most $n$ parts. A graph structure on $Y$ is given by declaring that two partitions are connected with an edge if their partition diagrams differ by one cell. Define the graph $G$ by glueing two copies $Y_+$ and $Y_-$ of $Y$ together such that, if ${\lambda}\in Y$ is such that ${\lambda}_n=0$, then ${\lambda}\in Y_+$ and ${\lambda}\in Y_-$ are identified.
Then $P \cong G$ where the weight $(\mu_1,\dotsc,\mu_n)$ is identified with the partition $(\mu_1,\mu_2,\dotsc,\mu_{n-1},0)$ if $\mu_n=0$, with the “positive" partition $(\mu_1,\dotsc,\mu_{n-1},\mu_n)\in Y_+$ if $\mu_n>0$, and with the “negative" partition $(\mu_1,\dotsc,\mu_{n-1},-\mu_n)\in Y_-$ if $\mu_n<0$.
Let $\mu$ and ${\lambda}$ be adjacent in $P$, and $x\in B({\overline{\Lambda}}_1)$ such that ${\lambda}-\mu={\mathrm{wt}}(x)$. We think of this as walking from $\mu$ to ${\lambda}$ by the step $x$. In terms of partitions, if $x=i$ for $1\le
i\le n-1$ then a cell is added to the $i$-th row. If $x={\overline{i}}$ for $1\le i\le n-1$ then a cell is removed from the $i$-th row. If $i=n$ then the above rules hold provided that ${\lambda},\mu\in Y_+$. If ${\lambda},\mu\in Y_-$ then the roles of $n$ and ${\overline{n}}$ are reversed.
Let $B=B({\overline{\Lambda}}_1)^{\otimes L}$. Let $b=b_L\dotsm b_1\in P(B)$ with $b_j\in B({\overline{\Lambda}}_1)$. In the usual way, $b$ can be regarded as a path in the set of dominant weights: the $i$-th weight is given by the weight of $b_i\dotsm b_1$. Alternatively $b$ describes a walk in $G$ from the empty partition to the element of $G$ corresponding to the weight of $b$.
\[ex:path\] Let $n=4$. Consider $b={\overline{4}}{\overline{4}}41321\in P(B)$ where $B=B({\overline{\Lambda}}_1)^{\otimes 7}$. The element $b$ corresponds to the walk in $G$ given by
$$\Yboxdim{8pt} \varnothing \rightarrow \yng(1) \rightarrow
\yng(1,1)\rightarrow \yng(1,1,1) \rightarrow \yng(2,1,1) \rightarrow
\young({{\hphantom{x}}}{{\hphantom{x}}},{{\hphantom{x}}},{{\hphantom{x}}},+) \rightarrow \yng(2,1,1) \rightarrow
\young({{\hphantom{x}}}{{\hphantom{x}}},{{\hphantom{x}}},{{\hphantom{x}}},-)$$
where the $+$ and $-$ markings on a partition indicate membership in $Y_+$ and $Y_-$ respectively.
In the following proposition, for weights ${\lambda},\mu\in P$, we write $\mu\subset{\lambda}$ if the corresponding elements of $G$ are both in $Y_+$ or both in $Y_-$ and the diagram of the partition associated with $\mu$ is contained in that of ${\lambda}$.
\[pp:weight\] Suppose $b\in P(B^1\otimes B\otimes B^1,{\lambda})$, ${\mathrm{rh}}(b)\in P(B^1\otimes B,\alpha)$, ${\mathrm{lh}}(b)\in P(B\otimes
B^1,\beta)$ and ${\mathrm{rh}}({\mathrm{lh}}(b))={\mathrm{lh}}({\mathrm{rh}}(b))\in P(B,\gamma)$. Then $\alpha$ is uniquely determined by ${\lambda}$, $\beta$, and $\gamma$. More precisely,
1. If $|{\lambda}|=|\gamma|+2$:
1. If the cells $\lambda/\beta$ and $\beta/\gamma$ are in different rows and different columns, then $\alpha={\lambda}-\{\beta/\gamma\}$.
2. If ${\lambda}/\beta$ and $\beta/\gamma$ are in the same row or in the same column, then $\alpha=\beta$.
2. If $|{\lambda}|=|\gamma|-2$:
1. If the cells $\beta/{\lambda}$ and $\gamma/\beta$ are in different rows and different columns, then $\alpha={\lambda}\cup\{\gamma/\beta\}$.
2. If $\beta/\lambda$ and $\gamma/\beta$ are in the same row or the same column, then $\alpha=\beta$.
3. If $|{\lambda}|=|\gamma|$ and ${\lambda}\not=\gamma$:
1. If ${\lambda}\supset \beta$ then $\alpha={\lambda}\cup\{\gamma/\beta\}$.
2. If ${\lambda}\subset \beta$ then $\alpha={\lambda}-\{\beta/\gamma\}$.
4. If ${\lambda}=\gamma$:
1. If ${\lambda}\subset \beta$:
1. If $\beta/{\lambda}$ is in the first column of ${\lambda}$:
1. If $\beta/{\lambda}$ is in the $n$-th row, for $\beta\in Y_\pm$ let $\alpha\in Y_{\mp}$ be the corresponding partition.
2. Otherwise let $\alpha=\beta$.
2. Else $\alpha\subset {\lambda}$ and $\alpha$ is obtained from ${\lambda}$ by removing the corner cell in the column to the left of $\beta/{\lambda}$.
2. If ${\lambda}\supset\beta$ then $\alpha$ is obtained from ${\lambda}$ by adjoining a cell to the column to the right of ${\lambda}/\beta$.
The rule for the weight $\alpha$ is given by van Leeuwen [@vL:1998 Rule 4.1.1]: $\alpha$ is the unique dominant element in the Weyl group orbit of the weight ${\lambda}+\gamma-\beta$. Using this rule the proof is straightforward.
\[rem:rhshape\] The two operations ${\mathrm{rh}}\circ {\mathrm{lh}}$ and ${\mathrm{lh}}\circ {\mathrm{rh}}$ define a pair of two-step walks in the graph $G$ from ${\lambda}$ to $\gamma$, whose intermediate vertices are $\beta$ and $\alpha$ respectively. If there is only one such walk then $\alpha=\beta$; this occurs in cases (1b) and (2b). If there are exactly two such walks then $\alpha$ is always chosen to be the intermediate vertex not equal to $\beta$; this occurs in cases (1a), (2a), (3a), and (3b). In the case that ${\lambda}=\gamma$ there may be many such walks; the proper choice of $\alpha$ given $\beta$ is described in the proposition.
Let $b$ be as in Example \[ex:path\]. Then ${\mathrm{lh}}(b)={\overline{4}}41321$, ${\mathrm{rh}}(b)=2{\overline{4}}3121$, and ${\mathrm{rh}}({\mathrm{lh}}(b))={\mathrm{lh}}({\mathrm{rh}}(b))={\overline{4}}3121$. Therefore ${\lambda}$ is the weight $(2,1,1,-1)$ or the partition $(2,1,1,1)\in Y_-$, $\beta$ is the weight and partition $(2,1,1,0)$, $\alpha$ is the weight $(2,2,1,-1)$ and the partition $(2,2,1,1)\in Y_-$, and $\gamma$ is the weight $(2,1,1,-1)$ and the partition $(2,1,1,1)\in Y_-$. Since ${\lambda}=\gamma$ (as elements of $P$ or $G$) and $\beta \subset {\lambda}$ as partitions, Case (4b) applies. The cell ${\lambda}/\beta$ is in the first column; therefore $\alpha$ should be obtained from $\gamma$ by adjoining a cell at the end of the second column, which agrees with the example.
In $D_4^{(1)}$ let $b=4{\overline{4}}321\in
(B^{1,1})^{\otimes 5}$. Then ${\mathrm{lh}}(b)={\overline{4}}321$, ${\mathrm{rh}}(b)=4321$, ${\mathrm{rh}}({\mathrm{lh}}(b))={\mathrm{lh}}({\mathrm{rh}}(b))=321$. Therefore ${\lambda}=(1,1,1,0)$, $\beta$ is the weight $(1,1,1,-1)$ or the partition $(1,1,1,1)\in Y_-$, $\gamma=(1,1,1,0)$, and $\alpha$ is the weight $(1,1,1,1)$ or the partition $(1,1,1,1)\in Y_+$. This is case (4a1A).
Rigged Configurations {#sec:RC}
=====================
In this section it is assumed that ${\mathfrak{g}}$ is nonexceptional and simply-laced, that is, ${\mathfrak{g}}=A_n^{(1)}$ or ${\mathfrak{g}}=D_n^{(1)}$.
Definition {#sec:rc def}
----------
Let $B\in {\mathcal{C}}$ for type $D_n^{(1)}$ and $B\in {{\mathcal{C}}^A}$ for type $A_n^{(1)}$. Recall the notation in subsection \[ss:tensorcats\], where $L=(L_i^{(a)}\mid (a,i)\in {\mathcal{H}})$ is the multiplicity array of $B$. The sequence of partitions $\nu=\{\nu^{(a)}\mid a\in {\bar{I}}\}$ is a **$(L,{\lambda})$-configuration** if $$\sum_{(a,i)\in{\mathcal{H}}} i m_i^{(a)} \alpha_a = \sum_{(a,i)\in{\mathcal{H}}} i
L_i^{(a)} {\overline{\Lambda}}_a- {\lambda},$$ where $m_i^{(a)}$ is the number of parts of length $i$ in partition $\nu^{(a)}$. A $(L,{\lambda})$-configuration is **admissible** if $p_i^{(a)}\ge 0$ for all $(a,i)\in{\mathcal{H}}$, where $p_i^{(a)}$ is the **vacancy number** $$p_i^{(a)}=\sum_{j\ge 1} \min(i,j) L_j^{(a)}
- \sum_{b\in {\bar{I}}} (\alpha_a | \alpha_b) \sum_{j\ge 1}
\min(i,j)m_j^{(b)}.$$ Here $(\cdot | \cdot )$ is the normalized invariant form on $P$ such that $(\alpha_i | \alpha_j)$ is the Cartan matrix. Let ${\mathrm{C}}(L,{\lambda})$ be the set of admissible $(L,{\lambda})$-configurations. A **rigged configuration** $(\nu,J)$ consists of a configuration $\nu\in {\mathrm{C}}(L,{\lambda})$ together with a double sequence of partitions $J=\{J^{(a,i)}\mid (a,i)\in{\mathcal{H}}\}$ such that the partition $J^{(a,i)}$ is contained in a $m_i^{(a)}\times p_i^{(a)}$ rectangle. The set of rigged configurations is denoted by ${\mathrm{RC}}(L,{\lambda})$.
The partition $J^{(a,i)}$ is called **singular** if it has a part of size $p_i^{(a)}$. The partition $J^{(a,i)}$ is called **cosingular** if it has a part of size zero, or equivalently, its complement in the rectangle of size $m_i^{(a)}\times p_i^{(a)}$ has a part of size $p_i^{(a)}$.
It is often useful to view a rigged configuration $(\nu,J)$ as a sequence of partitions $\nu$ where the parts of size $i$ in $\nu^{(a)}$ are labeled by the parts of $J^{(a,i)}$. The pair $(i,x)$ where $i$ is a part of $\nu^{(a)}$ and $x$ is a part of $J^{(a,i)}$ is called a **string** of the $a$-th rigged partition $(\nu,J)^{(a)}$. The label $x$ is called a **rigging** or **quantum number**. The corresponding **coquantum number** is $p_i^{(a)}-x$.
\[ex:rc\] Let ${\mathfrak{g}}=D_4^{(1)}$, $B=B^1\otimes B^2\otimes B^2\otimes B^3$ and ${\lambda}=2{\overline{\Lambda}}_1$. Then the following three sequences of partitions are admissible $(L,{\lambda})$-configurations $$\begin{aligned}
& \yngrc(3,2,2,1,1,0) \quad \yngrc(3,0,3,0) \quad \yngrc(3,0) \quad \yngrc(3,0)\\[2mm]
& \yngrc(2,1,2,1,1,0,1,0) \quad \yngrc(2,0,2,0,1,0,1,0)
\quad \yngrc(2,0,1,0) \quad \yngrc(2,0,1,0)\\[2mm]
& \yngrc(3,2,3,2) \quad \yngrc(3,0,3,0) \quad \yngrc(3,0) \quad \yngrc(3,0)\end{aligned}$$ where the corresponding vacancy numbers are written next to each part. Hence, writing the parts of $J^{(a,i)}$ next to the parts of size $i$ of partition $\nu^{(a)}$ the following would be a particular rigged configuration $$(\nu,J)\quad=\quad
\yngrc(3,0,2,1,1,0) \quad \yngrc(3,0,3,0) \quad \yngrc(3,0) \quad \yngrc(3,0).$$
Quantum number complementation
------------------------------
Let ${\theta}={\theta}_L:{\mathrm{RC}}(L,{\lambda})\rightarrow{\mathrm{RC}}(L,{\lambda})$ be the involution that preserves configurations and complements riggings with respect to the vacancy numbers. More precisely, each partition $J^{(a,i)}$ is replaced by the partition that is complementary to it within the $m_i^{(a)}\times p_i^{(a)}$ rectangle.
The RC reduction steps ${\overline{\delta}}$ and ${\widetilde{\delta}}$ {#ss:deltas}
-----------------------------------------------------------------------
Suppose $L_1^{(1)}>0$. Let ${{\mathrm{lh}}(L)}$ and ${{\mathrm{rh}}(L)}$be obtained from $L$ by removing one tensor factor $B^1$. In particular, if $B$ has $B^1$ as its left (resp. right) tensor factor, then ${{\mathrm{lh}}(L)}$ (resp. ${{\mathrm{rh}}(L)}$) is the multiplicity array for ${\mathrm{lh}}(B)$ (resp. ${\mathrm{rh}}(B)$). In [@OSS:2003b] a quantum number bijection ${\overline{\phi}}:P(B)\rightarrow {\mathrm{RC}}(L)$ was defined when $B$ is a tensor power of $B^1$. The key step in the definition of ${\overline{\phi}}$ is an algorithm that defines a map ${\overline{\delta}}:{\mathrm{RC}}(L)\rightarrow{\mathrm{RC}}({{\mathrm{lh}}(L)})$. The same algorithm defines such a map for the current case.
For $(\nu,J)\in{\mathrm{RC}}(L,{\lambda})$, the algorithm produces a smaller rigged configuration ${\overline{\delta}}(\nu,J)\in{\mathrm{RC}}({{\mathrm{lh}}(L)},\mu)$ for some $\mu\in{\lambda^-}$ and an element ${\mathrm{rk}}(\nu,J)\in B^1$ such that $$\label{eq:deltawt}
\mu + {\mathrm{wt}}({\mathrm{rk}}(\nu,J)) = {\lambda}.$$
We recall the algorithm for ${\overline{\delta}}$ explicitly for type $A_n^{(1)}$ and $D_n^{(1)}$. Although we do not use them here, the explicit algorithms exists for the other nonexceptional affine types and can be found in [@OSS:2002a].
### String selection for type $A_n^{(1)}$ {#string-selection-for-type-a_n1 .unnumbered}
Set $\ell^{(0)}=1$ and repeat the following process for $a=1,2,\ldots,n$ or until stopped. Find the smallest index $i\ge \ell^{(a-1)}$ such that $J^{(a,i)}$ is singular. If no such $i$ exists, set ${\mathrm{rk}}(\nu,J)=a$ and stop. Otherwise set $\ell^{(a)}=i$ and continue with $a+1$.
### String selection for type $D_n^{(1)}$ {#string-selection-for-type-d_n1 .unnumbered}
Set $\ell^{(0)}=1$ and repeat the following process for $a=1,2,\ldots,n-2$ or until stopped. Find the smallest index $i\ge
\ell^{(a-1)}$ such that $J^{(a,i)}$ is singular. If no such $i$ exists, set ${\mathrm{rk}}(\nu,J)=a$ and stop. Otherwise set $\ell^{(a)}=i$ and continue with $a+1$. Set all yet undefined $\ell^{(a)}$ to $\infty$.
If the process has not stopped at $a=n-2$, find the minimal indices $i,j\ge \ell^{(n-2)}$ such that $J^{(n-1,i)}$ and $J^{(n,j)}$ are singular. If neither $i$ nor $j$ exist, set ${\mathrm{rk}}(\nu,J)=n-1$ and stop. If $i$ exists, but not $j$, set $\ell^{(n-1)}=i$, ${\mathrm{rk}}(\nu,J)=n$ and stop. If $j$ exists, but not $i$, set $\ell^{(n)}=j$, ${\mathrm{rk}}(\nu,J)=\overline{n}$ and stop. If both $i$ and $j$ exist, set $\ell^{(n-1)}=i$, $\ell^{(n)}=j$ and continue with $a=n-2$.
Now continue for $a=n-2,n-3,\ldots,1$ or until stopped. Find the minimal index $i\ge {\bar{\ell}}^{(a+1)}$ where ${\bar{\ell}}^{(n-1)}
=\max(\ell^{(n-1)},\ell^{(n)})$ such that $J^{(a,i)}$ is singular (if $i=\ell^{(a)}$ then there need to be two parts of size $p_i^{(a)}$ in $J^{(a,i)}$). If no such $i$ exists, set ${\mathrm{rk}}(\nu,J)=\overline{a+1}$ and stop. If the process did not stop, set ${\mathrm{rk}}(\nu,J)=\overline{1}$. Set all yet undefined $\ell^{(a)}$ and ${\bar{\ell}}^{(a)}$ to $\infty$.
### The new rigged configuration {#the-new-rigged-configuration .unnumbered}
The rigged configuration $(\tilde{\nu},\tilde{J})={\overline{\delta}}(\nu,J)$ is obtained by removing a box from the selected strings and making the new strings singular again. Explicitly (ignoring the statements about ${\bar{\ell}}^{(a)}$ for type $A_n^{(1)}$) $$m_i^{(a)}(\tilde{\nu})=m_i^{(a)}(\nu)+\begin{cases}
1 & \text{if $i=\ell^{(a)}-1$}\\
-1 & \text{if $i=\ell^{(a)}$}\\
1 & \text{if $i={\bar{\ell}}^{(a)}-1$ and $1\le a\le n-2$}\\
-1 & \text{if $i={\bar{\ell}}^{(a)}$ and $1\le a \le n-2$}\\
0 & \text{otherwise.} \end{cases}$$ The partition $\tilde{J}^{(a,i)}$ is obtained from $J^{(a,i)}$ by removing a part of size $p_i^{(a)}(\nu)$ for $i=\ell^{(a)}$ and $i={\bar{\ell}}^{(a)}$, adding a part of size $p_i^{(a)}(\tilde{\nu})$ for $i=\ell^{(a)}-1$ and $i={\bar{\ell}}^{(a)}-1$, and leaving it unchanged otherwise.
For the rigged configuration $(\nu,J)$ of example \[ex:rc\], we have $${\overline{\delta}}(\nu,J)\quad=\quad
\yngrc(3,0,2,1) \quad \yngrc(2,0,2,0) \quad \yngrc(2,0) \quad \yngrc(2,0)$$ with ${\mathrm{rk}}(\nu,J)=\overline{2}$.
The next proposition was proved in [@KSS:2002; @OSS:2002a].
\[pp:boxbij\] The map ${\overline{\delta}}:{\mathrm{RC}}(L,{\lambda})\rightarrow \bigcup_{\mu\in {\lambda^-}} {\mathrm{RC}}({{\mathrm{lh}}(L)},\mu)$ is injective.
Note that for simply-laced type, knowing ${\lambda}$ and $\mu$ uniquely determines ${\mathrm{rk}}(\nu,J)$ by .
We may define the inverse of ${\overline{\delta}}$. To this end, let $${\lambda^+}=\{ \mu\in{\overline{P}}^+ \mid \text{$B(\mu)$ occurs in $B^1\times B({\lambda})$} \}.$$ Denote by ${\widetilde{{\mathrm{RC}}}}(L,{\lambda})$ the subset of ${\mathrm{RC}}(L,{\lambda})\times B^1$ given by $((\nu,J),b)$ such that ${\lambda}+{\mathrm{wt}}(b)\in {\overline{P}}^+$. By abuse of notation define $${\overline{\delta}}^{-1}:{\widetilde{{\mathrm{RC}}}}(L,{\lambda})\to \bigcup_{\beta\in {\lambda^+}}{\mathrm{RC}}({{\mathrm{lh}}^{-1}(L)},\beta)$$ by the following algorithm, where ${{\mathrm{lh}}^{-1}(L)}$ is obtained from $L$ by replacing $L_1^{(1)}$ by $L_1^{(1)}+1$.
### String selection for type $A_n^{(1)}$ {#string-selection-for-type-a_n1-1 .unnumbered}
In this case ${\mathrm{wt}}(b)=\epsilon_r$ for some $1\le r\le n+1$, where $\epsilon_r$ is the $r$-th canonical unit vector in ${\mathbb{Z}}^{n+1}$. Set $s^{(r)}=\infty$ and repeat the following process for $a=r-1,r-2,\ldots,1$. Find the largest index $i\le s^{(a+1)}$ such that $J^{(a,i)}$ is singular and set $s^{(a)}=i$; if no such $i$ exists set $s^{(a)}=0$. Set all undefined $s^{(a)}$ to infinity.
### String selection for type $D_n^{(1)}$ {#string-selection-for-type-d_n1-1 .unnumbered}
In this case ${\mathrm{wt}}(b)=\epsilon_r$ or ${\mathrm{wt}}(b)=-\epsilon_r$ for $1\le r\le n$, where $\epsilon_r$ is the $r$-th canonical unit vector in ${\mathbb{Z}}^{n}$. In the first case proceed exactly as for type $A_n^{(1)}$. Throughout the whole algorithm, if an index $i$ does not exist, set $i=0$.
If ${\mathrm{wt}}(b)=-\epsilon_n$, find the largest index $i$ such that $J^{(n,i)}$ is singular and set $s^{(n)}=i$. Find the largest index $i\le s^{(n)}$ such that $J^{(n-2,i)}$ is singular and set $s^{(n-2)}=i$. Then proceed as in type $A_n^{(1)}$.
If ${\mathrm{wt}}(b)=-\epsilon_{n-1}$, find the largest indices $i$ and $j$ such that $J^{(n-1,i)}$ and $J^{(n,j)}$ are singular and set $s^{(n-1)}=i$ and $s^{(n)}=j$. Then find the largest index $i\le \min\{s^{(n-1)},s^{(n)}\}$ such that $J^{(n-2,i)}$ is singular and set $s^{(n-2)}=i$. After this proceed as in type $A_n^{(1)}$.
Finally, if ${\mathrm{wt}}(b)=-\epsilon_r$ for $1\le r\le n-2$, set ${\bar{s}}^{(r-1)}=\infty$ and proceed for $a=r,r+1,\ldots,n-2$ as follows. Find the largest index $i\le {\bar{s}}^{(a-1)}$ such that $J^{(a,i)}$ is singular and set ${\bar{s}}^{(a)}=i$. Then find the largest indices $i\le {\bar{s}}^{(n-2)}$ and $j\le {\bar{s}}^{(n-2)}$ such that $J^{(n-1,i)}$ and $J^{(n,j)}$ are singular and set $s^{(n-1)}=i$ and $s^{(n)}=j$. After this proceed as for the case ${\mathrm{wt}}(b)=-\epsilon_{n-1}$.
Set all yet undefined $s^{(a)}$ and ${\bar{s}}^{(a)}$ to $\infty$.
### The new rigged configuration {#the-new-rigged-configuration-1 .unnumbered}
The rigged configuration $(\tilde{\nu},\tilde{J})={\overline{\delta}}^{-1}(\nu,J)$ is obtained by adding a box to the selected strings and making the new strings singular again.
Define ${\widetilde{\delta}}:{\mathrm{RC}}(L)\rightarrow{\mathrm{RC}}({{\mathrm{lh}}(L)})$ by ${\theta}_{{{\mathrm{lh}}(L)}}\circ{\overline{\delta}}\circ{\theta}_L$. Alternatively, ${\widetilde{\delta}}$ is defined by a coquantum number version of the map ${\overline{\delta}}$. Instead of selecting singular strings it selects cosingular strings and keeps coquantum numbers constant for unselected strings. It also produces an element ${\widetilde{{\mathrm{rk}}}}(\nu,J)\in B^1$. If $(\nu,J)\in{\mathrm{RC}}(L,{\lambda})$ and ${\widetilde{\delta}}(\nu,J)\in
{\mathrm{RC}}({{\mathrm{lh}}(L)},\mu)$ then $$\mu + {\mathrm{wt}}({\widetilde{{\mathrm{rk}}}}(\nu,J)) = {\lambda}.$$
Splitting on RCs
----------------
Let $s\ge2$. Suppose $B$ contains a distinguished tensor factor $B^{r,s}$, which is the case when we consider the maps ${\mathrm{ls}}$ and ${\mathrm{rs}}$. Let $L$ be the multiplicity array of $B$ and ${{\mathrm{ls}}(L)}$ that which is obtained from $L$ by replacing $B^{r,s}$ by $B^{r,1}$ and $B^{r,s-1}$.
\[pp:ljdef\] Let $L$ be such that $L_s^{(r)}\ge1$ for a particular $(r,s)\in{\mathcal{H}}$ with $s\ge2$ and let ${{\mathrm{ls}}(L)}$ be defined with respect to $(r,s)$. Then ${\mathrm{C}}(L,{\lambda})\subset {\mathrm{C}}({{\mathrm{ls}}(L)},{\lambda})$. Under this inclusion map, the vacancy number $p_i^{(a)}$ for $\nu$ increases by $\delta_{a,r} \chi(i<s)$ where $\chi(P)=1$ if $P$ is true and $\chi(P)=0$ otherwise. Hence there are well-defined injective maps ${\overline{j}},{\widetilde{j}}:{\mathrm{RC}}(L)\rightarrow {\mathrm{RC}}({{\mathrm{ls}}(L)})$ given by:
1. ${\overline{j}}(\nu,J)=(\nu,J)$.
2. ${\widetilde{j}}(\nu,J)=(\nu,J')$ where $J'$ is obtained from $J$ by adding $1$ to the rigging of each string in $(\nu,J)^{(r)}$ of length strictly less than $s$.
In particular, ${\overline{j}}$ preserves quantum numbers, ${\widetilde{j}}$ preserves coquantum numbers, and $$\label{eq:rj-lj}
{\widetilde{j}}= {\theta}_{{{\mathrm{ls}}(L)}} \circ {\overline{j}}\circ {\theta}_L.$$
Immediate from the definitions.
Box-splitting for RCs
---------------------
Suppose $r\ge2$ and $B\in{{\mathcal{C}}^A}$ has a distinguished tensor factor $B^{r,1}$. Let $L$ be the multiplicity array for $B$ and ${{\mathrm{lb}}(L)}$ that for the crystal obtained from $B$ by replacing $B^{r,1}$ by $B^{1,1}$ and $B^{r-1,1}$.
\[pp:rcbs\] Let $L$ be such that $L_1^{(r)}\ge1$ for some $r\ge2$. Let ${{\mathrm{lb}}(L)}$ be defined with respect to $r$. Then there are injections ${\overline{i}},{\widetilde{i}}:{\mathrm{RC}}(L,{\lambda})\rightarrow
{\mathrm{RC}}({{\mathrm{lb}}(L)},{\lambda})$ defined by adding singular (resp. cosingular) strings of length $1$ to $(\nu,J)^{(a)}$ for $1\le a < r$. Moreover the vacancy numbers stay the same.
Fermionic formula $M$ {#s:fermionic}
=====================
In this section we state the fermionic formula $M$ associated with rigged configurations for simply-laced algebras as introduced in [@HKOTY:1999] and virtual fermionic formulas for nonsimply-laced algebras (see [@OSS:2003a; @OSS:2003b]).
Fermionic formula $M$ {#fermionic-formula-m}
---------------------
Let $(q)_m=(1-q)(1-q^2)\cdots (1-q^m)$ and define the $q$-binomial coefficient for $m,p\in {\mathbb{Z}}_{\ge 0}$ as $${\genfrac{[}{]}{0pt}{}{m+p}{m}}=\frac{(q)_{m+p}}{(q)_m(q)_p}.$$ The fermionic formula for types $A_n^{(1)}$ and $D_n^{(1)}$ is given by [@HKOTY:1999] $$\label{eq:fermionic}
M_{L,{\lambda}}(q)=\sum_{\nu\in {\mathrm{C}}(L,{\lambda})} q^{{cc}(\nu)}
\prod_{(a,i)\in{\mathcal{H}}} {\genfrac{[}{]}{0pt}{}{m_i^{(a)}+p_i^{(a)}}{m_i^{(a)}}}$$ with $m_i^{(a)}$, $p_i^{(a)}$ and ${\mathrm{C}}(L,{\lambda})$ as in section \[sec:rc def\] and $${cc}(\nu)=\frac{1}{2} \sum_{a,b\in {\bar{I}}} \sum_{j,k\ge 1} (\alpha_a \mid
\alpha_b) \min(j,k) m_j^{(a)} m_k^{(b)}.$$ The fermionic formula can be restated solely in terms of rigged configurations. To this end recall that the $q$-binomial coefficient ${\genfrac{[}{]}{0pt}{}{m+p}{m}}$ is the generating function of partitions in a box of width $p$ and height $m$. Hence $$\label{eq:M rc}
M_{L,{\lambda}}(q)=\sum_{(\nu,J)\in {\mathrm{RC}}(L,{\lambda})} q^{{cc}(\nu,J)},$$ where ${cc}(\nu,J)={cc}(\nu)+\sum_{(a,i)\in {\mathcal{H}}} |J^{(a,i)}|$.
Virtual fermionic formula
-------------------------
Fermionic formulae for nonsimply-laced algebras were defined in [@HKOTT:2001 Section 4]. For $A_{2n}^{(2)\dagger}$ it was defined in [@OSS:2003a]. Here we recall virtual rigged configurations in analogy to virtual crystals as defined in [@OSS:2003b].
\[def:VRC\] Let $X$ and $Y$ be as in section \[ss:VX\], and ${\lambda}$, $B$ and $L$ as in section \[sec:rc def\] for type $X$. Let ${\Psi}:B\to{\widehat{V}}$ be the corresponding virtual $Y$-crystal and ${\widehat{L}}$ the multiplicity array corresponding to ${\widehat{V}}$. For $X\not\in\{A_{2n}^{(2)},A_{2n}^{(2)\dagger}\}$, ${{\mathrm{RC}}^v}(L,{\lambda})$ is the set of elements $({\widehat{\nu}},{\widehat{J}})\in {\mathrm{RC}}({\widehat{L}},{\Psi}({\lambda}))$ such that:
1. For all $i\in {\mathbb{Z}}_{>0}$, ${\widehat{m}}_i^{(a)}={\widehat{m}}_i^{(b)}$ and ${\widehat{J}}^{(a,i)}={\widehat{J}}^{(b,i)}$ if $a$ and $b$ are in the same ${\sigma}$-orbit in $I^Y$.
2. For all $i\in {\mathbb{Z}}_{>0}$, $a\in {\bar{I}}^X$, and $b\in {\iota}(a)\subset {\bar{I}}^Y$, we have ${\widehat{m}}_j^{(b)}=0$ if $j \not\in {\gamma}_a {\mathbb{Z}}$ and the parts of ${\widehat{J}}^{(b,i)}$ are multiples of ${\gamma}_a$.
For $X=A_{2n}^{(2)}$ the following changes must be made:
1. ${\widehat{m}}_j^{(n)}$ may be positive for any $j\ge1$.
For $X=A_{2n}^{(2)\dagger}$ one makes the exception (A2) and the additional condition that
1. The parts of ${\widehat{J}}^{(n,i)}$ must have the same parity as $i$.
[@OSS:2003b Theorem 4.2] \[th:M=VM\] There is a bijection $\Psi:{\mathrm{RC}}(L,{\lambda})\rightarrow {{\mathrm{RC}}^v}(L,{\lambda})$ sending $(\nu,J)\mapsto({\widehat{\nu}},{\widehat{J}})$ given as follows. For all $a\in
{\bar{I}}^X$, $b\in{\iota}(a)\subset {\bar{I}}^Y$, and $i\in{\mathbb{Z}}_{>0}$, $$\begin{aligned}
{\widehat{m}}_{{\gamma}_a i}^{(b)} &= m_i^{(a)} \\
{\widehat{J}}^{(b,{\gamma}_a i)}&={\gamma}_a J^{(a,i)},\end{aligned}$$ except when $X=A_{2n}^{(2)}$ or $X=A_{2n}^{(2)\dagger}$ and $a=n$, in which case $$\begin{split}
{\widehat{m}}_i^{(n)} &= m_i^{(n)} \\
{\widehat{J}}^{(n,i)} &= 2 J^{(n,i)}.
\end{split}$$ The cocharge changes by ${cc}({\widehat{\nu}},{\widehat{J}}) = {\gamma}_0 \,{cc}(\nu,J)$.
Defining the virtual fermionic formula as $$VM_{L,{\lambda}}(q)=\sum_{({\widehat{\nu}},{\widehat{J}})\in{{\mathrm{RC}}^v}(L,{\lambda})} q^{{cc}({\widehat{\nu}},{\widehat{J}})/{\gamma}_0}$$ we obtain as a corollary:
\[cor:M=VM\] ($M=VM$) $M_{L,{\lambda}}(q)=VM_{L,{\lambda}}(q)$.
Bijection {#sec:bij}
=========
Quantum number bijection
------------------------
The following result defines the bijection from paths to rigged configurations. It is valid for both $B\in {\mathcal{C}}$ and $B\in {{\mathcal{C}}^A}$.
\[pp:phi\] There exists a unique family of bijections ${\overline{\phi}}:P(B,{\lambda})\rightarrow {\mathrm{RC}}(L,{\lambda})$ such that the empty path maps to the empty rigged configuration, and:
1. Suppose $B=B^1 \otimes B'$. Let ${\mathrm{lh}}(B)=B'$ with multiplicity array ${{\mathrm{lh}}(L)}$. Then the diagram $$\label{eq:hatdiag}
\begin{CD}
P(B,{\lambda}) @>{{\overline{\phi}}}>> {\mathrm{RC}}(L,{\lambda}) \\
@V{{\mathrm{lh}}}VV @VV{{\overline{\delta}}}V \\
\displaystyle{\bigcup_{\mu\in{\lambda^-}} P({\mathrm{lh}}(B),\mu)} @>>{{\overline{\phi}}}> \displaystyle{\bigcup_{\mu\in{\lambda^-}}
{\mathrm{RC}}({{\mathrm{lh}}(L)},\mu)}
\end{CD}$$ commutes.
2. Suppose $B=B^{r,s} \otimes B'$ with $s\ge 2$ (and $r=1$ for ${\mathcal{C}}$). Let ${\mathrm{ls}}(B)=B^{r,1}\otimes B^{r,s-1}\otimes B'$ with multiplicity array ${{\mathrm{ls}}(L)}$. Then the following diagram commutes: $$\label{eq:splitdiag}
\begin{CD}
P(B,{\lambda}) @>{{\overline{\phi}}}>> {\mathrm{RC}}(L,{\lambda}) \\
@V{{\mathrm{ls}}}VV @VV{{\overline{j}}}V \\
P({\mathrm{ls}}(B),{\lambda}) @>>{{\overline{\phi}}}> {\mathrm{RC}}({{\mathrm{ls}}(L)},{\lambda})
\end{CD}$$
3. For ${{\mathcal{C}}^A}$, suppose $B=B^{r,1} \otimes B'$ with $r\ge2$. Let ${\mathrm{lb}}(B)=B^{1,1}\otimes B^{r-1,1}\otimes B'$ and ${{\mathrm{lb}}(L)}$ its multiplicity array. Then the following diagram commutes: $$\label{eq:boxdiag}
\begin{CD}
P(B,{\lambda}) @>{{\overline{\phi}}}>> {\mathrm{RC}}(L,{\lambda}) \\
@V{{\mathrm{lb}}}VV @VV{{\overline{i}}}V \\
P({\mathrm{lb}}(B),{\lambda}) @>>{{\overline{\phi}}}> {\mathrm{RC}}({{\mathrm{lb}}(L)},{\lambda})
\end{CD}$$
For type $A_n^{(1)}$ the existence of ${\overline{\phi}}$ was proven in [@KSS:2002]. The proof in case (1) for other nonexceptional types is essentially done in [@OSS:2002a]. It remains to prove case (2) for type $D_n^{(1)}$.
\[lem:resls\] Let $B=B^s\otimes B'$ with $s\ge 2$. For type $D_n^{(1)}$, the map ${\overline{\phi}}:P({\mathrm{ls}}(B),{\lambda})\to {\mathrm{RC}}({{\mathrm{ls}}(L)},{\lambda})$ restricts to a bijection ${\overline{\phi}}:{\mathrm{ls}}(P(B,{\lambda}))\to {\overline{j}}({\mathrm{RC}}(L,{\lambda}))$.
Let $b=x\otimes b_2 \otimes b'\in B^1\otimes B^{s-1}\otimes B'$ and ${\mathrm{ls}}(b_2)=y\otimes b_3\in B^1\otimes B^{s-2}$. Then $b\in
{\mathrm{Im}}({\mathrm{ls}})$ if and only if $x\le y$. (Note that this implies in particular that $n$ and $\bar{n}$ cannot appear in the same one-row crystal element).
By Proposition \[pp:ljdef\], $(\nu,J)\in {\mathrm{RC}}({{\mathrm{ls}}(L)},{\lambda})$ is in the image of ${\overline{j}}$ if and only if $(\nu,J)^{(1)}$ has no singular strings of length smaller than $s$.
Let us first show that if $b\in{\mathrm{Im}}({\mathrm{ls}})$ then ${\overline{\phi}}(b)\in {\mathrm{Im}}({\overline{j}})$. Hence assume that $b=x\otimes b_2\otimes b'$ with $x\le y$ with $y$ as defined above. By induction $(\nu',J')=
{\overline{\phi}}(y\otimes b_3\otimes b')$ has no singular strings in the first rigged partition of length smaller than $s-1$. Denote the lengths of the strings selected by ${\overline{\delta}}$ associated with the letter $y$ by $\ell_y^{(k)}$ and ${\bar{\ell}}_y^{(k)}$. Then in particular $\ell_y^{(1)}\ge s-1$. “Unsplitting” yields on the paths side $b_2\otimes b'$ and on the rigged configuration side $(\nu',J')$ with a change in the vacancy numbers by $-\delta_{a,1}\chi(i<s-1)$. Since $x\le y$ it follows that $\ell_x^{(k)}>\ell_y^{(k)}$ and ${\bar{\ell}}_x^{(k)}>{\bar{\ell}}_y^{(k)}$, where $\ell_x^{(k)}$ and ${\bar{\ell}}_x^{(k)}$ are the lengths of the strings selected by ${\overline{\delta}}$ associated with $x$. This shows in particular that $\ell_x^{(1)}\ge s$, and from the change in vacancy numbers from ${\overline{\phi}}(b_2\otimes b')$ to ${\overline{\phi}}(x\otimes b_2\otimes b')$ it follows that there are no singular strings in the first rigged partition of ${\overline{\phi}}(x\otimes b_2\otimes b')$ of length smaller than $s$.
Conversely, assume that $(\nu,J)\in {\mathrm{RC}}({{\mathrm{ls}}(L)},{\lambda})$ is in the image if ${\overline{j}}$. We need to show that then $b={\overline{\phi}}^{-1}(\nu,J)$ has the property that $x\le y$ in the above notation. Call the strings selected by ${\overline{\delta}}$ in $(\nu,J)$ $\ell_x^{(k)}$ and ${\bar{\ell}}_x^{(k)}$. By assumption $(\nu,J)^{(1)}$ has no singular string of length smaller than $s$. Hence $\ell_x^{(1)}\ge s$. By the definition of ${\overline{j}}$, we have that the first rigged partition of $(\nu',J')={\overline{j}}\circ {\overline{\delta}}(\nu,J)$ has no singular strings of length smaller than $s-1$. Hence $s-1\le \ell_y^{(1)}<\ell_x^{(1)}$, where $\ell_y^{(k)}$ and ${\bar{\ell}}_y^{(k)}$ are the lengths of the strings selected by ${\overline{\delta}}$ on $(\nu',J')$. The algorithm of ${\overline{\delta}}$ implies that $\ell_y^{(k)}<\ell_x^{(k)}$ and ${\bar{\ell}}_y^{(k)}<{\bar{\ell}}_x^{(k)}$, so that $x\le y$ as desired.
Coquantum number bijection
--------------------------
Let ${\widetilde{\phi}}={\theta}\circ{\overline{\phi}}$; it can be characterized as follows.
There exists a unique family of bijections ${\widetilde{\phi}}:P(B,{\lambda})\rightarrow {\mathrm{RC}}(L,{\lambda})$ with the same properties as in Proposition \[pp:phi\] except that ${\overline{\delta}}$, ${\overline{j}}$ and ${\overline{i}}$ are replaced by ${\widetilde{\delta}}$, ${\widetilde{j}}$ and ${\widetilde{i}}$ in , and , respectively.
Commutations of the basic steps
-------------------------------
We record the commutations among the basic steps of the path-RC bijection. Here $r=1$ for ${\mathcal{C}}$.
\[th:RCcomm\]
1. \[pt:dbdt\] $[{\overline{\delta}},{\widetilde{\delta}}]=0$.
2. \[pt:jdelta\] $[{\widetilde{j}},{\overline{\delta}}]=0$ and $[{\overline{j}},{\widetilde{\delta}}]=0$.
3. \[pt:ljrj\] $[{\overline{j}},{\widetilde{j}}]=0$.
4. \[pt:bsdelta\] $[{\widetilde{i}},{\overline{\delta}}]=0$ and $[{\overline{i}},{\widetilde{\delta}}]=0$.
5. \[pt:bslj\] $[{\widetilde{i}},{\overline{j}}]=0$ and $[{\overline{i}},{\widetilde{j}}]=0$.
The proof of part \[pt:dbdt\] for type $A_n^{(1)}$ is given in [@KSS:2002 Appendix A]. The proof of part \[pt:dbdt\] for type $D_n^{(1)}$ is quite technical and follows similar arguments as [@KSS:2002 Appendix A] (see also [@Sch:2004 Appendix C]). Details are available upon request. Parts \[pt:jdelta\] and \[pt:ljrj\] follow easily from the definitions. Parts \[pt:bsdelta\] through \[pt:bslj\] only apply for ${{\mathcal{C}}^A}$ and follow from [@KSS:2002].
For type $D_n^{(1)}$, there is an analogue of Proposition \[pp:weight\] for the commutation of ${\overline{\delta}}$ and ${\widetilde{\delta}}$. Let $(\nu,J)\in {\mathrm{RC}}(L,{\lambda})$, ${\widetilde{\delta}}(\nu,J)\in {\mathrm{RC}}({\mathrm{rh}}(L),\alpha)$, ${\overline{\delta}}(\nu,J)\in {\mathrm{RC}}({\mathrm{lh}}(L),\beta)$ and ${\widetilde{\delta}}({\overline{\delta}}(\nu,J))={\overline{\delta}}({\widetilde{\delta}}(\nu,J))\in {\mathrm{RC}}({\mathrm{lh}}({\mathrm{rh}}(L)),\gamma)$. Then $\alpha$ is uniquely determined by ${\lambda}$, $\beta$, and $\gamma$.
\[pp:weight rc\] For ${\lambda}$, $\alpha$, $\beta$, and $\gamma$ as above the statements of Proposition \[pp:weight\] hold.
The proof is an easy consequence of the commutation $[{\overline{\delta}},{\widetilde{\delta}}]=0$ and is available upon request.
The bijection and the various operations
----------------------------------------
\[th:bijops\] Under the family of bijections ${\overline{\phi}}$ the following operations correspond:
1. \[pt:ls\] ${\mathrm{ls}}$ with ${\overline{j}}$.
2. \[pt:lh\] ${\mathrm{lh}}$ with ${\overline{\delta}}$.
3. \[pt:rs\] ${\mathrm{rs}}$ with ${\widetilde{j}}$.
4. \[pt:rh\] ${\mathrm{rh}}$ with ${\widetilde{\delta}}$.
5. \[pt:\*\] $*$ with ${\theta}$.
6. \[pt:R\] $R$ with the identity.
7. \[pt:bs\] ${\mathrm{lb}}$ with ${\overline{i}}$ and ${\mathrm{rb}}$ with ${\widetilde{i}}$.
To illustrate point \[pt:rh\] of the above Theorem, take $$b=\begin{array}{|c|}\hline \overline{3}\\ \hline \end{array} \otimes
\young(23) \otimes \young(12) \otimes \young(1)$$ of type $D_4^{(1)}$. Then $${\mathrm{rh}}(b)=\young(3) \otimes \young(22) \otimes \young(11)$$ and $$\begin{aligned}
{\overline{\phi}}(b)&= \quad \yngrc(2,0,1,0,1,0) \quad \yngrc(1,1,1,0) \quad
\yngrc(1,0)
\quad \yngrc(1,0)\\[2mm]
{\overline{\phi}}({\mathrm{rh}}(b))={\widetilde{\delta}}({\overline{\phi}}(b))&= \quad \yngrc(2,0,1,0) \quad \yngrc(1,0) \quad {\varnothing}\quad
\qquad {\varnothing}.\end{aligned}$$
Everything is proved for ${{\mathcal{C}}^A}$ in [@KSS:2002], including part \[pt:bs\], which only applies in that case. We assume that $B\in{\mathcal{C}}$ for type $D_n^{(1)}$. Parts \[pt:ls\] and \[pt:lh\] hold by Proposition \[pp:phi\]. We prove parts \[pt:rs\], \[pt:rh\], and \[pt:\*\] simultaneously by induction. The induction is based first on the quantity $\sum_i s_i$ for the crystal $\bigotimes_i B^{s_i}$, and then by decreasing induction on the number of tensor factors.
Consider part \[pt:rs\]. Suppose first that $B=B^s$ for some $s\ge
2$. Then $P(B)$ has only one element $1^s$. It is easy to show that ${\overline{\phi}}(1^s)$ is the empty RC and that \[pt:rs\] holds. Suppose next that $B=B^1 \otimes B' \otimes B^s$. Consider the diagram $$\xymatrix{
P(B) \ar[rrr]^{{\mathrm{rs}}} \ar[dr]_{{\overline{\phi}}} \ar[ddd]_{{\mathrm{lh}}} & & & P({\mathrm{rs}}(B)) \ar[dl]^{{\overline{\phi}}} \ar[ddd]^{{\mathrm{lh}}} \\
& {\mathrm{RC}}(L) \ar[r]^{{\widetilde{j}}} \ar[d]_{{\overline{\delta}}} & {\mathrm{RC}}({{\mathrm{rs}}(L)}) \ar[d]^{{\overline{\delta}}} & \\
& {\mathrm{RC}}({{\mathrm{lh}}(L)}) \ar[r]_{{\widetilde{j}}} & {\mathrm{RC}}({\mathrm{rs}}({{\mathrm{lh}}(L)})) & \\
P({\mathrm{lh}}(B)) \ar[ur]^{{\overline{\phi}}} \ar[rrr]_{{\mathrm{rs}}} & & & P({\mathrm{rs}}({\mathrm{lh}}(B)))
\ar[ul]_{{\overline{\phi}}} }$$ Here $L$, ${{\mathrm{rs}}(L)}$, ${{\mathrm{lh}}(L)}$, ${\mathrm{rs}}({{\mathrm{lh}}(L)})$ are the multiplicities arrays corresponding to $B$, ${\mathrm{rs}}(B)$, ${\mathrm{lh}}(B)$, ${\mathrm{rs}}({\mathrm{lh}}(B))$, respectively. We shall view such a diagram as a cube in which the small square is in the background. The left and right faces commute by Proposition \[pp:phi\]. The front and back faces commute by Proposition \[pp:cryscomm\] part \[pt:lhrs\] and Theorem \[th:RCcomm\] part \[pt:jdelta\] respectively. The bottom face commutes by induction. It follows that the top face “commutes up to ${\overline{\delta}}$", that is, ${\overline{\delta}}\circ {\widetilde{j}}\circ {\overline{\phi}}= {\overline{\delta}}\circ {\overline{\phi}}\circ {\mathrm{rs}}$. But all maps in the top face preserve the highest weight. By Proposition \[pp:boxbij\] it follows that the top face commutes.
The remaining case is $B=B^{s'} \otimes B' \otimes B^s$ for $s,s'\ge
2$. Consider the diagram below, where ${\mathrm{rs}}({\mathrm{ls}}(L))$is obtained from ${{\mathrm{ls}}(L)}$ by splitting a $B^{s'}$ into $B^{s'-1}$ and $B^1$.
$$\xymatrix{
P(B) \ar[rrr]^{{\mathrm{rs}}} \ar[dr]_{{\overline{\phi}}} \ar[ddd]_{{\mathrm{ls}}} & & & P({\mathrm{rs}}(B)) \ar[dl]^{{\overline{\phi}}} \ar[ddd]^{{\mathrm{ls}}} \\
& {\mathrm{RC}}(L) \ar[r]^{{\widetilde{j}}} \ar[d]_{{\overline{j}}} & {\mathrm{RC}}({\mathrm{rs}}(L)) \ar[d]^{{\overline{j}}} & \\
& {\mathrm{RC}}({\mathrm{ls}}(L)) \ar[r]_{{\widetilde{j}}} & {\mathrm{RC}}({\mathrm{rs}}({\mathrm{ls}}(L)) & \\
P({\mathrm{ls}}(B)) \ar[ur]^{{\overline{\phi}}} \ar[rrr]_{{\mathrm{rs}}} & & & P({\mathrm{rs}}({\mathrm{ls}}(B)))
\ar[ul]_{{\overline{\phi}}} }$$ The left and right faces commute by Proposition \[pp:phi\]. The front and back faces commute by Proposition \[pp:cryscomm\] part \[pt:lsrs\] and Theorem \[th:RCcomm\] part \[pt:ljrj\] respectively. The bottom face commutes by induction. Since ${\overline{j}}$ is injective, it follows that the top face commutes. This finishes the proof of part \[pt:rs\].
We now prove part \[pt:rh\]. The proof is trivial for the base case $B=B^1$. Suppose next that $B=B^1 \otimes B' \otimes B^1$. $$\xymatrix{
P(B) \ar[rrr]^{{\mathrm{rh}}} \ar[dr]_{{\overline{\phi}}} \ar[ddd]_{{\mathrm{lh}}} & & & P({\mathrm{rh}}(B)) \ar[dl]^{{\overline{\phi}}} \ar[ddd]^{{\mathrm{lh}}} \\
& {\mathrm{RC}}(L) \ar[r]^{{\widetilde{\delta}}} \ar[d]_{{\overline{\delta}}} & {\mathrm{RC}}({{\mathrm{lh}}(L)}) \ar[d]^{{\overline{\delta}}} & \\
& {\mathrm{RC}}({{\mathrm{lh}}(L)}) \ar[r]_{{\widetilde{\delta}}} & {\mathrm{RC}}({\mathrm{rh}}({{\mathrm{lh}}(L)})) & \\
P({\mathrm{lh}}(B)) \ar[ur]^{{\overline{\phi}}} \ar[rrr]_{{\mathrm{rh}}} & & & P({\mathrm{rh}}({\mathrm{lh}}(B)))
\ar[ul]_{{\overline{\phi}}} }$$ The left and right faces commute by Proposition \[pp:phi\]. The front and back faces commute by Proposition \[pp:cryscomm\] part \[pt:lhrh\] and Theorem \[th:RCcomm\] part \[pt:dbdt\] respectively. The bottom face commutes by induction. Thus the top face commutes up to ${\overline{\delta}}$. By Proposition \[pp:boxbij\] it suffices to show that both ways around the top face, result in elements with the same highest weight. This follows from Propositions \[pp:weight\] and \[pp:weight rc\]. The remaining case is $B=B^s \otimes B' \otimes B^1$. Consider the diagram $$\xymatrix{
P(B) \ar[rrr]^{{\mathrm{rh}}} \ar[dr]_{{\overline{\phi}}} \ar[ddd]_{{\mathrm{ls}}} & & & P({\mathrm{rh}}(B)) \ar[dl]^{{\overline{\phi}}} \ar[ddd]^{{\mathrm{ls}}} \\
& {\mathrm{RC}}(L) \ar[r]^{{\widetilde{\delta}}} \ar[d]_{{\overline{j}}} & {\mathrm{RC}}({{\mathrm{rh}}(L)}) \ar[d]^{{\overline{j}}} & \\
& {\mathrm{RC}}({{\mathrm{ls}}(L)}) \ar[r]_{{\widetilde{\delta}}} & {\mathrm{RC}}({\mathrm{rh}}({{\mathrm{ls}}(L)})) & \\
P({\mathrm{ls}}(B)) \ar[ur]^{{\overline{\phi}}} \ar[rrr]_{{\mathrm{rh}}} & & & P({\mathrm{rh}}({\mathrm{ls}}(B)))
\ar[ul]_{{\overline{\phi}}} }$$ The left and right faces commute by Proposition \[pp:phi\]. The front and back faces commute by Proposition \[pp:cryscomm\] part \[pt:rhls\] and Theorem \[th:RCcomm\] part \[pt:jdelta\] respectively. The bottom face commutes by induction. Since ${\overline{j}}$ is injective it follows that the top face commutes. This proves part \[pt:rh\].
For part \[pt:\*\] the proof of the base case $B=B^s$ is trivial. Suppose next that $B=B^1 \otimes B' \otimes B^1$. Consider the diagram $$\xymatrix{
P(B) \ar[rrr]^{*} \ar[dr]_{{\overline{\phi}}} \ar[ddd]_{{\mathrm{rh}}} & & & P(B^*) \ar[dl]^{{\overline{\phi}}} \ar[ddd]^{{\mathrm{lh}}} \\
& {\mathrm{RC}}(L) \ar[r]^{{\theta}} \ar[d]_{{\widetilde{\delta}}} & {\mathrm{RC}}(L) \ar[d]^{{\overline{\delta}}} & \\
& {\mathrm{RC}}({{\mathrm{rh}}(L)}) \ar[r]_{{\theta}} & {\mathrm{RC}}({{\mathrm{rh}}(L)}) & \\
P({\mathrm{rh}}(B)) \ar[ur]^{{\overline{\phi}}} \ar[rrr]_{*} & & & P({\mathrm{rh}}(B)^*)
\ar[ul]_{{\overline{\phi}}} }$$ The right face commutes by Proposition \[pp:phi\]. The left commutes by part \[pt:rh\] which was just proved above. The back face commutes by the definition of ${\widetilde{\delta}}$. The commutation of the front face is given by Proposition \[pp:cryscomm\] part \[pt:hat\*\]. The bottom face commutes by induction. It follows that the top face commutes up to ${\overline{\delta}}$. Again it suffices to show that both ways around the top face produce elements of the same highest weight. But this holds since ${\overline{\phi}}$, ${\theta}$, and $*$ preserve the highest weight. Here we are using the fact that for ${\lambda}\in {\overline{P}}^+$, $V_{\lambda}^*\cong V_{\lambda}$.
Next let $B=B' \otimes B^s$ with $s\ge2$. $$\xymatrix{
P(B) \ar[rrr]^{*} \ar[dr]_{{\overline{\phi}}} \ar[ddd]_{{\mathrm{rs}}} & & & P(B^*) \ar[dl]^{{\overline{\phi}}} \ar[ddd]^{{\mathrm{ls}}} \\
& {\mathrm{RC}}(L) \ar[r]^{{\theta}} \ar[d]_{{\widetilde{j}}} & {\mathrm{RC}}(L) \ar[d]^{{\overline{j}}} & \\
& {\mathrm{RC}}({{\mathrm{ls}}(L)}) \ar[r]_{{\theta}} & {\mathrm{RC}}({{\mathrm{ls}}(L)}) & \\
P({\mathrm{rs}}(B)) \ar[ur]^{{\overline{\phi}}} \ar[rrr]_{*} & & & P({\mathrm{rs}}(B)^*)
\ar[ul]_{{\overline{\phi}}} }$$ The right face commutes by Proposition \[pp:phi\]. The left face commutes by part \[pt:rs\] which was proved above. The back face commutes by the definition of ${\widetilde{j}}$. The commutation of the front face is given by the definition of ${\mathrm{ls}}$ in . The bottom face commutes by induction. Since ${\overline{j}}$ is injective, the top face commutes.
For $B=B^s \otimes B'$ with $s\ge2$ the proof is similar to the previous case.
This concludes the proof of part \[pt:\*\].
For the proof of part \[pt:R\], let $B=B_k\otimes B_{k-1}\otimes \cdots \otimes B_1$. We may assume that $R=R_j$ is the R-matrix being applied at tensor positions $j$ and $j+1$ (from the right). By induction we may assume that $j=k-1$, that is, $R$ acts at the leftmost two tensor positions. By part \[pt:\*\] and Proposition \[pp:R\*\] we may assume that $j=1$. Again by induction we may assume that $k=2$. Let $B=B^t \otimes B^s$ (of type $D_n^{(1)}$). By Lemma \[lem:D2P\] $B$ is multiplicity-free as a $U_q(D_n)$-crystal. Since $R$ preserves weights it follows that $R(v_{p,q}^{t,s})=v_{p,q}^{s,t}$. A direct computation shows that ${\overline{\phi}}(v_{p,q}^{t,s})={\overline{\phi}}(v_{p,q}^{s,t})$.
$X=M$ for types $A_n^{(1)}$ and $D_n^{(1)}$
-------------------------------------------
In this subsection we will show that $X_{B,{\lambda}}=M_{L,{\lambda}}$ for $B\in{{\mathcal{C}}^A}$ for type $A_n^{(1)}$ and $B\in{\mathcal{C}}$ for type $D_n^{(1)}$. By Proposition \[pp:phi\] there is a bijection between the sets $P(B,{\lambda})$ and ${\mathrm{RC}}(L,{\lambda})$. Hence it remains to show that the statistics is preserved.
\[th:statistics\] Let $B\in{{\mathcal{C}}^A}$ be a crystal of type $A_n^{(1)}$ or $B\in{\mathcal{C}}$ a crystal of type $D_n^{(1)}$ and ${\lambda}$ a dominant integral weight. The coquantum number bijection ${\widetilde{\phi}}$ preserves the statistics, that is $D_B(b)={cc}({\widetilde{\phi}}(b))$ for all $b\in P(B,{\lambda})$.
For type $A_n^{(1)}$ the theorem follows from [@KSS:2002 Theorem 9.1]. Hence assume that $B\in{\mathcal{C}}$ of type $D_n^{(1)}$. By Theorem \[th:bijops\] part \[pt:rs\] and equations and the maps ${\mathrm{rs}}$ and ${\overline{j}}$ correspond under ${\widetilde{\phi}}$. By Theorem \[th:splitrow\] we have $D({\mathrm{rs}}(b))=D(b)$. Similarly, it follows immediately from the definition of ${\overline{j}}$ in Proposition \[pp:ljdef\] that ${cc}({\overline{j}}(\nu,J))={cc}(\nu,J)$. The maps $R$ and the identity also correspond under ${\widetilde{\phi}}$ by Theorem \[th:bijops\] part \[pt:R\], and neither of them changes the statistics.
There exists a sequence $\mathcal{S}_P$ of maps ${\mathrm{rs}}$ and $R$ which transforms a path $b\in P(B,{\lambda})$ into a path of single boxes. By Theorem \[th:bijops\] there exists a corresponding sequence $\mathcal{S}_{{\mathrm{RC}}}$ of maps ${\overline{j}}$ and the identity. Since neither of these maps changes the statistics it follows that $$D(\mathcal{S}_P(b))={cc}(\mathcal{S}_{{\mathrm{RC}}}({\widetilde{\phi}}(b))) \qquad \text{implies that}
\qquad D(b)={cc}({\widetilde{\phi}}(b)).$$ The theorem for the case $B=(B^{1,1})^{\otimes N}$ has already been proven in [@OSS:2002a].
\[cor:X=M AD\] For $B\in{{\mathcal{C}}^A}$ of type $A_n^{(1)}$ or $B\in{\mathcal{C}}$ of type $D_n^{(1)}$, $L$ the corresponding multiplicity array and ${\lambda}$ a dominant integral, we have $$X_{B,{\lambda}}(q)=M_{L,{\lambda}}(q).$$
This follows from Theorem \[th:statistics\], and .
Type $A_n^{(1)}$ dual bijection
===============================
For this section we assume type $A_n^{(1)}$. We define and study the properties of a dual analogue ${{\overline{\delta}}^\vee}$ of the ${\overline{\delta}}$ map that corresponds to removing a tensor factor $B^{1\vee}$ from the left. This is used to prove a duality symmetry (Theorem \[th:dualpath\]) for the path-RC bijection in type $A_n^{(1)}$. This in turn is useful for establishing the virtual bijections in section \[sec:virtualbij\].
Let ${{\mathcal{C}}^{A\vee}}\subset {{\mathcal{C}}^A}$ be the category of tensor products of crystals of the form $B^{1,s}$ and $B^{1,s\vee}$.
One goal of this section is to give a simpler way to compute ${\overline{\phi}}$ for $B\in {\mathcal{C}}^{A\vee}$. Since ${{\mathcal{C}}^{A\vee}}\subset{{\mathcal{C}}^A}$, Proposition \[pp:phi\] gives the definition of ${\overline{\phi}}$. By $B^{1,s\vee}$ is isomorphic to $B^{n,s}$. The definition of ${\overline{\phi}}$ involves left-splitting $B^{n,s}$, which produces columns $B^{n,1}$, each of which have to be “box split" into boxes $B^{1,1}$ and removed by ${\mathrm{lh}}$.
We introduce a dual analogue ${{\overline{\delta}}^\vee}$ of ${\overline{\delta}}$, which removes an entire column $B^{n,1}$ in a single step whose computation is entirely similar to a single ${\overline{\delta}}$ (rather than $n$ of them).
Using ${{\overline{\delta}}^\vee}$, we can compute ${\overline{\phi}}$ for $B\in {{\mathcal{C}}^{A\vee}}$ using essentially single row techniques.
Dual left hat {#ss:dual left hat}
-------------
Suppose that $B=B^{1\vee} \otimes B'$. In this particular case we write ${{\mathrm{lh}}^\vee}(B)=B'$. By Lemma \[lem:hwvhat\] there is a map ${{\mathrm{lh}}^\vee}:P(B)\rightarrow P({{\mathrm{lh}}^\vee}(B))$ given by removing the left tensor factor. Let ${{{\mathrm{lh}}^\vee}(L)}$ be the multiplicity array of ${{\mathrm{lh}}^\vee}(B)$.
The following algorithm is the same as ${\overline{\delta}}$ except that it starts from large indices instead of small. The map ${{\overline{\delta}}^\vee}:RC(L)\rightarrow
RC({{{\mathrm{lh}}^\vee}(L)})$ is defined as follows. Let $(\nu,J)\in RC(L)$. Initialize $\ell^{(n+1)}=0$ and $\ell^{(0)}=\infty$. For $i$ from $n$ down to $1$, assuming that $\ell^{(i+1)}$ has already been defined, let $\ell^{(i)}$ be the smallest integer such that $(\nu,J)^{(i)}$ has a singular string of length $\ell^{(i)}$ and $\ell^{(i)} \ge
\ell^{(i+1)}$. If no such singular string exists, let $\ell^{(j)}=\infty$ for $1\le j\le i$. Let ${\mathrm{rk}}^\vee(\nu,J)=(i+1)^\vee\in B^{1\vee}$ where $i$ is the maximum index $i$ such that $\ell^{(i)}=\infty$.
For $B=B^{1\vee}\otimes (B^1)^{\otimes 2} \otimes (B^2)^{\otimes 3} \otimes
(B^3)^{\otimes 2}$ of type $A_5^{(1)}$ and ${\lambda}={\overline{\Lambda}}_1+{\overline{\Lambda}}_2+2{\overline{\Lambda}}_3+{\overline{\Lambda}}_4+{\overline{\Lambda}}_6$ the rigged configuration $$(\nu,J)= \quad \yngrc(3,0,2,0,2,0,2,0) \quad \yngrc(2,0,2,0,1,0) \quad
\yngrc(1,0,1,0) \quad \yngrc(1,1) \quad \yngrc(1,0)$$ is in ${\mathrm{RC}}(L,{\lambda})$ with $L$ the multiplicity array corresponding to $B$. The same configuration now written with the vacancy number next to each part is $$\qquad \yngrc(3,1,2,1,2,1,2,1) \quad \yngrc(2,0,2,0,1,0) \quad
\yngrc(1,0,1,0) \quad \yngrc(1,1) \quad \yngrc(1,0)$$ Then $${{\overline{\delta}}^\vee}(\nu,J)=\quad \yngrc(3,0,2,0,2,0,2,0) \quad \yngrc(2,0,2,0) \quad \yngrc(1,0)
\quad {\varnothing}\quad {\varnothing}$$ and ${\mathrm{rk}}^\vee(\nu,J)=2^\vee$.
Given $\mu\in{\lambda^-}$, there is also an inverse of the dual algorithm ${{\overline{\delta}}^\vee}$ associated with the weight $({\lambda}-\mu)^\vee$ similar to the inverse of ${\overline{\delta}}$ as defined in section \[ss:deltas\].
\[pp:dbv\] ${{\overline{\delta}}^\vee}:RC(L)\rightarrow RC({{{\mathrm{lh}}^\vee}(L)})$ is a well-defined injective map such that the diagram commutes: $$\label{eq:dualhatdiag}
\begin{CD}
P(B) @>{{\overline{\phi}}}>> RC(L) \\
@V{{{\mathrm{lh}}^\vee}}VV @VV{{{\overline{\delta}}^\vee}}V \\
P({{\mathrm{lh}}^\vee}(B)) @>>{{\overline{\phi}}}> RC({{{\mathrm{lh}}^\vee}(L)}).
\end{CD}$$ Moreover, if ${\overline{\phi}}(b_1\otimes b)=(\nu,J)$ then ${\overline{\phi}}(b)={{\overline{\delta}}^\vee}(\nu,J)$ and $b_1={\mathrm{rk}}^\vee(\nu,J)$.
The map ${{\mathrm{lh}}^\vee}$ removes $B^{1\vee}\cong B^{n,1}$. This may be achieved by $n$ applications of ${\mathrm{lh}}\circ {\mathrm{lb}}$, which splits a box from a column and then removes it. Let $\Delta$ be the corresponding $n$-fold composition of maps ${\overline{\delta}}\circ {\overline{i}}$. It must be shown that $\Delta(\nu,J)={{\overline{\delta}}^\vee}(\nu,J)$.
Let $a^\vee=b_1$. The letters $1\le a_1<a_2<\cdots<a_n\le n+1$ in $b_1$ (where $b_1$ is viewed as an element of $B^{n,1}$) satisfy $$a_i=\begin{cases} i & \text{for $1\le i<a$}\\
i+1 & \text{for $a\le i\le n$.} \end{cases}$$ It is clear from Proposition \[pp:rcbs\] and the algorithm for ${\overline{\delta}}$ of section \[ss:deltas\] that the composition ${\overline{\delta}}\circ
{\overline{i}}$ corresponding to the letter $a_i$ for $a\le i\le n$ in $b_1$ removes a box from one string of length $s^{(i)}$ in the $i$-th rigged partition and leaves all other strings unchanged. It also follows from the algorithms and change of vacancy numbers that $s^{(i)}\ge s^{(i+1)}$ and that $s^{(i)}$ is the length of the smallest singular string in $(\nu,J)^{(i)}$ with this property. For $1\le i<a$ the composition ${\overline{\delta}}\circ {\overline{i}}$ leaves the rigged configuration unchanged with $s^{(i)}=\infty$. It follows by induction that $s^{(i)}=\ell^{(i)}$, where $\ell^{(i)}$ as in the definition of ${{\overline{\delta}}^\vee}$, and hence that $\Delta(\nu,J)={{\overline{\delta}}^\vee}(\nu,J)$.
Dual left split
---------------
We restate left splitting for a special case. Suppose $B=B^{s\vee} \otimes B'$ for $s\ge 2$. Define ${\mathrm{ls}}^\vee:B^{s\vee}\rightarrow {\mathrm{ls}}^\vee(B):=B^{1\vee}\otimes
B^{s-1\vee}$ to be the composite map $$\begin{CD}
B^{s\vee} @>{\sim}>> B^{n,s} @>{{\mathrm{ls}}}>> B^{n,1} \otimes B^{n,s-1}
@>{\sim}>> B^{1\vee} \otimes B^{s-1\vee}.
\end{CD}$$ By Example \[ex:dualArow\] we may write $b\in B^{s\vee}$ as a word of length $s$ in the dual alphabet. Computing ${\mathrm{ls}}^\vee$ using Example \[ex:dualA\], it is seen that ${\mathrm{ls}}(b)=b_2\otimes b_1$ where $b_2$ is the leftmost dual letter in $b$ and $b_1$ is the remaining word of length $s-1$ in the dual alphabet.
Let ${{{\mathrm{ls}}^\vee}(L)}$ be the multiplicity array for ${\mathrm{ls}}^\vee(B)$. Let us denote by ${\overline{j}}^\vee$ the map on RCs which corresponds to ${\mathrm{ls}}^\vee$ under the path-RC bijection ${\overline{\phi}}$. It is the map ${\overline{j}}$ with respect to $B^{n,s}$ and is therefore inclusion (with some changes in vacancy numbers). With these definitions the following diagram commutes by Proposition \[pp:phi\] for $A_n^{(1)}$: $$\label{eq:dualsplitdiag}
\begin{CD}
P(B,{\lambda}) @>{{\overline{\phi}}}>> {\mathrm{RC}}(L,{\lambda}) \\
@V{{\mathrm{ls}}^\vee}VV @VV{{\overline{j}}^\vee}V \\
P({\mathrm{ls}}^\vee(B),{\lambda}) @>>{{\overline{\phi}}}> {\mathrm{RC}}({{{\mathrm{ls}}^\vee}(L)},{\lambda}).
\end{CD}$$
The bijection ${\overline{\phi}}$ for ${{\mathcal{C}}^{A\vee}}$
---------------------------------------------------------------
The results of this section to this point may be summarized as follows.
\[pp:singledualbij\] There is a unique bijection ${\overline{\phi}}:P(B)\rightarrow {\mathrm{RC}}(L)$ satisfying the following properties. It sends the empty path to the empty rigged configuration, and if the leftmost tensor factor in $B$ is:
1. $B^1$: holds.
2. $B^s$ for $s\ge2$: holds.
3. $B^{1\vee}$: holds.
4. $B^{s\vee}$ for $s\ge2$: holds.
Duality on paths and the bijection
----------------------------------
Let $B\in {{\mathcal{C}}^{A\vee}}$.
Let $L$ and $L^\vee$ be the multiplicity arrays for $B$ and $B^\vee$ respectively. Explicitly, $L^{\vee(a)}_i = L^{(n+1-a)}_i$ for $1\le
a \le n$. Given a classical highest weight ${\lambda}$, let ${\lambda}^\vee=-w_0{\lambda}$ be the highest weight of the contragredient dual module to the $A_n$-module highest weight ${\lambda}$. There is a bijection $\vee:{\mathrm{RC}}(L,{\lambda})\rightarrow {\mathrm{RC}}(L^\vee,{\lambda}^\vee)$ given by $(\nu,J)\mapsto (\nu',J')$ where ${\nu'}^{(a)}=\nu^{(n+1-a)}$ and ${J'}^{(a,i)}$ is obtained from $J^{(n+1-a,i)}$ by complementation within the $m_i^{(n+1-a)}(\nu) \times p_i^{(n+1-a)}(\nu)$ rectangle.
\[th:dualpath\] [@OSS:2003a] Let $B\in{{\mathcal{C}}^A}$, $B^\vee$ its contragredient dual, and $L$ and $L^\vee$ their respective multiplicity arrays. The diagram commutes: $$\begin{CD}
P(B) @>{{\overline{\phi}}}>> RC(L) \\
@V{\vee}VV @VV{\vee}V \\
P(B^\vee) @>>{{\overline{\phi}}}> RC(L^\vee).
\end{CD}$$
Virtual bijection {#sec:virtualbij}
=================
In this section we will prove $X=M$ for the category ${\mathcal{C}}$ for the nonsimply-laced algebras. For the simply-laced types $A_n^{(1)}$ and $D_n^{(1)}$ this was proved in Corollary \[cor:X=M AD\]. For the non-simply-laced affine families it suffices to prove the following theorem.
\[th:virtualbij\] For $B\in {\mathcal{C}}$, let $\Psi:B\rightarrow {\widehat{V}}$ be the virtual crystal embedding, $L$ and $\widehat{L}$ the multiplicity arrays for $B$ and ${\widehat{V}}$ respectively. Then the simply-laced bijection ${\overline{\phi}}_{{\widehat{L}}}:P({\widehat{V}})\rightarrow {\mathrm{RC}}({\widehat{L}})$ restricts to a bijection ${{\overline{\phi}}^v}:{P^v}(B)\rightarrow {{\mathrm{RC}}^v}(L)$.
As an immediate corollary we obtain:
\[cor:X=VX=VM=M\] For ${\lambda}\in{\overline{P}}^+$, $B\in{\mathcal{C}}$ and $L$ the corresponding multiplicity array we have $$X_{B,{\lambda}}(q)=VX_{B,{\lambda}}(q)=VM_{L,{\lambda}}(q)=M_{L,{\lambda}}(q).$$
The left and right equalities were proven in Theorem \[th:X=VX\] and Corollary \[cor:M=VM\], respectively. The middle equality follows from Theorems \[th:statistics\] and \[th:virtualbij\].
The remainder of this section is occupied with the proof of Theorem \[th:virtualbij\].
Virtual ${\mathrm{lh}}$ {#sec:virtual lh}
-----------------------
Suppose $B=B_X=B^1_X \otimes B'_X\in {\mathcal{C}}$ with virtual crystal embeddings ${\Psi}:B_X\rightarrow {\widehat{V}}$ and ${\Psi}':B'_X\rightarrow
{\widehat{V}}'$. By abuse of notation we write ${\widehat{{\mathrm{lh}}}}({\widehat{V}})={\widehat{V}}'$. The map ${\widehat{{\mathrm{lh}}}}:{\widehat{V}}\rightarrow{\widehat{V}}'$ is defined by
1. If $Y=A_{2n-1}^{(1)}$ then ${\widehat{{\mathrm{lh}}}}:B^{1\vee}_Y \otimes B^1_Y
\otimes {\widehat{V}}'\rightarrow{\widehat{V}}'$ is defined by ${\widehat{{\mathrm{lh}}}}= {\mathrm{lh}}\circ {{\mathrm{lh}}^\vee}$, which drops the two leftmost factors in ${\widehat{V}}$.
2. If $Y=D_{n+1}^{(1)}$ and $X=B_n^{(1)}$ then ${\widehat{{\mathrm{lh}}}}:B_Y^2 \otimes
{\widehat{V}}'\rightarrow {\widehat{V}}'$ is defined by ${\widehat{{\mathrm{lh}}}}= {\mathrm{lh}}\circ {\mathrm{lh}}\circ {\mathrm{ls}}$. This accomplishes the same thing as deleting the tensor factor $B_Y^2$.
3. If $Y=D_{n+1}^{(1)}$ and $X=A_{2n-1}^{(2)}$ then ${\widehat{{\mathrm{lh}}}}:B_Y^1
\otimes {\widehat{V}}'\rightarrow{\widehat{V}}'$ is defined by ${\widehat{{\mathrm{lh}}}}={\mathrm{lh}}$.
Note that in each case the total effect of the map ${\widehat{{\mathrm{lh}}}}:{\widehat{V}}^1
\otimes {\widehat{V}}'\rightarrow {\widehat{V}}'$ is to drop the tensor factor ${\widehat{V}}^1$. Therefore the following diagram commutes trivially: $$\begin{CD}
B_X^1 \otimes B_X' @>{{\Psi}\otimes{\Psi}}>> {\widehat{V}}^1 \otimes {\widehat{V}}' \\
@V{{\mathrm{lh}}}VV @VV{{\widehat{{\mathrm{lh}}}}}V \\
B_X' @>>{{\Psi}}> {\widehat{V}}'.
\end{CD}$$
Virtual ${\mathrm{ls}}$ {#sec:virtual ls}
-----------------------
Let $s\ge2$. Recall the virtual ${\mathrm{rs}}$ map ${\widehat{{\mathrm{rs}}}}:{\widehat{V}}^s\rightarrow
{\widehat{V}}^{s-1} \otimes {\widehat{V}}^1$ defined in the proof of Proposition \[pp:vrs\]. Define the virtual ${\mathrm{ls}}$ map ${\widehat{{\mathrm{ls}}}}:{\widehat{V}}^s \rightarrow
{\widehat{V}}^1 \otimes {\widehat{V}}^{s-1}$ by $$\label{eq:vlsdef}
{\widehat{{\mathrm{ls}}}}=*\circ {\widehat{{\mathrm{rs}}}}\circ *.$$
\[pp:xvls\] The map ${\widehat{{\mathrm{ls}}}}:{\widehat{V}}^s\rightarrow {\widehat{V}}^1\otimes {\widehat{V}}^{s-1}$ is described explicitly as follows.
1. If $Y=A_{2n-1}^{(1)}$ then ${\widehat{{\mathrm{ls}}}}:B_Y^{s\vee}\otimes B_Y^s\rightarrow
B_Y^{1\vee} \otimes B_Y^1 \otimes B_Y^{s-1\vee}\otimes B_Y^{s-1}$ is the composition $$\begin{split}
&B_Y^{s\vee} \otimes B_Y^s \stackrel{{\mathrm{ls}}_Y^\vee\otimes 1}{\longrightarrow}
B_Y^{1\vee} \otimes B_Y^{s-1\vee} \otimes
B_Y^s\stackrel{R}{\longrightarrow}
B_Y^s \otimes B_Y^{1\vee} \otimes B_Y^{s-1\vee}\\
\stackrel{{\mathrm{ls}}_Y\otimes 1\otimes 1}{\longrightarrow}& B_Y^1 \otimes
B_Y^{s-1} \otimes B_Y^{1\vee} \otimes B_Y^{s-1\vee}
\stackrel{R}{\longrightarrow} B_Y^{1\vee}\otimes B_Y^1 \otimes
B_Y^{s-1\vee} \otimes B_Y^{s-1}.
\end{split}$$
2. If $Y=D_{n+1}^{(1)}$ and $X=B_n^{(1)}$ then ${\widehat{{\mathrm{ls}}}}:B_Y^{2s}\rightarrow B_Y^2\otimes B_Y^{2s-2}$ is the map that splits off the first two symbols, that is, $uv\mapsto u\otimes v$ where $uv\in B_Y^{2s}$, $u\in B_Y^2$, and $v\in B_Y^{2s-2}$.
3. If $Y=D_{n+1}^{(1)}$ and $X=A_{2n-1}^{(2)}$ then define ${\widehat{{\mathrm{ls}}}}={\mathrm{ls}}_Y:B_Y^s\rightarrow B_Y^1\otimes B_Y^{s-1}$.
It is enough to check these on classical highest weight vectors. This is easy because the various crystals are multiplicity-free as classical crystals.
\[rem:vls\] Let $B=B_X=B^s \otimes B'$ and let $\Psi:B\rightarrow {\widehat{V}}$ and $\Psi':B'\rightarrow {\widehat{V}}'$ be the virtual crystal realizations. By abuse of notation we write ${\widehat{{\mathrm{ls}}}}({\widehat{V}})={\widehat{V}}^1 \otimes {\widehat{V}}^{s-1}
\otimes {\widehat{V}}'$. We also use the notation ${\widehat{{\mathrm{ls}}}}$ for the map ${\widehat{{\mathrm{ls}}}}\otimes 1_{{\widehat{V}}'}:{\widehat{V}}^s\otimes {\widehat{V}}'\rightarrow {\widehat{V}}^1\otimes{\widehat{V}}^{s-1}
\otimes {\widehat{V}}'$. It also satisfies .
Virtual ${\overline{\delta}}$ and ${\overline{j}}$
--------------------------------------------------
Given the virtual crystal embedding ${\Psi}:B_X\rightarrow{\widehat{V}}$, let $L$ and ${\widehat{L}}$ be the multiplicity arrays for $B_X$ and ${\widehat{V}}$ respectively. The maps ${\widehat{\delta}}$ and ${\widehat{j}}$ are defined to be the maps on rigged configurations which correspond under ${\overline{\phi}}$ to the maps ${\widehat{{\mathrm{lh}}}}$ and ${\widehat{{\mathrm{ls}}}}$. More precisely, since ${\overline{\phi}}$ is a bijection for type $Y$ there are unique maps ${\widehat{\delta}}$ and ${\widehat{j}}$ defined by the commutation of the diagrams $$\label{eq:vdelta vj}
\begin{CD}
P({\widehat{V}}) @>{{\overline{\phi}}_{{\widehat{L}}}}>> RC({\widehat{L}}) \\
@V{{\widehat{{\mathrm{lh}}}}}VV @VV{{\widehat{\delta}}}V \\
P({\widehat{{\mathrm{lh}}}}({\widehat{V}})) @>>{{\overline{\phi}}_{{\widehat{{\mathrm{lh}}}}({\widehat{L}})}}> RC({\widehat{{\mathrm{lh}}}}({\widehat{L}}))
\end{CD}
\qquad\text{and}\qquad
\begin{CD}
P({\widehat{V}}) @>{{\overline{\phi}}_{{\widehat{L}}}}>> RC({\widehat{L}}) \\
@V{{\widehat{{\mathrm{ls}}}}}VV @VV{{\widehat{j}}}V \\
P({\widehat{{\mathrm{ls}}}}({\widehat{V}})) @>>{{\overline{\phi}}_{{\widehat{{\mathrm{ls}}}}({\widehat{L}})}}> RC({\widehat{{\mathrm{ls}}}}({\widehat{L}}))
\end{CD}$$ where ${\widehat{{\mathrm{lh}}}}({\widehat{L}})$ and ${\widehat{{\mathrm{ls}}}}({\widehat{L}})$ are the multiplicity arrays for ${\widehat{{\mathrm{lh}}}}({\widehat{V}})$ and ${\widehat{{\mathrm{ls}}}}({\widehat{V}})$ respectively.
For ${\lambda}\in{\overline{P}}^+(X)$ let ${\widehat{{\mathrm{rk}}}}:{\mathrm{RC}}({\widehat{L}},{\Psi}({\lambda}))\rightarrow
{\widehat{V}}^1$ be the map which gives the tensor product of the ranks of the sequence of rigged configurations that occur during the computation of ${\widehat{\delta}}$.
\[lem:vdb\] ${\widehat{\delta}}$ maps ${{\mathrm{RC}}^v}(L)$ into ${{\mathrm{RC}}^v}({{\mathrm{lh}}(L)})$ and ${\widehat{{\mathrm{rk}}}}$ maps ${{\mathrm{RC}}^v}(L)$ into ${\mathrm{Im}}({\Psi}:B_X^1\rightarrow {\widehat{V}}^1)$.
The proof proceeds by cases.
### $X=C_n^{(1)}$ and $Y=A_{2n-1}^{(1)}$. {#xc_n1-and-ya_2n-11. .unnumbered}
According to Definition \[def:VRC\] the elements $({\widehat{\nu}},{\widehat{J}})\in{{\mathrm{RC}}^v}(L)$ have the following properties:
1. \[sym\]${\widehat{m}}_i^{(a)}={\widehat{m}}_i^{(2n-a)}$ and ${\widehat{J}}^{(a,i)}={\widehat{J}}^{(2n-a,i)}$;
2. \[m even\]${\widehat{m}}^{(n)}_i=0$ if $i$ is odd;
3. \[J even\] The parts of ${\widehat{J}}^{(n,i)}$ are even.
From and Proposition \[pp:dbv\] it is clear that ${\widehat{\delta}}={\overline{\delta}}\circ {{\overline{\delta}}^\vee}$. It must be shown that ${\widehat{\delta}}({\widehat{\nu}},{\widehat{J}})$ also possesses the three properties -. Let $\ell^{\vee (a)}$ the lengths of the strings selected by ${{\overline{\delta}}^\vee}$ and $\ell^{(a)}$ be the lengths of the strings selected by the subsequent application of ${\overline{\delta}}$. Let ${\mathrm{rk}}^\vee({\widehat{\nu}},{\widehat{J}})=(2n+1-r)^\vee$ for some $1\le r\le 2n$. If $r\le
n$, it is clear from the definitions that $\ell^{(a)}=\ell^{\vee
(2n-a)}$ for $1\le a<r$, so that points - still hold. Here ${\widehat{{\mathrm{rk}}}}({\widehat{\nu}},{\widehat{J}})=(2n+1-r)^\vee \otimes r={\Psi}(r)$. For $r=n+1$, we must have $\ell^{\vee(n+1)}<\ell^{\vee(n)}$ since otherwise by the symmetry $\ell^{\vee(n-1)}=\ell^{\vee(n)}=\ell^{\vee(n+1)}<\infty$ which contradicts the assumption that $r=n+1$. However, this implies that $\ell^{(a)}=\ell^{\vee (2n-a)}$ for $1\le a<n$ and $\ell^{
(n)}=\ell^{\vee(n)}-1$. Since the vacancy numbers are all even - remain valid. One has ${\widehat{{\mathrm{rk}}}}({\widehat{\nu}},{\widehat{J}})=n^\vee \otimes (n+1)={\Psi}(\overline{n})$. Finally let $r>n+1$ and let $r'\le n$ be minimal such that $\ell^{\vee(2n-r')}=\ell^{\vee(n)}$. By symmetry we have $\ell^{\vee(a)}=\ell^{\vee(n)}$ for all $r'\le a\le 2n-r'$. By the algorithms for ${{\overline{\delta}}^\vee}$ and ${\overline{\delta}}$ and properties - for $({\widehat{\nu}},{\widehat{J}})$ it follows that $\ell^{(a)}=\ell^{\vee
(2n-a)}$ for $1\le a<r'$ and $2n-r'<a<r$, and $\ell^{(a)}=\ell^{\vee
\
(2n-a)}-1$ for $r'\le a\le 2n-r'$. Again this implies that properties - hold for ${\widehat{\delta}}({\widehat{\nu}},{\widehat{J}})$. Then ${\widehat{{\mathrm{rk}}}}({\widehat{\nu}},{\widehat{J}})=(2n+1-r)^\vee \otimes r={\Psi}(\overline{2n+1-r})$.
### $X=A_{2n}^{(2)}$ and $Y=A_{2n-1}^{(1)}$. {#xa_2n2-and-ya_2n-11. .unnumbered}
The elements in ${{\mathrm{RC}}^v}(L)$ are characterized by points and . Everything goes through as for the case $X=C_n^{(1)}$ except that, since ${\widehat{\nu}}^{(n)}$ may contain odd parts, it is possible that $\ell^{\vee(n)}=1$. In this case $\ell^{\vee(a)}=1$ for all $1\le
a\le 2n-1$ by point . Then $\ell^{(a)}=\infty$ for all $1\le a\le 2n-1$, so that ${\widehat{\delta}}({\widehat{\nu}},{\widehat{J}})$ again satisfies and . Then ${\widehat{{\mathrm{rk}}}}({\widehat{\nu}},{\widehat{J}})=1^\vee\otimes
1={\Psi}({{\varnothing}})$.
### $X=D_{n+1}^{(2)}$ and $Y=A_{2n-1}^{(1)}$. {#xd_n12-and-ya_2n-11. .unnumbered}
The elements in ${{\mathrm{RC}}^v}(L)$ are characterized by point . The proof goes through as before except that ${\widehat{J}}^{(n,i)}$ could have an odd part. This could only change the computation of ${\overline{\delta}}\circ{{\overline{\delta}}^\vee}$ if such an odd part were selected. Recall that $p_i^{(n)}$ is even for all $i$. Therefore the odd part cannot be selected by ${{\overline{\delta}}^\vee}$. It can only be selected by ${\overline{\delta}}$ if ${\mathrm{rk}}^\vee({\widehat{\nu}},{\widehat{J}})=(n+1)^\vee$ and the odd part has size $p_i^{(n)}-1$ for some $i\ge \ell^{\vee(n+1)}$. By point and the fact that $({\widehat{\nu}},{\widehat{J}})^{(a)}$ is unchanged by ${{\overline{\delta}}^\vee}$ for $1\le a\le n-1$, we have $\ell^{(a)}=\ell^{\vee(2n-a)}$ for $1\le a \le n-1$ and $\ell^{(n)}$ is the odd (now singular) part. Thus after applying ${\overline{\delta}}\circ{{\overline{\delta}}^\vee}$ point still holds. ${\widehat{{\mathrm{rk}}}}({\widehat{\nu}},{\widehat{J}})=(n+1)^\vee \otimes (n+1)={\Psi}(0)$. Note that $\ell^{(n+1)}=\infty$ since ${{\overline{\delta}}^\vee}$ caused the strings in the $(n+1)$-th rigged partition that were longer than $\ell^{\vee(n+1)}$, to become nonsingular.
### $X=A_{2n}^{(2)\dagger}$ and $Y=A_{2n-1}^{(1)}$. {#xa_2n2dagger-and-ya_2n-11. .unnumbered}
The elements in ${{\mathrm{RC}}^v}(L)$ are characterized by and
1. The parts of ${\widehat{J}}^{(n,i)}$ have the same parity as $i$.
Let ${\mathrm{rk}}^\vee({\widehat{\nu}},{\widehat{J}})=(2n+1-r)^\vee$ for some $1\le r\le 2n$. If $r\le n$, we have as for the case $X=C_n^{(1)}$ that $\ell^{(a)}=\ell^{\vee (2n-a)}$ for $1\le a<r$, so that , and (3’) still hold and ${\widehat{{\mathrm{rk}}}}({\widehat{\nu}},{\widehat{J}})=(2n+1-r)^\vee \otimes
r={\Psi}(r)$.
If $r=n+1$, note that $\ell^{\vee(n)}\in 2{\mathbb{Z}}$ since all vacancy numbers $p_i^{(n)}$ are even, so that by (3’) only the riggings for $i$ even can possibly be singular. As in case $C_n^{(1)}$ we must have $\ell^{\vee(n+1)}<\ell^{\vee(n)}$. By symmetry we have $\ell^{(a)}=\ell^{\vee(2n-a)}$ for $1\le a<n$. The application of ${{\overline{\delta}}^\vee}$ changes the vacancy numbers in the $n$-th rigged partition corresponding to the strings of length $i$ for $\ell^{\vee(n+1)}\le
i<\ell^{\vee(n)}$ by $-1$, which makes these vacancy numbers odd. In particular, the rigging of the new string of length $\ell^{\vee(n)}-1$ is odd. In addition, $\ell^{\vee(n+1)}\le
\ell^{(n)}<\ell^{\vee(n)}$ and by (3’) $\ell^{(n)}$ must be odd. By the change in vacancy number after the application of ${\overline{\delta}}$, the new rigging of the string of length $\ell^{(n)}-1$ must be even. Hence and (3’) hold for ${\widehat{\delta}}({\widehat{\nu}},{\widehat{J}})$ and ${\widehat{{\mathrm{rk}}}}({\widehat{\nu}},{\widehat{J}})=(n+1)^\vee \otimes (n+1)={\Psi}(0)$.
If $r>n+1$, let $r'\le n$ be defined as for the case $C_n^{(1)}$. As before $\ell^{\vee(a)}=\ell^{\vee(n)}$ for $r'\le a\le 2n-r'$. If $r'<n$ everything goes through as in case $C_n^{(1)}$. If $r'=n$ (which means that $\ell^{\vee(n+1)}<\ell^{\vee(n)}$), by the same arguments as for $r=n+1$, we have $\ell^{(a)}=\ell^{\vee(2n-a)}$ for $a\neq n$, $\ell^{\vee(n+1)}\le \ell^{(n)}<\ell^{\vee(n)}$ and (3’) holds for the new riggings. Hence properties and (3’) hold for ${\widehat{\delta}}({\widehat{\nu}},{\widehat{J}})$ and ${\widehat{{\mathrm{rk}}}}({\widehat{\nu}},{\widehat{J}})=(2n+1-r)^\vee \otimes
r={\Psi}(\overline{2n+1-r})$.
### $X=B_n^{(1)}$ and $Y=D_{n+1}^{(1)}$. {#xb_n1-and-yd_n11. .unnumbered}
The elements in ${{\mathrm{RC}}^v}(L)$ are characterized by
1. \[sym D\]$m_i^{(n)}=m_i^{(n+1)}$ and $J^{(n,i)}=J^{(n+1,i)}$ for all $i>0$;
2. \[even D\]$\nu^{(a)}$ and $J^{(a,i)}$ have only even parts for $1\le a<n$.
By section \[sec:virtual lh\] and we have ${\widehat{\delta}}={\overline{\delta}}\circ{\overline{\delta}}\circ{\overline{j}}$. Let $\ell^{(a)}$ and ${\bar{\ell}}^{(a)}$ (resp. $s^{(a)}$ and ${\bar{s}}^{(a)}$) be the length of the selected strings for the right (resp. left) ${\overline{\delta}}$. Then it follows from the definition of ${\overline{j}}$, ${\overline{\delta}}$ and point that $s^{(a)}=\ell^{(a)}-1$ for $1\le a<n$. Furthermore from point we obtain that $\ell^{(n)}=\ell^{(n+1)}>s^{(n)}=s^{(n+1)}$, and again by point that ${\bar{s}}^{(a)}={\bar{\ell}}^{(a)}-1$ for $1\le a<n$. This implies that points and hold for ${\widehat{\delta}}({\widehat{\nu}},{\widehat{J}})$. Moreover, let $x={\mathrm{rk}}({\widehat{\nu}},{\widehat{J}})$ and $y={\mathrm{rk}}({\overline{\delta}}({\widehat{\nu}},{\widehat{J}}))$. Note that $x,y\neq n+1,\overline{n+1}$ because of point (1). Also $x=y$ except possibly $x=n$ and $y=\overline{n}$. Then ${\widehat{{\mathrm{rk}}}}({\widehat{\nu}},{\widehat{J}})=xx={\Psi}(x)$ if $x=y$ or ${\widehat{{\mathrm{rk}}}}({\widehat{\nu}},{\widehat{J}})=n\overline{n}={\Psi}(0)$ if $x\neq y$.
### $X=A_{2n-1}^{(2)}$ and $Y=D_{n+1}^{(1)}$. {#xa_2n-12-and-yd_n11. .unnumbered}
The elements in ${{\mathrm{RC}}^v}(L)$ are characterized by point . It is obvious from its definition that ${\widehat{\delta}}={\overline{\delta}}$ preserves this property. Let $x={\mathrm{rk}}({\widehat{\nu}},{\widehat{J}})$. As before $x\neq n+1,\overline{n+1}$ because of point (1). Then ${\widehat{{\mathrm{rk}}}}({\widehat{\nu}},{\widehat{J}})=x={\Psi}(x)$.
Thus we may define the virtual rank map ${{\mathrm{rk}}^v}:{{\mathrm{RC}}^v}(L)\rightarrow
B_X^1$ by ${{\mathrm{rk}}^v}({\widehat{\nu}},{\widehat{J}})=x$ where ${\Psi}(x)={\widehat{{\mathrm{rk}}}}({\widehat{\nu}},{\widehat{J}})$ for all $({\widehat{\nu}},{\widehat{J}})\in{{\mathrm{RC}}^v}(L)$. Then we have:
\[pp:vboxbij\] The map $({\widehat{\delta}},{{\mathrm{rk}}^v}):{{\mathrm{RC}}^v}(L,{\lambda})\rightarrow \bigcup_{\mu\in{\lambda^-}}
{{\mathrm{RC}}^v}({{\mathrm{lh}}(L)},\mu) \times B_X^1$ is injective.
For the proof of Theorem \[th:virtualbij\] we also need the inverse to Lemma \[lem:vdb\] which involves the inverse of ${\widehat{\delta}}$. Let ${\lambda}\in {\overline{P}}^+_X$, $L=(L_1,L_2,\ldots)$ a multiplicity array and ${{\mathrm{lh}}^{-1}(L)}=(L_1+1,L_2,L_3,\ldots)$. Denote by ${{\widetilde{{\mathrm{RC}}}}^v}(L,{\lambda})$ the subset of ${{\mathrm{RC}}^v}(L,{\lambda})\times B^1$ given by $((\nu,J),b)$ such that ${\lambda}+{\mathrm{wt}}(b)\in{\overline{P}}^+$ and if $b=0$ then also ${\lambda}_n>0$. Let $\hat{b}={\Psi}(b)$. By abuse of notation we define $${\widehat{\delta}}^{-1}:{{\widetilde{{\mathrm{RC}}}}^v}(L,{\lambda})\to\bigcup_{\beta\in{\lambda^+}}{\mathrm{RC}}(\widehat{{{\mathrm{lh}}^{-1}(L)}},{\Psi}(\beta)).$$ If $Y=A_{2n-1}^{(1)}$, let $\hat{b}=b_1\otimes b_2$. Then ${\widehat{\delta}}^{-1}((\nu,J),b)={{{\overline{\delta}}^\vee}}^{-1}({\overline{\delta}}^{-1}((\nu,J),b_2),b_1)$, with ${\overline{\delta}}^{-1}$ as defined in section \[ss:deltas\] and ${{{\overline{\delta}}^\vee}}^{-1}$ as defined in section \[ss:dual left hat\]. If $Y=D_{n+1}^{(1)}$ and $X=B_n^{(1)}$, let $\hat{b}=xy$. Then ${\widehat{\delta}}^{-1}((\nu,J),b)={{\overline{\delta}}}^{-1}({\overline{\delta}}^{-1}((\nu,J),y),x)$. Finally for $Y=D_{n+1}^{(1)}$ and $X=A_{2n-1}^{(2)}$, let $\hat{b}=x$. Then ${\widehat{\delta}}^{-1}((\nu,J),b)={\overline{\delta}}^{-1}((\nu,J),x)$.
\[lem:vdb inv\] Given ${\lambda}$, $L$, ${{\mathrm{lh}}^{-1}(L)}$, $b$ and $\hat{b}$ as above the map ${\widehat{\delta}}^{-1}$ maps ${{\widetilde{{\mathrm{RC}}}}^v}(L,{\lambda})$ into $\bigcup_{\beta\in{\lambda^+}}{{\mathrm{RC}}^v}({{\mathrm{lh}}^{-1}(L)},\beta)$.
The proof is very similar to the proof of Lemma \[lem:vdb\].
\[lem:vls\] ${\widehat{j}}$ maps ${{\mathrm{RC}}^v}(L)$ into ${{\mathrm{RC}}^v}({{\mathrm{ls}}(L)})$.
Let $Y=A_{2n-1}^{(1)}$. By and section \[sec:virtual ls\] we have ${\widehat{j}}={\overline{j}}\circ {{\overline{j}}^\vee}$. Both ${\overline{j}}$ and ${{\overline{j}}^\vee}$ are inclusions that do not change the rigged configuration (only certain vacancy numbers). Hence if $({\widehat{\nu}},{\widehat{J}})\in{{\mathrm{RC}}^v}(L)$ has the characterization as stated in the previous lemma, then so does ${\widehat{j}}({\widehat{\nu}},{\widehat{J}})$.
Let $Y=D_{n+1}^{(1)}$. If $X=A_{2n-1}^{(2)}$, we have ${\widehat{j}}={\overline{j}}$. For $X=B_n^{(1)}$, let $B=B^s\otimes B'$ for $s\ge2$ and embeddings $\Psi:B\rightarrow{\widehat{V}}$ and $\Psi':B'\rightarrow{\widehat{V}}'$ with ${\widehat{V}}=B_Y^{2s} \otimes {\widehat{V}}'$. It can be shown (using $*$ and properties of ${\mathrm{rs}}$) that if $x,y\in B_Y^1$ and $u\in B_Y^{2s-2}$ are such that $xyu\in B_Y^{2s}$ then for any $b'\in {\widehat{V}}'$ one has ${\widehat{{\mathrm{ls}}}}(xyu\otimes b')=xy \otimes u \otimes b'$. One may show that the corresponding operation on RCs is inclusion. This may be seen by observing that ${\mathrm{ls}}\circ {\widehat{{\mathrm{ls}}}}:B_Y^{2s}\otimes {\widehat{V}}' \rightarrow
B_Y^1 \otimes B_Y^1\otimes B_Y^{2s-2} \otimes {\widehat{V}}'$, which sends $xyu\otimes b'$ to $x\otimes y\otimes u\otimes b'$, can also be computed by a composition of ${\mathrm{ls}}$ maps and $R$-matrices, whose corresponding maps on RCs are inclusions. This proves that ${\widehat{j}}({\widehat{\nu}},{\widehat{J}})\in{{\mathrm{RC}}^v}({{\mathrm{ls}}(L)})$.
Proof of Theorem \[th:virtualbij\]
----------------------------------
It must be shown that the bijection ${\overline{\phi}}_{{\widehat{L}}}:P({\widehat{V}})\rightarrow {\mathrm{RC}}({\widehat{L}})$ maps ${P^v}(B)$ (1) into and (2) onto ${{\mathrm{RC}}^v}(L)$, thereby defining a bijection ${{\overline{\phi}}^v}_L:{P^v}(B)\rightarrow{{\mathrm{RC}}^v}(L)$ by restriction. Let $B=B^s
\otimes B'$ with ${\Psi}:B\rightarrow {\widehat{V}}^s\otimes {\widehat{V}}'$.
### The case $s=1$: {#the-case-s1 .unnumbered}
For (1) consider a typical element of ${P^v}(B,{\lambda})$, given by ${\Psi}(b)$ with $b\in P(B,{\lambda})$. Write $b=x\otimes b'$ with $x\in
B_X^1$ and $b'\in P(B',\mu)$. Then ${\Psi}(b')\in{P^v}({\mathrm{lh}}(B),\mu)$. Let $({\widehat{\nu}},{\widehat{J}})={\overline{\phi}}_{{\widehat{L}}}({\Psi}(b))\in{\mathrm{RC}}({\widehat{L}})$. It must be shown that $({\widehat{\nu}},{\widehat{J}})\in {{\mathrm{RC}}^v}(L,{\lambda})$. By and induction one has ${\widehat{\delta}}({\widehat{\nu}},{\widehat{J}})\in {{\mathrm{RC}}^v}({{\mathrm{lh}}(L)},\mu)$ and ${\widehat{{\mathrm{rk}}}}({\widehat{\nu}},{\widehat{J}})={\Psi}(x)$. By Lemma \[lem:vdb inv\] we can conclude that $({\widehat{\nu}},{\widehat{J}})\in{{\mathrm{RC}}^v}(L,{\lambda})$.
For (2) let $({\widehat{\nu}},{\widehat{J}})\in{{\mathrm{RC}}^v}(L)$. Let ${\widehat{b}}={\widehat{x}}\otimes {{\widehat{b}}'}\in
P({\widehat{V}})$ (with ${\widehat{x}}\in{\widehat{V}}^1$ and ${{\widehat{b}}'}\in {\widehat{V}}'$) be such that ${\overline{\phi}}_{{\widehat{L}}}({\widehat{b}})=({\widehat{\nu}},{\widehat{J}})$. It must be shown that ${\widehat{b}}\in
{P^v}(B)$. By we have ${\overline{\phi}}_{{\widehat{{\mathrm{lh}}}}({\widehat{L}})}({\widehat{{\mathrm{lh}}}}({\widehat{b}}))= {\widehat{\delta}}({\overline{\phi}}_{{\widehat{L}}}({\widehat{b}}))=
{\widehat{\delta}}({\widehat{\nu}},{\widehat{J}})\in{{\mathrm{RC}}^v}({\widehat{{\mathrm{lh}}}}({\widehat{L}}))$. By induction ${{\widehat{b}}'}={\widehat{{\mathrm{lh}}}}({\widehat{b}})\in
{P^v}({\mathrm{lh}}(B))$; write ${{\widehat{b}}'}={\Psi}(b')$ for some $b'\in B'$. By Lemma \[lem:vdb\] and , ${\widehat{x}}={\Psi}(x)$ for $x={{\mathrm{rk}}^v}({\widehat{\nu}},{\widehat{J}})$. Let $b=x \otimes b'\in B$. By definition ${\Psi}(b)={\Psi}(x)\otimes {\Psi}(b')={\widehat{x}}\otimes {{\widehat{b}}'}={\widehat{b}}$. Therefore ${\widehat{b}}\in {P^v}(B)$ as desired.
### The case $s\ge2$: {#the-case-sge2 .unnumbered}
For (1), a typical element of ${P^v}(B)$ has the form ${\Psi}(b)$ for $b\in P(B)$. Let ${\overline{\phi}}_{{\widehat{L}}}({\Psi}(b))=({\widehat{\nu}},{\widehat{J}})\in {\mathrm{RC}}({\widehat{L}})$. It must be shown that $({\widehat{\nu}},{\widehat{J}})\in {{\mathrm{RC}}^v}(L)$. Note that ${\widehat{j}}({\widehat{\nu}},{\widehat{J}})={\widehat{j}}({\overline{\phi}}_{{\widehat{L}}}({\Psi}(b)))=
{\overline{\phi}}_{{\widehat{L}}^s}({\widehat{{\mathrm{ls}}}}({\Psi}(b)))\in{{\mathrm{RC}}^v}({{\mathrm{ls}}(L)})$ by and induction. But ${\widehat{j}}({\widehat{\nu}},{\widehat{J}})=({\widehat{\nu}},{\widehat{J}})$ and $({\widehat{\nu}},{\widehat{J}})\in{\mathrm{RC}}({\widehat{L}})$. It follows that $({\widehat{\nu}},{\widehat{J}})\in{{\mathrm{RC}}^v}(L)$.
For (2), let $({\widehat{\nu}},{\widehat{J}})\in{{\mathrm{RC}}^v}(L)$. Let ${\widehat{b}}\in P({\widehat{V}})$ be such that ${\overline{\phi}}_{{\widehat{L}}}({\widehat{b}})=({\widehat{\nu}},{\widehat{J}})$. It must be shown that ${\widehat{b}}\in
{P^v}(B)$. By and induction, ${\widehat{j}}({\widehat{\nu}},{\widehat{J}})={\widehat{j}}({\overline{\phi}}_{{\widehat{L}}}({\widehat{b}}))={\overline{\phi}}_{{\widehat{L}}^s}({\widehat{{\mathrm{ls}}}}({\widehat{b}}))\in
{{\mathrm{RC}}^v}({{\mathrm{ls}}(L)})$. Therefore ${\widehat{{\mathrm{ls}}}}({\widehat{b}})\in {P^v}({\mathrm{ls}}(B))$. We conclude that ${\widehat{b}}\in {P^v}(B)$ by , Proposition \[pp:vrs\] point \[it:vres\], and Proposition \[pp:\*emb\].
This concludes the proof of Theorem \[th:virtualbij\].
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|
---
author:
- Rahul Pandharipande$^1$
date: 11 March 1996
title: 'The Chow Ring of the Non-Linear Grassmannian'
---
$\bold{Summary}$ {#intro}
================
Let ${\Bbb{C}}$ be the ground field of complex numbers. Let $1\leq k \leq r$ be integers. The Grassmannian ${\bold G}({\bold P}^k,{\bold P}^r)$ of projective $k$-planes in ${\bold P}^r$ can be viewed as the moduli space of (unparameterized) regular maps from ${\bold P}^k$ to ${\bold P}^r$ of degree $1$. Let $M_{{\bold P}^k}({\bold P}^r,d)$ be the coarse moduli space of (unparameterized) regular maps $\mu:{\bold P}^k {\rightarrow}{\bold P}^r$ satisfying $\mu^{*}({{\cal{O}}}_{{\bold P}^r}(1)){\stackrel{\sim}{=}}{{\cal{O}}}_{{\bold P}^k}(d)$. Two maps $$\mu:{\bold P}^k {\rightarrow}{\bold P}^r, \ \mu':{\bold P}^k {\rightarrow}{\bold P}^r$$ are equivalent for the moduli problem if there exists an element $\sigma\in \bold{PGL}_{k+1}$ satisfying $\mu' \circ \sigma = \mu$. If $\mu:{\bold P}^k {\rightarrow}{\bold P}^r$ is a non-constant regular map, it is easy to show that $dim(Im(\mu))=k$ and $\mu: {\bold P}^k {\rightarrow}Im(\mu)$ is a [*finite*]{} morphism. The space $M_{{\bold P}^k}({\bold P}^r,d)$ is a natural non-linear generalization of the Grassmannian.
In section (\[construct\]), $M_{{\bold P}^k}({\bold P}^r,d)$ will be constructed via Geometric Invariant Theory. $M_{{\bold P}^k}({\bold P}^r,d)$ is an irreducible, normal, quasi-projective variety with finite quotient singularities. Let $A_i\big(M_{{\bold P}^k}({\bold P}^r,d)\big)
\otimes {\Bbb Q}$ be the Chow group (tensor ${\Bbb Q}$) of $i$-cycles modulo linear equivalence. Since the space $M_{{\bold P}^k}({\bold P}^r,d) $ has finite quotient singularities, the Chow groups $\bigoplus
(A_i\otimes
{\Bbb Q})$ naturally form a graded ring via intersection. Since ${{\Bbb Q}}$-coefficients are required for the intersection theory, all Chow groups considered here will be taken with ${{\Bbb Q}}$-coefficients. Let $Ch(k,r,d)$ denote the Chow ring of $M_{{\bold P}^k}({\bold P}^r,d) $. The ring $Ch(k,r,1)$ is simply the Chow ring of the linear Grassmannian ${\bold G}({\bold P}^k,{\bold P}^r)$. The main result of this paper is a determination of $Ch(k,r,d)$.
\[mainn\] There is a [*canonical*]{} isomorphism of graded rings $$\lambda: Ch(k,r,d) {\rightarrow}Ch(k,r,1).$$
Let ${\overline}{M}_{0,n}({\bold P}^r,d)$ be the coarse moduli space of $n$-pointed Kontsevich stable maps from a genus $0$ curve to ${\bold P}^r$. Let ${M}_{0,n}({\bold P}^r,d) \subset {\overline}{M}_{0,n}({\bold P}^r,d) $ denote the non-empty open set corresponding to $n$-pointed, stable maps from ${\bold P}^1$ to ${\bold P}^r$. The complement of ${M}_{0,n}({\bold P}^r,d)$ in ${\overline}{M}_{0,n}({\bold P}^r,d)$ consists of the stable maps with reducible domains. A foundational treatment of these moduli spaces of pointed stable maps of genus $0$ curves can be found in \[K\], \[KM\], and \[FP\]. The spaces $M_{0,0}({\bold P}^r,d)$ and $M_{{\bold P}^1}({\bold P}^r,d)$ are identical. The following corollary is therefore a special case of Theorem (\[mainn\]).
\[corr\] The Chow ring (with ${{\Bbb Q}}$-coefficients) of $M_{0,0}({\bold P}^r,d)$ is canonically isomorphic to the Chow ring of the Grassmannian ${\bold G}({\bold P}^1, {\bold P}^r)$.
Corollary (\[corr\]) is related by loose analogy to results and conjectures on the Chow ring of $M_g$. C. Faber has studied the subring of the Chow ring of $M_g$ generated by certain geometric classes. Faber has conjectured a presentation of this subring (which may be the entire Chow ring of $M_g$). The conjectured ring looks like the cohomology ring of a compact manifold – for example, it satisfies Poincaré duality. $M_{0,0}({\bold P}^r,d)\subset
{\overline}{M}_{0,0}({\bold P}^r,d)$ is a zero-pointed open cell analogous to $M_g\subset {\overline}{M}_g$. Corollary (\[corr\]), then, is analogous to Faber’s conjectures.
In \[GP\], the Poincaré polynomial of ${\overline}{M}_{0,n}({\bold P}^r,d)$ is computed. The [*virtual*]{} Poincaré polynomial of ${M}_{0,0}({\bold P}^r,d)$ is needed as a preliminary result. It was found the virtual Poincaré polynomial of ${M}_{0,0}({\bold P}^r,d)$ is essentially the Poincaré polynomial of $\bold{G}({\bold P}^1, {\bold P}^r)$. This observation provided the starting point for Theorem (\[mainn\]). Thanks are especially due to E. Getzler for many discussions about the geometry of the space $M_{0,0}({\bold P}^r,d)$. The theory of equivariant Chow groups (\[EG\], \[T\]) plays an essential role in the proof of Theorem (\[mainn\]). The author wishes to thank D. Edidin, W. Graham, and B. Totaro for the long discussions in which this theory was explained. The author has also benefitted from conversations with P. Belorouski, W. Fulton, and H. Tamvakis.
$M_{{\bold P}^k}({\bold P}^r,d)$ {#construct}
================================
A family of degree $d$ maps of ${\bold P}^k$ to ${\bold P}^r$ consists of the data $(\pi:\cal{P}{\rightarrow}S, \ \mu: \cal{P}{\rightarrow}{\bold P}^r)$ where:
1. $S$ is a noetherian scheme of finite type over ${\Bbb{C}}$.
2. $\pi:\cal{P}{\rightarrow}S$ is a flat projective morphism with geometric fibers isomorphic to ${\bold P}^k$.
3. The restriction of $\mu^{*}({{\cal{O}}}_{{\bold P}^r}(1))$ to each geometric fiber of $\pi$ is isomorphic to ${{\cal{O}}}_{{\bold P}^k}(d)$.
Two families of maps over $S$, $$(\pi:\cal{P}{\rightarrow}S, \ \mu), \ (\pi':\cal{P}'{\rightarrow}S, \ \mu' )$$ are isomorphic if there exists an isomorphism of schemes $\sigma: \cal{P} {\rightarrow}\cal{P}'$ such that $$\mu=\mu' \circ \sigma, \ \pi = \pi' \circ \sigma.$$ Let $\cal{M}_{{\bold P}^k}({\bold P}^r,d)$ be the contravariant functor from schemes to sets defined as follows. $\cal{M}_{{\bold P}^k}({\bold P}^r,d) \ (S)$ is the set of isomorphism classes of families over $S$ of degree $d$ maps from ${\bold P}^k$ to ${\bold P}^r$.
A coarse moduli space $M_{{\bold P}^k}({\bold P}^r,d)$ is easily obtained via Geometric Invariant Theory. Care is taken here to exhibit $M_{{\bold P}^k}({\bold P}^r,d)$ as a quotient of a proper $\bold{GL}_{k+1}$-action with finite stabilizers. In section (\[genarg\]), the equivariant Chow groups of this $\bold{GL}_{k+1}$-action will be analyzed.
Let $$U(k,r,d) \subset \bigoplus_{0}^{r}H^0({\bold P}^k, {{\cal{O}}}_{{\bold P}^k}
(d))$$ be the Zariski open locus of basepoint free $(r+1)$-tuples of polynomials. There is a natural $\bold{GL}_{k+1}$-action on $\bigoplus_{0}^{r}H^0({\bold P}^k, {{\cal{O}}}_{{\bold P}^k}
(d))$ obtained from the naturally linearized action of $\bold{GL}_{k+1}$ on ${\bold P}^k$. This $\bold{GL}_{k+1}$-action leaves $U(k,r,d)$ invariant. Note, since every regular map $\mu: {\bold P}^k {\rightarrow}{\bold P}^r$ is finite onto its image, $\bold{GL}_{k+1}$ acts with [*finite stabilizers*]{} on $U(k,r,d)$.
Let $\bold{1}{\stackrel{\sim}{=}}{\Bbb{C}}$ be a $1$ dimensional complex vector space with the trivial $GL_{k+1}$-action. Let $Det$ be the $1$ dimensional determinant representation of $GL_{k+1}$. For convenience, let $Z$ denote $\bigoplus_{0}^r
H^0({\bold P}^k, {{\cal{O}}}_{{\bold P}^k}(d))$. There is a $GL_{k+1}$-equivariant inclusion $$U(k,r,d) \subset {\bold P}\big( Det \otimes\
(\bold{1} \oplus Sym^q( Z) \oplus Z ) \big)$$ obtained by the following equation: $$\label{contort}
\xi \in U(k,r,d) {\rightarrow}[\ 1\otimes\
(1\oplus (\xi\otimes \cdots \otimes \xi) \oplus \xi)\ ].$$ The representations $Det$ and $Sym^q(Z)$ in (\[contort\]) occur to obtain the correct G.I.T. linearization. The final $Z$ factor occurs to insure (\[contort\]) is an inclusion (consider scaling $U(k,r,d)$ by a constant $q^{th}$-root of unity).
\[geintth\] Consider the naturally linearized action of $GL_{k+1}$ on $${\bold P}\big( Det \otimes\
(\bold{1} \oplus Sym^q( Z) \oplus Z ) \big).$$ Then, for $q>k+1$, $$U(k,r,d) \subset
{\bold P}\big( Det \otimes\
(\bold{1} \oplus Sym^q( Z) \oplus Z ) \big)
^{stable}.$$
The Lemma is a consequence of the Numerical Criterion of stability. A development of Geometric Invariant Theory can be found in \[MFK\] and \[N\]. Let $V_{k+1}$ be a $(k+1)$-dimensional ${\Bbb{C}}$-vector space such that ${\bold P}^k= {\bold P}(V_{k+1})$. Let ${\overline}{v}=v_0, \ldots, v_{k}$ be a basis of $V_{k+1}$ with integer weights $w_0, \ldots, w_{k}$ (not all zero). Let $\xi \in U(k,r,d)$ correspond to a basepoint free map determined (in the basis ${\overline}{v}$) by $[f_0, \ldots , f_r]$ where each $f_l$ is an an element of $Sym^d(V_{k+1}^*)$. The diagonal coordinates of $$\xi \in
{\bold P}\big( Det \otimes\
(\bold{1} \oplus Sym^q( Z) \oplus Z ) \big)$$ with respect to the ${\Bbb{C}}^*$-action determined by the weights and basis ${\overline}{v}$ are the following: $$1\otimes 1 \in Det\otimes \bold{1},$$ $$\label{termmm}
1 \otimes (\xi \otimes \cdots \otimes
\xi) \in Det \otimes Sym^q\big(
\bigoplus_{0}^{r} Sym^d(V_{k+1}^*)\big),$$ $$1 \otimes \xi \in Det \otimes \big(\bigoplus_{0}^r
Sym^d(V_{k+1}^*)\big).$$ The weight of $1\otimes 1 \in Det\otimes \bold{1}$ is $\sum_{0}^{k} w_i$. Since the polynomials $\{f_l\}$ do not simultaneously vanish at $[1,0, \ldots, 0]\in {\bold P}^k$, one of the coefficients of $v_0^{*d}$ among the polynomials $\{ f_l\}$ must be non-zero. Similarly non-zero coefficients of $v_1^{*d}, \ldots, v_{k}^{*d}$ can be found among the polynomials $\{f_l\}$. Therefore, the terms $$\label{jayy}
1\otimes (v_j^{*d} \otimes \cdots \otimes v_j^{*d})$$ occur in (\[termmm\]) and have weight $-qd\cdot w_j + \sum_{0}^{k}
w_i$.
There are now two cases. First assume $\sum_{0}^{k} w_i >0$, then $1\otimes 1$ has positive weight. If $\sum_{0}^{k} w_i \leq 0$, there must exist $j$ such that $w_j <0$. Let $w_j$ be the negative weight of greatest absolute value. Hence, for all $i$, if $w_i<0$, then $-w_j +w_i \geq 0$. Finally, since $q>k+1$, $$-qd \cdot w_j + \sum_{0}^{k} w_i >0.$$ The term (\[jayy\]) therefore has positive weight. The Numerical Criterion implies $\xi$ is a stable point for the $\bold{GL}_{k+1}$-action.
As a consequence of Lemma (\[geintth\]), $U(k,r,d)/\bold{GL}_{k+1}$ exists as a quasi-projective variety. Standard arguments show that the space $$M_{{\bold P}^k}({\bold P}^r,d) {\stackrel{\sim}{=}}U(k,r,d)/\bold{GL}_{k+1}$$ has the desired functorial properties. Note: the family of maps $$\label{fammy}
(\pi:\cal{P} {\rightarrow}S, \ \mu:
\cal{P}{\rightarrow}{\bold P}^r)$$ may not be a Zariski locally trivial ${\bold P}^k$-bundle over $S$. A Galois cover construction is required to obtain the canonical algebraic morphism $$\label{cann}
S {\rightarrow}M_{{\bold P}^k}({\bold P}^r, d)$$ induced by the family (\[fammy\]). Alternatively, one can define a map to $M_{{\bold P}^k}({\bold P}^r,d)$ locally in the étale topology on $S$. The morphism (\[cann\]) is then obtained via [*descente*]{}. Since $M_{{\bold P}^1}({\bold P}^r,d)$ and $M_{0,0}({\bold P}^r,d)$ coarsely represent the same functor, these spaces are canonically isomorphic.
Since $U(k,r,d)$ is nonsingular, contained in the stable locus, and the $GL_{k+1}$-action has finite stabilizers, Luna’s Etale Slice Theorem can be applied to conclude $M_{{\bold P}^k}
({\bold P}^r,d)$ has finite quotient singularities (see \[L\]). Luna’s Theorem requires a characteristic zero hypothesis.
Finally, since $U(k,r,d)$ is equivariant and contained in a G.I.T. stable locus, the group action $$\label{proppp}
GL_{k+1} \times U(k,r,d) {\rightarrow}U(k,r,d)$$ is a proper action. This is established in \[MFK\], Corollary (2.5). The properness of the action (\[proppp\]) is needed in section (\[genarg\]).
**[The Homomorphism $\lambda:Ch(k,r,d) {\rightarrow}Ch(k,r,1)$]{}** {#homlam}
===================================================================
Let $\nu: {\bold P}^r {\rightarrow}{\bold P}^r$ be a regular map satisfying $\nu^{*}({{\cal{O}}}_{{\bold P}^r}(1)) {\stackrel{\sim}{=}}{{\cal{O}}}_{{\bold P}^r}(d)$. The map $\nu$ induces a canonical morphism $\tau_\nu: {\bold G}({\bold P}^k, {\bold P}^r){\rightarrow}M_{{\bold P}^k}({\bold P}^r,d)$ by the following considerations. Let $\pi: \cal{P} {\rightarrow}\bold{G}({\bold P}^k, {\bold P}^r)$ be the tautological ${\bold P}^k$-bundle over the Grassmannian. Since $$\label{famm1}
\cal{P} \subset {\bold G}({\bold P}^k,{\bold P}^r) \times
{\bold P}^r,$$ there is a canonical projection $\eta: \cal{P} {\rightarrow}{\bold P}^r$. Let $\mu: \cal{P} {\rightarrow}{\bold P}^r$ be determined by $\mu= \nu \circ \eta$. The family $$\label{famm}
(\pi: \cal{P} {\rightarrow}\bold{G}({\bold P}^k, {\bold P}^r),\
\mu: \cal{P} {\rightarrow}{\bold P}^r)$$ is a family over ${\bold G}({\bold P}^k, {\bold P}^r)$ of degree $d$ maps from ${\bold P}^k$ to ${\bold P}^r$. Since $M_{{\bold P}^k}({\bold P}^r,d)$ is a coarse moduli space, the family (\[famm\]) induces a morphism from the base to moduli: $$\tau_{\nu}:\bold{G}({\bold P}^k, {\bold P}^r) {\rightarrow}M_{{\bold P}^k}({\bold P}^r,d).$$ Let $\tau_{\nu}^*$ be the ring homomorphism induced by pull-back: $$\tau^*_{\nu}: Ch(k,r,d) {\rightarrow}Ch(k,r,1).$$ Since $M_{{\bold P}^k}({\bold P}^r,d)$ has finite quotient singularities, the pull-back map $\tau^*_{\nu}$ is well defined (see \[V\]).
\[homomor\] The homomorphism $\tau_{\nu}^*$ does not depend upon $\nu$ and is a graded ring isomorphism.
Let $\lambda: Ch(k,r,d) {\rightarrow}Ch(k,r,1)$ be the ring isomorphism $\tau_{\nu}^*$ for any regular map $\nu$. Theorem (\[mainn\]) is a consequence of Proposition (\[homomor\]).
The proof of Proposition (\[homomor\]) will be undertaken in several steps. First the independence result will be established in Lemma (\[indy5\]). A surjectivity Lemma will be also be proven in this section. The injectivity of $\tau_{\nu}^*$ will be proven in section (\[genarg\]).
\[indy5\] The homomorphism $\tau_{\nu}^*$ does not depend upon $\nu$.
Let $U(r,r,d) \subset \bigoplus_{0}^{r} H^0({\bold P}^r,
{{\cal{O}}}_{{\bold P}^r}(d))$ be the Zariski open locus of basepoint free $(r+1)$-tuples of polynomials as defined in section (\[construct\]). There is a tautological morphism $$\nu_U: U(r,r,d) \times {\bold P}^r {\rightarrow}{\bold P}^r.$$ The tautological family (\[famm1\]) over the Grassmannian pulls-back to a tautological family $\cal{P}_U$ over $${\bold G}({\bold P}^k, {\bold P}^r) \times U(r,r,d).$$ $\cal{P}_U$ is equipped with a canonical projection $$\eta_U: \cal{P}_U {\rightarrow}U(r,r,d) \times {\bold P}^r.$$ Let $\mu_U= \nu_U \circ \eta_U$. The map $\mu_U$ defines a family of degree $d$ maps from ${\bold P}^k$ to ${\bold P}^r$ over ${\bold G}({\bold P}^k, {\bold P}^r) \times U(r,r,d)$. There is an induced map $$\tau_{U}:{\bold G}({\bold P}^k, {\bold P}^r) \times U(r,r,d)
{\rightarrow}M_{{\bold P}^k}({\bold P}^r,d).$$ The morphism $\tau_{\nu}$ is induced by the composition of the inclusion $$i_{\nu}:{\bold G}({\bold P}^k, {\bold P}^r)
{\rightarrow}{\bold G}({\bold P}^k, {\bold P}^r) \times [\nu]
\subset {\bold G}({\bold P}^k, {\bold P}^r) \times U(r,r,d)$$ with $\tau_{U}$. Hence, $\tau_{\nu}^*= i_{\nu}^* \circ \tau_U^*$ Since $U(r,r,d)$ is an open set in affine space, $i_{\nu}^*= i_{\nu'}$ for any two maps $[\nu], [\nu '] \in U(r,r,d)$.
If $k=r$, then ${\bold G}({\bold P}^r, {\bold P}^r)$ is a point and $\tau_{\nu}^*$ is surjective. Assume $k<r$. Let $1\leq j \leq r-k$. Let $H_j \subset {\bold P}^r$ be a linear subspace of codimension $k+j$. Define an algebraic subvariety $C(H_j)\subset M_{{\bold P}^k}({\bold P}^r,d)$ by the following condition. $C(H_j)$ is the set of maps that meet $H_j$. $C(H_j)$ is easily seen to be an irreducible subvariety of codimension $j$ in $M_{{\bold P}^k}({\bold P}^r,d)$. There is a natural $\bold{GL}_{r+1}$-action on $M_{{\bold P}^k}
({\bold P}^r,d)$ obtained from the symmetries of ${\bold P}^r$. Let $\xi \in \bold{GL}_{r+1}$. Certainly $$\xi( \ C(H_j)\ )= C(\ \xi(H_j)\ ).$$ Since $\bold{GL}_{r+1}$ is a connected rational group, the class $\sigma_j$ of $C(H_j)$ in the Chow ring $Ch(k,r,d)$ is well-defined (independent of $H_j$).
\[abcd\] The pull-back of the class $\sigma_j$ for $1\leq j \leq r-k$ is determined by: $$\tau_{\nu}^*(\sigma_j)= d^{k+j}\cdot \sigma_j.$$
Let $\nu: {\bold P}^r {\rightarrow}{\bold P}^r$ be a fixed morphism satisfying $\nu^*({{\cal{O}}}_{{\bold P}^r}(1)){\stackrel{\sim}{=}}{{\cal{O}}}_{{\bold P}^r}(d)$. Let $H_j\subset {\bold P}^r$ be a general (with respect to $\nu$) linear space. By Bertini’s Theorem, $\nu^{-1}(H_j)$ is a nonsingular complete intersection of $k+j$ hypersurfaces of degree $d$ in ${\bold P}^r$. The set theoretic inverse image $\tau_{\nu}^{-1}( C(H_j))$ is the set of $k$-planes of ${\bold P}^r$ meeting $\nu^{-1}(H_j)$. A simple tangent space argument shows that the scheme theoretic inverse image $\tau_{\nu}^{-1}(C(H_j))$ is generically reduced. Hence, $$\tau_{\nu}^* (\sigma_j) = [\tau_{\nu}^{-1}(C(H_j))]\in Ch(k,r,1).$$ It remains to determine $[\tau_{\nu}^{-1}(C(H_j))]\in Ch(k,r,1).$
Recall $\pi:\cal{P}{\rightarrow}{\bold G}({\bold P}^k,{\bold P}^r)$ is the tautological ${\bold P}^k$-bundle over the Grassmannian. Let $L$ be the Chern class of the line bundle $\eta^*({{\cal{O}}}_{{\bold P}^r}(1))$ on $\cal{P}$. The following equations hold: $$\pi_*(L^{k+j})= \sigma_j,$$ $$\pi_*( (d\cdot L)^{k+j})= [\tau_{\nu}^{-1}(C(H_j))].$$ These equations imply $\tau_{\nu}^*(\sigma_j)=d^{k+j}\cdot
\sigma_j.$
Consider the $d=1$ case, ${\bold G}({\bold P}^k, {\bold P}^r){\stackrel{\sim}{=}}M_{{\bold P}^k}({\bold P}^r,1)$. There is a tautological bundle sequence on ${\bold G}({\bold P}^k, {\bold P}^r)$: $$0 {\rightarrow}S {\rightarrow}{\Bbb{C}}^{r+1} {\rightarrow}Q {\rightarrow}0.$$ $Q$ is a bundle of rank $r-k$. For $1 \leq j \leq r-k$, let $c_j(Q) \in Ch(k,r,1)$ be the $j^{th}$ Chern class of $Q$. It is well known that $$c_j(Q)=\sigma_j.$$ Also, the classes $c_j(Q)\in Ch(k,r,1)$ generate $Ch(k,r,1)$ as ring. These facts can be found, for example, in \[F\]. Therefore, the following Lemma is a consequence of Lemma (\[abcd\]).
The homomorphism $\tau_{\nu}^*: Ch(k,r,d) {\rightarrow}Ch(k,r,1)$ is surjective.
In fact, the subring of $Ch(k,r,d)$ generated by $\sigma_1, \ldots, \sigma_{r-k}$ surjects onto $Ch(r,k,1)$ via $\tau_{\nu}^*$.
**[Generation of $Ch(1,r,d)$]{}** {#genn}
=================================
In order to complete the proof of Proposition (\[homomor\]), results on the generation of $Ch(k,r,d)$ are needed. In this section, a special argument in the $k=1$ case is developed. In sections (\[eqchg\])-(\[genarg\]), a general generation argument using the theory of equivariant Chow groups is established. The general argument also covers the $k=1$ case. The special method for the $k=1$ case involves a natural stratification of $M_{{\bold P}^1}({\bold P}^r,d)$. Unfortunately, this stratification does not easily generalize when $k>1$.
Let $0\leq j\leq r-1$. Let $\sigma_0\in Ch(1,r,d)$ be the unit (the fundamental class). Let $\sigma_{j\neq 0}$ be the class defined in section (\[homlam\]).
\[gen\] The elements $\sigma_i \cdot \sigma_j$ ($0\leq i \leq j \leq r-1$) span a ${{\Bbb Q}}$-basis of $Ch(1,r,d)$.
The proof of Proposition (\[gen\]) uses the 3-pointed moduli space of maps $M_{0,3}({\bold P}^r,d)$. Let $1,2,\infty \in {\bold P}^1$ be three marked points. There is a natural isomorphism: $$\label{udef}
M_{0,3}({\bold P}^r,d){\stackrel{\sim}{=}}{\bold P}(U)=U(1,r,d)/{\Bbb{C}}^*
\subset {\bold P}(\bigoplus_0^r H^0({\bold P}^1,
{{\cal{O}}}_{{\bold P}^1}(d)))$$ where $U(1,r,d)$ is the basepoint free locus (see section (\[construct\])). An element of ${\bold P}(U)$ corresponds to a degree $d$ map from ${\bold P}^1$ to ${\bold P}^r$ with the three markings $1,2,\infty\in {\bold P}^1$. A map $[\mu]\in M_{0,3}({\bold P}^r,d)$ corresponds to a point in ${\bold P}(U)$ by identifying the three markings of $[\mu]$ with the points $1,2,\infty\in {\bold P}^1$. A tangent space argument shows this identification is an isomorphism of schemes (both are non-singular varieties).
The proof of Proposition (\[gen\]) is a refinement of the methods that appear in \[P\]. For $0\leq j \leq r-1$, let $H_j \subset {\bold P}^r$ be a linear space of codimension $1+j$. For $0 \leq a,b \leq r-1$, let $C(H_a,H_b)\subset M_{0,0}
({\bold P}^r,d)$ be the subvariety of maps meeting $H_a$ and $H_b$ (where $H_a$ and $H_b$ are in general position). A simple argument shows the equation $$[C(H_a,H_b)]=\sigma_a \cdot \sigma_b$$ holds in $Ch(1,r,d)$. Note: intersection with the hyperplane $H_0$ imposes no condition on the maps. In particular, $C(H_0, H'_0)= M_{0,0}({\bold P}^r,d)$.
\[maxx\] Let $0\leq a,b \leq r-1$. Assume $(a,b) \neq (r-1,r-1)$. Let $H_a, H_b \subset {\bold P}^r$ be linear spaces of codimension $1+a, 1+b$ in general position. Let $H_{a+1}\subset H_a$, $H_{b+1}\subset H_b$ be linear spaces of codimension $1$. The natural map $$\label{mlem}
C(H_{a+1}, H_b)\
\cup \ C(H_a, H_{b+1})\ \cup \
C(H_0,H_{a}\cap H_{b})
{\rightarrow}C(H_a,H_b)$$ yields a surjection on Chow groups of proper codimension in $C(H_a, H_b)$. If the linear spaces $H_{a+1}$, $H_{b+1}$, or $H_a \cap H_b$ are empty, the corresponding cycle on the left in (\[mlem\]) is taken to be empty. By the assumption $(a,b)\neq (r-1,r-1)$, not all cycles are empty.
Let $F$ be a hyperplane in general position with respect to $H_a$ and $H_b$. Let ${\overline}{N}={\overline}{M}_{0,3}({\bold P}^r,d)$ and ${\overline}{M}={\overline}{M}_{0,0}({\bold P}^r,d)$. Let $N$, $M$ be the unbarred moduli spaces. Let $e_i:{\overline}{N} {\rightarrow}{\bold P}^r$ be the natural evaluation maps for the markings $1\leq i \leq 3$. Let $$X= e_1^{-1}(F) \cap e_2^{-1}(H_a) \cap
e_3^{-1}(H_b).$$ $X$ is closed subvariety of ${\overline}{N}$. The natural forgetful morphism $\rho: X{\rightarrow}{\overline}{M}$ is proper. Also $\rho(X) \cap M=C(H_a,H_b)$. Let $Z\subset C(H_a,H_b)$ be the open set of $\rho(X)$ corresponding to Kontsevich stable maps satisfying the following conditions:
1. The domain curve is ${\bold P}^1$.
2. The map meets $H_a$ and $H_b$.
3. The map does not pass through $F\cap H_a$, $F\cap H_b$, or $H_a \cap H_b$.
Let $[\mu] \in Z$ be a element. By condition (iii), the image $Im(\mu)\subset {\bold P}^r$ can not be contained in $F$, $H_a$, or $H_b$. Moreover, by (iii), $\rho^{-1}(Z) \subset N$. Hence, the map $\rho^{-1}(Z){\rightarrow}Z$ has finite fibers. Since $\rho^{-1}(Z){\rightarrow}Z$ is a proper morphism with finite fibers, it is a finite morphism. Therefore, if $A_i(\rho^{-1}(Z)) \otimes {{\Bbb Q}}=0$, then $A_i(Z) \otimes {{\Bbb Q}}=0$.
The set $\rho^{-1}(Z)\subset N {\stackrel{\sim}{=}}{\bold P}(U)$ (see (\[udef\]) above) is isomorphic to a quasi-projective variety in ${\bold P}(\bigoplus_0^r H^0({\bold P}^1,
{{\cal{O}}}_{{\bold P}^1}(d)))$. The quasi-projective subvariety $$\rho^{-1}(Z)\subset
{\bold P}(\bigoplus_0^r H^0({\bold P}^1,
{{\cal{O}}}_{{\bold P}^1}(d)))$$ can be identified as follows. Let $L_1\subset {\bold P}(U)$ correspond to maps $\mu:{\bold P}^1 {\rightarrow}{\bold P}^r$ satisfying $\mu(1)\in F$. Let $L_2$, $L_{\infty}$ be the linear subspaces in ${\bold P}(U)$ where $\mu(2)\in H_a$, $\mu(\infty)\in H_b$. Let $L_1\cap L_2 \cap L_{\infty}= I\subset {\bold P}(U)$. Let $D\subset I$ be the union of the three hypersurfaces of maps meeting the linear spaces $F\cap H_a$, $F\cap H_b$, and $H_a\cap H_b$ respectively. Since $(a,b) \neq (r-1,r-1)$, $F\cap H_a$ or $F\cap H_b$ is non-empty. Therefore, $D\subset I$ is a subvariety of codimension $1$. Then $$\rho^{-1}(Z)= I \setminus D.$$ $I$ is an open set of a linear subspace of ${\bold P}(\bigoplus_0^r H^0({\bold P}^1,
{{\cal{O}}}_{{\bold P}^1}(d)))$. Since $D$ is of codimension $1$ in $I$, all the Chow groups of $\rho^{-1}(Z)$ of proper codimension vanish. Hence all the Chow groups (tensor ${{\Bbb Q}}$) of $Z$ of proper codimension also vanish.
By definition, $Z\subset C(H_a, H_b)$. The complement of $Z$ in $C(H_a, H_b)$ is the set of maps meeting $F\cap H_a$, $F\cap H_b$, or $H_a\cap H_b$. Therefore, the complement of $Z$ in $C(H_a,H_b)$ is the union of three cycles: $$\label{onion3}
C(F\cap H_a, H_b)\ \cup\ C(H_a,F\cap H_b)\ \cup\
C(H_0,H_a\cap H_b).$$ Since the Chow groups of $Z$ vanish in proper codimension, the Chow groups of the union (\[onion3\]) surject onto the Chow groups of $C(H_a, H_b)$ (in proper codimension).
A vanishing result is also required.
\[four4\] Chow groups in proper codimension of $C(H_{r-1}, H'_{r-1})$ vanish.
Let $F$ be a hyperplane in general position with respect to two distinct points $p=H_{r-1}$ and $q=H'_{r-1}$. The notation $N\subset{\overline}{N}$, $M\subset {\overline}{M}$ of Lemma (\[maxx\]) will be used. Let $$X=e_1^{-1}(F) \cap e_2^{-1}(p)\cap e_3^{-1}(q).$$ Let $\rho: X {\rightarrow}{\overline}{M}$ be the proper forgetful morphism. Again, $\rho(X) \cap M=C(p,q)$. Let $Z\subset C(p,q)$ be the open set of $\rho(X)$ corresponding to Kontsevich stable maps satisfying the following conditions:
1. The domain curve is ${\bold P}^1$.
2. The map meets $p$ and $q$.
Note $F\cap p$, $F\cap q$, and $p\cap q$ are empty. By these conditions on $Z$, the map $\rho^{-1}(Z){\rightarrow}Z$ is finite and proper. Therefore, if $A_i(\rho^{-1}(Z)) \otimes {{\Bbb Q}}=0$, then $A_i(Z) \otimes {{\Bbb Q}}=0$. Also, $\rho^{-1}(Z) \subset N$.
The quasi-projective subvariety $$\rho^{-1}(Z)\subset
{\bold P}(\bigoplus_0^r H^0({\bold P}^1,
{{\cal{O}}}_{{\bold P}^1}(d)))$$ can be identified as follows. Let $L_1\subset {\bold P}(U)$ correspond to maps $\mu:{\bold P}^1 {\rightarrow}{\bold P}^r$ satisfying $\mu(1)\in F$. Let $L_2$, $L_{\infty}$ be the linear subspaces in ${\bold P}(U)$ where $\mu(2)\in p$, $\mu(\infty)\in q$. Let $L_1\cap L_2 \cap L_{\infty}= I\subset {\bold P}(U)$. Then $$\rho^{-1}(Z)= I$$ $I$ is an open set of a linear subspace of ${\bold P}(\bigoplus_0^r H^0({\bold P}^1,
{{\cal{O}}}_{{\bold P}^1}(d)))$. Let ${\overline}{I}$ be the closure of $I$. It will be shown that ${\overline}{I}\setminus I$ has codimension $1$ in ${\overline}{I}$. The Chow groups of $\rho^{-1}(Z)$ of proper codimension therefore vanish. Hence all the Chow groups of $Z$ of proper codimension also vanish.
Let $[A_0, \ldots, A_r]$ be homogeneous coordinates on ${\bold P}^r$. Let $$F=(A_0-A_r),\ p=[1,0, \ldots,0], \ q=[0,\ldots,0,1].$$ Let $[S,T]$ be homogeneous coordinates on ${\bold P}^1$. Let $1,2,\infty\in {\bold P}^1$ be the points $[1,1]$, $[1,0]$, $[0,1]$ respectively. An element $[\mu]\in {\bold P}(\bigoplus_0^r H^0({\bold P}^1,
{{\cal{O}}}_{{\bold P}^1}(d)))$ is given by an $r$-tuple of degree $d$ homogeneous polynomials in $S$ and $T$ : $[f_0, \ldots, f_r]$. The element $[\mu]$ is in $I$ if and only if
1. $f_0,\ldots, f_r$ span a basepoint free linear system on ${\bold P}^1$.
2. $f_0(1,1)=f_r(1,1)$.
3. $T$ divides $f_1, \ldots, f_r$.
4. $S$ divides $f_0, \ldots, f_{r-1}$.
The additional condition $$S \ \ divides\ \ f_r$$ is a codimension $1$ condition contained in the set ${\overline}{I}\setminus I$. Hence ${\overline}{I}\setminus I$ has codimension $1$ in $I$.
Repeated application of Lemma (\[maxx\]) shows the ring $Ch(1,r,d)$ is generated (as a ${{\Bbb Q}}$-vector space) by the classes $[C(H_a, H_b)]$ and the Chow groups of $C(H_{r-1}, H'_{r-1})$. Lemma (\[four4\]) shows the Chow groups of $C(H_{r-1}, H'_{r-1})$ vanish in proper codimension. Hence the classes $[C(H_a,H_b)]=\sigma_a \cdot \sigma_b$ generate $Ch(1,r,d)$.
Via the classical Schubert calculus, the classes $\sigma_a \cdot \sigma_b$ for $0\leq a,b \leq r-1$ are easily seen to span a [*basis*]{} of the Chow ring of the linear Grassmannian $Ch(1,r,1)$. Consider the ring homomorphism $$\tau^*_{\nu}: Ch(1,r,d) {\rightarrow}Ch(1,r,1)$$ defined in section (\[homlam\]). By Lemma (\[abcd\]), $$\tau^*_{\nu}(\sigma_0)=\sigma_0,$$ $$\forall a> 0, \ \ \tau^*_{\nu}(\sigma_a)= d^{1+a}\sigma_a,$$ $$\forall a,b> 0, \ \ \tau^*_{\nu}(\sigma_a
\cdot \sigma_b)= d^{2+a+b}\sigma_a\cdot \sigma_b,$$ Therefore, the elements $\sigma_a \cdot \sigma_b$ for $0\leq a,b \leq r-1$ are independent in $Ch(1,r,d)$. Since generation was established above, Proposition (\[gen\]) is proven. In case $k=1$, the injectivity of $\tau_{\nu}^*$ has been proven.
Equivariant Chow Groups {#eqchg}
=======================
Let $G$ be a group. Let $G\times X {\rightarrow}X$ be a left group action. In topology, the $G$-equivariant cohomology of $X$ is defined as follows. Let $EG$ be a contractable topological space equipped with a free left $G$-action and quotient $EG/G=BG$. Consider the left action of $G$ on $X\times EG$ defined by: $$g(x,b)= (g(x), g(b)).$$ $G$ acts freely on $X\times EG$. Let $X\times_{G} EG$ be the (topological) quotient. The $G$-equivariant cohomology of of $X$, $H_G^*(X)$, is defined by: $$H_G^*(X) = H^*_{sing}(X\times_{G} EG).$$ If $X$ is a a locally trivial principal $G$-bundle, then $X\times_{G} EG$ is a locally trivial fibration of $EG$ over the quotient $X/G$. In this case, $X\times_{G} EG$ is homotopy equivalent to $X/G$ and $$H_G^*(X) = H^*_{sing}(X\times_{G} EG) {\stackrel{\sim}{=}}H^*_{sing}(X/G).$$ For principal bundles, computing the equivariant cohomology ring is equivalent to computing the cohomology of the quotient.
There is an analogous equivariant theory of Chow groups developed by D. Edidin, W. Graham, and B. Totaro in \[EG\], \[T\]. Let $G$ be a linear algebraic group. Let $G\times X {\rightarrow}X$ be a linearized, algebraic $G$-action. The algebraic analogue of $EG$ is attained by approximation. Let $V$ be a ${\Bbb{C}}$-vector space. Let $G\times V {\rightarrow}V$ be an algebraic representation of of $G$. Let $W\subset V$ be a $G$-invariant open set satisfying:
1. The complement of $W$ in $V$ is of codimension greater than q.
2. $G$ acts on $W$ with trivial stabilizers.
3. There exists a geometric quotient $W{\rightarrow}W/G$.
$W$ is an approximation of $EG$ up to codimension $q$. By (iii) and the assumption of linearization, a geometric quotient $X\times _{G} W$ exits as an algebraic variety. Let $d=dim(X)$, $e=dim( X\times _{G} W)$. The equivariant Chow groups are defined by: $$\label{defff}
A^{G}_{d-j}(X)= A_{e-j}(X\times _{G} W)$$ for $0\leq j \leq q.$ An argument is required to check these equivariant Chow groups are well-defined (see \[EG\]). The basic functorial properties of equivariant Chow groups are established in \[EG\]. In particular, if $X$ is nonsingular, there is a natural intersection ring structure on $A_i^{G}(X)$.
Let $Z$ be a variety of dimension $z$. For notational convenience, a superscript will denote the Chow group codimension: $$A^{G}_{z-j}(Z) = A^j_G(Z), \ A_{z-j}(Z)=A^j(Z).$$ In particular, equation (\[defff\]) becomes: $$\forall\ 0\leq j \leq q, \ \
A_{G}^{j}(X)= A^j(X\times _{G} W).$$ The following result of \[EG\] will be used in section (\[genarg\]).
\[dane\] Let ${\Bbb{C}}$ be the ground field of complex numbers. Let $X$ be a quasi-projective variety. Let $G$ be a reductive group. Let $G\times X {\rightarrow}X$ be a linearized, proper, $G$-action with finite stabilizers. Let $X{\rightarrow}X/G$ be a quasi-projective geometric quotient. Then, there are natural isomorphisms for all $j$: $$A^{j}_G(X) \otimes {{\Bbb Q}}{\stackrel{\sim}{=}}A^{j}(X/G) \otimes {{\Bbb Q}}.$$
**[The Chow Ring of the Grassmannian and]{} $A_*^{GL}(pt)$** {#grdwk}
============================================================
Let $0 {\rightarrow}S {\rightarrow}{\Bbb{C}}^{r+1} {\rightarrow}Q {\rightarrow}0$ be the tautological sequence on ${\bold G}({\bold P}^k, {\bold P}^r)$. The following presentation of the Chow ring will be used in section (\[genarg\]). Let $$c_1, \ldots, c_{k+1} \in Ch(k,r,1)$$ be the Chern classes of the rank $k+1$ bundle $S$. These classes generate $Ch(k,r,1)$. Let $$c(S)= 1+ c_1 \ t+ c_2 \ t^2 + \cdots + c_{k+1}
\ t^{k+1},$$ $${1 \over c(S)} = 1+p_1(c_1) \ t+ p_2(c_1,c_2)
\ t^2+ p_3(c_1,c_2,c_3)\ t^3+
\cdots$$ where the latter is the formal inverse in the formal power series ring ${\Bbb{C}}[c_1, \ldots, c_{k+1}][[t]]$. The ideal of relations among $c_1, \ldots, c_{k+1}$ in $Ch(k,r,1)$ is generated by the polynomials $$\{p_j \ | \ j> r-k\}.$$ Geometrically, these relations are obtained from the vanishing of the $j^{th}$ Chern class of the rank $r-k$ bundle $Q$ for $j>r-k$.
In section (\[genarg\]), a basic result on push-forwards is needed.
\[bozo\] Let $\pi:{\bold P}(S) {\rightarrow}{\bold G}({\bold P}^k,{\bold P}^r)$ be the canonical projection. Let ${{\cal{O}}}_{{\bold P}(S)}(1)$ be the canonical line bundle on ${\bold P}(S)$. Then, for $l\geq k$, $$\label{ecc}
\pi_*( \ c_1^l({{\cal{O}}}_{{\bold P}(S)}(1))\ )= p_{l-k} \in Ch(k,r,1).$$
Let $\xi= c_1({{\cal{O}}}_{{\bold P}(S)}(1))$. Certainly, $\pi_*(\xi^k)= 1= p_0$. The equation $$\xi^{k+1} + c_1\ \xi^k+ \cdots +c_k \ \xi+ c_{k+1}=0$$ recursively yields (\[ecc\]).
The equivariant Chow ring $A_*^{\bold{GL}_{k+1}}(pt)$ is computed to motivate the construction in (\[genarg\]). The notation of section (\[construct\]) will be used here. Let $V_{k+1}$ be a fixed $k+1$- dimensional complex vector space such that ${\bold P}(V_{k+1})={\bold P}^k$, Let $U(k,n,1) \subset \bigoplus_{0}^{n}V^*_{k+1}$ be the basepoint free locus. The codimension of the complement of $U(k,n,1)$ is easily found to be $n-k+1$. $\bold{GL}(V_{k+1})$ acts on $U(k,n,1)$ with trivial stabilizers. As determined in section (1), there is a geometric quotient $$U(k,n,1)/\bold{GL}(V_{k+1}) {\stackrel{\sim}{=}}{\bold G}({\bold P}^k,{\bold P}^n).$$ By the definition of the equivariant Chow ring, $$A^{j}_{\bold{GL}_{k+1}} (pt)= A^{j}({\bold G}({\bold P}^k,{\bold P}^n))$$ for $0\leq j \leq n-k$. By the presentation of the Chow ring of ${\bold G}({\bold P}^k,{\bold P}^n)$ given above, the relations among the generators $c_1, \ldots, c_{k+1}$ start in codimension $n-k+1$. Hence, $A^{*}_{\bold{GL}_{k+1}}(pt)$ is freely generated (as a ring) by $c_1, \ldots, c_{k+1}$ where $c_j \in A^{j}_{\bold{GL}_{k+1}}(pt)$.
**[The Generation Argument]{}** {#genarg}
===============================
Again, let $U(k,n,1) \subset \bigoplus_{0}^{n}V^*_{k+1}$ be the basepoint free open set. As $n{\rightarrow}\infty$, $U(k,n,1)$ approximates $E\bold{GL}_{k+1}$. By the definitions, $$A^{j}_{\bold{GL}_{k+1}}(U(k,r,d)) {\stackrel{\sim}{=}}A^{j}\big( U(k,r,d) \times_{\bold{GL}_{k+1}} U(k,n,1)\big)$$ for $0 \leq j \leq n-k$. Recall $$U(k,r,d) \subset \bigoplus_{0}^{r} Sym^d(V_{k+1}^*)$$ is the basepoint free locus. There is a natural $\bold{GL}(V_{k+1})$-equivariant open inclusion, $$U(k,r,d) \times U(k,n,1) \subset \bigoplus_{0}^{r} Sym^d(V_{k+1}^*)
\times U(k,n,1),$$ which yields an open inclusion $$U(k,r,d) \times_{\bold{GL}_{k+1}}
U(k,n,1) \subset \bigoplus_{0}^{r} Sym^d(V_{k+1}^*)
\times_{\bold{GL}_{k+1}} U(k,n,1).$$ Let $0 {\rightarrow}S {\rightarrow}{\Bbb{C}}^{n+1} {\rightarrow}Q {\rightarrow}0$ be the tautological sequence on ${\bold G}({\bold P}^k,{\bold P}^n)$. It is routine to verify $$\bigoplus_{0}^{r} Sym^d(V_{k+1}^*)
\times_{\bold{GL}_{k+1}} U(k,n,1) = \bigoplus_{0}^{r} Sym^d(S^*)$$ where the latter is the total space of the bundle $\bigoplus_{0}^{r} Sym^d(S^*)$ over ${\bold G}({\bold P}^k,{\bold P}^n)$. Let $D$ be the complement of $U(k,r,d) \times_{\bold{GL}_{k+1}}
U(k,n,1)$ in $\bigoplus_{0}^{r}Sym^d(S^*)$.
The Chow ring of $\bigoplus_{0}^{r} Sym^d(S^*)$ is isomorphic to $Ch(k,n,1)$ via pull back. Let $dim$ be the dimension of the variety $ \bigoplus_{0}^{r} Sym^d(S^*)$. Let $$i_D: A_{dim-j}(D) {\rightarrow}A^j(\bigoplus_{0}^{r}
Sym^d(S^*))$$ be the map obtained by the inclusion $D\subset \bigoplus_{0}^{r} Sym^d(S^*)$. There are exact sequences of Chow groups $$A_{dim-j}(D) {\rightarrow}A^{j}(\bigoplus_{0}^{r}Sym^d(S^*)) {\rightarrow}A^{j}\big(U(k,r,d) \times_{\bold{GL}_{k+1}}
U(k,n,1)\big){\rightarrow}0.$$ Let $c_1, \ldots, c_{k+1}$ be the classes of $Ch(k,n,1)$ defined in section (\[grdwk\]).
\[bilt\] $p_j(c_1, \ldots, c_{k+1})
\in Im(i_D) \subset A^{j}(\bigoplus_{0}^{r}Sym^d(S^*))$ for all $j>r-k$.
The proof involves an auxiliary construction. Let $\pi:{\bold P}(S) {\rightarrow}{\bold G}({\bold P}^k,{\bold P}^n)$ be the projective bundle associated to $S$. Let $T= \pi^{*}\big( \bigoplus_{0}^{r}
Sym^d(S^*) \big)$. Denote the total space of the bundle $T$ also by $T$. There is a commutative diagram. $$\begin{CD}
T@>>> {\bold P}(S) \\
@V{\pi}VV @V{\pi}VV \\
\bigoplus_{0}^{r}Sym^d(S^*) @>>> {\bold G}({\bold P}^k,{\bold P}^n) \\
\end{CD}$$ There is a tautological rational evaluation map $$\gamma: T \ - \ - \ {\rightarrow}{\bold P}^r.$$ A point $\tau \in T$ is a triple $$\tau=(v, V\subset {\Bbb{C}}^{n+1}, (f_0, f_1, \ldots, f_r))$$ where $v$ is element of the $k$-dimensional projective space ${\bold P}(V)$ and $f_i \in Sym^d(V^*)$. The rational map $\gamma$ is obtained by $$\gamma (\tau)= [f_0(v), \ldots, f_r(v)].$$ Let ${\overline}{D}$ be the set of elements $\tau\in T$ such that all the $f_i$ vanish at $v$. ${\overline}{D}$ is the locus where $\gamma$ is undefined. The important fact is $$\pi({\overline}{D})= D \subset \bigoplus_{0}^{r}Sym^d(S^*)$$ and $\pi: {\overline}{D} {\rightarrow}D$ is a projective, birational morphism. The Lemma will be proved by finding the class of ${\overline}{D}$ in $A_*(T)$ and pushing forward.
Let $L$ be the line bundle ${{\cal{O}}}_{{\bold P}(S)}(d)$ on ${\bold P}(S)$. Let $L$ also denote the pull-back of ${{\cal{O}}}_{{\bold P}(S)}(d)$ to $T$. The rational map $\gamma$ is obtained from $r+1$ tautological sections of $L$ on $T$. There is a natural equivalence $$H^0({\bold G}({\bold P}^k,
{\bold P}^n), Sym^d(S^*)) {\stackrel{\sim}{=}}H^0({\bold P}(S),L).$$ Also, there is a natural inclusion $$H^0({\bold G}({\bold P}^k,{\bold P}^n),\
Sym^d(S^*) \otimes \bigoplus_{0}^{r} Sym^d(S)\ ) \subset H^0(T,L).$$ Since the bundle $Sym^d(S^*) \otimes Sym^d(S)$ has a canonical identity section, the bundle $Sym^d(S^*)\otimes \bigoplus _{0}^{r} Sym^d(S))$ has $r+1$ canonical sections. It is straightforward to verify these $r+1$ sections $z_0, \ldots, z_r$ of $L$ on $T$ yield the rational map $\gamma$. The base locus ${\overline}{D}$ is the common zero locus of the sections $z_0, \ldots, z_{r}$. In fact, ${\overline}{D}$ is a nonsingular variety of pure codimension $r+1$. Explicit equations show ${\overline}{D}$ is nonsingular complete intersection. Hence $[{\overline}{D}]= c_1(L)^{r+1} \in A_*(T)$.
Certainly $\pi_*(\ c_1(L)^{r+1}\ ) \in Im (i_D)$. Also, for all $\alpha \geq 0$, $$\pi_*(\ c_1(L)^{r+1+\alpha}\ )=
\pi_*([{\overline}{D}] \cap c_1(L)^{\alpha})\in Im(i_D).$$ It remains to compute $\pi_*(c_1(L)^{r+1+\alpha})
\in A_*(\bigoplus_{0}^{r} Sym^d(S^*))$. But since push-forward commutes with flat pull-back, it suffices to consider $c_1(L)^{r+1+\alpha} \in A_*({\bold P}(S))$ and compute $\pi_*(c_1(L)^{r+1+\alpha})\in Ch(k,n,1)$. By Lemma (\[ecc\]), since $c_1(L)= d\cdot c_1({{\cal{O}}}_{{\bold P}(S)}(1))$, $$\pi_*(c_1(L)^{r+1+\alpha})= d^{r+1+\alpha} \cdot
p_{r-k+1+\alpha}.$$ Hence $p_{j} \in Im(i_D)$ for all $j>r-k$.
By Lemma (\[bilt\]) and the presentation of $Ch(k,r,1)$ given in section (\[grdwk\]), the following inequality is obtained: $$\label{donn}
dim_{{{\Bbb Q}}} \ A^{j}_{\bold{GL}_{k+1}}(U(k,r,d)) \leq dim_{{{\Bbb Q}}}\
A^{j}({\bold G}({\bold P}^k,{\bold P}^r)).$$ Recall $\bold{GL}_{k+1} \times U(k,r,d) {\rightarrow}U(k,r,d)$ is a proper group action with finite stabilizers and geometric quotient $M_{{\bold P}^k}({\bold P}^r,d)$. Hence, by Proposition (\[dane\]) and inequality (\[donn\]), $$dim_{{{\Bbb Q}}} \ A^{j}(M_{{\bold P}^k}({\bold P}^r,d) \leq dim_{{{\Bbb Q}}}
\ A^{j}({\bold G}({\bold P}^k,{\bold P}^r)).$$ The surjectivity of $\tau_{\nu}^*:Ch(k,r,d) {\rightarrow}Ch(k,r,1)$ obtained in section (\[homlam\]) implies $$dim_{{{\Bbb Q}}} \ A^{j}(M_{{\bold P}^k}({\bold P}^r,d) \geq dim_{{{\Bbb Q}}}
\ A^{j}({\bold G}({\bold P}^k,{\bold P}^r)).$$ Therefore $\tau_{\nu}^*$ is injective. The proofs of Proposition (\[homomor\]) and Theorem (\[mainn\]) are complete. Since the subring of $Ch(r,k,d)$ generated by the classes $\sigma_1, \ldots, \sigma_{r-k}$ surjects onto $Ch(r,k,1)$ via $\tau_{\nu}^*$, $Ch(r,k,d)$ is generated (as a ring) by the classes $\sigma_1, \ldots, \sigma_{r-k}$.
[\[MFK\]]{} D. Edidin and W. Graham, [*Equivariant Intersection Theory*]{}, in preparation. W. Fulton, [*Intersection Theory*]{}, Springer-Verlag: Berlin, 1984. W. Fulton and R. Pandharipande, [*Notes on Stable Maps and Quantum Cohomology for Convex Spaces*]{}, in preparation. E. Getzler and R. Pandharipande, [*The Poincaré Polynomial of $\ {\overline}{M}_{0,n}({\bold P}^r,d)$*]{}, in preparation. M. Kontsevich, [*Enumeration of Rational Curves Via Torus Actions*]{}, 1994. M. Kontsevich and Y. Manin, [*Gromov-Witten Classes, Quantum Cohomology, and Enumerative Geometry*]{}, 1994. D. Luna, [*Slices Etale*]{}, Bull. Soc. Math. France [**[33]{}**]{} (1973), 81-105. D. Mumford, J. Fogarty, and F. Kirwan, [ *Geometric Invariant Theory*]{}, Springer-Verlag: Berlin, 1994. P. Newstead, [*Introduction to Moduli Problems and Orbit Spaces*]{}, Tata Institute Lecture Notes, Springer-Verlag, Berlin: 1978. R. Pandharipande, [*Intersections of ${{\Bbb Q}}$-Divisors on $M_{0,n}({\bold P}^r,d)$ and Enumerative Geometry*]{}, 1995. B. Totaro, [*The Chow Ring of the Symmetric Group*]{}, 1994. A. Vistoli, [*Intersection Theory on Algebraic Stacks and their Moduli*]{}, Inv. Math. [**97**]{} (1989), 613-670.
Department of Mathematics\
University of Chicago\
5743 S. University Ave\
Chicago, IL 60637\
rahul@@math.uchicago.edu
|
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abstract: 'Recently we have discussed realization of an exact chiral symmetry in theories with self-interacting fermions on the lattice, based upon an auxiliary field method. In this paper we describe construction of the lattice chiral symmetry and discuss its structure in more detail. The antifield formalism is used to make symmetry consideration more transparent. We show that the quantum master equation in the antifield formalism generates all the relevant Ward-Takahashi identities including a Ginsparg-Wilson relation for interacting theories. Solutions of the quantum master equation are obtained in a closed form, but the resulting actions are found to be singular. Canonical transformations are used to obtain four types of regular actions. Two of them may define consistent quantum theories. Their Yukawa couplings are the same as those obtained by using the chiral decomposition in the free field algebra. Inclusion of the complete set of the auxiliary fields is briefly discussed.'
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‘@=11 ‘@=12
c i ł v
Ł ¶
ł
June, 2002
hep-lat/0206006
NIIG-DP-02-5
[**Lattice Chiral Symmetry in Fermionic Interacting Theories and the Antifield Formalism**]{}\
[Yuji Igarashi$^a$, Hiroto So and Naoya Ukita]{}
[*PACS:*]{} 11.15.Ha; 11.30.Rd; 11.30.-j
[*Keywords:*]{} Lattice field theory; Chiral symmetry; Ginsparg-Wilson relation
Introduction
============
Recently much attention has been paid to new realization of symmetries which are not compatible, in ordinary sense, with regularizations. The discovery of the exact chiral symmetry on the lattice [@gw]-[@her] may be regarded as the prototype of such realization. (See, for example, ref.[@reviews] for reviews of recent development.) The need for considering regularization-dependent symmetries is not restricted to lattice theories: A closely related issue in continuum theories is to realize gauge symmetries in the Wilsonian renormalization group (RG) [@wk], which introduces an infrared cutoff to define the Wilsonian effective action.
In the new realization of a symmetry, such as the lattice chiral symmetry or the gauge symmetry along the RG flow, symmetry transformation inevitably depends on cutoff parameter so as to make reconciliation with a regularization. For such a theory, neither an action nor path integral measure may remain invariant under the transformation. One has to determine the action and the symmetry transformation at the same time in such a way that total change arising from the action and the measure vanish. It is therefore difficult in general to give a nonperturbative formulation of the symmetry in the presence of interactions.
For the chiral symmetry on the lattice, the gauge interactions were incorporated into the Dirac operator in vector gauge theories. Recently, chiral gauge theories with anomaly-free fermion multiplets have been constructed in the abelian case [@lu2]. In these theories, the Ginsparg-Wilson (GW) relation [@gw], the crucial algebraic identity in formulation of the symmetry, takes the same form as that in free field theories. The significance and consequences of this identity has been extensively discussed [@neu1]-[@reviews][@lu2]-[@fi]. However, it seems to be not clear how the GW relation should be generalized for describing other interactions.
In a previous work [@isu], referred to as I hereafter, we discussed a lattice chiral symmetry in theories with fermionic self-interactions, introducing a scalar and a pseudoscalar fields as auxiliary fields. A GW relation for these theories was given. It depends on the Yukawa couplings to the auxiliary fields. The Yukawa interactions we obtained take different form from those considered in the literature [@reviews][@Naya][@chand][@in][@fi]. The chiral transformation obtained from the GW relation becomes nonlinear. Under the transformation, none of the kinetic term of the Dirac fields, the Yukawa coupling term and the functional measure remains invariant. Although we gave chiral invariant partition function, these peculiar properties of the chiral symmetry remained to be studied.
The purpose of this paper is twofold: First, we consider construction of the lattice chiral symmetry discussed in I in more detail, using a framework of the antifield formalism originally developed by Batalin and Vilkovisky [@bv]. Second, besides the peculiar properties of the chiral symmetry described above, we argue that the the actions obtained in I are singular. In order to have regular actions, we introduce new dynamical variables, and it enables us to reconstruct the chiral symmetry.
The reason why we use the antifield formalism to describe the lattice chiral symmetry is as follows: The formalism has been recognized as the most general and powerful method of BRS quantization for theories with local as well as global symmetries. For regularization-dependent symmetries, it is a nontrivial problem to derive the Ward-Takahashi (WT) identities, and is certainly desirable to develop a general method for doing this task both in lattice and continuum theories. We believe that the antifield formalism is a strong candidate for the method. Actually, it has been applied to regularization-dependent symmetries realized along the RG flow [@iis1][@iis2]. In the antifield formalism, the presence of exact symmetries, irrespective of whether they are regularization-dependent or not, is expressed by the quantum master equation (QME) [@bv], $\Sigma =0$.
For the fermionic theories we consider, antifields are introduced at the stage of the block transformation [@gw] from a microscopic theory on a fine lattice to its macroscopic counterpart on a coarse lattice. In performing the transformation, we also use the auxiliary field method discussed in I. A scalar and a pseudo-scalar fields are introduced again to make an effective description of the fermionic self-interactions. Introduction of antifields for the Dirac and the auxiliary fields makes it possible to encode the chiral transformation in a natural way. This should be compared with the somehow ad hoc way of postulating the transformation rule in the conventional approach.
Since the fermionic interaction terms are replaced by the Yukawa term and a potential of the auxiliary fields, the fermionic sector in the macroscopic action is linearized. The price for the use of the auxiliary field method is that we only obtain a more weaker condition, vanishing of the expectation value $<\Sigma> =0$, rather than the QME $\Sigma =0$. Although the condition $<\Sigma> =0$ allows a wide class of solutions, we consider here only the solutions to the QME. We show that the QME generates all the relevant WT identities including the GW relation obtained in I. Under suitable assumptions, the QME can be solved in a closed form. The solutions are used to construct four types of chiral invariant partition functions.
The actions constructed with these sets of solutions turn out to have singularities in the Yukawa couplings as well as in the auxiliary field potential. The singularities[^1] in the Yukawa couplings arise at the momentum regions where would-be species doublers appear. In order to remove the singularities, we perform canonical transformations in the space of fields and antifields. Among four sets of the transformed regular actions, two of them contain massless doubler modes which decouple to the auxiliary fields. This decoupling occurs at tree level but could not be stable due to the quantum corrections. In other two sets of the actions, the doubler modes become massive, and decoupling to the auxiliary fields is ensured by the chiral symmetry. In these actions, the kinetic term of the Dirac fields, the Yukawa coupling term and the functional measure are all chiral invariant. The chiral transformation of the new variables takes the same form as the one for the free field theory. The Yukawa couplings coincide with those discussed in [@reviews][@chand][@in][@fi]. They can be obtained by using the chiral decomposition in the free field algebra [@reviews][@Naya][@iis2]. The actions obey the classical master equation rather than the QME.
As for the auxiliary fields, we restrict ourselves mostly to a scalar and a pseudoscalar fields. We may include other auxiliary fields in a similar manner. We try here to give a formal argument of inclusion of the complete set of the fields with multi-flavor, though the constructed actions suffer from the singularities discussed above.
This paper is organized as follows. In section 2, we describe construction of macroscopic action for fermionic system introducing scalar and pseudoscalar auxiliary fields. Then, the block transformation [@gw] is reconsidered in the antifield formalism. In section 3, the QME is derived. Under suitable assumptions, we show that it yields the WT identities discussed in I. Two sets of particular solutions of the QME are given. Using these non-perturbative solutions, we give chiral invariant partition functions on the coarse lattice in section 4. In section 5, reconstruction of the chiral symmetry using canonical transformations is discussed. We perform inclusion of the complete set of the auxiliary fields in section 6. The section 7 is devoted to summary and discussion. Derivation of some formulae is given in Appendix.
Macroscopic action in the antifield formalism
=============================================
In this section, we first construct a macroscopic action without introducing the antifields. Although this was discussed in I, a brief summary of the construction is given to make the present work self-contained. We then reconsider our microscopic as well as macroscopic theories in the antifield formalism, and give a phase-space extension of the block transformation.
Construction of macroscopic action
----------------------------------
Let $A_{\rm c}[\v_{\rm c},\bar{\v}_{\rm c}]$ be a microscopic action of the Dirac fields $\v_{\rm c}(x),\bar{\v}_{\rm c}(x)$. The fields are defined on a $d$ (even) dimensional fine lattice whose positions are labeled by $x$. For simplicity, they are assumed to carry a single flavor. The microscopic action describes a certain class of fermionic self-interactions. Let $A[\V,\bar{\V}]$ be an effective action of the Dirac fields $\V_{n},\bar{\V}_{n}$ defined on a coarse lattice. Indices $n, m$ are used for labeling sites of the lattice. The macroscopic action is obtained from the microscopic action via the block transformation $$\begin{aligned}
e^{-A[\V,\bar{\V}]} & \equiv &
\int{\cal D}\v_{\rm c}{\cal D}\bar{\v}_{\rm c}\
e^{- A_{\rm c}[\v_{\rm c},\bar{\v}_{\rm c}]
- \sum_{n}(\bar{\V}_{n}-\bar{B}_{n})\a(\V_{n}-B_{n})},
\label{2.3}\end{aligned}$$ where $\a$ is a constant parameter proportional to inverse of the coarse lattice spacing $a$, $\a \propto a^{-1}$. The gaussian integral in (\[2.3\]) relates the macroscopic fields $\V_{n},\bar{\V}_{n}$ to the block variables defined by $$\begin{aligned}
\left\{
\begin{array}{ccl}
B_n & \equiv &
\int d^{d}x\ f_n(x) \v_{\rm c}(x) \\
\bar{B}_n & \equiv &
\int d^{d}x\ \bar{\v}_{\rm c}(x) f_{n}^{\ast}(x)
\end{array}
\right. ,
\label{2.4}\end{aligned}$$ where $f_{n}(x)$ is an appropriate function for coarse graining. It is normalized as $\int d^{d}x\ f_{n}^{\ast}(x) f_{m}(x) = \d_{nm}$.
The path-integral over the microscopic fields in (\[2.3\]) will generate fermionic self-interaction terms in the macroscopic action $A[\V,\bar{\V}]$. Instead of dealing with such terms directly, we introduce some auxiliary fields on the coarse lattice to describe the fermionic interactions. The maximal number of the auxiliary fields to be introduced is equal to the dimension of the Clifford algebra, i.e., $2^d$ for the $d$ (even)-dimensional Dirac fields. In section 6, we discuss the inclusion of the complete set of the auxiliary fields. For simplicity, we restrict ourselves here to a scalar $\s_n$ and a pseudoscalar field $\p_n$, because the scalar and pseudoscalar interactions are recognized as the most important couplings to describe chiral symmetry and its spontaneous breaking in the effective theory. The macroscopic action we consider then takes of the form $$\begin{aligned}
A[\V,\bar{\V}] &=& \sum_{nm}\biggl\{\bar{\V}_{n} (D_{0})_{nm} \V_{m} \nonumber
\\
&{}& +
V[\bar{\V}_{n} (\d_{nm} + h(\nabla)_{nm})\V_{m}, \bar{\V}_{n} \gamma_{5}
(\d_{nm} + h(\nabla)_{nm})\V_{m}]\biggr\} ,
\label{2.5}\end{aligned}$$ where $D_{0}$ is the Dirac operator for the kinetic term, and $ V$ denotes fermionic interactions which consist of contact terms as well as non-contact ones with the difference operators $h(\nabla)_{nm}$. We may obtain the action (\[2.5\]) by performing integration over the auxiliary fields in a new macroscopic action: $$\begin{aligned}
e^{ -A[\V,\bar{\V}]} &\equiv&
\int{\cal D}\p{\cal D}\s \nonumber\\
&{}&\times ~
e^{- \sum_{nm}\bar{\V}_{n}\left(D_{0} + \d + h(\nabla)\right)_{nm}
(i\g_5\p + \s )_{m}\V_{m}- A_{X}[\p,\s]}.
\label{2.6}\end{aligned}$$ It is noted that the Dirac fields appear only bilinearly in the new action. All the fermionic interactions are cast into the Yukawa couplings with the auxiliary fields and the potential term $A_{X}[\p,\s]$. In summary, the block transformation is given by $$\begin{aligned}
\lefteqn{\int{\cal D}\p{\cal D}\s \
e^{- \sum_{nm}\bar{\V}_{n}{\tilde D}(\p,\s)_{nm} \V_{m}
- A_{X}[\p,\s]}} & & \nonumber \\
& \hspace{2cm} = &
\int{\cal D}\v_{\rm c}{\cal D}\bar{\v}_{\rm c}\
e^{- A_{\rm c}[\v_{\rm c},\bar{\v}_{\rm c}]
- \sum_{n}(\bar{\V}_{n}-\bar{B}_{n})\a(\V_{n}-B_{n})} ,
\label{2.7}\end{aligned}$$ with the total Dirac operator $$\begin{aligned}
{\tilde D}(\p,\s)_{nm} &=& (D_{0})_{nm} + \left(\d + h(\nabla)\right)_{nm}
(i\g_5\p + \s)_{m} \nonumber\\
&\equiv& D(\p,\s)_{nm} + (i\g_5\p + \s)_{n}.
\label{2.8}\end{aligned}$$ Here ${\tilde D}(\p,\s)$ is assumed to be at most linear in $\pi$ and $\s$. We now reconsider the block transformation (\[2.7\]) in the antifield formalism.
The antifield formalism and the block transformation
----------------------------------------------------
The antifield formalism [@bv] describes any local or global symmetry as a “BRS” symmetry. It defines a kind of “canonical structure” for a given action of fields by adding their “momentum variables” called antifields. In our previous papers [@iis1][@iis2], the formalism has been used for realization of symmetries along the Wilsonian RG flow. The purpose of this subsection is to give a lattice version of the formalism in the context of chiral symmetry.
In the antifield formalism, the chiral transformation in the microscopic theory takes the form of BRS transformation: $$\begin{aligned}
\d_{B} \v_{\rm c}(x) &=& i~C~\g_{5} \v_{\rm c}(x), \nonumber \\
\d_{B} \bar{\v}_{\rm c}(x) &=& i~C~\bar{\v}_{\rm c}(x) \g_{5},
\label{3.1}\end{aligned}$$ where $C$ is a constant ghost. It is Grassmann odd, therefore $C^{2}=0$. For the Dirac fields $\phi^{a}\equiv \{\v_{\rm c}, \bar{\v}_{\rm c}\}$, one introduces anti-Dirac fields $\phi^{*}_{a} \equiv \{\v_{\rm c}^{*},
\bar{\v}_{\rm c}^{*}\}$. Although the antifields are unphysical, they play an important rôle for encoding chiral symmetry, and should be eliminated only at the final stage of our calculation. In order to include the antifields, one considers an extended microscopic action, $$\begin{aligned}
S_{\rm c}[\phi, \phi^{*}] \equiv A_{\rm c}[\v_{\rm c},\bar{\v}_{\rm c}] +
\int d^{d} x [\v_{\rm c}^{*}(x) \d_{B} \v_{\rm c}(x) +
\d_{B} \bar{\v}_{\rm c}(x) \bar{\v}_{\rm c}^{*}(x)].
\label{3.2}\end{aligned}$$ It is noted here that the BRS transformation operator $\d_{B}$ is Grassmann odd and carries one unit of ghost number. Therefore, the antifields $\phi^{*}_{a}$ regarded as source terms for $\d_{B} \phi^{a}$ are Grassmann even and carry ghost number $-1$. The canonical structure in the theory with antifields is specified by the antibracket. For any functions $F[\phi, \phi^{*}]$ and $G[\phi, \phi^{*}]$, it is defined by $$\begin{aligned}
\left(F,~G\right)_{\phi}
&=& \int d^{d} x\biggl[
\frac{{\partial}^{r} F}{\partial \psi_{\rm c}(x)}
\frac{{\partial}^{l} G}{\partial \psi_{\rm c}^{*}(x)}
-\frac{{\partial}^{r} F}{\partial \psi_{\rm c}^{*}(x)}
\frac{{\partial}^{l} G}{\partial \psi_{\rm c}(x)}\nonumber\\
&~~~&+ \frac{{\partial}^{r} F}{\partial \bar{\psi}_{\rm c}(x)}
\frac{{\partial}^{l} G}{\partial \bar{\psi}_{\rm c}^{*}(x)}
-\frac{{\partial}^{r} F}{\partial \bar{\psi}_{\rm c}^{*}(x)}
\frac{{\partial}^{l} G}{\partial \bar{\psi}_{\rm c}(x)}\biggr].
\label{3.3}\end{aligned}$$ The chiral transformation of $F$ is described as $$\begin{aligned}
\d_{B} F = (F, S_{c})_{\phi}.
\label{3.4}\end{aligned}$$ Note that this is an operation from the right. If the original action $A_{\rm c}[\v_{\rm c},\bar{\v}_{\rm c}]$ is chiral invariant $\d_{B} A_{\rm c}=0$, the extended action $S_{\rm c}[\phi,
\phi^{*}]$ is so, too. It is expressed by the classical master equation, $$\begin{aligned}
(S_{c},~ S_{c})_{\phi} =0.
\label{3.5}\end{aligned}$$
For the Dirac fields on the coarse lattice, we introduce their anti-fields $\V^{*}_{n}$ and $ \bar\V^{*}_{n}$. We also include anti-auxiliary fields: $\pi^{*}_{n}$ and ${\s}^{*}_{n}$. Let $\Phi^{A}
\equiv \{\V_{n}, \bar{\V}_{n}, \pi_{n}, \s_{n}\}$ be all the fields on the coarse lattice, and $\Phi^{*}_{A} \equiv \{\V^{*}_{n},
\bar{\V}^{*}_{n}, \pi^{*}_{n}, {\s}^{*}_{n}\}$ be their antifields. Then, a phase-space extension of (\[2.7\]) is given by $$\begin{aligned}
\lefteqn{
\int{\cal D}\pi {\cal D} \s{\cal D}\pi^{*} {\cal D}\s^{*}
\prod_{n}\d (\pi_n^{*})\d (\s_n^{*})~
e^{-S[\Phi, \Phi^{*}]}} {\nonumber}\\
&=&
\int{\cal D}\phi {\cal D}\phi^{*}
\prod_{x}\d
\biggl(\sum_{n}\V_{n}^{*}f_{n}(x)-\v_{c}^{*}(x)\biggr)
\d
\biggl(\sum_{n}\bar{\V}_{n}^{*}f_{n}^{*}(x)-{\bar\v}_{c}^{*}(x)\biggr)
{\nonumber}\\&{}&\times
e^{- S_{\rm c}^{\rm total}[\phi, \phi^{*}]},
\label{3.6}\end{aligned}$$ where the block transformation for the antifield sector is described by using the $\delta$ functions. The total microscopic action in (\[3.6\]), $$\begin{aligned}
S_{\rm c}^{\rm total}[\phi, \phi^{*}] &\equiv&
S_{\rm c}[\phi, \phi^{*}]
+ \sum_{n}(\bar{\V}_{n}-\bar{B}_{n})\a(\V_{n}-B_{n}),
\label{3.7}\end{aligned}$$ has terms linear in the microscopic Dirac fields: $$\begin{aligned}
\int d^{d}x \sum_{n}\bigl(f_{n}^{*}(x)\bar{\v}_{\rm c}(x) \a
({\V} + i~C\g_{5} {\a}^{-1} \bar\V^{*})_{n}
+ f_{n}(x) (\bar{\V}- \V^{*}i~C \g_{5}{\a}^{-1})_{n}
\a {\v}_{\rm c}(x) \bigr).
\nonumber\end{aligned}$$ It implies that the effective source terms for $\bar{\v}_{\rm c}$ and ${\v}_{\rm c}$ are proportional to $({\V} + i~C\g_{5} {\a}^{-1} \bar\V^{*})_{n}$ and $(\bar{\V}- \V^{*}i~C \g_{5}{\a}^{-1})_{n}$, respectively. Thus, we find that the macroscopic action takes of the form $$\begin{aligned}
S[\Phi, \Phi^{*}]& =& \sum_{nm}
(\bar{\V}- \V^{*}i~C \g_{5}{\a}^{-1})_{n}({\tilde D}
(\pi,\s) - \a)_{nm} ({\V} + i~C\g_{5} {\a}^{-1} \bar\V^{*})_{m}\nonumber\\
&{}& + \sum_{n} \bar{\V}_{n} \a {\V}_{n} + A_{X}[\pi,\s] +
\sum_{n}(\pi^{*}_{n} \d_{B} \pi_{n} + \s^{*}_{n} \d_{B} \s_{n}),
\label{3.8}\end{aligned}$$ where the anti-auxiliary fields are included. They are multiplied by the BRS transformed auxiliary fields, $\d_{B} \pi_{n}$ and $\d_{B} \s_{n}$, which are to be determined later. We have used the total Dirac operator ${\tilde D}(\pi,\s)$ given in (\[2.8\]).
It is noted that the chiral transformation for the Dirac fields $\V,~\bar{\V}$ is automatically encoded due to the presence of the anti-Dirac fields $\V^{*},~\bar\V^{*}$ in (\[3.8\]): $$\begin{aligned}
\d_{B} {\V}_{n} &=& \left({\V}_{n},~S[\Phi, \Phi^{*}]\right)_{\Phi}
= i~C\g_{5}\left(1 - {\a}^{-1}{\tilde D}\right)_{nm}{\V}_{m},\nonumber\\
\d_{B} \bar{\V}_{n} &=& \left(\bar{\V}_{n},~S[\Phi, \Phi^{*}]\right)_{\Phi}
= i~C \bar{\V}_{m}\left(1 - {\a}^{-1}{\tilde D}\right)_{mn}\g_{5},
\label{3.9}\end{aligned}$$ where $(~,~)_{\Phi}$ denotes the antibracket for the macroscopic sector. It is given by $$\begin{aligned}
\left(F,~G\right)_{\Phi}
&=& \left(F,~G\right)_{D} + \left(F,~G\right)_{X},\nonumber\\
\left(F,~G\right)_{D} &=& \sum_{n}\biggl[
\frac{{\partial}^{r} F}{\partial \Psi_{n}}
\frac{{\partial}^{l} G}{\partial \Psi_{n}^{*}}
-\frac{{\partial}^{r} F}{\partial \Psi_{n}^{*}}
\frac{{\partial}^{l} G}{\partial \Psi_{n}}\nonumber\\
&~~~&+ \frac{{\partial}^{r} F}{\partial \bar{\Psi}_{n}}
\frac{{\partial}^{l} G}{\partial \bar{\Psi}_{n}^{*}}
-\frac{{\partial}^{r} F}{\partial \bar{\Psi}_{n}^{*}}
\frac{{\partial}^{l} G}{\partial \bar{\Psi}_{n}}\biggr],\nonumber\\
\left(F,~G\right)_{X} &=& \sum_{n}\biggl[
\frac{{\partial}^{r} F}{\partial {\p}_{n}}
\frac{{\partial}^{l} G}{\partial \p_{n}^{*}}
- \frac{{\partial}^{r} F}{\partial \p_{n}^{*}}
\frac{{\partial}^{l} G}{\partial \p_{n}}\nonumber\\
&~~~&+\frac{{\partial}^{r} F}{\partial {\s}_{n}}
\frac{{\partial}^{l} G}{\partial \s_{n}^{*}}
- \frac{{\partial}^{r} F}{\partial \s_{n}^{*}}
\frac{{\partial}^{l} G}{\partial \s_{n}}
\biggr].
\label{3.10}\end{aligned}$$
In the antifield formalism, one can use canonical transformations for the phase-space variables. They are defined as transformations from $\{\Phi^{A},~\Phi^{*}_{A}\}$ to $\{\Phi^{\prime A},~\Phi^{*\prime }_{A}\}$ that render the antibrackets invariant: $(F,~G)_{\Phi}=(F,~G)_{\Phi'}$. One thing which should be remarked is that the path-integral measure ${\cal D}\Phi {\cal D}\Phi^{*}$ is not left invariant in general under the canonical transformations. Since there is no Liouville measure in the phase-space, one has to take account of the associated Jacobian factor in quantum theory.
We can rewrite the fermionic part of the action in (\[3.8\]) by performing a canonical transformation, $$\begin{aligned}
\V_{n}^{\prime} &=&\V_{n} + i~ C \g_{5} \a^{-1} \bar{\V}_{n}^{*},\nonumber\\
\bar{\V}_{n}^{\prime} &=& \bar{\V}_{n} +\V_{n}^{*} i~C \g_{5} \a^{-1},\nonumber\\
\V_{n}^{*\prime} &=& \V_{n}^{*}, \nonumber\\
\bar{\V}_{n}^{*\prime} &=& \bar{\V}_{n}^{*}.
\label{3.17}\end{aligned}$$ It is easy to see that the Jacobian factor associated with this canonical transformation is trivial. Hereafter, we use the new set of variables in construction of the lattice chiral symmetry, and represent it by $\{ \V, \bar{\V}, \V^{*}, \bar{\V}^{*}\}$ removing primes. Then, using these variables, the macroscopic extended action (\[3.8\]) is expressed as $$\begin{aligned}
S[\Phi,~\Phi^{*}]
&=& S_{D} + S_{X},\nonumber\\
S_{D}&=& \sum_{nm}\bar{\V}_{n}{\tilde D}(\pi,\s)_{nm}\V_{m} {\nonumber}\\ &&
+
\sum_{nm}\biggl[
\V_{n}^{*}i~C \g_{5}\bigl(1 -2{\a}^{-1} {\tilde D}\bigr)_{nm}
\V_{m} - \bar{\V}_{n} i~C \g_{5} \d_{nm}\bar{\V}_{m}^{*}
\biggr],\nonumber\\
S_{X} &=& A_{X}[\pi,\s] +
\sum_{n}(\pi^{*}_{n} \d_{B} \pi_{n} + \s^{*}_{n} \d_{B} \s_{n}).
\label{3.18}\end{aligned}$$ It leads to the asymmetric form of the chiral transformation: $$\begin{aligned}
\d_{B} \V_{n} &=& i~C \g_{5}\left(1 -2 {\a}^{-1}{\tilde D}\right)_{nm}
\V_{m},\nonumber\\
\d_{B} \bar{\V}_{n} &=& i~C \bar{\V}_{n} \g_{5}.
\label{3.19}\end{aligned}$$ Here $\bar{\V}$ obeys the standard chiral transformation, while $\V$ does not. Instead, we may consider the chiral transformation $$\begin{aligned}
\d_{B} \V_{n} &=& i~C \g_{5}\V_{n},\nonumber\\
\d_{B} \bar{\V}_{n} &=& i~C \bar{\V}_{m} \left(1 -2 {\a}^{-1}{\tilde D}\right)_{mn}\g_{5},
\label{3.191}\end{aligned}$$ where $\V$ obeys the standard chiral transformation, while $\bar{\V}$ does not. The Dirac action which leads to (\[3.191\]) is given by $$\begin{aligned}
S_{D}&=& \sum_{nm}\bar{\V}_{n}{\tilde D}(\pi,\s)_{nm}\V_{m} {\nonumber}\\ &&
+
\sum_{nm}\biggl[
\V_{n}^{*}i~C \g_{5}\d_{nm}
\V_{m} - \bar{\V}_{n} i~C \left(1 -2 {\a}^{-1}{\tilde D}\right)_{nm} \g_{5}
\bar{\V}_{m}^{*}
\biggr],
\label{3.181}\end{aligned}$$ which can be obtained from (\[3.8\]) via another canonical transformation.
In this section, we have given the block transformation (\[3.6\]) in the antifield formalism. It relates symmetry properties in the microscopic theory to those in the macroscopic theory. After a canonical transformation, the extended action (\[3.18\]) or that with the action for the Dirac fields (\[3.181\]) has been obtained. We consider below the WT identities for this action.
The QME and its solutions
=========================
In the antifield formalism, the basic object which detects the presence of symmetry in a given quantum system is the WT operator. For a path-integral $\int {\cal D} \vp e^{-W[\vp, \vp^{*}]}$ with an action $W[\vp, \vp^{*}]$, the WT operator is defined as $\Sigma[\vp, \vp^{*}] =
(W,~W)_{\vp}/2 - \Delta_{\vp}W = e^{W} \Delta_{\vp} e^{-W}$, where $\Delta$ denotes the “divergence” operator whose explicit expression is given below. The WT operator $\Sigma[\vp, \vp^{*}]$ can be interpreted as follows: Consider a change of variables $\vp \to \vp +
(\vp, W)_{\vp}$. It induces changes in the action by $(W,W)_{\vp}/2$ and those arising from the functional measure by $\Delta_{\vp}W$. Invariance of the path-integral requires cancellation of these two contributions: $\Sigma[\vp, \vp^{*}] = 0$. This is the QME, which ensures the presence of BRS symmetry in the quantum system. In this section, we derive the QME in our macroscopic theory, and then solve it.
The QME in the macroscopic theory
---------------------------------
For the microscopic action, the WT operator reads $$\begin{aligned}
\Sigma [\phi, \phi^{*}] \equiv \frac{1}{2}(S_{c}, S_{c})_{\phi}
- \Delta_{\phi} S_{c},
\label{4.1}\end{aligned}$$ where the $\Delta$-derivative is given by $$\begin{aligned}
\Delta_{\phi}= \int d^{d}x \biggl[\frac{{\partial}^{r} }
{\partial \psi_{c}(x)}
\frac{{\partial}^{r} }{\partial \psi^{*}_{c}(x)}
-\frac{{\partial}^{r} }
{\partial {\bar\psi}_{c}(x)}
\frac{{\partial}^{r} }{\partial {\bar\psi}^{*}_{c}(x)}
\biggr].
\label{4.2}\end{aligned}$$
We now discuss the relation between the WT operator for the microscopic action and that for the macroscopic action. To this end, we consider the functional average of the WT operator in the microscopic theory [@iis1] $$\begin{aligned}
\lefteqn{ \left<\Sigma[\phi,\phi^{*}]\right>_{\phi} } {\nonumber}\\
&=& \biggl[\int{\cal D}\phi {\cal D}\phi^{*}\prod_{x}\d
\bigl(\sum_{n}\V_{n}^{*}f_{n}(x)-\v_{c}^{*}(x)\bigr)
\d
\bigl(\sum_{n}\bar{\V}_{n}^{*}f_{n}^{*}(x)-{\bar\v}_{c}^{*}(x)\bigr)
\nonumber\\
&{}& \times ~e^{- (S_{\rm c}^{\rm total}- S_{c})}\Delta_{\phi}e^{ - S_{c}}
\biggr]
\biggl[\int{\cal D}\phi {\cal D}\phi^{*}\prod_{x}\d
\bigl(\sum_{n}\V_{n}^{*}f_{n}(x)-\v_{c}^{*}(x)\bigr)\nonumber\\
&{}& \times \d
\bigl(\sum_{n}\bar{\V}_{n}^{*}f_{n}^{*}(x)-{\bar\v}_{c}^{*}(x)\bigr)
e^{- S_{\rm c}^{\rm total}}\biggr]^{-1},
\label{4.3}\end{aligned}$$ where the actions $ S_{c}$ and $S_{\rm c}^{\rm total}$ are those defined in (\[3.2\]) and (\[3.7\]). Performing integration by parts in (\[4.3\]) and using (\[3.6\]), one obtains $$\begin{aligned}
\left<\Sigma [\phi, \phi^{*}]\right>_{\phi}
&=&\frac{
\Delta_{D} \int{\cal D}\pi {\cal D} \s {\cal D}\pi^{*} {\cal D}\s^{*}
\prod_{n}\d (\pi_n^{*})\d (\s_n^{*})
e^{-S[\Phi, \Phi^{*}]}}{
\int{\cal D}\pi {\cal D} \s {\cal D}\pi^{*} {\cal D}\s^{*}
\prod_{n}\d (\pi_n^{*})\d (\s_n^{*})
e^{-S[\Phi, \Phi^{*}]}},
\label{4.4}\end{aligned}$$ where the $\Delta_{D}$ is the $\Delta$-derivative for the Dirac-field sector: $$\begin{aligned}
\Delta_{D} &=& \sum_{n}\left(\frac{\partial^r}{\partial \V_{n}}\frac{\partial^r}{\partial \V_{n}^{*}}+\frac{\partial^r}{\partial \bar{\V}_{n}}\frac{\partial^r}{\partial \bar{\V}_{n}^{*}}
\right).
\label{4.5}\end{aligned}$$ In order to include the contributions from the auxiliary fields in the WT operator, we note that there is a trivial identity $$\begin{aligned}
\int{\cal D}\pi {\cal D} \s {\cal D}\pi^{*} {\cal D}\s^{*}
\prod_{n}\d (\pi_n^{*})\d (\s_n^{*})
\Delta_{X} e^{-S[\Phi, \Phi^{*}]}=0,
\label{4.6}\end{aligned}$$ where the $\Delta_{X}$ is the $\Delta$-derivative for the auxiliary-field sector: $$\begin{aligned}
\Delta_{X} &=& -\sum_{n}\left(\frac{\partial^r}{\partial \p_{n}}\frac{\partial^r}{\partial \p_{n}^{*}}+\frac{\partial^r}{\partial \s_{n}}\frac{\partial^r}{\partial \s_{n}^{*}}
\right).
\label{4.7}\end{aligned}$$ Let us define the WT operator for the macroscopic action $S[\Phi,~\Phi^{*}]$ in (\[3.18\]): $$\begin{aligned}
\Sigma[\Phi,~\Phi^{*}] = \frac{1}{2}(S,~S) - (\Delta_{D} + \Delta_
{X})S =\frac{1}{2}(S,~S) - \Delta_{\Phi} S.
\label{4.8}\end{aligned}$$ Adding (\[4.6\]) to (\[4.4\]) and using $\Delta_{\Phi}\equiv
\Delta_{D}+ \Delta_{X}$, one finds that $$\begin{aligned}
\left<\Sigma [\phi, \phi^{*}]\right>_{\phi}
&=& \frac{ \int{\cal D}\pi {\cal D} \s {\cal D}\pi^{*} {\cal D}\s^{*}
\prod_{n}\d (\pi_n^{*})\d (\s_n^{*})\Delta_{\Phi}
e^{-S[\Phi, \Phi^{*}]}}
{ \int{\cal D}\pi {\cal D} \s {\cal D}\pi^{*} {\cal D}\s^{*}
\prod_{n}\d (\pi_n^{*})\d (\s_n^{*})
e^{-S[\Phi, \Phi^{*}]}}\nonumber\\
&=& \frac{ \int{\cal D}\pi {\cal D} \s {\cal D}\pi^{*} {\cal D}\s^{*}
\prod_{n}\d (\pi_n^{*})\d (\s_n^{*})
e^{-S[\Phi, \Phi^{*}]}\Sigma[\Phi, \Phi^{*}]}{
\int{\cal D}\pi {\cal D} \s {\cal D}\pi^{*} {\cal D}\s^{*}
\prod_{n}\d (\pi_n^{*})\d (\s_n^{*})
e^{-S[\Phi, \Phi^{*}]}}\nonumber\\
&\equiv& \left<\Sigma[\Phi, \Phi^{*}]\right>_{X}.
\label{4.8} \end{aligned}$$ This is our fundamental relation between the WT operators in both theories.
For the microscopic theory, we have assumed that the original fermionic action $A_{\rm c}$ is chiral invariant. This leads to the classical master equation (\[3.5\]). Using (\[3.2\]) and (\[4.2\]), one can directly verify that $\Delta_{\phi}S_{c} =0$. Therefore, the microscopic action satisfies the QME, $$\begin{aligned}
\Sigma[\phi,\phi^{*}]=0.
\label{4.9} \end{aligned}$$ Using (\[4.8\]), we obtain $\left<\Sigma [\Phi,
\Phi^{*}]\right>_{X}=0$ for the macroscopic theory. This implies the integral of the WT operator $\Sigma [\Phi, \Phi^{*}]$ over the auxiliary fields gives zero. It is allowed a wide class of solutions for which the WT operator becomes $\pi$ or $\s$ derivative of something. We consider here more restrict class of solutions for which the macroscopic action obeys the QME $$\begin{aligned}
\Sigma [\Phi, \Phi^{*}]&=& \frac{1}{2}
\left(S_{D} + S_{X}, S_{D} + S_{X}\right)_{\Phi}\nonumber\\
&{}& - \left[\Delta_{D} + \Delta_{X}\right]\left[ S_{D} + S_{X}\right] =0.
\label{4.10} \end{aligned}$$
In order to further reduce (\[4.10\]), we assume that $\d_{B} \p_{n}$ and $\d_{B} \s_{n}$ are given by $C$ times functions only of $\p_{n}$ and $\s_n$. Since there appear no fermionic contributions in $\Delta_{\Phi} S$, the quantum master equation can be decomposed into two conditions: $$\begin{aligned}
&{}& \frac{1}{2}\left(S_{D}, ~S_{D}\right)_{D}
+ \left(S_{D}, S_{X}\right)_{X}=0,
\label{4.11}\\
&{}& \frac{1}{2} \left(S_{X},~ S_{X}\right)_{X}
- \left[\Delta_{D}S_{D}+\Delta_{X}S_{X}\right]
=0.
\label{4.12} \end{aligned}$$ For any macroscopic fields $\Phi^{A}$, the BRS transform $\d_{B}
\Phi^{A}$ is proportional to the ghost $C$. It is convenient here to introduce Grassmann even counterpart $\d$ of the odd operator $\d_{B}$. We may define it by $$\begin{aligned}
\d_{B} \Phi^{A} &=& - \d \Phi^{A}~ C,\nonumber\\
\Delta_{X} S_{X} &=& - \d J_{X}~ C,
\label{4.13}\end{aligned}$$ where $\d J_{X}$ is the change in the functional measure ${\cal
D}\p{\cal D}\s$ induced by the chiral transformation $\d \p$ and $\d
\s$. On the other hand, the change in the fermionic functional measure is calculated to be $$\begin{aligned}
\Delta_{D} S_{D} =
2i \sum_n{\rm Tr}\left(\g_5 - \g_5 {\a}^{-1}{\tilde{D}}\right)_{nn}~ C .
\label{4.14}\end{aligned}$$
The above relations (\[4.13\]) and (\[4.14\]) can be used to show that the QME leads to the WT identities given in I. Actually, one finds that (\[4.11\]) and (\[4.12\]) yield $$\begin{aligned}
&{}& \frac{1}{2}\left(S_{D}, ~S_{D}\right)_{D}
+ \left(S_{D}, ~ S_{X}\right)_{X} \nonumber\\
&{}&~~~ = -i \bar{\V}_n\left[
\left\{\g_5 , \tilde{D}\right\} - 2{\a}^{-1}\tilde{D}\g_5 \tilde{D}
- i\ \d \tilde{D}
\right]_{nm}\V_{m}~ C =0,
\label{4.15}\\
&{}& \frac{1}{2} \left(S_{X},~ S_{X}\right)_{X}
- \left[\Delta_{D}~ S_{D}+\Delta_{X}~ S_{X}\right]\nonumber\\
&{}&~~= - \left[\d A_{X} +2i \sum_n{\rm Tr}\left(\g_5 - \g_5 {\a}^{-1}
{\tilde{D}}\right)_{nn} - \d J_{X}\right]~C=0.
\label{4.16}\end{aligned}$$
Having obtained the WT identities from the QME, we are now in a position to solve them. For notational simplicity, we take below $\a =1$, unless otherwise stated.
Solutions to the QME
--------------------
Let us consider first (\[4.15\]) which reduces to $$\begin{aligned}
\left[\left\{\g_{5} , \tilde{D}\right\} - 2\tilde{D}\g_5 \tilde{D}
- i\ \d \tilde{D}
\right]_{nm}=0 ,
\label{5.1}\end{aligned}$$ where $$\begin{aligned}
\tilde{D}_{nm}&=&D_{nm}+\d_{nm}X_{n} {\nonumber}\\
X_{n}&=&(i\g_{5}\p +\s)_{n}.
\label{5.11}\end{aligned}$$ This is the GW relation for our system with auxiliary fields. As discussed in I, it is straightforward to determine $\d X_{n}$ owing to the locality assumption: $$\begin{aligned}
\left\{
\begin{array}{ccl}
\d X_{n}&=& -2i~\g_{5}\left(X_{n}- X_{n}X_{n}\right)\\
\d\p_n & \equiv &
-2\s_n + 2\left(\s_n^2 - \p_n^2\right) \\
\d\s_n & \equiv &
2\p_n - 4\s_n\p_n
\end{array}
\right. .
\label{5.2}\end{aligned}$$ Note that $X$ commutes with $\g_{5}$, and obeys is a nonlinear transformation.[^2] Using this result, the GW relation reduces to $$\begin{aligned}
\biggl[\left\{\g_5 ,D\right\}
- 2D\g_5 D - 2 D\g_5 X
- 2 X\g_5 D
- i\ \d D \biggr]_{nm}= 0 .
\label{5.3} \end{aligned}$$
In order to solve (\[5.3\]), we make an ansatz for the Dirac operator: $$\begin{aligned}
\tilde{D}
& \equiv &
D_{0} + \left(1+{\cal L}(D_{0})\right) X \left(1+{\cal R}(D_{0})\right)
\nonumber \\
& = & D + X ,\nonumber \\
D & = & D_{0} + {\cal L}(D_{0})X +X{\cal R}(D_{0})+{\cal L}(D_{0})X{\cal R}(D_{0}),
\label{5.4}\end{aligned}$$ where the $ D_{0}$ is the Dirac operator in the free theory. It satisfies the original GW relation $$\begin{aligned}
\{\g_5,D_{0}\} = 2 D_{0}\g_{5}D_{0}.
\label{5.5}\end{aligned}$$ Let us suppose that a solution for $D_0$ such as the Neuberger’s type [@neu1] is given, and determine the functions ${\cal L} ( D_{0})$ and ${\cal R}( D_{0})$. One substitutes (\[5.4\]) into (\[5.3\]) using (\[5.2\]) for $\d X$. Then, the resulting expression for l.h.s. of (\[5.3\]) can be expanded in powers of $X$: There appear linear and quadratic terms in $X$. As shown in Appendix, both of these terms vanish if the following conditions are satisfied: $$\begin{aligned}
\left(1-2 D_{0}\right) \g_5 \left(1+{\cal L}\right)
= \left(1+{\cal L}\right) \g_5,
\label{5.60} \\
\left(1+{\cal R}\right) \g_5\left(1-2 D_{0}\right)
= \g_5 \left(1+{\cal R}\right),
\label{5.61} \\
(1+{\cal R})\g_5 (1+{\cal L})\g_5 =\g_5 (1+{\cal L})\g_5 (1+{\cal R})=1 .
\label{5.6}\end{aligned}$$ We find two sets of solutions to these equations given by $$\begin{aligned}
{\cal L}(D_0) &=& - D_0 + \ \frac{1}{1- \g_5{}D_0{}\g_5}
\ \g_5{}D_0{}\g_5{}D_0{} \nonumber \\
&=& -\frac{1-2 \g_5{}D_0{}\g_5}{1- \g_5{}D_0{}\g_5}
{}D_0 , \nonumber\\
{\cal R}(D_0) &=& - D_0 ,
\label{5.7}\end{aligned}$$ or $$\begin{aligned}
{\cal L}(D_0) &=& -D_0 ,
\nonumber \\
{\cal R}(D_0) &=& - D_0 + {}D_0{}\g_5{}D_0{}\g_5 \
\frac{1}{1-\g_5{}D_0{}\g_5} \label{33} \nonumber\\
&=& - D_0 \
\frac{1-2 \g_5{}D_0{}\g_5}{1- \g_5{}D_0{}\g_5}.
\label{5.8} \end{aligned}$$ In Appendix, we show that ${\cal L}$ and ${\cal R}$ in (\[5.7\]) solve (\[5.60\]) $\sim $ (\[5.6\]). In summary, the Dirac operator which solves the GW relation (\[5.1\]) is given by $$\begin{aligned}
\tilde{D} &=&
D_{0} + \g_5\left(
\frac{1}{1- D_0}\right)\g_5
\ X \ \left(1- D_0\right),
\label{5.9}\end{aligned}$$ or $$\begin{aligned}
\tilde{D}
&=&
D_{0} + \left(1-D_0\right)~ X~
\g_5\left(
\frac{1}{1- D_0}\right)\g_5.
\label{5.10}\end{aligned}$$ Let us consider (\[5.9\]) and (\[5.10\]). Since a matrix notation is used, the singularities arising from $D_{0}=1$ cannot be eliminated with the factor $(1-D_{0})$. The momenta satisfying the condition $D_{0} =1$ are those at which the species doublers appear. Therefore, the Yukawa terms suffer from the singularities. This kind of singularities have been discussed in ref.[@fi] in a different context.
Let us postpone discussion on the above singularities, and turn to the other condition (\[4.16\]). It reduces to three equations: $$\begin{aligned}
\d A^{(0)}_{X}[\p,\s] &=& 0 ,
\label{5.11}\\
\d A^{(1)}_{X}[\p,\s] &=&
\d J_{X}[\p,\s] = -8\sum_{n}\p_n \nonumber \\
&=& i 2^{2-{\frac{d}{2}}}\sum_{n}{\rm Tr}\g_5
\left(X_{n}-X^{\dagger}_{n}\right),
\label{5.12} \\
\d A^{(2)}_{X}[\p,\s] &=& - 2i \sum_n{\rm Tr}\left(\g_5 - \g_5 \tilde{D}
\right)_{nn}\nonumber\\
&=&
2i \sum_n{\rm Tr}\left(\g_5(1+L)X(1+R)\right)_{nn}
{\nonumber}\\
&=& 2i \sum_n{\rm Tr}\left(\g_5 X\right)_{nn},
\label{5.13}\end{aligned}$$ where we have used (\[5.6\]) to obtain the last expression of (\[5.13\]). The factor $ 2^{-d/2}$ in (\[5.12\]) is needed to normalize the trace, denoted by ${\rm Tr}$, over the spinor indices. In (\[5.11\]), $A_{X}^{(0)}$ corresponds to an invariant potential. The terms $ A_{X}^{(1)}$ and $ A_{X}^{(2)}$ are counter terms needed to cancel $\Delta_{X} S_{X}$ and $\Delta_{D} S_{D}$. Solutions of the above conditions are given by $$\begin{aligned}
A_{X}^{(0)}[\p,\s] &=& \sum_{n}h\left(\frac{\p_{n}^2 + \s_{n}^2}
{1- 2\s_n + \p_{n}^2 + \s_{n}^2}\right) ,
\label{5.14}\\
A_{X}^{(1)}[\p,\s] &=& \sum_{n}2
\ln\left(1- 2\s_n + \p_{n}^2 + \s_{n}^2\right)
\nonumber\\
&=& 2^{1-\frac{d}{2}}\sum_{n}{\rm Tr}
\ln\left(\left(1- X_n \right)
\left(1- X_n^{\dag}\right)\right) ,
\label{5.15} \\
A_{X}^{(2)}[\p,\s] & = &
\sum_{n}{\rm Tr}\left(\ln\left(1- X\right)
\right)_{nn},
\label{5.16}\end{aligned}$$ where a function $h$ is introduced to describe the invariant potential. We notice that all these three terms become singular at $X=1$.
In this section, we have solved the QME (\[4.10\]) to determine the effective action on the coarse lattice. The solutions we have obtained are used to construct a lattice chiral symmetry. Let us discuss its structure in the next section.
Lattice chiral symmetry in the macroscopic theory
=================================================
Let us first summarize our results. In contrast to I, we discuss quantization of the system in the antifield formalism. The partition function for the macroscopic theory is given by $$\begin{aligned}
Z_{\rm MACRO}
& = & \int {\cal D}\Phi {\cal D}\Phi^{*}\prod_{A} \d (\Phi^{*}_{A})
\exp -S[\Phi,~\Phi^{*}],
\label{6.1}\end{aligned}$$ with the total macroscopic action $$\begin{aligned}
S[\Phi,~\Phi^{*}]&=& S_{D} + S_{X},\nonumber\\
S_{X} &=& A_{X}^{(0)} + A_{X}^{(1)}+ A_{X}^{(2)}\nonumber\\
&{}&+2
\sum_{n}\left[\pi^{*}_{n}\left\{\s_n - \left(\s_n^2 - \p_n^2\right)\right\}C
\right.{\nonumber}\\ && \qquad \quad \left.
+ \s^{*}_{n}\left\{ -\p_n + 2\s_n\p_n\right\}C
\right].
\label{6.2}\end{aligned}$$ There arise four types of the action $S_{D}$ for the Dirac fields: $$\begin{aligned}
(I)~~ S_{D} &=& \sum_{nm}\left[\bar{\V}_{n}{\tilde D}_{nm}
\V_{m} +
\V_{n}^{*}i~C \g_{5}\left(1 -2{\tilde D}\right)_{nm}
\V_{m} - \bar{\V}_{n} i~C \g_{5}\d_{nm} \bar{\V}_{m}^{*}
\right],{\nonumber}\\
{\tilde D}&=&
D_{0} +\g_5 \frac{1}{1-D_0} \g_5 \ X
\left(1-D_0\right) ,
\label{SDI}\end{aligned}$$ $$\begin{aligned}
(II)~~ S_{D} &=& \sum_{nm}\left[\bar{\V}_{n}{\tilde D}_{nm}
\V_{m} + \V_{n}^{*}i~C \g_{5}\d_{nm}
\V_{m} - \bar{\V}_{n} i~C \left(1 -2 {\tilde D}\right)_{nm} \g_{5}
\bar{\V}_{m}^{*} \right],{\nonumber}\\
{\tilde D}&=&
D_{0} +\g_5 \frac{1}{1-D_0} \g_5 \ X
\left(1-D_0\right) ,
\label{SDII}\end{aligned}$$ $$\begin{aligned}
(III)~~S_{D} & = & \sum_{nm}\left[\bar{\V}_{n}{\tilde D}_{nm}
\V_{m} + \V_{n}^{*}i~C \g_{5}\left(1 -2{\tilde D}\right)_{nm}
\V_{m} - \bar{\V}_{n} i~C \g_{5}\d_{nm} \bar{\V}_{m}^{*}
\right],{\nonumber}\\
{\tilde D}&=& D_{0} +\left(1-D_0\right) X
\g_5 \frac{1}{1-D_0}\g_5 ,
\label{SDIII}\end{aligned}$$ $$\begin{aligned}
(IV)~~S_{D} & = & \sum_{nm}\left[\bar{\V}_{n}{\tilde D}_{nm}
\V_{m} + \V_{n}^{*}i~C \g_{5}\d_{nm}
\V_{m} - \bar{\V}_{n} i~C \left(1 -2 {\tilde D}\right)_{nm} \g_{5}
\bar{\V}_{m}^{*}\right],{\nonumber}\\
{\tilde D}&=& D_{0} +\left(1-D_0\right) X
\g_5 \frac{1}{1-D_0}\g_5 .
\label{SDIV}\end{aligned}$$
The potential terms of the auxiliary fields, $ A_{X}^{(0)}$, $ A_{X}^{(1)}$ and $ A_{X}^{(2)}$ are given in (\[5.14\]) $\sim$ (\[5.16\]). Under the BRS transformation $$\begin{aligned}
\d_{B} \V_{n} &=& i~C \g_{5}\left(1 -2 {\tilde D}\right)_{nm}
\V_{m} , \nonumber\\
\d_{B} \bar{\V}_{n} &=& i~C \bar{\V}_{n} \g_{5},\nonumber\\
\d_{B} X_{n}&=& 2i~\g_{5}\left(X_n- X_{n}X_{n}\right)~C,
\label{6.4}\end{aligned}$$ or $$\begin{aligned}
\d_{B} \V_{n} &=& i~C \g_{5} \V_{n} , \nonumber\\
\d_{B} \bar{\V}_{n} &=& i~C \bar{\V}_{m}
\left(1 -2 {\tilde D}\right)_{mn}\g_{5},\nonumber\\
\d_{B} X_{n}&=& 2i~\g_{5}\left(X_n- X_{n}X_{n}\right)~C,
\label{6.41}\end{aligned}$$ the functional measure in (\[6.1\]) multiplies by the contributions from the counter action, $$\begin{aligned}
\lefteqn{
{\cal D}\Phi {\cal D}\Phi^{*}\prod_{A} \d (\Phi^{*}_{A})
e^{-A_{X}^{(1)}[\p,\s] -A_{X}^{(2)}[\p,\s]}} \nonumber\\
& = & \prod_{n}d \V_{n}d \bar{\V}_{n}d \p_{n} d \s_{n}
\prod_{m} d \V_{m}^{*} d \bar{\V}_{m}^{*}d \p_{m}^{*} d \s_{m}^{*}
\d (\V_{m}^{*})\d (\bar{\V}_{m}^{*}) \d (\p_{m}^{*})\d (\s_{m}^{*})
\nonumber \\
&{}& ~ \times e^{-A_{X}^{(1)}[\p,\s] -A_{X}^{(2)}[\p,\s]},
\label{6.5}\end{aligned}$$ remains invariant. The remaining part of the action $$\begin{aligned}
S_{D} + S_{X} - A_{X}^{(1)} - A_{X}^{(2)},
\nonumber\end{aligned}$$ is also left invariant under (\[6.4\]) or (\[6.41\]). In other words, the macroscopic action (\[6.2\]) solves the QME, $\Sigma[\Phi,~\Phi^{*}]= (S,~S)_{\Phi}/2 - \Delta_{\Phi}S =0$, which demonstrates the presence of an exact chiral symmetry in the quantum system on the coarse lattice. In (\[6.1\]), the chiral transformation of the macroscopic fields is encoded as the BRS transformation due to the presence of the antifields. Once the transformation rule is known, one may eliminate the antifields by the integration to obtain the partition function given in I.
We have shown that the QME is solved in a closed form. It should be noted, however, that the resulting actions (\[SDI\]) $\sim$ (\[SDIV\]) are singular as discussed above. Moreover, the chiral symmetry is realized in a peculiar way. The macroscopic fields (except for $\bar{\V}$ or ${\V}$) transform nonlinearly: $\d_{B} \V$ or $\d_{B}\bar{\V} $ contains $X$, and $\d_{B} X$ has a quadratic term of $X$. As a result, in the fermionic action $S_{D}$ given in (\[6.2\]), neither the kinetic term $\bar{\V}D_0\V$ nor the Yukawa coupling $\bar{\V}(1+{\cal L})X(1+{\cal R})\V$ is chiral invariant, while their sum $ (\bar{\V}D_0\V) + (\bar{\V}(1+{\cal L})X(1+{\cal R})\V)$ becomes invariant. Turn to the functional measure ${\cal D}\Phi $ of the macroscopic fields, it is not chiral invariant so that the counter terms should be included. Furthermore, the basic invariant made out of the auxiliary fields is nonpolynomial as in (\[5.14\]).
Because of these problems, we would like to reconstruct the chiral symmetry in such a way that (1) the actions are free from the singularities, (2) the kinetic term of the Dirac fields, the Yukawa coupling term and the functional measure are all chiral invariant, and (3) the auxiliary field potential becomes polynomial. The reconstruction of the symmetry satisfying the above conditions can be done by employing a new set of canonical variables for each type of actions (\[SDI\]) $\sim$ (\[SDIV\]). We argue that two of the transformed actions define consistent quantum theories but the remaining two cases may not.
Reconstruction of chiral symmetry in terms of new canonical variables
=====================================================================
Let $\hat{\Phi}^{A}= \{\T,~{\bar{\Theta}},~{\hat X}=i\g_{5}{\hat \p} + {\hat
\s}\}$ be new fields. In the antifield formalism, we obtain the new fields by considering a canonical transformation from $\{\Phi^{A}, \Phi_{A}^{*}\}$ to $\{{\hat \Phi}^{A}, \hat{\Phi}^{*}_{A}\}$, where $ \hat{\Phi}^{*}_{A}=\{\T^{*},~{\bar{\Theta}}^{*}, \hat{X}^{*}\}$. The generator is given by $$\begin{aligned}
G[\Phi, \hat{\Phi}^{*}]= \sum_{nm}\T^{*}_{n} Y(X)_{nm}\ \V_{m}
+\sum_{n}{\bar{\V}}_{n}U(X)_{nm}{\bar{\Theta}}_{m}^{*} + \sum_{n}{\rm Tr}
\left[\hat{X}^{*}_{n} W(X)_{n}\right],
\label{6.6}\end{aligned}$$ where $\hat{X}^{*}= 2^{-d/2}[-i\g_{5}\hat{\p}^{*} + \hat{\s}^{*}]$. The matrices $Y$, $U$ and $W$ are functions of $X$, and symmetric in the spinor indices. The new fields are obtained by ${\hat \Phi}^{A}=\partial
G/\partial \hat{\Phi}^{*}_{A}$, while the old set of antifields are given by $ \Phi_{A}^{*}=\partial G/\partial \Phi^{A}$ : $$\begin{aligned}
\T_{n} &=& \left[Y(X) \V\right]_{n} ,\nonumber\\
{\bar{\Theta}}_{n} &=& \left[{\bar\V}U(X)\right]_{n},\nonumber\\
{\hat X}_{n} &=& W(X)_{n}, \nonumber\\
{\V}^{*}_{n} &=& \left[\T^{*} Y(X)\right]_{n}, \nonumber\\
{\bar \V}^{*}_{n}&=& \left[U(X){\bar{\Theta}}^{*}\right]_{n},\nonumber\\
X^{*}_{n} &=& {\rm Tr} \left[
\hat{X}^{*}_{n}\frac{\partial W(X)_{n}}{\partial X_{n}}\right]
+ \sum_{ml}
\T^{*}_{m}\frac{\partial Y(X)_{ml}}{\partial X_{n}} \V_{l}{\nonumber}\\
&{}& +
\sum_{ml}{\bar \V}_{m}
\frac{\partial Y(X)_{ml}}{\partial X_{n}}{\bar{\Theta}}^{*}_{l} .
\label{6.7} \end{aligned}$$
There is a variety of choices for the matrices $Y$, $U$ and $W$ with which the transformed actions are free from the singularities. Among them, we discuss the following four cases corresponding to the actions (\[SDI\]) $\sim$ (\[SDIV\]).
: For (\[SDI\]), we take $$\begin{aligned}
Y(X)_{nm} &=& \left[\frac{1}{1 -D_{0}}(1 - X)(1 - D_{0})\right]_{nm},{\nonumber}\\
U(X)_{mn} &=& \d_{nm}, {\nonumber}\\
W(X)_{n} &=& \frac{X_{n}}{1 - X_{n}}.
\label{YUX1} \end{aligned}$$ One can confirm that the Jacobian factor associated with the change of variables from $\{\pi,~\s\}_{n}$ to $\{\hat{\p}_{n},~\hat{\s}_{n}\}$ exactly cancels the contribution from the counter term $A_{X}^{(1)}$: $$\begin{aligned}
{\cal D} \pi {\cal D}\s~e^{-A_{X}^{(1)}} &\equiv& \prod_{n}d
\pi_{n}d \s_{n}~e^{-A_{X}^{(1)}}{\nonumber}\\
&=& {\cal D}\hat\pi{\cal D}\hat\s.
\label{measure}\end{aligned}$$ Likewise, since $$\begin{aligned}
\sum_{n}{\rm Tr} (\ln ~Y)_{nn} -A_{X}^{(2)} &=& 0,
\label{6.161}\end{aligned}$$ one can also see that the fermionic measure with the counter term $A_{X}^{(2)}$ becomes $$\begin{aligned}
{\cal D} \V {\cal D}\bar{\V}~e^{-A_{X}^{(2)}} &\equiv& \prod_{n}d
\V_{n}d \bar{\V}_{n}~e^{-A_{X}^{(2)}}
={\cal D}\T{\cal D} {\bar{\Theta}}~{\rm Det}(Y)~e^{-A_{X}^{(2)}}\nonumber\\
&=& {\cal D}\T{\cal D} {\bar{\Theta}}\exp
[\sum_{n}{\rm Tr}(\ln Y)_{nn} -A_{X}^{(2)} ]\nonumber\\
&=& {\cal D}\T{\cal D} {\bar{\Theta}}.
\label{6.15}\end{aligned}$$ Therefore, the transformed theory is described by the partition function $$\begin{aligned}
Z_{\rm old }&=&
\int {\cal D}\Phi {\cal D}\Phi^{*}\prod_{A} \d (\Phi^{*}_{A})
\exp -(S[\Phi,~\Phi^{*}])\nonumber\\
&=& Z_{\rm new}
= \int {\cal D}{\hat\Phi} {\cal D}{\hat\Phi}^{*}
\prod_{A} \d ({\hat\Phi}^{*}_{A})\exp -({\cal S}[{\hat\Phi},~{\hat\Phi}^{*}]),
\label{6.18}\end{aligned}$$ where the new action is given by $$\begin{aligned}
{\cal S} [{\hat\Phi},~{\hat\Phi}^{*}]&=& {\cal S}_{D} + {\cal S}_{X} ,
\nonumber\\
{\cal S}_{D} &=& \sum_{nm} {\bar{\Theta}}_{n}
\left[D_{0}+
\hat{X}(1- D_{0})\right]_{nm}\T_{m}\nonumber\\
&{}& +
\sum_{nm}\T^{*}_{n}C~i\g_{5} \left(1 -2 D_{0}\right)_{nm}
\T_{m}- \sum_{n}{\bar{\Theta}}_{n}C~i\g_{5}{\bar{\Theta}}^{*}_{n} ,
\nonumber\\
{\cal S}_{X} &=& \sum_{n}h({\hat\p}_{n}^{2}+ {\hat\s}_{n}^{2})
+ 2~\sum_{n}({\hat\p}_{n}^{*}{\hat\s}_{n}-{\hat\s}_{n}^{*}{\hat\p}_{n})~C.
\label{6.21}\end{aligned}$$ We have used here the relations $$\begin{aligned}
\bar{\V}_{n}\tilde{D}_{nm}\V_{m} &=& {\bar{\Theta}}_{n} (\tilde{D} Y^{-1})_{nm}\T_{m}
\nonumber\\
&=& {\bar{\Theta}}_{n} \left[D_{0} + \hat{X}(1 - D_{0})\right]_{nm}\T_{m} \nonumber\\
\frac{\p_{n}^{2} + \s_{n}^{2}}{(1-\s_{n})^{2}
+\p_{n}^{2}}&=& {\hat\p}_{n}^{2}+ {\hat\s}_{n}^{2}.
\label{6.20}\end{aligned}$$ The partition function (\[6.18\]) is invariant under $$\begin{aligned}
\d_{B} \T_{n} &=& i~C \g_{5}\left(1 -2 D_{0}\right)_{nm}
\T_{m} , \nonumber\\
\d_{B} \bar{\T}_{n} &=& i~C \bar{\T}_{n} \g_{5},\nonumber\\
\d_{B} \hat{X}_{n}&=& 2i~\g_{5}\hat{X}_n~C.
\label{chiral1}\end{aligned}$$ It is noticed that the Yukawa couplings in the action (\[6.21\]) are the same as those discussed by many authors [@reviews][@Naya][@chand][@in][@fi]: $$\begin{aligned}
{\cal O}_{\rm Yukawa}&=&
\sum_{nm} {\bar{\Theta}}_{n}
\left[\left(i\g_5\hat{\p}+\hat{\s}\right)(1-D_{0})\right]_{nm}
\T_{m} \nonumber\\
&=& \sum_{nm} \bar{\V}_{n}
\left[\left(i\g_5{\p}+{\s}\right)(1-D_{0})\right]_{nm}
\V_{m}\nonumber\\
&=&
\sum_{nm} {\bar{\Theta}}_{n}
i\g_5(1-D_{0})_{nm}
\T_{m}~\hat{\p}_{n} {\nonumber}\\
&{}&
+ \sum_{nm} {\bar{\Theta}}_{n}
(1-D_{0})_{nm}
\T_{m}~\hat{\s}_{n}.
\label{6.23}\end{aligned}$$ is form invariant, and chiral invariant: $$\begin{aligned}
\d_{B} {\cal O}_{\rm Yukawa}= ~0.
\label{6.24}\end{aligned}$$
: For (\[SDII\]), we may choose $$\begin{aligned}
Y(X)_{nm} &=& \d_{nm}{\nonumber}\\
U(X)_{nm} &=& \left[\g_5 \frac{1}{1-D_0}(1-X)\g_5\right]_{nm}{\nonumber}\\
W(X)_{n} &=& \frac{X_{n}}{1 - X_{n}}.
\label{YUX2} \end{aligned}$$ In this case, the Jacobian factor for the Dirac fields generates an additional factor ${\rm Det}(1 - D_{0})^{-1}$, we may define the partition function as $$\begin{aligned}
Z_{\rm old }&=&
\int {\cal D}\Phi {\cal D}\Phi^{*}\prod_{A} \d (\Phi^{*}_{A})
\exp -(S[\Phi,~\Phi^{*}])\nonumber\\
&=& \frac{1}{{\rm Det} (1-D_0)}\ Z_{\rm new}{\nonumber}\\
Z_{\rm new}
&=& \int {\cal D}{\hat\Phi} {\cal D}{\hat\Phi}^{*}
\prod_{A} \d ({\hat\Phi}^{*}_{A})\exp -({\cal S}[{\hat\Phi},~{\hat\Phi}^{*}]),
\label{partfunc2}\end{aligned}$$ where $$\begin{aligned}
{\cal S} [{\hat\Phi},~{\hat\Phi}^{*}]&=& {\cal S}_{D} + {\cal S}_{X} ,
\nonumber\\
{\cal S}_{D} &=& \sum_{nm} {\bar{\Theta}}_{n}
\left[D_{0}(1-\g_5 D_0 \g_5)\right]_{nm}\T_{m} + \sum_{nm} {\bar{\Theta}}_{n}
\left[\hat{X}(1- \g_{5}D_{0}\g_{5}\right]_{nm}\T_{m}\nonumber\\
&{}& +
\sum_{n}\T^{*}_{n}C~i\g_{5}\T_{m}- \sum_{n}{\bar{\Theta}}_{n}C~i\g_{5}{\bar{\Theta}}^{*}_{n} ,
\nonumber\\
{\cal S}_{X} &=& \sum_{n}h({\hat\p}_{n}^{2}+ {\hat\s}_{n}^{2})
+ 2~\sum_{n}({\hat\p}_{n}^{*}{\hat\s}_{n}-{\hat\s}_{n}^{*}{\hat\p}_{n})~C.
\label{action2}\end{aligned}$$ The partition function $Z_{\rm new}$ is invariant under $$\begin{aligned}
\d_{B} \T_{n} &=& i~C \g_{5} \T_{n} , \nonumber\\
\d_{B} \bar{\T}_{n} &=& i~C \bar{\T}_{n} \g_{5},\nonumber\\
\d_{B} \hat{X}_{n}&=& 2i~\g_{5}\hat{X}_n~C,
\label{chiral2}\end{aligned}$$ which is the same as the standard form of the chiral transformation in continuum (or microscopic) theories.
: For the action (\[SDIII\]), the same results as (\[partfunc2\]), (\[action2\]) and (\[chiral2\]) for the case (ii) can be obtained with $$\begin{aligned}
Y(X)_{nm} &=& \left[\g_5 \frac{1}{1-D_0}(1-X)\g_5\right]_{nm}{\nonumber}\\
U(X)_{nm} &=& \d_{nm} {\nonumber}\\
W(X)_{n} &=& \frac{X_{n}}{1 - X_{n}}.
\label{YUX3} \end{aligned}$$
: For the action (\[SDIV\]), we consider the matrices similar to the case (i) as $$\begin{aligned}
Y(X)_{nm} &=& \d_{nm},{\nonumber}\\
U(X)_{mn} &=& \left[(1 - D_{0})(1 - X)\frac{1}{1 -D_{0}}\right]_{nm}, {\nonumber}\\
W(X)_{n} &=& \frac{X_{n}}{1 - X_{n}}.
\label{YUX1} \end{aligned}$$ Then, one obtains the partition function (\[6.18\]) with the Dirac action $$\begin{aligned}
{\cal S}_{D} &=& \sum_{nm} {\bar{\Theta}}_{n}
\left[D_{0}+
(1- D_{0})\hat{X}\right]_{nm}\T_{m}\nonumber\\
&{}& +
\sum_{n}\T^{*}_{n}C~i\g_{5}
\T_{n}- \sum_{nm}{\bar{\Theta}}_{n}C~i\left(1 -2 D_{0}\right)_{nm}\g_{5}{\bar{\Theta}}^{*}_{m} ,
\label{Diracaction4}\end{aligned}$$ The chiral transformation takes of the form $$\begin{aligned}
\d_{B} \T_{n} &=& i~C \g_{5} \T_{n} , \nonumber\\
\d_{B} \bar{\T}_{n} &=& i~C \bar{\T}_{n} (1-2 D_0)\g_{5},\nonumber\\
\d_{B} \hat{X}_{n}&=& 2i~\g_{5}\hat{X}_n~C.
\label{chiral4}\end{aligned}$$
Let us discuss some physical consequences for the cases (i) $\sim$ (iv) listed above. The four are classified into two two groups: [(i) and (iv)]{}, [(ii) and (iii)]{}. Actually, (ii) and (iii) share the same action (\[action2\]) and the chiral transformation (\[chiral2\]). The kinetic term of the Dirac fields as well as the Yukawa term in this action contains the factor $(1- \g_{5}D_{0}\g_{5})$ in front of the Dirac fields. Since $D_{0}=1$ at the momenta where the doubler modes appear, this factor vanishes. Thus, the doubler modes remain massless, and decouple with the auxiliary fields. However, this is the case only at tree level, and decoupling could not persist at quantum level: There are other chiral invariant Yukawa terms such as $$\begin{aligned}
\bar{{{\cal }}}_n \hat{X}_n {{\cal _}}n , \qquad
\bar{{{\cal }}}_n \hat{X}_n (\g_5D_{0}\g_5D_{0})_{nm} {{\cal _}}m , {\nonumber}\end{aligned}$$ in which the doublers couple with the auxiliary fields. There are no reasons to exclude these terms in the quantum corrections. Therefore, (ii) and (iii) cannot give a consistent theory.
Unlike these cases, the doubler modes in (i) and (iv) are massive and decouple with the auxiliary fields because of the factor $(1 - D_{0})$ in the Yukawa couplings. The chiral invariant Yukawa terms always contain this factor, and therefore the chiral invariance protects the couplings of the doublers to the auxiliary fields. This can be seen by use of the chiral decomposition [@reviews][@Naya][@iis2]: $$\begin{aligned}
\hat{\T}_R &=& \frac{1+\hat{\gamma}_5}{2} \T, {\nonumber}\\
\hat{\T}_L &=& \frac{1-\hat{\gamma}_5}{2} \T, {\nonumber}\\
\bar{\T}_R &=& \bar{\T}\frac{1+\gamma_5}{2}, {\nonumber}\\
\bar{\T}_L &=& \bar{\T}\frac{1-\gamma_5}{2},
\label{chiral-projection}\end{aligned}$$ where $$\begin{aligned}
\hat{\gamma}_{5} = \gamma_{5}(1 - 2 D_{0}).
\label{gamma5hat}\end{aligned}$$ Using $$\begin{aligned}
\delta \hat{\T}_R &=& i~C~ \hat{\T}_R,~~~~~~~
\delta \hat{\T}_L = -i~C~ \hat{\T}_L, {\nonumber}\\
\delta \bar{\T}_R &=& i~C~ \bar{\T}_R,~~~~~~~
\delta \bar{\T}_L = -i~C~\bar{\T}_L,
\label{chiral-charge}\end{aligned}$$ we may construct the the Yukawa term by using the chiral projection. One finds that resulting Yukawa term is exactly the same as ${\cal O}_{\rm
Yukawa}$ in (\[6.23\]): $$\begin{aligned}
{\cal F}_{\rm Yukawa} &\equiv& \bar{\T}_R~ \hat{\T}_R (\hat{\s} + i\hat{\pi}) +
\bar{\T}_L~ \hat{\T}_L (\hat{\s} - i\hat{\pi}) {\nonumber}\\
&=& \bar{\T}\hat{X}(1- D_{0})\T = {\cal O}_{\rm Yukawa}
\label{Yukawa}\end{aligned}$$
In the transformed theory, the integration over the new auxiliary fields can be performed explicitly. The last expression of the Yukawa term in (\[6.23\]) can be used to do it. One then obtains $$\begin{aligned}
\lefteqn{ {Z_{\rm MACRO}} } \nonumber\\
&=&
\int{\cal D}\T{\cal D} {\bar{\Theta}}\
\exp\left(-\sum_{nm} {\bar{\Theta}}_{n} \left(D_{0}\right)_{nm}\T_{m}
-h\left({\cal O}_{\rm 4\mbox{-}fermi}
(\T,~ {\bar{\Theta}})
\right)
\right) ,
\label{6.25}\end{aligned}$$ where the antifields are integrated, too. The four-fermi interaction operator ${\cal O}_{\rm 4\mbox{-}fermi}$ is given by $$\begin{aligned}
{\cal O}_{\rm 4\mbox{-}fermi} (\T,~ {\bar{\Theta}})
&\equiv&
\left( \sum_{nm} {\bar{\Theta}}_{n}
i\g_5 (1-D_{0})_{nm} \T_{m}
\right)^2 \nonumber \\ &{}&
+
\left( \sum_{nm} {\bar{\Theta}}_{n}
(1-D_{0})_{nm} \T_{m}
\right)^2 . \nonumber\\
\label{6.26}\end{aligned}$$ For the simplest case $h(x)=x$, we obtain the Nambu-Jona-Lasinio model.
In this section, we have reconstructed the lattice chiral symmetry using the canonical transformations. Since these transformations are singular, the transformed theories are only equivalent to the original ones up to the singularities.
Inclusion of the complete set of the auxiliary fields
=====================================================
In the above formulation of the lattice chiral symmetry, the auxiliary fields we have considered are restricted to a scalar and a pseudoscalar fields. We discuss in this section inclusion of the complete set of the auxiliary fields. Let $\G^{i}\ (i=1\sim 2^{d})$ be the complete set of the Clifford algebra in $d$ (=even) dimensional space. The Dirac fields carry $N_{F}$ flavors, and form the fundamental representation of $u(N_{F})$ algebra with a basis $T^{a}\ (a=1\sim N_F^2)$ satisfying ${T^{a}}^{\dag}=T^{a}$. Let $\lambda^{A}\ (A=1\sim {\cal N}\equiv 2^d N_F^2)$ be the direct product of the above two sets of the matrices, $$\begin{aligned}
\lambda^A \ = \ \lambda^{ia} \ = \ \G^{i} \otimes T^a ,
\label{dp}\end{aligned}$$ normalized by $$\begin{aligned}
{\rm Tr}({\lambda^A}^{\dag}\lambda^B) \ =\
{\rm tr}_{(\G)}{\rm tr}_{(T)}\left({\G^{i}}^{\dag}\G^{j}\right)
\otimes \left(T^a T^b\right)
\ = \ 2^{\frac{d}{2}}N_F\ \d^{ij} \d^{ab} \ = \
2^{\frac{d}{2}}N_F\ \d^{ab}.
\label{norm}\end{aligned}$$ Here ${\rm tr}_{(\G)}$ and ${\rm tr}_{(T)}$ denote the traces in the spinor and the flavor spaces, respectively.
We introduce the complete set of the auxiliary fields $x^{A}$, and define $$\begin{aligned}
X &\equiv& x^A \lambda^A ,{\nonumber}\\
x^A & =& 2^{-\frac{d}{2}} N_F^{-1}
{\rm Tr}\left({\lambda^{A}}^{\dag}X\right) ,
\label{comX}\end{aligned}$$ which is the extension of (\[5.11\]).
Let us consider the block transformation suppressing the antifields for simplicity. $$\begin{aligned}
\lefteqn{
\int \CD X\,
\exp\left({-\sum_{nm}\bar{\V}_n \tD(X)_{nm} \V_m -A_{X}[X]}\right)}
{\nonumber}\\
&=&
\int \CD \v\CD \bar{\v}\,
\exp\left(-A_{c}[\v,\bar{\v}]
-\sum_{n}(\bar{\V}_n-\bar{B}_n)\a\left(\V_n-B_n\right)\right) ,
\label{funcmes}\end{aligned}$$ where ${\displaystyle}\CD X = \prod_{A=1}^{\cal N}\prod_{n} dx_{n}^{A}$. The $\tD(X)$ is again assumed to be linear in $X$. Hereafter, we take again $\alpha =1$. We may define the chiral transformation of the macroscopic fields as $$\begin{aligned}
\d\V_n &=& i\g_5\left(1-2\tD\right)_{nm}\V_m ,{\nonumber}\\
\d\bar{\V}_n &=& \bar{\V}_n i\g_5 ,{\nonumber}\\
\d X_n &=& -i\left\{\g_5,X_n\right\} + 2i X_n\g_5 X_n .
\label{chi-X}\end{aligned}$$ The WT identities associated with (\[chi-X\]) are given by $$\begin{aligned}
&
\left(i\{\g_5,\tD\}
-2i \tD\g_5\tD +\d\tD\right)_{nm} \ =\ 0,
\label{Res-gwr1} \\ &
\d A_{X}^{(0)}[X]\ = \ 0,
\label{Res-gwr2} {\nonumber}\\ &
\d A_{X}^{(1)}[X] \ = \ \d J_{X}[X] =
4{\cal N}\,\frac{2^{-\frac{d}{2}}}{N_F}
\sum_{n}{\rm Tr}\left(i\g_5 X_n\right),
\label{Res-gwr3} \\ &
\d A_{X}^{(2)}[X] = \d J_{\V,\bar{\V}} =
-2i\sum_{n}{\rm Tr}\left(\g_5 -\g_5 \tD(X)\right)_{nn} ,
\label{Res-gwr4}\end{aligned}$$ where the potential term $A_{X}$ is decomposed as $A_{X}=A_X^{(0)}+A_X^{(1)}+A_X^{(2)}$. It should be noted that the chiral transformation and the WT identities essentially take the same form as those for the truncated case, except that $X$ does not commute with $\g_{5}$ here. For the Dirac operator $$\begin{aligned}
\tD(X) &=& D_0 + (1+{\cal L}(D_0))X(1+{\cal{R}}(D_0)),
\label{tD}\end{aligned}$$ the GW relation (\[Res-gwr1\]) can be solved with the $D_{0}$, ${\cal L}$ and ${\cal{R}}$ given in (\[5.6\]) and (\[5.7\]) or (\[5.8\]). Likewise, one obtains the counter terms $$\begin{aligned}
A_X^{(1)}[X] &=& 2{\cal N}\,\frac{2^{-\frac{d}{2}}}{N_F}
\sum_{n}{\rm Tr}\ln\left(1-X_n \right) , {\nonumber}\\
A_X^{(2)}[X] &=& \sum_{n}{\rm Tr}\ln\left(1-{X_n}\right)\end{aligned}$$
In order to construct the invariant potential $A_{X}^{(0)}$, we may define new set of the auxiliary fields: $$\begin{aligned}
\hat{X}_n &\equiv& \frac{X_n}{1- X_n } ,
\label{newX}\end{aligned}$$ which obeys a linear transformation as $$\begin{aligned}
\d \hat{X}_n &=& -i\left\{\g_5,\hat{X}_n\right\}.
\label{newdX}\end{aligned}$$ Since $\hat{X}= \hat{x}^{A} \lambda^{A}$ transforms linearly, it is easy to obtain a quadratic invariant: $$\begin{aligned}
f_{\rm inv}^{I}(\hat{x}^A_n) &=& G_{AB}^I\,\hat{x}^A_n\,\hat{x}^B_n ,
\qquad (I=1,2,\cdots ),
\label{invX}\end{aligned}$$ where $G_{AB}^I$ are suitable coefficients. Using these invariants, we have $$\begin{aligned}
A_X^{(0)}[X] &=& \sum_{n}h\left(f_{\rm inv}^{I}(\hat{x}^A_n)\right).
\label{A0}\end{aligned}$$
In summary, a chiral invariant partition function $Z$ is given by $$\begin{aligned}
{Z}
&=&
\int \left[\CD X\CD \V \CD \bar{\V}\ e^{-A_X^{(1)}-A_X^{(2)}}\right] \
e^{-\sum_{nm}\bar{\V}_n\tD_{nm}\V_m -A_X^{(0)}},
\label{calZ}\end{aligned}$$ where $$\begin{aligned}
\tD(X) &=& D_0 + (1+{\cal L})X(1+{\cal R}) ,{\nonumber}\\
A_X^{(0)}[X] &=& \sum_{n}h\left(f_{\rm inv}^{I}(\hat{x}^A_n)\right) ,{\nonumber}\\
A_X^{(1)}[X]+A_X^{(2)}[X] &=&
\left(2{\cal N}\,\frac{2^{-\frac{d}{2}}}{N_F}+1\right)
\sum_{n}{\rm Tr}\ln\left(1-\frac{X_n}{\a}\right) .
\label{calZ2}\end{aligned}$$
Let us give a special case of $d=2$ and $N_{F}=1$: $X=i\g_5\p +\s +i\g_{\m}V_{\m}$. The chiral transformation is given by $$\begin{aligned}
\d\p_n &=& -2\s_n -2\p_n^2 + 2\s_n^2 + 2 {V_n^{\m}}^2,{\nonumber}\\
\d\s_n &=& 2\p_n - 4 \p_n\s_n , {\nonumber}\\
\d V_n^{\m} &=& -4 V^{\mu}_{n}\p_{n} .
\label{d=2}\end{aligned}$$ The new fields $\hat{x}^{A}$ defined in (\[newX\]) and their transformation are given by $$\begin{aligned}
\hat{\p}_n &= \ \frac{\p_n}
{\left(1-{\s_n}\right)^2 +\left({\p_n}\right)^2
+\left({V_n^{\m}}\right)^2},
\quad &
\d\hat{\p}_n \ = \ -2\hat{\s}_n ,
{\nonumber}\\ {\nonumber}\\
\hat{\s}_n &= \ \frac{\s_n - \p_n^2 -\s_n^2 - {V_n^{\m}}^2}
{\left(1-{\s_n}\right)^2 +\left({\p_n}\right)^2
+\left({V_n^{\m}}\right)^2},
\quad &
\d\hat{\s}_n \ = \ 2\hat{\p}_n , {\nonumber}\\
\hat{V}_n^{\m} &= \ \frac{V_n^{\m}}
{\left(1-{\s_n}\right)^2 +\left({\p_n}\right)^2
+\left({V_n^{\m}}\right)^2},
\quad &
\d\hat{V}_n^{\m} \ = 0 .
\label{d=2hatX}\end{aligned}$$ We have two quadratic invariants: $$\begin{aligned}
f_{\rm inv}^1 \ = \
\hat{\p}_n^2 + \hat{\s}_n^2
&=&
\frac{\p_n^2 +\left(\s_n -\p_n^2 -\s_n^2 - {V_n^{\m}}^2\right)^2}
{\left[\left(1-{\s_n}\right)^2 +\left({\p_n}\right)^2
+\left({V_n^{\m}}\right)^2\right]^2},{\nonumber}\\
f_{\rm inv}^2 \ = \ (\hat{V}^{\m}_n)^2
&=&
\frac{{V_n^{\m}}^2}
{\left[\left(1-{\s_n}\right)^2 +\left({\p_n}\right)^2
+\left({V_n^{\m}}\right)^2\right]^2}.
\label{d=2inv}\end{aligned}$$
In this section, we have discussed an extension of the auxiliary method. We have obtained a formal expression of chiral invariant partition function (\[calZ2\]). Its action, however, is not free from the singularities discussed in section 3, and it seems to be difficult to construct the canonical transformation which removes these singularities.
Summary and discussion
======================
There have been known nontrivial examples where exact chiral symmetries are realized in interacting theories on the lattice. One was given by Lüscher who discussed chiral gauge theories. The lattice chiral symmetry in fermionic interacting system discussed in this paper may provide another example. For our fermionic system with auxiliary fields, there arise two sets of the WT identities : One is the GW relation which tells us how to define chiral transformation for the Dirac as well as the auxiliary fields on the coarse lattice. Under the suitable locality assumption for the auxiliary fields, we have determined the chiral transformation for the macroscopic fields. The transformation rule is used to construct chiral invariant actions. The other WT identity can be interpreted as an anomaly matching relation between the microscopic and the macroscopic theories. This identity contains contributions arising from the transformation of the functional measure, and is used to construct counter terms needed to make the functional measure on the coarse lattice chiral invariant. In the antifield formalism, these WT identities are obtained from the QME $\Sigma =0$.
Owing to the auxiliary field method, the fermionic sector of our system is linearized. The price for it is that the integration over the auxiliary fields remains in the condition $<\Sigma> =0$. We have considered in this paper the QME $\Sigma =0$, and found four types of actions which solve the QME. However, they are found to have singularities in the Yukawa couplings and the potential of the auxiliary fields. Those in the Yukawa couplings are related to the presence of doubler modes. Beside these singularities, none of the kinetic term of the Dirac fields, the Yukawa couplings and the functional measure becomes chiral invariant in the realization of the symmetry with the block variables. In order to avoid these problems, we have used more suitable sets of variables obtained by (singular) canonical transformations. We have discussed four types of the transformed actions. In all cases, the new fields transform linearly, and have chiral invariant functional measure. Among these actions, only two of them define consistent quantum theories. They are exactly equivalent to those obtained by using the representation method for chiral algebra arising from free field theory: One defines the chiral transformation in the free theory, and then constructs chiral invariant Yukawa couplings from the chiral decomposition. The chiral invariance in such systems is expressed by the classical master equation rather than the QME. Since the new auxiliary fields belong to the conventional SO(2) multiplet, they are readily integrated to give purely fermionic system. The Nambu-Jona-Lasinio model emerges as the simplest one. Our results may give justification of the representation method for formulating lattice chiral symmetry in theories with generic interactions.
Let us discuss some implications of our results for realization of regularization-dependent symmetry in lattice or continuum theory. The block transformation on the lattice or its continuum analog plays an important rôle in inheriting symmetry properties of the macroscopic theory from those of the microscopic theory. In general, the functional measure for the original block variables may not be invariant under the cutoff-dependent symmetry transformations. The induced change in the functional measure corresponds to the $\Delta$ derivative of the extended action, whose explicit expression depends on the UV regularization scheme. However, unless nontrivial anomaly is present in the given theory, one can always find the counter action needed to cancel the $\Delta$ derivative contribution. The counter action is regularization dependent, but is expected to be antifield independent. Then, a canonical transformation will be performed in such a way that new fields have invariant measure. This corresponds to the reduction of the QME to the classical master equation.
Acknowledgments
===============
This work is supported in part by the Grants-in-Aid for Scientific Research No. 12640258, 12640259 and 13135209 from the Japan Society for the Promotion of Science. We also thank Yukawa Institute for Theoretical Physics in Kyoto University for hospitality and providing its computer fascilities.
Derivation of some formulae
===========================
Let us first derive (\[5.60\]), (\[5.61\]) and (\[5.6\]). The reduced GW relation (\[5.3\]) is divided into the local terms and the nonlocal terms which depend on ${\cal L}$ and/or ${\cal R}$. The nonlocal terms becomes $$\begin{aligned}
\begin{array}{lll}
&{}&
\{\g_5 ,{\cal L}X+X{\cal R}+{\cal L}X{\cal R}\} \\ &&
-2 D_{0}\g_5 (X+{\cal L}X+X{\cal R}+{\cal L}X{\cal R}) \\ &&
-2 (X+{\cal L}X+X{\cal R}+{\cal L}X{\cal R})\g_5 D_{0} \\ &&
-2 X \g_5 ({\cal L}X+X{\cal R}+{\cal L}X{\cal R}) \\ &&
-2 \left({\cal L}X+X{\cal R}+{\cal L}X{\cal R}\right)\g_5 X \\ &&
-2\left({\cal L}X+X{\cal R}+{\cal L}X{\cal R})\g_5 ({\cal L}X+X{\cal R}+{\cal L}X{\cal R}\right) \\ &&
-2 \left(\, {\cal L}\g_5(X- X^2)+\g_5(X- X^2){\cal R} +{\cal L}\g_5(X- X^2){\cal R} \, \right) \\ &&
= {\cal K}X + X {\cal H} + {\cal K}X{\cal R} +{\cal L}X {\cal H}-2 X{\cal G}X
-2 X {\cal G} X{\cal R} -2 {\cal L}X {\cal G}X -2 {\cal L}X{\cal G}X{\cal R} =0.
\end{array}
\label{a1} $$ where $$\begin{aligned}
{\cal K} &=& \g_5 {\cal L} - {\cal L}\g_5 -2 D_0 \g_5 -2 D_0 \g_5 {\cal L}, {\nonumber}\\
{\cal H} &=& -\g_5 {\cal R} + {\cal R}\g_5 -2 \g_5 D_0 -2 {\cal R}\g_5 D_0, {\nonumber}\\
{\cal G} &=& \g_5 {\cal L} +{\cal R}\g_5 +{\cal R}\g_5 {\cal L}.
\label{a2}\end{aligned}$$ The conditions ${\cal K}={\cal H}={\cal G}=0$ lead to (\[5.60\]), (\[5.61\]) and (\[5.6\]).
Let us show that ${\cal L}$ and ${\cal R}$ in (\[5.7\]) are solutions of (\[5.60\]) $\sim$ (\[5.6\]). ${\cal K}$ with ${\cal L}$ in (\[5.7\]) is the GW relation is free theory (\[5.5\]). ${\cal H}$ with ${\cal R}$ in (\[5.7\]) becomes $$\begin{aligned}
{\cal H}
&=& -\g_5{}D_0 -D_0\g_5 +2{}D_0{}\g_5{}D_0
{\nonumber}\\
& & -{}\g_5{}D_0{}\g_5{}D_0{}\g_5 \
\frac{1}{1- \g_5{}D_0{}\g_5}
+{}D_0{}\g_5{}D_0{}\g_5 \
\frac{1}{1- \g_5{}D_0{}\g_5}\
\g_5 \left(1-2 D_0\right)
{\nonumber}\\
&=& -{}D_0{}\g_5{}D_0 \ \frac{1}{1- \g_5{}D_0{}\g_5}
+{}D_0{}\g_5{}D_0 \ \frac{1}{1-{}D_0} \
\left(1-2 D_0\right)
{\nonumber}\\
&=& {}D_0{}\g_5{}D_0 \ \frac{1}{1- \g_5{}D_0{}\g_5}
\left(-\left(1-{}D_0\right)
+\left(1-{}\g_5{}D_0{}\g_5\right)
\left(1-2{}D_0\right)\right)
\frac{1}{1-{}D_0}
{\nonumber}\\
&=& 0 ,
\label{a3}\end{aligned}$$ where we have used $\g_5{}D_0{}\g_5{}D_0{}\g_5=D_0{}\g_5{}D_0$. Finally, $\cal{G}$ turns to be $$\begin{aligned}
{\cal G}&=&
{-{}\g_5{}D_0{} + \left(-{}D_0
+{}D_0{}\g_5{}D_0{}\g_5 \
\frac{1}{1- \g_5{}D_0{}\g_5}\right)\g_5}
{\nonumber}\\
&{}&~ +{ \left(-{}D_0 +{}D_0{}\g_5{}D_0{}\g_5 \
\frac{1}{1- \g_5{}D_0{}\g_5}\right)
\g_5\left(-{}D_0\right)}
{\nonumber}\\
&=& {\left(-\g_5{}D_0 -D_0\g_5 +2{}D_0{}\g_5{}D_0
\right)}
{\nonumber}\\
& &-{}D_0{}\g_5{}D_0
+{}D_0{}\g_5{}D_0{}\g_5 \ \frac{1}{1- \g_5{}D_0{}\g_5} \
\g_5
-{}D_0{}\g_5{}D_0{}\g_5 \ \frac{1}{1- \g_5{}D_0{}\g_5} \
\g_5{}D_0
{\nonumber}\\
&=& {}D_0{}\g_5{}D_0{}\g_5
\left(-1+\frac{1}{1- \g_5{}D_0{}\g_5}\right)\g_5
-{}D_0{}\g_5{}D_0{}\g_5 \ \frac{1}{1- \g_5{}D_0{}\g_5} \
\g_5{}D_0
{\nonumber}\\
&=& {}D_0{}\g_5{}D_0{}\g_5 \ \frac{1}{1- \g_5{}D_0{}\g_5} \
\g_5{}D_0
-{}D_0{}\g_5{}D_0{}\g_5 \ \frac{1}{1- \g_5{}D_0{}\g_5} \
\g_5{}D_0
{\nonumber}\\
&=& 0 .
\label{a4}\end{aligned}$$
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[^1]: A similar kind of singularities has been discussed in different context in [@fi].
[^2]: The matrix $\g_{5}$ satisfies $\g_{5}^{2}=1$.
|
---
abstract: 'We study phonon Hall effect (PHE) for ionic crystals in the presence of static magnetic field. Using Green-Kubo formula, we present an exact calculation of thermal conductivity tensor by considering both positive and negative frequency phonons. Numerical results are shown for some lattices such as hexagonal lattices, triangular lattices, and square lattices. We find that the PHE occurs on the nonmagnetic ionic crystal NaCl, although the magnitude is very small which is due to the tiny charge-to-mass ratio of the ions. The off-diagonal thermal conductivity is finite for nonzero magnetic field and changes sign for high value of magnetic field at high temperature. We also found that the off-diagonal thermal conductivity diverges as $\pm{1/T}$ at low temperature.'
author:
- 'Bijay K. Agarwalla'
- Lifa Zhang
- 'Jian-Sheng Wang'
- Baowen Li
title: '**Phonon Hall effect in ionic crystals in the presence of static magnetic field**'
---
INTRODUCTION
============
The magnetic field dependence of heat conductivity in paramagnetic dielectric crystals has been found experimentally [@strohm05]. This effect is known as “phonon Hall effect”. Applying magnetic field in the direction normal to the temperature gradient, the authors have discovered the appearance of a transverse thermal current in Tb$_3$Ga$_5$O$_{12}$ and the effect was confirmed in Ref. . Due to the Lorentz force on the electrons, electronic Hall effect can be easily understood. However for phonons there is no such direct coupling with the magnetic field. The underlying mechanism that determines this effect is the spin-phonon interaction (SPI)[@old-sp-papers; @sp-book; @ioselevich95] of phonons and paramagnetic ions Tb$^{+3}$. This effect is an analog of the anomalous Hall effect (AHE) in paramagnetic phase. The theory of this phenomenon was proposed in Refs. , where the authors consider the spin-phonon interaction and solve the problem perturbatively. The same effect was also studied in four-terminal junctions using nonequilibrium Green’s function (NEGF) approach [@lifa09]. In Ref. the authors explained that molecules with rotary degree of freedom can also have the same effect as PHE through renormalization of acoustic waves. This effect for gases is knows as Senftleben-Beenakker effect. Very recently, the topological nature of PHE and an associated phase transition was found in the paramagnetic dielectrics [@zhang10].
All the previous work on PHE [@strohm05; @sheng06; @kagan08; @wang09; @lifa09; @zhang10] are based on the paramagnetic dielectrics; and the recent work [@onose10] about the magnon Hall effect in the ferromagnetic insulator could also include the PHE to the thermal Hall effect. We know that all the materials in the previous experimental or theoretical work on PHE are paramagnetic or ferromagnetic ionic crystals; therefore, a question arises promptly: whether the PHE can occur in a general nonmagnetic ionic crystal? From a theoretical point of view, it is highly desirable to study both the existence and properties of the PHE in general ionic crystals with an applied magnetic field, which could also guide us to do the further experimental study on PHE.
As there is no magnetization effect in the nonmagnetic ionic crystal, intuitively one can not imagine the existence of PHE in the sample. The Hamiltonian in the harmonic approximation for the ionic crystal in the presence of a magnetic field is quadratic and is very similar to the form in Refs. . Because of the quadratic nature of the problem it is possible to diagonalize the Hamiltonian in terms of creation and annihilation operators. In Ref. the authors consider the Hamiltonian which is not positive definite and omit the contribution of negative frequency to the phonon Hall conductivity which was incorrect. Here we carefully reinvestigate the Hamiltonian of the ionic system, the non-positive-definite problem is naturally solved and the phonon Hall conductivity is calculated exactly by applying the Green-Kubo formula in the ballistic thermal transport region. The paper is organized as follows. In sec. II we first introduce the model, then outline the derivation of second quantization and also of thermal conductivity. In sec. III we present numerical results some lattices, and also apply our method to the sodium chloride (NaCl) crystal in Sec. IV. Finally we conclude with a short discussion in sec. V.
THE HAMILTONIAN IN A UNIFORM MAGNETIC FIELD
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The Hamiltonian operator of a crystal lattice in the presence of a magnetic field is, according to the minimal energy principle, given by [@Holz] $$\begin{aligned}
H &=& \frac{1}{2} \sum_{{l,x,\alpha}}M_{x}^{-1}\bigl(P_{\alpha}(l,x)-eA_{\alpha}(l,x)\bigr)^{2} \nonumber \\
&+& \frac{1}{2}\sum_{ll'}\sum_{xx'}\sum_{{\alpha}{\beta}}U_{\alpha}(l,x)\phi_{{\alpha}{\beta}}(l,x;l',x')U_{\beta}(l',x').\end{aligned}$$ Here $M_{x}$ and $e$ are the mass and charge respectively of the $x$-th ion in the unit cell, which is assumed to have $r$ ions, $U_{\alpha}(lx)$ is the $\alpha$-th cartesian coordinate of the displacement of the $x$-th atom in the $l$-th unit cell from the equilibrium position $x_{\alpha}(lx)$, $P_{\alpha}(lx)$ is the corresponding linear momentum operator, $\phi_{{\alpha}{\beta}}(l,x;l',x')$ are the atomic force constants for the crystal and ${\bf A}$ is the magnetic vector potential. In the presence of Lorentz gauge ($\nabla \cdot{\bf A} =0$) the vector potential can be written as a linear combintation of equilibrium position of each ion seperately $A_{\alpha}(l,x)=\sum_{\beta} C_{{\alpha}{\beta}}U_{\beta}(l,x)$ where $C$ is an antisymmetric $3 \times 3$ matrix. $P_{\alpha}(l,x)$ and $U_{\alpha}(l,x)$ satisfy the commutation relation $$[U_{\alpha}(l,x),P_{\beta}(l',x')]=i \hbar \delta_{{\alpha}{\beta}} \delta_{ll'} \delta_{xx'}.$$ Using this commutation relation and substituting the expression for ${\bf A}$, the Hamiltonian of the system can be written in a compact form $$\label{eq-ham}
H = \frac{1}{2} p^T p + \frac{1}{2} u^T K u + u^T {\tilde A} p,$$ where $u$ denotes the column vector of displacements of the atoms away from the equilibrium positions for all the degrees of freedoms, multiplied by a corresponding square root of mass $\sqrt{M_{x}}$, $p$ is the associated conjugate momenta. Superscript $T$ stands for transpose. The $K$ matrix has the following form $$K_{{\alpha x},{\beta x'}}(l,x;l',x')=\frac{{\phi_{{\alpha}{\beta}}(l,x;l',x')}}{\sqrt{M_{x}M_{x'}}}-({\tilde A})_{{\alpha x},{\beta x'}}^{2}\delta_{ll'},$$ where the matrix elements of ${\tilde A}$ are the cyclotron frequency for different masses and is given by $${\tilde A}_{{\alpha x},{\beta x'}}=\frac{e}{M_{x}}C_{{\alpha}{\beta}}\delta_{xx'}.$$ If we take the magnetic field ${\bf B}$ along $z$ direction, ${\bf B}=B \hat{z}$, then the typical form of ${\tilde A}$ matrix would be $$\left(\begin{array}{rrr} 0 & \Omega & 0 \\
-\Omega & 0 & 0 \\
0 & 0 & 0 \\
\end{array} \right).$$ where $\Omega=$$eB$/$M$. The term $u^T {\tilde A} p$ can be treated as onsite potential which was introduced phenomenologically in all other models to describe PHE but it comes out naturally for ionic crystals. Also because of the special form of the $K$ matrix ($K=\phi-{\tilde A}^{2}$) and ${\tilde A}$ being antisymmetric matrix the system is indeed stable for large values of magnetic field.
Using the Heisenberg equation of motion and Bloch’s theorem we can write the equation of motion for $u$ and $p$ in the following matrix form $${\cal M} \chi = \omega \chi$$ where $${\cal M}= -i\left( \begin{array}{cc} {\tilde A} & D \\
-I & {\tilde A} \end{array} \right) ,
\label{eq-eigen}$$ $\chi_{\sigma}({\bf k}) = (\mu_{\sigma}({\bf k}), \epsilon_{\sigma}({\bf k}))^T$, $\mu_{\sigma}({\bf k})$ and $\epsilon_{\sigma}({\bf k})$ are the polarization vectors corresponding to $p$ and $u$, respectively, $D({\bf k})=\sum_{l'} K_{ll'} e^{i {\bf k}\cdot ({\bf R}_{l'}-{\bf R}_{l})}$ is the dynamical matrix, ${\bf R}_{l}$ is the real-space lattice vector and $I$ is an identity matrix. The displacement and momentum operators can be written in the following standard second quantization form $$\begin{aligned}
u_l &=& \sum_{{\bf k},\sigma} \epsilon_{\sigma}({\bf k}) e^{i {\bf R}_l \cdot {\bf k}} \sqrt{\frac{\hbar}{2\left|\omega_{\sigma}({\bf k})\right | N}} \, a_{\sigma}({\bf k}); \nonumber\\
p_l &=& \sum_{{\bf k},\sigma} \mu_{\sigma}({\bf k}) e^{i {\bf R}_l \cdot {\bf k}} \sqrt{\frac{\hbar}{2\left|\omega_{\sigma}(\bf k)\right| N}} \, a_{\sigma}({\bf k}).\end{aligned}$$ Where ${\bf k}$ is the wavevector, $N$ is the number of unit cells, $\sigma$ is the phonon branch index which runs over both positive and negative frequencies, $\left|\omega_{\sigma}({\bf k})\right |=\omega_{\sigma}({\bf k}) \rm{sign}(\sigma)$ and $a_{-\sigma}({\bf k})=a_{\sigma}^{\dagger}(-{\bf k})$. The reason that we have negative frequencies is because they are also the eigenvalues of Eq.(7). Only both the positive and negative frequency eigenmodes form a complete set. Using this complete set of eigenvalues and eigenvectors the Hamiltonian can be re-expressed as $$H=\frac{1}{2}\sum_{{\bf k},\sigma} \hbar \left|\omega_{\sigma}({\bf k})\right| a_{\sigma}^\dagger({\bf k}) a_{\sigma}({\bf k}).$$ Using the defination of local energy density and continuity equation $$\begin{aligned}
\nabla \cdot S(x)+\frac{\partial H(x)}{\partial t}=0, \\
S=\frac{1}{V}\int d^{3}x S(x),\end{aligned}$$ total heat current operator can be written as [@hardy63; @sheng06; @kagan08; @wang09] $$S^{a}=\frac{1}{2}\sum_{l,l'}(R_{l}^{a}-R_{l'}^{a})u_{l}^T K_{l,l'} \dot{u}_{l'},$$ where $\dot{u_{l}}=p_{l}-{\tilde A}u_{l}$. The index $a$ corrensponds to different cartesian axes. $V$ is the total volume of $N$ unit cells. In terms of creation and annihilation operators the above expression can be written in an exact form as $$S^{a}=\frac{\hbar}{4V}\sum_{k,\sigma,\sigma'} F_{\sigma,\sigma'}^{a}({\bf k})\frac{\omega_{\sigma'}({\bf k})}{\sqrt{\left|\omega_{\sigma}({\bf k})\right| \left|\omega_{\sigma'}({\bf k})\right|}} a_{\sigma}^\dagger({\bf k}) a_{\sigma'}({\bf k}),$$ where the $F$ function is defined as $$F_{\sigma\sigma'}^a({\bf k}) =
\epsilon_{\sigma}^\dagger({\bf k}) \frac{\partial D({\bf k})}{\partial{\bf k}^a} \epsilon_{\sigma'}({\bf k}).$$ $F$ is a Hermitian matrix with real diagonal elements and is related to the group velocity $v_{\sigma}({\bf k})$ as $ F_{\sigma\sigma}^a({\bf k}) =2 v_{\sigma}({\bf k}){\omega}_{\sigma}({\bf k})$ Off-diagonal elements are in general not zero and is responsible for carrying heat in Hall effect. The above expression of $S_{a}$ contains all four possible combinations of creation and annihilation operators which are $a_{\sigma}^{\dagger}({\bf k})a_{\sigma'}({\bf k})$, $a_{\sigma}({\bf k})a_{\sigma'}^{\dagger}({\bf k})$, $a_{\sigma}^{\dagger}({\bf k})a_{\sigma'}^{\dagger}(-{\bf k})$ and $a_{\sigma}(-{\bf k})a_{\sigma'}({\bf k})$. The measured heat current ${\bf J}=\rm{Tr}(\rho {\bf S})$ vanishes in equilibrium. The thermal conductivity tensor can be calculated from the Green-kubo formula [@mahan00; @allen93] $$\kappa_{ab} = \frac{V}{T} \int_0^{\beta}\!\!\!\! d\lambda
\int_0^\infty\! dt\, \bigl\langle S^a(-i\lambda \hbar) S^b(t) \bigr\rangle_{\rm eq},$$ where $\beta=1/(k_BT)$, the average is taken over the equilibrium ensemble with Hamiltonian $H$. Substituting the expression $S^a$ and using Wick’s theorem we obtain
$$\begin{aligned}
\kappa_{ab} &=& \frac{\hbar}{32VT} \sum_{{\bf k}, \sigma \ne \sigma'}
F_{\sigma'\sigma}^a({\bf k}) \times
F_{\sigma\sigma'}^b({\bf k})\times \frac{(\omega+\omega')^2}{\omega\omega'}\qquad\nonumber \\
&& \times\frac{n(\omega')-n(\omega)}{\omega'-\omega} \times
\frac{1}{\eta - i (\omega- \omega')},
\label{eq-main}\end{aligned}$$
where $n(\omega)=(e^{\beta \hbar \omega}-1)^{-1}$ is the Bose distribution function. We have used the notations $\omega= \omega_{\sigma}({\bf k})$, $\omega'=\omega_{\sigma'}({\bf k})$. $\eta$ comes from the damping term $e^{-\eta t}$ which we have added in order to integrate the oscillatory factor $e^{i (\omega-\omega')t}$. Same phonon branch ($\sigma=\sigma'$) does not contribute to the thermal conductivity. For high temperature ($\beta \hbar \omega<1$) thermal conductivity reaches a constant value but for low temperature ($T \rightarrow 0$) it diverges as $\pm{1/T}$. For a perfect crystal without the magnetic field($A$=0) the diagonal elements of thermal conductivity $\kappa_{aa}$ diverge in the form of $1/\eta$ , corresponds to infinite conductivity and the off-diagonal elements $\kappa_{ab}$ are zero which is consistent with Fourier’s law. For nonzero magnetic field ($A \ne 0$) the diagonal elements of $\kappa$ are still infinite but off-diagonal elements gives rise to finite value of conductivity. The form of Eq.(17) is very similar to the anomalous Hall effect for electrons where the summation is replaced by an integral of ${\bf k}$ [@nagosa09]. The same form was also derived for disordered harmonic crystal and for amorphous solids [@allen93; @allen89]. The formula can also be considered from a topological point of view [@zhang10]. The major difference of our result with others[@wang09; @kagan08; @allen93] is in the expression for $S^{a}$ where we have taken all possible combinations of creation and annhilation operators. As a result we need to sum over all positive and negative phonon branches. We numerically calculate the conductivity and found that the effect is almost twice in comparison with Ref. .
CALCULATION ON SOME LATTICES
============================
Before studying the real ionic-crystal material, we calculate the PHE on some lattices to get some basic properties of the PHE in the ionic lattices. In this section we show the calculation of the PHE on the hexagonal lattices, triangular lattices, and square lattices.
In Fig.1, we give results of dispersion relation for a hexagonal lattice with nearest neighbour couplings. The primitive vectors for the lattice are ($a$,0) and ($a$/2,$\sqrt{3} a$/2) with $a=1$ Å. The longitudinal and transverse force constants are chosen as $K_{L}\,=\,0.144$ eV/(uÅ$^2)$ and $K_{T}=0.036$ eV/(uÅ$^2)$ respectively[@wang09]. This typical choice is comparable with experimental values. Here $h=eB/m$ which has dimension of frequency. In the presence of magnetic field the phonon spectrum is shifted (see Fig.1(a)), and the shift is proportional to the applied magnetic field. It can also be shown that for lattices having inversion symmetry $\omega_{\sigma}(-\bf k)= \omega_{\sigma}(\bf k)=-\omega_{-\sigma}(\bf k)$ even for nonzero magnetic field. In Fig.1(b) we have given a plot both for positive and negative eigenfrequencies $\omega_{\sigma}({\bf k})$ as a function $h$ and we can see that all eigenvalues are real even for very large value of $h$.
(a) Phonon dispersion curve for hexagonal lattice. The angular frequency (both positive and negative) of one accoustic mode as a function of $k_{x}a$ with $k_{y}=0$. The solid curves are for $h=0$ and dotted curve are for $h=10.0 \times 10^{12}$ rad/s. (b) The eigenfrequencies (two accoustic and two optical) as a function of $h$ for fixed wave vector ${\bf k}a=(1,1)$. ](fig1.eps){width="\columnwidth"}
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Thermal Hall conductivity $\kappa_{xy}$ (a) as a function of the magnetic field $h$ (b) as a function of temperature $T$ (c) divergence of Hall conductivity at low temperature for a hexagonal lattice.](fig2.eps "fig:"){width="\columnwidth"}
Thermal Hall conductivity $\kappa_{xy}$ (a) as a function of the magnetic field $h$ (b) as a function of temperature $T$ (c) divergence of Hall conductivity at low temperature for a hexagonal lattice.](fig2_ext.eps "fig:"){width="0.8\columnwidth" height="1.9in"}
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In Fig.2(a), we show result for the off-diagonal thermal conductivity $\kappa_{xy}$ as a function of magnetic field for three different temperatures $T$=10K, 50K, and 100K. It is clear that, at low temperature for small values of $h$, $\kappa_{xy}$ increases linerly with the magnetic field but for large $h$, it grows slower than linear. For very large $h$ it can even goes to negative value. At low temperature phonon Hall conductivity can change sign with increasing magnetic field. This result is consistent with Ref. . With increasing temperature, the magnitude of negative Hall conductivity decreases because when the temperature is high, more high energy modes are excited and spin-phonon interaction can not easily turn around the phonons.
In Fig.2(b) we plot the temperature dependence of $\kappa_{xy}$ for three different values of $h$. For small values of $h$ phonon Hall effect is almost negligable. But for high value of $h$ it increases slowler than linear with $T$ and for very high temperature it reaches a constant value. For low temperature $\kappa_{xy}^{-1}\sim T^{1.009,1.005,1.012}$ for $h=(10.0,1.0,0.1) \times 10^{12}$ rad/s respectively(see Fig.2(c)) which is close to 1.0 and consistent with Eq.(17). The divergence of $\kappa_{xy}$ as $1/T$ for $T\rightarrow 0$ is not unphysical as it shows the ballistic behaviour for which the longitudinal conductivity $\kappa_{xx}$ is always infinite at any temperature. However if we measure current it will be zero at $T\rightarrow 0$. And from another point of view, when $T\rightarrow 0$, the value of $\kappa_{xy}T$ is constant; because of $\Delta T<T$ in linear response region, the measurable quantity of transverse heat current $S^y=-\kappa_{yx} \Delta T/ L$ ( we know $\kappa_{yx}=-\kappa_{xy}$ ) will not be divergent and tend to zero for the periodic system $L \rightarrow \infty$. Furthermore we know that for the ballistic thermal transport, because of the zero temperature gradient in the system, the thermal conductivity is infinite while the thermal conductance is finite and meaningful. Therefore, the $1/T$ divergence of $\kappa_{xy}$ indeed embodies the ballistic picture of PHE at very low temperature.
Thermal Hall conductivity $\kappa_{xy}$ (a) as a function of the magnetic field $h$ and (b) as a function of temperature $T$ for a triangular lattice.](fig3.eps){width="\columnwidth"}
Source $A$ ($ev$) $B$(Å) $C$ ($ev$)Å$^{6}$
-------- ------------ --------- -------------------
Na-Na 487 0.23768 1.05
Na-Cl 14514 0.23768 6.99
Cl-Cl 405774 0.23768 72.40
In Fig 3. we show results for a triangular lattice. We have taken the same values of the parameters for the calculation. In this case also the off-diagonal thermal conductivity shows similar behaviour as in hexagonal lattice. At very low temperature thermal conductivity changes sign. The temperature depedence is also similar. At low temperature conductivity goes as $-1/T$.
For a square lattice with only the nearest neighbour coupling there is no PHE. Generally the off-diagonal term of $\kappa$ will be zero if the reflection symmetry is not broken. If there exists a symmetry operation $S$ such that $S K S^{-1}=K $ and $S A S^{-1}=-A$ then $\kappa_{ab}=0$ for $a\ne b$. In this case the spring constant matrix $K$ is diagonal and is invariant under reflection in $x$ or $y$ direction, hence it satisfies the above relation. If we consider next-neighbour coupling then the matrix $K$ will not have mirror reflection symmetry and the PHE can be observed which is consistent with Ref. .
THE PHE ON THE SODIUM CHLORIDE CRYSTAL
======================================
After the properties of the PHE are well studied for the model lattices, we turn to the calculation for the real ionic crystal NaCl. NaCl crystal can be described as fcc lattice with primitive vectors ${\bf a_{1}}=\frac{a}{2}({\bf y}+{\bf z})$, ${\bf a_{2}}=\frac{a}{2}({\bf x}+{\bf z})$ and ${\bf a_{3}}=\frac{a}{2}({\bf x}+{\bf y})$ and two point basis consisting of a sodium ion at ${\bf 0}$ and a chlorine ion at the center of the conventional cubic cell, $\frac{a}{2}({\bf x}+{\bf y}+{\bf z})$. Here $a$ is the lattice constant which is 5.63 [Å]{} for NaCl.
We calculate the force constant matrix using “General Utility Lattice Program” (GULP)[@gale97] by considering a unit cell. The interatomic potential we used includes both Coulombic and non-Coulombic terms $$V_{ij}(r_{i},r_{j})=\frac{1}{4 \pi \epsilon_0} \frac{Z_i Z_j e^{2}}{r_{ij}} + A_{ij}exp(-r_{ij}/B_{ij})-\frac{C_{ij}}{r_{ij}^{6}},$$ where $r_{ij}$ is the interatomic distance between ions $i$ and $j$ and $Z_i$ is the effective charge of the $i$th ion. The non-Coulombic part of the potential is known as Buckingham potential. $A_{ij}$ and $B_{ij}$ are parameters for the repulsive interaction and $C_{ij}$ is van der Waals constant. The values of the parameters are given in Table 1.[@chin]. Cutoff distance 12 [Å]{} is used in the calculation. With this potential model we went upto the third nearest neighbours and cut the force constant matrix properly in order to get the dynamical matrix. The choice of this type of unit cell gives us dynamical matrix of size $6 \times 6$.
Thermal Hall conductivity $\kappa_{xy}$ (a) as a function of the magnetic field $B$ (b) and as a function of temperature $T$ for NaCl.](fig4.eps){width="\columnwidth"}
In Fig.4(a), we give the off-diagonal thermal conductivity $\kappa_{xy}$ as a function of $B$ for temperature $T=100\,K$ and $T=50\,K$. We note that the sign of $\kappa_{xy}$ is negative. The growth of $\kappa_{xy}$ in this case, for small values of magnetic field is very fast, then it reaches a maximum and then decreases to zero with increasing magnetic field and even goes to positive value for high value of magnetic field.
In Fig.4(b), we display the temperature dependence of the off-diagonal thermal conductivity $\kappa_{xy}$ for $B=0.3$ and $B=0.5$ MTesla. Upto $T$=40K the off diagonal thermal conductivity is close to zero, meaning there is no heat flow. When $T \rightarrow 0$ it also diverge like $1/T$. But it gradually increases and finally saturates at high temperature. In all our calculations we have used $\eta$ of order 10$^{-6}K_{L}$ to get rid of the sigularity due to the accoustic phonons at $\Gamma$ point.
From the above calculation in the nonmagnetic ionic crystal NaCl, we observe the PHE although there is no magnetization in the system. The existence of PHE is due to the nonzero term of $\tilde{A}$ in Eq. (\[eq-ham\]). Both the positive and negative ions in the crystal contribute to $\tilde{A}$, and they can not calcel each other. Since in the term $\tilde{A}$, the charge to mass ratio $ (e/m)$ is very small for nonmagnetic ionic crystals NaCl, a very strong magnetic field is required to observe PHE in this case. However in the paramagnetic crystal the contribution from electrons dominates the term ${\tilde A}$, which increases about $10^{4}$ times and thus the PHE can be comparable to the experimental data. For ferromagnetic insulating ionic crystal, it will be increased much more because of the large magnetization which will contribute to large spin-phonon interaction.
CONCLUSION
==========
A theory for phonon Hall effect in general ionic crystals based on a ballistic lattice dynamic model is developed. An exact formula of phonon Hall conductivity formula from Green-Kubo approach is derived. It is shown that the effect can be present in the presence of a very strong magnetic field for general ionic crystals without magnetization. Due to the tiny charge-to-mass ratio, the PHE is so small that it is difficult to be measured experimentally. However the theoretically confirmed existence of PHE can give us deep understanding of the PHE and guide us the direction to do the experiments. The ferromagnetic insulator is a good candidate for the PHE. We hope some artificial lattices such as cool atoms in optical trap can generate similar kind of effect. We believe that the result for $\kappa_{xy}$ is robust against nonlinear perturbation, but it is interesting to work out a theory where nonlinearity is taken into account.
ACKNOWLEDGEMENTS
================
we are greatful to Mr. Juzar Thingna, Dr. Jin-Wu Jiang and Meng Lee Leek for insightful discussions. This work is supported in part by URC grant R-144-000-257-112 and R-144-000-222-646 of National University of Singapore.
Second Quantization
===================
For the negative phonon branches we define the polarization vectors as following $$\begin{aligned}
\epsilon_{-\sigma}(-{\bf k})&=&\epsilon_{\sigma}^{\ast}({\bf k}),\\
\omega_{-\sigma}(-{\bf k}) &=& -\omega_{\sigma}({\bf k}).\end{aligned}$$ This defination is consistent with Eq. (7). Also the creation and annihilation operators are defined as for $\sigma>0$ : $$a_{-\sigma}({\bf k})=a_{\sigma}^{\dagger}(-{\bf k}),$$ and the time dependence is given by for $\sigma>0$ : $$\begin{aligned}
a_{\sigma}({\bf k})(t)&=& a_{\sigma}({\bf k})e^{-i \omega_{\sigma}({\bf k})t},\\
a_{\sigma}^{\dagger}({\bf k})(t)&=& a_{\sigma}^{\dagger}({\bf k})e^{i \omega_{\sigma}({\bf k})t}.\end{aligned}$$ The commutaion relation between the operators is $$[a_{\sigma}({\bf k}), a_{\sigma'}({\bf k'})]=\delta{{\bf (k+k')}} \delta{(\sigma+\sigma')} \rm{sign}(\sigma).$$
The matrix $\cal M$ in Eq.(8) is not a normal matrix. Hence we need to consider both the left and right eigenvectors. However it can be easily shown that they are not really independent. For a given value of $\omega_{\sigma}({\bf k})$ and corresponding right eigenvector $\chi_{\sigma}({\bf k})=(\mu_{\sigma}({\bf k}), \epsilon_{\sigma}({\bf k}))^T$ one can choose the left eigenvector as $\tilde \chi_{\sigma}^T({\bf k})=(\epsilon_{\sigma}^{\dagger}({\bf k})$/$(-2 i \omega_{\sigma}({\bf k})), -\mu_{\sigma}^{\dagger}({\bf k})$/$(-2 i \omega_{\sigma}({\bf k})))$. With this choice of eigenvectors the normalization can be done according to $$\epsilon_{\sigma}^\dagger({\bf k})\cdot \epsilon_{\sigma}({\bf k}) + \frac{i}{\omega_{\sigma}({\bf k})} \epsilon_{\sigma}^\dagger({\bf k})\cdot {\tilde A} \cdot \epsilon_{\sigma}({\bf k)} = 1.$$ The Hamiltonian in Eq. (3) can be written as $$H=\frac{1}{2}\sum_{l,l'} \tilde{\mathbb{X}}^T_{l} \left( \begin{array}{cc} {\tilde A} \delta_{l,l'} & K_{l,l'} \\
-I \delta_{l,l'} & {\tilde A} \delta_{l,l'} \end{array} \right) {\mathbb{X}}_{l'}$$ where $$\begin{aligned}
{\mathbb{X}}_{l}=\left( \begin {array}{rr}p_{l} \\ u_{l} \end{array} \right)&=&\sum_{{\bf k},\sigma} \chi_{\sigma}({\bf k}) e^{i {\bf R}_l \cdot {\bf k}} \sqrt{\frac{\hbar}{2\left|\omega_{\sigma}({\bf k})\right | N}}\; a_{\sigma}({\bf k}); \nonumber \\
\tilde{\mathbb{X}}_{l} = \left( \begin {array}{rr}u_{l} \\ -p_{l} \end{array} \right)&=&\sum_{{\bf k},\sigma}(-2i\omega_{\sigma}({\bf k})) \tilde \chi_{\sigma}({\bf k})e^{-i {\bf R}_l \cdot {\bf k}}\times\nonumber \\
&&\sqrt{\frac{\hbar}{2\left|\omega_{\sigma}(\bf k)\right| N}} a_{\sigma}^{\dagger}({\bf k}),\end{aligned}$$ where we have used the definition for $u_{l}$ and $p_{l}$ and also the Hermitian property of the operator. Now using the definition of dynamical matrix, the identity that $\sum_{l}e^{i({\bf k'}-{\bf k}) \cdot {\bf R}_l}=N\delta({\bf k'}-{\bf k})$, and the orthogonality relation between left and right eigenvector $\tilde \chi_{\sigma}^T({\bf k})\cdot \chi_{\sigma'}({\bf k})=\delta_{\sigma \sigma'}$, the Hamiltonian reduces to the form of Eq. (11). The canonical commutation relation can be shown to be satisfied using the completeness of $\chi$, as $$[{\mathbb{X}}_{l},\tilde{\mathbb{X}}^T_{l'}]=-i\hbar \delta_{ll'}I.$$
Positive definiteness of the Hamiltonian
========================================
To prove that the Hamiltonian is positive definite we can write the Hamiltonian as $$H=\frac{1}{2} \left( \begin {array}{ll}u & p \end{array} \right) \left( \begin{array}{cc} K & 2{\tilde A} \\
0 & I \end{array} \right) \left( \begin {array}{rr}u \\ p \end{array} \right),
\label{eq-positive}$$ with $K=\phi -{\tilde A}^{2}$. We assume that $\phi$ is positive definite. Since transpose of a positive definite matrix is positive definite we can as well consider the matrix $$\cal Q=\left( \begin{array}{cc} K & {\tilde A} \\
-{\tilde A} & I \end{array} \right) ,
\label{eq-positive1}$$ which is a symmetric matrix. Hence it can be diagonalized by similarity transformation and can be written as $\cal Q=S D S^T$ where $\cal D=$diag$(\lambda_1,\lambda_2, ......)$. Now if $\lambda_i \ge 0$ $\forall$ $i$ then $\cal D$$^ \frac{1}{2}$ exists and we can write $\cal Q=B^T B $ with $\cal B=D$$^ \frac{1}{2} S^T$. So a real symmetric matrix has non-negative eigenvalues iff it can be factorized as $\cal B^T B$. The above matrix $\cal Q$ can be factorized according to $$\cal B=\left( \begin{array}{cc} -{\tilde A} & I \\
\sqrt{(K+{\tilde A}^{2})^{T}} & 0 \end{array} \right).
\label{eq-positive}$$ Hence the Hamiltonian is positive definite.
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C. Q. Feng, C. L. Cang, D. S. Qing, C. D. Quan, Chinese Phys. **13**, 1091 (2004).
|
---
abstract: 'We consider the so-called one-dimensional forest fire process. At each site of $\mathbb{Z}$, a tree appears at rate $1$. At each site of $\mathbb{Z}$, a fire starts at rate ${\lambda}>0$, immediately destroying the whole corresponding connected component of trees. We show that when ${\lambda}$ is made to tend to $0$ with an appropriate normalization, the forest fire process tends to a uniquely defined process, the dynamics of which we precisely describe. The normalization consists of accelerating time by a factor $\log(1/{\lambda})$ and of compressing space by a factor ${\lambda}\log(1/{\lambda})$. The limit process is quite simple: it can be built using a graphical construction and can be perfectly simulated. Finally, we derive some asymptotic estimates (when ${\lambda}\to0$) for the cluster-size distribution of the forest fire process.'
address:
- |
Institut de Mathématiques de Toulouse\
Université Paul Sabatier\
F-31062 Toulouse Cedex 9\
France\
- |
LAMA, Faculté de Sciences et Technologie\
Université Paris-Est\
61, Avenue du Général de Gaulle\
94010 Créteil Cedex\
France\
author:
-
-
title: 'Asymptotics of one-dimensional forest fire processes'
---
and
.
.
Introduction and main results
=============================
The model
---------
Consider two independent families of independent Poisson processes, $N=(N_t(i))_{t\geq0,i\in\mathbb{Z}}$ and $M^{\lambda}=(M^{\lambda}_t(i))_{t\geq0,i\in\mathbb{Z}}$, with respective rates $1$ and ${\lambda}>0$. Define ${\mathcal F}_t^{N,M^{\lambda}}:=\sigma(N_s(i),M^{\lambda}_s(i),
s\leq t, i \in\mathbb{Z})$. For $a,b \in\mathbb{Z}$ with , we set ${[\![}a, b
{]\!]}= \{a,\ldots,b\}$.
\[dflaffp\] Consider a ${\{0,1\}^\mathbb{Z}}$-valued $({\mathcal
F}_t^{N,M^{\lambda}})_{t\geq0}$-adapted process$(\eta^{\lambda}_t)_{t\geq0}$ such that $(\eta^{\lambda
}_t(i))_{t\geq0}$ is a.s. càdlàg for all $i\in\mathbb{Z}$.
We say that $(\eta^{\lambda}_t)_{t\geq0}$ is a ${\lambda}$-FFP (forest fire process) if a.s., for all $t\geq0$ and all $i\in\mathbb{Z}$, $$\eta^{\lambda}_t(i)={\int_0^t}{\mathbf{1}}_{\{\eta^{\lambda
}_{{s-}}(i)=0\}} \,dN_s(i)
- \sum_{k\in\mathbb{Z}}{\int_0^t}{\mathbf{1}}_{\{k\in C^{\lambda
}_{{s-}}(i)\}} \,dM^{\lambda}_s(k),$$ where $C^{\lambda}_s(i)=\varnothing$ if $\eta_t^{\lambda}(i)=0$, while $C^{\lambda}_s(i)={[\![}l_s^{\lambda}(i),r_s^{\lambda
}(i){]\!]}$ if $\eta
^{\lambda}
_s(i)=1$, with $$l_s^{\lambda}(i)=\sup\{k< i; \eta_s^{\lambda}(k)=0\}+1
\quad\mbox{and}\quad
r_s^{\lambda}(i)=\inf\{k > i; \eta_s^{\lambda}(k)=0\}-1.$$
Formally, we say that $\eta_t^{\lambda}(i)=0$ if there is no tree at site $i$ at time $t$ and $\eta_t^{\lambda}(i)=1$ otherwise. $C_t^{\lambda
}(i)$ stands for the connected component of occupied sites around $i$ at time $t$. Thus, the forest fire process starts from an empty initial configuration, trees appear on vacant sites at rate $1$ (according to $N$) and a fire starts on each site at rate ${\lambda}>0$ (according to $M^{\lambda}$), immediately burning the corresponding connected component of occupied sites.
This process can be shown to exist and to be unique (for almost every realization of $N,M^{\lambda}$) by using a *graphical construction*. Indeed, to build the process until a given time $T>0$, it suffices to work between sites $i$ which are vacant until time $T$ \[because $N_T(i)=0$\]. Interaction cannot cross such sites. Since such sites are a.s. infinitely many, this allows us to handle a graphical construction. We refer to Van den Berg and Jarai [@vdbj] (see also Liggett [@l]) for many examples of graphical constructions. It should be pointed out that this construction only works in dimension $1$.
Motivation and references {#bibi}
-------------------------
The study of self-organized critical (SOC) systems has become rather popular in physics since the end of the 1980s. SOC systems are simple models which are supposed to shed light on temporal and spatial randomness observed in a variety of natural phenomena showing *long-range correlations*, like sand piles, avalanches, earthquakes, stock market crashes, forest fires, shapes of mountains, clouds, etc. Roughly, the idea, which appears in Bak, Tang and Wiesenfeld [@btw1] with regard to sand piles, is that of systems *growing* toward a *critical state* and relaxing through *catastrophic* events (avalanches, crashes, fires, etc.). The most classical model is the sand pile model introduced in 1987 in [@btw1], but many variants or related models have been proposed and studied more or less rigorously, describing earthquakes (Olami, Feder and Christensen [@ofc]) or forest fires (Henley [@h], Drossel and Schwabl [@ds]). For surveys on the subject, see Bak, Tang and Wiesenfeld [@btw1; @btw2], Jensen [@j] and the references therein.
From the point of view of SOC systems, the forest fire model is interesting in the asymptotic regime $\lambda
\to0$. Indeed, fires are less frequent, but when they occur, destroyed clusters may be huge. This model has been the subject of many numerical and heuristic studies; see Drossel, Clar and Schwabl [@dcs] and Grassberger [@g] for references. However, there are few rigorous results. Even existence of the (time-dependent) process for a multidimensional lattice and given $\lambda>0$ has been proven only recently [@du1; @du2] and uniqueness is known to hold only for ${\lambda}$ large enough. The existence and uniqueness of an invariant distribution (as well as other qualitative properties), even in dimension $1$, have been proven only recently in [@bf] for ${\lambda}=1$. These last results can probably be extended to the case where $\lambda\geq1$, but the method in [@bf] completely breaks down for small values of ${\lambda}$.
The asymptotic behavior of the ${\lambda}$-FFP as ${\lambda}\to0$ has been studied numerically and heuristically [@ds; @dcs; @d; @g]. To our knowledge, the only mathematically rigorous results are the following.
\(a) Van den Berg and Jarai [@vdbj] have proven that for $t\geq3$, ${\mathbb{P}}[\eta_{t\log(1/{\lambda})}^{\lambda
}(0)=0]\simeq1/\log
(1/{\lambda})$, thus giving some idea of the density of vacant sites. This result was conjectured by Drossel, Clar and Schwabl [@dcs].
\(b) Van den Berg and Brouwer [@vdbb] have obtained some results in the two-dimensional case concerning the behavior of clusters near the *critical time*. However, these results are not completely rigorous since they are based on a percolation-like assumption, which is not rigorously proved.
\(c) Brouwer and Pennanen [@bp] have proven the existence of an invariant distribution for each fixed ${\lambda}>0$, as well as a precise version of the following estimate which extends (a): for ${\lambda}\in(0,1)$, at equilibrium, ${\mathbb{P}}[\#(C^{\lambda}(0))=x ] \simeq c /[x \log(1/{\lambda})]$ for $x \in\{ 1,\ldots,(1/{\lambda})^{1/3}\}$. It was conjectured in [@dcs] that this actually holds for $x \in\{1,\ldots,1/({\lambda}\log(1/{\lambda}))\}$, but this was rejected in [@vdbj].
In this paper, we rigorously derive a limit theorem which shows that the ${\lambda}$-FFP converges, under rescaling, to some limit forest fire process (LFFP). We precisely describe the dynamics of the LFFP and show that it is quite simple: in particular, it is unique, can be built by using a *graphical construction* and can thus be *perfectly* simulated. Our result allows us to prove a very weak version of (c) for $x\in\{1,\ldots,(1/{\lambda})^{1-{\varepsilon}}\}$, for any ${\varepsilon}>0$; see Corollary \[coco\] below.
Notation {#ssnota}
--------
We denote by $\#(I)$ the number of elements of a set $I$.
For $a,b \in\mathbb{Z}$, with $a\leq b$, we set ${[\![}a, b
{]\!]}= \{a,\ldots,b\}\subset\mathbb{Z}$.
For $I={[\![}a,b{]\!]}\subset\mathbb{Z}$ and $\alpha>0$, we will set $\alpha I := [\alpha a, \alpha b]\subset
{\mathbb{R}}$. For $\alpha>0$, we naturally adopt the convention that $\alpha\varnothing=\varnothing$.
For $J=[a,b]$, an interval of ${\mathbb{R}}$, $|J|=b-a$ stands for the length of $J$ and for $\alpha>0$, we set $\alpha J = [\alpha a, \alpha b]$.
For $x\in{\mathbb{R}}$, $\lfloor x \rfloor$ stands for the integer part of $x$.
Heuristic scales and relevant quantities {#hscales}
----------------------------------------
Our aim is to find some time scale for which tree clusters experience approximately one fire per unit of time. However, for ${\lambda}$ very small, clusters will be very large immediately before they burn. We must thus also rescale space, in order that, immediately before burning, clusters have a size of order $1$.
### Time scale {#time-scale .unnumbered}
Consider the cluster $C_t^{\lambda}(x)$ around some site $x$ at time $t$. It is quite clear that for ${\lambda}>0$ very small and $t$ not too large, one can neglect fires so that, roughly, each site is occupied with probability $1-e^{-t}$ and, thus, $C^{\lambda}_t(x)\simeq{[\![}x-X,x+Y{]\!]}$, where $X,Y$ are geometric random variables with parameter $1-e^{-t}$. As a consequence, $\#(C_t^{\lambda}(x))\simeq e^{t}$ for $t$ not too large. On the other hand, the cluster $C_t^{\lambda}(x)$ burns at rate ${\lambda}\#
(C_t^{\lambda}(x))$ (at time $t$) so that we decide to accelerate time by a factor $\log(1/{\lambda})$. In this way, ${\lambda}\#(C_{\log
(1/{\lambda})}^{\lambda}
(x))\simeq1$.
### Space scale {#space-scale .unnumbered}
We now rescale space in such a way that during a time interval of order $\log(1/{\lambda})$, something like one fire starts per unit of (space) length. Since fires occur at rate ${\lambda}$, our space scale has to be of order ${\lambda}\log(1/{\lambda})$: this means that we will identify ${[\![}0,\lfloor1/({\lambda}\log(1/{\lambda}))\rfloor
{]\!]}\subset
\mathbb{Z}$ with $[0,1]\subset{\mathbb{R}}$.
### Rescaled clusters {#rescaled-clusters .unnumbered}
We thus set, for ${\lambda}\in(0,1)$, $t\geq0$ and $x\in{\mathbb{R}}$, recalling Section \[ssnota\], $$\label{dlambda}
D^{\lambda}_t(x):= {\lambda}\log(1/{\lambda}) C^{\lambda}_{t\log
(1/{\lambda})}\bigl(
\bigl\lfloor x /\bigl( {\lambda}\log(1/{\lambda})\bigr) \bigr\rfloor\bigr)
\subset{\mathbb{R}}.$$
However, this creates an immediate difficulty: recalling that $\#(C_t^{\lambda}(x)) \simeq e^t$ for $t$ not too large, we see that for each site $x$, $|D^{\lambda}_t(x)| \simeq{\lambda}\log(1/{\lambda}) e^{t \log
(1/{\lambda})}=
{\lambda}^{1-t} \log(1/{\lambda})$, of which the limit as ${\lambda
}\to0$ is $0$ for $t<1$ and $+\infty$ for $t\geq1$.
For $t\geq1$, there might be fires in effect and one hopes that this will make the possible limit of $|D^{\lambda
}_t(x)|$ finite. However, fires can only reduce the size of clusters so that for $t<1$, the limit of $|D^{\lambda}_t(x)|$ will really be $0$. Thus, for a possible limit $|D(x)|$ of $|D^{\lambda}(x)|$, we should observe some paths of the following form: $|D_t(x)|=0$ for $t<1$, $|D_t(x)|>0$ for some times $t\in(1,\tau)$, after which it might be killed by a fire and thus come back to $0$, at which time it remains at $0$ for a time interval of length $1$, and so on.
This cannot be a Markov process because $|D(x)|$ always remains at $0$ during a time interval of length exactly $1$. We thus need to keep track of more information in order to control when it exits from $0$.
### Degree of smallness {#degree-of-smallness .unnumbered}
As was stated previously, we hope that for $t<1$, $|D^{\lambda}_t(x)| \simeq{\lambda}^{1-t} \log(1/{\lambda})\simeq
{\lambda}^{1-t}$. Thus, we will try to keep in mind the degree of smallness. We will define, for ${\lambda}\in(0,1)$, $x\in{\mathbb{R}}$ and $t>0$, $$\label{zlambda}
Z^{\lambda}_t(x):= \frac{\log[1+\# ( C^{\lambda}_{t\log(1/{\lambda})}(
\lfloor x /( {\lambda}\log(1/{\lambda})) \rfloor) ) ]}
{\log(1/{\lambda})} \in[0,\infty).$$
### Final description {#final-description .unnumbered}
We will study the ${\lambda}$-FFP via $(D^{\lambda}_t(x),Z^{\lambda}_t(x))_{x\in{\mathbb{R}},t\geq0}$. The main idea is that for ${\lambda}>0$ very small:
if $Z^{\lambda}_t(x)=z\in(0,1)$, then $|D^{\lambda}_t(x)|\simeq
0$ and the (rescaled) cluster containing $x$ is microscopic, but we control its smallness, in the sense that $|D^{\lambda}_t(x)|\simeq{\lambda}^{1-z}$ (in a very unprecise way);
if $Z^{\lambda}_t(x)=1$ \[we will show below that $Z^{\lambda
}_t(x)$ will never exceed $1$ in the limit ${\lambda}\to0$\], then the (rescaled) cluster containing $x$ is automatically macroscopic and has a length equal to $|D^{\lambda}_t(x)|\in(0,\infty)$.
The limit process {#sslffp}
-----------------
We now describe the limit process. We want this process to be Markov and this forces us to add some variables.
We consider a Poisson measure $M(dt,dx)$ on $[0,\infty) \times
{\mathbb{R}}$ with intensity measure $dt \,dx$. Again, we define ${\mathcal F}_t^M=\sigma(M(A),A\in{\mathcal B}([0,t] \times{\mathbb{R}}))$. We also define ${\mathcal I}:=\{[a,b], a \leq b\}$, the set of all closed finite intervals of ${\mathbb{R}}$.
\[dflffp\] A $({\mathcal F}_t^M)_{t\geq0}$-adapted process $(Z_t(x),D_t(x),H_t(x))_{t\geq0,x\in{\mathbb{R}}}$ with values in ${\mathbb{R}}_+\times{\mathcal I}\times{\mathbb{R}}_+$ is a limit forest fire process (LFFP) if a.s., for all $t\geq0$ and all $x \in{\mathbb{R}}$, $$\label{eqlffp}
\cases{
\displaystyle Z_t(x)= {\int_0^t}{\mathbf{1}}_{\{Z_s (x) < 1\}}
\,ds -
{\int_0^t}\int_{\mathbb{R}}{\mathbf{1}}_{\{ Z_{{s-}}(x)=1,y \in
D_{{{s-}}}(x)\}
}M(ds,dy),\cr
\displaystyle H_t(x)=
{\int_0^t}Z_{{s-}}(x){\mathbf{1}}_{\{Z_{{s-}}(x)<1\}} M(ds\times\{x\})
- {\int_0^t}{\mathbf{1}}_{\{H_s (x) > 0 \}}\,ds,}$$ where $D_t(x) = [L_t(x),R_t(x)]$ with $$\begin{aligned}
L_t(x) &=& \sup\{ y\leq x; Z_t(y)<1 \mbox{ or } H_t(y)>0 \},\\
R_t(x) &=& \inf\{ y\geq x; Z_t(y)<1 \mbox{ or } H_t(y)>0 \}.\end{aligned}$$
A typical path of the finite box version of the LFFP (see Section \[ex\]) is drawn and commented on in Figure \[figure2\] and a simulation algorithm is explained in the proof of Proposition \[eufini\].
Let us explain the dynamics of this process. We consider $T>0$ fixed and set ${\mathcal B}_T= \{x \in{\mathbb{R}}; M([0,T]\times\{x\})>
0\}$. For each $t\geq0$ and $x\in{\mathbb{R}}$, $D_t(x)$ stands for the occupied cluster containing $x$. We call this cluster *microscopic* if $D_t(x)=\{x\}$. We also have $D_t(x)=D_t(y)$ for all $y$ in the interior of $D_t(x)$: if $D_t(x)=[a,b]$, then $D_t(y)=[a,b]$ for all $y\in(a,b)$.
1\. *Initial condition.* We have $Z_0(x)=H_0(x)=0$ and $D_0(x)=\{x\}$ for all $x\in{\mathbb{R}}$.
2\. *Occupation of vacant zones.* Here, we consider $x\in{\mathbb{R}}\setminus{\mathcal B}_T$. We then have $H_t(x)=0$ for all $t\in[0,T]$. If $Z_t(x)<1$, then $D_t(x)=\{x\}$ and $Z_t(x)$ stands for the *degree of smallness* of the cluster containing $x$. Then $Z_t(x)$ grows linearly until it reaches $1$, as described by the first term on the right-hand side of the first equation in (\[eqlffp\]). If $Z_t(x)=1$, then the cluster containing $x$ is macroscopic and is described by $D_t(x)$.
3\. *Microscopic fires.* Here, we assume that $x\in{\mathcal B}_T$ and that the corresponding mark of $M$ happens at some time $t$ where $z:=Z_{{t-}}(x)<1$. In such a case, the cluster containing $x$ is microscopic. We then set $H_t(x)=Z_{{t-}}(x)$, as described by the first term on the right-hand side of the second equation of (\[eqlffp\]), and we leave the value of $Z_t(x)$ unchanged. We then let $H_s(x)$ decrease linearly until it reaches $0$; see the second term on the right-hand side of the second equation in (\[eqlffp\]). At all times where $H_s(x)>0$, that is, during $[t,t+z)$, the site $x$ acts like a barrier (see point 5 below).
4\. *Macroscopic fires.* Here, we assume that $x\in{\mathcal B}_T$ and that the corresponding mark of $M$ happens at some time $t$ where $Z_{{t-}}(x)=1$. This means that the cluster containing $x$ is macroscopic and thus this mark destroys the whole component $D_{{t-}}(x)$. That is, for all $y\in D_{{t-}}(x)$, we set $D_t(y)=\{y\}$, $Z_t(y)=0$. This is described by the second term on the right-hand side of the first equation in (\[eqlffp\]).
5\. *Clusters.* Finally, the definition of the clusters $(D_t(x))_{x\in{\mathbb{R}}}$ becomes more clear: these clusters are delimited by zones with microscopic sites \[i.e., $Z_t(y)<1$\] or by sites where there has (recently) been a microscopic fire \[i.e., $H_t(y)>0$\].
Main results
------------
First, we must note that it is not entirely clear that the limit process exists.
\[existe\] For any Poisson measure $M$, there a.s. exists a unique LFFP; recall Definition \[dflffp\]. Furthermore, it can be constructed graphically and thus its restriction to any finite box $[0,T]\times[-n,n]$ can be perfectly simulated.
To describe the convergence of the ${\lambda}$-FFP to the LFFP, we will need some more notation. Let ${\mathbb{D}}([0,T],E)$ denote the space of right-continuous and left-limited functions from the interval $[0,T]$ to a topological space $E$.
\[nocv\] (i) For two intervals $[a,b]$ and $[c,d]$, we set $\delta([a,b],[c,d])=|a-c|+|b-d|$. We also set, by convention, $\delta([a,b],\varnothing)=|b-a|$.
\(ii) For $(x,I), (y,J)$ in ${\mathbb{D}}([0,T],
{\mathbb{R}}\times{\mathcal I}\cup\{\varnothing\})$, let $$\delta_T((x,I),(y,J))={\sup_{[0,T]}}|x(t)-y(t)|+ \int_0^T\delta
(I(t),J(t)) \,dt.$$
We are finally in a position to state our main result.
\[converge\] Consider, for all ${\lambda}>0$, the processes $(Z^{\lambda}_t(x),D^{\lambda}_t(x))_{t\geq0,x\in{\mathbb{R}}}$ associated with the ${\lambda}$-FFP; see Definition \[dflaffp\] and (\[dlambda\]), (\[zlambda\]). Let $(Z_t(x),D_t(x)$,$H_t(x))_{t\geq0,x\in{\mathbb{R}}}$ be an LFFP, as in Definition \[dflffp\].
For any $T>0$ and any finite subset $\{x_1,\ldots,x_p\}\subset
{\mathbb{R}}$, $(Z^{\lambda}_t(x_i)$,$D^{\lambda}_t(x_i))_{t\in[0,T],i=1,\ldots,p}$ goes in law to $(Z_t(x_i),D_t(x_i))_{t\in[0,T],i=1,\ldots,p}$ in ${\mathbb{D}}([0,T]$, ${\mathbb{R}}\times
{\mathcal I})^p$ as ${\lambda}$ tends to $0$. Here, ${\mathbb{D}}([0,\infty)$, ${\mathbb{R}}\times{\mathcal I})$ is endowed with the distance $\delta_T$; see Notation \[nocv\].
For any finite subset $\{(t_1,x_1),\ldots,(t_p,x_p)\}\subset
{\mathbb{R}}_+ \times{\mathbb{R}}$, $(Z^{\lambda}_{t_i}(x_i)$,$D^{\lambda}_{t_i}(x_i))_{i=1,\ldots,p}$ goes in law to $(Z_{t_i}(x_i),D_{t_i}(x_i))_{i=1,\ldots,p}$ in $({\mathbb{R}}\times
{\mathcal I})^p$.
Observe that the process $H$ does not appear in the limit since for each $x\in{\mathbb{R}}$, a.s., for all $t\geq0$, $H_t(x)=0$. \[Of course, it is not the case that a.s., for all $x\in
{\mathbb{R}}$, all $t\geq0$, $H_t(x)=0$.\] We obtain the convergence of $D^{\lambda}$ to $D$ only when integrating in time. We cannot hope for a Skorokhod convergence since the limit process $D(x)$ jumps instantaneously from $\{x\}$ to some interval with positive length, while $D^{\lambda}(x)$ needs many small jumps (in a very short time interval) to become macroscopic.
As a matter of fact, we will obtain a convergence in probability, using a coupling argument. Essentially, we will consider a Poisson measure $M(dt,dx)$, as in Section \[sslffp\], and set, for ${\lambda}\in(0,1)$ and $i \in\mathbb{Z}$, $$M^{\lambda}_t(i)=M \bigl([0,t/\log(1/{\lambda})]\times
\bigl[i{\lambda}\log(1/{\lambda}),(i+1){\lambda}\log(1/{\lambda})\bigr) \bigr).$$ Then $(M^{\lambda}_t(i))_{t\geq0, i\in\mathbb{Z}}$ is an i.i.d. family of Poisson processes with rate ${\lambda}$.
The i.i.d. family of Poisson processes $(N_t(i))_{t\geq0, i\in\mathbb{Z}}$ with rate $1$ can be chosen arbitrarily, but we will decide to choose the same family for all values of ${\lambda}\in(0,1)$.
Heuristic arguments
-------------------
We now explain roughly the reasons why Theorem \[converge\] holds. We consider a ${\lambda}$-FFP $(\eta^{\lambda}_t)_{t\geq0}$ and the associated process $(Z^{\lambda}_t(x)$, $D^{\lambda}_t(x))_{t\geq
0,x\in{\mathbb{R}}}$. We assume below that ${\lambda}$ is very small.
0\. *Scales.* With our scales, there are $1/({\lambda}\log(1/{\lambda}))$ sites per unit of length. Approximately one fire starts per unit of time per unit of length. A vacant site becomes occupied at rate $\log(1/{\lambda})$.
1\. *Initial condition.* We have, for all $x\in{\mathbb{R}}$, $(Z^{\lambda}_0(x),D^{\lambda}_0(x))= (0,\varnothing) \simeq(0,\{x\})$.
2\. *Occupation of vacant zones.* Assume that a zone $[a,b]$ (which corresponds to the zone ${[\![}\lfloor
a/({\lambda}\log(1/{\lambda}))\rfloor,b/({\lambda}\log(1/{\lambda
}))\rfloor{]\!]}$ before rescaling) becomes completely vacant at some time $t$ \[or $t\log(1/{\lambda})$ before rescaling\] because it has been destroyed by a fire.
For $s\in[0,1)$, and if no fire starts on $[a,b]$ during $[t,t+s]$, we have $D^{\lambda}_{t+s}(x) \simeq[x \pm{\lambda}^{1-s}]$ and thus $Z^{\lambda}_{t+s}(x)\simeq s$ for all $x\in[a,b]$.
Indeed, $D^{\lambda}_{t+s}(x)\simeq[x-{\lambda}\log(1/{\lambda})X,
x+{\lambda}\log(1/{\lambda}) Y]$, where $X$ and $Y$ are geometric random variables with parameter $1-e^{-s\log(1/{\lambda})}=1-{\lambda}^s$. This comes from the fact that each site of $[a,b]$ is vacant at time $t$ and becomes occupied at rate $\log(1/{\lambda})$.
If no fire starts on $[a,b]$ during $[t,t+1]$, then $Z^{\lambda}_{t+1}(x)\simeq1$ and all the sites in $[a,b]$ are occupied (with very high probability) at time $t+1$. Indeed, we have $(b-a)/({\lambda}\log(1/{\lambda}))$ sites and each of them is occupied at time $t+1$ with probability $1-e^{-\log(1/{\lambda})}=1-{\lambda}$ so that all of them are occupied with probability $(1-{\lambda})^{(b-a)/({\lambda}\log(1/{\lambda
}))}\simeq
e^{-(b-a)/\log(1/{\lambda})}$, which goes to $1$ as ${\lambda}\to0$.
3\. *Microscopic fires.* Assume that a fire starts at some location $x$ (i.e., $\lfloor
x/({\lambda}\log(1/{\lambda}))\rfloor$ before rescaling) at some time $t$ \[or $t\log(1/{\lambda})$ before rescaling\] with $Z_{t-}^{\lambda}(x)=z\in(0,1)$. The possible clusters on the left and right of $x$ cannot then be connected during (approximately) $[t,t+z]$, but they can be connected after (approximately) $t+z$. In other words, $x$ acts like a barrier during $[t,t+z]$.
Indeed, the fire makes vacant a zone $A$ of approximate length ${\lambda}^{1-z}$ around $x$, which thus contains approximately ${\lambda}^{1-z}/({\lambda}\log
(1/{\lambda}))
\simeq
{\lambda}^{-z}$ sites. The probability that a fire starts again in $A$ after $t$ is very small. Thus, using the same computation as in point 2(ii), we observe that ${\mathbb{P}}[A $ is completely occupied at time $ t+s]\simeq(1-{\lambda}^s)^{{\lambda}^{-z}}\simeq
e^{-{\lambda}^{s-z}}$. When ${\lambda}\to0$, this quantity tends to $0$ if $s<z$ and to $1$ if $s>z$.
4\. *Macroscopic fires.* Assume, now, that a fire starts at some place $x$ (i.e., $\lfloor
x/({\lambda}\log(1/{\lambda}))\rfloor$ before rescaling) at some time $t$ \[or $t\log(1/{\lambda})$ before rescaling\] and that $Z^{\lambda}_{t}(x) \simeq1$. Thus, $D^{\lambda}_{t}(x)$ is macroscopic (i.e., its length is of order $1$ in our scales). This will thus make vacant the zone $D^{\lambda
}_t(x)$. Such a (macroscopic) zone needs a time of order $1$ to be completely occupied, as explained in point 2(ii).
5\. *Clusters.* For $t\geq0$, $x\in{\mathbb{R}}$, the cluster $D_t^{\lambda}(x)$ resembles $[x\pm{\lambda}^{1-z}]\simeq\{x\}$ if $Z_t^{\lambda
}(x)=z\in(0,1)$. We then say that $x$ is microscopic. Now, macroscopic clusters are delimited either by microscopic zones or by sites where there has been a microscopic fire (see point 3).
Comparing the arguments above to the rough description of the LFFP (see Section \[sslffp\]), our hope is that the ${\lambda}$-FFP resembles the LFFP for ${\lambda}>0$ very small.
Decay of correlations
---------------------
A byproduct of our result is an estimate on the decay of correlations in the LFFP for finite times. We refer to Proposition \[plocla\] below for a precise statement. The main idea is that for all $T>0$, there are constants $C_T>0$, $\alpha_T>0$ such that for all ${\lambda
}\in
(0,1)$ and all $A>0$, the values of the ${\lambda}$-FFP inside $[-A/({\lambda}\log(1/{\lambda
})),A/({\lambda}\log(1/{\lambda}))]$ are independent of the values outside $[-2A/({\lambda}\log(1/{\lambda})),
2A/({\lambda}\log(1/{\lambda}))]$ during the time interval $[0,T\log(1/{\lambda})]$, up to a probability smaller that $C_Te^{-\alpha_T A}$. In other words, for times of order $\log(1/{\lambda})$, the range of correlations is at most of order $1/({\lambda}\log(1/{\lambda}))$.
Cluster size distribution
-------------------------
Finally, we give results on the cluster size distribution, which are to be compared with [@vdbj; @bp]; see Section \[bibi\] above.
\[coco\] For each ${\lambda}>0$, consider a ${\lambda}$-FFP process $(\eta
^{\lambda}_t)_{t\geq0}$.
For some $0<c<C$, all $t\geq5/2$ and all $0\leq a < b < 1$, $$c (b-a) \leq\lim_{{\lambda}\to0}
{\mathbb{P}}\bigl(\#\bigl(C^{\lambda}_{t\log(1/{\lambda})}(0)\bigr) \in
[{\lambda}^{-a},{\lambda}^{-b}] \bigr) \leq C (b-a).$$
For some $0<c<C$, some $0< \kappa_1 <\kappa_2$, all $t \geq3/2$ and all $B>0$, $$c e^{-\kappa_2 B} \leq\lim_{{\lambda}\to0}
{\mathbb{P}}\bigl(\#\bigl(C^{\lambda}_{t\log(1/{\lambda})}(0)\bigr)
\geq B / \bigl({\lambda}\log(1/{\lambda})\bigr) \bigr)
\leq C e^{-\kappa_1 B}.$$
Point (i) says, roughly, that for $t$ large enough (say at equilibrium) and for $x << 1/{\lambda}$ \[say for $x \leq(1/{\lambda})^{1-{\varepsilon}}$\], choosing $a=\log(x) / \log(1/{\lambda})$ and $b=\log(x+1) / \log
(1/{\lambda})$, we have $$\begin{aligned}
{\mathbb{P}}\bigl(\#(C^{\lambda}(0)) = x\bigr) &\simeq&{\mathbb{P}}\bigl(\#
(C^{\lambda}(0)) \in[x,x+1]\bigr)\simeq
{\mathbb{P}}\bigl(\#(C^{\lambda}(0)) \in[{\lambda}^{-a},{\lambda
}^{-b}]\bigr)\\
&\simeq&(b-a) \simeq\frac{1}
{x\log(1/{\lambda})}.\end{aligned}$$ It is thus a very weak form of the result of [@bp], but it holds for a much wider class of $x$: here, we allow $x \leq1/{\lambda}^{1-{\varepsilon}}$, while $x\leq
1/{\lambda}^{1/3}$ was imposed in [@bp]. Another advantage of our result is that we can prove that the limit exists in (i).
![Shape of the cluster size distribution. Here, ${\lambda}=0.0001$ and the critical size is thus $1/({\lambda}\log(1/{\lambda}))\simeq1085$. We have drawn the approximate value (computed roughly just after Corollary \[coco\]) of $\log({\mathbb{P}}(\#(C^{\lambda
}(0))=x))$ as a function of $\log(x)$ for $x=1,\ldots,54\mbox{,}250$. We have made the curve continuous around $x=1085$ (without justification). The curve is linear for $x=1,\ldots,1085$ and nonlinear for $x\geq1085$.[]{data-label="figure1"}](524f01.eps)
Point (ii) roughly describes the cluster size distribution of macroscopic components, that is, of components of which the size is of order $1/({\lambda}\log(1/{\lambda}))$. Here, again, rough computations show that for $x > {\varepsilon}/({\lambda}\log(1/{\lambda}))$ and for $t$ large enough (say at equilibrium), $${\mathbb{P}}\bigl(\#(C^{\lambda}(0)) = x\bigr) \simeq{\lambda}\log
(1/{\lambda}) e^{-\kappa x {\lambda}\log
(1/{\lambda})}.$$ Thus, there is clearly a phase transition near the *critical size* $1/({\lambda}\log(1/{\lambda}))$; see Figure \[figure1\] for an illustration.
Organization of the paper
-------------------------
The paper is organized as follows. In Section \[ex\], we give the proof of Theorem \[existe\]. In Section \[locla\], we show that, in some sense, the ${\lambda}$-FFP can be localized in a finite box, uniformly for ${\lambda}>0$. Section \[cv\] is devoted to the proof of Theorem \[converge\]. Finally, we prove Corollary \[coco\] in Section \[conseq\].
Existence and uniqueness of the limit process {#ex}
=============================================
The goal of this section is to show that the LFFP is well defined, unique and can be obtained from a graphical construction. First, we show that when working on a finite space interval, the LFPP is somewhat discrete.
We consider a Poisson measure $M(dt,dx)$ on $[0,\infty) \times
{\mathbb{R}}$ with intensity measure $dt\, dx$. We define ${\mathcal F}_t^{M,A}=\sigma(M(B),B\in{\mathcal B}([0,t] \times[-A,A]))$.
\[dflffpA\] A $({\mathcal F}_t^{M,A})_{t\geq0}$-adapted process $$(Z_t^A(x),D_t^A(x), H_t^A(x))_{t\geq0,x\in[-A,A]}$$ with values in ${\mathbb{R}}_+\times{\mathcal I}\times{\mathbb{R}}_+$ is called an $A$-LFFP if a.s., for all $t\geq0$ and all $x \in[-A,A]$, $$\cases{
\displaystyle Z_t^A(x)= {\int_0^t}{\mathbf{1}}_{\{Z_s^A (x) < 1\}}\,ds -
{\int_0^t}\int_{[-A,A]} {\mathbf{1}}_{\{ Z_{{s-}}^A(x)=1,y \in
D_{{{s-}}}^A(x)\}}M(ds,dy),\cr
\displaystyle H_t^A(x)=
{\int_0^t}Z_{{s-}}^A(x){\mathbf{1}}_{\{Z_{{s-}}^A(x)<1\}} M(ds\times
\{x\})
- {\int_0^t}{\mathbf{1}}_{\{H_s^A (x) > 0 \}}\,ds,}$$ where $D_t^A(x) = [L_t^A(x),R_t^A(x)]$ with $$\label{tructruc}
\cases{\displaystyle L_t^A(x) = (-A) \lor\sup\{ y \in[-A,x]; Z_t^A(y)<1
\mbox{ or } H_t^A(y)>0 \},\cr
\displaystyle R_t^A(x) = A \land\inf\{ y\in[x,A]; Z_t^A(y)<1 \mbox{ or } H_t^A(y)>0
\}.}$$
A typical path of $(Z_t^A(x),D_t^A(x),H_t^A(x))_{t\geq0,x\in[-A,A]}$ is drawn in Figure \[figure2\].
![Limit forest fire process in a finite box. The filled zones represent zones in which $Z_t^A(x)=1$ and $H_t^A(x)=0$, that is, macroscopic clusters. The plain vertical segments represent the sites where $H_t^A(x)>0$. In the rest of the space, we always have $Z_t^A(x)<1$. Until time $1$, all of the particles are microscopic. The first eight marks of the Poisson measure fall in that zone. As a consequence, at each of these marks, the process $H^A$ starts. Their lifetime is equal to the instant where they have started (e.g., the segment above $t_1,x_1$ ends at time $2t_1$). At time $1$, all of the clusters where there has been no mark become macroscopic and merge together. However, this is limited by vertical segments. Here, at time $1$, we have the clusters $[-A,x_6]$, $[x_6,x_4]$, $[x_4,x_8]$, $[x_8,x_5]$, $[x_5,x_7]$ and $[x_7,A]$. The segment above $(t_4,x_4)$ ends at time $2t_4$ and thus, at this time, the clusters $[x_6,x_4]$ and $[x_4,x_8]$ merge into $[x_6,x_8]$. The ninth mark falls in the (macroscopic) zone $[x_6,x_8]$ and thus destroys it immediately. This zone $[x_6,x_8]$ will become macroscopic again only at time $t_9+1$. A process $H^A$ then starts at $x_{12}$ at time $t_{12}$. Since $Z^A_{t_{12}-}(x_{12})=t_{12}-t_9$ \[because $Z^A_{t_9}(x_{12})$ has been set to $0$\], the segment above $(t_{12},x_{12})$ will end at time $2t_{12}-t_9$. On the other hand, the segment $[x_8,x_7]$ has been destroyed at time $t_{10}$ and will thus remain microscopic until $t_{10}+1$. As a consequence, the only macroscopic clusters at time $t_9+1$ are $[-A,x_{12}]$, $[x_{12},x_8]$ and $[x_7,A]$. The zone $[x_8,x_7]$ then becomes macroscopic (but there have been marks at $x_{13}, x_{14}$) so that at time $t_{10}+1$, we get the macroscopic clusters $[-A,x_{12}]$, $[x_{12},x_{14}]$, $[x_{14},x_{13}]$ and $[x_{13},A]$. These clusters merge by pairs, at times $2t_{12}-t_9$, $2t_{13}-t_{10}$ and $2t_{14}-t_{10}$, so that we have a unique cluster $[-A,A]$ just before time $t_{15}$, where a mark falls and destroys the whole cluster $[-A,A]$.With this realization, we have $0 \in(x_{11},x_{15})$ and, thus, $Z^A_t(0)= t$ for $t \in[0,1]$, then $Z^A_t(0)=1$ for $t \in
[1,t_{10})$, then $Z^A_t(0)= t-t_{10}$ for $t \in[t_{10},t_{10}+1)$, then $Z^A_t(0)=1$ for $t \in[t_{10}+1,t_{15})$, etc. We also see that $D^A_t(0)=\{0\}$ for $t \in[0,1)$, $D^A_t(0)=[x_8,x_5]$ for $t\in[1,2t_5)$, $D^A_t(0)=[x_8,x_7]$ for $t\in[2t_5,t_{10})$, $D^A_t(0)=\{0\}$ for $t\in[t_{10},t_{10}+1)$, $D^A_t(0)=[x_{12},x_{14}]$ for $t \in[t_{10}+1,2t_{12}-t_9)$, $D^A_t(0)=[-A,x_{14}]$ for $t \in[2t_{12}-t_9,2t_{14}-t_{10})$, etc. Of course, $H^A_t(0)=0$ for all $t\geq0$, but, for example, $H^A_t(x_{11})=0$ for $t\in[0, t_{11})$, $H^A_t(x_{11})=2t_{11}-t_{10}-t$ for $t\in[t_{11},2t_{11}-t_{10})$ and then $H^A_t(x_{11})=0$ for $t\in[2t_{11}-t_{10},\infty)$.[]{data-label="figure2"}](524f02.eps)
Although the following proposition is almost obvious, its proof shows the construction of the $A$-LFFP in an algorithmic way.
\[eufini\] Consider a Poisson measure $M(dt,dx)$ on $[0,\infty) \times{\mathbb
{R}}$ with intensity measure $dt\, dx$. For any $A>0$, there a.s. exists a unique $A$-LFFP which can be perfectly simulated.
We omit the superscript $A$ in this proof. We consider the marks $(T_i,X_i)_{i \geq1}$ of $M\vert_{[0,\infty
)\times[-A,A]}$, where $0<T_1<T_2<\cdots.$ We set $T_0=0$ for convenience. We describe the construction via an algorithm, which also shows uniqueness, in the sense that there is no choice in the construction.
*Step* 0. First, we set $Z_0(x)=H_0(x)=0$ and $D_0(x)=\{x\}$ for all $x\in[-A,A]$.
*Step* $n+1$. Assume that the process has been built until $T_n$ for some $n\geq0$, that is, we know the values of $(Z_t(x),D_t(x),H_t(x))_{t\in[0,T_n],x\in[-A,A]}$.
We build $(Z_t(x),D_t(x),H_t(x))_{t\in(T_n,T_{n+1}),x\in[-A,A]}$ in the following way: for $t\in(T_n,T_{n+1})$ and $x\in[-A,A]$, we set $Z_t(x)= \min(1,Z_{T_n}(x)+t-T_n)$, $H_t(x)=\max
(0,H_{T_n}(x)-(t-T_n))$ and define $D_t(x)=[L_t(x),R_t(x)]$, as in (\[tructruc\]).
Next, we build $(Z_{T_{n+1}}(x),D_{T_{n+1}}(x),H_{T_{n+1}}(x))_{x\in[-A,A]}$.
If $Z_{T_{n+1}-}(X_{n+1})=1$, then we set $H_{T_{n+1}}(x)
=H_{T_{n+1}-}(x)$ for all $x\in[-A,A]$ and consider $[a,b]:= D_{T_{n+1}-}(X_{n+1})$. Set $Z_{T_{n+1}}(x) = 0$ for all $x\in(a,b)$ and $Z_{T_{n+1}}(x) =
Z_{T_{n+1}-}(x)$ for all $x\in[-A,A] \setminus[a,b]$. Finally, set:$Z_{T_{n+1}}(a)=0$ if $Z_{T_{n+1}-}(a)=1$; $Z_{T_{n+1}}(a)=Z_{T_{n+1}-}(a)$ if $Z_{T_{n+1}-}(a)<1$;$Z_{T_{n+1}}(b)=0$ if $Z_{T_{n+1}-}(b)=1$; $Z_{T_{n+1}}(b)=Z_{T_{n+1}-}(b)$ if $Z_{T_{n+1}-}(b)<1$.
If $Z_{T_{n+1}-}(X_{n+1})<1$, then we set $H_{T_{n+1}}(X_{n+1})=Z_{T_{n+1}-}(X_{n+1})$,$Z_{T_{n+1}}(X_{n+1})=Z_{T_{n+1}-}(X_{n+1})$ and $(Z_{T_{n+1}}(x),H_{T_{n+1}}(x))
=(Z_{T_{n+1}-}(x)$,$H_{T_{n+1}-}(x))$ for all $x\in[-A,A]\setminus\{X_{n+1}\}$.
Using the values of $(Z_{T_{n+1}}(x),H_{T_{n+1}}(x))_{x\in[-A,A]}$, we finally compute the values of $(D_{T_{n+1}}(x))_{x\in[-A,A]}$.
In case (i) above, we explained precisely what is done at the boundary of burning macroscopic components. This is not so important: it does not affect the uniqueness statement, but corresponds to using a slightly different definition of the process; we could have made other choices for this.
We now prove a refined version of Theorem \[existe\].
\[loc\] Consider a Poisson measure $M(dt,dx)$ on $[0,\infty) \times{\mathbb
{R}}$ with intensity measure $dt \,dx$. For $A>0$, consider the $A$-LFFP $(Z^A_t(x),D^A_t(x)$,$H^A_t(x))_{t\geq0,x\in[-A,A]}$ constructed in Proposition \[eufini\] (using $M$).
There a.s. exists a unique LFFP $(Z_t(x),D_t(x),H_t(x))_{t\geq0,x\in{\mathbb{R}}}$ (corresponding to $M$) and, furthermore, it is such that for all $T>0$, there are constants $\alpha_T>0$ and $C_T>0$ such that for all $A\geq2$, $$\begin{aligned}
\label{mix}
&&{\mathbb{P}}\bigl[(Z_t(x),D_t(x),H_t(x))_{t\in[0,T],x\in[-A/2,A/2]}
\nonumber\\[-8pt]\\[-8pt]
&&\qquad= (Z_t^A(x),D_t^A(x),H_t^A(x))_{t\in[0,T],x\in[-A/2,A/2]} \bigr]
\geq1 - C_T e^{-\alpha_T A}.\nonumber\end{aligned}$$
We divide the proof into several steps. We fix $T>0$ and work on $[0,T]$.
*Step* 1. For $a \in\mathbb{Z}$, we define the event $\Omega_a$ in the following way (see Figure \[figure3\] for an illustration).
![The event $\Omega_a$ (proof of Theorem \[existe\]). In hatched zones, we cannot state the values of the LFFP because one would need to know what happens outside $[a,a+1]$. Microscopic fires start at $(T_1,X_1)$ and $({\tilde T}_1,{\tilde X}_1)$. Hence, at time $S_1$, the connected component $[X_1,{\tilde X}_1]$ is macroscopic because $S_1\geq1$ and because during $[1,S_1)$, this component has not been subject to fires starting outside $[a,a+1]$: it is protected by $X_1$ and ${\tilde X}_1$ until time $2\min(T_1,{\tilde T}_1) \geq S_1$. As a consequence, the component $[X_1,{\tilde X}_1]$ is entirely killed by $(S_1,Y_1)$. We then iterate the arguments until we reach the final time $T$. With such a configuration, there are always *microscopic* sites in $[a,a+1]$ during $[0,T]$. Indeed, during $[0,1)$, all of the sites are microscopic, during $[1,S_1)$, the sites $X_1$ and ${\tilde X}_1$ are microscopic, during $[S_1,S_1+1)$, all the sites in $[X_1,{\tilde X}_1]$ are microscopic, etc.[]{data-label="figure3"}](524f03.eps)
The Poisson measure $M$ has exactly $3n$ marks in $[0,T]\times[a,a+1]$ for some $n\geq1$ and it is possible to call them $(T_k,X_k)_{k=1,\ldots,n}$, $({\tilde T}_k,{\tilde X}_k)_{k=1,\ldots,n}$ and $(S_k,Y_k)_{k=1,\ldots,n}$ in such a way that we have the following properties for all $k=1,\ldots,n$ (we set $T_0={\tilde T}_0=S_0=0$ and $X_0=a$, ${\tilde X}_0=a+1$ for convenience):
$T_k$ and ${\tilde T}_k$ belong to $(S_{k-1}+1/2,S_{k-1}+1)$ and $X_{k-1}<X_k<{\tilde X}_k<{\tilde X}_{k-1}$;
$S_k\in(S_{k-1}+1,S_{k-1}+2(T_k\land{\tilde T}_k-S_{k-1}))$ and $Y_k \in(X_k,{\tilde X}_k)$;
$S_n > T-1$.
*Step* 2. We next observe that if the LFFP exists, then, necessarily, $$\Omega_a \subset\{\forall t\in[0,T], \exists x\in(a,a+1),
H_t(x)>0 \mbox{ or } Z_t(x)<1 \}.$$ Indeed, $Z_t(x) =t <1$ for all $t\in[0,1)$ and $x\in{\mathbb{R}}$. Then $H_{T_1}(X_1)=Z_{T_1}(X_1)=T_1$, whence $H_t(X_1)>0$ on $[T_1,
2T_1]$ and $H_t({\tilde X}_1)>0$ on $[{\tilde T}_1,2{\tilde T}_1]$. As a consequence, we know that for all $x\in(X_1,{\tilde X}_1)$ and $t\in[1,S_1)$, we have $D_t(x)=[X_1,{\tilde X}_1]$. Since, now, $1<S_1<2(T_1\land{\tilde
T}_1)$ and since $Y_1 \in(X_1,{\tilde X}_1)$, we deduce that $Z_{S_1}(x)=0$ for all $x\in(X_1,{\tilde X}_1)$ and, as a consequence, $Z_t(x)=t-S_1<1$ for all $t\in[S_1,S_1+1)$. However, we now have $H_t(X_2)>0$ on $[T_2,
T_2+(T_2-S_1))$ and $H_t({\tilde X}_2)>0$ on $[{\tilde T}_2,{\tilde T}_2+({\tilde
T}_2-S_1))$. As a consequence, we know for all $x\in(X_2,{\tilde X}_2)$ and $t\in[S_1+1,S_2)$ that $D_t(x)=[X_2,{\tilde X}_2]$. Since, now, $S_1+1<S_2<S_1+2(T_1\land
{\tilde T}_1-S_1)$ and $Y_2 \in(X_2,{\tilde X}_2)$, we deduce that $Z_{S_2}(x)=0$ for all $x\in(X_2,{\tilde X}_2)$ and thus $Z_t(x)=t-S_2<1$ for all $t\in[S_2,S_2+1)$, etc.
*Step* 3. We deduce that for all $a\in\mathbb{Z}$, conditionally on $\Omega_a$, clusters to the left of $a$ are never connected (during $[0,T]$) to clusters to the right of $a+1$. Thus, on $\Omega_a$, fires starting to the left of $a$ do not affect the zone $[a+1,\infty)$ and fires starting to the right of $a+1$ do not affect the zone $(-\infty,a]$. Since, further, $\Omega_a$ concerns the Poisson measure $M$ only in $[0,T]\times[a,a+1]$, we deduce that on $\Omega_a$, the processes $(Z_t(x),D_t(x),H_t(x))_{t\geq0,x\in[a+1,\infty)}$ and $(Z_t(x),D_t(x),H_t(x))_{t\geq0,x\in(-\infty, a]}$ can be constructed separately.
*Step* 4. Clearly, $q_T={\mathbb{P}}[\Omega_a]$ does not depend on $a$, by translation invariance (of the law of $M$), and obviously $q_T>0$. Thus, a.s. there are infinitely many $a\in\mathbb{Z}$ such that $\Omega_a$ is realized. This allows a graphical construction: it suffices to work between such $a$’s (i.e., in finite boxes), as in Proposition \[eufini\].
*Step* 5. Using the same arguments, we easily deduce that for $A\geq2$, the LFFP and the $A$-LFFP coincide on $[-A/2,A/2]$ during $[0,T]$, provided that there are $a_1 \in[-A,-A/2-1]$ and $a_2\in[A/2,A-1]$ with $\Omega_{a_1} \cap\Omega_{a_2}$ realized. Furthermore, since $M$ is a Poisson measure, $\Omega_a$ is independent of $\Omega_b$ for all $a\ne b$ (with ). Thus, the probability on the left-hand side of (\[mix\]) is bounded below, for $A\geq2$, by $$1- {\mathbb{P}}\biggl[\bigcap_{a\in\mathbb{Z}\cap[-A,-A/2-1]} \Omega_a^c\biggr] -
{\mathbb{P}}\biggl[\bigcap_{a\in\mathbb{Z}\cap[A/2, A-1]} \Omega_a^c\biggr] \geq
1 - 2 (1-q_T)^{A/2-2},$$ hence we have (\[mix\]) with $\alpha_T=-\log(1-q_T)/2>0$ and $C_T=2/(1-q_T)^2$.
Localization of the FFP {#locla}
=======================
We first introduce the $({\lambda},A)$-FFP. We consider two independent families of i.i.d. Poisson processes $N=(N_t(i))_{t\geq0,i\in\mathbb{Z}}$ and $M^{\lambda}=(M^{\lambda}_t(i))_{t\geq0,i\in\mathbb{Z}}$, with respective rates $1$ and ${\lambda}>0$. For $A>0$ and ${\lambda}>0$, we define $$\label{alaila}
A_{\lambda}:=\bigl\lfloor A/\bigl({\lambda}\log(1/{\lambda})\bigr)\bigr\rfloor
\quad\mbox{and}\quad
I_A^{\lambda}:={[\![}-A_{\lambda},A_{\lambda}{]\!]},$$ and we set ${\mathcal F}_t^{N,M^{\lambda},A}:=\sigma(N_s(i),M^{\lambda}_s(i),
s\leq t, i \in
I_A^{\lambda})$.
\[dflaffpA\] Consider an $({\mathcal F}_t^{N,M^{\lambda},A})_{t\geq0}$-adapted process $(\eta^{{\lambda},A}_t)_{t\geq0}$ with values in $\{0,1\}
^{I_A^{\lambda}}$, such that $(\eta^{{\lambda},A}_t(i))_{t\geq0}$ is a.s. càdlàg for all $i\in I_A^{\lambda}$.
We say that $(\eta^{{\lambda},A}_t)_{t\geq0}$ is a $({\lambda},A)$-FFP if a.s., for all $t\geq0$ and $i\in
I_A^{\lambda}$, $$\eta^{{\lambda},A}_t(i)={\int_0^t}{\mathbf{1}}_{\{\eta^{{\lambda
},A}_{{s-}}(i)=0\}} \,dN_s(i)
- \sum_{k\in I_A^{\lambda}}{\int_0^t}{\mathbf{1}}_{\{k\in
C^{{\lambda},A}_{{s-}}(i)\}}
\,dM^{\lambda}_s(k),$$ where $C^{{\lambda},A}_s(i)=\varnothing$ if $\eta_t^{{\lambda
},A}(i)=0$, while $C^{{\lambda},A}_s(i)={[\![}l_s^{{\lambda},A}(i),r_s^{{\lambda
},A}(i){]\!]}$ if $\eta^{{\lambda},A}_s(i)=1$, where $$\begin{aligned}
l_s^{{\lambda},A}(i)&=&(-A_{\lambda}) \lor\bigl(\sup\{k< i; \eta
_s^{{\lambda},A}(k)=0\}+1
\bigr),\\
r_s^{{\lambda},A}(i)&=&A_{\lambda}\land\bigl(\inf\{k > i; \eta
_s^{{\lambda},A}(k)=0\}-1 \bigr).\end{aligned}$$ For $x \in[-A,A]$ and $t\geq0$, we introduce $$\begin{aligned}
\label{dlambdaA}
D^{{\lambda},A}_t(x)&=&{\lambda}\log(1/{\lambda}) C^{{\lambda
},A}_t\bigl(\bigl\lfloor x/\bigl({\lambda}\log
(1/{\lambda})\bigr)\bigr\rfloor\bigr)
\subset[-A,A],
\\
\label{zlambdaA}
Z^{{\lambda},A}_t(x)&=&\frac{\log
[1+\# (C^{{\lambda},A}_t(\lfloor x/({\lambda}\log(1/{\lambda
}))\rfloor) ) ]}
{\log(1/{\lambda})} \geq0.\end{aligned}$$
We now prove the following result, which is similar to Proposition \[loc\] for the ${\lambda}$-FFP.
\[plocla\] Let $T>0$ and ${\lambda}\in(0,1)$. Consider two families of Poisson processes $N=(N_t(i))_{t\geq0,i\in\mathbb{Z}}$ and $M^{\lambda}=(M^{\lambda}_t(i))_{t\geq0,i\in\mathbb{Z}}$ with respective rates $1$ and ${\lambda}>0$. Let $(\eta^{\lambda}_t)_{t\geq0}$ be the corresponding ${\lambda}$-FFP and, for each $A>0$, let $(\eta^{{\lambda},A}_t)_{t\geq0}$ be the corresponding $({\lambda},A)$-FFP. Recall (\[dlambda\]), (\[zlambda\]) and (\[dlambdaA\]), (\[zlambdaA\]). There are constants $\alpha_T>0$ and $C_T>0$, not depending on ${\lambda}\in(0,1)$, $A\geq2$, such that \[recalling (\[alaila\])\] $$\begin{aligned}
&&{\mathbb{P}}\bigl[(\eta^{\lambda}_t(i))_{t\in[0,T\log(1/{\lambda
})],i\in I_{A/2}^{\lambda}}
=(\eta^{{\lambda},A}_t(i))_{t\in[0,T\log(1/{\lambda})],i\in
I_{A/2}^{\lambda}} \bigr]\\
&&\qquad
\geq1 - C_T e^{-\alpha_T A},\\
&&{\mathbb{P}}\bigl[(Z^{\lambda}_t(x),D^{\lambda}_t(x))_{t\in[0,T],x \in
[-A/2,A/2]}
=(Z^{{\lambda},A}_t(x),D^{{\lambda},A}_t(x))_{t\in[0,T],x \in
[-A/2,A/2]} \bigr] \\
&&\qquad \geq1 - C_T e^{-\alpha_T A}.\end{aligned}$$
The proof is similar (but more complicated) to that of Proposition \[loc\]. Consider the true ${\lambda}$-FFP $(\eta_t^{\lambda}(i))_{t\geq0, i
\in\mathbb{Z}}$. Temporarily assume that for $a \in{\mathbb{R}}$, there is an event $\Omega_a^{\lambda}$, depending only on the Poisson processes $N_t(i)$ and $M^{\lambda}_t(i)$ for $t\in
[0,T\log
(1/{\lambda})]$ and $i \in J_a^{\lambda}:= {[\![}\lfloor a/({\lambda}\log
(1/{\lambda}))\rfloor,
\lfloor(a+1)/({\lambda}\log(1/{\lambda}))\rfloor{]\!]}$, such that:
on $\Omega^{\lambda}_a$, a.s., for all $t\in[0,T\log(1/{\lambda
})]$, there is some $i \in J_a^{\lambda}$ such that $\eta^{\lambda}_t(i)=0$;
there exists $q_T>0$ such that for all $a \in{\mathbb{R}}$ and ${\lambda}\in
(0,1)$, we have ${\mathbb{P}}(\Omega^{\lambda}_a) \geq q_T$.
The proof is then concluded using arguments similar to Steps 3, 4, 5 of the proof of Proposition \[loc\].
Fix some $\alpha>0$ and some ${\varepsilon}_T>0$ small enough, say $\alpha=0.01$ and ${\varepsilon}_T = 1/(32T)$. Let ${\lambda}_T>0$ be such that for ${\lambda}\in(0,{\lambda}_T)$, we have $1<{\lambda}^{\alpha-1} < \epsilon_T /\break({\lambda}\log(1/{\lambda}))$.
For ${\lambda}\in[{\lambda}_T,1)$ and $a\in{\mathbb{R}}$, we set $\Omega_a^{\lambda}=\{N_{T\log(1/{\lambda})}(\lfloor a/({\lambda
}\log(1/{\lambda}
))\rfloor)=0\}$, on which, of course, $\eta_t^{\lambda}(i)=0$ for all $t\in[0,T\log(1/{\lambda})]$ with $i=\lfloor a/({\lambda}\log(1/{\lambda}))\rfloor\in J_a^{\lambda}$. We then observe that $q'_T= \inf_{{\lambda}\in[{\lambda}_T,1)}
P(\Omega_a^{\lambda})
= \inf_{{\lambda}\in[{\lambda}_T,1)} e^{-T\log(1/{\lambda})}=
({\lambda}_T)^T>0$.
For ${\lambda}\in(0,{\lambda}_T)$ and $a\in{\mathbb{R}}$, we define the event $\Omega^{\lambda}_a$ on which points 1, 2 and 3 below are satisfied.
1\. The family of Poisson processes $(M^{\lambda}_t(i))_{t \in[0,T
\log
(1/{\lambda})],
i \in J_a^{\lambda}}$ has exactly $3n$ marks for some $1 \leq n \leq\lfloor T \rfloor$ and it is possible to call them $(T^{\lambda}_k,X^{\lambda}_k)_{k=1,\ldots,n}$, $({\tilde T}^{\lambda}_k,{\tilde X}^{\lambda}_k)_{k=1,\ldots,n}$ and $(S^{\lambda}_k,Y^{\lambda}_k)_{k=1,\ldots,n}$ in such a way that we have the following properties for all $k=1,\ldots,n$ (we set $T^{\lambda}_0={\tilde T}^{\lambda}_0=S^{\lambda
}_0=0$ and $X^{\lambda}_0=\lfloor a/\break({\lambda}\log(1/{\lambda}))\rfloor$, ${\tilde X}^{\lambda}_0=\lfloor(a+1)/({\lambda}\log(1/{\lambda
}))\rfloor$):
(1a) $X^{\lambda}_{k-1} < X^{\lambda}_k < Y^{\lambda}_k< {\tilde
X}^{\lambda}_k< {\tilde X}^{\lambda}_{k-1}$ with $\min\{ X^{\lambda}_k-X^{\lambda}_{k-1}, Y^{\lambda}_k - X^{\lambda
}_k, {\tilde X}^{\lambda}_k -
Y^{\lambda}_k,
{\tilde X}^{\lambda}_{k-1} - {\tilde X}^{\lambda}_k\} \geq4 \epsilon
_T / ({\lambda}\log(1/{\lambda}))$;
(1b) $T^{\lambda}_k$ and ${\tilde T}^{\lambda}_k$ belong to $[S^{\lambda}_{k-1}+(\frac
{1}{2} +
\alpha) \log(1/{\lambda}), S^{\lambda}_{k-1}+(1-\alpha) \log
(1/{\lambda})]$;
(1c) $S^{\lambda}_k\in[S^{\lambda}_{k-1}+(1+ \alpha) \log
(1/{\lambda}), S^{\lambda}_{k-1}
+2(T^{\lambda}_k\land{\tilde T}^{\lambda}_k-S^{\lambda}_{k-1}) -
\alpha\log(1/{\lambda})]$;
(1d) $S^{\lambda}_n\geq(T-1+\alpha) \log(1/{\lambda})$.
2\. For $k=1,\ldots,n$, we now set $\tau_k^{\lambda}=(S^{\lambda}_k-S^{\lambda}_{k-1})/(2\log
(1/{\lambda}))$, which belongs to $[(1+\alpha)/2,1-\alpha]$, due to 1. We consider the intervals $$\begin{aligned}
I^{\lambda}_k&=&{[\![}X^{\lambda}_k- \lfloor{\lambda}^{-\tau
_k^{\lambda}} \rfloor,
X^{\lambda}_k
+ \lfloor{\lambda}^{-\tau_k^{\lambda}} \rfloor{]\!]},\nonumber\\
I^{\lambda}_{k,-}&=& {[\![}X^{\lambda}_k - \lfloor{\lambda
}^{-\tau_k^{\lambda}}
\rfloor
- \lfloor{\varepsilon}_T/{\lambda}\log(1/{\lambda}) \rfloor,
X^{\lambda}_k -
\lfloor{\lambda}^{-\tau_k^{\lambda}} \rfloor-1 {]\!]},\nonumber\\
I^{\lambda}_{k,+}&=& {[\![}X^{\lambda}_k + \lfloor{\lambda
}^{-\tau_k^{\lambda}}
\rfloor+1 ,
X^{\lambda}_k + \lfloor{\lambda}^{-\tau_k^{\lambda}} \rfloor+
\lfloor{\varepsilon
}_T/{\lambda}\log
(1/{\lambda})
\rfloor{]\!]},\nonumber\\
L^{\lambda}_k&=& {[\![}X^{\lambda}_k + \lfloor{\lambda}^{-\tau
_k^{\lambda}} \rfloor+
\lfloor
{\varepsilon}_T/{\lambda}\log(1/{\lambda}) \rfloor+1,\\
&&\hspace*{8.3pt}
{\tilde X}^{\lambda}_k - \lfloor{\lambda}^{-\tau_k^{\lambda}}
\rfloor-
\lfloor{\varepsilon}_T/{\lambda}\log(1/{\lambda}) \rfloor-1
{]\!]}\end{aligned}$$ and similar intervals ${\tilde I}^{\lambda}_k,{\tilde I}^{\lambda
}_{k,-},{\tilde I}^{\lambda}_{k,+}$, around ${\tilde X}^{\lambda}_k$. For all $k=1, \ldots, n$, the family of Poisson processes $(N_t(i))_{t\geq0, i \in J^{\lambda}_a}$ satisfies:
(2a) $\forall i \in I_k^{\lambda}, N_{T^{\lambda}_{k}}(i) -
N_{S^{\lambda}_{k-1}}(i)
>0$ and $\forall i \in{\tilde I}_k^{\lambda}, N_{{\tilde T}^{\lambda
}_{k}}(i) - N_{S^{\lambda}_{k-1}}(i) >0$;
(2b) $\exists i \in I_{k,-}^{\lambda}$ such that $ N_{T^{\lambda
}_{k}}(i) -
N_{S^{\lambda}_{k-1}}(i) =0 $, $\exists i \in I_{k,+}^{\lambda}$ such that $N_{T^{\lambda}_{k}}(i) -
N_{S^{\lambda}
_{k-1}}(i) =0 $, $\exists i \in{\tilde I}_{k,-}^{\lambda}$ such that $N_{{\tilde
T}^{\lambda}_{k}}(i) -
N_{S^{\lambda}
_{k-1}}(i) =0 $ and $\exists i \in{\tilde I}_{k,+}^{\lambda}$ such that $N_{{\tilde
T}^{\lambda}_{k}}(i) -
N_{S^{\lambda}_{k-1}}(i) =0$;
(2c) $\exists i \in I_{k}^{\lambda}$ such that $N_{S^{\lambda
}_{k}}(i) - N_{T^{\lambda}
_{k}}(i) =0$ and $\exists i \in{\tilde I}_{k}^{\lambda}$ such that $N_{S^{\lambda
}_{k}}(i) - N_{{\tilde T}
^{\lambda}
_{k}}(i) =0$;
(2d) $\forall i \in L_{k}^{\lambda}, N_{S^{\lambda}_{k}}(i) -
N_{S^{\lambda}_{k-1}}(i) >0$.
3\. We finally assume that $\exists i \in L_{n}^{\lambda}$ such that $N_{T\log(1/{\lambda})}(i) - N_{S^{\lambda}_{n}}(i)=0$.
To show that on $\Omega^{\lambda}_a$, a.s., for all $t\in[0,T\log
(1/{\lambda})]$, there is some $i \in J_a^{\lambda}$ such that $\eta^{\lambda}_t(i)=0$, we proceed recursively. At time $0$, all sites are vacant. Fix $k \in\{ 1,\ldots, n\}$. Assume that for $t \leq S^{\lambda}_{k-1}$, there is some $i \in J_a^{\lambda}$ such that $\eta^{\lambda}_t(i)=0$ and that at time $S^{\lambda}_{k-1}$, all sites in the interval $L_{k-1}^{\lambda}$ are vacant.
Then, for $S^{\lambda}_{k-1} \leq t < T^{\lambda}_k$ (resp., $ S^{\lambda
}_{k-1} \leq t < {\tilde T}
^{\lambda}_k$), (2b) shows that there are vacant sites in both $I_{k,+}^{\lambda}$ and $I_{k,-}^{\lambda}$ (resp., in both ${\tilde I}_{k,+}^{\lambda}$ and ${\tilde
I}_{k,-}^{\lambda}$). This, together with (2a), shows that at time $T^{\lambda}_k-$ (resp., ${\tilde
T}^{\lambda}_k-$), all of the sites in the intervals $I_k^{\lambda}$ and ${\tilde
I}_k^{\lambda}$ are occupied (no fire may burn those sites because they are protected by the vacant sites in $I_{k,+}^{\lambda}, I_{k,-}^{\lambda}, {\tilde
I}_{k,+}^{\lambda}, {\tilde I}_{k,-}^{\lambda}$). Hence, the interval $I_k^{\lambda}$ (resp., ${\tilde I}_k^{\lambda}$) becomes completely vacant at time $T^{\lambda}_k$ (resp., ${\tilde T}^{\lambda}_k$). Between time $T^{\lambda}_k$ (resp., ${\tilde T}
^{\lambda}_k$) and time $S^{\lambda}_k$, since $I_k^{\lambda}$ (resp., ${\tilde
I}_k^{\lambda}$) is completely vacant at time $T^{\lambda}_k$ (resp., ${\tilde T}^{\lambda}_k$), (2c) shows that there is a vacant site in $I_k^{\lambda}$ (resp., ${\tilde I}_k^{\lambda}$).
At time $S^{\lambda}_k-$, the interval $L_{k}^{\lambda}$ is completely occupied, by virtue of (2d) and the fact that it cannot be burnt because it is protected by vacant sites in $I_{k,+}^{\lambda}$ (resp., ${\tilde
I}_{k,-}^{\lambda}$) between $S^{\lambda}_{k-1}$ and $T^{\lambda}_{k}$ (resp., ${\tilde
T}^{\lambda}_k$), and in $I_k^{\lambda}$ (resp., ${\tilde I}_k^{\lambda}$) between $T^{\lambda}_k$ (resp., ${\tilde T}^{\lambda}_k$) and $S^{\lambda}_k$. As a consequence, since $Y^{\lambda}_k \in L_k^{\lambda}$, the interval $L_k^{\lambda}$ becomes completely vacant at time $S^{\lambda}_k-$.
All of this shows that on $\Omega^{\lambda}_a$, there are vacant sites in $J^{\lambda}_a$ for all $t \in[0,S_n^{\lambda}]$ and that $L_n^{\lambda}$ is completely vacant at time $S^{\lambda}_n$. Finally, 3 implies that there are vacant sites in $L_n^{\lambda}\subset J^{\lambda}_a$ during $[S_n^{\lambda},T\log
(1/{\lambda})]$.
It remains to prove that there exists $q''_T>0$ such that for all $a \in{\mathbb{R}}$ and ${\lambda}\in(0,{\lambda}_T)$, we have ${\mathbb{P}}(\Omega^{\lambda}_a) \geq q''_T$. We separately treat the conditions 1 on $M^{\lambda}$ and 2 on $N$ (conditionally on $M^{\lambda}$) and use independence of these two families of Poisson processes to complete the proof.
First, for ${\lambda}\in(0, {\lambda}_T)$, we observe that we can construct $M^{\lambda}$ using a Poisson measure $M$ on $[0,\infty
)\times
{\mathbb{R}}$ with intensity $dt\,dx$ by setting, for all $i \in\mathbb{Z}$, $$M_t^{\lambda}(i)=M \bigl([0,t/\log(1/{\lambda})]\times
\bigl[i{\lambda}\log(1/{\lambda}), (i+1){\lambda}\log(1/{\lambda})\bigr) \bigr).$$ Hence \[since ${\varepsilon}_T/({\lambda}\log(1/{\lambda})) > 1$\], the event on which $M^{\lambda}$ satisfies 1 contains the event $\Omega'_a$ on which $M$ has exactly $3n$ marks in $[0,T]\times[a,a+1]$, for some $1\leq n \leq\lfloor T \rfloor$, which can be called $(T_k,X_k)_{k=1,\ldots,n}$, $({\tilde T}_k,{\tilde X}_k)_{k=1,\ldots,n}$ and $(S_k,Y_k)_{k=1,\ldots,n}$ in such a way that we have the following properties (we set $T_0={\tilde T}_0=S_0=0$ and $X_0=a$, ${\tilde X}_0=a+1$ for convenience) for all $k=1,\ldots,n$:
$\bullet$ $\min(\{ X_k-X_{k-1}, Y_k - X_k, {\tilde X}_k - Y_k,
{\tilde X}_{k-1} -
{\tilde X}
_k\})
> 5 \epsilon_T $;
$\bullet$ $T_k$ and ${\tilde T}_k$ belong to $(S_{k-1}+ 1/2 + \alpha,
S_{k-1}+ 1-\alpha);$
$\bullet$ $S_k \in(S_{k-1}+ 1+ \alpha, S_{k-1}+2(T_k \land{\tilde
T}_k-S_{k-1})
- \alpha) $;
$\bullet$ $S_n\geq(T-1) + \alpha$.
We then have ${\mathbb{P}}(\Omega_a')>0$ (as in the proof of Proposition \[loc\] and since ${\varepsilon}_T$ and $\alpha$ are sufficiently small) and this probability does not depend on $a$ (by translation invariance of the law of $M$) nor on ${\lambda}\in(0,{\lambda}_T)$ (since it concerns only $M$).
We then use basic computations on i.i.d. Poisson processes with rate $1$ to show that there is a (deterministic) constant $c>0$ such that for all $k= 1,
\ldots, n$, all ${\lambda}\in(0,\lambda_T)$, conditionally on $M^{\lambda}$ (we write ${\mathbb{P}}_M$ for the conditional probability w.r.t. $M^{\lambda}$):
$\bullet$ since $T^{\lambda}_k - S^{\lambda}_{k-1}\geq(\tau
_k^{\lambda}+\alpha
/2)\log
(1/{\lambda})$, due to (1c), and since $\#(I_k^{\lambda})=2\lfloor{\lambda}^{-\tau
_k^{\lambda}
}\rfloor+1$, we have $$\begin{aligned}
{\mathbb{P}}_M\bigl( \forall i \in I_k^{\lambda}, N_{T^{\lambda}_{k}}(i)
- N_{S^{\lambda}
_{k-1}}(i) >0\bigr)
&=& \bigl(1-e^{-(T^{\lambda}_{k}-S^{\lambda}_{k-1})} \bigr)^{ 2\lfloor{\lambda
}^{-\tau_k^{\lambda}
}\rfloor
+1 }\\
&\geq&(1- {\lambda}^{\tau_k^{\lambda}+\alpha/2} )^{2\lfloor
{\lambda}^{-\tau_k^{\lambda}
}\rfloor+1}
\geq c\end{aligned}$$ (it tends to $1$ as ${\lambda}\to0$) and the same computation works for ${\tilde I}^{\lambda}_k$;
$\bullet$ since $T^{\lambda}_k-S^{\lambda}_{k-1} \leq(1-\alpha)
\log(1/{\lambda}
)$, by (1b), and since $\#(I_{k,+}^{\lambda})=\lfloor{\varepsilon}_T/({\lambda
}\times\break\log(1/{\lambda}
)) \rfloor$, we have $$\begin{aligned}
{\mathbb{P}}_M\bigl( \exists i \in I_{k,+}^{\lambda}, N_{T^{\lambda
}_{k}}(i) - N_{S^{\lambda}
_{k-1}}(i) =0\bigr)
&=& 1 - \bigl(1-e^{-(T^{\lambda}_k-S^{\lambda}_{k-1})}
\bigr)^{\lfloor{\varepsilon}_T /{\lambda}\log(1/{\lambda})\rfloor}
\nonumber\\
&\geq& 1- (1-{\lambda}^{1-\alpha} )^{\lfloor{\varepsilon
}_T/({\lambda}\log
(1/{\lambda}))\rfloor} \geq c\end{aligned}$$ and the same computation works for $I_{k,-}^{\lambda},{\tilde
I}_{k,+}^{\lambda},{\tilde I}
_{k,-}^{\lambda}$;
$\bullet$ since $S^{\lambda}_k - T^{\lambda}_k \leq(\tau_k^{\lambda}- \alpha
/2)\log(1/{\lambda}
)$, due to (1c) \[we use the fact that $S_k^{\lambda}\leq2T_k^{\lambda
}-S_{k-1}^{\lambda}-\alpha\log
(1/{\lambda})$, whence $2S_k^{\lambda}\leq2T_k^{\lambda}+S_k^{\lambda
}-S_{k-1}^{\lambda}-\alpha\log
(1/{\lambda})
=2T_k^{\lambda}+2 (\tau_k^{\lambda}-\alpha/2) \log(1/{\lambda})$\], and since $\#(I_k^{\lambda})=2\lfloor{\lambda}^{-\tau_k^{\lambda
}}\rfloor+1$, we have $$\begin{aligned}
{\mathbb{P}}_M\bigl( \exists i \in I_k^{\lambda}, N_{S^{\lambda
}_{k}}(i) - N_{T^{\lambda}_{k}}(i)
=0\bigr) &=&
1 - \bigl(1-e^{- (S^{\lambda}_k-T^{\lambda}_k)} \bigr)^{2\lfloor{\lambda
}^{-\tau_k^{\lambda}
}\rfloor
+1}\\
&\geq& 1- (1-{\lambda}^{\tau_k^{\lambda}-\alpha/2} )^{2\lfloor
{\lambda}^{-\tau
_k^{\lambda}
}\rfloor+1}
\geq c\end{aligned}$$ and this also holds for ${\tilde I}^{\lambda}_k$;
$\bullet$ since $S^{\lambda}_k - S^{\lambda}_{k-1} \geq(1+\alpha)
\log(1/{\lambda})$, thanks to (1c), and since $\#(L_k^{\lambda}) \leq\lfloor(1/{\lambda}\log
(1/{\lambda}))\rfloor$, we have $$\begin{aligned}
{\mathbb{P}}_M\bigl( \forall i \in L_{k}^{\lambda}, N_{S^{\lambda
}_{k}}(i) - N_{S^{\lambda}
_{k-1}}(i) >0\bigr)
&=& \bigl( 1- e^{ -(S^{\lambda}_k - S^{\lambda}_{k-1}) } \bigr)^{\#(L_k^{\lambda})}
\\
&\geq& ( 1- {\lambda}^{1+\alpha} )^{\lfloor1/{\lambda}\log
(1/{\lambda})\rfloor}
\geq c;\end{aligned}$$
$\bullet$ since $T\log(1/{\lambda})-S_n^{\lambda}\leq(1-\alpha
)\log(1/{\lambda})$, by (1d), and $\# (L_n^{\lambda}) \geq4{\varepsilon}_T/({\lambda
}\log(1/\break{\lambda}
))$, by (1a), we have $$\begin{aligned}
{\mathbb{P}}_M\bigl(\exists i \in L_{n}^{\lambda},
N_{T\log(1/{\lambda})}(i) - N_{S^{\lambda}_{n}}(i)=0\bigr)&=&
1- \bigl(1-e^{-(T\log(1/{\lambda})-S_n^{\lambda})} \bigr)^{\# (L_n^{\lambda
})}\nonumber\\
&\geq& 1- (1-{\lambda}^{1-\alpha} )^{4{\varepsilon}_T/({\lambda}\log
(1/{\lambda}))}
\geq c.\end{aligned}$$
We observe that the domains $I_k^{\lambda}\times(S^{\lambda}_{k-1},
T^{\lambda}_k]$, ${\tilde I}^{\lambda}_k\times(S^{\lambda}_{k-1}, {\tilde T}^{\lambda}_k]$, $I_{k,+}^{\lambda}\times(S^{\lambda}_{k-1}$,$T^{\lambda}_k]$, $I_{k,-}^{\lambda}\times(S^{\lambda}_{k-1}, T^{\lambda}_k]$, ${\tilde I}_{k,+}^{\lambda}\times(S^{\lambda}_{k-1}, {\tilde
T}^{\lambda}_k]$, ${\tilde I}_{k,-}^{\lambda}\times(S^{\lambda}_{k-1}, {\tilde
T}^{\lambda}_k]$, $I_k^{\lambda}\times(T^{\lambda}_{k}, S^{\lambda}_k]$, ${\tilde I}_k^{\lambda}\times({\tilde T}^{\lambda}_{k}, S^{\lambda}_k]$, $L_{k}^{\lambda}\times(S^{\lambda}_{k-1}, S^{\lambda}_k]$, for $k=1,\ldots, n$, and $L_{n}^{\lambda}\times(S^{\lambda}_{n}, T\log(1/{\lambda})]$ are pairwise disjoint, thanks to 1 and to the smallness of ${\varepsilon}_T$ and ${\lambda}_T$: we have $\lfloor{\lambda
}^{-\tau_k^{\lambda}
} \rfloor
\leq{\lambda}^{\alpha-1} \leq{\varepsilon}_T/({\lambda}\log
(1/{\lambda}))$.
Since $n\leq T$, we deduce from all of the previous estimates the existence of a $q''_T>0$ such that for all $a \in{\mathbb{R}}$ and ${\lambda}\in(0,{\lambda}_T)$, we have ${\mathbb{P}}(\Omega^{\lambda}_a) \geq q''_T$. We complete the proof by choosing $q_T = \min(q'_T,q''_T)$.
Convergence proof {#cv}
=================
The goal of this section is to prove Theorem \[converge\].
Coupling
--------
We introduce a coupling between the ${\lambda}$-FFP, the LFFP and their localized versions.
\[cpp\] We consider a Poisson measure $M(dt,dx)$ on $[0,\infty) \times
{\mathbb{R}}$ with intensity measure $dt\,dx$. We consider an independent family of Poisson processes $(N_t(i))_{t\geq0, i\in\mathbb{Z}}$ with rate $1$. For ${\lambda}\in(0,1)$ and $i \in\mathbb{Z}$, we set $$M_t^{\lambda}(i)=M \bigl([0,t/\log(1/{\lambda})]\times
\bigl[i{\lambda}\log(1/{\lambda}), (i+1){\lambda}\log(1/{\lambda})\bigr) \bigr).$$ Then $(M_t^{\lambda}(i))_{t\geq0, i\in\mathbb{Z}}$ is a family of independent Poisson processes with rate ${\lambda}$. For all ${\lambda}\in(0,1)$, we consider the ${\lambda}$-FFP $(\eta^{\lambda}_t)_{t\geq0}$ (see Definition \[dflaffp\]) and for all $A>0$, we consider the $({\lambda},A)$-FFP $(\eta^{{\lambda},A}_t)_{t\geq
0}$ (see Definition \[dflaffpA\]) constructed with $N,M^{\lambda}$. We also introduce the processes $(Z^{\lambda}_t(x),D^{{\lambda}}_t(x))_{t\geq0,x\in{\mathbb{R}}}$, as in (\[dlambda\]), (\[zlambda\]), and $(Z^{{\lambda},A}_t(x),D^{{\lambda},A}_t(x))_{t \geq0,x\in[-A,A]}$, as in (\[dlambdaA\]), (\[zlambdaA\]).
We denote by $(Z_t(x),D_t(x),H_t(x))_{t\geq0, x\in{\mathbb{R}}}$ the LFFP constructed with $M$ (see Definition \[dflffp\]) and by $(Z_t^A(x),D_t^A(x),H_t^A(x))_{t\geq0, x\in[-A,A]}$ the $A$-LFFP constructed with $M$ (see Definition \[dflffpA\]).
Localization
------------
Temporarily assume that the following result holds.
\[heart\] Adopt Notation \[cpp\] as well as Notation \[nocv\].
For any $T>0$, $A>0$ and $x_0\in(-A,A)$, in probability, as ${\lambda}\to0$, $$\delta_T ( (Z^{{\lambda},A}(x_0),D^{{\lambda},A}(x_0)),
(Z^{A}(x_0),D^A(x_0)) )\qquad
\mbox{tends to } 0.$$
For any $t\in[0,\infty)$, $A>0$ and $x_0\in(-A,A)$, in probability, as ${\lambda}\to0$, $$|Z^{{\lambda},A}_t(x_0)-Z^{A}_t(x_0)|+\delta(D^{{\lambda
},A}_t(x_0)),D^A_t(x_0) )\qquad
\mbox{tends to } 0.$$
We are now in a position to give the following proof.
[Proof of Theorem \[converge\]]{} We only prove point (a), (b) being similarly checked. Let $T>0$ and $\{x_1,\ldots,x_n\} \subset[-B,B]\subset
{\mathbb{R}}$ be fixed. Consider the coupling introduced in Notation \[cpp\]. Proposition \[heart\] ensures us that for any ${\varepsilon}>0$ and $A>B$, we have $$\lim_{{\lambda}\to0}{\mathbb{P}}\Biggl[\sum_{1}^n \delta_T (
(Z^{{\lambda},A}(x_i),D^{{\lambda},A}(x_i)),
(Z^{A}(x_i),D^A(x_i)) ) >{\varepsilon}\Biggr] =0.$$ Now, let $$\begin{aligned}
&&\Omega_{A,T}^{\lambda}:= \{\forall i =1,\ldots,n, \forall
t\in[0,T],\\
&&\hspace*{44.27pt}
(Z^{{\lambda}}_t(x_i),D^{{\lambda}}_t(x_i))=
(Z^{{\lambda},A}_t(x_i),D^{{\lambda},A}_t(x_i)) \\
&&\hspace*{44.27pt} \mbox{and }(Z_t(x_i),D_t(x_i))= (Z_t^A(x_i),D_t^A(x_i)) \}.\end{aligned}$$ For all $A>2B$, we now have $$\begin{aligned}
&&\Omega_{A,T}^{\lambda}\subset\bigl\{
(Z_t^{\lambda}(x),D_t^{\lambda}(x))_{t\in[0,T],x\in[-A/2,A/2]}
\\
&&\hspace*{42.5pt}=
(Z_t^{{\lambda},A}(x),D_t^{{\lambda},A}(x))_{t\in[0,T],x\in
[-A/2,A/2]} \\
&&\hspace*{42.5pt}\mbox{and } (Z_t(x),D_t(x))_{t\in[0,T],x\in[-A/2,A/2]}\\
&&\hspace*{42.5pt}\hspace*{14.29pt}=
(Z_t^A(x),D_t^A(x))_{t\in[0,T],x\in[-A/2,A/2]} \bigr\}.\end{aligned}$$ However, Propositions \[loc\] and \[plocla\] yield that ${\mathbb{P}}[(\Omega_{A,T}^{\lambda})^c] \leq2C_T e^{-\alpha_T
A}$. Thus, for any $A>2B$, $$\limsup_{{\lambda}\to0}{\mathbb{P}}\Biggl[\sum_{1}^n \delta_T
((Z^{{\lambda}}(x_i),D^{{\lambda}}(x_i)),
(D(x_i),Z(x_i)) ) >{\varepsilon}\Biggr] \leq0 + 2C_T e^{-\alpha_T A}.$$ Letting $A$ tend to infinity, we deduce that $\sum_{i=1}^n\delta_T((Z^{{\lambda}}(x_i),D^{{\lambda}}(x_i)),
(D(x_i)$,$Z(x_i)))$ tends to $0$ in probability as ${\lambda}\to0$, hence the result.
Core of the proof
-----------------
The aim of this subsection is to prove Proposition \[heart\]. We fix $T>0$ and $A>0$. We consider the $({\lambda},A)$-FFP and the $A$-LFFP coupled, as in Notation \[cpp\], and use the notation introduced in (\[alaila\]). Throughout this proof, we will omit the superscript $A$ and we do not take into account the possible dependencies in $A$ and $T$.
For $J=(a,b)$ \[an open interval of $(-A,A)$\], ${\lambda}\in(0,1)$ and $\mu
\in(0,1]$, we consider $$\begin{aligned}
\label{xxbb}
J_{{\lambda},\mu} &=& \biggl[\hspace*{-3.5pt}\biggl[\biggl\lfloor\frac{a}{{\lambda}\log
(1/{\lambda})}+\frac{\mu
}{{\lambda}\log
^2(1/{\lambda})}
\biggr\rfloor, \nonumber\\
&&\hspace*{9.9pt}\biggl\lfloor\frac{b}{{\lambda}\log(1/{\lambda})}-
\frac{\mu}{{\lambda}\log^2(1/{\lambda})} \biggr\rfloor\biggr]\hspace*{-3.5pt}\biggr]
\subset\mathbb{Z},
\\
{\tilde Z}^{{\lambda},\mu}_t(J)
&=& 1 - \frac{\log(1+\#\{k\in
J_{{\lambda},\mu}, \eta^{\lambda}_{t\log
(1/{\lambda}
)}(k)=0\})}
{\log(1+\#(J_{{\lambda},\mu}))}.\nonumber\end{aligned}$$ Observe that ${\tilde Z}^{{\lambda},\mu}_t(J)=1$ if and only if all the sites of $J_{{\lambda},\mu}$ are occupied at time $t\log(1/{\lambda})$. The quantity ${\tilde
Z}^{{\lambda},\mu}_t(J)$ is a function of the density of vacant clusters in the (rescaled) zone $J$. Under some exchangeability properties, it should be closely related to the size of occupied clusters in that zone, that is, to $Z^{\lambda}_t(x)$ for $x\in J$.
For $x\in(-A,A)$, ${\lambda}\in(0,1)$ and $\mu\in(0,1]$, we introduce $$\begin{aligned}
\label{xla}
x_{{\lambda},\mu} &=& \biggl[\hspace*{-3.5pt}\biggl[\biggl\lfloor\frac{x}{{\lambda}\log
(1/{\lambda})}-\frac{\mu
}{{\lambda}\log
^2(1/{\lambda})}
\biggr\rfloor+1,\nonumber\\
&&\hspace*{9.9pt} \biggl\lfloor\frac{x}{{\lambda}\log(1/{\lambda})}+
\frac{\mu}{{\lambda}\log^2(1/{\lambda})} \biggr\rfloor-1
\biggr]\hspace*{-3.5pt}\biggr]\subset\mathbb{Z},\\
{\tilde H}^{{\lambda},\mu}_t(x) &=& \frac{\log(1+\#\{k\in x_{{\lambda
},\mu}, \eta^{\lambda}_{t\log
(1/{\lambda}
)}(k)=0\})}
{\log(1+\#(x_{{\lambda},\mu}))}. \nonumber\end{aligned}$$ Here, again, ${\tilde H}^{{\lambda},\mu}_t(x)=0$ if and only if all the sites of $x_{{\lambda},\mu}$ are occupied at time $t\log(1/{\lambda})$. Assume that a microscopic fire starts at some $x$. The process ${\tilde H}^{{\lambda},\mu}_t(x)$ will then allow us to quantify the duration for which this fire will be in effect.
Observe that we always have $\log(1+\#(x_{{\lambda},\mu}))\sim\log
(1+\#(J_{{\lambda},\mu}))
\sim\log(1/{\lambda})$ as ${\lambda}\to0$. Also, observe that if ${\tilde Z}^{{\lambda},\mu}_t(J)=z$, then there are $(1+\#(J_{{\lambda},\mu}))^{1-z}-1 \simeq{\lambda}^{z-1}$ vacant sites in $J_{{\lambda},\mu}$ at time $t\log(1/{\lambda})$. In the same way, ${\tilde H}^{{\lambda},\mu}_t(x)=h$ says that there are $(1+\#(x_{{\lambda},\mu}))^{h}-1 \simeq{\lambda}^{-h}$ vacant sites in $x_{{\lambda},\mu}$ at time $t\log(1/{\lambda})$.
We work conditionally on $M$. We denote by ${\mathbb{P}}_M$ the conditionalprobability given $M$. We recall that, conditionally on $M$, $(Z_t(x),D_t(x),\break H_t(x))_{t\in[0,T],x\in[-A,A]}$ is deterministic. We set $n =M([0,T] \times[-A;A])$, which is a.s. finite. We set $T_0=0$ and consider the marks $(X_q,T_q)_{1\leq q \leq n}$ of $M$, ordered in such a way that $T_0 < T_1< \cdots< T_n<T$.
We set ${\mathcal B}_0=\varnothing$ and for $q=1,\ldots,n$, we consider ${\mathcal B}_q=\{X_1,\ldots,X_q\}$, as well as the set ${\mathcal C}_q$ of connected components of $(-A,A)\setminus{\mathcal
B}_q$ (sometimes referred to as *cells*).
Observe that, by construction, we have, for $c\in{\mathcal C}_q$ and $x,y\in c$, $Z_t(x)=Z_t(y)$ for all $t\in[0,T_{q+1})$. Thus, we can introduce $Z_t(c)$.
We consider ${\lambda}_\mu>0$ (which depends on $M$) such that for all ${\lambda}\in(0,{\lambda}_\mu)$, we have $(X_i)_{{\lambda},\mu}\ne\varnothing$ and $(X_i)_{{\lambda},\mu
}\cap(X_j)_{{\lambda},\mu}=\varnothing$ for all $i\ne j$ with $i,j \in\{1,\ldots,n\}$.
We then observe that for ${\lambda}\in(0,{\lambda}_\mu)$ and for each $q=0,\ldots,n$, $\{x_{{\lambda},\mu},x\in{\mathcal B}_q\}\cup\{c_{{\lambda},\mu},
c \in{\mathcal C}_q \}$ is a partition of ${[\![}-{\tilde A}_{{\lambda},\mu},{\tilde A}_{{\lambda},\mu
}{]\!]}$, where ${\tilde A}_{{\lambda},\mu}
=\lfloor
A/({\lambda}\log(1/{\lambda})) -
\mu/({\lambda}\log^2(1/{\lambda}))\rfloor$.
With our coupling, for the $({\lambda},A)$-FFP $(\eta^{\lambda
}_t)_{t\geq0}$, for each $i= 1,\ldots,n$, a fire starts at the site $\lfloor X_i/({\lambda}\log(1/{\lambda})) \rfloor$ at time $T_i\log(1/{\lambda})$ and this describes all of the fires during $[0,T\log
(1/{\lambda})]$.
The lemma below shows some exchangeability properties inside cells \[connected components of $(-A,A)\setminus{\mathcal B}_q$\]. This will allow us to prove that for $c$ a cell and $x\in c$, the size of the occupied cluster around $x$ \[described by $Z^{\lambda}(x)$\] is closely related to the global density of occupied clusters in $c$ \[described by ${\tilde Z}^{{\lambda},\mu}(c)$\].
\[exch\] For ${\lambda}\in(0,1)$ and $\mu\in(0,1]$, set ${\mathcal
E}^{{\lambda},\mu}_0=\Omega$, and for $q=1,\ldots,n$, consider the event \[recalling Definition \[dflaffpA\] and (\[xxbb\])\] $$\begin{aligned}
{\mathcal E}^{{\lambda},\mu}_q &=& \bigl\{\forall i=1,\ldots,q, \forall c \in
{\mathcal C}_{i},
\mbox{ either } c_{{\lambda},\mu}\subset C^{\lambda}_{T_i\log
(1/{\lambda})-}(X_i) \\
&&\hspace*{4.5pt}\mbox{or } \eta^{\lambda}_{T_i\log(1/{\lambda})-}(k)=0 \mbox{
for some }
\max c_{{\lambda},\mu}< k < \min C^{\lambda}_{T_i\log(1/{\lambda
})-}(X_i) \\
&&\hspace*{4.5pt}\mbox{or } \eta^{\lambda}_{T_i\log(1/{\lambda})-}(k)=0 \mbox{
for some }
\max C^{\lambda}_{T_i\log(1/{\lambda})-}(X_i)<k<\min c_{{\lambda
},\mu}\bigr\}.\end{aligned}$$ Conditionally on $M$ and ${\mathcal E}^{{\lambda},\mu}_q$, for all $c\in{\mathcal C}_q$, the random variables$(\eta^{\lambda}_{T_q\log(1/{\lambda})}(k))_{k\in
c_{{\lambda},\mu}}$ are exchangeable.
Let $c \in{\mathcal C}_q$, let $\sigma$ be a permutation of $c_{{\lambda},\mu}$ and set, for simplicity, $\sigma(i)=i$ for $i\in I^{\lambda
}_A\setminus
c_{{\lambda},\mu}
$ \[recall (\[alaila\])\].
Consider the $({\lambda},A)$-FFP process $(\eta_t^{\lambda})_{t\geq0}$ constructed with $M$ and the family of Poisson processes $(N(i))_{i\in I^{\lambda}_A}$. Also, consider the $({\lambda},A)$-FFP process $(\tilde\eta_t^{\lambda})_{t\geq0}$ constructed with $M$ and the family of Poisson processes $({\tilde N}(i))_{i\in I^{\lambda}_A}$ defined by ${\tilde N}(i)=N(\sigma(i))$.
Observe that ${\mathcal E}^{{\lambda},\mu}_{k+1}\subset{\mathcal
E}^{{\lambda},\mu}_k$. For all $k=0,\ldots,q$, $c\subset c_k$ for some $c_k\in{\mathcal C}_k$. We will prove the following claims by induction on $k=0,\ldots,q$:
if $\tilde{\mathcal E}^{{\lambda},\mu}_k$ is the same event as ${\mathcal E}^{{\lambda},\mu}_k$ corresponding to $(\tilde\eta^{\lambda}_t)_{t\geq0}$, then $\tilde{\mathcal E}^{{\lambda},\mu}_k={\mathcal E}^{{\lambda
},\mu}_k$;
on ${\mathcal E}^{{\lambda},\mu}_k$, for all $t\in[0,T_k\log
(1/{\lambda})]$, $\tilde\eta_t^{\lambda}(i)= \eta_t^{\lambda}(\sigma(i))$ for all $i\in I^{\lambda}_A$ \[in particular, $\tilde\eta_t^{\lambda}(i)= \eta_t^{\lambda}(i)$ for all $i\notin c_{{\lambda},\mu}]$.
Of course, (i) and (ii) with $k=q$ imply the lemma. Indeed, let$\varphi\dvtx\{0,1\}^{\#(c_{{\lambda},\mu})}\mapsto{\mathbb{R}}$. We have $${\mathbb{E}}_M\bigl[{\mathbf{1}}_{{\mathcal E}^{{{\lambda},\mu
}}_q}\varphi\bigl(\bigl(\eta^{\lambda}_{T_q\log(1/{\lambda}
)}(i)\bigr)_{i\in
c_{{\lambda},\mu}} \bigr)\bigr]
={\mathbb{E}}_M\bigl[{\mathbf{1}}_{\tilde{\mathcal E}^{{{\lambda},\mu}}_q}
\varphi\bigl(\bigl(\tilde\eta^{\lambda}_{T_q\log(1/{\lambda})}(i)\bigr)_{i\in
c_{{\lambda},\mu}} \bigr)\bigr].$$ Using (i) and (ii), we then deduce that $${\mathbb{E}}_M\bigl[{\mathbf{1}}_{{\mathcal E}^{{{\lambda},\mu
}}_q}\varphi\bigl(\bigl(\eta^{\lambda}_{T_q\log(1/{\lambda}
)}(i)\bigr)_{i\in
c_{{\lambda},\mu}} \bigr)\bigr]
={\mathbb{E}}_M\bigl[{\mathbf{1}}_{{\mathcal E}^{{{\lambda},\mu}}_q}
\varphi\bigl(\bigl(\eta^{\lambda}_{T_q\log(1/{\lambda})}(\sigma(i))\bigr)_{i\in
c_{{\lambda},\mu}} \bigr)\bigr],$$ which proves the lemma.
First, (i) and (ii) with $k=0$ are obviously satisfied. Assume, now, that for some $k\in\{0,\ldots,q-1\}$, we have (i) and (ii). Then, on ${\mathcal E}^{{\lambda},\mu}_k$, for all $t\in[0,T_{k+1}\log(1/{\lambda}))$, $\tilde\eta_t^{\lambda}(i)= \eta_t^{\lambda}(\sigma(i))$ for all $i\in I^{\lambda}_A$. Indeed, they are equal on $[0,T_{k}\log(1/{\lambda})]$, by assumption, and they use the same Poisson process ${\tilde N}(i)=N(\sigma(i))$ on the time interval $[T_{k}\log(1/{\lambda}),T_{k+1}\log(1/{\lambda}))$).
We now check that ${\mathcal E}_{k+1}^{{\lambda},\mu}=\tilde
{\mathcal E}_{k+1}^{{\lambda},\mu}$. We know that ${\mathcal E}_{k}^{{\lambda},\mu}=\tilde{\mathcal E}_{k}^{{\lambda
},\mu}$ and the additional condition \[at time $T_{k+1}\log(1/{\lambda})-$\] concerns:
$\bullet$ sites outside $c_{{\lambda},\mu}$, for which the values of $\eta^{\lambda}
$ and $\tilde\eta^{\lambda}$ at time $T_{k+1}\log(1/{\lambda})-$ are the same;
$\bullet$ the event $c_{{\lambda},\mu}\subset C^{\lambda
}_{T_{k+1}\log(1/{\lambda})-}$, which is the same for $\eta^{\lambda}$ and $\tilde\eta^{\lambda}$ (it can be realized only if there are no vacant sites in $c_{{\lambda},\mu}$, which occurs, or not, simultaneously for $\eta^{\lambda}$ and $\tilde\eta^{\lambda}$).
We now conclude that (ii) remains true at time $T_{k+1}\log(1/{\lambda
})$ since the zone subject to fire either:
$\bullet$ is disjoint with $c_{{\lambda},\mu}$ so that the values of $\eta^{\lambda},
\tilde\eta^{\lambda}$ are left invariant in $c_{{\lambda},\mu}$, while they are modified in the same way outside $c_{{\lambda},\mu}$; or
$\bullet$ contains the whole zone $c_{{\lambda},\mu}$, which is thus destroyed simultaneously for $\eta^{\lambda}$ and $\tilde\eta
^{\lambda}$, and the values of $\eta^{\lambda},\tilde\eta^{\lambda}$ are modified in the same way outside $c_{{\lambda},\mu}$.
The next lemma shows, in some sense, that if a cell is *almost* completely occupied at time $t$, then it will be *really* completely occupied at time $t+$; and, if the effect of a microscopic fire is *almost* ended at time $t$, then it will be *really* ended at time $t+$.
\[undoncplein\] Let $\mu\in(0,1]$. Consider $k\in\{0,\ldots,n\}$, $c\in{\mathcal
C}_k$, $x\in
{\mathcal B}
_k$ and $t\in[T_k,T_{k+1})$.
Assume that for all ${\varepsilon}>0$, $\lim_{{\lambda}\to
0}{\mathbb{P}}
_M({\tilde Z}^{{\lambda},\mu}_t(c)<1-{\varepsilon})=0$. Then, for all $s\in(t,T_{k+1})$, $\lim_{{\lambda}\to0}{\mathbb
{P}}_M({\tilde Z}^{{\lambda},\mu}
_{s}(c)=1)=1$.
Assume that for all ${\varepsilon}>0$, $\lim_{{\lambda}\to
0}{\mathbb{P}}
_M({\tilde H}^{{\lambda},\mu}_t(x)>{\varepsilon})=0$. Then, for all $s\in(t,T_{k+1})$, $\lim_{{\lambda}\to0}{\mathbb
{P}}_M({\tilde H}^{{\lambda},\mu}
_{s}(x)=0)=1$.
The proofs of (i) and (ii) are similar. Let us, for example, prove (i). Thus, let $T_k\leq t< t+{\varepsilon}=s<T_{k+1}$. We start with $${\mathbb{P}}_M\bigl({\tilde Z}^{{\lambda},\mu}_{t+{\varepsilon}}(c)=1 \bigr)
\geq
{\mathbb{P}}_M\bigl({\tilde Z}^{{\lambda},\mu}_{t+{\varepsilon}}(c)=1
{\mid}{\tilde Z}^{{\lambda},\mu}
_{t}(c)>1-{\varepsilon}/2\bigr) {\mathbb{P}}_M\bigl( {\tilde Z}
^{{\lambda},\mu}_{t}(c)>1-{\varepsilon}/2\bigr),$$ so that it suffices to check that $\lim_{{\lambda}\to0} {\mathbb{P}}_M({\tilde Z}^{{\lambda},\mu
}_{t+{\varepsilon}}(c)=1 {\mid}
{\tilde Z}^{{\lambda},\mu}
_{t}(c)>1-{\varepsilon}/2)=1$. Let $v_t^{{\lambda},\mu}$ denote the number of vacant sites in $c_{{\lambda},\mu}$ (for $\eta^{\lambda}_{t\log(1/{\lambda})}$). Then ${\tilde Z}^{{\lambda},\mu}_{t+{\varepsilon}}(c)=1$ is equivalent to $v^{{\lambda},\mu}
_{t+{\varepsilon}}=0$ and one can easily check that ${\tilde Z}^{{\lambda},\mu}_t(c)>1-{\varepsilon}/2$ implies that $v^{{\lambda},\mu}_t \leq(1+\#
(c_{{\lambda},\mu}))^{{\varepsilon}
/2}\leq
(1+2A/({\lambda}\log(1/{\lambda})))^{{\varepsilon}/2}$.
Since $M((t,s]\times[-A,A])=0$ by assumption, we deduce that $M^{\lambda}_{s\log(1/{\lambda})}(i)=M^{\lambda
}_{t\log(1/{\lambda})}(i)$ for all $i\in I^{\lambda}_A$: no fire starts during $(t\log
(1/{\lambda}),s\log
(1/{\lambda})]$. Hence, each occupied site at time $t\log(1/{\lambda})$ remains occupied at time $s\log(1/{\lambda})$ and each vacant site at time $t\log(1/{\lambda})$ becomes occupied at time $s\log(1/{\lambda})$ with probability $1-e^{(t-s)\log(1/{\lambda})}=1-{\lambda}^{\varepsilon}$. Thus, $${\mathbb{P}}_M\bigl({\tilde Z}^{{\lambda},\mu}_{t+{\varepsilon}}(c)=1
{\mid}{\tilde Z}^{{\lambda},\mu}
_{t}(c)>1-{\varepsilon}/2\bigr)
\geq(1-{\lambda}^{\varepsilon})^{(1+2A/({\lambda}\log(1/{\lambda
})))^{{\varepsilon}/2}},$$ which tends to $1$ as ${\lambda}\to0$.
We end our preliminaries with a last lemma, which deals with estimates concerning the time needed to occupy vacant zones.
\[binomiale\] Let $\mu\in(0,1]$. Let $(\zeta_0^{\lambda}(i))_{i\in I^{\lambda
}_A}\in\{0,1\}
^{I^{\lambda}_A}$ and consider a family of i.i.d. Poisson processes $(P^{\lambda}_t(i))_{t\geq0,i\in I^{\lambda}_A}$, with rate $\log
(1/{\lambda})$, independent of $\zeta^{\lambda}_0$. Set $\zeta^{\lambda}_t(i)=\min(\zeta_0^{\lambda}(i)+P^{\lambda}_t(i),1)$.
1. Let $J=(a,b)\subset(-A,A)$ and $h\in[0,1]$. Set $v_t^{{\lambda},\mu}=\#\{i\in J_{{\lambda},\mu}, \zeta
^{\lambda}_t(i)=0\}$. Assume that $$\forall{\varepsilon}>0\qquad
{\mathbb{P}}\biggl( \biggl|\frac{\log(1+v^{{\lambda},\mu}_0)}{\log(1+\#
(J_{{\lambda},\mu}))} - h
\biggr|\geq{\varepsilon}\biggr)=0.$$
(a) Then, for all $T>0$ and ${\varepsilon}>0$, $$\lim_{{\lambda}\to0}
{\mathbb{P}}\biggl(\sup_{[0,T]} \biggl|\frac{\log(1+v^{{\lambda},\mu
}_t)}{\log(1+\#(J_{{\lambda},\mu}))} - (h-t)_+
\biggr|\geq
{\varepsilon}\biggr)=0.$$
(b) If the family $(\zeta^{\lambda}_0(i))_{i\in J_{{\lambda},\mu}}$ is exchangeable, then, for all $x\in J$, $T>0$ and ${\varepsilon}>0$, $$\lim_{{\lambda}\to0}
{\mathbb{P}}\biggl(\sup_{[0,T]} \biggl|\frac{\log(1+\# (G^{\lambda
}_t(x)))}{\log(1/{\lambda})}
- \bigl(1-(h-t)_+\bigr) \biggr|
\geq{\varepsilon}\biggr)=0,$$ where $G^{\lambda}_t(x)$ is the connected component of occupied sites around $\lfloor x/{\lambda}\log(1/{\lambda})\rfloor$ in $\zeta^{\lambda}_t$.
2. Let $x\in(-A,A)$ and $h\in[0,1]$. Set $v_t^{{\lambda},\mu}=\#\{i\in x_{{\lambda},\mu}, \zeta
^{\lambda}_t(i)=0\}$. Assume that $$\forall{\varepsilon}>0\qquad
{\mathbb{P}}\biggl( \biggl|\frac{\log(1+v^{{\lambda},\mu}_0)}{\log(1+\#
(x_{{\lambda},\mu}))} - h \biggr|
\geq{\varepsilon}\biggr)=0.$$ Then, for all $T>0$ and ${\varepsilon}>0$, $$\lim_{{\lambda}\to0}
{\mathbb{P}}\biggl(\sup_{[0,T]} \biggl|\frac{\log(1+v^{{\lambda},\mu
}_t)}{\log(1+\#(x_{{\lambda},\mu}))}
- (h-t)_+ \biggr|\geq
{\varepsilon}\biggr)=0.$$
The proof of part 2 is the same as that of 1(a) because $\log(1+\#(J_{{\lambda},\mu}))\sim\log(1+\#(x_{{\lambda
},\mu}))\sim\log(1/{\lambda})$ as ${\lambda}\to0$. Thus, we only prove 1 and everywhere replace $\log(1+\#(x_{{\lambda},\mu}))$ by $\log(1/{\lambda})$ without difficulty. By assumption, for all ${\varepsilon}>0$, we have $\lim_{{\lambda
}\to0}
{\mathbb{P}}(v^{{\lambda},\mu}_0 \in({\lambda}^{{\varepsilon
}-h}-1,{\lambda}^{-{\varepsilon
}-h})) =1$. We define $h_t=(h-t)_+$, $V^{{\lambda},\mu}_t=\log(1+v^{{\lambda
},\mu}_t)/\log(1/{\lambda})$ and, finally, $\Gamma^{\lambda}_t=\log(1+\#(G^{\lambda}_t(x)))/\log
(1/{\lambda})$.
*Step* 1. Let $t\geq0$ be fixed. We first show that for all ${\varepsilon}>0$, $\lim_{{\lambda}\to0}{\mathbb{P}}(|V^{{\lambda},\mu}_t - h_t
|\geq{\varepsilon})=0$. Conditionally on $v^{{\lambda},\mu}_0$, the random variable $v^{{\lambda},\mu}_t$ follows a binomial distribution $B(v_0^{{\lambda},\mu},{\lambda}^t)$ because each vacant site at time $0$ remains vacant with probability $e^{-t\log(1/{\lambda})}={\lambda}^t$.
*Case $h_t>0$.* Let ${\varepsilon}\in(0,h_t)$. We have to prove that ${\mathbb{P}}(v^{{\lambda},\mu}_t\in({\lambda
}^{{\varepsilon}-h_t}$,${\lambda}
^{-{\varepsilon}-h_t})
)\to1$. We know that $\lim_{{\lambda}\to0} {\mathbb{P}}(
v^{{\lambda},\mu}_0 \in({\lambda}^{{\varepsilon}/2-h},{\lambda
}^{-{\varepsilon}/2-h}) )=1$. The Bienaymé–Chebyshev inequality implies that $$\begin{aligned}
&&P[|v^{{\lambda},\mu}_t-v_0^{\lambda}{\lambda}^t|\leq
(v_0^{{\lambda},\mu}{\lambda}^t)^{2/3}{\mid}
v_0^{{\lambda},\mu}
\in({\lambda}^{{\varepsilon}/2-h},{\lambda}^{-{\varepsilon
}/2-h})]\\
&&\qquad\geq1- {\mathbb{E}}[v_0^{{\lambda},\mu}{\lambda}^t (1-{\lambda
}^t) (v_0^{{\lambda},\mu}{\lambda}
^t)^{-4/3}{\mid}
v_0^{{\lambda},\mu}
\in({\lambda}^{{\varepsilon}/2-h},{\lambda}^{-{\varepsilon}/2-h})]
\\
&&\qquad\geq1 - {\mathbb{E}}[(v_0^{{\lambda},\mu}{\lambda}^t)^{-1/3}{\mid
}v_0^{{\lambda},\mu}
\in({\lambda}^{{\varepsilon}/2-h},{\lambda}^{-{\varepsilon}/2-h})]\\
&&\qquad\geq1- ({\lambda}^{{\varepsilon}/2-h+t})^{-1/3},\end{aligned}$$ which tends to $1$ since $h_t=h-t>{\varepsilon}$.
However, the events $$|v^{{\lambda},\mu}_t-v_0^{{\lambda},\mu
}{\lambda}^t|\leq(v_0^{{\lambda},\mu}{\lambda}
^t)^{2/3}
\quad\mbox{and}\quad
v_0^{{\lambda},\mu}\in({\lambda}^{{\varepsilon}/2-h}, {\lambda
}^{-{\varepsilon}/2-h})$$ imply that $v^{{\lambda},\mu}_t \in
({\lambda}^{{\varepsilon}/2-h_t}-({\lambda}^{-{\varepsilon
}/2-h_t})^{2/3},{\lambda}
^{-{\varepsilon}/2-h_t}+({\lambda}^{-{\varepsilon}
/2-h_t})^{2/3})
\subset({\lambda}^{{\varepsilon}-h_t}$,${\lambda}^{-{\varepsilon
}-h_t})$ for ${\lambda}$ small enough, hence the result.
*Case $h_t=0$.* We have to show that for all ${\varepsilon}>0$, $\lim_{{\lambda}\to0}{\mathbb{P}}(v^{{\lambda},\mu}_t>{\lambda
}^{-{\varepsilon}})=0$, and it suffices to check that $\lim_{{\lambda}\to0}{\mathbb{P}}(v^{{\lambda},\mu}_t>{\lambda
}^{-{\varepsilon}} {\mid}v^{{\lambda},\mu}
_0<{\lambda}^{-{\varepsilon}/2-h})=0$. However, $$\begin{aligned}
&&{\mathbb{P}}(v^{{\lambda},\mu}_t>{\lambda}^{-{\varepsilon}} {\mid
}v^{{\lambda},\mu}_0<{\lambda}
^{-{\varepsilon}/2-h})\\
&&\qquad\leq{\lambda}^{\varepsilon}
{\mathbb{E}}[v^{{\lambda},\mu}_t {\mid}v^{{\lambda},\mu
}_0<{\lambda}^{-{\varepsilon}/2-h}] = {\lambda}
^{\varepsilon}
{\mathbb{E}}[ v^{{\lambda},\mu}_0 {\lambda}^t {\mid}v^{{\lambda
},\mu}_0<{\lambda}^{-{\varepsilon}/2-h}]\\
&&\qquad\leq{\lambda}^{{\varepsilon}+t}{\lambda}^{-{\varepsilon}/2-h}
={\lambda}
^{{\varepsilon}/2+t-h},\end{aligned}$$ which tends to $0$ since, by assumption, $t-h\geq0$.
*Step* 2. We now prove that for all ${\varepsilon}>0$, $\lim
_{{\lambda}
\to0} {\mathbb{P}}
(|\Gamma^{\lambda}_t-(1-h_t)|\geq{\varepsilon})=0$. It suffices to check that $\lim_{{\lambda}\to0} {\mathbb{P}}
( \#(G^{\lambda}_t(x)) \in({\lambda}^{{\varepsilon
}+h_t-1}-1,{\lambda}
^{-{\varepsilon}+h_t-1}))=1$. However, we know from Step 1 that there are approximately $(1/{\lambda})^{h_t}$ vacant sites in $J_{{\lambda},\mu}$, and $\#(J_{{\lambda},\mu})\simeq(1/{\lambda}\log(1/{\lambda}))$. We also know that the family $(\zeta^{\lambda}_t(i))_{i\in J_{{\lambda},\mu}}$ is exchangeable so that the vacant sites are uniformly distributed in $J_{{\lambda},\mu}$ (this statement is slightly misleading: there cannot be two vacant sites at the same place). We conclude that $\#(G^{\lambda}_t(x)) \simeq(1/{\lambda}\log
(1/{\lambda}))/\break
(1/{\lambda})^{h_t} \simeq{\lambda}^{h_t-1}$. This can be done rigorously without difficulty.
*Step* 3. We now prove 1(a), which relies on Step 1 and an ad hoc version of Dini’s theorem. Let ${\varepsilon}>0$. Consider a subdivision $0=t_0<t_1<\cdots<t_l=T$ with $t_{i+1}-t_i<{\varepsilon}/2$. Using Step 1, we have $\lim_{{\lambda}\to0}{\mathbb{P}}[\max_{i=0,\ldots,l}|V^{{\lambda
},\mu}_{t_i} -
(h-t_i)_+|>{\varepsilon}/2]=0$.
Now, observe that $t\mapsto V_t^{{\lambda},\mu}$ and $t \mapsto(h-t)_+$ are a.s. nonincreasing and that $t \mapsto(h-t)_+$ is Lipschitz continuous with Lipschitz constant $1$.
We deduce that ${\sup_{[0,T]}} |V^{{\lambda},\mu}_t-(h-t)_+|
\leq{\varepsilon}/2+ \max_{i=0,\ldots,l}\{|V^{{\lambda},\mu}_{t_i} -
(h-t_i)_+|\}$. Thus, ${\mathbb{P}}({\sup_{[0,T]}} |V^{{\lambda},\mu}_t-(h-t)_+|
>{\varepsilon})
\leq{\mathbb{P}}[{\max_{i=0,\ldots,l}}|V^{{\lambda},\mu}_{t_i} -
(h-t_i)_+|>{\varepsilon}/2]$, which completes the proof of 1(a).
*Step* 4. Point 1(b) is deduced from Step 2 exactly as point 1(a) was deduced from Step 1, using the fact that $t\mapsto
\Gamma
^{\lambda}_t$ and $t \mapsto1-h_t$ are a.s. nondecreasing.
We may now finally tackle the following proof.
[Proof of Proposition \[heart\]]{} For $x\in(-A,A)$ and $t\geq0$, we introduce $Z_t(x-)=\lim_{y\to x, y<x}Z_t(y)$ and $Z_t(x+)=\lim_{y\to x, y>x}Z_t(y)$, which represent the values of $Z_t$ in the cells on the left and right of $x$. If $x \in{\mathcal B}_n$, it is at the boundary of two cells $c_-,c_+ \in{\mathcal C}_n$, and then $Z_t(x-)=Z_t(c_-)$ and $Z_t(x+)=Z_t(c_+)$.
For $x\in{\mathcal B}_n$ and $t\geq0$, we set ${\tilde H}_t(x)=\max
(H_t(x),1-Z_t(x),1-Z_t(x-),1-Z_t(x+))$. Observe that for the LFFP, $x$ is *microscopic* (or *acts like a barrier*) if and only if ${\tilde H}_t(x)>0$ and, if so, it will remain microscopic during exactly $[t,t+{\tilde H}_t(x))$. Note that, in fact, $Z_t(x)$ always equals either $Z_t(x-)$ or $Z_t(x+)$.
We consider the set of times ${\mathcal K}:=\{t\in\{0,T\}$: there exists $x\in(-A,A)$ such that ${\tilde H}_t(x)=0$ but ${\tilde
H}_{t-{\varepsilon}}(x)>0$ for all ${\varepsilon}>0$ small enough$\}$. By construction, we see that ${\mathcal K}\subset\{
1,T_i+1,T_i+Z_{T_i-}(X_i),i=1,\ldots,n\}
\subset\{1, T_i+1,T_i+(T_i-T_j), 0\leq j < i \leq n \} $.
We work conditionally on $M$, by induction on $q=0,\ldots,n$. Consider the following assumption.
For all $0< \mu\leq1$, $c\in{\mathcal C}_q$ and ${\varepsilon}>0$, $\lim_{{\lambda}\to0}{\mathbb{P}}_M(|{\tilde Z}^{{{\lambda},\mu}
}_{T_q}(c)-Z_{T_q}(c)|>{\varepsilon})=0$.
For all $x\in{\mathcal B}_q$, $0 < \mu\leq1$ and ${\varepsilon}>0$, $\lim_{{\lambda}\to
0}{\mathbb{P}}_M(|{\tilde H}^{{\lambda},\mu}_{T_q}(x)-{\tilde
H}_{T_q}(x)|>{\varepsilon})=0$.
For all $0 < \mu\leq1$, $\lim_{{\lambda}\to0}{\mathbb
{P}}_M({\mathcal E}^{{\lambda},\mu}_q)=1$ (recall Lemma \[exch\]).
First, $({\mathcal H}_0)$ is obviously satisfied because $T_0=0$, ${\mathcal C}_0=(-A,A)$, ${\tilde Z}^{{\lambda},\mu}_{0}((-A$,$A))=0=Z_{0}((-A,A))$, ${\mathcal
B}_0=\varnothing$ and ${\mathcal E}^{{\lambda},\mu}_0=\Omega$.
The proposition will essentially be proven if we check that for $q=0,\ldots,n-1$, $({\mathcal H}_{q})$ implies:
\(a) for $c\in{\mathcal C}_q$, $0<\mu\leq1$ and ${\varepsilon}>0$, $\lim_{{\lambda}\to0}{\mathbb{P}}_M(\sup_{[T_q,T_{q+1})}|{\tilde
Z}^{{\lambda},\mu}
_{t}(c)-Z_{t}(c)|>{\varepsilon})=0$;
\(b) for $x \in(-A,A)\setminus{\mathcal B}_q$, ${\varepsilon}>0$, $\lim_{{\lambda}\to0}{\mathbb{P}}_M({\sup
_{[T_q,T_{q+1})}}|Z^{\lambda}
_{t}(x)-Z_{t}(x)|>{\varepsilon})=0$; (c) for $x\in{\mathcal B}_q$, $t\in[T_q,T_{q+1})$, $0< \mu\leq1$ and ${\varepsilon}
>0$, $\lim_{{\lambda}\to0}{\mathbb{P}}_M(|{\tilde H}^{{\lambda
},\mu}_{t}(x)-{\tilde H}
_{t}(x)|>{\varepsilon})$; (d) for $x \in(-A,A)\setminus{\mathcal B}_q$, $t\in(T_q,T_{q+1})\setminus{\mathcal K}$ and ${\varepsilon}>0$, $\lim_{{\lambda}\to0}{\mathbb{P}}_M(\delta(D^{\lambda}_{t}(x)$,$D_{t}(x)
)>{\varepsilon})=0$;
\(e) for $x \in(-A,A) \setminus{\mathcal B}_q$, ${\varepsilon}>0$, $\lim_{{\lambda}\to0}{\mathbb{P}}_M(\int_{T_q}^{T_{q+1}}\delta
(D^{\lambda}_{t}(x) ,
D_{t}(x) ) \,dt
>{\varepsilon})=0$;
\(f) $({\mathcal H}_{q+1})$ holds.
We thus assume $({\mathcal H}_{q})$ for some fixed $q\in\{0,\ldots,n-1\}$ and prove points (a)–(f). Below, we repeatedly use the fact that on the time interval $[T_q,T_{q+1})$, there are no fires at all in $(-A,A)$ for the LFFP and no fires at all during $[T_{q}\log(1/{\lambda}),T_{q+1}\log(1/{\lambda}))$ for the ${\lambda}$-FFP.
Set $\zeta^{\lambda}_0(i)=\eta^{\lambda}_{T_q\log(1/{\lambda})}(i)$ and consider the i.i.d. Poisson processes $P^{\lambda
}_t(i)=N_{(T_q+t)\log
(1/{\lambda})}(i)
-N_{T_q\log(1/{\lambda})}(i)$ with rate $\log(1/{\lambda})$. Then, for $t\in
[T_q,T_{q+1})$, $\eta^{\lambda}_{t\log(1/{\lambda})}(i)=\min(\zeta
_0(i)+P^{\lambda}_{t-T_q}(i),1)$.
*Point* (a). Let $0< \mu\leq1$. Let $c \in{\mathcal C}_q$. Observe that $({\mathcal H}_{q})$(i) says precisely that with $h=1-Z_{T_q}(c)\in[0,1]$, $\log(1+\#\{k\in c_{{\lambda},\mu}, \zeta
^{\lambda}
_0(k)=0\})
/\log(1+\#(c_{{\lambda},\mu}))$ tends to $h$ in probability (for ${\mathbb{P}}_M$). Applying part 1(a) of Lemma \[binomiale\] (with $J=c$), we get that $\sup_{[T_q,T_{q+1})} |1-{\tilde Z}^{{\lambda},\mu
}_t(c)-(h-(t-T_q))_+|$ tends to $0$ in probability (for ${\mathbb{P}}_M$). However, for $t\in[T_q,T_{q+1})$, we have $Z_t(c)=\min(Z_{T_q}(c)+(t-T_q),1)=\min(1-h+(t-T_q),1)
=1-(h-(t-T_q))_+$. Point (a) then follows.
*Point* (b). Now, let $x\in(-A,A)\setminus{\mathcal B}_q$. Then $x \in c$, for some $c\in
{\mathcal C}_q$. Due to Lemma \[exch\], we know that $(\zeta_0^{\lambda}(i))_{i \in
c_{{\lambda},\mu}}$ are exchangeable on ${\mathcal E}^{{\lambda},1}_q$. The previous reasoning, using part 1(b) of part 1(a) of Lemma \[binomiale\], shows that for all ${\varepsilon}>0$, $\lim_{{\lambda}\to0}{\mathbb{P}}_M( {\mathcal E}^{{\lambda},1}_q
\cap\{
{\sup_{[T_q,T_{q+1})}}|Z^{\lambda}_{t}(x)-Z_{t}(x)|>{\varepsilon}\}
)=0$. Using $({\mathcal H}_{q})$(iii) for $\mu=1$, we are done.
*Point* (c). Let $0< \mu\leq1$. Let $x \in{\mathcal B}_q$ and set $h={\tilde H}_{T_q}(x)$. We know by $({\mathcal H}_q)$(ii) that ${\tilde H}_{T_q}^{{\lambda},\mu
}(x)$ tends to ${\tilde H}_{T_q}(x)=h$ in probability (for ${\mathbb{P}}_M$). Now, using part 2(a) of Lemma \[binomiale\], we deduce that ${\sup_{[T_q,T_{q+1})}}|{\tilde H}^{{\lambda},\mu
}_t(x)-(h-(t-T_q))_+ |$ tends to $0$ in probability (for ${\mathbb{P}}_M$). We conclude by observing that, by construction, ${\tilde H}_t(x)= (h-(t-T_q))_+$ for $t\in
[T_q,T_{q+1})$.
*Point* (d). Let $x \in(-A,A)\setminus{\mathcal B}_q$ and $t \in(T_q,T_{q+1})\setminus{\mathcal K}$ be fixed.
*Case $Z_t(x)<1$.* In this case, $D_t(x)=\{x\}$ so that $\delta(D_t(x),D^{\lambda}_t(x))=|D^{\lambda}_t(x)|$. However, from (\[dlambda\]), (\[zlambda\]), we get that $|D^{\lambda}_t(x)|
\leq
{\lambda}^{1-Z^{\lambda}_t(x)}
\log(1/{\lambda})$. Since we know from (b) that $Z^{\lambda}_t(x)$ goes to $Z_t(x)<1$ in probability (for ${\mathbb{P}}_M$), we easily deduce that $|D^{\lambda}_t(x)|$ goes to $0$ in probability (for ${\mathbb{P}}_M$).
*Case $Z_t(x)=1$.* In this case, $D_t(x)=[a,b]$ for some $a,b\in{\mathcal B}_q\cup\{-A,A\}$. We assume that $-A<a<b<A$ for simplicity, the other cases being treated in a similar way. We thus have $Z_t(c)=1$ for all $c\in{\mathcal C}_q$ with $c\subset(a,b)$, ${\tilde H}_t(y)=0$ for all $y\in{\mathcal
B}_q\cap(a,b)$ and ${\tilde H}_t(a){\tilde H}_t(b)>0$.
On the one hand, we prove that for any ${\varepsilon}>0$, $\lim_{{\lambda}\to0}{\mathbb{P}}_M(D^{\lambda}_t(x)\subset
[a-{\varepsilon
},b+{\varepsilon}])=1$. Let us consider, for example, the left boundary $a$ and prove that $\lim_{{\lambda}\to0}{\mathbb{P}}_M(D^{\lambda}_t(x)\subset
[a-{\varepsilon},A])=1$.
We have ${\tilde H}_t(a)=h_a>0$. We deduce from (c) that $\lim_{{\lambda}\to0} {\mathbb{P}}_M({\tilde H}^{{\lambda
},1}_t(a)\geq h_a/2)=1$, which implies that there are vacant sites in $a_{{\lambda},1}$, that is,$\lim_{{\lambda}\to0} {\mathbb{P}}_M(\exists i\in a_{{\lambda},1},
\eta_{t\log(1/{\lambda}
)}(i)=0)=1$. Recalling the definition of $a_{{\lambda},1}$ \[see (\[xla\])\], we see that this implies that $\lim_{{\lambda}\to0}{\mathbb{P}}_M(D^{\lambda}_t(x)\subset
[a-1/\log(1/{\lambda}),\break A])=1$, hence $\lim_{{\lambda}\to0}{\mathbb{P}}_M(D^{\lambda}_t(x)\subset
[a-{\varepsilon},A])=1$ for any ${\varepsilon}>0$.
On the other hand, we prove that $\lim_{{\lambda}\to0} {\mathbb
{P}}_M((a+1/\log
(1/{\lambda}
),b-1/\log(1/\break{\lambda}))
\subset D^{\lambda}_t(x))=1$. Since $t\notin{\mathcal K}$, we deduce that there exists $s\in
(T_q,t)$ such that $Z_s(c)=1$ for all $c\in{\mathcal C}_q$ with $c\subset(a,b)$ and ${\tilde H}_s(y)=0$ for all $y\in{\mathcal B}_q\cap(a,b)$. We deduce from (a) that for all $c\in{\mathcal C}_q$ with $c\subset(a,b)$, $\lim_{{\lambda}\to0} {\mathbb{P}}_M({\tilde Z}^{{\lambda
},1}_s(c)>1-{\varepsilon})=0$, whence, by Lemma \[undoncplein\](i), $\lim_{{\lambda}\to0} {\mathbb{P}}_M({\tilde
Z}^{{\lambda},1}_t(c)=1)=1$.Similarly, we deduce from (c) that for all $y\in{\mathcal B}_q$ with $y\in(a,b)$,$\lim_{{\lambda}\to0} {\mathbb{P}}_M({\tilde H}^{{\lambda
},1}_s(y)>{\varepsilon})=0$, whence, by Lemma \[undoncplein\](ii), $\lim_{{\lambda}\to0} {\mathbb
{P}}_M({\tilde H}^{{\lambda},1}_t(y)=0)=1$. As a consequence, $\lim_{{\lambda}\to0} {\mathbb{P}}_M((a+1/\log
(1/{\lambda}),b-1/\log
(1/{\lambda}))
\subset D^{\lambda}_t(x))=1$.
This completes the proof of point (d).
*Point* (e). Point (e) follows from (d). Indeed, observe that $\delta(I,J) \leq2A$ for any intervals $I,J \subset(-A,A)$. Thus, for $x\in(-A,A)\setminus{\mathcal B}_q$, (d) implies that for $t\in[T_q,T_{q+1})\setminus{\mathcal K}$, $\lim_{{\lambda}\to0}{\mathbb{E}}_M(\delta(D^{\lambda}_t(x),D_t(x)))=0$. Since ${\mathcal K}$ is now finite, we deduce from Lebesgue’s dominated convergence theorem that $\lim_{{\lambda}\to0}\int_{T_q}^{T_{q+1}} {\mathbb{E}}_M(\delta
(D^{\lambda}_t(x),D_t(x)))\,dt=0$, from which (e) follows.
*Point* (f). Here, we show that $({\mathcal H}_{q+1})$ holds. We set $z:=Z_{T_{q+1}-}(X_{q+1})$ and separately treat the cases $z\in(0,1)$ and $z=1$. We a.s. never have $z=0$ because $Z_{T_{q+1}-}(X_{q+1})=\min(Z_{T_q}(X_{q+1})+(T_{q+1}-T_q),1)$ with $Z_{T_q}(X_{q+1})\geq0$ and $T_{q+1}>T_q$.
*Case* $z\in(0,1)$. We fix $\mu\in(0,1]$. In that case, $D_{T_{q+1}-}(X_{q+1})=\{X_{q+1}\}$ and for all $c \in{\mathcal C}_{q+1}$ (thus $c\subset\tilde c$ for some $\tilde
c\in{\mathcal C}_q$), $Z_{T_{q+1}}(c)=Z_{T_{q+1}-}(c)$. We have ${\tilde H}_{T_{q+1}}(X_{q+1})=\max(z,1-z)$ and for all $x\in{\mathcal B}_q$, ${\tilde H}_{T_{q+1}}(x)={\tilde
H}_{T_{q+1}-}(x)$. Consider the event $\Omega^{\lambda}_\alpha=\{Z^{\lambda}_{T_{q+1}-}(X_{q+1})\leq
z+\alpha\}$ for some $\alpha\in(0,1-z)$. Point (b) implies that $\lim_{{\lambda}\to0}{\mathbb{P}}_M(\Omega^{\lambda}_\alpha)=1$ (because $X_{q+1}\notin{\mathcal B}_q$).
- On $\Omega^{\lambda}_\alpha$, we have $\#(C^{\lambda}_{T_{q+1}\log(1/{\lambda})-}(X_{q+1}))\leq
(1/{\lambda}
)^{z+\alpha}$ \[see (\[zlambda\])\]. Since $z+\alpha<1$, we deduce that on $\Omega^{\lambda}_\alpha$, we have $\#(C^{\lambda}_{T_{q+1}\log(1/{\lambda})-}(X_{q+1}))< \mu
/\break (2{\lambda}\log^2(1/{\lambda}))$ (for all $\mu$, provided that ${\lambda}>0$ is small enough). Thus, on $\Omega^{\lambda}_\alpha$, for all $c\in{\mathcal C}_{q+1}$, there is a vacant site (strictly) between $c_{{\lambda},\mu}$ and $C^{\lambda}_{T_{q+1}\log
(1/{\lambda})-}(X_{q+1})$. Hence, ${\mathcal E}^{{\lambda},\mu}_q\cap\Omega^{\lambda}_\alpha\subset
{\mathcal E}^{{\lambda},\mu}_{q+1}$. Using $({\mathcal H}_q)$(iii), we deduce that $\lim_{{\lambda}\to0} {\mathbb{P}}_M({\mathcal
E}^{{\lambda},\mu}_{q+1})=1$.
- This also implies that on $\Omega^{\lambda}_\alpha$, for all $c\in{\mathcal C}_{q+1}$, we have ${\tilde Z}^{{\lambda},\mu}_{T_{q+1}}(c)={\tilde Z}^{{\lambda},\mu
}_{T_{q+1}-}(c)$ and thus point (a) and $\lim_{{\lambda}\to0}{\mathbb{P}}_M(\Omega^{\lambda}_\alpha)=1$ imply that $\lim_{{\lambda}\to0}{\mathbb{P}}_M(|{\tilde
Z}^{{\lambda},\mu}
_{T_{q+1}}(c)-Z_{T_{q+1}}(c)|\geq{\varepsilon})=0$ for all ${\varepsilon}>0$.
- For $x\in{\mathcal B}_{q+1}\setminus\{X_{q+1}\}={\mathcal
B}_q$, still on $\Omega
^{\lambda}_\alpha$, we also have ${\tilde H}^{{\lambda},\mu}_{T_{q+1}}(x)={\tilde
H}^{{\lambda},\mu}_{T_{q+1}-}(x)$, thus point (c) allows us to conclude that $({\mathcal H}_{q+1})$(ii) holds for those points $x$.
We now show that $\lim_{{\lambda}\to0}{\mathbb{P}}_M(|{\tilde H}^{{\lambda},\mu
}_{T_{q+1}}(X_{q+1})-{\tilde H}
_{T_{q+1}}(X_{q+1})|\geq{\varepsilon})=0$ for all ${\varepsilon}>0$, which implies that $({\mathcal H}_{q+1})$(ii) holds for $x=X_{q+1}$. Recall that ${\tilde H}_{T_{q+1}}(X_{q+1})=\max(z,1-z)$. Consider $c\in{\mathcal C}_q$ such that $X_{q+1}\in c$ and denote by $v^{{\lambda},\mu}_t$ the number of vacant sites in $x_{{\lambda},\mu}$ at time $t\log(1/{\lambda})$. Point (a) implies that at time $T_{q+1}\log(1/{\lambda})-$, there are around $(1/{\lambda})^{1-z}$ vacant sites in $c_{{\lambda
},\mu}$. Thus, by exchangeability of the family $(\eta^{\lambda}_{T_{q+1}\log
(1/{\lambda}
)-}(i))_{i\in c_{{\lambda},\mu}}$ (on the event ${\mathcal E}^{{{\lambda},\mu}}_{q}$, see Lemma \[exch\]), since $x_{{\lambda},\mu}\subset c_{{\lambda},\mu}$ and $\#
(x_{{\lambda},\mu})/\#(c_{{\lambda},\mu})\simeq
1/\log
(1/{\lambda})$, we deduce that $v^{{\lambda},\mu}_{T_{q+1}-} \simeq(1/{\lambda
})^{1-z}/\log(1/{\lambda})
\simeq(1/{\lambda})^{1-z}$ on ${\mathcal E}^{{\lambda},\mu}_q$. On the other hand, recalling (\[zlambda\]), we have $\#(C^{\lambda}_{T_{q+1}\log(1/{\lambda})-}(X_{q+1})) \simeq
(1/{\lambda})^z$. At time $T_{q+1}\log(1/{\lambda})$, this component is destroyed. Thus, still on ${\mathcal E}^{{\lambda},\mu}_q$, $v^{{\lambda},\mu}_{T_{q+1}} = v^{{\lambda},\mu}_{T_{q+1}-}+\#
(C^{\lambda}_{T_{q+1}\log(1/{\lambda}
)}(X_{q+1}))
\simeq(1/{\lambda})^{1-z} +
(1/{\lambda})^z\simeq(1/{\lambda})^{\max(z,1-z)}$. We conclude that ${\tilde H}^{{\lambda},\mu}_{T_{q+1}}(X_{q+1})= \log(1+v^{{\lambda
},\mu}_{T_{q+1}})/\break \log(\#
((X_{q+1})_{{\lambda},\mu})) \simeq
\max(z,1-z)={\tilde H}_{T_{q+1}}(X_{q+1})$. All of this can be done rigorously without difficulty and we deduce that for ${\varepsilon}>0$ and all $\mu\in(0,1]$, $\lim_{{\lambda}\to0}
{\mathbb{P}}_M(
|{\tilde H}^{{\lambda},\mu}_{T_{q+1}}(X_{q+1})-{\tilde
H}_{T_{q+1}}(X_{q+1})|\geq
{\varepsilon})=0$.
*Case $z=1$.* Let $a,b \in{\mathcal B}_q \cup\{-A,A\}$ be such that $D_{T_{q+1}-}(X_{q+1})=[a,b]$. We assume that $a,b \in{\mathcal B}_q$, the other cases being treated in a similar way. We thus have $h_a:={\tilde H}_{T_{q+1}-}(a)>0$, $h_b:={\tilde H}_{T_{q+1}-}(b)>0$. We also have ${\tilde H}_{T_{q+1}}(x)={\tilde H}_{T_{q+1}-}(x)$ for all $x\in{\mathcal B}_q \setminus[a,b]$, ${\tilde H}_{T_{q+1}}(x)=1$ for all $x\in{\mathcal B}_q \cap(a,b)$, $Z_{T_{q+1}}(c)=Z_{T_{q+1}-}(c)$ for all $c\in{\mathcal C}_{q+1}$ with $c\cap(a,b)=\varnothing$ and $Z_{T_{q+1}}(c)=0$ for all $c\in{\mathcal C}_{q+1}$ with $c\subset(a,b)$.
Let $\mu\in(0,1]$. Now, consider ${\tilde\Omega}^{{\lambda},\mu}$, the event that for all $c\in{\mathcal C}_{q}$ such that $c \subset(a,b)$, we have ${\tilde Z}^{{\lambda},\mu}_{T_{q+1}-}(c)=1$, that ${\tilde
H}^{{\lambda},\mu}_{T_{q+1}-}(a)>0$, that ${\tilde H}^{{\lambda},\mu}_{T_{q+1}-}(b)>0$ and that for all $x \in{\mathcal B}_q \cap(a,b)$, ${\tilde H}^{{\lambda},\mu
}_{T_{q+1}-}(x)=0$. Then (a), (c) and Lemma \[undoncplein\] collectively imply that $\lim_{{\lambda}\to0} {\mathbb{P}}_M({\tilde\Omega}^{{\lambda
},\mu})=1$ for all $\mu\in(0,1]$.
- We can easily check that ${\mathcal E}^{{\lambda},\mu}_{q} \cap
{\tilde\Omega}^{{\lambda},\mu}
\subset
{\mathcal E}^{{\lambda},\mu}_{q+1}$ (because for $c \in{\mathcal C}_{q+1}$ with $c \subset[a$, $b]$, we have $c_{{\lambda},\mu}\subset C^{\lambda}_{T_{q+1}\log
(1/{\lambda})-}(X_{q+1})$, while for $c \in{\mathcal C}_{q+1}$ with $c \cap[a,b] = \varnothing$, the vacant sites in $a_{{\lambda},\mu}$ and $b_{{\lambda},\mu}$ separate $c_{{\lambda},\mu}$ from$ C^{\lambda}_{T_{q+1}\log(1/{\lambda})-}(X_{q+1})$). As a consequence, $({\mathcal H}_{q+1})$(iii) holds for all $\mu\in(0,1]$.
- On ${\tilde\Omega}^{{\lambda},\mu}$, we have ${\tilde
Z}^{{\lambda},\mu}
_{T_{q+1}}(c)=0=Z_{T_{q+1}}(c)$ for all $c \in{\mathcal C}_{q+1}$ with $c \subset[a,b]$, and ${\tilde Z}^{{\lambda},\mu}_{T_{q+1}}(c)={\tilde Z}^{{\lambda},\mu
}_{T_{q+1}-}(c)$ for $c \in{\mathcal C}_{q+1}$ with $c\cap(a,b)=\varnothing$, from which $({\mathcal H}_{q+1})$(i) easily follows \[using (a)\].
- We also have, still on ${\tilde\Omega}^{{\lambda},\mu}$, that ${\tilde H}^{{\lambda},\mu}_{T_{q+1}}(x)=1={\tilde H}_{T_{q+1}}(x)$ for all $x \in{\mathcal B}
_{q+1}$ with $x \in(a,b)$, and $({\mathcal H}_{q+1})$(ii) follows for those $x$. For $x \in{\mathcal B}_{q+1}$ with $x \notin[a,b]$, we have ${\tilde H}^{{\lambda},\mu}_{T_{q+1}}(x)={\tilde H}^{{\lambda},\mu
}_{T_{q+1}-}(x)$, hence $({\mathcal H}_{q+1})$(ii) follows by point (c).
Finally, we have to check that $({\mathcal H}_{q+1})$(ii) holds for $x=a$ and $x=b$. Consider, for example, the case of $a$. Here, we are in the situation where so that, of course, ${\tilde H}_{T_{q+1}}(a)
= 1$. Let $c$ be the cell containing $a+$. We know that ${\tilde Z}^{{\lambda},\mu/2}_{T_{q+1}-}(c) =1$ which, on ${\tilde\Omega}^{{\lambda},\mu/2}$, implies that all sites between $a+\frac{\mu}{2\log(1/{\lambda})}$ and $a+\frac
{\mu}{\log(1/{\lambda})}$, that is, on an interval of length $\frac
{\mu}{2
\log(1/{\lambda})}$, are empty at time $T_{q+1}$, showing that a fixed proportion of $a_{{\lambda},\mu}$ is empty. Recalling that $\lim_{{\lambda}\to0}{\mathbb{P}}_M( {\tilde\Omega
}^{{\lambda},\mu/2}) = 1$, it readily follows that for all ${\varepsilon}>0$, $\lim_{{\lambda}\to
0}{\mathbb{P}}_M( {\tilde H}
^{{\lambda},\mu}
_{T_{q+1}}(a) > 1-{\varepsilon}) =1$. Recalling that $ {\tilde H}^{{\lambda},\mu}_{T_{q+1}}(a) \leq1$, we conclude that $({\mathcal H}
_{q+1})$(ii) holds for $x=a$.
### Conclusion {#conclusion .unnumbered}
Using points (b) and (e) above (with $q=0,\ldots,n$), plus very similar arguments on the time interval $(T_n,T]$ (during which there are no fires), we deduce that for all $x_0 \in(-A,A)\setminus{\mathcal B}_n$ and ${\varepsilon}>0$, $$\lim_{{\lambda}\to0}{\mathbb{P}}_M \biggl({\sup_{[0,T]}} |Z^{\lambda
}_t(x_0) - Z_t(x_0)| +
\int_0^T \delta(D^{\lambda}_t(x_0),D_t(x_0))\,dt \geq{\varepsilon}\biggr) =0.$$ But, of course, for $x_0 \in(-A,A)$, we have ${\mathbb{P}}(x_0 \in
{\mathcal B}_n)=0$ so that $$\lim_{{\lambda}\to0}{\mathbb{P}}\biggl({\sup_{[0,T]}} |Z^{\lambda}_t(x_0)
- Z_t(x_0)| +
\int_0^T \delta(D^{\lambda}_t(x_0),D_t(x_0))\,dt \geq{\varepsilon}\biggr) =0.$$ It remains to prove that for $t \in[0,T]$ and $x_0 \in(-A,A)$, we have $$\lim_{{\lambda}\to0} {\mathbb{P}}(\delta(D^{\lambda
}_t(x_0),D_t(x_0)))=0.$$
*Case $t\ne1$.* We deduce from point (d) above that if $x_0 \notin{\mathcal B}_n$ and $t \notin{\mathcal K}$, then we have $\lim_{{\lambda}\to0} {\mathbb{P}}_M(\delta(D^{\lambda
}_t(x_0),D_t(x_0)))=0$. Since ${\mathbb{P}}(x_0 \in{\mathcal B}_n)=0$ and ${\mathbb{P}}(t \in
{\mathcal K})=0$ (because $t \ne1$, recalling the definition of ${\mathcal K}$), we easily arrive at the desired conclusion.
*Case $t=1$.* In this case, $t \in{\mathcal K}$, but the result still holds. Observe that , by construction. Consider $q\in\{0,\ldots,n\}$ such that $T_q<1<T_{q+1}$ (with the convention that $T_0=0$, $T_{n+1}=T$) and consider $a,b \in{\mathcal B}_q \cup\{-A,A\}$ such that $D_1(x_0)=[a,b]$. Using the same arguments as in the proof of (d) (see Step 1), we then easily check that $\lim_{{\lambda}\to0} {\mathbb{P}}_M( D^{\lambda}_{1}(x_0)
\subset[a-{\varepsilon
},b+{\varepsilon}])=1$ for all ${\varepsilon}>0$ (the set ${\mathcal K}$ was not considered there). We also check, as in the proof of (d) (see Step 2), that for all $y \in
{\mathcal B}_q$ with $y \in(a,b)$, $\lim_{{\lambda}\to0} {\mathbb{P}}_M( H^{{\lambda},1}_{1}(y)=0)=1$ \[the set under consideration there was ${\mathcal K}$, but the time $1$ was not useful since $1$ is a.s. not a time where some $H(x)$ reaches $0$ for the first time\]. Finally, we just have to prove that for all $c \in{\mathcal C}_q$ with $c \subset(a,b)$, $\lim_{{\lambda}\to0} {\mathbb{P}}_M({\tilde
Z}^{{\lambda},1}_1(c)=1)=1$. Thus, let $c \in{\mathcal C}_q$ with $c \subset(a,b)$ and recall that $\lim_{{\lambda}\to0}{\mathbb{P}}_M({\mathcal
E}^{{\lambda},1}_q)=1$. However, on ${\mathcal E}^{{\lambda},1}_q$, there are no death events in $c_{\lambda}$ during the time interval $[0,\log(1/{\lambda})]$, so each site of $c_{{\lambda},1}$ is occupied at time $\log(1/{\lambda})$ with probability $1- {\lambda}$ and, hence, all the sites of $c_{{\lambda},1}$ are occupied with probability $(1-{\lambda})^{\#
(c_{{\lambda},1})}$. Since $\#(c_{{\lambda},1}) \leq2A/({\lambda}\log(1/{\lambda}))$, we get ${\mathbb{P}}_M({\tilde Z}^{{\lambda},1}_1(c)=1 \vert{\mathcal
E}^{{\lambda},1}_q) \geq(1-{\lambda})^{2A/({\lambda}
\log
(1/{\lambda}))}$, which tends to $1$ as ${\lambda}$ tends to $0$. Since we know that $\lim_{{\lambda}\to0}{\mathbb{P}}_M({\mathcal
E}^{{\lambda},1}_q)=1$, we deduce that $\lim_{{\lambda}\to0} {\mathbb{P}}_M( [a+1/\log
(1/{\lambda}),
b-1/\log(1/{\lambda})] \subset D^{\lambda}_{1}(x_0))=1$.
Finally, $\lim_{{\lambda}\to0} {\mathbb{P}}_M( \delta(D^{\lambda
}_{1}(x_0),D_1(x_0))\geq
{\varepsilon})=0$ for all ${\varepsilon}>0$, which was our goal.
Cluster size distribution {#conseq}
=========================
The aim of this section is to prove Corollary \[coco\]. We will use Theorem \[converge\], which asserts that the ${\lambda}$-FFP behaves like the LFFP for ${\lambda}>0$ small enough. We start with preliminary results.
\[zunif\] Consider an LFFP $(Z_t(x),D_t(x),H_t(x))_{t\geq0, x\in{\mathbb{R}}}$. We then have the following:
for any $t\in(1,\infty)$, $x\in{\mathbb{R}}$ and $z\in[0,1)$, ${\mathbb{P}}[Z_t(x)=z]=0$;
for any $t\in[0,\infty)$, $B>0$ and $x\in{\mathbb{R}}$, $P[|D_t(x)|=B]=0$;
there are constants $C>0$ and $\kappa_1>0$ such that for all $t\in[0,\infty)$, $x\in{\mathbb{R}}$ and $B>0$, ${\mathbb{P}}[|D_t(x)|\geq B] \leq C e^{-\kappa_1 B}$;
there are constants $c>0$ and $\kappa_2>0$ such that for all $t\in[3/2,\infty)$, $x\in{\mathbb{R}}$ and $B>0$, ${\mathbb{P}}[|D_t(x)|\geq B] \geq c e^{-\kappa_2 B}$;
there exist constants $0<c<C$ such that for all $t\geq5/2$, $0\leq a < b < 1$ and $x\in{\mathbb{R}}$, $c(b-a) \leq{\mathbb{P}}(Z_t(x)\in[a,b]) \leq C(b-a)$.
By translation invariance, it suffices to treat the case $x=0$.
*Point* (i). By Definition \[dflffp\], we see that for $t\in[0,1]$, we have a.s. $Z_t(0)=t$. However, for $t> 1$ and $z\in[0,1)$, $Z_t(0)=z$ implies that the cluster containing $0$ has been killed at time $t-z$, so, necessarily, $M(\{t-z\}\times{\mathbb{R}})>0$. This happens with probability $0$ since $t-z$ is deterministic.
*Point* (ii). Recalling Definition \[dflffp\], we see that for any $t\in[0,T]$, $|D_t(0)|$ is either $0$ or of the form $|X_i-X_j|$ (with $i\ne j$), where $(T_i,X_i)_{i\geq1}$ are the marks of the Poisson measure $M$. As before, we easily conclude that for $B>0$, ${\mathbb{P}}(|D_t(0)|=B)=0$.
*Point* (iii). First, if $t \in[0,1)$, then we have a.s. $|D_t(0)|=0$ and the result is obvious. Next, consider $t\geq1$. Recalling Definition \[dflffp\], we see that $|D_t(0)|= |L_t(0)|
+R_t(0)$. Clearly, $|L_t(0)|$ and $R_t(0)$ have the same law. For $B>0$, $\{R_t(0)> B \} \subset
\{ M([t-1/4,t]\times[0,B])=0\}$. Indeed, on $\{ M([t-1/4,t]\times[0,B])>0\}$, denote by $(\tau,X)\in[t-1/4,t]\times[0,B]$ a mark of $M$. Then, either:
$\bullet$ $Z_{\tau-}(X)=1$, in which case this mark starts a macroscopic fire so that $Z_{\tau}(X)=0$ and $Z_{s}(X)=s-\tau<1$ for all $s\in[\tau,\tau+1)$ (since $\tau\in[t-1/4,t]$, we clearly have $t \in[\tau,\tau+1)$ so that $Z_t(X)<1$ and, as a consequence, $R_t(0)\leq X \leq B$); or
$\bullet$ $Z_{\tau-}(X) \in(1/4,1]$ so that $H_{\tau}(X)=Z_{\tau-}(X)$ and thus $H_{s}(X)=Z_{\tau-}(X) - (s-\tau) >0$ for all $s\in[\tau,
\tau+
Z_{\tau-}(X))$ (since $\tau\in[t-1/4,t]$ and $Z_{\tau-}(X)>1/4$, we have $t \in[\tau, \tau+Z_{\tau-}(X))$, so $H_t(X)>0$ and, hence, $R_t(0)\leq X \leq B$); or, finally,
$\bullet$ $Z_{\tau-}(X)\leq1/4$, in which case $Z_{s}(X)=Z_{\tau-}(X)+(s-\tau)<1$ for all $s\in(\tau, \tau+1-Z_{\tau-}(X))$ and, in particular, $Z_t(X)<1$, hence $R_t(0)\leq X \leq B$.
As a conclusion, for all $t\geq1$, ${\mathbb{P}}[R_t(0) > B] \leq
{\mathbb{P}}[ M([t-1/4,t]\times[0,B])=0]=e^{-B/4}$, so ${\mathbb{P}}[|D_t(0)|> B] \leq
{\mathbb{P}}[|L_t(0)|> B/2]+ {\mathbb{P}}[R_t(0) > B/2] \leq2 e^{-B/8}$.
*Point* (iv). We first observe that for all $(t_0,x_0)$ such that $M(\{t_0,x_0\})=1$, we have $\max(1-Z_{t}(x_0),H_{t}(x_0))>0$ for all $t\in[t_0,t_0+1/2)$.
Indeed, if $Z_{t_0-}(x_0)=1$, then $Z_{t_0+s}(x_0) \leq
s<1$ for all $s\in[0,1)$. If, now, $z=Z_{t_0-}(x_0)<1$, then $Z_{t_0+s}(x_0) = s+z<1$ for $s \in[0,1-z)$ and $H_{t_0+s}(x_0) = z-s>0$ for $s\in[0,z)$ so that $\max(1-Z_{t_0+s}(x_0),H_{t_0+s}(x_0))>0$ for all $s\in[0,1/2)$.
Once this is seen, fix $t\geq3/2$. Consider the event ${\tilde\Omega}_{t,B}={\tilde\Omega}^1_{t,B} \cap{\tilde\Omega
}^2_{t}\cap{\tilde\Omega}^3_{t,B}$, where:
$\bullet$ ${\tilde\Omega}^{1}_{t,B}=\{M([t-3/2,t]\times[0,B])=0\}$;
$\bullet$ ${\tilde\Omega}^2_{t}$ is the event that in the box $[t-3/2,t]\times[-1,0]$, $M$ has exactly four marks, $(S_i,Y_i)_{i=1,\ldots,4}$, with $Y_4<Y_3<Y_2<Y_1$, $t-3/2 < S_1 < t-1$, $S_1<S_2 < S_1+1/2$, $S_2<S_3< S_2+1/2$, $S_3<S_4< S_3+1/2$ and $S_4+1/2>t$.
$\bullet$ ${\tilde\Omega}^3_{t,B}$ is the event that in the box $[t-3/2,t]\times[B,B+1]$, $M$ has exactly four marks, $({\tilde S}_i,{\tilde Y}_i)_{i=1,\ldots,4}$, with ${\tilde Y}_1<{\tilde Y}_2<{\tilde Y}_3<{\tilde Y}_4$, $t-3/2 < {\tilde
S}_1 < t-1$, ${\tilde S}_1<{\tilde S}_2 <
{\tilde S}
_1+1/2$, ${\tilde S}_2<{\tilde S}_3< S_2+1/2$, ${\tilde S}_3<{\tilde S}_4< {\tilde S}_3+1/2$ and ${\tilde S}_4+1/2>t$.
Of course, we have $p:={\mathbb{P}}({\tilde\Omega}^2_{t})={\mathbb
{P}}({\tilde\Omega}^3_{t,B})>0$ and this probability does not depend on $t\geq3/2$ or on $B>0$. Furthermore, ${\mathbb{P}}({\tilde\Omega}^1_{t,B})=e^{-3B/2}$. These three events being independent, we conclude that ${\mathbb{P}}({\tilde\Omega}_{t,B}) \geq p^2 e^{-3B/2}$. To conclude the proof of (iv), it thus suffices to check that ${\tilde\Omega}_{t,B}\subset\{
[0,B]\subset
D_t(0)\}$. However, on ${\tilde\Omega}_{t,B}$, using the arguments described at the beginning of the proof of point (iv), we observe that:
$\bullet$ the fire starting at $(S_2,Y_2)$ cannot affect $[0,B]$ because at time $S_2 \in[S_1, S_1 +1/2)$, $H_{S_2}(Y_1)>0$ or $Z_{S_2}(Y_1)>0$, with $Y_2<Y_1<0$;
$\bullet$ then the fire starting at $(S_3,Y_3)$ cannot affect $[0,B]$ because at time $S_3 \in[S_2,S_2+1/2)$, $H_{S_3}(Y_2)>0$ or $Z_{S_3}(Y_2)>0$, with $Y_3<Y_2<0$;
$\bullet$ then the fire starting at $(S_4,Y_4)$ cannot affect $[0,B]$ because at time $S_4 \in[S_3,S_3+1/2)$, $H_{S_4}(Y_3)>0$ or $Z_{S_4}(Y_3)>0$, with $Y_4<Y_3<0$;
$\bullet$ furthermore, the fires starting to the left of $-1$ during $(S_1,t]$ cannot affect $[0,B]$ because for all $t\in(S_1,t]$, there is always a site $x_t \in\{Y_1,Y_2,Y_3,Y_4\} \subset[-1,0]$ with $H_t(x_t)>0$ or $Z_t(x_t)<1$;
$\bullet$ the same arguments apply on the right of $B$.
As a conclusion, the zone $[0,B]$ is not affected by any fire during $(S_1 \lor{\tilde S}_1,t]$. Since the length of this time interval is greater than $1$, we deduce that for all $x \in[0,B]$, $Z_t(x)=\min(Z_{S_1 \lor{\tilde S}_1} + t- S_1 \lor{\tilde S}_1,1)
\geq\min(t- S_1 \lor{\tilde S}_1,1)=1$ and $H_t(x)=\max(H_{S_1 \lor{\tilde S}_1} - (t- S_1 \lor{\tilde S}_1),0)
\leq\max(1 - (t- S_1 \lor{\tilde S}_1),0)=0$, hence that $[0,B]\subset D_t(0)$.
*Point* (v). We observe, recalling Definition \[dflffp\], that for $0\leq a < b < 1$ and $t\geq1$, we have $Z_t(0)\in[a,b]$ if and only there exists $\tau\in[t-b,t-a]$ such that $Z_\tau(0)=0$. This happens if and only if $X_{t,a,b}:=\int_{t-b}^{t-a}\int_{\mathbb{R}}{\mathbf{1}}_{\{y\in
D_{{s-}}(0)\}
}M(ds,dy)\geq1$. We deduce that $${\mathbb{P}}\bigl(Z_t(0)\in[a,b] \bigr)
={\mathbb{P}}(X_{t,a,b}\geq1 )
\leq{\mathbb{E}}[X_{t,a,b} ]
= \int_{t-b}^{t-a}{\mathbb{E}}[|D_s(0)|]\,ds \leq C(b-a),$$ where we have used point (iii) for the last inequality.
Next, we have $\{M([t-b,t-a]\times D_{t-b}(0))\geq1 \}\subset
\{X_{t,a,b}\geq1\}$: it suffices to note that a.s. $\{X_{t,a,b}=0\} \subset\{X_{t,a,b}=0, D_{t-b}(0)\subset D_s(0)$ for all $s\in
[t-b,t-a] \} \subset\{M([t-b,t-a]\times D_{t-b}(0))=0\}$. Now, since $D_{t-b}(0)$ is ${\mathcal F}^M_{t-b}$-measurable, we deduce that for $t\geq5/2$, $$\begin{aligned}
{\mathbb{P}}\bigl(Z_t(0)\in[a,b] \bigr)
&\geq&{\mathbb{P}}\bigl[M\bigl((t-b,t-a]\times D_{t-b}(0)\bigr)>0 \bigr]\\
&\geq&{\mathbb{P}}[ |D_{t-b}(0)|\geq1 ] \bigl(1-e^{-(b-a)}\bigr) \geq c \bigl(1-e^{-(b-a)}\bigr),\end{aligned}$$ where we have used point (iv) (here, $t-b\geq3/2$) to get the last inequality. This completes the proof since $1-e^{-x}\geq x/2$ for all $x\in[0,1]$.
We now may tackle the following proof.
[Proof of Corollary \[coco\]]{} We thus consider, for each ${\lambda}>0$, a ${\lambda}$-FFP $(\eta
^{\lambda}
_t)_{t\geq0}$. Also, let $(Z_t(x),D_t(x),H_t(x))_{t\geq0, x\in{\mathbb{R}}}$ be an LFFP.
*Point* (i). Using Lemma \[zunif\](v), we only need to prove that for all $0\leq a < b < 1$ and all $t\geq5/2$, $$\lim_{{\lambda}\to0} {\mathbb{P}}\bigl(\#\bigl(C^{\lambda}_{t\log
(1/{\lambda})}(0)\bigr)\in[{\lambda}^{-a},{\lambda}^{-b}]
\bigr)
={\mathbb{P}}\bigl(Z_t(0)\in[a,b] \bigr).$$ Recalling (\[zlambda\]), we observe that $${\mathbb{P}}\bigl(\#\bigl(C^{\lambda}_{t\log(1/{\lambda})}(0)\bigr) \in[{\lambda
}^{-a},{\lambda}^{-b}] \bigr) =
{\mathbb{P}}\bigl(Z^{\lambda}_t(0) \in[a+{\varepsilon}(a,{\lambda
}),b+{\varepsilon}(b,{\lambda}
)] \bigr),$$ where ${\varepsilon}(z,{\lambda})= \log(1+{\lambda}^z)/\log
(1/{\lambda})
\to0$ as ${\lambda}\to0$ (if $z\geq0$).
We arrive at the desired conclusion by using Theorem \[converge\] \[which asserts that $Z_t^{\lambda}(0)$ goes in law to $Z_t(0)$\] and Lemma \[zunif\](i) \[from which ${\mathbb{P}}(Z_t(0)=a)={\mathbb{P}}(Z_t(0)=b)=0 $\].
*Point* (ii). Using part (iv) of Lemma \[zunif\](iii) and recalling (\[dlambda\]), it suffices to check that for all $t\geq3/2$ and all $B>0$, we have $$\lim_{{\lambda}\to0} {\mathbb{P}}[|D^{\lambda}_t(0)|\geq B ] =
{\mathbb{P}}[|D_t(0)|\geq B ].$$ This follows from Theorem \[converge\] and the fact that ${\mathbb{P}}(|D_t(0)|=B)=0$, thanks to Lemma \[zunif\](ii).
Acknowledgment {#acknowledgment .unnumbered}
==============
We are grateful to the referee who helped us to make the proofs more readable and, indeed, correct.
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---
abstract: |
Bulge-disc decomposition is a valuable tool for understanding galaxies. However, achieving robust measurements of component properties is difficult, even with high quality imaging, and it becomes even more so with the imaging typical of large surveys.
In this paper we consider the advantages of a new, multi-band approach to galaxy fitting. We perform automated bulge-disc decompositions for 163 nearby galaxies, by simultaneously fitting multiple images taken in five photometric filters. We show that we are able to recover structural measurements that agree well with various other works, and confirm a number of key results. We additionally use our results to illustrate the link between total [Sérsic]{}index and bulge-disc structure, and demonstrate that the visually classification of lenticular galaxies is strongly dependent on the inclination of their disc component.
By simulating the same set of galaxies as they would appear if observed at a range of redshifts, we are able to study the behaviour of bulge-disc decompositions as data quality diminishes. We examine how our multi-band fits perform, and compare to the results of more conventional, single-band methods. Multi-band fitting improves the measurement of all parameters, but particularly the bulge-to total flux ratio and component colours. We therefore encourage the use of this approach with future surveys.
author:
- |
Marina Vika,$^{1}$ Steven P. Bamford,$^{2}$ Boris H[ä]{}u[ß]{}ler,$^{3,4}$ Alex L. Rojas$^{1}$\
$^1$Carnegie Mellon University in Qatar, Education City, PO Box 24866, Doha, Qatar\
$^2$School of Physics and Astronomy, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK\
$^3$Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK\
$^4$University of Hertfordshire, Hatfield, Hertfordshire, AL10 9AB, UK
bibliography:
- 'references.bib'
date: 'Accepted ... Received ...; in original form ...'
title: 'MegaMorph – multi-wavelength measurement of galaxy structure: physically meaningful bulge-disc decomposition of galaxies near and far'
---
\[firstpage\]
galaxies: photometry — galaxies: fundamental parameters — galaxies: structure — methods: data analysis — techniques: image processing
Introduction {#sec:intro}
============
The spatial distribution of light within a galaxy is a key observable, with which we can constrain models of galaxy formation and evolution. The typical sizes, surface-brightness profiles and ellipticities of galaxies have been essential in determining the main physical mechanisms at work in producing the galaxy population (e.g., @tex:SN81). We have grown to understand that these properties are the result of multiple competing processes, including rapid collapse, ongoing gas accretion, disk instabilities, and the merging of existing stellar systems (e.g., @tex:B10). As a consequence, galaxies are often separable, at least to a degree, into components with distinct spatial structure, kinematics and stellar populations.
Observations often integrate over these components, e.g. aperture photometry, to estimate overall properties for each galaxy. With such quantities, one can gain a general picture of the merger and star-formation history of a given galaxy. Typically, however, a variety of histories can produce similar integrated properties. Considering the properties of a galaxy’s components separately enables a much more detailed account of its lifetime to be constructed.
The simplest approach to separating the properties of the main galaxy structures is bulge-disc decomposition. This can be applied to imaging data alone, and hence to the largest samples of galaxies available. Although conceptually simple, bulge-disc decomposition remains a challenging task, due to the variety of structures that galaxies display, not to mention the usual observational limitations of resolution and signal-to-noise.
A common method is to study and model the one dimensional (1D) light profile, along the major or minor axis of the galaxy, or azimuthally averaged. These 1D profiles are usually obtained by fitting a set of ellipses to the isophotes in the (2D) image. However, 1D representations of the radial surface-brightness distribution suffer from strong systematic uncertainties since they neglect the differing intrinsic shapes of the disk and bulge components.
A solution to this problem is two-dimensional (2D) decomposition [@tex:BF95], which utilises all the spatial information in the images (for more details about standard 1D and 2D methods see @tex:PH10). On the other hand, fitting in 2D is usually more sensitive to features, such as bars and spiral arms, which are difficult to model. The usual procedure for 2D bulge-disc decomposition is to fit a parametric model to the image, accounting for the point spread function (PSF), pixelisation and noise properties of the image. The projected surface-brightness profile of each component is typically modelled using an analytic function, the most common choice being the [Sérsic]{}profile [@tex:S68].
Separating galaxy components is supposedly easier for small samples of nearby galaxies where a more interactive fitting process can be applied. Multiple studies have applied 2D decomposition to examine the correlations between bulge and disc properties at optical to infrared wavelengths (e.g., @tex:Nv07 [@tex:MA08; @tex:BW09; @tex:TW11]), to study the coevolution of supermassive black holes and their host galaxy (e.g., @tex:KH08 [@tex:VD12]), to investigate the evolution of structure over cosmic time (e.g., @tex:BD12 [@tex:BD14]) and environment (@tex:HS10 [@tex:HL14]), to study the structural properties of isolated late type galaxies (e.g., @tex:DS08), and to measure quasar host galaxy parameters (e.g., @tex:MD00).
Some studies go a step further and attempt to decompose a third component, usually a bar (e.g., @tex:LS05 [@tex:G09; @tex:WJ09; @tex:G11b]). In addition to providing measurements of bar properties for study, including a potential bar in the model helps to avoid any such feature from contaminating measurements of the bulge and disc.
A significant issue lies in identifying which components are present, and hence which model parameters are to be trusted. This amounts to choosing the appropriate complexity of model for a given galaxy. Fitting a more complex model usually results in a significantly improved goodness-of-fit statistic (e.g., chi-squared), irrespective of whether or not the model parameters are physically meaningful. This problem is complicated by the presence of galaxy features that are not included in the model, such as cores, non-elliptical and twisted isophotes, dust lanes, etc. Many studies ultimately resort to selecting the most appropriate model by visual inspection of the original images and their fit residuals.
Elliptical galaxies are usually regarded to be one-component systems, and hence they are usually chosen to be modelled by a single [Sérsic]{}profile. However, it is far from clear whether this is physically the best way to describe these systems. Taking a different approach, @tex:HH13 fitted three components to each member of a sample of elliptical galaxies, finding that these galaxies can be well described by the combination of three [Sérsic]{}profiles, each with low [Sérsic]{}index but different effective radii. @tex:HH13 argue that these components have physically meaningful interpretations. The intermediate-size component is the original, built from early collapse and major mergers. The largest component is comprised of stars accreted in more recent minor mergers. Finally, the most compact component is attributed to central star formation following the dissipative accretion of gas brought in by some of those recent minor mergers.
For large samples of galaxies, more automated approaches to deciding how many components a galaxy comprises are essential. For example, @tex:AD06 employed a logical filter to decide whether the results of fitting a bulge-disc model were physically plausible, or whether their single-[Sérsic]{}fit should be preferred. They showed that the routine structural decomposition is an important for understanding the bimodality of galactic properties. @tex:SM11 have created the largest catalogue of multi-component galaxy structure to date. They fit one million galaxies with three different models, and used F-tests with a calibrated probability threshold to choose the best model for each galaxy. @tex:LG12 expanded the model options five, selecting between them using a logical filter. These studies have provided the first complete estimates of the bulge and disc properties for the local Universe.
To date, most studies have measured structural properties of a galaxy using only one image, in a single waveband. However, modern surveys provide images of the same galaxies in many different bands. In some cases, models are fit to each band independently. This does not produce reliable colours, however, so more often an initial model is fit to one preferred band, then the structural parameters are fixed during fits to the other bands. @tex:SM11 (following @tex:SW02) take a more consistent approach by fitting their models to images in two bands simultaneously, while @tex:MS13 use a hybrid procedure to produce bulge and disc colours in five optical bands.
Until recently, no method was available that could fit models to an arbitrary number of images at different wavelengths. Driven by a determination to make more effective use of the multi-wavelength imaging available from modern surveys, the [[MegaMorph]{}]{}project (@tex:HB13 [@tex:VB13b] and Bamford, in prep.) developed and tested a new version of two-dimensional photometric analysis which constrains a single, wavelength-dependent model using multiple images simultaneously.
This paper is one of a series that investigates the benefits of this multi-wavelength approach to measuring galaxy structural properties. In Bamford et al. (in prep.) we present this new tool in detail, describing the new features and demonstrating its use through some specific examples. In @tex:VB13b [hereafter V13] we test our new method by fitting single-[Sérsic]{}models to original and artificially-redshifted image of 163 nearby galaxies. In @tex:HB13 we demonstrate our approach on a large dataset from the GAMA [@tex:DN09; @tex:HK10] survey, automating both the preparation of the data and the fitting process itself. The resulting measurements – in particular the variation of structural parameters with wavelength – are studied further in @tex:VB14. The objective of the present paper, is to investigate the ability of [[<span style="font-variant:small-caps;">galfitm</span>]{}]{}to perform bulge-disc decomposition on galaxy images with a wide range of resolution and signal-to-noise. This is achieved by analysing the same sample as V13: large, nearby galaxies in the Sloan Digital Sky Survey (SDSS; @tex:AA09), with both original images and versions that have been convolved and resampled in order to simulate the galaxies’ appearance at a range of redshifts. A complementary analysis of multi-band bulge-disc decomposition, using the same GAMA sample as @tex:HB13, will be presented in a forthcoming paper (Haeussler et al., in prep.).
This paper is structured as follows: in Section 2 we present our data set, give a brief description of [[<span style="font-variant:small-caps;">galfitm</span>]{}]{}, and then explain how we fit our sample and identify reliable components. In Section 3 we present the distributions of structural parameters obtained from the original SDSS imaging, and examine the stability of these distributions with respect to the effects of distance. In Section 4 we present correlation between structural parameters and a way of separating elliptical from lenticular galaxies in our sample. We provide a summary in Section 5.
Data
====
Sample selection and imaging
----------------------------
{height="13cm" width="13.0cm"}
In this paper we use the same set of 4026 galaxy images as in V13. These images comprise a sample of 163 nearby galaxies with imaging from SDSS in the $u$, $g$, $r$, $i$ and $z$ passbands. Our galaxies typically extend over more than one SDSS frame, so to create them we employ [[<span style="font-variant:small-caps;">Montage</span>]{}]{}[@tex:JK10], which performs the transformations, rebinning and background adjustment necessary to combine the individual frames into a single mosaic.
In addition to the original images of our galaxies, we use 3863 further images in which the galaxies have been artificially redshifted. We use [[<span style="font-variant:small-caps;">ferengi</span>]{}]{}[@tex:BJ08] to create a set of $ugriz$ images mimicking the appearance that each of the 163 nearby galaxies would have if they were observed by SDSS at a range of redshifts. The artificial redshifting algorithm applies cosmological changes in angular size, surface brightness and, optionally, shifting of the restframe passband (k-correction), to simulate the observation of a given galaxy at a greater distance. We produce images for redshifts $0.01$–$0.25$, in steps of $0.01$. Note that not all our galaxies have images for every one of these redshifts, either because they originally have a redshift higher than $0.01$, or because the galaxy effectively becomes a point source. As in V13, to avoid confusion with genuine redshift biases, we disable the k-correction feature of [[<span style="font-variant:small-caps;">ferengi</span>]{}]{}in this work.
Full details of the redshifting process and the data preparation for both original and redshifted images have been given in V13. In the present paper we use the same masks, PSFs and sky estimates.
The morphological breakdown of our galaxy sample is given in the top row of Table \[table:sample\]. All classes, except elliptical, also include barred types. Note that, while we ensure a broad range of morphologies are included, the distribution of Hubble types in this sample is not representative of the local Universe. As described in V13, our sample is comprised of galaxies which have had their structure carefully measured by previous studies. We can then compare these with our semi-automated, multi- and single-wavelength results.
{height="10.5cm" width="11.5cm"}
{height="10.5cm" width="8.5cm"} {height="10.5cm" width="8.5cm"}
{height="10.5cm" width="8.5cm"} {height="10.5cm" width="8.5cm"}
Structural parameters
---------------------
### Fitting galaxies with [[<span style="font-variant:small-caps;">galfitm</span>]{}]{}
We use a modified version of [[<span style="font-variant:small-caps;">galfit3</span>]{}]{}to fit two-dimensional analytic models to our galaxy images. A detailed description of [[<span style="font-variant:small-caps;">galfit</span>]{}]{}is given by @tex:PH02 [@tex:PH10]. We have adapted [[<span style="font-variant:small-caps;">galfit</span>]{}]{}(version 3.0.2) for the requirements of this project, as briefly described below. To differentiate it from the standard release, we refer to our modified version as [[<span style="font-variant:small-caps;">galfitm</span>]{}]{}[^1]. All the work in this paper uses version [[<span style="font-variant:small-caps;">galfitm</span>]{}]{}-0.1.3.1.
Standard [[<span style="font-variant:small-caps;">galfit3</span>]{}]{}accepts only a single input image with which to constrain the model fit. To utilise multi-band data it was therefore necessary to make a number of significant modifications. However, we have endeavoured to retain the original code unchanged, wherever possible. [[<span style="font-variant:small-caps;">galfitm</span>]{}]{}is therefore backward compatible and produces almost identical results to [[<span style="font-variant:small-caps;">galfit3</span>]{}]{}when used with single-band data (see section 4.1 in V13).
Our modified code can accept an arbitrary number of (pixel-registered) images of the same region of sky at different wavelengths. To these images, [[<span style="font-variant:small-caps;">galfitm</span>]{}]{}fits a single, wavelength-dependent, model. As for [[<span style="font-variant:small-caps;">galfit3</span>]{}]{}, this model may comprise one or more components, each with a number of parameters. For example, for a single [Sérsic]{}function, the parameters are: centre position ($x_{\rm c}$, $y_{\rm c}$), magnitude ($m$), effective radius (${r_{\rm e}}$), [Sérsic]{}index ($n$), axial ratio ($b/a$) and position angle (${P\!A}$). To enable these model components to vary with wavelength, each of their standard parameters are replaced by functions of wavelength. For convenience, these are chosen to be Chebyshev polynomials (see @tex:HB13 and Bamford et al. (in prep.) for more details). Instead of directly fitting the standard parameters, [[<span style="font-variant:small-caps;">galfitm</span>]{}]{}optimises the coefficients of these polynomials to best match all the multi-band data.
For each standard parameter, the user may select the order of the polynomial that describes its wavelength dependence, and thereby the freedom that parameter has to vary. Some parameters may be entirely fixed; for example in this paper, we set $n = 1$ for the second [Sérsic]{}function, in order to model an exponential disc component. Other parameters may be allowed to vary as a constant with wavelength; e.g., one might allow the central $x$ and $y$ coordinates to vary during the fit, but require that they are the same in every band. Further parameters may be permitted to vary with wavelength as linear, quadratic, or higher-order functions; e.g., one could choose to allow the axis ratio $b/a$ to vary linearly with wavelength, in order to account for changes in ellipticity at different wavelengths. Ultimately, specifying a polynomial with as many coefficients as there are input bands allows that parameter to vary freely with wavelength. The user therefore has great flexibility to achieve a compromise between the freedom of the model, physical expectations, and the number of free parameters that must be constrained by the data.
A key element of our technique is that parameter values at the wavelengths of low signal-to-noise images can be partially interpolated or extrapolated from the higher signal-to-noise data. However, any significant signal present in those images should have an appropriate influence on the fit. Systematic biases will be thus be reduced in comparison to an extrapolation based only on the high signal-to-noise bands. The risk of such systematics may be further reduced by giving the model more freedom to vary with wavelength (e.g. linear or quadratic variation of $r_{\rm e,b}$, $r_{\rm e,d}$ and $n_{\rm b}$). The cost is an increase in the statistical uncertainties of the parameters in the low signal-to-noise bands.
In this work we have held most structural parameters fixed versus wavelength. This corresponds to assuming a simplified picture of galaxy structure, in which galaxies comprise only bulge and exponential disc components, each without colour gradients and not departing from an elliptical projected [Sérsic]{}profile. Real galaxies may not obey these assumptions. Therefore, while our structural constraints increase the stability of the fits, there is also a risk of introducing systematic biases in cases where the true wavelength dependence of the profile does not correspond to that assumed. Lower signal-to-noise images would be most susceptible to such systematics, as their parameters will be influenced by any higher signal-to-noise data.
Our approach makes the assumed variation of galaxy structure with wavelength explicit, and allows one to relax these assumptions in a selective and gradual manner. This flexibility allows the user to balance systematic and statistical uncertainties, using independent observational results, physical insight and knowledge of their dataset.
We plan to explore the variation of structural parameters with wavelength in detail in a future paper. However, based on preliminary results we find that the vast majority of results present in this paper do not change by allowing small wavelength variation of the structural parameters.
### Model choices
Most galaxies are considered to be primarily two-component systems, comprising a disc, with an exponential ($n=1$) profile, and a bulge, typically well represented by a [Sérsic]{}function with $n\sim 0.5$–$4$ [@tex:G01]. On the other hand, elliptical galaxies are generally considered to be single-component systems, describable by a single [Sérsic]{}profile.
In V13 we performed single-[Sérsic]{}fits to all of our images. In this paper we supplement these with bulge-disc decompositions, performed using [[<span style="font-variant:small-caps;">galfitm</span>]{}]{}to fit two superimposed elliptical [Sérsic]{}models. For the first component we fit all standard parameters (effective radius $r_{\rm e}$, apparent magnitude $m$, [Sérsic]{}index $n$, axis ratio $b/a$, and position angle ${P\!A}$), while for the second, we fix the [Sérsic]{}index to be equal to one.
In order to carry out a blind test of our analysis, and allow us to draw conclusions relevant for large surveys, we assume that we do not know the Hubble classification for our galaxies. We therefore fit them all (even known ellipticals) with two functions. In Section \[sec:correlations\], we will explore what can be learned from this, including the potential for separating single-component systems, i.e. ellipticals and pure disc galaxies, from two-component systems, using structural parameters diagnostics.
We run two sets of fits, each of which is performed on the $u$, $g$, $r$, $i$, $z$ band images for all of our original and redshifted galaxies. For the first (single-band fitting; or [`SM`]{}, to reuse the nomenclature of V13) we fit each band individually. For the second (multi-band fitting; [`MM`]{}) we fit each galaxy using all five bands simultaneously.
We allow the magnitudes (for both bulge and disc) to vary completely freely between bands. For the multi-band method this amounts to setting the wavelength dependence of magnitude to be described by a quartic polynomial, with as many free coefficients as the number of bands. We allow full freedom as we wish to avoid any potential biases on the recovered magnitudes, and hence colours, which may result from assuming a lower-order polynomial dependence.
For the effective radius and [Sérsic]{}index, we choose to not permit any variation with wavelength. This effectively ignores colour gradients within each component, but keeps the overall number of free model parameters down, hopefully improving the reliability of the decomposition process. Any measurements of the wavelength dependence of individual components will be noisy and are unlikely to provide significant evidence to contradict the reasonable default position of a constant value. This is therefore what we assume. Our decision is supported by previous results in the literature. For instance, both @tex:MC03 and @tex:MC11 find that the [Sérsic]{}index of the bulge, as well as the effective radius of the bulge and the disc, show no significant variation (or a slightly linear relation in rare cases) across optical and NIR wavelengths.
In Figure \[fig:sixgal\] we show our results (the effective radius, [Sérsic]{}index and spectral energy distribution), for the bulges of six example galaxies fit in our original and artificially-redshifted images. For most of our galaxies, the results of the single-band fits (solid lines) show substantial fluctuations with wavelength, which worsen with increasing redshift. The results of our multi-band fits (dashed lines), with $n$ and $r_{\rm e}$ constant with wavelength, recover reasonable values that are close to the average of the higher signal-to-noise bands ($gri$) for single-band fitting. The multi-band results are more resilient for different redshifts (e.g. black, red and blue lines in different columns).
We also assume that the shapes of our galaxy components do not change with wavelength, so we set the axis ratio, position angle and galaxy centre to be constant with wavelength.
In both runs ([`SM`]{}, [`MM`]{}) we use the same initial parameters for galaxy center $(x_{\rm c}, y_{\rm c})$, magnitude $(m)$, [Sérsic]{}index $(n)$, effective radius ($r_{\rm e}$), position angle ($\theta$), axis ratio $(b/a)$ and sky background value (although different values are used for each galaxy image, see V13 for more discussion of the sky estimate). We experimented with various different schemes for choosing initial parameters values, before selecting the following approach.
The sample of 163 original images is fit first. The initial magnitudes are determined using the [`MM`]{}single-[Sérsic]{}model results found in V13. The initial value for the bulge magnitude was set to $m_{\rm ss} + 0.75$, where $m_{\rm ss}$ is the single-[Sérsic]{}magnitude. The initial disc magnitude was set to $m_{\rm ss} + 0.65$ in order to start with a slightly fainter bulge than disc. The initial effective radius of the bulge was chosen to be $0.5 r_{\rm e,ss}$, where $r_{\rm e,ss}$ is the effective radius from the single-[Sérsic]{}fit. Similarly, the initial effective radius for the disc was set to be equal to the $r_{\rm e,ss}$ We therefore use the observation that bulges are typically smaller than their host discs. We found that starting with an equal bulge and disc there are more chances the bulge to fit parts of the disc component.
The initial [Sérsic]{}index of the bulge was chosen to be $n_{\rm ss}$, while the [Sérsic]{}index of the disc was fixed to unity, for an exponential profile. The initial values of disc axis ratio, disc position angle, disc and the bulge center ($x,y$) were set equal to the equivalent single-[Sérsic]{}value. In cases where a parameter was variable with wavelength in the single-[Sérsic]{}fit, but constant in the current paper, we took the median of the five values. Finally, the initial value of axis ratio for the bulge was set equal to 0.8, and the initial value of position angle for the bulge was arbitrarily set to 10 degrees. We confirmed that the final results of the fits do not depend on the these values.
All the parameters are allowed to vary during the fitting process, with the exception of the disc [Sérsic]{}index and the sky background, which were kept fixed. [[<span style="font-variant:small-caps;">galfit</span>]{}]{}and [[<span style="font-variant:small-caps;">galfitm</span>]{}]{}give the option to constrain the range of values for each parameter in order to avoid unphysical results. We make use of this option by applying the following constraints. We require both magnitudes (bulge and disc) to vary within the range of $5$ to $35$ mag. Similarly, both the effective radius of the bulge and the disc were allowed to vary within the range of $0.04$ to $600$ arcsec. We constrain the freedom of the [Sérsic]{}index by allowing it to vary within the range of $0.1$ to $15$. However, we exclude any bulge with $n_{\rm b}<0.3$ from the final sample. Finally in the case of the center ($x$, $y$) we applied two constraints, one to fix the bulge and disc to have the same center, and a second to restrict their variation, with respect to the single-[Sérsic]{}fit, to be no more than $\sqrt{s/8}$, where $s$ is the size of the image.
For each artificially-redshifted image, we repeat the same procedure as above to estimate the initial parameter values, but use the [`MM`]{}single-[Sérsic]{}result obtained for the same redshifted image. In cases where a single-[Sérsic]{}magnitude was unphysically faint, we calculated the initial parameter values by cosmologically adjusting the values obtained for the lowest-redshift artificial image. We apply the same constraints as for with the original galaxies.
In addition to [`SM`]{}and [`MM`]{}, we perform another set of fits to the artificially-redshifted images, which we refer to as ‘aperture fits’. For these we take the structural parameters from the [`SM`]{}$r$-band results and keep these fixed while performing single-band fits to the $u$, $g$, $i$ and $z$ band images. Only the magnitudes are allowed to vary freely during the fit. In this way we apply an identical model in all the bands and ensure we only measure the variations in the flux for a fixed ‘aperture’. This approach is commonly applied to ensure meaningful colours.
[ccccccccc|c]{} & Band & E & S0 & Sa & Sb & Sc & Sd & Sm/Irr & Total\
Total & & 23 & 18 &8 &29 & 50 & 24 & 11 &163\
& u & 23 & 17 & 7 & 18 & 30 & 11 & 6 & 112\
Reliable & g & 23 & 17 & 7 & 20 & 32 & 11 & 8 & 118\
bulge & r & 23 & 17 & 7 & 21 & 32 & 12 & 8 & 120\
& i & 23 & 17 & 7 & 21 & 32 & 13 & 8 & 121\
& z & 23 & 17 & 7 & 21 & 32 & 14 & 8 & 122\
& u & 19 & 17 & 7 & 27 & 49 & 23 & 11 & 153\
Reliable & g & 19 & 17 & 7 & 27 & 49 & 23 & 11 & 153\
disc & r & 19 & 17 & 7 & 27 & 49 & 23 & 11 & 153\
& i & 19 & 17 & 7 & 27 & 49 & 23 & 11 & 153\
& z & 19 & 17 & 6 & 27 & 49 & 23 & 11 & 152\
& u & 19 & 16 & 7 & 18 & 30 & 11 & 6 & 107\
Reliable & g & 19 & 16 & 7 & 20 & 32 & 11 & 8 & 113\
bulge & r & 19 & 16 & 7 & 21 & 32 & 12 & 8 & 115\
and disc & i & 19 & 16 & 7 & 21 & 32 & 13 & 8 & 116\
& z & 19 & 16 & 6 & 21 & 32 & 14 & 8 & 116\
\[table:sample\]
Inspection of individual galaxies {#sec:inspection}
---------------------------------
Figure \[fig:modimag\] shows the original $ugriz$ images, residuals from the single-[Sérsic]{}and bulge-disc [`MM`]{}fits, and the separate [`MM`]{}bulge and disc model components, for an example spiral galaxy The figure also includes various useful numbers for each fit. Using similar figures we have visually examined the fitted models – and their residuals – for all of our 163 galaxies, to ensure that their shape and size correspond to the real galaxy.
In addition to checking the images, we also inspect all the recovered parameters for both the original and the artificially-redshifted images for each galaxy. In a similar manner to V13, Figs. \[fig:gal1\] and \[fig:gal3\] present a summary of the bulge and disc results for two example galaxies. Equivalent plots are available for all the 163 galaxies. The left panel shows the [`SM`]{}results and the right panel the [`MM`]{}results. At redshift zero we plot the results from the original images. The first row of panels shows the absolute magnitude ($M$), the second row shows the effective radius, the third row shows the [Sérsic]{}index in the case of the bulge panel and the bulge-to-total flux ratio in the case of the disc panels. The last two rows show the axis ratio and the position angle.
In these figures we determine the absolute magnitude and the effective radius assuming distances simply derived from the observed redshift and adopted cosmology. Therefore, the values shown for the original images in Figs. \[fig:gal1\] and \[fig:gal3\] could differ slightly from later figures, for which we use more directly determined distances when they exist.
Figs. \[fig:gal1\] and \[fig:gal3\] illustrate some of the behaviours seen for our fits as the galaxies are simulated as they would appear at higher redshifts. In Fig. \[fig:gal1\] we present the recovered structural parameters for the galaxy NGC2776. We can see that the single-band results fit different structures in each band, particularly for the bulge, and that the parameters of these structures vary substantially with small changes in the simulated redshift. In contrast, the multi-band results are much more stable as a function of redshift, although some small trends are seen before even these results become noisy and unreliable at $z \ga 0.15$. The systematic decline in [Sérsic]{}index with simulated redshift appears to be a consequence of diminishing spatial resolution, and was also a fairly common feature of the single-[Sérsic]{}component fits in V13. For both methods [`SM`]{}and [`MM`]{}the disc parameters are much more stable than those of the bulge, presumably due to the disc’s larger size and less steep inner profile. Note that, even though the bulge and disc magnitudes are completely free to vary between bands, constraining the wavelength variation of $n$, $r_{\rm e}$, ${P\!A}$ and $b/a$ through multi-band fitting leads to much more stable measurements of the magnitudes, and hence colours.
Figure \[fig:gal3\] presents another set of recovered structural parameters, this time for the galaxy NGC6314. The first thing to notice is that, for the single-band fit, the effective radius of the bulge is much larger than the disc for the $g$- and $r$-bands. This is an indication that, in these bands, the [Sérsic]{}function is fitting the disc and the exponential function fitting the bulge, especially given the $n$ behaviour. The usual solution to this problem is to apply constraints on the fit, e.g. insist that the bulge be smaller than the disc. However, using the same set of constraints for the whole sample may introduce biases in other galaxies. Turning to multi-band solves this problem without requiring constraints, now the same structural components are fit in all the bands. In addition, again we see a reduction in the variations caused by small changes in simulated redshift, and that the fit remains more reliable to higher redshifts.
So far we have shown, via specific examples, that that our multi-band approach can measure the fluxes and sizes of galaxy bulge and disc components more reliably than if each band is fit individually, at least when we allow no freedom for the $r_{\rm e,b}$, $r_{\rm e,d}$, $n_{\rm b}$, $b/a_{\rm b}$, $b/a_{\rm d}$, ${P\!A}_{\rm b}$ and ${P\!A}_{\rm d}$ parameters to vary with wavelength. Substantial variations in the recovered parameters with relatively small changes data quality (redshift) are dramatically reduced. The improved stability is particularly noticeable at low signal-to-noise (S/N). As a result, it increases the distance out to which meaningful bulge-disc information can be recovered for a galaxy of a given luminosity. In Section \[sec:stat\_trends\] we demonstrate these improvements in a more general manner, by considering the average trends of various parameters versus redshift, for our whole galaxy sample.
![The difference in $\chi^2$ between the single-[Sérsic]{}and the bulge-disc fits, as function of bulge-to-total flux ratio. Only galaxies with a significant bulge are shown in this figure.[]{data-label="fig:chisq"}](chisquare_BT.pdf){width="45.00000%"}
![Comparison of the single-[Sérsic]{}total magnitudes against the sum of the component magnitudes from our bulge -disc fits. Filled circles denote galaxies with both a reliable bulge and disk. Representative error bars for our measurement are displayed in the top part of each panel. See text for further discussion on the uncertainty measurements.[]{data-label="fig:ssbd_tmag"}](compare_SS_BD_mag.pdf){width="50.00000%"}
Obtaining reliable structural measurements {#sec:sample}
------------------------------------------
Before studying the distribution of the galaxy component parameters, we must select a sample with reliable bulge-disc measurements. In this section we describe the various controls we apply to determine if our fits are physical meaningful. For those galaxies where the fitted model is a poorly match to the original image, given our physical expectations, we repeat the fit again with different initial parameter values or additional model components. We aim for our procedure to be applicable in an automated manner, that could be used for large surveys. However, as we have a small sample of galaxies, we still use the tools described in Section \[sec:inspection\] to inform our choices, to check if the automated selection agrees with visual inspection, and to aid the interpretation of our results.
We start by identifying unphysical models where re-attempting the fit may produce a better outcome. Examining the results of the multi-band fits to the original images, we find that 50 galaxies (out of 163) have bulges with $r_{\rm e,b}/r_{\rm e,d}>0.9$. After further investigation we separate these galaxies into two cases. The first group consists of 17 galaxies for which the $n=1$ component is fitting the inner structure of the galaxy and the free-$n$ component is fitting the disc. For these the measured [Sérsic]{}index varies between $0.5$ and $1.4$. All these galaxies are late type spirals (Sc, Sd, Sm) and 13 of them have a bar. For these cases we believe that the initial parameter values were far from optimal. We choose to fit these galaxies again, using a different set of initial values, with a brighter flux and larger size for the intended disc and lower flux and smaller size for the bulge. The new fit corrects the problem for the vast majority of the 17 galaxies.
The second group consists of 33 galaxies where the bulge fits an inner structure, but also dominates the outer region of the galaxy. In this group we find 10 early-type galaxies (E/S0), where 9 have bulge [Sérsic]{}values $2.8$–$5.7$ and one, NGC4458, has $n_{\rm b}=11.3$. The remaining 23 galaxies are spiral galaxies (Sa, Sb, Sc) with bulge [Sérsic]{}values between 4 and 11, usually accompanied by a high bulge-to-total ratio. These $n_{\rm b}$ and $B/T$ values are unusually high for late type spirals. For this second group of galaxies, we initially re-fit the galaxies in the same way as for the first group. This approach corrects the fits for almost one-third of the cases. For the remaining galaxies we attempt another fit with an addition of a third component (in the form of a central point source), together with the second set of initial parameters. We choose to add a PSF function to account for any extra flux in the centre of these galaxies that could be responsible for the high values of [Sérsic]{}index and effective radius.
These new fits return smaller bulges with lower [Sérsic]{}indices for another one-third of the cases, and they reduce the [Sérsic]{}index without reducing the effective radius for a further four galaxies. For the remainder we do not adopt the fit results with additional-PSF component, either because the value of the bulge effective radius or [Sérsic]{}index was larger than before, or the PSF magnitude was negligible ($< 30$ mag). One case where the addition of a PSF component failed to improve the fit is the elliptical galaxy NGC4458 (see the Appendix \[sec:notes\] for more details on this galaxy).
{width="85.00000%"}
After refitting both groups we then select our final samples of trustworthy bulges and discs. We reject bulges in the multi-band, original-image results with fit parameters on any of the constraints, insignificant bulge components (at least 3 mag fainter than the disc or below the SDSS point source detection limit), and bulges with effective radius smaller than 5 pixels. We perform these checks in each individual band. The final numbers of galaxies with acceptable bulge measurements are shown in Table \[table:sample\]. A large fraction of late-type spiral galaxies do not possess a reliable bulge measurement. For all these cases we trust the disc parameters but exclude the bulge properties from further analysis.
In the case of the disc component, we find a few occasions where the effective radius of the disc has taken unreasonably large values or its total brightness is more than three magnitudes fainter than the bulge. For these cases the bulge component returns very similar results to the single-[Sérsic]{}fit (Paper II), the disc only accounts for a minor details in the residual. Four of these galaxies (NGC4360, NGC4378, NGC4486, NGC4621) are elliptical galaxies and consequently may indeed lack a disc. In addition, we fail to fit a disc component for NGC4459, even though it is classified as S0. For these objects we trust the bulge parameters but exclude the disc properties from further analysis.
Two problematic cases that were discovered through the above checks are the galaxies NGC4378 and NGC4450. For these two galaxies both the bulge and disc components have been removed. While both these galaxies are two component galaxies, some image distortions hinder the bulge-disc decomposition process.
Around 110 galaxies have both bulge and disc components that we deem as trustworthy, depending on the band. We should stress that this sample includes 19 elliptical galaxies that were fitted with two functions. For these galaxies, we do not know if these two components correspond to truly distinct structures with different kinematics, but we choose to keep them in the further analysis. So far we have not found any indication that their fits are inappropriate, other than their visual classification[^2], which would not be available for a large automated sample. The number of galaxies that have both trustworthy bulge and disc measurements are broken down by morphology and band in Table \[table:sample\].
We attempt to identify elliptical galaxies by comparing the goodness-of-fit of our one- and two-component fits. In Fig. \[fig:chisq\] we plot the difference in reduced-$\chi^2$ between the single-[Sérsic]{}and bulge-disc fits as a function of bulge-to-total flux ratio. The improvement in reduced-$\chi^2$ for the elliptical galaxies is in the same range as the other Hubble categories. We find that 18% of our elliptical galaxies and 33% of our lenticular have a dramatic reduction ($\chi_{SS}^2-\chi_{BD}^2>0.1$) in their reduced-$\chi^2$ by adding an extra exponential function. Only 13% of our elliptical galaxies and 17% of the S0 show a negligible change in reduced-$\chi^2$ ($\chi_{SS}^2-\chi_{BD}^2<0.01$). In cases where a PSF function has been included we use the $\chi_{BD}^2$ for that fit. The addition of the PSF function in all the cases improved the $\chi_{BD}^2$ by less than $0.01$.
Chi-squared should always decrease when a model is given more freedom. The Bayesian Information Criterion (BIC) is based on $\chi^2$, but penalises additional parameters, in an attempt to provide a guide to whether the additional freedom is warranted by the data. However, applying the BIC to our data finds only nine cases where the single-[Sérsic]{}fit is deemed better than the bulge-disc model. All these cases have already been identified as having an insignificant bulge or disc by our above selection criteria. We conclude that, by using the reduced-$\chi^2$ or BIC, we cannot select a clean sample of elliptical galaxies.
Quantifying the uncertainties of the bulge and disc structural measurements is a challenging task. In V13 we provided the following uncertainties for our single-[Sérsic]{}fits: ($u$, $g$, $r$, $i$, $z$) for $m$ ($\pm0.13$, $\pm0.09$, $\pm0.1$, $\pm0.11$, $\pm0.12$), $r_{\rm e}$ ($\pm12$%, $\pm11$%, $\pm12$%, $\pm14$%, $\pm15$% ) and $n$ ($\pm9$%, $\pm11$%, $\pm14$%, $\pm15$%, $\pm17$%). These were based on plausible systematic uncertainties in the sky estimation, which typically dominates the error budget. As bulge-disc decomposition is a more complicated task, we expect the uncertainties on our bulge and disc measurements to be even larger. Overall, as we will also see from the further analysis, the bulge parameters are more dependent on the initial conditions, while the disk parameters are more robust. We attempt to determine indicative uncertainties on our fit parameters by refitting a randomly selected sample of 10 galaxies with different sky values. We alter the sky values by our estimated systematic sky uncertainties, as before. We find that both $n_{\rm b}$ and $r_{\rm e, b}$ can change by up to $\sim 25$%. The parameters of the disc are less strongly affected and are on the same level as the single-[Sérsic]{}uncertainties.
![The distribution of bulge-to-total flux ratio for different morphological bins, measured in the $r$-band using our multi-band ([`MM`]{}) method. Only galaxies with a significant bulge are shown in this figure.[]{data-label="fig:BThist"}](B_T_hist.pdf){width="47.00000%"}
![The median value of the bulge-to-total distribution, as seen in Fig. \[fig:BThist\], as a function of apparent redshift. At redshift zero the results from the original images are plotted, while for higher redshifts we show the results from the artificially-redshifted images. []{data-label="fig:BTz"}](BT_z.pdf){width="47.00000%"}
As a simple check of our fits, Fig. \[fig:ssbd\_tmag\] shows the difference in the recovered $r$-band total magnitude between our single-[Sérsic]{}and two-component models. We colour-code the galaxies based on their Hubble classification. For the vast majority of the galaxies the difference between the total magnitudes is smaller than 0.2 mag. Bright galaxies $m<11$ show a small systematic trend, where the single-[Sérsic]{}magnitude ($m_{\rm ss}$) is brighter than the sum of the bulge and disc magnitude ($m_{\rm sum}$). The outliers in this plot are NGC5850, NGC4725, IC0724, NGC4636, NGC4303 and NGC5806 with $M_{\rm sum}-M_{\rm ss} = 0.58, 0.42, 0.32, 0.28, 0.25 $ and $ 0.25$ respectively.
We compare our derived bulge and disc parameters with those measured by previous studies, specifically those that analysed small samples of galaxies and carefully fitted each galaxy individually. Initially, we compare with @tex:FD08, which presents structural parameters for 18 galaxies common to our sample, using $V$-band imaging from various sources. For comparison we convert our magnitudes to $V$-band central surface brightnesses. For the remaining parameters, $r_{\rm e,b}$, $r_{\rm e,d}$ and $n_{\rm b}$ we compare our multi-band results, which do not vary with wavelength, directly with the V-band measurements from @tex:FD08. Figure \[fig:FD08\] shows these comparisons. The agreement is satisfactory for most of the structural parameters. Both sets of bulge measurements display large error bars. The discrepancy between the two different studies appears to be a result of the method used to model the galaxy (1D versus 2D) and of different masking theme. Fisher & Drory decomposition is based on the major axis of each galaxy while ours utilises all the spatial information of the image. Various studies, e.g. @tex:FD04 [@tex:PH10], have found that the parameters derived with 1D fitting methods and 2D are not always in agreement. Even different approaches of 1D fitting e.g. major vs minor axis fitting can change the results. The second possible reason is that we fit the entire galaxy while in @tex:FD08 they manually exclude the inner and outer part of the galaxy. This practice has the advantage that you can exclude for instance the inner part of the galaxy that may not follow the [Sérsic]{}function but has the disadvantage that the fitting process dependents on personal choices (or personal experience) of what should be included in the fitting and what should be excluded. The four galaxies that show significant offset are discussed further in Appendix \[sec:notes\].
Additionally, we compare our results with @tex:M04 for our 12 common galaxies, some of which are also in the @tex:FD08 comparison. In order to perform a sensible comparison, we first recalibrated the absolute magnitudes found in @tex:M04 to the distances used in this paper, and the AB zeropoint system. Finally we converted $UBVRI$ magnitudes to $ugriz$ using the transformations provided by @tex:BR07. Our results are similar to @tex:M04, with $\Delta M < 0.5$ mag, except for the u-band measurements of NGC2841, NGC3521, NGC4274 and NGC4303. Further information about some of these cases can be found in Appendix \[sec:notes\].
Structural properties versus morphological type and redshift {#sec:stat_trends}
============================================================
Our primary aim in this paper is to demonstrate that our multi-band decomposition method is able to determine physically meaningful bulge and disc parameters, both for nearby galaxies with high-quality imaging and more distant galaxies with noisier and less well-resolved images. In this section we therefore study the behaviour of various bulge and disc parameters for galaxies with different morphological types. For each parameter we first present the distributions as measured on our original SDSS images, using our multi-band ([`MM`]{}) approach. We do not show [`SM`]{}results for the original imaging because at such high resolution both methods return similar results (see V13 and below). We then investigate the stability of our measurements on the artificially-redshifted images, by examining how the median parameters for each morphological group vary with redshift, and comparing the multi-band ([`MM`]{}) and single-band ([`SM`]{}) methods.
In Figures \[fig:BThist\]–\[fig:coldiffm\] and \[fig:coldiff\]–\[fig:r\_ferengi\] we divide our galaxies into four morphological groups: E, S0–Sa, Sb–Sc, and Sd–Irr. Where we present the results of artificial redshifting, we only plot up to a redshift of $0.15$ (in contrast to $0.25$ in previous figures), as beyond this redshift neither approach produces useable structural measurements.
[ccccccc]{} Hubble-type& \# of galaxies & $B/T$ & $n_{\rm b}$ & \# of galaxies & $\left<\Delta(g-i)\right>$ & $r_{\rm e,b}$/$r_{\rm e,d}$\
bins & for $B/T$ & $n_{\rm b}$ & & & for $\left<\Delta(g-i)\right>$ & $r_{\rm e,b}$/$r_{\rm e,d}$ & (mag) &\
E & 23 & $0.7 \pm 0.04$ & $3.5 \pm 0.5$ & 19 & $0.03 \pm 0.04$ & $0.3 \pm 0.2$\
S0–Sa & 24 & $0.54 \pm 0.06$ & $1.9 \pm 0.4$ & 23 & $0.05 \pm 0.1$ & $0.29 \pm 0.08$\
Sb–Sc & 53 & $0.2 \pm 0.04$ & $1.8 \pm 0.3$ & 53 & $0.28 \pm 0.06$ & $0.24 \pm 0.08$\
Sb–Sc & 49 & $0.17 \pm 0.03$ & $1.8 \pm 0.2$ & 49 & $0.3 \pm 0.07$ & $0.23 \pm 0.03$\
Sd–Irr & 20 & $0.07 \pm 0.02$ & $0.9 \pm 0.2$ & 20 & $0.28 \pm 0.1$ & $0.31 \pm 0.05$\
\[table:avegval\]
![The distribution of the bulge$-$disc colour difference for the galaxies measured in the original imaging. Only galaxies with both a significant bulge and disc are shown in this figure.[]{data-label="fig:coldiffm"}](b-d_colour_hist.pdf){width="48.00000%"}
![The $g-i$ colour of the bulge (top panel) and disc (bottom panel) component as a function of $r$-band absolute magnitude. Representative error bars for our measurement are displayed in the top part of each panel. See text for further discussion on the uncertainty measurements.[]{data-label="fig:colbul"}](colour_gi_AbsMag.pdf){width="48.00000%"}
![The median value of the bulge$-$disc colour difference distribution, as seen in Fig. \[fig:coldiffm\], as a function of redshift. At redshift zero the results from fitting the original images are plotted, while for higher redshifts we use the results from the artificially redshifted imaged. The horizontal lines are the median values of the E and Sb–Sc groups, as measured in the top panel. []{data-label="fig:coldiff"}](b-d_colour_diff.pdf){width="48.00000%"}
Bulge-to-total
--------------
First we consider the relative fluxes of the components. We remind the reader that magnitude is the only parameter in the [`MM`]{}fits that is entirely free to vary with wavelength. In Fig. \[fig:BThist\] we show the distribution of $r$-band bulge-to-total flux ratio, using trustworthy measurements from [`MM`]{}fits to the original images.
Only a few elliptical galaxies have $B/T$ close to one, contrary to expectations. Instead, most have $B/T \sim 0.7$, while four have $B/T < 0.6$. The four cases with small $B/T$ are NGC4636 ($B/T=0.36$), NGC4649 ($B/T=0.49$), IC3653 ($B/T=0.55$), NGC4473 ($B/T=0.58$). The S0–Sa galaxies display a broad distribution with a peak around $B/T=0.5$. In the lower panel we show intermediate and late disc galaxies. Sb–Sc galaxies show an extended distribution with about three-quarters having $B/T < 0.5$, and with a noticeable peak at $B/T < 0.1$, while all Sd-Irr galaxies have $B/T < 0.3$.
In Table \[table:avegval\] we give the average values of the $B/T$ for each Hubble-type bin, as measured using the original images. The uncertainties on the median are estimated as $1.253 \sigma/\sqrt{N}$, where $\sigma$ is the standard deviation and $N$ is the number of galaxies in each Hubble-type bin. We also measure median values when excluding galaxies with $r_{\rm e,b}>r_{\rm e,d}$. For these galaxies we suspect that the bulge component fits part of the disk component leading to high $B/T$ values. We only show this second set of results for Sb–Sc galaxies. The remaining morphological bins either contain no galaxies with $r_{\rm e,b}>r_{\rm e,d}$ or the median values do not change.
In Fig. \[fig:BTz\] we investigate our ability to recover the bulge-to-total flux ratio as our sample becomes more distant. For easier readability of the plot, we only show the $r$-band results for all images created with [[<span style="font-variant:small-caps;">ferengi</span>]{}]{}. We include only those galaxies used in Fig \[fig:BThist\]. To facilitate the comparison with the results for the original images, we add the median value of the original images at redshift zero. Following the same colour coding of different Hubble classifications as in Fig. \[fig:BThist\], we plot the median $B/T$ value for each group.
Both [`SM`]{}and [`MM`]{}show similar behaviour; $B/T$ decreases at high redshifts for the elliptical and early-disc samples, whereas the later-discs show the opposing trend. As the image resolution decreases and our ability to distinguish two components diminishes, there appears to be a tendency for the two functions to split the total flux equally. However, the [`MM`]{}median values show smaller fluctuations and are more stable to significantly higher redshifts than [`SM`]{}. For instance, the late-types appear well-recovered out to $z\sim0.09$ with [`MM`]{}, while the [`SM`]{}results are only stable to $z\sim0.05$.
We can compare our average $B/T$ values (from the original images) with those found in @tex:WJ09 and @tex:LS10. Using a sample of 143 galaxies observed in the $H$-band, @tex:WJ09 found that $\sim 69$% of bright spirals have $B/T<0.2$ and $76$% of the bulges have $n<2$. We find that $67$% of our spiral sample (Sa–Sd) have $B/T<0.2$ and $n<2$. Similarly, @tex:LS10 used a sample of $\sim300$ S0–Sm galaxies with images in the $K$-band found to determine $B/T$ for each Hubble category: $B/T_{\rm S0} = 0.39\pm0.13$, $B/T_{\rm Sa}=0.26\pm0.12$, $B/T_{\rm Sb}=0.12\pm0.09$ and $B/T_{\rm Sd}=0.06\pm0.13$. Despite some differences in our sample selections and the wavelength considered, our $B/T$ distributions for different morphologies are highly consistent with these two independent studies.
Colours
-------
Galaxies display a range of optical colours, which correlate strongly with morphology and structure (e.g., see @tex:WL13). However, the total colour of a galaxy averages over any distinct stellar populations it may contain. The bulge and disc components of a galaxy may be expected to comprise contrasting stellar populations due to their different formation histories. Considering their colours individually is thus a sensible first step toward better understanding the distribution of stellar populations within galaxies. For example, quantifying the differences and correlations between the colours of bulge and disc components can help us differentiate between proposed bulge formation scenarios.
Previous work has suggested that there are substantial variations in the colours of bulges and disks between galaxies, while the colours of the two components within a given galaxy are often similar [@tex:PB96], though significantly offset [@tex:MC04; @tex:CD09]. Here we briefly present the results for our initial sample, and show the advantage of the multi-band technique method for measuring bulge and disc colours.
In Fig. \[fig:coldiffm\] we plot the distribution of the colour difference between the two components. In the top panel we see that early-type galaxies contain bulges and disks with similar colours. In contrast, the late-types possess bulges that are significantly redder than their discs. The average values of the component colour difference can be found in Table \[table:avegval\]. For all spiral galaxies (Sa–Sm), we find $\left<\Delta(g-i)\right>\sim 0.3$ mag, irrespective of their more detailed Hubble type. Even S0s, considered alone, typically possess bulges that are slightly redder than their discs, with $\left<\Delta(g-i)\right> = 0.05\pm0.05$ mag.
These values compare very well with previous measurements in the literature. @tex:MC04 found an average bulge$-$disk colour difference of $\left<\Delta(B - R)\right> = 0.29 \pm 0.17$ mag for a sample of 172 low-inclination disc galaxies (S0–Irr), while @tex:HS10 find $\left<\Delta(B - R)\right> = 0.23 \pm 0.02$ mag for $L_\ast$ discs in eight low-redshift clusters. Similarly, @tex:CD09 reported a colour difference of $\left<\Delta(u-r)\right>= 0.27\pm0.04$ mag (without their average dust correction) using $\sim1500$ two-component galaxies extracted from the Millennium Galaxy Catalogue. The bulge$-$disc colour difference we find for S0s is also consistent with the $\left<\Delta(g-i)\right> = 0.09\pm0.01$ mag found by @tex:HL14 for S0s in Coma.
To examine this behaviour in more detail, Fig. \[fig:colbul\] presents the colour-magnitude distribution for each component, colour-coded by Hubble-type. Elliptical and S0 galaxies typically have both their components on a red-sequence, with $(g-i) \sim 1.2$ mag, resulting in the distributions centred around zero in the top panels of Fig. \[fig:coldiffm\]. The disc colours of early-spirals (Sa/Sab) are also typically on this red-sequence, while the discs of late-spirals (Sb–Sm) inhabit a blue cloud, with later types being fainter (though our heterogeneous sample selection may be somewhat responsible for this). The bulge colours of spirals show a considerable scatter. Some, particularly those of types Scd–Sm, lie in the blue cloud, whereas the bulges of Sab–Sc galaxies are often above than the red-sequence. Dust extinction may be responsible for these very red bulges. However, we do not see any significant trend in bulge colour with disc inclination, as one might expect if this were the case.
We now consider te behaviour of the bulge$-$disc colour difference versus apparent redshift. In the top panel of Fig. \[fig:coldiff\], we show the [`MM`]{}results. The early-type galaxies have a median colour difference very close to zero, which remains almost constant out to $z \sim 0.15$. For Sb–Sc galaxies the median value is stable till redshift 0.03, after which it is overestimated with respect to the original measurement, but at least relatively stable and differentiated from the early-types. Sd–Irr galaxies show a greater degree of variation beyond $z \ga 0.05$, although note that this sample contains intrinsically fainter galaxies than the other sets.
In the middle panel of Fig. \[fig:coldiff\] we show the results of fitting using the aperture method, for which structural parameters are fixed to the $r$-band results and colours obtained by fitting for the bulge and disc fluxes in the each other band. The initial behaviour is similar to that in the top panel, but with greater variation, such that the different Hubble types are less clearly differentiated beyond $z \ga 0.06$. However, we notice that [`MM`]{}median colours for the Sb- Sc galaxies beyond redshift 0.08 are maintained to higher values, compared to the colour at redshift zero, while the aperture median colour drops again close to the dashed line.
Finally, in the bottom panel, we show the [`SM`]{}results, from independent fits to each band. In this case the variations in structural parameters between bands make the colours of each component very noisy, and sensible values cannot be obtained beyond $z \ga 0.03$. This emphasises that colours for the bulges and discs of individual galaxies cannot be directly obtained via independent fits to multiple bands. Even using such measurements in a statistical fashion (e.g. to estimate the average colours of bulges) would be highly unreliable.
![The distribution of bulge [Sérsic]{}index for the fits to the original images. Only galaxies with a significant bulge are shown in this figure. A more detailed presentation of the bulge [Sérsic]{}index distribution, particularly for small values ($n_{\rm b}<1$), can be found in Fig. \[fig:BTnb\]. []{data-label="fig:n_montage"}](n_hist.pdf){width="48.00000%"}
![The median value of the bulge [Sérsic]{}index distribution, as seen in Fig. \[fig:n\_montage\], as a function of apparent redshift. At redshift zero the results from the original images are plotted, while for higher redshifts we use the results for the artificially-redshifted images. Horizontal lines are plotted at $n_{\rm b}=1$, $2$ and $4$.[]{data-label="fig:n_ferengi"}](n_z.pdf){width="48.00000%"}
[Sérsic]{}index {#sec:sindex}
---------------
We now move on to study how the structural parameters ($n_{\rm b}$, $r_{\rm e,b}$, $r_{\rm e,d}$) are distributed for different Hubble types, and investigate the performance of the multi-band fitting in measuring these values. In Fig. \[fig:n\_montage\] we plot the distribution of the bulge [Sérsic]{}index, as measured in the original images using the [`MM`]{}method. Elliptical galaxies present a peak around $4$. S0–Sa galaxies display a bimodality, with peaks around $1$–$2$ and $4$. Intermediate spiral types (Sb–Sc) have a wide range of $n_{\rm b}$ values, mostly in the range $\sim 1$–$4$. The five Sb–Sc galaxies with $n_{\rm b}>4$ are NGC5430, NGC2841, NGC3521, NGC3642 and NGC4698, with $n_{\rm b} = 4.2$, $5.7$, $6.7$, $6.8$ and $7.1$, respectively. See Appendix \[sec:notes\] for further discussion of some of these galaxies. All our Sd–Irr galaxies have bulges with $n_{\rm b} < 2$, except for NGC4653 and NGC4108B, with $n_{\rm b} = 2.7$ and $4.3$. The average values are given in Table \[table:avegval\]. Typical spiral bulges with $n_{\rm b} \sim 2$ and a progression to lower bulge [Sérsic]{}index for later Hubble types corresponds very well to expectations from the literature (e.g., @tex:GW08 [@tex:LS10; @tex:MC11; @tex:HL14]).
Figure \[fig:n\_ferengi\] uses our artificially-redshifted images to examine how well we are able to recover the bulge [Sérsic]{}index with increasing redshift. This plot is complicated by the bimodal distribution of the S0–Sa class, which results in the median being unstable. Overall, both [`MM`]{}and [`SM`]{}methods recover similar median $n_{\rm b}$ values. For the elliptical galaxies the median $n_{\rm b}$ is well recovered at all redshifts probed. For the spiral classes, the median $n_{\rm b}$ is more variable, particularly for $z \ga 0.05$. However, in general, the [`MM`]{}fits appear to be rather more stable.
Effective radius {#sec:re}
----------------
In Fig. \[fig:r\_montage\] we investigate the relationship between the sizes of the bulge and disc and Hubble type, by plotting histograms of the ratio of bulge effective radius to disc effective radius. Note that we do not constrain our bulges to be smaller than our discs, and neither do we subsequently exclude galaxies based on $r_{\rm e,b}/r_{\rm e,d}$. Consequently, in Fig \[fig:r\_montage\] we find seven galaxies[^3] with $r_{\rm e,b}/r_{\rm e,d}>1$. Most of these galaxies have peculiarities that interfere with the fit. They are discussed individually in Appendix \[sec:notes\].
Disregarding the few galaxies with $r_{\rm e,b}/r_{\rm e,d}>1$, we find very little difference between the distributions for different Hubble types. Average values are listed in Table \[table:avegval\]. Other studies also tend to find little dependence of the bulge-to-disc size ratio on morphology (e.g., @tex:GW08).
We find that bulges are typically around one-quarter of the size of their accompanying discs. Rather than the ratio of effective radii, other studies typically quote $r_{\rm e,b}/h$, where $h=r_{\rm e,d}/1.678$ is the exponential disc scalelength. Furthermore, $h$ is often corrected for inclination-dependent projection and extinction effects, which complicates comparisons. Finally, given the strong wavelength dependence of galaxy effective radius found by @tex:VB14, measurements at optical versus near-infrared wavelengths may be expected to differ significantly, even when an average extinction correction is applied. Nevertheless, we attempt an approximate comparison.
Assuming some average corrections, our median optical $r_{\rm e,b}/r_{\rm e,d} \approx 0.25$ translates into an extinction-corrected, face-on $r_{\rm e,b}/h \approx 0.35$. This agrees well with the values found by @tex:KW00, @tex:Nv07 and @tex:MA08, but is a factor of $1.5$ larger than found by the careful analysis of multiple datasets from the literature @tex:GW08 and twice that found by @tex:LS10. The latter study, and some of the works that were included in @tex:GW08, included additional central components in their models, such as bars or nuclei. This may have led to the smaller bulge sizes they measure. Given the care taken in these studies, we suspect that our bulge $r_{\rm e,b}$ may be somewhat overestimated. However, remember that our aim in this work is to perform fits to our nearby galaxies in a simple, automated manner, suitable for large surveys of relatively distant galaxies, and ascertain the performance of this approach.
With this in mind, Fig. \[fig:r\_ferengi\] shows the median $r_{\rm e,b}/r_{\rm e,d}$ for several Hubble type bins as a function of simulated redshift. For the multi-band ([`MM`]{}) fits, we again see that for low redshift data there is little difference with morphology. The average size ratios remain fairly constant to $z \sim 0.04$. After this, as the data quality becomes substantially poorer, a strong trend to increasing $r_{\rm e,b}/r_{\rm e,d}$ is seen, particularly affecting galaxies with lower $B/T$. Single-band fits perform reasonably similarly (neglecting the ellipticals, for which the reality of the disc is unclear). However, they show a stronger and noisier bias, which sets in at slightly lower redshifts.
Generally, we observe that the lower the data quality, the harder it is to separate the two components and the more similar their properties become. However, it is usually the bulge fit which is most affected, and hence biased. The effective radius of the disc components tend to remain stable for almost the entire redshift range considered, particularly for our multi-band fits.
![The distribution of the ratio of bulge and disc effective radii, as measured from the original images. Only galaxies with both a significant bulge and disc are shown in this figure. In the top panel there is one galaxy, NGC4458, outside the axes, with $r_{\rm e,b}$/$r_{\rm e,d} = 5$. An alternative presentation of the $r_{\rm e,b}/r_{\rm e,d}$ distribution can be found in Fig. \[fig:BTrratio\].[]{data-label="fig:r_montage"}](rb_rd_hist.pdf){width="48.00000%"}
![The median value of the $r_{\rm e,b}/r_{\rm e,d}$ distribution, as seen in Fig. \[fig:r\_montage\], as a function of apparent redshift. At redshift zero the results from the original images are plotted, while for higher redshifts we use the results for galaxies fit in the artificially-redshifted images.[]{data-label="fig:r_ferengi"}](rb_rd_z.pdf){width="48.00000%"}
Correlation of structure parameters {#sec:correlations}
===================================
Bulge-to-Total
--------------
For disk galaxies, the overall [Sérsic]{}index (of a single-[Sérsic]{}model fit) is often regarded as an indication of the bulge-to-total ratio ($B/T$). Indeed, we adopt this interpretation in @tex:HB13 [@tex:VB13b; @tex:VB14]. With our bulge-disk decompositions, we are now in a position to test this.
In Fig. \[fig:BTn\] we plot the bulge-to-total ratio as a function of the single-[Sérsic]{}index ($n_{\rm SS}$) measured in V13. We see that, as the overall [Sérsic]{}index increases, the bulge is responsible for a greater proportion of the galaxy flux, confirming our expectations. Galaxies with a low overall [Sérsic]{}index typically contain two components, a bulge and a disc, and the more dominant the bulge component, the higher the overall [Sérsic]{}index.
Galaxies of type Scd and later generally have low $B/T$, while earlier spirals (Sa–Sc) span a wide range of $B/T$. Interestingly, earlier types tend to have greater $n_{\rm SS}$ at a given $B/T$, suggesting that $n_{\rm SS}$ is also dependent on other aspects of galaxy structure. Most galaxies with $n_{\rm SS}\sim1$ have $B/T<0.1$, with the exception of four galaxies[^4].
For the original images, the resolution and signal-to-noise is sufficiently good that fitting bands individually is comparable to our multi-band approach. However, for more distant galaxies our multi-band method gives more robust measurements of $B/T$. We demonstrate this in Fig. \[fig:BTn\_z\], where we again plot the bulge-to-total ratio as a function of the single-component [Sérsic]{}index ($n_{\rm SS}$), but now using galaxies artificially redshifted to $z=0.1$. This figure shows the results of fitting each band independently ([`SM`]{}), in addition to our multi-band measurements ([`MM`]{}). The clearer correlation for [`MM`]{}clearly illustrates the advantage of our multi-band method.
A number of studies have presented a correlation between bulge-to-total ratio and the [Sérsic]{}index of the bulge (e.g., @tex:G01), particularly as a diagnostic for distinguishing so-called pseudo- and classical-bulges [@tex:DS08; @tex:WJ09; @tex:LS10]. Figure \[fig:BTnb\] confirms this relationship for our measurements. The scatter is relatively large, probably as a result of the difficulty in constraining the bulge properties, as discussed in the previous section. Nevertheless, it is clear that the more bulge-dominated a galaxy is, the higher its bulge [Sérsic]{}index.
We also see a weak correlation between the ratio of bulge and disc sizes, $r_{\rm e,b}/r_{\rm e,d}$, and the bulge-to-total flux ratio, in Fig. \[fig:BTrratio\]. There are some indications that the relation depends on morphological type, but the scatter and incompleteness of our sample prevent us from making definitive conclusions.
Component axis ratios and the division of ellipticals and lenticulars
---------------------------------------------------------------------
Figure \[fig:axisratio\] explores the relationship between the axis ratio of the bulge (top panel) and disc (bottom panel) versus the bulge [Sérsic]{}index. As before, we include galaxies with elliptical morphologies in these plots for two reasons. First, recent work has blurred the lines between elliptical and lenticular galaxies, with many ellipticals found to contain faint disc components when studied in detail [@tex:KA13]. Second, our aim is to inform work on large surveys, which may not have morphological classifications available. For these ellipticals, and despite our nomenclature, we do not go so far as to assume that the exponential component of our model represents a disc, but consider it to be an indication of additional structure that cannot be well-modelled by a single [Sérsic]{}component.
Considering all the points in the top panel of Fig. \[fig:axisratio\], there is a clear correlation such that bulges with higher [Sérsic]{}index tend to be more circular ($b/a \sim 1$). The vast majority of galaxies with elliptical morphologies are found in the upper-right region, with $n_{\rm b} > 2$ and $b/a > 0.5$, as might be expected. Lenticulars slightly separate out from ellipticals in this plot, generally being limited to slightly lower $b/a$ and a wider range of $n_{\rm b} > 2$.
Moving our attention to the lower panel, we first see little correlation between disk axial ratio and bulge [Sérsic]{}index. Note that our sample of spiral galaxies is seriously incomplete for inclined systems, due to selection restrictions applied by the studies from which V13 obtained their sample.
Focussing on ellipticals and S0s, we see a surprisingly strong separation between the two morphologies in disc $b/a$. The vast majority of galaxies with classified as elliptical have $(b/a)_{\rm disc} > 0.5$, while the lenticulars have mostly $(b/a)_{\rm disc} < 0.5$. We also see an offset of the inclined lenticulars to higher bulge [Sérsic]{}indices. We are not certain whether this reflects reality, or is a bias in the decomposition process. Simulations suggest that only small variations in $n_{\rm b}$ are expected from decomposition effects [@tex:PP13b]. S0s are generally not expected to contain significant amounts of dust, so extinction should not play a significant role. In any case the effects are typically $\la 0.1$ in $n_{\rm b}$.
In order to explore the separation of ellipticals and lenticulars in Fig. \[fig:axisratio\] further, we highlight early-type galaxies using their kinematic classification from @tex:EC11. We note that almost all the early-type galaxies with low $(b/a)_{\rm disc}$ are fast rotators, including both of the elliptical galaxies which fall in this region of parameter space dominated by S0s. The early-types with $(b/a)_{\rm disc} > 0.5$ are a mixture of fast and slow rotators, however most (or all with $n_{\rm b} > 2$) have been classified as elliptical galaxies (see @tex:CE11b and @tex:KA13 for a more thorough study of this topic).
Obviously a plot of $(b/a)_{\rm disc}$ versus $n_{\rm b}$, or even just $(b/a)_{\rm disc}$ alone, is an effective, automated way of separating galaxies with elliptical and lenticular visual morphologies. However, this raises the question of whether such a separation is a physically meaningful thing to do.
The difficulty of distinguishing face-on S0s from ellipticals is a well known problem. The result in Fig. \[fig:axisratio\] illustrates this issue in terms of quantitative structural measurements. When a fast-rotating early-type galaxy appears to have an inclined disc ($(b/a)_{\rm disc} \la 0.5$, it is usually classified as S0. If the same galaxy were viewed closer to face-on, it would be classified as elliptical. In our (non-representative) sample, this amounts to about half of true S0 galaxies (discy, fast-rotators that that would have been visually classified as S0 if viewable from other angles) being misclassified as elliptical. We presently do not have a reliable morphological or structural way of recovering these objects, but instead must resort to kinematic information (e.g., @tex:KA13). However, we remain hopeful that with additional work we can make further progress on an image-based solution to this long-standing problem.
![Bulge-to-total flux ratio as a function of [Sérsic]{}index for our [`MM`]{}method. Only galaxies with significant bulge measurements are shown in this figure. Representative error bars for our measurement are displayed. See text for further discussion on the uncertainty measurements. []{data-label="fig:BTn"}](B_T_n.pdf){width="45.00000%"}
![Bulge-to-total flux ratio as a function of [Sérsic]{}index for our [`MM`]{}fits to the artificially-redshifted images. Only galaxies with significant bulge measurements are shown in this figure. []{data-label="fig:BTn_z"}](B_T_n_z.pdf){width="48.00000%"}
![Bulge-to-total flux ratio as a function of the bulge [Sérsic]{}index from our [`MM`]{}fits. Only galaxies with significant bulge measurements are shown in this figure. Representative error bars for our measurement are displayed. See text for further discussion on the uncertainty measurements[]{data-label="fig:BTnb"}](B_T_nb.pdf){width="45.00000%"}
![Bulge-to-total flux ratio as a function of the bulge-to-disc size ratio. Only galaxies with both a significant bulge and disc are shown in this figure. Representative error bars for our measurement are displayed. See text for further discussion on the uncertainty measurements[]{data-label="fig:BTrratio"}](B_T_rratio.pdf){width="45.00000%"}
Conclusions {#sec:conclusions}
===========
All previous studies that have utilised one- or two-dimensional photometric bulge-disc decompositions have performed their fits using, at most, two bands simultaneously. In this work, for the first time, we have performed bulge-disc decompositions simultaneously on five-band imaging. To achieve this, we have used [[<span style="font-variant:small-caps;">galfitm</span>]{}]{}, a modified version of [[<span style="font-variant:small-caps;">galfit3</span>]{}]{}which enables a single, wavelength-dependent model to be fit to multiple images of the same galaxy.
We have evaluated the performance of our multi-band method by applying it to SDSS $ugriz$ images of 163 nearby galaxies, as well as to another 3863 artificially-redshifted images of the same objects. For our models, we use a combination of two [Sérsic]{}profiles: one for the disc, with $n$ fixed to one, and another for the bulge, with free-$n$.
Using the original images, we have shown that our fitting results generally agree well with structural parameters obtained from the literature, both when we compare specific galaxies and averages for bins of Hubble type. We confirm the standard picture that disc galaxies of earlier morphology have a larger fraction of their flux in a bulge component. However, the sizes of these bulges, with respect to their accompanying disc, do not vary significantly with Hubble type. We find that bulges in spiral galaxies typically have a [Sérsic]{}index $n \sim 2$, except in the latest-types, where it is more usually exponential $n \sim 1$. The dominant component of ellipticals has $n \sim 3.5$. Puzzlingly early-type discs appear to display a bimodality with peaks around $n \sim 1$ and $4$.
The observed colours of disks display a classic colour-magnitude diagram, with a well defined red sequence, inhabited by E-Sb galaxies, and a blue cloud corresponding to later Hubble types. The colour-magnitude diagram for bulges is more complex. The bulges of E-Sa galaxies lie on a red-sequence similar to their disks, with only a small average difference in the colours of their bulges and discs. The bulges of Sb-Sc galaxies are often even redder than the early-type red-sequence, indicative of dust reddening. On the other hand, for many late-type disc galaxies we find bulges with fairly blue colours, suggestive of recent star-formation. Despite this complexity, the average difference in the colours of bulges and discs within the same galaxy is constant for all spiral galaxies, $\left<\Delta(g-i)\right> \sim 0.3$ mag.
Our fits permit a quantitative illustration of the notorious difficulty of distinguishing between galaxies with elliptical and lenticular morphology. Most early-types are well-fitted by a combination of a [Sérsic]{}profile and an exponential profile. It is not clear how often this exponential profile represents a genuine disc component, or whether it reveals the presence of an extended halo or some other substructure. Nevertheless, a significant inclined disc component is strong indication that an early-type galaxy will be visually classified as S0 and possess fast-rotator kinematics. Unfortunately, the lack of S0 morphologies among galaxies with face-on disc components, despite kinematic evidence indicating the discs are real, suggests that such systems are usually misclassified as ellipticals.
Using our artificially-redshifted images we have investigated the range of redshift over which our fit parameters remain stable, and hence are inferred to be reliable. We have demonstrated that the results of our multi-band fits show less variation and are reliable to higher redshifts than the results of fitting each band independently. This is true even in the common single-band approach where one first performs a full fit on a preferred band, then fixes the structural parameters in all subsequent fits to those obtained in that preferred band.
Our approach produces somewhat more reliable measurements of the bulge [Sérsic]{}index and effective radius, although these are both difficult quantities to measure with consistent accuracy, especially when it is not feasible to fit each galaxy interactively. More promising is our method’s performance in recovering the bulge-to-total flux ratio and in differentiating between the colours of the bulge and disc. We therefore recommend the adoption of multi-band bulge-disc decomposition, allowing studies to reliably probe to greater distances and lower-luminosity galaxies.
Acknowledgments {#acknowledgments .unnumbered}
===============
This publication was made possible by NPRP grant \# 08-643-1-112 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors. BH and MV were supported by the NPRP grant. SPB gratefully acknowledges an STFC Advanced Fellowship. We would like to thank Carnegie Mellon University in Qatar and The University of Nottingham for their hospitality. We would like to thank the referee for the constructive comments that improved the paper.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
![The axis ratio of bulge (upper panel) and disc (lower panel) as a function of the bulge [Sérsic]{}index. The red and blue circles indicate slow and fast rotators, respectively, as measured in @tex:EC11. Only galaxies with both a significant bulge and disc are shown. Representative error bars for our measurement are displayed in the top part of each panel. See text for further discussion on the uncertainty measurements. []{data-label="fig:axisratio"}](axis-ratio_nbulge.pdf){width="48.00000%"}
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Notes on Individual galaxies {#sec:notes}
============================
Here, we briefly describe some galaxies with peculiar results. In Figure \[fig:apend\] we present the images and models for some of the following galaxies in order to show the visual appearance of the galaxies that return peculiar results. In the same Figure we have also added three additional galaxies with what we consider good result in order to support a comparison.
NGC3521
: \[SAB(rs)bc\] A high-inclination galaxy with strong dust features that make it extremely difficult to fit a two-component model. Our two-component model measures a large bulge with $r_{\rm e,b}/r_{\rm e,d}= 1.94$ and $n_{\rm b}=6.7$. This may not be realistic, but it has been retained in the analysis.\
NGC3642
: \[SA(r)bc\] A late-type galaxy for which we measure $r_{\rm e,b}/r_{\rm e,d}= 2.5$ and $n_{\rm b} = 6.7$. However, by examining the equivalent of Fig. \[fig:gal1\] for NGC3642, we noticed that that both $r_{\rm e,b}/r_{\rm e,d}$ and $n_{\rm b}$ decrease with redshift until $z=0.03$. After that, the values remain constant until $z=0.12$. The recovered values for this range of redshifts are $r_{\rm e,b}/r_{\rm e,d}= 0.2$ and $n_{\rm b} = 1.5$, which seem much more reasonable. These dramatic changes in parameters could be due to the detailed substructure that is visible in the high-resolution images. As the galaxy becomes more distant these substructure are less pronounced and, as a result, the fit parameters better reflect the properties of the bulge and disc.\
NGC3893
: \[SAB(rs)c\] A similar case to NGC3642.\
NGC4123
: \[SB(r)c\] A late-type galaxy for which our two-component model measures $n_{\rm b} = 10.5$. However, if we add a PSF into the model, the bulge [Sérsic]{}index is reduced to $1.86$ and $r_{\rm b}$ decreases by $50$%. The corresponding change in disc effective radius is $\sim 2$%. We therefore use the model including the PSF component.\
NGC4452
: \[S0(9)\] An almost perfectly edge-on galaxy, which contains a very thin disc (for which our model measures $b/a = 0.09$). The bulge component fits a second elongated component ($b/a = 0.37$) with $n_{\rm b} = 1.08$ and effective radius $20$% larger than the thin disc. The properties of the second component are more consistent with a thick disc, rather than a bulge. However, in the spirit of avoiding specal cases, we retain both components in the analysis.\
NGC4458
: \[E\] An elliptical galaxy with an internal structure. As a result, the exponential component fits the inner part of the galaxy and the free-[Sérsic]{}component measures $n_{\rm b} = 11$. Due to the peculiar fitting results, we attempted to fit this galaxy with various initial parameter values. However, the final results always remained the same, and agree for both [`SM`]{}and [`MM`]{}methods.\
NGC4459
: \[S0\] After various attempts with different initial values, we failed to fit a significant second component for this lenticular galaxy. In analysis we therefore only include the bulge component.\
NGC4698
: \[SA(s)ab\] A spiral galaxy with peculiar structure: the bulge is elongated perpendicular to the main disc. We measure structural parameters of $r_{\rm e,b}/r_{\rm e,d}= 1.5$ and $n_{\rm b} = 7.1$, consistently using both the [`SM`]{}and [`MM`]{}methods, and the redshifted images. However, these are not in agreement with other studies which focus on the unique structure of this galaxy.\
UGC08041
: \[SB(s)d\] is a similar case to NGC4123. The two component model measures $n_{\rm b}=9.37$, while the inclusion of a PSF component reduces this to $n_{\rm b}=1.17$. Both the effective radius of the bulge and disc component change less than $10$%. We use the model with the PSF component.
\[lastpage\]
[^1]: [[<span style="font-variant:small-caps;">galfitm</span>]{}]{}is publicly available at <http://www.nottingham.ac.uk/astronomy/megamorph/>.
[^2]: The visual classifications have been taken from NED
[^3]: black: IC3653 and NGC4458; red: NGC4452; green: NGC3521, NGC3642, NGC3893 and NGC4698.
[^4]: orange: NGC5624, green: NGC1084, purple: NGC0428 and NGC0853
|
---
abstract: 'The primary purpose of this paper is to clarify the relation between previous results in [@Schr11], [@Bre17] and [@BD18]. Let $E$ be a sufficiently large finite extension of $\mathbb{Q}_p$ and $\rho_p$ be a $p$adic semi-stable representation $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\rightarrow\mathrm{GL}_3(E)$ such that the Weil–Deligne representation $\mathrm{WD}(\rho_p)$ associated with it has rank two monodromy operator $N$ and the Hodge filtration associated with it is non-critical. Then by a computation of extensions of rank one $(\varphi, \Gamma)$-modules we know that the Hodge filtration of $\rho_p$ depends on three invariants in $E$. We construct a family of locally analytic representations $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ of $\mathrm{GL}_3(\mathbb{Q}_p)$ depending on three invariants $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3 \in E$ with each of the representation containing the locally algebraic representation $\mathrm{Alg}\otimes\mathrm{Steinberg}$ determined by $\mathrm{WD}(\rho_p)$ via classical local Langlands correspondence for $\mathrm{GL}_3(\mathbb{Q}_p)$ and by the Hodge–Tate weights of $\rho_p$. When $\rho_p$ comes from an automorphic representation $\pi$ of $G(\mathbb{A}_{\mathbb{Q}_p})$ with a fixed level $U^p$ prime to $p$ for a suitable unitary group $G_{/\mathbb{Q}}$, we show (under some technical assumption) that there is a unique locally analytic representation in the above family that occurs as a subrepresentation of the associated Hecke-isotypic subspace in the completed cohomology with level $U^p$. We recall that [@Bre17] constructed a family of locally analytic representations depending on four invariants ( cf. (4) in [@Bre17]) and proved that there is a unique representation in the family that embeds into the fixed Hecke-isotypic space above. We prove that if a representation $\Pi$ in Breuil’s family embeds into a certain Hecke-isotypic subspace of completed cohomology, then it must equally embed into $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ for certain choices of $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$ determined explicitly by $\Pi$. This gives a purely representation theoretic necessary condition for $\Pi$ to embed into completed cohomology. Moreover, certain natural subquotients of $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ give a true complex of locally analytic representations that realizes the derived object $\Sigma(\lambda, \underline{\mathscr{L}})$ in (1.14) of [@Schr11]. Consequently, the locally analytic representation $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ gives a relation between the higher $\mathscr{L}$invariants studied in [@Bre17] as well as [@BD18] and the $p$adic dilogarithm function which appears in the construction of $\Sigma(\lambda, \underline{\mathscr{L}})$ in [@Schr11].'
address: 'Département de Mathématiques Batiment 425, Faculté des Sciences d’Orsay Université Paris-Sud, 91405 Orsay, France'
author:
- Zicheng Qian
title: 'Dilogarithm and Higher $\mathscr{L}$invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$'
---
Introduction {#3section: introduction}
============
Let $p$ be a prime number and $F$ an imaginary quadratic extension of $\mathbb{Q}$ such that $p$ splits in $F$. We fix a unitary algebraic group $G$ over $\mathbb{Q}$ which becomes $\mathrm{GL}_n$ over $F$ and such that $G(\mathbb{R})$ is compact and $G$ is split at all places of $F$ above $p$. Then to each finite extension $E$ of $\mathbb{Q}_p$ and to each primeto$p$ level $U^p$ in $G(\mathbb{A}_{\mathbb{Q}}^{\infty, p})$, one can associate the Banach space of $p$adic automorphic forms $\widehat{S}(U^p, E)$. One can also associate with $U^p$ a set of finite places $D(U^p)$ of $\mathbb{Q}$ and a Hecke algebra $\mathbb{T}(U^p)$ which is the polynomial algebra freely generated by Hecke operators at places of $F$ lying above $D(U^p)$. In particular, the commutative algebra $\mathbb{T}(U^p)$ acts on $\widehat{S}(U^p, E)$ and commutes with the action of $G(\mathbb{Q}_p)\cong\mathrm{GL}_n(\mathbb{Q}_p)$ coming from translations on $G(\mathbb{A}_{\mathbb{Q}}^{\infty})$.
If $\rho:\mathrm{Gal}(\overline{F}/F)\rightarrow\mathrm{GL}_n(E)$ is a continuous irreducible representation, one considers the associated Hecke isotypic subspace $\widehat{S}(U^p, E)[\mathfrak{m}_{\rho}]$, which is a continuous admissible representation of $G(\mathbb{Q}_p)\cong\mathrm{GL}_n(\mathbb{Q}_p)$ over $E$, or its locally $\mathbb{Q}_p$analytic vectors $\widehat{S}(U^p, E)[\mathfrak{m}_{\rho}]^{\rm{an}}$, which is an admissible locally $\mathbb{Q}_p$analytic representation of $\mathrm{GL}_n(\mathbb{Q}_p)$. We fix $w_p$ a place of $F$ above $p$ and it is widely wished that $\widehat{S}(U^p, E)[\mathfrak{m}_{\rho}]$ (and its subspace $\widehat{S}(U^p, E)[\mathfrak{m}_{\rho}]^{\rm{an}}$ as well) determines and depends only on $\rho_p:=\rho|_{\mathrm{Gal}(\overline{F_{w_p}}/F_{w_p})}$. The case $n=2$ is well-known essentially due to various results in [@Col10], [@Eme]. The case $n\geq 3$ is much more difficult and only some partial results are known. We are particularly interested in the case when the subspace of locally algebraic vectors $\widehat{S}(U^p, E)[\mathfrak{m}_{\rho}]^{\rm{alg}}\subsetneq \widehat{S}(U^p, E)[\mathfrak{m}_{\rho}]$ is non-zero, which implies that $\rho_p$ is potentially semi-stable. Certain cases when $n=3$ and $\rho_p$ is semi-stable and non-crystalline have been studied in [@Bre17] and [@BD18]. We are going to continue their work and obtain some interesting relation between results in [@Bre17], [@BD18] and previous results in [@Schr11] which involve the $p$adic dilogarithm function.
We use the notation $\lambda\in X(T)_+$ for a weight $\lambda=(\lambda_1, \lambda_2, \lambda_3)$ (of the diagonal split torus $T$ of $\mathrm{GL}_3$) which is dominant with respect to the upper-triangular Borel subgroup $\overline{B}$ and hence satisfies $\lambda_1\geq \lambda_2\geq\lambda_3$. Given two locally analytic representations $V, W$ of $\mathrm{GL}_3(\mathbb{Q}_p)$, we use the shorten notation $\begin{xy}
(0,0)*+{V}="a"; (10,0)*+{W}="b";
{\ar@{-}"a";"b"};
\end{xy}$ (resp. the shorten notation $\begin{xy}
(0,0)*+{V}="a"; (10,0)*+{W}="b";
{\ar@{--}"a";"b"};
\end{xy}$) for a locally analytic representation determined by a non-zero (resp. possibly zero) element in $\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(W, ~V\right)$.
\[3theo: construction introduction\]\[Proposition \[3prop: main dim\], Proposition \[3prop: criterion of existence\]\] For each choice of $\lambda\in X(T)_+$ and $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$, there exists a locally analytic representation $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ of $\mathrm{GL}_3(\mathbb{Q}_p)$ of the form: $$\begin{xy}
(0,0)*+{\mathrm{St}_3^{\rm{an}}(\lambda)}="a"; (20,5)*+{v_{P_1}^{\rm{an}}(\lambda)}="b"; (40,8)*+{C_{s_1,s_1}}="d"; (60,8)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="f"; (20,-5)*+{v_{P_2}^{\rm{an}}(\lambda)}="c"; (40,-8)*+{C_{s_2,s_2}}="e"; (60,-8)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="g"; (80,3)*+{\overline{L}(\lambda)}="h1"; (80,-3)*+{\overline{L}(\lambda)}="h2";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"e"}; {\ar@{-}"d";"f"}; {\ar@{-}"e";"g"}; {\ar@{-}"f";"h1"}; {\ar@{-}"b";"h1"}; {\ar@{-}"g";"h1"}; {\ar@{-}"f";"h2"}; {\ar@{-}"c";"h2"}; {\ar@{-}"g";"h2"};
\end{xy}$$ where $\mathrm{St}_3^{\rm{an}}(\lambda)$, $v_{P_1}^{\rm{an}}(\lambda)$, $v_{P_2}^{\rm{an}}(\lambda)$, $\overline{L}(\lambda)$ and $C^{\ast}_{w^{\prime},w}$ for $w,w^{\prime}\in \{s_1, s_2, s_1s_2, s_2s_1\}$ and $\ast\in \{\varnothing, 1, 2\}$ are various explicit locally analytic representations defined in Section \[3subsection: main notation\]. Moreover, different choices of $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$ give non-isomorphic representations.
We will see in Lemma \[3lemm: distinguish mult two\] and (\[3main picture for min\]) that $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ is *the minimal locally analytic representation that involves $p$adic dilogarithm*, hence explains the notation ‘[min]{}’. We also construct a locally analytic representation $\Sigma^{\rm{min}, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ of the form $$\begin{xy}
(0,0)*+{\mathrm{St}_3^{\rm{an}}(\lambda)}="a"; (20,6)*+{v_{P_1}^{\rm{an}}(\lambda)}="b"; (40,12)*+{C_{s_1,s_1}}="d"; (65,10)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="f"; (20,-6)*+{v_{P_2}^{\rm{an}}(\lambda)}="c"; (40,-12)*+{C_{s_2,s_2}}="e"; (65,-10)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="g"; (90,3)*+{\overline{L}(\lambda)}="h1"; (90,-3)*+{\overline{L}(\lambda)}="h2"; (65,18)*+{C^1_{s_2s_1,s_2s_1}}="i"; (65,-18)*+{C^1_{s_1s_2,s_1s_2}}="j"; (90,15)*+{C^2_{s_1,s_1s_2}}="k"; (90,-15)*+{C^2_{s_2,s_2s_1}}="l";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"e"}; {\ar@{-}"d";"f"}; {\ar@{-}"e";"g"}; {\ar@{-}"f";"h1"}; {\ar@{-}"b";"h1"}; {\ar@{-}"g";"h1"}; {\ar@{-}"f";"h2"}; {\ar@{-}"c";"h2"}; {\ar@{-}"g";"h2"}; {\ar@{-}"d";"i"}; {\ar@{-}"e";"j"}; {\ar@{-}"i";"k"}; {\ar@{-}"f";"k"}; {\ar@{-}"j";"l"}; {\ar@{-}"g";"l"};
\end{xy}$$ which contains and is uniquely determined by $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$.
\[3theo: main introduction\]\[Theorem \[3theo: main\]\] Assume that $p\geq 5$ and $n=3$. Assume moreover that
1. $\rho$ is unramified at all finite places of $F$ above $D(U^p)$;
2. $\widehat{S}(U^p, E)[\mathfrak{m}_{\rho}]^{\rm{alg}}\neq 0$;
3. $\rho_p$ is semi-stable with Hodge–Tate weights $\{k_1>k_2>k_3\}$ such that $N^2\neq 0$;
4. $\rho_p$ is non-critical in the sense of Remark 6.1.4 of [@Bre17];
5. only one automorphic representation contributes to $\widehat{S}(U^p, E)[\mathfrak{m}_{\rho}]^{\rm{alg}}$.
Then there exists a unique choice of $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$ such that $\widehat{S}(U^p, E)[\mathfrak{m}_{\rho}]^{\rm{an}}$ contains (copies of) the locally analytic representation $$\Sigma^{\rm{min}, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}$$ where $\lambda=(\lambda_1, \lambda_2, \lambda_3)=(k_1-2, k_2-1, k_3)$ and $\alpha\in E^{\times}$ is determined by the Weil–Deligne representation $\mathrm{WD}(\rho_p)$ associated with $\rho_p$. Moreover, we have $$\begin{gathered}
\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(\Sigma^{\rm{min},+}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}, \widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]\right)\\
\xrightarrow{\sim}\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}, \widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]\right).\\\end{gathered}$$
The assumptions of our Theorem \[3theo: main introduction\] are the same as that of Theorem 1.3 of [@Bre17]. We do not attempt to obtain any explicit relation between $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$ and $\rho_p$, which is similar in flavor to Theorem 1.3 of [@Bre17]. On the other hand, Theorem 7.52 of [@BD18] does care about the explicit relation between invariants of the locally analytic representation associated with $\rho_p$, under further technical assumptions such as $\rho_p$ is ordinary with consecutive Hodge–Tate weights and has an irreducible mod $p$ reduction but without assuming our condition (v). The improvement of our Theorem \[3theo: main introduction\] upon Theorem 1.3 of [@Bre17] will be explained in Section \[3subsection: criterion of global embedding\]. One can naturally wish that there is a common refinement or generalization of our Theorem \[3theo: main introduction\] and Theorem 7.52 of [@BD18] by removing as many technical assumptions as possible.
\[3rema: construction max\] It is actually possible to construct a locally analytic representation $\Sigma^{\rm{max}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ of $\mathrm{GL}_3(\mathbb{Q}_p)$ containing $\Sigma^{\rm{min}, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ which is characterized by the fact that it is maximal (for inclusion) among the locally analytic representations $V$ satisfying the following conditions:
1. $\mathrm{soc}_{\mathrm{GL}_3(\mathbb{Q}_p)}(V)=V^{\rm{alg}}=\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}$;
2. each constituent of $V$ is a subquotient of a locally analytic principal series
where $V^{\rm{alg}}$ is the subspace of locally algebraic vectors in $V$. Moreover, one can use an immediate generalization of the arguments in the proof of Theorem \[3theo: main introduction\] (and thus of Theorem 1.1 of [@Bre17]) to show that $$\begin{gathered}
\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(\Sigma^{\rm{max}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}, \widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]\right)\\
\xrightarrow{\sim}\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}, \widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]\right).\\\end{gathered}$$ We can also show that $$\Sigma^{\rm{max}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)/\overline{L}(\lambda)\otimes_E\mathrm{St}_3$$ is independent of the choice of $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$, which is compatible with the fact that $$\Sigma^{\rm{min}, \ast}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)/\overline{L}(\lambda)\otimes_E\mathrm{St}_3$$ is independent of the choice of $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$ for each $\ast\in \{\varnothing, +\}$ as mentioned in Remark \[3quotient independent of invariants\]. However, the full construction of $\Sigma^{\rm{max}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ is lengthy and technical and thus we decided not to put it in the present article.
Derived object and dilogarithm {#3subsection: dilog}
------------------------------
We consider the bounded derived category $$\mathcal{D}^b\left(\mathrm{Mod}_{D(\mathrm{GL}_3(\mathbb{Q}_p), E)}\right)$$ associated with the abelian category $\mathrm{Mod}_{D(\mathrm{GL}_3(\mathbb{Q}_p), E)}$ of abstract modules over the algebra $D(\mathrm{GL}_3(\mathbb{Q}_p), E)$ of locally $\mathbb{Q}_p$analytic distributions on $\mathrm{GL}_3(\mathbb{Q}_p)$. An object $$\Sigma(\lambda, \underline{\mathscr{L}})^{\prime}\in \mathcal{D}^b\left(\mathrm{Mod}_{D(\mathrm{GL}_3(\mathbb{Q}_p), E)}\right)$$ (one should not confuse this notation $\Sigma(\lambda, \underline{\mathscr{L}})^{\prime}$ borrowed directly from [@Schr11] with our notation $\Sigma^+(\lambda, \underline{\mathscr{L}})$ before Lemma \[3lemm: ext13\]) has been constructed in [@Schr11] and plays a key role in Theorem 1.2 of [@Schr11]. An interesting feature of [@Schr11] is the appearance of the $p$adic dilogarithm function in the construction of $\Sigma(\lambda, \underline{\mathscr{L}})^{\prime}$ in Definition 5.19 of [@Schr11]. Roughly, the object $\Sigma(\lambda, \underline{\mathscr{L}})^{\prime}$ was constructed from the choice of an element in $\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p), \lambda}\left(\overline{L}(\lambda),~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$ together with general formal arguments in triangulated categories ( cf. Proposition 3.2 of [@Schr11]). In particular, $\Sigma(\lambda, \underline{\mathscr{L}})^{\prime}$ fits into the following distinguished triangle: $$F_\lambda^{\prime}\longrightarrow ~\Sigma(\lambda, \underline{\mathscr{L}})^{\prime}\longrightarrow \Sigma(\lambda, \mathscr{L}, \mathscr{L}^{\prime})^{\prime}[-1]\xrightarrow{~+1}$$ as illustrated in (5.99) of [@Schr11]. However, it was not clear in [@Schr11] whether there is an explicit complex $\left[C_{\bullet}\right]$ of locally analytic representations of $\mathrm{GL}_3(\mathbb{Q}_p)$ such that the object $$\mathcal{D}^{\prime}\in\mathcal{D}^b\left(\mathrm{Mod}_{D(\mathrm{GL}_3(\mathbb{Q}_p), E)}\right)$$ associated with $\left[C^{\prime}_{-\bullet}\right]$ satisfies $$\mathcal{D}^{\prime}\cong \Sigma(\lambda, \underline{\mathscr{L}})^{\prime}\in \mathcal{D}^b\left(\mathrm{Mod}_{D(\mathrm{GL}_3(\mathbb{Q}_p), E)}\right).$$ Although our notation are slightly different from [@Schr11] in the sense that the notation $\Sigma(\lambda, \mathscr{L}, \mathscr{L}^{\prime})$ (resp. the notation $F_\lambda$) is replaced with $\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ (resp. with $\overline{L}(\lambda)$), we show that
\[3theo: complex introduction\]\[Proposition \[3prop: relation with derived object\], (\[3sign of L invariant\]) and Lemma \[3lemm: easy normalization of higher invariant\]\] The complex $$\label{3key complex}
\left[\left(\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}}="a"; (20,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)^{\prime}\longrightarrow\Sigma^{\sharp, +}_i(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime} \right]$$ realizes the object $\Sigma(\lambda, \underline{\mathscr{L}})^{\prime}$ where $\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}}="a"; (20,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}$ is the unique non-split extension of $\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}$ by $\overline{L}(\lambda)$ thanks to Proposition \[3prop: locally algebraic extension\], $\Sigma^{\sharp, +}_i(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ is the locally analytic subrepresentation of $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ of the form $$\begin{xy}
(0,0)*+{\mathrm{St}_3^{\rm{an}}(\lambda)}="a"; (20,4)*+{v_{P_i}^{\rm{an}}(\lambda)}="b"; (40,8)*+{C_{s_i,s_i}}="d"; (65,8)*+{\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}}="f"; (20,-4)*+{v_{P_{3-i}}^{\rm{an}}(\lambda)}="c"; (40,-8)*+{C_{s_{3-i},s_{3-i}}}="e"; (40,0)*+{\overline{L}(\lambda)}="h";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"e"}; {\ar@{-}"d";"f"}; {\ar@{-}"b";"h"}; {\ar@{-}"c";"h"};
\end{xy}$$ and the invariants $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$ are determined by the formula $$\mathscr{L}_1=-\mathscr{L}^{\prime},~\mathscr{L}_2=-\mathscr{L},~\mathscr{L}_3=\gamma(\mathscr{L}^{\prime\prime}-\frac{1}{2}\mathscr{L}\mathscr{L}^{\prime})$$ with the constant $\gamma\in E^{\times}$ defined in Lemma \[3lemm: ext2 prime\].
\[3rema: canonical isomorphism\] Strictly speaking, the complex (\[3key complex\]) realizes an object in $\mathcal{D}^b\left(\mathrm{Mod}_{D(\mathrm{GL}_3(\mathbb{Q}_p), E)}\right)$ characterized by an element in $$\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p), \lambda}\left(\overline{L}(\lambda),~\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$$ due to formal arguments from Proposition 3.2 of [@Schr11]. However, we can prove that there is a canonical isomorphism $$\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p), \lambda}\left(\overline{L}(\lambda),~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\xrightarrow{\sim}\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p), \lambda}\left(\overline{L}(\lambda),~\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$$ and hence we can equally say that (\[3key complex\]) realizes $\Sigma(\lambda, \underline{\mathscr{L}})^{\prime}$ for a suitable normalization of notation as $\Sigma(\lambda, \underline{\mathscr{L}})$ has been constructed by choosing a non-zero element in $\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p), \lambda}\left(\overline{L}(\lambda),~\Sigma(\lambda, \mathscr{L}, \mathscr{L}^{\prime})\right)$ via Proposition 3.2 of [@Schr11]. Note that we have $$\Sigma(\lambda, \mathscr{L}, \mathscr{L}^{\prime})\cong\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)$$ by (\[3normalization of notation\]).
Higher $\mathscr{L}$invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ {#3subsection: criterion of global embedding}
----------------------------------------------------------------
It follows from (\[3main picture for min\]) and (\[3main picture for Ext1\]) that $\Sigma^{\rm{min},+}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ can be described more precisely by the following picture: $$\begin{xy}
(-20,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="a1";
(-6,9)*+{C^2_{s_1,1}}="a2"; (13.5,22)*+{C^1_{s_2s_1,1}}="a4"; (51,21)*+{C^2_{s_2s_1,1}}="a6"; (20,14)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="b1"; (70,12)*+{C^1_{s_2,1}}="b2"; (42,30)*+{C_{s_1,s_1}}="d1"; (77,22)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="f1"; (-6,-9)*+{C^2_{s_2,1}}="a3";(13.5,-22)*+{C^1_{s_1s_2,1}}="a5"; (51,-21)*+{C^2_{s_1s_2,1}}="a7"; (20,-14)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="c1"; (70,-12)*+{C^1_{s_1,1}}="c2"; (42,-30)*+{C_{s_2,s_2}}="e1"; (77,-22)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="g1"; (100,5)*+{\overline{L}(\lambda)^1}="h1"; (100,-5)*+{\overline{L}(\lambda)^2}="h2"; (73,-0)*+{C_{s_1s_2s_1,1}}="a8"; (75,35)*+{C^1_{s_2s_1,s_2s_1}}="i1"; (75,-35)*+{C^1_{s_1s_2,s_1s_2}}="j1"; (100,30)*+{C^2_{s_1,s_1s_2}}="i2"; (100,-30)*+{C^2_{s_2,s_2s_1}}="j2";
{\ar@{-}"a1";"a2"}; {\ar@{-}"a2";"a4"}; {\ar@{-}"a2";"b1"}; {\ar@{-}"a2";"a7"}; {\ar@{--}"a4";"b2"}; {\ar@{-}"a4";"a6"}; {\ar@{-}"a4";"d1"}; {\ar@{--}"a6";"c2"}; {\ar@{--}"a6";"a8"}; {\ar@{-}"b1";"b2"}; {\ar@{-}"b1";"d1"}; {\ar@{-}"b2";"h1"}; {\ar@{-}"d1";"f1"}; {\ar@{-}"f1";"h2"}; {\ar@{-}"f1";"h1"}; {\ar@{-}"a1";"a3"}; {\ar@{-}"a3";"a5"}; {\ar@{-}"a3";"c1"}; {\ar@{-}"a3";"a6"}; {\ar@{--}"a5";"c2"}; {\ar@{-}"a5";"e1"}; {\ar@{-}"a5";"a7"}; {\ar@{--}"a7";"b2"}; {\ar@{--}"a7";"a8"}; {\ar@{-}"c1";"c2"}; {\ar@{-}"c1";"e1"}; {\ar@{-}"c2";"h2"}; {\ar@{-}"e1";"g1"}; {\ar@{-}"g1";"h1"}; {\ar@{-}"g1";"h2"}; {\ar@{-}"d1";"i1"}; {\ar@{-}"e1";"j1"}; {\ar@{-}"i1";"i2"}; {\ar@{-}"j1";"j2"}; {\ar@{-}"f1";"i2"}; {\ar@{-}"g1";"j2"};
\end{xy}$$ and therefore contains a unique subrepresentation of the form $$\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="a1";
(20,8)*+{C^2_{s_1,1}}="a2"; (45,12)*+{C^1_{s_2s_1,1}}="a4"; (45,4)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="b1"; (70,8)*+{C_{s_1,s_1}}="d1"; (95,4)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="f1"; (20,-8)*+{C^2_{s_2,1}}="a3";(45,-12)*+{C^1_{s_1s_2,1}}="a5"; (45,-4)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="c1"; (70,-8)*+{C_{s_2,s_2}}="e1"; (95,-4)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="g1"; (95,12)*+{C^1_{s_2s_1,s_2s_1}}="i1"; (95,-12)*+{C^1_{s_1s_2,s_1s_2}}="j1"; (120,8)*+{C^2_{s_1,s_1s_2}}="i2"; (120,-8)*+{C^2_{s_2,s_2s_1}}="j2";
{\ar@{-}"a1";"a2"}; {\ar@{-}"a2";"a4"}; {\ar@{-}"a2";"b1"}; {\ar@{-}"a4";"d1"}; {\ar@{-}"b1";"d1"}; {\ar@{-}"d1";"f1"}; {\ar@{-}"a1";"a3"}; {\ar@{-}"a3";"a5"}; {\ar@{-}"a3";"c1"}; {\ar@{-}"a5";"e1"}; {\ar@{-}"c1";"e1"}; {\ar@{-}"e1";"g1"}; {\ar@{-}"d1";"i1"}; {\ar@{-}"e1";"j1"}; {\ar@{-}"i1";"i2"}; {\ar@{-}"j1";"j2"}; {\ar@{-}"f1";"i2"}; {\ar@{-}"g1";"j2"};
\end{xy}$$ which is denoted by $$\label{3representation ext 1}
\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="a"; (30,4)*+{\Pi^1(\underline{k}, \underline{D})}="b"; (30,-4)*+{\Pi^2(\underline{k}, \underline{D})}="c";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"};
\end{xy}$$ in Theorem 1.1 of [@Bre17]. It follows from Theorem 1.2 of [@Bre17] that $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\Pi^i(\underline{k}, \underline{D}),~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)=3$$ for $i=1,2$, and therefore a locally analytic representation of the form (\[3representation ext 1\]) depends on four invariants. On the other hand, by a computation of extensions of rank one $(\varphi, \Gamma)$modules we know that $\rho_p$ depends on three invariants. As a result, Theorem 1.1 of [@Bre17] predicts that not all representations of the form (\[3representation ext 1\]) can be embedded into $\widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]$ for a certain pair of $U^p$ and $\rho_p$. This is actually the case as we show that
\[3theo: relation introduction\]\[Corollary \[3coro: criterion\]\] If a locally analytic representation $\Pi$ of the form (\[3representation ext 1\]) can be embedded into $\widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]$ for a certain pair of $U^p$ and $\rho_p$, then it can be embedded into $$\Sigma^{\rm{min},+}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$$ for a unique choice of $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$ determined by $\Pi$.
Sketch of content {#3subsection: sketch}
-----------------
Section \[3section: preliminary\] recalls various well-known facts around locally analytic representations and our notation for a family of specific irreducible subquotients of locally analytic principal series to be used in the rest of the article. We emphasize that our definition of various $\mathrm{Ext}$groups follows [@Bre17] closely and the only difference is that we use the dual notation compared to that of [@Bre17]. We also recall the $p$adic dilogarithm function from Section 5.3 of [@Schr11] which is part of the main motivation of this article to relate [@Schr11] with [@Bre17] and [@BD18].
Section \[3section: GL2Qp\] proves a crucial fact (Proposition \[3prop: key result\]) on the non-existence of locally analytic representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ of a certain specific form using arguments involving infinitesimal characters of locally analytic representations. We learn such arguments essentially from Y. Ding.
Section \[3section: computation\] is a collection of various computational results necessary for the applications in Section \[3section: exact sequence min\]. These computations essentially make use of the formula in Section 5.2 and 5.3 of [@Bre17].
Section \[3section: technial min\] serves as the preparation of Section \[3section: exact sequence min\] for the construction of $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$. It makes full use of the computational results from Section \[3section: computation\] to compute the dimension of various more complicated $\mathrm{Ext}$groups to be crucially used in various important long exact sequences in Section \[3section: exact sequence min\]( cf. Lemma \[3lemm: upper bound\] and Proposition \[3prop: main dim\]).
Section \[3section: exact sequence min\] finishes the construction of $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ as well as $\Sigma^{\rm{min},+}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$. Moreover, the construction of $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ leads naturally to the construction of an explicit complex as in Theorem \[3theo: complex introduction\] that realizes the derived object $\Sigma(\lambda, \underline{\mathscr{L}})$ constructed in [@Schr11].
Section \[3section: local-global\] finishes the proof of Theorem \[3theo: main\] by directly mimicking arguments from the proof of Theorem 6.2.1 of [@Bre17]. In particular, we give a purely representation theoretic criterion for a representation of the form (\[3representation ext 1\]) to embed into completed cohomology as mentioned in Theorem \[3theo: relation introduction\].
Acknowledgement {#3subsection: acknowledgement}
---------------
The author expresses his gratefulness to Christophe Breuil for introducing the problem of relating [@Schr11] with [@Bre17] and [@BD18] and especially for his interest on the role played by the $p$adic dilogarithm function. The author also benefited a lot from countless discussions with Y. Ding especially for Section \[3section: GL2Qp\] of this article. Finally, the author thanks B. Schraen for his beautiful thesis which improved the author’s understanding on the subject.
Preliminary {#3section: preliminary}
===========
Locally analytic representations {#3subsection: locally analytic rep}
--------------------------------
In this section, we recall the definition of some well-known objects in the theory of locally analytic representations of $p$-adic reductive groups.
We fix a locally $\mathbb{Q}_p$-analytic group $H$ and denote the algebra of locally $\mathbb{Q}_p$-analytic distribution with coefficient $E$ on $H$ by $\mathcal{D}(H,E)$, which is defined as the strong dual of the locally convex $E$-vector space $C^{\rm{an}}(H,E)$ consisting of locally $\mathbb{Q}_p$-analytic functions on $H$. We use the notation $\mathrm{Rep}^{\rm{la}}_{H, E}$ (resp. $\mathrm{Rep}^{\infty}_{H, E}$) for the additve category consisting of locally $\mathbb{Q}_p$-analytic representations of $H$ (resp. smooth representations of $H$) with coefficient $E$. Therefore taking strong dual induces a fully faithful contravariant functor from $\mathrm{Rep}^{\rm{la}}_{H, E}$ to the abelian category $\mathrm{Mod}_{\mathcal{D}(H,E)}$ of abstract modules over $\mathcal{D}(H,E)$. The $E$-vector space $\mathrm{Ext}^i_{\mathcal{D}(H,E)}(M_1,M_2)$ is well-defined for any two objects $M_1, M_2\in \mathrm{Mod}_{\mathcal{D}(H,E)}$, and therefore we define $$\mathrm{Ext}^i_H(\Pi_1,\Pi_2):=\mathrm{Ext}^i_{\mathcal{D}(H,E)}(\Pi_2^{\prime},\Pi_1^{\prime})$$ for any two objects $\Pi_1, \Pi_2\in \mathrm{Rep}^{\rm{la}}_{H, E}$ where $\cdot^{\prime}$ is the notation for strong dual. We also define the cohomology of an object $M\in \mathrm{Mod}_{\mathcal{D}(H,E)}$ by $$H^i(H, M):=\mathrm{Ext}^i_{\mathcal{D}(H,E)}(1,M)$$ where $1$ is the strong dual of the trivial representation of $H$. If $H^{\prime}$ is a closed locally $\mathbb{Q}_p$-analytic normal subgroup of $H$, then $H/H^{\prime}$ is also a locally $\mathbb{Q}_p$-analytic group. It follows from the fact $$D(H, E)\otimes_{D(H^{\prime},E)}E\cong D(H/H^{\prime},E)$$ (see Section 5.1 of [@Bre17] for example) that $H^i(H^{\prime}, M)$ admits a structure of $\mathcal{D}(H/H^{\prime},E)$-module for each $M\in \mathrm{Mod}_{\mathcal{D}(H,E)}$. We define the $H^{\prime}$-homology of $\Pi\in\mathrm{Rep}^{\rm{la}}_{H, E}$ as the object (if it exists up to isomorphism) $H_i(H^{\prime}, \Pi)\in\mathrm{Rep}^{\rm{la}}_{H/H^{\prime}, E}$ such that $$H_i(H^{\prime}, \Pi)^{\prime}\cong H^i(H^{\prime}, \Pi^{\prime}).$$ We emphasize that $H_i(H^{\prime}, \Pi)$ is well defined in the sense above only after we know its existence. We fix a subgroup $Z$ of the center of the group $H$, then the algebra $\mathcal{D}(Z,E)$ consisting of locally $\mathbb{Q}_p$-analytic distribution with coefficient $E$ on $Z$ is naturally contained in the center of $\mathcal{D}(H,E)$. For each locally $\mathbb{Q}_p$-analytic $E$-character $\chi$ of $Z$, we can define the abelian subcategory $\mathrm{Mod}_{\mathcal{D}(H,E), \chi^{\prime}}$ consisting of all the objects in $\mathrm{Mod}_{\mathcal{D}(H,E)}$ on which $\mathcal{D}(Z,E)$ acts by $\chi^{\prime}$. Then we consider the functors $\mathrm{Ext}^i_{\mathcal{D}(H,E)}(-,-)$ defined as $\mathrm{Ext}^i_{\mathrm{Mod}_{\mathcal{D}(H,E),\chi^{\prime}}}(-,-)$ which are extensions inside the abelian category $\mathrm{Mod}_{\mathcal{D}(H,E),\chi^{\prime}}$. Consequently we can define $$\mathrm{Ext}^i_{H, \chi}(\Pi_1,\Pi_2):=\mathrm{Ext}^i_{\mathcal{D}(H,E), \chi^{\prime}}(\Pi_2^{\prime},\Pi_1^{\prime})$$ for any two objects $\Pi_1, \Pi_2\in \mathrm{Rep}^{\rm{la}}_{H, E}$ such that $\Pi_1^{\prime}, \Pi_2^{\prime} \in \mathrm{Mod}_{\mathcal{D}(H,E),\chi^{\prime}}$. In particular, if $Z$ is the center of $H$ and acts on $\Pi\in \mathrm{Rep}^{\rm{la}}_{H, E}$ via the character $\chi$, then $\Pi^{\prime}\in \mathrm{Mod}_{\mathcal{D}(H,E),\chi^{\prime}}$, and we usually say that $\Pi$ admits a central character $\chi$.
Assume now $H$ is the set of $\mathbb{Q}_p$-points of a split reductive group over $\mathbb{Q}_p$. We recall the category $\mathcal{O}$ together with its subcategory $\mathcal{O}^{\mathfrak{p}}_{\rm{alg}}$ for each parabolic subgroup $P\subseteq H$ from Section 9.3 of [@Hum08] or [@OS15]. The construction by Orlik–Strauch in [@OS15] gives us a functor $$\mathcal{F}_P^H : \mathcal{O}^{\mathfrak{p}}_{\rm{alg}}\times \mathrm{Rep}^{\infty}_{L, E}\rightarrow \mathrm{Rep}^{\rm{la}}_{H, E}$$ for each parabolic subgroup $P\subseteq H$ with Levi quotient $L$. We use the notation $\mathrm{Rep}^{\mathcal{OS}}_{H, E}$ for the abelian full subcategory of $\mathrm{Rep}^{\rm{la}}_{H, E}$ generated by the image of $\mathcal{F}_P^H$ when $P$ varies over all possible parabolic subgroups of $H$. Here we say a full subcategory is generated by a family of objects if it is the minimal full subcategory that contains these objects and is stable under extensions.
Formal properties {#3subsection: formal devissage}
-----------------
In this section, we recall and prove some general formal properties of locally analytic representations of $p$-adic reductive groups.
We fix a split $p$-adic reductive group $H$ and a parabolic subgroup $P$ of $H$. We use the notation $N$ for the unipotent radical of $P$ and fix a Levi subgroup $L$ of $P$.
\[3lemm: cohomology devissage\] We have a spectral sequece $$\mathrm{Ext}^j_{L,\ast}\left(H_k(N,~\Pi_1),~\Pi_2\right)\Rightarrow\mathrm{Ext}^{j+k}_{H,\ast}\left(\Pi_1,~\mathrm{Ind}_P^H\left(\Pi_2\right)^{\rm{an}}\right).$$ which implies an isomorphism $$\mathrm{Hom}_{L,\ast}\left(H_0(N,~\Pi_1),~\Pi_2\right)\xrightarrow{\sim}\mathrm{Hom}_{H,\ast}\left(\Pi_1,~\mathrm{Ind}_P^H\left(\Pi_2\right)^{\rm{an}}\right)$$ and a long exact sequence $$\begin{gathered}
\mathrm{Ext}^1_{L,\ast}\left(H_0(N,~\Pi_1),~\Pi_2\right)\hookrightarrow\mathrm{Ext}^1_{H,\ast}\left(\Pi_1,~\mathrm{Ind}_P^H\left(\Pi_2\right)^{\rm{an}}\right)\\
\rightarrow\mathrm{Hom}_{L,\ast}\left(H_1(N,~\Pi_1),~\Pi_2\right)\rightarrow\mathrm{Ext}^2_{L,\ast}\left(H_0(N,~\Pi_1),~\Pi_2\right)\end{gathered}$$ for each $\Pi_1\in \mathrm{Rep}^{\rm{la}}_{H, E}$, $\Pi_2\in \mathrm{Rep}^{\rm{la}}_{L, E}$ satisfying the ($\mathrm{FIN}$) condition in Section 6 of [@ST05], $\ast\in\{\varnothing, \chi\}$ where $\chi$ is a locally analytic character of the center of $H$.
This follows directly from our definition of $\mathrm{Ext}^k$ and $H_k$ in Section \[3subsection: locally analytic rep\] for $k\geq 0$, the original dual version in (44) and (45) of [@Bre17].
We fix a Borel subgroup $B\subseteq H$ together with its opposite Borel subgroup $\overline{B}$. We fix an irreducible object $M\in\mathcal{O}^{\overline{\mathfrak{b}}}_{\rm{alg}}$. We choose a parabolic subgroup $P\subseteq H$ such that $P$ is maximal among all the parabolic subgroups $Q\subseteq H$ such that $M\in\mathcal{O}^{\overline{\mathfrak{q}}}_{\rm{alg}}$ where $\overline{\mathfrak{q}}$ is the Lie algebra of the opposite parabolic subgroup $\overline{Q}$ associated with $Q$. We fix a smooth irreducible representation $\pi^{\infty}$ of $L$ and a smooth character $\delta$ of $H$. We know that [@OS15] constructed an irreducible locally analytic representation $$\mathcal{F}_P^H(M,~\pi^{\infty})$$ of $H$.
\[3lemm: det twist\] The functor $$-\otimes_E\delta$$ induces an equivalence of category from $\mathrm{Rep}^{\rm{la}}_{H, E}$ to itself. Moreover, the restriction of $-\otimes_E\delta$ to the subcategory $\mathrm{Rep}^{\mathcal{OS}}_{H, E}$ is again an equivalence of category to itself and satisfies $$\label{3smooth det twist}
\mathcal{F}_P^H(M,~\pi^{\infty})\otimes_E\delta\cong \mathcal{F}_P^H(M,~\pi^{\infty}\otimes_E\delta|_L)$$ for each irreducible object $\mathcal{F}_P^H(M,~\pi^{\infty})\in \mathrm{Rep}^{\mathcal{OS}}_{H, E}$.
The functor $-\otimes_E\delta$ is clearly an equivalence of category from $\mathrm{Rep}^{\rm{la}}_{H, E}$ to itself with quasi-inverse given by $$-\otimes_E\delta^{-1}.$$ It is sufficient to prove the formula (\[3smooth det twist\]) to finish the proof. First of all, we notice by formal reason (equivalence of category) that $\mathcal{F}_P^H(M,~\pi^{\infty})\otimes_E\delta$ is an irreducible object in $\mathrm{Rep}^{\rm{la}}_{H, E}$ since $\mathcal{F}_P^H(M,~\pi^{\infty})$ is. We use the notation $\overline{\mathfrak{n}}$ for the Lie algebra associated with the unipotent radical $\overline{N}$ of the opposite parabolic subgroup $\overline{P}$ of $P$. We define $M_L$ as the (finite dimensional) algebraic representation of $L$ whose dual is isomorphic to $M^{\overline{\mathfrak{n}}}$ as a representation of $\mathfrak{l}$ and note that we have a surjection $$U(\mathfrak{h})\otimes_{U(\overline{\mathfrak{p}})}M^{\overline{\mathfrak{n}}}\twoheadrightarrow M.$$ We observe that $N$ acts trivially on $\delta$, and therefore we have $$H_0\left(N,~\mathcal{F}_P^H(M,~\pi^{\infty})\otimes_E\delta\right)\cong H_0\left(N,~\mathcal{F}_P^H(M,~\pi^{\infty})\right)\otimes_E\delta|_L\twoheadrightarrow M_L\otimes_E\pi^{\infty}\otimes_E\delta|_L$$ which induces by Lemma \[3lemm: cohomology devissage\] a non-zero morphism $$\mathcal{F}_P^H(M,~\pi^{\infty})\otimes_E\delta\rightarrow\mathrm{Ind}_P^H\left(M_L\otimes_E\pi^{\infty}\otimes_E\delta|_L\right)^{\rm{an}}\cong \mathcal{F}_P^H(U(\mathfrak{h})\otimes_{U(\overline{\mathfrak{p}})}M^{\overline{\mathfrak{n}}},~\pi^{\infty}\otimes_E\delta|_L).$$ We finish the proof by the fact that $\mathcal{F}_P^H(M,~\pi^{\infty})\otimes_E\delta$ is irreducible and that $$\mathcal{F}_P^H(M,~\pi^{\infty}\otimes_E\delta|_L)\cong\mathrm{soc}_H\left(\mathcal{F}_P^H(U(\mathfrak{h})\otimes_{U(\overline{\mathfrak{p}})}M^{\overline{\mathfrak{n}}},~\pi^{\infty}\otimes_E\delta|_L)\right).$$ due to Corollary 3.3 of [@Bre16].
We fix a finite length locally analytic representation $V\in\mathrm{Rep}^{\rm{la}}_{H, E}$ equipped with a increasing filtration of subrepresentations $\{\mathrm{Fil}_kV\}_{0\leq k\leq m}$ such that $$\mathrm{Fil}_0(V)=0,~\mathrm{Fil}_m(V)=V\mbox{ and }\mathrm{gr}_{k+1}V:=\mathrm{Fil}_{k+1}V/\mathrm{Fil}_kV\neq 0\mbox{ for all }0\leq k\leq m-1.$$ Note that the assumption above automatically implies that $$\ell(V)\geq m$$ where $\ell(V)$ is the length of $V$.
\[3prop: formal devissages\] Assume that $W$ is another object of $\mathrm{Rep}^{\rm{la}}_{H, E}$ and $\chi$ is a locally analytic character of the center of $H$.
1. If $\mathrm{Ext}^1_{H,\chi}\left(W, ~\mathrm{gr}_kV\right)=0$ for each $1\leq k\leq m$, then we have $$\mathrm{Ext}^1_{H,\chi}\left(W, ~V\right)=0.$$
2. If there exists $1\leq k_0\leq m$ such that $\mathrm{Ext}^1_{H,\chi}\left(W, ~\mathrm{gr}_kV\right)=0$ for each $1\leq k\neq k_0\leq m$ and $\mathrm{dim}_E\mathrm{Ext}^1_{H,\chi}\left(W, ~\mathrm{gr}_{k_0}V\right)=1$, then we have $$\mathrm{dim}_E\mathrm{Ext}^1_{H,\chi}\left(W, ~V\right)\leq1;$$ if moreover $\mathrm{Ext}^2_{H,\chi}\left(W, ~\mathrm{gr}_kV\right)=0$ for each $1\leq k\leq k_0-1$ and $\mathrm{Hom}_{H,\chi}\left(W, ~\mathrm{gr}_kV\right)=0$ for each $k_0+1\leq k\leq m$, then we have $$\mathrm{dim}_E\mathrm{Ext}^1_{H,\chi}\left(W, V\right)=1.$$
The short exact sequence $\mathrm{Fil}_kV\hookrightarrow\mathrm{Fil}_{k+1}V\twoheadrightarrow\mathrm{gr}_{k+1}V$ induces a long exact sequence $$\mathrm{Ext}^1_{H,\chi}\left(W, ~\mathrm{Fil}_kV\right)\rightarrow\mathrm{Ext}^1_{H,\chi}\left(W, ~\mathrm{Fil}_{k+1}V\right)\rightarrow\mathrm{Ext}^1_{H,\chi}\left(W, ~\mathrm{gr}_{k+1}V\right)$$ which implies $$\mathrm{dim}_E\mathrm{Ext}^1_{H,\chi}\left(W, ~\mathrm{Fil}_{k+1}V\right)\leq \mathrm{dim}_E\mathrm{Ext}^1_{H,\chi}\left(W, ~\mathrm{Fil}_kV\right)+\mathrm{dim}_E\mathrm{Ext}^1_{H,\chi}\left(W, ~\mathrm{gr}_{k+1}V\right).$$ Therefore we finish the proof of part (i) and the first claim of part (ii) by induction on $k$ and the fact that $\mathrm{gr}_1V=\mathrm{Fil}_1V$.
It remains to show the second claim of part (ii). The same method as in the proof of part (i) shows that $$\label{3formal devissage1}
\mathrm{Ext}^1_{H,\chi}\left(W, ~\mathrm{Fil}_{k_0-1}V\right)=\mathrm{Ext}^2_{H,\chi}\left(W, ~\mathrm{Fil}_{k_0-1}V\right)=0$$ and $$\label{3formal devissage2}
\mathrm{Ext}^1_{H,\chi}\left(W, ~V/\mathrm{Fil}_{k_0}V\right)=\mathrm{Hom}_{H,\chi}\left(W, ~V/\mathrm{Fil}_{k_0}V\right)=0$$ The short exact sequence $\mathrm{Fil}_{k_0-1}V\hookrightarrow\mathrm{Fil}_{k_0}V\twoheadrightarrow\mathrm{gr}_{k_0}V$ induces the long exact sequence $$\begin{gathered}
\mathrm{Ext}^1_{H,\chi}\left(W, ~\mathrm{Fil}_{k_0-1}V\right)\rightarrow\mathrm{Ext}^1_{H,\chi}\left(W, ~\mathrm{Fil}_{k_0}V\right)
\rightarrow\mathrm{Ext}^1_{H,\chi}\left(W, ~\mathrm{gr}_{k_0}V\right)\rightarrow \mathrm{Ext}^2_{H,\chi}\left(W, ~\mathrm{Fil}_{k_0-1}V\right)\end{gathered}$$ which implies that $$\label{3formal devissage3}
\mathrm{dim}_E\mathrm{Ext}^1_{H,\chi}\left(W, ~\mathrm{Fil}_{k_0}V\right)=1$$ by (\[3formal devissage1\]). The short exact sequence $\mathrm{Fil}_{k_0}V\hookrightarrow V\twoheadrightarrow V/\mathrm{Fil}_{k_0}V$ induces the long exact sequence $$\begin{gathered}
\mathrm{Hom}_{H,\chi}\left(W, ~V/\mathrm{Fil}_{k_0}V\right)\rightarrow\mathrm{Ext}^1_{H,\chi}\left(W, ~\mathrm{Fil}_{k_0}V\right)
\rightarrow\mathrm{Ext}^1_{H,\chi}\left(W, ~V\right)\rightarrow\mathrm{Ext}^1_{H,\chi}\left(W, ~V/\mathrm{Fil}_{k_0}V\right)\end{gathered}$$ which finishes the proof by combining (\[3formal devissage2\]) and (\[3formal devissage3\]).
Some notation {#3subsection: main notation}
-------------
In this section, we are going to recall some standard notation for the $p$-adic reductive groups $\mathrm{GL}_2(\mathbb{Q}_p)$ and $\mathrm{GL}_3(\mathbb{Q}_p)$ as well as notation for some locally analytic representations of these groups.
We denote the lower-triangular Borel subgroup (resp. the diagonal maximal split torus) of $\mathrm{GL}_{2/\mathbb{Q}_p}$ by $B_2$ (resp. by $T_2$) and the unipotent radical of $B_2$ by $N_{\mathrm{GL}_2}$. We use the notation $s$ for the non-trivial element in the Weyl group of $\mathrm{GL}_2$. We fix a weight $\nu\in X(T_2)$ of $\mathrm{GL}_2$ of the following form $$\nu=(\nu_1,\nu_2)\in\mathbb{Z}^2$$ which corresponds to an algebraic character of $T_2(\mathbb{Q}_p)$ $$\delta_{T_2,\nu}:=\left(\begin{array}{cc}
a&0\\
0&b\\
\end{array}\right)\mapsto a^{\nu_1}b^{\nu_2}.$$ We denote the upper-triangular Borel subgroup by $\overline{B_2}$. If $\nu$ is dominant with respect to $\overline{B_2}$, namely if $\nu_1\geq \nu_2$, we use the notation $\overline{L}_{\mathrm{GL}_2}(\nu)$ (resp. $L_{\mathrm{GL}_2}(-\nu)$) for the irreducible algebraic representation of $\mathrm{GL}_2(\mathbb{Q}_p)$ with highest weight $\nu$ (resp. $-\nu$) with respect to the positive roots determined by $\overline{B_2}$ (resp. $B_2$). In particular, $\overline{L}_{\mathrm{GL}_2}(\nu)$ and $L_{\mathrm{GL}_2}(-\nu)$ are the dual of each other. We use the shorten notation $$I^{\mathrm{GL}_2}_{B_2}(\chi_{T_2}):=\left(\mathrm{Ind}_{B_2(\mathbb{Q}_p)}^{\mathrm{GL}_2(\mathbb{Q}_p)}\chi_{T_2}\right)^{\rm{an}}$$ for any locally analytic character $\chi_{T_2}$ of $T_2(\mathbb{Q}_p)$ and set $$i^{\mathrm{GL}_2}_{B_2}(\chi_{T_2}):=\left(\mathrm{Ind}_{B_2(\mathbb{Q}_p)}^{\mathrm{GL}_2(\mathbb{Q}_p)}\chi^{\infty}_{T_2}\right)^{\infty}\otimes_E \overline{L}_{\mathrm{GL}_2}(\nu)$$ if $\chi_{T_2}=\delta_{T_2,\nu}\otimes_E\chi^{\infty}_{T_2}$ is locally algebraic where $\chi^{\infty}_{T_2}$ is a smooth character of $T_2(\mathbb{Q}_p)$. Then we define the locally analytic Steinberg representation as well as the smooth Steinberg representation for $\mathrm{GL}_2(\mathbb{Q}_p)$ as follows $$\mathrm{St}^{\rm{an}}_2(\nu):=I^{\mathrm{GL}_2}_{B_2}(\delta_{T_2,\mu})/\overline{L}_{\mathrm{GL}_2}(\nu), ~ \mathrm{St}^{\infty}_2:=i^{\mathrm{GL}_2}_{B_2}(1_{T_2})/1_2$$ where $1_2$ (resp. $1_{T_2}$) is the trivial representation of $\mathrm{GL}_2(\mathbb{Q}_p)$ (resp. of $T_2(\mathbb{Q}_p)$).
We denote the lower-triangular Borel subgroup (resp. the diagonal maximal split torus) of $\mathrm{GL}_{3/\mathbb{Q}_p}$ by $B$ (resp. by $T$) and the unipotent radical of $B$ by $N$. We fix a weight $\lambda\in X(T)$ of $\mathrm{GL}_3$ of the following form $$\lambda=(\lambda_1,\lambda_2, \lambda_3)\in\mathbb{Z}^3,$$ which corresponds to an algebraic character of $T(\mathbb{Q}_p)$ $$\delta_{T,\lambda}:=\left(\begin{array}{ccc}
a&0&0\\
0&b&0\\
0&0&c\\
\end{array}\right)\mapsto a^{\lambda_1}b^{\lambda_2}c^{\lambda_3}.$$ We denote the center of $\mathrm{GL}_3$ by $Z$ and notice that $Z(\mathbb{Q}_p)\cong\mathbb{Q}_p^{\times}$. Hence the restriction of $\delta_{T,\lambda}$ to $Z(\mathbb{Q}_p)$ gives an algebraic character of $Z(\mathbb{Q}_p)$: $$\delta_{Z,\lambda}:=\left(\begin{array}{ccc}
a&0&0\\
0&a&0\\
0&0&a\\
\end{array}\right)\mapsto a^{\lambda_1+\lambda_2+\lambda_3}.$$ We use the shorten notation $$\mathrm{Ext}^i_{\ast, \lambda}(-,-):=\mathrm{Ext}^i_{\ast, \delta_{Z,\lambda}}(-,-)$$ for $\ast\in\{T(\mathbb{Q}_p), L_1(\mathbb{Q}_p), L_2(\mathbb{Q}_p), \mathrm{GL}_3(\mathbb{Q}_p)\}.$ In particular, the notation $$\mathrm{Ext}^i_{\ast, 0}(-,-)$$ means (higher) extensions with the trivial central character. We denote the upper-triangular Borel subgroup of $\mathrm{GL}_3$ by $\overline{B}$. If $\lambda$ is dominant with respect to $\overline{B}$, namely if $\lambda_1\geq \lambda_2\geq\lambda_3$, we use the notation $\overline{L}(\lambda)$ (resp. $L(-\lambda)$) for the irreducible algebraic representation of $\mathrm{GL}_3(\mathbb{Q}_p)$ with highest weight $\lambda$ (resp. $-\lambda$) with respect to the positive roots determined by $\overline{B}$ (resp. $B$). In particular, $\overline{L}(\lambda)$ and $L(-\lambda)$ are dual of each other. We use the notation $P_1:=\left(\begin{array}{ccc}
\ast&\ast&0\\
\ast&\ast&0\\
\ast&\ast&\ast\\
\end{array}\right)$ and $P_2:=\left(\begin{array}{ccc}
\ast&0&0\\
\ast&\ast&\ast\\
\ast&\ast&\ast\\
\end{array}\right)$ for the two standard maximal parabolic subgroups of $\mathrm{GL}_3$ with unipotent radical $N_1$ and $N_2$ respectively, and the notation $\overline{P_i}$ for the opposite parabolic subgroup of $P_i$ for $i=1,2$. We set $$L_i:=P_i\cap \overline{P_i}$$ and set $s_i$ for the simple reflection in the Weyl group of $L_i$ for each $i=1,2$. In particular, the Weyl group $W$ of $\mathrm{GL}_3$ can be lifted to a subgroup of $\mathrm{GL}_3$ with the following elements $$\{1, s_1, s_2, s_1s_2, s_2s_1, s_1s_2s_1\}.$$ We will usually use the shorten notation $N_i$ ( cf. Section \[3section: computation\]) for its set of $\mathbb{Q}_p$-points $N_i(\mathbb{Q}_p)$ if it does not cause any ambiguity. We use the notation $M(-\lambda)$ for the Verma module in $\mathcal{O}^{\mathfrak{b}}_{\rm{alg}}$ with highest weight $-\lambda$ (with respect to $B$) and simple quotient $L(-\lambda)$ for each $\lambda\in X(T)$ (not necessarily dominant). Similarly, we use the notation $M_i(-\lambda)$ for the parabolic Verma module in $\mathcal{O}^{\mathfrak{p}_i}_{\rm{alg}}$ with highest weight $-\lambda$ with respect to $B$ ( cf. Section 9.4 of [@Hum08]). We define $\overline{L}_i(\lambda)$ as the irreducible algebraic representation of $L_i(\mathbb{Q}_p)$ with a highest weight $\lambda$ dominant with respect to $\overline{B}\cap L_i$. For example, if $\lambda\in X(T)_+$, then we know that $\lambda$, $s_i\cdot\lambda$ and $s_is_{3-i}\cdot\lambda$ are dominant with respect to $\overline{B}\cap L_{3-i}$ for $i=1,2$. We use the following notation for various parabolic inductions $$I^{\mathrm{GL}_3}_{B}(\chi):=\left(\mathrm{Ind}_{B(\mathbb{Q}_p)}^{\mathrm{GL}_3(\mathbb{Q}_p)}\chi\right)^{\rm{an}}, ~I^{\mathrm{GL}_3}_{P_i}(\pi_i):=\left(\mathrm{Ind}_{P_i(\mathbb{Q}_p)}^{\mathrm{GL}_3(\mathbb{Q}_p)}\pi_i\right)^{\rm{an}}$$ if $\chi$ is an arbitrary locally analytic character of $T(\mathbb{Q}_p)$ and $\pi_i$ is an arbitrary locally analytic representation of $L_i(\mathbb{Q}_p)$ for each $i=1,2$. Moreover, we use the notation $$i^{\mathrm{GL}_3}_{B}(\chi):=\left(\mathrm{Ind}_{B(\mathbb{Q}_p)}^{\mathrm{GL}_3(\mathbb{Q}_p)}\chi^{\infty}\right)^{\infty}\otimes_E \overline{L}(\lambda), ~i^{\mathrm{GL}_3}_{P_i}(\pi_i):=\left(\mathrm{Ind}_{P_i(\mathbb{Q}_p)}^{\mathrm{GL}_3(\mathbb{Q}_p)}\pi_i^{\infty}\right)^{\infty}\otimes_E \overline{L}(\lambda)$$ for $i=1,2$ if $\chi=\delta_{T,\lambda}\otimes_E\chi^{\infty}$ and $\pi_i=\overline{L}_i(\lambda)\otimes_E\pi_i^{\infty}$ are locally algebraic where $\chi^{\infty}$ (resp. $\pi_i^{\infty}$) is a smooth representation of $T(\mathbb{Q}_p)$ (resp. of $L_i(\mathbb{Q}_p)$). We will also use similar notation for parabolic induction to Levi subgroups such as $I_{B\cap L_i}^{L_i}$ and $i_{B\cap L_i}^{L_i}$ for $i=1,2$. Then we define the locally analytic (generalized) Steinberg representation as well as the smooth (generalized) Steinberg representation for $\mathrm{GL}_3(\mathbb{Q}_p)$ by $$\mathrm{St}^{\rm{an}}_3(\lambda):=I^{\mathrm{GL}_3}_{B}(\delta_{T,\lambda})/\left(I^{\mathrm{GL}_3}_{P_1}(\overline{L}_1(\lambda))+I^{\mathrm{GL}_3}_{P_2}(\overline{L}_2(\lambda))\right), ~ \mathrm{St}^{\infty}_3:=i^{\mathrm{GL}_3}_{B}(1)/\left(i^{\mathrm{GL}_3}_{P_1}(1_{L_1})+i^{\mathrm{GL}_3}_{P_2}(1_{L_2})\right)$$ and $$v^{\rm{an}}_{P_i}(\lambda):=I^{\mathrm{GL}_3}_{P_i}(\overline{L}_i(\lambda))/\overline{L}(\lambda), ~ v^{\infty}_{P_i}:=i^{\mathrm{GL}_3}_{P_i}(1_{L_i})/1_3$$ where $1_3$ (resp. $1_{L_i}$) is the trivial representation of $\mathrm{GL}_3(\mathbb{Q}_p)$ (resp. of $L_i(\mathbb{Q}_p)$ for each $i=1,2$). We define the following smooth representations of $L_1(\mathbb{Q}_p)$: $$\begin{array}{ccc}
\pi_{1,1}^{\infty}&:=&\mathrm{St}_2^{\infty}\otimes_E1\\
\pi_{1,2}^{\infty}&:=&i_{B_2}^{\mathrm{GL}_2}\left(1\otimes_E|\cdot|^{-1}\right)\otimes_E|\cdot|\\
\pi_{1,3}^{\infty}&:=&\left(\mathrm{St}_2^{\infty}\otimes_E(|\cdot|^{-1}\circ\mathrm{det}_2)\right)\otimes_E|\cdot|^2\\
\end{array}$$ and the following smooth representations of $L_2(\mathbb{Q}_p)$: $$\begin{array}{ccc}
\pi_{2,1}^{\infty}&:=&1\otimes_E\mathrm{St}_2^{\infty}\\
\pi_{2,2}^{\infty}&:=&|\cdot|^{-1}\otimes_E i_{B_2}^{\mathrm{GL}_2}\left(|\cdot|\otimes_E1\right)\\
\pi_{2,3}^{\infty}&:=&|\cdot|^{-2}\otimes_E\left(\mathrm{St}_2^{\infty}\otimes_E(|\cdot|\circ\mathrm{det}_2)\right)\\
\end{array}$$ Consequently, we can define the following locally analytic representations for $i=1,2$: $$\label{3irr rep I}
\begin{array}{cccccc}
C^1_{s_i,1}&:=&\mathcal{F}^{\mathrm{GL}_3}_{P_{3-i}}\left(L(-s_i\cdot\lambda),~1_{L_{3-i}}\right)&C^2_{s_i,1}&:=&\mathcal{F}^{\mathrm{GL}_3}_{P_{3-i}}\left(L(-s_i\cdot\lambda),~\pi_{i,1}^{\infty}\right)\\
C^1_{s_is_{3-i},1}&:=&\mathcal{F}^{\mathrm{GL}_3}_{P_{3-i}}\left(L(-s_is_{3-i}\cdot\lambda),~1_{L_{3-i}}\right)&C^2_{s_is_{3-i},1}&:=&\mathcal{F}^{\mathrm{GL}_3}_{P_{3-i}}\left(L(-s_is_{3-i}\cdot\lambda),~\pi_{i,1}^{\infty}\right)\\
C_{s_i,s_i}&:=&\mathcal{F}^{\mathrm{GL}_3}_{P_{3-i}}\left(L(-s_i\cdot\lambda),~\pi_{i,2}^{\infty}\right)&C_{s_is_{3-i},s_i}&:=&\mathcal{F}^{\mathrm{GL}_3}_{P_{3-i}}\left(L(-s_is_{3-i}\cdot\lambda),~\pi_{i,2}^{\infty}\right)\\
C^1_{s_i,s_is_{3-i}}&:=&\mathcal{F}^{\mathrm{GL}_3}_{P_{3-i}}\left(L(-s_i\cdot\lambda),~\mathfrak{d}_{P_{3-i}}^{\infty}\right)&C^2_{s_1,1}&:=&\mathcal{F}^{\mathrm{GL}_3}_{P_{3-i}}\left(L(-s_i\cdot\lambda),~\pi_{i,3}^{\infty}\right)\\
C^1_{s_is_{3-i},s_is_{3-i}}&:=&\mathcal{F}^{\mathrm{GL}_3}_{P_{3-i}}\left(L(-s_is_{3-i}\cdot\lambda),~\mathfrak{d}_{P_{3-i}}^{\infty}\right)&C^2_{s_is_{3-i},1}&:=&\mathcal{F}^{\mathrm{GL}_3}_{P_{3-i}}\left(L(-s_is_{3-i}\cdot\lambda),~\pi_{i,3}^{\infty}\right)
\end{array}$$ where $$\mathfrak{d}_{P_1}^{\infty}:=|\cdot|^{-1}\circ\mathrm{det}_2\otimes_E|\cdot|^2\mbox{ and }\mathfrak{d}_{P_2}^{\infty}:=|\cdot|^{-2}\otimes_E|\cdot|\circ\mathrm{det}_2.$$ We also define $$\label{3irr rep II}
C_{s_1s_2s_1,w}:=\mathcal{F}^{\mathrm{GL}_3}_B\left(L(-s_1s_2s_1\cdot\lambda),~\chi_w^{\infty}\right)$$ for each $w\in W$ where $$\begin{xy}
(0,0)*+{\chi_1^{\infty}}; (6,0)*+{:=}; (12,0)*+{1_T}; (50,0)*+{\chi_{s_1}^{\infty}}; (56,0)*+{:=}; (73,0)*+{|\cdot|^{-1}\otimes_E|\cdot|\otimes_E1}; (105,0)*+{\chi_{s_2}^{\infty}}; (112,0)*+{:=}; (130,0)*+{1\otimes_E|\cdot|^{-1}\otimes_E|\cdot|}; (0,-6)*+{\chi_{s_1s_2}^{\infty}}; (6,-6)*+{:=}; (25,-6)*+{|\cdot|^{-2}\otimes_E|\cdot|\otimes_E|\cdot|}; (50,-6)*+{\chi_{s_2s_1}^{\infty}}; (56,-6)*+{:=}; (77,-6)*+{|\cdot|^{-1}\otimes_E|\cdot|^{-1}\otimes_E|\cdot|^2}; (105,-6)*+{\chi_{s_1s_2s_1}^{\infty}}; (112,-6)*+{:=}; (130,-6)*+{|\cdot|^{-2}\otimes_E1\otimes_E|\cdot|^2};
\end{xy}$$ We notice that the representations considered in (\[3irr rep I\]) and (\[3irr rep II\]) are all irreducible objects inside $\mathrm{Rep}^{\mathcal{OS}}_{\mathrm{GL}_3(\mathbb{Q}_p), E}$ according to the main theorem of [@OS15]. We use the notation $\Omega$ for the set whose elements are listed as the following: $$\begin{array}{cccc}
\overline{L}(\lambda)&\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}&\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}&\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\\
C^1_{s_1,1}&C^2_{s_1,1}&C^1_{s_2,1}&C^2_{s_2,1}\\
C^1_{s_1s_2,1}&C^2_{s_1s_2,1}&C^1_{s_2s_1,1}&C^2_{s_2s_1,1}\\
C^1_{s_1,s_1s_2}&C^2_{s_1,s_1s_2}&C^1_{s_2,s_2s_1}&C^2_{s_2,s_2s_1}\\
C^1_{s_1s_2,s_1s_2}&C^2_{s_1s_2,s_1s_2}&C^1_{s_2s_1,s_2s_1}&C^2_{s_2s_1,s_2s_1}\\
C_{s_1,s_1}&C_{s_1s_2,s_1}&C_{s_2,s_2}&C_{s_2s_1,s_2}\\
C_{s_1s_2s_1,w}&w\in W&&
\end{array}$$
It is actually possible to show that $\Omega$ is the set of (isomorphism classes of) irreducible objects of the block inside $\mathrm{Rep}^{\mathcal{OS}}_{\mathrm{GL}_3(\mathbb{Q}_p), E}$ containing the object $\overline{L}(\lambda)$.
\[3lemm: structure of St\] The representation $v^{\rm{an}}_{P_i}(\lambda)$ fits into a non-split extension $$\label{3generalized St picture}
\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}\hookrightarrow v^{\rm{an}}_{P_i}(\lambda)\twoheadrightarrow C^1_{s_{3-i},1}$$ for $i=1,2$. On the other hand, the representation $\mathrm{St}^{\rm{an}}_3(\lambda)$ has the following form: $$\label{3St picture}
\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="a"; (20,6)*+{C^2_{s_1,1}}="b"; (20,-6)*+{C^2_{s_2,1}}="c"; (40,6)*+{C^1_{s_2s_1,1}}="d"; (40,-6)*+{C^1_{s_1s_2,1}}="e"; (60,6)*+{C^2_{s_2s_1,1}}="f"; (60,-6)*+{C^2_{s_1s_2,1}}="g"; (80,0)*+{C_{s_1s_2s_1,1}}="h";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"e"}; {\ar@{-}"a";"b"}; {\ar@{-}"d";"f"}; {\ar@{-}"e";"g"}; {\ar@{-}"b";"g"}; {\ar@{-}"c";"f"}; {\ar@{--}"f";"h"}; {\ar@{--}"g";"h"};
\end{xy}.$$
The non-split short exact sequence follows directly from (3.62) of [@BD18]. It follows easily from the definition of $\mathrm{St}^{\rm{an}}_3(\lambda)$ that $$\mathrm{JH}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(\mathrm{St}^{\rm{an}}_3(\lambda)\right)=\{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty},~C^2_{s_1,1},~C^2_{s_2,1},~C^1_{s_2s_1,1},~C^1_{s_1s_2,1},~C^2_{s_2s_1,1},~C^2_{s_1s_2,1},~C_{s_1s_2s_1,1}\}$$ and each Jordan–Hölder factor occurs with multiplicity one. It follows from Section 5.2 of [@Bre17] that $$H_0\left(N_i,~\mathcal{F}^{\mathrm{GL}_3}_{P_i}\left(L(-s_{3-i}s_i\cdot\lambda),~i_{B\cap L_i}^{L_i}(1_T)\right)\right)=\overline{L}_i(-s_{3-i}s_i\cdot\lambda)\otimes_Ei_{B\cap L_i}^{L_i}(1_T)$$ which together with $$\mathrm{JH}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(\mathcal{F}^{\mathrm{GL}_3}_{P_i}\left(L(-s_{3-i}s_i\cdot\lambda),~i_{B\cap L_i}^{L_i}(1_T)\right)\right)=\{C^1_{s_{3-i}s_i,1},~C^2_{s_{3-i}s_i,1}\}$$ imply that $\mathcal{F}^{\mathrm{GL}_3}_{P_i}\left(L(-s_{3-i}s_i\cdot\lambda),~i_{B\cap L_i}^{L_i}(1_T)\right)$ fits into a non-split extension $$\label{3non split ext in St}
C^1_{s_{3-i}s_i,1}\hookrightarrow\mathcal{F}^{\mathrm{GL}_3}_{P_i}\left(L(-s_{3-i}s_i\cdot\lambda),~i_{B\cap L_i}^{L_i}(1_T)\right)\twoheadrightarrow C^2_{s_{3-i}s_i,1}$$ for $i=1,2$. We also observe from Section 5.2 and 5.3 of [@Bre17] that $$H_2\left(N_{3-i},~\mathcal{F}^{\mathrm{GL}_3}_{P_i}\left(M_i(-s_{3-i}\cdot\lambda),~\pi_{i,1}^{\infty}\right)\right)\not\cong H_2(N_{3-i},~C^2_{s_{3-i},1})\oplus H_2(N_{3-i},~C^2_{s_{3-i}s_i,1})$$ which together with $$\mathrm{JH}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(\mathcal{F}^{\mathrm{GL}_3}_{P_i}\left(M_i(-s_{3-i}\cdot\lambda),~\pi_{i,1}^{\infty}\right)\right)=\{C^2_{s_{3-i},1},~C^2_{s_{3-i}s_i,1}\}$$ imply that $\mathcal{F}^{\mathrm{GL}_3}_{P_i}\left(M_i(-s_{3-i}\cdot\lambda),~\pi_{i,1}^{\infty}\right)$ fits into a non-split extension $$\label{3non split ext in St prime}
C^2_{s_{3-i},1}\hookrightarrow\mathcal{F}^{\mathrm{GL}_3}_{P_i}\left(M_i(-s_{3-i}\cdot\lambda),~\pi_{i,1}^{\infty}\right)\twoheadrightarrow C^2_{s_{3-i}s_i,1}$$ for $i=1,2$. We notice that both $\mathcal{F}^{\mathrm{GL}_3}_{P_i}\left(L(-s_{3-i}s_i\cdot\lambda),~i_{B\cap L_i}^{L_i}(1_T)\right)$ and $\mathcal{F}^{\mathrm{GL}_3}_{P_i}\left(M_i(-s_{3-i}\cdot\lambda),~\pi_{i,1}^{\infty}\right)$ are subquotients of $\mathrm{St}^{\rm{an}}_3(\lambda)$ by various properties of the functors $\mathcal{F}^{\mathrm{GL}_3}_{P_i}$ ( cf. main theorem of [@OS15]) and the definition of $\mathrm{St}^{\rm{an}}_3(\lambda)$. We finish the proof by combining (\[3non split ext in St\]) and (\[3non split ext in St prime\]) with the results before Remark 3.38 of [@BD18].
It is actually possible to show that all the possibly non-split extensions indicated in (\[3St picture\]) are non-split, although they are essentially unrelated to the $p$-adic dilogarithm function.
$p$adic logarithm and dilogarithm {#3subsection: log dilog}
---------------------------------
In this section, we recall $p$adic logarithm and dilogarithm function as well as their representation theoretic interpretations.
We recall the $p$adic logarithm function $\mathrm{log}_0: \mathbb{Q}_p^{\times}\rightarrow \mathbb{Q}_p$ defined by power series on a open subgroup of $\mathbb{Z}_p^{\times}$ and then extended to $\mathbb{Q}_p^{\times}$ by the condition $\mathrm{log}_0(p)=0$. We also recall the $p$adic valuation function $\mathrm{val}_p: \mathbb{Q}_p^{\times}\rightarrow \mathbb{Z}$ satisfying $|\cdot|=p^{-\mathrm{val}_p(\cdot)}$ (and in particular $\mathrm{val}_p(p)=1$). We notice that $$\{\mathrm{log}_0,~\mathrm{val}_p\}$$ forms a basis of the two dimensional $E$vector space $$\mathrm{Hom}_{\rm{cont}}\left(\mathbb{Q}_p^{\times},~E\right).$$ We define $\mathrm{log}_{\mathscr{L}}:=\mathrm{log}_0-\mathscr{L}\mathrm{val}_p$ for each $\mathscr{L}\in E$ and consider the following two dimensional locally analytic representation of $\mathbb{Q}_p^{\times}$ $$V_{\mathscr{L}}: \mathbb{Q}_p^{\times}\rightarrow B_2(E), ~a\mapsto
\left(\begin{array}{cc}
1&\mathrm{log}_{\mathscr{L}}(a)\\
0&1\\
\end{array}\right)$$ and therefore $$\label{3extension of two trivial reps}
\mathrm{soc}_{\mathbb{Q}_p^{\times}}(V_{\mathscr{L}})=\mathrm{cosoc}_{\mathbb{Q}_p^{\times}}(V_{\mathscr{L}})=1$$ where $1$ is the notation for the trivial character of $\mathbb{Q}_p^{\times}$. We notice that $$\mathrm{Ext}^1_{\mathbb{Q}_p^{\times}}(1,1)\cong \mathrm{Hom}_{\rm{cont}}\left(\mathbb{Q}_p^{\times},~E\right),$$ by a standard fact in (continuous) group cohomology and therefore the set $\{V_{\mathscr{L}}\mid \mathscr{L}\in E\}$ exhausts (up to isomorphism) all different two dimensional locally analytic non-smooth $E$-representations of $\mathbb{Q}_p^{\times}$ satisfying (\[3extension of two trivial reps\]). We observe that $V_{\mathscr{L}}$ can be viewed as a representation of $T_2(\mathbb{Q}_p)\cong\mathbb{Q}_p^{\times}\times\mathbb{Q}_p^{\times}$ by composing with the map $$\label{3easy map}
T_2(\mathbb{Q}_p)\rightarrow \mathbb{Q}_p^{\times}: ~\left(\begin{array}{cc}
a&0\\
0&b\\
\end{array}\right) \mapsto a^{-1}b.$$ As a result, we can consider the parabolic induction $$I_{B_2}^{\mathrm{GL}_2}\left(V_{\mathscr{L}}\otimes_E\delta_{T_2,\nu}\right)$$ which naturally fits into an exact sequence $$\label{3non split self extension}
I_{B_2}^{\mathrm{GL}_2}(\delta_{T_2,\nu})\hookrightarrow I_{B_2}^{\mathrm{GL}_2}\left(V_{\mathscr{L}}\otimes_E\delta_{T_2,\nu}\right)\twoheadrightarrow I_{B_2}^{\mathrm{GL}_2}(\delta_{T_2,\nu}).$$ Then we define $\Sigma_{\mathrm{GL}_2}(\nu,\mathscr{L})$ as the subrepresentation of $I_{B_2}^{\mathrm{GL}_2}\left(V_{\mathscr{L}}\otimes_E\delta_{T_2,\nu}\right)/\overline{L}_{\mathrm{GL}_2}(\nu)$ with cosocle $\overline{L}_{\mathrm{GL}_2}(\nu)$. It follows from (the proof of) Theorem 3.14 of [@BD18] that $\Sigma_{\mathrm{GL}_2}(\nu,\mathscr{L})$ has the form $$\label{3uniserial GL2}
\begin{xy}
(0,0)*+{\mathrm{St}^{\rm{an}}_2(\nu)}="a"; (20,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="b";
{\ar@{-}"a";"b"};
\end{xy}$$ and the set $\{\Sigma_{\mathrm{GL}_2}(\nu,\mathscr{L})\mid \mathscr{L}\in E\}$ exhausts (up to isomorphism) all different locally analytic $E$-representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ of the form (\[3uniserial GL2\]) that do not contain $$\begin{xy}
(0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}^{\infty}_2}="a"; (26,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="b";
{\ar@{-}"a";"b"};
\end{xy}$$ as a subrepresentation. We have the embeddings $$\iota_i: \mathrm{GL}_2\hookrightarrow L_i$$ for $i=1,2$ by identifying $\mathrm{GL}_2$ with a Levi block of $L_i$, which induce the embeddings $$\iota_{T,i}: T_2\hookrightarrow T$$ by restricting $\iota_i$ to $T_2\subsetneq\mathrm{GL}_2$. We use the notation $\iota_{T,i}(V_{\mathscr{L}})$ for the locally analytic representation of $T(\mathbb{Q}_p)\cong (\mathbb{Q}_p^{\times})^3$ which is $V_{\mathscr{L}}$ after restricting to $T_2$ via $\iota_{T,i}$ and is trivial after restricting to the other copy of $\mathbb{Q}_p^{\times}$. By a direct analogue of $\Sigma_{\mathrm{GL}_2}(\nu,\mathscr{L})$, we can construct $\Sigma_{L_i}(\lambda,\mathscr{L})$ as the subrepresentation of $I_{B\cap L_i}^{L_i}\left(\iota_{T,i}(V_{\mathscr{L}})\otimes_E\delta_{T,\lambda}\right)/\overline{L}_i(\lambda)$ with cosocle $\overline{L}_i(\lambda)$. In fact, if we have $\lambda|_{T_2, \iota_{T,i}}=\nu$, then we obviously know that $\Sigma_{L_i}(\lambda,\mathscr{L})|_{\mathrm{GL}_2, \iota_i}\cong \Sigma_{\mathrm{GL}_2}(\nu,\mathscr{L})$ where the notation $(\cdot)|_{\ast, \star}$ means the restriction of $\cdot$ to $\ast$ via the embedding $\star$. We observe that the parabolic induction $I_{P_i}^{\mathrm{GL}_3}\left(\Sigma_{L_i}(\lambda,\mathscr{L})\right)$ fits into the exact sequence $$\begin{xy}
(0,0)*+{v_{P_{3-i}}^{\rm{an}}(\lambda)}="a"; (20,0)*+{\mathrm{St}_3^{\rm{an}}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\hookrightarrow I_{P_i}^{\mathrm{GL}_3}\left(\Sigma_{L_i}(\lambda,\mathscr{L})\right)\twoheadrightarrow
\begin{xy}
(0,0)*+{\overline{L}(\lambda)}="a"; (16,0)*+{v_{P_i}^{\rm{an}}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}.$$ According to Proposition 5.6 of [@Schr11] for example, we know that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\mathrm{St}_3^{\rm{an}}(\lambda)\right)=0$$ and thus we can define $\Sigma_i(\lambda, \mathscr{L})$ as the unique quotient of $I_{P_i}^{\mathrm{GL}_3}\left(\Sigma_{L_i}(\lambda,\mathscr{L})\right)$ that fits into the exact sequence $$\mathrm{St}_3^{\rm{an}}(\lambda)\hookrightarrow \Sigma_i(\lambda, \mathscr{L})\twoheadrightarrow v_{P_i}^{\rm{an}}(\lambda).$$ The constructions of $\Sigma_i(\lambda, \mathscr{L})$ above actually induce canonical isomorphisms $$\label{3canonical isomorphism}
\mathrm{Hom}_{\rm{cont}}\left(\mathbb{Q}_p^{\times},~E\right)\cong\mathrm{Ext}^1_{\mathbb{Q}_p^{\times}}\left(1,~1\right)\xrightarrow{\sim}\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(v_{P_i}^{\rm{an}}(\lambda), ~\mathrm{St}_3^{\rm{an}}(\lambda)\right)$$ for $i=1,2$. We denote the image of $\mathrm{log}_0$ (resp, of $\mathrm{val}_p$) in $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(v_{P_i}^{\rm{an}}(\lambda), ~\mathrm{St}_3^{\rm{an}}(\lambda)\right)$$ by $b_{i,\mathrm{log}_0}$ (resp. by $b_{i,\mathrm{val}_p}$). We use the notation $1_T$ for the trivial character of $T(\mathbb{Q}_p)$. We use the same notation $b_{i,\mathrm{log}_0}$ and $b_{i,\mathrm{val}_p}$ for the image of $\mathrm{log}_0$ and $\mathrm{val}_p$ respectively under the embedding $$\mathrm{Ext}^1_{\mathbb{Q}_p^{\times}}\left(1,~1\right)\hookrightarrow\mathrm{Ext}^1_{T(\mathbb{Q}_p),0}\left(1_T,~1_T\right)$$ induced by the maps $$T(\mathbb{Q}_p)\xrightarrow{p_i} T_2(\mathbb{Q}_p)\xrightarrow{(\ref{3easy map})}\mathbb{Q}_p^{\times}$$ where $p_i$ is the section of $\iota_{T,i}$ which is compatible with the projection $L_i\twoheadrightarrow\mathrm{GL}_2$. Recall the elements $c_{i,\mathrm{log}}, c_{i,\mathrm{val}}\in \mathrm{Ext}^1_{T(\mathbb{Q}_p),0}(1_T,1_T)$ constructed after (5.24) of [@Schr11] and observe that $$\label{3relation of basis}
\left\{\begin{array}{ccc}
c_{1,\mathrm{log}}=b_{1,\mathrm{log}_0}+2b_{2,\mathrm{log}_0},&c_{1,\mathrm{val}}=b_{1,\mathrm{val}_p}+2b_{2,\mathrm{val}_p}&\\
c_{2,\mathrm{log}}=2b_{1,\mathrm{log}_0}+b_{2,\mathrm{log}_0},&c_{2,\mathrm{val}}=2b_{1,\mathrm{val}_p}+b_{2,\mathrm{val}_p}&.\\
\end{array}\right.$$ We notice that there exists canonical surjections $$\label{3canonical surjection for L invariant}
\mathrm{Ext}^1_{T(\mathbb{Q}_p),0}\left(1_T,~1_T\right)\twoheadrightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(v_{P_i}^{\rm{an}}(\lambda), ~\mathrm{St}_3^{\rm{an}}(\lambda)\right)$$ with kernel spanned by $\{c_{i, \mathrm{log}},~c_{i, \mathrm{val}}\}$, according to (5.70) and (5.71) of [@Schr11]. Therefore the relation (\[3relation of basis\]) reduces via the surjection (\[3canonical surjection for L invariant\]) to $$\label{3relation of basis prime}
c_{3-i,\mathrm{log}}=-3b_{i,\mathrm{log}_0},~c_{3-i,\mathrm{val}}=-3b_{i,\mathrm{val}_p}$$ inside the quotient $\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(v_{P_i}^{\rm{an}}(\lambda), ~\mathrm{St}_3^{\rm{an}}(\lambda)\right)$. We define $\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ as the amalgamate sum of $\Sigma_1(\lambda, \mathscr{L}_1)$ and $\Sigma_2(\lambda, \mathscr{L}_2)$ over $\mathrm{St}_3^{\rm{an}}(\lambda)$, for each $\mathscr{L}_1, \mathscr{L}_2\in E$. Consequently, $\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ has the following form $$\begin{xy}
(0,0)*+{\mathrm{St}_3^{\rm{an}}(\lambda)}="a"; (20,6)*+{v_{P_1}^{\rm{an}}(\lambda)}="b"; (20,-6)*+{v_{P_2}^{\rm{an}}(\lambda)}="c";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"};
\end{xy}$$ and we have $$\label{3normalization of notation}
\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\cong \Sigma(\lambda, \mathscr{L}, \mathscr{L}^{\prime})$$ if $$\label{3sign of L invariant}
\mathscr{L}_1=-\mathscr{L}^{\prime}, \mathscr{L}_2=-\mathscr{L}\in E,$$ where $\Sigma(\lambda, \mathscr{L}, \mathscr{L}^{\prime})$ is the locally analytic representation defined in Definition 5.12 of [@Schr11] using the element $$(c_{2,\mathrm{log}}+\mathscr{L}^{\prime}c_{2,\mathrm{val}},~c_{1,\mathrm{log}}+\mathscr{L}c_{1,\mathrm{val}})$$ in $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(v_{P_1}^{\rm{an}}(\lambda)\oplus v_{P_2}^{\rm{an}}(\lambda), ~\mathrm{St}_3^{\rm{an}}(\lambda)\right).$$
\[3rema: sign of L invariant\] The appearance of a sign in (\[3sign of L invariant\]) is essentially due to Remark 3.1 of [@Ding18], which implies that our invariants $\mathscr{L}_1$ and $\mathscr{L}_2$ can be identified with Fontaine–Mazur $\mathscr{L}$invariants of the corresponding Galois representation via local-global compatibility.
We have a canonical morphism by (5.26) of [@Schr11] $$\label{3canonical morphism of Ext2}
\kappa: \mathrm{Ext}^2_{T(\mathbb{Q}_p),0}(1_T,1_T)\rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\mathrm{St}_3^{\rm{an}}(\lambda)\right).$$ Note that we also have $$\mathrm{Ext}^2_{T(\mathbb{Q}_p),0}(1_T,1_T)\cong \wedge^2\left(\mathrm{Ext}^1_{T(\mathbb{Q}_p),0}(1_T,1_T)\right)$$ by (5.24) of [@Schr11] and thus the set $$\{b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{val}_p}, b_{1,\mathrm{log}_0}\wedge b_{2,\mathrm{val}_p}, b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{log}_0}, b_{1,\mathrm{log}_0}\wedge b_{2,\mathrm{log}_0}, b_{1,\mathrm{val}_p}\wedge b_{1,\mathrm{log}_0}, b_{2,\mathrm{val}_p}\wedge b_{2,\mathrm{log}_0}\}$$ forms a basis of $\mathrm{Ext}^2_{T(\mathbb{Q}_p),0}\left(1_T,~1_T\right)$. It follows from (5.27) of [@Schr11] and (\[3relation of basis\]) that the set $$\{\kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{val}_p}), \kappa(b_{1,\mathrm{log}_0}\wedge b_{2,\mathrm{val}_p}), \kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{log}_0}), \kappa(b_{1,\mathrm{log}_0}\wedge b_{2,\mathrm{log}_0})\}$$ forms a basis of the image of (\[3canonical morphism of Ext2\]).
We recall the $p$adic dilogarithm function $li_2: \mathbb{Q}_p\setminus\{0,1\}\rightarrow \mathbb{Q}_p$ defined by Coleman in [@Cole82] and we consider the function $$D_{\mathscr{L}}(z):=li_2(z)+\frac{1}{2}\mathrm{log}_{\mathscr{L}}(z)\mathrm{log}_{\mathscr{L}}(1-z)$$ as in (5.34) of [@Schr11]. We also define $$d(z):=\mathrm{log}_{\mathscr{L}}(1-z)\mathrm{val}_p(z)-\mathrm{log}_{\mathscr{L}}(z)\mathrm{val}_p(1-z)$$ as in (5.36) of [@Schr11] which is also a locally analytic function over $\mathbb{Q}_p\setminus\{0,1\}$ and is independent of the choice of $\mathscr{L}\in E$. Note by our definition that $$D_{\mathscr{L}}-D_0=\frac{\mathscr{L}}{2}d.$$ It follows from Theorem 7.2 of [@Schr11] that $\{D_0,d\}$ can be identified with a basis of $$\mathrm{Ext}^2_{\mathrm{GL}_2(\mathbb{Q}_p),0}\left(1,~\mathrm{St}_2^{\rm{an}}\right)$$ ( cf. (5.38) of [@Schr11]) which naturally embeds into $\mathrm{Ext}^2_{\mathrm{GL}_2(\mathbb{Q}_p)}\left(1,~\mathrm{St}_2^{\rm{an}}\right)$. Then the map $\iota_i: \mathrm{GL}_2\hookrightarrow L_i$ induces the isomorphisms $$\label{3sequence of isomorphisms}
\mathrm{Ext}^2_{\mathrm{GL}_2(\mathbb{Q}_p)}\left(1_2,~\mathrm{St}_2^{\rm{an}}\right)\xleftarrow{\sim}\mathrm{Ext}^2_{L_i(\mathbb{Q}_p),0}\left(1_{L_i},~\mathrm{St}_2^{\rm{an}}\right)\xleftarrow{\sim}\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),0}\left(1_3,~I_{P_i}^{\mathrm{GL}_3}(\mathrm{St}_2^{\rm{an}})\right)$$ where $L_i(\mathbb{Q}_p)$ acts on $\mathrm{St}_2^{\rm{an}}$ via the projection $p_i$. We abuse the notation for the composition $$\iota_i: \mathrm{Ext}^2_{\mathrm{GL}_2(\mathbb{Q}_p)}\left(1_2,~\mathrm{St}_2^{\rm{an}}\right)\xleftarrow{\sim}\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),0}\left(1_3,~I_{P_i}^{\mathrm{GL}_3}(\mathrm{St}_2^{\rm{an}})\right)\rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),0}\left(1_3, ~\mathrm{St}_3^{\rm{an}}\right)$$ given by (\[3sequence of isomorphisms\]) and the surjection $$I_{P_i}^{\mathrm{GL}_3}(\mathrm{St}_2^{\rm{an}})\twoheadrightarrow\mathrm{St}_3^{\rm{an}}.$$ Finally there is canonical isomorphism $$\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),0}\left(1_3, ~\mathrm{St}_3^{\rm{an}}\right)\cong\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\mathrm{St}_3^{\rm{an}}(\lambda)\right)$$ by (5.20) of [@Schr11].
\[3lemm: ext2.1\] We have $$\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\mathrm{St}_3^{\rm{an}}(\lambda)\right)=5$$ and the set $$\{\kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{val}_p}), \kappa(b_{1,\mathrm{log}_0}\wedge b_{2,\mathrm{val}_p}), \kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{log}_0}), \kappa(b_{1,\mathrm{log}_0}\wedge b_{2,\mathrm{log}_0}), \iota_i(D_0)\}$$ forms a basis of $\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\mathrm{St}_3^{\rm{an}}(\lambda)\right)$ for $i=1,2$.
This follows directly from (5.57) of [@Schr11] and (\[3relation of basis\]).
\[3lemm: ext2 prime\] There exists $\gamma\in E^{\times}$ such that $$\iota_1(d)=\iota_2(d)=\gamma\left(\kappa(b_{1,\mathrm{log}_0}\wedge b_{2,\mathrm{val}_p}+b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{log}_0}\right).$$
This follows directly from Lemma 5.8 of [@Schr11] and (\[3relation of basis\]) if we take $$\gamma:=-3\alpha$$ where $\alpha\in E^{\times}$ is the constant in the statement of Lemma 5.8 of [@Schr11].
\[3lemm: ext2\] We have $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=1\mbox{ and }\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=2.$$ Moreover, the image of $$\{\kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{val}_p}), \iota_i(D_0)\}$$ under $$\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\mathrm{St}_3^{\rm{an}}(\lambda)\right)\rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$$ forms a basis of $\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$ for $i=1$ or $2$.
This follows directly from Corollary 5.17 of [@Schr11] and (\[3relation of basis\]).
We recall from (5.55) of [@Schr11] that $$\label{3definition of basis}
c_0:=\alpha^{-1}\iota_1(D_0)-\frac{1}{2}\kappa(c_{1,\mathrm{log}}\wedge c_{2,\mathrm{log}})$$ where $\alpha$ is defined in Lemma 5.8 of [@Schr11].
\[3lemm: easy normalization of higher invariant\] Assume that $\mathscr{L}_3\in E$ satisfies the equality $$\begin{gathered}
\label{3equality of higher invariant}
E\left(\iota_1(D_0)+\mathscr{L}_3\kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{val}_p})\right)=E\left(c_0+\mathscr{L}^{\prime\prime}\kappa(c_{1,\mathrm{val}}\wedge c_{2,\mathrm{val}})\right)\\
\subsetneq \mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right).\end{gathered}$$ Then we have $$\mathscr{L}_3=\gamma(\mathscr{L}^{\prime\prime}-\frac{1}{2}\mathscr{L}_1\mathscr{L}_2)=\gamma(\mathscr{L}^{\prime\prime}-\frac{1}{2}\mathscr{L}\mathscr{L}^{\prime}).$$
All the equalities in this lemma are understood to be inside $$\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$$ without causing ambiguity. It follows from our assumption (\[3equality of higher invariant\]) that $$\iota_1(D_0)+\mathscr{L}_3\kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{val}_p})=\alpha\left(c_0+\mathscr{L}^{\prime\prime}\kappa(c_{1,\mathrm{val}}\wedge c_{2,\mathrm{val}})\right)$$ which together with (\[3definition of basis\]) imply that $$\label{3equality in ext2}
\mathscr{L}_3\kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{val}_p})=\frac{\alpha}{2}\kappa(c_{1,\mathrm{log}}\wedge c_{2,\mathrm{log}})+\alpha\mathscr{L}^{\prime\prime}\kappa(c_{1,\mathrm{val}}\wedge c_{2,\mathrm{val}}).$$ We know that $$\label{3relation from simple invariant}
\kappa(c_{1,\mathrm{log}}\wedge c_{2,\mathrm{log}})=\mathscr{L}\mathscr{L}^{\prime}\kappa(c_{1,\mathrm{val}}\wedge c_{2,\mathrm{val}})$$ from the proof of Corollary 5.17 of [@Schr11] and that $$\label{3relation of val}
\kappa(c_{1,\mathrm{val}}\wedge c_{2,\mathrm{val}})=-3\kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{val}_p})$$ from (\[3relation of basis\]). Therefore we finish the proof by combining (\[3equality in ext2\]), (\[3relation from simple invariant\]) and (\[3relation of val\]) with (\[3sign of L invariant\]) and the equality $\gamma=-3\alpha$ from Lemma \[3lemm: ext2 prime\].
\[3equality of dilog\] We emphasize that we do not know whether $$E \iota_1(D_0)=E \iota_2(D_0)$$ in $\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\mathrm{St}_3^{\rm{an}}(\lambda)\right)$ or not, which is of independent interest.
A key result for $\mathrm{GL}_2(\mathbb{Q}_p)$ {#3section: GL2Qp}
==============================================
In this section, we are going to prove Proposition \[3prop: key result\] which will be a crucial ingredient for the proof of Lemma \[3lemm: ext6\] and Proposition \[3prop: main dim\].
We use the following shorten notation $$I(\nu):=I_{B_2}^{\mathrm{GL}_2}(\delta_{T_2,\nu}),~\widetilde{I}(\nu):=I_{B_2}^{\mathrm{GL}_2}(\delta_{T_2,\nu}\otimes_E(|\cdot|^{-1}\otimes_E|\cdot|))$$ for each weight $\nu\in X(T_2)$.
\[3lemm: ext1\] We have $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_2(\mathbb{Q}_p)}\left(\widetilde{I}(s\cdot\nu),~ \Sigma_{\mathrm{GL}_2}(\nu,\mathscr{L})\right)=1.$$
This is essentially contained in the proof of Theorem 3.14 of [@BD18]. In fact, we know that $$\begin{array}{cccc}
\mathrm{Ext}^1_{\mathrm{GL}_2(\mathbb{Q}_p)}\left(\widetilde{I}(s\cdot\nu),~\begin{xy}
(0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="a"; (26,0)*+{I(s\cdot\nu)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)&=&0&\\
\mathrm{Ext}^2_{\mathrm{GL}_2(\mathbb{Q}_p)}\left(\widetilde{I}(s\cdot\nu),~\begin{xy}
(0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="a"; (26,0)*+{I(s\cdot\nu)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)&=&0&
\end{array}$$ and $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_2(\mathbb{Q}_p)}(\widetilde{I}(s\cdot\nu),~\overline{L}_{\mathrm{GL}_2}(\nu))=1$$ which finish the proof by a simple devissage induced by the short exact sequence $$\left(\begin{xy}
(0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="a"; (26,0)*+{I(s\cdot\nu)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)\hookrightarrow \Sigma_{\mathrm{GL}_2}(\nu,\mathscr{L})\twoheadrightarrow \overline{L}_{\mathrm{GL}_2}(\nu).$$
We fix a split $p$-adic reductive group $H$ and have a natural embedding $$U(\mathfrak{h})\hookrightarrow D(H, E)_{\{1\}}\hookrightarrow D(H, E)$$ where $D(H, E)_{\{1\}}$ is the closed subalgebra of $D(H, E)$ consisting of distributions supported at the identity element ( cf. [@Koh07]). The embedding above induces another embedding $$\label{3center of distribution}
Z(U(\mathfrak{h}))\hookrightarrow Z(D(H, E))$$ by the main result of [@Koh07] where $Z(\cdot)$ is the notation for the center of a non-commutative algebra. We say that $\Pi\in\mathrm{Rep}^{\rm{la}}_{\mathrm{GL}_2(\mathbb{Q}_p), E}$ has an infinitesimal character if $Z(U(\mathfrak{h}))$ acts on $\Pi^{\prime }$ via a character.
\[3lemm: existence of inf char\] If $V, W\in\mathrm{Rep}^{\rm{la}}_{H, E}$ have both the same central character and the same infinitesimal character and satisfy $$\mathrm{Hom}_{H}\left(V, ~W\right)=0,$$ then any non-split extension of the form $\begin{xy}
(0,0)*+{W}="a"; (10,0)*+{V}="b";
{\ar@{-}"a";"b"};
\end{xy}$ has both the same central character and the same infinitesimal character as the one for $V$ and $W$.
This is a direct analogue of Lemma 3.1 in [@BD18] and follows essentially from the fact that both $D(Z(H), E)$ and $Z(U(\mathfrak{h}))$ are subalgebras of $Z(D(H, E))$ by [@Koh07].
We fix a Borel subgroup $B_H\subseteq H$ as well as its opposite Borel subgroup $\overline{B_H}$. We consider the split maximal torus $T_H:=B_H\cap \overline{B_H}$ and use the notation $N_H$ (resp. $\overline{N_H}$) for the unipotent radical of $B_H$ (resp. of $\overline{B_H}$).
\[3lemm: harish chandra\] If $V\in\mathrm{Rep}^{\rm{la}}_{H, E}$ has an infinitesimal character, then $U(\mathfrak{t}_{\mathfrak{h}})^{W_H}$ (as a subalgebra of $U(\mathfrak{t}_{\mathfrak{h}})$) acts on $J_{\overline{B_H}}(V)$ via a character where $W_H$ is the Weyl group of $H$.
We know by our assumption that $Z(U(\mathfrak{h}))$ acts on $V^{\prime}$ (and hence on $V$ as well) via a character. We note from (\[3center of distribution\]) that $Z(U(\mathfrak{h}))$ commutes with $D(\overline{N_H}, E)\subseteq D(H, E)$ and thus the action of $Z(U(\mathfrak{h}))$ on $V$ commutes with that of $\overline{N_H}$, which implies that $Z(U(\mathfrak{h}))$ acts on $V^{\overline{N_H}^{\circ}}$ via a character for each open compact subgroup $\overline{N_H}^{\circ}\subseteq \overline{N_H}$. We use the notation $$\theta: Z(U(\mathfrak{h}))\xrightarrow{\sim} U(\mathfrak{t}_{\mathfrak{h}})^{W_H}$$ for the Harish-Chandra isomorphism ( cf. Section 1.7 of [@Hum08]) and the notation $j_1$ and $j_2$ for the embeddings $$j_1: Z(U(\mathfrak{h}))\hookrightarrow U(\mathfrak{h})\mbox{ and }j_2: U(\mathfrak{t}_{\mathfrak{h}})\hookrightarrow U(\mathfrak{h}).$$ We choose an arbitrary Verma module $M_H(\lambda_H)$ with highest weight $\lambda_H$, namely we have $$M_H(\lambda):=U(\mathfrak{h})\otimes_{U(\overline{\mathfrak{b}_H})}\lambda_H.$$ We use the notation $M_H(\lambda_H)_{\mu}$ for the subspace of $M_H(\lambda)$ with $\mathfrak{t}_{\mathfrak{h}}$-weight $\mu$ and note that $$\mathrm{dim}_EM_H(\lambda_H)_{\lambda_H}=1.$$ We easily observe that $$\label{3preserve highest weight}
Z(U(\mathfrak{h}))\cdot M_H(\lambda_H)_{\lambda_H}=M_H(\lambda_H)_{\lambda_H}\mbox{ and }U(\mathfrak{t}_{\mathfrak{h}})\cdot M_H(\lambda_H)_{\lambda_H}=M_H(\lambda_H)_{\lambda_H}.$$ It is well-known that the the direct sum decomposition $$\label{3decomposition lie}
\mathfrak{h}=\mathfrak{n}_H\oplus\mathfrak{t}_{\mathfrak{h}}\oplus\overline{\mathfrak{n}_H}$$ induces a tensor decomposition of $E$-vector space $$\label{3PBW decomposition}
U(\mathfrak{h})=U(\mathfrak{n}_H)\otimes_EU(\mathfrak{t}_{\mathfrak{h}})\otimes_EU(\overline{\mathfrak{n}_H}).$$ Hence we can write each element in $U(\mathfrak{h})$ as a polynomial with variables indexed by a standard basis of $\mathfrak{h}$ that is compatible with (\[3decomposition lie\]). It follows from the definition of $\theta$ as the restriction to $Z(U(\mathfrak{h}))$ of the projection $U(\mathfrak{h})\twoheadrightarrow U(\mathfrak{t}_{\mathfrak{h}})$ (coming from (\[3PBW decomposition\])) that $$j_1(z)-j_2\circ\theta(z)\in U(\mathfrak{h})\cdot\overline{\mathfrak{n}_H}+\mathfrak{n}_H\cdot U(\mathfrak{h})$$ for each $z\in Z(U(\mathfrak{h}))$. If a monomial $f$ in the decomposition (\[3PBW decomposition\]) of $j_1(z)-j_2\circ\theta(z)$ belongs to $$\mathfrak{n}_H\cdot U(\mathfrak{n}_H)\cdot U(\mathfrak{t}_{\mathfrak{h}}),$$ then we have $$f\cdot M_H(\lambda_H)_{\lambda_H}\subseteq M_H(\lambda_H)_{\mu}$$ for some $\mu\neq \lambda_H$, which contradicts the fact (\[3preserve highest weight\]). Hence we conclude that $$j_1(z)-j_2\circ\theta(z)\in U(\mathfrak{h})\cdot\overline{\mathfrak{n}_H}$$ and in particular $$j_1(z)=j_2\circ\theta(z)$$ on $V^{\overline{N_H}^{\circ}}$ for each $z\in Z(U(\mathfrak{h}))$. Hence we deduce that $U(\mathfrak{t}_{\mathfrak{h}})^{W_H}$ acts on $V^{\overline{N_H}^{\circ}}$ via a character. We note by the definition of $J_{\overline{B_H}}$ ( cf. [@Eme06]) that we have a $T_H^+$-equivariant embedding $$\label{3definition of Jacquet}
J_{\overline{B_H}}(V)\hookrightarrow V^{\overline{N_H}^{\circ}}$$ where $T_H^+$ is a certain submonoid of $T_H$ containing an open compact subgroup. As a result, (\[3definition of Jacquet\]) is also $U(\mathfrak{t}_{\mathfrak{h}})$-equivariant and thus $U(\mathfrak{t}_{\mathfrak{h}})^{W_H}$ acts on $J_{\overline{B_H}}(V)$ via a character which finishes the proof.
We set $H=\mathrm{GL}_2(\mathbb{Q}_p)$, $B_H=B_2$ and $\overline{B_H}=\overline{B_2}$ in the rest of this section. The idea of the following lemma which is closely related to Lemma 3.20 of [@BD18], owes very much to Y.Ding.
\[3lemm: no inf char\] A locally analytic representation of either the form $$\label{3first form}
\begin{xy}
(0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="a"; (26,0)*+{I(s\cdot\nu)}="b"; (45,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="c"; (72,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="d";
{\ar@{-}"a";"b"}; {\ar@{-}"b";"c"}; {\ar@{-}"c";"d"};
\end{xy}$$ or the form $$\label{3second form}
\begin{xy}
(0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="a"; (20,0)*+{\widetilde{I}(s\cdot\nu)}="b"; (45,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="c"; (72,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="d";
{\ar@{-}"a";"b"}; {\ar@{-}"b";"c"}; {\ar@{-}"c";"d"};
\end{xy}$$ does not have an infinitesimal character.
Assume that a representation $V$ of the form (\[3first form\]) has an infinitesimal character. Note that $V$ can be represented by an element in the space $\mathrm{Ext}^1_{\mathrm{GL}_2(\mathbb{Q}_p)}(\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}, \Sigma_{\mathrm{GL}_2}(\nu, \mathscr{L}))$ for certain $\mathscr{L}\in E$. We consider the upper-triangular Borel subgroup $\overline{B_2}$ and the diagonal split torus $T_2$. Then by the proof of Lemma 3.20 of [@BD18] we know that the Jacquet functor $J_{\overline{B_2}}$ ( cf. [@Eme06] for the definition) induces a injection $$\begin{gathered}
\label{3Jacquet injection}
\mathrm{Ext}^1_{\mathrm{GL}_2(\mathbb{Q}_p)}\left(\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}, ~\Sigma_{\mathrm{GL}_2}(\nu, \mathscr{L})\right)\\
\hookrightarrow\mathrm{Ext}^1_{T_2(\mathbb{Q}_p)}\left(\delta_{T_2,\nu}\otimes_E(|\cdot|\otimes_E|\cdot|^{-1}), ~\delta_{T_2,\nu}\otimes_E(|\cdot|\otimes_E|\cdot|^{-1})\right).\end{gathered}$$ By twisting $\delta_{T_2, -\nu}\otimes_E(|\cdot|^{-1}\otimes_E|\cdot|)$ we have an isomorphism $$\label{3twist isomorphism}
\mathrm{Ext}^1_{T_2(\mathbb{Q}_p)}\left(\delta_{T_2,\nu}\otimes_E(|\cdot|\otimes_E|\cdot|^{-1}), ~\delta_{T_2,\nu}\otimes_E(|\cdot|\otimes_E|\cdot|^{-1})\right)\cong \mathrm{Ext}^1_{T_2(\mathbb{Q}_p)}\left(1_{T_2}, ~1_{T_2}\right).$$ It follows from Lemma 3.20 of [@BD18] (up to changes on notation) that the image of the composition of (\[3twist isomorphism\]) and (\[3Jacquet injection\]) is a certain two dimensional subspace $\mathrm{Ext}^1_{T_2(\mathbb{Q}_p)}(1, 1)_{\mathscr{L}}$ of $\mathrm{Ext}^1_{T_2(\mathbb{Q}_p)}(1, 1)$ depending on $\mathscr{L}$. More precisely, if we use the notation $\epsilon_1$, $\epsilon_2$ for the two charaters $$\epsilon_1: T_2(\mathbb{Q}_p)\rightarrow\mathbb{Q}_p^{\times},~\left(\begin{array}{cc}\
a&0\\
0&b\\
\end{array}\right)\mapsto a\mbox{ and } \epsilon_2: T_2(\mathbb{Q}_p)\rightarrow\mathbb{Q}_p^{\times},~\left(\begin{array}{cc}\
a&0\\
0&b\\
\end{array}\right)\mapsto b,$$ then the set $$\{\mathrm{log}_0\circ \epsilon_1, \mathrm{val}_p\circ \epsilon_1, \mathrm{log}_0\circ \epsilon_2, \mathrm{val}_p\circ \epsilon_2\}$$ forms a basis of $\mathrm{Ext}^1_{T_2(\mathbb{Q}_p)}(1, 1)$, and the subspace $\mathrm{Ext}^1_{T_2(\mathbb{Q}_p)}(1, 1)_{\mathscr{L}}$ has a basis $$\{\mathrm{log}_0\circ \epsilon_1+\mathrm{log}_0\circ \epsilon_2, \mathrm{val}_p\circ \epsilon_1+\mathrm{val}_p\circ \epsilon_2, \mathrm{log}_0\circ \epsilon_1-\mathrm{log}_0\circ \epsilon_2+\mathscr{L}(\mathrm{val}_p\circ \epsilon_1-\mathrm{val}_p\circ \epsilon_2)\}.$$ It follows from Lemma \[3lemm: harish chandra\] that $U(\mathfrak{t}_2)^{W_{\mathrm{GL}_2}}$ acts on $J_{\overline{B_2}}(V)$ via a character where $W_{\mathrm{GL}_2}$ is the notation for the Weyl group of $\mathrm{GL}_2(\mathbb{Q}_p)$. Therefore we deduce by a twisting that the the subspace of $\mathrm{Ext}^1_{T_2(\mathbb{Q}_p)}(1, 1)$ corresponding to $J_{\overline{B_2}}(V)$ is killed by $U(\mathfrak{t}_2)^{W_{\mathrm{GL}_2}}$. We notice that the subspace $M$ of $\mathrm{Ext}^1_{T_2(\mathbb{Q}_p)}(1, 1)$ killed by $U(\mathfrak{t}_2)^{W_{\mathrm{GL}_2}}$ is two dimensional with basis $$\{\mathrm{val}_p\circ \epsilon_1, \mathrm{val}_p\circ \epsilon_2\}$$ and we have $$M\cap\mathrm{Ext}^1_{T_2(\mathbb{Q}_p)}(1, 1)_{\mathscr{L}}=E\left(\mathrm{val}_p\circ \epsilon_1+\mathrm{val}_p\circ \epsilon_2\right).$$ However, the representation given by the line $E(\mathrm{val}_p\circ \epsilon_1+\mathrm{val}_p\circ \epsilon_2)$ has a subrepresentation of the form $$\begin{xy}
(0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="a"; (34,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy}$$ which is a contradiction.
The proof of the second statement is a direct analogue as we observe that $J_{\overline{B_2}}$ also induces the following embedding $$\begin{gathered}
\mathrm{Ext}^1_{\mathrm{GL}_2(\mathbb{Q}_p)}\left(\overline{L}_{\mathrm{GL}_2}(\nu), ~\begin{xy}
(0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="a"; (20,0)*+{\widetilde{I}(s\cdot\nu)}="b"; (45,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="c"; (72,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="d";
{\ar@{-}"a";"b"}; {\ar@{-}"b";"c"}; {\ar@{-}"c";"d"};
\end{xy}\right)\\
\hookrightarrow\mathrm{Ext}^1_{T_2(\mathbb{Q}_p)}\left(\delta_{T_2,\nu}, ~\delta_{T_2,\nu}\right).\end{gathered}$$
We define $\Sigma_2^+(\nu,\mathscr{L})$ as the unique (up to isomorphism) non-split extension of $\Sigma_{\mathrm{GL}_2}(\nu,\mathscr{L})$ by $\widetilde{I}(s\cdot\nu)$ given by Lemma \[3lemm: ext1\].
\[3prop: key result\] We have $$\mathrm{Ext}^1_{\mathrm{GL}_2(\mathbb{Q}_p)}\left(
\begin{xy}
(0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="a"; (26,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="b";
{\ar@{-}"a";"b"};
\end{xy},~ \Sigma_2^+(\nu,\mathscr{L})\right)=0.$$
Assume on the contrary that $V$ is a representation given by a certain non-zero element inside $$\mathrm{Ext}^1_{\mathrm{GL}_2(\mathbb{Q}_p)}\left(\begin{xy}
(0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="a"; (26,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="b";
{\ar@{-}"a";"b"};
\end{xy},~ \Sigma_2^+(\nu,\mathscr{L})\right).$$ We deduce that $V$ has both a central character and an infinitesimal character from Lemma \[3lemm: existence of inf char\] and the fact $$\mathrm{Hom}_{\mathrm{GL}_2(\mathbb{Q}_p)}\left(\begin{xy}
(0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="a"; (26,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="b";
{\ar@{-}"a";"b"};
\end{xy},~ \Sigma_2^+(\nu,\mathscr{L})\right)=0.$$ Note that we have $$\mathrm{Ext}^1_{\mathrm{GL}_2(\mathbb{Q}_p)}(\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}, ~I(s\cdot\nu))=\mathrm{Ext}^1_{\mathrm{GL}_2(\mathbb{Q}_p)}(\overline{L}_{\mathrm{GL}_2}(\nu), ~\widetilde{I}(s\cdot\nu))=0,$$ $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_2(\mathbb{Q}_p)}\left(\overline{L}_{\mathrm{GL}_2}(\nu),~\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}\right)=1$$ and $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_2(\mathbb{Q}_p)}\left(\overline{L}_{\mathrm{GL}_2}(\nu),~I(s\cdot\nu)\right)=1$$ by a combination of Lemma 3.13 of [@BD18] with Lemma \[3lemm: cohomology devissage\], and thus $V$ has a subrepresentation of one of the three following forms
1. $\begin{xy}
(0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="a"; (34,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy}$;
2. $\begin{xy}
(0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="a"; (26,0)*+{I(s\cdot\nu)}="b"; (45,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="c"; (72,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="d";
{\ar@{-}"a";"b"}; {\ar@{-}"b";"c"}; {\ar@{-}"c";"d"};
\end{xy}$;
3. $\begin{xy}
(0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="a"; (26,0)*+{I(s\cdot\nu)}="b"; (45,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="c"; (64,0)*+{\widetilde{I}(s\cdot\nu)}="d"; (90,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="e"; (115,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="f";
{\ar@{-}"a";"b"}; {\ar@{-}"b";"c"}; {\ar@{-}"c";"d"}; {\ar@{-}"d";"e"}; {\ar@{-}"e";"f"};
\end{xy}.$
In the first case, we know from Proposition 4.7 of [@Schr11] and the main result of [@Or05] that $$\mathrm{Ext}^1_{\mathrm{GL}_2(\mathbb{Q}_p),\nu}\left(\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}, ~\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}\right)=0$$ and therefore this case is impossible due to the existence of central character for $V$ (and hence for its subrepresentations). In the second case, we deduce from Lemma \[3lemm: no inf char\] a contradiction as $V$ has an infinitesimal character. In the third case, we thus know that $V$ has a quotient representation of the form $$\begin{xy}
(0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="a"; (20,0)*+{\widetilde{I}(s\cdot\nu)}="b"; (45,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="c"; (72,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="d";
{\ar@{-}"a";"b"}; {\ar@{-}"b";"c"}; {\ar@{-}"c";"d"};
\end{xy}$$ which can not have an infinitesimal character due to Lemma \[3lemm: no inf char\], a contradiction again. Hence we finish the proof.
\[3rema: second version\] Note that the argument in Proposition \[3prop: key result\] actually implies that $$\mathrm{Ext}^1_{\mathrm{GL}_2(\mathbb{Q}_p)}\left(\begin{xy}
(0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="a"; (26,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="b";
{\ar@{-}"a";"b"};
\end{xy},~ \begin{xy}
(0,0)*+{I(s\cdot\nu)}="b"; (20,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="c"; (40,0)*+{\widetilde{I}(s\cdot\nu)}="d";
{\ar@{-}"b";"c"}; {\ar@{-}"c";"d"};
\end{xy}\right)=0$$ and we can show by the same method that $$\mathrm{Ext}^1_{\mathrm{GL}_2(\mathbb{Q}_p)}\left(\begin{xy}
(26,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="b"; (0,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)}="a";
{\ar@{-}"a";"b"};
\end{xy},~ \begin{xy}
(0,0)*+{\widetilde{I}(s\cdot\nu)}="b"; (26,0)*+{\overline{L}_{\mathrm{GL}_2}(\nu)\otimes_E\mathrm{St}_2^{\infty}}="c"; (50,0)*+{I(s\cdot\nu)}="d";
{\ar@{-}"b";"c"}; {\ar@{-}"c";"d"};
\end{xy}\right)=0.$$
Computations of $\mathrm{Ext}$ I {#3section: computation}
================================
In this section, we are going to compute various $\mathrm{Ext}$groups based on known results on group cohomology in Section 5.2 and 5.3 of [@Bre17].
\[3prop: locally algebraic extension\] The following spaces are one dimensional $$\begin{xy}
(0,0)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}\right)}; (70,0)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}, ~\overline{L}(\lambda)\right)};
(6.5,-6)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}, ~\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}\right)};
(76.5,-6)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}, ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)};
(0.8,-12)*+{\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}, ~\overline{L}(\lambda)\right)};
(70.8,-12)*+{\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)};
(6,-18)*+{\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}, ~\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}\right)};
(76,-18)*+{\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}, ~\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}\right)};
\end{xy}$$ for $i=1,2$. Moreover, we have $$\mathrm{Ext}^k_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(V_1, ~V_2\right)=0$$ in all the other cases where $1\leq k\leq 2$ and $V_1,V_2\in\{\overline{L}(\lambda), \overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}, \overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}, \overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\}$.
This follows from a special case of Proposition 4.7 of [@Schr11] together with the main result of [@Or05].
\[3lemm: vanishing locally algebraic\] We have $$\begin{xy}
(0,0)*+{\mathrm{Ext}^k_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(
\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="a"; (20,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy},~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)}; (50,0)*+{=0};
(0.35,-6)*+{\mathrm{Ext}^k_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="a"; (27,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy},~\overline{L}(\lambda)\right)}; (50,-6)*+{=0};
(0.95,-12)*+{\mathrm{Ext}^k_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\begin{xy}
(20,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="b"; (0,0)*+{\overline{L}(\lambda)}="a";
{\ar@{-}"a";"b"};
\end{xy},~\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}\right)}; (50,-12)*+{=0};
\end{xy}$$ for $i=1,2$ and $k=1,2$.
It is sufficient to prove that $$\label{3first algebraic vanishing}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="a"; (20,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy},~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)=0$$ and $$\label{3second algebraic vanishing}
\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="a"; (20,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy},~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)=0$$ as the other cases are similar. We observe that (\[3first algebraic vanishing\]) is equivalent to the non-existence of a uniserial representation of the form $$\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="a"; (27,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="b"; (47,0)*+{\overline{L}(\lambda)}="c";
{\ar@{-}"a";"b"}; {\ar@{-}"b";"c"};
\end{xy}$$ which is again equivalent to the vanishing $$\label{3third algebraic vanishing}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="a"; (27,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)=0$$ according to the fact $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)=0$$ due to Proposition \[3prop: locally algebraic extension\]. The short exact sequence $$\left(\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="a"; (27,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)\hookrightarrow\mathcal{F}_{P_i}^{\mathrm{GL}_3}\left(M_i(-\lambda),~\pi_{1,3}^{\infty}\right)\twoheadrightarrow C^2_{s_{3-i},s_{3-i}s_i}$$ induces an injection $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="a"; (27,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)\hookrightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_{P_i}^{\mathrm{GL}_3}\left(M_i(-\lambda),~\pi_{i,3}^{\infty}\right)\right).$$ Therefore (\[3third algebraic vanishing\]) follows from Lemma \[3lemm: cohomology devissage\] and the facts that $$\mathrm{Ext}^1_{L_i(\mathbb{Q}_p),\lambda}\left(H_0(N_i, \overline{L}(\lambda)),~\overline{L}_i(\lambda)\otimes_E\pi_{i,3}^{\infty}\right)=\mathrm{Hom}_{L_i(\mathbb{Q}_p),\lambda}\left(H_1(N_i, \overline{L}(\lambda)),~\overline{L}_i(\lambda)\otimes_E\pi_{i,3}^{\infty}\right)=0.$$ On the other hand, the short exact sequence $$\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}\hookrightarrow\left(\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="a"; (20,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)\twoheadrightarrow\overline{L}(\lambda)$$ induces a long exact sequence $$\begin{gathered}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\hookrightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="a"; (20,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy},~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\\ \rightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty},~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)
\rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\\ \rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="a"; (20,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy},~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right) \rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty},~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\end{gathered}$$ and thus we can deduce (\[3second algebraic vanishing\]) from Proposition \[3prop: locally algebraic extension\] and (\[3first algebraic vanishing\]).
We define $W_0$ as the unique locally algebraic representation of length three satisfying $$\mathrm{soc}_{\mathrm{GL}_3(\mathbb{Q}_p)}(W_0)=\overline{L}(\lambda)\otimes_E\left(v_{P_1}^{\infty}\oplus v_{P_2}^{\infty}\right)\mbox{ and }\mathrm{cosoc}_{\mathrm{GL}_3(\mathbb{Q}_p)}(W_0)=\overline{L}(\lambda).$$ We also define the (unique up to isomorphism) locally algebraic representation of the form $$W_i:=\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="a"; (20,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}$$ for each $i=1,2$
\[3lemm: vanishing locally algebraic main\] We have $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0,~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)=1$$ and $$\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0,~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)=0.$$
The short exact sequence $$\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}\hookrightarrow W_0\twoheadrightarrow W_2$$ induces a long exact sequence $$\begin{gathered}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty},~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\hookrightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0,~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\\ \rightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2,~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)
\rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty},~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\\ \rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0,~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right) \rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2,~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\end{gathered}$$ which finishes the proof by Proposition \[3prop: locally algebraic extension\], (\[3first algebraic vanishing\]) and (\[3second algebraic vanishing\]).
We define the following subsets of $\Omega$: $$\begin{xy}
(-6,0)*+{\Omega_1\left(\overline{L}(\lambda)\right)}; (20,0)*+{:=}; (57.5,0)*+{\{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty},~\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty},~C^1_{s_1,1},~C^1_{s_2,1}\}};
(0,-6)*+{\Omega_1\left(\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}\right)}; (20,-6)*+{:=}; (60,-6)*+{\{\overline{L}(\lambda),~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty},~C^2_{s_1,1},~C_{s_2,s_2},~C^1_{s_1,s_1s_2}\}};
(0,-12)*+{\Omega_1\left(\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}\right)}; (20,-12)*+{:=}; (60,-12)*+{\{\overline{L}(\lambda),~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty},~C^2_{s_2,1},~C_{s_1,s_1},~C^1_{s_2,s_2s_1}\}};
(0.5,-18)*+{\Omega_1\left(\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)}; (20,-18)*+{:=}; (61.5,-18)*+{\{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty},~\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty},~C^2_{s_1,s_1s_2},~C^2_{s_2,s_2s_1}\}};
(-6,-24)*+{\Omega_2\left(\overline{L}(\lambda)\right)}; (20,-24)*+{:=}; (60.5,-24)*+{\{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty},~C^2_{s_1,1},~C^2_{s_2,1},~C^1_{s_1s_2,1},~C^1_{s_2s_1,1}\}};
(0,-30)*+{\Omega_2\left(\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}\right)}; (20,-30)*+{:=}; (62.5,-30)*+{\{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty},~C^1_{s_1,1},~C^2_{s_1,s_1s_2},~C^2_{s_1s_2,1},~C_{s_2s_1,s_2}\}};
(0,-36)*+{\Omega_2\left(\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}\right)}; (20,-36)*+{:=}; (62.5,-36)*+{\{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty},~C^1_{s_2,1},~C^2_{s_2,s_2s_1},~C^2_{s_2s_1,1},~C_{s_1s_2,s_1}\}};
(0.5,-42)*+{\Omega_2\left(\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)}; (20,-42)*+{:=}; (62,-42)*+{\{\overline{L}(\lambda),~C^1_{s_1,s_1s_2},~C^1_{s_2,s_2s_1},~C^2_{s_1s_2,s_1s_2},~C^2_{s_2s_1,s_2s_1}\}};
\end{xy}$$
\[3prop: list of N coh\] We have all explicit formula for $$H_k\left(N_i,~\mathcal{F}_{P_j}^{\mathrm{GL}_3}(M,~\pi_j^{\infty})\right)$$ for each smooth admissible representation $\pi_j^{\infty}$ of $L_j(\mathbb{Q}_p)$, each $$M\in \{L(-\lambda),~M_j(-\lambda), L(-s_{3-j}\cdot\lambda), ~M_j(-s_{3-j}\cdot\lambda),~L(-s_{3-j}s_j\cdot\lambda)\}$$ and each $0\leq k\leq 2$, $i,j=1,2$.
This follows directly from Section 5.2 and 5.3 of [@Bre17].
\[3lemm: nonvanishing ext1\] For $$V_0\in\{\overline{L}(\lambda),~\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty},~\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty},~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\},$$ we have $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(V_0,~V\right)=1$$ if $V\in\Omega_1(V_0)$ and $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(V_0,~V\right)=0$$ if $V\in\Omega\setminus\Omega_1(V_0)$.
We only prove the statements for $V_0=\overline{L}(\lambda)$ as other cases are similar. If $$V\in\{\overline{L}(\lambda),~\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty},~\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty},~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\}$$ then the conclusion follows from Proposition \[3prop: locally algebraic extension\]. If $$V=\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}s_j\cdot\lambda),~\pi_j^{\infty})$$ for a smooth irreducible representation $\pi_j^{\infty}$ and $j=1$ or $2$, then it follows from Lemma \[3lemm: cohomology devissage\] that $$\begin{gathered}
\label{3explicit spectral sequence 1}
\mathrm{Ext}^1_{L_j(\mathbb{Q}_p),\lambda}\left(H_0(N_j,~\overline{L}(\lambda)),~\overline{L}_j(s_{3-j}s_j\cdot\lambda)\otimes_E\pi_j^{\infty}\right)\hookrightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~V\right)\\
\rightarrow \mathrm{Hom}_{L_j(\mathbb{Q}_p),\lambda}\left(H_1(N_j,~\overline{L}(\lambda)),~\overline{L}_j(s_{3-j}s_j\cdot\lambda)\otimes_E\pi_j^{\infty}\right)\\
\rightarrow\mathrm{Ext}^2_{L_j(\mathbb{Q}_p),\lambda}\left(H_0(N_j,~\overline{L}(\lambda)),~\overline{L}_j(s_{3-j}s_j\cdot\lambda)\otimes_E\pi_j^{\infty}\right).\end{gathered}$$ It follows from Proposition \[3prop: list of N coh\] and (\[3explicit spectral sequence 1\]) that $$\begin{gathered}
\label{3explicit spectral sequence 2}
\mathrm{Ext}^1_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(\lambda),~\overline{L}_j(s_{3-j}s_j\cdot\lambda)\otimes_E\pi_j^{\infty}\right)\hookrightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~V\right)\\
\rightarrow \mathrm{Hom}_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(s_{3-j}\cdot\lambda),~\overline{L}_j(s_{3-j}s_j\cdot\lambda)\otimes_E\pi_j^{\infty}\right).\end{gathered}$$ We notice that $Z(L_j(\mathbb{Q}_p))$ acts via different characters on $\overline{L}_j(\lambda)$, $\overline{L}_j(s_{3-j}\cdot\lambda)$ and $\overline{L}_j(s_{3-j}s_j\cdot\lambda)\otimes_E\pi_j^{\infty}$, and thus we have the equalities $$\begin{array}{cccc}
\mathrm{Ext}^1_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(\lambda),~\overline{L}_j(s_{3-j}s_j\cdot\lambda)\otimes_E\pi_j^{\infty}\right)&=&0&\\
\mathrm{Hom}_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(s_{3-j}\cdot\lambda),~\overline{L}_j(s_{3-j}s_j\cdot\lambda)\otimes_E\pi_j^{\infty}\right)&=&0&
\end{array}$$ which imply that $$\label{3vanishing of ext 1.1}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}s_j\cdot\lambda),~\pi_j^{\infty})\right)=0$$ for each $\pi_j^{\infty}$ and $j=1,2$. If $$V=\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}\cdot\lambda),~\pi_j^{\infty})$$ for a smooth irreducible representation $\pi_j^{\infty}$ and $j=1$ or $2$, then the short exact sequence $$\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}\cdot\lambda),~\pi_j^{\infty})\hookrightarrow\mathcal{F}_{P_j}^{\mathrm{GL}_3}(M_j(-s_{3-j}\cdot\lambda),~\pi_j^{\infty})\twoheadrightarrow\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}s_j\cdot\lambda),~\pi_j^{\infty})$$ induces a long exact sequence $$\begin{gathered}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~V\right)\hookrightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_{P_j}^{\mathrm{GL}_3}(M_j(-s_{3-j}\cdot\lambda),~\pi_j^{\infty})\right)\\
\rightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}s_j\cdot\lambda),~\pi_j^{\infty})\right)\end{gathered}$$ which implies an isomorphism $$\label{3isomorphism of ext 1.2}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~V\right)\xrightarrow{\sim}\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_{P_j}^{\mathrm{GL}_3}(M_j(-s_{3-j}\cdot\lambda),~\pi_j^{\infty})\right)$$ by (\[3vanishing of ext 1.1\]). It follows from Proposition \[3prop: list of N coh\] and Lemma \[3lemm: cohomology devissage\] that $$\begin{gathered}
\label{3explicit spectral sequence 3}
\mathrm{Ext}^1_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(\lambda),~\overline{L}_j(s_{3-j}\cdot\lambda)\otimes_E\pi_j^{\infty}\right)\hookrightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~V\right)\\
\rightarrow \mathrm{Hom}_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(s_{3-j}\cdot\lambda),~\overline{L}_j(s_{3-j}\cdot\lambda)\otimes_E\pi_j^{\infty}\right)
\rightarrow\mathrm{Ext}^2_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(\lambda),~\overline{L}_j(s_{3-j}\cdot\lambda)\otimes_E\pi_j^{\infty}\right).\end{gathered}$$ As $Z(L_j(\mathbb{Q}_p))$ acts via different characters on $\overline{L}_j(\lambda)$ and $\overline{L}_j(s_{3-j}\cdot\lambda)\otimes_E\pi_j^{\infty}$, we have the equalities $$\begin{array}{cccc}
\mathrm{Ext}^1_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(\lambda),~\overline{L}_j(s_{3-j}s_j\cdot\lambda)\otimes_E\pi_j^{\infty}\right)&=&0&\\
\mathrm{Ext}^2_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(\lambda),~\overline{L}_j(s_{3-j}s_j\cdot\lambda)\otimes_E\pi_j^{\infty}\right)&=&0&
\end{array}$$ which imply that $$\label{3isomorphism of ext 1.3}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~V\right)
\xrightarrow{\sim} \mathrm{Hom}_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(s_{3-j}\cdot\lambda),~\overline{L}_j(s_{3-j}\cdot\lambda)\otimes_E\pi_j^{\infty}\right).$$ It is then obvious that $$\mathrm{Hom}_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(s_{3-j}\cdot\lambda),~\overline{L}_j(s_{3-j}\cdot\lambda)\otimes_E\pi_j^{\infty}\right)=0$$ for each smooth irreducible $\pi_j^{\infty}\neq 1_{L_j}$, and therefore $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}\cdot\lambda),~1_{L_j})\right)=1$$ and $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}\cdot\lambda),~1_{L_j})\right)=0$$ for each smooth irreducible $\pi_j^{\infty}\neq 1_{L_j}$. Finally, similar methods together with Proposition \[3prop: list of N coh\] also show that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_B^{\mathrm{GL}_3}(L(-s_1s_2s_1\cdot\lambda),~\chi_w^{\infty})\right)=0$$ for each $w\in W$.
We define $$\Omega^-:=\Omega\setminus\{\overline{L}(\lambda), ~\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}, ~\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}, ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\}.$$ Then we define the following subsets of $\Omega^-$ for $i=1,2$: $$\begin{xy}
(0,0)*+{\Omega_1\left(C^1_{s_i,1}\right)}; (20,0)*+{:=}; (55.8,0)*+{\{C^1_{s_is_{3-i},1},~C^2_{s_{3-i}s_i,1},~C^2_{s_i,1},~C^1_{s_i,1}\}};
(0,-6)*+{\Omega_1\left(C^2_{s_i,1}\right)}; (20,-6)*+{:=}; (58,-6)*+{\{C^2_{s_is_{3-i},1},~C_{s_{3-i}s_i,s_{3-i}},~C^1_{s_i,1},~C^2_{s_i,1}\}};
(4,-12)*+{\Omega_1\left(C^1_{s_i,s_is_{3-i}}\right)}; (20,-12)*+{:=}; (68,-12)*+{\{C^1_{s_is_{3-i},s_is_{3-i}},~C_{s_{3-i}s_i,s_{3-i}},~C^2_{s_i,s_is_{3-i}},~C^1_{s_i,s_is_{3-i}}\}};
(4,-18)*+{\Omega_1\left(C^2_{s_i,s_is_{3-i}}\right)}; (20,-18)*+{:=}; (69.4,-18)*+{\{C^2_{s_is_{3-i},s_is_{3-i}},~C^1_{s_{3-i}s_i,s_{3-i}s_i},~C^1_{s_i,s_is_{3-i}},~C^2_{s_i,s_is_{3-i}}\}};
(0.65,-24)*+{\Omega_1\left(C_{s_i,s_i}\right)}; (20,-24)*+{:=}; (63.5,-24)*+{\{C_{s_is_{3-i},s_i},~C^1_{s_{3-i}s_i,1},~C^2_{s_{3-i}s_i,s_{3-i}s_i},~C_{s_i,s_i}\}};
\end{xy}$$
\[3lemm: nonvanishing ext3\] For $$V_0\in\{C^1_{s_i,1},~C^2_{s_i,1},~C^1_{s_i,s_is_{3-i}},~C^2_{s_i,s_is_{3-i}},~C_{s_i,s_i}\mid i=1,2\},$$ we have $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(V_0,~V\right)=1$$ if $V\in\Omega_1(V_0)$ and $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(V_0,~V\right)=0$$ if $V\in\Omega^-\setminus\Omega_1(V_0)$.
The proof is very similar to that of Lemma \[3lemm: nonvanishing ext3\].
\[3lemm: nonvanishing ext2\] For $$V_0\in\{\overline{L}(\lambda),~\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty},~\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty},~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\},$$ we have $$\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(V_0,~V\right)=1$$ if $V\in\Omega_2(V_0)$ and $$\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(V_0,~V\right)=0$$ if $V\in\Omega\setminus\Omega_2(V_0)$.
We only prove the statements for $V_0=\overline{L}(\lambda)$ as other cases are similar. If $$V\in\{\overline{L}(\lambda),~\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty},~\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty},~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\}$$ then the conclusion follows from Proposition \[3prop: locally algebraic extension\]. We notice that $Z(L_j(\mathbb{Q}_p))$ acts via different characters on $\overline{L}_j(\lambda)$, $\overline{L}_j(s_{3-j}\cdot\lambda)$ and $\overline{L}_j(s_{3-j}s_j\cdot\lambda)\otimes_E\pi_j^{\infty}$, and thus we have $$\label{3vanishing of ext 2.1}
\begin{xy}
(0,0)*+{\mathrm{Ext}^2_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(\lambda),~\overline{L}_j(s_{3-j}s_j\cdot\lambda)\otimes_E\pi_j^{\infty}\right)}; (50,0)*+{=0};
(4.6,-6)*+{\mathrm{Ext}^1_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(s_{3-j}\cdot\lambda),~\overline{L}_j(s_{3-j}s_j\cdot\lambda)\otimes_E\pi_j^{\infty}\right)}; (50,-6)*+{=0};
(0,-12)*+{\mathrm{Ext}^3_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(\lambda),~\overline{L}_j(s_{3-j}s_j\cdot\lambda)\otimes_E\pi_j^{\infty}\right)}; (50,-12)*+{=0};
\end{xy}$$ On the other hand, we notice that $$\label{3vanishing of ext 2.2}
\mathrm{Hom}_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(s_{3-j}s_j\cdot\lambda),~\overline{L}_j(s_{3-j}s_j\cdot\lambda)\otimes_E\pi_j^{\infty}\right)=0$$ for each smooth irreducible $\pi_j^{\infty}\neq 1_{L_j}$ and $$\label{3nonvanishing of ext 2.3}
\mathrm{dim}_E\mathrm{Hom}_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(s_{3-j}s_j\cdot\lambda),~\overline{L}_j(s_{3-j}s_j\cdot\lambda)\right)=1.$$ We combine (\[3vanishing of ext 2.1\]), (\[3vanishing of ext 2.2\]) and (\[3nonvanishing of ext 2.3\]) with Lemma \[3lemm: cohomology devissage\] and Proposition \[3prop: list of N coh\] and deduce that $$\label{3vanishing of ext 2.4}
\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}s_j\cdot\lambda),~\pi_j^{\infty})\right)=0$$ for each smooth irreducible $\pi_j^{\infty}\neq 1_{L_j}$ and $$\label{3nonvanishing of ext 2.5}
\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}s_j\cdot\lambda),~1_{L_j})\right)=1$$ which finishes the proof if $$V=\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}s_j\cdot\lambda),~\pi_j^{\infty}).$$ Similarly, we have $$\label{3vanishing of ext 2.6}
\begin{xy}
(0,0)*+{\mathrm{Ext}^2_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(\lambda),~\overline{L}_j(s_{3-j}\cdot\lambda)\otimes_E\pi_j^{\infty}\right)}; (50,0)*+{=0};
(7,-6)*+{\mathrm{Hom}_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(s_{3-j}s_j\cdot\lambda),~\overline{L}_j(s_{3-j}\cdot\lambda)\otimes_E\pi_j^{\infty}\right)}; (50,-6)*+{=0};
(0,-12)*+{\mathrm{Ext}^3_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(\lambda),~\overline{L}_j(s_{3-j}\cdot\lambda)\otimes_E\pi_j^{\infty}\right)}; (50,-12)*+{=0};
\end{xy}$$ On the other hand, we have $$\label{3vanishing of ext 2.8}
\mathrm{Ext}^1_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(s_{3-i}\cdot\lambda),~\overline{L}_j(s_{3-j}\cdot\lambda)\otimes_E\pi_j^{\infty}\right)=0$$ for each smooth irreducible $\pi_j^{\infty}\neq \pi_{j,1}^{\infty}$ and $$\label{3nonvanishing of ext 2.9}
\mathrm{dim}_E\mathrm{Ext}^1_{L_j(\mathbb{Q}_p),\lambda}\left(\overline{L}_j(s_{3-i}\cdot\lambda),~\overline{L}_j(s_{3-j}\cdot\lambda)\otimes_E\pi_{j,1}^{\infty}\right)=1.$$ We combine (\[3vanishing of ext 2.6\]), (\[3vanishing of ext 2.8\]) and (\[3nonvanishing of ext 2.9\]) with Lemma \[3lemm: cohomology devissage\] and Proposition \[3prop: list of N coh\] and deduce that $$\label{3vanishing of ext 2.10}
\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_{P_j}^{\mathrm{GL}_3}(M_j(-s_{3-j}\cdot\lambda),~\pi_j^{\infty})\right)=0$$ for each smooth irreducible $\pi_j^{\infty}\neq \pi_{j,1}^{\infty}$ and $$\label{3nonvanishing of ext 2.11}
\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_{P_j}^{\mathrm{GL}_3}(M_j(-s_{3-j}\cdot\lambda),~\pi_{j,1}^{\infty})\right)=1.$$ The short exact sequence $$\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}\cdot\lambda),~\pi_j^{\infty})\hookrightarrow\mathcal{F}_{P_j}^{\mathrm{GL}_3}(M_j(-s_{3-j}\cdot\lambda),~\pi_j^{\infty})\twoheadrightarrow\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}s_j\cdot\lambda),~\pi_j^{\infty})$$ induces a long exact sequence $$\begin{gathered}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}s_j\cdot\lambda),~\pi_j^{\infty})\right)\rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}\cdot\lambda),~\pi_j^{\infty})\right)\\
\rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_{P_j}^{\mathrm{GL}_3}(M_j(-s_{3-j}\cdot\lambda),~\pi_j^{\infty})\right)
\rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}s_j\cdot\lambda),~\pi_j^{\infty})\right)\end{gathered}$$ which finishes the proof if $$V=\mathcal{F}_{P_j}^{\mathrm{GL}_3}(L(-s_{3-j}\cdot\lambda),~\pi_j^{\infty}).$$ Finally, similar methods together with Proposition \[3prop: list of N coh\] also show that $$\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\mathcal{F}_B^{\mathrm{GL}_3}(L(-s_1s_2s_1\cdot\lambda),~\chi_w^{\infty})\right)=0$$ for each $w\in W$.
\[3lemm: special vanishing 1\] We have $$\begin{xy}
(-10,0)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(
\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="a"; (20,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy},~C^2_{s_i,1}\right)}; (50,0)*+{=0};
(0,-6)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="a"; (27,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy},~C^1_{s_i,s_is_{3-i}}\right)}; (50,-6)*+{=0};
(-10,-12)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\begin{xy}
(20,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="b"; (0,0)*+{\overline{L}(\lambda)}="a";
{\ar@{-}"a";"b"};
\end{xy},~C^1_{s_i,1}\right)}; (50,-12)*+{=0};
(0,-18)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="a"; (27,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy},~C^2_{s_i,s_is_{3-i}}\right)}; (50,-18)*+{=0};
\end{xy}$$ for $i=1,2$.
We only prove the first vanishing $$\label{3equation vanishing 1.1}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(
W_i,~C^2_{s_i,1}\right)=0$$ as the other cases are similar. The embedding $$C^2_{s_i,1}\hookrightarrow\mathcal{F}_{P_{3-i}}^{\mathrm{GL}_3}(M_{3-i}(-s_i\cdot\lambda),~\pi_{3-i,1}^{\infty})$$ induces an embedding $$\label{3equation vanishing 1.2}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(
W_i,~C^2_{s_i,1}\right)\hookrightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(
W_i,~\mathcal{F}_{P_{3-i}}^{\mathrm{GL}_3}(M_{3-i}(-s_i\cdot\lambda),~\pi_{3-i,1}^{\infty})\right).$$ It follows from Proposition \[3prop: list of N coh\] that $$\label{3equation vanishing 1.3}
\begin{array}{cccc}
H_0(N_{3-i},~W_i)&=&\overline{L}_{3-i}(\lambda)\otimes_E\left(i_{B\cap L_{3-i}}^{L_{3-i}}(\chi_{s_{3-i}}^{\infty})\oplus \mathfrak{d}_{P_{3-i}}^{\infty}\right)&\\
H_1(N_{3-i},~W_i)&=&\overline{L}_{3-i}(s_i\cdot\lambda)\otimes_E\left(i_{B\cap L_{3-i}}^{L_{3-i}}(\chi_{s_{3-i}}^{\infty})\oplus \mathfrak{d}_{P_{3-i}}^{\infty}\right)&
\end{array}.$$ We notice that $Z(L_{3-i}(\mathbb{Q}_p))$ acts on $\overline{L}_{3-i}(\lambda)$ and $\overline{L}_{3-i}(s_i\cdot\lambda)$ via different characters and that $$\mathrm{Hom}_{L_{3-i}(\mathbb{Q}_p),\lambda}\left(i_{B\cap L_{3-i}}^{L_{3-i}}(\chi_{s_{3-i}}^{\infty}),~\overline{L}_{3-i}(s_i\cdot\lambda)\otimes_E\pi_{3-i,1}^{\infty}\right)=0.$$ Therefore we deduce from (\[3equation vanishing 1.3\]) the equalities $$\begin{xy}
(0,0)*+{\mathrm{Ext}^1_{L_{3-i}(\mathbb{Q}_p),\lambda}\left(H_0(N_{3-i},~W_i),~\overline{L}_{3-i}(s_i\cdot\lambda)\otimes_E\pi_{3-i,1}^{\infty}\right)}; (47,0)*+{=0};
(1,-6)*+{\mathrm{Hom}_{L_{3-i}(\mathbb{Q}_p),\lambda}\left(H_1(N_{3-i},~W_i),~\overline{L}_{3-i}(s_i\cdot\lambda)\otimes_E\pi_{3-i,1}^{\infty}\right)}; (47,-6)*+{=0};
\end{xy}$$ which imply by Lemma \[3lemm: cohomology devissage\] that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(
W_i,~\mathcal{F}_{P_{3-i}}^{\mathrm{GL}_3}(M_{3-i}(-s_i\cdot\lambda),~\pi_{3-i,1}^{\infty})\right)=0.$$ Hence we finish the proof of (\[3equation vanishing 1.1\]) by the embedding (\[3equation vanishing 1.2\]).
\[3lemm: special vanishing 2\] We have for $i=1,2$: $$\begin{xy}
(0,0)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="a"; (22,0)*+{C_{s_i,s_i}}="b";
{\ar@{-}"a";"b"};
\end{xy},~C^2_{s_i,1}\right)}; (50,0)*+{=0};
(5.25,-6)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}}="a"; (27,0)*+{C^2_{s_i,s_is_{3-i}}}="b";
{\ar@{-}"a";"b"};
\end{xy},~C_{s_i,s_i}\right)}; (50,-6)*+{=0};
(-2.4,-12)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\begin{xy}
(20,0)*+{C^1_{s_i,s_is_{3-i}}}="b"; (0,0)*+{\overline{L}(\lambda)}="a";
{\ar@{-}"a";"b"};
\end{xy},~C^1_{s_i,1}\right)}; (50,-12)*+{=0};
(0,-18)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\begin{xy}
(7,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="a"; (22,0)*+{C^2_{s_i,1}}="b";
{\ar@{-}"a";"b"};
\end{xy},~C^2_{s_i,s_is_{3-i}}\right)}; (50,-18)*+{=0};
\end{xy}$$
We only prove that $$\label{3equation vanishing 2.1}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="a"; (22,0)*+{C_{s_i,s_i}}="b";
{\ar@{-}"a";"b"};
\end{xy},~C^2_{s_i,1}\right)=0$$ as the other cases are similar. The surjection $$\mathcal{F}_{P_{3-i}}^{\mathrm{GL}_3}(M_{3-i}(-\lambda),~\pi_{3-i,2}^{\infty})\twoheadrightarrow \begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="a"; (22,0)*+{C_{s_i,s_i}}="b";
{\ar@{-}"a";"b"};
\end{xy}$$ and the embedding $$C^2_{s_i,1}\hookrightarrow \mathcal{F}_{P_{3-i}}^{\mathrm{GL}_3}(M_{3-i}(-s_i\cdot\lambda),~\pi_{3-i,1}^{\infty})$$ induce an embedding $$\begin{gathered}
\label{3equation vanishing 2.2}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="a"; (22,0)*+{C_{s_i,s_i}}="b";
{\ar@{-}"a";"b"};
\end{xy},~C^2_{s_i,1}\right)\\
\hookrightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\mathcal{F}_{P_{3-i}}^{\mathrm{GL}_3}(M_{3-i}(-\lambda),~\pi_{3-i,2}^{\infty}),~\mathcal{F}_{P_{3-i}}^{\mathrm{GL}_3}(M_{3-i}(-s_i\cdot\lambda),~\pi_{3-i,1}^{\infty})\right).\end{gathered}$$ It follows from Proposition \[3prop: list of N coh\] that $$H_0(N_{3-i},~\mathcal{F}_{P_{3-i}}^{\mathrm{GL}_3}(M_{3-i}(-\lambda),~\pi_{3-i,2}^{\infty}))=\left(\overline{L}_{3-i}(\lambda)\oplus\overline{L}_{3-i}(s_i\cdot\lambda)\right)\otimes_E\pi_{3-i,2}^{\infty}$$ and $$\begin{gathered}
H_1(N_{3-i},~\mathcal{F}_{P_{3-i}}^{\mathrm{GL}_3}(M_{3-i}(-\lambda),~\pi_{3-i,2}^{\infty}))\\
=\left(\overline{L}_{3-i}(s_i\cdot\lambda)\oplus\overline{L}_{3-i}(s_is_{3-i}\cdot\lambda)\right)\otimes_E\pi_{3-i,2}^{\infty}\oplus I_{B\cap L_{3-i}}^{L_{3-i}}\left(\delta_{s_i\cdot\lambda}\right)\oplus I_{B\cap L_{3-i}}^{L_{3-i}}\left(\delta_{s_i\cdot\lambda}\otimes_E\chi_{s_1s_2s_1}^{\infty}\right).\end{gathered}$$ We notice that $Z(L_{3-i}(\mathbb{Q}_p))$ acts on each direct summand of $H_k(N_{3-i},~\mathcal{F}_{P_{3-i}}^{\mathrm{GL}_3}(M_{3-i}(-\lambda),~\pi_{3-i,2}^{\infty}))$ ($k=0,1$) via a different character, and the only direct summand that produces the same character as $\overline{L}_{3-i}(s_i\cdot\lambda)\otimes\pi_{3-i,1}^{\infty}$ is $I_{B\cap L_{3-i}}^{L_{3-i}}\left(\delta_{s_i\cdot\lambda}\right)$. However, we know that $$\mathrm{cosoc}_{L_{3-i}(\mathbb{Q}_p),\lambda}\left(I_{B\cap L_{3-i}}^{L_{3-i}}\left(\delta_{s_i\cdot\lambda}\right)\right)=I_{B\cap L_{3-i}}^{L_{3-i}}\left(\delta_{s_{3-i}s_i\cdot\lambda}\right)$$ and thus $$\mathrm{Hom}_{L_{3-i}(\mathbb{Q}_p),\lambda}\left(I_{B\cap L_{3-i}}^{L_{3-i}}\left(\delta_{s_{3-i}s_i\cdot\lambda}\right),~\overline{L}_{3-i}(s_i\cdot\lambda)\otimes\pi_{3-i,1}^{\infty}\right)=0.$$ As a result, we deduce the equalities $$\begin{xy}
(0,0)*+{\mathrm{Ext}^1_{L_{3-i}(\mathbb{Q}_p),\lambda}\left(H_0(N_{3-i},~\mathcal{F}_{P_{3-i}}^{\mathrm{GL}_3}(M_{3-i}(-\lambda),~\pi_{3-i,2}^{\infty})),~\overline{L}_{3-i}(s_i\cdot\lambda)\otimes_E\pi_{3-i,1}^{\infty}\right)}; (66,0)*+{=0};
(0.85,-6)*+{\mathrm{Hom}_{L_{3-i}(\mathbb{Q}_p),\lambda}\left(H_1(N_{3-i},~\mathcal{F}_{P_{3-i}}^{\mathrm{GL}_3}(M_{3-i}(-\lambda),~\pi_{3-i,2}^{\infty})),~\overline{L}_{3-i}(s_i\cdot\lambda)\otimes_E\pi_{3-i,1}^{\infty}\right)}; (66,-6)*+{=0};
\end{xy}$$ which imply by Lemma \[3lemm: cohomology devissage\] that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(
\mathcal{F}_{P_{3-i}}^{\mathrm{GL}_3}(M_{3-i}(-\lambda),~\pi_{3-i,2}^{\infty}),~\mathcal{F}_{P_{3-i}}^{\mathrm{GL}_3}(M_{3-i}(-s_i\cdot\lambda),~\pi_{3-i,1}^{\infty})\right)=0.$$ Hence we finish the proof of (\[3equation vanishing 2.1\]) by the embedding (\[3equation vanishing 2.2\]).
\[3lemm: existence of diamond\] There exists a unique representation of the form $$\begin{xy}
(0,0)*+{C^2_{s_i,1}}="a"; (25,6)*+{C^1_{s_{3-i}s_i,1}}="b"; (25,-6)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="c"; (50,0)*+{C_{s_i,s_i}}="d";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"d"};
\end{xy}$$ or of the form $$\begin{xy}
(0,0)*+{C_{s_i,s_i}}="a"; (27,6)*+{C^1_{s_{3-i}s_i,s_{3-i}s_i}}="b"; (27,-6)*+{\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}}="c"; (54,0)*+{C^2_{s_i,s_is_{3-i}}}="d";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"d"};
\end{xy}.$$
We only prove the first statement as the second is similar. It follows from Proposition 4.4.2 of [@Bre17] that there exists a unique representation of the form $$\begin{xy}
(0,0)*+{C^2_{s_i,1}}="a"; (25,6)*+{C^1_{s_{3-i}s_i,1}}="b"; (25,-6)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="c"; (50,0)*+{C_{s_i,s_i}}="d";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{--}"b";"d"}; {\ar@{-}"c";"d"};
\end{xy}$$ but it is not proven there whether its quotient $$\label{3equation split in diamond}
\begin{xy}
(0,0)*+{C^1_{s_{3-i}s_i,1}}="b"; (20,0)*+{C_{s_i,s_i}}="d";
{\ar@{--}"b";"d"};
\end{xy}$$ is split or not. However, If (\[3equation split in diamond\]) is split, then there exists a representation of the form $$\begin{xy}
(0,0)*+{C^2_{s_i,1}}="a"; (20,-0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="c"; (40,0)*+{C_{s_i,s_i}}="d";
{\ar@{-}"a";"c"}; {\ar@{-}"c";"d"};
\end{xy}$$ which contradicts the first vanishing in Lemma \[3lemm: special vanishing 2\], and thus we finish the proof.
\[3rema: existence of diamond\] Our method used in Lemma \[3lemm: special vanishing 2\] and in Lemma \[3lemm: existence of diamond\] is different from the one due to Y.Ding mentioned in part (ii) of Remark 4.4.3 of [@Bre17]. It is not difficult to observe that $$\label{3equation rema 1}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(C_{s_i,s_i},~\begin{xy}
(20,0)*+{C^2_{s_i,1}}="b"; (40,6)*+{C^1_{s_{3-i}s_i,1}}="c"; (40,-6)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="d";
{\ar@{-}"b";"c"}; {\ar@{-}"b";"d"};
\end{xy}\right)=1$$ and $$\label{3equation rema 2}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(C^2_{s_i,s_is_{3-i}},~\begin{xy}
(20,0)*+{C_{s_i,s_i}}="b"; (40,6)*+{C^1_{s_{3-i}s_i,s_{3-i}s_i}}="c"; (40,-6)*+{\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}}="d";
{\ar@{-}"b";"c"}; {\ar@{-}"b";"d"};
\end{xy}\right)=1$$ for $i=1,2$. Similar methods as those used in Proposition 4.4.2 of [@Bre17], in Lemma \[3lemm: special vanishing 2\] and in Lemma \[3lemm: existence of diamond\] also imply the existence of a unique representation of the form $$\begin{xy}
(0,0)*+{C^1_{s_i,1}}="a"; (25,6)*+{C_{s_{3-i}s_i,s_{3-i}}}="b"; (25,-6)*+{\overline{L}(\lambda)}="c"; (50,0)*+{C^1_{s_i,s_is_{3-i}}}="d";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"d"};
\end{xy}$$ or of the form $$\begin{xy}
(0,0)*+{C^2_{s_i,s_is_{3-i}}}="a"; (27,6)*+{C_{s_{3-i}s_i,s_{3-i}}}="b"; (27,-6)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="c"; (54,0)*+{C^2_{s_i,1}}="d";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"d"};
\end{xy}.$$
Computations of $\mathrm{Ext}$ II {#3section: technial min}
=================================
In this section, we are going to establish several computational results (most notably Lemma \[3lemm: ext6\]) which have crucial applications in Section \[3section: local-global\].
\[3lemm: ext3\] We have $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(C_{s_i,s_i},~\Sigma_i(\lambda, \mathscr{L}_i)\right)=1$$ for $i=1,2$.
We only prove that $$\label{3equation 3.1}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(C_{s_1,s_1},~\Sigma_1(\lambda, \mathscr{L}_1)\right)=1$$ as the other equality is similar. We note that $\Sigma_1(\lambda, \mathscr{L}_1)$ admits a subrepresentation of the form $$W:=\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="a"; (20,0)*+{C^2_{s_1,1}}="b"; (40,6)*+{C^1_{s_2s_1,1}}="c"; (40,-6)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="d";
{\ar@{-}"a";"b"}; {\ar@{-}"b";"c"}; {\ar@{-}"b";"d"};
\end{xy}$$ due to Lemma 3.34, Lemma 3.37 and Remark 3.38 of [@BD18]. Therefore $\Sigma_1(\lambda, \mathscr{L}_1))$ admits a filtration such that $W$ appears as one term of the filtration and the only reducible graded piece is $$V_1:=\begin{xy}
(20,0)*+{C^2_{s_1,1}}="b"; (40,6)*+{C^1_{s_2s_1,1}}="c"; (40,-6)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="d";
{\ar@{-}"b";"c"}; {\ar@{-}"b";"d"};
\end{xy}.$$ It follows from Lemma 4.4.1 and Proposition 4.2.1 of [@Bre17] as well as our Lemma \[3lemm: nonvanishing ext3\] that $$\label{3equation 3.2}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(C_{s_1,s_1},~V\right)=0$$ for all graded pieces $V$ such that $V\neq V_1$. On the other hand, we have $$\label{3equation 3.4}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(C_{s_1,s_1},~V_1\right)=1$$ due to (\[3equation rema 1\]) and $$\label{3equation 3.3}
\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(C_{s_1,s_1},~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)=0$$ by Proposition 4.6.1 of [@Bre17]. Hence we finish the proof by combining (\[3equation 3.2\]), (\[3equation 3.4\]), (\[3equation 3.3\]) and part (ii) of Proposition \[3prop: formal devissages\].
\[3lemm: ext4\] We have $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}, ~\Sigma^+_i(\lambda,\mathscr{L}_i)\right)=3$$ for $i=1,2$.
By symmetry, it suffices to prove that $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}, ~\Sigma^+_1(\lambda,\mathscr{L}_1\right)=3.$$ This follows immediately from Lemma 3.42 of [@Bre17] as our $\Sigma^+_1(\lambda,\mathscr{L}_1)$ can be identified with the locally analytic representation $\widetilde{\Pi}^1(\lambda, \psi)$ defined before (3.76) of [@Bre17] up to changes on notation.
We define $\Sigma^+_1(\lambda, \mathscr{L}_1)$ (resp. $\Sigma^+_2(\lambda, \mathscr{L}_2)$) as the unique non-split extension given by a non-zero element in $\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(C_{s_1,s_1}, ~\Sigma_1(\lambda, \mathscr{L}_1)\right)$ (resp. in $\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(C_{s_2,s_2}, ~\Sigma_2(\lambda, \mathscr{L}_2)\right)$). Hence we may consider the amalgamate sum of $\Sigma^+_1(\lambda, \mathscr{L}_1)$ and $\Sigma^+_2(\lambda, \mathscr{L}_2)$ over $\mathrm{St}_3^{\rm{an}}(\lambda)$ and denote it by $\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)$. In particular, $\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ has the following form $$\begin{xy}
(0,0)*+{\mathrm{St}_3^{\rm{an}}(\lambda)}="a"; (20,4)*+{v_{P_1}^{\rm{an}}(\lambda)}="b"; (20,-4)*+{v_{P_2}^{\rm{an}}(\lambda)}="c"; (40,4)*+{C_{s_1,s_1}}="d"; (40,-4)*+{C_{s_2,s_2}}="e";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"e"};
\end{xy}.$$
\[3lemm: ext5\] We have $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}, ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=2$$ for $i=1,2$.
The short exact sequence $$\Sigma^+_2(\lambda, \mathscr{L}_2)\hookrightarrow\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\twoheadrightarrow \begin{xy}
(0,0)*+{v_{P_1}^{\rm{an}}(\lambda)}="a"; (18,0)*+{C_{s_1,s_1}}="b";
{\ar@{-}"a";"b"};
\end{xy}$$ induces the following long exact sequence $$\begin{gathered}
\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}, ~\begin{xy}
(0,0)*+{v_{P_1}^{\rm{an}}(\lambda)}="a"; (18,0)*+{C_{s_1,s_1}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)\\
\hookrightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}, ~\Sigma^+_2(\lambda, \mathscr{L}_2)\right)\\
\rightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}, ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\\
\rightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}, ~\begin{xy}
(0,0)*+{v_{P_1}^{\rm{an}}(\lambda)}="a"; (18,0)*+{C_{s_1,s_1}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right).\end{gathered}$$ As a result, we can deduce $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}, ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=2$$ from Lemma \[3lemm: ext4\] and the facts $$\mathrm{dim}_E\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}, ~\begin{xy}
(0,0)*+{v_{P_1}^{\rm{an}}(\lambda)}="a"; (18,0)*+{C_{s_1,s_1}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)=1$$ and $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}, ~\begin{xy}
(0,0)*+{v_{P_1}^{\rm{an}}(\lambda)}="a"; (18,0)*+{C_{s_1,s_1}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)=0$$ by Proposition \[3prop: locally algebraic extension\] and Lemma \[3lemm: nonvanishing ext1\]. The proof for $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}, ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=2$$ is similar.
\[3lemm: ext6\] We have $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(W_i, ~\Sigma^+_i(\lambda, \mathscr{L}_i))=0$$ and in particular $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_i, ~\Sigma_i(\lambda, \mathscr{L}_i)\right)=0$$ for $i=1,2$.
We only need to show the vanishing $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(W_2,~\Sigma^+_1(\lambda, \mathscr{L}_1))=0$$ as the others are similar or easier. We define $\nu:=\lambda_{T_2, \iota_{T,1}}$ (which is the restriction of $\lambda$ from $T$ to $T_2$ via the embedding $\iota_{T,1}: T_2\hookrightarrow T$) and view $\Sigma^+_{\mathrm{GL}_2}(\nu, \mathscr{L}_1)$ (which is defined before Proposition \[3prop: key result\]) as a locally analytic representation of $L_1(\mathbb{Q}_p)$ via the projection $p_1: L_1\twoheadrightarrow\mathrm{GL}_2$ and denote it by $\Sigma^+_{L_1}(\lambda, \mathscr{L}_1)$. We note by definition by of $\Sigma_1(\lambda, \mathscr{L}_1)$ that we have an isomorphism $$\Sigma_1(\lambda, \mathscr{L}_1)\xrightarrow{\sim} I_{P_1}^{\mathrm{GL}_3}\left(\Sigma_{L_1}(\lambda, \mathscr{L}_1)\right)/\left(
\begin{xy}
(0,0)*+{v_{P_2}^{\rm{an}}(\lambda)}="a"; (16,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right).$$ Therefore we can deduce from the short exact sequence $$\Sigma^+_{\mathrm{GL}_2}(\nu, \mathscr{L}_1)\hookrightarrow \Sigma^+_{\mathrm{GL}_2}(\nu, \mathscr{L}_1)\twoheadrightarrow \widetilde{I}(s\cdot\nu)$$ and the fact (up to viewing $\widetilde{I}(s\cdot\nu)$ as a locally analytic representation of $L_1(\mathbb{Q}_p)$ via the projection $p_1$) $$C_{s_1,s_1}\cong\mathrm{soc}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(I_{P_1}^{\mathrm{GL}_3}\left(\widetilde{I}(s\cdot\nu)\right)\right)$$ that we have an injection $$\Sigma^+_1(\lambda, \mathscr{L}_1)\hookrightarrow I_{P_1}^{\mathrm{GL}_3}\left(\Sigma^+_{L_1}(\lambda, \mathscr{L}_1)\right)/\left(
\begin{xy}
(0,0)*+{v_{P_2}^{\rm{an}}(\lambda)}="a"; (16,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)$$ which induces an injection $$\label{3reduce via injection}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~\Sigma^+_1(\lambda, \mathscr{L}_1)\right)\hookrightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~V\right)$$ where we use the shorten notation $$V:=I_{P_1}^{\mathrm{GL}_3}\left(\Sigma^+_{L_1}(\lambda, \mathscr{L}_1)\right)/\left(\begin{xy}
(0,0)*+{v_{P_2}^{\rm{an}}(\lambda)}="a"; (18,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right).$$ Note that we have an exact sequence $$\begin{gathered}
\label{3key long exact sequence for vanishing}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~I_{P_1}^{\mathrm{GL}_3}\left(\Sigma^+_{L_1}(\lambda, \mathscr{L}_1)\right)\right)\\ \rightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~V\right) \rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2,~\begin{xy}
(0,0)*+{v_{P_2}^{\rm{an}}(\lambda)}="a"; (18,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)\end{gathered}$$ It follows from Proposition \[3prop: list of N coh\] that $$\begin{array}{cccc}
H_0(N_1, ~W_2)&=&\overline{L}_1(\lambda)\otimes_Ei_{B\cap L_1}^{L_1}(\chi_{s_1}^{\infty})&\\
H_1(N_1, ~W_2)&=&\overline{L}_1(s_2\cdot \lambda)\otimes_E\otimes_Ei_{B\cap L_1}^{L_1}(\chi_{s_1}^{\infty})&.
\end{array}$$ Therefore we observe that $$\mathrm{Hom}_{L_1(\mathbb{Q}_p),\lambda}\left(H_1(N_1, ~W_2),~\Sigma^+_{L_1}(\lambda, \mathscr{L}_1)\right)=0$$ from the action of $Z(L_1(\mathbb{Q}_p))$ and $$\mathrm{Ext}^1_{L_1(\mathbb{Q}_p),\lambda}\left(H_0(N_1, ~W_2),~\Sigma^+_{L_1}(\lambda, \mathscr{L}_1)\right)=0$$ according to Proposition \[3prop: key result\] and the natural identification $$\mathrm{Ext}^1_{L_1(\mathbb{Q}_p),\lambda}(-,-)\cong \mathrm{Ext}^1_{\mathrm{GL}_2(\mathbb{Q}_p)}(-,-).$$ As a result, we deduce $$\label{3intermediate vanishing}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~I_{P_1}^{\mathrm{GL}_3}\left(\Sigma^+_{L_1}(\lambda, \mathscr{L}_1)\right)\right)=0$$ from Lemma \[3lemm: cohomology devissage\]. We know that $$\label{3a vanishing ext2}
\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2,~\begin{xy}
(0,0)*+{v_{P_2}^{\rm{an}}(\lambda)}="a"; (18,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)=0$$ due to Proposition \[3prop: locally algebraic extension\], Lemma \[3lemm: nonvanishing ext2\] and a simple devissage, and thus we finish the proof by (\[3reduce via injection\]), (\[3key long exact sequence for vanishing\]), (\[3intermediate vanishing\]) and (\[3a vanishing ext2\]).
\[3lemm: ext6.1\] We have $$\label{3first key dim}
\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^+_i(\lambda, \mathscr{L}_i)\right)=3$$ for each $i=1,2$, $$\label{3second key dim}
\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=2$$ and $$\label{3third key dim}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=1.$$
The equalities (\[3second key dim\]) and (\[3third key dim\]) follow directly from Lemma \[3lemm: ext2\] and the fact that $$\label{3vanishing of simple special ext}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~C_{s_i,s_i}\right)=\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~C_{s_i,s_i}\right)=0$$ by Lemma \[3lemm: nonvanishing ext1\] and Lemma \[3lemm: nonvanishing ext2\] using a long exact sequence induced from the short exact sequence $$\Sigma_i(\lambda, \mathscr{L}_i)\hookrightarrow\Sigma^+_i(\lambda, \mathscr{L}_i)\twoheadrightarrow C_{s_i,s_i}.$$ Due to a similar argument using (\[3vanishing of simple special ext\]), we only need to show that $$\label{3equation ext 6.1.1}
\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma_i(\lambda, \mathscr{L}_i)\right)=3$$ to finish the proof of (\[3first key dim\]). The short exact sequence $$\mathrm{St}_3^{\rm{an}}(\lambda)\hookrightarrow\Sigma_i(\lambda, \mathscr{L}_i)\twoheadrightarrow v_{P_i}^{\rm{an}}(\lambda)$$ induces a long exact sequence $$\begin{gathered}
\label{3equation ext 6.1.2}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma_i(\lambda, \mathscr{L}_i)\right)\rightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~v_{P_i}^{\rm{an}}(\lambda)\right)\\
\rightarrow \mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\mathrm{St}_3^{\rm{an}}(\lambda)\right)\rightarrow
\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma_i(\lambda, \mathscr{L}_i)\right)\rightarrow
\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~v_{P_i}^{\rm{an}}(\lambda)\right).\end{gathered}$$ We know that $$\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\mathrm{St}_3^{\rm{an}}(\lambda)\right)=5$$ by Lemma \[3lemm: ext2.1\]. It follows from Proposition \[3prop: locally algebraic extension\], Lemma \[3lemm: nonvanishing ext1\], Lemma \[3lemm: nonvanishing ext2\] and a simple devissage that $$\label{3equation ext 6.1.2 prime}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~v_{P_i}^{\rm{an}}(\lambda)\right)=2$$ and $$\label{3equation ext 6.1.2 primeprime}
\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~v_{P_i}^{\rm{an}}(\lambda)\right)=0.$$ Hence it remains to show that $$\label{3equation ext 6.1.3}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma_i(\lambda, \mathscr{L}_i)\right)=0$$ to deduce (\[3equation ext 6.1.1\]) from (\[3equation ext 6.1.2\]). The short exact sequence $$\begin{xy}
(0,0)*+{v_{P_{3-i}}^{\rm{an}}(\lambda)}="a"; (16,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\hookrightarrow I_{P_i}^{\mathrm{GL}_3}\left(\Sigma_{L_i}(\lambda, \mathscr{L}_i)\right)\twoheadrightarrow\Sigma_i(\lambda, \mathscr{L}_i)$$ induces $$\begin{gathered}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\begin{xy}
(0,0)*+{v_{P_{3-i}}^{\rm{an}}(\lambda)}="a"; (16,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)\\
\hookrightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~I_{P_i}^{\mathrm{GL}_3}\left(\Sigma_{L_i}(\lambda, \mathscr{L}_i)\right)\right)\twoheadrightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma_i(\lambda, \mathscr{L}_i)\right)\end{gathered}$$ by the vanishing $$\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), \begin{xy}
(0,0)*+{v_{P_{3-i}}^{\rm{an}}(\lambda)}="a"; (16,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)=0$$ using Proposition \[3prop: locally algebraic extension\] and Lemma \[3lemm: nonvanishing ext2\]. Therefore we only need to show that $$\label{3equation ext 6.1.4}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\begin{xy}
(0,0)*+{v_{P_{3-i}}^{\rm{an}}(\lambda)}="a"; (16,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)=1$$ and $$\label{3equation ext 6.1.5}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~I_{P_i}^{\mathrm{GL}_3}\left(\Sigma_{L_i}(\lambda, \mathscr{L}_i)\right)\right)=1.$$ The equality (\[3equation ext 6.1.5\]) follows from Lemma \[3lemm: cohomology devissage\] and the facts $$\mathrm{dim}_E\mathrm{Ext}^1_{L_i(\mathbb{Q}_p),\lambda}\left(H_0(N_i,~\overline{L}(\lambda)),~\Sigma_{L_i}(\lambda, \mathscr{L}_i)\right)=1,~\mathrm{Hom}_{L_i(\mathbb{Q}_p),\lambda}\left(H_1(N_i,~\overline{L}(\lambda)),~\Sigma_{L_i}(\lambda, \mathscr{L}_i)\right)=0$$ where the first equality essentially follows from Lemma 3.14 of [@BD18] and the second equality follows from checking the action of $Z(L_i(\mathbb{Q}_p))$. On the other hand, (\[3equation ext 6.1.4\]) follows from (\[3equation ext 6.1.2 prime\]) and Proposition \[3prop: locally algebraic extension\] by an easy devissage. Hence we finish the proof.
\[3prop: isomorphism from cup product\] The short exact sequence $$\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}\hookrightarrow W_i\twoheadrightarrow \overline{L}(\lambda)$$ induces the following isomorphisms $$\label{3first key isomorphism}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}, ~\Sigma^+_i(\lambda, \mathscr{L}_i)\right)\xrightarrow{\sim}\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^+_i(\lambda, \mathscr{L}_i)\right)$$ and $$\label{3second key isomorphism}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}, ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\xrightarrow{\sim}\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$$ for $i=1,2$.
The vanishing from Lemma \[3lemm: ext6\] implies that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}, ~\Sigma^+_i(\lambda, \mathscr{L}_i)\right)\rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^+_i(\lambda, \mathscr{L}_i)\right)$$ is an injection and hence an isomorphism as both spaces have dimension three according to Lemma \[3lemm: ext4\] and Lemma \[3lemm: ext6.1\]. The proof of (\[3second key isomorphism\]) is similar. We emphasize that both (\[3first key isomorphism\]) and (\[3second key isomorphism\]) can be interpreted as the isomorphism given by the cup product with the one dimensional space $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}\right).$$
We define $$\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2):=\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)/\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\mbox{ and }\Sigma^{\flat}_i(\lambda, \mathscr{L}_i):=\Sigma_i(\lambda, \mathscr{L}_i)/\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}$$ for $i=1,2$.
\[3lemm: ext7\] We have $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=1.$$
We define $\Sigma^{\flat,-}(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ as the subrepresentation of $\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ that fits into the following short exact sequence $$\label{3equation 7.1}
\Sigma^{\flat,-}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\hookrightarrow\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\twoheadrightarrow C^1_{s_2,1}\oplus C^1_{s_1,1},$$ ( cf. (\[3irr rep I\]) for the definition of $C^1_{s_2,1}$, $C^1_{s_1,1}$, $C^2_{s_2,1}$ and $C^2_{s_1,1}$ ) and then define $\Sigma^{\flat,--}(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ as the subrepresentation of $\Sigma^{\flat,-}(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ that fits into $$\label{3equation 7.1 prime}
\Sigma^{\flat,--}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\hookrightarrow\Sigma^{\flat,-}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\twoheadrightarrow \left(\begin{xy}
(0,0)*+{C^2_{s_1,1}}="a"; (20,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)\oplus \left(\begin{xy}
(0,0)*+{C^2_{s_2,1}}="a"; (20,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right).$$ It follows from Lemma \[3lemm: nonvanishing ext1\] that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~V)=0$$ for each $V\in\mathrm{JH}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(\Sigma^{\flat,--}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$ and therefore $$\label{3equation 7.1.1}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\flat,--}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=0$$ by part (i) of Proposition \[3prop: formal devissages\]. On the other hand, we know from Lemma \[3lemm: nonvanishing ext1\] and Lemma \[3lemm: special vanishing 1\] that there is no uniserial representation of the form $$\begin{xy}
(0,0)*+{C^2_{s_i,1}}="a"; (20,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="b"; (40,0)*+{\overline{L}(\lambda)}="c";
{\ar@{-}"a";"b"}; {\ar@{-}"b";"c"};
\end{xy}$$ which implies that $$\label{3equation 7.1.2}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\begin{xy}
(0,0)*+{C^2_{s_i,1}}="a"; (20,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)=0$$ for $i=1,2$. Hence we deduce from (\[3equation 7.1 prime\]), (\[3equation 7.1.1\]), (\[3equation 7.1.2\]) and Proposition \[3prop: formal devissages\] that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\flat,-}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=0.$$ Therefore (\[3equation 7.1\]) induces an injection $$\label{3equation 7.2}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\hookrightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~C^1_{s_2,1}\oplus C^1_{s_1,1}\right).$$ Assume first that (\[3equation 7.2\]) is a surjection, then we pick a representation $W$ represented by a non-zero element in $\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$ lying in the preimage of $\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~C^1_{s_2,1}\right)$ under (\[3equation 7.2\]). We note that there is a short exact sequence $$\Sigma^{\flat}_1(\lambda, \mathscr{L}_1)\hookrightarrow\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\twoheadrightarrow v_{P_2}^{\rm{an}}(\lambda).$$ We observe that $\overline{L}(\lambda)$ lies above neither $C^1_{s_1,1}$ nor $\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}$ inside $W$ by our definition and (\[3equation 7.1.2\]), and thus $W$ is mapped to zero under the map $$f:\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\rightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~v_{P_2}^{\rm{an}}(\lambda)\right)$$ which means that $W$ comes from an element in $$\mathrm{Ker}(f)=\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\flat}_1(\lambda, \mathscr{L}_1)\right)$$ and in particular $$\label{3equation 7.6}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\flat}_1(\lambda, \mathscr{L}_1)\right)\neq 0$$ The short exact sequence $$\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}\hookrightarrow W_2\twoheadrightarrow\overline{L}(\lambda)$$ induces an injection $$\label{3equation 7.3}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\flat}_1(\lambda, \mathscr{L}_1)\right)\hookrightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~\Sigma^{\flat}_1(\lambda, \mathscr{L}_1)\right).$$ On the other hand, the short exact sequence $$\label{3equation 7.3.1}
\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\hookrightarrow\Sigma_1(\lambda, \mathscr{L}_1)\twoheadrightarrow\Sigma^{\flat}_1(\lambda, \mathscr{L}_1)$$ induces a long exact sequence $$\begin{gathered}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\rightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~\Sigma_1(\lambda, \mathscr{L}_1)\right)\\ \rightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~\Sigma^{\flat}_1(\lambda, \mathscr{L}_1)\right)\rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\end{gathered}$$ which implies $$\label{3equation 7.4}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~\Sigma_1(\lambda, \mathscr{L}_1)\right)\xrightarrow{\sim}\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~\Sigma^{\flat}_1(\lambda, \mathscr{L}_1)\right)$$ as we have $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)=\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)=0$$ from Lemma \[3lemm: vanishing locally algebraic\]. We combine Lemma \[3lemm: ext6\], (\[3equation 7.3\]) and (\[3equation 7.4\]) and deduce that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\flat}_1(\lambda, \mathscr{L}_1)\right)=0$$ which contradicts (\[3equation 7.6\]). In all, we have thus shown that $$\label{3equation 7.5}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)<\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~C^1_{s_2,1}\oplus C^1_{s_1,1}\right)=2$$ by combining Lemma \[3lemm: nonvanishing ext1\]. Finally, the vanishing $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)=0$$ from Proposition \[3prop: locally algebraic extension\] implies an injection $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\hookrightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$$ which finishes the proof by combining Lemma \[3lemm: ext2\] and (\[3equation 7.5\]).
\[3lemm: ext8\] We have $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=2.$$
The short exact sequence $$\Sigma_i^{\flat}(\lambda, \mathscr{L}_i)\hookrightarrow\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\twoheadrightarrow v_{P_{3-i}}^{\rm{an}}(\lambda)$$ induces a long exact sequence $$\begin{gathered}
\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E v_{P_{3-i}}^{\infty}, ~v_{P_{3-i}}^{\rm{an}}(\lambda)\right)\hookrightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E v_{P_{3-i}}^{\infty}, ~\Sigma_i^{\flat}(\lambda, \mathscr{L}_i)\right)\\
\rightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E v_{P_{3-i}}^{\infty}, ~\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\rightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E v_{P_{3-i}}^{\infty}, ~v_{P_{3-i}}^{\rm{an}}(\lambda)\right).\end{gathered}$$ It is easy to observe that $$\mathrm{dim}_E\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E v_{P_{3-i}}^{\infty}, ~v_{P_{3-i}}^{\rm{an}}(\lambda)\right)=1$$ and $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E v_{P_{3-i}}^{\infty}, ~v_{P_{3-i}}^{\rm{an}}(\lambda)\right)=0$$ from Proposition \[3prop: locally algebraic extension\] and Lemma \[3lemm: nonvanishing ext1\]. We can actually observe from Lemma \[3lemm: nonvanishing ext1\] that the only $V\in\mathrm{JH}_{\mathrm{GL}_3(\mathbb{Q}_p)}(\Sigma_i^{\flat}(\lambda, \mathscr{L}_i))$ such that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E v_{P_{3-i}}^{\infty}, ~V\right)\neq 0$$ is $V=C^2_{s_{3-i}, 1}$ and $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E v_{P_{3-i}}^{\infty}, ~C^2_{s_{3-i}, 1}\right)=1.$$ Hence we deduce that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E v_{P_{3-i}}^{\infty}, ~\Sigma_i^{\flat}(\lambda, \mathscr{L}_i)\right)\leq 1$$ and therefore $$\label{3equation 8.1}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E v_{P_{3-i}}^{\infty}, ~\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=0$$ for $i=1,2$. The short exact sequence $$\overline{L}(\lambda)\otimes_E \left(v_{P_1}^{\infty}\oplus v_{P_2}^{\infty}\right)\hookrightarrow W_0\twoheadrightarrow \overline{L}(\lambda)$$ induces $$\begin{gathered}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\hookrightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\\ \rightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E \left(v_{P_{1}}^{\infty}\oplus v_{P_2}^{\infty}\right), ~\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\end{gathered}$$ which implies $$\label{3equation 8.2}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\xrightarrow{\sim} \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$$ by (\[3equation 8.1\]). Finally, the short exact sequence (\[3equation 7.3.1\]) induces $$\begin{gathered}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\hookrightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\\
\rightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma^{\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\end{gathered}$$ which finishes the proof by $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)=1\mbox{ and }\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)=0$$ from Lemma \[3lemm: vanishing locally algebraic main\], and by Lemma \[3lemm: ext7\] as well as (\[3equation 8.2\]).
\[3lemm: ext9\] We have the inequality $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\begin{xy}
(0,0)*+{v_{P_i}^{\rm{an}}(\lambda)}="a"; (18,0)*+{C_{s_i,s_i}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)\leq 2$$ for $i=1,2$.
We know that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_j}^{\infty}, ~C^1_{s_i,1}\right)=\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_j}^{\infty}, ~\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}\right)=0$$ for $i,j=1,2$ from Proposition \[3prop: locally algebraic extension\] and Lemma \[3lemm: nonvanishing ext1\], and thus $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_j}^{\infty}, ~v_{P_i}^{\rm{an}}(\lambda)\right)=0$$ for $i,j=1,2$ which together with (\[3equation ext 6.1.2 prime\]) imply that $$\begin{gathered}
\label{3equation 9.1}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~v_{P_i}^{\rm{an}}(\lambda)\right)\leq \mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_i, ~v_{P_i}^{\rm{an}}(\lambda)\right)\\
\leq \mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~v_{P_i}^{\rm{an}}(\lambda)\right)-\mathrm{dim}_E\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}, ~v_{P_i}^{\rm{an}}(\lambda)\right)\\
=2-1=1.\end{gathered}$$ On the other hand, note that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~C_{s_i,s_i}\right)=\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}, ~C_{s_i,s_i}\right)=0$$ by Lemma \[3lemm: nonvanishing ext1\] and thus we have $$\label{3equation 9.2}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~C_{s_i,s_i}\right)\leq \mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}, ~C_{s_i,s_i}\right)=1$$ where the last equality follows again from Lemma \[3lemm: nonvanishing ext1\]. We finish the proof by combining (\[3equation 9.1\]) and (\[3equation 9.2\]) with the inequality $$\begin{gathered}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\begin{xy}
(0,0)*+{v_{P_i}^{\rm{an}}(\lambda)}="a"; (18,0)*+{C_{s_i,s_i}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)\\
\leq \mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~v_{P_i}^{\rm{an}}(\lambda)\right)+\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~C_{s_i,s_i}\right).\end{gathered}$$
Key exact sequences {#3section: exact sequence min}
===================
\[3lemm: upper bound\] We have the inequality $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\leq 3.$$
The short exact sequence $$\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\hookrightarrow\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\twoheadrightarrow C_{s_1,s_1}\oplus C_{s_2,s_2}$$ induces the exact sequence $$\begin{gathered}
\label{3first key exact sequence}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\hookrightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\\ \rightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~C_{s_1,s_1}\oplus C_{s_2,s_2}\right).\end{gathered}$$ We know that $$\begin{gathered}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~C_{s_1,s_1}\oplus C_{s_2,s_2}\right)\\=\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~C_{s_1,s_1}\right)+\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~C_{s_2,s_2}\right)=1+1=2\end{gathered}$$ by Lemma \[3lemm: nonvanishing ext1\] and Lemma \[3lemm: nonvanishing ext2\]. We also know that $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=2$$ by Lemma \[3lemm: ext8\], and thus we obtain the following inequality: $$\begin{gathered}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\\ \leq \mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)+\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~C_{s_1,s_1}\oplus C_{s_2,s_2}\right)
=2+2=4.\end{gathered}$$ Assume first that $$\label{3assumption upper bound}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=4.$$ The short exact sequence $$\Sigma^+_1(\lambda,\mathscr{L}_1)\hookrightarrow \Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\twoheadrightarrow \left(\begin{xy}
(0,0)*+{v_{P_2}^{\rm{an}}(\lambda)}="a"; (18,0)*+{C_{s_2,s_2}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)$$ induces a long exact sequence $$\begin{gathered}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0,~\Sigma^+_1(\lambda,\mathscr{L}_1)\right)\hookrightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0,~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\\ \rightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0,~\begin{xy}
(0,0)*+{v_{P_2}^{\rm{an}}(\lambda)}="a"; (18,0)*+{C_{s_2,s_2}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)\end{gathered}$$ which implies $$\label{3lower bound}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0,~\Sigma^+_1(\lambda,\mathscr{L}_1)\right)\geq 2$$ by (\[3assumption upper bound\]) and Lemma \[3lemm: ext9\]. We observe that $\Sigma^+_1(\lambda,\mathscr{L}_1)$ admits a filtration whose only reducible graded piece is $$\begin{xy}
(0,0)*+{C^2_{s_1,1}}="a"; (20,0)*+{\overline{L}(\lambda)\otimes_E v_{P_1}^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy}$$ and thus it follows from Lemma \[3lemm: nonvanishing ext1\] and $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E v_{P_1}^{\infty},~C^2_{s_1,1}-\overline{L}(\lambda)\otimes_E v_{P_1}^{\infty}\right)=0$$ (coming from Proposition \[3prop: locally algebraic extension\], Lemma \[3lemm: nonvanishing ext1\] together with a simple devissage) that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E v_{P_1}^{\infty},~V\right)=0$$ for all graded pieces of such a filtration except the subrepresentation $\overline{L}(\lambda)\otimes_E \mathrm{St}_3^{\infty}$. Hence we deduce by part (ii) of Proposition \[3prop: formal devissages\] an isomorphism of one dimensional spaces $$\label{3easy computation}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E v_{P_1}^{\infty}, ~\overline{L}(\lambda)\otimes_E \mathrm{St}_3^{\infty}\right)\xrightarrow{\sim}\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E v_{P_1}^{\infty}, ~\Sigma^+_1(\lambda,\mathscr{L}_1)\right).$$ Then the short exact sequence $$\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}\hookrightarrow W_0\twoheadrightarrow W_2$$ induces a long exact sequence $$\begin{gathered}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~\Sigma^+_1(\lambda,\mathscr{L}_1)\right)\hookrightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma^+_1(\lambda,\mathscr{L}_1)\right)\\
\rightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E v_{P_1}^{\infty}, ~\Sigma^+_1(\lambda,\mathscr{L}_1)\right)\end{gathered}$$ which together with (\[3lower bound\]) and (\[3easy computation\]) implies that $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2,~\Sigma^+_1(\lambda,\mathscr{L}_1)\right)\geq 1$$ which contradicts Lemma \[3lemm: ext6\]. Hence we finish the proof.
\[3prop: main dim\] We have $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=3.$$
The short exact sequence $$\overline{L}(\lambda)\otimes_E\left(v_{P_2}^{\infty}\oplus v_{P_1}^{\infty}\right)\hookrightarrow W_0\twoheadrightarrow \overline{L}(\lambda)$$ induces a long exact sequence $$\begin{gathered}
\label{3main long exact sequence}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\hookrightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\\
\rightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_E \left(v_{P_2}^{\infty}\oplus v_{P_1}^{\infty}\right), ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\rightarrow \mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\end{gathered}$$ and thus we have $$\begin{gathered}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(W_0, ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2))\\
\geq \mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2))+\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda)\otimes_E\left(v_{P_2}^{\infty}\oplus v_{P_1}^{\infty}\right), ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2))\\-\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2))=1+4-2=3\end{gathered}$$ due to Lemma \[3lemm: ext5\] and Lemma \[3lemm: ext6.1\], which finishes the proof by combining with Lemma \[3lemm: upper bound\].
We define $\Sigma^{\sharp}(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ as the unique non-split extension of $\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ by $\overline{L}(\lambda)$ ( cf. Lemma \[3lemm: ext2\]) and then set $\Sigma^{\sharp,+}(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ to be the amalgamate sum of $\Sigma^{\sharp}(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ and $\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ over $\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)$. Hence $\Sigma^{\sharp}(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ has the form $$\begin{xy}
(0,0)*+{\mathrm{St}_3^{\rm{an}}(\lambda)}="a"; (20,4)*+{v_{P_1}^{\rm{an}}(\lambda)}="b"; (20,-4)*+{v_{P_2}^{\rm{an}}(\lambda)}="c"; (40,0)*+{\overline{L}(\lambda)}="d";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"d"};
\end{xy}$$ and $\Sigma^{\sharp,+}(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ has the form $$\begin{xy}
(0,0)*+{\mathrm{St}_3^{\rm{an}}(\lambda)}="a"; (20,4)*+{v_{P_1}^{\rm{an}}(\lambda)}="b"; (20,-4)*+{v_{P_2}^{\rm{an}}(\lambda)}="c"; (40,8)*+{C_{s_1,s_1}}="d"; (40,-8)*+{C_{s_2,s_2}}="e"; (40,0)*+{\overline{L}(\lambda)}="f";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"e"}; {\ar@{-}"b";"f"}; {\ar@{-}"c";"f"};
\end{xy}.$$ It follows from Lemma \[3lemm: ext2\], Proposition \[3prop: locally algebraic extension\], (\[3vanishing of simple special ext\]) and an easy devissage that $$\label{3vanishing of ext}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^{\sharp}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^{\sharp,+}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=0.$$ Then we set $$\Sigma^{\ast,\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2):=\Sigma^{\ast}(\lambda, \mathscr{L}_1, \mathscr{L}_2)/\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}$$ for $\ast=\{+\}, \{\sharp\}$ and $\{\sharp, +\}$. It follows from Lemma \[3lemm: ext7\], (\[3vanishing of simple special ext\]) and an easy devissage that $$\label{3vanishing of ext prime}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^{\sharp, \flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^{\sharp,+, \flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=0.$$
\[3lemm: ext10\] We have $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp}(\lambda, \mathscr{L}_1,\mathscr{L}_2)\right)=\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+}(\lambda, \mathscr{L}_1,\mathscr{L}_2)\right)=0$$ and $$\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=2$$
It follows from (\[3vanishing of simple special ext\]) that we only need to show that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp}(\lambda, \mathscr{L}_1,\mathscr{L}_2)\right)=0\mbox{ and }\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=2.$$ These results follow from combining the long exact sequence $$\begin{gathered}
\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\overline{L}(\lambda)\right)\hookrightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma(\lambda, \mathscr{L}_1,\mathscr{L}_2)\right)\\ \rightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp}(\lambda, \mathscr{L}_1,\mathscr{L}_2)\right) \rightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\overline{L}(\lambda)\right)\\ \rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma(\lambda, \mathscr{L}_1,\mathscr{L}_2)\right)\rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp}(\lambda, \mathscr{L}_1,\mathscr{L}_2)\right) \\ \rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\overline{L}(\lambda)\right),\end{gathered}$$ with Lemma \[3lemm: ext2\] and the equalities $$\begin{array}{cccc}
\mathrm{dim}_E&\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\overline{L}(\lambda))&=&1\\
&\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\overline{L}(\lambda))&=&0\\
&\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\overline{L}(\lambda))&=&0
\end{array}$$ due to Proposition \[3prop: locally algebraic extension\].
\[3lemm: ext11\] We have $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\Sigma^{\sharp,\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2))=\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\Sigma^{\sharp,+,\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2))=0$$ and $$\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\Sigma^{\sharp,\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2))= \mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\Sigma^{\sharp,+,\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2))\geq 1.$$
It follows from (\[3vanishing of simple special ext\]) that we only need to show that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\Sigma^{\sharp,\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2))=0\mbox{ and }\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\Sigma^{\sharp,\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2))\geq 1,$$ which follow from combining (\[3vanishing of ext prime\]), Lemma \[3lemm: ext10\] and the long exact sequence $$\begin{gathered}
\label{3equation 11.1}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty})\rightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\Sigma^{\sharp}(\lambda, \mathscr{L}_1, \mathscr{L}_2))\\ \rightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\Sigma^{\sharp,\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2)) \rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty})\\ \rightarrow \mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\Sigma^{\sharp}(\lambda, \mathscr{L}_1, \mathscr{L}_2))\rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\Sigma^{\sharp,\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2))\end{gathered}$$ with the equalities $$\begin{array}{cccc}
&\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty})&=&0\\
\mathrm{dim}_E&\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty})&=&1\\
\end{array}$$ due to Proposition \[3prop: locally algebraic extension\].
We use the shorten notation $\underline{\mathscr{L}}:=(\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_1^{\prime}, \mathscr{L}_2^{\prime})$ for a tuple of four elements in $E$. We recall from Proposition \[3prop: isomorphism from cup product\] an isomorphism of two dimensional spaces $$\label{3isomorphism for normalization}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty},~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\xrightarrow{\sim}\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right).$$ We emphasize that the isomorphism (\[3isomorphism for normalization\]) can be naturally interpreted as the cup product map $$\begin{gathered}
\label{3cup product interpretation}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty},~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)~\cup~ \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}\right)\\
\rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\end{gathered}$$ where $\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}\right)$ is one dimensional by Proposition \[3prop: locally algebraic extension\]. We recall from the proof of Lemma \[3lemm: ext6.1\] that there is a canonical isomorphism $$\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\xrightarrow{\sim}\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$$ which together with Lemma \[3lemm: ext2\] implies that $\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$ admits a basis of the form $$\{\kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{val}_p}), \iota_1(D_0)\},$$ and therefore the element $$\iota_1(D_0)+\mathscr{L}\kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{val}_p})$$ generates a line in $\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$ for each $\mathscr{L}\in E$. We define $\Sigma^+_i(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_i^{\prime})$ as the representation represent by the preimage of $$\iota_1(D_0)+\mathscr{L}_i^{\prime}\kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{val}_p})$$ in $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty},~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$$ via (\[3isomorphism for normalization\]) for $i=1,2$. Then we define $\Sigma^+(\lambda,\underline{\mathscr{L}})$ as the amalgamate sum of $\Sigma^+_1(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_1^{\prime})$ and $\Sigma^+_2(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_2^{\prime})$ over $\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)$, and therefore $\Sigma^+(\lambda,\underline{\mathscr{L}})$ has the form $$\begin{xy}
(0,0)*+{\mathrm{St}_3^{\rm{an}}(\lambda)}="a"; (20,4)*+{v_{P_1}^{\rm{an}}(\lambda)}="b"; (20,-4)*+{v_{P_2}^{\rm{an}}(\lambda)}="c"; (40,4)*+{C_{s_1,s_1}}="d"; (40,-4)*+{C_{s_2,s_2}}="e"; (60,4)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="g"; (60,-4)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="h";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"e"}; {\ar@{-}"d";"g"}; {\ar@{-}"e";"h"};
\end{xy}.$$ We define $\Sigma^{\sharp,+}(\lambda,\underline{\mathscr{L}})$ as the amalgamate sum of $\Sigma^{+}(\lambda,\underline{\mathscr{L}})$ and $\Sigma^{\sharp}(\lambda,\mathscr{L}_1, \mathscr{L}_2)$ over $\Sigma(\lambda,\mathscr{L}_1, \mathscr{L}_2)$, and thus $\Sigma^{\sharp,+}(\lambda,\underline{\mathscr{L}})$ has the form $$\begin{xy}
(0,0)*+{\mathrm{St}_3^{\rm{an}}(\lambda)}="a"; (20,4)*+{v_{P_1}^{\rm{an}}(\lambda)}="b"; (20,-4)*+{v_{P_2}^{\rm{an}}(\lambda)}="c"; (40,8)*+{C_{s_1,s_1}}="d"; (40,-8)*+{C_{s_2,s_2}}="e"; (40,0)*+{\overline{L}(\lambda)}="f"; (60,8)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="g"; (60,-8)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="h";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"e"}; {\ar@{-}"b";"f"}; {\ar@{-}"c";"f"}; {\ar@{-}"d";"g"}; {\ar@{-}"e";"h"};
\end{xy}.$$ We also need the quotients $$\Sigma^{+,\flat}(\lambda,\underline{\mathscr{L}}):=\Sigma^+(\lambda,\underline{\mathscr{L}})/\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty},~\Sigma^{\sharp,+,\flat}(\lambda,\underline{\mathscr{L}}):=\Sigma^{\sharp,+}(\lambda,\underline{\mathscr{L}})/\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}.$$
\[3lemm: ext13\] We have the inequality $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+,\flat}(\lambda,\underline{\mathscr{L}})\right)\leq 1.$$
The short exact sequence $$\Sigma^{\sharp,+,\flat}(\lambda,\mathscr{L}_1,\mathscr{L}_2)\hookrightarrow\Sigma^{\sharp,+,\flat}(\lambda,\underline{\mathscr{L}})\twoheadrightarrow \overline{L}(\lambda)\otimes_E\left(v_{P_2}^{\infty}\oplus v_{P_1}^{\infty}\right)$$ induces an injection $$\label{3equation 13.1}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+,\flat}(\lambda,\underline{\mathscr{L}})\right)\hookrightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\overline{L}(\lambda)\otimes_E\left(v_{P_2}^{\infty}\oplus v_{P_1}^{\infty}\right)\right)$$ by Lemma \[3lemm: ext11\]. Note that we have $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\overline{L}(\lambda)\otimes_E\left(v_{P_2}^{\infty}\oplus v_{P_1}^{\infty}\right)\right)=2$$ by Proposition \[3prop: locally algebraic extension\]. Assume first that (\[3equation 13.1\]) is a surjection, and thus we can pick a representation $W$ represented by a non-zero element lying in the preimage of $\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}$ under (\[3equation 13.1\]). We observe that the very existence of $W$ implies that $$\label{3nonvanishing of ext 13.1}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2,~\Sigma^{\sharp,+,\flat}(\lambda,\mathscr{L}_1,\mathscr{L}_2)\right)\neq 0.$$ We define $$\Sigma_i^{+,\flat}(\lambda,\mathscr{L}_i):=\Sigma_i^+(\lambda,\mathscr{L}_i)/\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}$$ and thus we have an embedding $$\Sigma_i^{+,\flat}(\lambda,\mathscr{L}_i)\hookrightarrow\Sigma^{\sharp,+,\flat}(\lambda,\mathscr{L}_1,\mathscr{L}_2)$$ for each $i=1,2$. We notice that the quotient $\Sigma^{\sharp,+,\flat}(\lambda,\mathscr{L}_1,\mathscr{L}_2)/\Sigma_1^{+,\flat}(\lambda,\mathscr{L}_1)$ fits into a short exact sequence $$\left(\begin{xy}
(0,0)*+{v_{P_2}^{\rm{an}}(\lambda)}="a"; (16,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)\hookrightarrow\Sigma^{\sharp,+,\flat}(\lambda,\mathscr{L}_1,\mathscr{L}_2)/\Sigma_1^{+,\flat}(\lambda,\mathscr{L}_1)\twoheadrightarrow C_{s_2,s_2}.$$ Hence it remains to show the equality $$\label{3equation 13.3}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2,~\begin{xy}
(0,0)*+{v_{P_2}^{\rm{an}}(\lambda)}="a"; (16,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)=0$$ and the equality $$\label{3equation 13.4}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~C_{s_2,s_2}\right)=0$$ to finish the proof of $$\label{3equation 13.2}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2, ~\Sigma^{\sharp,+,\flat}(\lambda,\mathscr{L}_1,\mathscr{L}_2)/\Sigma_1^{+,\flat}(\lambda,\mathscr{L}_1)\right)=0.$$ The vanishing (\[3equation 13.4\]) follows from Lemma \[3lemm: nonvanishing ext1\] and part (i) of Proposition \[3prop: formal devissages\]. It follows from Proposition \[3prop: locally algebraic extension\], Lemma \[3lemm: nonvanishing ext1\] and a simple devissage that $$\label{3equation 13.5}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty},~C^1_{s_1,1}\right)=\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\begin{xy}
(0,0)*+{C^1_{s_1,1}}="a"; (16,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)=0.$$ Hence if $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2,~\begin{xy}
(0,0)*+{C^1_{s_1,1}}="a"; (16,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)\neq0$$ then there exists a uniserial representation of the form $$\begin{xy}
(0,0)*+{C^1_{s_1,1}}="a"; (16,0)*+{\overline{L}(\lambda)}="b"; (35,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="c";
{\ar@{-}"a";"b"}; {\ar@{-}"b";"c"};
\end{xy}$$ which contradicts (\[3equation 13.5\]) and Lemma \[3lemm: special vanishing 1\]. As a result, we have shown that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2,~\begin{xy}
(0,0)*+{C^1_{s_1,1}}="a"; (16,0)*+{\overline{L}(\lambda)}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)=0$$ which together with Proposition \[3prop: locally algebraic extension\] and part (i) of Proposition \[3prop: formal devissages\] implies (\[3equation 13.3\]) and hence (\[3equation 13.2\]) as well concerning (\[3equation 13.4\]). Therefore we can combine (\[3equation 13.2\]) with Lemma \[3lemm: ext6\] and conclude that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2,~\Sigma^{\sharp,+,\flat}(\lambda,\mathscr{L}_1,\mathscr{L}_2)\right)=0$$ which contradicts (\[3nonvanishing of ext 13.1\]). Consequently, the injection (\[3equation 13.1\]) must be strict and we finish the proof.
According to Lemma \[3lemm: ext11\], the short exact sequence $$\Sigma^{\sharp,+}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\hookrightarrow\Sigma^{\sharp,+}(\lambda,\underline{\mathscr{L}})\twoheadrightarrow \overline{L}(\lambda)\otimes_E(v_{P_2}^{\infty}\oplus v_{P_1}^{\infty})$$ induces a long exact sequence: $$\begin{gathered}
\label{3second key sequence}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+}(\lambda,\underline{\mathscr{L}})\right)\hookrightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\overline{L}(\lambda)\otimes_E(v_{P_2}^{\infty}\oplus v_{P_1}^{\infty})\right)\\
\xrightarrow{f}\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+}(\lambda,\mathscr{L}_1,\mathscr{L}_2)\right)\end{gathered}$$
\[3prop: restriction\] We have $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}(\overline{L}(\lambda), ~\Sigma^{\sharp,+,\flat}(\lambda,\underline{\mathscr{L}}))=1$$ and the image of $f$ is not contained in the image of the natural injection $$\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\hookrightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+}(\lambda,\mathscr{L}_1,\mathscr{L}_2)\right).$$
We use the shorten notation for the two dimensional space $$M:=\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\overline{L}(\lambda)\otimes_E(v_{P_2}^{\infty}\oplus v_{P_1}^{\infty})\right).$$ We actually have the following commutative diagram $$\begin{xy}
(0,0)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+}(\lambda,\underline{\mathscr{L}})\right)}="a"; (40,0)*+{M}="b"; (80,0)*+{\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+}(\lambda,\mathscr{L}_1,\mathscr{L}_2)\right)}="c"; (0,-15)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+,\flat}(\lambda,\underline{\mathscr{L}})\right)}="d"; (40,-15)*+{M}="e"; (80,-15)*+{\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+,\flat}(\lambda,\mathscr{L}_1,\mathscr{L}_2)\right)}="f"; (32,3)*+{i}="g"; (47,3)*+{f}="h"; (32,-12)*+{j}="i"; (47,-12)*+{g}="j"; (-3,-7.5)*+{h}="k"; (77,-7.5)*+{k}="l";
{\ar@{^{(}->}"a";"b"}; {\ar@{->}"b";"c"}; {\ar@{^{(}->}"a";"d"}; {\ar@{=}"b";"e"}; {\ar@{->}"c";"f"}; {\ar@{^{(}->}"d";"e"}; {\ar@{->}"e";"f"};
\end{xy}$$ where the middle vertical map is just an equality. We know that $h$ is injective by the vanishing $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)=0$$ and $k$ has a one dimensional image by (\[3equation 11.1\]). Both $i$ and $j$ are injective due to (\[3vanishing of ext\]) and (\[3vanishing of ext prime\]). Therefore by a simple diagram chasing we have $$\begin{gathered}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+,\flat}(\lambda,\underline{\mathscr{L}})\right)\\
=\mathrm{dim}_EM-\mathrm{dim}_E\mathrm{Im}(g)\geq \mathrm{dim}_EM-\mathrm{dim}_E\mathrm{Im}(k)=2-1=1\end{gathered}$$ by Lemma \[3lemm: ext11\] and therefore $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+,\flat}(\lambda,\underline{\mathscr{L}})\right)=1$$ by Lemma \[3lemm: ext13\]. Moreover, the map $g$ has a one dimensional image and hence $k\circ f$ has one dimensional image, meaning that the image of $f$ has dimension one or two and is not contained in $\mathrm{Ker}(k)$, which is exactly the image of $$\label{3degenerate ext2}
\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\rightarrow\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+}(\lambda,\mathscr{L}_1,\mathscr{L}_2)\right)$$ by (\[3equation 11.1\]). In fact, the restriction of $f$ to the direct summand $\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}\right)$ is given by the cup product map with a non-zero element in the line of $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty},~\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$$ given by the preimage of $$E\left(\iota_1(D_0)+\mathscr{L}_i^{\prime}\kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{val}_p})\right)$$ via (\[3isomorphism for normalization\]) by our definition of $\Sigma^{\sharp,+}(\lambda,\underline{\mathscr{L}})$ and it is obvious that $\iota_1(D_0)+\mathscr{L}_i^{\prime}\kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{val}_p})$ does not lie in the image of (\[3degenerate ext2\]) which is exactly the line $E\kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{val}_p})$.
\[3prop: criterion of existence\] We have $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+}(\lambda,\underline{\mathscr{L}})\right)=1$$ if and only if $\mathscr{L}_1^{\prime}=\mathscr{L}_2^{\prime}=\mathscr{L}_3$ for a certain $\mathscr{L}_3\in E$.
It follows from (\[3second key sequence\]) that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+}(\lambda,\underline{\mathscr{L}})\right)=1$$ if and only if the image of $f$ is one dimensional. Then we notice by the interpretation of $f$ as cup product in Proposition \[3prop: restriction\] that the image of $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}\right)$$ under $f$ is the line of $$\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+}(\lambda,\mathscr{L}_1,\mathscr{L}_2)\right)$$ generated by $$\iota_1(D_0)+\mathscr{L}_i^{\prime}\kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{val}_p})$$ for each $i=1,2$. Therefore the image of $f$ is one dimensional if and only if the two lines for $i=1,2$ coincide which means that $$\mathscr{L}_1^{\prime}=\mathscr{L}_2^{\prime}=\mathscr{L}_3$$ for a certain $\mathscr{L}_3\in E$.
We use the notation $\Sigma^{\sharp,+}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ for the representation $\Sigma^{\sharp,+}(\lambda,\underline{\mathscr{L}})$ when $$\underline{\mathscr{L}}=(\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3, \mathscr{L}_3).$$ We define $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ as the unique representation (up to isomorphism) given by a non-zero element in $\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\sharp,+}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\right)$ according to Proposition \[3prop: criterion of existence\]. Therefore by our definition $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ has the following form $$\label{3naive picture for min}
\begin{xy}
(0,0)*+{\mathrm{St}_3^{\rm{an}}(\lambda)}="a"; (20,6)*+{v_{P_1}^{\rm{an}}(\lambda)}="b"; (20,-6)*+{v_{P_2}^{\rm{an}}(\lambda)}="c"; (40,8)*+{C_{s_1,s_1}}="d"; (40,-8)*+{C_{s_2,s_2}}="e"; (40,0)*+{\overline{L}(\lambda)}="f"; (60,8)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="g"; (60,-8)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="h"; (80,0)*+{\overline{L}(\lambda)}="i";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"e"}; {\ar@{-}"b";"f"}; {\ar@{-}"c";"f"}; {\ar@{-}"d";"g"}; {\ar@{-}"e";"h"}; {\ar@{-}"g";"i"}; {\ar@{-}"h";"i"}; {\ar@{--}"b";"i"}; {\ar@{--}"c";"i"};
\end{xy}.$$ It follows from Proposition \[3prop: locally algebraic extension\], Proposition \[3prop: criterion of existence\], the definition of $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ and an easy devissage that $$\label{3vanishing of ext1 for min}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\right)=0.$$
\[3rema: noncanonical\] The definition of the invariant $\mathscr{L}_3\in E$ of $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ obviously relies on the choice of a special $p$adic dilogarithm function $D_0$ which is non-canonical. This is similar to the definition of the invariants $\mathscr{L}_1, \mathscr{L}_2\in E$ which relies on the choice of a special $p$adic logarithm function $\mathrm{log}_0$.
\[3lemm: preparation for global\] We have $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma^{\sharp,+}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)=2.$$ Moreover, if $V$ is a locally analytic representation determined by a line $$M_V\subsetneq \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma^{\sharp,+}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$$ satisfying $$M_V\neq \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\hookrightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~\Sigma^{\sharp,+}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right),$$ then there exists a unique $\mathscr{L}_3\in E$ such that $$V\cong \Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3).$$
The short exact sequence $$\overline{L}(\lambda)\otimes_E\left(v_{P_1}^{\infty}\oplus v_{P_2}^{\infty}\right)\hookrightarrow W_0\twoheadrightarrow \overline{L}(\lambda)$$ together with Lemma \[3lemm: ext10\] induce a commutative diagram $$\begin{xy}
(0,0)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~V^+\right)}="a"; (55,0)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(V_1^{\rm{alg}}\oplus V_2^{\rm{alg}}, ~V^+\right)}="b"; (110,0)*+{\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~V^+\right)}="e";
(0,-12)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~V^{\sharp, +}\right)}="c"; (55,-12)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(V_1^{\rm{alg}}\oplus V_2^{\rm{alg}}, ~V^{\sharp, +}\right)}="d"; (110,-12)*+{\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~V^{\sharp, +}\right)}="f"; (23,2)*+{g_1}; (23,-10)*+{g_2}; (-3,-6)*+{h_1}; (52,-6)*+{h_2}; (107,-6)*+{h_3}; (85,2)*+{k_1}; (85,-10)*+{k_2};
{\ar@{->}"a";"b"}; {\ar@{->}"a";"c"}; {\ar@{^{(}->}"c";"d"}; {\ar@{^{(}->}"b";"d"}; {\ar@{->>}"b";"e"}; {\ar@{->}"d";"f"}; {\ar@{->}"e";"f"};
\end{xy}$$ where we use shorten notation $V_i^{\rm{alg}}$ for $\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}$, $V^+$ for $\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ and $V^{\sharp, +}$ for $\Sigma^{\sharp,+}(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ to save space. We observe that $g_2$ is an injection due to Lemma \[3lemm: ext10\], $k_1$ is a surjection by the proof of Proposition \[3prop: main dim\], $h_3$ is an isomorphism by Proposition \[3prop: locally algebraic extension\] and an easy devissage and finally $h_2$ is an injection. Assume that $h_2$ is not surjective, then any representation given by a non-zero element in $\mathrm{Coker}(h_2)$ admits a quotient of the form $$\label{3special uniserial}
\begin{xy}
(0,0)*+{C^1_{s_i,1}}="a"; (16,0)*+{\overline{L}(\lambda)}="b"; (32,0)*+{V_i^{\rm{alg}}}="c";
{\ar@{-}"a";"b"}; {\ar@{-}"b";"c"};
\end{xy}$$ for $i=1$ or $2$ due to Lemma \[3lemm: nonvanishing ext1\]. However, it follows from Lemma \[3lemm: special vanishing 1\] that there is no uniserial representation of the form (\[3special uniserial\]), which implies that $h_2$ is indeed an isomorphism, and hence $k_2$ is surjective by a diagram chasing. Therefore we conclude that $$\begin{gathered}
\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0, ~V^{\sharp, +}\right)\\
=\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(V_1^{\rm{alg}}\oplus V_2^{\rm{alg}}, ~V^{\sharp, +}\right)-\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~V^{\sharp, +}\right)\\
=\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(V_1^{\rm{alg}}\oplus V_2^{\rm{alg}}, ~V^+\right)-\mathrm{dim}_E\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~V^+\right)=4-2=2.\end{gathered}$$ The final claim on the existence of a unique $\mathscr{L}_3$ follows from Proposition \[3prop: criterion of existence\], our definition of $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ and the observation that the restriction of $k_2$ to the direct summand $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(V_i^{\rm{alg}}, ~V^{\sharp, +}\right)$$ induces isomorphisms $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(V_i^{\rm{alg}}, ~V^{\sharp, +}\right)\xrightarrow{\sim}\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda), ~V^{\sharp, +}\right)$$ which can be interpreted as the cup product morphism with the one dimensional space $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~V_i^{\rm{alg}}\right)$$ for $i=1,2$.
We define $\Sigma^{\sharp, +}_i(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ as the subrepresentation of $\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ that fits into the short exact sequence $$\Sigma^{\sharp, +}_i(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\hookrightarrow\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\twoheadrightarrow \overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}$$ for each $i=1,2$. We use the notation $\mathcal{D}_i(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}$ for the object in the derived category $\mathcal{D}^b\left(\mathrm{Mod}_{D(\mathrm{GL}_3(\mathbb{Q}_p), E)}\right)$ associated with the complex $$\left[W_{3-i}^{\prime}\longrightarrow\Sigma^{\sharp, +}_i(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}\right].$$
\[3prop: relation with derived object\] The object $$\mathcal{D}_i(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}\in \mathcal{D}^b\left(\mathrm{Mod}_{D(\mathrm{GL}_3(\mathbb{Q}_p), E)}\right)$$ fits into the distinguished triangle $$\label{3distinguished triangle}
\overline{L}(\lambda)^{\prime}\longrightarrow \mathcal{D}_i(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}\longrightarrow \Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)^{\prime}[-1]\xrightarrow{~+1}$$ for each $i=1,2$. Moreover, the element in $$\begin{gathered}
\label{3canonical translation}
\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\\
\xrightarrow{\sim}\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)\\
\cong \mathrm{Hom}_{\mathcal{D}^b\left(\mathrm{Mod}_{D(\mathrm{GL}_3(\mathbb{Q}_p), E)}\right)}\left(\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)^{\prime}[-2],~\overline{L}(\lambda)^{\prime}\right)\end{gathered}$$ associated with the distinguished triangle (\[3distinguished triangle\]) is $$\label{3explicit element}
\iota_1(D_0)+\mathscr{L}_3\kappa(b_{1,\mathrm{val}_p}\wedge b_{2,\mathrm{val}_p}).$$
It follows from Proposition 3.2 of [@Schr11] that there is a unique (up to isomorphism) object $$\mathcal{D}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}\in \mathcal{D}^b\left(\mathrm{Mod}_{D(\mathrm{GL}_3(\mathbb{Q}_p), E)}\right)$$ that fits into a distinguished triangle $$\label{3formal triangle}
\overline{L}(\lambda)^{\prime}\longrightarrow \mathcal{D}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\longrightarrow \Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)[-1]\xrightarrow{~+1}$$ such that the element in $\mathrm{Ext}^2_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$ associated with (\[3formal triangle\]) via (\[3canonical translation\]) is (\[3explicit element\]). It follows from $\mathbf{TR 2}$ ( cf. Section 10.2.1 of [@Wei94]) that $$\label{3formal triangle 2}
\mathcal{D}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}\longrightarrow\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)^{\prime}[-1]\longrightarrow\overline{L}(\lambda)^{\prime}[1]\xrightarrow{~+1}$$ is another distinguished triangle. The isomorphism (\[3isomorphism for normalization\]) can be reinterpreted as the isomorphism $$\begin{gathered}
\mathrm{Hom}_{\mathcal{D}^b\left(\mathrm{Mod}_{D(\mathrm{GL}_3(\mathbb{Q}_p), E)}\right)}\left(\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)^{\prime}[-1],~\left(\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}\right)^{\prime}\right)\\
\xrightarrow{\sim}\mathrm{Hom}_{\mathcal{D}^b\left(\mathrm{Mod}_{D(\mathrm{GL}_3(\mathbb{Q}_p), E)}\right)}\left(\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)^{\prime}[-1],~\overline{L}(\lambda)^{\prime}[1]\right)\end{gathered}$$ induced by the composition with $\mathrm{Hom}_{\mathcal{D}^b\left(\mathrm{Mod}_{D(\mathrm{GL}_3(\mathbb{Q}_p), E)}\right)}\left(\left(\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}\right)^{\prime},~\overline{L}(\lambda)^{\prime}[1]\right)$. As a result, each morphism $$\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)^{\prime}[-1]\rightarrow\overline{L}(\lambda)^{\prime}[1]$$ uniquely factors through a composition $$\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)^{\prime}[-1]\rightarrow\left(\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}\right)^{\prime}\rightarrow\overline{L}(\lambda)^{\prime}[1]$$ which induces a commutative diagram with four distinguished triangles $$\begin{xy}
(0,0)*+{\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)^{\prime}[-1]}="a"; (20,30)*+{\left(\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}\right)^{\prime}}="b"; (40,20)*+{\overline{L}(\lambda)^{\prime}[1]}="c"; (40,60)*+{\Sigma^{\sharp, +}_i(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}}="d"; (60,30)*+{\mathcal{D}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}}="e"; (80,0)*+{W_{3-i}^{\prime}[1]}="f"; (50,75)*+{}="g"; (80,40)*+{}="h"; (100,-10)*+{}="i"; (95,-22.5)*+{}="j"; (42.5,70)*+{+1}="k"; (70,40)*+{+1}="l"; (92,-2)*+{+1}="m"; (90,-10)*+{+1}="n";
{\ar@{->}"a";"b"}; {\ar@{->}"a";"c"}; {\ar@{->}"b";"c"}; {\ar@{->}"b";"d"}; {\ar@{->}"c";"e"}; {\ar@{->}"c";"f"}; {\ar@{->}"d";"e"}; {\ar@{->}"e";"f"}; {\ar@{->}"d";"g"}; {\ar@{->}"e";"h"}; {\ar@{->}"f";"i"}; {\ar@{->}"f";"j"};
\end{xy}$$ by $\mathbf{TR 4}$. Hence we deduce that $$\Sigma^{\sharp, +}_i(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}\longrightarrow \mathcal{D}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}\longrightarrow W_{3-i}^{\prime}[1]\xrightarrow{~+1}$$ or equivalently $$W_{3-i}^{\prime}\longrightarrow\Sigma^{\sharp, +}_i(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}\longrightarrow \mathcal{D}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}\xrightarrow{~+1}$$ is a distinguished triangle. On the other hand, it is easy to see that $\mathcal{D}_i(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}$ fits into the distinguished triangle $$W_{3-i}^{\prime}\longrightarrow\Sigma^{\sharp, +}_i(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}\longrightarrow \mathcal{D}_i(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}\xrightarrow{~+1}$$ and thus we conclude that $$\mathcal{D}_i(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}\cong\mathcal{D}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)^{\prime}\in\mathcal{D}^b\left(\mathrm{Mod}_{D(\mathrm{GL}_3(\mathbb{Q}_p), E)}\right)$$ by the uniqueness in Proposition 3.2 of [@Schr11]. Hence we finish the proof.
We define $\Sigma^{\rm{min},-}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ as the unique subrepresentation of $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ of the form $$\begin{xy}
(0,0)*+{\mathrm{St}_3^{\rm{an}}(\lambda)}="a"; (20,4)*+{v_{P_1}^{\rm{an}}(\lambda)}="b"; (20,-4)*+{v_{P_2}^{\rm{an}}(\lambda)}="c"; (40,4)*+{C_{s_1,s_1}}="d"; (40,-4)*+{C_{s_2,s_2}}="e"; (60,4)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="g"; (60,-4)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="h";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"e"}; {\ar@{-}"d";"g"}; {\ar@{-}"e";"h"};
\end{xy}$$ that fits into the short exact sequence $$\label{3short exact sequence definition}
\Sigma^{\rm{min},-}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\hookrightarrow\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\twoheadrightarrow\overline{L}(\lambda)^{\oplus2}$$ and $\Sigma^{\rm{min},--}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ as the unique subrepresentation of $\Sigma^{\rm{min},-}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ of the form $$\begin{xy}
(0,0)*+{\mathrm{St}_3^{\rm{an}}(\lambda)}="a"; (30,4)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="b"; (30,-4)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="c"; (60,4)*+{C_{s_1,s_1}}="d"; (60,-4)*+{C_{s_2,s_2}}="e";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"e"};
\end{xy}$$ that fits into the short exact sequence $$\label{3short exact sequence for min}
\Sigma^{\rm{min},--}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\hookrightarrow\Sigma^{\rm{min},-}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\twoheadrightarrow \left(\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}\right)\oplus \left(\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}\right)\oplus C^1_{s_2,1}\oplus C^1_{s_1,1}.$$ The short exact sequence (\[3short exact sequence definition\]) induces a long exact sequence $$\begin{gathered}
\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\overline{L}(\lambda)^{\oplus 2}\right)\hookrightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^{\rm{min},-}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\right)\\
\rightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\right)\rightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\overline{L}(\lambda)^{\oplus 2}\right)\end{gathered}$$ which easily implies that $$\mathrm{dim}_E\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^{\rm{min},-}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\right)=2$$ by Proposition \[3prop: locally algebraic extension\] and (\[3vanishing of ext1 for min\]). On the other hand, we notice that $\Sigma^{\rm{min},--}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ admits a filtration whose only reducible graded piece is $$\begin{xy}
(0,0)*+{C^1_{s_i,1}}="a"; (20,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy}$$ and $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~V\right)=0$$ for all graded pieces $V$ of such a filtration by Lemma \[3lemm: nonvanishing ext1\] and Lemma \[3lemm: special vanishing 1\], which implies that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^{\rm{min},--}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\right)=0.$$ Therefore (\[3short exact sequence for min\]) induces an injection of a two dimensional space into a four dimensional space $$\begin{gathered}
M^{\rm{min}}:=\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^{\rm{min},-}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\right)\\
\hookrightarrow M^+:=\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\left(\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}\right)\oplus \left(\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}\right)\oplus C^1_{s_2,1}\oplus C^1_{s_1,1}\right).\end{gathered}$$ It follows from the definition of $\Sigma^{\rm{min},-}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ that we have embeddings $$\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\hookrightarrow\Sigma^+(\lambda, \mathscr{L}_1, \mathscr{L}_2)\hookrightarrow\Sigma^{\rm{min},-}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$$ which allow us to identify $$M^-:=\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$$ with a line in $M^{\rm{min}}$. We use the number $1,2,3,4$ to index the four representations $\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}$, $\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}$, $C^1_{s_2,1}$ and $C^1_{s_1,1}$ respectively, and we use the notation $M_I$ for each subset $I\subseteq \{1,2,3,4\}$ to denote the corresponding subspace of $M^+$ with dimension the cardinality of $I$. For example, $M_{\{1,2\}}$ denotes the two dimensional subspace $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\left(\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}\right)\oplus \left(\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}\right)\right)$$ of $M^+$.
\[3lemm: distinguish mult two\] We have the following characterizations of $M^{\rm{min}}$ inside $M^+$: $$M^{\rm{min}}\cap M_{\{i,j\}}=0\mbox{ for }\{i,j\}\neq \{3,4\},$$ $$M^{\rm{min}}\cap M_{\{1,3,4\}}=M^{\rm{min}}\cap M_{\{2,3,4\}}=M^{\rm{min}}\cap M_{\{3,4\}}=M^-,$$ and $$M^{\rm{min}}=(M^{\rm{min}}\cap M_{\{1,2,3\}})\oplus (M^{\rm{min}}\cap M_{\{1,2,4\}}).$$
As $C^1_{s_1,1}$ and $C^1_{s_2,1}$ are in the cosocle of $\Sigma(\lambda, \mathscr{L}_1, \mathscr{L}_2)$, it is immediate that $$M^-\subseteq M_{\{3,4\}}.$$ It follows from (\[3naive picture for min\]) that $$M^{\rm{min}}\not\subseteq M_{\{3,4\}}$$ and thus $M^{\rm{min}}\cap M_{\{3,4\}}$ is one dimensional which must coincide with $M^-$. The proof of Lemma \[3lemm: upper bound\] implies that $M\not\subseteq M_{\{i,3,4\}}$ for $i=1,2$ and therefore $M\cap M_{\{i,3,4\}}$ is one dimensional, which implies that $$M^{\rm{min}}\cap M_{\{i,3,4\}}=M^-$$ by the inclusion $$M^{\rm{min}}\cap M_{\{3,4\}}\subseteq M^{\rm{min}}\cap M_{\{i,3,4\}}$$ for $i=1,2$. We observe ( cf. Lemma \[3lemm: ext6\]) that $$M^-\cap M_{\{3\}}=M^-\cap M_{\{4\}}=0$$ and thus $$M^{\rm{min}}\cap M_{\{i,j\}}=M^-\cap M_{\{i,j\}}=0$$ for each $\{i,j\}\neq \{3,4\}, \{1,2\}$. We define $\Sigma^{\rm{min},-,\prime}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ as the unique subrepresentation of $\Sigma^{\rm{min},-}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ that fits into the short exact sequence $$\Sigma^{\rm{min},-,\prime}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\hookrightarrow\Sigma^{\rm{min},-}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\twoheadrightarrow C^1_{s_1,1}\oplus C^1_{s_2,1}\oplus C_{s_1s_2s_1,1}$$ and then define $$\Sigma^{\rm{min},-,\prime,\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3):=\Sigma^{\rm{min},-,\prime}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)/\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}.$$ It is obvious that $M^{\rm{min}}\cap M_{\{1,2\}}\neq 0$ if and only if $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^{\rm{min},-,\prime}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\right)\neq 0$$ which implies that $$\label{3nonvanishing contradiction}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma^{\rm{min},-,\prime,\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\right)\neq 0$$ as $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)=0$$ due to Proposition \[3prop: locally algebraic extension\]. We notice that we have a direct sum decomposition $$\Sigma^{\rm{min},-,\prime,\flat}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)=V_1\oplus V_2$$ where $V_i$ is a representation of the form $$\begin{xy}
(0,0)*+{C^2_{s_i,1}}="a"; (30,4)*+{C^1_{s_{3-i}s_i,1}}="b"; (60,8)*+{C^2_{s_{3-i}s_i,1}}="c"; (30,-4)*+{\overline{L}(\lambda)\otimes_Ev_{P_i}^{\infty}}="d"; (60,0)*+{C_{s_i,s_i}}="e"; (90,4)*+{\overline{L}(\lambda)\otimes_Ev_{P_{3-i}}^{\infty}}="f";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"d"}; {\ar@{-}"b";"c"}; {\ar@{-}"b";"e"}; {\ar@{-}"d";"e"}; {\ar@{-}"e";"f"};
\end{xy}.$$ Switching $V_1$ and $V_2$ if necessary, we can assume by (\[3nonvanishing contradiction\]) that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~V_1\right)\neq 0.$$ On the other hand, we have an embedding $$V_1\hookrightarrow
\begin{xy}
(0,0)*+{\Sigma_1^{+,\flat}(\lambda, \mathscr{L}_1)}="a"; (25,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy}$$ which induces an embedding $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~V_1\right)\hookrightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\begin{xy}
(0,0)*+{\Sigma_1^{+,\flat}(\lambda, \mathscr{L}_1)}="a"; (25,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)$$ and in particular $$\label{3nonvanishing contradiction prime}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\begin{xy}
(0,0)*+{\Sigma_1^{+,\flat}(\lambda, \mathscr{L}_1)}="a"; (25,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)\neq 0.$$ The short exact sequences $$\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\hookrightarrow\Sigma_1(\lambda, \mathscr{L}_1)\twoheadrightarrow\Sigma_1^{\flat}(\lambda, \mathscr{L}_1),~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\hookrightarrow\Sigma_1^+(\lambda, \mathscr{L}_1)\twoheadrightarrow\Sigma_1^{+,\flat}(\lambda, \mathscr{L}_1)$$ induce isomorphisms $$\label{3isomorphism from flat}
\begin{xy}
(0,0)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma_1(\lambda, \mathscr{L}_1)\right)}; (30,0)*+{\xrightarrow{\sim}}; (60,0)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma_1^{\flat}(\lambda, \mathscr{L}_1)\right)}; (0.4,-6)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma_1^+(\lambda, \mathscr{L}_1)\right)}; (30,-6)*+{\xrightarrow{\sim}}; (62,-6)*+{\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma_1^{+,\flat}(\lambda, \mathscr{L}_1)\right)};
\end{xy}$$ by Lemma \[3lemm: vanishing locally algebraic\]. Hence we deduce that $$\label{3vanishing for flat}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2,~\Sigma_1^{\flat}(\lambda, \mathscr{L}_1)\right)=\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2,~\Sigma_1^{+,\flat}(\lambda, \mathscr{L}_1)\right)=0$$ from Lemma \[3lemm: ext6\] and (\[3isomorphism from flat\]). The surjection $W_2\twoheadrightarrow\overline{L}(\lambda)$ induces an embedding $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma_1^{\flat}(\lambda, \mathscr{L}_1)\right)\hookrightarrow \mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_2,~\Sigma_1^{\flat}(\lambda, \mathscr{L}_1)\right)$$ which together with (\[3vanishing for flat\]) imply that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma_1^{\flat}(\lambda, \mathscr{L}_1)\right)=0$$ and hence $$\label{3vanishing for flat prime}
\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\Sigma_1^{+,\flat}(\lambda, \mathscr{L}_1)\right)=0$$ by (\[3vanishing of simple special ext\]) and an easy devissage. It follows from (\[3vanishing for flat\]) and (\[3vanishing for flat prime\]) that $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(\overline{L}(\lambda),~\begin{xy}
(0,0)*+{\Sigma_1^{+,\flat}(\lambda, \mathscr{L}_1)}="a"; (25,0)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)=0$$ which contradicts (\[3nonvanishing contradiction prime\]). A a result, we have shown that $$M^{\rm{min}}\cap M_{\{1,2\}}=0.$$ As $M^-\not\subseteq M_{\{1,2,i\}}$ for $i=3,4$, we deduce that both $M^{\rm{min}}\cap M_{\{1,2,3\}}$ and $M^{\rm{min}}\cap M_{\{1,2,4\}}$ are one dimensional. On the other hand, since we know that $$(M^{\rm{min}}\cap M_{\{1,2,3\}})\cap(M^{\rm{min}}\cap M_{\{1,2,4\}})=M^{\rm{min}}\cap M_{\{1,2\}}=0,$$ we deduce the following direct sum decomposition $$M^{\rm{min}}=(M^{\rm{min}}\cap M_{\{1,2,3\}})\oplus (M^{\rm{min}}\cap M_{\{1,2,4\}}).$$
We use the notation $\overline{L}(\lambda)^i$ for copy of $\overline{L}(\lambda)$ inside $\overline{L}(\lambda)^{\oplus2}$ corresponding to the one dimensional space $M^{\rm{min}}\cap M_{\{1,2,i+2\}}$ inside $M^{\rm{min}}$, and therefore we have a surjection $$\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\twoheadrightarrow \left(
\begin{xy}
(0,0)*+{C^1_{s_2,1}}="a"; (16,0)*+{\overline{L}(\lambda)^1}="b";
{\ar@{-}"a";"b"};
\end{xy}\right)\oplus \left(\begin{xy}
(0,0)*+{C^1_{s_1,1}}="a"; (16,0)*+{\overline{L}(\lambda)^2}="b";
{\ar@{-}"a";"b"};
\end{xy}\right).$$
As a result, the representation $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ has the following form: $$\label{3simple picture for min}
\begin{xy}
(0,0)*+{\mathrm{St}_3^{\rm{an}}(\lambda)}="a"; (20,5)*+{v_{P_1}^{\rm{an}}(\lambda)}="b"; (40,9)*+{C_{s_1,s_1}}="d"; (60,9)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="f"; (20,-5)*+{v_{P_2}^{\rm{an}}(\lambda)}="c"; (40,-9)*+{C_{s_2,s_2}}="e"; (60,-9)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="g"; (90,3)*+{\overline{L}(\lambda)^1}="h1"; (90,-3)*+{\overline{L}(\lambda)^2}="h2";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"e"}; {\ar@{-}"d";"f"}; {\ar@{-}"e";"g"}; {\ar@{-}"f";"h1"}; {\ar@{-}"b";"h1"}; {\ar@{-}"g";"h1"}; {\ar@{-}"f";"h2"}; {\ar@{-}"c";"h2"}; {\ar@{-}"g";"h2"};
\end{xy}.$$ If we clarify the internal structure of $\mathrm{St}_3^{\rm{an}}(\lambda)$, $v_{P_1}^{\rm{an}}(\lambda)$ and $v_{P_2}^{\rm{an}}(\lambda)$ using Lemma \[3lemm: structure of St\], then $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ has the following form: $$\label{3main picture for min}
\begin{xy}
(-20,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="a1";
(-6,9)*+{C^2_{s_1,1}}="a2"; (13.5,22)*+{C^1_{s_2s_1,1}}="a4"; (51,21)*+{C^2_{s_2s_1,1}}="a6"; (20,14)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="b1"; (70,12)*+{C^1_{s_2,1}}="b2"; (42,30)*+{C_{s_1,s_1}}="d1"; (77,22)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="f1"; (-6,-9)*+{C^2_{s_2,1}}="a3";(13.5,-22)*+{C^1_{s_1s_2,1}}="a5"; (51,-21)*+{C^2_{s_1s_2,1}}="a7"; (20,-14)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="c1"; (70,-12)*+{C^1_{s_1,1}}="c2"; (42,-30)*+{C_{s_2,s_2}}="e1"; (77,-22)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="g1"; (100,5)*+{\overline{L}(\lambda)^1}="h1"; (100,-5)*+{\overline{L}(\lambda)^2}="h2"; (73,-0)*+{C_{s_1s_2s_1,1}}="a8";
{\ar@{-}"a1";"a2"}; {\ar@{-}"a2";"a4"}; {\ar@{-}"a2";"b1"}; {\ar@{-}"a2";"a7"}; {\ar@{--}"a4";"b2"}; {\ar@{-}"a4";"a6"}; {\ar@{-}"a4";"d1"}; {\ar@{--}"a6";"c2"}; {\ar@{--}"a6";"a8"}; {\ar@{-}"b1";"b2"}; {\ar@{-}"b1";"d1"}; {\ar@{-}"b2";"h1"}; {\ar@{-}"d1";"f1"}; {\ar@{-}"f1";"h2"}; {\ar@{-}"f1";"h1"}; {\ar@{-}"a1";"a3"}; {\ar@{-}"a3";"a5"}; {\ar@{-}"a3";"c1"}; {\ar@{-}"a3";"a6"}; {\ar@{--}"a5";"c2"}; {\ar@{-}"a5";"e1"}; {\ar@{-}"a5";"a7"}; {\ar@{--}"a7";"b2"}; {\ar@{--}"a7";"a8"}; {\ar@{-}"c1";"c2"}; {\ar@{-}"c1";"e1"}; {\ar@{-}"c2";"h2"}; {\ar@{-}"e1";"g1"}; {\ar@{-}"g1";"h1"}; {\ar@{-}"g1";"h2"};
\end{xy}.$$
It is actually possible to show that all the possibly split extensions illustrated in (\[3main picture for min\]) are non-split. However, the proof is quite technical and not related to the $p$adic dilogarithm function, and thus we decided not to include the proof here.
We observe that $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ admits a unique subrepresentation $\Sigma^{\mathrm{Ext}^1, -}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ of the form $$\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="a1";
(20,8)*+{C^2_{s_1,1}}="a2"; (45,12)*+{C^1_{s_2s_1,1}}="a4"; (45,4)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="b1"; (70,8)*+{C_{s_1,s_1}}="d1"; (95,4)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="f1"; (20,-8)*+{C^2_{s_2,1}}="a3";(45,-12)*+{C^1_{s_1s_2,1}}="a5"; (45,-4)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="c1"; (70,-8)*+{C_{s_2,s_2}}="e1"; (95,-4)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="g1";
{\ar@{-}"a1";"a2"}; {\ar@{-}"a2";"a4"}; {\ar@{-}"a2";"b1"}; {\ar@{-}"a4";"d1"}; {\ar@{-}"b1";"d1"}; {\ar@{-}"d1";"f1"}; {\ar@{-}"a1";"a3"}; {\ar@{-}"a3";"a5"}; {\ar@{-}"a3";"c1"}; {\ar@{-}"a5";"e1"}; {\ar@{-}"c1";"e1"}; {\ar@{-}"e1";"g1"};
\end{xy}$$ which can be uniquely extend to a representation $\Sigma^{\mathrm{Ext}^1}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ of the form: $$\label{3main picture for Ext1}
\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="a1";
(20,8)*+{C^2_{s_1,1}}="a2"; (45,12)*+{C^1_{s_2s_1,1}}="a4"; (45,4)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="b1"; (70,8)*+{C_{s_1,s_1}}="d1"; (95,4)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="f1"; (20,-8)*+{C^2_{s_2,1}}="a3";(45,-12)*+{C^1_{s_1s_2,1}}="a5"; (45,-4)*+{\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}}="c1"; (70,-8)*+{C_{s_2,s_2}}="e1"; (95,-4)*+{\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}}="g1"; (95,12)*+{C^1_{s_2s_1,s_2s_1}}="i1"; (95,-12)*+{C^1_{s_1s_2,s_1s_2}}="j1"; (120,8)*+{C^2_{s_1,s_1s_2}}="i2"; (120,-8)*+{C^2_{s_2,s_2s_1}}="j2";
{\ar@{-}"a1";"a2"}; {\ar@{-}"a2";"a4"}; {\ar@{-}"a2";"b1"}; {\ar@{-}"a4";"d1"}; {\ar@{-}"b1";"d1"}; {\ar@{-}"d1";"f1"}; {\ar@{-}"a1";"a3"}; {\ar@{-}"a3";"a5"}; {\ar@{-}"a3";"c1"}; {\ar@{-}"a5";"e1"}; {\ar@{-}"c1";"e1"}; {\ar@{-}"e1";"g1"}; {\ar@{-}"d1";"i1"}; {\ar@{-}"e1";"j1"}; {\ar@{-}"i1";"i2"}; {\ar@{-}"j1";"j2"}; {\ar@{-}"f1";"i2"}; {\ar@{-}"g1";"j2"};
\end{xy}$$ according to Section 4.4 and 4.6 of [@Bre17] together with our Lemma \[3lemm: existence of diamond\]. Finally, we define $\Sigma^{\rm{min},+}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ as the amalgamate sum of $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ and $\Sigma^{\mathrm{Ext}^1}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ over $\Sigma^{\mathrm{Ext}^1, -}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$.
\[3quotient independent of invariants\] It is actually possible to prove (by several technical computations of $\mathrm{Ext}$groups) that the quotient $$\Sigma^{\rm{min},+}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)/\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}$$ and the quotient $$\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)/\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}$$ are independent of the choices of $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$.
Local-global compatibility {#3section: local-global}
==========================
We are going to borrow most of the notation and assumptions from Section 6 of [@Bre17]. We fix embeddings $\iota_{\infty}:\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$, $\iota_p: \overline{\mathbb{Q}}$, an imaginary quadratic CM extension $F$ of $\mathbb{Q}$ and a unitary group $G/\mathbb{Q}$ attached to the extension $F/\mathbb{Q}$ such that $G\times_{\mathbb{Q}}F\cong\mathrm{GL}_3$ and $G(\mathbb{R})$ is compact. If $\ell$ is a finite place of $\mathbb{Q}$ which splits completely in $F$, we have isomorphisms $\iota_{G,w}: G(\mathbb{Q}_{\ell})\xrightarrow{\sim}G(F_w)\cong\mathrm{GL}_3(F_w)$ for each finite place $w$ of $F$ over $\ell$. We assume that $p$ splits completely in $F$, and we fix a finite place $w_0$ of $F$ dividing $p$ and therefore $G(\mathbb{Q}_p)\cong G(F_{w_0})\cong \mathrm{GL}_3(\mathbb{Q}_p)$.
We fix an open compact subgroup $U^p\subsetneq G(\mathbb{A}^{\infty,p}_{\mathbb{Q}})$ of the form $U^p=\prod_{\ell\neq p}U_{\ell}$ where $U_{\ell}$ is an open compact subgroup of $G(\mathbb{Q}_{\ell})$. For each finite extension $E$ of $\mathbb{Q}_p$ inside $\overline{\mathbb{Q}_p}$, we consider the following $\mathcal{O}_E$-lattice inside a $p$adic Banach space: $$\widehat{S}(U^p,\mathcal{O}_E):=\{f : G(\mathbb{Q})\backslash G(\mathbb{A}^{\infty}_{\mathbb{Q}})/U^p\rightarrow \mathcal{O}_E, ~f \text{ continuous}\}$$ and note that $\widehat{S}(U^p,E):=\widehat{S}(U^p,\mathcal{O}_E)\otimes_{\mathcal{O}_E}E$. The right translation of $G(\mathbb{Q}_p)$ on $G(\mathbb{Q})\backslash G(\mathbb{A}^{\infty}_{\mathbb{Q}})/U^p$ induces a $p$adic continuous action of $G(\mathbb{Q}_p)$ on $\widehat{S}(U^p,\mathcal{O}_E)$ which makes $\widehat{S}(U^p,E)$ an admissible Banach representation of $G(\mathbb{Q}_p)$ in the sense of [@ST02]. We use the notation $\widehat{S}(U^p,E)^{\rm{alg}}\subseteq\widehat{S}(U^p,E)^{\rm{an}}$ following Section 6 of [@Bre17] for the subspaces of locally $\mathbb{Q}_p$-algebraic vectors and locally $\mathbb{Q}_p$-analytic vectors inside $\widehat{S}(U^p,E)$ respectively. Moreover, we have the following decomposition: $$\label{3global decomposition}
\widehat{S}(U^p,E)^{\rm{alg}}\otimes_E\overline{\mathbb{Q}_p}\cong\bigoplus_{\pi}(\pi_f^{v_0})^{U_p}\otimes_{\overline{\mathbb{Q}}}(\pi_{v_0}\otimes_{\overline{\mathbb{Q}}}W_p)$$ where the direct sum is over the automorphic representations $\pi$ of $G(\mathbb{A}_{\mathbb{Q}})$ over $\mathbb{C}$ and $W_p$ is the $\mathbb{Q}_p$-algebraic representation of $G(\mathbb{Q}_p)$ over $\overline{\mathbb{Q}_p}$ associated with the algebraic representation $\pi_{\infty}$ of $G(\mathbb{R})$ over $\mathbb{C}$ via $\iota_p$ and $\iota_{\infty}$. In particular, each distinct $\pi$ appears with multiplicity one ( cf. the paragraph after (55) of [@Bre17] for further references).
We use the notation $D(U^p)$ for the set of finite places $\ell$ of $\mathbb{Q}$ that are different from $p$, split completely in $F$ and such that $U_{\ell}$ is a maximal open compact subgroup of $G(\mathbb{Q}_{\ell})$. Then we consider the commutative polynomial algebra $\mathbb{T}(U^p):=E[T^{(j)}_w]$ generated by the variables $T^{(j)}_w$ indexed by $j\in\{1,\cdots,n\}$ and $w$ a finite place of $F$ over a place $\ell$ of $\mathbb{Q}$ such that $\ell\in D(U^p)$. The algebra $\mathbb{T}(U^p)$ acts on $\widehat{S}(U^p,E)$, $\widehat{S}(U^p,E)^{\rm{alg}}$ and $\widehat{S}(U^p,E)^{\rm{an}}$ via the usual double coset operators. The action of $\mathbb{T}(U^p)$ commutes with that of $G(\mathbb{Q}_p)$.
We fix now $\alpha\in E^{\times}$, hence a Deligne–Fontaine module $\underline{D}$ over $\mathbb{Q}_p=F_{w_0}$ of rank three of the form $$\label{3Deligne Fontaine}
\underline{D}=Ee_2\oplus Ee_1\oplus Ee_0,\mbox{ with }
\left\{\begin{array}{ccc}
\varphi(e_2)&=&\alpha e_2\\
\varphi(e_1)&=&p^{-1}\alpha e_1\\
\varphi(e_0)&=&p^{-2}\alpha e_0\\
\end{array}\right.
\mbox{ and }
\left\{\begin{array}{ccc}
N(e_2)&=&e_1\\
N(e_1)&=&e_0\\
N(e_0)&=&0\\
\end{array}\right.$$ and finally a tuple of Hodge–Tate weights $\underline{k}=(k_1>k_2>k_3)$. If $\rho: \mathrm{Gal}(\overline{F}/F)\rightarrow \mathrm{GL}_3(E)$ is an absolute irreducible continuous representation which is unramified at each finite place $w$ lying over a finite place $\ell\in D(U^p)$, we can associate to $\rho$ a maximal ideal $\mathfrak{m}_{\rho}\subseteq \mathbb{T}(U^p)$ with residual field $E$ by the usual method described in the middle paragraph on Page 58 of [@Bre17]. We use the notation $\star_{\mathfrak{m}_\rho}$ for spaces of localization and $\star[\mathfrak{m}_\rho]$ for torsion subspaces where $\star\in\{\widehat{S}(U^p,E), \widehat{S}(U^p,E)^{\rm{alg}}, \widehat{S}(U^p,E)^{\rm{an}}\}$.
We assume that there exists $U^p$ and $\rho$ such that
1. $\rho$ is absolutely irreducible and unramified at each finite place $w$ of $F$ over a place $\ell$ of $\mathbb{Q}$ satisfying $\ell\in D(U^p)$;
2. $\widehat{S}(U^p,E)^{\rm{alg}}[\mathfrak{m}_{\rho}]\neq 0$ (hence $\rho$ is automorphic and $\rho_{w_0}:=\rho|_{\mathrm{Gal}(\overline{F_{w_0}}/F_{w_0})}$ is potentially semi-stable);
3. $\rho_{w_0}$ has Hodge–Tate weights $\underline{k}$ and gives the Deligne–Fontaine module $\underline{D}$.
By identifying $\widehat{S}(U^p,E)^{\rm{alg}}$ with a representation of $\mathrm{GL}_3(\mathbb{Q}_p)$ via $\iota_{G,w_0}$, we have the following isomorphism up to normalization from (\[3global decomposition\]) and [@Ca14]: $$\widehat{S}(U^{v_0},E)^{\rm{alg}}[\mathfrak{m}_{\rho}]\cong\left(\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}\right)^{\oplus d(U^p, \rho)}$$ for all $(U^p, \rho)$ satisfying the conditions (i), (ii) and (iii), where $\lambda=(\lambda_1,\lambda_2,\lambda_3)=(k_1-2,k_2-1,k_3)$ and $d(U^p,\rho)\geq 1$ is an integer depending only on $U^p$ and $\rho$.
\[3theo: main\] We consider $U^p=\prod_{\ell\neq p}U_{\ell}$ and $\rho: \mathrm{Gal}(\overline{F}/F)\rightarrow\mathrm{GL}_3(E)$ such that
1. $\rho$ is absolutely irreducible and unramified at each finite place $w$ of $F$ lying above $D(U^p)$;
2. $\widehat{S}(U^p, E)^{\rm{alg}}[\mathfrak{m}_{\rho}]\neq 0$;
3. $\rho$ has Hodge–Tate weights $\underline{k}$ and gives the Deligne–Fontaine module $\underline{D}$ as in (\[3Deligne Fontaine\]);
4. the filtration on $\underline{D}$ is non-critical in the sense of (ii) of Remark 6.1.4 of [@Bre17];
5. only one automorphic representation $\pi$ contributes to $\widehat{S}(U^p, E)^{\rm{alg}}[\mathfrak{m}_{\rho}]$.
Then there exists a unique choice of $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$ such that: $$\begin{gathered}
\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(\Sigma^{\rm{min},+}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}, ~\widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]\right)\\
\xrightarrow{\sim}\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}, ~\widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]\right).\\\end{gathered}$$
We recall several useful results from [@Bre17] and [@BH18].
\[3prop: injectivity\] Suppose that $U^p=\prod_{\ell\neq p}U_{\ell}$ is a sufficiently small open compact subgroup of $G(\mathbb{A}_{\mathbb{Q}}^{\infty,p})$, $\widehat{S}(U^p, E)^{\rm{an}}\hookrightarrow \Pi\twoheadrightarrow \Pi_1$ a short exact sequence of admissible locally analytic representations of $\mathrm{GL}_3(\mathbb{Q}_p)$, $\chi: T(\mathbb{Q}_p)\rightarrow E^{\times}$ a locally analytic character and $\eta: U(\mathfrak{t})\rightarrow E$ its derived character, then we have $T(\mathbb{Q}_p)^+$-equivariant short exact sequences of finite dimensional $E$-spaces $$(\widehat{S}(U^p, E)^{\rm{an}})^{\overline{N}(\mathbb{Z}_p)}[\mathfrak{t}=\eta]\hookrightarrow \Pi^{\overline{N}(\mathbb{Z}_p)}[\mathfrak{t}=\eta]\twoheadrightarrow \Pi_1^{\overline{N}(\mathbb{Z}_p)}[\mathfrak{t}=\eta]$$ and $$(\widehat{S}(U^p, E)^{\rm{an}})^{\overline{N}(\mathbb{Z}_p)}[\mathfrak{t}=\eta]_{\chi}\hookrightarrow \Pi^{\overline{N}(\mathbb{Z}_p)}[\mathfrak{t}=\eta]_{\chi}\twoheadrightarrow \Pi_1^{\overline{N}(\mathbb{Z}_p)}[\mathfrak{t}=\eta]_{\chi}$$ where $T(\mathbb{Q}_p)^+$ is a submonoid of $T(\mathbb{Q}_p)$ defined by $$T(\mathbb{Q}_p)^+:=\{t\in T(\mathbb{Q}_p)\mid t\overline{N}(\mathbb{Z}_p)t^{-1}\subseteq \overline{N}(\mathbb{Z}_p)\}.$$
This is Proposition 6.3.3 of [@Bre17] and Proposition 4.1 of [@BH18].
\[3prop: socle\] We fix $U^p$ and $\rho$ as in Theorem \[3theo: main\]. For a locally analytic character $\chi: T(\mathbb{Q}_p)\rightarrow E^{\times}$, we have $$\mathrm{Hom}_{T(\mathbb{Q}_p)^+}\left(\chi\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}, ~(\widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}])^{\overline{N}(\mathbb{Z}_p)}\right)\neq 0$$ if and only if $\chi=\delta_{\lambda}$.
This is Proposition 6.3.4 of [@Bre17].
We recall the notation $i_{B}^{\mathrm{GL}_3}(\chi_w^{\infty})$ for a smooth principal series for each $w\in W$ from Section \[3subsection: main notation\]. Given three locally analytic representations $V_i$ for $i=1,2,3$ and two surjections $V_1\twoheadrightarrow V_2$ and $V_3\twoheadrightarrow V_2$, we use the notation $V_1\times_{V_2}V_3$ for the representation given by the fiber product of $V_1$ and $V_3$ over $V_2$ with natural surjections $V_1\times_{V_2}V_3\twoheadrightarrow V_1$ and $V_1\times_{V_2}V_3\twoheadrightarrow V_3$. We also use the shorten notation $V^{\rm{alg}}$ for the maximally locally algebraic subrepresentation of a locally analytic representation $V$. We recall that $U^p$ is *sufficiently small* if there exists $\ell\neq p$ such that $U_{\ell}$ has no non-trivial element with finite order.
\[3prop: adjunction\] We fix $U^p$ and $\rho$ as in Theorem \[3theo: main\] and assume moreover that $U^p$ is a sufficiently small open compact subgroup of $G(\mathbb{A}_{\mathbb{Q}}^{\infty,p})$. We also fix a non-split short exact sequence $V_1\hookrightarrow V_2\twoheadrightarrow V_3$ of finite length representations inside the category $\mathrm{Rep}^{\mathcal{OS}}_{\mathrm{GL}_3(\mathbb{Q}_p), E}$ such that $V_1\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}$ embeds into $\widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]$. We conclude that:
1. if $V_3$ is irreducible and not locally algebraic, then we have an embedding $$V_2\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}\hookrightarrow\widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}];$$
2. if there is a surjection $$\overline{L}(\lambda)\otimes_Ei_{B}^{\mathrm{GL}_3}(\chi_w^{\infty})\twoheadrightarrow V_3$$ for a certain $w\in W$, then there exists a certain quotient $V_4$ of $V_2\times_{V_3}\left(\overline{L}(\lambda)\otimes_Ei_{B}^{\mathrm{GL}_3}(\chi_w^{\infty})\right)$ satisfying $$\mathrm{soc}_{\mathrm{GL}_3(\mathbb{Q}_p)}(V_4)=V_4^{\rm{alg}}=\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}$$ such that we have an embedding $$V_4\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}\hookrightarrow\widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}].$$
This is an immediate generalization (or rather formalization) of Section 6.4 of [@Bre17]. More precisely, part (i) (resp. (ii)) generalizes the Étape 1 (resp. the Étape 2) of Section 6.4 of [@Bre17].
We may assume that $\alpha=1$ for simplicity of notation thanks to Lemma \[3lemm: det twist\]. According to the Étape 1 and 2 of Section 6.2 of [@Bre17], we may assume without loss of generality that $U^p$ is sufficiently small and it is sufficient to show that there exists a unique choice of $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$ such that $$\label{3global embedding}
\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(\Sigma^{\rm{min}, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}, ~\widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]\right)\neq 0.$$ We borrow the notation $\Pi^i(\underline{k}, \underline{D})$ from Theorem 6.2.1 of [@Bre17]. We observe from (\[3main picture for min\]) that $\Sigma^{\rm{min},+}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ contains a unique subrepresentation $\Sigma^{\mathrm{Ext}^1}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ of the form $$\label{3special form ext 1}
\begin{xy}
(0,0)*+{\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}}="a"; (30,6)*+{\Pi^1(\underline{k}, \underline{D})}="b"; (30,-6)*+{\Pi^2(\underline{k}, \underline{D})}="c";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"};
\end{xy}.$$ Moreover, $\Sigma^{\rm{min},+}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ is uniquely determined by $\Sigma^{\mathrm{Ext}^1}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ up to isomorphism. It is known by Étape 3 of Section 6.2 of [@Bre17] that there is at most one choice of $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$ such that $$\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(\Sigma^{\mathrm{Ext}^1}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}, ~\widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]\right)\neq 0,$$ and thus there is at most one choice of $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$ such that (\[3global embedding\]) holds. As a result, it remains to show the existence of $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$ that satisfies (\[3global embedding\]). We notice that $\Sigma^{\rm{min},+}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ admits an increasing filtration $\mathrm{Fil}_{\bullet}$ satisfying the following conditions
1. the representations $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ and $\Sigma^{\sharp,+}(\lambda, \mathscr{L}_1, \mathscr{L}_2)$ ( cf. their definition after Proposition \[3prop: main dim\] and Proposition \[3prop: criterion of existence\]) appear as two consecutive terms of the filtration;
2. each graded piece is either locally algebraic or irreducible.
As a result, the only reducible graded pieces of this filtration is the quotient $$\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)/\Sigma^{\sharp,+}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\cong W_0.$$ Then we can prove the existence of $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$ satisfying (\[3global embedding\]) by reducing to the isomorphism $$\begin{gathered}
\label{3induction on filtration}
\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(\mathrm{Fil}_{k+1}\Sigma^{\rm{max}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}, ~\widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]\right)\\
\xrightarrow{\sim} \mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(\mathrm{Fil}_k\Sigma^{\rm{max}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}, ~\widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]\right)\end{gathered}$$ for each $k\in\mathbb{Z}$. If $$\mathrm{Gr}_k:=\mathrm{Fil}_{k+1}\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)/\mathrm{Fil}_k\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$$ is not locally algebraic, then (\[3induction on filtration\]) is true in this case by part (i) of Proposition \[3prop: adjunction\]. The only locally algebraic graded pieces of the filtration except $\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}$ are $\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}$, $\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}$ and $W_0$. The isomorphism (\[3induction on filtration\]) when the graded piece $\mathrm{Gr}_k$ equals $\overline{L}(\lambda)\otimes_Ev_{P_1}^{\infty}$ or $\overline{L}(\lambda)\otimes_Ev_{P_2}^{\infty}$ has been treated in Étape 2 of Section 6.4 of [@Bre17]. As a result, it remains to show that $$\begin{gathered}
\label{3last isomorphism}
\mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}, ~\widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]\right)\\
\xrightarrow{\sim} \mathrm{Hom}_{\mathrm{GL}_3(\mathbb{Q}_p)}\left(\Sigma^{\sharp,+}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\otimes_E(\mathrm{ur}(\alpha)\otimes_E\varepsilon^2)\circ\mathrm{det}, ~\widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]\right)\end{gathered}$$ to finish the proof of Theorem \[3theo: main\]. It follows from results in Section 5.3 of [@Bre17] ( cf. (53) of [@Bre17]) that $i_{B}^{\mathrm{GL}_3}(\chi_{s_1s_2s_1}^{\infty})$ has the form $$\begin{xy}
(0,0)*+{\mathrm{St}_3^{\infty}}="a"; (20,4)*+{v_{P_1}^{\infty}}="b"; (20,-4)*+{v_{P_2}^{\infty}}="c"; (40,0)*+{1_3}="d";
{\ar@{-}"a";"b"}; {\ar@{-}"a";"c"}; {\ar@{-}"b";"d"}; {\ar@{-}"c";"d"};
\end{xy}$$ and thus there is a surjection $$\overline{L}(\lambda)\otimes_Ei_{B}^{\mathrm{GL}_3}(\chi_{s_1s_2s_1}^{\infty})\twoheadrightarrow W_0.$$ According to part (ii) of Proposition \[3prop: adjunction\], we only need to show that any quotient $V$ of $$V^{\diamond}:=\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\times_{W_0}\left(\overline{L}(\lambda)\otimes_Ei_{B}^{\mathrm{GL}_3}(\chi_{s_1s_2s_1}^{\infty})\right)$$ such that $$\label{3necessary condition}
\mathrm{soc}_{\mathrm{GL}_3(\mathbb{Q}_p)}(V)=V^{\rm{alg}}=\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}$$ must have the form $$\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}^{\prime}_3)$$ for certain $\mathscr{L}^{\prime}_3\in E$. We recall from Proposition \[3prop: criterion of existence\] and our definition of $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ afterwards that $\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$ fits into a short exact sequence $$\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\hookrightarrow\Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\twoheadrightarrow W_0$$ and thus $V^\diamond$ fits (by definition of fiber product) into a short exact sequence $$\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\hookrightarrow V^{\diamond}\twoheadrightarrow i_{B}^{\mathrm{GL}_3}(\chi_{s_1s_2s_1}^{\infty})$$ and in particular $$\mathrm{soc}_{\mathrm{GL}_3(\mathbb{Q}_p)}(V^{\diamond})=\left(\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)^{\oplus2}.$$ Hence the condition (\[3necessary condition\]) implies that $V$ fits into a short exact sequence $$\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\xrightarrow{j} V^{\diamond}\twoheadrightarrow V$$ and that $$j\left(\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\cap \Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)=0\subseteq V^{\diamond}$$ which induces an injection $$\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\hookrightarrow V.$$ Therefore $V$ fits into a short exact sequence $$\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\hookrightarrow V\twoheadrightarrow W_0$$ and thus corresponds to a line $M_V$ inside $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0,~\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right)$$ which is two dimensional by Lemma \[3lemm: preparation for global\]. Moreover, the condition (\[3necessary condition\]) implies that $M_V$ is different from the line $$\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0,~\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\right)\hookrightarrow\mathrm{Ext}^1_{\mathrm{GL}_3(\mathbb{Q}_p),\lambda}\left(W_0,~\Sigma^{\sharp, +}(\lambda, \mathscr{L}_1, \mathscr{L}_2)\right).$$ Hence it follows from Lemma \[3lemm: preparation for global\] that there exists $\mathscr{L}^{\prime}_3\in E$ such that $$V\cong \Sigma^{\rm{min}}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}^{\prime}_3).$$
\[3coro: criterion\] If a locally analytic representation $\Pi$ of the form (\[3special form ext 1\]) is contained in $\widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]$ for a certain $U^p$ and $\rho$ as in Theorem \[3theo: main\], then there exists $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$ uniquely determined by $\Pi$ such that $$\Pi\hookrightarrow \Sigma^{\rm{min},+}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3).$$
We fix $U^p$ and $\rho$ such that the embedding $$\label{3embedding as assumption}
\Pi\hookrightarrow \widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]$$ exists. Then (\[3embedding as assumption\]) restricts to an embedding $$\overline{L}(\lambda)\otimes_E\mathrm{St}_3^{\infty}\hookrightarrow \widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]$$ which extends to an embedding $$\label{3embedding as conclusion}
\Sigma^{\rm{min},+}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\hookrightarrow \widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]$$ for a unique choice of $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$ according to Theorem \[3theo: main\]. The embedding (\[3embedding as conclusion\]) induces by restriction an embedding $$\Sigma^{\mathrm{Ext}^1}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)\hookrightarrow \widehat{S}(U^p, E)^{\rm{an}}[\mathfrak{m}_{\rho}]$$ and therefore we have $$\Pi\cong \Sigma^{\mathrm{Ext}^1}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$$ by Theorem 6.2.1 of [@Bre17]. In particular, we deduce an embedding $$\Pi\hookrightarrow \Sigma^{\rm{min},+}(\lambda, \mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3)$$ for certain invariants $\mathscr{L}_1, \mathscr{L}_2, \mathscr{L}_3\in E$ determined by $\Pi$.
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abstract: 'We analyze the relation between earning forecast accuracy and expected profitability of financial analysts. Modeling forecast errors with a multivariate Gaussian distribution, a complete characterization of the payoff of each analyst is provided. In particular, closed-form expressions for the probability density function, for the expectation, and, more generally, for moments of all orders are obtained. Our analysis shows that the relationship between forecast precision and trading profitability need not to be monotonic, and that, for any analyst, the impact on his expected payoff of the correlation between his forecasts and those of the other market participants depends on the accuracy of his signals. Furthermore, our model accommodates a unique full-communication equilibrium in the sense of [@Radner1979]: if all information is reflected in the market price, then the expected payoff of all market participants is equal to zero.'
author:
- |
Carlo Marinelli\
Department of Mathematics\
University College London, United Kingdom\
- |
Alex Weissensteiner\
School of Economics and Management\
Free University of Bolzano, Italy\
bibliography:
- 'CorrError.bib'
date: 28 January 2013
title: On the relation between forecast precision and trading profitability of financial analysts
---
Introduction
============
Many empirical studies indicate that financial analysts differ in their forecast accuracy [see, e.g., @Stickel1992; @Sinha1997], and that these differences are persistent over time [see @mikhail2004]. Therefore it is natural to ask how the forecast ability of an analyst translates into the profitability of a trading strategy based on his advice. This question is addressed, e.g., in the works of [@Loh2006] and [@Ertimur2007]. Both papers rank analysts according to their earnings forecast accuracy and show that the difference between factor-adjusted returns resulting from recommendations in the highest accuracy quintile and in the lowest accuracy quintile is significantly positive. However, as noted by [@Ertimur2007], since both papers focus only on the contemporaneous relationship between accuracy and profitability, the reported abnormal excess returns among analysts cannot be considered as evidence for the existence of an implementable ex-ante trading strategy.
Furthermore, a strand of literature reports that earning abnormal trading returns based on the recommendations of financial analysts is by no means an easy task: [@Bradshaw2004] shows that although earning forecasts have the highest explanatory power for recommendations, these projections have the least association with future excess returns; [@Barber2001] and [@mikhail2004] conclude that, after trading costs are taken into consideration, the differences in trading performance among analysts become insignificant; [@Brown2008] argue that reported abnormal returns might be spurious due to the fact that forecast errors are scaled by share prices.
To explain the absence of a clear positive relationship between forecast precision and trading profitability one can simply invoke the efficient market hypothesis: if market prices reflect correctly all available information, then, due to the level playing field, no market participant can earn abnormal returns. The paradox of the efficient market hypothesis is that, if every investor believed in the efficiency of the market, then the market would not be efficient because no one would have an incentive to process information. This issue is addressed by [@Grossman1980], who argues that the strong-form efficient market hypothesis is not a meaningful assumption and that gathering information makes sense up to the point where its marginal cost equals its marginal benefit.
Another natural way to analyze the problem is to consider inefficient markets. Different simulation studies by [@Schredelseker1984; @Schredelseker2001] and [@Pfeifer2009] show that for markets out of equilibrium and with asymmetric information the relationship between forecast accuracy and trading profitability might be non-monotonic. This could be an explanation of the fact that linear or monotonic statistical measures, such as Pearson or Spearman correlation, detect only weak or no dependence in the empirical data. The main findings were confirmed also in experimental market settings, see [@huber2007] and [@Huber2008]. [@Lawrenz2012] propose a theoretical model where agents learn, i.e. improve their forecast abilities in a Bayesian sense. They use numerical integration to calculate the expected profit of the single agents, and with a regression analysis they explain the result by forecast errors of the single analyst, by forecast errors of the others and by correlation effects among market participants. Surprisingly, existing empirical studies in this field seem to neglect the covariance of the forecast errors as an important explanatory variable.
Let us now turn to the objectives of this work: we propose a one-period model with asymmetric information, in the context of which we derive closed-form expressions for the probability density function, the expectation, as well as all higher moments of the payoff obtained by each market participant. We are not aware of other contributions in the literature where such complete characterizations in a model with asymmetric information have been obtained. Our analysis yields that any market participant benefits from a reduction (increase) in the volatility of his own signal (of the signals of other market participants) whenever his own signal is negatively correlated with the aggregated (weighted) signal of all other market participants. That is, the expected payoff of any agent improves if his signal becomes more accurate, provided his signal is negatively correlated to the aggregated signal of all other agents. This observation can be interpreted saying that, in this case, agents have an incentive to improve their forecast skills (on this issue cf., e.g., [@Mikhail1999], [@BravHeaton02] or [@Markov2006]). On the other hand, if his signal is positively correlated with the aggregated forecasts of all other market participants, a $J$-shaped relationship between forecast accuracy and expected payoff exists. Such a non-monotonic relationship was first reported in simulation studies by [@Schredelseker1984; @Schredelseker2001] and empirically confirmed by [@huber2007] and [@Huber2008]. Our model provides a rigorous explanation for the emergence of such effects.
Another implication of our analysis is that, for any agent, the impact of the correlation between his own signal and the aggregated signal of all other participants on his expected payoff depends on the accuracy of the signal: if the relative accuracy of this signal is above (below) a certain threshold, the impact of an increase in correlation is positive (negative), while for intermediate levels of accuracy the impact of correlation is non-monotonic. In this sense our model offers an explanation of two (apparently) contradicting empirical observations that have appeared in the literature, i.e. that analysts with a high reputation tend to issue similar predictions [known as herding effect, see, e.g., @Graham1999], as well as to produce recommendations that deviate significantly from the consensus forecasts [see, e.g. @Lamont2002], or to follow an anti-herding strategy [see, e.g. @Bernhardt2006].
Finally, our model accommodates a full communication equilibrium in the sense of [@Radner1979], and this equilibrium is unique. If all available information is correctly reflected in the market price, then the expected trading payoff of all analysts in our model is equal to zero.
The remaining content is organized as follows: in Section \[s:model\] we introduce the model (in particular, we obtain an expression for the expected payoff of each agent), discuss some of its implications through an accurate sensitivity analysis, and prove the existence and uniqueness of a full communication equilibrium. In Section \[s:joint\] we obtain the probability density function and we compute moments of all orders for the payoff of each agent. A numerical example is provided in Section \[s:num\], and Section \[s:con\] concludes.
**Notation.** We denote the Euclidean scalar product by ${\langle \cdot,\cdot \rangle}$. The Gaussian law on ${\mathbb{R}}^n$ with mean $m$ and covariance matrix $Q$ is denoted by $N_{{\mathbb{R}}^n}(m,Q)$, and we omit the subscript if the space is clear. The distribution and density functions of the standard Gaussian law on ${\mathbb{R}}$ are denoted by $\Phi$ and $\phi$, respectively.
Model {#s:model}
=====
In order to model incomplete information we assume that $n$ risk-neutral market participants do not know the true (or fair) value of a company. However, they process available information (e.g., accounting statements) and try to infer the true value of the firm [see, e.g., @Barron1998; @Markov2006]. Although we assume that each market participant $i$, $i=1,2,\ldots,n$, has an unbiased signal (i.e., forecast) $\xi_i$ about the fair value, agents are heterogeneous along two dimensions. In particular, agents differ in the precision of their estimates. In this way we capture the idea that analysts might have distinct skills and/or data at their disposal. Moreover, we assume that the correlation between individual forecasts may differ among analysts [see, e.g., @Fischer1998]. We assume that $\xi=(\xi_1,\dots,\xi_n)$ is a vector of centered jointly Gaussian random variables with non-singular covariance matrix $Q$.
According to their price forecasts, analysts submit conditional buy and sell orders. Whenever the price is below his own estimate ($p<\xi_i$), analyst $i$ is willing to buy and therefore takes a long position, otherwise ($p>\xi_i$) he takes a short position. The true value is revealed immediately after the trade. Following [@Fischer1998], we consider a dealer market, i.e. we assume that a market maker clears the market by buying and selling for his own account. We assume that this market maker sets the price $p$ equal to a weighted average of the different estimates, that is, $$p := \sum_{i=1}^n w_i\xi_i = {\langle w,\xi \rangle},
\label{eq:p}$$ where $w=(w_1,\ldots,w_n) \in {\mathbb{R}}^n$, $\sum_i w_i=1$. In order to avoid degenerate (and trivial) cases, we assume that at least two elements of $w$ are not zero. Of course, setting the price equal to the mean of the single estimates is a special case of this pricing mechanism. If the market maker exploited additional information, then he could assign more weight to more accurate analysts. This is in line with empirical studies which report higher price reactions to forecast revisions of analysts with a higher reputation, see, e.g., [@Gleason2003]. Furthermore, as we will show in Subsection \[s:ree\], the linear pricing function is flexible enough to incorporate correctly all signals revealed by the market participants and to lead to a full communication equilibrium in the sense of [@Radner1979].
We assume, without loss of generality, that the true value of the asset is zero, and, in analogy to [@BravHeaton02], we calculate the expected performance of all agents. For notational convenience we focus on agent 1, but it is clear that the extension of our analysis to the generic agent $i$, $i\in\{1,\ldots,n\}$, is just a matter of relabeling. The expected trading payoff of agent 1 is given by $$\mu_1 := {\mathop{{}\mathbb{E}}}\bigl[ (0-p)\operatorname{sgn}(\xi_1-p) \bigr]
= -{\mathop{{}\mathbb{E}}}\bigl[ p\,\operatorname{sgn}(\xi_1-p) \bigr],$$ where $\operatorname{sgn}: x \mapsto 1_{\left]0,+\infty\right[}(x) -
1_{\left]-\infty,0\right[}(x)$ stands for the signum function. In particular, if the forecast $\xi_1$ is above the market price $p$, then agent $1$ takes a long position with a payoff equal to $(0-p)$, otherwise, if his forecast $\xi_1$ is below the market price $p$, his payoff is equal to $(p-0)$.
As a first step, we determine the joint distribution of $\xi_1$ and $p$.
\[lm:S\] Let $Q_1$ denote the first row of the matrix $Q$. One has $(\xi_1,p)
\sim N_{{\mathbb{R}}^2}(0,S)$, where $$S =
\begin{bmatrix}
q_{11} & {\langle Q_1,w \rangle}\\
{\langle Q_1,w \rangle} & {\langle Qw,w \rangle}
\end{bmatrix}$$ and $\det S>0$. In particular, one has $(\xi_1,p) = (aX+bY,cY)$ in law, where $(X,Y) \sim N_{{\mathbb{R}}^2}(0,I)$ and $$a := \sqrt{q_{11}-b^2} >0, \qquad
b := \frac{{\langle Q_1,w \rangle}}{\sqrt{{\langle Qw,w \rangle}}}, \qquad
c := \sqrt{{\langle Qw,w \rangle}} >0.$$
Since $(\xi_1,p) = A\xi$, where $A: {\mathbb{R}}^n \to {\mathbb{R}}^2$ is the linear map represented by the matrix (which we denote by the same symbol, with an innocuous abuse of notation) $$\label{eq:map}
A=
\begin{bmatrix}
1 & 0 & \cdots & 0 \\
w_1 & w_2 & \cdots & w_n
\end{bmatrix},$$ well-known results on Gaussian laws imply that $(\xi_1,p) \sim
N_{{\mathbb{R}}^2}(0,S)$, where $$S=AQA'=
\begin{bmatrix}
q_{11} & \sum_{j=1}^n w_j q_{1j}\\
\sum_{i=1}^n w_i q_{i1} & \sum_{i,j=1}^n w_iw_jq_{ij}
\end{bmatrix}
=
\begin{bmatrix}
q_{11} & {\langle Q_1,w \rangle}\\
{\langle Q_1,w \rangle} & {\langle Qw,w \rangle}
\end{bmatrix}.
\label{eq:cov1}$$ Note that, due to our assumptions on $w$, $A$ has full rank, hence $S$ is non-singular. In particular, there exists an upper-triangular matrix $B$ of the type $$B=
\begin{bmatrix}
a & b\\
0 & c
\end{bmatrix}$$ such that $S=BB'$. Elementary computations show that one has $$a = \sqrt{q_{11}-b^2}, \qquad
b = \frac{{\langle Q_1,w \rangle}}{\sqrt{{\langle Qw,w \rangle}}}, \qquad
c = \sqrt{{\langle Qw,w \rangle}}.$$ Note that $a$ is well defined (and strictly positive) because $$q_{11} - b^2 = \frac{q_{11}{\langle Qw,w \rangle}-{\langle Q_1,w \rangle}^2}{{\langle Qw,w \rangle}}
= \frac{\det S}{{\langle Qw,w \rangle}},$$ where $\det S>0$ since, as remarked above, $S$ is non-singular, and ${\langle Qw,w \rangle}>0$ because $Q$ is strictly positive definite and $w \neq
0$. Let $Z:=(X,Y) \sim N_{{\mathbb{R}}^2}(0,I)$. Then the law of the random vector $BZ=(aX+bY,cY)$ is $N(0,BB')=N(0,S)$, i.e. it coincides with the law of $(\xi_1,p)$.
Writing $$S =
\begin{bmatrix}
\operatorname{Var} \xi_1 & \operatorname{Cov}(\xi_1,p)\\
\operatorname{Cov}(\xi_1,p) & \operatorname{Var} p
\end{bmatrix}
=:
\begin{bmatrix}
\sigma^2_1 & \sigma_{1p}\\
\sigma_{1p} & \sigma^2_p
\end{bmatrix},$$ it is immediately seen that the following identities hold: $$a = \sqrt{\sigma^2_1-\left(\frac{\sigma_{1p}}{\sigma_p}\right)^2},
\qquad
b = \frac{\sigma_{1p}}{\sigma_p},
\qquad
c = \sigma_p.$$ As the main result of this section, in the following we derive a closed-form expression for the expected payoff of the single agent.
\[prop:22\] Let $a$, $b$, $c$ be defined as in Lemma \[lm:S\], and $\beta:=(c-b)/a$. One has $$\mu_1 = -{\mathop{{}\mathbb{E}}}\bigl[ p \operatorname{sgn}(\xi_1-p) \bigr] =
\frac{2c\beta}{\sqrt{2\pi} \sqrt{1+\beta^2}}.$$
Taking into account that $a>0$, we can write $$\mu_1=-{\mathop{{}\mathbb{E}}}\bigl[ p\,\operatorname{sgn}(\xi_1-p) \bigr]
= -c{\mathop{{}\mathbb{E}}}\bigl[ Y \operatorname{sgn}(aX+(b-c)Y) \bigr]
= -c {\mathop{{}\mathbb{E}}}\bigl[ Y \operatorname{sgn}(X - \beta Y) \bigr].$$ The independence of $X$ and $Y$ yields $${\mathop{{}\mathbb{E}}}\bigl[ Y \operatorname{sgn}(X - \beta Y) \bigr] =
{\mathop{{}\mathbb{E}}}{\mathop{{}\mathbb{E}}}\bigl[Y \operatorname{sgn}(X - \beta Y) \big| Y \bigr] =
\int_{\mathbb{R}}y {\mathop{{}\mathbb{E}}}\bigl[\operatorname{sgn}(X - \beta y)\bigr] \phi(y)\,dy.$$ Observing that one has $$\begin{aligned}
{\mathop{{}\mathbb{E}}}\bigl[\operatorname{sgn}(X - \beta y)\bigr] &=
{\mathop{{}\mathbb{E}}}1_{\{X>\beta y\}} - {\mathop{{}\mathbb{E}}}1_{\{X<\beta y\}}
= {\mathbb{P}}(X>\beta y) - {\mathbb{P}}(X<\beta y)\\
&= 1 - 2{\mathbb{P}}(X<\beta y) = 1 - 2\Phi(\beta y),
\end{aligned}$$ we get, recalling that $\int_{\mathbb{R}}y\phi(y)\,dy=0$, $${\mathop{{}\mathbb{E}}}\bigl[ Y \operatorname{sgn}(X - \beta Y) \bigr] =
\int_{\mathbb{R}}y\bigl( 1-2\Phi(\beta y) \bigr) \phi(y)\,dy =
-2 \int_{\mathbb{R}}\Phi(\beta y) \, y \, \phi(y)\,dy.$$ We thus have $$\mu_1 = -c {\mathop{{}\mathbb{E}}}\bigl[ Y \operatorname{sgn}(X-\beta Y) \bigr] =
2c \int_{\mathbb{R}}\Phi(\beta y) y \phi(y)\,dy.$$ Since $\phi'(y)=-y\phi(y)$ and $\Phi'(\beta y)=\beta \phi(\beta y)$, integration by parts yields, taking into account that $\Phi$ is bounded and $\phi$ is rapidly decreasing at infinity, $$\mu_1 = 2 c \beta \int_{\mathbb{R}}\phi(\beta y) \, \phi(y)\,dy
= \frac{2c\beta}{\sqrt{2\pi}} \int_{\mathbb{R}}\frac{1}{\sqrt{2\pi}}
\exp\left( -(1+\beta^2)y^2/2 \right)\,dy.$$ By the change of variable $y=x/\sqrt{1+\beta^2}$, one finally obtains $$\mu_1 = \frac{2c\beta}{\sqrt{2\pi}\sqrt{1+\beta^2}}
\int_{\mathbb{R}}\frac{1}{\sqrt{2\pi}} e^{-x^2/2}\,dx
= \frac{2c\beta}{\sqrt{2\pi} \sqrt{1+\beta^2}}.
\qedhere$$
As shown in Section \[s:joint\] below, it is possible to provide a complete probabilistic characterization of the trading profit as a random variable, determining its density in closed form.
Recalling the definitions of $a$, $b$, $c$ and $\beta$, the expected payoff $\mu_1$ can also be written as $$\label{eq:expgain}
\mu_1 = \sqrt{2/\pi} \,
\frac{\sigma^2_p-\sigma_{1p}}{\sqrt{\sigma^2_1+\sigma^2_p-2\sigma_{1p}}}
= \sqrt{2/\pi} \,
\frac{\sigma^2_p-\sigma_1\sigma_p\rho_{1p}} {\sqrt{\sigma^2_1+\sigma^2_p-2\sigma_1\sigma_p\rho_{1p}}},$$ where $\left]-1,1\right[ \ni \rho_{1p} := \sigma_{1p}/(\sigma_1
\sigma_p)$ denotes the correlation between $\xi_1$ and $p$. Compared to previous (numerical) simulation studies, this closed-form solution allows for a thorough comparative static analysis. As can be seen from , $(\sigma_1,\sigma_r) \mapsto
\mu_1(\sigma_1,\sigma_r)$ is homogeneous of order $1$. Therefore, without loss of generality, in Figure \[f:eg\] we set $\sigma_p=1$ and show the effect of different levels of $\sigma_1$ and $\rho$.
\[f:eg\]
Sensitivity analysis {#s:imp}
--------------------
Let us introduce the random variable $r:=p - w_1\xi_1 = \sum_{k=2}^n
w_k\xi_k$, and define $$\sigma_r^2 := \operatorname{Var}(r), \qquad
\rho_{1r} := \frac{1}{\sigma_1\sigma_r} \operatorname{Cov}(\xi_1,r).$$ In this section we analyze the impact on the expected payoff $\mu_1$ due to variations in the parameters $\sigma_1$, $\sigma_r$, and $\rho_{1r}$. Note that one has $$\begin{aligned}
\sigma_p^2 &= \operatorname{Var}(w_1\xi_1 + r)
= w_1^2\sigma_1^2 + \sigma_r^2 + 2w_1\sigma_{1r}
= w_1^2\sigma_1^2 + \sigma_r^2 + 2w_1\sigma_1\sigma_r\rho_{1r},\\
\sigma_{1p} &= \operatorname{Cov}(\xi_1,w_1\xi_1+r)
= w_1 \sigma_1^2 + \sigma_{1r}
= w_1 \sigma_1^2 + \sigma_1\sigma_r\rho_{1r},\end{aligned}$$ which implies, by , $$\label{eq:mu-alt}
\mu_1 = \mu_1(\sigma_1,\sigma_r,\rho_{1r}) =
- \sqrt{2/\pi} \;
\frac{w_1(1-w_1)\sigma_1^2 - \sigma_r^2 + (1-2w_1)\rho_{1r}\sigma_r\sigma_1} {\sqrt{(w_1-1)^2\sigma_1^2+\sigma_r^2-2(1-w_1)\rho_{1r}\sigma_r\sigma_1}}.$$
\[p:sigma1\] The following properties of the function $\sigma_1 \mapsto
\mu_1(\sigma_1,\sigma_r,\rho_{1r})$ hold:
- it is strictly decreasing if $\rho_{1r} \leq 0$;
- if $\rho_{1r} > 0$, there exists $\bar{\sigma}_1>0$ such that it is strictly increasing on $\left]0,\bar{\sigma}_1\right]$ and strictly decreasing on $\left[\bar{\sigma}_1,\infty\right[$.
\(a) Let us write $$\mu_1
= - \sqrt{2/\pi} \; \frac{a\sigma_1^2+b\sigma_1-c} {\sqrt{\alpha\sigma_1^2-\beta\sigma_1+c}},$$ where $$\begin{aligned}
a &:= w_1(1-w_1) > 0, & b &:= (1-2w_1)\rho_{1r}\sigma_r,\\
c &:= \sigma_r^2 > 0, & \alpha &:= (1-w_1)^2 > 0,\\
\beta &:= 2(1-w_1)\rho_{1r}\sigma_r.
\end{aligned}$$ After a few tedious but straightforward calculations, one gets $$\frac{\partial\mu_1}{\partial\sigma_1}
= - \sqrt{2/\pi} \bigl( \alpha\sigma_1^2-\beta\sigma_1+c \bigr)^{-3/2}
\, g(\sigma_1),$$ where $$\begin{aligned}
g(x) &= a \alpha x^3
- \frac32 a \beta x^2
+ \bigl( (2a+\alpha) c - b \beta/2 \big)x
+ (b-\beta/2)c\\
&=: a_3x^3 + a_2x^2 + a_1x + a_0
\end{aligned}$$ and $$\begin{aligned}
a_3 &= w_1(1-w_1)^3 >0,\\
a_2 &= -3w_1(1-w_1)^2\sigma_r \rho_{1r},\\
a_1 &= \sigma_r^2 (1-w_1)
\bigl( (1+2\rho_{1r}^2)w_1 + 1-\rho_{1r}^2 \bigr) >0,\\
a_0 &= -w_1 \sigma_r^3 \rho_{1r}.
\end{aligned}$$ It is thus immediately seen that, if $\rho_{1r} \leq 0$, then $g(x) > 0$ for all $x \geq 0$, which proves (a).
\(b) We are going to show that, independently of the value of $\rho_{1r}>0$, the function $g$ is strictly increasing. In fact, one has $g'(x)=3a_3x^2 + 2a_2x + a_1$, whose discriminant $a_2^2 -
3a_1a_3$ has the same sign of $(1+w_1)(\rho_{1r}^2-1)$, which is clearly negative. Therefore, $g'(x)>0$ for all $x \geq 0$, hence $g$ is strictly increasing. In particular, since $g(0)=a_0<0$ and $\lim_{x\to+\infty} g(x)=+\infty$, it follows that $g$ has only one positive root $\bar{\sigma}_1$, with $g(x)<0$ for all $x \in
\left[0,\bar{\sigma}_1\right[$ and $g(x)>0$ for all $x \in
\left]\bar{\sigma}_1,+\infty\right[$, which proves (b).
Therefore, if the signal is negatively correlated with the aggregated signal of the others, then – ceteris paribus – a reduction in the volatility of the signal always increases the expected payoff. Our model is therefore at least partially consistent with empirical results reported in [@Mikhail1999], where single market participants have an incentive to acquire and process information in order to improve their forecast ability. On the other hand, it should be stressed that an improvement of the precision of the signal (i.e. a reduction of $\sigma_1$) does not always imply a higher expected payoff. In particular, if the signal of agent $1$ is positively correlated with the aggregated signal of all other agents, the effect of a decrease in $\sigma_1$ on $\mu_1$ depends, in a complex way, on all parameters of the model. We shall see that an analogous non-monotonic relationship holds with respect to variations in $\sigma_r$.
\[p:sigmar\] The following properties of the function $\sigma_r \mapsto
\mu_1(\sigma_1,\sigma_r,\rho_{1r})$ hold:
- it is strictly increasing if $\rho_{1r} \leq 0$;
- if $\rho_{1r} > 0$, there exists $\bar{\sigma}_r>0$ such that it is strictly decreasing on $\left]0,\bar{\sigma}_r\right]$ and strictly increasing on $\left[\bar{\sigma}_r,\infty\right[$.
Since $(\sigma_1,\sigma_r) \mapsto \mu_1(\sigma_1,\sigma_r)$ is homogeneous of order $1$, Euler’s theorem yields $$\label{eq:eule}
\sigma_1 \frac{\partial\mu_1}{\partial\sigma_1}(\sigma_1,\sigma_r)
+ \sigma_r \frac{\partial\mu_1}{\partial\sigma_r}(\sigma_1,\sigma_r)
= \mu_1(\sigma_1,\sigma_r),$$ hence $$\begin{aligned}
\frac{\partial\mu_1}{\partial\sigma_r} &= \frac{1}{\sigma_r}
\Bigl( \mu_1 - \sigma_1 \frac{\partial\mu_1}{\partial\sigma_1} \Bigr)\\
&= \frac{\sqrt{2/\pi}}{\sigma_r}
\bigl( \alpha\sigma_1^2-\beta\sigma_1+c \bigr)^{-3/2}
\Bigl( \sigma_1 g(\sigma_1) - \bigl( a\sigma_1^2 + b\sigma_1 - c\bigr)
\bigl( \alpha\sigma_1^2 - \beta\sigma_1 + c \bigl) \Bigr)\\
&= \frac{\sqrt{2/\pi}}{\sigma_r}
\bigl( \alpha\sigma_1^2-\beta\sigma_1+c \bigr)^{-3/2} \Bigl(
(-a\beta/2-b\alpha) \sigma_1^3
+ \bigl( c(a+2\alpha) + b\beta/2 \bigr) \sigma_1^2 +\\
&\hspace{3em} -\frac32 c\beta \sigma_1 + c^2 \Bigr)\\
&= \sqrt{2/\pi} \bigl( \alpha\sigma_1^2-\beta\sigma_1+c \bigr)^{-3/2}
\bigl( \sigma_r^3 + a_1 \sigma_1\sigma_r^2 + a_2\sigma_1^2\sigma_r
+ a_3 \sigma_1^3 \bigr)\\
&= \sqrt{2/\pi} \bigl( \alpha\sigma_1^2-\beta\sigma_1+c \bigr)^{-3/2}
h(\sigma_r),
\end{aligned}$$ where $h(x):=x^3 + a_1\sigma_1x^2 + a_2\sigma_1^2x + a_3\sigma_1^3$ and $$\begin{aligned}
a_3 &:= \frac{-a\beta/2-b\alpha}{\sigma_r} = -(1-w_1)^3 \rho_{1r},\\
a_2 &:= \frac{c(a+2\alpha) + b\beta/2}{\sigma_r^2}
= w_1^2(1+2\rho_{1r}^2) - 3w_1(1+\rho_{1r}^2)
+2+\rho_{1r}^2,\\
a_1 &:= \frac{-3c\beta/2}{\sigma_r^3} = -3(1-w_1) \rho_{1r}.
\end{aligned}$$ Let us first show that $a_2>0$: in fact, looking at the above definition of $a_2$ as a polynomial in $w_1$, its roots are $$\frac{3(1+\rho_{1r}^2) \pm (1-\rho_{1r}^2)}{2+4\rho_{1r}^2} \geq 1.$$ (a) Since $a_1, a_2, a_3>0$, Descartes’ rule of signs implies that $h$ has no positive roots. Moreover, since $h(0)=\sigma_1^3a_3>0$, we conclude that $h(x)>0$ for all $x>0$, i.e. $\partial\mu_1/\partial\sigma_r>0$.
\(b) Note that $h$ is strictly increasing: in fact, one has $h'(x) =
3x^2 +2a_1\sigma_1 x + a_2\sigma_1^2$, and the discriminant of this polynomial is proportional to $$a_1^2-3a_2 = 3(1-\rho_{1r}^2)(1-w_1)(w_1-2)<0.$$ Since $a_3<0$ implies that $h(0)<0$, we conclude that $h$ admits exactly one positive root $\bar{\sigma}_r$, as well as that $h$ is negative on $[0,\bar{\sigma}_r]$ and positive on $\left]\bar{\sigma}_r,+\infty\right[$.
Note that the first statement of the previous Proposition simply says that, if $\rho_{1r} \leq 0$, the expected payoff $\mu_1$ improves as $\sigma_r$ increases, i.e. agent $1$ obtains a higher payoff as the relative accuracy of his own signal improves. On the other hand, if $\rho_{1r} > 0$, in analogy to Proposition \[p:sigma1\], the relationship between $\mu_1$ and $\sigma_r$ is no longer monotonic and is determined in a complex way by all parameters of the model. This $J$-shaped relationship between forecast precision and trading profitability was first reported in simulation studies by [@Schredelseker1984; @Schredelseker2001] and also documented in experimental works by [@huber2007] and [@Huber2008]. The authors use a cumulative information setting which implies a positive pairwise correlation. In the numerical example of Section \[s:num\] we adopt this setting, thus reproducing the behavior observed in the above mentioned papers.
It should be remarked that, even though Propositions \[p:sigma1\] and \[p:sigmar\] are qualitatively very similar, it is not true in general that $\partial\mu_1/\partial\sigma_1(\sigma_1,\sigma_r)>0$ implies $\partial\mu_1/\partial\sigma_r(\sigma_1,\sigma_r)<0$ for all $\sigma_1$, $\sigma_r$, as it can be immediately realized looking at . More precise information on the values of the parameters determining the signs of $\mu_1$ and of its partial derivatives can be easily obtained.
Let $\sigma_1$, $\sigma_r$ be fixed positive constants, and $\alpha_+=\alpha_+(\rho_{1r})$ be defined as in below. The function $\rho_{1r} \mapsto
\mu_1(\sigma_1,\sigma_r,\rho_{1r})$ is locally decreasing if $\sigma_r < \alpha_+ \sigma_1$ and locally increasing if $\sigma_r >
\alpha_+ \sigma_1$.
One has, setting $\alpha:=\sigma_r/\sigma_1$, $$\begin{aligned}
\frac{\partial\mu_1}{\partial\rho_{1r}} &= \sqrt{2/\pi} \, A^{-3}
\Bigl[ (1-w_1)(1-2w_1) \sigma_1^2 \sigma_r^2 \rho +
w_1 \sigma_1 \sigma_r^3 -(1-w_1)^3\sigma_1^3 \sigma_r \Bigr]\\
&= \sqrt{2/\pi} \, A^{-3} \sigma_1^3\sigma_r \Bigl[ w_1\alpha^2 +
(1-w_1)(1-2w_1)\rho_{1r}\alpha - (1-w_1)^3 \Bigr],
\end{aligned}$$ where $A$ denotes the denominator of the fraction appearing in . Then the sign of $\partial\mu_1/\partial\rho_{1r}$ is equal to the sign of the polynomial in $\alpha$ $$w_1\alpha^2 + (1-w_1)(1-2w_1)\rho_{1r}\alpha - (1-w_1)^3,$$ whose roots $\alpha_-< 0 < \alpha_+$ are $$\label{eq:alpha}
\alpha_\pm := (1-w_1)
\frac{-(1-2w_1)\rho_{1r} \pm \sqrt{(1-2w_1)^2\rho_{1r}^2+4w_1(1-w_1)}}{2w_1}.$$ In particular, if $\sigma_r/\sigma_1 < \alpha_+$, then $\partial\mu_1/\partial\rho_{1r}$ is negative; if $\sigma_r/\sigma_1 >
\alpha_+$, then $\partial\mu_1/\partial\rho_{1r}$ is positive.
Let $\sigma_1$, $\sigma_r>0$ be given, and define $$\begin{aligned}
\ell &:= \frac{1-w_1}{2w_1} \Bigl(
\sqrt{(1-2w_1)^2+4w_1(1-w_1)} - \lvert 1-2w_1 \rvert \Bigr),\\
u &:= \frac{1-w_1}{2w_1} \Bigl( \sqrt{(1-2w_1)^2+4w_1(1-w_1)} +
\lvert 1-2w_1 \rvert \Bigr).
\end{aligned}$$ If $\sigma_r/\sigma_1 \in [\ell,u]$, then
- if $w_1<1/2$, then there exists $\bar{\rho}$ such that $\rho_{1r} \mapsto \mu_1(\rho_{1r})$ is decreasing on $[-1,\bar{\rho}]$ and increasing on $[\bar{\rho},1]$;
- if $w_1>1/2$, then there exists $\bar{\rho}$ such that $\rho_{1r} \mapsto \mu_1(\rho_{1r})$ is increasing on $[-1,\bar{\rho}]$ and decreasing on $[\bar{\rho},1]$;
Otherwise, if $\sigma_r/\sigma_1 < \ell$, then $\rho_{1r} \mapsto
\mu_1(\rho_{1r})$ is decreasing on $[-1,1]$; if $\sigma_r/\sigma_1 > u$, then $\rho_{1r} \mapsto \mu_1(\rho_{1r})$ is increasing on $[-1,1]$.
Note that one has $$\frac{\partial\alpha_+}{\partial\rho_{1r}} = \frac{1-w_1}{2w_1} \,
\frac{-a\sqrt{a^2\rho_{1r}^2+b}+a^2\rho_{1r}}{\sqrt{a^2\rho_{1r}^2+b}},$$ where $$a:=1-2w_1, \qquad b := 4w_1(1-w_1)>0,$$ therefore $\partial\alpha_+/\partial\rho_{1r}>0$ if and only if $$a^2\rho_{1r} > a \sqrt{a^2\rho_{1r}^2+b}.$$ Simple calculations immediately reveal that this inequality is always satisfied if $a=1-2w_1<0$, and never satisfied if $a=1-2w_1>0$. Equivalently, $\rho_{1r} \mapsto \alpha_+(\rho_{1r})$ is increasing if $w_1>1/2$, and decreasing if $w_1<1/2$. To complete the proofs it is enough to observe that one has $$\begin{aligned}
\alpha_+(-1) &= \frac{1-w_1}{2w_1} \Bigl(
(1-2w_1) + \sqrt{(1-2w_1)^2+4w_1(1-w_1)} \Bigr)\\
\alpha_+(1) &= \frac{1-w_1}{2w_1} \Bigl( -(1-2w_1) +
\sqrt{(1-2w_1)^2+4w_1(1-w_1)} \Bigr). \qedhere
\end{aligned}$$
To describe the relationship between the expect payoff of agent $1$ and the correlation coefficient $\rho_{1r}$ we can thus distinguish three regimes: if the relative accuracy of his signal is low ($\sigma_r/\sigma_1<\ell$), then, for any $\rho_{ir} \in [-1,1]$, agent $1$ gains from a decline in the correlation between his signal and the aggregated signal of all other agents; if the relative accuracy of his signal is high ($\sigma_r/\sigma_1>u$), then, for any $\rho_{ir}\in [-1,1]$, agent 1 gains from an increase in correlation; if the (inverse) relative accuracy $\sigma_r/\sigma_1$ falls within the interval $[\ell,u]$, then the dependence of $\mu_1$ on $\rho_{1r}$ turns out to be non-monotonic. In the first two cases, i.e. when the (inverse) relative accuracy $\sigma_r/\sigma_1$ falls above $u$ or below $\ell$, the result can be heuristically motivated as follows: if agent 1 overestimates (underestimates) the true value, then the market price will be even higher (lower) than his signal. According to his decision rule (buying when the signal is above the market price and selling when the signal is below the market price), he will then take the correct trading position. Moreover, note that, in the third regime, if $w_1<1/2$ the function $\rho_{1r} \mapsto \mu_1(\rho_{1r})$ has exactly one global minimum, which implies a higher expected payoff for extreme (either positive or negative) rather than for intermediate levels of correlation. This result hence offers a complete explanation of the contradicting empirical observations according to which analysts with a high reputation tend to herd [see, e.g. @Graham1999], as well as to deviate more drastically from the consensus forecast [see, e.g., @Lamont2002; @Bernhardt2006].
Full Communication Equilibrium\[s:ree\]
---------------------------------------
In Section \[s:imp\] we consider naive investors who trade on the basis of their individual signals. We propose a one-period model where the true value of the firm, and therefore gains and losses of each analyst, are revealed immediately after the trade. Of course, in the long run the different agents will participate to the market only under the condition that the expected payoff is not negative. This holds also for the market marker, whose expected payoff – by definition of a dealer market – is given by the negative sum of the single $\mu_i$’s. In order to avoid a breakdown of trading as described by the theory of lemon markets and given the zero-sum property of the game, an equilibrium implies that the expected payoff of each agent must be equal to zero (i.e, $\mu_i = 0$ $\forall
i=1,\ldots,n$). A natural question is whether our model, where prices are equal to a weighted average of the single signals, see , is able to accommodate such an equilibrium.
In order to keep the market alive, the market maker will have an interest to ensure a level playing field for all participants. He could use the observed covariance matrix $Q$ and the single signals $\xi_i$ revealed by the conditional orders to set the price. Using the model of [@Black1992] with non-informative priors, one obtains the following choice for the price: $p^*:={\langle Q^{-1}
{\mathbf{1}},{\mathbf{1}}\rangle}^{-1}{\langle Q^{-1} {\mathbf{1}},\xi \rangle}$, which corresponds in to $$w^*:={\langle Q^{-1} {\mathbf{1}},{\mathbf{1}}\rangle}^{-1}(Q^{-1} {\mathbf{1}}).
\label{eq:w}$$ If market prices are fully revealing all available information, then the market is said to be in a full communication equilibrium [see @Radner1979]. With this choice of $w$ one has, for any $k
\in \{1,2,\ldots,n\}$, $$\sigma_{kp} = {\langle Q_k,w^* \rangle} = \frac{{\langle Q^{-1}Q_1,{\mathbf{1}}\rangle}}{{\langle Q^{-1}{\mathbf{1}},{\mathbf{1}}\rangle}}
= \frac{{\langle e_k,{\mathbf{1}}\rangle}}{{\langle Q^{-1}{\mathbf{1}},{\mathbf{1}}\rangle}} = \frac1{{\langle Q^{-1}{\mathbf{1}},{\mathbf{1}}\rangle}}
= {\langle Qw^*,w^* \rangle} = \sigma_p^2,$$ hence, by , in equilibrium the expected payoff of all market participants is zero. As a consequence, the expected payoff of the market marker is also equal to zero. According to [@Radner1979] any fully-revealing communication equilibrium is also a fully-revealing rational expectation equilibrium. Let us show that, in fact, such equilibrium is unique, in the sense of the following Proposition.
There exists one and only one vector $w \in {\mathbb{R}}^n$, with $\sum_{k=1}^n w_k=1$, such that $\mu_i=0$ for all $i=1,2,\ldots,n$.
It is enough to show that there exists a unique vector $w \in
{\mathbb{R}}^n $ with $\sum_{k=1}^n w_k=1$, such that $$\label{eq:ww}
{\langle Qw,w \rangle} = {\langle Q_k,w \rangle} \qquad \forall k=1,\ldots,n.$$ Existence has already been proved by explicitly constructing a solution $w^*$. It is thus enough to prove uniqueness. Condition implies ${\langle Q_1,w \rangle} = {\langle Q_k,w \rangle}$ for all $k>1$, hence ${\langle Q_1-Q_k,w \rangle} = 0$ for all $k>1$. This in turn implies $w
\in V^\perp$, where $$V = \operatorname{span} (Q_1-Q_2, \ldots, Q_1-Q_n),$$ and $V^\perp$ stands for the orthogonal complement of $V$ in ${\mathbb{R}}^n$, so that ${\mathbb{R}}^n = V \oplus V^\perp$. Since $Q$ is assumed to be non-singular, the vectors $Q_1,\ldots,Q_n$ are linearly independent, hence $\dim V = n-1$, which implies that $\dim
V^\perp = 1$. Since $w^* \neq 0$, then $w^*$ is a generator of $V^\perp$. In particular, $0 \neq w \in V^\perp$ implies that there exists $\alpha \neq 0$ such that $w=\alpha w^*$. Then $1 =
\sum_{k=1}^n w_k=\alpha \sum_{k=1}^n w^*_k = \alpha$ yields the uniqueness of $w^*$.
Density function of the payoff {#s:joint}
==============================
In the previous section we have presented a closed-form solution for the expected trading payoffs of the single agents, see . As a matter of fact, we can give a complete characterization of trading payoffs as random variables. In this section we provide a closed-form expression for the density of the payoff for each agent. Throughout the section we adopt the notation introduced in Lemma \[lm:S\] and Proposition \[prop:22\].
We begin with an auxiliary result, which might be interesting in its own right.
\[prop:joint\] Let $$F(x_1,x_2) := {\mathbb{P}}\bigl( p \leq x_1,\, \operatorname{sgn}(\xi_1-p) = x_2 \bigr),
\qquad x_1 \in {\mathbb{R}}, \; x_2 \in \{-1,1\},$$ denote the joint distribution of $p$ and $\operatorname{sgn}(\xi_1-p)$. One has $$F(x_1,x_2;\beta) = \Phi(x_1/c) \Phi(-\beta x_1x_2/c)
- \frac{x_2}{2\pi} \arctan(1/\beta)
+ x_2 \operatorname{sgn}\beta \biggl( \frac14
+ \int_0^{\frac{|\beta|}{c}x_1} \Phi(z/|\beta|) \phi(z)\,dz \biggr).$$
By Lemma \[lm:S\], there exist $a>0$, $b\in{\mathbb{R}}$ and $c>0$ such that $p=cY$ and $\xi_1=aX+bY$ in law, where $X$ and $Y$ are independent standard Gaussian random variables. Setting $\beta=(c-b)/a$, one has $$F(x_1,x_2) = {\mathbb{P}}\bigl(Y \leq x_1/c,\; \operatorname{sgn}(X-\beta Y) = x_2\bigr).$$ Let us record, for later use, the following obvious observation: $$\begin{aligned}
\Phi(x_1/c) &= {\mathbb{P}}(Y \leq x_1/c) = {\mathbb{P}}\bigl( Y \leq x_1/c,\,
\operatorname{sgn}(X-\beta Y) = -1 \bigr)
+ {\mathbb{P}}\bigl( Y \leq x_1/c,\, \operatorname{sgn}(X-\beta Y) = 1 \bigr)\\
&= F(x_1,-1) + F(x_1,1).
\end{aligned}$$ Let us consider first the case $\beta>0$: one has $$F(x_1,-1) = {\mathbb{P}}\bigl( Y \leq x_1/c,\, \operatorname{sgn}(X-\beta Y) = -1 \bigr) =
{\mathbb{P}}\bigl( Y \leq x_1/c,\, X<\beta Y \bigr),$$ where we have used the fact that $a>0$, $c>0$. Hence $$\begin{aligned}
F(x_1,-1) &= {\mathbb{P}}\bigl(X/\beta \leq Y \leq x_1/c \bigr) \\
&= \int_{-\infty}^{\frac{\beta}{c}x_1} {\mathbb{P}}(z/\beta \leq Y \leq
x_1/c)
\, \phi(z)\,dz\\
&= \Phi(x_1/c) \Phi(\beta x_1/c)
- \int_{-\infty}^{\frac{\beta}{c}x_1} \Phi(z/\beta) \phi(z)\,dz.
\end{aligned}$$ Appealing to Lemma \[lm:atan\], we end up with $$F(x_1,-1) = \Phi(x_1/c) \Phi(\beta x_1/c)
- \frac14 + \frac{1}{2\pi} \arctan(1/\beta) - \int_0^{\frac{\beta}{c}x_1}
\Phi(z/\beta) \phi(z)\,dz.$$ The expression for $F(x_1,1)$ is obtained as follows: $$\begin{aligned}
F(x_1,1) &= \Phi(x_1/c) - F(x_1,-1)\\
&= \Phi(x_1/c) \bigl( 1 - \Phi(\beta x_1/c) \bigr) +\frac14 -
\frac{1}{2\pi} \arctan(1/\beta)
+ \int_0^{\frac{\beta}{c}x_1} \Phi(z/\beta) \phi(z)\,dz\\
&= \Phi(x_1/c) \Phi(-\beta x_1/c) + \frac14 - \frac{1}{2\pi}
\arctan(1/\beta)
+ \int_0^{\frac{\beta}{c}x_1} \Phi(z/\beta) \phi(z)\,dz\\
\end{aligned}$$ Let us now consider the case $\beta<0$: by a reasoning completely analogous to the one used above, we can write $$\begin{aligned}
F(x_1,1) &= {\mathbb{P}}\bigl(Y \leq x_1/c,\; X/\beta \leq Y\bigr)
= {\mathbb{P}}(X/\beta \leq Y \leq x_1/c)\\
&= \int_{\frac{\beta}{c}x_1}^\infty
{\mathbb{P}}(z/\beta \leq Y \leq x_1/c)\,\phi(z)\,dz\\
&= \Phi(x_1/c) \Phi(-\beta x_1/c)
- \int_{-\infty}^{-\frac{\beta}{c}x_1} \Phi(-z/\beta)\phi(z)\,dz\\
&= \Phi(x_1/c) \Phi(-\beta x_1/c) - \frac14 - \frac{1}{2\pi}
\arctan(1/\beta)
- \int_0^{\frac{|\beta|}{c}x_1} \Phi(z/|\beta|) \phi(z)\,dz,
\end{aligned}$$ hence also $$\begin{aligned}
F(x_1,-1) &= \Phi(x_1/c) - F(x_1,1)\\
&= \Phi(x_1/c) \bigl(1 - \Phi(-\beta x_1/c) \bigr) + \frac14 +
\frac{1}{2\pi} \arctan(1/\beta)
+ \int_0^{\frac{|\beta|}{c}x_1} \Phi(z/|\beta|) \phi(z)\,dz\\
&= \Phi(x_1/c) \Phi(\beta x_1/c) + \frac14 + \frac{1}{2\pi}
\arctan(1/\beta) + \int_0^{\frac{|\beta|}{c}x_1} \Phi(z/|\beta|)
\phi(z)\,dz.
\end{aligned}$$ We may thus write $$\begin{aligned}
F(x_1,-1) &= \Phi(x_1/c) \Phi(\beta x_1/c) + \frac{1}{2\pi}
\arctan(1/\beta) - \operatorname{sgn}\beta \left( \frac14 +
\int_0^{\frac{|\beta|}{c}x_1}
\Phi(z/\beta) \phi(z)\,dz\right),\\
F(x_1,1) &= \Phi(x_1/c) \Phi(-\beta x_1/c) - \frac{1}{2\pi}
\arctan(1/\beta) + \operatorname{sgn}\beta \left(\frac14 +
\int_0^{\frac{|\beta|}{c}x_1} \Phi(z/\beta) \phi(z)\,dz\right),
\end{aligned}$$ that are equivalent to the claim.
The joint distribution of $p$ and $\operatorname{sgn}(\xi_1-p)$ can alternatively be expressed in terms of the bivariate Gaussian law. Consider, for instance, the case $\beta>0$ and $x_2=-1$: Lemma \[lm:phi2\] yields $$\begin{aligned}
F(x_1,-1) &= \Phi(x_1/c) \Phi(\beta x_1/c)
- \int_{-\infty}^{\frac{\beta}{c}x_1} \Phi(z/\beta) \phi(z)\,dz\\
&= \Phi(x_1/c) \Phi(\beta x_1/c) - \Phi_2\left(\beta x_1/c,0;
-\frac{1}{\sqrt{1+\beta^2}}\right),
\end{aligned}$$ where $\Phi_2(\cdot,\cdot;\rho)$ denotes the distribution function of a bivariate Gaussian random variable with correlation coefficient $\rho$.
The following Proposition is the main result of this section and of the whole paper.
Let $M=p \operatorname{sgn}(\xi_1-p)$ denote the negative payoff of agent $1$. The random variable $M$ has a (smooth) density $f_M$ given by $$f_M(z) = \frac2c \phi(z/c) \Phi(-\beta z/c).$$
Let us first compute the distribution $F_M(z):={\mathbb{P}}(M \leq z)$ of the random variable $M$. One has $$\begin{aligned}
F_M(z) &= {\mathbb{P}}( p \operatorname{sgn}(\xi_1-p) \leq z)\\
&= {\mathbb{P}}\bigl( p \operatorname{sgn}(\xi_1-p) \leq z,\; \operatorname{sgn}(\xi_1-p)=-1 \bigr)
+ {\mathbb{P}}\bigl( p \operatorname{sgn}(\xi_1-p) \leq z,\; \operatorname{sgn}(\xi_1-p)=1 \bigr)\\
&= {\mathbb{P}}\bigl( p \geq -z,\; \operatorname{sgn}(\xi_1-p)=-1 \bigr)
+ {\mathbb{P}}\bigl( p \leq z,\; \operatorname{sgn}(\xi_1-p)=1 \bigr)\\
&= {\mathbb{P}}(\operatorname{sgn}(\xi_1-p)=-1) - F(-z,-1) + F(z,1),
\end{aligned}$$ where $F$ denotes the joint distribution of $p$ and $\operatorname{sgn}(\xi_1-p)$, and $${\mathbb{P}}(\operatorname{sgn}(\xi_1-p)=-1) = {\mathbb{P}}(X \leq \beta Y) = 1/2,$$ which follows immediately because $X$ and $Y$ are independent, symmetric, and have a continuous distribution: ${\mathbb{P}}(X \leq \beta Y) +
{\mathbb{P}}(X \geq \beta Y) = 1$, but ${\mathbb{P}}(X \geq \beta Y) = {\mathbb{P}}(-X \geq -\beta
Y) = {\mathbb{P}}(X \leq \beta Y)$.
Thanks to Proposition \[prop:joint\], one has $$F_M(z) = C_\beta + \bigl(2\Phi(z/c)-1\bigr) \Phi(-\beta z/c)
+ \operatorname{sgn}\beta \left( \int_0^{-\frac{|\beta|}{c}z} \Phi(y/|\beta|) \phi(y)\,dy
+ \int_0^{\frac{|\beta|}{c}z} \Phi(y/|\beta|) \phi(y)\,dy \right),$$ where $C_\beta$ denotes a constant that depends only on $\beta$, hence $$\begin{aligned}
f_M(z) = \frac{d}{dz} F_M(z) &= \frac2c \phi(z/c) \Phi(-\beta z/c)
-\frac{\beta}{c}\phi(\beta z/c) \bigl(2\Phi(z/c)-1\bigr)\\
&\qquad + \operatorname{sgn}\beta \, \Bigl( \frac{|\beta|}{c}
\Phi(z/c)\phi(|\beta|z/c)
- \frac{|\beta|}{c} \Phi(-z/c)\phi(-|\beta|z/c) \Bigr)\\
&= \frac2c \phi(z/c) \Phi(-\beta z/c) + \frac{\beta}{c}\phi(\beta
z/c)
-2\frac{\beta}{c} \Phi(z/c) \phi(\beta z/c)\\
&\qquad + \frac{\beta}{c} \Phi(z/c) \phi(\beta z/c)
- \frac{\beta}{c} \Phi(-z/c) \phi(\beta z/c)\\
&= \frac2c \phi(z/c) \Phi(-\beta z/c) + \frac{\beta}{c}\phi(\beta
z/c)
- \frac{\beta}{c}\phi(\beta z/c) \bigl( \Phi(z/c) + \Phi(-z/c) \bigr)\\
&= \frac2c \phi(z/c) \Phi(-\beta z/c).
\qedhere
\end{aligned}$$
It is immediately seen that the density of the random variable $-M$, which represents the trading payoff of agent $1$, is given by $$f_{-M}(z) = f_M(-z) = \frac2c \phi(z/c) \Phi(\beta z/c).$$
Higher moments {#s:mom}
--------------
Given the complete probabilistic characterization of the (negative) trading profit $M$ just obtained, one can compute any moment of $M$ simply integrating against the density $f_M$. It is easier, however, to proceed differently.
For an odd number $2k-1$, $k\in\mathbb{N}$, we define $(2k-1)!! :=
\prod_{j=1}^k (2j-1)$, and set, by convention, $(-1)!!:=1$.
Let $k \in \mathbb{N}$. One has $$\begin{aligned}
{\mathop{{}\mathbb{E}}}M^{2k} &= (2k-1)!! \, {\langle Qw,w \rangle}^k,\\
{\mathop{{}\mathbb{E}}}M^{2k-1} &= -\frac{2^k (k-1)! \, c^{2k-1} \beta}{\sqrt{2\pi(1+\beta^2)}} \,
\sum_{j=0}^{k-1} \frac{(2j-1)!!}{(2j)!!} \, \frac{1}{(1+\beta^2)^j}.
\end{aligned}$$
Since $M^{2k} = p^{2k}$, and $p={\langle w,\xi \rangle}$ is a centered Gaussian random variable with variance ${\langle Qw,w \rangle}$, standard formulas for moments of Gaussian laws give $${\mathop{{}\mathbb{E}}}M^{2k} = {\mathop{{}\mathbb{E}}}p^{2k} = {\mathop{{}\mathbb{E}}}{\langle w,\xi \rangle}^{2k}
= (2k-1)!! \, {\langle Qw,w \rangle}^k.$$ Moreover, by Lemma \[lm:S\], one has $$\begin{aligned}
{\mathop{{}\mathbb{E}}}M^{2k+1} &= {\mathop{{}\mathbb{E}}}p^{2k+1}\operatorname{sgn}(\xi_1-p)
= c^{2k+1} {\mathop{{}\mathbb{E}}}Y^{2k+1} \operatorname{sgn}(X-\beta Y)\\
&= c^{2k+1} \int_{\mathbb{R}}y^{2k+1} {\mathop{{}\mathbb{E}}}\left[\operatorname{sgn}(X-\beta y)\right] \, \phi(y)\,dy,
\end{aligned}$$ where ${\mathop{{}\mathbb{E}}}\left[\operatorname{sgn}(X-\beta y)\right] = 1-2\Phi(\beta y)$, hence $$\label{eq:dispari}
{\mathop{{}\mathbb{E}}}M^{2k+1} = -2c^{2k+1} \int_{\mathbb{R}}y^{2k+1} \Phi(\beta y)
\,\phi(y)\,dy.$$ Note that, setting $v(y)=y^{2k}\phi(y)$, one has $$v'(y) = \bigl( 2k y^{2k-1} - y^{2k+1}\bigr) \phi(y),$$ hence, integrating by parts in , $$\label{eq:ric}
\begin{split}
\int_{\mathbb{R}}\Phi(\beta y) \, y^{2k+1} \phi(y)\,dy &=
\int_{\mathbb{R}}\Phi(\beta y)
\bigl( y^{2k+1} - 2k y^{2k-1} \bigr)\phi(y)\,dy\\
&\qquad + 2k \int_{\mathbb{R}}\Phi(\beta y) \, y^{2k-1}\phi(y)\,dy\\
&= \beta \int_{\mathbb{R}}y^{2k} \phi(\beta y) \phi(y)\,dy
+ 2k \int_{\mathbb{R}}\Phi(\beta y) \, y^{2k-1}\phi(y)\,dy,
\end{split}$$ where, setting $\sigma_\beta := (1+\beta^2)^{-1/2}$ for convenience of notation, and recalling again standard formulas for moments of Gaussian laws, $$\begin{aligned}
\int_{\mathbb{R}}y^{2k} \phi(\beta y) \phi(y)\,dy &=
\bigl( 2\pi(1+\beta^2) \bigr)^{-1/2}
\frac{1}{\sigma_\beta \sqrt{2\pi}}
\int_{\mathbb{R}}y^{2k} e^{-y^2/(2\sigma_\beta^2)}\,dy\\
&= \bigl( 2\pi(1+\beta^2) \bigr)^{-1/2} \, (2k-1)!! \,
(1+\beta^2)^{-k}.
\end{aligned}$$ Therefore, setting $$\begin{aligned}
f_k &:= \int_{\mathbb{R}}\Phi(\beta y) \, y^{2k-1}\phi(y)\,dy,\\
a_k &:= \frac{\beta}{\sqrt{2\pi(1+\beta^2)}} \, (2k-1)!! \,
(1+\beta^2)^{-k},
\end{aligned}$$ can be written as $$f_{k+1} - 2k f_k = a_k, \qquad
f_1=\frac{\beta}{\sqrt{2\pi(1+\beta^2)}}.$$ Dividing both sides of this difference equation by $2^k k!$, we are left with $$\frac{f_{k+1}}{2^k\,k!} - \frac{f_k}{2^{k-1}\,(k-1)!}
= \frac{a_k}{2^k\,k!},$$ hence, setting $$g_k := \frac{f_k}{2^{k-1}\,(k-1)!},
\qquad b_k := \frac{a_k}{2^k\,k!},$$ the previous difference equation is equivalent to $$g_{k+1}-g_k = b_k, \qquad g_1 = \frac{\beta}{\sqrt{2\pi(1+\beta^2)}},$$ which can be easily solved, writing the telescoping sum $$\begin{aligned}
g_k - g_1 &= (g_k - g_{k-1}) + (g_{k-1} - g_{k-2}) + \cdots
+ (g_2 - g_1)\\
&= b_{k-1} + b_{k-2} + \cdots + b_1,
\end{aligned}$$ which yields $$\begin{aligned}
f_k &= 2^{k-1} (k-1)! \, g_k
= 2^{k-1} (k-1)! \, \biggl( g_1 + \sum_{j=1}^{k-1} b_j \biggl)\\
&= 2^{k-1} (k-1)! \, \biggl( \frac{\beta}{\sqrt{2\pi(1+\beta^2)}}
+ \frac{\beta}{\sqrt{2\pi(1+\beta^2)}} \sum_{j=1}^{k-1}
\frac{(2j-1)!!}{2^jj!} \, \frac{1}{(1+\beta^2)^j} \biggr)\\
&= \frac{2^{k-1} (k-1)! \, \beta}{\sqrt{2\pi(1+\beta^2)}} \,
\sum_{j=0}^{k-1} \frac{(2j-1)!!}{(2j)!!} \, \frac{1}{(1+\beta^2)^j}.
\end{aligned}$$ By definition of $f_k$ and one finally obtains $${\mathop{{}\mathbb{E}}}M^{2k-1} = -2c^{2k-1} f_k
= -\frac{2^k (k-1)! \, c^{2k-1} \beta}{\sqrt{2\pi(1+\beta^2)}} \,
\sum_{j=0}^{k-1} \frac{(2j-1)!!}{(2j)!!} \, \frac{1}{(1+\beta^2)^j}.
\qedhere$$
It is immediately seen that the moments of the payoff $-M$ of even degree coincide with those of $M$, while the moments of odd degree are equal in absolute value, but with opposite sign. Moreover, centered moments can easily be obtained by the non-central one just derived.
Numerical example {#s:num}
=================
In this section we illustrate our main results with a numerical example. We assume that four agents, labeled by $i \in
\{1,\ldots,4\}$, participate to the market. The covariance matrix $Q$ of their signals is displayed in Table \[t:signal\] below. The analysts are labeled according to their forecast precision (measured by the standard deviation $\sigma_i$ of their signals), in reverse order. The market maker assigns equal weight ($w_i=0.25)$ to each signal. The numerical values chosen here try to mimic models with a cumulative information structure [see @Schredelseker1984; @Schredelseker2001], where analysts with an intermediate accuracy implicitly face the highest correlation (note that the signal of agent $1$ is less correlated with other agents’ signals than the signal of agent $2$).
$i$ $\sigma_i$
----- ------------ ----- ----- ----- ----- ------- ------- ------- -------
1 1.3 1 0.9 0.6 0.3 1.690 1.404 0.858 0.390
2 1.2 0.9 1 0.8 0.6 1.404 1.440 1.056 0.720
3 1.1 0.6 0.8 1 0.7 0.858 1.056 1.210 0.770
4 1 0.3 0.6 0.7 1 0.390 0.720 0.770 1.000
: Signal structure of the numerical example: $C$ and $Q$ denote the correlation matrix and the covariance matrix, respectively.[]{data-label="t:signal"}
In the following we compare the payoffs of all analysts. We use the formulas of the previous section to calculate the first four central moments.[^1] Table \[t:mom\] shows that, although analyst $1$ has a less accurate signal than analyst $2$ (with a resulting higher variance and kurtosis in the payoffs), his expected payoff is better. Obviously, analyst $2$ suffers from his high correlation with all other market participants, which is also reflected in a more negative skewness.
Therefore, our model reproduces the non-monotone relationship between forecast precision and trading profitability observed in [@Schredelseker1984; @Schredelseker2001].
$i$ $\mu_i$ $\sigma^2_i$ $\varsigma_i$ $\kappa_i$
----- --------- -------------- --------------- ------------
1 -0.115 0.970 -0.001 2.825
2 -0.406 0.819 -0.029 2.018
3 0.016 0.983 0.000 2.900
4 0.285 0.902 0.010 2.444
: Higher moments of the payoff for analysts $i=1,\ldots,4$: $\varsigma_i$ and $\kappa_i$ denote the skewness and kurtosis of agent $i$, respectively.[]{data-label="t:mom"}
Figure \[f:density\] shows the density function of the payoff for all analysts.
![Density function for the payoffs of analysts $i=1,\ldots,4$.[]{data-label="f:density"}](density_4.png)
Figure \[f:mu1\] shows the dependence of the expected payoff of agent $1$ on the accuracy of his own signal (as measured by the standard deviation $\sigma_1$) and on the correlation between his signal $\xi_1$ and the aggregated signal of all other agents $r$. The value of $\sigma_r$ is equal to $0.739$. The non-monotonic relationship between $\mu_1$ and $\sigma_1$ is clearly displayed for some positive values of $\rho_{1r}$. Moreover, while for large (small) values of $\sigma_1$ the expected payoff is decreasing (increasing) as $\rho_{1r}$ increases, for intermediate values of $\sigma_1$ a non-monotonic behaviour can be observed. Given $w_1=0.25<1/2$, $\mu_1$ is first decreasing and then increasing in $\rho_{1r}$.
![Expected payoff of analysts $1$ as function of $\sigma_1$ and $\rho_{1r}$, with $\sigma_r=0.739$.[]{data-label="f:mu1"}](fig_mu_exe.png)
Conclusion {#s:con}
==========
We propose a model in which analysts differ in the precision of their signals as well as in the correlation between each other. We provide a complete characterization of the trading payoff of each agent, obtaining its probability density function in closed form, as well as explicit expressions for its moments of all orders. Such precise description allows us to perform a detailed sensitivity analysis, with the following implications: If the forecasts of an analyst are negatively correlated with the aggregated forecasts of the others, then he always takes advantage of improving his forecast skills. If the correlation between the forecasts is positive, then a non-monotonic relationship between forecast precision and trading profitability exists. Furthermore, the impact of correlation on the expected payoff of an analyst depends on the relative accuracy of his signal: if the relative accuracy is below (above) a certain threshold, then he suffers (benefits) from an increasing correlation, while for intermediate levels of relative accuracy the relationship is non-monotonic.
This is the first time, to the best of our knowledge, that an analytical model is proposed, which fully explains the non-trivial interplay between forecast accuracy and trading performance. Our model recovers as special cases previous (partial) results from simulation studies, and is able to explain the different – sometimes contradicting – empirical observations reported in the literature. Finally, we provide strong evidence that empirical studies on the relationship between forecast precision and trading profitability need to take into account the correlation structure of the forecasts.
Appendix
========
\[lm:atan\] Let $a \in {\mathbb{R}}$. One has $$\begin{aligned}
\int_0^\infty \Phi(ax)\,\phi(x)\,dx &= \frac14 + \frac{1}{2\pi}\arctan a,\\
\int_{-\infty}^0 \Phi(ax)\,\phi(x)\,dx &= \frac14 -
\frac{1}{2\pi}\arctan a.
\end{aligned}$$
For $a>0$ one has $$\Phi(ax) = \frac12 + \int_0^{ax} \phi(y)\,dy
\qquad \forall x \geq 0,$$ hence $$\int_0^\infty \Phi(ax)\,\phi(x)\,dx =
\int_0^\infty \left(\frac12 + \int_0^{ax} \phi(y)\,dy \right)
\phi(x)\,dx = \frac14 + \int_{D_a} \phi_2(x,y)\,dx\,dy,$$ where ${\mathbb{R}}^2 \supset D_a := \bigl\{ 0 \leq x < \infty, \; 0 \leq y
\leq ax\bigr\}$ and $\phi_2$ stands for the density of the standard Gaussian measure on ${\mathbb{R}}^2$. Since $D_a$ is a cone of ${\mathbb{R}}^2$ with vertex at the origin and aperture equal to $\arctan a$, taking the rotational invariance of $\phi_2$ into account (or, equivalently, passing to radial coordinates), we have $$\int_{D_a} \phi_2(x,y)\,dx\,dy = \frac{\arctan a}{2\pi}.$$ Let us now assume $a<0$. Then $\Phi(ax) = 1-\Phi(|a|x)$ for all $x
\geq 0$, hence, recalling that $\arctan$ is odd, $$\int_0^\infty \Phi(ax)\,\phi(x)\,dx = \frac12
- \int_0^\infty \Phi(|a|x)\,\phi(x)\,dx = \frac12 - \frac14
- \frac{\arctan |a|}{2\pi} = \frac14 + \frac{\arctan a}{2\pi}.$$ The first identity is thus proved. The second follows immediately: $$\int_{-\infty}^0 \Phi(ax)\,\phi(x)\,dx =
\int_0^\infty \Phi(-ax)\,\phi(x)\,dx =
\int_0^\infty \bigl(1-\Phi(ax)\bigr)\,\phi(x)\,dx =
\frac12 - \frac14 - \frac{1}{2\pi}\arctan a.
\qedhere$$
\[lm:phi2\] One has $$\int_{-\infty}^y \Phi(bx) \phi(x)\,dx =
\Phi_2\bigl(y,0;-b(1+b^2)^{-1/2}\bigr)$$
It is enough to write $$\Phi(bx) = \int_{-\infty}^{bx} \phi(z)\,dz =
\int_{-\infty}^0 \phi(z+bx)\,dz =
\sqrt{b^2+1} \int_{-\infty}^0 \phi(z\sqrt{b^2+1}+bx)\,dz,$$ which implies $$\int_{-\infty}^y \Phi(bx) \phi(x)\,dx =
\frac{\sqrt{b^2+1}}{2\pi} \, \int_{\Xi_y} \exp\Bigl(
- \frac12\bigl( (b^2+1)x^2 + 2b(b^2+1)^{1/2}xz + (b^2+1)z^2 \bigr)
\Bigr)\,dx\,dz,$$ where ${\mathbb{R}}^2 \supset \Xi_y:=\left]-\infty,y\right] \times
\left]-\infty,0\right]$. Writing $$\begin{aligned}
(b^2+1)x^2 + 2b(b^2+1)^{1/2}xz + (b^2+1)z^2 &=
(b^2+1) \left( x^2 + z^2 - 2 \frac{-b}{\sqrt{b^2+1}}xz \right)\\
&= \frac{1}{1-\rho^2} \bigl( x^2 - 2\rho xz + z^2 \bigr),
\end{aligned}$$ with $\displaystyle \rho:= -\frac{b}{(b^2+1)^{1/2}}$, we are left with $$\int_{-\infty}^y \Phi(bx) \phi(x)\,dx = \int_{\Xi_y} f(x,z;R)\,dx\,dz,$$ where $f(\cdot,\cdot;R)$ denotes the density of $N_{{\mathbb{R}}^2}(0,R)$, with $$R =
\begin{bmatrix}
1 & \rho\\
\rho & 1
\end{bmatrix} =
\begin{bmatrix}
1 & \displaystyle -\frac{b}{\sqrt{b^2+1}}\\
\displaystyle -\frac{b}{\sqrt{b^2+1}} & 1
\end{bmatrix}.
\qedhere$$
[^1]: Note that central moments can be immediately obtained from the non-central moments of Section \[s:mom\].
|
---
author:
- Gianfranco Brunetti
title: 'Non-thermal emission from Galaxy Clusters and future observations with the FERMI gamma-ray telescope and LOFAR'
---
Introduction {#sec:intro}
============
Clusters of galaxies are the largest gravitationally bound objects in the present universe, containing $\approx 10^{15}$ M$_{\odot}$ of hot ($10^8$ K) gas, galaxies and dark matter. The thermal gas, that is the dominant component in the Inter-Galactic-Medium (IGM), is mixed with non-thermal components such as magnetic fields and relativistic particles, as proved by radio observations. Non-thermal components play a key role by controlling transport processes in the IGM (e.g. Narayan & Medvedev 2001; Lazarian 2006a) and are sources of additional pressure (e.g. Ryu et al. 2003), thus their origin and evolution are important ingredients to understand the physics of the IGM.
The bulk of present information on the non-thermal components comes from radio telescopes that discovered an increasing number of Mpc-sized diffuse radio sources in a fraction of galaxy clusters (e.g. Feretti 2003; Ferrari et al. 2008). Cluster mergers are the most energetic events in the universe and are believed to power the mechanisms responsible for the origin of the non-thermal components in galaxy clusters. A fraction of the energy dissipated during these mergers is expected to be channelled into the amplification of the magnetic fields (e.g. Dolag et al. 2002; Subramanian et al. 2006) and into the acceleration of particles via shocks and turbulence that lead to a complex population of primary electrons and protons in the IGM (e.g. Ensslin et al. 1998; Sarazin 1999; Blasi 2001; Brunetti et al. 2001, 2004; Petrosian 2001; Miniati et al. 2001; Ryu et al. 2003; Dolag 2006; Brunetti & Lazarian 2007; Pfrommer 2008). Theoretically relativistic protons are expected to be the dominant non-thermal particles component since they have long life-times and remain confined within galaxy clusters for a Hubble time (e.g. Blasi et al. 2007 and ref. therein). Confinement enhances the probability to have p-p collisions that in turns give gamma ray emission via decay of $\pi^o$ produced during these collisions and inject secondary particles that emit synchrotron and inverse Compton (IC) radiation whose intensity depends on the energy density of protons. Only upper limits to the gamma ray emission from galaxy clusters have been obtained so far (Reimer et al. 2003), however the FERMI Gamma-ray telescope[^1] (formely GLAST) will shortly allow a step forward having a chance to obtain first detections of galaxy clusters or to put stringent constraints to the energy density of the relativistic protons.
The IGM is expected to be turbulent at some level and MHD turbulence may re-accelerate both primary and secondary particles during cluster mergers via second order Fermi mechanisms. Turbulence is naturally generated in cluster mergers (Roettiger et al. 1999; Ricker & Sarazin 2001; Dolag et al. 2005; Vazza et al. 2006; Iapichino & Niemeyer 2008) and the resulting particle re-acceleration process should enhance the synchrotron and IC emission by orders of magnitude (e.g. Brunetti 2004; Petrosian & Bykov 2008). In a few years the Low Frequency Array (LOFAR) and the Long Wavelength Array (LWA) will observe galaxy clusters at low radio frequencies (40-80 MHz and 120-240 MHz in the case of LOFAR[^2] and 10-88 MHz in the case of LWA[^3]) catching the bulk of their synchrotron cluster-scale emission and testing the different scenarios proposed for the origin of the non-thermal particles. Finally, the emerging pool of future hard X-ray telescopes (e.g. NuSTAR, Simbol-X, Next) should provide crucial constraints on the level of IC emission from clusters and on the strength of the magnetic field in the IGM.
Facts from radio observations suggest that MHD turbulence may play an important role in the acceleration of the electrons responsible for the cluster-scale radio emission. After briefly discussing observations (Sect.2), in Sect.3 we will focus on re-acceleration models showing expectations from the radio band to the gamma rays.
Giant Radio Halos {#sec:halos}
=================
Origin of the relativistic electrons in the IGM {#sec:origin}
-----------------------------------------------
The most prominent examples of diffuse non-thermal sources in galaxy clusters are the giant Radio Halos. These are low surface brightness, Mpc-scale diffuse sources found at the centre of a fraction of massive and merging galaxy clusters and are due to synchrotron radiation from relativistic electrons diffusing in $\approx \mu$G magnetic fields frozen in the IGM (Feretti 2003; Ferrari et al. 2008). The origin of Radio Halos is still unclear, the starting point is that the timescale necessary for the emitting electrons to diffuse over Radio Halo size-scales, of the order of a Hubble time, is much longer than the electrons’ radiative lifetime, $\approx 10^8$ years (Jaffe 1977). This implies the existence of mechanisms of either [*in situ*]{} particle acceleration or injection into the IGM. Understanding the physics of these mechanisms is crucial to model the non-thermal components in galaxy clusters and their evolution.
Two main possibilities have been proposed to explain Radio Halos : [*i)*]{} the so-called [*re-acceleration*]{} models, whereby relativistic electrons injected in the IGM are re-energized [*in situ*]{} by various mechanisms associated with the turbulence in massive merger events (e.g. Brunetti et al. 2001; Petrosian 2001; Fujita et al. 2003), and [*ii)*]{} the [*secondary electron*]{} models, whereby the relativistic electrons are secondary products of proton-proton collisions between relativistic and thermal protons in the IGM (e.g. Dennison 1980; Blasi & Colafrancesco 1999; Dolag & Ensslin 2000).
Extended and fairly regular radio emission is expected in the case of a secondary origin of the emitting electrons, since the parent primary protons can diffuse on large scales. Still several properties of Radio Halos cannot be simply reconciled with this scenario. Two “[*historical*]{}” points are :
- Since all clusters have suffered mergers (hierarchical scenario) and relativistic protons are mostly confined within clusters, extended radio emission should be basically observed in almost all clusters. On the other hand, Radio Halos are not common and, although a fairly large number of clusters has an adequate radio follow up, they are presently detected only in a fraction of massive and merging clusters (Giovannini et al. 1999; Buote 2001; Cassano et al. 2008; Venturi et al. 2008; Sect.\[sec:radio\]).
- Since the spectrum of relativistic protons (and of secondary electrons) is expected to be a simple power law over a large range of particle-momentum, the synchrotron spectrum of Radio Halos should be a power law. On the other hand (although the spectral shape of Radio Halos is still poorly known) the spectrum of the best studied Radio Halo, in the Coma cluster, shows a cut off at GHz frequencies implying a corresponding cut off in the spectrum of the emitting electrons at $\approx$ GeV energy that was interpreted in favour of [*in situ*]{} acceleration models (Schlickeiser et al. 1987; Thierbach et al. 2003).
Inputs from new radio observations {#sec:radio}
----------------------------------
The two models in Sect.\[sec:origin\] have different basic expectations in terms of statistical properties of Radio Halos and of their evolution. In particular, as already pointed out, in the context of [*secondary electron*]{} models, Radio Halos should be common and very long–living phenomena, on the other hand, in the context of [*re-acceleration*]{} models, due to the finite dissipation time-scale of the turbulence in the IGM, they should be [*transient phenomena*]{} with life–time $\approx$1 Gyr (or less) .
Radio pointed observations of a complete sample of about 50 X–ray luminous ($L_x \geq 5 \times 10^{44}$ erg s$^{-1}$) clusters at redshift $z = 0.2 - 0.4$ have been recently carried out at 610 MHz with the GMRT–[*Giant Metrewave Radio Telescope*]{} (Venturi et al. 2007, 2008). These observations were specifically designed to avoid problems in the detection of the cluster-scale emission due to the missing of short-baselines in the interferometric observations and to image, at the same time, both compact and extended sources in the selected clusters. Thus they allowed a fair analysis of the occurrence of Radio Halos in galaxy clusters. One of the most relevant findings of these GMRT observations is that only a fraction ($\leq$ 30 %) of the X-ray luminous clusters hosts Radio Halos (Venturi et al. 2008); all Radio Halos being in merging–clusters. Remarkably, the fraction of clusters with Radio Halos is also found to depend on the cluster X-ray luminosity (and mass) decreasing at $\leq$ 10% in the case of clusters at $z= 0.05
- 0.4$ with $L_X \approx 3\cdot 10^{44}-8\cdot 10^{44}$ erg s$^{-1}$ and suggesting the presence of some threshold in the mechanism for the generation of these sources (Cassano et al. 2008). Although these studies demonstrate that no cluster-scale emission is detected in the majority of cases, potentially synchrotron radio emission may be present in all clusters at a level just below the sensitivity of radio observations and this implies the importance to combine radio upper limits and detections. Fig.\[fig:brunetti07\] shows the distribution of GMRT clusters in the radio power ($P_{1.4}$)– cluster X-ray luminosity ($L_x$) plane. The important point is that clusters with similar $L_x$ (and redshift) have a bimodal distribution, with the upper limits to the synchrotron luminosity of clusters with no hint of Radio Halos that lie about one order of magnitude below the region of the clusters with Radio Halos ($P_{1.4}$–$L_x$ correlation). It is probably useful to stress that these upper limits are solid (conservative) being evaluated through injection of fake Radio Halos into the observed datasets in order to account for the sensitivity of the different observations to the cluster-scale emission (Brunetti et al. 2007; Venturi et al. 2008).
Cluster bimodality and the connection of Radio Halos with cluster mergers, suggest that Halos are [*transient phenomena*]{} that develop in merging clusters. From the lack of clusters in the region between Radio Halos and upper limits, in the case that Halo-clusters evolve into radio-quiet clusters (and vice versa), the evolution should be fast, in a timescale of $\approx$0.1-0.2 Gyr (Brunetti et al. 2007). This observational picture suggests that turbulent re-acceleration of relativistic electrons may trigger the formation of Radio Halos in merging clusters, in which case the acceleration timescale of the emitting electrons is indeed $\approx$0.1-0.2 Gyr. On the other hand, unless we admit the possibility of [*ad hoc*]{} fast (on timescale of $\approx$0.1-0.2 Gyr) dissipation of the magnetic field in clusters, the bimodal distribution in Fig.\[fig:brunetti07\] cannot be easily reconciled with [*secondary*]{} models. In this case - indeed - Radio Halos should be common and some general $P_{1.4}$–$L_x$ trend is predicted for all clusters (e.g. Miniati et al. 2001; Dolag 2006; Pfrommer 2008; see discussion in Brunetti et al. 2007 for more details).
{width="\columnwidth"}{width="\columnwidth"}
{width="2\columnwidth"}
Limits on cosmic ray protons and future with the FERMI gamma-ray telescope {#sec:CR}
--------------------------------------------------------------------------
Gamma ray observations with EGRET limit the energy of CR protons in a number of nearby clusters to $\leq$20 % of the energy of the IGM (Reimer et al. 2003; Pfrommer & Ensslin 2004).
The upper limits obtained for the clusters with no Radio Halos in Fig.\[fig:brunetti07\] imply that, regardless of the origin of the radio emission, the synchrotron emission from secondary electrons must be below the value of the upper limits. This allows to put a corresponding limit to the energy density of the primary protons from which secondaries are generated (Brunetti et al. 2007).
The limits are reported in Fig.\[fig:brunetti07\](right) as a function of the magnetic field strength in the IGM and for different spectra of the primary protons. By assuming $> \mu$G field, the upper limits to the energy density of relativistic protons in the IGM are about one order of magnitude more stringent than those obtained from EGRET upper limits (at least for relatively flat proton spectra). As a matter of fact, under these assumptions, no more than 1 % of the energy density of the IGM in X–ray luminous clusters can be in the form of relativistic protons; this would imply that even the FERMI Gamma-ray telescope might not be sensitive enough to detect the $\pi^o$–decay from these clusters. On the other hand, for steeper spectra of relativistic protons, or lower values of the field, the synchrotron constraints in Fig.\[fig:brunetti07\](right) become gradually less stringent and the energy content of relativistic protons permitted by observations is larger. Since efficient particle acceleration mechanisms would naturally produce flat proton spectra, it is unlikely that a population of protons with steep spectrum stores a relevant fraction of the IGM energy and Fig.\[fig:brunetti07\](right) reasonably implies that FERMI may detect the $\pi^o$ decay in a fairly large number of clusters provided that the IGM is magnetised at $\approx \mu$G level (or lower). In this case, since $E_{CR}/E_{th}$ (and also the spectrum of relativistic protons) can be constrained, the combination of gamma ray measurements with deep radio upper limits in clusters without Radio Halos will provide a novel tool to measure the magnetic field strength in galaxy clusters.
Re-acceleration models {#sec:reacceleration}
======================
Basics
------
The properties of the Mpc-scale radio emission in galaxy clusters suggest that turbulence, generated in cluster mergers, may play an important role in the acceleration of the emitting particles.
The physics of collisionless turbulence and of stochastic particle acceleration is complex and rather poorly understood, still several calculations from [*first principles*]{} have shown that there is room for efficient turbulent acceleration in the IGM.
One of the first points that have been realised is that turbulent re-acceleration is not efficient enough to extract electrons from the thermal IGM (e.g. Petrosian 2001) and thus a necessary assumption in the re-acceleration scenario is that seed particles, relativistic electrons with Lorentz factors $\gamma \sim 100-500$, are already present in the IGM. These seeds could be provided by the past activity of active galaxies in the IGM or by the past merger history of the cluster (e.g., Brunetti et al. 2001), alternatively they could be secondary electrons from p-p collisions (Brunetti & Blasi 2005).
Particle re-acceleration in the IGM may be due to small scale Alfvén modes (Ohno et al. 2002; Fujita et al. 2003; Brunetti et al. 2004) or due to compressible (magnetosonic) modes (Cassano & Brunetti 2005; Brunetti & Lazarian 2007). In the case of Alfvénic models, the acceleration of relativistic protons increases the damping of the modes and this severely limits the acceleration of relativistic electrons when the energy budget of relativistic protons is typically $\geq$ 5-7% of that of the IGM (Brunetti et al. 2004). The injection process of Alfvén modes at small, resonant, scales in the IGM and the assumption of isotropy of these modes represent the most important sources of uncertainty in this class of models (see discussion in Brunetti 2006 and Lazarian 2006b). Alternatively, compressible (magnetosonic) modes might re-accelerate fast particles via resonant Transit Time Damping (TTD) and non–resonant turbulent compression (e.g., Cho et al. 2006; Brunetti & Lazarian 2007). The main source of uncertainty in this second class of re-acceleration models is our ignorance on the viscosity in the IGM. Viscosity could indeed severely damp compressible modes on large scales and inhibit particle acceleration, although the super-Alfvénic (and sub-sonic) nature of the turbulence in the IGM is expected to suppress viscous dissipation (see discussion in Brunetti & Lazarian 2007).
The non-thermal spectrum of galaxy clusters
-------------------------------------------
The main goal of this Section is to suggest that the non-thermal emission from clusters is a mixture of two main spectral components: a long-living one that is emitted by secondary particles (and by $\pi^o$ decay) continuously generated during p-p collisions in the IGM, and a transient component that may be due to the re-acceleration of relativistic particles by MHD turbulence generated (and then dissipated) in cluster mergers. We model the re-acceleration of relativistic particles by MHD turbulence in the most simple situation in which only relativistic protons, accumulated during cluster lifetime, are initially present in a turbulent IGM. These protons generate secondary electrons via p-p collisions and in turns these secondaries (as well as protons) are re-accelerated by MHD turbulence. More specifically we adopt the Alfvenic model of Brunetti & Blasi (2005) and do not consider the possibility that [*relic*]{} primary electrons in the IGM can be re-accelerated. Also we do not consider the contribution to the cluster emission from fast electrons accelerated at shock waves that develop during cluster mergers and accretion of matter (see Ensslin, this conference).
An example of the expected broad band emission is reported in Fig.\[fig:broadspectrum\] for a Coma-like cluster (curves in Figure are not fits to the data).
Upper panels show the non-thermal emission (synchrotron, IC, $\pi^o$ decay) generated during a cluster merger, while the spectra in lower panels are calculated after 1 Gyr from the time at which turbulence is dissipated, and thus they only rely with the long-living component of the non-thermal cluster emission. As a first approximation the non thermal emission in the lower panels does not depend on the dynamics of the clusters but only on the energy content (and spectrum) of relativistic protons in the IGM (and on the magnetic field in the case of the synchrotron radio emission). On the other hand, the comparison between the spectra in upper and lower panels highlights the transient emission that is generated in connection with the injection of turbulence during cluster mergers.
The results reported in Fig.\[fig:broadspectrum\] have the potential to reproduce the radio bimodality observed in galaxy clusters (Fig.\[fig:brunetti07\]) : Radio Halos develop in connection with particle re-acceleration due to MHD turbulence in cluster mergers where the cluster-synchrotron emission is considerably boosted up (upper panel), while a fainter long-living radio emission from secondary electrons is expected to be common in clusters (lower panel); the level of this latter component must be consistent with the radio upper limits from radio observations (Fig.\[fig:brunetti07\], right).
An important point is that some level of gamma ray emission is expected. However, we do not expect a direct correlation between giant Radio Halos and the level of gamma ray emission from clusters, indeed this level only depends on the content of protons in the IGM (see Fig.\[fig:broadspectrum\] caption). FERMI is expected to shortly obtain crucial information on that. By taking into account the constraints on the energy content of relativistic protons obtained from the radio upper limits of clusters without Radio Halos (Sect.2.3, Fig.\[fig:brunetti07\]) the gamma ray emission from a Coma–like cluster (assuming $E_{CR} \propto E_{th}$ and B$\approx \mu$G as in Fig.\[fig:broadspectrum\]) is expected $\approx$10-20 times below present EGRET upper limits and no substantial amplification of this signal is expected in clusters with Radio Halo. On the other hand, in the case of smaller magnetic fields in the IGM, the gamma ray luminosity of clusters can be larger because a larger proton-content is permitted by radio upper limits (a more detailed discussion on the gamma ray properties of clusters in the re-acceleration scenario is given in Brunetti et al. in prep.).
Fig.\[fig:broadspectrum\] shows that a direct correlation is expected between Radio Halos and IC emission in the hard X-rays since the two spectral components are emitted by (essentially) the same population of relativistic electrons. Because the ratio between IC and radio luminosity depends on the magnetic field in the IGM, future hard X-ray telescopes (Simbol-X, NuSTAR and Next) are expected to detect a fairly large number of clusters with Radio Halos in the case that the IGM is magnetised at $\approx 1 \mu$G level (or lower).
Low frequency radio emission from galaxy clusters {#sec:radiolow}
-------------------------------------------------
The maximum energy at which electrons can be re-accelerated by turbulence in the IGM, and ultimately the frequencies at which the spectra of Radio Halos cut off (e.g. Fig.\[fig:broadspectrum\]), depend on the acceleration efficiency that essentially increases with the level of turbulence in the IGM. The spectral cut off affects our ability to detect Radio Halos in the universe, introducing a strong bias against observing them at frequencies substantially larger than the cut off frequency. In the context of the re-acceleration scenario, presently known Radio Halos must result from the rare, most energetic merging events and therefore be hosted only in the most massive and hot clusters (Cassano & Brunetti 2005). On the other hand, it has been realised that a large number of Radio Halos should be formed during much more common but less energetic mergers and should be visible only at lower frequencies because of their ultra steep spectral slope (Fig. \[fig:schema\]). Thus the fraction of clusters with Radio Halos increases at lower observing frequencies and LOFAR and LWA, that will observe galaxy clusters with unprecedented sensitivity at low radio frequencies, have the potential to catch the bulk of these sources in the universe (Cassano et al. 2006).
This is a unique expectation of the re-acceleration scenario. In particular, following Cassano et al. (2008), it has been calculated that observations at 150 MHz have the potential to increase the number of giant Radio Halos observed in galaxy clusters with mass $M\leq 10^{15}\,M_{\odot}$ by almost one order of magnitude with respect to observations at 1.4 GHz, while this increase is expected to be smaller for clusters with larger mass.
Summary {#sec:summary}
=======
Recent observations show that Radio Halos are not common in galaxy clusters suggesting that they are transient sources in cluster mergers. These facts support the idea that MHD turbulence generated in cluster mergers may play an important role in the re-acceleration of particles in the IGM.
We have suggested that in the context of the re-acceleration scenario the non-thermal emission from galaxy clusters is a combination of two components : a long-living one that is emitted by secondary particles (and by $\pi^o$ decay) continuously generated during p-p collisions in the IGM, and a transient component due to the re-acceleration of relativistic particles by MHD turbulence generated (and then dissipated) in cluster mergers.
The latter component may naturally explain Radio Halos in merging and massive clusters and their statistical properties, on the other hand FERMI may detect the long-living gamma ray emission from clusters due to the decay of $\pi^o$ produced via collisions between relativistic and thermal protons. In particular, given the constraints on the relativistic protons obtained from upper limits to the synchrotron emission due to secondaries in X-ray luminous clusters, we conclude that the detection of a fairly large number of clusters with FERMI would imply that the magnetic field in the IGM is at $\approx \mu$G level (or lower). In this case, the pool of future hard X-ray detectors (NuSTAR, Simbol-X and Next) should detect IC emission from a fairly large number of merging clusters with Radio Halos.
An important expectation of the re-acceleration scenario is the presence of a population of Radio Halos in the universe that should emerge only at low radio frequencies, and this can be easily tested in a few years with LOFAR and LWA. The detection of Radio Halos with ultra-steep spectral slope will thus provide compelling support to particle re-acceleration due to turbulence in the IGM.
![Picture showing the connection between acceleration efficiency and spectrum of Radio Halos in the re-acceleration scenario. Boxes indicate the frequency range in which Radio Halos can be studied with present radio telescopes and with LOFAR.[]{data-label="fig:schema"}](brunetti_f3){width="\columnwidth"}
[*Acknowledgment*]{}: The author is very grateful to the organizing committee of the Conference and also thanks P.Blasi, R.Cassano, V.Dogiel, S.Gabici, A.Lazarian and F.Vazza for the ongoing theoretical collaborations. This work is partially supported by PRIN-INAF2008 and ASI-INAF I/088/06/0.
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[^1]: http://fermi.gsfc.nasa.gov/
[^2]: http://www.lofar.org/
[^3]: http://lwa.nrl.navy.mil/
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---
abstract: |
A filling Dehn surface in a $3$-manifold $M$ is a generically immersed surface in $M$ that induces a cellular decomposition of $M$. Given a tame link $L$ in $M$ there is a filling Dehn sphere of $M$ that “trivializes” (*diametrically splits*) it. This allows to construct filling Dehn surfaces in the coverings of $M$ branched over $L$. It is shown that one of the simplest filling Dehn spheres of $S^3$ (Banchoff’s sphere) diametrically splits the trefoil knot. Filling Dehn spheres, and their Johansson diagrams, are constructed for the coverings of $S^3$ branched over the trefoil. The construction is explained in detail. Johansson diagrams for generic cyclic coverings and for the simplest locally cyclic and irregular ones are constructed explicitly, providing new proofs of known results about cyclic coverings and the $3$-fold irregular covering over the trefoil.\
\
*Dedicated to Prof. Maite Lozano on her 70th anniversary.*
address: 'Centro Universitario de la Defensa Zaragoza, Academia General Militar Carretera de Huesca s/n. 50090 Zaragoza, Spain — IUMA, Universidad de Zaragoza'
author:
- 'Álvaro Lozano-Rojo'
- Rubén Vigara
title: 'Banchoff’s sphere and branched covers over the trefoil'
---
[^1]
Introduction
============
Filling Dehn surfaces and their Johansson diagrams were introduced in [@Montesinos], following ideas of [@Haken1], as a new way to represent closed orientable $3$-manifolds. After [@Montesinos] some works have appeared on the subject [@Amendola09; @racsam; @spectrum; @peazolibro; @Anewproof; @RHomotopies; @tesis].
In [@knots], the authors propose a general framework in which filling Dehn surfaces can be applied to knot theory. Any knot (or link) in any $3$-manifold can be nicely intersected (*split*) by a filling Dehn sphere. This filling Dehn sphere appears to be an interesting tool for studying the branched coverings over the knot, because the splitting sphere has “nice lifts” to these branched coverings, in a similar way as the Heegaard surface of a $(g,1)$-decomposition of the knot [@Cattabriga-Mulazzani; @cristoforietal; @doll]. In [@knots], this is exemplified with the simplest of all knots: the unknot. The techniques of [@knots] are applied here to the next knot in increasing complexity after the unknot: the trefoil knot.
In Section \[sec:Dehn-surfaces-Johansson-diagrams\] we introduce the basic definitions and notation about filling Dehn surfaces. Section \[sec:splitting\] recalls the tools introduced in [@knots]. We refer to [@knots; @peazolibro] and references therein for more details on this subject.
In Section \[sec:trefoil\], it is shown that one of the simplest filling Dehn spheres of $S^3$ (*Banchoff’s sphere*) splits the trefoil knot. This is used in the subsequent sections to study covers of $S^3$ branched over the trefoil. In Section \[sec:branched-covers\] we study the cyclic branched covers, obtaining a new proof of Theorem 2.1 of [@CHK] that asserts that the $3$-manifolds introduced in [@sieradski] coincide with the cyclic branched covers of the trefoil knot. Section \[sec:other-examples\] gives some information about other type of coverings, as the locally cyclic (Section \[sub:trefoil-locally-cyclic\]) and irregular ones (Section \[sub:trefoil-non-cyclic\]). In particular, in Section \[sub:trefoil-non-cyclic\] we give another proof of the well known result that asserts that the irregular $3$-fold covering of $S^3$ branched over the trefoil is $S^3$ [@Burde; @GH; @MontesinosTesis].
Dehn surfaces and their Johansson’s diagrams {#sec:Dehn-surfaces-Johansson-diagrams}
============================================
Throughout the paper all $3$-manifolds are assumed to be closed and orientable, that is, compact connected and without boundary. On the contrary, surfaces are assumed to be compact, orientable and without boundary, but they could be disconnected. All objects are assumed to be in the smooth category: manifolds have a differentiable structure and all maps are smooth.
Let $M$ be a 3-manifold.
A subset $\Sigma\subset M$ is a *Dehn surface* in $M$ [@Papa] if there exists a surface $S$ and a general position immersion $f:S\rightarrow M$ such that $\Sigma=f\left(S\right)$. If this is the case, the surface $S$ is the *domain* of $\Sigma$ and it is said that $f$ *parametrizes* $\Sigma$. If $S$ is a $2$-sphere, then $\Sigma$ is a *Dehn sphere*.
Let $\Sigma$ be a Dehn surface in $M$ and consider a parametrization $f:S\to M$ of $\Sigma$. The *singularities* of $\Sigma$ are the points $x\in\Sigma$ such that $\#f^{-1}(x)>1$, and they are divided into *double points* where two sheets of $\Sigma$ intersect transversely ($\#f^{-1}(x)=2$), and *triple points* where three sheets of $\Sigma$ intersect transversely ($\#f^{-1}(x)=3$). The singularities of $\Sigma$ form the *singularity set* $S(\Sigma)$ of $\Sigma$. We denote by $T(\Sigma)$ the set of triple points of $\Sigma$. The connected components of $S(\Sigma)-T(\Sigma)$, $\Sigma-S(\Sigma)$ and $M-\Sigma$ are the *edges*, *faces* and *regions* of $\Sigma$, respectively.
In the following a *curve* in $S$, $\Sigma$ or $M$ is the image of an immersion from $\mathbb{S}^1$ or $\mathbb{R}$ into $S$, $\Sigma$ or $M$, respectively. A *double curve* of $\Sigma$ is a curve in $M$ contained in $S(\Sigma)$.
The preimage under $f$ of the singularity set of $\Sigma$, together with the information about how its points become identified by $f$ in $\Sigma$ is the *Johansson diagram* ${\mathcal{D}}$ of $\Sigma$, see [@Johansson1; @Montesinos]. Two points of $S$ are *related* by $f$ if they project onto the same point of $\Sigma$.
Because $S$ is compact and without boundary, double curves are closed and there is a finite number of them, and the number of triple points is also finite. Since $S$ and $M$ are orientable, the preimage under $f$ of a double curve of $\Sigma$ is the union of two different closed curves in $S$. These two curves are *sister curves* of ${\mathcal{D}}$. Thus, the Johansson diagram of $\Sigma$ is composed by an even number of different closed curves in $S$. We identify ${\mathcal{D}}$ with the set of different curves that compose it. For any curve $\alpha\in{\mathcal{D}}$ we denote by $\tau \alpha$ the sister curve of $\alpha$ in ${\mathcal{D}}$. This defines a free involution $\tau:{\mathcal{D}}\rightarrow{\mathcal{D}}$, the *sistering* of ${\mathcal{D}}$, that sends each curve of ${\mathcal{D}}$ into its sister curve in ${\mathcal{D}}$.
The curves of ${\mathcal{D}}$ transversely meet others or themselves at the *crossings* of ${\mathcal{D}}$. The crossings of ${\mathcal{D}}$ are the preimage under $f$ of the triple points of $\Sigma$. If $P$ is a triple point of $\Sigma$, the three crossings of ${\mathcal{D}}$ in $f^{-1}(P)$ form *the triplet of* $P$.
The Dehn surface $\Sigma\subset M$ *fills* $M$ if it defines a cell-decomposition of $M$ whose $0$-, $1$- and $2$-dimensional skeletons are $T(\Sigma)$, $S(\Sigma)$, and $\Sigma$ respectively [@Montesinos]. If $\Sigma$ fills $M$ and the domain $S$ of $\Sigma$ is connected, then it is possible to build $M$ out of the Johansson diagram ${\mathcal{D}}$ of $\Sigma$. Since every $3$-manifold has a filling Dehn sphere [@Montesinos], Johansson diagrams of filling Dehn spheres represent all closed orientable $3$-manifolds.
A special case of filling Dehn surfaces is when the domain $S$ of the filling Dehn surface $\Sigma$ is a disjoint union of $2$-spheres. In this case, we say that $\Sigma$ is a *filling collection of spheres*.
Splitting knots with filling Dehn spheres {#sec:splitting}
=========================================
In the following paragraphs we summarize some definitions and results from [@knots]. Results are stated without proof.
Let $L$ be a tame knot or link in a $3$-manifold $M$, and let $\Sigma$ be a filling Dehn surface of $M$.
The Dehn surface $\Sigma$ *splits* $L$ if:
1. $L$ intersects $\Sigma$ transversely in a finite set of non-singular points of $\Sigma$;
2. for each region $R$ of $\Sigma$, if the intersection $R\cap L$ is non-empty it is exactly one arc, unknotted in $R$; and
3. for each face $F$ of $\Sigma$, the intersection $F\cap L$ contains at most one point.
The Dehn surface $\Sigma$ *diametrically splits* $L$ if it splits $L$ and it intersects each connected component of $L$ exactly twice.
There is a filling Dehn sphere of $M$ that diametrically splits $L$.
Assume that $\Sigma$ splits $L$.
Our interest in filling Dehn surfaces that (diametrically) split knots relies on the following result. Let $p:\widehat{M}\to M$ be a finite sheeted branched covering with downstairs branching set $L$, and take $\widehat{\Sigma}=p^{-1}(\Sigma)$.
\[thm:filling-lift-to-filling\] The Dehn surface $\widehat{\Sigma}$ fills $\widehat{M}$. Moreover, if $L$ is a knot and $\Sigma$ diametrically splits $L$ then $\widehat{\Sigma}$ is a filling collection of spheres in $\widehat{M}$, and it is a Dehn sphere if and only if $p$ is locally cyclic.
Recall that a $n$-fold covering $p$ branched over a knot $L$ is *locally cyclic* if its monodromy map $\rho$ sends knot meridians onto $n$-cycles [@ST p. 209]. This is equivalent to say that $p:p^{-1}(L)\to L$ is a homeomorphism.
For the study of branched coverings over $L$ it is essential to know its group, i.e. the fundamental group of $M-L$. This can be done also using $\Sigma$. If $R_1,\ldots,R_m$ are all the different regions of $\Sigma$ disjoint from $L$ and we take a point $Q_i$ in each of these regions, $\Sigma-L$ is a strong deformation retract of $M-\left(L\cup \{Q_1,\ldots,Q_m\}\right)$. Hence
The fundamental groups of $M-L$ and $\Sigma- L$ are isomorphic.$\qed$
If $f:S\to M$ is a parametrization of $\Sigma$, the pair $({\mathcal{D}},f^{-1}(L))$, where ${\mathcal{D}}$ is the Johansson diagram of $\Sigma$, is a *Johansson diagram of* $L$.
\[prop:diagram-represent-knots\] The pair $(M,L)$ can be recovered from a Johansson diagram of $L$. In particular, if $L'$ is a link in a $3$-manifold $M'$ such that $L$ and $L'$ have identical Johansson diagrams, there is a homeomorphism between $M$ and $M'$ that maps $L$ onto $L'$.
Thus, all the information about $L$ is codified in its Johansson diagram.
A presentation of the fundamental group of a Dehn surface in terms of its Johansson diagram was introduced in [@spectrum] (cf. [@peazolibro]). Although the presentation given there is stated for Dehn surfaces of genus $g$, it is valid also in a more general context, including the case of $\Sigma-L$ where the domain surface is a punctured sphere. The generators of this presentation are of two kinds: *surfacewise* generators and ${\mathcal{D}}$-*dual* generators.
Set $M_L := M-L$ and $S_L := S-f^{-1}(L)$. Take a non-singular point $x$ of $\Sigma$ as the base point of the fundamental group $\pi_L:=\pi_1(M_L,x)$ of $M_L$. We also denote by $x$ the preimage of $x$ under $f$, and we choose it as the base point of the fundamental group $\pi_{S_L}:=\pi_1(S_L,x)$ of $S_L$.
The surfacewise generators of $\pi_L$ are obtained by pushing forward a generating set of $\pi_{S_L}$ to $M$ through $f$: if $\gamma_1,\ldots,\gamma_k$ are representatives of a set of generators of $\pi_{S_L}$, then $f\circ\gamma_1,\ldots,f\circ\gamma_k$ are representatives of a set of surfacewise generators of $\pi_L$.
Let $\alpha$ and $\beta$ be sister curves of the diagram ${\mathcal{D}}$. Consider two paths $a$ and $b$ in $S_L$ starting at $x$ and ending at points on $\alpha$ and $\beta$ respectively. Assume that the endpoints of $a$ and $b$ are related by $f$ and that they are not crossings of ${\mathcal{D}}$. In this case, $\alpha^{\#}=(f\circ a)\,(f\circ b)^{-1}$ is a loop in $\Sigma-L$ which is said to be *dual to* $\alpha$. The inverse loop $\beta^{\#}=(\alpha^{\#})^{-1}=(f\circ b)\,(f\circ a)^{-1}$ is dual to $\beta$. After repeating this construction for each pair of sister curves of ${\mathcal{D}}$ we obtain the set ${\mathcal{D}}^{\#}$ of ${\mathcal{D}}$-dual generators of $\pi_L$.
Surfacewise generators and ${\mathcal{D}}$-dual generators generate $\pi_L$.$\qed$
Thus, surfacewise and ${\mathcal{D}}$-dual generators lead to a presentation of $\pi_L$. The relators associated to this presentation are detailed in [@spectrum].
If $p:\widehat{M}\to M$ is an $n$-fold ($n<\infty$) covering of $M$ branched over $L$, according to Theorem \[thm:filling-lift-to-filling\], $\Sigma$ lifts to a filling Dehn surface $\widehat{\Sigma}$ of $\widehat{M}$. We want to construct the Johansson diagram $\widehat{{\mathcal{D}}}$ of $\widehat{\Sigma}$. This construction is specified in Algorithm 3.4 of [@knots]. The presentation of $\pi_L$ in terms of surfacewise and ${\mathcal{D}}$-dual generators fits quite well to this purpose. In short form:
- surfacewise generators of $\pi_L$ allow to construct the domain surface $\widehat{S}$ of $\widehat{\Sigma}$; and
- ${\mathcal{D}}$-dual generators allow to decide the sistering between the curves of $\widehat{{\mathcal{D}}}$.
By [@spectrum], there is a commutative diagram $$\begin{tikzcd}
\widehat{S} \rar{\widehat{f}} \dar[swap]{p_S} & \widehat{M} \dar{p} \\
S \rar[swap]{f} & M
\end{tikzcd}$$ where $\widehat{f}$ is a parametrization of $\widehat{\Sigma}$, and $p_S$ is an $n$-fold branched covering with branching set $f^{-1}(L)$.
Take $p^{-1}(x)=\{x_1,\ldots,x_n\}$, and also denote by $\{x_1,\ldots,x_n\}$ the corresponding points in $\widehat{S}$. If $\rho:\pi_L\to \Omega_n$ is the monodromy homomorphism associated with $p$, where $\Omega_n$ is the group of permutations of the set $\{x_1,\ldots,x_n\}$, the monodromy homomorphism $\rho_S:\pi_{S_L}\to \Omega_n$ associated with $p_S$ verifies $\rho_S=\rho \circ f_*$, where $f_*: \pi_{S_L}\to \pi_L$ is the homomorphism induced by $f$.
Since $f_*$ sends a set of generators of $\pi_{S_L}$ onto the surfacewise generators of $\pi_L$, $\rho_S$ is essentially the same as $\rho$ restricted to the surfacewise generators of $\pi_L$. Hence, the knowledge of $\rho$ allows to construct $\widehat{S}$.
Once $\widehat{S}$ is constructed, the curves of $\widehat{{\mathcal{D}}}$ are the lifts to $\widehat{S}$ of the curves of ${\mathcal{D}}$. Consider the pair of sister curves $\alpha$ and $\beta$ of ${\mathcal{D}}$ and their associated paths $a$ and $b$ as before. Let $a_i$ and $b_i$ be the lifts of $a$ and $b$ respectively to $\widehat{S}$ based at $x_i$, and let $\alpha_i$ and $\beta_i$ be the lifts of $\alpha$ and $\beta$ passing through the endpoint of $a_i$ and $b_i$ respectively, with $i=1,\ldots,n$. The monodromy map $\rho$ assigns to $\alpha^{\#}$ the permutation $\rho(\alpha^{\#})$ of $\{x_1,\ldots,x_n\}$ given by $$\rho(\alpha^{\#})(x_i)=x_{j}\iff
\text{\emph{the lift of $\alpha^{\#}$ starting at $x_i$ ends at $x_{j}$}.}$$ By the construction of $\alpha^{\#}$, the right-hand side of the previous equivalence is indeed equivalent to say that the endpoints of $a_i$ and $b_j$ are related by $\widehat{f}$. Therefore, $\alpha_i$ and $\beta_j$ are sister curves in $\widehat{{\mathcal{D}}}$ and they must be identified in such a way that the endpoints of $a_i$ and $b_j$ are related by $\widehat{f}$. Hence, $\rho(\alpha^{\#})$ tells us how the lifts of $\alpha$ to $\widehat{S}$ must be identified with the lifts of $\beta$ to $\widehat{S}$. Repeating the same argument for the rest of ${\mathcal{D}}$-dual generators, all the identifications between the curves of $\widehat{{\mathcal{D}}}$ are established.
Banchoff’s sphere and the trefoil knot {#sec:trefoil}
======================================
![Constructing Banchoff’s sphere: (a) starting from a bunch of three $2$-spheres, (b, c) we add tubes to build a Dehn sphere. In the right column we can see how the corresponding diagram changes.[]{data-label="fig:figs_banchoffTrefoil"}](banchoffTrefoil3.pdf)
![(a) Banchoff’s sphere diagram with the loops selected to generate the fundamental group of the trefoil knot complement. (b) The fan $\Delta$ obtained cutting previous diagram along the thin horizontal line between point $A$ and $B$.[]{data-label="fig:banchoff"}](banchoffGens "fig:") ![(a) Banchoff’s sphere diagram with the loops selected to generate the fundamental group of the trefoil knot complement. (b) The fan $\Delta$ obtained cutting previous diagram along the thin horizontal line between point $A$ and $B$.[]{data-label="fig:banchoff"}](banchoffFan "fig:")
Let $K$ be the trefoil knot lying in $S^3$ with a $2\pi/3$ rotational symmetry as in Figure \[fig:trefoil-banchoff\](a). A filling Dehn sphere in $S^3$ that diametrically splits $K$ can be constructed as follows. Look at the embedded $2$-sphere of Figure \[fig:trefoil-banchoff\](a), whose interior intersects $K$ in an unknotted arc, and take another two copies of it, each one located at each “petal” of the trefoil knot. These three embedded $2$-spheres in $S^3$ intersect themselves and $K$ as in Figure \[fig:trefoil-banchoff\](b), and they form a filling collection of spheres $\Sigma$ in $S^3$. Two Banchoff *type 1 surgeries* [@Banchoff] between the $2$-spheres transform this filling collection of spheres into a filling Dehn sphere $\Sigma_B$ of $S^3$ (see [@Shima]) which is called *Banchoff’s sphere* in [@peazolibro; @tesis]. According to [@peazolibro; @tesis], it is one of the three unique filling Dehn spheres of $S^3$ with only two triple points. Moreover, if the two surgeries are taken following the knot $K$ as it is indicated in Figure \[fig:trefoil-banchoff\](c), then $\Sigma_B$ diametrically splits $K$. Figure \[fig:figs\_banchoffTrefoil\] shows how its Johansson diagram ${\mathcal{D}}$ is obtained from the singular set of $\Sigma$. The dots in the right-hand side of the picture represent the intersection of the surface with the trefoil knot. Figure \[fig:banchoff\](a) shows the common representation of ${\mathcal{D}}$ where one point of the sphere has been sent to infinity.
Let $f:S^2\to S^3$ be a parametrization of $\Sigma_B$. The two curves $\alpha,\beta$ of the diagram ${\mathcal{D}}$ verify $\beta=\tau\alpha$, and they must be identified following the arrows in the obvious way. The preimages by $f$ of the two intersection points $A,B$ of $\Sigma_B$ with $K$ are the points also denoted by $A,B$ in $S^2$, see Figure \[fig:banchoff\](a). When the notation does not lead to confusion, we will use the same names to the objects in $\Sigma_B$ and their preimages in $S^2$.
The fundamental group $\pi_1(\Sigma_B- \{A,B\},x)\simeq\pi_K$ based at the point $x$ is generated by the loops $m$ and $c$, where:
- $m$ is the generator of $\pi_1(S^2-\{A,B\})$ depicted in Figure \[fig:banchoff\](a); and
- the loop $c=a\,b^{-1}=\alpha^{\#}$ dual to the curve $\alpha$ of ${\mathcal{D}}$, where $a$ and $b$ are the paths depicted in Figure \[fig:banchoff\](a) joining $x$ with related points on $\alpha$ and $\beta$ respectively.
Note that the loop $m$ in $\Sigma_B$ is homotopic to a meridian of $K$.
After computing for $\Sigma_B-\{A,B\}$ the presentation of its fundamental group given in [@spectrum], we obtain $$\label{ec:pi_1(M-Trefoil)}
\pi_K\simeq \langle\, m,c\mid mcm=cm^{-1}c \,\rangle.$$ It is straightforward to see that this group is isomorphic to the standard presentations of the trefoil knot group.
Cyclic branched covers over the trefoil knot {#sec:branched-covers}
============================================
Johansson diagrams and fundamental group
----------------------------------------
Let $p:\widehat{M}_n\to S^3$ be the $n$-fold ($n<\infty$) cyclic covering of $S^3$ branched over $K$. By Theorem \[thm:filling-lift-to-filling\], Banchoff’s sphere $\Sigma_B\subset S^3$ lifts to a filling Dehn sphere $\widehat{\Sigma}_B$ of $\widehat{M}_n$.
We use the same notation as in Section \[sec:splitting\]. Take $p^{-1}(x)=\{x_1,\ldots,x_n\}$, and denote also by $\{x_1,\ldots,x_n\}$ the corresponding points in $\widehat{S}$. Let $\rho:\pi_K\to \Omega_n$ be the monodromy homomorphism associated with $p$. Since $p$ is cyclic, $\rho$ sends $\pi_K$ onto a cyclic subgroup $C_n$ of $\Omega_n$. The fact that $C_n$ is abelian and the relation in imply that $\rho(c)=\rho(m)^3$. Therefore $C_n=\langle\rho(m)\rangle$. Since $C_n$ must act transitively on $\{x_1,\ldots, x_n\}$, $\rho(m)$ must be a cycle of order $n$. If we identify $\Omega_n$ with the permutation group of the subscripts $\{1,\ldots,n\}$ in the natural way, renaming $\{x_1,\ldots,x_n\}$ if necessary, we can assume that $\rho(m)=(1,2,\ldots, n)$. In the following paragraphs all the subscripts are considered modulo $n$.
The loop $m$ generates the fundamental group of $S-\{A,B\}$, which is infinite cyclic. Therefore, the monodromy homomorphism $\rho_S$ is given by $m\mapsto (1,2,\ldots, n)$. If $m_i$ is the lift of $m$ to $\widehat{S}$ based at $x_i$, by the election of $\rho(m)$, the lifted path $m_i$ starting at $x_i$ must have its endpoint at $m_{i+1}$.
![Building the diagram of a cyclic branched covering. The lifts of $m$ are the thick paths marked with triangle arrows. Those of $a$ are the thick paths marked with triangle empty arrows. The lifts of $b$ are unlabelled, but the path $b_i^{-1}$ is the one ending at $x_i$ also marked with an empty triangle arrow.[]{data-label="fig:general_cyclic"}](banchoffSieradskiFans)
After: (i) cutting the diagram of Figure \[fig:banchoff\](a) along the line that connects $A$ and $B$ in the same figure; and (ii) sending $B$ to infinity; the *fan* $\Delta$ of Figure \[fig:banchoff\](b) is obtained. The domain $\widehat{S}$ of $\widehat{\Sigma}_B$ is obtained by cyclically gluing $n$ copies of $\Delta$. If $\Delta_i$ is the copy of $\Delta$ that contains $x_i$, $i=1,\ldots,n$, the diagram $\widehat{{\mathcal{D}}}$ of $\widehat{\Sigma}_B$ is built up with $\Delta_1,\ldots,\Delta_n$ glued together counterclockwise (the direction does not matter, but we choose it according to the direction of $m$ in the diagram to visualize it better, see Figure \[fig:general\_cyclic\]).
As it is explained in Section \[sec:splitting\], the lifts of $c$ to $\widehat{\Sigma}_B$ describe how the curves of the diagram $\widehat{{\mathcal{D}}}$ become identified in $\widehat{\Sigma}_B$. Let $a_i$, $b_i$ and $c_i$ be the lifts of $a$, $b$ and $c$ respectively based at $x_i$. Let $\alpha_i$ be the lift of $\alpha$ to $\widehat{S}$ at which $a_i$ has its endpoint. In the same way, let $\beta_i$ be the lift of $\beta$ to $\widehat{S}$ at which $b_i$ has its endpoint. With this notation, the situation in $\widehat{S}$ is as depicted in Figure \[fig:general\_cyclic\]. The path $c_i$ connects $x_i$ with $x_{\rho(c)(i)}=x_{i+3}$, crossing a double curve of $\widehat{\Sigma}_B$. Hence, the curve at which $a_i$ ends must be identified with the one at which $b_{i+3}^{-1}$ starts and therefore $\tau\alpha_i = \beta_{i+2}$ (see Figure \[fig:general\_cyclic\]). This allows us to proof:
The fundamental group of $\widehat{M}_n$ is isomorphic to the Sieradski group $$\mathcal{S}(n) = \langle\, g_1,\ldots,g_n \mid
\text{$g_i = g_{i-1}\,g_{i+1}$ for $i=1,\ldots,n$}\,\rangle,$$ where the indices are taken modulo $n$.
The fundamental group of $\widehat{M}_n$ coincides with that of $\widehat{\Sigma}_B$. Since $\widehat{S}$ is a $2$-sphere, we can use the presentation given in [@peazolibro Chp. 4]. In this presentation, the fundamental group is generated by the dual loops to $\alpha_i$ and $\beta_i$, $i=1,\ldots,n$, and the relators are given by the triplets of $\widehat{{\mathcal{D}}}$. By construction, dual loops to sister curves are inverse to each other, and so $\pi_1(\widehat{M}_n)$ is generated just by the dual loops $\beta_1^{\#},\ldots,\beta_n^{\#}$.
Let us determine the triplets of $\widehat{{\mathcal{D}}}$. Take the point $P_1$ in the $i$-th fan $\Delta_i$ of Figure \[fig:general\_cyclic\]. The curves $\beta_{i-1}$ and $\beta_i$ intersect at $P_1$. Since $P_1$ is the third crossing after the arrow in $\beta_i$, it must be identified with the third crossing after the arrow in $\tau\beta_{i} = \alpha_{i-2}$, which is the point $P_2$ in $\Delta_{i-2}$. In the same way, since $P_2$ is the fifth crossing after the arrow in $\beta_{i-2}$, it must be identified with the fifth crossing after the arrow in $\tau\beta_{i-2}=\alpha_{i-4}$, which is the point $P_3$ in $\Delta_{i-3}$. Finally, since $P_3$ is the second crossing after the arrow in $\alpha_{i-3}$, it must be identified with the second crossing after the arrow in $\tau\alpha_{i-3}=\beta_{i-1}$, which is, as expected, $P_1$.
For each $j=1,2,3$ take a small path $\delta_j$ near $P_j$ as the dotted arcs in Figure \[fig:general\_cyclic\], in such a way that the endpoint of $\delta_j$ is related by $\widehat{f}$ with the starting point of $\delta_{j+1}$, where the subscripts are taken modulo $3$. Then the loop $(\widehat{f}\circ \delta_1)\,
(\widehat{f}\circ \delta_2)\,(\widehat{f}\circ \delta_3)$ is contractible in $\widehat{\Sigma}_B$. Hence, the product of $\widehat{{\mathcal{D}}}$-dual loops $\beta_{i}^{\#}\,\beta_{i-2}^{\#}\,\alpha_{i-3}^{\#} =
\beta_{i}^{\#}\,\beta_{i-2}^{\#}\,\left(\beta_{i-1}^{\#}\right)^{-1}$ is also contractible (full details in [@peazolibro Chp. 4]). Therefore $\beta_{i-1}^{\#}=\beta_{i}^{\#}\,\beta_{i-2}^{\#}$ in $\pi_1(\widehat{M}_n,x)$.
It is straightforward to see that all the relations are of this form. In [@peazolibro Chp. 4] it is proved that these are *all* the nontrivial relations. Taking $g_i=\beta_{n-i}^{\#}$, $i=1,\ldots,n$, the presentation of $\mathcal{S}(n)$ of the statement is obtained.
![Diagrams for the first cyclic coverings branched over the trefoil: (a) The lens space $L(3,1)$ as the cyclic $2$-fold covering of $S^3$ branched over $K$; (b) the $3$-fold covering; and (c) the $4$-fold covering. []{data-label="fig:cyclic"}](cyclicCoverings)
Consider $p:\widehat{M}_2\to S^3$ given by the presentation $m\mapsto(1,2)$, $c\mapsto(1,2)$, see Figure \[fig:cyclic\](a). The previous descriptions allows us to conclude that $\pi_1(M_2)={\mathbb{Z}}_3$, in fact $\widehat{M}_2$ is $L(3,1)$. Figure \[fig:cyclic\](b) shows the diagram constructed for $\widehat{M}_3$ given by the presentation $m\mapsto(1,2,3)$, $c\mapsto 1_{\Omega_3}$. The fundamental group is isomorphic to the group of the quaternions $Q_8$. Hence, $\widehat{M}_3$ is a prism manifold, which are characterized by their fundamental group. According to the notation of [@HKMR] this is $M(2,1)$, also called the Quaternionic Space [@montetese]. Consider now the presentation $m\mapsto(1,2,3,4)$, $c\mapsto (1,4,3,2)$, the fundamental group of $\widehat{M}_4$ is $SL_2({\mathbb{Z}}_3) \cong {\mathbb{Z}}_3\rtimes Q_8$. This is a tetrahedral manifold, which are also characterized by their fundamental group. In this case the manifold is the Octahedral space [@montetese].
The Sieradski complex
---------------------
![The Sieradski complex on the sphere $S^2=\partial D^3$.[]{data-label="fig:sieradskiCn"}](sieradskiComplex2D "fig:") ![The Sieradski complex on the sphere $S^2=\partial D^3$.[]{data-label="fig:sieradskiCn"}](Sieradski-bipyramid3D "fig:")
The family of polyhedra with identified faces depicted in Figure \[fig:sieradskiCn\] was introduced in [@sieradski]. The quotient spaces of these polyhedra is a family of $3$-manifolds $M_n$, with $n\geq 2$, whose fundamental groups are the Sieradski groups $\mathcal{S}(n)$. In [@CHK] it is proved that $M_n$ is the $n$-fold cyclic branched cover over the trefoil knot $\widehat{M}_n$. As it has been shown above, Sieradski groups naturally appear in our construction, so it is natural to expect that the lifts of Banchoff sphere have some relation with Sieradski polyhedra (see Figure \[fig:sieradskiCn\]). In fact, Banchoff sphere allows us to give an alternative proof of Theorem 2.1 of [@CHK].
For each $n=2,3,\ldots$ the $3$-manifold $M_n$ is the $n$-fold cyclic cover of $S^3$ branched over the trefoil.
Fix an $n=2,3,\ldots$. For the Sieradski polyhedron $\mathcal{P}_n$, consider the $2\pi/n$ rotation $r$ around the vertical central axis $J$ connecting the “north and south poles” $N$ and $S$ of $\mathcal{P}_n$. By the symmetry of the identification of points on $\partial \mathcal{P}_n$, $r$ preserves these identifications ($r$ sends identified points to identified points on $\partial \mathcal{P}_n$), and therefore $r$ defines a homeomorphism of $M_n$. Moreover, the group generated by $r$ is cyclic of order $n$. The quotient space of the pair $(M_n,J)$ under the action of $\langle r \rangle$ is the pair $(M_1,L)$, where $M_1$ is the manifold obtained after identifying the faces of the tetrahedron $\mathcal{P}_1$ of Figure \[fig:Sieradski-bipyramid\](a) in the following way: the two vertical faces must be identified by a rotation around the vertical edge; and the other two faces must be identified in the unique way in which the boundary edges become identified as indicated by the arrows. Let $L$ denote the image of the vertical edge of the tetrahedron after the identification.
The pair $(M_1,L)$ is homeomorphic to $(S^3,K)$.
Take the immersed surface $\Sigma$ in $M_1$ which is the projection of the surface depicted inside the tetrahedron in Figure \[fig:Sieradski-bipyramid\](b, c). The four pieces of this surface become glued by the identification as it is indicated in Figure \[fig:Sieradski-bipyramid\](d, e), and the intersection of $L$ with $\Sigma$ corresponds to the points $A$ and $B$ of the same figure. After some ambient isotopies, the diagram of $(\Sigma,L)$ becomes the one of Figure \[fig:banchoff\](a). It is not difficult to check that $\Sigma$ fills $M_1$. Therefore, $M_1$ is $S^3$ and $\Sigma$ is Banchoff’s sphere. Since the intersection of $L$ and $\Sigma$ coincides with the intersection of the trefoil and $\Sigma_B$, by Proposition \[prop:diagram-represent-knots\] we conclude also that $L$ is the trefoil knot.
The unique points of $M_n$ that become fixed by $r$ are those on $J$. Therefore, the covering of $M_n$ over $M_1$ defined by $\langle r\rangle$ is a $n$-fold covering of $S^3$ branched over $K$. Since the group $\langle r\rangle$ of deck transformations is cyclic of order $n$, it turns out that $M_n$ is the $n$-fold cyclic covering of $S^3$ branched over the trefoil knot.
The lift of Banchoff’s sphere to $M_n$ can be built inside $\mathcal{P}_n$ by cyclically gluing $n$ copies of the pieces of the surface of Figure \[fig:Sieradski-bipyramid\](b) around the axis $J$ (Figure \[fig:sieradskiCn\](b)).
Other examples {#sec:other-examples}
==============
Locally cyclic branched covers {#sub:trefoil-locally-cyclic}
------------------------------
Since all the locally cyclic coverings of 2 and 3 sheets of $S^3$ branched over $K$ are in fact cyclic, the first non-cyclic example $p:M\to S^3$ is the one given by the representation of $\pi_k$ into $\Omega_4$ defined by $m\mapsto(1,2,3,4)$ and $ c\mapsto(1,2)$. The construction of the diagram is as in the cyclic case. Since the image of $m$ is a cycle of maximal length the lift of the Dehn surface has $S^2$ as domain, and $\rho(c)$ describes how to identify the curves of the diagram. The final diagram is the one of Figure \[fig:locallycyclic\], which gives $$\pi_1(M) \cong \langle\, \alpha_1,\alpha_2,\alpha_3,\alpha_4 \mid
\alpha_1\alpha_2^{-2},
\alpha_1\alpha_3\alpha_4^{-1},
\alpha_1\alpha_3^{-1}\alpha_4,
\alpha_4\alpha_2^{-1}\alpha_3\,
\rangle\cong{\mathbb{Z}}_3\rtimes{\mathbb{Z}}_8.$$ Therefore $M$ is the prism manifold $M(3,2)$ in the notation of [@HKMR]. Compare it with the $4$-fold cyclic covering depicted in Figure \[fig:cyclic\](c).
![The diagram for the locally cyclic covering given by the presentation $m\mapsto (1,2,3,4)$, $a\mapsto (1,2)$.[]{data-label="fig:locallycyclic"}](locallyCyclicCovering)
The $3$-fold irregular cover {#sub:trefoil-non-cyclic}
----------------------------
Let $p:\widehat{M}\to S^3$ be an $n$-fold branched covering over the trefoil $K$ with $n<\infty$. In the previous sections we have seen how to construct the Johansson diagram of a filling Dehn sphere of $\widehat{M}$ when $\rho(m)$ is a cycle of order $n$, where $\rho$ is the monodromy of $p$. It is also possible to construct the Johansson diagram of a filling Dehn sphere of $\widehat{M}$ in the general case, even if $\rho(m)$ is not a cycle of maximal length. We illustrate this case with the simplest example: the $3$-fold irregular covering over the trefoil. We use the same notation as in previous sections.
![The first non-locally cyclic cover of $S^3$ branched over $K$ is $S^3$.[]{data-label="fig:irregular"}](irregularCover_A)
![The first non-locally cyclic cover of $S^3$ branched over $K$ is $S^3$.[]{data-label="fig:irregular"}](irregularCover_B "fig:") ![The first non-locally cyclic cover of $S^3$ branched over $K$ is $S^3$.[]{data-label="fig:irregular"}](irregularCover_C "fig:")
![The first non-locally cyclic cover of $S^3$ branched over $K$ is $S^3$.[]{data-label="fig:irregular"}](irregularCover_D "fig:") ![The first non-locally cyclic cover of $S^3$ branched over $K$ is $S^3$.[]{data-label="fig:irregular"}](irregularCover_E "fig:")
Assume that $n=3$, and that $\rho(m)=(12)$. The lifts $m_1$, $m_2$ and $m_3$ of $m$ to the domain surface $\widehat{S}$ of $\widehat{\Sigma}=p^{-1}(\Sigma_B)$ verify
- $m_1$ connects $x_1$ with $x_2$;
- $m_2$ connects $x_2$ with $x_1$; and
- $m_3$ connects $x_3$ with itself.
Therefore, $\widehat{S}$ is a disjoint union of two $2$-spheres. One of them, $S_{12}$, contains $x_1$ and $x_2$ and can be obtained by gluing two copies $\Delta_1$ and $\Delta_2$ of the fan $\Delta$. The other one, containing $x_3$ must be a copy of the domain surface of $\Sigma_B$. The restriction of $p_S$ to $S_{12}$ is a $2$-fold branched covering with branching set $\{A,B\}$, and $p_S|_{S_3}$ is a $1$-fold branched covering with branching set $\{A,B\}$, hence a homeomorphism. The diagram $\widehat{{\mathcal{D}}}$ of $\widehat{\Sigma}$ is the lift to $\widehat{S}=S_{12}\sqcup S_3$ of the diagram ${\mathcal{D}}$ through $p_S$. By the same arguments of previous sections, $\widehat{{\mathcal{D}}}$ in $S_{12}$ looks like the left-hand side of Figure \[fig:irregular\](a). The diagram $\widehat{{\mathcal{D}}}$ in $S_{3}$ looks exactly like the diagram ${\mathcal{D}}$ in $S$, except for the identification of the curves, that will be determined by the element $\rho(c)$ given by the monodromy homomorphism. Set $\Sigma_{12}=\widehat{f}(S_{12})$ and $\Sigma_{3}=\widehat{f}(S_{3})$.
Since $\rho(m)$ and $\rho(c)$ verify the identity $$\rho(m)\,\rho(c)\,\rho(m)=\rho(c)\,\rho(m)^{-1}\,\rho(c)$$ and the subgroup of $\Omega_3$ generated by $\rho(m)$ and $\rho(c)$ acts transitively on the set $\{1,2,3\}$, then $\rho(c)=(1,3)$ or $\rho(c)=(2,3)$. Assume that $\rho(c)=(2,3)$ (the case $\rho(c)=(1,3)$ is equivalent). The endpoint of $a_1$ is related by $\widehat{f}$ with the endpoint of $b_1$, the endpoint of $a_2$ is related by $\widehat{f}$ with with the endpoint of $b_3$, and the endpoint of $a_2$ is related by $\widehat{f}$ with the endpoint of $b_3$. The resulting sistering of $\widehat{{\mathcal{D}}}$ is indicated in Figure \[fig:irregular\](a) (in this figure we depict $S_3$ as the fan $\Delta_3$ glued with itself, compare it with Figure \[fig:banchoff\]).
Now, we modify $\widehat{\Sigma}$ by a surgery operation near a triple point where a sheet of $\Sigma_{12}$ and a sheet of $\Sigma_3$ intersect. We perform a “piping” between $\Sigma_{12}$ and $\Sigma_3$ around the triple point $P$ as indicated in the top part of Figure \[fig:irregular-piping\]. The local effect of this operation on the diagram is depicted in the bottom part of the same picture. The resulting Dehn surface $\widetilde{\Sigma}$ has as domain the connected sum of $S_{12}$ and $S_3$, hence it is also a Dehn sphere. Its Johansson diagram appears in Figure \[fig:irregular\](b).
The Dehn sphere $\widetilde{\Sigma}$ fills $\widehat{M}$.
In order to check that $\widetilde{\Sigma}$ fills $\widehat{M}$ it must be proved that all its edges, faces and regions are open $1$-, $2$- and $3$-dimensional disks, respectively. The diagram of $\widetilde{\Sigma}$ implies that the edges and faces of $\widetilde{\Sigma}$ verify this requirement.
The embedded $2$-sphere $\widehat{\Sigma}_3$ is nulhomotopic because $\Sigma_B$ is nulhomotopic, and therefore $\widehat{\Sigma}_3$ is separating in $\widehat{M}$. This implies that the surgery that transforms $\widehat{\Sigma}$ into $\widetilde{\Sigma}$ connects two regions $R_1$ and $R_2$ of $\widehat{\Sigma}$ on one connected component of $\widehat{M}-\widehat{\Sigma}_3$ with another two regions $R_3$ and $R_4$ of $\widehat{\Sigma}$ on the other connected component of $\widehat{M}-\widehat{\Sigma}_3$, creating two regions of $\widetilde{\Sigma}$. If $R_1=R_2$, there would be a loop $\lambda$ in $\widehat{M}$ that intersects $\widehat{\Sigma}$ transversely only at one non-singular point of $\widehat{\Sigma}$, and in this case $p\circ \lambda$ would intersect $\Sigma_B$ transversely only at one non-singular point of $\Sigma_B$; but this cannot happen because $\Sigma_B$ is nulhomotopic (as any Dehn sphere in $S^3$). Hence, $R_1\neq R_2$, and the same argument gives $R_3 \neq R_4$. Therefore, the four regions of $\widehat{\Sigma}$ that become connected in pairs by the surgery are all different, and this implies that all the regions of $\widetilde{\Sigma}$ are open $3$-balls.
The Johansson diagrams of two filling Dehn spheres of the same $3$-manifold are related by a sequence of ***f**-moves* [@peazolibro; @RHomotopies; @tesis], provided that both filling Dehn spheres are nulhomotopic (an equivalent set of moves is proposed in [@Amendola09]). It is natural to ask if ***f***-moves suffice to prove the following well-known result about the $3$-manifold $\widehat{M}$ (see [@peazolibro] for more details on ***f***-moves).
The $3$-fold irregular branched covering of the trefoil is the $3$-sphere.
Starting from the diagram $\widetilde{{\mathcal{D}}}$ of Figure \[fig:irregular\](b), after a *saddle move* the diagram ${\mathcal{D}}_1$ of Figure \[fig:irregular\](c) is obtained. After a *finger move* $-1$ (and ambient isotopies) we arrive to the diagram ${\mathcal{D}}_2$ of Figure \[fig:irregular\](d), and another finger move $-1$ finally gives the diagram ${\mathcal{D}}_3$ of Figure \[fig:irregular\](e), which coincides with the diagram of *Johansson’s sphere*, another well known filling Dehn sphere of $S^3$ (see [@peazolibro; @knots]). Since ${\mathcal{D}}_3$ is filling, ${\mathcal{D}}_2$ is also filling because it is obtained applying a finger move $+1$ to ${\mathcal{D}}_3$ (see [@peazolibro Lem. 5.6 and Thm. 5.20]). The same argument applies to conclude that ${\mathcal{D}}_1$ is also filling. Since $\widetilde{{\mathcal{D}}}$ and ${\mathcal{D}}_1$ are filling diagrams, the saddle move that relate them is an ***f***-move. By [@peazolibro Thm. 5.20], $\widehat{M}$ is $S^3$.
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[^1]: Partially supported by the European Social Fund and Diputación General de Aragón (Grant E15 Geometr[í]{}a) and by MINECO grants MTM2013-46337-C2, MTM2016-77642-C2, MTM2013-45710-C2 and MTM2016-76868-C2-2-P.
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abstract: 'We study the relation between gaseous absorbing column density ([$N_{\rm H}$]{}), infrared colors and detectability of the broad lines in a large sample of Seyfert 2 galaxies(Sy2s). We confirm that Sy2s without polarized broad lines tend to have cooler 60$\mu$m/25$\mu$m colors; this correlation was previously ascribed to the effect of obscuration towards the nuclear region. We find some evidence that Sy2s without polarized broad lines have larger absorbing column density ([$N_{\rm H}$]{}) and that a fraction of them are characterized by dust lanes crossing their nuclei. However, we find that the IR colors do not correlate with [$N_{\rm H}$]{}, in disagreement with the obscuration scenario. Also, Sy2s without polarized broad lines follow the same radio-FIR relation as normal and starburst galaxies, at variance with Sy2s with polarized broad lines. These results indicate that the lack of broad lines in the polarized spectrum of Sy2s is mostly due to the contribution/dilution from the host galaxy or from a circumnuclear starburst, though at a lower extent the obscuration toward the nuclear region also plays a role.'
author:
- Qiusheng Gu
- Roberto Maiolino
- 'Deborah Dultzin-Hacyan'
date: 'Received — ; accepted — '
title: Nuclear obscuration and scattering in Seyfert 2 galaxies
---
Introduction
============
According to the standard unification model, Seyfert 1 and 2 galaxies (Sy1s and Sy2s hereafter) are intrinsically the same objects and the absence of broad lines in Sy2s is ascribed to the obscuration by a pc-scale dusty torus oriented along the line of sight (see the reviews by Antonucci [@antonucci_rev] and Véron-Cetty & Véron [@veron]). The observational evidence for this model includes the detection of polarized broad emission lines in some Seyfert 2 galaxies (Antonucci & Miller [@antonucci]; Tran [@tran]; Moran et al. [@moran]), the detection of broad lines in the infrared spectrum of some Sy2s (Ruiz et al. [@ruiz]; Veilleux et al. [@veilleux]; Rix et al. [@rix]) and the detection of a prominent photoelectric cutoff in the X-ray spectra of Sy2s indicating the presence of large columns of gas along the line of sight (Koyama et al. [@koyama]; Awaki et al. [@awaki]; Maiolino et al. [@maiolino98]; Risaliti et al. [@risaliti]).
The spectropolarimetric observations of different samples of Seyfert 2 galaxies indicate that [*only*]{} about 40 % of Sy2s show broad lines in their polarized spectra (eg. Heisler et al. [@heisler]), although such surveys are probably biased since pre-selection was done according to the broad-band polarization.
According to the suggestion of Heisler et al. ([@heisler]), the detectability of a hidden BLR through spetropolarimetry in Sy2s is related to the inclination of the torus which, in turn, is related to the 60 $\mu$m to 25 $\mu$m flux ratio, $s_{60\mu m}/s_{25\mu m}$. More specifically, in those Sy2s showing polarized broad lines (PBL) the torus should be oriented more face-on so that the scattering medium is less obscured by the torus itself. This model would also explain the correlation between IR colors ($s_{60\mu m}/s_{25\mu m}$) and detectability of the PBL. In particular, when the torus is observed close to pole-on, the PBL should be more easy to detect and the IR color should be hotter since we are observing the hotter dust emitting region, in agreement with what is observed.
However, more recently Alexander ([@alexander]) compared the absorbing column densities inferred from the hard X-rays with the detectability of the PBL and found no correlation. This result is in contrast with Heisler’s et al. ([@heisler]) model, which would predict a higher absorbing column for Sy2s without PBL. Alexander ([@alexander]) suggests that the relation between detectability of the PBL and IR colors is indirect: the contribution from the host galaxy would both make the IR color cooler and would also dilute the nuclear optical spectrum making more difficult the detection of scattered polarized light.
The simple model of the obscuring torus has been subject to various modifications. In particular, while the pc-scale torus is probably responsible for the huge absorbing columns ([$N_{\rm H}$ ]{}$> 10^{24}$ cm$^{-2}$) observed in several Sy2s, observational evidence was also found for a larger scale ($\sim$ 100 pc) obscuring medium with lower absorbing column density ([$N_{\rm H}$ ]{}$\sim$ a few times $10^{22}$ cm$^{-2}$, Granato et al. [@granato]; Matt [@matt]; Maiolino [@maiolino] and references therein).
In this paper we expand the work done by Alexander ([@alexander]) by enlarging the sample of Sy2s for which information on both the absorbing [$N_{\rm H}$ ]{}and on the detection of PBL is available. We also seek additional constraints on the nature of the circumnuclear scattering and absorbing medium by comparing the mid- and far-IR colors with [$N_{\rm H}$ ]{}and the detectability of PBL with the nuclear morphology and with the radio power.
The paper is organized as follows. In Section 2 we present our sample of Seyfert 2 galaxies with spectropolarimetric observations, the results are given in Section 3. We discuss our results and their implications in Section 4 and summarize our conclusions in Section 5.
The Sample
==========
We collected all Seyfert 2 galaxies from the recent literature (from 1985 to 2000), for which both spectropolarimetric data and an estimate of [$N_{\rm H}$ ]{}from the X-rays are available. Within this sample, we got 22 Seyfert 2 galaxies with PBL, and 18 Sy2s without detection of PBL, which are presented in Table 1 and 2, respectively.
In Table 1 and 2, we report the following information: galaxy name (column 1); column density ([$N_{\rm H}$]{}) taken from Bassani et al. ([@bassani]); Risaliti et al. ([@risaliti]); and Alexander ([@alexander]) (column 2); the IRAS colors s$_{25\mu m}/s_{12\mu m}$ and s$_{60\mu m}/s_{25\mu m}$ in columns 3 and 4, where the IRAS fluxes are taken from Moshir et al. ([@moshir]); and the flux between 42.5 and 122.5 $\mu$m, FIR, where FIR $\rm = 1.26\times 10^{-14} (2.58\times s_{60\mu m} + s_{100\mu m})$, in column 5; the 1.49 GHz radio emission from the NRAO/VLA Sky Survey (NVSS) (Condon et al. [@condon]) in column 6; and the corresponding reference for PBL in column 7.
Name $N_{\rm H}$$^{\rm a}$ s$_{25\mu m}/s_{12\mu m}$ s$_{60\mu m}/s_{25\mu m}$ FIR$^{\rm b}$ F$_{\rm 1.49 GHz}$$^{\rm c}$ Ref
--------------- ----- ---------------------------- --------------------------- --------------------------- --------------- ------------------------------ -----------
Circinus 43000$^{+19000}_{ -11000}$ 3.640 3.634 12.065 ${\rm h}$
ESO 434$-$G40 162$^{+ 23}_{ - 21}$ 14.6 ${\rm h}$
F05189$-$2524 490$^{+ 10}_{ - 16}$ 4.632 3.960 0.600 29.1 ${\rm h}$
F09104$+$4109 24$^{+ 6}_{ - 6}$ 1.520 1.395 0.030 ${\rm j}$
F13197$-$1627 7943 3.135 2.047 0.259 275.3 ${\rm d}$
F20460$+$1925 250$^{+ 34}_{ - 32}$ 1.385 1.685 0.042 18.9 ${\rm h}$
F23060$+$0505 840$^{+ 190}_{ - 250}$ 1.438 2.500 0.051 6.8 ${\rm h}$
IC 3639 $>$ 100000 3.538 3.130 0.374 ${\rm i}$
IC 5063 2400$^{+ 200}_{ - 200}$ 3.310 1.557 0.250 ${\rm k}$
Mark 3 11000$^{+ 1500}_{ - 2500}$ 4.057 1.377 0.171 1100.9 ${\rm e}$
Mark 348 1060$^{+ 310}_{ - 260}$ 2.484 1.870 0.070 292.7 ${\rm e}$
Mark 463E 1600$^{+ 800}_{ - 800}$ 2.825 1.354 0.094 381.0 ${\rm e}$
Mark 477 $>$ 10000 2.160 2.500 0.067 60.8 ${\rm f}$
Mark 1210 $>$ 10000 3.800 0.880 0.079 114.9 ${\rm f}$
NGC 1068 $>$ 100000 2.267 2.140 9.070 4849.0 ${\rm g}$
NGC 2110 289$^{+ 21}_{ - 29}$ 2.378 5.068 0.224 299.4 ${\rm h}$
NGC 2273 $>$ 100000 2.957 4.654 0.335 63.4 ${\rm i}$
NGC 3081 6600$^{+ 1800}_{ - 1600}$ 5.7 ${\rm o}$
NGC 4388 4200$^{+ 600} _{- 1000}$ 3.550 3.070 0.578 120.4 ${\rm h}$
NGC 4507 2920$^{+ 230} _{- 230}$ 3.065 3.248 0.219 67.4 ${\rm o}$
NGC 5506 340$^{+ 26} _{- 12}$ 2.800 2.420 0.409 339.4 ${\rm h}$
NGC 7674 $>$ 100000 2.667 2.901 0.288 221.4 ${\rm e}$
Name $N_{\rm H}$$^{\rm a}$ s$_{25\mu m}/s_{12\mu m}$ s$_{60\mu m}/s_{25\mu m}$ FIR$^{\rm b}$ F$_{\rm 1.49 GHz}$$^{\rm c}$ Ref
--------------- ----- --------------------------- --------------------------- --------------------------- --------------- ------------------------------ -----------
F19254$-$7245 1995$^{\rm d}$ 5.462 3.690 0.249 ${\rm l}$
Mark 1066 $>$ 10000 4.620 4.524 0.505 ${\rm o}$
NGC 34 $>$ 1000$^{\rm d}$ 5.667 7.155 0.779 67.5 ${\rm l}$
NGC 1143 $>$ 100$^{\rm d}$ 2.692 7.600 0.319 ${\rm l}$
NGC 1386 $>$ 100000 2.880 4.090 0.316 37.8 ${\rm o}$
NGC 1667 $>$ 10000 1.763 8.731 0.375 77.3 ${\rm o}$
NGC 3281 7980$^{+ 1900} _{- 1500}$ 2.909 2.641 0.317 80.9 ${\rm o}$
NGC 3393 $>$ 100000 2.840 3.352 0.127 81.5 ${\rm m}$
NGC 4941 4500$^{+ 2500} _{- 1400}$ 2.280 2.351 0.095 20.3 ${\rm o}$
NGC 5128 2250$^{+ 1250} _{- 1250}$ 1.346 11.441 9.829 ${\rm n}$
NGC 5135 $>$ 10000 3.701 6.524 0.914 1.5 ${\rm l}$
NGC 5347 $>$ 10000 3.172 1.565 0.081 ${\rm o}$
NGC 5643 $>$ 100000 3.895 5.585 1.165 ${\rm o}$
NGC 7130 $>$ 10000 3.397 7.790 0.873 190.6 ${\rm l}$
NGC 7172 861$^{+ 79} _{- 33}$ 1.696 7.641 0.356 37.6 ${\rm l}$
NGC 7496 501$^{\rm d}$ 5.630 5.625 0.471 ${\rm l}$
NGC 7582 1240$^{+ 60} _{- 80}$ 4.689 7.585 2.477 ${\rm l}$
NGC 7590 $<$ 9.2 1.615 8.798 0.467 ${\rm l}$
$^{\rm a}$ Absorbing column density in unit of 10$^{20}$ cm$^{-2}$; $^{\rm b}$ In unit of 10$^{-12}$ W m$^{-2}$; $^{\rm c}$ In unit of mJy; $^{\rm d}$ Alexander [@alexander]; $^{\rm e}$ Miller & Goodrich [@miller]; $^{\rm f}$ Tran et al. [@tran92]; $^{\rm g}$ Antonucci & Miller [@antonucci]; $^{\rm h}$ Véron-Cetty & Véron [@veronsy]; $^{\rm i}$ Kay [@kay]; $^{\rm j}$ Hines & Wills [@hines]; $^{\rm k}$ Inglis et al. [@inglis]; $^{\rm l}$ Heisler et al. [@heisler]; $^{\rm m}$ Nagao et al. [@nagao]; $^{\rm n}$ Alexander et al. [@alexander99]; $^{\rm o}$ Moran et al. [@moran].
The Results
===========
[$N_{\rm H}$ ]{}versus detectability of polarized broad lines
-------------------------------------------------------------
We show the histogram distribution of column densities for Sy2s without and with PBL in Figs. 1a and 1b, respectively. Since there are 6 censored data (lower limits) among Sy2s with PBL and 11 among Sy2s without PBL, we need to use the survival analysis methods (ASURV Rev 1.2, Isobe, Feigelson & Nelson [@isobe]) to study the similarity of these two samples. We find that the probability for these two samples to be extracted from the same parent population is about 12 %, and the mean values of log [$N_{\rm H}$ ]{}(in units of cm$^{-2}$) are 23.6 $\pm$ 0.2 and 24.2 $\pm$ 0.2, respectively. This result suggests that Sy2s with PBL are affected by lower obscuration than Sy2s without PBL, but the statistical significance of the result is not high, and therefore not conclusive.
Infrared colors versus [$N_{\rm H}$ ]{}
---------------------------------------
In Figure 2, we show the plot of column density ([$N_{\rm H}$]{}) versus far-IR color ($\rm s_{60\mu m}/s_{25\mu m}$) for Seyfert 2 galaxies with PBL (filled circles) and Sy2s without PBL (open circles). Sy2s with PBL have warmer FIR colors than Sy2s without PBL, which confirms the result obtained by Heisler et al. ([@heisler]) with a higher statistical significance, the probability for these two samples from the same parent population is less than 0.0001 % and the mean values of $\log
(\rm s_{60\mu m}/s_{25\mu m})$ are 0.368 $\pm$ 0.044 and 0.722 $\pm$ 0.053, respectively. However, we do not find any correlation between the $\rm s_{60\mu m}/s_{25\mu m}$ color and the absorbing [$N_{\rm H}$]{}, at variance with what is expected from the model proposed by Heisler et al. ([@heisler]): according to the latter model higher [$N_{\rm H}$ ]{}should correspond to cooler IR colors.
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Fig. 3 is a plot similar to Fig. 2, but where the IR color is sampled at shorter wavelengths and, more specifically, the ratio $\rm s_{60\mu m}/s_{25\mu m}$ is replaced by $\rm s_{25\mu m}/s_{12\mu m}$. This ratio should be more sensitive to absorption of the inner (hotter) dust component. In this case there is a marginal evidence that Sy2s with [$N_{\rm H}$ ]{}$ < 10^{23}$ cm$^{-2}$ have warmer colors. Yet, the relation between $\rm s_{25\mu m}/s_{12\mu m}$ color and [$N_{\rm H}$ ]{}is not the one expected from the absorption by a dusty screen associated to the gaseous column observed in the X-rays, assuming a Galactic gas-to-dust ratio and extinction curve. Such a relation is given by the following formula:
$$\log (\frac{s_{25}}{s_{12}})_{obs} = \log (\frac{s_{25}}{s_{12}})_{int} + 0.042 \times N_{\rm H} \times 10^{-22}$$
where $\rm (s_{25\mu m}/s_{12\mu m})_{int}$ is the intrinsic color prior to absorption and where we assumed
$$\rm
A_{12} = \frac{1}{27} A_{\rm V} ~~~ and ~~~A_{25} = \frac{1}{55} A_{\rm V}$$
and $$A_{\rm v} = 5.0 \times 10^{-22} N_{\rm H}$$
(Bohlin et al. [@bohlin]). To estimate the intrinsic $\rm s_{25\mu m}/s_{12\mu m}$ color, we selected all of the Seyfert 1 galaxies listed in the Véron-Cetty & Véron ([@veronsy]) catalog with available IRAS fluxes and we derived a mean value of $\langle \log (\frac{s_{25}}{s_{12}}) \rangle = 0.11$. Therefore, within the framework of the unified model we assigned this value to $\log (\frac{s_{25}}{s_{12}})_{int}$ in equation 1. The resulting curve is shown in Fig. 3 with a dot-dashed line, which may match the observational data for low [$N_{\rm H}$ ]{}($< 10^{23}$ cm$^{-2}$), but fails to account for the majority of the sources at higher columns. This finding indicates that the observed distribution of IR colors is not to be ascribed to different degrees of absorption (at least for [$N_{\rm H}$ ]{}$ > 10^{23}$ cm$^{-2}$).
Detectability of polarized broad lines versus nuclear morphology
----------------------------------------------------------------
HST images of a large sample of Seyfert galaxies have shown that Seyfert 2 galaxies are characterized by dust lanes or irregular dust distribution crossing the nuclear region more often than Sy1s. Based on these results, Malkan et al. ([@malkan]) suggested that 100pc-scale dusty structures may play a role in the obscuration that generally affects Sy2s. As summarized in Maiolino ([@maiolino]) the presence of a 100pc-scale obscuring medium in Sy2s is supported by various pieces of evidence, but most likely such a large scale medium contributes to the absorption only with “moderate” gas columns (less than a few times 10$^{23}$ cm$^{-2}$).
Among all of the Seyfert 2 galaxies imaged by Malkan et al. ([@malkan]) we searched those observed in spectropolarimetry and found 32 of them. Out of these 32 Sy2s, 12 have PBL and 20 do not have PBL. 30% of the Sy2s without PBL are characterized by (large scale) dust lanes crossing the nuclear region or irregular nuclear dust distribution (according to the classification given in Malkan et al. [@malkan]) while none of the Sy2s with PBL show evidence for dusty nuclear features. This suggests that, at least in some cases, obscuration due to 100pc-scale dusty structures is responsible for hiding the mirror which reflects the broad lines.
Detectability of the polarized broad lines on the Radio-FIR plane
-----------------------------------------------------------------
It is well known that a tight correlation exists between radio and far-IR (FIR) emission for normal, starburst and Seyfert galaxies (the latter with larger scatter) (Helou et al. [@helou]). More recently, Ji et al. ([@ji]) have studied the radio–FIR relation of LINERs and found that the AGN– and starburst–supported LINERs can be distinguished on this diagram, with the AGN–dominated ones being scattered in the region with higher radio fluxes with respect to the starburst/normal galaxies correlation. In Fig. 4 we show the radio vs. FIR diagram for the Sy2s observed in spectropolarimetry. The Sy2s with PBL are spread mostly above the standard starburst correlation (dashed line), indicating the presence of an extra contribution to the radio emission due to the AGN. Instead, Sy2s without PBL follow more tightly the starburst correlation, suggesting that in these objects the starburst component dominates both the FIR and the radio emission.
Discussion and Implications
===========================
Our finding that the absorbing column density ([$N_{\rm H}$]{}) is marginally lower in Sy2s with PBL than in Sy2s without PBL is tentatively in favor of Heisler’s et al. ([@heisler]) model, which associates the visibility of the PBL with the amount of obscuration along the line of sight, though the statistical significance of the result is not high. Yet, the interpretation of the correlation between detectability of PBL and IR colors given by Heisler et al. is not supported by our results. They ascribe the colder IR colors observed in objects without PBL to the larger obscuration affecting the mid-IR emitting region. The mismatch between the expected and the observed IR colors in Figs. 2–3 indicates that the IR colors are unrelated to the obscuration affecting the active nucleus. There are various possible scenarios (not necessarily alternative) to explain such a mismatch. For a Galactic gas-to-dust ratio and extinction curve, at 12$\mu$m the extinction implied by a column larger than $10^{23}$ cm$^{-2}$ is higher than 2 mag, implying that in this case the 12$\mu$m radiation is heavily suppressed and, therefore, the observed radiation is probably dominated by the host galaxy (especially in the large IRAS beam). It is worth noting that at [$N_{\rm H}$ ]{}$ \le 10^{23}$ cm$^{-2}$ the 25$\mu$m/12$\mu$m color is lower (i.e. hotter) and follows the relation expected by the reddened Sy1 curve (dot-dashed curve in Fig. 3). Indeed, in this range of low [$N_{\rm H}$ ]{} the absorbing medium is more transparent to the 12$\mu$m radiation and might well dominate over the emission from the host galaxy. At longer wavelengths (25$\mu$m and 60$\mu$m) the dust extinction is much reduced and the emitting region is much more extended. In particular, at large gaseous columns ([$N_{\rm H}$ ]{}$> 10^{24}$ cm$^{-2}$) the medium responsible for absorption must be very compact ($<$ 10 pc), not to violate constraints on the gas mass given by the dynamical mass (Risaliti et al. [@risaliti], Maiolino [@maiolino]); as a consequence, in many cases (at least in the Compton thick sources) the obscuring medium is smaller than the dusty emitting region responsible for the 25$\mu$m and 60$\mu$m radiation (10-100 pc). Therefore, the scatter in the 60$\mu$m/25$\mu$m color probably reflects mostly variations in the relative contribution of the AGN (hotter) and starburst/galactic (colder) component to the IR radiation, as suggested by Alexander ([@alexander]), although some obscuration effect on the 25$\mu$m emission might be present.
These findings confirm and strengthen the result obtained by Alexander ([@alexander]) that the relation between visibility of PBL and IR colors is mostly due to the relative dominance of AGN and starburst/galactic component; in the sense that the latter both makes the IR colors cooler and dilutes the optical light making more difficult the detection of the PBL. This scenario is further supported by the finding that Sy2s without PBL follow the same radio-FIR correlation as starbursts (sect.3.4).
Yet, we find that, although the relation between PBL detectability and IR colors is mostly related to the relative contribution of the starburst/galactic component, the detectability of the PBL is also affected, to a lower extent, by the obscuration toward the nuclear region. This is indicated by the larger average [$N_{\rm H}$ ]{} and by the higher incidence of nuclear dusty features in Sy2s without PBL. Given the limited statistical significance of these findings ($\sim$ 90% for the difference in [$N_{\rm H}$ ]{}distribution) it is not surprising that such a trend was not found by Alexander ([@alexander]) which used a much smaller sample of objects.
Within the context of the relation between detectability of PBL and dominance of the starburst component, there is a possible explanation, alternative to the dilution of the optical light by the host galaxy suggested by Alexander ([@alexander]), and which might apply to some of the objects without PBL. Some authors have suggested the existence of a dichotomy in Sy2s where, at variance with the commonly accepted unified scenario, a fraction of the Sy2s do not host a hidden Sy1 nucleus but are intrinsically different, such objects would be “pure” Seyfert 2 galaxies (Hutchings & Neff [@hutchings]; Neff & Hutchings [@neff]; Heckman et al. [@heckman95]; Dultzin$-$Hacyan et al. [@deborah]; Gu et al. [@gu]). In particular the finding that the bolometric luminosity of some objects with a Sy2 spectrum is dominated by a nuclear starburst prompted Heckman et al. ([@heckman95], [@heckman97]) to argue that some extreme starburst population might mimic the narrow line spectrum of AGNs (partly supporting the model of Terlevich et al. [@terlevich]). Within this scenario, the relation between detectability of PBL and IR colors is trivial: some of the Sy2s with starburst-like cooler IR colors do not show evidence for PBL because the broad line region is absent.
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Conclusions
===========
In this paper, we collect 40 Seyfert 2 galaxies having both spectropolarimetric observations and a measure of the absorbing [$N_{\rm H}$ ]{}obtained by means of their X-ray spectrum. Out of these 40 objects 22 show broad lines in polarization (most likely ascribed to scattering of the broad line region) and 18 do not. For these objects we also analyzed the relation of [$N_{\rm H}$ ]{}and detectability of the broad lines with the mid- and far-IR colors. We confirm previous claims that Sy2s without polarized broad lines have cooler IR colors. We studied additional diagrams to test two scenarios proposed to explain such a correlation and that, more specifically, ascribe this effect either to obscuration of the nuclear region or to the contribution/dilution from circumnuclear starburst activity and from the host galaxy.
We found a marginal evidence for Sy2s showing broad lines in polarization to have lower [$N_{\rm H}$ ]{}. We also find that about 30% of the Sy2s without polarized broad lines show dust lanes crossing the nucleus, while none of the Sy2s with polarized broad lines show evidence for such nuclear dusty structures. These results suggest that the obscuration towards the nuclear regions (hence towards the scattering mirror) plays a role in hiding the polarized broad lines, at least in some objects.
On the other hand, we find that the absorbing column density does not correlate with the IR colors and, in particular, these quantities do not follow the relation expected in the case of absorption of the IR emitting region by the dust associated to the observed [$N_{\rm H}$]{}, therefore indicating that the distribution of IR colors is not to be ascribed to obscuration effects. Also, we find that Sy2s without detection of polarized broad lines follow the same radio–FIR relation as normal and starburst galaxies, at variance with Sy2s with polarized broad lines which tend to spread towards higher radio luminosities. These findings support previous claims that the relation between far-IR colors and detectability of the polarized broad lines in Sy2s is mostly related to dilution of the IR and optical light by a circumnuclear starburst or by the host galaxy.
Summarizing, our results indicate that the lack of broad lines in the polarized spectrum of Sy2s is mostly due to the contribution/dilution from the stellar component, though at a lower extent the obscuration towards the nuclear region also plays a role.
We would like to thank the anonymous referee for his/her careful reading the manuscript and valuable comments, which improved the paper a lot. A significant fraction of this work was done during the Gullermo Haro Workshop 2000, we are grateful to the organizers of the workshop who made possible this collaboration and enabled us to perform this research. QSGU acknowledges support from UNAM postdoctoral program (Mexico) and from National Natural Science Fundation of China and the National Major Project for Basic Research of the State Scientific Commission of China. RM acknowledges partial support by the Italian Space Agency (ASI) under grant ARS–99–15 and by the Italian Ministry for University and Research (MURST) under grant Cofin98–02–32. And DD-H acknowledges support from grant IN 115599 from PAPIIT-UNAM. This research has made use of NASA’s Astrophysics Data System Abstract Service and the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
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---
abstract: 'The mid-infrared region is well suited for exoplanet detection thanks to the reduced contrast between the planet and its host star with respect to the visible and near-infrared wavelength regimes. This contrast may be further improved with Vector Vortex Coronagraphs (VVCs), which allow us to cancel the starlight. One flavour of the VVC is the AGPM (Annular Groove Phase Mask), which adds the interesting properties of subwavelength gratings (achromaticity, robustness) to the already known properties of the VVC. In this paper, we present the optimized designs, as well as the expected performances of mid-IR AGPMs etched onto synthetic diamond substrates, which are considered for the E-ELT/METIS instrument.'
author:
- |
Brunella Carlomagno, Christian Delacroix, Olivier Absil, Pontus Forsberg, Serge Habraken, Aïssa Jolivet, Mikael Karlsson, Dimitri Mawet, Pierre Piron, Jean Surdej and Ernesto Vargas Catalan Department of Astrophysics, Geophysics and Oceanography, University of Liège, 17 allée du Six Août, B-4000 Sart Tilman, Belgium;\
ngström Laboratory, Uppsala University, Lägerhyddsvägen 1, SE-751 21 Uppsala, Sweden;\
European Southern Observatory, Alonso de Córdova 3107, Vitacura 7630355, Santiago, Chile
bibliography:
- 'article\_Carlomagno\_SPIE2014.bib'
title: 'Mid-IR AGPMs for ELT applications'
---
=1
INTRODUCTION {#sec:intro}
============
Direct detection of exoplanets presents two important challenges: the presence of the extremely brighter host star and the small angular separation between the latter and its companion. However, disentangling the signal of an exoplanet from its star allows us to obtain its spectrum, which can be analyzed to infer the atmospheric composition and eventually search for biosignatures. A simple way to alleviate the contrast problem is to observe in the mid-infrared ($3-13\mu m$) regime, where the contrast between the star and the planet is reduced. In order to improve this contrast, a coronagraphic technique can also be implemented. A mask stops the light of the star, while allowing the light of the companion (as the reflected or the emitted light of the planet) to pass through the optical system[@Lyot]. While an amplitude mask blocks directly the light, preventing from observing regions close to the star, the phase mask rejects the starlight by destructive interferences, using (spatially-distributed) phase shifts. In particular, the vortex phase mask produces a continuous helical phase ramp, with a singularity at the center, which creates an optical vortex and hence nulls the light locally. One flavour of the vortex phase mask is the $\textit{vector vortex}$[@Mawet2009; @Mawet11], which consists in a rotationally symmetric halfwave plate (HWP), with an optical axis orientation that rotates about the center.
![**Left**: Illustration of the spatial variation of the optical axis orientation of a vector vortex coronagraph (VVC) of topological charge 2, obtained with a rotationally symmetric HWP. **Right**: Computed $2\times2\pi$ phase ramp created by a charge-2 vortex.[]{data-label="pola_phase"}](pola_phase.png){width="15cm"}
The effect of this vector vortex is to make the incoming horizontal polarization turn twice as fast as the azimuthal coordinate. When the vector vortex has completed the rotation with a phase ramp of $2\times2\pi$, the produced optical vortex is of charge 2 (see Fig. \[pola\_phase\]). These systems are called Vector Vortex Coronagraphs (VVCs). They present interesting properties: a small inner working angle (down to $0.9\lambda/D$), high throughput, clear off-axis 360$\degree$ discovery space and simplicity. One type of VVC is the Annular Groove Phase Mask (AGPM), which can be rendered achromatic over an appreciable spectral range thanks to the use of subwavelength gratings[@Mawet05; @Delacroix12a]. This technology would be well suited for applications at mid-infrared wavelengths on future extremely large telescopes (ELTs).
DESIGN OPTIMIzation PROCEDURE {#sec:design}
=============================
Grating parameters {#gratparam}
------------------
The design optimization technique uses an algorithm based on the $\textit{Rigorous Coupled Wave Analysis}$ (RCWA)[@Moharam82] and coded in MATLAB[@Mawet2005]. The RCWA solves Maxwell’s equations without simplifying assumptions. The principal optimization parameters of the subwavelength grating profile are the filling factor $F$ and the depth $h$ (see Fig. \[grating\], left).
Thanks to its great optical, mechanical, thermal and chemical characteristics, synthetic diamond turns out to be a very well suited material to manufacture such AGPMs, based on an advanced micro-fabrication technique using nano-imprint lithography and reactive ion etching[@Forsberg13; @Forsberg14]. Due to fabrication issues, the walls of the etched grooves are not perfectly vertical, showing a slope $\alpha \sim 3 \degree$ in the present case (see Fig. \[grating\], right).
![**Left**: Schematic diagram of a trapezoidal grating. The filling factor $F$ is such that $F\Lambda$ corresponds to the line width on top of the walls. **Right**: Cross sectional view of a diamond AGPM dedicated to the L band. The grating sidewalls have an angle $\alpha \sim 3\degree$ and an average width $F_{\rm{equiv}}\Lambda \sim 0.5 \mu m$.[]{data-label="grating"}](grating.png){width="16cm"}
The period $\Lambda$ of the subwavelength grating is kept constant during the optimization. Its value is determined by the subwavelength limit: $$\Lambda < \frac{\lambda_{\rm{min}}}{n(\lambda_{\rm{min}})}$$ where $\lambda_{\rm{min}}$ is the shortest wavelength of the considered band (with a $100nm$ security margin), and $n(\lambda_{\rm{min}})$ is the refractive index of the substrate (diamond in our case) calculated at this wavelength. The refractive index was computed considering polynomial regressions and Sellmeier equations. The final values of the refractive index and the period are shown in Table \[refractive\_period\], for several considered bands.\
\[h!\]
Band Bandwidth \[$\mu m$\] $\lambda_{\rm{min}} [\mu m$\] $n(\lambda_{\rm{min}})$ Period $\Lambda$ \[$\mu m$\]
--------- ----------------------- ------------------------------- ------------------------- ------------------------------
L 3.5 – 4.1 3.4 2.3814 1.42
M 4.6 – 5 4.5 2.3810 1.89
lower N 8 – 11.3 7.9 2.3806 3.32
upper N 11 – 13.2 10.9 2.3805 4.58
: Calculated diamond refractive indices and subwavelength grating periods, for several considered mid-IR spectral windows.
\[refractive\_period\]
Null depth definition {#nulldef}
---------------------
Theoretically, a vortex coronagraph should provide a perfect on-axis light cancellation, but imperfections prevent it. The metric used for the optimization is the $\textit{null depth}$ $N(\lambda)$, defined as the contrast, integrated over the whole point spread function (PSF)[^1] $$N(\lambda)_{\rm{theo}} = \frac{I_{\rm{coro}}(\lambda)}{I_{\rm{off}}(\lambda)} = \frac{[1-\sqrt{q(\lambda)}]^2 + \epsilon(\lambda)^2 \sqrt{q(\lambda)}}{[1+\sqrt{q(\lambda)}]^2}$$ where $I_{\rm{coro}}$ is the signal intensity when the input beam is centered on the mask, while $I_{\rm{off}}$ is the signal intensity when the input beam is far from the mask center, $\epsilon (\lambda)$ is the phase error with respect to $\pi$, and $q(\lambda)$ is the flux ratio between the polarization components transverse electric (TE) and transverse magnetic (TM), respectively. It is in this equation that all geometrical parameters (filling factor $F$, depth $h$ and sidewall angle $\alpha$) are taken into account, via $\epsilon(\lambda)$ and $q(\lambda)$.
The optimization process follows a precise procedure. First, the RCWA algorithm provides a two-dimensional map of the theoretical null depth as a function of the optimization parameters (filling factor $F$ and depth $h$), then the same algorithm is used for a more precise optimization around the optimal parameters obtained after the first step. In this paper, only a sidewall angle of 3$\degree$ has been considered. The results of the optimization are presented in Table \[results\].\
Influence of the ghost {#exp_perfor}
----------------------
The optimized parameters and null depths obtained from RCWA simulations assume that only the zeroth order of the subwavelength grating is transmitted. In practice, the presence of a ghost signal has been confirmed by laboratory measurements. It is the result of multiple incoherent reflections within the substrate. The AGPM pattern etched on the frontside of the substrate (Fig. \[fabrication\], left) partially reduces the reflections, acting as an antireflective layer, to some degree. Most of the ghost signal is actually caused by the reflection on the flat interface, on the other side of the substrate. Therefore, an anti-reflective grating (ARG) needs to be etched on the backside of the component (Fig. \[fabrication\], right) to avoid these reflections. Typically, a binary square-shaped structure is used for the antireflection. For the diamond in L band, the raw backside reflection is $17 \%$, while the ARG reduces it to $1-2\%$ [@Forsberg13].
![SEM pictures of an L-band AGPM fabricated at the Ångström Laboratory using reactive ion etching and nanoimprint lithography. **Left**: Annular grooves etched on the frontside of the component. **Right**: Antireflective structure etched on the backside.[]{data-label="fabrication"}](diamond.png){width="17cm"}
The total null depth is then the sum of two components: $$N_{\rm{total}}(\lambda) = N_{\rm{theo}}(\lambda) + N_{\rm{ghost}}(\lambda)$$ where $$N_{\rm{ghost}} (\lambda)= \frac{I_{\rm{ghost}}(\lambda)}{I_{\rm{off}}(\lambda)} \; .$$ Even though the final parameters come from a perfect optimization, where the only metric is the theoretical null depth, the expected performances for a total null depth (considering the ghost) are improved by the use of the ARG. When no ARG is present, the performances deteriorate very fast, as shown in section \[sec:optimal\].\
Simulation results and laboratory validation {#sec:optimaldesign}
============================================
0.5cm
Spectral band selection {#sec:select}
-----------------------
We have applied our optimization procedure to the mid-infrared bands L, M and N. These three bands are expected to be addressed by the METIS instrument on the future E-ELT, for which this optimization study was proposed. Our AGPM could however be used for any other high contrast imaging instrument working at these wavelengths.\
Because the N band is very wide ($8 - 13.2 \mu m$), it is difficult to cover it with a single component. Therefore, we divided it into two sub-bands (lower and upper), on which the AGPM design was optimized. The upper N band was defined to suit the VISIR mid infrared camera of the VLT: $11-13.2 \mu m$. Recently, we have etched a few AGPMs covering this band. One of these is now installed on VISIR and shows promising performance[@Delacroix12b]. Concerning the lower N band, we followed a different path. A constraint was imposed on the mean null depth, which should be smaller than $10^{-3}$. We could then calculate the maximal width of the spectral band, that we have arbitrarily started at $8 \mu m$, to define the lower N band: $8-11.3 \mu m$ (see Fig. \[evoluzione\]).
![Definition of the lower N band spectral window. Optimized mean null depth for a minimal wavelength of $8\mu m$, function of the maximal wavelength. The desired mean null depth $10^{-3}$ is obtained for the bandwidth $8-11.3 \mu m$ .[]{data-label="evoluzione"}](evoluzione_paper.eps){width="14cm"}
Optimal AGPM parameters and null depth at L, M and N bands {#sec:optimal}
----------------------------------------------------------
The results of our RCWA simulations are presented in Table \[results\], with the optimized filling factor $F$, grating depth $h$, and mean null depth $N$ over the whole bandwidth.\
The AGPM mean null depth (over each spectral band) is shown in Fig. \[rcwa\] (left), function of the profile parameters $F$ and $h$. The dark blue region in the center of the figure corresponds to the optimal parameters region, providing the best contrast expressed here on a logarithmic scale. In Fig. \[rcwa\] (right), we see that the theoretical optimal mean null depth $N$ is comprised between $10^{-4}$ and $10^{-3}$. When the ghost is taken into account, the somewhat degraded null depth value is still close to $10^{-3}$, showing the excellent performance of the ARG.\
\[h!\]
Band Bandwidth $F$ $h [\mu m]$ $N$
--------- ----------- ------ ------------- -----------------------
L 3.5 – 4.1 0.45 5.22 $4.1 \times 10^{-4}$
M 4.6 – 5 0.41 6.07 $3.4 \times 10^{-4}$
lower N 8 – 11.3 0.49 15.77 $ 10^{-3}$
upper N 11 – 13.2 0.45 16.55 $4.1 \times 10^{-4}$
: Optimized filling factor $F$, grating depth $h$, and mean null depth $N$, for several considered mid-IR spectral windows.
\[results\]
\[t!\] ![RCWA multiparametric simulations for the L, M and N bands. **Left**: Mean null depth map, function of depth and filling factor, showing the optimal design values. For the N band, the upper right corner does not correspond to a possible geometrical solution, because of the merging of the sidewalls. **Right**: Computed coronagraphic performance of the AGPM, showing the benefits of etching an ARG on the backside of the component.[]{data-label="rcwa"}](rcwa.png "fig:"){height="19cm"}
L-band AGPM laboratory demonstration {#sec:labdemo}
------------------------------------
To validate the performance of the L-band AGPM manufactured at the Uppsala University, we carried out laboratory coronagraphic performance tests[@Delacroix13] on the YACADIRE optical bench at the Observatoire de Paris. Two series of frames were recorded. The first one was obtained with the AGPM placed at the optimal position (*coronagraphic frames*). The second one was obtained with the AGPM far from this position (*off-axis frames*), in order to measure a reference PSF without coronagraphic effect, but still propagating into the diamond substrate. Since the presence of the AGPM slightly modifies the PSF profile near the axis, we used the *raw null depth* to quantify the coronagraphic performance (rather than peak-to-peak attenuation). It is defined as the ratio between the integrated flux over a certain area around the centre of the final coronagraphic image, and the integrated flux of the same area in the off-axis image. While considering the full PSF would lead to a pessimistic result, because of all the background and high frequency artefacts, the sole central peak seems to be more appropriate. The raw null depth is then defined as: $$N_{\rm{AGPM}} = \frac{\int_{0}^{\rm{FWHM}}\int_{0}^{2\pi}\tilde{I}_{\rm{coro}}(r,\theta)r\,dr\,d\theta}{\int_{0}^{\rm{FWHM}}\int_{0}^{2\pi}\tilde{I}_{\rm{off}}(r,\theta)r\,dr\,d\theta}$$ where we use the FWHM of the PSF as a boundary for our integral and $\tilde{I}_{\rm{coro}}$ and $\tilde{I}_{\rm{off}}$ are the medians of the reduced coronagraphic and off-axis images. The measured raw null depth was $2\times10^{-3}$ over the band, which is only within a factor 2 of the expected performance $N\sim10^{-3}$ (Fig. \[rcwa\], right). Our L-band AGPMs were also recently tested and validated on sky, with two infrared cameras: VLT-NACO and LBT-LMIRCam[@Mawet13; @Absil13; @Defrere14].\
Finally, let us mention that no M-band AGPM has been etched so far. This will be done in the coming year and we plan to test these new AGPMs on a dedicated optical bench[@Jolivet14].\
CONCLUSIONS {#sec:conclusions}
===========
In this paper, we have presented the design and the expected performances of the AGPM in the mid-infrared regime, in particular the L, M and N bands. Our simulations and laboratory tests show that a null depth better than $10^{-3}$ can be achieved on most mid-infrared atmospheric windows, assuming a perfect input wavefront. This would translate into a contrast of about $5\times 10^{-6}$ at $2\lambda/D$ from the optical axis, which looks very promising for future ELT applications.
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (ERC Grant Agreement n.337569) and from the French Community of Belgium through an ARC grant for Concerted Research Actions.
[^1]: This null depth is equal to the peak-to-peak attenuation if the coronagraph is only limited by chromatism (in which case the coronagraph PSF is just a scaled-down version of the original PSF).
|
---
abstract: 'In this paper we study the meromorphic open-string vertex algebras and their modules over the two-dimensional Riemannian manifolds that are complete, connected, orientable, and of nonzero constant sectional curvature. For the MOSVA and the modules generated by eigenfunctions of the Laplace-Beltrami operator, we explicitly determine the basis vectors and discuss several properties.'
author:
- Fei Qi
title: 'Meromorphic open-string vertex algebras and modules over two-dimensional orientable space forms'
---
Introduction
============
Vertex algebras are algebraic structures formed by vertex operators satisfying commutativity and associativity. In mathematics, they arose naturally in the study of representations of infinite-dimensional Lie algebras and the Monster group (see [@B] and [@FLM]). In physics, they are used in the study of two-dimensional conformal field theory (see [@BPZ] and [@MS]). The commutativity and associativity allow us to view vertex algebras as analogues to the commutative associative algebras.
Meromorphic open-string vertex algebras (MOSVAs hereafter) are algebraic structures formed by meromorphic vertex operators satisfying only associativity without commutativity. They were introduced by Huang in 2012 (see [@H-MOSVA]), as special cases of the open-string vertex algebras introduced by Huang and Kong in 2003 (see [@HK-OSVA]) where all correlation functions are rational functions. Similar to the case of vertex algebras, MOSVAs can be viewed as analogues of associative algebras that are not necessarily commutative.
As a nontrivial example, Huang also introduced the MOSVA associated with a Riemannian manifold in 2012 (see [@H-MOSVA-Riemann]), using the parallel sections of the tensor algebra of the affinized tangent bundle. Given a complex-valued smooth function $f$ on an open subset $U$ of the manifold, Huang also constructed the module generated by $f$, and proved that association of $U$ with the sum of modules generated by all smooth functions over $U$ gives a presheaf of modules for the MOSVA.
Of particular interest are the modules generated by eigenfunctions of the Laplace-Beltrami operator. As such functions can be understood as quantum states in quantum mechanics, the modules they generate can be understood as the string-theoretic excitement to the quantum states. It is Huang’s idea that the modules for the MOSVA generated by the eigenfunctions and the yet-to-be-defined intertwining operators among these modules may lead to a mathematical construction of the quantum two-dimensional nonlinear $\sigma$-model.
In this paper, using the knowledge of Riemannian geometry, we explicitly determine a basis of the MOSVA over the two-dimensional orientable space forms, i.e., two-dimensional Riemannian manifold $M$ that are complete, connected, orientable, and of nonzero constant curvature $K$. The graded dimension of the MOSVA turns out to be related to the hypergeometric function ${}_2F_1$. We also explicitly determine the basis of the modules generated by any local eigenfunction over any open subset. The association of an open subset $U$ of $M$ with the sum of all modules generated by local eigenfunctions over $U$ turns out to be giving a sheaf.
The paper is organized as follows:
In Section 2, we briefly review the previously known results in [@H-MOSVA] and [@H-MOSVA-Riemann]. The axioms for the MOSVA and modules are slightly modified using the results in [@Q-Mod] in order to make it easier to verify.
In Section 3, we write down explicitly the parallel sections of related vector bundles. With this information from geometry, we give a basis of the MOSVA associated with $M$ constructed in [@H-MOSVA-Riemann] and compute its graded dimension. The exposition is improved in the following aspects: we work directly on the associative algebra of parallel sections of the related bundles. The MOSVA is constructed from inducing a certain module generated by the constant function. Replacing the constant function by generic complex-valued smooth functions, we obtain the modules.
In Section 4, using a lemma in higher covariant derivatives, we prove that all the zero-modes in the MOSVA act as a scalar on any local eigenfunction over any open subset. This allows us to give a basis of the module generated by such a function. We also study the graded dimension of the MOSVA and the module, which turns out to be related to the hypergeometric function. Finally, we note that the presheaf of modules generated by all eigenfunctions is indeed a sheaf.
**Acknowledgements.** The author would like to thank Yi-Zhi Huang for his long-term support and patient guidance. The author would also like to thank Robert Bryant for his guidance on holonomy groups over the tensor powers of tangent bundle and his correction on the statement of Lemma \[HolTensor\]. The author would also like to thank Igor Frenkel, Eric Schippers and Nolan Wallach for helpful discussion.
Previously known results
========================
Axioms of the meromorphic open-string vertex algebra and its left module
------------------------------------------------------------------------
\[DefMOSVA\]
A [*meromorphic open-string vertex algebra*]{} (hereafter MOSVA) is a ${{\mathbb Z}}$-graded vector space $V=\coprod_{n\in{{\mathbb Z}}} V_{(n)}$ (graded by [*weights*]{}) equipped with a [*vertex operator map*]{} $$\begin{aligned}
Y_V: V\otimes V &\to & V[[x,x^{-1}]]\\
u\otimes v &\mapsto& Y_V(u,x)v
\end{aligned}$$ and a [*vacuum*]{} ${\mathbf{1}}\in V$, satisfying the following axioms:
1. Axioms for the grading:
1. [*Lower bound condition*]{}: When $n$ is sufficiently negative, $V_{(n)}=0$.
2. [*${\mathbf{d}}$-commutator formula*]{}: Let ${\mathbf{d}}_{V}: V\to V$ be defined by ${\mathbf{d}}_{V}v=nv$ for $v\in V_{(n)}$. Then for every $v\in V$ $$[{\mathbf{d}}_{V}, Y_{V}(v, x)]=x\frac{d}{dx}Y_{V}(v, x)+Y_{V}({\mathbf{d}}_{V}v, x).$$
2. Axioms for the vacuum:
1. [*Identity property*]{}: Let $1_{V}$ be the identity operator on $V$. Then $Y_{V}(\mathbf{1}, x)=1_{V}$.
2. [*Creation property*]{}: For $u\in V$, $Y_{V}(u, x)\mathbf{1}\in V[[x]]$ and $\lim_{x\to 0}Y_{V}(u, x)\mathbf{1}=u$.
3. [*$D$-derivative property and $D$-commutator formula*]{}: Let $D_V: V\to V$ be the operator given by $$D_{V}v=\lim_{x\to 0}\frac{d}{dx}Y_{V}(v, x){\mathbf{1}}$$ for $v\in V$. Then for $v\in V$, $$\frac{d}{dx}Y_{V}(v, x)=Y_{V}(D_{V}v, x)=[D_{V}, Y_{V}(v, x)].$$
4. [*Weak associativity with pole-order condition*]{}: For every $u_1, v\in V$, there exists $p\in \mathbb{N}$ such that for every $u_2\in V$, $$(x_0+x_2)^p Y_V(u_1, x_0+x_2)Y_V(u_2, x_2)v = (x_0+x_2)^p Y_V(Y_V(u_1, x_0)u_2, x_2)v.$$
This definition is slightly more special than the definition given by Huang in [@H-MOSVA], where $p$ is not necessarily depending only on $u_1$ and $v$. This dependence is called pole-order condition and can be used to simplify the verification of axioms. Please see [@Q-Mod] for a detailed discussion.
\[DefMOSVA-L\] Let $V$ be a meromorphic open-string vertex algebra. A *left $V$-module* is a ${{\mathbb C}}$-graded vector space $W=\coprod_{m\in {{\mathbb C}}}W_{[m]}$ (graded by *weights*), equipped with a *vertex operator map* $$\begin{aligned}
Y_W^L: V\otimes W & \to & W[[x, x^{-1}]]\\
u\otimes w & \mapsto & Y_W^L(u, x)w,\end{aligned}$$ an operator ${\mathbf{d}}_{W}$ of weight $0$ and an operator $D_{W}$ of weight $1$, satisfying the following axioms:
1. Axioms for the grading:
1. *Lower bound condition*: When $\text{Re}{(m)}$ is sufficiently negative, $W_{[m]}=0$.
2. *$\mathbf{d}$-grading condition*: for every $w\in W_{[m]}$, ${\mathbf{d}}_W w = m w$.
3. *$\mathbf{d}$-commutator formula*: For $u\in V$, $$[\mathbf{d}_{W}, Y_W^L(u,x)]= Y_W^L(\mathbf{d}_{V}u,x)+x\frac{d}{dx}Y_W^L(u,x).$$
2. The *identity property*: $Y_W^L({\mathbf{1}},x)=1_{W}$.
3. The *$D$-derivative property* and the *$D$-commutator formula*: For $u\in V$, $$\begin{aligned}
\frac{d}{dx}Y_W^L(u, x)
&=&Y_W^L(D_{V}u, x) \\
&=&[D_{W}, Y_W^L(u, x)].\end{aligned}$$
4. [*Weak associativity with pole-order condition*]{}: For every $v_1\in V, w\in W$, there exists $p\in \mathbb{N}$ such that for every $v_2\in V$, $$(x_0+x_2)^p Y_W^L(v_1, x_0+x_2)Y_W^L(v_2, x_2)w = (x_0+x_2)^p Y_W^L(Y_V(v_1, x_0)v_2, x_2)w.$$
Example: Noncommutative Heisenberg
----------------------------------
The first nontrivial example of MOSVA is constructed by Huang in [@H-MOSVA]. We should recall the construction here.
Let $\mathfrak{h}$ be a finite-dimensional Euclidean space over ${{\mathbb R}}$. We define a vector space $$\hat{\mathfrak{h}} = \mathfrak{h} \otimes_{{\mathbb R}}{{\mathbb C}}[t, t^{-1}] \oplus {{\mathbb C}}\mathbf{k},$$ which is the ambient vector space of the Heisenberg Lie algebra. Note that $$\hat{\mathfrak{h}} = \hat{\mathfrak{h}}_- \oplus \hat{\mathfrak{h}}_0 \oplus \hat{\mathfrak{h}}_+,$$ where $$\begin{aligned}
\hat{\mathfrak{h}}_+ &= \mathfrak{h}\otimes_{{\mathbb R}}t{{\mathbb C}}[t],\\
\hat{\mathfrak{h}}_0 &= \mathfrak{h}\otimes_{{\mathbb R}}{{\mathbb C}}\oplus {{\mathbb C}}\mathbf{k},\\
\hat{\mathfrak{h}}_+ &= \mathfrak{h}\otimes_{{\mathbb R}}t^{-1}{{\mathbb C}}[t^{-1}].\end{aligned}$$
Let $N(\hat{\mathfrak{h}})$ be the quotient of the tensor algebra $T(\hat{h})$ of $\hat{h}$ modulo the two-sided ideal $I$ generated by $$\begin{aligned}
&(a\otimes t^{m})\otimes (b\otimes t^{n})
- (b\otimes t^{n})\otimes (a\otimes t^{m})
-m(a, b)\delta_{m+n, 0}\mathbf{k},& \nonumber\\
&(a\otimes t^{k})\otimes (b\otimes t^{0})
-(b\otimes t^{0})\otimes (a\otimes t^{k}),\nonumber&\\
&(a\otimes t^{k})\otimes \mathbf{k}-\mathbf{k}\otimes (a\otimes t^{k})& \label{IdealT(h)}\end{aligned}$$ for $a, b\in \mathfrak{h}$, $m\in {{\mathbb Z}}_{+}$, $n\in -{{\mathbb Z}}_{+}$, $k\in {{\mathbb Z}}$. Note that in the quotient, there are no relations between $X \otimes t^m $ and $Y\otimes t^n$ for $m, n\in Z_+$ and for $m, n\in {{\mathbb Z}}_-$. This is the main difference to the usual construction of Bosonic Fock space, where $$(a\otimes t^m)\otimes (b\otimes t^n)-(b\otimes t^n)\otimes (a\otimes t^m)$$ is also included in the generators of $I$, for each $a, b\in \mathfrak{h}, m, n\in {{\mathbb Z}}_{\pm}$. Nevertheless, the PBW structure still holds for this quotient: $N(\hat{\mathfrak{h}}) \simeq T(\hat{\mathfrak{h}}_-) \otimes T(\hat{\mathfrak{h}}_+) \otimes T(\mathfrak{h})\otimes T({{\mathbb C}}\mathbf{k})$ as vector spaces (see [@H-MOSVA], Proposition 3.1)
Let ${{\mathbb C}}= {{\mathbb C}}\mathbf{1}$ be a one-dimensional vector space on which $\mathfrak{h}$ acts by 0. Define the action of $\mathbf{k}$ by 1 and $\hat{\mathfrak{h}}_+$ by 0. One can prove that the induced module $N(\hat{\mathfrak{h}}) \otimes_{N(\hat{\mathfrak{h}}_+\oplus\hat{\mathfrak{h}}_0)} {{\mathbb C}}$ is isomorphic to $T(\hat{\mathfrak{h}}_-)$ as a vector space. We regard $T(\hat{\mathfrak{h}}_-)$ now as an $N(\hat{\mathfrak{h}})$-module and denote the action of $h\otimes t^j$ by $h(j)$. Then $T(\hat{\mathfrak{h}}_-)$ is spanned by $h_1(-m_1) \cdots h_k(-m_k)\mathbf{1}$ for $k \in \mathbb{N}, h_1, ..., h_k \in \mathfrak{h}, m_1, ..., m_k \in {{\mathbb Z}}_+$.
Huang proved the following theorem in [@H-MOSVA].
\[NoncomHeis\] The left $N(\hat{\mathfrak{h}})$-module $T(\hat{\mathfrak{h}}_-)$ forms a grading-restricted MOSVA with the following vertex operator action: $$\begin{aligned}
& Y(h_1(-m_1) \cdots h_k(-m_k)1, x) \\
& \qquad= {\mbox{\scriptsize ${\circ\atop\circ}$}}\frac 1 {(m_1-1)!} \frac {d^{m_1-1}}{dx^{m_1-1}}h_1(x) \cdots \frac 1 {(m_k-1)!} \frac {d^{m_k-1}}{dx^{m_k-1}}h_k(x) {\mbox{\scriptsize ${\circ\atop\circ}$}}, \end{aligned}$$ where $h_i(x) = \sum_{n\in {{\mathbb Z}}} h_i(n)x^{-n-1}$. The normal ordering is defined in the usual sense: $${\mbox{\scriptsize ${\circ\atop\circ}$}}h_1(m_1)\cdots h_k(m_k){\mbox{\scriptsize ${\circ\atop\circ}$}}= h_{\sigma(1)}(m_{\sigma(1)}) \cdots h_{\sigma(k)}(m_{\sigma(k)}),$$ where $\sigma\in S_k$ is the unique permutation such that $$\begin{aligned}
\sigma(1) < \cdots < \sigma(\alpha), \sigma(\alpha+1)< \cdots < \sigma(\beta), \sigma(\beta)<\cdots < \sigma(k),\quad\, \nonumber\\
m_{\sigma(1)}, ..., m_{\sigma(\alpha)}< 0, m_{\sigma(\alpha+1)}, ..., m_{\sigma(\beta)} >0, m_{\sigma(\beta+1)}, ..., m_{\sigma(k)}= 0. \label{Shuffle}\end{aligned}$$
MOSVA of a Riemannian manifold
------------------------------
In [@H-MOSVA-Riemann], Huang further constructed a MOSVA for any Riemannian manifold $M$, using the parallel sections of a certain bundle. We recall the construction here.
Let $M$ be a Riemannian manifold. Let $p\in M$. We consider the affinization of tangent bundle $$\widehat{TM} = TM \otimes_{{\mathbb R}}(M \times {{\mathbb C}}[t, t^{-1}]) \oplus (M\times {{\mathbb C}}\mathbf{k})$$ where $M\times {{\mathbb C}}[t,t^{-1}]$ and $M\times {{\mathbb C}}\mathbf{k}$ are trivial bundles over $M$. The fiber of this bundle at $p$ is nothing but $$\widehat{T_pM} = T_pM. \otimes_{{\mathbb R}}{{\mathbb C}}[t, t^{-1}]\oplus {{\mathbb C}}\mathbf{k}$$ Analogous to the constructions in the previous subsection, we have $$\begin{aligned}
\widehat{TM} & = \widehat{TM}_+ \oplus \widehat{TM}_0 \oplus \widehat{TM}_-,\end{aligned}$$ with $$\begin{aligned}
\widehat{TM}_\pm &= TM\otimes_{{\mathbb R}}(M \times t^{\pm 1}{{\mathbb C}}[t^{\pm 1}]), \\
\widehat{TM}_0 &= TM \otimes_{{\mathbb R}}(M \times {{\mathbb C}}t^0) \oplus TM \otimes_{{\mathbb R}}(M \times {{\mathbb C}}\mathbf{k}).\end{aligned}$$ Now we look into the tensor algebra bundle $T(\widehat{TM})$. We similarly construct a bundle $N(\widehat{TM})$, whose fiber at each point is obtained by taking the quotient of $T(T_pM)$ versus the two-sided ideal generated by elements in (\[IdealT(h)\]), for $a, b \in T_pM$. Likewise, $N(\widehat{TM})$ is isomorphic to $T(\widehat{TM}_-) \otimes T(\widehat{TM}_+)\otimes T(\widehat{TM})\otimes T({{\mathbb C}}\mathbf{k})$.
We now consider the space $\Pi(T(\widehat{TM}_-))$ consisting of parallel sections of the tensor algebra bundle $T(\widehat{TM}_-)$ of $\widehat{TM}_-$. It is well known that $\Pi(T(\widehat{TM}_-))$, as a vector space, is isomorphic to subspace of fixed points $T(\widehat{T_pM}_-)^{{\text{Hol}}(T(\widehat{TM}_-)}$, where Hol means the holonomy group of the bundle. Recall that Theorem \[NoncomHeis\] endows the space $T(\widehat{T_pM}_-)$ with a MOSVA structure. In [@H-MOSVA-Riemann], Huang proved that the elements of the holonomy group could be realized as automorphisms MOSVA, and on the fixed point subspace of automorphism there is a MOSVA structure. Thus Huang proved the following theorem:
\[MOSVA-Riemann\] On the space $\Pi(T(\widehat{TM}_-))$ of parallel sections there is a MOSVA structure, which is a subalgebra of the MOSVA of $T(\widehat{T_pM}_-)$ in Theorem \[NoncomHeis\].
Moreover, by an analogous argument to the Segal-Sugawara construction, Huang also showed that the Laplace-Beltrami operator on the manifold $M$ is realized as a component of some vertex operator.
Modules generated by smooth functions
-------------------------------------
In [@H-MOSVA-Riemann], Huang considered that the action of the parallel sections $\Pi(T(TM^{{\mathbb C}}))$ on the space of smooth functions via the covariant derivatives. More precisely, let $U$ be an open subset of $M$; let $f$ be a smooth function on $U$. Define a parallel section $X \in \Pi((TM^{{\mathbb C}})^{\otimes k})$ acts on $f$ by $$\psi_U(X)f:=(\sqrt{-1})^k(\nabla^k f)(X).$$ Huang proved (Theorem 4.1, [@H-MOSVA-Riemann]) that the action respects the associative algebra structure defined by $\otimes$, namely, for $X, Y\in \Pi(T(TM)^{{\mathbb C}})$, $$\psi_U(X\otimes Y) = \psi_U(X)\psi_U(Y).$$ Thus the space of complex-valued smooth functions can be viewed as a module for the associative algebra $\Pi(T(TM)^{{\mathbb C}})$.
Let $p\in U$. Identify $\Pi(T(TM)^{{\mathbb C}})$ with $T(T_pM^{{\mathbb C}})^{{\text{Hol}}(T(T_pM^{{\mathbb C}}))}$, which can be viewed as a subalgebra of $T(T_pM^{{\mathbb C}})$. Thus the module $C^\infty(U)$ can be induced, defining $$C_p(U) = T(T_pM^{{\mathbb C}})\otimes _{T(T_pM^{{{\mathbb C}}})^{{\text{Hol}}(T(T_pM)^{{{\mathbb C}}}}}C^\infty(U).$$ By Theorem 6.5 of [@H-MOSVA], on the vector space $$T(\widehat{T_pM}_-)\otimes C_p(U)$$ there is a natural module structure for the MOSVA $T(\widehat{T_pM}_-)$. In particular, it is a module for the MOSVA $T(\widehat{T_pM}_-)^{{\text{Hol}}(\widehat{T_pM}_-)} = \Pi(T(\widehat{TM}_-))$. We can thus consider the $\Pi(T(\widehat{TM}_-))$-submodule generated by elements $$1\otimes (1\otimes f)$$ for some $f\in C^\infty(U)$. Since the Laplace-Beltrami operator is a component of some vertex operator, it would be natural to consider the submodule generated by its eigenfunction. As the Laplace-Beltrami operator plays the role of energy operator in quantum mechanics, the submodule can then be interpreted as string-theoretical excitement of the quantum states.
Basis of the MOSVA and modules
==============================
Let $M$ be a two-dimensional Riemannian manifold with constant sectional curvature $K$. For convenience, we assume $M$ is orientable, connected and complete. We will also focus on the case $K\neq 0$.
In this section, we will identify a vector space $V_U(l, f)$ for an connected open subset $U$ of $M$, a complex number $l$ and a smooth function $f: U \to {{\mathbb C}}$. In case $f={\mathbf{1}}$, the constant function sending every point of $U$ to 1, $V_U(l, {\mathbf{1}})$ is a MOSVA. All $V_U(l, f)$ forms a module for this MOSVA.
Holonomy of the tensor powers of the complexified tangent bundle.
------------------------------------------------------------------
Recall that the holonomy group of a bundle $E$ based at a point $p\in M$ is the subgroup generated by all the parallel translations along piecewise smooth contractible loops based on $p$. We will simply denote the holonomy group by ${\text{Hol}}(E)$ since holonomy groups based at different points are isomorphic.
If $K\neq 0$, then the holonomy group ${\text{Hol}}(TM)$ of the tangent bundle $TM$ is $SO(2, {{\mathbb R}})$.
Since $M$ is orientable, we know that ${\text{Hol}}(TM) \subset SO(2,{{\mathbb R}})$ (see [@P]). Fix $p, q, r\in M$. Let $\gamma_1, \gamma_2, \gamma_3$ be geodesics connecting $pq, qr$ and $rp$. Let $\alpha_p, \alpha_q, \alpha_r$ be the angles of geodesic triangle $pqr$. Let $v\in T_pM$ be a unit vector. One sees easily that that composition of parallel transport along the concatenation of $\gamma_1, \gamma_2$ and $\gamma_3$ ends up with a unit vector $w\in T_pM$ that is obtained from rotating $v$ by the angle $3\pi - (\alpha_p+\alpha_q+\alpha_r)$, as shown in the following two figures.
By Gauss-Bonnet theorem (geodesic triangle version, see [@DoCarmo]), $$\alpha_p+\alpha_q+\alpha_r = \pi + K\cdot \text{Area}(pqr).$$ Now let $q, r$ vary near $p$, so that the area of the geodesic triangle varies continuously within some interval $[0, s]$. Then we see that rotations by angles between $2\pi - Ks$ and $2\pi$ are all included in the holonomy group. These rotations generate all $SO(2, {{\mathbb R}})$.
We focus on the complexified tangent bundle ${{\mathbb C}}\otimes_{{\mathbb R}}TM $, where ${{\mathbb C}}$ is regarded as a trivial bundle over $M$.
From now on, we shall denote the complexified tangent bundle ${{\mathbb C}}\otimes_{{\mathbb R}}TM$ by $E$. We will also omit the $\otimes_{{\mathbb R}}$ symbol when writing the smooth sections in $\Gamma(E)$. All $\otimes$ symbol will mean $\otimes_{{\mathbb C}}$ by default unless otherwise stated.
The connection on $E$ is related to that on $TM$ by $$\nabla (X+iY) = \nabla(X) + \sqrt{-1}\nabla(Y), X, Y \in \Gamma(TM).$$
${\text{Hol}}(E) = {\text{Hol}}(TM) = SO(2, {{\mathbb R}})$.
This follows from the observation that the bundle $E$ is essentially the direct sum $TM\oplus \sqrt{-1}\cdot TM$.
We will also consider the tensor bundle $E^{\otimes k}$ for each $k\in {{\mathbb Z}}_+$.
There is a natural surjective homomorphism ${\text{Hol}}(E) \to {\text{Hol}}(E^{\otimes k})$ of holonomy groups, where $g\in {\text{Hol}}(E)$ acts on each fiber $E^{\otimes k}_p$ by $$g(v_1\otimes \cdots \otimes v_n) = gv_1\otimes \cdots \otimes gv_n.$$ for any $v_1, ..., v_n\in E_p$.
For any piecewise smooth path $\gamma: [0, 1]\to M$ with $\gamma(0)=p$, let $P_{\gamma(t)}: E_p \to E_{\gamma(1)}$ be the parallel transport along $\gamma$ on the bundle $E$; let $P_{\gamma(t)}^k: E_p^{\otimes k} \to E_{\gamma(1)}^{\otimes k}$ be the parallel transport along $\gamma$ with respect to the bundle $E^{\otimes k}$. Then from the definition of the connection on $E^{\otimes k}$: $$\nabla(X_1 \otimes \cdots \otimes X_n) = \sum_{i=1}^n X_1 \otimes \cdots \otimes \nabla(X_i) \otimes \cdots \otimes X_n,$$ it follows that $$P_{\gamma(t)}^k(v_1\otimes \cdots \otimes v_k) = P_{\gamma(t)}v_1 \otimes \cdots \otimes P_{\gamma(t)}v_k.$$ In case $\gamma(t)$ is a loop based at $p$, this essentially realizes every element of $h\in {\text{Hol}}(E^{\otimes k})$ as $g^{\otimes k}$ for $g\in {\text{Hol}}(E)$. So the map $g\mapsto g^{\otimes k}$ gives a natural surjective homomorphism ${\text{Hol}}(E) \to {\text{Hol}}(E^{\otimes k})$.
\[HolTensor\] The holonomy group of $E^{\otimes k}$ is determined by $${\text{Hol}}(E^{\otimes k}) = \left\{\begin{aligned}
& SO(2, {{\mathbb R}}) & &\text{if }k\text{ is odd}\\
& SO(2, {{\mathbb R}})/\{\pm 1\} & & \text{if }k\text{ is even}
\end{aligned}\right.$$
We analyze the kernel of the homomorphism $SO(2, {{\mathbb R}}) \to {\text{Hol}}(E^{\otimes k})$. Fix $p\in M$. For the matrix $$M(\alpha)=\begin{bmatrix} \cos\alpha & -\sin\alpha \\
\sin\alpha & \cos\alpha
\end{bmatrix},$$ let $v_{\pm1}\in E_p^{{\mathbb C}}$ be an eigenvector of $M(\alpha)$ with eigenvalue $e^{ \pm i\alpha}$. The conclusion then follows from $$M(\alpha)(v_{i_1}\otimes \cdots \otimes v_{i_k}) = e^{i(m-n)\alpha}(v_{i_1}\otimes \cdots \otimes v_{i_k})$$ where $m=\#\{j: i_j = 1\}, n = \#\{j: i_j = -1\}$. In greater detail, if $k$ is odd then there is no way to make $e^{i(m-n)\alpha}=1$ for every choice of $i_1, ..., i_k$ unless $\alpha =0$; if $k$ is even, then there is no way to make $e^{i(m-n)\alpha}=1$ for every choice of $i_1, ..., i_k$ unless $\alpha = 0$ or $\alpha = \pi$.
Parallel sections of the tensor powers of the complexified tangent bundle
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$\Pi(E^{\otimes k}) = 0$ if $k$ is odd.
It follows directly from $-1\in{\text{Hol}}(E^{\otimes k})$.
To describe the parallel sections of the even tensor powers of $E$, we introduce the following notations:
\[sections\] Fix $p\in M$, let $\{e_1, e_2\}$ be an orthonormal basis of $T_p M$. Then for some neighborhood $V$ of $p$, we define local sections $X_1, X_2: V \to TV$ by parallel transporting $e_1, e_2$ to every $q\in V$. Finally, we introduce the following local sections of $E$: $$h_+ = X_1 - i X_2, h_- = X_1 + iX_2$$
Let $M$ be the unit sphere $M = \{(x, y, z)\in {{\mathbb R}}^3: x^2 + y^2 + z^2 = 1\}$. For $p = (1, 0, 0)$, we take the polar coordinate $$x = \cos \theta \sin\phi , y = \sin \theta \sin \phi, z = \cos \phi,$$ where $\theta\in [0, 2\pi), \phi\in (0, \pi)$. Then on the neighborhood $V = M \setminus \{(0, 0, \pm 1)\}$ of $p$, the metric is of the form $$ds^2 = d\phi^2 + \sin^2\phi d\theta^2.$$ We then take $$X_1 = \partial_\phi, X_2 = \frac 1 {\sin \phi}\partial_\theta$$ In this case, $$h_+ = \partial_\phi - \frac {\sqrt{-1}} {\sin \phi}\partial_\theta, h_- = \partial_\phi + \frac {\sqrt{-1}} {\sin \phi}\partial_\theta.$$
Let $M$ be a complete hyperbolic surface of genus $g$. From the discussion in [@JS], $M$ is realized as the orbit space $H/ \Gamma$, where $H$ is the Poincaré disk $$H = \{(x,y)\in {{\mathbb R}}^2: x^2+y^2 < 1\}$$ and $\Gamma$ is a Fuchsian group (with no fixed points on $H$). For $p = (0,0)$, let $V$ be the interior of the fundamental region of $\Gamma$ containing $p$. In other words, $V$ is the interior of a hyperbolic polygon with $2g$ sides. Viewed as a coordinate chart of $M$ near $p$, $V$ can be endowed with a metric of the form $$ds^2 = \frac {4(dx^2 + dy^2)}{(1-x^2-y^2)^2}$$ We then take $$X_1 = \frac {1-x^2-y^2} 2\partial_x, X_2 = \frac {1-x^2-y^2} 2\partial_y$$ In this case, $$h_+ = \frac {1-x^2-y^2} 2(\partial_x - \sqrt{-1}\partial_y), h_-= \frac {1-x^2-y^2} 2(\partial_x + \sqrt{-1}\partial_y)$$
\[Parallel-Tensors\] Each element in the following set $$\left\{h_{i_1} \otimes \cdots \otimes h_{i_k} \left|\begin{aligned} & i_1, ..., i_k \in \{+, -\}, \\
&\#\{j:i_j = +\} =
\#\{j:i_j = -\}\end{aligned}\right. \right\}$$ extend to global sections $M \to E^{\otimes k}$ and is parallel. The set form a basis of $\Pi(E^{\otimes k})$.
Fix $p\in M$. Let $M(\alpha)\in SO(2, {{\mathbb R}})$ be the matrix as in Lemma \[HolTensor\], which acts on $e_1, e_2\in T_pM$ by $$\begin{aligned}
M(\alpha)e_1 &= \cos(\alpha) e_1 + \sin(\alpha) e_2\\
M(\alpha)e_2 &= -\sin(\alpha) e_1 + \cos(\alpha) e_2.\end{aligned}$$ With this action, $M(\alpha)$ acts on $h_+|_p, h_-|_p$ in the fiber $E_p$ by $$M(\alpha) h_+|_p = e^{i\alpha} h_+|_p, M(\alpha)h_-|_p = e^{-i\alpha} h_-|_p.$$ In other words, $(h_\pm)_p$ is an eigenvector of $M(\alpha)$ with eigenvalue $e^{\pm i\alpha}$.
To determine the parallel sections $\Pi(E^{\otimes k})$, it suffices to study the fixed point subspace of $(E_p)^{\otimes k}$ under the action of $SO(2, {{\mathbb R}})/\{\pm 1\}$. By the same computation shown in Lemma \[HolTensor\], we know that $$M(\alpha)(h_{i_1}|_p\otimes \cdots \otimes h_{i_k}|_p) = h_{i_1}|_p \otimes \cdots \otimes h_{i_k}|_p$$ if and only if the number of $+$ appearing in $i_1, ..., i_k$ coincides with the number of $-$. Then by simple linear algebra, the set of vectors $$\{h_{i_1}|_p\otimes \cdots \otimes h_{i_k}|_p: \#\{j: i_j=+\} = \#\{j: i_j = -\}\}$$ form a basis of the fixed point subspace $((E_p^{{\mathbb C}})^{\otimes k})^{{\text{Hol}}(E^{\otimes k})}$.
Since $M$ is complete, every point on $M$ are connected to $p$ by some geodesic path. Every element in this fixed point subspace $E_p^{{\text{Hol}}(E^{\otimes k})}$ thus defines a parallel section globally on $M$ via parallel transport along geodesic paths. Moreover, the parallel section defined by $h_{i_1}|_p\otimes \cdots \otimes h_{i_k}|_p$ coincides with the local section $h_{i_1}\otimes \cdots \otimes h_{i_k}$ on $V$, since the latter is also defined via parallel transports. Thus the latter extends to global sections.
Parallel sections of the tensor algebra of the affinization
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Consider the following affinization of $E$: $$\begin{aligned}
\widehat{E}&= TM\otimes_{{\mathbb R}}(M\times {{\mathbb C}}[t, t^{-1}])\oplus (M\times {{\mathbb C}}k) \\
&= E\otimes(M\times {{\mathbb C}}[t, t^{-1}])\oplus (M\times {{\mathbb C}}k)\\
&= \widehat{E}_- \oplus \widehat{E}_0 \oplus \widehat{E}_+,\end{aligned}$$ where $$\begin{aligned}
\widehat{E}_\pm &= E \otimes (M \times t^{\pm 1}{{\mathbb C}}[t^{\pm 1}])\\
&= \bigoplus_{k=1}^\infty E \otimes (M \times {{\mathbb C}}t^{\pm k}),\\
\widehat{E}_0 &= E\otimes (M\times {{\mathbb C}}t^0) \oplus (M\times {{\mathbb C}}\mathbf{k}).\end{aligned}$$ We will need to use the tensor algebra bundle $$T(\hat{E}) = (M \times {{\mathbb C}}) \oplus \hat{E} \oplus \hat{E}^{\otimes 2} \oplus \cdots.$$ For each $p\in M$, the fiber $T(\widehat{E})_p$ is simply the tensor algebra of the vector space $E_p$: $$T(\widehat{E})_p = {{\mathbb C}}\oplus \widehat{E}_p \oplus (\widehat{E}_p)^{\otimes 2}\oplus \cdots.$$ Also note that for two smooth sections $X, Y \in \Gamma(T(\widehat{E}))$ , $$\label{SectionMult}
(X\otimes Y)_p = X_p \otimes Y_p, p\in M$$ defines an associative algebra structure on $\Gamma(T(\widehat{E}))$.
We construct the bundles $T(\hat{E}_\pm), T(E) $ and $T(M\times {{\mathbb C}}\mathbf{k})$ similarly. We will rewrite these bundles in the following way: $$\begin{aligned}
T(\widehat{E}_\pm) &= \bigoplus_{n=0}^\infty \bigoplus_{k=1}^n\bigoplus_{\substack{m_1+\cdots+ m_k=n\\ m_1, ..., m_k \in {{\mathbb Z}}_+}} (E \otimes_{{{\mathbb C}}} (M \times {{\mathbb C}}t^{\pm m_1})) \otimes_{{\mathbb C}}\cdots \otimes_{{\mathbb C}}(E \otimes_{{\mathbb C}}({M \times {{\mathbb C}}t^{\pm m_k}}))\\
T(E) &= \bigoplus_{n=0}^\infty (E\otimes {{\mathbb C}}t^0)^{\otimes n} = \bigoplus_{n=0}^\infty E^{\otimes n}. \\
T(M \times {{\mathbb C}}\mathbf{k}) &= \bigoplus_{n=0}^\infty (M\times {{\mathbb C}}\mathbf{k})^{\otimes n} = \bigoplus_{n=0}^\infty (M\times {{\mathbb C}}\mathbf{k}^{n}).\end{aligned}$$
We will need to use the parallel sections of all these bundles. Note that essentially $T(\widehat{E}_\pm)$ and $T(E)$ are direct sums of $E^{\otimes n}$. The following lemma will then apply to determine the parallel section of these bundles.
Let $B_1, B_2, ...$ be a sequence of vector bundles on $M$. Let $B = \bigoplus_{i=1}^\infty B_i$. Then the parallel sections of $B$ is the direct sum of the parallel sections of $B_i$, i.e., $\Pi(B) = \bigoplus_{i=1}^\infty \Pi(B_i)$.
Obviously $\bigoplus_{i=1}^\infty \Pi(B_i) \subseteq \Pi(B)$. We show the inverse inclusion here. Let $X$ be a parallel section of $B$. Fix any $p\in M$ and piecewise smooth loop $\gamma$ based on $p$. Consider $X_p = \sum\limits_{i \text{ finite}} (X_i)_p$, which is a finite sum of components in $(B_1)_p, (B_2)_p$, ... The parallel transport $T_{\gamma(1)}^{B}$ applied on $X$ amounts to the sum of the action of $T_{\gamma(1)}^{B_i}$ on $X_i$. Since it is a direct sum, we necessarily have $T_{\gamma(1)}^{B_i}(X_i)_p = (X_i)_p$. Since $\gamma$ is arbitrarily chosen, we see that $(X_i)_p \in (B_i)_p^{\text{Hol}(B)}$. That is to say, $X_p$ is a finite sum of elements in $(B_i)_p^{\text{Hol}(B)}$. Thus $X$ is a finite sum of parallel sections in $B_i$. So we proved that $\Pi(B)\subseteq \bigoplus_{i=1}^\infty \Pi(B_i). $
The lemma thus determines the parallel sections of these bundles:
\[ParallelSec\] $$\begin{aligned}
\Pi(T(\widehat{E}_\pm)) &=\bigoplus_{n=0}^\infty \bigoplus_{k=1}^{n}\bigoplus_{\substack{m_1+\cdots+ m_k=n\\ m_1, ..., m_k \in {{\mathbb Z}}_+}} \Pi((E \otimes_{{{\mathbb R}}} (M \times {{\mathbb C}}t^{\pm m_1})) \otimes_{{\mathbb C}}\cdots \otimes_{{\mathbb C}}(E \otimes_{{\mathbb R}}({M \times {{\mathbb C}}t^{\pm m_k}}))\\
&=\bigoplus_{n=0}^\infty \bigoplus_{k=1}^{n}\bigoplus_{\substack{m_1+\cdots+ m_k=n\\ m_1, ..., m_k \in {{\mathbb Z}}_+}} \text{span} \left\{ (h_{i_1} \otimes t^{\pm m_1}) \otimes \cdots \otimes (h_{i_k} \otimes t^{\pm m_k}): \begin{aligned} & i_1, ..., i_k \in \{+, -\},\\
& \#\{j: i_j=+\}=\#\{j: i_j=-\} \}\end{aligned}\right\}\\
\Pi(T(E)) &= \bigoplus_{n=0}^\infty \Pi((E\otimes (M\times {{\mathbb C}}t^0))^{\otimes n}) \\
&= \bigoplus_{n=0}^\infty \text{span}\left\{(h_{i_1}\otimes t^0 )\otimes \cdots \otimes (h_{i_n}\otimes t^0): \begin{aligned} & i_1, ..., i_n \in \{+, -\}, \\
& \#\{j: i_j=+\}=\#\{j: i_j=-\} \}\end{aligned}\right\}\\
\Pi(M \times {{\mathbb C}}\mathbf{k}) & = \bigoplus_{n=0}^\infty \Pi(M\times {{\mathbb C}}\mathbf{k}^n) = \bigoplus_{n=0}^\infty {{\mathbb C}}\mathbf{k}^n \end{aligned}$$
The structure of $\Pi(T(\widehat{E}_\pm))$ and $\Pi(T(E))$ follows directly from the lemma. The structure of $\Pi(M \times {{\mathbb C}}\mathbf{k})$ follows from the fact that any smooth function $f$ satisfying $\nabla_{\dot\gamma}f = 0$ for every path $\gamma$ is constant.
Note that for $X\in \Gamma(E)$, $X\otimes t^m$ is naturally a section for the bundle $E\otimes {{\mathbb C}}t^m$. We will consider the associative algebra $\Gamma(T(\widehat{E}))/I$, where $I$ is the two-sided ideal generated by $$\begin{aligned}
&(X\otimes t^{m})\otimes (Y\otimes t^{n})
- (Y\otimes t^{n})\otimes (X\otimes t^{m})
-m(X, Y)\delta_{m+n, 0}\mathbf{k},& \nonumber\\
&(X\otimes t^{k})\otimes (Y\otimes t^{0})
-(Y\otimes t^{0})\otimes (X\otimes t^{k}),\nonumber&\\
&(X\otimes t^{k})\otimes \mathbf{k}-\mathbf{k}\otimes (X\otimes t^{k})& \label{IdealN(E)}\end{aligned}$$ for $X, Y\in \Gamma(E), m\in {{\mathbb Z}}_+, n \in -{{\mathbb Z}}_+, k\in {{\mathbb Z}}$. Note that the inner product $(X,Y)$ defines a smooth complex-valued function on $M$. When $X, Y\in \{h_+, h_-\}$, the function defined by $(X, Y)$ is indeed a constant function.
The following spaces $$\Pi(T(\widehat{E}_-)) \otimes \Pi(T(\widehat{E}_+)) \otimes \Pi(T(E)) \otimes \Pi(T(M \times {{\mathbb C}}\mathbf{k})$$ and $$\Pi(T(\widehat{E}_+))\otimes \Pi(T(E)) \otimes \Pi(T({{\mathbb C}}\mathbf{k}))$$ of sections embed as a subalgebra of $\Gamma(T(\widehat{E}))/I$.
Let $A$ denote subalgebra generated by $\Pi(T(\widehat{E}_-)) \otimes \Pi(T(\widehat{E}_+)) \otimes \Pi(T(E)) \otimes \Pi(T(M \times {{\mathbb C}}\mathbf{k})$ in $\Gamma(T(\widehat{E}))$. Consider the two-sided ideal $I$ of $\Gamma(T(\widehat{E}))$ generated by elements in (\[IdealN(E)\]). Since for every $X, Y\in \{h_+, h_-\}$, $(X, Y)$ is a constant function, and no sections in $E$ other than $h_+, h_-$ are involved in space $A$, we see that $A \cap I \subseteq A$. Thus $A/I$ embeds into $\Gamma(T(\widehat{E}))/I$. It remains to show that $A/I$ as a vector space is $\Pi(T(\widehat{E}_-)) \otimes \Pi(T(\widehat{E}_+)) \otimes \Pi(T(E)) \otimes \Pi(T(M \times {{\mathbb C}}\mathbf{k})$. This is easily seen from noticing that $I$ allows us to rearrange the product of two sections in the desired order.
In the same way, we see that $\Pi(T(\widehat{E}_+))\otimes \Pi(T(E)) \otimes \Pi(T({{\mathbb C}}\mathbf{k}))$ embed associative subalgebra.
Construction of MOSVAs and modules {#3-5}
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Let $U$ be an open subset of $M$. Let $f: U\to {{\mathbb C}}$ be a complex-valued smooth function. We define the action of $X\in \Pi(T(E))$ by $$\psi_U(X)f = (\sqrt{-1})^n (\nabla^n f)(X), X \in \Pi(E^{\otimes n}).$$ It was shown in [@H-MOSVA-Riemann], Theorem 4.1 that for $X, Y\in \Pi(T(E))$, $$\psi_U(X\otimes Y)f = \psi_U(X)\psi_U(Y)f$$ We further define the action of $\Pi(T(M\times {{\mathbb C}}\mathbf{k}))$ by having $\mathbf{k}$ act by a scalar multiplication $l$ for some $l\in {{\mathbb C}}$, and the action of $\Pi(T(\widehat{E}_{+})$ by zero. This way, the space of complex-valued smooth functions becomes a module for the associative algebra $\Pi(T(\widehat{E}_+))\otimes \Pi(T(E))\otimes \Pi(T({{\mathbb C}}\mathbf{k}))$. Any complex-valued smooth function $f$ generates a submodule.
Now for a fixed function $f$, we induce the $\Pi(T(\widehat{E}_+))\otimes \Pi(T(E))\otimes \Pi(T({{\mathbb C}}\mathbf{k}))$-module generated by $f$ to a module for the larger algebra $\Pi(T(\widehat{E}_-))\otimes\Pi(T(\widehat{E}_+))\otimes \Pi(T(E))\otimes \Pi(T({{\mathbb C}}\mathbf{k}))$. We denote this module by $V_U(l, f)$. As a vector space, $$V_U(l, f) = \Pi(T(\widehat{E}_-)) \otimes \Pi(T(E))f$$ In case $\Pi(T(E))f = {{\mathbb C}}f$, we will omit the tensor symbol.
Denote by ${\mathbf{1}}$ the constant function that sends every point on $M$ to 1. Since all the covariant derivatives vanishes, $\Pi(T(E)){\mathbf{1}}= {{\mathbb C}}{\mathbf{1}}$. We will omit the tensor product sign and simply write $$V_U(l, {\mathbf{1}}) = \Pi(T(\widehat{E}_-)) {\mathbf{1}}$$ Denote the actions of $h_\pm \otimes t^k$ by $h_{\pm}(k)$. For a formal variable $x$, let $$h_{\pm}(x) = \sum_{n\in{{\mathbb Z}}} h_{\pm}(n)x^{-n-1}.$$ Then we have the following theorem
The vectors $$h_{i_1}(-m_1)\cdots h_{i_k}(-m_k){\mathbf{1}}, i_1, ..., i_k\in \{+, -\}, \#\{j: i_j=+\}=\#\{j: i_j=-\}$$ form a basis for $V_U(l, {\mathbf{1}})$. Together with the following vertex operator action $$\begin{aligned}
& Y(h_{i_1}(-m_1) \cdots h_{i_k}(-m_k){\mathbf{1}}, x) \\
& \qquad= {\mbox{\scriptsize ${\circ\atop\circ}$}}\frac 1 {(m_1-1)!} \frac {d^{m_1-1}}{dx^{m_1-1}}h_{i_1}(x) \cdots \frac 1 {(m_k-1)!} \frac {d^{m_k-1}}{dx^{m_k-1}}h_{i_k}(x) {\mbox{\scriptsize ${\circ\atop\circ}$}}\end{aligned}$$ defines a MOSVA structure on $V_U(l, {\mathbf{1}})$.
For $l=1$ the theorem has been proved in [@H-MOSVA-Riemann], Proposition 3.3. The generalization to $l\in {{\mathbb C}}$ is a trivial modification of the whole process. For exposition purposes, we will sketch a direct proof using the computational results in [@H-MOSVA] modified by the general central charge.
For convenience, we use $V$ to denote the space $V_U(l, {\mathbf{1}})$. The grading of $V$ is given by specifying $V_n$ to be the span of the vectors $h_{i_1}(-m_1)\cdots h_{i_k}(-m_k){\mathbf{1}}$, with $i_1, ..., i_k$ satisfying the conditions in the statement, and $m_1+\cdots + m_k = n$. From Proposition \[ParallelSec\], $n\geq 0$. So the grading is lower bounded.
Let $u = a_1(-m_1) \cdots a_{k_1}(-m_{k_1}){\mathbf{1}}$ and $v = b_1(-n_1) \cdots b_{k_2}(-n_{k_2}){\mathbf{1}}$, $a_1, ..., a_{k_1}, b_1, ..., b_{k_2}\in \{h_+, h_-\}, m_1, ..., m_{k_1}, n_1, ..., n_{k_2}\in {{\mathbb Z}}_+$, the coefficient of each power of $x$ in $Y(u, x)v$ is in $V$. In fact, the coefficients of each power of $x$ is a sum of elements of the form $${\mbox{\scriptsize ${\circ\atop\circ}$}}a_1(p_1)\cdots a_{k_1}(p_{k_1}){\mbox{\scriptsize ${\circ\atop\circ}$}}b_1(-n_1)\cdots b_{k_2}(-n_{k_2}){\mathbf{1}}$$ For every such $p_1, ..., p_k$, let $\sigma$ be the unique element in $S_k$ satisfying the condition $$\begin{aligned}
\sigma(1) < \cdots < \sigma(\alpha), & \sigma(\alpha+1)< \cdots< \sigma(\beta), & \sigma(\beta)<\cdots < \sigma(k_1) \nonumber\\
p_{\sigma(1)}, ..., p_{\sigma(\alpha)}< 0,& p(\sigma_{\alpha+1}), ..., p(\sigma_\beta) >0, & p(\sigma_{\beta+1}), ..., p(\sigma_{k_1})= 0,\nonumber \end{aligned}$$ so that the element can be written as $$\begin{aligned}
& a_{\sigma(1)}(p_{\sigma(1)}) \cdots a_{\sigma(\alpha)} (p_{\sigma(\alpha)}))
a_{\sigma(\alpha+1)}(p_{\sigma(\alpha+1)}) \cdots a_{\sigma(\beta)} (p_{\sigma(\beta)}))\\
& \quad \cdot a_{\sigma(\beta+1)}(0) \cdots a_{\sigma(k_1)} (0))b_1(-n_1)\cdots b_{k_2}(-n_{k_2}){\mathbf{1}}\end{aligned}$$ From the relations of the algebra, for $i,j\in \{+, -\}, n>0$, $h_i(0)$ commutes with all $h_j(-n)$, while $h_i(0){\mathbf{1}}= 0$. Thus if there exists some $j$ such that $p_j = 0$, the element is simply zero, which is certainly in $V$.
Otherwise, if $p_1, ..., p_{k_1}\neq 0$, then $\beta = k_1$. The element in question would then be $$\begin{aligned}
& a_{\sigma(1)}(p_{\sigma(1)}) \cdots a_{\sigma(\alpha)} (p_{\sigma(\alpha)}))
a_{\sigma(\alpha+1)}(p_{\sigma(\alpha+1)}) \cdots a_{\sigma(\beta)} (p_{\sigma(\beta)})) b_1(-n_1)\cdots b_{k_2}(-n_{k_2}){\mathbf{1}}\end{aligned}$$ From the relation of the algebra, for $i,j\in \{+, -\}$, $p, q>0$, the commutator of $h_i(p)$ and $h_j(-n)$ is $l p\delta_{p,n} (h_i, h_j)$. Notice that every time we swap the position of $h_i(p)$ and $h_j(-n)$, the number of $+$ and $-$ in the commutator term are both lowered by 1. So at the end of the day when all $h_i(p)$ with $p>0$ are positions before ${\mathbf{1}}$, the number of $+$ and $-$ in all the extra commutator terms are still kept the same. This shows that the elements are all in $V$. Thus we proved that $Y(u, x)v \in V[[x]]$.
Now we argue the weak associativity. From Corollary 4.9 in [@H-MOSVA], for every $a_1, ..., a_{k_1}, b_1, ..., b_{k_2} \in \{h_+, h_-\}$, $m_1, ..., m_{k_1}, n_1, ..., n_{k_2} \in {{\mathbb Z}}_+$, $$\begin{aligned}
& Y(a_1(-m_1)\cdots a_{k_1}(-m_{k_1}){\mathbf{1}}, x_1)Y(b_1(-n_1)\cdots b_{k_2}(-n_{k_2}){\mathbf{1}}, x_2) \\
& \quad = \sum_{i=0}^{\min\{k_1,k_2\}} \sum_{\substack{k_1\geq p_1 > \cdots > q_i \geq 1\\ 0\leq q_1 < \cdots < q_i \leq k_2}} l^i n_{q_1}\cdots n_{q_i}(a_{p_1}, b_{q_1})\cdots (a_{p_i}, b_{q_i})\\
& \qquad \quad \cdot \binom{-n_{q_1}-1}{m_{p_1}-1}\cdots \binom{-n_{q_i}-1}{m_{p_i}-1} (x_1-x_2)^{-m_{p_1}-n_{q_1}-\cdots - m_{p_i}-n_{q_i}}\\
& \qquad \quad \cdot {\mbox{\scriptsize ${\circ\atop\circ}$}}\left(\prod_{p\neq p_1, ..., p_i}\frac 1 {(m_p-1)!} \frac{\partial^{m_p-1}}{\partial x_1^{m_p-1}}a_p(x_1)\right)\left(\prod_{q\neq q_1, ..., q_i}\frac 1 {(n_q-1)!} \frac{\partial^{n_q-1}}{\partial x_2^{n_q-1}}b_q(x_2)\right){\mbox{\scriptsize ${\circ\atop\circ}$}}\end{aligned}$$ where the extra $l^i$ factor comes from a trivial generalization of Lemma 4.1 in [@H-MOSVA].
From Formula (5.29) of [@H-MOSVA], we see that $$\begin{aligned}
& Y(Y(a_1(-m_1)\cdots a_{k_1}(-m_{k_1}){\mathbf{1}}, x_0)b_1(-n_1)\cdots b_{k_2}(-n_{k_2}){\mathbf{1}}, x_2) \\
& \quad = \sum_{i=0}^{\min\{k_1,k_2\}} \sum_{\substack{k_1\geq p_1 > \cdots > q_i \geq 1\\ 0\leq q_1 < \cdots < q_i \leq k_2}} l^i n_{q_1}\cdots n_{q_i}(a_{p_1}, b_{q_1})\cdots (a_{p_i}, b_{q_i})\\
& \qquad \quad \cdot \binom{-n_{q_1}-1}{m_{p_1}-1}\cdots \binom{-n_{q_i}-1}{m_{p_i}-1} x_0^{-m_{p_1}-n_{q_1}-\cdots - m_{p_i}-n_{q_i}}\\
& \qquad \quad \cdot {\mbox{\scriptsize ${\circ\atop\circ}$}}\left(\prod_{p\neq p_1, ..., p_i}\frac 1 {(m_p-1)!} \frac{\partial^{m_p-1}}{\partial x_0^{m_p-1}}a_p(x_2+x_0)\right)\left(\prod_{q\neq q_1, ..., q_i}\frac 1 {(n_q-1)!} \frac{\partial^{n_q-1}}{\partial x_2^{n_q-1}}b_q(x_2)\right){\mbox{\scriptsize ${\circ\atop\circ}$}}\end{aligned}$$ where negative powers of $x_2+x_0$ are expanded as a formal series in $x_2, x_0$ with lower truncated powers of $x_0$.
To show the weak associativity, we fix $i$ and $p_1, ..., p_i$ and compute the lower bound of power of $x_1$ of the series from the action of $${\mbox{\scriptsize ${\circ\atop\circ}$}}\left(\prod_{p\neq p_1, ..., p_i}\frac 1 {(m_p-1)!} \frac{\partial^{m_p-1}}{\partial x_0^{m_p-1}}a_p(x_1)\right)\left(\prod_{q\neq q_1, ..., q_i}\frac 1 {(n_q-1)!} \frac{\partial^{n_q-1}}{\partial x_2^{n_q-1}}b_q(x_2)\right){\mbox{\scriptsize ${\circ\atop\circ}$}}$$ on an element $v = c_1(-r_1)\cdots c_{k_3}(-r_{k_3}){\mathbf{1}}$. For each $p \neq p_1, ..., p_i$, we compute the singular part of the term with respect to $a_p(x_1)$, namely, $$\left(\frac{\partial^{m_p-1}}{\partial x_1^{m_p-1}}a_p(x_1)\right)^-= \sum_{s_p \geq 0}a_p (s_p)( - s_p - 1) \cdots ( - s_p - {m_p} + 1)x_1^{ - s_p - {m_p}}$$ The term with the lowest power of $x_1$ will contain no $b_q(t)$ with $t>0$. Thus all such $a_p(s_p)$ would have the priority acting $v$, resulting in $$\prod_{p\neq p_1, ..., p_i} a_p(s_p)c_1(-r_1)\cdots c_{k_3}(-r_{k_3}){\mathbf{1}}$$ which is zero when $$\sum_{p\neq p_1, ..., p_i} s_p > r_1 + \cdots + r_{k_3}$$ So the power of $x_1$ is $$\begin{aligned}
\sum_{p\neq p_1, ..., p_i} (-s_p-m_p) &\geq -r_1 -\cdots - r_{k_3} - \sum_{p\neq p_1, ..., p_i} m_p\\
& \geq -r_1 -\cdots - r_{k_3} - m_1 - \cdots - m_{k_1}. \end{aligned}$$ The lower bound we obtained at the right-hand-side works for all possible choices of $i$ and $p_1, ..., p_i$. Moreover, it depends only on the element $a_1(-m_1)\cdots a_{k_1}(-m_{k_1}) {\mathbf{1}}$ and $c_1(-r_1)\cdots c_{k_3}(-r_{k_2}){\mathbf{1}}$. So the pole-order condition is verified.
Other axioms are verified similarly as in [@H-MOSVA].
Let $W = V_U(l, f)$. Define $D_W: W \to W$ by $$D_W(h_{i_1}(-m_1)\cdots h_{i_k}(-m_k) \otimes X f = m_1 \cdots m_k h_{i_1}(-m_1-1)\cdots h_{i_k}(-m_k-1) \otimes Xf$$ for any $i_1, ..., i_k \in \{+, -\}$ with $\#\{j: i_j=+\}=\#\{j: i_j=-\}$, any $X\in \Pi(T(E))$ and any smooth function $f: U\to {{\mathbb C}}$. Then with any lower bounded grading assigned on the vector subspace $1\otimes \Pi(T(E))f$, $(W, Y_W, D_W)$ forms a module for the MOSVA $V_U(l, {\mathbf{1}}). $
This is proved in [@H-MOSVA-Riemann], Theorem 5.1. A direct proof can also be constructed using the computational results in [@H-MOSVA] similarly as the theorem above. We will not repeat it.
If we understand $h_+(-m)$ and $h_-(-m)$ as the creation operators of certain physical objects (particles or strings), the conclusions in this section amounts to say that these physical objects are always created in pairs. There does not exist one-object states in the MOSVA $V_U(l, {\mathbf{1}})$ and the module $V_U(l, f)$.
Modules generated by eigenfunctions of the Laplace-Beltrami operator
====================================================================
In this section we study the module $V_U(l, f)$ for a smooth function $f: U\to {{\mathbb C}}$ satisfying $$-\Delta f = \lambda f$$ where $\Delta$ is the Laplace-Beltrami operator on $U$, $\lambda\in {{\mathbb C}}$. In physics, eigenfunctions correspond to quantum states. So the module $V_U(l, f)$ can be understood as string-theoretic excitements of the quantum state corresponding to $f$.
A lemma on higher order covariant derivatives
---------------------------------------------
In order to study the actions of $\Pi(T(E))$ on $f$, we will use the following lemma:
Let $f: U\to {{\mathbb C}}$ be a ${{\mathbb C}}$-valued smooth function. Then for $n \geq 3$, we have $$(\nabla^n f)(Z_1, ..., Z_{n-1}, Z_n ) - (\nabla^n f)(Z_1, ..., Z_n, Z_{n-1}) = 0$$ And for $i = 1, ..., n-2$, $$\begin{aligned}
& (\nabla^n f)(Z_1, ..., Z_i, Z_{i+1}, ..., Z_n) - (\nabla^n f)(Z_1, ..., Z_{i+1}, Z_{i}, ..., Z_n)\\
= & \sum_{j=i+2}^n(\nabla^{n-2} f)(Z_1, ..., -R(Z_i, Z_{i+1})Z_{j}, ..., Z_n) \\
= & (\nabla^{n-2} f)(Z_1, ..., -R(Z_i, Z_{i+1})Z_{i+2}, Z_{i+3} ..., Z_n) \\
& + (\nabla^{n-2} f)(Z_1, ..., Z_{i+2}, -R(Z_i, Z_{i+1})Z_{i+3}, ..., Z_n) + \\
& + \cdots \cdots \\
& + (\nabla^{n-2} f)(Z_1, ..., Z_{i+2}, Z_{i+3}, ..., -R(Z_i, Z_{i+1})Z_n)\end{aligned}$$
We prove the first equation by induction on $n$. For $n=3$, we have $$\begin{aligned}
(\nabla^3 f)(Z_1, Z_2, Z_3) &= (\nabla_{Z_1} (\nabla^2 f))(Z_2, Z_3) \\
& = \nabla_{Z_1} ((\nabla^2 f)(Z_2, Z_3)) - (\nabla^2 f)(\nabla_{Z_1}Z_2, Z_3) - (\nabla^2 f)(Z_2, \nabla_{Z_1}Z_3) \\
& \quad \text{(note that $\nabla^2 f(X, Y) = \nabla^2 f(Y, X)$)} \\
& = \nabla_{Z_1} ((\nabla^2 f)(Z_3, Z_2)) - (\nabla^2 f)(Z_3, \nabla_{Z_1}Z_2) - (\nabla^2 f)(\nabla_{Z_1}Z_3, Z_2) = (\nabla^3 f)(Z_1, Z_3, Z_2)\end{aligned}$$ Assume the equation holds for $n-1$: $$\begin{aligned}
(\nabla^n f)(Z_1, ..., Z_{n-1}, Z_n) & = (\nabla_{Z_1}(\nabla^{n-1}f))(Z_2, ..., Z_{n-1}, Z_n) \\
& = \nabla_{Z_1}((\nabla^{n-1}f)(Z_2, ..., Z_{n-1}, Z_n)) - (\nabla^{n-1}f)(\nabla_{Z_1}Z_2, ..., Z_{n-1}, Z_n) \\
& \quad - (\nabla^{n-1}f)(Z_2, ..., \nabla_{Z_1}Z_{n-1}, Z_n) - (\nabla^{n-1}f)(Z_2, ..., Z_{n-1}, \nabla_{Z_1}Z_n)\\
& \quad \text{(by induction hypothesis)}\\
& = \nabla_{Z_1}((\nabla^{n-1}f)(Z_2, ..., Z_n, Z_{n-1})) - (\nabla^{n-1}f)(\nabla_{Z_1}Z_2, ..., Z_n, Z_{n-1}) \\
& \quad - (\nabla^{n-1}f)(Z_2, ..., Z_n, \nabla_{Z_1}Z_{n-1}) - (\nabla^{n-1}f)(Z_2, ..., \nabla_{Z_1}Z_n, Z_{n-1})\\
& = (\nabla^n f)(Z_1, ..., Z_n, Z_{n-1})\end{aligned}$$ So the first equation is proved.
For the second equation, we first consider the case $i=1$: $$\begin{aligned}
&\quad (\nabla^n f)(Z_1, Z_2, Z_3, \cdots, Z_n)= (\nabla_{Z_1} (\nabla^{n-1}f))(Z_2, Z_3, \cdots, Z_n)\nonumber\\
&= \nabla_{Z_1}((\nabla^{n-1}f) (Z_2, Z_3 ..., Z_n)) - (\nabla^{n-1}f)(\nabla_{Z_1}Z_2, Z_3, ..., Z_n) - \sum_{j=3}^n (\nabla^{n-1}f)(Z_2, ..., \nabla_{Z_1}Z_j, ..., Z_n) \nonumber\\
&= \nabla_{Z_1}\nabla_{Z_2}((\nabla^{n-2}f) (Z_3, ..., Z_n)) - \sum_{j=3}^n \nabla_{Z_1}((\nabla^{n-2}f)(Z_3, ..., \nabla_{Z_2} Z_j, ..., Z_n))\label{Line1}\\
& \quad -\nabla_{\nabla_{Z_1}Z_2} ((\nabla^{n-2}f)(Z_3, ..., Z_n))+\sum_{j=3}^n(\nabla^{n-2}f)(Z_3,..., \nabla_{\nabla_{Z_1}Z_2}Z_j, ..., Z_n)\label{Line2}\\
& \quad
- \sum_{j=3}^n \left(\nabla_{Z_2}((\nabla^{n-2}f)(Z_3, ..., \nabla_{Z_1}Z_j, ..., Z_n) - \sum_{k=3}^{j-1}(\nabla^{n-2}f)(Z_3,..., \nabla_{Z_2}Z_k, ..., \nabla_{Z_1}Z_j, ..., Z_n)\right) \label{Line3} \\
& \quad - \sum_{j=3}^n \left(-(\nabla^{n-2}f)(Z_3, ..., \nabla_{Z_2}\nabla_{Z_1}Z_j, ..., Z_n)-\sum_{k=j+1}^{n} (\nabla^{n-2}f)(Z_3, ..., \nabla_{Z_1} Z_j , ..., \nabla_{Z_2}Z_k, ..., Z_n)\right)\label{Line4}\end{aligned}$$ Similarly, $$\begin{aligned}
&\quad (\nabla^n f)(Z_2, Z_1, Z_3, \cdots, Z_n)= (\nabla_{Z_2} (\nabla^{n-1}f))(Z_1, Z_3, \cdots, Z_n)\nonumber\\
& = \nabla_{Z_2}\nabla_{Z_1}((\nabla^{n-2}f) (Z_3, ..., Z_n)) - \sum_{j=3}^n \nabla_{Z_2}((\nabla^{n-2}f)(Z_3, ..., \nabla_{Z_1} Z_j, ..., Z_n))\label{Line5}\\
& \quad -\nabla_{\nabla_{Z_2}Z_1} ((\nabla^{n-2}f)(Z_3, ..., Z_n))+\sum_{j=3}^n(\nabla^{n-2}f)(Z_3,..., \nabla_{\nabla_{Z_2}Z_1}Z_j, ..., Z_n)\label{Line6}\\
& \quad
- \sum_{j=3}^n \left(\nabla_{Z_1}((\nabla^{n-2}f)(Z_3, ..., \nabla_{Z_2}Z_j, ..., Z_n) - \sum_{k=3}^{j-1}(\nabla^{n-2}f)(Z_3,..., \nabla_{Z_1}Z_k, ..., \nabla_{Z_2}Z_j, ..., Z_n)\right) \label{Line7}\\
& \quad - \sum_{j=3}^n \left(-(\nabla^{n-2}f)(Z_3, ..., \nabla_{Z_1}\nabla_{Z_2}Z_j, ..., Z_n)-\sum_{k=j+1}^{n} (\nabla^{n-2}f)(Z_3, ..., \nabla_{Z_2} Z_j , ..., \nabla_{Z_1}Z_k, ..., Z_n)\right)\label{Line8}\end{aligned}$$ Then in the difference, the second sum in (\[Line1\]) cancels out with the first term in the sum of (\[Line7\]); the first term in the sum of (\[Line3\]) cancels out with the second sum in (\[Line5\]); the second term in the sum of (\[Line3\]), together with second term in the sum of (\[Line4\]), cancel out those in (\[Line7\]) and (\[Line8\]). So the difference is $$\begin{aligned}
& \quad (\nabla^n f)(Z_1, Z_2, Z_3, ..., Z_n) - (\nabla^n f)(Z_2, Z_1, Z_3, ..., Z_n)\\
& = (\nabla_{Z_1}\nabla_{Z_2}-\nabla_{Z_2}\nabla_{Z_1})((\nabla^{n-2}f)(Z_3, ..., Z_n)) - \nabla_{\nabla_{Z_1}Z_2 - \nabla_{Z_2}Z_1} ((\nabla^{n-2}f)(Z_3, ..., Z_n) \\
& \quad+ \sum_{j=3}^n (\nabla^{n-2}f)(Z_3, ..., \nabla_{\nabla_{Z_1}Z_2 -\nabla_{Z_2}Z_1} Z_j, ..., Z_n) + \sum_{j=3}^n (\nabla^{n-2}f)(Z_3, ..., (\nabla_{Z_2}\nabla_{Z_1}-\nabla_{Z_1}\nabla_{Z_2})Z_j, ..., Z_n)\\
& = \sum_{j=3}^n (\nabla^{n-2}f)(Z_3, ..., (\nabla_{Z_2}\nabla_{Z_1}-\nabla_{Z_1}\nabla_{Z_2}+ \nabla_{\nabla_{Z_1}Z_2 - \nabla_{Z_2}Z_1})Z_j, ..., Z_n)\\
&=\sum_{j=3}^n (\nabla^{n-2}f)(Z_3, ..., -R(Z_1, Z_2)Z_j, ..., Z_n).\end{aligned}$$ So the case $i=1$ is proved for arbitrary $n$.
We proceed by induction of $i$. The base case has been proved above. Now we proceed with the inductive step. $$\begin{aligned}
& \quad (\nabla^n f)(Z_1, ..., Z_i, Z_{i+1}, ..., Z_n) = (\nabla_{Z_1}(\nabla^n f))(Z_2, ..., Z_i, Z_{i+1}, ..., Z_n) \\
&= \nabla_{Z_1}((\nabla^{n-1} f)(Z_2, ..., Z_i, Z_{i+1}, ..., Z_n)) - \sum_{k=2}^{i-1}(\nabla^{n-1} f)(Z_2, ..., \nabla_{Z_1}Z_k, ..., Z_i, Z_{i+1}, ..., Z_n) \\
& \quad - (\nabla^{n-1} f)(Z_2, ..., \nabla_{Z_1} Z_i, Z_{i+1}, ..., Z_n) - (\nabla^{n-1} f)(Z_2, ..., Z_i, \nabla_{Z_1} Z_{i+1}, ..., Z_n) \\
& \quad - \sum_{k=i+2}^n (\nabla^{n-1} f)(Z_2, ..., Z_i, Z_{i+1}, ..., \nabla_{Z_1}Z_k, ..., Z_n)\end{aligned}$$ Similarly, $$\begin{aligned}
& \quad (\nabla^n f)(Z_1, ..., Z_{i+1}, Z_i, ..., Z_n) = (\nabla_{Z_1}(\nabla^n f))(Z_2, ..., Z_{i+1}, Z_i, ..., Z_n) \\
&= \nabla_{Z_1}((\nabla^{n-1} f)(Z_2, ..., Z_{i+1}, Z_i, ..., Z_n)) - \sum_{k=2}^{i-1}(\nabla^{n-1} f)(Z_2, ..., \nabla_{Z_1}Z_k, ..., Z_{i+1}, Z_i, ..., Z_n) \\
& \quad - (\nabla^{n-1} f)(Z_2, ..., \nabla_{Z_1} Z_{i+1}, Z_i, ..., Z_n) - (\nabla^{n-1} f)(Z_2, ..., Z_{i+1}, \nabla_{Z_1} Z_i, ..., Z_n) \\
& \quad - \sum_{k=i+2}^n (\nabla^{n-1} f)(Z_2, ..., Z_{i+1}, Z_i, ..., \nabla_{Z_1}Z_k, ..., Z_n)\end{aligned}$$ We use the induction hypothesis to see that the difference is expressed as $$\begin{aligned}
& \nabla_{Z_1}\left(\sum_{j=i+2}^n (\nabla^{n-3}f)(Z_2, ..., -R(Z_i, Z_{i+1})Z_j, ..., Z_n) \right) \\
& - \sum_{j=i+2}^n\sum_{k=2}^{i-1}(\nabla^{n-3}f)(Z_2, ..., \nabla_{Z_1}Z_k, ..., -R(Z_i, Z_{i+1})Z_j,..., Z_n) \\
& - \sum_{j=i+2}^n(\nabla^{n-3}f)(Z_2, ..., -R(\nabla_{Z_1}Z_i, Z_{i+1})Z_j, ..., Z_n)\\
& - \sum_{j=i+2}^n(\nabla^{n-3}f)(Z_2, ..., -R(Z_i, \nabla_{Z_1}Z_{i+1})Z_j, ..., Z_n)\\
& - \sum_{k=i+2}^n\sum_{j=i+2}^{k-1} (\nabla^{n-3}f)(Z_2, ..., -R(Z_i, Z_{i+1})Z_j, ..., \nabla_{Z_1}Z_k, ..., Z_n) \\
& -\sum_{k=i+2}^n (\nabla^{n-3}f)(Z_2, ..., Z_{i+2},..., -R(Z_i, Z_{i+1})\nabla_{Z_1}Z_k, ..., Z_n)\\
& - \sum_{k=i+2}^n\sum_{j=k+1}^n(\nabla^{n-3}f)(Z_2, ..., Z_{i+2},..., \nabla_{Z_1}Z_k, ..., -R(Z_i, Z_{i+1})Z_j, ..., Z_n)\end{aligned}$$ which is equal to the right-hand-side.
Zero-mode actions
-----------------
Now we use the lemma to compute $\Pi(T(E))f$. Recall the definition of the sectional curvature $$K = \frac{(R(U, V)V, U)}{(U, U)(V, V) - (U, V)^2}$$ where $U, V$ are any vector fields. Then by our assumption in Definition \[sections\] $$(X_1, X_1) = (X_2, X_2) = 1, (X_1, X_2) = 0$$ We compute directly that $$\begin{aligned}
(R(X_1, X_2)X_2, X_1) = K,\quad & (R(X_1, X_2)X_2, X_2) = 0\\
(R(X_1, X_2)X_1, X_1) = 0,\quad & (R(X_1, X_2)X_1, X_2) = -K\end{aligned}$$ In other words, $$R(X_1, X_2)X_2 = KX_1, R(X_1, X_2)X_1 = -KX_2$$ Therefore, $$\begin{aligned}
R(h_+,h_-)h_+ = 2K h_+,
R(h_+,h_-)h_- = -2K h_-\end{aligned}$$
\[ConstMult\] If $\nabla^2(h_+, h_-)f$ is a multiple of $f$, then for every choice of $\sigma_1, ..., \sigma_n \in \{+, -\}, $ with equal number of $+$ and $-$, $\nabla^n(h_{\sigma_1}, ..., h_{\sigma_n})f$ is also a multiple of $f$.
We will use Theorem 4.1 in [@H-MOSVA-Riemann]: for $X$ and $Y$ are two parallel tensors of degree $m$ and $n$, $$\psi_U(X\otimes Y) = \psi_U(X) \psi_U(Y)$$ In other words, $$(\nabla^{m+n} f)(X, Y) = (\nabla^n (\nabla^m f(Y)))(X)$$
We argue by induction. We will compute the action of $h_+\otimes\cdots\otimes h_+\otimes h_-\otimes\cdots\otimes h_-$ explicitly. Other cases are similar.
Using the previous lemma, we write $$\begin{aligned}
& (\nabla^n f)(h_+, ..., h_+, h_+, h_-, h_-, ..., h_-)\\
=& (\nabla^n f)(h_+, ...,h_+, h_-, h_+, h_-, ..., h_-) + (\nabla^{n-2}f)(h_+, ..., h+, -R(h_+, h_-)h_-, ..., h_-) \\
& \hskip 6cm +\cdots+(\nabla^{n-2}f)(h_+, ..., h_+, h_-, ..., -R(h_+, h_-)h_-)\\
= & (\nabla^n f)(h_+, ...,h_+, h_-, h_+, h_-, ..., h_-) + 2K (\nabla^{n-2}f)(h_+, ..., h+, h_-, ..., h_-) \\
& \hskip 6cm +\cdots+2K(\nabla^{n-2}f)(h_+, ..., h_+, h_-, ..., 2 h_-)\end{aligned}$$ By induction hypothesis, all the extra terms are multiples of $f$. So it suffices to focus on the first term.
We will again try to move $h_-$ one spot ahead. Likewise, the extra terms, by induction hypothesis, will also be multiples of $f$.
Repeat the process until we arrive at $$(\nabla^n f)(h_+, h_-, h_+, ..., h_+, h_-, ..., h_-)$$ which, by Theorem 4.1 in [@H-MOSVA-Riemann], is equal to $$(\nabla^{n-2} (\nabla^2 f)(h_+, h_-))(h_+, ..., h_+, h_-, ..., h_-)$$ One sees that this is also a multiple of $f$, as $(\nabla^2 f)(h_+, h_-)$ is a multiple of $f$, as well as $(\nabla^{n-2}f)(h_+, .., h_+, h_-, ..., h_-)$.
\[ParallelScalar\] Let $f$ be an eigenfunction of the Laplace-Beltrami operator. Then as a vector space, $\Pi(T(E))f = {{\mathbb C}}f$.
It suffices to notice that $(\nabla^2 f)(h_+, h_-) = \Delta f$. Note that $\Delta f$ is defined by $(\nabla^2 f)(X_1,X_1) + (\nabla^2f)(X_2, X_2)$ where $X_1, X_2$ are vector fields in Definition \[sections\]. Since $h_+ = X_1 - \sqrt{-1} X_2, h_-= X_1 + \sqrt{-1} X_2$. Thus $$\begin{aligned}
(\nabla^2f)(h_+, h_-) &= (\nabla^2 f)(X_1-\sqrt{-1}X_2, X_1+\sqrt{-1} X_2) \\
&= (\nabla^2 f)(X_1, X_1) - \sqrt{-1}(\nabla^2 f)(X_2, X_1) + \sqrt{-1}(\nabla^2 f)(X_1, X_2) + (\nabla^2 f)(X_2, X_2)\end{aligned}$$ The conclusion then follows from $(\nabla^2f)(X_1,X_2)=(\nabla^2f)(X_2,X_1)$.
Consequences
------------
From Theorem \[ParallelScalar\], $\Pi(T(E))f$ is one-dimensional. We use the eigenvalue $\lambda$ of $f$ for the weight of this subspace in $V_U(l, f)$, thereby making $V_U(l, f)$ a module for $V_U(l, {\mathbf{1}})$. Then we conclude the following.
Let $f$ be an eigenfunction for the Laplace-Beltrami operator of eigenvalue $\lambda$. Then the following vectors $$h_{i_1}(-m_1)\cdots h_{i_k}(-m_k) f, m_1, ..., m_k \in {{\mathbb Z}}_+, i_1, ..., i_k\in \{+, -\}, \#\{j: i_j=+\}= \#\{j: i_j = -\},$$ form a basis of the module $V_U(l, f)$, where the weight of each basis vector is specified as $m_1 + \cdots + m_k + \lambda$.
$V_U(l, f)$ is irreducible.
The action of $h_{i_k}(m_k)\cdots h_{i_1}(m_1)$ on the basis vector $h_{i_1}(-m_1)\cdots h_{i_i}(-m_k)f$ sends it back to a constant multiple of $f$.
With the knowledge of the basis, we can compute the graded dimension of the modules $V_U(l, f)$.
The graded dimension of the MOSVA $V_U(l, {\mathbf{1}})$ is $$1 + \sum_{n=2}^\infty 2(n-1) \cdot {}_2F_1 \left(1-\frac n 2, \frac{3-n} 2; 2 ; 4\right)q^{n}$$ The graded dimension of $V_U(l, f)$ is $$q^{\lambda}+\sum_{n=2}^\infty 2(n-1) \cdot {}_2F_1 \left(1 -\frac n 2, \frac{3-n} 2; 2 ; 4\right)q^{n+\lambda}$$
It suffices to argue the formula for $V_U(l, {\mathbf{1}})$. Let $n\in {{\mathbb Z}}_+$. We compute $\dim V_{(2n)}$ and $\dim V_{(2n+1)}$ separately.
For each fixed $k\in {{\mathbb Z}}_+$, the set of ordered $k$-tuples $(m_1, ..., m_k)$ of positive numbers such that $m_1 + \cdots + m_k = 2n$ is well-known to be $\binom{2n-1}{k-1}$. Also within $i_1, ..., i_k$, the number of $+$ must be the same as the number of $-$. Thus $k$ must be an even number. Write $k = 2p$. Then the number of possible assignments of $+$ and $-$ to $i_1, ..., i_{2p}$ is simply $\binom{2p}p$. Summing up all possible choices of $p$, we have $$\dim V_{(2n)} = \sum_{p=1}^n \binom{2p} p \binom{2n-1}{2p-1} = \sum_{p=1}^n \frac{2(2n-1)!}{p!(p-1)!(2n-2p)!}$$ Recall that in general, $${}_2F_1(a,b;c;z) = \sum \limits_{q = 0}^\infty \frac{{a(a + 1) \cdots (a + q - 1)b(b + 1) \cdots (b + q - 1)}}{{c(c + 1) \cdots (c + q - 1)}}\frac{{{z^q}}}{{q!}}$$ Putting in $a=1-n, b=3/2-n, c=2, z=4$, we have $$\begin{aligned}
& {}_2F_1(1-n, \frac 3 2 - n; 2; 4) \\
& \quad = \sum_{q=0}^\infty \frac{(1-n)(1-n+1)\cdots (1-n+q-1)\cdot (\frac 3 2 - n)(\frac 3 2 - n + 1) \cdots (\frac 3 2 - n + q -1)}{2(2+1)\cdots (2+q-1)}\frac{2^q}{q!}\\
& \quad = \sum_{q=0}^{n-1} \frac{(-1)^q(n-1)(n-2)\cdots (n-q)\cdot (3 - 2n)(3 - 2n + 2) \cdots (3 - 2n + 2q -2)}{(q+1)!}\frac{2^q}{q!}\\
& \quad = \sum_{q=0}^{n-1} \frac{(2n-2)(2n-4)\cdots (2n-2q)\cdot(2n-3)(2n-5) \cdots (2n-2q-1)}{(q+1)!q!}\\
& \quad = \sum_{q=0}^{n-1} \frac{(2n-2)!}{(2n-2q-2)!(q+1)!q!} = \sum_{p=1}^{n} \frac{(2n-2)!}{(2n-2p)!p!(p-1)!} \end{aligned}$$ Thus $$\dim V_{(2n)}= 2(2n-1)\cdot {}_2F_1(1-n, \frac 3 2 - n; 2; 4)$$ Similarly, we compute that $$\begin{aligned}
\dim V_{(2n+1)} &= 2(2n)\cdot \sum_{p=1}^n \frac{(2n-1)!}{(2n-2p+1)!p!(p-1)!}\\
&= 2(2n)\cdot {}_2 F_1(\frac 1 2 - n, 1-n; 2; 4)\end{aligned}$$ The conclusion then follows.
Huang proved that $U\mapsto V_U(1, {\mathbf{1}})$ is indeed a sheaf. Huang considered the sum of modules $V_U(1, f)$ over all smooth $f: U\to {{\mathbb C}}$. He proved that the correspondence of open subset $U$ to this sum of $V_U(1, f)$’s form a presheaf (see details in [@H-MOSVA-Riemann], Section 5).
The generalization of these results to $V_U(l, {\mathbf{1}})$ and $V_U(l, f)$ for $l\in {{\mathbb C}}$ is trivial. Moreover, if we consider the module $$W_U = \bigoplus\{V_U(l, f): f: U\to {{\mathbb C}}\text{ smooth, with }\Delta f= -\lambda f\text{ for some }\lambda \in {{\mathbb C}}\}$$ Then we have the following proposition.
$U\mapsto W_U$ forms a sheaf over $M$.
In a nutshell, $W_U$ is the tensor product of the sheaf $U\mapsto V_U(l, {\mathbf{1}})$ with the sheaf of eigenfunctions over $M$.
Concluding remarks {#concluding-remarks .unnumbered}
==================
We have identified the MOSVA and modules over any two-dimensional complete connected orientable Riemannian manifolds of nonzero sectional curvature. It turns out algebraically they are all isomorphic. The geometric information is carried on the generators of the modules, namely the eigenfunctions of Laplace-Beltrami operator. We expect the yet-to-be-defined intertwining operators between these modules will be reflecting more geometric properties. Indeed, Huang observed that lattice intertwining operator algebras (genus-zero conformal field theories associated with tori) can be constructed using suitable modules for the MOSVA on tori $(K=0)$ generated by eigenfunctions that is well defined on the tori together with the intertwining operators among them. Huang further conjectured that on the MOSVA $V_U(l, {\mathbf{1}})$ there exists certain vertex Hopf algebra structure (to be appropriately defined) that allows us to study the tensor categories formed by its modules. These problems will be studied in future work.
[KWAK2]{}
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M. P. do Carmo, *Differential Geometry of Curves and Surfaces*, Prentice-Hall Inc., New Jersey, 1976
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Y.-Z. Huang, Meromorphic open string vertex algebras, *J. Math. Phys.* **54** (2013), 051702.
Y.-Z. Huang, Meromorphic open-string vertex algebras and Riemannian manifolds, arXiv:1205.2977.
Y.-Z. Huang, L. Kong, Open-string vertex algebras, tensor categories and operads, [*Comm. Math. Phys.*]{}, **250** (2004), 433–471
G. Jones, D. Singerman, *Complex functions: an algebraic and geometric viewpoint*, Cambridge University Press, Cambridge, 1987.
G. Moore, N. Seiberg, Classical and quantum conformal field theory, *Comm. Math. Phys.* **123** (1989), 177–254.
P. Petersen, *Riemannian Geometry*, third edition, Graduate Texts in Mathematics 171, Springer-Verlag, New York (2016)
F. Qi, On modules for meromorphic open-string vertex algebras, *Journal of Mathematical Physics* **60**, 031701 (2019)
|
---
author:
- |
C.M. Harris$^\dag$, M.J. Palmer$^\dag$, M.A. Parker$^\dag$, P. Richardson$^{\ddag}$, A. Sabetfakhri$^\dag$ and B.R. Webber$^{\dag}$\
\
$^\dag$ Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge, CB3 0HE, UK.\
$^\ddag$ Institute for Particle Physics Phenomenology, University of Durham, DH1 3LE, UK.
bibliography:
- 'BH.bib'
title: Exploring Higher Dimensional Black Holes at the Large Hadron Collider
---
Introduction
============
In recent theories with extra dimensions the fundamental Planck mass, [$M_{\rm PL}$]{}, can be as low as the TeV scale [@Arkani-Hamed:1998rs; @Antoniadis:1998ig; @Randall:1999ee; @Randall:1999vf] making the trans-Planckian regime accessible for future high energy colliders. These models aroused great theoretical interest because they address the weak-Planck scale hierarchy. In these theories microscopic quantum black holes could be produced at energies higher than the Planck mass at the LHC [@Argyres:1998qn; @Banks:1999gd; @Emparan:1999wa; @Giddings:2001bu; @Dimopoulos:2001hw; @Hossenfelder:2001dn; @Voloshin:2001vs; @Lester:Talk]. Once produced, the black hole would decay very rapidly to a spectrum of particles by Hawking radiation. Assuming that all the Standard Model matter and gauge fields are confined to the physical three-branes in a higher dimensional space, it has been shown that most of the black hole decay products are Standard Model quanta emitted on the brane [@Emparan:1999wa] and are therefore visible experimentally as very spectacular events.
We stress that quantum extra dimensional black holes in no way constitute any threat, being distinguished from the more familiar astrophysical variety by being much lighter and highly unstable. The astrophysical variety is much too heavy to be produced in current or planned collider experiments. Hereafter, all discussion of black holes relates only to the extra dimensional variety.
In the large extra dimensions scenario [@Arkani-Hamed:1998rs; @Antoniadis:1998ig] at distances small compared with the size of the extra dimensions and in the warped scenario [@Randall:1999ee; @Randall:1999vf] at distances small compared to the curvature scale of the geometry associated with the extra dimensions, black holes with a horizon radius, $r_{\rm BH}$, smaller than the size of the extra dimensions can be treated as higher-dimensional objects located on the brane and extending along the extra dimensions. It has been shown that these small black holes have modified properties, e.g. they are larger and colder compared to a 4-dimensional black hole with exactly the same mass [@Argyres:1998qn].
The black hole discovery potential is critically dependent on the value of [$M_{\rm PL}$]{}. Short scale gravity experiments and particle collider experiments provide limits on the fundamental Planck scale. However for smaller values of the number of extra dimensions, $n$, the more stringent constraints come from astrophysical and cosmological data, albeit with larger uncertainties. It is widely agreed that the one large extra dimension scenario is ruled out by such data. The present collider limits[^1] on the Planck scale range from 1.3 TeV for $n = 2$ to 0.3 TeV for $n = 6$ arising from the production of real (from missing energy signatures) or virtual Kaluza-Klein gravitons at the Tevatron Run I and LEP II [@Abreu:2000vk; @Abbiendi:2000hh; @Acosta:2002eq; @Abazov:2003gp]. For a comprehensive recent review of these constraints see, for example, [@Hewett:2002hv]. The LHC, with a centre of mass energy of 14 TeV, offers a good opportunity for black hole production if ${\ensuremath{M_{\rm PL}}}\sim$ TeV. The very large cross section for production of black holes not too much heavier than the fundamental Planck scale corresponds to a production rate of a few Hertz at the LHC design luminosity.
In the following sections, the process of the black hole production and decay is reviewed (section \[sec:bhdecay\]), followed by a description of the [[`CHARYBDIS`]{}]{} [@Harris:2003db; @Harris:2003eg] event generator (section \[sec:event\]). We then present a review of the principal theoretical uncertainties (section \[sec:ModelUncertainties\]) before moving on to experimental discussions. The characteristics of black hole decays are presented in section \[sec:character\] followed by a discussion of the measurement of the black hole mass in section \[sec:mass\]. We then discuss ways of determining the Planck mass (section \[sec:MPlanck\]) and finally, in section \[sec:ExtraD\], we study methods of determining the number of extra dimensions. Throughout, we have used the ATLAS fast simulation software [@Std:Atlfast2.0] to give a description of a typical detector and we have used the full simulation [@Std:FullSim] to verify the main results.
Black hole production and decay {#sec:bhdecay}
===============================
In the black hole event generator [[`CHARYBDIS`]{}]{}, which has been used in these studies, the black hole production is treated as a semi-classical process (black hole mass, ${\ensuremath{M_{\text{BH}}}}\gg {\ensuremath{M_{\rm PL}}}$) and it is assumed that the extra dimensions are large ($\gg r_{\rm BH}$). For black hole masses close to [$M_{\rm PL}$]{} this semi-classical approximation is not valid and a theory of quantum gravity would be required to calculate the cross section. To be within the semi-classical domain we restrict the mass of the black hole to be ${\ensuremath{M_{\text{BH}}}}\ge 5 \, {\ensuremath{M_{\rm PL}}}$. By geometrical arguments the semi-classical parton-level cross section for black hole production would be [@Banks:1999gd] $$\label{eq:geometry}
\hat{\sigma} (\hat{s} = {\ensuremath{M_{\text{BH}}}}^2)\approx \pi \, r^2_{\text{BH}}$$ where $\sqrt{\hat{s}}$ is the centre-of-mass energy of the colliding particles (see [@Harris:2003db equation 2.4]).
The radius for a Schwarzschild black hole is $$r_{\rm BH} = \frac{1}{\sqrt{\pi}{\ensuremath{M_{\rm PL}}}}\left(\frac{{\ensuremath{M_{\text{BH}}}}}{{\ensuremath{M_{\rm PL}}}}\right)^{\frac{1}{n+1}}\left(\frac{8\Gamma\left(\frac{n+3}{2}\right)}{n+2}\right)^{\frac{1}{n+1}}
\label{eq:radius}$$ where we have used the convention ${\ensuremath{M_{\rm PL}}}^{n+2} = 1/G_{(n+4)}$ where $G_{(n+4)}$ denotes the $n+4$ dimensional Newton’s constant [@Dimopoulos:2001hw], so for a fixed black hole mass, the cross section is lower for a higher number of extra dimensions. This convention has been used throughout this paper.
The decay of a spinning black hole comprises three phases [@Giddings:2001bu]: 1) the *balding phase*, in which the black hole loses its ‘hair’ (associated with the multipole moments) by the emission of radiation; 2) the *evaporation phase*, which starts with a brief spin-down phase, shedding away its angular momentum, followed by the *Schwarzschild phase*, emitting a large number of quanta which reduce the mass of the black hole; 3) finally the *Planck phase* (also called the *remnant decay*), when [$M_{\text{BH}}$]{} approaches [$M_{\rm PL}$]{}, in which the final decay takes place by the emission of a few quanta.
[[`CHARYBDIS`]{}]{} only models the Schwarzschild phase which is expected [@Giddings:2001bu] to account for the greatest proportion of the mass loss. The energy spectrum of decay products is approximately black body with corrective ‘grey-body’ factors [@Harris:2003eg], $\gamma$, which the generator includes. The spectrum for a *fixed* temperature black hole is $$\frac{dN}{dE} \propto \frac{E^2 \gamma}{\left(e^{E/T_H} \mp 1\right) T_H^{n+6}}
\label{eq:Spectrum}$$ the denominator includes a spin-statistics term which is $-1$ for bosons and $+1$ for fermions. The energy spectrum has a characteristic Hawking temperature, $T_H$, which is given by $$T_H = \frac{n+1}{4\pi r_{\text{BH}}}
\label{eq:Temperature}$$ and is thus related to the black hole mass and the number of extra dimensions by $$\log T_H = \frac{-1}{n+1}\log {\ensuremath{M_{\text{BH}}}}+ \text{constant}
\label{eq:TempMass}$$ where the constant is dependent only on $n$ and [$M_{\rm PL}$]{}. The generator can also model the time dependence in which case $T_H$ is recalculated after each emission (so as the black hole decays it gets hotter). Otherwise the initial $T_H$ is used throughout the decay.
Due to their high mass, black hole decays are very spectacular events with a large visible transverse energy, large multiplicity, and high sphericity with many hard jets and leptons. Since most of the black hole decay products result from the evaporation phase, as visible Standard Model particles, the ratio of the total hadronic to leptonic activity is expected to be roughly 5:1 [@Giddings:2001bu]. A few hard quanta with energy a sizable fraction of the [$M_{\rm PL}$]{} are also expected from the final Planck decay phase [@Giddings:2001bu].
The theoretical work to date has been done in the semi-classical approximation. This approximation is only valid if ${\ensuremath{M_{\text{BH}}}}\gg {\ensuremath{M_{\rm PL}}}$, ${\ensuremath{M_{\text{BH}}}}\gg T_H$ and the average multiplicity is large, $\langle N \rangle \gg 1$. This approximation can only be valid at the LHC if the Planck mass is low and even then, there will be problems if the number of dimensions is large (since this gives a temperature close to the Planck mass and thus low multiplicity).
Event generation and detector simulation {#sec:event}
========================================
[[`CHARYBDIS`]{}]{} has been used to generate Monte Carlo event samples. It is interfaced, via the Les Houches accord [@Boos:2001cv], to [`HERWIG`]{} [@Corcella:2000bw; @Corcella:2002jc] to perform the parton shower evolution of the partons produced in the decay and their hadronization. The generated events are then passed through the ATLAS fast simulation, [`ATLFAST`]{} [@Std:Atlfast2.0], in order to give a reasonable description of detector resolution and efficiency.
Unless otherwise mentioned, the [[`CHARYBDIS`]{}]{} options were set as follows:
- [Time variation of the black hole temperature was on ([`TIMVAR=TRUE`]{}).]{}
- [Grey-body effects were on ([`GRYBDY=TRUE`]{}).]{}
- [The black hole was allowed to decay to all Standard Model particles including Higgs particles ([`MSSDEC=3`]{}).]{}
- [Kinematic cut-off was turned off ([`KINCUT=FALSE`]{}).]{}
- [The number of particles in the remnant decay was 2 ([`NBODY=2`]{}).]{}
This set of options together with the Planck mass set to 1 TeV is called the ‘test case’ and we have used this to illustrate our techniques. Several samples have been generated, so to avoid confusion the number of dimensions is always specified. If a mass is given, then the generator was forced to produce black holes with a fixed mass, otherwise the range was set to 5000–14000 GeV.
Table \[tab:crossbh\] summarises the black hole production cross sections at the LHC for $n = 2, \,4$, and 6 with ${\ensuremath{M_{\rm PL}}}=1$ TeV.[^2]
Model uncertainties {#sec:ModelUncertainties}
===================
The theory of black hole production and decay contains many uncertainties and assumptions, particularly at LHC energies. A clear understanding of these is therefore essential in order for our analyses to be as widely applicable as possible. In this section we review these uncertainties.
Production cross section {#sec:CrossSection}
------------------------
The process of black hole production in hadron collisions is subject to a number of basic uncertainties. The order of magnitude of the parton-level cross section should be given by equation \[eq:geometry\], but the form factor relating the left- and right-hand sides is uncertain and would be expected to be $n$-dependent. Classical numerical simulations [@Yoshino:2002tx] suggest values in the range 0.5–2, increasing with $n$. These values are not included in the [[`CHARYBDIS`]{}]{} generator, but we take them into account when cross section data are used in our analysis (in sections \[sec:MPlanck\] and \[sec:ExtraD\]).
More fundamentally, the transition from the parton-level to the hadron-level cross section is based on the factorization formula $$\sigma(S) = \int dx_1\,dx_2\,f(x_1)f(x_2)\hat\sigma(\hat s=x_1 x_2 S)$$ where $f(x)$ is the parton distribution function (PDF) summed over parton flavours. The validity of this formula in the trans-Planckian energy region is unclear. Even if factorization remains valid, the extrapolation of the PDFs into this region based on Standard Model evolution from present energies is questionable. Also, comparison to Standard Model processes in the trans-Planckian regime would be difficult since perturbative physics would be suppressed.
The first stages of decay {#sec:EarlyStages}
-------------------------
[[`CHARYBDIS`]{}]{} does not model the initial balding or spin-down phases of the black hole decay. The amount of energy emitted from the black hole during these phases is expected to be small [@Giddings:2001bu] so such an omission should not be significant. However, it is probable that the energy spectrum will be modified at low energies.
Deposition on the brane
-----------------------
Estimates vary as to how much energy is expected to be emitted into the bulk via graviton emission, but it could be significant. One estimate suggests that the fraction of energy emitted into the bulk could be as high as 20% for $n=2\text{--}4$ rising to nearly 50% for $n=7$ [@Harris:Thesis]. Any energy emitted into the bulk will make an accurate measurement of the mass extremely difficult. Although this effect could in principle be observed as a change in the expected shape of the cross section as a function of black hole mass, determining this would be experimentally challenging. In these studies we have assumed that all the energy is deposited on our brane. A modified generator and a more detailed study would be necessary to understand the full impact of this assumption.
Kinematic limit {#sec:KinematicLimit}
---------------
A black hole can only emit a particle with an energy up to half of its mass in order to conserve energy–momentum. However, the grey-body distribution used to describe the Hawking radiation extends to infinite energy. Although the distribution is only valid for very massive (${\ensuremath{M_{\text{BH}}}}\gg {\ensuremath{M_{\rm PL}}}$) black holes it is still necessary to deal with the black holes as they become lighter. It is expected that given a full theory, the distribution would be modified to take this into account, but we have no such theory.
Figure \[fig:GenCorr\] shows the energy of the primary generator level decay products in the rest frame of the black hole. As can be seen, the kinematic limit affects most of the decays. This greatly modifies the energy spectrum and also leaves open the question of what to do when the generator chooses an unphysical decay, i.e. when it samples from the energy spectrum above the kinematic limit. Two options are implemented in [[`CHARYBDIS`]{}]{}. In the first case ([`KINECUT=FALSE`]{}), if an unphysical decay is chosen, it is thrown away and a new one is chosen. This process continues until the black hole has a mass less than the Planck mass at which point the decay moves to the final, remnant, stage. In the other option ([`KINECUT=TRUE`]{}), when an unphysical decay is chosen, the black hole decay moves straight to the final stage. The final stage of the decay is dealt with in the next section. It should be noted that for high temperature black holes, where the probability of an unphysical decay is large, the different choices implemented in the generator will lead to a large difference in the multiplicities and will have a significant impact of the energy distributions.
Remnant decay {#sec:RemnantDecay}
-------------
At the end of the evaporation phase, a light Planck scale black hole, called a remnant, remains which the generator must decay. How this would happen is unknown and will only be predicted by a quantum theory of gravity. [[`CHARYBDIS`]{}]{} implements this ‘remnant’ decay as an isotropic decay into 2–5 bodies (the number is an option). When the remnant decay occurs depends on the option chosen for handling the kinematic limit as described in section \[sec:KinematicLimit\]. It should be noted therefore, that the uncertainty here can easily affect the multiplicity and energy spectra. One example of an affected experimental observable is the photon energy spectrum. Figure \[fig:nbody\] shows the photon energy distributions (of all photons in the event) for 2-body and 4-body remnant decays for two values of $n$. Even for $n=2$ there is a noticeable effect, but for $n=4$, the effect is large.
Time-variation and black hole recoil {#sec:TimeVar}
------------------------------------
It has been argued [@Dimopoulos:2001hw] that due to the speed of the decay, the black hole does not have enough time to equilibrate between emissions and therefore that the time variation of the temperature can be ignored. Therefore, the initial Hawking temperature might be measured by fitting Planck’s formula for black-body radiation to the energy spectrum of the decay products for different bins in the initial black hole mass. Using equation \[eq:TempMass\] the number of dimensions can then be extracted. This is the approach taken at a theoretical level in [@Dimopoulos:2001hw].
To illustrate this procedure, we have used the test case with $n = 2$. Events were generated without grey-body factors in 500 GeV mass bins between 5000 and 10000 GeV. For each mass bin we have fitted the black-body spectrum to the generator level electron energy. Figure \[fig:tempnot\]a shows the result of this together with the fit using equation \[eq:TempMass\] from which we determine $n=1.7\pm 0.3$. Figure \[fig:tempnot\]b shows the result of the same procedure and the same test case but with time dependence turned on. In this case we determine $n=3.8 \pm 1.0$ which is well away from the model value. Time dependence is therefore a systematic effect with a strong impact on any measurement of $n$.
Another effect that has not been taken into account in previous studies is the recoil of the black hole. When a particle is emitted from the black hole, the black hole recoils against it. Therefore the next emission is in a boosted frame. Even in the case of a fixed temperature decay, the effects of recoil become more significant as the decay progresses and the black hole gets lighter. This is exacerbated in the time varying case since the black hole also gets hotter as it decays. Any analysis which makes use of the energy spectrum should therefore account for this.
Characteristics of the black hole decay {#sec:character}
=======================================
Black hole decays in the semi-classical limit have high multiplicity. However at LHC energies black holes would be on the edge of the semi-classical limit (depending on $n$) which can reduce the multiplicity and make predictions uncertain. This effect can be seen in figure \[fig:multiplicity\] which shows that the multiplicity decreases significantly with $n$. This is due to fact that $T_H$ is higher for larger $n$ at the same mass.
A black hole decay is also characterised by a large total transverse energy (figure \[fig:sumet\]) which increases as the black hole mass increases. Even the low multiplicity events tend to be rather spherical with high multiplicity events more so. These characteristics are very different from standard model and SUSY events which do not have the same access to very high energies and tend to produce less spherical events. Therefore, we believe that selecting events with high [$\sum\!p_T$]{}, high multiplicity ($>4$) and high sphericity will give a pure set of black hole events. In addition, it should be noted that the already small Standard Model background will be suppressed by the black hole production [@Dimopoulos:2001hw]. There are two further characteristics which will be interesting to measure and confirm the nature of the events: the missing $p_T$ ([${\cancel{p}}_T$]{}) distribution and the charge asymmetry.
[${\cancel{p}}_T$]{} distribution
---------------------------------
Although not all black hole decays contain neutrinos, some will have one or more with energies that can be as high as half the black hole mass. The missing energy can be even larger than for much of SUSY parameter space. In contrast, most of the Standard Model processes tend to have much lower missing transverse momenta. Figure \[fig:missptcom\] presents the distribution of the [${\cancel{p}}_T$]{} for Standard Model QCD events (with generator level cut $p_T >$ 600 GeV), SUSY events (at LHCC SUGRA point 5 [@LHCC:SUGRA]), and 5 TeV black holes with $n=2$ and 6.
Black hole charge
-----------------
Black holes are typically formed from valence quarks, so it is expected that the black holes would be charged. The average charge is somewhat energy dependent, but should be $\sim +2/3$. The rest of the charge from the protons is expected to disappear down the beam pipes or at very high $|\eta|$. The average black hole charge, $\langle Q_{\text{BH}}\rangle$, can be measured by determining the average charge of the charged leptons, $\langle Q_{\text{Lept}}\rangle$, which should be equal to the black hole charge times the probability of emitting a charged lepton. Figure \[fig:leptonCharge\] shows such a measurement for the test case with $n=2$ which gives $\langle Q_{\text{Lept}}\rangle = 0.1266\pm0.002$ and thus $\langle Q_{\text{BH}}\rangle = 0.654\pm0.008$ using the expected charged lepton emission probability of 0.1936.
Kinematic distributions {#sec:spectra}
-----------------------
The authors of [@Mocioiu:2003gi] have studied the hadronic decay of a black hole and found that the transverse momentum distribution of charged hadrons depends weakly on the number of large extra dimensions. In addition to the event multiplicity and transverse momentum distribution, figure \[fig:pt8\], we have also looked at the average $p_T$ of the events, jets, leptons, and the ratio of the difference and sum of the $i^{th}$ and the $j^{th}$ highest $p_T$ jet ($i \, ,j = 1, \, 2, \, 3, \, 4$) and found that these variables also depend only weakly on $n$. It is therefore not possible to get a constraint on $n$ using these distributions.
Event shape variables {#sec:shape}
---------------------
In addition to the event multiplicity and spectra, we have studied the following event shape variables: the sphericity [@Bjorken:1970wi], thrust [@Brandt:1964sa], and the Fox-Wolfram moment ratios [@Fox:1979id]. Since the sphericity ($S$) and thrust ($T$) are sensitive to underlying event and longitudinal motion, we have used the corresponding quantities for transverse momenta only.
Defining the Fox-Wolfram moments [@Fox:1979id] $$\label{eq:fox}
H_l = \sum_{i,j} \frac{\left| \mathbf{p}_i \right|
\left| \mathbf{p}_j \right|}{E^2_{\rm vis}}
P_l\left( \cos \theta_{ij} \right) \; , \; \; \; \; \;
l = 1, \, 2, \, 3, \, ...$$ the Fox-Wolfram moment ratios can be expressed as $H_l/H_0$, where $\theta_{ij}$ is the opening angle between particles $i$ and $j$, $E_{\rm vis}$ is the total visible energy of the event, and $P_l(x)$ are the Legendre polynomials.
Figures \[fig:eventshape\] and \[fig:eventfox\] show the distribution of the event shape variables for 5 and 8 TeV black holes with $n=2 - 6$. The distributions are relatively similar for higher values of $n$, making it hard to distinguish them from one another. Events are relatively spherical with a high transverse energy of a few [$M_{\rm PL}$]{}. For higher dimensions, the events become significantly less spherical and are more susceptible to variations in the treatment of the remnant decay.
Measurement of the black hole mass {#sec:mass}
==================================
The black hole 4-momentum is reconstructed simply by summing the 4-momenta of all of the particles in the event. We have illustrated this procedure for selected black hole mass points 5 and 8 TeV with $n = 2$ to 6 which were generated with masses ranging 200 GeV above and below the selected mass point. Events were selected by requiring at least 4 jets, all within the acceptance of the ATLAS tracking detector ($|\eta|<2.5$). Requiring the multiplicity to be greater than 4 ensures that the remnant decay will not be too dominant. The transverse momenta of the three highest $p_T$ jets was required to be above 500, 400, and 300 GeV respectively.[^3] In order to improve the reconstructed mass resolution, events were rejected if the missing transverse momentum was greater than 100 GeV.
The reconstructed Gaussian mass resolution and the overall signal efficiency (the fraction of accepted events) after the selection cuts for 5 and 8 TeV black hole in $n = 2$, 4 and 6 are given in table \[tab:efftable\] with sample plots in figure \[fig:massRes\]. The mass resolution can be improved slightly by raising the threshold of the jet $p_T$, but at the cost of a sharp drop in overall signal efficiency.
Measurement of the Planck mass {#sec:MPlanck}
==============================
Some authors [@Dimopoulos:2001hw] have suggested that since $n$ can be determined from the $T_H$–[$M_{\text{BH}}$]{} relationship (equation \[eq:TempMass\]), [$M_{\rm PL}$]{} can be measured from the normalisation of the temperature. For reasons outlined in the next section, we choose not to use this method but instead to follow the suggestion of [@ATL-PHYS-2003-037] and determine [$M_{\rm PL}$]{} from the cross section. In the convention used in this paper (and also in [@ATL-PHYS-2003-037]), the cross section is largely independent of $n$. Figure \[fig:CrossSection\] shows the parton-level cross section including the corrective form factors calculated in [@Yoshino:2002tx]. As can be seen, there is very little variation with $n$.
Due to the very high statistics available, the measurement of the parton-level cross section will be dominated by the various systematic errors. The main experimental error will be the luminosity which should be measured to 5% or better, together with some uncertainty in the efficiency. This is however likely to be small compared to the theoretical uncertainties discussed in section \[sec:CrossSection\]. We therefore conservatively estimate that the parton-level cross section could be determined to 20% which, for our test case of ${\ensuremath{M_{\rm PL}}}=1$ TeV, gives an error in [$M_{\rm PL}$]{} of about 10%. Obviously, the optimal approach is to fit the cross section and temperature data simultaneously and this will be demonstrated at the end of section \[sec:KLAnalysis\]. It is also possible that other processes and observations of new physics at the Planck scale may provide independent measurements of [$M_{\rm PL}$]{}.
Determination of the number of extra dimensions {#sec:ExtraD}
===============================================
Measuring the number of extra dimensions is not a straight-forward task given the uncertainties outlined in section \[sec:ModelUncertainties\]. One technique that has been suggested [@Dimopoulos:2001hw] uses the energy spectrum of electrons and photons below ${\ensuremath{M_{\text{BH}}}}/2$. However the authors of [@Dimopoulos:2001hw] ignore the likely effects on the low energy spectrum from the initial parts of the decay (section \[sec:EarlyStages\]), the effect of the remnant decay (section \[sec:RemnantDecay\]) and the recoil of the black hole (section \[sec:TimeVar\]). Their analysis is particularly sensitive to these effects because they were attempting to use the variation of $T_H$ with [$M_{\text{BH}}$]{} to measure $n$. To give some numerical estimates, the expected variation, $T_H(10~\text{TeV})-T_H(5~\text{TeV})$ is about 40 GeV for $n=2$ and 20 GeV for $n=5$ given $T_H(5~\text{TeV}) \sim 200$ GeV. This is clearly not a large variation to attempt to measure.
It should also be noted that the differences between different number of dimensions become less significant for higher $n$ due to the power law nature of equation \[eq:TempMass\]. We have investigated a number of variables and techniques whilst considering the effects of the many uncertainties.
Correlations
------------
Notwithstanding the comment above, we initially tried to make maximum use of the ${\ensuremath{M_{\text{BH}}}}$–$T_H$ relationship. In the case that $T_H$ varies with time, there is one ${\ensuremath{M_{\text{BH}}}}$–$T_H$ point per emission, rather than just one per event. We therefore developed a technique that used all the information. The method is:
1. [Reconstruct the black hole from all the particles in the event.]{}
2. [Determine the first(next) particle to be emitted.]{}
3. [Use the measured properties of this particle to determine the temperature of the black hole.]{}
4. [Record this mass–temperature point.]{}
5. [Reconstruct the black hole for the next stage using all the particles except for those that have been emitted]{}
6. [Repeat steps 2–5 until there are no particles left.]{}
There are two key parts to this algorithm: determining the order of the emitted particles and using the particle properties to determine the temperature. It was hoped that a method might be found which when averaged over many events would give the correct ${\ensuremath{M_{\text{BH}}}}$–$T_H$ relation.
The method used to determine the order was to assume that the softest particles were emitted first. This is because as the black hole decays it gets hotter, and so the average energy of the emitted particles should increase.
The most probable energy and thus $p_T$ for the emitted particle is proportional to the temperature of the black hole. Since the expected ${\ensuremath{M_{\text{BH}}}}$–$T_H$ relation is a power law, the ${\ensuremath{M_{\text{BH}}}}$–$p_T$ relation should have the same dependence on $n$. Thus we have plotted the average $p_T$ of the emitted particles for each mass bin.
Figure \[fig:corrSim\] shows the result of this method. The shape of the graph is very different from that expected, but does show a separation between the numbers of dimensions. This is perhaps not surprising since all black hole decays will at some point be affected by the kinematic limit problem to which this technique is particularly sensitive. Unfortunately this means that extracting the number of dimensions is extremely difficult to do without relying purely on Monte Carlo data. Further details on this method are given in [@Palmer:Thesis].
Kinematic limit {#sec:KLAnalysis}
---------------
In this section we present a new idea that shows a strong variation with the number of dimensions and is valid for many different scenarios. It may therefore allow $n$ to be determined despite the many uncertainties that affect black hole decays.
If a particle is emitted with an energy close to the kinematic limit (i.e. $E\sim {\ensuremath{M_{\text{BH}}}}/2$), then that particle is probably the first to be emitted. In particular, it is possible to measure the fraction of events, $p$, where the highest energy particle has an energy, $E_{max} > E_{cut}$ where $E_{cut} = {\ensuremath{M_{\text{BH}}}}/2 - E_d$ and $E_d$ is a parameter of the analysis that can be chosen, but should be small. The probability of the first emission being greater than $E_{cut}$ can also be calculated from integrating the Planck spectrum (equation \[eq:Spectrum\]) thus there is a reasonably direct connection between the experimental measurement and theory.
This method has many advantages: since it deals with the first emission, the probability of $E_{max}>E_{cut}$ will be the same regardless of whether the black hole temperature is time-varying or not. Also, since we restrict ourselves to black hole with masses much larger than the Planck mass, this measurement will not be affected by Planck scale effects or the remnant decay. Any effects which modify the low energy part of the spectrum should also have no effect. Finally, the effect of the black hole boost can be taken into account by determining $E_{max}$ in the black hole rest frame.
This technique is however strongly affected by the uncertainty in dealing with the kinematic limit (see section \[sec:KinematicLimit\]) since we do not know what the shape of the energy spectrum would be near the kinematic limit. Fortunately this uncertainty can be controlled by putting upper and a lower bounds on the theoretical estimate of $p$. The lower bound is set by assuming that none of the unphysical region of the Planck spectrum corresponds to an emission with energy greater than $E_{cut}$ in the actual energy spectrum. The lower bound is therefore $$p_{lower} = k\int^{{\ensuremath{M_{\text{BH}}}}/2}_{E_{cut}} P(E) dE$$ where $k$ is a normalising constant, $P(E)$ is the Planck spectrum (equation \[eq:Spectrum\]) and we have ignored the effects of the ‘grey-body’ factors (which are largest at low energy). The difference between fermions and bosons is small here. The upper bound uses the opposite assumption to the lower bound: that is, that all of the unphysical region in the Planck spectrum would correspond to an emission with energy greater than $E_{cut}$ in the actual energy spectrum. The upper bound is therefore $$p_{upper} = k\int^{\infty}_{E_{cut}} P(E) dE\,.$$ It should be noted that at very high energies and in the semi-classical limit, $p\rightarrow0$ as ${\ensuremath{M_{\text{BH}}}}\rightarrow\infty$ for any fixed $E_d$. These equations are therefore only valid when $p$ is small.
There are two competing effects: we would like to set $E_{cut}$ as low as possible so that the upper and lower bound are similar (note that they move apart as $E_{cut} \rightarrow {\ensuremath{M_{\text{BH}}}}/2$). However, the lower we set the cut, the greater the chance that the highest emission is not the first one. Whilst this is not in itself a problem, it does mean that the differences between the time varying and non-time varying cases become more pronounced. In the rest of this study we have set $E_d = 400$ GeV — setting it lower is difficult as experimental resolution effects start to dominate. The chance of a soft emission is strongly temperature dependent. Points have been omitted if this chance is greater than 50% which affects only $n=2$, 5 TeV black holes, but would be more significant at lower [$M_{\rm PL}$]{}.
We have included an approximate corrective factor for the emission of a first soft particle. This was calculated as $$k_{cor} = k\int^{x E_{cut}}_0 P(E) dE$$ giving $p_{cor} = (1+k_{cor}) p$. It is possible for a first emission of any energy to be followed by an emission of energy up to the kinematic limit, but the probability of this drops sharply if the first emission has an energy greater than $2 E_{cut}$. The correction factor, $x$, was therefore set to 2 for the lower bound and 3 for the upper bound.
A plot of the upper and lower bounds on $p$ for ${\ensuremath{M_{\rm PL}}}{}=1$ TeV is shown in figure \[fig:KLBands\]. As can be seen, if [$M_{\rm PL}$]{} is known, there is good potential for extracting $n$ if it is below 5 and could be constrained if $n$ is larger.
One experimentally tricky aspect of this measurement is that since the black hole mass is measured by adding all the particles in the event, the maximum energy in the reconstructed black hole rest frame must be less than ${\ensuremath{M_{\text{BH}}}}/2$. This introduces a bias to low energies in the measurement of $E_{max}$ and thus reduces the measurement of $p$. This has been corrected for by increasing $E_d$ by 100 GeV which is an estimate for the average mis-measurement of $E_{max}$. It should be noted that not boosting into the black hole rest frame would give a very significant over-measurement of $p$. In addition, to ensure that the black hole mass was well measured, events were excluded if they had ${\ensuremath{{\cancel{p}}_T}}{} > 100$ GeV, had any particles with $|\eta|>2.5$ or if the three highest energy particles accounted for more than 95% of the total energy in the black hole rest frame.
Figure \[fig:KLModelInvariant\] shows the results of this analysis for four different models all with $n=4$. The theoretical upper and lower limits are also shown. For models which have the kinematic cut off (see section \[sec:KinematicLimit\]), we would expect them to be consistent with the lower limit; this is the case for plots a, b and d in figure \[fig:KLModelInvariant\]. However, if the kinematic limit is on and the remnant decay is set to 2-body (plot c), we would expect the data to be consistent with the upper limit. This is because if an energy is chosen in the forbidden region (which is the additional region included in the upper limit integral), it will cause the decay to be terminated and the black hole to split into two. One of the two remnant decay particles must pass the cut. As can be seen from the figure, the plots do agree with these expectations.
Figure \[fig:KLNDependant\] shows similar plots to figure \[fig:KLModelInvariant\], but for the test case (see section \[sec:event\]) with different values of $n$. This emphasises that this technique is sensitive to $n$ whilst being largely model independent. Note that at high $n$, the data start to drop below the lower limit. This is due to the high temperature here ($T_H\sim470$ GeV for $n=5$) which significantly reduces the multiplicity. This result suggests that this analysis has an upper limit of validity in the region of $T_H\sim450$–500 GeV. Also, from figure \[fig:KLNDependant\]a, it can be seen that at low temperatures, this analysis will not measure the temperature (unless lighter black holes are seen, or very high statistics are available), but would instead place an upper limit on it. This may be enough to constrain $n$, or alternatively, measuring the energy distribution may be more successful here and these analyses could be combined to give a temperature measurement.
At higher temperatures, this technique becomes strongly sensitive to the temperature of the black hole. Indeed the plots above can easily be converted into temperature as a function of [$M_{\text{BH}}$]{}. This has been done for the test case with $n=4$ in figure \[fig:KinematicLimitTemp\] which also includes a band equivalent to the upper and lower limits on $p$. The band includes a systematic uncertainty on the measurement of the black hole mass of $\pm200$ GeV. Note that as suggested at the beginning of this section, the temperature variation with [$M_{\text{BH}}$]{} is not determined sufficiently to constrain $n$ without first measuring [$M_{\rm PL}$]{}. So instead, we fix the normalisation of $T_H$ at the black hole mass at which it is best measured. In this case, we take $T_H=340\pm30$ GeV at ${\ensuremath{M_{\text{BH}}}}=7000$ GeV. This measurement has been taken together with the parton-level cross section with an error of 20% (see section \[sec:MPlanck\]) and used to determine the model parameters $n$ and [$M_{\rm PL}$]{} in figure \[fig:KinematicLimitResult\]. In this case, this gives an error on the determination of $n$ of 0.75 and an error on [$M_{\rm PL}$]{} of 150 GeV. These results are indicative of how well this analysis can do. If the cross section error were reduced to 10%, the error on [$M_{\rm PL}$]{} would be 70 GeV and on $n$, 0.6, showing that, as expected, the cross section dominates the determination of [$M_{\rm PL}$]{}.
Any improvement in our understanding of how the distribution above the kinematic limit should be handled, or how the remnant would decay, would greatly improve this analysis by reducing the width of the bands in figures \[fig:KLModelInvariant\], \[fig:KLNDependant\] and \[fig:KinematicLimitTemp\].
Conclusions
===========
We have discussed the many theoretical uncertainties that can affect black hole decays and shown that in at least one case, these can lead to systematic mis-measurements of the number of extra dimensions if the analyses previously suggested are used. We have then shown the characteristics of black hole decays as they would be measured in the ATLAS detector. A number of different attempts to determine the model parameters have been discussed and a new technique has been introduced. This new technique has been shown to control many of the theoretical uncertainties and can be used to measure the black hole temperature. We have applied this technique to our test case with four extra dimensions and found the temperature to be $340\pm30$ GeV for a black hole mass of 7 TeV. This was combined with the parton-level cross section, assumed to be known to 20%, to give estimates of the Planck mass and the number of extra dimensions. We conclude that in this case the Planck mass can be determined to 15% and the number of extra dimensions to $\pm$0.75, with strongly correlated errors.
We thank members of the Cambridge SUSY Working Group; in particular C. G. Lester, for many useful suggestions and discussions. BRW thanks the CERN Theory Group for hospitality during part of this work. We thank the ATLAS Collaboration for the use the ATLAS physics analysis framework and tools which are the result of collaboration-wide efforts. This work was funded by the U.K. Particle Physics and Astronomy Research Council.
[^1]: Limits are given in the convention of [@Dimopoulos:2001hw] which is used throughout this paper (see section \[sec:bhdecay\]).
[^2]: In this analysis, we have used the MRSD-’ (DIS) parton distribution function (PDF) set [@Martin:1993zi] with $Q^2 = 1 / r^2_{\text{BH}}$, where $Q^2$ is the momentum scale squared at which a PDF is evaluated.
[^3]: A reconstructed jet was required to have a minimum momentum of 10 GeV within an $\eta-\phi$ cone of radius 0.4.
|
---
abstract: 'We consider the differential equations $y''''=\lambda_0(x)y''+s_0(x)y,$ where $\lambda_0(x), s_0(x)$ are $C^{\infty}-$functions. We prove (i) if the differential equation, has a polynomial solution of degree $n >0$, then $\delta_n=\lambda_n s_{n-1}-\lambda_{n-1}s_n=0,$ where $ \lambda_{n}= \lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1}\hbox{ and }\quad s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1},\quad n=1,2,\dots.$ Conversely (ii) if $\lambda_n\lambda_{n-1}\ne 0$ and $\delta_n=0$, then the differential equation has a polynomial solution of degree at most $n$. We show that the classical differential equations of Laguerre, Hermite, Legendre, Jacobi, Chebyshev (first and second kind), Gegenbauer, and the Hypergeometric type, etc, obey this criterion. Further, we find the polynomial solutions for the generalized Hermite, Laguerre, Legendre and Chebyshev differential equations.'
address:
- '$^1$ Department of Mathematics and Statistics, University of Prince Edward Island, 550 University Avenue, Charlottetown, PEI, Canada C1A 4P3.'
- '$^2$ Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard West, Montréal, Québec, Canada H3G 1M8'
- '$^3$ Gazi Universitesi, Fen-Edebiyat Fakültesi, Fizik Bölümü, 06500 Teknikokullar, Ankara, Turkey.'
author:
- 'Nasser Saad$^1$, Richard L. Hall$^2,$ and Hakan Ciftci$^3$'
title: Criterion for polynomial solutions to a class of linear differential equation of second order
---
CUQM-117\
math-ph/0609035\
\#1
0.2in [*Keywords*]{}: classical orthogonal polynomials, asymptotic iteration method, polynomial solutions of differential equation.
Introduction
============
The question as to whether a second order linear homogeneous differential equation has a polynomial solution has attracted much interest since the early classification of Bochner of othogonal polynomials [@bo]. In 1929, Bochner posed a problem of determining all families of orthogonal polynomials that are solutions of the differential equation $$\label{eq1}
(ax^2+bx+c) y''_n(x)+(dx+e)y_n'(x)-\mu_ny_n(x)=0.$$ Bochner found that, up to a linear change of variable, only the classical polynomials of Jacobi, Laguerre and Hermite and the Bessel polynomials satisfied a second-order differential equation [@gh]-[@wa] of the form (\[eq1\]). In general, the question as to which second order linear homogeneous differential equation has polynomial solutions (not necessary a sequence of orthogonal polynomials) is not easily answered, since it would involve studying a wide variety of equations, including those with regular and irregular singular points. In this article, we present a simple criterion for the existence of polynomial solutions of a differential equation of the form $$\label{eq2}
y''=\lambda_0 y'+s_0 y$$ where $\lambda_0, s_0$ are $C^{\infty}-$functions. A key feature of the present work is to note the invariant structure of the right-hand side of (\[eq2\]) under further differentiation. Indeed, if we differentiate (\[eq2\]) with respect to $x$, we find that $$\label{eq3}
y^{\prime\prime\prime}=\lambda_1 y^\prime+s_1 y$$ where $\lambda_1= \lambda_0^\prime+s_0+\lambda_0^2$ and $s_1=s_0^\prime+s_0\lambda_0.$ If we find the second derivative of equation (\[eq3\]), we obtain $$\label{eq4}
y^{(4)}=\lambda_2 y^\prime+s_2 y$$ where $\lambda_2= \lambda_1^\prime+s_1+\lambda_0\lambda_1\quad\hbox{ and }\quad s_2=s_1^\prime+s_0\lambda_1.$ Thus, for $(n+1)^{th}$ and $(n+2)^{th}$ derivative, $n=1,2,\dots$, we have $$\label{eq5}
y^{(n+1)}=\lambda_{n-1}y^\prime+s_{n-1}y$$ and $$\label{eq6}
y^{(n+2)}=\lambda_{n}y^\prime+s_{n}y$$ respectively, where $$\label{eq7}
\lambda_{n}= \lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1}\hbox{ ~~and~~ } s_{n}=s_{n-1}^\prime+s_0\lambda_{n-1}.$$ From (\[eq5\]) and (\[eq6\]) we have $$\label{eq8}
\lambda_n y^{(n+1)}- \lambda_{n-1}y^{(n+2)} = \delta_ny {\rm ~~~where~~~}\delta_n=\lambda_n s_{n-1}-\lambda_{n-1}s_n.$$ In an earlier paper [@cs] we proved the principal theorem of the Asymptotic Iteration Method (AIM), namely 0.1in [**Theorem 1:**]{} *Given $\lambda_0$ and $s_0$ in $C^{\infty}(a,b),$ the differential equation (\[eq2\]) has the general solution $$\label{eq9}
y(x)= \exp\left(-\int\limits^{x}\alpha(t) dt\right)
\left[C_2 +C_1\int\limits^{x}\exp\left(\int\limits^{t}(\lambda_0(\tau) + 2\alpha(\tau)) d\tau \right)dt\right]$$ if for some $n>0$ $$\label{eq10}
{s_{n}\over \lambda_{n}}={s_{n-1}\over \lambda_{n-1}} \equiv \alpha.$$* The present article is not about a classification of orthogonal polynomials which is well-established problem in the literature [@bo]-[@at]. Rather, the principal goal of the present paper is to characterize when Eq.(\[eq2\]) has a polynomial solution. In the next section we shall show that the differential equation (\[eq2\]) has a polynomial solution of degree $n$, if for some $n>0$, $\delta_n = 0.$ In Section 3, we show through detailed analysis that the classical differential equations of Laguerre, Hermite, Legendre, Jacobi, Chebyshev (first and second kind), Gegenbauer, and the Hypergeometric type, etc., obey this criterion. In Section 4, we apply the criterion to obtain polynomial solutions to the generalized Hermite, Laguerre, Legendre and Chebyshev differential equations. As we shall show, the criterion presented here works whether or not the differential equation (\[eq2\]) has a set of orthogonal polynomial solutions, or a class of orthogonal polynomial solutions in the quasi-definite sense [@ch].
A criterion for polynomial solutions
====================================
The existence of polynomial solutions is characterized by the vanishing of $\delta_n.$ This is the principal theoretical result of this paper. We have: 0.1in [**Theorem 2:**]{} [*(i) If the second-order differential equation (\[eq2\]) has a polynomial solution of degree $n,$ then $$\label{eq11}
\lambda_n s_{n-1} - \lambda_{n-1} s_n \equiv \delta_n = 0.$$ Conversely (ii) if $\lambda_n\lambda_{n-1}\ne 0,$ and $\delta_n=0,$ then the differential equation (\[eq2\]) has a polynomial solution whose degree is at most $n$.*]{} 0.1in [**Proof:**]{} (i) For the given differential equation (\[eq2\]), if $y$ is a polynomial of degree at most $n$ we have $y^{(n+1)} = y^{(n+2)} = 0.$ consequently we conclude from (\[eq8\]) that $\delta_n = 0.$ (ii) Conversely, if $\delta_n = 0$ and $\lambda_n\lambda_{n-1}\ne 0,$ then we have $s_{n-1}/\lambda_{n-1} = s_{n}/\lambda_{n}\equiv \alpha,$ and, from Theorem 1, we conclude that a solution is given by $y=\exp(-\int\limits^x \alpha(t) dt).$ Therefore, in particular, $y' = -\alpha y = -{s_{n-1}\over \lambda_{n-1}}y$. Consequently, from $y^{(n+1)}=\lambda_{n-1}y'+s_{n-1}y,$ we infer that $y^{(n+1)}=0,$ or, equivalently, that $y$ is a polynomial of degree at most $n$.0.1in Theorem 2 gives us the condition under which the given differential equation has a polynomial solution. Theorem (1), in particular (\[eq9\]), provides a tool for the explicit computation of these polynomials. In the next section, we apply these results to a variety of classes of differential equation: in each case we provide the explicit condition which yields polynomial solutions.
Some differential equations with polynomial solutions
=====================================================
In this section, we apply Theorem 2 to the classical differential equations of mathematical physics. First, we give an alternative proof to Bochner’s results (\[eq1\]), using the criterion developed in Theorem 2. 0.1in [**Theorem 3:**]{} [*The second-order differential equation (\[eq1\]) has a polynomial solution of degree $n$ if $$\label{eq12}
\mu_n=n(d+(n-1)a),\quad\quad n=0,1,2,\dots.$$ The corresponding polynomial solutions are $$\begin{aligned}
y_0&=&1\\
y_1&=&dx+e\\
y_2&=&(d+a)(d+2a)x^2+2(b+e)(d+a)x+e(b+e)+c(d+2a)\\
y_3&=&(d+2a)(d+3a)(d+4a)x^3+3(d+2a)(d+3a)(e+2b)x^2\\
&+&3(d+2a)(b(3e+2b)+c(4a+d)+e^2)x\\
&+&4dbc+e^3+3dec+10aec+2eb^2+3e^2b\\
\dots&=&\dots\end{aligned}$$* ]{} 0.1in [**Proof:**]{} By means of Theorem 2, we find for $\lambda_0=-{dx+e\over ax^2+bx+c}$ and $s_0={\mu\over ax^2+bx+c}$ that the termination condition $\delta_n=\lambda_n s_{n-1}-\lambda_{n-1} s_n =0$ yields $$\label{eq13}
\delta_n=-{1\over (ax^2+bx+c)^{n+1}}\prod_{k=0}^n(k(d+(k-1)a)-\mu_k),\quad n=1,2,\dots$$ which yields for $\delta_n=0$ that $\mu_n=n(d+(n-1)a)$ as required. For $n=0,1,2,\dots$ i.e. $\mu_0=0,\mu_1=d, \mu_2=2(d+a),\dots$, we obtian by $$\label{eq14}
y_n=\exp\bigg(-\int\limits^x {s_{n}(t)\over \lambda_{n}(t)}dt\bigg),\quad n=0,1,2,\dots,$$ the polynomial solutions just mentioned.0.1in
In Table I, we summarize the well-known differential equations which have polynomial solutions (as eigenfunctions). In each case, we give the explicit criterion, $\delta_n=0$, of Theorem 2. 0.2in 0.1in
DE $\lambda_0$ $s_0$ $\delta_n$ $\delta_n=0,\ n=0,1,\dots$
------------------ ----------------------------------------------- --------------------------- ----------------------------------------------------------------------------- ------------------------------
Cauchy-Euler$^1$ ${\alpha(x-b)\over (x-a)^2}$ ${\beta\over (x-a)^2}$ ${(-1)^{n+1}\over (a-x)^{2n+2}}\prod\limits_{i=1}^{n}(\beta+i(1-i+\alpha))$ $\beta=n(n-1-\alpha)$
Hermite$^{2a}$ $2x$ $-2k$ $2^{n+1}\prod\limits_{i=1}^{n}(k-i)$ $k=n$
Hermite$^{2b}$ $ax+b$ $c$ $(-1)^{n+1}\prod\limits_{i=0}^n (c+ia)$ $c=- na$
Laguerre $(1-{1\over x})$ ${a\over x}$ ${(-1)^{n+1}\over x^{n+1}}\prod\limits_{i=0}^{n} (i+a)$ $a=-n$
Confluent$^3$ $(b-{c\over x})$ ${a\over x}$ ${(-1)^{n+1}\over x^{n+1}}\prod\limits_{i=0}^{n} (ib+a)$ $a=-nb$
Hypergeometric ${(a+b+1)x-c\over x(1-x)}$ ${ab\over x(1-x)}$ ${1\over x^{n+1}(x-1)^{n+1}}\prod\limits_{i=0}^{n} (a+i)(b+i)$ $a=-n~ or~ b=-n$
Legendre ${2x\over 1-x^2}$ ${m(m+1)\over x^2-1}$ ${(-1)^n\over (x^2-1)^{n+1}}\prod\limits_{i=0}^{n} {(m^2-i^2)}$ $m=n$
Jacobi ${(\alpha+\beta+2)x+\beta+\alpha\over 1-x^2}$ ${-\gamma\over 1-x^2}$ $\prod\limits_{i=0}^n (i(i+1+\alpha+\beta)-\gamma)$ $\gamma=n(n+\alpha+\beta+1)$
Chebyshev$^{4a}$ ${x\over 1-x^2}$ ${-m\over 1-x^2}$ ${(-1)^{n+1}\over (x^2-1)^{n+1}}\prod\limits_{i=0}^{n} (m-i^2)$ $m=n^2$
Chebyshev$^{4b}$ ${3x\over 1-x^2}$ ${-m\over 1-x^2}$ ${-1\over (x^2-1)^{n+1}}\prod\limits_{i=0}^{n} (i((i+2)-m)$ $m=n(n+2)$
Gegenbauer ${(1+2k)x\over (1-x^2)}$ ${-\lambda\over (1-x^2)}$ ${-1\over (x^2-1)^{n+1}}\prod\limits_{i=0}^n(i(i+2k)-\lambda)$ $\lambda=n(n+2k)$
hyperspherical ${2(1+k)x\over (1-x^2)}$ ${-\lambda\over (1-x^2)}$ ${-1\over (x^2-1)^{n+1}}\prod\limits_{i=0}^n(i(i+1+2k)-\lambda)$ $\lambda=n(n+1+2k)$
${-2(x+1)\over x^2}$ ${\gamma\over x^2}$ ${(-1)^{n+1}\over x^{2n+2}}\prod\limits_{i=0}^n(\gamma-i(i+1))$ $\gamma=n(n+1)$
Generalized
Bessel$^{5b}$ ${-(ax+b)\over x^2}$ ${\gamma\over x^2}$ ${(-1)^{n+1}\over x^{2n+2}}\prod\limits_{i=0}^n(\gamma-i(i-1+a))$ $\gamma=n(n+a-1)$
0.2in [*[**Table I**]{}: Application of AIM to classical differential equations. For each differential equation which give the condition under which it have polynomial solutions.*]{}
Some remarks on Table I
-----------------------
1. This differential equation is a generalization of the Cauchy-Euler linear equation $$\label{eq15}
x^2y''+\alpha xy'+\beta y=0.$$ It is possible, however, to apply AIM to the differential equation (\[eq15\]). The termination condition yields in this case $$\label{eq16}
\delta_n = {(-1)^{n+1}\over x^{2n+2}}\prod\limits_{i=1}^{n}(\beta+i(1-i+\alpha))=0~\quad\hbox{or}\quad \beta=n(n-1-\alpha)$$ while the corresponding polynomials, as given by (\[eq14\]), are $y_0=1$, $y_1= x$, $y_2=x^2,\ \dots, y_n=x^n$. It is clear that these polynomials cannot form an orthogonal-polynomial sequence [@ch]. 0.1in
2. This differential equation can be regarded as a generalization of the well-known Hermite differential equation$^{2a}$. It is an elementary example of differential equation with non-rational coefficients (i.e. with $s_0$ and $\lambda_0$ non-rational) which has nonconstant polynomial solutions for $c\neq 0$. 0.1in
3. This is known as the confluent hypergeometric differential equation. It is also known as Kummer’s differential equation or Pochhammer-Barnes equation [@ed].
4. This differential equation is known as Chebyshev’s differential equation of the first kind and Chebyshev’s differential equation of the second kind, respectively. It is interesting to note that these differential equations are special cases of $$\label{eq17}
(1-x^2)y''-axy'+\mu y =0.$$ If we apply AIM directly to (\[eq17\]), we have by means of the termination condition (\[eq11\]) that $$\delta_n=-{1\over (x^2-1)^{n+1}}\prod\limits_{k=0}^{n} (i(i+a-1)-\mu_i)$$ thus, for $\delta_n=0$, we must have $\mu_n=n(n+a-1)$. The corresponding polynomial solutions, for $n=0,1,2,\dots$, are $y_0=1,y_1=x,y_2=(a+1)x^2-1,y_3=(a+1)x^3-3x,\dots,$ and in general $$y_n= {}_2F_1(-n,n+a-1,{a\over 2},{1-x\over 2})$$ up to a constant. Here, ${}_2F_1$, Gauss’ hypergeometric function, is defined by $$\label{eq18}
{}_2F_1(-n,b;c;x)=\sum\limits_{k=0}^n {(-n)_k (b)_k\over (c)_k k!} x^k,$$ where the Pochhammer symbol $(a)_k$ defined by $$(a)_0=1,\quad (a)_k=a(a+1)(a+2)\dots(a+k-1)={\Gamma(a+k)\over \Gamma(a)}.$$
5. The polynomial solution of these differential equations were studied by Krall and Frink [@kf]. The corresponding (Bessel) polynomial solutions are orthogonal in the quasi-definite sense [@ch].
In Table II we find the corresponding polynomial solutions for each differential equation mentioned in Table I. As an elementary application to quantum mechanics, we consider the Schrödinger equation $$-{d^2\psi\over dr^2}+\bigg(-{A\over r}+{\gamma(\gamma+1)\over r^2}\bigg)\psi =E\psi.$$ Writing $\psi(r)=r^{\gamma+1}e^{-\alpha r}y(r)$, we easily find that $y(r)$ must satisfy, for $E=-\alpha^2,$ the confluent hypergeometric differential equation $$y''(r)=2\bigg(\alpha-{\gamma+1\over r}\bigg)y'(r)+\bigg({-A+2\alpha(\gamma+1)\over r}\bigg)y(r).$$ The termination condition, mentioned in Table I, then yields $E=-\alpha^2= -{A^2\over 4(n+\gamma+1)^2}$, the eigenvalues of Schrödinger’s equation for the Kratzer potential. Furthermore, the corresponding (un-normalized) eigenfunctions are given, by means of Table II, as $$\psi_n(r)=(-1)^nr^{\gamma+1}e^{-\sqrt{-E} r}(2\gamma+2)_n{}_1F_1(-n;2\gamma+2;2\sqrt{-E}r).$$
[*[**Table II**]{}: The corresponding polynomial solutions for each differential equation mentioned in Table I.*]{} 0.1in
DE $y_n,~n=0,1,2,\dots$
---------------- -----------------------------------------------------------------------------------------------------------
Cauchy-Euler $y_0= 1$
$y_1=x-b$
$y_2=(\alpha-1)(\alpha-2)x^2+2(\alpha-1)(2a-\alpha b)x$
+ $\alpha^2b^2-a(2b+a)\alpha+2a^2$
$\dots$
Hermite $y_0(x)=1$
$y_1(x)=x$
$y_2(x)=2x^2-1$
$\dots$
$y_{2n}(x)=(-1)^n 2^n \left({1/ 2}\right)_n\ {}_1F_1\left(-n;\ {1/ 2};\ x^2\right),$
$y_{2n+1}(x)=(-1)^n 2^n \left({3/ 2}\right)_n\ x\ {}_1F_1\left(-n;\ {3/ 2};\ x^2\right)$
Hermite $y_0(x)=1$
$y_1(x)=ax+b$
$y_2(x)=(ax+b)^2-a$
$\dots$
$y_{2n}(x)=(-1)^n (2a)^n\left({1/2}\right)_n\ {}_1F_1\left(-n;\ {a/2};\ {(ax+b)^2/2}\right),$
$y_{2n+1}(x)=(-1)^n (2a)^n \left({3/2}\right)_n\ (ax+b)\ {}_1F_1\left(-n;\ {3a/2};\ {(ax+b)^2/2}\right).$
Laguerre $y_0=1$
$y_1=x-1$
$y_2=x^2-4x+2$
$\dots$
$y_n=(-1)^n n!\ {}_1F_1(-n,1,x)$
Confluent $y_0=1$
$y_1=bx-c$
$y_2=(1 + c)c -2 b(1 + c) x + b^2 x^2$
$\dots$
$y_n=(-1)^n (c)_n\ {}_1F_1(-n,c,bx)$
Hypergeometric $y_0=1$
$y_1=x+c$
$y_2=2x^2+4(c+1)x+c(c+1)$
$\dots$
$y_n=(c)_n~{}_2F_1(-n,-n; c, x)$
DE $y_n,~n=0,1,2,\dots$
-------------------- ------------------------------------------------------------------------------------------
Legendre $y_0=1$
$y_1=x$
$y_2=-1+x^2$
$\dots$
$y_n={}_2F_1(-n,1+n; 1, {(1-x)/2}).$
Jacobi $y_0=1$
$y_1=(\alpha-\beta)+(2+\alpha+\beta)x$
$y_2=(3+\alpha+\beta)(4+\alpha+\beta)x^2+2(\alpha-\beta)(3+\alpha+\beta)x-4-c-d+(c-d)^2$
$\dots$
$y_n={(\alpha+1)_n/ n!}~{}_2F_1(-n,n+\alpha+\beta+1;\alpha+1;{(1-x)/ 2}).$
Chebyshev $y_0=1$
$y_1=x$
$y_2=2x^2-1$
$\dots$
$y_n={}_2F_1(-n,n,{1\over 2},{(1-x)/2})$
Chebyshev $y_0=1$
$y_1=x$
$y_2=4x^2-1$
$\dots$
$y_n=(n+1){}_2F_1(-n,n+2,{3\over 2},{(1-x)/2})$
Gegenbauer $y_0=1$
$y_1=x$
$y_2=2(k+1)x^2-1$
$\dots$
$y_n={(2k)_n}~{}_2F_1(-n,n+2k;k+{1/2};{(1-x)/2})$
hyperspherical $y_0=1$
$y_1=x$
$y_2=(2k+3)x^2-1$
$\dots$
$y_n={(2k+1)_n}~{}_2F_1(-n,n+2k+1;k+1;{(1-x)/2})$
Bessel $y_1(x)=1+x$
polynomials $y_2(x)=1+3x+3x^2$
$y_3(x)=1+6x+15x^2+15x^3$
$\dots$
$y_n(x)={}_2F_0(-n,n+1;-;-{x/2})$
Generalized Bessel $y_1(x)=ax+b$
polynomials $y_2(x)=(a+1)(a+2)x^2+2b(a+1)x+b^2$
$y_3(x)=(a+2)(a+3)(a+4)x^3+3b(a+2)(a+3)x^2+3b^2(2+a)x+b^3$
$\dots$
$y_n(x)=b^n{}_2F_0(-n,n+a-1;-;-{x/b})$
The case of $\lambda_0=0$
-------------------------
In the early development of the asymptotic iteration method [@cs], one get the impression that the method is not applicable in the case of $\lambda_0=0$. This impression naturally arises because of the condition ${s_n\over \lambda_n}={s_{n-1}\over \lambda_{n-1}}$, $n=0,1,2,\dots$. If $\lambda_0=0$, however, we may have using (\[eq5\]) and (\[eq6\]) that ${y_{n+2}\over y_{n+1}}={s_n({\lambda_n\over s_n}y'+y)\over s_{n-1} ({\lambda_{n-1}\over s_{n-1}}y'+y)}$ for which the corresponding asymptotic condition now reads $$\label{eq19}
{\lambda_{n}\over s_{n}}={\lambda_{n-1}\over s_{n-1}} \equiv \alpha, \quad\quad n=1,2,\dots$$ This leads to the essentially equivalent termination condition (\[eq11\]) $$\delta_n={\lambda_{n} s_{n-1}}-{\lambda_{n-1} s_{n}}=0, \quad\quad n=1,2,\dots$$ A simple example which show the use of AIM in case of $\lambda_0=0$ is the differential equation $x^2y''-2y=0$. Direct use of AIM implies that $\delta_2=0$ and a polynomial solution by mean of (\[eq14\]) is $y=x^2.$
Application to generalized Hermite, Laguerre, Legendre and Chebyshev differential equations
===========================================================================================
[**Theorem 4:**]{} [*For $N$ a positive integer and $a,b\neq 0$, the second-order linear differential equation (known as the generalized Laguerre differential equation) $$\label{eq20}
u''=(ax^N-{b\over x})u'-acx^{N-1}u,$$ has a polynomial solution if $$\label{eq21}
c=n(N+1),\quad n=0,1,2,\dots.$$ The corresponding polynomial solutions are $$\label{eq22}
u_n(x)=(N+1)^n\bigg({b+N\over 1+N}\bigg)_n{}_1F_1(-n;{b+N\over 1+N};{ax^{N+1}\over 1+N})$$*]{} [Proof:]{} For $N=1$, the termination condition (\[eq11\]) yields $c=2n$, $n=0,1,2,\dots$ while (\[eq14\]) implies $$u_n(x)=\cases{1,&if $n=0$ (or $c=0$) \cr
1+b-ax^2,&if $n=1$ (or $c=2$)\cr
3+4b+b^2-2a(3+b)x^2+a^2x^4,&if $n=2$ (or $c=4$)\cr
\dots\cr
2^n\bigg({b+1\over 2}\bigg)_n{}_1F_1(-n;{b+1\over 2};{ax^2\over 2}),&for $n=0,1,2,\dots$ (or $c=2n$)
\cr}$$ For $N=2$, the termination condition (\[eq11\]) yields $c=3n$, $n=0,1,2,\dots$ while (\[eq14\]) implies $$u_n(x)=\cases{1,&if $n=0$ (or $c=0$) \cr
2+b-ax^3,&if $n=1$ (or $c=3$)\cr
10+7b+b^2-2a(5+b)x^3+a^2x^6,&if $n=2$ (or $c=6$)\cr
\dots\cr
3^n\bigg({b+2\over 3}\bigg)_n{}_1F_1(-n;{b+2\over 3};{ax^3\over 3}),&for $n=0,1,2,\dots$ (or $c=3n$)
\cr}$$ Similarly, for $N=3$, the termination condition (\[eq11\]) yields $c=4n$, $n=0,1,2,\dots$ while (\[eq14\]) implies $$u_n(x)=\cases{1,&if $n=0$ (or $c=0$) \cr
3+b-ax^4,&if $n=1$ (or $c=4$)\cr
21+10b+b^2-2a(7+b)x^4+a^2x^8,&if $n=2$ (or $c=8$)\cr
\dots\cr
4^n\bigg({b+3\over 4}\bigg)_n{}_1F_1(-n;{b+3\over 4};{ax^4\over 4}),&for $n=0,1,2,\dots$ (or $c=4n$)
\cr}$$ Similar expressions can be obtain for $N=4,5,\dots$. These results can be generalized by (\[eq22\]). 0.1in [**Theorem 5:**]{} [*If $b=0$, the second-order linear differential equation (\[eq20\]), known as generalized Hermite differential equation, has a polynomial solution if $$\label{eq23}
c=n(N+1),\quad n=0,1,2,\dots$$ or $$\label{eq24}
c=n(N+1)+1\quad n=0,1,2,\dots$$ In case of $c=n(N+1)$, the polynomial solutions are (for $n=0,1,2,\dots$) $$\label{eq25}
u_n(x)=(-1)^n (N+1)^n\bigg({N\over 1+N}\bigg)_n{}_1F_1(-n;{N\over 1+N};{ax^{N+1}\over 1+N}).$$ In case of $c=n(N+1)+1$, the polynomial solutions are (for $n=0,1,2,\dots$) $$\label{eq26}
u_n(x)=(-1)^n (N+1)^n\bigg({2+N\over 1+N}\bigg)_n~x~{}_1F_1(-n;{2+N\over 1+N};{ax^{N+1}\over 1+N}).$$* ]{} Proof: Similarly to the proof of Theorem 4, the conditions (\[eq23\]) and (\[eq24\]) follow directly by means of the termination condition (\[eq11\]), with $\lambda_0=ax^N$ and $s_0=-acx^{N-1}.$ Eqs (\[eq25\]) and (\[eq26\]) follow from (\[eq14\]). 0.1in [**Theorem 6:**]{} [*For $N$ a positive integer, the differential equation $$\label{eq27}
u''=\bigg({ax^N\over 1-sx^{N+1}}-{b\over x}\bigg)u'-{wx^{N-1}\over 1-sx^{N+1}}u,$$ has polynomial solutions $$\begin{aligned}
\label{eq28}
u_n(x)&=&{(-1)^n\over (N+1)^{-n}}\bigg({N+b\over N+1}\bigg)_n\nonumber\\
&\times& {}_2F_1(-n,{b-1\over N+1}+{a\over (N+1)s}+n;{b+N\over 1+N};sx^{N+1})\end{aligned}$$ if $$\label{eq29}
w=n(N+1)(s(b-1+n(N+1))+a),\quad n=0,1,2,\dots$$ where ${}_2F_1$ is Gauss’s hypergeometric function (\[eq18\]). If $b=0, s=1$, then the cases of $a=2$ and $a=1$ corresponding to differential equations known as the generalized Legendre and Chebyshev differential equations, respectively.* ]{} 0.1in [**Proof:**]{} Using AIM, condition (\[eq29\]) for polynomial solutions follows by means of the termination condition $\delta_n=0$ in a similar fashion to the proof of Theorem 4. Equation (\[eq28\]) then follows by means of (\[eq14\]) as generalization of the polynomial solutions for each of $n=0,1,2,\dots$ and $N=1,2,\dots$. 0.1in
Conclusion
==========
We have presented a simple criterion for the existence of polynomial solutions of second-order linear differential equations. Many of the classical differential equations that appear in mathematical physics can be analysed with this theory. Apart from its theoretical interest, the criterion can be used in a practical way to look for and to obtain polynomial solutions to eigenvalue problems of Schrödinger-type [@cs]-[@bb], and similarly for polynomial solutions of quasi-exact solvable models in quantum mechanics [@sh].
Acknowledgments {#acknowledgments .unnumbered}
===============
Partial financial support of this work under Grant Nos. GP3438 and GP249507 from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged by two of us (respectively \[RLH\] and \[NS\]).
References {#references .unnumbered}
==========
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|
---
abstract: |
We identify new structures in the halo of the Milky Way Galaxy from positions, colors and magnitudes of five million stars detected in the Sloan Digital Sky Survey. Most of these stars are within $1.26^\circ$ of the celestial equator. We present color-magnitude diagrams (CMDs) for stars in two previously discovered, tidally disrupted structures. The CMDs and turnoff colors are consistent with those of the Sagittarius dwarf galaxy, as had been predicted. In one direction, we are even able to detect a clump of red stars, similar to that of the Sagittarius dwarf, from stars spread across $110$ square degrees of sky. Focusing on stars with the colors of F turnoff objects, we identify at least five additional overdensities of stars.
Four of these may be pieces of the same halo structure, which would cover a region of the sky at least $40^\circ$ in diameter, at a distance of 11 kpc from the Sun (18 kpc from the center of the Galaxy). The turnoff is significantly bluer than that of thick disk stars, and closer to the Galactic plane than a power-law spheroid. We suggest two models to explain this new structure. One possibility is that this new structure could be a new dwarf satellite of the Milky Way, hidden in the Galactic plane, and in the process of being tidally disrupted. The other possibility is that it could be part of a disk-like distribution of stars which is metal-poor, with a scale height of approximately 2 kpc and a scale length of approximately 10 kpc.
The fifth overdensity, which is 20 kpc away, is some distance from the Sagittarius dwarf streamer orbit and is not associated with any known structure in the Galactic plane. We have tentatively identified a sixth overdensity in the halo. If this sixth structure is instead part of a smooth distribution of halo stars (the spheroid), then the spheroid must be very flattened, with axial ratio $q = 0.5$. It is likely that there are many smaller streams of stars in the Galactic halo.
author:
- 'Heidi Jo Newberg, Brian Yanny, Connie Rockosi, Eva K. Grebel, Hans-Walter Rix, Jon Brinkmann, Istvan Csabai, Greg Hennessy, Robert B. Hindsley, Rodrigo Ibata, Zeljko Ivezić, Don Lamb, E. Thomas Nash, Michael Odenkirchen, Heather A. Rave, D. P. Schneider, J. Allyn Smith, Andrea Stolte, Donald G. York'
title: The Ghost of Sagittarius and Lumps in the Halo of the Milky Way
---
Introduction\[intro\]
=====================
There is a growing body of evidence which shows that at least part of the halo of the Milky Way Galaxy was formed through the accretion of smaller satellite galaxies, and is not a relic of the initial collapse of the Milky Way. In the last decade, studies have convincingly identified moving groups and substructure in the halo by identifying groups of stars which are coherent in velocity [@mmh96; @hwzz99]. Simulations have predicted the existence of many halo streamers [@ll95; @jhb96; @jzsh99; @jsh99]. Most recently, studies have identified halo substructure and tidal stripping through spatial information alone [@ietal00; @ynetal00; @oetal01]. A striking example of substructure in the halo is the identification of the Sagittarius dwarf galaxy [@igi94], and its associated stream of tidally stripped stars, which appears to circle the Galaxy [@jsh95; @iwgis97; @il98; @jmsrk99; @hw01; @magc01].
The detection of substructure in the halo is important for our understanding of the formation of our galaxy, and also as a test of cold dark matter (CDM) and hierarchical clustering scenarios for structure formation in the Universe. For example, @bkw01 argue that the CDM scenario generically predicts large numbers of tidally disrupted streams in the halo of the Milky Way - perhaps enough to account for the stellar halo in its entirety. They also suggest that the amount of halo substructure could distinguish among proposed solutions to the “dwarf satellite problem,” the tendency of CDM N-body simulations to predict too many satellites in the halos of galaxies like the Milky Way and M31 [@klypin99; @moore99].
In @ynetal00 (hereafter Paper I), we used a large sample of faint blue stars from the Sloan Digital Sky Survey (SDSS) to discover two diffuse structures of stars in the halo. Their inferred density indicated to us that these structures were disrupted remnants of a previously bound structure, such as a dwarf galaxy. We were not able to see the full extent of either structure. @iils01 explained these two structures as two slices through the same great stream which completely circles the galaxy. The positions and distances of the stars in our structures exactly matched those expected from the tidal disruption of the Sagittarius dwarf spheroidal galaxy. Other pieces of this same stream have been recently reported by @detal01 and a simple model of the Sagittarius breakup is given in @hw01.
In this paper, we present additional observations of the equatorial ($-1.26^\circ < \delta_{2000} < 1.26^\circ$) data from the SDSS which probe a significantly larger angle of right ascension along the equatorial ring than in Paper I. We extend the methods of Paper I, which used faint blue stars with A-type colors to trace structure, to include the much larger sample of turnoff or near turnoff stars with F-type colors. These new data contain strong evidence for further halo substructure. The key figure in this paper is a 2D polar density histogram $(\theta,r) = (\rm RA, g')$ of stars in the plane of the celestial equator with F colors (primarily F dwarf stars) shown in Figure \[Fwedge\] and described in detail in §5.
We expect to detect these streams of stars in addition to, or as part of, the individual stellar components of the Milky Way galaxy. @bs84 published the “standard galaxy model," which contained two components: a thin disk modeled with a double exponential profile with scale height 0.325 kpc, and a halo modeled with a slightly flattened power-law spheroid with axial ratio 0.80. In the solar neighborhood, the spheroid stars were outnumbered by the thin disk stars by a factor of 1 in 500. An additional component, a thick disk, was proposed by @gr83. Since then, the popularity of models with a thick disk component has grown. The thick disk is typically modeled as a double exponential with a scale height of about $1 \pm 0.5$ kpc and a stellar frequency, compared with the thin disk, in the solar neighborhood of between 1:8 and 1:50 [@rr01; @rm93; @obrcm96; @rhcob96; @brk99; @cetal01; @kjs01]. Stars in the thick disk component dominate the star counts 2 to 5 kpc above the plane, and have chemical and kinematic properties intermediate to the thin disk and halo populations. See @n99 for a review of the status of the thick disk. See @gwk89; @m93; and @w99 for reviews of Galactic components.
The literature on the subject of Galactic components is vast; studies include star count analyses, kinematics, chemical properties of stars, and comparisons with other galaxies. We have summarized only the most basic structures, which may themselves have substructure, and which some authors may break into parts or name differently. We have not discussed stellar populations in the Galactic center, such as the bulge population. See @f88, @f99 for reviews of the Galactic bulge.
Observations
============
The observations are from several time-delay and integrate (TDI) CCD scans obtained under photometric conditions in good seeing (FWHM $< 1.9''$) on twelve nights between 1998 September 19 (run 94) and 2001 February 20 (run 2126) with the Sloan Digital Sky Survey (SDSS) mosaic imaging camera [@getal98]. See @yetal00 for a technical overview of the survey.
A single ‘run’ scans six 0.23 degree wide swathes of sky (‘scanlines’) separated by gaps of about 0.2 degrees. The gaps are filled by a second ‘strip’, containing six scanlines, which completes a filled ‘stripe’ on the sky. The SDSS survey area is divided into 48 numbered stripes, each 2.5$^\circ$ wide. Each stripe is an arc of 6 to 10 hours in length that follows a great circle which passes through the survey poles, ($\alpha, \delta$) = ($275^\circ, 0^\circ$) and ($\alpha, \delta$) = ($95^\circ, 0^\circ$). Equinox J2000 is implied throughout this paper. Most of the survey area is in the North Galactic cap. The few stripes in the South Galactic cap have separate stripe designations from their northern counterparts on the same great circle. In particular, the celestial equator ($\delta = 0$) is designated stripe 10 above the Galactic equator, and stripe 82 below. See @setal01 for further details of the survey conventions.
The star count work of this paper requires that the sampling of stars be quite uniform over a large area of sky. Because of the way the SDSS map of the sky is obtained, and pieced together in a mosaic fashion, it is important for the purposes of this paper to select a sample which does not unintentionally ‘double count’ objects in the boundary regions of overlapping survey pieces. The full SDSS database contains multiple copies of many objects. We select single copies of the objects using several flags stored with the object in the database. During image processing, objects are extracted from each scanline one ‘field’ at a time, where the breaks between fields are imposed somewhat arbitrarily every 1361 rows. So that objects which lie on these breakpoints are not lost, overlaps are processed with adjacent ‘fields.’ Objects which fall in an overlap may be in the catalog twice. Also, there are overlaps between the interleaved strips which make up a stripe. The flag ‘OK\_SCANLINE’ is assigned to only one copy of each object in an individual scanline, and uses astrometric declination limits (on the equator) to flag only the non-overlapping areas of the two strips in each stripe. If you take all objects from two interleaved runs which have ‘OK\_SCANLINE’ set, you will get one instance of each object from the combined two runs. Two instances of the same object may both have OK\_SCANLINE set if they are in overlapping runs covering exactly the same part of the sky; however, since the dataset used in this paper was constructed with only non-overlapping portions of runs, selecting with the OK\_SCANLINE flag produces only one instance of each object in a given stripe. Objects can also lie on overlaps between different stripes.
The flag ‘PRIMARY’ is assigned to one copy of each object in the entire database. Each numbered stripe is assigned a region in the sky over which its objects will be PRIMARY. Each numbered ‘run’ is assigned a region of the stripe over which it is PRIMARY. Since the stripes overlap more toward the survey poles, the area of sky over which the stripe is PRIMARY decreases towards its ends. To keep a sample of objects on a single stripe uniformly sampled in declination, one selects those with the OK\_SCANLINE flag set (rather than the PRIMARY flag). Most of the data used in this paper are located on stripes 10 and 82 on the celestial equator ($-1.26^\circ < \delta < 1.26^\circ$). We also use data from stripe 11, 2.5$^\circ$ above the equator, stripe 12, at $\delta \sim
+5^\circ$, and stripe 37, which follows part of an arc of a great circle tilted 67.5$^\circ$ relative to the equator. Table 1 presents details of the strips, the stripes, and the sky coverage of the data used in this paper. Not all sections of the equator scanned have both strips filled. In particular the data from the ends of runs 752, 756 and 1755 don’t have a corresponding filling strip. In order to have uniform star count statistics at all azimuths in these cases, double copies of the single strips were made to normalize the number counts to those areas of sky where two filled stripes were available. This is indicated by a “2” in the multiplicity column of Table 1.
The photometric system for the SDSS includes five filters, $u'\> g'\> r'\>
i'\> z'$ [@figdss96]. The system is approximately $AB_{\nu}$ normalized, with central wavelengths for the filters of 3543Å, 4770Å, 6231Å, 7625Å, and 9134Å, respectively, and effective widths of typically 1000Å. Since the precise calibration for the SDSS filter system is still in progress, magnitudes in this paper are quoted in the $u^* g^*
r^* i^* z^*$ system, which approximates the final SDSS system [@setal]. These systems differ absolutely (with small color terms) by only a few percent in $g^* r^* i^* z^*$, and no more than 10% in $u^*$.
The data were reduced with PHOTO [@l01] versions 5.1 and 5.2, and astrometrically calibrated with the ASTROM pipeline described in @pmk01.
Data Reduction\[datared\]
=========================
The SDSS software generates a database of measured object parameters and flags, including information on deblended ‘children’ of sources whose profiles overlap. One must select from this database a list of interesting objects to be used as input to analysis routines. We selected from the photometric catalog only those objects which were marked as stellar, unsaturated, and not too near the edge of the frame (too near is generally about $8''$). In order to ensure that only one instance of each object appears in the final object tables, we selected objects which were marked as OK\_SCANLINE as explained above.
Using these criteria, we generated a catalog of 4.3 million stars to $g^* \sim 23.5$ on the equator. The total area covered is approximately 560 square degrees. Data on off-equatorial stripes add an additional 0.7 million stars over approximately $70$ square degrees. Completeness vs. magnitude is discussed below, but we note here that for objects with $g^* > 22.5$, the star-galaxy separation results in most stellar objects being classified as galaxies and these thus are not prominent in our subsample.
The SDSS software measures object flux in a variety of ways. Since we are measuring stars only, we use magnitudes calculated from a fit of modeled stellar profiles (point-spread-function, or PSF, magnitudes) to each object. We correct these magnitudes for reddening using $E(B-V)$ from @sfd98, which has spatial resolution of 0.1 degrees, and the standard extinction curve [@ccm89], which for SDSS filters yields: $A_{u^*} =
5.2 E(B-V); A_{g^*} = 3.8 E(B-V); A_{r^*} = 2.8 E(B-V)$.
The flux of objects are presented in an inverse hyperbolic sine (asinh) representation of @lgs99. This definition has the feature (unlike a magnitude) that it is well defined for zero or negative fluxes, which can result from measurement of no flux at the position of an object detected in a different filter. Asinh numbers and magnitudes are the same to better than 0.1% for objects with $g^* < 21$, differ by 0.1 mag at $g^* = 23.9$. At zero flux, the asinh numbers go through $g^* \sim 25$. The $r^*$ asinh flux shifts in the same way as the $g^*$, and thus there is negligibly little change in $g^*-r^*$ color due to using these asinh numbers instead of magnitudes. For the magnitude ranges of interest here ($g^* < 22.5$), the difference is unimportant. In the remainder of this paper, we will refer to asinh numbers as magnitudes.
Figure \[redvsalpha\] shows reddening ($10\times E(B-V)$ in magnitudes) from @sfd98 around the celestial equator. We also plot number counts in 10 degree bins for a sample of color selected stellar objects with $18 < g^* < 21.5, 0 < u^*-g^* <0.3,
0.1 < g^*-r^* < 0.3$ versus right ascension around the equator. These objects, which are primarily quasars (see Figures 1, 4, and 5 of Paper I), should have a constant number density independent of Galactic latitude. Figure \[redvsalpha\] shows that the selection of stars of similar colors and magnitudes as a function of $\alpha$ around the sky is mostly unbiased. Near $\alpha = 60^\circ$ the amount of intervening interstellar dust is quite large and the errors in the reddening corrected magnitudes are larger than at most other $\alpha$. For $310^\circ < \alpha < 350^\circ$, only one of two SDSS strips of data is present, and thus the counts have less S/N than the rest of equatorial data. The counts have been normalized upwards by a factor of two in the figure and the error bars appropriately increased. Even with this normalization, it is apparent that the counts fall systematically slightly below that of those at, for example, $150^\circ < \alpha < 230^\circ$. We are uncertain of the reason for this.
From inter-comparison of objects detected twice in overlapping scans, we find the rms error for stellar sources with $g^* < 19$ is typically $\sim 2$%. For objects with $20 < g^* < 21$, typical errors are 5%, growing to 20% at $g^* = 23.5$ near the detection limit. For reference, blue stars with $0 < B-V < 0.2$ have an SDSS $g^*$ magnitude approximately equal to their Johnson $V$ magnitude. A theoretical color transformation is given by [@figdss96]: $g^*-r^* = 1.05(B-V) - 0.23$.
We plot in Figure \[limmag\] the rms dispersion of the difference in $g^*-r^*$ color for matched objects between two runs as a function of magnitude. For bright magnitudes this dispersion reflects the photometric errors of 2% in g and r (about $2\times \sqrt{2}\sim $3% in uncorrelated color). The photometric error increases to nearly 20% for objects near $g^* \sim 22.5$.
For some of our analyses, it is crucial to know the limiting magnitude, or more precisely the magnitude limit at which the survey can be considered complete, as a function of color and position in the sky. In the same Figure \[limmag\], we show the fraction of stars matched between two overlapping runs which make up the equatorial stripes. The matched fraction is calculated at two positions around the equator. One segment of matched data are at lower latitude, averaging $b \sim 30^\circ$ with $125^\circ < \alpha < 145^\circ$, while the other are at higher latitude with average $b \sim 50^\circ$, and $145^\circ < \alpha <
230^\circ$. Stars in three color ranges, $0.1 < g^*-r^* < 0.3, 0.3 <
g^*-r^* < 0.4$ and $0.6 < g^*-r^* < 0.7$ (all with $u^*-g^* > 0.4$) in one strip are matched to the full list of stars in the overlapping strip. The fraction which are matched is recorded as a function of $g^*$ magnitude. Figure \[limmag\] shows that for all three color bins of the high latitude matched set (the low latitude set is identical within the errors), the matched fraction is constant to about $g^* \sim 22.5$, after which it drops off steeply, and somewhat more quickly for objects of bluer color.
One notes that for bright objects, the matching fraction is not 100%. This is in part due to the fact that edge overlaps were used between the interleaving stripes (so that the same exact area of sky is not sampled by each overlapping patch). Also, the reduction software doesn’t resolve all objects around bright stars into separate detections. Independent matches against external catalogs indicate that the detection software detects over 99% of objects at these magnitudes ($18 < g^* < 22$).
Both segments of data at higher and lower latitudes give the same results for $g^* < 22.5$. Thus any selection we do based on magnitude $g^* < 22.5$ is free of significant color or completeness bias to this limit. The variation in density of objects with quasar colors indicates there is possibly some small variation in completeness limits or problems with reddening corrections at $60^\circ < \alpha < 76^\circ$ and $318^\circ < \alpha < 325^\circ$.
The imaging pipeline separates detected objects into stars and galaxies based on goodness of fit to PSFs and model galaxy profiles. For the seeing conditions under which these data were obtained, this separation produces excellent results to approximately $g^* \sim 21$. We show in Figure \[galaxies\] a color magnitude image of $\approx
100,000$ objects typed as galaxies, selected around the celestial equator, and binned as a Hess diagram [@h24]. Nearly all galaxies have colors redder than $g^*-r^* > 0.4$, significantly redder than the turnoff stars we are interested in at $g^*-r^* \sim 0.3$. There is some leakage of stars into the galaxy population for $g^*-r^* \sim 0.3$ at $g^* > 22.5$, again below the limits set by Figure \[limmag\]. The galaxy population’s localization in color-magnitude space affects none of the conclusions made here about turnoff-color star counts.
The Ghost of Sagittarius\[ghostsag\]
====================================
Since the positions and colors of the giant branches and horizontal branches of dwarf galaxy companions to the Milky Way differ considerably as a function of the dwarf’s metallicity, age and stellar population mix, these features can be used as identifying signatures of a given dwarf galaxy or cluster. In this section, we explore the color-magnitude distribution of stars in previously detected clumps. This discussion will motivate our use of F-colored stars to detect spatial structure in the next section.
Using a technique similar to that of @mskrjtlp99, we will construct a color-magnitude diagram of the two concentrations of stars from Paper I. To avoid confusion in referencing overdensities of stars, we will name them S$l\pm b-g$ where ($l\pm b$) are the Galactic coordinates of the approximate center of the structure (where it intersects an SDSS stripe), and $g$ is the approximate $g^*$ magnitude of the ‘turnoff’ F dwarf stars in that structure. If the structure is identified with a known halo component, such as the Sagittarius dwarf, then the identification may appear in parentheses after the structure name. Under this naming convention, the two structures identified in Paper I are given the names S341$+$57$-$22.5 and S167$-$54$-$21.5, and may be clearly seen in Figure\[Fwedge\].
For structure S341$+$57$-$22.5, we used stars marked as ‘PRIMARY’ in stripes 10 and 11 with $200^\circ < \alpha < 225^\circ$ and $u^*-g^* > 0.5$. The cut in $u^*-g^*$ eliminates blue quasars which would otherwise dominate the faint blue edge of the color-magnitude diagram. The ‘PRIMARY’ stars come from non-intersecting portions of stripes. Since the stripes are parts of great circle arcs, the non-overlapping portion of the stripe is thinner towards the survey poles than it is on the survey equator. On the celestial equator at $\alpha = 200^\circ$, the width of stripe 10 and 11 together is 4.8 degrees. On the celestial equator at $\alpha = 225^\circ$, the width of the two stripes is 3.8 degrees. The total area covered is 110 square degrees.
Due to the large number of stars in this area of sky, we generated an image of counts-in-cells of the color-magnitude diagram, with a bin width of 0.02 in $g^*-r^*$ and 0.05 in $g^*$. In order to reduce the number of field stars in the image, we subtracted a similarly generated color-magnitude image of stars in a similar portion of the sky which does not contain the Sagittarius dwarf. The subtracted stars are from stripe 10 and 11, $170^\circ < \alpha < 180^\circ$, plus twice stripe 10, $230^\circ < \alpha < 235^\circ$ (SDSS has not yet processed data in the $230^\circ < \alpha < 235^\circ$ range for stripe 11).
For stars at S341$+$57$-$22.5, the resulting Hess color-magnitude diagram image (with a greyscale stretch proportional to the square root of the number of stars in each bin) is shown in Figure \[sagsubn\]. One can clearly see the turnoff at ($g^*-r^*,g^*)= (0.2,22.5)$, the giant branch at $(g^*-r^*,g^*) = (0.5, 22)$ running to $(g^*-r^*,g^*) = (0.6, 20.5)$, blue stragglers at ($g^*-r^*, g^*)=(-0.1, 21.5)$, a blue horizontal branch at $(g^*-r^*,g^*)=(-0.15, 19.2)$ and a clump of red stars at ($g^*-r^*, g^*)=(0.55, 19.6)$. For comparison, we show in Figure \[sagcmd\] the identical plot for the Sagittarius dwarf itself, with data from @mbcimpp98. Since these Sagittarius dwarf data were taken with V and I filters, and the dwarf is much closer than the dispersed clump (23 vs. 45 kpc), we arbitrarily aligned the clump of red stars and applied a linear correction in the color direction so that the distance between the clump of red stars and the point where the horizontal branch and the upper main sequence (as traced by blue stragglers) meet is the same in each image. The adopted transformation equations are: $g^* = V + 1.39$ and $(g^*-r^*) = 0.9(V-I) - 0.41$. The relation between $g^*$ and $V$ includes both the difference in distance modulus and the filter transformation. Any differences in reddening or errors in reddening correction for either data set are implicitly included in the transformation. We then compared this empirical transformation with one derived from theoretical SDSS filter curves as given in @figdss96, and find that they match to within about 5%. The distance compensation, $g^* = V + 1.39$ is also within a few percent of the expected theoretical value, $g^* = V + 5 log(45/23) = V + 1.45$.
The similarity of color-magnitude diagrams can be judged from the color of the turnoff, the distance from the horizontal branch to the turnoff, the slope and degree of population of the red giant branch, the presence or absence of blue stragglers, and the color distribution of stars along the horizontal branch. Though there are some differences (most notably - the Sagittarius dwarf photometry shows few if any blue horizontal branch stars), the agreement between the color magnitude diagrams for the Sagittarius dwarf and our 110 square degree patch of sky on the equator is striking, and leaves little doubt that the tidally disrupted clumps of stars discovered in @ietal00 and Paper I are in fact pieces of the Sagittarius dwarf stream, in exactly the positions predicted by @ilitq01.
It is interesting to estimate the fraction of the Sagittarius dwarf which is present in our observed piece of its orbit. One measure is the number counts in the clump of red stars. We estimate that there are $500
\pm 50$ stars in the 110 square degree patch of sky with $19.30 < g^* < 19.65$ and $0.52 < g^*-r^* <
0.66$. For this estimate, we measured the clump of red stars in the unsubtracted color-magnitude image to reduce the statistical noise in the measurement (the background was determined by linear interpolation). @igi95 find $17,000$ horizontal branch stars in a patch of sky thought to contain half the mass of the Sagittarius dwarf. We detected about $500/34,000 = 1.5\%$ as many red stars in the clump as are present in the dwarf itself in a portion of its orbit extending $4.4^\circ$ on the sky. The orbit is roughly perpendicular to our scan line and presumably extends over $360^\circ$ on the sky. If the stellar density along the stream were constant (admittedly a naive assumption), and the stream only wraps around on itself (so as not to produce multiple streams at other positions on the sky), then this implies about as many stars (1.2 times as many in this calculation) in the stream as in the undisrupted dwarf. This number is interesting, but has a large error bar as the fraction of stars in the clump of red stars could easily differ between or within the dwarf and the stream.
Using exactly the same procedure as for S341$+$57$-$22.5 (Sagittarius), we generate a color-magnitude image for S167$-$54$-$21.5, which has also been tentatively identified as a piece of the Sagittarius stream by @iils01. Since we have no data adjacent to stripe 82, we used the full width of the collected data (there is no change in the $\delta$-width of the stripe on the sky as a function of right ascension). We used all stars with $15^\circ < \alpha < 50^\circ$ to make the clump color-magnitude image. We then subtracted a color-magnitude image of all stars with $10^\circ < \alpha < 15^\circ$ plus double-counting all of the stars with $0^\circ < \alpha < 5^\circ$ and $50^\circ < \alpha <
55^\circ$. The resulting color-magnitude image is shown in Figure \[sagsubs\]. One can see a clear turnoff, a giant branch, and blue straggler stars. The horizontal branch and clump of red stars, though possibly faintly present, are not compelling. There are definitely blue horizontal branch stars present, since this clump was originally detected in A-colored stars in Paper I. The number of blue horizontal branch stars in the Sagittarius south stream is only a third the number in the northern stream. This color-magnitude diagram is consistent with that of the Sagittarius dwarf, but does not present as strong a case for identification as that in the North.
Halo structure in F stars
=========================
Distribution on the celestial equator
-------------------------------------
Our detection of halo structure in Paper I relied on the standard candle characteristics of A-colored blue horizontal branch stars. The results of the previous section demonstrate our ability to examine the structure of the Milky Way using stars as faint as F dwarfs. If it is possible to use the much larger numbers of these main sequence stars, one anticipates that much more tenuous halo structures could be discerned, though not as far out into the halo. We pursue such a path.
From stripes 10 and 82, we generate a catalog of $4,270,645$ stars with $u^*-g^* > 0.4$ and $-1.0 < g^*-r^* < 2.5$. The $g^*-r^*$ color range is wide enough to include essentially all stars. In this section, we will be using not just the PRIMARY flagged stars, but all of the OK\_SCANLINE flagged stars from the runs used to fill in stripes 10 and 82 as detailed in Table 1. This way, the width of the stripe in declination does not change as a function of right ascension. The $u^*-g^*$ color cut removes primarily low redshift QSOs.
In Figure \[s10230240cmd\] we show a color-magnitude image of all of the stars centered in the direction (l,b) = (5,40), $230^\circ <
\alpha < 240^\circ$ in stripe 10. The stars with $g^*-r^* \sim 0.5$ and $g^* \sim 18$ are thought to be associated with the thick disk of the Milky Way. The stars with $g^*-r^* \sim 1.3$ are M stars in the thin disk, the thick disk, and, at $g^* > 22$, the halo. The bluer stars ($g^*-r^* \sim 0.3$) are generally ascribed to the halo. The clear separation in turnoff color between the “thick disk" and “halo" was described by @cetal01. The stars we are interested in are the bluer stars with $g^*-r^* \sim 0.3$, at $g^* > 19$, which are associated with the halo population. It is important to note that using a color separation to distinguish between “thick disk” and “halo” populations, though it appears to work well, is an empirical one, and is a separate distinction from a kinematic separation of the populations.
To separate the thick disk stars from halo stars, we select $334,066$ stars in stripes 10 and 82 (which are both on the celestial equator) with $0.1 < g^*-r^* < 0.3$, keeping the $u^*-g^* > 0.4$ color cut. This cut includes only the bluer “halo" stars; we have so many stars that we have the luxury of throwing half of them away to reduce thick disk contamination and keep a much smaller range of dwarf star absolute magnitudes in the sample. We plot this sample of stars in a 2D polar density histogram in Figure \[Fwedge\]. This figure is similar to the wedge plots of Paper I (see Figure 3 of that paper), but is displayed in an image by binning all the stars that would have appeared as individual dots within each pixel. The Sun is located at the center of the plot. Stars of the same apparent magnitude are at the same radial distance from the center of the plot, with $g^*=11$ at the center of the plot and $g^*=24$ at the edge (though the data cuts off at $g^* = 23.5$). If each star has the same intrinsic magnitude (roughly the magnitude of an F main sequence star), then the radius from the center of the diagram scales as the logarithm of the distance from us. Typical distances probed with turnoff stars range from a few kpc to about 60 kpc at the edge of the plot ($g^* = 23$).
The shading of each box indicates the relative number of F stars within the pixel’s azimuth and magnitude ranges, with 7.69 pixels/magnitude. It is generated by calculating for each star in the sample the (x, y) position the star should go in Figure \[Fwedge\], and then incrementing the count in the pixel which covers that spot.
Figure \[Fwedge\] does not show the smooth distribution of stars expected from a power-law spheroid or exponential disk stellar density distribution. The overdensities of stars at $(\alpha, g^*) = (210^\circ, 22)$ and $(\alpha, g^*) = (40^\circ, 21)$ are F dwarfs associated with S341$+$57$-$22.5 and S167$-$54$-$21.5 (Sagittarius). The dark radial line at $\alpha = 229^\circ$ is the main sequence turnoff of the globular cluster Pal 5.
The feature at $\alpha = 60^\circ$ is exactly coincident with a large interstellar dust cloud at that position in the sky, and does not represent halo structure. When the reddening correction is that large, one must worry about the distance to the source(s) of reddening, and the accuracy of the maps. Small differences in applied reddening change the intrinsic colors of the selected objects. The counts in this direction are consistent with over-correction for reddening, which moves redder stars into the color selection box. As is apparent in the color-magnitude Hess diagrams, the redder turnoff stars are more prevalent at brighter magnitudes.
Locations of other interesting overdensities are labeled in Figure \[Fwedge\] and summarized in Table 2. These and other overdensities will be discussed in detail below.
Spheroid models\[spheroidmodels\]
---------------------------------
What do typical Galactic stellar component models, such as an exponential thick disk or a power-law spheroid, look like in a wedge image such as that of Figure \[Fwedge\]? We will use the term ‘spheroid’ to describe any smooth distribution of stars (in excess of the known thin and thick disk populations) in the halo of the Milky Way, regardless of its density profile. The halo of the Milky Way is a region of space containing gravitationally bound matter. The halo stars of the Milky Way are a combination of dwarf galaxies, globular clusters, streamers, and a smooth component (spheroid). The usual density profile for the Galactic spheroid is a power-law, or alternatively a flattened power law with flattening parameter $q$, given by: $$\rho = \rho_o (X^2 + Y^2 + Z^2/q^2)^{\alpha/2},$$ where X, Y, and Z are the usual Galactocentric coordinates with Z perpendicular to the Galactic plane, and $\rho_o$ sets the density scale. If $q=1$, the model is spherically symmetric. $\alpha$ is thought to be about $-3.2\pm 0.3$ (see Paper I).
To generate the wedge image, we must transform to a heliocentric coordinate system, $(l, b, R)$, where $R$ is the distance from the Sun. In this coordinate system, the number of stars per magnitude bin is given by: $$\frac{dN}{dm} = \frac{dR}{dm} \frac{dN}{dR} = (\frac{R}{5}) (\Omega R^2 \rho_o r^\alpha),$$ where $$r^2 = R_o^2 + Q R^2 - 2 R_o R \cos(l) \cos(b),$$ $$Q= \cos^2(b) + \sin^2(b) /q^2.$$
In our simulations, we assume the distance to the center of the galaxy, $R_o$, is 8.0 kpc. In our plots, the number of pixels in a given apparent magnitude annulus of width $dm$ is proportional to $(m-11)$, and the width of the data in declination is constant. Therefore, $$\Omega \propto (m-11)^{-1},$$ as $dm$ is approximately constant for each pixel in the wedge image of Figure \[Fwedge\]. In this way we correct for the angular size of each pixel.
In order to construct the simulation, we need to relate the distance $R$ to the apparent magnitude $m$. For this we need to know the approximate absolute magnitude of the stars in Figure \[Fwedge\]. Clearly, there will be a spread in stellar magnitudes, which should result in a broader distribution in the data than in the simulated image. We find an estimate of the magnitudes of turnoff stars by using the distance in magnitudes from the horizontal branch of the S167$-$54$-$21.5 (Sagittarius, from Paper I) to the turnoff of the Sagittarius stars in the Figure \[Fwedge\], stripe 82. Figure \[Fstarmag\] shows the distribution of apparent magnitudes for stars with $30^\circ < \alpha < 45^\circ$. In each magnitude bin, we have subtracted the number of stars in a similar region of the equator that does not include the Sagittarius stream ($20^\circ < \alpha < 25^\circ$ and $45^\circ < \alpha < 55^\circ$). The range of apparent magnitudes at the peak of the distribution is $21.1 < g^* < 21.8$. Assuming a horizontal branch absolute magnitude $M_{g^*} \sim 0.7$ and $g^* = 18$ (Paper I), the absolute magnitudes of the stars in the image are estimated to be in the $3.8 < M_{g^*} < 4.5$ range, quite typical for F dwarfs. We adopt $M_{g^*} = 4.2$ as the typical magnitude of a turnoff star. This value is consistent with that estimated from SDSS magnitudes and known distances of the globular cluster Palomar 5.
We would like to draw your attention to some special azimuthal directions on the wedge plot of Figure \[Fwedge\]. The Galactic plane intersects the plane of the plot at $\alpha = 103^\circ$, and goes straight through the plot center to $\alpha = 283^\circ$. The place where $l=0^\circ$ is at $\alpha = 228^\circ$, almost in the direction of Pal 5 at $\alpha =
229^\circ$. In the celestial equatorial plane, this is not the direction of the Galactic center, but above the Galactic center in the direction which will intersect the Z-axis of the Galaxy. If the Galactic spheroid were very prolate ($q \rightarrow \infty$), then the highest density of stars intersected by the celestial equator would be at $l = 0^\circ$. If the spheroid were flattened into a pancake ($q \rightarrow 0$), the highest density of stars would be in the Galactic plane at $\alpha = 283^\circ$ ($b=0^\circ, l=32^\circ$), and there would be another high density of stars at $\alpha = 103^\circ$ ($b=0^\circ, l=212^\circ$); the relative densities would depend on how quickly the power-law drops off. If the spheroid is spherical, the highest density will be in the direction of the closest approach to the center of the Galaxy. This direction does not depend on the slope of the power-law or the inferred absolute magnitude of the stars. We show in Figures \[models\]a, \[models\]b, and \[models\]c images of the wedge image simulations which result from $q = 0.5, 1.0,$ and $1.5$ power-law spheroid models with $\alpha = -3.5$.
Using a similar procedure to that for the spheroidal models, we can generate a simulated exponential disk as seen in the cross section of the celestial equator. The relevant equations for the disk density profile models are: $$\rho = \rho_o e^{-r/s_l} e^{- |Z|/s_h},$$ $$\frac{dN}{dm} = \frac{dR}{dm} \frac{dN}{dR} = (\frac{R}{5}) (\Omega R^2 \rho_o e^{-r/s_l} e^{-|Z|/s_h}),$$ where $r^2 \equiv X^2 + Y^2$ and X, Y, Z are standard Galactocentric coordinates with the Sun at (X,Y,Z) = (-8.0,0,0). $\Omega$ is the same as in the spheroid model. A simulated exponential disk with scale length $s_l = 3$ kpc and scale height $s_h = 1$ kpc is shown in Figure \[models\]d. We used the same absolute magnitude for the simulated stars. Exponential disks generally put concentrations of stars at $b = 0$ ($s_h << s_l$). They result in concentrations of stars at $l = 0$ if $s_h >> s_l$.
Overdensity at $\alpha = 190^\circ$ - S297$+$63$-$20.0
------------------------------------------------------
Armed with these results, we turn our attention again to the data in the wedge plot of Figure \[Fwedge\]. Neither an exponential disk model nor a power-law model can put a density peak at $180^\circ < \alpha < 195^\circ$, where $(l,b)=(297^\circ,63^\circ)$. We tentatively identify the concentration at about $g^* = 20.5$ in Figure \[Fwedge\] as a stream or other diffuse concentration of stars in the halo, and name it S297$+$63$-$20.0. Figure \[image-S10-180-195\] shows the color-magnitude diagram for stars with $180^\circ < \alpha < 195^\circ$.
A recent paper by @v01 present corroborating evidence for this stream from observations of 5 clumped RR Lyraes at $\alpha=197^\circ$, at a similar inferred distance from the Sun (20 kpc).
Overdensity near $\alpha = 125^\circ$ - S223$+$20$-$19.4
--------------------------------------------------------
We now turn our attention to the concentration of stars near $\alpha
= 125^\circ$ at $g^* \sim 19.5$. As we noted above, it is possible to put concentrations of stars near the Galactic plane near the anti-center ($l \sim 180^\circ$) with either an exponential disk or a flattened spheroidal power-law model ($q < 0.6$). These stars are too faint to be produced by thin disk or thick disk stars from the double exponential profiles of any standard models. As we will show in §5.7, it would be necessary to postulate an unusually flattened power-law distribution, or an unusually large scale length exponential disk distribution, to put enough stars this far away from the Galactic center and still fit the number counts towards the Galactic center.
A color-magnitude image for stars in this direction is shown in Figure \[cmd125\]. Most of the brighter, bluer stars ($g^* < 18.5$, $0.4 < g^*-r^* < 0.6$), presumably thick and thin disk, are part of a distribution with a redder turnoff than the fainter stars at ($g^*
> 20$). What is stunning about the color-magnitude image in this direction is that the fainter, bluer stars appear to follow a main sequence, as if the stars are all at about the same distance from the Sun. In any kind of exponential disk or power-law distribution, one expects a much broader distribution of distances, which spreads the stars in the vertical direction on the color-magnitude diagram. See Figure 6a of @ls00 for an example of how a dwarf spheroidal in the field looks in such a CMD.
We show the shallow depth of the structure quantitatively in Figure \[dwarfthickness\]. Figure \[dwarfthickness\](a) shows power-law models for a variety of slopes and flattenings in the direction $(\alpha,\delta) = (125^\circ,0^\circ)$. Figure \[dwarfthickness\](b) shows a variety of exponential disks in the same direction. Compare the widths of the peaks of these models with the width (in magnitude) of the main sequence at $\alpha \sim 125^\circ$ shown in Figure \[dwarfthickness\](c). The black line gives star counts vs. magnitude in the color range $0.1 <
g^*-r^* < 0.3$. Note the peak centered at $g^* = 19.4$. If the data were broader than the model, we might expect that the data represented stars with a range of absolute magnitudes. Since the data are narrower than all of the models, no power-law or exponential disk model is a good to fit the data. The spheroid models are all very poor fits to the data. The only exponential model with any hope of fitting the data has a scale height of 2 kpc and a scale length of 10 kpc. Even this model produces a peak which is a little wide for comfort.
The red line in Figure \[dwarfthickness\](c) shows the magnitude distribution of all stars with $117^\circ < \alpha < 130^\circ$, $u^*-g^* > 0.4$, and $0.44 < g^*-r^* < 0.48$. The peak in this plot occurs at fainter magnitudes, since at the redder colors the stars are intrinsically fainter. The peak is narrower because these stars are on the main sequence rather than at the turnoff, where there is a broader range of intrinsic brightnesses of stars. The actual width of the stellar group must correspond to significantly less than one magnitude in distance modulus. Figure \[dwarfthickness\](c) also shows a sample model counts of a dwarf spheroidal galaxy at the distance of the stellar excess, offset from the Galactic center, at $(l,b,R_{GC}) = (212^\circ,0^\circ,18\> \rm kpc)$. The fact that this model nominally fits the SDSS stars counts allows the possibility of a newly discovered dwarf galaxy in the Galactic plane, though this is not the only possible interpretation (see §5.7).
One also verifies that incompleteness at the faint end as a function of color is not responsible for the turnoff-like feature in Figure \[cmd125\]. The tests of §\[datared\] indicate that this is not the case for stars with $g^* < 22$, well below the turnoff and main sequence seen here at $19.4 < g^* < 21.5$. We adopt the name S223$+$20$-$19.4 for this structure.
Overdensity near $\alpha = 75^\circ$ - S200$-$24$+$19.8
-------------------------------------------------------
Now we look at stars at $\alpha = 75^\circ$ in Figure \[Fwedge\] and see if those stars can be explained by smooth components. See Figure \[dwarfthickness2\] for evidence that the excess on this side of the plane is also thinner than expected for a power-law or exponential disk model. For this figure, we tightened up the color range plotted, to reduce contamination from the thick disk. As was evident in Figure \[redvsalpha\], the data in this region are of lower quality. In §\[properties\], we will show that the measured thick disk turnoff is much redder in this data, and may indicate a calibration error or incorrect reddening correction applied here. Although the absolute photometry is suspect in this region, we still detect an unexpectedly tight magnitude peak at $g \sim 19.8$. Figure \[cmd75\] shows a CMD of stars in stripe 82, $ 70^\circ < \alpha < 77^\circ$. This structure is named S200$-$24$-$19.8.
Other SDSS data near the Galactic plane - S218$+$22$-$19.5, S183$+$22$-$19.4
----------------------------------------------------------------------------
We looked through other SDSS data sets to see if the excess of stars near the plane showed up in other runs. We have data at low Galactic latitude in stripes 12 and 37. These data include $153,286$ stars of all colors in stripe 12 with $122^\circ < \alpha < 135^\circ$ and $252,099$ stars in stripe 37 with $112^\circ < \alpha < 125^\circ$. Figure \[mondwarfimages\] shows wedge plots for the ends of stripe 12 and 37.
The large increase in stars near the Galactic plane at Galactic latitude about $+20^\circ$ and $g^* \sim 19.5$ is apparent in stripes 12 and 37 as well. The magnitudes of the stars near the ends of stripe 12 and 37 is similarly as narrow, and peaked at a similar magnitude, as those at the end of stripe 10. We adopt the labels S218$+$22$-$19.5 and S183$+$22$-$19.4 for these apparent overdensities.
Figure \[image37\] shows a CMD of stars in stripe 37, separated by 40 degrees in Galactic longitude from those of Fig \[cmd125\] (at about the same $b$ = +20$^\circ$). The similarity between Figure \[image37\] and Figure \[cmd125\] is remarkable. The color-magnitude diagram for S218$+$22$-$19.5 looks the same as well.
Fits to the Galactic spheroid
-----------------------------
Now that we have identified several large features in the data which are not consistent with a smooth distribution of stars, we will attempt to constrain spheroid models. In Figure \[dwarfthickness3\], we show the distribution in magnitude for stars with $0.1 < g^*-r^* < 0.3$, $u^*-g^* < 0.4$, and $230^\circ < \alpha < 240^\circ$. We would be very surprised if these stars were not consistent with a spheroid population. The figure shows a very broad distribution in magnitude, most consistent with the a flattened ($q \sim 0.5$) power-law with slope of $\alpha \sim -3$, though one could imagine fitting other power-law distributions. Notice that the magnitude distribution in this direction is not consistent with an exponential disk with large scale lengths, especially for $g^*>20$.
We selected all of the stars with $u^*-g^* > 0.4$, $19.0 < g^* < 20.0$, and $0.1 < g^*-r^* < 0.3$. The number of these stars as a function of right ascension is shown in black in Figure \[dwarfboxplot2\]. We then attempted to fit spheroid models (also shown in Figure \[dwarfboxplot2\]) to the data. We don’t expect many thick disk stars in this plot, since we have selected only stars bluer than the nominal turnoff. We expect very few thin disk stars at these faint magnitudes. The models are generated by integrating the models of §\[spheroidmodels\] over our apparent magnitude range, $19 < g^* < 20$. With only angular information, there is very little difference between models with different power-law slopes or different assumed absolute magnitudes for the stars. There is greater sensitivity to the flattening of the spheroid. The only way to fit the star counts on both sides of $\alpha = 280^\circ$ is with a very large flattening, such as $q = 0.5$. More spherical models can fit the slope near $\alpha = 250^\circ$, but do not put enough stars at $\alpha = 320^\circ$. Note that we did not attempt to fit a triaxial halo, which has been used by other authors [@lh96] to explain an interesting asymmetry in the blue star counts around the Galactic center.
As a class, the power-law models cannot put enough stars around $\alpha \sim 100^\circ$ to explain the high counts there. We already encountered difficulty fitting this feature with a power-law in Figures \[dwarfthickness\] and \[dwarfthickness2\], but it is good to see the discrepancy in this plot as well. In order to produce anything close, one would need a flattening more like $q = 0.2$, which is quite a bit lower than anyone has previously considered and still doesn’t fit well.
Since a power-law spheroid does not seem a good fit to these stars, we look to other proposed Galactic components to fit the data. Evidence for a metal weak thick disk has been given by @mff90 and @n94. @cb00, using kinematics of faint blue stars, find a scale length for this component of 4.5 kpc. An exponential disk with a scale height of 2 kpc and a scale length of 10 kpc produces our best fit to the angular data near the Galactic plane. We need the large scale length to put enough stars out at $\alpha \sim 100^\circ$. This model gives a surprisingly good fit to the data. It is important to remember that the stars fit here are not the ones routinely assigned to the thick disk; they have a bluer turnoff. This would imply that we are seeing an ‘even-thicker-disk’ of different metallicity (or age) from the ‘thick disk.’ The only data that this model contradicts is the narrow magnitude profiles in Figures \[dwarfthickness\] and \[dwarfthickness2\]. It does not explain the fainter star counts in Figure \[dwarfthickness3\]. The center of the peak near the anticenter is slightly shifted between the exponential disk model and the data. This shift cannot be reduced by small changes in the scale lengths, assumed stellar absolute magnitudes, or the Sun’s distance from the Galactic center. It could be reduced by moving the Sun to 0.2 kpc above the Galactic plane (or by tilting or lowering the extended exponential disk model below the plane of the thin disk by a similar amount). We placed the Sun 20 pc above the Galactic plane in our standard model, in keeping with recent estimates of this parameter [@hgmc95; @c95; @bgs97; @hl95; @mv98; @cetal01].
We have proposed two possible explanations for the overdensity S223+20-19.4. The first possibility is that it is a previously undiscovered dwarf galaxy (probably in the process of tidally disrupting), or a stream from a dwarf galaxy. The other is that it is part of a smooth, metal-poor Galactic component with a double exponential profile with about 2 kpc scale height and 10 kpc scale length. One cannot produce this many star counts near the Galactic anticenter with a power-law distribution of stars. Based on the results of @cetal01, we do not expect stars with these blue colors in the thick disk. We will now show that the star counts do not fit exponential density models with the previously measured scale heights and scale lengths of the thin and thick disks.
There are two issues that were not addressed in the previous model fits in Figure \[dwarfboxplot2\]. We did not consider that the metallicity of the thick disk could have changed as a function of scale height. We also did not use our knowledge of the local normalization of stars from the various Galactic components to check whether the stellar distributions could be reasonably attributed to a known and measured disk component.
To address these issues, we selected all of the stars in a broader color range ($0.2 < g^*-r^* < 0.5$) and in five magnitude ranges: $15.5 < g^* < 16.5$, $16.5 < g^* < 17.5$, $17.5 < g^* < 18.5$, $18.5 < g^* < 19.5$, and $19.5 < g^* < 20.0$. This should contain nearly all turnoff stars in the thick disk and halo populations, and at the bright end the older turnoff stars in the thin disk. The densities of these distributions as a function of right ascension are shown in Figure \[five\]. We can now fit models to these data plots as a set. Since the stars are redder than in the previous plot, we assume they are fainter, using $M_{g^*} \sim 5.0$, which is about the absolute magnitude of the Sun. First, we fit the standard thin disk (scale height 0.25 kpc and scale length 2.5 kpc) and thick disk (scale height 1.0 kpc and scale length 3.0 kpc, with a local ratio of turnoff stars of 1:30 of the thin disk stars), shown as a black line in Figure \[five\]. The models are empirically normalized to the data at only one point. They are forced to match the data at $\alpha = 240^\circ$ for stars near $g^* = 17$. This fit sets the number of stars in this color range in the thick disk in the solar neighborhood to a reasonable value.
The star counts for standard thin and thick disk models do not fit. The discrepancy is most pronounced for the fainter star counts, where we see too many stars near the Galactic center and too few stars near the anticenter. We cannot move stars from the center to the anticenter by tweaking the assumed absolute magnitudes of the stars, the distance from the Sun to the center of the Galaxy, or the ratio of thin disk to thick disk stars. As we have seen, adding a power-law component will also not help add star counts near the anticenter. As we saw in Figure \[dwarfboxplot2\], the way to significantly increase the number of stars at faint magnitudes is to add a component with larger scale height and scale length.
We could take the thin disk and thick disk models and add an additional exponential disk model to attempt to fit the data. Since we now have so many adjustable parameters, we decided to fit only a ‘thin disk’ and an ‘metal-weak thick disk (MWTD),’ and were able to fit the data about as well as if standard ‘thick disk’ or ‘spheroid’ components were also included in the mix. We used a thin disk with a scale height of 300 pc and a scale length of 2.8 kpc, and a MWTD with a scale height of 1.8 kpc, a scale length of 8 kpc, and a thin disk to MWTD ratio of 100:1 in the solar neighborhood (for stars of this color range). We also used separate assumed absolute magnitudes for the two components: $g^* = 5.0$ for the thin disk and $g^* = 4.2$ for the MWTD. Note that the brighter stars, assumed to be part of the thinner disk, have $g^* - r^* \sim 0.25$, while the fainter stars associated with the MWTD structure have $g^* - r^* \sim 0.40$. Although one might expect the absolute magnitude of a more metal poor population to be fainter at the same color, the color difference (i.e. the thin disk samples stars further down the main sequence) is more important in this case. The models are fairly insensitive to our assumed absolute magnitudes. Figure \[five\] shows the model in red, with the thin disk and MWTD components in green and blue, respectively. With this model, the brighter star counts are dominated by the thin disk, and the fainter star counts are dominated by the MWTD.
We do not claim to show from this demonstration that a thick disk is ruled out. We have done the exercise of adding the third, thick disk exponential to the model, and it makes little difference. If we adjust slightly all of the parameters, a thick disk with reasonable properties can be easily added to the model. We refrain from quoting numbers for this, since there are so many correlated parameters in this model that the individual values of each parameter may have little meaning. The thick disk may help adjust the relative numbers of redder and bluer turnoff stars in detail. For example, look at the relative number of redder and bluer turnoff stars at $\alpha \sim 235^\circ$, $g^* = 17$ in Figure \[s10230240cmd\]. Most of them are the redder population. Now look the model for $16.5 < r^* < 17.5$ in Figure \[five\]. At $\alpha \sim 235^\circ$, somewhat more than half of the stars are MWTD, or the bluer population. We do note that qualitatively in the color-magnitude diagrams we see only two distinct turnoff colors, except in Figure \[cmd125\] where there is a set of very bright stars with a very blue turnoff. We do not see a turnoff that gets steadily redder with increasing magnitude, or which widely varies as a function of position in the Galaxy (see, for example, stars with $0.2 < g^*-r^* < 0.5$ in Figure \[s10230240cmd\]).
As with our model fits to Figure \[dwarfboxplot2\], the peak near the anticenter is not well centered on the model fits near $g^* = 18.5$. If we attempt to adjust our height above the plane to center the model, then the model becomes a very poor fit at the bright end. One could imagine that adding a warp to the MWTD might fix this discrepancy. Note in Figure \[Fwedge\] that there are multiple apparent overdensities of stars near the Galactic plane at the anticenter. At $\alpha \sim 125^\circ$, the overdensities are near $g^* = 15.5$ and $g^* = 19.5$. At $\alpha \sim 77^\circ$, the overdensities are near $g^* = 18$ and $g^* = 20$. We have no ready explanation for this.
Could all of this be explained without a disrupted dwarf galaxy or MWTD, but by warps or flares of the thick (or thin) disk? A warp in the disk means that the highest density of disk stars, which is generally in the Galactic plane ($b = 0^\circ$), shifts a little - to higher or lower Galactic latitude, depending on the direction. Such a warp has been detected in HI [@bt86] and possibly in stars as well [@cs93; @a01]. The measured shifts amount to less than half a kiloparsec deviation from the $b = 0^\circ$ plane. Shifting the thick disk up or down by this amount could throw more or fewer stars into our dataset at any given location, but would not explain why the stars had a bluer turnoff than the supposed thick disk stars.
Disk flaring occurs if the scale height of the disk increases with cylindrical radius from the center of the Galaxy. This effect has also been seen in the Milky Way [@a01] and perhaps in Andromeda [@gcr00]. But again, increasing the scale height with Galactocentric radius also does not put stars all at the same distance from us, and also does not explain the bluer turnoff of these stars. We would expect to see the flare putting stars into our sample at even larger scale heights at slightly higher Galactic latitudes. We do not see an excess of stars at $g^* > 20$ and $b > 20^\circ$ near the anti-center. The excess stops at $g^* \sim 19.5$.
What we would need is a thick (or thin) disk which goes out to 14 to 18 kpc from the center of the Galaxy, then warps sharply perpendicular to the plane, goes up to about $20^\circ$ Galactic latitude, and then ends abruptly. This would put stars all at the same distance from us (since we would be looking straight through the ‘disk’). The stars in this ‘disk’ must also have a turnoff with the same color as the spheroid population of stars in order to match the observations of Figure \[turnoffs\]. Alternatively, one could construct a flare model which flares up 18 kpc from the center of the Galaxy, and then decreases in scale height with distance.
S52$-$32$-$20.4
---------------
A look at Figure \[dwarfboxplot2\] in directions near the Galactic center shows that one could not fit the observed density of stars at $\alpha = 240^\circ$ and $\alpha = 320^\circ$ without assuming a very flattened spheroid ($q=0.5$). The exponential model is also a very flattened structure. The only way to avoid a very flattened spheroid is to postulate a large structure around $\alpha = 300^\circ$ which accounts for the star counts in this direction. A look at the distribution of magnitudes in this direction (Figure \[dwarfthickness4\]) shows that the magnitude distribution does not match our expectations for a power-law spheroid, and thus there is the possibility of yet more structure at $\alpha=320^\circ$. We identify this possible structure as S52$-$32$-$20.4.
We see below and in Table 2 that the stars in S52$-$32$-$20.4, which were originally assumed to be part of the spheroid population as well, appear to have significantly bluer turnoff stars than those of S6$+$41$-$20.0. This is a further indication that at least some of the stars in this direction are members of another Milky Way structure, and would release us from the need for a very flattened spheroid population.
If there is a stream at S52$-$32$-$20.4 that is not part of a smooth spheroidal distribution, then it is possible that we could fit a rounder $q=0.8$ model, also shown in Figure \[dwarfboxplot2\].
Properties of the halo structures\[properties\]
-----------------------------------------------
From positions of the turnoffs in color-magnitude diagrams in the vicinity of the identified structures, selection criteria were chosen which were intended to favor each of the structures mentioned above. The specific selections shown in Figure \[dwarfboxplot\] are: S223$+$20$-$19.4 and S200$-$24$-$19.8, black, ($7.05 (g^*-r^*) + 17.24 < g^* < 21$ and $g^*-r^* > 0.1$); S341$+$57$-$22.5 (Sagittarius), blue, ($21.5 < g^* < 23.5$ and $-0.1 < g^*-r^* < 0.7$); S167$-$54$-$21.5 (Sagittarius), red, ($20.5 < g^* < 22.5$ and $0.0 < g^*-r^* < 0.6$); and S297$+$63$-$20.0, green, ($20.0 < g^* < 21.5$ and $0.1 < g^*-r^* < 0.4$). The star counts per area as a function of right ascension for each of these selections are shown. Each point on the plot represents the number of stars with the selection criteria in a region of the sky 2.5 degrees wide in declination by 0.5 degrees wide in right ascension. The curves were normalized to match near the Galactic center; the scale factor is indicated in the figure legend.
We can use this plot to estimate the number of turnoff stars in S167$-$54$-$21.5 (Sagittarius), for example. The peak of the red curve in the plot is at 1000 stars. Subtracting off a background of 400 stars, that leaves 600 stars at the peak. But the curve has been multiplied by a scale of 2, so there were only really 300 stars at the peak. The width of the structure is about 50 degrees, or 100 bins. Multiplying $0.5 \times 100 \times 300$ for the area of a triangle gives $15,000$ turnoff stars spread over a $2.5 \times 50 = 125$ square degree area of sky.
The data curves in Figure \[dwarfboxplot\] also show a possible peak in the stellar density at $\alpha \sim 10^\circ$. This is faintly evident in Figure \[Fwedge\], but is not distinguished from an extension of S167$-$54$-$21.5 (Sagittarius).
In addition to counting stars in stellar streams, it is interesting to look for directional information on the angle at which the streams cross the celestial equator. We split the equatorial data into three roughly equal declination bins: $\delta < -0.4^\circ, -0.4^\circ < \delta < 0.4^\circ,$ and $\delta > 0.4^\circ$. The star counts in these bins are plotted as a function of right ascension in Figure \[streamdir\]. The center of S167$-$54$-$21.5 (Sagittarius) moves from $\alpha = 33^\circ$ to $\alpha = 36^\circ$ when the average declination goes from $\delta = -0.8^\circ$ to $\delta = 0.8^\circ$. The slope of this shift, $\Delta \delta/\Delta \alpha = 1.6/3 = 0.53$ is in excellent agreement with that predicted for the orbit of a Sagittarius stream at this position by @ilitq01, where $\Delta \delta/\Delta \alpha \sim 0.5$. The northern Sagittarius stream structure, S341$+$57$-$22.5, appears to move towards lower right ascensions as the declination increases, which is the expected sign, though the magnitude of the shift is smaller than predicted, suggesting some overlapping of streams may be present here. The direction of S297$+$63$-$20.0 cannot be distinguished from a track perpendicular to the equator. The structures S223$+$20$-$19.4 and S200$-$24$-$19.8 shift slightly towards lower right ascensions as the declination increases.
The color of the turnoffs of the various overdensities provide an illuminating check on their identities. Figure \[turnoffs\] shows a the number of stars near the turnoff as a function of color for stars in the various identified structures. The structures plotted are: S167$-$54$-$21.5 (Sagittarius), $30^\circ < \alpha < 45^\circ$, $21 < g^* < 21.75$; S297$+$63$-$20.0, $180^\circ < \alpha < 195^\circ$, $20 < g^* < 20.75$; S223$+$20$+$19.4, $120^\circ < \alpha < 130^\circ,
19.5 < g^* < 20.25$; S167$-$54$-$21.5, $70^\circ < \alpha < 80^\circ$, $19.25 < g^* < 20.0$; S52$-$32$-$20.4, $320^\circ < \alpha < 330^\circ$, $20.0 < g^* < 20.75$; S183$+$22$-$19.4, $100^\circ < \alpha < 125^\circ$, $19.5 < g^* < 20.25$, and the stars of the Sagittarius dwarf itself from Figure \[sagcmd\] with $22.25 < g^* < 23.0$. The counts for all curves have been normalized to peak at 1400. We chose the magnitude limits for each structure to produce the bluest possible turnoff. Table 2 lists the colors of the turnoffs of the structures in $g^*-r^*$.
The black line shows the stars which are most likely from the spheroidal population of stars in the Galactic halo. (According to our definition, a MWTD would qualify as a spheroid population.) It is interesting that the turnoff of the Sagittarius dwarf, whose photometry has been scaled to match that of S341$+$57$-$22.5 (Sagittarius), is bluer than that of the spheroid population. Likewise, the turnoff of S167$-$54$-$21.5 is blue - further evidence that this overdensity is a part of the tidal stream of the Sagittarius dwarf galaxy. None of the other identified structures have turnoffs as blue as these.
S223$+$20$-$19.4, S218$+$22$-$19.5 and S183$+$22$-$19.4 have the same color turnoffs as the spheroid distributions. However, S200$-$24$-$19.8, which one could imagine might belong to the same halo structure as S223$+$20$-$19.4 and S183$+$22$-$19.4, has a much bluer turnoff. We believe the reason for this can be found in Figure \[tdturnoffs\], which shows the number counts of stars near the thick disk turnoff ($16.0 < g^* < 16.75$). Note that the turnoff $g^*-r^*$ color of presumed thick disk stars in all directions are within a couple of tenths of 0.4, except in the direction of S200$-$24$-$19.8, which is much bluer than the rest. (There are very few thick disk stars in the field of the Sagittarius dwarf itself, which accounts for the apparently poor statistics of this curve.) If one adjusted the colors of the S200$-$24$-$19.8 turnoff stars by the amount needed for the thick disk turnoff in this direction to match all other directions, then the turnoff of S200$-$24$-$19.8 would more closely match the spheroid (S6+41-20.0) and S223$+$20$-$19.4, S183$+$22$-$19.4. We are uncertain as to the reason for the discrepancy, but believe that there is either a calibration error in these data, or the reddening correction could have been over-applied. If it is an error in the reddening correction which produced too blue a color by 0.05 magnitudes, then the $g^*$ and $r^*$ magnitudes should be shifted fainter by 0.15 and 0.10 magnitudes, respectively.
It is interesting that S297$+$63$-$20.0 appears to be intermediate in color between Sagittarius and the spheroid, as does S52$-$32$-$20.4. The latter is also a candidate for a flattened spheroid population.
Discussion
==========
What have we learned about the halo of the Milky Way from all of this? The most important lesson is that at distances of 20 kpc from the center of the Galaxy, the stellar density is not at all smoothly varying, as a power-law density distribution would be. It includes dwarf galaxies, globular clusters, and streamers of tidally stripped stars. With sufficiently large sky coverage, and good color photometry, these streamers can be identified by their density in space, and not just by kinematic techniques which have been previously used to identify moving groups in the solar neighborhood.
The prevalence and ambiguity of clumped stars in our data frustrate our attempts to fit any smoothly varying ‘spheroidal’ distribution. The only direction in the sky in which the stellar distribution looks at all like our expectations for a power-law distribution is at right ascension $240^\circ < \alpha < 250^\circ$, where the stars are less than 10 kpc from the Galactic center. In all other directions, the stellar distribution appears to be dominated by large structures with scale lengths of 10 kpc or greater, or by stars that do not fit neatly into the standard Galactic components. Even the stars near $\alpha = 245^\circ$ may not be identified as part of a presumed smooth power-law distribution in the halo.
All of our attempts to fit a power-law to the spheroid population of stars, both in magnitude and right ascension, produced best fits for a very flattened ($q \sim 0.5$) spheroid. We do not rule out a $q = 0.8$ spheroid, however, since we cannot be sure which of our data, if any, represents the spheroid distribution. We would also like to be clear that even if the spheroid is flattened, that does not imply a flattened halo dark matter distribution. One expects that stars and their associated dark matter are falling into the Galaxy from all directions.
The paper of @ksk94 is a recent work which found evidence for a flattened distribution of blue horizontal branch stars amongst other more spherically distributed populations of objects in the halo. This effect is discussed in @cb01, and references therein, and suggests that a flattened halo dominates at $R< 15$ kpc, while the outer halo is spherical. We imagine that if one averaged over all of the streams, the distribution could be spherical at large Galactocentric radii. Close to the plane the MWTD or a flattened spheroid could dominate, and the distribution would appear to be flattened.
The overdensities which we have so far identified, and the smooth distributions of stars which we have assigned to some of the overdensities, do not account for all of the stars in the dataset. For example, look at the star counts in Figure \[dwarfboxplot\] at $\alpha \sim 160^\circ$. There are significantly more stars here than in any model smooth model fit in Figure \[dwarfboxplot2\]. We could try to construct a stream profile for S297$+$63$-$20.0 that put stars out this far. The spreading of tidal streams in the disruption process is not unexpected, especially if the mass distribution of the halo is not spherical. However, in this case the profile would seem contrived to fill in gaps between the assumed spheroid distribution at $\alpha \sim 250^\circ$ and the structure at $\alpha \sim 75^\circ$.
There are enormous streams of stars in the halo. The tidal stream from the Sagittarius dwarf galaxy is one of them. We may have found additional large streams as described in this paper. Since our color cut is relatively blue, we are biased against finding older, or more metal rich streams with redder turnoff stars. We are also less sensitive to smaller, lower stellar density streams. One might expect that there are streams from smaller infalling stellar associations, or more disbursed streams from dwarf galaxies which were consumed by our Galaxy at earlier times in its history. These smaller or more disbursed streams might more naturally explain the difference in star counts between the models and the data at, for instance, $\alpha \sim 160^\circ$.
Debris from the Sagittarius dwarf \[debris\]
---------------------------------------------
We have shown evidence supporting the identification of S341$+$57$-$22.5 and S167$-$54$-$21.5 with the tidal stream of the Sagittarius dwarf galaxy. The color-magnitude diagrams for the stars in these structures match that of the dwarf itself. Also, the $g^*-r^*$ color of the turnoff is consistent with that of the Sagittarius dwarf. We find no reason to doubt the identification of the structures at $\alpha \sim 210^\circ$ and $\alpha \sim 35^\circ$ as pieces of the stream of the Sagittarius dwarf galaxy.
One might ask whether any of the other structures identified in this paper could be part of the Sagittarius stream as well.
We assume from its low stellar density that the structure S297$+$63$-$20.0 has undergone tidal disruption in the Milky Way. It is possible that it could be a part of the tidal stream of the Sagittarius dwarf galaxy. Figure 2 of @iils01 shows how debris from the Sagittarius dwarf is found off the main Sagittarius streamer orbit. If the F-stars in S297$+$63$-$20.0 are related to this off-stream debris, it implies a more disbursed stream than their $q=0.9$ model predicts. A model closer to $q=0.7$ is needed to explain the star density relative to that of S341$+$57$-$22.5 (Sagittarius) in terms of a single tidally precessed stream. Note, however, that the turnoff color of the stars in S297$+$63$-$20.0 do not support the idea that they originated in the Sagittarius dwarf galaxy. The turnoff color also does not rule out an identification with the Sagittarius stream; if they are associated it would imply that the stellar populations changed along a stream. It is also noteworthy that the S297$+$63$-$20.0 structure lies exactly on the plane of the Fornax-Leo-Sculptor dwarf galaxies [@m94], though it is much closer to the Galactic center than any of these dwarfs.
The Monoceros – Canis Major structure \[moncan\]
-------------------------------------------------
The most tantalizing structures we have identified are S223$+$20$-$19.4, S218$+$22$-$19.5 and S183$+$22$-$19.4, which may be two sides of the same contiguous structure. S200$-$24$-$19.8 could also belong to this structure, but its relationship is more difficult to establish, due to the lower data quality in this region. Though it is possible that we have found three independent, similar structures of stars in the halo, we find that possibility unlikely. In §5.7 we explored the possibility that this structure was part of a metal-weak thick disk. In this section, we explore the possibility that it is a tidally disrupted dwarf galaxy spread across $45^\circ$ on the sky and 11 kpc from us (18 kpc from the center of the Galaxy). We believe this would not have been identified previously because it is so large and close, and it is hidden by the plane of the Milky Way. Figure \[dwarfcenter\] shows our knowledge of the edge of this stellar structure.
Since we do not probe the full extent of any structure in this area of the sky due to the intervening Galactic plane, it is difficult to distinguish a disrupted galaxy residual stream from a dwarf galaxy. Without kinematic information it is difficult for us to identify streams with possible parent dwarf galaxies. Since we do not see much of the perimeter of this structure, we cannot distinguish very easily between a dwarf galaxy and a gravitationally unbound streamer which circles the entire Galaxy at an inclination $i < 20^\circ$ to the Galactic plane (or something in between). Most other orbital directions are ruled out because it is only evident at the ends of stripes 10, 11, 12, 37, and 82.
A dwarf galaxy or disrupted galaxy stream of stars in the Galactic halo provides a simpler model which produces stars all at about the same distance from us. With a distance modulus of about 15.2 (from Figure \[dwarfthickness\] and an assumed absolute magnitude of $g^* = 4.2$), the distance to S223$+$20$-$19.4 is 11 kpc from the Sun. The same distance is derived for S183$+$22$-$19.4. These two overdensities are separated by $40^\circ$ on the sky. If they indeed belong to the same structure, the structure is at least 8 kpc across. This is of the same scale as other large structures identified in the halo, including the Sagittarius dwarf spheroidal galaxy and the Sagittarius dwarf streamer. If S200$-$24$-$19.8 is part of the same structure, it is 8 kpc in the declination direction.
The width of the main sequence of S223$+$20$-$19.4 (see Figure \[dwarfthickness\]) is only about one magnitude wide at $g^* = 21.1$. From the errors in color alone (multiply the expected dispersion in $g^*-r^*$ color at $g^* = 21.1$, from Figure \[limmag\], by the slope of the main sequence in Figure \[s10230240cmd\]), we could explain this entire width. To gain an upper limit on the thickness of the structure, we assume the entire one magnitude dispersion is due to depth of the structure, and obtain an upper limit for the depth of the structure of 6 kpc. We obtain a similar measurement for the depth of S183$+$22$-$19.4.
We now ask what the mass of a satellite in the Galactic plane would have to be in order to remain tidally bound. A simple tidal analysis can be done following, for example, equation 7.84 in @bt87. The mass of the satellite within the tidal radius is given by: $$m_{sat} = 3 M_{MW} (\frac{r_{tidal}}{D})^3,$$ where $r_{tidal}$ is the tidal radius of the dwarf, $D$ is the distance of the satellite from the center of the Milky Way, and $M_{MW}$ is the mass of the Milky Way within a radius of $D$. This equation holds for $m_{sat} << M_{MW}$ and $r_{tidal} << D$. Plugging in $r_{tidal} = 4$ kpc and $D = 18$ kpc, we find that the satellite would have to have a mass equal to $3\%$ of the mass of the Milky Way. Estimating the mass of the Milky Way interior to $D$ from $M_{MW} = v_{MW}^2 D / G$ with $v_{MW} = 220$ km/sec, we find $M_{MW} = 2 \times 10^{11}$ M$_{\odot}$, and an inferred satellite mass of $6 \times 10^9$ M$_\odot$.
For reference, the dynamically estimated initial mass of the Sagittarius dwarf galaxy is between $10^9$ and $10^{11}$ M$_\odot$ [@il98; @jb00], and it is currently about $10^9 M_\odot$ [@jmsrk99]. Sagittarius is located 16 kpc from the Galactic center, and prolate with axis ratios 3:1:1 and a major axis of at least 9 kpc [@iwgis97]. So far, our observations could be explained by a dwarf galaxy, similar in size to the Sagittarius galaxy, and hiding in the plane of the Milky Way 18 kpc from the Galactic center.
We now ask whether the star counts support the existence of so massive a structure in the halo. For stars of this turnoff color, $g^*-r^* = 0.28$, a relatively metal-poor, spheroidal type population with $[Fe/H] = -1.7\pm 0.3$ is implied. An isochrone analysis like that for the Sagittarius stream of §\[ghostsag\] then indicates that these turnoff stars typically would have masses near $0.75 M_\odot$ and approximate ages of 13 Gyr.
It is difficult to estimate the total number of stars in the proposed structure. If it is a dwarf galaxy, we have only detected the tails of the distribution. Star counts must be estimated by extrapolation. The highest detected stellar density is about 1500 F and G stars above background in a 1.25 square degree region of the sky (Figure \[dwarfboxplot\]). A structure with constant stellar density over a $40^\circ \times 40^\circ$ area of the sky would contain $2 \times 10^6$ stars. This is a lower limit. If, instead, one fits to a model power-law distribution ($\alpha = -3.5$) of stars centered half way between our two detections, such as that shown in Figure \[dwarfthickness\]c and Figure \[dwarfboxplot\], we calculate $1 \times 10^7$ F and G stars in the whole structure. If we put the center of the dwarf galaxy in the plane of the Milky Way, the inferred star count is several times higher. One could increase or decrease the inferred mass in stars by suitably adjusting the axial ratios or density profiles of the models. A mass in stars of a few times $10^8$ solar masses is feasible, though by no means proven.
The total number of stars in the structure could easily be larger than these estimates if it is part of a stream which circles the Galaxy. If the stream contains at least $2 \times 10^6$ stars in the $40^\circ$ section of sky where we detected it, and if it extends all the way around the Galaxy with similar density, it must contain at least $2 \times 10^7$ stars. The actual stellar and dynamical masses are likely to be much higher, since these estimates use the lowest possible extent and stellar densities.
Thus, the overdensity could indicate a dwarf galaxy in the constellation Monoceros or in nearby Canis Major to the South. However, even if it is a dwarf galaxy, the high tidal mass calculated above suggests that it would be in the process of disrupting, just as the Sagittarius dwarf galaxy is. One could go a step further, and suggest that what we have detected is not a dwarf galaxy at all, but is instead a gravitationally unbound stream of stars. This conclusion might be preferred, since it frees us from explaining the coincidence of having found the very ends of the structure by chance in stripes 10, 82 and 37. Figure \[dwarfboxplot\] shows that even as a stream, this structure is significantly denser than the Sagittarius stream where it crosses zero declination. It is not only denser where we detect it, but it is also steeply rising as we run out of data.
If it is the result of the complete disruption of a gravitationally bound group of stars, the original mass of the infalling matter was probably quite large. The stream must contain at least $1 \times 10^6$ stars in the $40^\circ$ section of sky where we detected it. If it extends all the way around the Galaxy with similar density, it must contain at least $1 \times 10^7$ stars. The actual stellar and dynamical masses are likely to be much higher, since these estimates use the lowest possible extent and stellar densities.
Conclusions
===========
From stars in the Sloan Digital Sky Survey, we have shown that we can detect large ($\sim 10$ kpc) structures of stars in the halo of the Milky Way. In Paper I, we showed that substructure in the Galactic halo could be identified from photometric data for blue stars. In this paper we extended the technique to identify large structures directly from turnoff stars. The color-magnitude diagrams of the stars in the structures should resemble the color-magnitude diagrams of the original dwarf galaxies or clusters which fell into the Milky Way. Features of the diagrams can be used to constrain the origins of each detected overdensity.
As more data are collected from the Sloan Digital Sky Survey, we will be able to trace each structure through space, and connect the overdensities in each stripe to each other to build up a large scale map of large stellar streams in the halo of our Galaxy. For now, we must be content to identify and name each overdensity, and only to estimate their full extent and origin. In this paper, we studied the $g^*-r^*$ colors and $g^*$ magnitude distributions of seven overdensities of halo stars in the equatorial plane. We also show overdensities in three off-equatorial stripes, since they appear to be associated with the equatorial structures.
We emphasize these conclusions:
1\. We show additional evidence that the overdensities S341$+$57$-$22.5 and S167$-$54$-$21.5 are in fact part of the tidal stream of the Sagittarius dwarf galaxy. These structures were discovered Paper I, and were interpreted by @iils01 as two slices through the tidal stream of the Sagittarius dwarf galaxy.
The color-magnitude diagram of S341$+$57$-$22.5 bears striking resemblance to the color-magnitude diagram of the Sagittarius dwarf, including similar clumps of red stars. The color-magnitude diagram of S167$-$54$-$21.5 is consistent with that of the Sagittarius dwarf galaxy. In addition, the two overdensities are shown to have the same color turnoff stars as the Sagittarius dwarf galaxy; the turnoff of the Sagittarius dwarf is 0.08 to 0.1 magnitudes bluer in $g^*-r^*$ than the assumed Galactic spheroid stars, and substantially bluer than any other structure we have identified.
A comparison of the number of stars detected in the clump of red stars of S341$+$57$-$22.5 and S344$+$58$-$22.5 (in the adjacent stripe 11) with the number of similar stars in the Sagittarius dwarf indicate that we see about $1.5\%$ of the present stellar mass of the Sagittarius dwarf in this 110 square degree area of the sky. This result assumes a constant clump star to stellar mass ratio between the Sagittarius dwarf and the stream.
2\. From the spatial and magnitude distribution of turnoff stars in the spheroid, there is clear evidence for a diffuse structure S297$+$63$-$20.0, at a distance of about 20 kpc, extending over tens of degrees. Other evidence for this structure is a possible group of clustered RR Lyraes noted at the same distance and position by @v01. This structure is very close in position to the Sagittarius dwarf tidal stream at S341$+$57$-$22.5, but two magnitudes brighter.
Although its proximity to the Sagittarius stream suggests that it might be another part of this same disrupted galaxy, the color of its turnoff ($g^*-r^* = 0.26$) is not the same. It is intermediate between that of the Sgr dwarf ($g^*-r^*=0.22$) and that of the spheroid ($g^*-r^* = 0.28$). Surprisingly, the turnoff of S297$+$63$-$20.0 is nearly the same color as the turnoff of S52$-$32$-$20.4.
3\. We observe many more stars at low Galactic latitudes near the Galactic anticenter than standard models predict at $g^* \sim 19.5$. These stars were selected to be bluer than the turnoff of the thick disk stars. Several of our identified structures lie in this general direction, and may be part of the same physical structure in the Galaxy. The structures S223$+$20$-$19.4, S218$+$22$-$19.5, S183$+$22$-$19.4, and with less significance S200$-$24$-$19.8, have similar color-magnitude diagrams, turnoff colors, and inferred distances. The similarity between the color-magnitude diagrams for S223$+$20$-$19.4 and S183$+$22$-$19.4 is particularly striking. The narrow main sequence seen in the color-magnitude diagrams is consistent with stars all at the same distance, about 11 kpc from the Sun. S223$+$20$-$19.4 and S183$+$22$-$19.4 are separated by $40^\circ$ in right ascension. These are both separated from S200$-$24$-$19.8 by $40^\circ$ in declination. The inferred spatial extent of the structure is 8 kpc in declination, centered approximately on the Galactic plane, by at least 8 kpc in right ascension. Since it would seem coincidental to have detected the structure exactly at its ends, we expect the structure is substantially longer than 8 kpc. The inferred distance from magnitudes of turnoff stars is 11-16 kpc from the Sun. From the magnitude distribution, the structure is less than 6 kpc thick along the line of sight. The turnoff stars of this structure have colors of spheroid stars ($g^*-r^* = 0.28$), rather than colors of thick disk stars ($g^*-r^* = 0.40$).
We propose two possible explanations for the unexpectedly high concentrations of blue stars near the Galactic anticenter. One of the possibilities is that they are stars associated with a tidally disrupted dwarf galaxy. The other is that these stars are part of an ‘even thicker disk’ population which has a bluer turnoff than the thick disk, a scale height of about 2 kpc, and a scale length around 10 kpc. Though neither explanation explains all of the data, either model could be reasonably extended to work. We do not propose that these possibilities exclude all other models - they are merely the most reasonable explanations we could find.
The tidally disrupted dwarf galaxy model neatly explains a distribution of stars all at the same apparent distance. The presence of the disrupted Sagittarius dwarf galaxy proves that such structures can and do exist in the Milky Way halo, and that they can be detected by these techniques. The inferred physical parameters for such a structure, though large, are not prohibitive; the projected mass of the original dwarf galaxy could be of similar size to the Sagittarius dwarf galaxy. This model does not explain why the stars we see towards the Galactic center show an unexpectedly large flattening (other additional streams or Galactic components are required to explain this), or why the turnoff of this proposed dwarf has the same color as the stars towards the Galactic center.
The ‘even thicker’ double exponential disk model uses large scale lengths to put the peak of the stellar density at faint enough magnitudes. This model is appealing because it naturally explains why such a structure is found over at least $40^{\circ}$ of right ascension in the Galactic plane, and may correspond to the ‘metal-weak thick disk’ proposed by previous authors. The negatives of this model are that it does not fit the faint star counts near the Galactic center (Figure \[dwarfthickness3\]), and consumes all of the stars brighter than 20th magnitude which we expected were part of the power-law spheroid part of the halo. It also is rather broader in magnitude, spreading stars over a larger distance range, than the data suggest. This model would reduce the significance of, or eliminate, a power-law distribution of halo stars. The distribution in magnitude of the concentration of stars near the anticenter is somewhat narrower than expected for an exponential disk (Figure \[dwarfthickness\]).
Neither of these models explains the excess of stars at $15^{th}$ and $17^{th}$ magnitude near the plane at the Galactic anticenter. A stream model might introduce additional streams to explain this, whereas a disk model might introduce warping to explain this.
4\. On the other side of the Milky Way at $(l,b) = (52^\circ,-32^\circ)$, in a direction not far from the Galactic Center S52$-$32$-$20.4, there is evidence for stars distinct from a smooth spheroidal distribution of stars at $g^* = 20.8$. The distribution in magnitude is not consistent with a power-law spheroid, although it could be fit with an exponential disk with large scale length. This is in contrast to the structure S6$+$41$-$20.0, which is the only observed concentration of halo stars that shows the spatial distribution, both in right ascension and apparent magnitude, expected for a power-law Galactic spheroid. Additionally, the turnoff color of the stars in S52$-$32$-$20.4 is not the same as the presumed spheroid stars at S6$+$41$-$20.0, but rather intermediate between the spheroid and the Sagittarius dwarf.
If we try to fit a power-law to both the stars in S6$+$41$-$20.0 and S52$-$32$-$20.4, then we must have $q < 0.6$ (and there is a poor fit with distance in the direction of S52$-$32$-$20.4). If this structure is regarded as distinct from the spheroid, then the remaining spheroidal stars towards the Galactic center could be fit with a rounder model, $q=0.8$.
We do not present a single, coherent proposal for the components of the Milky Way, since it is not clear to what component each identified overdensity should be assigned. As more SDSS data is analyzed, and the extent of each structure is better known, we hope to generate a more coherent, defendable model.
5\. Aside from the obvious large overdensities in the halo, there is tantalizing evidence for further, smaller structures, for example at at $\alpha = 10^\circ$. One could imagine that there are even smaller structures which are not spatially resolved, which make up the difference between the observed star counts and the model fits to the spheroid population.
In this paper and in Paper I we identified seven or eight large overdensities which we believe might be associated with three or more halo structures. In view of these results, one must take seriously the possibility that there are many such previously unidentified structures in the halo. It is also probable that there are many smaller or more disrupted structures which might be better detected from kinematics than spatial information. Models of structure formation which have produced “too many galaxies per halo" may actually be predicting correct numbers of smaller halo structures. It appears we may be able to solve the problem by observationally finding more disrupted satellites in each halo. One cannot help but wonder many things about the results presented in this paper. We conclude our discoveries with a list of questions for which we do not yet have answers. Is there a previously undiscovered dwarf galaxy hidden in the plane of the Milky Way? If there is a massive streamer or dwarf galaxy which orbits our Milky Way in the Galactic plane, could this disrupt the disk at about 18 kpc from the Galactic center? Would it cause disk warping or flaring? Are there any dynamical models that could put a sheet of stars in a ring around the Galaxy without an infalling dwarf? If there is a structure with a scale length of many kiloparsecs which is only 11 kpc from us, can we detect any stars from this structure in the solar neighborhood kinematically or photometrically? Is there a metal-weak thick disk? Is there a model for the metal-weak thick disk which could explain why the stars seem to be shifted towards lower Galactic latitudes at $g^* \sim 18$? Why don’t we see a break, or at least a gradient, in the turnoff color between stars which would nominally be assigned to the thin disk and those which would be assigned to the thick disk? Finally, are there any halo stars which form a well-mixed, smooth, spheroidal distribution, and were any of them formed during the initial collapse of our Galaxy, as was proposed by @els62?
We acknowledge useful comments from Hugh Harris, Daniel Eisenstein, Amina Helmi, Doug Whittet and David Weinberg. We appreciate the close reading and extensive comments of the anonymous referee, which substantially improved the paper.
The Sloan Digital Sky Survey (SDSS) is a joint project of The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Princeton University, the United States Naval Observatory, and the University of Washington. Apache Point Observatory, site of the SDSS telescopes, is operated by the Astrophysical Research Consortium (ARC).
Funding for the project has been provided by the Alfred P. Sloan Foundation, the SDSS member institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http://www.sdss.org.
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[rcclrrrc]{}\
& & & & & & &\
& & & & & & &\
94 & 82 &N & 1998 Sep 19 & 350 & 56 & 1.7 & 1\
125 & 82 & S & 1998 Sep 25 & 350 & 77 & 1.9 & 1\
752 & 10 &S & 1999 Mar 21 & 145 & 233 & 1.4 & 1\
752 & 10 &S & 1999 Mar 21 & 234 & 250 & 1.4 & 2\
756 & 10 &N & 1999 Mar 22 & 117 & 121 & 1.4 & 2\
756 & 10 &N & 1999 Mar 22 & 122 & 235 & 1.4 & 1\
1350 & 37 &S & 2000 Apr 6 & 112 & 125 & 1.5 & 1\
1402 & 37 &S & 2000 Apr 27 & 112 & 125 & 1.5 & 1\
1450 & 37 &S & 2000 May 3 & 112 & 125 & 1.5 & 1\
1462 & 11 & S& 2000 May 5 & 120 & 137 & 1.5 & 1\
1752 & 82 &N & 2000 Oct 1 & 56 & 77 & 1.5 & 1\
1755 & 82 &S & 2000 Oct 2 & 319 & 350 & 1.4 & 2\
1907 & 11 &N & 2000 Nov 30 & 120 & 137 & 1.4 & 1\
2125 & 12 &S & 2001 Feb 20 & 122 & 135 & 1.5 & 1\
2126 & 12 &N & 2001 Feb 20 & 122 & 135 & 1.5 & 1\
[lrrcc]{}\
& & & &\
& & & &\
S6$+$41$-$20.0 & 10 & 235 & 0.30 & 0.40\
S167$-$54$-$21.5 & 82 & 37 & 0.22 & 0.40\
S297$+$63$-$20.0& 10 & 190 & 0.26 & 0.40\
8200$-$24$-$19.8 & 82 & 75 & 0.25 & 0.32\
S223$+$20$-$19.4 & 10 & 125 & 0.28 &0.38\
S183$+$22$-$19.4 & 37 & 135 & 0.28 &0.39\
S52$-$32$-$20.4 & 82 & 320 & 0.26 &0.40\
S218$+$22$-$19.5 & 12 & 125 & 0.28 & 0.39\
S341$+$57$-$22.5 & 10 & 213 & — & —\
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1.3cm
[**One–Loop Finiteness of the Four-Dimensional .1in Donaldson-Nair-Schiff Non-Linear Sigma-Model [^2]**]{}\
.3in
Sergei V. Ketov [^3]
[*Institut für Theoretische Physik, Universität Hannover*]{}\
[*Appelstraße 2, 30167 Hannover, Germany*]{}\
[*[email protected]*]{}
.2in
[**Abstract**]{}
The most general four-dimensional non-linear sigma-model, having the second-order derivatives only and interacting with a background metric and an antisymmetric tensor field, is constructed. Despite its apparent non-renormalizability, just imposing the one-loop UV-finiteness conditions determines the unique model, which may be finite to all orders of the quantum perturbation theory. This model is known as the four-dimensional Donaldson-Nair-Schiff theory, which is a four-dimensional analogue of the standard two-dimensional Wess-Zumino-Novikov-Witten model, and whose unique finiteness properties and an infinite-dimensional current algebra have long been suspected.
[**1**]{} [*Introduction*]{}. The two-dimensional Wess-Zumino-Novikov-Witten (WZNW) model [@wznw] is the particular non-linear sigma-model (NLSM) whose target space is a group manifold, and the NLSM torsion to be represented by the WZ term parallelizes the group manifold. The WZNW model is a conformally invariant quantum field theory and, hence, it is finite to all orders of the quantum perturbation theory. It possesses on-shell the conserved affine currents which satisfy an infinite-dimensional affine algebra. [^4]
It is quite natural to investigate to what extent those nice properties can be generalized to four dimensions, which would allow one to generalize some familiar concepts of two-dimensional conformal field theory up to four dimensions. Recently, some progress along these lines was reported by Losev, Moore, Nekrasov and Shatashvili [@lmns]. They mostly discussed the algebraic geometry aspects of a possible four-dimensional generalization of the WZNW model, while the issues of its renormalization and anticipated UV-finiteness remained open. In this Letter, I investigate the one-loop renormalization of the general four-dimensional NLSM coupled to a four-dimensional metric and a two-form, and determine the unique class of models which are one-loop finite. The relevant NLSM is essentially the one first introduced by Donaldson [@d], and later studied by Nair and Schiff [@ns] in the context of the three-dimensional Kähler-Chern-Simons theory. .2in
[**2**]{} [*The general action, and the background field method*]{}. Let $R_4$ be a four-dimensional manifold of Euclidean signature, which is parameterized by the coordinates $x^{\m}$, $\m=1,2,3,4$, and is equipped with a metric $h_{\m\n}(x)$ [*and*]{} a 2-form $\o=\o_{\m\n}(x) dx^{\m}\wedge dx^{\n}$, $\o_{\m\n}=-\o_{\n\m}$. Let $\F$ be a map from $R_4$ to another $n$-dimensional manifold $\cm$, and $\F^a$, $a=1,2,\ldots,n$, be the coordinates on $\cm$.
The most general NLSM action which is invariant under the reparametrizations of both $R_4$ and $\cm$, and has only second-order derivatives of the fields $\F^a$, is given by $$I[\F;h,\o] = \fracmm{1}{2\l^2} \int d^4x\,\sqrt{h}h^{\m\n}\pa_{\m}
\F^a\pa_{\n}
\F^b g_{ab}(\F) + \fracmm{\k}{2\l^2}\int d^4 x\,\ve^{\m\n\l\r}\o_{\m\n}\pa_{\l}
\F^a \pa_{\r}\F^b b_{ab}(\F)~,\eqno(1)$$ where the NLSM target space metric $g_{ab}$ and a 2-form $b=b_{ab}(\F) d\F^a
\wedge d\F^b$, $b_{ab}=-b_{ba}$, on $\cm$ have been introduced. In our notation, $dx^{\m}$ carries dimension one. Accordingly, the NLSM coupling constant $\l$ is of dimension one too, whereas another coupling constant $\k$ and all the fields are dimensionless.
In order to make our quantum calculations covariant with respect to the target space metric, we use the covariant background field method suitable for the NLSM [@he; @k2]. Let $\r^a(x,s)$ be the geodesic connecting $\F^a(x)$ with $\F^a(x)
+\p^a(x)$, such that $\r^a(x,s=0)=\F^a(x)$, $\r^a(x,s=1)=\F^a(x)+\p^a(x)$, and $$\fracmm{d^2}{ds^2}\r^a + \G^a_{bc}[\r]\fracmm{d}{ds}\r^b\fracmm{d}{ds}\r^c
=0~, \eqno(2)$$ where $\G^a_{bc}$ are the Christoffel symbols with respect to the metric $g_{ab}$. Let $\vec{\x}$ be the tangent vector to the geodesic, i.e. $$\x^a_{\rm s} =\fracmm{d}{ds}\r^a~,\qquad{\rm and}\qquad
\left. \x^a_{\rm s}\right|_{s=0}= \x^a~.\eqno(3)$$ The fields $\x^a(x)$ will be considered as the fundamental quantum fields in our theory whereas $\F^a$, $h_{\m\n}$ and $\o_{\m\n}$ as the background fields.
Let us now expand $I[\F+\p(\x)]$ in terms of the $\x$-fields, which ensures that all the coefficients of the expansion are tensorial quantities on $\cm$, $$I [ \F+\p(\x )]= I[ \F ] + I_1 + I_2 + \ldots~,\quad {\rm where}\quad
I_n=\left. \fracmm{1}{n!}\fracmm{d^n}{ds^n}I[\r(s)]\right|_{s=0}~.\eqno(4)$$ In practice, it is more convenient to use the derivative $D(s)$ to be defined with the covariant completion, instead of $d/ds$, i.e. $$\eqalign{
D(s) S[\r]=~&~\fracmm{d}{ds} S[\r]~,\cr
D(s)W_a[\r]=~&~\fracmm{d}{ds} W_a[\r] - \G^c_{ab}\x^b_{\rm s}W_c[\r]~,\cr
D(s)W^a[\r]=~&~\fracmm{d}{ds}W^a[\r] + \G^a_{bc}\x^b_{\rm s}W^c[\r],\cr}
\eqno(5)$$ Obviously, if $D(s)$ acts on a tensor function of $\r(s)$ only, we have $D(s)=\x^a_{\rm s}D_a$. Here are some useful identities: $$\fracmm{d}{ds}\pa_{\m}\r^a=\pa_{\m}\x^a_{\rm s}~,\quad
\fracmm{d}{ds}g_{ab}=\x^c_{\rm s}\pa_cg_{ab}~,\quad D(s)g_{ab}=0~,\quad
D(s)\x^a_{\rm s}=0~,\eqno(6)$$ and $$D(s)\pa_{\m}\r^a=(D_{\m}\x_{\rm s})^a~,\quad D^2(s)\pa_{\m}\r^a=R\ud{a}{bcd}
[\r]\x^b_{\rm s}\x^c_{\rm s}\pa_{\m}\r^d~,\eqno(7)$$ where $$(D_{\m}\x_{\rm s})^a \equiv
\pa_{\m}\x^a_{\rm s} +\G^a_{bc}[\r]\pa_{\m}\F^b\x^c_{\rm s}~.
\eqno(8)$$
The first term in the expansion (4) (and, hence, all the other) can only be expressed in terms of the totally antisymmetric field strength (3-form) $H_{abc}\equiv\frac{3}{2}\pa_{[a}b_{bc]}$ of the potential (2-form) $b_{ab}$ if we require the 2-form $\o$ to be [*closed*]{}, after integrating by parts in the second term of eq. (1). A calculation yields $$I_1=\fracmm{1}{\l^2}\int d^4 x\,\sqrt{h}h^{\m\n}(D_{\m}\x)^a\pa_{\n}
\F^bg_{ab}
(\F) +\fracmm{\k}{\l^2}\int d^4x\,\ve^{\m\n\r\l}\o_{\m\n}\pa_{\r}\F^a\pa_{\l}
\F^b\x^c H_{abc}(\F)~.\eqno(9)$$
Similarly, we find $$\eqalign{
I_2 =~&~ \fracmm{1}{2\l^2} \int d^4x\,\sqrt{h}h^{\m\n}\left[ g_{ab}(D_{\m}\x)^a
(D_{\n}\x)^b + R_{abcd}\pa_{\m}\F^a\pa_{\n}\F^d\x^b\x^c\right] \cr
~&~+ \fracmm{\k}{2\l^2}\int d^4x\,\ve^{\m\n\l\r}\o_{\m\n}\left[
2H_{abc}(\F)\pa_{\l}\F^a(D_{\r}\x)^b\x^c +D_bH_{adc}\pa_{\l}
\F^a\pa_{\r}\F^d\x^b\x^c\right]~.\cr} \eqno(10)$$
Let $V^i_a(\F)$ be the [*vielbein*]{} associated with the NLSM target space metric $g_{ab}(\F)$, $$g_{ab}(\F)=V^i_a(\F)V^j_b(\F)\d_{ij}~,\qquad V^{ai}=g^{ab}V^i_b~.\eqno(11)$$ Then, we can define the new derivative $\de_{\m}$ (without torsion) as $$V^i_a(D_{\m}\x)^a \equiv (\de_{\m}\x)^i = (\d^{ij}\pa_{\m}
+A^{ij}_{\m})\x^j~,\eqno(12)$$ where we have also introduced the vector $\x^i$ associated with $\x^a$ as follows: $$\x^i=V^i_a\x^a~,\qquad \x^a=V^{ai}\x^i~.\eqno(13)$$ It now allows us to rewrite eq. (10) to the form $$\eqalign{
I_2 =~&~\fracmm{1}{2\l^2}\int d^4x\,\left\{ \sqrt{h}h^{\m\n}(\de_{\m}\x)^i
(\de_{\n}\x)^i +2\k\ve^{\m\n\l\r}\o_{\m\n}H_{aij}\pa_{\l}\F^a(\de_{\r}\x)^i\x^j
\right.\cr
~&~\left. + (\sqrt{h}h^{\m\n}R_{aijb}\pa_{\m}\F^a\pa_{\n}\F^b
+\k\ve^{\m\n\l\r}\o_{\m\n}D_iH_{abj}\pa_{\l}\F^a\pa_{\r}\F^b)\x^i\x^j\right\}~,
\cr }\eqno(14)$$ where the kinetic terms for the quantum fields $\x^i$ have the standard form, thus defining the usual ($R_4$-covariant) $1/p^2$ propagator.
We are now going to redefine the connection in $\de_{\m}$ to $\hat{\de}_{\m}$, in order to hide in it the $(\de\x)\x$ term appearing in eq. (14). It suffices to take $$\hat{\de}_{\m}\equiv \de_{\m} +X_{\m}~,\eqno(15)$$ where $$X_{\m}^{ij} = \fracmm{\k}{\sqrt{h}}\ve^{\r\l\s\n}h_{\m\n}\o_{\r\l}\pa_{\s}
\F^a H^{ij}_a~.\eqno(16)$$
Finally, we have $$\eqalign{
I_2=~&~\fracmm{1}{2\l^2}\int d^4x\,\sqrt{h} \left\{
h^{\m\n}(\hat{\de}_{\m}\x)^i(\hat{\de}_{\n}\x)^i \right.\cr
~&~\left. + ( h^{\m\n}R_{aijb} +2\k\tilde{\o}^{\m\n}D_iH_{abj}
+4\k^2H_{aki}H_{bkj}
\tilde{\o}^{\m\r}h_{\r\l}\tilde{\o}^{\l\n})\pa_{\m}\F^a\pa_{\n}\F^b\x^i\x^j
\right\}~,\cr}\eqno(17)$$ where we have introduced the dual $\tilde{\o}$ of $\o$ as $$\tilde{\o}^{\m\n}\equiv \fracmm{1}{2\sqrt{h}}\ve^{\m\n\l\r}\o_{\l\r}~.
\eqno(18)$$
It is straightforward to calculate the other terms in the background-quantum field expansion (4) at any given order, although no simple reccursion formula is known. The $n$-th order term has the structure $$\eqalign{
I_n=~&~\fracmm{1}{2\l^2}\int d^4x\,\sqrt{h}\left\{
\left( h^{\m\n}\P^{(n,2)}_{(i_1\cdots i_n)(ab)}
+ \tilde{\o}^{\m\n}E^{(n,2)}_{(i_1\cdots i_n)[ab]}\right)
\x^{i_1}\x^{i_2}\cdots\x^{i_n}\pa_{\m}\F^a\pa_{\n}\F^b \right. \cr
~&~ + \left(h^{\m\n}\P^{(n,1)}_{(i_1\cdots i_{n-1})ja} +\tilde{\o}^{\m\n}
E^{(n,1)}_{(i_1\cdots i_{n-1})ja}\right) \x^{i_1}\x^{i_2}\cdots\x^{i_{n-1}}
(\hat{\de}_{\m}\x)^j\pa_{\n}\F^a \cr
~&~ \left. +\left(h^{\m\n}\P^{(n,0)}_{(i_1\cdots i_{n-2})(j_1j_2)}
+ \tilde{\o}^{\m\n}E^{(n,0)}_{(i_1\cdots i_{n-2})[j_1j_2]}\right)
\x^{i_1}\x^{i_2}\cdots\x^{i_{n-2}}
(\hat{\de}_{\m}\x)^{j_1}(\hat{\de}_{\n}\x)^{j_2}
\right\}~, \cr}\eqno(19)$$ where the $\P$’s and $E$’s are certain tensors to be constructed in terms of the curvature $R_{ijkl}$, the torsion $H_{ijk}$ and the covariant derivatives $\hat{D}_i$ on $\cm$, the metric $h$ and the two-form $\tilde{\o}$ on $R_4$. [^5] .2in
[**3**]{} [*One-loop finiteness conditions and their solution*]{}. We are now in a position to investigate the NLSM [*one-loop*]{} finiteness conditions, since eq. (17) is already sufficient for that purpose. We temporarily put aside the issues of the $R_4$-background dependence of the quantum effective action and the wave-function renormalization, i.e. consider on-shell contributions to the NLSM beta-functions first.
As far as the one-loop finiteness is concerned, the $(\pa\F)^2\x^2$ contribution to the $I_2$ has to vanish since, otherwise, it would inevitably contribute to the beta-functions. In the absence of the second term in the action (1), it would lead to the vanishing NLSM curvature $R_{ijkl}$ and, hence, to the linear NLSM action. The well-known statement about the non-renormalizabilty of the standard four-dimensional NLSM is therefore recovered this way. In the presence of the second term in the action (1), the situation is different since we now have the two additional resources — the 2-form $\o$ on $R_4$ and the torsion 3-form $H$ on $\cm$ — and they both can be adjusted in such a way that a cancellation between the first and the third terms in front of the $(\pa\F)^2\x^2$ contribution becomes possible. Clearly, it can only happen if we constrain the two-form $\o$ to satisfy the equation $$h^{\m\n} = -\tilde{\o}^{\m\r}h_{\r\l}\tilde{\o}^{\l\n}~.\eqno(20)$$ Given eq. (20), we still need the relation $$R_{ijkl} + 4\k^2 H_{ijm}H_{lkm}=0~.\eqno(21)$$ The minus sign in eq. (20) is important since, otherwise, there is no solution (see eq. (26) below).
The second term in front of the $(\pa\F)^2\x^2$ in eq. (17) has to vanish separately, since it represents an independent structure. It adds another condition, $$D_iH_{jkl}=0~,\eqno(22)$$ i.e. the torsion on $\cm$ has to be covariantly constant.
The remaining contribution to the $I_2$ now amounts to the [*minimal*]{} coupling of the quantum $\x$-fields to the external composite gauge field $X_{\m}$. Because of the gauge invariance, the on-shell invariant counterterms are to be constructed in terms of the generalized field strength and its covariant derivatives. In the case under consideration, the one-loop finiteness requires a [*parallelizable*]{} manifold $\cm$. The parallelizability condition for the generalized curvature $\hat{R}_{ijkl}$ is also sufficient since it is equivalent to eqs. (21) and (22) at the certain value of the real parameter $\k$ (see below).
Since the only parallelizable manifolds are group manifolds and seven-sphere, we have to choose our torsion to be represented by the group structure constants, [^6] $$H_{ijk}=f_{ijk}~,\eqno(23)$$ which automatically fulfils the condition (22). Indeed, for a group manifold, the vielbein is known to satisfy the Maurer-Cartan equation $$\fracmm{\pa V^i_a}{\pa\F^b} - \fracmm{\pa V^i_b}{\pa\F^a}
+2f^{ijk}V^j_aV^k_b=0~,\eqno(24)$$ whereas both the torsion and the curvature components, $H_{ijk}$ and $R_{ijkl}$, are all constants. Taking into account the explicit formula for the group curvature in terms of the structure constants, [^7] $$R_{ijkl}=-f^m_{ij}f^m_{kl}~,\eqno(25)$$ eq. (21) amounts to the relation $$4\k^2=1,\quad {\rm or} \quad \k=\pm \fracm{1}{2}~,\eqno(26)$$ which determines the dimensionless parameter $\k$ up to a sign. The dimensionful coupling constant $\l$ remains arbitrary, and it does not play an essential role in our considerations.
The crucial equation (20) has very simple geometrical meaning. Let us define the new tensor $$\tilde{\o}^{\m\r}h_{\r\l}\equiv J\ud{\m}{\l}~.\eqno(27)$$ Eq. (20) can now be rewritten to the form $$J\ud{\m}{\l}J\ud{\l}{\r}= -\d\ud{\m}{\r}~,\eqno(28)$$ i.e. $J$ is nothing but an (almost) [*complex structure*]{} on $R_4$ ! Eq. (27) also implies that $J^{\m\n}$ is antisymmetric, which means that $R_4$ has to be a [*hermitian*]{} manifold. Note that the one-loop finiteness conditions do [*not*]{} imply $\de_{\m}J\ud{\l}{\r}=0$ or $\hat{\de}_{\m}J\ud{\l}{\r}=0$, so that $R_4$ does not necessarily need to be a Kähler manifold. If, however, $R_4$ is Kählerian, then the closure of $\o$ follows automatically.
To summarize, the one-loop on-shell finiteness conditions for the four-dimensional NLSM of eq. (1) are:\
(i) the four-dimensional ‘spacetime’ $R_4$ has to be a [*hermitian*]{} manifold, equipped with a [*closed*]{} two-form $\o_{\m\n}$ defined by eq. (24), i.e. [*dual*]{} to $J^{\m\n}$;\
(ii) the NLSM target space manifold $\cm$ has to be a parallelizable [*group*]{} manifold (or, maybe, a seven-sphere).
The alternative derivation of the one-loop counterterm for the theory (17) can be performed by the (generalized) Schwinger-de Witt method [@bav]. The relevant action (17) can be represented in the form $$I_2 = \fracmm{1}{2\l^2}\int d^4x\,\sqrt{h}\,
\x^i\hat{F}_{ij}(\hat{\de})\x^j~,\eqno(29)$$ where the minimal second-order differential operator $\hat{F}(\hat{\de})$ can be easily read off from eq. (17). It is now straightforward to extract the one-loop counterterm for our case from the general results of ref. [@bav]. Having used dimensional regularization with the regularization parameter $2\e=4-d$, we find $$\eqalign{
\left.-\ha\Tr\ln \hat{F}(\hat{\de})\right|_{\rm div.}=~&~
\fracmm{i}{32\p^2\e}\int d^4x\,\sqrt{h}\,
\tr\left\{\fracm{1}{180}\left(R_{\m\n\l\r}
R^{\m\n\l\r}-R_{\m\n}R^{\m\n}+\bo R\right)\hat{\bf 1}\right.\cr
~&~ \left.+\fracm{1}{2}\hat{P}^2
+\fracm{1}{12}\hat{\car}_{\m\n}\hat{\car}^{\m\n}
+\fracm{1}{6} \bo \hat{P}\right\}~,\cr}\eqno(30)$$ where $\hat{P}$ just represents the $(\pa\F)^2$ term in eq. (17), whereas the generalized gauge field-strength $\hat{\car}^2$ is proportional to the generalized curvature with torsion.
Eq. (30) actually tells us something more, namely, the dependence of the one-loop counterterm from the four-dimensional background ‘spacetime’ metric $g_{\m\n}$ also. Because of the Gauss-Bonnet theorem valid in four dimensions, the curvature-squared term is reducible to the Ricci-tensor-dependent terms. Hence, in order to cancel all the $R_4$ background curvature dependent terms in the one-loop counterterm, it is necessary and sufficient to impose the [*Ricci-flateness*]{} condition on the $R_4\,$.
As regards possible (non-linear) field-renormalization effects, they should [*not*]{} be relevant for our results. As far as a NLSM is concerned, there exists a quantum field parametrization in which the field renormalization is absent (since we were not imposing any restrictions on allowed quantum field parametrizations, the parametrization required is just the one determineed by actual renormalization) [@k2].
For any complex manifold $R_4$ one can choose complex coordinates $(z^i,z^{\bar{i}})$, where $z^{\bar{i}}=(z^i)^*$ and $i=1,2$, in such a way that the complex structure $J$ takes the canonical constant form. Given such complex coordinates, the action (1) with $\cm=G$ takes the form of the Donaldson-Nair-Schiff (DNS) action [@d; @ns] $$I_{\rm DNS}[g] =-\,\fracmm{i}{4\p}\int_{R_4}\,\o\wedge\tr(g^{-1}\pa g
\wedge g^{-1}\bar{\pa}g) +\fracmm{i}{12\p}\int_{R_5}\,
\o\wedge\tr(g^{-1}dg)^3~,\eqno(31)$$ where we have introduced $R_5=R_4\otimes[0,1]$ and the $G$-valued fields [^8] $$g(x)=\exp\left[i\F^i(x)t^i\right]~.\eqno(32)$$
In accordance with the one-loop finiteness conditions, the 2-form $$\o=\fracmm{i}{2\l^2}h_{i\bar{j}}dz^i\wedge dz^{\bar{j}}\eqno(33)$$ has to be closed. If $R_4$ is a Kähler manifold, there exists the natural 2-form $\o$ which satisfies all our conditions – the so-called [*Kähler*]{} form [@cw]. .2in
[**4**]{} [*All-loop finiteness of the DNS action ?*]{} To prove the all-loop (on-shell, or $S$-matrix) finiteness of the DNS action, let us return back to the background field expansion specified by eqs. (4) and (19). Under the conditions (i) and (ii) given in sect. 3, only the last terms in the third line of eq. (19) survive, i.e. $$\P^{(n,2)}=\P^{(n,1)}=E^{(n,2)}=E^{(n,1)}=0~,\eqno(34)$$ the non-vanishing tensors $\P^{(n,0)}$ and $E^{(n,0)}$ being the products of the group structure constants [@k2]. Eq. (34) means, in particular, that [*all*]{} the $\pa\F$ dependence in the background-quantum field expansion (4) can be hidden inside the covariant derivatives (with torsion). Then, as far as the $l$-loop counterterms are concerned, the $\pa\F$-dependent contributions can only show up via the generalized field strength which is vanishing in our case. Hence, no covariant counterterms actually appear modulo the ones coming from the vacuum diagrams and depending upon the ‘spacetime’ metric only. The rigorous finiteness proof would require taking into account possible non-covariant divergences, if any. I believe that they all can be removed by a wave-function renormalization. Finally, in order to make sure of the absence of one-loop vacuum divergences. we already know that our complex ‘spacetime’ should be Ricci-flat. If it is also a Kählerian manifold, it is then a [*hyper-Kählerian*]{} one. It is the well-known theorem in four dimensions that any hyper-Kählerian manifold is actually [*self-dual*]{} [@ah]. But, for the self-dual four-dimensional backgrounds there can be no vacuum counterterms at all ! It can be proved e.g., by using the gravitational background field expansion for the self-dual gravity [@gr], or by invoking the relation which exists between the self-dual gravity and critical N=2 strings [@ov; @kbook]. The self-dual gravity is the [*exact*]{} effective field theory of the closed N=2 strings in four dimensions, [^9] while all the N=2 string scattering amplitudes with more than three legs vanish [@bv]. Therefore, there cannot be any renormalization of the self-dual gravity in four dimensions, at any loop order.
Above, both the Ricci-flatness and the Kählerness conditions for $R_4$ were explicitly used. It may well be possible to require only the Ricci-flatness for the hermitian ‘spacetime’, in order to get rid of the vacuum divergences, after switching to the (N=1) supersymmetric version of the theory. I believe that the DNS action can be supersymmetrized up to $N=4$.
The connection of the DNS action to the theory of N=2 strings becomes explicit when considering the equations of motion following from the action (31): $$\o\wedge \bar{\pa}(g^{-1}\pa g)=0~.\eqno(35)$$ These equations describe the coupling of the self-dual gravity to be represented by $\o$ to the principal NLSM fields associated with the self-dual Yang-Mills theory. Indeed, the so-called [*Yang*]{} equations $\bar{\pa}(g^{-1}\pa g)=0$ following from eq. (35) are known to be equivalent to the self-dual Yang-Mills equations in a particular gauge [@yang]. Since the self-dual Yang-Mills theory is the [*exact*]{} effective field theory for the open N=2 strings, ${}^8$ eq. (35) can be interpreted as the exact effective equation describing the interaction of open and closed N=2 strings. Invoking now the vanishing theorems for the open [*and*]{} closed N=2 string amplitudes, an anticipated finiteness of the DNS theory does not seem to be very surprising. .2in
[**5**]{} [*Conclusion*]{}. A possible finiteness of the DNS theory is consistent with its classical integrability (the Yang equations are solvable like that of the self-dual Yang-Mills !). After an explicit ‘space-time’ splitting of the Euclidean manifold $R_4=R_3\otimes R_1$, and introducing the phase space $\{P^i(x),g(x)\}$ for the theory (31), where the momenta $P^i$ are defined with respect to the symplectic form [@lmns] $$\O_{\o}=\int_{R_3}\,\tr\left[\d P\wedge g^{-1}\d g
- (I+\fracm{1}{4\p}\o\wedge g^{-1}dg)(g^{-1}\d g)^2\right]~,\eqno(36)$$ one finds the commutation relations $$\eqalign{
\[P^i(x),P^j(y)\]_{\o}=~&~f\du{k}{ij}(I+\fracm{1}{4\p}\o\wedge
g^{-1}dg)^k\d^{(3)} (x-y)~,\cr
\[P^i(x),g(y)\]_{\o}=~&~g(y)t^i\d^{(3)}(x-y)~.\cr}\eqno(37)$$ The infinite-dimensional symmetry algebra of the DNS theory can now be elegantly expressed in terms of the charges $$Q(\ve)=\int_{R_3}\,\ve^i\left[ I^i-\fracm{1}{4\p}\o\wedge
(g^{-1}dg)^i\right]~, \eqno(38)$$ where the Lie algebra-valued functional parameters $\ve^i(x)$ have been introduced. The charges $ Q(\ve)$ satisfy the algebra [@lmns] $$\{Q(\ve_1),Q(\ve_2)\}=Q(\[\ve_1,\ve_2\])+\int_{R_3}\,\o\wedge\tr
(\ve_1d\ve_2)~,\eqno(39)$$ which is the four-dimensional analogue of the affine algebra in the WZNW theory.
Given such remarkable properties of the DNS theory, there should exist its free field representation to be obtained by a field redefinition of the fields $g(x)$, which is yet to be found. Also, I expect the DNS action to be connected to a theory of $2+2$ dimensional membranes, or the M-theory. Even if the DNS theory is not finite beyond one loop, its supersymmetric version may appear to be finite to all orders. .2in
[**Acknowledgements**]{}
I would like to thank the Theory Group of the Institute of Experimental and Theoretical Physics (ITEP), where this work was initiated, for hospitality extended to me during my visit to Moscow, as well as to thank I. Buchbinder, A. Deriglazov, A. Losev and A. Morozov for useful discussions. .2in
[99]{} J. Wess and B. Zumino, ;\
S. Novikov, Sov. Math. Dokl. [**24**]{} (1981) 222, Usp. Mat. Nauk [**37**]{} (1982) 3;\
E. Witten, . S. V. Ketov, [*Conformal Field Theory*]{}, Singapore: World Scientific, 1995. A. Losev, G. Moore, N. Nekrasov and S. Shatashvili, [*Four-Dimensional Avatars od Two-Dimensional RCFT*]{}, Princeton, ITEP and Yale preprint, PUTP–1564, ITEP–TH.5/95 and YCTP-P15/95, September 1995; hep-th/9509151. S. Donaldson, Proc. London Math. Soc. [**50**]{} (1985) 1. V. P. Nair and J. Schiff, , ; . J. Honerkamp and G. Ecker, ;\
J. Honerkamp, . S. V. Ketov, ;\
S. V. Ketov, A. A. Deriglazov and Ya. S. Prager, . A. O. Barvinsky and G. A. Vilkovisky, Phys. Rep. [**119**]{} (1985) 1. P. Candelas, G. Horowitz, A. Streminger and E. Witten, ;\
P. Candelas and X. C. de la Ossa, . S. M. Christensen, S. Deser, M. J. Duff and M. T. Grisaru, . H. Ooguri and C. Vafa, . N. Berkovits and C. Vafa, ;\
J. Bischoff, S. V. Ketov and O. Lechtenfeld, ;\
S. V. Ketov and O. Lechtenfeld, . C. N. Yang, Phys. Rev. Lett. [**38**]{} (1977) 1377. M. F. Atiyah and N. J. Hitchin, [*The Geometry and Dynamics of Magnetic Monolopes*]{}, Princeton: Princeton Univeristy Press, 1988.
[^1]: =6.5in [\#1]{}
[^2]: Supported by the ‘Volkswagen Stiftung’
[^3]: On leave of absence from: High Current Electronics Institute of the Russian Academy of Sciences, ${~~~~~}$ Siberian Branch, Akademichesky 4, Tomsk 634055, Russia
[^4]: See, e.g. ref. [@kbook] for a review.
[^5]: In general, it is [*not*]{} possible to express the full expansion solely in terms of the generalized (with ${~~~~~}$ torsion) curvature $\hat{R}$ and the generalized (with torsion) covariant derivative $\hat{D}_i$, i.e. without an ${~~~~~}$ explicit appearance of the torsion $H$.
[^6]: The case of the seven-sphere deserves a separate study, and it is not considered here.
[^7]: We assume that our group is semi-simple and compact, for simplicity.
[^8]: The generators $t^i$ of the Lie algebra of $G$ with the structure constants $f_{ijk}$ satisfy the relations ${~~~~~}$ $\[t^i,t^j\]=2if^{ijk}t^k$ and $\tr(t^it^j)=2\d^{ij}$.
[^9]: Perhaps, modulo non-perturbative N=2 string corrections.
|
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abstract: 'A general input-output modelling technique for aperiodic-sampling linear systems has been developed. The procedure describes the dynamics of the system and includes the sequence of sampling periods among the variables to be handled. Some restrictive conditions on the sampling sequence are imposed in order to guarantee the validity of the model. The particularization to the periodic case represents an alternative to the classic methods of discretization of continuous systems without using the Z-transform. This kind of representation can be used largely for identification and control purposes.'
author:
- |
Amparo Fúster-Sabater and J.M. Guillén\
[Instituto de Electrónica de Comunicaciones, C.S.I.C.]{}\
[Serrano 144, 28006 Madrid, Spain]{}\
[[email protected]]{}
title: 'New modelling technique for aperiodic-sampling linear systems'
---
Introduction
============
Aperiodic sampling is a very interesting technique for improving the solution of several problems in control and identification. In particular, aperiodic sampling systems have been found useful in signal adaptation and compensation [@Hsia74; @Dormido80] and in optimal transmission of measuring errors in problems involving the solutions of systems of linear equations, such as observability [@Dormido79; @Troch73], controllability [@Sen80] and identifiability [@Sen79; @Sen80].
In all these cases it is essential to find a general model that
(*a*) describes the dynamics of the system;
(*b*) is adapted to real experimentation conditions (external representation against internal-state representation);
(*c*) includes the sequence of sampling periods among the variables to be handled.
A well-known input-output formulation is used successfully for linear systems sampled periodically. The scalar output of the plant at an arbitrary instant is described by means of input and output samples taken at previous instants. It seems natural to consider the aperiodic case in the same way, except that the finite difference equation coefficients, depending on the fixed sampling period, would have to be replaced by multivariable functions, depending on the aperiodic sampling sequence. Such a formulation would satisfy the above conditions.
There must be restrictions on the aperiodic sampling sequence in order that the model be valid, whereas in the periodic case any sampling interval is valid.
The systems considered are characterized by their impulse response (weighting function), which describes the output of the plant through a convolution expression. Consequently, the properties and results obtained are always given in terms of that function.
The paper is organized as follows. In §1 several basic assumptions and general considerations, which will be assumed through the work, are given. Section 2 is devoted to a modelling technique for aperiodic-sampling linear systems. This also includes the action of a sampler and zeroth-order hold preceding the plant. Section 3 contains some restrictive conditions on the sampling sequence. The problem of the selection of the aperiodic sequence is strictly considered in geometric terms and the results found in the literature are improved. In §4 a particularization to the periodic case has been made. The proposed technique represents an alternative to the classical methods of discretization of continuous systems, in order to obtain the discrete-model coefficients, without using the Z-transform. Finally conclusions in §5 end the paper.
Basic assumptions
=================
This discussion is restricted to:
\(i) linear time-invariant single-input/single-output differential systems of finite order $n$;
\(ii) systems whose transfer function $G(S)$ is a strictly proper rational function, and whose impulse response $h(t)$ is therefore a particular solution of
$$\label{eq:1}
h^{(n)}(t) + a_1h^{(n-1)}(t)+ \ldots + a_nh(t) = 0, \qquad t \geq 0$$
an *n*th-order homogeneous linear differential equation with constant coefficients $(a_i \in \mathbb{R})$.
$h(t)$ can be then be written as
$$\label{eq:2}
h(t)= \sum\limits_{i=1}^{n} C_i \, \varphi_i(t), \qquad t \geq 0$$
where $C_i \in \mathbb{C}$ are constant coefficients and $\varphi_i; \mathbb{R} \rightarrow \mathbb{C} \;\; (i = 1,\ldots ,
n)$ is the fundamental system of solutions of eqn. (\[eq:1\]).
We conclude this preliminary section with the following statement. Let $(G_i)$ be a family of vector functions $$G_i: \mathbb{R}^n \rightarrow \mathbb{R}^n
\qquad (i=0,1, \ldots, n)$$ $$G_i \in C^\infty(\mathbb{R}^n, \mathbb{R}^n)$$
$C^\infty(\mathbb{R}^n, \mathbb{R}^n)$ being the set of infinitely differentiable functions on $\mathbb{R}^n$. It then follows that if there exist
(*a*) an integer $r \leq n$ such that the elements $(G_0(z), \ldots$ $, G_r(z))$ are linearly independent for all $z \in \mathbb{R}^n$, and
(*b*) an integer $k>r$ such that $G_k(z)$ depends linearly on $(G_0(z), \ldots$ $, G_r(z))$, then there are functions
$$f_0, f_1, \ldots , f_n \in C^\infty(\mathbb{R}^n, \mathbb{R})$$ such that
$$\label{eq:3}
\sum\limits_{i=0}^{n}f_{i}(z) \; G_i(z)=0 \qquad \forall z \in \mathbb{R}^n$$
Modelling technique
===================
Generalities and key ideas of the methodology
---------------------------------------------
The class of linear time-invariant SISO systems is characterized by the impulse response, which describes for zero initial conditions the output of the plant through a convolution expression
$$\label{eq:4}
y_k=\sum\limits_{i=0}^{k}h(t_{k}-t_i) \, u_i$$
where $y_k$ is the output of the plant at time $t_k$, $u_i$ is the impulse input at time $t_i$, and $h(t)$ is the impulse response of the plant. According to (\[eq:1\]), the impulse response can also be written in matrix form by means of the equivalent linear system.
$$\label{eq:5}
\dot{X}(t) = AX(t)$$
where $$\label{eq:6}
X(t) = [\,h(t), \dot{h}(t), \ldots , h^{(n-1)}(t)\,]'$$ the symbol $'$ denotes the transpose and $A$ is an $n \times n$ bottom-companion matrix.
$$\label{eq:7}
A =
\left[\begin{array}{ccccc}
0 & 1 & 0 & \ldots & 0 \\
0 & 0 & 1 & \ldots & 0 \\
\vdots & \vdots & \vdots & \, & \vdots \\
-a_n & -a_{n-1} & -a_{n-2} & \ldots & -a_1 \\
\end{array}\right]$$
The solution of the linear system $$\label{eq:8}
\dot{X}(t) = exp(At) X_0 \qquad (X_0 = X(0))$$ is related to the impulse response through the expression
$$\label{eq:9}
h(t) = c \; exp(At) X_0$$
with $$\label{eq:10}
c = [\,1, 0, \ldots , 0\,]$$
Note that the components of the vector $X_0$ $$\label{eq:11}
X_0 = [\,h_1, h_2, \ldots , h_n\,]'$$ correspond to the first *n* Markov parameters
$$\label{eq:12}
h_{i+1} = \frac{d^ih(t)}{dt^i}\bigg |_{t=0} \qquad (i = 0, 1, \ldots, n-1)$$
It should also be noted that the triad $(A, X_0, c)$ leads us naturally to the observability canonical realization from the scalar impulse response.
From these matrices $A, X_0, c$ we are going to define a family of vectors functions
$$(G_0, G_1, \ldots, G_n)$$ $$G_i: \mathbb{R}^n \rightarrow \mathbb{R}^n
\qquad (i=0,1, \ldots, n)$$
given by $$\label{eq:13}
G_i(z_1, \ldots, z_n)= exp(A(z_1+ \ldots + z_i)) X_0 \qquad (i=1, \ldots, n)$$
$$\label{eq:14}
G_0(z_1, \ldots, z_n)= X_0$$
with $$z = [\, z_1, z_2, \ldots, z_n \,]' \in \mathbb{R}^n$$
The functions $G_i$ and the impulse response are related by means of the expression $$\label{eq:15}
h(z_1 + ... + z_i) =c \,G_i(z) \qquad z_1 + ... + z_i \geq 0$$
Let us now consider the statement given in §1 for the kind of function defined above.
From an analytical viewpoint, the functions $G_i$ belong to $C^\infty(\mathbb{R}^n, \mathbb{R}^n)$, as compositions of $C^\infty$ functions.
Let $I^n \subset \mathbb{R}^n$ be an open subset of $\mathbb{R}^n$ such that the vectors $(G_0(z), \ldots$ $, G_{n-1}(z))$ are linearly independent for all $z \in I^n$. In this case, it is easy to see that for the new domain $I^n$ the conditions *(a)* and *(b)* in the previous statement hold. In fact, condition *(a)* holds by definition of the subset $I^n$, and condition *(b)* holds by the dimensionality of the vector $G_i$. Hence there will be functions $$f_i(z) \in C^\infty(I^n, \mathbb{R}) \qquad (i=0, \ldots, n)$$
such that
$$\label{eq:16}
\sum\limits_{i=0}^{n}f_{n-i}(z) \; G_i(z)=0 \qquad \forall z \in I^n$$
Rewriting (\[eq:16\]) in matrix form, with $$G_i(z) = [\,G_{1i}(z) \ldots G_{ni}(z)\,]' \qquad (i=0, \ldots, n-1)$$
we get $$\label{eq:17}
\left[
\begin{array}{ccc}
G_{10}(z) & \ldots & G_{1,n-1}(z) \\
\vdots & \, & \vdots \\
G_{n0}(z) & \ldots & G_{n,n-1}(z) \\
\end{array}
\right]
\left[
\begin{array}{c}
f_n(z) \\
\vdots \\
f_1(z) \\
\end{array}
\right] =
\left[
\begin{array}{c}
G_{1n}(z) \\
\vdots \\
G_{nn}(z) \\
\end{array}
\right] (-f_0(z))$$
($f_0(z) = - 1$ for simplicity) and the functions $(f_1(z),
\ldots, f_n(z))$ can be obtained by solving a compatible system of linear equations.
The general form of the functions $f_i(z)$ is
$$\label{eq:18}
f_i(z) = \frac{Det[\,G_{0}(z) \ldots G_{n}(z) \ldots G_{n-1}(z)\,]}{Det[\,G_{0}(z) \ldots G_{n-1}(z)\,]}$$
where the numerator is the determinant obtained from the matrix $$[G_0(z), \ldots, G_{n-1}(z)]$$ by replacing the *i*th column by the column vector $G_n(z)$.
At this point, we identify the components of $z$ with the elements of the sampling period sequence
$$\label{eq:19}
\left \{ \begin{array}{r}
z_n=t_k - t_{k-1} = T_k \\
z_{n-1}=t_{k-1} -t_{k-2}=T_{k-1} \\
\vdots \qquad \qquad \;\;\; \;\;\; \vdots \;\;\; \qquad \;\;\; \vdots \;\;\;\\
z_1=t_{k-n+1}-t_{k-n}=T_{k-n+1} \\
\end{array} \right \}$$
We multiply both sides of (\[eq:16\]) by $c \, exp(Az^*)$, with $z^*$ successively taking the values
$$\label{eq:20}
z^* = - \sum\limits_{l=1}^{i} z_l \qquad (i=1, \ldots, n)$$
and we get in each case $$\label{eq:21}
\left \{ \begin{array}{rl}
c \, exp(A(z_{i+1}+ \ldots + z_n)) X_0 = & c \, \sum\limits_{l=1}^{n-i}f_{l}(z)\, exp(A(z_{i+1}+ \ldots + z_{n-l})) X_0 +\\
\, & c \, \sum\limits_{l=0}^{i-1}f_{n-l}(z)\, exp(-A(z_{l+1}+ \ldots + z_{i})) X_0 \\
\, & (i=1, \ldots, n)
\end{array}
\right \}$$
We define $$\label{eq:22}
g_{n-i}(z)= c \, \sum\limits_{l=0}^{i-1}f_{n-l}(z)\, exp(A(z_{l+1}+ \ldots + z_{i})) X_0 \qquad (i=1, \ldots, n)$$
From (\[eq:22\]) the functions $g_{n-i}(z)$ are of class $C^\infty$ as compositions of $C^\infty$ functions.
It can be checked that the functions $g_{n-i}(z)$ correspond to the first component of linear combinations of vectors $G_i$ with negative argument.
If we write
$$\left \{ \begin{array}{r}
f_{i}^k =f_{i}(z) \\
g_{j}^k =g_{j}(z)
\end{array}
\right \}$$
at time $t_k$, we can condense the preceding expressions into two sets of equations involving the functions $f_{i}^k, g_{j}^k$ and $h(t)$:
$$\label{eq:23}
\sum\limits_{i=0}^{n}f_{i}^k \,h(t_{k-i}-t{j})+g_{k-j}^k =0 \;\; (j=k, k-1, \ldots, k-n+1)$$
$$\label{eq:24}
\sum\limits_{i=0}^{n}f_{i}^k \,h(t_{k-i}-t_{j}) =0 \;\; (k \geq n ; \, j = k-n, \ldots, 0)$$
Multiplying every equation by $u_j$ $(j = k, k - 1, \ldots ,
1,0)$ (impulse inputs at the sampling instants) respectively and summing, we get $$f_1^k \,\sum\limits_{
l=0}^{k-1}u_{l}\, h(t_{k-1}-t_{l}) + \ldots + f_n^k \,\sum\limits_{
l=0}^{k-n}u_{l} \,h(t_{k-n}-t_{l})$$ $$\label{eq:25}
+\sum\limits_{j=0}^{n-1}g_{j}^k u_{k-j} = \sum\limits_{
l=0}^{k}u_{l} \,h(t_{k}-t_{l})$$
and, according to (\[eq:4\]), the preceding expression becomes
$$\label{eq:26}
y_k = \sum\limits_{i=1}^{n}f_{i}^k y_{k-i} + \sum\limits_{j=0}^{n-1}g_{j}^k u_{k-j}$$
which is called the input-output model for linear time-invariant aperiodic-sampling systems.
Simplified form of the functions $f_i, g_j$
-------------------------------------------
Companion matrices are an important example of what are called cyclic (or nonderogatory) matrices, which have only one (normalized) eigenvector associated with each distinct eigenvalue. This means that
\(i) the Jordan canonical form is clearly simplified (there is only one Jordan block for each distinct eigenvalue);
\(ii) the similarity transformation of the given matrix $A$ to the Jordan canonical form can be obtained in a standard way.
Indeed, $$\label{eq:27}
A = BJB^{-1}$$
where $J$ is the Jordan canonical form of the matrix $A$, and $B$ is an invertible matrix of a well-known general form.
In this way, (\[eq:16\]) becomes $$\label{eq:28}
\sum\limits_{i=0}^{n}f_{n-i}(z) \, exp(J\alpha_i) Y_0 = 0$$
with $$\label{eq:29}
\alpha_i=z_1+ \ldots + z_i \qquad (i=1, \ldots, n)$$
$$\label{eq:30}
\alpha_0 = 0$$
$$\label{eq:31}
Y_0 = B^{-1} X_0$$
Factorizing $det[exp (J \alpha_0) Y_0, \ldots, (J \alpha_{n-1})
Y_0]$ by means of Laplace’s expansion by minors and cancelling common factors in the numerator and denominator of (\[eq:18\]), the functions $f_i^k$ can be written as
$$\label{eq:32}
f_i^k = \frac{\Delta_i}{\Delta} \qquad (i=1, \ldots, n)$$
where
$$\label{eq:33}
\left \{ \begin{array}{cc}
\Delta = | \varphi_l (\alpha_j)| & (j=0, 1, \ldots, n-1) \\
\Delta_i = | \varphi_l (\alpha_j)| & (j=0, \ldots, n, \ldots, n-1) \\
\, & (l=1, \ldots, n)
\end{array} \right \}$$
The linear independence of the vectors $G_i$ implies the non-nullity of the determinant $\Delta$. We can check that the functions $f_i$ depend exclusively on the poles of the transfer function and on the sampling sequence. The result could be expected and agrees for the periodic case with the direct correspondence $$\lambda_i \rightarrow exp(\lambda_i T_0)$$
between the poles of the continuous and pulse transfer function.
The functions $g_j^k$ are given by
$$\label{eq:34}
g_{k-j}^k = -\, \sum\limits_{i=0}^{n}f_{i}^k \, h(t_{k-i}-t_{j}) \qquad (j=k, k-1, \ldots, k-n+1)$$
and depend on the poles and zeros of the transfer function, since they are obtained as a linear combination of the impulse response for specific arguments.
General formulation for systems with zeroth-order hold
------------------------------------------------------
If the sampler at the input is followed by a zeroth-order hold the input-output model is modified in the following way: $$\label{eq:35}
y_k=\sum\limits_{i=0}^{k}h(t_{k}-t_i) \, x_i$$
with
$$\label{eq:36}
x_i = u_i - u_{i-1}$$
$$\label{eq:37}
h(t)= \mathcal{L}^{-1} \left[
\begin{array}{c}
\frac{G(s)}{s}
\end{array}\right]$$
As $x_i$ represents the difference between two consecutive values of the input signal, the expressions (\[eq:23\]) and (\[eq:24\]) become $$\label{eq:38}
\left \{ \begin{array}{cc}
\sum\limits_{i=0}^{n}f_{i}^k[\, h(t_{k-i}-t_j) - h(t_{k-i}-t_{j+1})\,] + g_{k-j}^k=0 & (j=k, k-1, \ldots, k-n) \\
\sum\limits_{i=0}^{n}f_{i}^k[\, h(t_{k-i}-t_j) - h(t_{k-i}-t_{j+1})\,] =0 & (j=k-n-1, \ldots, 0) \\
\, & (k>n)
\end{array} \right \}$$
Briefly, the presence of a zeroth-order hold implies that
(*a*) the functions $f_i$ are the same as before;
(*b*) the functions $g_j$ are modified: the number functions $g_j$ is increased by one (so that now $0\leq j \leq n$), and their general expression is similar to (\[eq:34\]) but with $$h(t_{k-i}-t_j) \rightarrow h(t_{k-i}-t_j)- h(t_{k-i}-t_{j+1})$$ for $h(.)$ defined in (\[eq:37\]).
The results obtained were to be expected, since the functions $f_i$ depend only on the poles of the transfer function, which reflect the internal coupling in the system and its autonomous behaviour. However, the functions $g_j$ reflect the internal plant coupling to the input signal, which has been affected by the presence of the zeroth-order hold.
Main results
------------
We recall briefly the main results we have obtained.
(*a*) We have shown that the functions $f_i$, $g_j$ are infinitely differentiable.
(*b*) We have obtained a general and systematic formulation for every function $f_i$, $g_j$ in contrast with the procedure found in [@Mellado70], where all these functions are computed globally.
(*c*) We have developed a procedure to impose restrictive conditions on the sampling sequence in order to guarantee the linear independence of the vectors $(G_0(z), G_1 (z), \ldots , G_{n - 1}(z))$.
In the next section we shall determine the set of vectors $I^n \subset \mathbb{R}^n$ whose elements $z$ satisfy the above condition.
Choice of the sampling-period sequence. A geometric interpretation
==================================================================
The problem of the choice of the sampling sequence has been treated in [@Troch73] analytically. Making use of the concept of a Chebyshev system, some intervals of the real line are selected in which the sampling instants can be chosen freely. In this way, the non-nullity of the determinant $\Delta$ is guaranteed, and consequently the difference equation for the aperiodic case can be obtained (via (\[eq:26\]), (\[eq:32\]) and (\[eq:34\])).
In the present paper the same problem is considered geometrically. The choice of sampling-period sequence is directly related to the properties of certain vectors in the space $\mathbb{R}^n$. In this way, the intervals found by Troch [@Troch73] are largely increased for low-order models and some general considerations for higher-order models are made.
Second-order model (n = 2)
--------------------------
We are going to consider a second-order model with a pair of complex eigenvalues $$\label{eq:39}
a + jb \in \mathbb{C} \qquad (b > 0)$$
The problem depends on an adequate choice of the sampling periods in such a way that the linear independence of the vectors $$\label{eq:40}
[Y_0, Y_1] = [exp (J\alpha_0)\,z_0, exp (J\alpha_1)\,z_0]$$ will be preserved, where $$\label{eq:41}
\alpha_0 = 0$$ $$\label{eq:42}
\alpha_1 = t_{k-1} - t_{k-2}= T_{k-1}$$ $J_r$ is the real canonical form of the matrix $A$:
$$\label{eq:43}
J_r = \left[
\begin{array}{cc}
a & -b \\
b & a \\
\end{array}
\right]
\qquad (b > 0)$$
and $T$ is the (invertible) matrix governing the change of basis: $$\label{eq:44}
z_0 = T^{-1}X_0$$
As we are in $\mathbb{R}^2$, the geometric interpretation is very simple. The generic operator $exp (J_r\alpha)$ applied to the vector $z_0$ can be viewed as follows. It is a counterclockwise rotation through $b\alpha$ radians, followed by a stretching (or shrinking) of the length of $z_0$ by a factor $exp (a\alpha)$ [@Hirsch74].
From this interpretation $Y_0$ and $Y_1$, will be linearly independent if and only if
$$\label{eq:45}
b\alpha_1 = b\,T_{k-1}\neq {\dot{\pi}}$$
where ${\dot{\pi}}$ denotes an integral multiple of $\pi$. Otherwise the vectors will be colinear.
Comparing these results to those of Troch [@Troch73] we have the following. According to [@Troch73], given $t_{k-2}$, $t_{k-1}$ can be fixed so that $$\label{eq:46}
t_{k-1} \in (t_{k-2}, t_{k-2} + \frac{\pi}{b})$$
According to (\[eq:45\]), given $t_{k-2}$, $t_{k-1}$ can be fixed so that $$\label{eq:47}
t_{k-1} \in I = (t_{k-2}, \infty ) - (t_{k-2} + \frac{ {\dot{\pi}} }{b})$$
Only point values will be rejected.
It can also be checked that such a formulation is concerned with differences between sampling instants but not with the actual position of such instants on the real axis. This agrees perfectly with the kind of time-invariant systems we are dealing with.
If the condition (\[eq:45\]) is violated, the sampling process resonates with the system dynamics and the formulation then obtained no longer affords a faithful representation of the system.
We can also see that, for the second-order model, (\[eq:45\]) is a necessary and sufficient condition to guarantee the linear independence of the vectors $Y_i$ whereas the condition given by Troch is only sufficient.
Third-order model (n = 3)
-------------------------
We are going to consider a third-order model with a real pole and a complex pair
$$\label{eq:48}
\lambda \in \mathbb{R}, \;\; a + jb \in \mathbb{C} \;\, (b > 0)$$
The point is to choose the sampling instants in the right way; that is, such that the vectors
$$\label{eq:49}
[Y_0, Y_1, Y_2] = [exp (J_r\alpha_0)\,z_0, exp (J_r\alpha_1)\,z_0, exp (J_r\alpha_2)\,z_0]$$
are linearly independent, where $$\label{eq:50}
\alpha_0 = 0$$ $$\label{eq:51}
\alpha_1 = t_{k-2} - t_{k-3}= T_{k-2}$$ $$\label{eq:52}
\alpha_2 = t_{k-1} - t_{k-3}= T_{k-1} + T_{k-2}$$
$$\label{eq:53}
J_r = \left[
\begin{array}{ccc}
a & -b & 0 \\
b & a & 0 \\
0 & 0 & \lambda \\
\end{array}
\right]
\qquad (b>0)$$
with $z_0, T$ defined as before.
The problem is treated in three-dimensional space, and the geometric structure can be described as follows. The generic vector $Y(\alpha)$ is written as $$\label{eq:54}
Y(\alpha) = [exp (a \alpha)\, cos(b \alpha), exp (a \alpha) \, sin(b \alpha), exp (\lambda \alpha)]'$$
It is therefore, a spiral on the surface of revolution $$\label{eq:55}
z = (x^2+ y^2)^{\lambda/2a}$$
whose form is determined by the real part of the system eigenvalues.
The vectors $Y_0, Y_1, Y_2$ have their origin at the point $(0, 0, 0)$ and their ends at the points $Y_(\alpha_0), Y_(\alpha_1), Y_(\alpha_2)$ of the parametric curve.
Heuristically we can imagine the same vectors as before ($n = 2$) pointing upwards from the $(X, Y)$ plane as they have a third component on the $Z$ axis.
From this geometric interpretation we can study some interesting cases of linear dependence.
(*a*) If the sampling-period sequence is chosen in such a way that $$\label{eq:56}
b \alpha_i = {\dot{\pi}} \qquad (i = 0, 1, 2)$$ according to Fig. 1, the vectors $Y_0, Y_1, Y_2$ will be coplanar (in a plane containing the $Z$ axis) and therefore they will be linearly dependent since every vector can be written as a linear combination of the other two. In physical terms, this can be regarded as a resonance of the sampling sequence with a pair of complex eigenvalues. This situation can be generalized to higher-order models if the equation (\[eq:56\]) holds for the imaginary part of any pair of complex eigenvalues.
(*b*) If the real eigenvalue and the real part of the complex pair are equal then the revolution surface is a cone and there will be linear dependence if at least two vectors are on the same generator (see Fig. 2). Analytically, $$\label{eq:57}
b (\alpha_j - \alpha_i) = 2{\dot{\pi}}$$
for any $i,j$ such that $0\leq i < j \leq 2$.
Physically, this situation can be regarded as a resonance of all the eigenvalues with the time interval between two sampling instants, not necessarily consecutive.
This situation can be generalized to higher-order models if the eigenvalues are such that
\(i) the real parts are the same, and
\(ii) all the imaginary parts $b_j \; (j = 1, \ldots,
q)$ satisfy (\[eq:57\]).
(*c*) This is the general case where there is linear dependence because the vectors $Y_i$ are contained in an arbitrary plane, which obliquely intersects the surface of revolution passing through the origin.
The intersection of the oblique plane and the surface of revolution is a closed curve $\Gamma$. Therefore the spiral will intersect such a curve in only two points for each rotation of $2\pi$ radians. From this interpretation, different geometric structures can be obtained, depending on the parameter values. In fact, we have the following possibilities: $$\begin{array}{cc}
\lambda > 0 &
\left \{ \begin{array}{c}
a>0 \\
a<0 \end{array} \right \} \\
\, & \, \\
\lambda < 0 &
\left \{ \begin{array}{c}
a>0 \\
a<0 \end{array} \right \}
\end{array}$$
Now $\lambda > 0 \, (\lambda < 0)$ implies that the spiral goes up (down) as far as $\alpha \rightarrow \infty$; and $ a > 0 \, (a <
0)$ implies that the surface of revolution expands (shrinks) when we move up the $Z$ axis.
*Case 1:* $\lambda >0, a>0$
From examination of $P_0$ and $P_1$ the normalized projections of $Y_0$ and $Y_1$ on the $(X, Y)$ plane in Fig. 4, it is clear that if $Y_2$ is contained in the oblique plane then its normalized projection on the $(X, Y)$ plane is situated on the $arc P_0 P_1 =
b\alpha_1$ (here $P_0$ and $P_1$ denote the ends of the vectors $P_0$ and $P_1$ respectively). Therefore a sufficient condition that guarantees the linear independence of the three vectors is that the rotation $b \alpha_2$ is not part of such an arc.
In this way, $P_0 P_1$ and the corresponding multiples of $2\pi$ are the forbidden intervals, and consequently $P_1 P_0$ and the corresponding multiples of $2\pi$ are the allowed ones.
It has to be pointed out that inside the forbidden intervals there would only be two intersection points of the spiral with the curve $\Gamma$ for each rotation through angle $2\pi$. However, we reject the whole interval since we intend to find sampling intervals in which the linear independence of the vectors $Y_i$ is automatically guaranteed without analytical computations.
The most favourable case corresponds to a rotation $P_0 P_1$ as short as possible, because this would increase the length of the allowed interval. Reciprocally, the most disfavourable case corresponds to a rotation $P_0 P_1$, with angle close to $\pi$.
For values of $\alpha$ such that
$$\label{eq:58}
|Y(\alpha)| > |Y(\alpha^*)|$$
where $Y(\alpha^*)$ is the vector of maximum modulus with its origin at the point $(0, 0, 0)$ and its end on the curve $\Gamma$, the choice of the third vector is completely arbitrary, since $Y_2$ can never be on the oblique plane. Therefore the number of forbidden intervals is finite and depends on the previous relation.
For the case $\lambda > 0, a < 0$ the allowed intervals are the same as before.
*Case 2:* $\lambda < 0, a < 0$
We proceed in the same way as before. Figure 5 shows the projections of $Y_0, Y_1$ and the curve $\Gamma$ on the $(X, Y)$ plane. The diameter $R_1R_0$ is parallel to the chord joining the end of the normalized projections $P_0$ and $P_1$. For each rotation through angle $2\pi$, the spiral will intersect the curve $\Gamma$ in only one point at $P_1 R_1$, and the same will happen at $R_0 P_0$. Therefore a sufficient condition to guarantee the linear independence of the three vectors in that the rotation $b\alpha_2$ is not part of such an arc. Consequently, $P_0 P_1$ and $R_1 R_0$ and the corresponding multiples of $2\pi$ radians would be the allowed intervals.
It should be pointed out that the arc $P_1 R_1$ is allowed only for the first rotation. The most favourable case corresponds to a rotation $P_0 P_1 = b \alpha_1$, with angle as close to $\pi$ as possible. Conversely, the most unfavourable case corresponds to a rotation with angle as small as possible.
In this situation the number of forbidden intervals is not finite, since there are always two intersection points for each rotation of angle $2 \pi$.
For the case $\lambda < 0, a> 0$ the allowed intervals are the same as before.
Comparing these results with those of [@Troch73], we can see that for the latter the allowed interval, in rotational form, can be expressed as
$$\label{eq:59}
b\alpha_2 < \pi$$
which, according to the geometric interpretation, is clearly a shorter interval. In both cases the conditions are sufficient.
The only interesting cases for second- and third-order models are those considered in $\S\S$ 4.1 and 4.2.
For systems without complex eigenvalues, the linear independence of the vectors $Y_i$ is automatically guaranteed for any arbitrary choice of the sampling instants [@Troch73].
General considerations for higher-order models
----------------------------------------------
From the fourth-order upwards, the situation becomes more and more complex because we do not have the geometrical insight given by the plane or three-dimensional space. However, from the basic structures developed for $\mathbb{R}^2$ and $\mathbb{R}^3$, we can make some general considerations that simplify the choice of sampling instants in higher-order models.
In order to clarify these ideas, we are going to consider a 4th-order model ($n = 4$) with one pair of complex eigenvalues and one pair of real eigenvalues: $$\label{eq:60}
\lambda_1, \lambda_2 \in \mathbb{R}, \;\; a + jb \in \mathbb{C}$$
The parametric curve is $$\label{eq:61}
Y(\alpha) = [exp (a \alpha) \,cos(b \alpha), exp (a \alpha) \,sin(b \alpha), exp (\lambda_1 \alpha), exp (\lambda_2 \alpha)]'$$
which can be decomposed into $$\label{eq:62}
c_1(\alpha) = [exp (a \alpha) \,cos(b \alpha), exp (a \alpha) \,sin(b \alpha), exp (\lambda_1 \alpha)]$$
$$\label{eq:63}
c_2(\alpha) = [exp (a \alpha) \,cos(b \alpha), exp (a \alpha) \,sin(b \alpha), exp (\lambda_2 \alpha)]$$
and we can study the evolution of each curve separately in the same way as before.
The normalized projections of the vectors $$(c_1(\alpha_i)), (c_2(\alpha_i)) \qquad (i= 0, \ldots, 3)$$
on the $(X, Y)$ plane will coincide, since the only element varying is the third component and not the angle of rotation.
Take $(\alpha_0, \alpha_1,\alpha_2,\alpha_3)$ such that each subset of three elements in either of the sets $(c_1(\alpha_i))$ $(c_2(\alpha_i))$ is linearly independent. Then we have that each subset of three elements in $Y_i$ $(i = 0, \ldots, 3)$ is linearly independent. Therefore it suffices to check the linear dependence of any one vector with respect to the other three.
In analytical terms, this means that $$\label{eq:64}
M_1 \,c_1(\alpha_j)= M_2 \,c_2(\alpha_j)$$
where $M_1$ and $M_2$ are $3 \times 3$ matrices of general form
$$\label{eq:65}
\left \{ \begin{array}{c}
M_{1}= (c_1(\alpha_i)) \\
M_{2}= (c_2(\alpha_i))
\end{array} \right \} \qquad (i=0, \ldots, 3), \; i \neq j$$
If (\[eq:64\]) does not hold then the linear independence of the vectors $Y_i$ $(i = 0, \ldots, 3)$, obtained through the manipulation of 3-dimensional vectors, is guaranteed. Therefore we have been able to reduce the problem dimension by one.
The method can be generalized to arbitrary order $n$, although the problem becomes more and more complicated as the order of the model is increased.
If the multiplicity of the poles is greater than 1, the third component of the vectors $c_i(\alpha)$ is a polynomial expression with trigonometric functions, which will give a more complicated curve.
If there is more than one pair of complex eigenvalues, that with the greatest imaginary part (maximum eigenfrequency) will be the main pair; since it determines the rotation with greater angle.
When the dimension of the model is very large, it is more convenient to use the shorter intervals found in the literature or to reduce the order of the model and then apply the techniques developed to the reduced model.
Particularization to the periodic case
======================================
The aperiodic model developed in previous sections is also valid for the periodic case. In this situation, the general expressions and the geometric interpretation can be simplified.
Periodic model
--------------
Once the sampling period $T_0$ has been fixed, the multivariable functions $f_i, g_j$ are reduced to constant coefficients $a_i, b_j$.
It can be demonstrated by induction that for the one-variable case the coefficients $a_i$ can be written as
$$\label{eq:66}
a_i= (-1)^{i-1} \, \Sigma \phi_{j1} \ldots \phi_{ji}$$
where the sum is taken for $1 \leq jp \leq n$ and $1 \leq p \leq i
\; (i = 1, ldots, n)$ $$\label{eq:67}
\phi_{ji}(T_0)= exp(\lambda_i T_0)$$
($\lambda_i$ is an eigenvalue of the continuous system). If the multiplicity of $\lambda_i$ is $m_i > 1$ then there are $m_i$ functions $\phi_{i}$ which are all equal. The $a_i$ correspond to coefficients with opposite sign of the polynomial $A(z)$ in the $Z$-transfer function $$\label{eq:68}
G(z)= \frac{B(z)}{A(z)}$$
Indeed, $$\label{eq:69}
A(z) = 1+ a'_1 z^{-1} + \ldots + a'_n z^{-n}$$
with $$\label{eq:70}
a'_i = -a_i$$
It is well known that between the coefficients and the roots of a polynomial there is a variational relation like that given by (\[eq:66\]).
According to (\[eq:34\]), the coefficients $b_j$ are $$\label{eq:71}
b_{k-j}= - \sum\limits_{i=0}^{n}a_{i} \,h(t_{k-i}-t_{j}) \;\; (j = k, k-1, \ldots, k-n+1)$$
with $a_0= -1,\, k \geq n$. Equations (\[eq:66\]) and (\[eq:71\]) represent an alternative approach to that presented by the $Z$-transform in order to compute discrete-model coefficients from the weighting function.
The formulation corresponding to the case with zeroth-order hold can be also transcribed to the periodic version by means of (\[eq:38\]).
Influence of sampling period on the discrete model parameters
-------------------------------------------------------------
To discuss the effect of sampling period on the absolute values of parameters we give the following example.
We consider the continuous transfer function $$\label{eq:72}
G(s)= \frac{K}{(1+T_1s) \, (1+T_2s)\, (1+T_3s)}$$
with zeroth-order hold. The external representation is $$\label{eq:73}
y_k = \sum\limits_{i=1}^{3}a_{i} y_{k-i} + \sum\limits_{j=0}^{3}b_{j}^k u_{k-j}$$
where $a_i$ and $b_j$ can be obtained according to (\[eq:66\]) and (\[eq:71\]) with $$\label{eq:74}
\phi_i(T_0) = exp(-\frac{1}{T_i}T_0) \qquad (i=1, 2, 3)$$
For the following values of the continuous model parameters $$K = 1, \; T_1 = 10 s, \; T_2 = 7.5 s, \; T_3 = 5 s$$
the values of the discrete-model coefficients are shown in Table (\[table:headings1\]) for different sampling periods $T_0$.
$T_0$ $2$ $4$ $6$ $8$ $10$ $12$
------------------------------ ---------- ---------- ---------- ---------- ---------- ---------- -- -- --
$b_1$ $0.0026$ $0.0186$ $0.0510$ $0.0989$ $0.1586$ $0.2260$
$b_2$ $0.0092$ $0.0486$ $0.1086$ $0.1718$ $0.2257$ $0.2643$
$b_3$ $0.0018$ $0.0078$ $0.0139$ $0.0174$ $0.0181$ $0.0167$
$a_1$ $2.2549$ $1.7063$ $1.2993$ $0.9953$ $0.7668$ $0.5938$
$a_2$ $-1.689$ $-0.958$ $-0.547$ $-0.314$ $-0.182$ $-0.106$
$a_3$ $0.4203$ $0.1767$ $0.0742$ $0.0312$ $0.0131$ $0.0055$
$\Sigma b_i= 1 + \Sigma a_i$ $0.0139$ $0.0750$ $0.1736$ $0.2882$ $0.4025$ $0.5071$
: Values of the parameters for different sampling periods[]{data-label="table:headings1"}
These results agree with those obtained by Isermann [@Isermann81]. Because of the sign of $T_i$ and the general form of $a_i,\, b_j$, the magnitudes of the $a_i$ parameters decrease (in absolute value) and those of $b_j$ increase with increasing sampling period $T_0$.
If $T_i$ had the opposite sign the parameter behaviour would be entirely different.
For a small sampling period $T_0 = 1s$
$$b_i\ll |a_i|, \; \Sigma b_i \ll |a_i|$$
This is why small errors in the estimated parameters can have a significant influence on the input-output behaviour of the model. Indeed, $\Sigma b_i$ depends on the 4th or 5th place of $b_i$ after the decimal point.
If the sampling period is chosen too small, ill-conditioned matrices result, which leads to numerical problems. On the other hand, if the sampling period is chosen too large then the dynamical behaviour is described inexactly. For $T_0 = 10$ s the model is practically reduced to second order because
$$a_3\ll 1 + \Sigma|a_i|, \; b_3\ll \Sigma b_i$$
and for even greater sampling periods we get a first-order model.
A proper choice of sampling interval in most cases is not critical, because the range between too-small and too-large values is relatively wide [@Isermann81].
Geometric particularization
---------------------------
The geometric interpretation of §3 can be particularized to the periodic case. We are going to consider again the third order model because it is the most significant. For the cases previously presented, we get the following results.
(*a*) The vectors $Y_i$ are coplanar if and only if $$\label{eq:75}
bT_0 = {\dot{\pi}}$$
(*b*) This is included in the previous case.
(*c*) This situation never occurs, since different rotations of the vectors $Y_i$ would imply different sampling periods, which is not possible in the periodic case.
We can see that the transition from the aperiodic case to the periodic case is a simple particularization. In the opposite sense, the procedure is much more complicated, because there are many aperiodic situations that have no equivalent in the periodic version.
In analytical terms, the coefficients $a_i$, and consequently the $b_j$, are perfectly defined for all sampling periods $T_0$, according to (\[eq:66\]) and (\[eq:71\]). We recall that the $Z$-transform imposes no constraints on the choice of $T_0$. Conversely, in the aperiodic formulation there are some sampling sequences that are forbidden for the model developed. This confirms the complexity of the aperiodic case compared with the periodic one.
Coefficients of the discrete model for systems with dead time
-------------------------------------------------------------
We are going to consider the same system with zeroth-order hold as before, but also with dead time $T_d$. The convolution expression that reflects this situation is
$$\label{eq:76}
y_k=\sum\limits_{i=0}^{k}h(t_{k}-t_i-T_d) \, x_i$$
with $x_i$ and $h(t)$ defined as before. For
$$\label{eq:77}
(p-1)T_0 < T_d \leq p T_0$$
with $p \in Z$, the general input-output model can be written as
$$\label{eq:78}
y_k = \sum\limits_{i=1}^{n}a_{i} y_{k-i} + \sum\limits_{j=0}^{n}b_{j} u_{k-j-p}$$
In order to determine the coefficients $a_i$ and $b_j$, the procedure is similar to that previously developed. Indeed, now we have the expressions
$$\label{eq:79}
\sum\limits_{i=0}^{n}a_{i}[\, h(t_{k-i}-t_j-T_d) - h(t_{k-i}-t_{j+1}-T_d)\,] + b_{k-j}=0$$
with $(j=k-p, k-p-1, \ldots, k-p-n)$. $$\label{eq:80}
\sum\limits_{i=0}^{n}a_{i}[\, h(t_{k-i}-t_j-T_d) - h(t_{k-i}-t_{j+1}-T_d)\,] =0 \; (j=k-p-n-1, \ldots, 0)$$
and we see that for $j = k, k - 1, \ldots,k - p + 1$ $$\label{eq:81}
h(t_k-t_j-T_d)=0$$
There are two different cases.
\(i) $T_d = pT_0$, which implies that the coefficients $a_i$ are not modified by the dead time; the coefficients $b_j$ are not modified either, except that these coefficients multiply the impulse inputs $u_{k-j-p}\;\; (j
=0,1, \ldots , n)$, rather than the inputs $u_{k-j}\;\; (j
=0,1, \ldots , n)$ for the case without dead time.
\(ii) $(p - 1)T_0 < T_d < pT_0$, which implies that the coefficients $a_i$ are not modified by the dead time; the coefficients $b_j$ are now functions of $T_d$ whose general expression is given by (\[eq:79\]) and which multiply the inputs $u_{k-j-p}\;\; (j =0,1, \ldots , n)$.
Conclusions
===========
A general aperiodic model for linear time-invariant SISO systems has been developed. The model also covers more general cases such as systems with zeroth-order hold and dead time. The formulation considered stresses the importance of the sampling period against other system parameters. In this way, such systems have an additional element for analysis and manipulation. The periodic-sampling case appears as a simple particularization of the general procedure. The results obtained are simplified and the use of tables of $Z$-transforms are avoided. For every sampled system, in a periodic or aperiodic way, there will always be sampling-period sequences more or less adequate according to the general characteristics of the process under study. In the aperiodic case, there will also be some restrictive conditions on these sequences, although it has also been possible to give strategies for some special cases.
[9]{} Dormido, S., and de la Sen, M., 1979, *IEEE Trans. Autom. Control*, **24**, 634.
Dormido, S., de la Sen, M., and Mellado, M., 1980, *Advances in Control*, edited by D. G. Lainiotis and N. S. Tzannes (Dordrecht: Reidel), p. 37.
Hirsch, M. W., and Smale, S., 1974, *Differential Equations, Dynamic Systems, and Linear Algebra*. (London: Academic Press).
Hsia, T. C., 1974, *IEEE Trans. Autom. Control*, **19**, 39.
Isermann, R., 1981, *Digital Control Systems*. (Berlin: Springer-Verlag).
Mellado, M., Cartujo, P., and Guillen, J. M., 1970, *Anales Real Sociedad Española de Física y Química (Física)*, **66**, 39, (in Spanish).
de la Sen, M., and Dormido, S., 1979, *Proc. 17th Annual Conf.*, Department of Electrical Engineering and Coordinated Science Laboratory of the University of Illinois, p 206; 1981, *Electron. Lett.*, **17**, 922.
de la Sen, M., Dormido, S., and Mellado, M., 1980, *Revista de Informática y Automática*, **43**, 12, (in Spanish).
Troch, I., 1973, *Automatica*, **9**, 117.
|
---
abstract: 'We present an application of the recently developed ergodic theoretic machinery on scenery flows to a classical geometric measure theoretic problem in Euclidean spaces. We also review the enhancements to the theory required in our work. Our main result is a sharp version of the conical density theorem, which we reduce to a question on rectifiability.'
address: |
Department of Mathematics and Statistics\
P.O. Box 35 (MaD)\
FI-40014 University of Jyväskylä\
Finland
author:
- Antti Käenmäki
title: 'Scenery flow, conical densities, and rectifiability'
---
[^1]
Introduction
============
We survey a recent advance in the study of scenery flows and show how it can be applied in a classical question in geometric measure theory which a priori does not involve any dynamics. The reader is prompted to recall the expository article of Fisher [@Fisher2004] where it was discussed how the scenery flow is linked to rescaling on several well-studied structures, such as geodesic flows, Brownian motion, and Julia sets. The purpose of this note is to continue that line of introduction.
The idea behind the scenery flow has been examined in many occasions. Authors have considered the scenery flow for specific sets and measures arising from dynamics; see e.g. [@BedfordFisher1996; @BedfordFisher1997; @BedfordFisherUrbanski2002; @FergusonFraserSahlsten2013; @Zahle88]. Abstract scenery flows have also been studied with a view on applications to special sets and measures, again arising from dynamics or arithmetic; see e.g. [@Gavish2011; @Hochman2010; @HochmanShmerkin2013]. The main innovation of the recent article by Käenmäki, Sahlsten, and Shmerkin [@KaenmakiSahlstenShmerkin2014b] is to employ the general theory initiated by Furstenberg [@Furstenberg2008], greatly developed by Hochman [@Hochman2010] and extended by Käenmäki, Sahlsten, and Shmerkin [@KaenmakiSahlstenShmerkin2014a], to classical problems in geometric measure theory.
One of the most fundamental concepts of geometric measure theory is that of rectifiability. It is a measure-theoretical notion for smoothness and to a great extend, geometric measure theory is about studying rectifiable sets. The foundations of geometric measure theory were laid by Besicovitch [@Besicovitch1928; @Besicovitch1929]. For various characterizations and properties of rectifiability the reader is referred to the book of Mattila [@Mattila1995]. In conical density results, the idea is to examine how a measure is distributed in small balls. Finding conditions that guarantee the measure to be effectively spread out in different directions is a classical question going back to Besicovitch [@Besicovitch1938] and Marstrand [@Marstrand1954]. For an account of the development on conical density results the reader is referred to the survey of Käenmäki [@Kaenmaki2010].
The scenery flow is a well-suited tool to address problems concerning conical densities. The cones in question do not change under magnification and this allows to pass information between the original measure and its tangential structure. In fact, we will see that there is an intimate connection between rectifiability and conical densities.
This exposition comes in two parts. In the first part, we review dynamical aspects of the scenery flow and in the second part, we focus on geometric measure theory.
Dynamics of the scenery flow
============================
Let $(X,{\mathcal{B}},P)$ be a probability space. We shall assume that $X$ is a metric space and ${\mathcal{B}}$ is the Borel $\sigma$-algebra on $X$. Write ${\mathbb{R}}_+ = [0,\infty)$. A (one-sided) *flow* is a family $(F_t)_{t \in {\mathbb{R}}_+}$ of measurable maps $F_t \colon X \to X$ for which $$F_{t+t'} = F_{t} \circ F_{t'}, \quad t,t' \in {\mathbb{R}}_+.$$ In other words, $(F_t)_{t \in {\mathbb{R}}_+}$ is an additive ${\mathbb{R}}_+$ action on $X$. We also assume that $(x,t) \mapsto F_t(x)$ is measurable.
We say that a set $A \in {\mathcal{B}}$ is $F_t$ *invariant* if $P(F_t^{-1} A \triangle A)=0$ for all $t \ge 0$. If $F_t P = P$ for all $t \ge 0$, then we say that $P$ is $F_t$ *invariant*. In this case, we call $(X,{\mathcal{B}},P,(F_t)_{t \in {\mathbb{R}}_+})$ a *measure preserving flow*. Furthermore, a measure preserving flow is *ergodic*, if for all $t \geq 0$ the measure $P$ is ergodic with respect to the transformation $F_t \colon X \to X$, that is, for all $F_t$ invariant sets $A \in {\mathcal{B}}$ we have $P(A) \in \{ 0,1 \}$.
If $(X,{\mathcal{B}},P,(F_t)_{t \in {\mathbb{R}}_+})$ is an ergodic measure preserving flow, then for a $P$ integrable function $f \colon X \to {\mathbb{R}}$ we have $$\lim_{T \to \infty} \frac{1}{T} \int_0^T f(F_t x) {\,\mathrm{d}}t = \int f {\,\mathrm{d}}P$$ for $P$-almost all $x \in X$.
We write $\omega \sim P$ to indicate that $\omega$ is chosen randomly according to the measure $P$.
Any $F_t$ invariant measure $P$ can be decomposed into ergodic components $P_\omega$, $\omega \sim P$, such that $$P = \int P_\omega {\,\mathrm{d}}P(\omega).$$ This decomposition is unique up to $P$ measure zero sets.
Let us next define the scenery flow. We equip ${\mathbb{R}}^d$ with the usual Euclidean norm and the induced metric. Denote the closed unit ball by $B_1$. Let ${\mathcal{M}}_1 := {\mathcal{P}}(B_1)$ be the collection of all Borel probability measures on $B_1$ and ${\mathcal{M}}_1^* := \{ \mu \in {\mathcal{M}}_1 : 0 \in \operatorname{spt}(\mu) \}$. Here $\operatorname{spt}(\mu)$ is the support of $\mu$. To avoid any confusion, measures on measures will be called *distributions*. We define the *magnification* $S_t\mu$ of $\mu \in {\mathcal{M}}_1^*$ at $0$ by setting $$S_t\mu(A) := \frac{\mu(e^{-t}A)}{\mu(B(0,e^{-t}))}, \quad A \subset B_1.$$ In other words, the measure $S_t\mu$ is obtained by scaling $\mu|_{B(0,e^{-t})}$ into the unit ball and normalizing. Due to the exponential scaling, $(S_t)_{t \in {\mathbb{R}}_+}$ is a flow in the space ${\mathcal{M}}_1^*$ and we call it the *scenery flow* at $0$. An $S_t$ invariant distribution $P$ on ${\mathcal{M}}_1^*$ is called *scale invariant*. Although the action $S_t$ is discontinuous (at measures $\mu$ with $\mu(\partial B(0,r)) > 0$ for some $0<r<1$) and the set ${\mathcal{M}}_1^* \subset {\mathcal{M}}_1$ is not closed, we shall witness that the scenery flow behaves in a very similar way to a continuous flow on a compact metric space.
With the scenery flow we are now able to define tangent measures and distributions. Let $\mu$ be a Radon measure and $x \in \operatorname{spt}(\mu)$. We want to consider the scaling dynamics when magnifying around $x$. Let $T_x\mu(A) := \mu(A+x)$ and define $\mu_{x,t} := S_t(T_x\mu)$. Then the one-parameter family $(\mu_{x,t})_{t \in {\mathbb{R}}_+}$ is called the *scenery flow* at $x$. Accumulation points of this scenery in ${\mathcal{M}}_1$ will be called *tangent measures* of $\mu$ at $x$ and the family of tangent measures of $\mu$ at $x$ is denoted by ${\operatorname{Tan}}(\mu,x) \subset {\mathcal{M}}_1$. However, we are not interested in a single tangent measure, but the whole statistics of the scenery $\mu_{x,t}$ as $t \to \infty$. We remark that we have slightly deviated from Preiss’ original definition of tangent measures, which corresponds to taking weak limits of unrestricted blow-ups; see [@Preiss1987].
A *tangent distribution* of $\mu$ at $x \in \operatorname{spt}(\mu)$ is any weak limit of $${\langle}\mu {\rangle}_{x,T} := \frac{1}{T} \int_0^T \delta_{\mu_{x,t}} {\,\mathrm{d}}t$$ as $T \to \infty$. The family of tangent distributions of $\mu$ at $x$ is denoted by ${\mathcal{TD}}(\mu,x) \subset {\mathcal{P}}({\mathcal{M}}_1^*)$.
If the limit above is unique, then, intuitively, it means that the collection of views $\mu_{x,t}$ will have well-defined statistics when zooming into smaller and smaller neighbourhoods of $x$. The integration above makes sense since we are on a convex subset of a topological linear space. We emphasize that tangent distributions are measures on measures. Notice that the set ${\mathcal{TD}}(\mu,x)$ is non-empty and compact at $x \in \operatorname{spt}(\mu)$. Moreover, the support of each $P \in {\mathcal{TD}}(\mu,x)$ is contained in ${\operatorname{Tan}}(\mu,x)$.
According to Preiss’ well-known principle, tangent measures to tangent measures are tangent measures; see [@Preiss1987 Theorem 2.12]. We shall define an analogous condition for distributions. We say that a distribution $P$ on ${\mathcal{M}}_1$ is *quasi-Palm* if for any Borel set ${\mathcal{A}}\subset {\mathcal{M}}_1$ with $P({\mathcal{A}}) = 1$ it holds that for $P$-almost every $\nu \in {\mathcal{A}}$ and for $\nu$-almost every $z \in {\mathbb{R}}^d$ there exists $t_z > 0$ such that for $t \ge t_z$ we have $B(z, e^{-t}) \subset B_1$ and $$\nu_{z,t} \in {\mathcal{A}}.$$ This version of the quasi-Palm property actually requires that the unit sphere of the norm is a $C^1$ manifold and does not contain line segments; see [@KaenmakiSahlstenShmerkin2014b Lemma 3.23]. The Euclidean norm we use of course satisfies this requirement. If we were considering unrestricted blow-ups, then the requirement for $B(z,e^{-t})$ to be contained in $B_1$ could be dropped. Roughly speaking, the quasi-Palm property guarantees that the null sets of the distributions are invariant under translations to a typical point of the measure.
A distribution $P$ on ${\mathcal{M}}_1$ is a *fractal distribution* if it is scale invariant and quasi-Palm. A fractal distribution is an *ergodic fractal distribution* if it is ergodic with respect to $S_t$.
It follows from the Besicovitch density point theorem that ergodic components of a fractal distribution are ergodic fractal distributions; see [@Hochman2010 Theorem 1.3].
A general principle is that tangent objects enjoy some kind of spatial invariance. For tangent distributions, a very powerful formulation of this principle is the following theorem of Hochman [@Hochman2010 Theorem 1.7]. The result is analogous to a similar phenomenon discovered by Mörters and Preiss [@MortersPreiss1998 Theorem 1].
\[thm:hochman\] For any Radon measure $\mu$ and $\mu$-almost every $x$, all tangent distributions of $\mu$ at $x$ are fractal distributions.
Notice that as the action $S_t$ is discontinuous, even the scale invariance of tangent distributions or the fact that they are supported on ${\mathcal{M}}_1^*$ are not immediate, though they are perhaps expected. The most interesting part in the above theorem is that a typical tangent distribution satisfies the quasi-Palm property.
Hochman’s result is proved by using CP processes which are Markov processes on the dyadic scaling sceneries of a measure introduced by Furstenberg [@Furstenberg1970; @Furstenberg2008]. Let ${\mathcal{D}}$ be a partition of $[-1,1]^d$ into $2^d$ cubes of side length $1$. Given $x \in [-1,1]^d$, let $D(x)$ be the only element of ${\mathcal{D}}$ containing it. If $D \in {\mathcal{D}}$, then we write $T_D$ for the orientation preserving homothety mapping from $\overline{D}$ onto $[-1,1]^d$. Define the *CP magnification* $M$ on $\Omega := {\mathcal{P}}([-1,1]^d) \times [-1,1]^d$ by setting $$M(\mu,x) := \bigl( T_{D(x)}\mu/\mu(D(x)), T_{D(x)}(x) \bigr).$$ This is well-defined whenever $\mu(D(x))>0$. Note that, since zooming in is done dyadically, it is important to keep track of the orbit of the point that is being zoomed upon. A distribution $Q$ on $\Omega$ is *adapted* if there is a disintegration $$\int f(\nu,x) {\,\mathrm{d}}Q(\nu,x) = {\int\hspace{-0.09in}\int}f(\nu,x) {\,\mathrm{d}}\nu(x) {\,\mathrm{d}}\overline{Q}(\nu)$$ for all $f \in C(\Omega)$. Here $\overline{Q}$ is the projection of $Q$ onto the measure component. In other words, $Q$ is adapted if choosing a pair $(\mu,x)$ according to $Q$ can be done in a two-step process, by first choosing $\mu$ according to $\overline{Q}$ and then choosing $x$ according to $\mu$. A distribution on $\Omega$ is a *CP distribution* if it is $M$ invariant and adapted.
The *micromeasure distribution* of $\mu$ at $x \in \operatorname{spt}(\mu)$ is any weak limit of $${\langle}\mu,x {\rangle}_N := \frac{1}{N} \sum_{k=0}^{N-1} \delta_{M^k(\mu,x)}.$$ By compactness of ${\mathcal{P}}(\Omega)$, the family of micromeasure distributions is non-empty and compact, and by [@Hochman2010 Proposition 5.4], each micromeasure distribution is adapted. Furthermore, if the *intensity measure* of a micromeasure distribution $Q$ defined by $$[Q](A) := \int \mu(A) {\,\mathrm{d}}\overline{Q}(\mu), \quad A \subset [-1,1]^d,$$ is the normalized Lebesgue measure, then $Q$ is M invariant. By adaptedness, this is the case for any weak limit of ${\langle}\mu+z,x+z {\rangle}_N$ for Lebesgue almost all $z \in [-1/2,1/2]^d$; see [@Hochman2010 Proposition 5.5(2)]. In other words, by slightly adjusting the dyadic grid, a micromeasure distribution can be seen to be a CP distribution. The family of CP distributions having Lebesgue intensity is compact; see [@KaenmakiSahlstenShmerkin2014a Lemma 3.4].
If $Q$ is a CP distribution, then the system $(\Omega,M,Q)$ is a stationary one-sided process $(\xi_n)_{n \in {\mathbb{N}}}$ with $\xi_1 \sim Q$ and $M\xi_n = \xi_{n+1}$. Considering its two-sided extension, we see that there exists a natural extension ${\widehat}Q$ supported on the Cartesian product of all Radon measures and $[-1,1]^d$. A *centering* of ${\widehat}Q$ is a push-down of the suspension flow of ${\widehat}Q$ under the unrestricted magnification of $\mu$ at $x$. For a precise definition, see [@Hochman2010 Definition 1.13]. By [@Hochman2010 Theorem 1.14], a centering of ${\widehat}Q$ is an unrestricted fractal distribution. We remark that [@Hochman2010] and [@KaenmakiSahlstenShmerkin2014a] use $L^\infty$ norm to allow an easier link between CP processes and fractal distributions. By [@KaenmakiSahlstenShmerkin2014a Appendix A], the results are independent of the choice of the norm and hence, our use of the Euclidean norm is justified.
Relying on the above, we are now able to give an outline for the proof of Theorem \[thm:hochman\]. If $P = \lim_{k \to \infty} {\langle}\mu {\rangle}_{x,N_k}$ is a tangent distribution, then, passing to a subsequence, define a micromeasure distribution $Q = \lim_{i \to \infty} {\langle}\mu,x {\rangle}_{N_{k(i)}}$. Slightly adjusting the dyadic grid, we see that $Q$ is a CP distribution with Lebesgue intensity. Thus, by [@Hochman2010 Proposition 5.5(3)], $P$ is the restriction of the centering of ${\widehat}Q$ and hence, $P$ is a fractal distribution.
Although fractal distributions are defined in terms of seemingly strong geometric properties, the family of fractal distributions is in fact very robust. The following theorem is due to Käenmäki, Sahlsten, and Shmerkin [@KaenmakiSahlstenShmerkin2014a Theorem A].
\[thm:compact\] The family of fractal distributions is compact.
The result may appear rather surprising since the scenery flow is not continuous, its support is not closed, and, more significantly, the quasi-Palm property is not a closed property. The proof of this result is also based on the interplay between fractal distributions and CP processes. We have already seen that each CP distribution defines a fractal distribution. The converse is also true. Let us first assume that $P$ is an ergodic fractal distribution. If $f$ is a continuous function defined on ${\mathcal{P}}({\mathcal{M}}_1)$, then, by the Birkhoff ergodic theorem, we have $$\lim_{T \to \infty} \frac{1}{T} \int_0^T f(S_t\mu) {\,\mathrm{d}}t = \int f {\,\mathrm{d}}P$$ for $P$-almost all $\mu$. Considering a countable dense set of continuous functions $f$ and applying the quasi-Palm property, it follows that $$\label{eq:erg_usm}
\lim_{T \to \infty} {\langle}\mu {\rangle}_{x,T} = P$$ for $P$-almost all $\mu$ and for $\mu$-almost all $x$; see [@Hochman2010 Theorem 3.9]. As we already have seen, any tangent distribution can be expressed as the restriction of the centering of an extended CP distribution having Lebesgue intensity. Thus, by , the same holds for ergodic fractal distributions. Relying on the ergodic decomposition, this observation can be extended to non-ergodic fractal distributions; see [@Hochman2010 Theorem 1.15]. Therefore, since the family of CP distributions with Lebesgue intensity is compact, to prove Theorem \[thm:compact\], it suffices to show that the centering is a continuous operation. This is done in [@KaenmakiSahlstenShmerkin2014a Lemmas 3.5 and 3.6].
Together with convexity and the uniqueness of the ergodic decomposition, Theorem \[thm:compact\] implies that the family of fractal distributions is a Choquet simplex. Recall that a Poulsen simplex is a Choquet simplex in which extremal points are dense. Note that the set of extremal points is precisely the collection of ergodic fractal distributions. The following theorem is proved by Käenmäki, Sahlsten, and Shmerkin [@KaenmakiSahlstenShmerkin2014a Theorem B].
\[thm:poulsen\] The family of fractal distributions is a Poulsen simplex.
The proof is again based on the interplay between fractal distributions and CP processes. We prove that ergodic CP processes are dense by constructing a dense set of distributions of random self-similar measures on the dyadic grid. This is done by first approximating a given CP process by a finite convex combination of ergodic CP processes, and then, by splicing together those finite ergodic CP processes, constructing a sequence of ergodic CP processes converging to the convex combination. Roughly speaking, splicing of measures consists in pasting together a sequence of measures along dyadic scales. Splicing is often employed to construct measures with a given property based on properties of the component measures. For details, the reader is referred to [@KaenmakiSahlstenShmerkin2014a §4].
In geometric considerations, we usually construct a fractal distribution satisfying certain property. We often want to transfer that property back to a measure. This leads us to the concept of generated distributions.
We say that a measure $\mu$ *generates* a distribution $P$ at $x$ if $${\mathcal{TD}}(\mu,x) = \{ P \}.$$ If $\mu$ generates $P$ for $\mu$-almost all $x$, then we say that $\mu$ is a *uniformly scaling measure*.
One can think that the uniformly scaling property is an ergodic-theoretical notion of self-similarity. Hochman proved the striking fact that generated distributions are always fractal distributions. The following result of Käenmäki, Sahlsten, and Shmerkin [@KaenmakiSahlstenShmerkin2014a Theorem C] is a converse to this.
\[thm:usm\] If $P$ is a fractal distribution, then there exists a uniformly scaling measure $\mu$ generating $P$.
Recall that if $P$ is an ergodic fractal distribution, then, by , $P$-almost every measure is uniformly scaling. Thus, by Theorems \[thm:compact\] and \[thm:poulsen\], it suffices to show that the collection of fractal distributions satisfying the claim is closed. Let $(P_i)_i$ be a sequence of ergodic fractal distributions converging to $P$ and let $\mu_i$ be a uniformly scaling measure generating $P_i$. The proof is again based on the interplay between fractal distributions and CP processes. The rough idea to obtain a uniformly scaling measure generating $P$ is to splice the measures $\mu_i$ together. For the full proof, the reader is referred to [@KaenmakiSahlstenShmerkin2014a §5].
Geometry of measures
====================
Let $G(d,d-k)$ denote the set of all $(d-k)$-dimensional linear subspaces of ${\mathbb{R}}^d$. For $x \in {\mathbb{R}}^d$, $r>0$, $V \in G(d,d-k)$, and $0<\alpha\le 1$ define $$X(x,r,V,\alpha) = \{ y \in B(x,r) : {\operatorname{dist}}(y-x,V) < \alpha|y-x| \}.$$ Conical density results aim to give conditions on a measure which guarantee that the cones $X(x,r,V,\alpha)$ contain a large portion of the mass from the surrounding ball $B(x,r)$ for certain proportion of scales. For example, a lower bound on some dimension often is such a condition. Recall that the *lower local dimension* of a Radon measure $\mu$ at $x \in {\mathbb{R}}^d$ is $$\label{eq:lower_local_dim}
\operatorname{\underline{dim}_{loc}}(\mu,x) = \liminf_{r \downarrow 0} \frac{\log\mu(B(x,r))}{\log r}$$ and the *lower Hausdorff dimension* of $\mu$ is $$\begin{aligned}
\operatorname{\underline{dim}_H}(\mu) &= \operatorname*{ess\,inf}_{x \sim \mu} \operatorname{\underline{dim}_{loc}}(\mu,x) \\
&= \inf\{ \operatorname{dim_H}(A) : A \subset {\mathbb{R}}^d \text{ is a Borel set with } \mu(A)>0 \}.\end{aligned}$$ Here $\operatorname{dim_H}(A)$ is the Hausdorff dimension of the set $A \subset {\mathbb{R}}^d$. A measure $\mu$ is *exact-dimensional* if the limit in exists and is $\mu$-almost everywhere constant. In this case, the common value is simply denoted by $\dim(\mu)$.
Intuitively, the local dimension of a measure should not be affected by the geometry of the measure on a density zero set of scales. Thus one could expect that tangent distributions should encode all information on dimensions.
The *dimension of a fractal distribution* $P$ is $$\dim(P) = \int \dim(\mu) {\,\mathrm{d}}P(\mu).$$
The dimension above is well defined by the fact that if $P$ is a fractal distribution, then $P$-almost every measure is exact-dimensional; see [@Hochman2010 Lemma 1.18]. The dimension of fractal distributions has also other convenient properties. While the Hausdorff dimension is highly discontinuous on measures, the function $P \mapsto \dim(P)$ defined on the family of fractal distributions is continuous; see [@KaenmakiSahlstenShmerkin2014b Lemma 3.20]. The usefulness of the definition is manifested in the following result of Hochman [@Hochman2010 Proposition 1.19]. Recall Theorem \[thm:hochman\].
\[thm:fd\_dim\] If $\mu$ is a Radon measure, then $$\operatorname{\underline{dim}_{loc}}(\mu,x) = \inf\{ \dim(P) : P \in {\mathcal{TD}}(\mu,x) \}$$ for $\mu$-almost all $x$. Furthermore, if $\mu$ is a uniformly scaling measure generating a fractal distribution $P$, then $\mu$ is exact-dimensional and $\dim(\mu) = \dim(P)$.
It turns out that tangent distributions are well suited to address problems concerning conical densities. The cones in question do not change under magnification and this allows to pass information between the original measure and its tangent distributions. Let $${\mathcal{A}}_{\varepsilon}:= \{ \nu \in {\mathcal{M}}_1 : \nu(X(0,1,V,\alpha)) \le {\varepsilon}\text{ for some } V \in G(d,d-k) \}$$ for all ${\varepsilon}\ge 0$. It is straightforward to see that ${\mathcal{A}}_{\varepsilon}$ is closed for all ${\varepsilon}\ge 0$; see [@KaenmakiSahlstenShmerkin2014b Lemma 4.2]. The key observation is that $${\mathcal{A}}_0 = \{ \nu \in {\mathcal{M}}_1 : \operatorname{spt}(\nu) \cap X(0,1,V,\alpha) = \emptyset \text{ for some } V \in G(d,d-k) \},$$ where the defining property concerns only sets, is $S_t$ invariant.
The following conical density result is proved by Käenmäki, Sahlsten, and Shmerkin [@KaenmakiSahlstenShmerkin2014b Proposition 4.3]. Roughly speaking, it claims that if the dimension of the measure is large, then there are many scales in which the cones contain a relatively large portion of the mass. A slightly more precise version is that there exists ${\varepsilon}> 0$ such that if $\operatorname{\underline{dim}_H}(\mu) > k$, then for many scales $e^{-t}>0$ we have $$\inf_{V \in G(d,d-k)} \frac{\mu(X(x,e^{-t},V,\alpha))}{\mu(B(x,e^{-t}))} > {\varepsilon}$$ for $\mu$-almost all $x$. The precise formulation of the theorem is as follows.
\[thm:conical\] If $k \in \{ 1,\ldots,d-1 \}$, $k<s\le d$, and $0<\alpha\le 1$, then there exists ${\varepsilon}> 0$ satisfying the following: For every Radon measure $\mu$ on ${\mathbb{R}}^d$ with $\operatorname{\underline{dim}_H}(\mu) \ge s$ it holds that $$ \liminf_{T \to \infty} {\langle}\mu {\rangle}_{x,T}({\mathcal{M}}_1 \setminus {\mathcal{A}}_{\varepsilon}) \ge \frac{s-k}{d-k}$$ for $\mu$-almost all $x \in {\mathbb{R}}^d$.
The proof is based on showing that there cannot be “too many” rectifiable tangent measures. This means that, perhaps surprisingly, most of the known conical density results are, in some sense, a manifestation of rectifiability.
A set $E \subset {\mathbb{R}}^d$ is called *$k$-rectifiable* if there are countably many Lipschitz maps $f_i \colon {\mathbb{R}}^k \to {\mathbb{R}}^d$ so that $${\mathcal{H}}^k\Bigl( E \setminus \bigcup_i f_i({\mathbb{R}}^k) \Bigr) = 0.$$
Here ${\mathcal{H}}^k$ is the $k$-dimensional Hausdorff measure. Observe that a $k$-rectifiable set $E$ has $\operatorname{dim_H}(E) \le k$. A sufficient condition for a set $E \subset {\mathbb{R}}^d$ to be $k$-rectifiable is that for every $x \in E$ there are $V \in G(d,d-k)$, $0 < \alpha < 1$, and $r > 0$ such that $E \cap X(x,r,V,\alpha) = \emptyset$; see [@Mattila1995 Lemma 15.13]. Thus, if a fractal distribution $P$ satisfies $P({\mathcal{A}}_0) = 1$, then the quasi-Palm property implies that the support of $P$-almost every $\nu$ is $k$-rectifiable and hence $\dim(P) \le k$.
To prove Theorem \[thm:conical\], let $p,\delta>0$ be such that $p < (s-\delta-k)/(d-k) < (s-k)/(d-k)$. Suppose to the contrary that there is $0<\alpha\le 1$ so that for each $0<{\varepsilon}<{\varepsilon}(d,k,\alpha)$ there exists a Radon measure $\mu$ with $\operatorname{\underline{dim}_H}(\mu) \ge s$ such that the claim fails to hold for $p$, that is, $$\limsup_{T \to \infty}\,{\langle}\mu {\rangle}_{x,T}({\mathcal{A}}_{\varepsilon}) > 1-p$$ on a set $E_{\varepsilon}$ of positive $\mu$ measure. By Theorems \[thm:hochman\] and \[thm:fd\_dim\], we may assume that at points $x \in E_{{\varepsilon}}$, all tangent distributions of $\mu$ are fractal distributions and $$\inf\{\dim(P) : P \in {\mathcal{TD}}(\mu,x)\} = \operatorname{\underline{dim}_{loc}}(\mu,x) > s-\delta.$$ Fix $x \in E_{{\varepsilon}}$. For each $0<{\varepsilon}<{\varepsilon}(d,k,\alpha)$, as ${\mathcal{A}}_{\varepsilon}$ is closed, we find a tangent distribution $P_{{\varepsilon}} \in {\mathcal{TD}}(\mu,x)$ so that $P_{{\varepsilon}}({\mathcal{A}}_{\varepsilon}) \ge 1-p$. Since the sets ${\mathcal{A}}_{\varepsilon}$ are also nested, we get $$P({\mathcal{A}}_0) = \lim_{{\varepsilon}\downarrow 0} P({\mathcal{A}}_{\varepsilon}) \ge 1-p,$$ where $P$ is a weak limit of a sequence formed from $P_{\varepsilon}$ as ${\varepsilon}\downarrow 0$. Furthermore, since the collection of all fractal distributions is closed by Theorem \[thm:compact\] and the dimension is continuous, the limit distribution $P$ is a fractal distribution with $$\dim(P) \ge s-\delta.$$ Let $P_\omega$, $\omega \sim P$, be the ergodic components of $P$. By the invariance of ${\mathcal{A}}_0$, we have $P_\omega({\mathcal{A}}_0) \in \{ 0,1 \}$ for $P$-almost all $\omega$. If $P_\omega({\mathcal{A}}_0) = 0$, then we use the trivial estimate $\dim(P_\omega) \le d$, and if $P_\omega({\mathcal{A}}_0) = 1$, then the rectifiability argument gives $\dim(P_\omega) \le k$. Since $P(\{ \omega : P_\omega({\mathcal{A}}_0) = 1 \}) = P({\mathcal{A}}_0) \ge 1-p$ we estimate $$\begin{aligned}
s-\delta \le \dim(P) = \int \dim(P_\omega) {\,\mathrm{d}}P(\omega)
\le P({\mathcal{A}}_0)k + (1-P({\mathcal{A}}_0))d \le (1-p)k + pd\end{aligned}$$ yielding $p \ge (s-\delta-k)/(d-k)$. But this contradicts the choice of $\delta$. Thus the claim holds.
Relying on the existence of uniform scaling measures, we are able to study the sharpness of Theorem \[thm:conical\]. The following result is proved by Käenmäki, Sahlsten, and Shmerkin [@KaenmakiSahlstenShmerkin2014b Proposition 4.4].
If $k \in \{ 1,\ldots,d-1 \}$, $k < s \le d$, and $0<\alpha\le 1$, then there exists a Radon measure $\mu$ on ${\mathbb{R}}^d$ with $\dim(\mu) = s$ such that $$\lim_{T \to \infty} {\langle}\mu {\rangle}_{x,T}({\mathcal{M}}_1 \setminus {\mathcal{A}}_{\varepsilon}) =
\begin{cases}
(s-k)/(d-k), &\text{if } 0 < {\varepsilon}< {\varepsilon}(d,k,\alpha), \\
0, &\text{if } {\varepsilon}> {\varepsilon}(d,k,\alpha),
\end{cases}$$ for $\mu$-almost all $x \in {\mathbb{R}}^d$.
Here, for $k \in \{ 1,\ldots,d-1 \}$, $0 < \alpha \le 1$, and $V \in G(d,d-k)$, we have defined $${\varepsilon}(d,k,\alpha) := \frac{{\mathcal{L}}^d(X(0,1,V,\alpha))}{{\mathcal{L}}^d(B(0,1))}.$$ It follows from the rotational invariance of the Lebesgue measure ${\mathcal{L}}^d$ that ${\varepsilon}(d,k,\alpha)$ does not depend on the choice of $V$.
The measure $\mu$ above is just a uniform scaling measure generating $$P = \frac{s-k}{d-k} \delta_{\mathcal{L}}+ \Bigl( 1-\frac{s-k}{d-k} \Bigr) \delta_{\mathcal{H}},$$ where ${\mathcal{L}}$ is the normalization of ${\mathcal{L}}^d|_{B_1}$ and ${\mathcal{H}}$ is the normalization of ${\mathcal{H}}^k|_{W \cap B_1}$ for a fixed $W \in G(d,k)$. Since $P$ is a convex combination of two fractal distributions, it is a fractal distribution. The existence of $\mu$ is guaranteed by Theorem \[thm:usm\]. Recalling Theorem \[thm:fd\_dim\], we see that $\mu$ is exact-dimensional and $$\dim(\mu) = \dim(P) = \frac{s-k}{d-k}\,d + \Bigl( 1 - \frac{s-k}{d-k} \Bigr)k = s.$$ The goal is to verify that $\mu$ has the claimed properties.
Fix $0 < {\varepsilon}< {\varepsilon}(d,k,\alpha)$. Since ${\mathcal{L}}(X(0,1,V,\alpha)) = {\varepsilon}(d,k,\alpha) > {\varepsilon}$ for all $V \in G(d,d-k)$ and ${\mathcal{H}}(X(0,1,W^\bot,\alpha)) = 0$ we have $P({\mathcal{M}}_1 \setminus {\mathcal{A}}_{\varepsilon}) = (s-k)/(d-k)$. Thus, by the weak convergence, it follows that $$\lim_{T \to \infty} {\langle}\mu {\rangle}_{x,T}({\mathcal{M}}_1 \setminus {\mathcal{A}}_{\varepsilon}) = \frac{s-k}{d-k}.$$ In the case ${\varepsilon}> {\varepsilon}(d,k,\alpha)$ we can reason similarly.
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[^1]: *Date*: December 1, 2014.
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abstract: 'We present realistic equations of state for QCD matter at vanishing net-baryon density which embed recent lattice QCD results at high temperatures combined with a hadron resonance gas model in the low-temperature, confined phase. In the latter, we allow an implementation of partial chemical equilibrium, in which particle ratios are fixed at the chemical freeze-out, so that a description closer to the experimental situation is possible. Given the present uncertainty in the determination of the chemical freeze-out temperature from first-principle lattice QCD calculations, we consider different values within the expected range. The corresponding equations of state can be applied in the hydrodynamic modeling of relativistic heavy-ion collisions at the LHC and at the highest RHIC beam energies. Suitable parametrizations of our results as functions of the energy density are also provided.'
author:
- 'M. Bluhm'
- 'P. Alba'
- 'W. Alberico'
- 'A. Beraudo'
- 'C. Ratti'
title: 'Lattice QCD-based equations of state at vanishing net-baryon density'
---
Introduction \[sec:1\]
======================
In the relativistic heavy-ion collisions at RHIC (Relativistic Heavy-Ion Collider) and LHC (Large Hadron Collider), a hot deconfined state of strongly interacting matter is transiently created, the Quark-Gluon Plasma (QGP). This form of QCD matter is believed to have existed in the very first moments of our universe. As the produced hot and dense system cools down during its expansion, matter undergoes a transition from the QGP phase into a state dominated by color-confined, massive hadronic degrees of freedom. The nature of this phase transformation has been determined at vanishing baryon-chemical potential by first-principle lattice QCD simulations: it is an analytic crossover, taking place over a broad region of temperatures $T$ [@Aoki06]. The value of the (pseudo-) critical temperature $T_c$ associated with this confinement transition depends to some extent on the considered order-parameter. For example, the [*Wuppertal-Budapest*]{} (WB) and [*hotQCD*]{} collaborations found comparable values for chiral symmetry restoration: $T_c=$($155\pm 6$) MeV in [@Borsanyi09] and $T_c=$($154\pm 9$) MeV in [@Bazavov12], respectively.
The collective flow dynamics of the bulk of matter created in heavy ion collisions can be successfully modeled by means of relativistic hydrodynamics (cf. e.g. the reviews in [@Kolb03; @Gale13]), starting from a stage immediately after thermalization until the kinetic freeze-out of final state hadrons. Assuming local thermal equilibrium, the conservation equations for energy, momentum and for the additionally conserved charges (net-baryon number $N_B$, net-electric charge $N_Q$ and net-strangeness $N_S$) drive the evolution of the system. An essential ingredient for this modeling is the equation of state (EoS), which provides locally a relation between energy density $\epsilon$, pressure $p$ and the densities $n_B$, $n_Q$ and $n_S$ of the conserved charges. The parameter controlling the acceleration of the fluid collective flow due to pressure gradients is the speed of sound, $c_s=\sqrt{\partial p/\partial\epsilon}$.
A quantitative comparison of hydrodynamic simulations with the observed collective flow behavior revealed that the evolution of the system can be described by nearly ideal hydrodynamics, cf. e.g. [@Romatschke07; @Luzum08; @Song09; @Song11; @Schenke11; @Song13; @Luzum13]. In these studies, a uniquely small ratio of shear viscosity $\eta$ to entropy density $s$ of the hot matter was determined, cf. also the reviews in [@Schafer09; @Teaney10]. This led to our current understanding of the QGP as a strongly coupled, nearly perfect fluid [@Gyulassy05; @Shuryak05; @Heinz05]. Assuming the conservation of entropy, i.e. neglecting the viscous entropy production associated with such a small $\eta/s$ [@Shen10], one needs to know the EoS only along adiabatic paths. In this work, we concentrate on the situation of a vanishing $n_B$, i.e. we consider the path $n_B/s=0$. We note that in the thermal system created in a heavy-ion collision one always has $n_S=0$, while $n_Q$ (in the case of a partial stopping at the lowest center-of-mass energies) is related to $n_B$.
A rigorous determination of the equation of state in thermal and chemical equilibrium for $n_B=0$ in the non-perturbative regime of QCD can be achieved with lattice gauge theory simulations. These reach nowadays unprecedented levels of accuracy. A basic quantity for the EoS is the interaction measure $I=\epsilon-3p$, which has been calculated in [@Bazavov09; @Cheng10] and in [@Aoki05; @Borsanyi10; @Borsanyi11]. The numerical results for $I(T)/T^4$ in [@Bazavov09; @Cheng10] show significant differences from those in [@Borsanyi10; @Borsanyi11] in the transition region. In this work, we opt for utilizing the recent, continuum-extrapolated lattice QCD data from the WB-collaboration presented in [@Borsanyi11], corresponding to a system of 2+1 quark flavors with physical quark masses. By combining a suitable parametrization of these lattice QCD results with a hadron resonance gas (HRG) model in thermal and chemical equilibrium, we construct a baseline QCD equation of state for $n_B=0$.
The focus of our work lies, however, on the implementation of partial chemical equilibrium, i.e. a non-equilibrium situation, in the hadronic phase. In this way, one can properly account for the actual chemical composition in the confined phase, an issue which is not addressed within equilibrium lattice QCD thermodynamics. This is known to be of importance in order to reproduce not only the experimentally observed flow and $p_T$-spectra, but also the correct particle ratios [@Hirano02].
Because of the present uncertainty in the exact value of the chemical freeze-out temperature $T_{ch}$, cf. [@Borsanyi13], we consider various values for $T_{ch}$ within the expected range, below which the HRG is assumed to be in partial chemical equilibrium. In this way, different realistic QCD equations of state are obtained, which can be used in the hydrodynamic simulations of relativistic heavy-ion collisions for LHC and RHIC top beam energies at mid-rapidity when net-baryon density effects can be neglected. Having such QCD equations of state at hand will allow to study the possible impact of a variation of $15$ MeV in $T_{ch}$ on particle spectra as well as a more controlled determination of the QGP transport properties, as for example the shear and bulk viscosity coefficients. At smaller beam energies, effects of a non-vanishing net-baryon density become important. Corresponding QCD equations of state will be presented in a forthcoming publication.
The equation of state of QCD matter has been the subject of numerous studies in the literature. Among different other approaches, we mention combinations of the HRG model with an effective theory of QCD [@Laine06], with a phenomenological model for QCD thermodynamics [@Bluhm07] and with various parametrizations [@Chojnacki07; @Chojnacki08; @Song08; @Huovinen09] of lattice QCD results. Developments in using a parametrization of lattice QCD results for finite $n_B$ were recently reported in [@Huovinen11; @Huovinen12]. Moreover, in [@Steinheimer10] an EoS, describing both the QGP and the hadronic phase based on one effective model approach, was constructed and applied in finite-$n_B$ hydrodynamics studies (see also further developments in [@Steinheimer11]).
The paper is organized as follows: in section \[sec:2\], we discuss briefly the employed lattice QCD results [@Borsanyi11] and their combination with a HRG model in thermal and chemical equilibrium. Section \[sec:3\] deals with the inclusion of partial chemical equilibrium in the description of the hadronic phase. In section \[sec:4\], we discuss the obtained QCD equations of state and provide practical parametrizations of our results.
Construction of a Lattice QCD-based EoS\[sec:2\]
================================================
{width="45.00000%"} {width="45.00000%"}
In [@Borsanyi11], continuum-extrapolated lattice gauge theory results of QCD thermodynamics for 2+1 quark flavors with physical mass parameters were presented. The corresponding results for the scaled interaction measure $I(T)/T^4$ and for the scaled pressure $p(T)/T^4$ are depicted in Fig. \[Fig:1\] panels (a) and (b), respectively. A suitable parametrization of the results for $I(T)/T^4$ as a function of $T$, which provides an accurate description within the given error-band, can be found by employing a similar fit function as the one used in [@Borsanyi10]. From this, the other thermodynamic quantities follow via thermodynamic identities: the scaled pressure is determined by a definite $T$-integral of $I(T)/T^5$, while $\epsilon(T)=3p(T)+I(T)$ and $s(T)=(\epsilon(T)+p(T))/T$. This yields for $s(T)$ extrapolated to $T=800$ MeV a value of about $82.5\%$ of the [*Stefan-Boltzmann*]{} limit for a non-interacting gas of 3 massless quark flavors.
The thermodynamics of QCD matter in the hadronic phase can be well accounted for by the HRG model describing hadronic matter in thermal and chemical equilibrium, cf. e.g. [@Karsch03; @Tawfik05]. The pressure of the model in the thermodynamic limit is given by $$\begin{aligned}
\nonumber
p(T,\{\mu_k\}) & = & \sum_k (-1)^{B_k+1} \frac{d_kT}{(2\pi)^3} \int d^3\vec{p} \,\,\ln\Big[1 +\\
\label{equ:pressureHRG}
& & \,\,\, (-1)^{B_k+1} e^{-(\sqrt{\vec{p}^{\,2}+m_k^2}-\mu_k)/T}\Big] \,,\end{aligned}$$ where the sum is taken over all hadronic (including resonances) states $k$ (baryons and anti-baryons being summed independently) included in the model. In Eq. (\[equ:pressureHRG\]), $d_k$ and $m_k$ denote the degeneracy factor and the mass, and $\mu_k$ is the chemical potential of the hadron-species $k$. In chemical equilibrium, the latter reads $\mu_k=B_k\mu_B+Q_k\mu_Q+S_k\mu_S$, where $B_k$, $Q_k$ and $S_k$ are the respective quantum numbers of baryon charge, electric charge and strangeness, while $\mu_B$, $\mu_Q$ and $\mu_S$ denote the chemical potentials associated with $n_B$, $n_Q$ and $n_S$.
Other thermodynamic quantities follow from standard relations, e.g. $s\!=\!(\partial p/\partial T)_{\mu_k}$. The particle number density of species $k$, $n_k\!=\!(\partial p/\partial\mu_k)_T$, is given by the momentum-integral $$\label{equ:particledensity}
n_k(T,\mu_k)=\frac{d_k}{(2\pi)^3}\int d^3\vec{p} \frac{1}{(-1)^{B_k+1}+e^{(\sqrt{\vec{p}^{\,2}+m_k^2}-\mu_k)/T}}$$ and the net-baryon density follows from $n_B=\sum_k B_k n_k$. Since we consider $n_B=0$, all $\mu_k$ are set to zero in the chemical equilibrium case.
In this work, we employ a HRG model containing states up to a mass of $2$ GeV as, for example, listed in the edition [@PDG05] of the Particle Data Book. Such a list is also included in the EoS-package provided along with the work in [@Huovinen09]. As evident from Fig. \[Fig:1\], this choice is sufficient to describe the available lattice QCD data fairly well for temperatures below $175$ MeV, where HRG and lattice QCD results mostly overlap. In fact, the relative deviation of the HRG model from the lattice QCD data [@Borsanyi11] in this overlap region, taking the error-bars in the data into account, is at most $9\%$ in $I(T)/T^4$ and $5\%$ in $p(T)/T^4$.
Given the reasonable agreement between lattice QCD data and the HRG model, we construct an equation of state, which serves as a baseline EoS for the chemical equilibrium case: we utilize our suitable parametrization of the lattice QCD results from [@Borsanyi11] at high $T$ and change the prescription to the above discussed HRG model at low $T$ around a switching temperature of $172$ MeV. Generically, such an approach can introduce discontinuities in the thermodynamic quantities. We improve this situation by employing a straightforward interpolation procedure between the two parts in the interval $165$ MeV $\leq T\leq 180$ MeV, which ensures that the pressure and its first and second derivatives with respect to $T$ are continuous. In this way, the speed of sound remains a smooth function for all temperatures. A similar strategy was applied for the construction of the QCD equation of state in [@Huovinen09].
Hadron resonance gas in partial chemical equilibrium \[sec:3\]
==============================================================
In heavy-ion collisions, the time scales for inelastic particle number changing processes, which are responsible for the chemical equilibration of the hadronic matter, are typically much larger than the lifetime of the hadronic stage [@Teaney02]. Thus, it is more realistic to assume that the hadronic phase is not in complete chemical equilibrium. This was first discussed in [@Bebie92] and then considered in numerous works, cf. e.g. [@Hirano02; @Rapp02; @KolbRapp03; @Huovinen07; @ECHO-QGP]: according to this idea, hadronic matter is formed at the hadronization temperature $T_c$ in chemical equilibrium. However, for temperatures below the chemical freeze-out temperature $T_{ch}$, where $T_{ch}\leq T_c$, the inelastic processes become suppressed, while the elastic interactions mediated by frequent strong resonance formations and decays (e.g. $\pi\pi\to\rho\to\pi\pi$, $K\pi\to K^*\to K\pi$, $p\pi\to\Delta\to p\pi$ etc) continue to occur. Consequently, the experimentally observed ratios of particle multiplicities of those species $i$, which are stable against strong decays within the lifetime of the system, are fixed at $T_{ch}$. This is to say that for $T<T_{ch}$ the corresponding effective particle numbers $\bar{N}_i=N_i+\sum_r d_{r\to i}\, N_r$ are frozen. Here, $N_i$ denotes the actual particle number of the stable hadron $i$, $N_r$ the actual particle number of resonance $r$ and $d_{r\to i}$ gives the average number of hadrons $i$ produced in the decay of resonance $r$. For example, the conserved quantity in the process $\pi\pi\to\rho\to\pi\pi$ is the effective pion number $\bar{N}_\pi\!=\!N_\pi+2N_\rho$. The above sum has to be taken over all the states (resonances) that decay into hadron $i$ within the lifetime of the hadronic stage. As their effective number is fixed at $T_{ch}$, but $T$ decreases during the expansion of matter, each stable particle species $i$ acquires an effective, $T$-dependent chemical potential $\mu_i(T)$. The chemical potentials of the resonances, instead, can be written as a combination $\mu_r=\sum_i d_{r\to i}\,\mu_i$ of the effective chemical potentials of the stable hadrons. The hadronic phase is, thus, in a state of partial chemical equilibrium below $T_{ch}$.
[|c|c|c|c|c|c|c|c|c|]{}
------------------------------------------------------------------------
& & & &\
------------------------------------------------------------------------
species & $a_i$ & $b_i$/GeV$^{-1}$ & $a_i$ & $b_i$/GeV$^{-1}$ & $a_i$ & $b_i$/GeV$^{-1}$ & $a_i$ & $b_i$/GeV$^{-1}$\
------------------------------------------------------------------------
$\pi^0$ & 1.745 & -8.607 & 1.785 & -8.438 & 1.816 & -8.220 & 1.839 & -7.960\
$\pi^+$, $\pi^-$ & 1.766 & -8.520 & 1.803 & -8.334 & 1.835 & -8.140 & 1.853 & -7.836\
$K^+$, $K^-$ & 3.156 & -0.992 & 3.080 & -1.125 & 3.008 & -1.233 & 2.938 & -1.307\
$K^0$, $\overline{K}^0$ & 3.191 & -1.131 & 3.114 & -1.246 & 3.044 & -1.393 & 2.973 & -1.440\
$\eta$ & 3.545 & -2.127 & 3.467 & -2.296 & 3.396 & -2.465 & 3.324 & -2.538\
------------------------------------------------------------------------
$p$ & 6.104 & 1.504 & 5.893 & 1.489 & 5.694 & 1.446 & 5.507 & 1.396\
$n$ & 6.113 & 1.530 & 5.899 & 1.525 & 5.701 & 1.465 & 5.513 & 1.420\
$\Lambda^0$ & 6.914 & 4.951 & 6.642 & 4.730 & 6.389 & 4.466 & 6.153 & 4.202\
$\Sigma^+$ & 7.393 & 2.185 & 7.145 & 1.827 & 6.915 & 1.466 & 6.700 & 1.149\
$\Sigma^0$ & 7.420 & 2.160 & 7.170 & 1.806 & 6.940 & 1.453 & 6.723 & 1.145\
$\Sigma^-$ & 7.460 & 2.161 & 7.211 & 1.770 & 6.977 & 1.444 & 6.760 & 1.135\
$\Xi^0$ & 7.939 & 5.461 & 7.634 & 5.055 & 7.349 & 4.670 & 7.084 & 4.302\
$\Xi^-$ & 7.981 & 5.551 & 7.673 & 5.149 & 7.387 & 4.748 & 7.121 & 4.373\
$\Omega^-$ & 10.409 & 3.193 & 10.052 & 2.719 & 9.722 & 2.249 & 9.411 & 1.878\
The freeze-out of the chemical composition of the system at $T_{ch}$ implies, in addition to the conservation of energy, momentum and of the charges $N_B$, $N_Q$ and $N_S$, also the conservation of the effective number $\bar{N}_i$ of each stable particle species $i$ below $T_{ch}$. This makes the EoS a highly-involved relation between $p$, $\epsilon$ and all charge densities. For conserved entropy, the ratio between the effective particle number density and the entropy density $\bar{n}_i/s$ is fixed at $T_{ch}$. This provides a practical tool to conserve all the $\bar{N}_i$ and to determine all the $\mu_i(T)$ for $T<T_{ch}$ from the conditions $$\label{equ:conservation}
\frac{\bar{n}_i(T,\{\mu_{i'}(T)\})}{s(T,\{\mu_{i'}(T)\})}=\frac{\bar{n}_i(T_{ch},\{0\})}{s(T_{ch},\{0\})} \,,$$ which imply that each $\bar{n}_i$ depends, in general, on all the effective chemical potentials $\mu_{i'}(T)$ (including $\mu_i(T)$). The knowledge of all the $\mu_i(T)$ is, apart from knowing the EoS, necessary for determining the final state hadron abundances. We note that the above conditions entail also that the particle ratios of stable hadrons are fixed at $T_{ch}$: $\bar{n}_{i1}(T,\{\mu_i\})/\bar{n}_{i2}(T,\{\mu_i\})=\bar{n}_{i1}(T_{ch},\{0\})/\bar{n}_{i2}(T_{ch},\{0\})$.
In this work, we consider as stable particle species the mesons $\pi^0$, $\pi^+$, $\pi^-$, $K^+$, $K^-$, $K^0$, $\overline{K}^0$ and $\eta$ and the baryons $p$, $n$, $\Lambda^0$, $\Sigma^+$, $\Sigma^0$, $\Sigma^-$, $\Xi^0$, $\Xi^-$ and $\Omega^-$ as well as their respective anti-baryons, i.e. in total $26$ different states. Correspondingly, we consider different isospin states individually. In general, this becomes important only when considering non-vanishing net-densities $n_B$, $n_Q$ and/or $n_S$. In the $n_B=0$ case studied in this work, however, particles and their corresponding anti-particles develop the same effective chemical potentials. For the chemical freeze-out temperature, we consider different values, namely $T_{ch}/$MeV$=145,\,150,\,155$ and $160$. These are within the range of the $T_c$-values determined in lattice QCD [@Borsanyi09; @Bazavov12].
![\[Fig:2\] Temperature-dependence of the effective chemical potentials for selected hadronic states, considering a chemical freeze-out temperature of $T_{ch}=150$ MeV. The solid curves depict $\mu_i(T)$ for the baryons $\Omega^-$, $\Xi^-$, $\Sigma^-$, $\Lambda^0$ and $p$ from top to bottom, while the dashed curves show $\mu_i(T)$ for the mesons $\eta$, $K^-$ and $\pi^+$ from top to bottom.](chempots1.eps){width="45.00000%"}
In Fig. \[Fig:2\], we exhibit the temperature-dependence of the effective chemical potentials $\mu_i(T)$ of some representative particle species as determined from Eq. (\[equ:conservation\]) for $T_{ch}=150$ MeV. As can be seen from Fig. \[Fig:2\], the $\mu_i(T)$ increase with decreasing $T$. The $T$-dependence of $\mu_i(T)$ for species $i$ can be parametrized conveniently by the quadratic fit function $$\label{equ:ParaOfChemPots}
\mu_i(T)=a_i(T_{ch}-T)+b_i(T_{ch}-T)^2 \,.$$ Here, the parameters $a_i$ and $b_i$ depend on the value of $T_{ch}$. Since for a complete EoS the knowledge of all $\mu_i(T)$ is required, we summarize the corresponding parameter-values in Tab. \[Tab:1\]. With these, $\mu_i(T)$ is obtained in units of GeV for $T_{ch}$ and $T$ given in units of GeV.
{width="45.00000%"} {width="45.00000%"}
These parametrizations provide excellent fits for all $\mu_i(T)$ in the temperature range $70$ MeV $\leq T\leq T_{ch}$ with a maximal $\chi^2=9\cdot 10^{-6}$. We note that overall, for the large $T$-range explored in a hydrodynamic simulation, cubic fit functions for the $\mu_i(T)$ yield more accurate descriptions of the full numerical results obtained from Eq. (\[equ:conservation\]) than the quadratic functions, in particular for small $T$. In the interesting interval $70$ MeV $\leq T\leq T_{ch}$, however, the quadratic ansatz Eq. (\[equ:ParaOfChemPots\]) provides fits, which are comparable in accuracy with the cubic-fits for the baryons and anti-baryons, while they are even slightly better for the mesons.
Discussion and Conclusions \[sec:4\]
====================================
We obtain various equations of state by combining our parametrization of the lattice QCD data [@Borsanyi11] as a function of $T$ with the HRG model either in chemical equilibrium or in partial chemical equilibrium in the hadronic phase with various $T_{ch}$-values in the latter case. For the use in a hydrodynamic simulation, however, the EoS is usually given in the form $p(\epsilon,n_B)$, i.e. as a function of $\epsilon$ and $n_B$, together with the results for the effective chemical potentials $\mu_i$ required for determining the particle abundances. In Fig. \[Fig:3\], we show our results for the different equations of state $p(\epsilon)$ supplemented by the corresponding $T(\epsilon)$ for $n_B=0$. We concentrate in Fig. \[Fig:3\] on a visualization of the energy density regions, in which the confinement transition and the chemical freeze-out take place.
As it is evident from Fig. \[Fig:3\] panel (a), differences in $p(\epsilon)$ between chemical equilibrium (solid curve) and partial chemical equilibrium (dashed curves) in the hadronic phase are small. The $\epsilon$-dependence of $T$, instead, is visibly influenced for $\epsilon<\epsilon_{ch}$ by the chemical freeze-out (cf. panel (b) in Fig. \[Fig:3\]), where the value of $\epsilon_{ch}$ depends on $T_{ch}$.
Our results are collected in tabulated form and made available along with this publication [@webpage]. Moreover, for practical convenience we also provide parametrizations as a function of $\epsilon$ of these numerical results, similar to Ref. [@Shen10]. In the chemical equilibrium case, the relevant thermodynamic quantities can be parametrized in the following way: $$\begin{aligned}
\nonumber
p(\epsilon) & = & a_0\epsilon +\frac{a_1}{(a_2+1)}\,\epsilon^{a_2+1}+\frac{a_3}{a_4}\exp\left[a_4\epsilon\right] \\
\label{equ:ParamPressure}
& & -\frac{a_5}{(-a_7)^{a_6+1}}\,\Gamma(a_6+1,-a_7\epsilon) + a_8 \,, \\
\nonumber
s^{4/3}(\epsilon) & = & a_0+a_1\epsilon^{a_2} +a_3\exp\left[a_4\epsilon\right] \\
\label{equ:ParamEntropy}
& & + a_5\epsilon^{a_6}\exp\left[a_7\epsilon\right]\end{aligned}$$ and $$\label{equ:ParamTemperature}
T(\epsilon) = \frac{\epsilon+p(\epsilon)}{s(\epsilon)} \equiv \frac{1}{ds(\epsilon)/d\epsilon} \,.$$ Here, $\Gamma(s,x)=\int_x^\infty t^{s-1} \exp[-t]\,dt$ denotes the upper incomplete $\Gamma$-function. These ansatz-functions can provide excellent descriptions of our numerical EoS-results with proper choices for the entering parameters. We stress, that the parameters $a_i$ in $p(\epsilon)$ and in $s^{4/3}(\epsilon)$ in Eqs. (\[equ:ParamPressure\]) and (\[equ:ParamEntropy\]) are not meant to be the same: we use the same symbols only for practical purposes.
[|c|c|c|c|c|c|c|c|c|c|c|]{}
------------------------------------------------------------------------
quantity & $\epsilon$-region & $a_0$ & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $a_5$ & $a_6$ & $a_7$ & $a_8$\
------------------------------------------------------------------------
$p$ & $\epsilon<\epsilon_0$ & 0.275255 & 2524790 & 2711.84 & -0.275255 & -274.84 & 0.487526 & 0.0956908 & -388.771 & -0.000326\
& $\epsilon_0<\epsilon<\epsilon_1$ & 0.843569 & -60.3954 & 3.203 & -0.601971 & 2.06599 & -739.605 & 2.15326 & -122.409 & 0.2909065\
& $\epsilon_1<\epsilon<\epsilon_2$ & 4.7406 & -4.1849 & 0.1807 & -5.4941 & -1.8539 & 4.3735 & 0.1003 & -2.3275 & -1.321345\
& $\epsilon_2<\epsilon$ & $1/3$ & -0.1310034 & -0.4179 & -0.0230894 & -0.2797 & -0.0774 & 5.7231 & -3.3064 & -0.018477\
------------------------------------------------------------------------
$s^{4/3}$ & $\epsilon<\epsilon_0$ & 2.12885 & 12.50217 & 1.07208 & -2.12885 & 1.0032 & 0.0084625 & 1.42094 & 1345.43 &\
& $\epsilon_0<\epsilon<\epsilon_1$ & -0.000165 & 9.1583717 & 1.0786 & 0. & 1. & 0.5649 & 1.0959 & 9.9955 &\
& $\epsilon_1<\epsilon<\epsilon_2$ & -0.0003645 & 5.763101 & 1.3863 & -0.0000745 & 0.3105 & 6.7934 & 1.0337 & -0.0976 &\
& $\epsilon_2<\epsilon<\epsilon_3$ & -0.655216 & 18.36345 & 0.9912019 & 2.02343 & 0.00355427 & -7.78303 & 0.5725142 & 0.0039527 &\
& $\epsilon_3<\epsilon<\epsilon_4$ & 1.49791 & 14.83324 & 1.02947504 & 3.244652 & -0.0372865 & -7.96072 & 0.257059 & -0.056146 &\
& $\epsilon_4<\epsilon<\epsilon_5$ & -19.025 & 16.25163 & 1.012862 & 33.989528 & -0.00763319 & -18.69299 & 0.179563 & -0.009929 &\
& $\epsilon_5<\epsilon$ & -33.07911 & 16.978858 & 1.00512 & 0. & 1. & 0. & 1. & 1. &\
It turns out that, for an accurate description of the thermodynamic quantities, it becomes mandatory to split the parametrizations into different regions in $\epsilon$ and to fit the parameters for each $\epsilon$-region individually. We define as the splitting-points $\epsilon_0=0.001538$ GeV$/$fm$^3$, $\epsilon_1=0.032084$ GeV$/$fm$^3$, $\epsilon_2=0.567420$ GeV$/$fm$^3$, $\epsilon_3=1.2$ GeV$/$fm$^3$, $\epsilon_4=9.9$ GeV$/$fm$^3$ and $\epsilon_5=100$ GeV$/$fm$^3$. The points $\epsilon_3$, $\epsilon_4$ and $\epsilon_5$ are only of relevance for the parametrization of $s^{4/3}(\epsilon)$ in Eq. (\[equ:ParamEntropy\]) and, therefore, influence $T(\epsilon)$, but play no role for the parametrization of $p(\epsilon)$. The parameter-values entering $p(\epsilon)$ and $s^{4/3}(\epsilon)$ in the different $\epsilon$-regimes are summarized in Tab. \[Tab:2\]. With these, one obtains $p$ in units of GeV$/$fm$^3$, $s$ in units of $1/$fm$^3$ and $T$ from Eq. (\[equ:ParamTemperature\]) in units of GeV for $\epsilon$ given in units of GeV$/$fm$^3$. The high precision in the provided parametrizations is motivated by our goal to maintain thermodynamic consistency and continuity in the second derivatives at the splitting-points $\epsilon_i$ up to a high numerical accuracy.
The squared speed of sound $c_s^2(\epsilon)$ as a function of $\epsilon$ can be determined from $p(\epsilon)$ given in Eq. (\[equ:ParamPressure\]) as $$\label{equ:ParamSpeed}
c_s^2(\epsilon) = a_0+a_1\epsilon^{a_2} +a_3\exp\left[a_4\epsilon\right] + a_5\epsilon^{a_6}\exp\left[a_7\epsilon\right] \,.$$ By employing the parameter-values for $p(\epsilon)$ from Tab. \[Tab:2\], we find an excellent agreement between Eq. (\[equ:ParamSpeed\]) and the $c_s^2$-result obtained by numerically differentiating our tabulated $p(\epsilon)$-results within the temperature interval $30$ MeV $\leq T\leq 300$ MeV. Outside of this range, the quantitative agreement is still good, where $c_s^2$ exhibits the same qualitative behavior as our numerical results with asymptotics $c_s^2(\epsilon)\to 0$ for $\epsilon\to 0$ and $c_s^2(\epsilon)\to 1/3$ for $\epsilon\to\infty$.
The temperature-dependence of $c_s^2$ obtained by numerical differentiation is shown in Fig. \[Fig:4\] (solid curve) and confronted with the lattice QCD results available from the WB-collaboration [@Borsanyi10]. Our curve agrees with the lattice QCD data within error-bars: we also find a rather large $c_s^2(T)$ in the confinement transition region. This indicates that our EoS is rather stiff compared to some previously considered equations of state, as e.g. in [@Song08], but comparable in stiffness with the equation of state presented in [@Huovinen09].
![\[Fig:4\] (Color online) Temperature-dependence of the squared speed of sound $c_s^2(T)$. The solid curve shows the result obtained from a numerical differentiation of our tabulated $p(\epsilon)$-results for the EoS, in which the HRG is in chemical equilibrium. For comparison, the symbols depict available equilibrium lattice QCD data from [@Borsanyi10]. The dashed curves highlight $c_s^2(T)$ when instead partial chemical equilibrium is assumed in the hadronic phase. We consider $T_{ch}/$MeV$\,=145,\,150,\,155$ and $160$ (from top to bottom, respectively).](EoS4_11b.eps){width="45.00000%"}
[|c|c|c|c|c|c|c|c|]{}
------------------------------------------------------------------------
$T_{ch}/$GeV & quantity & $b_0$ & $b_1$ & $b_2$ & $b_3$ & $b_4$ & $b_5$\
------------------------------------------------------------------------
0.145 & $p$ & & 0.20421265 & 1.2147 & -0.006941 & -10.9535 &\
& $s^{4/3}$ & -5.727246 & 17.2995 & 1.0833 & & &\
& $T$ & -0.0570889 & 0.5799 & 0.1412 & -0.3744 & 0.0923 & -0.3201\
------------------------------------------------------------------------
0.150 & $p$ & & 0.19620877 & 1.2200 & -0.007064 & -10.4533 &\
& $s^{4/3}$ & -4.855376 & 16.5519 & 1.0928 & & &\
& $T$ & -0.838232 & 1.9880 & 0.1842 & -1.7244 & 0.1689 & -0.5845\
------------------------------------------------------------------------
0.155 & $p$ & & 0.18633308 & 1.2194 & -0.00730055 & -9.6563 &\
& $s^{4/3}$ & -4.4274 & 16.2165 & 1.0998 & & &\
& $T$ & -1.4628554 & 2.5136 & 0.2741 & -2.1268 & 0.2575 & -0.8329\
------------------------------------------------------------------------
0.160 & $p$ & & 0.17665061 & 1.2121 & -0.00726805 & -9.0082 &\
& $s^{4/3}$ & -4.1890264 & 16.0463 & 1.1055 & & &\
& $T$ & -1.2085036 & 2.5073 & 0.2522 & -2.1677 & 0.2371 & -0.6475\
When including partial chemical equilibrium into the EoS, the parametrizations discussed above have to be modified only for $\epsilon<\epsilon_{ch}$. The different values of $\epsilon_{ch}$, depending on the considered value for the chemical freeze-out temperature, are listed in the caption of Tab. \[Tab:3\]. For $\epsilon<\epsilon_{ch}$, we modify our parametrizations to $$\begin{aligned}
\label{equ:ParamPressureFO}
p(\epsilon) & = & b_1\epsilon^{b_2} + b_3\left(\exp\left[b_4\epsilon\right]-1\right) \,, \\
\label{equ:ParamEntropyFO}
s^{4/3}(\epsilon) & = & b_0\epsilon + b_1\epsilon^{b_2} \end{aligned}$$ and $$\label{equ:ParamTemperatureFO}
T(\epsilon) = b_0\epsilon + b_1\epsilon^{b_2} + b_3\epsilon^{b_4}\exp\left[b_5\epsilon\right] \,.$$ Correspondingly, the squared speed of sound follows now from Eq. (\[equ:ParamPressureFO\]) as $$\label{equ:ParamSpeedFO}
c_s^2(\epsilon) = b_1b_2\epsilon^{b_2-1}+b_3b_4\exp\left[b_4\epsilon\right] \,.$$ We stress that the parameters $b_i$ entering $p(\epsilon)$, $s^{4/3}(\epsilon)$ and $T(\epsilon)$ in Eqs. (\[equ:ParamPressureFO\]) - (\[equ:ParamTemperatureFO\]) are also here not meant to be the same.
By fitting the parametrizations in Eqs. (\[equ:ParamPressureFO\]) - (\[equ:ParamTemperatureFO\]) to our tabulated, numerical results for $\epsilon<\epsilon_{ch}$, we find quite accurate descriptions of the thermodynamic quantities for the parameter-values summarized in Tab. \[Tab:3\]. Again, the high precision in the parameters is given in order to maintain consistency in our approach at $\epsilon_{ch}$ with a high-level of accuracy. With these parameters, $p$, $s$ and $T$ are obtained in units of GeV$/$fm$^3$, $1/$fm$^3$ and GeV, respectively, for $\epsilon$ given in units of GeV$/$fm$^3$. Moreover, the above parametrizations satisfy the physical conditions $T(\epsilon)\to 0$, $p(\epsilon)\to 0$ and $s(\epsilon)\to 0$ for $\epsilon\to 0$. The qualitative behavior of the squared speed of sound is also nicely reproduced with, however, different asymptotics (in fact one obtains positive $c_s^2(\epsilon\to 0)=b_3b_4<1/3$, while our numerical results tend toward $0$).
The temperature-dependence of $c_s^2$ as determined from a numerical differentiation of our tabulated results is shown in Fig. \[Fig:4\] (dashed curves). One observes a discontinuity in $c_s^2(T)$ at $T=T_{ch}$, which is characteristic for the chemical freeze-out. Evidently, as expected the behavior of $c_s^2(T)$ in the non-equilibrium situation is different from the trend seen in equilibrium lattice QCD thermodynamics.
In summary, we constructed QCD equations of state for vanishing net-baryon density based at high $T$ on recent continuum-extrapolated lattice QCD results in the physical quark mass limit [@Borsanyi11], which were continuously combined with a HRG model at low $T$. The latter was considered to be either in chemical equilibrium or in partial chemical equilibrium. In the chemical equilibrium case, our baseline EoS in terms of the squared speed of sound shows only minor deviations of at most a few percent from the EoS presented in [@Huovinen09] such that, presumably, significant differences in the standard observables studied in hydrodynamic simulations are not to be expected from this EoS. Nevertheless, the focus of our work lay on the inclusion of partial chemical equilibrium for a more accurate description of the experimental situation, where we studied different values for the chemical freeze-out temperature $T_{ch}$ within the range presently expected from first-principle approaches [@Borsanyi13]. In view of the non-negligible differences in the temperature and chemical potential evolution of the system for different $T_{ch}$-values our work, thus, allows one to study the possible influence of deviations in $T_{ch}$ on the particle spectra in more detail compared to the work presented in [@Shen10; @Huovinen09].
Our results, being available in a tabulated form [@webpage], can be directly applied in the hydrodynamic modeling of high-energy heavy-ion collisions at the LHC and at RHIC for top beam energies. For convenience, we also provided practical parametrizations of our results, in particular, for the effective chemical potentials $\mu_i(T)$ of the stable hadrons in the partial chemical equilibrium case and for the temperature. Their knowledge is crucial for a determination of final state hadron abundances and spectra.
We have restricted ourselves to the $n_B=0$ case in this work. In general, however, our approach allows for respecting the conservation of finite values for $n_B/s$ and $n_Q/s$ (while $n_S/s=0$) as relevant for heavy-ion collisions. Corresponding results for non-zero (although not too large) values of the associated chemical potentials will be reported in a forthcoming publication.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge valuable discussions with P. Huovinen and U. Heinz. The work of C. Ratti and M. Bluhm is supported by funds provided by the Italian Ministry of Education, Universities and Research under the Firb Research Grant RBFR0814TT.
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[Equation\_of\_State/Home.html](Equation_of_State/Home.html).
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abstract: 'We present the [*asing the dentifiation of SCA alactic bjects*]{} () survey, which is designed to identify the unknown X-ray sources discovered during the Galactic Plane Survey (AGPS). Little is known about most of the AGPS sources, especially those that emit primarily in hard X-rays ($2-10$ keV) within the $F_{x} \sim 10^{-13} \mathrm{~to~} 10^{-11}$ X-ray flux range. In ChIcAGO, the subarcsecond localization capabilities of have been combined with a detailed multi-wavelength follow-up program, with the ultimate goal of classifying the $>100$ unidentified sources in the AGPS. Overall to date, 93 unidentified AGPS sources have been observed with as part of the survey. A total of 253 X-ray point sources have been detected in these observations within $3''$ of the original positions. We have identified infrared and optical counterparts to the majority of these sources, using both new observations and catalogs from existing Galactic plane surveys. X-ray and infrared population statistics for the X-ray point sources detected in the observations reveal that the primary populations of Galactic plane X-ray sources that emit in the $F_{x} \sim 10^{-13} \mathrm{~to~} 10^{-11}$ flux range are active stellar coronae, massive stars with strong stellar winds that are possibly in colliding-wind binaries, X-ray binaries, and magnetars. There is also a fifth population that is still unidentified but, based on its X-ray and infrared properties, likely comprise partly of Galactic sources and partly of active galactic nuclei.'
author:
- 'Gemma E. Anderson, B. M. Gaensler, David L. Kaplan, Patrick O. Slane, Michael P. Muno, Bettina Posselt, Jaesub Hong, Stephen S. Murray, Danny T. H. Steeghs Crystal L. Brogan, Jeremy J. Drake, Sean A. Farrell, Robert A. Benjamin, Deepto Chakrabarty, Janet E. Drew, John P. Finley, Jonathan E. Grindlay, T. Joseph W. Lazio, Julia C. Lee, Jon C. Mauerhan, Marten H. van Kerkwijk'
title: 'Chasing the Identification of *ASCA* Galactic Objects (ChIcAGO) - An X-ray Survey of Unidentified Sources in the Galactic Plane. I: Source Sample and Initial Results'
---
Introduction
============
From 1996 to 1999, the *Advanced Satellite for Cosmology and Astrophysics* () performed the Galactic plane survey (AGPS), which was designed to study 40 deg$^{2}$ of the X-ray sky, over the Galactic coordinates $|l| \lesssim 45^{\circ}$ and $|b| \lesssim 0^{\circ}.4$, in the $0.7-10$ keV energy range [@sugizaki01]. This survey resulted in a catalog of 163 discrete X-ray sources with X-ray fluxes between $F_{x} \sim 10^{-13} \mathrm{~and~} 10^{-11}$ , many of which are much harder and more absorbed than any other X-ray source previously detected in the Galactic plane. While the AGPS yielded the first ever distribution of hard ($2-10$ keV) Galactic plane X-ray sources, ’s limited spatial resolution ($3'$) and large positional uncertainty ($1'$) left $>100$ of the AGPS sources unidentified. Even in the era of the *Chandra X-ray Observatory* and the *XMM-Newton* telescope, a substantial fraction of the AGPS source catalog, and therefore a large fraction of the Galactic plane X-ray population, still remain unidentified.
For the last few years, new and archival multi-wavelength data have been used to improve the general understanding of the Galactic X-ray sources detected in the AGPS. Recent work has demonstrated that unidentified sources represent a whole range of unusual objects. For example @gelfand07 used new and archival and observations to identify the AGPS source (also known as ) as a magnetar sitting at the center of a faint and small, previously unidentified, radio supernova remnant (SNR) called G327.24–0.13. Investigations of archival data allowed @kaplan07 to identify the AGPS source as a likely symbiotic X-ray binary (SyXB) comprising of a late-type giant or supergiant and a neutron star (NS) with a 112s pulse period. @gaensler08 identified the AGPS source as the X-ray emission associated with the radio . The X-ray spectrum, combined with the presence of non-thermal, polarized, radio emission, showed G350.1–0.3 to be a very young and luminous SNR. A central compact object was also resolved in these X-ray observations and identified as a NS. [@funk07] and [@lemiere09] observations have demonstrated that the AGPS source is an X-ray pulsar wind nebula (PWN) located at the center of the radio . results, discussed in @anderson11, have revealed that two AGPS sources, and , are massive stars in colliding wind binaries (CWBs). New , , and ATCA observations have also been used to identify the AGPS source as the radio and X-ray emitting magnetar, , and have exposed the likely X-ray transient nature of this source [@anderson12]. These identifications over the last 8 years have therefore demonstrated that many of the unidentified AGPS sources are unusual and rare Galactic plane X-ray objects.
The most comprehensive X-ray survey to-date, in terms of area coverage, was performed by the *ROSAT* X-ray Satellite [for example see @voges99], which mapped the soft X-ray source population ($0.1-2.4$ keV) down to a flux sensitivity of a few $10^{-13}$ . Projects that focused on the *ROSAT* data covering the Galactic plane [e.g. the ROSAT Galactic Plane Survey; @motch97; @motch98] demonstrated that stars and AGN dominate the soft X-ray sky. However, performing a similar Galactic plane survey to include those sources with energies up to $10$ keV, sensitive to the $F_{x} \sim 10^{-13} \mathrm{~to~} \sim10^{-11}$ flux range, would be impractical to achieve with the current X-ray telescopes and due to their limited fields of view. Astronomers have therefore had to rely upon characterizing the distribution of the harder X-ray source populations within much smaller regions of the Galactic plane [e.g. @hands04; @ebisawa05; @grindlay05]. For example, @motch10 used the XGPS [@hands04] to determine the contributions of active stellar coronae and accreting X-ray source populations in the Galactic plane for $F_{x} \lesssim 10^{-12}$ . The Chandra Multiwavelength Plane survey [ChaMPlane; @grindlay05] has now surveyed 7 deg$^{2}$ of the Galactic plane and bulge with [@vandenberg12], identifying the contributions of magnetic cataclysmic variables (CVs) to the Galactic ridge X-ray emission [@hong12b].
The key to obtaining a complete understanding of the Galactic plane X-ray source populations, from $0.3-10$ keV, that make up the $F_{x} \sim 10^{-13} \mathrm{~to~} 10^{-11}$ X-ray flux range is to identify the unidentified AGPS sources, as covered a much larger area of the Galactic plane ($\sim40$ deg$^{2}$) than other X-ray surveys [for example the XGPS and ChaMPlane; @hands04; @motch10; @grindlay05; @vandenberg12]. In order to identify the AGPS sources, the [*asing the dentifiation of SCA alactic bjects*]{} () survey was conceived. In this survey the subarcsecond capabilities of are used to localize the unidentified AGPS sources listed by @sugizaki01. Once the positions of these sources have been determined, an extensive multi-wavelength program is activated, which is aimed at determining the identities of the sources and the nature of their X-ray emission.
In this paper, we present the results of observations of 93 unidentified AGPS sources, along with the multi-wavelength follow-up that has allowed the identification of optical, infrared and radio counterparts. Section 2 explains the observing strategy employed to localize the unidentified AGPS sources. To begin the identification process, we automated the data analysis and preliminary multi-wavelength follow-up, which involves comparisons with existing optical, near-infrared (NIR) and infrared (IR) surveys. X-ray spectral modeling using “quantile analysis” [@hong04] and @cash79 statistics, and further multi-wavelength observations in the optical, infrared and radio bands required to ultimately classify each source, are also described. Section 3 details the results of each AGPS position observed with . These results include details on the individual X-ray sources detected, the parameters of their likely X-ray spectral shapes, and the names and magnitudes of their infrared, optical and radio counterparts. The possibility of short term variability or periodicity is also explored. In Section 4, we discuss the AGPS sources that have been identified through a visual inspection of radio Galactic plane surveys. The X-ray fluxes and NIR and IR magnitudes of the remaining unidentified sources, reported in Section 3, are then used to conduct X-ray and infrared population statistics. Resulting flux and color-color diagrams allow the identification of likely Galactic plane X-ray populations with infrared counterparts. This analysis is followed by a discussion of particularly interesting individual sources that have been identified as a result of this work. The final part of this section includes a tabulated summary of all the 163 AGPS sources along with their confirmed identifications (obtained from the literature and the present paper) or their tentative identifications that are based on our survey statistical results. In Section 5 we summarize the results from this paper, with a particular focus on our statistical findings.
Method
======
Observations
-------------
The main goal of the survey is to localize the positions of the unidentified AGPS sources, so that multi-wavelength follow-up can be used to identify them. It is therefore necessary to design an experiment that will allow each source to be localized precisely enough to identify counterparts in the crowded Galactic plane. can provide subarcsecond localization as it has an intrinsic astrometric precision accuracy of $0\farcs6$ at 90% confidence within $2'$ of the aim-point [@weisskopf03].
For all but the brightest targets, ’s Advanced CCD Imaging Spectrometer [ACIS; @garmire03] was used as it provides simultaneous positional, temporal and spectroscopic information. The ACIS-S configuration was chosen as it fully encompasses the AGPS positional uncertainties of up to $3'$ [@sugizaki01]. (The aimpoint of the ACIS-I configuration is near a chip gap.) The High Resolution Camera [HRC; @murray00] in the I focal plane array was used to observe those sources with a predicted ACIS count-rate $>0.2$ counts s$^{-1}$, to avoid positional, spectral, and temporal degradation associated with pile-up [@davis01].
At ’s high angular resolution, only a small number of X-ray counts are required to localize each source sufficiently to overcome confusion from IR field stars in the Galactic plane. We first considered the number density of such stars in the $K_{s}$-band at low Galactic latitudes. Figure 24 of @kaplan04 shows that $0.2$ stars arcsec$^{-2}$ are expected with a magnitude $K_{s} \lesssim 19$. For there to be a $<25\%$ chance of random alignment of the source with an infrared field star of this magnitude, a total astrometric error $<0\farcs7$ is required. Using 2MASS as a guide, given its extremely high positional precision ($0\farcs1$ $1\sigma$ error) and ’s 90% absolute astrometry error of $0\farcs6$, a centroiding error of $<0\farcs4$ with 95% accuracy is required for . The Interactive Analysis of Observations (`CIAO`)[^1] software tool `wavdetect` [@freeman02] was chosen to detect the point sources in our fields. Equation (5) of @hong05 provides the 95% confidence position error circle of a point source detected with `wavdetect` for a given number of source counts at a given off axis angle. At the maximum off-axis angle expected for an source localization ($<3'$), $\sim100$ X-ray counts are required to ensure that a source’s centroiding error is below $0\farcs4$.
Using the count rates and power-law spectral fits calculated by @sugizaki01 for each AGPS source and the Proposal Planning Toolkit (PIMMS)[^2], the exposure time required to detect $\sim100$ counts with for each source was estimated. For those AGPS sources that were too faint for @sugizaki01 to calculate spectral fit parameters, an absorbed power-law model with a photon index $\Gamma=2$ and an absorption $N_{H}=10^{22}$ cm$^{-2}$ was used, which are representative values of a non-thermal X-ray source and typical Galactic plane absorption.
In order to select the AGPS source candidates to be observed with , each source was investigated individually. First those AGPS sources that have already been conclusively identified, either by @sugizaki01 or by other groups in the literature, were removed from the target list. Based on this criteria, a total of 43 AGPS sources were identified and therefore rejected for follow-up (these sources are described in Appendix A). The images of each of the remaining unidentified AGPS sources were then studied to determine if any sources appeared to be too extended for to successfully localize in a short amount of time. These sources were also rejected for follow-up. The remaining, unidentified AGPS sources, were then prioritized for follow-up based on their absorbed X-ray flux or count-rate that was listed by @sugizaki01.
A total of 93 AGPS sources have been observed with as part of the survey, of which 84 were imaged with ACIS-S and 9 were imaged with HRC-I. The observations took place over a three and a half year period, from 2007 January to 2010 July. The exposure times ranged from $\sim1-10$ ks. All the details of these observations are listed in Table \[Tab1\]. The initial automated analysis of these observations was conducted using the Multi-wavelength Analysis Pipeline, described in Section 2.2. We then performed a more detailed X-ray analysis and counterpart study for those 74 sources with $>20$ X-ray counts, as such sources are approximately within the original AGPS sources X-ray flux range (see Sections 3.2 and 3.3).
ChIcAGO Multi-wavelength Analysis Pipeline (MAP)
------------------------------------------------
It is crucial to the efficiency of the project to automate the analysis of the observations, such as the detection and extraction of sources, as well as the search for multi-wavelength counterparts. We therefore created the Multi-wavelength Analysis Pipeline (MAP) for this task. MAP takes the ACIS-S or HRC-I observation of an AGPS source field and detects and analyzes all point sources within $3'$, equivalent to the largest likely position error, for the original AGPS source positions supplied by @sugizaki01. From hereon we refer to all point sources detected in the observations of the AGPS fields as “ sources”. The X-ray analysis component of this pipeline uses the `CIAO` software, version 4.3, with CALDB version 4.5.5, and follows standard reduction recipes given in the online `CIAO` 4.3 Science Threads.[^3]
MAP carries out the following steps, all of which are explained in more detail below. These steps apply to both ACIS-S and HRC-I datasets unless otherwise stated.
- An image of the original detection of the AGPS source is created (for example see the top image of Figure \[Fig1\]).
- The `CIAO` tool `chandra_repro` is run to reprocess the data.
- The new event file is filtered to only include photons with energies in the range $0.3-8.0$ keV.
- The `CIAO` tool `wavdetect` is used to detect all X-ray point sources ( sources) within $3'$ of the AGPS position. A image is then created for each source (for example see the bottom image of Figure \[Fig1\]). If no sources are detected, MAP ends.
- The position, source counts, and associated errors are calculated for each source detected. If the dataset is an ACIS-S observation then the total counts are obtained in the $0.3-8.0$, $0.5-2.0$, and $2.0-8.0$ keV energy ranges, and the energy quartiles ($E_{25}$, $E_{50}$, and $E_{75}$), which are used in quantile analysis (see Section 2.3.1), are calculated.[^4]
- The `CIAO` tool `specextract` is run on ACIS-S datasets to obtain the source and background spectrum files and their corresponding redistribution matrix file (RMF) and the ancillary response file (ARF) for each source. These files are used in quantile analysis and spectral modelling (see Section 2.3).
- A timing analysis is conducted on each source (detected with either the ACIS or HRC instruments) to search for short term variability and periodicity.
- A light-curve with 8 bins is constructed using the `CIAO` tool `dmextract`. The $\chi^{2}$ is calculated for this light-curve in order to test for short-term variability (e.g. Figure \[Fig2\]).
- The $Z^{2}_{1}$ statistic is calculated to search for sinusoidal periodicity [@buccheri83]. This process creates a power spectrum and folded light curve (e.g. Figure \[Fig3\]), which predicts the most likely pulsed frequency, its corresponding power, and the probability that the power is random noise.
- Multi-wavelength follow-up and catalog searches are conducted to identify likely optical, infrared, and radio counterparts to each source.
- The USNO B1 [@monet03], 2MASS PSC [@skrutskie06], and GLIMPSE I and II [@benjamin03] catalogs are accessed to obtain a list of all optical and infrared sources within $4''$ of the source’s `wavdetect` position.
- Small sized ($6'$ by $6'$) image cutouts, centered on the source’s `wavdetect` position, are obtained from the 2nd Digitized Sky Survey [Red: DSS2R and Blue: DSS2B; @mclean00], 2MASS, and the GLIMPSE I and II surveys. A $30'$ by $30'$ radio image cutout is obtained from the Sydney University Molonglo Sky Survey [SUMSS; @bock99]. Examples of all the image cutouts can be found in Figures \[Fig4\] and \[Fig5\].
MAP first generates an image of the AGPS source as originally detected by the Gas Imaging Spectrometer [GIS; @ohashi96] onboard . The GIS has a circular field-of-view with a $50'$ diameter. The top image of Figure \[Fig1\] shows the GIS detection of the AGPS source . We have chosen AX J144701–5919 as the example source for illustrating the output of MAP because it was the first AGPS source observed with as part of the survey, and it is also an interesting source with a bright counterpart [as demonstrated by its identification as an X-ray emitting WR star in @anderson11].
MAP reprocesses all of the observations, both ACIS and HRC, using the `CIAO` `chandra_repro` script, which creates a new level=2 event file and bad pixel file. The ACIS data are filtered to only include events with energies in the range $0.3-8.0$ keV in order to avoid high-energy and cosmic ray particle backgrounds.
We have chosen to optimize MAP for the detection of point sources as our short observations ($<10$ ks) are not very sensitive to extended sources. MAP uses the `CIAO` wavelet detection algorithm, `wavdetect`, which we have set to search for all sources with wavelet scales of radii 1, 2, 4, 8 and 16 pixels. This does, however, introduce a source selection bias as it is possible that AGPS sources that were unresolved with the PSF could be extended, and therefore resolvable with . Re-running MAP using `wavdetect` with larger scales more appropriate for extended sources, or using the `CIAO` Voronoi Tessellation and Percolation source detection algorithm `vtdetect` [@ebeling93], could be conducted in the future to detect extended sources in the images.
ChIcAGO MAP utilizes `wavdetect` to detect all sources within $3'$ of the original position (for example see the detection of AX J144701–5919 in the bottom image of Figure \[Fig1\]). This search radius is based on the position accuracy and spatial resolution of and therefore designed to ensure that the majority of contributing X-ray sources are encompassed. (However, it is also possible that there are associated X-ray sources beyond $3'$ from the AGPS position. This could be due to the inaccuracy of the positions of those sources that are blended or near the edge of the field-of-view. This is further explored in Section 3.1 and Appendix B.) In many cases more than one X-ray source could have been contributing to the total X-ray flux from an AGPS source originally detected with . (Note this is not the case for the example of AX J144701–5919.) The positional accuracy of `wavdetect` has been well investigated and tested in previous surveys [i.e. ChaMPlane; @hong05].
The position of each source as output by `wavdetect` is obtained and Equation (5) of @hong05 is used to calculate the 95% confidence position error circle.[^5] This `wavdetect` error is then added in quadrature to the absolute astrometry error of to obtain the total position error of each source.
The source regions used to calculate the total number of source counts, which are centered on the source position as output by `wavdetect`, have a radius equivalent to 95% of the point-spread function (PSF) at 1.5 keV. The background subtraction is performed using an annulus whose size is between two and five times the above 95% PSF radius, centered on the source position. Given the low number of counts detected, usually $<100$, the 1$\sigma$ lower and upper confidence limits of the total number of counts are calculated using @gehrels86 statistics. The following energy based analysis is then performed on the ACIS-S observations. The total number of counts are calculated for the $0.3-8.0$, $0.5-2.0$, and $2.0-8.0$ keV energy ranges. ChIcAGO MAP then uses the extracted ACIS-S counts and corresponding energies to calculate the 25%, 50% and 75% photon fractions ($E_{25}$, $E_{50}$ and $E_{75}$), the energies below which 25%, 50% and 75% of the photon energies are found, respectively. These median ($E_{50}$) and quartile ($E_{25}$ and $E_{75}$) values can immediately characterize the hardness of a source without using conventional hardness ratios, which are not versatile enough to account for diverse X-ray spectral types in the Galactic plane. These quartile fractions can then be used to employ quantile analysis [@hong04], which uses a quantile based color-color diagram to classify spectral features and shapes of low count sources (see Section 2.3.1).
The `CIAO` tool, `specextract`, is also run on the ACIS-S detected sources to generate source and background spectrum files and their corresponding RMF and the ARF. These files are used to perform spectral interpolation with quantile analysis and to conduct spectral modelling and fitting with the `CIAO` package `Sherpa`. Further details on the spectral investigations of the sources can be found in Section 2.3
MAP then performs a timing analysis on all the sources, detected with either the ACIS or HRC instruments, to search for evidence for short term variability and periodicity. In each case a light-curve is extracted using `dmextract` where the counts are divided into 8 bins (for example see Figure \[Fig2\]). The Gehrels approximation to confidence limits for a Poisson distribution is used to estimate the errors as there are $<20$ X-ray counts in each bin. The $\chi^{2}$ statistics are adopted to test for variability [for example see @gaensler00]. (It should be noted that as `dmextract` uses the upper (larger) Gehrels confidence limit to estimate the count-rate errors, the resulting $\chi^{2}$ output by MAP may be underestimated.) For 7 degrees of freedom, $\chi^{2} \gtrsim 24.3$ is required for a source to be considered variable at 99.9% confidence.
![The ACIS-S light-curve of the source AX J144701–5919 as output by MAP. This source does not exhibit evidence for short-term variability as its resulting $\chi^{2} = 3.4$ for 7 degrees of freedom is below the confidence threshold of 99.9%.[]{data-label="Fig2"}](f2.eps){width="40.00000%"}
After correcting the photon arrival times to the Solar System barycenter, the target source is then investigated for evidence of periodicity using the $Z_{n}^{2}$ test [@buccheri83], equivalent to the Rayleigh statistic when $n$, the chosen number of harmonics, is set to 1. MAP uses $n=1$ for the sake of simplicity and because such a test is sensitive to sinusoidal distributions. $Z_{1}^{2}$ has a probability density function equivalent to the $\chi^{2}$ statistic with $2$ degrees of freedom. ChIcAGO MAP searched for periodicity down to 6.48s [twice the frame-time resolution; @weisskopf03] for sources detected with ACIS, and down to 0.01s for sources detected with HRC.[^6] A power spectrum (power vs. frequency) and folded light curve (counts per bin vs. phase) of each source is generated, predicting the pulsed frequency of the highest $Z_{1}^{2}$ power, and the probability that this power was random noise for a given number of trials (for example see Figure \[Fig3\]). A 99.9% confidence was required for a source to have a significant level of periodicity, which corresponds approximately to $Z_{1}^{2} \gtrsim 26$ and $Z_{1}^{2} \gtrsim 36$ for ACIS and HRC observations, respectively.
It is also possible that many of the AGPS sources are transient or undergo long-term variability, so have changed significantly in flux since the original observations. The detailed analysis of any periodic, variable, and transient sources is beyond the scope of this paper. We only flag those source that may fit into one of the above categories for the purpose of future investigations.
The next step in the identification process is to search for multi-wavelength counterparts to the sources. MAP accesses the US Naval Observatory B Catalog, version 1.0 [USNO B1, visual magnitude bands $B$, $R$, and $I$; @monet03], the Two Micron All Sky Survey Point Source Catalog [2MASS PSC, near-infrared magnitude bands $J$, $H$, and $K_{s}$; @skrutskie06], and the Galactic Legacy Infrared Mid-Plane Survey Extraordinaire I and II Spring 07’ Catalogs (highly reliable) and Archives (more complete, less reliable) [GLIMPSE, infrared magnitude bands 3.6, 4.5, 5.8, and 8.0 $\mu$m; @benjamin03] to obtain a list of all the optical and infrared sources within $4''$ of the `wavdetect` position of each source. The information extracted from these surveys includes the position of the source, the offset from the position and the magnitudes listed in the given survey or dataset.[^7]
Small sized images cutouts ($6'$ by $6'$) from optical and infrared surveys, centered on the source `wavdetect` position, are also downloaded to enable a visual inspection of likely counterparts and their surrounding environments. The $B$ and $R$ magnitude band images are obtained from the 2nd Digitized Sky Survey [Blue: DSS2B and Red: DSS2R; @mclean00] and the $J$, $H$, and $K$ magnitude band images are obtained from 2MASS. Image cutouts of the infrared magnitude bands 3.6, 4.5, 5.8 and 8.0 $\mu$m are obtained from the GLIMPSE I and II, version 3.5, surveys.[^8] A $30'$ by $30'$ image cutout of the 843 MHz radio sky is also generated from SUMSS, which is a survey conducted with the Molonglo Observatory Synthesis Telescope (MOST) and has a resolution of $43'' \times 43'' cosec |\mathrm{dec}|$ [@bock99].[^9] An example of all the image cutouts generated by MAP for AX J144701–5919 can be seen in Figures \[Fig4\] and \[Fig5\].
Sources Spectral Investigation
-------------------------------
Deducing the best spectral model fit to the sources is difficult using standard X-ray spectral fitting techniques due to the small number of X-ray source counts detected in the observations (usually $<100$). We therefore implement two different techniques for predicting the best spectral parameters for the brighter ($>20$ X-ray counts) sources. The first technique is “quantile analysis" [@hong04], which has been recently developed to address some of the problems associated with spectral modeling of sources with low number statistics. Quantile analysis allows for the interpolation of likely spectral shapes of X-ray sources with as few as $10$ counts. The second technique is utilizing the `CIAO` spectral fitting tool `Sherpa` to obtain best fit spectral parameters using @cash79 statistics. These statistics are based on Poisson distributed data and are therefore ideal for modeling spectra with a limited number of source counts. Both methods are described in detail below.
### Quantile Analysis
Quantile analysis uses predetermined fractions of the total number of energy source counts, such as the median ($E_{50}$) and quartile energies ($E_{25} \mathrm{~and~} E_{75}$), to construct a quantile-based phase space that can be overlayed with grid patterns of common spectral models. This quantile phase space is more sensitive to the wide range of Galactic X-ray sources than those constructed from conventional hardness ratios. This is because there is no count-dependent sensitivity bias toward any spectral type, which is inherent to the choice of sub-energy bands in the conventional spectral hardness or X-ray color analysis. In the survey, quantile analysis is adopted to calculate potential spectral shapes of the sources.
The ACIS-S detected sources selected for quantile analysis have $>20$ (net) X-ray counts in the $0.3-8.0$ keV energy band, as these are likely to be bright enough to be the original AGPS sources (see Section 2.1). The net counts were obtained by summing the total number of counts inside a source region that is $6$ times the error radius in size and subtracting the background normalized counts calculated from an annulus that has an outer radius of $\sim15''$ with the source region subtracted. The three quartiles ($E_{25}, E_{50}, \mathrm{~and~} E_{75}$) and their corresponding errors were calculated as outlined by @hong04.
The quantile phase space, suggested by @hong09 [see their Figure 4], used to calculate the likely spectral shapes, was constructed for each source. This phase space consists of the normalized logarithmic median $(\mathrm{log}(E_{50}/E_{low})/\mathrm{log}(E_{high}/E_{low}))$ and the normalized quartile ratio ($3 \times (E_{25}-E_{low})/(E_{75}-E_{low}))$, where $E_{low} = 0.3$ keV and $E_{high}=8.0$ keV, equivalent to the energy range explored. The logarithmic median phase space takes advantage of the higher sensitivity at low energies in typical X-ray telescope CCDs, while keeping the spectral discernibility more or less uniform throughout the full range of the phase space. In order to compensate for the spatial change of the detector response in Chandra/ACIS, the RMF and the ARF, appropriate to the observed location of each source in the CCD, are calculated using the `CIAO` tools.
The data points of the sources in the quantile diagram were compared with simple power-law and thermal bremsstrahlung spectral models to extract the most plausible spectral parameter values for each source. It should be noted that quantile analysis cannot evaluate which model is more likely unless the estimated parameters of the model turn out to be unphysical (e.g. $\Gamma>4$). The estimate of the spectral parameters are limited to $-2 \leq \Gamma \leq 4$ for a power law model and $0.1 \mathrm{~keV} \leq kT \leq 10 \mathrm{~keV}$ for a bremsstrahlung model. The explored extinction ($N_{H}$) covers the range $0.01 - 100 \times 10^{22}$ cm$^{-2}$. If the data point for a source in quantile phase space sits outside of the model grid set by these parameter ranges, the model is considered incompatible with the observed spectrum of the source. The quantile errors allow the spectral parameter uncertainties to be calculated for each source [@hong04].
### Spectral Modeling
The spectral modeling of the sources was conducted using the `CIAO` 4.5 spectral fitting package `Sherpa` with the statistics set to the `XSpec` [@dorman01] implementation of @cash79 statistics. Cash statistics apply a maximum likelihood ratio test that can be performed on sources with a low number of source counts per bin. We chose to restrict all spectral modeling to those sources with $>50$ X-ray counts as attempted modeling of those sources with less counts usually did not converge.
Cash statistics cannot be performed on a background subtracted spectrum. The source and background spectrum must instead be modeled simultaneously. However, in the case of the observations of sources the background is extremely low and cannot be described by a generic spectral model. As a result attempting to model the background spectrum does not improve the overall spectral fit. We therefore only model the source spectrum of the sources.
Both an absorbed power law and an absorbed thermal bremsstrahlung model are applied to the source spectra so that they can be directly compared to the quantile analysis spectral interpolation results. The parameter errors are calculated using the `Sherpa` “projection" function, which estimates the $1\sigma$ confidence intervals. The absorbed and unabsorbed flux (plus errors) are calculated using the `Sherpa` function “sample\_flux", which is new to `CIAO` 4.5. The overall goodness of fit measure is defined by the value of the Cash statistic divided by the number of degrees of freedom and should be of order 1.
Multi-wavelength Follow-up Observations
---------------------------------------
While the X-ray morphology and spectrum can provide information on the nature of a source, the key to identification is usually through an extensive multi-wavelength follow-up campaign. MAP identifies the possible optical and infrared counterparts in existing multi-wavelength Galactic plane surveys. There are, however, many cases where the counterparts are too faint to be detected in these surveys due to the high absorption in the Galactic plane. It is also possible that the high object density in the Galactic plane may result in confusion with nearby sources. In these cases, further optical and infrared photometric observations were conducted with large telescopes to obtain detections of faint counterpart candidates and to separate likely blends. If the first photometric observing attempt was unsuccessful at detecting or separating a counterpart candidate then deeper imaging using longer exposure times was conducted. Those X-ray sources that remain undetected at optical and infrared wavelengths will need to be further investigated in the X-ray band or at other wavelengths.
The radio wavelength band is also a useful diagnostic for identifying X-ray sources in the Galactic plane. Comparing the X-ray source positions with radio surveys can indicate if there is a likely radio counterpart or whether the X-ray source lies in a diffuse region of radio emission in the Galactic plane. Interferometric radio observations of a small subset of sources were obtained in order to resolve confusing regions of radio emission and allow for the detection or confirmation of compact radio counterparts.
### Optical and Near Infrared Observations with Magellan
The optical and NIR photometric observations presented in this paper were obtained using instruments on the twin 6.5m, Baade and Clay, Magellan telescopes, located at Las Campanas Observatory, Chile. The NIR photometry, $1-2.5 \mu$m, was obtained using the Persson’s Auxiliary Nasmyth Infrared Camera [PANIC; @martini04; @osip08] on Baade. PANIC was used primarily to detect the counterparts of sources with no cataloged counterparts or to utilize this instrument’s high angular resolution to eliminate possible blendings. Observations were obtained in the $J$, $H$ and $K_{s}$ photometric bands, using short exposures ($10-30$s) that were dithered to account for the high sky background inherent to NIR observations.
The PANIC NIR imaging data were reduced using the Image Reduction and Analysis facility [IRAF; @tody86; @tody93] and the PANIC Data Reduction Package for `IRAF` [@martini04], taking into account the corresponding darks, sky flats and bad pixel maps obtained for each respective night. The absolute astrometry for the PANIC observations was derived using the 2MASS PSC and the Graphical Astronomy and Image Analysis Tool [`GAIA`; @draper09]. The positional accuracy of the 2MASS PSC is $0\farcs1$ [$1\sigma$, @skrutskie06], and since there are usually many 2MASS sources in the field, a similar order of astrometric accuracy at the target positions was reached. `SExtractor` [@bertin96] was used for source detection, and the calibration of the photometry was performed by applying 2MASS PSC photometry to known 2MASS sources in the target fields. No correction to the atmospheric extinction was applied as this effect is very small in the NIR ($\sim0.06$ magnitudes in the $H$-band). The errors were obtained by comparing the NIR-magnitudes with 2MASS PSC magnitudes, and reflected average deviations from the 2MASS catalog magnitudes.
Optical photometric observations of the counterparts were obtained utilizing several Magellan instruments depending on availability. These included the Inamori Magellan Areal Camera and Spectrograph [IMACS; @dressler06; @osip08] and the Raymond and Beverly Sackler Magellan Instant Camera [MagIC; @osip08] on the Baade telescope. These instruments provide access to different photometric filters including Bessel $B$, $V$ and $R$, and CTIO $I$.
The IMACS $B$, $V$, $R$ and $I$ imaging data were reduced using `IRAF`, in which the data were trimmed, overscan-corrected and flat-fielded. A dark current subtraction was not applied since the dark images showed it to be negligible. Standard stars were observed throughout in several bands, although the weather during some of the observing nights varied. The absolute astrometry was again computed using `GAIA` with comparisons to the USNO B1 Catalog or the 2MASS PSC. The photometry of each counterpart was calculated using `SExtractor` and was calibrated using the USNO B1 Catalog.
MagIC observations were obtained in the $V$, $R$, and $I$ bands. Short 30s exposures were obtained in each filter, which were later combined to make sure that the brighter stars (used for astrometric referencing) were not saturated. These data were then reduced following standard procedures in `IRAF`: overscan subtraction for each amplifier, flatfielded using dome flats, and the separate exposures combined. The astrometry was applied by referencing the observations to the 2MASS PSC, resulting in an rms residual of $0\farcs1$ in each coordinate. Photometric calibration was derived using observations of the @stetson00 standard fields. The measured photometry for the airmass terms appropriate to Las Campanas Observatory were corrected and the zero-points with scatters of $0.02$ magnitudes in each filter were obtained. Similarly to the PANIC and IMACS observations, the photometry of each counterpart was then measured using `SExtractor`.
### Radio follow-up and ATCA Observations
Radio survey data already exist for all of the AGPS source regions, via the first and second epoch Molonglo Galactic Plane Surveys at 843 MHz [MGPS1 and MGPS2 respectively, $43'' \times 43'' cosec |\mathrm{dec}|$ resolution; @green99; @murphy07], the 90cm Multi-configuration Very Large Array Survey of the Galactic Plane [$42''$ resolution; @brogan06], The Multi-Array Galactic Plane Imaging Survey at 1.4 GHz [MAGPIS, $6''$ resolution; @helfand06], and the Very Large Array (VLA) Galactic Plane Survey at 1.4 GHz [VGPS, $1'$ resolution; @stil06]. To search for possible radio counterparts we first visually inspected the above surveys at the position of each source. Any coincident radio emission was then categorized (see Section 3.4).
Follow-up Australia Telescope Compact Array (ATCA) observations were conducted to identify the nature of any radio counterparts to the sources found through this visual inspection. The ATCA was also used to resolve any diffuse radio emission surrounding a source, allowing the detection of any underlying, compact radio counterpart. The high resolution ATCA observations were instrumental in confirming a positional coincidence between the radio counterpart and the position of the source, to measure accurate radio fluxes and spectral indices, to broadly characterize variability with respect to earlier epochs, and to constrain the object’s spatial extent. The positions of ten sources were observed with the ATCA on 2008 January 21 and on 2008 April 11. Each source was observed for $\sim1$ hour at each of 1.4, 2.4, 4.8 and 8.6 GHz over a 12 hour period with a 6km baseline configuration.
Results
=======
Results
--------
We present results on 93 APGS sources that have been observed with as part of the survey. In many cases more than one source was detected within $3'$ of the original AGPS source positions so, as mentioned in Section 2.2, we will refer to these all as sources. We therefore detected a total of 253 sources in these 93 observations. The naming convention we have adopted is to call each source by the AGPS coordinate name, using ChI as the prefix, with a suffix between 1 to $i$, where $i$ is equal to the number of sources detected in the $3'$ field. (The source name order is based on the order that `wavdetect` detected and output the sources in MAP.) For example, two sources, ChI J165646–4239\_1 and ChI J165646–4239\_2, were detected with in the vicinity of the AGPS source . These two sources are the black dots in Figure \[Fig6\]. The GIS detection of AX J165646–4239 is overlayed on Figure \[Fig6\] in the form of contours, demonstrating that both ChI J165646–4239\_1 and ChI J165646–4239\_2 may have contributed to the X-ray emission originally detected for this source in the AGPS.
![The $3'$ radius field-of-view of the observation of the AGPS source AX J165646–4239, which is centered on the position of AX J165646–4239 as listed by @sugizaki01. Two sources, ChI J165646–4239\_1 and ChI J165646–4239\_2, which are black dots indicated by arrows, were detected with and are labeled on this Figure. The black contours represent the smoothed GIS detection of AX J165646–4239 at 65%, 75%, 85%, and 95% of the peak count-rate.[]{data-label="Fig6"}](f6.eps){width="40.00000%"}
The observations of the 93 AGPS sources are summarized in Table \[Tab1\]. This information includes the observation identification, the date of the observation, instrument (ACIS-S or HRC-I), and the exposure time. Table \[Tab1\] also lists the sources detected by MAP within $3'$ of the position and their corresponding `wavdetect` position, the offset of this position from the original AGPS position, the 95% `wavdetect` and total source position error, the background corrected net counts in the $0.3-8.0$, $0.5-2.0$ and $2.0-8.0$ keV energy ranges,[^10] and the median energy quartile ($E_{50}$) and quartile ratio ($3\times(E_{25}-0.3)/(E_{75}-0.3)$) used in quantile analysis (see Section 2.3.1).
The variability analysis performed by MAP detected only one source, ChI J170444–4109\_1, with a $\chi^{2}$ exceeding the 99.9% short term variability threshold ($\chi^{2}=80.6$). However, this variability is not real as the ACIS-S detection of ChI J170444–4109\_1 fell in the chip gap between the CCDs. This resulted in its light-curve displaying the spacecraft’s built-in dither as the source moved on and off the chip at regular intervals.[^11] No other source had a $\chi^{2}$ exceeding the 99.9% confidence threshold for short term variability or periodicity. This does not, however, rule out the possibility that some of these sources are periodic as the $Z^{2}_{1}$ period search technique is extremely limited for $<200$ X-ray counts.
Several faint sources were detected with `wavdetect` in the observations that are not suspected of being significant contributors to the counts detected in the AGPS. (All these faint sources are included in Table \[Tab1\].) In order to determine the detection significance of these sources, we independently calculated the probability of a false detection (Pfd) for each source based on the Poisson statistics. We expect there to be $\sim100000$ trials in a $3'$ radius ACIS CCD field-of-view if a detection cell size of a $1''$ radius circle is assumed with 2D Nyquist sampling [@weisskopf07]. (A generic detection cell size of $1''$ for this detection significance calculation was suggested by @weisskopf07 since the `CIAO` source detection algorithms usually use a detection cell size of between $40-80\%$ of the PSF at a given energy.[^12]) Pfd $=10^{-5}$ implies that there is one false source in the $3'$ radius ACIS field. The probability of a false detection was calculated for each source [see footnote 13 in @kashyap10]. Only 11 sources (out of 253) have a Pfd $>10^{-5}$ but all have Pfd $<7 \times 10^{-4}$. Several of these sources also have a likely optical and/or infrared counterpart, increasing the significance of these detections. As this technique for calculating source significance is highly theoretical (e.g. the effective detection cell size of ACIS is likely to be slightly smaller than the $1''$ radius circles), without any consideration for possible counterparts, we will include these 11 marginally significant sources in Table \[Tab1\] (denoted by a \* symbol in the first column).
Several of the sources detected in these observations have also been listed in the Source Catalog [CSC; @evans10]. We assume that a CSC source and a source are the same if the separation between their two positions is less than the quadratic sum of their 95% error radii plus a constant term of $0\farcs7$, which accounts for the 95% absolute astrometry error of assuming the errors follow a Gaussian distribution[^13] [see Equation (6) of @hong05]. We have conducted the same comparison with the fifth public release of the Second *XMM-Newton* Serendipitous Source Catalogue [2XMMi-DR3; @watson09][^14]. The CSC and 2XMMi-DR3 names that correspond to any sources are listed in Table \[Tab1\]. Further analysis of the observations will be conducted in future work.
The purpose of the survey is to study those X-ray sources detected in the observations that fall within the flux range of the AGPS sources ($F_{x} \sim 10^{-13}\mathrm{~to~}10^{-12}$ ). In 62 out of the 93 AGPS sources observed with , MAP found 74 sources with $>20$ X-ray counts that fall within the investigated $3'$ field-of-view that is centered on the AGPS positions. For the purposes of this study we will focus on these 74 sources, listed in Table \[Tab2\], as they are approximately within the original AGPS flux range. The detailed analysis of those 179 sources with $<20$ counts will be deferred to future work. There were 6 AGPS sources where no source was detected by : , , , , , and . At least three of these AGPS sources, AX J180816–2021, AX J185905+0333, and AX J191046+0917, may be transient given how bright they were in the original detections [@sugizaki01]. The remaining 25 AGPS fields observed with only have faint sources with $<20$ X-ray counts. There are also 6 AGPS sources, , , , , , and , where multiple sources detected in each region (with $\lesssim30$ X-ray counts) sum to $\gtrsim60$ counts, which are close to the number of X-ray counts that were expected to be detected with from each AGPS source (see Section 2.1). Many of the sources in these field also have optical and/or infrared counterparts and may therefore be members of star clusters (see Section 4.1 for further discussions).
The contribution of X-ray emission beyond the $3'$ search radius was also investigated using `wavdetect` to identify all the point sources with $>20$ X-ray counts in the observations that lie between $3'-5'$ from the original AGPS position. Only 14 X-ray point sources were found in the $3'-5'$ annulus surrounding 11 AGPS sources, which demonstrates that the $3'$ search radius used by MAP is reasonable and has likely allowed us to identify the majority of sources. These 14 X-ray point sources are further discussed in Appendix B where we explore the likelihood of whether they could be associated with their nearby AGPS source.
Quantile Analysis and Spectral Modeling Results
-----------------------------------------------
Quantile analysis and spectral modeling using Cash statistics were both techniques used to infer the most likely absorbed power-law and absorbed thermal bremsstrahlung spectrum of the bright source detected by ACIS-S.[^15] The spectral parameters and the absorbed and unabsorbed X-ray fluxes for an absorbed power law and absorbed thermal bremsstrahlung model are listed in Tables \[Tab3\] and \[Tab4\], respectively. There are also six HRC-I detected sources with $>20$ counts (ChI J144042–6001\_1, ChI J153818–5541\_1, ChI J163252–4746\_2, ChI J163751–4656\_1, ChI J165420–4337\_1, and ChI J172642–3540\_1) that are not included in Tables \[Tab3\] and \[Tab4\] as their X-ray observations do not contain any spectral information. The possible counterparts to these six sources are, however, explored in detail in Section 3.3.
There is no measure of goodness for the quantile analysis derived absorbed power-law and absorbed bremsstrahlung spectral parameters listed for the sources in Tables \[Tab3\] and \[Tab4\]. Instead a spectral interpolation is classified as unreasonable if the resulting parameters are outside $-2<\Gamma<4$ for a power-law model and $0.1< kT <10$ keV for a bremsstrahlung model. These parameter limits can allow for the grouping of the sources into existing categories that are based on the physical understanding of X-ray sources that emit in the energy range ($0.3-8$ keV). While these selection criteria explicitly excludes the identification of new types of X-ray sources with unusual spectra, such an investigation is beyond the scope of this work. In all cases if one of the quantile analysis spectral interpolations was disregarded for a given source, the other is reasonable when considering the above criteria.
As a way to further explore the goodness of the spectral interpolations, the `CIAO` spectral fitting tool `Sherpa` was used to fit the X-ray spectrum of each source with spectral parameters derived from quantile analysis in Tables \[Tab3\] and \[Tab4\]. The $\chi^{2}_{red}$ was calculated for each source where $>40$ X-ray counts was detected with ACIS-S. In many cases we found $\chi^{2}_{red} < 1$, which is not unexpected given the low number of X-ray counts detected, indicating that any reasonable model is a decent fit. However, there are also a few cases where $\chi^{2}_{red} > 2$ indicating that the spectral parameters do not adequately describe the spectrum. The values of $\chi^{2}_{red}$ corresponding to the quantile interpolated spectral parameters are included in Tables \[Tab3\] and \[Tab4\].
The parameters and fluxes resulting from the absorbed power law and absorbed thermal bremsstrahlung spectral modeling of the sources have also been included in Tables \[Tab3\] and \[Tab4\] so that they can be directly compared to the quantile analysis results. Best fit absorbed power law parameters were obtained for all the ACIS-S detected sources with $>50$ X-ray counts (see Table \[Tab3\]). However, there were several cases where the absorbed thermal bremsstrahlung fitting was unsuccessful as the `Sherpa` modeling algorithm hit the hard maximum limit on the $kT$ parameter ($10$ keV). These unsuccessful fits were not included in Table \[Tab4\].
All of the parameters and absorbed fluxes derived from the Cash statistics spectral fitting agree within $3\sigma$ of those derived from the quantile analysis spectral interpolation, the majority of which agree within the $1\sigma$ errors. The only exception is the $kT$ parameter value for ChI J171910–3652\_2. However, the unabsorbed fluxes were far less agreeable between the two techniques as there were several sources for which this value differed by $>3\sigma$. These include ChI J183356–0822\_2 and ChI J184738–0156\_1 from the power law spectral analysis and ChI J144547–5931\_1, ChI J144701–5919\_1, ChI J165646–4239\_1, ChI J171910–3652\_2, ChI J172050–3710\_1, and ChI J185608+0218\_1 from the bremsstrahlung spectral analysis. Both these techniques therefore appear to be successful in constraining the spectral parameters and absorbed fluxes for each of the investigated sources but less successful in constraining the unabsorbed fluxes. It should also be noted that the majority of the reduced Cash statistics from the spectral modeling were systematically higher than the corresponding $\chi^{2}_{red} $ derived from the spectral parameters interpolated through quantile analysis.
Infrared and Optical Counterparts
---------------------------------
Infrared and optical follow-up were primarily performed on those sources with $>20$ X-ray counts (see Table \[Tab2\]). In order to determine which optical and infrared sources are counterparts to sources, we used a similar technique to that described by @zhao05, using their Equation (11). If the separation between a source’s `wavdetect` position and its possible counterpart is less than the quadratic sum of their 3$\sigma$ position errors and the 3$\sigma$ pointing error[^16], then the X-ray and optical (or infrared) sources are likely to be associated. The 1$\sigma$ position errors for all sources in 2MASS PSC and the GLIMPSE[^17] catalogs are $0\farcs1$ [@skrutskie06] and $0\farcs3$, respectively. USNO B has an astrometric accuracy of $<0\farcs25$ [@monet03]. We have assumed that the error distributions of the observations, pointing, and USNO B Catalog are all Gaussian for the purposes of identifying possible counterparts to the sources. While this assumption is not necessarily correct in every case, the full examination required to obtain the Gaussian errors would involve a very complicated approach. However, other Galactic plane X-ray surveys that search for multi-wavelength counterparts assuming Gaussian errors for cross-correlation purposes have had successful results [e.g. ChaMPlane; @zhao05]. Based on these results we feel that Gaussian errors are an acceptable assumption for the purpose of identifying optical and infrared counterparts to the sources.
The infrared properties, such as the names and magnitudes of any likely 2MASS or GLIMPSE counterparts, together with the NIR magnitudes ($J,~H,~\mathrm{and}~K$) obtained from Magellan PANIC observations, are listed in Table \[Tab2\] along with the date of each observation. We assume that those sources with no listed 2MASS (or PANIC) counterpart have 2MASS PSC limiting magnitudes $J>15.8$, $H>15.1$, and $K>14.3$ [@skrutskie06]. In Table \[Tab5\], optical magnitudes have also been provided for those 44 sources with $>20$ X-ray counts that have optical counterparts in the USNO B1 Catalog or in one of the IMACS or MagIC Magellan observations. Two other sources (ChI J170017–4220\_1 and ChI J181213–1842\_7) with magnitude limits obtained with either IMACS or MagIC are also included in Table \[Tab5\].
Of the 74 sources with $>20$ X-ray counts listed in Table \[Tab2\], 59 have a NIR counterpart, 44 of which are 2MASS sources and 15 of which were detected in PANIC observations. Looking into the mid-infrared wavelength bands, we find that 41 of these 2MASS sources and 3 of the PANIC sources also have GLIMPSE counterparts. NIR magnitude limits were obtained for 4 other PANIC-observed sources, since any possible counterparts were too faint to be detected or, in the case of ChI J181116–1828\_2 and ChI J185643+0220\_2, the counterpart appeared to be blended. (ChI J181116–1828\_2 does, however, have a unique GLIMPSE counterpart.) If we include ChI J181116–1828\_2 and ChI J185643+0220\_2, for which we have detected a counterpart but the magnitudes are only an upper limit due to blending, then 89% of our PANIC observations of sources have yielded a detection in one of more of the $J$, $H$, or $K$ filter bands. There are also a few sources where the 2MASS counterpart magnitudes are listed as 95% confidence upper limits due to a non-detection or inconsistent deblending. We were therefore able to use PANIC to obtain more accurate magnitudes for 4 sources for which have limited 2MASS magnitude information in one or more bands. (These 4 sources can be identified as those with three letters listed in the “Data” column of Table \[Tab2\].)
All of the 46 sources listed in Table \[Tab5\] (44 sources with optical counterparts and 2 with limiting magnitudes obtained with Magellan instruments) have NIR counterparts detected with either 2MASS or PANIC. Of the 44 with optical counterparts, 41 have USNO B1 counterparts, for 4 of which extra magnitude measurements were obtained with one of the two Magellan optical imagers. A further 3 sources, uncataloged in USNO B1, were also detected in the optical with these Magellan instruments. Of the 74 sources with $>20$ X-ray counts, 14 do not have a detected optical, NIR or IR counterpart.
We conducted an experiment similar to that outlined in @kaplan04 to quantify the probability that the optical and infrared survey counterparts quoted in Tables \[Tab2\] and \[Tab5\], are a random chance association with the $>20$ X-ray count source of which they are coincident. We searched for all 2MASS, GLIMPSE and USNO B1 sources brighter than the possible counterpart listed in Tables \[Tab2\] and \[Tab5\] within $10'$ of each source position. (We only searched for survey sources that have a brighter $K_{s}$, $3.6 \mu$m, and second epoch $R$ band magnitude for the 2MASS, GLIMPSE and USNO B1 surveys, respectively.) We then used the resulting statistics to determine the number of survey sources brighter than the listed counterpart that are likely to be detected within a region the same size as the source’s 95% position error circle. We did this for each source individually as the density of sources can vary dramatically across the Galactic plane. In most cases the resulting chance of a random association is very low ($<0.01$). Those sources that have a random chance of association $>0.01$ in either 2MASS or GLIMPSE are listed in Table \[Tab6\]. For each source the chance of a random association with a USNO B1 is $<0.01$.
We refer to the sources that have $<20$ counts as “secondary” sources. In Table \[Tab7\] we list the names of any USNO B1, 2MASS and GLIMPSE sources that appear to be coincident with a secondary source based on our position agreement criteria outlined in Section 2.4.1. This Table also includes the offset in arcseconds between the secondary source’s `wavdetect` position and the position of the coincident survey source. (Only those secondary sources that have a coincident survey source have been included in Table \[Tab7\].) A summary of the fraction of sources with a coincident source in the 2MASS, GLIMPSE and USNO B1 catalogs, can be found in Table \[Tab8\]. This Table includes the fraction of the total number of sources, as well the fraction of just the $>20$ X-ray count sources, with a coincident survey source. (Note that we assume the coincident survey sources to the $>20$ X-ray count sources are counterparts based on the very low chance of random associations as demonstrated by Table \[Tab6\].)
Radio Counterparts
------------------
As mentioned in Section 2.4.2, the position of each source (above and below $20$ X-ray counts) was visually inspected for any possibly associated radio emission in the MGPS, MAGPIS, VGPS and the 90cm Multi-configuration Very Large Array Survey of the Galactic Plane. The results of this inspection are listed in Table \[Tab9\]. source position comparisons were also made with the @green09 catalog of Galactic supernova remnants. Any possibly counterparts that are known objects, such as supernova remnants (SNRs), H [ii]{} regions, infrared dark clouds [IRDCs; @peretto09], colliding-wind binaries (CWBs) or massive stars, are listed in Table \[Tab9\] under “type”. However, if the radio sources are uncataloged, they have instead been flagged as being possibly compact, diffuse, or arc/shell structured diffuse emission. Two sources, ChI J181116–1828\_5 and ChI J184741–0219\_3, appear to be previously unidentified AGN as their coincident radio sources show core-lobe morphologies in the MAGPIS 1.4 GHz survey images.
Each of the sources with a possible radio association is then listed in Table \[Tab9\] as being either coincident to, adjacent to, or on the limb of the radio source (such as on the limb of a SNR, diffuse emission or H [ii]{} region). If a source is listed as either coincident or on the limb of a SNR then this means it is within the extent of the SNR based on the SNR’s size quoted in the @green09 catalog. In the cases of the two candidate AGN, these sources appear to be directly coincident with the core of the AGN. The name of each radio source and the corresponding reference are listed in Table \[Tab9\].
The 10 sources observed with the ATCA are also included in Table \[Tab9\]. The ATCA-detected compact radio counterparts to both ChI J144701–5919\_1 and ChI J163252–4746\_2, aided in their identification as X-ray emitting massive stars or CWBs [see @anderson11]. No radio counterparts were detected for the other 8 sources observed with the ATCA.
In summary, Table \[Tab9\] shows there to be 16 sources, from the observations of 8 different AGPS source regions, that are coincident, or on the limb, of 9 SNRs. There are 54 sources, from 7 different AGPS source regions, that fall within the extent of 6 H [ii]{} regions. There are also 4 massive stars, all of which are confirmed or candidate CWBs [see @anderson11; @motch10], with radio counterparts. Only two sources, both toward the same single AGPS target, are coincident with an IRDC.
Several sources also fall within regions of uncataloged extended radio emission. These include 15 sources, from 5 different AGPS source regions, falling within the extent of 5 regions of diffuse radio emission. Of these 15 sources, there are 6 (from 3 different AGPS source regions) that are coincident with uncataloged diffuse emission with an arc or shell structure. Excluding the two AGN candidates, there is only one other source coincident with an unidentified compact radio source.
There are 74 sources (14 sources with $>20$ X-ray counts and 60 with $<20$ X-ray counts), out of the 253 detected, with no optical or infrared counterparts, making them possible compact object candidates and therefore potentially detectable in the radio. We therefore searched for any possible pulsar counterparts in the Australia Telescope National Facility Pulsar Catalogue [Version 1.44[^18] @manchester05] but no known pulsars exist within $0\farcm6$ of the `wavdetect` position of any of the 74 sources.
Discussion
==========
Unidentified Sources with Radio Counterparts
---------------------------------------------
Table \[Tab9\] lists 16 sources that fall within the extent of 9 SNRs. X-ray point sources within SNRs could be associated compact objects. Identification of an optical or infrared counterpart discounts such a possibility, since the optical/IR counterparts to neutron stars and other compact objects are expected to be very faint [for the details on this approach see @kaplan04]. Those SNR coincident sources with a random chance of association $>0.01$ in the USNO B1 and GLIMPSE catalogs are listed in Table \[Tab10\]. (The chance of a random association between one of these 16 sources and a 2MASS catalog source is $<0.01$.)
Of the 16 sources inside SNRs, only 5 have no optical or infrared counterparts. These 5 X-ray sources are all very faint, with $<8$ X-ray counts detected for each in the observations. These are ChI J145732–5901\_2 in SNR G318.2+0.1 [@whiteoak96], ChI J182435–1311\_2,3,4 in SNR G18.1–0.1 [@helfand06; @brogan06] and ChI J184447–0305\_1 in SNR G29.3667+0.1000 [@helfand06]. @bocchino01 has already reported on three X-ray sources within SNR G318.2+0.1 but not at the position of ChI J145732–5901\_2. Both SNR G18.1–0.1 and SNR G29.3667+0.1000 are newly discovered SNRs so little X-ray analysis has been done on these objects. Further investigation is required to determine if any of these 5 sources are compact objects and if they are associated with the surrounding SNRs.
There are 54 sources coincident with 6 different H [ii]{} regions that were detected in the observations of 7 AGPS sources (see Table \[Tab9\]). Based on the results from X-ray observations of other H [ii]{} regions [for example see @broos07] it is possible that many of these 54 sources could be pre-main sequence (PMS) stars, massive OB and WR stars, and CWBs. @simpson04 also found optically obscured star clusters in 2MASS images within $1'$ of 3 of these AGPS sources; AX J144519–5949, AX J151005–5824, and AX J162208–5005, which supports a H [ii]{} region and young and massive star interpretation for these sources. Of these 54 sources, 43 have optical and/or infrared counterparts supporting a possible stellar origin (see Tables \[Tab2\], \[Tab5\] and \[Tab7\]). In most cases the chance of random association with a field source is low ($<0.01$). Those H [ii]{} region coincident sources for which the chance of random association with a 2MASS or GLIMPSE source is $>0.01$ are listed in Table \[Tab11\].
The possible nature of these H [ii]{} region coincident sources could be further investigated by comparing their luminosities to those of PMS stars, massive OB and WR stars, and CWBs. A spectral analysis of the 54 sources is extremely difficult given that they all have $\le32$ X-ray counts. However, the primary goal is to obtain a wide-band flux that can then be converted into a luminosity. Quantile analysis was therefore performed to obtain absorbed Mewe-Kaastra-Liedahl [Mekal; @mewe85; @mewe86; @kaastra92; @liedahl95] spectral interpolations of all the H [ii]{} region coincident sources with $\ge5$ X-ray counts. A Mekal model was chosen as thin thermal plasma emission is expected from hot X-ray emitting stars [for example see @wolk05; @sana06]. The absorbed Mekal spectral interpolations can be found in Table \[Tab11\].
The luminosity of these sources were calculated using kinematic distance estimates to the H [ii]{} regions of which they are coincident. Kinematic distances calculated by @russeil03 were used to calculate luminosities for the sources coincident with G320.3–0.3, G333.6–0.2, and G59.5–0.2. (Note that there is a more distant kinematic distance estimate of 6.3 kpc to G59.5–0.2 calculated by @kuchar94 but we have decided to use the more recent estimate from @russeil03.) The kinematic distances used in the luminosity calculations for G326.96+0.03 and G62.9+0.1 were obtained from @mcclure01 and @fich84, respectively. The kinematic distance to the massive young stellar object G316.8112–00.0566 [@busfield06], likely embedded within GAL 316.8–00.1, was used in the luminosity calculations for those sources coincident with this H [ii]{} region. (The distance from @busfield06 agrees reasonably well with the near kinematic distance to GAL 316.8–00.1 calculated by @caswell87 when revised for a modern Galactic center distance of 8.5 kpc.) Table \[Tab11\] lists the kinematic distance and corresponding absorbed and unabsorbed luminosities calculated for each H [ii]{} region coincident source.
The range of absorbed luminosities calculated for all the sources coincident with the 6 H [ii]{} regions span the range $30.6 \mathrm{~erg s}^{-1} <\mathrm{Log~}L_{x,abs}<32.4 \mathrm{~erg s}^{-1}$ ($0.3-8$ keV). This absorbed luminosity range is similar to that observed from the H [ii]{} region M17 [$29.3 \mathrm{~erg s}^{-1} <\mathrm{Log~}L_{x,abs}<32.8 \mathrm{~erg s}^{-1}$ ($0.5-8$ keV); @broos07]. With the exception of ChI J151005–5824\_8, the H [ii]{} region coincident sources in Table \[Tab11\] have unabsorbed luminosities between $L_{x,unab} \sim 10^{31} \mathrm{~to~} 10^{35}$ erg s$^{-1}$. The unabsorbed luminosities of these sources cover the ranges of what has been observed from flaring PMS stars [$L_{x,unab} \sim 10^{30} \mathrm{~to~} 10^{33}$ erg s$^{-1}$; @favata05; @wolk05], single and binary massive O type stars [$L_{x,unab} \sim 10^{31} \mathrm{~to~} 10^{33}$ erg s$^{-1}$; @oskinova05; @sana06], WR stars [$L_{x,unab} \sim 10^{31} \mathrm{~to~} 10^{34}$ erg s$^{-1}$; @oskinova05; @mauerhan10], and CWBs [$L_{x,unab} \sim 10^{32} \mathrm{~to~} 10^{34}$ erg s$^{-1}$; @oskinova05; @mauerhan10]. In fact, we were able to identify ChI J194310+2318\_5, which is within G59.5–0.2, as the O7V((f)) type star [@walborn73] using the SIMBAD Astronomical Database. It is therefore likely that the AGPS sources AX J144519–5949, AX J151005–5824, AX J154905–5420, AX J162208–5005, AX J194310+2318, AX J194332+2323, and AX J195006+2628 are young and massive stars within H [ii]{} regions. Deeper X-ray, radio and IR observations are required to determine the precise nature of the individual sources.
Unidentified Sources with Infrared Counterparts
------------------------------------------------
The X-ray and infrared population statistics performed in this section are just conducted using those sources with $>20$ X-ray counts, therefore concentrating on the persistent populations that fall within the AGPS flux range ($F_{x} \sim 10^{-13} \mathrm{~to~} 10^{-11}$ ). However, discarding the sources with $<20$ X-ray counts limits our analysis as we are excluding the populations of sources that exhibit long term variability or transient behavior. Sources with $<20$ X-ray counts will need to be investigated in future work using archival X-ray observations at different epochs.
### X-ray and Infrared Populations Statistics
Using the X-ray and infrared properties of the unidentified sources, it is possible to classify some of the sources detected in the AGPS into possible populations. We chose to focus on the IR counterparts as this waveband is less affected by interstellar extinction when compared to the optical band. This group of sources therefore makes up a larger subset of the unidentified sources than those with optical counterparts (see Section 3.3). For those unidentified sources observed with the HRC instrument (for which there is no spectral information), we generated fake spectra, using `XSpec` and the `CIAO` spectral fitting tool `Sherpa`. These spectra are based on the absorbed power-law fits reported in @sugizaki01, allowing the absorbed X-ray flux and median energy ($E_{50}$) of the unidentified source in question to be calculated. These values are used in the statistical plots described below.
To help identify possible distinct populations in the statistical plots, we have also included both archival sources [those AGPS source identified by @sugizaki01 or in the literature and so were not observed with as part of the survey] and the previously identified sources [for example those investigated by @anderson11; @anderson12]. Fake spectra were generated using `Sherpa` and `XSpec` for these archival sources, using spectral fits in the literature, to determine their absorbed X-ray fluxes and $E_{50}$ values for the energy ranges investigated. All archival AGPS sources are summarized and tabulated in Section 4.5 (see Table \[Tab12\]) and are individually described in Appendix A.
The identified sources were divided into the following categories: AGN, CVs, CWBs, HMXBs, magnetars, massive stars and stars. The “AGN” category includes ChI J184741–0219\_3, which we identified by positional comparison with MAGPIS radio data (for the radio identification and further details on this source see Sections 3.4 and 4.3.15, respectively). The “CWB” category includes AX J163252-4746 and AX J184738-0156, which were identified in @anderson11 [listed as ChI J163252–4746\_2 and ChI J184738–0156\_1 in Table \[Tab1\], respectively], and and , which were identified in the XGPS [@motch10 listed as ChI J183116-1008\_1 and ChI J183206-0938\_1 in Table \[Tab1\], respectively]. The “HMXB” category includes the archival AGPS sources that are supergiant HMXBs [@mcclintock06], supergiant fast X-ray transients [SFXTs; @sguera06], and SyXBs [@masetti07]. The infrared and X-ray fluxes from magnetars are variable and correlated [@durant05] so the fluxes we used in the “magnetar” category are from infrared and X-ray observations that occurred close together in time. PSR J1622–4950, which was determined through the observation to be the main contributor to AX J162246–4946 has also been included as an identified magnetar [see @anderson12 and Section 4.3.6 for further details]. The sources included in the “massive star” category are massive stars that are Wolf-Rayet (WR), luminous blue variable (LBV) stars, and massive O-type stars, which emit X-rays through instability-driven wind-shocks [@lucy80; @lucy82] and possibly through colliding-winds in a CWB. These include and AX J144701-5919, which were identified by @anderson11 [listed as ChI J144547-5931\_1 and ChI J144701-5919\_1 in Table \[Tab1\], respectively]. All other non-degenerate stars are in the “star” category, and most likely correspond to active stellar coronae or PMS stars.
We first investigated the relationship between the X-ray and infrared flux of the sources by comparing these properties to those of known stars and AGN. Figure 5 of @gelfand07 shows the X-ray versus $K_{s}$-band flux of sources from the Orion Ultradeep Project [COUP; @getman05] and XBootes Survey [@jannuzi04; @kenter05], which are stars (predominantly in the PMS) and AGN, respectively. In Figure \[Fig7\] we create a similar plot that includes the unidentified sources (U ChIcAGO; red data points) along with the COUP stars (blue crosses) and the XBootes Survey AGN (magenta diamonds) [for further details on the data from these surveys see @gelfand07 and references therein]. The X-ray flux ($F_{x,2-7keV}$) is over the $2.0-7.0$ keV energy range and the $K_{s}$-band flux ($F_{Ks} = \lambda F_{\lambda,Ks}$) is derived from $F_{\lambda,Ks}$ ($\mu$m$^{-1}$) where the effective wavelength is $\lambda=2.159$ $\mu$m. The archival sources and the identified sources with $K$-band counterparts have also been included in Figure \[Fig7\] in order to further distinguish between possible X-ray populations.
{width="80.00000%"}
Two groups and two outliers are apparent in Figure \[Fig7\] among the unidentified sources: group 1, those coincident with the COUP stars, and group 2 that sits to the right of the AGN detected in the XBootes survey. The two outliers are ChI J154557-5443\_3, on the very left of this Figure with a hard X-ray flux limit of $F_{x,2-7keV} \lesssim 3 \times 10^{-19}$ ,[^19] and the source on the bottom right, ChI J153818-5541\_1, with an X-ray flux of $F_{x,2-7keV} \sim 1 \times 10^{-11}$ , located near the identified magnetars.
Group 1 is distributed similarly to the COUP stars, following a “track” indicating an increase in $K$-band flux with X-ray flux. This comparison demonstrates that a large number of the stellar population could be PMS stars. While the overall Galactic X-ray population is dominated by field stars, we would expect that the ChIcAGO survey (and the AGPS) is bias towards PMS stars as such objects are brighter and harder X-ray emitters [for example see @wolk05]. The identified and archival sources that have been categorized as massive stars (light pink dots) and CWBs (green data points) congregate near the top right of group 1, beyond most of the COUP stars. These massive stars and CWBs are composed of massive late-type WR stars in the nitrogen sequence that are hydrogen rich [WNH: @smith08], or their massive O star progenitors [Of: @crowther95], all of which are expected to be bright in the infrared and potentially harder in X-rays than other types of X-ray emitting stars (yellow data points). [See @anderson11 for further details on these WR and massive O stars.] It is therefore possible that the unidentified sources located near these massive stars and CWBs in Figure \[Fig7\] could be similar objects. It should also be noted that the identified AGPS HMXBs sit to the right of group 1, with no unidentified sources near their positions to draw comparison with.
Group 2 sits at a similar $K_{s}$-band flux but higher X-ray flux than the AGN from the XBootes survey. The identified and archival AGN are coincident with the unidentified sources in group 2, indicating that at least part of this group could also be AGN. Such AGN would have to be very X-ray bright in order to be detected through the high foreground column density in the Galactic plane. The identified archival CVs sit adjacent to group 2, at a slightly higher X-ray flux, indicating another possible population identification.
The unidentified source ChI J154557-5443\_3, has a similar $K$-band flux to the COUP stars but has an extremely faint hard X-ray flux. No identified or archival sources are located near ChI J154557-5443\_3 in Figure \[Fig7\] that suggest a clue to its nature. ChI J154557-5443\_3 is discussed further in Section 4.3.4.
The unidentified source that sits at the bottom right of Figure \[Fig7\], ChI J153818-5541\_1, is very faint in the near-infrared but bright in high energy X-rays. It is therefore very similar in both X-ray and $K_{s}$-band flux to the identified archival magnetars, suggesting a similar identification that warrants further investigation. ChI J153818-5541\_1 is further discussed in Section 4.3.3.
In order to further identify the possible populations that make up the above described groups, we created a statistical plot that also takes into account the hardness of each of the unidentified sources. Figure \[Fig8\] plots the X-ray-to-$K_{s}$-band flux ratio ($F_{x,0.3-8.0 \mathrm{keV}} / F_{Ks}$) versus the median energy ($E_{50}$ keV), in the $0.3-8.0$ keV energy band, of each unidentified source (“U ;” red data points). We further separated the sources into the categories “low”, “medium” and “high”, based on their ACIS-S and HRC-I X-ray count rates, using symbol sizes to distinguish these categories. The smallest data points (low) have count rates $<15$ counts s$^{-1}$, the medium sized data points (medium) have count rates between 15 and 40 counts s$^{-1}$, and the largest data points (high) have count rates $>40$ counts s$^{-1}$. The identified and archival sources, also divided into categories based on the X-ray count-rate expected from a ACIS-S observation, have also been included in Figure \[Fig8\]. As the X-ray fluxes and median energy errors are fundamentally based on the number of X-ray counts detected in the observations, we show two representative error bars for the low count sources ($20-60$ counts) and the high count sources ($80-120$ counts). The error associated with the $K_{s}$-band flux is greater for PANIC magnitudes than for 2MASS magnitudes. We therefore adopt the PANIC $K_{s}$-band flux errors so that the vertical error bars represent the maximum possible error in the $F_{x}/F_{Ks}$ ratio for low- and high-count unidentified sources.
{width="80.00000%"}
In Figure \[Fig8\], group 1 plus the source ChI J154557-5443\_3 from Figure \[Fig7\] have flux ratios $F_{x}/ F_{Ks} < 0.1$ but have a wide range of median energies. Group 2 is harder than group 1, with $E_{50} > 1.5$ keV but with a flux ratio $0.1 < F_{x} / F_{Ks} < 20$. The unidentified source ChI J153818-5541\_1, from Figure \[Fig7\] is quite hard ($E_{50} > 3$ keV), with $F_{x} / F_{Ks} \approx 300$. Using Figure \[Fig7\] as a guide, as well as the relative positions between the unidentified sources and the identified and archival sources in Figure \[Fig8\], we divided the groups and isolated sources from Figure \[Fig7\] into six different population regions. These regions are marked by dashed lines and labeled with Roman numerals in Figure \[Fig8\]. The region boundaries in this plot are based on the observed X-ray properties and $K_{s}$ band flux. If these values were extinction and absorption corrected, both the $F_{x} / F_{Ks}$ and $E_{50}$ region boundaries would lower.
Region i ($E_{50} < 1.5$ keV and $F_{x} / F_{Ks} < 0.1$) in Figure \[Fig8\] contains unidentified sources with low, medium and high count rates. The majority of the low count rate unidentified sources included in this Figure fall into this region. These low count rate sources could be either very nearby objects that were only detected due to their close proximity to the Solar System or they are simply a more distant version of the medium and high count rate sources in Region i. All the unidentified sources in this region are unlikely to be extragalactic given their softer X-ray spectra and bright $K_{s}$-band counterparts compared to the XBootes AGN in Figure \[Fig7\]. Within this region are three of the archival X-ray stars (yellow data points), of which there is an RS CVn star, a PMS star and a multiple system. The unidentified sources in Region i are therefore likely to be soft X-ray stars with active stellar coronae or PMS stars, similar to the three archival stars, but at a variety of distances in the Galaxy.
The sources in Region ii ($E_{50} > 1.5$ keV and $F_{x} / F_{Ks} < 0.1$) of Figure \[Fig8\] have similar $F_{x}/F_{Ks}$ ratios, but slightly harder X-ray emission, to those in Region i. This region also has fewer unidentified sources than Region i, implying that it may contain a slightly rarer X-ray source population. The majority of the unidentified sources in Region ii have low or medium count-rates, with only one in the high count rate category. The identified and archival sources in this region are CWBs (green data points) and massive stars (pink dots), all of which sit at the top of the stellar track in Figure \[Fig7\]. We defined $E_{50} > 1.5$ keV as the lower energy cutoff of Region ii by looking at the closest unidentified sources to the CWBs and massive stars in Figure \[Fig7\]. The CWBs and massive stars are WNH and Of stars, which produce X-rays through instability-driven wind-shocks but can also, in the case of CWBs, produce hard X-rays due to colliding-winds [for example see @anderson11]. It is therefore likely that many of the unidentified sources in Region ii are WR and Of stars, some of which may also be CWBs.
Region iii ($E_{50} > 4.0$ keV and $1 \times 10^{-3} < F_{x} / F_{Ks} < 20$) in Figure \[Fig8\] encompasses the archival HMXBs (grey dots) and the archival and identified AGN (cyan dots). The only two unidentified sources in this region are ChI J170017-4220\_1 (high) and ChI J172550-3533\_1 (low), and are therefore quite hard X-ray sources, making HMXB or AGN identifications a strong possibility.
Region iv ($1.5 < E_{50} < 4.0$ keV and $0.1 < F_{x} / F_{Ks} < 10$) in Figure \[Fig8\] contains 12 medium and high count rate unidentified sources. However, there are no identified or archival sources in this region that could indicate any likely source populations. The best clue comes from Figure \[Fig7\], which shows that these unidentified sources are in the same region of this plot as the identified and archival AGN. In the Galactic plane, the relation of X-ray sources in the $2.0-10.0$ keV energy range [see Figure 15 of @hands04], demonstrates that within the X-ray flux range $1 \times 10^{-13} < F_{x} < 2 \times 10^{-12}$ of Region iv, between 0.06 and 8 extragalactic sources are expected per square degree. These number densities are consistent with more recent modeling conducted by @mateos08 who used 1129 observations at $|b| > 20^{\circ}$ to demonstrate that sources in the $2-10$ keV energy range, at high Galactic latitudes, agree with AGN models to better than 10%. As the sources in Region iv have a number density $< 8$ deg$^{-2}$ it is not unreasonable to speculate that the many of the unidentified sources in this region could be AGN. However, we compared the $N_{H}$ values of the Region iv sources calculated from the power law quantile analysis and spectral fits in Table \[Tab3\] to the Galactic column densities in their direction from surveys conducted by @kalberla05 and @dickey90.[^20] In all by four cases the sources have $N_{H}$ values an order of a magnitude lower than the Galactic $N_{H}$, suggesting a possible Galactic origin. The sources ChI J181116–1828\_2, ChI J181213–1842\_7, ChI J190749+0803\_1, and ChI J194152+2251\_2 have $N_{H}$ values of the same order as the Galactic column density [$N_{H} = 1.2 \times 10^{22}, 1.3 \times 10^{22}, 1.5\times10^{22}, \rm{~and~} 1.1\times10^{22} \rm{~cm}^{-2}$, respectively, @kalberla05], which is more indicative of an extragalacitc origin and therefore an AGN identification. Further investigation is required to confirm the nature of the Region iv population.
Region v ($1.5 < E_{50} < 4.0$ keV and $10 < F_{x} / F_{Ks} < 50$) of Figure \[Fig8\] encompasses the two identified archival CVs and the lower $F_{x} / F_{Ks}$ ratio limit of unidentified source ChI J181852-1559\_2. As its flux ratio is only a lower limit (corresponding to an upper-limit on the $K$-band flux), it is possible that ChI J181852-1559\_2 may instead be a member of Region vi (described below).
Region vi ($1.3 < E_{50} < 4.0$ keV and $F_{x} / F_{Ks} > 1 \times 10^{2}$) contains four identified archival magnetars. The only unidentified source that sits within this region is ChI J153818-5541\_1. Based on its proximity to these magnetars, it is possible that ChI J153818-5541\_1 could also be a magnetar, however, its X-ray emission is much harder in comparison. Regardless, the position of ChI J153818-5541\_1 in Figure \[Fig8\] indicates that this source is definitely worthy of further study.
### Infrared Population Statistics
In order to further refine the possible populations within Figure \[Fig8\], we investigated the infrared colors of the unidentified sources, once again drawing comparisons with the identified and archival sources. As mentioned above, there is strong evidence in Figure \[Fig8\] that some of the unidentified sources in Region ii could be massive stars such as WR and Of stars, and perhaps even CWBs. It is also possible that many of the sources, particularly in Region i, are PMS stars given their similar X-ray-to-infrared flux ratio to the COUP stars in Figure \[Fig8\]. However, for the purposes of this paper we have chosen to concentrate on the selection criteria for the hard X-ray emitting massive stars. We therefore leave the investigation of the PMS star population in the survey for future work [for infrared selection criteria for PMS stars see @favata05; @maercker05; @maercker06].
@hadfield07 created a selection criterion for WR stars using GLIMPSE and 2MASS magnitudes, which was further refined by @mauerhan11. Figures \[Fig9a\] and \[Fig9b\] are recreations of the @hadfield07 \[3.6\]-\[4.5\] vs \[3.6\]-\[8.0\] and $J-K_{s}$ vs $K_{s} -$ \[8.0\] color-color plots, showing the unidentified sources. (The numbers in brackets correspond to the effective wavelength in $\mu$m of the GLIMPSE magnitude bands.) The dashed lines in Figures \[Fig9a\] and \[Fig9b\] indicate the color space used by @mauerhan11 to select WR candidates. These color spaces indicate where WR stars are expected to fall, in comparison to field stars, due to the infrared excess of WR stars resulting from free-free emission generated in their strong, dense stellar winds [see @mauerhan11 and references therein].
In Figures \[Fig9a\] and \[Fig9b\], the unidentified sources (UChI) have been separated into different colored symbols based on whether they are in Region i (red), Region ii (blue), or Region iii (black). In both these Figures, the majority of unidentified sources, particularly from Region i, do not have an infrared excess so cluster together near the origin where @hadfield07 state the general stellar locus is located. These unidentified sources are therefore unlikely to have strong stellar winds, so are more likely to have active stellar coronae. There are, however, several Region ii sources, and three Region i sources, that have more unusual colors, falling within, very close or below the indicated WR color spaces. (It should be noted that many of these unusually colored unidentified sources only have magnitude lower limits in one or more of the filter bands, making their positions in these color-color diagrams uncertain. See Figures \[Fig9a\] and \[Fig9b\] for details on these magnitude limited sources.)
Rather than keeping the original identified source categories we have instead separated the identified and archival massive stars and CWBs into categories based on their dominant stellar components. These categories are WR and LBVs (green data points), massive O type stars (pink data points), and all other non-degenerate stars (yellow data points). The HMXBs are still indicated by grey data points.
The identified WR stars fall within one or both of the @mauerhan11 WR color spaces in Figures \[Fig9a\] and \[Fig9b\]. Based on their positions within one or both WR color spaces, it is possible that the unidentified Region ii sources ChI J181915–1601\_2, ChI J180857–2004\_2 and ChI J183345–0828\_1 may also be WR stars. In fact, using this color space technique @hadfield07 identified the 2MASS counterpart of ChI J181915–1601\_2, , as a WR star and further classified it as a WN7o star using spectroscopy. There are also three Region i unidentified sources, ChI J154557–5443\_1, ChI J154557–5443\_3, and ChI J172550–3533\_2, whose magnitude limits fall within the WR color space of Figure \[Fig9b\] and below the WR color space of Figure \[Fig9a\], which may also be worth further investigation.
The identified source AX J144547-5931 [an Of type star, which is also shown as a pink dot in Figures \[Fig7\] and \[Fig8\], see @anderson11] sits to the left of both WR color spaces in Figures \[Fig9a\] and \[Fig9b\]. The unidentified Region ii sources ChI J182435–1311\_1, ChI J182651–1206\_4, ChI J183356–0822\_3, and ChI J184652–0240\_1 are also located in this same vicinity as AX J144547–5931 in one or both Figures (within $0.2<[3.6]-[8.0]<0.5$ in Figure \[Fig9a\] and within $0.5<K_{s}-[8.0]<1.2$ in Figure \[Fig9b\]). While these four unidentified sources have unusual infrared colors, they are not as extreme as those of typical WR stars. Given their proximity in Figure \[Fig9\] to the Of star AX J144547-5931, it is possible that these unidentified sources could be similar massive O-type stars. The remaining Region ii sources lie within the general stellar locus of Figures \[Fig9a\] and \[Fig9b\], so could be massive stars, active stellar coronae, or PMS stars.
The archival HMXBs also have unusual infrared colors as they fall within or above the top edge of the WR color spaces in Figures \[Fig9a\] and \[Fig9b\]. These infrared colors may be due to intrinsic absorption [for example see @rodriguez06]. Several unidentified Region ii sources also lie close to the HMXBs in these Figures, but as they have much softer median X-ray energies (see Figure \[Fig8\]), such an identification is unlikely. ChI J170017–4220\_1 is the only Region iii source with sufficient infrared magnitude information to be displayed in Figures \[Fig9a\] and \[Fig9b\]. This unidentified source falls within the vicinity of the HMXBs in Figure \[Fig8\] but has much more extreme infrared colors, and lies above the WR color space in both color-color plots. Further investigation is required to confirm the X-ray binary nature of ChI J170017–4220\_1.
None of the 11 unidentified Region iv sources have been detected at 8 $\mu$m, with only three, ChI J163751–4656\_1, ChI J165420–4337\_1, and ChI J190818+0745\_1, detected with GLIMPSE at 3.6 and 4.5 $\mu$m, all of which are likely to be Galactic in origin (see Section 4.2.1). In Figure \[Fig9b\] we have plotted these three unidentified sources using their 4.5 $\mu$m magnitude in place of 8 $\mu$m magnitude. Only ChI J165420-4337\_1 falls within the WR color space. Given the extreme uncertainties associated with the $K_{s} - [8.0]$ color of ChI J165420-4337\_1, no conclusions can be drawn about its possible nature.
Further Discussion of Interesting and AGPS Sources
---------------------------------------------------
We have flagged several of the unidentified sources as interesting and worthy of future follow-up based on their X-ray and infrared properties explored in Section 4.2. These population statistics have also allowed us to make some tentative identifications. The details of these sources are described below. There are several unidentified sources that are not detailed in this Section but are listed, along with their tentative identification based on the population statistics, in Table \[Tab12\].
### ChI J144519–5949\_2
ChI J144519–5949\_2 is one of the six sources detected within the H [ii]{} region GAL 316.8–00.1 (see Table \[Tab9\]). As it does not have a GLIMPSE counterpart, it is not included in Figures \[Fig9a\] and \[Fig9b\]. However, its 2MASS counterpart, 2MASS 14452143-5949251, has unusual colors and falls within the WR color space of the $J-H$ vs $H-K_{s}$ plot in Figure 1 of @mauerhan11. We therefore tentatively identify ChI J144519–5949\_2 as a candidate WR star. Given the location of ChI J144519–5949\_2 in Region ii of Figure \[Fig8\], particularly near two identified CWBs, it is very likely that this star has very strong winds that are generating X-rays through instability-driven wind-shocks. This object could even possibly be a CWB [for example see @anderson11].
### ChI J150436-5824\_1
Several unidentified AGPS sources have been investigated by @degenaar12 using the X-ray telescope (XRT) on the *Swift* satellite [@gehrels04]. @degenaar12 suggest that ChI J150436-5824\_1 (which they publish under the AGPS name ) could possibly be associated with a main sequence star that falls within the 90% XRT error circle. However, the observation of ChI J150436-5824\_1 demonstrates that there is no cataloged counterpart within the $<1''$ position error circle of this X-ray source, arguing against this tentative identification.
### ChI J153818–5541\_1
In Figure \[Fig7\], ChI J153818–5541\_1 has properties that are clearly not consistent with the stellar or AGN populations. It sits near the bottom right-hand corner of Figure \[Fig7\], having similar X-ray and infrared fluxes to the archival magnetars (black data points). ChI J153818–5541\_1 also falls within Region vi in Figure \[Fig8\] along with these same magnetars, further indicating a possible magnetar nature. However, during the writing of this paper, recent studies with *Swift* have indicated that ChI J153818–5541\_1 (published under its AGPS name ) may in fact be a low mass X-ray binary [LMXB; @degenaar12]. If this is the case, then X-ray and infrared statistical analysis is not a foolproof diagnostic. That being said, this same statistical analysis has allowed us to show that interesting and unusual Galactic X-ray sources fall within Region vi of Figure \[Fig8\]. (It should be noted that a magnetar identification for this source has not been completely ruled out.)
### ChI J154557–5443\_3
ChI J154557–5443\_3 is a soft X-ray source as demonstrated by its isolated position in Figure \[Fig7\]. Only 2 X-ray counts, from a total of 22 detected with , had an energy $>2$ keV, but otherwise the source’s median energy ($E_{50} = 0.8$ keV) is consistent with the other sources in Region i of Figure \[Fig8\]. The X-ray-to-optical flux ratio for this source is log\[$F_{(x,0.3-8\mathrm{keV})}/F_{R}]=-2.5$ (where $F_{R}=\lambda F_{R,\lambda}$ using the $R$ magnitude band), which is consistent with what is expected from stars and normal galaxies [@mainieri02]. An unusual property of this source is its position in WR color space in Figure \[Fig9b\], but this may result from using its $5.8$ $\mu$m GLIMPSE magnitude in place of a lacking $8.0~\mu$m magnitude. An optical or infrared spectrum would likely reveal the identity of this source. ChI J154557–5443\_3 is therefore unlikely to be related to as only detected hard counts ($>2$ keV) from this AGPS source.
### ChI J162011–5002\_1
@degenaar12 tentatively identified ChI J162011–5002\_1 (published under its AGPS name AX J1620.1–5002) as a candidate accreting magnetized white dwarf based on its hard X-ray spectrum described by a flat power law, and lack of any cataloged optical/infrared counterpart. The power-law index obtained using quantile spectral interpolation (see Table \[Tab3\]) agrees with the result from @degenaar12 and is consistent with the spectra of magnetically accreting white dwarfs [$\Gamma < 1$; @muno03]. However, our analysis resulted in a higher column density than the value calculated by @degenaar12 [$N_{H} \lesssim 3 \times 10^{21}$ cm$^{-2}$]. Deep imaging in the $H$-band with PANIC resulted in the detection of a faint counterpart ($H=17.01 \pm 0.14$). Further follow-up is required to confirm this classification.
### AX J162246–4946
The newly discovered radio and X-ray emitting magnetar, PSR J1622–4950 [also known as CXOU J162244.8–495054, @evans10; @levin10], was not cataloged by MAP. This is because it lies $4'$ from the position of AX J162246–4946, and is therefore outside the $3'$ radius for which MAP searches for X-ray point sources. However, PSR J1622–4950 was detected in this observation and further investigation in @anderson12 demonstrates that this magnetar may have contributed up to $75\%$ of the X-ray emission originally detected by from AX J162246–4946. We therefore identify AX J162246–4946 as PSR J1622–4950, and this source has been included as an identified magnetar in Figures \[Fig7\] and \[Fig8\]. (For further discussion on X-ray point sources detected beyond the $3'$ search radius surrounding the AGPS position see Appendix B.)
### AX J165951–4209
While @sugizaki01 reported an absorbed flux of $F_{x}=4.04 \times 10^{-12}$ from AX J165951–4209 in the $0.7-10.0$ keV band, this AGPS source was not detected in the observation on 2008 June 21, with an upper limit on any X-ray flux of $F_{x} \sim 5 \times 10^{-14}$ ($0.3-8$ keV). AX J165951–4209 was also not detected with the *Swift* XRT on 2008 January 23 [@degenaar12] at a flux level of $F_{x} \sim 3 \times 10^{-13}$ ($0.3-10$ keV). While variability of several orders of magnitude (in this case a factor of $\sim100$) is typical behavior of both Be X-ray binaries [BeXs; @reig11] and black hole transients [@mcclintock06], the hard power law index calculated from the spectrum of this source [$\Gamma \sim 0.47$; @sugizaki01] is quite hard. This power law index is harder than what is usually observed from black hole transients [@mcclintock06] but is consistent with spectra from BeXs [for example see @haberl08]. We therefore suggest that AX J165951–4209 could be a transient BeX.
### ChI J170017–4220\_1
ChI J170017–4220\_1 (also known as ) has long been assumed to be a HMXB [see @liu06; @bird07; @krivonos07; @bird10]. This is, however, unconfirmed and is partly based upon the assumption that the Be star (2MASS J17002524-4219003) is the counterpart to ChI J170017–4220\_1 [@negueruela07]. The observation shows that HD 153295 is actually the counterpart to ChI J170017–4220\_2 (see Table \[Tab7\]) and that ChI J170017–4220\_1 has no 2MASS counterpart. The main clue to the nature of ChI J170017–4220\_1 comes from @markwardt10, who detected a 54s X-ray pulse period and 44 day orbital period using archival and *Rossi X-ray Timing Explorer* data. These period values are suggestive of a Be HMXB [@markwardt10].
We obtained PANIC NIR observations of ChI J170017–4220\_1 and identified its GLIMPSE counterpart, allowing this source to be represented in Figures \[Fig7\], \[Fig8\], and \[Fig9\]. ChI J170017–4220\_1 is part of Group 2 in Figure \[Fig7\], and is therefore consistent with an AGN, but this source is also situated in the same region as the archival HMXBs in Figure \[Fig8\]. Its placement on Figures \[Fig9a\] and \[Fig9b\] demonstrates that it has very unusual infrared colors, even more extreme than the archival HMXBs, which could indicate strong winds or a large amount of circumstellar absorption. Further investigation is required to determine the true nature of ChI J170017–4220\_1.
### ChI J172050–3710\_1
@degenaar12 suggested that ChI J172050–3710\_1 (also known as ) may be associated with the NIR source 2MASS J17205180-3710371, which has colors similar to a main sequence star. They also suggested that the low absorption inferred from a power-law X-ray fit indicates that ChI J172050–3710\_1 may be a foreground star. We agree that this 2MASS source is the likely counterpart to ChI J172050–3710\_1 based on the position obtained with the observation (see Table \[Tab2\]). We also agree that the counterpart colors are unremarkable (see Figures \[Fig9a\] and \[Fig9b\]), given that they are consistent with the general stellar locus depicted in Figure 1 of @hadfield07. Our quantile thermal bremsstrahlung spectral interpolation of ChI J172050–3710\_1 (see Table \[Tab4\]) also predicts a low value of $N_{H}$. This, combined with the unremarkable colors of 2MASS J17205180-3710371, and the fact that this source is situated in Region i of Figure \[Fig8\] with the other soft X-ray emitting stars, supports the claim by @degenaar12 that ChI J172050–3710\_1 is likely a foreground main sequence star. We therefore argue that the likely source of X-ray emission from ChI J172050–3710\_1 is generated in an active stellar corona.
### ChI J180857–2004\_2
ChI J180857–2004\_2 falls within the inner edge of the infrared dust bubble CN 148 [@churchwell07]. Such bubbles are formed by the stellar winds of young hot stars impacting the interstellar medium [@churchwell06]. The morphology of the dust cloud immediately surrounding ChI J180857–2004\_2 demonstrates a shell-type structure. This could indicate that its stellar winds are impacting the environment and generating a small secondary bubble within CN 148. If this is the case, then ChI J180857–2004\_2 could be a young massive star with strong stellar winds. This is already indicated by its position within the WR color spaces in Figures \[Fig9a\] and \[Fig9b\], and by the detection of hard X-ray emission with , indicated by its position within Region ii of Figure \[Fig8\]. ChI J180857–2004\_2 could therefore be a WR star, for which the X-ray emission is generated through instability-driven wind-shocks or in a CWB [see @anderson11].
### ChI J181116–1828\_5
ChI J181116–1828\_5 is a faint point source detected in the observation of so it is unlikely to be the main contributor of X-ray emission to this AGPS source. This source is, however, quite hard, with all but one count having an energy $>2$ keV, and is coincident with a MAGPIS radio source with a core-lobe morphology (see Section 3.4). Given the morphology of its likely radio counterpart, along with the detection of predominately hard X-rays ($E_{50}=3.4$ keV), we tentatively identify ChI J181116-1828\_5 as an AGN.
### ChI J181852–1559\_2
During the writing of this paper, ChI J181852–1559\_2 (also known as ) was proposed as a magnetar candidate through the analysis of several X-ray observations [@mereghetti12]. This proposed identification is encouraging as the lower limit of ChI J181852–1559\_2 in Figure \[Fig8\] is consistent with the archival magnetars in Region vi, demonstrating the usefulness of this statistical plot for identifying interesting sources.
### ChI J181915–1601\_2
As mentioned in Section 4.2.2, @hadfield07 identified (the counterpart to ChI J181915–1601\_2, see Table \[Tab2\]) as a WR star of type WN7o using the same selection criteria that we have adopted in Figures \[Fig9a\] and \[Fig9b\]. The mechanism behind the production of X-ray emission is still unknown, but is likely caused by massive stellar winds as demonstrated by the position of ChI J181915–1601\_2 within Region ii of Figure \[Fig8\]. Further investigation is required to determine if this WR star is mainly producing X-ray emission through instability-driven wind-shocks, or if it is part of a CWB.
### ChI J183345–0828\_1
@kargaltsev12 identified as the counterpart to ChI J183345–0828\_1 (also known as CXOU J183340.3–082830), but the nature of this X-ray source is still unknown. ChI J183345–0828\_1 falls within an extended X-ray source in SNR G23.5+0.1, which @kargaltsev12 have tentatively identified as the PWN powered by . It is possible that both this PWN and ChI J183345–0828\_1 may have contributed to the X-ray emission originally detected by from . While ChI J183345–0828\_1 is unlikely to be associated with the PWN, given the relative brightness of its IR counterpart, it does fall within the WR color space in Figure \[Fig9b\]. ChI J183345–0828\_1 could therefore be a massive star with strong stellar winds.
### ChI J184741–0219\_3
As mentioned in Section 3.4, ChI J184741–0219\_3 is coincident with a MAGPIS radio source that has a core-lobe morphology. Deep PANIC observations also allowed the detection of its infrared counterpart ($K=18.46$), which places it in Region iii of Figure \[Fig8\] (represented by the smallest cyan data point) along with the other archival AGN [@bassani09], which is the middle sized cyan data point. (We obtained the NIR magnitudes $J>20.6 \pm 0.3$, $H=17.3 \pm 0.2$ and $K_{s}=14.3 \pm 0.2$ on 2007 July 29 for AX J183039–1002 using PANIC.) As ChI J184741–0219\_3 had by far the highest count-rate of all sources detected in the $3'$ region around the aim-point for this target (see Table \[Tab1\]), it is likely the main contributor to the X-ray emission detected from in the AGPS.
ChI J184741–0219\_3 has a hard X-ray spectrum (see Table \[Tab3\]) where the resulting absorbed X-ray flux from the power-law spectral interpolation is $F_{x}=2.6 \pm 0.5 \times 10^{-13}$ in the $0.3-8$ keV energy band. Using the same spectral interpolation parameters and the count-rates [@sugizaki01], PIMMS estimates that ChI J184741–0219\_3 had an absorbed X-ray flux of $F_{x} \approx 1 \times 10^{-12}$ during the 1999 AGPS observation. @degenaar12 also observed the field of ChI J184741–0219\_3 with the *Swift* XRT in 2007 March but only obtained an absorbed X-ray flux upper-limit of $F_{x} < 1 \times 10^{-13}$ ($0.3-10$ keV). These flux measurements suggest likely long-term variability from this source. We therefore tentatively identify ChI J184741–0219\_3 as an X-ray variable AGN.
### AX J185905+0333
*Suzaku* observations have demonstrated that AX J185905+0333 is likely an X-ray luminous cluster of galaxies behind the Galactic Plane [@yamauchi11]. The extended nature of this source is the likely reason it was not detected in the short observation.
### AX J191046+0917
AX J191046+0917 (also known as AX J1910.7+0917), while not detected in the observation, has been detected intermittently in a small number of , , and *Integral* observations, allowing @pavan11 to identify it as a likely HMXB candidate. Further X-ray observations are required to confirm such an identification and further refine its class.
### ChI J194939+2631\_1
During the writing of this paper, ChI J194939+2631\_1 (also known as AX J194939+2631) was identified as a CV by @zolotukhin11 using the observation and the Isaac Newton Telescope Photometric H$\alpha$ Survey of the northern Galactic plane [IPHAS; @drew05]. Further follow-up is required to confirm their tentative intermediate polar (IP) classification.
Sources Identified with SIMBAD
-------------------------------
We were also able to identify four of the bright ($>20$ X-ray counts) sources using the SIMBAD Astronomical Database. These four sources are described below. We also discuss the secondary source ChI J165420–4337\_2.
### ChI J144042–6001\_1
The position obtained with the observation confirms the @sugizaki01 identification of ChI J144042-6001\_1 (also known as ) as the PMS star . This source appears as an identified star (yellow data point) in group 1 of Figure \[Fig7\], in Region i of Figure \[Fig8\], and is consistent with the general stellar locus in Figures \[Fig9a\] and \[Fig9b\].
### ChI J165420–4337\_2
@sugizaki01 identified as the K5 type star . After observing the position of AX J165420–4337 with HRC, we detected two sources ChI J165420–4337\_1 (123 X-ray counts detected) and ChI J165420–4337\_2 (16 X-ray counts detected). As ChI J165420–4337\_1 is by far the brightest of the sources detected in this observation, it is therefore likely to be the main source of X-ray emission originally detected in the AGPS. The fainter (secondary) source in the field, ChI J165420–4337\_2, is in fact the X-ray counterpart to HD 326426 based on position comparisons using SIMBAD. Therefore the identification of AX J165420–4337 as HD 326426 by @sugizaki01 is incorrect.
### ChI J172550–3533\_2
Using the position obtained with the observation, we identified ChI J172550–3533\_2 as the dwarf Nova V478 Sco [@vogt82]. ChI J172550–3533\_2 has a soft X-ray spectrum, as indicated by its position within Region i of Figure \[Fig8\], but has quite unusual infrared colors based on its position below the WR color-spaces in Figures \[Fig9a\] and \[Fig9b\].
### ChI J172642–3540\_1
Using SIMBAD, we identified ChI J172642–3540\_1 as the X-ray counterpart to the K2V type star CD-35 11565 [@torres06]. ChI J172642–3540\_1 is soft, like the other stars in Region i of Figure \[Fig8\]. Its position in Figures \[Fig9a\] and \[Fig9b\] is also consistent with the general stellar population.
### ChI J194310+2318\_5
ChI J194310+2318\_5 is one of the 18 sources detected within the H [ii]{} region G59.5–0.2 (see Section 4.1 and Table \[Tab9\]). Using SIMBAD and the position for this source, we determined that ChI J194310+2318\_5 is the X-ray counterpart to the O7V((f)) type star [@walborn73]. This source is quite soft (within Region i of Figure \[Fig8\]) and has unremarkable infrared colors (see Figures \[Fig9a\] and \[Fig9b\]). The placement of ChI J194310+2318\_5 in the aforementioned Figures, combined with its stellar classification, make this source compatible with an active stellar corona identification.
Confirmed and Tentative Identifications of the AGPS Sources
-----------------------------------------------------------
Table \[Tab12\] reproduces the original list of the 163 AGPS sources from @sugizaki01, but now including the corresponding sources with $>20$ X-ray counts in column two. The confirmed or tentative identification of these sources are listed in the third column, where the abbreviations are given in the Table caption. Those AGPS sources and corresponding sources with confirmed identifications were either obtained from the literature, and are therefore called “archival AGPS” sources, or were obtained through the survey’s observations [i.e. this paper and @anderson11; @anderson12]. The tentative identifications were made through the multi-wavelength follow-up and population statistics conducted on the sources in Section 4.2. The forth column gives the most common name for the AGPS and/or source while column five lists the references from which the X-ray and infrared properties of a given source was obtained for use in the statistical plots (Figures \[Fig7\], \[Fig8\] and \[Fig9\]).
The tentative identifications of the sources are based on the statistical plots in Section 4.2. For example, if a source falls within Region i of Figure \[Fig8\], then its type in Table \[Tab12\] has been listed as being either an active stellar corona (ASC) or PMS star (ASC/PMS). If the source falls within Region ii of Figure \[Fig8\], and also within one or both of the WR color spaces in Figures \[Fig9a\] and \[Fig9b\], then its type has been classified as a WR star. Those sources that fall within Region ii of Figure \[Fig8\] and near AX J144547-5931 in Figures \[Fig9a\] and \[Fig9b\] (see text in Section 4.2.2) have been classified as massive O-type stars (MS-O). All Region ii stars that fall within the general stellar loci of Figures \[Fig9a\] and \[Fig9b\] could be either massive stars (MS) or ASC so are listed as MS/ASC. (A PMS star interpretation is also possible for these Region ii sources.)
Those Region ii sources listed as Wolf-Rayet (WR), massive stars (MS), or massive O-type stars (MS-O), still need to be properly investigated for evidence of X-ray emission emanating from colliding-winds in a CWB. Such an identification requires further X-ray spectroscopic follow-up to identify plasma temperatures between 1 and 10 keV [@usov92]. There are also 7 AGPS sources that are listed as H [ii]{} regions. Those AGPS sources identified as “H [ii]{}” are actually made up of many X-ray point sources that were detected in our observations (see Section 4.1) and are therefore young and massive stars in the H [ii]{} region listed. If there is an identified X-ray star that is a significant contributor to the X-ray emission, the stellar type is listed before the H [ii]{} abbreviation in column 3, along with its source name in column 2. The suffixes of the other sources coincident with the H [ii]{} region are listed in parentheses. All other AGPS and sources for which there is no identification information are classified as unknown (U). If no sources were detected in a observation, then the AGPS source falls in the no detection (ND) category. A detailed description of some of the AGPS sources identified through the survey and of the archival AGPS sources (i.e. those sources identified in the literature and therefore not observed with as part of the survey) can be found in Section 4.3 and Appendix A, respectively.
The final column in Table \[Tab12\] contains a flag for columns one and two that indicates whether an AGPS and/or source has a confirmed identification (I) obtained through work in the survey or a tentative identification (T) using the population statistics in Section 4.2. Those AGPS sources for which no sources were detected within $3'$ of the position, or only faint ($<20$ X-ray counts) sources were detected (F), are also indicated. (This flag excludes those AGPS sources that have been identified as H [ii]{} regions.) The unidentified population of sources that fall in Region iv of Figure \[Fig8\] (R) are also included. There is also a flag for column three that indicates whether the type identification for a source is unconfirmed (N) due to it being based on its tentative statistical identification in Section 4.2.
Conclusions
===========
The main aim of the survey is to identify the Galactic plane X-ray source populations that make up the $F_{x} \sim 10^{-13} \mathrm{~to~} 10^{-11}$ flux range. To achieve this we have used new observations from the X-ray telescope, along with extensive multi-wavelength follow-up, to identify sources from the Galactic Plane Survey [@sugizaki01]. We have reported observations of the positions of 93 unidentified AGPS sources with , from which a total of 253 X-ray point sources, termed “ sources”, have been detected.
Through visual inspection of Galactic plane radio surveys, we have found 5 sources within supernova remnants that have no cataloged optical or infrared counterparts. These sources could potentially be compact objects associated with their surrounding SNRs. Further radio analysis has also demonstrated that the sources detected in the observations of the AGPS sources AX J144519–5949, AX J151005–5824, AX J154905–5420, , AX J194310+2318, AX J194332+2323, and AX J195006+2628 are all coincident with H [ii]{} regions. Table \[Tab11\] demonstrates that the range of luminosities, which are calculated using kinematic distances to the H [ii]{} regions, are consistent with the luminosities we expect from flaring PMS stars, massive stars, and CWBs. We therefore identify the $54$ separate sources seen in these observations as young and massive stars within H [ii]{} regions.
Of the 93 -observed AGPS fields, 62 have one or more sources with $>20$ X-ray counts, resulting in the detailed study of 74 sources in this paper. The multi-wavelength follow-up of these sources demonstrates the need for ’s subarcsecond localization capabilities to correctly identify likely infrared and optical counterparts. The main focus of this paper has been on those unidentified sources with $>20$ X-ray counts and with near-infrared or infrared counterparts. This has allowed us to perform population statistics to identify some of the likely objects that make up the $F_{x} \sim 10^{-13} \mathrm{~to~} 10^{-11}$ Galactic plane X-ray source populations.
We have developed a new statistical diagnostic for identifying likely populations of X-ray emitting sources using $K$-band fluxes and upper-limits (see Figure \[Fig8\]). The unidentified sources in Region i of Figure \[Fig8\] have soft X-ray emission and low X-ray-to-infrared flux ratios, making them consistent with many of the archival and identified stars. Their X-ray-to-infrared flux ratios are also similar to the COUP stars (see Figure \[Fig7\]), which are predominantly PMS stars. The majority of the Region i sources also fall within the general stellar locus that is expected in Figures \[Fig9a\] and \[Fig9b\] [@hadfield07]. They are therefore likely to be active stellar coronae, which is consistent with the main soft X-ray populations expected in the Galactic plane [@hong05], or PMS stars.
Many of the sources in Region ii of Figure \[Fig8\] have infrared colors similar to known Wolf-Rayet stars, as demonstrated in Figure \[Fig9\], which indicate the presence of excess infrared emission resulting from strong, dense stellar winds. These sources are therefore likely to be massive stars generating X-rays through instability-driven wind shocks or even colliding winds in CWBs [for example see @anderson11].
Only two unidentified sources are located within Region iii of Figure \[Fig8\], along with the archival high-mass and symbiotic X-ray binaries (and AGN). As such X-ray binaries (XRBs) are rare, only a few unidentified sources are expected to fall within this group. This result therefore demonstrates that Figure \[Fig8\] may be a very useful diagnostic for identifying XRBs.
Region vi contains four identified magnetars and a candidate LMXB. Even though there are likely two different source populations in Region vi, Figure \[Fig8\] demonstrates that hard X-ray sources ($E_{50}>1.3$ keV), with an X-ray-to-infrared flux ratio $F_{x}/F_{Ks} > 10^{2}$, are very rare and interesting Galactic X-ray sources.
The population of sources in Region iv of Figure \[Fig8\] remains unidentified. Based on their position relative to the identified AGN in Figure \[Fig7\] and they high $N_{H}$ values compared to the Galactic column density, we suggest that the 4 sources ChI J181116–1828\_2, ChI J181213–1842\_7, ChI J190749+0803\_1, and ChI J194152+2251\_2 could be background AGN. The remaining 8 unidentified Region iv sources have far lower $N_{H}$ values than the Galactic column densities indicating that they could be located in our own Galaxy. Optical and infrared spectroscopic follow-up is required to identify the true nature of this population.
With further source identifications, a full model of the hard ($2-10$ keV) Galactic plane X-ray populations between $F_{x} \sim 10^{-13} \mathrm{~and~} 10^{-11}$ will be able to be constructed. This will be more complete than those constructed from previous X-ray surveys in the same flux range, as it will be representative of 40 deg$^{2}$ of the Galactic plane. It will also show individual contributions from different Galactic X-ray source populations including non-accretion powered sources such as CWBs, SNRs, PWNe, and magnetars, which have not been a focus of previous work. Using the distribution and distance estimates, it will then be possible to construct luminosity functions and three-dimensional spatial distributions of each class of X-ray source in the Galactic plane.
Source Descriptions of the Archival AGPS Sources
================================================
As mentioned in Section 1, approximately one third of the AGPS sources were identified by @sugizaki01 or were classified by other research groups prior to the survey. It is these identified AGPS sources, referred to as “archival sources” in Section 4.2.1, which have been used to narrow down the possible unidentified source populations. The archival AGPS sources are listed in Table \[Tab12\] and are briefly described in below.
AX J143416–6024
: RS CVn type variable star, HD 127535, with spectral type K1IIIe [@sugizaki01].
AX J155052–5418
: X-ray and radio emitting magnetar 1E 1547.0-5408, associated with the possible radio SNR G327.24-0.13 [@gelfand07].
AX J155644-5325
: KOIIIe type star, TYC 8697–1438-1 [@torres06].
AX J161929–4945
: SFXT, a sub-class of HMXBs that display fast X-ray outbursts [@sguera06; @tomsick06].
AX J162155–4939
: K3III type star, HD 147070 [@sugizaki01]; this identification needs to be confirmed by follow-up X-ray observations
AX J163159–4752
: Accretion driven 1300s X-ray pulsar in a supergiant HMXB [@rodriguez06; @walter06]. This system is one of the highly absorbed HMXBs identified by *Integral* [@negueruela07].
AX J163351–4807
: The magnetic Of?p star HD 148937 [@naze12].
AX J163555–4719
: The X-ray emission associated with SNR G337.2+0.1 and its PWN [@combi06]. This system is also associated with the *Fermi*-LAT source 1FGL J1635.7–4715 [@abdo10].
AX J163904–4642
: Originally identified as a 912s pulsating, heavily-absorbed, HMXB [@bodaghee06; @thompson06], this source has now been reclassified as a SyXB [@nespoli10].
AX J164042–4632
: X-ray PWN associated with the radio SNR G338.3–0.0 and the very high energy $\gamma$-ray source HESS J1640–465 [@funk07; @lemiere09]. This system is also associated with the *Fermi*-LAT source 1FGL J1640.8–4634 [@abdo10; @slane10].
AX J165437–4333
: The F7V type star HD 152335 [@sugizaki01].
AX J165904–4242
: Herbig Be star V921 Sco, where the X-ray emission may arise from magnetic activity [@hamaguchi05]. [@sugizaki01 incorrectly assigned this star as the counterpart to AX J165901–4208.]
AX J170006–4157
: Magnetized CV, likely of the IP class, with 715s X-ray pulsations [@torii99; @kaur10].
AX J170047–4139
: 38s pulsating HMXB with an Ofpe/WNL type mass donor [@chakrabarty02; @mason09].
AX J170349–4142
: SNR G344.7–0.1 and its possible central compact object [CCO; @combi10]. There is a possible $\gamma$-ray counterpart, HESS J1702–420 [@giacani11].
AX J171804–3726
: SNR G349.7+0.2 and its possible CCO [@slane02a; @lazendic05]. This remnant is also associated with the *Fermi*-LAT source 1FGL J1717.9–3729 [@castro10].
AX J172105–3726
: SNR G350.1–0.3 and its CCO [@gaensler08; @lovchinsky11].
AX J172743–3506
: SNR G352.7–0.1 [@giacani09].
AX J173441–3234
: Colliding-wind binary (CWB) HD 159176 (07V+07V) in the young open cluster NGC 6383. The short period of this binary implies that the winds likely collide well before reaching their terminal velocities, limiting the hardness of the resulting thermal X-ray emission [@debecker04].
AX J173518–3237
: SNR G355.6–0.0 [@yamauchi08].
AX J180225–2300
: X-ray emission associated with the OB type and pre-main-sequence stars in the Trifid Nebula. The main X-ray contributor is the HD 164492 multiple system of OB stars [@rho04].
AX J180838–2024
: Magnetar SGR 1806–20 [@kouveliotou98].
AX J180902–1948
: SNR G10.5–0.0. Other than this detection [@sugizaki01], no other X-ray papers exist on this source. The radio SNR was discovered by @brogan06.
AX J180948–1918
: PSR J1809–1917 and its PWN, which are likely associated with HESS J1809–193 [@kargaltsev07; @aharonian07].
AX J180951–1943
: The X-ray and radio emitting magnetar XTE J1810–197 [@ibrahim04; @halpern05].
AX J181211–1835
: SNR G12.0–0.1 [@yamauchi08].
AX J182104–1420
: SNR G16.7+0.1 and its central PWN [@helfand03].
AX J183039–1002
: A Compton-thick active galactic nucleus [@bassani09]. A $K_{s}$-band magnitude of $14.3 \pm 0.2$ was obtained for this source with PANIC on 2007 July 29 (see Section 4.3.15).
AX J183221–0840
: Magnetized CV, likely of the IP class, with 1549.1s X-ray period pulsations [@sugizaki00; @kaur10].
AX J183528–0737
: The 112s pulse period X-ray binary, Scutum X-1. This system is likely to be a SyXB [@kaplan07].
AX J183800–0655
: The 70.5 ms pulsar, PSR J1838–0655, and its PWN. This system is possibly associated with HESS J1837–069 [@gotthelf08; @kargaltsev12].
AX J183931–0544
: The luminous blue variable (LBV) candidate G26.47+0.02. This source is possibly in a CWB [@paron12]. It is assumed that the 2MASS J18393224–0544204 is the correct NIR counterpart for the purpose of the statistical analysis in Section 4.4.
AX J184121–0455
: The magnetar 1E 1841–045 and its associated SNR G27.4+0.0 [Kes 73; @gotthelf97; @morii03].
AX J184355–0351
: X-ray emission associated with the non-thermal SNR G28.6–0.1/AX J1843.8–0352 and the thermal source CXO J184357–035441 [which may or may not be part of SNR G28.6–0.1, @bamba01; @ueno03].
AX J184629–0258
: X-ray emission from the SNR G29.7–0.3 (Kes 75), its central pulsar PSR J1846–0258 and associated PWN [@helfand03b].
AX J184848–0129
: X-ray sources in the Galactic globular cluster GLIMPSE–C01 [@pooley07] and the nearby diffuse source CXOU J184846.3–013040 [either a PWN or the globular cluster’s bow shock; @mirabal10]. These sources may also be associated with the *Fermi*-LAT source 0FGL J1848.6–0138 [@luque09].
AX J184930–0055
: X-ray emission associated with the thermal composite SNR G31.9+0.0 [3C 391; @chen01; @chen04]. This SNR is likely associated with 1FGL J1849.0–0055 [@castro10].
AX J185015–0025
: The X-ray synchrotron-dominated SNR G32.4+0.1 [@yamaguchi04].
AX J185240+0038
: X-ray emission associated with the SNR G33.6+0.1 (Kes 79) and the 105 ms pulsar PSR J1852+0040 [there is no detectable PWN; @gotthelf05]. This pulsar has been described as an “anti-magnetar” [@halpern10].
AX J185551+0129
: X-ray emission from SNR G34.7–0.4 (also known as W44 and 3C392) and the PWN associated with its central pulsar, PSR B1853+0.1 [@petre02]. This system may be associated with the *Fermi*-LAT source 0FGL J1855.9+0126/1FGL 1856.1+0122 [@abdo09; @abdo10].
AX J190734+0709
: SNR G41.1–0.3 [3C 397; @safiharb05].
AX J191105+0906
: SNR G43.3–0.2 [W49B; @hwang00].
AX J194649+2512
: X-ray emission likely associated with the H$\alpha$ emission line star VES 52 [@kohoutek97; @sugizaki01 this identification still needs to be properly confirmed by follow-up X-ray observations].
*CHANDRA* detected X-ray point sources beyond the $3'$ search region
====================================================================
Table \[Tab13\] lists those 14 detected X-ray point sources with $>20$ X-ray counts that lie within $3'-5'$ of the position of 11 AGPS sources and therefore outside the MAP default search radius. This Table includes the name of the AGPS sources for which the above applies, the position of the X-ray source, the offset from the original position, the net number of counts, and the most likely 2MASS counterpart.
A thorough analysis of these 14 X-ray point sources is beyond the scope of this paper, however, we have done a preliminary investigation of their possible contribution to the fluxes originally measured in the AGPS [@sugizaki01]. Using a similar technique to that described in Section 2.1 we entered the power law spectral fit of the AGPS source measured by @sugizaki01 into PIMMS in order to estimate the number of source counts expected to be detected in the corresponding observation. (Once again the photon index and absorption were set to $\Gamma=2$ and $N_{H}=10^{22}\mathrm{cm}^{-2}$ if no power law fit was provided.) These count predictions are listed in Table \[Tab13\]. The total number of counts detected from all the X-ray sources within $3'$ of the AGPS position are also included alongside these values. In the case of AX J154557–5420, AX J154905–5420, AX J172642–3504, AX J181915–1601, AX J184741–0129, AX J194332+2323, the total number of X-ray counts detected within $3'$ of the AGPS position contribute $\leq65\%$ of the predicted number of counts. It is therefore possible that the 8 X-ray point sources detected between $3'-5'$ from these AGPS sources could have contributed to the X-ray flux originally detected with . Such a result would not be unexpected for AX J154905–5420 and AX J194332+2323 as their corresponding resolved point sources are stars in H [ii]{} regions, which were not individually resolved with . This analysis demonstrates that few X-ray sources beyond $3'$ of the position contributed to the overall flux of a given AGPS source. The $3'$ search radius used in MAP is therefore reasonable for detecting the majority of AGPS associated point sources detected with . [Note that this Table does not include the magnetar PSR J1622–4950 that was detected in the observation of AX J162246–4946, $4'$ for its position, as it was well investigated in @anderson12].
G.E.A acknowledges the support of an Australian Postgraduate Award. B.M.G. acknowledges the support of an Australian Laureate Fellowship through ARC grant FL100100114. P.O.S. acknowledges partial support from NASA Contract NAS8-03060. D.T.H.S. acknowledges a STFC Advanced Fellowship. J.J.D was supported by NASA contract NAS8-39073 to the Chandra X-ray Center (CXC). Support for this work was also provided by NASA through Award Number GO9-0155X issued by the CXC, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA. The access to major research facilities program is supported by the Commonwealth of Australia under the *International Science Linkages program*. This research makes use of data obtained with the *X-ray Observatory*, and software provided by the CXC in the application packages `CIAO`. The ATCA is part of the Australia Telescope, funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO. The MOST is operated with the support of the Australian Research Council and the Science Foundation for Physics within the University of Sydney. Observing time on the 6.5m Clay and Baade Magellan Telescopes, located at Las Campanas Observatory, was allocated through the Harvard-Smithsonian Center for Astrophysics and the Massachusetts Institute of Technology. 2MASS is a joint project of the University of Massachusetts and the IPAC/Caltech, funded by the NASA and NFS. GLIMPSE survey data are part of the Spitzer Legacy Program. The *Spitzer Space Telescope* is operated by JPL/Caltech under a contract with NASA. This research has made use of NASA’s Astrophysics Data System.
[*Facilities:*]{} , , , , , , , , ,
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, M. C., [Wu]{}, K., [Trimble]{}, V., [O’Dell]{}, S. L., [Elsner]{}, R. F., [Zavlin]{}, V. E., & [Kouveliotou]{}, C. 2007, , 657, 1026
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[llccccccccccccccc]{}\
& J143148-6021\_1 & 14:31:50.18 & $-$60:22:08.3 & 0.97 & 0.6 & 0.9 & & $4.0_{-1.9}^{+3.2}$ & $4.0_{-1.9}^{+3.2}$ & $0.0_{-0.0}^{+1.9}$ & & & & &\
& J143148-6021\_2 & 14:31:48.56 & $-$60:21:44.5 & 0.56 & 0.5 & 0.8 & & $10_{-3}^{+4}$ & $7.0_{-2.6}^{+3.8}$ & $2.0_{-1.3}^{+2.7}$ & & $1.1 \pm 0.4$ & $1.4 \pm 0.9$ & &\
& J143148-6021\_3 & 14:31:48.06 & $-$60:19:59.5 & 1.19 & 0.3 & 0.8 & & $146_{-12}^{+13}$ & $51_{-7}^{+8}$ & $95_{-10}^{+11}$ & & $2.7 \pm 0.2$ & $1.1 \pm 0.1$ & 143148.0-601959 &\
& J143148-6021\_4 & 14:31:41.52 & $-$60:19:55.2 & 1.56 & 0.6 & 0.9 & & $5.9_{-2.4}^{+3.6}$ & $2.9_{-1.6}^{+2.9}$ & $0.0_{-0.0}^{+1.9}$ & & $0.6 \pm 0.3$ & $0.5 \pm 0.4$ & &\
& J143148-6021\_5 & 14:32:02.80 & $-$60:18:57.4 & 2.81 & 0.7 & 1.0 & & $20_{-4}^{+6}$ & $0.0_{-0.0}^{+1.9}$ & $20_{-4}^{+6}$ & & $4.4 \pm 0.5$ & $1.8 \pm 0.2$ & 143202.8-601857 &\
\
& J144042-6001\_1 & 14:40:38.40 & $-$60:01:36.8 & 0.56 & 0.3 & 0.8 & & $326_{-18}^{+19}$ & & & & & & 144038.4-600136 &\
\
& J144519-5949\_1 & 14:45:15.08 & $-$59:49:29.1 & 0.61 & 0.7 & 1.0 & & $3.0_{-1.7}^{+2.9}$ & $2.0_{-1.3}^{+2.7}$ & $1.0_{-0.9}^{+2.3}$ & & & &\
& J144519-5949\_2 & 14:45:21.39 & $-$59:49:25.3 & 0.47 & 0.4 & 0.8 & & $27_{-5}^{+6}$ & $4.0_{-1.9}^{+3.2}$ & $23_{-5}^{+6}$ & & $2.8 \pm 0.3$ & $1.7 \pm 0.3$ & 144521.3-594925 &\
& J144519-5949\_3 & 14:45:12.73 & $-$59:48:38.4 & 1.42 & 0.5 & 0.8 & & $13_{-4}^{+5}$ & $0.0_{-0.0}^{+1.9}$ & $13_{-4}^{+5}$ & & $3.8 \pm 0.4$ & $2.1 \pm 0.3$ & 144512.7-594838 &\
& J144519-5949\_4 & 14:45:19.31 & $-$59:48:00.7 & 1.79 & 0.8 & 1.1 & & $4.0_{-1.9}^{+3.2}$ & $4.0_{-1.9}^{+3.2}$ & $0.0_{-0.0}^{+1.9}$ & & & & &\
& J144519-5949\_5 & 14:44:57.87 & $-$59:50:39.0 & 2.81 & 1.0 & 1.2 & & $7.0_{-2.6}^{+3.8}$ & $3.0_{-1.7}^{+2.9}$ & $4.0_{-1.9}^{+3.2}$ & & $1.7 \pm 0.5$ & $1.1 \pm 0.4$ & &\
& J144519-5949\_6 & 14:45:05.79 & $-$59:49:32.2 & 1.71 & 0.8 & 1.1 & & $3.0_{-1.6}^{+2.9}$ & $1.0_{-0.8}^{+2.3}$ & $2.0_{-1.3}^{+2.7}$ & & & & &\
\
& J144547-5931\_1 & 14:45:43.67 & $-$59:32:05.2 & 0.46 & 0.3 & 0.8 & & $69_{-8}^{+9}$ & $31_{-6}^{+7}$ & $38_{-6}^{+7}$ & & $2.3 \pm 0.2$ & $1.8 \pm 0.1$ & 144543.6-593205 &\
\
& J144701-5919\_1 & 14:46:53.59 & $-$59:19:38.3 & 1.04 & 0.3 & 0.8 & & $81_{-9}^{+10}$ & $36_{-6}^{+7}$ & $45_{-7}^{+8}$ & & $2.3 \pm 0.1$ & $1.7 \pm 0.2$ & 144653.5-591938 &\
\
& J145732-5901\_1 & 14:57:34.93 & $-$58:59:53.9 & 1.45 & 0.4 & 0.8 & & $19_{-4}^{+5}$ & $16_{-4}^{+5}$ & $2.8_{-1.6}^{+2.9}$ & & $1.5 \pm 0.1$ & $1.9 \pm 0.4$ & &\
& J145732-5901\_2 & 14:57:42.16 & $-$58:59:04.5 & 2.56 & 0.8 & 1.1 & & $5.9_{-2.4}^{+3.6}$ & $0.0_{-0.0}^{+1.9}$ & $5.9_{-2.4}^{+3.6}$ & & $4.8 \pm 0.8$ & $2.1 \pm 0.4$ & &\
\
& J150436-5824\_1 & 15:04:30.99 & $-$58:24:11.3 & 0.89 & 0.3 & 0.8 & & $141_{-12}^{+13}$ & $36_{-6}^{+7}$ & $104_{-10}^{+11}$ & & $3.7 \pm 0.2$ & $1.1 \pm 0.1$ & &\
& J150436-5824\_2 & 15:04:31.40 & $-$58:23:05.8 & 1.75 & 0.8 & 1.1 & & $4.8_{-2.1}^{+3.4}$ & $5.0_{-2.2}^{+3.4}$ & $0.0_{-0.0}^{+1.9}$ & & & & &\
& J150436-5824\_3 & 15:04:51.09 & $-$58:24:58.8 & 1.93 & 0.6 & 0.9 & & $7.8_{-2.7}^{+4.0}$ & $8.0_{-2.8}^{+4.0}$ & $0.0_{-0.0}^{+1.9}$ & & $1.1 \pm 0.1$ & $2.2 \pm 0.8$ & &\
\
& J151005-5824\_1 & 15:10:15.36 & $-$58:25:42.1 & 1.49 & 0.6 & 0.9 & & $6.9_{-2.6}^{+3.8}$ & $1.0_{-0.9}^{+2.3}$ & $5.9_{-2.4}^{+3.6}$ & & $2.8 \pm 0.6$ & $1.9 \pm 0.5$ & &\
& J151005-5824\_2 & 15:09:54.87 & $-$58:24:37.1 & 1.45 & 0.6 & 0.9 & & $6.8_{-2.6}^{+3.8}$ & $1.0_{-0.9}^{+2.3}$ & $5.8_{-2.4}^{+3.6}$ & & $4.0 \pm 1.5$ & $1.1 \pm 0.4$ & &\
& J151005-5824\_3 & 15:10:07.69 & $-$58:23:19.8 & 1.59 & 0.7 & 1.0 & & $6.6_{-2.5}^{+3.8}$ & $7.0_{-2.6}^{+3.8}$ & $0.0_{-0.0}^{+1.9}$ & & $1.2 \pm 0.2$ & $2.0 \pm 0.4$ & &\
& J151005-5824\_4 & 15:10:07.13 & $-$58:26:09.0 & 1.26 & 0.6 & 0.9 & & $5.5_{-2.3}^{+3.6}$ & $1.9_{-1.3}^{+2.7}$ & $3.7_{-1.9}^{+3.2}$ & & $3.8 \pm 1.2$ & $1.0 \pm 0.4$ & &\
& J151005-5824\_5 & 15:10:06.84 & $-$58:25:56.8 & 1.06 & 0.5 & 0.9 & & $5.3_{-2.2}^{+3.6}$ & $1.7_{-1.2}^{+2.7}$ & $3.6_{-1.8}^{+3.2}$ & & $3.5 \pm 1.4$ & $0.9 \pm 0.4$ & &\
& J151005-5824\_6 & 15:09:51.83 & $-$58:25:03.1 & 1.83 & 0.6 & 0.9 & & $11_{-3}^{+4}$ & $9.0_{-3.0}^{+4.1}$ & $1.8_{-1.2}^{+2.7}$ & & $1.4 \pm 0.2$ & $2.1 \pm 0.9$ & &\
& J151005-5824\_7 & 15:10:11.48 & $-$58:23:36.7 & 1.49 & 0.4 & 0.8 & & $16_{-4}^{+5}$ & $0.0_{-0.0}^{+1.9}$ & $16_{-4}^{+5}$ & & $4.9 \pm 0.7$ & $1.8 \pm 0.2$ & &\
& J151005-5824\_8 & 15:10:03.45 & $-$58:26:45.6 & 1.88 & 0.6 & 0.9 & & $7.9_{-2.8}^{+4.0}$ & $0.0_{-0.0}^{+1.9}$ & $7.9_{-2.8}^{+4.0}$ & & $2.9 \pm 0.2$ & $2.3 \pm 0.2$ & &\
& J151005-5824\_9 & 15:09:55.60 & $-$58:26:22.0 & 1.98 & 0.8 & 1.0 & & $4.7_{-2.1}^{+3.4}$ & $4.9_{-2.2}^{+3.4}$ & $0.0_{-0.0}^{+1.9}$ & & $1.4 \pm 0.4$ & $1.1 \pm 0.3$ & &\
& J151005-5824\_10 & 15:09:52.35 & $-$58:27:12.2 & 2.90 & 0.9 & 1.1 & & $6.3_{-2.5}^{+3.8}$ & $0.8_{-0.7}^{+2.3}$ & $5.6_{-2.3}^{+3.6}$ & & $2.9 \pm 1.2$ & $0.9 \pm 0.5$ & &\
& J151005-5824\_11 & 15:10:05.09 & $-$58:26:02.1 & 1.14 & 0.6 & 0.9 & & $4.6_{-2.1}^{+3.4}$ & $1.9_{-1.3}^{+2.7}$ & $2.7_{-1.6}^{+2.9}$ & & $2.0 \pm 1.6$ & $0.9 \pm 0.3$ & &\
\
& J153751-5556\_1$\dagger$ & 15:37:59.32 & $-$55:54:23.3 & 2.16 & 0.5 & 0.9 & & $13_{-4}^{+5}$ & $11_{-3}^{+4}$ & $0.0_{-0.0}^{+1.9}$ & & $0.8 \pm 0.1$ & $1.6 \pm 0.8$ & &\
& J153751-5556\_2 & 15:37:35.15 & $-$55:56:50.1 & 2.40 & 0.9 & 1.1 & & $5.8_{-2.3}^{+3.6}$ & $6.0_{-2.4}^{+3.6}$ & $0.0_{-0.0}^{+1.9}$ & & & & &\
\
& J153818-5541\_1 & 15:38:14.00 & $-$55:42:13.6 & 1.02 & 0.4 & 0.8 & & $49_{-7}^{+8}$ & & & & & & 153813.9-554213 &\
\
& J154122-5522\_1 & 15:41:15.79 & $-$55:21:28.7 & 1.05 & 0.3 & 0.8 & & $83_{-9}^{+10}$ & $65_{-8}^{+9}$ & $16_{-4}^{+5}$ & & $1.2 \pm 0.1$ & $1.2 \pm 0.1$ & 154115.7-552128 &\
& J154122-5522\_2 & 15:41:33.14 & $-$55:21:12.5 & 1.77 & 0.8 & 1.1 & & $3.0_{-1.6}^{+2.9}$ & $0.0_{-0.0}^{+1.9}$ & $3.0_{-1.7}^{+2.9}$ & & & & &\
& J154122-5522\_3 & 15:41:10.72 & $-$55:22:51.4 & 1.82 & 0.8 & 1.1 & & $6.0_{-2.4}^{+3.6}$ & $4.0_{-1.9}^{+3.2}$ & $2.0_{-1.3}^{+2.7}$ & & & & &\
\
& J154557-5443\_1 & 15:45:58.23 & $-$54:44:24.8 & 1.27 & 0.4 & 0.8 & & $26_{-5}^{+6}$ & $23_{-5}^{+6}$ & $3.0_{-1.6}^{+2.9}$ & & $1.2 \pm 0.2$ & $1.4 \pm 0.2$ & &\
& J154557-5443\_2 & 15:45:44.38 & $-$54:43:28.3 & 1.97 & 0.5 & 0.8 & & $20_{-4}^{+6}$ & $0.0_{-0.0}^{+1.9}$ & $21_{-5}^{+6}$ & & $4.4 \pm 0.4$ & $2.2 \pm 0.3$ & &\
& J154557-5443\_3 & 15:46:12.16 & $-$54:43:10.9 & 2.07 & 0.4 & 0.8 & & $22_{-5}^{+6}$ & $20_{-4}^{+6}$ & $1.7_{-1.2}^{+2.7}$ & & $0.8 \pm 0.1$ & $1.8 \pm 0.6$ & &\
& J154557-5443\_4 & 15:45:52.21 & $-$54:42:44.8 & 0.90 & 0.5 & 0.8 & & $11_{-3}^{+4}$ & $1.0_{-0.9}^{+2.3}$ & $10_{-3}^{+4}$ & & $3.9 \pm 0.7$ & $1.5 \pm 0.4$ & &\
& J154557-5443\_5 & 15:45:55.39 & $-$54:43:29.1 & 0.49 & 0.6 & 0.9 & & $4.7_{-2.1}^{+3.4}$ & $0.0_{-0.0}^{+1.9}$ & $4.9_{-2.1}^{+3.4}$ & & $4.2 \pm 0.5$ & $2.1 \pm 0.2$ & &\
& J154557-5443\_6 & 15:45:55.41 & $-$54:42:39.8 & 0.59 & 0.5 & 0.9 & & $5.8_{-2.3}^{+3.6}$ & $5.9_{-2.4}^{+3.6}$ & $0.0_{-0.0}^{+1.9}$ & & $0.8 \pm 0.1$ & $1.5 \pm 0.4$ & &\
& J154557-5443\_7 & 15:46:10.81 & $-$54:41:35.2 & 2.43 & 0.9 & 1.1 & & $5.9_{-2.4}^{+3.6}$ & $4.0_{-1.9}^{+3.2}$ & $1.9_{-1.3}^{+2.7}$ & & $1.7 \pm 0.3$ & $1.9 \pm 0.5$ & &\
\
& J154905-5420\_1 & 15:48:58.53 & $-$54:21:47.7 & 1.61 & 0.6 & 0.9 & & $8.9_{-2.9}^{+4.1}$ & $0.0_{-0.0}^{+1.9}$ & $8.9_{-2.9}^{+4.1}$ & & $2.9 \pm 0.4$ & $2.0 \pm 0.2$ & &\
& J154905-5420\_2 & 15:49:02.83 & $-$54:17:54.6 & 2.63 & 0.7 & 1.0 & & $14_{-4}^{+5}$ & $2.8_{-1.6}^{+2.9}$ & $12_{-3}^{+5}$ & & $3.4 \pm 0.6$ & $1.4 \pm 0.3$ & &\
& J154905-5420\_3 & 15:48:52.31 & $-$54:19:12.9 & 2.30 & 0.9 & 1.1 & & $4.7_{-2.1}^{+3.4}$ & $0.0_{-0.0}^{+1.9}$ & $4.8_{-2.1}^{+3.4}$ & & $3.3 \pm 0.5$ & $2.1 \pm 0.3$ & &\
& J154905-5420\_4 & 15:49:04.92 & $-$54:19:30.2 & 1.02 & 0.6 & 0.9 & & $5.4_{-2.3}^{+3.6}$ & $5.8_{-2.4}^{+3.6}$ & $0.0_{-0.0}^{+1.9}$ & & $1.1 \pm 0.2$ & $1.5 \pm 0.4$ & &\
& J154905-5420\_5 & 15:49:11.59 & $-$54:22:27.7 & 2.15 & 0.6 & 0.9 & & $10_{-3}^{+4}$ & $5.9_{-2.4}^{+3.6}$ & $4.0_{-1.9}^{+3.2}$ & & $1.5 \pm 0.8$ & $0.7 \pm 0.2$ & &\
& J154905-5420\_6 & 15:49:05.63 & $-$54:22:07.7 & 1.61 & 0.7 & 1.0 & & $3.9_{-1.9}^{+3.2}$ & $0.0_{-0.0}^{+1.9}$ & $3.9_{-1.9}^{+3.2}$ & & & & &\
& J154905-5420\_7 & 15:48:55.72 & $-$54:19:54.5 & 1.52 & 0.5 & 0.9 & & $8.1_{-2.8}^{+4.1}$ & $2.7_{-1.6}^{+2.9}$ & $5.6_{-2.3}^{+3.6}$ & & $2.8 \pm 1.5$ & $0.9 \pm 0.3$ & &\
& J154905-5420\_8 & 15:49:09.17 & $-$54:19:27.9 & 1.20 & 0.6 & 0.9 & & $4.8_{-2.1}^{+3.4}$ & $1.0_{-0.8}^{+2.3}$ & $3.9_{-1.9}^{+3.2}$ & & $2.9 \pm 1.5$ & $1.0 \pm 0.3$ & &\
& J154905-5420\_9 & 15:49:09.32 & $-$54:19:17.0 & 1.37 & 0.6 & 0.9 & & $6.9_{-2.6}^{+3.8}$ & $4.0_{-1.9}^{+3.2}$ & $2.0_{-1.3}^{+2.7}$ & & $1.3 \pm 0.6$ & $1.0 \pm 0.5$ & &\
& J154905-5420\_10 & 15:48:48.45 & $-$54:19:00.5 & 2.88 & 1.1 & 1.3 & & $4.8_{-2.1}^{+3.4}$ & $4.0_{-1.9}^{+3.2}$ & $0.9_{-0.8}^{+2.3}$ & & $1.1 \pm 1.8$ & $0.7 \pm 0.7$ & &\
& J154905-5420\_11 & 15:49:07.39 & $-$54:18:53.5 & 1.66 & 0.5 & 0.9 & & $12_{-3}^{+5}$ & $8.9_{-2.9}^{+4.1}$ & $2.9_{-1.6}^{+2.9}$ & & $1.6 \pm 0.4$ & $1.3 \pm 0.6$ & &\
\
& J154951-5416\_1 & 15:49:51.66 & $-$54:16:29.8 & 0.12 & 0.4 & 0.8 & & $16_{-4}^{+5}$ & $8.0_{-2.8}^{+4.0}$ & $8.0_{-2.8}^{+4.0}$ & & $2.0 \pm 0.7$ & $0.6 \pm 0.2$ & 154951.6-541629 & J154951.7-541630\
\
& J155035-5408\_1 & 15:50:21.23 & $-$54:09:54.3 & 2.41 & 0.5 & 0.9 & & $21_{-5}^{+6}$ & $19_{-4}^{+5}$ & $0.0_{-0.0}^{+1.9}$ & & $0.9 \pm 0.1$ & $1.3 \pm 0.3$ & &\
& J155035-5408\_2 & 15:50:33.73 & $-$54:09:16.2 & 0.63 & 0.4 & 0.8 & & $18_{-4}^{+5}$ & $5.0_{-2.2}^{+3.4}$ & $13_{-4}^{+5}$ & & $3.0 \pm 0.7$ & $1.1 \pm 0.2$ & & J155033.7-540914\
& J155035-5408\_3 & 15:50:37.60 & $-$54:07:21.9 & 1.37 & 0.3 & 0.8 & & $203_{-14}^{+15}$ & $152_{-12}^{+13}$ & $51_{-7}^{+8}$ & & $1.6 \pm 0.1$ & $1.5 \pm 0.1$ & & J155037.5-540722\
\
& J155331-5347\_1 & 15:53:36.01 & $-$53:48:26.9 & 1.03 & 0.3 & 0.8 & & $71_{-8}^{+9}$ & $60_{-8}^{+9}$ & $9.0_{-3.0}^{+4.1}$ & & $1.1 \pm 0.0$ & $1.8 \pm 0.3$ & 155336.0-534826 &\
& J155331-5347\_2 & 15:53:26.06 & $-$53:47:58.6 & 0.90 & 0.8 & 1.1 & & $2.9_{-1.6}^{+2.9}$ & $0.0_{-0.0}^{+1.9}$ & $3.0_{-1.6}^{+2.9}$ & & & & &\
\
& J155831-5334\_1 & 15:58:28.38 & $-$53:35:20.8 & 0.67 & 0.3 & 0.8 & & $53_{-7}^{+8}$ & $32_{-6}^{+7}$ & $20_{-4}^{+6}$ & & $1.8 \pm 0.1$ & $1.1 \pm 0.2$ & 155828.3-533520 &\
& J155831-5334\_2 & 15:58:47.31 & $-$53:34:06.6 & 2.47 & 0.7 & 1.0 & & $10_{-3}^{+4}$ & $8.0_{-2.8}^{+4.0}$ & $2.0_{-1.3}^{+2.7}$ & & $1.3 \pm 0.2$ & $1.8 \pm 0.6$ & &\
\
& J162011-5002\_1 & 16:20:06.69 & $-$50:01:59.0 & 0.80 & 0.3 & 0.8 & & $53_{-7}^{+8}$ & $6.0_{-2.4}^{+3.6}$ & $47_{-7}^{+8}$ & & $3.7 \pm 0.4$ & $1.6 \pm 0.2$ & 162006.7-500159 &\
& J162011-5002\_2 & 16:20:06.05 & $-$50:01:40.8 & 1.00 & 0.4 & 0.8 & & $13_{-4}^{+5}$ & $11_{-3}^{+4}$ & $2.0_{-1.3}^{+2.7}$ & & $1.2 \pm 0.3$ & $1.2 \pm 0.8$ & 162006.0-500140 &\
\
& J162046-4942\_1 & 16:20:48.82 & $-$49:42:14.8 & 0.65 & 0.3 & 0.8 & & $117_{-11}^{+12}$ & $80_{-9}^{+10}$ & $37_{-6}^{+7}$ & & $1.6 \pm 0.1$ & $1.0 \pm 0.2$ & & J162048.8-494214\
&\
& J162208-5005\_1 & 16:22:09.73 & $-$50:06:00.6 & 0.38 & 0.4 & 0.8 & & $18_{-4}^{+5}$ & $4.8_{-2.1}^{+3.4}$ & $14_{-4}^{+5}$ & & $3.2 \pm 0.3$ & $1.4 \pm 0.3$ & 162209.7-500600 &\
& J162208-5005\_2 & 16:22:08.55 & $-$50:05:55.1 & 0.22 & 0.7 & 1.0 & & $3.9_{-1.9}^{+3.2}$ & $0.0_{-0.0}^{+1.9}$ & $3.9_{-1.9}^{+3.2}$ & & & & &\
& J162208-5005\_3 & 16:22:06.98 & $-$50:04:58.4 & 0.76 & 0.6 & 0.9 & & $7.0_{-2.6}^{+3.8}$ & $2.0_{-1.3}^{+2.7}$ & $5.0_{-2.2}^{+3.4}$ & & $3.7 \pm 1.0$ & $1.2 \pm 0.4$ & &\
\
& J162246-4946\_1 & 16:22:48.03 & $-$49:48:54.6 & 2.00 & 0.8 & 1.1 & & $4.0_{-1.9}^{+3.2}$ & $2.0_{-1.3}^{+2.7}$ & $2.0_{-1.3}^{+2.7}$ & & & & &\
& J162246-4946\_2 & 16:22:51.68 & $-$49:46:37.9 & 0.80 & 0.4 & 0.8 & & $17_{-4}^{+5}$ & $16_{-4}^{+5}$ & $0.0_{-0.0}^{+1.9}$ & & $0.9 \pm 0.0$ & $2.2 \pm 0.4$ & 162251.6-494637 &\
& J162246-4946\_3\* & 16:22:49.63 & $-$49:44:47.5 & 2.17 & 0.8 & 1.1 & & $3.7_{-1.8}^{+3.2}$ & & & & & & &\
& J162246-4946\_4 & 16:22:58.77 & $-$49:46:11.1 & 2.03 & 0.6 & 0.9 & & $5.9_{-2.4}^{+3.6}$ & $5.0_{-2.2}^{+3.4}$ & $0.9_{-0.8}^{+2.3}$ & & $1.1 \pm 0.7$ & $1.0 \pm 0.7$ & &\
\
& J163252-4746\_1 & 16:32:59.75 & $-$47:47:13.6 & 1.65 & 0.9 & 1.1 & & $4.6_{-2.1}^{+3.4}$ & & & & & & &\
& J163252-4746\_2 & 16:32:48.55 & $-$47:45:06.2 & 1.22 & 0.3 & 0.8 & & $65_{-8}^{+9}$ & & & & & & 163248.5-474506 & J163248.6-474505\
\
& J163524-4728\_1\* & 16:35:20.72 & $-$47:27:55.5 & 0.88 & 0.6 & 0.9 & & $3.6_{-1.8}^{+3.2}$ & & & & & & &\
& J163524-4728\_2\* & 16:35:32.89 & $-$47:27:44.6 & 1.51 & 0.9 & 1.1 & & $2.6_{-1.5}^{+2.9}$ & & & & & & &\
& J163524-4728\_3 & 16:35:20.86 & $-$47:30:56.7 & 2.60 & 0.6 & 0.9 & & $13_{-4}^{+5}$ & $4.0_{-1.9}^{+3.2}$ & $7.7_{-2.7}^{+4.0}$ & & $2.6 \pm 0.9$ & $0.9 \pm 0.3$ & &\
& J163524-4728\_4 & 16:35:20.54 & $-$47:28:31.6 & 0.75 & 0.5 & 0.8 & & $7.8_{-2.7}^{+4.0}$ & $7.0_{-2.6}^{+3.8}$ & $0.9_{-0.8}^{+2.3}$ & & $1.2 \pm 0.6$ & $1.4 \pm 1.6$ & 163520.5-472830 & J163520.5-472832\
\
& J163751-4656\_1 & 16:37:50.82 & $-$46:55:45.3 & 0.89 & 0.3 & 0.8 & & $72_{-8}^{+10}$ & & & & & & 163750.8-465545 & J163750.8-465544\
\
& J165105-4403\_1 & 16:51:11.63 & $-$44:02:38.8 & 1.34 & 0.3 & 0.8 & & $150_{-12}^{+13}$ & $56_{-7}^{+9}$ & $94_{-10}^{+11}$ & & $2.4 \pm 0.2$ & $1.0 \pm 0.1$ & 165111.6-440238 &\
\
& J165217-4414\_1 & 16:52:19.02 & $-$44:14:01.7 & 0.28 & 0.3 & 0.8 & & $107_{-10}^{+11}$ & $53_{-7}^{+8}$ & $54_{-7}^{+8}$ & & $2.0 \pm 0.1$ & $1.1 \pm 0.1$ & 165219.0-441401 &\
\
& J165420-4337\_1 & 16:54:23.44 & $-$43:37:43.7 & 0.55 & 0.3 & 0.8 & & $123_{-11}^{+12}$ & & & & & & 165423.4-433743 &\
& J165420-4337\_2 & 16:54:34.86 & $-$43:38:02.4 & 2.64 & 0.6 & 0.9 & & $16_{-4}^{+5}$ & & & & & & 165434.8-433802 &\
\
& J165646-4239\_1 & 16:56:49.80 & $-$42:38:49.4 & 1.01 & 0.3 & 0.8 & & $265_{-16}^{+17}$ & $228_{-15}^{+16}$ & $14_{-4}^{+5}$ & & $0.9 \pm 0.0$ & $1.5 \pm 0.1$ & 165649.7-423849 &\
& J165646-4239\_2 & 16:56:50.64 & $-$42:38:12.1 & 1.61 & 0.4 & 0.8 & & $57_{-8}^{+9}$ & $34_{-6}^{+7}$ & $23_{-5}^{+6}$ & & $1.8 \pm 0.1$ & $1.1 \pm 0.2$ & 165650.6-423812 &\
\
& J165707-4255\_1 & 16:57:03.74 & $-$42:54:42.0 & 0.85 & 0.3 & 0.8 & & $61_{-8}^{+9}$ & $50_{-7}^{+8}$ & $6.0_{-2.4}^{+3.6}$ & & $0.9 \pm 0.1$ & $1.2 \pm 0.2$ & 165703.7-425442 &\
& J165707-4255\_2 & 16:57:19.97 & $-$42:56:20.4 & 2.58 & 1.0 & 1.2 & & $5.0_{-2.2}^{+3.4}$ & $5.0_{-2.2}^{+3.4}$ & $0.0_{-0.0}^{+1.9}$ & & & & &\
\
& J165901-4208\_1 & 16:59:05.65 & $-$42:07:26.0 & 0.96 & 0.4 & 0.8 & & $20_{-4}^{+6}$ & $0.0_{-0.0}^{+1.9}$ & $20_{-4}^{+6}$ & & $4.4 \pm 0.4$ & $1.7 \pm 0.3$ & 165905.6-420725 &\
& J165901-4208\_2 & 16:59:11.97 & $-$42:09:34.8 & 2.38 & 0.5 & 0.9 & & $16_{-4}^{+5}$ & $4.0_{-1.9}^{+3.2}$ & $12_{-3}^{+5}$ & & $2.6 \pm 0.7$ & $1.2 \pm 0.3$ & &\
& J165901-4208\_3$\dagger$ & 16:59:10.07 & $-$42:08:45.2 & 1.65 & 0.5 & 0.9 & & $8.7_{-2.9}^{+4.1}$ & $8.0_{-2.8}^{+4.0}$ & $0.8_{-0.7}^{+2.3}$ & & $1.0 \pm 0.4$ & $1.3 \pm 1.7$ & &\
& J165901-4208\_4$\dagger$ & 16:59:17.62 & $-$42:07:38.3 & 2.95 & 0.5 & 0.9 & & $11_{-3}^{+5}$ & $0.0_{-0.0}^{+1.9}$ & $12_{-3}^{+5}$ & & $4.0 \pm 0.6$ & $1.6 \pm 0.3$ & &\
& J165901-4208\_5 & 16:58:54.24 & $-$42:05:49.0 & 2.69 & 0.7 & 1.0 & & $10_{-3}^{+4}$ & $5.8_{-2.4}^{+3.6}$ & $4.6_{-2.1}^{+3.4}$ & & $1.7 \pm 0.8$ & $0.9 \pm 0.3$ & &\
\
& J165951-4209 & &\
\
& J170017-4220\_1 & 17:00:19.29 & $-$42:20:19.1 & 0.25 & 0.3 & 0.8 & & $66_{-8}^{+9}$ & $1.0_{-0.8}^{+2.3}$ & $65_{-8}^{+9}$ & & $5.1 \pm 0.3$ & $1.9 \pm 0.1$ & 170019.2-422019 & J170019.2-422019\
& J170017-4220\_2 & 17:00:25.25 & $-$42:19:00.3 & 1.93 & 0.6 & 0.9 & & $8.0_{-2.8}^{+4.0}$ & $3.0_{-1.6}^{+2.9}$ & $5.0_{-2.2}^{+3.4}$ & & $3.1 \pm 0.7$ & $1.1 \pm 0.4$ & &\
\
& J170052-4210\_1 & 17:00:58.27 & $-$42:10:37.0 & 0.98 & 0.4 & 0.8 & & $36_{-6}^{+7}$ & $34_{-6}^{+7}$ & $0.0_{-0.0}^{+1.9}$ & & $0.9 \pm 0.1$ & $1.5 \pm 0.3$ & 170058.2-421037 & J170058.1-421036\
\
& J170112-4212\_1 & 17:01:17.92 & $-$42:13:38.0 & 1.72 & 0.6 & 0.9 & & $8.0_{-2.8}^{+4.0}$ & $0.0_{-0.0}^{+1.9}$ & $8.0_{-2.8}^{+4.0}$ & & $4.5 \pm 0.7$ & $1.7 \pm 0.2$ & & J170117.9-421337\
\
& J170444-4109\_1$\dagger$ & 17:04:40.62 & $-$41:11:27.4 & 2.00 & 0.3 & 0.8 & & $148_{-12}^{+13}$ & $122_{-11}^{+12}$ & $25_{-5}^{+6}$ & & $1.1 \pm 0.1$ & $1.4 \pm 0.1$ & 170440.6-411127 & J170440.6-411126\
& J170444-4109\_2 & 17:04:42.53 & $-$41:09:19.4 & 0.48 & 0.4 & 0.8 & & $22_{-5}^{+6}$ & $1.0_{-0.9}^{+2.3}$ & $21_{-5}^{+6}$ & & $3.9 \pm 0.6$ & $1.5 \pm 0.2$ & 170442.5-410919 & J170442.6-410920\
\
& J170536-4038\_1 & 17:05:39.82 & $-$40:38:12.4 & 0.85 & 0.3 & 0.8 & & $113_{-11}^{+12}$ & $89_{-9}^{+10}$ & $24_{-5}^{+6}$ & & $1.2 \pm 0.1$ & $1.3 \pm 0.2$ & 170539.8-403812 &\
\
& J170555-4104\_1 & 17:05:45.50 & $-$41:04:02.7 & 2.13 & 0.6 & 0.9 & & $9.0_{-2.9}^{+4.1}$ & $8.0_{-2.8}^{+4.0}$ & $1.0_{-0.9}^{+2.3}$ & & $0.9 \pm 0.3$ & $1.3 \pm 0.3$ & & &\
& J170555-4104\_2 & 17:05:45.62 & $-$41:04:17.5 & 2.02 & 0.5 & 0.9 & & $12_{-3}^{+5}$ & $10_{-3}^{+4}$ & $2.0_{-1.3}^{+2.7}$ & & $1.5 \pm 0.3$ & $1.3 \pm 0.4$ & 170545.6-410417 & &\
\
& J171715-3718\_1 & 17:17:07.59 & $-$37:16:14.3 & 2.87 & 0.6 & 0.9 & & $19_{-4}^{+5}$ & $19_{-4}^{+5}$ & $0.0_{-0.0}^{+1.9}$ & & $1.0 \pm 0.1$ & $1.9 \pm 0.2$ & 171707.6-371614 &\
& J171715-3718\_2 & 17:17:23.99 & $-$37:17:15.8 & 2.32 & 0.6 & 0.9 & & $11_{-3}^{+4}$ & $0.0_{-0.0}^{+1.9}$ & $11_{-3}^{+4}$ & & $6.2 \pm 0.4$ & $2.3 \pm 0.4$ & 171723.9-371715 &\
\
& J171910-3652\_1 & 17:19:06.24 & $-$36:51:31.8 & 1.56 & 0.4 & 0.8 & & $20_{-4}^{+6}$ & $5.0_{-2.2}^{+3.4}$ & $15_{-4}^{+5}$ & & $2.9 \pm 0.5$ & $1.3 \pm 0.2$ & 171906.2-365131 &\
& J171910-3652\_2 & 17:19:13.10 & $-$36:51:24.4 & 1.56 & 0.3 & 0.8 & & $114_{-11}^{+12}$ & $95_{-10}^{+11}$ & $11_{-3}^{+4}$ & & $0.9 \pm 0.0$ & $1.7 \pm 0.2$ & 171913.1-365124 &\
& J171910-3652\_3 & 17:19:24.07 & $-$36:52:09.7 & 2.84 & 0.7 & 1.0 & & $12_{-3}^{+5}$ & $8.9_{-2.9}^{+4.1}$ & $2.0_{-1.3}^{+2.7}$ & & $1.1 \pm 0.2$ & $1.1 \pm 0.6$ & 171924.0-365209 &\
\
& J171922-3703\_1 & 17:19:20.88 & $-$37:01:55.2 & 1.16 & 0.3 & 0.8 & & $143_{-12}^{+13}$ & $9.0_{-2.9}^{+4.1}$ & $134_{-12}^{+13}$ & & $4.5 \pm 0.2$ & $1.7 \pm 0.1$ & &\
& J171922-3703\_2$\dagger$ & 17:19:29.90 & $-$37:01:12.7 & 2.25 & 0.6 & 0.9 & & $8.7_{-2.9}^{+4.1}$ & $8.8_{-2.9}^{+4.1}$ & $0.0_{-0.0}^{+1.9}$ & & $0.9 \pm 0.1$ & $1.9 \pm 0.3$ & &\
& J171922-3703\_3 & 17:19:27.08 & $-$37:00:10.6 & 2.94 & 0.6 & 0.9 & & $15_{-4}^{+5}$ & $0.0_{-0.0}^{+1.9}$ & $15_{-4}^{+5}$ & & $4.2 \pm 0.3$ & $2.2 \pm 0.4$ & &\
& J171922-3703\_4 & 17:19:21.43 & $-$37:03:22.2 & 0.49 & 0.5 & 0.9 & & $6.8_{-2.6}^{+3.8}$ & $0.0_{-0.0}^{+1.9}$ & $6.8_{-2.6}^{+3.8}$ & & $3.9 \pm 0.5$ & $2.6 \pm 0.4$ & &\
\
& J172050-3710\_1 & 17:20:51.80 & $-$37:10:37.4 & 0.26 & 0.3 & 0.8 & & $314_{-18}^{+19}$ & $280_{-17}^{+18}$ & $28_{-5}^{+6}$ & & $1.0 \pm 0.0$ & $1.6 \pm 0.1$ & &\
\
& J172550-3533\_1 & 17:25:49.58 & $-$35:33:41.4 & 0.22 & 0.3 & 0.8 & & $43_{-7}^{+8}$ & $1.0_{-0.9}^{+2.3}$ & $42_{-6}^{+8}$ & & $4.7 \pm 0.3$ & $1.9 \pm 0.2$ & &\
& J172550-3533\_2 & 17:25:58.53 & $-$35:32:31.3 & 1.97 & 0.4 & 0.8 & & $44_{-7}^{+8}$ & $35_{-6}^{+7}$ & $7.9_{-2.8}^{+4.0}$ & & $1.2 \pm 0.1$ & $1.4 \pm 0.2$ & &\
& J172550-3533\_3 & 17:25:50.24 & $-$35:30:54.7 & 2.75 & 0.4 & 0.8 & & $40_{-6}^{+7}$ & $34_{-6}^{+7}$ & $5.9_{-2.4}^{+3.6}$ & & $1.3 \pm 0.1$ & $1.5 \pm 0.2$ & &\
\
& J172623-3516\_1 & 17:26:25.52 & $-$35:16:22.9 & 0.62 & 0.3 & 0.8 & & $43_{-7}^{+8}$ & $35_{-6}^{+7}$ & $8.0_{-2.8}^{+4.0}$ & & $1.4 \pm 0.1$ & $1.5 \pm 0.4$ & 172625.5-351622 &\
& J172623-3516\_2 & 17:26:25.89 & $-$35:15:22.5 & 1.52 & 0.6 & 0.9 & & $5.9_{-2.4}^{+3.6}$ & $5.0_{-2.2}^{+3.4}$ & $0.9_{-0.8}^{+2.3}$ & & $1.2 \pm 0.4$ & $1.7 \pm 0.8$ & &\
\
& J172642-3504\_1 & 17:26:39.21 & $-$35:04:18.4 & 0.68 & 0.3 & 0.8 & & $87_{-9}^{+10}$ & $7.9_{-2.8}^{+4.0}$ & $79_{-9}^{+10}$ & & $3.7 \pm 0.2$ & $1.5 \pm 0.1$ & & J172639.2-350418\
& J172642-3504\_2 & 17:26:37.20 & $-$35:05:00.7 & 1.43 & 0.6 & 0.9 & & $5.7_{-2.3}^{+3.6}$ & $1.0_{-0.9}^{+2.3}$ & $4.7_{-2.1}^{+3.4}$ & & $2.5 \pm 0.8$ & $1.4 \pm 0.3$ & &\
& J172642-3504\_3 & 17:26:47.45 & $-$35:04:56.7 & 1.41 & 0.7 & 1.0 & & $4.6_{-2.1}^{+3.4}$ & $1.9_{-1.3}^{+2.7}$ & $2.8_{-1.6}^{+2.9}$ & & $2.4 \pm 0.9$ & $0.7 \pm 0.4$ & &\
& J172642-3504\_4$\dagger$ & 17:26:45.22 & $-$35:02:05.6 & 2.02 & 0.5 & 0.8 & & $19_{-4}^{+5}$ & $16_{-4}^{+5}$ & $1.0_{-0.8}^{+2.3}$ & & $1.0 \pm 0.1$ & $1.1 \pm 0.5$ & & J172645.3-350205\
& J172642-3504\_5$\dagger$ & 17:26:36.84 & $-$35:02:09.5 & 2.16 & 0.9 & 1.2 & & $3.9_{-1.9}^{+3.2}$ & $0.0_{-0.0}^{+1.9}$ & $4.0_{-1.9}^{+3.2}$ & & & & &\
\
& J172642-3540\_1 & 17:26:41.73 & $-$35:40:52.5 & 0.22 & 0.3 & 0.8 & & $81_{-9}^{+10}$ & & & & & & 172641.7-354052 &\
& J172642-3540\_2\* & 17:26:55.49 & $-$35:41:48.1 & 2.78 & 1.3 & 1.5 & & $2.7_{-1.6}^{+2.9}$ & & & & & & &\
\
& J173548-3207\_1 & 17:35:46.28 & $-$32:07:9.8 & 0.37 & 0.3 & 0.8 & & $45_{-7}^{+8}$ & $29_{-5}^{+6}$ & $16_{-4}^{+5}$ & & $1.4 \pm 0.2$ & $1.2 \pm 0.2$ & 173546.2-320709 &\
\
& J173628-3141\_1 & 17:36:27.31 & $-$31:41:23.4 & 0.28 & 0.4 & 0.8 & & $10_{-3}^{+4}$ & $8.9_{-2.9}^{+4.1}$ & $0.0_{-0.0}^{+1.9}$ & & $0.9 \pm 0.1$ & $1.7 \pm 0.5$ & &\
& J173628-3141\_2 & 17:36:25.82 & $-$31:41:09.8 & 0.48 & 0.6 & 0.9 & & $5.0_{-2.2}^{+3.4}$ & $0.0_{-0.0}^{+1.9}$ & $5.0_{-2.2}^{+3.4}$ & & $5.7 \pm 1.4$ & $1.4 \pm 0.5$ & &\
& J173628-3141\_3 & 17:36:35.48 & $-$31:39:30.1 & 2.29 & 1.1 & 1.3 & & $3.9_{-1.9}^{+3.2}$ & $0.0_{-0.0}^{+1.9}$ & $3.9_{-1.9}^{+3.2}$ & & & & &\
\
& J175331-2538 & &\
\
& J175404-2553\_1$\dagger$ & 17:54:01.73 & $-$25:54:58.0 & 1.84 & 0.7 & 1.0 & & $5.8_{-2.3}^{+3.6}$ & $0.9_{-0.8}^{+2.3}$ & $4.9_{-2.1}^{+3.4}$ & & $3.8 \pm 0.9$ & $1.6 \pm 0.4$ & &\
& J175404-2553\_2\* & 17:54:14.42 & $-$25:53:54.4 & 2.37& 1.0 & 1.2 & & $2.6_{-1.5}^{+2.9}$ & & & & & & &\
& J175404-2553\_3 & 17:54:02.52 & $-$25:53:07.1 & 0.42 & 0.4 & 0.8 & & $28_{-5}^{+6}$ & $0.0_{-0.0}^{+1.9}$ & $28_{-5}^{+6}$ & & $4.9 \pm 0.2$ & $2.2 \pm 0.2$ & &\
\
& J180816-2021 & &\
\
& J180857-2004\_1 & 18:08:59.25 & $-$20:05:11.5 & 0.77 & 0.4 & 0.8 & & $43_{-7}^{+8}$ & $4.8_{-2.1}^{+3.4}$ & $39_{-6}^{+7}$ & & $3.3 \pm 0.3$ & $1.7 \pm 0.2$ & &\
& J180857-2004\_2 & 18:08:59.11 & $-$20:05:08.2 & 0.70 & 0.3 & 0.8 & & $77_{-9}^{+10}$ & $15_{-4}^{+5}$ & $63_{-8}^{+9}$ & & $3.2 \pm 0.2$ & $1.7 \pm 0.1$ & &\
& J180857-2004\_3 & 18:09:08.97 & $-$20:04:22.2 & 2.73 & 0.9 & 1.2 & & $6.7_{-2.5}^{+3.8}$ & $6.9_{-2.6}^{+3.8}$ & $0.0_{-0.0}^{+1.9}$ & & $1.2 \pm 0.1$ & $2.2 \pm 0.3$ & & J180909.0-200422\
& J180857-2004\_4 & 18:08:58.76 & $-$20:01:36.3 & 2.97 & 0.6 & 0.9 & & $13_{-4}^{+5}$ & $9.0_{-2.9}^{+4.1}$ & $3.9_{-1.9}^{+3.2}$ & & $1.2 \pm 0.4$ & $1.1 \pm 0.3$ & &\
\
& J181033-1917\_1 & 18:10:32.80 & $-$19:17:54.6 & 0.36 & 0.5 & 0.8 & & $9.0_{-3.0}^{+4.1}$ & $8.0_{-2.8}^{+4.0}$ & $1.0_{-0.9}^{+2.3}$ & & $1.3 \pm 0.3$ & $1.6 \pm 0.5$ & 181032.7-191755 &\
\
& J181116-1828\_1 & 18:11:20.83 & $-$18:28:19.9 & 0.96 & 0.5 & 0.9 & & $6.9_{-2.6}^{+3.8}$ & $7.0_{-2.6}^{+3.8}$ & $0.0_{-0.0}^{+1.9}$ & & $1.3 \pm 0.3$ & $1.1 \pm 0.6$ & &\
& J181116-1828\_2 & 18:11:13.11 & $-$18:27:58.6 & 0.92 & 0.3 & 0.8 & & $50_{-7}^{+8}$ & $15_{-4}^{+5}$ & $35_{-6}^{+7}$ & & $2.8 \pm 0.3$ & $1.3 \pm 0.2$ & 181113.1-182758 & J181113.2-182800\
& J181116-1828\_3 & 18:11:09.73 & $-$18:27:47.0 & 1.74 & 0.5 & 0.9 & & $10_{-3}^{+4}$ & $9.0_{-3.0}^{+4.1}$ & $1.0_{-0.9}^{+2.3}$ & & $1.1 \pm 0.1$ & $2.3 \pm 0.7$ & 181109.7-182746 &\
& J181116-1828\_4 & 18:11:19.08 & $-$18:30:48.4 & 2.60 & 1.1 & 1.3 & & $3.8_{-1.9}^{+3.2}$ & $3.9_{-1.9}^{+3.2}$ & $0.0_{-0.0}^{+1.9}$ & & & & &\
& J181116-1828\_5 & 18:11:05.79 & $-$18:28:21.7 & 2.61 & 0.7 & 1.0 & & $7.9_{-2.8}^{+4.0}$ & $0.9_{-0.8}^{+2.3}$ & $7.0_{-2.6}^{+3.8}$ & & $3.4 \pm 0.6$ & $1.9 \pm 0.5$ & &\
\
& J181120-1913\_1 & 18:11:23.27 & $-$19:16:15.4 & 2.71 & 0.9 & 1.2 & & $5.9_{-2.4}^{+3.6}$ & $0.0_{-0.0}^{+1.9}$ & $6.0_{-2.4}^{+3.6}$ & & & & &\
\
& J181213-1842\_1$\dagger$ & 18:12:20.87 & $-$18:44:03.0 & 2.31 & 0.6 & 0.9 & & $9.0_{-3.0}^{+4.1}$ & $9.0_{-3.0}^{+4.1}$ & $0.0_{-0.0}^{+1.9}$ & & $1.0 \pm 0.2$ & $2.4 \pm 0.6$ & &\
& J181213-1842\_2 & 18:12:14.94 & $-$18:39:37.8 & 2.80 & 0.8 & 1.1 & & $6.8_{-2.6}^{+3.8}$ & $7.0_{-2.6}^{+3.8}$ & $0.0_{-0.0}^{+1.9}$ & & $0.9 \pm 0.2$ & $1.5 \pm 0.3$ & &\
& J181213-1842\_3 & 18:12:12.54 & $-$18:43:27.6 & 1.09 & 0.6 & 0.9 & & $4.9_{-2.2}^{+3.4}$ & $2.0_{-1.3}^{+2.7}$ & $3.0_{-1.6}^{+2.9}$ & & $2.3 \pm 0.5$ & $1.6 \pm 0.3$ & &\
& J181213-1842\_4 & 18:12:12.87 & $-$18:41:26.1 & 1.02 & 0.7 & 1.0 & & $3.9_{-1.9}^{+3.2}$ & $1.9_{-1.3}^{+2.7}$ & $2.0_{-1.3}^{+2.7}$ & & & & &\
& J181213-1842\_5 & 18:12:08.65 & $-$18:40:34.8 & 2.22 & 1.1 & 1.4 & & $3.0_{-1.7}^{+2.9}$ & $2.0_{-1.3}^{+2.7}$ & $1.0_{-0.9}^{+2.3}$ & & & & &\
& J181213-1842\_6 & 18:12:05.06 & $-$18:43:25.1 & 2.32 & 0.9 & 1.2 & & $4.7_{-2.1}^{+3.4}$ & $4.9_{-2.2}^{+3.4}$ & $0.0_{-0.0}^{+1.9}$ & & $1.1 \pm 0.2$ & $1.6 \pm 0.4$ & &\
& J181213-1842\_7 & 18:12:10.38 & $-$18:42:08.8 & 0.88 & 0.3 & 0.8 & & $113_{-11}^{+12}$ & $17_{-4}^{+5}$ & $96_{-10}^{+11}$ & & $3.8 \pm 0.2$ & $1.3 \pm 0.1$ & 181210.3-184208 & J181210.5-184208\
& J181213-1842\_8 & 18:12:24.74 & $-$18:42:02.8 & 2.59 & 0.9 & 1.2 & & $4.9_{-2.1}^{+3.4}$ & $5.0_{-2.2}^{+3.4}$ & $0.0_{-0.0}^{+1.9}$ & & $1.3 \pm 0.1$ & $2.5 \pm 0.5$ & &\
& J181213-1842\_9 & 18:12:09.14 & $-$18:41:46.3 & 1.31 & 0.4 & 0.8 & & $22_{-5}^{+6}$ & $18_{-4}^{+5}$ & $4.0_{-1.9}^{+3.2}$ & & $1.0 \pm 0.2$ & $1.2 \pm 0.3$ & 181209.1-184146 & J181209.2-184149\
\
& J181705-1607\_1 & 18:17:04.38 & $-$16:10:03.5 & 2.15 & 1.1 & 1.3 & & $3.0_{-1.6}^{+2.9}$ & $0.0_{-0.0}^{+1.9}$ & $3.0_{-1.6}^{+2.9}$ & & & & &\
& J181705-1607\_2 & 18:17:03.79 & $-$16:06:27.4 & 1.51 & 0.7 & 1.0 & & $5.0_{-2.2}^{+3.4}$ & $4.0_{-1.9}^{+3.2}$ & $1.0_{-0.9}^{+2.3}$ & & & & &\
& J181705-1607\_3 & 18:17:03.41 & $-$16:09:04.4 & 1.24 & 0.7 & 1.0 & & $3.8_{-1.9}^{+3.2}$ & $1.0_{-0.8}^{+2.3}$ & $2.9_{-1.6}^{+2.9}$ & & & & &\
& J181705-1607\_4 & 18:16:56.60 & $-$16:08:23.5 & 2.14 & 0.8 & 1.1 & & $3.9_{-1.9}^{+3.2}$ & $3.0_{-1.6}^{+2.9}$ & $1.0_{-0.8}^{+2.3}$ & & & & &\
& J181705-1607\_5$\dagger$ & 18:17:05.84 & $-$16:05:57.6 & 1.96 & 0.5 & 0.9 & & $11_{-3}^{+4}$ & $1.0_{-0.9}^{+2.3}$ & $10_{-3}^{+4}$ & & $3.3 \pm 0.6$ & $1.7 \pm 0.3$ & 181705.8-160557 &\
& J181705-1607\_6 & 18:17:03.81 & $-$16:10:11.8 & 2.30 & 0.8 & 1.1 & & $6.9_{-2.6}^{+3.8}$ & $0.0_{-0.0}^{+1.9}$ & $7.0_{-2.6}^{+3.8}$ & & $4.6 \pm 0.8$ & $1.7 \pm 0.4$ & & J181703.8-161012\
& J181705-1607\_7 & 18:16:59.70 & $-$16:06:18.1 & 2.10 & 0.9 & 1.2 & & $3.8_{-1.9}^{+3.2}$ & $2.9_{-1.6}^{+2.9}$ & $0.9_{-0.8}^{+2.3}$ & & & & &\
\
& J181852-1559\_1 & 18:18:51.76 & $-$15:59:40.2 & 0.29 & 0.5 & 0.9 & & $6.0_{-2.4}^{+3.6}$ & $6.0_{-2.4}^{+3.6}$ & $0.0_{-0.0}^{+1.9}$ & & $1.0 \pm 0.1$ & $1.6 \pm 0.3$ & 181851.8-155940 &\
& J181852-1559\_2 & 18:18:51.38 & $-$15:59:22.4 & 0.30 & 0.3 & 0.8 & & $41_{-6}^{+7}$ & $2.0_{-1.3}^{+2.7}$ & $39_{-6}^{+7}$ & & $3.8 \pm 0.3$ & $1.6 \pm 0.2$ & 181851.3-155922 &\
& J181852-1559\_3 & 18:18:50.09 & $-$15:59:16.3 & 0.62 & 0.7 & 1.0 & & $3.0_{-1.7}^{+2.9}$ & $3.0_{-1.7}^{+2.9}$ & $0.0_{-0.0}^{+1.9}$ & & & & &\
\
& J181915-1601\_1 & 18:19:12.98 & $-$16:03:40.6 & 2.41 & 1.0 & 1.2 & & $3.0_{-1.7}^{+2.9}$ & $0.0_{-0.0}^{+1.9}$ & $3.0_{-1.7}^{+2.9}$ & & & & &\
& J181915-1601\_2$\dagger$ & 18:19:22.19 & $-$16:03:12.5 & 2.35 & 0.4 & 0.8 & & $54_{-7}^{+8}$ & $25_{-5}^{+6}$ & $29_{-5}^{+6}$ & & $2.3 \pm 0.3$ & $1.1 \pm 0.1$ & 181922.1-160309 & J181922.1-160310\
& J181915-1601\_3 & 18:19:13.52 & $-$16:01:21.8 & 0.62 & 0.3 & 0.8 & & $75_{-9}^{+10}$ & $27_{-5}^{+6}$ & $48_{-7}^{+8}$ & & $2.4 \pm 0.3$ & $1.1 \pm 0.1$ & &\
\
& J181917-1548\_1 & 18:19:08.54 & $-$15:49:27.3 & 2.48 & 0.6 & 0.9 & & $13_{-4}^{+5}$ & & & & & & 181907.8-154928 &\
\
& J182216-1425\_1 & 18:22:24.24 & $-$14:26:13.1 & 2.12 & 0.5 & 0.9 & & $16_{-4}^{+5}$ & $0.0_{-0.0}^{+1.9}$ & $16_{-4}^{+5}$ & & $4.9 \pm 0.2$ & $2.3 \pm 0.3$ & &\
& J182216-1425\_2 & 18:22:17.16 & $-$14:22:32.6 & 2.91 & 0.7 & 1.0 & & $8.9_{-2.9}^{+4.1}$ & $0.0_{-0.0}^{+1.9}$ & $8.9_{-2.9}^{+4.1}$ & & $4.3 \pm 0.5$ & $2.0 \pm 0.3$ & &\
\
& J182435-1311\_1 & 18:24:37.61 & $-$13:10:36.7 & 0.71 & 0.3 & 0.8 & & $91_{-10}^{+11}$ & $76_{-9}^{+10}$ & $15_{-4}^{+5}$ & & $1.5 \pm 0.1$ & $1.8 \pm 0.1$ & &\
& J182435-1311\_2 & 18:24:32.90 & $-$13:12:07.7 & 1.19 & 0.7 & 1.0 & & $4.5_{-2.1}^{+3.4}$ & $3.9_{-1.9}^{+3.2}$ & $0.7_{-0.6}^{+2.3}$ & & $1.2 \pm 1.0$ & $0.9 \pm 0.6$ & &\
& J182435-1311\_3 & 18:24:33.72 & $-$13:09:38.4 & 1.60 & 0.6 & 0.9 & & $5.6_{-2.3}^{+3.6}$ & $0.0_{-0.0}^{+1.9}$ & $5.9_{-2.4}^{+3.6}$ & & $4.0 \pm 0.4$ & $2.2 \pm 0.3$ & &\
& J182435-1311\_4 & 18:24:39.38 & $-$13:08:50.2 & 2.48 & 0.6 & 0.9 & & $8.5_{-2.9}^{+4.1}$ & $4.7_{-2.1}^{+3.4}$ & $3.9_{-1.9}^{+3.2}$ & & $1.7 \pm 0.3$ & $1.6 \pm 0.8$ & &\
\
& J182509-1253\_1 & 18:25:08.73 & $-$12:53:31.6 & 0.33 & 0.3 & 0.8 & & $90_{-9}^{+11}$ & $78_{-9}^{+10}$ & $11_{-3}^{+4}$ & & $1.2 \pm 0.1$ & $1.5 \pm 0.1$ & &\
& J182509-1253\_2 & 18:25:03.16 & $-$12:52:11.1 & 2.00 & 0.5 & 0.9 & & $10_{-3}^{+4}$ & $10_{-3}^{+4}$ & $0.0_{-0.0}^{+1.9}$ & & $0.9 \pm 0.2$ & $1.6 \pm 0.5$ & &\
& J182509-1253\_3 & 18:25:11.59 & $-$12:51:57.6 & 1.45 & 0.4 & 0.8 & & $47_{-7}^{+8}$ & $0.9_{-0.8}^{+2.3}$ & $46_{-7}^{+8}$ & & $4.4 \pm 0.2$ & $1.8 \pm 0.2$ & &\
\
& J182530-1144\_1\* & 18:25:26.77 & $-$11:45:28.7 & 1.09 & 0.8 & 1.1 & & $2.8_{-1.6}^{+2.9}$ & & & & & & &\
& J182530-1144\_2 & 18:25:24.77 & $-$11:45:24.7 & 1.50 & 0.3 & 0.8 & & $236_{-15}^{+16}$ & $54_{-7}^{+8}$ & $182_{-13}^{+15}$ & & $3.5 \pm 0.2$ & $1.2 \pm 0.1$ & &\
\
& J182538-1214\_1 & 18:25:33.69 & $-$12:14:50.8 & 1.21 & 0.4 & 0.8 & & $32_{-6}^{+7}$ & $26_{-5}^{+6}$ & $5.0_{-2.2}^{+3.4}$ & & $1.0 \pm 0.1$ & $1.6 \pm 0.4$ & & J182533.6-121452\
\
& J182651-1206\_1 & 18:26:55.40 & $-$12:05:14.1 & 1.58 & 0.5 & 0.9 & & $7.9_{-2.8}^{+4.0}$ & $2.9_{-1.6}^{+2.9}$ & $5.0_{-2.2}^{+3.4}$ & & $2.6 \pm 0.6$ & $1.6 \pm 0.5$ & &\
& J182651-1206\_2$\dagger$ & 18:26:55.83 & $-$12:04:42.2 & 2.07 & 0.9 & 1.2 & & $3.0_{-1.6}^{+2.9}$ & $0.0_{-0.0}^{+1.9}$ & $0.0_{-0.0}^{+1.9}$ & & & & &\
& J182651-1206\_3 & 18:26:58.59 & $-$12:08:43.4 & 2.94 & 0.7 & 1.0 & & $13_{-4}^{+5}$ & $0.0_{-0.0}^{+1.9}$ & $14_{-4}^{+5}$ & & $5.1 \pm 0.4$ & $2.3 \pm 0.1$ & &\
& J182651-1206\_4 & 18:26:46.88 & $-$12:07:05.7 & 1.24 & 0.4 & 0.8 & & $22_{-5}^{+6}$ & $15_{-4}^{+5}$ & $5.9_{-2.4}^{+3.6}$ & & $1.6 \pm 0.2$ & $1.2 \pm 0.3$ & &\
& J182651-1206\_5 & 18:26:48.75 & $-$12:06:32.2 & 0.59 & 0.5 & 0.8 & & $10_{-3}^{+4}$ & $2.0_{-1.3}^{+2.7}$ & $7.9_{-2.8}^{+4.0}$ & & $3.8 \pm 1.5$ & $1.1 \pm 0.4$ & &\
\
& J183116-1008\_1 & 18:31:16.52 & $-$10:09:24.8 & 0.84 & 0.3 & 0.8 & & $86_{-9}^{+10}$ & $5.9_{-2.4}^{+3.6}$ & $80_{-9}^{+10}$ & & $3.4 \pm 0.2$ & $1.7 \pm 0.1$ & 183116.4-100924 & J183116.5-100926\
& J183116-1008\_2\* & 18:31:16.44 & $-$10:09:10.6 & 0.60 & 0.7 & 1.0 & & $2.8_{-1.6}^{+2.9}$ & & & & & & &\
& J183116-1008\_3 & 18:31:21.00 & $-$10:08:00.1 & 1.29 & 0.6 & 0.9 & & $5.9_{-2.4}^{+3.6}$ & $2.9_{-1.6}^{+2.9}$ & $2.0_{-1.3}^{+2.7}$ & & $1.8 \pm 0.9$ & $1.1 \pm 0.7$ & &\
\
& J183206-0938\_1 & 18:32:08.95 & $-$09:39:05.7 & 0.84 & 0.3 & 0.8 & & $153_{-12}^{+13}$ & $32_{-6}^{+7}$ & $121_{-11}^{+12}$ & & $3.3 \pm 0.2$ & $1.3 \pm 0.1$ & 183208.9-093905 &\
\
& J183345-0828\_1 & 18:33:40.39 & $-$08:28:30.8 & 1.34 & 0.4 & 0.8 & & $40_{-6}^{+7}$ & $11_{-3}^{+4}$ & $29_{-5}^{+6}$ & & $2.6 \pm 0.3$ & $1.4 \pm 0.2$ & &\
& J183345-0828\_2 & 18:33:45.30 & $-$08:25:18.9 & 2.83 & 1.0 & 1.2 & & $3.8_{-1.9}^{+3.2}$ & $4.0_{-1.9}^{+3.2}$ & $0.0_{-0.0}^{+1.9}$ & & & & & J183345.1-082520\
\
& J183356-0822\_1$\dagger$ & 18:33:50.37 & $-$08:23:03.3 & 1.56 & 0.5 & 0.8 & & $10_{-3}^{+4}$ & $7.9_{-2.8}^{+4.0}$ & $1.8_{-1.2}^{+2.7}$ & & $1.4 \pm 0.2$ & $1.6 \pm 0.4$ & & J183350.4-082300\
& J183356-0822\_2 & 18:33:59.51 & $-$08:22:26.8 & 0.85 & 0.3 & 0.8 & & $66_{-8}^{+9}$ & $0.0_{-0.0}^{+1.9}$ & $66_{-8}^{+9}$ & & $4.5 \pm 0.3$ & $2.1 \pm 0.1$ & & J183359.6-082225\
& J183356-0822\_3 & 18:33:55.97 & $-$08:21:40.2 & 1.26 & 0.4 & 0.8 & & $22_{-5}^{+6}$ & $6.0_{-2.4}^{+3.6}$ & $16_{-4}^{+5}$ & & $3.1 \pm 0.4$ & $1.3 \pm 0.3$ & & J183356.1-082138\
& J183356-0822\_4 & 18:33:52.12 & $-$08:22:22.6 & 1.24 & 0.4 & 0.8 & & $13_{-4}^{+5}$ & $0.9_{-0.8}^{+2.3}$ & $12_{-3}^{+5}$ & & $3.4 \pm 0.6$ & $1.6 \pm 0.3$ & &\
& J183356-0822\_5 & 18:33:49.44 & $-$08:21:01.6 & 2.60 & 0.7 & 1.0 & & $7.8_{-2.7}^{+4.0}$ & $0.0_{-0.0}^{+1.9}$ & $7.8_{-2.7}^{+4.0}$ & & $5.0 \pm 0.5$ & $2.2 \pm 0.4$ & &\
& J183356-0822\_6 & 18:34:05.42 & $-$08:20:48.4 & 3.03 & 0.9 & 1.1 & & $5.7_{-2.3}^{+3.6}$ & $0.0_{-0.0}^{+1.9}$ & $5.9_{-2.4}^{+3.6}$ & & $4.1 \pm 0.9$ & $1.8 \pm 0.5$ & &\
& J183356-0822\_7 & 18:34:05.65 & $-$08:21:03.2 & 2.91 & 0.9 & 1.1 & & $6.6_{-2.5}^{+3.8}$ & $5.0_{-2.2}^{+3.4}$ & $1.8_{-1.2}^{+2.7}$ & & $1.6 \pm 0.5$ & $1.5 \pm 0.6$ & &\
\
& J183518-0754\_1 & 18:35:24.44 & $-$07:54:38.2 & 1.54 & 0.6 & 0.9 & & $9.1_{-3.0}^{+4.3}$ & & & & & & &\
\
& J183607-0756 & &\
\
& J184008-0543\_1 & 18:40:04.88 & $-$05:44:15.5 & 0.96 & 0.5 & 0.8 & & $8.9_{-2.9}^{+4.1}$ & $0.0_{-0.0}^{+1.9}$ & $8.9_{-2.9}^{+4.1}$ & & $3.8 \pm 0.3$ & $2.3 \pm 0.3$ & 184004.8-054415 &\
& J184008-0543\_2 & 18:40:03.45 & $-$05:44:03.4 & 1.25 & 0.6 & 0.9 & & $6.0_{-2.4}^{+3.6}$ & $0.0_{-0.0}^{+1.9}$ & $6.0_{-2.4}^{+3.6}$ & & $4.6 \pm 0.8$ & $1.9 \pm 0.3$ & 184003.4-054403 &\
\
& J184400-0355\_1 & 18:44:03.29 & $-$03:58:18.6 & 2.81 & 0.8 & 1.1 & & $11_{-3}^{+4}$ & $11_{-3}^{+4}$ & $0.0_{-0.0}^{+1.9}$ & & $1.0 \pm 0.2$ & $1.7 \pm 0.6$ & 184403.2-035819 &\
\
& J184447-0305\_1 & 18:44:41.84 & $-$03:05:51.2 & 1.47 & 0.7 & 1.0 & & $3.9_{-1.9}^{+3.2}$ & $1.0_{-0.8}^{+2.3}$ & $3.0_{-1.6}^{+2.9}$ & & & & & J184441.9-030551\
\
& J184652-0240\_1 & 18:46:47.68 & $-$02:41:20.1 & 1.61 & 0.3 & 0.8 & & $86_{-9}^{+10}$ & $47_{-7}^{+8}$ & $39_{-6}^{+7}$ & & $1.9 \pm 0.1$ & $1.5 \pm 0.2$ & 184647.6-024120 &\
& J184652-0240\_2 & 18:46:56.27 & $-$02:39:53.4 & 0.98 & 0.7 & 1.0 & & $3.9_{-1.9}^{+3.2}$ & $3.0_{-1.6}^{+2.9}$ & $1.0_{-0.8}^{+2.3}$ & & & & &\
\
& J184738-0156\_1 & 18:47:36.65 & $-$01:56:33.3 & 0.50 & 0.3 & 0.8 & & $120_{-11}^{+12}$ & $3.0_{-1.7}^{+2.9}$ & $117_{-11}^{+12}$ & & $4.2 \pm 0.2$ & $2.0 \pm 0.1$ & 184736.6-015633 & J184736.6-015633\
\
& J184741-0219\_1\* & 18:47:48.01 & $-$02:21:25.4 & 2.93 & 1.5 & 1.7 & & $2.6_{-1.5}^{+2.9}$ & & & & & & &\
& J184741-0219\_2 & 18:47:36.52 & $-$02:19:32.2 & 1.30 & 0.6 & 0.9 & & $5.8_{-2.4}^{+3.6}$ & $4.9_{-2.2}^{+3.4}$ & $0.9_{-0.8}^{+2.3}$ & & $1.7 \pm 0.5$ & $1.3 \pm 0.7$ & &\
& J184741-0219\_3 & 18:47:37.28 & $-$02:18:49.4 & 1.02 & 0.4 & 0.8 & & $32_{-6}^{+7}$ & $1.0_{-0.8}^{+2.3}$ & $31_{-6}^{+7}$ & & $5.0 \pm 0.3$ & $1.9 \pm 0.2$ & &\
& J184741-0219\_4 & 18:47:42.99 & $-$02:18:22.2 & 0.78 & 0.5 & 0.9 & & $8.8_{-2.9}^{+4.1}$ & $6.9_{-2.6}^{+3.8}$ & $1.9_{-1.3}^{+2.7}$ & & $1.3 \pm 0.3$ & $1.3 \pm 0.3$ & &\
& J184741-0219\_5 & 18:47:44.30 & $-$02:19:16.5 & 0.80 & 0.8 & 1.0 & & $4.0_{-1.9}^{+3.2}$ & $4.0_{-1.9}^{+3.2}$ & $0.0_{-0.0}^{+1.9}$ & & & & &\
\
& J185608+0218\_1 & 18:56:09.55 & +02:17:44.0 & 0.61 & 0.3 & 0.8 & & $192_{-14}^{+15}$ & $165_{-13}^{+14}$ & $20_{-4}^{+6}$ & & $1.0 \pm 0.0$ & $1.6 \pm 0.1$ & 185609.5+021743 &\
& J185608+0218\_2 & 18:56:05.53 & +02:18:22.1 & 0.62 & 0.5 & 0.8 & & $8.0_{-2.8}^{+4.0}$ & $1.0_{-0.9}^{+2.3}$ & $7.0_{-2.6}^{+3.8}$ & & $4.9 \pm 0.9$ & $1.8 \pm 0.4$ & &\
\
& J185643+0220\_1 & 18:56:48.00 & +02:18:36.6 & 2.06 & 0.6 & 0.9 & & $11_{-3}^{+4}$ & $9.0_{-2.9}^{+4.1}$ & $1.9_{-1.3}^{+2.7}$ & & $1.3 \pm 0.2$ & $1.4 \pm 0.5$ & &\
& J185643+0220\_2 & 18:56:43.61 & +02:19:21.5 & 0.93 & 0.3 & 0.8 & & $69_{-8}^{+9}$ & $4.0_{-1.9}^{+3.2}$ & $65_{-8}^{+9}$ & & $3.9 \pm 0.2$ & $1.4 \pm 0.2$ & &\
& J185643+0220\_3 & 18:56:43.98 & +02:20:35.9 & 0.37 & 0.6 & 0.9 & & $4.8_{-2.1}^{+3.4}$ & $0.9_{-0.8}^{+2.3}$ & $3.9_{-1.9}^{+3.2}$ & & $2.9 \pm 0.5$ & $1.9 \pm 0.3$ & &\
\
& J185750+0240\_1 & 18:57:53.65 & +02:40:10.8 & 0.71 & 0.3 & 0.8 & & $44_{-7}^{+8}$ & $11_{-3}^{+4}$ & $33_{-6}^{+7}$ & & $3.2 \pm 0.4$ & $1.1 \pm 0.2$ & 185753.6+024010 &\
\
& J185905+0333 & &\
\
& J190534+0659\_1 & 19:05:32.78 & +06:59:33.6 & 0.72 & 0.4 & 0.8 & & $22_{-5}^{+6}$ & $18_{-4}^{+5}$ & $4.0_{-1.9}^{+3.2}$ & & $1.6 \pm 0.2$ & $1.3 \pm 0.4$ & 190532.7+065933 &\
\
& J190749+0803\_1 & 19:07:48.84 & +08:03:42.9 & 0.27 & 0.3 & 0.8 & & $91_{-10}^{+11}$ & $9.0_{-3.0}^{+4.1}$ & $82_{-9}^{+10}$ & & $3.7 \pm 0.2$ & $1.7 \pm 0.1$ & 190748.8+080342 &\
\
& J190814+0832\_1 & 19:08:11.18 & +08:29:53.1 & 2.70 & 1.0 & 1.2 & & $3.8_{-1.9}^{+3.2}$ & $1.0_{-0.8}^{+2.3}$ & $2.9_{-1.6}^{+2.9}$ & & & & &\
& J190814+0832\_2 & 19:08:09.95 & +08:31:47.7 & 1.29 & 0.4 & 0.8 & & $43_{-7}^{+8}$ & $31_{-6}^{+7}$ & $12_{-3}^{+5}$ & & $1.6 \pm 0.1$ & $1.4 \pm 0.2$ & 190809.9+083147 &\
& J190814+0832\_3 & 19:08:15.90 & +08:32:40.6 & 0.43 & 0.6 & 0.9 & & $4.9_{-2.2}^{+3.4}$ & $5.0_{-2.2}^{+3.4}$ & $0.0_{-0.0}^{+1.9}$ & & $0.9 \pm 0.1$ & $1.9 \pm 0.3$ & &\
& J190814+0832\_4 & 19:08:19.00 & +08:33:43.8 & 1.71 & 0.8 & 1.1 & & $3.8_{-1.9}^{+3.2}$ & $0.0_{-0.0}^{+1.9}$ & $3.9_{-1.9}^{+3.2}$ & & & & &\
\
& J190818+0745\_1 & 19:08:13.20 & +07:43:57.7 & 2.00 & 0.3 & 0.8 & & $138_{-12}^{+13}$ & $79_{-9}^{+10}$ & $58_{-8}^{+9}$ & & $1.8 \pm 0.1$ & $1.0 \pm 0.1$ & 190813.2+074357 &\
& J190818+0745\_2 & 19:08:25.46 & +07:44:28.3 & 1.92 & 0.5 & 0.9 & & $12_{-3}^{+5}$ & $4.0_{-1.9}^{+3.2}$ & $7.9_{-2.8}^{+4.0}$ & & $3.7 \pm 1.3$ & $0.8 \pm 0.3$ & 190825.4+074428 &\
& J190818+0745\_3 & 19:08:22.73 & +07:47:48.9 & 2.59 & 1.1 & 1.3 & & $3.0_{-1.6}^{+2.9}$ & $1.0_{-0.9}^{+2.3}$ & $0.0_{-0.0}^{+1.9}$ & & & & &\
\
& J191046+0917 & &\
\
& J194152+2251\_1\* & 19:41:56.53 & +22:49:06.9 & 2.73 & 1.3 & 1.5 & & $2.9_{-1.6}^{+2.9}$ & & & & & & &\
& J194152+2251\_2 & 19:41:54.60 & +22:51:12.5 & 0.63 & 0.3 & 0.8 & & $72_{-8}^{+10}$ & $13_{-4}^{+5}$ & $59_{-8}^{+9}$ & & $3.4 \pm 0.3$ & $1.2 \pm 0.1$ & 194154.6+225112 &\
& J194152+2251\_3 & 19:41:52.95 & +22:54:18.3 & 2.59 & 1.1 & 1.3 & & $2.9_{-1.6}^{+2.9}$ & $3.0_{-1.7}^{+2.9}$ & $0.0_{-0.0}^{+1.9}$ & & & & &\
\
& J194310+2318\_1 & 19:43:06.77 & +23:16:12.5 & 2.76 & 0.6 & 0.9 & & $19_{-4}^{+5}$ & $15_{-4}^{+5}$ & $4.0_{-1.9}^{+3.2}$ & & $1.1 \pm 0.2$ & $1.4 \pm 0.4$ & &\
& J194310+2318\_2\* & 19:43:07.78 & +23:17:03.1 & 1.88 & 0.8 & 1.1 & & $3.7_{-1.8}^{+3.2}$ & & & & & & &\
& J194310+2318\_3 & 19:43:04.94 & +23:18:33.8 & 1.27 & 0.8 & 1.1 & & $4.0_{-1.9}^{+3.2}$ & $2.0_{-1.3}^{+2.7}$ & $2.0_{-1.3}^{+2.7}$ & & & & &\
& J194310+2318\_4$\dagger$ & 19:43:11.33 & +23:20:38.9 & 1.82 & 0.7 & 1.0 & & $4.9_{-2.1}^{+3.4}$ & $4.0_{-1.9}^{+3.2}$ & $0.9_{-0.8}^{+2.3}$ & & $0.9 \pm 0.5$ & $1.0 \pm 0.4$ & &\
& J194310+2318\_5 & 19:43:10.98 & +23:17:45.5 & 1.09 & 0.4 & 0.8 & & $31_{-6}^{+7}$ & $30_{-5}^{+7}$ & $0.0_{-0.0}^{+1.9}$ & & $1.0 \pm 0.1$ & $1.3 \pm 0.2$ & &\
& J194310+2318\_6 & 19:43:07.07 & +23:18:31.7 & 0.81 & 0.4 & 0.8 & & $20_{-4}^{+6}$ & $17_{-4}^{+5}$ & $3.0_{-1.6}^{+2.9}$ & & $1.6 \pm 0.1$ & $1.8 \pm 0.4$ & &\
& J194310+2318\_7$\dagger$ & 19:43:14.53 & +23:20:16.4 & 1.73 & 0.6 & 0.9 & & $7.0_{-2.6}^{+3.8}$ & $6.0_{-2.4}^{+3.6}$ & $1.0_{-0.8}^{+2.3}$ & & $1.2 \pm 0.4$ & $2.0 \pm 1.0$ & &\
& J194310+2318\_8 & 19:43:13.06 & +23:17:25.2 & 1.55 & 0.7 & 1.0 & & $4.9_{-2.2}^{+3.4}$ & $2.0_{-1.3}^{+2.7}$ & $3.0_{-1.6}^{+2.9}$ & & $2.7 \pm 1.5$ & $0.6 \pm 0.4$ & &\
& J194310+2318\_9 & 19:43:12.58 & +23:18:15.8 & 0.78 & 0.6 & 0.9 & & $5.8_{-2.4}^{+3.6}$ & $6.0_{-2.4}^{+3.6}$ & $0.0_{-0.0}^{+1.9}$ & & $1.1 \pm 0.1$ & $2.1 \pm 0.3$ & &\
& J194310+2318\_10 & 19:43:08.44 & +23:18:17.3 & 0.70 & 0.6 & 0.9 & & $4.0_{-1.9}^{+3.2}$ & $2.0_{-1.3}^{+2.7}$ & $2.0_{-1.3}^{+2.7}$ & & & & &\
\
& J194332+2323\_1 & 19:43:27.18 & +23:23:31.2 & 1.30 & 0.4 & 0.8 & & $15_{-4}^{+5}$ & $12_{-3}^{+5}$ & $3.0_{-1.6}^{+2.9}$ & & $1.3 \pm 0.2$ & $1.4 \pm 0.4$ & &\
& J194332+2323\_2 & 19:43:31.97 & +23:24:55.6 & 1.06 & 0.6 & 0.9 & & $3.9_{-1.9}^{+3.2}$ & $4.0_{-1.9}^{+3.2}$ & $0.0_{-0.0}^{+1.9}$ & & & & &\
& J194332+2323\_3 & 19:43:43.32 & +23:24:56.4 & 2.67 & 0.8 & 1.0 & & $6.8_{-2.6}^{+3.8}$ & $6.9_{-2.6}^{+3.8}$ & $0.0_{-0.0}^{+1.9}$ & & $1.5 \pm 0.2$ & $2.1 \pm 0.3$ & &\
& J194332+2323\_4 & 19:43:29.79 & +23:25:09.7 & 1.44 & 0.7 & 1.0 & & $3.9_{-1.9}^{+3.2}$ & $3.0_{-1.7}^{+2.9}$ & $0.9_{-0.8}^{+2.3}$ & & & & &\
& J194332+2323\_5$\dagger$ & 19:43:33.02 & +23:26:24.6 & 2.53 & 0.5 & 0.9 & & $14_{-4}^{+5}$ & $13_{-4}^{+5}$ & $1.0_{-0.9}^{+2.3}$ & & $1.5 \pm 0.1$ & $1.6 \pm 1.1$ & &\
& J194332+2323\_6 & 19:43:23.05 & +23:25:17.5 & 2.61 & 1.0 & 1.2 & & $6.8_{-2.6}^{+3.8}$ & $5.9_{-2.4}^{+3.6}$ & $0.9_{-0.8}^{+2.3}$ & & $1.1 \pm 0.6$ & $1.3 \pm 1.0$ & &\
& J194332+2323\_7 & 19:43:28.11 & +23:21:13.6 & 2.85 & 0.8 & 1.1 & & $8.7_{-2.9}^{+4.1}$ & $8.0_{-2.8}^{+4.0}$ & $0.0_{-0.0}^{+1.9}$ & & $0.8 \pm 0.1$ & $1.6 \pm 0.5$ & &\
& J194332+2323\_8 & 19:43:23.24 & +23:23:26.4 & 2.20 & 0.9 & 1.1 & & $4.8_{-2.1}^{+3.4}$ & $4.9_{-2.2}^{+3.4}$ & $0.0_{-0.0}^{+1.9}$ & & & & &\
\
& J194622+2436\_1 & 19:46:24.99 & +24:35:46.8 & 1.17 & 0.8 & 1.1 & & $3.0_{-1.7}^{+2.9}$ & $1.0_{-0.9}^{+2.3}$ & $2.0_{-1.3}^{+2.7}$ & & & & &\
& J194622+2436\_2 & 19:46:34.33 & +24:37:05.9 & 2.75 & 1.1 & 1.3 & & $4.0_{-1.9}^{+3.2}$ & $4.0_{-1.9}^{+3.2}$ & $0.0_{-0.0}^{+1.9}$ & & & & &\
& J194622+2436\_3 & 19:46:15.28 & +24:38:32.1 & 2.37 & 1.0 & 1.2 & & $3.0_{-1.6}^{+2.9}$ & $2.0_{-1.3}^{+2.7}$ & $1.0_{-0.8}^{+2.3}$ & & & & &\
\
& J194939+2631\_1 & 19:49:38.38 & +26:31:49.0 & 0.33 & 0.3 & 0.8 & & $114_{-11}^{+12}$ & $17_{-4}^{+5}$ & $97_{-10}^{+11}$ & & $3.8 \pm 0.1$ & $1.6 \pm 0.1$ & 194938.3+263149 &\
\
& J194951+2534\_1\* & 19:49:48.84 & +25:35:53.1 & 1.63 & 0.9 & 1.1 & & $2.7_{-1.6}^{+2.9}$ & & & & & & &\
\
& J195006+2628\_1 & 19:50:16.16 & +26:27:17.1 & 2.40 & 0.7 & 1.0 & & $6.9_{-2.6}^{+3.8}$ & $7.0_{-2.6}^{+3.8}$ & $0.0_{-0.0}^{+1.9}$ & & $1.0 \pm 0.1$ & $2.1 \pm 0.6$ & &\
& J195006+2628\_2 & 19:50:04.86 & +26:28:21.8 & 0.37 & 0.4 & 0.8 & & $28_{-5}^{+6}$ & $25_{-5}^{+6}$ & $2.7_{-1.6}^{+2.9}$ & & $1.1 \pm 0.1$ & $1.5 \pm 0.4$ & &\
& J195006+2628\_3 & 19:50:03.54 & +26:27:43.3 & 0.89 & 0.6 & 0.9 & & $5.7_{-2.3}^{+3.6}$ & $3.0_{-1.7}^{+2.9}$ & $2.7_{-1.6}^{+2.9}$ & & $1.4 \pm 0.5$ & $1.4 \pm 0.5$ & &\
& J195006+2628\_4 & 19:50:05.25 & +26:28:17.5 & 0.28 & 0.5 & 0.8 & & $6.8_{-2.6}^{+3.8}$ & $7.0_{-2.6}^{+3.8}$ & $0.0_{-0.0}^{+1.9}$ & & $1.5 \pm 0.2$ & $1.9 \pm 0.4$ & &\
& J195006+2628\_5 & 19:50:01.10 & +26:29:34.2 & 1.74 & 0.5 & 0.9 & & $12_{-3}^{+5}$ & $12_{-3}^{+5}$ & $0.0_{-0.0}^{+1.9}$ & & $1.1 \pm 0.1$ & $2.0 \pm 0.4$ & &
[lccccccccccc]{} J143148-6021\_3 & P & & $17.1 \pm 0.1 $ & $16.3 \pm 0.1 $ & $15.6 \pm 0.1 $ & 2005-05-12 & & & & &\
J144042-6001\_1 & M & 14403847-6001368 & $8.28 \pm 0.02 $ & $7.88 \pm 0.02 $ & $7.70 \pm 0.02 $ & & 316.1785+00.0060 & $ 7.65 \pm 0.06 $ & $ 7.63 \pm 0.05 $ & $ 7.64 \pm 0.03 $ & $ 7.59 \pm 0.03 $\
J144519-5949\_2 & M & 14452143-5949251 & $14.12 \pm 0.04 $ & $10.95 \pm 0.03 $ & $9.30 \pm 0.03 $ & & & & & &\
J144547-5931\_1 & M & 14454369-5932050 & $9.23 \pm 0.04 $ & $8.10 \pm 0.03 $ & $7.52 \pm 0.02 $ & & 316.9656+00.1864 & $ 7.07 \pm 0.05 $ & $ 6.89 \pm 0.04 $ & $ 6.708 \pm 0.03 $ & $ 6.63 \pm 0.02 $\
J144701-5919\_1 & M & 14465358-5919382 & $8.93 \pm 0.02 $ & $7.60 \pm 0.04 $ & $6.82 \pm 0.02 $ & & 317.1882+00.3106 & $ 6.70 \pm 0.09^{\dagger} $ & $ 6.33 \pm 0.09^{\dagger} $ & $ 5.574 \pm 0.03 $ & $ 5.32 \pm 0.03 $\
J150436-5824\_1 & & & & & & & & & & &\
J153818-5541\_1 & P\*\* & & $>21.9 \pm 0.3 $ & $>19.5 \pm 0.15 $ & $18.0 \pm 0.5 $ & 2009-04-13 & & & & &\
J154122-5522\_1 & M & 15411580-5521286 & $9.60 \pm 0.02 $ & $8.93 \pm 0.06 $ & $8.68 \pm 0.02 $ & & 325.4962-00.1082 & $ 8.61 \pm 0.05 $ & $ 8.68 \pm 0.06 $ & $ 8.51 \pm 0.04 $ & $ 8.47 \pm 0.03 $\
J154557-5443\_1 & M & 15455824-5444247 & $>13.52 \pm 0.0 $ & $>12.82 \pm 0.0 $ & $12.92 \pm 0.05 $ & & 326.4069-00.0270 & $ 12.78 \pm 0.07 $ & $ 12.91 \pm 0.12 $ & & $ 11.30 \pm 0.14 $\
J154557-5443\_2 & & & & & & & & & & &\
J154557-5443\_3 & M & 15461218-5443106 & $13.82 \pm 0.02 $ & $13.12 \pm 0.02 $ & $12.93 \pm 0.04 $ & & 326.4461-00.0314 & $ 12.54 \pm 0.08^{\dagger} $ & $ 12.67 \pm 0.17^{\dagger} $ & $ 11.75\pm 0.22^{\dagger} $ &\
J155035-5408\_1 & M & 15502123-5409540 & $9.90 \pm 0.02 $ & $9.71 \pm 0.02 $ & $9.59 \pm 0.02 $ & & 327.2633+00.0289 & $ 9.60 \pm 0.04 $ & $ 9.59 \pm 0.05 $ & $ 9.51 \pm 0.05 $ & $ 9.58 \pm 0.05 $\
J155035-5408\_3 & M & 15503759-5407217 & $11.02 \pm 0.03 $ & $10.13 \pm 0.05 $ & $9.82 \pm 0.06 $ & & 327.3210+00.0369 & $ 9.50 \pm 0.05 $ & $ 9.65 \pm 0.06 $ & $ 9.42 \pm 0.06 $ & $ 9.45 \pm 0.05 $\
J155331-5347\_1 & M & 15533602-5348270 & $10.82 \pm 0.02 $ & $10.19 \pm 0.03 $ & $9.98 \pm 0.03 $ & & 327.8592+00.0053 & $ 9.97 \pm 0.06 $ & $ 9.95 \pm 0.06 $ & $ 9.80 \pm 0.08 $ & $ 9.72 \pm 0.05 $\
J155831-5334\_1 & M,P,P & 15582833-5335205 & $15.20 \pm 0.06 $ & $14.74 \pm 0.11 $ & $14.58 \pm 0.11 $ & 2009-04-14 & & & & &\
J162011-5002\_1 & P & & & $17.01 \pm 0.14 $ & & 2009-04-14 & & & & &\
J162046-4942\_1 & M & 16204880-4942141 & $9.02 \pm 0.03 $ & $8.05 \pm 0.03 $ & $7.64 \pm 0.03 $ & & 333.7304+00.2208 & $ 7.47 \pm 0.05 $ & $ 7.61 \pm 0.05 $ & $ 7.45 \pm 0.03 $ & $ 7.39 \pm 0.05 $\
J163252-4746\_2 & M,P,M & 16324851-4745062 & $13.36 \pm 0.04 $ & $13.4 \pm 0.1 $ & $>9.27 \pm 0.0 $ & 2008-05-12 & 336.5085+00.1571 & $ 8.18 \pm 0.06 $ & $ 7.43 \pm 0.06 $ & $ 7.27 \pm 0.03 $ & $ 6.76 \pm 0.03 $\
J163751-4656\_1 & M,M,P & 16375079-4655452 & $14.69 \pm 0.03 $ & $14.57 \pm 0.06 $ & $14.07 \pm 0.11 $ & 2007-06-25 & 337.6914+00.0830 & $ 14.31 \pm 0.07^{\dagger} $ & $ 14.03 \pm 0.15^{\dagger} $ & &\
J165105-4403\_1 & & & & & & & & & & &\
J165217-4414\_1 & P & & $16.38 \pm 0.11 $ & $15.6 \pm 0.11 $ & $15.34 \pm 0.11 $ & 2008-05-12 & & & & &\
J165420-4337\_1 & P & & $15.9 \pm 0.2 $ & $15.4 \pm 0.2 $ & $15.2 \pm 0.2 $ & 2008-05-13 & 342.0899+00.0150 & $ 14.05 \pm 0.13^{\dagger} $ & $ 13.32 \pm 0.16^{\dagger} $ & &\
J165646-4239\_1 & M & 16564984-4238479 & $9.52 \pm 0.03 $ & $8.86 \pm 0.03 $ & $8.61 \pm 0.03 $ & & 343.1338+00.2844 & $ 8.40 \pm 0.08 $ & $ 8.38 \pm 0.08 $ & $ 8.35 \pm 0.05 $ & $ 8.30 \pm 0.04 $\
J165646-4239\_2 & P & & $19.01 \pm 0.11 $ & $18.27 \pm 0.11 $ & $18.23 \pm 0.11 $ & 2007-07-29 & & & & &\
J165707-4255\_1 & M & 16570375-4254413 & $10.35 \pm 0.03 $ & $9.67 \pm 0.02 $ & $9.46 \pm 0.02 $ & & 342.9535+00.0859 & $ 9.39 \pm 0.03 $ & $ 9.38 \pm 0.05 $ & $ 9.19 \pm 0.04 $ & $ 9.35 \pm 0.04 $\
J170017-4220\_1 & P & & $21.5 \pm 0.5 $ & $17.2 \pm 0.2 $ & $14.7 \pm 0.2 $ & 2008-05-12 & 343.7749-00.0290 & $ 12.60 \pm 0.06 $ & $ 11.97 \pm 0.09 $ & $ 11.51 \pm 0.10 $ & $ 11.77 \pm 0.17 $\
J170052-4210\_1 & M & 17005828-4210369 & $11.02 \pm 0.03 $ & $10.25 \pm 0.02 $ & $10.00 \pm 0.02 $ & & 343.9762-00.0242 & $ 9.88 \pm 0.04 $ & $ 9.87 \pm 0.04 $ & $ 9.813 \pm 0.06 $ & $ 9.75 \pm 0.04 $\
J170444-4109\_1 & M & 17044059-4111271 & $11.78 \pm 0.03 $ & $11.19 \pm 0.02 $ & $10.91 \pm 0.03 $ & & 345.1795+00.0300 & $ 10.54 \pm 0.06 $ & $ 10.49 \pm 0.06 $ & $ 10.56\pm 0.06 $ & $ 10.58 \pm 0.06 $\
J170444-4109\_2 & & & & & & & & & & &\
J170536-4038\_1 & M & 17053981-4038123 & $9.65 \pm 0.03 $ & $9.27 \pm 0.03 $ & $9.11 \pm 0.03 $ & & 345.7335+00.2171 & $ 9.01 \pm 0.05 $ & $ 9.05 \pm 0.05 $ & $ 8.97 \pm 0.05 $ & $ 8.93 \pm 0.03 $\
J171910-3652\_2 & M & 17191309-3651241 & $10.26 \pm 0.03 $ & $9.57 \pm 0.03 $ & $9.42 \pm 0.03 $ & & 350.3441+00.3074 & $ 9.33 \pm 0.04 $ & $ 9.32 \pm 0.05 $ & $ 9.30 \pm 0.03 $ & $ 9.25 \pm 0.03 $\
J171922-3703\_1 & & & & & & & & & & &\
J172050-3710\_1 & M & 17205180-3710371 & $9.90 \pm 0.02 $ & $9.55 \pm 0.02 $ & $9.44 \pm 0.02 $ & & 350.2693-00.1451 & $ 9.40 \pm 0.03 $ & $ 9.39 \pm 0.04 $ & $ 9.37 \pm 0.04 $ & $ 9.33 \pm 0.02 $\
J172550-3533\_1 & P\*\* & & $>21.6 \pm 0.2 $ & $20.87 \pm 0.32 $ & $19.0 \pm 0.24 $ & 2009-04-14 & & & & &\
J172550-3533\_2 & M & 17255853-3532315 & $14.26 \pm 0.04 $ & $13.71 \pm 0.04 $ & $13.59 \pm 0.05 $ & & 352.1998-00.0707 & $ 13.10 \pm 0.08^{\dagger} $ & $ 12.84 \pm 0.15^{\dagger} $ & $ 11.59 \pm 0.14^{\dagger} $ &\
J172550-3533\_3 & M & 17255031-3530553 & $10.02 \pm 0.03 $ & $9.64 \pm 0.02 $ & $9.56 \pm 0.02 $ & & 352.2063-00.0327 & $ 9.44 \pm 0.04 $ & $ 9.44 \pm 0.04 $ & $ 9.47 \pm 0.04 $ & $ 9.47 \pm 0.04 $\
J172623-3516\_1 & M & 17262553-3516229 & $9.46 \pm 0.02 $ & $8.70 \pm 0.04 $ & $8.32 \pm 0.03 $ & & 352.4741+00.0035 & $ 8.12 \pm 0.03 $ & $ 8.21 \pm 0.04 $ & $ 8.11 \pm 0.03 $ & $ 8.08 \pm 0.02 $\
J172642-3504\_1 & & & & & & & & & & &\
J172642-3540\_1 & M & 17264172-3540523 & $9.33 \pm 0.02 $ & $8.83 \pm 0.04 $ & $8.74 \pm 0.03 $ & & 352.1660-00.2698 & $ 8.63 \pm 0.03 $ & $ 8.73 \pm 0.04 $ & $ 8.59 \pm 0.03 $ & $ 8.56 \pm 0.02 $\
J173548-3207\_1 & M & 17354627-3207099 & $14.19 \pm 0.04 $ & $>13.44 \pm 0.0 $ & $>12.71 \pm 0.0 $ & & 356.1769+00.1101 & $ 13.23 \pm 0.11 $ & $ 13.33 \pm 0.10 $ & &\
J175404-2553\_3 & & & & & & & & & & &\
J180857-2004\_1 & & & & & & & & & & &\
J180857-2004\_2 & M & 18085910-2005084 & $11.63 \pm 0.04 $ & $9.69 \pm 0.06 $ & $8.61 \pm 0.04 $ & & 010.3179-00.1519 & $ 8.03 \pm 0.14 $ & $ 7.70 \pm 0.20 $ & $ 7.43 \pm 0.10 $ &\
J181116-1828\_2 & P\* & & & $>15.4 \pm 0.0 $ & $>15.1 \pm 0.0 $ & 2008-05-13 & 011.9905+00.1688 & $ 13.71 \pm 0.12 $ & $ 13.56 \pm 0.18 $ & &\
J181213-1842\_7 & P\* & & $18.5 \pm 0.2 $ & $17.1 \pm 0.2 $ & $16.1 \pm 0.2 $ & 2008-05-12 & & & & &\
J181213-1842\_9 & M & 18120913-1841459 & $11.73 \pm 0.02 $ & $11.48 \pm 0.03 $ & $11.40 \pm 0.02 $ & & 011.8954-00.1354 & $ 11.35 \pm 0.15 $ & $ 11.27 \pm 0.12 $ & &\
J181852-1559\_2 & P & & $>20.3 \pm 0.3 $ & $>18.8 \pm 0.3 $ & $>18.0 \pm 0.3 $ & 2008-05-13 & & & & &\
J181915-1601\_2 & M & 18192219-1603123 & $9.09 \pm 0.03 $ & $8.3 \pm 0.06 $ & $7.76 \pm 0.02 $ & & 015.0392-00.3903 & $ 7.22 \pm 0.05 $ & $ 6.84 \pm 0.04 $ & $ 6.66 \pm 0.03 $ & $ 6.25 \pm 0.03 $\
J181915-1601\_3 & & & & & & & & & & &\
J182435-1311\_1 & M & 18243761-1310369 & $7.95 \pm 0.02 $ & $7.38 \pm 0.05 $ & $7.06 \pm 0.02 $ & & 018.1759-00.1606 & $ 6.89 \pm 0.04 $ & $ 6.61 \pm 0.05 $ & $ 6.60 \pm 0.03 $ & $ 6.47 \pm 0.03 $\
J182509-1253\_1 & M & 18250874-1253318 & $9.51 \pm 0.03 $ & $8.96 \pm 0.02 $ & $8.81 \pm 0.02 $ & & 018.4867-00.1392 & $ 8.75 \pm 0.04 $ & $ 8.75 \pm 0.06 $ & $ 8.65 \pm 0.03 $ & $ 8.60 \pm 0.04 $\
J182509-1253\_3 & & & & & & & & & & &\
J182530-1144\_2 & P & & $>21.4 \pm 0.0 $ & $>19.2 \pm 0.2 $ & & 2009-04-12 & & & & &\
J182538-1214\_1 & M & 18253369-1214505 & $11.27 \pm 0.02 $ & $10.76 \pm 0.02 $ & $10.59 \pm 0.02 $ & & 019.1042+00.0722 & $ 10.47 \pm 0.07 $ & $ 10.42 \pm 0.06 $ & $ 10.40 \pm 0.09 $ &\
J182651-1206\_4 & M & 18264689-1207058 & $12.75 \pm 0.03 $ & $12.10 \pm 0.03 $ & $11.86 \pm 0.03 $ & & 019.3575-00.1313 & $ 11.46 \pm 0.06 $ & $ 11.41 \pm 0.08 $ & $ 11.25 \pm 0.10 $ & $ 11.33 \pm 0.20 $\
J183116-1008\_1 & M & 18311653-1009250 & $9.09 \pm 0.03 $ & $8.29 \pm 0.04 $ & $7.63 \pm 0.02 $ & & 021.6064-00.1970 & $ 7.17 \pm 0.04^{\dagger} $ & $ 6.83 \pm 0.06^{\dagger} $ & $ 6.66 \pm 0.03^{\dagger} $ & $ 6.32 \pm 0.03^{\dagger} $\
J183206-0938\_1 & M & 18320893-0939058 & $7.05 \pm 0.02 $ & $5.40 \pm 0.02 $ & $4.34 \pm 0.03 $ & & & & & &\
J183345-0828\_1 & M,P,P & 18334038-0828304 & $15.71 \pm 0.10 $ & $14.4 \pm 0.11 $ & $13.68 \pm 0.11 $ & 2009-04-14 & 023.3713+00.0549 & $ 13.04 \pm 0.09 $ & $ 13.00 \pm 0.13 $ & $ 12.49 \pm 0.25 $ &\
J183356-0822\_2 & & & & & & & & & & &\
J183356-0822\_3 & M & 18335598-0821402 & $14.32 \pm 0.03 $ & $12.11 \pm 0.02 $ & $11.18 \pm 0.03 $ & & 023.5020+00.0505 & $ 10.44 \pm 0.04 $ & $ 10.39 \pm 0.06 $ & $ 10.12 \pm 0.05 $ & $ 10.13 \pm 0.07 $\
J184652-0240\_1 & M & 18464768-0241199 & $9.32 \pm 0.03 $ & $8.21 \pm 0.05 $ & $7.64 \pm 0.02 $ & & 030.0097-00.1937 & $ 7.22 \pm 0.05 $ & $ 7.09 \pm 0.05 $ & $ 6.89 \pm 0.03 $ & $ 6.95 \pm 0.03 $\
J184738-0156\_1 & M & 18473666-0156334 & $14.85 \pm 0.07 $ & $11.01 \pm 0.06 $ & $8.66 \pm 0.04 $ & & 030.7667-00.0346 & & & $ 5.48 \pm 0.19^{\dagger} $ &\
J184741-0219\_3 & P\*\* & & $>21.0 \pm 0.3 $ & $>20.3 \pm 0.3 $ & $18.46 \pm 0.21 $ & 2009-04-14 & & & & &\
J185608+0218\_1 & M & 18560956+0217438 & $8.44 \pm 0.02 $ & $8.02 \pm 0.04 $ & $7.90 \pm 0.02 $ & & 035.5129-00.0030 & $ 7.78 \pm 0.05 $ & $ 7.90 \pm 0.04 $ & $ 7.87 \pm 0.03 $ & $ 7.80 \pm 0.02 $\
J185643+0220\_2 & P\* & & & $>19.1 \pm 0.2 $ & & 2009-04-13 & & & & &\
J185750+0240\_1 & P\* & & $16.61 \pm 0.11 $ & $14.66 \pm 0.11 $ & $13.74 \pm 0.11 $ & 2007-07-29 & & & & &\
J190534+0659\_1 & M & 19053278+0659334 & $8.58 \pm 0.03 $ & $>8.04 \pm 0.0 $ & $>7.51 \pm 0.0 $ & & 040.7595+00.0667 & $ 7.19 \pm 0.04 $ & $ 7.26 \pm 0.05 $ & $ 7.14 \pm 0.03 $ & $ 7.09 \pm 0.03 $\
J190749+0803\_1 & P & & $19.76 \pm 0.11 $ & $18.39 \pm 0.11 $ & $17.77 \pm 0.11 $ & 2007-07-29 & & & & &\
J190814+0832\_2 & M & 19080997+0831478 & $11.29 \pm 0.03 $ & $10.49 \pm 0.03 $ & $10.21 \pm 0.03 $ & & 042.4230+00.1975 & $ 10.06 \pm 0.04 $ & $ 10.06 \pm 0.06 $ & $ 9.99 \pm 0.06 $ & $ 9.87 \pm 0.04 $\
J190818+0745\_1 & M & 19081320+0743576 & $15.17 \pm 0.04 $ & $14.36 \pm 0.04 $ & $14.04 \pm 0.05 $ & & 041.7215-00.1814 & $ 13.71 \pm 0.09 $ & $ 13.54 \pm 0.12 $ & &\
J194152+2251\_2 & P\* & & $18.09 \pm 0.11 $ & $17.05 \pm 0.11 $ & $16.7 \pm 0.12 $ & 2007-07-29/31 & & & & &\
J194310+2318\_5 & M & 19431096+2317454 & $8.03 \pm 0.02 $ & $7.89 \pm 0.02 $ & $7.84 \pm 0.02 $ & & 059.4021-00.1526 & $ 7.84 \pm 0.04 $ & $ 7.83 \pm 0.05 $ & $ 7.80 \pm 0.03 $ & $ 7.84 \pm 0.03 $\
J194939+2631\_1 & P & & & $17.38 \pm 0.13 $ & & 2009-04-13 & & & & &\
J195006+2628\_2 & P & & $14.43 \pm 0.11 $ & $13.79 \pm 0.11 $ & $13.5 \pm 0.11 $ & 2009-04-13 & 062.9345+00.0934 & $ 13.28 \pm 0.10 $ & $ 13.36 \pm 0.13 $ & &
[lccccccccccc]{} J143148-6021\_3 & $0.91_{-0.35}^{+0.44}$ & $0.61_{-0.34}^{+0.54}$ & $9.12 \pm 0.75$ & $10.93 \pm 0.90$ & 0.3 & & $0.70_{-0.25}^{+0.26}$ & $0.42_{-0.17}^{+0.20}$ & $9.85_{-1.15}^{+1.35}$ & $11.17_{-1.31}^{+1.29}$ & 0.5\
J144547-5931\_1 & & & & & & & $4.98_{-0.94}^{+1.06}$ & $4.53_{-1.07}^{+1.25}$ & $2.38_{-0.59}^{+0.45}$ & $1802.71_{-1388.60}^{+4414.31}$ & 1.0\
J144701-5919\_1 & & & & & & & $3.39_{-0.61}^{+0.67}$ & $2.70_{-0.64}^{+0.73}$ & $5.55_{-1.90}^{+1.15}$ & $177.46_{-139.42}^{+157.69}$ & 0.6\
J150436-5824\_1 & $-0.37_{-0.00}^{+0.17}$ & $0.01_{-0.00}^{+0.06}$ & $8.45 \pm 0.71$ & $8.47 \pm 0.71$ & 0.9 & & $-0.05_{-0.27}^{+0.28}$ & $0.49_{-0.27}^{+0.32}$ & $8.46_{-0.84}^{+1.07}$ & $9.03_{-0.98}^{+1.05}$ & 1.2\
J154122-5522\_1 & $2.53_{-0.49}^{+0.62}$ & $0.14_{-0.11}^{+0.14}$ & $2.50 \pm 0.27$ & $4.46 \pm 0.49$ & 0.6 & & $2.35_{-0.39}^{+0.43}$ & $0.15_{-0.09}^{+0.10}$ & $2.86_{-0.46}^{+0.38}$ & $4.82_{-1.22}^{+0.99}$ & 1.0\
J154557-5443\_1 & $2.98_{-0.70}^{+1.30}$ & $0.29_{-0.17}^{+0.19}$ & $0.17 \pm 0.04$ & $0.56 \pm 0.12$ & & & & & & &\
J155035-5408\_3 & $2.73_{-0.46}^{+0.55}$ & $0.67_{-0.21}^{+0.24}$ & $2.90 \pm 0.20$ & $11.47 \pm 0.80$ & 0.9 & & $2.72_{-0.27}^{+0.28}$ & $0.61 \pm 0.11$ & $2.88_{-0.33}^{+0.36}$ & $11.33_{-3.55}^{+3.17}$ & 1.8\
J155831-5334\_1 & $1.70_{-0.63}^{+0.72}$ & $0.43_{-0.34}^{+0.45}$ & $3.62 \pm 0.50$ & $5.45 \pm 0.76$ & 0.4 & & $1.65_{-0.49}^{+0.54}$ & $0.38_{-0.22}^{+0.28}$ & $3.31_{-0.63}^{+0.74}$ & $5.99_{-2.16}^{+1.39}$ & 1.0\
J162011-5002\_1 & $0.88_{-1.00}^{+1.00}$ & $2.50_{-1.56}^{+2.8}$ & $8.17 \pm 1.11$ & $11.73 \pm 1.6$ & 0.6 & & $0.75_{-0.75}^{+0.83}$ & $2.59_{-1.53}^{+2.02}$ & $7.62_{-2.68}^{+2.09}$ & $12.82_{-3.94}^{+1.50}$ & 1.1\
J162046-4942\_1 & $1.42_{-0.31}^{+0.35}$ & $0.12_{-0.10}^{+0.15}$ & $3.35 \pm 0.31$ & $3.84 \pm 0.35$ & 1.4 & & $1.71_{-0.29}^{+0.31}$ & $0.31_{-0.11}^{+0.13}$ & $3.32_{-0.53}^{+0.61}$ & $5.01_{-0.85}^{+0.72}$ & 2.7\
J165105-4403\_1 & $0.84_{-0.31}^{+0.37}$ & $0.34_{-0.25}^{+0.36}$ & $12.6 \pm 1.03$ & $14.19 \pm 1.15$ & 0.8 & & $0.97_{-0.27}^{+0.28}$ & $0.58_{-0.19}^{+0.22}$ & $13.42_{-1.83}^{+1.59}$ & $16.43_{-1.68}^{+2.16}$ & 1.1\
J165217-4414\_1 & $1.63_{-0.49}^{+0.47}$ & $0.64_{-0.34}^{+0.41}$ & $5.97 \pm 0.57$ & $9.32 \pm 0.9$ & 0.7 & & $1.29_{-0.29}^{+0.31}$ & $0.38_{-0.15}^{+0.18}$ & $6.48_{-1.06}^{+1.14}$ & $8.66_{-1.56}^{+1.01}$ & 0.9\
J165646-4239\_1 & & & & & & & $3.93_{-0.36}^{+0.40}$ & $0.22 \pm 0.06$ & $4.99_{-0.39}^{+0.36}$ & $27.34_{-13.30}^{+18.76}$ & 2.8\
J165646-4239\_2 & $1.49_{-0.62}^{+0.79}$ & $0.31_{-0.30}^{+0.48}$ & $2.27 \pm 0.30$ & $2.98 \pm 0.39$ & 0.4 & & $1.28_{-0.41}^{+0.44}$ & $0.22_{-0.17}^{+0.21}$ & $2.37_{-0.36}^{+0.49}$ & $3.18_{-0.56}^{+0.35}$ & 0.7\
J165707-4255\_1 & $3.91_{-1.25}^{+1.07}$ & $0.22_{-0.20}^{+0.21}$ & $2.5 \pm 0.32$ & $13.51 \pm 1.73$ & 2.1 & & $3.17_{-0.60}^{+0.79}$ & $0.07_{-0.07}^{+0.12}$ & $3.05_{-0.59}^{+0.53}$ & $7.05_{-2.67}^{+4.82}$ & 2.6\
J170017-4220\_1 & $-1.89_{-0.10}^{+1.14}$ & $1.00_{-0.99}^{+5.60}$ & $19.25 \pm 2.37$ & $19.91 \pm 2.45$ & 1.0 & & $0.24_{-0.98}^{+1.10}$ & $14.01_{-6.85}^{+8.49}$ & $11.36_{-7.08}^{+5.90}$ & $29.98_{-8.94}^{+6.53}$ & 1.0\
J170052-4210\_1 & & & & & & & $4.89_{-1.45}^{+1.98}$ & $0.37_{-0.23}^{+0.31}$ & $0.81_{-0.22}^{+0.23}$ & $32.64_{-26.84}^{+152.63}$ & 3.6\
J170444-4109\_1 & $3.38_{-0.65}^{+0.86}$ & $0.16_{-0.13}^{+0.18}$ & $9.01 \pm 0.73$ & $27.22 \pm 2.21$ & 1.7 & & $3.17_{-0.35}^{+0.38}$ & $0.23 \pm 0.09$ & $8.35_{-1.16}^{+1.04}$ & $28.45_{-8.65}^{+14.82}$ & 2.1\
J170444-4109\_2 & $0.31_{-1.37}^{+1.60}$ & $1.70_{-1.69}^{+3.10}$ & $2.10 \pm 0.50$ & $2.51 \pm 0.59$ & & & & & & &\
J170536-4038\_1 & $2.88_{-0.63}^{+0.79}$ & $0.26_{-0.15}^{+0.19}$ & $6.54 \pm 0.61$ & $19.03 \pm 1.79$ & 1.0 & & $2.09_{-0.31}^{+0.33}$ & $0.14 \pm 0.08$ & $8.50_{-1.15}^{+1.05}$ & $12.81_{-1.93}^{+2.24}$ & 1.7\
J171910-3652\_2 & & & & & & & $3.25_{-0.44}^{+0.49}$ & $0.14_{-0.07}^{+0.08}$ & $2.64_{-0.29}^{+0.38}$ & $6.78_{-2.33}^{+3.77}$ & 3.3\
J171922-3703\_1 & $-0.75_{-0.55}^{+0.49}$ & $1.15_{-1.03}^{+1.85}$ & $6.83 \pm 0.56$ & $7.27 \pm 0.60$ & 0.6 & & $-0.26_{-0.39}^{+0.41}$ & $2.19_{-0.92}^{+1.12}$ & $5.79_{-1.32}^{+1.10}$ & $7.09_{-1.65}^{+0.84}$ & 1.4\
J172050-3710\_1 & & & & & & & $4.70_{-0.37}^{+0.39}$ & $0.54_{-0.07}^{+0.08}$ & $2.07_{-0.18}^{+0.11}$ & $67.55_{-32.05}^{+49.73}$ & 1.8\
J172550-3533\_1 & $0.14_{-1.40}^{+1.93}$ & $6.60_{-5.3}^{+11.9}$ & $2.8 \pm 0.43$ & $4.11 \pm 0.63$ & 0.4 & & & & & &\
J172550-3533\_2 & $3.28_{-1.00}^{+1.49}$ & $0.38_{-0.25}^{+0.38}$ & $0.44 \pm 0.06$ & $2.28 \pm 0.34$ & 0.6 & & & & & &\
J172550-3533\_3 & $3.17_{-0.79}^{+1.11}$ & $0.28_{-0.22}^{+0.30}$ & $0.55 \pm 0.09$ & $2.11 \pm 0.33$ & & & & & & &\
J172623-3516\_1 & $2.94_{-1.28}^{+2.04}$ & $0.46_{-0.40}^{+0.84}$ & $1.07 \pm 0.16$ & $4.44 \pm 0.68$ & 0.6 & & $1.63_{-0.47}^{+0.52}$ & $0.13_{-0.13}^{+0.17}$ & $1.64_{-0.35}^{+0.61}$ & $2.49_{-0.54}^{+0.41}$ & 1.4\
J172642-3504\_1 & $0.84_{-0.70}^{+0.75}$ & $2.30_{-1.25}^{+2.00}$ & $2.86 \pm 0.31$ & $4.01 \pm 0.43$ & 0.2 & & $0.64_{-0.51}^{+0.55}$ & $2.40_{-0.98}^{+1.18}$ & $3.02_{-1.03}^{+0.57}$ & $4.28_{-1.61}^{+0.52}$ & 0.6\
J173548-3207\_1 & $2.04_{-0.47}^{+0.67}$ & $0.20_{-0.15}^{+0.22}$ & $1.65 \pm 0.24$ & $2.54 \pm 0.37$ & 3.5 & & $2.17_{-0.55}^{+0.60}$ & $0.32_{-0.19}^{+0.22}$ & $1.64_{-0.48}^{+0.35}$ & $3.80_{-1.69}^{+1.86}$ & 3.3\
J180857-2004\_1 & $2.84_{-1.28}^{+1.86}$ & $5.40_{-2.80}^{+4.60}$ & $1.16 \pm 0.17$ & $14.45 \pm 2.18$ & & & & & & &\
J180857-2004\_2 & $3.45_{-0.93}^{+1.28}$ & $6.20_{-2.00}^{+3.00}$ & $1.94 \pm 0.22$ & $80.67 \pm 9.02$ & 0.7 & & $2.71_{-0.67}^{+0.72}$ & $4.35_{-1.15}^{+1.32}$ & $1.96_{-0.59}^{+0.40}$ & $23.39_{-13.91}^{+17.88}$ & 1.7\
J181116-1828\_2 & $2.00_{-1.09}^{+1.38}$ & $2.20_{-1.50}^{+2.20}$ & $2.77 \pm 0.39$ & $7.66 \pm 1.07$ & 1.6 & & $1.52_{-0.61}^{+0.68}$ & $1.13_{-0.58}^{+0.75}$ & $2.24_{-0.88}^{+0.58}$ & $5.14_{-2.33}^{+1.53}$ & 2.3\
J181213-1842\_7 & $-0.16_{-0.52}^{+0.55}$ & $0.52_{-0.50}^{+0.73}$ & $10.49 \pm 0.98$ & $11.03 \pm 1.03$ & 0.5 & & $0.37_{-0.39}^{+0.42}$ & $1.62_{-0.65}^{+0.77}$ & $10.12_{-1.96}^{+1.67}$ & $12.47_{-1.92}^{+1.70}$ & 0.5\
J181213-1842\_9 & $3.63_{-1.07}^{+1.35}$ & $0.20_{-0.19}^{+0.31}$ & $0.35 \pm 0.09$ & $1.48 \pm 0.36$ & & & & & & &\
J181852-1559\_2 & $0.77_{-0.82}^{+1.11}$ & $2.50_{-1.59}^{+3.20}$ & $6.00 \pm 0.94$ & $8.34 \pm 1.30$ & 1.1 & & $-0.18_{-0.77}^{+0.88}$ & $1.24_{-1.24}^{+2.19}$ & $5.22_{-3.62}^{+3.02}$ & $6.48_{-3.06}^{+3.14}$ & 0.7\
J181915-1601\_2 & $1.32_{-0.37}^{+0.49}$ & $0.70_{-0.33}^{+0.50}$ & $3.79 \pm 0.52$ & $5.20 \pm 0.71$ & 1.5 & & $2.14_{-0.63}^{+0.69}$ & $1.46_{-0.58}^{+0.71}$ & $2.97_{-0.94}^{+0.89}$ & $9.40_{-4.37}^{+7.92}$ & 2.0\
J181915-1601\_3 & $1.21_{-0.34}^{+0.46}$ & $0.67_{-0.31}^{+0.48}$ & $5.35 \pm 0.62$ & $7.03 \pm 0.81$ & 1.2 & & $1.61_{-0.48}^{+0.52}$ & $1.38_{-0.50}^{+0.59}$ & $5.25_{-1.29}^{+1.04}$ & $10.99_{-4.17}^{+2.77}$ & 1.6\
J182435-1311\_1 & & & & & & & $4.77_{-0.70}^{+0.78}$ & $1.44_{-0.30}^{+0.34}$ & $0.45_{-0.13}^{+0.09}$ & $60.30_{-51.18}^{+99.63}$ & 3.4\
J182509-1253\_1 & & & & & & & $3.40_{-0.52}^{+0.57}$ & $0.44_{-0.13}^{+0.14}$ & $0.57_{-0.09}^{+0.12}$ & $4.49_{-2.67}^{+3.81}$ & 2.2\
J182509-1253\_3 & $-0.05_{-0.84}^{+1.65}$ & $3.00_{-2.66}^{+7.50}$ & $1.96 \pm 0.28$ & $2.40 \pm 0.34$ & 1.9 & & $-0.08_{-0.99}^{+1.15}$ & $2.98_{-2.55}^{+4.05}$ & $1.25_{-0.96}^{+0.79}$ & $1.84_{-0.69}^{+0.74}$ & 2.5\
J182530-1144\_2 & $0.14_{-0.37}^{+0.38}$ & $0.55_{-0.31}^{+0.33}$ & $9.44 \pm 0.61$ & $10.13 \pm 0.66$ & 1.0 & & $0.43_{-0.23}^{+0.24}$ & $1.01_{-0.28}^{+0.32}$ & $9.37_{-1.22}^{+0.71}$ & $10.83_{-1.16}^{+1.04}$ & 1.7\
J182651-1206\_4 & $1.80_{-0.70}^{+0.89}$ & $0.28_{-0.27}^{+0.42}$ & $0.34 \pm 0.08$ & $0.49 \pm 0.12$ & & & & & & &\
J183116-1008\_1 & $2.28_{-0.55}^{+0.70}$ & $4.80_{-1.30}^{+1.80}$ & $5.66 \pm 0.61$ & $28.9 \pm 3.13$ & 1.4 & & $3.28_{-0.79}^{+0.86}$ & $8.15_{-1.97}^{+2.26}$ & $5.33_{-2.67}^{+1.11}$ & $373.72_{-273.72}^{+629.26}$ & 1.7\
J183206-0938\_1 & $0.91_{-0.46}^{+0.47}$ & $1.50_{-0.71}^{+0.90}$ & $29.85 \pm 2.40$ & $39.86 \pm 3.20$ & 0.7 & & $0.86_{-0.35}^{+0.36}$ & $1.81_{-0.51}^{+0.58}$ & $33.04_{-7.94}^{+3.69}$ & $47.45_{-11.41}^{+5.54}$ & 1.4\
J183345-0828\_1 & $2.63_{-1.03}^{+1.40}$ & $2.60_{-1.35}^{+2.00}$ & $0.65 \pm 0.10$ & $4.04 \pm 0.64$ & & & & & & &\
J183356-0822\_2 & $2.38_{-1.75}^{+2.60}$ & $18.0_{-10.20}^{+17.00}$ & $2.06 \pm 0.25$ & $26.08 \pm 3.14$ & 1.0 & & $0.82_{-1.13}^{+1.30}$ & $10.46_{-5.41}^{+7.15}$ & $1.65_{-1.21}^{+0.90}$ & $5.80_{-1.38}^{+0.95}$ & 1.2\
J183356-0822\_3 & $1.28_{-1.11}^{+1.28}$ & $1.80_{-1.61}^{+2.40}$ & $0.43 \pm 0.11$ & $0.69 \pm 0.17$ & & & & & & &\
J184652-0240\_1 & $2.88_{-0.70}^{+0.96}$ & $1.50_{-0.56}^{+0.80}$ & $4.14 \pm 0.45$ & $28.0 \pm 3.02$ & 1.0 & & $2.92_{-0.51}^{+0.55}$ & $1.51_{-0.38}^{+0.44}$ & $3.93_{-0.71}^{+0.79}$ & $35.73_{-19.70}^{+24.49}$ & 1.8\
J184738-0156\_1 & $2.35_{-1.14}^{+1.79}$ & $12.0_{-5.20}^{+9.00}$ & $19.56 \pm 1.76$ & $178.36 \pm 16.08$ & 1.2 & & $0.53_{-0.53}^{+0.57}$ & $5.13_{-1.75}^{+2.14}$ & $21.11_{-8.53}^{+4.10}$ & $35.70_{-12.23}^{+3.87}$ & 2.2\
J184741-0219\_3 & $-1.62_{-0.38}^{+1.86}$ & $1.00_{-0.99}^{+12.00}$ & $2.57 \pm 0.46$ & $2.67 \pm 0.47$ & & & & & & &\
J185608+0218\_1 & & & & & & & $3.83_{-0.40}^{+0.44}$ & $0.33 \pm 0.08$ & $6.97_{-0.74}^{+0.66}$ & $50.56_{-22.77}^{+47.69}$ & 1.8\
J185643+0220\_2 & $0.00_{-0.44}^{+0.63}$ & $0.97_{-0.81}^{+1.53}$ & $3.27 \pm 0.39$ & $3.59 \pm 0.43$ & 1.3 & & $0.75_{-0.64}^{+0.70}$ & $3.22_{-1.47}^{+1.83}$ & $3.03_{-1.35}^{+0.53}$ & $4.97_{-0.80}^{+0.63}$ & 1.6\
J185750+0240\_1 & $0.25_{-0.58}^{+0.74}$ & $0.32_{-0.31}^{+1.08}$ & $4.20 \pm 0.63$ & $4.43 \pm 0.66$ & 0.5 & & $0.31_{-0.52}^{+0.60}$ & $0.30_{-0.30}^{+0.61}$ & $4.01_{-1.45}^{+0.78}$ & $4.28_{-0.97}^{+1.01}$ & 0.4\
J190534+0659\_1 & $2.28_{-0.96}^{+1.35}$ & $0.52_{-0.43}^{+0.68}$ & $0.73 \pm 0.19$ & $1.70 \pm 0.45$ & & & & & & &\
J190749+0803\_1 & $1.74_{-0.75}^{+0.93}$ & $4.80_{-2.00}^{+3.00}$ & $6.12 \pm 0.64$ & $17.21 \pm 1.80$ & 0.2 & & $0.75_{-0.51}^{+0.55}$ & $2.73_{-1.04}^{+1.24}$ & $6.79_{-1.89}^{+0.99}$ & $10.52_{-2.06}^{+1.11}$ & 0.6\
J190814+0832\_2 & $2.66_{-0.56}^{+0.62}$ & $0.73_{-0.27}^{+0.37}$ & $0.9 \pm 0.14$ & $3.41 \pm 0.53$ & 1.0 & & $3.22_{-0.74}^{+0.86}$ & $0.74_{-0.29}^{+0.36}$ & $0.73_{-0.21}^{+0.18}$ & $5.83_{-3.54}^{+6.28}$ & 0.7\
J190818+0745\_1 & $1.32_{-0.25}^{+0.28}$ & $0.20_{-0.12}^{+0.15}$ & $10.45 \pm 0.89$ & $12.38 \pm 1.05$ & 0.7 & & $1.35_{-0.25}^{+0.26}$ & $0.22_{-0.10}^{+0.11}$ & $10.35_{-1.38}^{+1.30}$ & $12.91_{-1.30}^{+1.55}$ & 1.2\
J194152+2251\_2 & $0.38_{-0.69}^{+0.56}$ & $0.82_{-0.63}^{+0.58}$ & $6.07 \pm 0.72$ & $6.84 \pm 0.81$ & 0.3 & & $0.36_{-0.46}^{+0.50}$ & $0.91_{-0.55}^{+0.72}$ & $5.96_{-2.02}^{+1.36}$ & $7.02_{-1.88}^{+1.39}$ & 0.3\
J194939+2631\_1 & $0.67_{-0.49}^{+0.69}$ & $2.30_{-1.15}^{+1.90}$ & $16.71 \pm 1.57$ & $22.28 \pm 2.09$ & 0.9 & & $-0.10_{-0.35}^{+0.38}$ & $0.67_{-0.43}^{+0.57}$ & $18.24_{-4.08}^{+2.62}$ & $20.06_{-3.87}^{+2.18}$ & 1.5
[lccccccccccc]{} J144519-5949\_2 & $1.38_{-0.65}^{+2.15}$ & $4.80_{-2.2}^{+3.60}$ & $2.17 \pm 0.44$ & $21.59 \pm 4.37$ & & & & & & &\
J144547-5931\_1 & $0.84_{-0.21}^{+0.30}$ & $4.10_{-0.9}^{+1.20}$ & $2.56 \pm 0.31$ & $57.74 \pm 6.95$ & 0.9 & & $1.01_{-0.23}^{+0.37}$ & $3.27_{-0.74}^{+0.86}$ & $1.93_{-1.24}^{+0.68}$ & $37.28_{-22.51}^{+9.98}$ & 0.9\
J144701-5919\_1 & $1.00_{-0.39}^{+0.70}$ & $3.50_{-1.2}^{+1.90}$ & $5.49 \pm 0.60$ & $74.01 \pm 8.12$ & 0.6 & & $2.04_{-0.54}^{+0.95}$ & $1.93_{-0.43}^{+0.49}$ & $5.78_{-2.03}^{+1.16}$ & $23.52_{-7.51}^{+4.04}$ & 0.9\
J154122-5522\_1 & $1.88_{-0.70}^{+0.96}$ & $0.04_{-0.03}^{+0.09}$ & $2.35 \pm 0.26$ & $2.72 \pm 0.3$ & 0.6 & & $2.62_{-0.80}^{+1.68}$ & $0.05_{-0.05}^{+0.06}$ & $2.46_{-0.47}^{+0.45}$ & $3.08_{-0.70}^{+0.47}$ & 1.0\
J154557-5443\_1 & $1.38_{-0.73}^{+1.32}$ & $0.14_{-0.1}^{+0.15}$ & $0.16 \pm 0.03$ & $0.25 \pm 0.05$ & & & & & & &\
J154557-5443\_2 & $1.77_{-1.33}^{+96.43}$ & $24.0_{-15.8}^{+46.0}$ & $0.63 \pm 0.16$ & $17.65 \pm 4.49$ & & & & & & &\
J154557-5443\_3 & $0.10_{-0.00}^{+0.30}$ & $0.97_{-0.88}^{+1.23}$ & $0.11 \pm 0.03$ & $224.95 \pm 51.46$ & & & & & & &\
J155035-5408\_1 & $0.35_{-0.18}^{+0.18}$ & $0.14_{-0.14}^{+0.38}$ & $0.18 \pm 0.05$ & $0.56 \pm 0.14$ & & & & & & &\
J155035-5408\_3 & $2.14_{-0.72}^{+0.91}$ & $0.43_{-0.1}^{+0.21}$ & $2.73 \pm 0.19$ & $5.29 \pm 0.37$ & 0.9 & & $2.17_{-0.38}^{+0.55}$ & $0.42_{-0.07}^{+0.08}$ & $2.81_{-0.37}^{+0.28}$ & $5.43_{-0.85}^{+0.45}$ & 1.8\
J155331-5347\_1 & $0.31_{-0.16}^{+0.31}$ & $0.70_{-0.39}^{+0.85}$ & $1.25 \pm 0.15$ & $30.92 \pm 3.67$ & 2.2 & & & & & &\
J155831-5334\_1 & $9.04_{-5.99}^{+89.16}$ & $0.34_{-0.22}^{+0.30}$ & $3.54 \pm 0.49$ & $4.73 \pm 0.66$ & 0.4 & & & & & &\
J162046-4942\_1 & & & & & & & $8.37_{-3.50}^{+16.53}$ & $0.24_{-0.08}^{+0.09}$ & $3.48_{-0.61}^{+0.50}$ & $4.48_{-0.89}^{+0.56}$ & 2.7\
J165646-4239\_1 & $0.35_{-0.10}^{+0.13}$ & $0.25_{-0.10}^{+0.15}$ & $4.06 \pm 0.25$ & $20.59 \pm 1.26$ & 1.8 & & $0.78_{-0.11}^{+0.14}$ & $0.03 \pm 0.03$ & $5.06_{-0.64}^{+0.53}$ & $6.62_{-0.99}^{+0.49}$ & 2.8\
J165707-4255\_1 & $0.71_{-0.31}^{+0.21}$ & $0.05_{-0.04}^{+0.16}$ & $2.53 \pm 0.32$ & $3.39 \pm 0.43$ & 2.1 & & $0.87_{-0.16}^{+0.24}$ & $<0.02$ & $2.69_{-0.53}^{+0.24}$ & $2.78_{-0.57}^{+0.24}$ & 3.3\
J170052-4210\_1 & $0.31_{-0.16}^{+0.35}$ & $0.25_{-0.22}^{+0.45}$ & $0.84 \pm 0.14$ & $4.93 \pm 0.82$ & & & $0.51_{-0.19}^{+0.42}$ & $0.13_{-0.13}^{+0.17}$ & $0.84_{-0.27}^{+0.18}$ & $1.98_{-0.80}^{+0.69}$ & 3.9\
J170444-4109\_1 & $0.97_{-0.29}^{+0.28}$ & $0.01_{0.00}^{+0.10}$ & $9.28 \pm 0.75$ & $9.85 \pm 0.80$ & 1.8 & & $1.54_{-0.30}^{+0.42}$ & $0.03_{-0.03}^{+0.06}$ & $8.51_{-0.88}^{+0.66}$ & $10.48_{-1.02}^{+0.92}$ & 2.4\
J170536-4038\_1 & $1.46_{-0.55}^{+1.11}$ & $0.13_{-0.10}^{+0.14}$ & $6.09 \pm 0.57$ & $9.33 \pm 0.88$ & 1.1 & & $3.95_{-1.29}^{+2.79}$ & $0.05_{-0.05}^{+0.06}$ & $8.65_{-1.65}^{+0.81}$ & $9.68_{-1.40}^{+1.46}$ & 1.8\
J171910-3652\_2 & $0.17_{-0.06}^{+0.07}$ & $0.61_{-0.24}^{+0.39}$ & $1.81 \pm 0.17$ & $136.22 \pm 12.59$ & 1.1 & & $1.23_{-0.20}^{+0.25}$ & $<0.03$ & $2.59_{-0.26}^{+0.30}$ & $2.71_{-0.20}^{+0.25}$ & 4.00\
J172050-3710\_1 & $0.38_{-0.06}^{+0.08}$ & $0.37_{-0.09}^{+0.09}$ & $1.77 \pm 0.10$ & $12.78 \pm 0.72$ & 1.1 & & $0.67_{-0.08}^{+0.09}$ & $0.24_{-0.04}^{+0.05}$ & $2.03_{-0.28}^{+0.13}$ & $6.06_{-0.98}^{+0.75}$ & 2.1\
J172550-3533\_2 & $1.14_{-0.55}^{+1.35}$ & $0.20_{-0.16}^{+0.25}$ & $0.41 \pm 0.06$ & $0.81 \pm 0.12$ & 0.6 & & & & & &\
J172550-3533\_3 & $1.28_{-0.56}^{+1.14}$ & $0.10_{-0.09}^{+0.21}$ & $0.55 \pm 0.09$ & $0.79 \pm 0.12$ & & & & & & &\
J172623-3516\_1 & $1.56_{-1.10}^{+6.01}$ & $0.28_{-0.26}^{+0.69}$ & $1.0 \pm 0.15$ & $1.93 \pm 0.29$ & 0.6 & & & & & &\
J173548-3207\_1 & $4.00_{-2.30}^{+6.00}$ & $0.12_{-0.09}^{+0.21}$ & $1.58 \pm 0.23$ & $1.95 \pm 0.29$ & 3.6 & & $3.67_{-1.59}^{+5.98}$ & $0.21_{-0.13}^{+0.15}$ & $1.57_{-0.70}^{+0.42}$ & $2.13_{-0.59}^{+0.68}$ & 3.3\
J175404-2553\_3 & $2.56_{-1.85}^{+95.64}$ & $33.00_{-18.00}^{+40.00}$ & $1.34 \pm 0.27$ & $28.13 \pm 5.69$ & & & & & & &\
J180857-2004\_1 & $3.19_{-1.77}^{+19.41}$ & $4.40_{-2.00}^{+3.20}$ & $1.12 \pm 0.17$ & $4.19 \pm 0.63$ & & & & & & &\
J180857-2004\_2 & $2.21_{-0.82}^{+1.68}$ & $4.90_{-1.40}^{+1.90}$ & $1.88 \pm 0.21$ & $10.23 \pm 1.14$ & 0.6 & & $4.03_{-1.50}^{+4.75}$ & $3.37_{-0.78}^{+0.88}$ & $1.78_{-1.13}^{+0.55}$ & $5.52_{-3.43}^{+1.58}$ & 1.9\
J181116-1828\_2 & $6.52_{-4.45}^{+91.68}$ & $1.85_{-0.85}^{+1.55}$ & $2.7 \pm 0.38$ & $5.36 \pm 0.75$ & 1.5 & & & & & &\
J181213-1842\_9 & $0.84_{-0.49}^{+0.79}$ & $0.04_{-0.03}^{+0.27}$ & $0.35 \pm 0.09$ & $0.45 \pm 0.11$ & & & & & & &\
J181915-1601\_2 & & & & & & & $5.44_{-2.61}^{+17.68}$ & $1.15_{-0.40}^{+0.49}$ & $3.25_{-0.99}^{+0.55}$ & $5.92_{-1.37}^{+1.18}$ & 2.1\
J182435-1311\_1 & $0.84_{-0.17}^{+0.19}$ & $0.82_{-0.15}^{+0.18}$ & $0.47 \pm 0.05$ & $2.54 \pm 0.27$ & 1.7 & & $0.74_{-0.15}^{+0.21}$ & $0.99_{-0.21}^{+0.24}$ & $0.44_{-0.21}^{+0.08}$ & $3.90_{-2.11}^{+0.91}$ & 3.3\
J182509-1253\_1 & $0.76_{-0.23}^{+0.38}$ & $0.26_{-0.12}^{+0.17}$ & $0.49 \pm 0.05$ & $1.4 \pm 0.15$ & 1.6 & & $1.37_{-0.34}^{+0.56}$ & $0.20_{-0.08}^{+0.09}$ & $0.54_{-0.19}^{+0.09}$ & $1.03_{-0.22}^{+0.21}$ & 2.7\
J182538-1214\_1 & $0.38_{-0.26}^{+0.90}$ & $0.37_{-0.36}^{+1.13}$ & $0.52 \pm 0.10$ & $3.86 \pm 0.72$ & & & & & & &\
J182651-1206\_4 & $6.31_{-4.17}^{+91.89}$ & $0.20_{-0.19}^{+0.28}$ & $0.32 \pm 0.08$ & $0.42 \pm 0.10$ & & & & & & &\
J183116-1008\_1 & $5.26_{-2.28}^{+7.89}$ & $4.10_{-1.00}^{+1.30}$ & $5.55 \pm 0.60$ & $15.35 \pm 1.67$ & 1.5 & & $2.96_{-0.96}^{+2.26}$ & $6.37_{-1.35}^{+1.54}$ & $5.18_{-2.86}^{+1.52}$ & $26.82_{-13.83}^{+7.42}$ & 1.9\
J183345-0828\_1 & $3.26_{-1.77}^{+13.04}$ & $2.00_{-0.90}^{+1.40}$ & $0.62 \pm 0.10$ & $1.64 \pm 0.26$ & & & & & & &\
J183356-0822\_2 & $6.31_{-4.64}^{+91.89}$ & $16.0_{-6.6}^{+13.0}$ & $2.05 \pm 0.25$ & $10.44 \pm 1.26$ & 1.0 & & & & & &\
J184652-0240\_1 & $2.21_{-0.86}^{+2.00}$ & $1.10_{-0.37}^{+0.55}$ & $3.9 \pm 0.42$ & $10.29 \pm 1.11$ & 0.9 & & $2.21_{-0.59}^{+1.06}$ & $1.14_{-0.27}^{+0.31}$ & $3.66_{-1.46}^{+0.79}$ & $10.87_{-3.90}^{+1.81}$ & 1.8\
J184738-0156\_1 & $5.47_{-3.33}^{+92.73}$ & $11.0_{-4.20}^{+6.00}$ & $19.34 \pm 1.74$ & $83.9 \pm 7.56$ & 1.2 & & & & & &\
J185608+0218\_1 & $0.42_{-0.11}^{+0.17}$ & $0.31_{-0.12}^{+0.15}$ & $5.66 \pm 0.41$ & $30.15 \pm 2.16$ & 1.3 & & $0.96_{-0.16}^{+0.21}$ & $0.10_{-0.04}^{+0.05}$ & $6.82_{-0.83}^{+0.72}$ & $11.02_{-2.17}^{+1.91}$ & 2.0\
J190534+0659\_1 & $3.12_{-1.91}^{+41.53}$ & $0.37_{-0.30}^{+0.45}$ & $0.69 \pm 0.18$ & $1.12 \pm 0.30$ & & & & & & &\
J190814+0832\_2 & $2.21_{-0.72}^{+1.75}$ & $0.52_{-0.19}^{+0.27}$ & $0.84 \pm 0.13$ & $1.71 \pm 0.26$ & 0.8 & & $1.46_{-0.48}^{+1.00}$ & $0.50_{-0.21}^{+0.25}$ & $0.77$ & & 0.4\
J194310+2318\_5 & $0.55_{-0.25}^{+0.29}$ & $0.12_{-0.1}^{+0.22}$ & $0.69 \pm 0.13$ & $1.42 \pm 0.26$ & & & & & & &\
J195006+2628\_2 & $0.48_{-0.32}^{+1.01}$ & $0.29_{-0.28}^{+0.81}$ & $0.22 \pm 0.04$ & $0.97 \pm 0.20$ & & & & & & &
[lccccccc]{} J143148-6021\_3 & 0296-0521275 & U,A,A,A & 19.71 $\pm$ 0.3 & 20.49 $\pm$ 0.03 & 19.31 $\pm$ 0.03 & 18.27 $\pm$ 0.03 & 2008-05-13\
J144042-6001\_1 & 0299-0503336 & U & 10.26 $\pm$ 0.3 & & 9.36 $\pm$ 0.3 & 8.99 $\pm$ 0.3 &\
J144547-5931\_1 & 0304-0487344 & U & 20.54 $\pm$ 0.3 & & 15.79 $\pm$ 0.3 & 12.30 $\pm$ 0.3 &\
J144701-5919\_1 & 0306-0492632 & U & & & 16.61 $\pm$ 0.3 & 12.95 $\pm$ 0.3 &\
J154122-5522\_1 & 0346-0518164 & U & 13.17 $\pm$ 0.3 & & 11.87 $\pm$ 0.3 & 10.87 $\pm$ 0.3 &\
J154557-5443\_1 & 0352-0535858 & U & 16.16 $\pm$ 0.3 & & 15.10 $\pm$ 0.3 & 14.67 $\pm$ 0.3 &\
J154557-5443\_3 & 0352-0535969 & U & 17.91 $\pm$ 0.3 & & 16.51 $\pm$ 0.3 & 15.42 $\pm$ 0.3 &\
J155035-5408\_1 & 0358-0537979 & U & 11.40 $\pm$ 0.3 & & 10.71 $\pm$ 0.3 & 10.43 $\pm$ 0.3 &\
J155035-5408\_3 & 0358-0538167 & U & 15.81 $\pm$ 0.3 & & 13.96 $\pm$ 0.3 & 13.09 $\pm$ 0.3 &\
J155331-5347\_1 & 0361-0529851 & U & 14.38 $\pm$ 0.3 & & 12.93 $\pm$ 0.3 & 12.74 $\pm$ 0.3 &\
J155831-5334\_1 & 0364-0530109 & U & 16.99 $\pm$ 0.3 & & 16.37 $\pm$ 0.3 & 16.15 $\pm$ 0.3 &\
J162046-4942\_1 & 0402-0531543 & U & 15.03 $\pm$ 0.3 & & 12.48 $\pm$ 0.3 & 10.71 $\pm$ 0.3 &\
J163252-4746\_2 & & A & & $>25$ & $>25$ & 21.93 $\pm$ 0.05 & 2008-05-13\
J163751-4656\_1 & 0430-0573746 & U & 15.27 $\pm$ 0.3 & & 14.94 $\pm$ 0.3 & 15.07 $\pm$ 0.3 &\
J165217-4414\_1 & 0457-0509597 & U,A,A,A & 19.03 $\pm$ 0.3 & 19.73 $\pm$ 0.03 & 18.74 $\pm$ 0.03 & 17.80 $\pm$ 0.03 & 2008-05-13\
J165420-4337\_1 & 0463-0473777 & U,A,A,A & 17.62 $\pm$ 0.3 & 19.88 $\pm$ 0.03 & 19.01 $\pm$ 0.03 & 17.82 $\pm$ 0.03 & 2008-05-13\
J165646-4239\_1 & 0473-0602564 & U & 14.49 $\pm$ 0.3 & & 12.46 $\pm$ 0.3 & 10.89 $\pm$ 0.3 &\
J165646-4239\_2 & & A & & 22.69 $\pm$ 0.06 & 21.76 $\pm$ 0.05 & 20.83 $\pm$ 0.04 & 2008-05-13\
J165707-4255\_1 & 0470-0567106 & U & 14.15 $\pm$ 0.3 & & 12.37 $\pm$ 0.3 & 11.04 $\pm$ 0.3 &\
J170017-4220\_1 & & I,A,A,A & $>23.5 \pm$ 0.3 & $>25$ & $>25$ & $>23.4 \pm$ 0.3 & I:2008-07-30,A:2008-05-13\
J170052-4210\_1 & 0478-0576746 & U & 15.02 $\pm$ 0.3 & & 13.57 $\pm$ 0.3 & 11.83 $\pm$ 0.3 &\
J170444-4109\_1 & 0488-0491640 & U & 16.68 $\pm$ 0.3 & & 15.07 $\pm$ 0.3 & 13.01 $\pm$ 0.3 &\
J170536-4038\_1 & 0493-0497837 & U & 11.65 $\pm$ 0.3 & & 10.89 $\pm$ 0.3 & 10.33 $\pm$ 0.3 &\
J172050-3710\_1 & 0528-0624586 & U & 11.97 $\pm$ 0.3 & & 11.62 $\pm$ 0.3 & 11.47 $\pm$ 0.3 &\
J172550-3533\_2 & 0544-0498651 & U & 16.04 $\pm$ 0.3 & & 14.84 $\pm$ 0.3 & 14.97 $\pm$ 0.3 &\
J172550-3533\_3 & 0544-0498592 & U & 11.82 $\pm$ 0.3 & & 11.16 $\pm$ 0.3 & 10.89 $\pm$ 0.3 &\
J172623-3516\_1 & 0547-0493410 & U & 14.26 $\pm$ 0.3 & & 11.86 $\pm$ 0.3 & 10.82 $\pm$ 0.3 &\
J172642-3540\_1 & 0543-0502139 & U & 11.62 $\pm$ 0.3 & & 9.98 $\pm$ 0.3 & 9.16 $\pm$ 0.3 &\
J173548-3207\_1 & 0578-0723476 & U & 15.19 $\pm$ 0.3 & & 14.67 $\pm$ 0.3 & 14.45 $\pm$ 0.3 &\
J181116-1828\_2 & & A(blend) & & 20.08 $\pm$ 0.03 & 18.93 $\pm$ 0.03 & 17.84 $\pm$ 0.03 & 2008-05-13\
J181213-1842\_7 & & I & & $>23.5 \pm$ 0.5 & & & 2008-07-30\
J181213-1842\_9 & 0713-0551349 & U & 13.43 $\pm$ 0.3 & & & 12.76 $\pm$ 0.3 &\
J181915-1601\_2 & 0739-0563199 & U & 16.15 $\pm$ 0.3 & & 13.66 $\pm$ 0.3 & 11.23 $\pm$ 0.3 &\
J182435-1311\_1 & 0768-0532899 & U & 13.08 $\pm$ 0.3 & & 11.01 $\pm$ 0.3 & 9.92 $\pm$ 0.3 &\
J182509-1253\_1 & 0771-0546066 & U & 12.64 $\pm$ 0.3 & & 11.36 $\pm$ 0.3 & 10.74 $\pm$ 0.3 &\
J182538-1214\_1 & 0777-0567961 & U & 14.33 $\pm$ 0.3 & & 13.00 $\pm$ 0.3 & 12.89 $\pm$ 0.3 &\
J182651-1206\_4 & 0778-0569793 & U & 16.62 $\pm$ 0.3 & & 13.97 $\pm$ 0.3 & 14.10 $\pm$ 0.3 &\
J183116-1008\_1 & 0798-0402571 & U & 16.79 $\pm$ 0.3 & & 13.38 $\pm$ 0.3 & &\
J183206-0938\_1 & 0803-0434604 & U & & & 16.59 $\pm$ 0.3 & 11.35 $\pm$ 0.3 &\
J184652-0240\_1 & 0873-0550754 & U & 19.50 $\pm$ 0.3 & & 15.44 $\pm$ 0.3 & 12.41 $\pm$ 0.3 &\
J185608+0218\_1 & 0922-0521590 & U & 10.36 $\pm$ 0.3 & & 9.48 $\pm$ 0.3 & 9.12 $\pm$ 0.3 &\
J190534+0659\_1 & 0969-0463844 & U & 14.67 $\pm$ 0.3 & & 11.98 $\pm$ 0.3 & 10.46 $\pm$ 0.3 &\
J190814+0832\_2 & 0985-0428114 & U & 16.05 $\pm$ 0.3 & & 14.32 $\pm$ 0.3 & 12.78 $\pm$ 0.3 &\
J190818+0745\_1 & 0977-0515881 & U & 18.04 $\pm$ 0.3 & & 16.74 $\pm$ 0.3 & 16.54 $\pm$ 0.3 &\
J194310+2318\_5 & 1132-0447077 & U & 9.63 $\pm$ 0.3 & & 9.08 $\pm$ 0.3 & 8.85 $\pm$ 0.3 &\
J194939+2631\_1 & 1165-0456015 & U,I,I,NA & 20.20 $\pm$ 0.3 & 20.1 $\pm$ 0.1 & 19.1 $\pm$ 0.1 & & 2008-07-30
[lcc]{} J155831–5334\_1 & 1.76 &\
J163751–4656\_1 & 1.67 & 3.16\
J165420–4337\_1 & & 2.29\
J172550–3533\_2 & 1.08 & 1.71\
J173548–3207\_1 & 1.22 & 3.02\
J181116–1828\_2 & & 3.15\
J183345–0828\_1 & 1.25 &\
J190818+0745\_1 & & 1.17
[lcccccc]{} J143148-6021\_1 & 0296-0521300 & 0.4 & 14315021-6022087 & 0.56 & G315.0382+00.1212 & 0.41\
J143148-6021\_2 & 0296-0521279 & 0.35 & 14314859-6021444 & 0.25 & G315.0377+00.1287 & 0.36\
J143148-6021\_4 & 0296-0521208 & 0.32 & 14314161-6019552 & 0.7 & G315.0358+00.1622 & 0.56\
J144519-5949\_1 & & & 14451506-5949286 & 0.51 & G316.7879-00.0504 & 0.61\
J144519-5949\_4 & 0301-0471052 & 0.36 & 14451937-5948007 & 0.47 & G316.8065-00.0321 & 0.64\
J144519-5949\_5 & & & 14445787-5950387 & 0.3 & G316.7471-00.0527 & 0.66\
J144519-5949\_6 & & & 14450584-5949317 & 0.64 & G316.7701-00.0429 & 0.78\
J145732-5901\_1 & & & 14573495-5859538 & 0.18 & G318.5603-00.0084 & 0.22\
J150436-5824\_2 & & & 15043142-5823048 & 0.98 & G319.6454+00.1008 & 1.11\
J150436-5824\_3 & 0315-0471854 & 0.84 & 15045112-5824582 & 0.57 & G319.6676+00.0522 & 0.67\
J151005-5824\_4 & & & 15100715-5826089 & 0.23 & G320.2556-00.3078 & 0.62\
J151005-5824\_5 & & & 15100685-5825567 & 0.14 & &\
J151005-5824\_9 & & & 15095562-5826220 & 0.16 & G320.2324-00.2984 & 0.74\
J151005-5824\_10 & & & 15095233-5827126 & 0.52 & G320.2189-00.3069 & 0.46\
J151005-5824\_11 & & & 15100507-5826021 & 0.15 & G320.2527-00.3040 & 0.39\
J153751-5556\_1 & 0340-0505998 & 0.41 & 15375937-5554232 & 0.45 & G324.7982-00.2732 & 0.29\
J153751-5556\_2 & 0340-0505789 & 0.37 & 15373513-5556504 & 0.38 & G324.7284-00.2727 & 0.13\
J154122-5522\_2 & & & & & G325.5315-00.1292 & 0.55\
J154122-5522\_3 & & & & & G325.4728-00.1194 & 0.49\
J154557-5443\_6 & 0352-0535841 & 0.33 & & & &\
J154557-5443\_7 & 0353-0529489 & 1.13 & & & &\
J154905-5420\_1 & & & 15485853-5421473 & 0.43 & G326.9824-00.0001 & 0.54\
J154905-5420\_10 & 0356-0527966 & 0.53 & 15484845-5419001 & 0.26 & G326.9922+00.0514 & 0.45\
J154905-5420\_11 & 0356-0528195 & 0.62 & 15490739-5418534 & 0.09 & G327.0294+00.0242 & 0.2\
J154905-5420\_3 & & & 15485233-5419128 & 0.26 & G326.9974+00.0428 & 0.19\
J154905-5420\_4 & 0356-0528174 & 0.54 & 15490492-5419300 & 0.14 & G327.0183+00.0200 & 0.19\
J154905-5420\_5 & & & 15491154-5422275 & 0.43 & G327.0002-00.0285 & 0.54\
J154905-5420\_6 & & & & & G326.9922-00.0152 & 0.78\
J154905-5420\_8 & & & & & G327.0269+00.0141\* & 0.62\
J154951-5416\_1 & 0357-0538762 & 0.27 & 15495168-5416297 & 0.28 & G327.1384-00.0116\* & 0.46\
J155035-5408\_2 & 0358-0538121 & 0.35 & 15503372-5409160 & 0.14 & G327.2937+00.0181 & 0.37\
J155831-5334\_2 & 0364-0530387 & 0.28 & 15584724-5334064 & 0.59 & G328.6013-00.3046 & 0.71\
J162011-5002\_2 & 0399-0531568 & 0.16 & 16200603-5001406 & 0.23 & G333.4207+00.0715 & 0.36\
J162208-5005\_3 & & & 16220699-5004580 & 0.38 & G333.6115-00.1955 & 0.53\
J162246-4946\_2 & & & 16225165-4946377 & 0.27 & G333.9125-00.0647 & 0.4\
J162246-4946\_4 & & & 16225873-4946109 & 0.33 & G333.9312-00.0730 & 0.69\
J163524-4728\_3 & 0424-0650841 & 0.26 & 16352088-4730567 & 0.23 & G336.9720+00.0028 & 0.07\
J163524-4728\_4 & 0425-0649089 & 0.39 & 16352057-4728317 & 0.44 & G337.0012+00.0305 & 0.39\
J165420-4337\_2 & 0463-0473953 & 0.73 & 16543480-4338027 & 0.65 & G342.1074-00.0150 & 0.71\
J165707-4255\_2 & 0470-0567346 & 0.62 & 16571994-4256206 & 0.33 & G342.9629+00.0301 & 0.44\
J165901-4208\_1 & & & & & G343.8034+00.2822 & 0.34\
J165901-4208\_2 & & & 16591198-4209350 & 0.34 & G343.7873+00.2447 & 0.44\
J165901-4208\_3 & 0478-0574898 & 0.2 & 16591006-4208451 & 0.08 & G343.7945+00.2579 & 0.13\
J170017-4220\_2 & 0476-0587932 & 0.03 & 17002524-4219003 & 0.08 & G343.8034-00.0300 & 0.19\
J170112-4212\_1 & & & & & G343.9738-00.1029 & 0.23\
J170555-4104\_1 & 0489-0494967 & 0.4 & 17054550-4104024 & 0.28 & G345.4011-00.0573 & 0.17\
J170555-4104\_2 & 0489-0494969 & 0.28 & 17054564-4104172 & 0.37 & G345.3981-00.0601 & 0.41\
J171715-3718\_1 & 0527-0619568 & 0.72 & 17170761-3716139 & 0.38 & G349.7655+00.4099\* & 0.59\
J171715-3718\_2 & & & 17172402-3717160 & 0.5 & G349.7829+00.3556 & 0.71\
J171910-3652\_3 & 0531-0542255 & 0.11 & 17192403-3652095 & 0.41 & G350.3546+00.2705 & 1.02\
J171922-3703\_2 & 0529-0602946 & 0.11 & 17192991-3701128 & 0.22 & G350.2423+00.1677 & 0.33\
J171922-3703\_4 & & & 17192143-3703225 & 0.3 & G350.1967+00.1702 & 0.44\
J172623-3516\_2 & & & 17262588-3515229 & 0.48 & G352.4885+00.0118 & 0.73\
J172642-3504\_3 & 0549-0511010 & 0.62 & 17264746-3504569 & 0.28 & &\
J172642-3504\_4 & 0549-0510994 & 0.71 & 17264522-3502047 & 0.8 & G352.7090+00.0808 & 0.47\
J172642-3504\_5 & & & & & G352.6923+00.1040 & 0.08\
J173628-3141\_1 & 0583-0582872 & 0.43 & 17362729-3141234 & 0.2 & G356.6169+00.2196 & 0.1\
J180857-2004\_3 & 0699-0558263 & 0.25 & 18090897-2004218 & 0.41 & G010.3480-00.1793 & 0.35\
J181033-1917\_1 & & & 18103278-1917550 & 0.49 & G011.1845-00.0929 & 0.66\
J181116-1828\_1 & 0715-0555726 & 0.14 & 18112081-1828199 & 0.17 & G011.9999+00.1394 & 0.3\
J181116-1828\_3 & & & 18110974-1827473 & 0.44 & G011.9868+00.1823 & 0.3\
J181116-1828\_4 & 0714-0538112 & 0.39 & 18111909-1830479 & 0.47 & G011.9605+00.1256 & 0.39\
J181213-1842\_1 & & & 18122087-1844028 & 0.12 & G011.8843-00.1943 & 0.24\
J181213-1842\_2 & 0713-0551671 & 0.24 & & & G011.9377-00.1384 & 0.26\
J181213-1842\_4 & 0713-0551557 & 0.45 & 18121282-1841259 & 0.66 & G011.9073-00.1456 & 0.29\
J181213-1842\_5 & 0713-0551321 & 0.4 & & & &\
J181213-1842\_6 & 0712-0536482 & 0.4 & 18120510-1843248 & 0.76 & G011.8637-00.1347 & 1.09\
J181213-1842\_8 & 0712-0537442 & 0.58 & 18122472-1842024 & 0.47 & G011.9210-00.1916 & 0.36\
J181705-1607\_2 & 0738-0565615 & 0.57 & 18170377-1606272 & 0.3 & G014.7290+00.0720 & 0.12\
J181705-1607\_4 & 0738-0565492 & 0.63 & 18165657-1608235 & 0.36 & G014.6869+00.0820 & 0.39\
J181705-1607\_5 & 0739-0559943 & 0.14 & 18170585-1605576 & 0.2 & G014.7402+00.0686 & 0.38\
J181705-1607\_7 & 0738-0565550 & 0.44 & 18165970-1606182 & 0.18 & G014.7235+00.0876 & 0.38\
J181852-1559\_1 & 0740-0564778 & 0.27 & 18185174-1559403 & 0.22 & G015.0335-00.2550 & 0.14\
J181852-1559\_3 & 0740-0564744 & 0.23 & 18185004-1559157 & 0.8 & G015.0364-00.2457 & 0.84\
J181917-1548\_1 & & & 18190851-1549270 & 0.45 & G015.2154-00.2336 & 0.19\
J182509-1253\_2 & 0771-0545940 & 0.28 & 18250314-1252109 & 0.27 & G018.4960-00.1086 & 0.09\
J182651-1206\_1 & 0779-0556414 & 0.48 & & & G019.4011-00.1473 & 0.31\
J183116-1008\_2 & 0798-0402569 & 0.37 & 18311645-1009105 & 0.3 & G021.6098-00.1949 & 0.07\
J183116-1008\_3 & 0798-0402703 & 0.31 & 18312100-1007598 & 0.34 & G021.6358-00.2024 & 0.19\
J183345-0828\_2 & 0815-0416246 & 0.31 & 18334527-0825192 & 0.51 & G023.4277+00.0616 & 0.42\
J183356-0822\_1 & 0816-0437498 & 0.42 & 18335034-0823032 & 0.39 & G023.4708+00.0605 & 0.35\
J183356-0822\_6 & 0816-0437875 & 0.81 & & & &\
J183356-0822\_7 & 0816-0437882 & 0.72 & 18340563-0821033 & 0.32 & &\
J183518-0754\_1 & 0820-0496843 & 0.22 & 18352446-0754383 & 0.38 & G024.0701-00.0659 & 0.7\
J184008-0543\_1 & & & & & G026.5331-00.0987 & 0.13\
J184400-0355\_1 & 0860-0388202 & 0.42 & 18440331-0358188 & 0.5 & G028.5561-00.1712 & 0.33\
J184652-0240\_2 & 0873-0550942 & 0.5 & 18465626-0239530 & 0.27 & G030.0475-00.2144 & 0.41\
J184741-0219\_2 & 0876-0581370 & 0.34 & 18473651-0219323 & 0.18 & G030.4257-00.2088 & 0.24\
J184741-0219\_4 & & & 18474299-0218227 & 0.51 & G030.4552-00.2240 & 0.38\
J184741-0219\_5 & 0876-0581506 & 0.24 & 18474431-0219167 & 0.34 & G030.4444-00.2357 & 0.14\
J185643+0220\_1 & 0923-0533662 & 0.17 & 18564801+0218361 & 0.49 & G035.5989-00.1389 & 0.55\
J190814+0832\_1 & & & 19081119+0829537 & 0.67 & G042.3972+00.1784 & 0.73\
J190814+0832\_3 & 0985-0428173 & 0.46 & 19081590+0832408 & 0.24 & G042.4474+00.1826 & 0.31\
J190818+0745\_3 & & & & & G041.7964-00.1868 & 0.78\
J194152+2251\_1 & 1128-0498149 & 0.81 & & & &\
J194152+2251\_3 & 1129-0484063 & 0.26 & 19415297+2254177 & 0.67 & G058.9142-00.0873 & 0.73\
J194310+2318\_1 & 1132-0446999 & 0.24 & 19430677+2316122 & 0.25 & G059.3716-00.1515 & 0.48\
J194310+2318\_2 & 1132-0447018 & 0.8 & 19430779+2317032 & 0.26 & G059.3858-00.1479 & 0.33\
J194310+2318\_3 & 1133-0422179 & 0.52 & 19430494+2318339 & 0.16 & G059.4023-00.1258 & 0.22\
J194310+2318\_4 & 1133-0422272 & 0.5 & 19431133+2320391 & 0.27 & G059.4446-00.1298 & 0.15\
J194310+2318\_6 & 1133-0422204 & 0.21 & 19430704+2318318 & 0.33 & G059.4058-00.1331 & 0.19\
J194310+2318\_7 & 1133-0422314 & 0.17 & 19431451+2320165 & 0.21 & G059.4452-00.1435 & 0.02\
J194310+2318\_8 & & & 19431304+2317250 & 0.26 & G059.4011-00.1623 & 0.33\
J194310+2318\_10 & & & 19430845+2318172 & 0.16 & G059.4050-00.1398 & 0.29\
J194332+2323\_1 & 1133-0422494 & 0.51 & 19432717+2323314 & 0.26 & G059.5163-00.1585 & 0.3\
J194332+2323\_2 & 1134-0415530 & 0.45 & & & &\
J194332+2323\_3 & & & 19434329+2324562 & 0.4 & G059.5674-00.2002 & 0.29\
J194332+2323\_4 & 1134-0415508 & 0.19 & 19432977+2325098 & 0.25 & G059.5450-00.1535 & 0.16\
J194332+2323\_5 & 1134-0415545 & 0.27 & 19433302+2326247 & 0.14 & G059.5692-00.1539 & 0.18\
J194332+2323\_6 & 1134-0415461 & 0.2 & 19432305+2325174 & 0.08 & G059.5340-00.1301 & 0.23\
J194332+2323\_8 & & & & & G059.5078-00.1462 & 0.91\
J194622+2436\_1 & 1145-0435437 & 0.25 & 19462497+2435462 & 0.58 & G060.8974-00.1421 & 0.69\
J194622+2436\_2 & 1146-0434793 & 0.48 & 19463435+2437058 & 0.41 & G060.9345-00.1617 & 0.58\
J194622+2436\_3 & 1146-0434550 & 1.05 & 19461529+2438323 & 0.39 & G060.9190-00.0872 & 0.52\
J195006+2628\_1 & 1164-0455522 & 0.46 & 19501612+2627172 & 0.43 & G062.9404+00.0480 & 0.23\
J195006+2628\_3 & & & 19500352+2627436 & 0.39 & G062.9228+00.0922 & 0.15\
J195006+2628\_4 & 1164-0455331 & 0.89 & 19500525+2628178 & 0.36 & G062.9343+00.0915 & 0.4\
J195006+2628\_5 & 1164-0455237 & 0.28 & 19500110+2629342 & 0.11 & G062.9447+00.1156 & 0.24
[lccc]{} Total (253 sources) & 57 & 61 & 47\
$>20$ X-ray counts (74 sources) & 59 & 61 & 55
[lccccc]{} J144042–6001\_1 & MGPS & SNR G316.3–0.0 & SNR & L & 1\
J144519–5949\_(1-6) & MGPS & GAL 316.8-00.1 & H [ii]{} & C & 2\
J144701–5919\_1 & ATCA & AX J144701–5919 & MS & C & 3\
J145732–5901\_(1-2) & MGPS & SNR G318.2+0.1 & SNR & L & 4,5\
J151005–5824\_(1-11) & MGPS & G320.3–0.3 & H [ii]{} & C & 6\
J153818–5541\_1 & ATCA & & ND & & 7\
J154905–5420\_(1-11) & MGPS & G326.96+0.03 & H [ii]{} & A & 8\
J155331–5347\_1 & ATCA & & ND & & 7\
J155331–5347\_2 & ATCA & & ND & & 7\
J162208–5005\_(1-3) & MGPS & G333.6–0.2 & H [ii]{} & C & 6\
J163252–4746\_1 & ATCA & & ND & & 7\
J163252–4746\_2 & ATCA & AX J163252–4746 & CWB & C & 3\
J163751–4656\_1 & ATCA & & ND & & 7\
J165217–4414\_1 & MGPS & uncataloged & Arc & C &\
J165420–4337\_1 & ATCA & & ND & & 7\
J165420–4337\_2 & ATCA & & ND & & 7\
J165646–4239\_(1-2) & MGPS & SDC G343.306+0.161 & IRDC & C & 9\
J165707–4255\_(1-2) & MGPS & uncataloged & Arc & C &\
J170052–4210\_1 & ATCA & & ND & & 7\
J172550–3533\_(1-3) & MGPS & uncataloged & Arc & L &\
J180857–2004\_1,2 & 90, MAGPIS & MAGPIS 10.3139–01417 & D & C & 10\
J181116–1828\_5 & MAGPIS & MAGPIS 11.97095+0.19155 & AGN & K & 10\
J181213–1842\_1,8 & 90, MAGPIS & SNR G11.8–0.2 & SNR & L & 11\
J181213–1842\_2,5 & 90, MAGPIS & SNR G12.0–0.1 & SNR & A & 12\
J181705–1607\_(1-7) & 90, MAGPIS & MAGPIS 14.6167+0.0667 & D & L & 10\
J182435–1311\_(1-4) & 90, MAGPIS & SNR G18.1–0.1 & SNR & C & 10,11\
J182538–1214\_1 & 90, MAGPIS & SNR G19.1+0.2 & SNR & L & 11\
J183206–0938\_1 & MAGPIS & MAGPIS 22.15394–0.15414 & CWB & C & 10,13\
J183356–0822\_6,7 & MAGPIS & SNR 23.5667–0.0333 & SNR & L & 10\
J184400–0355\_1 & MAGPIS & SNR G28.6–0.1 & SNR & L & 14\
J184447–0305\_1 & MAGPIS & SNR 29.3667+0.1000 & SNR & C & 10\
J184738–0156\_1 & MAGPIS & NVSS 184736–015632 & CWB & C & 3,15\
J184741–0219\_3 & MAGPIS & MAGPIS 30.43741–0.20625 & AGN & K & 10\
J194310+2318\_(1-10) & VGPS & G59.5–0.2 & H [ii]{} & C & 6\
J194332+2323\_(1-8) & VGPS & G59.5–0.2 & H [ii]{} & C & 6\
J194622+2436\_1 & VGPS & NVSS 194620+243514 & Com & L & 15\
J195006+2628\_(1-5) & VGPS & G62.9+0.1 & H [ii]{} & C & 6,16
[lcc]{} J181213-1842\_1 & & 1.15\
J181213-1842\_2 & 3.05 & 7.15\
J181213-1842\_5 & 4.04 &\
J182435-1311\_1 & & 3.17\
J183356-0822\_6 & 1.68 &
[lccccccccccccc]{} J144519-5949\_2 & $1.52_{-0.73}^{+1.38}$ & $5.00_{-2.10}^{+3.60}$ & $23.70 \pm 4.80$ & $218.44 \pm 44.20$ & & 2.8 & 1 & & 32.35 & 33.31 & & &\
J144519-5949\_3 & $0.78_{-0.57}^{+2.69}$ & $22.00_{-15.00}^{+51.00}$ & $13.05 \pm 4.73$ & $7207.60 \pm 2610.31$ & & 2.8 & 1 & & 32.09 & 34.83 & & &\
J144519-5949\_5 & $>1.94$ & $0.14_{-0.14}^{+0.68}$ & $3.37 \pm 2.29$ & $3.98 \pm 2.71$ & & 2.8 & 1 & & 31.50 & 31.57 & & &\
J151005-5824\_1 & $0.90_{-0.77}^{+1.80}$ & $7.60_{-4.60}^{+27.40}$ & $0.74 \pm 0.45$ & $47.76 \pm 28.64$ & & 4.7 & 2 & & 31.29 & 33.10 & & &\
J151005-5824\_2 & $82.45$ & $2.80_{-2.07}^{+0.00}$ & $1.72 \pm 0.85$ & $2.92 \pm 1.45$ & & 4.7 & 2 & & 31.66 & 31.89 & & &\
J151005-5824\_3 & $0.21_{-0.11}^{+1.42}$ & $1.60_{-1.35}^{+1.30}$ & $0.18 \pm 0.12$ & $65.24 \pm 44.33$ & & 4.7 & 2 & & 30.68 & 33.24 & & &\
J151005-5824\_4 & $82.45$ & $2.20_{-1.53}^{+-0.10}$ & $1.21 \pm 0.73$ & $1.96 \pm 1.18$ & & 4.7 & 2 & & 31.51 & 31.71 & & &\
J151005-5824\_5 & $82.45$ & $1.80_{-1.50}^{+0.00}$ & $1.14 \pm 0.68$ & $1.76 \pm 1.06$ & & 4.7 & 2 & & 31.48 & 31.67 & & &\
J151005-5824\_6 & $>0.1$ & $2.50_{-2.49}^{+10.50}$ & $0.44 \pm 0.18$ & $452.64 \pm 182.20$ & & 4.7 & 2 & & 31.07 & 34.08 & & &\
J151005-5824\_7 & $>2.21$ & $21.00_{-17.00}^{+11.00}$ & $6.46 \pm 1.94$ & $36.97 \pm 11.10$ & & 4.7 & 2 & & 32.23 & 32.99 & & &\
J151005-5824\_8 & $0.17_{-0.07}^{+0.33}$ & $33.00_{-21.00}^{+28.00}$ & $0.96 \pm 0.48$ & $218762449.44 \pm 108233831.76$ & & 4.7 & 2 & & 31.41 & 39.76 & & &\
J151005-5824\_9 & $>2.49$ & $0.01_{0.00}^{+0.25}$ & $0.41 \pm 0.28$ & $0.42 \pm 0.28$ & & 4.7 & 2 & & 31.04 & 31.04 & & & 3.15\
J151005-5824\_10 & $>5.26$ & $1.20_{-1.16}^{+1.00}$ & $1.39 \pm 0.69$ & $2.00 \pm 0.99$ & & 4.7 & 2 & & 31.56 & 31.72 & & & 1.71\
J151005-5824\_11 & $>0.1$ & $0.38_{-0.38}^{+0.87}$ & $0.64 \pm 0.43$ & $0.79 \pm 0.54$ & & 4.7 & 2 & & 31.23 & 31.32 & & &\
J154905-5420\_1 & $0.61_{-0.33}^{+0.81}$ & $11.50_{-5.60}^{+12.50}$ & $1.49 \pm 0.68$ & $743.78 \pm 340.69$ & & 3.7 & 3 & & 31.39 & 34.09 & & &\
J154905-5420\_2 & $>2.63$ & $2.30_{-0.80}^{+3.30}$ & $4.07 \pm 1.35$ & $6.63 \pm 2.20$ & & 3.7 & 3 & & 31.82 & 32.04 & & &\
J154905-5420\_3 & $0.74_{-0.56}^{+1.54}$ & $15.50_{-9.50}^{+39.50}$ & $0.99 \pm 0.67$ & $397.68 \pm 270.25$ & & 3.7 & 3 & & 31.21 & 33.81 & & 2.40 & 2.21\
J154905-5420\_4 & $1.80_{-1.69}^{+3.04}$ & $0.01_{0.00}^{+1.29}$ & $0.40 \pm 0.24$ & $0.41 \pm 0.25$ & & 3.7 & 3 & & 30.82 & 30.83 & & 1.62 & 1.88\
J154905-5420\_5 & $>0.1$ & $0.01_{0.00}^{+0.42}$ & $1.66 \pm 0.71$ & $1.69 \pm 0.72$ & & 3.7 & 3 & & 31.44 & 31.44 & & &\
J154905-5420\_7 & $>16.3$ & $1.05_{-1.04}^{+0.35}$ & $1.89 \pm 0.87$ & $2.66 \pm 1.22$ & & 3.7 & 3 & & 31.49 & 31.64 & & &\
J154905-5420\_8 & $>4.63$ & $1.20_{-1.19}^{+1.10}$ & $1.09 \pm 0.74$ & $1.57 \pm 1.07$ & & 3.7 & 3 & & 31.25 & 31.41 & & & 3.02\
J154905-5420\_9 & $>0.1$ & $0.01_{0.00}^{+0.60}$ & $0.70 \pm 0.38$ & $0.71 \pm 0.38$ & & 3.7 & 3 & & 31.06 & 31.07 & & &\
J154905-5420\_10 & $>0.1$ & $0.01_{0.00}^{+0.66}$ & $0.46 \pm 0.31$ & $0.47 \pm 0.32$ & & 3.7 & 3 & & 30.88 & 30.89 & & 3.25 & 5.29\
J154905-5420\_11 & $>0.63$ & $0.46_{-0.45}^{+1.39}$ & $1.24 \pm 0.47$ & $2.05 \pm 0.78$ & & 3.7 & 3 & & 31.31 & 31.53 & & &\
J162208-5005\_1 & $>1.91$ & $2.40_{-1.10}^{+3.60}$ & $23.73 \pm 6.53$ & $42.69 \pm 11.74$ & & 3.1 & 2 & & 32.44 & 32.69 & & &\
J162208-5005\_3 & $>6.52$ & $2.60_{-1.40}^{+2.30}$ & $6.75 \pm 4.59$ & $11.30 \pm 7.68$ & & 3.1 & 2 & & 31.89 & 32.11 & & &\
J194310+2318\_1 & $2.28_{-1.86}^{+6.97}$ & $0.01_{0.00}^{+1.04}$ & $5.31 \pm 1.46$ & $5.47 \pm 1.50$ & & 2.6 & 2 & & 31.63 & 31.65 & & &\
J194310+2318\_4 & $0.12_{-0.02}^{+19.33}$ & $0.67_{-0.66}^{+0.00}$ & $1.06 \pm 0.72$ & $253.21 \pm 172.07$ & & 2.6 & 2 & & 30.93 & 33.31 & & & 1.06\
J194310+2318\_5 & $0.14_{-0.03}^{+3.33}$ & $0.76_{-0.75}^{+-0.03}$ & $5.97 \pm 1.08$ & $1446.16 \pm 262.87$ & & 2.6 & 2 & & 31.68 & 34.07 & & &\
J194310+2318\_6 & $0.76_{-0.62}^{+1.38}$ & $1.75_{-1.05}^{+3.85}$ & $5.33 \pm 1.41$ & $94.55 \pm 24.98$ & & 2.6 & 2 & & 31.63 & 32.88 & & &\
J194310+2318\_7 & $>0.19$ & $1.60_{-1.59}^{+10.90}$ & $1.70 \pm 0.92$ & $909.74 \pm 491.76$ & & 2.6 & 2 & & 31.14 & 33.87 & & &\
J194310+2318\_8 & $82.45$ & $0.94_{-0.93}^{+0.26}$ & $3.77 \pm 2.56$ & $5.22 \pm 3.55$ & & 2.6 & 2 & & 31.48 & 31.63 & & &\
J194310+2318\_9 & $0.10_{0.00}^{+1.28}$ & $1.70_{-1.69}^{+1.40}$ & $0.99 \pm 0.59$ & $18914.51 \pm 11342.64$ & & 2.6 & 2 & & 30.90 & 35.18 & & &\
J194332+2323\_1 & $2.91_{-2.56}^{+7.09}$ & $0.13_{-0.12}^{+1.52}$ & $1.65 \pm 0.55$ & $2.10 \pm 0.69$ & & 2.6 & 2 & & 31.13 & 31.23 & & &\
J194332+2323\_3 & $0.22_{-0.11}^{+0.33}$ & $3.00_{-1.30}^{+4.60}$ & $0.64 \pm 0.35$ & $1053.73 \pm 569.60$ & & 2.6 & 2 & & 30.72 & 33.93 & & &\
J194332+2323\_5 & $>0.1$ & $1.15_{-1.14}^{+10.35}$ & $3.81 \pm 1.32$ & $24.79 \pm 8.57$ & & 2.6 & 2 & & 31.49 & 32.30 & & &\
J194332+2323\_6 & $>0.1$ & $0.01_{0.00}^{+2.49}$ & $0.70 \pm 0.38$ & $0.73 \pm 0.39$ & & 2.6 & 2 & & 30.76 & 30.77 & & 1.35 & 1.38\
J194332+2323\_7 & $0.15_{-0.05}^{+0.55}$ & $0.58_{-0.57}^{+.15}$ & $0.64 \pm 0.29$ & $51.52 \pm 23.60$ & & 2.6 & 2 & & 30.72 & 32.62 & & &\
J195006+2628\_1 & $0.10_{0.00}^{+0.04}$ & $1.25_{-0.73}^{+2.35}$ & $0.70 \pm 0.38$ & $3815.10 \pm 2062.27$ & & 2.3 & 4 & & 30.65 & 34.38 & & &\
J195006+2628\_2 & $1.74_{-1.62}^{+2.02}$ & $0.01_{0.00}^{+1.49}$ & $2.74 \pm 0.55$ & $2.83 \pm 0.57$ & & 2.3 & 4 & & 31.24 & 31.25 & & &\
J195006+2628\_3 & $2.70_{-2.59}^{+16.75}$ & $0.23_{-0.22}^{+1.87}$ & $0.61 \pm 0.42$ & $.88 \pm .60$ & & 2.3 & 4 & & 30.59 & 30.75 & & &\
J195006+2628\_4 & $0.40_{-0.29}^{+1.23}$ & $2.10_{-1.37}^{+3.50}$ & $0.63 \pm 0.34$ & $58.70 \pm 31.73$ & & 2.3 & 4 & & 30.60 & 32.57 & & &\
J195006+2628\_5 & $0.11_{-0.01}^{+1.76}$ & $1.55_{-1.54}^{+1.35}$ & $0.74 \pm 0.28$ & $5807.00 \pm 2211.85$ & & 2.3 & 4 & & 30.67 & 34.57 & & &\
[llcccc]{} AX J143148–6021 & ChI J143148–6021\_3 & U & & 1 & n,R,n\
AX J143416–6024 & & RS CVn & HD 127535 & 2,3\* & n,n,n\
AX J144042–6001 & ChI J144042-6001\_1 & PMS & HD 128696 & 2\* & n,n,n\
AX J144519–5949 & ChI J144519-5949\_2(1,3-6) & WR and H [ii]{} & GAL 316.8–00.1 & 1 & T,T,N\
AX J144547–5931 & ChI J144547-5931\_1 & OIf$^{+}$ & & 4,4 & I,I,n\
AX J144701–5919 & ChI J144701-5919\_1 & WR & & 4,4 & I,I,n\
AX J145605–5913 & & U & & & n,n,n\
AX J145732–5901 & & U & & & F,n,n\
AX J150436–5824 & ChI J150436–5824\_1 & U & & 1 & n,n,n\
AX J151005–5824 & ChI J151005–5824\_(1-11) & H [ii]{} & G320.3–0.3 & 1 & I,n,N\
AX J153751–5556 & & U & & & F,n,n\
AX J153818–5541 & ChI J153818-5541\_1 & LMXB & & 5 & n,n,N\
AX J153947–5532 & & U & & & n,n,n\
AX J154122–5522 & ChI J154122–5522\_1 & ASC/PMS & & 1 & T,T,N\
AX J154233–5519 & & U & & & n,n,n\
AX J154557–5443 & ChI J154557–5443\_1 & U & & 1 & n,n,n\
& ChI J154557–5443\_2 & U & & 1 & n,n,n\
& ChI J154557–5443\_3 & U & & 1 & n,n,n\
AX J154905–5420 & ChI J154905–5420\_(1-11) & H [ii]{} & G326.96+0.03 & 1 & I,n,N\
AX J154951–5416 & & U & & & F,n,n\
AX J155035–5408 & ChI J155035–5408\_1 & ASC/PMS & & 1 & n,T,N\
& ChI J155035–5408\_3 & MS/ASC & & 1 & n,T,N\
AX J155052–5418 & & Magnetar & 1E 1547.0–5408 & 6,7 & n,n,n\
AX J155331–5347 & ChI J155331–5347\_1 & ASC/PMS & & 1 & T,T,N\
AX J155644–5325 & & KOIIIe & TYC 8697–1438-1 & 8 & n,n,n\
AX J155831–5334 & ChI J155831–5334\_1 & U & & 1\*\* & n,R,n\
AX J161929–4945 & & HMXB & IGR J16195–4945 & 9,3\*\* & n,n,n\
AX J162011–5002 & ChI J162011-5002\_1 & CV & & 5 & n,n,N\
AX J162046–4942 & ChI J162046–4942\_1 & MS/ASC & & 1\*\* & T,T,N\
AX J162125–4933 & & U & & & n,n,n\
AX J162138–4934 & & U & & & n,n,n\
AX J162155–4939 & & K3III & HD 147070 & 2\* & n,n,n\
AX J162208–5005 & ChI J162208–5005\_(1-3) & H [ii]{} & G333.6–0.2 & 1 & I,n,N\
AX J162246–4946 & & Magnetar & PSR J1622–4950 & 10,10 & I,n,n\
AX J163159–4752 & & HMXB & IGR J16320–4751 & 11,12 & n,n,n\
AX J163252–4746 & ChI J163252-4746\_2 & CWB & & 4,4 & I,I,n\
AX J163351–4807 & & Of?p & HD 148937 & 13,3\* & n,n,n\
AX J163524–4728 & & U & & & F,n,n\
AX J163555–4719 & & SNR/PWN & SNR G337.2+0.1 & 14 & n,n,n\
AX J163751–4656 & ChI J163751–4656\_1 & U & & 1 & n,R,n\
AX J163904–4642 & & SyXB & IGR J16393-4643 & 15,15 & n,n,n\
AX J164042–4632 & & PWN & SNR G338.3–0.0 & 16\* & n,n,n\
AX J165105–4403 & ChI J165105–4403\_1 & U & & 1 & n,n,n\
AX J165217–4414 & ChI J165217–4414\_1 & U & & 1 & n,R,n\
AX J165420–4337 & ChI J165420–4337\_1 & U & & 1\*\* & n,R,n\
AX J165437–4333 & & Binary Star & HD 152335 & 2,3\* & n,n,n\
AX J165646–4239 & ChI J165646–4239\_1 & ASC/PMS & & 1 & n,T,N\
& ChI J165646–4239\_2 & U & & 1 & n,R,n\
AX J165707–4255 & ChI J165707–4255\_1 & ASC/PMS & & 1 & T,T,N\
AX J165723–4321 & & U & & & n,n,n\
AX J165901–4208 & & U & & \*\* & F,n,n\
AX J165904–4242 & & Be star & V921 Sco & 17,3 & n,n,n\
AX J165922–4234 & & U & & & n,n,n\
AX J165951–4209 & & ND & & & F,n,n,\
AX J170006–4157 & & CV & AX J1700.1–4157 & 18,18 & n,n,n\
AX J170017–4220 & ChI J170017-4220\_1 & HMXB & & 19 & T,T,N\
AX J170047–4139 & & HMXB & OAO 1657–415 & 20,20\* & n,n,n\
AX J170052–4210 & ChI J170052–4210\_1 & ASC/PMS & & 1 & T,T,N\
AX J170112–4212 & & U & & & F,n,n\
AX J170349–4142 & & SNR & SNR G344.7–0.1 & 21\* & n,n,n\
AX J170444–4109 & ChI J170444–4109\_1 & ASC/PMS & & 1 & n,T,N\
& ChI J170444–4109\_2 & U & & 1 & n,n,n\
AX J170506–4113 & & U & & & n,n,n\
AX J170536–4038 & ChI J170536–4038\_1 & ASC/PMS & & 1 & T,T,N\
AX J170555–4104 & & U & & & F,n,n\
AX J171715–3718 & & U & & & F,n,n\
AX J171804–3726 & & SNR & SNR G349.7+0.2 & 22\* & n,n,n\
AX J171910–3652 & ChI J171910–3652\_2 & ASC/PMS & & 1 & T,T,N\
AX J171922–3703 & ChI J171922–3703\_1 & U & & 1 & n,n,n\
AX J172050–3710 & ChI J172050-3710\_1 & ASC & & 5 & T,T,N\
AX J172105–3726 & & SNR & SNR G350.1–0.3 & 23 & n,n,n\
AX J172550–3533 & ChI J172550–3533\_1 & AGN or HMXB & & 1 & n,T,N\
& ChI J172550–3533\_2 & CV & V478 Sco & 1 & n,I,n\
& ChI J172550–3533\_3 & ASC/PMS & & 1 & n,T,N\
AX J172623–3516 & ChI J172623–3516\_1 & ASC/PMS & & 1\*\* & T,T,N\
AX J172642–3504 & ChI J172642–3504\_1 & U & & 1 & n,n,n\
AX J172642–3540 & ChI J172642-3540\_1 & K2V & CD-35 11565 & 1 & I,I,n\
AX J172743–3506 & & SNR & SNR G352.7–0.1 & 24\* & n,n,n\
AX J173441–3234 & & CWB & HD 159176 & 25,3\* & n,n,n\
AX J173518–3237 & & SNR & SNR G355.6–0.0 & 26\* & n,n,n\
AX J173548–3207 & ChI J173548–3207\_1 & U & & 1 & n,n,n\
AX J173628–3141 & & U & & & F,n,n\
AX J175331–2538 & & ND & & & F,n,n\
AX J175404–2553 & ChI J175404–2553\_3 & U & & 1 & n,n,n\
AX J180225–2300 & & OB and PMS stars & Triffid Nebula (HD 164492) & 27 & n,n,n\
AX J180800–1956 & & U & & & n,n,n\
AX J180816–2021 & & ND & & & F,n,n\
AX J180838–2024 & & Magnetar & SGR 1806–20 & 28,29\*\* & n,n,n\
AX J180857–2004 & ChI J180857–2004\_1 & U & & 1 & n,n,n\
& ChI J180857–2004\_2 & WR & & 1 & n,T,N\
AX J180902–1948 & & SNR & SNR 10.5–0.0 & 30 & I,n,n\
AX J180948–1918 & & PWN & PSR J1809–1917 & 31\*\* & n,n,n\
AX J180951–1943 & & Magnetar & XTE J1810–197 & 32,33 & n,n,n\
AX J181033–1917 & & U & & & F,n,n\
AX J181116–1828 & ChI J181116–1828\_2 & U & & 1 & n,R,n\
AX J181120–1913 & & U & & \*\* & F,n,n\
AX J181211–1835 & & SNR & SNR G12.0–0.1 & 26\* & n,n,n\
AX J181213–1842 & ChI J181213–1842\_7 & U & & 1 & n,R,n\
& ChI J181213–1842\_9 & ASC/PMS & & 1 & n,T,N\
AX J181705–1607 & & U & & & F,n,n\
AX J181848–1527 & & U & & & n,n,n\
AX J181852–1559 & ChI J181852–1559\_2 & Magnetar & & 34 & n,n,N\
AX J181915–1601 & ChI J181915-1601\_2 & WR & & 1\*\* & n,T,n\
& ChI J181915-1601\_3 & U & & 1 & n,n,n\
AX J181917–1548 & & U & & & F,n,n\
AX J182104–1420 & & SNR/PWN & SNR G16.7+0.1 & 35\* & n,n,n\
AX J182216–1425 & & U & & & F,n,n\
AX J182435–1311 & ChI J182435–1311\_1 & MS-O & & 1 & T,T,N\
AX J182442–1253 & & U & & & n,n,n\
AX J182509–1253 & ChI J182509–1253\_1 & ASC/PMS & & 1 & T,T,N\
& ChI J182509–1253\_3 & U & & 1 & n,n,n\
AX J182530–1144 & ChI J182530–1144\_2 & U & & 1 & n,n,n\
AX J182538–1214 & ChI J182538–1214\_1 & ASC/PMS & & 1 & T,T,N\
AX J182651–1206 & ChI J182651–1206\_4 & MS-O & & 1 & T,T,N\
AX J182846–1116 & & U & & & n,n,n\
AX J183039–1002 & & AGN & & 36,1 & n,n,n\
AX J183114–0943 & & U & & & n,n,n\
AX J183116–1008 & ChI J183116-1008\_1 & CWB & & 37,37 & n,n,n\
AX J183206–0938 & ChI J183206-0938\_1 & CWB & & 37,37 & n,n,N\
AX J183206–0940 & & U & & & n,n,n\
AX J183221–0840 & & CV & AX J1832.3–0840 & 18,18 & n,n,n\
AX J183231–0916 & & U & & & n,n,n\
AX J183345–0828 & & PWN & PSR B1830–08 & 38\* & n,n,N\
& ChI J183345–0828\_1 & WR & & 1 & n,T,N\
AX J183356–0822 & ChI J183356–0822\_2 & U & & 1 & n,n,n\
& ChI J183356–0822\_3 & MS-O & & 1 & n,T,N\
AX J183440–0801 & & U & & & n,n,n\
AX J183506–0806 & & U & & & n,n,n\
AX J183518–0754 & & U & & & F,n,n\
AX J183528–0737 & & SyXB & Scutum X-1 & 39,39 & n,n,n\
AX J183607–0756 & & ND & & & F,n,n\
AX J183618–0647 & & U & & & n,n,n\
AX J183800–0655 & & PSR/PWN & PSR J1838–0655 & 40 & n,n,n\
AX J183931–0544 & & LBV & G26.47+0.02 & 41,41 & n,n,n\
AX J183957–0546 & & U & & & n,n,n\
AX J184004–0552 & & U & & & n,n,n\
AX J184008–0543 & & U & & \*\* & F,n,n\
AX J184024–0544 & & U & & & n,n,n\
AX J184121–0455 & & Magnetar/SNR & 1E 1841–045/SNR G27.4+0.0 (Kes 73) & 42\* & n,n,n\
AX J184355–0351 & & SNR & SNR G28.6–0.1 & 43 & n,n,n\
AX J184400–0355 & & U & & & F,n,n\
AX J184447–0305 & & U & & & F,n,n\
AX J184600–0231 & & U & & & n,n,n\
AX J184610–0239 & & U & & & n,n,n\
AX J184629–0258 & & SNR/PWN & SNR G29.7-0.3(Kes 75)/PSR J1846–0258 & 44\* & n,n,n\
AX J184652–0240 & ChI J184652–0240\_1 & MS-O & & 1 & T,T,N\
AX J184738–0156 & ChI J184738–0156\_1 & CWB & WR121a & 4,4 & I,I,n\
AX J184741–0219 & ChI J184741–0219\_3 & AGN & & 1 & I,I,N\
AX J184848–0129 & & Globular cluster & GLIMPSE–C01 & 45 & n,n,n\
AX J184930–0055 & & SNR & SNR G31.9+0.0 (3C 391) & 46\* & n,n,n\
AX J185015–0025 & & SNR & SNR G32.4+0.1 & 47 & n,n,n\
AX J185240+0038 & & SNR/PSR & SNR G33.6+0.1 (Kes 79)/PSR J1852+0040 & 48\* & n,n,n\
AX J185551+0129 & & SNR/PWN & SNR G34.7–0.4 (W44)/PSR B1853+0.1 & 49 & n,n,n\
AX J185608+0218 & ChI J185608+0218\_1 & ASC/PMS & & 1\*\* & T,T,N\
AX J185643+0220 & ChI J185643+0220\_2 & U & & 1 & n,n,n\
AX J185651+0245 & & U & & & n,n,n\
AX J185721+0247 & & U & & & n,n,n\
AX J185750+0240 & ChI J185750+0240\_1 & U & & 1 & n,R,n\
AX J185905+0333 & & Galaxy cluster & AX J185905+0333 & 50 & n,n,n\
AX J190007+0427 & & U & & & n,n,n\
AX J190144+0459 & & U & & & n,n,n\
AX J190534+0659 & ChI J190534+0659\_1 & MS/ASC & & 1 & T,T,N\
AX J190734+0709 & & SNR & SNR G41.1–0.3 (3C 397) & 51\* & n,n,n\
AX J190749+0803 & ChI J190749+0803\_1 & U & & 1 & n,R,n\
AX J190814+0832 & ChI J190814+0832\_2 & MS/ASC & & 1 & T,T,N\
AX J190818+0745 & ChI J190818+0745\_1 & U & & 1 & n,R,n\
AX J191046+0917 & & HMXB & AX J1910.7+0917 & 52 & n,n,N\
AX J191105+0906 & & SNR & SNR G43.3–0.2 (W49B) & 53\* & n,n,n\
AX J194152+2251 & ChI J194152+2251\_2 & U & & 1 & n,R,n\
AX J194310+2318 & ChI J194310+2318\_5(1-4,6-10) & O7V((f)) and H [ii]{} & G59.5–0.2 & 1\* & I,I,N\
AX J194332+2323 & ChI J194332+2323\_(1-8) & H [ii]{} & G59.5–0.2 & 1\* & I,n,N\
AX J194622+2436 & & U & & & F,n,n\
AX J194649+2512 & & H$\alpha$ Star & EM\* VES 52 & 54\* & n,n,n\
AX J194939+2631 & ChI J194939+2631\_1 & CV & & 55 & n,n,N\
AX J194951+2534 & & U & & & F,n,n\
AX J195006+2628 & ChI J195006+2628\_(1-5) & H [ii]{} & G62.9+0.1 & 1 & I,n,N
[ccccccccc]{} J150436–5824 & 15:04:13.54 & -58:25:07.4 & 0.84 & 3.0 & 47.7 $\pm$ 8.0 & 139 & 154 &\
J154557–5443 & 15:46:09.08 & -54:39:12.9 & 0.89 & 4.2 & 48.0 $\pm$ 8.1 & 147 & 96 & 15460913–5439128\
& 15:45:54.60 & -54:38:49.6 & 0.94 & 4.3 & 38.9 $\pm$ 7.4 & & &\
J154905–5420 & 15:49:12.40 & -54:16:30.2 & 0.94 & 4.1 & 31.8 $\pm$ 6.8 & 178 & 84 & 15491237–5416301\
& 15:49:14.67 & -54:24:54.6 & 1.03 & 4.6 & 20.2 $\pm$ 5.7 & & & 15491463–5424549\
J155035–5408 & 15:50:19.25 & -54:11:27.3 & 0.89 & 3.6 & 35.5 $\pm$ 7.1 & 152 & 242 & 15501925–5411271\
& 15:50:05.97 & -54:07:23.1 & 1.07 & 4.5 & 22.8 $\pm$ 6.0 & & &\
J170444–4109 & 17:04:26.88 & -41:07:34.9 & 0.95 & 3.9 & 26.7 $\pm$ 6.3 & 95 & 170 & 17042687–4107351\
J172642–3504 & 17:26:27.79 & -35:07:27.2 & 0.88 & 4.5 & 72.1 $\pm$ 9.6 & 206 & 120 & 17262781–3507281\
J181213–1842 & 18:12:29.26 & -18:45:21.6 & 1.02 & 4.7 & 21.9 $\pm$ 5.8 & 92 & 172 & 18122923–1845222\
J181915–1601 & 18:19:29.65 & -16:04:33.3 & 0.95 & 4.6 & 30.0 $\pm$ 6.5 & 201 & 132 &\
J184738–0156 & 18:47:52.81 & -01:59:57.9 & 0.89 & 4.8 & 96.5 $\pm$ 10.9& 150 & 120 & 18475281–0159575\
J184741–0219 & 18:47:32.53 & -02:22:23.8 & 0.93 & 4.0 & 34.2 $\pm$ 7.0 & 211 & 53 &\
J194332+2323 & 19:43:29.97 & +23:20:52.3 & 0.88 & 3.1 & 28.4 $\pm$ 6.5 & 143 & 64 & 19432997+2320524
[^1]: http://cxc.harvard.edu/ciao/index.html
[^2]: http://cxc.harvard.edu/toolkit/pimms.jsp
[^3]: http://cxc.harvard.edu/ciao4.3/threads/index.html
[^4]: The HRC instrument has very poor spectral resolution so this, and the following step, are not conducted on these datasets.
[^5]: This equation was constructed by running `wavdetect` on ACIS-I and ACIS-S data. It is therefore unknown whether this position error equation is applicable to the positions calculated from running `wavdetect` on HRC data sets. However, since the `wavdetect` algorithm is not instrument specific, it is likely that this equation estimates reasonable errors for sources detected in HRC observations.
[^6]: The lower-limit of 0.01s for the HRC observations is chosen because a wiring error in the detector degrades the time resolution accuracy from $16\mu s$ to the mean time between events. See http://cxc.harvard.edu/proposer/POG/html/chap7.html\#sec:hrc\_anom
[^7]: The catalog information is downloaded via a generic URL from the VizieR Service (http://vizier.cfa.harvard.edu/viz-bin/VizieR) and the NASA/IPAC Infrared Science Archive (http://irsa.ipac.caltech.edu/).
[^8]: http://irsa.ipac.caltech.edu/data/SPITZER/GLIMPSE/
[^9]: The images from the DSS2R, DSS2B, 2MASS and SUMSS surveys are downloaded using a generic URL from the virtual observatory SkyView, which is a service of the Astrophysics Science Division at NASA/GSFC and the High Energy Astrophysics Division of the Smithsonian Astrophysical Observatory (SAO), http://skyview.gsfc.nasa.gov/
[^10]: The net counts were calculated separately for each energy range so there are several cases where the total number of counts in the $0.3-8.0$ keV energy range is slightly different to the sum of counts detected in the $0.5-2.0$ and $2.0-8.0$ keV energy ranges. These differences are due to rounding errors, as well as the exclusion of very soft counts that were detected between $0.3-0.5$ keV. (The $0.3-0.5$ keV energy range was excluded as the $0.5-2$ keV energy band is more commonly used than the $0.3-2$ keV energy band.)
[^11]: see http://cxc.harvard.edu/ciao/why/dither.html
[^12]: See the `CIAO` detect manual http://cxc.harvard.edu/ciao/download/doc/detect\_manual/
[^13]: The astrometric error is included in this calculation as the source positions listed in the CSC can be obtained from more than one dataset.
[^14]: http://xmmssc-www.star.le.ac.uk/Catalogue/2XMMi-DR3/
[^15]: sources with $>20$ X-ray counts for quantile analysis and $>50$ X-ray counts for spectral modeling.
[^16]: http://cxc.harvard.edu/cal/ASPECT/celmon/
[^17]: See GLIMPSE documents http://www.astro.wisc.edu/sirtf/docs.html
[^18]: http://www.atnf.csiro.au/research/pulsar/psrcat
[^19]: The majority of X-ray counts that detected from ChI J154557-5443\_3 were soft ($<2$ keV). This hard flux is therefore an upper-limit.
[^20]: The Galactic column densities were obtained using the online HEASARC calculator http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl
|
---
abstract: |
The Test Function Conjecture due to Haines and Kottwitz predicts that the geometric Bernstein center is a source of test functions required by the Langlands-Kottwitz method for expressing the local semisimple Hasse-Weil zeta function of a Shimura variety in terms of automorphic L-functions. Haines and Rapoport found an explicit formula for such test functions in the Drinfeld case with pro-$p$ Iwahori level structure.
This article generalizes the Haines-Rapoport formula for the Drinfeld case to a broader class of split groups. The main theorem presents a new formula for test functions with pro-$p$ Iwahori level structure, which can be computed through some combinatorics on Coxeter groups. Explicit descriptions of the test function in certain low-rank general linear and symplectic group examples are included.
author:
- Marc Horn
bibliography:
- 'thesis\_bibliography.bib'
title: |
A Combinatorial Formula for Test Functions\
with Pro-$p$ Iwahori Level Structure
---
|
---
abstract: |
In this article, we define operator algebras internal to a rigid C\*-tensor category $\cC$. A C\*/W\*-algebra object in $\cC$ is an algebra object $\bfA$ in $\ind$-$\cC$ whose category of free modules $\FreeMod_{\cC}(\bfA)$ is a $\cC$-module C\*/W\*-category respectively. When $\cC=\fdHilb$, the category of finite dimensional Hilbert spaces, we recover the usual notions of operator algebras.
We generalize basic representation theoretic results, such as the Gelfand-Naimark and von Neumann bicommutant theorems, along with the GNS construction. We define the notion of completely positive maps between C\*-algebra objects in $\cC$ and prove the analog of the Stinespring dilation theorem.
As an application, we discuss approximation and rigidity properties, including amenability, the Haagerup property, and property (T) for a connected W\*-algebra $\bfM$ in $\cC$. Our definitions simultaneously unify the definitions of analytic properties for discrete quantum groups and rigid C\*-tensor categories.
author:
- Corey Jones and David Penneys
title: 'Operator algebras in rigid C\*-tensor categories'
---
|
---
abstract: 'In this paper we study optimal stopping problems for nonlinear Markov processes driven by a McKean-Vlasov SDE and aim at solving them numerically by Monte Carlo. To this end we propose a novel regression algorithm based on the corresponding particle system and prove its convergence. The proof of convergence is based on perturbation analysis of a related linear regression problem. The performance of the proposed algorithms is illustrated by a numerical example.'
author:
- 'Denis Belomestny[^1] and John Schoenmakers[^2]\'
bibliography:
- 'perturbation\_analysis.bib'
- 'particles.bib'
- 'MV\_optimal\_stop.bib'
title: '[Optimal stopping of McKean-Vlasov diffusions via regression on particle systems]{}'
---
Introduction
============
Numerical solution of multidimensional optimal stopping problems remains an important and active area of research with applications in finance, operations research and control. As the underlying models are getting more and more complex, the computational issues are becoming more relevant than ever. As a matter of fact, analytic and usual finite difference methods for solving optimal stopping problems deteriorate in high-dimensional problems. Therefore attention has turned to probabilistic approaches, based on Monte Carlo based approximative dynamic programming. Historically, one of the first motivating examples was the pricing of an American (or Bermudan) put option in a Black-Scholes model. Not much later, many American options that showed up involved high dimensional underlying processes, which has led to the development of several Monte Carlo based methods in the last decades (see e.g. [@Gl]). Pricing American derivatives, hence solving optimal stopping problems, via Monte Carlo has always been viewed as a challenging task, because it requires backward dynamic programming that seems to be incompatible with the forward structure of Monte Carlo methods. In particular much research was focused on the development of fast methods to compute approximations to the optimal exercise policy. The seminal paper of Longstaff and Schwartz [@J_LS2001] proposed to use a cross-sectional regression over a Monte Carlo sample to compute the conditional expectations involved in the dynamic programming algorithm. The key innovation in [@J_LS2001] was the use of the estimated conditional expectations for decision-making, rather than quantification of expected gains. This corresponds to replacing a regression problem with a classification one, see [@egloff2005monte]. Originally designed for the setup of American option pricing, this algorithm, which we term RMC (Regression Monte Carlo), has become widely accepted in financial mathematics, insurance and dynamic programming settings. It has also been implemented in many proprietary valuation systems employed by the financial industry. The great success of RMC is due to its flexible and simple implementation as well as its strong empirical performance. Other eminent examples of fast approximation methods include the functional optimization approach of [@A], the mesh method of [@BG], the regression-based approaches of [@Car], [@J_TV2001], [@egloff2005monte] and [@B1].
In this paper we propose and study a simulation based method for solving an optimal stopping problem for nonlinear Markov processes of the McKean-Vlasov type. In spite of extensive literature on stochastic particle systems corresponding to MV-SDEs, generic numerical methods for solving optimal stopping problems in this context are hard to find (to the best of our knowledge). Our study is motivated by the recent theoretical developments in control problems for MV-SDEs and recent applications of MV-SDEs in financial mathematics. In this respect we mention the recent work of [@pham2017dynamic] (see also the references therein), where a general form of the Bellman principle is derived for the optimal control problems under McKean-Vlasov dynamics.
Because of dependence of the transition kernels on the marginal distributions, nonlinear Markov processes can not be sampled like standard diffusion processes. Instead the so-called interacting particle method combined with time discretisation is used to approximate them. However, unlike the standard Monte Carlo, the simulated particles are not independent. The key result in the theory of interacting particle systems is the so-called propagation of chaos result showing that the particles fulfil a kind of law of large numbers. In particular, one can prove strong convergence of the interacting particle system to the solution of the original McKean-Vlasov equation. Here we propose a fast approximation method for optimal stopping problems related to (generally multidimensional) MV- SDEs in spirit of the celebrated RMC method by Longstaff and Schwartz, which is based on the underlying particle system of essentially dependent particles, rather than a Monte Carlo sample of independent trajectories as in [@J_LS2001]. In this respect one can speak about the Particle-Regression-Monte-Carlo (PRMC) method. The convergence of this method is proved via a perturbation analysis of a related linear regression problem due to an i.i.d. sample of the original MV-SDE, and a recursive error analysis of the backward induction algorithm (in the spirit of [@J_BelKolSch] and [@Z] for the case of independent samples). From a mathematical point of view, this analysis may be considered as the main contribution of the present paper. Summing up, we provide a generic simulation based numerical approach for solving optimal stopping problems with respect to a reward that depends on a process following multidimensional MV-SDE. Although this problem may be relevant in a financial context, we note that financial terminology used in this paper is merely chosen for illustrative purposes, and that the application scope of the developed methods is not restricted to finance.
The structure of the paper is as follows. The general setup for optimal stopping in a Markovian environment is presented in Section \[mainsetup\]. In this section we also give a concise recap of the Longstaff-Schwartz and Tsitsiklis van Roy method developed in [@J_LS2001] and [@J_TV2001], respectively. Section \[secMV\] introduces Mckean-Vlasov equations and their connection with particle systems, while Section \[ApprDP\] introduces a particle version of regression based backward dynamic programming in the spirit of [@J_LS2001]. In Section \[regpart\] we present one of our main results, Theorem \[mainth\], that deals with the convergence of the regression approach applied to (generally dependent) particles. The convergence of the PRMC algorithm is studied in Section \[conver\]. Before proceeding to a rather general perturbation analysis in Section \[pertan\] and proving Theorem \[mainth\] and Theorem \[thm\_main\] in Section \[proofs\], we present some numerical experiments in Section \[numsec\].
Optimal stopping, backward dynamic program {#mainsetup}
==========================================
Let us explain the issue of optimal stopping in the context of American options as a popular illustration. An American option grants its holder the right to select time at which she exercises the option, i.e., calls a pre-specified reward or cash-flow. This is in contrast to a European option that may be exercised only at a fixed date. A general class of American option pricing problems, i.e., optimal stopping problems respectively, can be formulated with respect to an underlying $\mathbb{R}^{d}$-valued Markov process $X$ $:=$ $\{X_{t},\,0\leq t\leq T\}$ defined on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{0\leq t\leq T},\mathrm{P})$ . The process $X$ is assumed to be adapted to a filtration $(\mathcal{F}_{t})_{0\leq t\leq T}$ in the sense that each $X_{t}$ is $\mathcal{F}_{t}$ measurable. Recall that each $\mathcal{F}_{t}$ is a $\sigma$ -algebra of subsets of $\Omega$ such that $\mathcal{F}_{s}\subseteq\mathcal{F}_{t}$ for $s\leq t.$ In general, $X$ may describe any underlying physical or economical quantity of interest such as, for example, the outside temperature, some (not necessarily tradable) market index or price. Henceforth we restrict our attention to the case where only a finite number $\mathcal{J}$ of stopping (exercise) opportunities $0<t_{1}<t_{2}<\ldots<t_{\mathcal{J}}=T$ are allowed (Bermudan options in financial terms). In this respect it should be noted that a continuous exercise (American) option can be approximated by such a Bermudan option with arbitrary accuracy, and so this is not a huge restriction. We now consider a pre-specified reward $g_{j}(Z_{j})$ in terms of the discrete time Markov chain $$Z_{i}:=X_{t_{i}},\quad i=0,\ldots,\mathcal{J},$$ with $Z_{0}:=X_{t_{0}}:=X_{0},$ for some given functions $g_{1},\ldots
,g_{\mathcal{J}}$ mapping $\mathbb{R}^{d}$ into $[0,\infty).$ Note that for technical convenience and without loss of generality we exclude $t_{0}=0$ from the set of exercise dates. In a financial context we may assume that the reward $g_{j}(Z_{j})$ is expressed in units of some (tradable) pricing numéraire that has initial value $1$ Euro, say. That is, if exercised at time $t_{j},\,j=1,\ldots,\mathcal{J}$, the option pays cash equivalent to $g_{j}(Z_{j})$ units of the numéraire. Let $\mathcal{T}_{j}$ denote for $j=0,...,\mathcal{J}$ the set of stopping times taking values in $\{j,j+1,\ldots,\mathcal{J}\}\setminus\{0\}.$ In the spirit of contingent claim pricing in finance we then assume that, for $j=0,...,\mathcal{J},$ a fair price $V_{j}^{\ast}(z)$ of the corresponding Bermudan option at time $t_{j}$ in state $z,$ given that the option was not exercised prior to $t_{j}$, is its value under the optimal exercise policy, $$V_{j}^{\ast}(z)=\sup_{\tau\in\mathcal{T}_{j}}\mathsf{E}[g_{\tau}(Z_{\tau
})|Z_{j}=z]=\mathsf{E}[g_{\tau_{j}^{\ast}}(Z_{\tau_{j}^{\ast}})|Z_{j}=z],\quad
z\in\mathbb{R}^{d}, \label{stop}$$ hence the solution to an optimal stopping problem. In (\[stop\]) we introduced an optimal stopping family (policy), which can be also expressed in the form $$\begin{aligned}
\tau_{j}^{\ast} :=\min\left\{ j\leq l\leq\mathcal{J}:g_{l}(Z_{l})\geq
C_{l}^{\ast}(Z_{l})\right\} , \label{sf}$$ where $$\begin{aligned}
C_{j}^{\ast}(z) :=\mathsf{E}[V_{j+1}^{\ast}(Z_{j+1})|Z_{j}=z],\quad
j=1,\ldots,\mathcal{J}-1,\text{ \ \ and \ \ }C_{\mathcal{J}}^{\ast}(z)\equiv0,
\label{cf}$$ are so-called continuation functions. The process $V_{j}^{\ast}(Z_{j})$ is called the *Snell envelope* of the reward process $(g_{k}(Z_{k})).$ Both the stopping family (\[sf\]) and the set of continuation functions (\[cf\]) satisfy the well-known *Dynamic Programming Principle*. In particular, for (\[sf\]) we have the backward recursion $$\begin{aligned}
\tau_{\mathcal{J}}^{\ast} & =\mathcal{J},\\
\tau_{j}^{\ast} & =j\,1_{\left\{ g_{j}(Z_{j})\geq C_{j}^{\ast}(Z_{j})\right\} }+\tau_{j+1}^{\ast}1_{\left\{ g_{j}(Z_{j})<C_{j}^{\ast
}(Z_{j})\right\} },\quad j=1,\ldots,\mathcal{J}-1,\end{aligned}$$ and for (\[cf\]) it holds, $$\begin{aligned}
C_{\mathcal{J}}^{\ast}(z) & \equiv0,\label{dyn}\\
C_{j}^{\ast}(z) & =\mathsf{E}[g_{\tau_{j+1}^{\ast}}(Z_{\tau_{j+1}^{\ast}})|Z_{j}=z]\nonumber\\
& =\mathsf{E}[\max(g_{j+1}(Z_{j+1}),C_{j+1}^{\ast}(Z_{j+1}))|Z_{j}=z],\quad
j=1,\ldots,\mathcal{J}-1.\nonumber\end{aligned}$$ A common feature of almost all existing fast approximation algorithms is that they deliver estimates $C_{N,1}(z),\ldots,C_{N,\mathcal{J}-1}(z)$ for the continuation functions $C_{1}^{\ast}(z),$ $...,$ $C_{\mathcal{J}-1}^{\ast
}(z).$ Here the index $N$ indicates that the above estimates are based on the set of independent training trajectories $\bigl(Z_{0}^{(i)},\ldots,Z_{\mathcal{J}}^{(i)}\bigr),$ $i=1,\ldots,N,$ all starting from one point $Z_{0}$ i.e., $Z_{0}$ $=$ $Z_{0}^{(1)}=\ldots
=Z_{0}^{(N)}.$ In the case of regression methods the estimates for the continuation values are obtained via regression based Monte Carlo. For example, at step $\mathcal{J}-j$ one may estimate the expectation $$\mathsf{E}[\max(g_{j+1}(Z_{j+1}),C_{N,j+1}(Z_{j+1}))\bigr)Z_{j}=z]
\label{regr_aim}$$ via linear regression based on the set of paths $$\bigl(Z_{j}^{(i)},\max\{g_{j+1}(Z_{j+1}^{(i)}),C_{N,j+1}(Z_{j+1}^{(i)})\}\bigr),\quad i=1,\ldots,N, \label{TVr}$$ where $C_{N,j+1}(z)$ is the estimate for $C_{j+1}^{\ast}(z)$ obtained in the previous step, and $C_{N,\mathcal{J}}(z):=0.$ This approach is basically the method by Tsitsiklis and van Roy [@J_TV2001]. Alternatively, in the so-called Longstaff-Schwarz algorithm, one mixes the estimates of the continuation values with the corresponding estimates of the stopping times. More precisely, on each trajectory $i=1,...,N,$ approximate stopping times $\tau_{N,j}^{(i)},$ $j=1,\ldots,\mathcal{J},$ are recursively constructed by first initializing $\tau_{N,\mathcal{J}}^{(i)}=\mathcal{J},$ for all $i.$ Then, once $\tau_{N,j+1}^{(i)},$ $i=1,...,N,$ is constructed for $j<\mathcal{J},$ one computes from the sample $$\bigl(Z_{j}^{(i)},g_{\tau_{N,j+1}^{(i)}}(Z_{\tau_{N,j+1}^{(i)}}^{(i)})\bigr),\quad i=1,\ldots,N, \label{LSr}$$ an estimate $C_{N,j}(z)$ of the continuation function $C_{j+1}^{\ast}(z)$ by projection on the linear span of a set of basis functions. Subsequently, the approximate stopping times corresponding to exercise date $j$ on trajectories $i=1,...,N,$ are defined via, $$\tau_{N,j}^{(i)}=j\,1_{\bigl\{g_{j}(Z_{j}^{(i)})\geq C_{N,j}(Z_{j}^{(i)})\bigr\}}+\tau^{(i)}_{N,j+1}1_{\bigl\{g_{j}(Z_{j}^{(i)})<C_{N,j}(Z_{j}^{(i)})\bigr\}}. \label{tau}$$ Working all the way back, we thus obtain again a set of approximate continuation functions $C_{N,1}(z),\ldots,C_{N,\mathcal{J}-1}(z)$ (note that $C_{N,\mathcal{J}}(z)\equiv0$).
It should be noted that the algorithm based on (\[LSr\]) is more popular than the one based on (\[TVr\]), because it behaves more stable in practice, particularly when the number of exercise dates is getting very large. Intuitively, this can be explained by the fact that the regression (\[LSr\]) is always carried out on actually realized cash-flows, rather than on approximative value functions as in (\[TVr\]) which may become quite unprecise because of error propagation due to a huge number of exercise dates, see for example [@B1] for more rigorous arguments.
Given the estimates $C_{N,1},\ldots,C_{N,\mathcal{J}-1}$, constructed by one of the fast approximation methods above, we can construct a lower bound (low biased estimate) for $V_{0}^{\ast}$ using the (generally suboptimal) stopping rule: $$\tau_{N}=\min\bigl\{1\leq j\leq\mathcal{J}:g_{j}(Z_{j})\geq C_{N,j}(Z_{j})\bigr\}$$ with $C_{N,\mathcal{J}}\equiv0$ by definition. Indeed, fix a natural number $N_{\text{test}}$ and simulate $N_{\text{test}}$ new independent trajectories of the process $Z.$ A low-biased estimate for $V_{0}^{\ast}$ can then be defined as $$V_{0}^{N_{\text{test}},N}=\frac{1}{N_{\text{test}}}\sum_{r=1}^{N_{\text{test}}}g_{\tau_{N}^{(r)}}\bigl(Z_{\tau_{k}^{(r)}}^{(r)}\bigr)$$ with $$\tau_{N}^{(r)}=\inf\Bigl\{1\leq j\leq\mathcal{J}:g_{j}(Z_{j}^{(r)})\geq
C_{N,j}(Z_{j}^{(r)})\Bigr\},\quad r=1,\ldots,N_{\text{test}}.$$
McKean-Vlasov equations and particle systems {#secMV}
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Let $[0,T]$ be a finite time interval and $(\Omega,\mathcal{F},\mathrm{P})$ be a complete probability space, where a standard $m$-dimensional Brownian motion $W$ is defined. We consider a class of McKean-Vlasov stochastic differential equations (MVSDE), i.e., stochastic differential equations whose drift and diffusion coefficients may depend on the current distribution of the process, of the form: $$\left\{
\begin{array}
[c]{ll}X_{t} & =\xi+\int_{0}^{t}\int_{\mathbb{R}^{d}}a(X_{s},y)\mu_{s}(dy)ds+\int
_{0}^{t}\int_{\mathbb{R}^{d}}b(X_{s},y)\mu_{s}(dy)dW_{s}\\
\mu_{t} & =\mathrm{Law}(X_{t}),\quad t\geq0,\quad X_{0}\sim\mu_{0}\end{array}
\right. \label{eq:sde}$$ where $\mu_{0}$ is a distribution in $\mathbb{R}^{d},$ $a:\,\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ and $b:\,\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d\times m}.$ A popular way of simulating the MVSDE (\[eq:sde\]) is to sample the so-called particle system $\mathbf{X}_{t}^{N}=\bigl(X_{t}^{1,N},\ldots,X_{t}^{N,N}\bigr)\in
\mathbb{R}^{d\times N}$ from $N\times d$-dimensional SDE $$X_{t}^{i,N}=\xi^{i}+\frac{1}{N}\sum_{j=1}^{N}\int_{0}^{t}a(X_{s}^{i,N},X_{s}^{j,N})\,ds+\frac{1}{N}\sum_{j=1}^{N}\int_{0}^{t}b(X_{s}^{i,N},X_{s}^{j,N})\,dW_{s}^{i} \label{eq:par}$$ for $i=1,\ldots,N,$ where $\xi^{i},$ $i=1,\ldots,N,$ are i.i.d copies of a r.v. $\xi,$ distributed according the law $\mu_{0},$ and $W^{i},$ $i=1,...,N,$ are independent copies of $W.$ Under suitable assumptions (see, for example [@antonelli2002rate]) one has that$$\left\Vert \sup_{0\leq r\leq T}\left\vert X_{r}^{\cdot,N}-X_{r}^{\cdot
}\right\vert \right\Vert _{p}\leq C_{p}N^{-1/2}. \label{Koh}$$ In practice, $N\times d$-dimensional SDE system (\[eq:par\]) cannot be solved analytically either and one has to approximate its solution by a suitable numerical integration scheme such as the Euler method, leading to a next approximation $\mathbf{X}_{t}^{N,\delta}=\bigl(X_{t}^{1,N,\delta},\ldots,X_{t}^{N,N,\delta}\bigr)$ with $\delta$ being the size of the each Euler time step. Following [@antonelli2002rate], one then has$$\left\Vert \sup_{0\leq r\leq T}\left\vert X_{r}^{\cdot,N,\delta}-X_{r}^{\cdot,N}\right\vert \right\Vert _{p}\lesssim\sqrt{\delta}, \label{Euler}$$ where $\lesssim$ involves a constant that does not depend on $N$ and $\delta.$
\[exact\] In order to fix the main ideas and to avoid a notational blow up, we assume in this paper that the system $\mathbf{X}_{t}^{N}$ (cf. (\[eq:par\])) is constructed exactly, hence we neglect the numerical integration error (\[Euler\]). On the other hand, due to (\[Euler\]) it will be clear how several results in this paper have to be adapted in the case where (\[eq:par\]) is approximated using the Euler scheme.
[In fact, the solution to the MVSDE (\[eq:sde\]) may be considered as a usual non-autonomous, Markovian diffusion, since $\left\{ \mu_{s}:0\leq s\leq
T\right\} $ is some deterministic flow of distributions, although not explicitly known beforehand. Therefore, we may consider the stopping problem (\[stop\]) with respect to the solution (\[eq:sde\]), while the standard notions of the Snell envelope and the Dynamic Programming Principle still apply. However, in contrast to the standard diffusion processes $X,$ where independent trajectories of $X$ may be simulated straightforwardly by Monte Carlo, simulating of independent copies of (\[eq:sde\]) is not directly possible. As a way out, we will work with the particle system (\[eq:par\]) of *dependent* particles, instead of an ensemble of independent trajectories of (\[eq:sde\]).]{}
Dynamic programming on particle systems {#ApprDP}
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In this section we describe a particle version of the Longstaff-Schwarz regression algorithm due to (\[LSr\]) and (\[tau\]). First we run the particle system $\mathbf{X}_{t}^{N}=\bigl(X_{t}^{1,N},\ldots,X_{t}^{N,N}\bigr)\in\mathbb{R}^{d\times N}$ as described above and set $$Z_{j}^{i,N}=X_{j\Delta}^{i,N},\quad j=0,\ldots,\mathcal{J},\quad i=1,\ldots,N,
\label{Zt}$$ with $\mathcal{J}=\lfloor T/\Delta\rfloor.$ It should be noted that, unlike Monte Carlo, the trajectories (\[Zt\]) are generally *dependent*. We now consider an approximative dynamic programming algorithm based on the (generally dependent) paths $\mathbf{Z}^{N}=(Z_{j}^{i,N},\,i=1,\ldots,N,$ $j=0,\ldots,\mathcal{J}).$ In the spirit of the Longstaff-Schwarz algorithm we compute sequentially for $j=\mathcal{J},\ldots,1,$ approximate continuation functions $C_{N,j}$ and approximate stopping times $\tau_{N,j}^{(i)},$ $i=1,...,N.$ That is, we initialize $\tau_{N,\mathcal{J}}^{(i)}=\mathcal{J},$ $i=1,...,N,$ and $C_{N,\mathcal{J}}=0,$ and once $\tau_{N,j+1}^{(i)},$ $i=1,...,N,$ and $C_{N,j+1},$ $\,j<\mathcal{J},$ are constructed, $C_{N,j}$ is obtained from solving the minimization,$$C_{N,j}:=\underset{h\in\mathcal{H}_{K}}{\arg\min}\left\{ \frac{1}{N}\sum_{i=1}^{N}\left( g_{\tau_{N,j+1}^{(i)}}(Z_{\tau_{N,j+1}^{(i)}}^{i,N})-h(Z_{j}^{i,N})\right) ^{2}\right\} \label{Cd}$$ ($C_{N,\mathcal{J}}:=0$). Next $\tau_{N,j}^{(i)}$ is updated analogue to the scheme (\[tau\]), i.e.,$$\tau_{N,j}^{(i)}=j\,1_{\bigl\{g_{j}(Z_{j}^{i,N})\geq C_{N,j}(Z_{j}^{i,N})\bigr\}}+\tau_{N,j+1}1_{\bigl\{g_{j}(Z_{j}^{i,N})<C_{N,j}(Z_{j}^{i,N})\bigr\}}.$$ Note that, for fixed $j,$ the pairs$$\left( Z_{j}^{i,N},g_{\tau_{N,j+1}^{(i)}}(Z_{\tau_{N,j+1}^{(i)}}^{i,N})\right) ,\text{ \ \ }i=1,...,N, \label{pas}$$ are generally dependent, but, have the same distribution for each $i.$ As such (\[Cd\]) is indeed can be viewed as an estimator of $C_{j}^{\ast}$ (cf. (\[dyn\])). Usually the space $\mathcal{H}_{K}$ is taken to be the linear span of some given set of basis functions, so that the minimization (\[Cd\]) boils down to a linear least squares problem that can be solved via straightforward linear algebra. Let us refer to the above algorithm as the PRMC algorithm.
Regression on interacting particle systems {#regpart}
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In this section we consider for some fixed $t\geq0$ and time $T>t$ a generic problem of computing the functionals of the form $$w(x)=\mathsf{E}\left[ f\left( X_{T}\right) \mid X_{t}=x\right] ,
\label{ef}$$ where $(X_{t})$ is the solution of . In (\[ef\]), $T$ may in general be any random time. Let $\mathbf{X}_{t}^{N}=(X_{t}^{1,N},\ldots
,X_{t}^{N,N})$ be a particle system (\[eq:par\]). Furthermore, let for each $K\in\mathbb{N},$ $\mathcal{H}_{K}$ be a $K$-dimensional linear space of functions $h:\mathbb{R}^{d}\rightarrow\mathbb{R}$ and consider the estimate $$\widetilde{w}_{N}:=\underset{h\in\mathcal{H}_{K}}{\arg\min}\left\{ \frac
{1}{N}\sum_{i=1}^{N}\left( f(X_{T}^{i,N})-h(X_{t}^{i,N})\right)
^{2}\right\} ,$$ where the dimension $K$ may depend on $N.$ Let $\left( \psi_{k}\right)
_{k=1,2,...}$ be a sequence of linearly independent basis functions and set $\mathcal{H}_{K}:=\mathsf{span}\left\{ \psi_{1},\ldots,\psi_{K}\right\} .$ In this section we are going to analyze the properties of the estimate $\widetilde{w}_{N}.$ Note that the random variables $X_{T}^{1,N},\ldots
,X_{T}^{N,N}$ are generally dependent, so that the known results from regression analysis (see, e.g. [@gyorfi2002distribution]) can not be applied directly. Consider the truncated version of the estimate $\widetilde{w}_{N}$ defined as $T_{M}\widetilde{w}_{N},$ where $T_{M}$ is a truncation operator defined for a generic function $h$ and a threshold $M$ as $$T_{M}h=\begin{cases}
M, & h>M,\\
h, & -M\leq f\leq M,\\
-M, & h<-M.
\end{cases}
\label{trunc}$$ We have the following theorem.
\[mainth\] Assume that all functions $\psi_{k},$ $k=1,2,\ldots$ and $f$ are globally bounded and Lipschitz continuous. That is, there exist constants $M_{f},$ $L_{f},$ $L_{k},$ $M_{k},\ell_{k},$ $k=1,2,\ldots.$ such that for all $x,y\in\mathbb{R}^{d},$ $$\begin{aligned}
|f(x)| & \leq M_{f},\quad\frac{1}{K}\sum_{k=1}^{K}|\psi_{k}(x)|^{2}\leq M_{K}^{2}, \label{eq:bound_psi}\\
\quad\left\vert \psi_{k}(x)-\psi_{k}(y)\right\vert & \leq L_{k}\left\vert
x-y\right\vert ,\quad\left\vert f(x)-f(y)\right\vert \leq L_{f}\left\vert
x-y\right\vert. \label{eq:psi_lip}
$$ Further suppose that $$0<\varkappa_{\circ}\leq\lambda_{\min}\left( \Sigma_{K}\right) <\lambda
_{\max}\left( \Sigma_{K}\right) \leq\varkappa^{\circ}<\infty
\label{eq: lambda_ass}$$ for all $K\in\mathbb{N},$ where $$\Sigma_{K}=\left( \int\psi_{k}\left( x\right) \psi_{l}\left( x\right)
\mu_{t}(dx),\,k,l=1,\ldots,K\right) .$$ Then, it holds $$\begin{gathered}
\Vert T_{M_{f}}\widetilde{w}^{N}(x)-w\left( x\right) \Vert_{L_{2}(\mu_{t})}\\
\lesssim\frac{M_{K}\sqrt{K}}{\sqrt{N}}\Bigl[d_{1}M_{f}\ell_{K}+d_{2}L_{f}\Bigr]+\frac{M_{f}}{\sqrt{N}}\Bigl[d_{3}\ell_{K}+\sqrt{1+\log N}\sqrt
{K}\Bigr]\\
+M_{f}\sqrt{K}\exp\left[ -d_{4}\frac{N}{KM_{K}^{2}}\right] +\inf
_{h\in\mathcal{H}_{K}}\Vert h-w\Vert_{L_{2}(\mu_{t})}\label{res}$$ with $$\begin{aligned}
\ell_{K}^{2} :=\sum_{k=1}^{K}L_{k}^{2}. \label{lk}\end{aligned}$$ In the constants $d_{1,2,3,4}$ depend on $\varkappa_{\circ},$ $\varkappa^{\circ}$ only, $\lesssim$ denotes $\leq$ up to a universal constant for each term.
Theorem \[mainth\] will be proved in Section \[proofs\].
Convergence analysis of the PRMC algorithm {#conver}
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In this section we investigate the convergence properties of the PRMC regression algorithm. To this end, we modify the PRMC algorithm in such a way that our fundamental result, Theorem \[mainth\], may be applied. In fact, we follow an approach in the spirit of [@Z] (cf. [@J_BelKolSch]), and assume that instead of one particle sample $\mathbf{Z}^{N},$ we have at hand for $j=1,\ldots,\mathcal{J}-1,$ independent particle samples $\mathbf{Z}^{j;N}:=(Z_{r}^{j;i,N},\,i=1,\ldots,N,$ $r=0,\ldots,\mathcal{J}),$ all starting at $Z_{0}=X_{0}.$ We next assume that $C_{N,j}$ and $\tau_{N,j}$ are constructed in the following backward recursive way: As initialization we set $\tau_{N,\mathcal{J}}=\mathcal{J}$ and $C_{N,\mathcal{J}}\equiv0.$ Once $C_{N,j+1}$ and $\tau_{N,j+1},$ $j+1\leq\mathcal{J},$ are determined based on the samples $\mathbf{Z}^{j+1;N},\ldots,\mathbf{Z}^{\mathcal{J}-1;N},$ then $C_{N,j}$ is constructed based on the samples $\mathbf{Z}^{j;N},\ldots
,\mathbf{Z}^{\mathcal{J}-1;N},$ via$$C_{N,j}=\underset{h\in\mathcal{H}_{K}}{\arg\min}\sum_{i=1}^{N}\left(
g_{\tau^{(i)}_{N,j+1}}\bigl( Z_{\tau^{(i)}_{N,j+1}}^{j;i,N}\bigr)
-h\bigl( Z_{j}^{j;i,N}\bigr) \right) ^{2},$$ and, subsequently, $\tau_{N,j}$ is defined by $$\tau_{N,j}:=j\,1_{\bigl\{g_{j}(Z_{j})\geq C_{N,j}(Z_{j})\bigr\}}+\tau
_{N,j+1}1_{\bigl\{g_{j}(Z_{j})<C_{N,j}(Z_{j})\bigr\}},$$ for a generic dummy trajectory $\left( Z_{l}\right) _{l=0,\ldots
,\mathcal{J}}$ corresponding to the (exact) solution of (\[eq:sde\]) independent of $$\mathcal{G}_{j}:=\sigma\left\{ \mathbf{Z}^{j;N},\ldots,\mathbf{Z}^{\mathcal{J}-1;N}\right\} .$$ Let us further define $$\overline{C}_{N,j}(z):=\mathsf{E}_{\mathcal{G}_{j+1}}\left[ \left.
g_{\tau_{N,j+1}}\left( Z_{\tau_{N,j+1}}\right) \right\vert Z_{j}=z\right] .
\label{cbar}$$ Note that the random function $\overline{C}_{N,j}$ is $\mathcal{G}_{j+1}$-measurable while its estimate $C_{N,j}$ is $\mathcal{G}_{j}$-measurable. By running this procedure all the way down to $j=1,$ we so end up with a sequence of approximative continuation functions $C_{N,j}\left( \cdot\right) ,$ and the corresponding conditional expectations $\overline{C}_{N,j}(z).$ The following lemma holds.
\[lem23\] For the conditional expectations (\[cbar\]) we have that,$$\left\Vert \overline{C}_{N,j} -C_{j}^{\ast} \right\Vert _{L_{p}(\mu_{j})}\leq\sum_{l=j+1}^{\mathcal{J}-1}\left\Vert C_{N,l} -C_{l}^{\ast} \right\Vert
_{L_{p}(\mu_{l})} \label{eq:bound_1}$$ with $p\geq1$ by slightly abusing notation and using $\mu_{j}=\mu_{t_{j}}.$ Note that the inequality (\[eq:bound\_1\]) involves $\mathcal{G}_{j+1}$-measurable objects.
It is interesting to compare the estimate with similar ones in Lemma 2.3 of [@Z].
The following theorem states the convergence of the approximate continuation functions in the PRMC algorithm to the exact ones, respectively.
\[thm\_main\] Assume that the conditions , , are fulfilled with $f$ replaced by $g_{j}$ uniformly in $j=1,\ldots,\mathcal{J}$. By denoting the norm$$\left\Vert \cdot\right\Vert _{L_{2}(\mu_{j},\mathbb{P})}^{2}:=\mathsf{E}\left[ \left\Vert \cdot\right\Vert _{L_{2}(\mu_{j})}^{2}\right] ,$$ due to the unconditional expectation with respect to the all in probability measure $\mathbb{P},$ one has for $j=1,\ldots,\mathcal{J}-1,$$$\left\Vert C_{N,\mathcal{J}-j}-C_{\mathcal{J}-j}^{\ast}\right\Vert _{L_{2}(\mu_{\mathcal{J}-j},\mathbb{P})}\leq\Delta_{N,K}(2+c)^{j},\label{eq: main}$$ where $\Delta_{N,K}=c\bigl(\epsilon_{N,K}+\max_{j}\inf_{h\in\mathcal{H}_{K}}\left\Vert C_{j}^{\ast}-h\right\Vert _{L_{2}(\mu_{j})}\bigr)$ for some $c>0$ with $$\epsilon_{N,K}=\sqrt{\frac{K}{N}}\left( M_{K}\ell_{K}+\eta_{1}\sqrt{1+\log
N}\right) +\eta_{2}\sqrt{K}\exp\left[ -\eta_{3}\frac{N}{KM_{K}^{2}}\right]$$ for some constants $\eta_{1,2,3}>0$ not depending on $K$ and $N.$
The Hermite polynomial of order $j$ is given, for $j\geq0$, by: $$H_{j}(x)=(-1)^{j}e^{x^{2}}\frac{d^{j}}{dx^{j}}(e^{-x^{2}}).$$ Hermite polynomials are orthogonal with respect to the weight function $e^{-x^{2}}$ and satisfy: $\int_{{\mathbb{R}}}H_{j}(x)H_{\ell}(x)e^{-x^{2}}dx=2^{j}j!\sqrt{\pi}\delta_{j,\ell}.$ The Hermite function of order $j$ is given by: $$\psi_{j}(x)=c_{j}H_{j}(x)e^{-x^{2}/2},\quad c_{j}=\left( 2^{j}j!\sqrt{\pi
}\right) ^{-1/2}.\label{fhermite}$$ The sequence $(\psi_{j},j\geq0)$ is an orthonormal basis of ${L}_{2}({\mathbb{R}})$. The infinite norm of $\psi_{j}$ satisfies (see Szegö [@szego1975orthogonal] p.242): $$\,\Vert\psi_{j}\Vert_{\infty}\leq M_{0},\quad\quad M_{0}\simeq1,086435/\pi
^{1/4}\simeq0.8160.\label{borneh}$$ Furthermore, since $\psi_{k}^{\prime}(x)=\sqrt{k/2}\psi_{k-1}(x)-\sqrt
{(k+1)/2}\psi_{k+1}(x)$ we derive that the condition is fulfilled with $L_{k}=2M_{0}\sqrt{k}.$
Numerical experiment {#numsec}
====================
As a simple illustration of the proposed methodology, let us consider optimal stopping problem in the so-called Shimizu-Yamada model$$dX_{t}=\left( a\mathsf{E}\left[ X_{t}\right] +bX_{t}\right) \, dt+\sigma
dW_{t},\quad X_{0}=x_{0}, \quad t\in[0,T] \label{SY}$$ (see [@frank2004stochastic], Section 3.10), which has the explicit solution$$X_{t}=x_{0}e^{\left( a+b\right) t}+\sigma\int_{0}^{t}e^{b(t-s)}dW_{s}
\label{Gs}$$ that in turn solves the ordinary SDE$$dX_{t}=(x_{0}ae^{\ \left( a+b\right) t}+bX_{t})\,dt+\sigma dW_{t}
\label{oSDE}$$ (cf. [@belomestny2017projected]). It is straightforward to show that the conditional mean and variance of (\[Gs\]) are given by $$\begin{aligned}
\mathsf{E}\left[ X_{t}|X_{s}\right] & =e^{b\left( t-s\right) }X_{s}+x_{0}e^{bt}\left( e^{at}-e^{as}\right) \text{ \ \ and}\\
\mathrm{Var}\left[ X_{t}|X_{s}\right] & =\sigma^{2}\frac{e^{2b\left(
t-s\right) }-1}{2b},\text{ \ \ }0\leq s\leq t,\end{aligned}$$ respectively. That is, for $s=0$ we have that$$\mathsf{E}\left[ X_{t}\right] =x_{0}e^{\left( a+b\right) t}\text{ \ \ and
\ \ }\mathrm{Var}\left[ X_{t}\right] =\sigma^{2}\frac{e^{2bt}-1}{2b}.
\label{EV}$$ In the particular case $b=-a,$ $a>0$ one so has $$\begin{aligned}
\mathsf{E}\left[ X_{t}|X_{s}\right] & =e^{-a\left( t-s\right) }X_{s}+x_{0}\left( 1-e^{-a\left( t-s\right) }\right) ,\label{CondD}\\
\mathrm{Var}\left[ X_{t}|X_{s}\right] & =\sigma^{2}\frac{1-e^{-2a\left(
t-s\right) }}{2a},\text{ \ \ }0\leq s\leq t,\text{ \ \ }a>0,\nonumber\end{aligned}$$ and (\[EV\]) yields $\mathsf{E}\left[ X_{t}\right] =x_{0}$ for all $t.$ For this case the particle system (\[eq:par\]) reads$$X_{t}^{i,N}=x_{0}+\frac{a}{N}\sum_{j=1}^{N}\int_{0}^{t}X_{s}^{j,N}ds-a\int
_{0}^{t}X_{s}^{i,N}ds+\sigma W_{t}^{i},\quad t\in[0,T]. \label{POU}$$ We now consider the optimal stopping problem$$V_{0}^{\ast}=\sup_{\tau\in\mathcal{T}_{j}}\mathsf{E}[g_{\tau}(X_{t_{\tau}})],
\label{stopn}$$ for some reward functions $g_{j}:\mathbb{R}\rightarrow\mathbb{R}_{\geq0},$ $t_{j}=jT/\mathcal{J},$ where $(X_{t})$ solves (\[SY\]) with $b=-a,$ $a>0.$ Since $X$ follows an ordinary SDE (\[oSDE\]), we may compute an approximation to (\[stopn\]) numerically by the Longstaff-Schwarz method [@J_LS2001] based on independent trajectories of (\[oSDE\]), and then compare it to the particle based Longstaff-Schwarz algorithm proposed in Section \[ApprDP\]. Let us further consider, for illustration a Bermudan put option (in financial terms),$$g_{j}(x)=e^{-rt_{j}}(x-K)^{+},\quad j=0,\ldots,\mathcal{J},$$ for some $K>0,$ where $r$ can be interpreted as interest rate. We take the following parameters $d=1,$ $x_{0}=1,$$K=0.1,$ $a=1,$ $T=1,$ $\mathcal{J}=100$ and implement the following two phase algorithm. In the first stage we run $N_{\mathrm{tr}}$ trajectories either of the particle system or of the process . Using these trajectories, we estimate the corresponding continuation functions using linear regression with quadratic polynomials and reward functions as basis functions. In the second stage we use the estimated continuation values on a new set of $N_{\mathrm{test}}=5000$ testing trajectories to construct a suboptimal stopping rule and consequently a lower bound for $V_{0}^{*}$ by averaging over the testing paths. We also compute dual upper bounds using the estimated continuation functions and $N_{\mathrm{in}}=100$ inner paths to approximate one step conditional expectations, see, e.g. Chapter 3 in [@belomestny2018book]. The results for different values of $N_{\mathrm{tr}}$ are shown in Table \[TablePRMC\].
\[c\][|c|c|c|]{}$N_{\mathrm{tr}}$ & RMC & PRMC\
10 & \[0.9393(0.0079), 1.2742(0.0076)\] & \[0.9287(0.0058), 1.1750(0.0038)\]\
50 & \[1.0047(0.0082), 1.0942(0.0019)\] & \[0.9829(0.0072), 1.1745(0.0041)\]\
100 & \[1.0144(0.0073), 1.0871(0.0013)\] & \[1.0079(0.0080), 1.0978(0.0023)\]\
300 & \[1.0342(0.0077), 1.0718(0.0009)\] & \[1.0330(0.0070), 1.0700(0.0010)\]\
1000 & \[1.0575(0.0075), 1.0699(0.0007)\] & \[1.0546(0.0078), 1.0689(0.0008)\]\
As can be seen from Table \[TablePRMC\], the PRMC (Particle Regression MC) performs a bit worse than RMC (Regression MC), but the difference becomes smaller as $N_{\mathrm{tr}}$ increases.
Perturbation analysis for linear regression {#pertan}
===========================================
Consider a least squares problem of the form $$\beta^{\circ}=\underset{\beta\in\mathbb{R}^{d}}{\arg\min}\sum_{i=1}^{N}(Y_{i}-\beta^{\top}U_{i})^{2}, \label{eq:least_squares}$$ where for $i=1,...,N,$ $\left( Y_{i},U_{i}\right) $ are i.i.d. pairs of a random variable $Y_{i}$ and a random (column) vector $U_{i}\in$ $\mathbb{R}^{d}.$ With $U:=(U_{1},\ldots,U_{N})\in\mathbb{R}^{d\times N},$ $Z=N^{-1/2}U^{\top},$ and $V=N^{-1/2}\left( Y_{1},\ldots,Y_{N}\right) ^{\top},$ the solution of the problem (\[eq:least\_squares\]) can be written in terms of pseudo inverses (denoted with $\dag$),$$\beta^{\circ}=\left( UU^{\top}\right) ^{-1}UY=\left( Z^{\top}Z\right)
^{-1}Z^{\top}V=Z^{\dag}V. \label{bet}$$ Consider now the least squares problem (\[eq:least\_squares\]) due to a perturbation $\left( \widetilde{Y}_{i},\widetilde{U}_{i}\right) $ of the pairs $\left( Y_{i},U_{i}\right) ,$ and define $\widetilde{Z}$ and $\widetilde{V}$ accordingly. We so consider (cf. (\[bet\]))$$\widetilde{\beta}^{\circ}=\left( \widetilde{Z}^{\top}\widetilde{Z}\right)
^{-1}\widetilde{Z}^{\top}\widetilde{V}=\widetilde{Z}^{\dag}\widetilde{V}
\label{bett}$$ and set$$\widetilde{Z}=Z+E,\quad\widetilde{V}=V+F. \label{pert}$$ While the rows of $Z$ and the components of $V$ are independent, the rows of the perturbation matrix $E$ and the components of the perturbation vector $F$ are generally dependent. Also we note that we don’t assume any kind of independence between the perturbations $E$ and $F$ and the matrix $Z$ and vector $V,$ respectively.
\[thm:perturbation\] Consider the least squares problem (\[eq:least\_squares\]) with solution (\[bet\]), and its perturbation due to (\[pert\]) with solution (\[bett\]), respectively. Assume that $U_{1},\ldots,U_{N}$ in (\[eq:least\_squares\]) are i.i.d. random vectors in $\mathbb{R}^{d}$ such that for some $M>0,$ $\left\Vert U_{1}\right\Vert \leq
M$ a.s. Set $$\mathsf{E}\left[ U_{1}U_{1}^{\top}\right] =\Sigma,$$ so that $$Z^{\top}Z=\frac{1}{N}UU^{\top}=\frac{1}{N}\sum_{i=1}^{N}U_{i}U_{i}^{\top}.$$ Let $\lambda_{\min}(\Sigma)$ be the smallest eigenvalue, and $\lambda_{\max
}(\Sigma)$ be the largest eigenvalue of $\Sigma,$ respectively. Then for any $\rho\in(0,\lambda_{\min}(\Sigma))$ and $\varepsilon\in(0,\lambda_{\min
}\left( \Sigma\right) -\rho)$ we have on the set $\mathcal{C}:=\mathcal{C}_{1}\cap\mathcal{C}_{2}\cap\mathcal{C}_{3}\cap\mathcal{C}_{4}$ with$$\begin{aligned}
\mathcal{C}_{1} & :\text{ \ \ }\lambda_{\mathrm{\max}}\left( Z^{\top
}Z\right) <\lambda_{\mathrm{\max}}\left( \Sigma\right) +\varepsilon,\\
\mathcal{C}_{2} & :\text{ \ \ }\lambda_{\min}\left( Z^{\top}Z\right)
>\lambda_{\mathrm{\min}}\left( \Sigma\right) -\varepsilon,\\
\mathcal{C}_{3} & :\text{ \ \ }\lambda_{\mathrm{\min}}\left( \Sigma\right)
-\left( 2\sqrt{\lambda_{\mathrm{\max}}\left( \Sigma\right) +\varepsilon
}+1\right) \Vert E\Vert>\rho+\varepsilon,\\
\mathcal{C}_{4} & :\text{ \ \ }\Vert E\Vert<1.\end{aligned}$$ that$$\Vert\widetilde{\beta}^{\circ}-\beta^{\circ}\Vert\leq c_{1}(\Sigma
,\varepsilon,\rho)\Vert E\Vert\Vert V\Vert+c_{2}(\Sigma,\varepsilon,\rho)\Vert
F\Vert,$$ where$$\begin{aligned}
c_{1}(\Sigma,\varepsilon,\rho) & :=\frac{1}{\rho}+\frac{2\left(
\lambda_{\mathrm{\max}}\left( \Sigma\right) +\varepsilon\right)
+\sqrt{\lambda_{\mathrm{\max}}\left( \Sigma\right) +\varepsilon}}{\rho^{2}}\text{ \ \ and }\\
c_{2}(\Sigma,\varepsilon,\rho) & :=c_{1}(\Sigma,\varepsilon,\rho
)+\frac{\sqrt{\lambda_{\mathrm{\max}}\left( \Sigma\right) +\varepsilon}}{\lambda_{\mathrm{\min}}\left( \Sigma\right) -\varepsilon}.\end{aligned}$$ Furthermore, for any $\delta\in(0,1),$ and $N$ such that$$\varepsilon=\varepsilon_{\delta,N}=M\sqrt{\,\frac{\log(2d/\delta)}{NC}}\frac{\lambda_{\max}^{3/2}(\Sigma)}{\lambda_{\min}(\Sigma)}\leq\lambda
_{\mathrm{\min}}(\Sigma)-\rho\label{eN}$$ (cf. (\[epdel\])), one has for the probability of $\mathcal{C},$ $$\mathsf{P}\left[ \mathcal{C}\right] \geq1-\delta-C_{p}\left( \left(
\frac{2\sqrt{\lambda_{\mathrm{\max}}\left( \Sigma\right) +\varepsilon}+1}{\lambda_{\mathrm{\min}}\left( \Sigma\right) -\varepsilon-\rho}\right)
^{p}+1\right) ,$$ provided $\delta$ and $C_{p}:=\mathsf{E}\left[ \left\vert E\right\vert
^{p}\right] $ are small enough (such that the above bound is positive).
Note that $\mathcal{C}:=\mathcal{C}_{1}\cap\mathcal{C}_{2}\cap\mathcal{C}_{3}\cap\mathcal{C}_{4}$ implies (\[cond\]) in Lemma \[App\] and so by this Lemma,$$\begin{aligned}
\Vert(Z+E)^{\dagger}-Z^{\dagger}\Vert & \leq\frac{\Vert E\Vert}{\rho}\left[
1+\frac{\left( 2\Vert Z\Vert+1\right) \Vert Z\Vert}{\rho}\right] \\
& \leq\frac{\Vert E\Vert}{\rho}\left[ 1+\frac{2\left( \lambda
_{\mathrm{\max}}\left( \Sigma\right) +\varepsilon\right) +\sqrt
{\lambda_{\mathrm{\max}}\left( \Sigma\right) +\varepsilon}}{\rho}\right] \\
& =c_{1}(\Sigma,\varepsilon,\rho)\Vert E\Vert\end{aligned}$$ Thus, on $\mathcal{C}$ one has also,$$\begin{aligned}
\Vert(Z+E)^{\dagger}\Vert & \leq\Vert(Z+E)^{\dagger}-Z^{\dagger}\Vert+\Vert
Z^{\dagger}\Vert\\
& \leq c_{1}(\Sigma,\varepsilon,\rho)+\frac{\sqrt{\lambda_{\mathrm{\max}}\left( \Sigma\right) +\varepsilon}}{\lambda_{\mathrm{\min}}\left(
\Sigma\right) -\varepsilon}=c_{2}(\Sigma,\varepsilon,\rho),\end{aligned}$$ using that $\Vert Z^{\dagger}\Vert\leq\Vert(Z^{\top}Z)^{-1}\Vert\Vert Z\Vert.$ So on $\mathcal{C}$ we get,$$\begin{aligned}
\Vert\widetilde{\beta}^{\circ}-\beta^{\circ}\Vert & =\Vert\widetilde
{Z}^{\dagger}\widetilde{V}-Z^{\dagger}V\Vert=\Vert(Z+E)^{\dagger
}(V+F)-Z^{\dagger}V\Vert\\
& \leq c_{1}(\Sigma,\varepsilon,\rho)\Vert E\Vert\Vert V\Vert+c_{2}(\Sigma,\varepsilon,\rho)\Vert F\Vert.\end{aligned}$$ For the probability of $\mathcal{C}$ one has$$\mathsf{P}\left[ \mathcal{C}\right] \geq1-\mathsf{P}\left[ \Omega
\backslash\mathcal{C}_{1}\cup\Omega\backslash\mathcal{C}_{2}\right]
-\mathsf{P}\left[ \Omega\backslash\mathcal{C}_{3}\right] -\mathsf{P}\left[
\Omega\backslash\mathcal{C}_{4}\right] . \label{esp}$$ For the term $\mathsf{P}\left[ \Omega\backslash\mathcal{C}_{1}\cup
\Omega\backslash\mathcal{C}_{2}\right] $ we are going to apply Lemma \[lem:spectrum\_subgaussian\]. It is easy to see that, since $0<\lambda
_{\mathrm{\min}}(\Sigma)\leq\lambda_{\mathrm{\max}}(\Sigma),$ (\[eN\]) implies (\[epdelc\]) in Lemma \[lem:spectrum\_subgaussian\]. So, due to this lemma, we have that $$\mathsf{P}\left[ \Omega\backslash\mathcal{C}_{1}\cup\Omega\backslash
\mathcal{C}_{2}\right] \leq\delta.$$ Furthermore,$$\begin{aligned}
\mathsf{P}\left[ \Omega\backslash\mathcal{C}_{3}\right] & =\mathsf{P}\left[ \lambda_{\mathrm{\min}}\left( \Sigma\right) -\left( 2\sqrt
{\lambda_{\mathrm{\max}}\left( \Sigma\right) +\varepsilon}+1\right) \Vert
E\Vert\leq\rho+\varepsilon\right] \\
& =\mathsf{P}\left[ \frac{\lambda_{\mathrm{\min}}\left( \Sigma\right)
-\varepsilon-\rho}{2\sqrt{\lambda_{\mathrm{\max}}\left( \Sigma\right)
+\varepsilon}+1}\leq\Vert E\Vert\right] \\
& \leq\left( \frac{2\sqrt{\lambda_{\mathrm{\max}}\left( \Sigma\right)
+\varepsilon}+1}{\lambda_{\mathrm{\min}}\left( \Sigma\right) -\varepsilon
-\rho}\right) ^{p}\mathsf{E}\left[ \Vert E\Vert^{p}\right] ,\end{aligned}$$ and$$\mathsf{P}\left[ \Omega\backslash\mathcal{C}_{4}\right] =\mathsf{P}\left[
\Vert E\Vert\geq1\right] \leq\mathsf{E}\left[ \Vert E\Vert^{p}\right] .$$ The statement now follows from (\[esp\]).
\[cor9\] Let us take $\varepsilon=\rho=\lambda_{\mathrm{\min}}\left(
\Sigma\right) /4.$ Then with$$\begin{aligned}
c_{1}(\Sigma) & :=\frac{1}{\lambda_{\min}\left( \Sigma\right) /4}\\
& +\frac{2\left( \lambda_{\mathrm{\max}}\left( \Sigma\right)
+\lambda_{\min}\left( \Sigma\right) /4\right) +\sqrt{\lambda_{\mathrm{\max
}}\left( \Sigma\right) +\lambda_{\min}\left( \Sigma\right) /4}}{\left(
\lambda_{\min}\left( \Sigma\right) /4\right) ^{2}},\\
c_{2}(\Sigma) & :=c_{1}(\Sigma)+\frac{\sqrt{\lambda_{\mathrm{\max}}\left(
\Sigma\right) +\lambda_{\min}\left( \Sigma\right) /4}}{\lambda
_{\mathrm{\min}}\left( \Sigma\right) -\lambda_{\min}\left( \Sigma\right)
/4},\end{aligned}$$ we have on $\mathcal{C},$$$\left\Vert \widetilde{\beta}^{\circ}-\beta^{\circ}\right\Vert \leq
c_{1}(\Sigma)\left\Vert E\right\Vert \left\Vert V\right\Vert +c_{2}(\Sigma)\left\Vert F\right\Vert ,$$ with probability$$\begin{aligned}
\mathsf{P}\left[ \mathcal{C}\right] & \geq1-2d\exp\left[ -N\frac
{C\lambda_{\min}^{4}(\Sigma)}{16M^{2}\lambda_{\max}^{3}(\Sigma)}\right] \\
& -C_{p}\left( \left( \frac{2\sqrt{\lambda_{\mathrm{\max}}\left(
\Sigma\right) +\lambda_{\mathrm{\min}}\left( \Sigma\right) /4}+1}{\lambda_{\mathrm{\min}}\left( \Sigma\right) /2}\right) ^{p}+1\right) .\end{aligned}$$
Proofs
======
Proof of Theorem \[mainth\] {#proofs}
---------------------------
Let $\mathbf{X}_{t}=\left( X_{t}^{1},\ldots,X_{t}^{N}\right) $ be a vector of i.i.d. copies of the exact solution to (\[eq:sde\]), and define for fixed $t,$ $$w_{N}:=\underset{h\in\mathcal{H}_{K_{N}}}{\arg\min}\left\{ \frac{1}{N}\sum_{i=1}^{N}\left( f(X_{T}^{i})-h(X_{t}^{i})\right) ^{2}\right\} .$$ Further let us denote by $V,\widetilde{V}\in\mathbb{R}^{N}$ the column vectors with coordinates $$V_{i}=\frac{f(X_{T}^{i})}{\sqrt{N}},\quad\widetilde{V}_{i}:=\frac
{f(X_{T}^{i,N})}{\sqrt{N}},\quad i=1,\ldots,N,$$ respectively, and consider the $\mathbb{R}^{N\times K}$ matrices $$\begin{aligned}
\widetilde{Z} & =\left( \psi_{k}\left( X_{t}^{i,N}\right) /\sqrt
{N},\,i=1,\ldots,N,\,k=1,\ldots,K\right) ,\\
Z & =\left( \psi_{k}\left( X_{t}^{i}\right) /\sqrt{N},\,i=1,\ldots
,N,\,k=1,\ldots,K\right) ,\end{aligned}$$ respectively. Then we have $$\widetilde{w}_{N}=\widetilde{\beta}_{N}^{\top}\boldsymbol{\psi}_{K}\left(
\cdot\right) ,\quad\widetilde{\beta}_{N}=\left( \widetilde{Z}^{\top
}\widetilde{Z}\right) ^{-1}\widetilde{Z}^{\top}\widetilde{V}=\widetilde
{Z}^{\dagger}\widetilde{V}$$ and $$w_{N}=\beta_{N}^{\top}\boldsymbol{\psi}_{K}\left( \cdot\right) ,\quad
\beta_{N}=\left( Z^{\top}Z\right) ^{-1}Z^{\top}{V}=Z^{\dagger}V\text{ }$$ with $\boldsymbol{\psi}_{K}=\left( \psi_{1},\ldots,\psi_{K}\right) ^{\top}.$ By using that $$\left\vert T_{M}\widetilde{w}^{N}\left( x\right) -T_{M}w^{N}\left(
x\right) \right\vert \leq\left\vert \widetilde{w}^{N}\left( x\right)
-w^{N}\left( x\right) \right\vert$$ almost surely, one has for any event $\mathcal{C}\in\mathcal{F}$ $$\begin{gathered}
\left( \mathsf{E}\left[ \int\left( T_{M}\widetilde{w}^{N}\left( x\right)
-w\left( x\right) \right) ^{2}\mu_{t}(dx)\right] \right) ^{1/2}\leq\\
\left( \mathsf{E}\left[ \int1_{\mathcal{C}}\left( \widetilde{w}^{N}\left(
x\right) -w^{N}\left( x\right) \right) ^{2}\mu_{t}(dx)\right] \right)
^{1/2}+2M_{f}\left( \mathsf{P}\left[ \Omega\backslash\mathcal{C}\right]
\right) ^{1/2}\\
+\left( \mathsf{E}\left[ \int\left( T_{M}w^{N}\left( x\right) -w\left(
x\right) \right) ^{2}\mu_{t}(dx)\right] \right) ^{1/2}\\
\leq M_{K}\sqrt{K}\left( \mathsf{E}\left[ \left\Vert \widetilde{\beta}_{N}-\beta_{N}\right\Vert ^{2}1_{\mathcal{C}}\right] \right) ^{1/2}\\
+2M_{f}\left( \mathsf{P}\left[ \Omega\backslash\mathcal{C}\right] \right)
^{1/2}\\
+\left( \mathsf{E}\left[ \int\left( T_{M}w^{N}\left( x\right) -w\left(
x\right) \right) ^{2}\mu_{t}(dx)\right] \right) ^{1/2}.\end{gathered}$$ Set $$\begin{aligned}
U & =\left( \psi_{k}(X_{t}^{i}),\,i=1,\ldots,N,\,k=1,\ldots,K\right)
^{\top}\in\mathbb{R}^{N\times K},\\
V & =\left( (f(X_{t}^{i,N})/\sqrt{N},\,i=1,\ldots,N\right) ^{\top}\in\mathbb{R}^{N},\\
E & =\left( (\psi_{k}(X_{t}^{i,N})-\psi_{k}(X_{t}^{i}))/\sqrt
{N},\,i=1,\ldots,N,\,k=1,\ldots,K\right) \in\mathbb{R}^{N\times K},\\
F & =\left( (f(X_{t}^{i,N})-f(X_{t}^{i}))/\sqrt{N},\,i=1,\ldots,N\right)
^{\top}\in\mathbb{R}^{N},\end{aligned}$$ then, with $\Sigma=\mathsf{E}\left[ UU^{\top}\right] ,$ $d=K,$ and $\left\Vert U_{i}\right\Vert \leq\sqrt{K}M_{K},$ Corollary \[cor9\] implies$$\left\Vert \widetilde{\beta}^{\circ}-\beta^{\circ}\right\Vert ^{2}\leq
2c_{1}^{2}M_{f}^{2}\left\Vert E\right\Vert ^{2}+2c_{2}^{2}\left\Vert
F\right\Vert ^{2},$$ on a set $\mathcal{C}$ with probability$$\begin{aligned}
\mathsf{P}\left[ \mathcal{C}\right] & \geq1-2K\exp\left[ -N\frac
{C\varkappa_{\circ}^{4}}{16KM_{K}^{2}\left( \varkappa^{\circ}\right) ^{3}}\right] \\
& -\mathsf{E}[\Vert E\Vert^{p}]\left( \left( \frac{\sqrt{5\varkappa^{\circ
}}+1}{\varkappa_{\circ}/2}\right) ^{p}+1\right) ,\end{aligned}$$ where constants $c_{1},$ $c_{2}$ only depend on $\varkappa_{\circ},$ $\varkappa^{\circ}.$ In particular we may take$$\begin{aligned}
c_{1} & :=\frac{44+8\sqrt{5/\varkappa^{\circ}}}{\varkappa_{\circ}},\text{
\ \ and}\\
c_{2} & :=d_{1}+\frac{2\sqrt{5\varkappa^{\circ}}}{3\varkappa_{\circ}}=\frac{132+2\sqrt{5\varkappa^{\circ}}+24\sqrt{5/\varkappa^{\circ}}}{3\varkappa_{\circ}}.\end{aligned}$$ As a consequence, $$\begin{aligned}
\mathsf{E}\left[ \Vert\widetilde{\beta}_{N}-\beta_{N}\Vert^{2}1_{\mathcal{C}}\right] & \leq2c_{1}^{2}M_{f}^{2}\mathsf{E}\left[ \Vert E\Vert
^{2}\right] +2c_{2}^{2}\mathsf{E}\left[ \Vert F\Vert^{2}\right] \\
& \leq2c_{1}^{2}M_{f}^{2}\left( \mathsf{E}\left[ \frac{1}{N}\sum_{i=1}^{N}\sum_{k=1}^{K}\left( \psi_{k}\left( X_{t}^{i,N}\right) -\psi_{k}\left(
X_{t}^{i}\right) \right) ^{2}\right] \right) \\
& +2c_{2}^{2}\mathsf{E}\left[ \frac{1}{N}\sum_{i=1}^{N}\left( f\left(
X_{t}^{i,N}\right) -f\left( X_{t}^{i}\right) \right) ^{2}\right] \\
& \leq\left( 2c_{1}^{2}M_{f}^{2}\sum_{k=1}^{K}L_{k}^{2}+2c_{2}^{2}L_{f}^{2}\right) \mathsf{E}\left[ \left\vert X_{t}^{\cdot,N}-X_{t}^{\cdot
}\right\vert ^{2}\right]\end{aligned}$$ We further have for $p\geq2,$ $$\begin{aligned}
\left( \mathsf{E}\left[ \left\vert E\right\vert ^{p}\right] \right)
^{1/p} & \leq\left( \mathsf{E}\left[ \left( \frac{1}{N}\sum_{i=1}^{N}\sum_{k=1}^{K}\left( \psi_{k}\left( X_{t}^{i,N}\right) -\psi_{k}\left(
X_{t}^{i}\right) \right) ^{2}\right) ^{p/2}\right] \right) ^{1/p}\\
& =\left( \left( \mathsf{E}\left[ \left( \frac{1}{N}\sum_{i=1}^{N}\sum_{k=1}^{K}\left( \psi_{k}\left( X_{t}^{i,N}\right) -\psi_{k}\left(
X_{t}^{i}\right) \right) ^{2}\right) ^{p/2}\right] \right) ^{2/p}\right)
^{1/2}\\
& \leq\left( \frac{1}{N}\sum_{i=1}^{N}\sum_{k=1}^{K}\left( \mathsf{E}\left[ \left( \psi_{k}\left( X_{t}^{i,N}\right) -\psi_{k}\left( X_{t}^{i}\right) \right) ^{p}\right] \right) ^{2/p}\right) ^{1/2}\\
& \leq\left( \frac{1}{N}\sum_{i=1}^{N}\sum_{k=1}^{K}L_{k}^{2}\left(
\mathsf{E}\left[ \left\vert X_{t}^{i,N}-X_{t}^{i}\right\vert ^{p}\right]
\right) ^{2/p}\right) ^{1/2}\\
& =\sqrt{\sum_{k=1}^{K}L_{k}^{2}}\left( \mathsf{E}\left[ \left\vert
X_{t}^{\cdot,N}-X_{t}^{\cdot}\right\vert ^{p}\right] \right) ^{1/p}.\end{aligned}$$ Combining the latter bounds with (\[Koh\]) and Theorem 11.3 from [@gyorfi2002distribution], and taking $p=2$ for simplicity, we get (using subadditivity of the squareroot) $$\begin{gathered}
\left( \mathsf{E}\left[ \int\left( T_{M}\widetilde{w}^{N}\left( x\right)
-w\left( x\right) \right) ^{2}\mu_{t}(dx)\right] \right) ^{1/2}\leq\\
\leq M_{K}\sqrt{2K}\left( c_{1}M_{f}\sqrt{\sum_{k=1}^{K}L_{k}^{2}}+c_{2}L_{f}\right) \frac{C_{2}}{\sqrt{N}}\\
+2M_{f}\sqrt{2K}\exp\left[ -N\frac{C\varkappa_{\circ}^{4}}{32KM_{K}^{2}\left( \varkappa^{\circ}\right) ^{3}}\right] \\
2M_{f}\sqrt{\sum_{k=1}^{K}L_{k}^{2}}\left( \frac{\sqrt{5\varkappa^{\circ}}+1}{\varkappa_{\circ}/2}+1\right) \frac{C_{2}}{\sqrt{N}}\\
+c_{3}M_{f}\frac{\sqrt{1+\log N}\sqrt{K}}{\sqrt{N}}\\
+c_{4}\inf_{h\in\mathcal{H}_{K}}\left( \int\left( h(x)-w(t,x)\right)
^{2}\mu_{t}(dx)\right) ^{1/2}$$ for universal constants $c_{3},c_{4}.$ Summarizing, and using (\[lk\]), yields (\[res\]).
Proof of Lemma \[lem23\]
------------------------
Let us observe that for $j<\mathcal{J},$$$\begin{gathered}
g_{\tau_{j+1}^{\ast}}(Z_{\tau_{j+1}})-g_{\tau_{N,j+1}}(Z_{\tau_{N,j+1}})=\left( g_{j+1}(Z_{j+1})-g_{\tau_{N,j+1}}(Z_{\tau_{N,j+1}})\right)
1_{\{\tau_{j+1}^{\ast}=j+1,\tau_{N,j+1}>j+1\}}\\
+\left( g_{\tau_{j+1}^{\ast}}(Z_{\tau_{j+1}^{\ast}})-g_{j}(Z_{j})\right)
1_{\{\tau_{j+1}^{\ast}>j+1,\tau_{N,j+1}=j+1\}}\\
+\left( g_{\tau_{j+1}^{\ast}}(Z_{\tau_{j+1}^{\ast}})-g_{\tau_{N,j+1}}(Z_{\tau_{N,j+1}})\right) 1_{\{\tau_{j+1}^{\ast}>j+1,\tau_{N,j+1}>j+1\}}.\end{gathered}$$ By abbreviating temporarily in this proof $\mathsf{E:=E}_{\mathcal{G}_{j+1}},$ and denoting $\mathcal{R}_{N,j}:=\mathsf{E}\left[ \left. g_{\tau_{j+1}^{\ast}}(Z_{\tau_{j+1}^{\ast}})-g_{\tau_{N,j+1}}(Z_{\tau_{N,j+1}})\right\vert
Z_{j}\right] ,$ we have $\mathcal{R}_{N,j}\geq0$ almost surely, and$$\begin{aligned}
\mathcal{R}_{N,j} & =\mathsf{E}\left[ \left. \left( g_{j+1}(Z_{j+1})-\mathsf{E}\left[ \left. g_{\tau_{N,j+2}}(Z_{\tau_{N,j+2}})\right\vert Z_{j+1}\right] \right) 1_{\{\tau_{j+1}^{\ast}=j+1,\tau
_{N,j+1}>j+1\}}\right\vert Z_{j}\right] \nonumber\\
& +\mathsf{E}\left[ \left. \left( \mathsf{E}\left[ \left. g_{\tau
_{j+2}^{\ast}}(Z_{\tau_{j+2}^{\ast}})\right\vert Z_{j+1}\right]
-g_{j+1}(Z_{j+1})\right) 1_{\{\tau_{j+1}^{\ast}>j+1,\tau_{N,j+1}=j+1\}}\right\vert Z_{j}\right] \nonumber\\
& +\mathsf{E}\left[ \left. \mathsf{E}\left[ \left. g_{\tau_{j+2}^{\ast}}(Z_{\tau_{j+2}^{\ast}})-g_{\tau_{N,j+2}}(Z_{\tau_{N,j+2}})\right\vert
Z_{j+1}\right] 1_{\{\tau_{j+1}^{\ast}>j+1,\tau_{N,j+1}>j+1\}}\right\vert
Z_{j}\right] \nonumber\\
& =T_{1}+T_{2}+\mathsf{E}\left[ \left. \mathcal{R}_{N,j+1}1_{\{\tau
_{j+1}^{\ast}>j+1,\tau_{N,j+1}>j+1\}}\right\vert Z_{j}\right] . \label{c1}$$ For $T_{1}$ we have$$\begin{aligned}
T_{1} & =\mathsf{E}\left[ \left. \left( g_{j+1}(Z_{j+1})-\mathsf{E}\left[ \left. g_{\tau_{j+2}^{\ast}}(Z_{\tau_{j+2}^{\ast}})\right\vert
Z_{j+1}\right] \right) 1_{\{\tau_{j+1}^{\ast}=j+1,\tau_{N,j+1}>j+1\}}\right\vert Z_{j}\right] \\
& +\mathsf{E}\left[ \left. \left( \mathsf{E}\left[ \left. g_{\tau
_{j+2}^{\ast}}(Z_{\tau_{j+2}^{\ast}})\right\vert Z_{j+1}\right]
-\mathsf{E}\left[ \left. g_{\tau_{N,j+2}}(Z_{\tau_{N,j+2}})\right\vert
Z_{j+1}\right] \right) 1_{\{\tau_{j+1}^{\ast}=j+1,\tau_{N,j+1}>j+1\}}\right\vert Z_{j}\right] ,\end{aligned}$$ and since $$\begin{aligned}
C_{N,j+1}(Z_{j+1}) & \geq g_{j+1}(Z_{j+1})\geq\mathsf{E}\left[ \left.
g_{\tau_{j+2}^{\ast}}(Z_{\tau_{j+2}^{\ast}})\right\vert Z_{j+1}\right] \\
& =C_{j+1}^{\ast}(Z_{j+1})\geq\mathsf{E}\left[ \left. g_{\tau_{N,j+2}}(Z_{\tau_{N,j+2}})\right\vert Z_{j+1}\right]\end{aligned}$$ on $\{\tau_{j+1}^{\ast}=j+1,\tau_{N,j+1}>j+1\},$ we get $$\begin{aligned}
0 & \leq T_{1}\leq\mathsf{E}\left[ \left. \left( C_{N,j+1}(Z_{l+1})-C_{j+1}^{\ast}(Z_{j+1})\right) 1_{\{\tau_{j+1}^{\ast}=j+1,\tau
_{N,j+1}>j+1\}}\right\vert Z_{j}\right] \nonumber\\
& +\mathsf{E}\left[ \left. \mathcal{R}_{N,j+1}1_{\{\tau_{j+1}^{\ast
}=j+1,\tau_{N,j+1}>j+1\}}\right\vert Z_{j}\right] . \label{c2}$$ Similarly, for $T_{2}$ we have $$0\leq T_{2}\leq\mathsf{E}\left[ \left. \left( C_{j+1}^{\ast}(Z_{j+1})-C_{N,j+1}(Z_{j+1})\right) 1_{\{\tau_{j+1}^{\ast}>j+1,\tau_{N,j+1}=j+1\}}\right\vert Z_{j}\right] . \label{c3}$$ Combining (\[c1\]), (\[c2\]), and (\[c3\]), yields$$\mathcal{R}_{N,j}\leq\mathsf{E}\left[ \left. \left\vert C_{N,j+1}(Z_{j+1})-C_{j+1}^{\ast}(Z_{j+1})\right\vert \right\vert Z_{j}\right]
+\mathsf{E}\left[ \left. \mathcal{R}_{N,j+1}\right\vert Z_{j}\right] .$$ By straightforward induction, using the tower property and the final condition $\mathcal{R}_{N,\mathcal{J}-1}=0,$ we so obtain$$0\leq C_{j}^{\ast}\left( Z_{j}\right) -\overline{C}_{N,j}\left(
Z_{j}\right) \leq\sum_{l=j+1}^{\mathcal{J}-1}\mathsf{E}\left[ \left.
\left\vert C_{N,l}(Z_{l})-C_{l}^{\ast}(Z_{l})\right\vert \right\vert
Z_{j}\right] .$$ Taking on both sides the $L_{p}$-norm, applying the triangle inequality, and using that$$\mathsf{E}\left[ \mathsf{E}\left[ \left. \left\vert C_{N,l}(Z_{j})-C_{l}^{\ast}(Z_{l})\right\vert \right\vert Z_{j}\right] ^{p}\right]
\leq\mathsf{E}\left[ \left\vert C_{N,l}(Z_{l})-C_{l}^{\ast}(Z_{l})\right\vert
^{p}\right] ,$$ finally gives (\[eq:bound\_1\]).
Proof of Theorem \[thm\_main\]
------------------------------
Theorem \[mainth\] implies,$$\mathsf{E}_{\mathcal{G}_{j+1}}\left[ \left\Vert C_{N,j}-\overline{C}_{N,j}\right\Vert _{L_{2}(\mu_{j})}^{2}\right] \leq c_{1}^{2}\epsilon
_{N,K}^{2}+c_{2}^{2}\inf_{h\in\mathcal{H}_{K}}\left\Vert \overline{C}_{N,j}\left( \cdot\right) -h\right\Vert _{L_{2}(\mu_{j})}^{2},$$ almost surely, for some $\eta,c_{1},c_{2}>0,$ which do not depend on $j,K,$ and $N.$ Hence, for the unconditional expectation we get,$$\mathsf{E}\left[ \left\Vert C_{N,j}-\overline{C}_{N,j}\right\Vert _{L_{2}(\mu_{j})}^{2}\right] \leq c_{1}^{2}\epsilon_{N,K}^{2}+c_{2}^{2}\inf
_{h\in\mathcal{H}_{K}}\mathsf{E}\left[ \left\Vert \overline{C}_{N,j}\left(
\cdot\right) -h\right\Vert _{L_{2}(\mu_{j})}^{2}\right]$$ and so $$\left\Vert C_{N,j}-\overline{C}_{N,j}\right\Vert _{L_{2}(\mu_{j},\mathbb{P})}\leq c_{1}\epsilon_{N,K}+c_{2}\inf_{h\in\mathcal{H}_{K}}\left\Vert
\overline{C}_{N,j}\left( \cdot\right) -h\right\Vert _{L_{2}(\mu
_{j},\mathbb{P})} .\label{gy}$$ By using (\[gy\]) and the unconditional expectation applied to Lemma \[lem23\] with $p=2,$ we get$$\begin{aligned}
\left\Vert C_{N,j}-C_{j}^{\ast}\right\Vert _{L_{2}(\mu_{j},\mathbb{P})} &
\leq\left\Vert C_{N,j}-\overline{C}_{N,j}\right\Vert _{L_{2}(\mu
_{j},\mathbb{P})}+\left\Vert \overline{C}_{N,j}-C_{j}^{\ast}\right\Vert
_{L_{2}(\mu_{j},\mathbb{P})}\\
& \leq c_{1}\epsilon_{N,K}+c_{2}\inf_{h\in\mathcal{H}_{K}}\left\Vert
C_{j}^{\ast}-h\right\Vert _{L_{2}(\mu_{j})}\\
& +(1+c_{2})\left\Vert \overline{C}_{N,j}-C_{j}^{\ast}\right\Vert _{L_{2}(\mu_{j},\mathbb{P})}\\
& \leq\Delta_{N,K}+(1+c_{2})\sum_{l=j+1}^{\mathcal{J}}\left\Vert
C_{N,l}-C_{l}^{\ast}\right\Vert _{L_{2}(\mu_{l},\mathbb{P})}$$ with $c=\max\{c_{1},c_{2}\}.$ We prove the statement by induction. Suppose that the inequality holds for $j=k,$ then $$\begin{aligned}
\left\Vert C_{N,\mathcal{J}-k-1}-C_{\mathcal{J}-k-1}^{\ast}\right\Vert
_{L_{2}(\mu_{\mathcal{J}-k-1},\mathbb{P})} & \leq\Delta_{N,K}\\
& +(1+c_{2})\sum_{l=0}^{k}\left\Vert C_{N,\mathcal{J}-l}-C_{\mathcal{J}-l}^{\ast}\right\Vert _{L_{2}(\mu_{\mathcal{J}-l},\mathbb{P})},\\
& \leq\Delta_{N,K}+\Delta_{N,K}(1+c)\sum_{l=0}^{k}(2+c)^{l}\\
& =\Delta_{N,K}(1+((2+c)^{k+1}-1))\\
& =\Delta_{N,K}(2+c)^{k+1}$$ and holds also for $j=k+1.$
Appendix
========
In this section we present two auxiliary lemmas that were needed in Section \[pertan\].
\[App\] Let $\rho>0$ and the matrix $Z\in\mathbb{R}^{N\times d}$ be of full rank with $N>d.$ Let $Z$ and $E\in\mathbb{R}^{N\times d}$ be such that $$\lambda_{\mathrm{min}}\left( Z^{\top}Z\right) -\left( 2\Vert Z\Vert
+1\right) \Vert E\Vert>\rho,\text{ \ \ }\Vert E\Vert<1. \label{cond}$$ Then we have $$\Vert(Z+E)^{\dagger}-Z^{\dagger}\Vert\leq\frac{\Vert E\Vert}{\rho}\left[
1+\frac{\left( 2\Vert Z\Vert+1\right) \Vert Z\Vert}{\rho}\right] .
\label{onA}$$
Denote $$\Delta=Z^{\top}E+E^{\top}Z+E^{\top}E,$$ then using the identity $$\begin{aligned}
\left( (Z+E)^{\top}(Z+E)\right) ^{-1}-\left( Z^{\top}Z\right) ^{-1} &
=-\left( (Z+E)^{\top}(Z+E)\right) ^{-1}\Delta\left( Z^{\top}Z\right)
^{-1}\\
& =-\left( Z^{\top}Z+\Delta\right) ^{-1}\Delta\left( Z^{\top}Z\right)
^{-1},\end{aligned}$$ we derive$$\begin{gathered}
\Vert(((Z+E)^{\top}(Z+E))^{-1}-(Z^{\top}Z)^{-1})Z^{\top}\Vert\\
\leq\Vert(Z^{\top}Z+\Delta)^{-1}\Vert\Vert(Z^{\top}Z)^{-1}\Vert(2\Vert
Z\Vert+1)\Vert E\Vert\Vert Z\Vert\\
\leq\frac{(2\Vert Z\Vert+1)\Vert E\Vert\Vert Z\Vert}{\rho^{2}}, \label{onA11}$$ since we have $\Vert(Z^{\top}Z)^{-1}\Vert=\lambda_{\mathrm{min}}^{-1}\left(
Z^{\top}Z\right) <\rho^{-1}$ and $$\begin{aligned}
\lambda_{\mathrm{min}}\left( Z^{\top}Z+\Delta\right) & =\underset
{\left\vert x\right\vert =1}{\inf}x^{\top}\left( Z^{\top}Z+\Delta\right)
x\geq\underset{\left\vert x\right\vert =1}{\inf}x^{\top}Z^{\top}Zx+\underset{\left\vert x\right\vert =1}{\inf}x^{\top}\Delta x\\
& \geq\lambda_{\mathrm{min}}\left( Z^{\top}Z\right) -\Vert\Delta\Vert
\geq\lambda_{\mathrm{min}}\left( Z^{\top}Z\right) -\left( 2\Vert
Z\Vert+1\right) \Vert E\Vert\\
& >\rho>0.\end{aligned}$$ Analogously we have $$\left\Vert \left( (Z+E)^{\top}(Z+E)\right) ^{-1}E\right\Vert =\left\Vert
\left( Z^{\top}Z+\Delta\right) ^{-1}E\right\Vert \leq\frac{\left\Vert
E\right\Vert }{\rho}, \label{onA2}$$ and then (\[onA\]) follows by (\[onA11\]), (\[onA2\]), and the triangle inequality.
\[lem:spectrum\_subgaussian\] Let $X_{1},\ldots,X_{N}$ be independent identically distributed random vectors in $\mathbb{R}^{d}$ such that$$\mathsf{E}\left[ X_{1}X_{1}^{\top}\right] =\Sigma$$ and for some $M>0,$ $\Vert X_{1}\Vert\leq M$ almost surely. Then for all $\delta\in(0,1),$ $$\begin{gathered}
\mathbb{P}\left( \left\{ \lambda_{\max}\left( \frac{1}{N}\sum_{i=1}^{N}X_{i}X_{i}^{\top}\right) >\lambda_{\max}\left( \Sigma\right)
+\varepsilon_{\delta,N}\right\} \cup\right. \label{lemed}\\
\left. \left\{ \lambda_{\min}\left( \frac{1}{N}\sum_{i=1}^{N}X_{i}X_{i}^{\top}\right) <\lambda_{\min}\left( \Sigma\right) -\varepsilon
_{\delta,N}\right\} \right) \leq\delta,\end{gathered}$$ where $$\varepsilon_{\delta,N}=M\sqrt{\frac{\log(2d/\delta)}{NC}}\frac{\lambda_{\max
}^{3/2}(\Sigma)}{\lambda_{\min}(\Sigma)} \label{epdel}$$ for some absolute constant $C>0,$ provided $N$ is large enough such that $$M\sqrt{\frac{\log(2d/\delta)}{NC}}\leq\lambda_{\max}^{1/2}(\Sigma).
\label{epdelc}$$
For $z>0$ we have$$\begin{gathered}
\mathrm{P}\left[ \lambda_{\max}\left( \frac{1}{N}\sum_{i=1}^{N}X_{i}X_{i}^{\top}\right) -\lambda_{\max}\left( \Sigma\right) >z\right]
\label{oth1}\\
\leq\mathrm{P}\left[ \left\Vert \frac{1}{N}\sum_{i=1}^{N}X_{i}X_{i}^{\top
}-\Sigma\right\Vert >z\right] .\nonumber\end{gathered}$$ On the other hand, since for positive matrices $A,B,$$$\left\vert \lambda_{\min}\left( A\right) -\lambda_{\min}\left( B\right)
\right\vert \leq\left\Vert B^{-1}\right\Vert \left\Vert B\right\Vert
\left\Vert B-A\right\Vert ,$$ we have for $0<z<\lambda_{\min}\left( \Sigma\right) ,$$$\begin{aligned}
& \mathrm{P}\left[ \lambda_{\min}\left( \frac{1}{N}\sum_{i=1}^{N}X_{i}X_{i}^{\top}\right) <\lambda_{\min}\left( \Sigma\right) -z\right]
\label{oth}\\
& \leq\mathrm{P}\left[ \left\Vert \frac{1}{N}\sum_{i=1}^{N}X_{i}X_{i}^{\top
}-\Sigma\right\Vert >z\frac{\lambda_{\min}(\Sigma)}{\lambda_{\max}(\Sigma
)}\right] .\nonumber\end{aligned}$$ Theorem 5.44 in [@MR2963170] implies that for any $s>0,$ $$\mathrm{P}\left[ \left\Vert \frac{1}{N}\sum_{i=1}^{N}X_{i}X_{i}^{\top}-\Sigma\right\Vert >\max\left\{ \Vert\Sigma\Vert^{1/2}\sqrt{\frac{s^{2}M^{2}}{N}},\frac{s^{2}M^{2}}{N}\right\} \right] \leq d\cdot\exp(-Cs^{2}),$$ where $C$ is an absolute constant. For $N$ such that $s^{2}M/N\leq\Vert
\Sigma\Vert$ and $s=\sqrt{C^{-1}\log(2d/\delta)},$ we so obtain $$\begin{gathered}
\mathrm{P}\left[ \left\Vert \frac{1}{N}\sum_{i=1}^{N}X_{i}X_{i}^{\top}-\Sigma\right\Vert >M\sqrt{\frac{\Vert\Sigma\Vert\log(2d/\delta)}{NC}}\right]
\leq\delta/2,\text{ \ \ for }N\text{ such that}\label{oth2}\\
M\sqrt{\frac{\log(2d/\delta)}{NC}}\leq\Vert\Sigma\Vert^{1/2}=\lambda_{\max
}^{1/2}\left( \Sigma\right) .\nonumber\end{gathered}$$ Thus, (\[lemed\]) follows from (\[oth2\]) and taking $z=$ $\varepsilon
_{\delta,N}$ given by (\[epdel\]) in (\[oth1\]) and (\[oth\]), respectively.
[^1]: [Duisburg-Essen University. Email: `[email protected]`]{}
[^2]: [Weierstrass Institute of Applied Mathematics. Email: `[email protected]`]{}
|
---
author:
- |
T. Harmark and N.A. Obers[^1]\
Niels Bohr Institute and Nordita, Blegdamsvej 17, DK-2100 Copenhagen, Denmark\
E-mail: ,
title: Thermodynamics of Spinning Branes and their Dual Field Theories
---
Introduction and conclusion
===========================
In the early 1970s, two important discoveries were made which have played a dominant role in theoretical physics ever since. The first discovery, by Bekenstein [@Bekenstein:1973ur] and Hawking [@Hawking:1974df], was that four-dimensional black holes have thermodynamic properties due to Hawking radiation. Thus, by studying thermodynamics of black holes one probes the nature of quantum gravity. In the framework of string and M-theory, this discovery has been one of the main motivations to consider the thermodynamics of black $p$-branes [@Horowitz:1991cd; @Gubser:1996de; @Klebanov:1996un]. The second discovery, by ’t Hooft [@'tHooft:1974jz], was that non-Abelian gauge theories simplify in the ’t Hooft limit. In this limit the planar diagrams dominate and the theories thus become more tractable.
More recently, it has become clear that these two discoveries are in fact connected through the conjectured correspondence between the near-horizon limit of brane solutions in string/M-theory and certain quantum field theories in the large $N$ limit [@Maldacena:1997re; @Itzhaki:1998dd]. As a consequence of this correspondence, studying the thermodynamic properties of black $p$-branes not only probes quantum gravity, but can in addition provide information about the thermodynamics of quantum field theories in the large $N$ limit.
In particular, for the non-dilatonic branes (D3,M2,M5) the near-horizon limit of the supergravity solutions has been conjectured [@Maldacena:1997re] to be dual to a certain limit of the corresponding conformal field theories (see also Refs. [@Gubser:1998bc; @Witten:1998qj] for an elaboration of the conjecture at the level of the partition function and correlation functions). In these AdS/CFT correspondences, the near-horizon background geometry is of the form $AdS_{p+2} \times
S^{d-1}$ and the dual field theories are conformal. Moreover, for the more general dilatonic branes of type II string theory preserving 16 supersymmetries, similar duality relations have also been obtained [@Itzhaki:1998dd], which may be characterized more generally as Domain Wall/QFT correspondences [@Boonstra:1998mp; @Behrndt:1999mk]. See Ref. [@Aharony:1999ti] for a comprehensive review and Refs. [@Klebanov:1999ku; @Petersen:1999zh; @DiVecchia:1999yr] for some introductory lectures on the AdS/CFT correspondence and field theories in the large $N$ limit.
A common feature of these dualities between near-horizon backgrounds and field theories, is that the supergravity black $p$-brane solution exhibits an $SO(d)$ isometry (where $d=D-p-1$ is the dimension of the transverse space) which manifests itself as the R-symmetry of the dual field theory. As a consequence, by considering black $p$-brane solutions that rotate in the transverse space, we expect on the one hand to learn more about the field theory side, and on the other hand, be able to perform further non-trivial tests of the duality conjectures that include the dependence on this R-symmetry group. In particular, as will be reviewed below, the thermodynamics on the two sides provides a useful starting point for such a comparison.
The first construction of spinning branes solutions, rotating in the transverse space, can be found in Refs. [@Horowitz:1996tm; @Cvetic:1996xz; @Cvetic:1996dt; @Cvetic:1996ek] from which, in principle the most general black $p$-brane solution [@Cvetic:1999xp] can be derived by oxidization. Also, various spinning brane solutions [@Russo:1998mm; @Csaki:1998cb; @Kraus:1998hv; @Russo:1998by; @Csaki:1999vb; @Correia:1999bt; @Brandhuber:1999jr] have recently been constructed and employed with the purpose to provide extra dimensionfull parameters in the decoupling of the unwanted KK modes in the context of obtaining QCD in various dimensions via the AdS/CFT correspondence [@Witten:1998zw]. Other examples of spinning brane solutions include the spinning NS5-brane [@Sfetsos:1999pq] and rotating Kaluza Klein black holes [@Larsen:1999pp]. Spinning branes have also been used [@Cai:1999ad] in the study of D-brane probes [@Tseytlin:1998cq; @Kiritsis:1999ke; @Kiritsis:1999tx]. Many aspects of the case in which the rotation does not lie in the transverse space [@Hawking:1998ct; @Hawking:1998kw; @Berman:1999mh; @Caldarelli:1999xj; @Hawking:1999dp], generally referred to as the Kerr-AdS type, have also been considered in view of the AdS/CFT correspondence but will not be considered in this paper.
It is interesting in its own right to study the thermodynamic properties of black $p$-branes and rotating versions thereof, since this may teach us more about black brane physics. In particular, the near-horizon solution is thermodynamically much better behaved than the asymptotically-flat solution, which along with its relevance to a certain limit of the dual field theories, makes it very interesting to study the thermodynamics in this case. For the non-dilatonic branes the study of the thermodynamic stability has been initiated[^2] in a number of recent papers [@Gubser:1998jb; @Cai:1998ji; @Cvetic:1999ne; @Cai:1999hg; @Cvetic:1999rb]. The stability for the D3-brane with one non-zero angular momentum was addressed in [@Gubser:1998jb] followed by an analysis both in the grand canonical and canonical ensemble for all non-dilatonic branes [@Cai:1998ji]. It was found that these two ensembles are not equivalent. An analysis of the critical behavior near these boundaries was also performed and shown to obey scaling laws of statistical physics. The case of multiple angular momenta was considered in Ref. [@Cvetic:1999rb] for both ensembles.
Furthermore, in order to compare with the field theories, a regularization method [@Gubser:1998jb; @Cvetic:1999rb] has been proposed and used in order to compare the stability behavior obtained from the supergravity solution with that of the corresponding field theory in the weakly coupled limit. The angular momenta take values in the isometry group of the sphere, and hence map onto the R-charges in the dual field theory. Therefore, the angular velocities on the brane correspond in the field theory to voltages under the R-symmetry. In the presence of these voltages, a regularization is required, since for massless bosons with non-zero R-charge negative thermal occupation numbers occur. For the D3-brane case, it was found that the regulated field theory analysis predicts a similar upper bound on the angular momentum (or R-charge) density as obtained from the near-horizon brane solution. The critical exponents obtained from the supergravity solution are, however, not reproduced, though a mean field theory analysis has been suggested to cure this discrepancy. Finally, Ref. [@Cvetic:1999rb] also presents evidence for localization of angular momentum on the brane outside the region of stability, and the occurrence of a first order phase transition.
The comparison of boundaries of stability in the two dual sides is one way to obtain evidence and predictions of the correspondence between near-horizon brane solutions and field theories. Another route, that also uses thermodynamic quantities is consideration of the free energy which, in the non-rotating case, has been computed from the Euclidean action by a suitable regularization method [@Hawking:1983dh; @Witten:1998zw]. For non-rotating D3, M2 and M5-branes it has been conjectured that there exists a smooth interpolating function connecting the two limits [@Gubser:1998nz]. In particular for the D3-brane one finds that for the near-horizon $AdS_5 \times S^5$ limit the free energy differs by a factor 3/4 from the weakly coupled N=4 SYM expression. Since the former limit corresponds to the strong ’t Hooft coupling limit, it can be envisaged that higher derivative string corrections on the supergravity modify this result in such a way that minus the free energy increases towards the weak coupling limit. This conjecture was tested [@Gubser:1998nz; @Pawelczyk:1998pb] by computing the correction to the free energy arising from the tree level $R^4$ term in the type IIB effective action, and shown to be in agreement. In this spirit, the study of such corrections is also interesting to perform in the presence of rotation.
In this paper, we will address various issues related to the developments described above, with emphasis on a general treatment for all black $p$-branes that are 1/2 BPS solutions of string and M-theory in the extremal and non-rotating limit. This includes the M2 and M5-branes of M-theory and the D and NS-branes of string theory. We will first write down the general asymptotically-flat solution of these spinning black $p$-brane in $D$ dimensions. Since the transverse space is $d = D-p-1$ dimensional, these spinning solutions are characterized by a set of angular momenta $l_i$, $i=1 \ldots n$ where $n={\rm rank} (SO(d))$, along with the non-extremality parameter $r_0$ and another parameter $\alpha$ related to the charge. Using standard methods of black hole thermodynamics we compute the relevant thermodynamic quantities of the general solution and show that the conventional Smarr formula is obeyed. (see Section \[secrotbra\]).
Our main interest, however, will be in the near-horizon limit of these spinning branes, which we will also compute in generality. The corresponding thermodynamics that results in this limit will also be obtained. In the near-horizon limit the charge and chemical potential become constant and are not thermodynamic parameters anymore, so that the thermodynamic quantities are given in terms of the $n+1$ supergravity parameters $(r_0,l_i)$. In particular, we derive and check a modified Smarr law for the near-horizon background which is due to a different scaling of the solution as compared to the asymptotically-flat case. One also finds a simple formula for the Gibbs free energy for any near-horizon spinning black $p$-brane solution with $d$ transverse dimensions $$\label{gibbs}
F = - \frac{V_p V(S^{d-1})}{16\pi G} \frac{d-4}{2}
r_0^{d-2}$$ In a low angular momentum expansion, we rewrite this expression in terms of the intensive thermodynamic quantities, the temperature $T$ and the angular velocities $\Omega_i$. For comparison, we then use the correspondence with field theory in the large $N$ limit (and appropriate limit of the ’t Hooft coupling limit in the case of D-branes), to write this free energy in terms of the field theory variables. (see Section \[secnhlimit\]).
We proceed with presenting a general analysis of the boundaries of stability in both the grand canonical ensemble (with thermodynamic variables $(T,\Omega_i)$) and the canonical ensemble (with thermodynamic variables $(T,J_i)$) of the near-horizon spinning branes. While there is a one-to-one correspondence between the $n+1$ supergravity variables and the extensive quantities $(S,J_i)$, the map to the intensive ones $(T,\Omega_i)$ or the mixed combination $(T,J_i)$ involves a non-invertible function, a fact which is crucial to the stability analysis. We will show in particular that for general $d$, the two ensembles are not equivalent and that increasing the number of equal-valued angular momenta enlarges the stable region[^3]. For one non-zero angular momentum we find, in the grand canonical ensemble, that the region of stability (for $d \geq 5$) is determined by the condition $$J \leq \sqrt{\frac{d-2}{d-4}} \frac{S}{2 \pi}$$ so that there is an upper bound on the amount of angular momentum the brane can carry in order to be stable. Put another way, at a critical value of the angular momentum density (which equals the R-charge density in the dual field theory) a phase transition occurs. The supergravity description also determines an upper bound on the angular velocity, $$\Omega \leq \frac{2 \pi}{ \sqrt{(d-2)(d-4)}} T$$ which is saturated at the critical value of the angular momentum. As a byproduct of the analysis we obtain an exact expression of the Gibbs free energy in terms of $(T,\Omega)$ for all branes in the case of one non-zero angular momentum[^4]. We will also comment on the nature of the instability and discuss the setup to be solved in order to determine whether there is some region of parameter space in which phase mixing is thermodynamically favored, so that the angular momentum localizes on the brane. Finally, we give a uniform treatment of the critical exponents for all spinning branes in both ensembles and show that all of these are 1/2, a value which satisfies scaling laws in statistical physics. (see Section \[seccritbeh\]).
An important question is to what extent do we observe the above stability phenomena in the large $N$ limit of the dual field theory, also at weak coupling. To this end, extending the method of Ref. [@Gubser:1998jb], we obtain in an ideal gas approximation the free energies of the field theories for the case of the M-branes and the D-branes of type II string theory. We review and extend the interpolation conjecture, stating that the free energy smoothly interpolates between the weak and strong coupling limit. The free energies in the weakly coupled regime enable us to corroborate these conjectures by computing the boundaries of stability in this regime. The corresponding critical values of the dimensionless quantity $\Omega/T$ for the D2, D3, D4, M2 and M5-branes are remarkably close in the weak and strong coupling limit. (see Section \[secfieldtheory\]).
Finally, we also test the interpolation conjecture by considering the free energy. We first establish that for all near-horizon spinning branes the on-shell Euclidean action reproduces the thermodynamically obtained Gibbs free energy . For the spinning D3-brane we then calculate, in a weak angular momentum expansion, the correction to the free energy due to the tree-level $R^4$ term in the type IIB effective action. This order $\lambda^{-3/2}$ correction is positive (in the range of validity) and hence supports the conjectured existence of a smooth interpolating function between the free energy in the weak and strong coupling limit. (see Section \[secfefromaction\]).
A number of appendices are included: Appendix \[appenergy\] gives a general discussion of (non-rotating) black $p$-branes, including those that preserve a lower amount of supersymmetry. We also find the thermodynamic quantities and Smarr formula for both the asymptotically-flat and near-horizon solutions. Appendix \[sphcoor\] reviews spheroidal coordinates which are relevant for the explicit form of spinning brane backgrounds. Appendix \[appeuclnhsol\] shows how the Euclidean spinning brane solution can be obtained from the Minkowskian solution. Appendix \[basisch\] discusses the change of variables from the supergravity variables to the intensive thermodynamic variables in a weak angular momentum expansion. Finally, Appendix \[apppoly\] gives various useful expressions for the polylogarithms which are used in Section \[secfieldtheory\] to compute the free energies of the weakly coupled field theories in the presence of voltage under the R-symmetry.
General spinning p-branes \[secrotbra\]
========================================
More than forty years after the discovery of the Schwarzschild black hole metric, Kerr presented in 1963 the first metric for a rotating black hole [@Kerr:1963ud]. About twenty years later, this was generalized to neutral rotating black holes of arbitrary dimensions in [@Myers:1986un]. In [@Horowitz:1996tm; @Cvetic:1996xz; @Cvetic:1996dt] these were further generalized to charged rotating black hole solutions of the low-energy effective action of toroidally compactified string theory. In [@Cvetic:1996ek] the first spinning brane solutions appeared and recently spinning brane solutions of type II string theory and M-theory have been presented in [@Russo:1998mm; @Csaki:1998cb; @Kraus:1998hv; @Sfetsos:1999pq; @Cvetic:1999xp].
In this section we consider the general spinning brane solutions of string theory and M-theory. The general solution is presented in Section \[subsecspinsol\] and in Section \[secthermdyn\] we derive the thermodynamic quantities of the general spinning brane solutions.
The spinning black $p$-brane solutions \[subsecspinsol\]
---------------------------------------------------------
In this section we present the general charged spinning black $p$-brane solution with a maximal number of angular momenta for branes of string theory and M-theory. These solutions have the property that they are 1/2 BPS states in the extremal and non-rotating limit. Thus, they include the D- and NS-branes of 10-dimensional string theory[^5] and the M-branes of 11-dimensional M-theory, as well as the branes living in toroidal compactifications of these theories. The solutions can be derived by oxidizing spinning charged black hole solutions in a $D-p$ dimensional space-time [@Horowitz:1996tm; @Cvetic:1996xz; @Cvetic:1996dt; @Cvetic:1996ek; @Russo:1998mm; @Csaki:1998cb; @Kraus:1998hv; @Sfetsos:1999pq; @Cvetic:1999xp].
We only write the solutions for electric branes, since the magnetic solutions can easily be obtained by the standard electromagnetic duality transformation. In our conventions the coordinate system is taken to be $(t,y^i,x^a)$, where $t$ is the time, $y^i$, $i=1\ldots p$ the spatial world-volume coordinates and $x^a$ the transverse coordinates. The space-time dimension is denoted by $D$, so that $d = D - p - 1$ is the dimension of the transverse space. The spinning brane solutions given below are solutions of the action $$\label{pbact}
I = \frac{1}{16\pi G} \int d^D x \sqrt{g} \Big( R
- \frac{1}{2} \partial_\mu \phi \partial^\mu \phi
- \frac{1}{2 (p+2)!} e^{a\phi} F_{p+2}^2 \Big)$$ where $ F_{p+2} $ is the $(p+2)$-form electric field strength. This action arises as part of the 10-dimensional string effective action in the Einstein frame or the 11-dimensional supergravity action, and toroidal compactifications of these actions[^6]. The value of $a$ is a characteristic number for each brane, and for branes that are 1/2 BPS in the extremal and non-rotating limit, one has the relation $$\label{aeq}
2(D-2) = (p+1)(d-2) + \frac{1}{2}a^2 (D-2)$$ Appendix \[appenergy\] reviews[^7] more general black brane solutions that do not fulfill this identity and preserve a smaller amount of supersymmetry (See also Table \[tabbranes\] for the values of $a$ for each of the branes that we consider).
As described in Appendix \[sphcoor\], the spinning solutions depend on a set of angular momentum parameters $ l_1,l_2,...,l_n
$ where $n= [\frac{d}{2}]$ is the rank of $SO(d)$. Two further parameters that characterize the solutions are the non-extremality parameter $r_0$ and a dimensionless parameter $\alpha$, related to the charge. In particular, $r_0=0$ corresponds to the extremal $p$-brane solution, while $\alpha =0$ corresponds to a neutral brane. The relation of these parameters to the thermodynamic quantities of the solution will be discussed in Section \[secthermdyn\]. We restrict ourselves to the cases for which the transverse dimensions lie in the range $ 3 \leq d \leq 9$ so that the brane solutions are asymptotically flat.
The metric of a charged spinning $p$-brane solution of the action then takes the form $$\begin{aligned}
\label{solmet}
ds^2 &=& H^{-\frac{d-2}{D-2}}
\Big( - f dt^2 + \sum_{i=1}^p (dy^i)^2 \Big)
+ H^{\frac{p+1}{D-2}} \Big( \bar{f}^{-1} K_d dr^2
+ \Lambda_{\alpha \beta} d\eta^\alpha d\eta^\beta \Big)
{\nonumber}\\ &&
+ H^{-\frac{d-2}{D-2}} \frac{1}{K_d L_d} \frac{r_0^{d-2}}{r^{d-2}}
\Big( \sum_{i,j=1}^n l_i l_j \mu_i^2 \mu_j^2 d\phi_i d\phi_j
- 2 \cosh \alpha \sum_{i=1}^n l_i \mu_i^2 dt d\phi_i \Big)\end{aligned}$$ The electric dilaton is $$\label{soldil}
e^\phi = H^{\frac{a}{2}}$$ and the electric potential $A_{p+1}$ (with field strength $F_{p+2}=dA_{p+1}$) is given by $$\label{solpot} A_{p+1} = (-1)^p \frac{1}{\sinh \alpha} \Big( H^{-1}
-1 \Big) \Big( \cosh \alpha dt - \sum_{i=1}^n l_i \mu_i^2 d\phi_i
\Big) \wedge dy^1 \wedge \cdots \wedge dy^p$$ Here, we have used spheroidal coordinates for the flat transverse space metric $$\label{flattr}
\sum_{a=1}^d (dx^a)^2
= K_d dr^2 + \Lambda_{\alpha \beta} d\eta^\alpha d\eta^\beta$$ the explicit form of which can be found in Appendix \[sphcoor\], which also gives the angular dependence of the quantities $\mu_i$. Moreover, we have defined
\[defs\] $$\label{Ldeq}
L_d = \prod_{i=1}^n \Big( 1 + \frac{l_i^2}{r^2} \Big)
{\ ,\qquad}H = 1 + \frac{1}{K_d L_d} \frac{r_0^{d-2} \sinh^2 \alpha}{r^{d-2}}$$ $$f = 1 - \frac{1}{K_d L_d} \frac{r_0^{d-2}}{r^{d-2}} {\ ,\qquad}\bar{f} = 1 - \frac{1}{L_d} \frac{r_0^{d-2}}{r^{d-2}}$$
and we note that the harmonic function $H$ is such that the branes are asymptotically flat.
The physical situation that this solution describes is a charged black $p$-brane rotating in the angles $\phi_1,\phi_2,...,\phi_n$ (see Appendix \[sphcoor\]). The rotation is static, meaning that the points of the $p$-brane move with time, but that the total set of points of the brane in the embedding space does not change with time. Thus, the solution describes a spinning charged black $p$-brane.
Thermodynamics of spinning branes \[secthermdyn\]
--------------------------------------------------
We proceed with describing some general physical properties of the solution given in (\[solmet\]), (\[soldil\]) and (\[solpot\]), and the computation of its relevant thermodynamic quantities.
The horizon is at $r=r_H$ where $r_H$ is the highest root of the equation $\bar{f} (r) = 0$, so that $$\label{horizon}
L_d (r_H) r_H^{d-2} = r_0^{d-2}$$ where $L_d$ is defined in . On the other hand, the solution of the equation $ f (r) = 0 $ with the maximal possible value of $r$ describes the so-called ergosphere, which coincides with the horizon for special values of the angles $ \theta $ and $ \psi_1,\psi_2,...,\psi_{d-n-2} $. It is useful to find a coordinate transformation to a system in which these two hyper-surfaces coincide. To this end we write $$\label{newcoor} \tilde{t} = t {\ ,\qquad}\tilde{\phi}_i = \phi_i -
\Omega_i t {\ ,\qquad}i=1 \ldots n$$ with all other coordinates unchanged. Thus, we want to find $ \{\Omega_i,i=1\ldots n \} $ so that $$\label{eqkill}
g_{\tilde{t} \tilde{t}} \Big|_{r=r_H} = 0$$ In the transformed frame one has $$\label{newgtt} g_{\tilde{t}\tilde{t}} = g_{tt} + \sum_{i,j=1}^n
\Omega_i \Omega_j g_{\phi_i \phi_j} + 2 \sum_{i=1}^n \Omega_i g_{t
\phi_i}$$ so that (\[eqkill\]) can be written as $$\label{eqkill2}
g_{tt}\Big|_{r=r_H}
+ \sum_{i,j=1}^n \Omega_i \Omega_j g_{\phi_i \phi_j}\Big|_{r=r_H}
+ 2 \sum_{i=1}^n \Omega_i g_{t \phi_i}\Big|_{r=r_H} = 0$$ Since Eq. (\[eqkill2\]) should hold for all angles, we can consider the special choice for which $\mu_i = 1$ and $\mu_{j\neq
i} = 0$. Then (\[eqkill2\]) becomes $$\label{eqkill3} g_{tt}\Big|_{r=r_H} + \Omega_i^2 g_{\phi_i
\phi_j}\Big|_{r=r_H} + 2 \Omega_i g_{t \phi_i}\Big|_{r=r_H} = 0
{\ ,\qquad}i = 1 \ldots n$$ which is satisfied by $$\Omega_i = \frac{l_i}{(l_i^2 + r_H^2) \cosh \alpha} {\ ,\qquad}i=1 \ldots
n$$ The new coordinate system defined in (\[newcoor\]) can be seen as comoving coordinates on the horizon, i.e. coordinates for which the points on the brane in the embedding space do not move with time[^8]. From the definition (\[newcoor\]) it then follows that $
\Omega_i $ is the angular velocity of a particle on the horizon with respect to the angle $ \phi_i $. Thus, $ \Omega_i$ is the angular velocity of the black $p$-brane with respect to the angle $ \phi_i $. Moreover, in the new coordinate system the off-diagonal metric component $$\label{newgtphi}
g_{\tilde{t} \tilde{\phi}_i}
= g_{t\phi_i} + \sum_{j=1}^n \Omega_j g_{\phi_j \phi_i }$$ has the property that it vanishes at the horizon $$\label{extrakill}
g_{\tilde{t} \tilde{\phi}_i}\Big|_{r=r_H} = 0$$ For completeness we also mention that the new coordinate system has the Killing vector $$\label{nullkill}
V \equiv \frac{\partial}{\partial \tilde{t}} =
\frac{\partial}{\partial t} + \sum_{i=1}^n \Omega_i
\frac{\partial}{\partial \phi_i}$$ with norm $ V^2 = g_{\tilde{t}\tilde{t}} $. It then follows from that at the horizon this is a null Killing vector.
The new frame (\[newcoor\]) also enables us to compute the temperature. By construction, both the metric components $ g_{\tilde{t}
\tilde{t}} $ and $ g^{rr} $ are zero for $ r=r_H $. As a consequence, the standard procedure of transforming to Euclidean space can be employed to find the temperature of the $p$-brane: First we go to the Euclidean signature by a Wick rotation $ \tau
= i\tilde{t} $, and reinterpret the path-integral partition function as a partition function for a statistical system in $D-1$ dimensions with the temperature $T = 1/ \beta$, where $\beta$ is the periodicity of $\tau$. This periodicity is determined by avoiding a singularity in space time (see for example [@Hawking:1983dh]). In the case at hand, the Euclidean metric near the horizon can be written as $$ds^2 = - \partial_r g_{\tilde{t}\tilde{t}}|_{r=r_H} ( r-r_H)d\tau^2
+ \frac{1}{\partial_r g^{rr}|_{r=r_H} ( r-r_H)} dr^2 + \cdots
= \rho^2 d\Theta^2 + d\rho^2 + \cdots$$ with $$\rho = 2 \sqrt{ \frac{r-r_H}{\partial_r g_{rr}|_{r=r_H}}},\ \
\Theta = \frac{1}{2} \sqrt{-\partial_r g_{\tilde{t}\tilde{t}}|_{r=r_H}
\partial_r g^{rr}|_{r=r_H}} \tau$$ To avoid a conical singularity we need to require that $ \Theta
$ is periodic with period $ 2\pi $, which determines $$\label{temp} \beta = \frac{1}{T} = 4\pi
\frac{1}{\sqrt{-\partial_r g_{\tilde{t}\tilde{t}}|_{r=r_H}
\partial_r g^{rr}|_{r=r_H}}}$$ Using and the formula for $T$ can also be written as $$T = \frac{1}{4\pi} \lim_{r \rightarrow r_H} \sqrt{
\frac{g^{\mu \nu} \partial_\mu V \partial_\nu V}{-V^2} }$$ One can then proceed to calculate the temperature $T$ from (\[temp\]) using the particular choice of angles $ \theta =
\frac{\pi}{2}$, so that $ \mu_1 = 1 $ and $ \mu_{i \neq 1} =
0 $. With this choice, we also have $ \partial_\theta \mu_i^2
=
\partial_{\psi_j} \mu_i^2 = 0 $ for all $i$ and $j$. After a tedious calculation, one obtains $$T = \frac{d-2-2\kappa }{4\pi r_H \cosh \alpha}$$ where we have defined $$\label{kappa} \kappa = \sum_{i=1}^n \frac{l_i^2}{l_i^2 + r_H^2 }$$ In fact, it follows[^9] from that $d-2-2 \kappa \geq 0$ and hence $T \geq 0$. For $r_0 > 0$ it is thus possible to have $T=0$ and this in turn defines a boundary on the region of possible values of $(l_1,\ldots,l_n)$ in units of $r_0$.
Besides $\Omega_i$ and $T$, the chemical potential $$\mu = - A_{y^1 y^2 \cdots y^p \tilde{t}} \Big|_{r=r_H} = \tanh \alpha$$ is also determined by the solution at the horizon. The ADM mass $M$ and the charge $Q$ of a spinning black $p$-brane are the same as for the non-rotating black $p$-brane, since we can measure these physical quantities in the asymptotic region of the space-time. In the asymptotic region, one can check that the metric (\[solmet\]) does not contain the angular momenta $ l_i $ to leading order in $ 1/r $. To calculate the ADM mass $M$ one can therefore use the prescription given in Ref. [@Lu:1993vt]. The angular momenta $J_i$, $i=1 \ldots n$, can be read from the asymptotic expansion of (\[solmet\]) using the formula [@Myers:1986un] $$g_{t\phi_i} = - \frac{8 \pi G}{V_p V(S^{d-1})} \frac{\mu_i^2}{r^{d-2}} J_i
+ {{\cal{O}}}\Big(\frac{\mu_i^2}{r^d} \Big)$$ where $$\label{volsphe}
V(S^{d-1}) = \frac{2 \pi^{d/2}}{\Gamma( d/2 )}$$ is the volume of the $d-1$ dimensional unit sphere. Finally, the entropy $S$ can be calculated from the Bekenstein-Hawking formula $$S = \frac{A_H}{4 G }$$ where $A_H$ is the area of the outer horizon. Alternatively, $S$ can be found using the integrated Smarr formula reviewed below which follows from the 1st law of black-hole thermodynamics.
Summarizing, we list the complete set of thermodynamic quantities for a general spinning black $p$-brane
\[pbranethermo\] $$\label{admmass}
M = \frac{V_p V(S^{d-1})}{16\pi G} r_0^{d-2}
\Big( d-1 + (d-2)\sinh^2 \alpha \Big)$$ $$\label{tands}
T = \frac{d-2-2\kappa }{4 \pi r_H \cosh \alpha} {\ ,\qquad}S = \frac{V_p V(S^{d-1}) }{4G} r_0^{d-2} r_H \cosh \alpha$$ $$\label{muandq}
\mu = \tanh \alpha {\ ,\qquad}Q = \frac{V_p V(S^{d-1})}{16\pi G} r_0^{d-2} (d-2) \sinh \alpha \cosh \alpha$$ $$\label{angmom}
\Omega_i = \frac{l_i}{(l_i^2 + r_H^2)\cosh \alpha} {\ ,\qquad}J_i = \frac{V_p V(S^{d-1})}{8\pi G} r_0^{d-2} l_i \cosh \alpha$$
where $\kappa$ is defined in , $V_p$ is the worldvolume of the $p$-brane and $V(S^{d-1})$ is the volume of the (unit radius) transverse $(d-1)$-sphere given in .
In further detail, the internal energy of a spinning charged black $p$-brane is the mass $M$. The other extensive thermodynamic parameters are the entropy $S$, the charge $Q$ and the angular momenta $\{J_i \}$, and the first law of thermodynamics is $$\label{firstlaw} dM = T dS + \mu dQ + \sum_{i=1}^n \Omega_i dJ_i
{\ ,\qquad}M=M(S,Q,\{J_i \})$$ Under the canonical scaling $$\label{scaling}
r_0 \rightarrow \lambda r_0 {\ ,\qquad}l_i \rightarrow \lambda l_i {\ ,\qquad}\alpha \rightarrow \alpha$$ we have the transformations $$M \rightarrow \lambda^{d-2} M,\ \
S \rightarrow \lambda^{d-1} S,\ \
Q \rightarrow \lambda^{d-2} Q,\ \
J_i \rightarrow \lambda^{d-1} J_i$$ It then follows from Euler’s theorem that $$\label{smarr}
(d-2) M = (d-1) TS + (d-2) \mu Q + (d-1) \sum_{i=1}^n \Omega_i J_i$$ which is known as the integrated Smarr formula [@Smarr:1973]. One can also reverse the logic and derive this formula using Killing vectors [@Myers:1986un], and then use the scaling (\[scaling\]) to find . As an important check we note that the quantities listed in indeed satisfy .
As an aid to the reader and for use below, we also give here the explicit expressions for the relevant parameters entering the solution and the corresponding thermodynamics for the M-branes in $D=11$ and the D-branes in $D=10$. To this end, it is useful to define the parameter $ h
$ via the relation $$\label{hdef}
h^{d-2} = r_0^{d-2} \cosh \alpha \sinh \alpha$$ Then, for the branes of M-theory we have the relations $$\label{Mrel}
16 \pi G = (2\pi)^8 l_p^9
{\ ,\qquad}h^6 = 2^5 \pi^2 N l_p^6 \;\; \; (\mbox{M2})
{\ ,\qquad}h^3 = \pi N l_p^3 \;\;\; (\mbox{M5})$$ where $l_p$ is the 11-dimensional Planck length and $N$ the number of coincident branes. In parallel, for the D$p$-branes of type II string theory we record $$\label{IIrel}
16 \pi G = (2\pi)^7 g_s^2 l_s^8 {\ ,\qquad}h^{d-2} =
\frac{(2\pi)^{d-2} N g_s l_s^{d-2}}{(d-2) V(S^{d-1})} \;\; \;
(\mbox{D}p)$$ where $l_s$ is the string length and $g_s$ the string coupling.
Near-horizon limit of general spinning branes \[secnhlimit\]
=============================================================
In this section we examine the near-horizon limit of the spinning $p$-brane solutions considered in Section \[secrotbra\]. This provides further insights into the thermodynamics of black branes. More importantly, this is relevant since according to the correspondence [@Maldacena:1997re; @Itzhaki:1998dd] between near-horizon brane solutions and field theories, this gives information about the strongly coupled regime of these field theories in the presence of non-zero voltages under the R-symmetry.
In Section \[subsecnhsol\] we take the near-horizon limit while in Section \[nhtherm\] we find the relevant thermodynamic quantities of the solution. Finally, in Section \[subsecdual\] we discuss the map between the supergravity solutions and the dual field theories, obtaining in particular the free energies of these field theories.
The near-horizon solution \[subsecnhsol\]
------------------------------------------
To find the near-horizon solution, one has to take an appropriate limit of the solution that is specified as follows: We introduce a dimensionfull parameter $\ell$ and perform the rescaling
$$\begin{aligned}
\label{rescal}
&& r = \frac{{r_{\mathrm{old}}}}{\ell^2} {\ ,\qquad}r_0 = \frac{{(r_0)_{\mathrm{old}}}}{\ell^2} {\ ,\qquad}l_i = \frac{{(l_i)_{\mathrm{old}}}}{\ell^2} {\ ,\qquad}h^{d-2} = \frac{{h_{\mathrm{old}}}^{d-2}}{\ell^{2d-8}} \\
\label{solrescal}
&& ds^2 = \frac{{(ds^2)_{\mathrm{old}}}}{\ell^{4(d-2)/(D-2)}}
{\ ,\qquad}e^{\phi} = \ell^{2a} e^{\phi_{\rm old}} {\ ,\qquad}A = \frac{A_{\rm old}}{\ell^4} {\ ,\qquad}G = \frac{{G_{\mathrm{old}}}}{\ell^{2(d-2)}}\end{aligned}$$
where the new quantities on the left hand side are expressed in terms of the old quantities labelled with a subscript “old”, and we recall that ${h_{\mathrm{old}}}$ is defined in . Note that the rescaling in leaves the action invariant due to the relation . The near-horizon limit is defined as the limit $ \ell \rightarrow 0 $ keeping all the new quantities in (\[rescal\]) fixed. In particular, implies that in this limit we have $\frac{1}{4} e^{2\alpha} \rightarrow \ell^{-4} (h/r_0)^{d-2}$.
Using - the corresponding near-horizon solution then becomes
\[nhsol\] $$\begin{aligned}
\label{nearmet}
ds^2 &=& H^{-\frac{d-2}{D-2}}
\Big( - f dt^2 + \sum_{i=1}^p (dy^i)^2 \Big)
+ H^{\frac{p+1}{D-2}} \Big( \bar{f}^{-1} K_d dr^2
+ \Lambda_{\alpha \beta} d\eta^\alpha d\eta^\beta \Big)
{\nonumber}\\ &&
-2 H^{-\frac{d-2}{D-2}}
\frac{1}{K_d L_d} \frac{h^{\frac{d-2}{2}}r_0^{\frac{d-2}{2}}}{r^{d-2}}
\sum_{i=1}^n l_i \mu_i^2 dt d\phi_i\end{aligned}$$ $$\label{neardil}
e^\phi = H^{\frac{a}{2}}$$ $$\label{nearpot} A_{p+1} = (-1)^p \Big( H^{-1} dt +
\frac{r_0^{\frac{d-2}{2}}}{h^{\frac{d-2}{2}}} \sum_{i=1}^n l_i
\mu_i^2 d\phi_i \Big) \wedge dy^1 \wedge \cdots \wedge dy^p$$
where the harmonic function is now $$H = \frac{1}{K_d L_d} \frac{h^{d-2}}{r^{d-2}}$$ and the functions $L_d$, $K_d$, $f$, $\bar f$ are as defined before in , , since the scale factor drops out in these expressions.
Thermodynamics in the near-horizon limit \[nhtherm\]
-----------------------------------------------------
We now turn to the thermodynamics of the near-horizon spinning $p$-brane solution obtained in the previous subsection. Using the rescaling (\[rescal\]) in the expressions (\[tands\]) and (\[angmom\]) for ($T,S$) and ($\Omega_i,J_i$) one finds in the near-horizon limit $\ell \rightarrow 0$ the following quantities,
\[limthermo\] $$\label{tandsnhlim}
T = \frac{d-2-2\kappa}{4 \pi r_H}
\frac{r_0^{\frac{d-2}{2}}}{h^{\frac{d-2}{2}}} {\ ,\qquad}S = \frac{V_p V(S^{d-1}) }{4G} r_0^{\frac{d-2}{2}} h^{\frac{d-2}{2}} r_H$$ $$\Omega_i = \frac{l_i}{(l_i^2 +r_H^2)}
\frac{r_0^{\frac{d-2}{2}}}{h^{\frac{d-2}{2}}} {\ ,\qquad}J_i = \frac{V_p V(S^{d-1})}{8\pi G} r_0^{\frac{d-2}{2}} h^{\frac{d-2}{2}}
l_i$$
From we see that the chemical potential $ \mu = 1 $ and that the charge $ Q $ is constant. Thus, in the near-horizon limit the chemical potential and the charge are not anymore thermodynamic parameters.
In Appendix \[appenergy\] we derive the internal energy of a black $p$-brane in the near-horizon limit by defining this energy to be the energy above extremality $E = M - Q $. Since $M$ and $Q$ are not affected by the rotation of the brane, it follows from that $$\label{nhenergy}
E = \frac{V_p V(S^{d-1})}{16\pi G} \frac{d}{2} r_0^{d-2}$$
The first law of thermodynamics for a spinning $p$-brane in the near-horizon limit is $$\label{nearfl}
dE = T dS + \sum_{i=1}^n \Omega_i dJ_i
{\ ,\qquad}E = E(S,\{ J_i \} )$$ Under the canonical rescaling $$h \rightarrow h,\ \
r_0 \rightarrow \lambda r_0,\ \
l_i \rightarrow \lambda l_i$$ we have the transformation properties $$\label{limscale}
E \rightarrow \lambda^{d-2} E,\ \
S \rightarrow \lambda^{d/2} S,\ \
J_i \rightarrow \lambda^{d/2} J_i$$ as follows from and . The scalings imply with Euler’s theorem the integrated Smarr formula for the near-horizon solution $$\label{smarrnh}
(d-2) E = \frac{d}{2} TS + \frac{d}{2} \sum_{i=1}^n \Omega_i J_i$$ Remark that this conservation law for the near-horizon solution is different from the Smarr formula of the asymptotically-flat solution, due to the different scaling behavior. It is not difficult to obtain the energy function of the microcanonical ensemble in terms of the extensive variables using the horizon equation and , yielding $$E^{d/2} = \left(\frac{d}{2} \right)^{d/2} \left( \frac{V_p
V(S^{d-1})}{16\pi G} \right)^{-(d-4)/2} h^{-(d-2)^2/2} \left(
\frac{S}{4 \pi} \right)^{d-2} \prod_i \left( 1 + \left( \frac{2
\pi J_i}{S} \right)^2 \right)$$
For later use we also calculate the Gibbs free energy $$\label{nhgibbs}
F = E - TS - \sum_{i=1}^n \Omega_i J_i = - \frac{d-4}{d} E
= - \frac{V_p V(S^{d-1})}{16\pi G} \frac{d-4}{2} r_0^{d-2}$$ which satisfies the thermodynamic relation $$\label{dfeq}
dF = - S dT - \sum_{i=1}^n J_i d\Omega_i {\ ,\qquad}F= F(T,\{ \Omega_i \} )$$ A remark is in order here for the special case $d=4$, which includes the D5 and NS5-brane in 10 dimensions, since in that case it follows from that $F=0$. From we observe that since the partial derivatives of $F$ with respect to $T$ and $\{ \Omega_i \}$ are nonzero, these variables cannot be independent. Thus the phase diagram in terms of these variables degenerates into a submanifold with at least one dimension less. This point will be further illustrated in Section \[subsecgrandcan\] where we discuss the phase diagram for one non-zero angular momentum. For the non-rotating case, one immediately deduces from that the temperature must be constant for $d=4$.
Since the Gibbs free energy is properly given in terms of the intensive quantities $T$ and $\{\Omega_i \}$ we need to write the expression in terms of these variables[^10]. The change of variables from the supergravity variables $(r_0,\{l_i\})$ to these thermodynamic variables is given in Appendix \[basisch\] in a low angular momentum expansion $$\frac{l_i}{r_0} \ll 1$$ through order ${{{\cal{O}}}} (l_i^4)$. Using the result we find that $$\begin{aligned}
\label{freeexp} F &=& - \frac{V_p V(S^{d-1})}{16\pi G}
\frac{d-4}{2} \tilde{T}^{(2d-4)/(d-4)} h^{(d-2)^2/(d-4)} \left[ 1
+ \frac{2}{d-4} \sum_i \tilde \omega_i^2 \right. {\nonumber}\\ && \left.
- \frac{2(d-6)}{(d-2)(d-4)^2} \left( \sum_{i} \tilde \omega_i^2
\right)^2 + \frac{1}{d-4} \sum_i \tilde \omega_i^4
+ \ldots \right]\end{aligned}$$ for $d \neq 4$ where we have defined $ \tilde{T} = 4\pi T/(d-2)$ and $\tilde \omega_i =\Omega_i/\tilde{T}$.
As will be explained in Section \[subsecdual\], $F$ is the free energy for the field theory living on the brane in the strongly coupled large $N$ limit. In Section \[secfieldtheory\] we compare this expression with the corresponding expressions in the weakly coupled field theory. Moreover, in Section \[secfreeenergy\] we show that the free energy is reproduced by calculating the (regularized) Euclidean action of the solution.
The dual field theories \[subsecdual\]
---------------------------------------
In the remainder of this paper, we will restrict ourselves to the spinning brane solutions of type II string theory and M-theory. For these $p$-branes we can map [@Maldacena:1997re; @Itzhaki:1998dd] the near-horizon limit to a dual Quantum Field Theory (QFT) with 16 supercharges, namely the field theory that lives on the particular $p$-brane in the low-energy limit. As explained in Refs. [@Maldacena:1997re; @Itzhaki:1998dd], in the near-horizon limit the bulk dynamics decouples[^11] from the field theory living on the $p$-brane, so that the supergravity solution in the near-horizon limit describes the strongly coupled large $N$ limit of the dual QFT.
The fact that the branes are spinning, introduces the new thermodynamic parameters $ \Omega_i $ and $ J_i $ on the supergravity side which need to be mapped to the field theory side, where they are conjectured to correspond to voltage and charge for the field theory R-symmetry group. Thus, the validity of this correspondence requires the R-symmetry groups to be $SO(d)$, with the charges $J_i$ taking values in the Cartan subgroup $SO(2)^n$ of $SO(d)$ and their Legendre transforms corresponding to the voltages $\Omega_i$. In Section \[secfieldtheory\] we analyze the field theory in the weakly coupled regime using the R-charge quantum numbers of the massless degrees of freedom.
Indeed, for the $p$-branes with $p \leq 6$ it has been noted in [@Itzhaki:1998dd; @Boonstra:1998mp] that the dual QFTs have the correct R-symmetry groups. In particular, in Ref. [@Itzhaki:1998dd] it was noted that D$p$-branes have an $ ISO(1,p) \times SO(d) $ symmetry in the near-horizon limit, where $ SO(d) $ corresponds to the R-symmetry group of the dual field theory and $ ISO(1,p) $ corresponds to the Poincaré symmetry of the dual field theory. Moreover, in the dual frame, as considered in Ref. [@Boonstra:1998mp] (see also [@Behrndt:1999mk]) the near-horizon solutions under consideration can be written as $ DW_{p+2} \times S^{d-1} $ with a linear dilaton field, where $ DW_{p+2} $ is the $p+2$ dimensional Domain-Wall. Also in this language, the isometry group $ SO(d) $ of $ S^{d-1} $ translates into the R-symmetry group of the dual field theory.
As explained in [@Itzhaki:1998dd; @Behrndt:1999mk] we can trust the supergravity description of the dual field theory, when the string coupling $ g_s \ll 1 $ and the curvatures of the geometry are small. This implies in all cases that the number of coincident $p$-branes $N \gg 1$. For the M2- and M5-brane this is the only requirement since there is no string coupling in 11-dimensional M-theory. For the D$p$-branes in 10 dimensions one must further demand that [@Itzhaki:1998dd] $$\label{geff}
1 \ll g_{\mathrm{eff}}^2 \ll N^{\frac{4}{7-p}} {\ ,\qquad}g_{\mathrm{eff}}^2 = g_{\mathrm{YM}}^2 N r^{p-3}$$ where $ g_{\mathrm{eff}}^2 $ is the effective coupling, $ g_{\mathrm{YM}}^2 $ the coupling of the Yang-Mills theory on the D$p$-brane and $ r $ is the rescaled radial coordinate in the near-horizon limit (the distance to the D-brane probe) and the Higgs expectation value in the dual QFT[^12]. Thus, the near-horizon limit describes the dual QFT in the large $N$ and strongly coupled limit. Note that the thermodynamic expressions are valid when $r$ is replaced by $r_H$ in Eq. . For larger values of the effective coupling the D1 and D5-brane flow to the NS1 and NS5-brane respectively, while the self-dual D3-brane flows to itself [@Itzhaki:1998dd]. In particular, the NS1-brane description is valid for $ N^{2/3} \ll g_{\mathrm{eff}}^2 \ll N $ and the NS5-brane description for $ N^{2} \ll g_{\mathrm{eff}}^2 $ (see also Refs. [@Maldacena:1997cg; @Sfetsos:1999pq] for further details on the type II NS5-branes).
In view of this correspondence, we can write the Gibbs free energy and other thermodynamic quantities in terms of field theory variables[^13]. In particular we need to specify the relation between the parameter $\ell$ entering the near-horizon limit and the relevant length scale of the theory and compute the rescaled quantities in . In the following $N$ is the number of coincident branes and we have defined the quantity $\omega_i =\Omega_i/T$.
For the M2-brane we need the relations $$\label{M2rel}
\ell = l_p^{3/4} \co 16 \pi G = (2\pi)^8 {\ ,\qquad}h^6 = 2^5 \pi^2 N$$ where $ l_p $ is the 11-dimensional Planck length and we have used . Using this in gives the Gibbs free energy $$\label{strongm2fe} F_{\rm M2} = -\frac{2^{7/2} \pi^2}{3^4} N^{3/2}
V_2 T^3 \left[ 1 + \frac{9}{8 \pi^2} \sum_{i=1}^4 \omega_i^2
- \frac{27}{128 \pi^4 }
\left( \sum_{i=1}^4 \omega_i^2 \right)^2
+ \frac{81}{64 \pi^4 }
\sum_{i=1}^4 \omega_i^4 + \ldots \right]$$ For the M5-brane we have $$\label{M5rel}
\ell = l_p^{3/2} \co 16 \pi G = (2\pi)^8 {\ ,\qquad}h^3 = \pi N$$ giving $$\label{strongm5fe} F_{\rm M5} = -\frac{2^6 \pi^3}{3^7} N^3 V_5
T^6 \left[ 1 + \frac{9}{8\pi^2 } \sum_{i=1}^2 \omega_i^2
+ \frac{27}{128\pi^4 }
\left( \sum_{i=1}^2 \omega_i^2 \right)^2
+ \frac{81}{256\pi^4 }
\sum_{i=1}^2\omega_i^4 + \ldots \right]$$ For the D$p$-brane of type II string theory we have from $$\label{dprel}
\ell = l_s \co h^{d-2} = \frac{(2\pi)^{2d-9} }{(d-2)V(S^{d-1})}
\lambda {\ ,\qquad}\frac{V(S^{d-1}) h^{2(d-2)} }{16\pi G} =
\frac{(2\pi)^{2d-11}}{(d-2)^2 V(S^{d-1}) } N^2$$ where $ l_s $, $g_s$ are the string length and coupling and $\lambda= g_{\mathrm{YM}}^2 N$ is the ’t Hooft coupling with the Yang-Mills coupling given by $
g_{\mathrm{YM}}^2 = (2\pi)^{p-2} g_s l_s^{p-3} $. Using these relations in we obtain (for $ p \neq 5 $) $$\label{strongdpfe} F_{{\rm D}p} = - c_p V_p N^2
\lambda^{-\frac{p-3}{p-5}} T^{\frac{2(7-p)}{5-p}} \left[ 1 +
\frac{S^1_p}{\pi^2 }
\sum_i \omega_i^2
+ \frac{S^2_p}{\pi^4 }
\left( \sum_{i} \omega_i^2 \right)^2
+ \frac{S^3_p}{\pi^4 } \sum_i
\omega_i^4
+ \ldots \right]$$ where $c_p$, $S^1_p$, $S^2_p$ and $S^3_p$ are listed in Table \[tabfrdata\] and we recall that $p = 9-d$ for $D=10$.
Under type IIB S-duality we have $\tilde g_s = 1/g_s$ and $\tilde l_s = l_s g_s^{1/2}$, so that for the type IIB NS1 and NS5-brane we need $\ell = \tilde l_s$ and the thermodynamics is exactly the same as for the D1 and D5-brane when expressed in terms of $\lambda$ and $N$. Note that the free energies for the D$p$-branes with $p \leq 4$ are negative, while the D5 and NS5-brane have zero free energy and the D6-brane positive free energy.
In the discussion above, we have chosen to explicitly write down the free energies in terms of the variables of the dual field theories, since these expressions will play an important role below. Of course, the same can be done for the other thermodynamic quantities listed in using , and . Note also that for the special value of $\kappa = \frac{1}{2} (d-2)$ the temperature vanishes, implying that besides the usual extremal limit describing zero temperature field theory, we also have a limit in which the temperature is zero, accompanied by non-zero R-charges.
$p$ $c_p$ $S^1_p$ $S^2_p$ $S^3_p$
----- -------------------------------------------- ----------------- ---------------------- ---------------------
0 $(2^{21} 3^2 5^7 7^{-19} \pi^{14})^{1/5} $ $\frac{49}{40}$ $-\frac{1029}{3200}$ $\frac{2401}{1280}$
1 $2^4 3^{-4} \pi^{5/2} $ $\frac{9}{8}$ $-\frac{27}{128}$ $\frac{81}{64}$
2 $(2^{13} 3^5 5^{13} \pi^{8})^{1/3} $ $\frac{25}{24}$ $-\frac{125}{1152}$ $\frac{625}{768}$
3 $2^{-3} \pi^2 $ $1 $ $0$ $\frac{1}{2}$
4 $2^5 3^{-7} \pi^{2} $ $\frac{9}{8}$ $\frac{27}{128}$ $\frac{81}{256}$
6 $-2^3 \pi^4$ $-\frac{1}{8}$ $0 $ $\frac{5}{256}$
: Relevant coefficients for the free energy of D$p$-branes. \[tabfrdata\]
Stability analysis of near-horizon spinning branes \[seccritbeh\]
==================================================================
In this section we analyze the critical behaviour of the near-horizon limit of spinning $p$-branes, using the thermodynamics obtained in Section \[nhtherm\]. Using the mapping between the supergravity solutions and the dual QFTs, as described in Section \[subsecdual\], we can find the critical behaviour for the strongly coupled dual field theories with non-zero voltages under the R-symmetry.
Section \[subsecbound\] presents a general discussion of boundaries of stability in the grand canonical and canonical ensemble. These two ensembles are then considered in more detail in Sections \[subsecgrandcan\] and \[subseccan\] respectively. In Section \[subseccritexp\] we finally consider the critical exponents in the two ensembles.
Boundaries of stability \[subsecbound\]
----------------------------------------
There are two different settings in which we can study the stability of near-horizon spinning branes. In the first one, to which we refer as the grand canonical ensemble, we imagine the system to be in equilibrium with a reservoir of temperature $T$ and angular velocities $\Omega_i$. Thermodynamic stability then requires negativity of the eigenvalues of the Hessian of the Gibbs free energy. In particular, a boundary of stability occurs when the determinant of the Hessian is zero or infinite. In the second one, referred to below as the canonical ensemble, we have constant angular momenta $J_i$ and a heat reservoir with temperature $T$. Stability in this situation demands positivity of the heat capacity $C_J$ and the boundaries of stability occur when this specific heat is zero or infinite.
In the case at hand, we have a system in which the thermodynamic quantities are given in terms of the supergravity variables, so that the boundaries of stability will crucially depend on the change of variables between these two descriptions. We will therefore repeatedly need the determinants of the Jacobians, and we define $ D_{T \Omega} $ as the determinant $ \frac{
\partial(T,\Omega_1,\Omega_2,...,\Omega_n)}
{\partial(r_H,l_1,l_2,...,l_n)} $ and likewise for $ D_{T J} $, $ D_{S \Omega} $ and $ D_{S J} $.
In the grand canonical ensemble we need the determinant of the Hessian of the Gibbs free energy, which can be written as[^14] $$\det {\mbox{Hes}}(-F) = \frac{D_{S J }} {D_{T\Omega}}$$ so that the zeroes of the two determinants $D_{S J}$, $D_{T\Omega}$ determine the boundaries of stability. For completeness and use below we also give the specific heat $$C_\Omega = T \Big( \frac{\partial S}{\partial T}
\Big)_{\Omega_1,...,\Omega_n} = T\frac{D_{S \Omega }}
{D_{T\Omega}}$$ showing that $\det {\mbox{Hes}}(F)$ and $C_\Omega$ may have different zeroes. In the canonical ensemble, on the other hand, we need the specific heat $$C_J = T \Big( \frac{\partial S}{\partial T} \Big)_{J_1,...,J_n}
= T \frac{D_{S J}}{D_{T J}}$$ and the boundaries of stability are determined by the determinants $D_{SJ}$ and $D_{TJ}$.
In further detail, using the thermodynamic quantities in it then follows that
$$\det {\mbox{Hes}}(-F) = 8 \pi^2
\left( \frac{V_p V(S^{d-1}) h^{d-2} }{8 \pi G} \right)^{n+1}
r_H^4 \prod_i (1+x_i)^2 \frac{\Delta_{S J} }{\Delta_{T\Omega}}$$
$$\label{com} C_\Omega = \frac{V_p
V(S^{d-1})}{4G} h^{\frac{d-2}{2}} r_0^{\frac{d-2}{2}} r_H
(d-2-2\kappa) \frac{\Delta_{S\Omega}} {\Delta_{T\Omega}}$$
$$\label{cj}
C_J
= \frac{V_p V(S^{d-1})}{4G} h^{\frac{d-2}{2}} r_0^{\frac{d-2}{2}} r_H
(d-2-2\kappa ) \frac{\Delta_{S J}}{\Delta_{T J}}$$
where the functions $\Delta$ are related to the determinants $D$ up to positive define functions, and given by
\[Dfun\] $$\begin{aligned}
\label{dto}
\Delta_{T \Omega} &=& (d-4) \Big[ d-2 - (d-4) \sum_i
x_i + (d-6) \sum_{i < j} x_i x_j {\nonumber}\\ && -(d-8) \sum_{i < j <
k} x_i x_j x_k
+(d-10) x_1 x_2 x_3 x_4 \Big]\end{aligned}$$ $$\begin{aligned}
\label{dso}
\Delta_{S \Omega} &=& d - (d-4) \sum_i x_i +
(d-8)\sum_{i
< j} x_i x_j {\nonumber}\\ && -(d-12)\sum_{i < j < k} x_i x_j x_k
+(d-16) x_1 x_2 x_3 x_4\end{aligned}$$ $$\begin{aligned}
\label{dtj}
\Delta_{T J} &=&
(d-4)(d-2)
- 2(d-8) \sum_i \frac{x_i}{1+x_i} {\nonumber}\\ && - 4(d-2) \sum_{i}
\frac{x_i^2}{(1+x_i)^2} + 16 \sum_{i<j}
\frac{x_i}{1+x_i}\frac{x_j}{1+x_j}\end{aligned}$$ $$\label{dsj}
\Delta_{S J} = d$$
Here, we have defined the dimensionless ratios $$\label{xdef} x_i = \frac{l_i^2}{r_H^2}$$ The expressions are written for the case of maximal possible number of angular momenta $n=4$, but hold also for $n<4$ by setting the appropriate $x_i=0$. One observes that, as seen for the free energy in , the case $ d=4 $ is special since $ \Delta_{T \Omega} = 0 $ identically, implying that the coordinates $ (T,\{\Omega_i\}) $ are not independent.
Since $\Delta_{SJ} \neq 0$, the boundaries of stability in the two ensembles can thus be determined as follows: In the grand canonical ensemble a boundary is reached when $\Delta_{T
\Omega}=0$ and the Hessian of the Gibbs free energy diverges. In the canonical ensemble on the other hand, we have a boundary of stability when $\Delta_{TJ}=0$, in which case the specific heat $C_J$ diverges. More precisely, the boundaries of stability are $n$-dimensional submanifolds in the $(n+1)$-dimensional phase diagram, where one of these two determinants vanish. In the following subsections we study these conditions for the special case of $m \leq n$ equal angular momenta, supplemented with a detailed discussion for the simplest case of one non-zero angular momentum. For the non-dilatonic branes this analysis was performed in Refs. [@Gubser:1998jb; @Cai:1998ji; @Cvetic:1999rb].
It should be remarked that, at first sight, the analysis shows that we do not have any first-order phase transitions, since all first derivatives of the thermodynamic potentials are continuous everywhere. The phase transitions are instead second-order, though this result should be taken with care, since [@Cvetic:1999rb] has given evidence for a first-order phase transition. We will comment on this possibility in the next subsection. Another general result of the analysis is that the boundaries of stability are distinct in the two ensembles that we consider, with a larger region of stability in the canonical ensemble as compared to the grand canonical ensemble, in accordance with standard thermodynamics.
We emphasize here that although we have phrased the analysis in terms of the variables $(r_H,x_i)$, these are in one-to-one correspondence with the thermodynamic extensive variables $(S,J_i)$ through the relations $$\label{sjrel} \sqrt{x_i} = \frac{2\pi J_i}{S} {\ ,\qquad}r_H^{d/2} =
\left( \frac{V_p V(S^{d-1})}{16 \pi G} \right)^{-1} h^{-(d-2)^2/2}
\frac{S}{4\pi}\prod_i \sqrt{1 + x_i}$$ which follow from and . Hence, conditions on $x_i$ can be directly translated into conditions on the ratio $J_i/S$. Alternatively, one may rephrase the stability conditions in terms of the dimensionless ratios \[chidef\] $$\chi_i
= \frac{E^{d/2}}{J_i^{d-2}} = d^{d/2} 2^{4-3d} \left(\frac{V_p
V(S^{d-1})}{16 \pi G}\right)^{-(d-4)/2} h^{-(d-2)^2/2}
\left(
\frac{r_0}{l_i} \right)^{d-2}$$ where $$\left(
\frac{r_0}{l_i} \right)^{d-2} = x_i^{(d-2)/2} \prod_{j=1}^n
(1+x_j)$$ For the case of one angular momentum, $\chi$ is up to a numerical constant the variable used in the D3-brane analysis of Refs. [@Gubser:1998jb; @Cvetic:1999rb].
Grand canonical ensemble \[subsecgrandcan\]
--------------------------------------------
We consider the case of $m \leq n$ equal non-zero angular momentum, so that $ x_i = x=l^2/r_H^2$, $i=1 \ldots m$, in which case the relevant quantity $\Delta_{T \Omega}$ in simplifies to $$\label{com1} \Delta_{T\Omega} = (d-4) \Big(d-2 - (d-2 -2m) x \Big)
(1-x)^{m-1}
$$ We first note that for $d=3$ (which includes the D6-brane) $\det
{\mbox{Hes}}(-F)$ is less than zero for all $x$, and hence corresponds to an unstable situation. The case $d=4$ (which includes the D5 and NS5-brane), for which $T$ and $\Omega_i$, $i=1 \ldots n$ are not independent will be treated separately at the end of this subsection, so in the following we assume $d \geq 5$. In this case, we know that for zero angular momentum, i.e. $x=0$, the branes are stable. We will be concerned only with the first instability that occurs as $x$ is increased, which is hence determined by the first zero of $\Delta_{T \Omega}$.
It follows from that there is a boundary of stability at the value $$\label{critgce}
x_c^{(m)} = \left\{ \begin{array}{ll}
\frac{d-2}{d-4} & {\ ,\qquad}m =1 \\
1 & {\ ,\qquad}m > 1 \end{array}\right.$$ In further detail, stability requires $x \leq x_c^{(m)}$ or equivalently, using this becomes $$J \leq \sqrt{x_c^{(m)}}
\frac{S}{2\pi}$$ One may also calculate from that for $m$ equal angular momenta $$\label{mgentom} \tilde \omega =
\frac{\sqrt{x}}{1+ x/x^{(m)}_\star }$$ where we recall the definitions $\tilde T = 4 \pi T/(d-2) $, $\tilde \omega = \Omega/\tilde T$ and we have defined $$\label{critvalgce}
x^{(m)}_\star = \frac{d-2}{d-2-2m}$$ If for instance $S(T,\{ \Omega_i\})$ is known, Eq. can be viewed as an equation of state using $\sqrt{x} = 2 \pi J/S$. With the critical values of $x$ in the corresponding critical values of $\tilde \omega$ are determined by substitution in so that $$\tilde \omega_c^{(m)} = \left\{ \begin{array}{ll}
\frac{1}{2} \sqrt{\frac{d-2}{d-4}} & {\ ,\qquad}m =1 \\
\frac{d-2}{2(d-2-m)} & {\ ,\qquad}m > 1 \end{array}\right.$$ summarized together with $x_c^{(m)}$ in Table \[tabcrit\]. As seen from the table, the critical values $\tilde \omega_c^{(m)}$ increase as the number of non-zero angular momenta increases, so turning on more equal-valued angular momenta has a stabilizing effect.
[ ]{}
It is also interesting to examine the behavior of the specific heat $C_\Omega $ in , for which we need in addition to ,
\[com2\] $$\Delta_{S\Omega} = \Big( d - (d-4 m ) x \Big)(1-x)^{m-1}$$ $$(d-2-2 \kappa) = \frac{1}{(1+x)^m}[d-2 + (d-2 -2 m) x]$$
which follows from and $\kappa$ in . Besides a diverging specific heat at $x_c^{(m)}$, we see that $C_\Omega$ vanishes, on the other hand, for the values $$\label{specheat} x_0^{(m)}=\frac{d}{d-4m}
{\ ,\qquad}{\ ,\qquad}x_T^{(m)} =- x^{(m)}_\star$$ obtained from the zero of $\Delta_{S \Omega}/\Delta_{T \Omega}$ and the temperature $T$ using [^15].
[ ]{}
In the remainder of this subsection we restrict to the case of one non-zero angular momentum, which by itself exhibits various interesting physical phenomena. The stable region is $x \leq x_c$, with the critical value given by $$x_c \equiv x_c^{(1)} = \frac{d-2}{d-4}$$ or using , $$J \leq \sqrt{\frac{d-2}{d-4}} \frac{S}{2\pi}$$ The stability requirement thus sets an upper bound on the angular momentum, and a phase transition occurs at the critical value of the angular momentum density. In the dual field theory, this corresponds to a critical value of the R-charge density. From eqs. one can also derive the general formulae
$$\label{gentom} \tilde \omega = \frac{\sqrt{x}}{1+ x/x_c }$$
$$\label{trel} \frac{1}{\tilde T} = \frac{\sqrt{1+x}}{1+ x/x_c }
r_H^{(4-d)/2} h^{(d-2)/2}$$
It is not difficult to see that at the boundary of stability $x=x_c $ where the Hessian diverges, the ratio $\tilde \omega$ is maximized[^16], so that the supergravity description sets an upper bound on this quantity, $$\label{crittom} \tilde \omega \leq \tilde \omega_c = \frac{1}{2}
\sqrt{x_c}$$ Moreover, as easily seen from , for each value of $\tilde \omega$ below this maximum there are two values of $x$, one corresponding to a stable and the other to an unstable configuration. In particular the two supergravity descriptions with $(r_H,x)$ and $(\tilde r_H, \tilde x)$ related by $$\label{sheets}
\tilde x = \frac{x_c^2}{x} {\ ,\qquad}\tilde r_H = r_H \left(
\frac{x^2}{x_c^2} \frac{1 + \frac{x_c^2}{x} }{1 + x }
\right)^{1/(d-4)}$$ give the same values of $T$, $\Omega$. The phase diagram therefore consists of two sheets, a stable one and an unstable one.
As an illustration consider a process in which one starts with a non-rotating non-extremal brane at given $r_0 = r_H $ and turn on the angular momentum $l$ adiabatically, while keeping the horizon radius constant. When the critical value $\tilde \omega_c$ is reached the configuration becomes unstable and for the D3-brane two scenarios have been proposed [@Cvetic:1999rb]: D-brane fragmentation, in which the branes fly apart in the transverse dimension, and phase mixing in which angular momentum localizes on the brane. The latter possibility will be briefly discussed below for the general $p$-brane. Note also that for vanishing horizon radius but non-zero angular momentum we have that $x \rightarrow
\infty$, so that the brane is unstable in this situation, and turning on adiabatically the horizon radius would not cure this instability. Note, however, that $C_\Omega$ is positive not only for $x
<x_c$ but also for $x > x_0 \equiv x_0^{(1)}$ in , so the specific heat will be positive in this situation.
----- ----- --------------- -------------------------- -------------------------------------- -------------------- -------------
$d$ $m$ $x_c^{(m)}$ $\tilde \omega_c^{(m)} $ $\hat x_c^{(m)}$ $\tilde j_c^{(m)}$ $x_T^{(m)}$
5 1 3 $\frac{\sqrt{3}}{2}$ $2 + \sqrt{5}$ 7.238
2 1 $\frac{3}{2}$ 3
6 1 2 $\frac{\sqrt{2}}{2}$ $\frac{5+\sqrt{33}}{2} $ 14.12
2 1 1
3 1 2 2
7 1 $\frac{5}{3}$ $\frac{\sqrt{15}}{6}$ $\frac{16+\sqrt{301}}{3} $ 49.59
2 1 $\frac{5}{6}$ $\frac{17}{5}+\frac{2}{5}\sqrt{91}$ 165.5
3 1 $\frac{5}{4}$ 5
8 1 $\frac{3}{2}$ $\frac{\sqrt{6}}{4}$
2 1 $\frac{3}{4}$ $3+2\sqrt{3}$ 211.5
3 1 1
4 1 $\frac{3}{2}$ 3
9 1 $\frac{7}{5}$ $\frac{\sqrt{35}}{10}$
2 1 $\frac{7}{10}$ $\frac{11}{3}+\frac{2}{3}\sqrt{39}$ 522.2
3 1 $\frac{7}{8}$ $\frac{32}{7}+\frac{3}{7}\sqrt{141}$ 4589
4 1 $\frac{7}{6}$ 7
----- ----- --------------- -------------------------- -------------------------------------- -------------------- -------------
: Boundaries of stability in the grand canonical ensemble (GCE) and canonical ensemble (CE) for $m \leq n$ equal non-zero angular momenta. The values in the last column give zero temperature. \[tabcrit\]
[ ]{}
It is possible to obtain a closed form expression for the free energy on the two branches $x \leq x_c$ and $x \geq x_c$ respectively. To this end we solve for $x$ yielding the two solutions $$\label{xsol} x_{\pm} = 8 \frac{\tilde \omega_c^4}{\tilde
\omega^2} \left( 1 - \frac{1}{2} \left( \frac{\tilde
\omega}{\tilde \omega_c} \right)^2 \pm \sqrt{ 1- \left(
\frac{\tilde \omega}{\tilde \omega_c} \right)^2 } \right)$$ where we have used the value of $\tilde \omega_c$ in . It is easy to check that the solution $x_-$ has the property that $x_- \rightarrow 0$ when $\omega \rightarrow 0$, whereas the other solution $x_+$ goes to infinity in that limit. Thus, $x_-$ describes the stable branch $0
< x\leq x_c$ and $x_+$ the unstable branch $x > x_c$. To obtain the explicit expression for the free energy we use to express $r_H$ in terms of $T$ and $x$, as well as , which together with the horizon equation implies $F \sim (1+x) r_H^{d-2}$. The resulting free energy for each of the two branches is then $$F_{\pm} = - \frac{V_p V(S^{d-1})}{16\pi G} h^{(d-2)^2/(d-4)}
\frac{d-4}{2} \tilde{T}^{(2d-4)/(d-4)}
(1+x_{\pm})^{2(d-3)/(d-4)} \left( 1 + \frac{x_{\pm} }{x_c}
\right)^{-2\frac{d-2}{d-4}}$$ with $x_{\pm}$ given in . As a check, we note that expanding $F_-$ for small $\tilde \omega$ reproduces the expansion given in , as it should. Differentiating $F_-$ with respect to $T$ gives the entropy $S(T,\Omega)$, so that we can use $\sqrt{x}
= 2 \pi J/S$ in to determine the exact form of the equation of state for one non-zero angular momentum. As a curiosity we also mention the expansion of the free energy $F_+$ on the unstable branch, $$F_+ \sim T^2 \Omega^{4/(d-4)} \left[ 1 + {{{\cal{O}}}} \left(\frac{\Omega}{
T} \right) \right]$$ exhibiting a universal $T^2$ dependence for all branes, but due to the unstable nature of this branch the relevance of this expression is presently unclear.
[ ]{}
In Ref. [@Cvetic:1999rb], it was shown that for the spinning D3-brane there exists a possibility that a mixing of these two phases is thermodynamically favored (maximizing entropy), so that as a consequence of the instability angular momentum is localized on the brane. To carry out this analysis for the general case $d
>4$, one needs to work in the microcanonical ensemble and consider the mixed states determined by . Thus the problem is to maximize the entropy $$S_{\rm av} = \mu S (r_H,x) + (1 - \mu ) S (\tilde r_H, \tilde x)$$ for given energy $E_{\rm av}$ and angular momentum $J_{\rm av}$, subject to the constraints $$E_{\rm av} = \mu E (r_H,x) + (1 - \mu ) E (\tilde r_H, \tilde x)
{\ ,\qquad}J_{\rm av} = \mu J (r_H,x) + (1 - \mu ) J (\tilde r_H, \tilde x)$$ with $(\tilde r_H, \tilde x)$ expressed in $(r_H,x)$ through . We have not carried out this analysis but expect that the features observed for $d=6$ (including a first-order phase transition) in [@Cvetic:1999rb], will persist for the other cases $d
> 4$. We thus expect that there will be a mixed state and first-order phase transition at some critical value of $x < x_c$.
[ ]{}
As pointed out before, the case $d=4$ needs a special treatment, since from we have that the free energy vanishes, so that $d F=0$. This is due to the fact that the $n+1$ variables $(T,\{\Omega_i \})$ are not independent anymore. Indeed, for one non-zero angular momentum we read off from that $$\label{curve} ( 2 \pi T)^2 + \Omega^2 = h^{-2}$$ which characterizes the phase space. For zero angular momentum we recover the known fact that the temperature is constant for the NS5 and D5-branes. As a further check, using also $\Omega/T = (2 \pi)^2 J/S$ in this case, the curve implies that $S d T + J d \Omega =0$, in accord with the thermodynamic relation with $dF=0$.
Canonical ensemble \[subseccan\]
---------------------------------
We also discuss the canonical ensemble for $m$ non-zero equal angular momenta, for which the relevant quantity $\Delta_{TJ}$ takes the form, $$\begin{aligned}
\label{cj1} \Delta_{TJ} & = \frac{1}{(1+x)^2} & \Big[ (d-2)(d-4)
+ 2\Big( (d-2)(d-4) -( d-8)m \Big) x \nonumber \\
&& + (d-2-2m)(d-4-4m)x^2 \Big]\end{aligned}$$ and we recall that the specific heat $C_J$ also vanishes at the zeroes of the temperature, i.e. $x_T^{(m)}$ given in . The case $d=4$ is stable for any $m$ and for $d=3$ we find the curious behavior that there is lower bound on $x$ namely $(-4 +\sqrt{21})/5$, so that e.g. the non-rotating D6-brane is unstable, but becomes stable in the canonical ensemble when the angular momentum is large enough. In the following we will restrict again to $d \geq 5$.
The positive solutions of the quadratic equation are listed as $\hat x_c^{(m)}$ in Table \[tabcrit\], and correspond to the boundaries of stability, with the property that for $x\leq
\hat x_c^{(m)}$ the branes are stable. From the table we infer a number of observations: For one non-zero angular momentum the branes with $d=8,9$ are stable for any value of $x$, but when more angular momenta are switched on a boundary of stability emerges. Moreover, for maximal number of non-zero angular momenta all branes are stable.
To further examine the boundary of stability we use to construct the dimensionless ratio
$$\begin{aligned}
&\tilde j = \frac{J^{d-4}}{\tilde T^d \zeta_d} = (2\sqrt{x})^{d-4}
(1+x)^{d-2m} \Big(1+x/x^{(m)}_\star \Big)^{-d} \label{jtrat}
\\ &\zeta_d \equiv \Big(\frac{V_p V(S^{d-1}) }{16 \pi G}
\Big)^{d-4} h^{(d-2)^2} &\end{aligned}$$
where $x^{(m)}_\star$ is defined in . The numerical values of the relevant ratio $\tilde j_c $ on the boundary are also listed in the Table \[tabcrit\]. Note that, in analogy with the quantity $\tilde \omega$ relevant for the grand canonical ensemble, in this case the boundary of stability occurs also precisely at the maximum of the ratio $\tilde j$ in . One can also easily obtain the corresponding critical values of $\tilde \omega$ using and the critical values $\hat x_c^{(m)}$. In parallel with the grand canonical ensemble the phase diagram for the cases with a boundary of stability consists again of a stable and unstable sheet. It would be interesting to examine the possibility of phase mixing along the lines described in the previous subsection.
Critical exponents \[subseccritexp\]
-------------------------------------
We conclude this section with a general analysis of the critical exponents in both the ensembles for the case of one non-zero angular momentum. To this end, we note that besides the specific heats and , one also has the response functions
$$\chi_T = \Big( \frac{\partial J}{\partial \Omega} \Big)_T =
\frac{V_p V(S^{d-1}) h^{d-2} }{8 \pi G} r_H^2 (1+x)^2
\frac{\Delta_{TJ}}{\Delta_{T\Omega}}$$
$$\alpha_\Omega = \Big( \frac{\partial J}{\partial T} \Big)_\Omega =
\frac{V_p V(S^{d-1}) h^{d-2} }{G} r_H^2 \sqrt{x} \
\frac{2-(d-4)x }{\Delta_{T\Omega}}$$
$$\alpha_J = \Big( \frac{\partial \Omega}{\partial T} \Big)_J =
- \frac{8 \pi \sqrt{x} } { (1+x)^2} \frac{2-(d-4)x }{\Delta_{T J}}$$
where $\chi_T$ is the isothermal capacitance. The following discussion pertains to the cases in which a boundary of stability was found in the one angular momentum case, i.e. $d \geq 5$ in the grand canonical ensemble, and $d=3,5,6,7$ in the canonical ensemble.
Starting with the grand canonical ensemble, we consider a point $(T_c,\Omega_c)$ on the boundary of stability[^17]. Following a similar analysis as in [@Cai:1998ji], we show that this point behaves as a critical point in ordinary thermodynamics. The stable region has $T \geq T_c$ and $\Omega \leq \Omega_c$, so we define the quantities $$\epsilon_T = \frac{T-T_c}{T_c},\ \ \epsilon_\Omega =
\frac{\Omega_c-\Omega}{\Omega_c}$$ and consider a function $f(T,\Omega)$ near the point $
(T_c,\Omega_c) $. The critical exponents $n_T$ and $n_\Omega$ for $f(T,\Omega)$ are then defined as
$$\label{ntdef}
n_T = - \lim_{\epsilon_T \rightarrow 0} \frac{\ln
f}{\ln \epsilon_T} \Big|_{\epsilon_\Omega=0} = - \lim_{\epsilon_T
\rightarrow 0} \frac{d \ln f|_{\epsilon_\Omega=0}}{d \ln
\epsilon_T}$$
$$n_\Omega = - \lim_{\epsilon_\Omega \rightarrow 0} \frac{\ln f}{\ln
\epsilon_\Omega} \Big|_{\epsilon_T=0} = - \lim_{\epsilon_\Omega
\rightarrow 0} \frac{d \ln f|_{\epsilon_T=0}}{d \ln
\epsilon_\Omega}$$
We assume that the function satisfies $$\label{funcform}
f(T,\Omega_c) = \frac{g(T)}{h_T (x)} {\ ,\qquad}f(T_c,\Omega) = \frac{\tilde g(\Omega )}{h_\Omega (x)}$$ where $g(T_c)$ and $\tilde g(\Omega_c)$ are finite and different from zero and both $h_T$ and $h_\Omega$ satisfy $ h |_{x=x_c}=0 $ and $ \frac{dh}{dx} |_{x=x_c} \neq 0 $. This is indeed true for the response functions $C_\Omega$, $\chi_T$, $\alpha_\Omega$ and the quantities $(S-
S_c)^{-1}$ and $(J-J_c)^{-1}$.
We first approach the critical point by varying the temperature, and hence put $ \Omega = \Omega_c $. Using (\[gentom\]) and (\[crittom\]) one obtains $$\epsilon_T (x) = \frac{1}{2}
\sqrt{ \frac{x_c}{x} } \left(1+\frac{x}{x_c} \right) -1$$ and substituting in one finds $$\begin{aligned}
n_T &=& - \lim_{\epsilon_T \rightarrow 0} \frac{d \ln
f|_{\epsilon_T=0}}{d \ln \epsilon_T} = - \lim_{\epsilon_T
\rightarrow 0} \frac{1}{f} \frac{d f}{d \ln \epsilon_T} = -
\lim_{x \rightarrow x_c} \frac{\epsilon_T}{f} \frac{d f}{d x}
\Big( \frac{d \epsilon_T}{dx} \Big)^{-1} {\nonumber}\\ &=& - \lim_{x
\rightarrow x_c} \frac{\epsilon_T}{h_T} \frac{d h_T}{d x} \Big(
\frac{d \epsilon_T}{dx} \Big)^{-1} = \frac{1}{2}\end{aligned}$$ Here, the last step follows from $ h_T |_{x=x_c} = 0$, $ \frac{dh_T}{dx} |_{x=x_c} \neq 0$, $ \frac{d\epsilon_T}{dx} |_{x=x_c} = 0$ and $ \frac{d^2
\epsilon_T}{dx^2} |_{x=x_c} \neq 0$. On the other hand, approaching the critical line by varying $\Omega$, we need to put $ T=T_c $ and have $$\epsilon_\Omega (x) = 1- 2 \sqrt{ \frac{x}{x_c} } \frac{1}{ 1 + x/x_c}$$ Since $ \frac{d\epsilon_\Omega}{dx} |_{x=x_c} = 0$ and $
\frac{d^2 \epsilon_\Omega}{dx^2} |_{x=x_c} \neq 0$ we also find that $ n_\Omega = \frac{1}{2} $.
Since each of the functions $C_\Omega$, $\chi_T$, $\alpha_\Omega$, $(S-S_c)^{-1}$ and $(J-J_c)^{-1}$ is of the form (\[funcform\]), one immediately concludes that for each of these quantities the critical exponents is equal to $\frac{1}{2}$. The common value $\frac{1}{2}$, which was earlier found [@Cai:1998ji] for the non-dilatonic branes[^18], apparently persists and this value has been shown to be in agreement with scaling laws in statistical physics [@Cai:1998ji].
The critical analysis in the canonical ensemble proceeds along the same lines. In this case we consider a point $(T_c,J_c)$ on the boundary of stability. Repeating the analysis above essentially with the replacement $\Omega \rightarrow J$, and using the fact that $C_J$, $\alpha_J$, $(S-S_c)^{-1}$ and $(\Omega-\Omega_c)^{-1}$ are all of the form (\[funcform\]) (with $\Omega \rightarrow J$), it follows that also in this case all critical exponents are $
\frac{1}{2} $.
Field theory analysis \[secfieldtheory\]
=========================================
In this section we consider the quantum field theories living on the D and M-branes in the limit where they are free field theories. Using the ideal gas approximation we compute in Section \[subsecweakfe\] the free energies with non-zero R-voltage under the R-symmetry. We do this to compare the thermodynamic behaviour in the weak coupling limit with the strong coupling limit. In Section \[subsecinterpo\] we discuss the interpolation between weak and strong coupling, while in Section \[subsecweaktherm\] we find the stability behaviour at weak coupling and compare this to the strong coupling limit.
The free energy for weakly coupled field theory \[subsecweakfe\]
-----------------------------------------------------------------
In this section we calculate the free energies for the extremely weakly coupled limit of the dual field theories extending the regularization method used in [@Gubser:1998jb; @Cvetic:1999rb] for the D3, M2 and M5-brane.
We start by writing the free energy with all R-charge voltages $
\{ \Omega_i \} $ turned off. As we shall see, this depends only on the spatial dimension $p$ of the field theory, and on the number of massless bosonic and fermionic degrees of freedom. In particular, the field theories that we consider have 16 supercharges so that for $N=1$ these theories have 8 bosonic and 8 fermionic degrees of freedom. Using the ideal gas approximation, where particles are assumed to have negligible interaction, we get the free energy[^19] $$F = T V_p \int \frac{d^p q}{(2\pi)^p}
\left[ 8 \log \Big( 1 - e^{-\beta |q|} \Big)
- 8 \log \Big( 1 + e^{-\beta |q|} \Big) \right]
= - k_p V_p T^{p+1}$$ with $$k_p = 2^{4-p} (2 - 2^{-p})
\frac{(p-1)! }{\Gamma(p/2) \pi^{p/2}} \zeta(p+1)$$ where $p \geq 1$.
If we consider non-zero R-voltage, we must replace $ \beta |q| $ with $ \beta |q| + \beta \sum_{i=1}^n \alpha_i \Omega_i $ in the partition function, where $ \vec{\alpha} = {(\alpha_1,\alpha_2,...,\alpha_n)} $ is the $SO(d)$ weight vector of the particle. The resulting free energy is $$\label{free1}
F = T V_p \int \frac{d^p q}{(2\pi)^p} \sum_{\vec{\alpha}} s_{\vec{\alpha}}
\log \left[ 1 - s_{\vec{\alpha}} \exp \Big( - \beta |q|
- \beta \sum_{i=1}^n \alpha_i \Omega_i \Big) \right]$$ where $ \vec{\alpha} $ runs over the 16 different particles and $ s_{\vec{\alpha}} $ is $ +1 $ for bosons and $ -1 $ for fermions. The weights $ \vec{\alpha} $ for the different $ SO(d) $ R-charge groups are listed in Table \[tabweights\]. Note that the weights for $ SO(2n) $ and $ SO(2n+1) $ are the same, and that branes with identical R-symmetry group have the same weights.
$n$ Bosons Fermions
----- ----------------------------------------------------- ---------------------------------------------------------------------
1 $6(0),\ (\pm 1)$ $4(\pm \frac{1}{2})$
2 $4(0),\ (\pm 1 ,0),\ (0,\pm 1)$ $2(\pm \frac{1}{2},\pm \frac{1}{2})$
3 $2(0),\ (\pm 1 ,0,0),\ (0,\pm 1 ,0),\ (0,0, \pm 1)$ $(\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2})$
4 $(\pm 1 ,0,0,0),\ (0,\pm 1 ,0,0),$ $(\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2})$
$(0,0, \pm 1,0),\ (0,0,0, \pm 1)$ number of pluses = even
: Weights for the 8 bosons and 8 fermions in the four possible cases labeled by $n=[\frac{d}{2}]$, corresponding to $3
\leq d \leq 8$. Numbers in front of the weights denote the degeneracy of the spectrum with respect to this weight. In the $n=4$ case the 8 fermions all have same chirality under the $SO(8)$. \[tabweights\]
The integrals for the 8 bosons in are clearly divergent since $ \beta \Omega_i $ is real. In Ref. [@Gubser:1998jb] it was proposed to perform an analytic continuation by considering $ \beta \Omega_i $ to be complex, so that using the free energies can be expressed in terms of polylogarithms, $$\label{free2}
F = - \frac{\Gamma(p)}{2^{p-1} \pi^{p/2} \Gamma(p/2) } V_p T^{p+1}
\sum_{\vec{\alpha}} {\mathrm{Li}}_{p+1} \left[ s_{\vec{\alpha}}
\exp \Big( - \sum_{i=1}^n \alpha_i \omega_i \Big) \right]$$ where $ \omega_i = \beta \Omega_i $. The polylogarithms are not defined for real numbers greater than one, but in Appendix \[apppoly\] we discuss the continuation to this region, along with some general properties of polylogarithms.
Using the exact functions $ B_n(x) $ and $ F_n(x) $ for $ x
\in \mathbb{R} $ of Appendix \[apppoly\], we can in principle write all the free energies for the different $p$-branes exactly. To save space, we restrict ourselves to write the energies with odd $p$ exactly, and write the energies with even $p$ to fourth order in $\omega_i$. In Section \[subsecweaktherm\], however, we use the fact that all the free energies are known to all orders in $ \omega_i $.
For the M-branes, it is believed that $N=1$ corresponds to a free field theory, while for $N>1$ the field theories are interacting. Thus, for a single M2-brane and M5-brane we have
\[weakmfe\] $$\begin{aligned}
F_{\mathrm{M2}} &=& - V_2 T^3 \frac{1}{\pi} \left[ 7\zeta(3) -
\frac{1}{2} \sum_{i=1}^4 \log( \omega_i ) \omega_i^2 + \Big(
\frac{1}{2}\log(2) + \frac{3}{4} \Big) \sum_{i=1}^4 \omega_i^2
\right. {\nonumber}\\ && \left. + \frac{1}{128} \Big( \sum_{i=1}^4
\omega_i^2 \Big)^2 - \frac{10}{1152} \sum_{i=1}^4 \omega_i^4 +
\frac{1}{16} \omega_1 \omega_2 \omega_3 \omega_4 +
{{\cal{O}}}(\omega_i^6) \right]\end{aligned}$$ $$\begin{aligned}
F_{\mathrm{M5}} &=& - V_5 T^6 \left[ \frac{\pi^3}{30}
+ \frac{\pi}{24}(\omega_1^2+\omega_2^2)
+ \frac{1}{96\pi}(\omega_1^2+\omega_2^2)^2
+ \frac{1}{48\pi}(\omega_1^4+\omega_2^4) \right.
{\nonumber}\\ &&
\left.
+ \frac{1}{1152\pi^3} (\omega_1^2+\omega_2^2)^3
-\frac{1}{288\pi^3} (\omega_1^6+\omega_2^6) \right]\end{aligned}$$
The ideal gas approximation is valid for the D-branes when $
\lambda = 0 $. In this limit, the free energies for $N$ D$p$-branes take the form
\[weakdfe\] $$F_{\mathrm{D1}} = - 2 \pi N^2 V_1 T^2$$ $$\begin{aligned}
F_{\mathrm{D2}} &=& - N^2 V_2 T^3 \frac{1}{\pi} \left[ 7\zeta(3) -
\frac{1}{2} \sum_{i=1}^3 \log( \omega_i ) \omega_i^2 + \Big(
\frac{1}{2} \log(2) + \frac{3}{4} \Big) \sum_{i=1}^3 \omega_i^2
\right. {\nonumber}\\ && \left. + \frac{1}{128} \Big( \sum_{i=1}^3
\omega_i^2 \Big)^2 - \frac{10}{1152} \sum_{i=1}^3 \omega_i^4 +
{{\cal{O}}}(\omega_i^6) \right]\end{aligned}$$ $$\label{weakD3free}
F_{\mathrm{D3}} = - N^2 V_3 T^4 \left[ \frac{\pi^2}{6}
+ \frac{1}{4} \sum_{i=1}^3 \omega_i^2
+ \frac{1}{32\pi^2} \Big( \sum_{i=1}^3 \omega_i^2 \Big)^2
- \frac{1}{16\pi^2} \sum_{i=1}^3 \omega_i^4 \right]$$ $$\begin{aligned}
F_{\mathrm{D4}} &=& - N^2 V_4 T^5 \frac{1}{\pi^2} \left[
\frac{93}{8}\zeta(5) + \frac{21}{16}\zeta(3)
(\omega_1^2+\omega_2^2) + \Big( \frac{25}{192} +
\frac{1}{64}\log(2) \Big) (\omega_1^4 + \omega_2^4) \right. {\nonumber}\\
&& \left. + \frac{3}{32}\log(2) \omega_1^2 \omega_2^2 +
\frac{1}{16} \Big( -\log(\omega_1) \omega_1^4 - \log(\omega_2)
\omega_2^4 \Big) + {{\cal{O}}}(\omega_i^6) \right]\end{aligned}$$ $$\begin{aligned}
F_{\mathrm{D5}} &=& - N^2 V_5 T^6 \left[ \frac{\pi^3}{30}
+ \frac{\pi}{24}(\omega_1^2+\omega_2^2)
+ \frac{1}{96\pi}(\omega_1^2+\omega_2^2)^2
+ \frac{1}{48\pi}(\omega_1^4+\omega_2^4) \right.
{\nonumber}\\ &&
\left.
+ \frac{1}{1152\pi^3} (\omega_1^2+\omega_2^2)^3
-\frac{1}{288\pi^3} (\omega_1^6+\omega_2^6) \right]\end{aligned}$$ $$F_{\mathrm{D6}} = - N^2 V_6 T^7 \frac{1}{\pi^3} \left[
\frac{1905}{64}\zeta(7) + \frac{465}{128}\zeta(5) \omega^2 +
\frac{95}{512}\zeta(3) \omega^4 + {{\cal{O}}}(\omega^5) \right]$$
Interpolation between weakly and strongly coupled theories \[subsecinterpo\]
-----------------------------------------------------------------------------
While the expressions represent the free energies of the M-branes for $N=1$, the supergravity results and are the corresponding free energies in the $N \rightarrow \infty$ limit. As discussed in [@Gubser:1998nz] (without R-voltage $\Omega_i$) it is expected that there is a smooth interpolating function $f(N,\{\omega_i\})$ so that the free energy for all $N$ is given by $$\label{interm}
F_N (T,\{\Omega_i\}) = f(N,\{\omega_i\}) F_{N=1} (T,\{\Omega_i\})$$ where $ F_{N=1} (T,\{\Omega_i\}) $ is the free energy for $N=1$, given in . In other words, one can conjecture that if all higher derivative terms in the effective 11-dimensional supergravity action were known, and if one could find the spinning black M-brane solution in this effective action, one could compute the free energies for all $N$, and in particular the free energies for $N=1$. The status of this conjecture, however, is not clear since there could very well be a phase transition obstructing the smooth interpolation to the free theory limit.
Turning to the D-branes, the expressions in represent the free energies for $N \gg 1$ and $ \lambda = 0 $ (which in particular means that $ \lambda \ll r_H^{3-p} $), while the supergravity results are valid for $N \gg
1$ and large ’t Hooft coupling, $ \lambda \gg r_H^{3-p} $ (for the D$p$-branes with $p\neq 3$ there is also has an upper bound on $\lambda$, see Eq. ). Thus, we can consider the free energies of D-branes with $N$ fixed but with $\lambda$ varying between the two limits just described. One can then conjecture that for fixed $N \gg 1$ there exists a smooth interpolation function $f(\lambda,T,\{ \Omega_i \} )$ so that we can write $$\label{interd}
F_\lambda (T,\{\Omega_i\}) = f(\lambda,T,\{ \Omega_i \} )
F_{\lambda=0} (T,\{\Omega_i\})$$ where $ F_{\lambda=0} (T,\{\Omega_i\}) $ is the free energy for $ \lambda = 0 $. Moreover, for the D3-brane the field theory is conformal, so that the function is expected to depend on dimensionless quantities only, i.e. $f(\lambda,\{ \omega_i \})$. The possibility of such a smooth interpolation has previously been discussed in [@Gubser:1998nz] for the D3-brane without the R-voltage. An important first check of this conjecture is the fact that the free energies of the D-branes in the two limits show the same $N^2$ factor in front. This implies that only string loop corrections, which carry factors of $\frac{1}{N^2}$ would modify this behavior, and thus do not have to be considered in the large $N$ limit. Comparing the $\lambda$-dependence on the other hand, one sees that only for the D3-brane the same form is found in the two limits. Again, if one knew the higher derivative terms of the effective action of type II string theory and one could solve the equations of motion for a spinning black D-brane, one could presumably find the smooth interpolation between the two limits of $\lambda$. Again, this conjecture assumes that there is not a phase transition between weak and strong coupling. This assumption has been challenged in [@Li:1998kd] for the D3-brane.
We will return to this issue in Section \[secd3corr\], where we calculate the leading order correction in $\lambda^{-3/2}$ to the D3-brane free energy, due to the $l_s^6 R^4$ term in the type IIB effective action. In Section \[subsecweaktherm\] we take another path and compare the thermodynamic behavior of the field theory in the two limits, including a study of the thermodynamic stability.
Stability behavior at weak coupling \[subsecweaktherm\]
--------------------------------------------------------
We analyze the thermodynamic stability of the weakly coupled QFTs using the free energies and . For simplicity, we restrict ourselves to the case of one non-zero voltage $\Omega_1=\Omega$. As a consequence, branes with equal number of spatial dimensions $p$ have the same thermodynamics, since our analysis is not affected by the overall dependence on $N$.
From the Gibbs free energy $F=F(T,\Omega)$ we compute the heat capacity $$C_\Omega = - T \Big( \frac{\partial^2 F}{\partial T^2} \Big)_\Omega$$ and in the cases we consider, one can check that $ C_\Omega $ is always positive. Instead we can extract the stability behaviour by considering the Hessian matrix $$\label{eqhes}
{\mbox{Hes}}(F) = \left( \begin{array}{cc} \frac{\partial^2 F}{\partial T^2}
& \frac{\partial^2 F}{\partial T \partial \Omega} \\
\frac{\partial^2 F}{\partial T \partial \Omega}
& \frac{\partial^2 F}{\partial \Omega^2} \end{array} \right)$$ of the free energy $F=F(T,\Omega)$. For a stable point, the Hessian should be negative definite. Since $ C_\Omega $ is positive for $ \Omega = 0 $, the boundary of stability is reached when one of the eigenvalues of the Hessian changes sign. Since there are no singularities this occurs when $\det({\mbox{Hes}}(F) ) = 0$, so that the boundary of stability is characterized by a certain critical value of $\omega = \Omega/T$. The results of the analysis for the various values of $p$ are given in Table \[tabweakcrit\]. To analyze the cases of even $p$, we use that we know the free energy to all orders in $\omega$. Thus, one can take an appropriate number of terms in order to ensure that the value of $\omega$ that one finds has the required accuracy.
$p$ $\omega_c$
----- ------------
1 Stable
2 $1.5404$
3 $2.4713$
4 $3.3131$
5 $4.1458$
6 $4.9948$
: The boundary of stability for the various $p$-branes in the weakly coupled field theory limit. \[tabweakcrit\]
Considering Table \[tabweakcrit\] we see that most branes have a boundary of stability at a certain value of $\omega$, as also seen in the stability analysis of Section \[subsecgrandcan\]. We also remark that all the values of $\omega$ in Table \[tabweakcrit\] have the same orders of magnitude as the values in Table \[tabcrit\], and thus it seems plausible that the conjectured interpolation between the two limits of the QFTs described in Section \[subsecweakfe\] should connect the values of $\omega$. Table \[tabcompomega\] summarizes the values of $\omega$ for the weak and strong coupling limits of the D, and M-branes, and for the D2, D3, D4, M2 and M5-branes the critical values of $\omega$ in the two limits are seen to be remarkably close. We also note that $\omega$ is increasing with $p$ in both limits.
Brane $\omega_{\mathrm{weak}}$ $\omega_{\mathrm{strong}}$
------- -------------------------- ----------------------------
D1 Stable $1.2825$
D2 $1.5404$ $1.6223$
D3 $2.4713$ $2.2214$
D4 $3.3131$ $3.6276$
D5 $4.1458$ Not defined
D6 $4.9948$ Unstable
M2 $1.5404$ $1.2825$
M5 $4.1458$ $3.6276$
: Comparison between the boundaries of stability for the type II D$p$-branes n the weak and strong coupling limits of $\lambda$ and for the M-branes in the $N=1$ and $N\rightarrow \infty$ limits. \[tabcompomega\]
The D1-brane and D6-brane, however, are seen from Table \[tabcompomega\] to have qualitatively different stability behaviour in the weak and strong coupling limits, so that in this case the interpolation should somehow create or destroy a boundary of stability at some special point between the two limits. If we for definiteness think about the QFT living on $N$ D1-branes on top of each other, we see that for $\lambda = 0$ it is stable, also with R-voltage turned on, while for $\lambda$ large it should exhibit a boundary of stability. Thus, at some value of $\lambda$ the QFT makes a transition from being completely stable to being potentially unstable. It would be interesting to study how this mechanism works in detail.
For the D5-branes we also have completely different qualitative behaviour. At weak coupling we can vary the thermodynamic parameters $ T $ and $ \{ \Omega_i \} $ freely, while at strong coupling, they are constrained (see Section \[subsecgrandcan\]). Thus, somehow the phase space must expand as one moves away from strong coupling. One could also try to study this phenomenon for non-rotating branes, here the temperature is constant at strong coupling.
If we instead consider the canonical ensemble, with variables $T$ and $J$, one must consider the heat capacity $$C_J = T \Big( \frac{\partial S}{\partial T} \Big)_J
= T \det( {\mbox{Hes}}(F) )
\left[ \Big( \frac{\partial J}{\partial \Omega} \Big)_T \right]^{-1}$$ Thus, we see that $C_J$ is zero whenever $\det({\mbox{Hes}}(F) )$ is, i.e. the canonical ensemble has the same stability behaviour as the grand canonical ensemble. This result should not be surprising since we in fact have used standard statistical physics to derive our thermodynamic relations, so that it is expected that general results, such as the equivalence of ensembles, should hold. Nevertheless, it would be interesting to test this by computing corrections to the stability behaviour from the weakly coupled field theory side, or the supergravity side, since the results of Section \[seccritbeh\] show that the thermodynamic ensembles are not equivalent in the strongly coupled large $N$ limit. If we consider a D-brane, the expectation would be that the boundaries of stability in the two different ensembles start for $\lambda=0$ at the same value, move away from each other as $\lambda$ increases and finally reach the values given in Section \[seccritbeh\].
Finally we note that, with respect to the critical exponents there is also a qualitative difference between the weak and strong coupling limit of the QFT. As discussed in Section \[subseccritexp\] the heat capacities $C_\Omega$ and $C_J$ both behave as $1/\sqrt{T-T_c}$ near the boundary of stability. But, in the weak coupling limit one can easily check that $C_\Omega$ and $C_J$ both behave as $(T-T_c)^\alpha$ with $\alpha \geq 0$, $\alpha$ being different for the two heat capacities. Another way to see this, is to note that while the heat capacities in the strong coupling limit have singularities on the boundary of stability, they are continuous in the weak coupling limit.
In conclusion we see that there are many similarities between the thermodynamics at weak and at strong coupling (or small and large $N$ for the M-branes), but also important qualitative differences that are non-trivial to connect by an interpolation between the two limits. In the next section we make the first step towards a quantitative understanding of this conjectured interpolation for spinning branes.
Free energies from the supergravity action {#secfefromaction}
==========================================
It is well known that one can obtain the free energy of a field theory by Wick rotation and evaluating the path integral partition function with the boundary condition that time is periodic with the inverse temperature as period. For general relativity, this method has been applied in order to compute the free energy of a black hole, but, with some difficulty since one need to think of ways to circumvent the problems of quantizing the gravitational field. In Anti-de Sitter space though, this has proved surprisingly easy since one can just evaluate the action on the background geometry [@Hawking:1983dh]. Moreover, the free energy for the near-horizon limit of the D3, M2 and M5-brane has been reproduced with this method [@Witten:1998zw], which is not surprising since these branes all have Anti-de Sitter space times a sphere as their near-horizon geometry.
In Section \[secfreeenergy\] we will extend these results to spinning branes in the near-horizon limit, showing that we are able to reproduce the free energy found in Section \[nhtherm\]. In Section \[secd3corr\] we compute the first correction in $1/\lambda$ to the free energy of the spinning D3-brane found in Section \[secfreeenergy\]. This will then be used to test the conjecture that there exists a smooth interpolating function between $\lambda=0$ and $\lambda
=\infty$, as discussed in Section \[subsecinterpo\].
Free energies from the low-energy effective action \[secfreeenergy\]
--------------------------------------------------------------------
The Euclidean low-energy supergravity action is $$\label{euclact}
I_E = I_E^{\mathrm{bulk}} + I_E^{\mathrm{bd}}$$ where the bulk term is given by $$\label{bulkaction}
I_E^{\mathrm{bulk}} = - \frac{1}{16\pi G} \int_{\cal{M}} d^D x \sqrt{g}
\Big( R-\frac{1}{2} \partial_\mu
\phi \partial^\mu \phi - \frac{1}{2(p+2)!} e^{a\phi} F_{p+2}^2 \Big)$$ and the boundary term is [@Gibbons:1977ue] $$\label{boundaryterm}
I_E^{\mathrm{bd}} = - \frac{1}{8\pi G}
\int_{\partial {\cal{M}} } d^{D-1} x \sqrt{h} K$$ with $h_{\mu \nu}$ the boundary metric and $K$ the extrinsic curvature. In this section we obtain the free energy as $I_E/ \beta$, where $\beta=1/T$ is the inverse temperature and $I_E$ is the regularized value of the Euclidean action evaluated on the Wick rotated spinning $p$-brane solution in the near-horizon limit. We restrict ourselves to one non-zero angular momentum, since we expect that because the free energy is independent of the angular momentum, more non-zero angular momenta will not alter our final result.
In Appendix \[appeuclnhsol\] we perform a Wick rotation of the near-horizon solution in the presence of one non-zero angular momentum, to obtain the Euclidean spinning brane solution . Starting with the bulk term , we substitute and integrate over the time $\tau$ and the angles to arrive at the following general expression $$\label{angint}
\frac{L (r) }{\beta} =
\frac{V_p V(S^{d-1})}{16 \pi G} \frac{(d-2)^3}{D-2}
r^{d-3}\left[ 1 + \sum_{s=1}^{\infty}
\left(\tilde v_s + \tilde w_s \left( \frac{r_0}{r} \right)^{d-2} \right)
\Big( \frac{{{\tilde{l}}}}{r} \Big)^{2s} \right]$$ where the $\beta=1/T$ factor is the period of the time $\tau$. Here the coefficients $\tilde v$ and $\tilde w$ can be computed in principle through any desired order for any brane solution.
To evaluate the final integral over $r$ we need to introduce a regularization method along the lines of [@Witten:1998zw; @Gubser:1998nz]. In this prescription we first perform the integral up to a cutoff radius $\rma$ and subtract the contribution of the extremal brane with a temperature equal to the original brane at the cutoff radius $\rma$. Thus, integrating the expression from the horizon radius $r_H$ to the cutoff radius $\rma$ we arrive at $$\label{rint}
\left.
\frac{I_E^{\rm bulk} }{\beta} =
\frac{V_p V(S^{d-1}) }{16 \pi G} \frac{(d-2)^2}{D-2}
r^{d-2}\left[ 1 + \sum_{s=1}^{\infty}
\left( v_s + w_s \left( \frac{r_0}{r} \right)^{d-2} \right)
\Big( \frac{{{\tilde{l}}}}{r} \Big)^{2s } \right] \right\vert_{r_H}^{\rma}$$ In particular, for the near-horizon spinning solutions in 10-dimensional type II string theory and 11-dimensional M-theory, one finds by explicit evaluation that the expansion coefficients $v_s, w_s$ can be uniformly written [^20] as
$$v_1 = -\frac{d-2}{d} {\ ,\qquad}v_s =
-\frac{4}{(2s+ d-4 )(2s + d-2) } {\ ,\qquad}s \geq 2$$
$$w_s = -\frac{2}{2s+ d-2 } {\ ,\qquad}s \geq 1$$
with $d$ the transverse dimension. Note that these coefficients satisfy the recursive relations $$\label{recrel}
v_s = w_{s-1} - w_{s} {\ ,\qquad}v_0 = 1 {\ ,\qquad}w_0 = -1$$ the importance of which will become apparent below. Continuing with we find after some algebra that $$\begin{aligned}
\label{exp1}
\frac{I_E^{\rm bulk} }{\beta} &=&
\frac{V_p V(S^{d-1})}{16 \pi G} \frac{(d-2)^2}{D-2}
\left[ \rma^{d-2} - r_0^{d-2}
+ \sum_{s=1}^{\infty} \left(v_s \rma^{d-2}
+ w_s r_0^{d-2} \right) \left( \frac{{{\tilde{l}}}}{\rma} \right)^{2s}
\right.
{\nonumber}\\ &&
\left.
- \sum_{s=0}^{\infty} ( v_s r_H^{d-2} + w_s r_0^{d-2} )
\left( \frac{{{\tilde{l}}}}{r_H} \right)^{2s } \right]\end{aligned}$$ Using the relation to write $r_0^{d-2} = r_H^{d-2} ( 1- ({{\tilde{l}}}/r_H)^2)$ the recursion relation implies that the last term in cancels, giving $$\label{exp2}
\frac{I_E^{\rm bulk} }{\beta}
=\frac{V_p V(S^{d-1})}{16 \pi G} \frac{(d-2)^2}{D-2}
\left[ \rma^{d-2} - r_0^{d-2} + \sum_{s=1}^{\infty}
(v_s \rma^{d-2}
+ w_s r_0^{d-2} ) \left( \frac{{{\tilde{l}}}}{\rma} \right)^{2s} \right]$$ The regularized bulk contribution to the free energy is $$\label{regfree0}
F_{\rm bulk} = \lim_{\rma \rightarrow \infty} \left[
\frac{I_E^{\rm bulk} }{\beta} - \left. \frac{I_E^{\rm bulk}}{\beta'} \right\vert_{r_0=0} \right]
= \lim_{\rma \rightarrow \infty} \left[
\frac{I_E^{\rm bulk} }{\beta} - \left. \frac{\beta}{\beta'}
\frac{I_E^{\rm bulk}}{\beta} \right\vert_{r_0=0} \right]$$ where the ratio of the temperatures is given by $$\label{tempratio}
\frac{\beta}{\beta'} = f^{1/2}\vert_{r=\rma} = 1 - \frac{1}{2}
\left( \frac{r_0}{\rma} \right)^{d-2} + {{{\cal{O}}}} (\rma^{-d} )$$ We note that this expression is meaningful since there is no dependence on the angles to order ${{{\cal{O}}}} (\rma^{-d})$. Substituting and in we then find after taking the limit the result $$\label{regfree}
F_{\rm bulk} =-
\frac{V_p V(S^{d-1})}{16 \pi G} \frac{(d-2)^2}{2(D-2)} r_0^{d-2}$$ To find the boundary contribution we note that there are two boundaries, at $ r=r_H $ and $ r=\rma $ respectively. The boundary action then gives[^21] $$\begin{aligned}
\label{bdact}
\frac{I_E^{\mathrm{bd}}}{\beta}
&=& - \frac{1}{\beta} \frac{1}{8 \pi G} \int_{ \partial {\cal{M}}}
d^{D-1} x \Big(\partial_r (\sqrt{g}\sqrt{g^{rr}})\Big)
\sqrt{g^{rr}}
{\nonumber}\\
&=& \frac{V_p V(S^{d-1}) }{16 \pi G}
\left[ \left( \frac{(p+1)(d-2)}{D-2} - 2(d-1) \right)
( \rma^{d-2} - r_0^{d-2} ) \right.
{\nonumber}\\ && \left. \phantom{ \frac{(p+1)(d-2)}{D-2} }
- (d-2) r_0^{d-2} \right] \left(1 + {{\cal{O}}}(\rma^{-2}) \right) \end{aligned}$$ where we remark that only the boundary at $ r=\rma $ contributes. We note that here the $r_0$-dependent terms are either written explicitly or are of order ${{\cal{O}}}(\rma^{-2})$. From and one then obtains the regularized boundary contribution to the free energy $$\label{Fbd}
F_{\mathrm{bd}} = \lim_{\rma \rightarrow \infty} \left[
\frac{I_E^{\mathrm{bd}}}{\beta}
- \frac{\beta}{\beta'} \frac{I_E^{\mathrm{bd} }}{\beta}
\Big|_{r_0=0} \right]
= - \frac{V_p V(S^{d-1}) }{16 \pi G}
\left[ \frac{(p+1)(d-2)}{2(D-2)} - 1 \right] r_0^{d-2}$$ which we note vanishes for non-dilatonic branes, as seen using . Adding the two free energy contributions and we get $$F = F_{\mathrm{bulk}} + F_{\mathrm{bd}}
= - \frac{V_p V(S^{d-1}) }{16 \pi G}
\frac{d-4}{2} r_0^{d-2}$$ which precisely reproduces the thermodynamically computed Gibbs free energy . This fact will be used implicitly when we calculate string corrections to the free energy of the spinning D3-brane in Section \[secd3corr\].
Corrections from higher derivative terms \[secd3corr\]
-------------------------------------------------------
In this section we test the conjecture that there exists smooth interpolation functions between strong and weak ’t Hooft coupling $\lambda=g_{\mathrm{YM}}^2 N$ for the D-branes, as discussed in Section \[subsecweakfe\]. The idea is to compute the correction to the free energy from the $ l_s^6 R^4 $ term in type II string theory since this gives us the first correction in $ 1/\lambda $.
We restrict ourselves to the case of the D3-brane, since the constant dilaton for the non-corrected solution makes the computation considerably easier. The extremal non-rotating D3-brane has the geometry $AdS_5 \times S^5$ in the near-horizon limit and the dual field theory is the N=4 $D=4$ SYM [@Maldacena:1997re]. For the spinning D3-brane the dual field theory is N=4 $D=4$ SYM at finite temperature with the R-voltage turned on. The free energy of the spinning D3-brane in the strong and weakly coupled limits has previously been discussed in [@Gubser:1998jb; @Kraus:1998hv; @Cvetic:1999rb].
We furthermore restrict ourselves to one non-zero angular momentum only but the methods we use can easily be extended to more angular momenta. To simplify the computations, we work in the limit $ \omega \ll \pi $ with $ \omega = \Omega / T $. This corresponds to the limit $ l \ll r_0 $. We develop all series in $\omega$ to order $\omega^4$, with the next corrections coming from a $\omega^6$ term. With this, our results are accurate for $\omega < 1 $ up to about $1\%$.
Thus, we will test the interpolation between the free energy $$\label{weakD3fe}
F_{\lambda=0} (T,\Omega) = - N^2 V_3 T^4 \left( \frac{\pi^2}{6}
+ \frac{1}{4} \omega^2 - \frac{1}{32\pi^2} \omega^4 \right)$$ for weak coupling, obtained from , and the free energy $$\label{strongD3fe}
F_{\lambda=\infty} (T,\Omega) = - N^2 V_3 T^4 \left( \frac{\pi^2}{8}
+ \frac{1}{8} \omega^2 + \frac{1}{16\pi^2} \omega^4 + {{\cal{O}}}( \omega^6 )
\right)$$ for strong coupling, obtained from . In this case we can write the interpolation conjecture as $$\label{interD3}
F_\lambda(T,\Omega) = f(\lambda,\omega) F_{\lambda=0}(T,\Omega)$$ where $f=f(\lambda,\omega)$ is the interpolation function. To zeroth order in $ 1/\lambda $ we thus have $$\label{flambdainf}
f(\lambda=\infty,\omega)
= \frac{3}{4} - \frac{3}{8\pi^2} \omega^2 + \frac{69}{64\pi^4}\omega^4
+ {{\cal{O}}}( \omega^6 )$$ Comparing and we see that we should expect $f(\lambda,\omega)$ to be smaller than one, and we also expect it to be decreasing with $\lambda$, for fixed $\omega < 1$, since $$\label{inequal}
-F_{\lambda=\infty} < -F_{\lambda=0} \mbox{ for } \omega < 1$$ i.e. the absolute value of the free energy for $\lambda=\infty$ is less than the one for $\lambda=0$ for $\omega < 1$. In Ref. [@Gubser:1998nz] the interpolation function $f(\lambda,\omega)$ was studied for $\omega=0$, and it was found that $$\label{zerofcomp}
f(\lambda,0) = \frac{3}{4} + \frac{45}{32}
\zeta(3) (2 \lambda)^{-3/2} + \ldots$$ by computation of the correction from the $ l_s^6 R^4 $ term in type IIB string theory. The computation clearly supports the conjecture that there is a monotonous smooth interpolation function, since the $ \lambda^{-3/2} $ correction is positive[^22].
As previously stated, the higher derivative correction term[^23] in the supergravity action for type IIB string theory that we want to consider is the $ l_s^6 R^4 $ term. In the Euclidean case, the term is given in the Einstein frame by $$\label{coract}
\delta I_E = - \frac{1}{16 \pi G} \int d^{10} x \sqrt{g} \, \gamma
e^{-\frac{3}{2} \phi} W$$ with $ \gamma = \frac{1}{8} \zeta (3) l_s^6 $ and $$\begin{aligned}
W &=& C^{\mu_1 \mu_2 \mu_3 \mu_4} C_{\nu_1 \mu_2 \mu_3 \nu_4}
C_{\mu_1}{}^{\nu_2 \nu_3 \nu_1} C^{\nu_4}{}_{\nu_2 \nu_3 \mu_4}
{\nonumber}\\ &&
+ \frac{1}{2}
C^{\mu_1 \mu_2 \mu_3 \mu_4} C_{\nu_1 \nu_4 \mu_3 \mu_4}
C_{\mu_1}{}^{\nu_2 \nu_3 \nu_1} C^{\nu_4}{}_{\nu_2 \nu_3 \mu_2}\end{aligned}$$ where $ C_{\mu \nu \rho \sigma} $ is the Weyl tensor. In the near-horizon limit with $ \ell = l_s \rightarrow
0 $ we have the same term in terms of the rescaled quantities, but with $ \gamma $ rescaled to $$\gamma = \frac{1}{8} \zeta(3)$$ From we furthermore have the relations $$\label{D3scal}
h^4 = 2 \lambda {\ ,\qquad}\frac{V(S^5) h^8}{16 \pi G} =
\frac{N^2}{8\pi^2}$$
String theory admits two different kinds of expansions, the loop expansion in $ g_s $ and the derivative expansion in $\alpha' = l_s^2 $. In particular, for the type IIB $R^4$ term there is also a one-loop term of the form $ g_s^2 l_s^6
R^4 $, and an infinite sum of D-instanton corrections. Through the AdS/CFT correspondence this translates into a $ 1/N $ and $
1/\lambda $ expansion (see e.g. [@Banks:1998nr] which also discusses the instanton corrections). The $ l_s^6 R^4 $ tree-level term becomes then a $ \lambda^{-3/2} R^4 $ term, while the $ g_s^2
l_s^6 R^4 $ one loop term becomes an $ N^{-2} \lambda^{1/2} R^4
$ term. The $ N^{-2} \lambda^{1/2} R^4 $ term is clearly not interesting for this computation, since we want to keep $N$ fixed and large. It is also subleading since we keep $ g_s $ small in the limit we consider.
We now compute the $ 1/\lambda $ correction to the free energy by inserting the non-corrected Wick rotated solution for the Euclidean spinning D3-brane in the near-horizon limit, into the higher derivative term as the background geometry. Substituting the solution we find for the first three terms[^24] in a weak angular momentum expansion $$\begin{gathered}
W = \frac{180}{h^8} \left( \frac{r_0}{r} \right)^{16} \left[ 1 +
\frac{2}{3} \left[ 10 \cos^2 \theta
+ (4 - 5\cos^2 \theta )
\left( \frac{r}{r_0} \right)^{4} \right] \left( \frac{{{\tilde{l}}}}{r} \right)^2
\right.
\\
+ \frac{1}{120} \left[ 3617 \cos^4 \theta
+ ( -3032 \cos^2 \theta + 2288 ) \cos^2 \theta
\left( \frac{r}{r_0} \right)^{4} \right.
\\
\left. \left.
+ ( 512 - 904 \cos^2 \theta + 644 \cos^4 \theta )
\left( \frac{r}{r_0} \right)^{8}
\right]
\left( \frac{{{\tilde{l}}}}{r} \right)^4 + \ldots \right]\end{gathered}$$ while the volume element is $$\sqrt{g} = h^2 r^3 \cos^3 \theta \sin \theta
\cos \psi_1 \sin \psi_1 \left[ 1- \frac{1}{2} \cos^2 \theta
\left( \frac{{{\tilde{l}}}}{r} \right)^2
-\frac{1}{8} \cos^4 \theta \left( \frac{{{\tilde{l}}}}{r} \right)^4 + \ldots \right]$$ Substituting this in , integrating over the angles, the world-volume, the Euclidean time $\tau$ from 0 to $\beta$, and $r$ from $r_H$ to infinity, we arrive at $$\label{corfree}
\delta F = \frac{\delta I_E}{\beta}
= - \frac{V_3 V(S^5)}{16 \pi G}
\frac{\gamma}{h^6}
r_0^4 \left[
15 + \frac{111}{7} \left( \frac{{{\tilde{l}}}}{r_0} \right)^2 +
\frac{5885}{96} \left( \frac{{{\tilde{l}}}}{r_0} \right)^4 + \ldots \right]$$ Here we have also used the expansion for the horizon radius $$r_H = r_0\left[ 1 + \frac{1}{4} \left( \frac{{{\tilde{l}}}}{r_0} \right)^2
+ \frac{1}{32} \left( \frac{{{\tilde{l}}}}{r_0} \right)^4
+ \ldots \right]$$ which follows from with one angular momentum turned on only and the replacement $l = i {{\tilde{l}}}$.
Finally, we write the result in terms of the thermodynamic parameters $T$, $\Omega$ using , which imply,
$$r_0^4 = (T h^2)^4
\left[ 1 + \frac{1}{\pi^2} \omega^2
+ \frac{1}{2\pi^4} \omega^4 + {{\cal{O}}}(\omega^6) \right]$$
$$-\left( \frac{{{\tilde{l}}}}{r_0} \right)^2
= \frac{1}{\pi^2} \omega^2
+ \frac{1}{2\pi^4} \omega^4 + {{\cal{O}}}(\omega^6)$$
We also use the relations that enable to transform to the field theory parameters. Then we obtain the final form of the correction $$\label{deltaF}
\delta F =
- \frac{\zeta(3) \pi^2}{64} (2\lambda)^{-3/2} N^2 V_3 T^4
\left[ 15 - \frac{6}{7\pi^2} \omega^2
+\frac{28151}{672\pi^4} \omega^4 + {{\cal{O}}}(\omega^6) \right]$$ This gives the interpolation function $$\label{generalf}
f(\lambda,\omega)
= \frac{3}{4} - \frac{3}{8\pi^2} \omega^2 + \frac{69}{64\pi^4}\omega^4
+ \frac{\zeta(3)}{64} (2\lambda)^{-3/2}
\left( 90 - \frac{981}{7\pi^2} \omega^2 + \frac{7655}{16\pi^4} \omega^4 \right)
+ \cdots$$ which includes and as special cases. Because of we expect the correction term in to be positive for $ \omega < 1 $ and this is indeed the case. Since the corrections away from $\lambda=\infty$ behave as expected, we consider this as further evidence for the interpolation conjecture in the case of spinning D3-branes,
It is not a priori apparent that the method used to compute gives the full result to first order in $\gamma$, since this semi-classical method does not consider induced perturbations of the geometry. However, we note that for the non-rotating D3-brane case, the perturbed metric induced by the correction term was shown [@Gubser:1998nz; @Pawelczyk:1998pb] to yield the same correction to the free energy as the semiclassical approximation in which the correction is evaluated for the original unperturbed metric[^25]. Thus, it seems that the thermodynamics somehow disregards perturbations of the geometry. We expect this to hold also for non-dilatonic spinning branes. It would be interesting, though technically difficult, to find the actual perturbed metric for the spinning D3-brane and determine whether the result is still given by . Another interesting check on the interpolation function would be to compute the $ 1/\lambda $ correction to the boundary of stability. From the values in Table \[tabcompomega\] we would expect this correction to be positive.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank J. Ambj[ø]{}rn, E. Cheung, P. Damgaard, P. Di Vecchia, E. Kiritsis, F. Larsen, R. Marotta, J. L. Petersen, B. Pioline, K. Savvidis, K. Sfetsos and R. Szabo for useful discussions and correspondence, and we thank J. Correia for useful discussions and early participation on the subjects of Section 2. NO thanks the CERN Theory Division for support and hospitality during completion of this work. This work is supported in part by TMR network ERBFMRXCT96-0045.
Thermodynamics of black p-branes in the near-horizon limit {#appenergy}
==========================================================
In this appendix we consider the thermodynamics, and in particular the energy above extremality in the near-horizon limit of a general non-rotating $p$-brane. We also derive the Smarr formula and check that it is fulfilled. This means that the first law of thermodynamics is obeyed in the near-horizon limit, and that there are no extra thermodynamic parameters related to the charge. We begin by giving a short review of the classification of $p$-branes that preserve a certain fraction of the supersymmetry (in the extremal limit).
A black $p$-brane solution of the action is characterized by a particular value of $a$. If we define $$b \equiv \frac{2(D-2)}{(p+1)(d-2)+\frac{1}{2}a^2(D-2)}$$ then the non-rotating black $p$-brane background takes the form
\[pbrasol\] $$ds^2 = H^{-\frac{d-2}{D-2}b} \Big( - f dt^2 + \sum_{i=1}^p (dy^i)^2 \Big)
+ H^{\frac{p+1}{D-2}b} \Big( f^{-1} dr^2 + r^2 d\Omega_{d-1}^2 \Big)$$ $$e^\phi = H^{\frac{a}{2}b}$$ $$A_{p+1} = (-1)^p \sqrt{b} \coth \alpha \Big( H^{-1} -1 \Big)
dt \wedge dy^1 \wedge dy^2 \wedge \cdots \wedge dy^p$$
with $$H = 1 + \frac{r_0^{d-2} \sinh^2 \alpha}{r^{d-2}}
{\ ,\qquad}f = 1 - \frac{r_0^{d-2}}{r^{d-2}}$$ This $p$-brane solution is a $1/2^b$-BPS state [@Behrndt:1999mk] for $ r_0 = 0 $, so that the spinning solutions discussed in the text correspond to $b=1$. Table \[tabbranes\] lists the most common branes together with the corresponding values of $D$, $a$ and $b$ (a more extensive list with other values of $b$ can be found in Ref. [@Behrndt:1999mk]).
Brane Theory $D$ $a$ $b$
------- --------------- ----- ----------- -----
M2 M 11 0 1
M5 M 11 0 1
D$p$ string 10 $(3-p)/2$ 1
NS1 string 10 $-1$ 1
NS5 string 10 1 1
d$p$ little string 6 $1-p$ 2
: The characteristic numbers $a$ and $b$ for branes with $b$ equal to 1 or 2.\[tabbranes\]
The thermodynamic quantities of the background are
\[thermgenb\] $$M = \frac{V_p V(S^{d-1})}{16 \pi G} r_0^{d-2}
\Big[ d-1 + b (d-2) \sinh^2 \alpha \Big]$$ $$\label{tseqs}
T = \frac{d-2}{4\pi r_0} (\cosh \alpha)^{-b}
{\ ,\qquad}S = \frac{V_p V(S^{d-1})}{4G} r_0^{d-1} ( \cosh \alpha )^{b}$$ $$\mu = \tanh \alpha
{\ ,\qquad}Q = \frac{V_p V(S^{d-1})}{16 \pi G} b (d-2) r_0^{d-2}
\cosh \alpha \sinh \alpha$$
satisfying the Smarr formula $$\label{smarrr}
(d-2) M = (d-1) TS + (d-2) \mu Q$$ and the first law of thermodynamics $$dM = T dS + \mu dQ {\ ,\qquad}M = M(S,Q)$$ The energy above extremality is $$\label{inten}
E = M - Q = \frac{V_p V(S^{d-1})}{16 \pi G} r_0^{d-2}
\Big[ d-1 + b (d-2) (\sinh^2 \alpha - \cosh \alpha \sinh \alpha ) \Big]$$ The near-horizon limit is defined via the rescaling
\[rescal2\] $$r = \frac{{r_{\mathrm{old}}}}{\ell^2} {\ ,\qquad}r_0 = \frac{{(r_0)_{\mathrm{old}}}}{\ell^2} {\ ,\qquad}h^{d-2} = \frac{{h_{\mathrm{old}}}^{d-2}}{\ell^{2d-4-\frac{4}{b}}}$$ $$ds^2 = \frac{{(ds^2)_{\mathrm{old}}}}{\ell^{4(d-2)b/(D-2)}} {\ ,\qquad}e^{\phi} = \ell^{2a} e^{\phi_{\rm old}} {\ ,\qquad}A = \frac{A_{\rm old}}{\ell^4} {\ ,\qquad}G = \frac{{G_{\mathrm{old}}}}{\ell^{2(d-2)}}$$
and taking $ \ell \rightarrow 0 $, keeping the old quantities fixed. Note that the near-horizon limit depends on the fraction of supersymmetries that is preserved and that we recover for $b =1$ . Using this limit in and we obtain
\[tseqsnh\] $$T = \frac{d-2}{4\pi r_0} \Big( \frac{r_0}{h} \Big)^{\frac{d-2}{2} b}
{\ ,\qquad}S = \frac{V_p V(S^{d-1})}{4G} r_0^{d-1}
\Big( \frac{h}{r_0} \Big)^{\frac{d-2}{2} b}$$ $$\label{intennh}
E = \frac{V_p V(S^{d-1})}{16 \pi G}
\Big[ d-1 - \frac{b}{2} (d-2) \Big]r_0^{d-2}$$
To derive the Smarr formula we consider the canonical rescaling $$h \rightarrow h
{\ ,\qquad}r_0 \rightarrow \lambda r_0$$ under which we have the transformation $$E \rightarrow \lambda^{d-2}
{\ ,\qquad}S \rightarrow \lambda^{(1-\frac{1}{2}b) d +b-1 } S$$ This gives the Smarr formula $$\label{smarrrr}
(d-2) E = \Big( d-1 - \frac{b}{2} (d-2) \Big) TS$$ corresponding to the first law of thermodynamics $$dE = T dS {\ ,\qquad}E=E(S)$$ The Smarr formula is indeed satisfied with . We note that is qualitatively different from the asymptotically-flat black brane Smarr formula since it exhibits a dependence on the amount of unbroken supersymmetry that the brane has in the extremal limit.
The free energy is given by $$F = E - TS
= - \frac{\frac{b}{2}(d-2) - 1}{d-1 - \frac{b}{2} (d-2)} E
= - \frac{V_p V(S^{d-1})}{16 \pi G} \Big[ \frac{b}{2} (d-2) -1 \Big] r_0^{d-2}$$ In particular, for $b=1$ we have $$\label{eandfb1}
E = \frac{V_p V(S^{d-1})}{16 \pi G} \frac{d}{2} r_0^{d-2}
{\ ,\qquad}F = - \frac{d-4}{d} E
= - \frac{V_p V(S^{d-1})}{16 \pi G} \frac{d-4}{2} r_0^{d-2}$$ while for $b=2$ the result reads $$E = \frac{V_p V(S^{d-1})}{16 \pi G} r_0^{d-2}
{\ ,\qquad}F = - (d-3) E
= -\frac{V_p V(S^{d-1})}{16 \pi G} (d-3) r_0^{d-2}$$
Spheroidal coordinates \[sphcoor\]
==================================
In this appendix we define the spheroidal coordinates for a $d$-dimensional Euclidean space with Cartesian coordinates $x^a,
a=1 \ldots d$. We define the metric $$(ds_d)^2 = \sum_{a=1}^d (dx^a)^2$$ and treat the cases $d$ even and odd separately.
[ ]{}
The spheroidal coordinates are the “radius” $r$ and the angles $ \theta , \psi_1 ,..., \psi_{n-2} ,$, $ \phi_1 ,..., \phi_n $. Define the quantities $$\begin{aligned}
&& \mu_1 = \sin \theta,\ \ \mu_2 = \cos \theta \sin \psi_1,\ \
\mu_3 = \cos \theta \cos \psi_1 \sin \psi_2 {\ ,\qquad}\ldots {\ ,\qquad}{\nonumber}\\
&& \mu_{n-1} = \cos \theta \cos \psi_1 \cdots \cos \psi_{n-3} \sin
\psi_{n-2},\ \ \mu_n = \cos \theta \cos \psi_1 \cdots \cos
\psi_{n-2}\end{aligned}$$ which satisfy $$\sum_{i=1}^n \mu_i^2 =1$$ The spheroidal coordinates are then defined by $$x^{2i-1} = \sqrt{r^2+l_i^2} \mu_i \cos \phi_i {\ ,\qquad}{\ ,\qquad}x^{2i} =
\sqrt{r^2+l_i^2} \mu_i \sin \phi_i {\ ,\qquad}i = 1 \ldots n$$ The coordinates $ \phi_1,...,\phi_n $ are the rotation angles and $ l_1,...,l_n $ correspond to the angular momenta in these angles. We have $$\sum_{a=1}^d (x^a)^2 = r^2 + \sum_{i=1}^n l_i^2 \mu_i^2$$ and the ranges of the angles are given by $$0 \leq \theta , \psi_1 ,..., \psi_{n-2} \leq \frac{\pi}{2},\ \ 0
\leq \phi_1 ,..., \phi_n \leq 2\pi$$
[ ]{}
The spheroidal coordinates are the “radius” $r$ and the angles $ \theta , \psi_1 ,..., \psi_{n-1}$, $\phi_1 ,..., \phi_n $. Define the quantities $$\begin{aligned}
&& \mu_1 = \sin \theta,\ \ \mu_2 = \cos \theta \sin \psi_1,\ \
\mu_3 = \cos \theta \cos \psi_1 \sin \psi_2 {\ ,\qquad}\ldots {\ ,\qquad}{\nonumber}\\
&& \mu_n = \cos \theta \cos \psi_1 \cdots \cos \psi_{n-2} \sin
\psi_{n-1},\ \ \mu_{n+1} = \cos \theta \cos \psi_1 \cdots \cos
\psi_{n-1}\end{aligned}$$ which satisfy $$\sum_{i=1}^{n+1} \mu_i^2 =1$$ The spheroidal coordinates are then defined by
$$x^{2i-1} = \sqrt{r^2+l_i^2} \mu_i \cos \phi_i {\ ,\qquad}x^{2i} =
\sqrt{r^2+l_i^2} \mu_i \sin \phi_i {\ ,\qquad}i = 1 \ldots n$$
$$x^d = r \mu_{n+1}$$
The coordinates $ \phi_1,...,\phi_n $ are the rotation angles and $ l_1,...,l_n $ correspond to the angular momenta in these angles. In this case, we have $$\sum_{a=1}^d (x^a)^2 = r^2 + \sum_{i=1}^n l_i^2 \mu_i^2$$ The ranges of the angles are, for $ d \geq 5 $, given by $$0 \leq \theta , \psi_1 ,..., \psi_{n-2} \leq \frac{\pi}{2},\ \ 0
\leq \psi_{n-1} \leq \pi,\ \ 0 \leq \phi_1 ,..., \phi_n \leq 2\pi$$ and for $d=3$ we have $$0 \leq \theta \leq \pi,\ \ 0 \leq \phi_1 \leq 2\pi$$ [ ]{}
The metric in spheroidal coordinates takes the form $$\sum_{a=1}^d (dx^a)^2 = K_d dr^2 + \Lambda_{\alpha \beta}
d\eta^\alpha d\eta^\beta$$ where $\eta^{\alpha}$ denote the set of angular coordinates. For the radial coordinate the metric component takes the form $$g_{rr} = K_d (r,\theta,\psi_1,...,\psi_{d-n-2}) \equiv \left\{
\begin{array}{ll} \sum_{i=1}^n \mu_i^2 \Big( 1+\frac{l_i^2}{r^2}
\Big)^{-1}
& {\ ,\qquad}d=2n \\
\sum_{i=1}^n \mu_i^2 \Big( 1+\frac{l_i^2}{r^2} \Big)^{-1} +
\mu_{n+1}^2
& {\ ,\qquad}d=2n+1
\end{array} \right.$$ and the general form of the remaining non-zero components is [$$\begin{aligned}
g_{\theta \theta} &=& r^2 + l_1^2 \cos^2 \theta + \tan^2 \theta
\Big( \mu_2^2 l_2^2 + \cdots + \mu_n^2 l_n^2 \Big)
\\
g_{\psi_1 \psi_1} &=& \cos^2 \theta \Big( r^2 + l_2^2 \cos^2
\psi_1 \Big) + \tan^2 \psi_1 \Big( \mu_3^2 l_3^2 + \cdots +
\mu_n^2 l_n^2 \Big)
\\
g_{\psi_2 \psi_2} &=& \cos^2 \theta \cos^2 \psi_1 \Big( r^2 +
l_3^2 \cos^2 \psi_2 \Big) + \tan^2 \psi_2 \Big( \mu_4^2 l_4^2 +
\cdots + \mu_n^2 l_n^2 \Big)
\\
g_{\psi_3 \psi_3} &=& \cos^2 \theta \cos^2 \psi_1 \cos^2 \psi_2
\Big( r^2 + l_4^2 \cos^2 \psi_3 \Big) + \tan^2 \psi_3 \Big(
\mu_5^2 l_5^2 + \cdots + \mu_n^2 l_n^2 \Big)
\\
g_{\psi_{n-2} \psi_{n-2}} &=& \cos^2 \theta \cos^2 \psi_1 \cdots
\cos^2 \psi_{n-3} \Big( r^2 + l_{n-1}^2 \cos^2 \psi_{n-2}
+ l_n^2 \sin^2 \psi_{n-2} \Big)
\\
g_{\theta \psi_1} &=& - \tan \theta \cot \psi_1 \mu_2^2 l_2^2 +
\tan \theta \tan \psi_1 \Big( \mu_3^2 l_3^2 + \cdots + \mu_n^2
l_n^2 \Big)
\\
g_{\theta \psi_2} &=& - \tan \theta \cot \psi_2 \mu_3^2 l_3^2 +
\tan \theta \tan \psi_2 \Big( \mu_4^2 l_4^2 + \cdots + \mu_n^2
l_n^2 \Big)
\\
g_{\psi_1 \psi_2} &=& - \tan \psi_1 \cot \psi_2 \mu_3^2 l_3^2 +
\tan \psi_1 \tan \psi_2 \Big( \mu_4^2 l_4^2 + \cdots + \mu_n^2
l_n^2 \Big)
\\
g_{\phi_i \phi_i} &=& \mu_i^2 (r^2 + l_i^2) {\ ,\qquad}i=1 \ldots n \end{aligned}$$ ]{} for both $d=2n$ and $d=2n+1$. As an aid to the reader we list below the angles and explicit expressions for the spheroidal metric when $ 3 \leq d \leq 9$.
$$d=3 \co \theta,\phi_1$$
$${\nonumber}(ds_3)^2 = K_3 dr^2
+ \Big(r^2 + l_1^2 \cos^2 \theta \Big) d\theta^2 + \sin^2 \theta
\Big(r^2 + l_1^2\Big) d\phi_1^2$$
$$d=4 \co \theta,\phi_1,\phi_2$$
[$$\begin{aligned}
&& (ds_4)^2 = K_4 dr^2 + \Big( r^2 + l_1^2 \cos^2 \theta +
l_2^2 \sin^2 \theta \Big) d\theta^2 + \sin^2 \theta \Big(r^2
+l_1^2\Big) d\phi_1^2 \hfill
\\ &&
+ \cos^2 \theta \Big(r^2 +l_2^2\Big) d\phi_2^2 \end{aligned}$$ ]{} $$d=5 \co \theta, \psi_1,\phi_1,\phi_2$$ [$$\begin{aligned}
&& (ds_5)^2 = K_5 dr^2 + \Big( r^2 + l_1^2 \cos^2 \theta +
l_2^2 \sin^2 \theta \sin^2 \psi_1 \Big) d\theta^2 + \cos^2 \theta
\Big( r^2 + l_2^2 \cos^2 \psi_1 \Big) d\psi_1^2
\\ &&
- 2 l_2^2 \cos \theta \sin \theta \cos \psi_1 \sin \psi_1 d\theta
d\psi_1 + \sin^2 \theta \Big(r^2+l_1^2\Big) d\phi_1^2 + \cos^2
\theta \sin^2 \psi_1 \Big(r^2+l_2^2\Big) d\phi_2^2 \end{aligned}$$ ]{} $$d=6 \co \theta, \psi_1,\phi_1,\phi_2,\phi_3$$ [$$\begin{aligned}
&& (ds_6)^2 = K_6 dr^2 + \Big( r^2 + l_1^2 \cos^2 \theta +
l_2^2 \sin^2 \theta \sin^2 \psi_1 + l_3^2 \sin^2 \theta \cos^2
\psi_1 \Big) d\theta^2
\\ &&
+ \cos^2 \theta \Big( r^2 + l_2^2 \cos^2 \psi_1 + l_3^2 \sin^2
\psi_1 \Big) d\psi_1^2 + 2 \cos \theta \sin \theta \cos \psi_1
\sin \psi_1 \Big( -l_2^2 + l_3^2 \Big) d\theta d\psi_1
\\ &&
+ \sin^2 \theta \Big(r^2 +l_1^2\Big) d\phi_1^2 + \cos^2 \theta
\sin^2 \psi_1 \Big(r^2 +l_2^2\Big) d\phi_2^2 + \cos^2 \theta
\cos^2 \psi_1 \Big(r^2 +l_3^2\Big) d\phi_3^2 \end{aligned}$$ ]{} $$d=7 \co \theta,\psi_1,\psi_2,\phi_1,\phi_2,\phi_3$$ [$$\begin{aligned}
&& (ds_7)^2 = K_7 dr^2 + \Big( r^2 + l_1^2 \cos^2 \theta +
l_2^2 \sin^2 \theta \sin^2 \psi_1 + l_3^2 \sin^2 \theta \cos^2
\psi_1 \sin^2 \psi_2 \Big) d\theta^2
\\ &&
+ \cos^2 \theta \Big( r^2 + l_2^2 \cos^2 \psi_1 + l_3^2 \sin^2
\theta \sin^2 \psi_2 \Big) d\psi_1^2 + \cos^2 \theta \cos^2 \psi_1
\Big( r^2 + l_3^2 \cos^2 \psi_2 \Big) d\psi_2^2
\\ &&
+ 2\cos \theta \sin \theta \cos \psi_1 \sin \psi_1 \Big( -l_2^2 +
l_3^2 \sin^2 \psi_2 \Big) d\theta d\psi_1
\\ &&
- 2 \cos \theta \sin \theta \cos^2 \psi_1 \cos \psi_2 \sin \psi_2
l_3^2 d\theta d\psi_2
\\ &&
- 2 l_3^2 \cos \psi_1 \sin \psi_1 \cos^2 \psi_1 \cos \psi_2 \sin
\psi_2 d\psi_1 d\psi_2 + \sin^2 \theta \Big(r^2 +l_1^2\Big)
d\phi_1^2
\\ &&
+ \cos^2 \theta \sin^2 \psi_1 \Big(r^2 +l_2^2\Big) d\phi_2^2 +
\cos^2 \theta \cos^2 \psi_1 \sin^2 \psi_2 \Big(r^2 +l_3^2\Big)
d\phi_3^2 \end{aligned}$$ ]{} $$d=8 \co \theta,\psi_1,\psi_2,\phi_1,\phi_2,\phi_3,\phi_4$$ [$$\begin{aligned}
&& (ds_8)^2 = K_8 dr^2 + \Big( r^2 + l_1^2 \cos^2 \theta +
l_2^2 \sin^2 \theta \sin^2 \psi_1 + l_3^2 \sin^2 \theta \cos^2
\psi_1 \sin^2 \psi_2
\\ &&
+ l_4^2 \sin^2 \theta \cos^2 \psi_1 \cos^2 \psi_2 \Big) d\theta^2
\\ &&
+ \cos^2 \theta \Big( r^2 + l_2^2 \cos^2 \psi_1 + l_3^2 \sin^2
\psi_1 \sin^2 \psi_2 + l_4^2 \sin^2 \psi_1 \cos^2 \psi_2 \Big)
d\psi_1^2
\\ &&
+ \cos^2 \theta \cos^2 \psi_1 \Big( r^2 + l_3^2 \cos^2 \psi_2 +
l_4^2 \sin^2 \psi_2 \Big) d\psi_2^2
\\ &&
+ 2 \cos \theta \sin \theta \cos \psi_1 \sin \psi_1 \Big( -l_2^2 +
l_3^2 \sin^2 \psi_2 + l_4^2 \cos^2 \psi_2 \Big) d\theta d\psi_1
\\ &&
+ 2 \cos \theta \sin \theta \cos^2 \psi_1 \cos \psi_2 \sin \psi_2
\Big( -l_3^2 + l_4^2 \Big) d\theta d\psi_2
\\ &&
+ 2 \cos^2 \theta \cos \psi_1 \sin \psi_1 \cos \psi_2 \sin \psi_2
\Big( - l_3^2 + l_4^2 \Big) d\psi_1 d\psi_2 + \sin^2 \theta
\Big(r^2 +l_1^2 \Big) d\phi_1^2
\\ &&
+ \cos^2 \theta \sin^2 \psi_1 \Big(r^2 +l_2^2 \Big) d\phi_2^2 +
\cos^2 \theta \cos^2 \psi_1 \sin^2 \psi_2 \Big(r^2 +l_3^2 \Big)
d\phi_3^2
\\ &&
+ \cos^2 \theta \cos^2 \psi_1 \cos^2 \psi_2 \Big(r^2 +l_4^2 \Big)
d\phi_4^2 \end{aligned}$$ ]{} $$d=9 \co \theta,\psi_1,\psi_2,\psi_3,\phi_1,\phi_2,\phi_3,\phi_4$$ [$$\begin{aligned}
&& (ds_9)^2 = K_9 dr^2 + \Big( r^2 + l_1^2 \cos^2 \theta +
l_2^2 \sin^2 \theta \sin^2 \psi_1 + l_3^2 \sin^2 \theta \cos^2
\psi_1 \sin^2 \psi_2
\\ &&
+ l_4^2 \sin^2 \theta \cos^2 \psi_1 \cos^2 \psi_2 \sin^2 \psi_3
\Big) d \theta^2 + \cos^2 \theta \Big( r^2 + l_2^2 \cos^2 \psi_1 +
l_3^2 \sin^2 \psi_1 \sin^2 \psi_2
\\ &&
+ l_4^2 \sin^2 \psi_1 \cos^2 \psi_2 \sin^2 \psi_3 \Big) d \psi_1^2
+ \cos^2 \theta \cos^2 \psi_1 \Big( r^2 + l_3^2 \cos^2 \psi_2 +
l_4^2 \sin^2 \psi_2 \sin^2 \psi_3 \Big) d\psi_2^2
\\ &&
+ \cos^2 \theta \cos^2 \psi_1 \cos^2 \psi_2 \Big( r^2 + l_4^2
\cos^2 \psi_3 \Big) d\psi_3^2
\\ &&
+ 2 \cos \theta \sin \theta \cos \psi_1 \sin \psi_1 \Big( -l_2^2 +
l_3^2 \sin^2 \psi_2 + l_4^2 \cos^2 \psi_2 \sin^2 \psi_3 \Big) d
\theta d \psi_1
\\ &&
+ 2 \cos \theta \sin \theta \cos^2 \psi_1 \cos \psi_2 \sin \psi_2
\Big( -l_3^2 + l_4^2 \sin^2 \psi_3 \Big) d \theta d \psi_2
\\ &&
- 2 l_4^2 \cos \theta \sin \theta \cos^2 \psi_1 \cos^2 \psi_2 \cos
\psi_3 \sin \psi_3 d \theta d \psi_3
\\ &&
+ 2\cos^2 \theta \cos \psi_1 \sin \psi_1 \cos \psi_2 \sin \psi_2
\Big( -l_3^2 + l_4^2 \sin^2 \psi_3 \Big) d \psi_1 d \psi_2
\\ &&
- 2 l_4^2 \cos^2 \theta \cos^2 \psi_1 \cos \psi_2 \sin \psi_2 \cos
\psi_3 \sin \psi_3 d \psi_2 d \psi_3 + \sin^2 \theta \Big(r^2
+l_1^2 \Big) d\phi_1^2
\\ &&
+ \cos^2 \theta \sin^2 \psi_1 \Big(r^2 +l_2^2 \Big) d\phi_2^2 +
\cos^2 \theta \cos^2 \psi_1 \sin^2 \psi_2 \Big(r^2 +l_3^2 \Big)
d\phi_3^2
\\ &&
+ \cos^2 \theta \cos^2 \psi_1 \cos^2 \psi_2 \sin^2 \psi_3 \Big(r^2
+l_4^2 \Big) d\phi_4^2 \end{aligned}$$ ]{}
[ ]{}
As an aid to the reader we also give the explicit expressions for the spheroidal metric when only one angular momentum $ l_1 = l $ is non-zero,
$$(ds_3)^2 = \Big( 1 - \frac{l^2 \sin^2 \theta }{l^2 + r^2 } \Big)
dr^2 + \Big(r^2 + l^2 \cos^2 \theta \Big) d\theta^2 + \sin^2
\theta \Big(r^2 + l^2\Big) d\phi_1^2$$
$$(ds_4)^2 = \Big( 1 - \frac{l^2 \sin^2 \theta }{l^2 + r^2 } \Big)
dr^2 + \Big( r^2 + l^2 \cos^2 \theta \Big) d\theta^2 + \sin^2
\theta \Big(r^2 +l^2\Big) d\phi_1^2 + r^2 \cos^2 \theta d\phi_2^2$$
$$\begin{aligned}
&& (ds_5)^2 = \Big( 1 - \frac{l^2 \sin^2 \theta }{l^2 + r^2 }
\Big) dr^2 + \Big( r^2 + l^2 \cos^2 \theta \Big) d\theta^2 + r^2
\cos^2 \theta d\psi_1^2 + \sin^2 \theta \Big(r^2+l^2\Big)
d\phi_1^2
{\nonumber}\\ &&
+ r^2 \cos^2 \theta \sin^2 \psi_1 d\phi_2^2\end{aligned}$$
$$\begin{aligned}
&& (ds_6)^2 = \Big( 1 - \frac{l^2 \sin^2 \theta }{l^2 + r^2 }
\Big) dr^2 + \Big( r^2 + l^2 \cos^2 \theta \Big) d\theta^2 + r^2
\cos^2 \theta d\psi_1^2 + \sin^2 \theta \Big(r^2 +l^2\Big)
d\phi_1^2 {\nonumber}\\ && + r^2 \cos^2 \theta \sin^2 \psi_1 d\phi_2^2 +
r^2 \cos^2 \theta \cos^2 \psi_1 d\phi_3^2\end{aligned}$$
$$\begin{aligned}
&& (ds_7)^2 = \Big( 1 - \frac{l^2 \sin^2 \theta }{l^2 + r^2
} \Big) dr^2 + \Big( r^2 + l^2 \cos^2 \theta \Big) d\theta^2 + r^2
\cos^2 \theta d\psi_1^2 + r^2 \cos^2 \theta \cos^2 \psi_1
d\psi_2^2 {\nonumber}\\ && + \sin^2 \theta \Big(r^2 +l^2\Big) d\phi_1^2 +
r^2 \cos^2 \theta \sin^2 \psi_1 d\phi_2^2 + r^2 \cos^2 \theta
\cos^2 \psi_1 \sin^2 \psi_2 d\phi_3^2\end{aligned}$$
$$\begin{aligned}
&& (ds_8)^2 = \Big( 1 - \frac{l^2 \sin^2 \theta }{l^2 + r^2 }
\Big) dr^2 + \Big( r^2 + l^2 \cos^2 \theta \Big) d\theta^2 + r^2
\cos^2 \theta d\psi_1^2 + r^2 \cos^2 \theta \cos^2 \psi_1
d\psi_2^2 {\nonumber}\\ && + \sin^2 \theta \Big(r^2 +l^2 \Big) d\phi_1^2
+ r^2 \cos^2 \theta \sin^2 \psi_1 d\phi_2^2 + r^2 \cos^2 \theta
\cos^2 \psi_1 \sin^2 \psi_2 d\phi_3^2 {\nonumber}\\ && + r^2 \cos^2
\theta \cos^2 \psi_1 \cos^2 \psi_2 d\phi_4^2\end{aligned}$$
$$\begin{aligned}
&& (ds_9)^2 = \Big( 1 - \frac{l^2 \sin^2
\theta }{l^2 + r^2 } \Big) dr^2 + \Big( r^2 + l^2 \cos^2 \theta
\Big) d \theta^2 + r^2 \cos^2 \theta d \psi_1^2 + r^2 \cos^2
\theta \cos^2 \psi_1 d\psi_2^2 {\nonumber}\\ && + r^2 \cos^2 \theta
\cos^2 \psi_1 \cos^2 \psi_2 d\psi_3^2 + \sin^2 \theta \Big(r^2
+l^2 \Big) d\phi_1^2 + r^2 \cos^2 \theta \sin^2 \psi_1 d\phi_2^2
{\nonumber}\\ && + r^2 \cos^2 \theta \cos^2 \psi_1 \sin^2 \psi_2
d\phi_3^2 + r^2 \cos^2 \theta \cos^2 \psi_1 \cos^2 \psi_2 \sin^2
\psi_3 d\phi_4^2\end{aligned}$$
The Euclidean near-horizon solution \[appeuclnhsol\]
=====================================================
In this appendix we give the Euclidean version of the near-horizon solution for one non-zero angular momentum. This comes into play in Sections \[secfreeenergy\] and \[secd3corr\] when calculating the value of the action and of the corrected action. The Euclidean solutions can simply be obtained by performing the Wick rotation $$\tau = it,\ \ {{\tilde{l}}}_i = -i l_i$$ This induces the replacement $l_i^2 \rightarrow -{{\tilde{l}}}_i^2$ in the definitions of $L_d$ of and $K_d$, $\Lambda_{\alpha \beta}$ of the spheroidal metric . In addition, we find that in the metric we have $ -f d t^2 \rightarrow f d\tau^2$ as well as $ l_i d t d\phi_i \rightarrow {{\tilde{l}}}_i d \tau d \phi_i$ in the off-diagonal terms. Finally, in the electric $(p+1)$-form potential we replace $$\Big( H^{-1} d\tau + \frac{r_0^{\frac{d-2}{2}}}{h^{\frac{d-2}{2}}}
\sum_{i=1}^n l_i \mu_i^2 d\phi_i \Big)
\rightarrow
-i
\Big( H^{-1} d\tau - \frac{r_0^{\frac{d-2}{2}}}{h^{\frac{d-2}{2}}}
\sum_{i=1}^n {{\tilde{l}}}_i \mu_i^2 d\phi_i \Big)$$ The above substitution rules should enable the reader to easily write down the general Euclidean case, and we confine ourselves with the explicit form for the case of one angular momentum ${{\tilde{l}}}\equiv {{\tilde{l}}}_1 \neq 0$ only
\[nheucl\] $$\begin{aligned}
\label{euclmet}
ds^2 &=& H^{-\frac{d-2}{D-2}}
\Big( f d\tau^2
+ \sum_{i=1}^p (dy^i)^2 \Big)
+ H^{\frac{p+1}{D-2}} \Big( \bar{f}^{-1} \frac{r^2 -{{\tilde{l}}}^2 \cos^2 \theta}{r^2 -{{\tilde{l}}}^2} dr^2
+ \Lambda_{\alpha \beta} d\eta^\alpha d\eta^\beta \Big)
{\nonumber}\\ &&
- 2 H^{-\frac{d-2}{D-2}}
\frac{1}{ 1 - \frac{{{\tilde{l}}}^2 \cos^2 \theta}{r^2} } \frac{h^{\frac{d-2}{2}}
r_0^{\frac{d-2}{2}}}{r^{d-2}}
{{\tilde{l}}}\sin^2 \theta d\tau d\phi_1\end{aligned}$$ $$\label{eucldil}
e^\phi = H^{\frac{a}{2}}$$ $$A_{p+1} = -i (-1)^p
\Big( H^{-1} d \tau - \frac{r_0^{\frac{d-2}{2}}}{h^{\frac{d-2}{2}}}
{{\tilde{l}}}\sin^2 \theta d\phi_1 \Big)
\wedge dy^1 \wedge dy^2 \wedge \cdots \wedge dy^p$$
where $$H =
\frac{1}{ 1 - \frac{{{\tilde{l}}}^2 \cos^2 \theta}{r^2} } \frac{h^{d-2}}{r^{d-2} }
{\ ,\qquad}f = 1 - \frac{1}{ 1 - \frac{{{\tilde{l}}}^2 \cos^2 \theta}{r^2} } \frac{r_0^{d-2}}{r^{d-2} }{\ ,\qquad}\bar f = 1 - \frac{1}{ 1 - \frac{{{\tilde{l}}}^2 }{r^2} } \frac{r_0^{d-2}}{r^{d-2} }$$ and the expressions for $\Lambda_{\alpha \beta}$ in the one-angular momentum case can be found in Appendix \[sphcoor\].
Change of variables from $(r_0,\{l_i\})$ to $(T,\{ \Omega_i \})$ \[basisch\]
============================================================================
In this appendix we give the formulae needed to go from the supergravity variables $(r_0,\{l_i\})$ to the thermodynamic quantities $(T,\{ \Omega_i \})$. Since it is not possible to obtain closed expressions (for general $d$) for this change of variables, we perform this analysis in a weak angular momentum expansion $$\frac{l_i}{r_0} \ll 1$$ keeping the first three terms only, which suffices for the applications of the text.
We use expressions for $(T,\{\Omega_i\})$, $$\label{TOrel} T = \frac{d-2-2\kappa}{4 \pi r_H}
\frac{r_0^{\frac{d-2}{2}}}{h^{\frac{d-2}{2}}} {\ ,\qquad}\Omega_i =
\frac{l_i}{(l_i^2 +r_H^2)}
\frac{r_0^{\frac{d-2}{2}}}{h^{\frac{d-2}{2}}}$$ to compute the quantities $(r_0,\{l_i\})$ in terms of the former. For this we first need to use the relation determining the horizon radius $r_H$ in terms of these, and we find $$\begin{gathered}
\label{horradius} r_H = r_0\left[ 1 - \frac{1}{d-2} \sum_i \left(
\frac{l_i}{r_0} \right)^2 - \frac{3}{2(d-2)^2} \left( \sum_i
\left( \frac{l_i}{r_0} \right)^2 \right)^2 \right. \\ \left. +
\frac{1}{2(d-2)} \sum_i \left( \frac{l_i}{r_0} \right)^4
+ \ldots \right]\end{gathered}$$ Substituting this in we obtain the expressions
$$\begin{gathered}
T= \frac{d-2}{4 \pi} \frac{r_0^{(d-4)/2}}{h^{(d-2)/2}} \left[ 1
-\frac{1}{d-2} \sum_i \left( \frac{l_i}{r_0} \right)^2
- \frac{7}{2(d-2)^2} \left( \sum_i \left( \frac{l_i }{r_0}
\right)^2 \right)^2 \right. \\ \left. + \frac{3}{2(d-2)} \sum_i
\left( \frac{l_i}{r_0} \right)^4 +\ldots \right]\end{gathered}$$
$$\Omega_i^2 = \frac{r_0^{d-4}}{h^{d-2}}
\left( \frac{l_i}{r_0} \right)^2
\left[ 1 + \frac{4}{d-2} \sum_j \left( \frac{l_j}{r_0} \right)^2
-2 \left( \frac{l_i}{r_0} \right)^2 +\ldots \right]$$
which can be inverted to give
$$\begin{gathered}
r_0 = \left( \tilde T h^{(d-2)/2} \right)^{2/(d-4)} \left[ 1 +
\frac{2}{(d-4)(d-2)} \sum_i \tilde \omega_i^2 \right.
\\
\left. - \frac{2(2d-9)}{(d-2)^2(d-4)^2}
\left( \sum_i \tilde \omega_i^2
\right)^2
+ \frac{1}{(d-2)(d-4)}
\sum_i \tilde \omega_i^4 + \ldots
\right]\end{gathered}$$
$$\label{lexp} \left( \frac{l_i}{r_0} \right)^2
=
\tilde \omega_i^2 \left[ 1 -
\frac{6}{d-2} \sum_j \tilde \omega_j^2
+2
\tilde \omega_i^2
+ \ldots \right]$$
where we have defined $$\tilde T \equiv \frac{4 \pi T}{d-2}
{\ ,\qquad}\tilde \omega_i =
\frac{\Omega_i}{\tilde T}$$ Finally, we also give the expression $$\begin{gathered}
\label{r0exp} r_0^{d-2} = \left( \tilde T h^{(d-2)/2}
\right)^{2(d-2)/(d-4)} \left[ 1 + \frac{2}{d-4} \sum_i
\tilde \omega_i^2 \right.
\\
\left. - \frac{2(d-6)}{(d-2)(d-4)^2} \left( \sum_i
\tilde \omega_i^2 \right)^2 + \frac{1}{d-4}
\sum_i \tilde \omega_i^4 + \ldots
\right]\end{gathered}$$ which enters the free energy .
Polylogarithms \[apppoly\]
===========================
In this appendix we define the polylogarithm functions and give some general properties. We also discuss a continuation of the polylogarithms to real numbers greater than one, which is used in Section \[subsecweakfe\]. The $n$th polylogarithm function is defined as $${\mathrm{Li}}_n (z) = \sum_{k=1}^\infty \frac{z^k}{k^n}$$ for $ z \in \mathbb{C} - \{ u \in \mathbb{R}+ 2\pi i \mathbb{Z} |
{\mathrm{Re}}(u) > 1 \} $, where $ {\mathrm{Re}}(u) $ means the real part of $ u
$. satisfying $${\mathrm{Li}}_n ( 1 ) = \zeta(n)\ \mathrm{for}\ n \neq 1$$ We also have the relation $${\mathrm{Li}}_n(-1) = \tilde{\zeta}(n)$$ where we have defined $$\tilde{\zeta} (n) = \left\{ \begin{array}{ll} (1-2^{1-n})\zeta(n)\
& {\ ,\qquad}n \neq 1 \\ -\log(2) & {\ ,\qquad}n = 1
\end{array} \right.$$ The polylogarithm satisfies the integral formula $$\label{liint} \int_0^\infty dx\, x^{n-2} \log( 1 - e^{z-x} ) = -
\Gamma(n-1) {\mathrm{Li}}_n (e^z)$$ for $ z \in \{ u \in \mathbb{C} | {\mathrm{Im}}(u) \not\in 2\pi \mathbb{Z}
\} $.
We also define $$B_n(z) = \frac{1}{2} \left( {\mathrm{Li}}_n(e^z) + {\mathrm{Li}}_n(e^{-z}) \right)$$ for $ z \in \{ u \in \mathbb{C} | {\mathrm{Im}}(u) \in [-\pi,\pi ]-\{0\} \}
$, and $$F_n(z) = \frac{1}{2} \left( {\mathrm{Li}}_n(-e^z) + {\mathrm{Li}}_n(-e^{-z}) \right)$$ for $ z \in \{ u \in \mathbb{C} | {\mathrm{Im}}(u) \in (-\pi,\pi ) \} $.
For even $n$ we have $$B_n(z) = \sum_{k=0}^{n/2} \zeta(n-2k) \frac{z^{2k}}{(2k)!} \pm
\frac{i \pi}{2} \frac{z^{n-1}}{(n-1)!}$$ where the optional sign is the sign of $ {\mathrm{Im}}(z) $. For odd $n$ we have $$\begin{aligned}
B_n(z) &=& \sum_{k=0}^{\frac{n-3}{2}} \zeta( n-2k )
\frac{z^{2k}}{(2k)!} + \left( \pm \frac{i \pi}{2} +
\sum_{k=1}^{n-1} \frac{1}{k} - \log(z) \right)
\frac{z^{n-1}}{(n-1)!} {\nonumber}\\ && + \sum_{k=\frac{n+1}{2}}^{\infty}
\zeta( n-2k ) \frac{z^{2k}}{(2k)!}\end{aligned}$$ where the optional sign is the sign of $ {\mathrm{Im}}(z) $. These functions satisfy $$B_{n-2} (z) = \frac{d^2}{dz^2} B_n (z)$$ for any $n$. Considering $F_n$, we find that for even $n$ we have $$F_n(z) = \sum_{k=0}^{n/2} \tilde{\zeta}(n-2k) \frac{z^{2k}}{(2k)!}$$ and for odd $n$ we have $$F_n(z) = \sum_{k=0}^{\infty} \tilde{\zeta}(n-2k)
\frac{z^{2k}}{(2k)!}$$ It is easy to see that $$F_{n-2} (z) = \frac{d^2}{dz^2} F_n (z)$$ for any $n$.
If we want to define $ B_n(z) $ also for $ z \in \mathbb{R} $ we can note that while the imaginary part of $ B_n(z) $ changes sign when crossing the real line, the real part is continuous. Thus, it is natural to define $ B_n $ on the real line as the limit of the real part of $ B_n $. This is also what the principal value prescription gives, since this is $ \frac{1}{2} (
B(x+i\epsilon) + B(x-i\epsilon) ) $ for $ \epsilon \rightarrow
0^+ $ with $ x \in \mathbb{R} $. Thus, let $ x \in \mathbb{R}
$, then for even $n$ we write $$B_n(x) = \sum_{k=0}^{n/2} \zeta(n-2k) \frac{x^{2k}}{(2k)!}$$ and for odd $n$ we write $$B_n(x) = \sum_{k=0}^{\frac{n-3}{2}} \zeta( n-2k )
\frac{x^{2k}}{(2k)!} + \left( \sum_{k=1}^{n-1} \frac{1}{k} -
\log(x) \right) \frac{x^{n-1}}{(n-1)!} +
\sum_{k=\frac{n+1}{2}}^{\infty} \zeta( n-2k ) \frac{x^{2k}}{(2k)!}$$ Again, these functions satisfy $$B_{n-2} (x) = \frac{d^2}{dx^2} B_n (x)$$ for any $n$.
[10]{}
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[^1]: Work supported in part by TMR network ERBFMRXCT96-0045.
[^2]: Other aspects of the thermodynamics in relation to the AdS/CFT correspondence and holography were studied e.g. in Refs. [@Barbon:1998ix; @Barbon:1998cr; @Chamblin:1999tk; @Chamblin:1999hg; @Martinec:1999bf].
[^3]: Except for the cases $d=8,9$, which have no stability boundary for one non-zero angular momentum.
[^4]: The case $d=4$, which can be seen from to be special since the free energy vanishes, will be treated separately. In this case, the temperature and angular velocity are not independent, so that the phase diagram is degenerate.
[^5]: Note that the D-branes of type I string theory and the NS-branes of heterotic string theory are included in this class of branes. When discussing the dual field theories in the near-horizon limit we restrict to type II string theory and M-theory only.
[^6]: Toroidal compactifications of the supergravity introduce more scalars in addition to the dilaton, which can be ignored since these moduli do not affect the background solution and its resulting thermodynamics.
[^7]: This appendix also gives the general thermodynamic relations for all non-rotating black $p$-branes, including those which preserve less than half of the supersymmetries.
[^8]: In Ref. [@Youm:1997rz] it was argued that the comoving frame is the natural frame for studying thermodynamics of rotating black holes and that the statistical analysis of rotating black holes is simplified in this frame.
[^9]: To see this, define the function $h(x) =
x^{d-2} \prod_{i=1}^n ( 1 + (l_i/x)^2) - r_0^{d-2} $ and compute $h'(x)= [d-2 - 2 \sum_{i=1}^n l_i^2/(l_i^2 + x^2)] x^{d-3}
\prod_{i=1}^n ( 1 + (l_i/x)^2) $. Using the fact that $h$ and $h'$ are both positive for large $x$, it follows that $r_H$ cannot be the highest root of $h(x)=0$ if $d-2- 2 \kappa <0 $.
[^10]: In Section \[subsecgrandcan\] we obtain the exact expressions for one non-zero angular momentum.
[^11]: With the exception of the D6-brane, as discussed for example in [@Itzhaki:1998dd].
[^12]: See e.g. [@Itzhaki:1998dd; @Tseytlin:1998cq; @Kiritsis:1999ke; @Kiritsis:1999tx; @Cai:1999ad] for discussions of D-brane probes, including thermal and spinning D-branes.
[^13]: See [@Behrndt:1999mk] for a more detailed explanation of the mapping between the near-horizon supergravity solutions and QFTs, including a description of the cases with $D <
10$.
[^14]: One could also use the determinant of the Hessian of the internal energy $E(S,J)$, which is the inverse of the Hessian of the Gibbs free energy.
[^15]: Note also that for $d=3$ the specific heat vanishes at $x_T =1$.
[^16]: As a consequence, one could have determined this boundary of stability by maximizing $\Omega/T$, providing an alternative method without having to resort to computing Jacobians.
[^17]: The boundary of stability does not have any special point other than $(T=0,\Omega=0)$ so we take a generic point different from that.
[^18]: These critical exponents should be related to the corresponding exponents in correlation functions of the field theory. In the field theory analysis for the D3-brane case [@Gubser:1998jb] no agreement was found, but a mean field treatment was suggested to cure this discrepancy.
[^19]: For a confining theory, it is understood that the temperature is above the confining temperature.
[^20]: We have checked these relations up to fourth order in $(\tilde l/r)^2$ for all 10-dimensional and 11-dimensional brane solutions, and we believe that they are generally valid. Using these values one can also write a closed form expression for in terms of logarithms.
[^21]: We thank J. Correia for useful discussions about this computation.
[^22]: On the weak coupling side, a two loop calculation [@Fotopoulos:1998es] has shown that the leading correction in $\lambda$ is negative, giving further evidence for the interpolation conjecture. In [@Kim:1999sg] further corrections in $\lambda$ from the weak coupling side are considered and also found to support the interpolation conjecture.
[^23]: For dual field theories with a smaller amount of supersymmetry, Refs.[@Nojiri:1999uh; @Nojiri:1999ji] consider analogous higher derivative corrections to obtain the modification of the thermodynamics.
[^24]: We have obtained the exact result but refrain from giving this rather lengthy expression.
[^25]: This feature has been shown to persist as well for the corrections to the near-horizon M2 and M5-brane backgrounds originating from the $R^4$ term in M-theory [@Caldarelli:1999ar].
|
---
abstract: '[**In superconductors with unconventional pairing mechanisms, the energy gap in the excitation spectrum often has nodes, which allow quasiparticle excitations at low energies. In many cases, e.g. $d$-wave cuprate superconductors, the position and topology of nodes are imposed by the symmetry, and thus the presence of gapless excitations is protected against disorder. Here we report on the observation of distinct changes in the gap structure of iron-pnictide superconductors with increasing impurity scattering. By the successive introduction of nonmagnetic point defects into BaFe$_2$(As$_{1-x}$P$_x$)$_2$ crystals via electron irradiation, we find from the low-temperature penetration depth measurements that the nodal state changes to a nodeless state with fully gapped excitations. Moreover, under further irradiation the gapped state evolves into another gapless state, providing bulk evidence of unconventional sign-changing $s$-wave superconductivity. This demonstrates that the topology of the superconducting gap can be controlled by disorder, which is a strikingly unique feature of iron pnictides.** ]{}'
author:
- 'Y. Mizukami$^{1,2}$'
- 'M. Konczykowski$^{3}$'
- 'Y. Kawamoto$^1$'
- 'S. Kurata$^{1,2}$'
- 'S. Kasahara$^1$'
- 'K. Hashimoto$^{1,4}$'
- 'V. Mishra$^5$'
- 'A. Kreisel$^6$'
- 'Y. Wang$^6$'
- 'P.J. Hirschfeld$^6$'
- 'Y. Matsuda$^1$'
- 'T. Shibauchi$^{1,2}$'
title: 'Disorder-induced topological change of the superconducting gap structure in iron pnictides'
---
When repulsive electron-electron interactions are strong, a sign change in the superconducting order parameter (or the gap function) often leads to some energy gain for electron pairing [@Scalapino95; @Hirschfeld11]. The positions of the gap nodes in momentum ${\bf k}$ space, at which the order parameter changes sign, are determined by the superconducting pairing interactions. In the high transition temperature ($T_{\rm c}$) cuprates, it has been established that the lines of nodes are located along $k_x=\pm k_y$, as expected in the $d_{x^2-y^2}$ symmetry of the order parameter [@Tsuei00]. In iron pnictides, on the other hand, the multiband Fermi surface structure leads to several candidates for the superconducting symmetry, including sign-changing $s_\pm$ with and without nodes, sign-preserving $s_{++}$, $d$-wave and time-reversal symmetry breaking $s+{\rm i}d$ [@Hirschfeld11; @Mazin08; @Kuroki09; @Graser09; @Kontani10; @Thomale11]. Different symmetries and details of higher order momentum dependence of the gap yield topologically different nodal structures, such as vertical or horizontal lines of nodes, nodal loops or no nodes.
The effects of nonmagnetic impurities on unconventional superconductivity has been one of the central issues in condensed matter physics [@Balatsky06; @Alloul09]. For the conventional BCS superconductors, it has been established that the effect is essentially null: the transition temperature is robust against nonmagnetic impurity scattering (the so-called Anderson theorem) and so is the superconducting gap. For the unconventional superconductors, the important aspect of the impurity effects is to mix gaps on different parts of the Fermi surface and thereby smear out the momentum dependence [@Mishra09]. In the case of superconducting gap with symmetry protected nodes such as $d$-wave, this averaging mechanism leads to the suppression of the gap amplitude, which enhances the low-lying quasiparticle excitations near the nodal positions. In addition to this, the sign change in the order parameter gives rise to impurity-induced Andreev bound states, which lead to additional quasiparticle excitations [@Hirschfeld93]. Such pair-breaking effects of impurities have been observed, e.g. in Zn-doped YBa$_2$Cu$_3$O$_7$ in the bulk measurements of magnetic penetration depth, where the $T$-linear temperature dependence in the clean-limit $d$-wave superconductivity gradually changes to a $T^2$ dependence at low temperatures with increasing Zn concentrations [@Bonn93].
In sharp contrast, when the nodal positions are not symmetry protected, as in the nodal $s$-wave case, the averaging mechanism of impurity scattering can displace the nodes, and at a certain critical impurity concentration the nodes may be lifted if intraband scattering dominates [@Mishra09], eliminating the low-energy quasiparticle excitations. In the fully gapped state after the node lifting, we have two cases in the multiband superconductors. If the signs of the order parameter on different bands are opposite, residual interband scattering can give rise to midgap Andreev bound states localized at nonmagnetic impurities that can contribute to the low-energy excitations, provided that the concentration of impurities is enough to create such states. If there is no sign change, Anderson’s theorem will be recovered, no Andreev states will be created, and thus no significant further change is expected. Indeed such a difference between nodal sign-changing $s_\pm$ and sign-preserving $s_{++}$ cases has been theoretically suggested by the recent calculations for multiband superconductivity considering the band structure of iron pnictides [@Wang13]. Therefore, the impurity effects on the gap nodes and low-energy excitations can be used as a powerful probe for the pairing symmetry of superconductors.
![[**Particle irradiation and created defects.**]{} [**a**]{}, Contour plot of maximum energy transferred to Fe atoms (recoil energy) for irradiated particles with the rest mass $m$ and incident energy $E$. Typical threshold energy $E_d$ for the displacement of Fe atoms from the lattice is marked by thick orange line. 2.5MeV electron irradiation used in this study (red square) has orders of magnitude smaller recoil energy than other particle irradiation owing to the small mass. Typical energies for neutron (green diamond), proton (purple circle), $\alpha$ particle (black triangle) and heavy-ion Pb (blue diamond) irradiation are indicated. [**b-d**]{}, Schematic illustrations for different types of defects created by particle irradiation. Columnar defects can be created by heavy-ion irradiation ([**b**]{}). Particle irradiation with relatively large recoil energies tends to have cascades of point defects due to successive collisions of atoms ([**c**]{}). Electron irradiation with a small recoil energy is the most reliable way to obtain uniform point defects ([**d**]{}). []{data-label="irradiation"}](Fig1.eps){width="1.0\linewidth"}
In this study, we focus on isovalently substituted system BaFe$_2$(As$_{1-x}$P$_x$)$_2$ close to the optimum composition with $T_{\rm c}\sim30$K [@Kasahara10]. This system is particularly suitable for the study of the impurity effect on gap structure, because several experiments have indicated that the pristine crystals are very clean and exhibit nodes in the superconducting gap [@Shibauchi14]. To detect changes in the low-energy quasiparticle excitations, we measure the magnetic penetration depth $\lambda$ at low temperatures, a fundamental property of superconductors whose $T$ dependence is directly related to the excited quasiparticles. The tunnel diode oscillators (TDOs) in $^3$He and dilution refrigerators operating at 13MHz are used to measure the temperature dependence of $\lambda$ down to 0.4K and 80mK, respectively [@Hashimoto10; @Hashimoto12]. To introduce the impurity scattering in a controllable way, we employ electron irradiation with the incident energy of 2.5MeV, for which energy transfer from impinging electron to the lattice is above the threshold energy for the formation of vacancy-interstitial (Frenkel) pairs that act as point defects. The long attenuation length and the small recoil energy due to the small mass of electrons is important to create uniformly distributed point defects over the entire crystal with the width of $\sim 30\,\mu$m (Figs.\[irradiation\]a-d). If the recoil energy is too large, one may expect creation of complex defects such as clusters and cascades of point defects and, in the extreme case, columnar tracks, which has been realized by heavy-ion irradiation [@Nakajima09] (Figs.\[irradiation\]b and c). Another advantage of electron irradiation is that, unlike chemical substitutions, the defects can be introduced without changing lattice constants, which is quite important as the gap structure may be sensitive to the lattice parameters in iron-based superconductors [@Hirschfeld11].
![[**Effect of electron irradiation on the superconducting transition in BaFe$_2$(As$_{1-x}$P$_x$)$_2$ single crystals.**]{} [**a**]{}, Temperature dependence of ac susceptibility at 13MHz for crystals (\#28A and \#28B) from a batch of $T_{\rm c0}=28$K with irradiated doses of 0, 1.0, 2.0, 4.0, 6.0 C/cm$^2$ with decreasing $T_{\rm c}$. [**b**]{}, Similar plot for crystals (\#29A, \#29B, \#29C, \#29D, and \#29E) from a batch of $T_{\rm c0}=29$K with irradiated doses of 0, 1.5, 2.7, 4.7, 4.9, 8.3C/cm$^2$ with decreasing $T_{\rm c}$. [**c**]{}, Temperature dependence of resistivity for a crystal (\#30A) with $T_{\rm c0}=30$K with irradiated doses of 0, 1.1, 2.3, 3.8, 4.5C/cm$^2$ with decreasing $T_{\rm c}$. Dotted lines are linear extrapolations to zero temperature to estimate changes in residual resistivity $\Delta\rho_0$. [**d**]{}, Transition temperature $T_{\rm c}$ normalized by the pristine value $T_{\rm c0}=30$K as a function of $\Delta\rho_0$, determined by the resistivity measurements in crystal \#30A. []{data-label="Tc_dose"}](Fig2.eps){width="0.75\linewidth"}
By successive electron irradiation into clean BaFe$_2$(As$_{1-x}$P$_x$)$_2$ ($x=0.33-0.36$) single crystals with the initial transition temperatures $T_{c0}$ of 28, 29, and 30K, we observe a systematic downward shift of $T_{\rm c}$ with increasing defect dose (Figs.\[Tc\_dose\]a-d). The transition width in the ac susceptibility data measured by the TDO frequency change (Figs.\[Tc\_dose\]a and b) remains almost unchanged after irradiation, which implies a good homogeneity of the point defects introduced. The temperature dependence of in-plane resistivity $\rho(T)$ measured by the van der Pauw configuration (Fig.\[Tc\_dose\]c) shows parallel shifts of $\rho(T)$ after each irradiation in a single crystal. The parallel shifts imply that point defects increase impurity (elastic) scattering with little changes of carrier concentrations and inelastic scattering, which is closely related to the electron correlations. We estimate the change in the residual resistivity $\Delta\rho_0$ by extrapolating the normal state data linearly to the zero temperature. The $T_{\rm c}$ reduction rate with respect to $\Delta\rho_0$ is about $-0.3$K$\mu\Omega^{-1}$cm$^{-1}$ (Fig.\[Tc\_dose\]d), which is comparable to the similar electron irradiation measurements in Ru-substituted BaFe$_2$As$_2$ [@Prozorov14]. Previous studies of the $T_{\rm c}$ reduction rate in iron-pnictide superconductors by chemical substitutions [@Sato10; @Li12; @Kirshenbaum12] and by particle irradiation [@Prozorov14; @Tarantini10; @Nakajima10; @Taen13] focus on the comparisons with theoretical calculations for $s_\pm$ and $s_{++}$-wave superconductivity [@Onari09; @Wang13], but they report various values of the suppression rate. Here we instead focus on changes of low-energy excitations induced by disorder from penetration depth measurements.
![[**Effect of electron irradiation on the low-temperature penetration depth in BaFe$_2$(As$_{1-x}$P$_x$)$_2$ single crystals.**]{} [**a,b**]{}, Change in the magnetic penetration depth $\Delta\lambda$ plotted against $(T/T_{\rm c})^2$ for the same $T_{\rm c0}=28$K samples as Fig.\[Tc\_dose\]a ([**a**]{}) and for the $T_{\rm c0}=29$K samples as Fig.\[Tc\_dose\]a ([**b**]{}). The same colours are used as Figs.2a and b for the corresponding irradiated doses. Each curve is shifted vertically for clarity. Lines are the $T^2$ dependence fits at high temperatures. [**c,d**]{}, Temperature dependence of $\Delta\lambda$ for sample \#28B with 2.0C/cm$^2$ ([**c**]{}) and for sample \#29C with 2.7C/cm$^2$ ([**d**]{}). Red lines are the fits to the exponential dependence expected in fully gapped superconductors. [**e**]{}, Temperature dependence of $\Delta\lambda$ for sample \#29A before (red) and after irradiation (blue) of 4.9C/cm$^2$ dose. Inset is an expanded view at the lowest temperatures. []{data-label="lambda"}](Fig3.eps){width="1.0\linewidth"}
In pristine crystals of BaFe$_2$(As$_{1-x}$P$_x$)$_2$, the penetration depth shows a strong temperature dependence at low temperatures (Figs.\[lambda\]a and b), as reported previously [@Hashimoto10; @Hashimoto12]. The temperature dependence of the change in the penetration depth $\Delta\lambda(T)=\lambda(T)-\lambda(0)$ can be fitted to a power law $T^n$ with the exponent $n<1.5$, indicating that this system has lines nodes in the energy gap. After irradiation, we first find that $\Delta\lambda(T)$ at the lowest temperatures becomes more gentle, or the exponent $n$ increases, and in $T_{\rm c0}=28$K crystal we have almost $T^2$ dependence (Fig.\[lambda\]a). Further increase of the defect density results in a flat temperature dependence below $T/T_{\rm c}\sim0.06-0.1$, indicating that the system is changed to a fully gapped state (Figs.\[lambda\]c and d). A fitting to the exponential dependence gives the gap size $2\Delta\gtrsim k_{\rm B}T_{\rm c}$, which is a substantial portion of the BCS value. Such flat temperature dependence of $\Delta\lambda$ has been reproduced in three crystals measured in this study, one of which has been measured down to 80mK (Fig.\[lambda\]e). The data represent completely temperature-independent behaviour below $T/T_{\rm c}\sim0.05$ within a precision of $\sim1$[Å]{} (Fig.\[lambda\]e, inset). The fact that we do not observe any Curie upturn at low temperatures is a strong indication that the point defects are essentially nonmagnetic [@Cooper96]. In fact, if we use the reported estimate of the defect density [@Beek13], our precision gives an upper limit of magnetic moment of $\sim0.2\mu_{\rm B}$ per Fe defect, which is much smaller than the moment expected in the magnetic spin states of Fe.
Further increase of the defect density leads to another change of $\Delta\lambda(T)$. The $T$-dependence at the lowest temperatures gets steeper with increasing defect dose, which is an opposite trend to the initial change at low doses. This second stage of the changes clearly indicates that by irradiation we create the low-energy excitations again inside the formed gap. At the highest doses we measured, we observe the $T^2$ dependence, which is a manifestation of the formation of the Andreev bound states expected for the sign-changing order parameter with impurity scattering. The impurity band corresponding to these bound states must overlap the Fermi level in order to cause this effect.
![[**Theoretical calculations of impurity effects in iron-based superconductors.**]{} [**a**]{}, Schematic of possible $s_\pm$ and $s_{++}$ states. Large circles and small ellipsoids (black lines) are hole and electron Fermi surfaces, respectively. Red and blue represents the superconducting order parameter with different signs. [**b**]{}, Schematic of $s_\pm$ order parameter vs. azimuthal angle $\phi$ (top row) and density of states $N$ vs. energy $\omega$ (bottom row) with increasing irradiation dosage (from left to right). Dotted lines (top row) are zero lines. [**c**]{}, Density of states at Fermi level $N(0)$ for nodal band plotted as false color vs. inter/intraband scattering ratio $\alpha$ and irradiation-induced residual resistivity $\Delta \rho_0$ for the model defined in Ref.. [**d**]{}, Penetration depth change $\Delta\lambda$ vs. reduced temperature $T/T_{\rm c}$ for different values of $\Delta \rho_0$ as marked in legend, and inter/intraband scattering ratio $\alpha =0.6$ (arrows in [**c**]{}). Each curve is shifted vertically for clarity. []{data-label="theory"}](Fig4.eps){width="1.0\linewidth"}
The observation of the impurity-induced fully gapped state provides one of the strongest pieces of evidence from bulk measurements that the nodes are not symmetry protected in this system. Moreover, the two-stage changes in the low energy excitations observed here most likely come from the peculiar band-dependent gap functions of multiband iron-pnictide superconductors. Owing to the well separated hole and electron Fermi surface sheets, the fully gapped $s_\pm$ superconductivity with opposite signs of the order parameter on different bands has been predicted by the theories based on spin fluctuations with the antiferromagnetic vector $\bf{Q}=(\pi,\pi)$ in the 2-Fe zone notation [@Mazin08; @Kuroki09]. The presence of nodes in the pristine samples implies some competing additional mechanisms, such as spin fluctuations with a different $\bf{Q}$ vector [@Kuroki09] or orbital fluctuations which prefer the same sign in both bands [@Kontani10]. To reproduce the observed two-stage changes, we have made calculations for the low-energy excitations and the low-temperature behaviour of $\lambda(T)$ by using a two-band model [@Wang13] (Figs.\[theory\]a-d). We assume that one band is fully gapped, and the other has accidental nodes whose positions are not symmetry protected. Two possible realizations with $s_\pm$ and $s_{++}$ structure are shown in Fig.\[theory\]a. Figure\[theory\]b shows schematically the evolution of the gap function in the $s_\pm$ case as a function of increasing disorder of dominant intraband and subdominant interband character. The corresponding densities of states are also plotted. Only in the last panels, where the subgap bound state appears, do the $s_\pm$ and $s_{++}$ cases differ qualitatively.
Figures\[theory\]c and d show concrete theoretical calculations using the model of Ref. in support of this scenario. The variation of the density of states at the Fermi level with scattering rate and the ratio $\alpha$ of inter- and intraband scattering (Fig.\[theory\]c), clearly shows the nonmonotonic behaviour of the density of states with scattering rate at a fixed $\alpha$. The temperature dependence of $\Delta\lambda$ depends directly on the residual density of states at the Fermi level, and therefore changes with increasing scattering as follows: (1) $T$-linear, (2) $T^2$, (3) exponential, and (4) $T^2$ (Fig.\[theory\]d). It is the overlap of the bound state in the last panel – not present in the $s_{++}$ case – with the Fermi level that gives rise to the “re-entrant” final $T^2$ dependence of the penetration depth. Thus the observed sequential changes of $\Delta\lambda(T)$ in this system are fully consistent with a sign changing $s$-wave superconducting gap of $s_\pm$ type.
Our results demonstrates that the gap topology and the low-energy excitations can be changed by controlling disorder. Such an impurity effect is unprecedented among superconductors, highlighting a unique aspect of iron-based superconductors. Our study also shows that the impurity effects on gap structure can provide phase information on the superconducting order parameter in bulk measurements, in contrast to other phase sensitive experiments [@Tsuei00; @Hanaguri10], most of which require excellent surfaces or interfaces.
Methods {#methods .unnumbered}
=======
The single crystals of BaFe$_2$(As$_{1-x}$P$_x$)$_2$ were grown by the self-flux method [@Kasahara10] and characterised by several techniques as reported previously [@Shibauchi14]. The observation of the quantum oscillations in this series of crystals and the sharp superconducting transition indicate the very high quality of our pristine crystals. We used several crystals from three batches, which exhibit slightly different $T_{\rm c0}$ values of 28, 29, and 30K. Electron irradiation experiments were performed on SIRIUS platform operated by LSI at Ecole Polytechnique, composed by Pelltron type NEC accelerator and closed cycle cryocooler maintaining sample immersed in liquid hydrogen at 20-22K during irradiation. The low-temperature environment is important to prevent defect migration and agglomeration. Partial annealing of introduced defects occurs upon warming to room temperature and sample transfer [@Prozorov14]. The resistivity measurements were performed at LSI, Ecole Polytechnique by the van der Pauw method with four contacts on corners of crystals to minimize the possible effect of the unirradiated area due to contacts. The penetration depth measurements by using 13MHz TDOs [@Hashimoto10; @Hashimoto12] were performed at Kyoto University before and after irradiation.
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Acknowledgments {#acknowledgments .unnumbered}
===============
We thank C.J. van der Beek, A. Carrington, H. Kontani, and R. Prozorov for fruitful discussion. We also thank B. Boizot, J. Losco, and V. Metayer for technical assistance. This work was supported by Grants-in-Aid for Scientific Research (KAKENHI) from Japan Society for the Promotion of Science (JSPS), and by the “Topological Quantum Phenomena” (No.25103713) Grant-in Aid for Scientific Research on Innovative Areas from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. Irradiation experiments were supported by EMIR network, proposal No.11–10–8071.
Author contributions {#author-contributions .unnumbered}
====================
T.S. conceived the project. S.Kasahara carried out sample preparation. Y.Mizukami, M.K., S.Kasahara, and T.S. performed irradiation experiments. M.K. performed resistivity measurements. Y.Mizukami, Y.K., S.Kurata, K.H., and T.S. performed penetration depth measurements and analysed data. V.M., A.K., Y.W., and P.J.H. performed theoretical calculations. Y.Matsuda and T.S. directed the research. T.S. wrote the manuscript with inputs from Y.Mizukami, M.K., K.H., V.M., A.K., Y.W., P.J.H., and Y.Matsuda.
|
---
abstract: 'We study the effects of low-energy nodal quasiparticles on the classical phase fluctuations in a two-dimensional $d$-wave superconductor. The singularities of the phase-only action at $T\to 0$ are removed in the presence of disorder, which justifies using an extended classical $XY$-model to describe phase fluctuations at low temperatures.'
address: 'Department of Physics, Brock University, St.Catharines, Ontario, Canada L2S 3A1'
author:
- 'K. V. Samokhin and B. Mitrović'
title: 'Classical phase fluctuations in $d$-wave superconductors'
---
Introduction {#sec:Intro}
============
The spectacular successes of the Bardeen-Cooper-Schrieffer (BCS) mean-field theory of superconductivity are based on the fact that the so-called Ginzburg-Levanyuk number $Gi_{(D)}$, which controls the size of fluctuation effects in $D$-dimensional samples, is very small in bulk conventional materials.[@LarVar05] On the other hand, the order parameter fluctuations become more pronounced, even dominant, in low-dimensional systems with a small Fermi energy, e.g. in quasi-two-dimensional cuprate superconductors.[@KBCC88] In particular, the fluctuations of the order parameter phase in the underdoped cuprates are enhanced due to a low value of the superfluid density, leading to the large deviations from the BCS picture, including the pseudogap phenomenon.[@TS99] According to Emery and Kivelson,[@EK95-2] the Cooper pairs survive in underdoped cuprates far above the critical temperature $T_c$, but without global phase coherence, which is destroyed by thermal phase fluctuations through the Berezinskii-Kosterlitz-Thouless mechanism.[@BKT] This idea has been further elaborated by many authors, for a review see, e.g., Ref. [@LQS01]. As temperature is lowered, a long-range phase coherence sets in, but the phase fluctuations continue to play important role in the superconducting state, remaining predominantly classical well below $T_c$.[@EK95-1; @CKEM99] One might expect however that quantum phase fluctuations eventually take over at the lowest temperatures. The crossover temperature $T_{cl}$ between the classical and quantum regimes is quite high (of the order of $T_c$) in clean charged systems, but can be significantly reduced in the presence of dissipation. While the estimates of $T_{cl}$ in cuprates obtained by different groups vary considerably, see e.g. Refs. [@EK95-1; @KDH01; @BCCPR01], here we adopt the view that the dissipation is strong enough to make the quantum effects negligible.
In this brief review we develop an effective long-wavelength theory of the classical phase fluctuations in $d$-wave superconductors, both with and without elastic disorder. At the Gaussian level, the lowest-order term in the gradient energy expansion is simply $\rho_s\bvs^2/2$, where $\rho_s$ is the superfluid mass density and $\bvs$ is the superfluid velocity. We show that the higher-order gradient terms, which contain $\nabla\bvs$, are singular in a clean system at low temperatures due to the presence of gap zeros,[@SM03] and also discuss the effects of disorder on those singularities. As a by-product of our theory, we address the question whether using the classical $XY$-model in $d$-wave superconductors can be justified from microscopic theory.
The article is organized as follows. In Sec. \[sec:Derivation\], the general field-theoretical description of the bosonic excitations in superconductors and a phase-only effective action are derived in the clean case. In Sec. \[sec:2D\], we focus on the case of a two-dimensional neutral $d$-wave superconductor, which is treated in the nodal approximation. The microscopic expressions for the energy of fluctuations are compared to the predictions of the classical $XY$-model. In Sec. \[sec:Disorder\], we derive the $d$-wave phase-only action in the presence of elastic impurity scattering. Sec. \[sec:Conclusion\] concludes with a discussion of our results.
Effective field theory: Clean case {#sec:Derivation}
==================================
The starting point of our analysis is the tight-binding Hamiltonian $$\label{Hamilt}
H=\sum\limits_{\br\br'}\xi_{\br\br'}c^\dagger_{\br\sigma}
c_{\br'\sigma}+\sum\limits_{\br}U_{\br}c^\dagger_{\br\sigma}
c_{\br\sigma}
-g\sum\limits_{\langle\br\br'\rangle}B^\dagger_{\br\br'}B_{\br\br'}
+\frac{1}{2}\sum\limits_{\br\br'}(n_{\br}-n_0)
V_{\br\br'}(n_{\br'}-n_0),$$ where $\br$ label the sites of a tetragonal lattice. The hopping amplitude $t_{\br\br'}$ and the chemical potential $\mu$ are combined into the band-dispersion matrix $\xi_{\br\br'}=-t_{\br\br'}-\mu\delta_{\br\br'}$, which is real and symmetric in the absence of external magnetic field. The second term describes impurity scattering. The third term is the BCS interaction in the $d$-wave channel, $g>0$ is the coupling constant, and the operator $B_{\br\br'}=(c_{\br'\downarrow}
c_{\br\uparrow}-c_{\br'\uparrow}c_{\br\downarrow})/\sqrt{2}$ destroys a singlet pair of electrons at the nearest-neighbor sites $\langle\br\br'\rangle$ in the $xy$ plane. The last term describes the repulsive interaction between electrons, $n_{\br}=c^\dagger_{\br\sigma}c_{\br\sigma}$ is the particle number density, $n_0=\langle n_{\br}\rangle$ is the average number of particles per site (which is equal to the ionic background density, thus ensuring the overall charge neutrality of the system), and $V_{\br\br'}$ is the interaction matrix.
Let us first look into the clean case. Setting $U_{\br}=0$ in Eq. (\[Hamilt\]), the partition function can be written as a functional integral over Grassmann fields $c_{\br\sigma}(\tau)$ and $\bar c_{\br\sigma}(\tau)$: $$\label{Zcc}
Z=\Tr\, e^{-\beta H}=\int{\cal D}c\,{\cal D}\bar c\;e^{-S[\bar c,c]},$$ where $S=\int_0^\beta d\tau\left[\sum_{\br}\bar
c_{\br\sigma}\partial_\tau c_{\br\sigma}+H(\tau)\right]$, $\beta=1/T$ (in our units $k_B=\hbar=1$). Using the Hubbard-Stratonovich transformation to decouple the interaction terms, we end up with the representation of $Z$ as a functional integral over $c,\bar c$ and two bosonic fields: a complex field $\Delta_{\br\br'}(\tau)$, which describes the superconducting order parameter fluctuations and is non-zero only on the bonds between the nearest neighbors, and a real scalar potential field $\varphi_{\br}(\tau)$. The fermionic part of the action then becomes $S=\Tr(\bar C{\cal G}^{-1}C)$, where $$\label{Nambu C}
C_{\br}=\left(%
\begin{array}{c}
c_{\br\uparrow} \\
\bar c_{\br\downarrow} \\
\end{array}\right),\quad
\bar C_{\br}=\left(%
\begin{array}{cc}
\bar c_{\br\uparrow} & c_{\br\downarrow} \\
\end{array}%
\right)$$ are the Nambu spinors, and ${\cal G}^{-1}$ is the inverse Green’s operator: $$\label{Gdef}
{\cal G}^{-1}(\br,\tau;\br',\tau')
=\delta(\tau-\tau')
\left(\begin{array}{cc}
\delta_{\br\br'}[-\partial_{\tau}-i\varphi_{\br}(\tau)]-\xi_{\br\br'} &
-\Delta_{\br\br'}(\tau)\\
-\Delta^*_{\br\br'}(\tau) &
\delta_{\br\br'}[-\partial_{\tau}+i\varphi_{\br}(\tau)]
+\xi_{\br\br'}
\end{array}\right).$$ We use the notation “$\Tr$” for the full operator trace with respect to both the space-time coordinates and the Nambu matrix indices, reserving “$\tr$” for a $2\times 2$ matrix trace in the Nambu space. Integrating out the fermionic fields, we obtain $$\label{Z}
Z=\int{\cal D}\Delta^*\,{\cal D}\Delta\,{\cal D}\varphi\,
\;e^{-S_{eff}[\Delta^*,\Delta,\varphi]},$$ with the effective action $$\label{Seff gen}
S_{eff}=-\Tr\ln {\cal G}^{-1}
+\int\limits_0^\beta d\tau\left(\frac{1}{g}
\sum\limits_{\br\br'}|\Delta_{\br\br'}|^2
+\frac{1}{2}\sum\limits_{\br\br'}\varphi_{\br}
V_{\br\br'}^{-1}\varphi_{\br'}
-in_0\sum\limits_{\br}\varphi_{\br}\right).$$
The mean-field BCS theory corresponds to a stationary and uniform saddle point of the effective action (\[Seff gen\]), which is found from the equations $\delta S_{eff}/\delta\Delta^*=\delta
S_{eff}/\delta\varphi=0$. The solution describing $d$-wave pairing is given by $\varphi_{0,\br}=0$ and $$\label{Delta dwave}
\Delta_{0,\br\br'}=\left\{\begin{array}{cc}
+\Delta_0, & \quad\mbox{if}\quad\br'=\br\pm a\hat{\bm{x}}, \\
-\Delta_0, & \quad\mbox{if}\quad\br'=\br\pm a\hat{\bm{y}}.
\end{array}\right.$$ In the momentum representation, $\Delta_{\bk}=\Delta_0(T)\phi_{\bk}$, where $\phi_{\bk}=2(\cos
k_xa-\cos k_ya)$ is the $d$-wave symmetry factor. The temperature dependence of the gap amplitude $\Delta_0$ is determined by the standard BCS self-consistency equation, generalized to the case of an anisotropic order parameter.[@Book]
Inverting the operator (\[Gdef\]), we find the mean-field matrix Green’s function: $$\label{G0}
{\cal G}_0(\bk,\omega_n)=
\left(%
\begin{array}{cc}
G_0(\bk,\omega_n) & -F_0(\bk,\omega_n) \\
-F_0(\bk,\omega_n) & -G_0(-\bk,-\omega_n) \\
\end{array}%
\right)=
-\frac{i\omega_n\tau_0+\xi_{\bk}\tau_3+\Delta_{\bk}\tau_1}{\omega_n^2
+\xi_{\bk}^2+\Delta_{\bk}^2}.$$ Here $\omega_n=(2n+1)\pi T$ is the fermionic Matsubara frequency, $G_0$ and $F_0$ are the usual normal and anomalous Gor’kov’s functions of the superconductor,[@AGD] $\xi_{\bk}=\xi_{-\bk}$ is the band dispersion of free electrons, and $\tau_i$ are Pauli matrices in the Nambu space. The Green’s function (\[G0\]) determines the single-particle properties in the mean-field approximation. In particular, after its analytical continuation to the real frequency axis, $i\omega_n\to\omega+i0$, one obtains the energies of elementary fermionic excitations, or the Bogoliubov quasiparticles: $$\label{E_k}
E_{\bk}=\sqrt{\xi_{\bk}^2+\Delta_{\bk}^2}.$$ For the $d$-wave order parameter, the gap in the excitation energy vanishes along the diagonals of the Brillouin zone. These zeros, or the gap “nodes”, are responsible for many peculiar thermodynamical and transport properties in the superconducting state on the mean-field level.[@Book] Below we show that the gap nodes also have dramatic effects on the long-wavelength behavior of phase fluctuations.
Fluctuations {#sec:Fluct}
------------
Deviations from the mean-field solution can be represented in terms of the amplitude and phase fluctuations: $\Delta_{\br\br'}(\tau)=[\Delta_{0,\br\br'}+\delta\Delta_{\br\br'}(\tau)]
e^{i\Theta_{\br\br'}(\tau)}$, where $\delta\Delta$ and $\Theta$ are real. We neglect the amplitude fluctuations because they are gapped, see e.g. Ref. [@BTCC02], and therefore make a negligible contribution at low temperatures. Since the number of bonds in a square lattice is twice the number of sites, one needs two on-site phase fields $\theta_{\br}(\tau)$ and $\tilde\theta_{\br}(\tau)$ to describe the phase degrees of freedom. One possible parametrization is $$\Theta_{\br\br'}=\left\{\begin{array}{cc}
\theta_{\br}, & \mbox{if }\br'=\br+a\hat{\bm{x}} \\
\theta_{\br}+\tilde\theta_{\br}, & \mbox{if }
\br'=\br+a\hat{\bm{y}}.
\end{array}\right.$$ The fluctuations of $\tilde\theta$, which describe a change in the symmetry of the order parameter from a pure $d$-wave to a $d+is$-wave, can be neglected.[@PRRM00]
If $\theta_{\br}$ changes slowly over the lattice constant, then one can make the replacement $$\label{average phase}
\Delta_{0,\br\br'}e^{i\theta_{\br}}\to\Delta_{0,\br\br'}
e^{i(\theta_{\br}+\theta_{\br'})/2},$$ (recall that $\br$ and $\br'$ are nearest neighbors). The next step is to perform a gauge transformation to make the off-diagonal elements of ${\cal G}^{-1}$ in Eq. (\[Gdef\]) real: $$\label{tilde G}
U^\dagger(\br,\tau){\cal G}^{-1}(\br,\tau;\br',\tau')U(\br',\tau')=
\tilde {\cal G}^{-1}(\br,\tau;\br',\tau'),$$ where $U(\br,\tau)=\exp[i\tau_3\theta_{\br}(\tau)/2]$. This transformation leaves the operator trace in the effective action (\[Seff gen\]) invariant. Although the order parameter (\[average phase\]) is no longer invariant under local phase rotations $\theta_{\br}\to\theta_{\br}+2\pi$, our results are not affected since we consider only small fluctuations of the phase in the low-temperature limit, where the contribution of vortices can be safely neglected.
The operator $\tilde{\cal G}^{-1}$ can be represented in the form $\tilde {\cal G}^{-1}={\cal G}_0^{-1}-\Sigma$, where ${\cal G}_0$ is the mean-field matrix Green’s function, whose Fourier transform is given by Eq. (\[G0\]), and $$\label{Sigma}
\Sigma(\br,\tau;\br',\tau')
=\delta(\tau-\tau')\tau_3\biggl\{i\delta_{\br\br'}\left[\frac{1}{2}
\frac{\partial\theta_{\br}(\tau)}{\partial\tau}+\varphi_{\br}(\tau)
\right]
+\xi_{\br\br'}\left[e^{-i\tau_3[\theta_{\br}(\tau)-\theta_{\br'}(\tau)]/2}
-1\right]\biggr\}$$ is the self-energy correction due to fluctuations. At slow temporal and spatial variations of $\theta$ and small $\varphi$, one can expand the effective action (\[Seff gen\]) in powers of $\Sigma$, keeping only the two lowest orders in the expansion, with the following result: $$\label{Seff theta phi}
S_{eff}[\theta,\varphi]=S_0
+\Tr({\cal G}_0\Sigma)+\frac{1}{2}\Tr({\cal G}_0\Sigma{\cal
G}_0\Sigma)+O(\Sigma^3)
+\frac{1}{2}\int\limits_0^\beta d\tau
\sum\limits_{\br\br'}\varphi_{\br}(\tau)
V_{\br\br'}^{-1}\varphi_{\br'}(\tau)-in_0\int\limits_0^\beta
d\tau\sum\limits_{\br}\varphi_{\br}(\tau),$$ where $$S_0=-\Tr\ln{\cal G}_0^{-1}+\beta\frac{1}{g}
\sum\limits_{\br\br'}|\Delta_{0,\br\br'}|^2=\beta{\cal E}_0$$ is the saddle-point action, ${\cal E}_0$ is the total mean-field energy of the superconductor, and ${\cal G}_0$ is the saddle-point Green’s function, see Eq. (\[G0\]). For non-interacting fluctuations we keep only the terms of the first and second order in $\Sigma$.
Calculating the traces in Eq. (\[Seff theta phi\]), we obtain the Gaussian action $$S_{eff}[\theta,\varphi]=S_0+S_1+S_2,$$ where $$S_1=\frac{in_0}{2}\int\limits_0^\beta d\tau\;\sum\limits_{\br}
\frac{\partial\theta_{\br}(\tau)}{\partial\tau}=
i\pi
n_0\sum\limits_{\br}\frac{\theta_{\br}(\beta)-\theta_{\br}(0)}{2\pi}$$ is the topological term containing the phase winding numbers, and $$\label{S2}
S_2 = \frac{1}{2}\sum_Q\left[L_{\varphi\varphi}(Q)|\varphi(Q)|^2
+L_{\theta\varphi}(Q)\theta^*(Q)\varphi(Q)
+\frac{1}{4}L_{\theta\theta}(Q)|\theta(Q)|^2\right].$$ Here we use the shorthand notations $Q=(\bq,\nu_m)$ and $$\sum_Q (...)=T\sum\limits_m\sum\limits_{\bq} (...),$$ where $\nu_m=2m\pi T$ is the bosonic Matsubara frequency and the momentum summation goes over the first Brillouin zone. The lattice Fourier transforms of the fields are defined by the usual expressions: $\theta_{\br}(\tau)={\cal
N}^{-1/2}\sum_{\bq}\theta(\bq,\tau)e^{i\bq\tau}$ *etc*, where ${\cal N}$ is the number of lattice sites. Since both $\varphi$ and $\theta$ are real, they satisfy $\theta^*(Q)=\theta(-Q)$, $\varphi^*(Q)=\varphi(-Q)$. The coefficients in Eq. (\[S2\]) are given by $$\begin{aligned}
&& L_{\varphi\varphi}(Q)=V^{-1}(\bq)+\Pi_0(Q),\nonumber \\
&&\label{Ls} L_{\theta\varphi}(Q)=i\nu_m\Pi_0(Q)+q_i\Pi^i_1(Q),\\
&& L_{\theta\theta}(Q)=\nu_m^2\Pi_0(Q)-
2i\nu_mq_i\Pi^i_1(Q)+q_i
q_j\Pi^{ij}_2(Q).\nonumber\end{aligned}$$ Here $V(\bq)$ is the Fourier transform of the interaction matrix $V_{\br\br'}$, and $$\begin{aligned}
&&\label{Pi0}\Pi_0(Q)=-\sum_K\tr[{\cal G}_0(K+Q)\tau_3{\cal G}_0(K)\tau_3],\\
&&\label{Pi1}
\Pi^i_1(Q)=\sum_K v_i\,\tr[{\cal G}_0(K+Q)\tau_3{\cal G}_0(K)\tau_0],\\
&&\label{Pi2}\Pi^{ij}_2(Q)=\sum_K m^{-1}_{ij}\tr[{\cal G}_0(K)\tau_3]+
\sum_K v_iv_j\,\tr[{\cal G}_0(K+Q)\tau_0{\cal G}_0(K)\tau_0].\end{aligned}$$ In these expressions, $K=(\bk,\omega_n)$, $$\sum_K (...)=T\sum\limits_n\frac{1}{\cal N}\sum\limits_{\bk}(...)
\stackrel{{\cal N}\to\infty}{\longrightarrow}
T\sum\limits_n\Omega\int\frac{d^D\bk}{(2\pi)^D} (...),$$ $m^{-1}_{ij}(\bk)=\partial^2\xi_{\bk}/\partial k_i\partial k_j$ is the inverse effective mass tensor, $\bm{v}(\bk)=\partial\xi_{\bk}/\partial\bk$ is the quasiparticle band velocity, and $\Omega$ is the unit cell volume (to simplify the notations, below we set $\Omega=1$). Eqs. (\[Ls\]) are obtained in the limit of small $\bq$ from more general expressions, using the gradient expansion $\xi_{\bk+\bq}=\xi_{\bk}+\bm{v}\bq+(1/2)m^{-1}_{ij}q_i
q_j+O(q^3)$. Integrating out the field $\varphi$, we finally arrive at the phase-only effective action: $$\label{Seff theta}
S_{eff}[\theta]=S_0+S_1+\frac{1}{8}\sum_Q
\biggl[L_{\theta\theta}(Q)+
\frac{L^2_{\theta\varphi}(Q)}{L_{\varphi\varphi}(Q)}\biggr]|\theta(Q)|^2.$$
In this article, we focus on the case of classical phase fluctuations and neglect all interactions other than those responsible for the Cooper pairing, which corresponds to neglecting the $\tau$-dependence of $\theta$ and setting $V(\bq)=0$. Then the topological term vanishes and only the $\Pi_2^{ij}$ contribution survives in the third term in Eq. (\[Seff theta\]), so that the effective action becomes $$\label{Seff cl}
S_{eff}[\theta]=\beta{\cal E}_0+\beta{\cal E}[\theta],$$ where ${\cal E}$ is the energy of fluctuations in the Gaussian approximation. Calculating the Matsubara sums in Eq. (\[Pi2\]) and introducing the superfluid velocity $\bvs=(1/2m)\nabla\theta$, where $m$ is the electron mass, we obtain $$\label{E gen}
{\cal E}=\frac{1}{2}\sum\limits_{\bq}{\cal K}_{ij}(\bq)
v^*_{s,i}(\bq)v_{s,j}(\bq),$$ with the kernel $$\label{kernel}
{\cal K}_{ij}(\bq)\equiv m^2\Pi^{ij}_2(\bq,0)
={\cal K}^{(0)}_{ij}+{\cal I}_{ij}(\bq),$$ where $$\begin{aligned}
&&\label{cal K0} {\cal K}^{(0)}_{ij}=
2m^2T\sum\limits_n\int\frac{d^D\bk}{(2\pi)^D}\,
m^{-1}_{ij}(\bk)G_0(\bk,\omega_n),\\
&&\label{cal I} {\cal I}_{ij}(\bq)=-m^2\int\frac{d^D\bk}{(2\pi)^D}
\,v_i(\bk)v_j(\bk)\biggl[C_-(\bk,\bq)\frac{\tanh\frac{E_{\bk+\bq}}{2T}+
\tanh\frac{E_{\bk}}{2T}}{E_{\bk+\bq}+E_{\bk}}
+C_+(\bk,\bq)\frac{\tanh\frac{E_{\bk+\bq}}{2T}-
\tanh\frac{E_{\bk}}{2T}}{E_{\bk+\bq}-E_{\bk}}\biggr],\end{aligned}$$ and $$\label{Cpm}
C_\pm(\bk,\bq)=\frac{1}{2}\left(1\pm
\frac{\xi_{\bk}\xi_{\bk+\bq}+\Delta_{\bk}\Delta_{\bk+\bq}}{E_{\bk}E_{\bk+\bq}}\right)$$ are the coherence factors.
An important characteristic of the superconductor is the superfluid density tensor, which is defined as $$\rho_{s,ij}(T)\equiv{\cal K}_{ij}(\bm{0}).$$ Its temperature dependence can be easily found in two limiting cases. In the normal state, $\Delta_{\bk}=0$, and one can use the identity $\partial G_0/\partial\bk=\bm{v}G_0^2$ in Eq. (\[Pi2\]) to obtain $\rho_{s,ij}(T>T_c)=0$. On the other hand, at zero temperature we have $\rho_{s,ij}(0)={\cal K}^{(0)}_{ij}$, since ${\cal I}_{ij}(\bm{0})=0$ at $T=0$.
The expressions (\[kernel\],\[cal K0\],\[cal I\]) are valid for arbitrary band structure and gap symmetry. In a Galilean-invariant system, i.e. for $\xi_{\bk}=\bk^2/2m-\mu$, the tensor (\[cal K0\]) takes a particularly simple form: ${\cal
K}^{(0)}_{ij}=\rho_0\delta_{ij}$, where $\rho_0=2mT\sum_n\int_{\bk}G_0(\bk,\omega_n)$ is the mass density of electrons. Therefore, the superfluid density tensor at zero temperature is $\rho_{s,ij}(0)=\rho_0\delta_{ij}$, i.e. all electrons are superconducting. In general, there is no such simple relation in a crystal.
Two-dimensional case {#sec:2D}
====================
In this section, we apply the general theory developed above to a two-dimensional $d$-wave superconductor. In this case, the low-energy physics at $T\to 0$ can be conveniently described using the so-called “nodal approximation”,[@Lee93] which takes advantage of the fact that the excitation energy (\[E\_k\]) for the $d$-wave order parameter can be linearized in the vicinity of the four gap nodes located at $\bk_n=k_F\hat\bk_n$ ($n=1,2,3,4$) on the Fermi surface. Here $$\displaystyle\hat\bk_1=\frac{\hat{\bm{x}}+\hat{\bm{y}}}{\sqrt{2}},\
\displaystyle\hat\bk_2=\frac{-\hat{\bm{x}}+\hat{\bm{y}}}{\sqrt{2}},\
\displaystyle\hat\bk_3=\frac{-\hat{\bm{x}}-\hat{\bm{y}}}{\sqrt{2}},\
\displaystyle\hat\bk_4=\frac{\hat{\bm{x}}-\hat{\bm{y}}}{\sqrt{2}}.$$ For the nodal quasiparticles in the vicinity of the $n$th node we have $\bk=\bk_n+\delta\bk$, and $$\label{nodal_approx}
\xi_{\bk}=v_F\delta k_\perp,\quad
\Delta_{\bk}=v_\Delta\delta k_\parallel,\quad
E_{\bk}=\sqrt{v_F^2\delta k_\perp^2+v_\Delta^2
\delta k_\parallel^2},$$ where $\delta k_\perp$ and $\delta k_\parallel$ are the momentum components perpendicular and parallel to the Fermi surface, $v_F$ is the Fermi velocity at the nodes, and $v_\Delta=|\partial\Delta_{\bk}/\partial\bk_\parallel|$ is the slope of the superconducting gap function near the nodes. Thus, the excitation spectrum near the gap nodes is described by an anisotropic Dirac cone. The anisotropy ratio $v_F/v_\Delta$ is an important characteristic of the high-$T_c$ cuprates, which depends on the material and the doping level, e.g. $v_F/v_\Delta\simeq 14$ and $19$ in the optimally doped YBCO and Bi-2212, respectively.[@ratio]
In the nodal approximation, the momentum integration over the whole Brillouin zone is replaced by a sum of four integrals over the small regions in $\bk$-space around the nodes: $$\label{nodal int}
\int\frac{d^2\bk}{(2\pi)^2}\,(...)\to
\sum\limits_{n=1}^4\int\frac{d\delta k_\perp d\delta
k_\parallel}{(2\pi)^2}\,(...)
=\frac{1}{2\pi v_Fv_\Delta}\sum\limits_{n=1}^4
\int\limits_0^{\epsilon_{max}}d\epsilon\,
\epsilon\int\limits_0^{2\pi}\frac{d\alpha}{2\pi}\,(...).$$ In the last integral, we changed to the polar coordinates: $v_F\delta k_\perp=\epsilon\cos\alpha$, $v_\Delta\delta
k_\parallel=\epsilon\sin\alpha$, and $E_{\bk}=\epsilon$. The ultraviolet cutoff $\epsilon_{max}\simeq\Delta_0$ is introduced to make sure that the area of the integration region is equal to the area of the original Brillouin zone.[@DL00] In most calculations in this article this cutoff can be extended to infinity.
Classical phase fluctuations {#sec:2D-class}
----------------------------
We can now calculate the energy of the classical phase fluctuations, see Eq. (\[E gen\]). The nodal approximation cannot be applied to the momentum integral in Eq. (\[cal K0\]) because it contains contributions from all electrons, including those far from the Fermi surface. Assuming that the band dispersion can be treated in the effective mass approximation, which amounts to the replacement $m^{-1}_{ij}\to(1/m^*)\delta_{ij}$, one obtains $$\label{cal K0 eff mass}
{\cal K}^{(0)}_{ij}=\frac{m}{m^*}\rho_0\delta_{ij},$$ where $\rho_0$ is the average mass density of electrons.
In contrast, the second term in the kernel ${\cal K}$ can be calculated in the nodal approximation. Using Eqs. (\[nodal\_approx\],\[nodal int\]), we have $$\label{cal I sum}
{\cal I}_{ij}(\bq)=-\frac{m^2}{2\pi}\frac{v_F}{v_\Delta}
\sum\limits_{n=1}^4\hat k_{n,i}\hat k_{n,j}S_n(\bq),$$ where $$S_1(\bq)=S_3(\bq)=Ts\left(\frac{\gamma_{1}}{T}\right),\quad
S_2(\bq)=S_4(\bq)=Ts\left(\frac{\gamma_{2}}{T}\right).$$ Here $$\label{gamma12}
\gamma_{1,2}(\bq)=\frac{1}{\sqrt{2}}\sqrt{v_F^2(q_x\pm q_y)^2+
v_\Delta^2(q_x\mp q_y)^2}$$ are the energies of the nodal quasiparticles with $\delta\bk=\bq$, and the scaling function $s(x)$ is defined by an integral: $$\label{s}
s(x)=\int\limits_0^\infty dy\,y
\int\limits_0^{2\pi}\frac{d\alpha}{2\pi}\,
[f_+(x,y,\alpha)+f_-(x,y,\alpha)],$$ where $$f_\pm=\frac{1}{2}\left(1\pm\frac{y+
x\cos\alpha}{\sqrt{x^2+y^2+2xy\cos\alpha}}\right)\frac{\tanh(\sqrt{x^2+y^2+2xy\cos\alpha}/2)\mp
\tanh(y/2)}{\sqrt{x^2+y^2+2xy\cos\alpha}\mp y}.$$ Note that the cutoff energy $\epsilon_{max}$ has been replaced by infinity, due to the rapid convergence of the integrals. One can show that the function $s(x)$ has the following asymptotics: $$\label{asymp}
s(x)=\left\{\begin{array}{ll}
\displaystyle 2\ln 2+\frac{x^2}{24} &,\ \mbox{at }x\to 0\\
\displaystyle \frac{\pi x}{8} &,\ \mbox{at }x\to\infty.
\end{array}\right.$$
After the summation over the four nodes in Eq. (\[cal I sum\]), one finally obtains $$\label{K final}
{\cal K}_{ij}(\bq)=\frac{m}{m^*}\rho_0\delta_{ij}-
\frac{m^2}{2\pi}\frac{v_F}{v_\Delta}T\left(\begin{array}{cc}
F_+(\bq) & F_-(\bq)\\
F_-(\bq) & F_+(\bq)
\end{array}\right)_{ij},$$ where $$F_\pm(\bq)=s\left[\frac{\gamma_{1}(\bq)}{T}\right]\pm
s\left[\frac{\gamma_{2}(\bq)}{T}\right].$$ The expression (\[K final\]) is exact in the nodal approximation, i.e. for the conical quasiparticle spectrum. In terms of $\bq$, the applicability region of the nodal approximation is $\gamma_{1,2}(\bq)\ll\Delta_0$. At higher energies of quasiparticles, the deviations of the spectrum from the linearized form (\[nodal\_approx\]) should be taken into account.
We would like to note that in the nodal approximation, $\Pi_1^i(\bq,0)=0$ and therefore $L_{\theta\varphi}(\bq,0)=0$, see Eqs. (\[Ls\]), (\[Pi1\]). This means that the classical fluctuation energy in two dimensions has the form (\[E gen\]) with the kernel (\[K final\]), even if the Coulomb interaction is taken into account.
At $T=0$, using the large-$x$ asymptotics of $s(x)$ in Eq. (\[asymp\]), the kernel takes the form $$\label{K zero T}
{\cal K}_{ij}(\bq)=\frac{m}{m^*}\rho_0\delta_{ij}-
\frac{m^2}{16}\frac{v_F}{v_\Delta}
\left(\begin{array}{ccc}
\gamma_1(\bq)+\gamma_2(\bq) &\ & \gamma_1(\bq)-\gamma_2(\bq)\\
\gamma_1(\bq)-\gamma_2(\bq) &\ & \gamma_1(\bq)+\gamma_2(\bq)
\end{array}\right)_{ij}.$$ Setting $\bq=0$ here, we find the superfluid density: $\rho_s(0)=(m/m^*)\rho_0$. We also see that the kernel is a non-analytical function of $\bq$, which means that no gradient expansion of the energy exists.
At finite temperatures and small $\bq$, such that $\gamma_{1,2}(\bq)\ll T$, the small-$x$ asymptotics of $s(x)$ yields $$\label{K nonzero T}
{\cal K}_{ij}(\bq)=\left(\frac{m}{m^*}\rho_0
-\frac{2\ln 2}{\pi}\frac{v_F}{v_\Delta}m^2T\right)\delta_{ij}
-\frac{m^2}{48\pi}\frac{v_F}{v_\Delta}\frac{1}{T}
\left(\begin{array}{ccc}
(v_F^2+v_\Delta^2)(q_x^2+q_y^2) && 2(v_F^2-v_\Delta^2)q_xq_y\\
2(v_F^2-v_\Delta^2)q_xq_y && (v_F^2+v_\Delta^2)(q_x^2+q_y^2)
\end{array}\right)_{ij}.$$ The first term describes the depletion of the superfluid density due to the thermal excitation of quasiparticles: $$\label{sf density T}
\rho_s(T)=\rho_s(0)-\frac{2\ln 2}{\pi}\frac{v_F}{v_\Delta}m^2T,$$ see also Refs. [@Lee93; @LW97], which explains a linear in $T$ increase of the magnetic penetration depth $\lambda(T)$ at low temperatures,[@Gross86; @AGR91] observed in high-$T_c$ cuprates.[@Hardy93]
The quadratic $\bq$-dependence of the expression (\[K nonzero T\]) implies that the kernel ${\cal K}_{ij}(\bR)$ in real space is proportional to $\exp(-|\bR|/\tilde\xi)$, with the length $\tilde\xi$ given by $$\label{tilde xi}
\tilde\xi(T)=\left(\frac{m^2v_F^3}{48\pi
v_\Delta\rho_{s}T}\right)^{1/2}\sim\left(\frac{T_c}{T}
\right)^{1/2}\xi_0,$$ where $\xi_0=v_F/2\pi T_c$ is the BCS coherence length (we assumed that $v_F\gg v_\Delta$). This behavior is similar to that of the electromagnetic response function in conventional $s$-wave superconductors, see e.g. Ref. [@Tink96]. An important difference however is that the characteristic length $\tilde\xi$ is temperature-dependent: $\tilde\xi(T)\sim T^{-1/2}$ at $T\to 0$. It is because of the divergence of $\tilde\xi$ that the gradient expansion of the classical energy of fluctuations breaks down at $T\to 0$. Note also that the length $\tilde\xi$ is different from other characteristic lengths discussed in the literature: $\xi_0$ – the coherence length, or the correlation length of the gap amplitude fluctuations, which remains constant at $T\to 0$, and $\xi_{pair}$ – the size of a Cooper pair, which is infinite in the $d$-wave case.[@BTCC02] In a conventional $s$-wave superconductor, all three lengths are of the same order.
The physical interpretation of our findings is the same as that of the non-local Meissner effect:[@non-loc] since the gap function $\Delta_{\bk}$ has nodes on the Fermi surface, then the anisotropic coherence length $v_F/|\Delta_{\bk}|$ exceeds the London penetration depth $\lambda_L$ close to the nodes, and the local electrodynamics breaks down.
One can expect that the nodal quasiparticles also affect the non-Gaussian terms in the effective action (\[Seff theta phi\]). As shown in the Appendix, indeed no expansion of the classical fluctuation energy in powers of $\bvs$ exists at $T=0$.
Failure of the classical $XY$-model {#sec:2D-XY}
-----------------------------------
The effects of the phase fluctuations in superconductors are most often studied using either the classical or the quantum versions of the $XY$-model. The energy of the classical $XY$-model in the absence of external fields has the form $$\label{XYdef}
{\cal E}_{XY}=\sum\limits_{\bR\bR'}J_{\bR\bR'}
[1-\cos(\theta_{\bR}-\theta_{\bR'})],$$ where $\theta_{\bR}$ is the phase of the order parameter at site $\bR$ of a coarse-grained square lattice, whose lattice spacing $d$ is of the order of the superconducting correlation length $\xi$. The summation goes over all bonds in the lattice, and the coupling constants $J_{\bR\bR'}=J(\bm{\rho})$, where $\bm{\rho}=\bR-\bR'$, are called the phase stiffness coefficients. While in the Gaussian approximation, see below, the coupling constants are temperature-independent, the interaction of fluctuations leads to a thermal renormalization of the $J$s, and eventually to a phase transition into the disordered state.
The $XY$-model is believed to provide a correct description of any system with broken U(1) symmetry if the amplitude fluctuations of the order parameter are negligible. Typical examples are classical Heisenberg magnets, superfluids and superconductors. The experimental systems to which the lattice model (\[XYdef\]) has been applied include granular superconductors and fabricated arrays of Josephson junctions, see, e.g., Ref. [@GW84] and the references therein. In those cases $d$ is given by the distance between grains. Although the simplicity of the $XY$-model is physically appealing, its rigorous microscopic derivation for homogeneous high-$T_c$ superconductors does not exist. The usual way of justification, see, e.g., Ref. [@PRRM00], involves expanding the cosine in Eq. (\[XYdef\]) and matching the expansion coefficients with those in the Gaussian phase-only action. That the microscopic theory fails to reproduce the quantum generalization of the $XY$-model has already been noticed in Ref. [@BTC04]. Here we show, following Ref. [@SM03], that even for the classical phase fluctuations in a $d$-wave superconductor at low temperatures, the long-wavelength limit of the microscopic theory is not consistent with Eq. (\[XYdef\]).
For slow variations of the phase, the energy (\[XYdef\]) takes a Gaussian form: $${\cal E}_{XY}=\frac{1}{2}\sum\limits_{\bR\bR'}J_{\bR\bR'}
(\theta_{\bR}-\theta_{\bR'})^2=\frac{1}{2}\sum\limits_{\bq}|\theta(\bq)|^2
\sum\limits_{\bm{\rho}}J(\bm{\rho})(1-\cos\bq\bm{\rho}).$$ In terms of the superfluid velocity, we have $$\label{E XY Gauss}
{\cal E}_{XY}=\frac{1}{2}\sum\limits_{\bq}{\cal K}^{XY}_{ij}(\bq)
v^*_{s,i}(\bq)v_{s,j}(\bq).$$ The kernel here has a well-defined Taylor expansion in powers of $\bq$: $$\label{XY kernel}
{\cal K}^{XY}_{ij}(\bq)=\rho^{XY}_{s,ij}+\Lambda_{ij,kl}q_kq_l+O(\bq^4),$$ where $$\rho^{XY}_{s,ij}=2m^2\sum\limits_{\bm{\rho}}J(\bm{\rho})\rho_i\rho_j$$ is the superfluid mass density tensor (for example, if the only non-zero coupling is between the nearest-neighbor sites, then $\rho^{XY}_{s,ij}=8m^2d^2J\delta_{ij}$), and $$\label{Lambda}
\Lambda_{ij,kl}=-\frac{m^2}{6}
\sum\limits_{\bm{\rho}}J(\bm{\rho})\rho_i\rho_j\rho_k\rho_l.$$
Comparing Eqs. (\[XY kernel\]) and (\[K zero T\]), we see that the effective long-wavelength theory of the classical phase fluctuations at $T=0$ does not have the form of the $XY$-model, since the momentum dependence of the two energies is clearly different. At $T>0$, although the expression (\[K nonzero T\]) is quadratic in $\bq$, the coefficients diverge as $T\to 0$, which is not the case for $\Lambda_{ij,kl}$ above. Thus, the microscopic theory fails to reproduce the long-wavelength structure of the classical $XY$-model in a clean $d$-wave superconductor.
Disordered case {#sec:Disorder}
===============
In the presence of impurities, a full effective field theory for the disordered interacting system described by the Hamiltonian (\[Hamilt\]) would include the fluctuations of the order parameter and of the scalar potential coupled with the disorder-induced soft modes (the diffusons and the Cooperons). Such theories, usually having the form of a non-linear $\sigma$-model, have been developed, see, e.g. Ref. [@sigma-models] and the references therein, to study the effects that are beyond the scope of the present work, for instance the suppression of $T_c$ due to the interplay of disorder and interactions in $s$-wave superconductors. Our goal here is to check if the elastic impurity scattering removes the divergencies in the gradient expansion of the classical phase-only action discussed above. The disorder is treated essentially in the saddle-point approximation and the Coulomb interaction is neglected.
As a bookkeeping device to obtain an effective action for the order parameter fluctuations, we use the replica trick: $\langle\ln Z\rangle=\lim_{n\to 0}(\langle Z^n\rangle-1)/n$ (the angular brackets denote averaging with respect to disorder). From Eq. (\[Z\]) we have $$\label{Zcc n}
Z^n=\int\prod\limits_{a=1}^n{\cal D}c^a{\cal D}\bar c^a\, e^{-S[\bar c,c]},$$ where $S=S_0+S_{int}$, $$\begin{aligned}
\label{tilde S}
&&S_0=\sum\limits_a\int\limits_0^\beta d\tau\biggl(\sum_{\br}\bar
c^a_{\br\sigma}\partial_\tau c^a_{\br\sigma}+
\sum\limits_{\br\br'}\xi_{\br\br'}\bar c^a_{\br\sigma}c^a_{\br'\sigma}
+\sum\limits_{\br}U_{\br}\bar
c^a_{\br\sigma}c^a_{\br\sigma}\biggr),\\
&&S_{int}=-g\sum\limits_a\int\limits_0^\beta d\tau
\sum\limits_{\langle\br\br'\rangle}\bar B^a_{\br\br'}B^a_{\br\br'}.\end{aligned}$$ The impurity potential here is assumed to be Gaussian-distributed, with zero mean and the correlator $$\langle U_{\br}U_{\br'}\rangle=\frac{1}{2\pi
N_F\tau}\delta_{\br\br'},$$ where $N_F$ is the density of states at the Fermi level, and $\tau$ is the electron mean-free time due to elastic scattering. The next step is to use an incomplete Hubbard-Stratonovich transformation to decouple only the interaction terms $S_{int}$ in each replica, before disorder averaging.[@BK94] Proceeding as in Sec. \[sec:Derivation\], we have $$\label{Zn}
\langle Z^n\rangle=\int\prod\limits_{a=1}^n{\cal D}\Delta^{a,*}\,{\cal D}\Delta^a\,
e^{-S_{eff}[\Delta^*,\Delta]},$$ with the effective action $$\label{Seff a}
S_{eff}=-\ln\langle\Det{\cal G}^{-1}\rangle+
\frac{1}{g}\sum_a\int\limits_0^\beta d\tau
\sum\limits_{\br\br'}|\Delta^a_{\br\br'}|^2,$$ instead of Eq. (\[Seff gen\]). Here $$\label{Gab}
{\cal G}_{ab}^{-1}(\br,\tau;\br',\tau')
=\delta_{ab}\delta(\tau-\tau')\left(\begin{array}{cc}
\delta_{\br\br'}(-\partial_{\tau}-U_{\br})-\xi_{\br\br'} &
-\Delta^a_{\br\br'}(\tau)\\
-\Delta^{a,*}_{\br\br'}(\tau) &
\delta_{\br\br'}(-\partial_{\tau}+U_{\br})
+\xi_{\br\br'}
\end{array}\right),$$ and “Det” stands for the full operator determinant with respect to the space-time coordinates and the Nambu and the replica indices. The disorder averaging in the first term in $S_{eff}$ generates effective coupling between different replicas, resulting in a non-linear bosonic field theory.
While the saddle point of the effective action (\[Seff a\]) has the same structure as in the clean case: $\Delta^a_{0,\br\br'}(\tau)=\Delta_{0,\br\br'}$, see Eq. (\[Delta dwave\]), the temperature dependence of the gap amplitude is different, in particular, the critical temperature is suppressed by impurities.[@Book] The mean-field Green’s function is the unity matrix in the replica space: ${\cal
G}_{0,ab}^{-1}=\delta_{ab}{\cal G}_0^{-1}$, where $$\label{cal G0 dis}
{\cal G}_0^{-1}(\br,\br';\omega_n)=
\left(\begin{array}{cc}
\delta_{\br\br'}(i\omega_n-U_{\br})-\xi_{\br\br'} &
-\Delta_{0,\br\br'}\\
-\Delta_{0,\br\br'} &
\delta_{\br\br'}(i\omega_n+U_{\br})
+\xi_{\br\br'}\end{array}\right)$$ in the Matsubara frequency representation. The disorder averaging can be done using the standard diagram technique.[@AGD] In the so-called self-consistent Born approximation, which assumes a sufficiently weak impurity scattering and also neglects the diagrams with crossed impurity lines, the average mean-field Green’s function has the form $$\label{G0 dis}
\langle{\cal G}_0(\bk,\omega_n)\rangle=
-\frac{i\tilde\omega_n\tau_0+\xi_{\bk}\tau_3+\Delta_{\bk}\tau_1}{\tilde\omega_n^2
+\xi_{\bk}^2+\Delta_{\bk}^2},$$ where $\tilde\omega_n$ satisfies the equation $i\tilde\omega_n=i\omega_n+(1/2\tau)
\langle(i\tilde\omega_n/\sqrt{\tilde\omega_n^2+\Delta_{\bk}^2})\rangle_{\bk}$.[@Book]
Assuming replica-symmetric phase fluctuations, the order parameter can be written in the form $\Delta^a_{\br\br'}(\tau)=\Delta_{0,\br\br'}e^{i[\theta_{\br}(\tau)+\theta_{\br'}(\tau)]/2}$, see Eq. (\[average phase\]). Performing the gauge transformation (\[tilde G\]) in each replica, we have $\tilde{\cal
G}_{ab}^{-1}=\delta_{ab}({\cal G}_0^{-1}-\Sigma)$, where the self-energy operator has the form (\[Sigma\]) with $\varphi_{\br}(\tau)=0$. Therefore, $$\ln\langle\Det{\cal G}^{-1}\rangle=
\ln\langle\Det{\tilde{\cal G}}^{-1}\rangle
=n\langle\Tr\ln{\cal G}_0^{-1}\rangle+
n\langle\Tr\ln(1-{\cal G}_0\Sigma)\rangle+O(n^2),$$ in the limit $n\to 0$. Substituting this expansion in Eq. (\[Zn\]), we see that the replica index can be omitted, and the effective action (\[Seff theta phi\]) gets replaced by its disorder average: $$\label{Seff phase dis}
S_{eff}[\theta]=\beta{\cal E}_0-\langle\Tr\ln(1-{\cal G}_0\Sigma)\rangle,$$ where ${\cal E}_0$ is the mean-field energy of a disordered superconductor. Considering only the static fluctuations and expanding the operator trace in powers of the phase gradients, we obtain the effective action in the form (\[Seff cl\]), where the energy of fluctuations is now given by $$\label{cal E dis}
{\cal E}=\frac{1}{2}
\sum\limits_{\bq}\;\langle{\cal K}_{ij}(\bq)\rangle
v^*_{s,i}(\bq)v_{s,j}(\bq),$$ with $$\label{kernel av}
\langle{\cal K}_{ij}(\bq)\rangle=
\langle{\cal K}^{(0)}_{ij}\rangle+\langle{\cal I}_{ij}(\bq)\rangle.$$ Before proceeding with the calculation of the disorder averages, we would like to note that Eq. (\[cal E dis\]) could also be derived using a less formal approach, without introducing replicas. Assuming that electrons move in the presence of the random potential and a given order parameter field $\Delta_{\br\br'}=\Delta_{0,\br\br'}e^{i(\theta_{\br}+\theta_{\br'})/2}$, one can define the energy functional of phase fluctuations: $$\label{GL func}
{\cal E}[\theta]=-\frac{1}{\beta}\langle\ln Z[\theta]\rangle+
\frac{1}{\beta}\langle\ln Z[0]\rangle,$$ where $Z[\theta]=\Det({\cal G}_0^{-1}-\Sigma)$ is the partition function. Expanding $\ln Z[\theta]$ in powers of the phase gradients, followed by averaging each expansion coefficient with respect to disorder, we arrive at Eq. (\[cal E dis\]).
The disorder averaging of the first term in Eq. (\[kernel av\]) is straightforward: $$\label{cal K0 dis}
\langle{\cal K}^{(0)}_{ij}\rangle=
2m^2T\sum\limits_n\int\frac{d^D\bk}{(2\pi)^D}\,
m^{-1}_{ij}(\bk)\langle G_0(\bk,\omega_n)\rangle=
\frac{m}{m^*}\rho_0\delta_{ij},$$ where we used the effective mass approximation, as in Eq. (\[cal K0 eff mass\]).
The average of the product of two Green’s functions in the second term includes the impurity vertex corrections. However, since we are interested in the behavior of the kernel in the long-wavelength limit $\bq\to 0$ in a sufficiently clean system (the precise criterion will be discussed below), the vertex corrections are negligible. Thus $$\label{cal I dis}
\langle{\cal I}_{ij}(\bq)\rangle=m^2
T\sum\limits_n\int\frac{d^D\bk}{(2\pi)^D}v_i(\bk)
v_j(\bk)\,\tr[\langle{\cal
G}_0(\bk+\bq,\omega_n)\rangle\tau_0
\langle{\cal G}_0(\bk,\omega_n)\rangle\tau_0].$$ In order to calculate the Matsubara sum we use the spectral representation $$\label{spec rep}
{\cal G}_0(\bk,\omega_n)=\int\limits_{-\infty}^{\infty}\frac{d\epsilon}{\pi}
\frac{\im{\cal G}_0^R(\bk,\epsilon)}{\epsilon-i\omega_n},$$ where the retarded matrix Green’s function is obtained from Eq. (\[G0 dis\]) by analytical continuation: $$\label{G R}
{\cal G}_0^R(\bk,\epsilon)=\frac{t(\epsilon)\tau_0+\xi_{\bk}\tau_3+
\Delta_{\bk}\tau_1}{t^2(\epsilon)-\xi_{\bk}^2-\Delta_{\bk}^2},$$ and $t(\epsilon)=(i\tilde\omega_n)|_{i\omega_n\to\epsilon+i0}$ satisfies the equation $t=\epsilon+(i/2\tau)\langle(t/\sqrt{t^2-\Delta_{\bk}^2})\rangle_{\bk}$. Instead of finding the exact energy dependence of $t(\epsilon)$ for the $d$-wave gap, we use below a simplified expression $$\label{t}
t(\epsilon)=\epsilon+i\Gamma,$$ which captures the essential qualitative effects of the impurity broadening of the single-electron states, with $\Gamma$ being an energy-independent effective scattering rate. The exact solution shows that the scattering rate indeed approaches a constant value at $\epsilon\to 0$, i.e. in the so-called universal limit.[@Book]
Inserting the representation (\[spec rep\]) in Eq. (\[cal I dis\]), one obtains $$\begin{aligned}
\langle{\cal I}_{ij}(\bq)\rangle=-\frac{m^2}{2}
\int\limits_{-\infty}^\infty d\epsilon_1\int\limits_{-\infty}^\infty d\epsilon_2
\frac{\tanh\frac{\epsilon_1}{2T}-\tanh\frac{\epsilon_2}{2T}}{\epsilon_1-\epsilon_2}
\int\frac{d^D\bk}{(2\pi)^D}\,v_i(\bk)v_j(\bk)\nonumber\\
\times\bigl\{C_-(\bk,\bq)[d_+(\bk+\bq,\epsilon_1)d_-(\bk,\epsilon_2)+
d_-(\bk+\bq,\epsilon_1)d_+(\bk,\epsilon_2)]\nonumber\\
\label{cal I dis gen} +C_+(\bk,\bq)[d_+(\bk+\bq,\epsilon_1)d_+(\bk,\epsilon_2)+
d_-(\bk+\bq,\epsilon_1)d_-(\bk,\epsilon_2)]\bigr\},\end{aligned}$$ where $$d_\pm(\bk,\epsilon)=\frac{1}{\pi}\frac{\Gamma}{
(\epsilon\pm E_{\bk})^2+\Gamma^2}.$$ and $C_\pm$ are the coherence factors (\[Cpm\]). The expressions (\[cal I\]) are recovered from Eq. (\[cal I dis gen\]) in the clean limit $\Gamma\to 0$, when $d_\pm(\bk,\epsilon)\to\delta(\epsilon\pm E_{\bk})$.
We focus on the case of zero temperature, when the kernel is a non-analytical function of $\bq$ in the absence of impurities, see Eq. (\[K zero T\]). At $T=0$, the integrals over $\epsilon_{1,2}$ can be calculated, giving $$\langle{\cal I}_{ij}(\bq)\rangle=
-\frac{2m^2}{\pi}\int\frac{d^D\bk}{(2\pi)^D}\,v_i(\bk)v_j(\bk)
\biggl[C_-(\bk,\bq)\frac{\arctan\frac{E_{\bk+\bq}}{\Gamma}+
\arctan\frac{E_{\bk}}{\Gamma}}{E_{\bk+\bq}+E_{\bk}}
+C_+(\bk,\bq)\frac{\arctan\frac{E_{\bk+\bq}}{\Gamma}-
\arctan\frac{E_{\bk}}{\Gamma}}{E_{\bk+\bq}-E_{\bk}}\biggr].$$ Comparing this to Eq. (\[cal I\]), we see that the energy of classical phase fluctuations in the disordered case at zero temperature has exactly the same form as in the clean case at a finite temperature, if one formally replaces $\tanh(E/2T)\to(2/\pi)\arctan(E/\Gamma)$. Therefore one can expect that the disorder will affect the phase fluctuations in the same way as temperature does, i.e. the singularities of the effective action will be washed out.
To check this conclusion quantitatively, let us evaluate the long-wavelength asymptotics of $\langle{\cal I}_{ij}(\bq)\rangle$ in the nodal approximation. Using Eqs. (\[nodal\_approx\],\[nodal int\]), we have $$\label{cal I dis sum}
\langle{\cal I}_{ij}(\bq)\rangle=-\frac{m^2}{2\pi}\frac{v_F}{v_\Delta}
\sum\limits_{n=1}^4\hat k_{n,i}\hat k_{n,j}\tilde S_n(\bq),$$ where $$\tilde S_1(\bq)=\tilde S_3(\bq)=\Gamma\tilde s\left(\frac{\gamma_{1}}{\Gamma}\right),\quad
\tilde S_2(\bq)=\tilde S_4(\bq)=\Gamma\tilde
s\left(\frac{\gamma_{2}}{\Gamma}\right),$$ and $\gamma_{1,2}(\bq)$ are given by Eq. (\[gamma12\]). The function $\tilde s(x)$ is defined by $$\label{s dis}
\tilde s(x)=\int\limits_0^{2\pi}\frac{d\alpha}{2\pi}
\int\limits_0^{y_{max}} dy\,y[\tilde f_+(x,y,\alpha)+\tilde f_-(x,y,\alpha)],$$ where $y_{max}=\epsilon_{max}/\Gamma$, and $$\tilde f_\pm=\frac{1}{\pi}\left(1\pm\frac{y+
x\cos\alpha}{\sqrt{x^2+y^2+2xy\cos\alpha}}\right)
\frac{\arctan(\sqrt{x^2+y^2+2xy\cos\alpha})\mp
\arctan(y)}{\sqrt{x^2+y^2+2xy\cos\alpha}\mp y}.$$ In contrast to the clean case, see Eq. (\[s\]), the energy cutoff here cannot be extended to infinity, due to the logarithmic divergence of the integrals.
The clean-case expression (\[K zero T\]) is recovered from Eq. (\[s dis\]) at $x\gg 1$, i.e. at $\Gamma\ll\gamma_{1,2}(\bq)\ll\epsilon_{max}$. In the opposite limit $x\ll 1$, one finds $$\label{tilde s asymp}
\tilde s(x)=\frac{2}{\pi}\ln y_{max}+\frac{1}{6\pi}x^2,$$ which is valid at $y_{max}\gg 1$. Inserting this into Eq. (\[cal I dis sum\]), we finally obtain $$\label{K Gamma}
\langle{\cal K}_{ij}(\bq)\rangle=\left(\frac{m}{m^*}\rho_0
-\frac{2}{\pi^2}\frac{v_F}{v_\Delta}m^2
\Gamma\ln\frac{\epsilon_{max}}{\Gamma}\right)\delta_{ij}
-\frac{m^2}{12\pi^2}\frac{v_F}{v_\Delta}\frac{1}{\Gamma}
\left(\begin{array}{cc}
(v_F^2+v_\Delta^2)(q_x^2+q_y^2) & 2(v_F^2-v_\Delta^2)q_xq_y\\
2(v_F^2-v_\Delta^2)q_xq_y & (v_F^2+v_\Delta^2)(q_x^2+q_y^2)
\end{array}\right)_{ij}.$$ The first term describes the depletion of the superfluid density at zero temperature due to the impurity scattering: $$\label{sf density Gamma}
\rho_s(T=0,\Gamma)=\rho_s(T=0,0)-\frac{2}{\pi^2}\frac{v_F}{v_\Delta}m^2
\Gamma\ln\frac{\Delta_0}{\Gamma},$$ see also Ref. [@HG93] (here we used $\epsilon_{max}\simeq\Delta_0$). Upon increasing $\Gamma$, the superfluid density decreases and eventually vanishes at some critical disorder strength. One cannot reach the critical point using the expression (\[sf density Gamma\]), since its validity is limited to the case of weak impurity scattering $\Gamma\ll\Delta_0$.
The second term in Eq. (\[K Gamma\]) shows that, in contrast to the clean case, the energy is a non-singular function of momentum, even at $T=0$. Comparing it to the expression (\[XY kernel\]), we see that the microscopic theory, at least at the Gaussian level, has the same long-wavelength behavior as the classical $XY$-model. The non-zero elements of the tensor $\Lambda$ are given by $$\label{Lambda micro}
\begin{array}{l}
\displaystyle\Lambda_{xx,xx}=\Lambda_{xx,yy}=-
\frac{m^2}{12\pi^2}\frac{v_F}{v_\Delta}\frac{1}{\Gamma}(v_F^2+v_\Delta^2),\\
\displaystyle \Lambda_{xy,xy}=
-\frac{m^2}{12\pi^2}\frac{v_F}{v_\Delta}\frac{1}{\Gamma}(v_F^2-v_\Delta^2), \\
\end{array}$$ with other elements obtained by symmetry. One can see that keeping only the nearest-neighbor phase stiffness coefficient in Eq. (\[Lambda\]) is not sufficient to reproduce the tensor structure (\[Lambda micro\]), therefore one has to consider an extended $XY$-model, in which the couplings between next-nearest neighbors *etc*, are also taken into account. This was first noticed in Ref. [@KC02] for an $s$-wave superconductor.
Conclusions {#sec:Conclusion}
===========
To summarize, we have shown that the effective field theory for classical phase fluctuations in a clean $d$-wave superconductor suffers from singularities which make the gradient expansion of the fluctuation energy impossible. This means, in particular, that the physics of classical phase fluctuations cannot be described by the $XY$-model at low temperatures. In the presence of disorder, a well-defined gradient expansion of the Gaussian phase-only action is restored, and has the same form as that of an extended classical $XY$-model.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to S. Sharapov for useful discussions. This work was supported by the Natural Sciences and Engineering Research Council of Canada.
Condensate energy and non-linear Meissner effect {#app:Int}
================================================
In this Appendix we calculate the higher-order terms in the expansion of the effective action $$\label{Seff diff}
S_{eff}=S_0-\Tr\ln\tilde{\cal G}^{-1}+\Tr\ln {\cal G}_0^{-1},$$ in powers of the phase gradients. We consider only the limit of uniformly-moving condensate, when the superfluid velocity $\bvs$ is constant, so that $\theta_{\br}-\theta_{\br'}=2m\bvs(\br-\br')$. In the absence of the scalar potential, the self-energy (\[Sigma\]) becomes translationally invariant: $$\Sigma(\br,\tau;\br',\tau')
=\delta(\tau-\tau')\tau_3\xi_{\br\br'}
\bigl[e^{-i\tau_3m\bvs(\br-\br')}-1\bigr].$$ The gauge-transformed Green’s function (\[tilde G\]) becomes diagonal in the momentum space: $$\tilde{\cal G}^{-1}(\bk,\omega_n)=\left(\begin{array}{cc}
i\omega_n-\xi^+_{\bk} & -\Delta_{\bk} \\
-\Delta_{\bk} & i\omega_n+\xi^-_{\bk}
\end{array}\right),$$ where $\xi^{\pm}_{\bk}=\xi_{\bk\pm m\bvs}$, which allows one to calculate the operator traces: $$\Tr\ln\tilde{\cal G}^{-1}=\ln\Det\tilde{\cal G}^{-1}
=\beta{\cal V} T\sum\limits_n\int\frac{d^D\bk}{(2\pi)^D}\,
\ln\det\tilde {\cal G}^{-1}(\bk,\omega_n)$$ where ${\cal V}$ is the system volume and “det” denotes a $2\times 2$ matrix determinant in the electron-hole space. Inserting this in Eq. (\[Seff diff\]), we obtain $S_{eff}=S_0+\beta{\cal E}$, where $$\label{Seff nl}
{\cal E}=-{\cal V}T\sum\limits_n\int\frac{d^D\bk}{(2\pi)^D}
\ln\frac{(i\omega_n-\tilde E_{\bk,+})
(i\omega_n+\tilde E_{\bk,-})}{(i\omega_n-E_{\bk})(i\omega_n+E_{\bk})}$$ has the meaning of the kinetic energy of uniformly moving condensate, and $$\tilde E_{\bk,\pm}=\pm\frac{\xi^+_{\bk}-\xi^-_{\bk}}{2}+
\sqrt{\left(\frac{\xi^+_{\bk}+\xi^-_{\bk}}{2}\right)^2+\Delta_{\bk}^2}$$ are the quasiparticle energies affected by the superflow. Using the identity $$T\sum\limits_n\ln\frac{i\omega_n-\tilde E}{i\omega_n-E}
=T\ln\frac{\cosh\frac{\tilde E}{2T}}{\cosh\frac{E}{2T}},$$ the energy density can be written in the form $$\label{E int}
\frac{\cal E}{\cal V}=-T\int\frac{d^D\bk}{(2\pi)^D}\;
\ln\frac{\cosh\frac{\tilde E_{\bk,+}}{2T}
\cosh\frac{\tilde E_{\bk,-}}{2T}}{\cosh^2\frac{E_{\bk}}{2T}}.$$ At $T=0$, this becomes $$\label{E zeroT}
\frac{\cal E}{\cal
V}=-\frac{1}{2}\int\frac{d^D\bk}{(2\pi)^D}\,
\bigl(|\tilde E_{\bk,+}|+|\tilde E_{\bk,+}|-2E_{\bk}\bigr).$$
We focus now on the case of a two-dimensional $d$-wave superconductor. Assuming for simplicity a Galilean-invariant case, characterized by the parabolic dispersion $\xi_{\bk}=\bk^2/2m-\mu$, with the effective mass equal to the bare electron mass $m$, we have $$\tilde E_{\bk,\pm}=\sqrt{(\xi_{\bk}+\zeta)^2+
\Delta_{\bk}^2}\pm\bk\bvs=E_{\zeta,\bk}\pm\bk\bvs,$$ where $\bk\bvs$ is the so-called Doppler shift of the quasiparticle energy in the presence of moving condensate, and $\zeta=m\bvs^2/2$. The expression (\[E zeroT\]) can then be written as ${\cal E}/{\cal V}=({\cal E}/{\cal V})_{reg}+({\cal
E}/{\cal V})_{Dopp}$. The first term, $$\label{E reg}
\left(\frac{\cal E}{\cal V}\right)_{reg}=-\int\frac{d^2\bk}{(2\pi)^2}\;
(E_{\zeta,\bk}-E_{\bk})
=-\zeta\int\frac{d^2\bk}{(2\pi)^2}\;\frac{\xi_{\bk}}{E_{\bk}}+
O(\zeta^2)=\frac{\rho_0\bvs^2}{2}+O(\bvs^4),$$ has a well-defined Taylor expansion in powers of $\bvs$. Note that the nodal approximation cannot be used here because of the contributions from the regions far from the Fermi surface, see Sec. \[sec:2D-class\]. In contrast, the second term, $$\label{E Dopp}
\left(\frac{\cal E}{\cal V}\right)_{Dopp}=-\frac{1}{2}
\int\frac{d^2\bk}{(2\pi)^2}\;\Bigl[|E_{\zeta,\bk}+\eta|+
|E_{\zeta,\bk}-\eta|-2E_{\zeta,\bk}\Bigr],$$ can be calculated in the nodal approximation. Using Eqs. (\[nodal\_approx\],\[nodal int\]), we obtain $$\left(\frac{\cal E}{\cal V}\right)_{Dopp}=-\frac{1}{12\pi
v_Fv_\Delta}\sum\limits_{n=1}^4|\bk_n\bvs|^3,$$ which is a non-analytical function of $\bvs$. Putting all pieces together, we arrive at the following result: $$\label{E result}
\frac{\cal E}{\cal V}=\frac{\rho_0\bvs^2}{2}-
\frac{m^3v_F^2}{12\sqrt{2}\pi v_\Delta}\bigl(|v_{s,x}+v_{s,y}|^3+
|v_{s,x}-v_{s,y}|^3\bigr)+O(\bvs^4).$$ We see that the nodal quasiparticles make the kinetic energy of the condensate a non-analytical function of the superfluid velocity.[@Vol97] This singular behavior is closely related to the so-called non-linear Meissner effect:[@non-lin] the screening supercurrent acts as a pair-breaker in $d$-wave superconductors, creating a finite density of normal excitations even at $T=0$. The depletion of the supercurrent by those excitations leads to a non-analytical dependence of the electromagnetic response functions on $\bvs$ and therefore the external magnetic field.
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abstract: 'We show that the conditions which originate the spin and pseudospin symmetries in the Dirac equation are the same that produce equivalent energy spectra of relativistic spin-1/2 and spin-0 particles in the presence of vector and scalar potentials. The conclusions do not depend on the particular shapes of the potentials and can be important in different fields of physics. When both scalar and vector potentials are spherical, these conditions for isospectrality imply that the spin-orbit and Darwin terms of either the upper component or the lower component of the Dirac spinor vanish, making it equivalent, as far as energy is concerned, to a spin-0 state. In this case, besides energy, a scalar particle will also have the same orbital angular momentum as the (conserved) orbital angular momentum of either the upper or lower component of the corresponding spin-1/2 particle. We point out a few possible applications of this result.'
author:
- 'P. Alberto'
- 'A. S. de Castro'
- 'M. Malheiro'
title: 'Spin and pseudospin symmetries and the equivalent spectra of relativistic spin-1/2 and spin-0 particles'
---
When describing some strong interacting systems it is often useful, because of simplicity, to approximate the behavior of relativistic spin-1/2 particles by scalar spin-0 particles obeying the Klein-Gordon equation. An example is the case of relativistic quark models used for studying quark-hadron duality because of the added complexity of structure functions of Dirac particles as compared to scalar ones. It turns out that some results (e.g., the onset of scaling in some structure functions) almost do not depend on the spin structure of the particle [@jeschonnek]. In this work we will give another example of an observable, the energy, whose value may not depend on the spinor structure of the particle, i.e., whether one has a spin-1/2 or a spin-0 particle. We will show that when a Dirac particle is subjected to scalar and vector potentials of equal magnitude, it will have exactly the same energy spectrum as a scalar particle of the same mass under the same potentials. As we will see, this happens because the spin-orbit and Darwin terms in the second-order equation for either the upper or lower spinor component vanish when the scalar and vector potentials have equal magnitude. It is not uncommon to find physical systems in which strong interacting relativistic particles are subject to Lorentz scalar potentials (or position-dependent effective masses) that are of the same order of magnitude of potentials which couple to the energy (time components of Lorentz four-vectors). For instance, the scalar and vector (hereafter meaning time-component of a four-vector potential) nuclear mean-field potentials have opposite signs but similar magnitudes, whereas relativistic models of mesons with a heavy and a light quark, like D- or B-mesons, explain the observed small spin-orbit splitting by having vector and scalar potentials with the same sign and similar strengths [@gino_mesons].
It is well-known that all the components of the free Dirac spinor, *i.e.*, the solution of the free Dirac equation, satisfy the free Klein-Gordon equation. Indeed, from the free Dirac equation $$\label{free_Dirac} (i\hbar\gamma^\mu\partial_\mu-mc)\Psi=0$$ one gets $$\label{free_KG_Dirac}
(-i\hbar\gamma^\nu\partial_\nu-mc)(i\hbar\gamma^\mu\partial_\mu-mc)\Psi=
(\hbar^2\partial^\mu\partial_\mu+m^2c^2)\Psi=0 \ ,$$ where use has been made of the relation $\gamma^\mu\gamma^\nu\partial_\mu\partial_\nu=\partial_\mu\partial^\mu$. In a similar way, for the time-independent free Dirac equation we would have $$\label{free_t_ind_Dirac} (c\,\bmath\alpha\cdot\bmath p +\beta mc^2
)\psi=(-i\hbar c\,\bmath\alpha\cdot\nabla+\beta mc^2)\psi=E\psi \ ,$$ where, as usual, $\psi(\bmath r)=\Psi(\bmath r,t)\,\exp{(i\,E\,t/\hbar)}$, $\bmath\alpha=\gamma^0\bmath\gamma$ and $\beta=\gamma^0$. Then, by left multiplying Eq. (\[free\_t\_ind\_Dirac\]) by $c\bmath\alpha\cdot\bmath p+\beta mc^2$, one gets the time-independent free Klein-Gordon equation $$\label{free_t_ind_KG}
(c^2\bmath{p}^2+m^2c^4)\psi=(-\hbar^2c^2\nabla^2+m^2c^2)\psi=E^2\psi\
,$$ where the relation $\{\beta,\bmath\alpha\}=0$ was used. This all means that the free four-component Dirac spinor, and of course all of its components, satisfy the Klein-Gordon equation. This is not surprising, because, after all, both free spin-1/2 and spin-0 particles obey the same relativistic dispersion relation, $E^2={\bmath p}^2c^2+m^2c^4$, in spite of having different spinor structures and thus different wave functions. Since there is no spin-dependent interaction, one expects both to have the same energy spectrum.
We consider now the case of a spin-1/2 particle subject to a Lorentz scalar potential $V_s$ plus a vector potential $V_v$. The time-independent Dirac equation is given by $$\label{dirac_V_S} [c\,\bmath\alpha\cdot\bmath p
+\beta(mc^2+V_s)]\psi=(E-V_v)\psi$$ It is convenient to define the four-spinors $\psi_\pm=P_\pm\psi=[(I\pm\beta)/2]\,\psi$ such that $$\psi_+=\left(\begin{array}{c}
\phi\\[2mm] 0
\end{array}\right)\qquad \psi_-=\left(\begin{array}{c}
0\\[2mm] \chi
\end{array}\right) \ ,$$ where $\phi$ and $\chi$ are respectively the upper and lower two-component spinors. Using the properties and anti-commutation relations of the matrices $\beta$ and $\bmath\alpha$ we can apply the projectors $P_\pm$ to the Dirac equation (\[dirac\_V\_S\]) and decompose it into two coupled equations for $\psi_+$ and $\psi_-$: $$\begin{aligned}
\label{psi-1}
c\,\bmath\alpha\cdot\bmath p\,\psi_-+(mc^2+V_s)\psi_+ &=&(E-V_v)\psi_+ \\
\label{psi+1}
c\,\bmath\alpha\cdot\bmath
p\,\psi_+-(mc^2+V_s)\psi_-&=&(E-V_v)\psi_- \ .\end{aligned}$$ Applying the operator $c\,\bmath\alpha\cdot\bmath p$ on the left of these equations and using them to write $\psi_+$ and $\psi_-$ in terms of $\bmath\alpha\cdot\bmath p\,\psi_-$ and $\bmath\alpha\cdot\bmath p\,\psi_+$ respectively, we finally get second-order equations for $\psi_+$ and $\psi_-$: $$\begin{aligned}
c^2\bmath p^2\,\psi_++c^2\,\frac{[\bmath\alpha\cdot\bmath p\,\Delta]
\bmath\alpha\cdot\bmath p\, \psi_+}{E-\Delta+mc^2}
&=&(E-\Delta+mc^2)(E-\Sigma-mc^2)\psi_+ \\
c^2\bmath p^2 \,\psi_-+c^2\,\frac{[\bmath\alpha\cdot\bmath p\,\Sigma]
\bmath\alpha\cdot\bmath p\, \psi_-}{E-\Sigma-mc^2}
&=&(E-\Delta+mc^2)(E-\Sigma-mc^2)\psi_-\end{aligned}$$ where the square brackets $[\ ]$ mean that the operator $\bmath\alpha\cdot\bmath p$ only acts on the potential in front of it and we defined $\Sigma=V_v+V_s$ and $\Delta=V_v-V_s$. The second term in these equations can be further elaborated noting that the Dirac $\alpha_i$ matrices satisfy the relation $
\alpha_i\alpha_j=\delta_{ij}+\frac2\hbar\,i\epsilon_{ijk} S_k$ where $S_k$, $k=1,2,3$, are the spin operator components. The second-order equations read now $$\begin{aligned}
\label{2nd_psi_+}
c^2 \,\bmath p^2\,\psi_++c^2\,\frac{[\bmath p\,\Delta]
\cdot\bmath p\, \psi_+ + \frac{2i}{\hbar}\,[\bmath
p\,\Delta]\times\bmath p\cdot \bmath S\, \psi_+}{E-\Delta+mc^2}\,
&=&(E-\Delta+mc^2)(E-\Sigma-mc^2)\psi_+ \\
\label{2nd_psi_-}
c^2\,\bmath p^2 \,\psi_-+c^2\,\frac{[\bmath p\,\Sigma]
\cdot\bmath p\, \psi_- + \frac{2i}{\hbar}\,[\bmath
p\,\Sigma]\times\bmath p\cdot \bmath S\, \psi_-}{E-\Sigma-mc^2}\,
&=&(E-\Delta+mc^2)(E-\Sigma-mc^2)\psi_- .\end{aligned}$$
Now, if $\bmath p\,\Delta=0$, meaning that $\Delta$ is constant or zero (if $\Delta$ goes to zero at infinity, the two conditions are equivalent), then the second term in eq. (\[2nd\_psi\_+\]) disappears and we have $$c^2\,\bmath p^2 \psi_+
=(E-\Delta+mc^2)(E-\Sigma-mc^2)\psi_+=[(E-V_v)^2-(mc^2+V_s)^2]\psi_+\ ,$$ which is precisely the time-independent Klein-Gordon equation for a scalar potential $V_s$ plus a vector potential $V_v$[^1]. Since the second-order equation determines the eigenvalues for the spin-1/2 particle, this means that when $\bmath p\,\Delta=0$, a spin-1/2 and a spin-0 particle with the same mass and subject to the same potentials $V_s$ and $V_v$ will have the same energy spectrum, including *both* bound and scattering states. This last sufficient condition for isospectrality can be relaxed to demand that just the combination $mc^2+V_s$ be the same for both particles, *allowing them to have different masses*. This is so because this weaker condition does not change the gradient of $\Delta$ and $\Sigma$ and therefore the condition $\bmath p\,\Delta=0$ will still hold. On the other hand, if the scalar and vector potentials are such that $\bmath p\,\Sigma=0$, we would obtain a Klein-Gordon equation for $\psi_-$, and again the spectrum for spin-0 and spin-1/2 particles would be the same, provided they are subjected to the same vector potential and $mc^2+V_s$ is the same for both particles. If both $V_s$ and $V_v$ are central potentials, *i.e.*, only depend on the radial coordinate, then the numerators of the second terms in equations (\[2nd\_psi\_+\]) and (\[2nd\_psi\_-\]) read $$\begin{aligned}
[\bmath p\,\Delta] \cdot\bmath p\, \psi_+ +
\frac{2i}{\hbar}\,[\bmath p\,\Delta]\times\bmath p\cdot \bmath S\,
\psi_+&=&-\hbar^2\Delta'\,\frac{\partial\psi_+}{\partial r}+\frac
2r\,\Delta'\bmath L\cdot \bmath S\,\psi_+ \\
\lbrack\bmath p\, \Sigma\rbrack \cdot\bmath p\, \psi_- +
\frac{2i}{\hbar}\,\lbrack\bmath p\,\Sigma\rbrack\times\bmath p\cdot
\bmath S\, \psi_-&=&-\hbar^2\Sigma'\,\frac{\partial\psi_-}{\partial
r}+\frac 2r\,\Sigma'\bmath L\cdot \bmath S\,\psi_-\ ,\end{aligned}$$ where $\Delta'$ and $\Sigma'$ are the derivatives with respect to $r$ of the radial potentials $\Delta(r)$ and $\Sigma(r)$, and $\bmath L=\bmath r\times\bmath p$ is the orbital angular momentum operator. From these equations ones sees that these terms, which set apart the Dirac second-order equations for the upper and lower components of the Dirac spinor from the Klein-Gordon equation and thus are the origin of the different spectra for spin-1/2 and spin-0 particles, are composed of a derivative term, related to the Darwin term which appears in the Foldy-Wouthuysen expansion, and a $\bmath
L\cdot \bmath S$ spin-orbit term. If $\Delta'=0$ ($\Sigma'=0$), then there is no spin-orbit term for the upper (lower) component of the Dirac spinor. In turn, since the second-order equation determines the energy eigenvalues, this means that the orbital angular momentum of the respective component is a good quantum number of the Dirac spinor. This can be a bit surprising, since one knows that in general the orbital quantum number is not a good quantum number for a Dirac particle, since $\bmath L^2$ does not commute with a Dirac Hamiltonian with radial potentials. The reason why this does not happen in these cases was reported in Refs. [@levi; @gin_rep], and we now review it in a slight different fashion. Let us consider in more detail the case of spherical potentials such that $\Delta'=0$. One knows that a spinor that is a solution of a Dirac equation with spherically symmetric potentials can be generally written as $$\label{psi_jm}
\psi_{jm}(\mbox{\boldmath $r$})=\left(
\begin{array}{c}
\displaystyle {\rm i}\frac{g_{j\,l}(r)}{r} \mathcal{Y}_{j\,l\,m}(\mbox{\boldmath $\hat{r}$}) \\[.2cm]
\displaystyle\frac{f_{j\,\tilde l}(r)}{r} \mathcal{Y}_{j\,\tilde l\,m}(\mbox{\boldmath $\hat{r}$})
\end{array}
\right)\, .$$ where $\mathcal{Y}_{j\,l\,m}$ are the spinor spherical harmonics. These result from the coupling of spherical harmonics and two-dimensional Pauli spinors $\chi_{m_s}$, $\mathcal{Y}_{j\,l\,m}=\sum_{m_s}\sum_{m_l}{ \langle \, l \, m_l \, ; \, 1/2 \, m_s \, | \,
j \, m \, \rangle }
Y_{l\,m_l}\chi_{m_s}$, where ${ \langle \, l \, m_l \, ; \, 1/2 \, m_s \, | \,
j \, m \, \rangle }$ is a Clebsch-Gordan coefficient and $\tilde l=l\pm1$, the plus and minus signs being related to whether one has aligned or anti-aligned spin, *i.e.*, $j=l\pm1/2$. The spinor spherical harmonics for the lower component satisfy the relation $\mathcal{Y}_{j\,\tilde l\,m}=-\bmath\sigma\cdot\bmath{\hat r}\,
\mathcal{Y}_{j\,l\,m}$. The fact that the upper and lower components have different orbital angular momenta is related to the fact, mentioned before, that $\bmath L^2$ does not commute with the Dirac Hamiltonian $$\label{H_P+_P-} H=c\,\bmath\alpha\cdot\bmath
p+\beta(V_s+mc^2)+V_v=c\,\bmath\alpha\cdot\bmath p+\beta mc^2+\Sigma
P_++\Delta P_-\ ,$$ where $P_\pm$ are the projectors defined above. However, when $\Delta'=0$, there is an extra SU(2) symmetry of $H$ (so-called “spin symmetry") as first shown by Bell and Ruegg [@bell]. When we have spherical potentials, Ginocchio showed that there is an additional SU(2) symmetry (for a recent review see [@gin_rep]). The generators of this last symmetry are $$\label{cal_L}
\bmath{\mathcal{L}}=\bmath L
P_++\frac1{p^2}\bmath\alpha\cdot\bmath p\,\bmath
L\,\bmath\alpha\cdot\bmath p\,P_-=
\left(\begin{array}{cc} \bmath L&0\\
0&U_p\,\bmath L\,U_p\end{array}\right)\ ,$$ where $U_p=\bmath\sigma\cdot\bmath p/(\sqrt{p^2})$ is the helicity operator. One can check that $\bmath{\mathcal{L}}$ commutes with the Dirac Hamiltonian, $$\begin{aligned}
\nonumber[H,\bmath{\mathcal{L}}]&=& [c\,\bmath\alpha\cdot\bmath
p,\bmath L P_++\frac1{p^2}\bmath\alpha\cdot\bmath p\,\bmath
L\,\bmath\alpha\cdot\bmath
p\,P_-]+[\Delta,\frac1{p^2}\bmath\alpha\cdot\bmath p\,\bmath
L\,\bmath\alpha\cdot\bmath p]+[\Sigma,\bmath L] \\
&=&[\Delta,\frac1{p^2}\bmath\alpha\cdot\bmath p\,\bmath
L\,\bmath\alpha\cdot\bmath p\, ]=0 \ ,\end{aligned}$$ where the last equality comes from the fact that $\Delta'=0$. The Casimir $\bmath{\mathcal{L}}^2$ operator is given by $\ds\bmath{\mathcal{L}}^2=\bmath L^2
P_++\frac1{p^2}\bmath\alpha\cdot\bmath p\,\bmath
L^2\,\bmath\alpha\cdot\bmath p\,P_-$. Applying this operator to the spinor $\psi_{jm}$ (\[psi\_jm\]), we get $$\begin{aligned}
\nonumber\bmath{\mathcal{L}}^2\psi_{jm}&=&\bmath L^2\psi_{jm}^+
+\frac1{p^2}\bmath\alpha\cdot\bmath p\,\bmath
L^2\,\bmath\alpha\cdot\bmath p\,\psi_{jm}^-
=\hbar^2l(l+1)\psi_{jm}^++ \frac{\bmath\alpha\cdot\bmath p\,c{\bmath
L}^2\,\psi_{jm}^+}{E-\Delta+mc^2}\\
\label{eigen_L^2}
&=&\hbar^2l(l+1)\psi_{jm}^++\hbar^2l(l+1)\psi_{jm}^-=\hbar^2l(l+1)\psi_{jm}
\ ,\end{aligned}$$ where $\psi_{jm}^{\pm}=P_{\pm}\psi_{jm}$ and we used the relation, valid when $\Delta'=0$, $\ds\psi_{jm}^+=(E-\Delta+mc^2)\frac{\bmath\alpha\cdot\bmath p}{c
p^2}\psi_{jm}^-$. From (\[eigen\_L\^2\]) we see that $\psi_{jm}$ is indeed an eigenstate of $\bmath{\mathcal{L}}^2$. Thus the orbital quantum number of the upper component $l$ is a good quantum number of the system when the spherical potentials $V_s(r)$ and $V_v(r)$ are such that $V_v(r)=V_s(r)+C_\Delta$, where $C_\Delta$ is an arbitrary constant. Also, according to we have said before, there is a state of a spin-0 particle subjected to these same spherical potentials (or, at least, with a scalar potential such that the sum $V_s+mc^2$ is the same) that has the same energy and the same orbital angular momentum as $\psi_{jm}$. In addition, the wave function of this scalar particle would be proportional to the spatial part of the wave function of the upper component.
Note that the generator of the “spin symmetry” $\bmath{\mathcal{S}}$ is given by a similar expression as (\[cal\_L\]) just replacing $\bmath L$ by $\hbar/2\,\bmath\sigma$ [@bell; @gin_rep], meaning that $\bmath{\mathcal{S}}^2\equiv
\bmath{S}^2=3/4\,\hbar^2 I$ so that spin is also a good quantum number, as would be expected. Actually, one can show that the total angular momentum operator $\bmath J$ can be written as $\bmath{\mathcal{L}}+\bmath{\mathcal{S}}$, so that $l$, $m_l$ (eigenvalue of $\mathcal{L}_z$), $s=1/2$, $m_s$ (eigenvalue of $\mathcal{S}_z$) are good quantum numbers. Then, of course, $j$ and $m=m_l+m_s$ are also good quantum numbers, but only in a trivial way, because there is no longer spin-orbit coupling. Therefore, in the spinor (\[psi\_jm\]) one could just replace the spinor spherical harmonic $\mathcal{Y}_{j\,l\,m}$ by $Y_{l\,m_l}\chi_{m_s}$ and $\mathcal{Y}_{j\,\tilde l\,m}$ by $-\bmath\sigma\cdot\bmath{\hat
r}\, Y_{l\,m_l}\chi_{m_s}$. Note that if $\Delta$ is a nonrelativistic potential, $\Delta\ll mc^2$ and $\Delta'\ll
m^2c^4/(\hbar c)$, *i.e.*, it is slowly varying over a Compton wavelength. In this case, the spin-orbit term will also get suppressed. In fact, the derivative of the $\Delta$ potential is the origin of the well-known relativistic spin-orbit effect which appears as a relativistic correction term in atomic physics or in the $v/c$ Foldy-Wouthuysen expansion (only the derivative of $V_v$ appears because usually no Lorentz scalar potential $V_s$ is considered, and therefore $\Delta=V_v$).
When $\Sigma'=0$, or $V_v(r)=-V_s(r)+C_\Sigma$, with $C_\Sigma$ an arbitrary constant, there is again a $SU(2)$ symmetry, usually called pseudospin symmetry ([@bell; @gino]) which is relevant for describing the single-particle level structure of several nuclei. This symmetry has a dynamical character and cannot be fully realized in nuclei because in Relativistic Mean-field Theories the $\Sigma$ potential is the only binding potential for nucleons [@alb1; @alb2]. For harmonic oscillator potentials this is no longer the case, since $\Delta$, acting as an effective mass going to infinity, can bind Dirac particles [@gino_oh; @nosso], even when $\Sigma=0$. As before, in the special case of spherical potentials, there is another SU(2) symmetry whose generators are $$\bmath{\tilde \mathcal{L}}=\frac1{p^2}\bmath\alpha\cdot\bmath
p\,\bmath L\,\bmath\alpha\cdot\bmath p\,P_++\bmath L P_-=
\left(\begin{array}{cc} U_p\,\bmath L\,U_p&0\\
0&\bmath L\end{array}\right)\ .$$ In the same way as before, applying $\bmath{\tilde \mathcal{L}}^2$ to $\psi_{jm}$, we would find that $\bmath{\tilde \mathcal{L}}^2\psi_{jm}=\hbar^2\tilde l(\tilde
l+1)\psi_{jm}$, that is, this time it is the orbital quantum number of the lower component $\tilde l$ which is a good quantum number of the system and can be used to classify energy levels. Again, provided the vector and scalar potentials are adequately related, there would be a corresponding state of a spin-0 particle with the same energy and same orbital angular momentum $\tilde l$, and, furthermore, its wave function would be proportional to the spatial part of the wave function of the lower component. As before, the pseudospin symmetry generator $\bmath{\tilde \mathcal{S}}$ can be obtained from $\bmath{\tilde \mathcal{L}}$ by replacing $\bmath L$ by $\hbar/2\,\bmath\sigma$. The good quantum numbers of the system would be, besides $\tilde l$, $m_{\tilde l}$, $\tilde s\equiv s=1/2$ and $m_{\tilde s}$. Again, $\bmath
J=\bmath{\tilde\mathcal{L}}+\bmath{\tilde\mathcal{S}}$. It is interesting that, as has been noted by Ginocchio [@gino_oh], the generators of spin and pseudospin symmetries are related through a $\gamma^5$ transformation since $\bmath{\tilde\mathcal{S}}=\gamma^5\bmath{\mathcal{S}}\gamma^5$ and $\bmath{\tilde\mathcal{L}}=\gamma^5\bmath{\mathcal{L}}\gamma^5$. This property was used in a recent work to relate spin symmetric and pseudospin symmetric spectra of harmonic oscillator potentials [@Castro_OH1+1]. There it was shown that for massless particles (or ultrarelativistic particles) the spin- and pseudo-spin spectra of Dirac particles are the same. In addition, this means that spin-symmetric massless eigenstates of $\gamma^5$ would be also pseudo-spin symmetric and vice-versa. Since in this case $\Delta=\Sigma=0$, or $V_v=V_s=0$, this is, of course, just another way of stating the well-known fact that free massless Dirac particles have good chirality.
Naturally, for free spin-1/2 particles described by spherical waves, *both* $l$ and $\tilde l$ are good quantum numbers, which just reflects the fact that one can have free spherical waves with any orbital angular momentum for the upper or lower component and still have the same energy, as long as their linear momentum magnitude is the same, or, put in another way, the energy of a free spin-1/2 particle cannot depend on its direction of motion.
In summary, we showed that when a relativistic spin-1/2 particle is subject to vector and scalar potentials such that $V_v=\pm
V_s+C_\pm$, where $C_\pm$ are constants, its energy spectrum does not depend on their spinorial structure, being identical to the spectrum of a spin-0 particle which has no spinorial structure. This amounts to say that if the potentials have these configurations there is no spin-orbit coupling and Darwin term. If the scalar and vector potentials are spherical, one can classify the energy levels according to the orbital angular momentum quantum number of either the upper or the lower component of the Dirac spinor. This would then correspond to having a spin-0 particle with orbital angular momentum $l$ or $\tilde l$, respectively. This spectral identity can of course happen only with potentials which do not involve the spinorial structure of the Dirac equation in an intrinsic way. For instance, a tensor potential of the form ${\rm i}\beta\sigma^{\mu\nu}(\partial_\mu A_\nu-\partial_\nu A_\mu)$ does not have an analog in the Klein-Gordon equation, so that one could not have a spin-0 particle with the same spectrum as a spin-1/2 particle with such a potential. This is the case of the so-called Dirac oscillator [@dirac_osc] (see [@nosso] for a complete reference list), in which the Dirac equation contains a potential of the form ${\rm i}\beta\sigma^{0i}m\omega r_i={\rm i}
m\omega\beta\bmath\alpha\cdot\bmath r$. Another important potential, the electromagnetic vector potential $\bmath A$, which is the spatial part of the electromagnetic four-vector potential, can be added via the minimal coupling scheme to both the Dirac and the Klein-Gordon equations. Since $\bmath\alpha\cdot(\bmath p-e\bmath
A)\bmath\alpha\cdot(\bmath p-e\bmath A)= (\bmath p-e\bmath
A)^2+2e\hbar\nabla\times\bmath A\cdot\bmath S$, the spectra of spin-0 and spin-1/2 particles cannot be identical as long as there is a magnetic field present, even though the condition $V_v=\pm
V_s+C_\pm$ is fulfilled. It is important also to remark that, since for an electromagnetic interaction $V_v$ is the time-component of the electromagnetic four-vector potential, this last condition is gauge invariant in the present case, in which we are dealing with stationary states, *i.e*, time-independent potentials. So, in the absence of a external magnetic field (allowing, for instance, an electromagnetic vector potential $\bmath A$ which is constant or a gradient of a scalar function), a spin-0 and spin-1/2 particle subject to the same electromagnetic potential $V_v$ and a Lorentz scalar potential fulfilling the above relation would have the same spectrum.
The remark made above about the similarity of spin-0 and spin-1/2 wave functions can be relevant for calculations in which the observables do not depend on the spin structure of the particle, like some structure functions. One such calculation was made by Paris [@Paris] in a massless confined Dirac particle, in which $V_v=V_s$. It would be interesting to see how a Klein-Gordon particle would behave under the same potentials. More generally, this spectral identity can also have experimental implications in different fields of physics, since, should such an identity be found, it would signal the presence of a Lorentz scalar field having a similar magnitude as that of a time-component of a Lorentz vector field, or at least differing just by a constant.
We acknowledge financial support from CNPQ, FAPESP and FCT (POCTI) scientific program.
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[^1]: There are some authors who introduce a scalar potential ${\cal V}_s$ in the Klein-Gordon equation by making the replacement $m^2c^4\to
m^2c^4+{\cal V}_s^2$. Here we introduce it, as most authors do, as an effective mass $m^{*\,2}=(m+V_s/c^2)^2$, since it is the way that it is introduced in the Dirac equation. The two potentials are related by ${\cal V}_s^2=(mc^2+V_s)^2-m^2c^4$.
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---
author:
- 'F. Yan'
- 'N. Casasayas-Barris'
- 'K. Molaverdikhani'
- 'F. J. Alonso-Floriano'
- 'A. Reiners'
- 'E. Pallé'
- 'Th. Henning'
- 'P. Mollière'
- 'G. Chen'
- 'L. Nortmann'
- 'I. A. G. Snellen'
- 'I. Ribas, A. Quirrenbach, J. A. Caballero, P. J. Amado, M. Azzaro, F. F. Bauer, M. Cortés Contreras, S. Czesla, S. Khalafinejad, L. M. Lara, M. López-Puertas, D. Montes, E. Nagel, M. Oshagh, A. Sánchez-López, M. Stangret'
- 'M. Zechmeister'
bibliography:
- 'CaII-refer.bib'
date: 'Received 29 July 2019; Accepted 30 October 2019'
title: 'Ionized calcium in the atmospheres of two ultra-hot exoplanets WASP-33b and KELT-9b'
---
[Ultra-hot Jupiters are emerging as a new class of exoplanets. Studying their chemical compositions and temperature structures will improve the understanding of their mass loss rate as well as their formation and evolution. We present the detection of ionized calcium in the two hottest giant exoplanets – KELT-9b and WASP-33b. By utilizing transit datasets from CARMENES and HARPS-N observations, we achieved high confidence level detections of using the cross-correlation method. We further obtain the transmission spectra around the individual lines of the H&K doublet and the near-infrared triplet, and measure their line profiles. The H&K lines have an average line depth of 2.02 $\pm$ 0.17 $\%$ (effective radius of 1.56 $R_\mathrm{p}$) for WASP-33b and an average line depth of 0.78 $\pm$ 0.04 $\%$ (effective radius of 1.47 $R_\mathrm{p}$) for KELT-9b, which indicates that the absorptions are from very high upper atmosphere layers close to the planetary Roche lobes. The observed lines are significantly deeper than the predicted values from the hydrostatic models. Such a discrepancy is probably a result of hydrodynamic outflow that transports a significant amount of into the upper atmosphere. The prominent detection with the lack of significant detection implies that calcium is mostly ionized in the upper atmospheres of the two planets. ]{}
Introduction
============
Ultra-hot Jupiters (UHJs) are a new class of exoplanets emerging in the recent years. They are highly irradiated gas giants with day-side temperatures that are typically $\gtrsim$ 2200 K [@Parmentier2018]. Most of these planets orbit very close to A- or F-type stars. Their extremely high day-side temperatures cause thermal dissociation of molecules and ionization of atoms [@Arcangeli2018; @Lothringer2018]. Depending on the heat transport efficiency, different chemical components can form at their night-sides as well as terminators [@Parmentier2018; @Bell2018; @Helling2019]. Furthermore, the strong stellar ultraviolet (UV) and/or extreme-ultraviolet irradiation causes significant mass loss, affecting the planetary atmospheric composition and evolution [@Bisikalo2013; @Fossati2018].
Observations of UHJs have revealed peculiar properties of their atmospheres. For example, [@Kreidberg2018] found the absence of $\mathrm{H_2O}$ features at the day-side atmosphere of WASP-103b and they attributed this to the thermal dissociation of $\mathrm{H_2O}$. [@Arcangeli2018] analyzed the day-side spectrum of WASP-18b and found that molecules are thermally dissociated while the $\mathrm{H^-}$ ion opacity becomes important. [@Yan2018] detected strong hydrogen $\mathrm{H\alpha}$ absorption in the transmission spectrum of KELT-9b, which indicates that the planet has a hot escaping hydrogen atmosphere. The $\mathrm{H\alpha}$ line was also detected in two other UHJs: MASCARA-2b [@Casasayas-Barris2018] and WASP-12b [@Jensen2018]. Some atomic/ionic metal lines are also detected in UHJs, for instance, [@Fossati2010] detected in WASP-12b using UV transmission spectroscopy with the *Hubble Space Telescope* and various metal elements (including Fe, Ti, Mg, and Na) have been discovered in KELT-9b [@Hoeijmakers2018; @Cauley2019; @Hoeijmakers2019].
Theoretically, calcium should exist and probably get ionized into in the upper atmosphere of UHJs. [@Khalafinejad2018] analyzed the near-infrared triplet during the transit of the hot gas giant WASP-17b but did not detect any signals. Very recently, the near infrared triplet lines were detected for the first time in MASCARA-2b, an UHJ with equilibrium temperature $T_\mathrm{eq}$ $\sim$ 2260K [@Casasayas-Barris2019]. can also exist in the exospheres of rocky planets [@Mura2011]. For example, is detected in the exosphere of Mercury [@Vervack2010]. [@Ridden-Harper2016] searched for in the exosphere of the hot rocky planet 55 Cancri e and they found a tentative signal of in one of the four transit datasets. [@Guenther2011] attempted to detect calcium in the exosphere of another hot rocky planet, Corot-7b, but were only able to derive an upper limit of the amount of calcium in the exosphere.
Here we report the detections of in two UHJs: KELT-9b and WASP-33b. KELT-9b ($T_\mathrm{eq}$ $\sim$ 4050 K) is the hottest exoplanet discovered so far [@Gaudi2017] and its host star is a fast-rotating early A-type star. Hydrogen Balmer lines and several kinds of metals [@Yan2018; @Hoeijmakers2018; @Cauley2019], but not , have been detected in its atmosphere. WASP-33b ($T_\mathrm{eq}$ $\sim$ 2710 K) is the second hottest giant exoplanet, and it orbits a fast-rotating A5-type star [@Cameron2010]. The host star is a $\mathrm{\delta}$ Scuti variable [@Herrero2011; @Essen2014]. A temperature inversion, as well as TiO and evidence of AlO, have been detected in the planet [@Hayne2015; @Nugroho2017; @Essen2019].
The paper is organized as follows. We present the transit observations of the two planets in Section 2. In Section 3, we describe the method to obtain the transmission spectrum of the five lines – the two H&K doublet lines and the three near-infrared triplet (IRT) lines. In Section 4, we present the results and discussions including the cross-correlation signal, transmission spectra of individual lines, mixing ratios of and , and comparison with models. Conclusions are presented in Section 5.
Observations
============
For each of the two planets, we analyzed one transit dataset from CARMENES, which covers the IRT lines and one transit dataset from HARPS-North (HARPS-N), which covers the H&K lines.
WASP-33b observations
---------------------
We observed two transits of WASP-33b. The first transit was observed on 5 January 2017 with the CARMENES [@Quirrenbach2018] spectrograph, installed at the 3.5 m telescope of the Calar Alto Observatory. The CARMENES visual channel has a high spectral resolution (R $\sim$ 94600) and a wide spectral coverage (520 – 960nm). A continuous observing sequence of $\sim$ 4.5 hours was performed, covering 0.7 hour before transit and 0.8 hour after transit. The exposure time was set to 120 s, but the first 19 spectra had shorter exposure times (ranging from 65 s to 120 s). The data reduction of the raw spectra was performed with the CARACAL pipeline [@Caballero2016], which includes bias, flat and cosmic ray corrections, and wavelength calibration. The spectrum produced by the pipeline is at vacuum wavelength and in the Earth’s rest frame. We converted the wavelengths into air wavelengths in our study.
The second transit was observed on 8 November 2018 with the HARPS-N spectrograph mounted on the Telescopio Nazionale Galileo. The instrument has a resolution of R $\sim$ 115000 and a wavelength coverage of 383 – 690nm. The observation lasted for 9 hours, and we obtained 141 spectra. The raw data were reduced with the HARPS-N pipeline (Data Reduction Software). The pipeline produces order-merged, one-dimensional spectra with a re-sampled wavelength step of 0.01 $\mathrm{\AA}$. The barycentric Earth radial velocity was already corrected by the pipeline, but we converted it back into the Earth’s rest frame in order to be consistent with the CARMENES data. The observation logs of the two transits are summarized in Table \[obs\_log\].
KELT-9b observations
--------------------
We used archival data of one transit from CARMENES observations and one transit from HARPS-N observations. The details of the two observations were described in [@Yan2018] and [@Hoeijmakers2018], respectively (see Table \[obs\_log\] for summaries). The CARMENES observation was performed under a partially cloudy weather, thus the spectral signal-to-noise ratio (SNR) was relatively low, and part of the observation was lost due to clouds passing by.
Instrument Wavelength coverage Date Exposure time \[s\] $N_\mathrm{spectra}$
---------- ------------ ----------------------------------- ------------ --------------------- ---------------------- --
WASP-33b CARMENES 520 – 960 nm (contains IRT lines) 2017-01-05 120 93
HARPS-N 383 – 690 nm (contains H&K lines) 2018-11-08 200 141
KELT-9b CARMENES 520 – 960 nm (contains IRT lines) 2017-08-06 300, 400 48
HARPS-N 383 – 690 nm (contains H&K lines) 2017-07-31 600 49
The first 19 spectra had exposure times shorter than 120 s.
Method
======
We used two different methods to search for and study the lines: the cross-correlation method and the direct transmission spectrum of individual lines. The cross-correlation of high-resolution spectroscopic observations with theoretical model spectra has proven to be a powerful and robust technique to detect molecular and atomic species in exoplanet atmospheres [e. g. @Snellen2010; @Alonso-Floriano2019]. The direct transmission spectrum method allows detailed study of the line profiles and direct comparison with models [e. g. @Wyttenbach2015; @Yan2018]. In this work, we focused on the H&K lines (K line 3933.66 $\AA$, H line 3968.47 $\AA$) and the IRT lines (8498.02 $\AA$, 8542.09 $\AA$, 8662.14 $\AA$). These five lines are the strongest lines in the observed wavelength range.
Obtaining the transmission spectral matrix
------------------------------------------
The transmission spectrum of each line was retrieved separately. We firstly normalized the spectrum and then removed the telluric and stellar lines.
The telluric removal was performed in the Earth’s rest frame. The telluric lines in the wavelength range of the lines are mostly $\mathrm{H_2O}$ lines. We employed a theoretical $\mathrm{H_2O}$ transmission model described in [@Yan2015b]. The model was used to fit and remove the $\mathrm{H_2O}$ lines (Fig. \[telluric-remove\]).
The removal of the stellar lines was performed by dividing each spectrum with the out-of-transit master spectrum. The master spectrum was obtained by averaging all the observed out-of-transit spectra with their continuum SNRs as weights. Before obtaining the master spectrum, we aligned all the spectra to the stellar rest frame by correcting the barycentric Earth radial velocity and systemic velocity (–3.0 kms$^{-1}$ for WASP-33b and –20.6 kms$^{-1}$ for KELT-9b). By performing such a division, the residual spectra during transit should contain the transmission signal from the planetary atmosphere, while the spectra out-of-transit should be normalized to unity.
![Example of telluric removal. The upper line is the normalized spectrum around two of the IRT lines of KELT-9. The lower line is the telluric absorption corrected spectrum (shifted down by 0.3 for clarity).[]{data-label="telluric-remove"}](telluric-remove.png){width="49.00000%"}
Correction of stellar RM and CLV effects
----------------------------------------
During the exoplanet transit, the stellar line profile changes due to the Rossiter-McLaughlin (RM) effect and the center-to-limb variation (CLV) effect. The RM effect [@Rossiter1924; @McLaughlin1924; @Queloz2000] causes line profile distortions due to the stellar rotation. The CLV effect is the variation of stellar lines across the stellar disk’s center to limb [@Yan2015a; @Czesla2015; @Yan2017], and the effect is mostly the result of differential limb darkening between the stellar line and the adjacent continuum.
These two effects are encoded in the obtained transmission spectral matrix and the strength of these effects depends on the actual stellar and planetary parameters. The evaluation and correction of the RM and CLV effects are required for transmission spectrum studies. The effects of different lines and different planets have been evaluated, for example, by [@Casasayas-Barris2018], [@Yan2018], [@Salz2018], [@Nortmann2018], and [@Keles2019].
We modeled and corrected the RM and CLV effects simultaneously following the method in [@Yan2018]. The details of the CLV-only model are described in [@Yan2017] and we included the RM effect by assigning the rotational RV to each of the stellar surface elements as done in [@Yan2018]. The RM effect is generally stronger than the CLV effect for these fast rotating stars (c.f. Fig. \[Seperate-CLV+RM\] for RM-only and CLV-only models). Actually, the correction of the RM effect is a crucial step in performing the transmission spectroscopy of UHJs because their host stars are normally early-type stars that are fast rotators. The stellar and planetary parameters used for the systems of WASP-33b and KELT-9b are listed in Tables \[paras\_W33\] and \[paras\_K9\], respectively.
The planetary orbit of WASP-33b undergoes a nodal precession as discovered by [@Johnson2015]. They measured the changes of orbital inclination and the spin-orbit misalignment angle at two different epochs (2008 and 2014) using Doppler tomography. Measuring the current spin-orbit misalignment during our observations would require additional data reduction procedures such as filtering the stellar pulsation [@Johnson2015], which is beyond the scope of this paper. Therefore, we adopted the change rates and calculated the expected orbital inclination and spin-orbit angle at the dates of our observations.
As noted by [@Yan2018], the actual effective radius at a given spectral line is larger than 1 $R_\mathrm{p}$ because of the planetary atmospheric absorption. Consequently, the model of the RM + CLV effects should be built with a larger effective radius. We introduced a factor $f$ to account for such an effect by assuming that the actual stellar line profile change is $f$ times the simulated RM + CLV effects with 1 $R_\mathrm{p}$. In addition, our model has intrinsic errors. For example, our model is in 1D local thermodynamic equilibrium (LTE), while the actual stellar profile is better characterized by a 3D non-LTE stellar model. Thus, such an $f$ factor can also account for the errors of the model. By fitting the observed line profile change with the models using a Markov chain Monte Carlo analysis [@Mackey2013], we obtained $f = 2.1 \pm 0.1$ for the K line and $f = 1.6 \pm 0.2$ for the H line. For all the other lines, we simply used the 1 $R_\mathrm{p}$ scenario as the data do not have sufficient quality to obtain an $f$ value.
Fig. \[CaIIK-SME\] presents the model result of the 3933.66 $\AA$ line of KELT-9b as an example. The stellar RM and CLV effects dominate the line profile change. After correcting these stellar effects, we were able to detect the planetary absorption clearly. The effects of the lines behave differently from the effects of the $\mathrm{H\alpha}$ line [Supplementary Fig. 2 in @Yan2018], demonstrating the importance of modeling the RM and CLV effects for individual lines.
Parameter Symbol \[unit\] Value
------------------------- ----------------------------------- ---------------------------
*The star*
Effective temperature $T_\mathrm{eff} [K]$ 7430 $\pm$ 100
Radius $R_\star$ \[$R_\odot$\] 1.509$_{-0.027}^{+0.016}$
Mass $M_\star$ \[$M_\odot$\] 1.561$_{-0.079}^{+0.045}$
Metallicity \[Fe/H\] \[dex\] –0.1 $\pm$ 0.2
Rotational velocity $v$ sin $i_\star$ \[kms$^{-1}$\] 86.63$_{-0.32}^{+0.37}$
Systemic velocity $v_\mathrm{sys}$ \[kms$^{-1}$\] –3.0 $\pm$ 0.4
*The planet*
Radius $R_\mathrm{p}$ \[$R_\mathrm{J}$\] 1.679$_{-0.030}^{+0.019}$
Mass $M_\mathrm{p}$ \[$M_\mathrm{J}$\] 2.16 $\pm$ 0.20
Orbital semi-major axis $a$ \[$R_\star$\] 3.69 $\pm$ 0.05
Orbital period $P$ \[d\] 1.219870897
Transit epoch (BJD) $T_\mathrm {0}$ \[d\] 2454163.22449
Transit depth $\delta$ \[%\] 1.4
RV semi-amplitude $K_\mathrm{p}$ \[kms$^{-1}$\] 231 $\pm$ 3
Equilibrium temperature $T_\mathrm{eq}$ \[K\] 2710 $\pm$ 50
*2017 January 5*
Orbital inclination $i$ \[deg\] 89.50
Spin-orbit inclination $\lambda$ \[deg\] –114.05
*2018 November 8*
Orbital inclination $i$ \[deg\] 90.14
Spin-orbit inclination $\lambda$ \[deg\] –114.93
: Parameters of the WASP-33b system.[]{data-label="paras_W33"}
Adopted from [@Lehmann2015] with parameters form [@Kovacs2013].
Adopted from [@Johnson2015].
Adopted from [@Nugroho2017].
Adopted from [@Maciejewski2018].
Predicted value using parameters in [@Johnson2015].
Parameter Symbol \[unit\] Value
------------------------- ----------------------------------- ---------------------------
*The star*
Effective temperature $T_\mathrm{eff} [K]$ 10170 $\pm$ 450
Radius $R_\star$ \[$R_\odot$\] 2.362$_{-0.063}^{+0.075}$
Mass $M_\star$ \[$M_\odot$\] 2.52$_{-0.20}^{+0.25}$
Metallicity \[Fe/H\] \[dex\] -0.03 $\pm$ 0.2
Rotational velocity $v$ sin $i_\star$ \[kms$^{-1}$\] 111.4 $\pm$ 1.3
Systemic velocity $v_\mathrm{sys}$ \[kms$^{-1}$\] -20.6 $\pm$ 0.1
*The planet*
Radius $R_\mathrm{p}$ \[$R_\mathrm{J}$\] 1.891$_{-0.053}^{+0.061}$
Mass $M_\mathrm{p}$ \[$M_\mathrm{J}$\] 2.88 $\pm$ 0.84
Orbital semi-major axis $a$ \[$R_\star$\] 3.15 $\pm$ 0.09
Orbital period $P$ \[d\] 1.4811235
Transit epoch (BJD) $T_\mathrm {0}$ \[d\] 2457095.68572
Transit depth $\delta$ \[%\] 0.68
RV semi-amplitude $K_\mathrm{p}$ \[kms$^{-1}$\] 254$_{-10}^{+12}$
Equilibrium temperature $T_\mathrm{eq}$ \[K\] 4050 $\pm$ 180
Orbital inclination $i$ \[deg\] 86.79 $\pm$ 0.25
Spin-orbit inclination $\lambda$ \[deg\] -84.8 $\pm$ 1.4
: Parameters of the KELT-9b system.[]{data-label="paras_K9"}
All the parameters are adopted from [@Gaudi2017].
![Models of separate CLV and RM effects of the K 3933.66 $\AA$ line for KELT-9b. The upper panel is the CLV effect-only model, the middle panel is the RM effect-only model, and the bottom panel is the model combing both effects. For fast rotating stars like KELT-9, the RM effect is stronger than the CLV effect. The simulation here is for the 1 $R_\mathrm{p}$ case. []{data-label="Seperate-CLV+RM"}](App-1.png){width="45.00000%"}
![RM + CLV effects of the K 3933.66 $\AA$ line for KELT-9b. *Upper panel:* observed transmission spectral matrix. *Middle panel:* simulated stellar line profile changes due to the RM + CLV effects with an $f$ factor $f = 2.1$ (see text). Separate models of the CLV effect and RM effect are presented in Fig. \[Seperate-CLV+RM\]. *Bottom panel:* transmission spectrum after the correction of the RM + CLV effects. The white horizontal lines label ingress and egress. The obvious shadow with a RV drift from – 90 kms$^{-1}$ at ingress to + 90 kms$^{-1}$ at egress is the planetary absorption. []{data-label="CaIIK-SME"}](CaIIK-SME.png){width="45.00000%"}
Cross-correlation with simulated template
-----------------------------------------
We simulated the transmission spectra using the *petitRADTRANS* code [@Molliere2019]. We assumed that the atmosphere has a solar abundance and an isothermal temperature with a value close to the equilibrium temperature (2700 K for WASP-33b and 4000K for KELT-9b). We also assumed that is completely ionized into . We used a mean molecular weight of 1.3 which is the value for an atomic atmosphere with solar abundance. According to [@Hoeijmakers2019], the transmission spectral continuum due to $\mathrm{H^-}$ absorption in UHJs is normally between 1 mbar and 10 mbar, and the atmosphere below the continuum level cannot be probed. Thus, we set a continuum level of 1 mbar when simulating the transmission spectrum. This was done by adding an absorber of infinite strength for P > 1 mbar. The template spectra were subsequently convolved with the instrument profiles. We established a grid of template spectra with radial velocity (RV) shifts from – 500 kms$^{-1}$ to + 500 kms$^{-1}$ with a step of 1 kms$^{-1}$.
Before cross-correlation, we filtered the residual spectra using a Gaussian filter with a $\sigma$ of $\sim$ 1.5 $\AA$. In this way, we filtered out large scale spectral features, which could be caused by the blaze function variation and the stellar pulsation. We cross-correlated the residual spectra with the simulated template spectra as in [@Snellen2010]. The cross-correlations of the H&K from HARPS-N observations and the IRT from CARMENES observations were performed independently. For each observed spectrum, we obtained one cross-correlation function (CCF; Fig. \[map-W33\]b). We then combined all the in transit CCFs by shifting them to the planetary rest frame for a given $K_\mathrm{p}$ (RV semi-amplitude of planetary orbital motion). In this way, we generated the $K_\mathrm{p}$-map with $K_\mathrm{p}$ ranging from 0 to 400 kms$^{-1}$ with a step of 1 kms$^{-1}$ (Fig. \[map-W33\]c). This two-dimensional CCF map has been widely used in previous cross-correlation studies, as in Figure 8 of [@Birkby2017] and Figure 14 of [@Nugroho2017]. In order to estimate the SNR, we measured the noise of the $K_\mathrm{p}$ map as the standard deviation of CCF values with RV ranges of –200 to –100 kms$^{-1}$ and +100 to +200 kms$^{-1}$. Each $K_\mathrm{p}$-map was then divided by the corresponding noise value.
Results and discussion
======================
Detection of using the cross-correlation method
-----------------------------------------------
The cross-correlation results of the two planets are presented in Figs. \[map-W33\] and \[map-K9\]. Panel *b* in the figures are cross-correlation maps between the model spectrum and the residual spectrum with the RM+CLV effects corrected. We also calculated the cross-correlation maps between the model spectrum and the original residual spectrum for comparison (panel *a*).
The H&K and the IRT lines are detected in both planets. We further added the $K_\mathrm{p}$-maps of H&K and IRT to obtain the combined $K_\mathrm{p}$-map of the five lines (right panel in the figures). Here we added directly the H&K and IRT $K_\mathrm{p}$-maps divided by their corresponding noise values. The combined $K_\mathrm{p}$ maps show strong cross-correlation signals at the expected $K_\mathrm{p}$ values ($231\pm3$ kms$^{-1}$ for WASP-33b and 254$_{-10}^{+12}$ kms$^{-1}$ for KELT-9b). These $K_\mathrm{p}$ values are calculated using Kepler’s third law with orbital parameters from the literature (Tables \[paras\_W33\] and \[paras\_K9\]). The peak SNR value in the $K_\mathrm{p}$-map is located at $K_\mathrm{p}$=224 kms$^{-1}$ for WASP-33b and $K_\mathrm{p}$=266 kms$^{-1}$ for KELT-9b. For KELT-9b, [@Yan2018] derived $K_\mathrm{p}$=$269\pm6$ kms$^{-1}$ using H$\mathrm{\alpha}$ absorption and [@Hoeijmakers2019] obtained a $K_\mathrm{p}$ value of $234.2\pm0.9$ kms$^{-1}$ using the planetary absorption lines. These $K_\mathrm{p}$ values derived from planetary absorption are different, but broadly consistent, with the expected $K_\mathrm{p}$ values derived from orbital parameters. Considering that the planetary atmosphere may have additional RV components originated from dynamics, we decided to use the expected $K_\mathrm{p}$ values from Kepler’s third law in the rest of the paper.
The bottom panel in the figures presents the CCFs at the expected $K_\mathrm{p}$ values. Since we already corrected for the systemic velocity as described in Section 3.1, the planetary signal is expected to be located at RV $\sim$ 0 kms$^{-1}$.
For WASP-33b, the IRT signal is very clear and can be seen in the CCF map directly (middle panel in Fig. \[map-W33\]b). The H&K signal is also strong but is less significant than the IRT lines, probably because the deep stellar H&K lines significantly reduce the flux level. The combined cross-correlation function of the five lines yields a 11 $\sigma$ detection.
For KELT-9b, the CCF map of the H&K lines shows a clear signal. However, the IRT signal is less significant, which we attribute to the bad weather conditions during the CARMENES observation. Nevertheless, the IRT signal is still detected at a 4 $\sigma$ level as shown in Fig. \[map-K9\]d. The combined CCF shows a 7 $\sigma$ detection.
The correction of the stellar RM and CLV effects plays an important role in the detection. By comparing the CCF maps with and without the stellar correction in Fig. \[map-W33\] and Fig. \[map-K9\], the improvement after the correction is significant. [@Hoeijmakers2019] searched for in KELT-9b using the same HARPS-N data as in this work, however, they were not able to detect the H&K lines. Probably, the different treatment of the RM and CLV effects between our works could be responsible for the different results obtained. They used an empirical model to fit the stellar residuals presented in the CCF map, which did not properly correct the stellar RM and CLV effects of the H&K lines (see Fig. \[CaIIK-SME\]).
Stellar chromospheric activity can potentially affect the planetary detection but is not expected to pose a serious problem in early A-type stars [e. g. @Schmitt1997]. [@Cauley2018] and [@Khalafinejad2018] investigated the effect of stellar activity on transmission spectra of calcium lines. One prominent distinction between stellar activity and planetary absorption is the radial velocity difference [@Barnes2016]. In our work, the detected signals follow the expected orbital velocities of KELT-9b and WASP-33b, which strongly support their planetary origin.
The host star of WASP-33b is a variable star and stellar pulsations could potentially change the stellar line profile [@Cameron2010]. However, since the feature occurs at the planetary velocity and only during transit (c. f. the middle figure in Fig. \[map-W33\]b), the detected signal is unlikely to be the result of stellar pulsation. Although the planetary atmosphere feature is unambiguous, the stellar pulsation features probably affect the CCF map. For instance, there is a feature in the IRT CCF map at around –30 kms$^{-1}$ (see middle panel in Fig. \[map-W33\]b). Such a pulsation feature produces a dark region next to the planetary atmosphere signal on the $K_\mathrm{p}$-map (middle panel in Fig. \[map-W33\]c).
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Transmission spectrum of individual lines
-----------------------------------------
In addition to the detection using the cross-correlation technique, we also obtained the transmission spectrum of each individual line in order to study them in detail (Figs. \[tran-W33\] and \[tran-K9\]). We added up all the residual spectra observed in transit (but excluding the ingress and egress). Here the stellar RM and CLV effects are already corrected in the residual spectra. Before adding up, we shifted the spectra to the planetary rest frame using the literature $K_\mathrm{p}$ values.
In order to study the absorption line profile, we averaged the H&K lines as well as the triplet lines (see Fig. \[fig-profiles\]). We performed the averaging based on the facts that the line depths of the two H&K lines are similar and the line depths of the triplet lines are also similar. Subsequently, a Gaussian function is fitted to the average spectrum. The fit results are presented in Table \[fit\_result\]. We measured the standard deviation of the spectrum in the ranges of –200 to –100 kms$^{-1}$ and +100 to +200 kms$^{-1}$ and assigned this value as the error of each data point.
The average line depth of the H&K lines is significantly larger than that of the IRT lines. The line depth ratio between them is $3.0\pm0.3$ for WASP-33b and $2.7\pm0.5$ for KELT-9b. This is because the H&K lines correspond to resonant transitions from the ground state of , while the IRT lines are not. For both planets, the 8542 $\AA$ line is the strongest and the 8498 $\AA$ line is the weakest among the three IRT lines. These relative line strengths are consistent with the transmission spectral model.
We calculated the effective radius at the line center ($R_\mathrm{eff}$) and compared it with the effective Roche lobe radius at the planetary terminator [@Ehrenreich2010]. $R_\mathrm{eff}$ is obtained using the equation: $\pi R_\mathrm{eff}^2 / \pi R_\mathrm{p}^2 = (\delta + h) / \delta$, where $\delta$ is the optical photometric transit depth (c. f. Tables \[paras\_W33\] and \[paras\_K9\]) and $h$ is the observed line depth. For WASP-33b, the $R_\mathrm{eff}$ of H&K lines is 1.56 $R_\mathrm{p}$ which is very close to the effective Roche radius ($1.71_{-0.07}^{+0.08}~R_\mathrm{p}$). For KELT-9b, the $R_\mathrm{eff}$ of H&K lines is 1.47 $R_\mathrm{p}$ and the effective Roche radius is $1.91_{-0.26}^{+0.22}~R_\mathrm{p}$. Therefore, the effective radii of both planets are close but below the Roche radii. As a result, we infer that the ionized calcium detected here mostly originates from the extended atmospheric envelope within the Roche lobe instead of from the already escaped material beyond the Roche lobe. Escaped material beyond the Roche lobe can potentially form a comet-like tail as detected in some exoplanets using the hydrogen Lyman-$\mathrm{\alpha}$ and the helium 1083 nm absorptions [e. g. @Ehrenreich2015; @Nortmann2018].
The full width at half maximum (FWHM) of the average line profile is also presented in Table \[fit\_result\]. The FWHM values of the lines in KELT-9b agree in general with the values of other metal lines in KELT-9b measured by [@Cauley2019] and [@Hoeijmakers2019]. The observed line width is probably a combined result of thermal broadening, rotational broadening, and hydrodynamic escape velocity.
The measured line centers have blue- or red-shifted RVs of several kms$^{-1}$ ($v_\mathrm{center}$ values in Table \[fit\_result\]). However, we do not claim the detection of atmospheric winds considering the large errors. The residuals of the RM and CLV effects as well as stellar pulsation can potentially affect the obtained transmission line profile. Furthermore, the uncertainty of the stellar systemic velocity also affects the measurement of $v_\mathrm{center}$. For example, [@Gaudi2017] reported $v_\mathrm{sys}$ of KELT-9 to be –20.6 $\pm$ 0.1 kms$^{-1}$. However, [@Hoeijmakers2019] obtained a value of –17.7 $\pm$ 0.1 kms$^{-1}$ using HARPS-N observations. Later, [@Borsa2019] measured $v_\mathrm{sys}$ of KELT-9 as –19.819 $\pm$ 0.024 kms$^{-1}$ also using HARPS-N data. For WASP-33, [@Nugroho2017] found that their measured $v_\mathrm{sys}$ deviates from the values in other works by $\sim$ 1kms$^{-1}$. The discrepancy between the $v_\mathrm{sys}$ measurements could be due to different instrument RV zero-points, stellar templates, and methods to measure RV, as well as to the relatively large stellar variability of WASP-33. In principle, radial velocity measurement of fast rotating early type stars like WASP-33 and KELT-9 is intrinsically difficult because of the lack of sufficient stellar lines and the broad line profile. Therefore, one should be cautious when interpreting the measured shift of the line center as the signature of atmospheric winds.
In order to investigate the time series of the absorption, we measured the relative fluxes of each line in the residual spectra with an 1 $\AA$ band centered at the line core. Subsequently, we averaged the obtained light curves of the H&K lines and the IRT lines and binned the data points with a phase step of 0.01 (Fig. \[fig-LC\]). The light curves of the WASP-33b IRT lines and the KELT-9b H&K lines have clear absorption signals during transit. However, the absorption signals are less prominent in the light curves of the WASP-33b H&K lines and the KELT-9b IRT lines, which is probably due to lower data quality, stellar pulsation noise, and residuals of the RM + CLV effects.
{width="88.00000%"}
{width="95.00000%"}
Lines $v_\mathrm{center}$ \[kms$^{-1}$\] FWHM \[kms$^{-1}$\] Line depth Model line depth $R_\mathrm{eff}$ \[$R_\mathrm{p}$\]
-------------- ------- ------------------------------------ --------------------- ----------------------- ------------------ -------------------------------------
**WASP-33b** H&K –1.9 $\pm$ 0.7 15.2 $\pm$ 1.5 (2.02 $\pm$ 0.17)$\%$ 0.235$\%$ 1.56 $\pm$ 0.04
IRT 2.0 $\pm$ 0.7 26.0 $\pm$ 1.7 (0.67 $\pm$ 0.04)$\%$ 0.064$\%$ 1.22 $\pm$ 0.01
**KELT-9b** H&K 3.2 $\pm$ 0.7 27.5 $\pm$ 1.8 (0.78 $\pm$ 0.04)$\%$ 0.162$\%$ 1.47 $\pm$ 0.02
IRT –2.2 $\pm$ 2.2 24.4 $\pm$ 5.1 (0.29 $\pm$ 0.05)$\%$ 0.076$\%$ 1.19 $\pm$ 0.03
**Notes.** Here $v_\mathrm{center}$ is the measured RV shift of the line center compared to the theoretical line center. The model line depth is the average line depth from the transmission model. $R_\mathrm{eff}$ is the effective radius at the line center calculated using the observed line depth value and the photometric transit depth.
Mixing ratios of and
---------------------
We calculated the equilibrium chemistry between atomic calcium () and singly ionized calcium () using the chemical module in the *petitCode* [@Molliere2015; @Molliere2017]. Figure \[Ca-temperature\] shows the mixing ratio variation with temperature. For an atmosphere with a solar metallicity and at a pressure of 10 mbar, ionized calcium becomes dominant at temperatures higher than 3000 K. In general, the calcium ionization rate is higher with higher temperature and lower pressure.
As calculated by [@Hoeijmakers2019], the spectral continuum level for UHJ is typically located between 1 mbar and 10 mbar, and transmission spectroscopy probes only the atmosphere above the continuum level. Therefore, we expect to be the dominant calcium feature in the transmission spectroscopy of UHJs. can be probed at lower altitudes if the planetary atmosphere is cooler than 3000 K. For example, atomic calcium was detected in the relatively cool hot-Jupiter HD 209458b, which has a $T_\mathrm{{eq}}$ of 1460 K [@Astudillo-Defru2013]. Here we did not include photo-ionization in the chemical model. The ionization fraction will be higher when including photo-ionization.
Figure \[Mixing-ratio\] shows the and mixing ratios of the two planets assuming isothermal temperature distributions. For WASP-33b, is dominant at high altitudes with pressures < 1 mbar assuming solar metallicity; for KELT-9b, is dominant at pressures < 10 bar. According to our simulations, the average $\mathrm{H^-}$ continuum level is located at $\sim$ 8 mbar for WASP-33b and $\sim$ 4 mbar for KELT-9b in the wavelength region studied here and assuming solar metallicity. Thus, for KELT-9b, is the dominant species in the region probed by transmission spectroscopy; for WASP-33b, can be probed at lower altitudes but only within a small altitude range from 1 mbar to $\sim$ 8 mbar. We assumed isothermal temperature with $T_\mathrm{{eq}}$ values, but the actual temperature at the planetary terminators may deviate from $T_\mathrm{{eq}}$ depending on the 3D atmospheric circulation as well as other mechanisms such as temperature inversion. Therefore, the actual mixing ratio profiles could be different from the ones presented in Figure \[Mixing-ratio\].
We also searched for in the two planets using the cross-correlation method. Since most of the lines are in the HARPS-N wavelength region, we only cross-correlated the modeled spectrum with the HARPS-N dataset. The simulated stellar RM and CLV effects were corrected before the cross-correlation process. We were not able to detect signals in the two planets. [@Hoeijmakers2019] searched for in KELT-9b using two transit observations from HARPS-N and there was only tentative evidence of . These observational results imply that the atmospheres of the two planets are extremely hot and is dominant.
![The (solid line) and (dashed line) mixing ratios as a function of temperature. Here we assumed chemical equilibrium without photo-ionization and a pressure of 10 mbar. Different colors label different metallicities (\[Fe/H\] in unit of dex). []{data-label="Ca-temperature"}](Ca-temperature.pdf){width="45.00000%"}
![Neutral and ionized calcium profiles for WASP-33b (*left*) and KELT-9b (*right*). Here we assumed isothermal temperatures as indicated in the top left corner of each panel. At solar metallicity (orange lines), is dominant at high altitudes with pressure < 1 mbar for WASP-33b and with pressure < 10 bar for KELT-9b (denoted with horizontal dashed lines). []{data-label="Mixing-ratio"}](Mixing-ratio.png){width="45.00000%"}
Model of transmission spectrum
------------------------------
The transmission spectra of both planets are significantly stronger than the model predictions that are calculated assuming equilibrium temperature and solar abundance (c.f. Figure \[fig-profiles\]). The second last column of Table \[fit\_result\] lists the line depths from the models. Here we considered the rotational broadening by assuming tidal locking using the method of [@Brogi2016]. The observed line depths of WASP-33b are 8.6 and 10 times stronger than the depths from the model for the H&K and IRT lines, respectively. For KELT-9b, the observed line depths are 4.8 and 3.8 times stronger for the H&K and IRT lines, respectively. When increasing the isothermal temperature in the models, we obtained larger line depths because of the increased scale heights; the line depth also increases with an increasing calcium abundance. Figure \[fig-K9-models\] compares different models for the H line in KELT-9b. Even with temperature as high as 10000 K and a Ca abundance 100 times the solar Ca abundance, the observed line depth is still stronger than the model.
When simulating the transmission spectrum, we set a cut-off level of 1 mbar to account for possible continuum opacities. The major continuum opacity for the two planets is $\mathrm{H^-}$ absorption. The $\mathrm{H^-}$ mixing ratio and the $\mathrm{H^-}$ transmission spectrum are presented in Figure \[H-mixing-ratio\] and Figure \[H-spec\], respectively. The absorption of $\mathrm{H^-}$ peaks at $\sim$ 0.85 $\mathrm{\mu m}$. The average $\mathrm{H^-}$ continuum level in the optical wavelength range is around 4 mbar and 8 mbar for KELT-9b and WASP-33b, respectively. In order to evaluate the impact of choosing different continuum levels, we simulated transmission spectra using cut-off levels from 0.1 mbar to 10 mbar and found that the line depth increased by less than 20$\%$. Therefore, we concluded that setting different continuum levels can not explain the strong lines observed.
Such a strong absorption in the two planets can be caused by the hydrodynamic escape that brings up calcium ions and, as a result, significantly enhances its density at high altitudes. Compared to hydrostatic models, the hydrodynamic outflow increases significantly the density of materials at altitudes close to the Roche lobe [@Vidal-Madjar2004]. [@Hoeijmakers2019] also found that the observed line depths are stronger than the modeled values and they attributed this discrepancy to the hydrodynamic escape that transports materials to high altitudes. Theoretical models predict that the atmospheres of UHJ are prone to strong atmospheric escape because they receive large amounts of stellar ultraviolet/extreme-ultraviolet radiation [@Fossati2018]. [@Yan2018] observed a strong $\mathrm{H\alpha}$ absorption in KELT-9b, which is evidence of substantial escape of the atmosphere. Further line modeling work with hydrodynamic escape included will be able to constrain the temperature profile and mass loss rate of the planets [e. g. @Odert2019].
![Different models for the H line absorption for KELT-9b. The black points are the observed transmission spectrum (binned every 7 points). The blue line is the model with a temperature of 4000 K. The red line is the model with rotational broadening included. The green line is the model with a temperature of 10000 K and the yellow line is with increased Ca abundance. The observed absorption is stronger than the model predictions. Such a strong absorption indicates a hydrodynamic outflow of the material. []{data-label="fig-K9-models"}](K9-models.png){width="49.00000%"}
![Mixing ratios of hydrogen species calculated assuming solar metallicity, equilibrium chemistry, and isothermal temperature. The upper panel is for WASP-33b (2700K) and the lower panel is for KELT-9b (4000K). In both planets, $\mathrm{H^-}$ exists at low altitudes and is the main opacity source of the spectral continuum. The calcium profiles are also plotted as dashed lines. []{data-label="H-mixing-ratio"}](H-+W33.png "fig:"){width="49.00000%"} ![Mixing ratios of hydrogen species calculated assuming solar metallicity, equilibrium chemistry, and isothermal temperature. The upper panel is for WASP-33b (2700K) and the lower panel is for KELT-9b (4000K). In both planets, $\mathrm{H^-}$ exists at low altitudes and is the main opacity source of the spectral continuum. The calcium profiles are also plotted as dashed lines. []{data-label="H-mixing-ratio"}](H-+K9.png "fig:"){width="49.00000%"}
![Modeled transmission spectrum of $\mathrm{H^-}$ for KELT-9b.[]{data-label="H-spec"}](H-+spec-K9.png){width="49.00000%"}
Conclusions
===========
We have detected singly ionized calcium in KELT-9b and WASP-33b – the two hottest hot-Jupiters discovered so far. Together with the very recent detection in MASCARA-2b, these three UHJs are the only exoplanets with detected in their atmospheres. Our detections and lack of detections demonstrate that calcium is probably mostly ionized into in the upper atmosphere of UHJs.
In addition to the detection using the cross-correlation method, we obtained the transmission spectra from the full set of the five lines (H&K doublet and near-infrared triplet). The effective radii of the H&K lines are close to the Roche lobes of the planets, indicating that the calcium ions are from the very upper atmospheres where mass loss is underway. The obtained line depths are significantly stronger than predictions by hydrostatic models assuming an isothermal temperature of $T_\mathrm{{eq}}$. This is probably because the upper atmosphere is hotter than $T_\mathrm{{eq}}$ and hydrodynamic outflow brings up to the high altitudes. Further modeling work with hydrodynamic escape included is thought to be required to fit the line profile and retrieve the temperature structure.
Due to the high ionization rate of calcium in the upper atmospheres of UHJs and the strong opacities of the H&K and near-infrared triplet lines, the transmission spectrum is especially suitable for probing the high altitude atmospheres and revealing the properties of this peculiar class of exoplanets. These lines have great potential for the study of planet-star interaction, such as atmospheric escape and the impact of stellar wind.
We are grateful to the anonymous referee for his/her report. F. Y. acknowledges the support of the DFG priority program SPP 1992 “Exploring the Diversity of Extrasolar Planets (RE 1664/16-1)”. CARMENES is an instrument for the Centro Astronómico Hispano-Alemán de Calar Alto (CAHA, Almería, Spain). CARMENES is funded by the German Max-Planck-Gesellschaft (MPG), the Spanish Consejo Superior de Investigaciones Científicas (CSIC), the European Union through FEDER/ERF FICTS-2011-02 funds, and the members of the CARMENES Consortium (Max-Planck-Institut für Astronomie, Instituto de Astrofísica de Andalucía, Landessternwarte Königstuhl, Institut de Ciències de l’Espai, Institut für Astrophysik Göttingen, Universidad Complutense de Madrid, Thüringer Landessternwarte Tautenburg, Instituto de Astrofísica de Canarias, Hamburger Sternwarte, Centro de Astrobiología and Centro Astronómico Hispano-Alemán), with additional contributions by the Spanish Ministry of Economy, the German Science Foundation through the Major Research Instrumentation Programme and DFG Research Unit FOR2544 “Blue Planets around Red Stars”, the Klaus Tschira Stiftung, the states of Baden-Württemberg and Niedersachsen, and by the Junta de Andalucía. Based on data from the CARMENES data archive at CAB (INTA-CSIC). This work is based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. P. M and I. S. acknowledge support from the European 82 Research Council under the European Union’s Horizon 2020 research and innovation program under grant agreement No. 694513.
|
[**Martin Lambertsen,**]{} .1cm [**Werner Vogelsang**]{}\
Institute for Theoretical Physics, Tübingen University, Auf der Morgenstelle 14,\
72076 Tübingen, Germany\
[**Abstract**]{}
We present a comprehensive comparison of the available experimental data for the Drell-Yan lepton angular coefficients $\lambda$ and $\nu$ to calculations at leading and next-to-leading order of perturbative QCD. To obtain the next-to-leading order corrections, we make use of publicly available numerical codes that allow us to compute the Drell-Yan cross section at second order in perturbation theory and from which the contributions we need can be extracted. Our comparisons reveal that perturbative QCD is able to describe the experimental data overall rather well, especially at colliders, but also in the fixed-target regime. On the basis of the angular coefficients alone, there appears to be little (if any) convincing evidence for effects that go beyond fixed-order collinear factorized perturbation theory, although the presence of such effects is not ruled out.
Introduction
============
It has been known for a long time [@Collins:1977iv; @Lam:1978pu] that leptons produced in the Drell-Yan process $H_1H_2\to \ell\bar\ell X$ may show nontrivial angular distributions. We denote the momentum of the intermediate virtual boson $V=\gamma^*,Z$ that decays into the lepton pair by $q$. In a specific rest frame of the virtual boson (for our purposes, the Collins-Soper frame [@Collins:1977iv]) we can define polar and azimuthal lepton decay angles $\theta$ and $\phi$, respectively. Considering, for simplicity, a situation where contributions by $Z$-bosons are negligible and only the exchange of an intermediate virtual photon is relevant, one can show that the cross section differential in $d^4q$ and $d\Omega\equiv d\cos\theta d\phi$ may be written as = ( W\_[T]{}[(1+\^2)]{}+W\_[L]{}[(1-\^2)]{} + W\_2+ W\_\^22) \[eqn: diff cross section\], where $\alpha$ is the fine structure constant, $N_c=3$ the number of colors in QCD, $Q^2=q^2$ and $s$ the c.m.s. energy squared of the incoming hadrons $H_1$ and $H_2$. The structure functions $W_T$, $W_L$, $W_\Delta$, $W_{\Delta\Delta}$ are functions of $q$. They parametrize the hadronic tensor as \^ = - (g\^- T\^T\^) (W\_T+ W\_) - 2 X\^X\^W\_ + Z\^Z\^(W\_L -W\_T - W\_ ) - (X\^Z\^+ Z\^X\^) W\_ , \[WTLDDD\] where $X$, $Y$, $Z$ and $T$ are a set of orthonormal axes that one introduces in the Collins-Soper frame. If also $Z$-bosons contribute, there are additional angular terms and structure functions in the cross section formula. For details of the derivation of the cross section (also for discussion of other related reference frames), see Refs. [@Collins:1977iv; @Lam:1978pu; @Boer:2006eq; @Berger:2007si; @Peng:2014hta].
From the differential cross section one easily derives an expression for the normalized decay angle distribution ( )\^[-1]{} \[angdef\] in terms of the structure functions. Using Eq. (\[eqn: diff cross section\]) we obtain = . \[sigmaW\] One usually writes this as & = & , \[DNdOmega\] where = , = , = . \[lmnWrel\] Much effort has gone into studies of these angular coefficients $\lambda,\mu,\nu$, both experimentally and theoretically. On the experimental side, measurements of the coefficients are by now available over a wide range of kinematics, from fixed-target energies [@Guanziroli:1987rp; @Conway:1989fs; @Zhu:2006gx; @Zhu:2008sj] all the way to the Tevatron [@Aaltonen:2011nr] $p\bar{p}$ and the LHC $pp$ colliders [@Khachatryan:2015paa]. In the fixed-target regime various combinations of beams and targets are available; data have been taken with pion beams off nuclear (tungsten) targets [@Guanziroli:1987rp; @Conway:1989fs] and also for $pp$ and $pd$ collisions [@Zhu:2006gx; @Zhu:2008sj]. The experimental results are typically given as functions of the transverse momentum $q_T$ of the virtual boson, in a certain range of the lepton pair mass, $Q\equiv\sqrt{Q^2}$. For the fixed-target data, $q_T$ is limited to a few GeV and $Q$ is usually around . This is very different for the high-energy collider measurements which are carried out around $Q=m_Z$, where $m_Z$ is the $Z$-boson mass. The range in $q_T$ explored here is much larger and reaches to almost at the Tevatron and even much beyond that at the LHC.
The lowest-order (LO) partonic channel $q\bar{q}\to V\,(\to \ell\bar\ell)$ with collinear incoming partons leads to the prediction $\lambda=1$, $\mu=\nu=0$. However, for this process the virtual photon has vanishing transverse momentum, $q_T=0$, so it cannot contribute to the cross section at finite $q_T$. The situation changes when “intrinsic” parton transverse momenta are taken into account. The coefficient $\nu$, especially, which corresponds to a $\cos2\phi$ dependence in azimuthal angle, has received a lot of attention in this context since it was discovered [@Boer:1999mm] that it may probe interesting novel parton distribution functions of the nucleon, known as Boer-Mulders functions [@Boer:1997nt]. These functions represent a transverse-polarization asymmetry of quarks inside an unpolarized hadron and are “T-odd” and hence related to nontrivial (re)scattering effects in QCD (see [@Collins:2002kn]). Detailed phenomenological [@Barone:2010gk; @Lu:2011mz] or model-based [@Pasquini:2014ppa] studies have been presented that confront the fixed-target experimental data with theoretical expectations based on the Boer-Mulders functions.
Already the early theoretical studies [@Lam:1978zr; @Collins:1978yt; @Kajantie-78; @Cleymans-78; @Lindfors-79] revealed that also plain perturbative-QCD radiative effects lead to departures from the simple prediction $\lambda=1$, $\mu=\nu=0$, starting from ${\cal O}(\alpha_s)$ with the processes $q\bar{q}\to Vg$ and $qg\to Vq$. At $q_T\neq 0$ in fact the latter processes become the LO ones. A venerable result of [@Lam:1978pu; @Lam:1980uc] obtained on the basis of these LO reactions is the [*Lam-Tung relation*]{}, \[LT\] 1--2=0, which holds separately for both partonic channels in the Collins-Soper frame [@Collins:1977iv]. Next-to-leading order (NLO) corrections to the cross sections relevant for the angular coefficients have first been derived in Refs. [@Mirkes:1992hu; @Mirkes:1994dp]. These suggest overall modest ${\cal O}(\alpha_s^2)$ effects on $\lambda$, $\mu$, $\nu$, so that also the Lam-Tung relation, although found to be violated at NLO, still holds to fairly good approximation. The data from the fixed-target experiment E615 [@Guanziroli:1987rp] indicate a violation of the Lam-Tung relation, while the other fixed-target sets are overall consistent with it, as are the Tevatron data [@Aaltonen:2011nr]. A clear violation of the Lam-Tung relation, on the other hand, was observed recently at the highest energies, in $pp$ collisions at the LHC [@Khachatryan:2015paa].
In the present paper, we take a fresh look at the Drell-Yan angular dependences in the framework of perturbative QCD. Specifically, we present an exhaustive comparison of the LO and NLO QCD predictions for the parameters $\lambda$ and $\nu$ with the experimental data, over the whole energy range available. Rather than attempting to retrieve the results of [@Mirkes:1992hu; @Mirkes:1994dp], we determine new NLO predictions. For this purpose, we use the publicly available codes [<span style="font-variant:small-caps;">fewz</span> ]{}(version 3.1) [@Li:2012wna] and [<span style="font-variant:small-caps;">dynnlo</span> ]{} [@Catani:2009sm]. These allow us to compute the full Drell-Yan cross section at next-to-next-to-leading (NNLO) order of QCD, when $q\bar{q}\to V$ is the LO process. As discussed above, the contributions to the angular coefficients that we are interested in are at nonvanishing $q_T$, so that the order $\alpha_s^2$ in this case is only NLO. Since all ${\cal O}(\alpha_s^2)$ contributions are included in the [<span style="font-variant:small-caps;">fewz</span> ]{}and [<span style="font-variant:small-caps;">dynnlo</span> ]{}codes, we can therefore use these codes to extract the angular coefficients $\lambda$, $\mu$, $\nu$ at NLO, providing a new and entirely independent calculation.
To our knowledge, such a comprehensive analysis has never been performed in the past. Our study was very much inspired by the recent work [@Peng:2015spa], in which the LHC results for the angular coefficients were analyzed on general theoretical grounds, attributing the observed violation of the Lam-Tung relation to a “noncoplanarity” of the axis of the incoming partons with respect to the hadron plane, which may be constrained by the combined Tevatron and LHC data. As the authors of [@Peng:2015spa] pointed out, the most likely physical explanation for the LHC result on the violation of the Lam-Tung relation is QCD radiative effects at NLO (or beyond). We indeed confirm this in our study.
We push the purely perturbative framework also to the fixed-target regime, where there have been hardly any phenomenological analyses of the Drell-Yan angular coefficients in the context of hard-scattering QCD. Reference [@Brandenburg:1993cj] presents results at the energy of the NA10 experiment; however the kinematics relevant at NA10 was not properly implemented. Of course, in the fixed-target regime $q_T$ can become quite small, smaller than, say, or so. For such low values one does not expect fixed-order perturbation theory to provide reliable results for cross sections, even if $Q$ is relatively large. Intrinsic transverse momenta of the initial partons may become relevant, among them precisely the Boer-Mulders functions mentioned earlier. The possible role of higher-twist contributions has been discussed as well [@Brandenburg:1994wf; @Zhou:2009rp]. Furthermore, as is well known, large logarithmic perturbative corrections of the form $\alpha_s^k \log^m(Q^2/q_T^2)/q_T^2\;$ ($m=1,\ldots,
2k-1$) appear in calculations at fixed perturbative order $k$, as a result of soft-gluon emission. In order to describe the cross sections, one needs to resum these corrections to all orders in the strong coupling and also implement nonperturbative contributions (see especially [@Konychev:2005iy], and references therein). As was discussed in Refs. [@Boer:2006eq; @Berger:2007si], such corrections will likely cancel to a significant degree in the angular coefficients $\lambda$ and $\nu$, since the same type of leading logarithms occur in the numerator and denominator for both quantities. Also, it is expected [@Berger:2007si] that the Lam-Tung relation will remain essentially untouched by the soft-gluon effects.
Thus, although clearly collinear perturbation theory at fixed-order (NLO) that we will use here cannot provide a completely adequate framework for describing cross sections in all kinematic regimes of interest for the angular coefficients, our results to be presented below yield important benchmarks, in our view. In the light of the observations concerning the soft-gluon effects mentioned above, it appears likely that fixed-order perturbation theory will work much better for ratios of cross sections than for the cross sections themselves. In fact, we will find that we can describe all data sets quite well, and that we do not find any clear-cut evidence for nontrivial additional contributions to be attributed to parton intrinsic momenta. We stress again that QCD radiative effects are typically not considered at all when for example Boer-Mulders functions are extracted from data for $\nu$ (although the conceptual framework for such a combined analysis is available [@Bacchetta:2008xw]). At the very least, our results establish the relevance of the radiative effects for phenomenological studies of the Drell-Yan angular dependences.
Our paper is organized as follows. In Sec. \[sec2\] we explain how we extract the angular coefficients from the available Drell-Yan NNLO codes. Section \[sec3\] shows our phenomenological results, and in Sec. \[sec4\] we conclude our work.
Extraction of angular coefficients at NLO \[sec2\]
==================================================
It is actually relatively straightforward to use the [<span style="font-variant:small-caps;">fewz</span> ]{} [@Li:2012wna] and [<span style="font-variant:small-caps;">dynnlo</span> ]{} [@Catani:2009sm] codes to determine the angular coefficients $\lambda,\mu,\nu$. The programs allow us to compute cross sections over suitable ranges of any kinematic variable, providing full control over the four-momenta of the produced particles. As already pointed out in [@Lam:1978pu], the structure functions $W_T$, $W_L$, $W_\Delta$, $W_{\Delta\Delta}$ may be projected out by computing the following combinations of cross sections: $$\begin{aligned}
\label{eqn: seperate Ws}
2W_{T}+W_{L} &= {\cal N}\, \frac{d\sigma}{d^{4}q} \,, \nn \\[2mm]
W_{T}-W_{L} &= \frac{8}{3}\,{\cal N}\, {\left[\frac{d\sigma}{d^{4}q}
{\left({\left|\cos\theta\right|}>\frac{1}{2}\right)} - \frac{d\sigma}{d^{4}q}{\left({\left|\cos\theta\right|}<\frac{1}{2}\right)}\right]}\,,\nn \\[2mm]
W_{\Delta} &= \frac{\pi}{2}\,{\cal N}\, {\left[\frac{d\sigma}{d^{4}q}{\left(\sin 2\theta\cos\phi>0\right)} -
\frac{d\sigma}{d^{4}q}{\left(\sin 2\theta\cos\phi<0\right)}\right]} \nn \,,\\[2mm]
W_{\Delta\Delta} &= \frac{\pi}{2}\,{\cal N}\, {\left[\frac{d\sigma}{d^{4}q}{\left(\cos 2\phi>0\right)} -
\frac{d\sigma}{d^{4}q}{\left(\cos 2\phi<0\right)}\right]} \,,\end{aligned}$$ where ${\cal N}=12 \pi^3 (Qs/\alpha)^2$. Using Eqs. (\[lmnWrel\]), the angular coefficients follow immediately from these expressions. We note that Eqs. (\[eqn: seperate Ws\]) are valid both for exchanged photons and $Z$ bosons. As mentioned earlier, in cases where $Z$ bosons contribute the cross section has additional angular pieces; however these do not survive the integrations in Eqs. (\[eqn: seperate Ws\]).
The remaining task is to determine the kinematical variables that appear in Eqs. (\[eqn: seperate Ws\]) from the momenta of the outgoing leptons given in the Monte Carlo integration codes of [@Li:2012wna; @Catani:2009sm]. To this end, we use that the momentum of one lepton, written in the Collins-Soper frame as $\ell^\mu_{\mathrm{CS}}=\frac{Q}{2}(1,\sin\theta\cos\phi,
\sin\theta\sin\phi,\cos\theta)$, becomes in the hadronic c.m.s. [@Arnold:2008kf] $$\begin{aligned}
\notag \ell_{\mathrm{cm}}^\mu &= \frac{1}{2}
\begin{pmatrix}
q_{0}{\left(1+\sin\alpha \sin\theta\cos\phi\right)} + q_L \cos\alpha \cos\theta \\[2mm]
q_{T}\cos\varphi + Q \frac{\sin\theta}{\cos\alpha} ( \cos\phi\cos\varphi-\cos\alpha\sin\phi\sin\varphi) \\[2mm]
q_{T}\sin\varphi + Q \frac{\sin\theta}{\cos\alpha} ( \cos\phi\sin\varphi+\cos\alpha\sin\phi\cos\varphi) \\[2mm]
q_L{\left(1+\sin\alpha \sin\theta\cos\phi\right)} + q_0 \cos\alpha \cos\theta
\end{pmatrix}\,,\end{aligned}$$ where ,, and where $q_0$ and $q_L$ are the energy and the longitudinal component (with respect to the collision axis) of the virtual boson in the hadronic c.m.s., so that $q^\mu_{\mathrm{cm}}=(q_0,q_T \cos\varphi,q_T\sin\varphi,q_L)$. To project out the combinations of trigonometric functions we need, we introduce $$\begin{aligned}
\mathcal{P}_{1}^\mu\equiv
\begin{pmatrix}
q_L \\[1mm] 0 \\[1mm] 0 \\[1mm] q_{0}
\end{pmatrix}, \qquad
\mathcal{P}_{2}^\mu \equiv q_{T}
\begin{pmatrix}
0 \\ \cos\varphi \\ \sin\varphi \\ 0
\end{pmatrix} ,
\qquad \mathcal{P}_{3}^\mu \equiv q_{T}
\begin{pmatrix}
0 \\ \sin\varphi \\ -\cos\varphi \\ 0
\end{pmatrix}\,.\end{aligned}$$ We then have $$\begin{aligned}
\cos\theta &= - \frac{2 \,\ell_{\mathrm{cm}}\cdot \mathcal{P}_1}{(Q^2+q_T^2)\cos\alpha}\,, \nn\\[2mm]
\sin2\theta\cos\phi &= \frac{4\,\ell_{\mathrm{cm}}\cdot \mathcal{P}_1}{Q^{2}+q_{T}^{2}}
{\left[\frac{q_{T}}{Q} + \frac{2\,\ell_{\mathrm{cm}}\cdot \mathcal{P}_2}{q_{T}Q}\right]}\,, \nn\\[2mm]
\cos2\phi &= 1 - \frac{2{\left(\ell_{\mathrm{cm}}\cdot \mathcal{P}_3\right)}^{2}}{q_{T}^{2}
{\left[\frac{Q^2}{4}-\frac{{\left(\ell_{\mathrm{cm}}\cdot \mathcal{P}_1\right)}^{2}}{Q^{2}+q_{T}^{2}}\right]}}\,.\label{ellP}\end{aligned}$$ The four-momentum of the lepton in the hadronic c.m.s. is provided in the Monte Carlo integration codes, while that of the virtual boson is fixed by the external kinematics. Writing $\ell^\mu_{\mathrm{cm}}=(\ell_{\mathrm{cm}}^0,\ell_{\mathrm{cm}}^1,\ell_{\mathrm{cm}}^2,\ell_{\mathrm{cm}}^3)$, we have $$\begin{aligned}
\ell_{\mathrm{cm}}\cdot \mathcal{P}_1 &= q_L \,\ell_{\mathrm{cm}}^0-q_0\, \ell_{\mathrm{cm}}^3 \,,\nn \\[2mm]
\ell_{\mathrm{cm}}\cdot \mathcal{P}_2 &= -q_{T}{\left(\ell_{\mathrm{cm}}^1\,\cos\varphi + \ell_{\mathrm{cm}}^2\,\sin\varphi\right)} \,,\nn\\[2mm]
\ell_{\mathrm{cm}}\cdot \mathcal{P}_3 &= q_{T}{\left(\ell_{\mathrm{cm}}^2\,\cos\varphi - \ell_{\mathrm{cm}}^1\,\sin\varphi\right)}\,.\end{aligned}$$ Inserting these expressions into Eqs. (\[ellP\]), one can now easily implement the appropriate cuts in the codes so that the structure functions $W_T$, $W_L$, $W_\Delta$, $W_{\Delta\Delta}$ can be extracted via Eqs. (\[eqn: seperate Ws\]).
Comparison to data \[sec3\]
===========================
We now present comparisons of the theoretical predictions at LO and NLO to the available experimental data for the angular coefficients $\lambda$ and $\nu$. We do not show any results for the coefficient $\mu$ which comes out always extremely small and in fact usually consistent with zero both in the theoretical calculation and in experiment, within the respective uncertainties. We first note that we have validated our technique for extracting the Drell-Yan angular coefficients from the [<span style="font-variant:small-caps;">fewz</span> ]{}(version 3.1) [@Li:2012wna] and [<span style="font-variant:small-caps;">dynnlo</span> ]{} [@Catani:2009sm] codes by writing a completely independent LO code. We have found perfect agreement between this code and the LO results we extracted from [<span style="font-variant:small-caps;">fewz</span> ]{}and [<span style="font-variant:small-caps;">dynnlo</span> ]{}. In the figures below, the LO curves will always refer to those from our own code. We also note that the NLO results we show in the following have all been obtained with the [<span style="font-variant:small-caps;">fewz</span> ]{}code. We have compared to the results of [<span style="font-variant:small-caps;">dynnlo</span> ]{}and found excellent consistency of the two codes both at LO and NLO.
![[*Same as Fig. \[fig1\], but for a more forward/backward rapidity interval $1<|\eta|< 2.1$.*]{}[]{data-label="fig2"}](cms_lowrap.pdf){width="90.00000%"}
![[*Same as Fig. \[fig1\], but for a more forward/backward rapidity interval $1<|\eta|< 2.1$.*]{}[]{data-label="fig2"}](cms_midrap.pdf){width="90.00000%"}
Although the implementation of Eqs. (\[eqn: seperate Ws\]) and the relevant kinematics into the [<span style="font-variant:small-caps;">fewz</span> ]{}or [<span style="font-variant:small-caps;">dynnlo</span> ]{}codes is relatively straightforward, the computational load for performing a comprehensive comparison of the data with NLO theory is very large. To obtain the NLO results presented below, we have run an equivalent of one Intel Quad-Core i5-3470 CPU using all of its cores for about 2 years. In order to collect sufficiently high statistics at very high values of $q_T$, where the cross section drops very rapidly, we have performed dedicated runs for which we have implemented cuts on the low-$q_{T}$ region, forcing the Monte Carlo integration to sample high $q_{T}$. We also note that typically the result for the lowest-$q_T$ bin is unreliable, since this bin contains the (NNLO) contributions at $q_T=0$. Nonetheless, our results are sufficiently accurate in all regions of interest and thus allow us to derive solid conclusions. We mention that we also had to modify the codes to accommodate pion beams and nuclear (deuteron/tungsten) targets. This implementation was always checked against our own LO code.
![[*Comparison of LO (lines) and NLO ([<span style="font-variant:small-caps;">fewz</span> ]{} [@Li:2012wna], histograms) theoretical results to the CDF data [@Aaltonen:2011nr] for the angular coefficients $\lambda$ and $\nu$ taken in $p\bar{p}$ scattering at $\sqrt{s}=\SI{1960}{GeV}$. We have integrated over $\num{66}\leq Q\leq \SI{116}{GeV}$ and over $|\eta|<3.6$ of the virtual boson.*]{}[]{data-label="fig3"}](cms_midrap_expbins.pdf){width="90.00000%"}
![[*Comparison of LO (lines) and NLO ([<span style="font-variant:small-caps;">fewz</span> ]{} [@Li:2012wna], histograms) theoretical results to the CDF data [@Aaltonen:2011nr] for the angular coefficients $\lambda$ and $\nu$ taken in $p\bar{p}$ scattering at $\sqrt{s}=\SI{1960}{GeV}$. We have integrated over $\num{66}\leq Q\leq \SI{116}{GeV}$ and over $|\eta|<3.6$ of the virtual boson.*]{}[]{data-label="fig3"}](cdf.pdf){width="90.00000%"}
Throughout this paper, we use the parton distribution functions of the proton of Ref. [@Martin:2009iq], adopting their NLO (LO) set for the NLO (LO) calculation. The choice of parton distributions has a very small effect on the Drell-Yan angular coefficients. When dealing with nuclear targets (tungsten was used for all of the pion scattering experiments and deuterons for one set of E866 measurements) we compute the parton distributions of the nucleus just by considering the relevant isospin relations for protons and neutrons, averaging over the appropriate proton and neutron number. We do not add any other nuclear effects. For the parton distributions of the pion, we use the set in [@Sutton:1991ay]; the set in [@Gluck:1991ey] would give very similar results. Finally, our choice for the factorization and renormalization scales will always be $\mu=Q$. We have checked that other possible scale choices such as $\mu=\sqrt{Q^2+q_T^2}$ do not change the results for the angular coefficients significantly even at LO, making an impact of at most a few percent, and only at high values of $q_T$. Here we have simultaneously varied the scales in the cross sections appearing in the numerators [*and*]{} in the denominators of the angular coefficients; relaxing this condition one would likely be able to generate a larger dependence on the choice of scale. On the other hand, as is known from previous calculations [@Li:2012wna; @Catani:2009sm], the scale dependence of the Drell-Yan cross section is overall much reduced at higher orders anyway.
We present our results essentially in the order of decreasing energy, starting with a comparison to the high-energy collider data from the LHC [@Khachatryan:2015paa] and Tevatron [@Aaltonen:2011nr]. The reason is that for these data sets $Q$ is very large, $Q\sim m_Z$, so that perturbative methods should be well justified. The transverse momentum $q_T$ varies over a broad range, taking low values as well as values of order $Q$. At the lower end, where $q_T\ll Q$, it may well be necessary to perform an all-order resummation of perturbative double logarithms in $q_T/Q$ in order to describe the Drell-Yan cross section properly. However, as mentioned in the Introduction, such logarithms are expected to cancel to a large extent in the angular coefficients [@Boer:2006eq; @Berger:2007si]. Thus, if ever fixed-order perturbative QCD predictions are able to provide an adequate description of the angular coefficients, it should be in the kinematic regimes explored at the LHC and Tevatron.
Figures \[fig1\] and \[fig2\] show our results for $\lambda$ and $\nu$ compared to the CMS data [@Khachatryan:2015paa], for two separate bins in the rapidity of the virtual boson, . We note that CMS presents their data in terms of a different set of angular coefficients termed $A_0$, $A_1$, $A_2$, $A_3$, which are directly related to the coefficients we use here. In particular, we have $\lambda=(2-3A_0)/(2+A_0)$ and $\nu=2A_2/(2+A_0)$. As in Ref. [@Peng:2015spa], in order to present a full comparison in terms of $\lambda$ and $\nu$, we transform the experimental data correspondingly. Here we have propagated the experimental uncertainties, albeit without taking into account any correlations. The lines in the figures show our LO results for the coefficients. As one can see, they qualitatively follow the trend of the data, but for the coefficient $\nu$ a clear deviation between data and LO theory is observed. This is precisely the finding also emphasized in Ref. [@Peng:2015spa] where it was argued (without explicit NLO calculation) that the discrepancy ought to be related to higher-order QCD effects. Indeed, this is what we find. The NLO results (histograms) show a markedly better agreement with the data, which in fact is nearly perfect. The coefficient $\lambda$, on the other hand, changes only marginally from LO to NLO. As is visible in the figures, the results at very high values of $q_T$ are numerically less accurate, as shown by the somewhat erratic behavior of the histograms. In order to collect higher statistics, we have also performed runs for which we integrated over only eight $q_T$ bins, choosing exactly the ones used in the experimental analysis. The corresponding results are shown in Fig. \[fig1X\] for the range $1<|\eta|< 2.1$. Our goal was to make sure that the numerical uncertainty for these bins is much smaller than the experimental one [*even*]{} in the bin at highest $q_T$. The figure once more impressively shows how NLO theory leads to an excellent description of the CMS data.
It is interesting to note that NLO [<span style="font-variant:small-caps;">fewz</span> ]{}results were also shown in the CMS paper [@Khachatryan:2015paa]. However, the agreement with the data for the coefficient $A_2$ (which multiplies the $\cos 2\phi$ dependence of the cross section) reported there appears to be not quite as good as the one we find for our coefficient $\nu$. It is conceivable that our computation of the coefficients via Eqs. is numerically more stable.
We next turn to the comparison to the CDF data [@Aaltonen:2011nr] taken in $p\bar{p}$ collisions at $\sqrt{s}=\SI{1960}{GeV}$ at the Tevatron. The results are shown in Fig. \[fig3\]. We observe that both the LO and the NLO results are in good agreement with the data, NLO doing a bit better overall. Both coefficients $\lambda$ and $\nu$ decrease slightly when going to NLO. For $\nu$, this effect is less pronounced than for the LHC case, which may be attributed to a much stronger contribution by the $q\bar{q}$ channel in the present $p\bar{p}$ case, which receives smaller radiative corrections. Again, this feature was predicted phenomenologically in Ref. [@Peng:2015spa].
We now consider the fixed-target regime, where we start with a comparison to the Fermilab E866/NuSea data taken with an proton beam in $pp$ [@Zhu:2008sj] and $pd$ [@Zhu:2006gx] scattering. The comparisons to the two data sets are shown in Figs. \[fig4\] and \[fig5\]. We first note that the $pp$ data are overall in much better agreement with the theoretical curves than the $pd$ ones. For $pp$ scattering, the coefficient $\lambda$ is well described, given the relatively large experimental uncertainties. There is a slight trend in the data for the coefficient $\nu$ to be lower than the theoretical prediction. The NLO corrections in fact provide a slight improvement here. For $pd$ scattering, the two data points for $\nu$ at the highest $q_T$ are clearly below theory even at NLO. The coefficient $\lambda$ is not well described, neither at LO nor at NLO. An important point to note in this context is the positivity constraint [@Lam:1978pu] W\_L0, which immediately implies \[constraint\] 1 . This condition is completely general and relies only on the hermiticity of the neutral current. It is interesting to observe that the $pd$ data shown in Fig. \[fig4\] are only in borderline agreement with this positivity constraint.
![[*Same as Fig. \[fig4\], but for $pd$ scattering. Data are from Ref. [@Zhu:2006gx]*]{}[]{data-label="fig5"}](e866_pp.pdf){width="90.00000%"}
![[*Same as Fig. \[fig4\], but for $pd$ scattering. Data are from Ref. [@Zhu:2006gx]*]{}[]{data-label="fig5"}](e866_pd.pdf){width="90.00000%"}
Going further down in energy, we finally discuss the data from the $\pi+$tungsten scattering experiments NA10 [@Guanziroli:1987rp] and E615 [@Conway:1989fs]. NA10 used three different energies for the incident pions, $E_\pi=\SIlist{286;194;140}{GeV}$, while E615 operated a pion beam with energy . Figures \[fig6\]–\[fig8\] show the comparisons of our LO and NLO results for $\lambda$ and $\nu$ to the NA10 data. The NLO corrections are overall small for $\nu$, but for $\lambda$ they become more pronounced toward larger $q_T$. We note that NLO results for one of the NA10 energies were also reported in Ref. [@Brandenburg:1993cj], where however not the appropriate kinematical regime in $Q$ was chosen, leading to an underestimate of $\nu$ which has unfortunately given rise to the general notion in the literature that perturbative QCD cannot describe the Drell-Yan angular coefficients. We also note that for the kinematics used in [@Brandenburg:1993cj] the NLO corrections appear to be somewhat smaller than the ones we find here. The three cases shown in Figs. \[fig6\]–\[fig8\] have in common that the data for $\nu$ are well described, perhaps slightly less so for the pion energy . The experimental uncertainties for the coefficient $\lambda$ are very large, and it is not possible to draw solid conclusions from the comparison. We note that wherever there are tensions between data and theory concerning $\lambda$, the data tend to lie uncomfortably close to (or even above) the positivity constraint $\lambda \leq 1$.
![[*Same as Fig. \[fig6\], but at pion energy $E_\pi=\SI{194}{GeV}$ and integrated over $Q\geq \SI{4.05}{GeV}$.*]{}[]{data-label="fig7"}](na10_286gev.pdf){width="90.00000%"}
![[*Same as Fig. \[fig6\], but at pion energy $E_\pi=\SI{194}{GeV}$ and integrated over $Q\geq \SI{4.05}{GeV}$.*]{}[]{data-label="fig7"}](na10_194gev.pdf){width="90.00000%"}
![[*Comparison of LO (lines) and NLO ([<span style="font-variant:small-caps;">fewz</span> ]{} [@Li:2012wna], histograms) theoretical results to the $\pi+$tungsten scattering data from E615 [@Conway:1989fs] taken with pion beam energy $E_\pi=\SI{252}{GeV}$. We have integrated over the mass range $\num{4.05}\leq Q\leq \SI{8.55}{GeV}$. We have also implemented the cuts $0\leq x_F\leq 1$ and $\num{0.2}\leq x_\pi\leq 1$, where $x_\pi=\frac{1}{2}(x_F+\sqrt{x_F^2+4Q^2/s})$ with $x_F=2q_L/\sqrt{s}$ the Feynman variable, which is counted as positive in the forward direction of the pion beam.*]{}[]{data-label="fig9"}](na10_140gev.pdf){width="90.00000%"}
![[*Comparison of LO (lines) and NLO ([<span style="font-variant:small-caps;">fewz</span> ]{} [@Li:2012wna], histograms) theoretical results to the $\pi+$tungsten scattering data from E615 [@Conway:1989fs] taken with pion beam energy $E_\pi=\SI{252}{GeV}$. We have integrated over the mass range $\num{4.05}\leq Q\leq \SI{8.55}{GeV}$. We have also implemented the cuts $0\leq x_F\leq 1$ and $\num{0.2}\leq x_\pi\leq 1$, where $x_\pi=\frac{1}{2}(x_F+\sqrt{x_F^2+4Q^2/s})$ with $x_F=2q_L/\sqrt{s}$ the Feynman variable, which is counted as positive in the forward direction of the pion beam.*]{}[]{data-label="fig9"}](e615.pdf){width="90.00000%"}
In case of E615, we find the results shown in Fig. \[fig9\]. We observe that neither the description of $\lambda$ nor that of $\nu$ is good. The NLO corrections are overall small and thus do not change this picture. It is clear that on the basis of the data one would derive a significant violation of the Lam-Tung relation (\[LT\]), since $\lambda$ and $\nu$ both enter the relation with the same sign, and the data for both $\lambda$ and $\nu$ are higher than theory (the latter satisfying the relation at LO). It is worth pointing out, however, that the experimental uncertainties are large and, more importantly, again the data show a certain tension with respect to the positivity limit (\[constraint\]).
Conclusions \[sec4\]
====================
We have presented detailed and exhaustive comparisons of data for the Drell-Yan lepton angular coefficients $\lambda$ and $\nu$ to LO and NLO perturbative-QCD calculations. To obtain NLO results, we have employed public codes that allow us to compute the full Drell-Yan cross section at NNLO, and in which the angular pieces we are interested in are contained.
Our numerical results show that overall perturbative QCD is able to describe the experimental data quite well. For the recent LHC data the agreement is very good, when the NLO corrections are taken into account. This finding is in line with arguments made in the recent literature [@Peng:2015spa]. Also the Tevatron data are very well described at NLO. Toward the fixed-target regime, we again find an overall good agreement, with possible exceptions for the E866 $pd$ data set for $\nu$ at high $q_T$ and for the E615 data. We remark that the latter data set carries large uncertainties and also hints at tensions with the positivity constraint $\lambda\leq 1$.
To be sure, the description of the cross sections that enter the angular coefficients requires input beyond fixed-order QCD perturbation theory, notably in terms of resummations of logarithms in $q_T/Q$ and of transverse-momentum dependent parton distributions. On the other hand, based on the angular coefficients alone, in our view there is no convincing evidence for any effects other than the ones we have considered here. In particular, we argue that one should dispel the myth that perturbative QCD is not able to describe the Drell-Yan angular coefficients, which in fact has been iterated over and over in the literature. While we most certainly do not wish to exclude the presence of contributions by the Boer-Mulders effect in the $\cos2\phi$ part of the angular distribution, it is also clear from our study that future phenomenological studies of the effect should incorporate the QCD radiative effects.
We finally stress that our results clearly make the case for new precision data for the Drell-Yan angular coefficients that would allow to convincingly establish whether there are departures from the “plain” QCD radiative effects we have considered here. We hope that such data will be forthcoming from measurements at the COMPASS [@Chiosso:2015naa] or E906 [@Nakano:2016jdm] experiments, or possibly at RHIC.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Jen-Chieh Peng for useful discussions and for communications on the E866 data. We also acknowledge helpful discussions with Peter Schweitzer and Marco Stratmann. This work was supported in part by the Deutsche Forschungsgemeinschaft (Grant No. VO 1049/1). The authors acknowledge support by the state of Baden-Württemberg through bwHPC.
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|
---
abstract: 'We introduce the analogue of the metric tensor in case of $q$-deformed differential calculus. We analyse the consequences of the existence of the metric, showing that this enforces severe restrictions on the parameters of the theory. We discuss in detail the examples of the Manin plane and the $q$-deformation of $SU(2)$. Finally we touch the topic of relations with the Connes’ approach.'
---
[**Metric on Quantum Spaces**]{}\
\
\
Andrzej Sitarz [^1] [^2]\
Department of Field Theory\
Institute of Physics\
Jagiellonian University\
Reymonta 4, 30-059 Kraków, Poland
Introduction
============
Quantum groups and quantum spaces are an interesting non-trivial generalization of Lie groups and manifolds [@qg]. The deformation parameter $q$ allows to recover the latter in the continuous limit $q \to 1$, which suggests that the noncommutativity of space could possibly provide a regularization mechanism [@ma]. Therefore one may expect that $q$-deformations could be an interesting basis for physical theories, in particular for gravity and gauge theories [@qgth]. The natural language for such studies is the $q$-deformed differential calculus (see [@zum] for a review) constructed within the framework of noncommutative differential geometry [@con]. The construction, however is not unique, and even after imposing bicovariance in the situation of quantum groups, in general we are left with many choices of possible theories.
The additional problem, which arises in the course of constructing physical $q$-deformed theories is the question of metric. In the classical situation, $q=1$, the metric is given by the metric tensor. The latter could be equivalently defined as a bilinear functional from $\Omega^1 \times \Omega^1$ to $\a$, where $\a$ is the original commutative algebra, and $\Omega^1$ is the bimodule of one-forms over $\a$. Now, we may generalize it to the case of noncommutative geometry and define metric as a middle-linear functional $\eta: \Omega^1 \times \Omega^1 \to \a$, i.e. $\eta$ is linear with respect to addition in $\Omega^1$ and satisfies: $$\eta( a \omega b, \rho c) \; = \; a \eta(\omega, b \rho) c,
\label{ml}$$ for every $a,b,c \in \a$ and $\omega, \rho \in \Omega^1$.The middle linearity naturally replaces the bilinearity condition in this case, however, we shall see that it is far more restrictive. The above definition proved to be suitable in our studies of discrete geometries [@ja].
We may also introduce the notion of hermitian metric, which could be defined on the differential algebra with involution. We say that metric $\eta$ is hermitian, if for all one-forms $u,v$ the following identity holds:
$$\eta( u, v) \; = \; ( \eta(v^\star, u^\star) )^\star.
\label{her}$$
In this paper we shall briefly discuss the consequences of the introduction of the metric to the analysis of $q$-deformed theories. We shall concentrate on two simple examples, leaving the general case for future studies [@jedr]. Finally, we shall discuss the relations between the above definition of the metric and the approach of Connes [@con].
Metric on the Manin Plane
=========================
Let us remind that the Manin plane is defined by replacing the commutativity of the generators of $\c[x,y]$ by the relation: $$x y \; = \; q y x,
\label{mp1}$$ where $q$ is a complex unitary number, $q\bar{q}=1$. We restrict our considerations here to the algebra obtained as a quotient of the free algebra generated by $y$ and $y$ by the ideal set by the relation (\[mp1\]).
Now, let us consider a $GL(2)_q$ invariant differential calculus [@brzez]. It has one free parameter $s$ and its multiplication rules are as follows:
$$\begin{aligned}
x dx &=& s dx x, \\
x dy &=& (s-1) dx y + q dy x, \\
y dx &=& s q^{-1} dx y, \\
y dy &=& s dy y.\end{aligned}$$
The metric, due to the middle-linearity, is completely determined by its values on the forms $dx$ and $dy$. If we call them $\eta^{xx}$ for $\eta(dx,dx)$, and $\eta^{xy},\eta^{yx},\eta^{yy}$ for other combinations respectively, we find the following set of constraints: $$\begin{aligned}
x \eta^{xx} & = & s^2 \eta^{xx} x, \label{r1} \\
y \eta^{xx} & = & s^2 q^{-2} \eta^{xx} y \\
x \eta^{xy} & = & s(s-1) \eta^{xx} y + sq \eta^{xy} x \\
y \eta^{xy} & = & s^2 q^{-1} \eta^{xy} y \\
x \eta^{yx} & = & s(s-1) q^{-1} \eta^{xx} y + sq \eta^{yx} x \\
y \eta^{yx} & = & s^2 q^{-1} \eta^{yx} y \\
x \eta^{yy} & = & s(s-1) \eta^{xy} y + (s-1) q \eta^{yx} y + q^2
\eta^{yy} x \\
y \eta^{yy} & = & s^2 \eta^{yy} y, \label{r2}\end{aligned}$$
Now, if we analyse them we find restrictions on the parameters $s$ and $q$. Since only monomials satisfy the commutation relations of the type (\[mp1\]), we come to conclusion that in the relation (\[r1\]) $s^2$ must be equal to $q^n$ for some $n
\geq 0$. Analogously, from (\[r2\]) we see that $s^2$ must be $q^{-m}$ for $m \geq 0$. Therefore, either $n=m=0$ and hence $s^2=1$ or $q^{n+m}=1$ and $s$ is any of the powers of $q$. In either case the possible values of $s$ are restricted to a finite set. In particular, we have found that out of infinitely many models of $q$-deformed, $GL(2)_q$ invariant differential calculus on the Manin plane, only two of them admit a metric for every value $q$.
Metric on $SU(2)_q$
===================
The Hopf algebra of $SU(2)_q$ is generated by two elements (and their conjugates), satisfying the following relations [@Woro]:
[[c]{}@[c]{}]{} $a^\star a + b^\star b = 1$ & $a a^\star + q^2 b^\star b = 1$
[[c]{}@[c]{}@[c]{}]{} $b^\star b = b b^\star$ & $ab = q ba $ & $a b^\star = q b^\star a$
where $q \in [-1,1]$.
Following Woronowicz [@Woro] we introduce the bimodule of one-forms, to be a free right-module generated by three elements $c^0,c^1,c^2$, with the following rules of left multiplication by the generator of $SU(2)_q$:\
[[l]{}@[l]{}@[l]{}]{} $c^0 a = q^{-1} a c^0$ & $c^1 a = q^{-2} a c^1$ & $c^2 a = q^{-1}
a
c^2$\
$c^0 b = q^{-1} b c^0$ & $c^1 b = q^{-2} b c^1$ & $c^2 b = q^{-1}
b
c^2$\
$c^0 a^\star = q a^\star c^0$ & $c^1 a^\star = q^{2} a^\star c^1$ & $c^2 a^\star = q a^\star c^2$\
$c^0 b^\star = q b^\star c^0$ & $c^1 b^\star = q^{2} b^\star c^1$ & $c^2 b^\star = q b^\star c^2.$
\
Additionally, the involution is extended to the bimodule of one forms and we have:\
[[c]{}[c]{}[c]{}]{} $(c^0)^\star = q c^0$ & $(c^1)^\star = - c^1$ & $(c^2)^\star =
q^{-1} c^2.$
\
Suppose now that we introduce the metric $\eta$, as proposed in the first section. Since the module of one-forms is free, the metric is completely determined by its values on the one-forms forming the basis. Let us call $\eta(c^i,c^j)$ by $\eta^{ij}$, $i=0,1,2$. Then, if we impose the condition of middle-linearity, we obtain the following set of relations for the elements $\eta^{ij}$:
$$\begin{aligned}
a \eta^{ij} &=& q^{\phi(i,j)} \eta^{ij} a, \\
a^\star \eta^{ij} &=& q^{-\phi(i,j)} \eta^{ij} a^\star, \\
b \eta^{ij} &=& q^{\phi(i,j)} \eta^{ij} b, \label{rr1} \\
b^\star \eta^{ij} &=& q^{-\phi(i,j)} \eta^{ij} b^\star,
\label{rr2}\end{aligned}$$
where $\phi(i,j)$ is defined as:
$$\phi(i,j) = \cases{ 4 & if $i=j=1$ \cr
3 & if $i \not= j =1$ or $j \not= i =1$ \cr
2 & if $i \not= 1$ and $j \not=1$}.$$
One may easily verify that in the considered algebra such constraints may be satisfied only if $q^2=1$. Indeed, from the relations (\[rr1\]) and (\[rr2\]) we obtain that $q^{2 \phi(i,j)}=1$. By taking all possible values of $i$ and $j$ we recover the above condition $q^2=1$. In the only non-trivial case $q=-1$ we could have, for instance, the following metric: $$\eta^{ij} \; = \; ab , \;\;\; \hbox{for} \;\; i \not= j =1 \;\;
\hbox{or} \;\; j \not= i =1,$$ and the other components taken as constants. Of course, we could scale each component by an arbitrary element of the center of the algebra.
In the case of $SU(2)_q$ the existence of the metric is a very strong requirement, which practically determines the value of the deformation parameter $q$ in the considered example of the differential calculus.
Conclusions
===========
As we have shown in two previous sections, the existence of a non-trivial metric is, in general, a very strong assumption. We have demonstrated that the noncommutativeness of the original algebra as well as of the differential calculus, enforce severe restrictions on the possible metrics. They could not be satisfied in general and lead to the constraints on the free parameters of theory. Therefore some models of differential calculus seem to be selected in a natural way by admitting the existence of the metric. It should be therefore interesting to determine such relations for other models, in particular for the general case of the bicovariant differential calculus on quantum groups.
Having defined the metric one could also use the construction to pursue the physical aspect of $q$-deformed theories. The natural next step should be the introduction of linear connections and $q$-deformed gravity, which is the topic of our current investigation [@jedr].
Appendix
========
In Connes approach, the basic object is a $K$-cycle, defined by the algebra ${\cal A}$, its representation $\pi$ on a Hilbert space and the Dirac operator $D$. The differential algebra could be derived from this construction by extending the definition of $\pi$ to the universal differential algebra $\Omega({\cal A})$: $$\pi(a^0 da^1 \ldots da^n) \; = \; a^0 [D,a^1] \ldots [D,a^n],$$ and by dividing $\Omega({\cal A})$ by the differential ideal $\pi^{-1}(0) + d \pi^{-1}(0)$.
Now, introducing a Dixmier trace, we have the integration on ${\cal A}$: $$\int a \; = \; \hbox{Tr} \, {\pi}(a),$$ as well as a complex valued functional on the bimodule of one- forms: $$\langle u, v \rangle \; = \; \hbox{Tr}( \pi(u) \pi(v) ).
\label{ff}$$
Let us turn back to the situation we were analysing. If we have a hermitian metric $\eta$ and a positive trace (integration $\int$) on the algebra ${\cal A}$ (which could be equivalent to the Dixmier trace), we can recover the functional of the type (\[ff\]) as follows: $$\langle u, v \rangle \; = \; \int \eta(u,v),$$
Now, the interesting question is whether the existence of the metric for a given differential calculus over ${\cal A}$ is equivalent to the existence of the corresponding $K$-cycle over ${\cal A}$. If so, we could use the results of our studies of the metric tensor in noncommutative geometry also in the broader context. This should provide us with a link, which would enable to extend the discrete geometry formalism of the Standard model [@CL] to include also the gravitational component. Additionally, we could then proceed with the introduction of $q$-deformed spinors, attempting to deform physical models of fundamental matter fields.
[99]{}
v\#1[[**\#1**]{}]{} V.Drinfeld, Sov.Math.Dokl. 32 (1985) 254,\
M.Jimbo, Lett. Math.Phys. 10 (1985) 63,\
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S.Majid, Int.J.Mod.Phys. Ǎ5 (1990) 4689.
L.Castellani, preprint DFTT 19/92,\
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T.Brzezinski, S.Majid, preprint DAMPT/92-27\
T.Brzezinski, S.Majid, Phys.Lett. B298 (1993) 339,\
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A.Sitarz, preprint TPJU 7/92\
A.Sitarz, preprint TPJU 4/93, to appear in Phys.Lett.B,
A.Sitarz, in preparation
A.Connes, Publ.Math. IHES Vol. 62 (1986) 41,\
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[^1]: Partially supported by KBN grant 2P 302 168 4
[^2]: E-mail: [email protected]
|
---
abstract: 'We analyze the single-particle spectra of a bi-layered electron system near a stripe instability and compare the results with ARPES experiments on the Bi2212 cuprate superconductor near optimum doping, addressing also the issue of the puzzling absence of bonding-antibonding splitting.'
address: |
Istituto Nazionale per la Fisica della Materia, UdR Roma 1 and Dipartimento di Fisica, Università di Roma “La Sapienza”,\
Piazzale Aldo Moro 2, I-00185 Roma, Italy
author:
- 'S. Caprara, C. Di Castro, and M. Grilli'
title: 'Single-particle spectra near a stripe instability'
---
[2]{}
It was proposed that the anomalous normal-state properties of the cuprate superconductors at optimum doping may result from the mixing of the doped holes with the collective charge and spin fluctuations near a stripe instability [@cdg]. To investigate the corresponding quasiparticle spectra, and to compare with experiments on the bi-layered Bi2212 [@saini], we introduce the Hamiltonian $$\begin{aligned}
H&=&\sum_{k;\sigma}\sum_\ell\left(\xi_k-\delta\mu\right)
c^{+}_{k\sigma\ell}c_{k\sigma\ell}\\
&-&t_\perp \sum_{k;\sigma} \gamma_k
\left( c^+_{k\sigma1}c_{k\sigma2} + h.c.\right)\\
&+&
\sum_{k,q;\sigma,\rho}\sum_\ell \sum_i g_i c^{+}_{k+q\sigma\ell}c_{k\rho\ell}
\tau^i_{\sigma\rho}S^i_{-q\ell}\end{aligned}$$ where $\xi_k=-2t(c_x+c_y)+4t'c_x c_y-\mu$, with $c_{x,y}=\cos(k_{x,y}a)$, is the tight-binding dispersion for electrons on a square lattice with nearest and next-to-nearest neighbors hopping, $a$ is the lattice spacing, $\mu$ is the bare chemical potential and $\delta\mu$ is the shift in the interacting system. The planes are labelled by $\ell=1,2$ and $t_\perp$ is the interplane matrix element modulated by $\gamma_k={1\over 2}|c_x-c_y|$ to account for the suppression at $k_x\simeq \pm k_y$ [@and]. The constants $g_i$ couple electrons to charge ($i=0$) and spin ($i=1,2,3$) fluctuations. The spin structure of the generalized density coupled to $S_{-q\ell}^i$ is accounted for by the Pauli matrix $\tau^i$. The counterterm $\delta\mu\sim O(g^2)$ is determined, order by order in perturbation theory, to fix the number of electrons. The fluctuating fields $S_{-q\ell}^i$ are characterized by the susceptibilities $\chi_{ij\ell\ell'}(q,\omega)=\delta_{ij}
\delta_{\ell\ell'} A_i/[\kappa^2+\eta_{q-Q_{i\ell}}-{\rm i}{\bar \tau}
\omega]^{-1}$, where $A_i$ are constants, the mass $\kappa^2$ vanishes at criticality, $\eta_k=2-c_x-c_y$ reproduces the $k^2$ behavior at $ka\ll 1$, preserving the lattice periodicity, $Q_{i\ell}$ are the critical wave-vectors, ${\bar \tau}$ is a characteristic time scale, and the dimensionless coupling constants are $\lambda_i=g_i^2 A_i/t$. In the paramagnetic phase the parameters are the same for $i=1,2,3$, and we label charge and spin fluctuations with $c,s$. The direction of the charge modulation is debated. As a matter of illustration we analyze the case $Q_{c1}= 0.4(\pi/a,-\pi/a)$, $Q_{s1}=(\pi/a,\pi/a)$, suggested by the ARPES experiments on Bi2212 [@saini]. We allow for a mismatch of the charge modulation pattern on the two planes and take $Q_{c2}= 0.4(\pi/a,\pi/a)$, while $Q_{s2}=Q_{s1}$. The Hamiltonian is written in the simpler “plane representation”, in which the interaction is diagonal, and the decoupling of the two planes as $t_\perp\to 0$ is evident. The alternative “band representation” is obtained by diagonalization of the fermionic part of the Hamiltonian, but the interaction is not diagonal, and the role of $t_\perp$ is less transparent. The two representations are, of course, equivalent.
The $O(\lambda)$ perturbation theory accounts for the main dressing of the electrons. An average over $\pm Q_c$ is performed to maintain inversion symmetry, leading to the self-energy $\Sigma_\ell=<\Sigma_{c\ell}>+3\Sigma_s
-\delta\mu$. Previous analysis for $t_\perp=0$ [@pap] showed that spectral weight is transferred from the quasiparticle peak to incoherent shadow resonances. The changes in the single-particle spectra and in the distribution of low-lying spectral weight are in agreement with the experiment [@saini]. The suppression of spectral weight at the Fermi level near the $M$ points \[i.e. $(0,\pm\pi/a)$ and $(\pm\pi/a,0)$\], due to spin fluctuations, is modulated by charge fluctuations.
Here we address the absence of bi-layer splitting, despite a sizable calculated $t_\perp$ [@and]. The contribution of the plane $\ell$ to the spectral density is $$A_\ell =-{1\over \pi}\sum_{\alpha=1,2}
{\rm Im} \left. {1+(-1)^{\alpha+\ell}\Sigma_- /{\tilde t}_\perp
\over \omega - \xi_k-\Sigma_+ +(-1)^\alpha {\tilde t}_\perp}
\right|_{R}$$ where $\Sigma_\pm={1\over 2}(\Sigma_{\ell=1}\pm\Sigma_{\ell=2})$, ${\tilde t}_\perp=(t_\perp^2\gamma_k^2+\Sigma_-^2)^{1/2}$ and ${R}$ means retarded. For $\lambda_i=0$ the quasiparticle FS consists of two branches, corresponding to the bonding and antibonding band (Fig. 1, left panel), well separated near the M points, where $\gamma_k$ is larger. However, a moderate coupling between the electrons and the critical fluctuations is sufficient to eliminate the FS splitting. Indeed, the suppression of spectral weight is stronger near the $M$ points where the splitting is expected to be larger, and is weaker where the splitting is naturally suppressed by $\gamma_k$. The effect is enhanced in the case of a mismatch between the fluctuation patterns on the two different planes. The resulting FS, projected onto a single plane (e.g. $\ell=1$) as it is suitable to interpret ARPES results, is essentially the same as in the absence of interlayer coupling (Fig. 1, right panel). Thus the absence of the band splitting in ARPES spectra of bi-layered materials can be due to the enhancement of charge and spin fluctuations scattering the quasiparticles near a stripe instability.
[ Acknowledgments.]{} Part of this work was carried out with the financial support of the I.N.F.M. - P.R.A. 1996.
C. Castellani, C. Di Castro, M. Grilli, Phys. Rev. Lett. 75 (1995) 4650. N. L. Saini [*et al.*]{}, Phys. Rev. Lett. 79 (1997) 3467. O. K. Andersen [*et al.*]{}, Phys. Rev. B 49 (1994) 4145. S. Caprara [*et al.*]{}, Phys. Rev. B 59 (1999) 14980.
(5.0,15.0) (-4.0,-8.0)
\[fig1\]
|
---
abstract: 'Over the course of three hours on 27 December 2008 we obtained optical (R-band) observations of the blazar S5 0716$+$714 at a very fast cadence of 10 s. Using several different techniques we find fluctuations with an approximately 15-minute quasi-period to be present in the first portion of that data at a $>3 \sigma$ confidence level. This is the fastest QPO that has been claimed to be observed in any blazar at any wavelength. While this data is insufficient to strongly constrain models for such fluctuations, the presence of such a short timescale when the source is not in a very low state seems to favor the action of turbulence behind a shock in the blazar’s relativistic jet.'
author:
- 'Bindu Rani, Alok C. Gupta, U. C. Joshi, S. Ganesh and Paul J. Wiita'
title: 'Quasi-Periodic Oscillations of $\sim$15 minutes in the Optical Light Curve of the BL Lac S5 0716$+$714'
---
Introduction
============
Characteristic timescales of variability provide an important way to probe the sub-parsec scale central engines in active galactic nuclei (AGN) by providing information about the sizes and locations of emission regions. In blazars, i.e., BL Lacertae objects (BL Lacs) and flat spectrum radio quasars (FSRQs), Doppler boosted emission from a relativistic jet has long been recognized to provide the only feasible explanation for their non-thermal spectra and radio morphologies on small-scales [e.g., @blandford1978; @urry1995]. Still, the question of just where emission at different frequencies originates remains somewhat uncertain [e.g., @marscher2008]. Rapid fluctuations have long been known to characterize blazars, with the prototype, BL Lac, seen to flicker over just a few minutes in early single channel photometry with 15-second temporal resolution [@racine1970].
The bright, high declination BL Lac, S5 0716$+$714, at redshift $z = 0.31 \pm 0.08$ [@nilsson2008] has been extensively studied across the electromagnetic spectrum and exhibits strong variability on a wide range of timescales, ranging from minutes to years [e.g., @gupta2008a; @gupta2008b; @gupta2009 and references therein]. The optical duty cycle of S5 0716$+$714 is nearly unity, indicating that the source is always in an active state in the visible [@wagner1995]. This blazar was recently shown to be a strong source in the high energy gamma-ray band by Fermi-LAT [@abdo2009].
There is good evidence for the presence of quasi-periodic oscillations (QPOs) in the emission of just a few blazars [@espaillat2008; @gupta2009; @rani2009; @lachowicz2009]. The blazar S5 0716$+$714 is among these rare exceptions: a possible QPO on the timescale of $\sim$1 day may have been observed simultaneously in an optical and a radio band [@quirrenbach1991]. On another occasion, quasi-periodicity with a time scale of $\sim$4 days appeared to be present in its optical emission [@heidt1996]. Five major optical outbursts between 1995 and 2007 have occurred at intervals of $\sim$ 3.0 $\pm$ 0.3 years [e.g., @raiteri2003; @gupta2008a and references therein]. Recently, @gupta2009 used a wavelet analysis on the 20 best nights of over 100 nights of high quality optical data taken by @montagni2006, and found high probabilities that S5 0716$+$714 showed quasi-periodic components to its variations on several nights that ranged between $\sim$25 and $\sim$73 minutes.
Among the other blazars, PKS 2155$-$304 possibly showed a quasi-periodicity around 0.7 days during 5 days of observations at UV and optical wavelengths [@urry1993]. Very recently, somewhat better evidence for a QPO of $\sim$4.6 h in the XMM-Newton X-ray light curve of PKS 2155$-$304 has been reported by @lachowicz2009. An XMM-Newton light curve of the quasar 3C 273 appears to have a quasi-periodic component with a timescale of about 3.3 ks [@espaillat2008]. Using the $\sim$13 year long data taken by the All Sky Monitor on the Rossi X-ray Timing Explorer satellite, @rani2009 reported good evidence of nearly periodic variations of $\sim$ 17.7 days in the blazar AO 0235$+$164 and $\sim$ 420 days in the blazar 1ES 2321$+$419. The narrow line Seyfert 1 galaxy, RE J1034$+$396, while not a blazar, strongly indicated the presence of a $\sim$1 hour periodicity during a 91 ks observation by the X-ray satellite XMM-Newton [@gierlinski2008].
In this Letter, we exhibit evidence for a QPO of $\sim$15 minutes in a single densely sampled optical light curve of the blazar S5 0716$+$714. We first used a structure function (SF) analysis to find a hint of such a QPO and we then quantified the strength of this signal using Lomb-Scargle Periodogram (LSP) and Power Spectral Density (PSD) methods. We find this to be a strong case for the discovery of the shortest nearly periodic variation seen for any blazar, or for that matter, any AGN, in any waveband.
Observations and Data Reduction
===============================
Our observations of S5 0716$+$714 were carried out with an Andor EMCCD (Electron Multiplying Charge Coupled Device) camera mounted at the f/13 Cassegrain focus of the 1.2 m telescope operated by the Physical Research Laboratory (PRL) at Gurushikhar, Mt. Abu, India. We observed this source on 23, 27 and 28 December 2008 and 3 January 2009; the total amount of data collected over those four nights was 9.6 hours. The 1k $\times$ 1k EMCCD has square pixels with sides of 13 $\mu$m size. With electron multiplication technology, the read noise in the system is expected to be negligible compared to normal CCD cameras [@mackay2001] and the performance approaches near photon counting efficiency. The camera was thermoelectrically cooled to $-$80 C$^{\circ}$ for our observations and had negligible dark current. An R filter and a temporal resolution of only 10 seconds were employed. The typical seeing was $\sim$1.6 arcsec. On each night, we took several bias frames and twilight sky flats in the R band. To improve the S/N ratio, we performed these observations in 2 pixel $\times$ 2 pixel binning mode so that 4 pixels work as a single super-pixel.
The image pre-processing was done using the standard routines in Image Reduction and Analysis Facility[^1] (IRAF) software. Data analysis, or processing of the data, involved performing aperture photometry using Dominion Astronomical Observatory Photometry (DAOPHOT II) software [@stetson1992]. We first carried out aperture photometry with four different aperture radii, i.e., 1$\times$FWHM, 2$\times$FWHM, 3$\times$FWHM and 4$\times$FWHM. We discovered that aperture radii of 3$\times$FWHM usually provided the best S/N ratio and we adopted it for our work. The standard stars 8 and 11 [@nicolas2001] whose apparent brightnesses were close to that of the source and were always observed in the same field as the blazar were used to check that the variability was intrinsic to the blazar. The standard star 11 was used to calibrate the blazar’s magnitude. Only on the night of 27 December 2008 did we detect interesting rapid variability and that light curve is displayed in Fig. 1. We note that over the past 15 years S5 0716$+$714 has varied between $\sim 12.3$ and $\sim 15.6$ R-band magnitudes, though it was even fainter earlier [@raiteri2003; @nesci2005; @gupta2008a].
Analysis and Results
====================
In order to be certain the apparent variability of S5 0716$+$714 is significant we used the F-test, shown by @deigo2010 to be superior to commonly used methods. The F-statistic is the ratio of the sample variances, or $F = {s^{2}_{Q}}/{s^{2}_{S}},$ where the variance for the quasar differential light curve is $s^{2}_{Q}$, while that for the standard star is $s^{2}_{S}$. We used the F-test code available in R[^2] and find $F = 18.3748$, with a significance level of 0.9999998, or $>5 \sigma$.
We have also calculated the variability amplitude parameter, $A$ [@heidt1996], to see the percentage variation in the light curve of source. For S5 0716$+$714 we find $A = 16.9$$\%$. The calculated fractional rms variability amplitude for the LC [@vaughan2003] is $F_{var} = 15.45$.
A visual inspection of the light curve for the first two hours shown in Fig. 1(a) indicates a possible periodic modulation of the variability at about 900 s, along with a hint of even faster modulations at the very beginning of the observation. The calibrated light curve for the entire 3 hours of measurements taken at a 10 s cadence, along with the differential instrumental magnitudes of standard stars 8 and 11, are displayed in Fig. 1(b). The light curve averaged over 30 s intervals folded at a putative period of 900 s is Fig. 2(a).
Structure Function
------------------
The first order structure function (SF) is a simple way to search for periodicities and timescales of variability in time series data trains [e.g., @simonetti1985]. Here we give only a very brief summary of the method; for details refer to @rani2009. The first order SF for a data train, $a$, is defined as $$\begin{aligned}
D^{1}_{a}(k) = {\frac{1}{N^{1}_{a}(k)}} {\sum_{i=1}^N} w(i)w(i+k){[a(i+k) - a(i)]}^{2},\end{aligned}$$ where $k$ is the time lag, ${N^{1}_{a}(k)} = \sum w(i)w(i+k)$, and the weighting factor, $w(i)$, is 1 if a measurement exists for the $i^{th}$ interval, and 0 otherwise. For a time series containing a periodic pattern, the SF curve shows minima at time lags equal to the period and its subharmonics [e.g., @lachowicz2006], although dips and wiggles in SFs are not always reliable indicators of timescales [@emmano2010]. The SF analysis curve of the whole data set is displayed in Fig. 2(c). The first dip and the 7 cycles of its subsequent subharmonics correspond to a possible period of 927$\pm$30 seconds.
Lomb$-$Scargle Periodogram
--------------------------
The Lomb-Scargle Periodogram (LSP), introduced by @lomb1976 and extended later by @scargle1982, is an excellent technique for searching time series, as long as white-noise, $P_N(f) \propto f^0$, is the dominant noise process. @press1989 provided a more practical mathematical formulation. For the for details of method and formulae see @rani2009 and references therein.
We used an online available R-language code for the LSP[^3]. The LSP analysis of the whole data set is displayed in Fig. 2(b). The LSP analysis revealed the detection of significant frequency corresponding to a period of 904 seconds with a significance level of 0.999999977. Two questions usually arise concerning the validity of a periodogram result [@scargle1982]; the first is statistical and the second is spectral leakage. The statistical difficulty is mitigated by the good S/N ratio of $\sim$35 in our case. Spectral leakage, which is also known as aliasing, involves the spreading of periodogram power to other frequencies that are actually not present in the data. Since our data is uniformly sampled there might be chances of aliasing. But as essentially the same period is confirmed by SF and PSD analyses the strong signal is very unlikely to arise in this fashion.
However, as LCs of most AGN contain red-noise as well as white-noise, a more robust test is required to quantify the presence of a QPO.
Power Spectral Density
----------------------
The power spectral density (PSD) is a powerful tool to search for periodic signals in time series, including those contaminated by white- and/or red-noise [e.g., @vaughan2005]. We employed a PSD analysis method [@vaughan2003; @vaughan2005] that is suitable for these types of LCs. First, as shown in Fig. 3, we fit a single power-law (SPL) to the calculated PSD, assuming it to have a form $P(f) \propto f^{\alpha}$ at low frequencies and then examined the significance of the frequency peak using the method of @vaughan2005. This analysis indicates the presence of a QPO signal with peak frequency $\simeq$0.001077 Hz (or period $\simeq$928 s), with a 3.4$\sigma$ significance level. The calculated significance is global, i.e., corrected for the number of frequencies tested. The range of frequencies used for calculating the global significance of the QPO is $0.0002 \le f \le 0.002$ which amounts to 28 frequency bins. This range excludes frequencies that are significantly dominated by white noise.
We next checked the statistical significance of this QPO using Monte Carlo simulations. We generated a series of $10^4$ simulated LCs following a given SPL having the same number of bins, mean and variance as the observed LC [@timmer1995] using an IDL code available on-line[^4]. The PSD analysis resulting from the simulated LCs using a SPL with the index from the best fit to our data are compatible with the results shown in Fig. 2 and indicate an average significance of $3.2\sigma$. We also considered the alternative null hypothesis of a broken power-law (BPL). The best fitting BPL indices are, respectively, $+0.39$ and $-0.8$ above and below the break frequency of $\simeq 0.0011$Hz. The nominal statistical significance of the QPO frequency in this case is $\sim 3.1\sigma$. Finally, we performed PSD analyses of simulated LCs generated from BPLs and calculated periodograms for each of them, finding that the periodic signal was still significant at $3 \sigma$. Hence we conclude that the observed QPO at a frequency of $\sim$0.001077 Hz is statistically significant, irrespective of the assumed model of continuum power.
Discussion and Conclusions
==========================
This discovery of a nearly periodic signal of $\sim$900 seconds in the optical R passband light curve of the blazar S5 0716$+$716 adds a unique new point to the variability studies of blazars at intraday timescales. The presence of 7 cycles with a $> 3 \sigma$ significance level allows us to make a strong claim for the shortest optical QPO detected so far.
The simplest possible explanation for such a short period might be the flux arising from hot spots or some other non-axisymmetric phenomenon related to the orbital motions that are close to the innermost stable circular orbit around a supermassive black hole (SMBH) [e.g., @zhang1991]. Adopting $z = 0.31$ for S5 0716$+$714 [@nilsson2008], means that a 900 second period at the inner edge of a corotating disk corresponds to a SMBH mass of 1.5 $\times$10$^{6}$ M$_{\odot}$ for a non-rotating BH and 9.6 $\times$10$^{6}$ M$_{\odot}$ for a maximally rotating BH [@gupta2009]. If the source arises somewhat further out in the accretion disk, then the BH mass would be even less than these modest values.
However, since blazar jets are pointing very close to the line-of-sight of the observer [e.g., @urry1995] the emerging flux, particularly in active phases, is dominated by emission from jets. Turbulence behind a shock propagating down a jet [e.g. @marscher1992] is a very plausible way to produce dominant eddies whose turnover times can yield short-lived, quasi-periodic fluctuations in emission at different wavelengths. Since Doppler boosting will greatly amplify the very weak intrinsic flux variations produced by small changes in the magnetic field or relativistic electron density, these intrinsically weak fluctuations can be raised to the level at which they can be detected [e.g., @qian1991]. This same Doppler boosting reduces the time-scale at which these fluctuations are observed compared to the time-scale they possess in the emission frame. Although it is difficult to quantify these effects precisely, this mechanism does seem to provide an excellent way to understand the type of short-lived optical intra-night variability with periods of tens of minutes seen here.
It is also possible that QPOs originate from a relativistic shock propagating down a jet that possesses a helical structure, as can be induced by magnetohydrodynamical instabilities [@hardee1999] or even through precession. Indeed, in some cases where radio jets can be resolved transversely using Very Long Baseline Interferometry, edge-brightened and non-axisymmetric structures are seen (e.g., M87, [@ly2007]; Mkn 501, [@piner2009]). A relativistic shock propagating down such a perturbed jet will induce significantly increased emission at the locations where the shock intersects with a region of enhanced magnetic field and/or electron density corresponding to such a non-axisymmetric structure. Because Doppler boosting is a sensitive function of viewing angle substantial changes in amplitude of jet emission can be seen by the observer [@camenzind1992; @gopal1992]. Therefore, the observed periodic component in the optical light curve of S5 0716$+$714 might be attributed to the intersections of a relativistic shock with successive twists of a non-axisymmetric jet structure, though they would have to be surprisingly tight to yield such a short period.
We have analyzed the optical R passband light curve of the well-known BL Lac S5 0716$+$714 observed on 27 December 2008 with a 10 second cadence that provided the best time resolution so far obtained for a blazar. Different analyses all indicate this light curve contains a periodic component to its fluctuations of about 15 min. Although this particular BL Lac showed earlier evidence of periodic variations in radio through X-ray wavebands ranging from tens of minutes to several years, our new data provides the shortest known quasi-period yet detected in a blazar.
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[^1]: 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.
[^2]: R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org.
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[^4]: http://astro.uni-tuebingen.de/software/idl/aitlib/timing/timmerlc.html
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---
abstract: 'It is well known that under generic $C^r$ smooth perturbations, the phenomenon of Arnold diffusion exists in the *a priori* unstable Hamiltonian systems. In this paper, by using variational methods, we will prove that under generic Gevrey smooth perturbations, Arnold diffusion still exists in the *a priori* unstable and time-periodic Hamiltonian systems with multiple degrees of freedom.'
address:
- '$^\dag$ Morningside center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China'
- '$^\dag$ Department of Mathematics, Nanjing University, Nanjing 210093, China'
- '\* Department of Mathematics, Nanjing University, Nanjing 210093, China'
author:
- 'Qinbo Chen$^\dag$'
- 'Chong-Qing Cheng\*'
title: 'Gevrey genericity of Arnold diffusion in *a priori* unstable Hamiltonian systems'
---
Introduction
============
In this paper, we denote by $\T^n\times\R^n$ the cotangent bundle $T^*\T^n$ of the torus $\T^n$ with $\T=\R/\Z$, and endow $\T^n\times\R^n$ with its usual coordinates $(q,p)$ where $q=(q_1,\cdots,q_n)$ and $p=(p_1,\cdots,p_n)$. We also endow the phase space with its canonical symplectic form $\Omega=\sum\limits_{i=1}^ndq_i\wedge dp_i$. A Hamiltonian system is usually a dynamical system governed by the following Hamilton’s equation $$\dot{q}=\frac{\partial H}{\partial p}, \quad \dot{p}=-\frac{\partial H}{\partial q},\quad (q,p)\in\T^n\times\R^n$$ where $H=H(q, p, t)$ is a Hamiltonian and the dependence on the time $t$ is $1$-periodic, so $t\in\T$.
One of the most important problems is to describe the time evolution of the nearly-integrable Hamiltonian systems $$H=H_0(p)+\varepsilon H_1(q, p, t)$$ where $H_0$ is strictly convex. This problem was considered by Poincaré to be the fundamental problem of Hamiltonian dynamics. Notice that such systems do not admit any instability phenomenon when $\varepsilon=0$. For $0<\varepsilon\ll 1$, the classical KAM theory asserts that a set of nearly full measure in phase space consists of invariant tori carrying quasi-periodic motions, and the oscillation of the action variables $p$ on each KAM torus is at most $O(\sqrt{\varepsilon})$. For $n\geq2$, the complement to the union of all KAM tori is connected, so it is natural to ask whether there are orbits whose action variables $p$ can change by a quantity of order $1$. In the 1960s, Arnold [@Ar1964] first gave an example for $n=2$ which shows that there are orbits along which the action variables $p$ slowly drift to a distance of order $1$, he also conjectured that such an evolution also occurs in generic nearly-integrable systems for $n\geq 2$. Since then this phenomenon was called “Arnold diffusion ", and lots of works have been devoted to this study.
Arnold’s special example contains a normally hyperbolic invariant cylinder (NHIC) foliated by a family of hyperbolic invariant tori, where the unstable manifold of one torus transversally intersects the stable manifold of another nearby torus. These tori constitute a transition chain along which the diffusion orbits drift. Thus, this inspired a large number of studies to the Hamiltonians which already possess normally hyperbolic cylinders, by considering $H=H_0+\varepsilon H_1$ where $H_0(q, p)=h(I)+P(x, y)$ with $q=(\theta,x)$ and $p=(I,y)$, and $P(x,y)$ has some a priori hyperbolicity. Such a system is usually called “*a priori* unstable" to be distinguished from the nearly-integrable systems (i.e. “*a priori* stable"). There have been many works devoted to these *a priori* unstable systems based on Arnold’s original geometric mechanism, see [@BBB2003; @BB2002; @Cr2003; @FP2001; @KL2008; @KS2012; @LM2005; @Zh2011], etc. However, for general *a priori* unstable systems the transition chain is not formed by a continuous family of tori but a Cantorian family, and the gaps on the family of tori are larger than the size of the intersections of the stable and unstable manifolds, which is known as the *large gap problem*.
For the large gap problem, essential progress has been made by Mather in [@Ma1993] where he introduced a variational construction of connecting orbits for positive definite time-periodic systems, and in an unpublished manuscript [@Ma1995] he also showed the existence of orbits with unbounded energy in perturbations of a geodesic flow on $\T^2$ by a generic time-periodic potential. Based on these ideas, the authors in [@CY2004] constructed the diffusion orbits crossing the gaps by a variational mechanism and proved the genericity of Arnold diffusion for the *a priori* unstable systems with two and a half degrees of freedom. Meanwhile, by using geometric methods, substantial progress has also been made in [@DLS2003; @DLS2006] as well as in [@Tr2002; @Tr2004] for the *a priori* unstable systems with $2+1/2$ degrees of freedom. More specifically, the authors in [@DLS2003; @DLS2006] defined the so-called scattering map based on the transversal intersection of the stable and unstable manifolds, and overcame the large gap problem by a detailed analysis of the dynamics restricted on the NHIC and the scattering map. One advantage of this method is that it can be applied to some non-convex systems; In [@Tr2002] the author defined the so-called separatrix map near the normally hyperbolic invariant cylinder, and then he showed the existence of diffusion orbits by analyzing the dynamics of this map and even estimated the optimal diffusion speed of order $\varepsilon/|\log \varepsilon|$ [@Tr2004]. Later, the case for the *a priori* unstable system with higher degrees of freedom has also been fully studied, by both variational and geometric methods, see for instance [@Be2008; @CY2009; @DLS2016; @DT2018; @GKZ2016; @LC2010; @Tr2012] and so on.
For *a priori* stable case, Mather first made an announcement in [@Ma2004] (see also [@Ma2012]) for systems with two degrees of freedom in the time-periodic case or with three degrees of freedom in the autonomous case, under so-called cusp residual condition. So the diffusion problem in this situation was thought to possess only cusp-residual genericity. This first complete proof is the preprint [@Ch2012] which deals with the autonomous systems with three degrees of freedom, and the main ingredients have been published in the recent works [@CZh2016; @Ch2017Uniform; @Ch2017; @Ch2018]. The main difficulty for *a priori* stable case is the dynamics around strong double resonance. In fact, away from the strong double resonance, one could find some pieces of normally hyperbolic invariant cylinders along which the local diffusion orbits can be constructed as in the a priori unstable case, see for instance [@BKZ2016]. However, the first study about the dynamics around strong double resonances in details is [@Ch2017] where the author presented a variational mechanism of the diffusion orbits passing through strong double resonance, and which eventually gives rise to the proof of Arnold diffusion in the sense of cusp-residual genericity [@Ch2018]. Besides, we refer the reader to the announcement [@KZ2015] and the preprint [@KZ2013] for systems with 2.5 degrees of freedom by variational methods, and refer to the preprints [@Marco2016Arnold; @Marco2016chains; @GM2017] for systems with 3 degrees of freedom by geometric methods, and also refer to the preprint [@CX2015] for arbitrarily higher degrees of freedom. Anyway, there have been many other works related to the problem of Arnold diffusion but we cannot list all of them, see for example [@Bessi1996; @BCV2001; @BT1999; @DH2011; @DLS2000; @GT2008; @GR2007; @KZ2014; @ZC2014].
However, so far as we know, the genericity (or cusp-residual genericity) of Arnold diffusion has only been proved for the perturbations in the $C^r$ smooth topology, not yet for the analytic topology, or the Gevrey smooth topology that was first introduced by Gevrey [@Gev1918]. The goal of this paper is to deal with the Gevrey genericity of Arnold diffusion. Given $\alpha\geq 1$, a Gevrey-$\alpha$ function is an ultra-differentiable function whose $k$-th order partial derivatives are bounded by $O(M^{-|k|}k!^{\alpha})$. For the case $\alpha=1$, it is exactly a real-analytic function. Thus the Gevrey class is intermediate between the $C^\infty$ class and the real analytic class. Besides, one crucial element of the Gevrey class is that it allows the existence of bump functions.
To consider the Arnold diffusion problem in the Gevrey category, we would adopt the Gevrey norm introduced by Marco and Sauzin in [@MS2003] during a collaboration with Herman (see Definition \[def of gev\]). Apart from the theory of PDE where they have been widely used, the Gevrey class was also studied in the field of Dynamical Systems. For example, we refer to ([@Bo2011; @Bo2013; @BF2017; @BM2011; @LDG2017; @Po2004], etc) for the stability theory, such as KAM theory and Nekhoroshev theory. We also refer the reader to ([@BK2005; @LMS2015; @MS2004; @Wa2015], etc) for the relevant results on instability. All these studies make us believe that one can also consider the genericity problem in the Gevrey case.
So in this paper, we decide to focus on the *a priori* unstable, Gevrey-$\alpha$ ($\alpha>1$) Hamiltonian systems. The case $\alpha=1$ (i.e. the analytic genericity) is more complicated and still open, a recent work [@GT2017] has made some progress in this direction, and they considered *a priori* chaotic symplectic map which has a normally hyperbolic invariant cylinder and an associated transverse homoclinic cylinder.
Now, we first give a brief introduction to the concept of Gevrey function and collect some basic properties for these functions.
\[def of gev\] Let $\alpha\geq 1, \bL>0$ and $K$ be a $n$-dimensional domain. A real-valued $C^\infty$ function $f(x)$ defined on $K$ is said to be Gevrey-($\alpha,\bL$) if $$\| f\|_{\alpha,\bL}:=\sum\limits_{k\in\N^n}\frac{\bL^{|k|\alpha}}{k!^\alpha}\|\partial^kf\|_{C^0(K)}<+\infty,$$ with the standard notations $k=(k_1,\cdots,k_n)\in\N^n$, $|k|=k_1+\cdots+k_n$, $k!=k_1!\cdots k_n!$ and $\partial^k=\partial^{k_1}_{x_1}\cdots\partial^{k_n}_{x_n}$.
Let $\bG^{\alpha,\bL}(K):=\{ f\in C^\infty(K)~:~\|f\|_{\alpha,\bL}<+\infty\}$, it’s easy to verify that $\bG^{\alpha,\bL}(K)$ is a Banach space with the norm $\|\cdot\|_{\alpha,\bL}$. Sometimes we also set $\bG^{\alpha}(K):=\bigcup\limits_{\bL>0}\bG^{\alpha,\bL}(K)$.
For example, if $K\subset\R^n$, then any real analytic function defined in $K$ which admits an analytic extension in the complex domain $\{z\in\C^n:\textup{dist}(z,K)\leq \bL^\alpha\}$, belongs to $\bG^{\alpha,\bL}(K)$. In particular, for $\alpha=1$, $K\subset\R^n$, $\bG^{1,\bL}(K)$ is exactly the space of all real analytic functions defined in $K$ that admit an analytic extension in the complex domain $\{z\in\mathbb{C}^n~|~\mathrm{d}(z,K)<\bL\}$. However, for $\alpha>1$, $f\in\bG^{\alpha,\bL}(K)$ is not analytic anymore. Therefore, the Gevrey-smooth category is intermediate between the $C^\infty$ category and the analytic category.
Gevrey functions have the following useful properties which have been already proved in [@MS2003]:
1. \[algebra norm\] The norm $\|\cdot\|_{\alpha,\bL}$ is an algebra norm, namely $\|fg\|_{\alpha,\bL}\leq\|f\|_{\alpha,\bL}\|g\|_{\alpha,\bL}$.
2. \[derivative Gevrey\]Suppose $0<\lambda<\bL$ and $f\in\bG^{\alpha,\bL}(K)$, then all partial derivatives of $f$ belong to $\bG^{\alpha,\bL-\lambda}(K)$ and $\sum\limits_{k\in\N^n,|k|=l} \|\partial^kf\|_{\alpha,\bL-\lambda}\leq l!^\alpha\lambda^{-l\alpha}\|f\|_{\alpha,\bL}.$
3. \[composition\]Let $f\in\bG^{\alpha,\bL}(K_m)$ where $K_m$ is a $m$-dimensional domain and let $g=(g_1,\cdots,g_m)$ be a mapping whose component $g_i\in\bG^{\alpha,\bL_1}(K_n)$ . If $g(K_n)\subset K_m$ and $\|g_i\|_{\alpha,\bL_1}-\|g_i\|_{C^0(K_n)}\leq\bL^\alpha/n^{\alpha-1}$ for all $1\leq i\leq m,$ then $f\circ g\in\bG^{\alpha,\bL_1}(K_n)$ and $\|f\circ g\|_{\alpha,\bL_1}\leq\|f\|_{\alpha,\bL}$.
Statement of the result
-----------------------
Our settings are as follows: throughout this paper, we fix, once and for all, a constant $R>1$. Set $$\sD_R=\T^n\times\bar{B}_R(0)\times\T,$$ where $B_R(0)\subset\R^n$ is the open ball of radius $R$ centered at 0 and $\bar{B}_R(0)$ is the closure. By Definition \[def of gev\], the space $\bG^{\alpha,\bL}(\sD_R)$ consists of all real-valued functions $f(q,p,t)$ satisfying $$\label{gevrey norm}
\| f\|_{\alpha,\bL}=\sum\limits_{k\in\N^{2n+1}}\frac{\bL^{|k|\alpha}}{k!^\alpha}\|\partial^kf\|_{C^0(\sD_R)}<+\infty,$$ Let $C^\omega_d(\sD_R)$ be the space of all real-valued analytic functions defined on $\sD_R$, admitting an analytic extension in the complex domain $\{(q,p,t)\in(\C/\Z)^n\times\C^n\times(\C/\Z):\|\textup{Im}q\|_{\infty}< d,~\textup{dist}(p,\bar{B}_R(0)) < d, \|\textup{Im}t\|_{\infty}<d\}$. Set $C^\omega(\sD_R)=\bigcup\limits_{d>0} C^\omega_d(\sD_R)$ , it is well known that
- For $\alpha\geq 1$, $\bL>0$ and any $d>\bL^\alpha$, $$C^\omega_d(\sD_R)\subset \bG^{\alpha,\bL}(\sD_R)\subset C^\infty(\sD_R),\quad C^\omega(\sD_R)\subset \bG^{\alpha}(\sD_R)\subset C^\infty(\sD_R).$$
- $C^\omega(\sD_R)=\bG^{1}(\sD_R).$
Now we begin to introduce the *a priori* unstable and time-periodic Hamiltonian system with arbitrary $n\geq 2$ degrees of freedom: let $q=(q_1,\htq)\in\T\times\T^{n-1}$ and $p=(p_1,\htp)\in\R\times\R^{n-1}$ where $\htq$ denotes $(q_2,\cdots,q_n)$ and $\htp$ denotes $(p_2,\cdots,p_n)$, we consider the Hamiltonian $$\label{hamiltonian}
\begin{aligned}
H(q, p, t)=H_0(q, p)+H_1(q, p, t),
\end{aligned}$$ where $H_0(q, p)=h_1(p_1)+h_2(\htq,\htp)$ and $H_1$ is a small perturbation which is 1-periodic in time. We assume that $H_0$ satisfies:
1. *Convexity and superlinearity*: for every point $(q,t)$, the Hessian $\partial_{pp}H_0(q,p,t)$ is positive definite and $\lim_{\|p\|\rightarrow +\infty} H_0(q,p,t)/\|p\|=+\infty.$
2. *A priori hyperbolicity*: the Hamiltonian flow $\phi^t_{h_2}$, determined by $h_2$, has a non-degenerate hyperbolic fixed point $(\htq,\htp)=(x^*,y^*)$ and $h_2(\htq,y^*): \T^{n-1} \rightarrow \R$ attains its unique maximum at $\htq=x^*$. Without loss of generality, we can assume $(x^*,y^*)=(\bfo,\bfo)$.
Throughout this paper, we use the boldface $\bfo$ to denote the multidimensional vector $(0,\cdots,0)$. The assumptions on the unperturbed $H_0$ are in the same spirit as in [@CY2009] while the main results and proofs have some differences. A typical example of such kind of systems is the coupling of a rotator and pendulums $$H=\frac{p_1^2}{2}+\sum_{i=2}^n\big(\frac{p_i^2}{2}+(\cos2\pi q_i-1)\big)+H_1(q, p, t),$$ which has been considered many times in the literature. Keep this example in mind will help the readers better understand our results and methods.
Let $\fB^{\bL}_{\varepsilon,R}$$\subset\bG^{\alpha,\bL}(\sD_R)$ denote the open ball of radius $\varepsilon$ centered at the origin (under the norm $\|\cdot\|_{\alpha,\bL}$).
\[main theorem\] Given $\alpha>1, R>1, s>0 $ and let $B_s(y_1),\cdots, B_s(y_k)\subset\R^n$ be open balls with $y_\ell\in[-R+1,R-1]\times\{\bfo\}\subset\R^n$ for each $\ell=1,\cdots, k$. Assume that $H_0$ in is $C^r$ $( r= 3,4,\cdots,\infty,\omega)$ or $\bG^{\alpha}(\sD_R)$, then there exists $\bL_0=\bL_0(H_0,\alpha,R,s)$ such that:
For any $0<\bL\leq\bL_0$, there exist $\varepsilon_0=\varepsilon_0(H_0,\alpha,R,s,\bL)$ and an open and dense subset $\fS^{\bL}_{\varepsilon_0,R}\subset\fB^{\bL}_{\varepsilon_0,R}$ in the Banach space $\bG^{\alpha,\bL}(\sD_R)$ such that for each perturbation $H_1\in\fS^{\bL}_{\varepsilon_0,R}$, the system $H=H_0+H_1$ admit an orbit $(q(t),p(t))$ and times $t_1<t_2<\cdots<t_k$ such that the action variables $p(t)$ pass through the ball $B_s(y_\ell)$ at the time $t=t_\ell$.
Just as J. Mather did in [@Ma2004; @Ma2012], the smoothness of the unperturbed Hamiltonian $H_0$ could differ from that of the perturbation term $H_1$. Notice that the constant $\bL_0$ is a posteriori estimate, the reason why $\bL_0$ can not be arbitrary is because we need to use the Gevrey approximation (see Theorem \[Gevrey approx\]) during our proof.
As we know, Mather’s cohomology equivalence is trivial for an autonomous system (cf. [@Be2002]). The problem is that, unlike the time-periodic case, there is no canonical global transverse section of the flow in an autonomous system. In [@LC2010], this difficulty was overcame by taking local transverse sections, which generalized Mather’s cohomology equivalence. Thus we believe that the Gevrey genericity is still true for the *a priori* unstable autonomous Hamiltonians. However, in this paper we only consider the non-autonomous case.
Compared with the results of genericity in other papers, we can prove the genericity not only in the usual sense, but also in the sense of Mañé, which means it is a typical phenomenon when perturbed by potential functions. More specifically, let $\mathbf{B}^{\bL}_{\varepsilon}$$\subset\bG^{\alpha,\bL}(\T^n\times\T)$ denote the open ball of radius $\varepsilon$ centered at the origin (under the norm $\|\cdot\|_{\alpha,\bL}$), we have
\[main thm2\] Under the same assumptions as in Theorem \[main theorem\], there exists $\bL_0=\bL_0(H_0,\alpha,R,s)$ such that: For any $0<\bL\leq\bL_0$, there exist $\varepsilon_0=\varepsilon_0(H_0,\alpha,R,s,\bL)$ and an open and dense subset $\mathbf{S}^{\bL}_{\varepsilon_0}\subset\mathbf{B}^{\bL}_{\varepsilon_0}$ in the Banach space $\bG^{\alpha,\bL}(\T^n\times\T)$ such that for each potential perturbation $H_1\in\mathbf{S}^{\bL}_{\varepsilon_0}$, the system $H=H_0+H_1$ admit an orbit $(q(t),p(t))$ and times $t_1<t_2<\cdots<t_k$ such that the action variable $p(t)$ pass through the ball $B_s(y_\ell)$ at the time $t=t_\ell$.
Outline of this paper
---------------------
In this paper, we decide to adopt variational methods to construct diffusion orbits, so it requires us to transform into Lagrangian formalism. We still denote by $\T^n\times\R^n$ the tangent bundle $T\T^n$, and endow $\T^n\times\R^n$ with its usual coordinates $(q, v)$. The Lagrangian $L:\T^n\times\R^n\times\T \rightarrow \R$ associated with $H$ is defined as follows: $$\label{lagrangian}
L(q,v,t):=\max\limits_{p}\{\langle p,v\rangle-H(q,p,t)\}=L_0(q,v)+L_1(q,v,t),\quad L_0=l_1(v_1)+l_2(\htq,\htv).$$ where $\htv$ denotes the vector $(v_2,\cdots,v_n)$. By condition (**H**1), the Legendre transformation $$\begin{aligned}
\sL: T^*\T^n\times\T &\rightarrow T\T^n\times\T,\\
(q,p,t) &\mapsto(q,\frac{\partial H}{\partial p}(q,p,t),t)
\end{aligned}$$ is a diffeomorphism, then we also have $$L(q,v,t)=\langle \pi_p\circ\sL^{-1}(q,v,t),v\rangle-H\circ\sL^{-1}(q,v,t),$$ where $\pi_p$ denotes the projection $(q,p,t)\mapsto p$. Thus the Hamilton’s equation $\dot{q}=\frac{\partial H}{\partial p}$, $\dot{p}=-\frac{\partial H}{\partial q}$ is equivalent to the Euler-Lagrange equation $$\frac{d}{dt}(\frac{\partial L}{\partial v})-\frac{\partial L}{\partial q}=0.$$
On the other hand, by the Legendre transformation and condition (**H**2), we have $$h_2(\htq,\bfo)+l_2(\htq,\htv)\geq 0,\quad h_2(\bfo,\bfo)+l_2(\bfo,\bfo)=0,\quad (\htq,\htv)\in T\T^{n-1},$$ and because $\htq=\bfo$ (mod 1) is the unique maximum point of the function $h_2(\cdot,\bfo):\T^{n-1}\to\R$, $$\label{weiyimin}
l_2(\bfo,\bfo)=-h_2(\bfo,\bfo)\leq-h_2(\htq,\bfo)\leq l_2(\htq,\htv),\quad (\htq,\htv)\in T\T^{n-1}.$$ Thus, $(\bfo,\bfo)$ is the unique minimum point of $l_2$, which means that $(\htq,\htv)=(\bfo,\bfo)$ is a hyperbolic fixed point for the Euler-Lagrange flow $\phi^t_{l_2}$.
Compared with the proof of $C^r$-genericity of the *a priori* unstable systems in [@CY2004; @CY2009], the methods of this paper contain some new techniques. Indeed, the technique in [@CY2004; @CY2009] which perturbs the generating functions to create genericity, seems not applicable for the Gevrey genericity. The reason is, when we estimate the Gevrey smoothness of a Hamiltonian flow, we cannot avoid the decrease of Gevrey coefficient $\bL$ during the switch from a generating function to a Hamiltonian, or the switch from a Lagrangian to a Hamiltonian (see property (G\[derivative Gevrey\]) above). Thus in this paper, inspired by the ideas in [@Ch2017], we decide to directly perturb a Hamiltonian system by the Gevrey potential functions, one advantage of this is that the Lagrangian associated with a perturbed Hamiltonian $H+V(q,t)$ is exactly $L-V(q,t)$. In order to do so, we need to give some technical estimation such as the Gevrey approximations and the corresponding inverse function theorem. It is also worth mentioning that we can establish the genericity not only in the usual sense but also in the sense of Mañé. Besides, we also believe that our results could also be obtained by geometric methods, such as the geometric tools (scattering maps) developed in [@DLS2006; @DH2009; @DLS2016] , or that (separatrix maps) in [@Tr2004; @Tr2012; @DT2018] .
Recall that the regularity of barrier function is crucial in the variational proof of genericity. To achieve this, [@CY2004; @CY2009] reparameterized each invariant curve on the normally hyperbolic invariant cylinder by an “area" parameter $\sigma$. However, by using weak KAM theory of Fathi [@Fa2008], one could show that this “area" parameter $\sigma$ is exactly the cohomology class $c$ (See section \[sub holder\]), and which, to some extent, will help us simplify the proof.
Our paper is organized as follows. Section \[Mathertheory\] gives some basic notions and results in Mather theory. Section \[sec EWS\] considers the elementary weak KAM solutions, and a special “barrier function" whose minimal points correspond to the semi-static orbits connecting two different Aubry classes. In section \[sec local and global\], we introduce the concept of generalized transition chain and then give the variational mechanism of constructing diffusing orbits along this chain. In Section \[some properties Gev\], we will show some properties of Gevrey functions which are crucial for us. Section \[sec proof main\] is the main part of this paper. First, we generalize the genericity of uniquely minimal measure in the Gevrey (or analytic) topology. Second, we obtain certain regularity of the elementary weak KAM solutions and verify how to choose suitable Gevrey space. Finally, by proving the total disconnectedness, we establish the genericity of generalized transition chain, which completes the proof of Theorem \[main theorem\] and Theorem \[main thm2\].
The authors would like to thank the referees for their careful reading and useful suggestions, which help us a lot to revise this paper.
Preliminaries: Mather theory {#Mathertheory}
============================
\[sec\_Preliminaries\] In this section, we recall some useful results in Mather theory which are necessary for the purpose of this paper, the main references are Mather’s original papers [@Ma1991; @Ma1993]. Let $M$ be a connected and compact smooth manifold without boundary, equipped with a smooth Riemannian metric $g$. Let $TM$ denote the tangent bundle, a point of $TM$ will be denoted by $(q,v)$ with $q\in M$ and $v\in T_qM$. We shall denote by $\|\cdot\|_q$ the norm induced by $g$ on the fiber $T_qM$. A time-periodic $C^2$ function $L=L(q, v, t):TM\times\T\rightarrow \R$ is called a *Tonelli Lagrangian* if it satisfies:
1. *Convexity*: $L$ is strictly convex in each fiber, i.e., the second partial derivative $\partial^2 L/\partial v^2(q, v, t)$ is positive definite, as a quadratic form, for any $(q, t) \in M\times\T$;
2. *Superlinear growth*: $L$ is superlinear in each fiber, i.e., $$\lim\limits_{\|v\|_q\rightarrow +\infty}\frac{L(q,v,t)}{\|v\|_q}=+\infty,~\text{uniformly on~} (q,t)\in M\times\T;$$
3. *Completeness*: All solutions of the Euler-Lagrange equation are well defined for all $t\in\R$.
Obviously, the classical mechanical systems shall satisfy the Tonelli conditions. In our problem, the manifold is assumed to be the torus $\T^n$ and the Lagrangian $L$ in only satisfies Tonelli conditions (1) and (2). As for condition (3), one could always introduce a new convex Lagrangian $\tilde{L}$ such that $L=\tilde{L}$ on the set $\{\|v\|_q\leq K\}$, and $\tilde{L}$ is Riemannian at infinity (i.e. there exists $K>0$ such that $\tilde{L}=\frac{\|v\|_q}{2}$ for $\|v\|_q\geq K$), see for instance [@CIPP2000], so it is easy to verify that $\tilde{L}$ is a Tonelli Lagrangian. Since $\tilde{L}$ and $L$ generate the same Euler-Lagrange flow when restricted in the region $\{\|v\|_q\leq K\}$, by suitably choosing $K$ we can assume, without loss of generality, the function $L$ in is a Tonelli Lagrangian.
Let $I=[a,b]$ be an interval and $\gamma:I\rightarrow M$ be any absolutely continuous curve. Given a cohomology class $c\in H^1(M,\R)$, we choose and fix a closed 1-form $\eta_c$ with $[\eta_c]=c$. Denote by $$A_c(\gamma):=\int_a^b L(d\gamma(t),t)-\eta_c(d\gamma(t))\, dt$$ the action of $\gamma$ where $d\gamma(t)=(\gamma(t),\dot\gamma(t))$. A curve $\gamma: I\rightarrow M$ is called *$c$-minimal* if $$A_c(\gamma)=\min_{\substack{\xi(a)=\gamma(a),\xi(b)=\gamma(b)\\ \xi\in C^{ac}(I,M)}}\int_a^b L(d\xi(t),t)-\eta_c(d\xi(t))\, dt,$$ where $C^{ac}(I,M)$ denote the set of all absolutely continuous curves on $M$. It is worth mentioning that any minimal curve would satisfy the Euler-Lagrange equation. A curve $\gamma:$ $\R$ $\rightarrow$ $M$ is called *globally $c$-minimal* if for any $a<b\in\R$, the curve $\gamma:[a,b]\to M$ is a $c$-minimal. So we can introduce the *globally minimal set* $$\tilde{\cG}(c):=\bigcup\big\{ (d\gamma(t),t)~:~ \gamma:\R\rightarrow M~ \text{is ~} c\text{-minimal} \big\}.$$
Let $\phi^t_L$ be the Euler-Lagrange flow on $TM\times\T$, and $\fM_L$ be the space of all $\phi^t_L$-invariant probability measures on $TM\times\T$. To each $\mu\in\fM_L$, Mather has proved that $\int_{TM\times\T} \lambda \,d\mu$=0 holds for any exact 1-form $\lambda$, which yields that $\int_{TM\times\T} L-\eta_c \,d\mu=\int_{TM\times\T} L-\eta^\prime_c \, d\mu$ if $\eta_c-\eta_c^\prime$ is exact. This leads us to define *Mather’s $\alpha$ function*, $$\alpha(c):=-\inf\limits_{\mu\in\fM_L}\int_{TM\times\T} L-\eta_c \,d\mu.$$ To some extent, the value $\alpha(c)$ is a minimal average action for $L-\eta_c$. It’s easy to be checked that $\alpha:H^1(M,\R)\rightarrow\R$ is finite everywhere, convex and superlinear.
For each $\mu\in\fM_L$, the *rotation vector* $\rho(\mu)$ associated with $\mu$ is the unique element in $H_1(M,\R)$ that satisfies $$\langle\rho(\mu),[\eta_c]\rangle=\int_{TM\times\T}\eta_c\, d\mu,\quad \text{~ for all closed 1-form~} \eta_c,$$ here $\langle\cdot,\cdot\rangle$ denotes the dual pairing between homology and cohomology. Then we can define *Mather’s $\beta$ function* as follows: $$\beta(h):=\inf\limits_{\mu\in\fM_L, \rho(\mu)=h}\int_{TM\times\T} L\,d\mu.$$ Similarly, $\beta:H_1(M,\R)\rightarrow\R$ is also finite everywhere, convex and superlinear. In fact, the function $\beta$ is the Legendre-Fenchel dual of the function $\alpha$, i.e. $\beta(h)=\max\limits_{c}\{\langle h,c\rangle-\alpha(c)\}$.
Denote $$\fM^c:=\bigg\{ \mu: \int_{TM\times\T}L-\eta_c \,d\mu=-\alpha(c) \bigg\},~ \fM_h:=\bigg\{ \mu : \rho(\mu)=h, \int_{TM\times\T}L \,d\mu=\beta(h) \bigg\}.$$ We call each element $\mu\in\fM^c$ a *$c$-minimal measure*, and the so-called *Mather set* is defined by $$\tilde{\cM}(c):=\bigcup_{\mu\in\fM^c} \text{supp}\mu$$
To study more dynamical properties, we need to find some “larger" minimal invariant sets and discuss their topology structures. For fixed time $t^\prime> t\in\R$, we define the action function $h^{t,t^\prime}_c:M\times M\rightarrow \R$ by $$h^{t,t^\prime}_c(x,x^\prime):=\min_{\substack{\gamma(t)=x, \gamma(t^\prime)=x^\prime\\ \gamma\in C^{ac}([t,t'],M)}}\int_t^{t^\prime} (L-\eta_c)(d\gamma(s),s)\,ds+\alpha(c)\cdot(t^\prime-t).$$ Then we define a real-valued function $\Phi_c:(M\times\T)\times(M\times\T)\rightarrow \R$ by $$\Phi_c((x,\tau),(x^\prime, \tau^\prime)):=\inf_{\substack{t^\prime>t,~t\equiv\tau \text{mod}~1 \\ t^\prime\equiv \tau^\prime \text{mod}~1}}h^{t,t^\prime}_c(x,x^\prime).$$ and a real-valued function $h_c^\infty:(M\times\T)\times(M\times\T)\rightarrow \R $ by $$h_c^\infty((x,\tau),(x^\prime, \tau^\prime))=\liminf\limits_{\substack{t\equiv\tau \text{mod}~1\\ t^\prime\equiv \tau^\prime \text{mod}~1,t^\prime-t\rightarrow+\infty}}h^{t,t^\prime}_c(x,x^\prime)$$ In the literature, $h_c^\infty$ and $\Phi_c$ are called the *Peierls barrier function* and *Mañé’s potential* respectively. A minimal curve $\gamma:\R\rightarrow M$ is called $c$-semi static if $$\label{semi static}
A_c(\gamma|_{[t,t^\prime]})+\alpha(c)\cdot(t^\prime-t)=\Phi_c\big(~(\gamma(t),t\text{~mod~} 1),~(\gamma(t^\prime),t^\prime\text{~mod~} 1)~\big).$$ A minimal curve $\gamma:\R\rightarrow M$ is called $c$-static if $$\label{static}
A_c(\gamma|_{[t,t^\prime]})+\alpha(c)\cdot(t^\prime-t)=-\Phi_c\big(~(\gamma(t^\prime),t^\prime\text{~mod~} 1),~(\gamma(t),t\text{~mod~} 1)~\big).$$ This leads us to introduce the so-called *Aubry set* $\tilde{\cA}(c)$ and the *Mañé set* $\tilde{\cN}(c)$ in $TM\times\T$ as follows: $$\begin{split}
\tilde{\cA}(c):=&\bigcup \big\{(d\gamma(t),t\text{~mod~}1)~:~\gamma ~\text{is}~ c\text{-static}\big\}\\
\tilde{\cN}(c):=&\bigcup \big\{(d\gamma(t),t\text{~mod~}1)~:~\gamma ~\text{is}~ c\text{-semi static}\big\}.
\end{split}$$ The $\alpha$-limit and $\omega$-limit set of any $c$-minimal curve $(d\gamma(t),t)$ have to be contained in $\tilde{\cA}(c)$, see for instance [@Be2002]. The Mather set, the Aubry set and the Mañé set are also symplectic invariants [@Be2007symp]. In addition, with the canonical projection $\pi:TM\times\T\rightarrow M\times\T$, one could define the *projected Aubry set* $\cA(c)=\pi\tilde{\cA}(c)$, the *projected Mather set* $\cM(c)=\pi\tilde{\cM}(c)$, the *projected Mañé set* $\cN(c)=\pi\tilde{\cN}(c)$ and the *projected globally minimal set* $\cG(c)=\pi\tilde{\cG}(c)$. Then the following inclusion relations hold (see [@Be2002]): $$\tilde{\cM}(c)\subset\tilde{\cA}(c)\subset\tilde{\cN}(c)\subset\tilde{\cG}(c),\quad \cM(c)\subset\cA(c)\subset \cN(c)\subset\cG(c).$$
Next, we present some key properties of these minimal invariant sets, which will be fully exploited in the construction of diffusion orbits. Property (1) below is a classical result which was first proved by J. Mather in [@Ma1991], and the proof of property (2) could be found in [@Be2002; @CY2004].
\[upper semi\] For the Tonelli Lagrangian $L$, we have:
1. The restriction $\pi|_{\tilde{\mathcal{A}}(c)}$ is injective and its inverse $(\pi|_{\tilde{\cA}(c)})^{-1}:\cA(c)\rightarrow\tilde{\cA}(c)$ is Lipschitz continuous.
2. The set-valued map $(c,L)\rightarrow\tilde{\cG}(c,L)$ and the set-valued map $(c,L)\rightarrow\tilde{\cN}(c,L)$ are both upper semi-continuous.
We set $$d_c((x,\tau),(x^\prime, \tau^\prime)):=h_c^\infty((x,\tau),(x^\prime, \tau^\prime))+h_c^\infty((x^\prime, \tau^\prime),(x,\tau)).$$ By definition , one can easily obtain that $$h^\infty_c((x,\tau),(x, \tau))=0\Longleftrightarrow(x,\tau)\in\cA(c),$$ hence $d_c$ is a pseudo-metric on the projected Aubry set $\cA(c)$. Two points $(x,\tau), (x^\prime, \tau^\prime)\in\cA(c)$ are said to be in the same *Aubry class* if $d_c((x,\tau),(x^\prime, \tau^\prime))=0$. If the Aubry class is unique, then $\tilde{\cA}(c)=\tilde{\cN}(c)$. It is worth mentioning that, for generic Lagrangians $L$, there are only a finite number of Aubry classes [@BC2008].
To describe the Mañé set in another point of view, we define the following real-valued function $$B_c^*(x,\tau):=\min\limits_{\substack{(x_\ell,\tau_\ell)\in\cA(c)\\\ell=1,2}}\{h_c^\infty((x_1,\tau_1),(x, \tau))+h_c^\infty((x,\tau),(x_2, \tau_2))-h_c^\infty((x_1,\tau_1),(x_2, \tau_2)) \}$$ Mather has proved in [@Ma2004] that $\min B_c^*=0$ and the set of all minimal points equals $\cN(c)$, i.e. $$\label{manebarrier}
B_c^*(x,\tau)=0 \Longleftrightarrow (x,\tau)\in\cN(c).$$
To prove Theorem \[generic G1\] in Section \[sec proof main\], we need another description of the minimal measure which is due to Ma[ñ]{}[é]{} [@Mane1996]. He realized that the minimal measures can be obtained through a variational principle without *a priori* flow invariance. Let $\rm C$ be the set of all continuous functions $f:TM\times\T\to\R$ having linear growth at most, i.e. $$\|f\|_l:=\sup_{(v,t)}\frac{|f(q,v,t)|}{1+\|v\|_q}<+\infty,$$ and endow $\rm C$ with the norm $\|\cdot\|_l$. Let $\rm C^*$ be the vector space of all continuous linear functionals $\nu: \rm C\to\R$ provided with the weak-$*$ topology, namely, $$\lim\limits_{k\to+\infty}\nu_k=\nu \Longleftrightarrow \lim\limits_{k\to+\infty}\int_{TM\times\T}f\, d\nu_k=\int_{TM\times\T}f \,d\nu,\quad \forall f\in \rm C.$$
For any $N\in\Z^+$ and any $N$-periodic absolutely continuous curve $\gamma:\R\rightarrow M$, one can define a probability measure $\mu_\gamma$ as follows: $$\label{holonomic}
\int_{TM\times\T}f \,d \mu_\gamma :=\frac{1}{N}\int_0^N f(d\gamma(t),t)\,dt,\quad \forall f\in \rm C.$$ Let $$\Gamma=\bigcup_{N\in\Z^+}\big\{~\mu_\gamma~:~\gamma\in C^{ac}(\R,M) \textup{~is~} N \textup{-periodic} ~\big\}\subset \rm C^*.$$ and let $\mathcal{H}$ be the closure of $\Gamma$ in $\rm C^*$. Obviously, the set $\mathcal{H}$ is convex.
$\mu_\gamma$ defined in has a naturally associated homology class $\rho(\mu_\gamma)=\frac{1}{N}[\gamma]\in H_1(M,\R),$ where $[\gamma]$ denotes the homology class of $\gamma$, then one could continuously extend to $\mathcal{H}$ so that $\rho:\mathcal{H}\rightarrow H_1(M,\R)$ is a surjective map. Similar with Mather, Mañé defined the sets of minimal measures as follows: $$\label{definition of mane}
\begin{split}
\mathfrak{H}^c&:=\bigg\{\mu\in\mathcal{H} ~:~ \int L-\eta_c \,d\mu=\min\limits_{\nu\in\mathcal{H}}\int L-\eta_c\, d\nu \bigg\}, \\
\mathfrak{H}_h&:=\bigg\{\mu\in\mathcal{H} ~:~\rho(\mu)=h, \int L\,d\mu=\min\limits_{\nu\in\mathcal{H},\rho(\mu)=h}\int L\,d\nu \bigg\}.
\end{split}$$
The following proposition implies that Mañé’s definition of minimal measure is equivalent to Mather’s definition.
[([@Mane1996])]{}\[Mather and Mane\] The sets $\fM^c=\mathfrak{H}^c, \fM_h=\mathfrak{H}_h .$
Elementary weak KAM solutions and heteroclinic orbits {#sec EWS}
=====================================================
Weak KAM solutions {#sub weakkam}
------------------
Weak KAM solution is the basic element in weak KAM theory which builds a link between Mather theory and the theory of viscosity solutions of Hamilton-Jacobi equation. However, here we only introduce some basic concepts and properties which help us better understand Mather theory. For more details, we refer the reader to Fathi’s book [@Fa2008] for time-independent systems, and to [@Be2008; @CIS2013; @WY2012] for time-dependent systems.
\[weakKAM\]A continuous function $u_c^-:M\times\T\rightarrow\R$ is called a *backward weak KAM solution* if
1. For any absolutely continuous curve $\gamma:[a,b]\to M$, $$u_c^-(\gamma(b),b)-u_c^-(\gamma(a),a)\leq \int_{a}^{b} (L-\eta_c)(d\gamma(s),s)+\alpha(c)\,ds.$$
2. For each $(x,t)\in M\times\R$, there exists a *backward calibrated curve* $\gamma^-:(-\infty,t]\rightarrow M$ with $\gamma^-(t)=x$ such that for all $a<b\leq t$, $$u_c^-(\gamma^-(b),b)-u_c^-(\gamma^-(a),a)=\int_{a}^{b}(L-\eta_c)(d\gamma^-(s),s)+\alpha(c)\,ds.$$
Similarly, a continuous function $u_c^+:M\times\T\rightarrow\R$ is called a *forward weak KAM solution* if
1. For any absolutely continuous curve $\gamma:[a,b]\to M$, $$u_c^+(\gamma(b),b)-u_c^+(\gamma(a),a)\leq \int_{a}^{b} (L-\eta_c)(d\gamma(s),s)+\alpha(c)\,ds.$$
2. For each $(x,t)\in M\times\R$, there exists a *forward calibrated curve* $\gamma^+:[t,+\infty)\rightarrow M$ with $\gamma^+(t)=x$ such that for all $t\leq a<b$, $$u_c^+(\gamma^+(b),b)-u_c^+(\gamma^+(a),a)=\int_{a}^{b}(L-\eta_c)(d\gamma^+(s),s)+\alpha(c)\,ds.$$
For example, it is well known in weak KAM theory that for any $(x_0,t_0)\in M\times\T$, the barrier function $ h_c^\infty((x_0,t_0),\cdot): M\times\T\to\R$ is a backward weak KAM solution and $-h_c^\infty(\cdot,(x_0, t_0)):M\times\T\to\R$ is a forward weak KAM solution. If there is only one Aubry class, then function $ h_c^\infty((x_0,t_0),\cdot)$ is the unique backward weak KAM solution up to an additive constant, the function $ -h_c^\infty(\cdot,(x_0,t_0))$ is the unique forward weak KAM solution up to an additive constant.
By definition any backward (forward) calibrated curve is $c$-semi static. The following properties for weak KAM solutions are well known and the proof can be found in [@Fa2008] or [@CIS2013]:
\[properties weak KAM\] $u^-_c$ is Lipschitz continuous and differentiable on $\cA(c)$. If $u^-_c$ is differentiable at $(x_0,t_0)\in M\times\T$, then $$\partial_tu^-_c(x_0,t_0)+H(x_0,c+\partial_xu^-_c(x_0,t_0),t_0)=\alpha(c).$$ It also determines a unique $c$-semi static curve $\gamma^-_c:(-\infty,t_0]\rightarrow M$ with $\gamma^-_c(t_0)=x_0$, and such that $u_c^-$ is differentiable at each point $(\gamma^-_c(t),t)$ with $t\leq t_0$, namely $c+\partial_xu^-_c(\gamma^-_c(t),t)=\frac{\partial L}{\partial v}(d\gamma^-_c(t),t)$\
$u^+_c$ is Lipschitz continuous and differentiable on $\cA(c)$. If $u^+_c$ is differentiable at $(x_0,t_0)\in M\times\T$, then $$\partial_tu^+_c(x_0,t_0)+H(x_0,c+\partial_xu^+_c(x_0,t_0),t_0)=\alpha(c).$$ It also determines a unique $c$-semi static curve $\gamma^+_c:[t_0,+\infty)\rightarrow M$ with $\gamma^+_c(t_0)=x_0$, and such that $u_c^+$ is differentiable at each point $(\gamma^+_c(t),t)$ with $t\geq t_0$, namely $c+\partial_xu^+_c(\gamma^+_c(t),t)=\frac{\partial L}{\partial v}(d\gamma^+_c(t),t)$.
Elementary weak KAM solutions
-----------------------------
It is a generic property that a Lagrangian only admits a finite number of Aubry classes [@BC2008]. We have already know that the weak KAM solution is unique if the Aubry class is unique. If two or more Aubry classes exists, there are infinitely many weak KAM solutions, among which we are only interested in the elementary weak KAM solutions. Assume that for certain cohomology class $c$ the Aubry classes are $\{\cA_{c,i}: i=1,2,\cdots,k\}$, hence the projected Aubry set $\cA(c)=\bigcup\limits_i\cA_{c,i}$. The concept of elementary weak KAM solution first appeared in the paper [@Be2008] by Bernard. However, to be better applied to our problem, here we decide to use an analogue concept defined in [@Ch2012].
\[def of EWS\] Fix $i$ and perturb the Lagrangian $L\rightarrow L+\varepsilon V(x,t)$ where $\varepsilon>0$ and $V$ is a non-negative $C^r$ function satisfying $\textup{supp}V\cap\cA_{c,i}=\emptyset$ and $V\big|_{\cA_{c,j}}>0$ for each $j\neq i$. Then for the cohomology class $c$, the perturbed Lagrangian has only one Aubry class $\cA_{c,i}$ and its backward weak KAM solution, denoted by $u^-_{c,i,\varepsilon}$, is unique up to an additive constant. If the limit $$\label{def ofews}
u^-_{c,i}:=\lim\limits_{\varepsilon\rightarrow 0^+}u^-_{c,i,\varepsilon},$$ exists, we call $u^-_{c,i}$ a *backward elementary weak KAM solution*. Similarly one can define a *forward elementary weak KAM solution* $u^+_{c,i}$.
In the following theorem, we will prove the existence of elementary weak KAM solutions and give explicit representation formulas as well.
\[representation of EWS\] The backward (resp. forward) elementary weak KAM soltution $u^-_{c,i}$ (resp. $u^+_{c,i}$) always exists and is unique up to an additive constant. More precisely, let $(x_i, \tau_i)$ be any point in $\cA_{c,i}$, then there exists a constant $C$ (resp. $C'$) depending on $(x_i, \tau_i)$, such that $$u^-_{c,i}(x,\tau)=h^\infty_c((x_i,\tau_i),(x,\tau))+C \qquad (resp. ~ u^+_{c,i}(x,\tau)=-h^\infty_c((x,\tau),(x_i,\tau_i))+C'.)$$
We only give the proof for $u^-_{c,i}$ since $u^+_{c,i}$ is similar. Denote by $\alpha(c)$ and $\alpha_\varepsilon(c)$ the value of Mather’s $\alpha$- function at the cohomology class $c$ for the Lagrangians $L-\eta_c$ and $L-\eta_c+\varepsilon V$ respectively, and denote by $h^\infty_c((x_i,\tau_i),(x,\tau))$ and $h^\infty_{c,\varepsilon}((x_i,\tau_i),(x,\tau))$ the corresponding Peierls barrier functions. We first claim that $$\label{limit of barrierfun}
h^\infty_c((x_i,\tau_i),(x,\tau))=\lim\limits_{\varepsilon\to0}h^\infty_{c,\varepsilon}((x_i,\tau_i),(x,\tau)).$$ Indeed, as $V\geq 0$ and its support does not intersect with $\cA_{c,i}$, we have $\alpha_\varepsilon(c)=\alpha(c)$ and $$h^\infty_c((x_i,\tau_i),(x,\tau))\leq \liminf\limits_{\varepsilon\to0} h^\infty_{c,\varepsilon}((x_i,\tau_i),(x,\tau)).$$ On the other hand, for $(x,\tau)\in M\times\T$, there exist a sequence of positive numbers $\{t_n\}_{n\in\Z}$ with $t_n\equiv\tau$(mod 1) and $t^*\in[0, 1]$ with $t^*\equiv\tau_i$(mod 1) such that $t_n-t^*\to+\infty$ and $$h^\infty_c((x_i,\tau_i),(x,\tau))=\lim\limits_{n\to\infty}h^{t^*,t_n}_c(x_i,x).$$ For each $n$, we take the minimal curve $\gamma^n:[t^*, t_n]\to M$ connecting $x_i$ to $x$ and $$h^{t^*,t_n}_c(x_i,x)=\int_{t^*}^{t_n}L(d\gamma^n(t),t)-\langle \eta_c, d\gamma^n(t) \rangle+\alpha(c) \,dt.$$ Then, $$\begin{split}
\limsup\limits_{\varepsilon\to 0}h^\infty_{c,\varepsilon}((x_i,\tau_i),(x,\tau))&\leq \limsup\limits_{\varepsilon\to 0}\liminf\limits_{n\to\infty}\int_{t^*}^{t_n}(L+\varepsilon V)(d\gamma^n(t),t)-\langle \eta_c, d\gamma^n(t) \rangle+\alpha(c) \,dt\\
&=\liminf\limits_{n\to\infty}\limsup\limits_{\varepsilon\to 0}\int_{t^*}^{t_n}(L+\varepsilon V)(d\gamma^n(t),t)-\langle \eta_c, d\gamma^n(t)
\rangle+\alpha(c)\, dt\\
&=\liminf\limits_{n\to\infty}\int_{t^*}^{t_n}L(d\gamma^n(t),t)-\langle \eta_c, d\gamma^n(t) \rangle+\alpha(c) \,dt\\
&=\liminf\limits_{n\to\infty}h^{t^*,t_n}_c(x_i,x)=h^\infty_c((x_i,\tau_i),(x,\tau)).
\end{split}$$ This yields equality $\eqref{limit of barrierfun}$. As $\cA_{c,i}$ is the unique Aubry class for $L-c+\varepsilon V$ ($\varepsilon>0$) and $h^\infty_{c,\varepsilon}((x_i,\tau_i),\cdot)$ is a backward weak KAM solution, then it is well known in weak KAM theory that, $$u^-_{c,i,\varepsilon}(\cdot)-h^\infty_{c,\varepsilon}((x_i,\tau_i),\cdot)=\textup{constant},$$ which means $u^-_{c,i,\varepsilon}(\cdot)=h^\infty_{c,\varepsilon}((x_i,\tau_i),\cdot)+C$. By Definition \[def of EWS\], we finish the proof.
Fix a point $(x_i, \tau_i)\in\cA_{c,i}$ for each $i\in\{1,\cdots,k\}$, we conclude from Theorem \[representation of EWS\] that the set of all backward elementary weak KAM solutions is exactly $\big\{h^\infty_c((x_i,\tau_i),\cdot)+C~:~C\in\R, ~ i=1,\cdots,k\big\}$, and the set of all forward elementary weak KAM solutions is exactly $\big\{-h^\infty_c(\cdot,(x_i,\tau_i))+C ~: ~C\in\R, ~ i=1,\cdots,k\big\}$.
Heteroclinic orbits between Aubry classes
-----------------------------------------
To study the heteroclinic orbits from a variational viewpoint, we will use another kind of barrier function. Indeed, let $u^-_{c,i}(x,\tau)$ and $u^+_{c,j}(x,\tau)$ be some backward and forward elementary weak KAM solution respectively. Define $$\label{anotherkind barr}
B_{c,i,j}(x,\tau):=u^-_{c,i}(x,\tau)-u^+_{c,j}(x,\tau), ~\forall (x,\tau)\in M\times\T,$$ roughly speaking, it measures the action along curves which joining the Aubry class $\cA_{c,i}$ to $\cA_{c,j}$, we refer the reader to [@Be2008; @CY2009; @Ch2012] for more discussions.
Let $\arg\min f:=\{a ~|~f(a)=\min f\}$, we have
\[manejifenlei\] Suppose that the projected Aubry set $\cA(c)=\bigcup\limits_{i=1}^k\cA_{c,i}$ consists of $k$ $(k\geq 2)$ Aubry classes, then the projected Mañé set $$\cN(c)=\bigcup\limits_{i,j=1}^k \arg\min B_{c,i,j},$$
We first prove $ \cN(c)\supseteq\arg\min B_{c,i,j}$ for each $i, j$. Take two points $(x_i,\tau_i)\in\cA_{c,i}$ and $(x_j,\tau_j)\in\cA_{c,j}$, by Theorem \[representation of EWS\] there exist two constants $C_i$ and $C_j$ such that $$u^-_{c,i}(x,\tau)=h^\infty_c((x_i,\tau_i),(x,\tau))+C_i,\quad u^+_{c,j}(x,\tau)=-h^\infty_c((x,\tau),(x_j,\tau_j))+C_j.$$ So it’s easy to compute that $$\min B_{c,i,j}(x,\tau)=h^\infty_c((x_i,\tau_i),(x_j,\tau_j))+C_i-C_j.$$ If $(\tilde{x},\tilde{\tau})\in\arg\min B_{c,i,j}$, then $$h^\infty_c((x_i,\tau_i),(\tilde{x},\tilde{\tau}))+C_i+h^\infty_c((\tilde{x},\tilde{\tau}),(x_j,\tau_j))-C_j=h^\infty_c((x_i,\tau_i),(x_j,\tau_j))+C_i-C_j,$$ namely $h^\infty_c((x_i,\tau_i),(\tilde{x},\tilde{\tau}))+h^\infty_c((\tilde{x},\tilde{\tau}),(x_j,\tau_j))-h^\infty_c((x_i,\tau_i),(x_j,\tau_j))=0$. By one obtains $(\tilde{x},\tilde{\tau})\in\cN(c)$.
Next, we need to prove $\cN(c)\subset\bigcup\limits_{i,j=1}^k \arg\min B_{c,i,j}$. For each $(\bar{x},\bar{\tau})\in\cN(c)$, one deduces from that there always exist $m, n\in\{1,2,\cdots,k\}$, and two points $(x_m,\tau_m)\in\cA_{c,m}$, $(x_n,\tau_n)\in\cA_{c,n}$ such that $$h_c^\infty((x_m,\tau_m),(\bar{x},\bar{\tau}))+h_c^\infty((\bar{x},\bar{\tau}),(x_n, \tau_n))=h_c^\infty((x_m,\tau_m),(x_n, \tau_n)).$$ Combining with Theorem \[representation of EWS\], one gets that for each $(x,\tau)\in M\times\T$ $$\begin{split}
&u_{c,m}^-(\bar{x},\bar{\tau})-u_{c,n}^+(\bar{x},\bar{\tau})-\Big(u^-_{c,m}(x,\tau)-u^+_{c,n}(x,\tau)\Big)\\
=&h_c^\infty((x_m,\tau_m),(\bar{x},\bar{\tau}))+h_c^\infty((\bar{x},\bar{\tau}),(x_n, \tau_n))-\Big(h_c^\infty((x_m,\tau_m),(x,\tau))+h_c^\infty((x,\tau),(x_n, \tau_n))\Big)\\
=&h_c^\infty((x_m,\tau_m),(x_n, \tau_n))-\Big(h_c^\infty((x_m,\tau_m),(x,\tau))+h_c^\infty((x,\tau),(x_n, \tau_n))\Big)\\
\leq& 0,
\end{split}$$ hence $(\bar{x},\bar{\tau})\in \arg\min B_{c,m,n}$. This completes the proof.
From now on, we denote by $\cN_{i,j}(c)$ the set of $c$-semi static curves which are negatively asymptotic to $\cA_{c,i}$ and positively asymptotic to $\cA_{c,j}$, i.e., $$\label{ij maneorbit}
\cN_{i,j}(c):=\{(x,\tau): \exists \textup{~a~} c\textup{-semi static curve~} \gamma \textup{~with~} \gamma(\tau)=x \textup{~and~} \alpha(\gamma(t),t)\subset\cA_{c,i},\omega(\gamma(t),t)\subseteq\cA_{c,j} \}.$$ Obviously $\cN_{i,j}(c)\subset\cN(c)$, and each $(x,\tau)\in\cN_{i,j}(c)$ satisfies $$h^\infty_c((x_i,\tau_i),(x,\tau))+h^\infty_c((x,\tau),(x_j,\tau_j))=h^\infty_c((x_i,\tau_i),(x_j,\tau_j)),$$ so $\cN_{i,j}(c)\subset\arg\min B_{c,i,j}$ by Theorem \[representation of EWS\]. Further, $\cA_{c,i}\cup\cA_{c,j}\cup\cN_{i,j}(c)\subseteq\arg\min B_{c,i,j}$.
Conversely, the equality $\arg\min B_{c,i,j}\setminus\cA(c)=\cN_{i,j}(c)$ would not hold in general. For instance, the pendulum Lagrangian $L=\frac{v^2}{2}-(\cos8\pi x-1)$ has four Aubry classes for the cohomology class $c=0\in H^1(\T,\R)$: $$\tilde{\cA}_1=(0,0), \tilde{\cA}_2=(\frac{1}{4},0), \tilde{\cA}_3=(\frac{1}{2},0), \tilde{\cA}_4=(\frac{3}{4},0)$$ which are all hyperbolic fixed points. By symmetry, it’s easy to calculate that $\arg\min B_{c,1,3}=\T$ but $\cN_{1,3}(c)=\emptyset.$
However, for the case that only two Aubry classes exist, we can give a precise description.
\[double description\] Suppose that the projected Aubry set $\cA(c)=\cA_{c,1}\cup\cA_{c,2}$ has only two Aubry classes, then $$\arg\min B_{c,1,2}= \cA_{c,1}\cup\cA_{c,2}\cup\cN_{1,2}(c)\quad\textup{and}\quad \arg\min B_{c,2,1}=\cA_{c,1}\cup\cA_{c,2}\cup\cN_{2,1}(c).$$
We only prove $\arg\min B_{c,1,2}= \cA_{c,1}\cup\cA_{c,2}\cup\cN_{1,2}(c)$ and another case is similar. By the analysis above, it only remains for us to verify $\arg\min B_{c,1,2}\subset\cA_{c,1}\cup\cA_{c,2}\cup\cN_{1,2}(c)$. Indeed, for $(x,\tau)\in\arg\min B_{c,1,2}$ where one can take $\tau=0$ for simplicity, $$\label{Cor1}
B_{c,1,2}(x,0)=u^-_{c,1}(x,0)-u^+_{c,2}(x,0)=\min B_{c,1,2}$$ and by Proposition \[manejifenlei\] there exists a $c$-semi static curve $\gamma:\R\to M$ with $\gamma(0)=x$ such that $\gamma$ is calibrated by $u^-_{c,1}$ on $(-\infty, 0]$ and is calibrated by $u^+_{c,2}$ on $(0, +\infty]$.
Next, there exist two points $(\alpha,0)$, $(\omega, 0)\in\cA(c)$ and a sequence of positive integers $\{m_k\}_{k}, \{n_k\}_{k}$ $\subset \Z^+$ such that $$\lim_{k\to\infty}\gamma(-m_k)=\alpha\textup{~and~}\lim_{k\to\infty}\gamma(n_k)=\omega.$$ Recall that $\gamma$ is $c$-semi static, then $$\begin{aligned}
&u^-_{c,1}(\gamma(0),0)-u^-_{c,1}(\gamma(-m_k),0)+u^+_{c,2}(\gamma(n_k),0)-u^+_{c,2}(\gamma(0),0)\\
=&\int_{-m_k}^{n_k}L(d\gamma(t),t)-\langle \eta_c,d\gamma(t)\rangle+\alpha(c)\, dt.\end{aligned}$$ Let $\lim\inf k\to\infty$, we have $$\label{Cor2}
B_{c,1,2}(x,0)=u^-_{c,1}(\alpha,0)-u^+_{c,2}(\omega,0)+h_c^\infty\big((\alpha,0),(\omega,0)\big).$$
On the other hand, without loss of generality, we could assume $u^-_{c,1}(x,0)=h_c^\infty\big((x_1,0),(x,0)\big)$ with $(x_1,0)\in\cA_{c,1}$ and $u^+_{c,2}(x,0)=-h_c^\infty\big((x,0),(x_2,0)\big)$ with $(x_2,0)\in\cA_{c,2}$ by Theorem \[representation of EWS\]. Thus, one deduces from and that $$h_c^\infty\big((x_1,0),(x_2,0)\big)=h_c^\infty\big((x_1,0),(\alpha,0)\big)+h_c^\infty\big((\omega,0),(x_2,0)\big)+h_c^\infty\big((\alpha,0),(\omega,0)\big).$$ this could happen only if either $(\alpha,0), (\omega,0)$ belong to the same Aubry class or $(\alpha,0)\in\cA_{c,1}$, $(\omega,0)\in\cA_{c,2}$, which completes the proof.
Proposition \[double description\] will be useful in Section \[sec proof main\] where we extend the Lagrangian to a double covering space such that the lift of the Aubry set contains two Aubry classes.
Variational mechanism of diffusion orbits {#sec local and global}
=========================================
In this section, we aim to give a brief introduction to our variational construction of diffusion orbits for a Tonelli Lagrangian $L: TM\to\R$ with $M=\T^n$. The advantage lies in that it requires less information about the geometric structure. Our diffusion orbits are constructed by shadowing a sequence of local connecting orbits, along each of them the Lagrangian action attains “local minimum". Among them there are two types of local connecting orbit, one is based on Mather’s variational mechanism constructing orbits with respect to the cohomology equivalence [@Ma1993; @Ma1995], the other one is based on Arnold’s geometric mechanism [@Ar1964] whose variational version was first studied by Bessi [@Bessi1996] for Arnold’s original example, and be generalized to more general systems in [@CY2004; @CY2009; @Be2008].
Given the cohomology class $c\in H^1(M,\R) $, we define $$\mathbb{V}_{c}=\bigcap_U\{i_{U*}H_1(U,\R): U\, \text{is a neighborhood of}\ \cN_0(c) \},$$ where $i_{U*}:H_1(U,\R)\to H_1(M,\R)$ is the mapping induced by the inclusion map $i_U$: $U\to M$, $\cN_0(c)$ denotes the time-0 section of the Mañé set. Let $\mathbb{V}_{c}^{\bot}\subset H^1(M,\mathbb{R})$ denotes the annihilator of $\mathbb{V}_{c}$, i.e. $c'\in \mathbb{V}_{c}^{\bot}$ if and only if $\langle c',h \rangle =0$ for all $h\in \mathbb{V}_c$. Clearly, $$\mathbb{V}_{c}^{\bot}=\bigcup_U\{\ker i_{U}^*: U\, \text{is a neighborhood of}\, \cN_0(c)\}.$$ In fact, Mather has proved that there exists a neighborhood $U$ of $\cN_0(c)$ in $M$ such that $\mathbb{V}_{c}=i_{U^*}H_1(U,\R)$ and $\mathbb{V}^\bot_{c}=\ker i^*_U$ (see [@Ma1993]). Then we can introduce the cohomology equivalence (or $c$-equivalence) due to J. Mather.
\[def\_c\_equivalenve\] We say that $c,c'\in H^1(M,\R)$ are $c$-equivalent if there exists a continuous curve $\Gamma$: $[0,1]\to H^1(M,\R)$ such that $\Gamma(0)=c$, $\Gamma(1)=c'$ and for all $s_0\in [0,1]$, $\exists$ $\varepsilon>0$ such that $\Gamma(s)-\Gamma(s_0)\in \mathbb{V}_{{\Gamma}(s_0)}^{\bot}$ whenever $|s-s_0|<\varepsilon$ and $s\in [0,1]$.
When $c$ is equivalent to $c'$, there would be an orbit of the Euler-Lagrange flow connecting the Aubry set $\tilde{\cA}(c)$ to the Aubry set $\tilde{\cA}(c')$ [@Ma1993]. Next, let us recall Arnold’s famous example in [@Ar1964]: when the stable and unstable manifold of an invariant circle intersect transversally each other, then the unstable manifold of this circle would also intersects the stable manifold of other invariant circles nearby. To understand this mechanism in a variational viewpoint, we let $\check{\pi}:\check{M}\rightarrow \T^n$ be a finite covering of $\T^n$. Denote by $\tilde{\cN}(c,\check{M}), \tilde{\cA}(c,\check{M})$ the corresponding Mañé set and Aubry set with respect to $\check{M}$. $\tilde{\cA}(c,\check{M})$ may have several Aubry classes even if $\tilde{\cA}(c)$ is unique. Notice that it is not necessary to work always in nontrivial finite covering space, in fact, one can choose $\check{M}=M$ if the Aubry set already contains more than one classes. So for Arnold’s famous example, the intersection of the stable and unstable manifold would imply that the set $\check{\pi}\cN(c,\check{M})\big|_{t=0}\setminus\big(\cA(c)\big|_{t=0}+\delta\big)$ is discrete.
This leads us to introduce the concept of *generalized transition chain*. It could be found in [@CY2009 Definition 5.1] as a generalization of Arnold’s transition chain [@Ar1964]. In this paper, we adopt the definition as in [@Ch2017 Definition 4.1] (see also [@Ch2018 Definition 2.2]) which is more general.
\[transition chain\] Two cohomology classes $c, c'\in H^1(M,\R)$ are joined by a generalized transition chain if a continuous path $\Gamma: [0,1]\to H^1(M,\R)$ exists such that $\Gamma(0)=c, \Gamma(1)=c'$, and for each $s\in[0,1]$ at least one of the following cases takes place:
(1) There is $\delta_s>0$, for each $s'\in (s-\delta_s, s+\delta_s)\bigcap [0,1]$, $\Gamma(s')$ is $c$-equivalent to $\Gamma(s)$.
(2) There exist a finite covering $\check{\pi}:\check{M}\to M$ and a small $\delta_s>0$ such that the set $\check{\pi}\cN(\Gamma(s),\check{M})\big|_{t=0}$ $\setminus$ $\big(\cA(\Gamma(s))\big|_{t=0}+\delta_s\big)$ is non-empty and totally disconnected, and $\cA(\Gamma(s'))$ lies in a neighborhood of $\cA(\Gamma(s))$ provided $|s'-s|$ is small.
The statement “ $\cA(\Gamma(s'))$ lies in a neighborhood of $\cA(\Gamma(s))$ provided $|s'-s|$ is small " in condition (2) could be guaranteed by the upper semi-continuity of Aubry sets. Since the number of Aubry classes in our model is only finite (in fact, two at most), such upper semi-continuity is always true (cf. [@Ma2014Example]). Condition (2) appears weaker than the condition of transversal intersection of stable and unstable manifolds, because it still works when the intersection is only topologically transversal. Condition (2) is usually applied to the case where the Aubry set $\cA(\Gamma(s))$ is contained in a neighborhood of some lower dimensional torus, while condition (1) is usually applied to the case where the Mañé set $\cN(\Gamma(s))$ is homologically trivial.
Along a generalized transition chain, one could construct diffusing orbits:
\[generalized transition thm\] If $c$, $c'\in H^1(M,\R)$ are connected by a generalized transition chain $\Gamma$, then
1. there exists an orbit $d\gamma(t):\R\to TM$ of the Euler-Lagrange flow $\phi^t_L$ connecting the Aubry set $\tilde{\cA}(c)$ to $\tilde{\cA}(c')$, which means the $\alpha$-limit set $\alpha(d\gamma(t),t)\subset\tilde{\cA}(c)$ and the $\omega$-limit set $\omega(d\gamma(t),t)\subset\tilde{\cA}(c')$.
2. for any $c_1,\cdots, c_k\in \Gamma$ and small $\varepsilon>0$, there exist an orbit $d\gamma(t):\R\to TM$ of the Euler-Lagrange flow $\phi^t_L$ and times $t_1<\cdots<t_k$, such that the orbit $(d\gamma(t),t)$ passes through the $\varepsilon$-neighborhood of $\tilde{\cA}(c_\ell)$ at time $t=t_\ell$.
The proof of Theorem \[generalized transition thm\] is similar to that of [@CY2009 Section 5] and can also be found in [@Ch2012 Section 7]. This kind of variational mechanism of connecting orbits has also been used in [@Ch2017; @Ch2018]. However, for the reader’s convenience, we provide a proof of the theorem in appendix \[sec\_proof\_of\_connectingthm\]. We end this section by a simple illustration of the diffusion orbits in geometry, such orbits constructed by us in Theorem $\ref{main theorem}$ and Theorem \[main thm2\] would drift near the normally hyperbolic cylinder (see figure \[picture1\]).
![A global connecting orbit shadowing the generalized transition chain []{data-label="picture1"}](pic1.eps){width="7cm"}
Technical estimates on Gevrey functions {#some properties Gev}
=======================================
In this part, we will prove some important properties of Gevrey functions defined on the torus $\T^n=\R^n/\Z^n$, which are useful in the choice of Gevrey space (see section \[determin of coeff\]). We will present this section in a self-contained way for the reader’s convenience.
The variational proof of the genericity of Arnold diffusion depends on the existence of bump function. The reason why we have not proven the genericity in analytic topology so far is because there does not exist any analytic bump functions. However, the Gevrey bump function exists. Here we give a modified Gevrey bump function which is based on the one constructed in [@MS2004].
\[Gevrey bumpfunction\] Let $\alpha>1, \bL>0$, $D=[a_1,b_1]\times\cdots\times[a_n,b_n]$$\varsubsetneq\T^n$ be a $n$-dimensional cube and $U$ be an open neighborhood of $D$. Then there exists $f\in\bG^{\alpha,\bL}(\T^n)$ such that $0\leq f\leq 1$ and $$f(x)=1 \Longleftrightarrow x\in D, \quad \textup{supp}f\subset U.$$
We first claim that for $0<d<d'<\frac{1}{2}$, there exists a function $g\in\bG^{\alpha,\bL}(\T)$ such that $0\leq g\leq 1$ and $$g(x)=1 \Longleftrightarrow x\in [-d,d],\quad\textup{supp}g\subset [-d',d'].$$ Indeed, let $\alpha=1+\frac{1}{\sigma}$ ($\sigma>0$) and define a non-negative function $h\in C^\infty(\R)$ as follows: $h(x)=0$ for $x\leq 0$, $h(x)=$$\exp(-\frac{\lambda}{x^\sigma})$ for $x>0$. Then $h\in\bG^{\alpha,\bL}(\R)$ if the constant $\lambda>(2\bL^\alpha/\sin a)^\sigma/\sigma$ with $a=\frac{\pi}{4}\min\{1,\frac{1}{\sigma}\}$ (cf. [@MS2003 Lemma A.3]). Next, we define $\psi(x)=\int_{-\infty}^xh\big(t+\frac{d'-d}{2}\big)h\big(-t+\frac{d'-d}{2}\big)~dt.$ It’s easy to compute that $\psi\geq 0$ is non-decreasing and $$\psi(x)=\left\{
\begin{array}{ll}
0, & x\leq-\frac{d'-d}{2} \\
K, & x\geq \frac{d'-d}{2}
\end{array}
\right.$$ where $$K=\int^{\frac{d'-d}{2}}_{-\frac{d'-d}{2}}h\big(t+\frac{d'-d}{2}\big)h\big(-t+\frac{d'-d}{2}\big)~dt>0.$$ Then we define the function $$g(x)=\frac{1}{K^2}\psi\big(x+\frac{d'+d}{2}\big)\psi\bigg(-x+\frac{d'+d}{2}\big).$$ Obviously, $0\leq g\leq 1$, $\textup{supp}g\subset[-d',d']$ and $g(x)=1$ $\Longleftrightarrow$ $x\in[-d,d]$. It can be viewed as a function defined on $\T$. Hence by property (G\[algebra norm\]) in Section \[introduction\], $g\in\bG^{\alpha,\bL}(\T)$, which proves our claim.
Next, without loss of generality we assume $D=[-d_1,d_1]\times\cdots\times[-d_n,d_n]$ with $0<d_i<\frac{1}{2}$. By assumption, we can find another cube $D'=[-d'_1,d'_1]\times\cdots\times[-d'_n,d'_n]$ such that $D\subset D'\subset U\subset\T^n$. By the claim above, for each $i\in\{1,\cdots,n\}$ there exists a function $f_i\in\bG^{\alpha,\bL}(\T)$ such that $0\leq f_i\leq 1$, $\textup{supp}f_i\subset[-d_i',d_i']$, $f_i(x)=1$ $\Longleftrightarrow$ $x\in[-d_i,d_i]$. Thus we define $$f(x_1,\cdots,x_n):=\prod\limits_{i=1}^nf_i(x_i),$$ which meets our requirements.
Next, we prove that the inverse of a Gevrey map is still Gevrey smooth. For each high dimensional map $\varphi=(\varphi_1,\cdots,\varphi_n): V\to\R^n$ where $\varphi_i\in\bG^{\alpha,\bL}(V)$, its norm could be defined as follows: $$\|\varphi\|_{\alpha,\bL}:=\sum\limits_{i=1}^n\|\varphi_i\|_{\alpha,\bL}.$$ In what follows, $(0,1)^n$ denotes the unit domain $(0,1)\times\cdots\times(0,1)$ in $\R^n$. We also refer the reader to [@Kom1979] for the inverse function theorem of the general ultra-differentiable mappings.
\[inverse thm\] Let $X,Y$ be two open sets in $(0,1)^n$ and let $f: X\to Y$ be a Gevrey-$(\alpha,\bL)$ map with $\alpha\geq 1$. If the Jacobian matrix $Jf$ is non-degenerate at $x_0\in X$, then there exist an open set $U$ containing $x_0$, an open set $V$ containing $f(x_0)$, a constant $\bL_1<\bL$, and a unique inverse map $f^{-1}:V\to U$ such that $f^{-1}\in\bG^{\alpha,\bL_1}(V)$.
For simplicity we suppose the Jacobian matrix $J_{x_0}f=I_n$ where $I_n=\textup{diag}(1,1,\cdots,1)$, otherwise we can replace $f$ by $f\circ(J_{x_0}f)^{-1}$. We also suppose $f(x_0)=x_0$, otherwise we can replace $f$ by $f+x_0-f(x_0)$. If we write $f=id+h$ in a neighborhood of $x_0$, then $h(x_0)=0$, $J_{x_0}h=0$. For $0<\varepsilon\ll 1$ there exist $d>0$ and an open ball $B_{d}(x_0)=$$\{x\in X:\|x-x_0\|<d\}$ such that $$\label{derofh}
\quad \|h\|_{C^1(B_d(x_0))}\leq\varepsilon.$$ By classical Inverse Function Theorem, there exist two small open sets $U, V\subset B_{d/2}(x_0)$ containing $x_0$ and a unique $ C^\infty$ inverse map $f^{-1}:V\to U$ where $f^{-1}(x_0)=x_0$. Let $\bL_1=\varepsilon^{\frac{2}{3\alpha}}$, next we will prove $f^{-1}\in\bG^{\alpha,\bL_1}(V)$ by Contraction Mapping Principle.
we can write $f^{-1}=id+g$ , so $g\in C^\infty(V)$ and the equality $$g(y)=-h(y+g(y)),~\forall y\in V$$ holds. Define the set $E=\{\varphi=(\varphi_1,\cdots,\varphi_n): \varphi(x_0)=0,~\varphi\in\bG^{\alpha,\bL_1}(V),~\|\varphi\|_{\alpha,\bL_1}\leq \varepsilon^{\frac{3}{4}}\}$ with the norm $\|\cdot\|_{\alpha,\bL_1}$, it’s a non-empty, closed and convex set in the Banach space $\bG^{\alpha,\bL_1}(V)$. Define the operator $$(T\varphi)(y):=-h(y+\varphi(y)), \forall y\in V.$$
$\bullet$ We first claim that the mapping $T\varphi\in E,$ $\forall \varphi\in E.$ In fact, for each $\varphi\in E$, $(T\varphi)(x_0)=0$. For $y\in V\subset B_{d/2}(x_0)$, we have $\|y+\varphi(y)-x_0\|\leq\|y-x_0\|+\|\varphi(y)-\varphi(x_0)\|\leq\frac{d}{2}+\|J\varphi\|\|y-x_0\|<d$ hence $(id+\varphi)(V)$$\subset B_d(x_0)$. Moreover, let $\bL_2:=\bL\varepsilon^{\frac{1}{2\alpha}}$ and $\varepsilon$ be suitably small. For each $i\in\{1,\cdots,n\}$, $$\begin{split}
\|x_i+\varphi_i\|_{\alpha,\bL_1}-\|x_i+\varphi_i\|_{C^0}=&\sum\limits_{j=1}^n\bL_1^{\alpha}\|\delta_{ij}+\partial_{x_j}\varphi_i\|_{C^0}+\sum\limits_{k\in\N^n,|k|\geq2}\frac{\bL_1^{|k|\alpha}}{(k!)^\alpha}\|\partial^k\varphi_i\|_{C^0}\\
\leq&n\bL_1^\alpha(1+\frac{\varepsilon^\frac{3}{4}}{\bL_1^\alpha})+\|\varphi_i\|_{\alpha,\bL_1}\leq2n\varepsilon^{\frac{2}{3}}+\varepsilon^{\frac{3}{4}}\leq\frac{\bL_2^\alpha}{n^{\alpha-1}},
\end{split}$$ where $\delta_{ij}=1$ for $i=j$ and $\delta_{ij}=0$ for $i\neq j$. Hence by property (G\[composition\]) in Section \[introduction\], $\|T\varphi\|_{\alpha,\bL_1}=\|h\circ(id+\varphi)\|_{\alpha,\bL_1}\leq\|h\|_{\alpha,\bL_2,B_d(x_0)}$ since $(id+\varphi)(V)$$\subset B_d(x_0)$. Now it only remains to verify that $$\|h\|_{\alpha,\bL_2,B_d(x_0)}\leq\varepsilon^{\frac{3}{4}}.$$ Recall that for $|k|\geq 2$ and $x\in B_d(x_0)$, $\partial^k f_i(x)=\partial^k h_i(x)$. By using , we have $$\label{hi esti}
\begin{split}
\|h_i\|_{\alpha,\bL_2,B_d(x_0)}&=\|h_i\|_{C^0(B_d(x_0))}+\sum\limits_{k\in\N^n,|k|=1}\bL_2^\alpha\|\partial^kh_i\|_{C^0(B_d(x_0))}+\sum\limits_{k\in\N^n,|k|\geq2}\frac{\bL_2^{|k|\alpha}}{k!^\alpha}\|\partial^kf_i\|_{C^0(B_d(x_0))}\\
&\leq (1+n\bL_2^\alpha)\varepsilon+\sum\limits_{k\in\N^n,|k|\geq2}\frac{\bL^{|k|\alpha}\varepsilon^{\frac{|k|}{2}}}{k!^\alpha}\|\partial^kf_i\|_{C^0(B_d(x_0))}\\
&\leq (1+n\bL^\alpha\varepsilon^\frac{1}{2})\varepsilon+\varepsilon\|f\|_{\alpha,\bL}\leq\frac{\varepsilon^\frac{3}{4}}{n},
\end{split}$$ which proves the claim.
$\bullet$ On the other hand, for $\varphi,\tilde\varphi\in E$ and $i\in\{1,\cdots,n\}$, by the Newton-Leibniz formula we have $$\begin{split}
h_i(x+\varphi(x))-h_i(x+\tilde{\varphi}(x))=&\bigg(\int_0^1Jh_i\big(x+s\varphi(x)+(1-s)\tilde{\varphi}(x)\big)ds\bigg)\bigg(\varphi(x)-\tilde{\varphi}(x)\bigg)\\
=&F(x)\big(\varphi(x)-\tilde{\varphi}(x)\big)
\end{split}$$ where $Jh_i$ is the Jacobian matrix. It follows from property (G\[derivative Gevrey\]) in Section \[introduction\] and that $$\|Jh_i\|_{\alpha,\frac{\bL_2}{2},B_d(x_0)}\leq \frac{\|h_i\|_{\alpha,\bL_2,B_d(x_0)}}{(\bL_2-\bL_2/2)^\alpha}\sim O(\varepsilon^\frac{1}{4})<\frac{1}{2n}$$ provided $\varepsilon$ is suitably small. By property (G\[composition\]), $\|F\|_{{\alpha,\bL_1,V}}\leq \|Jh_i\|_{\alpha,\frac{\bL_2}{2},B_d(x_0)}\leq\frac{1}{2n}$.
Finally, we deduce from (G\[algebra norm\]) that $$\|h_i\circ(id+\varphi)-h_i\circ(id+\tilde{\varphi})\|_{\alpha,\bL_1}\leq\|F\|_{\alpha,\bL_1}\|\varphi-\tilde{\varphi}\|_{\alpha,\bL_1}\leq\frac{1}{2n}\|\varphi-\tilde{\varphi}\|_{\alpha,\bL_1}.$$ Hence $\|h\circ(id+\varphi)-h\circ(id+\tilde{\varphi})\|_{\alpha,\bL_1}\leq\frac{1}{2}\|\varphi-\tilde{\varphi}\|_{\alpha,\bL_1}$, namely $$\|T\varphi-T\tilde\varphi\|_{\alpha,\bL_1}\leq\frac{1}{2}\|\varphi-\tilde\varphi\|_{\alpha,\bL_1}.$$
In conclusion, $T: E\to E$ is a contraction mapping. By contraction mapping principle, $T$ has a unique fixed point, hence the fixed point must be $g$. Therefore, $f^{-1}=id+g\in\bG^{\alpha,\bL_1}(V)$.
Sometimes we need to approximate a continuous function by Gevrey smooth ones. Convolution provides us with a systematic technique. More specifically, for any $\alpha>1, \bL>0$, by Lemma \[Gevrey bumpfunction\] there exists a non-negative function $\eta\in\bG^{\alpha,\bL}(\R^n)$ such that $\textup{supp}\eta$ $\subset[\frac{1}{4},\frac{3}{4}]^n$ and $\int_{\R^n}\eta(x) dx=1$. Next we set $\eta_\varepsilon(x)=\frac{1}{\varepsilon^n}\eta(\frac{x}{\varepsilon})$ $(0<\varepsilon<1, x\in\R^n)$ which is called the mollifier. So we can define the convolution of $\eta_\varepsilon$ and $f\in C^0(\T^n)$ by $$\label{mollifier}
\eta_\varepsilon*f(x)=\int_{\T^n}\eta_\varepsilon(x-y)f(y)dy,~\forall~x\in\T^n.$$
1. Let $\alpha>1$, and $U\subset\T^n$, $V\subsetneq(0,1)^n$ be two open sets. If $f:U\to V$ is a continuous map, then there exists a sequence of maps $f^\varepsilon:U\to(0,1)^n$ such that $f^\varepsilon\in\bG^{\alpha,\bL_\varepsilon}(U)$. Furthermore, $\bL_\varepsilon\to 0$ and $\|f^\varepsilon-f\|_{C^0}\to 0$ as $\varepsilon$ tends to 0.
2. Let $\alpha>1$, $U, V$ be connected open sets satisfying $\bar{U},\bar{V}\varsubsetneq\T^n$ and $f: U\to V$ be a continuous map. Then there exists a sequence of maps $f^\varepsilon:U\to \T^n$ such that $f^\varepsilon\in\bG^{\alpha,\bL_\varepsilon}(U)$, $\bL_\varepsilon\to 0$ and $\|f^\varepsilon-f\|_{C^0}\to 0$ as $\varepsilon$ tends to 0. Specifically, if $f$ is a diffeomorphism and the determinant $\det(Jf)$ ($Jf$ is the Jacobian matrix) has a uniform positive distance away from zero, then the Gevrey map $f^\varepsilon:U\to V^\varepsilon$ with $V^\varepsilon=f^\varepsilon(U)$ will also be a diffeomorphism provided that $\varepsilon$ is small enough.
\[Gevrey approx\]
(1): Let $f=(f_1,\cdots,f_n)$ and $f_i$ ($1\leq i\leq n$) be continuous, we only need to prove that each $f_i$ can be approximated by the Gevrey smooth functions. Indeed, let $f_i^\varepsilon=\eta_\varepsilon*f_i$ ($0<\varepsilon<1$) as where $\eta\in\bG^{\alpha,\bL}(\R^n)$. It’s easy to check that $f_i^\varepsilon:U\to(0,1)$ since $\int_{\R^n}\eta_\varepsilon(x) dx=1$ and $\textup{supp}\eta_\varepsilon$$\subset[\frac{\varepsilon}{4},\frac{3\varepsilon}{4}]^n$. By the classical properties of convolutions, one obtains $f^\varepsilon_i\in C^\infty$ and $$\|f_i^\varepsilon-f_i\|_{C^0}\to 0, \textup{~as~} \varepsilon\to 0.$$ $$\partial^kf^\varepsilon_i=\partial^k\eta_\varepsilon*f_i=\int_{\T^n}\partial^k\eta_\varepsilon(x-y)f_i(y)dy,~\forall~k=(k_1,\cdots,k_n)\in\Z^n, k_i\geq 0.$$ It only remains to prove $f_i^\varepsilon$ is Gevrey smooth. In fact, if we set $\bL_\varepsilon=\bL\varepsilon^{\frac{1}{\alpha}}$, then $$\begin{split}
\|f^\varepsilon_i\|_{\alpha,\bL_\varepsilon}
&\leq\sum\limits_{k}\frac{\bL_\varepsilon^{|k|\alpha}}{k!^\alpha}\|\partial^k\eta_\varepsilon\|_{C^0}\|f_i\|_{C^0}\\
&\leq \frac{\|f_i\|_{C^0}}{\varepsilon^n}\sum\limits_{k}\frac{\bL_\varepsilon^{|k|\alpha}\varepsilon^{-|k|}}{k!^\alpha}\|\partial^k\eta\|_{C^0}\\
&=\frac{\|f_i\|_{C^0}}{\varepsilon^n}\sum\limits_{k}\frac{\bL^{|k|\alpha}}{k!^\alpha}\|\partial^k\eta\|_{C^0}=\frac{\|f_i\|_{C^0}}{\varepsilon^n}\|\eta\|_{\alpha,\bL}.
\end{split}$$ Obviously, $\bL_\varepsilon\to 0$ as $\varepsilon\to 0$. This completes the proof of (1).
(2): The first part is not hard to get by the techniques in (1). Furthermore, if $f$ is a diffeomorphism from $U$ to $V$, then by using $\partial^kf^\varepsilon=\eta_\varepsilon*\partial^kf$ one gets $$\label{gevapp}
\|f^\varepsilon-f\|_{C^1}\to 0,\quad \varepsilon\to 0.$$ Since $\det(Jf)$ has a uniform positive distance away from zero, it concludes from and Theorem \[inverse thm\] that $f^\varepsilon: U\to f^\varepsilon(U)$ would also be a diffeomorphism.
Proof of the main results {#sec proof main}
=========================
This section mainly focuses on the proof of Theorem \[main theorem\] and Theorem \[main thm2\]. Before that, we need to do some preparations.
Genericity of uniquely minimal measure in Gevrey or analytic topology
---------------------------------------------------------------------
Recall the definition of rotation vector in section \[sec\_Preliminaries\], fix some $h\in H_1(M,\R)$, it is well known that generically in the $C^r$ ($r\geq 2$ or $\infty$) topology, the Lagrangian has a unique minimal measure $\mu$ with the rotation vector $\rho(\mu)=h$ (see [@Mane1996]). Now we are going to prove that such kind of property still holds in the Gevrey topology. In a Gevrey space $\bG^{\alpha,\bL}(M\times\T)$ with $\alpha\geq 1, \bL>0$, we say a property is *generic in the sense of Mañé* if, for each Tonelli Lagrangian $L: TM\times\T\to\R$, there exists a residual subset $\mathcal{O}\subset\bG^{\alpha,\bL}(M\times\T)$ such that the property holds for each Lagrangian $L+\phi$, $\phi\in\mathcal{O}$.
\[generic G1\] Let $h\in H_1(M,\R)$, $\alpha\geq 1, \bL>0$ and $L:TM\times\T\to\R$ be a Tonelli Lagrangian, then there exists a residual subset $\mathcal{O}\subset\bG^{\alpha,\bL}(M\times\T)$ such that $\forall \phi\in\mathcal{O}$, the Lagrangian $L+\phi$ has a unique minimal measure with the rotation vector $h$.
Recall Mañé’s equivalent definition of minimal measure in Section \[Mathertheory\], we are going to prove the theorem in the following setting based on the idea of Mañé.
(a) Set $E:=\bG^{\alpha,\bL}(M\times\T).$ Obviously, it is a Banach space.
(b) Denote by $F\subset \mathrm{C}^*$ the vector space spanned by the set of probability measures $\mu\in\mathcal{H}$ with $\int_{M\times\T}Ld\,\mu<\infty$, the definitions of the sets $\mathcal{H}$ and $\mathrm{C}$ are in Section \[Mathertheory\]. Recall that for $\mu_k, \mu\in F$, $$\lim\limits_{k\to+\infty}\,\mu_k=\mu \Longleftrightarrow \lim\limits_{k\to+\infty}\int_{TM\times\T}f\, d\mu_k=\int_{TM\times\T}f d\mu,\quad\forall f\in \mathrm{C}.$$
(c) Let $\mathcal{L}: F\to \R$ be a linear map such that $\mathcal{L}(\mu)=\int L\, d\mu$, for every $\mu\in F$.
(d) $\varphi: E\to F^*$ is a linear map such that for each $\phi\in E$, $\varphi(\phi)\in F^*$ is defined as follows $$\langle\varphi(\phi),\mu\rangle:=\int \phi\, d\mu,~ \mu\in F.$$
(e) $K:=\{\mu\in F~|~\rho(\mu)=h\}$. It’s easy to check that $K$ is a separable metrizable convex subset.
For $\phi\in E$, we denote $$\arg\min(\phi)=\{~\mu\in K~|~\mathcal{L}(\mu)+\langle \varphi(\phi),\mu \rangle=\min\limits_{\nu\in K}(\mathcal{L}(\nu)+\langle \varphi(\phi),\nu \rangle )~\}.$$ It’s easy to verify that our setting satisfies all conditions of that in [@Mane1996 Proposition 3.1], then there exists a residual subset $\mathcal{O}\subset E$ such that $\phi\in\mathcal{O}$ implies $$\#\arg\min(\phi)=1.$$ Since $\arg\min(\phi)=\mathfrak{H}_h(L+\phi)$, see , it follows from Proposition \[Mather and Mane\] that the Lagrangian $L+\phi$ admits only one invariant minimal measure with the rotation vector $h$.
For $\alpha=1$, $\bG^{1,\bL}$ is the space of analytic functions, so the above genericity also holds in the analytic topology.
Since the intersection of countably many residual sets is still residual, we have
\[corgeneric G1\] Let $L: T\T^n\times\T$ be a Tonelli Lagrangian, $\alpha\geq 1$, $\bL>0$, then there exists a residual set $\mathcal{O}_1\subset\bG^{\alpha,\bL}(\T^n\times\T)$ such that for all rational $h=(h_1,\cdots,h_n)\in H_1(\T^n,\R)$ with $h_i\in\Q$ and all $V\in\mathcal{O}_1$, the Lagrangian $L+V: T\T^n\times\T\to\R$ admits only one invariant minimal measure with the rotation vector $h$.
Hölder regularity {#sub holder}
-----------------
In the following text, we will establish the Hölder regularity of elementary weak KAM solutions with respect to the cohomology classes. This property is crucial for our proof of Theorem \[generic G2\].
Before that, we need to do some preliminaries. Let us go back to the Hamiltonian and let $$\Sigma(0)=\{(q_1,\bfo,p_1,\bfo): q_1\in\T, |p_1|\leq R+1\}\subset\T^n\times\R^n$$ denote the standard cylinder restricted on the time-$0$ section, where $R$ is the constant fixed in Section \[introduction\]. By condition (H2), $\Sigma(0)$ is a normally hyperbolic invariant cylinder (NHIC) for the time-1 map of the Hamiltonian flow $\Phi_{H_0}^t$. Since the Hamiltonian $H_0$ is integrable when restricted in the cylinder $\Sigma(0)$, the rate $\mu$ in is 1 and $\log\mu=0$, so it follows from Theorem \[persistence\] that there exists $$\label{diyigeepsilon}
\varepsilon_1=\varepsilon_1(H_0,R)>0$$ such that if $\|H_1\|_{C^r(\sD_R)}\leq\varepsilon_1$ $(r\geq 3)$, the time-1 map $\Phi_H^1$ still admits a $C^{r-1}$ normally hyperbolic invariant cylinder $\Sigma_H(0)$, which is a small deformation of $\Sigma(0)$ and can be considered as the image of the following diffeomorphism (see figure \[crumpled cylin\]) $$\label{graph of cylinder}
\begin{split}
\psi:\Sigma(0)&\to \Sigma_H(0)\subset \T^n\times\R^n, \\
(q_1,\bfo,p_1,\bfo)&\mapsto(q_1,\hat{\bq}(q_1,p_1),p_1, \hat{\bp}(q_1,p_1)).
\end{split}$$ where the multidimensional functions $\hat{\bq}=(\bq_2,\cdots,\bq_n)$ and $\hat{\bp}=(\bp_2,\cdots,\bp_n)$. Then $\psi$ induces a 2-form $\psi^*\Omega$ on the standard cylinder $\Sigma(0)$ with $\Omega=\sum\limits_{i=1}^ndq_i\wedge dp_i$, $$\psi^*\Omega=\bigg(1+\sum_{j=2}^{n}\frac{\partial(\bq_{j},\bp_{j})}{\partial(q_1,p_1)}\bigg)dq_1\wedge dp_1.$$ Since the second de Rham cohomology group $H^2(\Sigma(0) ,\R)=\{0\}$, by using Moser’s trick on the isotopy of symplectic forms, we could find a diffeomorphism $\psi_1:\Sigma(0)\to\Sigma(0)$ such that $$\psi_1^*\psi^*\Omega=dq_1\wedge dp_1.$$
![$\Sigma_H(0)$ is a small deformation of $\Sigma(0)$ []{data-label="crumpled cylin"}](cylinder.eps "fig:"){height="2.4cm"}\
Recall that $\Sigma_H(0)$ is invariant under $\Phi_H^1$ and $(\Phi_H^1)^*\Omega=\Omega$, $$\big( (\psi\circ\psi_1)^{-1}\circ\Phi_H^1\circ (\psi\circ\psi_1) \big)^*dq_1\wedge dp_1=dq_1\wedge dp_1.$$ Combining with the fact that $(\psi\circ\psi_1)^{-1}\circ\Phi_H^1\circ (\psi\circ\psi_1)$ is a small perturbation of $\Phi_{H_0}^1$, the map $(\psi\circ\psi_1)^{-1}\circ\Phi_H^1\circ (\psi\circ\psi_1)$ is an exact twist map, hence one can apply the classical Aubry-Mather theory to describe the minimal orbits on $\Sigma(0)$: given any $\rho\in\R$, there exists an Aubry-Mather set with rotation number $\rho$ such that
1. if $\rho\in\Q$, the set consists of periodic orbits.
2. if $\rho\in\R\setminus \Q$, the set is either an invariant circle or a Denjoy set.
For simplicity, we denote by $$\Sigma_H(s)=\Phi^s_H(\Sigma_H(0),0), \quad \Sigma(s)=\Phi^s_{H_0}(\Sigma(0),0)$$ the 2-dimensional manifolds and denote by $$\tilde{\Sigma}_H=\bigcup_{s\in\T}\Sigma_H(s),\quad\tilde{\Sigma}=\bigcup_{s\in\T}\Sigma(s)$$ the 3-dimensional manifolds in $T^*\T^n\times\T$. By using the Legendre transformation $\sL$, the set $\sL\tilde{\Sigma}_H$ is $\phi_L^t$-invariant in $T\T^n\times\T$. Given a cohomology class $c=(c_1,\bfo)\in H^1(\T^n,\R)$ with $|c_1|\leq R-1$, the following lemma proves that the Aubry set $\tilde{\cA}(c)$ lies inside $\sL\tilde{\Sigma}_H$.
\[minimal set on cylinder\] Let $H$ be the Hamiltonian and $L$ be the associated Lagrangian , there exists $\varepsilon_1=\varepsilon_1(H_0,R)>0$ such that if $\|H_1\|_{C^3(\sD_R)}\leq \varepsilon_1$ , then for each $c=(c_1, 0)$ with $|c_1|\leq R-1$, the globally minimal set $\tilde{\cG}_L(c)\subset\sL\tilde{\Sigma}_H$.
We first consider the autonomous Lagrangian $l_2$. It follows from that $(\bfo,\bfo)$ is the unique minimal point of $l_2$, so the globally minimal set of the Lagrangian $l_2$ is $$\tilde{\cG}_{l_2}=(\bfo,\bfo)\times\T\subset T\T^{n-1}\times\T.$$ Thus, for all $c=(c_1,\bfo)$ with $|c_1|\leq R$, the globally minimal set of $L_0=l_1(v_1)+l_2(\htq,\htv)$ is $$\tilde{\cG}_{L_0}(c)=\{(q_1,\bfo,\nabla h_1(c_1),\bfo,t)~:~ q_1,t\in\T\} \textup{~and~} \tilde{\cG}_{L_0}(c)\subset\sL\tilde{\Sigma}.$$
Next, we take a small neighborhood $U$ of $\sL\tilde{\Sigma}$ in the space $T\T^n\times\T$ and let $\varepsilon_1=\varepsilon_1(H_0,R)$ be the constant defined in . Since $\|H_1\|_{C^3(\sD_R)}\leq\varepsilon_1$, the corresponding $\|L_1\|_{C^2(\sD_R)}$ is sufficiently small. So by the upper semi-continuity in Proposition \[upper semi\], $\tilde{\cG}_{L}(c)\subset U$ for all $c\in[-R+1,R-1]\times\{\bfo\}$, furthermore, $$\sL^{-1}\tilde{\cG}_{L}(c)\subset \sL^{-1} U.$$
On the other hand, we deduce from Theorem \[persistence\] that as long as $\varepsilon_1$ is small enough, then $$\tilde{\Sigma}_H\subset \sL^{-1} U.$$ Finally, it follows from the normal hyperbolicity that $\tilde{\Sigma}_H$ is the unique $\phi^t_L$-invariant set in the neighborhood $\sL^{-1} U$, which means $\sL^{-1}\tilde{\cG}_{L}(c)\subset\tilde{\Sigma}_H$ since $\sL^{-1}\tilde{\cG}_{L}(c)$ is a $\phi^t_L$-invariant set.
In the remainder of this paper, we will use the following notations for simplicity.
**Notation:**
1. In what follows, $M$ denotes the manifold $\T^n=\R^n/\Z^n$. We denote by $$\check{M}=\T\times2\T\times\T^{n-2}=\R/\Z\times\R/2\Z\times\R^{n-2}/\Z^{n-2},\quad \check{\pi}:\check{M}\to M$$ the double covering of $M$. We use such double covering to distinguish between $0$ and $1$ in the $q_2$-coordinate, and identify 0 with 2 in the $q_2$-coordinate.
The Hamiltonian $H: T^*M\times\T\to\R$ and the Lagrangian $L:TM\times\T\to\R$ could naturally extend to $T^*\check{M}$ and $T\check{M}$ respectively. So by abuse of notation, we continue to write $H: T^*\check{M}\times\T\to\R$ and $L: T\check{M}\times\T\to\R$ for the new Hamiltonian and Lagrangian respectively. In this setting, the lift of the normally hyperbolic invariant cylinder $\Sigma_H(0)$ would have two copies $$\check{\pi}^{-1}\Sigma_H(0)=\Sigma_{H,l}(0)\cup\Sigma_{H,u}(0),$$ where the subscripts $l, u$ are introduced to indicate “lower" and “upper" respectively.
2. For simplicity, let $\pi_q$ be the natural projection from $T\check{M}$ (resp. $TM$) to $\check{M}$ (resp. $M$) or from $T^*\check{M}$ (resp. $T^*M$) to $\check{M}$ (resp. $M$).
3. Let $\kappa>0$ be small, we denote by $\mathrm{U}_\kappa=\mathrm{U}_{\kappa,l}\cup\mathrm{U}_{\kappa,u}$ the disconnected subset of $\check{M}$ where $$\mathrm{U}_{\kappa,l}=\T\times[\kappa, 1-\kappa]\times\T^{n-2},\quad\mathrm{U}_{\kappa,u}=\T\times[1+\kappa, 2-\kappa]\times\T^{n-2}.$$ Let $\rN_{\kappa}=\check{M}\setminus\mathrm{U}_\kappa=\mathrm{N}_{\kappa,l}\cup\mathrm{N}_{\kappa,u}$ where $$\rN_{\kappa,l}=\T\times(-\kappa, \kappa)\times\T^{n-2},\quad\rN_{\kappa,u}=\T\times(1-\kappa, 1+\kappa)\times\T^{n-2}.$$ The subscripts $l, u$ are also introduced to indicate the “lower" and the “upper" respectively (See figure \[picture2\]). The number $\kappa$ should be chosen such that $$\pi_q\circ\Sigma_{H,l}(0)\subset\rN_{\kappa,l}\textup{~and~}\pi_q\circ\Sigma_{H,u}(0)\subseteq\rN_{\kappa,u}.$$
4. For $c=(c_1, \bfo)$$\in H^1(M,\R)$, if the Aubry set $\tilde{\cA}_{L}(c,M)|_{t=0}$ is an invariant circle, we denote by $$\Upsilon_c=\sL^{-1}\tilde{\cA}_{L}(c,M)|_{t=0}\subset T^*M\times\{t=0\}$$ the invariant curve in the cotangent space. This leads us to set $$\label{buianquandeshangtongdiao}
\mathbb{S}:=\{ (c_1,\bfo)\in [-R+1,R-1]\times\{\bfo\} : \Upsilon_c\subset \Sigma_H(0) \textup{~is an invariant curve}\}.$$
Now we focus on $c=(c_1, \bfo)\in\mathbb{S}$. Let $W^{s,loc}_{\Upsilon_c}=\bigcup\limits_{q\in\Upsilon_c}W^{s,loc}_q$ and $ W^{u,loc}_{\Upsilon_c}=\bigcup\limits_{q\in\Upsilon_c}W^{u,loc}_q $ be the local stable and unstable manifold of $\Upsilon_c$ respectively. By Theorem \[property of NHIM\], the leaf $W^{s,loc}_q$ ($W^{u,loc}_q$) has smooth dependence on the base point $q\in\Sigma_H(0)$. Thus, although each leaf is smooth, $W^{s,loc}_{\Upsilon_c}$ ( $W^{u,loc}_{\Upsilon_c}$) is only Lipschitz since $\Upsilon_c$ is only a Lipschitz curve in general. Besides, the local stable (unstable) manifold can be viewed as a “horizontal" graph above $\check\pi\circ\rN_\kappa$, namely $$\begin{split}
W^{s,loc}_{\Upsilon_c}&=\{ \big(q_1,\htq, \bp_1^s(q_1,\htq), \hat{\bp}^s(q_1,\htq)\big)\in T^*M\times\{t=0\}: (q_1,\htq)\in\check\pi\circ\rN_\kappa \}\\
W^{u,loc}_{\Upsilon_c}&=\{ \big(q_1,\htq, \bp_1^u(q_1,\htq), \hat{\bp}^u(q_1,\htq)\big)\in T^*M\times\{t=0\}: (q_1,\htq)\in\check\pi\circ\rN_\kappa \}
\end{split}$$ where $\bp_1^{s,u}, \hat{\bp}^{s,u}$ are Lipschitz functions defined on $\check\pi\circ\rN_\kappa$.
On the other hand, in the covering space $\check{M}$, the Aubry set $\tilde{\cA}_{L}(c,\check{M})$ is the union of two disjoint copies of $\tilde{\cA}_{L}(c,M)$ satisfying $\check{\pi}\tilde{\cA}_{L}(c,\check{M})=\tilde{\cA}_{L}(c,M)$. More precisely, $\sL^{-1}\tilde{\cA}_{L}(c,\check{M})\big|_{t=0}=\Upsilon_{c,l}\cup\Upsilon_{c,u}$, where $\Upsilon_{c,\imath}$ lies on $\Sigma_{H,\imath}(0)$ and its stable and unstable manifolds are $$\begin{split}
W^{s,loc}_{\Upsilon_{c,\imath}}&=\{ \big(q_1,\htq, \bp_1^s(q_1,\htq), \hat{\bp}^s(q_1,\htq)\big)\in T^*\check{M}\times\{t=0\}: (q_1,\htq)\in\rN_{\kappa,\imath} \}\\
W^{u,loc}_{\Upsilon_{c,\imath}}&=\{ \big(q_1,\htq, \bp_1^u(q_1,\htq), \hat{\bp}^u(q_1,\htq)\big)\in T^*\check{M}\times\{t=0\}: (q_1,\htq)\in\rN_{\kappa,\imath} \}
\end{split}$$ with $\imath=l, u$. The following lemma gives the relation between the elementary weak KAM solutions and the local stable (unstable) manifolds.
\[local manifolds representation\] There exists $\kappa>0$ such that for all $c=(c_1, \bfo)\in\mathbb{S}$, we have
1. for each backward elementary weak KAM solution $u^-_{c, \imath}(q,t)$, the function $u^-_{c, \imath}(q,0)$ is $C^{1,1}$ in the domain $\rN_{\kappa,\imath}$ and generates the local unstable manifold of $\Upsilon_{c,\imath}$, i.e. $$W^{u,loc}_{\Upsilon_{c,\imath}}=\{ \big(q, c+\partial_qu^-_{c, \imath}(q,0)\big): q\in\rN_{\kappa,\imath} \},\qquad \imath=l, u.$$
2. for each forward elementary weak KAM solution $u^+_{c, \imath}(q,t)$, the function $u^+_{c, \imath}(q,0)$ is $C^{1,1}$ in the domain $\rN_{\kappa,\imath}$ and generates the local stable manifold of $\Upsilon_{c,\imath}$, i.e. $$W^{s,loc}_{\Upsilon_{c,\imath}}=\{ \big(q, c+\partial_qu^+_{c, \imath}(q,0)\big): q\in\rN_{\kappa,\imath} \},\qquad \imath=l, u.$$
We only prove for the case $u^-_{c, l}$ since the other cases are similar.\
$\bullet$ Firstly, we claim that there exists a neighborhood $V$ of $\pi_q\circ\Upsilon_{c,l}$ in $\check{M}$ such that for each $\xi^-: (-\infty, 0]\to \check{M}$ calibrated by $u^-_{c,l}$ with $\xi^-(0)\in V$, the $\alpha$-limit set of the backward minimal configuration $\{\xi^-(-i)\}_{i\in\Z^+}$ must be contained in $\pi_q\circ\Upsilon_{c,l}$.
We prove it by contradiction, then there exist a sequence of backward calibrated curves $\xi^-_k:(-\infty, 0]\to\check{M}$ with $\xi_k^-(0)=x_k$, and a sequence $\alpha_k$ which belongs to the $\alpha$-limit set of the backward minimal configuration $\{\xi_k^-(-i)\}_{i\in\Z^+}$ satisfying $$\label{class1}
\lim\limits_{k\to\infty}x_k=x^*\in\pi_q\circ\Upsilon_{c,l}\quad\textup{and}\quad \lim\limits_{k\to\infty}\alpha_k=\alpha^*\notin\pi_q\circ\Upsilon_{c,l}.$$ This implies $\alpha^*\in\pi_q\circ\Upsilon_{c,u}$. By Theorem \[representation of EWS\], each $\xi^-_k: (-\infty, 0]\to\check{M}$ is $c$-semi static and calibrated by $h^\infty_c((x^*,0),\cdot)$, namely $$h^\infty_c\big((x^*,0),(\xi^-_k(0),0)\big)-h^\infty_c\big((x^*,0),(\xi^-_k(-t),-t)\big)=h_c^{-t,0}\big(\xi^-_k(-t),\xi^-_k(0)\big),\quad\forall t\in\Z^+.$$ Let $\liminf t\to+\infty$, we get $h^\infty_c\big((x^*,0),(x_k,0)\big)-h^\infty_c\big((x^*,0),(\alpha_k,0)\big)\geq h^\infty_c\big((\alpha_k,0),(x_k,0)\big),$ hence the equality $h^\infty_c((x^*,0),(x_k,0))-h^\infty_c((x^*,0),(\alpha_k,0))= h^\infty_c((\alpha_k,0),(x_k,0))$ holds. Let $\lim k\to\infty$, yields $$0=h^\infty_c\big((x^*,0),(x^*,0)\big)=h^\infty_c\big((x^*,0),(\alpha^*,0)\big)+ h^\infty_c\big((\alpha^*,0),(x^*,0)\big),$$ which means $(x^*,0)$ and $(\alpha^*,0)$ belong to the same Aubry class, this contradicts to .\
$\bullet$ Next, let $\kappa<\frac{1}{4}$, we claim that there exists $\kappa>0$ such that $\rN^-_{\kappa,l}\subset V$, and each $u^-_{c,l}$-calibrated curve $\gamma^-: (-\infty, 0]\to \check{M}$ with $\gamma^-(0)\in\rN_{\kappa,l}$ satisfies $\gamma^-(-i)\in\rN_{\frac 14,l}$, $\forall~i\in\N$.
Let’s prove it by contradiction, then there exist a sequence of $u^-_{c,l}$-calibrated $\gamma^-_j: (-\infty, 0]\to\check{M}$ and positive integers $T_j$ such that $\gamma_j^-(-T_j)\notin\rN_{\frac{1}{4},l},$ $\gamma_j^-(-k)\in\rN_{\frac{1}{4},l},$ $k\in\{0,1,\cdots,T_j-1\}$ and $\lim\limits_{j\to\infty}\textup{dist}(\gamma^-_j(0), \pi_q\circ\Upsilon_{c,l})=0$.
We set $\eta^-_j(t):=\gamma_j^-(t-T_j)$, then $\eta_j: (-\infty, T_j]\to \check{M}$ is still a calibrated curve and $$\label{contain1}
\eta^-_j(0)\notin\rN_{\frac{1}{4},l},~ \eta_j^-(k)\in\rN_{\frac{1}{4},l}, ~ k\in\{1,2,\cdots,T_j\}$$ and $$\label{a sequence of curves}
\lim\limits_{j\to\infty}\textup{dist}(\eta^-_j(T_j), \pi_q\circ\Upsilon_{c,l})=0.$$ Extracting a subsequence if necessary, we suppose $(\eta^-_j(t), \dot{\eta}^-_j(t))$ converges uniformly to a limit curve $(\eta^-(t), \dot{\eta}^-(t)): I\to\check{M}$ on any compact intervals of $\R$, the interval $I$ is either $(-\infty, T]$ or $\R$ ($T$ is a positive integer). Obviously, $\eta^-(t)$ is still calibrated by $u^-_{c,l}$ and $$\label{initial value of limit curve}
\eta^-(0)\notin\rN_{\frac{1}{4},l}.$$
In the case $I=(-\infty, T]$, $\eta^-(T)\in\pi_q\circ\Upsilon_{c,l}$ by , hence $\{\eta^-(i)\}_{i\in\Z}\subset\pi_q\circ\Upsilon_{c,l}$, which contradicts to . In the case $I=\R$, it follows from that the $\omega$-limit set of $\{\eta^-(i)\}_{i\in\Z}$ lies in $\pi_q\circ\Upsilon_{c,l}$. So $\{\eta^-(i)\}_{i\in\Z}\subset\pi_q\circ\Upsilon_{c,l}$ since the $\omega$- and $\alpha$-limit set belong to the same Aubry class, which contradicts to .\
$\bullet$ To sum up, for each $u^-_{c,l}$-calibrated curve $\gamma^-$ with $\gamma^-(0)\in\rN_{\kappa,l}$, $\{d\gamma^-(-i)\}_{i\in\Z^+}$ would always stay in a neighborhood of the cylinder $\sL\Sigma_{H,l}(0)$ where the unstable manifold $W^{u}_{\Upsilon_{c,l}}$ keeps horizontal, and the $\alpha$-limit set of $\{d\gamma^-(-i)\}_{i\in\Z^+}$ lies in $\Upsilon_{c,l}$. By normal hyperbolicity, $\{d\gamma^-(-i)\}_{i\in\Z^+}\subset W^{u,loc}_{\Upsilon_{c,l}}$. Hence for each $q\in \rN_{\kappa,l}$, there exists a unique $u^-_{c,l}$-calibrated curve $\gamma^-: (-\infty, 0]\to\check{M}$ with $\gamma^-(0)=q$ since $W^{u,loc}_{\Upsilon_{c,l}}$ is a Lipschitz graph over $\rN_{\kappa,l}$.
Therefore, by weak KAM theory $u^-_{c,l}$ is $C^{1,1}$ in $\rN_{\kappa,l}$, and it follows from Proposition \[properties weak KAM\] that $$c+\partial_qu^-_{c,l}(q,0)=\sL^{-1}\big(d{\gamma}^-(0),0\big)\in W^{u,loc}_{\Upsilon_{c,l}},$$ which completes the proof.
In [@CY2004], the authors introduced the “area" parameters $\sigma$ to parameterize the invariant curves lying on the NHIC so that the invariant circle $\Gamma_\sigma$ is $\frac{1}{2}$-Hölder continuous in $\sigma$, namely $$\|\Gamma_{\sigma_1}-\Gamma_{\sigma_2}\|\leq C|\sigma_1-\sigma_2|^{\frac{1}{2}}$$ However, this result can be strengthened by the techniques in weak KAM theory. Roughly speaking, the “area" parameter $\sigma$ is, to some extent, the cohomology class $c$. Similar results could also be found in [@BKZ2016]. For more details, see the following Lemma \[local11\] and Theorem \[global regularity of elementary solutions\].
Recall that the invariant curve $\Upsilon_{c,\imath}$ with $c\in\mathbb{S}$ and $\imath=l,u$ can be viewed as a Lipschitz graph over $q_1$. More precisely, by abuse of notation, we continue to write $\Upsilon_{c,\imath}$ for this Lipschitz function $$\Upsilon_{c,\imath}: \T\rightarrow\Sigma_{H,\imath}(0)$$ $$q_1\mapsto(q_1, ~\pi_{\htq}\circ\Upsilon_{c,\imath}(q_1), ~\pi_{p_1}\circ\Upsilon_{c,\imath}(q_1),~ \pi_{\htp}\circ\Upsilon_{c,\imath}(q_1))$$ with $\pi_{q_1}\circ\Upsilon_{c,\imath}(q_1)=q_1$ and $\imath=l, u$. Then, we have:
\[local11\] There exists $C>0$ such that for all $c, c'\in\mathbb{S}$,
1. $\max\limits_{q_1}\|\Upsilon_{c,l}(q_1)-\Upsilon_{c',l}(q_1)\|\leq C\|c-c'\|^{\frac{1}{2}},$
2. $\max\limits_{q_1}\|\Upsilon_{c,u}(q_1)-\Upsilon_{c',u}(q_1)\|\leq C\|c-c'\|^{\frac{1}{2}}.$
We only prove (1) and the other is similar. Proposition \[properties weak KAM\] and Proposition \[upper semi\] together yeild that any weak KAM solution $u^-_{c,l}$ is $C^{1,1}$ on $\pi_q\circ\Upsilon_{c,l}$. Define $\omega_1$ $=\big(c_1+\partial_{q_1}u^-_{c,l}(q,0)\big)dq_1$ $+\sum\limits_{i=2}^n\partial_{q_i}u^-_{c,l}(q,0)dq_i$ and $\omega_2=p_1dq_1+\sum\limits_{i=2}^np_i dq_i$, which are both 1-forms in the manifold $T^*\check{M}=\check{M}\times\R^n$. Obviously, $\omega_1|_{\Upsilon_{c,l}}=\omega_2|_{\Upsilon_{c,l}}$. Then, $$\int_{\Upsilon_{c,l}}\omega_2=\int_{\Upsilon_{c,l}}\omega_1=\int_{\Upsilon_{c,l}}c_1dq_1+du^-_{c,l}(q,0)=\int_{\Upsilon_{c,l}}c_1dq_1=c_1.$$ For $c, c'\in\mathbb{S}$, we assume $c'_1>c_1$ and $D$ is the region on $\Sigma_{H,l}(0)$ between $\Upsilon_{c,l}$ and $\Upsilon_{c',l}$ (see figure \[regionD\]). By Stoke’s theorem, $$\label{stoke formula}
\int_{D}\sum\limits_{i=1}^n dp_i\wedge dq_i=\int_{\Upsilon_{c,l}}\omega_2-\int_{\Upsilon_{c',l}}\omega_2=c_1-c'_1.$$
![The region $D$ is bounded by two invariant circles[]{data-label="regionD"}](holder.eps){width="4.4cm"}
Then and together imply $$\label{area form estimate}
\begin{split}
|c_1-c_1'|&=\big|\int_{D}\sum\limits_{i=1}^n dp_i\wedge dq_i\big|=\big|\int_{D}\bigg(1+\sum_{j=2}^{n}\frac{\partial(\bp_{j},\bq_{j})}{\partial(p_1,q_1)}\bigg)dp_1\wedge dq_1\big|\\
&\geq\frac{1}{4}\big|\int_{D}dp_1\wedge dq_1\big|=\frac{1}{4}\big|\int_{\Upsilon_{c,l}}p_1dq_1-\int_{\Upsilon_{c',l}}p_1dq_1\big|\\
&=\frac{1}{4}\big|\int_{\T}\pi_{p_1}\circ\Upsilon_{c,l}(q_1)-\pi_{p_1}\circ\Upsilon_{c',l}(q_1)~dq_1\big|
\end{split}$$ As the functions $\pi_{p_1}\circ\Upsilon_{c,l}$, $\pi_{p_1}\circ\Upsilon_{c',l}:\T\rightarrow\R$ satisfy $\pi_{p_1}\circ\Upsilon_{c',l}> \pi_{p_1}\circ\Upsilon_{c,l}$, a direct calculation shows $$\label{holder regularity of Upsilon}
\int_{\T}\pi_{p_1}\circ\Upsilon_{c',l}(q_1)-\pi_{p_1}\circ\Upsilon_{c,l}(q_1)~dq_1\geq\frac{1}{4C_L}\big(\max\limits_{q_1}|\pi_{p_1}\circ\Upsilon_{c',l}(q_1)-\pi_{p_1}\circ\Upsilon_{c,l}(q_1)|\big)^2,$$ where $C_L$ is the Lipschitz constant of $\pi_{p_1}\circ\Upsilon_{c,l}$ and $\pi_{p_1}\circ\Upsilon_{c',l}$.
On the other hand, as the function $\hat{\bp}(q_1,p_1)$ in is at least $C^1$, some constant $K>0$ exists such that $$\label{norm estimate}
\begin{split}
&\|\pi_{p}\circ\Upsilon_{c,l}(q_1)-\pi_{p}\circ\Upsilon_{c',l}(q_1)\|\\
=& |\pi_{p_1}\circ\Upsilon_{c,l}(q_1)-\pi_{p_1}\circ\Upsilon_{c',l}(q_1)|+\|\pi_{\htp}\circ\Upsilon_{c,l}(q_1)-\pi_{\htp}\circ\Upsilon_{c',l}(q_1)\|\\
=&|\pi_{p_1}\circ\Upsilon_{c,l}(q_1)-\pi_{p_1}\circ\Upsilon_{c',l}(q_1)|+\|\hat{\bp}(q_1,\pi_{p_1}\circ\Upsilon_{c,l}(q_1))-\hat{\bp}(q_1,\pi_{p_1}\circ\Upsilon_{c',l}(q_1))\|\\
\leq&(1+K)|\pi_{p_1}\circ\Upsilon_{c,l}(q_1)-\pi_{p_1}\circ\Upsilon_{c',l}(q_1)|,
\end{split}$$ Thus, combining , with , we get the estimate $$\begin{split}
\|c-c'\|&\geq|c_1-c_1'|\geq\frac{1}{16C_L}\big(\max\limits_{q_1}|\pi_{p_1}\circ\Upsilon_{c,l}(q_1)-\pi_{p_1}\circ\Upsilon_{c',l}(q_1)|\big)^2\\
&\geq\frac{1}{16C_L(1+K)^{2}}\big(\max\limits_{q_1}\|\pi_{p}\circ\Upsilon_{c,l}(q_1)-\pi_{p}\circ\Upsilon_{c',l}(q_1)\|\big)^2
\end{split}$$ Namely, $$\max\limits_{q_1}\|\pi_{p}\circ\Upsilon_{c,l}(q_1)-\pi_{p}\circ\Upsilon_{c',l}(q_1)\|\leq4\sqrt{C_L}(1+K)\|c-c'\|^\frac{1}{2}.$$
Finally, as the function $\hat{\bq}$ in is Lipschitz continuous in $q_1, p_1$, we obtain $$\max\limits_{q_1}\|\pi_{\htq}\circ\Upsilon_{c,l}(q_1)-\pi_{\htq}\circ\Upsilon_{c',l}(q_1)\|\leq \tilde{C}\max\limits_{q_1}|\pi_{p_1}\circ\Upsilon_{c,l}(q_1)-\pi_{p_1}\circ\Upsilon_{c',l}(q_1)|\leq4\sqrt{C_L}\tilde{C}(1+K)\|c-c'\|^\frac{1}{2}.$$
By setting $C=4\sqrt{C_L}(1+\tilde{C})(1+K)$, we complete the proof.
Now, we will give a result which is similar to [@CY2009 Lemma 6.4]. Recall that any elementary weak KAM solution plus any constant is still a elementary weak KAM solution, so we have
\[global regularity of elementary solutions\] Let $\kappa>0$ be as shown in Lemma \[local manifolds representation\] and fix two points $z_l\in\rN_{\kappa,l}, z_u\in\rN_{\kappa,u}$. Let $u^\pm_{c,l}(q,t)$, $u^\pm_{c,u}(q,t)$ be the elementary weak KAM solutions satisfying $u^{\pm}_{c,l}(z_l,0)=u^{\pm}_{c,l}(z_u,0)\equiv \textup{constant}$ for all $c\in\mathbb{S}$. Then there exists $C_h>0$ such that for all $c, c'\in\mathbb{S}$ $$|u^{\pm}_{c,l}(q,0)-u^\pm_{c',l}(q,0)|\leq C_h(\|c'-c\|^{\frac{1}{2}}+\|c'-c\|),~\forall q\in\check{M}\setminus \rN_{\kappa,u}$$ and $$|u^{\pm}_{c,u}(q,0)-u^\pm_{c',u}(q,0)|\leq C_h(\|c'-c\|^{\frac{1}{2}}+\|c'-c\|), ~\forall q\in\check{M}\setminus \rN_{\kappa,l}.$$
We only prove the case for $u^-_{c,l}$ and the others are similar. Normal hyperbolicity guarantees the smooth dependence of the unstable leaves $W_q^{u,loc}$ on the base points $q\in\Sigma_H(0)$, so one deduces from Lemma \[local11\] that the local unstable manifold of $\Upsilon_{c,l}$ is also $\frac{1}{2}-$Hölder continuous in $c\in\mathbb{S}$. By Lemma \[local manifolds representation\], some constant $C_1>0$ exists such that $$\| \big(c+\partial_qu_{c,l}^-(q,0)\big)-\big(c'+\partial_qu_{c',l}^-(q,0)\big) \|\leq C_1\|c-c'\|^{\frac{1}{2}}, ~\forall q\in\rN_{\kappa,l}, ~\forall c, c'\in\mathbb{S}.$$ Thus, we obtain by integral calculation that $\forall$$c, c'\in\mathbb{S}$ and $\forall q\in\rN_{\kappa, l}$ $$\begin{split}
\big| \big( u^{-}_{c,l}(q,0)- u^{-}_{c,l}(z_l,0)+\langle c, q-z_l\rangle \big)-\big( u^-_{c',l}(q,0)-u^-_{c',l}(z_l,0)+\langle c', q-z_l\rangle \big) \big|\leq C_1\|c-c'\|^{\frac{1}{2}}.
\end{split}$$ Since we have chosen $u^{-}_{c,l}(z_l,0)\equiv \textup{constant}$ for all $c\in\mathbb{S}$, we get that $\forall$$c, c'\in\mathbb{S}$ and $\forall q\in\rN_{\kappa, l}$ $$\label{local regularity of sol}
\begin{split}
\big| u^{-}_{c,l}(q,0)- u^-_{c',l}(q,0) \big|\leq C_1\|c-c'\|^{\frac{1}{2}}+\|c-c'\|.
\end{split}$$
Next, for each $q\in\check{M}\setminus \rN_{\kappa,u}$, there exists a backward calibrated curve $\gamma^-_{c,l}$ with $\gamma_{c,l}^-(0)=q$, which is asymptotic to $\pi_q\circ\Upsilon_{c,l}$ as $t\to-\infty$. Since the duration of $\gamma^-_{c,l}$ staying outside of $\rN_{\kappa,l}$ is uniformly bounded, denoted by $T_l\in\Z^+$, we have $\gamma^-_{c,l}(-k)\in \rN_{\kappa,l}$ for $k\geq T_{l}$, $k\in\Z$. Thus, $$u^-_{c,l}(\gamma^-_{c,l}(0),0)-u^-_{c,l}(\gamma^-_{c,l}(-T_l),-T_l)=\int^0_{-T_l}L(\gamma^-_{c,l}(s),\dot{\gamma}^-_{c,l}(s),s)-\langle c, \dot{\gamma}^-_{c,l}(s)\rangle+\alpha(c)\, ds,$$ $$u^-_{c',l}(\gamma^-_{c,l}(0),0)-u^-_{c',l}(\gamma^-_{c,l}(-T_l),-T_l)\leq\int^0_{-T_l}L(\gamma^-_{c,l}(s),\dot{\gamma}^-_{c,l}(s),s)-\langle c', \dot{\gamma}^-_{c,l}(s)\rangle+\alpha(c')\,ds.$$ Subtract the first formula from the second one and combine with inequality , $$\begin{split}
u^-_{c',l}(q,0)-u^-_{c,l}(q,0)\leq & u^-_{c',l}(\gamma^-_{c,l}(-T_l),-T_l)-u^-_{c,l}(\gamma^-_{c,l}(-T_l),-T_l)\\
&+\int^0_{-T_l}\langle c-c', \dot{\gamma}^-_{c,l}(s)\rangle+\alpha(c')-\alpha(c)\, ds\\
\leq & u^-_{c',l}(\gamma^-_{c,l}(-T_l),0)-u^-_{c,l}(\gamma^-_{c,l}(-T_l),0)+C_2\|c'-c\|\\
\leq &C_1\|c'-c\|^{\frac 12}+\|c'-c\|+C_2\|c'-c\|
\end{split}$$ The second inequality follows from the well known facts that $\|\dot{\gamma}^-_{c,l}\|$ is uniformly bounded and Mather’s alpha function is Lipschitzian. So we conclude that there exists $C_h>0$ such that $$u^-_{c',l}(q,0)-u^-_{c,l}(q,0)\leq C_h(\|c'-c\|^{\frac{1}{2}}+\|c'-c\|), ~\forall q\in\check{M}\setminus \rN_{\kappa,u}.$$ In a similar way, we can prove $u^-_{c,l}(q,0)-u^-_{c',l}(q,0)$$\leq C_h(\|c'-c\|^{\frac{1}{2}}+\|c'-c\|),$ which completes the proof.
Choice of the Gevrey space {#determin of coeff}
---------------------------
In what follows, we assume $\alpha>1$. The generic existence of Arnold diffusion is not always true for all Gevrey space $\bG^{\alpha,\bL}$ ($\bL>0$), but only for $\bG^{\alpha,\bL}$ with $\bL$ less than some positive constant $\bL_0$. This is caused by Gevrey approximation, we will explain it and show how to choose $\bL_0$ in the following paragraph.
First, we take the Lagrangian $L_0=l_1(v_1)+l_2(\htq,\htv)$ in . For each $c=(c_1,\bfo)$, $|c_1|\leq R-1$, $$\tilde{\cA}_{L_0}(c,M)|_{t=0}=\tilde{\cN}_{L_0}(c,M)|_{t=0}=\{(q_1, \bfo, \nabla h_1(c_1), \bfo)\in TM: q_1\in\T\}$$ which is a $\phi_{L_0}^1$-invariant circle. Next, we work in the covering space $\check{M}$ and consider $L_0: T\check{M}\to\R$. Restricted on the time section $\{t=0\}$, the lift of the Aubry set has two copies $$\begin{split}
\tilde{\cA}_{L_0,l}(c,\check{M})|_{t=0}&=\{ (q_1,\mathbf{d_0},\nabla h_1(c_1),0)\in T\check{M}: q_1\in\T \}\\
\tilde{\cA}_{L_0,u}(c,\check{M})|_{t=0}&=\{ (q_1,\mathbf{d_1},\nabla h_1(c_1),0)\in T\check{M}: q_1\in\T \}
\end{split}$$ with $\mathbf{d}_0=(0,0,\cdots,0)$, $\mathbf{d}_1=(1,0,\cdots,0)\in 2\T\times\T^{n-2}$ and they lie on the following two invariant cylinders respectively $$\begin{split}
\sL\Sigma_l(0)=&\{(q_1,\mathbf{d_0},\nabla h_1(p_1),\bfo)\in T\check{M}: q_1\in\T, |p_1|\leq R \} \\
\sL\Sigma_u(0)=&\{(q_1,\mathbf{d_1},\nabla h_1(p_1),\bfo)\in T\check{M}: q_1\in\T, |p_1|\leq R \}.
\end{split}$$ Notice that $\pi_q\circ\sL\Sigma_l(0)=\T\times\mathbf{d_0}$ and $\pi_q\circ\sL\Sigma_u(0)=\T\times\mathbf{d_1}$.
![$V_{c,l}$ (blue) in a fundamental domain of $\check{M}\times\T$ for the case $n=2$ []{data-label="tubular"}](tubular.eps){width="8cm"}
Denote by $u^\pm_{c,l,L_0}, u^\pm_{c,u,L_0}$ the elementary weak KAM solutions of $L_0$. For each $x\in \mathrm{U}_{\kappa,l}$, there exists a unique $u^-_{c,l,L_0}$-calibrated curve $\xi^-_{x,c}(t): (-\infty, 0]\to\check{M}$ such that $\xi^-_{x,c}(0)=x$ and approaches $\cA_{L_0,l}(c)$ as $t\to-\infty$. Take $T_c=T_c(\kappa,L_0)>0$ small enough, we obtain a local tubular neighborhood $$V_{c,l}=\{(\xi_{x,c}^-(t),t)\in\check{M}\times\T: x\in \mathrm{U}_{\kappa,l}, -T_c\leq t\leq 0\}$$ which is diffeomorphic to $\mathrm{U}_{\kappa,l}\times [-T_c, 0]$ (see figure \[tubular\] for an illustration), namely there is a diffeomorphism $$f: \mathrm{U}_{\kappa,l}\times [-T_c, 0]\to V_{c,l}$$ such that $f(x,t)=(\xi^-_{x,c}(t),t)$, this is guaranteed by $T_c\ll 1$. Notice that $V_{c,l}$ would vary in $c$.
Recall that $\check{M}=\T\times[0,2]\times\T^{n-2}/\sim$, where the equivalent relation $\sim$ is defined by identifying 0 with 2 in the $q_2$-coordinate. Below we will fix, once and for all, a constant $\delta\ll 1$, one deduces from Theorem \[Gevrey approx\] that there exists a Gevrey-$(\alpha, \lambda_c)$ diffeomorphism $$\Psi_{c,l}: \mathrm{U}_{\kappa,l}\times [-T_c,0]\to \text{\uj V}_{c,l}$$ such that $\|\Psi_{c,l}-f\|_{C^0(\mathrm{U}_{\kappa,l}\times [-T_c,0])}\leq \delta/2$, where $\text{\uj V}_{c,l}\varsubsetneq\T\times(0,1)\times\T^{n-2}\times\T$ and $\lambda_c=\lambda_c(\kappa,L_0,\delta)\ll 1$. It means $\Psi_{c,l}(x,\cdot)$ remains $\delta/2-$close to $\xi^-_{x,c}(\cdot)$ in the sense that $$\textup{dist}(\Psi_{c,l}(x,t), \xi^-_{x,c}(t))<\delta/2, \forall~(x,t)\in\mathrm{U}_{\kappa,l}\times[-T_c, 0].$$ Thus let the number $\varepsilon_1$ defined in Lemma \[minimal set on cylinder\] be suitably small, one could find an interval $I_c=\{(c'_1,0): c'_1\in (c_1-\tau, c_1+\tau) \}$ depending on $\kappa, L_0, \delta$, such that if $\|L_1\|_{C^2}<\varepsilon_1$, then the Lagrangian $L=L_0+L_1$ satisfies: for each $c'\in I_c$, $x\in\rU_{\kappa,l}$,
$\bullet$ the $u^-_{c',l,L}$-calibrated curve $\gamma^-_{x,c',L}(t):(-\infty,0]\to\check{M}$ with $\gamma^-_{x,c',L}(0)=x$ approaches $\cA_{L,l}(c',\check{M})$ as $t\to-\infty$.
$\bullet$ $\gamma^-_{x,c',L}(\cdot)$ is still $\delta-$close to $\Psi_{c,l}(x,\cdot)$ in the sense that $$\label{tubular approximation}
\textup{dist}(\gamma^-_{x,c',L}(t),\Psi_{c,l}(x,t) )<\delta,\quad \forall -T_c\leq t\leq 0.$$ These properties are guaranteed by the upper semi-continuity. By the finite covering theorem, there exist finitely many intervals $\{I_{c^i}\}_{i=0}^m$ such that $$\label{interval decomp}
\bigcup_{0\leq i\leq m} I_{c^i}\supset [-R+1,R-1]\times\{\bfo\},$$ the corresponding diffeomorphism $\Psi_{c^i,l}: \mathrm{U}_{\kappa,l}\times[-T_{c^i}, 0]\to \text{\uj V}_{c^i,l}$ is Gevrey-$(\alpha, \lambda_{c^i})$, and the positive number $T_{c^i}\ll 1$. By Theorem \[inverse thm\], some constant $\lambda'_{c^i}<\lambda_{c^i}$ exists such that $\Psi_{c^i,l}^{-1}$ is Gevrey-$(\alpha, \lambda'_{c^i})$ smooth.
In what follows, we set $$\bL_0=\min\{\lambda'_{c^i}: i=0,\cdots,m\},$$ hence $$\Psi_{c^i,l}^{-1}: \text{\uj V}_{c^i,l}\to\mathrm{U}_{\kappa,l}\times[-T_{c^i}, 0]$$ is Gevrey-$(\alpha, \bL)$ smooth, $\forall~\bL\leq\bL_0$.
Similarly, these procedures can be carried out for the region $\mathrm{U}_{\kappa,u}$ and one can get the corresponding Gevrey diffeomorphism $\Psi_{c,u}:\mathrm{U}_{\kappa,u}\times[0, T_{c^i}]\to\text{\uj V}_{c^i,u}$. For simplicity, we still assume the same interval decomposition $\bigcup\limits_{i=0}^m I_{c^i}$ as and $\Psi_{c^i,u}^{-1}:$$\text{\uj V}_{c^i,u}\to\mathrm{U}_{\kappa,u}\times[0, T_{c^i}]$ is Gevrey-$(\alpha, \bL)$ smooth for $\forall~\bL\leq\bL_0$, where each $\Psi_{c^i,u}$ ($i=0,\cdots,m$) possesses the similar property as .
Total disconnectedness {#total discon}
----------------------
Let $\alpha>1$, we are going to study the topology structure of the minimal points of $B_{c,l,u}=u^{-}_{c,l}-u^{+}_{c,u}$ and $B_{c,u,l}=u^{-}_{c,u}-u^{+}_{c,l}$ defined in , where $u^{\pm}_{c,\imath}$ ($\imath=l, u$) are the elementary weak KAM solutions. Inspired by the techniques in [@Ch2017 Section 4.2], we will directly perturb a Lagrangian by potential functions. Compared with the methods in [@CY2004] or [@CY2009] which perturb the generating functions, our current methods provide more information, it can show the genericity not only in the usual sense but also in the sense of Mañé.
Recall the interval decomposition $\cup_{0\leq i\leq m} I_{c^i}$ in section \[determin of coeff\], one can always suppose that the length of each interval $I_{c^i}$ is less than 1. Then Theorem \[global regularity of elementary solutions\] implies that for all $c, c'\in I_{c^i}\cap\mathbb{S}$ and $q\in\mathrm{U}_{\kappa}=\mathrm{U}_{\kappa,l}\cup\mathrm{U}_{\kappa,u},$ $$\label{simplicity of global regularity}
\begin{split}
|u^{\pm}_{c,l}(q,0)-u^\pm_{c',l}(q,0)|&\leq 2C_h\|c'-c\|^{\frac{1}{2}}, \\
|u^{\pm}_{c,u}(q,0)-u^\pm_{c',u}(q,0)|&\leq 2C_h\|c'-c\|^{\frac{1}{2}}.
\end{split}$$
Fix $\bL\leq \bL_0$ and $\varepsilon_0\ll 1$, let $$\label{space1}
\mathfrak{P}:=\{P\in\bG^{\alpha,\bL}(M\times\T): \|P\|_{\alpha,\bL}\leq\varepsilon_0,~\text{supp}P\cap\check{\pi}\circ\rN_{\kappa}=\emptyset\}$$ be a set in $\bG^{\alpha,\bL}(M\times\T)$. Obviously, $\mathfrak{P}$ is a closed and convex set, any potential perturbation $P\in\mathfrak{P}$ to the Hamiltonian $H$ would not affect the cylinder $\Sigma_H(0)$. It is worth mentioning that, by natural extension, any function in $\bG^{\alpha,\bL}(M\times\T)$ can be viewed as a function defined on $\check{M}\times\T$. As usual, let $L$ be the Lagrangian associated with $H$.
\[generic G2\] Let $\alpha>1$, $\bL\leq\bL_0$. There exists a residual set $\mathcal{W}$$\subset\mathfrak{P}$ such that for each Gevrey potential function $P\in\mathcal{W}$, the Lagrangian $L+P: T\check{M}\times\T\to\R$ satisfies: $\forall$ $c\in\mathbb{S}$, the sets $$\arg\min B_{c,l,u}\big|_{\rU_{\kappa,l}\cup\rU_{\kappa,u}},\quad\arg\min B_{c,u,l}\big|_{\rU_{\kappa,l}\cup\rU_{\kappa,u}}$$ are both totally disconnected.
We only need to prove this theorem for $\bL=\bL_0$. For the intervals $\{I_{c^0},\cdots,I_{c^m}\}$, we first consider $I_{c^0}$. Let $x$ denote $(x_1, x_2,,\cdots,x_n)$, we choose a $n$-dimensional disk $$D=\{(x_1, x_2,,\cdots,x_n, t)\in\check{M}\times\T:~t=0, |x_i-x_{i,0}|\leq d, i=1,\cdots,n\}\subset\mathrm{U}_{\kappa,l}$$ which is centered at the point $(x_{1,0}, x_{2,0},\cdots, x_{n,0})$ and $d$ is small. We also set $$D+d_1:=\{(x_1, x_2,\cdots,x_n,t)\in\check{M}\times\T:~t=0, |x_i-x_{i,0}|\leq d+d_1,i=1,\cdots,n\}\subset\rU_{\kappa,l}$$ with $0<d_1\ll 1$ (see Figure \[picture2\]).
![A fundamental domain of $\check{M}\times\{t=0\}$ for the case $n=2$[]{data-label="picture2"}](pic2.eps "fig:"){width="7.5cm"}\
Let $\mu\ll 1$, for $i=1,2,\cdots,n$, we take the sets $$\begin{aligned}
\mathfrak{V}_{i}:=\bigg\{\mu\Big(\sum\limits_{\ell=1,2}a_{i,\ell}\cos2\ell\pi(x_i-x_{i,0}) +b_{i,\ell}\sin2\ell\pi(x_i-x_{i,0})\Big)~:~a_{i,\ell}, b_{i,\ell}\in[1,2]\bigg\},\end{aligned}$$ Obviously, $\mathfrak{V}_{1},\cdots, \mathfrak{V}_{n}\subset C^\omega(M)$. In the following text, we will construct perturbations based on the potential functions in $\mathfrak{V}_{i}$.
In what follows, we will use some notations defined in section \[determin of coeff\]. Fix a sufficiently large constant $\mathfrak{L}\gg\bL_0$, by Lemma \[Gevrey bumpfunction\] one can construct a function $\rho(x,t)=\rho_1(t)\rho_2(x):\check{M}\times\T\to\R$ such that $\rho_1:\T\to\R$ and $\rho_2(x):\check{M}\to\R$ are both non-negative Gevrey-$(\alpha, \mathfrak{L})$ functions, and $$\rho_1(t)=\left\{
\begin{array}{ll}
>0, &~t\in (-T_{c^0}, 0)\\
=0, &~t\in\T\setminus(-T_{c^0}, 0)
\end{array}
\right.$$ where $ T_{c^0}\ll 1$ is chosen in section \[determin of coeff\]. $\rho_2|_D\equiv1$ and supp$\rho_2\subset D+d_1$$\subset\mathrm{U}_{\kappa,l}$. We set $$\text{\uj C}:=\{ \Psi_{c^0,l}(x,t)~|~(x,t)\in (D+d_1)\times[-T_{c^0},0] \},$$ then $\text{\uj C}\subset\text{\uj V}_{c^0,l}\subset\T\times[0, 1]\times\T^{n-2}\times\T$.
$\bullet$ With $V\in\mathfrak{V}_{i}$, which can also be viewed as a function on $\check{M}$, one can define $\tilde{V}\in C^\infty(\check{M}\times\T)$ as follows: on the “lower" domain $\T\times[0, 1]\times\T^{n-2}\times\T\subset\check{M}\times\T$, $$\tilde{V}(z)=\begin{cases}
(\rho V)\circ\Psi_{c^0,l}^{-1}(z)=\rho(x,t)V(x), & ~\Psi_{c^0,l}(x,t)=z\in\text{\uj C}\\
0, & z\in(\T\times[0, 1]\times\T)\setminus\text{\uj C}
\end{cases}$$ Then we symmetrically extend the definition to the “upper" domain $\T\times[1, 2]\times\T^{n-2}$ such that $$\tilde{V}(y,t)=\tilde{V}(y-\mathbf{e}_2,t)$$ with $\mathbf{e}_2=(0, 1,0,\cdots,0).$ Thus the support of $\tilde{V}$ satisfies $$\textup{supp}\tilde{V}\subset \text{\uj C}\cup(\text{\uj C}+\mathbf{e}_2).$$ The properties (G\[algebra norm\]), (G\[composition\]) in Section \[introduction\] and $\mathfrak{L}\gg\bL_0$ together yield,$$\label{construction of V}
\tilde{V}\in\bG^{\alpha, \bL_0}(\check{M}\times\T).$$
$\bullet$ Conversely, by the symmetry of $\tilde{V}\in\bG^{\alpha,\bL_0}(\check{M}\times\T)$, it can also be viewed as a function defined on $M\times\T$, namely $\tilde{V}\in\bG^{\alpha,\bL_0}(M\times\T)$. This observation is crucial in the following proof. From the construction above, some constant $C_1$ exists such that $$\label{integration of V}
\int_{-T_{c^0}}^{0}\tilde{V}(\Psi_{c^0,l}(x,t))\,dt=C_1V(x),\quad \text{for all~} x\in D.$$
For each $i=1,\cdots,n$, let $\Pi_i$ be the standard projections to the $i$-th coordinate of $\check{M}$. For the Lagrangian $L:T\check{M}\times\T\to\R$, we denote by $u^-_{c,l}(q,t)$, $u^+_{c,u}(q,t)$ the elementary weak KAM solutions of $L$ and denote by $u^-_{c,l,\tilde V}(q,t)$, $u^+_{c,u,\tilde V}(q,t)$ the elementary weak KAM solutions of the perturbed Lagrangian $ L(x,v,t)+\tilde V(x,t)$.
\[diameter compare\] There exists an open and dense set $\mathcal{U}_{D}\subset\mathfrak{P}$ (see ) such that for each $\tilde{V}\in\mathcal{U}_{D}$ and $c\in I_{c^0}\cap\mathbb{S}$, $$\label{banjingxiao}
\Pi_i\arg\min\big(u^-_{c,l,\tilde V}(x,0)-u^+_{c,u,\tilde V}(x,0)\big)\big|_{D}\subsetneq [x_{i,0}-d,x_{i,0}+d],\quad i=1,2,\cdots,n.$$
We begin with the perturbation $\tilde{V}$ of the form , where $V\in\bigcup_{i=1}^n\mathfrak{V}_{i}$. Note that under this potential perturbation, the cylinders $\Sigma_{H,l}(0)$ and $\Sigma_{H,u}(0)$ remain unchanged, hence the Aubry set $\tilde{\cA}_{L+\tilde{V}}(c,\check{M})$$=$$\tilde{\cA}_{L}(c,\check{M})$.
**Step 1:** For $c\in I_{c^0}\cap\mathbb{S}$ and $x\in D$, let $D$ be sufficiently small if necessary, each $u^+_{c,u}$-calibrated curve $\gamma^+_{x,c}:[0,+\infty)\to\check{M}$ with $\gamma^+_{x,c}(0)=x$ is asymptotic to $\pi_q\circ\Upsilon_{c,u}$, and $$\textup{supp}\tilde{V}~\bigcap~ \big(\bigcup\limits_{t>0}(\gamma^+_{x,c}(t),t)\big)=\emptyset,$$ hence $u^+_{c,u,\tilde V}\big|_D=u^+_{c,u}\big|_D$. But the function $u^-_{c,l,\tilde V}$ would undergo small perturbation. Indeed, for $x\in D$, we take a $u^-_{c,l,\tilde{V}}$-calibrated curve $\gamma^-_{x,c,\tilde{V}}:(-\infty,0]\to\check{M}$ with $\gamma^-_{x,c,\tilde{V}}(0)=x$, then for $ k\in\Z^+$, $$u^-_{c,l,\tilde{V}}(\gamma^-_{x,c,\tilde{V}}(0),0)-u^-_{c,l,\tilde{V}}(\gamma^-_{x,c,\tilde{V}}(-k),-k)= \int_{-k}^0 (L-\eta_c+\tilde{V})(d\gamma^-_{x,c,\tilde{V}}(t),t)+\alpha(c)\,dt,$$ For another perturbation $\tilde{V}'$, we have $$u^-_{c,l,\tilde V'}(\gamma^-_{x,c,\tilde{V}}(0),0)-u^-_{c,l,\tilde V'}(\gamma^-_{x,c,\tilde{V}}(-k),-k)\leq \int_{-k}^0(L-\eta_c+\tilde V')(d\gamma^-_{x,c,\tilde{V}}(t),t)+\alpha(c)\,dt.$$ By normal hyperbolicity, there exists a uniform upper bound $T\in\Z^+$ such that for $k\geq T$, $\gamma^-_{c,l,\tilde{V}}(-k)$ would enter into a small neighborhood of $\pi_q\circ\Upsilon_{c,l}$. Restricted on this neighborhood, $u^-_{c,l,\tilde{V}}$ is equal to $u^-_{c,l,\tilde V'}$ since $\tilde{V}, \tilde{V}'$ are both bump functions. Thus we deduce from the last two formulas that $$u^-_{c,l,\tilde V'}(x,0)-u^-_{c,l,\tilde{V}}(x,0)\leq \int_{-T}^0 (\tilde V'-\tilde{V})(\gamma^-_{x,c,\tilde{V}}(t),t)\,dt.$$ Conversely, we can similarly prove that $$u^-_{c,l,\tilde V'}(x,0)-u^-_{c,l,\tilde{V}}(x,0)\geq \int_{-T}^0 (\tilde V'-\tilde{V})(\gamma^-_{x,c,\tilde{V}'}(t),t)\,dt,$$ where $\gamma^-_{x,c,\tilde V'}$ denotes the backward $u^-_{c,l,\tilde V'}$-calibrated curve with $\gamma^-_{x,c,\tilde V'}(0)=x$. Since $x$ lies in the region where $u^-_{c,l,\tilde V}$ is differentiable, one has $\|\gamma^-_{x,c,\tilde V'}(t)-\gamma^-_{x,c,\tilde V}(t)\|\to 0$ as $\|\tilde{V}' -\tilde V\|\to 0$, which is guaranteed by the upper semi-continuity. Therefore, for $c\in I_{c^0}\cap\mathbb{S}$ and $x\in D$, $$\label{potential perturbation}
\begin{split}
u^-_{c,l,\tilde V'}(x,0)-u^-_{c,l,\tilde V}(x,0)=\sK_c(\tilde V'-\tilde{V})+\sR_c(\tilde V'-\tilde{V})
\end{split}$$ where $$\sR_c(\tilde V'-\tilde V)=o(\| V'-V\|_{C^0})$$ since $V, V'\in\bigcup_{i=1}^n\mathfrak{V}_{i}$ are finitely linear combination of the trigonometric functions, and the operator $\sK_c$ is $$\label{linear operator}
\sK_{c}\tilde V(x)=\int_{-T}^0\tilde V(\gamma_{x,c,\tilde{V}}^-(t), t)\,dt.$$
**Step 2:** For $i=1$, we prove that there exists arbitrarily small perturbation $\tilde{V}\in\mathfrak{P}$ of the form , such that $$\label{touying1}
\Pi_1\arg\min\big(u^-_{c,l,\tilde V}(x,0)-u^+_{c,u,\tilde V}(x,0)\big)\big|_{D}\subsetneq [x_{1,0}-d,x_{1,0}+d]$$
We construct a grid for the parameters $(a_{1,\ell},b_{1,\ell})$ in $\mathfrak{V}_1$ by splitting the domain $[1,2]^4$ equally into 4-dimensional cubes whose side length is $\mu^2$, namely $$\Delta a_{1,\ell}=\Delta b_{1,\ell}=\mu^{2},\quad \ell=1,2.$$ There are as many as $[\mu^{-8}]$ cubes.
Let $\textrm{Osc}_{x\in D}f$ denote the oscillation of $f$, it describes the difference between the supremum and infimum of $f$ on $D$. Let the interval $I_{c^o}$ and the constant $\delta$ in suitably small, then for all $c\in I_{c^0}\cap\mathbb{S}$ and $x\in D$, as long as $\mu$ is small enough, the backward $c$-semi static curve $\gamma^-_{x,c,\tilde{V}}(t)$ would stay in the $\delta$-neighborhood of the curve $\Psi_{c^0,l}(t)$ for $t\in[-T_{c^0},0]$, and some constant $C_2>0$ exists such that $$\begin{aligned}
\label{zhendang1}
\text{Osc}_{x\in D}(\sK_c(\tilde V-\tilde V'))>\frac{1}{2}C_1\text{Osc}_{x\in D}(V-V')>C_2\mu\Delta\end{aligned}$$ where $\Delta=\max\{|a_{1,\ell}-a'_{1,\ell}|,|b_{1,\ell}-b'_{1,\ell}|:~\ell=1,2\}$. This is guaranteed by and the fact that $V$ is a finitely linear combination of $\{\sin2\ell\pi x_1,\cos2\ell\pi x_1:\ell =1,2\}$.
Next, we split the interval $I_{c^0}$ equally into $[K_s\mu^{-6}]$ subintervals, where $K_s=L_s(\frac{24C_h}{C_2})^2$ and $L_s$ is the length of $I_{c^0}$. We pick up the subinterval that has non-empty intersection with $\mathbb{S}$, and denote all these kinds of subintervals by $\{\text{\uj J}_i\}_{i\in\mathbb{J}}$. Obviously, the cardinality of the set $\mathbb{J}$ is less than $[K_s\mu^{-6}]$.
Fix some $c^*\in\text{\uj J}_i\cap\mathbb{S}$, if for some parameter $(a^*_{1,\ell}, b^*_{1,\ell}), \ell=1,2$ and its corresponding potential perturbation $V^*\in\mathfrak{V}_1$, formula does not hold, then $$\label{zhendang2}
\textup{Osc}_{x\in D}\min\limits_{x_2,\cdots,x_n}\big( u^-_{c^*,l,\tilde V^*}(x,0)-u^+_{c^*,u,\tilde V^*}(x,0)\big)=0.$$ Next, for other $V'=\mu\Big(\sum\limits_{\ell=1,2}a'_{1,\ell}\cos2\ell\pi(x_1-x_{1,0}) +b'_{1,\ell}\sin2\ell\pi(x_1-x_{1,0})\Big)\in\mathfrak{V}_1$ and the associated perturbation $\tilde {V}'$, it follows from that for all $c\in \text{\uj J}_i\cap\mathbb{S}$ and $x\in D$, $$\begin{split}
u^-_{c,l,\tilde V'}(x,0)-u^+_{c,u,\tilde V'}(x,0)=&\big(u^-_{c,l,\tilde V'}(x,0)-u^-_{c^*,l,\tilde V'}(x,0)\big)-\big(u^+_{c,u,\tilde V'}(x,0)-u^+_{c^*,u,\tilde V'}(x,0)\big) \\
+&\big( u^-_{c^*,l, \tilde V^*}(x,0)-u^+_{c^*,u,\tilde V^*}(x,0)\big)+\big(\sK_{c^*}+\sR_{c^*}\big)(\tilde V'-\tilde V^*).
\end{split}$$ As the length of $\text{\uj J}_i$ is $\frac{L_s}{[K_s\mu^-6]}$ and $c,c^*\in\text{\uj J}_i\cap\mathbb{S}$, one deduces from that $$\begin{split}
\|\big(u^-_{c,l,\tilde V'}(x,0)-u^-_{c^*,l,\tilde V'}(x,0)\big)-\big(u^+_{c,u,\tilde V'}(x,0)-u^+_{c^*,u,\tilde V'}(x,0)\big)\|&\leq4C_h\|c-c^*\|^{\frac 12}\\
&\leq 4C_h\big(\frac{L_s}{[K_s\mu^{-6}]}\big)^{\frac{1}{2}}
\leq\frac{C_2\mu^3}{6}.
\end{split}$$ Since $\mu\ll 1$, one has $\|\tilde V'-\tilde V^*\|\ll 1$ and $$\label{frac13}
\|\sR_{c^*}(\tilde V'-\tilde V^*)\|=o(\|\tilde V'-\tilde V^*\|_{C^0})\leq\frac{1}{6}\|\sK_{c^*}(\tilde V'-\tilde V^*)\|.$$ Note that $V^*, V'\in\mathfrak{V}_{1}$ are independent of $x_2,\cdots,x_n$, hence if the parameter $(a'_{1,\ell}, b'_{1,\ell}), \ell=1, 2$ satisfies $$\label{relation of coefficients}
\max\{|a^*_{1,\ell}-a'_{1,\ell}|, |b^*_{1,\ell}-a'_{1,\ell}| : \ell=1,2\}\geq \mu^2,$$ then inequalities , and give rise to $$\textup{Osc}_{x\in D}\min\limits_{x_2,\cdots,x_n}\big(u^-_{c,l,\tilde V'}(x,0)-u^+_{c,u,\tilde V'}(x,0)\big)\geq\frac{C_2}{2}\mu^3>0.$$
Therefore, for each $c\in\text{\uj J}_i\cap\mathbb{S}$ and $V'\in\mathfrak{V}_1$ satisfying , we have $$\label{reduce}
\begin{split}
\textup{Osc}_{x\in D}\min\limits_{x_2,\cdots,x_n}\big(u^-_{c,l,\tilde V'}(x,0)-u^+_{c,u,\tilde V'}(x,0)\big)
%=\textup{Osc}_{x\in D}\min\limits_{x_2}\big( u^-_{c,l}(x,0)-u^+_{c,u}(x,0)+(\sK_c+\sR_c)\tilde V'\big)
>0.
\end{split}$$ It implies that for each $\text{\uj J}_i$, we only need to cancel out at most $2^4$ cubes from the grid $\{\Delta a_{1,\ell}, \Delta b_{1,\ell}: \ell=1, 2\}$ so that formula holds for all other cubes. Let $i$ ranges over $\mathbb{J}$, we obtain a set $\text{\uj P}_1\subseteq\{(a_{1,1}, a_{1,2}, b_{1,1}, b_{1,2}): a_{1,\ell}\in[1,2], b_{1,\ell}\in[1, 2], \ell=1, 2\}$ with Lebesgue measure $$\textup{meas}\text{\uj P}_1\geq1-2^4(\mu^2)^4|\mathbb{J}|\geq1-2^4K_s\mu^2>0,$$ such that formula holds for $\forall$ $(a'_{1,1}, a'_{1,2}, b'_{1,1}, b'_{1,2})\in\text{\uj P}_1$, $\forall$ $c\in I_{c^0}\cap\mathbb{S}$. From we know the perturbation $\tilde{V}\in \bG^{\alpha,\bL_0}(\check{M}\times\T)$ constructed by us has symmetry, so $\tilde{V}\in \bG^{\alpha,\bL_0}(M\times\T)$. As $\mu$ is arbitrarily small, we have $\tilde{V}\in\mathfrak{P}$, which completes the proof of .
**Step 3:** We have proved property has density in $\mathfrak{P}$. The openness is obvious, so there is an open and dense set $\mathcal{U}_{D,1}$ such that holds for each $\tilde{V}\in\mathcal{U}_{D,1}$.
Similarly, for $i=2,\cdots,n$, we consider the potential function $V\in\mathfrak{V}_i$ and the associated perturbation $\tilde{V}$. By repeating the same procedures as in Step 2, we obtain an open and dense set $\mathcal{U}_{D,i}\subset\mathfrak{P}$, such that for each $\tilde{V}\in\mathcal{U}_{D,i}$, $$\Pi_i\arg\min\big(u^-_{c,l,\tilde V}(x,0)-u^+_{c,u,\tilde V}(x,0)\big)\big|_{D}\subsetneq [x_{i,0}-d,x_{i,0}+d]$$ Set $\mathcal{U}_D=\bigcap_{i=1}^n\mathcal{U}_{D,i}$, it is an open and dense set in $\mathfrak{P}$, which proves Lemma \[diameter compare\].
Now we continue to prove Theorem \[generic G2\].
$\bullet$ From Lemma \[diameter compare\] we know that for each small disk $D\subseteq\rU_{\kappa,l}$, there exists an open and dense set $\mathcal{U}_D\subset\mathfrak{P}$ such that holds for each Lagrangian $L+\tilde V$ with $\tilde V\in\mathcal{U}_D$. Next, we take a countable topology basis $\bigcup\limits_jD_j$ for $\rU_{\kappa,l}$ where the diameter of $D_j$ approaches to $0$ as $j\to\infty$. Thus $\mathcal{U}_{I_{c^0}}=\bigcap_j\mathcal{U}_{D_j}$ is a residual set in $\mathfrak{P}$, and the set $\arg\min\big(u^-_{c,l,P}(x,0)-u^+_{c,u,P}(x,0)\big)\big|_{\rU_{\kappa,l}}$ is totally disconnected for each $P\in\mathcal{U}_{I_{c^0}}$ and $c\in\mathbb{S}\cap I_{c^0}$.
By repeating the procedures above for other intervals $I_{c^i}, i=1,\cdots,m$, we can also obtain the corresponding residual sets $\mathcal{U}_{I_{c^i}}, i=1,\cdots,m$. So the intersection $\mathcal{U}_l=\bigcap\limits_{i=0}^m\mathcal{U}_{I_{c^i}}$ is residual, and the set $$\arg\min\big(u^-_{c,l,P}(x,0)-u^+_{c,u,P}(x,0)\big)\big|_{\rU_{\kappa,l}}$$ is totally disconnected for each $P\in\mathcal{U}_l$ and $c\in\mathbb{S}$.
$\bullet$ Similarly, one can prove that there exists a residual set $\mathcal{U}_u\subset\mathfrak{P}$, such that the set $$\arg\min\big(u^-_{c,l,P}(x,0)-u^+_{c,u,P}(x,0)\big)\big|_{\rU_{\kappa,u}}$$ is totally disconnected for each $P\in\mathcal{U}_u$ and $c\in\mathbb{S}$.
$\bullet$ Conversely, by applying the techniques above to $u^-_{c,u,P}(x,0)-u^+_{c,l,P}(x,0)$, we can also obtain two residual sets $\mathcal{V}_l$ and $\mathcal{V}_u$ in $\mathfrak{P}$, such that the set $\arg\min\big(u^-_{c,u,P}(x,0)-u^+_{c,l,P}(x,0)\big)\big|_{\rU_{\kappa,l}}$ is totally disconnected for each $c\in\mathbb{S}$ and $P\in\mathcal{V}_l$, and the set $\arg\min\big(u^-_{c,u,P}(x,0)-u^+_{c,l,P}(x,0)\big)\big|_{\rU_{\kappa,u}}$ is totally disconnected for each $ c\in\mathbb{S}$ and $P\in\mathcal{V}_u$.
Therefore, set $\mathcal{W}=\mathcal{U}_l\cap\mathcal{U}_u\cap\mathcal{V}_l\cap\mathcal{V}_u$, which completes the proof of Theorem \[generic G2\].
Proof of Theorem \[main theorem\] and \[main thm2\]
---------------------------------------------------
Finally, we can begin to prove the main results in this paper. Let $R>1$, $\alpha>1$ and $0<\bL\leq\bL_0$.
In our problem $M=\T^n$, $s>0$, $y_\ell\in[-R+1,R-1]\times\{\bfo\}$, $\ell\in\{1,\cdots,k\}$. We set $$\varepsilon_0=\min\{1,\bL^{\alpha},\frac{\bL^{2\alpha}}{2!^\alpha},\frac{\bL^{3\alpha}}{3!^\alpha}\}\varepsilon_1$$ where $\varepsilon_1$ is chosen as in Lemma \[minimal set on cylinder\]. $\|H_1\|_{\alpha,\bL}\leq\varepsilon_0$ gives rise to $\|H_1\|_{C^3}\leq\varepsilon_1$, hence the *a priori* unstable Hamiltonian $H=H_0+H_1$ has a deformed normally hyperbolic invariant cylinder (NHIC). For each $c=(c_1,\bfo)$ with $|c_1|\leq R-1$, the globally minimal set $\tilde{\cG}(c,L)$ lies on this NHIC. Next, let $L=L_0+L_1$ be the Lagrangian associated with $H$, then $$\pi_p\circ\sL^{-1}\tilde{\cN}_{L_0}(c)=c, ~ \forall c=(c_1,\bfo), |c_1|\leq R.$$ So let $\varepsilon_0$ be suitably small if necessary, we deduce from the upper semi-continuity that $$\label{dist in thm}
\textup{dist}(\sL^{-1}\tilde{\cN}_{L}(c),\sL^{-1}\tilde{\cN}_{L_0}(c))\leq s/2.$$
**Density:** For any *a priori* unstable Hamiltonian $H=H_0+H_1$ with $\|H_1\|_{\alpha,\bL}<\varepsilon_0,$ we will prove that there exists an arbitrarily small perturbation $V\in\bG^{\alpha,\bL}(M\times\T)$ such that $\|H_1+V\|_{\alpha,\bL}<\varepsilon_0$, and $H=H_0+H_1+V$ admit an orbit $(q(t),p(t))$ of the flow $\Phi^t_H$ and times $t_1<\cdots<t_k$ such that $p(t)$ passes through the ball $B_s(y_\ell)$ at the time $t=t_\ell$. Indeed, we will establish a generalized transition chain along which one is able to construct diffusion orbits.
Let $0<d<\varepsilon_0-\|H_1\|_{\alpha,\bL}$ be arbitrarily small.
$\bullet$ Firstly, by Aubry-Mather theory we could know that for any irrational homology class $h=(h_1,\bfo)$ with $h_1\in\R\setminus\Q$, the corresponding minimal set has only one Aubry class. Next, by Theorem \[generic G1\] and Corollary \[corgeneric G1\] we could find a $\phi\in\bG^{\alpha,\bL}(M\times\T)$ with $\|\phi\|_{\alpha,\bL}<\frac{d}{2}$, such that for any rational homology class $h=(\frac{p}{q},\bfo)\in H^1(M,\R)$, the perturbed Lagrangian $L_0+L_1+\phi$ has a uniquely minimal measure, namely only one minimal periodic orbit with rotation number $h$. By the Legendre transformation, the associated Hamiltonian is exactly $H_0+H_1-\phi$. As $\|H_1-\phi\|_{\alpha,\bL}<\varepsilon_0$, the NHIC persists and the Mañé set $\tilde{\cN}(c)$ lies on the NHIC for all $c=(c_1,\bfo)\in H^1(M,\R)$, $|c_1|\leq R-1$. Thus the Aubry class is unique, which means $$\tilde{\cA}(c)=\tilde{\cN}(c),$$ $\forall c=(c_1,\bfo)$, $|c_1|\leq R-1$.
Recall that $\tilde{\cN}(c)\big|_{t=0}$ lies on the NHIC. If the set $\cN(c)\big|_{t=0}$ is homologically trivial, then the $c$-equivalence holds inside $(c_1-\delta_c,c_1+\delta_c)\times\{\bfo\}$ with some $\delta_c>0$, which satisfies condition (1) in Definition \[transition chain\]. Otherwise, $\tilde{\cN}(c)\big|_{t=0}$ must be an invariant curve since $\tilde{\cA}(c)=\tilde{\cN}(c)$ and $\tilde{\cA}(c)\big|_{t=0}$ is a Lipschitz graph. In this case we define as the set$$\mathbb{S}:=\{ (c_1,\bfo)\in [-R+1,R-1]\times\{\bfo\} : \Upsilon_c\textup{~is an invariant curve on the NHIC}\}.$$ By Theorem \[generic G2\] we could always find a potential perturbation $P\in\bG^{\alpha,\bL}(M\times\T)$ with $\|P\|_{\alpha,\bL}<\frac{d}{2}$, such that the Lagrangian $L_0+L_1+\phi+P: T\check{M}\times\T\to\R$ defined in the double covering space satisfies: for all $c\in\mathbb{S}$, $$\arg\min B_{c,l,u}\big|_{\rU_{\kappa,l}\cup\rU_{\kappa,u}},\quad\arg\min B_{c,u,l}\big|_{\rU_{\kappa,l}\cup\rU_{\kappa,u}}$$ are both totally disconnected. Then Proposition \[manejifenlei\] and \[double description\] together yield that for each $c\in\mathbb{S}$, there exists $ \delta_c>0$, $$\check{\pi}\cN(c,\check{M})\big|_{t=0}\setminus(\cA(c,\check{M})\big|_{t=0}+\delta_c)$$ is totally disconnected, which satisfies condition (2) in Definition \[transition chain\]. By the Legendre transformation, the corresponding Hamiltonian is exactly $H_0+H_1-\phi-P$ where $\|\phi+P\|_{\alpha,\bL}\leq d$ and $\|H_1-\phi-P\|_{\alpha,\bL}<\varepsilon_0$.
$\bullet$ Set $V=-\phi-P$. By the analysis above, we have constructed a generalized transition chain inside $[-R+1,R-1]\times\{\bfo\}$$\subset H^1(M,\R)$ for the Lagrangian $L_0+L_1-V$. We conclude from Theorem \[generalized transition thm\] and that the Hamiltonian $H=H_0+H_1+V$ admit an orbit $(q(t),p(t))$ and times $t_1<\cdots<t_k$ such that $p(t)$ pass through the ball $B_s(y_\ell)$ at the time $t=t_\ell$.
Finally, as a result of $\|V\|_{\alpha,\bL}<d$ and the arbitrariness of $d>0$, we complete the proof of density in $\fB^{\bL}_{\varepsilon_0,R}$.
**Openness:** Since the time for the above orbit $(q(t),p(t))$ passing through the balls $B_s(y_1)$, $\cdots$, $B_s(y_k)$ is finite, the smooth dependence of solutions of ODEs on parameters guarantees the openness in $\fB^{\bL}_{\varepsilon_0,R}$. This completes the proof of Theorem \[main theorem\].
Notice that in the proof of density above, the perturbation we constructed is Gevrey potential function. Combining with the obvious openness property, Theorem \[main thm2\] is also true.
Normally hyperbolic theory
==========================
A remarkable feature for the *a priori* unstable Hamiltonian systems is the existence of normally hyperbolic invariant cylinder (NHIC), along which the diffusion orbits would drift. So in this appendix, we will present some classic results in the normally hyperbolic theory, here we only give a less general introduction which is better applied to our problem, and refer the reader to [@DLS2000; @Fen1971; @Fen1977; @HPS1977; @Wi1994] for the proof and more detailed introductions.
\[Def NHIM\] Let $M$ be a smooth manifold and $f: M\to M$ be a $C^r (r>1)$ diffeomorphism. Let $N\subset M$ be a submanifold (probably with boundary) which is invariant under $f$. Then $N$ is called a normally hyperbolic invariant manifold (NHIM) if there is an $f$-invariant splitting for every $x\in N$ $$T_xM=T_xN\oplus E_x^s\oplus E_x^u$$ and constant $C>0$, rates $0<\lambda<1<\mu$ with $\lambda\mu<1$ such that $$\label{hyp splitting}
\begin{split}
v\in T_xN & \Longleftrightarrow |Df^k(x)v|\leq C\mu^{|k|}|v| , \quad k\in\Z,\\
v\in E^s_x & \Longleftrightarrow |Df^k(x)v|\leq C\lambda^k|v|, \quad k\geq0,\\
v\in E^u_x & \Longleftrightarrow |Df^k(x)v|\leq C\lambda^{|k|}|v|, \quad k\leq0.\\
\end{split}$$
It is possible to choose a Riemann metric on $M$ such that the constant $C=1$, possibly need to modify the rates $\lambda, \mu.$
For any sufficiently small $\delta>0$, we could take an neighborhood $U$ of $N$, and define the local stable and unstable sets of $N$ $$\label{definition of local stable}
\begin{split}
W_N^{s,loc} &=\{y\in U ~|~ \textup{dist}(f^k(y),N)\leq C_\delta(\lambda+\delta)^k,~k\geq0\}, \\
W_N^{u,loc} &=\{y\in U ~|~ \textup{dist}(f^k(y),N)\leq C_\delta(\lambda+\delta)^{|k|},~k\leq0\}.
\end{split}$$ For each $x\in N$, the stable and unstable leaves are defined as follows: $$\label{definition of local stable leaf}
\begin{split}
W_x^{s,loc} &=\{y\in U ~|~ \textup{dist}(f^k(x),f^k(y))\leq C_\delta(\lambda+\delta)^k,~k\geq0\}, \\
W_x^{u,loc} &=\{y\in U ~|~ \textup{dist}(f^k(x),f^k(y))\leq C_\delta(\lambda+\delta)^{|k|},~k\leq0\}.
\end{split}$$ Then we have the following properties:
\[property of NHIM\] Let $N$ be a compact NHIM shown as in and $\delta>0$ be a sufficiently small number. Suppose $1<l=\min(r, \frac{|\log\lambda|}{\log\mu}-\delta)$ , then
1. $N$, $W_N^{s,loc}$ and $W_N^{u,loc}$ are $C^{l}$ manifolds. For each $x\in N$, the manifolds $W_x^{s,loc}$ and $W_x^{u,loc}$ are $C^{r}$ and $T_xW_x^{s,loc}=E_x^s$, $T_xW_x^{u,loc}=E_x^u$.
2. $W_N^{s,loc}, W_N^{u,loc}$ are foliated by the stable and unstable leaves, i.e. $$W_N^{s,loc}=\bigcup\limits_{x\in N} W_x^{s,loc}, W_N^{u,loc}=\bigcup\limits_{x\in N} W_x^{u,loc}.$$ Moreover, $x\neq x'\Longrightarrow W_x^{s,loc}\bigcap W_{x'}^{s,loc}=\emptyset, W_x^{u,loc}\bigcap W_{x'}^{u,loc}=\emptyset.$
3. The map $x\to W_x^{s,loc} ~(W_x^{u,loc})$ is $C^{l-j}$ in $x\in N$ when $W_x^{s,loc} ~( W_x^{u,loc})$ is given in the $C^j$ topology.
[(1)]{}: We could also define the global stable (unstable) sets $W_N^{s,u}$ and the leaves $W_x^{s,u}$, just by replacing $U$ with $M$ in , . But $W_N^{s,u}, W_x^{s,u}$ may fail to be embedded manifolds.\
[(2)]{}: The manifolds $N, W_N^{s,loc}, W_N^{u,loc}$ may fail to be $C^\infty$ even if $f$ is analytic.
The normal hyperbolicity has stability under perturbations. Roughly speaking, the normally hyperbolic invariant manifold may persist under perturbations.
\[persistence\] Suppose that $N$ is a NHIM and $\varepsilon>0$ is sufficiently small, then for any $C^r$ $(r>1)$ diffeomorphism $f_\varepsilon: M\to M$ satisfying $\|f_\varepsilon-f\|_{C^1}<\varepsilon$, there exists a NHIM $N_\varepsilon$ that is $C^l$ diffeomorphic and close to $N$ where $1<l=\min(r, \frac{|\log\lambda|}{\log\mu}-\delta)$ with a small number $\delta >0$.
Variational construction of global connecting orbits {#sec_proof_of_connectingthm}
====================================================
This section aims to prove Theorem \[generalized transition thm\], which could be obtained by modifying the arguments and techniques in [@CY2009], and we also refer the reader to [@Ch2012] or [@Ch2018] for more details. Throughout this section, we assume $M=\T^n$. Our diffusion orbits are constructed by shadowing a sequence of local connecting orbits, along each of them the Lagrangian action attains a “local minimum".
Local connecting orbits {#sec localconnect}
-----------------------
An orbit $(d\gamma(t),t):\R\to TM\times\T$ is said connecting one Aubry class $\tilde{\cA}(c)$ to another one $\tilde{\cA}(c')$ if the $\alpha$-limit set of the orbit is contained in $\tilde{\cA}(c)$ and the $\omega$-limit set is contained in $\tilde{\cA}(c')$. We will introduce two types of local connecting orbits: type-$c$ and type-$h$, the former one corresponds to Mather’s cohomology equivalence, while the later one corresponds to Arnold’s mechanism in a variational viewpoint. Before that, we need to some preparations.
### Time-step Lagrangian and upper semi-continuity
Both types are strongly depend on the upper semi-continuity of minimal curves of a modified Lagrangian $L^*:T\T^n\times\R\rightarrow\R$ which is defined as follows: let $L^+,L^-$ be two time-1 periodic Tonelli Lagrangians and $$L^*(\cdot,t):=\begin{cases}
L^-(\cdot,t), & ~t\in(-\infty,0]\\
L^+(\cdot,t), & ~t\in[1,+\infty).
\end{cases}$$ Notice that $L^*$ is not periodic in time $t$, instead, it is periodic when restricted on either $(-\infty,0]$ or $[1,+\infty)$. We call such a modified Lagrangian $L^*$ a *time-step* Lagrangian.
For a *time-step* Lagrangian $L^*$, a curve $\gamma:\R\rightarrow\T^n$ is called minimal if for any $t<t^\prime\in\R$, $$\int_t^{t^\prime}L^*(\gamma(s),\dot{\gamma}(s),s)\,ds=\min\limits_{\substack{\zeta(t)=\gamma(t),\zeta(t^\prime)=\gamma(t^\prime)\\ \zeta\in C^{ac}([t,t^\prime],\T^n)}}\int_t^{t^\prime}L^*(\zeta(s),\dot{\zeta}(s),s)\,ds$$ So we denote by $\sG(L^*)$ the set of all minimal curves and $\tilde{\sG}(L^*)=\bigcup_{\gamma\in\sG}(\gamma(t),\dot{\gamma}(t),t)$.
Let $\alpha^\pm$ denote Mather’s minimal average action of $L^\pm$. For $m_0, m_1\in \T^n$ and $T_0,T_1\in\Z_+$, we define $$h^{T_0,T_1}_{L^*}(m_0,m_1):=\inf\limits_{\substack{\gamma(-T_0)=m_0,\gamma(T_1)=m_1\\ \gamma\in C^{ac}([-T_0,T_1],\T^n)}}\int_{-T_0}^{T_1}L^*(\gamma(t),\dot{\gamma}(t),t)\,dt+T_0\alpha^-+T_1\alpha^+$$ and $$h_{L^*}^{\infty}(m_0,m_1):=\liminf_{T_0,T_1\to+\infty}h_{L^*}^{T_0,T_1}(m_0,m_1)$$ which are bounded. We take any two sequences of positive integers $\{T_0^i\}_{i\in\Z_+}$ and $\{T_1^i\}_{i\in\Z_+}$ with $T_\ell^i\rightarrow+\infty$ ($\ell=0,1$) as $i\to+\infty$ and the associated minimal curve $\gamma_i(t,m_0,m_1)$: $[-T^i_0,T^i_1]\to \T^n$ connecting $m_0$ to $m_1$ such that $$h_{L^*}^{\infty}(m_0,m_1)=\lim_{i\to\infty}h_{L^*}^{T_0^i,T_1^i}(m_0,m_1)=\lim_{i\to\infty}\int_{-T_0^i}^{T_1^i}L^*(\gamma_i(t),\dot{\gamma}_i(t),t)\,dt +T^i_0\alpha^-+T^i_1\alpha^+.$$
The following lemma shows that any accumulation point $\gamma$ of $\{\gamma_i\}_i$ is a pseudo curve playing the similar role as a semi-static curve. For the proof, see [@CY2004] or [@CY2009].
\[pseudo curve\] Let $\gamma$: $\R\to \T^n$ be an accumulation point of $\{\gamma_i\}_i$ shown as above. Then for any $s\geq 0$, $t\geq 1$, $$\label{pseudo curve formula}
\begin{split}
\int_{-s}^{t} L^*(\gamma(\tau),\dot{\gamma}(\tau),\tau)\,d\tau+s\alpha^-+t\alpha^+=\inf\limits_{\substack{\xi(-s_1)=\gamma(-s)\\\xi(t_1) =\gamma(t) \\ s_1-s\in\Z, ~t_1-t\in\Z\\ s_1\geq 0,~t_1\geq 1}}\int_{-s_1}^{t_1} L^*(\xi(\tau),\dot{\xi}(\tau),\tau)\,d\tau+s_1\alpha^-+t_1\alpha^+,
\end{split}$$ where the minimum is taken over all absolutely continuous curves.
This leads us to define the set of *pseudo connecting curves* $$\sC(L^*):=\{\gamma |~\gamma\in\sG(L^*) \text{~and~} \eqref{pseudo curve formula}\text{~holds}\}.$$ Clearly, for each $\gamma\in\sC(L^*)$ the orbit $(\gamma(t),\dot\gamma(t),t)$ would approach the Aubry set $\tilde{\cA}(L^-)$ of the Lagrangian $L^-$ as $t\to -\infty$ and approach $\tilde{\cA}(L^+)$ of $L^+$ as $t\to +\infty$. This is why we call it a pseudo connecting curve. Define the following sets $$\tilde{\cC}(L^*):=\bigcup_{\gamma\in\sC(L^*)}(\gamma(t),\dot\gamma(t),t),\qquad \cC(L^*):=\bigcup_{\gamma\in\sC(L^*)}(\gamma(t),t).$$ Notice that if $L^*$ is a time-1 periodic , then $\tilde{\cC}(L^*)$ is exactly the Mañé set and $\cC(L^*)$ is exactly the projected Mañé set. So we have the following property:
\[uppersemi of pseudo curves\] The set-valued map $L^*\to\sC(L^*)$ is upper semi-continuous, namely if $L^*_i\to L^*$ in the $C^2$ topology, then we have $$\limsup_i \sC(L^*_i)\subset \sC(L^*).$$ Consequently, the map $L^*\to\tilde{\cC}(L^*)$ is also upper semi-continuous.
Let the time-step Lagrangian $L^*_i\to L^*$ in the $C^2$ topology. If $\{\gamma_i\}_{i}$ converges $C^0$-uniformly to a curve $\gamma$ on each compact interval of $\R$ with $\gamma_i\in\sC(L^*_i)$. We claim that $\gamma\in\sC(L^*)$.
Indeed, there exists $K>0$ such that $\|\dot{\gamma}_i(t)\|\leq K$ for all $t\in\R$, so the set $\{\gamma_i\}_{i}$ is compact in the $C^1$ topology. Since each $\gamma_i$ satisfies the Euler-Lagrange equation, by using the positive definiteness of $L^*_i$, one can write the Euler-Lagrange equation in the form of $\ddot{x}=f_i(x,\dot{x},t)$ for some $f_i$, which implies $\{\gamma_i\}_{i}$ is compact in the $C^2$ topology. By the Arzelà-Ascoli theorem, extracting a subsequence if necessary, we can assume that $\gamma_i$ converges $C^1$-uniformly to a $C^1$ curve $\tilde{\gamma}$ on each interval of $\R$. Obviously, $\tilde{\gamma}=\gamma.$
Next, if $\gamma\notin\sC(L^*)$, there would be some $s\geq 0, t\geq 1$, a curve $\tilde{\gamma}:[-s-n_1, t+n_2]\to M$ and $\delta>0$ such that the action $$\int_{-s-n_1}^{t+n_2}L^*(\tilde{\gamma}(\tau),\dot{\tilde{\gamma}}(\tau),\tau)\,d\tau+n_1\alpha^-+n_2\alpha^+\leq \int_{-s}^{t}L^*(\gamma(\tau),\dot{\gamma}(\tau),\tau)\,d\tau-\delta$$ where $s, s+n_1\geq 0$, $t, t+n_2\geq 1$ and $\tilde{\gamma}(-s-n_1)=\gamma(-s)$, $\tilde{\gamma}(t+n_2)=\gamma(t)$. Since $\gamma$ is an accumulation point of $\gamma_i$, for any small $\varepsilon>0$, there would be a sufficiently large $i$ such that $\|\gamma-\gamma_i\|_{C^1[s,t]}\leq\varepsilon$ and a curve $\tilde{\gamma}_i: [-s-n_1, t+n_2]\to M$ with $\tilde{\gamma}_i(-s-n_1)=\gamma_i(-s)$, $\tilde{\gamma}_i(t+n_2)=\gamma_i(t)$ such that $$\int_{-s-n_1}^{t+n_2}L^*(\tilde{\gamma}_i(\tau),\dot{\tilde{\gamma}}_i(\tau),\tau)\,d\tau+n_1\alpha^-+n_2\alpha^+\leq \int_{-s}^{t}L^*(\gamma_i(\tau),\dot{\gamma}_i(\tau),\tau)\,d\tau-\frac{\delta}{2},$$ By , $\gamma_i\notin\sC(L^*_i)$, which is a contradiction. This proves $\gamma\in\sC(L^*)$.
Consequently, the proof of the upper semi-continuity for $L^*\to\tilde{\cC}(L^*)$ is directly obtained by the arguments above.
### Local connecting orbits of type-[*c*]{} {#sub localc}
In condition of the cohomology equivalence (see definition \[def\_c\_equivalenve\]), we will show how to construct local connecting orbits based on Mather’s variational mechanism. This idea of construction is first proposed by J. Mather in [@Ma1993].
\[clemma connect\] Let $L:T\T^n\times\T\to\R$ be a Tonelli Lagrangian and $c, c'\in H^1(\T^n,\R)$ are cohomology equivalent through a path $\Gamma:[0,1]\to H^1(\T^n,\R)$. Then there would exist $c=c_0, c_1, \dots, c_k=c'$ on the path $\Gamma$, closed 1-forms $\eta_i$ and $\bar{\mu}_i$ on $M$ with $[\eta_i]=c_i$, $[\bar{\mu}_i]=c_{i+1}-c_i$ and a smooth function $\rho_i(t):\R\to[0,1]$ for $i=1,\cdots,k$, such that the time-step Lagrangian $$L_{\eta_i,\mu_i}=L-\eta_i-\mu_i,\quad \mu_i=\rho_i(t)\bar{\mu}_i$$ possesses the following properties:
For each curve $\gamma\in \sC(L_{\eta_i,\mu_i})$, it determines an orbit $(d\gamma(t),t)$, connecting $\tilde{\cA}(c_i)$ to $\tilde{\cA}(c_{i+1})$, of the Euler-Lagrange flow $\phi^t_L$.
By definition \[def\_c\_equivalenve\], it is obvious that there exist $c=c_0, c_1, \dots, c_k=c'$ on the path $\Gamma$, closed 1-forms $\eta_i$ and $\bar{\mu}_i$ on $M$ with $[\eta_i]=c_i$, $[\bar{\mu}_i]=c_{i+1}-c_i$$\in \mathbb{V}^{\bot}_{c_i}$ for each $i=1,\cdots,k$. By the arguments in section \[sec local and global\], there is also a neighborhood $U_i$ of the projected Mañé set $\cN_0(c_i)$ such that $\mathbb{V}_{c_i}=i_{*U_i}H_1(U_i,\R)$.
In particular, we can suppose $\bar{\mu}_i=0$ on $U_i$. Indeed, as $[\bar{\mu}_i]\in \mathbb{V}^{\bot}_{c_i}$, $\bar{\mu}_i$ is exact when restricted on $U_i$ and there is a smooth function $f:M\to\R$ satisfying $df=\bar{\mu}_i$ on $U_i$, hence we can replace $\bar{\mu}_i$ by $\bar{\mu}_i-df$.
As $\cN_0(c_i)\subset U_i$, there exists $\delta_i\ll 1$ such that $\cN_t(c_i)\subset U_i$ for all $t\in[0, \delta_i]$. Let $\rho_i:\R\to [0,1]$ be a smooth function such that $\rho_i(t)=0$ for $t\in(-\infty, 0]$, $\rho_i(t)=1$ for $t\in[\delta_i,+\infty)$. We set $\mu_i=\rho_i(t)\bar{\mu}_i$ and introduce a *time-step* Lagrangian $$L_{\eta_i,\mu_i}=L-\eta_i-\mu_i: T\T^n\times\R\to\R.$$ For each orbit $\gamma\in\sC(L_{\eta_i,\mu_i})$, by the upper semi-continuity of Proposition \[uppersemi of pseudo curves\], $$\label{belong typec}
\gamma(t)\in U_i,~\forall~t\in[0,\delta_i]$$ holds provided $|\bar{\mu}_i|$ is small enough.
Clearly, $(\gamma,\dot{\gamma})$ solves the Euler-Lagrange equation of $L_{\eta_i,\mu_i}$. To verify it solves the Euler-Lagrange equation of $L$, we see that $\gamma\big|_{[0,\delta_i]}\subset U_i$ and $L_{\eta_i,\mu_i}=L-\eta_i$ on $U_i$ where $\eta_i$ is a closed 1-form, so $\gamma(t)$ solves the Euler-Lagrange equation of $L$ for $t\in[0,\delta_i]$. On the other hand, for $t\in(-\infty, \delta_i]$ we have $L_{\eta_i,\mu_i}=L-\eta_i$, then $\gamma(t)$ is a $c_{i}$-semi static curve $L$ on the interval $(-\infty, \delta_i]$. Similarly, $\gamma(t)$ is a $c_{i+1}$-semi static curve of $L$ for $t\in[\delta_i,+\infty)$. Thus, $(\gamma,\dot{\gamma}): \R\to T\T^n$ solves the Euler-Lagrange equation of $L$, and by section \[sec\_Preliminaries\], this orbit would connect $\tilde{\cA}(c_i)$ and to $\tilde{\cA}(c_{i+1})$ .
### Local connecting orbits of type-[*h*]{} {#sub localh}
Next, we will discuss the so-called local connecting orbits of type-$h$, it can be thought of as a variational version of Arnold’s mechanism, the condition of geometric transversality is extended to the total disconnectedness of minimal points of barrier function. It is used to handle the situation where the cohomology equivalence does not always exist. Usually, it is applied to the case where the Aubry set lies in a neighborhood of some lower dimensional torus, in that case, we let $\check{\pi}:\check{M}\rightarrow \T^n$ be a finite covering of $\T^n$. Denote by $\tilde{\cN}(c,\check{M}), \tilde{\cA}(c,\check{M})$ the Mañé set and Aubry set with respect to $\check{M}$, then $\tilde{\cA}(c,\check{M})$ would have more than one Aubry classes. In fact, for the construction of type-$h$ local connecting orbits in our proof of Theorem \[main theorem\] and \[main thm2\], it only involves two Aubry classes (see section \[sec proof main\]).
Thus, we only need to deal with the situation where the Tonelli Lagrangian $L: T\T^n\times\T\rightarrow \R$ contains more than one Aubry classes. Let $\cA(c)\big|_{t=0} $ denote the time-0 section of the projected Aubry set $\cA(c)$, i.e. $\cA(c)\bigcap(\T^n\times\{t=0\})$, then we can obtain the local connecting oribits of type-$h$ as follows:
\[connecting type h\] Let the projected Aubry set $\cA(c)=\cA_{c,1}\cup\cdots \cup\cA_{c,k}$ consists of $k$ $(k\geq 2)$ Aubry classes. If there exists an open set $U\subset \T^n\setminus\cA(c)|_{t=0}$ such that $U\bigcap (\cN(c)|_{t=0})$ is non-empty and totally disconnected. Then for any $c'$ sufficiently close to $c$, there exists an orbit of the Euler-Lagrange flow $\phi^t_L$ whose $\alpha$-limit set lies in $\tilde{\cA}(c)$ and $\omega$-limit set lies in $\tilde{\cA}(c')$.
As the number of Aubry class of $\cA(c)$ is finite, it is well known that the map $c\mapsto \cA(c)$ is upper semi-continuous, which means that if $c'$ is sufficiently close to $c$, the projected Aubry set $\cA(c')$ will be contained in a small neighborhood of $\cA(c)$.
Since each Aubry class is compact and they are disjoint, we have $\text{\rm dist}(\cA_{c,i},\cA_{c,i'})>0$ for each $i\neq i'$, and there exist open neighborhoods $N_1,\cdots,N_k \subset M$ such that $\cA_{c,i}\big|_{t=0}\subset N_i$ for each $1\leq i\leq k$ and $\textup{dist}(N_i,N_{i'})>0$ for $i\neq i'$. So by the analysis above, $\cA(c')|_{t=0}\subset \bigcup_i N_i$. From Proposition \[manejifenlei\] and definition we know $\cN(c)=\cup_{i,i'}\cN_{i,i'}(c)$, hence there is a pair $(j, j')$ such that $\cA(c')|_{t=0}\cap N_{j'}\neq\emptyset$ and $U\cap \cN_{j,j'}(c')\neq\emptyset$.
By assumption, we could also find simply connected open sets $F$ and $O$ such that $F\subset O\subset U$, $\textup{dist}(O,\bigcup\limits_{i=1}^k\bar{N}_i)>0$ and $\emptyset\neq O\bigcap\big(\cN_{j,j'}(c)\big|_{t=0}\big)\subset F$. Then some $\delta>0$ exists such that $$\label{belongrelation}
O\bigcap\big(\cN_{j,j'}(c)\big|_{0\leq t\leq\delta}\big)\subset F.$$ Let $\eta$ and $\bar{\mu}$ be closed 1-forms such that $[\eta]=c$, $[\bar{\mu}]=c'-c$ and let $\rho:\R\to [0, 1]$ be a smooth function such that $\rho(t)=0$ for $t\leq 0$, $\rho=1$ for $t\geq\delta$. Notice that by the simple connectedness, we are able to choose $\bar{\mu}$ such that $\textup{supp}\bar{\mu}\cap \bar{O}=\emptyset$. Next, we construct a smooth function $\psi(x,t)=\varepsilon\psi_1(x)\psi_2(t):M\times\T\to [-1, 1]$ ($\varepsilon>0$) satisfying the following conditions: $$\psi_1(x)\left\{
\begin{array}{ll}
=1, & x\in \bar{F}, \\
<1, & x\in O\setminus F,\\
<0, & x\in\bigcup\limits_{i\neq j,j'}N_i, \\
=0, & \textup{elsewhere.}
\end{array}
\right.$$ and $$\psi_2(t)\left\{
\begin{array}{ll}
>0, & t\in (0,\delta), \\
=0, & t\in(-\infty,0]\cup[\delta,+\infty).
\end{array}
\right.$$ Then we set $\mu=\rho(t)\bar{\mu}$ and introduce a time-step Lagrangian $$L_{\eta,\mu,\psi}=L-\eta-\mu-\psi: T\T^n\times\R\to\R.$$
If $\mu=0$. Since $\psi(x,t)=0$ for $(x,t)\in\cA_{c,j}\cup\cA_{c,j'}$ and $\psi(x,t)<0$ for $(x,t)\in\bigcup\limits_{i\neq j,j'}N_i$, the Lagrangian $L_{\eta,0,\psi}$ contains only two Aubry classes which are exactly $\cA_{c,j}$ and $\cA_{c,j'}$ provided the positive number $\varepsilon$ in $\psi$ is small enough. The set $\sC(L_{\eta,0,\psi})$ satisfies:
(a) $\cA_{c,j}\cup\cA_{c,j'}\subset\sC(L_{\eta,0,\psi})$.
(b) $\sC(L_{\eta,0,\psi})\setminus\big(\cA_{c,j}\cup\cA_{c,j'}\big)$ is non-empty and for each pseudo connecting curve $\xi\in\sC(L_{\eta,0,\psi})\setminus\big(\cA_{c,j}\cup\cA_{c,j'}\big)$, we have $\xi(t)\in F$ for $0\leq t\leq\delta$, but its integer translation $K^*\xi(t):=\xi(t-K)$ with $K\in\Z\setminus{0}$ does not belong to $\sC(L_{\eta,0,\psi})$ since $L_{\eta,0,\psi}$ is not periodic in $t$.
(c) $\sC(L_{\eta,0,\psi})$ does not contain any other curves,
these properties follow directly from , the fact $\psi(x,0)$ attains its maximum if and only if $x\in\bar{F}$, and the upper semi-continuity of $(\eta,0,\mu)\mapsto \sC(L_{\eta,0,\psi})$.
If $\mu\neq 0$. For $m_0\in\cA_{c,j}\big|_{t=0}, m_1\in \cA_{c,j'}\big|_{t=0}$, let $T_0^k, T_1^k\to+\infty$ be the sequences of positive integers such that $$\lim\limits_{k\to\infty}h_{L_{\eta,\mu,\psi}}^{T_0^k,T_1^k}(m_0,m_1)=h_{L_{\eta,\mu,\psi}}^{\infty}(m_0,m_1).$$ Let $\gamma_k(t,m_0,m_1): [-T^k_0, T^k_1]\to M$ be a minimizer associated with $h_{L_{\eta,\mu,\psi}}^{T_0^k,T_1^k}(m_0,m_1)$ and $\gamma$ be any accumulation point of $\{\gamma_k\}_k$, then $\gamma\in\sC(L_{\eta,\mu,\psi})$ and if $\mu$ and $\varepsilon$ are small enough, we deduce from the properties $(a),(b),(c)$ and the upper semi-continuity of $(\eta,\mu,\psi)\to \sC(L_{\eta,\mu,\psi})$ that $$\label{constant region}
\gamma(t)\in F,\quad\forall t\in[0, \delta].$$
Obviously, $(\gamma,\dot{\gamma})$ satisfies the Euler-Lagrangian equation of $L_{\eta,\mu,\psi}$, but we still need to verify that it solves the Euler-Lagrangian equation of $L$. In fact, $L_{\eta,\mu,\psi}=L-\eta$ for $t\leq 0$ and $L_{\eta,\mu,\psi}=L-\eta+\bar{\mu}$ for $t\geq\delta$, as $\eta$, $\bar{\mu}$ are closed 1-forms, so $\gamma(t)$ solves the Euler-Lagrangian equation of $L$ for $t\in(-\infty, 0]\cup[\delta, +\infty)$, which also implies that $\gamma:(-\infty,0]\to \T^n$ is a $c$-semi static curve of $L$ and $\gamma:[\delta,+\infty)\to M$ is a $c'$-semi static curve of $L$, thus $$\alpha(d\gamma(t),t)\subset\tilde{\cA}(c),\quad \omega(d\gamma(t),t)\subset\tilde{\cA}(c').$$ Besides, for $t\in[0, \delta]$, we deduce from that the Euler-Lagrange equation $(\frac{d}{dt}\partial_v-\partial_x)L_{\eta,\mu,\psi}=0$ is equivalent to $(\frac{d}{dt}\partial_v-\partial_x)L=0$ along the curve $\gamma(t)$ within $0\leq t\leq\delta$, which shows that $(\gamma,\dot{\gamma})$ also solves the Euler-Lagrange equation of $L$ for $t\in[0,\delta]$. This completes our proof.
From the proof of Theorem \[connecting type h\], the connecting orbit $(\gamma,\dot{\gamma})$ obtained in this theorem is locally minimal in the following sense:
[**Local minimum**]{}: [ *There are two open balls $V^-,V^+\subset M$ and $k^-,k^+\in\Z+$ such that $\bar{V}^-\subset N_j\setminus \cA(c)\big|_{t=0}$ and $\bar{V}^+\subset N_{j'}\setminus \cA(c')\big|_{t=0}$, $\gamma(-k^-)\in V^-$, $\gamma(k^+)\in V^+$ and $$\label{local minimal property}
\begin{aligned}
&h_c^{\infty}(x^-,m_0)+h_{L_{\eta,\mu,\psi}}^{k^-,k^+}(m_0,m_1)+h_{c'}^{\infty}(m_1,x^+)\\
>&\liminf_{k^-_i, k_i^+\to\infty}\int_{-k^-_i}^{k^+_i}
L_{\eta,\mu,\psi}(\gamma(t),\dot{\gamma}(t),t)\,dt+k^-_i\alpha(c)+k^+_i\alpha(c')
\end{aligned}$$ holds for all $(m_0,m_1)\in \partial(V^-\times V^+)$, $x^-\in N_i\cap \alpha(\gamma)|_{t=0}$, $x^+\in N_{j'}\cap\omega(\gamma)|_{t=0}$, where $k_i^-, k_i^+$ are the sequences such that $\gamma(-k_i^-)\to x^-$ and $\gamma(k_i^+)\to x^+$.*]{}
The set of curves starting from $V_i^-$ and reaching $V_{i'}^+$ within time $k^-+k^+$ would make up a neighborhood of the curve $\gamma$ in the space of curves. If it touches the boundary of this neighborhood, the action of $L_{\eta,\mu,\psi}$ along a curve $\xi$ will be larger than the action along $\gamma$. Besides, the connecting orbit of type-$c$ also has some local minimal property. In this case, the modified Lagrangian has the form $L_{\eta,\mu}$. The local minimality is crucial in the variational construction of global connecting orbits.
Global connecting orbits {#sec variationconst}
------------------------
After the preparations above, we will explain how to prove Theorem \[generalized transition thm\] from a variational viewpoint. Intrinsically, we construct a global connecting orbit by shadowing a sequence of local connecting orbits.
Actually, the proof parallels to that of [@CY2009] by a small modification. We only give a sketch of the idea here, and refer the reader to [@CY2009 Section 5], [@Ch2018] or [@Ch2012] for more details. For the generalized transition chain $\Gamma:[0,1]\to H^1(\T^n,\R)$ with $\Gamma(0)=c$ and $\Gamma(1)=c'$, by definition there exists a sequence $0=s_0<s_1<\cdots<s_m=1$ such that $s_i$ is sufficiently close to $s_{i+1}$ for each $0\leq i\leq m-1$, and $\cA(\Gamma(s_i))$ could be connected to $\cA(\Gamma(s_{i+1}))$ by a local minimal orbit of either type-$c$ (as Theorem \[clemma connect\]) or type-$h$ (as Theorem \[connecting type h\]). Then the global connecting orbits are just constructed by shadowing these local ones.
For each $i\in \{0,1,\cdots,m-1\}$, we take $\eta_i,\mu_i$, $\psi_i$ and $\delta_i>0$ as that in the proof of Theorem \[clemma connect\] and \[connecting type h\], where $\psi_i=0$ in the case of type-$c$. Then we choose $k_i\in\Z_+$ with $k_0=0$ and $k_{i+1}-k_i$ is suitably large for each $i\in \{0,1,\cdots,m-1\}$, and introduce a modified Lagrangian $$L^*:=L-\eta_0-\sum\limits_{i=0}^{m-1}k_i^*(\mu_i+\psi_i),$$ where $k_i^*$ denotes a time translation operator such that $k^*_if(x,t)=f(x, t-k_i)$, and $\psi_i= 0$ in the case of type-$c$. By this definition, we see that $L^*=L-\eta_0$ on $t\leq k_0=0$, $L^*=L-\eta_m$ on $t\geq k_{m-1}+\delta_{m-1}$, and for each $i\in\{0, 1,\cdots,m-2\}$, $L^*=L-\eta_i-k_i^*(\mu_i+\psi_i)$ on $t\in [k_i, k_i+\delta_i]$ and $L^*=L-\eta_{i+1}$ for $t\in [k_i+\delta_i, k_{i+1}]$.
For integers $T_0, T_m\in\Z+$ and $x_0,x_m\in \T^n$, we define $$h^{T_0,T_m}(x_0,x_m)=\inf_{\xi}\int_{-T_0}^{T_m+k_{m-1}}L^*(\xi(s),\dot{\xi}(s),s)\,ds+\sum\limits_{i=1}^{m-1}(k_i-k_{i-1})\alpha(c_i)+T_0\alpha(c_0)+T_m\alpha(c_m),$$ where the infimum is taken over all absolutely continuous curves $\xi$ defined on the interval $[-T_0,T_m+k_{m-1}]$ under some boundary conditions. By carefully setting boundary conditions and using standard arguments in variational methods, one could find that the minimizer $\gamma(t; T_0, T_m,m_0,m_1)$ of the action $h^{T_0,T_m}(x_0,x_m)$ is smooth everywhere, along which the term $k_i^*(\mu_i+\psi_i)$ would not contribute to the Euler-Lagrange equation. Hence the minimizer produces an orbit of the flow $\phi^t_L$, which passes through the $\varepsilon$-neighborhood of $\tilde{\cA}(\Gamma(s_i))$ at some time $t=t_i$. Let $T_0, T_m\to+\infty$, we could also get an accumulation curve $\gamma(t):\R\to \T^n$ of the sequence $\{\gamma(t; T_0, T_m,m_0,m_1)\}$ such that the $\alpha$-limit set of $(d\gamma(t),t)$ lies in $\tilde{\cA}(c)$ and the $\omega$-limit set of $(d\gamma(t),t)$ lies in $\tilde{\cA}(c')$. This completes the proof.
[**Acknowledgments**]{} The authors were supported by National Basic Research Program of China (973 Program)(Grant No. 2013CB834100), National Natural Science Foundation of China (Grant No. 11631006) and a program PAPD of Jiangsu Province, China. The authors also would like to thank the referees for their careful reading and useful suggestions.
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---
abstract: 'A concentration-saturated helium mixture at the melting pressure consists of two liquid phases and one or two solid phases. The equilibrium system is univariant, whose properties depend uniquely on temperature. Four coexisting phases can exist on singular points, which are called quadruple points. As a univariant system, the melting pressure could be used as a thermometric standard. It would provide some advantages compared to the current reference, namely pure $^3$He, especially at the lowest temperatures below 1 mK. We have extended the melting pressure measurements of the concentration-saturated helium mixture from 10 mK to 460 mK. The density of the dilute liquid phase was also recorded. The effect of the equilibrium crystal structure changing from hcp to bcc was clearly seen at $T=294$ mK at the melting pressure $P=2.638$ MPa. We observed the existence of metastable solid phases around this point. No evidence was found for the presence of another, disputed, quadruple point at around 400 mK. The experimental results agree well with our previous calculations at low temperatures, but deviate above 200 mK.'
author:
- 'J. Rysti'
- 'M. S. Manninen'
- 'J. Tuoriniemi'
bibliography:
- 'refs.bib'
title: 'Measurements on Melting Pressure, Metastable Solid Phases, and Molar Volume of Univariant Saturated Helium Mixture'
---
Introduction {#Sec:Intro}
============
At any pressure, liquid $^3$He–$^4$He mixture separates into two phases below $T\sim 0.9$ K, provided enough of both components are present. The so-called dilute phase is rich in $^4$He and has a finite solubility of $^3$He even down to the zero temperature. The $^3$He-rich phase becomes pure $^3$He in the limit $T\to 0$. Increasing the pressure of a phase-separated liquid mixture results eventually in the appearance of a solid phase whereupon the system reaches the melting curve of the concentration-saturated helium mixture. It consists of three or four simultaneous phases. The latter situation defines a quadruple point, which fixes all the intensive thermodynamic variables. The saturated system at the melting pressure is a univariant and its properties are uniquely determined by temperature. It therefore offers the possibility to use it as a thermometric standard in a similar manner as pure $^3$He. The provisional low temperature scale from 0.9 mK to 1 K (PLTS-2000) is based on the melting pressure of pure $^3$He [@PLTS2000]. Below its nuclear magnetic ordering temperature $T_\mathrm{N}\approx0.9$ mK, the melting pressure of pure $^3$He is proportional to $T^4$ due to the entropy of solid $^3$He. However, the melting pressure of saturated helium mixture is proportional to $T^2$ because of the entropy of the fermionic normal $^3$He component in the dilute liquid phase. The slope of the melting pressure curve of the helium mixture system thus remains larger than that of the pure system at very low temperatures and has the potential to offer better resolution there. A differential pressure gauge with $^4$He as reference can be used to increase accuracy [@LT24paper; @LT25paper]. Admittedly, operating with a mixture system is more involved than the pure system. However, in some applications, such as the adiabatic melting refrigeration of $^3$He–$^4$He mixtures, this type of thermometer is very natural [@TuoriniemiAdiabatic].
Since the melting pressure of pure $^4$He is lower than that of pure $^3$He, the solid mixture always contains $^4$He. In the limit of zero temperature, the solid phase becomes pure $^4$He [@EdwardsBalibar]. In this case the melting pressure is that of pure $^4$He plus an additional contribution due to the osmotic pressure in the dilute liquid [@Osmotic]. As temperature is increased, the melting pressure first increases quadratically, since it is mostly determined by the entropy of $^3$He in the degenerate Fermi liquid phases with practically constant solubilities [@Solubility]. At higher temperatures, $^3$He dissolves into the solid at larger quantities. This turns the melting pressure down, while its maximum is obtained at around 300 mK. The melting pressure falls below that of pure $^4$He at about 400 mK and reaches a minimum in the range of 1 K. During increasing concentration of the solid phase, the preferable crystal structure changes from hexagonal close-packed (hcp) of pure $^4$He to body-centered cubic (bcc), which is the same structure as in pure $^3$He. This change occurs at the maximum melting pressure, which then is a quadruple point.
In this paper we report the results of measurements on the melting pressure of the concentration-saturated helium mixture between 10 mK and 460 mK and the observed nucleation pressures for creating new crystals at various temperatures. This work extends the previous measurements from below 1 mK to 10 mK [@EliasMeltingPressure]. The molar volume of the dilute liquid phase at the melting pressure is given as well. We also report the existence of metastable solid phases around the maximum pressure.
Experimental arrangement
========================
The experimental setup on the nuclear refrigerator [@Weijun2000] was designed for studies of helium crystals at submillikelvin temperatures [@MasaPaper]. However, the experiment described here concerned temperatures only above 10mK and thus the nuclear stage with the cell was always thermally connected to the mixing chamber of the dilution refrigerator via an aluminium heat switch.
The sample cell consisted of two volumes, the lower bellows chamber of a hydraulic press and the actual experimental cell, see Fig. \[Fig:Cell\].
![(Color online) The experimental setup. The central cuboid volume of the experimental cell was surrounded by an annular volume and contained the interdigital capacitors (IDC) and quartz tuning forks. Pressure in the experimental cell was varied with a hydraulic press (bellows chambers) and measured with a capacitive pressure gauge (PG). The cell was bolted on a copper nuclear demagnetization stage and its walls were covered with sintered silver powder. The solid phase usually nucleated into the central cuboid volume.[]{data-label="Fig:Cell"}](fig_cellall.ps)
Since the melting pressure of $^3$He–$^4$He mixtures has a deep minimum at around 1K it is not possible to pressurize helium mixtures at low temperatures above that pressure through a filling line, which will be blocked by solid helium. However, the volume, and thus the pressure of the lower bellows chamber could be altered by regulating the $^4$He pressure in the upper operating bellows chamber. The pressure in the upper bellows was always far below the melting pressure of $^4$He because of the upper to lower bellows compressing area ratio of about $6.7~\mathrm{cm}^2/2.0~\mathrm{cm}^2$.
The experimental cell consisted of a central cuboid volume ($1\,\textrm{cm} \times 1\,\textrm{cm}\times 2\,\textrm{cm}$) and an outer annular volume which provided a thermal guard. Interdigital capacitors for density measurements and quartz tuning forks for thermometry were located in the cuboid volume. The helium sample was thermalized with porous sintered silver powder on the cell walls. Photographs of the experimental cell are shown in Ref.[@MasaPaper].
The capacitive pressure gauge used in these experiments was of a Straty-Adams type made of beryllium copper. It had the sensitivity $dC/dP = 20~\mathrm{pF/MPa}$ in the range of helium mixture melting pressures. It was calibrated seven months earlier using pure $^3$He in the cell and a mechanical pressure gauge at room temperature. The single point calibration accuracy was about 2 kPa. The pressure gauge suffered from slight drifting over time. The scale was therefore further fixed to the zero temperature value of the saturated mixture melting pressure (2.566 MPa [@EliasMeltingPressure; @Grilly1973]), when used for the present experiments. The capacitances of the $^3$He calibration had to be scaled by 1.0052 to correct for the difference accumulated over the seven months, during which the cryostat was kept mainly below 0.1 K but also warmed up to 4 K. This resulted in approximately 11 kPa increase of the reading at the melting pressure, when comparing new and old calibrations. Further minor changes in the pressure gauge were observed during the present measurements after depressurization-pressurization cycles, which were performed several times to increase the amount of $^3$He in the cell. The reading shifted by an amount between 1 kPa to 4 kPa after each cycle. These were corrected in the data by simply shifting the points so that the values for the solubility saturated system before and after the re-pressurizations matched. The additional small drift with time was of the order of few hundred pascals over the experiment. The pressure gauge calibration is shown in Fig. \[Fig:PGcalibration\]. The fitted function is $P/\mathrm{MPa} = 5.6334-190.01(C/\mathrm{pF})^{-1}$.
![Pressure gauge calibration. In the main frame the difference between the calibration data and the resulting fit is given. The inset shows the absolute data. The capacitive pressure gauge was calibrated against a mechanical pressure gauge at room temperature. The calibration here was performed only to pressures between $P=2.25$ MPa and $P=2.55$ MPa, since the present studies were restricted to the melting pressure. Modest extrapolation to higher pressures, up to $P=2.638$ MPa, is needed to cover the complete range of interest.[]{data-label="Fig:PGcalibration"}](PGcalibration.eps){width="98.00000%"}
Two interdigital capacitors (IDC) on opposite walls of the cell cavity were used to monitor the local helium density. In practice they remained in the dilute liquid phase, except for times when the solid was grown to large enough size. The capacitors were calibrated using pure liquid $^4$He and experimental data from literature [@Watson1969; @Tanaka2000]. Such calibrations had been performed with pure $^3$He also, when experiments on that were done, but it was observed that between the $^3$He and mixture experiments both capacitors had somewhat inconsistent values at a given nominal helium density. The discrepancy was slightly different for the two IDC’s. This was observed when the capacitances of the two capacitors were plotted against each other and results from different liquids were compared. The origin of this effect remains unknown. The difference between the molar polarizabilities [@MolarPolarizability] of the two helium isotopes is much too small to account for this and the magnitude of the effect was not exactly the same for the two capacitors. Eventually pure $^4$He over the entire pressure range and a few $^3$He points adjusted to fit the $^4$He data were used for the calibrations. The maximum differences between pure $^3$He and pure $^4$He calibrations were 0.18 cm$^3/\mathrm{mol}$ and 0.12 cm$^3/\mathrm{mol}$ for the two IDC’s in the mixture molar volume range. The maximum relative discrepancy was thus 0.7%. The variation in molar volumes for the two calibrations as functions of capacitances were very similar for both capacitors, see Fig. \[Fig:IDCcalibration\].
![(Color online) Interdigital capacitor calibrations. The readings of the two capacitors were fitted to a known functional form obtained from the Clausius-Mossotti relation and the molar polarizability given by Ref. [@MolarPolarizability]. The fit range was between $v_\mathrm{m}=23.3$ cm$^3/\mathrm{mol}$ and $v_\mathrm{m}=29.8$ cm$^3/$mol. The solid curves are the final calibrations using pure $^4$He and a few adjusted $^3$He points. The $^3$He data are indicated by triangles and $^4$He by circles. The dashed curves are the original calibrations done in pure $^3$He. The experimental data at various pressures used for the calibrations were from Refs. [@Watson1969; @Tanaka2000; @Greywall1983; @Greywall1986; @Abraham1971]. The insets display the difference between the calibration data and the obtained fits.[]{data-label="Fig:IDCcalibration"}](IDCcalibration.eps){width="98.00000%"}
This indicates that the majority the effect was either due to the different helium isotopes used for the calibrations or an inconsistency between the literature values of the two pure liquid molar volumes. The $^3$He calibration gave consistently smaller molar volumes at a given capacitance compared to the $^4$He, with the difference being larger at smaller capacitances (larger molar volumes).
The capacitances $C$ were transformed to molar volumes $v_\mathrm{m}$ by using the Clausius-Mossotti relation $$\frac{\epsilon -1}{\epsilon +2}=\frac{4\pi \alpha_\mathrm{m}}{3 v_\mathrm{m}},$$ experimental data for the molar polarizability $\alpha_\mathrm{m}$ of helium [@MolarPolarizability], and using the expected capacitance for an interdigital capacitor $$\label{Eq:IDC}
C\approx N\epsilon_0 l \frac{\epsilon + \epsilon_\mathrm{s}}{2} .$$ In Eq. (\[Eq:IDC\]) $N$ is the number of digits, $\epsilon_0$ is the vacuum permittivity, $l$ is the length of the digits, and $\epsilon$ and $\epsilon_\mathrm{s}$ are the relative dielectric constants of the medium and the sapphire substrate, respectively. Two fit parameters for each capacitor, $N\epsilon_0 l$ and $\epsilon_\mathrm{s}$, were used. The fitted values of these parameters were credible based on the actual geometry and materials of the capacitors. The calibrations for the capacitors are represented in Fig. \[Fig:IDCcalibration\]. The measurement accuracy was better than 0.1 fF, while the density sensitivity was $dC/dv_\mathrm{m}=-40~\mathrm{fF/(cm^3/mol)}$ in the range of saturated helium mixture molar volumes.
A small but systematic difference between the two IDC’s as a function of pressure was observed. This difference corresponded to less than 0.02 cm$^3/$mol over the pressure range from 14 kPa to 3.36 MPa. The IDC’s were also observed to feature unexpected temperature dependencies, which differed between the two devices. The temperature dependencies were measured with an empty cell up to a temperature $T=180$ mK and with pure $^4$He at the saturated vapor pressure up to $T=750$ mK, where the molar volume is still practically unchanged. The other capacitor showed some variation even below 10 mK. Above 10 mK its capacitance increased as a function of temperature, reaching a maximum at about 30 mK, where it was 0.4 fF above the 10 mK value. After this the capacitance decreased and leveled off around 300 mK to a value, which was 0.3 fF below the 10 mK reading. For this one the temperature dependence with an empty cell and pure $^4$He did not completely agree, although the general features were similar. The capacitance of the other device decreased monotonically above 10 mK also reaching a stable value at around 300 mK, being 0.5 fF below the 10 mK value. If untreated, these small effects would have changed the molar volumes by 0.02 cm$^3/$mol, or 0.06%, at most, but they were subtracted from the data assuming the same temperature dependence also in mixtures.
Temperature was measured with a thermal noise thermometer, which was compared against a commercial calibrated germanium resistance thermometer. The Ge-calibration was checked against the $^3$He vapor pressure (ITS-90) [@0026-1394-27-1-002]. The noise thermometer was attached to the dilution refrigerator’s mixing chamber.
Results {#Sec:Results}
=======
Melting pressure
----------------
The melting pressure was measured by performing continuous temperature sweeps with solid and liquid helium in the cell, as well as melting previous solids and nucleating new ones at fixed temperatures. The solid growth and melting at fixed temperatures were controlled by the bellows. A typical re-nucleation was performed by melting the old solid away and decreasing the pressure few tens of kilopascals further. The bellows flow was then reversed and the cell was pressurized at a rate between about $1-2$ kPa/min. Usually the flow to the bellows was halted within a few minutes after nucleation. The appearance of the solid phase was observed as a stop in the increase of pressure as the bellows was pushed at a constant rate. Sometimes the solids were grown to larger sizes, even to cover the IDC’s entirely.
The measured melting pressure of the concentration-saturated helium mixture is given in Fig. \[Fig:Pm\].
![(Color online) Melting pressure of the saturated helium mixture as a function of temperature. The open circles represent new crystals nucleated at constant temperatures. The grey circles indicate the required pressures for nucleations. The triangles are for the pressures of metastable solids, which after some time relaxed to the equilibrium pressures. The dashed curves for the nucleation pressure and metastable pressure are guides to the eye. The black dots (forming a nearly full line) represent the continuous temperature sweeps. The horizontal dashed line is the melting pressure of pure $^4$He. The dash-dotted lines are the results of earlier calculations for the hcp and bcc crystal structures [@MeltPresCalc]. The quadruple point, where hcp solid, bcc solid, dilute liquid, and rich liquid phases coexist, is indicated by an arrow.[]{data-label="Fig:Pm"}](Pm.eps){width="98.00000%"}
It shows the results of new nucleations at constant temperatures and the continuous temperature sweeps. The continuous data have been averaged to reduce scatter. The required pressures for nucleations are also indicated. The melting pressure of pure $^4$He is plotted for reference. The two dash-dotted curves are the results of earlier calculations for the hcp and bcc phases [@MeltPresCalc]. The melting pressures of new nucleations and the required nucleation pressures are also given in Table \[Tab:Pm\].
[lll|lll]{} $T$ (mK) & $P_\mathrm{m}$ (MPa) & $P_\mathrm{n}$ (MPa) & $T$ (mK) & $P_\mathrm{m}$ (MPa) & $P_\mathrm{n}$ (MPa)\
10 & 2.567 & 2.574 & 273 & 2.635 & 2.676\
39 & 2.569 & 2.576 & 288 & 2.638 & 2.674\
144 & 2.591 & 2.596 & 295 & 2.640 & 2.675\
174 & 2.602 & 2.606 & 302 & 2.638 & 2.674\
175 & 2.602 & 2.605 & 329 & 2.626 & 2.658\
187 & 2.608 & 2.612 & 380 & 2.587 & 2.608\
201 & 2.613 & 2.618 & 411 & 2.552 & 2.565\
216 & 2.618 & 2.627 & 421 & 2.540 & 2.549\
230 & 2.623 & 2.645 & 425 & 2.532 & 2.542\
232 & 2.623 & 2.651 & 440 & 2.511 & 2.516\
238 & 2.627 & 2.669 & 442 & 2.510 & 2.516\
244 & 2.628 & 2.673 & 451 & 2.496 & 2.500\
245 & 2.627 & 2.672 & 455 & 2.488 & 2.494\
259 & 2.632 & 2.676 & 457 & 2.488 & 2.494\
As seen in Fig. \[Fig:Pm\], the continuous temperature sweeps give results very similar to new nucleations, even though the equilibrium $^3$He concentration in the solid phase varies greatly with temperature. We thus note that the liquid-solid interface remains in equilibrium at all times either because concentration gradients remain trapped inside the solid phase and cause no problems, or for the more unlikely reason of very fast $^3$He concentration relaxation within the solid. This is important for the practical use of the mixture melting pressure as a thermometer.
Low temperature expansion for the melting pressure of the solubility saturated helium mixture can be obtained by fitting a quartic polynomial to the shown data. Restricting this to temperatures below $T=140$ mK gives $P_\mathrm{m}/\mathrm{MPa} = 2.5663 + 1.2846 (T/\mathrm{K})^2 - 2.0651 (T/\mathrm{K})^4$. The fit was first performed using a second order polynomial below $T=60$ mK and the second order term was kept constant in the quartic fit. Odd powers have been omitted, since this is a Fermi system. The differences between the measured values and the fits are plotted in Fig. \[Fig:PdiffLowT\] together with the difference to the calculated pressure of Ref. [@MeltPresCalc].
![(Color online) Difference between the measured melting pressure of the saturated helium mixture and a quadratic fit, a quartic fit, and the calculation of Ref. [@MeltPresCalc]. The quadratic fit has been done using data up to $T=60$ mK and the quartic up to $T=140$ mK.[]{data-label="Fig:PdiffLowT"}](PdiffLowT.eps){width="98.00000%"}
In previous experiments at lower temperatures, the value of the quadratic fit parameter, which in our present analysis is 1.28, has been found as 1.1 [@LT24paper] or 0.92 [@EliasMeltingPressure].
At temperatures below $T=200$ mK the calculated hcp curve of Ref. [@MeltPresCalc] coincides with the experimental one, see Fig. \[Fig:Pm\]. Above this temperature the calculated pressures are continually below the measured ones. This includes the entire bcc branch. The hcp-bcc crossing is also predicted to exist at a lower temperature and pressure. This discrepancy is not very surprising. The model interaction potential used to calculate the chemical potential of $^3$He in the dilute liquid phase had been constructed at low temperatures and small concentrations [@Potential]. It was known to exhibit some degree of uncertainty at higher temperatures.
Around the maximum melting pressure, metastable solids were observed nucleations. These are indicated as triangles in Fig. \[Fig:Pm\]. The system could remain in these states for several hours before spontaneously relaxing to the equilibrium melting pressures. The relaxation rate could be accelerated by increasing the experimental volume by operating the bellows and thus melting the metastable fraction of the solid. The same pattern continues on the high-temperature side of the maximum pressure as well, but unfortunately the amount of $^3$He in that set of measurements was not enough for a saturated mixture and thus these data points have been omitted from Fig. \[Fig:Pm\]. After addition of $^3$He into the cell, no nucleations were performed around the maximum, as the attention was toward the higher-temperature region.
The metastable states were also seen in the temperature sweeps. Starting from one side of the maximum, the system would continue on the metastable upper pressure to the other side before collapsing to the lower equilibrium value. This is depicted in Fig. \[Fig:PmCrossing\].
![(Color online) Melting pressure data obtained by continuously changing the temperature with solid mixture in the cell. No averaging has been performed on these data. The temperature gauge used to plot this figure was a carbon resistor, whose calibration was not as accurate as that of the thermal noise thermometer, but which offered readings more frequently. The temperature scale thus differs slightly from Fig. \[Fig:Pm\]. (Mind also that unlike the noise thermometer this resistor was located inside the dilution refrigerator mixing chamber and during temperature changes the temperature inside the mixing chamber is not exactly the same as outside.) Two different phases cross at around $T=300$ mK. The arrows indicate future in time, when reaching the metastable states. Some of the data to the right of the crossing point is for a homogeneous system, as there was not enough $^3$He in the system to maintain saturation. The dashed line separates the saturated and homogeneous equilibrium systems.[]{data-label="Fig:PmCrossing"}](PmCrossing.eps){width="98.00000%"}
The metastable solids extended further to the lower temperature side than the higher, but due to the steeper descent of the pressure on the high temperature side, the maximum pressure difference was about the same. Since the amount of $^3$He in this case was not enough for a saturated mixture above the crossing point ($T\gtrsim300$ mK), it is possible that the system entered a supersaturated state upon cooling of the homogeneous mixture. The metastable behavior demonstrated in Fig. \[Fig:PmCrossing\] is a clear indication that the crossing pressure is the crossing point of two different phases; in this case the crossing of hcp and bcc solids. This point occurs at the maximum equilibrium pressure, which according to our measurements is $P = 2.638$ MPa at $T=294$ mK. This is the quadruple point consisting of dilute and rich liquid phases and the two solid phases. Calculations by Edwards and Balibar placed this quadruple point at $T=283$ mK and $P=2.63$ MPa [@EdwardsBalibar], measurements of Lopatik at $T=0.28$ K and $P=2.63$ MPa [@Lopatik], van den Brandt *et al.* at $T=0.30$ K and $P=2.63$ MPa [@Brandt1982], and Tedrow and Lee at $T=0.37$ K and $P=2.63$ MPa [@Tedrow]. The pressures have been very consistent between different authors, but there has been some discrepancy in the temperature. Our obtained pressure is slightly higher than that of the others, but the temperature fits in between earlier measurements.
Some of the nucleation events are plotted in Fig. \[Fig:Nucleations\].
![Nucleations of solid from helium mixture at eight temperatures. The time scale is the same for all figures and the zero moment represents the initial nucleation. Each figure displays a pressure range of 50 kPa. Metastable solids are seen in the figures for temperatures $T=245$ mK and $T=288$ mK. The horizontal dashed lines in each plot indicate the equilibrium melting pressure. For the $T=288$ mK solid, the system reached the equilibrium state 24 hours after the initial nucleation.[]{data-label="Fig:Nucleations"}](Nucleations.eps){width="100.00000%"}
At low temperatures the nucleations were quite fast. The relaxation was markedly slower at higher temperatures. Increasing $^3$He concentration in the solid phase at higher temperatures may have been a contributing factor to this apparent lag in the solid relaxation. More $^3$He must flow from the liquid phases to the solid at higher temperatures. Metastable solids are seen in these figures at temperatures $T=245$ mK and $T=288$ mK. The solid at $T=288$ mK was left on its own and the pressure began to drop from the metastable state about 19 hours after the initial nucleation event. The system reached the equilibrium pressure, which is indicated by a horizontal dashed line, five hours later.
The locations of nucleations in the cell could not be determined most of the time as the solid did not nucleate on the capacitors. This is slightly surprising, since the electric fields of the capacitors should ease nucleation. Above $T=180$ mK the solid nucleated on an IDC only once (at $T=230$ mK). At low temperatures the solid always nucleated on one of the capacitors and if that particular capacitor was not in use, nucleation occurred on the other IDC. However, the nucleation process seemed not to be related directly to the measurement AC voltage nor to the provisionally applied DC voltage, but rather to the on and off switching of the measurement voltage. The capacitors had to be measured alternately, because the two bridges used the same measurement frequency and there was crosstalk between them if active simultaneously. It seems, though, that the nucleations always occurred inside the main sample volume, since reasonable amount of solid growth brought it into the view of the capacitors each time this was done. The location of the pure $^3$He phase could not be observed. If it existed in a volume outside the main experimental cavity, some $^3$He currents in the capillaries must have occurred during growth or melting of the solid.
Fig. \[Fig:OverP\] depicts the required nucleation overpressure relative to the melting pressure.
![Difference between the nucleation pressure and melting pressure. The circles are for the equilibrium pressures and the triangles for the metastable solids. The dashed curves are guides to the eye. The vertical dashed line indicates the quadruple point ($T=294$ mK), where dilute and rich liquids and hcp and bcc solids coexist.[]{data-label="Fig:OverP"}](OverP.eps){width="98.00000%"}
The dashed line is a guide to the eye. The difference in the pressures peaks at about 250 mK, in the neighborhood of the hcp-bcc crossing (294 mK). At its highest value, the difference is almost 50 kPa. The maximum occurs in the range, where the metastable solids were created upon nucleation. The strong dependence of the nucleation overpressure on temperature is very prominent, but remains without explanation at this time.
Solid solution of helium isotopes can experience a phase separation into a $^3$He-dilute and $^3$He-rich phases similarly as liquid mixtures. The calculations presented in Ref. [@MeltPresCalc] indicated that the bcc solid at the melting pressure might undergo a separation into dilute and rich phases around $T=380$ mK. This would constitute another quadruple point. Vvedenskii reported having observed this at $T=380$ mK and $P=2.60$ MPa [@Vvedenskii]. Tedrow and Lee observed a pressure drop and warming at $T=0.25$ K while cooling, which was attributed to supercooling of the system [@Tedrow]. On warming they had a kink in pressure and brief cooling at $T=0.37$ K. Edwards and Balibar concluded from their calculations that the quadruple point does not quite exist [@EdwardsBalibar].
We searched for such possible quadruple point (rich and dilute liquids and bcc solids) in the temperature range of the experiment. Nothing indicated a sudden change in the rate of $^3$He accumulation into a growing solid at any temperature. We must therefore conclude that according to our measurements, the disputed quadruple point does not exist. It is quite possible that the prediction of its existence in our calculations was due to the already-mentioned inaccuracy of the model at those temperatures.
Molar volume
------------
The measured molar volume of the saturated dilute liquid phase is given in Fig. \[Fig:Vm\].
![(Color online) Molar volume of liquid helium mixture at the melting pressure as a function of temperature. The black symbols represent saturated mixture, whereas the grey points are for homogeneous mixtures with different concentrations. Somewhat crude estimate gives $x^\mathrm{L}=15$% for the lower and $x^\mathrm{L}=24$% for the higher concentration (see text). The inset shows the same data as a log-log plot with the zero-temperature value subtracted. The dotted curve and line are the power law fit to the data. A reproducible transition to a different level of molar volume was observed in the more dilute homogeneous liquid at around $T=400$ mK. This feature is shown in more detail in the dashed circle, which zooms in the region of interest. The two levels are indicated by the black lines.[]{data-label="Fig:Vm"}](Vm.eps){width="98.00000%"}
The black symbols represent the saturated mixture and the grey dots are for homogeneous liquids. Adjacent points have been averaged to reduce scatter. At higher temperatures, data with larger crystals have been omitted, because then systems with large amount of solid have a lower homogenization temperature. This is because the crystal contains relatively more $^3$He than the liquid. We performed measurements with two different $^3$He quantities, which is seen as two different homogenization temperatures ($T=311$ mK and $T=460$ mK). $^3$He was added by pressurizing the cell with about $7\%/10\%/15\%$ mixtures and then decreasing the pressure multiple times at low temperature, which resulted in accumulation of $^3$He into the low temperature cell. Therefore, the overall concentrations were not known to very good precision. Assuming a quadratic temperature dependence for the liquid saturation concentration and using values from Ref. [@Solubility], we find the total concentrations to have been $x^\mathrm{L}\sim 11$% and $x^\mathrm{L}\sim 15$%. The actual concentrations were undoubtedly larger, since the following terms in the polynomial expansion for $x^\mathrm{L}$ are positive. We will return to the issue of the true $^3$He content below.
The molar volume obeys the power law $v_\mathrm{m} \propto T^{3.5}$ remarkably well over the entire temperature range of the saturated system. This is demonstrated in the inset of Fig. \[Fig:Vm\], which shows a log-log plot of the data. The fitted power law, which is $v_\mathrm{m} = 23.453 + 9.663~(T/\mathrm{K})^{3.477}$ $\mathrm{cm^3/mol}$, is also shown in the figure.
The molar volume of a dilute liquid helium mixture $v_\mathrm{m}^\mathrm{L}$ is often written in the form $$\label{Eq:MolarVolume}
v_\mathrm{m}^\mathrm{L} = v_{40}^\mathrm{L}(1 + \alpha x^\mathrm{L}) ,$$ where $v_{40}^\mathrm{L}$ is the molar volume of pure liquid $^4$He, $x^\mathrm{L}$ is the $^3$He concentration, and $\alpha$ is the so-called Bardeen-Baym-Pines parameter, or excess volume parameter. Using our measured molar volume, we can evaluate $\alpha$ in the zero temperature limit. For this we need the molar volume of $^4$He extrapolated to the melting pressure of mixtures ($v_{40}^\mathrm{L}=23.160~\mathrm{cm^3/mol}$ at $P=2.566~\mathrm{MPa}$) [@Watson1969; @Tanaka2000] and the solubility at the melting pressure $x^\mathrm{L} = 8.12$% [@Solubility]. The result is $\alpha = 0.15\pm0.04$, which corresponds well with the extrapolated value $\alpha \approx 0.16$ of Watson *et al.* [@Watson1969]. We could use our data to find $\alpha$ over the entire measured range of melting pressure, but the solubility of $^3$He in $^4$He at the melting pressure has not been determined as a function of temperature beyond some tens of millikelvins. We can, however, turn this consideration around and find better estimates for the maximum liquid concentrations in our experiments by using the molar volume data. We now use the obtained zero-temperature value of $\alpha$ and neglect its temperature dependence as a minor effect [@PhysRev.188.309; @PhysRevB.67.094503]. We further extrapolate the pressure dependence of this parameter from the data of Watson *et al.* [@Watson1969]. We find $x^\mathrm{L}=15$% for the more dilute system and $x^\mathrm{L}=24$% for the more concentrated one. These are in accordance with the supposition that the true concentrations were higher than obtained by assuming just a quadratic temperature dependence for the solubility.
From the slopes of the homogeneous cases $dv_\mathrm{m}/dT$, Eq. (\[Eq:MolarVolume\]), and the measured melting pressure we can compute $d x^\mathrm{L}/dT$. The result at $T=325$ mK is $dx^\mathrm{L}/dT \approx 3\%/\mathrm{K}$ and at $T=460$ mK it is $dx^\mathrm{L}/dT \approx -32\%/\mathrm{K}$. This means that at the lower temperature a growing solid takes more $^4$He compared to the concentration of the dilute liquid phase and thus tends to slightly concentrate the liquid. At the higher temperature the situation is reversed, and the solid takes much $^3$He, diluting the liquid. This is exactly what is expected, since the equilibrium concentration in the solid phase increases much faster as a function of temperature compared to the liquid [@MeltPresCalc].
On the lower homogeneous branch at approximately $T=400$ mK a transfer to a different level of molar volume is seen. It was reproducible when going both up and down in temperature. The significance of this effect is not clear.
Discussion
==========
We measured the melting pressure of the saturated helium mixture between $T=10$ mK and $T=460$ mK. For helium mixtures to be used as a thermometric standard, further measurements would be desirable for a consistency check, preferably with a differential pressure gauge using pure $^4$He as reference.
The effect of the equilibrium crystal structure changing between hcp and bcc was clearly observed in the melting pressure data producing a quadruple point on the curve. We found no evidence for the existence of another quadruple point consisting of dilute and rich liquid phases and dilute and rich bcc phases, which had been indicated by an earlier calculation and one other experiment.
Some features were observed, which are merely reported without explanation at this time. The nucleation overpressure has a strong dependence on temperature, attaining its maximum in the region of the maximum melting pressure, where the hcp-bcc coexistence point exists. On both sides of the maximum pressure, long-living metastable solids were first nucleated before an equilibrium pressure was reached. Finally, a small feature in the molar volume of homogeneous dilute liquid phase with coexisting solid phase seems to exist at around $T=400$ mK for $x^\mathrm{L}\approx 15$%.
This work has been supported in part by the EU 7th Framework Programme (FP7/2007-2013, Grant No. 228464 Microkelvin) and by the Academy of Finland through its LTQ CoE grant (project no. 250280). We also acknowledge the National Doctoral Programme in Materials Physics for financial support. We thank A. Sebedash and I. Todoshchenko for useful discussions.
|
---
abstract: 'We applied image stacking on empty-field Faint Images of the Radio Sky at Twenty-Centimeters (FIRST) survey maps centred on optically identified high-redshift quasars at $z\geq4$ to uncover the hidden $\mu$Jy radio emission in these active galactic nuclei (AGN). The median stacking procedure for the full sample of $2229$ optically identified AGN uncovered an unresolved point source with an integrated flux density of 52 $\mu$Jy, with a signal-to-noise ratio $\sim10$. We co-added the individual image centre pixels to estimate the characteristic monochromatic radio power at $1.4$ GHz considering various values for the radio spectral index, revealing a radio population with $P_\mathrm{1.4GHz}\sim10^{24}$ W Hz$^{-1}$. Assuming that the entire radio emission originates from star-forming (SF) activity in the nuclear region of the host galaxy, we obtained an upper limit on the characteristic star formation rate, $\sim4200$ M$_\odot$ yr$^{-1}$. The angular resolution of FIRST images is insufficient to distinguish between the SF and AGN origin of radio emission at these redshifts. However, a comparison with properties of individual sources from the literature indicates that a mixed nature is likely. Future very long baseline interferometry radio observations and ultra-deep Square Kilometre Array surveys are expected to be sensitive enough to detect and resolve the central $1-10$ kpc region in the host galaxies, and thus discriminate between SF and AGN related emission.'
author:
- |
Krisztina Perger,$^{1,2}$[^1] Sándor Frey,$^{2}$ Krisztina É. Gabányi,$^{3,2}$, L. Viktor Tóth$^1$\
$^{1}$Department of Astronomy, Eötvös Loránd University, Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary\
$^{2}$Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly Thege Miklós út 15-17, H-1121 Budapest, Hungary\
$^{3}$MTA-ELTE Extragalactic Astrophysics Research Group, Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary
date: 'Accepted 23 September 2019. Received 30 August 2019; in original form 7 June 2019'
title: 'Unveiling the weak radio quasar population at $z\geq4$'
---
\[firstpage\]
galaxies: active – radio continuum: galaxies – galaxies: star formation – galaxies: high-redshift – quasars: general – methods: data analysis
Introduction
============
Since the discovery of the first quasar at redshift $z\geq4$ [@1987Natur.325..131W], the number of active galactic nuclei (AGN) known at the highest redshifts is continuously increasing. As a result of extensive observing campaigns and surveys [e.g., @2000AJ....120.1579Y; @2010AJ....140.1868W; @2011AJ....142...72E; @2016arXiv161205560C; @2016MNRAS.460.1270D; @2017AJ....154...28B], there are nearly $3000$ sources identified between $z=4$ and $z=7.54$ to date, yet only $\sim6.5$ per cent of these exhibit radio emission [@2017FrASS...4....9P] detected at 1.4 GHz by either the Very Large Array (VLA) Faint Images of the Radio Sky at Twenty-Centimeters[^2] [FIRST, @1995ApJ...450..559B; @2015ApJ...801...26H] survey or the National Radio Astronomy Observatory (NRAO) VLA Sky Survey[^3] [NVSS, @1998AJ....115.1693C]. The remaining sources are either outside the footprint of both FIRST and NVSS, or are below the detection limit of the two surveys.
Several studies are available in the literature on the overall radio luminosity function (LF) for AGN . Many authors found evidence for bimodality in the distribution of the AGN population [e.g., @1980AnA....88L..12S; @1989AJ.....98.1195K; @1999ApJ...511..612G; @2007ApJ...654...99W], dividing the population to distinct samples of radio-loud and radio-quiet objects. The validity of this dichotomy was recently challenged and found to be the result of observational and mathematical bias effects [@2008MNRAS.387..856Z; @2012ApJ...754...12M; @2012ApJ...759...30B; @2013ApJ...768...37C]. The weak radio emitters with flux densities below the detection threshold ($\sim1$ mJy for FIRST and $\sim2.5$ mJy for NVSS) might also exhibit radio jets at parsec scales [e.g., @2017ApJ...835L..20W], but generally remain silent and hidden in flux density limited surveys due to their low radio power and large distance. Faint objects with radio flux densities below the FIRST detection limit can either host radio-quiet AGN, or harbour radio-loud quasars that reside at higher redshifts.
The lurking low- and modest-power members of the high-redshift AGN (hAGN) population can be uncovered by going below the survey threshold and estimating their typical flux densities by means of radio image stacking [e.g., @2007ApJ...654...99W]. Numerous works with stacking analysis are available in the literature on various samples of radio-weak quasars. Analysis of stacked FIRST images at $\sim 8000$ radio-quiet quasar positions from the 2dF QSO redshift survey at medium redshifts ($z\lesssim2.3$) resulted in median flux density levels between $20$ and $40~\mu$Jy [@2005MNRAS.360..453W]. @2008AJ....136.1097H also found 10s of $\mu$Jy flux densities for a mixed set of Sloan Digital Sky Survey (SDSS) and Luminous Red Galaxy samples. Stacking of extremely red quasars at $2\leq z\leq 4$ using VLA maps at $1.4$ and $6.2$ GHz yielded similar values [@2018MNRAS.477..830H].
There is a debate on the nature of physical processes responsible for the radio emission at the faint end ($P_\mathrm{1.4GHz}\leq10^{22.5-23}$ W Hz$^{-1}$) of the continuous radio LF, with possible contribution of AGN jets [@1997ARAnA..35..607Z; @2018ApJ...869..117R], AGN winds [@2010ApJ...711..125J; @2014MNRAS.442..784Z; @2016MNRAS.455.4191Z; @2018ApJ...869..117R], and enhanced star formation or starburst events in the host galaxy [@2004MNRAS.352..399J; @2013ApJ...768...37C; @2013ApJ...778...94R; @2018MNRAS.477..830H]. The origin of radio emission in radio-quiet AGN was also hypothesized to be driven by magnetically heated accreation disk coronae [@2008MNRAS.390..847L; @2019MNRAS.482.5513L]. Upper limits on star formation rate (SFR) determined in stacking studies vary between a few to $10$ M$_\odot$ yr$^{-1}$ [e.g., @2007AJ....134..457D; @2008AJ....136.1097H]. In other works, low-luminosity AGN jets and circumnuclear star-forming regions in the host galaxies together are considered responsible for the radio emission [e.g., @2011ApJ...730...61K; @2011ApJ...742...45P]. It was also found that the radio LF usually peaks at higher redshifts for radio-loud sources than for radio-quiet AGN [@1999ApJ...511..612G; @2017ApJ...842...87M]. A multi-wavelength study of a radio-selected Cosmic Evolution Survey (COSMOS) field sample ($S_\mathrm{1.4GHz}\geq37~\mu$Jy) at $z\leq6$ suggested star formation as the dominant process for the sub-mJy radio population, explaining the dichotomy with AGN activity modes and connection to AGN–host galaxy feedback [@2017AnA...602A...3D]. Another study using a sample with $S_\mathrm{1.4GHz}\geq11.5~\mu$Jy applied spectral energy distribution fitting and led to similar conclusions: the peak of the radio LF is at $z\sim2$ [@2018AnA...620A.192C]. The radio loudness dichotomy was addressed also at low radio frequencies [$120-168$ MHz, @2019AnA...622A..11G], concluding that low-power quasars are dominated by star formation.
In this paper, we utilised $2229$ empty-field FIRST radio maps centred at positions of hAGN (quasars) identified in the optical or near-infrared. We analysed them with mean and median stacking methods to uncover the underlying radio quasar population, and to examine the possibility of the radio emission originating from either AGN activity or star formation. The aspects of sample selection are detailed in Section \[sec:sample\]. We describe the stacking procedure in Section \[sec:stacking\]. Results and derived properties are discussed in Section \[sec:discussion\]. We summarise our findings and conclude the paper in Section \[sec:summary\].
Throughout this work, we assumed a standard $\Lambda$CDM cosmological model with $\Omega_\mathrm{m}=0.3, \Omega_{\Lambda}=0.7$, and $H_0=70$ km s$^{-1}$ Mpc$^{-1}$ for calculations.
Sample selection and data {#sec:sample}
=========================
The sample for the stacking analysis was defined by using the high-redshift ($z \geq 4$) AGN catalogue by @2017FrASS...4....9P[[^4]]{}. We selected optically identified AGN from the catalogue with positions falling into the FIRST survey footprint. The flux density of the objects selected is below the detection limit of the survey (typically $\sim1$ mJy). At the time of the analysis, the total number of such AGN was $2232$. Intensity fluctuations (rms noise) at the optical AGN positions in the FIRST images vary in the range from tens of $\mu$Jy to $\lesssim 1$ mJy. To test an underlying high-redshift radio AGN population, we stacked FIRST radio maps centred at these positions. The images in Flexible Image Transport System (FITS) format [@1981AnAS...44..363W] were obtained from the FIRST image cutout service[^5]. We downloaded images of $4\farcm5\times4\farcm5$ size for each individual AGN position. After excluding the largely incomplete or entirely ragged maps, we obtained $2229$ images for the stacking process. The redshift distribution of the stacked sample is shown in Fig. \[fig:histogram\], and the first 5 entries of the list of $2229$ AGN are given in Table \[tab:sample\]. The full list is available in the supplementary material.
![Redshift distribution of the $2229$ high-redshift active galactic nuclei used for the stacking analysis.[]{data-label="fig:histogram"}](histogram.eps){width="\linewidth"}
Name R.A. (h m s) Dec ($\degr~'~''$) Redshift Reference
---------------------------- --------------- -------------------- ---------- ----------------------
SDSS J000046.69+010951.2 $00~00~46.69$ $+01~09~51.24$ 4.25 @2012AnA...548A..66P
SDSS J000124.23+111212.6 $00~01~24.23$ $+11~12~12.69$ 4.30 @2017AnA...597A..79P
SDSS J000404.71+000039.0 $00~04~04.71$ $+00~00~39.08$ 4.31 @2012AnA...548A..66P
SDSS J000457.11$-$000538.7 $00~04~57.11$ $-00~05~38.78$ 4.05 @2012AnA...548A..66P
SDSS J000527.14+025813.2 $00~05~27.15$ $+02~58~13.29$ 4.11 @2012AnA...548A..66P
\
[*Notes.*]{} Column 1 – object name, Column 2 – right ascension, Column 3 – declination, Column 4 – spectroscopic redshift, Column 5 – literature reference of discovery. The full table of $2229$ AGN is available in the electronic version of the journal. \[tab:sample\]
Stacking analysis {#sec:stacking}
=================
Stacking of catalogue sources
-----------------------------
The sample of $2229$ objects were binned based on the redshift of each source into four subsamples with approximately equal number of images in each bin. There are $554$, $559$, $559$, and $557$ objects in the four bins with redshift boundaries of $4.0\leq z_1<4.1,4.1\leq z_2<4.3,4.3\leq z_3<4.7,$ and $4.7\leq z_4<7.6$, respectively.
We performed the stacking in both mean and median procedures. Both the mean [e.g. @2007ApJ...654...99W; @2007AJ....134..457D; @2008AJ....136.1097H] and median [e.g. @2007ApJ...654...99W; @2007AJ....134..457D; @2008AJ....136.1097H; @2018MNRAS.477..830H; @2018ApJ...869..117R] stacking methods are commonly used in the literature. Mean stacking determines the arithmetic average of intensities in the image pixels within the sample. However, this method is very sensitive to outlying values in the sample, as well as to the threshold set to avoid the contaminating point sources. In turn, median stacking deals with the median of intensity values corresponding to the image pixels in the sample.
To evaluate the applicability of maps resulted from both methods, we calculated root-mean-square (rms) noise, maximum of the intensity, and signal-to-noise ratio (SNR) for both the mean and median stacked images in each bin and for the full sample. Typical rms noise values in the separate redshift bins span the range $10-20~\mu$Jy beam$^{-1}$ for mean stacked images, while median stacking resulted in $\sim 7\mu$Jy beam$^{-1}$ noise levels. The full-sample maps have $8~\mu$Jy beam$^{-1}$ and $3~\mu$Jy beam$^{-1}$ rms noise for the mean and median methods, respectively.
Bin rms \[$\mu$Jy beam$^{-1}$\] $I_\mathrm{max} $ \[$\mu$Jy beam$^{-1}$\] SNR
----- ----------------------------- ------------------------------------------- -----
1 7 27 4
2 7 52 7
3 7 38 5
4 7 30 4
All 3 35 11
: Properties of median-stacked maps.[]{data-label="tab:data"}
\
[*Notes.*]{} Column 1 – redshift bin, Column 2 – image noise, Column 3 – maximum intensity, Column 4 – signal-to-noise ratio
Despite the high SNR values (up to $50$) found in each mean-stacked map, the method provides insignificant results in the search of a hidden central source. The maps are contaminated and dominated by peaks at random off-centre locations, caused by strong intensity peaks from the stacked individual maps. The presence of off-centre sources leads to an increase of the rms image noise in the field compared to median stacking. Therefore mean-stacked maps are not considered in the further study and calculations.
Median maps on the contrary are not sensitive to occasional bright off-centre sources. They show a nearly uniform noise level in each subsample ($7~\mu$Jy beam$^{-1}$), revealing a protruding radio peak at the image centre for the entire sample with SNR exceeding 10. Intensity maxima for the median-stacked images are found to be $27, 52, 38$ and $30~\mu$Jy beam$^{-1}$ for the redshift-binned data, and $35~\mu$Jy beam$^{-1}$ for the full sample. Radio maps obtained by median stacking are shown in Fig. \[fig:bins\], for which the calculated image properties are listed in Table \[tab:data\], while the radial profile of the full-sample median-stacked image is illustrated in Fig. \[fig:radial\].
The image rms noise reached by stacking is expected to decrease with respect to the single-image rms by $\sqrt N$, where $N$ is the number of stacked images. The original FIRST image noise levels are $\approx 0.15$ mJy. For the sample of $N=2229$ objects, this predicts the rms of $150~\mu$Jy beam$^{-1}$ divided by $\sqrt{2229}$, i.e. $\approx 3~\mu$Jy beam$^{-1}$ to be reached with stacking. Indeed, this is equal to the actual rms values obtained for the full-sample median-stacked images.
\
![Pixel-by-pixel radial plot of the full-sample median-stacked intensity values for the inner $15''$ radius area. 1 pixel equals $1.8''$. For the data points, the unbiased values are shown, after applying the bias correction factor $1.4$ [@2007ApJ...654...99W].[]{data-label="fig:radial"}](radial_all_in_unbiased.eps){width="\linewidth"}
Stacking of fake sources and the image noise
--------------------------------------------
To check the reliability of our results, we created four samples of the same size, with arbitrary source coordinates in the manner of adding or subtracting $1\degr$ to/from either or both the right ascensions and declinations of objects in the the original sample. Binning was based on the redshifts of AGN in the original (true) positions.
We did not find any radio emission in either of the four samples of fake sources, neither with mean, nor with median stacking methods. The mean images show outlier sources outside the central pixels, similarly to the results obtained for the true sample. The rms noise levels of the binned subsample maps are $10-20~\mu$Jy beam$^{-1}$ and $\sim7~\mu$Jy beam$^{-1}$ for the mean and median stacking, respectively. Stacking of all images resulted in rms values comparable to that of the real sample and consistent with the expected values from the stacking procedure, $8~\mu$Jy beam$^{-1}$ for mean and $3~\mu$Jy beam$^{-1}$ for median maps. The SNR values for the mean images vary from $16$ to $50$ in the separate redshift bins, and are below $\approx 30$ when the full fake samples were included in mean stacking. The image peaks are caused by occasional bright off-centre sources appearing in some fields, similarly to the real sample. For median stacking, the typical values in the fabricated samples and in their subsets are in the range of $4\lesssim \mathrm{SNR}\lesssim5$, and indicate no significant detection at the central location, as expected. Due to the similar SNR values in the samples of fake sources, the binned data were not considered for further analysis, only the full sample.
Model fit to the stacked data
-----------------------------
For further analysis, we fitted a circular Gaussian model component to the central region of the full-sample median-stacked image, to characterise the brightness distribution using the [imfit]{} task in the U.S. National Radio Astronomy Observatory (NRAO) Astronomical Image Processing System[^6] [[aips]{}, @2003ASSL..285..109G]. This resulted in an unresolved point source. By multiplying the fitted model component flux density with the bias correction factor of $1.4$ suggested by @2007ApJ...654...99W, we obtained an unbiased value of $S_\mathrm{1.4GHz}=52\pm1\,\mu{\mathrm Jy}$.
Characteristic 1.4-GHz radio power
-----------------------------------
To derive a characteristic value of the monochromatic rest-frame 1.4-GHz radio power for the hAGN population, we co-added the individual FIRST image cutouts around the positions of the full sample of $2229$ objects that are individually not detected in the FIRST survey. Motivated by the model-fitting results of the median stacking, we assumed that the radio emission originates from point sources unresolved in FIRST. Thus the flux density was derived from the cumulative brightness value of the central pixel for the following analysis. We consider that the co-added flux density of the $2229$ hAGN is the sum of the flux densities of the individual sources, and that all hAGN included in the stacking process have the same radio power at $1.4$ GHz, $P_\mathrm{1.4GHz}$. This latter assumption is obviously not true, but the value derived this way is characteristic to the hAGN sample as a whole.
We used the following relationship between flux density and power (including $K$-correction): $$S_\mathrm{1.4GHz,sum}=\sum_{i=1}^{N}\frac{P_\mathrm{1.4GHz}}{4\pi D_{\mathrm{L}i}^2(1+z_i)^{-\alpha-1}},$$ where $S_\mathrm{1.4GHz,sum}=77$ mJy is the cumulative flux density derived from co-adding the $N=2229$ FIRST map centres, $\alpha$ is the radio spectral index (using the convention $S_\nu\propto\nu^\alpha$, where $\nu$ is the frequency), and $D_{\mathrm{L}i}$ and $z_i$ are the luminosity distance and the redshift of the $i$-th individual source, respectively. Since the individual power-law spectral indices are unknown for the sources, we assumed a common value for the whole sample. Various radio spectral indices appropriate for galaxies are found in the literature [e.g. @2003ApJ...599..971H; @2007AnA...462..525H; @2007ApJ...654...99W; @2018MNRAS.477..830H]. Therefore we calculated the characteristic power considering several different values in the range $-0.5\leq\alpha\leq-1$, corresponding to steep radio spectra. The estimated values of the $1.4$-GHz power for each spectral index are listed in Col. 2 in Table \[tab:powsfr\].
Upper limit on star formation
-----------------------------
Radio emission from galaxies may originate from AGN activity in the nuclear region where synchrotron radiation is produced in powerful relativistic plasma jets driven by accretion onto the central supermassive black hole. On the other hand, synchrotron emission from relativistic electrons and free-free emission from ionized hydrogen regions may be widespread in the AGN host galaxy as a consequence of recent start formation .
We can assume that the radio emission in our stacked hAGN sample originates solely from star formation taking place around the central regions of the host galaxy, rather than from AGN activity. This way we can calculate an upper limit on the star formation rate (SFR) using the estimated characteristic $1.4$-GHz power value, applying various correlations between the radio power and SFR from the literature [@2003ApJ...586..794B; @2006ApJ...643..173S; @2017MNRAS.466.4917D; @2019MNRAS.482..560M]. The SFR values obtained are listed in Cols. 3–6 in Table \[tab:powsfr\]. The upper limits on the SFR span about an order of magnitude, depending on the radio spectral index assumed and the correlation used. But we can conclude that the typical SFR upper limits are in the order of $10^3$ M$_\odot$ yr$^{-1}$.
-------- ------- -------- -------- ------- --------
(1) (2) (3) (4)
$-0.5$ $2.9$ $1600$ $1800$ $370$ $1700$
$-0.6$ $3.4$ $1900$ $2100$ $420$ $2000$
$-0.7$ $4.1$ $2300$ $2500$ $480$ $2400$
$-0.8$ $4.8$ $2700$ $3000$ $540$ $2800$
$-0.9$ $5.7$ $3200$ $3500$ $610$ $3400$
$-1 $ $6.8$ $3700$ $4200$ $700$ $4000$
-------- ------- -------- -------- ------- --------
: Estimated characteristic 1.4-GHz radio powers and star formation rates for the radio sources associated with the central pixels of the co-added $2229$ FIRST maps.
\[tab:powsfr\]\
[*Notes.*]{} Column 1 – assumed spectral index, Column 2 – 1.4-GHz radio power, Column 3–6 – star formation rates calculated using various relationships by (1) @2003ApJ...586..794B, (2) @2006ApJ...643..173S, (3) @2017MNRAS.466.4917D, and (4) @2019MNRAS.482..560M
Discussion {#sec:discussion}
==========
Flux densities
--------------
The flux density of $52~\mu$Jy found in our median stacking analysis is comparable but somewhat higher than the values obtained in previous stacking works [e.g. @2005MNRAS.360..453W; @2007ApJ...654...99W; @2008AJ....136.1097H; @2018MNRAS.477..830H]. Considering that our sources are located at higher redshifts, this suggests that the hAGN are generally intrinsically more powerful than the sources studied in those samples.
For comparison, the ultraluminous quasar SDSS J010013.02+280225.8 at $z=6.3$ – not included in our stacking analysis due to its position outside the FIRST survey footprint – was detected with the VLA at a higher frequency, 3 GHz ($0.65''$ resolution and $\sim3~\mu$Jy beam$^{-1}$ sensitivity), with a flux density of $\sim100~\mu$Jy [@2016ApJ...830...53W]. Assuming a spectral index of $-0.7$, this scales to $\sim170~\mu$Jy at $1.4$ GHz. Further high-resolution observations performed with the Very Long Baseline Array (VLBA) at $1.5$ GHz revealed a radio structure partially resolved at $\sim 10$ milli-arcsecond (mas) level, with $\sim90~\mu$Jy flux density [@2017ApJ...835L..20W]. The detection of a mas-scale radio structure in this individual source raises the possibility that the FIRST-undetected sources in our stacked sample could also be targeted with sensitive very long baseline interferometry (VLBI) observations in the future, in a hope of revealing weak compact AGN-related radio emission. It also suggests that the radio emission in at least some of the sources in our $2229$-element sample does have synchrotron AGN jet contribution and is not associated with star formation only.
Radio power and AGN radio emission
----------------------------------
All estimated values for the $1.4$-GHz radio power are $P_\mathrm{1.4GHz}\sim10^{24}$ W Hz$^{-1}$ (Table \[tab:powsfr\]) in this study. If the radio emission in these objects would originate solely from AGN activity, the FIRST-undetected hAGN would be among the radio AGN population ($P_\mathrm{1.4GHz}\geq2\times10^{22.5}$ W Hz$^{-1}$), since powers exceed the threshold found in empirical studies between supernova-related and AGN activity [$P_\mathrm{1.4GHz}=2\times10^{21}$ W Hz$^{-1}$, @2000ApJ...530..704K; @2011AnA...526A..74M]. The estimated radio powers also surpass values measured for prominent starburst galaxies, e.g. Arp220, Arp229A, and Mrk273 [$P_\mathrm{1.4GHz}=2-4.5\times10^{22}$ W Hz$^{-1}$, @2012MNRAS.423.1325A].
Monochromatic powers found in this work are 2 to 5 orders of magnitude higher than those found for low-power sources at low redshifts ($z<0.3$), e.g. Lyman-break analogues (LBAs) including an LBA with dominant star formation (J0150+1308), an AGN (J1029+4829) and an AGN–SFR composite source (J0921+4509) [@2012MNRAS.423.1325A]. Also, low-redshift AGN SDSSJ1155+1507, SDSSJ2104$-$0009, SDSSJ2304$-$0933 [@2016ApJ...826..106G], and NGC3147 [@2004ApJ...603...42A] exhibit radio powers in the range of $5\times10^{21}$ W Hz$^{-1}$ to a few times $10^{22}$ W Hz$^{-1}$, or even lower powers between $10^{19}$ W Hz$^{-1}$ to a few times $10^{20}$ W Hz$^{-1}$ in the AGN Henize2-10 [@2012ApJ...750L..24R], NGC4203, NGC4535 [@2002ApJ...581..925U; @2004ApJ...603...42A], NGC864, and NGC4123 [@2002ApJ...581..925U]. It was discussed in e.g. @2005MNRAS.362...25B [@2008ASPC..399..413S; @2013MNRAS.433.2647S] and @2016MNRAS.455.2731R that the most luminous AGN have radio powers in the range from $10^{24}$ W Hz$^{-1}$ to a few times $10^{26}$ W Hz$^{-1}$, which is comparable with our results. These studies of low-redshift AGN found that host galaxies with $1.4$-GHz powers above $10^{23}$ W Hz$^{-1}$ usually host radio-loud AGN in their centres. This outlines that AGN activity represents a significant contribution to the 1.4 GHz radio flux densities in the stacked sample.
Deep radio surveys at high redshift, such as the Extended Chandra Deep Field-South [E-CDFS, @2008ApJS..179..114M; @2013ApJS..205...13M] survey and the VLA-COSMOS 3 GHz Large Project can provide additional information about the most distant galaxy centres. Our estimated radio powers ($\sim 10^{24}$ WHz$^{-1}$) are supported by the results achieved by e.g. @2017AnA...602A...3D, , and . In a multiwavelength analysis of $z\leq6$ radio sources in the COSMOS field at 3 GHz, @2017AnA...602A...3D divided the sample into three populations: radio-quiet (RQ) and radio-loud (RL) AGN, and star-forming galaxies (SFGs). The majority of RQ AGN hosts show enhanced star formation. Similar results were obtained by , revealing that radio sources at $1.4$ GHz with flux densities above $\sim200~\mu$Jy are predominantly AGN, and that with decreasing flux densities the SFGs take over as the dominant population in up to $\sim60$ per cent of the sample. However, the $1.4$ GHz luminosity functions determined for AGN with radio excess (with respect to the expected contribution of SF to the radio emission) using different procedures by showed that rest-frame radio powers span the range of $\sim 10^{24}-10^{26}~$WHz$^{-1}$ for AGN at the highest redshifts, when relying solely on observational data, and $10^{22}-10^{27}~$WHz$^{-1}$ with the application of evolution models. The fractional distribution found in the COSMOS field studies would imply an RQ–SFG dominance in the sample. Our estimated radio powers suggest that the objects in the stacking analysis are mostly radio-loud AGN. Another study of radio sources below $\sim100~\mu$Jy based on the E-CDFS survey concluded that besides the dominance of SFGs and the decrase of the RL AGN towards the lowest flux densities, the number of RQ AGN increases [@2013MNRAS.436.3759B]. They also determined radio power distributions for all three populations, and found median values of both RQ and RL AGN which coincide with our estimated radio powers, while the power at the peak of the distribution for SFGs is an order of magnitude lower.
Star formation rate
-------------------
SFR upper limits derived in this paper (Table \[tab:powsfr\]) are two to three orders of magnitude higher than SFR values found for individual AGN host galaxies in the literature, e.g. $0.5-2$ M$_\odot$ yr$^{-1}$ for SDSSJ2104$-$0009 and SDSSJ2304$-$0933 [@2016ApJ...826..106G], $5-8$ M$_\odot$ yr$^{-1}$ for J0150+1308, J0921+4509, and J1029+4829 [@2012MNRAS.423.1325A], and $\sim1$ M$_\odot$ yr$^{-1}$ with H$_\alpha$ star formation rates predicted by @2002ApJ...581..925U, as well as SFR with upper limits found in previous works using image stacking [$\leq10$ M$_\odot$ yr$^{-1}$, e.g. @2007AJ....134..457D; @2008AJ....136.1097H]. Even ultraluminous infrared galaxies were found to show star formation with rates varying between a couple of tens to a few hundreds of M$_\odot$ yr$^{-1}$ [e.g. @2010ApJ...715..572H], suggesting that our upper limits appreciably overestimate the contribution of star formation to the 1.4-GHz radio power.
However, turning to high-redshift objects, CO and \[C II\] line detection and dust emission in individual $z\sim6$ quasars indicate significant star formation in the central few kpc region in some host galaxies, with SFR in the order of $\sim1000$ M$_\odot$ yr$^{-1}$ [e.g. @2003AnA...409L..47B; @2013ApJ...770...13W; @2013ApJ...773...44W; @2019ApJ...876...99S]. Note that those AGN with an estimated SFR of a few thousand M$_\odot$ yr$^{-1}$ have not been included in our stacked sample because they are outside the FIRST survey coverage. For some individual objects that are included in our FIRST image stacking, molecular and atomic line observations determined a wide range of SFR values. These start from a few tens of M$_\odot$ yr$^{-1}$, e.g. $48$ M$_\odot$ yr$^{-1}$ for CFHQSJ0210$-$0456, $<40$ M$_\odot$ yr$^{-1}$ for CFHQSJ2329$-$0301 [@2013ApJ...770...13W], and $\sim80$ M$_\odot$ yr$^{-1}$ for CFHQSJ0055+0146 [@2015ApJ...801..123W]. SFR values of a couple of hundreds M$_\odot$ yr$^{-1}$ were found for VIKINGJ2348$-$3054 and VIKINGJ0109$-$3047 [$\sim700$ and $\sim900$ M$_\odot$ yr$^{-1}$, respectively, @2016ApJ...816...37V], ULASJ1120+0641 [$\sim200$ M$_\odot$ yr$^{-1}$, @2017ApJ...837..146V], and SDSSJ0100+2800 [$\sim650$ M$_\odot$ yr$^{-1}$, @2016ApJ...830...53W]. There are also examples of sources with SFR up to thousands of M$_\odot$ yr$^{-1}$ [@2004ApJ...615L..17W; @2019ApJ...874L..30V].
The difference between the SFR upper limits we estimated as characteristic for the stacked hAGN sample ($\lesssim4000$ M$_\odot$ yr$^{-1}$, Table \[tab:powsfr\]) and the values measured for individual sources using independent methods is naturally explained if we relax the assumption that the entire radio emission comes from star forming activity in the host galaxies, without AGN contribution. Moreover, we calculated with a single characteristic radio power for all sources. In reality, the physical conditions in the individual objects are clearly more complex than just assuming a constant power, and one or the other process solely responsible for the radio emission. In fact, radio emission originating from both AGN and star formation must be present in most objects, in various proportions.
In a hope to refine the results, one may take the AGN radio luminosity function (LF) into account, up to the highest redshifts . Improvements in determining the radio LF at the earliest cosmological epochs are expected from the currently ongoing $2-4$ GHz VLA Sky Survey [VLASS, @2015fers.confE...6M]. It will have higher sensitivity ($\sim70~\mu$Jy for the combined 3-epoch observations) and angular resolution ($2\farcs5$) compared to the FIRST survey. Also, repeating a stacking analysis similar to the one reported here, the deeper VLASS radio maps would allow for determining redshift-dependent properties of the FIRST-undetected AGN. Our study with binned sub-samples did not provide conclusive results because of the insufficient SNR in the stacked images.
Origin of the sub-mJy radio emission
------------------------------------
Ascertaining the role of SF and AGN contribution to the sub-mJy radio emission could be aided by the application of methods independent from the FIRST observations. Based on mid-infrared (MIR) polycyclic aromatic hydrocarbon (PAH) detections in low-redshift galaxies, PAH emission lines can be found very close to the galaxy core, at $\sim1-20$ kpc distances [e.g., @2007ApJ...669..841S; @2019ApJ...871..190M]. Since star formation indicators are found this close to the core, and our stacking analysis showed that the fitted Gaussian model component is unresolved in FIRST (corresponding to $26-35$ kpc linear size depending on the redshift), the angular resolution provided by FIRST is insufficient to distinguish between star formation and AGN related emission. Observations with the upcoming Square Kilometre Array (SKA), in cooperation with globally distributed VLBI arrays could provide an adequate mas or sub-mas resolution [@2012PASA...29...42G; @2015aska.confE.143P] and the thermal sensitivity of a few $\mu$Jy [@2015aska.confE.143P] sufficient for direct detection of compact AGN-related radio emission in the most powerful members of the radio-hAGN population.
The importance and contribution of SF to the radio emission can also be tackled using MIR observations. Considering the correlation between the MIR and $1.4$-GHz radio powers, and applying the fit parameters in @2005ApJ...632L..79W, we calculated the $8~\mu$m and $24~\mu$m powers that correspond to the characteristic $P_\mathrm{1.4GHz}$ values found in our study. Consistency of the MIR–radio correlation is considered invariant for 5 orders of magnitude of flux densities up to $z\sim3.5$ [@2001ApJ...554..803Y; @2008MNRAS.386..953I] and was found reliable even at $\mu$Jy levels [@2009MNRAS.394..105G]. Characteristic MIR flux densities were estimated for our hAGN sample, based on the characteristic power derived from the co-added 1.4-GHz FIRST radio maps. Values of $10-15$ mJy and $30-50$ mJy were found for the $8~\mu$m and $24~\mu$m flux densities, with mean values $14~$mJy and $45~$mJy, respectively. To estimate the level of AGN contamination contributing to the calculated values of SFR, we determined $q_{24}$ values using FIRST upper limits and $24~\mu$m flux densities derived using $22~\mu$m emission measured by the Wide-field Infrared Survey Explorer [WISE, @2010AJ....140.1868W] from the AllWISE catalog[^7] [@2014yCat.2328....0C]. We followed the analysis described by e.g. @2013MNRAS.436.3759B. On the one hand, all $124$ sources detected by WISE are above the theoretical limit separating RL AGN from RQ AGN and SFGs, so no further constraints could be obtained. On the other hand, the $24~\mu$m flux densities are in the range of $\sim2-11~\mu$Jy with a mean value of $4~\mu$Jy, which is an order of magnitude lower than the values calculated from the MIR–radio correlation, assuming all radio emission is SF-related. This indicates radio excess for the stacked objects (at least those with WISE detection), implying that the radio emission is AGN-related.
Summary {#sec:summary}
=======
We applied median stacking on 1.4-GHz VLA FIRST survey image cutouts centred on $2229$ optically identified but individually radio-undetected quasar positions. These objects populate the redshift range $4\leq z < 7.6$. Stacking of the full sample resulted in an unresolved point source with a bias-corrected flux density $52$ $\mu$Jy. Co-adding the radio map central pixels revealed a moderately radio-loud AGN population, with a characteristic 1.4-GHz radio power $P_\mathrm{1.4GHz}\sim10^{24}$ W Hz$^{-1}$. Under the simplifying assumption that the entire radio emission in the sample is produced by star forming activity in the quasar host galaxies, we obtained upper limits of the star formation rate in the order of a few $1000$ M$_\odot$ yr$^{-1}$. Based on literature data on individual AGN, we argue that the source of radio emission in the sample is rather a mixture of star formation and AGN-related activity. The spatial resolution of FIRST images is not sufficient to distinguish between the different mechanisms responsible for the radio emission. Future measurements with VLBI and SKA-VLBI could help determining the relative importance of the two emission types in individual objects. Stacking studies similar to the one presented here will benefit from the improved sensitivity and angular resolution of the ongoing VLASS whose radio images could provide data sufficient for determining redshift-dependent properties of high-redshift radio quasars.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful for the constructive comments by the anonymous referee that led to an improved discussion of our results. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. KÉG was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and by the Ministry of Human Capacities within the framework of the ÚNKP (New National Excellence Program).
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\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: <http://sundog.stsci.edu/>
[^3]: <https://www.cv.nrao.edu/nvss/>
[^4]: The regularly updated version of the catalogue can be accessed at <http://astro.elte.hu/~perger/catalog.html>
[^5]: <https://third.ucllnl.org/cgi-bin/firstcutout>
[^6]: http://www.aips.nrao.edu/index.shtml
[^7]: http://wise2.ipac.caltech.edu/docs/release/allwise/
|
---
abstract: 'Measurements of the large cellular flows on the Sun were made by local correlation tracking of features (supergranules) seen in full-disk Doppler images obtained by the Helioseismic and Magnetic Imager (HMI) instrument on the NASA Solar Dynamics Observatory (SDO) satellite. Several improvements made to the local correlation tracking method allowed for more precise measurements of these flows. Measurements were made hourly over the nearly ten years of the mission-to-date. A four-hour time lag between images was determined to give the best results as a compromise between increased feature displacement and decreased feature evolution. The hourly measurements were averaged over the 34 days that it takes to observe all longitudes at all latitudes to produce daily maps of the latitudinal and longitudinal velocities. Analyses of these flow maps reveal many interesting characteristics of these large cellular flows. While flows at all latitudes are largely in the form of vortices with left-handed helicity in the north and right-handed helicity in the south, there are key distinctions between the low latitude and high latitudes cells. The low latitude cells have roughly circular shapes, lifetimes of about one month, rotate nearly rigidly, do not drift in latitude, and do not exhibit any correlation between longitudinal and latitudinal flow. The high latitude cells have long extensions that spiral inward toward the poles and can wrap nearly completely around the Sun. They have lifetimes of several months, rotate differentially with latitude, drift poleward at speeds approaching 2 m s$^{-1}$, and have a strong correlation between prograde and equatorward flows. Spherical harmonic spectral analyses of maps of the divergence and curl of the flows confirm that the flows are dominated by the curl component with RMS velocities of about 12 m s$^{-1}$ at wavenumber $\ell$ = 10. Fourier transforms in time over 1024 daily records of the spherical harmonic spectra indicate two notable components - an $m = \pm\ell$ feature representing the low latitude component and an $m = \pm1$ feature representing the high latitude component. The dispersion relation for the low latitude component is well represented by that derived for Rossby waves or r-modes. The high latitude component has a constant temporal frequency for all $\ell$ indicating features advected by differential rotation at rates representative of the base of the convection zone high latitudes. The poleward motions of these features further suggest that the high latitude meridional flow at the base of the convection zone is poleward - not equatorward.'
author:
- 'David H. Hathaway'
- 'Lisa A. Upton'
bibliography:
- 'Hathaway.bib'
title: 'Hydrodynamic Properties of the Sun’€™s Giant Cellular Flows'
---
Introduction {#sec:intro}
============
The spectrum of the cellular convective motions at the surface of the Sun (and presumably at the surface other late-type stars) spans a broad range of cell sizes [@Hathaway_etal15]. The smallest are granules with diameters of $\sim1000$ km, lifetimes of $\sim10$ minutes, and flow velocities $\sim4000$ m s$^{-1}$ (approaching the local speed of sound). These convective cells are driven by radiative cooling at the Sun’s photosphere [@SteinNordlund00; @Nordlund_etal09]. The spectrum of cellular flows extends all the way to the global scale (100s of Mm) with one notable feature - a broad but distinct bump representing supergranules. Supergranules have diameters of $\sim30$ Mm, lifetimes of $\sim18$ hours, and flow velocities of $\sim500$ m s$^{-1}$. While there is no spectral evidence of any separation of scales associated with the giant cellular flows, there is a dynamical distinction. As shown in [@Hathaway_etal15] the cellular flows with diameters greater than $\sim 200$ Mm (spherical harmonic degrees less than 20-30) are dominated by the effects of the Sun’s rotation. The flows are toroidal in the sense that they consist of vortices with much weaker diverging/converging flow components. Here we examine the hydrodynamical properties of these giant cellular flows to ascertain their structures and dynamics, and to determine the roles they may play in producing the solar differential rotation, meridional circulation, and the magnetic dynamo responsible for the sunspot cycle.
The existence of giant cellular flows on the Sun was first proposed by [@SimonWeiss68] who coined the term “giant cells” for these structures. They suggested that the solar convection zone should support cells that span the 200 Mm depth of the convection zone itself. Early linear models of convection in rotating spherical shells (for both incompressible [@Gilman75] and compressible [@GlatzmaierGilman81] fluids) found that cells spanning the convection zone should be highly influenced by the Sun’s rotation. The cells tended to align with the rotation axis (the Taylor-Proudman Theorem) and to stretch north/south across the equator. The Coriolis force acting on the flows in these “banana” cells produces an equatorward flux of angular momentum (a Reynolds stress) that would drive/maintain solar differential rotation with a rapidly rotation equator.
Over the intervening years, numerical models allowed for highly nonlinear flows and encompassed more and more density scale heights so as to bring the modeled domain closer to spanning the entire convection zone [@Gilman79; @Miesch_etal00; @BrunToomre02; @Miesch_etal08; @Hotta_etal15; @Nelson_etal18]. As the modeled flows became more turbulent, the simple banana cell structure became less apparent, particularly in the near surface layers. Nonetheless, these models showed that the cellular flows produced Reynolds stresses which were critically important for maintaining the Sun’s differential rotation. In these simulations, the cells propagate prograde at low latitudes and retrograde at high latitudes. The cellular flows themselves have negative kinetic helicity (the dot product of velocity and the curl of velocity) in the north and positive kinetic helicity in the south throughout the bulk of the convection zone. (Note that in the axisymmetric mean field models of solar/stellar convection [@DurneySpruit79; @Hathaway84; @KichatinovRudiger93; @KichatinovRudiger05], the Reynolds stresses are parameterized and referred to as the “$\Lambda$-effect”.)
Observational evidence for giant cells was slow in coming. [@Bumba70] noted the existence of large magnetic structures that might be associated with giant cells but we now know that these structures are a consequence of the transport of magnetic flux by differential rotation, meridional flow, and supergranule diffusion [@DeVore_etal85]. The advent of continuous, full-disk observations of the Sun in the mid-1990s provided new data appropriate for observing and characterizing giant cells. [@Hathaway_etal96] used direct Doppler observations from the Global Oscillations Network Group (GONG) network of instruments and found that large-scale Doppler features were evidenced by long-lived features rotating at the Carrington rotation rate. [@Beck_etal98] used direct Doppler observations from the Michelson Doppler Investigation (MDI) instriment on the ESA/NASA Solar and Heliospheric Observatory (SOHO) mission [@Scherrer_etal95] and also found long-lived features that rotated at the Carrington rate. They noted that the low latitude features they found were extended in longitude but with narrow latitudinal extents - unlike the banana cells in theoretical models. [@Lisle_etal04] also used direct Doppler observations from MDI and found that supergranules tend to align with each other in a north-south direction over a wide range of latitudes - consistent with the banana cells in theoretical models.
[@Hathaway_etal13] tracked the motions of groups of supergranules seen in Doppler data from the Helioseismic and Magnetic Imager (HMI) instrument on the NASA Solar Dynamics Observatory (SDO) satellite [@Scherrer_etal12] and produced the first images of giant cellular flows. The hydrodynamic properties of these flows included kinetic helicity and Reynolds stresses like those in the simulations, but the cellular structure was quite different - dominated by long, narrow cells at high latitudes with peak flow velocities of $\sim20$ m s$^{-1}$. Shortly thereafter, [@Bogart_etal15] used the helioseismic technique of ring-diagram analysis and found nearly identical structures that coincided in space and time with those found by [@Hathaway_etal13], but with peak flow velocities of only $\sim0.5$ m s$^{-1}$.
Recently, [@Loptien_etal18] tracked the motions of granules seen in intensity images from SDO/HMI and found giant cellular structures in the equatorial region that had properties associated with Rossby waves [@Rossby39] or r-modes [@Saio82; @WolffBlizard86] - waves in which horizontal pressure gradients are balanced by the Coriolis force on the associated flows. Observations were limited to the lower latitudes but indicated flow velocities of $\sim10$ m s$^{-1}$ at $\pm50^\circ$ latitude. A ring-diagram analysis by [@Proxauf_etal20] has provided depth information indicating a slight decrease in amplitude with depth down to 8 Mm.
Here we improve and expand on the observations of [@Hathaway_etal13]. We track the motions of supergranules seen in Doppler observations from SDO/HMI. The Doppler signal associated with these largely horizontal flows becomes strongest near the observed limb of the Sun. This characteristic allows for flow measurements at high latitudes - latitudes largely unexplored by other methods (helioseismology, tracking granules in intensity images). We find that the giant cellular flows on the Sun have two distinct regimes - a high-latitude regime with long, narrow cells that spiral into the polar regions, and a low-latitude regime with more circular cells that propagate like Rossby waves or r-modes. The high latitude cells have hydrodynamic properties that impact the global dynamics of the Sun’s convection zone and the mechanisms associated with its magnetic dynamo.
Data and Data Preparation {#sec:data}
=========================
The data we used to measure the giant cell flows are full-disk Dopplergrams obtained by SDO/HMI. A set of individual filtergrams at 24 wavelengths/polarizations were obtained every 45 s, each was registered to a central time, and then averaged over 720 s with a tapered temporal filter by the instrument team to produce Doppler images largely free of signal from the 5-minute p-mode oscillations. We took these 720 s Dopplergrams and found and removed the signals fixed relative to the visible disk (spacecraft motion, solar rotation/differential rotation, meridional flow, and convective blue shift) see [@Hathaway_etal15] for further details concerning this process.
The resulting Dopplergrams are dominated by the signals from solar supergranules. Since these flows are largely horizontal, the Doppler signal peaks near the limb. This gives good signal at high latitudes (unlike the radial flow signal associated with the p-modes used in helioseismology).
These $4096^2$ full-disk Doppler images of the solar supergranulation were mapped to $4096^2$ pixels in equirectangular heliographic coordinates (4096 equispaced positions in longitude from $90^\circ$ east of the central meridian to $90^\circ$ west of the central meridian and 4096 equispaced position in latitude from the south pole to the north pole). The Doppler signals at targeted longitude/latitude positions were determined using bi-cubic interpolation from the $4\times4$ pixels surrounding the location in the full-disk image.
These $4096^2$ heliographic maps of the supergranule Doppler signal were convolved with a Gaussian blurring function and then resampled to produce $512^2$ maps for the local correlation tracking procedure. This provided sufficient spatial resolution to resolve the supergranules while minimizing the computational effort in tracking these features.
Local Correlation Tracking Procedure {#sec:LCT}
====================================
We measured the proper motions of supergranules using a local correlation tracking (LCT) method [@November_etal87] like that used in [@Hathaway_etal13]. The cross-correlation coefficient, $\chi$, is a number between $\pm 1$ given by
$$\begin{aligned}
\label{eqn:XC}
\chi(\theta,\phi,\Delta\theta,\Delta\phi,\Delta t) = & \int \int [V(\theta,\phi,t) - \overline {V(\theta,\phi,t)}] [V(\theta + \Delta\theta ,\phi + \Delta\phi,t + \Delta t) - \overline {V(\theta + \Delta\theta ,\phi + \Delta\phi,t + \Delta t)}] d\theta d\phi \nonumber \\ &
/ [\sigma(\theta,\phi,t) \sigma(\theta + \Delta\theta ,\phi + \Delta\phi,t + \Delta t)]\end{aligned}$$
where $\sigma$ is the RMS variation given by
$$\label{eqn:sigma}
\sigma^2(\theta,\phi,t) = \int \int [V(\theta,\phi,t) - \overline {V(\theta,\phi,t)}]^2 d\theta d\phi$$
with $V(\theta,\phi,t)$ being the Doppler velocity at co-latitude $\theta$, longitude $\phi$ and time $t$. The overbar indicates an average over the area within the correlation window while $\Delta t$ is the time difference and $\Delta\theta$ and $\Delta\phi$ are the spatial offsets between the observations.
The horizontal velocity vector at a given location was determined by the displacement giving the highest correlation for a patch of data within the correlation window in one map when correlated to a similar patch in a second map obtained at a predetermined time lag (e.g. 4-hr, 8-hr, or 16-hr). We made these determinations at a $256^2$ array of points equispaced on the Doppler velocity maps. While the LCT method closely follows that used in [@Hathaway_etal13], several improvements have been made based on a recent study [@Mahajan_etal20].
[@Hathaway_etal13] used a circular correlation window with a 21-pixel diameter ($\sim 90$ Mm at the equator). Here we used an oval window that becomes elongated in longitude with latitude by a factor of $\csc \theta$ so as to cover the same physical area on the surface of the Sun at all latitudes (up to $75^\circ$ latitude, beyond which the oval remains 21-by-81 pixels in size). We also expanded the search area (larger maximum $\Delta\theta$ and $\Delta\phi$) in our search for the best correlation. This assures a full sampling of possible displacements.
We modified the method for determining the location of the correlation maximum. We find an estimate for the location of the peak to a fraction of a pixel by calculating the locations of the peaks of the parabolas passing through the maximum correlation in both the latitudinal and longitudinal directions. This estimate is used to then shift (using a bi-cubic interpolator) the original patch of data by that fraction of a pixel to then repeat the process and better determine the fractional pixel shift.
A final revision to the LCT procedure was to use a 4-hr time-lag along with the 8-hr and 16-hr lime-lags used in [@Hathaway_etal13]. The different time-lags have important consequences. The shorter the time-lag the smaller the displacement and the larger the error in the measurement. The longer the time-lag the more the evolution of the supergranules within the correlation window and the larger the error in the measurement. Experiments with time-lags from 1-hr to 24-hr indicate, for this data, a minimum in measurement noise at a time-lag of 4-hr - the nominal time-lag used for the bulk of this study. Another consequence of using different time-lags is sensitivity to flows at different depths within the Sun. [@Hathaway12A] showed that tracking supergranules with longer time-lags gives results that are sensitive to longer-lived supergranules that tend to be larger and to extend to deeper depths. In fact, supergranules were found to be advected by flows at depths equal to their diameters and proportional to the time-lag used. Thus, we can use the different time-lags to obtain information about changes in the flow structure with depth.
An example of a raw LCT flow map obtained with the 4hr time-lag is shown in Figure \[fig:LCTsample\]. The latitudinal velocity, $V_\theta$, is shown in the left panel. The longitudinal velocity, $V_\phi$, is shown in the right panel. Both have a velocity scale of $\pm300$ m s$^{-1}$. The longitudinal velocity is dominated by the differential rotation of the supergranule pattern with large retrograde flow at high latitudes and prograde flow near the equator. (Note that all velocities are relative to the frame of reference rotating with the Carrington period of 27.2753 days synodic.) The latitudinal velocity shows a hint of the $\sim15$ m s$^{-1}$ poleward meridional flow. Both velocity components exhibit a mottled pattern associated with the large cellular flows. Similar maps are obtained with the 8hr and 16hr time-lags but with higher noise levels, with different differential rotation and meridional flow velocities, and slight differences in the mottled patterns. These raw 4-hr LCT flow maps were constructed at hourly intervals over the length-to-date of the SDO Mission (2010 May through 2020 March). Flow maps at 8-hr and 16-hr were obtained over more limited intervals of the mission.
![Latitudinal (left panel, northward in red, southward in blue) and longitudinal (right panel, prograde in red, retrograde in blue) velocities determined from local correlation tracking using a single pair of Doppler maps separated by 4-hr in time. Both have a velocity scale of $\pm300$ m s$^{-1}$.[]{data-label="fig:LCTsample"}](LCTmapImage.png){width="1.0\columnwidth"}
Giant Cell Flow Map Construction Procedure {#sec:GiantCellMaps}
==========================================
The raw LCT flow maps described in the previous section were averaged together to produce giant cell flow maps. (Typically some 800 raw hourly LCT flow maps obtained over 34 days are averaged together to make each of the daily giant cell flow maps.) Here, again, we improve upon the method used in [@Hathaway_etal13].
[@Hathaway_etal13] determined the differential rotation and meridional flow as functions of latitude only, and then removed those axisymmetic flow signals from the raw LCT flow maps before averaging. We have found that the differential rotation and meridional flow signals have components that appear to vary in longitude. This is illustrated in Figure \[fig:DiskAveragedFlows\] which shows the signal averaged over all raw LCT flow maps at the 4-hr time-lag. This long average removes the mottled signal due to the large cellular flows and reveals the signals due to meridional flow (left) and differential rotation (right). Both flow components exhibit some apparent variation with longitude relative to the central meridian. We attribute this spurious signal component to line-of-sight projection effects on the supergranule Doppler signal (cf. [@Hathaway_etal06]). It is this disk averaged signal, with its added longitudinal variation, that is removed from the raw LCT flow maps to better reveal the giant cell flows.
![Disk averaged flows. Latitudinal (left panel, northward in red, southward in blue, $\pm30$ m s$^{-1}$) and longitudinal (right panel, prograde in red, retrograde in blue, $\pm200$ m s$^{-1}$) velocity signals when averaged over all raw LCT flow maps at the 4-hr time-lag.[]{data-label="fig:DiskAveragedFlows"}](DiskAveragedFlows.png){width="1.0\columnwidth"}
After removing these disk averaged signals associated with the axisymmetric flows, the raw LCT flows maps are averaged together using the Carrington longitude of the central meridian associated with each map as a reference for positioning the data. [@Hathaway_etal13] constructed giant cell flows maps in the traditional manner for individual Carrington rotations - using the Carrington longitude for each raw LCT flow map as the longitude to use at all latitudes. This commonly used method neglects the Sun’s differential rotation. The method results in features near the equator being pinched together and features near the poles being stretched apart relative to where they actually are on the Sun at a given time. More than $360^\circ$ in longitude on the Sun are placed on these maps at the equator due to a rotation period three days shorter than the Carrington period. As little as $285^\circ$ in longitude on the Sun are stretched out near the poles due to a rotation period seven days longer than the Carrington period.
Here we average the raw LCT flow maps using longitudes based on differential rotation. The flow data at each latitude is shifted in longitude to its place on a giant cell flow map based on a measured differential rotation profile. This process allows us to construct giant cell flow maps of the entire solar surface for any given moment in time (assuming evolution of the flow pattern on a timescale longer than the rotation period). The process does, however, requires us to measure the differential rotation profile of the giant cell flow pattern itself.
We find the differential rotation profile for the giant cell features by cross-correlating longitudinal strips of velocity data from temporary giant cell flow maps constructed 28 days apart. (The temporary giant cell flow maps are constructed in the traditional manner used in [@Hathaway_etal13] but are made on a daily basis instead of a Carrington rotation period basis.) We cross-correlate data for the full range of longitude shifts and average together all the cross-correlations. The results are shown in Figure \[fig:Xcor28Days\]. A functional fit through the locations of the cross-correlation peak at each latitude gives the longitude shift, $\delta\phi$, for the giant cellular features as
$$\label{eqn:GCDR1}
\delta\phi(\theta,\Delta t) = (-3^\circ + 28^\circ \cos^2\theta - 220^\circ \cos^4\theta
+ 90^\circ \cos^6\theta) \Delta t/28^d$$
where $\Delta t$ is the time difference (relative to the target time for the giant cell flow map) of the raw LCT flow map being added to the average. (Aspects of this differential rotation profile, and the associated meridional flow profile will be discussed in Section \[sec:DRMF\].)
![Cross-correlation amplitude (red positive and blue negative) for a 28-day time-lag between temporary giant cell flow maps as a function of longitudinal shift and latitude. The three pixels surrounding the correlation peak at each latitude are colored blue. A polynomial fit to the peak locations (Eq. \[eqn:GCDR1\]) is indicated by a white line through the blue pixels.[]{data-label="fig:Xcor28Days"}](CrossCorrelationAt28Days.png){width="1.0\columnwidth"}
We use this longitude shift (Eq. \[eqn:GCDR1\]) to construct maps of the giant cell flows on a daily basis. Given the target day and time, we average some 800 raw LCT flow maps from $\pm 17d$ so as to include all longitudes even at the highest latitudes. The velocity data at each latitude is shifted in longitude according to Eq. \[eqn:GCDR1\] to where it would have been at the target time. An example of one of these daily giant cell flow maps is shown in Figure \[fig:GCFlowMaps\].
![Giant cell flow map example from $00^h:00^m:00^s$ UT on 2010 July 1. The upper panel shows the latitudinal velocity (red northward). The lower panel shows the longitudinal velocity (red prograde).[]{data-label="fig:GCFlowMaps"}](GiantCellVelocityMap.png){width="1.0\columnwidth"}
Giant Cell Morphologies and Lifetimes {#sec:Morphology}
=====================================
Inspecting the 3000+ daily giant cell maps over the nearly 10 years of the SDO mission-to-date reveals key morphological characteristics of the cellular flows. The low latitude cells are nearly circular in shape while the high latitude cells are elongated in longitude to form a spiral pattern centered on the poles.
This is born out by auto-correlating strips of data at different latitudes as shown in Figure \[fig:GCAutoCor\]. Vorticity maps were constructed from each set of daily giant cell flow maps (Figure \[fig:GCFlowMaps\]) by taking the curl of the horizontal flow vectors. Three fairly broad ($\pm 22.5^\circ$) strips (the full $360^\circ$ in longitude) from the vorticity maps, centered on the equator and at $\pm 60^\circ$, were auto-correlated with themselves to reveal the cell morphology at those locations. Auto-correlation maps were constructed by shifting each vorticity map strip through $\pm 180^\circ$ in longitude and $\pm 22.5^\circ$ in latitude.
The average auto-correlation maps in Figure \[fig:GCAutoCor\] show explicitly that the cells at high latitudes are long and narrow with characteristic tilts that produce a spiral pattern around the poles. The low latitude cells are nearly circular with no evidence of any tendency to be elongated north to south or east to west.
![Auto-correlation maps (red positive, blue negative correlation) of the vorticity at three latitudes - $60^\circ$ North (left panel), the equator (middle panel), and $60^\circ$ South (right panel). Cells at high latitudes are sheared out in longitude to form long narrow tilted cells. Cells at the equator are nearly circular.[]{data-label="fig:GCAutoCor"}](GiantCellVorticityAutoCorrelation.png){width="1.0\columnwidth"}
The lifetimes of the cells can be estimated from the strength of the cross-correlation shown in Figure \[fig:Xcor28Days\]. Similar cross-correlations were calculated for the longer time-lag of 56 days. The cross-correlation at the equator drops from its initial 1.0 to 0.3 after 28 days and to 0.07 after 56 days - indicating lifetimes between one and two months for the cells near the equator. The cross-correlation at latitudes of $\pm 60^\circ$ drops from its initial 1.0 to 0.5 after 28 days and to 0.3 after 56 days - indicating lifetimes of several months for the high latitude cells.
This difference in both morphology and lifetimes of the cells at low vs. high latitudes is the first indication of two different dynamical regimes for these large cellular flows.
Giant Cell Differential Rotation and Meridional Motion {#sec:DRMF}
======================================================
We measure the differential rotation and meridional motion of the giant cells by finding the displacement in longitude and latitude that gives the maximum correlation for narrow longitude strips from giant cell flow maps separated by 28 days in time. These measurements were made on six month long intervals of data so as to obtain estimates of uncertainty and variability.
![left Panel: Sidereal differential rotation rate of the giant cells measured over 19 6-month intervals. The rotation rate derived from the motions of the small magnetic elements by [@HathawayRightmire11] is shown in red for reference. The horizontal line is at the Carrington rate of 456 nHz. Corresponding sidereal rotation periods are indicated on the right-hand axis. Right Panel: Meridional motion of the giant cells measured over 19 6-month intervals. There is little or no meridional motion of the cells between $\pm25^\circ$. At higher latitudes the meridional motion is poleward at velocities approaching 2 m s$^{-1}$.[]{data-label="fig:GCDRMF"}](GiantCellDRMF.pdf){width="1.0\columnwidth"}
The differential rotation profiles for each of these intervals are shown in the left panel of Figure \[fig:GCDRMF\] along with, for reference, the rotation rate of the small magnetic elements as given by [@HathawayRightmire11]. A surprising feature of this profile is the relatively flat section within about $25^\circ$ of the equator with rotation slightly slower than the Carrington rate. Both the flatness and the slowness are not exhibited by other solar features at these latitudes at any depth within the Sun.
The meridional motion profiles for each of the six-month intervals are shown in the right panel of Figure \[fig:GCDRMF\]. Here again we find a flat section within about $25^\circ$ of the equator. The meridional motion of the giant cell pattern at higher latitudes is poleward in each hemisphere with a peak velocity approaching 2 m s$^{-1}$.
These axisymmetric flow measurements provide another indication of the two different dynamical regimes - one at low latitudes and another with different characteristics at high latitudes.
Giant Cell Kinetic Helicity and Reynolds Stress {#sec:KHRS}
===============================================
![Left panel: Kinetic helicity $<\nabla\cdot{V_H} ~\nabla\times{V_H}>$ of the giant cells measured over 19 6-month intervals. The kinetic helicity is negative (left-handed) in the north and positive (right-handed) in the south but with very different latitudinal variations at low vs. high latitudes. Right panel: Reynolds stress $<V_\theta V_\phi>$ of the giant cells measured over 19 6-month intervals. The angular momentum flux is near zero at the low latitudes but strongly toward the equator at high latitudes.[]{data-label="fig:GCKHRS"}](GiantCellKHRS.pdf){width="1.0\columnwidth"}
The effects of the Sun’s rotation on these large, long-lived cellular flows is expected to produce key signatures in both the kinetic helicity (the dot product of the velocity with the vorticity) and the component of the Reynolds stress given by the (anti) correlation of poleward flows and prograde flows ($<V_\theta V_\phi>$).
We produce a proxy, $<\nabla\cdot{V_H} ~\nabla\times{V_H}>$, for the radial component of the kinetic helicity by using the divergence of the horizontal flows as an estimate of the relative amplitude and direction of the radial flow vector at each location. (This association follows from the mass continuity equation.) The profiles of this kinetic helicity proxy are shown in the left panel of Figure \[fig:GCKHRS\]. The kinetic helicity is negative (left-handed) in the north and positive (right-handed) in the south. The low- and high-latitude flow regimes are also reflected in this quantity. The kinetic helicity increases linearly across the equator from about $25^\circ$ north to $25^\circ$ south. At higher latitudes it is relatively constant (albeit with a slight dip at about $50^\circ$ latitude in each hemisphere).
The Reynolds stress $<V_\theta V_\phi>$ is associated with the latitudinal transport of angular momentum and, as such, it is key to producing and maintaining the Sun’s differential rotation with its rapidly rotating equator. The latitudinal profiles of this component of the Reynolds stress are shown in the right panel of Figure \[fig:GCKHRS\]. The Reynolds stress is small (consistent with zero) at low latitudes. The Reynolds stress rises rapidly at high latitude to give a strong flux of angular momentum toward the equator.
Spherical Harmonic Spectra {#sec:Spectra}
==========================
We produced spherical harmonic spectra of the giant cell flows by first constructing maps of the curl and of the divergence of the horizontal velocity. This provides us with measures of the two spherical harmonic components - the toroidal and poloidal components.
Following [@Chandrasekhar61] we can fully represent the horizontal flows on the surface of the Sun using toroidal, $T_\ell^m$, and poloidal, $S_\ell^m$, spectral coefficients with
$$\label{eqn:V_theta}
V_\theta(\theta,\phi) = \sum_\ell \sum_{m=-\ell}^\ell \lbrack
S_\ell^m {\partial Y_\ell^m \over \partial\theta}
+ T_\ell^m {1 \over \sin\theta} {\partial Y_\ell^m \over \partial\phi} \rbrack$$
$$\label{eqn:V_phi}
V_\phi(\theta,\phi) = \sum_\ell \sum_{m=-\ell}^\ell \lbrack
S_\ell^m {1 \over \sin\theta} {\partial Y_\ell^m \over \partial\phi}
- T_\ell^m {\partial Y_\ell^m \over \partial\theta} \rbrack$$
where
$$Y_\ell^m(\theta,\phi) = \bar{P}_\ell^m(\cos\theta) e^{im\phi}$$
is a normalized spherical harmonic of degree $\ell$ and order $m$.
By choosing this description of the horizontal velocities, the divergence of the flow field gives
$$\nabla\cdot{V_H} = {1 \over r} \sum_\ell \sum_{m=-\ell}^\ell \ell (\ell + 1) S_\ell^m Y_\ell^m(\theta,\phi)$$
and the curl of the flow field gives
$$\nabla\times{V_H} = - {1 \over r} \sum_\ell \sum_{m=-\ell}^\ell \ell (\ell + 1) T_\ell^m Y_\ell^m(\theta,\phi)$$
The mean squared velocity on the surface is given by
$$V_{RMS}^2 = {1 \over 4 \pi} \int_0^\pi \int_0^{2\pi} [V_\theta^2 + V_\phi^2] d\phi \sin\theta d\theta$$
which becomes
$$V_{RMS}^2 = \sum_\ell \sum_{m=-\ell}^\ell \ell (\ell + 1) [{S_\ell^m}^2 + {T_\ell^m}^2]$$
With this description in mind, we find the spherical harmonic spectral coefficients for the giant cellular flows by first taking the divergence of the giant cell flow velocities (Figure \[fig:GCFlowMaps\]) to isolate the poloidal components and by taking the curl to isolate the toroidal components. Performing a spherical harmonic transform on the divergence and curl maps then gives the individual spectral coefficients.
The average spectral amplitudes are shown in Figure \[fig:Spectrum2D\] with the toroidal components in the top panel and the poloidal components in the bottom panel. The toroidal coefficients are clearly much stronger than the poloidal coefficients - indicating that the flows are dominated by vortices with significant curl and little divergence. (Note that, at each latitude, the longitudinal average of each component of the giant cellular flows was removed from each maps since these represent axisymmetric flows – hence the lack of any power at $m=0$ in Figure \[fig:Spectrum2D\].)
Figure \[fig:Spectrum2D\] also shows signatures for two different components within the toroidal coefficients - a strong component at $m=\pm1$ (the high latitude component) and a weaker but clearly evident component with $m=\pm\ell$ (the equatorial component).
![Spectral amplitudes as functions of spherical harmonic degree $\ell$ and azimuthal order $m$. The averages of the toroidal (vorticity) coefficients are shown in the upper panel. The averages of the poloidal (divergence) coefficients are shown in the lower panel. The toroidal coefficients dominate and show signatures of distinct high- and low-latitude components at $m=\pm1$ and $m=\pm\ell$ respectively.[]{data-label="fig:Spectrum2D"}](VelocitySpectrum2D.png){width="1.0\columnwidth"}
The characteristic (RMS) velocity associated with each component is shown as a function of spherical harmonic degree $\ell$ in Figure \[fig:Spectrum1D\]. The toroidal component dominates – as expected from the effects of the Sun’s rotation. (Note the characteristic velocity of $\sim12$ m s$^{-1}$ at $\ell=10$.) While the characteristic velocities seem to fall-off at wavenumbers above $\ell\sim20$, this is attributed to the drop in sensitivity at higher wavenumbers. While we make LCT measurements at 256 locations in latitude and longitude from each pair of Doppler maps, the 21-pixel (90 Mm) diameter correlation window used in the LCT step only gives 24 fully independent samples in latitude. Figure \[fig:Spectrum1D\] also includes, in red, the spectra from giant cell flow maps constructed from LCT measurements made with an 8-hr time lag and, in green, the spectra from giant cell flow maps constructed from LCT measurements made with a 16-hr time lag. These measurements represent deeper layers (22 Mm at 4-hr, 26 Mm at 8-hr, and 37 Mm at 16-hr time lags according to [@Hathaway12B]) within the surface shear layer but still exhibit virtually the same velocity spectra for the giant cell flows in those deeper layers.
The velocity spectra derived from direct Doppler measurements in [@Hathaway_etal15] are shown in blue in Figure \[fig:Spectrum1D\]. While these spectral are not subject to any decrease in amplitude due to loss of resolution at these wavenumbers, they are subject to contamination due to instrumental artifacts in the Doppler data itself (see discussion in [@Hathaway_etal15]). The LCT measurements are somewhat less than the direct Doppler measurements at the well resolved wavenumbers (below $\sim10$). We expect that the low wavenumber LCT spectral amplitudes are better representative of the actual flows on the Sun and that the direct Doppler measurements are contaminated by instrumental artifacts at these wavenumbers. It is also important to note that the direct Doppler spectra indicate equal parts toroidal and poloidal flow at $\ell=30$ while the LCT measurements indicate that this cross-over must occur at a higher wavenumber.
![Characteristic velocities as functions of spherical harmonic degree $\ell$. The toroidal components (solid lines) dominate the poloidal components (dotted lines) in this wavenumber range. Spectra derived from measurements with 4-hr time lags are shown in black, those derived from measurements with 8-hr time lags are shown in red, and those derived from measurements with 16-hr time lags are shown in green. Spectra derived from direct Doppler measurements by [@Hathaway_etal15] are shown in blue. The characteristic velocities increase monotonically – the turnover at wavenumbers above $\ell\sim20$ is due to the drop-off in sensitivity associated with the 90 Mm window diameter used in the LCT step. The direct Doppler measurements overestimate the spectral amplitudes at these wavenumbers.[]{data-label="fig:Spectrum1D"}](VelocitySpectrum1D.pdf){width="1.0\columnwidth"}
Equatorial Rossby Waves {#sec:RossbyWaves}
=======================
Wave-like phenomena are characterized by taking the Fourier transform in time of the individual spectral coefficients shown in Figure \[fig:Spectrum2D\]. Our giant cell flow maps and spherical harmonic spectra of the vorticity are generated at a cadence of 1 per day. We examine the temporal variation by taking the Fourier transform in time of 1024-day records sampled at 6-month intervals (14 such overlapping records are contained in the SDO/HMI mission to date). A cosine taper (of total length 256) was applied to each end of each spectral coefficient time series prior to taking the Fourier transform to reduce ringing due to end effects.
The toroidal spectral amplitudes as functions of temporal frequency and wavenumber, $\ell$, for modes with $m=\ell$ (equatorial modes) are shown in the left panel in Figure \[fig:GCk\_omega\]. These modes propagate retrograde with a dependence upon wavenumber very much like the dispersion relation given for Rossby waves or r-modes [@Saio82]
$$\label{eqn:k_omega}
\omega = {-2\Omega_\odot m \over [\ell(\ell + 1)]}$$
where $\Omega_\odot/2\pi = 456$ nHz is the Carrington rotation frequency. This functional form is indicated by the curved line in the left panel of Figure \[fig:GCk\_omega\]. This Rossby wave characteristic of the equatorial modes was first noted by [@Loptien_etal18].
Rossby waves, as derived by [@Rossby39], are waves that arise in shallow water or thin atmospheric layers when the characteristic time scale for the flows are much longer than the rotation period. Under these circumstances the flows become toroidal (geostrophic) with the Coriolis force acting on the flows balanced by the horizontal pressure gradients associated with the wave disturbance. They gain their propagation characteristics from the latitudinal variation in strength of the radial component of the rotation vector – the component that produces the Coriolis on the horizontal flows.
R-modes are extensions of these waves to deep spherical shells – initially the stably stratified radiative zones of massive early-type stars [@Saio82] . [@WolffBlizard86] explored their properties in the Sun in the absence of convective motions. Models of solar convection typically find that the convective structures propagate prograde without these r-mode characteristics (see [@Miesch05] and references therein).
![Vorticity spectral amplitudes as functions of spherical harmonic degree $\ell$ and temporal frequency for modes with $m=\ell$ (left) and with $m=1$ (right). The functional form of the dispersion relation for Rossby waves given by Eq. \[eqn:k\_omega\] is shown as the curved line at negative frequencies on the left.[]{data-label="fig:GCk_omega"}](GiantCellVorticity_k_omega.png){width="1.0\columnwidth"}
Polar Vortices {#sec:PolarVortices}
==============
The polar vortices are represented by the toroidal components with $m=\pm1$ in Figure \[fig:Spectrum2D\]. (Toroidal components with $m=\pm2$ and $m=\pm3$, while weaker, also contribute to the structure of the polar vortices.) The spectral amplitudes as functions of temporal frequency and wavenumber, $\ell$, for modes with $m=\pm1$ are shown in the right panel of Figure \[fig:GCk\_omega\]. These modes propagate retrograde with no variation in temporal frequency with wavenumber – indicating that these high latitude modes are advected retrograde by the differential rotation at a rate about 80 nHz slower than the 456 nHz Carrington reference rate. This matches the rotation rate at $\sim 60^\circ$ latitude as seen in Figure \[fig:GCDRMF\]. The same behavior is also seen for the $m=\pm2$ and $m=\pm3$ components but with nearly double and triple the temporal frequencies, giving similar rotation rates. Note that the $m=\pm1$ components are dominated by the odd $\ell$ modes (symmetric across the equator) while the $m=\pm3$ components are dominated by the even $\ell$ modes (anti-symmetric across the equator). The $m=\pm2$ components do not show any preference for symmetry across the equator.
Conclusions {#sec:Conclusions}
===========
We have improved upon the LCT method used in [@Hathaway_etal13] and, by including information on the differential rotation of the giant cellular flows, improved our maps of the giant cellular flows. We find strong evidence, from multiple characteristics, for the existence of two very different flow regimes – low latitudes Rossby waves and high latitude polar vortices. These flows are illustrated in Figure \[fig:GiantCells\] and the associated animation [SolarVortices.mp4](http://solarcyclescience.com/bin/SolarVortices.mp4).
[]{data-label="fig:GiantCells"}](GiantCellFlows.png){width="1.0\columnwidth"}
The low latitude Rossby waves are nearly circular in shape as indicated by visual inspection of the giant cell flow maps (Figure \[fig:GCFlowMaps\]) and by the auto-correlation study illustrated in Figure \[fig:GCAutoCor\]. They are not extended north to south as suggested by nearly every numerical model of solar convection zone dynamics (cf. [@Miesch05] and references therin) and by reports of north/south alignment of supergranules [@Lisle_etal04]. They are also not extended east to west as reported by [@Beck_etal98].
These low latitude Rossby waves have lifetimes only slightly longer than the 27 day rotation period of the Sun. This may be due the waves getting in and out of phase with each other as the low wavenumber waves propagate faster than the higher wavenumber waves (Eq. \[eqn:k\_omega\]).
The low latitude Rossby waves do exhibit the kinetic helicity signatures associated with the effects of the Sun’s rotation (Figure \[fig:GCKHRS\]) with left-handed helicity in the north and right-handed helicity in the south. They do not, however, exhibit any Reynolds stress like that needed to maintain the Sun’s differential rotation.
The low latitude Rossby waves propagate retrograde relative to the Carrington rotation frame of reference, do not exhibit substantial differential rotation with latitude, and do not propagate either poleward or equatorward (Figure \[fig:GCDRMF\]). Spectral analyses of these low latitude flows show clear evidence for a dispersion relation like that found for Rossby waves [@Rossby39] or r-modes [@Saio82]. This constitutes further confirmation of the discoveries of these waves by [@Loptien_etal18] using LCT of granules and by [@Proxauf_etal20] using helioseismology. Our observations extend the measurements of these waves to greater depths. The granules tracked by [@Loptien_etal18] extend to depths of $\sim1$ Mm while the acoustic waves used by [@Proxauf_etal20] reliably sample flows at depths down to $\sim8$ Mm. Our LCT measurements of supergranules at 4-hr, 8-hr, and 16-hr time lags represent flows at depths of 22 Mm, 26 Mm, and 37 Mm respectively. While [@Proxauf_etal20] conclude that the Rossby wave amplitudes decrease by 10% with depth over the outermost 8 Mm, we find that the spectral amplitudes are constant down to far greater depths.
The high latitude polar vortices differ morphologically and dynamically from the low latitude Rossby waves. The polar vortices are long, narrow features that spiral into the polar regions as indicated by visual inspection of the giant cell flow maps (Figure \[fig:GCFlowMaps\]), by the auto-correlation study (Figure \[fig:GCAutoCor\]), and by the flow structures shown in Figure \[fig:GiantCells\] and the associated animation ([SolarVortices.mp4](http://solarcyclescience.com/bin/SolarVortices.mp4)).
The polar vortices have lifetimes of several months. These features are not well represented in numerical models of the solar convection zone but they have been seen via measurements with other methods. [@Bogart_etal15] used the helioseismic method of ring diagram analysis and found high latitude features that match those found here in terms of shape, position, orientation, and lifetimes – but with far weaker flow velocities (only 0.5 m s$^{-1}$ instead of 12 m s$^{-1}$). [@Howe_etal15] also used ring diagram analysis and, while they didn’t show maps of the velocity structures, they did find velocity structures with velocities more commensurate with ours ($\sim20$ m s$^{-1}$).
While [@Hathaway_etal13] concluded that the flow velocities in these giant cellular flows increased with depth, we now find that the velocity spectra are unchanged with depth (Figure \[fig:Spectrum1D\]). The previous conclusion was based on increased velocities seen in the giant cell flow maps themselves and was influenced by the increase in velocity measurement noise at longer time lags. The velocity spectral amplitudes shown in Figure \[fig:Spectrum1D\] are somewhat smaller than those found by [@Hathaway_etal15] and they indicate that the cross-over from toroidal flows at low $\ell$ to poloidal flows at high $\ell$ must occur at a wavenumber substantial higher than $\ell=30$ as given by [@Hathaway_etal15]. The LCT measurements employed here are expected to be more reliable than the direct Doppler measurements of [@Hathaway_etal15] at these low wavenumbers due to contamination by instrumental artifacts in the direct Doppler data.
The polar vortices exhibit the kinetic helicity signatures associated with the effects of the Sun’s rotation (Figure \[fig:GCKHRS\]) with left-handed helicity in the north and right-handed helicity in the south. Unlike the low latitude Rossby waves, they exhibit a strong Reynolds stress giving an equatorward transport of angular momentum needed to help maintain the Sun’s differential rotation. This Reynolds stress can be attributed to the shape and orientation of the features themselves. The Sun’s rotation forces the flows themselves to be directed along the structures and the structures are oriented in a spiral pattern.
The polar vortices rotate differentially with 34 day periods above $75^\circ$ latitude, 30 days at $60^\circ$ latitude, and 28 days at $45^\circ$ latitude (Figure \[fig:GCDRMF\]). This differential rotation profile differs from that seen near the surface (the magnetic element rotation profile shown in red in Figure \[fig:GCDRMF\] is representative of the surface shear layer at a depth of about 20 Mm). The faster rotation rates of the polar vortices (relative to the surface) are much more representative of the rotation near the base of the convection zone (cf. [@Howe09]). The rotation rates of the polar vortices at $60^\circ$ and $75^\circ$ match the internal rotation rates at those latitudes at one and only one radius, $\sim0.78$ R$_\odot$ – the top of the tachocline at the base of the convection zone.
The polar vortices also exhibit a meridional flow – poleward at a peak velocity of about 2 m s$^{-1}$. If these vortices do extend to the base of the convection zone where we find similar rotation rates, then this meridional flow observation indicates a slow poleward meridional flow at the same depth.
The radial (and latitudinal) structure of the Sun’s meridional circulation has very important consequences for models of the Sun’s magnetic dynamo. Flux transport dynamos [@Choudhuri_etal95; @NandyChoudhuri02] assume that the poleward meridional flow seen near the surface sinks inward in the polar regions and returns equatorward at the base of the convection zone where they expect the sunspot magnetic fields to arise - thus giving rise to the equatorward drift of the sunspot zones. However, recent observations of an equatorward meridional flow at the base of the surface shear layer[@Hathaway12B; @Zhao_etal13] and now this observation suggesting poleward flow at the base of the convection zone (in agreement with the double cell meridional structure suggested by the observations of [@Zhao_etal13]) further challenge these Flux Transport Dynamo models.
The hydrodynamic properties of these giant cellular flows are not well represented (if at all) in current numerical models of the Sun’s convection zone dynamics (neither the high latitude spirals nor the low latitude Rossby waves are found in these models). This suggests that these models (which undoubtedly solve the proper hydrodynamic and MHD equations) might require further development of surface boundary conditions and/or initial conditions. It is well worth noting that the Sun previously rotated faster and had a lower luminosity - conditions that may have led to the solar differential rotation with flows like these polar vortices that continue to this day to maintain the solar differential rotation as a positive feedback mechanism. Another possibility is that the low latitude Rossby waves may be a rather shallow phenomena that hides underlying structures more representative in models simulations at low latitudes.
The HMI data used here are courtesy of the NASA/SDO and the HMI science teams. The authors benefited from discussions with Sushant Mahajan on improving the LCT method and from comments on the manuscript by Phil Scherrer, Todd Hokesema, Leif Svalgaard, and Adam Hathaway. DHH was supported by the HMI science team at Stanford University. L.A.U. was supported by the NSF Atmospheric and Geospace Sciences Postdoctoral Research Fellowship Program (Award Number:1624438) and the NASA Living with a Star Program (Grants:NNH16ZDA001N-LWS and NNH18ZDA001N-LWS).
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abstract: 'Dynes ansatz [@dynes] is a broadly utilized procedure to extract the superconducting energy gap from the tunnelling differential conductance, a quantity that is measured by scanning tunnelling microscopy (STM) techniques. In this letter we investigate the limit of applicability of this ansatz in nano-scale superconductors and propose a generalization that permits to study thermal and quantum fluctuations in STM experiments.'
author:
- 'Antonio M. García-García, Pedro Ribeiro'
date: 'Received: date / Accepted: date'
title: Quantifying fluctuations from the tunnelling differential conductance
---
Introduction
=============
Scanning tunneling microscopy techniques (STM) have revolutionized [@stm1] the field of experimental superconductivity for both conventional and high temperature superconductors. A typical observable in these experiments is the normalized tunnelling differential conductance, $$\begin{aligned}
\frac{dI}{dV} & = & \frac{1}{4T}\int_{-\infty}^{\infty}d\omega\, N_{s}(\omega,\Delta)\left[\frac{1}{\cosh^{2}\left(\frac{\omega+eV}{2k_{B}T}\right)}\right]\label{didv}\end{aligned}$$ where $N_s$ is the local density of states, $T$ is temperature, $V$ is the gate voltage. In order to make contact with properties of the superconductor, such as the superconducting energy gap $\Delta$, is required to fit the experimental data with a theoretical expression of $dI/dV$ that includes $\Delta$ as a fitting parameter. In principle this is not an easy task as for many materials the theoretical local density of states $N_s(\omega,\Delta)$ is not exactly known or its calculation involves the use of the rather cumbersome Eliashberg theory of superconductivity. However Dynes noted [@dynes] that for some metals the energy gap $\Delta \equiv \Delta_D$ obtained by a simple fitting function, $$\begin{aligned}
N_{D}(\omega) & = & \im\left(\frac{\omega+i\Gamma}{\sqrt{\Delta_D^{2}-\left(\omega+i\Gamma_D\right)^{2}}}\right) \label{DDOS}\end{aligned}$$ is in excellent agreement with the one obtained by other experimental techniques. $\Gamma_D(T)$ in (\[DDOS\]) is other fitting parameter that describes different sources of decoherence. In light of this success this procedure is now employed to obtain the energy gap in STM experiments involving a broad range of conventional and high $T_c$ superconductors. In case of a material with a non s-wave order parameter symmetry the Dynes ansatz is modified to account for the expected momentum dependence of $\Delta_D$.\
In this letter we investigate the limit of applicability of (\[DDOS\]) in nano-scale superconductors where deviations from mean-field predictions are well documented [@richardson; @nmat]. These were recently used in [@nmat1] to analyze STM data from Pb superconducting nano particles. We focus on sizes $L < 10$nm for which deviations from mean-field predictions start to be relevant. First we introduce a model capable to describing thermal fluctuations and provide theoretical expressions for the gap and $dI/dV$. Then we fit this $dI/dV$ by Dynes ansatz (\[DDOS\]),(\[didv\]) and compare the resulting gap with (\[SPAgap\]).Finally we propose a generalization of Dynes ansatz that broadens the information obtained from STM experiments.
Theoretical description of thermal fluctuations in zero dimensional superconductors
===================================================================================
In order to test Dynes procedure in the nanoscale we study the reduced BCS Hamiltonian by path integral techniques in the static path approximation (SPA) [@static] which provide a good description of thermal fluctuations in the zero dimensional limit corresponding to grain sizes much smaller than the superconducting coherence length. Here we only present the main results and refer to [@static; @nmat1] for details. The partition function $Z$ is given by, $$\begin{aligned}
\frac{Z}{Z_{0}} & = & \int d\left|\Delta\right|\,\left|\Delta\right|\, e^{-\beta\delta^{-1}\mathcal{A}(\left|\Delta\right|)}\end{aligned}$$ with $$\begin{aligned}
\mathcal{A}\left(\left|\Delta\right|\right) & = & \lambda^{-1}\left|\Delta\right|^{2}-\int_{-E_{D}}^{E_{D}}d\varepsilon\,\varrho\left(\varepsilon\right)\left[\left(\xi-\left|\varepsilon\right|\right)+\frac{2}{\beta}\log\left(\frac{e^{-\beta\xi}+1}{e^{-\beta\left|\varepsilon\right|}+1}\right)\right] \end{aligned}$$ where $\xi=\sqrt{\varepsilon^{2}+\left|\Delta\right|^{2}}$, $\beta = 1/T$ and $\varrho\left(\varepsilon\right)=\sum_{\alpha}\delta\left(\varepsilon-\varepsilon_{\alpha}\right)$ is the one particle density of states, $Z_{0}$ is the parition function for non-interacting electrons, $\lambda$ is the dimensionless coupling constant, $E_D$ is the Debye energy and $\delta$ is the mean level spacing. From the explicit knowledge of $Z$, the normalized density of states (DOS) $N_{s}(\omega)$ is, $$\begin{aligned}
N_{s}(\omega) & = & \left(\frac{Z}{Z_{0}}\right)^{-1}\int d\left|\Delta\right|\,\left|\Delta\right|e^{-\beta\delta^{-1}\mathcal{A}(\left|\Delta\right|)}\int_{-E_{D}}^{E_{D}}d\varepsilon\,\varrho \left(\varepsilon\right)\im\left[\frac{\omega + i \Gamma_S}{\sqrt{\Delta^2-(\omega - i \Gamma_S)^2}} \right].\label{SPADOS}\end{aligned}$$ where $\Gamma_S(T)$ is a phenomenological parameter that describes all decoherence mechanisms except thermal fluctuations. The superconducting gap $\bar \Delta$ in this formalism is given by, $$\begin{aligned}
{\bar \Delta}^{2} & = & \left(\frac{Z}{Z_{0}}\right)^{-1}\int d\left|\Delta\right|e^{-\beta\delta^{-1}\mathcal{A}(\left|\Delta\right|)}\left|\Delta\right|^{3}\left[\lambda\int_{-E_{D}}^{E_{D}}d\varepsilon\,\varrho\left(\varepsilon\right)\frac{\tanh\left(\frac{\beta\xi}{2}\right)}{2\xi}\right]^{2}\label{SPAgap}.\end{aligned}$$ We note that $\bar \Delta$ leads to the bulk gap $\Delta_0 = \dfrac{E_D}{\sinh{1/\lambda}}$ for $L \to \infty$ however finite size effects in $\bar \Delta$ might differ slightly from those in the spectral gap [@richardson].
Justification of Dynes ansatz in the nanoscale region
=====================================================
We now test Dynes ansatz by comparing $\Delta_D(T)$ (\[DDOS\]) with ${\bar \Delta}(T)$ (\[SPAgap\]) where the former is obtained by fitting the theoretical $dI/dV$ (obtained from (\[SPADOS\])) with the Dynes $dI/dV$ coming from (\[DDOS\]). Dynes ansatz is presumably applicable provided that the low energy excitations are well defined quasiparticles - Fermi liquid theory holds - and a gap do exist in the spectrum such that the quasiparticle dispersion relation is $E = \sqrt{\epsilon^2+\Delta^2}$ even if $\Delta$ is not the BCS prediction. We show in Fig. \[Dynes\_SPA\]a that Dynes expression fits very well the theoretical $dI/dV$ and in Fig. \[Dynes\_SPA\]b that the energy gap $\Delta_D(T)$, obtained from (\[DDOS\]), is in excellent agreement with the theoretical prediction (\[SPAgap\]) that includes thermal fluctuations. As $\delta$ increases a small difference is observed. It is therefore likely that, as fluctuations become dominant, (\[DDOS\]) is less accurate. It is therefore justified to employ Dynes ansatz to fit the experimental $dI/dV$ in the nanoscale region provided that fluctuations are not dominant $\delta \ll {\rm max}(\Delta_0,T_c)$. For larger fluctuations the generalized expressions (\[SPADOS\]), (\[SPAgap\]) provides a suitable generalization of Dynes expression in the nanoscale where now the fitting parameters are $\lambda$ and $\Gamma_S$ but $\Gamma_S \neq \Gamma_D$ since $\Gamma_D$ does not include the effect of thermal fluctuations. Therefore this generalization of the Dynes ansatz allows to single out the effect of fluctuations from other forms of decoherence.
 $dI/dV$ (\[didv\]) as a function of $eV$ for $\delta = 0.1\Delta_0$, $\Gamma = 0.1\Delta_0$, $E_D = 9\Delta_0$, $\lambda=0.34$. Red lines stand for the theoretical $dI/dV$ from (\[DDOS\]) and dots correspond to Dynes fitting (\[SPADOS\]). We assume $\rho(\epsilon) \approx 1/\delta$, (b) the order parameter for the same parameters as in (a). Red lines ${\bar \Delta(T)}$ (\[SPAgap\]) correspond to the SPA prediction and black dots $\Delta_{D}(T)$ stand for Dynes fitting, (c) $\Gamma_{\text{th}}(T)$ for the same parameters as in (b) (see text for more details). ](fig1.pdf){width="1\columnwidth"}
Quantifying fluctuations in STM experiments
===========================================
The analysis of experimental STM data by the SPA formalism (\[SPADOS\]), (\[SPAgap\]) developed above has clear advantages over (\[DDOS\]) even for sizes in which Dynes ansatz describes well the energy gap. We focus on the study of $\Gamma(T)$ which includes effects leading to a finite quasiparticle life-time and thermal fluctuations as well. We note that $\Gamma=\Gamma_{D}$ in (\[DDOS\]) englobes two contributions: $\Gamma_{D}=\tau^{-1}+\Gamma_{\text{th}}$ where the former accounts for the quasiparticle life time and the latter is related to thermal fluctuations arising due to finite size effects. It is therefore expected that $\Gamma_{\text{th}}$ has a peak around the mean-field critical temperature with a typical width that describes the interval of temperatures where fluctuations are important. A important drawback of Dynes approach is that it is not possible to disentangle these two contributions. The situation is different if the SPA formalism (\[SPADOS\]),(\[SPAgap\]) is employed. Here thermal fluctuations are included so that $\Gamma_S \equiv 1/\tau$ [*only*]{} accounts for the finite quasiparticle lifetime. This is an important difference. One can quantitatively estimate the role of thermal fluctuations by the quantity $\Gamma_{\text{th}}=\Gamma_{D}-\Gamma_S$ which can be obtained by fitting the experimental data by both Dynes ansatz (\[DDOS\]) and the SPA expression (\[SPADOS\]). This quantity has a maximum (see Fig. \[Dynes\_SPA\]c) near the critical temperature where thermal fluctuations are more important. Therefore it can be used to estimate: a)the importance of thermal fluctuations with respect to other sources of decoherence, b) the (would-be) critical temperature and c) the region around $T_c$ where fluctuations are relevant. Similar findings are applicable to $T \ll T_c$ provided that quantum fluctuations are also included in the theoretical formalism. In summary, the use of a theoretical framework that takes into account deviations from mean-field to analyse experimental data expands substantially the information that can be obtained from STM experiments.
[99]{} R. C. Dynes, et al., Phys Rev. Lett. 53, 2437 (1984). G. Binnig, H. Rohrer, IBM Journal of Research and Development 30: 4 (1986). J. von Delft and D.C. Ralph, Phys. Rep., 345, 61 (2001). S. Bose, et al., Nature Mat. 9, 550 (2010). I. Brihuega, et al., arXiv:0904.0354, Phys. Rev. B. in press. R. Denton, B. Mulschlegel, and D.J. Scalapino, Phys. Rev. B **7**, 3589 (1973).
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abstract: 'Exact Lagrangian in compact form is derived for planar internal waves in a two-fluid system with a relatively small density jump (the Boussinesq limit taking place in real oceanic conditions), in the presence of a background shear current of constant vorticity, and over arbitrary bottom profile. Long-wave asymptotic approximations of higher orders are derived from the exact Hamiltonian functional in a remarkably simple way, for two different parametrizations of the interface shape.'
author:
- 'V. P. Ruban'
title: 'Nonlinear interfacial waves in a constant-vorticity planar flow over variable depth '
---
Large-amplitude internal waves are important phenomena in near-coastal ocean dynamics (see, e.g., Refs.[@AHLT_1985; @LHA_1985; @BRAB_1997; @DR_1978; @VBR_2000; @VH_2002; @GPTK_2004; @VS_2006], and references therein). A number of theoretical models is used to study internal waves analytically, including the most popular two-fluid model [@B_1966; @B_1967; @O_1975; @J_1977], where the system is placed between fixed upper and bottom boundaries and consists of two fluid layers having different constant densities ($\rho_1=1$ in the upper layer, and $\rho_2=1+\epsilon$ in the lower layer). In the simplest variant, the rigid walls are both horizontal, and the flow is potential within each layer, but additional ingredients can be incorporated into the model, such as a background shear current and large bathymetric variations (see Refs.[@BBVG_1993; @SL_2002; @C_2006; @deZN_2008; @deZVNC_2009], and references therein). In the general case, an analytical study of this system requires quite lengthy calculations, especially when fully nonlinear dispersive approximations of higher orders are considered [@deZN_2008; @deZVNC_2009; @CC_1999; @OG_2003; @CGK_2005; @CCMRS_2006; @BLS_2008; @CBJ_2009; @DDK_2010]. In the present work, it will be shown that in the case $\epsilon\ll 1$ (the so called Boussinesq limit), there exists an elegant and remarkably short way how to derive fully nonlinear dispersive models of high orders. It should be noted that in real oceanic circumstances $\epsilon$ is indeed small, typically $\epsilon\sim 0.01$, and therefore the theory developed below is of practical importance. Moreover, our approach easily takes into account the presence of a shear background current of constant vorticity, as well as arbitrary variations of the bed profile, and that makes the model more adequate for studying the real-world processes of interaction of internal waves with tidal currents over non-uniform sea bed. We also concern in this work the problem of proper parametrization of steep wave profiles and, besides the usual single-valued $y=\eta(x,t)$ representation, we consider an alternative parametrization which is more suitable to describe arbitrary steep and even multi-valued wave profiles.
*Variational formalism. —* The method employed here is based on a variational formulation for the interface dynamics in a two-fluid planar system having constant vorticities $\Omega_{1,2}$ within each layer. Let the (unknown) interface profile be $y=\eta(x,t)$, where $y$ is the vertical coordinate, $x$ is the horizontal coordinate, and $t$ is the time variable. A two-dimensional (2D) velocity field can be represented as follows (subscripts $1,2$ numbering the layers are omitted), $${\bf v}(x,y,t)=\Omega {\bf S}(x,y) +\nabla\varphi(x,y,t),$$ where the stationary divergence-free field ${\bf S}(x,y)$ satisfies equation $\mbox{curl }{\bf S}=1$ and has zero normal component at the rigid boundaries. The unknown potentials $\varphi_{1,2}(x,y,t)$ satisfy the Laplace equation $(\partial_x^2+\partial_y^2)\varphi_{1,2}=0$, with the boundary conditions $(\nabla\varphi_{1,2}\cdot{\bf n})=0$ at a rigid wall (where ${\bf n}$ the normal unit vector), and $\varphi_{1,2}[x,\eta(x,t),t]=\psi_{1,2}(x,t)$ at the interface. Since there is a kinematic boundary condition at the interface, $$({\bf v}_1\cdot {\bf n})=({\bf v}_2\cdot {\bf n})=V_n=\eta_t/\sqrt{1+\eta_x^2},$$ functions $\psi_{1}(x,t)$ and $\psi_{2}(x,t)$ are related to each other by a linear integral operator depending on $\eta(x,t)$. Therefore there are only two basic unknown functions, $\eta(x,t)$ and, for instance, $$\psi(x,t)=\psi_1(x,t)-(1+\epsilon)\psi_2(x,t).$$ It can be proved that the Lagrangian functional for the system under consideration has the following structure, $$\label{L_gamma}
{\cal L}_{general}=-\int\psi\eta_t dx +\frac{\gamma}{2}\int\eta\partial_x^{-1}\eta_t dx
-{\cal H}\{\eta,\psi\},$$ where $\gamma=(1+\epsilon)\Omega_2-\Omega_1$, and the Hamiltonian functional is the total energy of the system — kinetic plus potential. The general idea of the proof is similar to Refs.[@W_2007; @W_2008; @R2010JETP], and it is based on a generalization of the Bernoulli equation for constant-vorticity 2D incompressible flows (subscripts $1,2$ are omitted), $$\partial_t\varphi+\Omega\Theta+{\bf v}^2/2+g y +P/\rho=0,$$ where $\Theta(x,y,t)$ is the total stream function in the layer, $g$ is the gravity acceleration, and $P(x,y,t)$ is the pressure. At the interface, $\Theta$ and $P$ should be continuous, and this requirement gives us the dynamic boundary condition $$\label{dyn_boundary_cond}
\Big[(1+\epsilon)\partial_t\varphi_2-\partial_t\varphi_1 +\gamma\Theta
+(1+\epsilon)\frac{{\bf v}_2^2}{2}-\frac{{\bf v}_1^2}{2}\Big]\Big|_{y=\eta}+\epsilon g \eta=0.$$ Since, by definition, the relation $$\partial_t\psi_{1,2}=[\partial_t\varphi_{1,2}+\eta_t \partial_y\varphi_{1,2}]|_{y=\eta}$$ takes place, and $\eta_t=\partial_x(\Theta|_{y=\eta})$, Eq.(\[dyn\_boundary\_cond\]) is equivalent to $$\label{Ham_dynamic}
\psi_t+\gamma\partial_x^{-1}\eta_t=\delta{\cal H}/\delta\eta,$$ while the kinematic boundary condition can be represented as $$\label{Ham_kinematic}
-\eta_t=\delta{\cal H}/\delta\psi.$$ Equations (\[Ham\_dynamic\]) and (\[Ham\_kinematic\]) are equivalent to Eq.(\[L\_gamma\]). Let us for example prove that $V_n\sqrt{1+\eta_x^2}=-\delta{\cal H}/\delta\psi$ (consideration of the dynamic boundary condition is slightly more involved but analogous). Since $\psi$ is present in the kinetic energy only, the corresponding variation of the Hamiltonian is $$\begin{aligned}
\delta{\cal H}&=&\!\int_{D1} \!({\bf v}_1\cdot\nabla\delta\varphi_1) dx dy
+(1+\epsilon)\!\int_{D2}\! ({\bf v}_2\cdot\nabla \delta\varphi_2) dx dy\nonumber\\
&=&\int V_n\sqrt{1+\eta_x^2}[(1+\epsilon)\delta\psi_2-\delta\psi_1]dx,\end{aligned}$$ where we have integrated by parts and used the boundary conditions and incompressibility. Taking into account the definition of $\psi$, we arrive at the required result.
*Exact Hamiltonian theory in Boussinesq limit. —* The main technical difficulty for application of the Hamiltonian formalism at finite $\epsilon$ is the absence of compact expression for the kinetic energy of the flow in terms of functions $\eta$ and $\psi$. Therefore, below we consider the case $\epsilon\ll 1$ and, for simplicity, $\Omega_1=\Omega_2=\Omega$, resulting in $\gamma\approx 0$. What is very essential, in this limit $\psi\approx\psi_1-\psi_2$, which means that $-\psi_x(x,t)dx$ is the strength of singular vorticity concentrated at the interface. Since the kinetic energy of the system in the Boussinesq limit is given by integral $
{\cal K}_{total}\approx(1/2)\int\Theta\Omega_{total}dxdy,
$ and $$(\partial_x^2+\partial_y^2)\Theta=-\Omega_{total}=-\Omega+\psi_x(x,t)\delta[y-\eta(x,t)],$$ the above observation allows us to express the kinetic energy in a closed form through the Green’s function of the 2D Laplace operator in the domain between the fixed boundaries, with zero boundary conditions. An explicit expression for the Green’s function is known in terms of curvilinear conformal coordinates $u$ and $v$, with $v=0$ at the upper boundary, $v=-1$ at the bottom, $(\partial_x^2+\partial_y^2)v=0$ inside the domain, and $u(x,y)$ being a harmonically conjugate for $v(x,y)$ [@R2004PRE; @R2010PRE]. In other words, $x+iy=z(u+iv)$, where $z(w)$ is an analytic function of complex variable $w=u+iv$. We can write the Lagrangian in terms of the interface profile $v=-q(u,t)$ and $\psi(u,t)$, $${\cal L}=\int\psi J(u,q)q_t dx -{\cal H}\{\psi,q\} ,$$ where $J(u,q)=|z'(u-iq)|^2$ is the Jacobian. The corresponding equations of motion are $$\label{Ham_eqs}
J(u,q)q_t=\delta{\cal H}/\delta\psi,\qquad -J(u,q)\psi_t=\delta{\cal H}/\delta q.$$ The Hamiltonian consists of three parts, ${\cal H}={\cal P}+{\cal K}+{\cal S}$. The potential energy is $$\label{P}
{\cal P}=\frac{\epsilon g}{2}\int[\mbox{Im }z(u-iq)]^2\mbox{Re}[\partial_uz(u-iq)]du.$$ ${\cal K}$ is the kinetic energy in the absence of external current, $$\begin{aligned}
{\cal K}&=&\frac{1}{4\pi}\iint\ln\Big|\frac{\sinh[(\pi/2)(u_2-u_1-i(q_2+q_1))]}
{\sinh[(\pi/2)(u_2-u_1-i(q_2-q_1))]}\Big|\nonumber\\
&&\qquad\qquad\times\psi'_1\psi'_2d u_1 du_2,\end{aligned}$$ where $q_1=q(u_1,t)$, $\psi'_1=[\partial_u\psi(u,t)]|_{u=u_1}$, and so on. The functional ${\cal S}$ takes into account the presence of a background shear current, $$\label{S}
{\cal S}=\int[Cq-\Omega\Theta_s(u,-q)]\psi' du +\mbox{const}.$$ Here $C$ is a total flux of the current, and $\Theta_s(u,v)$ is the function which satisfies zero boundary conditions and equation $(\partial_x^2+\partial_y^2)\Theta_s=-1$. Usually it is assumed that the upper rigid boundary is horizontal at $y=0$, and that results in $y(u,-v)=-y(u,v)$. Then the stream function $\Theta_s(u,v)$ can be represented as follows (the idea how to calculate $\Theta_s$ is similar to Ref.[@R2008PRE]), $$\label{Theta_s}
\Theta_s(u,v)=-\frac{1}{2}\Big\{y^2(u,v)+
\Big[\frac{\sinh\hat k v}{\sinh\hat k}\Big]y^2(u,-1)\Big\},$$ where $\hat k=-i\partial_u$ is the differential operator, which is diagonal in Fourier representation.
Let us say here that the case $\gamma\not= 0$, though more cumbersome, can be considered in a similar manner.
*Long-wave approximations. —* Thus, all the terms in exact Hamiltonian functional have been explicitly specified. However, this exact description cannot be applied easily for analytical and numerical studies in view of the strongly non-local character of the Hamiltonian. This situation with internal waves is in contrast with the exact description of 2D surface waves of constant vorticity [@R2008PRE], where at least numerical implementation is simple and efficient with a fast Fourier transform. That is why the problem of simplified description for moderately steep interfacial waves has attracted much attention in last years [@deZN_2008; @deZVNC_2009; @CC_1999; @OG_2003; @CGK_2005; @CCMRS_2006; @BLS_2008; @CBJ_2009; @DDK_2010]. Here we suggest a simple procedure for the Boussinesq case. We see that the only non-local part in the total Hamiltonian is ${\cal K}$. To derive simplified quasi-local approximations, we will essentially use a Fourier transform of the Green’s function, $$\frac{1}{2\pi}\ln\Big|\frac{\sinh[(\pi/2)(\tilde u-is)]}
{\sinh[(\pi/2)(\tilde u-ia)]}\Big|=\int F(k,a,s)e^{ik\tilde u}\frac{dk}{2\pi},$$ where $\tilde u=u_2-u_1$, $a=|q_2-q_1|$, $s=q_2+q_1$, and $$\label{F_kas}
F(k,a,s)=\frac{\cosh k(1-a)-\cosh k(1-s)}{2k\sinh k}.$$ This expression can be easily obtained by the known methods of contour integration of analytic functions. It is the simplicity of Eq.(\[F\_kas\]) that allows us to derive the mentioned dispersive models of high orders for nonlinear internal waves. The approximations correspond to expansion of $F$ in powers of small quantities $ka$ and $ks$. Therefore, the long-wave assumption actually includes three different asymptotic regimes: (i) deep-water theory, when $k\gg 1$ but $kq\ll 1$, (ii) finite-depth theory, when $q\ll 1$ and $k\sim 1$, (iii) shallow-water theory, when $q\sim 1$ and $k\ll 1$ \[for self-consistency, in this regime the bottom variations should be long-scaled as well, $|x''(u)|/x'(u)\ll 1$, where $x(u)=z(u+0i)$ is a purely real function\]. Thus, we write $$\begin{aligned}
F&=&\frac{s-a}{2}+\frac{a^2-s^2}{4}k\coth k+\cdots\nonumber\\
&+&\frac{(s^{2n-1}-a^{2n-1})}{2(2n-1)!}k^{2n-2}\nonumber\\
&+&\frac{(a^{2n}-s^{2n})}{2(2n)!}k^{2n-1}\coth k +\cdots.\end{aligned}$$ It is not difficult to understand that each term proportional to $a^{2m+1}k^{2m}$ gives zero contribution to the Hamiltonian. The other terms are easily transformed to finite sums of products as $\psi'_1q_1^{m_1}\psi'_2q_2^{m_2}$, thus resulting in the quasi-local expansion ${\cal K}={\cal K}_1+{\cal K}_2+{\cal K}_3+\cdots$, where $$\begin{aligned}
{\cal K}_1&=&\frac{1}{2}\int q\psi'^2 du,\\
{\cal K}_2&=&-\frac{1}{2}\int\psi'_1 q_1\psi'_2 q_2 [k\coth k]e^{ik(u_2-u_1)}
\frac{dk}{2\pi}du_1du_2
\nonumber\\
&=&-\frac{1}{2}\int q\psi'[\hat k\coth\hat k](q\psi')du,\end{aligned}$$ $$\begin{aligned}
{\cal K}_3&=&\frac{1}{4\cdot3!}\int(q_1+q_2)^3\psi'_1\psi'_2 e^{ik(u_2-u_1)}
\frac{dk}{2\pi}du_1du_2
\nonumber\\
&=&\frac{1}{12}\int [\psi''(q^3\psi')' +3(q^2\psi')'(q\psi')'] du,\end{aligned}$$ $$\begin{aligned}
{\cal K}_4&=&\frac{1}{4\cdot4!}\int [(q_1-q_2)^4-(q_1+q_2)^4]\psi'_1\psi'_2\nonumber\\
&&\qquad\qquad\times [k^3\coth k]e^{ik(u_2-u_1)}\frac{dk}{2\pi}du_1du_2
\nonumber\\
&=&-\frac{1}{6}\int (q\psi')'[\hat k\coth\hat k](q^3\psi')' du.\end{aligned}$$ Analogously, we derive $$\begin{aligned}
{\cal K}_5&=&\frac{1}{240}\int\Big[(q^5\psi')''\psi'''+5(q^4\psi')''(q\psi')''\nonumber\\
&&\qquad\qquad+10(q^3\psi')''(q^2\psi')''\Big]du,\\
{\cal K}_6&=&-\frac{1}{360}\int\Big[3(q^5\psi')'' [\hat k\coth\hat k](q\psi')''\nonumber\\
&&\qquad\qquad+10(q^3\psi')'' [\hat k\coth\hat k](q^3\psi')''\Big]du,\\
{\cal K}_7&=&\frac{1}{2\cdot 7!}\int\Big[(q^7\psi')'''\psi^{(4)}
+7(q^6\psi')'''(q\psi')'''\nonumber\\
&+&21(q^5\psi')'''(q^2\psi')'''\!+\!35(q^4\psi')'''(q^3\psi')'''\Big]du.\end{aligned}$$ It is a simple exercise to calculate the variational derivatives of the obtained approximate Hamiltonian and substitute them into Eqs.(\[Ham\_eqs\]). Hopefully, with appropriate regularization at short scales, the equations will appear to be convenient for future numerical implementation, since all the linear operators (the differentiations and $[\hat k\coth\hat k]$) are diagonal in Fourier representation.
The functionals ${\cal P}$ and ${\cal S}$ take a simplified form in the case of a slowly varying depth, when the series $$\label{z_expansion}
z(u-iq)=x(u)-i x'(u)q-x''(u)\frac{q^2}{2}+ix'''(u)\frac{q^3}{6} +\cdots$$ rapidly converges. Then substitution of Eq.(\[z\_expansion\]) into Eqs.(\[P\]) and (\[Theta\_s\]) gives us $${\cal P}\approx \frac{\epsilon g}{2}\int
\Big\{q^2x'^3+q^4\Big[\frac{(x'^2x'')'}{4}-\frac{5}{6}x'^2x'''\Big]\Big\}du,$$ $$\begin{aligned}
{\cal S}&\approx& \int \psi'
\Big\{Cq+\frac{\Omega}{2}\Big[q^2\Big(x'^2-\frac{x'x'''}{3}q^2\Big)\nonumber\\
&&-q\Big(x'^2-\frac{x'x'''}{3}\Big)+
\frac{(x'^2)''}{6}\Big(q^3-q\Big)\Big]\Big\}du.\end{aligned}$$ Note that the Jacobian is this case is $$J(u,q)\approx x'^2+[(x'')^2-x'x''']q^2.$$
In the shallow-water regime $k\coth k\approx1+k^2/3$, and ${\cal K}$ takes a purely local form, $$\begin{aligned}
{\cal K}_{local}&=&\frac{1}{2}\int q(1-q)\psi'^2 du\nonumber\\
&+&\frac{1}{12}\int\left\{(q\psi')'[(1-q)\psi']'-[q(1-q)\psi']\psi''\right\}du\nonumber\\
&-&\frac{1}{12}\int[q^2(1-q)\psi']'[(1-q)\psi']'du\nonumber\\
&-&\frac{1}{12}\int[(1-q)^2q\psi']'(q\psi')'du +{\cal O}\{\partial_u^6\},\end{aligned}$$ which is apparently symmetric with respect to change $q\to (1-q)$. This symmetry is present also in the exact expression (\[F\_kas\]), since if $a\to a$ and $s\to (2-s)$, then $F\to F$.
*Alternative parametrization. —* Strongly nonlinear internal waves are known to have the tendency toward overturning their profiles. The above theory can be generalized to admit more steep, and even multi-valued dependences $y=\eta(x,t)$. To explain the basic idea how to manage in that case, we consider here the deep-water regime in the absence of current. An arbitrary interface shape can be represented in a parametric form, $$x=X(\sigma,t)\equiv\sigma +\tilde X(\sigma,t),\qquad y=Y(\sigma,t)<0,$$ with a parameter $-\infty<\sigma<+\infty$ along the curve. What is important, smooth functions $\tilde X(\sigma,t)$ and $Y(\sigma,t)$ are able to represent quite steep wave profiles. The corresponding Lagrangian is $$\begin{aligned}
&&{\cal L}_{param}=\int[X' Y_t-X_t Y']\psi d\sigma-
\frac{\epsilon g}{2}\int Y^2X' d\sigma\nonumber\\
&&-\frac{1}{8\pi}\!\int\!\ln\Bigg[\frac{(X_2\!-\!X_1)^2\!+\!(Y_2\!+\!Y_1)^2}
{(X_2\!\!-X_1)^2\!+\!(Y_2\!-\!Y_1)^2}\Bigg]\psi'_1\psi'_2 d\sigma_1 d\sigma_2.\end{aligned}$$ To simplify the non-local term $\tilde{\cal K}$, we again use a Fourier transform of the Green’s function, but in a slightly different manner, $$\begin{aligned}
&&\frac{1}{4\pi}\ln\Bigg[\frac{(\sigma_2-\sigma_1+\tilde X_2-\tilde X_1)^2+(Y_2+Y_1)^2}
{(\sigma_2-\sigma_1+\tilde X_2-\tilde X_1)^2+(Y_2-Y_1)^2}\Bigg]\nonumber\\
&&=\int\tilde F(\kappa,\tilde X_2,Y_2,\tilde X_1,Y_1)
e^{i\kappa(\sigma_2-\sigma_1)}\frac{d\kappa}{2\pi},\end{aligned}$$ with \[note the appearance of factor $\exp\{i\kappa(\tilde X_2-\tilde X_1)\}$\] $$\label{tilde_F}
\tilde F=e^{i\kappa(\tilde X_2-\tilde X_1)}
\Big[\frac{e^{-|\kappa||Y_2-Y_1|}-e^{-|\kappa||Y_2+Y_1|}}{2|\kappa|}\Big].$$ Now we expand the exponents in Eq.(\[tilde\_F\]) in powers of the arguments, and take into account that terms proportional to $(\tilde X_2-\tilde X_1)^m|Y_2-Y_1|^{2n+1}k^{m+2n}$ give zero contribution to the Hamiltonian. As the result, we obtain $$\begin{aligned}
\tilde {\cal K}&=&-\frac{1}{2}\int Y\psi'^2d\sigma
-\frac{1}{2}\int(Y\psi')|\hat\kappa|( Y\psi')d\sigma\nonumber\\
&&-\frac{1}{2}\int[\tilde X Y\psi'\psi''+\tilde X \psi'(Y\psi')']d\sigma\nonumber\\
&&-\frac{1}{12}\int[\psi''(Y^3\psi')' +3(Y^2\psi')'(Y\psi')']d\sigma\nonumber\\
&&+\frac{1}{4}\int\Big[(Y\tilde X^2\psi')'\psi''
-2(Y\tilde X\psi')'(\tilde X\psi')'\nonumber\\
&&\qquad\qquad\qquad +(Y\psi')'(\tilde X^2\psi')'\Big]d\sigma\nonumber\\
&&+\int Y\psi'|\hat\kappa|(\tilde X Y\psi')'d\sigma+{\cal O}\{\partial_\sigma^5\},
\label{tilde_K}\end{aligned}$$ where $|\hat\kappa|=|\partial_\sigma|$ is a pseudo-differential operator.
It should be noted that if the parametrization of interface is arbitrary, without any relation between $\tilde X(\sigma,t)$ and $Y(\sigma,t)$, then the tangential component of the interface motion (the combination $X'X_t+Y'Y_t$) is not determined by variational equations of motion, and therefore it remains arbitrary. One can somehow fix the parametrization, for example by relation $\tilde X(\sigma)=-\hat H Y(\sigma)$, where $\hat H=i\,\mbox{sign }\hat\kappa$ is the Hilbert operator. Such parametrization is used in the theory of surface waves at the deep water (see Refs.[@DKSZ96; @DLZ95; @RD2005PRE], and references therein). For the present problem this choice has no special meaning, it is only important that arbitrary 2D curves can be represented in this way. Then equations of motion for the two basic functions $Y(\sigma,t)$ and $\psi(\sigma,t)$ can be written in a non-canonical form (for technical details, see Ref.[@RD2005PRE]), $$\begin{aligned}
Y_t&=&\mbox{Im}\Big\{iZ'(1+i\hat H )\Big[{(\delta\tilde{\cal K}/\delta\psi)}/
{|Z'|^2}\Big]\Big\},\nonumber\\
\psi_t&=&{\mbox{Im}\Big\{(1-i\hat H)
\left[2({\delta\tilde{\cal K}}/{\delta Z})Z'
+ ({\delta\tilde{\cal K}}/{\delta\psi})\psi'\right]\Big\}}/{|Z'|^2}\nonumber\\
&&-\epsilon g\,\mbox{Im\,}Z
-\psi'\hat H\left[{(\delta\tilde{\cal K}/\delta\psi)}/{|Z'|^2}\right],\end{aligned}$$ where $\psi'=\partial_\sigma\psi(\sigma,t)$, $Z'=\partial_\sigma Z(\sigma,t)$, and $$Z(\sigma,t)=X(\sigma,t)+iY(\sigma,t)=\sigma+(i-\hat H)Y(\sigma,t),$$ $$2(\delta\tilde{\cal K}/\delta Z)=(\delta\tilde{\cal K}/\delta \tilde X)
-i(\delta\tilde{\cal K}/\delta Y).$$ With appropriate regularization at very short scales, the above equations can be efficiently simulated on computer using fast Fourier transform routines. It should be noted that for correct treatment of quite steep waves, say when $\kappa Y\sim A$ with $A\gtrsim 1$, one has to expand $\tilde F$ in Eq.(\[tilde\_F\]) up to a sufficiently high order $N$ satisfying the condition ${(2A)^{N+1}}/{(N+1)!}\ll 1$. For example, the third-order approximation Eq.(\[tilde\_K\]) can be good only up to $\kappa Y\approx 1$.
*Summary and discussion. —* To summarize, in this work a simple method has been suggested for derivation of higher-order dispersive approximations in the theory of fully nonlinear planar interfacial waves with a small density jump, in a constant-vorticity flow over non-uniform bed. Explicit quasi-local expressions have been presented up to the 7th order. The method is based on a variational formulation of the interface dynamics, and it uses an expansion of a Fourier transform of the Green’s function entering a non-local part of the exact Hamiltonian.
It should be noted that analogous expansion of a Fourier-transformed Green’s function can be used also in three-dimensional case for the deep-water purely potential regime. Another possible application concerns the recently suggested two-layer compressible atmospheric model [@R2010JETP]. Details of the corresponding studies will be published elsewhere.
These investigations were supported by RFBR (grants 09-01-00631 and 07-01-92165), by the “Leading Scientific Schools of Russia” grant 6885.2010.2, and by the Program “Fundamental Problems of Nonlinear Dynamics” from the RAS Presidium.
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abstract: 'The microquasar V404 Cygni (also known as GS 2023+338) was previously reported to have possible GeV $\gamma$-ray emission in two days during its 2015 outburst. In order to provide more detailed information at the high energy range for this black hole binary system, we conduct detailed analysis to the data obtained with the Large Area Telescope (LAT) onboard [*the Fermi Gamma Ray Space Telescope (Fermi)*]{}. Both LAT database and source catalog used are the latest. From the analysis, we can not confirm the previous detection. Instead, we find one possible detection ($\sim 4\sigma$) of the source at the end of the outburst in the time period of 2015 Aug. 17–19, and one convincing detection ($\simeq 7\sigma$) in 2016 Aug. 23–25. The latter shows that the [$\gamma$-ray]{} emission of the source is soft with photon index $\Gamma\sim 2.9$, mostly detected below $\sim 1.3$ GeV with [*Fermi*]{} LAT. As [$\gamma$-ray]{} emission from microquasars is likely associated with their jet activity, we discuss the results by comparing with those well studied cases, namely Cyg X-3 and Cyg X-1. The detection establishes V404 Cygni as one of four microquasars with detectable $\gamma$-ray emission, and adds interesting features to the small group, or in a more general context to X-ray binaries with jets.'
author:
- Yi Xing
- Zhongxiang Wang
title: 'Detection of the microquasar V404 Cygni at $\gamma$-rays revisited: only one flaring event in quiescence'
---
Introduction
============
The X-ray binary [V404 Cygni]{} (or GS 2023+338) is a stellar-mass black hole system having a $\sim 1\ M_{\sun}$ low-mass companion star orbiting around a $\sim 9\ M_{\sun}$ black hole with an orbital period of 6.5 days [@ccn92; @cc94; @kfr10]. The binary has a distance of 2.39kpc [@mil+09]. Long after a previous outburst in 1989 [@mak89], the source underwent another outburst in 2015. Both events triggered extensive observations at multi-frequencies. Particularly for the second outburst, which lasted from 2015 Jun. 15 [@bar+15] to mid Aug. [@siv+15; @plo+17], monitoring observations from radio to $\gamma$-rays were carried out. Different aspects of the binary, with particular attention to physical processes related to the black hole accretion (see, e.g., @bm16), have been learned from the observations (e.g., @rod+15 [@plo+17; @wal+17; @tet+17; @mai+17] and references therein). In addition at the end of 2015, a mini-outburst was seen from the source, which lasted approximately a month and exhibited similar features as those in the main 2015 outburst (see @mun+17 and references therein).
This black hole binary also belongs to the microquasar category [@mr99], as jets associated with the black hole are believed to be present in its quiescence state (e.g., @gfh05 [@ran+16; @plo+19]) and jet ejection was observed in the 2015 outburst (e.g., @wal+17 [@tet+17; @tet+19]; however there are arguments that all black hole binaries may be considered as microquasars such as in @zha13). Since [$\gamma$-ray]{} emission is theoretically expected to arise from microqusars (e.g., @aa99 [@gak02; @rom+03]) and observationally the microqusars Cyg X-3 and Cyg X-1 were detected at $\gamma$-rays [@fer09; @tav+09; @sab+10; @bod+13; @mal+13], search for [$\gamma$-ray]{} emission from [V404 Cygni]{} was carried out. @loh+16 reported possible detection of the source ($\sim 4.5\sigma$) in a 12-h time bin on 2015 Jun. 26 during the second outburst, where the data were obtained with the Large Area Telescope (LAT) onboard the [*Fermi Gamma Ray Space Telescope (Fermi)*]{}. The detection was seemingly confirmed by the [*AGILE*]{} observation, as @pia+17 reported their detection of the source near the same time in the energy range of 50–400MeV (although the detection significance was $\sim$4.3$\sigma$). In addition, very-high-energy [$\gamma$-ray]{} observations were also conducted by the MAGIC telescopes in 2015 Jun., but no detection was found in the energy range of 200–1250 GeV [@ahn+17].
Given that [*Fermi*]{} LAT has been collecting data for nearly 12 years and the database and source catalog were updated several times since the previous work by @loh+16, it is necessary to re-analyze the [$\gamma$-ray]{} data for [V404 Cygni]{} for the purpose of confirming the previous detection results and searching for new detection. We thus conducted detailed analysis. Different results were obtained. In this paper, we report the results.
LAT Data Analysis and Results {#sec:obs}
=============================
LAT data and source model
-------------------------
LAT has been scanning the whole sky continuously and collecting data in GeV band since 2008 Aug. [@atw+09]. We selected 0.1–500 GeV LAT events inside a $20\arcdeg\times 20\arcdeg$ region (region of interest; RoI) centered at the position of [V404 Cygni]{}, over the time period from 2008-08-04 15:43:36 (UTC) to 2020-03-05 01:16:35 (UTC; approximately 11.5 yrs). The latest [*Fermi*]{} Pass 8 database was used. Following the recommendations of the LAT team[^1], the events with quality flags of ‘bad’ and zenith angles larger than 90 degrees were excluded; the latter is to prevent the Earth’s limb contamination.
Based on the very recently released [*Fermi*]{} LAT 10-year source catalog (4FGL-DR2), we constructed a source model (for 4FGL, see @4fgl20). The sources listed in 4FGL-DR2 that are within a 20 degree radius circular region from [V404 Cygni]{} were included in the source model. Their spectral forms are provided in the catalog. In our analysis, the sources within 5 degrees from the target were set to have free spectral parameters, and for the other sources, their spectral parameters were fixed at the values given in the catalog. The background Galactic and extragalactic diffuse spectral models (gll\_iem\_v07.fits and iso\_P8R3\_SOURCE\_V2\_v1.txt respectively) were also included in the source model. Their normalizations were set as free parameters.
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"}
2015 outburst
-------------
Since both [*Fermi*]{} LAT and [*AGILE*]{} possibly detected [V404 Cygni]{} during its 2015 outburst near Jun. 26 (MJD 57199; @pia+17), we repeated the analysis given in @loh+16. Standard binned likelihood analysis to the 0.1–100 GeV LAT data in each 12-h bins during the outburst were performed, where each time bin was shifted by 6h forward (instead of 12h) when constructing the time bins. Emission at the source position was assumed to have a power law. Since the outburst approximately ended during 2015 mid Aug. [@siv+15], we extended the analysis to Aug. (compared to Jul. 17 as the end of data in @loh+16; see also their Figure 1). The obtained Test Statistic (TS) values were shown in Figure \[fig:olc\]. There is a relatively high TS point at MJD 57199, but the value is $<$9 (i.e., the detection significance is $<$3$\sigma$), not as high as that given in @loh+16. Instead, we found two TS$\sim 15$ points at MJD 57251 and 57251.25 (Aug. 17). Therefore based on our analysis to the Pass 8 data with the latest source catalog used, the claimed detection of [V404 Cygni]{} is questionable. For the data point at MJD 57251, we presented detailed analysis and results in the following Section \[sec:3\].
Search for possible detection in the LAT time period
----------------------------------------------------
We analyzed the 11.5 yr LAT data to search for possible [$\gamma$-ray]{} emission from [V404 Cygni]{}. We first performed the binned likelihood analysis to the whole selected LAT data. The energy range of 0.3–500 GeV was used given the relatively large uncertainties of the instrument response function of LAT in the $<$0.3 GeV energy range. Also in the low energy range, there is strong background emission or possible contamination from nearby sources for the regions along the Galactic plane (Galactic latitude of [V404 Cygni]{} is $-$2). No emission was found at the position of the source from the analysis, as the obtained TS$\sim 0$ (see the left panel of Figure \[fig:ts\]).
Given that both microquasars Cyg X-3 and Cyg X-1 show variable $\gamma$-ray emission (e.g., @cor+12 [@bod+13]) and were significantly detected in short time periods such as in one-day time bins (e.g., @sab+10 [@zdz+18]), we focused on searching for detection in different short time bins from the likelihood analysis. We tested different time bins, such as 1-day, 3-day, or 6.5-day (i.e., the orbital period of [V404 Cygni]{}; @cc94) and found that the analysis to the data in 3-day time bins (Figure \[fig:3lc\]) well show possible detection over the LAT data time period.
First there were two $\sim$440 day time periods containing several points with TS$>$9 (marked as 1 and 2 respectively in Figure \[fig:3lc\]). When we changed to use 1-day or 6.5-day bins, no significantly better results were obtained for the data points in the two time periods. We tested to perform the likelihood analysis to the LAT data in each of the two $\sim 440$ time periods, but the TS values were still low ($\sim$3). As [$\gamma$-ray]{} emission from a microquasar may be orbitally modulated [@fer09; @zdz+17], we also tested to divide the data in each time period into four orbital phase ranges, centered at phase 0.0, 0.25, 0.5, and 0.75, where the orbital parameters of [V404 Cygni]{} given by @cc94 were used. The TS results from the likelihood analysis indicated no detection in any of the orbital phase ranges. We concluded that no detection was found in the data of these two time periods.
![3-day bin TS values ([*bottom*]{} panel) obtained at the position of [V404 Cygni]{}. For the TS$>$9 bins, the fluxes are shown in the [*upper*]{} panel. Two $\sim$440 day time periods (1 and 2) and two data points (3 and 4) are marked based on the TS values. The time periods of the 2015 outburst and mini-outburst are marked by the dashed (start time) and dash-dotted (end time) lines.[]{data-label="fig:3lc"}](f3.eps){width="47.00000%"}
However there are two 3-day bin data points with TS$>16$, for which we mark with point 3 and 4 in Figure \[fig:3lc\]. The first is contained in the above time period 2 and was from the data in MJD 57251–57253. It is at the end of the 2015 outburst and was found to have TS$\sim$15 in our 12-h bin analysis (Figure \[fig:olc\]). The second was from the data in MJD 57623–57625 and is more significant given TS$\sim$45. Therefore we conducted detailed analysis for these two data points, which are given in the following sections.
{width="42.00000%"} {width="42.00000%"}
Possible detection in MJD 57251–57253 {#sec:3}
-------------------------------------
In order to confirm this possible detection, we calculated a TS map of a $3\arcdeg\times 3\arcdeg$ region centered at the position of [V404 Cygni]{}. The obtained TS map is shown in the middle panel of Figure \[fig:ts\]. Excess emission at the position of our target is clearly seen, with the maximum TS$\simeq$17.7. There are four catalog sources in the region. We note that although their positions are close to [V404 Cygni]{}, they caused negligible contamination to our analysis. Firstly the time period is short, only 3 days, and secondly these sources were not bright. For example the detection significances given in 4FGL-DR2 for the two closest sources are 16$\sigma$ and 5$\sigma$. (Even we kept the sources when calculating the TS map, there were no notable changes to the obtained TS map.) We ran *gtfindsrc* in [Fermitools]{} to determine the position of the excess emission. The resulting 1$\sigma$ nominal uncertainty is 01, and [V404 Cygni]{} is within the error circle (see Figure \[fig:ts\]). We performed the likelihood analysis to the data with the fitted position and obtained a photon index of $\Gamma= 2.5\pm$0.4 and a 0.3–500 GeV flux of $F_{0.3-500}= 1.1\pm 0.4\times 10^{-7}$ photons s$^{-1}$cm$^{-2}$ (with a TS value of 18).
The [$\gamma$-ray]{} spectrum of the excess emission was extracted by performing maximum likelihood analysis of the LAT data in 10 evenly divided energy bands in logarithm from 0.1–500 GeV. The spectral normalizations of the sources within 5 degrees from it were set as free parameters, while all the other parameters of the sources in RoI were fixed at the values obtained from the above maximum likelihood analysis. For the obtained spectral data points, we kept those when TS is greater than 4 and derived 95% flux upper limits otherwise. The results are provided in Table \[tab:spectra\] and also shown in Figure \[fig:spectra\]. We note that there are only two data points with TS$\geq$4 at 0.2–0.5 GeV and 1.3–3.0 GeV respectively, and the second is more significant with TS$\simeq$20.
![Fluxes and TS values in one-day bins ([*top*]{} and [*bottom*]{} panel respectively), shifted forward by 0.25 day, around the MJD 57623–57625 detection. Only fluxes with TS$\geq$9 are shown. The two dotted lines indicate the time period of MJD 57623–57625.[]{data-label="fig:1lc"}](f5.eps){width="42.00000%"}
Detection in MJD 57623–57625
----------------------------
We performed the same analysis to the data in this 3-day bin as the above in Section \[sec:3\]. The TS map of the $3\arcdeg\times 3\arcdeg$ region is shown in the right panel of Figure \[fig:ts\], indicating the maximum TS$\simeq$46.6 at the target’s position. The determined position for the excess emission has a nominal uncertainty of 016, and [V404 Cygni]{} is in the error circle. The likelihood analysis gave $\Gamma= 2.9\pm$0.3 and $F_{0.3-500}= 2.7\pm 0.6\times 10^{-7}$ photons s$^{-1}$cm$^{-2}$ (with a TS value of 47). The [$\gamma$-ray]{} spectrum was obtained, with the results given in Table \[tab:spectra\] and shown in Figure \[fig:spectra\]. The emission is still soft, mostly detected in the energy range of 0.1–1.3 GeV.
Since the excess emission has relatively high TS value, we constructed a short time-bin light curve around the detection. To avoid the large uncertainties in the low energy end, we used the data in the 0.3–500 GeV energy range. We found that a one-day light curve, shifted forward by 0.25 day, may reveal the detailed variations. As shown in Figure \[fig:1lc\], the source appears to have a sudden brightening and then a relatively slow decay. Using the orbital parameters given in @cc94, we checked the orbital phases for the time period. The start and end times (MJD 57623.0 and 57626.0 respectively) correspond to the orbital phase 0.32 and 0.78 respectively (when the companion star was mostly behind the black hole).
Discussion
==========
Having used the latest [*Fermi*]{} LAT database and source catalog, we re-analyzed the LAT data for searching for possible detection of [V404 Cygni]{} at $\gamma$-rays. Different from that reported in @loh+16, we did not find similar possible detection near the peak of the 2015 outburst (or possible detection in its mini-outburst). Instead, we found one possible detection ($\sim 4\sigma$) at the end of the outburst and one at the time period of MJD 57623–57626 with a significance of $\sim 7\sigma$. The excess emission in each detection matches [V404 Cygni]{} in position. Because the positional uncertainties are relatively large (01 and 016), we checked the SIMBAD Astronomical Database for sources within the 016 error circle. There are only several galaxies, detected by [*NuSTAR*]{} at the hard X-ray range of 3–24 keV and identified at optical and infrared wavelengths [@lan+17], within the error circle, and no blazars, which are the dominant [$\gamma$-ray]{} sources in the sky [@4fgl20].
The excess emission at the position of [V404 Cygni]{} is soft. For the latter detection with sufficiently high significance, $\Gamma= 2.9\pm0.3$ and the dominant emission is in the energy range of $\leq 1.3$GeV. This emission is similar to that observed in the flaring events of Cyg X-3 and Cyg X-1 [@fer09; @zdz+18; @zdz+17]. In addition, [$\gamma$-ray]{} emission is seen at spots in the lobes of the jets from the microqusar SS 433 [@abe+18; @xin+19]. Although the origin of the GeV [$\gamma$-ray]{} emission is under discussion (e.g.,@ras+19 [@sun+19]), it is extremely soft with $\Gamma\sim 6$ and in the energy range of $\leq 1.8$ GeV [@xin+19]. These similarities support the association of the excess emission with [V404 Cygni]{}. In case that the detection was due to a flare from an unseen background blazar, we checked the properties of the identified blazars in the LAT source catalog. Most of them have emission much harder than the excess emission. For example, one of 1190 BL Lac type and 24 of 730 flat-spectrum-radio-quasar type blazars have $\Gamma \geq 2.9$. It is very unlikely to have a blazar in a random 016-radius circular region only showing a three-day, soft-emission flare.
Based on the likelihood analysis results for the MJD 57623–57625 detection, the observed 0.3–500 GeV (isotropic) luminosity is $\sim 1.9\times 10^{35}$ergs$^{-1}$. The jet luminosity (collimation corrected; see @lkp17) from [V404 Cygni]{} could be a factor of $\sim$20 larger, that is $\sim 4\times 10^{36}$ergs$^{-1}$. From fitting the optical and infrared broad-band spectrum of the source, the mass accretion rate was estimated to be $\sim 7\times 10^{16}$gs$^{-1}$ [@mm06], indicating an accretion power of $\sim 6\times 10^{37}\eta$ergs$^{-1}$ (where $\eta$ is the efficiency) in the binary system. Therefore there is sufficient energy to power such jet emission even in quiescence. Given these considerations, i.e., the match in position, the soft spectrum similar to those of the [$\gamma$-ray]{} microquasars, and sufficient accretion power, we conclude that we have detected [V404 Cygni]{} likely during MJD 57623–57625 and possibly during MJD 57251–57253.
Different models with jets considered have been proposed to explain [$\gamma$-ray]{} emission from microqusars (e.g., @aa99 [@gak02; @rom+03]). Based on detailed studies of Cyg X-1 (also Cyg X-3; see, e.g., @zdz+18), which are able to fit its broad-band spectrum from radio to $\gamma$-rays, the [$\gamma$-ray]{}emission is determined to likely contain the components due to jets’ synchrotron self-Compton radiation and upscattering of photons from the accretion disk and companion star [@zdz+14; @zdz+17]. [$\gamma$-ray]{} flares are due to jet activity, clearly shown in the case of Cyg X-3 from radio and [$\gamma$-ray]{} monitoring [@cor+12]. For [V404 Cygni]{} in the quiescent state, radio observations have shown both significant long-term and short-term flux variations, in the latter of which the flares were seen to exhibit a possible pattern of having fast rise and slow decay [@plo+19]. We note that the detection in MJD 57623–57625 shows a possibly similar pattern (Figure \[fig:1lc\]), although its time scale is much longer than hour-long time scales observed at radio frequencies. There were a few radio observations of [V404 Cygni]{} in quiescence [@plo+19], but none of them were conducted at times very close to the two [$\gamma$-ray]{} events we have found. Hopefully a close radio monitoring of the source over a long term may be possible in the near future with more advanced radio facilities, which may help reveal the connection between radio and [$\gamma$-ray]{} flares. We checked the [*Swift*]{} BAT data [@kri+13], but did not find any significant brightening around the LAT detection in the daily 15–50 keV light curve of [V404 Cygni]{}.
This detection of the [$\gamma$-ray]{} flaring events from [V404 Cygni]{} in its quiescent state establishes the source as another microquasar with detectable [$\gamma$-ray]{} emission, and moreover exhibits interesting features among [$\gamma$-ray]{} microquasars. For example, no detection was found during the outburst or mini-outburst when jet ejection was observed at radio and millimeter frequencies or indirectly derived from X-ray observations [@tet+17; @tet+19; @wal+17]. If the detection at the end of the outburst was true, it might provide a hint to the physical process, any particular jet activity, at that end phase. In order to understand how high-energy emission is related to jet activity in such a case, simultaneous $\gamma$-ray and radio detection will provide key information. However it is not easy to obtain such detection since based on our search, detectable [$\gamma$-ray]{} flaring events from [V404 Cygni]{} are rare. Finally, different from Cyg X-3 and Cyg X-1 (note that SS 433 is a peculiar case; @abe+18) that have a high-mass companion, this binary belongs to the more-general low-mass X-ray binary (LMXB) class. We thus may consider that LMXBs with jets (including neutron star LMXBs; e.g., @rus+06) might all be able to produce some sorts of [$\gamma$-ray]{} emission, and searches for short-term flaring events among them might produce interesting results.
We thank F. Xie for useful discussion about jet activity in different black-hole X-ray binaries. This research made use of the High Performance Computing Resource in the Core Facility for Advanced Research Computing at Shanghai Astronomical Observatory. This research was supported by the National Program on Key Research and Development Project (Grant No. 2016YFA0400804) and the National Natural Science Foundation of China (11633007, U1738131). Z.W. acknowledges the support by the Original Innovation Program of the Chinese Academy of Sciences (E085021002).
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-------- ------------- -------------------------- ---- -------------------------- ---- -- -- -- -- -- -- -- --
$E$ Band $F/10^{-10}$ TS $F/10^{-10}$ TS
(GeV) (GeV) (erg cm$^{-2}$ s$^{-1}$) (erg cm$^{-2}$ s$^{-1}$)
0.15 0.1–0.2 1.1 0 2$\pm$1 12
0.36 0.2–0.5 0.6$\pm$0.4 4 1.3$\pm$0.4 14
0.84 0.5–1.3 0.5 0 1.8$\pm$0.5 44
1.97 1.3–3.0 0.8$\pm$0.3 20 0.2 0
4.62 3.0–7.1 0.4 0 0.4$\pm$0.3 5
10.83 7.1–16.6 0.9 0 0.8 0
25.37 16.6–38.8 1.9 0 1.8 0
59.46 38.8–91.0 4.3 0 4.1 0
139.36 91.0–213.3 10.2 0 9.7 0
326.60 213.3–500.0 24.3 0 23.1 0
-------- ------------- -------------------------- ---- -------------------------- ---- -- -- -- -- -- -- -- --
: Flux measurements for [V404 Cygni]{} from two sets of 3-day LAT data[]{data-label="tab:spectra"}
[^1]: http://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/
|
---
abstract: 'Multiple sclerosis (MS) affects the central nervous system with a wide range of symptoms. MS can, for example, cause pain, changes in mood and fatigue, and may impair a person’s movement, speech and visual functions. Diagnosis of MS typically involves a combination of complex clinical assessments and tests to rule out other diseases with similar symptoms. New technologies, such as smartphone monitoring in free-living conditions, could potentially aid in objectively assessing the symptoms of MS by quantifying symptom presence and intensity over long periods of time. Here, we present a deep-learning approach to diagnosing MS from smartphone-derived digital biomarkers that uses a novel combination of a multilayer perceptron with neural soft attention to improve learning of patterns in long-term smartphone monitoring data. Using data from a cohort of 774 participants, we demonstrate that our deep-learning models are able to distinguish between people with and without MS with an area under the receiver operating characteristic curve of [0.88]{} (95% CI: 0.70, 0.88). Our experimental results indicate that digital biomarkers derived from smartphone data could in the future be used as additional diagnostic criteria for MS.'
author:
- 'Patrick Schwab and Walter Karlen[^1] [^2]'
bibliography:
- 'references.bib'
title: A Deep Learning Approach to Diagnosing Multiple Sclerosis from Smartphone Data
---
Artificial neural networks, digital biomarkers, medical diagnosis, multiple sclerosis, explainability
Introduction {#sec:introduction}
============
sclerosis (MS) is a neurological disease that affects around 2 million people worldwide [@vos2016global]. The neural lesions caused by MS reduce the capability of neurons to transmit information, which leads to a wide range of symptoms, such as changes in sensation, mobility, balance, vision, and cognition [@brownlee2017diagnosis]. Diagnosing MS requires objective evidence of two lesions in the central nervous system disseminated both in time and space [@mcdonald2001recommended; @polman2011diagnostic]. Physicians typically use a combination of clinical assessments of symptoms, blood tests, imaging, cerebrospinal fluid analysis and analysis of evoked potentials to rule out other diseases with similar symptoms [@mcdonald2001recommended; @polman2011diagnostic; @filippi2016mri]. Currently, no cure exists for MS, but there are treatments available that are effective at managing the symptoms of MS and may significantly improve long-term outcomes [@brownlee2017diagnosis; @marziniak2016variations; @scolding2015association]. To receive early access to these treatments, a timely diagnosis is of paramount importance for patients.
![Smartphone-based tests (top) can be used to assess cognition, movement and finger dexterity symptoms of multiple sclerosis (MS) and track their progression over time. We train machine-learning models to learn to produce a scalar diagnostic score $y$ (bottom) from the data collected during any number of those tests to learn to diagnose MS. []{data-label="fig:teaser"}](img/teaser_copy.pdf){width="\linewidth"}
Smartphone-based tests could potentially be used to quantify symptoms of MS in the wild over long periods of time (Figure \[fig:teaser\]). However, to date, it has not been established whether and to what degree smartphone monitoring data can be used to derive digital biomarkers for the diagnosis of MS. A particularly challenging aspect of using smartphone data to derive digital biomarkers for the diagnosis of MS is that smartphone monitoring yields large amounts of high-resolution data from multiple symptom categories. Identifying the salient input segments and reaching a clinically meaningful conclusion from raw sensor data in an accurate and timely manner is therefore challenging both for physicians and machines.
To address these issues, we present a machine-learning approach for distinguishing between people with and without MS from smartphone data. At its core, our method uses an attentive aggregation model (AAM) that integrates the results of multiple tests over time to produce a single diagnostic score. By integrating neural attention in our model, we are additionally able to quantify the importance of individual tests towards the model’s output. Our experiments on real-world smartphone monitoring data show that our method outperforms several strong baselines, identifies meaningful patterns, and that smartphone data could potentially be used to derive digital biomarkers for the diagnosis of MS.
Concretely, our contributions are as follows:
- We present a deep-learning approach to distinguishing between people with and without MS that integrates data from multiple types of smartphone-based tests performed over long time frames (up to more than 200 days).
- We extend our deep-learning approach with neural soft attention in order to quantify the importance of individual input features towards the final diagnostic score.
- We utilise real-world smartphone monitoring data from 774 subjects to evaluate, for the first time, the comparative performance of machine-learning models and several strong baselines in distinguishing between people with and without MS.
Related Work
============
Background
----------
Using machine learning to aid in medical tasks has attracted much research interest. Researchers have, for example, used machine learning for mortality modelling [@ghassemi2014unfolding], sepsis decision support [@horng2017creating], alarm reduction in critical care [@schwab2018not], to provide explainable decisions in medical decision-support systems [@lundberg2017unified; @lundberg2018explainable; @schwab2018granger], and to identify subtypes in autism spectrum disorders [@doshi2014comorbidity] and scleroderma [@schulam2015clustering]. Giving a reliable diagnosis is one of the most challenging tasks in medicine that requires significant domain knowledge in the disease being assessed, and the ability to integrate information from a large number of potentially noisy data sources. Machine learning is an attractive tool to perform automated diagnoses because it can draw upon the experience of millions of historical samples in its training data, and seamlessly integrate data from multiple disparate data sources. In previous studies, machine learning has, for example, been used to diagnose skin cancer from image data [@esteva2017dermatologist], glaucoma from standard automated perimetry [@chan2002comparison], and a large number of diseases from electronic health records [@lipton2015learning; @choi2016doctor] and lab test results [@razavian2016multi]. However, obtaining objective data about symptoms and symptom progression over time is challenging in many diseases. For some diseases, wearable devices and smartphones have emerged as viable tools for gathering diagnostic data in the wild. Smartphones and telemonitoring have, for example, been used to gather digital biomarkers for the diagnosis of melanomas [@wolf2013diagnostic], bipolar disorder [@faurholt2019objective], cognitive function [@piau2019current], and Parkinson’s disease [@tsanas2010enhanced; @tsanas2010accurate; @arora2015detecting; @zhan2018using; @schwab2019phonemd]. The use of machine learning on high-resolution health data often requires specialised approaches to effectively deal with missingness [@lipton2016directly; @che2018recurrent], long-term temporal dependencies [@choi2016doctor], noise [@schwab2017beat], heterogeneity [@libbrecht2015machine], irregular sampling [@lipton2015learning], sparsity [@lasko2013computational], and multivariate input data [@schwab2018not; @schwab2019phonemd; @ghassemi2015multivariate]. In this work, we build on these advances to develop a novel approach to learning to diagnose MS from smartphone-derived digital biomarkers that addresses the aforementioned challenges.
Monitoring and Diagnosis of MS with smartphones
-----------------------------------------------
The clinical state-of-the-art in monitoring symptoms and symptom progression in MS is based on a combination of clinical assessments, such as neurological exams, magnetic resonance imaging (MRI), and the Expanded Disability Status Scale (EDSS) [@kurtzke1983rating; @wattjes2015evidence]. However, these tests can only be performed at clinical centers by medical specialists. With dozens of mHealth apps available to manage MS on all major smartphone platforms, smartphone apps have recently emerged as a readily accessible alternative to non-invasively track symptoms of MS in the wild [@giunti2018supply; @boukhvalova2019smartphone]. Prior studies on the use mHealth in MS have, for example, evaluated telemedicine-enabled remote EDSS scoring [@bove2018toward], measurement devices for estimating walking ability [@dalla2017smart] and fatigue [@barrios2018msfatigue], and machine learning for assessing gait impairment in MS [@mcginnis2017machine].
In contrast to existing works that focused on single daily-life aspects of already diagnosed MS patients, we present an approach to diagnosing MS from smartphone-derived digital biomarkers, and verify this approach on a real-world dataset collected from a MS cohort. Our machine-learning approach addresses multiple challenges in learning from sensor-based smartphone tests that are self-administered multiple times over long periods of time. Most notably, with the integration of a global neural soft attention mechanism, we enable the quantification of the importance of individual smartphone tests towards the final diagnostic score, overcome the challenges of missingness, sparse data, long-term temporal dependencies between tests, and multivariate data with irregular sampling.
Methods and Materials
=====================
Smartphone Tests
----------------
We utilise data collected by the Floodlight Open study, a large smartphone-based observational study for MS [@montalban2018floodlight; @midaglia2019floodlight]. The de-identified dataset used in this work is openly available to researchers[^3]. In the study, participants were asked to actively perform a number of smartphone-based tests on a daily basis using their personal smartphones in the wild and without any clinical supervision (Figure \[fig:teaser\]). However, participants were free to choose when and if they performed the daily tests. Many participants therefore did not strictly adhere to the daily test protocol, and performed the tests irregularly. In addition to the manual tests, the app also passively collected movement data of the participants in order to determine their radius of living. The following tests were included in the study (Figure \[fig:teaser\]) [@montalban2018floodlight]:
- **Mood Questionnaire.** In the mood questionnaire, participants were asked a single question about their current well-being. The answers were mapped to a scalar mood score that was recorded for each answer. The score was used to track changes of participants’ mood over time.
- **Symbol Matching.** In the symbol matching test, participants were presented with a mapping of symbols to numbers. Participants were then prompted with a single symbol from this mapping, and asked to translate the shown symbol into the corresponding number using an on-screen virtual keyboard. Once the user entered their response, a new symbol would be shown. The goal was to translate the presented symbols as quickly and accurately as possible in a fixed amount of time. As metrics, the average response time and the number of correct responses were recorded. There was also a baseline version of this test in which participants simply had to input the presented numbers directly without any intermediate mapping for which the same metrics were recorded.
- **Walking.** In the walking test, participants were asked to take a walk for two minutes, where ever they saw fit. Their smartphones recorded the number of steps taken during this walk in order to capture whether the participants’ ability to walk was impaired.
- **U-turn.** In the U-turn test, participants were asked to walk as quickly as possible between two fixed points of their choice that should approximately have been four meters apart. Their smartphones recorded the number of turns and the average turn speed during this test in order to assess the participants’ ability to turn.
- **Balance.** In the balance test, participants were asked to stand still and hold their balance for a fixed amount of time. The app recorded their postural sway during this test in order to evaluate to what degree the participant was able to remain still. Impaired postural control and an increased risk of falling are symptoms commonly associated with MS [@cameron2010postural].
- **Mobility.** The mobility test recorded the daily life space of the participant using their smartphone’s location sensors. The mobility test was the only test that ran in the background and did not have to be manually activated.
- **Pinching.** In the pinching test, participants were presented with a series of virtual objects in varying locations on their smartphone screens. The participants were then asked to perform a pinching gesture using their fingers to squash the objects as quickly as possible. The app recorded the number of successfully squashed objects over a fixed amount of time, and which hand was used to perform the pinching gesture. The aim was to measure the participants’ pinching ability which may be impaired in people with MS [@chen2007hand].
- **Drawing.** In the drawing test, participants were asked to draw a sequence of shapes that was shown on their screens using their fingers - twice for each shape. The shapes represented a square, a circle, a figure eight, and a spiral. For each shape, the app recorded the best Hausdorff distance between the drawn and shown shapes.
Problem Statement
-----------------
We consider the setting in which we are given a number $k$ of tests, including test scores $s_i$, the time since the last test $t_i$, a one-hot encoded representation $m_i$ of the test metric with $i \in [0 {\mathrel{{.}\,{.}}\nobreak}k-1]$, and demographic data $d$, including age and sex, of the participant that performed the tests. Our goal is to train a predictive model $P$ that produces a scalar diagnostic score $y$ between $[0,1]$ that indicates the likelihood of the given set of test results belonging to a participant with or without MS. $$\begin{aligned}
\label{eq:y}
y &= P([0, t_1, ..., t_i], [m_0, ..., m_i], [s_0, ..., s_i], d).\end{aligned}$$ The primary challenge in this setting is to identify predictive patterns among the potentially large set of tests performed irregularly over long periods of time that provide evidence for or against a MS diagnosis.
Attentive Aggregation Model (AAM)
---------------------------------
As predictive model $P$, we use an AAM - a deep-learning model that utilises neural soft attention in order to integrate information from a potentially large number of smartphone test results. AAMs are based on evidence aggregation models (EAMs) [@schwab2019phonemd]. As a first step, we concatenate the time since the respective prior test $t_i$, result scores $s_i$, and test metric indicators $m_i$ from the smartphone tests into $k$ features $x_i$. $$\begin{aligned}
x_i &= \text{concatenate}(t_i, m_i, s_i)\end{aligned}$$ We then use a multilayer perceptron (MLP) with a configurable number $L$ of hidden layers and $N$ neurons per hidden layer to process each input feature into a $N$-dimensional high-level hidden feature representation $h_i$. $$\begin{aligned}
h_i &= \text{MLP}(x_i)\end{aligned}$$ Next, we aggregate the information from all $k$ high-level hidden representations into a single aggregated hidden representation $h_\text{all}$ that reflects all available tests for a given participant. To do so, we used a learned neural soft attention mechanism that weighs the individual hidden representations $h_i$ of each individual test instance by their respective importance $a_i$ towards the final diagnostic output score. $$\begin{aligned}
h_\text{all} &= \textstyle{\sum}_{i=1}^{k} a_i h_i\end{aligned}$$ Following [@schwab2018granger; @schwab2017beat; @xu2015show], we calculate the attention factors $a_i$ by first projecting the individual hidden representations $h_i$ into an attention space representation $u_i$ through a single-layer MLP with a weight matrix $W$ and bias $b$. $$\begin{aligned}
\label{eq:u_i}
u_{i} &= \text{activation}(Wh_{i} + b)\end{aligned}$$ We finalise the calculation of the attention factors $a_i$ by computing a softmax over the attention space representations $u_i$ of $h_i$ using the most informative hidden representation $u_\text{max}$. $W$, $b$ and $u_{\text{max}}$ are learned parameters and optimised together with the other parameters during training [@schwab2019phonemd]. $$\begin{aligned}
\label{eq:a_i}
a_{i} &= \text{softmax}(u_{i}^Tu_{\text{max}})\end{aligned}$$ We then calculate the final diagnostic score $y$ using a MLP with one sigmoid output node on a concatenation of the aggregated hidden state $h_\text{all}$ and the demographic data $d$. $$\begin{aligned}
y &= \text{MLP}(\text{concatenate}(h_\text{all}, d))\end{aligned}$$ AAMs rely solely on attention to aggregate test results over time. Using attention to perform the temporal aggregation has the advantage that attention mechanisms learn global input-output relations without regard to their distance in the input sequence, and in this manner improve learning of long-range temporal dependencies [@vaswani2017attention]. While global attention models are commonly employed in natural language processing [@vaswani2017attention; @devlin2018bert], we are not aware of any prior works that apply global attention to improve learning of long-term temporal dependencies on smartphone sensor data.
Experiments
===========
To evaluate the predictive performance of AAMs in diagnosing MS from smartphone data, we performed experiments that compared the diagnostic performance of AAMs and several baseline models on real-world smartphone monitoring data. Our experiments aimed to answer the following questions:
1. What is the predictive performance of AAMs in diagnosing MS from smartphone data?
2. What is the predictive performance of AAMs compared to other methods, such as Mean Aggregation and the demographic baseline?
3. Which test types were most informative for diagnosing MS, and to what degree?
4. To what degree does including more tests performed by subjects improve predictive performance of AAMs?
5. Does the neural attention mechanism identify meaningful patterns?
The following subsections outline the experimental details of the conducted empirical evaluations.
Dataset and Study Cohort
------------------------
We used data from the Floodlight Open study that recruited participants via their personal smartphones in, among others, the United States (US), Canada, Denmark, Spain, Italy, the Czech Republic, and Switzerland [@montalban2018floodlight; @midaglia2019floodlight]. To perform our experimental comparison, we used all of the available smartphone monitoring data from April 23$^\text{rd}$ 2018 to August 29$^\text{th}$ 2019. In addition to regularly performing the smartphone-based tests, participants also provided their demographic profiles upon sign-up. The demographic profile included age, sex, and whether or not they had an existing diagnosis of MS. To ensure that a minimal amount of data points are available for diagnosis, we excluded all participants that had produced fewer than 20 test results during the analysed time frame (312 participants). We chose the cutoff at a minimum of 20 tests as this corresponds roughly to two sets of the daily test suite, which would be the minimal amount of data needed to assess symptom progression over time. We assigned the included patients to three folds for training (70%), validation (10%) and testing (20%) randomised within strata of diagnostic status, app usage, sex, age, and number of tasks performed (Table \[tb:dataset\]). After stratification, the percentage of participants with MS was roughly 52% across the three folds, and the median app usage duration - the difference in time between a participant’s first to last performed test - was around 20 days (Table \[tb:dataset\]).
Property Training Validation Test
--------------- ------------------- ------------------- ------------------- --
Subjects (\#) 542 (70%) 77 (10%) 155 (20%)
MS (%) 51.9 52.0 51.6
Female (%) 60.3 59.7 60.7
Age (years) 41.0 (27.0, 59.0) 41.0 (26.5, 56.5) 41.0 (28.0, 57.5)
Usage (days) 22.4 (0.6, 203.5) 19.0 (0.8, 175.5) 18.8 (1.0, 130.0)
: Population Statistics. []{data-label="tb:dataset"}
Models
------
We trained a demographic baseline model, a Mean Aggregation baseline, and several ablations of AAMs. The AAMs used a fully-connected neural network as their base, and a single neuron with a sigmoid activation function as output. We trained one AAM version that received the demographic information (AAM + age + sex), and one that did not (AAM). For computational reasons, we limited the maximum number of test results per participant. To estimate the performance benefit of having access to information from more tests in the analysis, we trained AAMs and Mean Aggregation using up to the first 25, 30, 40, 50, 100, 150, 200, 250, 300 and 350 test results per participant, if available. For the demographic baseline, we used a random forest (RF) model that received as input only the age and sex of the participant (Age + sex). We used the demographic baseline to evaluate whether and to what degree data from the smartphone-based tests improves predictive performance, since the demographic baseline only had access to demographic data and did not include data from the smartphone-based tests. As a simple reference baseline, the Mean Aggregation utilised the mean normalised test result score to produce the final diagnostic score - this served to determine whether the use of more complex, learned aggregation methods, such as AAMs, is effective and warranted.
Hyperparameters
---------------
To ensure all models were given the same degree of hyperparameter optimisation, we used a standardised approach where each model was given exactly the same optimisation budget of 50 hyperparameter optimisation runs with hyperparameters chosen from pre-defined ranges (Table \[tb:hyperparameters\]). For the demographic baseline model, we used a RF with $T$ trees and a maximum tree depth of $D$. For the AAMs, we used an initial MLP with $L$ hidden layers, $N$ hidden units per hidden layer, a dropout percentage of $p$ between the hidden layers, and an L$2$ weight penalty of strength $s$. We trained AAMs to optimise binary cross entropy for a maximum of 300 epochs with a minibatch size of $B$ participants and a learning rate of 0.003. In addition, we used early stopping with an early stopping patience of 32 epochs on the validation set.
Hyperparameter Range / Choices
-- ---------------------------------- ----------------------
Number of hidden units $N$ 16, 32, 64, 128
Batch size $B$ 16, 32, 64
L$2$ regularisation strength $s$ 0.0001, 0.00001, 0.0
Number of layers $L$ (1, 3)
Dropout percentage $p$ (0%, 35%)
Tree depth $D$ 3, 4, 5
Number of trees $T$ 32, 64, 128, 256
: Hyperparameters. []{data-label="tb:hyperparameters"}
Model (max. 250 test results) AUC AUPR F$_1$ Sensitivity Specificity
------------------------------- -------------------------------------- -------------------------------------- -------------------------------------- -------------------------------------- --------------------------------------
AAM + age + sex **0.88** (0.70, 0.88) **0.90** (0.67, 0.90) **0.80** (0.65, 0.83) **0.83** (0.59, 0.86) (0.62, 0.89)
Mean Aggregation + age + sex [^$\dagger$^]{}0.77 (0.70, 0.82) [^$\dagger$^]{}0.76 (0.64, 0.84) [^$\dagger$^]{}0.71 (0.65, 0.78) [^$\dagger$^]{}0.68 (0.59, 0.85) [^$\dagger$^]{}[0.75]{} (0.59, 0.87)
Age + sex [^$\dagger$^]{}[0.76]{} (0.69, 0.84) [^$\dagger$^]{}[0.75]{} (0.65, 0.86) [^$\dagger$^]{}[0.69]{} (0.62, 0.79) [^$\dagger$^]{}[0.73]{} (0.55, 0.83) [^$\dagger$^]{}[0.61]{} (0.58, 0.89)
AAM [^$\dagger$^]{}[0.72]{} (0.56, 0.82) [^$\dagger$^]{}[0.67]{} (0.57, 0.84) [^$\dagger$^]{}[0.61]{} (0.53, 0.77) [^$\dagger$^]{}[0.63]{} (0.45, 0.79) [^$\dagger$^]{}[0.83]{} (0.53, 0.86)
Mean Aggregation [^$\dagger$^]{}0.56 (0.50, 0.67) [^$\dagger$^]{}0.61 (0.49, 0.74) [^$\dagger$^]{}0.39 (0.20, 0.54) [^$\dagger$^]{}0.28 (0.13, 0.43) [^$\dagger$^]{}**0.85** (0.70, 0.97)
Preprocessing
-------------
We normalised the time between two test results $t_i$ to the range $[0,1]$ using the highest observed $t_i$ on the training fold of the dataset. We additionally normalised all test result scores $x_i$ to the range of $[0,1]$ using the lowest and highest observed test result for each test metric on the training fold of the dataset.
Metrics
-------
### Predictive Performance
We evaluated all models in terms of their area under the receiver operating characteristic curve (AUC), the area under the precision recall curve (AUPR), and F$_1$ score on the test set of 155 participants. For the comparison of predictive performance, we additionally computed the sensitivity and specificity of the respective models. We also quantified the uncertainty of all the performance metrics by computing 95% confidence intervals (CIs) using bootstrap resampling with 1000 bootstrap samples. To assess the statistical significance of our results, we applied $t$-tests at significance level $\alpha = 0.05$ to the main comparisons. We additionally applied the Bonferroni correction to adjust for multiple comparisons.
![Performance comparison of AAM (dots, blue) and Mean Aggregation (triangles, orange) in terms of their Area Under the Precision-Recall Curve (AUPR, y-axis) when varying the maximum number of test results (x-axis) available to predict the MS diagnosis for each participant. \*\*\* = significant at p < 0.001.[]{data-label="fig:task_number"}](img/task_number){width="\linewidth"}
### Importance of Test Types
To quantify the importance of the various test types toward the diagnostic performance of the AAM, we retrained AAMs with the same hyperparameters after removing the test results from exactly one type of test. The reduction in predictive performance associated with removing the information of one test type can be seen as a proxy for the importance of that test type [@schwab2018granger; @schwab2019cxplain], since features that are associated with a higher reduction in prediction error carry more weight in improving the model’s ability to predict MS in participants.
### Neural Attention
In order to qualitatively inspect the patterns that were captured by the AAM in the data, we additionally plotted the attention assigned to the test results from a sample participant with MS over time.
![Performance comparison of AAM in terms of their F$_1$ score (y-axis) in predicting the MS diagnosis for each participant after removal of the information of all tests of a specific type (labelled test types, bottom) from the dataset. The reference baseline without removal of any test types (All Tests) is highlighted in orange. \*\*\* = significant at p < 0.001.[]{data-label="fig:test_performance"}](img/task_type){width="\linewidth"}
{width="\linewidth"}
Results
=======
Predictive Performance
----------------------
In terms of predictive performance for diagnosing MS, the AAMs models with demographic information (AAM + age + sex) achieved a significantly ($p < 0.05$) higher AUC, AUPR and F$_1$ than all the baselines we evaluated (Table \[tb:results\_all\]). In particular, we found that integrating the information from the smartphone tests was crucial, as AAMs that used both the demographic data and the smartphone test information significantly ($p < 0.05$) outperformed the demographic baseline (Age + sex) in terms of AUC, AUPR, and F$_1$. Mean Aggregation had a considerably lower performance than AAMs, and displayed the worst AUC, AUPR and F$_1$ of the compared models - demonstrating that the use of more sophisticated adaptive aggregation models for integrating information from smartphone-based tests over time, such as AAMs, is effective and warranted. We also found that AAMs achieved a high level of both sensitivity and specificity, whereas the demographic model emphasised sensitivity, and Mean Aggregation specificity. In addition, we found that, while the digital biomarkers contained significant signal towards predicting MS, they were by themselves (AAM) overall not more predictive than the demographic information alone.
Impact of Performing More Tests
-------------------------------
The results from comparing AAMs and Mean Aggregation in terms of their AUPR across a wide range of numbers of test results indicated that AAMs are better able to leverage an increasing number of tests performed (Figure \[fig:task\_number\]). In terms of predictive performance as measured by the AUPR, the AAM surpassed the Mean Aggregation baseline at all evaluated maximum numbers of test results. We also found that having access to a higher number of tests consistently and significantly (50 vs. 100 test results, $p<0.001$) improved the performance of AAMs up to a maximum of 250 test results per participant. For maximum numbers of test results higher than 250, AAMs stagnated in predictive performance - likely because (i) the additional information of performing more tests was marginal, or because (ii) they were not able to effectively integrate information from higher numbers of test results. In contrast, the predictive performance of Mean Aggregation only slightly increased with higher numbers of test results.
Importance of Test Types
------------------------
When comparing the marginal reduction in prediction error associated with removing a specific type of test from the set of available tests, we found that the drawing and mood tests were contributing significantly larger marginal reductions ($p<0.001$) in prediction error to the AAM (Figure \[fig:test\_performance\]) - indicating that the drawing and mood tests were more predictive of MS diagnosis than other test types. The removal of other test types did not lead to similarly considerable reductions in prediction error compared to the AAM that had access to information from all test types (All Tests). This result could indicate that the results of other test types were either (i) not highly predictive of MS, or (ii) correlated with other tests to a degree that strongly impacted their marginal contributions.
Neural Attention
----------------
Qualitatively, on the data from one sample participant, we found that the model focused on mood, drawing, and symbol matching tests to diagnose MS (Figure \[fig:atn\]). The focus on mood and drawing tests for this subject are in line with our findings on the overall importance of the test types (Table \[fig:test\_performance\]).
Discussion
==========
To the best of our knowledge, this work is the first to present a machine-learning approach to diagnosing MS from long-term smartphone monitoring data collected outside of the clinical environment. In order to derive a scalar diagnostic score for MS from smartphone monitoring data, we used a novel AAM to aggregate data from multiple types of tests over long periods of time. The AAMs used neural attention to quantify the degree to which individual tests contribute to the final diagnostic score, and to overcome the challenges of missingness, sparse data, long-term temporal dependencies and irregular sampling. Our experimental results indicate that our models outperform several strong baselines, identify meaningful data points within the set of performed tests, and that smartphone-based digital biomarkers could potentially be used to aid in diagnosing MS alongside existing clinical tests. Among the several potential advantages of using smartphone-derived digital biomarkers for diagnosing MS are that smartphone-based tests (i) can be administered remotely and therefore potentially expand access to underserved geographic regions, (ii) are inexpensive to distribute and could therefore potentially become a low-cost alternative to more expensive in-clinic tests, and (iii) are able to integrate information from long-term symptom monitoring, and therefore potentially better represent and quantify fluctuations in symptom burden over time. An additional benefit of using smartphone-based diagnostics for MS is that machine-learning models can, as demonstrated in this work, identify which symptom categories are most indicative of MS and how they interact over time, and could therefore potentially in the future be used to monitor disease progression and inform follow-up treatment decisions. Finally, our results show that information from smartphone-based biomarkers are to some degree orthogonal to more traditional measurements, such as demographic data, and could therefore potentially be integrated with information from existing clinical tests and other multimodal data sources, such as MRI, to further increase diagnostic accuracy.
![Comparison of the receiver operating characteristic (ROC) curves of AAM + age + sex (black, dot), Age + sex (orange, triangle), and Mean Aggregation + age + sex (blue, square) when using a maximum of 250 test results to predict the MS diagnosis for each test set participant. Symbols indicate the operating points (thresholds selected on the validation fold) presented in the comparison in Table \[tb:results\_all\]. []{data-label="fig:roc"}](img/roc){width="\linewidth"}
Limitations
-----------
While of respectable size, the studied cohort was restricted to residents of a limited number of countries and likely not representative of the global population. More importantly, the data originated from patients with a diagnosis at various stages of the disease, instead of pre-diagnosis. Therefore, our work cannot conclude whether people without clinical MS diagnosis could be identified before they receive their diagnosis. A prospective validation in a larger, clinically representative cohort will be necessary to conclusively establish the performance and utility of smartphone-derived biomarkers as a tool to aid in the diagnosis of MS, and their robustness when confronted with other disorders that have similar symptoms. Further work should investigate whether such biomarkers are also suitable to track disease progression, identify and predict relapses, enable the effective tuning of therapeutic options and medication dosages, and eventually also provide an accurate prognosis.
Conclusion
==========
We presented a novel machine-learning approach to distinguishing between people with and without MS based on long-term smartphone monitoring data. Our method uses an AAM to aggregate data from multiple types of tests over long periods of time in order to produce a scalar diagnostic score. AAMs use neural attention to quantify the degree to which individual tests contributed to the diagnostic score, and to overcome the challenges of missingness, sparse data, long-term temporal dependencies and irregular sampling. In an experimental evaluation on real-world smartphone monitoring data from a cohort of 774 people, we demonstrated that AAMs identify predictive and meaningful digital biomarkers for diagnosing MS. Our experiments show that smartphone-derived digital biomarkers could potentially be used to aid in diagnosing MS in the future alongside existing clinical tests. Smartphone-based tools for tracking symptoms and symptom progression in MS may improve clinical decision-making by giving clinicians access to objectively measured high-resolution health data collected outside the restricted clinical environment. In addition, our solution based on attention mechanisms further elucidates the basis of the model decisions and may enhance the clinician’s understanding of the provided diagnostic score. We therefore believe our initial results may warrant further research on how digital biomarkers could be integrated into clinical workflows.
Acknowledgments {#acknowledgments .unnumbered}
===============
The data used in this manuscript were contributed by users of the Floodlight Open mobile application developed by Genentech Inc: <https://floodlightopen.com>. Patrick Schwab is an affiliated PhD fellow at the Max Planck ETH Center for Learning Systems.
[^1]: This work was partially funded by the Swiss National Science Foundation (SNSF) project No. 167302 within NRP 75 “Big Data”. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPUs used for this research.
[^2]: P. Schwab and W. Karlen are with the Mobile Health Systems Lab, Institute of Robotics and Intelligent Systems, Department of Health Sciences and Technology, ETH Zurich, Switzerland (e-mails: [email protected], [email protected]).
[^3]: <https://floodlightopen.com>
|
---
author:
- |
George B. Mertzios , Mordechai Shalom[^1] ,\
Prudence W.H. Wong , Shmuel Zaks
bibliography:
- 'Optical.bib'
- 'Approximation.bib'
- 'Mordo.bib'
- 'Miscelaneous.bib'
title: 'Online Regenerator Placement[^2] [^3]'
---
**Keywords:** online algorithms, competitive ratio, optical networks, regenerators.\
[^1]: Corresponding Author: Mendel Singer 15/6, 34984, HAIFA-ISRAEL, Tel: +972(54)753 6550
[^2]: This work was supported in part by the Israel Science Foundation grant No. 1249/08 and British Council Grant UKTELHAI09.
[^3]: A preliminary version of this paper appeared in the *15th International Conference on Principles of Distributed Systems (OPODIS), Toulouse, France, December 2011, pp. 4–17.*
|
---
abstract: 'We compute the Konishi anomalous dimension perturbatively up to six loop using the finite set of functional equations (FiNLIE) derived recently in [@GKLV]. The recursive procedure can be in principle extended to higher loops, the only obstacle being the complexity of the computation.'
author:
- 'S. Leurent$^{a,b}$, D. Serban$^{c}$, D. Volin$^{d}$'
title: 'Six-loop Konishi anomalous dimension from the Y-system '
---
\[sec:int\]Introduction
=======================
Using integrability in conjunction with the AdS/CFT correspondence led to some of the main achievements in theoretical high energy physics of the last decade. One of them is determining the spectrum of the anomalous dimensions of the planar ${\cal N}=4$ SUSY gauge theory, or equivalently finding the spectrum of free string theory in $AdS_5\times S^5$ background. For a recent review on the subject see [@AdSReview]. For operators with large charges, the anomalous dimension is determined by a set of Bethe Ansatz equations, [@AdSReview], while for operators with small charges, the information about the spectrum is encapsulated into an infinite set of functional equations derivable from the thermodynamic Bethe Ansatz (TBA) equations [@Alyosha] and known under the name of $Y$-system. For the AdS/CFT integrable system, the $Y$-system was conjectured in [@GKV] and derived from the the TBA equations in [@TBA:GKKV] . The anomalous dimension of the Konishi operator, the shortest operator not protected by supersymmetry, is used as a testing ground for these equations, both for analytical and numerical computations. In particular, in perturbation four [@GKV] and five loop [@fiveTBA; @fiveBH] corrections to the Bethe Ansatz were computed using the $Y$-system, and they were found to coincide both with the corresponding Lüscher corrections [@JanikBaj; @fiveL] and with the four-[@Kpert] and five-loop [@Eden:2012fe] perturbative gauge theory computations. At strong coupling, the result of the extrapolation to short, Konishi-like operators [@KS:GSSV] agrees both with the numerical results [@GKVnum; @Fnum] and with the string predictions [@KS:RT]. Recently [@GKLV; @BHfin], the $Y$-system was reformulated in terms of a finite closed set of functional equations. In the present work, we set up a recursive procedure to solve in perturbation the equations of [@GKLV] and we perform the explicit computation up to six loops. The procedure can be in principle continued to higher loops, in particular to the double-wrapping order at eight loops. Double wrapping was already attained for the ground state energy in the twisted AdS/CFT [@doublewrap], however our case is significantly more complicated because of necessity to account the displacement of the Bethe roots.
\[sec:finlie\]The set of functional equations
=============================================
The wrapping corrections are encoded in a finite set of functional equations [@GKLV] which were derived from the AdS/CFT Y-system by solving the Hirota equation in the semi-infinite bands of the $T$-hook defined in [@GKV]. The different solutions are then glued together using analyticity constraints which reflect the physical properties the Y-system has to satisfy. We give below a brief summary of this set of functional equations.
#### Input data.
The functional equations depend on a specific operator through the position of the Bethe roots $u_j$, via the objects $
B_{(\pm)}=\prod_{j=1}^M\sqrt{\frac{g}{\hat x_j^\mp}}\left(\frac{1}{x}-\hat x_j^\mp\right)\!,R_{(\pm)}=\prod_{j=1}^M\sqrt{\frac{g}{\hat x_j^\mp}}\left({x}-\hat x_j^\mp\right)\,$ and the Baxter polynomial $Q^\pm=(-1)^MB_{(\pm)}R_{(\pm)}$. Here we use the conventions $f^{[a]}=f(u+ia/2)$, $f^\pm=f^{[\pm1]}$, and $x(u)={u}/{2g}+i\sqrt{1-{u^2}/{4g^2}}$ is the Zhukovsky variable in the so-called mirror regime, with a cut on $\check Z\equiv(-\infty,-2g]\cup[2g,\infty)$. For the Zhukovsky variable in the physical regime, with a branch cut on the interval $\hat Z\equiv [-2g,2g]$, we use the notation $\hat x(u)={u}/{2g}+\sqrt{{u}/{2g}-1}\sqrt{u/2g+1}$. By convention, we denote with a hat the quantities which depend on $\hat x(u)$, if we want to emphasize the position of the branch cut. For the Konishi operator there are two magnons, $M=2$, with $u_1=-u_2=1/\sqrt{12}+\CO(g^2)$ and $Q(u)=u^2-u_1^2$.
#### Parameterization of the $T$- and $q$-functions.
When considering a state from the $sl(2)$ sector, as it is the case for Konishi, the $Y$-system can be determined from only three functions: two real-valued densities with a finite support on $\hat Z$, $\rho$ and $\rho_2$, and a complex-valued function $U$ analytic in the upper half-plane. A relatively simple formulation of the set of functional equations can be obtained using $T$-functions in two different gauges denoted ${\CT}_{a,s}$ and ${{\mathscr T}}_{a,s}$. The first gauge gives a simple solution of the Hirota equation in the right band $ s\geq a$ \[cT\] \_[0,s]{}=1, \_[1,s]{}=s+\_s, \_[2,s]{}=\_[1,1]{}\^[\[+s\]]{}\_[1,1]{}\^[\[-s\]]{}, with $(\CK\ast f)(u)=\frac{1}{2\pi i}\int_{-\infty}^{\infty}dv\,{f(v)}/({v-u})$ and $\CK_s\equiv\CK^{[s]}-\CK^{[-s]}$. The second gauge gives a solution of the Hirota equation in the upper band of the $T$-hook, $a\geq|s|$, \[curlyT\] [[T]{}]{}\_[a,2]{}&=&q\_\^[\[+a\]]{} \_\^[\[-a\]]{}, [[T]{}]{}\_[a,-1]{}=(U\^[\[+a\]]{}|U\^[\[-a\]]{})\^2[[T]{}]{}\_[a,1]{},\
[[T]{}]{}\_[a,1]{}&=&q\_1\^[\[+a\]]{} \_2\^[\[-a\]]{}+q\_2\^[\[+a\]]{} \_1\^[\[-a\]]{}+q\_3\^[\[+a\]]{} \_4\^[\[-a\]]{}+q\_4\^[\[+a\]]{} \_3\^[\[-a\]]{},\
[[T]{}]{}\_[a,0]{}&=&q\_[12]{}\^[\[+a\]]{} \_[12]{}\^[\[-a\]]{}+q\_[34]{}\^[\[+a\]]{} \_[34]{}\^[\[-a\]]{}\
&-&q\_[14]{}\^[\[+a\]]{} \_[14]{}\^[\[-a\]]{}-q\_[23]{}\^[\[+a\]]{} \_[23]{}\^[\[-a\]]{}-q\_[13]{}\^[\[+a\]]{} \_[24]{}\^[\[-a\]]{}-q\_[24]{}\^[\[+a\]]{} \_[13]{}\^[\[-a\]]{}, We will also use the combinations ${{\mathscr T}}_{a,s}^c$ defined by (\[curlyT\]) for $a<|s|$. The $q$-functions related among themselves by the Plücker relations [@GKLV] are determined by $\rho_2$ and $U$ as follows
\[q1q2\]&&q\_1=1, q\_2=-iu+\_2-[[\_[pv]{}]{}]{}, \
&& q\_[12]{}=(u-u\_1-)(u+u\_1+|)Q+q\_[12]{},\
\[eqqij\]&&=\_[k=0]{}\^\^[\[2k+1\]]{},\
&&q\_[34]{}q\_[12]{}=q\_[13]{}q\_[24]{}-q\_[14]{}q\_[23]{}, q\_q\_[12]{}=q\_2\^–q\_2\^+,\
&&q\_3q\_[12]{}\^+=q\_2q\_[13]{}\^+-q\_[14]{}\^+, q\_[4]{}q\_[12]{}\^+=q\_2q\_[23]{}\^+-q\_[24]{}\^+,\
&&[[W]{}]{}\_[a]{}=q\_[3]{}\^[\[+a\]]{}|q\_[4]{}\^[\[-a\]]{}+q\_[4]{}\^[\[+a\]]{}|q\_[3]{}\^[\[-a\]]{}, [[W]{}]{}[[W]{}]{}\_0.
Above, $\hat W_{\rm pv}=\frac{1}{2}(\hat W^{[+0]}+\hat W^{[-0]})$. The definition of $\rho_2(u)$ differs slightly from the one in [@GKLV], so that here $\rho_2(u)$ is of the square root type, in the sense that $\rho_2(u)/\sqrt{4g^2-u^2}$ is analytic in the vicinity of the real axis.
#### Auxiliary integral equations.
In the intermediate steps of the computations one needs to compute 3 quantities, $Y_{1,1},Y_{2,2},\hat h$. $Y_{1,1},Y_{2,2}$ are determined from the following relations, considered at ${\rm Im}(u)>0$:
\[eqY\] \[eqYoY\] && $$\frac{{{\mathscr T}}_{2,1}{\CT}_{1,1}^{-}}{\CT_{1,2}{{\mathscr T}}_{1,1}^{-}}$$\^2=2i\^[\[1-0\]]{},\
\[eqYtY\]&&1[Y\_[1,1]{}Y\_[2,2]{}]{}=2\^[\[-0\]]{}.
The function ${{\hat h}}$ is found from equations (5.38) and (6.14) in [@GKLV]. Only its large-volume asymptotic solution (\[hathU\]) is needed for the six-loop computation.
#### Equations for $\rho,\rho_2,U$.
After $Y_{1,1},Y_{2,2},{{\hat h}}$ are found, the set of functional equations can be closed by finding $\rho,\rho_2$ from \[magic\] &=&, =, equations valid for $u\in \hat Z$, and determining $U$ from $$\label{eqU}
\hspace{-1.1em}
\left[\frac{U}{{{\hat h}}}
\frac{{{\hat h}}^{[2]}}{ U ^{[2]}}\right]^2=\frac{Y_{1,1}{{\mathscr T}}_{0,0}^-}{Y_{2,2}{{\mathscr T}}_{1,0}}
\!\[\frac{{{\mathscr T}}_{2,1}{\CT}_{1,1}^{-}}{\CT_{1,2}{{\mathscr T}}_{1,1}^{-}}\]^2 \[\frac{{Y_{1,1}Y_{2,2}}{{\mathscr T}}_{0,0}^-}{{{\mathscr T}}_{1,0}} \]^{[2]} \hspace{-1em}$$ for ${\rm Im}(u)>0$. The solution of (\[magic\]) has a one-parameter ambiguity fixed as explained below.
#### Supplementary constraints.
Let us now emphasize the role of the Bethe roots. Bethe roots appear as the zeroes of the following functions:
\[alphaconstraint\][[T]{}]{}\_[1,0]{}\^+(u\_j)=0,&&[ fixes the value of]{} ,\
\[T11constraint\][[T]{}]{}\_[1,1]{}(u\_j)=0,&&[ fixes the ambiguity in ]{}. Using them and the Hirota equation ${{\mathscr T}}_{1,0}^+{{\mathscr T}}_{1,0}^-={{\mathscr T}}_{0,0}{{\mathscr T}}_{2,0}+(U^+\bar U^-)^2{{\mathscr T}}_{1,1}^2$, we conclude that ${{\mathscr T}}_{0,0}{{\mathscr T}}_{2,0}$ should have a double zero at Bethe root which appears to be a double zero of ${{\mathscr T}}_{0,0}$.
The overall normalization of $U$ is not fixed by and should be defined from \[Unorm\] U|U= uZ.
#### Exact Bethe equations.
The set of equations above has a solution for a range of values of $u_1$. To get the correct answer for the energy, one has to insert the value of $u_1$ which is fixed by the exact Bethe equation, which can be written [@GKLV] in the form $$-\left[\frac{\hat h^-}{\hat
h^+}\right]^2\frac{Y_{2,2}^+}{Y_{2,2}^-}\frac{\CT_{1,2}^+}{\CT_{1,2}^-}\frac{\hat\CT_{1,1}^{[-2]}}{\hat\CT_{1,1}^{[+2]}}=1\ \ {\rm at}\ \ u=u_j\,.
\label{eq:BetheAn}$$
[**Asymptotic solution.**]{} The equations listed above depend on the parameter $L$ which sets the large $u$ behavior of ${{\hat h}}$. In the large volume limit $L\to\infty$ the Y-system, and hence the functional equations above, can be solved explicitly [@GKV]. All the functions $q_{ij}$, with the exception of $q_{12}$, and $q_3$ and $q_4$ are suppressed at least by a factor ${{\hat x}}^{-L}$ with respect to $q_1$, $q_2$ and $q_{12}$, so they are zero in the asymptotic expressions of ${{\mathscr T}}_{a,s}$. $q_2$ and $q_{12}$ are in this limit given by $
(q_2)_{\rm as}=-iu+\CK*\rho_2\;,
\
(q_{12})_{\rm as}=Q\,.$ The asymptotic values of $\rho$ and $\rho_2$ are ()\_[as]{}=4,(\_2)\_[as]{}=-4, where $E_{\rm as}$ is defined in (\[wrapen\]). One can check that both the equations (\[eqY\]) and (\[magic\]) are satisfied by $\left(Y_{1,1}Y_{2,2}\right)_{\rm as}\!\!=\!\frac{\Bm\Rp}{\Bp\Rm}$ and $\left(\frac{Y_{1,1}}{Y_{2,2}}\frac{{{\mathscr T}}_{2,1}^2}{{\CT}_{1,2}^2}\right)_{\rm
as}\!\!\!=\left(\!\frac{{{E}}_{\rm as}+2}{{{E}}_{\rm as}-2}\right)^2\frac {Q^+}{Q^-}\frac{\Bp^{[-2]}x^{[-2]}}{\Bm^{[+2]}x^{[+2]}}$. The asymptotic values of ${{\hat h}}$ and $\hat U$ are given by = ()\^= \_[j=1]{}\^2, \[hathU\] where for $\chi(x,y)$ one can use the BES perturbative expansion [@BES]. The shift operator $\D$ is defined such that $\D
f=f^D=f^+$. The overall normalization $\Lambda_h$ is irrelevant, [*cf.*]{} and , whereas $\Lambda_{U}=\frac 12\sqrt{E_{\rm as}(E_{\rm as}-2)}$ $\times \exp({6\gamma\,
g^2(-1\!+\!4g^2-28g^4)\!+\!18\zeta_3g^4-\!24\,g^6(3\zeta_3+5\zeta_5)})$ is fixed by . The constant $\gamma$ depends on the regularization scheme for diverging sum. We use the prescription $\frac{1}{1-\D^2}\frac 1{u}=-i\,\psi(-iu)$ for which $\gamma$ is the Euler constant.
Although for Konishi-like operators $L$ is not large, the asymptotic solution is still valid up to at least $L$ loops (four loops for Konishi), so the weak coupling is effectively a large volume limit. We use as a constraint that the full solution should reduce to the asymptotic one at weak coupling. In the following, we will continue to call ‘asymptotic’ the above quantities evaluated at the [*exact*]{} position of the Bethe roots $u_j$, determined by equations (\[eq:BetheAn\]). These quantities will therefore incorporate part of the wrapping corrections via the corrections to the Bethe roots.
\[sec:Weak\]Weak coupling expansion
===================================
In order to solve perturbatively the functional equations, our strategy is to subtract from the exact equations the asymptotic ones and to use the fact that the deviations of the $T$ functions from the asymptotic values are small. The resulting equations depend on the ratio of the exact and the asymptotic values, so we denote $(T)_{\rm r}\equiv T/(T)_{\rm as}$. The following quantities enter the functional equations H= ()\_[r]{}, r= ()\_[r]{}, r\_\*= ()\_[r]{}. All these quantities are small in perturbation, and this will allow performing the expansion of the functional equations. For example, up to seven loops, one has \[Happr\] H-+. The quantities $U$ and $q_2$ are determined asymptotically from (\[q1q2\]) and (\[hathU\]), and at the leading order they are equal to U\^2=-+…, q\_2=-iu-i[3]{}1u+…. As for the $q$-functions $q_{13}, q_{23}=q_{14}, q_{24}$, they are given by the sums in (\[eqqij\]) and in perturbation they have an array of equidistant poles which will be responsible for the appearance of the zeta functions in the final answer. For example, at the leading order, we have q\_[13]{}=&18g\^[4]{}(-iu+Q \^[(1)]{}(-iu+1/2)), where $\psi^{(n)}$ denotes the $n$-th derivative of the digamma function. The functions $q_3$ and $q_4$ have a similar structure. $q_{34}$ is given by a double sum and it contributes only at eight loops and higher. A priori, the infinite sum in (\[eqqij\]) creates also infinitely many poles at positions of the shifted Bethe roots. However, because of the equality $(\hat U^+/\hat
U^-)^2=-Q^{[+2]}/Q^{[-2]}$ which holds at the Bethe root at first four nontrivial orders and which is just the asymptotic Bethe equation, the poles in $q_{13}$ cancel out pairwise (except for the first one, which is cancelled by an overall factor $Q$). The cancellation mechanism still holds for $q_{14}$ and $q_{24}$ because $(\hat q_2^++\hat{\bar q}_2^-)_{\rm as}=(\hat{{\mathscr T}}_{1,1})_{\rm
as}$ and $\hat{{\mathscr T}}_{1,1}(u_1)=0$. At least up to seven loops, the $q$-functions are given by linear combination of the *multiple Hurwitz zeta functions,* which we define by $$\begin{aligned}
\label{MHZ}
\!\! \eta_{a_1,a_2,\cdots,a_n}(u)&=&\!\sum_{k=0}^{\infty}\left(\frac
1 {u+i k}\right)^{a_1}\!\!\! \!\eta_{a_2,\cdots,a_n}\!(u+i(k+1)),\no\\
\eta_a(u)&=&\,\frac{i^a}{(a-1)!}\;\psi^{(a-1)}(-i\,u)\;,\ \ a\geq 1,
$$ with coefficients which are rational functions of $u$ .
![\[fig:iterative\]Structure of the perturbative computation.](Iterative_Structure)
The algorithm for computing the perturbative computation is summarized in figure \[fig:iterative\]. The interior loop determines the densities $\rho$ and $\rho_2$, or rather their variations with respect to the asymptotic values, $\delta\rho$ and $\delta\rho_2$, from equations (\[eqY\]), analytically continued to $u\in\hat Z$, and from (\[magic\]). In order to preserve the square root structure of the two densities, we need to make a rescaling $z=u/2g$. When performing various integrations, we encounter two different situations. In the first, the integration is (parallel to) the real axis and we deform the contour to pass just below it and to avoid the possible singularities on the real axis. This allows us to perform uniform expansion in $g$. In the second situation the Cauchy kernel $\CK$ acts on a function with a finite support, like $\delta\rho$ and $\delta\rho_2$, then it can be expanded in terms of moments: \[momenta\] \^[\[s\]]{}=\_[n0]{}\_[-1]{}\^1z\^[2n]{}(z), except for $s=0$, where $
\CK^{[\pm 0]}\ast\rho=\pm\rho/2\ +\ \slash\!\!\!\CK\ast\rho\,,
$ the slash meaning principal value. The equations (\[eqY\]) and (\[magic\]) finally reduce to the linear system \[leadrhorho2\] (
[cc]{} 2 & 12\^\
1 & 3\
) (
[c]{} v\_1\
g\^[2 ]{}v\_2\
)=g\^9(
[c]{} C\_1\
C\_2\
), where $C_1$ and $C_2$ are Taylor series in $g^2$ and $z$ is known from the previous orders of perturbative expansion (at the leading order in $g$ they are constants) and v\_1=, v\_2=(-4iz / \_2). By definition of $\rho$ (and $\rho_2$), we know that $\rho(2gz)/\sqrt{1-z^2}$ is analytic in the vicinity of the real axis and hence can be Taylor-expanded. At a given order in $g$, this means that $\delta \rho$ and $\delta\rho_2$ are polynomial in $z$ times $\sqrt{1-z^2}$. Using this information, the equation (\[leadrhorho2\]) gives, at leading order, \_2=g\^7(A+Bz\^2), =g\^9C, with $6A=C_1-2C_2$, $2C=-C_1+4C_2$. The value of $B$ is unconstrained by (\[leadrhorho2\]), and it is fixed by .
The energy can be computed [@GKLV] from the behavior at large $u$ of $\ln Y_{1,1}Y_{2,2}$, $\ln Y_{1,1}Y_{2,2}\simeq {iE}/{u}$. Isolating the asymptotic and the wrapping part in the above expression, $E=
E_{\rm as}+E_{\rm wrap}$, one obtains E\_[as]{}=2-8g[Im]{}(1[x\_1\^+]{}), \[wrapen\] E\_[wrap]{}=\_[-i0]{}. The Asymptotic Bethe Ansatz [@BES] predicts up to 6 loops: &&E\_[BAE]{}=E\_[as]{}(u\_1u\_[1,BAE]{})=2+12g\^2-48g\^4+336g\^6\
&&-(2820+288\_3)g\^8+(26508+4320\_3+2880\_5)g\^[10]{}\
&&-(269148+55296\_3+44064\_5+30240\_7)g\^[12]{}. At finite volume ($L=2$ for the Konishi operator), the energy receives corrections both from ${{E}}_{\rm as}$ through the correction of the position of Bethe roots $u_{1}$, and from $E_{\rm wrap}$. The corrections to the Bethe equations leading to displacement of the Bethe roots start at $5$ loops, where they are due to correction of $Y_{2,2}$ only. At 6 loops one should take into account the first corrections to $\rho_2$, whereas effects from correction to $\rho$ and ${{\hat h}}$ are delayed at least up to 7 loops. $E_{\rm wrap}$ is non-zero starting from 4 loops. The single-wrapping corrections, up to seven loop, follow from computations within interior loop in figure \[fig:iterative\], whereas exterior loop has to be run only once, to find the explicit analytic expression (\[hathU\]). For $H$ one can use the approximation (\[Happr\]). The double wrapping effects, in particular correction to $(U)_{\rm as}$, are important starting from 8 loops.
We have performed the explicit perturbative expansions discussed in previous sections and computed $\delta E\equiv E-E_{\rm BAE}$ up to six loops, i.e. up to $g^{12}$ term. Intermediate expressions are too bulky to be presented here. We summarized them in the [*Mathematica*]{} notebook file [@link]. They contain $\eta$-functions (\[MHZ\]) and their residues at the Bethe root. However, the final expression is significantly simpler and is given in terms of zeta-functions:&&[[E]{}]{}\_[4&5 loop]{}=(324 + 864 \_[3]{} - 1440 \_[5]{})g\^8+\
&&(-11340+2592 \_3-11520 \_5 -5184 \_3\^2+30240 \_7)g\^[10]{},\
&&[[E]{}]{}\_[6 loop]{}=(261468 - [207360]{} \_3 - [20736]{} \_3\^2 + 156384 \_5\
&& + [155520]{} \_3 \_5 + 105840 \_7 - 489888 \_9 )g\^[12]{} . At four and five loops we reproduced the already known answers [@JanikBaj; @fiveL]. Our final expression for the Energy of the Konishi operator is $$\begin{aligned}
E=&2 + 12 g^2 - 48 g^4 + 336 g^6 +
(-2496 + 576 \zeta_3 \nonumber\\-& 1440 \zeta_5) g^8 +
(15168 + 6912 \zeta_3 - 5184 \zeta_3^2 - 8640 \zeta_5 \nonumber\\+&
30240 \zeta_7) g^{10}+
(-7680 - 262656 \zeta_3 - 20736 \zeta_3^2 \nonumber\\+& 112320 \zeta_5 +
155520 \zeta_3 \zeta_5 + 75600 \zeta_7 - 489888 \zeta_9) g^{12}\end{aligned}$$ We were informed that Z. Bajnok and R. Janik have obtained [@BJ6loop] six and seven-loop corrections using Lüscher’s method; our result coincides with their six loop result. Also, fitting the known numerical results [@GKVnum; @Fnum; @Gromovprivate] with a diagonal Padé approximant, we were able to fix the 6-loop energy with $5\%$ confidence. Our analytic result is compatible with this numerical estimation.
Conclusion
==========
We have computed the wrapping corrections of the Konishi operator up to 6-loop order using the functional equations proposed in [@GKLV]. We have adjusted the structure of the functional equations for a systematic perturbative expansion and we hope to be able to apply our methods to reach double-wrapping orders for the Konishi states. An interesting question to explore is whether the cancellation of poles at Bethe roots observed when computing $q_{ij}$ holds at any order and if it can be used as a regularity condition implying the exact Bethe equation. Another question is what type of functions appear in the final answer. So far the expression for the energy is reducing to Euler-Zagier sums and we believe that it will be always so.
The authors are indebted to Z. Bajnok, S. Frolov, V. Kazakov, I. Kostov and especially to N. Gromov for useful discussions and to I. Shenderovich and E. Sobko for involvement in the early stages of this project. D.S. and S.L. thank Nordita for hospitality, where part of this work was done.
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|
---
abstract: 'Transfer learning involves taking information and insight from one problem domain and applying it to a new problem domain. Although widely used in practice, theory for transfer learning remains less well-developed. To address this, we prove several novel results related to transfer learning, showing the need to carefully select which sets of information to transfer and the need for dependence between transferred information and target problems. Furthermore, we prove how the degree of probabilistic change in an algorithm using transfer learning places an upper bound on the amount of improvement possible. These results build on the algorithmic search framework for machine learning, allowing the results to apply to a wide range of learning problems using transfer.'
author:
- Jake Williams
- Abel Tadesse
- Tyler Sam
- Huey Sun
- |
\
George D. Monta[ñ]{}ez
bibliography:
- 'references.bib'
title: Limits of Transfer Learning
---
Introduction
============
Transfer learning is a type of machine learning where insight gained from solving one problem is applied to solve a separate, but related problem [@pan2009survey]. Currently an exciting new frontier in machine learning, transfer learning has diverse practical application in a number of fields, from training self-driving cars [@choi2018driving], where model parameters are learned in simulated environments and transferred to real-life contexts, to audio transcription [@wang2015transfer], where patterns learned from common accents are applied to learn less common accents. Despite its potential for use in industry, little is known about the theoretical guarantees and limitations of transfer learning.
To analyze transfer learning, we need a way to talk about the breadth of possible problems we can transfer from and to under a unified formalism. One such approach is the reduction of various machine learning problems (such as regression and classification) to a type of search, using the method of the algorithmic search framework [@Montanez2016TheFO; @montanez2017machine]. This reduction allows for the simultaneous analysis of a host of different problems, as results proven within the framework can be applied to any of the problems cast into it. In this work, we show how transfer learning can fit within the framework, and define affinity as a measure of the extent to which information learned from solving one problem is applicable to another. Under this definition, we prove a number of useful theorems that connect affinity with the probability of success of transfer learning. We conclude our work with applied examples and suggest an experimental heuristic to determine conditions under which transfer learning is likely to succeed.
Distinctions from Prior Work
============================
Previous work within the algorithmic search framework has focused on bias [@montanez2019fobfl; @lauw2020bias], a measure of the extent to which a distribution of information resources is predisposed towards a fixed target. The case of transfer learning carries additional complexity as the recipient problem can use not only its native information resource, but the learned information passed from the source as well. Thus, affinity serves as an analogue to bias which expresses this nuance, and enables us to prove a variety of interesting bounds for transfer learning.
Background
==========
Transfer Learning
-----------------
### Definition of Transfer Learning
Transfer learning can be defined by two machine learning problems [@pan2009survey], a source problem and a recipient problem. Each of these is defined by two parts, a domain and a task. The domain is defined by the feature space, ${\mathcal{X}}$, the label space, ${\mathcal{Y}}$, and the data, $D = \{(x_i, y_i), \dots, (x_n, y)\}$, where $x_i \in {\mathcal{X}}$ and $y_i \in {\mathcal{Y}}$. The task is defined by an objective function $P_f(Y|X)$, which is a conditional distribution over the label space, conditioned on an element of the feature space. In other words, it tells us the probability that a given label is correct for a particular input. A machine learning problem is “solved” by an algorithm $\mathcal{A}$, which takes in the domain and outputs a function $P_{\mathcal{A}}(Y|X)$. The success of an algorithm is its ability to learn the objective function as its output. Learning and optimization algorithms use a loss function ${\mathcal{L}}(p)$ to evaluate an output function to decide if it is worthy of outputting. Such algorithms can be viewed as black-box search algorithms [@Montanez2016TheFO], where the particular algorithm determines the behavior of the black box. For transfer learning under this view, the output is defined as the final element in the search history.
### Types of Transfer Learning
Pan and Yang separated transfer learning into four categories based on the type of information passed between domains [@pan2009survey]:
- [*Instance transfer*]{}: Supplementing the target domain data with a subset of data from the source domain.
- [*Feature-representation transfer*]{}: Using a feature-representation of inputs that is learned in the source domain to minimize differences between the source and target domains and reduce generalization error in the target task.
- [*Parameter transfer*]{}: Passing a subset of the parameters of a model from the source domain to the target domain to improve the starting point in the target domain.
- [*Relational-knowledge transfer*]{}: Learning a relation between knowledge in the source domain to pass to the target domain, especially when either or both do not follow i.i.d. assumptions.
The Search Framework
--------------------
![Black-box search algorithm. We add evaluated queries to the history according to the distribution iteratively. Reproduced from [@Montanez2016TheFO].[]{data-label="fig:searchFramework"}](img/black-box-algo.pdf)
To analyze transfer learning from a theoretical perspective, we take inspiration from previous work that views machine learning as a type of search. [Monta[ñ]{}ez]{} casts machine learning problems, including Vapnik’s general learning problem (covering regression, classification, and density estimation) into an algorithmic search framework [@montanez2017machine]. For example, classification is seen as a search through all possible labelings of the data, and clustering as a search through all possible ways to cluster the data [@montanez2017machine]. This framework provides a common structure which we can use to analyze different machine learning problems, as each of them can be seen as a search problem with a defined set of components. Furthermore, any result we prove about search problems applies to all machine learning problems we can represent within the framework.
Within the algorithmic search framework, the three components of a search problem are the search space $\mathrm{\Omega}$, target set $T$, and external information resource $F$. The search space, which is finite and discrete due to the finite precision representation of numbers on computers, is the set of elements to be examined. The target set is a nonempty subset of $\mathrm{\Omega}$ that contains the elements we wish to find. Finally, the external information resource is used to evaluate the elements of the search space. Usually, the target set and external information resource are related, as the external information resource guides the search to the target [@Montanez2016TheFO].
In this framework, an iterative algorithm searches for an element in the target set, depicted in Figure \[fig:searchFramework\]. The algorithm is viewed as a black box that produces a probability distribution over the search space from the search history. At each step, an element is sampled from $\mathrm{\Omega}$ according to the most recent probability distribution. The external information resource is then used to evaluate the queried element, and the element and its evaluation are added to the search history. Thus, the search history is the collection of all points sampled and all information gained from the information resource during the course of the search. Finally, the algorithm creates a new probability distribution according to its rules. Abstracting the creation of the probability distribution allows the search framework to work with many different search algorithms [@montanez2017machine].
Decomposable Probability-of-Success Metrics
-------------------------------------------
Working within the same algorithmic search framework [@Montanez2016TheFO; @montanez2017machine; @montanez2019fobfl], to measure the performance of search and learning algorithms, Sam et al. [@sam2020decomposable] defined [decomposable probability-of-success metrics]{} as $${\phi}(t,f) = \mathbf{t}^{\top}\mathbf{P}_{{\phi},f}=P_{{\phi}}(X \in t|f)$$ where $\mathbf{P}_{{\phi},f}$ is not a function of target set $t$ (with corresponding target function $\mathbf{t}$), being conditionally independent of it given information resource $f$. They note that one can view $\mathbf{t}^{\top}\mathbf{P}_{{\phi},f}$ as an expectation over the probability of successfully querying an element from the target set at each step according to an arbitrary distribution. In the case of transfer learning, the distribution we choose should place most or all of its weight on the last or last couple of steps – since we transfer knowledge from the source problem’s model after training, we care about our success at the last few steps when we’re done training, rather than the first few.
Casting Transfer Learning into the Search Framework
---------------------------------------------------
Let $\mathcal{A}$ denote a fixed learning algorithm. We cast the [source]{}, which consists of $\mathcal{X}_s, \mathcal{Y}_s, D_s,$ and $P_{f, s}(Y|X)$, into the algorithmic search framework as
1. $\mathrm{\Omega} = $ range$({\mathcal{A}})$;
2. $T = \{P(Y|X) \in \textrm{range(${\mathcal{A}}$)} \mid \Xi(P, P_{f, s}) < \epsilon \}$;
3. $F = \{D_s, \mathcal{L}_s \}$;
4. $F(\emptyset) = \emptyset; \textrm{ and }$
5. $F(\omega_i) = \mathcal{L}_s(\omega_i)$.
where $w_i$ is the $i$th query in the search process, $\mathcal{L}_s$ the loss function for the [source]{}, and $\Xi_s$ is an error functional on learned conditional distribution $P$ and the optimal conditional distribution $P_{f, s}$.
Generally, any information from the [source]{} can be encoded in a binary string, so we represent the knowledge transferred as a finite length binary string. Let this string be $L = \{0, 1\}^n$. Thus, we cast the [recipient]{}, which consists of $\mathcal{X}_r, \mathcal{Y}_r, D_r,$ and $P_{f, r}(Y|X)$, into the search framework as
1. $\mathrm{\Omega} = $ range$({\mathcal{A}})$;
2. $T = \{P(Y|X) \in \textrm{range(${\mathcal{A}}$)} \mid \Xi(P, P_{f, r}) < \epsilon \}$;
3. $F = \{D_r, \mathcal{L}_r \}$;
4. $F(\emptyset) = L; \textrm{ and }$
5. $F(\omega_i) = \mathcal{L}_r(\omega_i)$.
where $\mathcal{L}_r$ is a loss function, and $\Xi_t$ is an error functional on $P$ and the optimal conditional distribution $P_{f, r}$.
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Preliminaries
=============
Affinity
--------
In a transfer learning problem, we want to know how the [source]{} problem can improve the [recipient]{} problem, which it does through the information resource. So, we can think about how the bias of the [recipient]{} is changed by the [learned knowledge]{} that the [source]{} passes over. Recall that the bias is defined by a distribution over possible information resources. However, we know that the information resource will contain ${f_{R_o}}$, the original information resource from the recipient problem. Our distribution over information resources will therefore take that into account, and only care about the [learned knowledge]{} being passed over from the [source]{}.
To quantify this, we let ${\mathcal{D}}_L$ be the distribution placed over $L$, the possible [learning resources]{}, by the [source]{}. We can use it to make statements similar to bias in traditional machine learning by defining a property called *affinity*.
Consider a transfer learning problem with a fixed $k$-hot target vector $\mathbf{t}$, fixed [recipient]{} information resource ${f_{R_o}}$, and a distribution ${\mathcal{D}}_L$ over a collection of possible [learning resources]{}, with $L \sim {\mathcal{D}}_L$. The **affinity** between the distribution and the [recipient]{} problem is defined as $$\begin{aligned}
\text{Affin}({\mathcal{D}}_L, \textbf{t}, {f_{R_o}}) &= {\mathbb{E}}_{{\mathcal{D}}_L}[\textbf{t}^{\top}{\bf P}_{{\phi},{f_{R_{o}+L}}}] - \textbf{t}^{\top}{\bf P}_{{\phi},{f_{R_o}}} \\
&= \textbf{t}^{\top}{\mathbb{E}}_{{\mathcal{D}}_L}[{\bf P}_{{\phi},{f_{R_{o}+L}}}] - {\phi}(t , {f_{R_o}}) \\
&= \textbf{t}^{\top}\int_{\mathcal{B}} {\bf P}_{{\phi},{f_{R_{o}+L}}} {\mathcal{D}}_L(l) \text{d}l - {\phi}(t,{f_{R_o}}).\end{aligned}$$ Affinity can be interpreted as the expected increase or decrease in performance on the recipient problem when using a learning resource sampled from a set according to a given distribution.
Using affinity, we seek to prove bounds similar to existing bounds about bias, such as the Famine of Favorable Targets and Famine of Favorable Information Resources [@montanez2019fobfl].
Theoretical Results
===================
We begin by showing that affinity is a conserved quantity, implying that positive affinity towards one target is offset by negative affinity towards other targets.
[theorem]{}[consofaffin]{}\[thm:CONS-OF-AFFIN\] For any arbitrary distribution ${\mathcal{D}}$ and any ${f_{R_o}}$, $$\sum_{\mathbf{t}} {\mathrm{Affin}}({\mathcal{D}}, \mathbf{t}, {f_{R_o}}) = 0.$$
This result agrees with other no free lunch [@Wolpert1997NoFL] and conservation of information results [@SchafferConservation; @dembski2009conservation; @lauw2020bias], showing that trade-offs must always be made in learning.
Assuming the dependence structure of Figure \[DSTL\], we next bound the mutual information between our updated information resource and the recipient target in terms of the source and recipient information resources.
[theorem]{}[tld]{}\[TLuD\] Define $$\phi_{TL} := {\mathbb{E}}_{T_R,F_{R+L}}[\phi(T_R,F_{R+L})] = \Pr(\omega \in T_R; \mathcal{A})$$ as the probability of success for transfer learning. Then, $$\begin{aligned}
\phi_{TL} \leq \frac{I(F_S; T_R) + I(F_R; T_R) + D(P_{T_R} \| \mathcal{U}_{T_R}) + 1}{I_{\mathrm{\Omega}}}\end{aligned}$$ where $I_{\mathrm{\Omega}} = -\log |T_R|/|\mathrm{\Omega}|$ ($T_R$ being of fixed size), $D(P_{T_R} \| \mathcal{U}_{T_R})$ is the Kullback-Leibler divergence between the marginal distribution on $T_R$ and the uniform distribution on $T_R$, and $I(F; T)$ is the mutual information.
This theorem upper bounds the probability of successful transfer ($\phi_{TL}$) to show that transfer learning can’t help us more than our information resources allow. This point is determined by $I(F_S; T_R)$, the amount of mutual information between the source’s information resource and the recipient’s target, by $I(F_R; T_R)$ the amount of mutual information between the [recipient]{}’s information resource and the recipient’s target, and how much $P_{T_R}$ (the distribution over the [recipient]{}’s target) ‘diverges’ from the uniform distribution over the [recipient]{}’s target, $\mathcal{U}_{T_R}$. This makes sense in that
- the more dependent $F_S$ and $T_R$, the more useful we expect the source’s information resource to be in searching for $T_R$, in which case $q_{TL}$ can take on larger values.
- the more $P_{T_R}$ diverges from $\mathcal{U}_{T_R}$, the less helpless we are against the randomness (since the uniform distribution maximizes entropic uncertainty).
[theorem]{}[fofli]{} Let $\mathcal{B}$ be a finite set of learning resources and let $t \subseteq \Omega$ be an arbitrary fixed $k$-size target set. Given a [recipient]{} problem $(\Omega, t, {f_{R_o}})$, define $$\begin{aligned}
\mathcal{B}_{\phi_{\mathrm{min}}} &= \{l \in \mathcal{B} \mid {\phi}(t,{f_{R_o + l}}) \geq \phi_{\mathrm{min}} \},
\end{aligned}$$ where ${\phi}(t, {f_{R_o + l}})$ is the [decomposable probability-of-success metric]{} for algorithm ${\mathcal{A}}$ on search problem $(\Omega, t,{f_{R_o + l}})$ and $\phi_{\mathrm{min}} \in (0,1]$ represents the minimally acceptable probability of success under $\phi$. Then, $$\begin{aligned}
\frac{|{\mathcal{B}}_{\phi_{\mathrm{min}}}|}{|{\mathcal{B}}|} &\leq \frac{{\phi}(t, {f_{R_o}}) + {\mathrm{Affin}}(\mathcal{U}[{\mathcal{B}}], \mathbf{t}, {f_{R_o}})}{\phi_{\mathrm{min}}}
\end{aligned}$$ where $\phi(t, {f_{R_o}})$ is the [decomposable probability-of-success metric]{} with the [recipient]{}’s original information resource. \[thm:fofli\]
Theorem \[thm:fofli\] demonstrates the proportion of $\phi_{\mathrm{min}}$-favorable information resources for transfer learning is bounded by the degree of success without transfer, along with the affinity (average performance improvement) of the set of resources as a whole. Highly favorable transferable resources are rare for difficult tasks, within any neutral set of resources lacking high affinity. Unless a set of information resources is curated towards a specific transfer task by having high affinity towards it, the set will not and cannot contain a large proportion of highly favorable elements.
[theorem]{}[futility]{}\[thm:futilityafs\] For any fixed algorithm $\mathcal{A}$, fixed [recipient]{} problem $(\Omega, t, {f_{R_o}})$, where $t \subseteq \Omega$ with a corresponding target function $\mathbf{t}$, and distribution over information resources ${\mathcal{D}}_L$, if ${\mathrm{Affin}}({\mathcal{D}}_L, \mathbf{t}, {f_{R_o}}) = 0$, then $$\begin{aligned}
\Pr(\omega \in t; {\mathcal{A}}_L)
&= \phi(t, {f_{R_o}})
\end{aligned}$$ where $\Pr(\omega \in t; {\mathcal{A}}_L)$ represents the expected [decomposable]{} probability of successfully sampling an element of $t$ using ${\mathcal{A}}$ with transfer, marginalized over learning resources $L \sim {\mathcal{D}}_L$, and $\phi(t, {f_{R_o}})$ is the probability of success without $L$ under the given [decomposable]{} metric.
Theorem \[thm:futilityafs\] tells us that transfer learning only helps in the case that we have a favorable distribution on learning resources, tuned to the specific problem at hand. Given a distribution *not* tuned in favor of our specific problem, we can perform no better than if we had not used transfer learning. **This proves that transfer learning is not inherently beneficial in and of itself**, unless it is accompanied by a favorably tuned distribution over resources to be transferred. A natural question is how rare such favorably tuned distributions are, which we next consider in Theorem \[thm:fofad\].
[theorem]{}[fofad]{}\[thm:fofad\] Given a fixed target function $\mathbf{t}$ and a finite set of learned information resources $\mathcal{B}$, let $$\mathcal{P} = \{ \mathcal{D} \mid \mathcal{D} \in \mathbb{R}^{|\mathcal{B}|}, \sum_{l \in {\mathcal{B}}} \mathcal{D}(l) = 1 \}$$ be the set of all discrete $|\mathcal{B}|$-dimensional simplex vectors. Then, $$\begin{aligned}
\frac{\mu (\mathcal{G}_{\mathbf{t}, \phi_{\mathrm{min}}})}{\mu (\mathcal{P})} &\leq \frac{\phi(t,{f_{R_o}}) + {\mathrm{Affin}}(\mathcal{U}[{\mathcal{B}}], \mathbf{t}, {f_{R_o}})}{\phi_{\mathrm{min}}}\end{aligned}$$where $\mathcal{G}_{\mathbf{t}, \phi_{\mathrm{min}}} = \{ \mathcal{D} \mid \mathcal{D} \in \mathcal{P},{\mathrm{Affin}}(\mathcal{D}, \mathbf{t}, {f_{R_o}}) \geq \phi_{\mathrm{min}} \}$ and $\mu$ is Lebesgue measure.
We find that highly favorable distributions are quite rare for problems that are difficult without transfer learning, unless we restrict ourselves to distributions over sets of highly favorable learning resources. (Clearly, finding a favorable distribution over a set of good options is not a difficult problem.) Additionally, note that we have recovered the same bound as in Theorem \[thm:fofli\].
[theorem]{}[sa]{} Given the performance of a search algorithm on the recipient problem in the transfer learning case, ${\phi}_{TL}$, and without the learning resource, ${\phi}_{NoTL}$, we can upperbound the absolute difference as $$\begin{aligned}
|{\phi}_{TL} - {\phi}_{NoTL}| &\leq |T|\sqrt{\frac{1}{2}D_{KL}(\mathbf{P}_{TL}||\mathbf{P}_{NoTL})}.\end{aligned}$$
This result shows that unless using the learning resource significantly changes the resulting distribution over the search space, the change in performance from transfer learning will be minimal.
Examples and Applications
=========================
Examples
--------
We can use examples to evaluate our theoretical results. To demonstrate how Theorem \[TLuD\] can apply to an actual case of machine learning, we can construct a pair of machine learning problems in such a way that we can properly quantify each of the terms in the inequality, allowing us to show how the probability of successful search is directly affected by the transfer of knowledge from the source problem.
Let $\mathrm{\Omega}$ be a $16 \times 16$ grid and $|T| = k = 1$. In this case, we know that the target set is a single cell in the grid, so choosing a target set is equivalent to choosing a cell in the grid. Let the distribution on target sets $P_T$ be uniformly random across the grid. For simplicity, we will assume that there is no information about the target set in the information resource, and that any information will have to come via transfer from the source problem. Thus, $I(F_R; T_R) = 0$.
First, suppose that we provide no information through transfer, meaning that a learning algorithm can do no better than randomly guessing. The probability of successful search will be $1/256$. We can calculate the bound from our theorem using the known quantities:
- $I(F_S; T_R) = 0$;
- $I(F_R; T_R) = 0$;
- $H(T) = 8$ (because it takes 4 bits to specify a row and 4 bits to specify a column)
- $D(P_{T_R} \| \mathcal{U}_{T_R}) = \log_2 \binom{256}{1} - H(T) = 8 - 8 = 0$;
- $I_{\mathrm{\Omega}} = -\log 1/256 = 8$;
Thus, we upper bound the probability of successful search at $1/8$.
Now, suppose that we had an algorithm which had been trained to learn which half, the top or bottom, our target set was in. This is a relatively easier task, and would be ideal for transfer learning. Under these circumstances, the actual probability of successful search doubles to $1/128$. We can examine the effect that this transfer of knowledge has on our probability of success.
- $I(F_S; T_R) = H(T_R) - H(T_R | F_S) = 8 - 7 = 1$;
- $I(F_R; T_R) = 0$;
- $H(T) = 8$;
- $D(P_{T_R} \| \mathcal{U}_{T_R}) = 0$;
- $I_{\mathrm{\Omega}} = 8$;
The only change is in the mutual information between the the recipient target set and the source information resource, which was able to perfectly identify which half the target set was in. This brings the probability of successful search to $1/4$, exactly twice as high as without transfer learning.
This result is encouraging, because it demonstrates that the upper bound for transfer learning under dependence is able to reflect changes in the use of transfer learning and their effects. The upper bound being twice as high when the probability of success is doubled is good. However, the bound is very loose. In both cases, the bound is 32 times as large as the actual probability of success. Tightening the bound may be possible; however, as seen in this example, the bound we have can already serve a practical purpose.
Transferability Heuristic
-------------------------
Our theoretical results suggest that we cannot expect transfer learning to be successful without careful selection of transferred information. Thus, it is imperative to identify instances in which transferred resources will raise the probability of success. In this section, we explore a simple heuristic indicating conditions in which transfer learning may be successful, motivated by our theorems. Theorem \[TLuD\] shows that source information resources with strong dependence on the recipient target can raise the upper bound on performance. Thus, given a source problem and a recipient problem, our heuristic uses the success of an algorithm on the recipient problem after training solely on the source problem and **not** the recipient problem as a way of assessing potential for successful transfer. Using a classification task, we test whether this heuristic reliably identifies cases where transfer learning works well.
We focused on two similar image classification problems, classifying tigers versus wolves[^1] (TvW) and classifying cats versus dogs[^2] (CvD). Due to the parallels in these two problem, we expect that a model trained for one task will be able to help us with the other. In our experiment, we used a generic deep convolutional neural network image classification model (VGG16 [@simonyan2014very], using Keras[^3]) to evaluate the aforementioned heuristic to see whether it correlates with any benefit in transfer learning. The table below contains our results:
[|c|P[1.5cm]{}|P[2.5cm]{}|P[1.5cm]{}|P[2cm]{}|P[3cm]{}|]{} Run & Source Problem & Source Testing Accuracy & Recipient Problem & Additional Training & Recipient Testing Accuracy\
1 & CvD & 84.8% & TvW & N & 74.24%\
2 & CvD & 84.8% & TvW & Y & 95.35%\
3 & TvW & 92.16% & CvD & N & 48.36%\
4 & TvW & 92.16% & CvD & Y & 82.44%\
The [Source Problem]{} column denotes the problem we are transferring from, and the [Recipient Problem]{} column denotes the problem we are transferring to. The [Source Testing Accuracy]{} column contains the image classification model’s testing accuracy on the source problem after training on its dataset, using a disjoint test dataset. The [Additional Training]{} column indicates whether we did any additional training before testing the model’s accuracy on the recipient problem’s dataset — [N]{} indicates no training, which means that the following entry in the second [Recipient Testing Accuracy]{} column contains the results of the heuristic, while [Y]{} indicates an additional training phase, which means that the following entry in the [Recipient Testing Accuracy]{} column contains the experimental performance of transfer learning. In each run we start by training our model on the source problem.
Consider Runs 1 and 2. Run 1 is the heuristic run for the CvD $\rightarrow$ TvW transfer learning problem. When we apply the trained CvD model to the TvW problem without retraining, we get a testing accuracy of 74.24%. This result is promising, as it’s significantly above a random fair coin flip, indicating that our CvD model has learned something about the difference between cats and dogs that can be weakly generalized to other images of feline and canine animals. Looking at Run 2, we see that taking our model and training additionally on the TvW dataset yields a transfer learning testing accuracy of 95.35%, which is higher than the testing accuracy when we train our model solely on TvW (92.16%). This is an example where transfer learning improves our model’s success, suggesting that the pre-training step is helping our algorithm generalize.
When we look at Runs 3 and 4, we see the other side of the picture. The heuristic for the TvW $\rightarrow$ CvD transfer learning problem in Run 3 is a miserable 48.36%, which is roughly how well we would do randomly flipping a fair coin. It’s important to note that this heuristic is not symmetric, which is to be expected — for example, if the TvW model is learning based on the background of the images and not the animals themselves, we would expect a poor application to the CvD problem regardless of how well the CvD model can apply to the TvD problem. Looking at Run 4, the transfer learning testing accuracy is 82.44%, which is below the testing accuracy when we train solely on the CvD dataset (84.8%). This offers some preliminary support for our heuristic — when the success of the heuristic is closer to random, it may be the case that pre-training not only fails to benefit the algorithm, but can even hurt performance.
Let us consider what insights we can gain from the above results regarding our heuristic. A high value means that the algorithm trained on the source problem is able to perform well on the recipient problem, which indicates that the algorithm is able to identify and discriminate between salient features of the recipient problem. Thus, when we transfer what it learns (e.g., the model weights), we expect to see a boost in performance. Conversely, a low value (around 50%, since any much lower would allow us to simply flip the labels to obtain a good classifier) indicates that the algorithm is unable to learn features useful for the recipient problem, so we would expect transfer to be unsuccessful. It’s important to note that this heuristic is heavily algorithm independent, which is not the case for our theoretical results — problems with a large degree of latent similarity can receive poor values by our heuristic if the algorithm struggles to learn the underlying features of the problem.
These results offer preliminary support for the suggested heuristic, which was proposed to identify information resources that would be suitable for transfer learning. More research is needed to explore how well it works in practice on a wide variety of problems, which we leave for future work.
Conclusion
==========
Transfer learning is a type of machine learning that involves a source and recipient problem, where information learned by solving the source problem is used to benefit the process of solving the recipient problem. A popular and potentially lucrative avenue of application is in transferring knowledge from data-rich problems to more niche, difficult problems that suffer from a lack of clean and dependable data. To analyze the bounds of transfer learning, applicable to a large diversity of source/recipient problem pairs, we cast transfer learning into the algorithmic search framework, and define affinity as the degree to which learned information is predisposed towards the recipient problem’s target. In our work, we characterize various properties of affinity, show why affinity is essential for the success of transfer learning, and prove results connecting the probability of success of transfer learning to elements of the search framework.
Additionally, we introduce a heuristic to evaluate the likelihood of success of transfer, namely, the success of the source algorithm applied directly to the recipient problem without additional training. Our results show that the heuristic holds promise as a way of identifying potentially transferable information resources, and offers additional interpretability regarding the similarity between the source and recipient problems.
Much work remains to be done to develop theory for transfer learning. Through the results presented here, we learn that there are limits to when transfer learning can be successful, and gain some insight into what powers successful transfer between problems.
Appendix: Proofs {#appendix-proofs .unnumbered}
================
Note that $\sum_{\mathbf{t}}\mathbf{t}$ is the sum of all target vectors definable on $\mathrm{\Omega}$, which themselves correspond to the nonempty subsets of $\mathrm{\Omega}$. Thus, the sum equals a constant vector, $c \cdot \bm{1} = [c, c, \ldots, c]^{\top}$ where $c = 2^{|\mathrm{\Omega}|-1}$.
By the definition of affinity and the linearity of expectation, we have $$\begin{aligned}
\sum_{\mathbf{t}} {\mathrm{Affin}}({\mathcal{D}}, \mathbf{t}, {f_{R_o}})
&= \sum_{\mathbf{t}}[ {\mathbb{E}}_{\mathcal{D}}[\mathbf{t}^{\top}\mathbf{P}_{\phi, {f_{R_{o}+L}}}]- \mathbf{t}^{\top}\mathbf{P}_{\phi, {f_{R_o}}}] \\
&= \sum_{\mathbf{t}} {\mathbb{E}}_{\mathcal{D}}[\mathbf{t}^{\top}\mathbf{P}_{\phi, {f_{R_{o}+L}}}]- \sum_{\mathbf{t}}\mathbf{t}^{\top}\mathbf{P}_{\phi, {f_{R_o}}} \\
&= \left(\sum_{\mathbf{t}}\mathbf{t}^{\top}\right){\mathbb{E}}_{\mathcal{D}}[\mathbf{P}_{\phi, {f_{R_{o}+L}}}]- \left(\sum_{\mathbf{t}}\mathbf{t}^{\top}\right)\mathbf{P}_{\phi, {f_{R_o}}} \\
&= (c \cdot \bm{1}^{\top}){\mathbb{E}}_{\mathcal{D}}[\mathbf{P}_{\phi, {f_{R_{o}+L}}}]- (c \cdot \bm{1}^{\top})\mathbf{P}_{\phi, {f_{R_o}}} \\
&= c \cdot (\bm{1}^{\top}{\mathbb{E}}_{\mathcal{D}}[\mathbf{P}_{\phi, {f_{R_{o}+L}}}])- c \cdot (\bm{1}^{\top}\mathbf{P}_{\phi, {f_{R_o}}}) \\
&= c - c = 0
\end{aligned}$$ where the third equality follows from the fact that neither ${\mathbb{E}}_{\mathcal{D}}[\mathbf{P}_{\phi, {f_{R_{o}+L}}}]$ nor $\mathbf{P}_{\phi, {f_{R_o}}}$ is a function of $\mathbf{t}$, allowing both to be pulled out of their sums, and the penultimate equality follows from the linearity of expectation and the fact that $\bm{1}^{\top}\mathbf{P} = 1$ for any probability mass vector $\mathbf{P}$.
[lemma]{}[mutualInfoLem]{}\[lem:MI\] If $I( {F_{R_{o}+L}}; T_R) \leq I( {F_{R_o}}, L; T_R)$ then $$I({F_{R_{o}+L}}; T_R) \leq I(F_S; T_R) + I({F_{R_o}}; T_R).$$
$$\begin{aligned}
I({F_{R_{o}+L}}; T_R) &\leq I({F_{R_o}}, L; T_R) \\
&= I(L; T_R \mid {F_{R_o}}) + I({F_{R_o}}; T_R) \\
&= H(L \mid {F_{R_o}}) - H(L \mid {F_{R_o}}, T_R) + I({F_{R_o}}; T_R) \\
&= H(L \mid {F_{R_o}}) - H(L \mid T_R) + I({F_{R_o}}; T_R) \\
&\leq H(L) - H(L \mid T_R) + I({F_{R_o}};T_R)\\
&= I(L;T_R) + I({F_{R_o}};T_R) \\
&\leq I(F_S; T_R) + I({F_{R_o}}; T_R)\end{aligned}$$
where the first equality follows from application of the chain rule for mutual information, the second and fourth equalities follow from the definition of mutual information, the third equality follows from the conditional independence assumption, and the final inequality follows by application of the Data Processing Inequality [@cover2012elements].
By d-separation of the graphical model structure in Figure \[DSTL\] and the Data Processing Inequality [@cover2012elements], we have that $I({F_{R_{o}+L}}; T_R) \leq I({F_{R_o}}, L; T_R)$. Applying the result from Lemma \[lem:MI\] to the Learning Under Dependence theorem [@sam2020decomposable], we obtain $$\begin{aligned}
\phi_{TL} &\leq \frac{I(F_{R+L}; T_R) + D(P_{T_R} \| \mathcal{U}_{T_R}) + 1}{I_{\mathrm{\Omega}}}\\
&\leq \frac{I(F_S; T_R) + I(F_R; T_R) + D(P_{T_R} \| \mathcal{U}_{T_R}) + 1}{I_{\mathrm{\Omega}}}.
\end{aligned}$$
We seek to bound the proportion of successful search problems for which ${\phi}(t, f) \geq \phi_{\mathrm{min}}$ for any threshold $\phi_{\mathrm{min}} \in (0, 1]$. Then, $$\begin{aligned}
\frac{|\mathcal{B}_{q_{\mathrm{min}}}|}{|\mathcal{B}|} &= \frac{1}{ |\mathcal{B}|} \sum_{l \in \mathcal{B}} \mathds{1}_{{\phi}(t,{f_{R_o + l}}) \geq \phi_{\mathrm{min}}}\\
&= \mathbb{E}_{\mathcal{U}[\mathcal{B}]}[\mathds{1}_{{\phi}(t,{f_{R_{o}+L}}) \geq \phi_{\mathrm{min}}}] \\
&= \Pr({\phi}(t, {f_{R_{o}+L}}) \geq \phi_{\mathrm{min}})\\
&=\Pr(\mathbf{t}^{\top} \mathbf{P}_{{\phi}, {f_{R_{o}+L}}} \geq \phi_\mathrm{min})
\end{aligned}$$ where the final equality follows from the definition of [decomposable probability-of-success metrics]{}.
Note that all of the randomness in ${f_{R_{o}+L}}$ comes from the learned information, $L$, and not the fixed [recipient]{} information resource $R_o$. Applying Markov’s Inequality and the definition of ${\mathrm{Affin}}(\mathcal{D}_L,{\mathcal{A}}, \mathbf{t})$, we obtain $$\begin{aligned}
\frac{|\mathcal{B}_{\phi_{\mathrm{min}}}|}{|\mathcal{B}|} &\leq \frac{\mathbb{E}_{\mathcal{U}[\mathcal{B}]} [\mathbf{t}^{\top} \mathbf{P}_{{\phi}, {f_{R_{o}+L}}}]}{\phi_{\mathrm{min}}} \\
&= \frac{{\phi}(t, {f_{R_o}}) + {\mathrm{Affin}}(\mathcal{U}[{\mathcal{B}}],\mathbf{t}, {f_{R_o}})}{\phi_{\mathrm{min}}}.
\end{aligned}$$
Let $\mathcal{L}$ be the space of possible learning resources. Then, $$\begin{aligned}
\Pr(\omega \in t; \mathcal{A}_L)
&= \int_\mathcal{L} \Pr(\omega \in t, l; \mathcal{A}) {\text{d}}l\\
&= \int_\mathcal{L} \Pr(\omega \in t \mid l; \mathcal{A})\Pr(l) {\text{d}}l.
\end{aligned}$$ Since we are considering the general $\phi$ probability of success for algorithm $\mathcal{A}$ on $t$ using learning resource $l$, but with a fixed [recipient]{} information resource ${f_{R_o}}$, we have $$\Pr(\omega \in t \mid l; \mathcal{A}) = P_{{\phi}, {f_{R_o}}}(\omega \in t \mid l) = P_{{\phi}, {f_{R_o + l}}}(\omega \in t).$$ Also note that $\Pr(l) = \mathcal{D}_L(l)$ because our information resources are drawn from the distribution $\mathcal{D}_L$. Making these substitutions, we obtain $$\begin{aligned}
\Pr(\omega \in t; \mathcal{A}_L)
&= \int_\mathcal{L} P_{{\phi}, {f_{R_o + l}}}(\omega \in t)\mathcal{D}_L(l) {\text{d}}l\\
&= \mathbb{E}_{\mathcal{D}_L}\left[P_{{\phi}, {f_{R_{o}+L}}}(\omega \in t)\right]\\
&= \mathbb{E}_{\mathcal{D}_L}\left[\mathbf{t}^{\top}\textbf{P}_{{\phi},{f_{R_{o}+L}}}\right]\\
&= {\mathrm{Affin}}(\mathcal{D}_L, \bm{t}, {f_{R_o}}) + \mathbf{t}^{\top}\textbf{P}_{{\phi},{f_{R_o}}}\\
&= \phi(t,{f_{R_o}}).
\end{aligned}$$
[lemma]{}[equivAffin]{}\[lem:equiaf\] Given a fixed [recipient]{} problem ($\Omega, t, {f_{R_o}}$), where $t$ has corresponding target function $\mathbf{t}$, a finite set of learning resources ${\mathcal{B}}$, and a set $\mathcal{P} = \{\mathcal{D}\mid \mathcal{D}\in \mathbb{R}^{|{\mathcal{B}}|}, \sum_{l \in {\mathcal{B}}} \mathcal{D}(l) = 1\}$ of all discrete $|{\mathcal{B}}|$-dimensional simplex vectors, $$\begin{aligned}
{\mathbb{E}}_{{\mathcal{U}[\mathcal{P}]}}[{\mathrm{Affin}}(\mathcal{D},\mathbf{t},{f_{R_o}})] = {\mathrm{Affin}}(\mathcal{U}[{\mathcal{B}}],\mathbf{t},{f_{R_o}})\end{aligned}$$ where $\mathcal{D} \sim {\mathcal{U}[\mathcal{P}]}$.
Let $L \sim {\mathcal{D}}$. Then, $$\begin{aligned}
& {\mathbb{E}}_{{\mathcal{U}[\mathcal{P}]}}[{\mathrm{Affin}}({\mathcal{D}},\textbf{t},{f_{R_o}})]\\
&\phantom{+++}= {\mathbb{E}}_{{\mathcal{U}[\mathcal{P}]}}[{\mathbb{E}}_{{\mathcal{D}}}[\mathbf{t}^{\top}\textbf{P}_{{\phi},{f_{R_{o}+L}}}] - \phi(t, {f_{R_o}})] \\
&\phantom{+++}= {\mathbb{E}}_{{\mathcal{U}[\mathcal{P}]}}\left[\sum_{l \in {\mathcal{B}}} {\mathcal{D}}(l)\mathbf{t}^{\top}\textbf{P}_{{\phi},{f_{R_o + l}}}\right] - \phi(t, {f_{R_o}})\\
& \phantom{+++}= \sum_{l \in {\mathcal{B}}}\mathbf{t}^{\top}\textbf{P}_{{\phi},{f_{R_o + l}}}{\mathbb{E}}_{{\mathcal{U}[\mathcal{P}]}}[{\mathcal{D}}(l)] - \phi(t, {f_{R_o}})\end{aligned}$$ The quantity ${\mathbb{E}}_{{\mathcal{U}[\mathcal{P}]}}[{\mathcal{D}}(l)]$ is a uniform expectation on the amount of mass that the random distribution ${\mathcal{D}}$ places on resource $l$. Since $\mathcal{P}$ contains all possible distributions over ${\mathcal{B}}$, under uniform expectation the same amount of probability mass gets placed on each information resource. So, ${\mathbb{E}}_{{\mathcal{U}[\mathcal{P}]}}[{\mathcal{D}}(i)] = {\mathbb{E}}_{{\mathcal{U}[\mathcal{P}]}}[{\mathcal{D}}(j)]$ for any $i, j \in {\mathcal{B}}$. Since the probability mass on any two learning resources is equivalent and the total probability mass must sum to one, by the Expectation of Simplex Vectors is Simplex [@montanez2019fobfl], we have ${\mathbb{E}}_{{\mathcal{U}[\mathcal{P}]}}[{\mathcal{D}}(f)] = \frac{1}{|{\mathcal{B}}|}$. Continuing, $$\begin{aligned}
{\mathbb{E}}_{{\mathcal{U}[\mathcal{P}]}}[{\mathrm{Affin}}({\mathcal{D}},\textbf{t},{f_{R_o}})] &= \frac{1}{|{\mathcal{B}}|}\sum_{l \in {\mathcal{B}}} \mathbf{t}^{\top}\textbf{P}_{{\phi},{f_{R_{o}+L}}} - \phi(t, {f_{R_o}}) \\
&= {\mathrm{Affin}}(\mathcal{U}[{\mathcal{B}}], \textbf{t}, {f_{R_o}}).\end{aligned}$$
Let ${\mathcal{D}}\sim {\mathcal{U}[\mathcal{P}]}$. Then, $$\begin{aligned}
\frac{\mu (\mathcal{G}_{\mathbf{t}, \phi_{\mathrm{min}}})}{\mu (\mathcal{P})} &= \Pr({\mathrm{Affin}}({\mathcal{D}}, \mathbf{t}, {f_{R_o}}) \geq \phi_{\mathrm{min}}) \\
&= \Pr[\phi(t,{f_{R_o}}) + {\mathrm{Affin}}({\mathcal{D}}, \mathbf{t}, {f_{R_o}}) \geq \phi(t,{f_{R_o}}) + \phi_{\mathrm{min}}] \\
&= \Pr[{\mathbb{E}}_{{\mathcal{D}}}[\textbf{t}^{\top}{\bf P}_{{\phi},{f_{R_{o}+L}}}] \geq \phi(t,{f_{R_o}}) + \phi_{\mathrm{min}}].
\end{aligned}$$
Applying Markov’s Inequality and Lemma \[lem:equiaf\], we obtain $$\begin{aligned}
\frac{\mu (\mathcal{G}_{\mathbf{t}, \phi_{\mathrm{min}}})}{\mu (\mathcal{P})} &\leq \frac{{\mathbb{E}}_{{\mathcal{U}[\mathcal{P}]}}[{\mathbb{E}}_{{\mathcal{D}}}[\textbf{t}^{\top}{\bf P}_{{\phi},{f_{R_{o}+L}}}]]}{\phi(t,{f_{R_o}}) + \phi_{\mathrm{min}}} \\
&= \frac{\phi(t,{f_{R_o}}) + {\mathbb{E}}_{{\mathcal{U}[\mathcal{P}]}}[{\mathrm{Affin}}(\mathcal{D}, \mathbf{t}, {f_{R_o}})]}{\phi(t,{f_{R_o}}) + \phi_{\mathrm{min}}} \\
&= \frac{\phi(t,{f_{R_o}}) + {\mathrm{Affin}}(\mathcal{U}[{\mathcal{B}}], \mathbf{t}, {f_{R_o}})}{\phi(t,{f_{R_o}}) + \phi_{\mathrm{min}}} \\
&\leq \frac{\phi(t,{f_{R_o}}) + {\mathrm{Affin}}(\mathcal{U}[{\mathcal{B}}], \mathbf{t}, {f_{R_o}})}{\phi_{\mathrm{min}}}.
\end{aligned}$$
$$\begin{aligned}
|{\phi}_{TL} - {\phi}_{NoTL}| &= |\textbf{t}^{\top}(\mathbf{P}_{TL}-\mathbf{P}_{NoTL})|\\
&= |\sum_{\omega}\mathds{1}_{\omega \in T}(\mathbf{P}_{TL}(\omega)-\mathbf{P}_{NoTL}(\omega))|\\
& \leq |T| \sup_{w \in T} |\mathbf{P}_{TL}(\omega)-\mathbf{P}_{NoTL}(\omega)|\\
&\leq |T|\sqrt{\frac{1}{2}D_{KL}(\mathbf{P}_{TL}||\mathbf{P}_{NoTL})}
\end{aligned}$$
where the first equality follows form the definition of decomposable probability of success metrics and the final inequality follows by application of Pinsker’s Inequality.
[^1]: <http://image-net.org/challenges/LSVRC/2014/browse-synsets>
[^2]: <https://www.kaggle.com/c/dogs-vs-cats-redux-kernels-edition/data>
[^3]: <https://keras.io/applications/#vgg16>
|
---
abstract: |
We prove that almost any pair of real numbers ${\alpha},{\beta}$, satisfies the following inhomogeneous uniform version of Littlewood’s conjecture: $$\label{main equation}
\forall {\gamma},{\delta}\in{\mathbb{R}},\quad \liminf_{|n|\to\infty} {\left|n\right|}{\langlen{\alpha}-{\gamma}\rangle}{\langlen{\beta}-{\delta}\rangle}=0,$$ where ${\langle\cdot\rangle}$ denotes the distance from the nearest integer. The existence of even a single pair that satisfies , solves a problem of Cassels [@Ca] from the 50’s. We then prove that if $1,{\alpha},{\beta}$ span a totally real cubic number field, then ${\alpha},{\beta}$, satisfy . This generalizes a result of Cassels and Swinnerton-Dyer, which says that such pairs satisfies Littlewood’s conjecture. It is further shown that if ${\alpha},{\beta}$ are any two real numbers, such that $1,{\alpha},{\beta}$, are linearly dependent over ${\mathbb{Q}}$, they cannot satisfy . The results are then applied to give examples of irregular orbit closures of the diagonal groups of a new type. The results are derived from rigidity results concerning hyperbolic actions of higher rank commutative groups on homogeneous spaces.
author:
- Uri Shapira
title: A solution to a problem of Cassels and Diophantine properties of cubic numbers
---
[^1]
Introduction
============
Notation
--------
We first fix our notation and define the basic objects to be discussed in this paper. Let $X_d$ denote the space of $d$-dimensional unimodular lattices in ${\mathbb{R}}^d$ and let $Y_d$ denote the space of translates of such lattices. Points of $Y_d$ will be referred to as *grids*, hence for $x\in X_d$ and $v\in{\mathbb{R}}^d$, $y=x+v\in Y_d$ is the grid obtained by translating the lattice $x$ by the vector $v$. We denote by $\pi$ the natural projection $$\label{projections}
Y_d \stackrel{\pi}{\longrightarrow} X_d,\quad x+v \mapsto x.$$ For each $x\in X_d$, we identify the fiber $\pi^{-1}(x)$ in $Y_d$ with the torus ${\mathbb{R}}_d/x$. Let $N:{\mathbb{R}}^d\to {\mathbb{R}}$ denote the function $N(w)=\prod_1^dw_i$. For a grid $y\in Y_d$, we define the *product set* of $y$ to be $$\label{prod set}
P(y)={\left\{N(w) : w\in y\right\}}$$ In this paper we shall study properties of the product set. We will mainly be interested in density properties and the values near zero. We denote $$\label{mu}
N(y)= \inf{\left\{{\left|N(w)\right|} : w\in y \right\}}.$$ The ambiguous use of the symbol $N$ both for a function on ${\mathbb{R}}^d$ and for a function on $Y_d$ and the lack of appearance of the dimension $d$ in the notation should not cause any confusion. The *inhomogeneous minimum* of a lattice $x\in X_d$ is defined by $$\label{inhomogeneous minimum}
\mu(x)=\sup{\left\{N(y) : y\in \pi^{-1}(x)\right\}}.$$ The *inhomogeneous Markov spectrum* (or just the spectrum) is defined by $$\label{spectrum}
{\mathcal{S}}_d={\left\{\mu(x) : x\in X_d\right\}}.$$ A more geometric way to visualize the above notions is the following: The *star body* of radius ${\epsilon}>0$ is the set $S_{\epsilon}={\left\{w\in{\mathbb{R}}^d : {\left|N(w)\right|}<{\epsilon}\right\}}$. In terms of star bodies, for a grid $y\in Y_d$, $N(y)=\inf{\left\{{\epsilon}: S_{\epsilon}\cap y\ne\emptyset\right\}}$ and for a lattice $x\in X_d$, $\mu(x)$ is the least number such that for any ${\epsilon}>\mu(x)$, the star body $S_{\epsilon}$ intersects all the grids of $x$ or equivalently, $S_{\epsilon}$ projects onto the torus $\pi^{-1}(x)$ under the natural projection.
Dimension 2 {#dim2}
-----------
In [@D] Davenport showed (generalizing a result of Khintchine) that for any $x\in X_2$ one has $\mu(x)>\frac{1}{128}$, hence the spectrum ${\mathcal{S}}_2$ is bounded away from zero. The constant $\frac{1}{128}$ is not optimal and much work has been done to improve Davenport’s lower bound of the spectrum (see [@Ca],[@Ca2],[@D] and the references therein). The set ${\left\{y\in\pi^{-1}(x): \mu(y)>0\right\}}$ (where $x\in X_2$ is arbitrary) has been investigated too and in a recent work [@Ts] it was shown that it has full Hausdorff dimension and in fact that it is a winning set for Schmidt’s game.
Cassels problem
---------------
In his book [@Ca] (p. 307), Cassels raised the following natural question:
\[Cassels problem\] In dimension $d\ge 3$, is the infemum of the spectrum ${\mathcal{S}}_d$ equal to zero.
We answer Cassels’ problem affirmatively. In fact we show that the infemum is attained and give explicit constructions of lattices attaining the minimum. The following theorem is a consequence of corollary \[almost any GDP\](1):
\[theorem 101\] For $d\ge 3$, almost any lattice $x\in X_d$ (with respect to the natural probability measure) satisfies $\mu(x)=0$.
Diophantine approximations {#DA}
--------------------------
Of particular interest to Diophantine approximations, are lattices of the following forms: Let $v\in{\mathbb{R}}^{d-1}$ be a column vector. Denote $$\label{hor sgrps}
h_v={\left(
\begin{array}{ll}
I_{d-1}&v\\
0 & 1
\end{array}
\right)},\quad
g_v={\left(
\begin{array}{ll}
1&v^t\\
0 & I_{d-1}
\end{array}
\right)}$$ where $I_{d-1}$ denotes the identity matrix of dimension $d-1$ and the $0$’s denote the corresponding trivial vectors. Let $x_{v},z_v\in X_d$, denote the lattices spanned by the columns of $h_v$ and $g_v$ respectively. For ${\gamma}\in{\mathbb{R}}$, denoting by ${\langle{\gamma}\rangle}$, the distance from ${\gamma}$ to the nearest integer, an easy calculation shows that the statements $$\label{mu(x)1}
\forall \vec{{\gamma}}\in{\mathbb{R}}^{d-1}\quad \liminf_{|n|\to\infty} \left|n\right|\prod_1^{d-1}{\langlenv_i-{\gamma}_i\rangle}=0,$$ $$\label{mu(z)2}
\forall{\gamma}\in {\mathbb{R}}\quad \liminf_{\prod {\left|n_i\right|}\to\infty}\prod_1^{d-1}{\left|n_i\right|}{\langle\sum_1^{d-1}n_iv_i-{\gamma}\rangle}=0,$$ imply that $\mu(x_{v})=0$ and $\mu(z_{v})=0$ respectively.
\[corollary 102\] Let $d\ge 3$
1. For almost any $v\in{\mathbb{R}}^{d-1}$ (with respect to Lebesgue measure) and are satisfied and in particular $\mu(x_{v})=\mu(z_{v})=0$.
2. Nonetheless, if $\dim span_{\mathbb{Q}}{\left\{1,v_1,\dots,v_{d-1}\right\}}\le 2$ then $\mu(x_{v}),\mu(z_{v})$ are positive.
Part (1) of the above theorem is a consequence of corollary \[almost any GDP\]. Part (2) follows from known results in dimension 2 and will be proved in §\[irregular orbits\].\
Perhaps the most interesting amongst the results in this paper is the following theorem which shows that certain pairs of algebraic numbers are generic. The proof follows from corollary \[cubics are good\] and corollary \[almost any GDP\](3).
\[my result\] If $1,{\alpha},{\beta}$ form a basis for a totally real cubic number field, then $$\label{my result 1}
\forall {\gamma},{\delta}\in{\mathbb{R}}\quad \liminf_{|n|\to\infty} |n|{\langlen{\alpha}-{\gamma}\rangle}{\langlen{\beta}-{\delta}\rangle}=0,$$ $$\label{my result 1'}
\forall {\gamma}\in{\mathbb{R}}\quad \liminf_{|nm|\to\infty} |nm|{\langlen{\alpha}+m{\beta}-{\gamma}\rangle}=0.$$
Remarks
-------
1. Cassels and Swinnerton-Dyer have shown [@CaSD] that any real pair ${\alpha},{\beta}$, belonging to the same cubic totally real field, satisfies Littlewood’s conjecture, i.e. satisfies with ${\gamma}={\delta}=0.$ Thus theorem \[my result\] together with theorem \[corollary 102\](2), can be viewed as a strengthening of their result.
2. As Cassels points out in his book [@Ca], problem \[Cassels problem\] belongs to a family of problems for various forms (other than $N$). Barnes [@Ba] solved an analogous problem with $N$ replaced by an indefinite quadratic form in $d\ge 3$ variables. Our method, when adapted appropriately, seem to give a different proof of Barnes’ result.
3. In a recent paper [@Bu], Y. Bugeaud raised (independently of Cassels) the question of existence of pairs ${\alpha},{\beta}\in{\mathbb{R}}$ which satisfy .
4. Our methods are dynamical and rely on rigidity results such as Ratner’s theorem [@R], the results and techniques appearing in [@LW] and the extension of Furstenberg’s times 2 times 3 theorem [@F] due to Berend [@B]. But, although the usual ergodic theoretic arguments provide existence only, our results provide us with concrete examples of numbers and lattices with nontrivial dynamical and Diophantine properties.
Acknowledgments
---------------
: I would like to express my gratitude to Elon Lindenstrauss for numerous useful conversations, especially for the reference to [@Ca] and for discussing theorem \[density\]. I thank Hillel Furstenberg and Yitzhak Katznelson for stimulating conversations concerning problem \[Cassels problem\]. Finally, I would like to express my deepest gratitude and appreciation to my adviser and teacher, Barak Weiss, for his constant help, encouragement and belief.
Basic notions, groups and homogeneous spaces {#Basic notions}
============================================
When $d\ge 2$ is fixed we denote $G=SL_d({\mathbb{R}})\ltimes{\mathbb{R}}^d$, $G_0=SL_d({\mathbb{R}})$ and $V={\mathbb{R}}^d$. We shall identify $G_0,V$ with the corresponding subgroups of $G$. Denote by $A<G_0$ the subgroup of diagonal matrices with positive diagonal entries. The Lie algebra of $A$ is identified with the Euclidean $d-1$ dimensional space $$\label{euclidean space}
{\mathfrak{a}}={\left\{\textbf{t}=(t_1,\dots,t_d)\in{\mathbb{R}}^d : \sum_1^dt_i=0\right\}}.$$ $A$ is isomorphic to the additive group ${\mathfrak{a}}$, via the exponent map $\exp:{\mathfrak{a}}\to A$ given by $\exp(\textbf{t})=diag(e^{t_1},\dots,e^{t_d}).$ We denote the inverse of $\exp$ by $\log$. The *roots* of $A$ are the linear functionals on ${\mathfrak{a}}$ of the following forms $$\label{roots of A}
\forall 1\le i\ne j\le d, \quad \textbf{t}\mapsto t_i-t_j ; \quad\forall 1\le k\le d,\quad \textbf{t}\mapsto t_k.$$ The set of roots will be denoted by $\Phi$. As suggested in , we say that a root ${\alpha}\in\Phi$ corresponds to a pair $1\le i\ne j\le d$ or to an index $1\le k\le d$. To each root ${\alpha}\in\Phi$, there corresponds a one parameter unipotent subgroup ${\left\{u_{\alpha}(t)\right\}}_{t\in{\mathbb{R}}}<G$, called the *root group*, for which the following equation is satisfied $$\label{alpha(a)}
au_{\alpha}(t)a^{-1}=u_{\alpha}(e^{{\alpha}(\log(a))}t).$$ When the root ${\alpha}$ corresponds to a pair $i\ne j$, we sometime denote $u_{\alpha}(t)=u_{ij}(t)$. In this case $u_{ij}(t)\in G_0$ is the matrix all of whose entries are zero, except for the $ij$’th which is equal to $t$ and the diagonal entries which are equal to $1$. When ${\alpha}$ corresponds to $1\le k\le d$, we sometime denote $u_{\alpha}(t)=u_k(t)$. In this case, $u_k(t)\in V$ is the vector, $te_k$, where $e_k$ is the $k$’th standard vector. We sometime abuse notation and write, for a root ${\alpha}\in\Phi$ and $a\in A$, ${\alpha}(a)$ instead of ${\alpha}(\log(a))$.
For an element $a\in A$ we define the *stable horospherical subgroup* of $G$ corresponding to it to be $U^-(a)=\left\{(g,v)\in G : a^n(g,v)a^{-n}\to_{n\to\infty} e\right\}$, and the *unstable horospherical subgroup* to be $U^+(a)=U^-(a^{-1})$. An element $b\in A$ is called *regular* if for any root ${\alpha}\in\Phi$, ${\alpha}(b)\ne 0$. For $b\in A$, any element $g\in G$ which is close enough to $e$, has a unique decomposition $g=cu^+u^-$, where $c$ centralizes $b$, $u^+\in U^+(b),u^-\in U^-(b)$ and $c,u^+,u^-$ lie in corresponding neighborhoods of $e$. If $b$ is regular then the centralizer of $b$ is $A$.\
The linear action of $G_0$ on ${\mathbb{R}}^d$ induces a transitive action of $G_0$ on $X_d$. The stabilizer of the lattice ${\mathbb{Z}}^d\in X_d$ is ${\Gamma}_0=SL_d({\mathbb{Z}})$. This enables us to identify $X_d$ with the homogeneous space $G_0/{\Gamma}_0$. For $g\in G_0$, we denote $\bar{g}=g{\Gamma}_0$. $\bar{g}\in X_d$ represents the lattice spanned by the columns of the matrix $g$. In a similar manner we identify $Y_d$ with $G/{\Gamma}$, where ${\Gamma}=SL_d({\mathbb{Z}})\ltimes {\mathbb{Z}}^d$. For $(g,v)\in G$, $(g,v){\Gamma}$ represents the grid $\bar{g}+v.$ ${\Gamma}$ (resp ${\Gamma}_0$) is a lattice in $G$ (resp $G_0$). The $G$ (resp $G_0$) invariant probability measure on $Y_d$ (resp $X_d$) will be referred to as the *Haar measure*. $G_0$ and its subgroups act on $X_d,Y_d$ and the action commutes with the projection $\pi:Y_d\to X_d$. Finally, we say that a grid $y=x+v$ is *rational*, if $v$ belongs to the ${\mathbb{Q}}$-span of the lattice $x$. This is equivalent to saying that $y\in\pi^{-1}(x)$ is a torsion element.
Compact $A$ orbits {#compact orbits}
==================
The following classification theorem, essentially goes back to [@Bac]. A modern proof can be found in [@LW] or [@Mc]. Before stating it, let us recall some notions from number theory. A *totally real number field* is a finite extension of ${\mathbb{Q}}$, all of whose embeddings into ${\mathbb{C}}$ are real. A *lattice* in a number field, is the ${\mathbb{Z}}$-span of a basis of the field over ${\mathbb{Q}}$. Let $K$ be a totally real number field of degree $d$ and let ${\sigma}_i, i=1\dots d$, be the different embeddings of $K$ into the reals. The map ${\varphi}=({\sigma}_1,\dots,{\sigma}_d)^t:K\to{\mathbb{R}}^d$ is called a *geometric embedding*. It is well known that if ${\Lambda}$ is a lattice in $K$, then ${\varphi}({\Lambda})$ is a lattice in ${\mathbb{R}}^d.$ The ring of integers in $K$ is denoted by ${\mathcal{O}}_K$ and the group of units of this ring is denoted by ${\mathcal{O}}_K^*$. The logarithmic embedding of ${\mathcal{O}}_K^*$ in ${\mathfrak{a}}$ (see ) is given by ${\omega}\mapsto(\log{\left|{\sigma}_1({\omega})\right|},\dots,\log{\left|{\sigma}_d({\omega})\right|}).$ We shall denote the image of ${\mathcal{O}}_K^*$ by ${\Omega}_K$. Dirichlet’s unit theorem implies that ${\Omega}_K$ is a lattice in ${\mathfrak{a}}$.
\[compact lattice orbits\] Let $x_0\in X_d$. $Ax_0$ is compact if and only if there exists $a\in A$ such that $ax_0$ is (up to multiplication by a normalizing scalar) the geometric embedding of a lattice in a totally real number field, $K$, of degree $d$. Moreover there exists some finite index subgroup ${\Omega}<{\Omega}_K$ such that ${\Omega}=\log{\left(Stab_A(x_0)\right)}$.
As a corollary we get a classification of the compact $A$-orbits in $Y_d$. The proof is left to the reader.
\[compact grid orbits\] A grid $y\in Y_d$ has a compact $A$-orbit, if and only if it is rational and $\pi(y)\in X_d$ has a compact orbit. In this case, $Stab_A(y)$ is of finite index in $Stab_A(\pi(y))$.
The following corollary of theorem \[compact lattice orbits\] is one of the places in which higher rank is reflected.
\[independence\] Let $d\ge 3$ and let $y\in Y_d$ be a grid with a compact $A$-orbit. Denote $A_0=Stab_A(y)$. Then, for any root ${\alpha}\in\Phi$, the set ${\left\{{\alpha}(a) : a\in A_0\right\}}$ is dense in the reals.
Let $K$ be the totally real number field of degree $d$ arising from theorem \[compact lattice orbits\] and corollary \[compact grid orbits\] and let ${\alpha}\in\Phi$ be a root. By corollary \[compact grid orbits\], $\log(A_0)$ is of finite index in ${\Omega}_K$. It follows that it is enough to justify why ${\alpha}({\Omega}_K)$ is dense in the reals. As ${\Omega}_K$ is a lattice in ${\mathfrak{a}}$, this is equivalent to ${\Omega}_K\cap ker({\alpha})$ not being a lattice in $ker({\alpha})$. If ${\alpha}$ corresponds to a pair $i\ne j$ (see ), then if ${\Omega}_K\cap ker({\alpha})<ker({\alpha})$ is a lattice, then there is a subfield of $K$ (the field ${\left\{\theta\in K : {\sigma}_i(\theta)={\sigma}_j(\theta)\right\}}$) with a group of units containing a copy of ${\mathbb{Z}}^{d-2}$. The degree of this subfield is at most $d/2$ and so by Dirichlet’s unit theorem the degree of the group of units in this subfield is at most $d/2-1$. This means that $d/2-1\ge d-2$ which is equivalent to $d\le 2$ a contradiction. If ${\alpha}$ corresponds to $k$, then the situation is even simpler as ${\Omega}_K\cap ker({\alpha})={\left\{0\right\}}$.
Dynamics and *GDP* lattices {#existence}
===========================
Inheritance
-----------
The reason that the action of $A$ on $X_d,Y_d$ is of importance to us is the invariance of the product set, namely $\forall a\in A, y\in Y_d$, $P(y)=P(ay)$.
\[GDP definition\]
1. A grid $y\in Y_d$ is called $DP$ (dense products) if $\overline{P(y)}={\mathbb{R}}$.
2. A lattice $x\in X_d$ is called $GDP$ if any grid $y\in\pi^{-1}(x)$ is $DP$.
The proofs of the next useful lemma and its corollary are left to the reader.
\[inheritance lemma\] If $y,y_0\in Y_d$ are such that $y_0\in\overline{Ay}$, then $\overline{P(y_0)}\subset \overline{P(y)}$.
\[inheritance 2\]
1. If $y,y_0\in Y_d$ are such that $y_0\in\overline{Ay}$ and $y_0$ is $DP$, then $y$ is $DP$ too.
2. If $x,x_0\in X_d$ are such that $x_0\in\overline{Ax}$ and $x_0$ is $GDP$, then $x$ is $GDP$ too.
\[two ways to prove DP\]
1. If $y\in Y_d$ is such that $\overline{Ay}\supset\pi^{-1}(x_0)$ for some $x_0\in X_d$ then $y$ is $DP$.
2. If $y\in Y_d$ is such that there exists $y_0\in Y_d$ and a root group ${\left\{u_{ij}(t)\right\}}_{t\in{\mathbb{R}}}<G_0$ such that $\overline{Ay}\supset {\left\{u_{ij}(t)y_0 : t\in I\right\}}$, where $I\subset {\mathbb{R}}$ is a ray, then $y$ is $DP$.
To see (1) note that from the inheritance lemma it follows that $\forall v\in{\mathbb{R}}^d$ $P(x_0+v)\subset \overline{P(y)}$. Clearly $\cup_{v\in{\mathbb{R}}^d}P(x_0+v)={\mathbb{R}}$. To see (2) note that it follows from [@R] theorem B, that $${\left\{u_{ij}(t)y_0:t\in{\mathbb{R}}\right\}}\subset \overline{{\left\{u_{ij}(t)y_0 : t\in I\right\}}}.$$ Let $w\in y_0$ be a vector all of whose coordinates are nonzero. By the inheritance lemma $$\overline{P(y)}\supset{\left\{N{\left(u_{ij}(t)w\right)} : t\in {\mathbb{R}}\right\}}={\left\{N(w){\left(\frac{w_j}{w_i}t+1\right)}:t\in{\mathbb{R}}\right\}}={\mathbb{R}}.$$
Existence of $GDP$ lattices for $d\ge 3$
----------------------------------------
The proof of the following theorem is based on the ideas presented in [@LW].
\[existence of GDP\] If $x,x_0\in X_d$ ($d\ge 3$), $Ax_0$ is compact and $x_0\in \overline{Ax}\setminus Ax$, then $x$ is $GDP$.
Let $y\in \pi^{-1}(x)$. Consider $F=\overline{Ay}$ and $F_0=F\cap\pi^{-1}(x_0)$. Note that from the compactness of the fibers of $\pi$ and the assumptions of the theorem it follows that $F_0\ne\emptyset$. In [@Sh] (lemma 4.8) it is shown that any irrational grid $y_0\in \pi^{-1}(x_0)$, satisfies $\overline{Ay_0}\supset \pi^{-1}(x_0)$. Hence by lemma \[two ways to prove DP\] (1), $y_0$ is $DP$. Hence, if $F_0$ contains an irrational grid then $y$ is $DP$ by corollary \[inheritance 2\] (1).\
Assume then that $F_0$ contains only rational grids and let $y_0\in F_0$ (this could happen for example if $y$ is a rational grid). By corollary \[compact grid orbits\], $Ay_0$ is compact. Denote $A_0=Stab_A{\left(y_0\right)}.$ Choose a regular element $b\in A_0$. Let $U^-=U^-(b),U^+=U^+(b),$ be the corresponding stable and unstable horospherical subgroups of $G$. Any point which is close enough to $y_0$ in $Y_d$, has a unique representation of the form $au^+u^-y_0$, where $a\in A,u^+\in U^+$ and $u^-\in U^-$ are in corresponding neighborhoods of the identity. Choose a sequence $y_n\to y_0$ from the orbit $Ay$. We may assume that $$\label{transverse convergence}
y_n=a_nu_n^+u_n^-y_0 \in F$$ where $a_n,u_n^+,u_n^-\to e$. We may further assume that $a_n=e$ for all $n$, for if not, replace $y_n$ by $a_n^{-1}y_n$. The fact that $y_0$ is not in $Ay$ implies that the pairs $(u_n^+,u_n^-)$ are nontrivial for any $n$. Our first goal is to show:\
**Claim** 1: *There exist a point in $F$ of the form $uy_0$, where $u\ne e$ is in $U^+$ or $U^-$.*\
If there exists an $n$ with one of $u_n^+$ or $u_n^-$, being trivial, then the claim follows. If not, we denote for any $n$ by $k_n$, the least integer such that the maximum of the absolute values of the entries of $b^{k_n}u_n^+b^{-k_n}$ is greater than $1$. It then follows that this absolute value lies in some interval of the form $[1,M]$ (where $M$ only depends on the choice of $b$). Since $u_n^+\to e$ we must have $k_n\to\infty$. It is easy to see that the convergence $b^{k_n}u^-b^{-k_n}\to e$, for $u^-\in U^-$, is uniform on compact subsets of $U^-$. Hence in particular, $b^{k_n}u_n^-b^{-k_n}\to e$. Thus after going to a subsequence and abusing notation, we may assume that $b^{k_n}u_n^+b^{-k_n}\to u$, where $e\ne u\in U^+.$ Hence $$\lim b^{k_n}y_n=\lim b^{k_n}u_n^+b^{-k_n}b^{k_n}u_n^-b^{-k_n}y_0=uy_0\in F$$ and claim 1 follows.\
**Claim** 2: *There exist a root ${\alpha}\in \Phi$ and $t_0\ne 0$ such that $u_{\alpha}(t_0)y_0\in F$.*\
Let $u$ be as in claim 1. We denote for $g\in G$, $$\Phi_g={\left\{{\alpha}\in\Phi :\textrm{ the entry corresponding to ${\alpha}$ in $g$ is nonzero}\right\}}.$$ If $\Phi_u$ contains only one root, claim 2 follows. If not, there exists a one parameter semigroup $\{a_t\}_{t\ge 0}<A$ such that $\Phi_u$ is the union of two non empty disjoint sets, $\Phi_u^-,\Phi_u^0$, such that for ${\alpha}\in\Phi_u^-$, ${\alpha}(a_1)<0$, while for ${\alpha}\in\Phi_u^0$, ${\alpha}(a_1)=0$ (see [@LW] step 4.5 for details). It follows that for any sequence $t_n\to\infty$, $a_{t_n}ua_{t_n}^{-1}\to u'$, where $\Phi_{u'}=\Phi_u^0$, which is strictly smaller then $\Phi_u$. Since $Ay_0\simeq A/A_0$ is a $(d-1)$-torus, we can always find a sequence $t_n\to\infty$, such that $a_{t_n}y_0\to y_0$. Thus $$\lim a_{t_n}uy_0=\lim a_{t_n}ua_{t_n}^{-1}a_{t_n}y_0=u'y_0\in F.$$ Repeating this process a finite number of times, we end up with a root ${\alpha}$ and some nonzero real number $t_0$, such that $u_{\alpha}(t_0)y_0\in F$ and claim 2 follows.\
**Claim** 3: *There exists a ray $I\subset {\mathbb{R}}$ such that ${\left\{u_{\alpha}(t)y_0 : t\in I\right\}}\subset F$*.\
By corollary \[independence\], we have that ${\left\{{\alpha}(a) : a\in A_0\right\}}$ is dense in ${\mathbb{R}}$. It follows that $I=\overline{{\left\{e^{{\alpha}(a)}t_0 : a\in A_0\right\}}}$ is a ray. We have $${\left\{au_{\alpha}(t_0)y_0 : a\in A_0\right\}} = {\left\{au_{\alpha}(t_0)a^{-1}y_0 : a\in A_0\right\}} = {\left\{u_{\alpha}(e^{{\alpha}(a)}t_0)y_0 : a\in A_0\right\}}\subset F.$$ Claim 3 now follows from the fact that $F$ is closed.\
Note that from our assumption that $F_0$ contain only rational grids, it follow that the root group $u_{\alpha}(t)$ is contained in $G_0$. It now follows from lemma \[two ways to prove DP\] (2) that $y$ is $DP$ and the theorem follows.
\[dense imply GDP\] For $d\ge 3$, any lattice with a dense $A$ orbit is $GDP$.
This is a consequence of theorem \[existence of GDP\], and corollary \[inheritance 2\](2).
The following lemma is well known. We give the outline of a proof.
\[expanding leaf generic\] For any $d\ge 2$ and almost any $v\in {\mathbb{R}}^{d-1}$ (with respect to Lebesgue measure) $\overline{Ax_{v}}=\overline{Az_{v}}=X_d$.
Let us consider lattices of the form $x_v$ for example. Denote $$a_t=diag(e^t,\dots,e^t,e^{(1-d)t}).$$ Note that for any positive $t$ the unstable horospherical subgroup of $a_t$ is (recall the notation of $\S\S$ \[DA\]) $U^+(a_t)={\left\{h_v:v\in{\mathbb{R}}^{d-1}\right\}}$. For any point $x\in X_d$ there exists neighborhoods $W^+_x,W^-_x,W^0_x$ of the identity elements in the groups $U^+(a_t),U^-(a_t)$ and the centralizer of $a_t$, such that the map $W^0_x\times W^-_x\times W^+_x\to X_d$ given by $(c,g,h_v)\mapsto cgh_vx$ is a diffeomorphism with a neighborhood, $W_x$, of $x$ in $X_d$. Note that if $x_i=c_ig_ih_vx, i=1,2$ are two points in $W_x$ having the same $U^+$ coordinate, then the trajectory ${\left\{a_tx_1\right\}}_{t\ge 0}$ is dense in $X_d$ if and only if ${\left\{a_tx_2\right\}}_{t\ge 0}$ is dense too. As the action of $a_t$ on $X_d$ is ergodic we know that for almost any $x'\in W_x$, ${\left\{a_tx'\right\}}_{t\ge 0}$ is dense in $X_d$ and from analyzing the structure of the Haar measure on $X_d$ restricted to $W_x$ we conclude that for almost any $v$ in the neighborhood of zero in ${\mathbb{R}}^{d-1}$ corresponding to $W^+_x$, ${\left\{a_th_vx\right\}}_{t\ge 0}$ is dense in $X_d$. We abuse notation and think of $W_x^+$ as contained in ${\mathbb{R}}^{d-1}$.\
To finish the argument we find a countable collection $v_i\in{\mathbb{R}}^{d-1}$ such that the neighborhoods $W_{x_{v_i}}$ satisfy ${\mathbb{R}}^{d-1}=\cup_i{\left(v_i+W_{x_{v_i}}\right)}$ and note that $W_{x_{v_i}}x_{v_i}={\left\{x_w:w\in v_i+W_{x_{v_i}}\right\}}.$
\[almost any GDP\] Let $d\ge 3$
1. Almost any lattice $x\in X_d$ (with respect to Haar measure) is $GDP$.
2. For almost any $v\in{\mathbb{R}}^{d-1}$ (with respect to Lebesgue measure), both $x_{v},z_{v}\in X_d$ are $GDP$.
3. If $v\in{\mathbb{R}}^{d-1}$ is such that $x_v$ (resp $z_v$) is $GDP$, then (resp ) is satisfied.
\(1) follows from the ergodicity of the $A$ action on $X_d$ which in particular means that almost any point has a dense orbit and corollary \[dense imply GDP\]. (2) follows from lemma \[expanding leaf generic\] and corollary \[dense imply GDP\]. (3) is left to be verified by the reader.
A density result {#density result}
================
Let $x_0\in X_3$ be a point with a compact $A$-orbit. We shall use the following facts: It follows from Lemma 4.1 of [@LW], that the orbit of $x_0$ under any root group $u_{ij}(t)$, is dense in $X_3$, moreover Theorem B of [@R], implies that in fact ${\left\{u_{ij}(t)x_0\right\}}_{t\in I}$ is dense in $X_3$, for any ray $I\subset {\mathbb{R}}$. It follows from corollary 1.4 in [@LW], that if $b\in G_0$ is lower or upper triangular but not diagonal, then $\overline{Abx_0}= X_3$. A more careful look yields the following theorem. The author is indebted to Elon Lindenstrauss, for valuable ideas appearing in the proof.
\[density\] Let $x_0\in X_3$ be a lattice with a compact $A$-orbit. If $p= {\left(
\begin{array}{lll}
{\alpha}&0&0\\
{\beta}&{\gamma}&{\delta}\\
\eta&\tau&\mu
\end{array}
\right)}\in G_0$ is such that both $\tau,\mu\ne 0$, then $\overline{Apx_0}= X_3$.
A straightforward computation shows $$\label{shrinking translate}
p={\left(
\begin{array}{lll}
1&0&0\\
0&1&\frac{{\delta}}{\mu}\\
0&0&1
\end{array}
\right)}
{\left(
\begin{array}{lll}
{\alpha}&0&0\\
{\beta}-\frac{{\delta}\eta}{\mu} & {\gamma}-\frac{{\delta}\tau}{\mu}&0\\
\eta&0&\mu
\end{array}
\right)}
{\left(
\begin{array}{lll}
1&0&0\\
0& 1&0\\
0&\frac{\tau}{\mu}&1
\end{array}
\right)}=u_{23}(t_0)b_1b_2,$$ where we denoted $t_0=\frac{{\delta}}{\mu}$ and the matrices appearing in the middle of by $u_{23}{\left(t_0\right)},b_1$ and $b_2$ according to appearance. Note that the matrix $b=b_1b_2$ is nondiagonal as $\tau\ne 0$, hence by the preceding discussion, if we denote $x_1=bx_0$, then $x_1$ has a dense $A$-orbit. Hence, it is enough to show that $x_1$ belongs to the orbit closure of $px_0=u_{23}{\left(t_0\right)}x_1$. This will follow from the existence of a recurrence sequence $a_n\in A$ for $x_1$ (i.e. a sequence such that $a_nx_1\to x_1$) which in addition satisfies $a_nu_{23}{\left(t_0\right)}a_n^{-1}\to e,$ for then $$\label{no name}
\lim a_n px_0=\lim a_n u_{23}(t_0)a_n^{-1}a_nx_1=x_1.$$ A sequence $a_n$ satisfies $a_nu_{23}(t_0)a_n^{-1}\to e$, if and only if $t_2^{(n)}-t_3^{(n)}\to-\infty$, where $\textbf{t}^{(n)}=\log(a_n)$. Thus it is enough to show that for any $m>0$, there exists a recurrence sequence for $x_1$ in $A_m=\exp{\left(R_m\right)}$, where $R_m={\left\{\textbf{t}\in {\mathfrak{a}}: t_2-t_3\le-m\right\}}$ is a half plane. Choose $m>0$. We shall show that in fact $A_mx_1$ is dense in $X_3$. Denote $$\label{a-t}
a_t=diag{\left(e^{2t},e^{-t},e^{-t}\right)}\textrm{ and }a'=diag{\left(e^{-m},1,e^m\right)}.$$ The line ${\left\{a'a_t\right\}}_{t\in{\mathbb{R}}}$ lies on the boundary of $A_m$. As $b=b_1b_2$, we write (emphasizing the desired partition into products) $$\label{c.1}
A_mx_1\supset a'a_tx_1=a'a_tbx_0={\left(a'a_tb_1{\left(a'a_t\right)}^{-1}\right)}\cdot{\left({\left(a'a_t\right)}b_2{\left(a'a_t\right)}^{-1}\right)}\cdot{\left({\left(a'a_t\right)}x_0\right)}.$$ We observe that for any sequence $t_n\to\infty$, $a'a_{t_n}b_1{\left(a'a_{t_n}\right)}^{-1}$ converges to the diagonal matrix $a''=diag{\left({\alpha},{\gamma}-\frac{{\delta}\tau}{\mu},\mu\right)}$, while at the same time ${\left(a'a_{t_n}\right)}b_2{\left(a'a_{t_n}\right)}^{-1}$ converges to $u_{32}{\left(s_0\right)},$ where $s_0=e^m\frac{\tau}{\mu}\ne 0$. Furthermore, because $Ax_0\simeq A/Stab_A(x_0)$, and because the line $a_t$ is irrational with respect to the lattice $Stab_A(x_0)$ (by theorem \[compact lattice orbits\]), any trajectory of ${\left\{a_t\right\}}_{t\ge 0}$ in $Ax_0$ is dense there. In particular any point in $Ax_0$ is a limit point of some sequence ${\left(a'a_{t_n}\right)}x_0$, for some sequence $t_n\to\infty$. It follows now from , that $$\label{c.2}
\overline{A_mx_1}\supset a''u_{32}{\left(s_0\right)}Ax_0=u_{32}(s_1)Ax_0,$$ for a suitable choice of $s_1\ne 0$. As $A_m$ is closed under multiplication, $\overline{A_mx_1}$ closed under the action of $A_m$. In particular, it follows from , that for any $a\in A_m$, $au_{32}(s_1)a^{-1}x_0\in\overline{A_mx_1}$, i.e. $u_{32}(s)x_0\in \overline{A_mx_1}$, where $s$ ranges over the set ${\left\{e^ts_1:t\ge m\right\}}$, which is a ray. The discussion preceding this proof now implies the density of $A_mx_1$ and in particular that $x_1\in\overline{A_mx_1}$ as desired.
\[cubics are good\] Let $K$ be a totally real cubic number field and let $1,{\alpha},{\beta}$ be a basis of $K$ over ${\mathbb{Q}}$. Denote $v=({\alpha},{\beta})^t\in{\mathbb{R}}^2$. Then the lattices $x_v,z_v$, have dense $A$ orbits in $X_3$ and in particular they are $GDP$ by corollary \[dense imply GDP\].
Let us denote ${\alpha}={\alpha}_1,{\beta}={\beta}_1$ and let ${\alpha}_i,{\beta}_i,i=2,3$, be the other two embeddings of ${\alpha}_1,{\beta}_1$ into the reals. Denote $
g_0=c{\left(\begin{array}{lll}
1&{\alpha}_1&{\beta}_1\\
1&{\alpha}_2&{\beta}_2\\
1&{\alpha}_3&{\beta}_3
\end{array}
\right)}$, where $c$ is chosen so that $\det{\left(g_0\right)}=1$. Then, $\bar{g}_0\in X_3$ has a compact $A$-orbit by theorem \[compact lattice orbits\]. It is easy to see that there exists a unique matrix $p\in G_0$ as in lemma \[density\], such that (recall the notation of $\S\S$ \[DA\]) $$\label{strudel}
pg_0=g_v.$$ The reader can easily check that the relevant entries of $p$ must be nonzero. We apply lemma \[density\], and conclude that $\bar{g}_v=z_v$ has a dense $A$-orbit in $X_3$. In order to see that $x_v$ has a dense orbit, we note that the involution $g\mapsto (g^t)^{-1}=g^*$ of $G_0$ , descends to a diffeomorphism of $X_3$. We denote this map by $\bar{g}\mapsto\bar{g}^*=\overline{g^*}$. This is the well known map which sends a lattice to its dual. Since the group $A$ is invariant under this involution, we have that for any lattice $x$, ${\left(\overline{Ax}\right)}^*=\overline{Ax^*}.$ In particular, $x$ has a dense orbit if and only if $x^*$ has. In a similar way to what we have already shown, one can show that the lattice spanned by the columns of $g_1={\left(
\begin{array}{lll}
1&0&0\\
0&1&0\\
-{\alpha}&-{\beta}&1
\end{array}
\right)},$ has a dense $A$-orbit, in $X_3$. As $g_1^*=h_v$, it follows that $\bar{h}_v=x_v$ has a dense orbit too, as desired.
Irregular $A$ orbits {#irregular orbits}
====================
In this section we use the existence of lattices $x\in X_d$ ($d\ge 3$) for which $\mu(x)=0$ (theorem \[theorem 101\]), and theorem \[corollary 102\](2) to give examples of lattices in $X_3$ having irregular $A$ orbit closures of a new type. This serves as a counterexample to conjecture 1.1 in [@Ma]. Our example proceeds the recent counterexample, F .Maucourant gave to this conjecture in [@Mau]. Our example is different in nature from Maucourant’s example. We use a maximal split torus whilst in [@Mau] the acting group does not “separate roots” which seem to be the reason for the abnormality. It still seems plausible that a slightly different version of that conjecture will be true.
We first note that if $x_1,x_2\in X_d$ are commensurable lattices (that is their intersection is of finite index in each), then $\mu(x_1)=0$ if and only if $\mu(x_2)=0$. Let $v\in{\mathbb{R}}^{d-1}$ satisfy $\dim_{\mathbb{Q}}span{\left\{1,v_1\dots,v_{d-1}\right\}}\le 2$. Then, there exists ${\alpha}\in{\mathbb{R}}$ and rationals $p_i,q_i, i=1\dots d-1$ such that $v_i=q_i{\alpha}+p_i$. Denote $v'=(q_1{\alpha},\dots,q_{d-1}{\alpha})^t$. It follows that $x_v,z_v$ are commensurable to $x_{v'},z_{v'}$ respectively.\
**Claim** 1: $\mu(x_{v'})>0$. Working with the definition of $\mu$ we see that it is enough to argue the existence of $d-1$ real numbers ${\gamma}_i$ for which $$\label{mu ind}
N{\left(x_{v'}+(-{\gamma}_1,\dots,-{\gamma}_{d-1},1/2)^t\right)}=\inf_{n\in{\mathbb{Z}}}{\left|n+1/2\right|}\prod_1^{d-1}{\langlenq_i{\alpha}-{\gamma}_i\rangle}>0.$$ From Davenport’s result described in $\S\S$ \[dim2\] it follows that there exists ${\gamma}_i\in{\mathbb{R}}$ such that for each $i$, $\inf_{n\in{\mathbb{Z}}} {\left|n+1/2\right|}{\langlenq_i{\alpha}-{\gamma}_i\rangle}>0$. Moreover, if we denote by $m$, a common denominator for the $q_i$’s , then by [@Ca2] (theorem 1), we can choose the ${\gamma}_i$’s such that for any $i\ne j$, $\frac{{\gamma}_i}{q_i}-\frac{{\gamma}_j}{q_j}\notin\frac{1}{m}{\mathbb{Z}}$. Denote for $r,s\in{\mathbb{R}}$ by ${\langler-s\rangle}_m$ the distance modulo $\frac{1}{m}{\mathbb{Z}}$ from $r$ to $s$. Denote $\rho=\min_{i\ne j}{\langle\frac{{\gamma}_i}{q_i}-\frac{{\gamma}_j}{q_j}\rangle}_m$. Note that for ${\epsilon}>0$, ${\langlenq_i{\alpha}-{\gamma}_i\rangle}<{\epsilon}\Rightarrow {\langlen{\alpha}-\frac{{\gamma}_i}{q_i}\rangle}_m<\frac{{\epsilon}}{{\left|q_i\right|}}$. Hence if $\max_i\frac{{\epsilon}}{{\left|q_i\right|}}<\rho/2$, then ${\langlenq_i{\alpha}-{\gamma}_i\rangle}<{\epsilon}$ for at most one index $i$. Let ${\epsilon}>0$ be such. Assume that the left hand side of is smaller than ${\epsilon}^{d-1}/2$. Then for some $k$, ${\langlenq_k{\alpha}-{\gamma}_k\rangle}<{\epsilon}$ which implies that the left hand side of is $>{\epsilon}^{d-2}\inf_{n\in{\mathbb{Z}}}{\left|n+1/2\right|}{\langlenq_k{\alpha}-{\gamma}_k\rangle}>0$ as desired.\
**Claim 2** $\mu(z_{v'})>0$. We use the notation as in Claim 1. From Davenport’s result, we know that there exists $0\ne{\gamma}\in{\mathbb{R}}$ such that $\inf_{k\in{\mathbb{Z}}}{\left|k+1/2\right|}{\langlek{\left({\alpha}/m\right)}-{\gamma}\rangle}>0.$ The reader will easily argue the existence of a constant $c>0$ for which $$\forall\vec{n}\in{\mathbb{Z}}^{d-1}\setminus{\left\{0\right\}},\quad \prod_1^{d-1}{\left|n_i+1/2\right|}\ge c{\left|m\vec{q}\cdot\vec{n}+1/2\right|}.$$ Working with the definition of $\mu$ we see that $$\begin{array}{lll}
\mu(z_{v'})&\ge&N{\left(z_{v'}+({\gamma},1/2,\dots,1/2)^t\right)}=\inf_{\vec{n}\in{\mathbb{Z}}^{d-1}}\prod{\left|n_i+1/2\right|}{\langle{\left(m\vec{q}\cdot\vec{n}\right)}{\alpha}/m-{\gamma}\rangle}\\
&\ge&\min{\left\{\inf_{\vec{n}\cdot\vec{q}\ne 0}c{\left|m\vec{q}\cdot\vec{n}+1/2\right|}{\langle{\left(m\vec{q}\cdot\vec{n}\right)}{\alpha}/m-{\gamma}\rangle}; \frac{{\left|{\gamma}\right|}}{2^{d-1}}\right\}}>0.
\end{array}$$
Observe that if $x_n\to x$ in $X_d$, then $\limsup\mu(x_n)\le\mu(x)$. It follows that for $x_0,x\in X_d$ $$\label{mu inheritance}
x_0\in\overline{Ax}\Rightarrow \mu(x_0)\ge\mu(x).$$ By theorem \[theorem 101\], if $x\in X_d$, ($d\ge 3$) and $\mu(x)>0$, then $Ax$ is not dense.\
The following statement is a special case of Conjecture 1.1 of [@Ma]:
\[Margulis conjecture\] For $x\in X_3$, one of the following three options occurs:
1. $Ax$ is dense.
2. $Ax$ is closed.
3. $Ax$ is contained in a closed orbit $Hx$ of an intermediate group $A<H<G$, where $H$ could be one of the following three subgroups of $G_0$: $$H_1=
\left(\begin{array}{lll}
*&*&0\\
*&*&0\\
0&0&*
\end{array}\right),
H_2=
\left(\begin{array}{lll}
*&0&0\\
0&*&* \\
0&*&*
\end{array}\right),
H_3=
\left(\begin{array}{lll}
*&0&* \\
0&*&0\\
*&0&*
\end{array}\right).$$
For $t\in{\mathbb{R}}$ denote $v_t=(t,t)^t\in{\mathbb{R}}^2$. We denote the one parameter group $h_{v_t}$ (recall the notation of §§ \[DA\]) simply by $h_t$ and the lattice $x_{v_t}$ by $x_t$.
There exists $t\in{\mathbb{R}}$ such that $x_t\in X_3$ violates conjecture \[Margulis conjecture\].
By theorem \[corollary 102\](2), $\mu(x_t)>0$ hence by theorem \[theorem 101\] and , possibility (1) is ruled out. To rule out possibilities (2) and (3), we note that if $H$ is either one of the groups $A,H_1,H_2,H_3$, then for any $g\in G_0$, $H\bar{g}$ is closed in $X_3$ if and only if $g^{-1}Hg$ is defined over ${\mathbb{Q}}$. Assume to get a contradiction that for any $t\in{\mathbb{R}}$, for $H$ equals one of the above, the group $h_t^{-1}Hh_t$ is defined over ${\mathbb{Q}}$. Two elements $g_1,g_2\in G_0$ conjugate $H$ to the same group if and only if $g_1g_2^{-1}$ normalizes $H$. All the above groups are of finite index in their normalizers in $G_0$ and so there exists some $k$ such that whenever $g$ normalizes $H$, then $g^k\in H$. As there are only countably many ${\mathbb{Q}}$-groups in $G_0$, there must exist some $t\ne s$ such that ${\left(h_th_s^{-1}\right)}^k=h_{k(t-s)}\in H$ which of course never happens.
[99]{} D. Berend, *Multi-invariant sets on tori*. Trans. A.M.S . 280 (1983), 509-532. E.S Barnes *The inhomogeneous minima of indefinite quadratic forms.* J. Austral. Math. Soc 2 (1961/62). P. Bachmann, *Die Arithmetik der quadratischen formen*. Zweite Abtheilung, especially Kap. 12 (Die zerlegbaren formen). Leipzig and Berlin: Teubner (1923). Y. Bugeaud, *Multiplicative Diophantine approximation.* To appear in Dynamical systems and Diophantine Approximation, Proceedings of the conference held at the Institut Henri Poincare, Seminaires et Congres, Societe mathematique de France. J. W. S Cassels, *An Introduction to the Geometry of Numbers*. Corrected reprint of the 1971 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1997. J. W. S Cassels, *The inhomogeneous minimum of binary quadratic ternary cubic and quaternary quartic forms.* Proc. Cambridge Phil. Soc. 48, 72-86, 519-520. J. W. S. Cassels and H. P. F. Swinnerton-Dyer, *On the product of three homogeneous linear forms and the indefinite ternary quadratic forms*, Philos. Trans. Roy. Soc. London. Ser. A. 248, (1955). 73–96. H. Davenport *Indefinite binary quadratic forms and Euclid’s algorithm in real quadratic fields* Proc. London Math. Soc. (2) 53, (1951). 65-82. H. Furstenberg, *Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation*. Math. Systems Theory, 1:1–49, 1967. E. Lindenstrauss, B. Weiss, *On sets invariant under the action of the diagonal group*. Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1481–1500. G. A. Margulis. *Problems and conjectures in rigidity theory*, Mathematics: frontiers and perspectives, pages 161-174. Amer. Math. Soc., Providence, RI, 2000. F. Maucourant. *A non-homogeneous orbit of a diagonal subgroup* arXiv:0707.2920v2 \[math.DS\] M. Ratner, *Raghunathan’s topological conjecture and distributions of unipotent flows.* Duke Math. J. 63 (1991), no. 1, 235–280. C. T. McMullen. *Minkowski’s conjecture, well-rounded lattices and topological dimension.* J.Amer.Math.Soc 18(2005), 711-734 U. Shapira. *On a generalization of Littlewood’s conjecture*, preprint (2008). M. Einsiedler and J. Tseng. *Badly approximable systems of affine forms.* Preprint.
[^1]: \* Part of the author’s Ph.D thesis at the Hebrew University of Jerusalem.\
Email: [email protected]
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abstract: 'We show that, for $n\geq 3$, $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ holds almost everywhere for all $f \in H^s (\mathbb{R}^n)$ provided that $s>\frac{n}{2(n+1)}$. Due to a counterexample by Bourgain, up to the endpoint, this result is sharp and fully resolves a problem raised by Carleson. Our main theorem is a fractal $L^2$ restriction estimate, which also gives improved results on the size of divergence set of Schrödinger solutions, the Falconer distance set problem and the spherical average Fourier decay rates of fractal measures. The key ingredients of the proof include multilinear Kakeya estimates, decoupling and induction on scales.'
address:
- |
University of Maryland\
College Park, MD
- |
University of Wisconsin-Madison\
Madison, WI
author:
- Xiumin Du
- Ruixiang Zhang
title: Sharp $L^2$ estimate of Schrödinger maximal function in higher dimensions
---
Introduction
============
The solution to the free Schrödinger equation $$\begin{cases}
iu_t - \Delta u = 0, &(x,t)\in \mathbb{R}^n \times \mathbb{R} \\
u(x,0)=f(x), & x \in \mathbb{R}^n
\end{cases}$$ is given by $$e^{it\Delta}f(x)=(2\pi)^{-n}\int e^{i\left(x\cdot\xi+t|\xi|^2\right)}\widehat{f}(\xi) \, d\xi.$$
In [@lC], Carleson proposed the problem of identifying the optimal $s$ for which $\lim_{t \to 0}e^{it\Delta}f(x)=f(x)$ almost everywhere whenever $f\in H^s(\mathbb{R}^n),$ and proved convergence for $s \geq \frac 1 4$ when $n=1$. Dahlberg and Kenig [@DK] then showed that this result is sharp. The higher dimensional case has since been studied by several authors [@aC; @mC; @pS; @lV; @jB; @MVV; @TV; @sL; @jB12; @LR17; @DG; @jB16; @LR17'; @DGL; @DGLZ]. In particular, almost everywhere convergence holds if $s>\frac 12-\frac{1}{4n}$ when $n\geq 2$ ($n=2$ due to Lee [@sL] and $n\geq 2$ due to Bourgain [@jB12]). Recently Bourgain [@jB16] gave counterexamples showing that convergence can fail if $s<\frac{n}{2(n+1)}$. Since then, Guth, Li and the first author [@DGL] improved the sufficient condition when $n=2$ to the almost sharp range $s>\frac{1}{3}$. In higher dimensions ($n\geq 3$), Guth, Li and the authors [@DGLZ] proved the convergence for $s>\frac{n+1}{2(n+2)}$.
In this article, we establish the following theorem, which is sharp up to the endpoint.
\[thm-pc\] Let $n\geq 3$. For every $f\in H^s(\mathbb R^n)$ with $s>\frac{n}{2(n+1)}$, $\lim_{t \to 0}e^{it\Delta}f(x)=f(x)$ almost everywhere.
We use $B^m(x, r)$ to represent a ball centered at $x$ with radius $r$ in ${\mathbb{R}}^m$. By a standard smooth approximation argument, Theorem \[thm-pc\] is a consequence of the following estimate of the Schrödinger maximal function:
\[thm-L2-0\] Let $n\geq 3$. For any $s > \frac{n}{2(n+1)}$, the following bound holds: for any function $f \in H^s(\mathbb{R}^n)$, $$\label{eq-L2-0}
\left\| \sup_{0 < t \le 1} | e^{it \Delta} f| \right\|_{L^2(B^n(0,1))} \le C_s \| f \|_{H^s(\mathbb{R}^n)}.$$
Via a localization argument, Littlewood-Paley decomposition and parabolic rescaling, Theorem \[thm-L2-0\] is reduced to the following theorem which we will prove in this paper:
\[thm-L2\] Let $n\geq 3$. For any ${\varepsilon}>0$, there exists a constant $C_{\varepsilon}$ such that $$\label{eq-L2}
\left\|\underset{0<t\leq R}\sup|e^{it\Delta}f|\right\|_{L^2(B^n(0,R))} \leq
C_{\varepsilon}R^{\frac{n}{2(n+1)}+{\varepsilon}} \|f\|_2$$ holds for all $R\geq 1$ and all $f$ with ${\rm supp}\widehat{f}\subset A(1)=\{\xi\in {\mathbb{R}}^n:|\xi|\sim 1\}$.
When $n=1,2$, our proof of Theorem \[thm-L2\] remains valid and recovers the almost sharp results of the pointwise convergence problem. However, the sharp $L^2$ estimates of the Schrödinger maximal function are not as strong as the previous sharp $L^p$ estimates in the cases $n=1,2$: $$\label{L4}
\left\| \sup_{t>0} | e^{it \Delta} f| \right\|_{L^4({\mathbb{R}})} \le C \| f \|_{H^{1/4}(\mathbb{R})}\,, \quad \text{\cite[Kenig-Ponce-Vega]{KPV}}\,,$$ and $$\label{L3}
\left\| \sup_{0 < t \le 1} | e^{it \Delta} f| \right\|_{L^3({\mathbb{R}}^2)} \le C_s \| f \|_{H^{s}(\mathbb{R}^2)}\,, \forall s>\frac 13, \quad \text{\cite[D.-Guth-Li]{DGL}} \footnote[1]{
The global $L^3$ estimate \eqref{L3} follows easily from the local $L^3$ estimate in \cite{DGL}, via a localization argument using wave packet decomposition.} \,.$$ Testing with the standard examples used in restriction theory seems to suggest that the following estimate holds for all $n\geq 1$: $$\label{Lp}
\left\| \sup_{0 < t \le 1} | e^{it \Delta} f| \right\|_{L^{\frac{2(n+1)}{n}}({\mathbb{R}}^n)} \le C \| f \|_{H^{\frac{n}{2(n+1)}}(\mathbb{R}^n)}\,.$$
From and we see that is true for $n=1$, and is true up to the endpoint for $n=2$. However, the estimate fails in higher dimensions. In a recent work of Kim, Wang and the authors [@DKWZ], by looking at Bourgain’s counterexample [@jB16] in every intermediate dimension, we showed that the following local estimate $$\label{Lp-loc}
\left\| \sup_{0 < t \le 1} | e^{it \Delta} f| \right\|_{L^p(B^n(0,1))} \le C_s \| f \|_{H^{s}(\mathbb{R}^n)}\,, \forall s>\frac {n}{2(n+1)}$$ fails if $p> p_0:= 2+\frac{4}{(n-1)(n+2)}$. Note that $\frac{2(n+1)}{n} > p_0$ when $n\geq 3$ and henceforth fails. To our best knowledge, the following two problems are still open when $n\geq 3$: determine the optimal $p=p(n)$ for which we can have (\[Lp-loc\]) and identify the optimal $s=s(n,p)$ for which with $p>2$ fixed holds.
In our proof of , no typical $L^2$ arguments such as Plancherel and $TT^*$ are invoked to take advantage of the particular use of the $L^2$ norm on the left hand side of . In fact, the $L^2$ norm will be converted to $L^p$ norm (see Proposition \[thm-main\]), where $p=\frac{2(n+1)}{n-1}$ is the sharp exponent for the $l^2$ decoupling theorem in dimension $n$. The $L^2$ is used on the left hand side of mostly because the numerology adds up favorably for that space.
By lattice $L$-cubes we mean cubes of the form $l+[0,L]^n$ with $l\in (L{\mathbb{Z}})^n$. Our main result is the following fractal $L^2$ restriction estimate, from which Theorem \[thm-L2\] follows.
\[thm-L2X\] Let $n\geq 1$. For any ${\varepsilon}>0$, there exists a constant $C_{\varepsilon}$ such that the following holds for all $R\geq 1$ and all $f$ with ${\rm supp}\widehat{f}\subset B^n(0,1)$. Suppose that $X=\bigcup_k B_k$ is a union of lattice unit cubes in $B^{n+1}(0,R)$ and each lattice $R^{1/2}$-cube intersecting $X$ contains $\sim \lambda$ many unit cubes in $X$. Let $1\leq \alpha\leq n+1$ and $\gamma$ be given by $$\label{ga-L2}
\gamma:=\max_{ \underset{x'\in {\mathbb{R}}^{n+1},r\geq 1}{B^{n+1}(x',r)\subset B^{n+1}(0,R)}} \frac{\#\{B_k : B_k \subset B(x',r)\}}{r^\alpha}\,.$$ Then $$\label{eq-L2X}
\|{e^{i t \Delta}}f\|_{L^2(X)} \leq C_{\varepsilon}\gamma^{\frac{2}{(n+1)(n+2)}} \lambda^{\frac{n}{(n+1)(n+2)}} R^{\frac{\alpha}{(n+1)(n+2)}+{\varepsilon}}\|f\|_2\,.$$
Note that in Theorem \[thm-L2X\], ${\lambda}\leq \gamma R^{\alpha/2}$. As a direct result of Theorem \[thm-L2X\], there holds a slightly weaker fractal $L^2$ restriction estimate. It has a relatively simpler statement:
\[cor-L2X\] Let $n\geq 1$. For any ${\varepsilon}>0$, there exists a constant $C_{\varepsilon}$ such that the following holds for all $R\geq 1$ and all $f$ with ${\rm supp}\widehat{f}\subset B^n(0,1)$. Suppose that $X=\bigcup_{k} B_k$ is a union of lattice unit cubes in $B^{n+1}(0,R)$. Let $1\leq {\alpha}\leq n+1$ and $\gamma$ be given by $$\label{ga-L2'}
\gamma:=\max_{ \underset{x'\in {\mathbb{R}}^{n+1},r\geq 1}{B^{n+1}(x',r)\subset B^{n+1}(0,R)}} \frac{\#\{B_k : B_k \subset B(x',r)\}}{r^\alpha}\,.$$ Then $$\label{eq-L2X'}
\|{e^{i t \Delta}}f\|_{L^2(X)} \leq C_{\varepsilon}\gamma^{\frac{1}{n+1}} R^{\frac{\alpha}{2(n+1)}+{\varepsilon}}\|f\|_2\,.$$
We will see that Corollary \[cor-L2X\] is sufficient to derive the sharp $L^2$ estimate of Schrödinger maximal function (Theorem \[thm-L2\]) and all other applications in Section \[sec-app\]. This corollary can also be proved directly by a slightly simpler argument. The case $n=1$ of Corollary \[cor-L2X\] can be recovered using the ingredients in Wolff’s paper [@W]. See Subsection \[sec-rmk\] for a discussion.
Nevertheless, Theorem \[thm-L2X\] has two advantages compared to Corollary \[cor-L2X\]. Firstly, it gives us a better $L^2$ restriction estimate if the set $X$ of unit cubes is fairly sparse at the scale $R^{1/2}$. Secondly, it tells us some geometric information about a set $X$ of unit cubes when $\|{e^{i t \Delta}}f\|_{L^2(X)}$ is comparable to $\|{e^{i t \Delta}}f\|_{L^2(B(0,R))}$. For example, taking $\alpha = n+1$ (hence $\gamma \lesssim 1$) we have:
\[cor-fillin\] Let $n\geq 1$. Suppose that $X=\bigcup_{k} B_k$ is a union of lattice unit cubes in $B^{n+1}(0,R)$ and each lattice $R^{1/2}$-cube intersecting $X$ contains $\sim \lambda$ many unit cubes in $X$. Suppose there is a function $f$ with ${\rm supp}{\widehat}f \subset B^n(0,1)$ and $\|f\|_2 \neq 0$ such that $\|{e^{i t \Delta}}f\|_{L^2(X)} \gtrsim R^{1/2}\|f\|_2$. Then $\lambda \gtrapprox R^{\frac{n+1}{2}}$.
As a remark, the scale $R^{1/2}$ in Corollary \[cor-fillin\] is the largest one can have. Indeed, with the assumption of the corollary, the unit cubes in $X$ do not have to almost fill $R^{\beta}$-cubes completely for $\beta > 1/2$. One can see this from the Knapp example where we only have one wave packet.
To prove our main result - Theorem \[thm-L2X\], we will use a broad-narrow analysis, which has similar spirit as the techniques in the work of Bourgain-Guth [@BG], Bourgain [@jB12], Bourgain-Demeter [@BD] and Guth [@G2].
In the broad case, we can exploit the transversality and apply the multilinear refined Strichartz estimate, which is a result obtained by Guth, Li and the authors in [@DGLZ] (see [@DGL; @DGOWWZ; @DGLZ] for applications of the refined Strichartz estimate). In the narrow case, we use the $l^2$ decoupling theorem of Bourgain-Demeter [@BD] in a lower dimension and perform induction on scales. The way we do induction has its roots in the proof of the linear refined Strichartz estimate, due to Guth, Li and the first author (essentially proved in [@DGL], see [@DGLZ] for the statement in the general setting).
Our method is related to Bourgain’s in [@jB12], where he has a similar broad-narrow analysis, (Here we have the size of the small ball being $K^2$ instead of $K$ as in [@jB12] for a technical issue similar to what one has in [@BD; @G2]). He applied multilinear restriction to control the broad part in the sharp range $s>\frac{n}{2(n+1)}$ (except the endpoint). He speculated from this that the above range of $s$ might be sharp (see the end of the introduction in [@jB16]). In [@jB12] the narrow part was handled following the general approach from [@BG], which gives non-sharp estimates. Historically, one could view the present non-endpoint solution to Carleson’s problem as building on [@jB12], providing a subtler way of handling the narrow part and proving Corollary \[cor-L2X\]. For the stronger Theorem \[thm-L2X\] and Corollary \[cor-fillin\], one needs a different ingredient, namely the multilinear refined Strichartz in [@DGLZ], to handle the broad part.
In Section \[sec-app\] we show how Corollary \[cor-L2X\] and Theorem \[thm-L2\] follow from Theorem \[thm-L2X\], and we also present applications of Theorem \[thm-L2X\] to other problems - bounding the size of the divergence set of Schrödinger solutions (Theorem \[thm-pc-f\]), the Falconer distance set problem (Theorem \[thm-falc\] and \[thm-falc1\]) and the spherical average Fourier decay rates of fractal measures (Theorem \[thm-avgdec\]). We prove Theorem \[thm-L2X\] in Section \[sec-pf\].
**Notation.** We write $A\lesssim B$ if $A\leq CB$ for some absolute constant $C$, $A \sim B$ if $A\lesssim B$ and $B\lesssim A$; $A\ll B$ if $A$ is much less than $B$; $A\lessapprox B$ if $A\leq C_{\varepsilon}R^{\varepsilon}B$ for any ${\varepsilon}>0, R>1$. Sometimes we also write $A\lesssim B$ if $A\leq C_{\varepsilon}B$ for some constant $C_{\varepsilon}$ depending on ${\varepsilon}$ (when the dependence on ${\varepsilon}$ is unimportant).
By an $r$-ball (cube) we mean a ball (cube) of radius (side length) $r$. An $r\times\cdots\times r \times L$-tube (box) means a tube (box) with radius (short sides length) $r$ and length $L$. For a set ${\mathcal S}$, $\#{\mathcal S}$ denotes its cardinality.
The authors would like to thank Larry Guth and Xiaochun Li for several discussions. They also thank Larry Guth for making some historical remarks, as well as sharing his lecture notes on decoupling online, from which they got much inspiration. The second author would like to thank Jean Bourgain and Zihua Guo who introduced the problem to him. The authors are also indebted to Daniel Eceizabarrena and Luis Vega for a discussion on the history of the Schrödinger maximal estimate in dimension $1+1$.
The material is based upon work supported by the National Science Foundation under Grant No. 1638352, the Shiing-Shen Chern Fund and the James D. Wolfensohn Fund while the authors were in residence at the Institute for Advanced Study during the academic year 2017-2018.
Applications of Theorem \[thm-L2X\] {#sec-app}
===================================
0
Sharp $L^2$ estimate of Schrödinger maximal function
----------------------------------------------------
In this subsection, we show how Corollary \[cor-L2X\] and Theorem \[thm-L2\] follow from Theorem \[thm-L2X\], via the dyadic pigeonholing argument and the locally constant property.
Given $X=\bigcup_{k} B_k$, a union of lattice unit cubes in $B^{n+1}(0,R)$ satisfying the assumptions of Corollary \[cor-L2X\], we sort the lattice $R^{1/2}$-cubes in ${\mathbb{R}}^{n+1}$ intersecting $X$ by the number $\lambda$ of unit cubes $B_k$ contained in it. Since $1\leq \lambda \leq R^{O(1)}$, there are only $O(\log R)$ choices for the dyadic number $\lambda$. So we can choose a dyadic number $\lambda$ and a subset ${\mathcal B}_\lambda$ of $\{B_k\}$ such that for each unit cube $B$ in ${\mathcal B}_\lambda$, the lattice $R^{1/2}$-cube containing it contains $\sim \lambda$ many unit cubes from ${\mathcal B}_\lambda$ and $$\|{e^{i t \Delta}}f\|_{L^2(X)} \lessapprox \|{e^{i t \Delta}}f\|_{L^2(\bigcup_{B\in {\mathcal B}_{\lambda}} B)}\,.$$ By applying Theorem \[thm-L2X\] to $\|{e^{i t \Delta}}f\|_{L^2(\bigcup_{B\in {\mathcal B}_{\lambda}} B)}$, we get $$\|{e^{i t \Delta}}f\|_{L^2(X)} \lessapprox \gamma^{\frac{2}{(n+1)(n+2)}} \lambda^{\frac{n}{(n+1)(n+2)}} R^{\frac{\alpha}{(n+1)(n+2)}}\|f\|_2\,,$$ and follows from the fact that $\lambda \leq \gamma R^{\alpha/2}$.
We will show that $$\label{linL2}
\left\|\sup_{0<t<R} |{e^{i t \Delta}}f|\right\|_{L^2(B^n(0,R))} \lessapprox R^{\frac{n}{2(n+1)}} \|f\|_2$$ holds for all $R\geq 1$ and all $f$ with Fourier support in $B^n(0,1)$.
By viewing $|{e^{i t \Delta}}f(x)|$ essentially as constant on unit balls, we can find a set $X$ described as follows: $X$ is a union of unit balls in $B^n(0,R)\times [0,R]$ satisfying the property that each vertical thin tube of dimensions $1\times \cdots \times 1 \times R$ contains exactly one unit ball in $X$, and $$\left\| \sup_{0<t< R} |{e^{i t \Delta}}f| \right\|_{L^2(B^n(0,R))} \lessapprox \|{e^{i t \Delta}}f\|_{L^2(X)}\,.$$ The desired estimate follows by applying Corollary \[cor-L2X\] to $\|{e^{i t \Delta}}f\|_{L^2(X)}$ with ${\alpha}=n$ and $\gamma \lesssim 1$.
Other applications {#sec-other}
------------------
By formalizing the locally constant property, from Corollary \[cor-L2X\] we derive some weighted $L^2$ estimates - Theorem \[thm-L2-al\] and \[thm-L2-al’\], which in turn have applications to several problems described below.
Let ${\alpha}\in(0,d]$. We say that $\mu$ is an *${\alpha}$-dimensional measure* in ${\mathbb{R}}^d$ if it is a probability measure supported in the unit ball $B^d(0,1)$ and satisfies that $$\mu(B(x,r))\leq C_\mu r^{\alpha}, \quad \forall r>0, \quad \forall x\in {\mathbb{R}}^d.$$
Denote $d\mu_R(\cdot):=R^{\alpha}d\mu(\frac \cdot R)$.
\[thm-L2-al\] Let $n\geq 1,{\alpha}\in(0,n]$ and $\mu$ be an ${\alpha}$-dimensional measure in ${\mathbb{R}}^n$. Then $$\label{L2-al}
\left\|\sup_{0<t<R} |{e^{i t \Delta}}f|\right\|_{L^2\left(B^n(0,R);d\mu_R(x)\right)} \lessapprox R^{\frac{{\alpha}}{2(n+1)}} \|f\|_2\,,$$ whenever $R\geq 1$ and $f$ has Fourier support in $B^n (0, 1)$.
\[thm-L2-al’\] Let $n\geq 1,{\alpha}\in(0,n+1]$ and $\mu$ be an ${\alpha}$-dimensional measure in ${\mathbb{R}}^{n+1}$. Then $$\label{L2-al'}
\left\|{e^{i t \Delta}}f\right\|_{L^2\left(B^{n+1}(0,R);d\mu_R(x,t)\right)} \lessapprox R^{\frac{{\alpha}}{2(n+1)}} \|f\|_2\,,$$ whenever $R\geq 1$ and $f$ has Fourier support in $B^{n} (0, 1)$.
We defer the proof of these weighted $L^2$ estimates to the end of this subsection. Let’s first see their applications. We omit history and various previous results on the following three problems and refer the readers to [@DGLZ; @DGOWWZ; @LR] and the references therein.
**(I) Hausdorff dimension of divergence set of Schrödinger solutions**
A natural refinement of Carleson’s problem was initiated by Sjögren and Sjölin [@SS]: determine the size of divergence set, in particular, consider $${\alpha}_n(s):= \sup_{f\in H^s({\mathbb{R}}^n)} {\rm dim} \left\{x\in {\mathbb{R}}^n: \lim_{t \to 0}e^{it\Delta}f(x)\neq f(x) \right\}\,,$$ where ${\rm dim}$ stands for the Hausdorff dimension.
The following theorem is a direct result of Theorem \[thm-L2-al\] (c.f. [@DGLZ; @LR]). When $n=2$, it recovers the corresponding result derived from the sharp $L^3$ estimate of the Schrödinger maximal function in D.-Guth-Li [@DGL]. When $n\geq 3$, it improves the previous best known result in D.-Guth-Li-Z. [@DGLZ].
\[thm-pc-f\] Let $n\geq 2$. Then $${\alpha}_n(s)\leq n+1-\frac{2(n+1)s}{n}, \quad
\frac{n}{2(n+1)}<s<\frac n 4\,.$$
**(II) Falconer distance set problem**
Let $E\subset\mathbb{R}^d$ be a compact subset, its distance set $\Delta(E)$ is defined by $$\Delta(E):=\{|x-y|:x,y\in E\}\,.$$
\[Falconer [@F]\] Let $d\geq 2$ and $E\subset\mathbb{R}^d$ be a compact set. Then $${\rm dim}(E)> \frac d 2 \Rightarrow |\Delta(E)|>0.$$ Here $|\cdot|$ denotes the Lebesgue measure and ${\rm dim}(\cdot)$ is the Hausdorff dimension.
Following a scheme due to Mattila (c.f. [@DGOWWZ Proposition 2.3]), Theorem \[thm-L2-al’\] implies the following result towards Falconer’s conjecture. When $d=2,3$, this recovers the previous best known results of Wolff (d=2, [@W]) and D.-Guth-Ou-Wang-Wilson-Z. (d=3, [@DGOWWZ]), via a different approach. In the case $d\geq 4$, this improves the previous best known result in [@DGOWWZ]:
\[thm-falc\] Let $d\geq 2$ and $E\subset\mathbb{R}^d$ be a compact set with $${\rm dim}(E)> \frac{d^2}{2d-1}=\frac d 2 +\frac 1 4 + \frac{1}{8d-4}\,.$$ Then $|\Delta(E)|>0$.
By applying a very recent work of Liu [@Liu Theorem 1.4], Theorem \[thm-L2-al’\] also implies the following result for the pinned distance set problem, with the same threshold:
\[thm-falc1\] Let $d\geq 2$ and $E\subset \mathbb{R}^d$ be a compact set with $${\rm dim}(E)> \frac{d^2}{2d-1}=\frac d 2 +\frac 1 4 + \frac{1}{8d-4}\,.$$ Then there exists $x\in E$ such that its pinned distance set $$\Delta_x(E):=\{|x-y|:\,y\in E\}$$has positive Lebesgue measure.
**(III) Spherical average Fourier decay rates of fractal measures**
Let ${\beta}_d({\alpha})$ denote the supremum of the numbers ${\beta}$ for which $$\label{eq:AvrDec}
\left\|\widehat \mu (R\cdot)\right\|_{L^2({\mathbb{S}}^{d-1})}^2 \leq C_{{\alpha},\mu} R^{-{\beta}}$$ whenever $R> 1$ and $\mu$ is an ${\alpha}$-dimensional measure in ${\mathbb{R}}^d$. The problem of identifying the precise value of $\beta_d({\alpha})$ was proposed by Mattila [@M04].
A lower bound of $\beta_d({\alpha})$ as in Theorem \[thm-avgdec\] follows from Theorem \[thm-L2-al’\] (c.f. [@DGOWWZ Remark 2.5]). When $d=2$, this recovers the sharp result of Wolff [@W]. When $d=3$ and ${\alpha}\in(\frac 32, 2]$, this recovers the previous best known result of D.-Guth-Ou-Wang-Wilson-Z. [@DGOWWZ]. In the case $d=3, {\alpha}\in(2,3)$ or $d\geq 4,\alpha\in(d/2,d)$, this improves the previous best known result in [@DGOWWZ].
\[thm-avgdec\] Let $d\geq 2$ and $\alpha\in (\frac d 2,d)$. Then $$\beta_d({\alpha}) \geq \frac{(d-1){\alpha}}{d}\,.$$
The proofs of Theorems \[thm-L2-al\] and \[thm-L2-al’\] are entirely similar and we only do the proof of the former here, which is slightly more involved.
Denote ${e^{i t \Delta}}f(x)$ by $Ef(x,t)$, and $(x,t)$ by $\tilde x$. Since $\mathrm{supp} {\widehat}{f} \subseteq B^n(0, 1)$, we have $\mathrm{supp} {\widehat}{Ef} \subseteq B^{n+1}(0, 1)$. Thus there exists a Schwartz bump function $\psi$ on ${\mathbb{R}}^{n+1}$ (we require ${\widehat}{\psi} \equiv 1$ on $B^{n+1} (0, 100)$) such that $(Ef)^2 = (Ef)^2 * \psi$.
The function $\max_{|\tilde y- \tilde x|\leq e^{100n}} |\psi (\tilde y)|$ is rapidly decaying. We call it $\psi_1 (\tilde x)$. Note also that any $(x, t)$ in ${\mathbb{R}}^{n+1}$ belongs to a unique integral lattice cube whose center we denote by $\tilde m =(m,m_{n+1})= (m_1, \ldots, m_{n+1}) = \tilde m(x, t)$.
Then we have $$\label{eqn1provingL2al}
\begin{split}
&\left\|\sup_{0<t<R} |{e^{i t \Delta}}f|\right\|_{L^2\left(B^n(0,R);d\mu_R\right)}^2\\
= & \int_{B^n(0,R)} \sup_{0 <t < R} |E f(x,t)|^2 d\mu_R(x)\\
\leq & \int_{B^n(0,R)} \sup_{0 <t < R} \left(|Ef|^2*|\psi|\right)(x,t) d\mu_R(x)\\
\leq & \int_{B^n(0,R)} \sup_{0 <t < R} \left(|E f|^2*\psi_1\right)(\tilde m(x, t)) d\mu_R(x)\\
\leq & \sum_{\underset{|m_i|\leq R}{m=(m_1, \ldots, m_n) \in {\mathbb{Z}}^n}, } \left(\int_{|x-m|\leq 10} d\mu_R(x)\right)
\cdot \sup_{\underset{0\leq m_{n+1}\leq R}{m_{n+1} \in {\mathbb{Z}}} } (|E f|^2*\psi_1)(m,m_{n+1}).
\end{split}$$
For each $m \in {\mathbb{Z}}^n$, let $b(m)$ be an integer in $[0, R]$ such that $$\sup_{\underset{0\leq m_{n+1}\leq R}{m_{n+1} \in {\mathbb{Z}}} } (|E f|^2*\psi_1)(m,m_{n+1}) = (|E f|^2*\psi_1)(m,b(m)).$$ Also we assume $\|f\|_2 = 1$ so $|{e^{i t \Delta}}f|$ is uniformly bounded pointwisely. For each $m\in{\mathbb{Z}}^n$ we define $$\nu_m := \int_{|x-m|\leq 10} d\mu_R(x) \lesssim 1\,.$$ By , we have $$\label{eqn2provingL2al}
\begin{split}
&\left\|\sup_{0<t<R} |{e^{i t \Delta}}f|\right\|_{L^2\left(B^n(0,R);d\mu_R\right)}^2 \\
\lesssim & \sum_{\underset{\nu \in [R^{-100n}, 1]} {\nu \text{ dyadic}} } \sum_{ \underset{\nu_m \sim \nu}{m\in {\mathbb{Z}}^n, |m_i|\leq R}} \nu \cdot (|E f|^2*\psi_1)(m, b(m)) + R^{-90n}.
\end{split}$$
For each dyadic $\nu$, denote $A_{\nu} = \{m \in {\mathbb{Z}}^n : |m_i|\leq R, \nu_m \sim \nu\}$. Performing a dyadic pigeonholing over $\nu$ we see that there exists a dyadic $\nu \in [R^{-100n}, 1]$ such that for any small $\varepsilon > 0$, $$\label{eqn3provingL2al}
\begin{split}
&\left\|\sup_{0<t<R} |{e^{i t \Delta}}f|\right\|_{L^2\left(B^n(0,R);d\mu_R\right)}^2\\
\lessapprox & \sum_{m \in A_{\nu}} \nu\cdot (|E f|^2*\psi_1)(m, b(m)) + R^{-89n}\\
\lesssim & \sum_{m \in A_{\nu}} \nu \cdot\left(\int_{B^{n+1} ((m,b(m)), R^{\varepsilon})}|E f|^2\right) + R^{-89n}\\
\lesssim & \nu \cdot \int_{\bigcup_{m \in A_{\nu}} B^{n+1} ((m,b(m)), R^{\varepsilon})}|E f|^2 + R^{-89n}.
\end{split}$$
Consider the set $X_{\nu} = \bigcup_{m \in A_{\nu}} B^{n+1} ((m,b(m)), R^{\varepsilon})$. It is a union of a collection of distinct $R^{\varepsilon}$-balls and at the same time, it is also a union of unit balls. These balls’ projection onto the $(x_1, \ldots, x_n)$-plane are essentially disjoint (a point can be covered $\lesssim R^{\varepsilon}$ times). For every $r> R^{2\varepsilon}$ by the definition of $A_{\nu}$, the intersection of $X_{\nu}$ and any $r$-ball can be contained in no more than $R^{10n\varepsilon} \nu^{-1} r^{\alpha}$ disjoint $R^{\varepsilon}$-balls. Hence we can apply Corollary \[cor-L2X\] to $X_{\nu}$ with $\gamma \lesssim R^{100 n \varepsilon} \nu^{-1}$ and $\alpha$. With (\[eqn3provingL2al\]) this gives $$\label{eqn4provingL2al}
\left\|\sup_{0<t<R} |{e^{i t \Delta}}f|\right\|_{L^2\left(B^n(0,R);d\mu_R\right)}^2 \lessapprox \nu^{\frac{n-1}{n+1}} R^{\frac{{\alpha}}{n+1}} \|f\|_2^2 \lesssim R^{\frac{{\alpha}}{n+1}} \|f\|_2^2.$$ This concludes the proof.
Main inductive proposition and proof of Theorem \[thm-L2X\] {#sec-pf}
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0
To prove Theorem \[thm-L2X\], we will use a broad-narrow analysis, which involves inductions. To make everything work we introduce another parameter $K$ and state the theorem in a slightly different way. We say that a collection of quantities are dyadically constant if all the quantities are in the same interval of the form $[2^j,2^{j+1}]$, where $j$ is an integer. This is our main inductive proposition:
\[thm-main\] Let $n\geq 1$. For any $0<{\varepsilon}<1/100$, there exist constants $C_{\varepsilon}$ and $0<\delta=\delta({\varepsilon}) \ll {\varepsilon}$ (e.g. $\delta={\varepsilon}^{100}$) such that the following holds for all $R\geq 1$ and all $f$ with ${\rm supp}\widehat{f}\subset B^n(0,1)$. Let $p=\frac{2(n+1)}{n-1}$ ($p=\infty$ when $n=1$). Suppose that $Y=\bigcup_{k=1}^M B_k$ is a union of lattice $K^2$-cubes in $B^{n+1}(0,R)$ and each lattice $R^{1/2}$-cube intersecting $Y$ contains $\sim \lambda$ many $K^2$-cubes in $Y$, where $K=R^\delta$. Suppose that $$\|{e^{i t \Delta}}f\|_{L^p(B_k)} \text{ is dyadically a constant in } k=1,2,\cdots,M.$$ Let $1\leq\alpha\leq n+1$ and $\gamma$ be given by $$\label{ga}
\gamma:=\max_{ \underset{x'\in {\mathbb{R}}^{n+1},r\geq K^2}{B^{n+1}(x',r)\subset B^{n+1}(0,R)}} \frac{\#\{B_k : B_k \subset B(x',r)\}}{r^\alpha}\,.$$ Then $$\label{eq-main}
\|{e^{i t \Delta}}f\|_{L^p(Y)} \leq C_{\varepsilon}M^{-\frac{1}{n+1}}\gamma^{\frac{2}{(n+1)(n+2)}} \lambda^{\frac{n}{(n+1)(n+2)}} R^{\frac{\alpha}{(n+1)(n+2)}+{\varepsilon}}\|f\|_2\,.$$
Theorem \[thm-L2X\] follows from Proposition \[thm-main\] by a dyadic pigeonholing argument:
Given $X=\bigcup_{k} B_k$, a union of lattice unit cubes satisfying the assumptions of Theorem \[thm-L2X\], we sort these unit cubes $B_k$ according to the value of $\|{e^{i t \Delta}}f\|_{L^p(B_k)}$. Assuming $\|f\|_2=1$, there are only $O(\log R)$ significant dyadic choices for this value. Therefore we can choose $X'\subset X$, a union of unit cubes $B$, such that $$\left\{\|{e^{i t \Delta}}f\|_{L^p(B)}: B\in X'\right\} \text{ are dyadically constant}$$ and $$\|{e^{i t \Delta}}f\|_{L^2(X)} \lessapprox \|{e^{i t \Delta}}f\|_{L^2(X')}\,.$$ Let $M$ be the total number of unit cubes $B$ in $X'$. Since $f$ has Fourier support in the unit ball, by locally constant property, $|{e^{i t \Delta}}f|$ is essentially constant on unit balls. Therefore, the estimate is equivalent to $$\label{eq-LpX}
\|{e^{i t \Delta}}f\|_{L^p(X')} \lessapprox M^{-\frac{1}{n+1}}\gamma^{\frac{2}{(n+1)(n+2)}} \lambda^{\frac{n}{(n+1)(n+2)}} R^{\frac{\alpha}{(n+1)(n+2)}}\|f\|_2\,,$$ where $p=\frac{2(n+1)}{n-1}$, and $\gamma,\lambda$ are as in the assumptions of Theorem \[thm-L2X\].
We further sort the unit cubes $B$ in $X'$ as follows:
1. Let $\beta$ be a dyadic number, and ${\mathcal B}_\beta$ a sub-collection of the unit cubes in $X'$ such that for each $B$ in ${\mathcal B}_\beta$, the lattice $K^2$-cube $\tilde B$ containing $B$ satisfies $$\|{e^{i t \Delta}}f\|_{L^p(\tilde B)} \sim \beta\,.$$ Denote the collection of relevant $K^2$-cubes by $\tilde{\mathcal B}_\beta$.
2. Fix $\beta$. Let $\lambda'$ be a dyadic number and ${\mathcal B}_{\beta,\lambda'}$ a sub-collection of ${\mathcal B}_{\beta}$ such that for each $B\in {\mathcal B}_{\beta,\lambda'}$, the lattice $R^{1/2}$-cube $Q$ containing $B$ contains $\sim \lambda'$ many $K^2$-cubes from $\tilde{\mathcal B}_\beta$. Denote the collection of relevant $K^2$-cubes by $\tilde{\mathcal B}_{\beta,\lambda'}$.
Since there are only $O(\log R)$ many significant choices for all dyadic numbers $\beta,\lambda'$, we can choose some $\beta$ and $\lambda'$ so that $\#{\mathcal B}_{\beta,\lambda'} \gtrapprox M$. Then it follows easily by definition that $$M':=\#\tilde {\mathcal B}_{\beta,\lambda'} \gtrapprox M, \quad \lambda'\leq \lambda\,,$$ and $$\gamma':= \max_{ \underset{x'\in {\mathbb{R}}^{n+1},r\geq K^2}{B^{n+1}(x',r)\subset B^{n+1}(0,R)}} \frac{\#\{\tilde B \in \tilde{\mathcal B}_{\beta,\lambda'} :\tilde B \subset B(x',r)\}}{r^\alpha} \leq \gamma\,.$$
Applying Proposition \[thm-main\] to $\|{e^{i t \Delta}}f\|_{L^p(Y)}$ with $Y=\bigcup_{\tilde B \in \tilde{\mathcal B}_{\beta,\lambda'}} \tilde B$ and parameters $M',\gamma',\lambda'$, we get $$\|{e^{i t \Delta}}f\|_{L^p(X)} \lessapprox \|{e^{i t \Delta}}f\|_{L^p(Y)} \lessapprox
M^{-\frac{1}{n+1}}\gamma^{\frac{2}{(n+1)(n+2)}} \lambda^{\frac{n}{(n+1)(n+2)}} R^{\frac{\alpha}{(n+1)(n+2)}}\|f\|_2\,,$$ as desired.
The rest of this section is devoted to a proof of Proposition \[thm-main\]. Note that when the radius $R$ is $\lesssim 1$, the estimate is trivial. So we can assume that $R$ is sufficiently large compared to any constant depending on ${\varepsilon}$. We will induct on radius $R$ in our proof.
In the proof, we will sometimes have paragraphs starting with **Intuition**. We hope that these will help the readers understand what we do next.
For our union $Y$ of $K^2$-cubes, we want to use decoupling theory on each $K^2$-cube. This will relate the whole ${e^{i t \Delta}}f$ to its contributions ${e^{i t \Delta}}f_{\tau}$ from various $1/K$-caps $\tau$ in the frequency space. Instead of doing decoupling in dimension $n+1$, we are going to do a broad-narrow analysis following Bourgain-Guth [@BG], Bourgain [@jB12], Bourgain-Demeter [@BD] and Guth [@G2]: for each $K^2$-cube, one of the following two has to happen:
\(i) It is broad in the sense that there are $n+1$ contributing caps that are transversal. In this case the function is controlled by multilinear estimates which are usually strong enough.
\(ii) It is narrow (i.e. not broad). In this case all the contributing caps have normal directions close to a hyperplane, which enables us to use decoupling in dimension $n$.
Either way we get better estimates than a direct $(n+1)$-dimensional decoupling. We control the broad part directly, and do an induction on the narrow part. Our induction has its roots in the proof of the refined Strichartz estimate in [@DGL; @DGLZ].
Throughout this section we fix $p=\frac{2(n+1)}{n-1}$. In the frequency space we decompose $B^n(0,1)$ into disjoint $K^{-1}$-cubes $\tau$. Denote the set of $K^{-1}$-cubes $\tau$ by ${\mathcal S}$. For a function $f$ with ${\rm supp}\widehat{f}\subset B^n(0,1)$ we have $f=\sum_\tau f_\tau$, where ${\widehat}{f_\tau}$ is ${\widehat}f$ restricted to $\tau$. Given a $K^2$-cube $B$, we define its **significant** set as $${\mathcal S}(B):=\left\{\tau \in {\mathcal S}: \|{e^{i t \Delta}}f_\tau\|_{L^p(B)} \geq \frac{1}{100(\#{\mathcal S})}\|{e^{i t \Delta}}f\|_{L^p(B)}\right\}\,.$$ Note that due to the triangle inequality $$\big\|\sum_{\tau\in {\mathcal S}(B)}{e^{i t \Delta}}f_\tau\big\|_{L^p(B)} \sim \|{e^{i t \Delta}}f\|_{L^p(B)}\,.$$ We say that a $K^2$-cube $B$ is **narrow** if there is an $n$-dimensional subspace $V$ such that for all $\tau \in {\mathcal S}(B)$ $${\rm Angle}(G(\tau),V) \leq \frac{1}{100nK}\,,$$ where $G(\tau)\subset S^n$ is a spherical cap of radius $\sim K^{-1}$ given by $$G(\tau):=\left\{\frac{(-2\xi,1)}{|(-2\xi,1)|}\in S^n:\xi\in\tau\right\}\,,$$ and ${\rm Angle}(G(\tau),V)$ denotes the smallest angle between any non-zero vector $v\in V$ and $v'\in G(\tau)$. Otherwise we say the $K^2$-cube $B$ is **broad**. It follows from this definition that for any broad $B$, there exist $\tau_1,\cdots \tau_{n+1} \in {\mathcal S}(B)$ such that for any $v_j \in G(\tau_j)$ $$\label{eq-trans}
|v_1 \wedge v_2\wedge \cdots \wedge v_{n+1}| \gtrsim K^{-n}\,.$$ Denote the union of broad $K^2$-cubes $B_k$ in $Y$ by $Y_{\rm broad}$, and the union of narrow $K^2$-cubes $B_k$ in $Y$ by $Y_{\rm narrow}$. We call it the broad case if $Y_{\rm broad}$ contains $\geq M/2$ many $K^2$-cubes, and the narrow case otherwise. We will deal with the broad case in Subsection \[sec-br\] using the multilinear refined Strichartz estimate from [@DGLZ]. And we handle the narrow case in Subsection \[sec-nr\] by an inductive argument via the Bourgain-Demeter $l^2$ decoupling theorem [@BD] and induction on scales.
Broad case {#sec-br}
----------
Recall that $K=R^\delta$. A key tool we are using in the broad case is the following multilinear refined Strichartz estimate from [@DGLZ], which is proved using $l^2$ decoupling, induction on scales and multilinear Kakeya estimates (see [@BCT; @G]).
\[c.f. Theorem 4.2 in [@DGLZ]\] \[multstr\] Let $q=\frac{2(n+2)}{n}$. Let $f$ be a function with Fourier support in $B^n(0,1)$. Suppose that $\tau_1,\cdots,\tau_{n+1} \in {\mathcal S}$ and holds for any $v_j\in G(\tau_j)$. Suppose that $Q_1, Q_2,\cdots, Q_N$ are lattice $R^{1/2}$-cubes in $B_R^{n+1}$, so that $$\| {e^{i t \Delta}}f_{\tau_i} \|_{L^{q}(Q_j)} \textrm{ is dyadically a constant in $j$, for each $i=1,2,\cdots,n+1$}.$$ Let $Y$ denote $\bigcup_{j=1}^N Q_j$. Then for any ${\epsilon}> 0$, $$\label{kRS}
\left\| \prod_{i=1}^{n+1} \left|{e^{i t \Delta}}f_{\tau_i}\right|^{\frac{1}{n+1}} \right\|_{L^{q}(Y)} \le C_{\varepsilon}R^{\varepsilon}N^{-\frac{n}{(n+1)(n+2)}} \|f\|_2\,.$$
Throughout the remainder of this subsection we will prove Proposition \[thm-main\] in the broad case. In the broad case, there are $\sim M$ many broad $K^2$-cubes $B$. Denote the collection of $(n+1)$-tuple of transverse caps by $\Gamma$: $$\Gamma:=\left\{\tilde\tau=(\tau_1,\cdots,\tau_{n+1}):\tau_j\in {\mathcal S}\text{ and } \eqref{eq-trans} \text{ holds for any } v_j\in G(\tau_j)\right\}\,.$$ Then for each broad $B$, $$\label{eq-br}
\left\|{e^{i t \Delta}}f\right\|^p_{L^p(B)} \leq K^{O(1)} \prod_{j=1}^{n+1}\left(\int_B \big|{e^{i t \Delta}}f_{\tau_j}\big|^p\right)^{\frac{1}{n+1}}\,,$$ for some $\tilde\tau=(\tau_1,\cdots,\tau_{n+1})\in \Gamma$. In order to exploit the transversality, we want to bound the above geometric average of integrals by an integral of geometric average up to a loss of $K^{O(1)}$. We can do this by using translations and locally constant property. Given a $K^2$-cube $B$, denote its center by $x_B$. We break $B$ into finitely overlapping balls of the form $B(x_B+v,2)$, where $v\in B(0,K^2)\cap {\mathbb{Z}}^{n+1}$. For each $\tau_j$, we can view $|{e^{i t \Delta}}f_{\tau_j}|$ essentially as constant on each $B(x_B+v,2)$. Choose $v_j\in B(0,K^2)\cap {\mathbb{Z}}^{n+1}$ such that $\|{e^{i t \Delta}}f_{\tau_j}\|_{L^\infty(B)}$ is attained in $B(x_B+v_j,2)$. Denote $v_j=(x_j,t_j)$ and define $f_{\tau_j,v_j}$ by $$\widehat{f_{\tau_j,v_j}} (\xi) := \widehat{f_{\tau_j}} (\xi) e^{i(x_j\cdot \xi +t_j|\xi|^2)}\,.$$ Then ${e^{i t \Delta}}f_{\tau_j,v_j} (x) = e^{i(t+t_j)\Delta} f_{\tau_j} (x+x_j)$ and $|{e^{i t \Delta}}f_{\tau_j,v_j} (x)|$ attains $\|{e^{i t \Delta}}f_{\tau_j}\|_{L^\infty(B)}$ in $B(x_B,2)$. Therefore $$\label{eq-rt}
\int_B \big|{e^{i t \Delta}}f_{\tau_j}\big|^p \leq K^{O(1)} \int_{B(x_B,2)} \big|{e^{i t \Delta}}f_{\tau_j,v_j}\big|^p\,.$$ Now for each broad $B$, we find some $\tilde\tau=(\tau_1,\cdots,\tau_{n+1})\in \Gamma$ and $\tilde v =(v_1,\cdots,v_{n+1})$ such that $$\label{eq-lcst}
\begin{split}
\left\|{e^{i t \Delta}}f\right\|^p_{L^p(B)} \leq &K^{O(1)} \prod_{j=1}^{n+1}\left(\int_{B(x_B,2)} \big|{e^{i t \Delta}}f_{\tau_j,v_j}\big|^p\right)^{\frac{1}{n+1}}\\
\leq & K^{O(1)} \int_{B(x_B,2)} \prod_{j=1}^{n+1}\left|{e^{i t \Delta}}f_{\tau_j,v_j}\right|^{\frac{p}{n+1}}\,.
\end{split}$$ Since there are only $K^{O(1)}$ choices for $\tilde \tau$ and $\tilde v$, we can choose some $\tilde \tau$ and $\tilde v$ such that holds for at least $K^{-C} M$ broad balls $B$. From now on, fix $\tilde \tau$ and $\tilde v$, and let $f_j$ denote $f_{\tau_j,v_j}$. Next we further sort the collection ${\mathcal B}$ of remaining broad balls as follows:
1. For a dyadic number $A$, let ${\mathcal B}_A$ be a sub-collection of ${\mathcal B}$ in which for each $B$ we have $$\left\|\prod_{j=1}^{n+1}\left|{e^{i t \Delta}}f_j\right|^{\frac{1}{n+1}}\right\|_{L^\infty(B(x_B,2))} \sim A\,.$$
2. Fix $A$, for dyadic numbers $\tilde \lambda,\iota_1,\cdots,\iota_{n+1}$, let ${\mathcal B}_{A,\tilde \lambda,\iota_1,\cdots,\iota_{n+1}}$ be a sub-collection of ${\mathcal B}_A$ in which for each $B$, the $R^{1/2}$-cube $Q$ containing $B$ contains $\sim \tilde \lambda$ cubes from ${\mathcal B}_A$ and $$\|{e^{i t \Delta}}f_j\|_{L^q(Q)} \sim \iota_j, \quad j=1,2,\cdots,n+1\,.$$ Here $q=\frac{2(n+2)}{n}$.
Recall that $p>q$, where $p=\frac{2(n+1)}{n-1}$ is the sharp exponent for decoupling in dimension $n$, and $q=\frac{2(n+2)}{n}$ is the exponent for which the multilinear refined Strichartz estimate in demension $n+1$ holds. The first dyadic pigeonholing together with the locally constant property enables us to dominate $L^p$-norm by $L^q$-norm using reverse Hölder. The second dyadic pigeonholing allows us to apply the multilinear refined Strichartz estimate to control the $L^q$-norm.
We can assume that $\|f\|_2=1$. Then all the above dyadic numbers making significant contributions can be assumed to be between $R^{-C}$ and $R^{C}$ for a large constant $C$. Therefore, there exist some dyadic numbers $A,\tilde \lambda,\iota_1,\cdots,\iota_{n+1}$ such that ${\mathcal B}_{A,\tilde \lambda,\iota_1,\cdots,\iota_{n+1}}$ contains $\geq K^{-C} M$ many cubes $B$. Fix a choice of $A,\tilde \lambda,\iota_1,\cdots,\iota_{n+1}$ and denote ${\mathcal B}_{A,\tilde \lambda,\iota_1,\cdots,\iota_{n+1}}$ by ${\mathcal B}$ for convenience (a mild abuse of notation). Then, in the broad case, it follows from and our choice of $A$ that $$\label{all'}
\begin{split}
\|{e^{i t \Delta}}f\|_{L^p(Y)} \leq &K^{O(1)} \left\| \prod_{j=1}^{n+1}\left|{e^{i t \Delta}}f_j\right|^{\frac{1}{n+1}} \right\|_{L^p(\cup_{B\in {\mathcal B}}B(x_B,2))}\\
\leq &K^{O(1)} M^{\frac 1p -\frac 1 q} \left\| \prod_{j=1}^{n+1}\left|{e^{i t \Delta}}f_j\right|^{\frac{1}{n+1}} \right\|_{L^q(\cup_{B\in {\mathcal B}}B(x_B,2))}\\
\leq &K^{O(1)} M^{-\frac {1}{(n+1)(n+2)}} \left\| \prod_{j=1}^{n+1}\left|{e^{i t \Delta}}f_j\right|^{\frac{1}{n+1}} \right\|_{L^q(\cup_{Q\in {\mathcal Q}} Q)}\,,
\end{split}$$ where ${\mathcal Q}$ is the collection of relevant $R^{1/2}$-cubes $Q$ when we define ${\mathcal B}$. Note that $$(\#{\mathcal Q}) \lambda \geq (\#{\mathcal Q}) \tilde \lambda \sim \#{\mathcal B}\geq K^{-C}M\,,$$ so $$\tilde N :=\#{\mathcal Q}\geq K^{-C}\frac{M}{\lambda}\,.$$ Applying Theorem \[multstr\], we get $$\left\| \prod_{j=1}^{n+1}\left|{e^{i t \Delta}}f_j\right|^{\frac{1}{n+1}} \right\|_{L^q(\cup_{Q\in {\mathcal Q}} Q)}
\leq K^{O(1)} \left(\frac M\lambda\right)^{-\frac{n}{(n+1)(n+2)}}
\|f\|_2\,,$$ and therefore by , $$\|{e^{i t \Delta}}f\|_{L^p(Y)} \leq K^{O(1)} M^{-\frac{1}{n+2}} \lambda^{\frac{n}{(n+1)(n+2)}}
\|f\|_2\,.$$ Note that $$M^{-\frac{1}{n+2}} \lambda^{\frac{n}{(n+1)(n+2)}} \leq K^{O(1)} M^{-\frac{1}{n+1}}\gamma^{\frac{2}{(n+1)(n+2)}} \lambda^{\frac{n}{(n+1)(n+2)}} R^{\frac{\alpha}{(n+1)(n+2)}}$$ holds if and only if $M\leq K^{O(1)}\gamma^2R^\alpha$. Indeed, by definition of $\gamma$, we have $M\leq \gamma R^\alpha$ and $\gamma \geq K^{-2\alpha}$. So the broad case is done.
Narrow case {#sec-nr}
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For each narrow ball, we have the following lemma which is a consequence of the $l^2$ decoupling theorem in dimension $n$ and Minkowski’s inequality. This argument is essentially contained in Bourgain-Demeter’s proof of the $l^2$ decoupling conjecture and we omit the details (see the proof of Proposition 5.5 in [@BD]).
\[lem-nr-dec\] Suppose that $B$ is a narrow $K^2$-cube in ${\mathbb{R}}^{n+1}$. Then for any ${\varepsilon}>0$, $$\|{e^{i t \Delta}}f\|_{L^p(B)} \leq C_{\varepsilon}K^{{\varepsilon}} \left(\sum_{\tau\in {\mathcal S}}\left\|{e^{i t \Delta}}f_\tau\right\|_{L^p(\omega_B)}^2\right)^{1/2} \,,$$ here $p=\frac{2(n+1)}{n-1}$, ${\mathcal S}$ denotes the set of $K^{-1}$-cubes which tile $B^n(0,1)$, and $\omega_B$ is a weight function which is essentially a characteristic function on $B$. More precisely, $\omega_B$ has Fourier support in $B(0,K^{-2})$ and satisfies $$1_B(\tilde x) \lesssim \omega_B(\tilde x) \leq \left(1+\frac{|\tilde x - C(B)|}{K^2}\right)^{-1000n}.$$
For each $\tau \in {\mathcal S}$, we will deal with ${e^{i t \Delta}}f_\tau$ by parabolic rescaling and induction on radius. In order to do so, we need to further decompose $f$ in physical space and perform dyadic pigeonholing several times to get the right setup for our inductive hypothesis at scale $R_1:=R/K^2$ after rescaling.
For each $1/K$-cap $\tau$, all wave packets associated with $f_\tau$ through a given point have to lie in a common box that has one side length $R$ and other side lengths $R/K$. Every single box of this type will become an $R/K^2$-ball if we perform a parabolic rescaling to transform $\tau$ into the standard $1$-cap. We want to use the inductive hypothesis for radius $R/K^2$ in an efficient way. A few dyadic pigeonholing steps will be needed.
First, we break the physical ball $B^n(0,R)$ into $R/K$-cubes $D$. For each pair $(\tau, D)$, let $f_{\Box_{\tau, D}}$ be the function formed by cutting off $f$ on the cube $D$ (with a Schwartz tail) in physical space and the cube $\tau$ in Fourier space. Note that $e^{it\Delta} f_{\Box_{\tau,D}}$, restricted to $B^{n+1}(R)$, is essentially supported on an $R/K \times \cdots \times R/K \times R$-box, which we denote by $\Box_{\tau, D}$. The box $\Box_{\tau, D}$ is in the direction given by $(-2c(\tau),1)$ and intersects ${t=0}$ at the cube $D$, where $c(\tau)$ is the center of $\tau$. For a fixed $\tau$, the different boxes $\Box_{\tau, D}$ tile $B^{n+1}(0,R)$. In particular, for each $\tau$, a given $K^2$-cube $B$ lies in exactly one box $\Box_{\tau, D}$. We write $f=\sum_{\Box} f_\Box$ for abbreviation. By Lemma \[lem-nr-dec\], for each narrow $K^2$-cube $B$, $$\label{eq-dec}
\|{e^{i t \Delta}}f\|_{L^p(B)} \lesssim K^{{\varepsilon}^4} \left(\sum_{\Box}\left\|{e^{i t \Delta}}f_\Box\right\|_{L^p(\omega_B)}^2\right)^{1/2} \,.$$
\[ht\] ![[]{data-label="fig-stri"}](refined_stri.png "fig:")
We will have a gain $\frac{1}{K^{2{\varepsilon}}}$ from induction on radius. Therefore, in we are allowed to lose a small power of $K$. This small power depends on ${\varepsilon}$ and should be smaller than $2{\varepsilon}$. It could be ${\varepsilon}^2,{\varepsilon}^3,{\varepsilon}^4,$ etc.
Next, we perform a dyadic pigeonholing to get our inductive hypothesis for each $f_{\Box}$. Recall that $K=R^{\delta}$, where $\delta={\varepsilon}^{100}$. Denote $$R_1:=R/K^2=R^{1-2\delta}, \quad K_1 := R_1^\delta=R^{\delta-2\delta^2}\,.$$ Tile $\Box$ by $KK_1^2\times \cdots \times KK_1^2 \times K^2K_1^2$-tubes $S$, and also tile $\Box$ by $R^{1/2}\times \cdots \times R^{1/2} \times KR^{1/2}$-tubes $S'$ (all running parallel to the long axis of $\Box$). To understand these scales, see Figure \[fig-stri\] for the change in physical space during the process of parabolic rescaling. In particular, after rescaling the $\Box$ becomes an $R_1$-cube, the tubes $S'$ and $S$ become lattice $R_1^{1/2}$-cubes and $K_1^2$-cubes respectively.
We apply the following to regroup tubes $S$ and $S'$ inside each $\Box$:
1. Sort those tubes $S$ which intersect $Y$ according to the value $\|{e^{i t \Delta}}f_\Box\|_{L^p(S)}$ and the number of narrow $K^2$-cubes contained in it. For dyadic numbers $\eta, \beta_1$, we use ${\mathbb{S}}_{\Box,\eta,\beta_1}$ to stand for the collection of tubes $S\subset \Box$ each of which containing $\sim \eta$ narrow $K^2$-cubes in $Y_{\rm narrow}$ and $\|{e^{i t \Delta}}f_\Box\|_{L^p(S)} \sim \beta_1$.
2. For fixed $\eta,\beta_1$, we sort the tubes $S'\subset \Box$ according to the number of tubes $S\in {\mathbb{S}}_{\Box,\eta,\beta_1}$ contained in it. For dyadic number $\lambda_1$, let ${\mathbb{S}}_{\Box,\eta,\beta_1,\lambda_1}$ be the sub-collection of ${\mathbb{S}}_{\Box,\eta,\beta_1}$ such that for each $S\in {\mathbb{S}}_{\Box, \eta,\beta_1,\lambda_1}$, the tube $S'$ containing $S$ contains $\sim \lambda_1$ tubes from ${\mathbb{S}}_{\Box,\eta, \beta_1}$.
3. For fixed $\eta, \beta_1,\lambda_1$, we sort the boxes $\Box$ according to the value $\|f_\Box\|_2$, the number $\#{\mathbb{S}}_{\Box, \eta, \beta_1, \lambda_1}$ and the value $\gamma_1$ defined below. For dyadic numbers $\beta_2, M_1,\gamma_1$, let ${\mathbb{B}}_{\eta,\beta_1,\lambda_1,\beta_2, M_1,\gamma_1}$ denote the collection of boxes $\Box$ each of which satisfying that $$\|f_\Box\|_2\sim \beta_2, \quad \#{\mathbb{S}}_{\Box,\eta,\beta_1,\lambda_1} \sim M_1$$ and $$\label{ga1}
\max_{T_{r}\subset \Box:r\geq K_1^2} \frac{\#\{S\in {\mathbb{S}}_{\Box,\eta,\beta_1,\lambda_1}: S\subset T_r\}}{r^\alpha} \sim \gamma_1\,,$$ where $T_r$ are $Kr \times \cdots \times Kr \times K^2 r$-tubes in $\Box$ running parallel to the long axis of $\Box$.
Let $Y_{\Box,\eta,\beta_1,\lambda_1}$ be the union of the tubes $S$ in ${\mathbb{S}}_{\Box,\eta,\beta_1,\lambda_1}$, and ${\raisebox{0.7ex}{\(\chi\)}}_{Y_{\Box,\eta,\beta_1,\lambda_1}}$ the corresponding characteristic function. Then on $Y_{\rm narrow}$ we can write $${e^{i t \Delta}}f = \sum_{\eta,\beta_1,\lambda_1,\beta_2, M_1,\gamma_1}\left( \sum_{\Box \in {\mathbb{B}}_{\eta,\beta_1,\lambda_1,\beta_2, M_1,\gamma_1}} {e^{i t \Delta}}f_{\Box} \cdot {\raisebox{0.7ex}{\(\chi\)}}_{Y_{\Box,\eta,\beta_1,\lambda_1}} \right) + O(R^{-1000n}) \|f\|_2\,.$$
The small error term $O(R^{-1000n}) \|f\|_2$ will prove to be harmless in our computations. We will neglect this term in the sequel. Again, to make the statement really rigorous one needs to increase the side lengths of $\Box$ by a tiny power of $R$, say $R^{{\delta}^{100}} \sim K^{\delta^{99}}$. As before, we choose to ignore this technicality in order to facilitate the main exposition.
In particular, on each narrow $B$ we have $$\label{fB}
{e^{i t \Delta}}f = \sum_{\eta,\beta_1,\lambda_1,\beta_2, M_1,\gamma_1}\left( \sum_{\underset{B\subset Y_{\Box,\eta,\beta_1,\lambda_1}}{\Box \in {\mathbb{B}}_{\eta,\beta_1,\lambda_1,\beta_2, M_1,\gamma_1}}} {e^{i t \Delta}}f_{\Box} \right)\,.$$
Without loss of generality, we assume that $\|f\|_2=1$. Then we can further assume that the dyadic numbers above are in reasonable ranges, say $$1\leq \eta \leq K^{O(1)}, \quad R^{-C}\leq \beta_1 \leq K^{O(1)}, \quad 1\leq \lambda_1 \leq R^{O(1)}$$ and $$R^{-C} \leq \beta_2 \leq 1, \quad 1\leq M_1\leq R^{O(1)},\quad K^{-2n}\leq \gamma_1\leq R^{O(1)}\,,$$ where $C$ is a large constant such that the contributions from those $\beta_1$ and $\beta_2$ less than $R^{-C}$ are negligible. Therefore, there are only $O(\log R)$ significant choices for each dyadic number. Because of and , by pigeonholing, we can choose $\eta,\beta_1,\lambda_1,\beta_2, M_1,\gamma_1$ so that $$\label{dya1}
\|{e^{i t \Delta}}f\|_{L^p(B)} \lesssim (\log R)^{6} K^{{\varepsilon}^4} \left(\sum_{\underset{B\subset Y_{\Box,\eta,\beta_1,\lambda_1}}{\Box \in {\mathbb{B}}_{\eta,\beta_1,\lambda_1,\beta_2, M_1,\gamma_1}}}\|{e^{i t \Delta}}f_\Box\|_{L^p(\omega_B)}^2\right)^{1/2}$$ holds for a fraction $\gtrsim (\log R)^{-6}$ of all narrow $K^2$-cubes $B$.
We fix $\eta,\beta_1,\lambda_1,\beta_2, M_1,\gamma_1$ for the rest of the proof. Let $Y_\Box$ and ${\mathbb{B}}$ stand for the abbreviations of $Y_{\Box,\eta,\beta_1,\lambda_1}$ and ${\mathbb{B}}_{\eta,\beta_1,\lambda_1,\beta_2, M_1,\gamma_1}$ respectively. Finally we sort the narrow balls $B$ satisfying by $
\#\{\Box\in {\mathbb{B}}: B\subset Y_{\Box}\}$. Let $Y'\subset Y_{\rm narrow}$ be a union of narrow $K^2$-cubes $B$ each of which obeying $$\label{dya1'}
\|{e^{i t \Delta}}f\|_{L^p(B)} \lesssim (\log R)^6 K^{{\varepsilon}^4} \left(\sum_{\Box\in {\mathbb{B}}: B\subset Y_\Box}\|{e^{i t \Delta}}f_\Box\|_{L^p(\omega_B)}^2\right)^{1/2}$$ and $$\label{dya2}
\#\{\Box\in {\mathbb{B}}: B\subset Y_{\Box}\} \sim \mu$$ for some dyadic number $1\leq \mu \leq K^{O(1)}$, moreover the number of $K^2$-cubes $B$ in $Y'$ is $\gtrsim$ $(\log R)^{-7} M$.
Now we are done with dyadic pigeonholing argument and let us put all these together. By our assumption that $\|{e^{i t \Delta}}f\|_{L^p(B_k)}$ is essentially constant in $k=1,2,\cdots,M$, in the narrow case we have $$\label{eq-Y'}
\|{e^{i t \Delta}}f\|_{L^p(Y)}^p \lesssim (\log R)^7 \sum_{B\subset Y'} \|{e^{i t \Delta}}f\|^p_{L^p(B)}\,.$$ For each $B\subset Y'$, it follows from , and Hölder’s inequality that $$\label{eq-H}
\|{e^{i t \Delta}}f\|^p_{L^p(B)} \lesssim (\log R)^{6p} K^{{\varepsilon}^4p} \mu^{\frac{p}{2}-1} \sum_{\Box\in {\mathbb{B}}: B\subset Y_{\Box}}\|{e^{i t \Delta}}f_\Box\|_{L^p(\omega_B)}^p\,.$$ Putting and together and as before omiting the rapidly decaying tails, $$\label{Y-Ybox}
\|{e^{i t \Delta}}f\|_{L^p(Y)} \lesssim (\log R)^{13} K^{{\varepsilon}^4} \mu^{\frac{1}{n+1}} \left(\sum_{\Box\in {\mathbb{B}}} \left\|{e^{i t \Delta}}f_\Box\right\|^p_{L^p(Y_\Box)}\right)^{1/p}\,.$$
Next, to each $\|{e^{i t \Delta}}f_\Box\|_{L^p(Y_\Box)}$ we apply parabolic rescaling and induction on radius. For each $1/K$-cube $\tau=\tau_{\Box}$ in $B^n(0,1)$, we write $\xi=\xi_0+K^{-1} \zeta \in \tau$, where $\xi_0$ is the center of $\tau$. Then $$|{e^{i t \Delta}}f_{\Box} (x)| =K^{-n/2} |e^{i\tilde t \Delta} g (\tilde x)|$$ for some function $g$ with Fourier support in the unit cube and $\|g\|_2=\|f_{\Box}\|_2$, where the new coordinates $(\tilde x,\tilde t)$ are related to the old coordinates $(x,t)$ by $$\label{coord}
\begin{cases}
\tilde x =K^{-1} x + 2 t K^{-1} \xi_0\,, \\
\tilde t = K^{-2} t \,.
\end{cases}$$ For simplicity, denote the above relation by $(\tilde x,\tilde t)=F(x,t)$. Therefore $$\label{Ybox-Ytilde}
\|{e^{i t \Delta}}f_\Box (x)\|_{L^p(Y_\Box)} = K^{\frac{n+2}{p}-\frac{n}{2}} \|e^{i\tilde t \Delta} g(\tilde x)\|_{L^p(\tilde Y)}=
K^{-\frac{1}{n+1}} \|e^{i\tilde t \Delta} g(\tilde x)\|_{L^p(\tilde Y)},$$ where $\tilde Y$ is the image of $Y_\Box$ under the new coordinates.
Note that we can apply our inductive hypothesis at scale $R_1=R/K^2$ to $\|e^{i\tilde t \Delta} g(\tilde x)\|_{L^p(\tilde Y)}$ with new parameters $M_1, \gamma_1, \lambda_1, R_1$. More precisely, $\tilde Y = F(Y_{\Box})$ consists of $\sim M_1$ distinct $K_1^2$-cubes $F(S)$ in an $R_1$-ball $F(\Box)$, and the $K_1^2$-cubes $F(S)$ are organized into $R_1^{1/2}$-cubes $F(S')$ such that each cube $F(S')$ contains $\sim \lambda_1$ cubes $F(S)$. Moreover, $\|e^{i\tilde t \Delta} g(\tilde x)\|_{L^p(F(S))}$ is dyadically a constant in $S \subset Y_{\Box}$. By our choice of $\gamma_1$, we have $$\max_{ \underset{x'\in {\mathbb{R}}^{n+1},r\geq K_1^2}{B^{n+1}(x',r)\subset F(\Box)} }\frac{\#\{F(S) : F(S) \subset B(x',r)\}}{r^\alpha} \sim \gamma_1\,.$$ Henceforth, by and inductive hypothesis at scale $R_1$ we have $$\label{ind}
\begin{split}
&\|{e^{i t \Delta}}f_\Box (x)\|_{L^p(Y_\Box)} \\ \lesssim &K^{-\frac{1}{n+1}} M_1^{-\frac{1}{n+1}}\gamma_1^{\frac{2}{(n+1)(n+2)}} \lambda_1^{\frac{n}{(n+1)(n+2)}} \left(\frac{R}{K^2}\right)^{\frac{\alpha}{(n+1)(n+2)}+{\varepsilon}}\|f_\Box\|_2\,.
\end{split}$$ From and we obtain $$\label{all}
\begin{split}
&\|{e^{i t \Delta}}f\|_{L^p(Y)}\\ \lesssim &K^{2{\varepsilon}^4} \mu^{\frac{1}{n+1}}
K^{-\frac{1}{n+1}} M_1^{-\frac{1}{n+1}}\gamma_1^{\frac{2}{(n+1)(n+2)}} \lambda_1^{\frac{n}{(n+1)(n+2)}} \left(\frac{R}{K^2}\right)^{\frac{\alpha}{(n+1)(n+2)}+{\varepsilon}} \left(\sum_{\Box\in {\mathbb{B}}}\|f_\Box\|_2^p\right)^{1/p} \\
\lesssim & K^{2{\varepsilon}^4} \left(\frac{\mu}{\#{\mathbb{B}}} \right)^{\frac{1}{n+1}}
K^{-\frac{1}{n+1}} M_1^{-\frac{1}{n+1}}\gamma_1^{\frac{2}{(n+1)(n+2)}} \lambda_1^{\frac{n}{(n+1)(n+2)}} \left(\frac{R}{K^2}\right)^{\frac{\alpha}{(n+1)(n+2)}+{\varepsilon}} \|f\|_2 \,,
\end{split}$$ where the last inequality follows from orthogonality $\sum_\Box\|f_\Box\|_2^2 \lesssim \|f\|_2^2$ and the assumption that $\|f_\Box\|_2 \sim$ constant in $\Box\in {\mathbb{B}}$.
To finish our inductive argument, we have to relate the old and new parameters. Our setup allows us to do this in a nice way: Given $M_1, \lambda_1$ and $\gamma_1$, if $\eta$ is small, i.e. each $S$ contains very few narrow $K^2$-cubes, then $M$ is relatively small; if $\eta$ is large, i.e. each $S$ contains a lot of narrow $K^2$-cubes, then $\lambda$ and $\gamma$ are relatively large. Both make the right-hand side of what we want to prove reasonably large. This is the reason why one could believe the numerology will work out.
Consider the cardinality of the set $\{(\Box, B): \Box \in {\mathbb{B}}, B\subset Y_\Box\cap Y'\}$. By our choice of $\mu$ as in , there is a lower bound $$\#\{(\Box, B): \Box \in {\mathbb{B}}, B\subset Y_\Box\cap Y'\} \gtrsim (\log R)^{-7} M \mu\,.$$ On the other hand, by our choices of $M_1$ and $\eta$, for each $\Box\in {\mathbb{B}}$, $Y_\Box$ contains $\sim M_1$ tubes $S$ and each $S$ contains $\sim \eta$ narrow cubes in $Y_{\rm narrow}$, so $$\#\{(\Box, B): \Box \in {\mathbb{B}}, B\subset Y_\Box\cap Y'\} \lesssim (\#{\mathbb{B}})M_1\eta\,.$$ Therefore, we get $$\label{muB}
\frac{\mu}{\#{\mathbb{B}}} \lesssim \frac{(\log R)^7 M_1\eta}{M}\,.$$
Next by our choices of $\gamma_1$ as in and $\eta$, $$\begin{split}
&\gamma_1 \cdot \eta\\
\sim &
\max_{T_{r}\subset \Box:r\geq K_1^2} \frac{\#\{S: S \subset Y_\Box\cap T_r\}}{r^\alpha} \cdot \#\{B: B \subset S\cap Y_{\rm narrow} \text{ for any fixed } S\subset Y_\Box\} \\
\lesssim & \max_{T_{r}\subset \Box:r\geq K_1^2} \frac{\#\{B\subset Y:B \subset T_r\}}{r^\alpha} \leq \frac{K\gamma (Kr)^\alpha}{r^\alpha}= \gamma K^{\alpha+1}\,,
\end{split}$$ where the last inequality follows from the definition of $\gamma$ and the fact that we can cover a $Kr \times \cdots \times Kr \times K^2 r$-tube $T_r$ by $\sim K$ finitely overlapping $Kr$-balls. Hence, $$\label{eta}
\eta \lesssim \frac{\gamma K^{\alpha+1}}{\gamma_1}\,.$$
Finally we relate $\lambda_1$ and $\lambda$ by considering the number of narrow $K^2$-balls in each relevant $R^{1/2}\times \cdots \times R^{1/2} \times KR^{1/2}$-tube $S'$. Recall that each relevant $S'$ contains $\sim \lambda_1$ tubes $S$ in $Y_\Box$ and each such $S$ contains $\sim \eta$ narrow balls. On the other hand, we can cover $S'$ by $\sim K$ finitely overlapping $R^{1/2}$-balls and by assumption each $R^{1/2}$-ball contains $\lesssim \lambda$ many $K^2$-cubes in $Y$. Thus it follows that $$\label{la-la1}
\lambda_1\lesssim \frac{K\lambda}{\eta}\,.$$
By inserting and into , $$\begin{split}
&\|{e^{i t \Delta}}f\|_{L^p(Y)} \\
\lesssim &\frac{K^{3{\varepsilon}^4}}{K^{2{\varepsilon}}}\left(\frac{\eta\gamma_1}{K^{\alpha+1}}\right)^{\frac{2}{(n+1)(n+2)}} M^{-\frac{1}{n+1}} \lambda^{\frac{n}{(n+1)(n+2)}} R^{\frac{\alpha}{(n+1)(n+2)}+{\varepsilon}}\|f\|_2 \\
\lesssim & \frac{K^{3{\varepsilon}^4}}{K^{2{\varepsilon}}} M^{-\frac{1}{n+1}}\gamma^{\frac{2}{(n+1)(n+2)}} \lambda^{\frac{n}{(n+1)(n+2)}} R^{\frac{\alpha}{(n+1)(n+2)}+{\varepsilon}}\|f\|_2\,,
\end{split}$$ where the last inequality follows from . Since $K=R^{\delta}$ and $R$ can be assumed to be sufficiently large compared to any constant depending on ${\varepsilon}$, we have $\frac{K^{3{\varepsilon}^4}}{K^{2{\varepsilon}}} \ll 1$ and the induction closes for the narrow case. This completes the proof of Proposition \[thm-main\].
Remark {#sec-rmk}
------
In Section \[sec-app\], we have seen that Corollary \[cor-L2X\] is a direct result of Theorem \[thm-L2X\], and they are equally useful in applications to the sharp $L^2$ estimate of Schrödinger maximal function. We can also prove Corollary \[cor-L2X\] from scratch using a similar argument as in this section, which is slightly easier in two aspects compared to that of Theorem \[thm-L2X\]. First, in the broad case, it is sufficient to use multilinear restriction estimates and not necessary to invoke the multilinear refined Strichartz. Secondly, because there is one parameter less, the dyadic pigeonholing argument in the narrow case would be slightly reduced, for example, see Figure \[fig-restr\] for tubes of different scales in the $\Box$ under the setting of Corollary \[cor-L2X\].
![ (in inductive argument for Corollary \[cor-L2X\])[]{data-label="fig-restr"}](multilinear_restr.png)
In fact, an adaptation of some arguments in the work [@W] of Wolff on the Falconer distance set problem in dimension $2$ can already imply Corollary \[cor-L2X\] when $n=1$. In the special case $n=1$, the broad versus narrow dichotomy becomes the one on bilinear versus linear. To handle the linear part, the idea of induction on scales and splitting the ball into rectangular boxes “$\Box$” of size $R \times R/K$ in our proof already existed in Wolff’s paper. We thank Hong Wang for pointing this out to us.
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---
abstract: 'We address the problem of identifying linear relations among variables based on noisy measurements. This is, of course, a central question in problems involving “Big Data.” Often a key assumption is that measurement errors in each variable are independent. This precise formulation has its roots in the work of Charles Spearman in 1904 and of Ragnar Frisch in the 1930’s. Various topics such as errors-in-variables, factor analysis, and instrumental variables, all refer to alternative formulations of the problem of how to account for the anticipated way that noise enters in the data. In the present paper we begin by describing the basic theory and provide alternative modern proofs to some key results. We then go on to consider certain generalizations of the theory as well applying certain novel numerical techniques to the problem. A central role is played by the Frisch-Kalman dictum which aims at a noise contribution that allows a maximal set of simultaneous linear relations among the noise-free variables –a rank minimization problem. In the years since Frisch’s original formulation, there have been several insights including trace minimization as a convenient heuristic to replace rank minimization. We discuss convex relaxations and certificates guaranteeing global optimality. A complementary point of view to the Frisch-Kalman dictum is introduced in which models lead to a min-max quadratic estimation error for the error-free variables. Points of contact between the two formalisms are discussed and various alternative regularization schemes are indicated.'
author:
- 'Lipeng Ning[^1]'
- 'Tryphon T. Georgiou[^2]'
- 'Allen Tannenbaum[^3]'
- '[Stephen P. Boyd]{}[^4]'
bibliography:
- 'IEEEabrv.bib'
- 'Frisch\_bib.bib'
title: |
Linear models based on noisy data\
and the Frisch scheme
---
Introduction
============
The standard paradigm in modeling is to postulate that measured quantities contain a contribution of “accidental deviation” [@Spearman] from the otherwise “uniformities” that characterize an underlying law. Therefore, a key issue when identifying dependencies between variables is how to account for the contribution of noise in the data. Various assumptions on the structure of noise and of the possible dependencies lead to a number of corresponding methodologies.
The purpose of the present paper is to consider from a modern computational point of view, the important situation where the noise components are assumed independent, and the consequences of this assumption –the data is typically abstracted into a corresponding (estimated) covariance statistic. This independence assumption underlies the errors-in-variables model [@Durbin; @KlepperLeamer] and factor analysis [@AndersonRubin; @Ledermann; @Harman1966; @Joreskog1969; @Shapiro], and has a century-old history [@Frisch2; @Reiersol; @Koopmans]; see also [@Kalman1982; @Kalman1985; @Los; @Woodgate1; @Guidorzi95; @Soderstrom2007errors; @Anderson2008; @Forni2000]. Accordingly, given the large classical literature on this problem, this paper will also have a tutorial flavor.
The precise formulation has its roots in the work of Ragnar Frisch in the 1930’s. The central assumption is that the noise components are independent of the underlying variables and are also mutually independent [@Kalman1982; @Kalman1985]. In addition, since several alternative linear relations are typically consistent with the data, a maximal set of simultaneous dependencies is sought as a means to limit uncertainty and to provide canonical models [@Kalman1982; @Kalman1985]. This particular dictum gives rise to a (non-convex) rank-minimization problem. Thus, it is somewhat surprising that the special case where the maximal number of possible simultaneous linear relations is equal to $1$ can be explicitly characterized –this was accomplished over half a century ago by Reiers[ø]{}l [@Reiersol]; see also [@Kalman1982; @KlepperLeamer]. To date no other case is known that admits a precise closed-form solution.
In recent years, emphasis has been shifting from hard, non-convex optimization to convex regularizations, which in addition scale nicely with the size of the problem. Following this trend we revisit the Frisch problem from several alternative angles. We first present an overview of the literature, and present several new insights and proofs. In the process, we also give an extension of Reiers[ø]{}l’s result to complex matrices. Our main interest is in exploring recently studied convex optimization problems that approximate rank minimization by use of suitable surrogates. In particular, we study iterative schemes for treating the general Frisch problem and focus on certificates that guarantee optimality. In parallel, we consider a viewpoint that serves as an alternative to the Frisch problem where now, instead of a maximal number of simultaneous linear relations, we seek a uniformly optimal estimator for the unobserved data under the independence assumption of the Frisch scheme. The optimal estimator is obtained as a solution to a min-max optimization problem. Rank-regularized and min-max alternatives are discussed and an example is given to highlight the potential and limitations of the techniques.
The remainder of this paper is organized as follows. We first introduce the errors-in-variables problem in Section \[sec:datastrcuture\]. In Section \[sec:Frisch\], we revisit the Frisch problem, and a related problem due to Shapiro, and provide a geometric interpretation of Reiers[ø]{}l’s result along with a generalization to complex-valued covariances. In Section \[sec:MinTrace\], we present an iterative trace-minimization scheme for solving the Frisch problem and provide computable lower-bounds for the minimum-rank. In Section \[correspondence\], we bring up the question of estimation in the context of the Frisch scheme and motivate a suitable a rank-regularized min-max optimization problem in Section \[sec:regularized\]. Some concluding remarks are provided in Section \[sec:conclusion\].
Notation
========
$\;$\
-------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------
${{\mathcal R}}(\cdot)$, ${{\mathcal N}}(\cdot)$ range space, null space
$\Pi_{\mathcal X}$ orthogonal projection onto ${\mathcal X}$
$>0\;\; (\geq 0)$ positive definite (resp., positive semi-definite)
${{\mathbf S}}_n$ $=\;\;\left\{M \mid M\in {{\mathbb R}}^{n\times n},\; M=M' \right\}$
${{\mathbf S}}_{n,+}$ $=\;\;\left\{M \mid M\in {{\mathbf S}}_n,\; M\geq0 \right\}$
${{\mathbf H}}_n$ $=\;\;\left\{M \mid M\in {{\mathbb C}}^{n\times n},\; M=M^* \right\}$
${{\mathbf H}}_{n,+}$ $=\;\;\left\{M \mid M\in {{\mathbf H}}_n,\; M\geq0 \right\}$
$[\cdot ]_{k\ell},\;\; ([\cdot ]_{k})$ $(k, \ell)$-th entry (resp., $k$-th entry)
$|M|$ determinant of $M\in {{\mathbb R}}^{n\times n}$
$n_+(\cdot)$ number of positive eigenvalues
$\diag: {{\mathbb R}}^{n\times n} \to {{\mathbb R}}^n: M\mapsto d$ where $[d]_i=[M]_{ii}$ for $i=1, \ldots n$
$\diag^*: {{\mathbb R}}^{n} \rightarrow {{\mathbb R}}^{n\times n}: d\mapsto D$ where $D$ is diagonal and $[D]_{ii}=[d]_{i}$ for $i=1,\ldots n$
$M{\succ_{\hspace*{-1pt}_e}}0\;({\succeq_{\hspace*{-1pt}_e}}0,\; {\prec_{\hspace*{-1pt}_e}}0,\;{\preceq_{\hspace*{-1pt}_e}}0)$ the off-diagonal entries are $>0$ (resp. $\geq 0$, $<0$, $\leq 0$),
or can be made so by changing the signs of selected
rows and corresponding columns
-------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------
Data and basic assumptions {#sec:datastrcuture}
==========================
Consider a Gaussian vector ${{\mathbf x}}$ taking values in ${{\mathbb R}}^{n\times 1}$ having zero mean and covariance $\Sigma$. We assume that it represents an additive mixture of a Gaussian “noise-free” vector ${{\hat{\mathbf x} }}$ and a “noise component” ${{\tilde{\mathbf x}}}$, thus $$\label{eq:xa}
{{\mathbf x}}={{\hat{\mathbf x} }}+{{\tilde{\mathbf x}}}.$$ The entries of ${{\tilde{\mathbf x}}}$ are assumed independent of one another and independent of the entries of ${{\hat{\mathbf x} }}$ with both vectors having zero mean and covariances $\hat\Sigma$ and $\tilde\Sigma$, respectively. Thus,
\[eq:firstsetofconstraints\] $$\begin{aligned}
&&{{\mathcal E}}({{\tilde{\mathbf x}}}{{\tilde{\mathbf x}}}') =: \tilde\Sigma \mbox{ is diagonal} \label{eq:xc}\\
&&{{\mathcal E}}({{\hat{\mathbf x} }}{{\tilde{\mathbf x}}}')=0. \label{eq:xb}\end{aligned}$$ Throughout ${{\mathcal E}}(\cdot)$ denotes the expectation operation and $0$ denotes the zero vector/matrix of appropriate size. The noise-free entries of ${{\hat{\mathbf x} }}$ are assumed to satisfy a set of $q$ simultaneous linear relations. Hence, $M'{{\hat{\mathbf x} }}=0$, with $M\in {{\mathbb R}}^{n\times q}$ and $n>\rank(M)=q>0$. The problem is mainly to infer these relations. Equivalently, ${{\mathcal E}}({{\hat{\mathbf x} }}{{\hat{\mathbf x} }}') =: \hat\Sigma$ has $$\begin{aligned}
&&\rank(\hat\Sigma)= n-q \label{eq:xd}\end{aligned}$$
and $\hat\Sigma M=0$. Statistics are typically estimated from observation records. To this end, consider a sequence $$x_t\in{{\mathbb R}}^{n\times 1},\; t=1,\ldots,T$$ of independent measurements (realizations) of ${{\mathbf x}}$ and, likewise, let $\hat x_t$ and $\tilde x_t$ represent the corresponding values of the noise-free variable and noise components. Denote by $$X=\left[\begin{matrix} x_1\;x_2\; \ldots\; x_T\end{matrix}\right]\in {{\mathbb R}}^{n\times T}$$ the matrix of observations of ${{\mathbf x}}$ and similarly denote by $\hat X$ and $\tilde X$ the corresponding matrices of the noise-free and noise entries, respectively. Data for identifying relations among the noise-free variables are typically limited to the observation matrix $X$ and, neglecting a scaling factor of $1/T$, the data is typically abstracted in the form of a sample covariance $XX^\prime$. For the most part we will assume that sample covariances are accurate approximations of true covariances, and hence the modeling assumptions amount to
$$\begin{aligned}
&& \tilde X \tilde X ^\prime \simeq \mbox{ diagonal}\label{eq:diagonal}\\
&& \hat X \tilde X ^\prime\simeq 0 \label{eq:orthogonality}\\
&&\rank(\hat X) =n-q \label{eq:rank}\end{aligned}$$
since $M^\prime \hat X=0$.
The number of possible linear relations among the noise free variables and the corresponding coefficient matrix need to be determined from either $X$ or $\Sigma$. This motivates the Frisch and Shapiro problems discussed in Section \[sec:Frisch\]. An alternative set of problems can be motivated by the need to determine $\hat X$ from $X$ via suitable decomposition $$\label{eq:decompose}
X=\hat X+\tilde X$$ in a way that is consistent with the existence of a set of $q$ linear relations. We will return to this in Section \[sec:min-max\].
The problems of Frisch and Shapiro {#sec:Frisch}
==================================
We begin with the Frisch problem concerning the decomposition of a covariance matrix $\Sigma$ that is consistent with the assumptions in Section \[sec:datastrcuture\]. The fact that, in practice, $\Sigma$ is an empirical sample covariance motivates relaxing (\[eq:xc\]-\[eq:xd\]) in various ways. In particular, relaxation of the constraint $\tilde \Sigma\geq 0$ leads to the Shapiro problem.
\[problem1\] Given $\Sigma\in{{\mathbf S}}_{n,+}$, determine $$\begin{aligned}
\nonumber
{{\operatorname{mr}}}_+(\Sigma)&:=&\min\{\rank(\hat\Sigma) \mid \Sigma=\tilde \Sigma+\hat \Sigma,\\&&\tilde\Sigma, \hat\Sigma\geq 0,\;\tilde\Sigma \mbox{ is diagonal}\}.\label{eq:mc}\end{aligned}$$
\[problemShapiro\] Given $\Sigma\in{{\mathbf S}}_{n,+}$, determine $$\begin{aligned}
\nonumber
{{\operatorname{mr}}}(\Sigma)&:=&\min\{\rank(\hat\Sigma) \mid \Sigma=\tilde \Sigma+\hat \Sigma,\\&& \hat\Sigma\geq 0,\;\tilde\Sigma \mbox{ is diagonal}\}.\label{eq:mc2}\end{aligned}$$
The Frisch problem was studied by several researchers, see e.g., [@Kalman1985; @Los; @Woodgate1; @woodgate2] and the references therein. On the other hand, Shapiro [@Shapiro] introduced the above relaxed version, removing the requirement that $\tilde \Sigma\geq 0$, in an attempt to gain understanding of the algebraic constraints imposed by the off-diagonal elements of $\Sigma$ on the decomposition. We refer to ${{\operatorname{mr}}}_+(\cdot)$ as the [*Frisch minimum rank*]{} and ${{\operatorname{mr}}}(\cdot)$ as the [*Shapiro minimum rank*]{}. The former is lower semicontinuous whereas the latter is not, as stated next. This difference is crucial if one wants to apply this type of methodology to real data, namely some sort of continuity is necessary.
\[lemma:lowersc\] ${{\operatorname{mr}}}_+(\cdot)$ is lower semicontinuous whereas ${{\operatorname{mr}}}(\cdot)$ is not.
Assume that for a given $\Sigma>0$ there exists a sequence $\Sigma_1,\,\Sigma_2,\,\ldots$ of positive definite matrices such that $\Sigma_i\rightarrow \Sigma$ while $${{\operatorname{mr}}}_+(\Sigma_i)<{{\operatorname{mr}}}_+(\Sigma)=r,\; \mbox{ for all }i=1,\,2,\,\dots.$$ Decompose $\Sigma_i=\hat\Sigma_i+D_i$ with $\rank(\hat\Sigma_i)<r$, $\Sigma_i\geq D_i\geq 0$ and $D_i$ diagonal. Then there exist convergent subsequences $\hat\Sigma_{i_k}\rightarrow \hat\Sigma$ and $D_{i_k}\rightarrow D$, as $k\to \infty$. Since $\Sigma_{i_k}\rightarrow \hat\Sigma+D=\Sigma$, by the lower semicontinuity of the rank, $$\rank(\hat\Sigma)\leq \lim_{k\rightarrow \infty}\inf \rank(\hat\Sigma_{i_k})<r={{\operatorname{mr}}}_+(\Sigma).$$ This is a contradiction. On the other hand, to see that ${{\operatorname{mr}}}(\cdot)$ is not lower semicontinuous consider $$\Sigma= \left[\begin{matrix} 3 & -1 &-1 \\ -1 &3 & 0\\ -1 & 0 &3\end{matrix}\right] \mbox{ and }\Sigma_\epsilon
=\left[\begin{matrix} 3 & -1 &-1 \\ -1 &3 & \epsilon \\ -1 & \epsilon &3\end{matrix}\right],\;\
{\hat\Sigma_\epsilon}=\left[\begin{matrix} \frac{1}{\epsilon} & -1 &-1 \\ -1 &\epsilon & \epsilon\\ -1 & \epsilon &\epsilon\end{matrix}\right]$$ for $\epsilon>0$. Clearly ${{\operatorname{mr}}}(\Sigma)=2$. Also $\lim_{\epsilon\to 0}\Sigma_\epsilon=\Sigma$. Yet $\Sigma_\epsilon=
\hat\Sigma_\epsilon+D_\epsilon$ while $\Sigma_\epsilon$ has rank $1$ and $D_\epsilon$ is diagonal ($\not\geq 0$). Hence ${{\operatorname{mr}}}(\Sigma_\epsilon)=1$.
Assuming that the off-diagonal entries of $\Sigma>0$ of size $n\times n$ are known with absolute certainty, any “minimum rank” (${{\operatorname{mr}}}_+(\cdot)$ and ${{\operatorname{mr}}}(\cdot)$) is bounded below by the so-called Lederman bound, i.e., $$\begin{aligned}
\label{ledermann}
\frac{2n+1-\sqrt{8n+1}}{2}\leq {{\operatorname{mr}}}(\Sigma)\leq {{\operatorname{mr}}}_+(\Sigma),\end{aligned}$$ which holds on a generic set of positive definite matrices $\Sigma$, that is, on a (Zariski open) subset of positive definite matrices. Equivalently, the set of matrices $\Sigma$ for which ${{\operatorname{mr}}}(\Sigma)$ is lower than the Lederman bound is non-generic –their entries satisfy algebraic equations which fail under small perturbation. To see this, consider any factorization $$\Sigma =FF^\prime,$$ with $F\in{{\mathbb R}}^{n\times r}$. There are $(n-r)r + \frac{r(r+1)}{2}$ independent entries in $F$ (when accounting for the action of a unitary transformation of $F$ on the right), whereas the value of the off-diagonal entries of $\Sigma$ impose $\frac{n(n-1)}{2}$ constraints. Thus, the number of independent entries in $F$ exceeds the number of constraints when $(n-r)^2\geq n+r$ which then leads to the inequality $\frac{2n+1-\sqrt{8n+1}}{2}\leq r$. The bound was first noted in [@Ledermann] while the independence of the constraints has been detailed in [@Bekker1997]. In general, the computation of the exact value for ${{\operatorname{mr}}}_+(\Sigma)$ and ${{\operatorname{mr}}}(\Sigma)$ is a non-trivial matter. Thus, it is rather surprising that an exact analytic result is available for both, in the special case when $r=n-1$. We review this next in the form of two theorems.
\[thm:Reiersol\] Let $\Sigma\in {{\mathbf S}}_{n,+}$ and $\Sigma>0$, then $${{\operatorname{mr}}}_+(\Sigma)=n-1 \Leftrightarrow \Sigma^{-1} {\succ_{\hspace*{-1pt}_e}}0.$$
\[thm:Shapiro\] Let $\Sigma\in{{\mathbf S}}_{n,+}$ and irreducible, $${{\operatorname{mr}}}(\Sigma)=n-1\Leftrightarrow \Sigma{\preceq_{\hspace*{-1pt}_e}}0.$$
The characterization of covariance matrices $\Sigma$ for which ${{\operatorname{mr}}}_+(\Sigma)=n-1$ was first recognized by T. C. Koopmans in 1937 [@Koopmans] and proven by Reiersøl [@Reiersol] who used the Perron-Frobenius theory to improve on Koopmans’ analysis. Later on, R. E. Kalman streamlined and completed the steps in [@Kalman1982] relying again on the Perron-Frobenius theorem (see also Klepper and Leamer [@KlepperLeamer] for a detailed analysis). Our treatment below takes a slightly different angle and provides some geometric insight by pointing as a key reason that the maximal number of vectors at an obtuse angle from one another can exceed the dimension of the ambient space by at most one (Corollary \[cor:numberofobtuseangles\]). We provide new proofs where we also utilize a dual formulation with an analogous decomposition of the inverse covariance.
A geometric insight
-------------------
We begin with two basic lemmas for irreducible matrices in $M\in{{\mathbf S}}_{n,+}$. Recall that a matrix is reducible if by permutation of rows and columns can be brought into a block diagonal form, otherwise it is irreducible.
\[lemma:previous\] Let $M>0$ and irreducible. Then, $$\begin{aligned}
\label{eq:first}
M{\preceq_{\hspace*{-1pt}_e}}0 &\Rightarrow & M^{-1}{\succ_{\hspace*{-1pt}_e}}0.\end{aligned}$$
\[lemma:next\] Let $M\geq 0$ and irreducible. Then, $$\begin{aligned}
\label{eq:nullitybound}
M{\preceq_{\hspace*{-1pt}_e}}0
&\Rightarrow & {\rm nullity}(M)\leq 1.\end{aligned}$$
It is easy to verify that for matrices of size $2\times 2$, (\[eq:first\]) holds true. Assume that the statement also holds true for matrices of size up to $k\times k$, for a certain value of $k$, and consider a matrix $M$ of size $(k+1)\times(k+1)$ with $M>0$ and $M{\preceq_{\hspace*{-1pt}_e}}0$. Partition $$M=\left[\begin{matrix}A &b\\b' &c\end{matrix}\right]$$ so that $c$ is a scalar and, hence, $A$ is of size $k\times k$. Partitioning conformably, $$M^{-1}=\left[\begin{matrix}F &g\\g' &h\end{matrix}\right]$$ where $$F=(A-bc^{-1}b')^{-1}, ~g=-A^{-1}bh, \mbox{ and }h=(c-b'A^{-1}b)^{-1}>0.$$
For the case where $A$ is irreducible, because $A$ has size $k\times
k$ and $A{\preceq_{\hspace*{-1pt}_e}}0$, invoking our hypothesis we conclude that $A^{-1}{\succ_{\hspace*{-1pt}_e}}0$. Now, since $b$ has only non-positive entries and $b\neq0$, $g=-A^{-1}bh$ has positive entries. Since $-bc^{-1}b'{\preceq_{\hspace*{-1pt}_e}}0$ and $A{\preceq_{\hspace*{-1pt}_e}}0$, then $A-bc^{-1}b'{\preceq_{\hspace*{-1pt}_e}}0$ is also irreducible. Thus $F=(A-bc^{-1}b')^{-1}$ has positive entries by hypothesis.
For the case where $A$ is reducible, permutation of columns and rows brings $A$ into a block-diagonal form with irreducible blocks. Thus, $A^{-1}$ is also block diagonal matrix with each block entry-wise positive. Because $M$ is irreducible, $b$ must have at least one non-zero entry corresponding to the rows of each diagonal blocks of $A$. Then $A-bc^{-1}b'$ is irreducible and ${\preceq_{\hspace*{-1pt}_e}}0$. Also $A^{-1}b$ has all of its entries negative. Therefore $F=(A-bc^{-1}b')^{-1}$ and $g=-A^{-1}bh$ have positive entries. Therefore $M^{-1}{\succ_{\hspace*{-1pt}_e}}0$.
Rearrange rows and columns and partition $$M=\left[\begin{matrix}A &B\\B' &C\end{matrix}\right]$$ so that $A$ is nonsingular and of maximal size, equal to the rank of $M$. Then $$\label{eq:equality}
C=B'A^{-1}B.$$
We first show that $B'A^{-1}B{\succeq_{\hspace*{-1pt}_e}}0$. Assume that $A$ is irreducible. Then $A^{-1}{\succ_{\hspace*{-1pt}_e}}0$. At the same time $B$ has negative entries and not all zero (since $M$ is irreducible). In this case, $B'A^{-1}B{\succ_{\hspace*{-1pt}_e}}0$. If on the other hand $A$ is reducible, Lemma \[lemma:previous\] applied to the (irreducible) blocks of $A$ implies that $A^{-1}{\succeq_{\hspace*{-1pt}_e}}0$. Therefore, in this case, $B'A^{-1}B{\succeq_{\hspace*{-1pt}_e}}0$.
Returning to and in view of the fact that $C{\preceq_{\hspace*{-1pt}_e}}0$ while $B'A^{-1}B{\succeq_{\hspace*{-1pt}_e}}0$ we conclude that, either $C$ is a scalar (and hence there are no off-diagonal negative entries), or both $C$ and $B'A^{-1}B$ are diagonal. The latter contradicts the assumption that $M$ is irreducible. Hence, the nullity of $M$ can be at most $1$.
Lemma \[lemma:next\] provides the following geometric insight, stated as a corollary.
\[cor:numberofobtuseangles\] In any Euclidean space of dimension $n$, there can be at most $n+1$ vectors forming an obtuse angle with one another.
The Grammian $M=[v_k'v_\ell]_{k,\ell=1}^{n+q}$ of a selection $\{v_k\mid k=1,\ldots, n+q\}$ of such vectors has off-diagonal entries which are negative. Hence, by Lemma \[lemma:next\], the nullity of $M$ cannot exceed $1$.
The necessity part of Theorem \[thm:Shapiro\] is also a direct corollary of Lemma \[lemma:next\].
\[prop:weaker\] Let $\Sigma\in{{\mathbf S}}_{n,+}$ and irreducible. Then $$\Sigma {\preceq_{\hspace*{-1pt}_e}}0 \Rightarrow{{\operatorname{mr}}}(\Sigma)=n-1.$$
Let $\Sigma =\hat \Sigma+\tilde\Sigma$, with $\tilde\Sigma$ diagonal and $\hat\Sigma\geq 0$. $\hat\Sigma$ is irreducible since $\Sigma$ is irreducible. From Lemma \[lemma:next\], the nullity of $\hat\Sigma$ is at most $1$. Thus ${{\operatorname{mr}}}(\Sigma)=n-1$.
A dual decomposition
--------------------
The matrix inversion lemma provides a correspondence between an additive decomposition of a positive-definite matrix and a decomposition of its inverse, albeit with a different sign in one of the summands. This is stated next.
\[lemma:decompositions\] Let $$\label{eq:first_decomposition}
\Sigma=D+FF'$$ with $\Sigma,D\in{{\mathbf S}}_{n,+}$, with $\Sigma,D>0$ and $F\in{{\mathbb R}}^{n\times r}$. Then $$\label{eq:second_decomposition}
S:=\Sigma^{-1} = E - GG'$$ for $E=D^{-1}$ and $G=D^{-1}F(I+F'D^{-1}F)^{-1/2}$. Conversely, if (\[eq:second\_decomposition\]) holds with $G\in{{\mathbb R}}^{n\times r}$, then so does (\[eq:first\_decomposition\]) for $D=E^{-1}$ and $F=E^{-1}G(I-G'E^{-1}G)^{-1/2}$.
This follows from the identity $(I\pm MM')^{-1}=I\mp M(I\mp M'M)^{-1}M'$.
Application of the lemma suggests the following variation to Frisch’s problem.
\[problem2\] Given a positive-definite $n\times n$ symmetric matrix $S$ determine the [*dual minimum rank*]{}: $$\begin{aligned}
\nonumber
{{\operatorname{mr_{dual}}}}(S)&:=&\min\{\rank(\hat S \mid S=E -\hat S,\\&& \hat S,E\geq 0,\;E \mbox{ is diagonal}\nonumber\}.\label{eq:mcdual}\end{aligned}$$
Clearly, if $S=\Sigma^{-1}=E-GG^\prime$ (as in (\[eq:second\_decomposition\])), then $E>0$. Furthermore, a decomposition of $S$ always gives rise to a decomposition $\Sigma=D+FF^\prime$ (as in (\[eq:first\_decomposition\])) with the terms $FF'$ and $GG'$ having the same rank. Thus, it is clear that $$\label{eq:inequality}
{{\operatorname{mr}}}_+(\Sigma)\leq {{\operatorname{mr_{dual}}}}(\Sigma^{-1}),$$ and that the above holds with equality when an optimal choice of $D\equiv\tilde \Sigma$ in (\[eq:mc\]) is invertible. However, if $D$ is allowed to be singular, the rank of the summands $FF'$ and $GG'$ may not agree. This is can be seen using the following example. Take $$\Sigma=\left[\begin{matrix}
2&1&1\\
1&2&1\\
1&1&1\end{matrix}\right].$$ It is clear that $\Sigma$ admits a decomposition $\Sigma=\tilde\Sigma+\hat\Sigma$, in correspondence with (\[eq:first\_decomposition\]), where $\tilde\Sigma=D=\diag\{1,1,0\}$ while $\hat\Sigma=FF'$ as well as $F'=[1,\,1,\,1]$ are of rank one. On the other hand, $$S=\Sigma^{-1}=\left[\begin{matrix}
\;\;1&\;\;0&-1\\
\;\;0&\;\;1&-1\\
-1&-1&\;\;3\end{matrix}\right].$$ Taking $E=\diag\{e_1,\;e_2,\;e_3\}$ in (\[eq:second\_decomposition\]), it is evident that the rank of $$GG'=E-S=\left[\begin{matrix}
e_1-1&0&1\\
0&e_2-1&1\\
1&1&e_3-3\end{matrix}\right]$$ cannot be less than $2$ without violating the non-negativity assumption for the summand $GG'$. The minimal rank for the factor $G$ is $2$ and is attained by taking $e_1=e_2=2$ and $e_3=5$.
On the other hand, in general, if we perturb $\Sigma$ to $\Sigma+\epsilon I$ and, accordingly, $D$ to $D+\epsilon I$, then $$\label{eq:inequality2}
{{\operatorname{mr_{dual}}}}((\Sigma+\epsilon I)^{-1})\leq {{\operatorname{mr}}}_+(\Sigma),~ \forall \epsilon>0.$$ Equality in holds for sufficiently small value of $\epsilon$. Thus, ${{\operatorname{mr}}}_+$ and ${{\operatorname{mr_{dual}}}}$ are closely related. However, it should be noted that ${{\operatorname{mr_{dual}}}}(\cdot)$ fails to be lower semi-continuous since a small perturbation of the off-diagonal entries can reduce ${{\operatorname{mr_{dual}}}}(\cdot)$. Yet, interestingly, an exact characterization of the ${{\operatorname{mr_{dual}}}}(S)=n-1$ can be obtained which is analogous to those for ${{\operatorname{mr}}}_+$ and ${{\operatorname{mr}}}$ being equal to $n-1$; the condition for ${{\operatorname{mr_{dual}}}}$ will be used to prove the Reiersøl and Shapiro theorems.
\[thm:dualreiersol\] For $S\in{{\mathbf S}}_{n,+}$, with $S>0$ and irreducible, $$\label{dualreiersol}
{{\operatorname{mr_{dual}}}}(S)=n-1 \Leftrightarrow S {\succeq_{\hspace*{-1pt}_e}}0.$$
If $S{\succeq_{\hspace*{-1pt}_e}}0$ and $E$ is diagonal satisfying $E\geq S>0$, then $E-S=GG'{\preceq_{\hspace*{-1pt}_e}}0$. By invoking Lemma \[lemma:next\] we deduce that if $E-S$ is singular, $\rank(G)=n-1$. Hence, ${{\operatorname{mr_{dual}}}}(S)=n-1$.
To establish that ${{\operatorname{mr_{dual}}}}(S)=n-1\Rightarrow S {\succeq_{\hspace*{-1pt}_e}}0$, we assume that the condition $S{\succeq_{\hspace*{-1pt}_e}}0$ fails and show that ${{\operatorname{mr_{dual}}}}(S)<n-1$. We first argue the case for a $3\times 3$ matrix $S=[s_{ij}]_{i,j=1}^3$. Provided $S\not{\succeq_{\hspace*{-1pt}_e}}0$ we can assume that it has strictly negative off-diagonal entries (which can be done by reflecting the signs of rows and columns). We now let $$\begin{aligned}
e_{i}&=&s_{ii}-\frac{s_{ij}s_{ki}}{s_{jk}}\end{aligned}$$ for $i\in\{1,2,3\}$ and $(i,j,k)$ being permutations of $(1,2,3)$. These are all positive. Let $\tilde S=\diag^*(e_1,e_2,e_3)$. It can be seen that $\tilde S-S\geq0$ while $\rank(\tilde S-S)=1$. To verify the latter observe that $\tilde S-S=vv'$ for $$v'=\left[\begin{matrix}
\sqrt{e_1-s_{11}},&
\sqrt{e_2-s_{22}},&
\sqrt{e_3-s_{33}}\end{matrix}\right].$$ This establishes the reverse implication for matrices of size $3\times 3$.
We now assume that the statement holds true for matrices of size up to $(n-1)\times (n-1)$ for some $n\geq 4$ and use induction. So let $S,\;\tilde S$ be of size $n\times n$ with $S\not{\succeq_{\hspace*{-1pt}_e}}0$ and $\tilde S$ diagonal. We need to prove that ${{\operatorname{mr_{dual}}}}(S)<n-1$. We partition $$S=\left[\begin{matrix} A&b\\ b'&c \end{matrix}
\right]
,\; \tilde S=\left[\begin{matrix} E&0\\ 0& e\end{matrix}\right]$$ with $A,\;E$ being $(n-1)\times (n-1)$. For any $\tilde S$ such that $\tilde S-S\geq 0$, $e$ cannot be equal to $c$, otherwise $b=0$ and $S$ is reducible. Further, $\tilde S-S\geq 0$ if and only if $e>c$ and $$M:=E-(A+b(e-c)^{-1}b')\geq0.$$ The nullity of $\tilde S-S$ coincides with that of $M$. To prove our claim, it suffices to show that $A_e:=A+b(e-c)^{-1}b'\not{\succeq_{\hspace*{-1pt}_e}}0$, or that $A_e$ is reducible for some $e>c$. (Since, in either case, by our hypothesis, the nullity of $M$ for a suitable $E$ exceeds $1$.)
We now consider two possible cases where $S{\succeq_{\hspace*{-1pt}_e}}0$ fails. First, we consider the case where already $A\not {\succeq_{\hspace*{-1pt}_e}}0$. Then obviously $A_e\not{\succeq_{\hspace*{-1pt}_e}}0$ for $e-c$ sufficiently large. The second possibility is $S\not {\succeq_{\hspace*{-1pt}_e}}0$ while $A{\succeq_{\hspace*{-1pt}_e}}0$. But if $A$ is (transformed into) element-wise nonnegative, then $bb'$ must have at least one pair of negative off-diagonal entries. Then, consider $A_e=A+\lambda bb'$ for $\lambda=(e-c)^{-1}\in(0,\infty)$. Evidently, for certain values of $\lambda$ entries of $A_e$ change sign. If a whole row becomes zero for a particular value of $\lambda$, then $A_e$ is reducible. In all other cases, there are values of $\lambda$ for which $A_e\not{\succeq_{\hspace*{-1pt}_e}}0$. This completes the proof.
Proof of Reiers[ø]{}l’s theorem (Theorem \[thm:Reiersol\]) {#sec:proofReiersol}
----------------------------------------------------------
We first show that $\Sigma^{-1}{\succ_{\hspace*{-1pt}_e}}0$ implies ${{\operatorname{mr}}}_+(\Sigma)=n-1$. From the continuity of the inverse, $(\Sigma+\epsilon I)^{-1}{\succ_{\hspace*{-1pt}_e}}0$ for sufficiently small $\epsilon>0$. Applying Theorem \[thm:dualreiersol\], we conclude that $${{\operatorname{mr_{dual}}}}((\Sigma+\epsilon I)^{-1})=n-1.$$ Since ${{\operatorname{mr}}}_+(\Sigma)\geq {{\operatorname{mr_{dual}}}}((\Sigma+\epsilon I)^{-1})$ as in , we conclude that ${{\operatorname{mr}}}_+(\Sigma)=n-1$.
To prove that ${{\operatorname{mr}}}_+(\Sigma) =n-1\Rightarrow \Sigma^{-1}{\succ_{\hspace*{-1pt}_e}}0$, we show that assuming $\Sigma^{-1}\not{\succ_{\hspace*{-1pt}_e}}0$ and ${{\operatorname{mr}}}_+(\Sigma)
=n-1$ together leads to a contradiction. From the continuity of the inverse and the lower semicontinuity of ${{\operatorname{mr}}}_+(\cdot)$ (Proposition \[lemma:lowersc\]), there exists a symmetric matrix $\Delta$ and an $\epsilon>0$ such that $$(\Sigma+\epsilon \Delta)^{-1} \not {\succeq_{\hspace*{-1pt}_e}}0, \text{~and~} {{\operatorname{mr}}}_+(\Sigma+\epsilon \Delta)=n-1.$$ Then, from Theorem \[thm:dualreiersol\], $
{{\operatorname{mr_{dual}}}}((\Sigma+\epsilon \Delta)^{-1})< n-1
$ while from $${{\operatorname{mr}}}_+(\Sigma+\epsilon \Delta) \leq {{\operatorname{mr_{dual}}}}((\Sigma+\epsilon \Delta)^{-1}).$$ Thus, we have a contradiction and therefore $\Sigma^{-1}{\succ_{\hspace*{-1pt}_e}}0$. $\Box$
Proof of Shapiro’s theorem (Theorem \[thm:Shapiro\]) {#sec:proofShapiro}
----------------------------------------------------
Given $\Sigma\geq 0$ consider $\lambda>0$ such that $\lambda I-\Sigma\geq0$, a diagonal $D$, and let $E:=\lambda I-D$. Since $\Sigma-D=E-(\lambda I -\Sigma)$, $$\begin{aligned}
\label{eq:mrmrdual}
{{\operatorname{mr}}}(\Sigma)={{\operatorname{mr_{dual}}}}(\lambda I-\Sigma).\end{aligned}$$ If $\Sigma$ is irreducible and $\Sigma{\preceq_{\hspace*{-1pt}_e}}0$, then $\lambda I-\Sigma$ is irreducible and $\lambda I-\Sigma{\succeq_{\hspace*{-1pt}_e}}0$. It follows (Theorem \[thm:dualreiersol\]) that ${{\operatorname{mr_{dual}}}}(\lambda I-\Sigma)=n-1$, and therefore ${{\operatorname{mr}}}(\Sigma)=n-1$ as well.
For the the reverse direction, if ${{\operatorname{mr}}}(\Sigma)=n-1$ then ${{\operatorname{mr_{dual}}}}(\lambda I-\Sigma)=n-1$, which implies that $\lambda I-\Sigma{\succeq_{\hspace*{-1pt}_e}}0$ and therefore that $\Sigma{\preceq_{\hspace*{-1pt}_e}}0$. $\Box$
The original proof in [@Shapiro1982b] claims that for any $\Sigma\geq 0$ of size $n\times n$ with $n>3$ and $\Sigma\not {\preceq_{\hspace*{-1pt}_e}}0$, there exists a $(n-1)\times (n-1)$ principle minor that is $\not{\preceq_{\hspace*{-1pt}_e}}0$. This statement fails for the following sign pattern $$\footnotesize{
\left[\begin{matrix}+&0&-&-\\0&+&-&+\\-&-&+&0\\-&+&0&+ \end{matrix} \right].}$$ This matrix can not transformed to have all nonpositive off-diagonal entries, yet all its $3\times 3$ principle minors ${\preceq_{\hspace*{-1pt}_e}}0$.
Parametrization of solutions under Reiers[ø]{}l’s and Shapiro’s conditions {#section:parametrization}
--------------------------------------------------------------------------
For either the Frisch or the Shapiro problem, a solution is not unique in general. The parametrization of solutions to the Frisch problem when ${{\operatorname{mr}}}_+(\Sigma)=n-1$ has been known and is briefly explained below (without proof). Interestingly, an analogous parametrization is possible for Shapiro’s problem and this is given in Proposition \[shapiro\_parametrization\] that follows, and both are presented here for completeness of the exposition.
\[lemma:parameter1\] Let $\Sigma\in{{\mathbf S}}_{n,+}$ with $\Sigma>0$ and $\Sigma^{-1}{\succ_{\hspace*{-1pt}_e}}0$. The following hold:
- For $D\geq0$ diagonal with $\Sigma-D\geq0$ and singular, there is a probability vector $\rho$ ($\rho$ has entries $\geq 0$ that sum up to $1$) such that $(\Sigma-D)\Sigma^{-1}\rho=0$.
- For any probability vector $\rho$, $$D=\diag^*\left(\left[\frac{[\rho]_i}{[\Sigma^{-1}\rho]_i}, i=1,\ldots, n\right] \right)$$ satisfies $\Sigma-D\geq0$ and $\Sigma-D$ is singular.
See [@Kalman1982; @KlepperLeamer].
Thus, solutions of Frisch’s problem under Reiers[ø]{}l’s conditions are in bijective correspondence with probability vectors. A very similar result holds true for Shapiro’s problem.
\[shapiro\_parametrization\] Let $\Sigma\in{{\mathbf S}}_{n,+}$ be irreducible and have $\leq 0$ off-diagonal entries. The following hold:
- For $D$ diagonal with $\Sigma-D\geq0$ and singular, there is a strictly positive vector $v$ such that $(\Sigma-D)v=0$.
- For any strictly positive vector $v\in {{\mathbb R}}^{n\times 1}$, $$\begin{aligned}
\label{eq:Dshapiro}
D=\diag^*\left(\left[\frac{[\Sigma v]_i}{[v]_i}, i=1,\ldots, n\right] \right)\end{aligned}$$ satisfies that $\Sigma-D\geq0$ and $\Sigma-D$ is singular.
To prove $(i)$, we note that if $(\Sigma-D)v=0$, then $v{\succ_{\hspace*{-1pt}_e}}0$. To see this consider $(\Sigma-D+\epsilon I)^{-1}$ for $\epsilon>0$. From Lemma \[lemma:previous\], $$(\Sigma-D+\epsilon I)^{-1}{\succ_{\hspace*{-1pt}_e}}0$$ and since $v$ is an eigenvector corresponding to its largest eigenvalue, a power iteration argument concludes that $v{\succ_{\hspace*{-1pt}_e}}0$.
To prove $ii)$, it is easy to verify that the diagonal matrix $D$ in for $v{\succ_{\hspace*{-1pt}_e}}0$ satisfies $(\Sigma-D)v=0$. We only need to prove that $\Sigma-D\geq0$. Without loss of generality we assume that all the entries of $v$ are equal. (This can always be done by scaling the entries of $v$ and scaling accordingly rows and columns of $\Sigma$.) Since $v$ is a null vector of $\Sigma-D$ and since $M:=\Sigma-D$ has $\leq 0$ off-diagonal entries $$[M]_{ii}=\sum_{j\neq i}|[M]_{ij}|.$$ Gersgorin Circle Theorem (e.g., see [@Varga2004]) now states that every eigenvalue of $M$ lies within at least one of the closed discs $\left\{{\rm Disk}\left([M]_{ii}, \sum_{j\neq i}|[M]_{ij}| \right), i=1, \ldots, n\right\}$. No disc intersects the negative real line. Therefore $\Sigma-D\geq0$.
Decomposition of complex-valued matrices
----------------------------------------
Complex-valued covariance matrices are commonly used in radar and antenna arrays [@vantrees]. The rank of $\Sigma-D$, for noise covariance $D$ as in the Frisch problem, is an indication of the number of (dominant) scatterers in the scattering field. If this is of the same order as the number of array elements (e.g., $n-1$), any conclusion about their location may be suspect. Thus, it is natural to seek conditions for ${{\operatorname{mr}}}_+(\Sigma)=n-1$ analogous to those given by Reiers[ø]{}l, for the case of complex covariances, as a possible warning. This we do next.
Consider complex-valued observation vectors $
x_t=y_t+ {{\rm i}}z_t,~ t=1,\ldots T,
$ where ${{\rm i}}=\sqrt{-1}$ and $y_t, z_t \in {{\mathbb R}}^{n\times 1}$, and set $$X=[x_1,\; \ldots x_T]=Y+ {{\rm i}}Z$$ with $Y=[y_1,\; \ldots y_T]$, $Z=[z_1,\; \ldots z_T]$. The (scaled) sample covariance is $$\begin{aligned}
\Sigma=XX^*
&=\Sigma_{\rm r}+{{\rm i}}\Sigma_{\rm i}\in{{\mathbf H}}_{n,+},\end{aligned}$$ where the real part $\Sigma_{\rm r}:=YY'+ZZ'$ is symmetric, the imaginary part $\Sigma_{\rm i}:=ZY'-YZ'$ is anti-symmetric, and “$*$” denotes complex-conjugate transpose. As before, we consider a decomposition $$\Sigma=\hat\Sigma+D$$ with $\hat\Sigma\geq 0$ singular and $D\geq 0$ diagonal. We refer to [@Anderson1988; @Deistler1989] for the special case where ${{\operatorname{mr}}}_+(\Sigma)=1$. In this section we present a sufficient condition for a Reiers[ø]{}l-case where ${{\operatorname{mr}}}_+(\Sigma)=n-1$.
Before we proceed we note that re-casting the problem in terms of the real-valued $$R:=\left[
\begin{array}{cc}
\Sigma_{\rm r} & \Sigma_{\rm i} \\
\Sigma_{\rm i}^\prime & \Sigma_{\rm r} \\
\end{array}
\right]\in{{\mathbf S}}_{2n,+}$$ does not allow taking advantage of earlier results. The structure of $R$ with antisymmetric off-diagonal blocks implies that if $[a',\;b']'$ is a null vector then so is $[-b',\;a']'$ (since, accordingly, $a+ {{\rm i}}b$ and ${{\rm i}}a - b$ are both null vectors of $\Sigma$). Thus, in general, the nullity of $R$ is not $1$ and the theorem of Reiers[ø]{}l is not applicable. Further, the corresponding noise covariance is diagonal with repeated blocks.
The following lemmas for the complex case echo Lemma \[lemma:previous\] and Lemma \[lemma:next\].
\[lemma:complexprevious\] Let $M\in{{\mathbf H}}_{n,+}$ be irreducible. If the argument of each non-zero off-diagonal entry of $-M$ is in $\left(-\frac{\pi}{2^n},~ \frac{\pi}{2^n} \right)$, then each entry of $M^{-1}$ has argument in $\left(-\frac{\pi}{2}+\frac{\pi}{2^n}, ~ \frac{\pi}{2}-\frac{\pi}{2^n}\right)$.
It is easy to verify the lemma for $2\times 2$ matrices. Assume that the statement holds for sizes up to $n\times n$ and consider an $(n+1)\times (n+1)$ matrix $M$ that satisfies the conditions of the lemma. Partition $$M=\left[
\begin{array}{cc}
A & b \\
b^* & c \\
\end{array}
\right]$$ with $A$ is of size $n\times n$, and conformably, $$M^{-1}=\left[
\begin{array}{cc}
F & g \\
g^* & h \\
\end{array}
\right].$$ By assumption non-zero entries of $-A$ and $-b$ have their argument in $\left(-\frac{\pi}{2^{n+1}}, ~\frac{\pi}{2^{n+1}}\right)$. Then, by bounding the possible contribution of the respective terms, it follows that for the argument of each of the entries of $-A+bc^{-1}b^*$ is in $\left(-\frac{\pi}{2^n}, ~\frac{\pi}{2^n}\right)$. Then, the argument of each entry of $F=(A-bc^{-1}b^*)^{-1}$ is in $\left(-\frac{\pi}{2}+\frac{\pi}{2^n}, ~ \frac{\pi}{2}-\frac{\pi}{2^n}\right)$; this follows by assumption since $F$ is $n\times n$. Clearly, $\left(-\frac{\pi}{2}+\frac{\pi}{2^n}, ~ \frac{\pi}{2}-\frac{\pi}{2^n}\right) \subset \left(-\frac{\pi}{2}+\frac{\pi}{2^{n+1}}, ~\frac{\pi}{2}-\frac{\pi}{2^{n+1}}\right)$. Regarding $g$, by bounding the possible contribution of respective terms, we similarly conclude that the argument of each of its non-zero entries is in $\left(-\frac{\pi}{2}+\frac{\pi}{2^{n+1}}, ~ \frac{\pi}{2}-\frac{\pi}{2^{n+1}}\right)$.
\[lemma:complexnext\] Let $M\in{{\mathbf H}}_{n,+}$ be irreducible. If the argument of each non-zero off-diagonal entry of $-M$ is in $\left(-\frac{\pi}{2^n},~\frac{\pi}{2^n}\right)$, then $\rank(M)\geq n-1$.
First rearrange rows and columns of $M$, and partition as $$M=\left[
\begin{array}{cc}
A & B \\
B^* & C \\
\end{array}
\right]$$ so that $A$ is nonsingular and of size equal to the rank of $M$, which we denote by $r$. Then $$\label{eq:CBAB}
C=B^*A^{-1}B$$ and has size equal to the nullity of $M$. We now compare the argument of the off-diagonal entries of $C$ and $B^*A^{-1}B$, and show they cannot be equal unless $C$ is a scalar. Since the off-diagonal entries of $-A$ have their argument in $\left(-\frac{\pi}{2^n}, ~\frac{\pi}{2^n}\right)\subset
\left(-\frac{\pi}{2^r}, ~\frac{\pi}{2^r}\right)$, the off-diagonal entries of $A^{-1}$ have their argument in $\left(-\frac{\pi}{2}+\frac{\pi}{2^r}, ~
\frac{\pi}{2}-\frac{\pi}{2^r}\right)$ from Lemma \[lemma:complexprevious\]. Now, the $(k,\ell)$ entry of $B^*A^{-1}B$ is $$\begin{aligned}
[B^*A^{-1}B]_{k\ell}=\sum_{i,j}[B^*]_{ki}[A^{-1}]_{ij}[B]_{j \ell}\end{aligned}$$ and the phase of each summand is $$\arg([B^*]_{ki}[A^{-1}]_{ij} [B]_{j \ell}) \in\left(-\frac{\pi}{2}+\frac{\pi}{2^r}-\frac{\pi}{2^{n-1}},~ \frac{\pi}{2}-\frac{\pi}{2^r}+\frac{\pi}{2^{n-1}}\right).$$ Thus, the non-zero off-diagonal entries of $B^*A^{-1}B$ have positive real part while $$\arg(-[C]_{k\ell})\in \left(-\frac{\pi}{2^n},~\frac{\pi}{2^n}\right) .$$ Hence, either the off-diagonal entries of $B^*A^{-1}B$ and $C$ are zero, in which case these are diagonal matrices and $M$ must be reducible, or $B^*A^{-1}B$ and $C$ are both scalars. This concludes the proof.
\[prop:complexReiersol\] Let $\Sigma\in{{\mathbf H}}_{n,+}$ be irreducible. If the argument of each non-zero off-diagonal entry of $-\Sigma$ is in $\left(-\frac{\pi}{2^n},~ \frac{\pi}{2^n}\right)$, then ${{\operatorname{mr}}}(\Sigma)=n-1$.
The matrix $\Sigma-D$ is irreducible since $D$ is diagonal. If $\Sigma-D\geq0$ and singular, and since the argument of each non-zero off-diagonal entry of $-(\Sigma-D)$ is in $\left(-\frac{\pi}{2^n},~ \frac{\pi}{2^n}\right)$, Lemma \[lemma:complexnext\] applies and gives that $\rank(\Sigma-D)=n-1$.
Clearly, since ${{\operatorname{mr}}}_+(\Sigma)\geq{{\operatorname{mr}}}(\Sigma)$, under the condition of Theorem \[prop:complexReiersol\], ${{\operatorname{mr}}}_+(\Sigma)=n-1$. It is also clear that for $S\in{{\mathbf H}}_{n,+}$ irreducible with all non-zero off-diagonal entries having argument in $\left(-\frac{\pi}{2^n},~ \frac{\pi}{2^n}\right)$, we also conclude that ${{\operatorname{mr_{dual}}}}(S)=n-1$.
Trace minimization heuristics {#sec:MinTrace}
=============================
The rank of a matrix is a non-convex function of its elements and the problem to find the matrix of minimal rank within a given set is a difficult one, in general. Therefore, certain heuristics have been developed over the years to obtain approximate solutions. In particular, in the context of factor analysis, trace minimization has been pursued as a suitable heuristic [@Ledermann1940; @Shapiro; @Shapiro1982b] thereby relaxing the Frisch problem into $$\begin{aligned}
\nonumber
&\min_{D: \Sigma\geq D\geq0} {{\operatorname{trace}}}(\Sigma-D),\end{aligned}$$ for a diagonal matrix $D$; with a relaxation of $D\geq 0$ corresponding to Shapiro’s problem. The theoretical basis for using the trace and, more generally, the nuclear norm for non-symmetric matrices, as a surrogate for the rank was provided by Fazel [*etal.*]{} [@Fazel2001] who proved that these constitute convex envelops of the rank function on bounded sets of matrices.
The relation between minimum trace factor analysis and minimum rank factor analysis goes back to Ledermann in [@Ledermann1939] (see [@Della1982] and [@Saunderson2012]). Herein we only refer to two propositions which characterize minimizers for the two problems, Frisch’s and Shapiro’s, respectively.
\[prop:mintrace\] Let $\Sigma=\hat\Sigma_1+D_1>0$ for a diagonal $D_1\geq0$. Then, $$\begin{aligned}
\label{trmc}
&(\hat\Sigma_1,D_1)=\arg\min\{ {{\operatorname{trace}}}(\hat\Sigma) \mid \Sigma=\hat\Sigma+D>0,\;\hat\Sigma\geq 0,\;\mbox{diagonal }D\geq 0\}\\
& \Leftrightarrow~ \exists~ \Lambda_1 \geq0 ~:~ \hat\Sigma_1 \Lambda_1=0 \text{~and~} \left\{
\begin{array}{ll}
[\Lambda_1]_{ii}=1, & \text{~if~} [D_1]_{ii}>0, \\
\left[ \Lambda_1\right]_{ii}\geq1, &\text{~if~}[D_1]_{ii}=0.\nonumber
\end{array}
\right.\end{aligned}$$
\[prop:MTFAShapiro\] Let $\Sigma=\hat\Sigma_2+D_2>0$ for a diagonal $D_2$. Then,$$\begin{aligned}
\label{trmc2}
&(\hat\Sigma_2,D_2)=\arg\min\{ {{\operatorname{trace}}}(\hat\Sigma) \mid \Sigma=\hat\Sigma+D>0,\;\hat\Sigma\geq 0,\;\mbox{diagonal }D\}\\
& \Leftrightarrow~\exists~ \Lambda_2 \geq0 ~:~ \hat\Sigma_2 \Lambda_2=0 \text{~and~} [\Lambda_2]_{ii}=1~ \forall i.\nonumber\end{aligned}$$
Evidently, when the solutions to these two problems differ and $D_1\neq D_2$, then there exists $k\in\left\{1, \ldots, n\right\}$ such that $$[D_2]_{kk}<0 \text{~and~} [D_1]_{kk}=0.$$ Further, the essence of Proposition \[prop:MTFAShapiro\] is that a singular $\hat\Sigma$ originates from such a minimization problem if and only if there is a correlation matrix in its null space. The matrices $\Lambda_1$ and $\Lambda_2$ appear as Lagrange multipliers in the respective problems.
[Factor analysis is closely related to [*low-rank matrix completion*]{} as well as to [*sparse and low-rank decomposition*]{} problems. Typically, low-rank matrix completion asks for a matrix $X$ which satisfies a linear constraint ${{\mathcal A}}(X)=b$ and has low/minimal rank (${{\mathcal A}}(\cdot)$ denotes a linear map ${{\mathcal A}}\,:\,{{{\mathbb R}}}^{n\times n}\rightarrow {{{\mathbb R}}}^p$). Thus, factor analysis corresponds to the special case where ${{\mathcal A}}(\cdot)$ maps $X$ onto its off-diagonal entries. In a recent work by Recht [*etal.*]{} [@Recht2010guaranteed], the nuclear norm of $X$ was considered as a convex relaxation of $\rank(X)$ for such problems and a sufficient condition for exact recovery was provided. However, this sufficient condition amounts to the requirement that the null space of ${{\mathcal A}}(\cdot)$ contains no matrix of low-rank. Therefore, since in factor analysis diagonal matrices are in fact contained in the null space of ${{\mathcal A}}(\cdot)$ and include matrices of low-rank, the condition in [@Recht2010guaranteed] does not apply directly. Other works on low-rank matrix completion (see, e.g., [@Recht2010guaranteed; @Candes2009exact]) mainly focus on assessing the probability of exact recovery and on constructing efficient computational algorithms for [*large-scale*]{} low-rank completion problems [@Keshavan2010matrix; @Keshavan2010noisy]. On the other hand, since diagonal matrices are sparse (most of their entries are zero), the work on matrix decomposition into sparse and low-rank components by Chandrasekaran [*etal.*]{} [@Chandrasekaran2011rank] is very pertinent. In this, the $\ell_1$ and nuclear norms were used as surrogates for sparsity and rank, respectively, and a sufficient condition for exact recovery was provided which captures a certain “rank-sparsity incoherence”; an analogous but stronger sufficient “incoherence” condition which applies to problem is given in [@Saunderson2012].]{}
Weighted minimum trace factor analysis
--------------------------------------
Both ${{\operatorname{mr}}}(\Sigma)$ and ${{\operatorname{mr}}}_+(\Sigma)$ in and , respectively, remain invariant under scaling of rows and the corresponding columns of $\Sigma$ by the same coefficients. On the other hand, the minimizers in and and their respective ranks are not invariant under scaling. This fact motivates weighted-trace minimization, $$\begin{aligned}
\label{eq:Dw}
\min\left\{ {{\operatorname{trace}}}(W\hat\Sigma) \mid \Sigma=\hat\Sigma+D,~\hat\Sigma\geq 0,~\mbox{diagonal }D\geq 0 \right\},\end{aligned}$$ given $\Sigma>0$ and a diagonal weight $W>0$. As before the characterization of minimizers relates to a suitable condition for the corresponding Lagrange multipliers:
\[prop:WMTFAShapiro\] Let $\Sigma=\hat\Sigma_0+D_0>0$ for a diagonal matrix $D_0\geq0$ and consider a diagonal $W>0$. Then, $$\begin{aligned}
\label{trmc3}
&(\hat\Sigma_0,D_0)=\arg\min\{ {{\operatorname{trace}}}(W\hat\Sigma) \mid \Sigma=\hat\Sigma+D>0,\;\hat\Sigma\geq
0,\;\mbox{diagonal }D\geq 0\}\\
& \Leftrightarrow~ \exists~ \Lambda_0 \geq0 ~:~
\hat\Sigma \Lambda_0=0 \text{~and~} \left\{
\begin{array}{ll}
[\Lambda_0]_{ii}=[W]_{ii}, & \text{~if~} [D_0]_{ii}>0, \\
\left[ \Lambda_0\right]_{ii}\geq [W]_{ii}, &\text{~if~}[D_0]_{ii}=0.\nonumber
\end{array}
\right.\nonumber\end{aligned}$$
A corresponding sufficient and necessary condition for $(\hat\Sigma, D)$ to be a minimizer in Shapiro’s problem is that there exists a Grammian in the null space of $\hat\Sigma$ whose diagonal entries are equal to the diagonal entries of $W$.
Minimum-rank solutions may be recovered as solutions to using suitable choices of weight. However, these choices depend on $\Sigma$ and are not known in advance –this motivates a selection of certain canonical $\Sigma$-dependent weight as well as iteratively improving the choice of weight. One should note that since $D$ is diagonal, letting $W$ be a not-necessarily diagonal matrix does not change the problem –only the diagonal entries of $W$ determine the minimizer.
We first consider taking $W=\Sigma^{-1}$. A rationale for this choice is that the minimal value in bounds ${{\operatorname{mr}}}_+(\Sigma)$ from below, since for any decomposition $\Sigma=\hat\Sigma+D$, $$\begin{aligned}
\nonumber
\rank(\hat \Sigma) =&~ {{\operatorname{trace}}}(\hat\Sigma^\sharp \hat\Sigma)\\\nonumber
\geq&~ {{\operatorname{trace}}}((\hat\Sigma+D)^{-1} \hat\Sigma)\\
=&~ {{\operatorname{trace}}}(\Sigma^{-1} \hat\Sigma)\label{eq:rankjustify}\end{aligned}$$ where $^\sharp$ denotes the Moore-Penrose pseudo inverse. Continuing with this line of analysis $$\begin{aligned}
\rank(\hat \Sigma) =&~ {{\operatorname{trace}}}(\hat\Sigma^\sharp \hat\Sigma)\nonumber\\
\geq&~ {{\operatorname{trace}}}((\hat\Sigma+\epsilon I)^{-1} \hat\Sigma)\label{eq:ranktrace}\end{aligned}$$ for any $\epsilon>0$, suggests the iterative re-weighting process $$\begin{aligned}
\label{minimizer_a}
D_{(k+1)}:=&~\arg\min_{D}{{\operatorname{trace}}}\left((\Sigma-D_{(k)} +\epsilon I)^{-1}(\Sigma-D)\right)\end{aligned}$$ for $k=1,\,2,\,\ldots$ and $D_{(0)}:=0$. In fact, as pointed out in [@Fazel2003], corresponds to minimizing $\log\det(\Sigma-D+\epsilon I)$ by local linearization.
Next we provide a sufficient condition for $\hat\Sigma$ to be such a stationary point , i.e., for $\hat\Sigma$ to satisfy $$\begin{aligned}
\label{stationary_a}
\arg\min_{D}{{\operatorname{trace}}}\left((\hat\Sigma+\epsilon I)^{-1}(\hat\Sigma-D)\right)=0.\end{aligned}$$ The notation $\circ$ used below denotes the element-wise product between vectors or matrices which is also known as *Schur product* [@Horn1990matrix] and, likewise, for vectors $a, b \in {{\mathbb R}}^{n\times 1}$, $a\circ b\in {{\mathbb R}}^{n\times 1}$ with $[a\circ b]_i=[a]_i[b]_i$.
\[prop:stationary\_a\] Let $\hat\Sigma\in{{\mathbf S}}_{n,+}$ and let the columns of $U$ form a basis of ${{\mathcal R}}(\hat\Sigma)$. If $$\begin{aligned}
\label{eq:stationary_a}
{{\mathcal R}}(U\circ U) \subset {{\mathcal R}}(\Pi_{{{\mathcal N}}(\hat\Sigma)}\circ\Pi_{{{\mathcal N}}(\hat\Sigma)} ),\end{aligned}$$ then $\hat\Sigma$ satisfies for all $\epsilon\in(0,\; \epsilon_1)$ and some $\epsilon_1>0$.
We first need the following result which generalizes [@Shapiro1985 Theorem 3.1].
\[lemma:trace\] For $A\in {{\mathbb R}}^{n\times p}$ and $B\in {{\mathbb R}}^{n\times q}$ having columns $a_1, \ldots, a_p$ and $b_1, \ldots, b_q$, respectively, we let $$\begin{aligned}
C&=[a_1\circ b_1, a_1\circ b_2, \ldots,a_2\circ b_1\dots a_p \circ b_q]\in {{\mathbb R}}^{n\times pq},\\
\phi &: ~{{\mathbb R}}^n \hspace*{.4cm}\rightarrow {{\mathbb R}}^n \hspace*{.57cm} d \mapsto \diag(AA'\diag^*(d)BB'), \mbox{ and}\\
\psi &: ~{{\mathbb R}}^{p\times q} \rightarrow {{\mathbb R}}^n \hspace*{.5cm} \Delta \mapsto \diag(A\Delta B').\end{aligned}$$ Then ${{\mathcal R}}(\phi)={{\mathcal R}}(\psi)={{\mathcal R}}((AA')\circ (BB'))={{\mathcal R}}(C)$.
Since $\diag(AA'\diag^*(d)BB')=((AA')\circ (BB'))d$, it follows that $${{\mathcal R}}(\phi)={{\mathcal R}}((AA')\circ (BB').$$ Moreover, $\diag(A\Delta B')= \sum_{i=1}^p\sum_{j=1}^q a_i\circ b_j [\Delta]_{ij}$, and then ${{\mathcal R}}(\psi)={{\mathcal R}}(C)$. We only need to show that ${{\mathcal R}}(C)={{\mathcal R}}((AA')\circ (BB'))$. This follows from $$\begin{aligned}
(AA')\circ (BB') =&~\sum_{i=1}^p\sum_{j=1}^q (a_ia_i')\circ (b_jb_j')\\
=&~\sum_{i=1}^p\sum_{j=1}^q (a_i\circ b_j) (a_i\circ b_j)'
=CC'.\end{aligned}$$ Thus ${{\mathcal R}}(C)={{\mathcal R}}((AA')\circ (BB'))$.
[*\[Proof of Proposition \[prop:stationary\_a\]:\]*]{} Assume that $\hat\Sigma$ satisfies . If $\rank(\hat\Sigma)=r$, let $\hat\Sigma=USU'$ be the eigendecomposition of $\hat\Sigma$ with $S=\diag^*(s)$ with $s\in {{\mathbb R}}^r$. Let the columns of $V$ be an orthogonal basis of the null space of $\hat\Sigma$, i.e., $\Pi_{{{\mathcal N}}(\hat\Sigma)}=VV'$. Then $$\begin{aligned}
(\hat\Sigma+\epsilon I)^{-1}=(\hat\Sigma+\epsilon \Pi_{{{\mathcal R}}(\hat\Sigma)}+\epsilon \Pi_{{{\mathcal N}}(\hat\Sigma)})^{-1} =(\hat\Sigma+\epsilon \Pi_{{{\mathcal R}}(\hat\Sigma)})^\sharp+\frac{1}{\epsilon} \Pi_{{{\mathcal N}}(\hat\Sigma)},\end{aligned}$$ and $$\begin{aligned}
\arg\min_{D:\hat\Sigma\geq D} {{\operatorname{trace}}}\left((\hat\Sigma+\epsilon I)^{-1}(\hat\Sigma-D)\right)& =\\
&\hspace*{-1.5cm}\arg\min_{D:\hat\Sigma\geq D} {{\operatorname{trace}}}\left(\left(\epsilon( \hat\Sigma+\epsilon \Pi_{{{\mathcal R}}(\hat\Sigma)})^\sharp+\Pi_{{{\mathcal N}}(\hat\Sigma)}\right)(\hat\Sigma-D)\right).\end{aligned}$$ From Proposition \[prop:WMTFAShapiro\], holds if there is $M\in {{\mathbf S}}_{r,+}$ such that $$\begin{aligned}
\label{stationaryaDiag}
\diag(VMV')=\diag\left( \epsilon( \hat\Sigma+\epsilon \Pi_{{{\mathcal R}}(\hat\Sigma)})^\sharp+\Pi_{{{\mathcal N}}(\hat\Sigma)}\right).\end{aligned}$$ Obviously, if $\epsilon=0$ $M=I$ satisfies the above equation. We consider the matrix $M$ of the form $M=I+\Delta$. For holds, we need $\diag((\hat\Sigma+\epsilon \Pi_{{{\mathcal R}}})^\sharp)$ to be in the range of $\psi$ for $$\psi: {{\mathbf S}}_n \rightarrow {{\mathbb R}}^n \hspace*{.57cm} \Delta \mapsto \diag(V\Delta V').$$ From Lemma \[lemma:trace\] that ${{\mathcal R}}(\psi)={{\mathcal R}}(\Pi_{{{\mathcal N}}(\hat\Sigma)}\circ\Pi_{{{\mathcal N}}(\hat\Sigma)})$. On the other hand, since $$\epsilon(\hat\Sigma+\epsilon \Pi_{{{\mathcal R}}(\hat\Sigma)})^\sharp=U\diag\left(\left[\frac{\epsilon}{[s]_1+\epsilon}, \ldots, \frac{\epsilon}{[s]_r+\epsilon} \right]\right)U',$$ then $\diag(\epsilon(\hat\Sigma+\epsilon
\Pi_{{{\mathcal R}}(\hat\Sigma)})^\sharp)\in {{\mathcal R}}(U\circ U)$. So if holds, there is always a $\Delta$ such that $M=I+\Delta$ satisfies . Morover, it is also required that $I+\Delta\geq0$. Since the map from $\epsilon$ to $\Delta$ is continuous, for small enough $\epsilon$, i.e. in a interval $(0, \epsilon_1)$ the condition $I+\Delta$ can always be satisfied.
We note that is a sufficient condition for $\hat\Sigma$ to be a stationary point of in both Frisch’s and Shapiro’s settings.
Certificates of minimum rank {#sec:CertifMinRank}
============================
We are interested in obtaining bounds on the minimal rank for the Frisch problem so as to ensure optimality when candidate solutions are obtained by the earlier optimization approach in .
The following two bounds were proposed in [@Woodgate1], and follow from Theorem \[thm:Reiersol\]. However, both of these bounds require exhaustive search which may be prohibitively expensive when $n$ is large.
\[cor1\] Let $\Sigma\in{{\mathbf S}}_{n,+}$ and $\Sigma>0.$ If there is an $s_1\times s_1$ principle minor of $\Sigma$ whose inverse is positive, then $$\begin{aligned}
{{\operatorname{mr}}}_+(\Sigma)&\geq s_1-1.
\end{aligned}$$ If there is an $s_2\times s_2$ principle minor of $\Sigma^{-1}$ which is element-wise positive, then $$\begin{aligned}
{{\operatorname{mr}}}_+(\Sigma)&\geq s_2-1.
\end{aligned}$$
Next we discuss three other bounds that are computationally more tractable –the first two were proposed by Guttman [@Guttman1954]. Guttman’s bounds are based on a conservative assessment for the admissible range of each of the diagonal entries of $D=\Sigma-\hat\Sigma$.
\[prop:Guttman\] Let $\Sigma\in {{\mathbf S}}_{n,+}$ and let $$\begin{aligned}
D_1&:=\diag^*(\diag(\Sigma))\\
D_2&:=\left(\diag^*(\diag(\Sigma^{-1}))\right)^{-1}.
\end{aligned}$$ Then the following hold, $$\begin{aligned}
&{{\operatorname{mr}}}_+(\Sigma)\geq n_+(\Sigma-D_1) \label{Guttman:bound1}\\
&{{\operatorname{mr}}}_+(\Sigma)\geq n_+(\Sigma-D_2). \label{Guttman:bound2}\\
\nonumber\end{aligned}$$ Further, $n_+(\Sigma-D_1)\leq n_+(\Sigma-D_2)$.
The proof follows from the fact that $\Sigma\geq D$ implies $D\leq D_2\leq D_1$. See [@Guttman1954] for details.
It is also easy to see that ${{\operatorname{mr}}}(\Sigma)\geq n_+(\Sigma-D_1)$ which provides a lower bound for the minimum rank in Shapiro’s problem. Next we return to a bound, which we noted earlier in .
\[tracebound\] Let $\Sigma\in{{\mathbf S}}_{n,+}$. Then the following holds: $$\begin{aligned}
\label{eq:lowerbound3}
{{\operatorname{mr}}}_+(\Sigma)\geq \min_{\Sigma\geq D\geq 0}{{\operatorname{trace}}}(\Sigma^{-1}(\Sigma-D)).\end{aligned}$$
The statement follows readily from .
Evidently an analogous statement holds for ${{\operatorname{mr}}}(\Sigma)$. We note that and remain invariant under scaling of rows and corresponding columns, whereas does not, hence these two cannot be compared directly.
Correspondence between decompositions {#correspondence}
=====================================
We now return to the decomposition of the data matrix $X=\hat
X+\tilde X$ as in and its relation to the corresponding sample covariances. The decomposition of $X$ into “noise-free” and “noisy” components implies a corresponding decomposition for the sample covariance, but in the converse direction, a decomposition $ \Sigma=\hat\Sigma+\tilde\Sigma $ leads to a family of compatible decompositions for $X$, which corresponds to the boundary of a matrix-ball. This is discussed next.
\[prop:decomposition\] Let $X\in{{\mathbb R}}^{n\times T}$, and $\Sigma:=XX^\prime$. If $$\label{eq:decompose2}
\Sigma=\hat \Sigma+\tilde \Sigma$$ with $\hat \Sigma$, $\tilde \Sigma$ symmetric and non-negative definite, there exists a decomposition
\[conditions\] $$\label{eq:Xdecompose}
X=\hat X+\tilde X$$ for which $$\begin{aligned}
\label{cond2}
&&\hat X \tilde X^\prime = 0,\\\label{cond3}
&&\hat \Sigma = \hat X \hat X^\prime,\\\label{cond4}
&&\tilde \Sigma = \tilde X\tilde X^\prime.\end{aligned}$$
Further, all pairs $(\hat X,\,\tilde X)$ that satisfy (\[eq:Xdecompose\]-\[cond4\]) are of the form $$\label{parametrization}
\hat X=\hat\Sigma \Sigma^{-1} X+R^{1/2}V,\;
\tilde X=\tilde\Sigma \Sigma^{-1} X-R^{1/2}V,$$ with
\[Rs\] $$\begin{aligned}
\label{R}
R&:=&\hat \Sigma - \hat \Sigma \Sigma^{-1} \hat \Sigma\\
&=&\tilde \Sigma - \tilde \Sigma \Sigma^{-1} \tilde \Sigma \label{R2}\\\nonumber
&=&\hat \Sigma \Sigma^{-1}\tilde \Sigma\\\nonumber
&=&\tilde \Sigma \Sigma^{-1}\hat \Sigma,\nonumber\end{aligned}$$
and $V\in{{\mathbb R}}^{n\times T}$ such that $VV'=I$, $XV'=0$.
The proof relies on a standard lemma ([@douglas Theorem 2]) which states that if $A\in{{\mathbb R}}^{n\times T}$, $B\in{{\mathbb R}}^{n\times m}$ with $m\leq T$ such that $A A^\prime = B B^\prime,$ then $A=BU$ for some $U\in{{\mathbb R}}^{m\times T}$ with $U U^\prime =I$. Thus, we let $A:=X$, $$S:=\left[\begin{matrix}\hat \Sigma & 0\\0&\tilde \Sigma\end{matrix}\right],$$ and $B:=\left[\begin{matrix}I&I \end{matrix}\right] S^{1/2}$, where $S^{1/2}$ is the matrix-square root of $S$. It follows that there exists a matrix $U$ as above for which $A=BU$, and therefore we can take $$\left[\begin{matrix}\hat X \\ \tilde X\end{matrix}\right]:=S^{1/2}U.$$ This establishes the existence of the decomposition .
In order to parameterize all such pairs $(\hat X,\,\tilde X)$, let $U_o$ be an orthogonal (square) matrix such that $$XU_o=[\Sigma^{1/2} \; 0].$$ Then $\hat X U_o$ and $\tilde X U_o$ must be of the form $$\label{UXs}
\hat X U_o=: \left[\begin{matrix}\hat X_1&\Delta \end{matrix}\right],\;
\tilde X U_o=: \left[\begin{matrix}\tilde X_1& -\Delta \end{matrix}\right],$$ with $\hat X_1$, $\tilde X_1$ square matrices. Since $$\left[\begin{matrix}\hat X\\\tilde X \end{matrix}\right]
\left[\begin{matrix}\hat X^\prime &\tilde X^\prime \end{matrix}\right]
=\left[\begin{matrix}\hat \Sigma& 0\\0&\tilde \Sigma \end{matrix}\right],$$ then
$$\begin{aligned}
\label{first}
&&\hat X_1\hat X_1^\prime+\Delta\Delta^\prime=\hat \Sigma\\\label{second}
&&\hat X_1\tilde X_1^\prime-\Delta\Delta^\prime=0\\\label{third}
&&\tilde X_1\tilde X_1^\prime+\Delta\Delta^\prime=\tilde \Sigma.\end{aligned}$$
Substituting $\hat X_1\tilde X_1^\prime$ for $\Delta\Delta^\prime$ into (\[first\]) and using the fact that $\tilde X_1=X_1-\hat X_1$ with $X_1=\Sigma^{1/2}$ we obtain that $$\begin{aligned}
&&\hat X_1=\hat \Sigma\Sigma^{-1/2}.\end{aligned}$$ Similarly, using (\[third\]) instead, we obtain that $$\begin{aligned}
&&\tilde X_1=\tilde \Sigma\Sigma^{-1/2}.\end{aligned}$$ Substituting into (\[second\]), (\[first\]) and (\[third\]) we obtain the following three relations $$\begin{aligned}
\Delta\Delta^\prime &=& \hat \Sigma \Sigma^{-1}\tilde \Sigma\\
&=&\hat \Sigma - \hat \Sigma \Sigma^{-1} \hat \Sigma\\
&=&\tilde \Sigma - \tilde \Sigma \Sigma^{-1} \tilde \Sigma.\end{aligned}$$ Since $\Delta\Delta^\prime$ and the $\Sigma$’s are all symmetric, $$\begin{aligned}
\Delta\Delta^\prime&=&\tilde \Sigma \Sigma^{-1}\hat \Sigma\end{aligned}$$ as well. Thus, $\Delta=R^{1/2}V_1$ with $V_1V_1^\prime=I$. The proof is completed by substituting the expressions for $\hat X_1$ and $\Delta$ into .
Interestingly, $$\rank(R)+\rank(\Sigma)=\rank \left(\left[
\begin{array}{cc}
\hat\Sigma & \hat\Sigma \\
\hat\Sigma & \Sigma \\
\end{array}
\right] \right)=\rank \left(\left[
\begin{array}{cc}
\hat\Sigma & 0 \\
0 & \tilde\Sigma \\
\end{array}
\right] \right)=\rank(\hat\Sigma)+\rank(\tilde\Sigma),$$ and hence, the rank of the “uncertainty radius” $R$ of the corresponding $\hat X$ and $\tilde X$-matrix spheres is $$\rank(R)= \rank(\hat\Sigma)+\rank(\tilde\Sigma)-\rank(\Sigma).$$ In cases where identifying $\hat X$ from the data matrix $X$, different criteria may be used to quantify uncertainty. One such is the rank of $R$ while another is its trace, which is the variance of estimation error in determining $\hat X$. This topic is considered next and its relation to the Frisch decomposition highlighted.
Uncertainty and worst-case estimation {#sec:min-max}
=====================================
The basic premise of the decomposition (\[eq:decompose2\]) is that, in principle, no probabilistic description of the data is needed. Thus, under the assumptions of Proposition \[prop:decomposition\], $R$ represents a deterministic radius of uncertainty in interpreting the data. On the other hand, when data and noise are probabilistic in nature and represent samples of jointly Gaussian random vectors ${{\mathbf x}},\;{{\hat{\mathbf x} }},\; {{\tilde{\mathbf x}}}$ as in (\[eq:xa\] - \[eq:xc\]), the conditional expectation of ${{\hat{\mathbf x} }}$ given ${{\mathbf x}}$ is $E\{{{\hat{\mathbf x} }}|{{\mathbf x}}\}=\hat\Sigma\Sigma^{-1} {{\mathbf x}}$, while the variance of the error $$\begin{aligned}
E\{({{\hat{\mathbf x} }}-\hat\Sigma \Sigma^{-1}{{\mathbf x}})({{\hat{\mathbf x} }}-\hat\Sigma \Sigma^{-1}{{\mathbf x}})^\prime\}&=&\hat\Sigma - \hat\Sigma\Sigma^{-1}\hat\Sigma\\
&=&R\end{aligned}$$ is the radius of the deterministic uncertainty set. Either way, it is of interest to assess how this radius depends on the decomposition of $\Sigma$.
Uniformly optimal decomposition
-------------------------------
Since the decomposition of $\Sigma$ in the Frisch problem is not unique, it is natural to seek a uniformly optimal choice of the estimate $K{{\mathbf x}}$ for ${{\hat{\mathbf x} }}$ over all admissible decompositions. To this end, we denote the mean-squared-error loss function $$\begin{aligned}
\label{eq:LossFunction}
L(K, \hat \Sigma, \tilde\Sigma)&:=&{{\operatorname{trace}}}\left({{\mathcal E}}\left( ({{\hat{\mathbf x} }}-K{{\mathbf x}})({{\hat{\mathbf x} }}-K{{\mathbf x}})^\prime\right)\right)\nonumber\\
&\;=&{{\operatorname{trace}}}\left(\hat\Sigma-K\hat\Sigma-\hat\Sigma K'+K(\hat\Sigma+\tilde\Sigma) K' \right),\label{eq:loss}\end{aligned}$$ and define $$\begin{aligned}
{{\mathcal S}}(\Sigma):= \{(\hat\Sigma, \tilde\Sigma) : &~\Sigma=\hat\Sigma+\tilde\Sigma,\; \hat\Sigma,\; \tilde\Sigma\geq0 \text{~and~} \tilde\Sigma \text{~is diagonal} \}\end{aligned}$$ as the set of all admissible pairs. Thus, a uniformly-optimal decomposition of $X$ into signal plus noise relates to the following min-max problem: $$\begin{aligned}
\label{prob:minmax}
\min_{K}\max_{(\hat\Sigma,\tilde\Sigma)\in{{\mathcal S}}(\Sigma)} L(K, \hat \Sigma, \tilde\Sigma).\end{aligned}$$ The minimizer of is the uniformly optimal estimator gain $K$. Analogous min-max problems, over different uncertainty sets, have been studied in the literature [@Eldar2004competitive]. In our setting
\[eq:concave\] $$\begin{aligned}
\min_{K}\max_{(\hat\Sigma,\tilde\Sigma)\in{{\mathcal S}}(\Sigma)} L(K, \hat \Sigma,\tilde\Sigma)&\geq&\max_{(\hat\Sigma,\tilde\Sigma)\in{{\mathcal S}}(\Sigma)}\min_{K} L(K, \hat \Sigma,\tilde\Sigma)\label{minmaxmaxmin}\\
&=&\max_{(\hat\Sigma,\tilde\Sigma)\in{{\mathcal S}}(\Sigma)} {{\operatorname{trace}}}\left(\hat\Sigma-\hat\Sigma\Sigma^{-1}\hat\Sigma\right)\label{eq:concave1}\\
&=&\max_{(\hat\Sigma,\tilde\Sigma)\in{{\mathcal S}}(\Sigma)} {{\operatorname{trace}}}\left(\tilde\Sigma-\tilde\Sigma\Sigma^{-1}\tilde\Sigma\right).\label{eq:concave2}\end{aligned}$$
The functions to maximize in and are both strictly concave in $\hat\Sigma$ and $\tilde\Sigma$. Therefore the maximizer is unique. Thus, we denote $$\label{optsolution}
(K_{\rm opt}, \hat\Sigma_{\rm opt}, \tilde\Sigma_{\rm opt}) :=\arg \max_{(\hat\Sigma,\tilde\Sigma)\in{{\mathcal S}}(\Sigma)}\min_{K} L(K, \hat \Sigma,\tilde\Sigma),$$ where, clearly, $K_{\rm opt}=\hat\Sigma_{\rm opt}\Sigma^{-1}$.
In general, the decomposition suggested by the uniformly optimal estimation problem does not lead to a singular signal covariance $\hat\Sigma$. The condition for when that happens is given next. Interestingly, this is expressed in terms of half the candidate noise covariance utilized in obtaining one of the Guttman bounds (Proposition \[prop:Guttman\]).
\[prop:maxmin\] Let $\Sigma>0$, and let $$\label{eq:D0}
D_0:=\frac12 \diag^*\left(\diag(\Sigma^{-1})\right)^{-1}$$ (which is equal to $\frac12 D_2$ defined in Proposition \[prop:Guttman\]). If $\Sigma-D_0\geq0$, then
\[Solution\] $$\label{InteriorSolution}
\tilde\Sigma_{\rm opt}=D_0 \text{~and~} \hat\Sigma_{\rm opt}=\Sigma-D_0.$$ Otherwise, $$\label{BoundarySolution}
\tilde\Sigma_{\rm opt}\leq D_0 \text{~and~} \hat\Sigma_{\rm opt} \text{~is singular}.$$
From , $$\begin{aligned}
L(K_{\rm opt}, \hat \Sigma_{\rm opt}, \tilde \Sigma_{\rm opt})&=&\max \left\{\tilde \Sigma-\tilde \Sigma\Sigma^{-1}\tilde \Sigma ~\mid~ \Sigma\geq\tilde\Sigma\geq0, \tilde\Sigma \text{~is diagonal} \right\}\nonumber\\
&\leq& \max \left\{\tilde \Sigma-\tilde \Sigma\Sigma^{-1}\tilde \Sigma ~\mid~\tilde\Sigma \text{~is diagonal} \right\}\label{relaxedD}\\
&=&\frac12 {{\operatorname{trace}}}(D_0)\nonumber\end{aligned}$$ with the maximum attained for $\tilde\Sigma=D_0$. Then follows. In order to prove , consider the Lagrangian corresponding to $${{\mathcal L}}(\tilde\Sigma,\Lambda_0, \Lambda_1) ={{\operatorname{trace}}}(\tilde\Sigma-\tilde\Sigma\Sigma^{-1}\tilde\Sigma+\Lambda_0(\Sigma-\tilde\Sigma)+\Lambda_1\tilde\Sigma)$$ where $\Lambda_0,\;\Lambda_1$ are Lagrange multipliers. The optimal values satisfy
$$\begin{aligned}
&&[I-2\Sigma^{-1}\tilde\Sigma_{\rm opt}-\Lambda_{0}+\Lambda_{1}]_{kk}=0, \;\forall\; k=1,\ldots, n,\label{condition1}\\
&& \Lambda_{0}\hat\Sigma_{\rm opt}=0,\; \Lambda_{0}\geq0,\label{condition2}\\
&& \Lambda_{1}\tilde\Sigma_{\rm opt}=0,\; \Lambda_{1}\geq0 \text{~and is diagonal}.\label{condition3}\end{aligned}$$
If $\Sigma- D_0\not\geq0$ we show that $\hat\Sigma_{\rm opt}$ is singular. Assume the contrary, i.e., that $\hat\Sigma_{\rm opt}>0$. From , we see that $\Lambda_{0}=0$, while from , $ [I-2\Sigma^{-1}\tilde\Sigma_{\rm
opt}]_{kk}\leq 0. $ This gives that $$[\tilde\Sigma_{\rm opt}]_{kk}\geq \frac{1}{2[\Sigma^{-1}]_{kk}}= [D_0]_{kk},$$ for all $k=1, \ldots, n$, which contradicts the fact that $\Sigma-D_0\not\geq0$. Therefore $\hat\Sigma_{\rm opt}$ is singular. We now assume that $\tilde\Sigma\not \leq D_0$. Then there exists $k$ such that $[\tilde\Sigma_{\rm opt}]_{kk}> [D_0]_{kk}$. From and , we have that $$[\Lambda_{1}]_{kk}=0 \text{~and~} [I-2\Sigma^{-1}\tilde\Sigma_{\rm opt}]_{kk}\geq0$$ which contradicts the assumption that $[\tilde\Sigma_{\rm opt}]_{kk}>
[D_0]_{kk}$. Therefore $\tilde\Sigma_{\rm opt}\leq D_0$ and has been established.
We remark that while $$\begin{aligned}
{{\mathcal E}}\left( ({{\hat{\mathbf x} }}-K{{\mathbf x}})({{\hat{\mathbf x} }}-K{{\mathbf x}})^\prime\right)&=&\hat\Sigma-K\hat\Sigma-\hat\Sigma K'+K\Sigma K'\\
&=&(\hat\Sigma\Sigma^{-\frac12}-K\Sigma^{\frac12})(\hat\Sigma\Sigma^{-\frac12}-K\Sigma^{\frac12})^\prime+\hat\Sigma-\hat\Sigma\Sigma^{-1}\hat\Sigma\end{aligned}$$ is matrix-convex in $K$ and a unique minimum for $K=\hat\Sigma\Sigma^{-1}$, the error covariance $\hat\Sigma-\hat\Sigma\Sigma^{-1}\hat\Sigma $ may not have a unique maximum in the positive semi-definite sense. To see this, consider $\Sigma=\left[
\begin{array}{cc}
2 & 1 \\
1 & 2 \\
\end{array}
\right]
$. In this case $D_0=\frac{3}{4}I$, $\hat\Sigma_{\rm opt}=\left[
\begin{array}{cc}
5/4 & 1 \\
1 & 5/4 \\
\end{array}
\right]$, and $$\label{eq:Ropt}
\hat\Sigma_{\rm opt}-\hat\Sigma_{\rm opt}\Sigma^{-1}\hat\Sigma_{\rm opt}=\left[
\begin{array}{cc}
3/8 & 3/16 \\
3/16 & 3/8 \\
\end{array}
\right].$$ On the other hand, for $\hat\Sigma=\left[
\begin{array}{cc}
3/2 & 1 \\
1 & 3/2 \\
\end{array}
\right]$, then $$\hat\Sigma-\hat\Sigma\Sigma^{-1}\hat\Sigma=\left[
\begin{array}{cc}
1/3 & 1/12 \\
1/12 & 1/3 \\
\end{array}
\right]$$ which is neither larger nor smaller than in the sense of semi-definiteness. This is a key reason for considering scalar loss functions of the error covariance as in .
Next we note that there is no gap between the min-max and max-min values in the two sides of .
\[prop:minmax\] For $\Sigma\in{{\mathbf S}}_{n,+}$, then $$\label{eq:equal}
\min_{K}\max_{(\hat\Sigma, \tilde\Sigma)\in{{\mathcal S}}(\Sigma)} L(K, \hat \Sigma, \tilde\Sigma)=\max_{(\hat\Sigma, \tilde\Sigma)\in{{\mathcal S}}(\Sigma)}\min_{K} L(K, \hat \Sigma, \tilde\Sigma).$$
We observe that for a fixed $K$, the function $L(K, \hat \Sigma,
\tilde\Sigma)$ is a linear function of $(\hat\Sigma, \tilde\Sigma)$. For fixed $(\hat\Sigma, \tilde\Sigma)$, the function is a convex function of $K$. Under this conditions it is standard that holds, see e.g. [@Boyd2004convex page 281].
We remark that when $D_0=\frac12
\diag^*\left(\diag(\Sigma^{-1})\right)^{-1}$ is admissible as noise covariance, i.e., $\Sigma- D_0\geq0$, the optimal signal covariance is $\hat\Sigma_{\rm opt}=\Sigma-D_0$, and the gain matrix $K_{\rm
opt}=\hat\Sigma_{\rm opt}\Sigma^{-1}=I-D_0\Sigma^{-1}$ has all diagonal entries equal to $\frac{1}{2}$. Thus, with $K_{\rm opt}$ in the mean-square-error loss is independent of $\hat\Sigma$ and equal to ${{\operatorname{trace}}}\left(K_{\rm opt}\Sigma K_{\rm
opt}^\prime\right)$ for any admissible decomposition of $\Sigma$.
We also remark that the key condition (Proposition \[prop:maxmin\]) $$\begin{aligned}
\label{InvariantCondition}
& \Sigma\geq\frac12 \diag^*\left(\diag(\Sigma^{-1}) \right)^{-1}\\
&\Leftrightarrow 2\diag^*\left(\diag(\Sigma^{-1}) \right)\geq \Sigma^{-1}
\end{aligned}$$ can be equivalently written as $\Sigma^{-1}\circ (2I-{\bf 1}{\bf
1}')\geq0$, and interestingly, amounts to the positive semi-definitess of a matrix formed by changing the signs of all off-diagonal entries of $\Sigma^{-1}$. The set of all such matrices, $\left\{S
\mid S\geq 0,~ S\circ (2I-{\bf 1}{\bf 1}')\geq0 \right\}$, is convex, invariant under scaling rows and corresponding columns, and contains the set of diagonally dominant matrices $\{S \mid S\geq 0,~
[S]_{ii}\geq \sum_{j\neq i} |[S]_{ij}| \text{~for all ~} i\}$.
We conclude this section by noting that ${{\operatorname{trace}}}(R_{\rm opt})$, with $$R_{\rm opt}:=\hat\Sigma_{\rm opt}-\hat\Sigma_{\rm opt}\Sigma^{-1}\hat\Sigma_{\rm opt},$$ quantifies the distance between admissible decompositions of $\Sigma$. This is stated next.
\[lemma:radius\] For $\Sigma>0$ and any pair $(\hat\Sigma, \tilde\Sigma)\in {{\mathcal S}}(\Sigma)$, $${{\operatorname{trace}}}\left( (\hat\Sigma-\hat\Sigma_{\rm opt})\Sigma^{-1}(\hat\Sigma-\hat\Sigma_{\rm opt})' \right)\leq {{\operatorname{trace}}}(R_{\rm opt}).$$
Clearly $0\leq{{\operatorname{trace}}}(\hat\Sigma-\hat\Sigma\Sigma^{-1}\hat\Sigma)$, while from Proposition \[prop:minmax\], $$\begin{aligned}
L(K_{\rm opt}, \hat\Sigma, \tilde\Sigma)
&=& {{\operatorname{trace}}}(\hat\Sigma-2\hat\Sigma_{\rm opt}\Sigma^{-1}\hat\Sigma+\hat\Sigma_{\rm opt}\Sigma^{-1}\hat\Sigma_{\rm opt}')\label{eqB}\\
&\leq& {{\operatorname{trace}}}(R_{\rm opt}).\nonumber\end{aligned}$$ Thus, ${{\operatorname{trace}}}(\hat\Sigma\Sigma^{-1}\hat\Sigma-2\hat\Sigma_{\rm opt}\Sigma^{-1}\hat\Sigma+\hat\Sigma_{\rm opt}\Sigma^{-1}\hat\Sigma_{\rm opt}')\leq {{\operatorname{trace}}}(R_{\rm opt})$.
Uniformly optimal estimation and trace regularization {#sec:regularized}
-----------------------------------------------------
A decomposition of $\Sigma$ in accordance with the min-max estimation problem of the previous section often produces an invertible signal covariance $\hat\Sigma$. On the other hand, it is often the case and it is the premise of factor analysis, that $\hat\Sigma$ is singular of low rank and, thereby, allows identifying linear relations in the data. In this section we consider combining the mean-square-error loss function with regularization term promoting a low rank for the signal covariance $\hat\Sigma$ [@Fazel2001]. More specifically, we consider $$\label{prob:minmaxRank}
J=\min_{K}\max_{(\hat\Sigma,\tilde\Sigma)\in{{\mathcal S}}(\Sigma)} \left(L(K,
\hat \Sigma, \tilde\Sigma)-\lambda\cdot {{\operatorname{trace}}}(\hat\Sigma)\right),$$ for $\lambda\geq0$, and properties of its solutions.
As noted in Proposition \[prop:minmax\] (see [@Boyd2004convex page 281]), here too there is no gap between the min-max and the max-min, which becomes
$$\begin{aligned}
&\max_{(\hat\Sigma,\tilde\Sigma)\in{{\mathcal S}}(\Sigma)}\min_{K} L(K, \hat \Sigma, \tilde\Sigma)-\lambda\cdot {{\operatorname{trace}}}(\hat\Sigma)\nonumber\\
&= \max_{(\hat\Sigma,\tilde\Sigma)\in{{\mathcal S}}(\Sigma)}\min_{K} {{\operatorname{trace}}}\left( (1-\lambda)\hat\Sigma-K\hat\Sigma-\hat\Sigma K'+K(\hat\Sigma+\tilde\Sigma)K' \right)\nonumber\\
&=\max_{(\hat\Sigma,\tilde\Sigma)\in{{\mathcal S}}(\Sigma)} {{\operatorname{trace}}}\left( (1-\lambda)\hat\Sigma-\hat\Sigma(\hat\Sigma+\tilde\Sigma)^{-1}\hat\Sigma \right) \label{eq:Kcanceled}\\
&=\max_{(\hat\Sigma,\tilde\Sigma)\in{{\mathcal S}}(\Sigma)} {{\operatorname{trace}}}\left( -\lambda\Sigma+ (1+\lambda)\tilde\Sigma-\tilde\Sigma(\hat\Sigma+\tilde\Sigma)^{-1}\tilde\Sigma \right). \label{eq:Sigtilde}\end{aligned}$$
Since and are strictly concave functions of $\hat\Sigma$ and $\tilde\Sigma$, respectively, there is a unique set of optimal values $(K_{\lambda, \rm opt}, \hat\Sigma_{\lambda,\rm opt}, \tilde\Sigma_{\lambda,\rm opt})$.
Let $\Sigma>0$, $D_0=\frac12 \left(\diag^*\diag(\Sigma^{-1})\right)^{-1},$ $\lambda_{\rm min}$ be the smallest eigenvalue of $D_0^{-\frac12}\Sigma D_0^{-\frac12}$, and $(K_{\lambda, \rm opt}, \hat\Sigma_{\lambda,\rm opt}, \tilde\Sigma_{\lambda,\rm opt})$ as above, for $\lambda\geq0$. For any $\lambda\geq\lambda_{\rm min}-1$, $\hat\Sigma_{\lambda,{\rm opt}}$ is singular.
The trace of $( -\lambda\Sigma+ (1+\lambda)\tilde\Sigma-\tilde\Sigma\Sigma^{-1}\tilde\Sigma )$ is maximal for the diagonal choice $\tilde \Sigma = (1+\lambda)D_0$. For any $\lambda \geq \lambda_{\rm min}-1$, $\Sigma-(1+\lambda) D_0$ fails to be positive semidefinite. Thus, the constraint $\Sigma-\tilde\Sigma\geq 0$ in is active and $\hat\Sigma_{\lambda, {\rm opt}}$ is singular.
Note that $\Sigma-2D_0\not\geq 0$ (unless $\Sigma$ is diagonal), and therefore $\lambda_{\rm min}<2$. Hence, for $\lambda\geq1$, $\hat\Sigma_{\lambda, {\rm opt}}$ is singular. When $\lambda\to 0$ we recover the solution in , whereas for $\lambda\to\infty$ we recover the solution in Proposition \[prop:mintrace\].
Accounting for statistical errors {#statisticalerrors}
=================================
From an applications standpoint $\Sigma$ represents an empirical covariance, estimated on the basis of a finite observation record in $X$. Hence and are only approximately valid, as already suggested in Section \[sec:datastrcuture\]. Thus, in order to account for sampling errors we can introduce a penalty for the size of $C:=\hat X\tilde X^\prime$, conditioned so that $$\Sigma=\hat\Sigma + \tilde\Sigma +C +C',$$ and a penalty for the distance of $\tilde \Sigma$ from the set $\{D \mid D\mbox{ diagonal}\}$.
Alternatively, we can use the Wasserstein 2-distance [@olkin1982; @ning2011] between the respective Gaussian probability density functions, which can be written in the form of a semidefinite program $$d(\hat\Sigma+D, \Sigma)=\min_{C_1}\left({{\operatorname{trace}}}(\Sigma+\hat\Sigma+D+C_1+C_1') \mid \left[
\begin{array}{cc} \hat\Sigma+D & C_1 \\
C_1' & \Sigma \\
\end{array}
\right]\geq0 \right).$$
Returning to the uncertainty radius of Section \[correspondence\] and the problem discussed in Section \[sec:min-max\], we note that the problem $$\nonumber
\max\min_{K} L(K, \hat \Sigma,D)\\
=\max {{\operatorname{trace}}}\left(\hat\Sigma-\hat\Sigma(\hat\Sigma+D)^{-1}\hat\Sigma\right)$$ can be expressed as the semidefinite program $$\nonumber
\max_Q \left\{ {{\operatorname{trace}}}\left(\hat\Sigma-Q \right)\mid
\left[
\begin{array}{cc}
Q & \hat\Sigma \\
\hat\Sigma & \hat\Sigma+D \\
\end{array}
\right]\geq 0
\right\}.$$ Thus, putting the above together, a formulation that incorporates the various tradeoffs between the dimension of the signal subspace, mean-square-error loss, and statistical errors is to maximize $$\label{eq:maxmin2}
{{\operatorname{trace}}}(\hat\Sigma -Q) - \lambda_1\, {{\operatorname{trace}}}(\hat\Sigma) -\lambda_2\, {{\operatorname{trace}}}(\hat\Sigma + D - C_1-C_1^\prime)$$ subject to $$\begin{aligned}
\left[
\begin{array}{cc}
Q & \hat\Sigma \\
\hat\Sigma & \hat\Sigma+D \\
\end{array}
\right]\geq 0,\;\left[
\begin{array}{cc} \hat\Sigma+D & C_1 \\
C_1' & \Sigma \\
\end{array}
\right]\geq0, \mbox{ with }D\geq 0 \mbox{ and diagonal.}\end{aligned}$$ The value of the parameters $\lambda_1$, $\lambda_2$ dictate the relative importance that we place on the various terms and determine the tradeoffs in the problem.
We conclude with an example to highlight the potential and limitations of the techniques. We generate data $X$ in the form $$X=FV+\tilde X$$ where $F\in{{\mathbb R}}^{n\times r}$, $V\in {{\mathbb R}}^{r\times T}$, and $\tilde X\in {{\mathbb R}}^{n\times T}$ with $n=50$, $r=10$, $T=100$. The elements of $F$ and $V$ are generated from normal distributions with mean zero and unit covariance. The columns of $\tilde X$ are generated from a normal distribution with mean zero and diagonal covariance, itself having (diagonal) entries which are uniformly drawn from interval $[1, 10]$. The matrix $\Sigma=XX'$ is subsequently scaled so that ${{\operatorname{trace}}}(\Sigma)=1$. We determine $$(\hat\Sigma,Q,D)={\rm arg}\max \left\{ {{\operatorname{trace}}}(\hat\Sigma-Q)-\lambda\cdot {{\operatorname{trace}}}(\hat\Sigma)\right\}$$ subject to $$\begin{aligned}
\left[\begin{matrix}Q &\hat\Sigma\\ \hat\Sigma & \hat\Sigma+D \end{matrix} \right]\geq0, ~d(\hat\Sigma+D, \Sigma)\leq \epsilon, \text{~with~} \hat\Sigma, D\geq0 \text{~and~} D \text{~diagonal},\end{aligned}$$ and tabulate below a typical set of values for the rank of $\hat\Sigma$ (Table 1) as a function of $\lambda$ and $\epsilon$. We observe a “plateau” where the rank stabilizes at $10$ over a small range of values for $\epsilon$ and $\lambda$. Naturally, such a plateau may be taken as an indication of a suitable range of parameters. Although the current setting where a small perturbation in the empirical covariance $\Sigma$ is allowed, the bounds for the rank in and are still pertinent. In fact, for this example, in $7/10$ instances where the $\rank(\hat\Sigma)=10$ the bound in (computed based on the perturbed covariance $\hat\Sigma+D$) has been tight and it thus a valid certificate. For the same range of parameters, the bound in has been lower than the actual rank of $\hat\Sigma$. In general, the bounds in and are not comparable as either one may be tighter than the other.\
$0$ $0.08$ $0.10$ $0.12$ $0.14$ $0.16$
------- ----- -------- -------- -------- -------- -------- --
$1$ 46 26 24 23 22 22
$5$ 46 17 14 10 10 9
$10$ 45 16 12 10 10 8
$20$ 45 15 12 10 10 8
$50$ 45 15 12 10 10 8
$100$ 45 15 11 10 10 8
\
[Table 1: $\rank(\hat\Sigma)$ as a function of $\lambda$ and $\epsilon$]{}\
Conclusions {#sec:conclusion}
===========
In this paper we considered the general problem of identifying linear relations among variables based on noisy measurements –a classical problem of major importance in the current era of “Big Data.” Novel numerical techniques and increasingly powerful computers have made it possible to successfully treat a number of key issues in this topic in a unified manner. Thus, the goal of the paper has been to present and develop in a unified manner key ideas of the theory of noise-in-variables linear modeling.
More specifically, we considered two different viewpoints for the linear model problem under the assumption of independent noise. From an estimation viewpoint, we quantify the uncertainty in estimating “noise-free” data based on noise-in-variables linear models. We proposed a min-max estimation problem which aims at a uniformly optimal estimator –the solution can be obtained using convex optimization. From the modeling viewpoint, we also derived several classical results for the Frisch problem that asks for the maximum number of simultaneous linear relations. Our results provide a geometric insight to the Reiersøl theorem, a generalization to complex-valued matrices, an iterative re-weighting trace minimization scheme for obtaining solutions of low rank along with a characterization of fixed points, and certain computational tractable lower bounds to serve as certificates for identifying the minimum rank. Finally, we consider regularized min-max estimation problems which integrate various objectives (low-rank, minimal worst-case estimation error) and explain their effectiveness in a numerical example.
In recent years, techniques such as the ones presented in this work are becoming increasingly important in subjects where one has very large noisy datasets including medical imaging, genomics/proteomics, and finance. It is our hope that the material we presented in this paper will be used in these topics. It must be noted that throughout the present work we emphasized independence of noise in individual variables. Evidently, more general and versatile structures for the noise statistics can be treated in a similar manner, and these may become important when dealing with large databases.
A very important topic for future research is that of dealing with statistical errors in estimating empirical statistics. It is common to quantify distances using standard matrix norms –as is done in the present paper as well. Alternative distance measures such as the Wasserstein distance mentioned in Section \[statisticalerrors\] and others (see e.g., [@ning2011]) may become increasingly important in quantifying statistical uncertainty.
Finally, we raise the question of the asymptotic performance of certificates such as those presented in Section \[sec:CertifMinRank\]. It is important to know how the tightness of the certificate to the minimal rank of linear models relates to the size of the problem.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by grants from NSF, NIH, AFOSR, ONR, and MDA. This work is part of the National Alliance for Medical Image Computing (NA-MIC), funded by the National Institutes of Health through the NIH Roadmap for Medical Research, Grant U54 EB005149. Information on the National Centers for Biomedical Computing can be obtained from http://nihroadmap.nih.gov /bioinformatics. Finally, this project was supported by grants from the National Center for Research Resources (P41-RR-013218) and the National Institute of Biomedical Imaging and Bioengineering (P41-EB-015902) of the National Institutes of Health.
[^1]: L. Ning is with the Dept. of Electrical & Comp. Eng., University of Minnesota, Minneapolis, Minnesota 55455, [[email protected]]{}
[^2]: T. T. Georgiou is with the Dept. of Electrical & Comp. Eng., University of Minnesota, Minneapolis, Minnesota 55455, [[email protected]]{}
[^3]: A. Tannenbaum is with the Comprehensive Cancer Center and Dept. of Electrical & Comp. Eng., University of Alabama, Birmingham, AL 35294, [[email protected]]{}
[^4]: S. P. Boyd is with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305, [[email protected]]{}
|
---
abstract: 'Contrary to popular misconception, the question in the title is far from simple. It involves sets of numbers on the first level, sets of sets of numbers on the second level, and so on, endlessly. The infinite hierarchy of the levels involved distinguishes the concept of “definable number” from such notions as “natural number”, “rational number”, “algebraic number”, “computable number” etc.'
author:
- Boris Tsirelson
title: 'Can each number be specified by a finite text?'
---
=1
Introduction {#sect1}
============
The question in the title may seem simple, but is able to cause controversy and trip up professional mathematicians. Here is a quote from a talk “Must there be numbers we cannot describe or define?” [@Ha1] by [J.D. Hamkins](http://en.wikipedia.org/wiki/Joel David Hamkins).\
\
Heard at a good math tea anywhere:\
\
Does this argument withstand scrutiny?\
See also “Maybe there’s no such thing as a random sequence” [@Do] by [P.G. Doyle](https://math.dartmouth.edu/~doyle/) (in particular, on pages 6,7 note two excerpts from [A. Tarski](http://en.wikipedia.org/wiki/Alfred Tarski) [@Tar2]). And on Wikipedia one can also find the flawed “math tea” argument on talk pages and obsolete versions of articles. And elsewhere on the Internet. I, the author, was myself a witness and accomplice. I shared and voiced the flawed argument in informal discussions (but not articles or lectures). Despite some awareness (but not professionalism) in mathematical logic, I was a small part of the problem, and now I try to become a small part of the solution, spreading the truth.
Careless handling of the concept “number specified by a finite text” leads to paradoxes; in particular, [Richard’s paradox](http://en.wikipedia.org/wiki/Richard's paradox).[^1] See also [“Definability paradoxes”](https://www.dpmms.cam.ac.uk/~wtg10/richardsparadox.html) by Timothy Gowers.
In order to ask (and hopefully solve) a well-posed question we have to formalize the concept “number specified by a finite text” via a well-defined mathematical notion “definable number”. What exactly is meant by “text”? And what exactly is meant by “number specified by text”? Does “specified” mean “defined”? Can we define such notions as “definition” and “definable”? Striving to understand definitions in general, let us start with some examples.
136 notable constants are collected, defined and discussed in the book “Mathematical constants” by Steven Finch [@Fi]. The first member of this collection is [“Pythagoras’ Constant, $\sqrt2$”](http://en.wikipedia.org/wiki/Square root
of 2); the second is [“The Golden Mean, $\varphi$”](http://en.wikipedia.org/wiki/Golden ratio); the third [“The Natural Logarithmic Base, $e$”](http://en.wikipedia.org/wiki/E (mathematical constant)); the fourth [“Archimedes’ Constant, $\pi$”](http://en.wikipedia.org/wiki/Pi); and the last (eleventh) in Chapter 1 “Well-Known Constants” is [“Chaitin’s Constant”](http://en.wikipedia.org/wiki/Chaitin's constant).
Each constant has several equivalent definitions. Below we take for each constant the first (main) definition from the mentioned book.
The first constant $\sqrt2$ is defined as the positive real number whose product by itself is equal to 2. That is, the real number $x$ satisfying $x>0$ and $x^2=2$.
The second constant $\varphi$ is defined as the real number satisfying $\varphi>0$ and $1+\frac1\varphi=\varphi$.
The third constant $e$ is defined as the limit of $\textstyle (1+x)^{1/x}$ as $x\to0$. That is, the real number satisfying the following condition:
> for every $\varepsilon>0$ there exists $\delta>0$ such that for every $x$ satisfying $-\delta<x<\delta$ and $x\ne0$ holds $\textstyle -\varepsilon<(1+x)^{1/x}-e<\varepsilon$.
The same condition in symbols:[^2] $$\begin{gathered}
\forall \varepsilon>0 \;\> \exists \delta>0 \;\> \forall x \;\; \(\, ( -\delta<x<\delta \,\,\land\,\, x\ne0 )
\imp \\
( -\varepsilon<(1+x)^{1/x}-e<\varepsilon ) \,\).\end{gathered}$$
We note that these three definitions are of the form “the real number $x$ satisfying $P(x)$” where $P(x)$ is a statement that may be true or false depending on the value of its variable $x$; in other words, not a statement when $x$ is just a variable, but a statement whenever a real number is substituted for the variable. Such $P(x)$ is called a property of $x$, or a [predicate](http://en.wikipedia.org/wiki/Predicate (mathematical logic)) (on real numbers).
Not all predicates may be used this way. For example, we cannot say “the real number $x$ satisfying $x^2=2$” (why “the”? two numbers satisfy, one positive, one negative), nor “the real number $x$ satisfying $x^2=-2$” (no such numbers). In order to say “the real number $x$ satisfying $P(x)$” we have to prove existence and uniqueness:
> existence: $\exists x \;\> P(x)$ (in words: there exists $x$ such that $P(x)$);
>
> uniqueness: $\forall x,y \;\> \(\, ( P(x) \land P(y) ) \imp (x=y) \,\)$\
> (in words: whenever $x$ and $y$ satisfy $P$ they are equal).
In this case one [says “there is one and only one such $x$” and writes “$\exists! x \; P(x)$”](http://en.wikipedia.org/wiki/Uniqueness quantification).
The road to definable numbers passes through definable predicates. We postpone this matter to the next section and return to examples.
The fourth constant $\pi$ is defined as the area enclosed by a circle of radius 1.
This definition involves geometry. True, a lot of equivalent definitions in terms of numbers are well-known; in particular, according to the mentioned book, this area is equal to $\textstyle \, 4\int_0^1 \sqrt{1-x^2} \, dx =
\lim_{n\to\infty} \frac4{n^2} \sum_{k=0}^n \sqrt{n^2-k^2} \, $. However, in general, every branch of mathematics may be involved in a definition of a number; existence of an equivalent definition in terms of (only) numbers is not guaranteed.
The last example is Chaitin’s constant. In contrast to the four constants (mentioned above) of evident theoretical and practical importance, Chaitin’s constant is rather of theoretical interest. Its definition is intricate. Here is a simplified version, sufficient for our purpose.
The last constant $\Om$ is defined as the sum of the series $\textstyle \Om = \sum_{N=1}^\infty 2^{-N}
A_N$ where $A_N$ is equal to 1 if there exist natural numbers $x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9$ such that $f(N,x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9)=0$, otherwise $A_N=0$; and $f$ is a polynomial in 10 variables, with integer coefficients, such that the sequence $A_1, A_2, \dots$ is uncomputable.
[Hilbert’s tenth problem](http://en.wikipedia.org/wiki/Hilbert’s tenth problem) asked for a general algorithm that could ascertain whether the [Diophantine equation](http://en.wikipedia.org/wiki/Diophantine equation) $f(x_0,\dots,x_k)=0$ has positive integer solutions $(x_0,\dots,x_k)$, given arbitrary polynomial $f$ with integer coefficients. It appears that no such algorithm can exist even for a single $f$ and arbitrary $x_0$, when $f$ is complicated enough. See Wikipedia: [computability theory](http://en.wikipedia.org/wiki/Computability theory), [Matiyasevich’s theorem](http://en.wikipedia.org/wiki/Diophantine set#Matiyasevich's_theorem); and Scholarpedia:[Matiyasevich theorem](http://www.scholarpedia.org/article/Matiyasevich_theorem).
The five numbers $\sqrt2, \varphi, e, \pi, \Om$ are defined, thus, should be definable according to any reasonable approach to definability. The first four numbers $\sqrt2, \varphi, e, \pi$ are computable (both theoretically and practically; in fact, trillions, that is, millions of millions, of decimal digits of $\pi$ are already computed), but the last number $\Om$ is uncomputable. How so? Striving to better understand this strange situation we may introduce approximations $A_{M,N}$ to the numbers $A_N$ as follows: $A_{M,N}$ is equal to 1 if there exist natural numbers $x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9$ less than $M$ such that $f(N,x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9)=0$, otherwise $A_N=0$; here $M$ is arbitrary. For each $N$ we have $A_{M,N}\uparrow A_N$ as $M\to\infty$; that is, the sequence $A_{1,N}, A_{2,N}, \dots$ is increasing, and converges to $A_N$. Also, this sequence $A_{1,N}, A_{2,N}, \dots$ is computable (given $M$, just check all the $(M-1)^9$ points $(N,x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9)$, $0<x_1<M, \dots,
0<x_9<M$). Now we introduce approximations $\om_M$ to the number $\Om$ as follows: $\textstyle \om_M = \sum_{N=1}^M
2^{-N} A_{M,N}$. We have $\om_M\uparrow\Om$ (as $M\to\infty$), and the sequence $\om_1, \om_2, \dots$ is computable. A wonder: a computable increasing sequence of rational numbers converges to a uncomputable number!
For every $N$ there exists $M$ such that $A_{M,N}=A_N$; such $M$ depending on $N$, denote it $M_N$ and get $\textstyle
\sum_{N=1}^\infty 2^{-N} A_{M_N,N}=\Om$; moreover, $\textstyle \Om-\sum_{N=1}^K 2^{-N} A_{M_N,N}\le 2^{-K}$ for all $K$. In order to compute $\Om$ up to $2^{-K}$ it suffices to compute $\textstyle \sum_{N=1}^K 2^{-N} A_{M_N,N}$. Doesn’t it mean that $\Om$ is computable? No, it does not, unless the sequence $M_1,M_2,\dots$ is computable. Well, these numbers need not be optimal, just large enough. Isn’t $\textstyle M_N=10^{1000N}$ large enough? Amazingly, no, this is not large enough. Moreover, $\textstyle M_N=10^{10^{1000N}}$ is not enough. And even the “power tower” $M_N=\underbrace{10^{10^{\cdot^{\cdot^{10}}}}}_{1000N}$ is still not enough!
Here is the first paragraph from a prize-winning article by Bjorn Poonen [@Po]:
> *Does the equation $\textstyle x^3+y^3+z^3=29$ have a solution in integers? Yes: $(3, 1,
> 1)$, for instance. How about $\textstyle x^3+y^3+z^3=30$? Again yes, although this was not known until 1999: the smallest solution is $(-283059965, -2218888517, 2220422932)$. And how about $\textstyle x^3+y^3+z^3=33?$ This is an unsolved problem.*
Given that the simple Diophantine equation $\textstyle N+x^3+y^3-z^3=0$ has solutions for $N=30$ but only beyond $10^9$ we may guess that the “worst case” Diophantine equation $f(N,x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9)=0$ needs *very* large $M_N$. In fact, the sequence $M_1,M_2,\dots$ has to be uncomputable (otherwise $\Om$ would be computable, but it is not). Some computable sequences grow fantastically fast. See Wikipedia: [“Ackermann function”](http://en.wikipedia.org/wiki/Ackermann function), [“Fast-growing hierarchy”](http://en.wikipedia.org/wiki/Fast-growing hierarchy). And nevertheless, no one of them bounds from above the sequence $M_1,M_2,\dots\,$ Reality beyond imagination!
Every computable number is definable, but a definable number need not be computable. Computability being another story, we return to definability.
From predicates to relations {#sect2}
============================
Recall the five definitions mentioned in the introduction. They should be special cases of a general notion “definition”. In order to formalize this idea we have to be more pedantic than in the introduction. *“Nothing but the hard technical story is any real good”* ([Littlewood](http://en.wikipedia.org/wiki/John Edensor Littlewood), [A Mathematician’s Miscellany](http://en.wikipedia.org/wiki/A Mathematician's Miscellany), page 70); exercises are waiting for you.
All mathematical objects (real numbers, limits, sets etc.) are treated in the framework of the mainstream mathematics, unless stated otherwise. Alternative approaches are sometimes mentioned in Sections \[sect9\], \[sect10\]. [Naive set theory](http://en.wikipedia.org/wiki/Naive_set_theory) suffices for Sections \[sect2\]–\[sect7\]; [axiomatic set theory](http://en.wikipedia.org/wiki/Set_theory#Axiomatic_set_theory) is used in Sections \[sect8\]–\[sect10\].
A definition is a text in a language. A straightforward formalization of such notions as “definition” and “definable” uses “formal language” (a formalization of “language”) and other notions of [model theory](http://en.wikipedia.org/wiki/Model
theory). Surprisingly, there is a shorter way. [Operations on sets](http://en.wikipedia.org/wiki/Set_(mathematics)#Basic_operations) are used instead of [logical symbols](http://en.wikipedia.org/wiki/First-order_logic#Logical_symbols), and [relations](http://en.wikipedia.org/wiki/Finitary relation) instead of [predicates](http://en.wikipedia.org/wiki/Predicate (mathematical logic)).
*“However, predicates have many different uses and interpretations in mathematics and logic, and their precise definition, meaning and use will vary from theory to theory.”* ([Quoted from Wikipedia.](http://en.wikipedia.org/wiki/Predicate
(mathematical logic))) Here we use predicates for informal explanations only; on the formal level they will be avoided (replaced with relations).
The number $\sqrt2$ was defined as the real number $x$ such that $P(x)$, where $P(x)$ is the predicate “$x>0$ and $x^2=2$”. This predicate is the conjunction $P_1(x)\land P_2(x)$ of two predicates $P_1(x)$ and $P_2(x)$, the first being “$x>0$”, the second “$x^2=2$”. The single-element set $A=\{x\in\R\mid P(x)\}=\{\sqrt2\}$ corresponding to the predicate $P(x)$ is the intersection $A=A_1\cap A_2$ of the sets $A_1=\{x\in\R\mid P_1(x)\}=(0,\infty)$ and $A_2=\{x\in\R\mid P_2(x)\}=\{-\sqrt2,\sqrt2\}$. (Here and everywhere, $\R$ is the set of all real numbers.)[^3]
This is instructive. *In order to formalize a definition of a number via its defining property, we have to deal with sets of numbers, and more generally, relations between numbers.*
Also, $x^2$ is the product $x\cdot x$, and $2$ is the sum $1+1$. But what is “product”, “sum”, “1” and “0”? The answer is given by the [axiomatic approach to real numbers](http://en.wikipedia.org/wiki/Real number#Axiomatic_approach): they are a complete totally ordered field. It means that addition, multiplication and order are defined and have the appropriate properties. Thus, 0 is defined as the real number $x$ satisfying the condition $\forall y \;
(x+y=y)$. Similarly, 1 is defined as the real number $x$ satisfying the condition $\forall y \; (x\cdot y=y)$.
Now we need predicates with two and more variables. The order is a binary (that is, with two variables) predicate “$x\le
y$”. Addition is a ternary (that is, with three variables) predicate “$x+y=z$”. Similarly, multiplication is a ternary predicate “$xy=z$” (denoted also “$x\cdot y=z$” or “$x\times y=z$”).
Each unary (that is, with one variable) predicate $P(x)$ on real numbers leads to a set $\{x\in\R\mid P(x)\}$ of real numbers, a subset of the real line $\R$. Likewise, each binary predicate $P(x,y)$ on reals leads to a set $\{(x,y)\in\R^2\mid P(x,y)\}$ of pairs of real numbers, a subset of the Cartesian plane $\R^2=\R
\times \R$, the latter being the [Cartesian product](http://en.wikipedia.org/wiki/Cartesian product#A_two-dimensional_coordinate_system) of the real line by itself. On the other hand, a [binary relation](http://en.wikipedia.org/wiki/Binary relation) on $\R$ is defined as an arbitrary subset of $\R^2$.
Thus, each binary predicate on reals leads to a binary relation on reals. If we swap the variables, that is, turn to another predicate $Q(x,y)$ that is $P(y,x)$, then we get another relation $\{(x,y)\mid Q(x,y)\}=\{(x,y)\mid
P(y,x)\}=\{(y,x)\mid P(x,y)\}$, [inverse](http://en.wikipedia.org/wiki/Inverse relation) (in other words, converse, or opposite) to the former relation (generally different, but sometimes the same).
Similarly, each ternary predicate on reals leads to a ternary relation on reals; and, changing the order of variables, we get $3!=6$ ternary relations (generally, different) corresponding to 6 [permutations](http://en.wikipedia.org/wiki/Permutation) of 3 variables. And generally, each predicate on reals leads to a relation on reals (a subset of $\R^n$); and, changing the order of variables, we get $n!$ such relations. The case $n=1$ is included (for unification); a unary relation on reals (called also property of reals) is a subset of $\R$.
Thus, on reals, the order is the binary relation $\{(x,y)\mid x\le y\}$, the addition is the ternary relation$\{(x,y,z)\mid x+y=z\}$, and the multiplication is the ternary relation $\{(x,y,z)\mid xy=z\}$. Still, we cannot forget predicates until we understand how to construct new relations out of these basic relations. For example, how to construct the binary relation $\{(x,y)\mid x+y=y\}$ and the unary relation $\{x\mid \forall y \; (x+y=y)\}\,$? We know that if a predicate $P(x)$ is the conjunction $P_1(x)\land P_2(x)$ of two predicates, then it leads to the intersection $A=A_1\cap A_2$ of the corresponding sets. Similarly, the disjunction $P_1(x)\lor P_2(x)$ leads to the union $A=A_1\cup
A_2$, and the negation $\neg P_1(x)$ leads to the complement $A=\R\setminus A_1$. Also, the implication $P_1(x)\imp
P_2(x)$ leads to $A=(\mathbb R\setminus A_1)\cup A_2$, and the equivalence $P_1(x)\equ P_2(x)$ leads to $A=\((\R\setminus A_1)\cap(\R\setminus A_2)\) \cup (A_1\cap A_2)$. The same holds for predicates; the disjunction $P_1(x,y)\lor P_2(x,y)$ still corresponds to the union $A=A_1\cup A_2$, the negation $\neg P_1(x,y,z)$ to the complement $A=\R^3\setminus A_1$, etc. But what to do when $P(x,y)$ is $P_1(x,y,y)$, or $P(x)$ is $\forall y \;
P_1(x,y)$, or $P(x,y,z)$ is $P_1(y,x)\land P_2(y,z)$, etc?
This question was answered, in context of axiomatic set theory, in the first half of the 20th century. A somewhat different answer, in the context of definability, was given by van den Dries in 1998 [@Dr1],[@Dr2] and slightly modified by Auke Bart Booij in 2013 [@Bo]; see also Macintyre 2016 [@Mac “Defining First-Order Definability”]. Here is the answer (slightly modified).
First, in addition to the Boolean operations (union and complement; intersection is superfluous, since it is complement of the union of complements) on subsets of $\R^n$, we introduce permutation of coordinates; for example ($n=3$), $A=\{(x,y,z)\in\R^3\mid(z,x,y)\in A_1\}$; and in general, $$A=\{(x_1,\dots,x_n)\in\R^n\mid(x_{i_1},\dots,x_{i_n})\in A_1\}$$ where $(i_1,\dots,i_n)$ is an arbitrary permutation of $(1,\dots,n)$.
In particular, permutation of coordinates in a binary relation gives the inverse relation. For example, the inverse to $\{(x,y)\mid x\le y\}$ is $\{(x,y)\mid y\le x\}=\{(x,y)\mid x\ge y\}$. And, by the way, the intersection of these two is the relation $\{(x,y)\mid x=y\}$ (corresponding to the predicate “$x=y$”).
Second, set multiplication, in other words, [Cartesian product](http://en.wikipedia.org/wiki/Cartesian product) by $\R$: $A=A_1\times\R$, that is, $$A=\{(x_1,\dots,x_{n+1})\in\R^{n+1}\mid(x_1,\dots,x_n)\in A_1\},$$ turns a relation to a relation that is formally , but the last variable is unrelated to others.
Now, returning to a predicate $P(x,y,z)$ of the form $P_1(y,x)\land P_2(y,z)$, we treat the corresponding ternary relation $A=\{(x,y,z)\in\R^3\mid P(x,y,z)\}$ as the intersection of two ternary relations $A_1=\{(x,y,z)\in\R^3\mid
P_1(y,x)\}$ and $A_2=\{(x,y,z)\in\R^3\mid P_2(y,z)\}$; and $A_1$ as the Cartesian product of the binary relation $B_1=\{(x,y)\in\R^2\mid P_1(y,x)\}$ by $\R$, $B_1$ being inverse to the relation $B_2=\{(x,y)\in\R^2\mid P_1(x,y)\}$ (corresponding to the given predicate $P_1(x,y)$); and $A_2$ as obtained (by permutation of coordinates) from the Cartesian product $\{(y,z,x)\in\R^3\mid P_2(y,z)\} = \{(y,z)\in\R^2\mid P_2(y,z)\}\times\R$ (by $\R$) of the relation corresponding to the given predicate $P_2(y,z)$.
Third, the projection; for example ($n=1$), $A=\{x\mid\exists y\in\R \; \((x,y)\in A_1\)\}$; and in general, $$A=\{(x_1,\dots,x_n)\mid\exists x_{n+1}\in\R \; \((x_1,\dots,x_{n+1})\in A_1\)\};$$ it turns a $(n+1)$-ary relation to a $n$-ary relation. For $n=1$ the set $A$ is also called the [domain of the binary relation](http://en.wikipedia.org/wiki/Binary relation#Formal_definition) $A_1$.
Now, returning to a predicate $P(x,y)$ of the form $P_1(x,y,y),\,$ we rewrite it as “$\exists z\; \(P_1(x,y,z) \land
y=z\)$” and treat the corresponding binary relation as the projection of the ternary relation $\{(x,y,z)\mid
P_1(x,y,z)\} \cap \{(x,y,z)\mid y=z\}$, and $\{(x,y,z)\mid y=z\}$ as a permutation of the Cartesian product $\{(y,z)\mid
y=z\}\times\R$.
What if $P(x)$ is “$\forall y \; P_1(x,y)$”? Then we rewrite it as “$\neg\exists y \; \neg P_1(x,y)$” and get the complement of the projection of the complement of the relation corresponding to $P_1(x,y)$.
So, we accept the 3 given relations (order, addition, multiplication) as “definable”, and we accept the 5 operations (complement, union, permutation, set multiplication, projection) for producing definable relations out of other definable relations. Thus we get infinitely many definable relations (unary, binary, ternary and so on).
More formally, these relations are called “first-order definable (without parameters) over $(\R; \le, +, \times)$”; but, less formally, “definable over” is often replaced with “definable in” (and sometimes “definable from”); “without parameters” is omitted throughout this essay; also “first order” and “over $(\R; \dots)$” are often omitted in this section. See Wikipedia: “Definable set”: [Definition](http://en.wikipedia.org/wiki/Definable
set#Definition); [The field of real numbers](http://en.wikipedia.org/wiki/Definable set#The_field_of_real_numbers).
Generally, starting from a set (not necessarily the real line) and some chosen relations on this set (including the equality relation if needed), and applying the 5 operations (complement, union, permutation, set multiplication, projection) repeatedly (in all possible combinations), one obtains an infinite collection of relations (unary, binary, ternary and so on) on the given set. Every such collection of relations is called a structure (Booij [@Bo]), or a VDD-structure ([Brian Tyrrell](https://www.maths.tcd.ie/~btyrrel/) [@Ty]) on the given set. According to Tyrrell [@Ty page 3], “The advantage of this definition is that no model theory is then needed to develop the theory”. The technical term “VDD structure” (rather than just “structure” used by van den Dries and Booij) is chosen by Tyrrell “to prevent a notation clash” (Tyrrell [@Ty page 2]), since many other structures of different kinds are widely used in mathematics. “VDD” apparently refers to van den Dries who pioneered this approach. But let us take a shorter term “”, where “D” refers to “definable” and “Dries” as well. The obtained (by the 5 operations) from the chosen relations, in other words, generated by these relations, is the smallest containing these relations.
Generality aside, we return to the special case, the of definable relations on the real line defined above (generated by order, addition and multiplication; though, the order appears to be superfluous).
Prove that a relation is definable in $(\R; \le, +, \times)$ if and only if it is definable in $(\R;
+, \times)$. *Hint:* $x\le y$ if and only if $\exists z\in\R \; (x+z^2=y)$.
We say that a number $x$ is definable, if the single-element set $\{x\}$ is a definable unary relation.
Prove that the numbers 0 and 1 are definable. *Hint:* recall “$\forall y \; (x+y=y)$” and “$\forall y \;
(x\cdot y=y)$”.
Prove that the sum of two definable numbers is definable. *Hint:* $\exists y\in\R \; \exists z\in\R
\;\, \((y\in A_1) \land (z\in A_2) \land (y+z=x)\)$.
Prove that the number $\frac{355}{113}$ is definable. *Hint:* $\exists y\in\R \; \exists z\in\R \;\,
(y=113 \land z=355 \land xy=z)$.
Prove that the number $\sqrt2$ is definable. *Hint:* $(x>0)\land(x\cdot x=2)$.
Prove that the golden ratio $\varphi$ is definable.
Prove that the binary relation “$y=|x|$” is definable. *Hint:* $(x^2=y^2 \land y\ge0)$.
In contrast, the ternary relation “$x^y=z$” is not definable. Moreover, the binary relation $\{(x,y)\mid y=2^x \land
0\le x\le1 \}$ is not definable. The problem is that all relations definable in $(\R; +, \times)$ are [semialgebraic sets](http://en.wikipedia.org/wiki/Semialgebraic set) over (the subring of) integers.[^4]
Thus, we cannot define the number $e$ via $(1+x)^{1/x}$ in this framework. Also, only [algebraic numbers](http://en.wikipedia.org/wiki/Algebraic number) are definable in this framework.
Each natural number is definable, which does not mean that the set $\N$ of all natural numbers is definable (in $(\R; +, \times)$). In fact, it is not![^5]
We could accept the set $\N$ of natural numbers as definable, that is, turn to definability in $(\R; +,\times,\N)$, but does it help to define the number $e$? Surprisingly, it does! “\[…\] then the situation changes drastically” (van den Dries [@Dr1 Example 1.3]). See also Booij [@Bo page 17]: “\[…\] if we add the seemingly innocent set $\mathbf
Z$ to the tame structure of semialgebraic sets, we get a wild structure \[…\]”
Beyond the algebraic {#sect3}
====================
**In this section, “definable” means “first order definable in $(\R; +,\times,\N)$”.** In other words, the real line is endowed with the generated by addition, multiplication, and the set of natural numbers. Good news: we’ll see that the five numbers $ \sqrt {2}, \varphi, e, \pi, \Om, $ discussed in Introduction, are definable. Bad news: in addition to their usual definitions we’ll use Diophantine equations, computability and Matiyasevich’s theorem (mentioned in Introduction in relation to Chaitin’s constant). The reader not acquainted with computability theory should rely on intuitive idea of computation (instead of formal proofs of computability), and consult the linked Wikipedia article for computability-related notions (“recursively enumerable”, “computable sequence”). Alternatively, the reader may skip to Section \[sect5\]; there, usual definitions will apply, no computability needed.
Every [Diophantine set](http://en.wikipedia.org/wiki/Diophantine set) $$\{(a_1,\dots,a_n)\in\N^n\mid \exists x_1,\dots,x_m\in\N \;\; p(a_1,\dots,a_n,\,x_1,\dots,x_m)=0 \}$$ (where $p(\dots)$ is a polynomial with integer coefficients), treated as a subset of $\R^n$, is a definable relation. And every [recursively enumerable](http://en.wikipedia.org/wiki/Recursively enumerable set) set is Diophantine.
For every [computable sequence](http://en.wikipedia.org/wiki/Integer sequence#Computable_and_definable_sequences) $(k_1,k_2,\dots)$ of natural numbers, the binary relation $\{(n,k)\in\N^2 \mid k=k_n\} = \{ (1,k_1), (2,k_2), \dots \}$ is recursively enumerable, therefore definable.
In particular, the binary relation “$x\in\N$ and $y=x^x$” is definable, as well as “$x\in\N$ and $y=(x+1)^x$”. Now (at last!) the number $e$ is definable, via $\lim_{n\to\infty} (1+\tfrac1n)^n = \lim_{n\to\infty} \tfrac{(n+1)^n}{n^n}$; more formally, $e$ is the real number $x$ satisfying the condition $$\forall \varepsilon>0 \; \exists n\in\N\; \forall m\in\N \;\;
\( m\ge n \imp -\varepsilon m^m < (m+1)^m-e m^m < \varepsilon m^m \).$$ This is not quite the definition mentioned in Introduction, but equivalent to it.
Similarly, for every convergent computable sequence of rational numbers, its limit is a definable number. In other words, every [limit computable](http://en.wikipedia.org/wiki/Computation in the
limit#Limit_computable_real_numbers) real number is definable.
Every computable real number is limit computable, therefore definable. In particular, the number $\pi$ is computable, therefore definable.
Chaitin’s constant is not computable, but still, limit computable (recall Introduction: it is the limit of a computable increasing sequence of rational numbers), therefore definable. So, all the five constants discussed in Introduction (taken from the book “Mathematical constants”) are definable. Moreover, all the constants discussed in that book are definable.
On the other hand, if we choose a number between 0 and 1 at random, according to the [uniform distribution](http://en.wikipedia.org/wiki/Uniform distribution (continuous)), we [almost surely](http://en.wikipedia.org/wiki/Almost surely) get an undefinable number, because the definable numbers are a [countable set](http://en.wikipedia.org/wiki/Countable set).
Of course, such a randomly chosen undefinable number is not an explicit example of undefinable number. It may seem that “explicit example of undefinable number” is a patent nonsense, just as “defined undefinable number”. But no, not quite nonsense, see Section \[sect4\].
An infinite sequence $(x_1,x_2,\dots)=(x_n)_n$ of real numbers is nothing but the binary relation $\{(n,x)\mid n\in\N
\land x=x_n\} =$ $\{(1,x_1),(2,x_2),\dots\}$; if this binary relation is definable, we say that the sequence is definable. If a sequence is definable, then all its members are definable numbers. However, a sequence of definable numbers is generally not definable.
If a definable sequence converges, then its limit is a definable number. Prove it.
A function $f:\R \to \R$ is nothing but the binary relation “$f(x)=y$”, that is, $A=\{(x,y)\mid f(x)=y\}$; if this binary relation is definable, we say that the function is definable. An arbitrary binary relation $A$ is a function if and only if for every $x$ there exists one and only one $y$ such that $(x,y)\in A$.
If $f$ is a definable function and $x$ is a definable number, then $f(x)$ is a definable number. Prove it.
However, a function that has definable values at all definable arguments is generally not definable.
If a definable function is differentiable, then its derivative is a definable function. Prove it. *Hint:* the derivative is the limit of…
If a definable function $f$ is continuous, then its [antiderivative](http://en.wikipedia.org/wiki/Antiderivative) $F$ is definable if and only if $F(0)$ is a definable number. Prove it. *Hint:* $F(x)=F(0)+\lim_{n\to\infty} \frac x n \sum_{k=1}^n f(\frac k n x)$.
Similarly to the number $e$ we can treat the exponential function $x\mapsto e^x$. First, the relation $
\big\{(n,p,q,u)\mid n\in\N \land p\in\N \land q\in\N \land u=\(1+\frac p q \cdot \frac 1n\)^n\big\}$ is definable (since $ \(1+\frac p q \cdot \frac 1n\)^n$ is a computable function of $n,p,q$). Second, the relation $\{(x,y)\mid y=e^x\}$ is definable, since $e^x$ is the limit of $ \(1+\frac p q \cdot \frac 1n\)^n$ as $n$ tends to infinity and $\tfrac p q$ tends to $x$; more formally (but still not completely formally…), $y=e^x$ if and only if $$\begin{gathered}
\forall \varepsilon>0 \; \exists \delta>0 \; \forall n\in\N\, \forall p\in\Z\, \forall q\in\N\,
\forall u \;\; \\
\bigg( \Big(n\ge\frac1\delta\Big) \land \Big(-\delta<x-\frac p q<\delta\Big) \land \Big(u=\(1+\tfrac p q \cdot
\tfrac 1n\)^n\Big) \imp \varepsilon < y-u < \varepsilon \bigg);\end{gathered}$$ here $\Z$ is the set of integers (evidently definable).
The cosine function may be treated via complex numbers and [Euler’s formula](http://en.wikipedia.org/wiki/Euler's formula) $e^{ix}=\cos x + i \sin x$. First, the real part of the complex number $ \(1+i\frac p q \cdot \frac
1n\)^n$ is a computable function of $n,p,q$. Second, its limit as $n$ tends to infinity and $\tfrac p q$ tends to $x$ is equal to the real part $\cos x$ of the complex number $e^{ix}$.
Note that the [exponential integral](http://en.wikipedia.org/wiki/Exponential integral) $\operatorname{Ei}(x)$ and the [sine integral](http://en.wikipedia.org/wiki/Trigonometric integral#Sine_integral) $\operatorname{Si}(x)$ are definable [nonelementary functions](http://en.wikipedia.org/wiki/Nonelementary integral).
Definable functions can be [pathological](http://en.wikipedia.org/wiki/Pathological (mathematics)) and disrespect dimension. In particular, there is a definable one-to-one correspondence between the (two-dimensional) square $(0,1)\times(0,1)$ and a subset of the (one-dimensional) interval $(0,1)$, which will be used in Section \[sect6\]. Here is a way to this fact.
Given two numbers $x,y\in(0,1)$, we consider their decimal digits: $ x=(0.\al_1\al_2\dots)_{10}=\sum_{n=1}^\infty
10^{-n}\al_n$ where $\al_n\in\{0,1,2,3,4,5,6,7,8,9\}$ for each $n$, and the set $\{n:\al_n\ne9\}$ is infinite (since we represent, say, $\tfrac12$ as $(0.5000\dots)_{10}$ rather than $(0.4999\dots)_{10}$); and similarly $y=(0.\be_1\be_2\dots)_{10}$. We interweave their digits, getting a third number $z=(0.\al_1\be_1\al_2\be_2\dots)_{10} \in (0,1)$. The ternary relation between such $x,y,z$ is a function $W_2:(0,1)\times(0,1)\to(0,1)$. Not all numbers of $(0,1)$ are of the form $W_2(x,y)$ (for example, $\frac{21}{1100}=(0.01909090\dots)_{10}$ is not), which does not matter. It does matter that $x,y$ are uniquely determined by $W_2(x,y)$, that is, $W_2(x_1,y_1)=W_2(x_2,y_2)$ implies $x_1=x_2 \land y_1=y_2$. In other words, $W_2$ is an [injection](http://en.wikipedia.org/wiki/Injective function) $(0,1)\times(0,1)\to(0,1)$.
Denoting by $D(n,x)$ the $n$-th decimal digit $\al_n$ of $x\in(0,1)$ we have $D(n,x)=\lfloor 10 \cdot
\operatorname{frac} (10^{n-1}x) \rfloor$; here $\lfloor a \rfloor$ is the [integer part](http://en.wikipedia.org/wiki/Floor and
ceiling functions) of $a$, and $\operatorname{frac}(a)=a-\lfloor a \rfloor$ is the [fractional part](http://en.wikipedia.org/wiki/Fractional part) of $a$.
The integer part function is definable. Prove it. *Hint:* $\{(x,\lfloor x \rfloor) \mid x>0\} = \{(x,n)\mid x>0
\land n+1\in\N \land n\le x<n+1\}$.
The function $D:\N\times(0,1)\to\R$ is definable. (See Booij [@Bo Lemma 3.4].) Prove it. *Hint:* $D(n,x)=d \equ \exists k\in\N \;\; \exists y\in\R \;\; \( k=10^{n-1} \land
y=\operatorname{frac}(kx) \land d=\lfloor 10y\rfloor \)$.
The function $W_2:(0,1)^2\to(0,1)$ is definable. Prove it. *Hint:* $z=W_2(x,y) \equ \forall
n\in\N \;\; \( D(2n,z)=D(n,y) \land D(2n-1,z)=D(n,x) \)$.
Generalize the previous exercise to $W_3:(0,1)^3\to(0,1)$. *Hint:* consider $D(3n-2,z)$, $D(3n-1,z)$, $D(3n,z)$.
Explicit example of undefinable number {#sect4}
======================================
We construct such example in two steps. First, we enumerate all numbers definable in $(\R; +,\times,\N)$ (“first order” is meant but omitted, as before); that is, we construct a sequence $(x_1,x_2,\dots)$ of real numbers that contains all numbers definable in $(\R; +,\times,\N)$ (and only such numbers). Second, we construct a real number not contained in this sequence.
The second step is well-known and simple, so let us do it now, for an arbitrary sequence $(x_1,x_2,\dots)$ of real numbers. We construct a real number $x$ via its decimal digits, as $ x=\sum_{n=1}^\infty \frac{\al_n}{10^n}$, and we choose each $\al_n$ to be different from the digit (after the decimal point) of the absolute value $|x_n|$ of $x_n$. To be specific, let us take $\al_n=3$ if $10k+7\le 10^n |x_n|<10k+8$ for some integer $k$, and $\al_n=7$ otherwise. Then $x\ne x_n$ since the integral part of $10^n|x|$, being of the form $10\ell+\al_n$ for integer $\ell$, is different from the integral part of $10^n|x_n|$, the latter being of the form $10k+\be_n$ for integer $k$, and $\al_n\ne\be_n$ (either $\be_n=7,\al_n=3$ or $\be_n\ne7,\al_n=7$). This is an instance of [Cantor’s diagonal argument](http://en.wikipedia.org/wiki/Cantor's diagonal argument).
Now we start constructing a sequence $(x_1,x_2,\dots)$ of real numbers that contains all numbers definable in $(\mathbb
R; +,\times,\N)$ (and only such numbers). These numbers being elements of single-element subsets of $\R$ definable in $(\R; +,\times,\N)$, and these subsets being unary relations, we enumerate all relations (unary, binary, …) definable in $(\R; +,\times,\N)$. These are obtained from the three given relations (addition, multiplication, “naturality”) via the 5 operations (complement, union, permutation, set multiplication, projection) applied repeatedly. We may save on permutations by restricting ourselves to [adjacent transpositions](http://en.wikipedia.org/wiki/Cyclic_permutation#Transpositions), that is, permutations that swap two adjacent numbers $k,k+1$ and leave intact other numbers of $\{1,\dots,n\}$; this is sufficient, since every permutation is a product of some adjacent transpositions. We start with the three given relations $$\begin{aligned}
A_1&=\{(x,y,z)\mid x+y=z\}, &&\text{``addition''} \\
A_2&=\{(x,y,z)\mid xy=z\}, &&\text{``multiplication''} \\
A_3&=\N, &&\text{``naturality''}\end{aligned}$$ and apply to them the five operations (whenever possible). The first operation “complement” gives $$\begin{aligned}
A_4&=\{(x,y,z)\mid x+y\ne z\}, \\
A_5&=\{(x,y,z)\mid xy\ne z\}, \\
A_6&=\{x\mid x\notin\N\}.\end{aligned}$$ The “union” operation gives $$A_7=\{(x,y,z)\mid (x+y=z) \lor (xy=z)\}.$$ The “permutation” operation (reduced to adjacent transpositions), applied to the ternary relation $A_1$, gives two relations $A_8,A_9$; namely, $A_8=\{(x,y,z)\mid y+x=z\}$ (equal to $A_1$ due to commutativity, but we do not bother) and $A_9=\{(x,y,z)\mid x+z=y\}$; we apply the same to $A_2$ getting $A_{10},A_{11}$. Further, “set multiplication” gives the 4-ary relation $$A_{12}=\{(x,y,z,w)\mid x+y=z\},$$ similarly $A_{13}$, and $A_{14}=\{(x,y)\mid x\in\N\}$. The most remarkable “projection” operation gives $$A_{15}=\{(x,y)\mid \exists z \; (x+y=z)\}$$ (in fact, $A_{15}=\R^2$), and similarly $A_{16}$.
The first 16 relations $A_1,\dots,A_{16}$ are thus constructed. On the next iteration we apply the 5 operations to these 16 relations (whenever possible; though, some are superfluous) and get a longer finite list. And so on, endlessly. A bit cumbersome, but really, a routine exercise in programming, isn’t it? Well, it is, provided however that the “programming language” stipulates the data type “relation over $\R$” and the relevant operations on relations. By the way, equality test for relations is not needed (unless we want to skip repetitions); but test of existence and uniqueness (for unary relations), and extraction of the unique element, are needed for the next step.
Now we are in position to construct $x_n$; for each $n$ we check, whether the relation $A_n$ is of the form $\{u\}$ for $u\in\R$ or not; if it is, we take $x_n=u$, otherwise $x_n=0$. (Note that $x_n=0$ whenever the relation $A_n$ is not unary.)
Applying the diagonal argument (above) to this sequence $(x_1,x_2,\dots)$ we construct a real number $x$ not contained in the sequence, therefore, not definable in $(\R; +,\times,\N)$.
This number $x$ is defined, but not in $(\R; +,\times,\N)$. Why not? Because the definition of $x$ involves a sequence of relations in $\R$. Sequences of numbers are used in Section \[sect3\], but sequences of relations are something new, beyond the first order. (See Wikipedia: [First-order logic](http://en.wikipedia.org/wiki/First-order logic), [Second-order logic](http://en.wikipedia.org/wiki/Second-order logic).)
Is there a better approach? Could we define in $(\R; +,\times,\N)$ the same sequence $(x_1,x_2,\dots)$, or maybe another sequence containing all computable numbers, by a clever trick? No, this is impossible. For every sequence definable in $(\R; +,\times,\N)$ the diagonal argument gives a number definable in $(\mathbb R; +,\times,\N)$ and not contained in the given sequence.
> $\bullet$ *there is no definable enumeration of definable reals* (Poincaré 1909), see [Stanford Encyclopedia of Philosophy: Paradoxes and Contemporary Logic](https://plato.stanford.edu/entries/paradoxes-contemporary-logic/).
Second order {#sect5}
============
We introduce second-order definability in $(\R; +, \times)$. The set $\N$ of natural numbers is second-order definable in $(\R; +, \times)$, as we’ll see soon. In contrast to the first order definability, usual definitions of mathematical constants will apply without recourse to computability and Diophantine sets.
A [second-order predicate](http://en.wikipedia.org/wiki/Second-order predicate) is a predicate that takes a first-order predicate as an argument. Likewise, a second-order relation is a relation between relations. For example, the binary relation $f'=g$ between a function $f$ and its derivative $g$ may be thought of as a relation between two binary relations: first, the relation $f(x)=y$ between real numbers $x,y$, and second, the relation $g(t)=v$ between real numbers $t,v$. First order definability of a real number involves definable first-order relations (between real numbers). Second order definability of a real number involves definable second-order relations (between first-order relations). Here is a possible formalization of this idea.
We introduce the set $$\! \mathbf S = \( \R \cup \R^2 \cup \R^3 \cup \dots \) \cup \( \oP(\R)
\cup \oP(\R^2) \cup \oP(\R^3) \cup \dots \) = \bigg( \bigcup_{n=1}^\infty
\R^n \bigg) \cup \bigg( \bigcup_{n=1}^\infty \oP(\R^n) \bigg)$$ that contains, on one hand, all tuples (finite sequences) $(x_1,\dots,x_n)\in\R^n$ of real numbers (for all $n$; here we do not distinguish 1-tuples from real numbers), and on the other hand, all $n$-ary relations $A\subset\mathbb R^n$ on $\R$ (for all $n$). Here $\oP(\R)$ is the set of all subsets (that is, the [power set](http://en.wikipedia.org/wiki/Power
set)) of the real line $\R$, in other words, of unary relations on $\R$; $\oP(\R^2)$ is the set of all subsets (that is, the [power set](http://en.wikipedia.org/wiki/Power set)) of the Cartesian plane $\R^2$, in other words, of binary relations on $\R$; and so on. On this set $\mathbf S$ we introduce two relations:
membership, the binary relation $\cup_{n=1}^\infty \{ \((x_1,\dots,x_n),A\)\mid (x_1,\dots,x_n)\in A \}
\subset \mathbf S^2 $; it says that the given $n$-tuple belongs to the given $n$-ary relation;
“appendment”, the ternary relation $\cup_{n=1}^\infty \{ \( x, (x_1,\dots,x_n), (x_1,\dots,x_n,x) \mid
x_1,\dots,x_n,x \in \R \} \subset \mathbf S^3 $; it says that the latter tuple results from the former tuple by appending the given real number.
The two ternary relations on $\R$, addition and multiplication, may be thought of as ternary relations on $\mathbf S$ (since $\R \subset \mathbf S$):
addition: $\{(x,y,z)\mid x,y,z\in\R, x+y=z\} \subset \mathbf S^3$;
multiplication: $\{(x,y,z)\mid x,y,z\in\R, xy=z\} \subset \mathbf S^3$.
We endow $\mathbf S$ with the generated by the four relations (membership, appendment, addition, multiplication). All relations on $\mathbf S$ that belong to this will be called second-order definable. In the rest of this section, “definable” means “second-order definable”, unless stated otherwise.
The set $\cup_{n=1}^\infty \R^n$ of all tuples and the set $\cup_{n=1}^\infty \oP(\R^n)$ of all relations are definable subsets of $\mathbf S$. Prove it. *Hint:* first, the set of all tuples is $\{ s\in
\mathbf S \mid \exists r,t \in \mathbf S \; (r,s,t)\in A \}$ where $A$ is the appendment relation; second, take the complement.
The set $\R$ of all real numbers is a definable subset of $\mathbf S$. Prove it. *Hint:* $\R = \{
r\in \mathbf S \mid \exists s,t\in \mathbf S \;(r,s,t)\in A \}$ where $A$ is the appendment relation.
Each $\R^n$ is a definable subset of $\mathbf S$. Prove it. *Hint:* $\R^{n+1} = \{ t\in \mathbf S
\mid \exists r \in \R \; \exists s \in \R^n \; (r,s,t)\in A \}$ where $A$ is the appendment relation.
The set $\oP(\R)$ of all sets of real numbers (that is, unary relations) is a definable subset of $\mathbf S$. Prove it. *Hint:* for $A\in\cup_{n=1}^\infty \oP(\R^n)$ we have $A\in \oP(\R) \equ A\subset\R \equ ( \forall a\in A
\;\> a\in\R) \equ \neg ( \exists a\in A \;\> a\notin\R)$; apply the projection to $\{(A,a) \mid A \in \cup_{n=1}^\infty
\oP(\R^n) \land a\in A \land a \notin \R \}$.
For each $n$ the set $\oP(\R^n)$ of all $n$-ary relations is a definable subset of $\mathbf S$. Prove it. *Hint:* similar to the previous exercise.
If $B\subset \oP(\R)$ is a definable set of subsets of $\R$, then the union $\cup_{A\in B} A$ of all these subsets is a definable set (of real numbers). Prove it. *Hint:* for $x\in\R$ we have $x\in\cup_{A\in B} A \equ \exists A \;
(A\in B \land x\in A)$; take the projection of $\{(x,A) \mid A\in B \land x\in A \} = (\R \times B) \cap \{(x,A)\mid
x\in A\}$.
Do the same for the intersection $\cap_{A\in B} A$. *Hint:* $\cap_{A\in B} A = \R \setminus \cup_{A\in B} (\R
\setminus A)$; consider $\{(x,A) \mid A\in B \land x\in \R \setminus A \} = (\R \times B) \cap \{(x,A)\mid x\notin
A\}$. (But what if $B$ is empty?)
Generalize the two exercises above to $B\subset \oP(\R^2)$, $B\subset \oP(\R^3)$ and so on. *Hint:* now $x$ is a tuple.
In particular, taking a single-element set $B=\{A\}$ we see that definability of $\{A\}$ implies definability of $A$. The converse holds as well (see below).
If a set $A\subset\R$ (of real numbers) is definable, then the set $\oP(A)$ (of all subsets of $A$) is definable. Prove it. *Hint:* for $A_1\in\oP(\R)$ we have $A_1\in\oP(A) \equ A_1 \subset A \equ (\forall x\in A_1\;\> x\in A)\equ \neg
( \exists x\in A_1\;\> x\notin A)$; consider $\{(A_1,x) \mid A_1\in\oP(\R) \land x\in\R \land x\in A_1 \land x\notin A\} =
\( \oP(\R) \times (\R\setminus A) \) \cap \{(A_1,x)\mid x\in A_1\}$.
Do the same for the set $\{A_1\in \oP(\R) \mid A_1\supset A\}$ (of all supersets of $A$). *Hint:* similarly to the previous exercise, consider $\( P(\R)\times A\) \cap \{(A_1,x)\mid x\notin A_1\}$.
Generalize the two exercises above to $A\subset\R^2$, $A\subset\R^3$ and so on.
*Remark.* These 11 exercises (above) do not use addition and multiplication, nor any properties of real numbers. They generalize readily to a more general situation. One may start with an arbitrary set $R$ (rather than the real line $\R$), consider the set $S$ constructed from $R$ as above (all tuples and all relations), endow $S$ by a such that the two relations on $S$, membership and appendment, are definable, and generalize the 11 exercises to this case.
Taking the intersection of the set of subsets and the set of supersets we see that definability of $A$ implies definability of the single-element set (called [singleton](http://en.wikipedia.org/wiki/Singleton (mathematics))) $B=\{A\}$. So, $A$ is definable if and only if $\{A\}$ is definable. And still (by convention, as before) a real number $x$ is definable if and only if $\{x\}$ is definable.
Does it mean that, for example, numbers $0$, $1$, $\sqrt2$ are definable (as well as every rational number and every algebraic number)? We know that they are first-order definable in $(\R; +,\times)$; does it follow that they are (second-order) definable in $\mathbf S?$
The answer is affirmative, but needs a proof. Here we face another general question. Let $S$ be a set and $R\subset S$ its subset. Every relation on $R$ is also a relation on $S$ (since $R\subset S \imp R^n\subset S^n \imp \oP(R^n) \subset \oP(S^n)$). Thus, given some relations on $R$, we get two s; first, the on $R$ generated by the given relations, and second, the on $S$ generated by the same relations.
**Lemma.** Assume that $R$ is a definable subset of $S$ (according to the second ). Then every relation on $R$ definable according to the first is also definable according to the second .[^6]
Now we are in position to prove definability of the set $\N$ of natural numbers. It is sufficient to prove definability of the set $B\subset\oP(\R)$ of all sets $A\subset\R$ satisfying the two conditions $1\in A$ and $\forall x\in A \;
(x+1\in A)$ (since the intersection of all these $A$ is $\N$). The complement $\oP(\R)\setminus B=\{A\in\oP(\R) \mid
\exists x\in\R \; (x\in A\land x+1\notin A)\}$ is the projection of the intersection of two sets, $\{(A,x)\in\oP(\mathbb
R)\times\R \mid x\in A\}$ and $\{(A,x)\in\oP(\R)\times\R \mid x+1\notin A\}$. The former results from the (permuted) membership relation; the latter is the projection of the projection of $\{(A,x,y,z)\in \oP(\R) \times \R^3 \mid
y\in\{1\} \land x+y=z \land z\notin A\}$, this set being the intersection of three sets: first, $\oP(\R) \times \R\times
\{1\} \times \R$; second, $\oP(\R)$ times the addition relation; third, $\oP(\R) \times \mathbb R\times\R \times
(\R\setminus A)$. It follows that $B$ is definable, whence $ \N = \cap_{A\in B} A$ is definable.
This is instructive. *In order to formalize a definition of a set via its defining property, we have to deal with sets of sets, and more generally, relations between sets.*
Using again the lemma above we see that all real numbers first-order definable in $(\R; +,\times,\N)$ are second-order definable. Section \[sect3\] gives many examples, including the five numbers $ \sqrt {2}, \varphi, e, \pi, \Om, $ discussed in Introduction. But second-order proofs of their definability are much more easy and natural.
The binary relation “$x\in\N \land y=x!$” is the sequence $(n!)_{n\in\N}$ of [factorials](http://en.wikipedia.org/wiki/Factorial), that is, the set $\{(1,1),(2,2),(3,6),(4,24),(5,120),\dots\}$. It is definable, similarly to $\N$, since it is the *least* subset $A$ of $\R^2$ such that $(1,1)\in A$ and $(x,y)\in A \imp \(x+1,(x+1)y\)\in A$. Alternatively, it is definable since it is the *only* subset $A$ of $\R^2$ with the following three properties: $$\begin{gathered}
\forall (x,y)\in A \;\; x\in\N, \\
\forall x\in\N \;\; \exists! y\in\R \;\; (x,y)\in A, \\
\forall (x,y)\in A \;\; \(x+1,(x+1)y\)\in A.\end{gathered}$$ That is, the factorial is the only function $\N \to \R$ satisfying the [recurrence relation](http://en.wikipedia.org/wiki/Recurrence relation) $(n+1)!=(n+1)n!$ and the initial condition $1!=1$.
Partial sums of the series $ \sum_{n=0}^\infty \frac1{n!}$ are a definable sequence. Prove it.
\[5.13\] The number $e$ is definable. Deduce it from the previous exercise.
In the first-order framework it is possible to treat many functions (for instance, the exponential function $x\mapsto
e^x$, the sine and cosine functions $\sin, \cos$, the exponential integral $\operatorname{Ei}$ and the sine integral $\operatorname{Si}$) and many relations between functions (for instance, derivative and antiderivative); arguments and values of these functions are arbitrary real numbers (not necessarily definable), but the functions are definable. Such notions as arbitrary functions (not necessarily definable), continuous functions (and their antiderivatives), differentiable functions (and their derivatives) need the second-order framework.
As was noted there, a function $f:\R \to \R$ is nothing but the binary relation “$f(x)=y$”, that is, $A=\{(x,y)\mid f(x)=y\}$. An arbitrary binary relation $A$ is such a function if and only if for every $x$ there exists one and only one $y$ such that $(x,y)\in A$ (existence and uniqueness). For functions defined on arbitrary subsets of the real line the condition is weaker: for every $x$ there exists at most one $y$ such that $(x,y)\in A$ (uniqueness).
\(a) All $A\in\oP(\R^2)$ satisfying the uniqueness condition are a definable subset of $\oP(\R^2)$; (b) the same holds for the existence and uniqueness condition. Prove it.
All continuous functions $\R \to \R$ are a definable subset of $\oP(\R^2)$. Prove it.
All differentiable functions $\R \to \R$ are a definable subset of $\oP(\R^2)$. Prove it.
The binary relation “$f'=g$” is definable. That is, the set of all pairs $(f,g)$ of functions $\R \to \R$ such that $\forall x\in\R \; \(f'(x)=g(x)\)$ is a definable subset of $\oP(\R^2)\times \oP(\R^2)$ (in other words, definable binary relation on $\oP(\R^2)$). Prove it.
Antiderivative can now be treated in full generality. In contrast, in the first-order framework it was treated via Riemann integral $ F(x)=F(0)+\lim_{n\to\infty} \frac x n \sum_{k=1}^n f(\frac k n x)$ for continuous definable $f$ only. In particular, now the exponential function $x\mapsto e^x$ may be treated via $f(e^x)-f(1)=x$ where $f'(x)=\tfrac1x$ for $x>0$; accordingly, the constant $e$ may be treated via $f(e)-f(1)=1$. Alternatively, the exponential function may be treated via the differential equation $f'=f$ (and initial condition $f(0)=1$). Trigonometric functions $\sin, \cos$ may be treated via the differential equation $f''=-f$; accordingly, the constant $\pi$ may be treated as the least positive number such that $(f''=-f) \imp \(f(\pi)=-f(0)\)$. Or, alternatively, as $ \pi = 4\int_0^1
\sqrt{1-x^2} \, dx$ (via antiderivative).
This is instructive. *In the second-order framework we may define functions (and infinite sequences) via their properties,* irrespective of computability, Diophantine equations and other tricks of the first-order framework.
Nice; but what about second-order definable real numbers? Are they all first-order definable, or not? Even if obtained from complicated differential equations, they are computable, therefore, first-order definable in $(\R; +,\times,\N)$. Probably, our only chance to find a second-order definable but first-order undefinable number is, to prove that the explicit example of (first-order) undefinable number, given in Section \[sect4\], is second-order definable; and our only chance to prove this conjecture is, to formalize that section within the second-order framework.
First-order undefinable but second-order definable {#sect6}
==================================================
Recall the infinite sequence of relations $(A_k)_{k=1}^\infty$ treated in Section \[sect4\]. Is it second-order definable? Each $A_k$ belongs to the set $\mathbf S$ (from Section \[sect5\]); their infinite sequence is a binary relation between $k$ and $A_k$ (namely, the set of pairs $\{(1,A_1),(2,A_2),\dots\}$), thus, a special case of a binary relation on $\mathbf S$; the question is, whether this relation is definable, or not. Like the sequence of factorials, it is defined by recursion. But factorials, being numbers, are first-order objects, which is why their sequence is second-order definable via its properties. In contrast, relations $A_k$ are second-order objects! Does it mean that third order is needed for defining their sequence by recursion?
True, the sequence of factorials is first-order definable (over $(\R; +,\times,\N)$) due to its computability, via Matiyasevich’s theorem. Could something like that be invented for second-order objects? Probably not.
Yet, these obstacles are surmountable. The sequence of relations may be replaced with a single relation by a kind of [currying](http://en.wikipedia.org/wiki/Currying) (or rather, uncurrying); the [disjoint union](http://en.wikipedia.org/wiki/Disjoint union) $\{1\}\times A_1 \cup \{2\}\times A_2 \cup \dots$ may be used instead of the set of pairs $\{(1,A_1),(2,A_2),\dots\}$. Further, relations $A_k$ of different [arities](http://en.wikipedia.org/wiki/Arity) may be replaced with unary relations (subsets of the real line), since two real numbers may be encoded into a single real number via an appropriate definable [injection](http://en.wikipedia.org/wiki/Injective function) $\R^2 \to \R$, and the same applies to three and more numbers (moreover, to infinitely many numbers, see Booij [@Bo Sect. 3.2]). In addition, tuples $(x_1,\dots,x_n)$ may be replaced (whenever needed) by finite sequences $(x_k)_{k=1}^n = \{(1,x_1),\dots,(n,x_n)\}$, which provides a richer assortment of definable relations.
The distinction between tuples and finite sequences is a technical subtlety that may be ignored in many contexts, but sometimes requires attention. It is tempting to say that an ordered pair $(a,b)$, a 2-tuple (that is, tuple of length 2), and a 2-sequence (that is, finite sequence of length 2) are just all the same. However, the 2-sequence is, by definition, a function on $\{1,2\}$, thus, the set of two ordered pairs $\{(1,a),(2,b)\}$. Surely we cannot define an ordered pair to be a set of two other ordered pairs! If sequences are defined via functions, and functions are defined via pairs, then pairs must be defined before sequences, and cannot be the same as 2-sequences. See Wikipedia: [sequence (formal definition)](http://en.wikipedia.org/wiki/Sequence#Formal_definition_and_basic_properties), [tuples (as nested ordered pairs)](http://en.wikipedia.org/wiki/Tuple#Definitions), and [ordered pair: Kuratowski’s definition](http://en.wikipedia.org/wiki/Ordered_pair#Defining_the_ordered_pair_using_set_theory). For convenience we’ll denote a finite sequence $(x_k)_{k=1}^n = \{(1,x_1),\dots,(n,x_n)\}$ by $[x_1,\dots,x_n]$; it is similar to, but different from, the tuple $(x_1,\dots,x_n)$.
We’ll construct again, this time in the second-order framework, the sequence $(x_1,x_2,\dots)$ of real numbers that contains all numbers first-order definable in $(\R; +,\times,\N)$, exactly the same sequence as in Section \[sect4\]. To this end we’ll construct first the disjoint union $\{1\}\times B_1
\cup \{2\}\times B_2 \cup \dots$ of unary relations $B_k$ on $\R$ similar to, but different from, relations $A_1,A_2,\dots$ (unary, binary, ...) constructed there (that exhaust all relations first-order definable in $(\mathbb
R; +,\times,\N)$).
Before the unary relations $B_k$ we construct 4-tuples $b_k$ of integers (call them “instructions”) imitating a program for a machine that computes $B_k$. Similarly to a [machine language instruction](http://en.wikipedia.org/wiki/Machine code), each $b_k$ contains an [operation code](http://en.wikipedia.org/wiki/Opcode), address of the first [operand](http://en.wikipedia.org/wiki/Operand#Computer_science), a parameter or address of the second operand (if applicable, otherwise 0), and in addition, the arity of $A_k$.
Recall Section \[sect4\]. Three relations $A_1,A_2,A_3$ of arities $3,3,1$ are given, and lead to the next $13$ relations $A_4,\dots,A_{16}$. In particular, $A_4$ is the complement of $A_1$. Accordingly, we let $b_4=(1,1,0,3)$; here, operation code $1$ means “complement...”, operand address $1$ means “...of $A_1$”, the third number $0$ is dummy, and the last number $3$ means that the relation $A_4$ is ternary. Similarly, $b_5=(1,2,0,3)$ and $b_6=(1,3,0,1)$.
Further, $A_7$ being the union of $A_1$ and $A_2$, we let $b_7=(2,1,2,3)$; operation code $2$ means “union…”, first operand address $1$ means “…of $A_1$”, second operand address $2$ means “…and $A_2$”, and again, $3$ is the arity of $A_7$.
Further, $A_8$ being a permutation of $A_1$, we let $b_8=(3,1,1,3)$; operation code $3$ means “permutation…”, operand address 1 means “…of $A_1$”, the parameter $1$ means “swap $1$ and $2$”, and $3$ is the arity of $A_8$. Similarly, $b_9=(3,1,2,3)$ (in $A_1$ swap $2$ and $3$), $b_{10}=(3,2,1,3)$ (in $A_2$ swap $1$ and $2$), $b_{11}=(3,2,2,3)$ (in $A_2$ swap $2$ and $3$).
Further, $A_{12}$ being $A_1\times\R$, we let $b_{12}=(4,1,0,4)$; operation code $4$ means “set multiplication”, $1$ refers to the operand $A_1$, and $4$ is the arity of $A_{12}$. Similarly, $b_{13}=(4,2,0,4)$ and $b_{14}=(4,3,0,2)$.
Further, $A_{15}$ being the projection of $A_1$, we let $b_{15}=(5,1,0,2)$; operation code 5 means “projection…”, $1$ means “…of $A_1$”, and $2$ means “…is binary”. Similarly, $b_{16}=(5,2,0,2)$.
This way, the finite sequence $[3,3,1]$ of natural numbers (interpreted as arities) leads to the finite sequence $[b_4,\dots,b_{16}]$ of 4-tuples (interpreted as instructions). Similarly, every finite sequence of natural numbers leads to the corresponding finite sequence of 4-tuples. The relation between these two finite sequences is definable; the proof is rather cumbersome, like a routine exercise in programming, but doable. Having this relation, we define an infinite sequence of 4-tuples $b_k$ (interpreted as the infinite “program”) together with an infinite sequence $(k_n)_{n=1}^\infty$ by the following defining properties:
$k_1=3; \;\;\; \forall n\in\N \;\; k_n<k_{n+1}$;
$b_1=b_2=(0,0,0,3); \; b_3=(0,0,0,1)$;
for every $n=1,2,\dots$ the finite sequence $[b_{k_n+1},\dots,b_{k_{n+1}}]$ of 4-tuples corresponds (according to the definable relation treated above) to the finite sequence of the natural numbers that are the last (fourth) elements of the 4-tuples $b_1,\dots,b_{k_n}$.
In particular, $k_1=3$, $k_2=16$; the third property for $n=1$ states that $[b_4,\dots,b_{16}]$ corresponds to $[3,3,1]$. And for $n=2$ it states that $[b_{17},\dots,b_{k_3}]$ corresponds to $[3,3,1,3,3,1,3,3,3,3,3,4,4,2,2,2]$. And so on.
The infinite program is ready. It could compute all relations $A_k$ if executed by a machine able to process relations of all arities. Is such machine available in our framework? The disjoint union $\{1\}\times A_1 \cup \{2\}\times A_2
\cup \dots$ could be used instead of the set of pairs $\{(1,A_1),(2,A_2),\dots\}$, but is not contained in (say) $\R^{100}$. True, in practice 100-ary relations do not occur in definitions; but we investigate definability in principle (rather than in practice). We encode all relation into unary relations as follows.
We recall the definable injective functions $W_2:(0,1)^2\to(0,1)$ and $W_3:(0,1)^3\to(0,1)$ treated in the end of Section \[sect3\]. The same works for any $(0,1)^m$. But we need to serve all dimensions $m$ by a single definable function. To this end we turn from sets $\R^m$ of $m$-tuples $(x_1,\dots,x_m)$ to sets, denote them $\mathbb
R^{[m]}$, of $m$-sequences $[x_1,\dots,x_m]$.
\[6.1\] The set of all finite sequences of real numbers, $\{ [x_1,\dots,x_m] \mid m\in\N,\,x_1,\dots,x_m\in\R\} =
\cup_{m=1}^\infty \R^{[m]} \subset \oP(\R^2)$, is definable, and the binary relation “length”, $\{ ([x_1,\dots,x_m],m) \mid
m\in\N,\,x_1,\dots,x_m\in\R\} = \cup_{m=1}^\infty \( \R^{[m]} \times \{m\} \) \subset \oP(\R^2)\times\R$, on $\mathbf
S$ is a definable function on that set. Prove it. *Hint:* start with the binary relation.
\[6.2\] The function $E:([x_1,\dots,x_m],k)\mapsto x_k$ (“evaluation”) is a definable real-valued function on the set $\cup_{m=1}^\infty \( \R^{[m]} \times \{1,\dots,m\} \)$. Prove it. *Hint:* for $s=[x_1,\dots,x_m]$, $k\in\N$ and $x\in\R$ we have $E(s,k)=x \equ (k,x)\in s \equ \exists p \( p\in s \land (k,x)=p \)$; use the two given relations on $\mathbf S$, membership and appendment (consider $n=1$ in the definition of appendment).
We choose a definable bijection $h:\R\to(0,1)$, for example, $ h(x)=\frac12(1+\frac{x}{1+|x|})$, and define a function $W:\cup_{m=1}^\infty \R^{[m]} \to \R$ by $W([x_1,\dots,x_m])=W_m\(h(x_1),\dots,h(x_m)\)$ for all $m\in\N$ and $x_1,\dots,x_m\in\R$.
The function $W$ is definable. Prove it. *Hint:* for all $m\in\N$, $x_1,\dots,x_m\in\R$ and $z\in\R$ we have $W([x_1,\dots,x_m])=z \equ W_m\(h(x_1),\dots,h(x_m)\)=z \equ \forall k\in\{1,\dots,m\} \;\; \forall n\in\N \;\;
D(m(n-1)+k,z) = D(n,h(x_k))$; that is, for all $m\in\N$, $s\in\R^{[m]}$ and $z\in\R$ holds $W(s)=z \equ \forall k\in\N
\;\; \( k\le m \imp \forall n\in\N \;\; D(m(n-1)+k,z)=D(n,h(E(s,k)) \)$.
At last, we are in position to “execute the infinite program” $(b_k)_{k=4}^\infty$, that is, to prove (second-order) definability of the set $B = \{1\}\times B_1 \cup \{2\}\times B_2 \cup \dots \subset \R^2$, the disjoint union of unary relations $B_k$ on $\R$ that encode (according to $W$) the relations $A_1,A_2,\dots$ (that exhaust all relations first-order definable in $(\R; +,\times,\N)$).
We extract $B_1=\{c\in\R \mid (1,c)\in B\}$, decode the ternary relation $\{[x,y,z]\mid W([x,y,z])\in B_1\} =
\{s\in\R^{[3]} \mid W(s)\in B_1\}$ and require it to be (like $A_1$) the addition relation $\{[x,y,z] \mid x+y=z\} =
\{s\in \R^{[3]} \mid E(s,1)+E(s,2)=E(s,3)\}$. That is, we require $$\forall s\in\R^{[3]} \;\; \( (1,W(s))\in B \equ E(s,1)+E(s,2)=E(s,3) \).$$ This condition fails to uniquely determine the set $B_1$, since the image of $\R^{[3]}$ under $W$ is not the whole $\R$ (not even the whole $(0,1)$). We prevent irrelevant points by requiring in addition that $\forall x\in\R \;\; \(
(1,x)\in B \imp \exists s\in \R^{[3]} \;\; W(s)=x \)$. We do not repeat such reservation below.
Similarly, $B_2$ must encode the multiplication relation, and $B_3$ must encode the set of natural numbers: $$\begin{gathered}
\forall s\in\R^{[3]} \;\; \( (2,W(s))\in B \equ E(s,1) E(s,2)=E(s,3) \); \\
\forall s\in\R^{[1]} \;\; \( (3,W(s))\in B \equ E(s,1) \in \N \).\end{gathered}$$ These first three requirements (above) are special. Other requirements should be formulated in general, like this: for every $k\ge4$, if the first element of $b_k$ (the operation code) equals 1, then (...), otherwise (...). But let us consider several examples before the general case.
According to the instruction $b_4$, the set $B_4$ must encode the complement of the set encoded by $B_1$: $$\forall s\in\R^{[3]} \;\; \( (4,W(s))\in B \equ (1,W(s))\notin B \);$$ similarly, $$\begin{aligned}
\forall s\in\R^{[3]} \;\; &\( (5,W(s))\in B \equ (2,W(s))\notin B \); \\
\forall s\in\R^{[1]} \;\; &\( (6,W(s))\in B \equ (3,W(s))\notin B \).\end{aligned}$$ According to the instruction $b_7$, the set $B_7$ must encode the union of the sets encoded by $B_1$ and $B_2$: $$\forall s\in\R^{[3]} \;\; \( (7,W(s))\in B \equ \( (1,W(s))\in B \lor (2,W(s))\in B \) \).$$ According to the instruction $b_8$, the set $B_8$ must encode the permutation of the set encoded by $B_1$: $$\forall [x,y,z]\in\R^{[3]} \;\; \( (8,W([x,y,z]))\in B \equ (1,W([y,x,z]))\in B \);$$ similarly, $$\begin{aligned}
\forall [x,y,z]\in\R^{[3]} \;\; &\( (9,W([x,y,z]))\in B \equ (1,W([x,z,y]))\in B \); \\
\forall [x,y,z]\in\R^{[3]} \;\; &\( (10,W([x,y,z]))\in B \equ (2,W([y,x,z]))\in B \); \\
\forall [x,y,z]\in\R^{[3]} \;\; &\( (11,W([x,y,z]))\in B \equ (2,W([x,z,y]))\in B \).\end{aligned}$$ According to the instruction $b_{12}$, the set $B_{12}$ must encode the Cartesian product (by $\R$) of the set encoded by $B_1$: $$\forall [x,y,z,u]\in\R^{[4]} \;\; \( (12,W([x,y,z,u]))\in B \equ (1,W([x,y,z]))\in B \);$$ similarly, $$\begin{aligned}
\forall [x,y,z,u]\in\R^{[4]} \;\; &\( (13,W([x,y,z,u]))\in B \equ (2,W([x,y,z]))\in B
\); \\
\forall [x,y]\in\R^{[2]} \;\; &\( (14,W([x,y]))\in B \equ (3,W([x]))\in B \).\end{aligned}$$ According to the instruction $b_{15}$, the set $B_{15}$ must encode the projection of the set encoded by $B_1$: $$\forall [x,y]\in\R^{[2]} \;\; \( (15,W([x,y]))\in B \equ \exists z\in\R \;\; (1,W([x,y,z]))\in B \);$$ similarly, $$\forall [x,y]\in\R^{[2]} \;\; \( (16,W([x,y]))\in B \equ \exists z\in\R \;\; (2,W([x,y,z]))\in B \).$$
Toward the general formulation. We observe that the first two cases (complement and union) are unproblematic, while the other three cases (permutation, set multiplication, and projection) need some additional effort. The informal quantifiers like “$\forall [x,y,z]$” should be replaced with “$\forall s$”, and the needed relations between finite sequences should be generalized (and formalized).
The binary relation of truncation $\{([x_1,\dots,x_{n+1}],[x_1,\dots,x_n]) \mid n\in\N,
x_1,\dots,x_{n+1}\in\R\}$ is definable. Prove it. *Hint:* use the evaluation function.
The ternary relation of appendment $\{(x,[x_1,\dots,x_n],[x_1,\dots,x_{n+1}]) \mid n\in\N,
x_1,\dots,x_n,x\in\R\}$ is definable. Prove it.
Now the reader should be able to compose himself the general formulation. Also the additional condition that prevents irrelevant points should be stipulated. We conclude that the set $B = \{1\}\times B_1 \cup \{2\}\times B_2 \cup \dots
\subset \R^2$ is definable. For each $n$ we check, whether the relation encoded by $B_n$ is of the form $\{u\}$ for $u\in\R$ or not; if it is, we take $x_n=u$, otherwise $x_n=0$. We get the definable sequence that contains all numbers first-order definable in $(\R; +,\times,\N)$. The next step (explained in Section \[sect4\]), readily formalized (via the function $D$ from Section \[sect3\]), provides a definable number not contained in this sequence.
Fast-growing sequences {#sect7}
======================
Looking at decimal digits of two real numbers, for example, $$\begin{aligned}
x &= 0.62831\,85307\,17958\,64769\,25286\,76655\,90057\,68394\,33879\,87502\,11641\dots
\\ % \,94988\,91846\,15632\,81257\,2417\,97256\,06965\,06842\,34135\dots
y &= 0.65465\,36707\,07977\,14379\,82924\,56246\,85835\,55692\,08082\,39542\,45575\dots
% \,15320\,30341\,52669\,17935\,39584\,09434\,80222\,78477\,78618\dots\end{aligned}$$ can you see, which one is “more definable”? Probably not. (Answer: $y=\sqrt{3/7}$ is [algebraic](http://en.wikipedia.org/wiki/Algebraic number), therefore first-order definable in $(\R;+,\times)$, while $x=\pi/5$ is not.) Surprisingly, a kind of visualization of definability is possible in an interesting special case. The number $$\sum_{n=0}^\infty 10^{-3^n} =
0.\mathbf10\mathbf100\,000\mathbf10\,00000\,00000\,00000\,0\mathbf1000\,00000\,00000\,00000\dots$$ is [transcendental](http://en.wikipedia.org/wiki/Transcendental number) (that is, not algebraic). Moreover, every number of the form $\sum_{n=1}^\infty 10^{-k_n}$ with $ k_n \in \N $, $
\lim_{n\to\infty} \frac{k_{n+1}}{k_n} > 2 $ is transcendental, which follows from [Roth’s theorem](http://en.wikipedia.org/wiki/Roth's theorem).
If $(k_n)_{n=1}^\infty$ is a definable sequence of natural numbers, strictly increasing (that is, $ k_1<k_2<\dots $), then the number $ \sum_{n=1}^\infty 10^{-k_n} $ is definable. Prove it both in the framework of Section \[sect3\] (first-order definability in $(\R; +,\times,\N)$) and the framework of Section \[sect5\] (second-order definability). *Hint:* $ \forall i\in\N \;\; \( D(i,x)=1 \equ \exists n\in\N \;\, i=k_n \) $, and $ \forall i\in\N \;\; D(i,x) \le 1 $.
\[\*\] If a number $ \sum_{n=1}^\infty 10^{-k_n} $ with $ k_n \in \N $, $ k_1<k_2<\dots $, is definable, then the sequence $(k_n)_{n=1}^\infty$ is definable. Prove it in the framework of Section \[sect5\] (second-order definability). *Hint:* for every $n$, $k_{n+1}$ is the least $k$ such that $ k>k_n \land D(k,x)=1 $.
Note that the sequence $(k_n)_{n=1}^\infty$ is defined by its property, which works only in the second-order framework. The first-order framework requires an explicit relation between $n$ and $k=k_n$. Nevertheless, the claim of Exercise \[\*\] holds also in the framework of Section \[sect3\].[^7]
Thus, in order to get a first-order undefinable but second-order definable real number, it is sufficient to find a first-order undefinable but second-order definable strictly increasing sequence of natural numbers. This can be made similarly to Sections \[sect4\], \[sect6\], replacing Cantor’s diagonal argument with the following fact:
For every sequence of sequences (of numbers) there exists a strictly increasing sequence (of numbers) that overtakes all the given sequences (of numbers).
The proof is immediate: take $ y_n = n+\max_{i,j\in\{1,\dots,n\}} x_{i,j} $ where the number $ x_{i,j} $ is the $i$-th element of the $j$-th given sequence; then clearly $ y_n > x_{n,m} $ whenever $ n \ge m $.
If the ternary relation $\{(i,j,x_{i,j})\mid i,j\in\N\}$ is definable, then the binary relation $\{(n,y_n)\mid
n\in\N\}$ is definable. Prove it. *Hint:* $ y=y_n \equ \( ( \exists i \;\, \exists j \;\; (i\le n\land
j\le n\land y-n=x_{i,j})) \land ( \forall i \;\, \forall j \;\; ( i\le n\land j\le n \imp y-n \ge x_{i,j} ) )
\) $.
Reusing the construction of Section \[sect6\], we enumerate all sequences of natural numbers, definable in the framework of Section \[sect3\], by enumeration definable in the framework of Section \[sect5\], and then overtake them all by a strictly increasing sequence of natural numbers, definable in the framework of Section \[sect5\].
To fully appreciate the incredible growth rate of this sequence, we note that it overtakes all computable sequences, as well as an extremely fast-growing sequence $(M_N)_{N=1}^\infty$ mentioned in Introduction. Recall $A_N$ and $A_{M,N}$ discussed there. In the framework of Section \[sect3\], the ternary relation $\{(M,N,A_{M,N}) \mid M,N \in \N\}$, being recursively enumerable (therefore Diophantine) is definable; and the binary relation $\{(N,A_N) \mid N \in \N\}$ is definable, since $ a=A_N \equ \( ( \exists M\in\N \;\; a=A_{M,N} ) \land ( \forall M\in\N \;\; a\ge A_{M,N} ) \)
$. Defining $M_N$ as the least $M$ such that $A_{M,N}=A_N$ we observe that the sequence $(M_N)_{N=1}^\infty$ is definable (since $ m=M_N \equ ( A_{m,N}=A_N \land A_{m-1,N}<A_N ) $). On the other hand, as noted in Introduction, this sequence cannot be bounded from above by a computable sequence.
More discussions of large numbers are available, see Scott Aaronson, John Baez and references therein. A quote from Aaronson (pages 11–12):
> *You defy him to name a bigger number without invoking Turing machines or some equivalent. And as he ponders this challenge, the power of the Turing machine concept dawns on him.*
Definability could be mentioned here along with Turing machines.
Definable but uncertain {#sect8}
=======================
Two sets are called [equinumerous](http://en.wikipedia.org/wiki/equinumerous) if there exists a one-to-one correspondence between them. In particular, two subsets $A,B$ of $\R$ are equinumerous if (and only if) $\exists f \in
\oP(\R^2) \;\; \( f \subset A\times B \land ( \forall x\in A \;\, \exists! y\in B \;\; (x,y)\in f ) \land (
\forall y\in B \;\, \exists! x\in A \;\; (x,y)\in f ) \) $. We see that the binary relation “equinumerosity” on $
\oP(\R) $ is second-order definable.
Some subsets of $\R$ are equinumerous to $\{1,\dots,n\}$ for some $n\in\N$ (these are finite sets). Others may be equinumerous to $\N$ (these are called countable, or countably infinite), or $\R$ (these are called sets of cardinality continuum), or… what else? Can a set be more than countable but less than continuum?
This seemingly innocent question is one of the most famous in set theory, the first among the [Hilbert’s problems](http://en.wikipedia.org/wiki/Hilbert's problems). The answer was expected to be “no such sets”, which is the [continuum hypothesis](http://en.wikipedia.org/wiki/continuum_hypothesis) (CH); Georg Cantor tried hard to prove it, in vain; Kurt Gödel proved in 1940 that CH cannot be disproved within [the axiomatic set theory called ZFC](http://en.wikipedia.org/wiki/Zermelo-Fraenkel set theory), and hoped that new axioms will disprove it; Paul Cohen proved in 1963 that CH cannot be proved within ZFC, and felt intuitively that it is *obviously* false. Nowadays some experts hope to find “the missing axiom”, others argue that this is hopeless.
A wonder: million published theorems in all branches of mathematics formally are deduced from the 9 axioms of ZFC; they answer, affirmatively or negatively, million mathematical questions; some questions remain [open](http://en.wikipedia.org/wiki/open problem), waiting for solutions in the ZFC framework; but the continuum hypothesis is an exception!
Back to definability. Consider the set $Z$ of all subsets of $\R$ that are more than countable but less than continuum. We do not know, whether $Z$ is empty or not, but anyway, we know that $Z$ is second-order definable. We define a number $z$ by the following property: $$(\, z=0 \;\land\; \exists A \;\; A\in Z \,) \;\;\lor\;\; (\, z=1 \;\land\; \forall A \;\; A\notin Z \,) \, .$$ That is, $z$ is $1$ if CH is true, and $0$ otherwise. This is a valid definition; $z$ is second-order definable; but we cannot know, is it $0$ or $1$. Each one of the two equalities, $z=0$ and $z=1$, could be added (separately!) to the axioms of ZFC without contradiction;[^8] according to the model theory, it means existence of two models of ZFC, one with $z=0$, the other with $z=1$. In this sense, $z$ is model dependent.
Is $z$ computable? Yes, it is, just because $0$ and $1$ are computable numbers, and $z$ is one of these. You might feel bothered, even outraged, but this is a valid argument. Compare it with the well-known proof that an irrational elevated to an irrational power may be rational: $\({\sqrt2}\)^{\sqrt2}$ is either rational (which gives the needed example), or irrational, in which case $\({\sqrt2}^{\sqrt2}\)^{\sqrt2}=\({\sqrt2}\)^2=2$ gives the needed example. Seeing this, some retreat to [intuitionism](https://en.wikipedia.org/wiki/Intuitionism), but almost all mathematics is [classical](https://en.wikipedia.org/wiki/Classical mathematics), it accepts the [law of excluded middle](https://en.wikipedia.org/wiki/Law of excluded middle) and cannot arbitrarily disallow it in some cases.
So, what is the algorithm for computing $z$? Surely the definition of an algorithm disallows such condition as “if CH holds, then” within an algorithm. However, it cannot disallow a model-dependent algorithm $ A = (\text{if CH holds
then } A_1 \text{ else } A_0) $, where $A_1$ is a (trivial) algorithm that computes the number 1, and $A_0$ computes $0$. The conditioning “if CH holds, then” is allowed outside the algorithms (similarly, the conditioning “if $\({\sqrt2}\)^{\sqrt2}$ is rational, then” is allowed outside the formulas). If you are unhappy with the affirmative answer to the question “is $z$ computable?”, ask a different question: “is $z$ computable by a model-independent algorithm?” The answer is negative (see below).
On the other hand, definability of the number $z$ is established by a kind of “generalized algorithm” able to process second-order objects (real numbers, relations between these, and relations between relations; recall the “program” $(b_k)_k$ in Section \[sect6\]). This “generalized algorithm” is model independent, but its output is model dependent.
In contrast, the number $\pi$ is model independent; for every rational number $r$ one of the two inequalities $\pi<r$, $\pi>r$ is provable in ZFC. The same applies to the numbers $ \sqrt{2}, \varphi, e $ discussed in Introduction, since each of these numbers can be computed by a model independent algorithm. If a number is computable by a model-independent algorithm, then this number is both model independent and computable.
What about Chaitin’s constant $\Om$? It is limit computable by a model-independent algorithm. Also, it is first-order definable (in $(\R; +,\times,\N)$), and the first-order framework disallows questions (such as CH) about arbitrary sets of numbers, thus, one might hope that $\Om$ is model independent. But it is not!
Here we need one more fact about $\Om$. The sequence $(A_N)_{N=1}^\infty$ of its binary digits is not just uncomputable, that is, the set $\{N\mid A_N=1\}$ is not just non-recursive, but moreover, this set belongs to “the most important class of recursively enumerable sets which are not recursive”, the so-called [creative sets](http://en.wikipedia.org/wiki/creative and productive sets), or equivalently, complete recursively enumerable sets. Basically, it means that this sequence contains answers to all questions of the form “does the natural number $n$ belong to the recursively enumerable set $A$?” And in particular(!), all questions of the form “can the statement $S$ be deduced from the theory ZFC?”, since in ZFC (and many other formal theories as well) the set of (numbers of) provable statements is recursively enumerable. Taking $S$ to be the negation of something provable (for instance, $0\ne0$) we get the question “is ZFC consistent?” answered by one of the binary digits $A_N$ of $\Om$, whose number $N_{\text{ZFC}}$ can be computed; if this $A_{N_{\text{ZFC}}}$ is $0$, then ZFC is consistent; if $A_{N_{\text{ZFC}}}$ is $1$, then ZFC is inconsistent. However, by a famous Gödel theorem, this question cannot be answered by ZFC itself! Assuming that ZFC is consistent we have $A_{N_{\text{ZFC}}}=0$, but this truth is not provable (nor refutable) in ZFC. (In fact, it is provable in [ZFC+large cardinal axiom](https://en.wikipedia.org/wiki/Large cardinal).) Therefore, in some models of ZFC we have $A_{N_{\text{ZFC}}}=0$, in others $A_{N_{\text{ZFC}}}=1$, which shows that $\Om$ is model dependent. Moreover, there are versions of $\Om$ such that *every* binary digit of $\Om$ is model dependent [@So], [@Ca].
Yet the (first-order) case of $A_{N_{\text{ZFC}}}$ is less bothering that the (second-order) case of $z$, since we still believe that $A_{N_{\text{ZFC}}}=0$. Adding the axiom “$A_{N_{\text{ZFC}}}=1$”, that is, “ZFC is inconsistent” to ZFC we get a theory that is consistent[^9] but not [$\om$-consistent](https://en.wikipedia.org/wiki/Omega-consistent). This strange theory claims existence of a proof of “$0\ne0$” in ZFC, of a finite length $N_{0\ne0}$, this length being a natural number. And nevertheless, this theory claims that $N_{0\ne0}>1$, $N_{0\ne0}>2$, $N_{0\ne0}>3$, and so on, endlessly.[^10] Every model of this strange theory contains [more natural numbers](https://en.wikipedia.org/wiki/Non-standard model of arithmetic) than the usual $1,2,3,\dots\,$ In mathematical logic we must carefully distinguish between two concepts of a natural number, one belonging to a theory, the other to its [metatheory](https://en.wikipedia.org/wiki/Metamathematics). In particular, when saying “for every rational number $r$ one of the two inequalities $\pi<r$, $\pi>r$ is provable in ZFC” we should mean that $|r|$ is the ratio of two metatheoretical natural numbers.
Using as binary digits an infinite sequence of *independent* “yes/no” parameters of models of ZFC we get a model dependent definable number $w$ whose possible values are *all* real numbers. More exactly, the following holds in the metatheory: for every real number $x$ there exists a model of ZFC[^11] whose natural numbers (and therefore rational numbers) are the same as in the metatheory, and for every rational number $r$ the inequality $ w>r $ holds in the model if and only if $ x>r $. Is this possible in the first or second order framework? I do not know. But in the third order framework this is possible, as suggested by the [generalized continuum hypothesis](https://en.wikipedia.org/wiki/Generalized continuum hypothesis).
Higher orders; set theory {#sect9}
=========================
Recall the transition from first-order definability to second-order definability (Section \[sect5\]); from the set $\R$ of all real numbers to the set $\mathbf S$ of all tuples and relations over $\R$, and the on $\mathbf S$ generated by the on $\R$ and two relations, membership and appendment, on $\mathbf S$. The next step suggests itself: the set $ \mathbf T = \( \mathbf S \cup \mathbf S^2 \cup \mathbf S^3 \cup \dots \) \cup \(
\oP(\mathbf S) \cup \oP(\mathbf S^2) \cup \oP(\mathbf S^3) \cup \dots \) $ of all tuples and relations over $\mathbf S$, with the on $\mathbf T$ generated by the on $\mathbf S$ and two relations, membership and appendment, on $\mathbf T$, formalizes third-order definability. This way we may introduce infinitely many orders of definability, $ \R \subset \mathbf S \subset \mathbf T \subset \dots $, or $ T_1
\subset T_2 \subset T_3 \subset \dots $ where $ T_1 = \R $, $ T_2 = \mathbf S $, $ T_3 = \mathbf T $ and so on. Similarly to Section \[sect6\] we can prove that each order brings new definable real numbers (and new, faster-growing sequences of natural numbers, recall Section \[sect7\]).
But this is only the tip of the iceberg. The union of all these sets, $ T_\infty = T_1 \cup T_2 \cup \dots $, endowed with the generated by the given s on all $T_n$, formalizes a new, transfinte order of definability, and starts a new sequence of orders. Should we denote them by $T_{\infty+1}, T_{\infty+2}, \dots \,$? What about $T_{\infty+\infty}\,$? How high is this hierarchy? Is it countable, or not?
Transfinite hierarchies are investigated by set theory (see Wikipedia:[Set theory](https://en.wikipedia.org/wiki/Set
theory), and Section “Some ontology” there). Surprisingly, set theory does not need the field $\R$ of real numbers as the starting point; not even the set $\N$ of natural numbers. A wonder: set theory is able to start from nothing and get everything!
[The cumulative hierarchy](https://en.wikipedia.org/wiki/Von Neumann universe) starts with the [empty set](https://en.wikipedia.org/wiki/Empty set), denoted by $\emptyset$ or $\{\}$, the number $0$ defined as just another name of the empty set, and stage zero, denoted by $V_0$ and defined as still another name of the empty set. On the next step we consider the set $\oP(V_0)$ of all subsets of $V_0$. There is only one subset of $\emptyset$, the empty set itself, thus $\oP(V_0) = \oP(\emptyset)=\{\emptyset\}=\{0\} $; we define the number $1$ to be $\{0\}$, and stage one $ V_1=\oP(V_0) $. Similarly, $ \oP(V_1) $ is the two-element set $ 2 = \{\emptyset,\{\emptyset\}\} = \{0,1\} = V_2 $, stage two. Somewhat dissimilarly, $
\oP(V_2) $ is the four-element set $ \{\emptyset,\{0\},\{1\},\{0,1\}\} = \{0,1,\{1\},2\} $, its three-element subset $ \{0,1,2\} $ is (by definition) the number $3$, and $ V_3 = \oP(V_2) $ is the third stage. More generally, $ n+1 = \{0,1,\dots,n\} \subset \oP(V_n) = V_{n+1} $ for $ n=1,2,3,4,\dots $. Thus, $ V_n $ is a set of $\underbrace{2^{2^{\cdot^{\cdot^{2}}}}}_{n-1}$ elements(!), while $n$ is its subset of $n$ elements.
$ V_0 \subset V_1 \subset V_2 \subset \dots \, $ Prove it. *Hint:* $ A \subset B \imp \oP(A) \subset \oP(B) $.
Here we face crossroads. One way is to treat the union $ V_0 \cup V_1 \cup V_2 \cup \dots $ of all $ V_n $ as the class $V$ of all sets (a [proper class](https://en.wikipedia.org/wiki/Class (set theory)), not a set). This way leads to the finite set theory (see Takahashi [@Tak], Baratella and Ferro [@Bar] and others; see also Wikipedia:[General set theory](https://en.wikipedia.org/wiki/General set theory)). The other way is to treat the union $ V_0 \cup V_1 \cup V_2 \cup \dots $ of all $ V_n $ as an infinite set, its infinite subset $\om = \{0,1,2,\dots\}$ as the first transfinite ordinal number, and $ V_\om =
\cup_{n=0}^\infty V_n $ as the first transfinite stage of the cumulative hierarchy. This way leads to the set theory widely accepted by the mainstream mathematics.
Finite set theory {#sect9.1}
-----------------
The finite set theory is equivalent (in some sense) to [arithmetic](https://en.wikipedia.org/wiki/Peano axioms) (Kaye and Wong [@Ka]); consistency of these theories is nearly indubitable, in contrast to the (full) set theory whose [axiom of infinity](https://en.wikipedia.org/wiki/Axiom of infinity) says basically that the class $V_\om$ is a set.
In the finite set theory, the class $\N\cup\{0\}$ of numbers $0,1,2,\dots$ may be defined as the class of all sets $x$ such that $x$ is transitive, that is, $ \forall y \; \forall z \;\> ( y\in x \land z\in y \impl z\in x ) $, and $x$ is totally ordered by membership, that is, $ \forall y \; \forall z \;\> \( ( y\in x \land z\in x ) \impl ( y=z \lor y\in z
\lor z\in y ) \) $. Adding the condition that $x$ is non-empty, that is, $ \exists y \; y \in x $, we get the class $\N$ of natural numbers $\{1,2,\dots\}$.
Each set $x$ is equinumerous to one and only one $n\in\N\cup\{0\}$; as before, “equinumerous” means existence of a set $f$ such that $ f \subset x\times n $, that is, $ \forall p\in f \; \exists y\in x \; \exists m\in n \;\> p=(y,m) $ where $ (y,m)=\{\{y\},\{y,m\}\} $), and $f$ is a one-to-one correspondence between $x$ and $n$, that is, $ ( \forall
y\in x \;\, \exists! m\in n \;\; (y,m)\in f ) \land ( \forall m\in n \;\, \exists! y\in x \;\; (y,m)\in f ) $. In this case we say that $n$ is the number of members of $x$.
The sum $m+n$ of $m,n\in\N\cup\{0\}$ may be defined as the number of members in the disjoint union $ \{1\}\times m \cup
\{2\}\times n $. The product $mn$ of $m,n\in\N\cup\{0\}$ may be defined as the number of members in the set product $ m \times n =
\{(k,\ell)\mid k\in m, \ell\in n\} $. The power $m^n$ for $m,n\in\N\cup\{0\}$ may be defined as the number of functions from $n$ to $m$ (that is, from $\{0,\dots,n-1\}$ to $\{0,\dots,m-1\}$).
A rational number could be defined as an equivalence class of triples $(p,n,q)$ of natural numbers $p,n,q\in\N$ w.r.t. such equivalence relation: $(p_1,n_1,q_1)\sim(p_2,n_2,q_2)$ when $ p_1 q_2 + n_2 q_1 = p_2 q_1 + n_1 q_2 $ (informally this means that $ \frac{p_1-n_1}{q_1} = \frac{p_2-n_2}{q_2} $, of course). However, in this case we cannot introduce the class of rational numbers (since a proper class cannot be member of a class). Thus, it is better to choose a single element in each equivalence class, and define a rational number as a triple $(p,n,q)$ of numbers $p,n,q\in\N\cup\{0\}$ such that $q\ne0$, at least one of the two numbers $p,n$ is $0$, and the other is [coprime](https://en.wikipedia.org/wiki/Coprime integers) to $q$ (or $0$). We get the class $\Q$ of all rational numbers. And, in order to treat natural numbers as a special case of rational numbers, we identify each natural number $n\in\N$ with the corresponding rational number $(n,0,1)\in\Q$.
Back to definability. We want to endow the class $V$ (of all sets in the finite set theory) with the generated by the membership relation $ \{ (x,y) \mid x\in y \} $. True, the notion of a on $V$ transcends the finite set theory, since a collection of classes is neither a set nor a class. But still, in the metatheory, a class may be called definable when it is obtainable from the membership relation by the 5 operations (complement, union, permutation, Cartesian product, projection) introduced in Section \[sect2\] for relations on the real line $\R$ in particular, and arbitrary set in general. However, a pair of real numbers is not a real number, while a pair of sets is a set! That is, $\R$ and $\R^2$ are disjoint; in contrast, $ V^2 \subset V $. The order relation “$x<y$” between real numbers $x,y\in\R$ is a subset of $\R^2$ (rather than $\R$). But what about the membership relation “$x\in y$” between sets $x,y\in V\,$? Should we treat it as a subclass of $V$ or $V^2\,$?
True, the plane $\R^2$ is not a subset of the line $\R$, but it can be injected into $\R$ by a definable function; recall the injection $ W_2 : \R^2 \to \R $ introduced in the end of Section \[sect3\] and used in Section \[sect6\] for encoding binary relations by unary relations. For example, the binary relation $ \{(x,y)\mid x<y\} $ is encoded by the unary relation $\{W_2(x,y) \mid x<y\} $.
Here is a general lemma basically applicable in both situations, $W_2:\R^2\to\R$ and $ V^2\subset V$ (though, in the latter case it needs some adaptation to the proper class).
**Lemma.** Let $R$ be a set endowed with a , and $ f : R^2 \to R $ a definable injection. Then a binary relation $ A \subset R^2 $ is definable if and only if the unary relation $ f(A) \subset R $ is definable.
Prove this lemma. *Hint:* “If”: $ (x,y)\in A \equ \exists z \;\> \( f(x,y)=z \land z\in f(A) \)
$. “Only if”: $ z\in f(A) \equ \exists x,y \;\> \( f(x,y)=z \land (x,y)\in A \) $.
All relations over $V$ mentioned above are definable classes. Prove it. *Hint:* the equality relation “$x=y$” is “$\forall z \;\> ( z\in x \equ z\in y ) $”; the relation “$\{x\}=y$” is “$\forall z \;\> ( z=x \equ z\in y ) $”; the ternary relation “$\{x,y\}=z$” is “$\forall u \;\> \( (u=x \lor u=y) \equ u\in z \) $”; the ternary relation “$(x,y)=z$” is “$\exists u,v,w \;\> ( \{x\}=u \land \{x,y\}=v \land \{u,v\}=z )$”; the lemma applies; further, $ \N\cup\{0\} $ is the intersection of the class of transitive sets and the class of sets totally ordered by membership, etc. etc., up to “$(p_1,n_1,q_1)\sim(p_2,n_2,q_2)$”.
Similarly, the basic relations between rational numbers are definable classes.
Real numbers cannot be represented by finite sets, but can be represented by classes (of finite sets) in several ways. In the spirit of [Dedekind cuts](https://en.wikipedia.org/wiki/Dedekind cut) we treat a real number as the class of all rational numbers smaller than this real number. More formally: a real number is a subclass $A$ of $\Q$ such that
$A$ is a [lower class](https://en.wikipedia.org/wiki/Upper set); that is, $\forall a,b\in\Q \;\> ( a<b
\land b\in A \impl a\in A ) $;
$A$ contains no greatest element; that is, $\forall a\in A \; \exists b\in A \;\> a<b $;
$A$ is not empty, and not the whole $\Q$; that is, $\exists a\in\Q \;\> a\in A $ and $\exists b\in\Q \;\>
b\notin A $.
And, in order to treat rational numbers as a special case of real numbers, we identify each rational number $a\in\Q$ with the corresponding real number $\{b\in\Q\mid b<a\} \in \R$.
Some examples. The real number $\sqrt2$ (“the Pythagoras’ constant”) is the class of all rational numbers $a$ such that $ a<0 \;\lor\; a^2<2 $. The golden mean $\varphi$ is the class of all rational numbers $a$ such that $ a\le 0 \;\lor\; 0<a<1+\frac1a $. The real number $e$ is the class of all rational numbers $a$ such that $ \exists n\in\N \;\>
\frac{(n+1)^n}{n^n} > a $.
Can we define $e$ via factorials, as in Exercise \[5.13\]? We can define factorials without recursion; $n!$ is the number of bijective functions from $n$ to itself (that is, from $\{0,\dots,n-1\}$ to itself; in other words, permutations). But still, we need recursion when defining partial sums of the series $ \sum_{n=1}^\infty \frac1{n!} $ for $e$. Generally, an infinite sequence of rational numbers $ (s_n)_{n=1}^\infty $ is the class of pairs $(n,s_n)$. Specifically, the sequence of partial sums of $\sum_n a_n$ is the class $S$ of pairs such that $ \forall
n\in\N \; \forall b\in\Q \;\> \( (n-1,b)\in A \impl (n,b+a_n)\in A \) $ (and $ \forall n\in\N \; \exists! b\in\Q \;\>
(n-1,b)\in A $, and $ (0,0)\in A $, of course). But we cannot define a class by its property! We deal with a on $V$. A class must be defined by a common property of all its members, not a property of the class. Otherwise it would be second-order definability in $V$ (thus, a transfinite level of the cumulative hierarchy). Can we formulate the appropriate property of a pair $(n,s_n)$ alone? Yes, we can. Here is the property: there exists a function $ f :
\{0,\dots,n\} \to \Q $ such that $ f(0)=0 $ and $ \forall k\in\{1,\dots,n\} \;\> f(k)=f(k-1)+a_k $. The clue is that a finite segment of the infinite sequence (of partial sums) is enough.
Similarly, an infinite sequence $(x_n)_{n=0}^\infty$ of sets $ x_n \in V $ is the class of pairs $(n,x_n)$, and it can be defined recursively, by a recurrence relation of the form $ \forall n\in\N \;\> (x_{n-1},x_n) \in B $ where $ B $ is a definable class of pairs, and an initial condition for $ x_0 $. (Use finite segments of the infinite sequence.)
Thus, every computable sequence of natural (or rational) numbers is a definable class of pairs. No need to use Diophantine sets. Rather, for every [Turing machine](https://en.wikipedia.org/wiki/Turing machine), all possible [“complete configurations”](https://en.wikipedia.org/wiki/Turing machine#Turing_machine_"state"_diagrams) (called also “situations” and “instantaneous descriptions”) may be treated as elements of a subclass of $V$, and the rule of transition from one complete configuration to the next complete configuration may be treated as a definable class of pairs (of complete configurations).
It follows that every computable real number, and moreover, every limit computable real number is definable. Having a convergent definable sequence $(a_n)_n$ of rational numbers, we define its limit as the class of rational numbers $ b $ such that $ \exists n \; \forall k \;\> ( k>n \impl a_k>b+\frac1n ) $. In particular, $\pi$ (the Archimedes’ constant) and $\Om$ (the Chaitin’s constant) are definable.
A sequence $(x_n)_n$ of real numbers cannot be treated as the class of pairs $(n,x_n)$ (since $x_n$ is not a set), but can be treated as the disjoint union $ \{1\}\times x_1 \cup \{2\}\times x_2 \cup \dots $, that is, the set of pairs $(n,a)$ where $a\in x_n$ (recall a similar workaround in Section \[sect6\]). Also, a continuous function $ f : \R \to \R
$ cannot be treated as the class of pairs $\(x,f(x)\)$, but can be treated as the class of pairs $(a,b)$ of rational numbers such that $ b < f(a) $. Such precautions allow us to translate basic calculus into the language of finite set theory. However, arbitrary functions $ \R \to \R $ and arbitrary subsets of $\R$ are unavailable. Thus, the continuum hypothesis makes no sense. Also, transferring [measure theory](https://en.wikipedia.org/wiki/Measure (mathematics)) and related topics (especially, [theory of random processes](https://en.wikipedia.org/wiki/Stochastic process)) to this ground (as far as possible) requires effort and ingenuity.
The finite set theory can provide a reliable alternative airfield for much (maybe most) of the mathematical results especially important for applications, in case of catastrophic developments in the transfinite hierarchy. Several possible such “alternative airfields” are examined by mathematicians and philosophers [@Fe1], [@Fe2], [@We2], [@Si2], [@Si], [@We3], [@St].
Informally, the finite set theory uses (for infinite classes) the idea of [potential infinity](https://en.wikipedia.org/wiki/Potential
infinity), prevalent before [Georg Cantor](https://en.wikipedia.org/wiki/Georg Cantor), while the transfinite hierarchy uses the idea of [actual (completed) infinity](https://en.wikipedia.org/wiki/Actual infinity), prevalent after Georg Cantor.
Transfinite hierarchy {#sect9.2}
---------------------
The transfinite part of the cumulative hierarchy begins with the first transfinite ordinal number $ \om =
\{0,1,2,\dots\} $ (an infinite set) and the first transfinite stage $ V_\om = \cup_{n\in\om} V_n $ of the hierarchy (an infinite set; $ \om \subset V_\om $). Note that $ x\in V_\om $ implies $ \oP(x) \in V_\om $, but $ x\subset
V_\om $ implies rather $ \oP(x) \subset V_{\om+1} $. We continue as before: $$\begin{aligned}
\om+1 &= \om \cup \{\om\} = \{0,1,2,\dots\} \cup \{\om\} \subset \oP(V_\om) = V_{\om+1} \, , \\
\om+2 &= (\om+1) \cup \{\om+1\} = \{0,1,2,\dots\} \cup \{\om,\om+1\} \subset \oP(V_{\om+1}) = V_{\om+2}\end{aligned}$$ and so on; we get the stages $ V_{\om+n} $ for all finite $n$, and again, $ V_{\om+n} \subset V_{\om+n+1} $. The union of all these stages is the stage $ V_{2\om} = V_\om \cup V_{\om+1} \cup V_{\om+2} \cup \dots = \cup_{\al<2\om} V_\al $ (still an infinite set), and $ 2\om = \{0,1,2,\dots\} \cup \{\om,\om+1,\om+2,\dots\} $ (an infinite subset of $V_{2\om}$). Again, $ x\in V_{2\om} $ implies $ \oP(x) \in V_{2\om} $. Let us dwell here before climbing higher.
Encoding of various mathematical objects by sets is somewhat arbitrary (see Wikipedia: [Equivalent definitions of mathematical structures](https://en.wikipedia.org/wiki/Equivalent definitions of mathematical structures); likewise, an image may be encoded by files of [type jpeg, gif, png etc.](https://en.wikipedia.org/wiki/Image file
formats)), and their places in the hierarchy vary accordingly. Treating a pair $(a,b)$ as $\{\{a\},\{a,b\}\}$ and a triple $(a,b,c)$ as $ \((a,b),c\) $ we get (for $0<n<\om$) $$\begin{gathered}
\forall a,b \in V_n \;\> (a,b) \in V_{n+2} \, ; \quad \forall a,b,c \in V_n \;\> (a,b,c) \in V_{n+4} \, ; \\
\forall a,b \in V_\om \;\> (a,b) \in V_\om \, ; \quad \forall a,b,c \in V_\om \;\> (a,b,c) \in V_\om \, ; \\
\forall a,b \in V_{\om+n} \;\> (a,b) \in V_{\om+n+2} \, ; \quad \forall a,b,c \in V_{\om+n} \;\> (a,b,c) \in
V_{\om+n+4} \, .\end{gathered}$$ Treating the set $\N$ of natural numbers as $\om\setminus\{0\}$ we get $ \N \subset V_\om $, $ \N \in V_{\om+1} $. Treating a rational number as an equivalence class of triples $(p,n,q)$ of natural numbers we get $ \Q \subset V_{\om+1}
$, $ \Q \in V_{\om+2} $, where $ \Q $ is the set of all rational numbers. Alternatively, treating an integer as an equivalence class of pairs of natural numbers, and a rational number as an equivalence class of pairs of integers, we get $$\Z \subset V_{\om+1} \, , \;\; \Z \in V_{\om+2} \, ; \quad
\Q \subset V_{\om+4} \, , \;\; \Q \in V_{\om+5} \, ;$$ here $ \Z $ is the set of all integers. Treating a real number as a set of rational numbers we get $$\R \subset V_{\om+n} \, , \quad \R \in V_{\om+n+1} \, ,$$ where $\R$ is the set of all real numbers, and $n$ is such that $ \Q \in V_{\om+n} $; be it $2$ or $5$, anyway, it follows that $ \R \in V_{2\om} $.
Taking into account that generally $ A\in V_{2\om} \impl \oP(A) \in V_{2\om} $, and $ A,B\in V_{2\om} \impl
A\times B\in V_{2\om} $ (since $ A,B \subset V_{\om+n} \impl A\times B \subset V_{\om+n+2} $), we get $ \R^n \in
V_{2\om} $ and $ \oP(\R^n) \in V_{2\om} $ for all $n\in\N$. Every subset of $\R^n$ belongs to $V_{2\om}$, and every set of subsets of $\R^n$ belongs to $V_{2\om}$; in particular, the [](https://en.wikipedia.org/wiki/Sigma-algebra) of all [Lebesgue measurable](https://en.wikipedia.org/wiki/Lebesgue
measure) subsets of $\R^n$ belongs to $V_{2\om}$. Also, every function $ \R^n \to \R^m $ belongs to $V_{2\om}$, and every set of such functions belongs to $V_{2\om}$; in particular, every equivalence class (under the relation of equality [almost everywhere](https://en.wikipedia.org/wiki/Almost everywhere)) of Lebesgue measurable functions $ \R^n \to \R^m $ belongs to $V_{2\om}$, and the set [$L^1(\R^n)$](https://en.wikipedia.org/wiki/Lp
space) of all equivalence classes of [Lebesgue integrable](https://en.wikipedia.org/wiki/Lebesgue integration) functions $ \R^n \to \R $ belongs to $V_{2\om}$. And the set of all [bounded linear operators](https://en.wikipedia.org/wiki/Bounded operator) $L^1(\R^n)\to L^1(\R^n)$ belongs to $V_{2\om}$. Clearly, a lot of notable mathematical objects belong to $V_{2\om}$.
Would something like $V_{\om+100}$ suffice for all the objects mentioned above? The answer is negative as long as $ \R^n
$ is defined as $ \R^{n-1} \times \R = \{ (x,y) \mid x\in\R^n, y\in\R \} $ where $ (x,y) $ means $ \{\{x\},\{x,y\}\}
$. For every $ n\in\N $ the relation $ \R^n \notin V_{\om+2n-1} $ is ensured by the two exercises below.
If $ A \times B \subset V_{\om+n+2} $, then $ A,B \subset V_{\om+n} $. Prove it. *Hint:* $ \{\{a\},\{a,b\}\} =
(a,b) \in V_{\om+n+2} \impl a,b \in V_{\om+n} $.
If $ A^{n+1} \subset V_{\om+2n} $ for some $n$, then $ A \subset V_\om $. Prove it. *Hint:* induction in $n\ge1$, and the previous exercise.
A more economical encoding is available (and was used in Section \[sect6\], see Exer. \[6.1\], \[6.2\]); instead of the set $ \R^n $ of all $n$-tuples $(x_1,\dots,x_n)$ we may use the set $ \R^{[n]} $ of all $n$-sequences $[x_1,\dots,x_n]$; as before, $ [x_1,\dots,x_n] = \{ (1,x_1), \dots, (n,x_n) \} $ is the set of pairs.
\[9.6\] If $ A \in V_{\om+m+1} $, then $ A^{[n]} \in V_{\om+m+4} $ for all $ n\in\N $. Prove it. *Hint:* $a_1,\dots,a_n \in
V_{\om+m} \impl [a_1,\dots,a_n] \in V_{\om+m+3} $.
A lot of theorems are published about real numbers, real-valued functions of real arguments, spaces of such functions etc. I wonder, is there at least one such theorem sensitive to the distinction between $ V_{\om+100} $ and $
V_{\om+200} \,$? That is, theorem that can be formulated and proved within $ V_{\om+200} $ but not $ V_{\om+100} \,$? I guess, the answer is negative. A seemingly similar question: is definability of real numbers sensitive to the distinction between $ V_{\om+100} $ and $ V_{\om+200} \,$? I mean, is there at least one real number definable in $
V_{\om+200} $ but not $ V_{\om+100} \,$? This time, the answer is affirmative, as explained below.
For each $ n \in \N\cup\{0\} $ we endow the set $ V_{\om+n} $ with the $ D_{\om+n} $ generated by the membership relation $ \{ (x,y) \mid x\in y \} $ (for $x,y \in V_{\om+n}$, of course).
Recall that, treating a real number as a set of rational numbers, and a rational number as an equivalence class of triples $(p,n,q)$ of natural numbers, we have $ \Q \in V_{\om+2} $ and $ \R \in V_{\om+3} $. That is, $ \Q \subset
V_{\om+1} $ and $ \R \subset V_{\om+2} $.
Similarly to the finite set theory, $ \N $ and $ \Q $ are definable subsets of $ V_{\om+n} $ (whenever $n\ge1$), and the basic relations between natural numbers are definable, as well as the basic relations between rational numbers. Dissimilarly to the finite set theory, $ \R $ is a definable subset of $ V_{\om+n} $ (whenever $n\ge2$), and the basic relations between real numbers are definable. An example: for $x,y\in\R $ we have $ x\le y \equ
\forall a\in\Q \;\> (a<x \impl a<y) \equ \forall a\in\Q \;\> (a\in x \impl a\in y) \equ x
\subset y $. Another example: for $x,y,z\in\R $ we have $ x+y=z \equ \forall c\in\Q \;\> \( c<z
\equ \exists a\in\Q \;\> ( a<x \land c-a<y) \) $. Also the relation “$x=\{b\in\Q\mid b<a\}$” between a rational number $a$ and the corresponding real number $x$ is definable, which implies definability of $\N$ embedded into $\R$. Thus, all real numbers first-order definable in $(\R;+,\times,\N)$ (as in Section \[sect3\]) are definable in $V_{\om+n}$ (whenever $n\ge2$).
What about second-order definability? It was treated in Section \[sect5\] as a on the set $ \(
\cup_{n=1}^\infty \R^n ) \cup \( \cup_{n=1}^\infty \oP(\R^n) \) $, but a more economically encoded set $ S = \(
\cup_{n=1}^\infty \R^{[n]} ) \cup \( \cup_{n=1}^\infty \oP(\R^{[n]}) \) $ may be used equally well.
\[9.7\] $ S \subset V_{\om+6} $. Prove it. *Hint:* use Exercise \[9.6\].
Moreover, for every $n\ge6$, $S$ is a definable subset of $ V_{\om+n} $; and the four relations (that generate the in Section \[sect5\]) are definable relations on $ V_{\om+n} $. Thus, all real numbers second-order definable as in Section \[sect5\] are definable in $V_{\om+n}$ whenever $n\ge6$. In particular, the “first-order undefinable but second-order definable” number of Section \[sect6\] is definable in $V_{\om+6}$. However, all said does not mean that it is undefinable in $V_{\om+2}$.
What we need is the second-order definability in $(V_{\om+2},D_{\om+2})$ rather than $(\R;+,\times,\N)$; that is, definability in the set $ W_{\om+2} = \( \cup_{n=1}^\infty V^n_{\om+2} ) \cup \( \cup_{n=1}^\infty \oP(V^n_{\om+2}) \)
$.
$ W_{\om+2} \subset V_{\om+6} $. Prove it. *Hint:* similar to Exercise \[9.7\].
Once again, $ W_{\om+2} $ is a definable subset of $ V_{\om+6} $, and all real numbers definable in $ W_{\om+2} $ are definable in $ V_{\om+6} $. That is, all real numbers second-order definable in $ V_{\om+2} $ are (first-order) definable in $ V_{\om+6} $.
A straightforward generalization of Section \[sect6\] gives a real number second-order definable in $ V_{\om+2} $ but first-order undefinable in $ V_{\om+2} $. This number is definable in $ V_{\om+6} $ but undefinable in $ V_{\om+2}
$. Similarly, for each $ n\ge2 $ there exist real numbers definable in $ V_{\om+n+4} $ but undefinable in $ V_{\om+n}
$. We observe an infinite hierarchy of definability orders within $ V_{2\om} $.
Climbing higher on the cumulative hierarchy we get stages $ V_\al $ for ordinal numbers $ \al $ such as $ 2\om+n,
3\om+n, \dots $ Still higher, $ \om \cdot \om = \om^2 $, then $ \om^3, \dots $, then $ \om^\om, \om^{(\om^2)},
\om^{(\om^3)}, \dots \om^{(\om^\om)}, \dots $ Everyone may continue until feeling too dizzy; see Wikipedia:[Ordinal notation](https://en.wikipedia.org/wiki/Ordinal notation), [Ordinal collapsing function](https://en.wikipedia.org/wiki/Ordinal collapsing function), [Large countable ordinal](https://en.wikipedia.org/wiki/Large countable ordinal). All these are countable ordinals. By the way, every countable ordinal $\al$ may be visualized by a set of rational numbers, using a strictly increasing function $ f : \al \to \Q $ (that is, $ f : \{\be\mid\be<\al\} \to \Q $). For example, $2\om$ may be visualized by $ \{ 1-\frac1n \mid n\in\N \} \cup \{ 2-\frac1n \mid n\in\N \} $.
For every countable ordinal $\al\ge\om+2$ there exist real numbers definable in $ V_{\al+4} $ but undefinable in $ V_\al
$. Moreover, some of these real numbers are of the form $ \sum_{k=1}^\infty 10^{-k_n} $ (recall Section \[sect7\]), since there exists an increasing sequence (of natural numbers) definable in $ V_{\al+4} $ that overtakes all sequences definable in $ V_\al $.
A wonder: stages $V_\al$ for $\al$ like $ \om^{\om^\om} $ are as far from ordinary mathematics as numbers like $
10^{10^{1000}} $ from ordinary engineering. Nevertheless these $ V_\al $ contribute to the supply of definable real numbers.
Still higher, the set of all countable ordinals is the first uncountable ordinal $ \om_1 $. It cannot be visualized by a set of rational or real numbers. Its cardinality is the first uncountable cardinality $ \aleph_1 $. The continuum hypothesis is equivalent to the equality between $ \aleph_1 $ and the cardinality continuum.
For every ordinal $ \al \ge \om+2 $ (countable or not) the set of all real numbers definable in $ V_\al $ is countable (and moreover, has an enumeration definable in $ V_{\al+4} $). In particular, the set of all real numbers definable in $
V_{\om_1} $ is countable. On the other hand, new definable real numbers emerge on all countable levels, and there are uncountably many such levels. A contradiction?!
No, this is not a contradiction. Denoting by $R_\al$ the set of all real numbers definable in $V_\al$, and by $\cO_\al$ the set of all ordinals definable in $V_\al$, we have $ R_\be \subset R_\al $ wherever $ \be \in \cO_\al $ (which follows from the lemma of Section \[sect5\]). For all countable ordinals mentioned before we have $ \cO_\al = \al
$ (that is, all ordinals below $\al$ are definable in $V_\al$). In contrast, $ \cO_{\om_1} \ne \om_1 $, since $
\cO_{\om_1} $ is countable. The union $ \ti\R = \cup_{\al<\om_1} R_\al $ contains all real numbers definable *with ordinal parameters* $ \al \in \om_1 $; but definability with parameters is outside the scope of this essay (recall Section \[sect2\]).
It is natural to ask, whether $ \cO_\al = \al $ for all countable ordinals $\al$, or not. Probably, we only know that the affirmative answer cannot be proved without the choice axiom, and do not know, which answer (if any) can be proved *with* the choice axiom.
If $\cO_\al = \al $ for all countable ordinals $\al$, then $R_\al \uparrow \ti R$, and $\ti R$ is uncountable (of the cardinality $\aleph_1$ of $\om_1$). Otherwise, there exists a countable ordinal $\al$ such that $ \cO_\al \ne \al $ while $ \cO_\be = \be $ for all $\be < \al$ (and therefore $
\cO_\al = \be $ for some $\be < \al$). In this case the transfinite sequence $ (\cO_\al)_{\al<\om_1} $ is not monotone, and we do not know, whether the union $\ti R$ is countable, or not.
Countability or uncountability of $\ti R$ matters for model dependence. There is a countable set of formulas (in the language of the set theory) that define real numbers on all levels $V_\al$. Some are model independent, others are model dependent. If $\ti R$ is countable, then each of these model dependent definable real numbers has at most countably many possible values. Otherwise, if $\ti R$ is uncountable, then at least one of these model dependent definable real numbers has uncountably many possible values.
This matter is closely related to the position of Laureano Luna [@Lu0] (see also [@Lu pages 19–20]):
> *“Pieces of language taken as mere* syntactical *expressions (letter-strings) should be distinguished from definitions, which are* semantical *objects, namely,* interpreted *letter-strings.”* (Page 61.)\
> *“The meanings of the letter-strings that express definitions of reals are context-dependent, the context being here the definability level on which they are used. \[…\] if some letter-strings express more than one definition of a real number, there is no reason to think there are only countably many such definitions and only countably many definable real numbers.”* (Page 64.)
Two objections arise. First, we did not prove that $\ti R$ is uncountable. Second, model dependence does not apply to $V_\al$, since $V_\alpha$ is not a model of ZFC. We’ll return to the second objection after climbing on the cumulative hierarchy to $V_{\om_1}$ and much higher.
The stage $V_{\om_1}$ of the cumulative hierarchy is vast; its cardinality is very large (much larger than the cardinality $\aleph_1$ of $\om_1$). Now consider the first ordinal $\al$ of this very large cardinality and the corresponding stage $V_\al$. Iterating this jump we get a slight idea of the class of all sets, the incredible universe $V_{\text{ZFC}}$ of the set theory ZFC. The whole $V_{\text{ZFC}}$ grows from a small seed, the first infinite ordinal $\om$, whose existence is just postulated (the axiom of infinity).
If you want to soar above $V_{\text{ZFC}}$, you need a new axiom of infinity that ensures existence of an ordinal $\al$ such that $V_\al$ is a model of ZFC; every such ordinal, being [initial](https://en.wikipedia.org/wiki/Initial
ordinal), is a cardinal, called a [worldly cardinal](https://en.wikipedia.org/wiki/Worldly
cardinal). For climbing still higher try the so-called [large cardinals](https://en.wikipedia.org/wiki/Large
cardinal). And be assured that these supernal stages do contribute to the supply of definable real numbers.
Assuming existence of large cardinals we get a transfinite hierarchy of worldly cardinals $\kappa_\al$, and the corresponding models $W_\al = V_{\kappa_\al}$ of ZFC, for all countable ordinals $\al$ (and more; but here we do not need uncountable $\al$). Using $W_\al$ instead of $V_\al$ we get new versions of $\cO_\al$, $R_\al$ and $\ti R$. Again, we do not know, whether this new $\ti R$ is countable, or not. If it is uncountable, then again, at least one model dependent definable real number has uncountably many possible values; and this time, model dependence applies.
Getting rid of undefinable numbers {#sect9.3}
----------------------------------
Climbing down to earth, is it possible to restrict ourselves to definable numbers and still use the existing theory of real numbers and related objects? An affirmative answer was found in 1952 [@My] and enhanced recently [@HLR].
Before climbing down we need to climb up to the first worldly cardinal $\al$ and the corresponding model $V_\al$ of ZFC. Within the model we consider the [constructible hierarchy](https://en.wikipedia.org/wiki/Constructible universe) $(L_\be)_{\be\le\al}$, take the least $\be$ such that $ L_\be $ is a model of ZFC, and get the so-called [minimal transitive model](https://en.wikipedia.org/wiki/Minimal model (set theory)) of ZFC. This model is pointwise definable [@Ha1 “Minimal transitive model”], that is, every member of this model is definable (in this model).
Accordingly, this model is countable (and $\be$ is countable). Nevertheless, every theorem of ZFC holds in every model of ZFC; in particular, [Cantor’s theorem](https://en.wikipedia.org/wiki/Georg Cantor's first set theory article) “$\R$ is uncountable” holds in the countable model $L_\be$. No contradiction; enumerations of $ \R \cap L_\be $ exist, but do not belong to $L_\be$. Likewise, [a well-known theorem of measure theory](https://en.wikipedia.org/wiki/Lebesgue measure) states that the interval $(0,1)$ cannot be covered by a sequence of intervals $(a_n,b_n)$ of total length $\sum_{n=1}^\infty (b_n-a_n) < 1 $. True, for every $\eps>0$ the set $(0,1)\cap L_\be$, being countable, can be covered by a sequence of intervals of total length $\eps$; but such sequences do not belong to $L_\be$ (even if endpoints $a_n,b_n$ do belong). Working in $L_\be$ we have to ensure that all relevant objects (not only real numbers) belong to $L_\be$.
> $\bullet$ *One often hears it said that since there are indenumerably many sets and only denumerably many names, therefore there must be nameless sets. The above shows this argument to be fallacious.* (Myhill 1952, see [@My the last paragraph].)
> $\bullet$ *In my opinion, an object is conceivable only if it can be defined with a finite number of words.* (Poincaré 1910, translated from German, see [@Lu0 page 58].)
Conclusion {#sect10}
==========
Each definition (of a real number, or another mathematical object) is a finite text in a language. The language may be formal (mathematical) or informal (natural). In both cases the text is composed of expressions that [refer](http://en.wikipedia.org/wiki/Reference) to objects and relations between objects. The [extension](http://en.wikipedia.org/wiki/Extension (semantics)) of an expression is the corresponding set of objects, or set of pairs (of objects), or triples, and so on. For a mathematical language, all objects are mathematical; a natural language may mention non-mathematical objects, and even itself, as in the phrase *“The preceding two paragraphs are an expression in English that unambiguously defines a real number $r$”* (recall Introduction, Richard’s paradox), which leads to a problem: the mentioned expression in English fails to define! *“So when we speak in English about English, the ‘English’ in the metalanguage is not exactly the same as the ‘English’ in the object-language.”* [@Lu p. 15]. A natural language, intended to be its own metalanguage, is burdened with paradoxes. A mathematical language avoids such (and hopefully, any) paradoxes at the expense of being different from its metalanguage. The metalanguage is able to enumerate all real numbers definable in the language and define more real numbers.
On one hand, a natural language itself is inappropriate for mathematics. On the other hand, a mother tongue is always a natural language. In order to avoid both restrictions of a fixed mathematical theory and paradoxes of a natural language we may get the best of both worlds by considering two-part texts. The first part, written in a natural language, introduces a mathematical theory. The second part, written in the (mathematical) language of this theory, defines some real number.
In this framework the question “is every real number definable?” falls out of mathematics, because the notion “mathematical theory” above cannot be formalized. Some may admit only potential infinity and stop on the finite set theory. Some may admit actual infinity and the transfinite hierarchy up to (exclusively) some preferred ordinal (sometimes $2\omega$ [@Tar2]; more often, something controversially believed to be the first undefinable ordinal). Or the whole universe of ZFC but no more (equivalently, up to the first worldly cardinal). Or the [Tarski-Grothendieck set theory](http://en.wikipedia.org/wiki/Tarski-Grothendieck set theory). Or higher, up to some preferred [large cardinal](http://en.wikipedia.org/wiki/Large cardinal). Or some more exotic [alternative set theory](http://en.wikipedia.org/wiki/Alternative set theory). Or something brand new, like a kind of [homotopy type theory](http://en.wikipedia.org/wiki/Homotopy type theory). Or even something not yet published. Most choices mentioned above were unthinkable in the first half of the 20th century. Who knows what may happen near year 2100? *“Mathematics has no generally accepted definition”* (from Wikipedia:[Definitions of mathematics](http://en.wikipedia.org/wiki/Definitions of mathematics)); the same can be said about “mathematical theory”. Admitting actual infinity we do not get rid of potential infinity; the latter returns, hardened, on a higher level [@Si], [@Li]. Maybe the recent, [hotly debated](http://logic.harvard.edu/EFI_Hamkins_Comments.pdf) conception of [multiverse](http://en.wikipedia.org/wiki/Multiverse) is another fathomable segment of the unfathomable potential infinity of mathematics. “The most general definition of a definition” appears to be as problematic as “the set of all sets”.
Another problem manifests itself as model dependence for a mathematical language, and context dependence for a natural language. In both cases a single text may refer to different objects (depending on the model or the context, respectively), which blurs the idea of definability. Recall Section \[sect8\] (the last paragraph): every real number is “hardwired” in some model of the set theory (ZFC). We may feel that definability of the number does not follow unless the model is definable; but what do we mean by definability of a model? Another case, recall Section \[sect9.2\] (the last paragraph): uncountable hierarchy of models indexed by countable ordinals leads to a set of real numbers, possibly uncountable (but maybe not). Nothing is “hardwired” here, except for these countable ordinals. Outside mathematics, a more or less similar case, treated by Luna [@Lu0], shows context dependence in a hierarchy of contexts (levels of definability) indexed by definable countable ordinals. A mathematical counterpart with uncountably many definable real numbers could exist in some alternative (to ZFC and alike; probably closer to [Kripke-Platek](http://en.wikipedia.org/wiki/Kripke–Platek set theory)) set theory such that the class of all definable countable ordinals is not a countable set, and preferably, not a set at all [@Lu0 p. 65].
Bad news: definability is a very subtle property of a real number. Good news: other properties, more relevant to applications, are unsubtle; and definability is rather of philosophical interest. *“Mathematicians, in general, do not like to deal with the notion of definability; their attitude toward this notion is one of distrust and reserve.”* (Tarski [@Tar2], the first phrase; now partially obsolete, partially actual.)
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Storer, Timothy James (2010). “[A defence of predicativism as a philosophy of mathematics](https://core.ac.uk/display/1322419)”. *Dissertation (Cambridge).*
Luna, Laureano (2017). [“Rescuing Poincaré from Richard’s paradox”](https://doi.org/10.1080/01445340.2016.1247322). *History and Philosophy of Logic* (Taylor & Francis) **38** (1) 57–71.
Luna, Laureano; Taylor, William (2010). [“Cantor’s proof in the full definable universe”](https://doi.org/10.26686/ajl.v9i0.1818). *The Australasian Journal of Logic* **9**: 10–26.
[Hamkins, Joel David](http://en.wikipedia.org/wiki/Joel David Hamkins); Linetsky, David; Reitz, Jonas (2013). [“Pointwise definable models of set theory”](https://doi.org/10.2178/jsl.7801090). *The Journal of Symbolic Logic* **78** (1): 139–156. Also [*arXiv:1105.4597*](https://arxiv.org/abs/1105.4597).
[Myhill, John](http://en.wikipedia.org/wiki/John Myhill) (1952). [“The hypothesis that all classes are nameable”](https://doi.org/10.1073/pnas.38.11.979). *Proceedings of the National Academy of Sciences* **38** (11): 979–981.
[Linnebo, Øystein](http://en.wikipedia.org/wiki/Øystein Linnebo) (2013). [“The potential hierarchy of sets”](https://doi.org/10.1017/S1755020313000014). *The Review of Symbolic Logic* **6** (2): 205–228.
[Tarski, Alfred](http://en.wikipedia.org/wiki/Alfred Tarski) (1956). “On definable sets of real numbers”. [*Logic, Semantics, Metamathematics: Papers from 1923 to 1938.*](https://books.google.co.il/books?id=2uhra9PEFZsC) Oxford, Clarendon Press. pp. 110–-142.
[^1]: *The paradox begins with the observation that certain expressions of natural language define real numbers unambiguously, while other expressions of natural language do not. For example, “The real number the integer part of which is 17 and the $n$th decimal place of which is 0 if $n$ is even and 1 if $n$ is odd” defines the real number 17.1010101... = 1693/99, while the phrase “the capital of England” does not define a real number.\
Thus there is an infinite list of English phrases (such that each phrase is of finite length, but lengths vary in the list) that define real numbers unambiguously. We first arrange this list of phrases by increasing length, then order all phrases of equal length [lexicographically](http://en.wikipedia.org/wiki/Lexicographical order) (in dictionary order), so that the ordering is [canonical](http://en.wikipedia.org/wiki/Canonical form). This yields an infinite list of the corresponding real numbers: $r_1,r_2,\dots$ Now define a new real number $r$ as follows. The integer part of $r$ is $0$, the $n$th decimal place of $r$ is $1$ if the $n$th decimal place of $r_n$ is not $1$, and the $n$th decimal place of $r$ is $2$ if the $n$th decimal place of $r_n$ is 1.\
The preceding two paragraphs are an expression in English that unambiguously defines a real number $r$. Thus $r$ must be one of the numbers $r_n$. However, $r$ was constructed so that it cannot equal any of the $r_n$. This is the paradoxical contradiction.* (Quoted from Wikipedia.)
[^2]: Logical notation: $\land$ “and” $\lor$ “or” $\imp$ “implies” $\neg$ “not” $\forall$ “for every” $\exists$ “there exists (at least one)” $\exists!$ “there exists one and only one” [(link to a longer list)](http://en.wikipedia.org/wiki/List of logic symbols).
[^3]: Set notation:\
$A=\{x\mid P(x)\}$ [“$A$ is the set of all $x$ such that $P(x)$”](http://en.wikipedia.org/wiki/Set-builder notation#Sets_defined_by_a_predicate) $x\in A$ “$x$ [belongs](http://en.wikipedia.org/wiki/Element (mathematics)) to $A$”\
$A\cup B$ [union](http://en.wikipedia.org/wiki/Union (set theory)) $A\cap B$ [intersection](http://en.wikipedia.org/wiki/Intersection (set theory)) $A\setminus B$ [set difference](http://en.wikipedia.org/wiki/Complement (set theory)#Relative_complement) $A\times B$ [Cartesian product](http://en.wikipedia.org/wiki/Cartesian product)\
$\R$ [real line](http://en.wikipedia.org/wiki/Real line) $\R^2$ [Cartesian plane](http://en.wikipedia.org/wiki/Real coordinate space) and [more](http://en.wikipedia.org/wiki/Set (mathematics)), [more](http://en.wikipedia.org/wiki/Algebra_of_sets), [more](http://en.wikipedia.org/wiki/Set-builder_notation).
[^4]: *Theorem.* All relations definable in $(\R; +, \times)$ are semialgebraic sets over integers.
*Proof.* The two relations “$+$”, “$\times$” are semialgebraic (evidently). Two operations, permutation and set multiplication, applied to semialgebraic relations, give semialgebraic relations (evidently). The third, projection operation, applied to a semialgebraic relation, gives a semialgebraic relation by the [Tarski-Seidenberg theorem](http://en.wikipedia.org/wiki/Tarski–Seidenberg theorem)[@Ba Theorem 2.76].
*Theorem.* If $a>1$ and $-\infty<b<c<\infty$, then the binary relation $\{(x,y)\mid (y=a^x) \land (b\le x\le c)
\}$ is not semialgebraic.
*Proof.* Assume the contrary. Then the function $x\mapsto a^x$ on $[b,c]$, being semialgebraic, must be algebraic.[@Ba Prop. 2.86], [@Mar Corollary 3.5]. It means existence of a polynomial $p(\cdot,\cdot)$ (not identically 0) such that $p(x,a^x)=0$ for all $x\in[b,c]$. It follows that $p(z,e^{z\log a})=0$ for all complex numbers $z$. Taking $z=\tfrac{2n\pi i}{\log a}$ we get $p\(\tfrac{2n\pi i}{\log a},1\)=0$ for all integer $n$. Therefore $p(z,1)=0$ for all complex $z$ (otherwise the polynomial $z\mapsto p(z,1)$ cannot have infinitely many roots). Similarly, taking $z=\tfrac{\log u+2n\pi i}{\log a}$ we get $p(z,u)=0$ for all complex $z$ and all $u>0$, therefore everywhere; a contradiction.
[^5]: Follows immediately from the lemma below.
*Lemma.* For every semialgebraic subset $A$ of $\R$ there exists $a\in\R$ such that either $(a,\infty)\subset A$ or $(a,\infty)\cap A=\emptyset$.
*Proof.* First, the claim holds for every set of the form $A=\{x\in\R\mid p(x)>0\}$ where $p(\cdot)$ is a polynomial, since either $p(x)\to+\infty$ as $x\to+\infty$, or $p(x)\to-\infty$ as $x\to+\infty$, or $p(x)$ is constant. Second, a Boolean operation (union, complement), applied to sets that satisfy the claim, gives a set that satisfies the claim (evidently).
[^6]: *Proof.* Denote the first by $D_R$ and the second by $D_S$. We know that $R\in D_S$. It follows (via set multiplication) that $R\times S \in D_S$, $R\times S\times S = R\times S^2 \in D_S$, and so on; by induction, $R\times S^n \in D_S$ for all $n$. Thus (via permutation), $S^n\times R \in D_S$.
In order to prove that $D_R \subset D_S$ we compare the five operations on relations (complement, union, permutation, set multiplication, projection) over $R$ (call them ) and over $S$ (). We have to check that each applied to relations on $R$ that belong to $D_S$ gives again a relation (on $R$) that belongs to $D_S$.
For the union we have nothing to check, since the of two relations is equal to their . Similarly, we have nothing to check for permutation and projection. Only set multiplication and complement need some attention.
Set multiplication. The applied to $A\in\oP(R^n)\cap D_S$ gives $A\times R$. We have $ A\times R = (A\times S) \cap (S^n\times R) \in D_S$ since $A\times S \in D_S$ and $S^n\times R
\in D_S$.
It follows (by induction) that $R^n\in D_S$ for all $n$.
Complement. The applied to $A\in\oP(R^n)\cap D_S$ gives $R^n\setminus A$. We note that the $S^n\setminus A$ belongs to $D_S$ (since $A\in D_S$), thus $R^n\setminus A = R^n \cap (S^n\setminus A) \in D_S$ (since $R^n\in D_S$).
[^7]: It is easy to obtain the sequence $(n_k)_{k=1}^\infty$ out of the sequence $(s_k)_{k=1}^\infty$ of sums $ s_k = \sum_{i=1}^k \al_i $ of the digits $ \al_i = D(i,x) $. The problem is that in the first-order framework we cannot define $(s_k)_{k=1}^\infty$ just by the property “$ \forall k \;\;
s_{k+1}=s_k+\al_{k+1} $”. Yet, this obstacle is surmountable; we can computably encode by natural numbers all tuples of natural numbers. (A similar trick was used in Section \[sect6\].)
[^8]: Assuming, of course, that ZFC itself is consistent.
[^9]: Assuming, of course, that ZFC itself is consistent.
[^10]: Beware of the elusive distinction between two phrases, “for each $n$ it claims $N_{0\ne0}>n$” and “it claims $\forall n \;\> N_{0\ne0}>n$”.
[^11]: Assuming, of course, that ZFC itself is consistent.
|
---
abstract: 'The augmentation powers in an integral group ring $\mathbb ZG$ induce a natural filtration of the unit group of $\mathbb ZG$ analogous to the filtration of the group $G$ given by its dimension series $\{D_n(G)\}_{n\ge 1}$. The purpose of the present article is to investigate this filtration, in particular, the triviality of its intersection.'
author:
- |
Sugandha Maheshwary[[^1] ]{}\
[*Indian Institute of Science Education and Research, Mohali,*]{}\
[*Sector 81, Mohali (Punjab)-140306, India.*]{}\
[*email: [email protected]*]{}
- |
Inder Bir S. Passi [[^2]]{}\
[*Centre for Advanced Study in Mathematics,*]{}\
[*Panjab University, Chandigarh-160014, India*]{}\
[& ]{}\
[*Indian Institute of Science Education and Research, Mohali,*]{}\
[*Sector 81, Mohali (Punjab)-140306, India.*]{}\
[*email: [email protected]* ]{}
title: '**Units and Augmentation Powers in Integral Group Rings**'
---
: integral group rings, augmentation powers, unit group, lower central series, dimension series.\
[**MSC2000 : 16U60, 20F14, 20K15**]{}
Introduction
============
Given a group $G$, the powers $\Delta^n(G),\ n\geq 1$, of the augmentation ideal $\Delta(G)$ of its integral group ring $\mathbb ZG$ induce a $\Delta$-adic filtration of $G$, namely, the one given by its dimension subgroups defined by setting $$D_n(G)=G\cap (1+\Delta^n(G)),\ n=1,2,3,\ldots$$ The dimension series of $G$ has been a subject of intensive research (see [@Pas79], [@GuP87], [@MP09]). This filtration of a group $G$ (group of trivial units in $\mathbb{Z}G$), suggests its natural extension to the full unit group $\mathcal V(\mathbb ZG)$ of normalized units, i.e., the group of units of augmentation one in $\mathbb ZG$, by setting $$\mathcal V_n(\mathbb ZG)=\mathcal V(\mathbb ZG)\cap(1+\Delta^n(G)),\ n=1,2,3\ldots$$ It is easy to see that $\{\mathcal V_n(\mathbb ZG)\}_{n\ge 1}$ is a central series in $\mathcal V(\mathbb ZG)$ and consequently, for every $n\ge 1$, the $n^\mathrm{th}$ term $\gamma_n(\mathcal V(\mathbb ZG))$ of the lower central series $\{\gamma_n(\mathcal V(\mathbb ZG))\}_{n=1}^\infty$ of $\mathcal V(\mathbb ZG)$ is contained in $\mathcal V_n(\mathbb ZG)$. Thus the triviality of the [*$\Delta$-adic residue*]{} of $\mathcal{V}(\mathbb{Z}G)$ $$\mathcal{V}_\omega(\mathbb{Z}G):=\cap_{n=1}^\infty\mathcal{V}_n(\mathbb{Z}G)$$ implies the residual nilpotence of $\mathcal{V}(\mathbb ZG)$, and so, in particular, motivates its further investigation.
The paper is structured as follows: We begin by collecting some results about the above stated filtration in Section 2. In Section 3, we take up the investigation of groups $G$ with $\mathcal{V}_\omega(\mathbb{Z}G)=\{1\}$. Section 4 deals with the groups having trivial $\mathcal D$-residue, i.e., with groups whose dimension series intersect in identity.
We now mention some of our main results.
If $G$ is a finite group, then the filtration $\{\mathcal{V}_n(\mathbb{Z}G)\}_{n\geq 1}$ terminates, i.e.,$\mathcal{V}_n(\mathbb{Z}G)=\{1\}$, for some $n\geq 1$, if and only if\
$(i)$ $G$ is an abelian group of exponent $2,\,3,\,4$ or $6$; or\
$(ii)$ $G=K_8\times E$, where $K_8$ denotes the quaternion group of order $8$ and $E$ denotes an elementary abelian 2-group (Theorem \[theo:Finite\_Terrminates\]). Further, $\mathcal{V}(\mathbb{Z}G)$ has trivial $\Delta$-adic residue, if and only if\
$(i)$ $G$ is an abelian group of exponent $6$; or\
$(ii)$ $G$ is a $p$-group (Theorem \[Finite\_TrivialResidue\]).\
It is interesting to compare the constraints obtained on finite groups satisfying the specified conditions.
For an arbitrary group $G$, if $\mathcal{V}(\mathbb{Z}G)$ has trivial $\Delta$-adic residue, then $G$ cannot have an element of order $pq$ with primes $p<q$, except possibly when $(p,\,q)= (2,\,3)$ (Theorem \[2,3\_only\]). Furthermore, if $G$ is a nilpotent group which is $\{2,\,3\}$-torsion-free, then for $\Delta$-adic residue of $\mathcal{V}(\mathbb{Z}G)$ to be trivial, the torsion subgroup $T$ of $G$ must satisfy one of the following conditions:\
$(i)$ $T=\{1\}$ i.e., $G$ is a torsion-free nilpotent group, or;\
$(ii)$ $T$ is a $p$-group with no element of infinite $p$-height (Theorem \[nil\]).\
In addition, if $G$ is abelian, then it turns out that $\mathcal{V}(\mathbb{Z}G)$ possesses trivial $\Delta$-adic residue, if and only if $\mathcal{V}(\mathbb{Z}T)$ does (Theorem \[abelian\]).
We briefly examine the class $\mathcal C$, of groups $G$ with $\mathcal{V}_\omega(\mathbb{Z}G)=\{1\}$, and prove that a group $G$ belongs to $\mathcal C$, if and only if all its quotients $G/\gamma_n(G)$ do so (Theorem \[nilpotentquotient\]), and that this class is closed under discrimination (Theorem \[disc\]). Finally, we examine the groups $G$ which have the property that the dimension series $\{D_{n,\,\mathbb Q}(G)\}_{n\ge 1}$ over the rationals has non-trivial intersection while $\{D_{n}(G)\}_{n\ge 1}$, the one overintergers has trivial intersection (Theorem \[dimension\]).
For basic results on units and augmentation powers in group rings, we refer the reader to [@Seh78] and [@Pas79].
The filtration $\mathcal{V}_n(\mathbb{Z}G)$
===========================================
As mentioned in the introduction, the filtration $\{\mathcal{V}_n(\mathbb{Z}G)\}_{n=1}^{\infty}$ of the group of the normalized units $\mathcal{V}(\mathbb{Z}G)$ is given by $$\mathcal V_n(\mathbb ZG)=\mathcal V(\mathbb ZG)\cap(1+\Delta^n(G)),\ n=1,2,3\ldots$$ and for every $n\geq 1$, $$\nonumber
\gamma_n(\mathcal V(\mathbb ZG))\subseteq \mathcal V_n(\mathbb ZG),$$ where $\gamma_n(\mathcal V(\mathbb ZG))$ denotes the $n^{\mathrm {th}}$ term of the lower central series $\{\gamma_n(\mathcal V(\mathbb ZG))\}_{n=1}^\infty$.\
In this section, we collect some results about this filtration.
(a)
: It is well-known that, for every group $G$, the map $$g\mapsto g-1+\Delta^2(G),\ g\in G,$$ induces an isomorphism $G/\gamma_2(G)\cong \Delta(G)/\Delta^2(G)$. This map extends to $\mathcal V(\mathbb ZG)$ to yield $$\label{v2}
\mathcal V(\mathbb ZG)=G\mathcal V_2(\mathbb ZG),\quad
\mathcal V(\mathbb ZG )/\mathcal V_2(\mathbb ZG)\cong G/\gamma_2(G)\cong \Delta(G)/\Delta^2(G).$$
(b)
: In case $G$ is an abelian group, then $$\label{T2}
\mathcal V(\mathbb ZG)=G\oplus \mathcal V_2(\mathbb ZG),$$ and $\mathcal V_{2}(\mathbb{Z}G)$ is torsion-free (see [@Seh93], Theorem 45.1).
(c)
: One of the generic constructions of units in $\mathcal{V}(\mathbb{Z}G)$ is that of bicyclic units. For $g,\,h \in G$, define $$u_{g\,,h}:=1+(g-1)h\hat{g},\quad \mathrm{where} ~\hat{g}=1+g+\cdots+g^{n-1},$$ $n$ being the order of $g$. The unit $u_{g\,,h}$ is called a [*[bicyclic unit]{}*]{} in $\mathbb{Z}G$; it is trivial, if and only if, $h$ normalizes $\langle g\rangle$, and is of infinite order otherwise. Since $u_{g,\,h}=1+(g-1)(h-1)\hat{g}\equiv 1\mod \Delta^2(G),$ it follows that all bicyclic units $$\nonumber
u_{g,\,h}\in \mathcal{V}_{2}(\mathbb{Z}G).$$ Moreover, if $g,\,h$ are of relatively prime orders, then $$\label{eq:bicyclic}
u_{g,\,h}\in \mathcal V_\omega(\mathbb ZG),$$ for, in that case, $(g-1)(h-1)\in \Delta^\omega(G):=\cap_{n=1}^\infty(\Delta^n(G)).$
Another generic construction of units in $\mathcal{V}(\mathbb{Z}G)$, is that of *Bass units*. Given $g\in G$ of order $n$ and $k$, $m$ positive integers such that $k^m\equiv 1 \mod n$, $$u_{k,\,m}(g): = \left(1 +g+\ldots+g^{k-1}\right)^m+\frac{1-k^m}{n}\left(1 +g+\ldots+g^{n-1}\right),$$ is a unit which is trivial, i.e., an element of $G$, if and only if $k\equiv \pm 1 ~\mathrm{mod} ~n$.\
We observe the following elementary but useful lemma.
\[TorsionOnly\] If $G$ is a finite group, and $ \mathcal V_n(\mathbb ZG)= \{1\}$ for some $n\geq 1$, then all units of $\mathcal V(\mathbb ZG)$ must be torsion. In particular, all Bass and bicyclic units must be trivial.
Let $u\in \mathcal V(\mathbb ZG)$. As $u \in 1 + \Delta(G)$, therefore, for any $n\geq 1$, there exists $m_n\in \mathbb{N}$ such that $u^{m_n}\in 1 +\Delta^n(G)$, i.e., $u^{m_n}\in \mathcal V_n(\mathbb ZG)$. Hence, if $u$ is non-trivial, it must be torsion.
A well known result in the theory of units in group rings states that for a finite group $G$, all central torsion units in $\mathbb ZG$ must be trivial (see e.g. [@Seh93], Corollary 1.7). Further, a group $G$ such that all central units in $\mathbb ZG$ are trivial, is termed as [*cut-group*]{}; this class of groups is currently a topic of active research ([@BMP17], [@Mah18], [@Bac18], [@BCJM], [@BMP19], [@Tre19]). In this article, we quite often use the characterization of abelian cut-groups, namely, an abelian group $G$ is a cut-group, if and only if its exponent divides $4$ or $6$.\
We now give a classification of finite groups for which this filtration terminates with identity after a finite number of steps.
\[theo:Finite\_Terrminates\] Let $G$ be a finite group. Then $\mathcal{V}_n(\mathbb{Z}G)=\{1\}$, for some $n\geq 1$, if and only if, either
(i)
: $G$ is an abelian cut-group; or
(ii)
: $G=K_8\times E$, where $K_8$ denotes quaternion group of order $8$ and $E$ denotes elementary abelian 2-group.
Let $G$ be a finite group such that $\mathcal{V}_n(\mathbb{Z}G)=\{1\}$, for some $n\geq 1$. Since $\gamma_n(\mathcal{V}(\mathbb{Z}G))\subseteq \mathcal{V}_n(\mathbb{Z}G)$, we have that $\mathcal{V}(\mathbb{Z}G)$ is nilpotent. By classification of finite groups $G$ with nilpotent unit group $\mathcal{V}(\mathbb{Z}G)$ ([@Mil76], Theorem 1), $G$ must be either an abelian group or $G=K_8\times E$, where $K_8$ denotes quaternion group of order $8$ and $E$ denotes elementary abelian 2-group. Now, if $G$ is a finite abelian group which is not a cut-group, then there exists a non-trivial Bass unit in $\mathcal{V}(\mathbb{Z}G)$, which in view of Lemma \[TorsionOnly\], contradicts the assumption.
Conversely, if $G$ is an abelian cut-group, then $\mathcal{V}_2(\mathbb{Z}G)=\{1\}$ (see (\[T2\])) and if $G=K_8\times E$, then by Berman-Higman theorem, we have $\mathcal{V}(\mathbb{Z}G)=G$ implying $D_n(G)=\mathcal{V}_n(\mathbb{Z}G),~n\geq 1$. Moreover, in this case $G$ is a nilpotent group of class $2$, i.e., $\gamma_3(G)=\{1\}$. Since $D_3(G)= \gamma_3(G)$ (see [@Pas79], Theorem 5.10, p.66), we obtain $\mathcal{V}_3(\mathbb{Z}G)=\{1\}$.
Groups $G$ with $\Delta$-adic residue of $\mathcal{V}(\mathbb{Z}G)$ trivial
===========================================================================
The triviality of $\Delta$-adic residue of $\mathcal{V}(\mathbb{Z}G)$ naturally restricts the structure of the group $G$, and consequently also of its subgroups, it being a subgroup closed property. In this section, we explore the structure of groups with this property.
Before proceeding further, observe that if for a group $G$, $\Delta^{\omega}(G)=\{0\}$, then trivially $\mathcal{V}_\omega(\mathbb{Z}G)=\{1\}$. A characterization for a group $G$ to have the property $\Delta^{\omega}(G)=\{0\}$ is known, which we recall for the convenience of reader. To this end, we need few definitions, which we give next.
A group $G$ is said to be [*discriminated by a class $\mathbf{C}$*]{} of groups if, for every finite subset $g_1,\,g_2,\,\ldots\,,\,g_n$ of distinct elements of $G$, there exists a group $H\in \mathbf{C}$ and a homomorphism $\varphi:G\rightarrow H$, such that $\varphi(g_i)\neq \varphi(g_j)$ for $i\neq j$. For a class of groups $\mathbf{C}$, a group $G$ is said to be residually in $\mathbf{C}$, if it satisfies: $1\neq x\in G \implies \exists$ a normal subgroup $N_x$ of $G$ such that $x\not\in N_{x}$ and $G/N_{x}\in \mathbf{C}$. Clearly, if $G\in \mathbf{C}$, then $G$ is residually in $\mathbf{C}$, with $N_x=\{1 \},~\forall~x\neq 1.$ Note that if a class $\mathbf{C}$ is closed under subgroups and finite direct sums, then to say that $G$ is [*residually in $ \mathbf{C}$*]{} is equivalent to saying that $G$ is discriminated by the class $\mathbf{C}$.
\[Delta0\]$\mathrm(\cite{Lic77},see~also~ \cite{Pas79},~ Theorem~2.30,~ p.\, 92)$ For a group $G$, $\Delta^{\omega}(G)=\{0\}$ if, and only if, either $G$ is residually ‘torsion-free nilpotent’ or $G$ is discriminated by the class of nilpotent $p_i$-groups, $i\in I$, of bounded exponents, where $\{p_i,i\in I\}$ is some set of primes.
We now characterise the finite groups $G$ for which $ \mathcal{V}(\mathbb{Z}G)$ has trivial $\Delta$-adic residue. For this, we first prove the following:
\[lem:No\_Nilpotent\] Let $G, H$ be finite groups of relatively prime orders. If $\mathbb{Z}G$ has a non-zero nilpotent element, then $\mathcal{V}(\mathbb{Z}[G\oplus H])$ does not have trivial $\Delta$-adic residue.
Let $G, H$ be finite groups of relatively prime order and let $\alpha(\neq 0)$ be a nilpotent element in $\mathbb{Z}G$, so that $\alpha^k=0$, for some $k\in \mathbb{N}$ and hence $\alpha\in \Delta(G).$ Further, if $h(\neq 1)\in H$, then $h-1 \in \Delta(H)$. By assumption on the orders of $G$ and $H$, we obtain $\alpha(h-1)\in \Delta^{\omega}(G\oplus H)$. Note that as $\alpha$ is nilpotent and as $\alpha$ and $h-1$ commute, $\alpha(h-1)$ is a non-zero nilpotent element in $\mathbb{Z}[G \oplus H]$. Consequently, $1+\alpha(h-1)$ is a non-trivial unit in $\mathcal{V}_\omega(\mathbb{Z}[G \oplus H]).$
\[Finite\_TrivialResidue\] Let $G$ be a finite group. Then $\mathcal V_\omega(\mathbb{Z}G)=\{1\}$, if and only if, either
(i)
: $G$ is an abelian group of exponent $6$, or;
(ii)
: $G$ is a $p$-group.
Let $G$ be a finite group with $\mathcal{V}_\omega(\mathbb{Z}G)=\{1 \}$. If $G$ is not a $p$-group, then let $z\in G$ be an element of order $pq$, with $p,\ q $ primes, $p<q$. The cyclic subgroup $H:=\langle z\rangle$ of $G$ satisfies $\mathcal{V}_\omega(\mathbb{Z}H)=\{1 \}$. On expressing $z$ as $z=xy$ with elements $x,y$ of orders $p$ and $q$ respectively, we have an exact sequence $$1\to \mathcal{V}_\omega(\mathbb{Z}H)\to \mathcal{V}(\mathbb{Z}H)\to \mathcal{V}(\mathbb{Z}\langle x\rangle)\oplus \mathcal{V}(\mathbb{Z}\langle y\rangle)\to 1,$$ induced by the natural projections $H\to \langle x\rangle,\ H\to \langle y\rangle.$ Recall that for a finite abelian group $A$, the torsion-free rank $\rho(\mathcal{V}(\mathbb{Z}A)) $ of the unit group $\mathcal{V}(\mathbb{Z}A)$ is given by the following formula (see [@Seh78], Theorem 3.1, p.54): $$\rho(\mathcal{V}(\mathbb{Z}A)) =\frac{1}{2}\{|A|+n_2-2c_A+1\},$$ where $n_2$ is the number of elements of order 2 in $A$ and $c_A$ is the number of cyclic subgroups in $A$. Thus,
$$\rho(\mathcal{V}(\mathbb{Z}\langle x\rangle ))=
\begin{cases}
0,~\mathrm{if}~p=2,\\
\frac{p-3}{2},~{otherwise}
\end{cases},~~~\rho(\mathcal{V}(\mathbb{Z}\langle y\rangle ))=\frac{q-3}{2}$$ and $$\rho(\mathcal{V}(\mathbb{Z}H))=\begin{cases}
q-3,~\mathrm{if}~p=2,\\\frac{pq-7}{2},~\mathrm{otherwise}.
\end{cases}$$ Therefore, $$\rho(\mathcal{V}(\mathbb{Z}H))>\rho(\mathcal{V}(\mathbb{Z}\langle x\rangle))+\rho(\mathcal{V}(\mathbb{Z}\langle y\rangle)),$$ except possibly when $p=2$ and $q=3$, implying that $G$ must be a (2,3)-group.
Moreover, as $\mathcal{V}(\mathbb{Z}G)$ has trivial $\Delta$-adic residue, it follows that it is residually nilpotent. Hence, by [@MW82], $G$ is a nilpotent group (with commutator group a $p$-group). Thus, if the exponent of $G$ is not $6$, then either $G$ (i) has an element $x$ of order 4 and an element $y$ of order 3; or (ii) has an element $x$ of order 2 and an element $y$ of order 9. In both the cases, rank considerations, as above yield $\mathcal{V}_\omega(\mathbb{Z}G)\neq\{1\}.$ Consequently, $G=E\oplus S$, where $E$ is an elementary abelian $2$-group and $S$ is a group of exponent $3$. As we assume $\mathcal{V}_\omega(\mathbb{Z}G)=\{1\}$, it follows from Lemma \[lem:No\_Nilpotent\] that neither $\mathbb{Z}E$ nor $\mathbb{Z}S$ can have non-zero nilpotent element. By the classification of finite groups whose integral group rings do not have non-zero nilpotent elements [@Seh75], $S$ must be abelian, i.e., $G$ is an abelian group of exponent $6$.
Conversely, if $G$ is an abelian group of exponent $6$, then $\mathcal{V}(\mathbb{Z}G)=G$ and therefore $\mathcal V_\omega (\mathbb{Z}G)=\{1\}$. Also, if $G$ is a $p$-group, then by Theorem \[Delta0\], $ \Delta^\omega(G)=\{0\}$, and hence $\mathcal V_\omega (\mathbb{Z}G)=\{1\}.$
As already observed in (\[eq:bicyclic\]), the existence of non-trivial bicyclic units $u_{g,\,h}\in \mathbb{Z}G$, with the elements $g,\,h\in G$ of relatively prime orders, implies that the $\Delta$-adic residue of $\mathcal{V}(\mathbb{Z}G)$ is non-trivial. Thus, if $G$ is a group with $\mathcal{V}_\omega(\mathbb{Z}G)=\{1\}$, then either $G$ does not have elements of relatively relatively prime orders, or every bicyclic unit $u_{g,\,h}$, with the elements $g,\, h$ of relatively prime orders, is trivial. We next observe that in general, the triviality of the $\Delta$-adic residue of $\mathcal{V}(\mathbb{Z}G)$, for an arbitrary (not necessarily finite) group $G$ has substantial impact on its torsion elements.\
Following the arguments of Theorem \[Finite\_TrivialResidue\], we first record the following result which further brings out the constraints on torsion elements of groups with the property under consideration.
\[2,3\_only\] If $G$ is a group with $\Delta$-adic residue of $\mathcal{V}(\mathbb{Z}G)$ trivial, then $G$ cannot have an element of order $pq$ with primes $p<q$, except possibly when $(p,\,q)= (2,\,3)$; in particular, if the group $G$ is either $2$-torsion-free or $3$-torsion-free, then every torion element of $G$ has prime-power order.
We now give necessary conditions for triviality of $\Delta$-adic residue. For proving the same, recall that an element $g$ of a group $G$ is said to have [*infinite $p$-height*]{} in $G$, if for every choice of natural numbers $i$ and $j$, there exist elements $x\in G$ and $y\in \gamma_j(G)$ such that $x^{p^i}=gy.$ The set of elements of infinite $p$-height in $G$ forms a normal sugroup, and we denote it by $G(p)$.
\[nil\] Let $G$ be a nilpotent group with $\mathcal{V}_\omega(\mathbb{Z}G)=\{1\}$, and let $T$ be its torsion subgroup. Then, one of the following holds:
(i)
: $T=\{1 \}, $
(ii)
: $T$ is a $(2,3)$ group of exponent 6.
(iii)
: $T$ is a $p$-group and $T(p)\neq T$ is an abelian $p$-group of exponent at most $4$.
In particular, if $G$ is a nilpotent group with its torsion subgroup being $\{2,\ 3\}$-torsion-free. Then, $\mathcal{V}(\mathbb{Z}G)$ has trivial $\Delta$-adic residue, only if either $G$ is a torsion-free group or its torsion subgroup is a $p$-group which has no element of infinite $p$-height.
Let $G$ be a nilpotent group with $\mathcal{V}_\omega(\mathbb{Z}G)=\{1\}$, and let $T$ be its torsion subgroup. If $G$ is not torsion-free, then in view of Theorem \[2,3\_only\], either $T$ is a $p$-group, or must be a $(2,\,3)$ group. Moreover, in the latter case, argunig as in Theorem \[Finite\_TrivialResidue\], we have that $T$ must be a $(2,\,3)$ group of exponent $6$.
Suppose $T(p) \neq \{1 \}$ and let $x,y\in T(p)$. Since $x$ is a $p$-element and $y$ is of infinite $p$-height, it follows from ([@Pas79], Theorem 2.3, p.97) that $$(x-1)(y-1)\in \Delta^\omega(T).$$ Consequently, $${\mathcal{V}_2(\mathbb{Z}[T(p)])\subseteq \mathcal{V}_\omega(\mathbb ZT).}$$ Since $\mathcal{V}_\omega(\mathbb ZT)=\{1 \}$, we have $$\mathcal{V}_2(\mathbb Z[T(p)])=\{1 \}.$$ Therefore, by (\[v2\]), $$\mathcal{V}(\mathbb Z[T(p)])=T(p).$$ Moreover, by ([@Mal49], Theorem 1), we have that elements of infinite $p$-height of a nilpotent group commute with all its $p$-elements, and hence $T(p)$ is a central subgroup of $T$. Thus $T(p)$ is a $p$-group which is an abelian cut-group and thus, its exponent is $2$, $3$ or $4$ and consequently $T\neq T(p)$.
Conversely, if $G$ is a nilpotent group which is torsion-free or finitely generated, then $\mathcal{V}(\mathbb{Z}G)$ has trivial $\Delta$-adic residue. For, if $G$ is a torsion-free nilpotent group, then $\Delta^\omega(G)=\{0\}$, and therefore $\mathcal{V}_\omega(\mathbb{Z}G)=\{1 \}.$ If $G$ is finitely generated, then it again follows that $\Delta^\omega(G)=\{0\}$. Further, if $T(p)=\{1 \}$, then $T$ is residually nilpotent $p$-groups of bounded exponent, and hence $\Delta^\omega(T)=\{0\}$ ([@Pas79], Theorem 2.11, p.84) and therefore $\mathcal{V}_\omega(\mathbb{Z}T)=\{1 \}$ i.e., $\mathcal{V}(\mathbb{Z}T)$ has trivial $\Delta$-adic residue.
The triviality of the $\Delta$-adic residue $\mathcal V_\omega(\mathbb ZG)$ does not, in general, imply the triviality of $\Delta^\omega(G)$. For instance, observe that\
$\mathcal{V}(\mathbb{Z}[C_2\oplus {Z}])= C_2\oplus {Z}$ ([[@Seh78], p.57]{}),
\
and therefore, $$\mathcal{V}(\mathbb{Z}[C_2\oplus Q])= C_2\oplus Q\simeq \mathcal{V}(\mathbb{Z}C_2)\oplus \mathcal{V}(\mathbb{Z} Q),$$ where $Z$ is an infinite cyclic group and $Q$ is the additive group of rationals.Consequently, $$\mathcal{V}_\omega(\mathbb{Z}[C_2\oplus Q])=\{1\}.$$ On the other hand, it is easy to see that $\Delta^\omega(C_2\oplus Q)\not=\{0\}$.
We next proceed to analyse the case of abelian groups.
\[abelian\]Let $G$ be an abelian group and let $T$ be its torsion subgroup. Then, $\mathcal V_\omega(\mathbb ZG)=\{1\}$ if, and only if, $\mathcal V_\omega(\mathbb ZT)=\{1\}$.
Let $G$ be an abelian group and let $T$ be its torsion subgroup. Clearly, if $\mathcal V_\omega(\mathbb ZG)=\{1\}$, then $\mathcal V_\omega(\mathbb ZT)=\{1\}$. For the converse, let $u\in \mathcal{V}_\omega(\mathbb ZG)$. Then, as $\mathcal V(\mathbb ZG)=\mathcal V(\mathbb ZT)G$ ([@Seh78], p.56), we have that $u=vg$, with $v\in \mathcal V(\mathbb ZT),\ g\in G$. Since $\mathcal{V}_\omega(\mathbb Z[G/T])=\{1\}$, $G/T$ being torsion-free, projecting $u$ onto $G/T$ yields $g\in T$, and hence $u\in \mathcal V(\mathbb ZT).$ The result now follows from the fact that, $$\mathcal V(\mathbb ZT)\cap \mathcal V_\omega(\mathbb ZG)=\mathcal V_\omega(\mathbb ZT).$$ This is because $$\mathcal V(\mathbb ZT)\cap \mathcal V_n(\mathbb ZG)=\mathcal V_n(\mathbb ZT),\ \text{for all}\ n\geq 1.$$ Note that for the last assertion, it is enough to prove for finitely generated abelian groups, and there it follows from the fact that the group splits over its torsion subgroup.
Theorems \[nil\] and \[abelian\] immediately imply the following:
\[Abelian\] For an abelian group $G$ which is $\{2,\,3\}$-torsion-free, $\mathcal V(\mathbb ZG)$ has trivial $\Delta$-adic residue if, and only if, it is either torsion-free or its torsion subgroup is a $p$-group which has no element of infinite $p$-height.
If $G$ is a nilpotent group and $T$ is the torsion subgroup of $G$ such that idempotents in $\mathbb QT$ are central in $\mathbb QG$, then $$\mathcal V(\mathbb ZG)=\mathcal V(\mathbb ZT)G.$$The above requirement on idempotents holds true, for instance, if $\mathbb QG$ has no non-zero nilpotent elements ([@Seh78], p.194).
Let $G$ be a nilpotent group and $T$ be its torsion subgroup. If $G/T$ is finitely generated, then $$\mathcal V_\omega(\mathbb ZG)\cap \mathbb ZT=\mathcal V_\omega(\mathbb ZT).$$
Let $G$ be a nilpotent group of class $c$ and let $1\to H\to G\to Z\to 1$ be a split exact sequence, so that $$G=HZ,\ H\lhd G,\ H\cap Z=\{1\}.$$ Regard the integral group ring $\mathbb ZH$ as a left $\mathbb ZG$-module with $Z$ acting on $H$ by conjugation and $H$ acting on $\mathbb ZH$ by left multiplication. Then, we have by Swan’s Lemma (see [@Pas79], Theorem 2.3, p.79), $$\nonumber\label{module}
^{\Delta^{m^{c}}(G)}\mathbb ZH\subseteq \Delta^m(H),\ \text{for all $m\geq 1$}.$$ Now if $u=n_1g_1+n_2g_2+\ldots n_rg_r\in \mathbb ZG$ is an element of augmentation $1$, and $g_i=h_iz_i$, for $h_i\in H,\ z_i\in Z$, then we have $$^{u-1}1=\sum n_i(h_i-1)\in \Delta(H).$$ Consequently, if $u\in \mathcal V_\omega(\mathbb ZG)$, then$$^{u-1}1\in \Delta^\omega(H).$$ It thus follows that we have $$\nonumber
\mathcal V_\omega(\mathbb ZG)\cap \mathbb ZH=\mathcal V_\omega(\mathbb ZH).$$
Next, let $T$ be torsion subgroup of $G$. If $G/T$ is finitely generated, then we have a series $$T\lhd H_1\lhd H_2\lhd \ldots\lhd H_n=G$$ with $H_{i+1}/H_i$ infinite cyclic, $1\leq i\leq n-1$. Induction yields the desired result, i.e., $$\nonumber
\mathcal V_\omega(\mathbb ZG)\cap \mathbb ZT=\mathcal V_\omega(\mathbb ZT).$$
We next shift our analysis to the quotients of a residually nilpotent group by elements of lower central series.
\[nilpotentquotient\] Let $G$ be a residually nilpotent group with lower central series $\{\gamma_{n}(G)\}_{n\geq 1}$. Then, the following statements are equivalent:
(i)
: $\mathcal V_\omega(\mathbb Z[G/\gamma_n(G)])=\{1\}$ for all $n\geq 1$;
(ii)
: $\mathcal V_\omega(\mathbb ZG)=\{1\}.$
Let $G$ be as in the statement of the theorem and let $\pi_n:G\to G/\gamma_n(G)$, $n\geq 1$, be the natural projection from the group $G$ to its quotient by $\gamma_n(G)$. Extend $\pi_n$ to the group ring $\mathbb ZG$ by linearity and let the extended map also be denoted by $\pi_n$. Note that $$\ker \pi_n\subseteq \Delta (G,\ \gamma_n(G)).$$ Since $\gamma_n(G)\subseteq 1+\Delta^n(G)$, it follows that $$\ker(\pi_n|_{\mathcal V(\mathbb ZG)})\subseteq\mathcal V_n(\mathbb ZG), \quad n\geq 1.$$Consequently, $$K:=\cap_{n\geq 1}\ker \pi_n|_{\mathcal V(\mathbb ZG)} \subseteq \mathcal V_\omega(\mathbb ZG).$$ Hence, if $\mathcal V_\omega(\mathbb ZG)=\{1\}$, then $K=\{1\}$, implying $\mathcal V_\omega(\mathbb Z[G/\gamma_n(G)])=\{1\}$, for all $n\geq 1$.\
Conversely, let $G$ be a residually nilpotent group and let $\mathcal V_\omega(\mathbb Z[G/\gamma_n(G)])=\{1\}$, for all $n\geq 1.$ In this case, let $u=\sum n_ig_i$ be an element of $\mathcal V_\omega(\mathbb ZG)$ with $n_i\in \mathbb Z$ and all $g_i\in G$ distinct. Choose $m\geq 1$ such that $g_i^{-1}g_j\not\in \gamma_m(G)$, for $i\not= j$. Under the above assumption, $\pi_n(u)=\bar{1},~ \text{for all}\ n\geq 1.$ The choice of $m$ implies that $u=g\in \gamma_\omega(G)=\{1\}$.
We next prove that the class of groups $G$ with $\mathcal{V}_\omega(\mathbb{Z}G)=\{1\}$ is closed with respect to discrimination.
\[disc\]Let $\mathcal C$ denote the class of groups which $G$ such that $\mathcal{V}(\mathbb{Z}G)$ has trivial $\Delta$-adic residue, and let Disc $\mathcal C$ denote the class of groups discriminated by the class $\mathcal{C}$, then $$\mathcal C=\operatorname{Disc}\,\mathcal C.$$
Clearly, by definition, $\mathcal C\subseteq\operatorname{Disc}\,\mathcal C$. Now, let $G\in \operatorname{Disc}\,\mathcal C$, and let $u\in \mathcal V_\omega(\mathbb ZG)$, so that $u=\alpha_1g_1+\ldots+\alpha_ng_n$, where $g_i\neq g_j,$ if $i\neq j$. Let $\varphi:G\to H$ be a homomorphism with $H\in \mathcal C$ such that $\varphi(g_1),\ldots,\varphi(g_n)$ are distinct elements of $H$. Extend $\varphi$ to $\mathbb ZG$ by linearity. Then, observe that $\varphi(u)\in \mathcal V_\omega(\mathbb ZH)=\{1\}$ and it thus follows that $u=g\in \ker\varphi$. In case $g\not=1$, we have a homomorphism $\psi:G\to K$ with $K\in \mathcal C$ such that $\psi(g)\not=1=\psi(1).$ Since $g-1\in \Delta^\omega(G)$, extension of $\psi$ to $\mathbb ZG $ shows that $\psi(g-1)=0$, with $K\in \mathcal C$. Hence, it follows that $G\in \mathcal C$.
Groups with trivial $\mathcal D$-residue
========================================
We next shift our focus to the study of groups $G$ whose $\mathcal D$-residue, namely, $$D_\omega(G):= \cap _{n=1}^\infty D_n(G)$$ is trivial. Since $$\gamma_{n}(G)\subseteq D_n(G)\subseteq \mathcal{V}_n(\mathbb{Z}G),$$ the triviality of the $\Delta$-adic residue of a group always implies that of its $\mathcal D$-residue.\
Further, over the field of rationals, we have the following equivalences known for an arbitrary group $G$ ([@Pas79], Theorem 2.26, p.90):
(i)
: $G$ is residually torsion-free nilpotent;
(ii)
: $\Delta^\omega_\mathbb Q(G)=\{0\};$
(iii)
: $D_{\omega,\,\mathbb Q}(G)=\{1\}$.
We prove the following:
\[dimension\] Let $G$ be a group such that
(i)
: $D_{\omega,\,\mathbb{Q}}(G)\neq \{1\}$; and
(ii)
: $D_{\omega}(G)=\{1\}$.
Then, for any finitely many elements $g_1,\,g_2,\,...\,,\,g_n\in G$, one of the following holds:
(i)
: the left-normed commuatator $[g_1,\,g_2,\,...\,,\,g_n]\in C_G(D_{\omega,\,\mathbb{Q}}(G))$, the centralizer of $D_{\omega,\, \mathbb{Q}}(G)$ in $G$; or
(ii)
: $g_1,\,g_2,\,...\,,\,g_n$ are discriminated by the class $$\mathcal{K}:=\cup_{p\ prime}\,\mathcal{K}_{p},$$ where $\mathcal{K}_{p}$ denotes the class of nilpotent $p$-groups of bounded exponent.
In order to prove the above result, we first prove the following lemma:
\[DiscriminatedOrCentralize\] Let $G$ be a group which is not residually “torsion-free nilpotent" and let $1\not= g\in D_{\omega,\, \mathbb{Q}}(G)$. If $1\not=g_1\in G$ is such that $1$ and $g_1$ are not discriminated by the class $\mathcal{K}$, then $[g_1,\,g]\in D_\omega(G).$
Let $G$, $g_1$ and $g$ be as in the statement of the lemma. Since $$[g_1,\,g]-1=g_1^{-1}g_2^{-1}[(g_1-1)(g-1)-(g-1)(g_1-1)],$$ it suffices to prove that, for all $n\geq 1$, $(g_1-1)(g-1)$ and $(g-1)(g_1-1) $ belong to $ \Delta^n(G)$.
Let $ n\geq 1$ be fixed. Note firstly that $1\not= g\in D_{n,\,\mathbb{Q}}(G)$ implies that $$m_n(g-1)\in \Delta^n(G),$$ for some $m_n\in \mathbb{N}$. Let $m_n=p_1^{\alpha_1}p_2^{\alpha_2}...p_r^{\alpha_r}$ be the prime factorization of $m_n$. Since $1$ and $g_1$ are not discriminated by the class $\mathcal{K}$, for any group $H\in \mathcal{K}$, and for any homomorphism $\phi:G\rightarrow H$, $\phi(g_1)=1_H$. In particular, for all natural numbers $l$, $k$ and primes $p$, since $G/\gamma_{l}(G)G^{p^k}\in {\mathcal{K}_p}$, we have that $$g_1\in \cap_{l,k}\gamma_{l}(G)G^{p^k}.$$ Consequently, $g_1-1\in \Delta^n(G)+p^\ell\Delta(G)$ for all primes $p$ and natural numbers $\ell \geq 1$. It then follows that both $(g-1)(g_1-1)$ and $(g_1-1)(g-1)$ lie in $\Delta^n(G)$.
*Proof of Theorem* \[dimension\]. We first prove the result for $n=1$. If $g_1\not\in C_G(D_{\omega,\,\mathbb{Q}}(G))$, then there exists $ 1\not= g\in D_{\omega,\mathbb{Q}}(G)$ such that $[g_1,\,g]\neq 1$. But then, if 1 and $g_1$ are not discriminated by the class $\mathcal{K}$, by Lemma \[DiscriminatedOrCentralize\], $1\not=[g_1,\,g]\in D_{\omega}(G)$, which contradicts the assumption. Hence,1 and $g_1$ must be discriminated by the class $\mathcal{K}$.
Next, let $w_n:=[g_1,\,g_2,\,...\,,\,g_n]\not=1$, for $n>1$, be a left normed commutator. If $w_n\not \in C_G(D_{\omega,\,\mathbb{Q}}(G))$, then by the above argument, 1 and $w_n$ are discriminated by the class $\mathcal{K}$, i.e., there exists $H \in \mathcal{K}$ and a homomorphism $\phi:G \rightarrow H$, such that $\phi(w_n)\not=1$. Since $\phi(w_n)=[\phi(g_1),\,\phi(g_2),\,...,\,\phi(g_n)]$, it follows that none of $\phi(g_i)$ is an identity element. Hence, the elements $g_1,\,g_2,\,...\,,\,g_n$ are discriminated by the class $\mathcal{K}$. $\Box$
Since for every torsion group $G$, we have $D_{\omega,\,\mathbb{Q}}(G)=G$, the above theorem yields the following:
Let $G$ be a torsion group with trivial $\mathcal{D}$-residue, then, every non-central element is discriminated from the identity element by the class $\mathcal{K}$.
For a group $G$, the above analysis done on $\mathcal V(\mathbb ZG)$ yields the following:
Let $G$ be a group satisfying
(i)
: $\mathcal{V}_\omega(\mathbb{Z}G)=\{1\}$; and
(ii)
: $\mathcal{V}_{\omega,\,\mathbb{Q}}(\mathbb{Z}G)\neq\{1\}$, where $\mathcal{V}_{\omega,\,\mathbb{Q}}(\mathbb{Z}G):=\mathcal{V}(\mathbb{Z}G)\cap(1+\Delta^\omega_\mathbb{Q}(G))$.
Then, for finitely many elements $v_1,\,v_2,\,...\,,\, v_n\in \mathcal{V}(\mathbb{Z}G)$, one of the following holds:
(i)
: the left-normed commutator $[v_1,\,v_2,\,...\,,\, v_n]\in C_{\mathcal{V}}(\mathcal{V}_{\omega,\mathbb{Q}}(\mathbb{Z}G))$, the centralizer of $\mathcal{V}_{\omega,\mathbb{Q}}(\mathbb{Z}G)$ in $\mathcal{V}:=\mathcal{V}(\mathbb{Z}G)$; or
(ii)
: there exists a prime $p$ and integers $k,\,\ell$ such that none of the elements $v_1,\,v_2,\,...\,,\, v_n$ belong to $\mathcal{V}_{p,\,k,\,\ell}(\mathbb{Z}G):=\{v\in \mathcal{V}(\mathbb{Z}G):v-1 \in \Delta^k(G)+p^\ell\mathbb{Z}G\}.$
If the group $G$ is such that $\mathcal{V}(\mathbb{Z}G)=\mathcal{V}_{\omega,\,\mathbb{Q}}(\mathbb{Z}G)$ and $\mathcal{V}_\omega(\mathbb{Z}G)=\{1\}$, then every non central unit $u\in \mathcal{V}(\mathbb{Z}G)$ does not belong to some $\mathcal{V}_{p,\,k,\,l}(\mathbb{Z}G)$.
In case $G$ is a finite group, then clearly $\mathcal V_{\omega,\,\mathbb Q}(\mathbb ZG)\not=\{1\}.$ Thus we immediately have the following:
If $G$ is a finite cut-group with $\Delta$-adic residue of $\mathcal{V}(\mathbb{Z}G)$ trivial, then $G$ is nilpotent and every non-central unit is missed by some $\mathcal{V}_{p,\,k,\,l}(\mathbb{Z}G)$.
**Acknowledgement**
The second author is thankful to Ashoka University, Sonipat, for making available their facilities.
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[^1]: Research supported by DST, India (INSPIRE/04/2017/000897) is gratefully acknowledged.
[^2]: Corresponding author
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abstract: 'Functional connections in the brain are frequently represented by weighted networks, with nodes representing locations in the brain, and edges representing the strength of connectivity between these locations. One challenge in analyzing such data is that inference at the individual edge level is not particularly biologically meaningful; interpretation is more useful at the level of so-called functional regions, or groups of nodes and connections between them; this is often called “graph-aware” inference in the neuroimaging literature. However, pooling over functional regions leads to significant loss of information and lower accuracy. Another challenge is correlation among edge weights within a subject, which makes inference based on independence assumptions unreliable. We address both these challenges with a linear mixed effects model, which accounts for functional regions and for edge dependence, while still modeling individual edge weights to avoid loss of information. The model allows for comparing two populations, such as patients and healthy controls, both at the functional regions level and at individual edge level, leading to biologically meaningful interpretations. We fit this model to a resting state fMRI data on schizophrenics and healthy controls, obtaining interpretable results consistent with the schizophrenia literature.'
author:
- Yura Kim and Elizaveta Levina
bibliography:
- 'references.bib'
title: 'Graph-aware Modeling of Brain Connectivity Networks'
---
Acknowledgments {#sec:acknowledgments .unnumbered}
===============
This research was supported in part by NSF grant DMS-1521551, ONR grant N000141612910, and a Dana Foundation grant to E. Levina, as well as by computational resources and services provided by Advanced Research Computing at the University of Michigan, Ann Arbor. We thank Prof. Stephan Taylor (Psychiatry, University of Michigan) and Prof. Chandra Sripada (Psychiatry and Philosophy, University of Michigan) and members of both of their labs for many useful discussions, and the Taylor lab for providing processed schizophrenia data. Special thanks to Daniel A. Kessler (formerly Psychiatry and now Statistics, University of Michigan) for his help with the draft and many useful discussions.
|
---
abstract: 'We define pointwise partial differential relations for holomorphic discs. Given a relative homotopy class, a relation, and a generic almost complex structure we provide the moduli space of discs which have an injective point with the structure of a smooth manifold. Applications to the local behaviour are given and an adjunction inequality for singularities is derived. Moreover we show that for a coordinate class of a monotone Lagrangian split torus generically the number of non-immersed holomorphic discs is even.'
address: 'Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931 Köln,Germany'
author:
- Kai Zehmisch
title: Holomorphic jets in symplectic manifolds
---
[^1]
Introduction\[intro\]
=====================
We consider holomorphic curves $C$ in a symplectic manifold $(M,\omega)$ w.r.t. a compatible almost complex structure $J$. The boundary $\partial C$ is contained in a Lagrangian submanifold $L$. We suppose $C$ to be parametrized by a smooth map $u\co (\Sigma,\Gamma){\rightarrow}(M,L)$ defined on a Riemann surface $\Sigma$ with complex structure ${\mathrm i}$ and boundary $\Gamma$ such that $J_u\circ Tu=Tu\circ{\mathrm i}$. The map $u$ is called [**holomorphic**]{}, see [@grom85].
Locally holomorphic curves $u$ exists and the partial derivatives $$\partial_xu(0),\ldots,\partial_x^ru(0)$$ ($z=x+iy,r=0,1,2\ldots$) can take any given value, cf. [@sik94]. This determines the $r$-th Taylor polynomial at $0\in{\mathbb C}$ because $u$ is a homogeneous solution of the non-linear Cauchy-Riemann equation $u_x+J(u)u_y=0$. Taking $r$-equivalence classes of germs of holomorphic maps which have the same partial derivatives $\partial_x$ up to order $r$ defines the space of holomorphic $r$-jets on $(\Sigma,\Gamma)\times(M,L)$. This space is a smooth fibration over the source space $(\Sigma,\Gamma)$, see Section \[gentang\]. We consider the product manifold of jet spaces with the same sources $z_1,\ldots,z_{m_0}\in\Sigma$ and $x_1,\ldots,x_{m_1}\in\Gamma$ of orders $r_1,\ldots,r_{m_0},s_1,\ldots,s_{m_0}$. A submanifold $R$ is called a [**holomorphic jet relation**]{}.
Examples are defined by higher order tangency and intersection relations such as holomorphic curves
- intersecting a holomorphic submanifold, see [@ciemoh07],
- intersecting a Lagrangian submanifold (or cycles in there),
- with double points or singularities, see [@bar99; @bar00; @ohzh09; @wen10; @ohzh11; @oh11].
A holomorphic curve is called [**somewhere injective**]{} if there is an immersed injective point on each connected component. The aim of this second paper of the programme begun in [@zehm13] is to show that the moduli space of somewhere injective holomorphic curves which represent a given homology class and jets in $R$ is a manifold provided the almost complex structure is chosen generically, see Section \[theunivjetsp\], \[wellknowncase\], and \[addenthm\]. The dimension is determined by the Maslov, resp. first Chern, number and the dimension of $R$.
In view of the work of Lazzarini [@lazz00; @lazz11] and McDuff-Salamon [@mcsa04] we study the local behaviour of holomorphic discs for generic almost complex structures. In \[aprioripertandlocb\] we prove that a somewhere injective holomorphic disc has a dense set of injective points in the interior and on the boundary, i.e. is [**simple**]{} and [**simple along the boundary**]{}, see [@zehm13]. In \[multicoverddiscs\] we prove Lazzarinis theorem [@lazz11 Theorem B] that generically any non-constant holomorphic disc is multiply covered in the remaining dimension $4$. In \[secadjineq\] and \[seconsing\] we estimate the number of double points and singularities (counted multiplicity) in terms of topological data. In \[secexample\] we give an example how to define Gromov-Witten type invariants counting discs with singular points. In \[subsecemb\] we discuss generic existence of immersed and embedded holomorphic curves.
Generalized holomorphic tangencies\[gentang\]
=============================================
Definition\[jetdef\]
--------------------
Let $\Sigma$ be a Riemann surface and $(M,J)$ a $2n$-dimensional almost complex manifold. We consider holomorphic maps $u\co\Sigma{\rightarrow}(M,J)$. A [**local representation**]{} with [**source**]{} $z\in\Sigma$ and [**target**]{} $u(z)\in M$ consists of a conformal chart $(U,k)$ about $z=k(0)$ and a chart $(V,h)$ about $u(z)=h(0)$ such that w.l.o.g. $(h_*J)(0)={\mathrm i}$ is the standard complex multiplication on ${\mathbb C}^n$. Usually we will suppress the localization in the notation such that the [**Cauchy-Riemann equation**]{} equals $$u_x+J(u)u_y=0$$ w.r.t. the conformal coordinates $z=x+{\mathrm i}y$.
Two germs of holomorphic maps $u,v$ are [**$r$-equivalent**]{} ($r=0,1,2,\ldots$) at $z\in\Sigma$ if $u(z)=v(z)$ and if in a local representation the partial derivatives $\partial_x$ at $0$ coincide up to order $r$, i.e. if $$\partial_x^{\ell}u(0)=\partial_x^{\ell}v(0)$$ for all $\ell=1,\ldots,r$.
With the knowledge of $\partial_xu(0),\ldots,\partial_x^ru(0)$ one can reconstruct the $r$-th Taylor polynomial at $0$ by taking partial derivatives of the Cauchy-Riemann equation $$u_y=J(u)u_x$$ as follows: $$\begin{aligned}
u_{xy}&=&J(u)u_{xx}+\big(DJ(u)\cdot u_x\big)u_x\\
u_{yy}&=&J(u)u_{xy}+\big(DJ(u)\cdot u_y\big)u_x\\
&\vdots&\end{aligned}$$ Therefore, [*all*]{} partial derivatives of $u$ and $v$ up to order $r$ coincide at $0$. By the chain rule this implies independence of the chosen local representation which we used in the definition. Therefore, the equivalence relation is well defined. The $r$-equivalence class $j_z^ru$ is called the [**holomorphic $r$-jet**]{}. We denote the space of all $r$-jets by $$(\Sigma\times M)_J^r\equiv\operatorname{\mathrm{Jet}}^r.$$
A representation
----------------
For a small open neighbourhood $V$ of $0\in{\mathbb R}^{2n}$ the space of holomorphic jets from ${\mathbb C}$ to $(V,J)$ can be identified with $$({\mathbb C}\times V)_J^r\equiv{\mathbb C}\times V\times ({\mathbb R}^{2n})^r.$$
\[repr\] Let $J$ be an almost complex structure on ${\mathbb R}^{2n}$ with $J(0)={\mathrm i}$. There exists an open neighbourhood $V$ of $0\in{\mathbb R}^{2n}$ such that for all $(z_0,p)\in{\mathbb C}\times V$ and $a_1,\ldots,a_r\in{\mathbb R}^{2n}$ there exists a germ of holomorphic maps $u\co{\mathbb C}{\rightarrow}({\mathbb R}^{2n},J)$ at $z_0$ satisfying $$\partial_xu(z_0)=a_1,\ldots,\partial_x^ru(z_0)=a_r.$$
Translations in ${\mathbb C}$ are conformal so that it is enough to prove the proposition for $z_0=0$. For some $p>2$ we consider the operator $$T(\xi)=\big(\xi_x+{\mathrm i}\xi_y;\,\xi(0),\partial_x\xi(0),\ldots,\partial_x^r\xi(0)\big).$$ The domain consists of all $\xi\in W^{r+1,p}(D,{\mathbb R}^{2n})$, where $D$ is the unit disc, such that $\xi({\mathrm e}^{2\pi {\mathrm i}\theta})\in{\mathrm e}^{(2r+1)\pi {\mathrm i}\theta}{\mathbb R}^n$ for all $\theta\in[0,1)$, with ${\mathbb R}^n$ identified with $({\mathbb R}\times\{0\})^n$. The target space is $W^{r,p}(D,{\mathbb R}^{2n})\times ({\mathbb R}^{2n})^{r+1}$. By [@mcsa04 Chapter C.4] the operator $T$ is invertible.
As on [@mcsa04 p. 627] (written with Lazzarini) we consider the map $$F(v)=\big(v_x+J(v)v_y;\,v(0),\partial_xv(0),\ldots,\partial_x^rv(0)\big),$$ whose linearization at zero $$DF(0)\cdot\xi=T(\xi)$$ is invertible. By the inverse function theorem $F$ is a diffeomorphism near zero so that the equation $F(v)=(0;\,a_0,\varepsilon a_1,\ldots,\varepsilon^ra_r)$ has a solution $v$ for $\varepsilon>0$ sufficiently small. The desired holomorphic germ is $u=v\circ 1/\varepsilon$.
Differentiable structure
------------------------
Consider local representations $(U,k)$ and $(V,h)$. By Proposition \[repr\] and the proof the maps $$\begin{array}{ccc}
(U\times V)_J^r&{\longrightarrow}&(kU\times hV)_{h_*J}^r
\\
j_z^ru&\longmapsto&j_{k(z)}^r(h\circ u\circ k^{-1})
\end{array}$$ define charts. Therefore, $(\Sigma\times M)_J^r$ is a manifold of dimension $2\big(1+n(r+1)\big)$. The projection onto $\Sigma\times M$ is an affine fibration, cf. [@geig03].
The case with boundary
----------------------
Consider a Riemann surface $\Sigma$ with boundary $\Gamma$. The conformal atlas of $\Sigma$ is enriched by boundary preserving conformal maps into the upper half-plane $H^+$. We consider holomorphic maps which take on $\Gamma$ values in a maximally totally real submanifold $L$ of $(M,J)$. By [@mcsa04 p. 539] there exists charts $h$ of $M$ about points of $L$ which take values in ${\mathbb R}^n$ along $L$ and satisfy $h_*J={\mathrm i}$ on ${\mathbb R}^n$. By a local representation such a choice of charts is understood.
Two germs $u,v\co (\Sigma,\Gamma){\rightarrow}(M,L)$ of holomorphic maps at $x\in\Gamma$ define the same [**holomorphic $s$-jet**]{} ($s=0,1,2,\ldots$) provided that $u_{|\Gamma},v_{|\Gamma}$ are $s$-equivalent in the sense of smooth functions. In other words, the $s$-tangency class $j_x^ru$ is characterized as in \[jetdef\] w.r.t. local representations preserving the boundary condition. The space of all holomorphic $s$-jets on the boundary is denoted by $$(\Gamma\times L)_J^s\equiv\operatorname{\mathrm{Jet}}^s.$$
\[holhalfrep\] Any $s$-jet of a smooth function $\Gamma{\rightarrow}L$ has a holomorphic representative. Moreover, $(\Gamma\times L)_J^s$ is a manifold of dimension $1+n(s+1)$, and an affine fibration over $\Gamma\times L$ induces by the source-target map.
Holomorphic half-disc representation
------------------------------------
We will show that $s$-jets of smooth maps $\Gamma{\rightarrow}L$ can be represented by the restriction of germs of $J$-holomorphic maps $(\Sigma,\Gamma){\rightarrow}(M,L)$ such that the representation depends smoothly on the data. This will prove Proposition \[holhalfrep\], see [@geig03].
We consider an almost complex structure $J$ on ${\mathbb R}^{2n}$ such that $J={\mathrm i}$ on ${\mathbb R}^n$. We claim that there exists an open neighbourhood $V'$ of $0\in{\mathbb R}^n$ such that for all $(x_0,p)\in{\mathbb R}\times V'$ and $a_1,\ldots,a_r\in{\mathbb R}^n$ there exists a germ of $J$-holomorphic maps $u\co(H^+,{\mathbb R}){\rightarrow}({\mathbb R}^{2n},{\mathbb R}^n)$ at $x_0$ satisfying $$\partial_xu(x_0)=a_1,\ldots,\partial_x^ru(x_0)=a_r.$$ It is enough to to show this for $x_0=0$.
[**Step $\mathbf{1}$:**]{} On the unit disc $D\subset{\mathbb C}$ we consider the operator $u\mapsto u_x+{\mathrm i}u_y$ defined on the space $V$ of all ${\mathbb C}^n$-valued functions of Sobolev-class $W^{s+1,p}$, $p>2$, subject to the boundary condition $u({\mathrm e}^{2\pi {\mathrm i}\theta})\in{\mathrm e}^{s\pi {\mathrm i}\theta}{\mathbb R}^n$, $\theta\in [0,1)$. The operator takes values in $W^{s,p}$. By [@mcsa04 Chapter C.4] this operator is onto whose $n(s+1)$-dimensional kernel is generated by $$p_{\ell,b}(z)=\frac{{\mathrm i}^{\ell}}{\ell !2^{s-\ell}}b(1+z)^{s-\ell}(1-z)^{\ell},$$ $b\in{\mathbb R}^n, \ell=0,1,\ldots,s$. Notice, that $$j_1^{\ell-1}p_{\ell,b}=0
\qquad\text{and}\qquad
\partial_y^{\ell}p_{\ell,b}(1)=b.$$ Set $$V_k=\big\{u\in V\;\big|\; D^{\alpha}u(1)=0\;\;\;\forall |\alpha|\leq k-1\big\}$$ and $$W_k=\big\{f\in W^{s,p}(D,{\mathbb C}^n)\;\big|\; D^{\beta}f(1)=0\;\;\;\forall |\beta|\leq k-2\big\}$$ for $k=0,1,\ldots,s$. Due to the pointwise constraints and the boundary condition the operator $$\begin{array}{rcc}
S_k\co V_k&{\longrightarrow}& W_k\times{\mathbb R}^n
\\
u&\longmapsto&\big(u_x+{\mathrm i}u_y,\partial_y^ku(1)\big)
\end{array}$$ is well defined, onto, and its $n(s-k)$-dimensional kernel is generated by $p_{\ell,b}$ for $\ell=k+1,\ldots,s$ and $b\in{\mathbb R}^n$.
[**Step $\mathbf{2}$:**]{} Let $\Omega$ be a domain in $H^+$ obtained by smoothing the corners of the unit half-disc such that $\partial\Omega\cap{\mathbb R}$ is an interval $I$ which contains $0$. Let $\varphi$ be a conformal diffeomorphism $(\Omega,0){\rightarrow}(D,1)$ up to the boundary. Let $V_{\Omega}$ be the space of all ${\mathbb C}^n$-valued functions of Sobolev-class $W^{s+1,p}$ on $\Omega$ subject to the boundary condition $$u(z)\in{\mathrm e}^{s\tfrac{{\mathrm i}}{2}\arg\varphi(z)}{\mathbb R}^n,$$ $z\in\partial\Omega$, where the argument is normalized by $\arg({\mathrm e}^{2\pi {\mathrm i}\theta})=2\pi\theta$. Abbreviate $W^{s,p}(\Omega,{\mathbb C}^n)$ by $W_{\Omega}$. Define $$V_{\Omega,k}=\big\{u\in V_{\Omega}\;\big|\; D^{\alpha}u(0)=0\;\;\;\forall |\alpha|\leq k-1\big\}$$ and $$W_{\Omega,k}=\big\{f\in W_{\Omega}\;\big|\; D^{\beta}f(0)=0\;\;\;\forall |\beta|\leq k-2\big\}$$ and $$\begin{array}{rcc}
S_{\Omega,k}\co V_{\Omega,k}&{\longrightarrow}& W_{\Omega,k}\times{\mathbb R}^n
\\
u&\longmapsto&\big(u_x+{\mathrm i}u_y,\partial_x^ku(0)\big).
\end{array}$$ The invertible maps $u\mapsto u\circ\varphi$ and $$(f,h)\mapsto\big((\varphi_x^1+{\mathrm i}\varphi_y^1)f\circ\varphi,c^kh\big)$$ (using $\varphi=\varphi^1+\varphi^2$ and $\partial_x\varphi(0)=c{\mathrm i}$ for a positive constant $c$) conjugate $S_{\Omega,k}$ to $S_k$. Therefore, $S_{\Omega,k}$ is onto and has $n(s-k)$-dimensional kernel.
[**Step $\mathbf{3}$:**]{} Let $L(z)$, $z\in\partial\Omega$, be a loop of totally real subspaces of ${\mathbb C}^n$ with Maslov index $s$ such that $L(x)={\mathbb R}^n$ for $x\in I$. By Arnol’ds theorem there exists a smooth function $A\co{\mathbb C}{\rightarrow}\operatorname{\mathrm{Gl}}(n,{\mathbb C})$ such that $A(0)=\mathbbm{1}$ and $$A(z)L(z)={\mathrm e}^{s\tfrac{{\mathrm i}}{2}\arg\varphi(z)}{\mathbb R}^n$$ for all $z\in\partial\Omega$, cf. [@mcsa04 p. 554]. Consider the operator $$\hat{Q}\co V_{\Omega}{\rightarrow}W_{\Omega}\times({\mathbb R}^n)^{s+1}$$ mapping $$u\longmapsto\big(u_x+{\mathrm i}u_y;\,u(0),A_x(0)u(0)+u_x(0),\ldots,\partial_x^s(Au)(0)\big).$$ In order to show surjectivity of $\hat{Q}$ let $f\in W_{\Omega}$ and $b_0,b_1,\ldots,b_s\in{\mathbb R}^n$ be arbitrarily given. Because $S_{\Omega,0}$ is onto we find $v^0\in V_{\Omega}$ such that $v_x^0+{\mathrm i}v_y^0=f$ and $v^0(0)=b_0$. Assume that $v^{\ell}\in V_{\Omega}$ is constructed recursively such that $v_x^{\ell}+{\mathrm i}v_y^{\ell}=f$ and $$v^{\ell}(0)=b_0,\ldots,\partial_x^{\ell}(Av^{\ell})(0)=b_{\ell}.$$ By an application of $S_{\Omega,\ell+1}$ we find a holomorphic $q\in V_{\Omega,\ell+1}$ such that $$\partial_x^{\ell+1}q(0)=b_{\ell+1}-\partial_x^{\ell+1}(Av^{\ell})(0).$$ Set $v^{\ell+1}=v^{\ell}+q$ and observe that $v_x^{\ell+1}+{\mathrm i}v_y^{\ell+1}=f$ and $$v^{\ell+1}(0)=b_0,\ldots,\partial_x^{\ell+1}(Av^{\ell+1})(0)=b_{\ell+1}.$$ Therefore, $u=v^s$ is a solution of $\hat{Q}(u)=\big(f;\,b_0,\ldots,b_s\big)$, i.e. $\hat{Q}$ is onto.
[**Step $\mathbf{4}$:**]{} We show that $\hat{Q}$ has trivial kernel. Consider $u\in\ker\hat{Q}$. If $u$ is not constant so at least one of the coordinate functions $u^j\in{\mathbb C}$. By [@geizeh10 Lemma 9.5] the sum of the orders of the zeros of $u^j$ on the boundary plus twice the orders of zeros of $u^j$ at the interior equals the Maslov index $s$. But $u^j$ vanishes at least to the $(s+1)$-st order at zero. This implies $u=0$. Therefore, $\hat{Q}$ is invertible.
[**Step $\mathbf{5}$:**]{} Let $V_L$ be the space of all ${\mathbb C}^n$-valued functions of Sobolev-class $W^{s+1,p}$ on $\Omega$ such that $u(z)\in L(z)$ for all $z\in\partial\Omega$. Consider the operator $$T\co V_L{\rightarrow}W_{\Omega}\times({\mathbb R}^n)^{s+1}$$ mapping $$u\longmapsto\big(u_x+{\mathrm i}u_y;\,u(0),\partial_xu(0),\ldots,\partial_x^su(0)\big).$$ The operator $$Q=(A^{-1}\times\mathbbm{1})\circ T\circ A$$ equals $$Q(u)=\hat{Q}(u)+\big(A^{-1}(A_x+{\mathrm i}A_y)u;\,0\ldots,0\big).$$ The second term is a compact perturbation. Because $\hat{Q}$ is invertible the operator $Q$ is Fredholm of index zero. Hence, $T$ is a Fredholm operator of index zero. Because the Maslov index of $L$ is $s$ an application of [@geizeh10 Lemma 9.5] as in Step $4$ shows triviality of $\ker T$. Hence, the operator $T$ is invertible.
[**Step $\mathbf{6}$:**]{} Taking the derivative of the map $F$ at $v=0$ as in Proposition \[repr\] proves the claim. Q.E.D.
The universal jet space\[theunivjetsp\]
=======================================
We consider a $2n$-dimensional symplectic manifold $(M,\omega)$, a Lagrangian submanifold $L$, and a compatible almost complex structure $J_0$ with the induced metric $\omega(\,.\,,J_0\,.\,)$.
Almost complex structures\[acstandfloer\]
-----------------------------------------
The space ${\mathcal J}$ of all compatible almost complex structures $J$ is a Fréchet manifold. The model is the space of symmetric endomorphism fields $Y$ on $M$ which anti-commute with $J_0$. The homeomorphism $$Y\mapsto J_0(\mathbbm{1}+Y)(\mathbbm{1}-Y)^{-1},$$ $\|Y\|_{C^0}<1$, serves as a global chart for ${\mathcal J}$. The inverse is $$H\co J\mapsto (J+J_0)^{-1}(J-J_0),$$ cf. [@aud94 Proposition 1.1.6].
For global considerations additionally we require the almost complex structures to coincide with $J_0$ in the complement of a relative compact subset of $M$, the so-called [**perturbation domain**]{}.
For a sequence $(\varepsilon_j)$ of positive real numbers we consider infinitesimal almost complex structures which are finite in the [**Floer-norm**]{} $$\sum_{j=0}^{\infty}\varepsilon_j\|Y\|_{C^j}.$$ We choose $(\varepsilon_j)$ such that the resulting open subset is separable and dense in $C^{\infty}$, cf. [@sch95]. The image under $H^{-1}$ is the separable Banach manifold ${\mathcal I}$ of compatible almost complex structures with the induced [**Floer-topology**]{}.
Definition
----------
We consider points $z_1,\ldots,z_{m_0}$ on $\Sigma$ and $x_1,\ldots,x_{m_1}$ on the boundary $\Gamma$. A [**source**]{} is a tuple $$(\mathbf{z},\mathbf{x})=(z_1,\ldots,z_{m_0},x_1,\ldots,x_{m_1})$$ of pairwise distinct points. The space of all sources is denoted by $${\mathcal S}\subset\Sigma^{m_0}\times\Gamma^{m_1}\equiv(\Sigma\times\Gamma)^{\mathbf{m}},$$ where we wrote $\mathbf{m}=(m_0,m_1)$ for the number of source points. The length by definition is $\|\mathbf{m}\|=2m_0+m_1$.
A germ of maps $u\co(\Sigma,\Gamma){\rightarrow}(M,L)$ at $\{\mathbf{z},\mathbf{x}\}$ is a germ of product maps $$u^{\mathbf{m}}=(u^1,\ldots,u^{m_0},v^1,\ldots,v^{m_1})\co(\Sigma,\Gamma)^{\mathbf{m}}{\longrightarrow}(M,L)^{\mathbf{m}}$$ at $(\mathbf{z},\mathbf{x})\in{\mathcal S}$, where we think of the $v^j$ to be extended to a neighbourhood of the sources. Similarly, a germ of almost complex structures $J\in{\mathcal J}$ at $$u(\mathbf{z},\mathbf{x})=\big(u^1(z_1),\ldots,v^{m_1}(x_{m_1})\big)$$ is understood. We say that $(u,J)$ is a [**holomorphic multi-germ**]{} of maps $u^{\mathbf{m}}$ at $(\mathbf{z},\mathbf{x})$ if $J\in{\mathcal J}$ is a germ at $u(\mathbf{z},\mathbf{x})$ and $u$ is a germ of $J$-holomorphic maps at $(\mathbf{z},\mathbf{x})$. Notice, that about equal targets of $u$ for different sources the almost complex structure may differ.
On the set of holomorphic multi-germs $(u,J)$ we define a [**$(\mathbf{r},\mathbf{s})$-tangency relation**]{}: Consider a vector of non-negative integers $$\mathbf{t}=(\mathbf{r},\mathbf{s})=(r_1,\ldots,r_{m_0},s_1,\ldots,s_{m_1}).$$ The length is $$\|\mathbf{t}\|=2(r_1,\ldots,r_{m_0})+(s_1,\ldots,s_{m_1}).$$ Two holomorphic multi-germs $(u_1,J_1)$ and $(u_2,J_2)$ are said to be [**$\mathbf{t}$-equivalent**]{} at $(\mathbf{z},\mathbf{x})$ if $u_1(\mathbf{z},\mathbf{x})=u_2(\mathbf{z},\mathbf{x})=\mathbf{p}$, the $(\mathbf{t}-\mathbf{1})$-jets of $J_1$ and $J_2$ at $\mathbf{p}$ coincide, and $j_{(\mathbf{z},\mathbf{x})}^{\mathbf{t}}u_1=j_{(\mathbf{z},\mathbf{x})}^{\mathbf{t}}u_2$, where we used the notation $$j_{(\mathbf{z},\mathbf{x})}^{\mathbf{t}}u=
(j_{\mathbf{z}}^{\mathbf{r}}u,j_{\mathbf{x}}^{\mathbf{s}}u)=
\big(j_{z_1}^{r_1}u,\ldots,j_{z_{m_0}}^{r_{m_0}}u,j_{x_1}^{s_1}u,\ldots,j_{x_{m_1}}^{s_{m_1}}u\big)$$ of holomorphic jets. As in \[jetdef\] one verifies that this is a correct definition. The equivalence class of a holomorphic multi-germ $(u,J)$ is denoted by $j_{(\mathbf{z},\mathbf{x})}^{\mathbf{t}}(u,J)$; the space of all $\mathbf{t}$-tangency classes holomorphic for some $J\in{\mathcal J}$ by $$\operatorname{{\mathcal J}\!\mathrm{et}}^{\mathbf{t}}.$$
We denote by $E^{(\mathbf{t}-\mathbf{1})}$ the space of jets of smooth sections into the bundle $E$ over $M$ whose fibres consists of $\omega$-compatible complex structures on the corresponding tangent spaces of $M$.
\[univdiffstr\] The natural map $$j_{(\mathbf{z},\mathbf{x})}^{\mathbf{t}}(u,J)\longmapsto\big((\mathbf{z},\mathbf{x}),j_{u(\mathbf{z},\mathbf{x})}^{\mathbf{t}-\mathbf{1}}J\big)$$ is a locally trivial fibration $$\operatorname{{\mathcal J}\!\mathrm{et}}^{\mathbf{t}}{\longrightarrow}{\mathcal S}\times E^{(\mathbf{t}-\mathbf{1})}$$ with affine fibre of dimension $n\|\mathbf{t}\|$.
With help of a Hermitian trivialization the fibre of $E{\rightarrow}M$ is diffeomorphic to the unite ball $W$ in the $n(n+1)$-dimensional vector space of symmetric $(2n\times 2n)$-matrices which anti-commute with ${\mathrm i}$. Hence, $E^{r-1}$ is modeled on $V\times W\times{\mathbb R}^{d_{r-1}}$ (resp. $V'\times W\times{\mathbb R}^{d_{r-1}}$) with $$d_{r-1}=\frac{(2n+r-1)!}{(2n)!(r-1)!}n(n+1),$$ where $V\subset{\mathbb R}^{2n}$ (resp. $V'\subset{\mathbb R}^n$) is an open subset. As in Section \[gentang\] the space of holomorphic $r$-jets (for fixed $J$) can be described locally by $U\times V\times {\mathbb R}^{2rn}$ (resp. $U'\times V'\times {\mathbb R}^{rn}$) for open subsets $U\subset\Sigma$ (resp. $U'\subset\Gamma$).
We claim that the local model for $\operatorname{{\mathcal J}\!\mathrm{et}}^{\mathbf{t}}$ is $$(U,U')^{\mathbf{m}}\times(V,V')^{\mathbf{m}}\times({\mathbb R}^{2n},{\mathbb R}^n)^{\mathbf{t}}\times(W\times{\mathbb R}^{d_{r_1-1}})\times\ldots\times(W\times{\mathbb R}^{d_{s_{m_1}-1}}).$$ Considering each $\mathbf{m}$-coordinate separately we represent jets of $J\in{\mathcal J}$ by local $\omega$-compatible almost complex structures. Represent a holomorphic jet by a germ of holomorphic maps w.r.t. the constructed local $J$’s as in Section \[gentang\]. Notice that a smooth variation of jets of $J\in{\mathcal J}$ is followed by a smooth variation of local $J$’s. A smooth variation of $\mathbf{a}=(a_0,a_1,\ldots,a_r)$ induces a smooth variation of local holomorphic maps:
In view of Propositions \[repr\] and \[holhalfrep\] we define $${\mathcal F}(\mathbf{b},v,J)=\big(v_x+J(v)v_y;\,v(0)-b_0,\partial_xv(0)-b_1,\ldots,\partial_x^rv(0)-b_r\big).$$ The partial derivative $$\frac{\partial{\mathcal F}}{\partial v}(\mathbf{b},0,J)\cdot\xi=T(\xi)$$ is invertible. By the implicit function theorem there exists a local smooth map $v=v(\mathbf{b},J)$ such that $${\mathcal F}\big(\mathbf{b},v(\mathbf{b},J),J\big)=0.$$ The local map $$u\big(a_0,a_1,\ldots,a_r,J\big)(z)=v\big(a_0,\varepsilon a_1,\ldots,\varepsilon^ra_r,J\big)(z/\varepsilon)$$ is $J$-holomorphic with $j_0^ru=\mathbf{a}$, which varies smoothly with the data (including the variation given by translations of the source $z_0$ (resp. $x_0$)).
The argument requires $J(0)={\mathrm i}$ (resp. $J={\mathrm i}$ on ${\mathbb R}^n$) for $J\in{\mathcal J}(V)$ and open subsets $V\subset{\mathbb R}^{2n}\equiv M$. In order to achieve this we take a symplectic Darboux chart (resp. Weinstein chart which is a symplectic embedding $(T^*{\mathbb R}^n,{\mathbb R}^n){\rightarrow}(T^*L,{\mathcal O}_{T*L})$ induced by a chart ${\mathbb R}^n{\rightarrow}L$ followed by a local symplectic embedding $(T^*L,{\mathcal O}_{T*L}){\rightarrow}(M,L)$ induced by a Weinstein neighbourhood.) Set $$\varphi_J(\mathbf{x},\mathbf{y})=x^j\partial_{x^j}+y^jJ(\mathbf{x},\mathbf{y})\partial_{x^j}.$$ Because $J$ is compatible with $\operatorname{{\mathrm d}\mathbf{x}\wedge{\mathrm d}\mathbf{y}}$ and the $\partial_{x^j}$’s span the Lagrangian ${\mathbb R}^n$ about $0$ (resp. on ${\mathbb R}^n$) $\partial_{x^j},J(\mathbf{x},\mathbf{y})\partial_{x^j}$, $j=1,\ldots,n$, is a base. The differential is $({\mathrm i},J)$-complex and invertible at $0$ (resp. on ${\mathbb R}^n$). Transforming the position and the almost complex structures via $$(\mathbf{x},\mathbf{y},J)\mapsto\big(\varphi_J(\mathbf{x},\mathbf{y}),(\varphi_J)_*J\big)$$ allows the use of the map ${\mathcal F}$. This shows smoothness of the transition function which are affine, see [@geig03].
\[tangentvector\] The proof allows a description of a tangent vector of $\operatorname{{\mathcal J}\!\mathrm{et}}^{\mathbf{t}}$ at a point $j_{(\mathbf{z},\mathbf{x})}^{\mathbf{t}}(u,J)$. Let $Y$ be a local infinitesimal almost complex structure in $T_J{\mathcal J}$ and $\xi$ a variation of local holomorphic maps. I.e. $\xi$ is a local vector field along $u$ such that $(\mathbf{v},\mathbf{w})\in T_{(\mathbf{z},\mathbf{x})}{\mathcal S}$, $\tilde{\xi}_{\mathbf{m}}=\xi_{\mathbf{m}}+Du_{\mathbf{m}}\cdot (\mathbf{v},\mathbf{w})$, and $\tilde{Y}=Y+DJ(u)\cdot\tilde{\xi}$ solve $$\tilde{\xi}_x+J(u)\tilde{\xi}_y+\tilde{Y}(u)u_y=0.$$ Hence, a tangent vector can be represented by $j_{(\mathbf{v},\mathbf{w})}^{\mathbf{t}}(\tilde{\xi},Y)$ ([*sic*]{}) understood analogously to the case of $(u,J)$’s.
Universal relations
-------------------
Denote by $\big((\Sigma,\Gamma)\times(M,L)\big)^{(\mathbf{t})}$ the smooth $\mathbf{t}$-jet space of maps $(\Sigma,\Gamma){\rightarrow}(M,L)$. The jet with sources in $\Gamma$ are computed via relative charts. Because the $r$-th Taylor polynomial of a $J$-holomorphic map is completely determined by the holomorphic $r$-jet and the $(r-1)$-jet of $J$ there is a well defined map $$j_{(\mathbf{z},\mathbf{x})}^{\mathbf{t}}(u,J)
\longmapsto
\big(j_{(\mathbf{z},\mathbf{x})}^{\mathbf{t}}u,j_{u(\mathbf{z},\mathbf{x})}^{\mathbf{t}-\mathbf{1}}J\big).$$ Because of Proposition \[univdiffstr\] (local description) and Remark \[tangentvector\] this map is an embedding $$\operatorname{{\mathcal J}\!\mathrm{et}}^{\mathbf{t}}{\longrightarrow}\big((\Sigma,\Gamma)\times(M,L)\big)^{(\mathbf{t})}\times E^{(\mathbf{t}-\mathbf{1})},$$ whose projection to the second factor is a surjective submersion.
A submanifold ${\mathcal R}$ of $\operatorname{{\mathcal J}\!\mathrm{et}}^{\mathbf{t}}$ is called a [**higher order intersection relation**]{} if the image under the natural map is a product manifold with second factor $E^{(\mathbf{t}-\mathbf{1})}$, i.e. ${\mathcal R}$ is $J$-independent. ${\mathcal R}$ is called [**source-free**]{} if the relation puts no restriction to the marked points, i.e. in a local fibre chart (as in the prove of Proposition \[univdiffstr\]) ${\mathcal R}$ is a product manifold with first factor $(U,U')^{\mathbf{m}}$ which corresponds to the source space ${\mathcal S}$.
\[codimofr\] The intersection $R$ of ${\mathcal R}$ with any fibre over $E^{(\mathbf{t}-\mathbf{1})}$ is a submanifold and the codimension of ${\mathcal R}$ is $$\|\mathbf{m}\|+n(\|\mathbf{m}\|+\|\mathbf{t}\|)-\dim R.$$ The intersection $R$ of a source-free relation ${\mathcal R}$ with any fibre over ${\mathcal S}\times E^{(\mathbf{t}-\mathbf{1})}$ is a submanifold and the codimension of ${\mathcal R}$ is $$n(\|\mathbf{m}\|+\|\mathbf{t}\|)-\dim R.$$
Universal moduli space\[univmodspaceisbanmanifold\]
---------------------------------------------------
Let $\Sigma$ be a Riemann surface which is closed or compact with boundary $\Gamma$. Let ${\mathcal R}$ be a intersection relation of order $\mathbf{t}$ and let $A$ be a relative singular integer $2$-homology class in $(M,L)$. The universal moduli space ${\mathcal U}$ by definition is the set of all tuples $(u,J,\mathbf{z},\mathbf{x})$, where $(\mathbf{z},\mathbf{x})\in{\mathcal S}$, $J\in{\mathcal I}$ (see \[acstandfloer\]), and $u$ is a $J$-holomorphic map $(\Sigma,\Gamma){\rightarrow}(M,L)$ representing $A$ such that
- $j_{(\mathbf{z},\mathbf{x})}^{\mathbf{t}}(u,J)\in{\mathcal R}$,
- $u$ is simple (the open set of injective points is dense) and simple along the boundary (the open set of injective points of $u_{|\Gamma}$ is dense in $\Gamma$), see [@zehm13 Corollary 8.5],
- $u(\Sigma)$ is contained in the perturbation domain, see \[acstandfloer\].
\[unimodspacerel\] The universal moduli space ${\mathcal U}$ is a separable Banach manifold.
The universal moduli space for the empty relation ${\mathcal U}_{\emptyset}$ (i.e. $\mathbf{m}=(0,0)$) is a separable Banach manifold, see [@mcsa04 Chapter 3]. In the presence of a relation ${\mathcal R}$ we consider the jet extension map $$(u,J,\mathbf{z},\mathbf{x})\longmapsto j_{(\mathbf{z},\mathbf{x})}^{\mathbf{t}}(u,J).$$ The preimage of ${\mathcal R}$ under $$j^{\mathbf{t}}\co{\mathcal U}_{\emptyset}\times{\mathcal S}{\longrightarrow}\operatorname{{\mathcal J}\!\mathrm{et}}^{\mathbf{t}}$$ is ${\mathcal U}$. We consider a tangent vector of $\operatorname{{\mathcal J}\!\mathrm{et}}^{\mathbf{t}}$ at $j_{(\mathbf{z},\mathbf{x})}^{\mathbf{t}}(u,J)$ which is transverse to ${\mathcal R}$. By $J$-independence of ${\mathcal R}$ we are free to assume that the tangent vector is taken w.r.t. the trivial infinitesimal almost complex structure, see Remark \[tangentvector\]. I.e. for $(\mathbf{v},\mathbf{w})\in T_{(\mathbf{z},\mathbf{x})}{\mathcal S}$ and a local holomorphic section $\xi$ of $(u^*TM,u^*TL)$ near $\{\mathbf{z},\mathbf{x}\}$ the tangent vector is given by $j_{(\mathbf{v},\mathbf{w})}^{\mathbf{t}}(\tilde{\xi},0)$. The holomorphic curve $u$ has the annulus property (see [@zehm13 Theorem 1.1]) and the half-annulus property, see [@zehm13 Theorem 1.3 and Corollary 9.5]. Therefore, as on [@mcsa04 p. 63] $\tilde{\xi}$ extends to a smooth global section $\xi$ such that $(\xi,Y)$ is tangent to ${\mathcal U}_{\emptyset}$ for an infinitesimal almost complex structure $Y\in{\mathcal I}$, see [@mcsa04 Exercise 3.4.5]. Because the universal $\mathbf{t}$-jet space has finite dimension the $\mathbf{t}$-jet extension map is transverse to ${\mathcal R}$. The claim follows with [@lang99 Section II.2].
\[wellknowncase\] If $\Gamma=\emptyset$, Proposition \[unimodspacerel\] follows from [@ciemoh07 Lemmata 6.5, 6.6] and [@mcsa04 Chapter 3]. Moreover, it is enough to require the holomorphic curves to intersect the perturbation domain non-trivially. If $\Gamma\neq\emptyset$ and $\|\mathbf{t}\|=0$, Proposition \[unimodspacerel\] follows with [@mcsa04 Chapter 3]. In that case the assumptions can be relaxed: Each connected component of $\Sigma$ has an injective point of $u$ which is mapped to the perturbation domain.
Generic perturbation
--------------------
By the argument in \[univmodspaceisbanmanifold\] ${\mathcal U}$ is a submanifold in ${\mathcal U}_{\emptyset}\times{\mathcal S}$ with co-dimension given by Proposition \[codimofr\]. Moreover, the projection ${\mathcal U}_{\emptyset}{\rightarrow}{\mathcal I}$ is a smooth Fredholm map of index $\mu(A)+n\chi(\Sigma)$, where $\mu$ denotes the Maslov index and $\chi$ the Euler characteristic, see [@mcsa04 Chapter 3 and Appendix C]. Therefore, the induced map ${\mathcal U}\subset{\mathcal U}_{\emptyset}\times{\mathcal S}{\rightarrow}{\mathcal I}$ is Fredholm of index $\mu(A)+n\chi(\Sigma)+\|\mathbf{m}\|-\operatorname{\mathrm{codim}}{\mathcal R}$. By Sard-Smales theorem the set of regular values is of second Baire category in ${\mathcal I}$ so that by the implicit function theorem the preimage ${\mathcal U}_J$ of a regular value $J\in{\mathcal I}$ is a manifold of dimension given by the index, cf. [@mcsa04 Appendix A]. We call ${\mathcal U}_J$ the ${\mathcal R}$[**-moduli space**]{} and write ${\mathcal U}_{\emptyset,J}$ in the case of the empty relation. Notice, that $J$ is a regular value if $J$ is regular in the sense of [@mcsa04 Definition 3.1.4], i.e. the on $u\in{\mathcal U}_{\emptyset,J}$ linearized Cauchy-Riemann operator is transverse to the zero-section for all $u\in{\mathcal U}_{\emptyset,J}$, and the jet extension map $j^{\mathbf{t}}\co{\mathcal U}_{\emptyset,J}\times{\mathcal S}{\rightarrow}\operatorname{\mathrm{Jet}}^{\mathbf{t}}$ is transverse to $R$. We call $J$ ${\mathcal R}$[**-regular**]{} or [**generic**]{}.
The set of all ${\mathcal R}$-regular almost complex structures is of second Baire category in ${\mathcal I}$ (and therefore dense in ${\mathcal J}$). For all ${\mathcal R}$-regular $J\in{\mathcal I}$ the ${\mathcal R}$-moduli space is a manifold of dimension $$\mu(A)+n\big(\chi(\Sigma)-\|\mathbf{m}\|-\|\mathbf{t}\|\big)+\dim R.$$ If ${\mathcal R}$ is source-free the dimension is $$\mu(A)+n\big(\chi(\Sigma)-\|\mathbf{t}\|\big)+(1-n)\|\mathbf{m}\|+\dim R.$$
After an additional perturbation of $J\in{\mathcal I}$ smoothness of ${\mathcal R}$-moduli spaces holds for holomorphic maps which have an injective point on each connected component of $\Sigma$, see Corollary \[transforallsomewinjcurves\] below.
Enumerative relations\[enumrel\]
================================
We assume the dimension of $M$ to be greater or equal than $4$.
A priori perturbation and local behaviour\[aprioripertandlocb\]
---------------------------------------------------------------
We consider holomorphic curves $u$ such that on each connected component there exists an injective point which is mapped by $u$ into the perturbation domain. By Remark \[wellknowncase\] the following moduli problem can be assumed to be transverse for a generic choice of $J\in{\mathcal I}$:
- $[u]=A$,
- $u$ has $n_0$ interior, $n_1$ boundary, and $\ell$ mixed double points, i.e. $u$ is subject to the intersection relation $$\big(\Delta_M\big)^{n_0}\times\big(\{(p,q)\in M\times L\;|\;p=q\}\big)^{\ell}\times\big(\Delta_L\big)^{n_1}$$ of order $\|\mathbf{t}\|=0$ and length $\|\mathbf{m}\|=(2n_0+\ell,2n_1+\ell)$.
Taking the dimension formula into account this leads as in [@bircor07; @lazz11] to the [**adjunction inequality**]{} $$(n-2)\|(n_0,n_1)\|+(2n-3)\ell\leq\mu(A)+n\chi(\Sigma).$$ Therefore, $u$ has finitely many double points if $n\geq3$. In particular, $u$ is simple and simple along the boundary. For $n=2$ there are finitely many mixed double points. With [@lazz00; @lazz11] $u$ is simple, cf. Remark \[aprioripertrem\] for an alternative argument.
\[aprioripert\] There exists a subset ${\mathcal I}_{\infty}$ of second Baire category in ${\mathcal I}$ such that for all $J\in{\mathcal I}_{\infty}$ any somewhere injective $J$-holomorphic curve is simple and simple along the boundary.
${\mathcal I}_{\infty}$ is the intersection of sets of second category, where the intersection is taken over all relative integral homology $2$-classes $A$ in $(M,L)$ (which is known to be countable) and all intersection relations as above counted by $n_0$, $n_1$, and $\ell$. This proves the corollary in dimension $2n\geq6$.
For $2n=4$ it suffices to establish simplicity along the boundary. We consider the following moduli problem: Let $N$ be a natural number. Let ${\mathcal P}_N$ be a partition of $\Gamma$ into $N$ segments (i.e. connected open subsets) of equal length. We equip each segment $S\in{\mathcal P}_N$ with pairwise distinct points $y_1,\ldots,y_N\in S$. The holomorphic curves $u$ are assumed to be somewhere injective in the sense of Remark \[wellknowncase\]. We require
- $[u]=A$,
- there are pairwise distinct points $x_1,\ldots,x_N\in\Gamma$ each different form the $y_j$’s such that $u(x_j)=u(y_j)$ for all $j=1,\ldots,N$.
Let ${\mathcal I}_{\infty}$ be the intersection of all regular values in ${\mathcal I}$ taken over all $A$ and $N$ corresponding to the moduli problems. I.e. the moduli spaces $\{(u,x_1,\ldots,x_N)\}$ are cut out transversely for $J\in{\mathcal I}_{\infty}$ and are of dimension $\mu(A)+2\chi(\Sigma)-N$.
Arguing by contradiction we suppose that $u$ is a somewhere injective holomorphic curve which is not simple along the boundary. By [@zehm13 Proposition 9.3] there exists a diffeomorphism $\varphi\co S_1{\rightarrow}S_2$ between two disjoint segments of $\Gamma$ such that $u(x)=u\big(\varphi(x)\big)$ for all $x\in S_1$. Let $N>\mu([u])+2\chi(\Sigma)$ be a natural number such that the length of a segment in ${\mathcal P}_N$ is smaller than $1/2$ times the length of $S_2$. Let $S\in{\mathcal P}_N$ be a segment which is contained in $S_2$. In particular, with $x_j:=\varphi^{-1}(y_j)$ we obtain $u(x_j)=u(y_j)$ for all $j=1,\ldots,N$. Therefore, $(u,x_1,\ldots,x_N)$ is an element of one of the above moduli spaces which has a negative dimension. This contradiction proves the claim.
\[aprioripertrem\] A similar argument shows simplicity of $u$. Replace ${\mathcal P}_N$ by a covering by open balls of radius $1/N$ each equipped with $N$ distinct points. Simplicity fails if there is a diffeomorphism $\varphi\co U_1{\rightarrow}U_2$ between two open disjoint subsets of $\Sigma$ such that $u(z)=u\big(\varphi(z)\big)$ for all $z\in U_1$, see [@zehm13 Proposition 2.7]. Choose $N>\tfrac12(\mu([u])+2\chi(\Sigma))$ such that there is a ball in ${\mathcal P}_N$ which is a subset of $U_2$.
For reasons of beauty we give one more argument. Because there are finitely many mixed double points the weak and the strong variant of simplicity along the boundary considered in [@zehm13] are the same. By [@zehm13 Proposition 6.4] the argument in the above proof, which shows simplicity along the boundary, suffices to show simplicity.
Generic multiply covered discs\[multicoverddiscs\]
--------------------------------------------------
A non-constant holomorphic disc $u$ (i.e. $\Sigma= D$) is called [**multiply covered**]{} if there exists a simple holomorphic disc $v$ and a holomorphic map $\pi\co (D,\partial D){\rightarrow}(D,\partial D)$ continuous up to the boundary with $\pi^{-1}(\partial D)=\partial D$ such that $u=v\circ\pi$. In [@lazz11 Theorem B] Lazzarini proved the remarkable fact that generically all non-constant holomorphic discs attached to $L$ are multiply covered provided $\dim M\geq6$. We drop the restriction to the dimension.
For generic $J\in{\mathcal I}$ each non-constant holomorphic discs which is contained in the perturbation domain is multiply covered.
Lazzarinis proof of [@lazz11 Theorem B] is based on his decomposition theorem [@lazz11 Theorem A]. Each non-constant holomorphic disc $u$ can be cut along an embedded graph in $D$ into finitely many simple holomorphic discs whose union has the same image as $u$ and whose homology classes weighted with positive multiples add up to $[u]$. On [@lazz11 p. 254/5] he shows that the graph equals $\partial D$ using ([@lazz11 Proposition 5.15], which can replaced by) the following two results:
- Let $v$ be a simple holomorphic disc. Then there exists no diffeomorphism $\varphi\co S_1{\rightarrow}S_2$ between two disjoint open segments on $\partial D$ such that $v=v\circ\varphi$.
- Let $v_1,v_2$ be simple holomorphic discs with $v_1(D)\not\subset v_2(D)$ and $v_2(D)\not\subset v_1(D)$. Then there exists no diffeomorphism $\varphi\co S_1{\rightarrow}S_2$ between two disjoint open segments on $\partial D$ such that $v_1=v_2\circ\varphi$.
Notice that $v_1,v_2$ define a somewhere injective holomorphic map on $D\sqcup D$ in the sense that each connected component has an injective point. An application of the argument from Corollary \[aprioripert\] proves both items. This establishes Lazzarinis theorem for $\dim M=4$.
Addendum to the Theorem\[addenthm\]
-----------------------------------
Using Corollary \[aprioripert\] the Theorem can be extended in the case the relation puts conditions on the holomorphic jets of order $\geq1$ along the boundary.
\[transforallsomewinjcurves\] For any ${\mathcal R}$-regular $J\in{\mathcal I}_{\infty}$ the conclusion of the Theorem holds for $J$-holomorphic curves which are contained in the perturbation domain and are somewhere injective.
Adjunction inequality\[secadjineq\]
-----------------------------------
The group of conformal automorphisms $G$ of $(\Sigma,\Gamma)$ acts on the ${\mathcal R}$-moduli space for any source-free relation ${\mathcal R}$ by $$g\cdot (u,\mathbf{z},\mathbf{x})=\big(u\circ g^{-1},g(\mathbf{z}),g(\mathbf{x})\big)$$ such that the quotient $${\mathcal M}_R={\mathcal U}_J/G$$ is a smooth manifold of dimension $$\mu(A)+n\big(\chi(\Sigma)-\|\mathbf{t}\|\big)+(1-n)\|\mathbf{m}\|+\dim R-d,$$ where $d$ is the dimension of $G$.
A consequence of the discussion in \[aprioripertandlocb\] we obtain:
\[coradineq\] For a generic choice of $J\in{\mathcal I}$ and all $J$-holomorphic curves $u$ which have an injective point on each connected component of $\Sigma$ which are mapped into the perturbation domain by $u$ we have: $$(n-2)\|(n_0,n_1)\|+(2n-3)\ell\leq\mu([u])+n\chi(\Sigma)-d.$$ Here $n_0$ (resp. $n_1$, $\ell$) is the number of interior (resp. boundary, mixed) double points.
In particular, if the Maslov index vanishes and $n\geq3$ generically somewhere injective holomorphic disc maps are injective.
Singularities\[seconsing\]
--------------------------
A non-constant holomorphic map $u$ has a [**singularity of order**]{} $k$ at $z$ if $j_z^ku=0$ and $D^{k+1}u(z)\neq 0$. By Carlemans similarity principle (cf. [@hoze94 Chapter A.6] and [@abb04 Theorem A.2]) $k$ is the greatest natural number such that all partial derivatives of $u$ at $z$ vanish up to order $k$.
For a generic choice of compatible almost complex structure the moduli space of unparametrized somewhere injective (in the sense of Remark \[wellknowncase\]) holomorphic curves with vanishing derivatives at $\mathbf{m}$ points up to order $\mathbf{k}$ has dimension $$\mu(A)+n\big(\chi(\Sigma)-\|\mathbf{k}\|\big)+\|\mathbf{m}\|-d.$$ As in Corollary \[coradineq\] we obtain, cf. [@wen10 Corollary 3.17] and [@ohzh09; @oh11]:
\[corsing\] For generic $J\in{\mathcal I}$ and all somewhere injective $J$-holomorphic curves $u$ which are contained in the perturbation domain we have: $$n\|\mathbf{k}\|-\|\mathbf{m}\|\leq\mu([u])+n\chi(\Sigma)-d.$$ Here $\mathbf{k}$ is the order of singularities of $u$ at $\mathbf{m}$ points.
For example, generically, all somewhere injective holomorphic discs with Maslov index less or equal than $1$ are immersed.
The map $(u,\mathbf{z},\mathbf{x})\mapsto u$ which is defined on the moduli space of curves as in Corollary \[corsing\] is an immersion. This is because the kernel of the linearization $(\xi,\mathbf{v},\mathbf{w})\mapsto\xi$ is given by all $(\mathbf{v},\mathbf{w})\in T{\mathcal S}$ satisfying $j_{(\mathbf{z},\mathbf{x})}^{\mathbf{k}}Tu(\mathbf{v},\mathbf{w})=0$.
Example\[secexample\]
---------------------
Consider a rational split Lagrangian submanifold $L=S^1\times L'$ in ${\mathbb R}^2\times{\mathbb R}^{2n-2}$ with area spectrum $\pi{\mathbb Z}$. For generic compatible almost complex structures which equal ${\mathrm i}$ outside a large ball we consider holomorphic discs representing the class $[D\times\{*\}]$, which are simple, see [@lazz00; @lazz11]. By Corollary \[corsing\] there is at most one singularity on each disc, which is simple and on the boundary. Hence the zero-dimensional moduli space of discs with a simple boundary singularity ${\mathcal M}_{\partial\text{-}\mathrm{sing}}$ embeds via $[u,\mathbf{z},\mathbf{x}]\mapsto [u]$ into the moduli space of all discs ${\mathcal M}$. By Gromov compactness ${\mathcal M}$ is compact, see [@fra08; @grom85]. Therefore, there are finitely many discs with singularities. The number is even as an argument using a cobordism with boundary ${\mathcal M}_{\partial\text{-}\mathrm{sing}}$ which corresponds to a generic path of almost complex structures with starting point ${\mathrm i}$ shows, cf. [@mcsa04 p. 43 and Remark 3.2.8].
Embeddings\[subsecemb\]
-----------------------
We consider somewhere injective holomorphic curves contained in the perturbation domain which represent a homology class $A$. Generically, the corresponding moduli space ${\mathcal M}$ is a manifolds of dimension $\mu(A)+n\chi(\Sigma)-d$. We can assume that the moduli spaces of curves which additionally have a singularity at the interior or on the boundary are manifolds of dimension
- $\dim{\mathcal M}_{\mathrm{sing}}=\dim{\mathcal M}+2(1-n)$
- $\dim{\mathcal M}_{\partial\text{-}\mathrm{sing}}=\dim{\mathcal M}+1-n$
these with interior, mixed, and boundary double points are manifolds of dimension
- $\dim{\mathcal M}_{\mathrm{inter}}=\dim{\mathcal M}+2(2-n)$
- $\dim{\mathcal M}_{\mathrm{mix}}=\dim{\mathcal M}+3-2n$
- $\dim{\mathcal M}_{\partial\text{-}\mathrm{inter}}=\dim{\mathcal M}+2-n$
and that the moduli space of curves intersection $L$ at an interior point is a manifold of dimension
- $\dim{\mathcal M}_L=\dim{\mathcal M}+2-n$.
Each of the above moduli spaces admits a smooth map into ${\mathcal M}$ induced by forgetting the marked points. The complement of the images is the subset of [**perfectly embedded**]{} curves. By Sards theorem the union of the critical values has measure zero. Because the closure of the images of the moduli spaces subject to a pure intersection relation is contained in the union with the images of ${\mathcal M}_{\mathrm{sing}}$ and ${\mathcal M}_{\partial\text{-}\mathrm{sing}}$ we obtain:
Generically, the set of all perfectly embedded holomorphic curves in ${\mathcal M}$ is open and dense provided $\dim M\geq6$.
An immersed holomorphic curve is called [**clean**]{} if there is no mixed double point and all double points and all interior intersections with $L$ are simple and transverse. With similar arguments we get:
If $\dim M=4$ the set of all clean immersions in ${\mathcal M}$ generically is open and dense.
The research presented in this article generalizes parts of my doctoral thesis [@zehm08]. I am grateful to Matthias Schwarz for suggesting the topic. I would like to thank Georgios Dimitroglou Rizell, Jonathan David Evans, and Samuel Lisi for pointing out the potential applications to (open) Gromov-Witten and (relative) symplectic field theory. This motivated parts of this work. Further, I would like to thank Hansjörg Geiges for explaining to me how to use jets and for illuminating discussions about the correct use of the contraposition.
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[^1]: The author is partially supported by DFG grant ZE 992/1-1.
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abstract: 'We have examined the metal enrichment of the intergalactic medium (IGM) based on a galactic wind model. A galactic wind driven by supernovae brings metallic gas to the IGM but not so far beyond the gravitational potential. The expanding velocity of the outflow depends on the star formation timescale. Examining 3D calculation for the IGM in CDM model, we find that only 10 % region has metallicity larger than $10^{-2}Z_{\odot}$ at $z=3$. Wide range of the IGM metallicity produces variety of CIV column densities for a fixed HI column density.'
address:
- 'National Institute for Fusion Science, Oroshi-cho, Toki, Gifu 509-52, Japan '
- 'Information Processing Center, Chiba University, Inage-ku, Chiba 263, Japan '
author:
- 'Izumi Murakami$^1$, Kazuyuki Yamashita$^2$'
---
Introduction
============
Recent observations have shown us that many Ly $\alpha$ clouds are contaminated with metals. Associated CIV absorption lines indicate that these clouds are likely to have about 1/100 solar metallicity [@Ty95], [@CS95], [@SC96]. This puzzles us how such intergalactic clouds are metal-enriched, since metals must be synthesized in stars, i.e. in galaxies. Another question is whether Ly $\alpha$ clouds with lower HI column densities are metal-enriched or not.
In this work we examine galactic wind driven by supernova explosions to provide metals to the IGM, in order to find answers for above questions.
MODELS
======
We have examined three models to investigate how a galactic wind propagate and bring metallic gas to the IGM on the minihalo model: a spherical cloud model, a grid toy model, and a 3D CDM model.
Assumptions over these models are summarized as follows: Gas and dark matter are in a flat universe ($\Omega=1$) with $H_0 = 50 \rm km \ s^{-1} Mpc^{-1}$ ($h=0.5$). Stars are formed at high density and low temperature region. Eleven percent of stars in mass become supernovae with releasing $10^{51}$erg and $3.2 M_{\odot}$ metallic gas per an average supernova which mass is $20.5 M_{\odot}$. Uniform UV background radiation is assumed with power-law spectrum of $\alpha=1$. The flux evolution model is assumed similarly to Haardt & Madau’s model [@HM] and $J_{21}= 0.72$ at $z=3$, where $J_{21}=
J(912{\rm \AA})/ 10^{-21} \rm erg \ s^{-1}cm^{-2}Hz^{-1}str^{-1}$. For the spherical model the constant flux model with $J_{21}=1$ is also considered. This will change the formation epoch of stars. Ionization equilibrium is assumed, and radiative cooling and UV heating are taken into account.
Spherical cloud model
---------------------
First we examine propagation of a galactic wind from a minihalo with a simple spherical model. A system of dark matter and gas ($\Omega_b=0.1$) evolves from a Gaussian density fluctuation [@BSS]. Star formation criterion is set as $ \rho_b > 1.67 \times 10^{-24} \rm g \ cm^{-3}$, and $T < 10^4 \rm K.$ Stars are formed with the timescale, $\tau_{SF}$.
An expanding shell due to supernova explosions propagates into the expanding region, being accumulated all gas which is collapsing onto the minihalo. The expansion is accelerated by the pressure gradient. The expanding velocity of the shell when it is propagating into the IGM region is $V_{exp} \simeq 150 - 300 $km/s ($\tau_{SF}=10^7 - 3 \times 10^7$yr). In this spherical model the metallic gas is confined in the hot cavity region. When $\tau_{SF}$ is long such as $10^8$yr, supernovae do not produce an expanding shell. Lasting period of the star formation is short $\sim (3-4) \tau_{SF}$. The stellar component ($M_{\ast} \lsim (0.03-0.05)M_g$) distributes at $R_{\ast} \lsim 3-10$ kpc (depending on the initial density fluctuation). Average metallicity of the system is $ Z= M_Z/M_g \simeq 0.06-0.005 Z_{\odot} $ at $z \sim 3$.
Grid geometry model
--------------------
To examine how metals expand in a space where voids (low density region) and gas sheets or wall (high density region) exist, we consider a toy model with grid geometry of walls and voids. Here we consider a $2h^{-1}$ Mpc box (comoving) with $64^3$ mesh and $64^3$ CDM particles. Walls separate 8 void regions in the simulation box. The void under-density is assumed as $\delta \rho_V= 0.80 $ and the wall over-density, $\delta \rho_W= 1.62 $. The wall width is put as $ 0.25h^{-1}$ Mpc. A galaxy ($M_{\ast}=5 \times 10^{10} M_{\odot}$) which is put near the center has supernova explosions at $z=4$ to spread out metallic gas into the IGM.
Shocked hot gas shell following cool and dense gas shell expands in the IGM. The wall prevents the shell expansion which propagates toward the voids with the expanding velocity, $\sim 200$km/s, at $z=2.75$. Most metallic gas is accumulated in the dense shell, where CIV number density is high as well as HI number density. The unperturbed walls are not metal enriched. The IGM metallicity is almost uniform from the central region to the dense shell. Beyond the shell the metallicity decreases with decreasing gas density.
3D IGM model in CDM model
--------------------------
We have performed 3D simulations in the CDM model to examine metal distribution of the IGM. We take a $20 h^{-1}$Mpc (comoving) box with $64^3$ mesh and $64^3$ CDM particles. We assume $\Omega_b=0.052 $, and $\sigma_8= 1.01 $ for the CDM model. Galaxies are assumed to form at a cell where $\rho_{tot} > 5 \rho_{crit}$, $\rho_b > \bar{\rho}_b$, $\nabla \mbox{\boldmath $v$} <0$, and $T < 2 \times 10^5$K, and all gas in the cell is converted to stars. (e.g. [@cen].) Supernovae are assumed to explode soon after galaxies are formed.
First we find that supernova feedback acts to suppress galaxy formation near galaxies. The galaxy formation rate (mass per unit redshift) becomes nearly constant at $z <10$, contrary to a model without supernova feedback, which rate show a peak around $z \sim 6$.
The IGM is heated by supernovae during $z \approx 5-12$, and by gravitational collapse at $z \lsim 5$, in average. The metallicity distribution mostly traces the high density regions where galaxies are formed and metals are produced. Some low density regions caused by galactic winds and bulk motion are found to have high metallicity. The metallicity is less than 1/100 $Z_{\odot}$ at more than 90 % region at $z=3$. Supernova explosions produce two trends in plots of the metallicity vs. the gas density. The metallicity keeps high near galaxies with wide range of the gas density. On the other hand, expanding gas towards void regions makes a trend of lower metallicity with lower density. These trends are clearly seen in the toy grid model in §2.2 and were not found by models without supernova feedback [@GO]. The CIV column density, N(CIV), spreads widely for a fixed HI column density, because of wide range of the metallicity. At $z=3$ all $ 10^{15} \leq $ N(HI)$ <10^{17} \rm cm^{-2}$ clouds seem to have N(CIV)$ \geq 10^{12} \rm cm^{-2},$ similarly to observations [@SC96].
DISCUSSION
===========
The results of these models show that: (1) galactic winds from minihalos and galaxies carry metallic gas to the IGM but are not powerful enough to pollute all the IGM. Even at $z=0$ only 30 % region has metallicity larger than $10^{-2}Z_{\odot}$; and (2) CIV absorption lines would be associated at high density regions. The CIV column densities and metallicity are likely to show similar tendency to the observations.
These models are still preliminary and we need more improvements. For example, simulations with higher spatial resolution would make a change of galaxy formation rate, because the formation criterion might allow to form more galaxies but increased number of supernovae would prevent succeeding formation of galaxies. The change of the galaxy formation rate will affect the precise metallicity distribution. Improved models are being planed to examine, which will give us more realistic information on the metallicity of the IGM.
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abstract: |
Supporting the programming of stateful packet forwarding functions in hardware has recently attracted the interest of the research community. When designing such switching chips, the challenge is to guarantee the ability to program functions that can read and modify data plane’s state, while keeping line rate performance and state consistency. Current state-of-the-art designs are based on a very conservative *all-or-nothing* model: programmability is limited only to those functions that are guaranteed to sustain line rate, with any traffic workload. In effect, this limits the maximum time to execute state update operations.
In this paper, we explore possible options to relax these constraints by using simulations on real traffic traces. We then propose a model in which functions can be executed in a larger but bounded time, while preventing data hazards with memory locking. We present results showing that such flexibility can be supported with little or no throughput degradation.
author:
- |
[ Carmelo Cascone$^{\ddagger}$, Roberto Bifulco$^{\ast}$, Salvatore Pontarelli$^{+}$]{}, Antonio Capone$^{\ddagger}$\
[ $^{\ddagger}$ Politecnico di Milano, $^{\ast}$ NEC Laboratories Europe, $^{+}$ CNIT/Univ. Roma Tor Vergata]{}\
[ [email protected], [email protected], [email protected], [email protected]]{}
bibliography:
- 'biblio.bib'
title: 'Relaxing state-access constraints in stateful programmable data planes'
---
Introduction
============
Processing large network traffic loads requires the realization of complex algorithms in the network data plane, both to implement smart traffic forwarding policies at line rate or to offload processing that used to happen on general purpose CPUs. To meet the required performance levels while still providing the ability to quickly adapt the algorithms to new emerging needs, a new generation of programmable network switches and interfaces has been proposed [@rmt; @flexnic].
In contrast with previous work on active networks [@activenet], network processors [@intelixp] and software network functions [@clickos], this new generation of programmable data planes can provide programmability without paying the cost of a lower forwarding performance. RMT [@rmt] is an example of a hardware design that can achieve high throughput while providing a programmable stateless Match-Action Table (MAT) abstraction, similar to the one provided by OpenFlow [@openflow], but with programmer-defined protocol fields and forwarding actions. Recently, Sivaraman et al. [@domino] demonstrated that it is also possible to implement a programmable high performance stateful data plane in hardware, provided that strict constraints on the per-packet execution time are met. Here, the challenge is to guarantee the ability to program stateful algorithms that read and modify data plane’s state, while keeping line rate performance and state consistency.
More specifically, the implementation of a high performance hardware data plane requires processing packets in parallel. Typically, this is achieved with a pipeline design. Each pipeline’s stage performs a few operations on a packet, and all the stages are executed in parallel. At each tick of the hardware’s clock, packets are moved to the next pipeline’s stage, the packet in the last stage exits the pipeline and a new packet enters in the first stage[^1]. The length of the pipeline finally defines the number of packets actually processed at the same time, in a given clock cycle.
When state read and write operations are quick enough to be executed in a single pipeline’s stage, i.e., in a clock cycle, the state consistency problem is inherently solved, while a *data hazard* arises when more complex computations are required. In fact, complex computations require longer time to be executed, therefore, they may not be completed within a single clock cycle and are instead split to be executed over multiple pipeline’s stages. Since state is typically read in the first stage and written back, after modification, in the last stage, there is a risk the first stage may read an inconsistent state when a new packet enters the pipeline. That is, the read state is going to be invalidated by a result written back in a later stage.
In [@domino], line rate performance are guaranteed by ensuring that state read and write operations happen within the same clock cycle, and that no state is shared between pipeline’s stages. The cost of this design is the inability to express complex operations that take longer to complete.
In this paper, we explore options to relax this constraint by making two observations. First, data planes are usually dimensioned for the worst case scenario, that is, processing minimum size packets at full line rate capacity. However, a data plane pipeline performs algorithms only acting on packets’ header. For a given line rate, larger packet sizes actually mean a lower rate of packet headers to process. Hence, more time per packet header can be used to execute the pipeline operations. Second, to process a given packet, many algorithms access only a subset of the overall data plane’s state. If the pipeline stages access different portions of the state, then the risk of a data hazard is limited to the risk of having in the pipeline two or more packets whose processing requires access to the same portion of the state.
To check the actual data hazard probability when taking into account our observations, we designed a trace-based simulator and run it using real traffic traces, from both carrier and data center networks. To model state accesses, we observe that a common practice for many data plane algorithms is to deal with a network flow-level abstraction. Thus, we assume that only the processing of packets belonging to the same flow requires access to a common portion of the state. Our findings confirm that, in most cases, there is just a little probability of incurring in a data hazard even if state read and write operations happen in different clock cycles. Of course, such probability depends on the packet size and network flows distributions, as well as on the aggregation level of the network flow definition.
Given our findings, we provide as second contribution a sketch of a pipeline design that avoids such data hazards by stalling the pipeline, when needed. Our simulations show that such a design is able to provide line rate throughput, despite the stalls, even extending the time between state read and write to several clock cycles. Furthermore, our design introduces little overhead in terms of memory and circuitry complexity when compared to state-of-the-art solutions. On the flip side of the coin, we are unable to provide line rate throughput in all the cases, therefore the data plane performance is dependent on the actual traffic load properties. Moreover, stalling the pipeline requires the introduction of small queues at the entrance of the pipeline stages. Dimensioning such queues introduces a new variable in the design space: small queues may provide lesser throughput, while big ones may introduce significant latency to the packet forwarding.
Background {#sec: background}
==========
This section presents the data plane architecture we use as our reference model throughout the paper.
![Programmable data plane architecture[]{data-label="fig:arch1"}](fig/fig1.pdf){width="\columnwidth"}
Data plane solutions such as RMT [@rmt], Intel’s FlexPipe [@flexpipe], and Cavium’s XPliant Packet Architecture [@cavium] implement a high level architecture similar to the one sketched in Figure \[fig:arch1\]. In such architectures, packets received from the input ports are stored (enqueued) in a per-port queue and served, with a round robin policy, by a mixer that feeds the packets to an ingress pipeline. After the ingress pipeline, the packets are stored in a common data memory. A scheduler selects which packets should be transmitted to the egress queues. A packet selected for forwarding is first processed by an egress pipeline, which is in principle similar to the ingress one, before being finally transmitted to the egress queues[^2].
Ingress and egress pipelines are composed by a programmable packet parser [@rmt] and by a variable number of Match-Action Table (MAT) elements. For each new packet, the parser extracts the *headers* that are then processed by the MATs elements. A MAT element implements itself a pipeline, whose architecture may sensibly change from one implementation to the other [@hpsr; @domino]. In this paper, we focus on the MAT element’s pipeline, since it is in here that stateful operations are executed.
For the sake of our discussion, we can ignore most of the details of the actual MAT element and model its internal pipeline using just a sequence of stages plus memory. Each stage performs a limited amount of operations, such as read and write from/to memory, or some sort of computation. Since a complex operation can be split in a number of simpler operations executed in multiple stages, the number of stages finally defines the complexity of the operations that can be implemented by the MAT element. Also, since each stage adds a clock cycle to the latency, the number of stages directly impacts the forwarding latency of a packet traversing the MAT element. Finally, and most importantly for the state consistency problem, the number of stages between a memory read and a memory write operations has important implications on the probability of incurring in a race condition. Intuitively, the longer is the time to process a value read from memory, before writing it back, the more probable is the reception of a new packet whose processing requires access to that same memory area. Without loss of generality and to simplify our exposition, in this paper we assume that the first MAT element’s pipeline’s stage reads from memory, while the last one writes back to it.
Rethinking design assumptions
-----------------------------
Recently proposed programmable stateful data planes, such as Banzai [@domino], eliminate data hazards by executing memory read and write operations within the same stage, i.e., in one clock cycle. Such a design derives from the worst case assumption that all packets have minimum size, that they arrive back-to-back, i.e., with no inter-packet gaps, and that they all need to access the same memory area. More specifically, let’s consider a data plane with a throughput of 640 Gb/s, with a chip clocked at 1 GHz, as it is the case of RMT [@rmt]. We can assume that packets are read in chunk of at most 80 bytes (i.e., 80 $\times$ 8 bit $\times$ 1 GHz $=$ 640 Gb/s) when entering the data plane’s pipeline. Consequently, it will take 1 clock cycle to read packets with minimum size $\le 80$ bytes, while it will take more cycles to read longer packets, e.g., 19 for 1500 bytes.
As mentioned earlier, the data plane’s pipeline is dimensioned to accept a new packet’s headers at each clock cycle. However, even when all packets arrive back-to-back, the variability of the packet size will cause the pipeline to experience one or more idle cycles. For example, in the case of maximum size packets of 1500 bytes, the pipeline will receive a new packet’s headers at intervals of 19 clock cycles[^3].
We observe that packets produced by today’s application have very variable size distributions. E.g., spanning from 64 bytes to 1500 bytes, in a typical case. Which could leave some space for relaxing the constraint on the memory read and write operations, when dealing with non corner-case traffic loads.
Per-flow concurrency
--------------------
The second observation is that data plane state can be categorized in two types: *global state* and *flow state*. The first type is state that is shared among all packets, with no distinction, while flow state is shared only by packets of the same flow. We do not put any restriction on the definition of a flow, for example it could be the TCP or UDP 5-tuple, the IP address source-destination pair, the destination IP address, a portion of the latter, e.g. the first 16 bits, or a switch-dependent metadata, such as the packet’s ingress or egress port.
Usually packet processing functions use a combination of the two, or only one. For example, a stateful firewall needs to maintain state for each TCP connection. A source NAT (SNAT) that dynamically translates the source IP address and port of outgoing connections, needs to maintain both flow state and global state: flow state for each L4 connection in order to distinguish between packets of new or existing connections, and in the case of new ones, it needs to pick a source address and port from a pool of available ones. Maintaining a pool of addresses and ports, e.g. using a stack, is an example of global state. Advanced load balancing schemes such as CONGA [@conga] maintain state at different aggregation levels: i) the 5-tuple in order to distinguish between flowlets, i.e. burst of packets, of the same L4 connection, ii) tunnel ID to maintain real-time utilization levels of several paths and iii) global state to maintain the best path among all available ones, to assign new flowlets to.
The notion of flow state is at the base of existing abstractions such as OpenState [@openstate] and FAST [@fast] which both extends the processing of a MAT with stateful capabilities. In both approaches, before the packet is processed by a MAT, a flow key is derived by looking solely at a subset of header fields decided by a programmer at configuration-time. Regardless of the specific operation to compute the flow key, in both approaches the operation on state that a programmer can define are limited to the portion of the memory associated to the packet’s flow key. Indeed, if we imagine the state as an array, where each cell contains the state of a specific flow, it follows that multiple packets accessing and modifying different cells can be processed in parallel, with no harm for consistency. If the flow key is used as an unique index to access the state array, then packets with different flow keys can be processed in parallel.
\[table:trace summary\]
![Packet size cumulative distribution.[]{data-label="fig:pkt size cdf"}](gp_fig/pkt_size-cdf.pdf){width="\columnwidth"}
Motivating experiments {#sec: motivating experiments}
======================
We start by evaluating the actual probability of generating data hazards when processing packets with stateful functions spanning many clock cycles. To do so we implemented a simulator that we feed we real traffic traces. The code of the simulator is available at [@opp-sim].
Traffic traces {#sec:pkt traces}
--------------
We used 4 publicly available traffic traces (Table \[table:trace summary\]). Three traces ([*chi-15*]{}, [*sj-12*]{}, [*mawi-15*]{}) are taken from backbone carrier networks and one from a datacenter ([*fb-web*]{}). Each trace presents different characteristics, in terms of packet size (Figure \[fig:pkt size cdf\]) and number of flows when using different aggregation keys (Table \[table:trace summary\]). CAIDA publishes 1-hour long traces 4 times per year. We select [*chi-15*]{}as one representative of usual conditions as the packet size and number of flows is close to the majority of the other traces published by CAIDA in the recent years [@caida-stats]. The packet size presents a bimodal distribution, 30% of packets have minimum size below 80 bytes, wile 50% have larger size close to 1500 bytes. On the other hand, [*sj-12*]{}and [*mawi-15*]{}represent an abnormal situation. In both traces there is a prevalence of smaller packets and, in the case of [*sj-12*]{}, also an unusual large number of 5-tuples w.r.t. the number of distinct IP destination address.
Finally, [*fb-web*]{}includes packets collected from a Facebook’s datacenter’s cluster that serves web requests. As such it presents a predominance of small packets (80% have size $<$ 200 bytes). It must be noted that this trace is the result of uniform sampling with rate 1:30k. As a consequence of the sampling, we were not able to count the number of distinct flows. The reason is that the probability that two consecutive packets belong to different flows is higher than the other traces, not because of the traffic characteristics but because packets are indeed distant in time (a distance potentially greater than the average flow life). Hence, we use [*fb-web*]{}only to measure the effects of the variable packet size, not the flow distribution.
First results: fraction of data hazards {#sec:data hazard}
---------------------------------------
We simulate the case of a stateful processing block comprising $N$ sequential (pipelined) stages, where each stage is executed in 1 clock cycle, and where the first stage reads from the memory while the last writes back. We call pipeline depth the length in clock cycles of the processing block, hence $N$. A data hazard is the event in which the first stage of the action pipeline processes a header, while another one is currently traveling in the same pipeline. Clearly, when $N=1$, there is no risk of data hazards.
We simulate the case when all packets are received at the switch back-to-back, hence line rate utilization is 100% with no inter-packet gaps. Packets are read in chunks of 80 bytes (as in RMT), hence taking 19 clock cycles to read 1 packet of 1500 bytes. For $N \leq 18$, there is no risk of data hazard. Conversely, with small packets the pipeline will experience shorter idle gaps. In the worst case, the headers of minimum size packets arriving back-to-back, will be also processed back-to-back in the pipeline, hence causing a data hazard for any $N>1$. When considering per-flow concurrency, if two headers belonging to *distinct* flows are processed back-to-back, this does *not* generate a data hazard.
In doing simulations we process traffic in batches of 100k packets. For each batch we compute the fraction of data hazards (FDH) over the total number of clock cycles needed to process 100k packets (which depends on the packet size). To reduce simulation time, for each trace we select batches at a rate of 1:100, in other words we evaluate one batch of 100k consecutive packets every 10m packets. The observation here is that traffic characteristics vary slowly in a period of 10m packets, hence multiple batches close in time will produce similar results. For each trace, we extract the 99th percentile from all FDH samples. As an example, if for a given trace the 99th percentile of the FDH is 0.3, it means that in the 99% of batches evaluated, the FDH was below 30%.
Figure \[fig:hazard 1F\] shows the results for all traces when all packets are considered belonging to the same flow, i.e. accessing global state. Instead, in Figure \[fig:hazard MF\] FDH values are plotted for each trace when aggregating packets with different flow keys. As expected, the FDH greatly depends on the packet size distribution and flow keys, with smaller probability of hazards for traces with higher prevalence of larger packets, and for longer, i.e. finer, flow aggregation keys. such a [*chi-15*]{}. In the second case, per-flow concurrency affects the results. For example, with [*sj-12*]{}when considering state associated to distinct 5-tuples, the risk on incurring in a data hazard is way below the case when state is associated to distinct destination IP addresses. This result follows the flow distribution showed in Table \[table:trace summary\]. For all traces, using 5-tuples performs better than other flow keys. For [*chi-15*]{}, in all cases the FDH is around 1%.
This result is important because it shows the probability of creating inconsistent state, and hence, if memory locking is a viable approach, the probability of incurring in such locking, thus affecting both throughput and latency.
\
![Architecture of a stateful processing block with memory locking. Headers are queued based on their flow key (FK). A scheduler compares the FK of the head of each queue with what is traveling in the function pipeline, admitting only one header per flow.[]{data-label="fig:locking arch"}](fig/locking-arch.pdf){width="1\columnwidth"}
Approach: memory locking {#sec:approach}
========================
We propose here an approach to perform memory locking among packets competing to access the same memory portion. Locking is implemented by stalling the pipeline. That is, if two packets of the same flow arrive back-to-back, processing is paused for the second packet until the first one has left the pipeline. This already affects throughput by a factor of $1/N$ per flow, hence aggregate throughput is maximized when at least $N$ flows are active. Clearly, stalling calls for buffering which also introduces additional latency. We are interested in measuring the impact on throughput and latency when such locking approach is implemented.
We present a simple but effective pipeline design that implements stalling in order to prevent data hazards (Figure \[fig:locking arch\]). In our design, a stateful processing function spanning N clock cycles, is preceded by few $Q$ queues and a scheduler. For each packet’s headers, a first block extracts a flow key (FK), then a dispatcher stores the headers in the $q$-th queue, where $q = hash(FK)\; mod\; Q$, thus preserving the processing order between packets of the same flow. Each queue can store at most $Q_{len}$ headers.
The scheduler decides which packet to admit in the processing pipeline by looking at the tip of each queue and comparing the head-of-line FK with the at most $N$ FKs currently traveling in the pipeline. The scheduler admits a header if its FK is *not* currently in the pipeline. We assume the scheduler is work-conserving, meaning that all non-empty queues are compared at the same time, if at least one header can be served it will do so. To avoid starvation of a queue, the scheduler serves queues in a round-robin fashion, i.e. with cyclic priority.
We assume a FK can have arbitrary length of $FK_{len}$ bits, depending on the number of state memory cells available, for example $FK_{len} = 32$ bits for $2^{32}$ memory cells. As $Q < 2^{FK_{len}}$, multiple flows will end up sharing the same queue. Such an event may generate head-of-line blocking, in which all packets in a queue are held by the first one. Clearly, such a problem can be reduced by adding more queues, which has a cost in terms of silicon needed to implement both the queues and the scheduler.
For the scheduler to be work-conserving, it needs to compare all queues at the same time, hence increasing the number of wires with i) the number of queues and ii) the number of bits to compare for each queue. For this reason, to simplify the implementation of the scheduler’s comparator, the FK is reduced to a smaller space of $W$ bits. This operation can be performed by the flow key extractor, which along the FK (needed later to access the state) extends the headers with a field $w$. In our experiments we compute $w = hash(FK)\; mod\; W$. For example with $W=4$, the scheduler is able to distinguish among $2^{4}=16$ flows. If $W < FK_{len}$ there will be different flows colliding onto the same value $w$, impacting performance. Flow collision also depends on the hash function, in all our experiments $hash() = crc16()$, which is a common feature in packet processing architectures. However, we do not investigate the impact of other hash functions on the distribution of FKs among the different queues and values of $w$.
[Maximum number of clock cycles (up to 30) per processing function, to sustain a given throughput. In all cases $W=4\; bits$. “Global” represents the case when packets need to access global state. Latency values are given for 1 Ghz clock frequency, i.e. 1 clock cycle = 1 ns.]{}\
\[table:cbudget\]
Silicon overhead
----------------
We evaluate now the requirements in terms of silicon area of the added hardware blocks. There are two types of blocks to consider: i) the logic needed to realize the proposed locking scheme and ii) the logic blocks that will realize the stateful processing function.
The combinatorial logic complexity of the locking scheme is basically that of $Q \times N$ comparators each one of $W$ bits, that corresponds to few thousands of logic gates. An ASIC chip nowadays has more than $10^8$ logic gates, hence we consider this header negligible for small values of $Q$, $N$ and $W$. The memory requirements to implement buffering of headers is $H_{len} \times Q \times Q_{len}$ bits, where $H_{len}$ is the length in bits of the data path. With $H_{len} = 88$ bytes (80 for the header and 8 for the metadata), $Q=4$ and $Q_{len}=100$ it requires 35,2 KB of memory overhead for the queues, that is approximately 3.5% of the memory overhead compared with the memory of a MAT stage in RMT [@rmt].
For the second type of block, silicon overhead depends on the actual function implemented, as such we are not able to provide numbers. However, in current programmable ASIC switching technology 80% of chip area is due to memory (TCAMs and the IO, buffer, and queue subsystem), and less than 20% area is due to logic [@rmt]. As a result, we expect that supporting more complex processing functions will not be constrained by chip area. In other words, it seems it is easier to add more logic than to add memory.
Trace-based results {#sec:results}
===================
We evaluated the proposed architecture using the same traffic traces presented in Section \[sec:pkt traces\]. When collecting performance metrics, we used the same approach described in Section \[sec:data hazard\]: traffic is processed in batches of 100k packets, with 10m packets distance between each batch; and 100% line rate utilization. For each batch of packets we compute the *throughput* as the fraction of packets served by the scheduler, over the total number of packets received; and the *latency* as the number of clock cycles from when the packet is completely received to when it is served by the scheduler, i.e. it enters the function pipeline. For simplicity we consider that when $N=1$, i.e. no locking required, latency is 0. Latency is computed for each packet, for each batch we take the 99th percentile among all latency values, finally we take the maximum among all batches for a given trace. For example, a latency value of 5 means that in the worst case, 99% of the packets experienced a latency of no more than 5 clock cycles, e.g. 5ns at 1Ghz.
We evaluated these metrics when varying the different parameters described in Section \[sec:approach\] for the different traces. We present here a subset of the results, a more detailed collection of results can be found at [@opp-sim].
Table \[table:cbudget\] shows results in terms of clock cycle budget, which is the maximum number of clock cycles allowed for a stateful function to complete execution, while sustaining a given throughput. For example, to sustain 100% throughput, using queues of size 10 (headers) does not provide any benefit, as the clock cycle budget is 1 for each trace and flow key. However, by adding more capacity to queues up to 100, budget improves even when using only 1 queue, allowing for functions spanning 20 clock cycles for all flow keys with [*chi-15*]{}, and 4 clock cycles with [*sj-12*]{}, but only when aggregating packets per 5-tuple. Clearly, long queues impact latency. Both clock cycle budget and latency improve if we can admit for a lower throughput of 99.9%, i.e. allowing for 0.1% drop probability. Clearly, reducing utilization (100% in our experiments) reduces further the risk of drop while maintaining the same cycle budget.
Discussion {#sec:discussion}
==========
**Issues with blocking architectures** While the proposed solution enables the execution of more complex operations directly in the data plane, it implements a blocking architecture. That is, for particular workloads, the data plane is unable to offer line rate forwarding throughput. As a consequence, the processing programmed in the data plane should be adapted to the expected network load characteristics, for the line rate to be achievable. Our work helps in defining the boundaries of the achievable performance, for a given workload and set of operations. An additional problem of the dependency on the workload is the possibility to exploit such dependency, e.g., to perform a denial of service attack on the data plane. However, the ability to program stateful algorithms in the data plane should help in detecting and mitigating such exploitations at little cost.
**What can we do more with more clock cycles?** We do not have yet a concrete example of application, however, if one can tolerate a blocking architecture, she can trade the complexity of investigating hardware circuit design for a specific function, e.g. to enforce atomic execution as in Banzai [@domino], with the possibility of using simpler but slower hardware blocks. To the far end, we envision the possibility of using a general purpose packet processor, carefully programmed to complete execution in a longer, but bounded, clock cycle budget, with predictable performance when the traffic characteristics are known. Finally, another option is that of supporting larger memories (e.g. DRAM) which have slower access times ($>$ 1 clock tick at 1GHz) to read and write values. The same multi-queue scheduling approach could be used to coordinate access to multiple parallel memory banks, where each bank is associated to a queue.
Conclusion {#sec:conclusion}
==========
This paper presented a model for a packet processing pipeline which allows execution of complex functions that read and write data plane’s state at line rate, when read and write operations are performed at different stages. Prevention of data hazards is performed by stalling the pipeline. By using simulations on real traffic traces from both carrier and datacenter networks, we show that such model can be applied with little or no throughput degradation. The exact clock cycle budget and latency depends on the packet size distribution (more with larger packets) and on the granularity of the flow key used to access state (more with longer flow keys, e.g. the 5-tuple). The code used for the simulations and additional results are available at [@opp-sim].
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been partly funded by the EU in the context of the H2020 “BEBA” project (Grant Agreement: 644122).
[^1]: While the data plane emits one packet per clock cycle, it is in fact processing a number of packets in parallel.
[^2]: While in this paper we focus on a store-and-forward mode of operation, our considerations are applicable also to data planes that work in a cut-through configuration.
[^3]: While in principle it is possible to widen the pipeline data-path to reduce the maximum number of clock cycles to process a packet, and to increase the throughput, the routing of such a big number of parallel wires prevents several technological challenges that actually limit the maximum data-path width. For example in RMT[@rmt] it is explicitly mentioned that the data-path is limited due to these technology constraints. Furthermore, the maximum achievable throughput is finally limited by the network interfaces speed.
|
---
abstract: 'We explore the conditions under which colloids can be stabilized by the addition of smaller particles. The largest repulsive barriers between colloids occur when the added particles repel each other with soft interactions, leading to an accumulation near the colloid surfaces. At lower densities these layers of mobile particles (nanoparticle halos) result in stabilization, but when too many are added, the interactions become attractive again. We systematically study these effects –accumulation repulsion, re-entrant attraction, and bridging – by accurate integral equation techniques.'
author:
- 'S. Karanikas'
- 'A. A. Louis'
title: Dynamic Colloidal Stabilization by Nanoparticle Halos
---
[^1]
Colloidal dispersions – solid particles with radii ranging from a few $nm$ to a few $\mu m$, suspended in a liquid solvent – are common in nature and widely used in industry. Blood, paint, ink and cement are typical examples. Because interatomic dispersion forces induce effective van der Waals interactions with large attractive values at contact, colloids will irreversibly aggregate, which is usually undesirable, unless their surfaces are prevented from approaching too closely. The two most common ways to achieve this are called [*steric*]{} and [*charge stabilization*]{}[@Hunt01]. Popular strategies for steric stabilization usually involve grafting a layer of polymers onto the colloid surface, resulting in dense repulsive brushes that prevent close contact. For charge stabilization, the route most common in nature, the colloids have surface charges of the same sign, leading to a double layer of microscopic co- and counter-ions. Adding this effective repulsion to the intrinsic van der Waals attraction results in the famous Derjaguin Landau Verwey Overbeek (DLVO) potential[@DLVO], with a metastable free-energy barrier preventing aggregation.
In an important recent development, a third strategy for colloidal stabilization, termed [*nanoparticle haloing*]{}, was introduced by Lewis and co-workers[@Tohv01]. By adding charged hydrous zirconia nanoparticles of average radius $3 nm$ to a suspension of (marginally charged) colloidal silica spheres of radius $285 nm$ in deionized water, the following behavior was observed: For low nanoparticle concentrations the silica spheres aggregate, driven by the generic van der Waals attractions. At intermediate nanoparticle concentrations, the dispersion becomes stable, whereas at higher concentrations the silica spheres aggregate again. The authors[@Tohv01] attribute the initial stabilization to layering of the small nanoparticles near the colloidal surfaces. These “halos” occur because it is advantageous for the charged nanoparticles to be near the uncharged colloid surfaces. When two colloids then approach each other, their respective halos repel, preventing aggregation. The re-entrant aggregated phase, observed at higher nanoparticle concentrations, was attributed to normal entropic depletion attraction[@Asak58].
Clearly, a new route to stabilize colloids would have many potential applications. Indeed, nanoparticle haloing has already been used to enhance the self-assembly of 3-D colloidal crystals on patterned surfaces[@Lee04]. Nevertheless, even though this novel stabilization strategy has been demonstrated by experiment, many questions remain about its generic applicability. To address these issues we carry out a systematic theoretical study of the effective interaction $ \beta V_{bb}^{eff}(r)$[@Liko01] between colloids, induced by (much) smaller particles ($\beta^{-1}$$=$$k_B T$ is the reduced temperature.). We find fairly large regime of parameter space where $\beta V_{bb}^{eff}(r)$ is repulsive enough for stabilization, but this is usually followed by re-entrant attraction at higher small particle packing fraction. The picture that emerges is considerably more subtle than that of a simple static layer of adsorbed particles akin to steric stabilization. Instead, the nanoparticle halos are dilute, and in dynamic equilibrium with the bulk solution. Furthermore, we observe no obvious change in their character when the re-entrant attraction kicks in, implying that this phenomenon is more complex than simple depletion attraction.
The key quantity we study is the effective interaction[@Liko01] $
\beta V_{bb}^{eff}(r)$ between two spheres of diameter $\sigma_{bb}$, induced by smaller spheres. Its properties are determined by the number density $\rho_s$$=$$N_s/V$ of small particles and by the interactions $\beta \Phi_{bs}(r)$ and $\beta \Phi_{ss}(r)$; it is independent of the intrinsic interaction $\beta \Phi_{bb}(r)$[@Loui02a]. If $\Phi_{bb}(r)$ is attractive and leads to aggregation, then introducing small particles that induce a $\beta V_{bb}^{eff}(r)$ repulsive enough to counteract $\beta \Phi_{bb}(r)$ will stabilize the colloids.
The basic big-small and small-small interactions $\beta \Phi_{ij}(r)$ are modeled by a hard-core Yukawa (HCY) form which is versatile without having too many parameters to vary[@Loui02a]. $\beta \Phi_{ij}(r) = \infty$ if $r$$<$$\sigma_{ij}$; $\beta\Phi_{ij}(r)=\phi_{ij}(r)$ for $r$$>$$\sigma_{ij}$, where in each case $r$ denotes the distance between the centers of the particles, and the Yukawa tail is $$\label{eq2.1}
\beta \phi_{ij}(r) = \frac{ \beta \epsilon_{ij} \sigma_{ij}}{r} \exp \left[ -
\frac{(r - \sigma_{ij})}{\lambda_{ij}} \right],$$ where $\sigma_{bs}$=$\frac12 (\sigma_{bb}$+$\sigma_{ss})$ with $\sigma_{ss}$ the small particle hard-core diameter. By varying the size ratio $q=\sigma_{ss}/\sigma_{bs}$, packing fraction $\eta_s=\frac16 \pi
\rho_s \sigma_{ss}^3$, and the 4 dimensionless potential parameters $\beta \epsilon_{ss}$, $\beta \epsilon_{bs}$, $\lambda_{ss}/\sigma_{ss}$ and $\lambda_{bs}/\sigma_{bs}$, a wide variety of different physical situations can be studied[@Loui02a]. For repulsive interactions, for example, Eq. (\[eq2.1\]) is a good model for charged suspensions[@Hans00].
![\[fig:hnc-simulations\] (a): Comparison of simulations (from [@Loui02a]) and HNC calculations of the effective potential $\beta
V_{bb}^{eff}(r)$. The size-ratio $q=0.2$, and the potential parameters $\lambda_{ss}/ \sigma_{ss} = 1/3$, $\lambda_{bs}/\sigma_{ss}=1/1.2$, and $\beta \epsilon_{ss} = 2.99$ are kept constant, while $\beta
\epsilon_{bs}$ is varied. (b) HNC and DFT[@Roth00a] calculations for small size-ratios $q$. To facilitate comparisons, we plot $q\beta
V_{bs}^{eff}(r)$ and shift the curve for $q=0.1$ up by $+0.1$. ](fig-hnc-simulations.eps){width="8cm"}
To restrict this vast parameter space somewhat, and inspired by the successful experiments[@Tohv01], we choose $\epsilon_{ss} >0$ and, initially, $\epsilon_{bs} =0$. The effective potentials $\beta V_{bb}^{eff}(r)$ are calculated by using the two-component Ornstein Zernike (OZ) equations in the $\rho_b\rightarrow 0$ limit where they decouple, together with the hypernetted chain (HNC) integral equation closure[@Hans86], leading to: $\beta V_{bb}^{eff}(r) = -\rho_s
\int d{\bf r'} h_{bs}(r) c_{bs}(|{\bf r'} - {\bf r}|)$, where $h_{bs}(r) = g_{bs}(r)-1$ with $g_{bs}(r)$ the pair correlation function between big and small particles, and $c_{bs}(r)$ is the direct correlation function, related to $h_{bs}(r)$ by the OZ equation[@Hans86]. In this limit, HNC has some important advantages[@Kara04] over other popular integral equations such as Percus Yevick (PY) or Rogers Young (RY)[@Hans86]. For example, it is exact for the AO model[@Asak58] at all densities $\rho_s$ (PY is not[@Loui01a; @Kara04]). Moreover, HNC is known to be particularly accurate for soft repulsive potentials of the type we are investigating[@Hans86]. To validate our method, we compare, in Fig. \[fig:hnc-simulations\], the performance of HNC with several simulations[@Loui02a] for $\epsilon_{ss} >0$, and find excellent agreement. Since we also want to study rather extreme size ratios, we compare, in Fig. \[fig:hnc-simulations\]b, to depletion potentials for hard spheres (HS) calculated with an accurate Density Functional Theory (DFT) approach[@Roth00a]. Again HNC performs remarkably well. These results provide the confidence that, even if HNC is not perfectly quantitative, the trends we uncover will be robust, provided we limit ourselves to soft repulsions and low packing fractions[@HNC]. Fortuitously, this appears to be the regime where the nanoparticle haloing mechanism operates most effectively.
![\[fig:effective-potential\] Two typical examples of the effective potentials. First there is initial stabilization, and then a re-entrant attraction. ](effective-potential.eps){width="8cm"}
In order to systematically investigate the conditions for which repulsive stabilization occurs, we calculated, with HNC, $\beta
V^{eff}_{bb}(r)$ for a large number of parameter combinations. Two typical examples are shown in Fig. \[fig:effective-potential\], demonstrating the common pattern we find: for increasing packing fractions a maximum first appears close to contact, and continues to increase until at higher $\eta_s$ a secondary minimum appears that grows with $\eta_s$ and rapidly moves to a separation of about one $\sigma_{ss}$. This sequence of initial stabilization followed by re-entrant attraction is similar to that seen in the experiments[@Tohv01], and is found throughout the parameter regime we investigated.
To further quantify the region of stability, we choose the following measure: For a given set $\beta \epsilon_{ss}$, $\lambda_{ss}/\sigma_{ss}$, and $q$, we calculate the effective potentials for different $\eta_s$, as done for Fig. \[fig:effective-potential\]. The “stability window” is defined as $\Delta_\eta = \eta_s^u/\eta_s^s$, where $\eta_s^s$ is the packing fraction above which the maximum of $\beta V_{bb}^{eff}(r)$ is $>5$ (leading to kinetic stabilization), and $\eta_s^u$ is the packing fraction below which the minimum of $\beta V_{bb}^{eff}(r)$ is $< -2$ (a conservative estimate of where short-range attractions induce aggregation[@Loui01a]). The way our stability measure $\Delta_\eta$ varies with potential parameters is depicted in Fig. \[fig:epsvslambda\], from which some general trends can be extracted: The size of the window increases with increasing $\lambda_{ss}/\sigma_{ss}$ and decreasing $\epsilon_{ss}$ and $q$. (In the HS limit ($\beta \epsilon_{ss}=0$) we find no window of stability). For a number of points (A-D) in Fig. \[fig:epsvslambda\], we show the values of $\eta_s^u$ and $\eta_s^u$. As expected, these packing fractions decrease with increasing $\beta \epsilon_{ss}$ and $\lambda_{ss}/\sigma_{ss}$ since the small particles repel each other more and have a larger effective “size”.
![\[fig:epsvslambda\] The “equi-$\Delta_\eta$” lines denote different values of the stability window $\Delta_\eta = \eta_s^u/\eta_s^s$ for size-ratios $q=0.01$ and $q=0.05$. For points A-D the stability window packing fractions, for $q=0.01$, are listed in the format ($\eta_s^s,\eta_s^u$). Inset: The 2D packing fraction $\eta^{2D}$ of a “halo” for different size-ratios $q$. The potential parameters are $\beta \epsilon_{ss}=2$, and $\lambda_{ss}/\sigma_{ss}=0.5$. The symbols denote bulk packing fractions $\eta_s^s$ and $\eta_s^u$ for $q=0.01$ (circles), $q=0.05$ (diamonds) and $q=0.1$ (squares). ](epsvslambda.eps){width="8cm"}
The effective repulsion is clearly related to the accumulation of particles near the colloids (nano-particle halos[@Tohv01]). We define the “halo”, as those particles between $r=\sigma_{bs}$ and $r=r_{min}$, the distance at which the pair correlation function $g_{bs}(r)$ has its first minimum. The number of particles $N_{halo}$ follows from integrating $g_{bs}(r)$ up to $r_{min}$. The 2-D packing fraction is given by $\eta^{2D}=\frac14 \pi \rho^{2D} \sigma_{ss}^2$ where $\rho^{2D} = N_{halo}/( 4 \pi \sigma_{bs}^2)$. For all the parameters studied we find the same behavior depicted in the inset of Fig. \[fig:epsvslambda\]: The halo packing fraction $\eta^{2D}$ is linear with $\eta_s$ and there is no change of slope or other obvious property marking either the beginning of stabilization at $\eta_s^s$ or re-entrant attraction at $\eta_s^u$. We have also investigated other surface properties such as the adsorption $\Gamma_s = -\int
h_{bs}(r) d{\bf r}$ and the related surface tension $\gamma_s$[@Loui02b]. In contrast to polymeric depletants, where $\gamma_s$ helps determine $\beta V_{bb}^{eff}(r)$[@Loui02b], we observe no clear signatures of $\eta_s^s$ or $\eta_s^u$ in the surface tension or the adsorption..
For small $q$ we expect the $g_{bs}(r)$ to be similar within corrections ${\cal O}(q^{-1})$, which explains why the $\eta^{2D}$ v.s. $\eta_s$ curves are so close for different $q$. However, the stability windows, shown by the symbols in the inset of Fig. \[fig:epsvslambda\], differ significantly: They are at lower $\eta_s$ for smaller $q$, something we observe more generally. This can be understood from an approximate Derjaguin[@Hunt01] argument valid in the small $q$ limit. The potentials scale as $1/q$ times the force between two plates, and so stabilization (and re-entrant attraction) are achieved at lower packing fractions. However, this doesn’t easily explain why the window size also grows with decreasing $q$.
The halos are very dilute at stabilization, and we have checked that all layer densities studied are well below that of any two-dimensional freezing transition. In fact, the layers shown in the inset of Fig. \[fig:epsvslambda\] are among the densest we investigated; for some $\Phi_{ss}$, $\eta_s^{2D}$ can easily be an order of magnitude lower at $\eta_s^s$. At these low packing fractions, the particles rapidly diffuse between halos and the bulk. In contrast to a steric stabilization mechanism, where the layers are static, we emphasize that this nanoparticle halo stabilization mechanism is [*dynamic*]{}.
Further evidence against a naive picture of static layers comes from the re-entrant attraction. If the halos would become saturated, so that additional small particles can no longer segregate to the colloidal surface and instead act as depletants, then one might expect a linear dependence of the minimum of $\beta V_{bb}^{eff}(r)$ on $\eta_s$ as in AO[@Asak58] or HS[@Roth00a] depletion. Instead, the minimum in $\beta V_{bb}(r)$ grows initially as $\eta_s^2$, closely resembling the behavior of the second minimum of HS systems[@Roth00a], which suggests that both minima have a similar more complex origin in correlation effects. In fact, they would be directly related if the potentials were interpreted in terms of a non-additive HS reference system with $\sigma_{bs} <
\frac12(\sigma_{bb}+\sigma_{ss})$, as explained in [@Loui01b]. The repulsive effective interactions found in many other theoretical studies of $\beta V_{bb}^{eff}(r)$ can also be qualitatively interpreted in this way (see e.g. [@Loui02a] and references therein for a discussion), suggesting that non-additivity may be fruitfully used to interpret the re-entrant attraction[@Kara04].
One might argue that since adding an attractive $\phi_{bs}(r)$ should increase the number of particles in a layer, this should enhance the stabilization effect. However, we find more subtle scenarios. If we choose $\lambda_{bs}=\lambda_{ss}$, to model residual charge on the large colloids, then for weak attractions the window indeed grows slightly. But, as $\beta\epsilon_{bs}$ becomes more negative, the potentials rapidly develop a large attractive component. This phenomenon, sometimes called [*bridging*]{} for polymeric additives[@Hunt01], results from configurations where the two bigger colloids are both attracted to the same set of smaller particles[@Kara04]. An example of bridging is demonstrated in Fig. \[fig:bridging\](a), and is representative of what we find more generally: the stability window $\Delta_\eta$ initially grows slightly, but then rapidly disappears, typically around $\beta \epsilon_{bs} \lesssim -1.5$.
On the other hand, a dramatic enhancement of the stabilization occurs for longer ranged colloid-nanoparticle attractions, as demonstrated in Fig. \[fig:bridging\] for $\lambda_{bs} = 3 \lambda_{ss}$. The bridging effect is bypassed and the first minimum shifts up to positive absolute values, in fact for these parameters we find no re-entrant attraction within the range where we trust HNC[@HNC]. In general we find this effect for $\lambda_{bs} > \lambda_{ss}$, but exactly where it kicks in depends $q$ and the other potential parameters. In all cases studied, the 2D layer densities are still very low so that the “halos” are dilute; typically for more negative $\beta
\epsilon_{bs}$ bridging sets in again[@Kara04]. Of course when $\beta \phi_{bs}(r)$ is attractive enough to induce static saturated layers, then the particles would be sterically stabilized. But technically this is a non-equilibrium effect: it only works if the colloidal particles are first isolated from each other on the timescale that the (saturated) layer forms.
![\[fig:bridging\] (a) Adding an attractive $\beta \phi_{bs}(r)$ generates a deep minimum due to bridging effects when $\lambda_{bs}=\lambda_{ss}$. (b) A dramatic stabilization effect occurs for weak longer ranged attractions: $\lambda_{bs} = 3 \lambda_{ss} = 3 \sigma_{ss}$, and $\beta \epsilon_{bs}=-0.5$. ](bridging.eps){width="8cm"}
Finally, having systematically explored the effect of different parameters on $\beta V_{bb}^{eff}(r)$, we make some recommendations for experiments. For $\epsilon_{bs}=0$ the best stabilization should occur for modestly charged small additives (nanoparticles) since the stability windows are largest for small $q$, modest $\beta
\epsilon_{ss}$ and large $\lambda_{ss}/\sigma_{ss}$. Small $\sigma_{ss}$ are needed to enhance the latter parameter, since the (Debye) screening length is typically fixed by solution conditions, e.g. $\lambda_{ss} \approx 30 nm$ for $0.1mM$ monovalent salt concentration. Another argument in favor of small particles concerns the [*dynamic*]{} nature of the layers. If the big particles are driven at velocities such that the smaller particles can no longer adiabatically follow, then the stabilization effect may disappear. Since the self-diffusion coefficients of the nanoparticles scale as $1/\sigma_{ss}$[@Hunt01], this again favors small particles. Moreover, larger windows $\Delta_\eta$ also enhance stability under halo fluctuations.
Whereas adding charge to the colloids can destroy the stabilization effect, a modest but longer ranged attraction $\beta
\phi_{bs}(r)$ can significantly enhance it. The latter effect could be induced by residual van der Waals attractions, although this recommendation must be tempered by the difficulty of adding van der Waals attractions between different species without simultaneously significantly increasing them between similar species. On the other hand, the advantage of small van der Waals attractions (which are independent of Debye screening length), is that the ratio $\lambda_{bs}/\lambda_{ss}$ can be tuned by changing salt concentration. This adds another handle for engineering effective potentials and concomitant phase behavior[@Kara04].
We observe the same general trends seen in the experiments of ref. [@Tohv01], such as lower $\eta_s^s$ with smaller $q$ and values of $\eta^{2D}$ well below saturation. A direct quantitative comparison, however, is hampered by their use of polydisperse small particles, and the difficulty in deriving accurate potentials $\beta\phi_{ij}(r)$. Taking their estimates($\beta \epsilon_{ss}
\approx 6$, $\lambda_{ss}/\sigma_{ss} \approx 0.6$) we find a smaller window $\Delta_\eta \approx 2$, at higher $\eta_s$ than what they observed. The difference could stem from a small attractive $\beta \phi_{bs}(r)$ or from polydispersity (preliminary calculations suggest that this lowers the effective $\eta_s^s$ and $\eta_s^u$ (P. Bryk, [*private communication*]{})).
In conclusion, we discovered a substantial parameter regime where the addition of small (nano) particles can stabilize bigger colloids. Fortuitously, this occurs where the flexible HNC integral equation is most reliable. We usually find a stability window of packing fractions, above or below which the colloids aggregate again. The effects are significantly enhanced for weak longer ranged attractive $\beta \phi_{bs}(r)$. Although the stabilization is clearly related to the formation of diffusive accumulation layers around the bigger particles, we find no simple relationship to layer properties. This suggests these effects are related to more complex correlations. Colloidal stabilization by dynamic nanoparticle halos should be widely applicable and complimentary to existing steric and charge stabilization techniques[@Hunt01; @DLVO]. This new mechanism may also have relevance to smaller scale biological interactions[@Leck01]. \[Note: upon completion of this work we became aware of a study by J. Liu and E. Luijten, cond-mat/0411278, which uses different techniques, but arrives at similar conclusions\]
We thank H Löwen for early discussions, and J. Dzubiella, R. Roth and P. Bryk for invaluable help with the calculations. SK thanks Schlumberger Cambridge Research and the EPSRC for a studentship, and AAL thanks the Royal Society (London) for financial support.
[99]{}
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E.J. Verwey and J. Th. G. Overbeek, [*Theory of the Stability of Lyophobic Colloids*]{}, Elsevier, Amsterdam (1948).
V. Tohver [*et al*]{}. Proc. Natl. Acad. Sci. [**98**]{}, 8950 (2001); V. Tohver, A. Chan, O. Sakurada, and J.A. Lewis, Langmuir [**17**]{}, 8414 (2001).
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J.P. Hansen and I.R. McDonald, [*Theory of Simple Liquids, 2nd Ed.*]{}, Academic Press, London (1986). A. A. Louis, Phil. Trans. Roy. Soc. A [**359**]{}, 939 (2001). S. Karanikas and A. A. Louis ( to be published).
R. Roth, R. Evans, and S. Dietrich, Phys. Rev. E. [**62**]{}, 5360 (2000). The Derjaguin limit is reached very slowly for HS systems (M. Oettel, Phys. Rev. E [**69**]{}, 041404 (2004)). Up to now no simulations of $V_{bb}^{eff}(r)$ have been performed for these extreme size ratios, see, however J. Liu and E. Luijten, Phys. Rev. Lett. [**92**]{}, 035504 (2004).
We stress that no claim is being made for the general accuracy of HNC for all parameter values. For example the predictions for HS begin to show important quantitative deviations for $\eta_s > 0.3$ and the comparison with simulations of ref [@Loui02a] is less accurate for attractions $(\epsilon_{ss} < 0)$.
A.A. Louis and R. Roth, J. Phys.: Condens. Matter [**33**]{} L777 (2001); R. Roth, R. Evans, and A. A. Louis, Phys. Rev. E [**64**]{}, 051202 (2001). A.A. Louis, P.G. Bolhuis, E.J. Meijer and J.-P. Hansen, J. Chem. Phys. [**117**]{}, 1893 (2002).
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[^1]: Author for correspondence: [email protected]
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\[ =1 \] = -1 cm = -.50 in =24.5 pt =16 cm =22.5 cm
=24.5pt 0 cm
**Lepton Flavour Mixing Matrix and CP Violation**
**from Neutrino Oscillation Experiments**
1 cm
M. Fukugita$^{1}$ and M. Tanimoto$^2$
$^1$ Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 2778582, Japan
$^2$ Department of Physics, Niigata University, Niigata 950-2181, Japan
4 cm [**Abstract**]{}
The measurement of the charged-current $^8$B solar neutrino reaction on deuterium at the Sudbury Neutrino Observatory confirms the neutrino oscillation hypothesis for the solar neutrino problem, and the result favours the solution with large neutrino mixing angles. We demonstrate that the current neutrino oscillation data (including atmospheric and reactor neutrinos) are sufficient to construct the lepton flavour mixing matrix with a reasonable accuracy. We also infer the maximum size of CP violation effects consistent with the current neutrino oscillation experiments.
The Sudbury Neutrino Observatory (SNO) experiment \[1\] measured the charged current rate for $^8$B solar neutrino reactions on deuterium. The derived electron neutrino flux is lower by $3.3\sigma$ than the neutrino flux obtained from the electron scattering event rate measured by the same experiment, and more accurately by Super-Kamiokande (SK) \[2\]. This difference is ascribed to the neutral-current induced reaction of three species of the left-handed neutrinos on electron scattering. The sum of the electron neutrino flux and that of other neutrinos, inferred from the difference, agrees with the electron neutrino generation rate expected in the sun \[3\], hereby confirming the neutrino oscillation hypothesis as the origin of the long-standing solar neutrino problem.
The SNO result also narrows the range of the neutrino oscillation parameters. The rate of the conversion of electron neutrinos to other neutrinos is somewhat larger than is expected in the optimal solutions within the neutrino oscillation hypothesis (e.g., \[2\],\[4\]) inferred from solar neutrino experiments available prior to the SNO experiment \[5\],\[2\],\[6\]. This indicates that the new optimal solution including the SNO data is located slightly inwards within the MSW \[7\] triangle. This makes the small angle solution (SMA) of the MSW effect disfavoured, since the absence of the distortion of the neutrino energy spectrum and of day-night effect observed at SK leaves only the small angle edge of the SMA to be allowed \[2\], contrary to the charged-current event at SNO indicates. For a similar reason the neutrino oscillation hypothesis [*in vacuo*]{} is now disfavoured. We are left with the large mixing angle solution (LMA) and the LOW solution \[4\] of the MSW conversion as the most likely solution to the solar neutrino problem. Detailed statistical analyses including the SNO data have already been made \[8\],\[9\],\[10\], and confirmed the picture we have sketched here.
In this [*Letter*]{} we present an analysis of the neutrino oscillation from a somewhat different view. We ask the question whether we can determine the lepton flavour mixing matrix using the presently available oscillation data, and answer this question positively. We also ask the magnitude of the CP violation effect allowed by the current neutrino oscillation experiment.
We write the neutrino mixing matrix
$$|\psi_\alpha\rangle = U_{\alpha i}|\psi_i\rangle ,$$
where $\alpha=e,\mu,\tau$ and $i=1,2,3$. The solar neutrino experiments tell us about $|U_{e1}|$ and $|U_{e2}|$, and the atmospheric \[11\],\[12\] and the K2K long base-line accelerator experiment \[13\] tell us about $|U_{\mu3}|$. Another constraint is imposed on the element $|U_{e3}|$ from the Chooz reactor experiment \[14\]. If errors are small this information should be sufficient to construct the full $3\times3$ lepton flavour mixing matrix (up to the phase) under the unitarity constraint on the matrix.
We consider all constraints at a 90% confidence level. We fix for simplicity the mass square difference between $\nu_\tau$ and $\nu_\mu$ to be $3.2\times 10^{-3}$ (eV)$^2$ as deduced from the atmospheric neutrino experiment of SK \[11\]. We determine allowed regions of the Kobayashi-Maskawa angles from the oscillation parameters and mapped them into physical mixing matrix elements. The present experimental information is not sufficient to constrain the phase factor. So we vary the phase $\phi$ between 0 and $\pi$ while searching for the allowed angles, adopting the matrix representation in [*Review of Particle Physics*]{} \[15\].
We take LMA for the solar neutrino mixing solution. The mixing matrix we derived reads
$$U= \left[ \matrix{0.74-0.90 & 0.45-0.65 & <0.16 \cr
0.22-0.61 & 0.46-0.77 & 0.57-1/\sqrt 2 \cr
0.14-0.55 & 0.36-0.68 & 1/\sqrt 2-0.82 \cr
} \right]\ ,$$
where we are confined to the case of $\nu_\mu<\nu_\tau$, and only the modulus of the elements are shown. If we allow for $\nu_\mu>\nu_\tau$ the (2,3) element takes $1/\sqrt 2 - 0.82$ and the (3,3) element is $0.57 - 1/\sqrt 2$, [*i.e.*]{}, the two elements are interchanged as the phase is unconstrained in our analysis. It is interesting to note that [*all*]{} matrix elements are reasonably constrained with the present neutrino oscillation data. This predicts the oscillation properties between any kinds of neutrinos. It is also interesting to notice that all elements except for $U_{e3}$ are sizable. A marginal disparity is seen between the $U_{e3}$ and $U_{\tau1}$ elements, but more accurate input of $U_{e2}$ is needed for a definitive conclusion. The small $U_{e3}$ is the characteristic that has been predicted in some phenomenological neutrino mass matrix models \[16\],\[17\] prior to the Chooz experiment[^1]. Once this matrix is determined one can infer the maximum size of CP violation in the lepton sector. In the last few years the feasibility of detecting CP violation has been studied by a number of authors in view of long-baseline neutrino experiments with strong neutrino beams \[18\],\[19\]. In these studies only a few representative values of neutrino oscillation parameters, as written in terms of the angle representation, are adopted to examine experimental feasibility, and systematic parameter searches are not made. A more general analysis is straightforward with our neutrino matrix. The factor that represents the net CP violation effect can be written as \[20\]:
$$J={|U_{e1}|~|U_{e2}|~|U_{\mu3}|~|U_{\tau3}|~|U_{e3}| \over
1-|U_{e3}|^2} \sin\phi\ .$$
This is evaluated using the lepton mixing matrix, but more conveniently expressed only with experimentally relevant quantities, as
$$J={1\over 4}{\sqrt{\sin^22\theta_{sol}}\sqrt{\sin^22\theta_{atm}}|U_{e3}| \over
1-|U_{e3}|^2} \sin\phi\ ,$$
where $\theta_{sol}$ is the mixing angle that directly comes into solar neutrino oscillation, $\theta_{atm}$ is that for the atmospheric neutrino oscillation, and $|U_{e3}|$ is directly constrained by the $\bar\nu_e\rightarrow \bar\nu_\tau$ oscillation experiment. Now it is easy to show that the maximum value of $J$ is given by
$$%J\leq {1\over 4}{\sqrt{0.93}\ \sqrt{1}\ 0.16\over 1-0.16^2}\sin\phi
J\leq 0.040 \sin\phi\ .
\label{eq:5}$$
Apart from the phase factor $\sin\phi$ the crucial factor that controls the feasibility of the CP violation experiment is $|U_{e3}|$. This demonstrates the importance of the measurement of $\bar\nu_e\rightarrow \bar\nu_\tau$ oscillation beyond the current limit set by the Chooz experiment.
We remark that if the modulus of the matrix elements (other than the four we used as input) is experimentally determined, we can predict the CP violation phase $\phi$ as,
$$\cos\phi=\frac{1}{2 |U_{e 1}| |U_{e2}||U_{e3}| |U_{\mu 3}||U_{\tau 3}|}
[(1-|U_{e3}|^2)^2 |U_{\mu 1}|^2 - |U_{e2}|^2 |U_{\tau 3}|^2
-|U_{e1}|^2 |U_{e3}|^2 |U_{\mu 3}|^2].$$
We do not repeat the discussion about actual CP violation effects given in the literature \[19\], but let us quote that the disparity of neutrino oscillation due to CP violation is given by \[21\]
$$\begin{aligned}
\Delta P&=& P(\bar\nu_\mu\rightarrow \bar\nu_e)-P(\nu_\mu\rightarrow \nu_e)\cr
&=& 4J f \leq 0.16 f\sin\phi\ ,\end{aligned}$$
where $
f=4\sin(\Delta_{12}/2)\sin(\Delta_{32}/2)\sin(\Delta_{31}/2)
$ with $\Delta_{ij}=\Delta m_{ij}^2 L/2E_\nu$ ($L$ is the length of the baseline, and $E_\nu$ is the neutrino beam energy). The upper limit of $J$ corresponds to about 2/3 the value assumed in the analysis of Arafune et al.\[19\], but if it takes a value close to this limit the CP violation effect is probably visible.
So far we have discussed the case of the LMA solution. The experiments also allow the LOW solution, which are located close to the bottom of the MSW triangle, albeit the parameter range allowed by a 90% confidence is narrow, If LOW is the solution, the mixing matrix is
$$U= \left[ \matrix{0.71-0.79 & 0.61-0.71 & <0.16 \cr
0.34-0.65 & 0.42-0.70 & 0.57-1/\sqrt 2 \cr
0.25-0.58 & 0.32-0.63 & 1/\sqrt 2-0.82 \cr
} \right]\ .$$
This matrix is quite similar to the one given for LMA, but the constraint is slightly tighter because of a smaller allowed region for $U_{e2}$. We obtain a CP violation $J$ factor similar to (5). The observable CP violation effect, however, contains the mass square difference, and the small $\Delta m_{12}^2$ of the LOW solution pushes the effect outside the range feasible with accelerator experiments. We should await the KamLAND experiment \[22\] for a decisive answer as to the selection between the two solutions.
[**Acknowledgements**]{}
We thank Tsutomu Yanagida for many valuable discussions over many aspects of neutrino physics, including the present work.
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[^1]: Note that (1,3) and (3,1) elements are reversely expressed in \[16\] by convention.
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abstract: 'In recent years we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and clocked type theory, and spatial and cohesive type theory. In this paper we study *modal dependent type theory*: dependent type theory with an operator satisfying (a dependent version of) the K axiom of modal logic. We investigate both semantics and syntax. For the semantics, we introduce categories with families with a *dependent right adjoint* ([CwDRA]{}) and show that the examples above can be presented as such. Indeed, we show that any category with finite limits and an adjunction of endofunctors gives rise to a [CwDRA]{} via the local universe construction. For the syntax, we introduce a dependently typed extension of Fitch-style modal lambda-calculus, show that it can be interpreted in any [CwDRA]{}, and build a term model. We extend the syntax and semantics with universes.'
bibliography:
- 'drat.bib'
title: Modal Dependent Type Theory and Dependent Right Adjoints
---
Introduction {#sec:intro}
============
Dependent types are a powerful technology for both programming and formal proof. In recent years we have seen several new models of dependent type theory extended with a type former resembling modal necessity[^1], such as nominal type theory [@PittsAM:deptta], guarded [@birkedal2011first; @BirkedalL:gdtt-conf; @GDTTmodel; @GCTT] and clocked [@CloTTmodel] type theory, and spatial and cohesive type theory [@shulman2018brouwer]. These examples all satisfy the K axiom of modal logic $$\label{axiom_K}
{\square}(A\to B)\to{\square}A\to{\square}B$$ but are not all (co)monads, the more extensively studied construction in the context of dependent type theory [@Krishnaswami:Integrating; @dePaiva:Fibrational; @vakar2017search; @shulman2018brouwer]. Motivated in part by these examples, in this paper we study *modal dependent type theory*: dependent type theory with an operator satisfying (a dependent generalisation of) the K axiom[^2] of modal logic. We investigate both semantics and syntax.
For the semantics, we introduce categories with families with a *dependent right adjoint* ([CwDRA]{}) and show that this dependent right adjoint models the modality in the examples mentioned above. Indeed, we show that any finite limit category with an adjunction of endofunctors[^3] gives rise to a [CwDRA]{} via the local universe construction [@LumsdainePL:locumo]. In particular, by applying the local universe construction to a locally cartesian closed category with an adjunction of endofunctors, we get a model of modal dependent type theory with $\Pi$- and $\Sigma$-types.
For the syntax, we adapt the simply typed Fitch-style modal lambda-calculus introduced by and , inspired by Fitch’s proof theory for modal logic [@Fitch:Symbolic]. In such a calculus ${\square}$ is introduced by ‘shutting’ a strict subordinate proof and eliminating by ‘opening’ one. For example the K axiom is inhabited by the term $$\label{eq:K_inhabited}
\lambda f.\lambda x.{\ensuremath{\mathsf{shut}}}(({\ensuremath{\mathsf{open}}}\,f)({\ensuremath{\mathsf{open}}}\,x))$$ The nesting of subordinate proofs can be tracked in sequent style by a special symbol in the context which we call a *lock*, and write ${\mbox{\faUnlock}}$; the open lock symbol is intended to suggest we have access to the contents of a box. Following , the lock can be understood as an operation on contexts *left adjoint* to ${\square}$; hence Fitch-style modal $\lambda$-calculus has a model in any cartesian closed category equipped with an adjunction of endofunctors. Here we show, in work inspired by Clocked Type Theory [@bahr2017clocks], that Fitch-style $\lambda$-calculus lifts with a minimum of difficulty to dependent types. In particular the term , where $f$ is a dependent function, has type $${\square}(\Pi y:A.\,B) \to \Pi x:{\square}A.\,{\square}{B[{\ensuremath{\mathsf{open}}}\,x/y]}$$ This dependent version of the K axiom, not obviously expressible without the ${\ensuremath{\mathsf{open}}}$ construct of a Fitch-style calculus, allows modalised functions to be applied to modalised data even in the dependent case. This capability is known to be essential in at least one example, namely proofs about guarded recursion [@BirkedalL:gdtt-conf][^4]. We show that our calculus can be soundly interpreted in any [CwDRA]{}, and construct a term model.
We also extend the syntax and semantics of modal dependent type theory with universes. Here we restrict attention to models based on (pre)sheaves, for which Coquand has proposed a particularly simple formulation of universes [@Coquand:CwU]. We show how to extend Coquand’s notion of a category with universes with dependent right adjoints, and observe that a construction encoding the modality on the universe, introduced for guarded type theory by , in fact arises for more general reasons.
Another motivation for the present work is that it can be understood as providing a notion of a *dependent* adjunction between endofunctors. An ordinary adjunction ${\mathsf{L}}\dashv \operatorname{\mathsf{R}}$ on a category ${{\mathbf{C}}}$ is a natural bijective correspondence ${{{{\mathbf{C}}}}({\mathsf{L}}A,B)} {\cong}{{{{\mathbf{C}}}}(A,\operatorname{\mathsf{R}}B)}$. With dependent types one might consider *dependent* functions from ${\mathsf{L}}A$ to $B$, where $B$ may depend on ${\mathsf{L}}A$, and similarly from $A$ to $\operatorname{\mathsf{R}}B$. Our notion of [CwDRA]{} then defines what it means to have an adjoint correspondence in this dependent case. Our Fitch-style modal dependent type theory can therefore also be understood as a term language for dependent adjoints.
**Outline**We introduce [CwDRAs]{} in Section \[sec:CwDRA\], and present the syntax of modal dependent type theory in Section \[sec:syntax\]. In Section \[sec:contruction-of-CwDRAs\] we show how to construct a [CwDRA]{} from an adjunction on a category with finite limits. In Section \[sec:examples\] we show how various models in the literature can be presented as [CwDRAs]{}. The extension with universes is defined in Section \[sec:universes\]. We end with a discussion of related and future work in Section \[sec:discussion\].
Categorical Semantics of Modal Dependent Type Theory {#sec:CwDRA}
====================================================
The notion of *category with families* (CwF) [@Dybjer1996; @hofmann1997syntax] provides a semantics for the development of dependent type theory which elides some difficult aspects of syntax, such as variable binding, as well as the coherence problems of simpler notions of model. It can be connected to syntax by a soundness argument and term model construction, and to more mathematical models via ‘strictification’ constructions [@hofmann1994interpretation; @LumsdainePL:locumo]. In this section we extend this notion to introduce *categories with a dependent right adjoint* (CwDRA). We first recall the standard definition:
\[def:cwf\] A [**CwF**]{} is specified by:
1. A category ${{\mathbf{C}}}$ with a terminal object ${\top}$. Given objects $\Gamma,\Delta\in{{\mathbf{C}}}$, write ${{\mathbf{C}}}(\Delta,\Gamma)$ for the set of morphisms from $\Delta$ to $\Gamma$ in ${{\mathbf{C}}}$. The identity morphism on $\Gamma$ is just written ${\mathsf{id}}$ with $\Gamma$ implicit. The composition of $\gamma\in{{\mathbf{C}}}(\Delta,\Gamma)$ with $\delta\in{{\mathbf{C}}}(\Phi,\Delta)$ is written $\gamma{\circ}\delta$.
2. For each object $\Gamma\in{{\mathbf{C}}}$, a set ${{{{\mathbf{C}}}}(\Gamma)}$ of *families* over $\Gamma$.
3. For each object $\Gamma\in{{\mathbf{C}}}$ and family $A\in{{{{\mathbf{C}}}}(\Gamma)}$, a set ${{{{\mathbf{C}}}}(\Gamma\mathbin{\scriptstyle\vdash}A)}$ of *elements* of the family $A$ over $\Gamma$.
4. \[item:2\] For each morphism $\gamma\in{{{{\mathbf{C}}}}(\Delta,\Gamma)}$, *re-indexing* functions $A\in{{{{\mathbf{C}}}}(\Gamma)} \mapsto A[\gamma]\in{{{{\mathbf{C}}}}(\Delta)}$ and $a\in{{{{\mathbf{C}}}}(\Gamma\mathbin{\scriptstyle\vdash}A)}\mapsto a[\gamma]\in{{{{\mathbf{C}}}}(\Delta\mathbin{\scriptstyle\vdash}A[\gamma])}$, satisfying $A[{\mathsf{id}}]=A$, $A[\gamma{\circ}\delta] = A[\gamma][\delta]$, $a[{\mathsf{id}}]=a$ and $a[\gamma{\circ}\delta] = a[\gamma][\delta]$.
5. \[item:1\] For each object $\Gamma\in{{\mathbf{C}}}$ and family $A\in{{{{\mathbf{C}}}}(\Gamma)}$, a *comprehension object* ${\Gamma{.}A}\in{{\mathbf{C}}}$ equipped with a *projection morphism* ${\mathsf{p}}_A\in{{\mathbf{C}}}({\Gamma{.}A},\Gamma)$, a *generic element* ${\mathsf{q}}_A\in{{{{\mathbf{C}}}}({\Gamma{.}A}\mathbin{\scriptstyle\vdash}A[{\mathsf{p}}_A])}$ and a *pairing operation* mapping $\gamma\in{{{{\mathbf{C}}}}(\Delta,\Gamma)}$ and $a\in{{{{\mathbf{C}}}}(\Delta\mathbin{\scriptstyle\vdash}A[\gamma])}$ to ${\left(\gamma,a\right)}\in{{\mathbf{C}}}(\Delta,{\Gamma{.}A})$ satisfying ${\mathsf{p}}_A{\circ}{\left(\gamma,a\right)}= \gamma$, ${\mathsf{q}}_A[{\left(\gamma,a\right)}] = a$, ${\left(\gamma,a\right)}{\circ}\delta =
{\left(\gamma{\circ}\delta,a[\delta]\right)}$ and ${\left({\mathsf{p}}_A,{\mathsf{q}}_A\right)} = {\mathsf{id}}$.
A *dependent right adjoint* then extends the definition of CwF with a functor on contexts ${\mathsf{L}}$ and an operation on families $\operatorname{\mathsf{R}}$, intuitively understood to be left and right adjoints:
\[def:cwf-a\] A *CwDRA* is a CwF ${{\mathbf{C}}}$ equipped with the following extra structure:
1. \[item:3\] An endofunctor ${\mathsf{L}}:{{\mathbf{C}}}{\rightarrow}{{\mathbf{C}}}$ on the underlying category of the CwF.
2. For each object $\Gamma\in{{\mathbf{C}}}$ and family $A\in{{{{\mathbf{C}}}}({\mathsf{L}}\Gamma)}$, a family $\operatorname{\mathsf{R}}_\Gamma A \in{{{{\mathbf{C}}}}(\Gamma)}$, stable under re-indexing in the sense that for all $\gamma\in{{\mathbf{C}}}(\Delta,\Gamma)$ we have $$\label{eq:11}
(\operatorname{\mathsf{R}}_\Gamma A)[\gamma] = \operatorname{\mathsf{R}}_\Delta(A[{\mathsf{L}}\gamma]) \in {{{{\mathbf{C}}}}(\Delta)}$$
3. For each object $\Gamma\in{{\mathbf{C}}}$ and family $A\in{{{{\mathbf{C}}}}({\mathsf{L}}\Gamma)}$ a bijection $$\label{eq:18}
{{{{\mathbf{C}}}}({\mathsf{L}}\Gamma\mathbin{\scriptstyle\vdash}A)}{\cong}{{{{\mathbf{C}}}}(\Gamma\mathbin{\scriptstyle\vdash}\operatorname{\mathsf{R}}_\Gamma A)}$$ We write the effect of this bijection on $a\in{{{{\mathbf{C}}}}({\mathsf{L}}\Gamma\mathbin{\scriptstyle\vdash}A)}$ as ${\overline{a}}\in{{{{\mathbf{C}}}}(\Gamma\mathbin{\scriptstyle\vdash}\operatorname{\mathsf{R}}_\Gamma A)}$ and write the effect of its inverse on $b\in{{{{\mathbf{C}}}}(\Gamma\mathbin{\scriptstyle\vdash}\operatorname{\mathsf{R}}_\Gamma A)}$ also as ${\overline{b}}\in{{{{\mathbf{C}}}}({\mathsf{L}}\Gamma\mathbin{\scriptstyle\vdash}A)}$. Thus $$\begin{aligned}
{\overline{{\overline{a}}}} &= a &&(a\in{{{{\mathbf{C}}}}({\mathsf{L}}\Gamma\mathbin{\scriptstyle\vdash}A)}) \label{eq:12}\\
{\overline{{\overline{b}}}} &= b &&(b\in{{{{\mathbf{C}}}}(\Gamma\mathbin{\scriptstyle\vdash}\operatorname{\mathsf{R}}_\Gamma A)})\label{eq:13}
\end{aligned}$$ The bijection is required to be stable under re-indexing in the sense that for all $\gamma\in{{\mathbf{C}}}(\Delta,\Gamma)$ we have $$\label{eq:14}
{\overline{a}}[\gamma] = {\overline{a[{\mathsf{L}}\gamma]}}$$
Note that equation (\[eq:14\]) is well-typed by (\[eq:11\]). Equation (\[eq:14\]) also implies that the opposite direction of the isomorphism (\[eq:18\]) is natural, i.e., that the equation $$\label{eq:15}
{\overline{b}}[{\mathsf{L}}\gamma] = {\overline{b[\gamma]}}$$ also holds, since ${\overline{b[\gamma]}} = {\overline{{\overline{{\overline{b}}}}[\gamma]}} =
{\overline{{\overline{{\overline{b}}[{\mathsf{L}}\gamma]}}}} = {\overline{b}}[{\mathsf{L}}\gamma]$.
Syntax of Modal Dependent Type Theory {#sec:syntax}
=====================================
In this section we extend Fitch-style modal $\lambda$-calculus [@Borghuis:Coming] to dependent types, and connect this to the notion of CwDRA via a soundness proof and term model construction. We define our dependent types broadly in the style of ECC [@Luo:ECC], as this is close to the implementation of some proof assistants [@norell:thesis].
We define the raw syntax of contexts, types, and terms as follows: $$\begin{aligned}
\Gamma &\;{\triangleq}\; {\diamond}\;\mid\; \Gamma,x:A \;\mid\; \Gamma,{\mbox{\faUnlock}}\\
A &\;{\triangleq}\; {\Pi x:A.\,B} \;\mid\; {\square}A \\
t &\;{\triangleq}\; x \;\mid\; \lambda x:A.\,t \;\mid\; t\,t \;\mid\; {\ensuremath{\mathsf{shut}}}\,t \;\mid\; {\ensuremath{\mathsf{open}}}\,t\end{aligned}$$ We omit the leftmost ‘${\diamond},$’ where the context is non-empty. We will usually omit the type annotation on the $\lambda$ for brevity. $\Pi$-types are included in the grammar as an example to show that standard type formers can be defined as usual, without reference to the locks in the context. One could similarly add an empty type, unit type, booleans, $\Sigma$-types, W-types, universes (of which more in Section \[sec:universes\]), and so forth.
Judgements have forms $$\begin{aligned}
{\Gamma\vdash} &\qquad\mbox{`$\Gamma$ is a well-formed context'} \\
{\Gamma\vdashA} &\qquad\mbox{`$A$ is a well-formed type in context $\Gamma$'} \\
{\Gamma\vdashA=B} &\qquad\mbox{`$A$ and $B$ are equal types in context $\Gamma$'} \\
{\Gamma\vdasht:A} &\qquad\mbox{`$t$ is a term with type $A$ in context $\Gamma$'} \\
{\Gamma\vdasht=u:A} &\qquad\mbox{`$t$ and $u$ are equal terms with type $A$ in context $\Gamma$'}\end{aligned}$$
Context formation rules:
Type formation rules:
Type equality rules are as standard, asserting equivalence, and congruence with respect to all type formers.
Term formation rules:
\
Term equality rules, omitting equivalence and congruence:
Figure \[fig:typing\_rules\] presents the typing rules of the calculus. The syntactic results below follow easily by induction on these rules. We remark only that exchange of variables with locks, and weakening of locks, are not admissible, and that the (lock-free) weakening $\Gamma'$ in the ${\ensuremath{\mathsf{open}}}$ rule is essential to proving variable weakening.
\[lem:syntactic\_sanity\] Let ${\mathcal{J}}$ range over the possible strings to the right of a turnstile in a judgement.
1. If $\Gamma,x:A,y:B,\Gamma'\vdash{\mathcal{J}}$ and $x$ is not free in $B$, then $\Gamma,y:B,x:A,\Gamma'\vdash{\mathcal{J}}$;
2. If $\Gamma,\Gamma'\vdash{\mathcal{J}}$, and $\Gamma\vdash A$, and $x$ is a fresh variable, then $\Gamma,x:A,\Gamma'\vdash{\mathcal{J}}$;
3. If $\Gamma,x:A,\Gamma'\vdash {\mathcal{J}}$ and $\Gamma\vdash u:A$, then $\Gamma,{\Gamma'[u/x]}\vdash{{\mathcal{J}}[u/x]}$;
4. If $\Gamma\vdash t:A$ then $\Gamma\vdash A$;
5. If ${\Gamma\vdasht=u:A}$ then ${\Gamma\vdasht:A}$ and ${\Gamma\vdashu:A}$.
Sound interpretation in [CwDRAs]{} {#sec:interpret}
----------------------------------
In this section we show that the calculus of Figure \[fig:typing\_rules\] can be soundly interpreted in any CwDRA. We wish to give meaning to contexts, types, and terms, but (via the type conversion rule) these can have multiple derivations, so it is not possible to work by induction on the formation rules. Instead, following e.g. , we define a partial map from *raw syntax* to semantics by induction on the grammar, then prove this map is defined for well-formed syntax. By ‘raw syntax’ we mean contexts, types accompanied by a context, and terms accompanied by context and type, defined via the grammar. The *size* of a type or term is the number of connectives and variables used to define it, and the size of a context is the sum of the sizes of its types.
Well-defined contexts $\Gamma$ will be interpreted as objects ${\llbracket\Gamma\rrbracket}$ in ${{{\mathbf{C}}}}$, types in context ${\Gamma\vdashA}$ as families in ${{{{\mathbf{C}}}}({\llbracket\Gamma\rrbracket})}$, and typed terms in context ${\Gamma\vdasht:A}$ as elements in ${{{{\mathbf{C}}}}({\llbracket\Gamma\rrbracket}\mathbin{\scriptstyle\vdash}{\llbracket{\Gamma\vdashA}\rrbracket})}$. Where there is no confusion we write ${\llbracket{\Gamma\vdashA}\rrbracket}$ as ${\llbracketA\rrbracket}$ and ${\llbracket{\Gamma\vdasht:A}\rrbracket}$ as ${\llbracket\Gamma\vdash t\rrbracket}$ or ${\llbrackett\rrbracket}$.
The partial interpretation of raw syntax is as follows, following the convention that ill-formed expressions (for example, where a subexpression is undefined) are undefined. We omit the details for $\Pi$-types and other standard constructions, which are as usual.
- ${\llbracket{\diamond}\rrbracket}={\top}$;
- ${\llbracket\Gamma,x:A\rrbracket}={{\llbracket\Gamma\rrbracket}{.}{\llbracketA\rrbracket}}$;
- ${\llbracket\Gamma,{\mbox{\faUnlock}}\rrbracket}={\mathsf{L}}{\llbracket\Gamma\rrbracket}$;
- ${\llbracket\Gamma\vdash{\square}A\rrbracket}=R_{{\llbracket\Gamma\rrbracket}}({\llbracketA\rrbracket})$;
- ${\llbracket{\Gamma,x:A,x_1:A_1,\ldots,x_n:A_n\vdashx:A}\rrbracket}=
{\mathsf{q}}_{{\llbracketA\rrbracket}}[{\mathsf{p}}_{{\llbracketA_1\rrbracket}}\circ\cdots\circ{\mathsf{p}}_{{\llbracketA_n\rrbracket}}]$;
- ${\llbracket{\Gamma\vdash{\ensuremath{\mathsf{shut}}}\,t:{\square}A}\rrbracket}={\overline{{\llbrackett\rrbracket}}}$;
- ${\llbracket{\Gamma,{\mbox{\faUnlock}},x_1:A_1,\ldots,x_n:A_n\vdash{\ensuremath{\mathsf{open}}}\,t:A}\rrbracket}=
{\overline{{\llbrackett\rrbracket}}}[{\mathsf{p}}_{{\llbracketA_1\rrbracket}}\circ\cdots\circ{\mathsf{p}}_{{\llbracketA_n\rrbracket}}]$.
In Figure \[fig:struct\_morphisms\] we define expressions ${\mathsf{P}(\Gamma;A;\Gamma')}$, ${\mathsf{E}(\Gamma;A;B;\Gamma')}$, and ${\mathsf{S}(\Gamma;A;\Gamma';t)}$ that, where defined, define morphisms in ${{\mathbf{C}}}$ corresponding respectively to weakening, exchange, and substitution in contexts.
------------------------------------------------------- ----- --------------------------------------------------------------------------------------------------------------------------------------------------
${\mathsf{P}(\Gamma;A;{\diamond})}$ $=$ ${\mathsf{p}}_{{\llbracketA\rrbracket}}$
${\mathsf{P}(\Gamma;A;\Gamma',y:B)}$ $=$ ${\left({\mathsf{P}(\Gamma;A;\Gamma')}\circ
{\mathsf{p}}_{{\llbracket{\Gamma,x:A,\Gamma'\vdashB}\rrbracket}},{\mathsf{q}}_{{\llbracket{\Gamma,x:A,\Gamma'\vdashB}\rrbracket}}\right)}$
${\mathsf{P}(\Gamma;A;\Gamma',{\mbox{\faUnlock}})}$ $=$ ${\mathsf{L}}\,{\mathsf{P}(\Gamma;A;\Gamma')}$
${\mathsf{E}(\Gamma;A;B;{\diamond})}$ $=$ $(({\mathsf{p}}_{{\llbracket{\Gamma\vdashB}\rrbracket}}\circ
{\mathsf{p}}_{{\llbracket{\Gamma,y:B\vdashA}\rrbracket}},
{\mathsf{q}}_{{\llbracket{\Gamma,y:B\vdashA}\rrbracket}}),{\mathsf{q}}_{{\llbracket{\Gamma\vdashB}\rrbracket}}
[{\mathsf{p}}_{{\llbracket{\Gamma,y:B\vdashA}\rrbracket}}])$
${\mathsf{E}(\Gamma;A;B;\Gamma',z:C)}$ $=$ $({\mathsf{E}(\Gamma;A;B;\Gamma')}\circ
{\mathsf{p}}_{{\llbracket{\Gamma,y:B,x:A,\Gamma'\vdashC}\rrbracket}},
{\mathsf{q}}_{{\llbracket{\Gamma,y:B,x:A,\Gamma'\vdashC}\rrbracket}})$
${\mathsf{E}(\Gamma;A;B;\Gamma',{\mbox{\faUnlock}})}$ $=$ ${\mathsf{L}}\,{\mathsf{E}(\Gamma;A;B;\Gamma')}$
${\mathsf{S}(\Gamma;A;{\diamond};t)}$ $=$ $({\mathsf{id}},{\llbrackett\rrbracket})$
${\mathsf{S}(\Gamma;A;\Gamma',y:B;t)}$ $=$ $({\mathsf{S}(\Gamma;A;\Gamma';t)}\circ
{\mathsf{p}}_{{\llbracket{\Gamma,{\Gamma'[t/x]}\vdash{B[t/x]}}\rrbracket}},
{\mathsf{q}}_{{\llbracket{\Gamma,{\Gamma'[t/x]}\vdash{B[t/x]}}\rrbracket}})$
${\mathsf{S}(\Gamma;A;\Gamma',{\mbox{\faUnlock}};t)}$ $=$ ${\mathsf{L}}\,{\mathsf{S}(\Gamma;A;\Gamma';t)}$
------------------------------------------------------- ----- --------------------------------------------------------------------------------------------------------------------------------------------------
\[lem:Proj\] Suppose ${\llbracket\Gamma,\Gamma'\rrbracket}$ and ${\llbracket\Gamma,x:A,\Gamma'\rrbracket}$ are defined. Then the following properties hold:
1. ${\llbracket\Gamma,x:A,\Gamma'\vdash X\rrbracket}\simeq
{\llbracket\Gamma,\Gamma'\vdash X\rrbracket}[{\mathsf{P}(\Gamma;A;\Gamma')}]$, where $\simeq$ is Kleene equality, and $X$ is a type or typed term;
2. ${\mathsf{P}(\Gamma;A;\Gamma')}$ is a well-defined morphism from ${\llbracket\Gamma,x:A,\Gamma'\rrbracket}$ to ${\llbracket\Gamma,\Gamma'\rrbracket}$;
The proof proceeds by mutual induction on the size of $\Gamma'$ (for statement 2) and the size of $\Gamma'$ plus the size of $X$ (for statement 1). We present only the cases particular to ${\square}$.
We start with statement 1. We use the mutual induction with statement 2 at the smaller size of $\Gamma'$ alone to ensure that ${\mathsf{P}(\Gamma;A;\Gamma')}$ is well-formed with the correct domain and codomain, then proceed by induction on the construction of $X$.
The ${\square}$ case follows because $$\begin{aligned}
{\llbracket\Gamma,x:A,\Gamma'\vdash{\square}B\rrbracket} &\simeq
\operatorname{\mathsf{R}}_{{\llbracket\Gamma,x:A,\Gamma'\rrbracket}}{\llbracket\Gamma,x:A,\Gamma',{\mbox{\faUnlock}}\vdash B\rrbracket} \\
&\simeq \operatorname{\mathsf{R}}_{{\llbracket\Gamma,x:A,\Gamma'\rrbracket}}({\llbracket\Gamma,\Gamma',{\mbox{\faUnlock}}\vdash B\rrbracket}
[{\mathsf{P}(\Gamma;A;\Gamma',{\mbox{\faUnlock}})}]) \tag{induction} \\
&\simeq R_{{\llbracket\Gamma,x:A,\Gamma'\rrbracket}}({\llbracket\Gamma,\Gamma',{\mbox{\faUnlock}}\vdash B\rrbracket}
[{\mathsf{L}}{\mathsf{P}(\Gamma;A;\Gamma')}]) \\
&\simeq (\operatorname{\mathsf{R}}_{{\llbracket\Gamma,\Gamma'\rrbracket}}{\llbracket\Gamma,\Gamma',{\mbox{\faUnlock}}\vdash B\rrbracket})
[{\mathsf{P}(\Gamma;A;\Gamma')}] \tag{\ref{eq:11}} \\
&= {\llbracket\Gamma,\Gamma'\vdash{\square}B\rrbracket}[{\mathsf{P}(\Gamma;A;\Gamma')}]\end{aligned}$$
The ${\ensuremath{\mathsf{shut}}}$ case follows immediately from and induction. For ${\ensuremath{\mathsf{open}}}$, the case where the deleted variable $x$ is to the right of the lock follows by Definition \[def:cwf\] part 5. Suppose instead it is to the left. Then $$\begin{aligned}
&{\llbracket\Gamma,\Gamma',{\mbox{\faUnlock}},y_1:B_1,\ldots,y_n:B_n\vdash{\ensuremath{\mathsf{open}}}\,t\rrbracket}
[{\mathsf{P}(\Gamma;A;\Gamma',{\mbox{\faUnlock}},y_1:B_1,\ldots,y_n:B_n)}] \\
&\simeq {\overline{{\llbrackett\rrbracket}}}[{\mathsf{p}}_{{\llbracketB_1\rrbracket}}\circ\cdots\circ{\mathsf{p}}_{{\llbracketB_n\rrbracket}}\circ
{\mathsf{P}(\Gamma;A;\Gamma',{\mbox{\faUnlock}},y_1:B_1,\ldots,y_n:B_n)}] \\
&\simeq {\overline{{\llbrackett\rrbracket}}}[{\mathsf{P}(\Gamma;A;\Gamma',{\mbox{\faUnlock}})}\circ
{\mathsf{p}}_{{\llbracketB_1\rrbracket}}\circ\cdots\circ{\mathsf{p}}_{{\llbracketB_n\rrbracket}}] \tag{Definition~\ref{def:cwf} part 5} \\
&\simeq {\overline{{\llbrackett\rrbracket}}}[{\mathsf{L}}{\mathsf{P}(\Gamma;A;\Gamma')}]
[{\mathsf{p}}_{{\llbracketB_1\rrbracket}}\circ\cdots\circ{\mathsf{p}}_{{\llbracketB_n\rrbracket}}] \\
&\simeq {\overline{{\llbrackett\rrbracket}[{\mathsf{P}(\Gamma;A;\Gamma')}]}}
[{\mathsf{p}}_{{\llbracketB_1\rrbracket}}\circ\cdots\circ{\mathsf{p}}_{{\llbracketB_n\rrbracket}}] \tag{\ref{eq:15}} \\
&\simeq {\overline{{\llbracket\Gamma,x:A,\Gamma'\vdash t\rrbracket}}}
[{\mathsf{p}}_{{\llbracketB_1\rrbracket}}\circ\cdots\circ{\mathsf{p}}_{{\llbracketB_n\rrbracket}}] \tag{induction} \\
&\simeq {{\llbracket\Gamma,x:A,\Gamma',{\mbox{\faUnlock}},y_1:B_1,\ldots,y_n:B_n\vdash {\ensuremath{\mathsf{open}}}\,t\rrbracket}}\end{aligned}$$
For statement 2, the lock case holds immediately by application of the functor ${\mathsf{L}}$. Other cases follow as standard; for example the base case holds because ${\mathsf{p}}_{{\llbracketA\rrbracket}}$ is indeed a morphism.
\[lem:exchange\] Suppose ${\llbracket\Gamma,x:A,y:B,\Gamma'\rrbracket}$ and ${\llbracket\Gamma\vdash B\rrbracket}$ are defined. Then the following properties hold:
1. ${\llbracket{\Gamma,y:B,x:A,\Gamma'\vdashX}\rrbracket}\simeq
{\llbracket{\Gamma,x:A,y:B,\Gamma'\vdashX}\rrbracket}
[{\mathsf{E}(\Gamma;A;B;\Gamma')}]$, where $X$ is a type or typed term;
2. ${\mathsf{E}(\Gamma;A;B;\Gamma')}$ is a well-defined morphism from ${\llbracket\Gamma,y:B,x:A,\Gamma'\rrbracket}$ to ${\llbracket\Gamma,x:A,y:B,\Gamma'\rrbracket}$;
The base case of statement 1 uses Lemma \[lem:Proj\]; the proof otherwise follows just as with Lemma \[lem:Proj\].
\[lem:Subst\] Suppose ${\llbracket\Gamma\vdash t:A\rrbracket}$ and ${\llbracket\Gamma,x:A,\Gamma'\rrbracket}$ are defined. Then the following properties hold:
1. ${\llbracket\Gamma,{\Gamma'[t/x]}\vdash{X[t/x]}\rrbracket}\simeq
{\llbracket\Gamma,x:A,\Gamma'\vdash X\rrbracket}[{\mathsf{S}(\Gamma;A;\Gamma';t)}]$, where $X$ is a type or typed term;
2. ${\mathsf{S}(\Gamma;A;\Gamma';t)}$ is a well-defined morphism from ${\llbracket\Gamma,{\Gamma'[t/x]}\rrbracket}$ to ${\llbracket\Gamma,x:A,\Gamma'\rrbracket}$;
As with Lemma \[lem:Proj\].
Where a context, type, or term is well-formed, its denotation is well-defined, and all types and terms identified by equations have the same denotation.
Most cases follow as usual, using Lemmas \[lem:Proj\], \[lem:exchange\], and \[lem:Subst\] as needed. The well-definedness of the formation rules for ${\square}$ are straightforward, so we present only the equations for ${\square}$:
Starting with ${\Gamma,{\mbox{\faUnlock}}\vdasht:A}$ we have ${\Gamma,
{\mbox{\faUnlock}},x_1:A_1,\ldots,x_n:A_n\vdash{\ensuremath{\mathsf{open}}}\,{\ensuremath{\mathsf{shut}}}\,t:A}$ and wish to prove its denotation is equal to that of $t$ (with the weakening $x_1,\ldots,x_n$). Then ${\llbracket{\ensuremath{\mathsf{open}}}\,{\ensuremath{\mathsf{shut}}}\,t\rrbracket}={\overline{{\overline{{\llbrackett\rrbracket}}}}}[{\mathsf{p}}_{{\llbracketA_1\rrbracket}}\circ
\cdots\circ{\mathsf{p}}_{{\llbracketA_n\rrbracket}}]={\llbrackett\rrbracket}[{\mathsf{p}}_{{\llbracketA_1\rrbracket}}\circ\cdots
\circ{\mathsf{p}}_{{\llbracketA_n\rrbracket}}]$, which is the weakening of $t$ by Lemma \[lem:Proj\].
The equality of ${\llbracket{\ensuremath{\mathsf{shut}}}\,{\ensuremath{\mathsf{open}}}\,t\rrbracket}$ and ${\llbrackett\rrbracket}$ is straightforward.
Term model {#sec:term_model}
----------
We now develop as our first example of a CwDRA, a term model built from the syntax of our calculus. The objects of this category are contexts modulo equality, which is defined pointwise via type equality. We define an arrow $\Delta\to\Gamma$ as a sequence of substitutions of an equivalence class of terms for each variable in $\Gamma$:
- the empty sequence is an arrow $\Delta\to\cdot$;
- Given $f:\Delta\to\Gamma$, type ${\Gamma\vdashA}$ and term ${\Delta\vdasht:A\,f}$, where $A\,f$ is the result of applying the substitutions $f$ to $A$, then $[t/x]\circ f$ modulo equality on $t$ is an arrow $\Delta\to\Gamma,x:A$;
- Given $f:\Delta\to\Gamma$ and a well-formed context $\Delta,{\mbox{\faUnlock}},
\Delta'$ with no locks in $\Delta'$, then $f$ is also an arrow $\Delta,{\mbox{\faUnlock}},\Delta'\to\Gamma,{\mbox{\faUnlock}}$;
We usually refer to the equivalence classes in arrows via representatives. Note that substitution respects these equivalence classes because of the congruence rules.
We next prove that this defines a category. Identity arrows are easily constructed:
\[lem:weak\_term\_arr\] If $f:\Delta\to\Gamma$ then $f:\Delta,x:A\to\Gamma$.
By induction on the construction on $f$. The base case is trivial.
Given $f:\Delta\to\Gamma$ and ${\Delta\vdasht:B\,f}$, by induction we have $f:\Delta,x:A\to\Gamma$ and by variable weakening we have ${\Delta,x:A\vdasht:B\,f}$ as required.
Supposing we have $f:\Delta\to\Gamma$ yielding $f:\Delta,{\mbox{\faUnlock}},\Delta'\to
\Gamma$, we could similarly get $f:\Delta,{\mbox{\faUnlock}},\Delta',x:A\to\Gamma$.
The identity on $\Gamma$ simply replaces all variables by themselves.
The identity on each $\Gamma$ is well defined as an arrow.
By induction on $\Gamma$. The identity on $\cdot$ is the empty sequence of substitutions. Given $id:\Gamma\to\Gamma$, we have $id:
\Gamma,x:A\to\Gamma$ by Lemma \[lem:weak\_term\_arr\], and ${\Gamma,x:A\vdashx:A}$ as required. $id:\Gamma\to\Gamma$ immediately yields $id:\Gamma,{\mbox{\faUnlock}}\to\Gamma,{\mbox{\faUnlock}}$.
The composition case is slightly more interesting:
\[Lem:arrow\_on\_judg\] Given $\Gamma,\Gamma'\vdash{\mathcal{J}}$ and $f:\Delta\to\Gamma$, we have $\Delta,\Gamma'\,f\vdash{\mathcal{J}}\,f$.
By induction on the construction on $f$. The base case requires that $\Gamma'\vdash{\mathcal{J}}$ implies $\Delta,\Gamma'\vdash{\mathcal{J}}$; this *left weakening* property is easily proved by induction on the typing rules.
Given $f:\Delta\to\Gamma$, ${\Delta\vdasht:A\,f}$ and $\Gamma,x:A,\Gamma'\vdash{\mathcal{J}}$, by induction $\Delta,x:A\,f,\Gamma'\,f\vdash{\mathcal{J}}\,f$. Then by Lemma \[lem:syntactic\_sanity\] part 3 we have $\Delta,{(\Gamma'\,f)[t/x]}\vdash{({\mathcal{J}}\,f)[t/x]}$ as required. The lock case is trivial.
The composition of $f:\Delta\to\Delta'$ and $g:\Delta'\to\Gamma$ involves replacing each $[t/x]$ in $g$ with $[t\,f/x]$.
The composition of two arrows $f:\Delta\to\Delta'$ and $g:\Delta'\to\Gamma$ is a well-defined arrow.
By induction on the definition of $g$. The base case is trivial, and extension by a new substitution follows via Lemma \[Lem:arrow\_on\_judg\].
Now suppose we have $g:\Delta'\to\Gamma$ yielding $g:\Delta',{\mbox{\faUnlock}},
\Delta''\to\Gamma,{\mbox{\faUnlock}}$. Now if we have $f:\Delta\to\Delta',{\mbox{\faUnlock}},\Delta''$ this must have arisen via some $f':\Delta_0\to\Delta'$ generating $f':\Delta_0,{\mbox{\faUnlock}},\Delta_1\to\Delta',{\mbox{\faUnlock}}$, where $\Delta=\Delta_0,{\mbox{\faUnlock}},
\Delta_1$. By induction we have well-defined $g\circ f':\Delta_0\to\Gamma$. Hence $g\circ f':\Delta\to\Gamma,{\mbox{\faUnlock}}$. But $g\circ f'=g\circ f$ because the variables of $\Delta''$ do not appear in $g$.
Checking the category axioms is straightforward. The category definitions then extend to a CwF in the usual way: the terminal object is ${\diamond}$, the families over $\Gamma$ are the types modulo equivalence well-defined in context $\Gamma$, the elements of any such type are the terms modulo equivalence, re-indexing is substitution, comprehension corresponds to extending a context with a new variable, the projection morphism is the replacement of variables by themselves, and the generic element is given by the variable rule.
Moving to the definition of a CwDRA, the endofunctor ${\mathsf{L}}$ acts by mapping $\Gamma\mapsto\Gamma,{\mbox{\faUnlock}}$, and does not change arrows. The family $\operatorname{\mathsf{R}}_\Gamma A$ is the type ${\Gamma\vdash{\square}A}$, which is stable under re-indexing by Lemma \[lem:syntactic\_sanity\] part 3. The bijections between families are supplied by the ${\ensuremath{\mathsf{shut}}}$ and ${\ensuremath{\mathsf{open}}}$ rules, with all equations following from the definitional equalities.
We do not attempt to prove that the term model is the *initial* CwDRA; such a result for dependent type theories appears to require syntax be written in a more verbose style than is appropriate for a paper introducing a new type theory [@Castellan:Dependent]. Nonetheless our type theory and notion of model are close enough that we conjecture that such a development is possible.
A general construction of CwDRAs {#sec:contruction-of-CwDRAs}
================================
In this section we show how to construct a CwDRA from an adjunction of endofunctors on a category with finite limits. We will refer to categories with finite limits more briefly as *cartesian* categories. We will use this construction in Section \[sec:examples\] to prove that the examples mentioned in the introduction can indeed be presented as [CwDRAs]{}. Our construction is an extension of the local universe construction [@LumsdainePL:locumo], which maps cartesian categories to categories with families, and locally cartesian closed categories to categories with families with $\Pi$- and $\Sigma$-types. The local universe construction is one of the known solutions to the problem of constructing a strict model of type theory out of a locally cartesian closed category (see [@hofmann1994interpretation; @LumsdainePL:locumo; @LumsdainePL:simmuf; @hofmann1997syntax] for discussions of alternative approaches to ’strictification’).
We first recall the local universe construction. Since it can be traced back to Giraud’s work on fibred categories [@Giraud:Cohomologie], we refer to it as the Giraud CwF associated to a cartesian category.
\[def:Giraud\] Let ${{\mathbf{C}}}$ be a cartesian category. The **Giraud CwF of ${{\mathbf{C}}}$** (${\mathscr{G}}{{\mathbf{C}}}$) is the CwF whose underlying category is ${{\mathbf{C}}}$, and where a family $A\in{{\mathscr{G}}{{\mathbf{C}}}(\Gamma)}$ is a pair of morphisms $$\label{eq:giraud:family}
\begin{split}
\xymatrix{& E\ar[d]^v\\\Gamma \ar[r]_u & U}
\end{split}$$ and an element of ${{\mathscr{G}}{{\mathbf{C}}}(\Gamma\mathbin{\scriptstyle\vdash}A)}$, for $A=(u,v)\in{{\mathscr{G}}{{\mathbf{C}}}(\Gamma)}$, is a map $a:\Gamma\to E$ such that $v\circ a= u$. Reindexing of $A=(u,v)\in{{\mathscr{G}}{{\mathbf{C}}}(\Gamma)}$ and $a\in{{\mathscr{G}}{{\mathbf{C}}}(\Gamma\mathbin{\scriptstyle\vdash}A)}$ along $\gamma\in{{{{\mathbf{C}}}}(\Delta,\Gamma)}$ are given by $$\begin{aligned}
A[\gamma] &{\triangleq}(u{\circ}\gamma,v)
\in{{\mathscr{G}}{{\mathbf{C}}}(\Delta)}\label{eq:8}\\
a[\gamma] &{\triangleq}a{\circ}\gamma \in{{\mathscr{G}}{{\mathbf{C}}}(\Delta\mathbin{\scriptstyle\vdash}A[\gamma])}
\end{aligned}$$ The comprehension ${\Gamma{.}A}\in{{\mathbf{C}}}$, for $A=(u,v)\in{{\mathscr{G}}{{\mathbf{C}}}(\Gamma)}$, is given by the pullback of diagram (\[eq:giraud:family\]), $$ \begin{split}
\xymatrix{{\Gamma{.}A} \ar[d]_{{\mathsf{p}}_A} \ar[r]^{{\mathsf{q}}_A} & E\ar[d]^v\\\Gamma \ar[r]_u & U}
\end{split}$$ with projection morphism ${\mathsf{p}}_A$ and generic element ${\mathsf{q}}_A$ as indicated in the diagram. Note that ${\mathsf{q}}_A$ is an element of $A[{\mathsf{p}}_A] = (u\circ {\mathsf{p}}_A, v)$ as required by commutativity of the pullback square. The pairing operation is obtained from the universal property of pullbacks.
Note that the local universe construction does indeed yield a category with families; in particular, reindexing in ${\mathscr{G}}{{\mathbf{C}}}$ is strict as required, simply because reindexing is given by composition.
\[Psh\] The name ‘local universe’ derives from the similarity to Voevodsky’s use of a (global) universe $U$ to construct strict models of type theory [@Voevodsky:csys; @LumsdainePL:simmuf] in which types in a context $\Gamma$ are modelled as morphisms $\Gamma \to U$. In the local universe construction, the universe varies from type to type.
In fact, the local universe construction is functorial; a precise statement requires a novel notion of CwF-morphism:
\[def:Gir-map\] A [**weak CwF morphism**]{} $\operatorname{\mathsf{R}}$ between CwFs consists of a functor $\operatorname{\mathsf{R}}: {{\mathbf{C}}}\to{{\mathbf{D}}}$ between the underlying categories preserving the terminal object, an operation on families mapping $A\in{{{{\mathbf{C}}}}(\Gamma)}$ to a family $\operatorname{\mathsf{R}}A\in{{{\mathbf{D}}}(\operatorname{\mathsf{R}}\Gamma)}$ and an operation on elements mapping $a\in{{{{\mathbf{C}}}}(\Gamma\mathbin{\scriptstyle\vdash}A)}$ to an element $\operatorname{\mathsf{R}}a\in{{{\mathbf{D}}}(\operatorname{\mathsf{R}}\Gamma\mathbin{\scriptstyle\vdash}\operatorname{\mathsf{R}}A)}$, such that
1. The functor $\operatorname{\mathsf{R}}: {{\mathbf{C}}}\to{{\mathbf{D}}}$ preserves terminal objects (up to isomorphism)
2. The operations on families and elements commute with reindexing in the sense that $\operatorname{\mathsf{R}}A [\operatorname{\mathsf{R}}\gamma] = \operatorname{\mathsf{R}}(A[\gamma])$ and $\operatorname{\mathsf{R}}t[\operatorname{\mathsf{R}}\gamma] = \operatorname{\mathsf{R}}(t[\gamma])$.
3. The maps ${\left(\operatorname{\mathsf{R}}{\mathsf{p}}_A,\operatorname{\mathsf{R}}{\mathsf{q}}_A\right)} : \operatorname{\mathsf{R}}({\Gamma{.}A}) \to {\operatorname{\mathsf{R}}\Gamma{.}\operatorname{\mathsf{R}}A}$ are isomorphisms for all $\Gamma$ and $A$. We write $\nu_{\Gamma,A}$ for the inverse.
We note the following equalities as consequences of the axioms above. $$\begin{aligned}
\operatorname{\mathsf{R}}({\mathsf{p}}_A) \circ \nu_{\Gamma,A} & = {\mathsf{p}}_{\operatorname{\mathsf{R}}A} \label{eq:nu:proj} \\
\operatorname{\mathsf{R}}({\mathsf{q}}_A)[\nu_{\Gamma,A}] & = {\mathsf{q}}_{\operatorname{\mathsf{R}}A} \label{eq:nu:gen} \\
\nu_{\Gamma,A}\circ {\left(\operatorname{\mathsf{R}}\gamma,\operatorname{\mathsf{R}}a\right)} & = \operatorname{\mathsf{R}}{\left(\gamma,a\right)} \label{eq:nu:pair} \end{aligned}$$ For example, the last of these is proved by postcomposing with the inverse of $\nu_{\Gamma,A}$ and noting $$\begin{aligned}
{\left(\operatorname{\mathsf{R}}{\mathsf{p}}_A,\operatorname{\mathsf{R}}{\mathsf{q}}_A\right)}\circ \operatorname{\mathsf{R}}{\left(\gamma,a\right)}
& = {\left(\operatorname{\mathsf{R}}{\mathsf{p}}_A \circ \operatorname{\mathsf{R}}{\left(\gamma,a\right)},\operatorname{\mathsf{R}}{\mathsf{q}}_A[\operatorname{\mathsf{R}}{\left(\gamma,a\right)}]\right)} \\
& = {\left(\operatorname{\mathsf{R}}({\mathsf{p}}_A \circ {\left(\gamma,a\right)}),\operatorname{\mathsf{R}}({\mathsf{q}}_A[{\left(\gamma,a\right)}])\right)} \\
& = {\left(\operatorname{\mathsf{R}}\gamma,\operatorname{\mathsf{R}}a\right)}\end{aligned}$$
Note that a weak CwF morphism preserves comprehension and the terminal object only up to isomorphism instead of on the nose, as required by the stricter notion of morphism of Dybjer [@Dybjer1996 Definition 2]. Weak CwF morphisms sit between strict CwF-morphisms and pseudo-CwF morphisms [@DBLP:journals/lmcs/CastellanCD17]. The latter allow substitution to be preserved only up to isomorphism satisfying a number of coherence conditions. Since weak CwF morphisms preserve substitution on the nose, these are not needed here.
\[functor-lex\] ${\mathscr{G}}$ extends to a functor from the category of cartesian categories and finite limit preserving functors, to the category of CwFs with weak morphisms.
Let $\operatorname{\mathsf{R}}:{{\mathbf{C}}}\to{{\mathbf{D}}}$ be a finite limit preserving functor. For each $\Gamma\in{{\mathbf{C}}}$ and $A=(u,v)\in{{\mathscr{G}}{{\mathbf{C}}}(\Gamma)}$, we simply let $\operatorname{\mathsf{R}}A {\triangleq}(\operatorname{\mathsf{R}}u,\operatorname{\mathsf{R}}v)$. Likewise, for an element $a\in{{\mathscr{G}}{{\mathbf{C}}}(\Gamma\mathbin{\scriptstyle\vdash}A)}$, we let $\operatorname{\mathsf{R}}a$ be the action of $\operatorname{\mathsf{R}}$ on the morphism $a$. Finally, since comprehension is defined by pullback and $\operatorname{\mathsf{R}}$ preserves pullbacks up to isomorphism, we obtain the required $\nu_{\Gamma,A}$.
We now embark on showing that if we apply the local universe construction to a cartesian category ${{\mathbf{C}}}$ with a pair of adjoint endofunctors, then the resulting CwF ${\mathscr{G}}{{\mathbf{C}}}$ is in fact a [CwDRA]{} (Theorem \[thm:giraud\]). To this end, we introduce the auxiliary notion of a category with families with an adjunction:
\[def:CwF+A\] A [**CwF+A**]{} consists of a CwF with an adjunction ${\mathsf{L}}{\dashv}\operatorname{\mathsf{R}}$ on the category of contexts, such that $\operatorname{\mathsf{R}}$ extends to a weak CwF endomorphism.
\[CwDRAfromCwF+A\] If ${{\mathbf{C}}}$ with the adjunction ${\mathsf{L}}\dashv \operatorname{\mathsf{R}}$ is a CwF+A, then there is a [CwDRA]{} structure on ${{\mathbf{C}}}$ with ${\mathsf{L}}$ as the required functor on ${{\mathbf{C}}}$.
We write $\eta$ for the unit of the adjunction. For a family $A\in{{{{\mathbf{C}}}}({\mathsf{L}}\Gamma)}$, we define $\operatorname{\mathsf{R}}_{\Gamma} A \in{{{{\mathbf{C}}}}(\Gamma)}$ to be $(\operatorname{\mathsf{R}}A)[\eta]$. For an element $a\in{{{{\mathbf{C}}}}({\mathsf{L}}\Gamma\mathbin{\scriptstyle\vdash}A)}$, we define its transpose ${\overline{a}}\in{{{{\mathbf{C}}}}(\Gamma\mathbin{\scriptstyle\vdash}\operatorname{\mathsf{R}}_\Gamma A)}$ to be $(\operatorname{\mathsf{R}}a)[\eta]$. For the opposite direction, suppose $b\in{{{{\mathbf{C}}}}(\Gamma\mathbin{\scriptstyle\vdash}\operatorname{\mathsf{R}}_\Gamma A)}$. Since ${\left(\eta,b\right)}:\Gamma \rightarrow {\operatorname{\mathsf{R}}{\mathsf{L}}\Gamma{.}\operatorname{\mathsf{R}}A}$, we have that ${\mathsf{L}}(\nu_{{\mathsf{L}}\Gamma,A}
\circ {\left(\eta,b\right)}):{\mathsf{L}}\Gamma\rightarrow {\mathsf{L}}\operatorname{\mathsf{R}}({{\mathsf{L}}\Gamma{.}A})$ and thus we can define ${\overline{b}}\in{{{{\mathbf{C}}}}({\mathsf{L}}\Gamma\mathbin{\scriptstyle\vdash}A)}$ to be the element ${\mathsf{q}}_A[\varepsilon\circ {\mathsf{L}}(\nu_{{\mathsf{L}}\Gamma,A}
\circ {\left(\eta,b\right)})]$. Note that this is well typed because ${\mathsf{q}}_A$ is an element of the family $A[{\mathsf{p}}_A]$ and so ${\overline{b}}$ is an element of $$\begin{aligned}
A[{\mathsf{p}}_A \circ \varepsilon\circ {\mathsf{L}}(\nu \circ {\left(\eta,b\right)})] & = A[\varepsilon\circ {\mathsf{L}}(\operatorname{\mathsf{R}}{\mathsf{p}}_A \circ \nu \circ {\left(\eta,b\right)})] \\
& = A[\varepsilon\circ {\mathsf{L}}({\mathsf{p}}_{\operatorname{\mathsf{R}}A} \circ {\left(\eta,b\right)})] \\
& = A[\varepsilon\circ {\mathsf{L}}(\eta)] \\
& = A\end{aligned}$$ using equation (\[eq:nu:proj\]) in the second equality. These operations can be proved inverses of each other using the equations (\[eq:nu:gen\]) and (\[eq:nu:pair\]).
Note that the conditions for a CwF+A are stronger than those for a [CwDRA]{}; for instance, a [CwDRA]{} does not require $\operatorname{\mathsf{R}}$ to be defined on the context category. We return to the relation between these constructions in Section \[sec:CwFAfromCwDRA\]
\[thm:giraud-CwF+A\] If ${{\mathbf{C}}}$ is a cartesian category and ${\mathsf{L}}\dashv \operatorname{\mathsf{R}}$ are adjoint endofunctors on ${{\mathbf{C}}}$, then ${\mathscr{G}}{{\mathbf{C}}}$ with the adjunction ${\mathsf{L}}\dashv \operatorname{\mathsf{R}}$ is a CwF+A.
We are already given an adjunction on the underlying category of ${\mathscr{G}}{{\mathbf{C}}}$. Theorem \[functor-lex\] constructs the weak CwF morphism.
\[thm:giraud\] If ${{\mathbf{C}}}$ is a cartesian category and ${\mathsf{L}}\dashv \operatorname{\mathsf{R}}$ are adjoint endofunctors on ${{\mathbf{C}}}$, then ${\mathscr{G}}{{\mathbf{C}}}$ has the structure of a [CwDRA]{}.
By Lemmas \[thm:giraud-CwF+A\] and \[CwDRAfromCwF+A\].
The above Theorem \[thm:giraud\] thus provides a general construction of [CwDRAs]{}. In Section \[sec:examples\] we use it to present examples from the literature. As mentioned earlier, the local universe construction interacts well with other type formers: If we start with a locally cartesian closed category ${{\mathbf{C}}}$ (with W-types, Id-types and a universe), then ${\mathscr{G}}{{\mathbf{C}}}$ also models dependent products $\Pi$ and sums $\Sigma$ (and W-types, Id-types and a universe); see . In Section \[sec:universes\] we consider universes.
CwF+A from a CwDRA {#sec:CwFAfromCwDRA}
------------------
In this subsection we show how to produce a CwF+A from a [CwDRA]{} under the assumption that the CwF is *democratic*. Intuitively, a democratic CwF is one where every context comes from a type, and hence it is not surprising that for a democratic CwDRA one can use the action of the dependent right adjoint on families to define a right adjoint on contexts.
A CwF is [**democratic**]{} [@Clairambault2011] if for every context $\Gamma$ there is a family $\widehat{\Gamma}\in{{{{\mathbf{C}}}}({\top})}$ and an isomorphism $\zeta_\Gamma:\Gamma \rightarrow {{\top}{.}\widehat{\Gamma}}$.
\[thm:giraud-converse\] Let ${{\mathbf{C}}}$ be a democratic [CwDRA]{}. The endofunctor ${{\mathsf{L}}}:{{\mathbf{C}}}\to{{\mathbf{C}}}$, part of the [CwDRA]{} structure, has a right adjoint $\operatorname{\mathsf{R}}$.
For $\Gamma\in{{\mathbf{C}}}$, we define $\operatorname{\mathsf{R}}\Gamma\in{{\mathbf{C}}}$ by $$\label{eq:17}
\operatorname{\mathsf{R}}\Gamma {\triangleq}{{\top}{.}\operatorname{\mathsf{R}}_{\top}(\widehat{\Gamma}[!_{{\mathsf{L}}{\top}}])}$$ We have a bijection, natural in $\Delta$ $$\begin{aligned}
{{\mathbf{C}}}(\Delta, \operatorname{\mathsf{R}}\Gamma) & \cong {{{{\mathbf{C}}}}(\Delta\mathbin{\scriptstyle\vdash}\operatorname{\mathsf{R}}_{\top}(\widehat{\Gamma} [!_{{\mathsf{L}}{\top}}])) [!_\Delta]))} \\
&\cong {{{{\mathbf{C}}}}(\Delta\mathbin{\scriptstyle\vdash}\operatorname{\mathsf{R}}_\Delta (\widehat{\Gamma} [!_{{\mathsf{L}}\Delta}])))} \\
&\cong {{{{\mathbf{C}}}}({\mathsf{L}}\Delta\mathbin{\scriptstyle\vdash}\widehat{\Gamma} [!_{{\mathsf{L}}\Delta}]))}\\
&\cong {{\mathbf{C}}}({\mathsf{L}}\Delta, {{\top}{.}\widehat{\Gamma})}\\
&\cong {{\mathbf{C}}}({\mathsf{L}}\Delta, \Gamma)
\end{aligned}$$ The last of the above bijections follows by composition with ${\zeta_\Gamma^{-1}}$.
Let $\gamma:\Gamma' \to \Gamma$ we have then an action $ \gamma^*:{{\mathbf{C}}}(-,\operatorname{\mathsf{R}}\Gamma')\to {{\mathbf{C}}}(-,\operatorname{\mathsf{R}}\Gamma)$ given by $${{\mathbf{C}}}(-,\operatorname{\mathsf{R}}\Gamma') \cong {{\mathbf{C}}}({\mathsf{L}}-,\Gamma') \xrightarrow{-\circ \gamma} {{\mathbf{C}}}({\mathsf{L}}-,\Gamma) \cong {{\mathbf{C}}}(-,\operatorname{\mathsf{R}}\Gamma)$$ Define $\operatorname{\mathsf{R}}\gamma = \gamma^*_{\operatorname{\mathsf{R}}\Gamma'} ({\mathsf{id}}_{\operatorname{\mathsf{R}}\Gamma'})$. Then the correspondence ${{\mathbf{C}}}(\Delta, \operatorname{\mathsf{R}}\Gamma) \cong {{\mathbf{C}}}({\mathsf{L}}\Delta, \Gamma)$ is natural in $\Gamma$, proving that $\operatorname{\mathsf{R}}$ is a right adjoint to ${\mathsf{L}}$.
Consider a democratic [CwDRA]{}, with ${{\mathbf{C}}}$ as the underlying category, and ${\mathsf{L}}\dashv \operatorname{\mathsf{R}}$ the adjunction obtained from the above theorem. We then extend $\operatorname{\mathsf{R}}$ to a weak CwF morphism by defining, for a family $A \in{{{{\mathbf{C}}}}(\Gamma)}$ and an element $a\in{{{{\mathbf{C}}}}(\Gamma\mathbin{\scriptstyle\vdash}A)}$, $$\begin{aligned}
&\operatorname{\mathsf{R}}A {\triangleq}\operatorname{\mathsf{R}}_{\operatorname{\mathsf{R}}\Gamma} (A[\varepsilon]) &\operatorname{\mathsf{R}}a {\triangleq}{\overline{a[\varepsilon]}}\end{aligned}$$ where $\varepsilon:{\mathsf{L}}\operatorname{\mathsf{R}}\Gamma \to \Gamma$ is the counit of the adjunction.
\[lem:R\_w\_morphism\] $\operatorname{\mathsf{R}}$ as defined above is a weak CwF morphism. In particular, for $A \in {{{{\mathbf{C}}}}(\Gamma)}$ we have an isomorphism $\nu_{\Gamma,A}:{\operatorname{\mathsf{R}}\Gamma{.}\operatorname{\mathsf{R}}A} \to \operatorname{\mathsf{R}}({\Gamma{.}A})$, inverse to ${\left(\operatorname{\mathsf{R}}{\mathsf{p}}_A,\operatorname{\mathsf{R}}{\mathsf{q}}_A\right)}$.
We will show a bijection ${{\mathbf{C}}}(\Delta, \operatorname{\mathsf{R}}\Gamma.\operatorname{\mathsf{R}}A) \cong {{\mathbf{C}}}(\Delta, \operatorname{\mathsf{R}}(\Gamma.A))$ natural in $\Delta$. We have $$\begin{aligned}
{{\mathbf{C}}}(\Delta,\operatorname{\mathsf{R}}\Gamma.\operatorname{\mathsf{R}}A) &\cong \prod_{\gamma:{{\mathbf{C}}}(\Delta,\operatorname{\mathsf{R}}\Gamma)} {{{{\mathbf{C}}}}(\Delta\mathbin{\scriptstyle\vdash}(\operatorname{\mathsf{R}}A) [\gamma])}\end{aligned}$$ We have a bijection $-^\top: {{\mathbf{C}}}(\Delta,\operatorname{\mathsf{R}}\Gamma) \cong {{\mathbf{C}}}({\mathsf{L}}\Delta,\Gamma)$. But $$(\operatorname{\mathsf{R}}A)[\gamma] = (\operatorname{\mathsf{R}}_{\operatorname{\mathsf{R}}\Gamma} A[\varepsilon])[\gamma] = \operatorname{\mathsf{R}}_\Delta (A[\varepsilon \circ {\mathsf{L}}\gamma]) = \operatorname{\mathsf{R}}_\Delta (A[\gamma^\top])$$ Hence we have a bijection ${{{{\mathbf{C}}}}(\Delta\mathbin{\scriptstyle\vdash}(\operatorname{\mathsf{R}}A)[\gamma])} \cong {{{{\mathbf{C}}}}({\mathsf{L}}\Delta\mathbin{\scriptstyle\vdash}A[\gamma^\top])}$. So $$\begin{aligned}
{{\mathbf{C}}}(\Delta,\operatorname{\mathsf{R}}\Gamma.\operatorname{\mathsf{R}}A) &\cong \prod_{\gamma:{{\mathbf{C}}}(\Delta,\operatorname{\mathsf{R}}\Gamma)} {{{{\mathbf{C}}}}(\Delta\mathbin{\scriptstyle\vdash}(\operatorname{\mathsf{R}}A) [\gamma])}\\
& \cong \prod_{\gamma':{{\mathbf{C}}}({\mathsf{L}}\Delta,\Gamma)} {{{{\mathbf{C}}}}({\mathsf{L}}\Delta\mathbin{\scriptstyle\vdash}A[\gamma'])}\\
& \cong {{\mathbf{C}}}({\mathsf{L}}\Delta, \Gamma.A) \\
& \cong {{\mathbf{C}}}(\Delta,\operatorname{\mathsf{R}}(\Gamma.A)) \end{aligned}$$ By the Yoneda lemma, this implies $\operatorname{\mathsf{R}}\Gamma.\operatorname{\mathsf{R}}A \cong {{\mathbf{C}}}(\Delta,\operatorname{\mathsf{R}}(\Gamma.A))$, and it is easy to check that the direction ${{\mathbf{C}}}(\Delta,\operatorname{\mathsf{R}}(\Gamma.A)) \to \operatorname{\mathsf{R}}\Gamma.\operatorname{\mathsf{R}}A$ is given by ${\left(\operatorname{\mathsf{R}}{\mathsf{p}}_A,\operatorname{\mathsf{R}}{\mathsf{q}}_A\right)}$.
\[cor:Dem\_cwFDRA\_is\_CwF+A\] A democratic CwDRA has the structure of CwF+A
For a category ${{\mathbf{C}}}$ with a terminal object, the CwF ${\mathscr{G}}{{\mathbf{C}}}$ is democratic with $\widehat{\Gamma}$ given by the diagram: $$\xymatrix{& \Gamma\ar[d]^{!_\Gamma}\\{1}\ar[r]_{!_{1}} & 1}$$
For ordinary dependent type theory, the term model is a democratic CwF [@DBLP:journals/lmcs/CastellanCD17 Section 4]. However, the term model for our modal dependent type theory is *not* democratic, since there is, for example, no type corresponding to the context ${\mbox{\faUnlock}}$ consisting of just one lock.
Examples {#sec:examples}
========
We now present concrete examples of [CwDRA]{}s generated from cartesian categories with an adjunction of endofunctors, including those mentioned in the introduction.
**$\Pi$ type with closed domain**Consider a CwF where the underlying category of contexts ${{\mathbf{C}}}$ is cartesian closed, and let $A$ be a closed type. We have then an adjunction of endofunctors ${-\times {{\top}{.}A} {\dashv}-^{{{\top}{.}A}}}$ on ${{\mathbf{C}}}$, and suppose that the right adjoint extends to a weak CwF endomorphism, giving the structure of a CwF+A. As we saw above, this happens e.g. when the CwF is of the form ${\mathscr{G}}{{\mathbf{C}}}$. In this case $\operatorname{\mathsf{R}}_\Gamma B$ behaves as a type of the form $\Pi(x:A) B$ since ${{\mathbf{C}}}(\Gamma \vdash \operatorname{\mathsf{R}}_\Gamma B) \cong {{\mathbf{C}}}(\Gamma \times {{\top}{.}A} \vdash B)
\cong {{\mathbf{C}}}({\Gamma{.}(A[!_{\Gamma}])} \vdash B)$. Thus, the notion of dependent right adjoint generalises $\Pi$ types with closed domain. This generalises to the setting where ${{\mathbf{C}}}$ carries the structure of a monoidal closed category, in which case the adjunction $-\otimes {{\top}{.}A} {\dashv}{{{\top}{.}A}} \multimap (-)$ extends to give a dependent notion of linear function space with closed domain. The next example is an instance of this.
**Dependent name abstraction**The notion of *dependent name abstraction* for families of nominal sets was introduced by Pitts et al. [@PittsAM:deptta Section 3.6] to give a semantics for an extension of Martin-Löf Type Theory with names and constructs for freshness and name-abstraction. It provides an example of a [CwDRA]{} that can be presented via Theorem \[thm:giraud\]. In this case ${{\mathbf{C}}}$ is the category ${\mathbf{Nom}}$ of nominal sets and equivariant functions [@PittsAM:nomsns]. Its objects are sets $\Gamma$ equipped with an action of finite permutations of a fixed infinite set of atomic names ${\mathbb{A}}$, with respect to which the elements of $\Gamma$ are finitely supported, and its morphisms are functions that preserve the action of name permutations. ${\mathbf{Nom}}$ is a topos (it is equivalent to the Schanuel topos [@PittsAM:nomsns Section 6.3]) and hence in particular is cartesian. We take the functor ${\mathsf{L}}:{\mathbf{Nom}}\rightarrow{\mathbf{Nom}}$ to be separated product [@PittsAM:nomsns Section 3.4] with the nominal set of atomic names. This has a right adjoint $\operatorname{\mathsf{R}}$ that sends each $\Gamma\in{\mathbf{Nom}}$ to the nominal set of name abstractions $[{\mathbb{A}}]\Gamma$ [@PittsAM:nomsns Section 4.2] whose elements are a generic form of $\alpha$-equivalence class in the case that $\Gamma$ is a nominal set of syntax trees for some language.
Applying Theorem \[thm:giraud\], we get a [CwDRA]{} structure on ${\mathscr{G}}{\mathbf{Nom}}$. In fact the CwF ${\mathscr{G}}{\mathbf{Nom}}$ has an equivalent, more concrete description in this case, in terms of *families of nominal sets* [@PittsAM:deptta Section 3.1]. Under this equivalence, the value $\operatorname{\mathsf{R}}_\Gamma A\in{\mathscr{G}}{\mathbf{Nom}}(\Gamma)$ of the dependent right adjoint at $A\in{\mathscr{G}}({\mathsf{L}}\Gamma)$ corresponds to the family of *dependent name abstractions* defined by . The bijection is given in one direction by the name abstraction operation [@PittsAM:deptta (40)] and in the other by concretion at a fresh name [@PittsAM:deptta (42)].
**Guarded and Clocked Type Theory**Guarded recursion [@Nakano:Modality] is an extension of type theory with a modal *later* operator, denoted ${\triangleright}$, on types, an operation ${\mathsf{next}}: A \to {\triangleright}A$ and a guarded fixed point operator ${\mathsf{fix}}: ({\triangleright}A \to A) \to A$ mapping $f$ to a fixed point for $f \circ {\mathsf{next}}$. The standard model of guarded recursion is the topos of trees [@birkedal2011first], i.e. the category of presheaves on $\omega$, with ${\triangleright}X(n+1) = X(n)$, ${\triangleright}X(0) = 1$. The later operator has a left adjoint ${\triangleleft}$, called *earlier*, given by ${\triangleleft}X(n) = X(n+1)$, so ${\triangleright}$ yields a dependent right adjoint on the induced [CwDRA]{}.
Birkedal et al. [@birkedal2011first Section 6.1] show that ${\triangleleft}$ in a dependently typed setting does not commute with reindexing. However it *does* have a left adjoint, namely the ‘stutter’ functor $!$ with $!X(0)=X(0)$ and $!X(n+1)=X(n)$, so ${\triangleleft}$ does give rise to a well-behaved modality in the setting of this paper. This apparent contradiction is resolved by the use of locks in the context: ${\Gamma\vdashA}$ does not give rise to a well-behaved ${\Gamma\vdash{\triangleleft}A}$, but ${\Gamma,{\mbox{\faUnlock}}\vdashA}$ does. This is an intriguing example of the Fitch-style approach increasing expressivity.
Guarded recursion can be used to encode coinduction given a *constant* modality [@clouston2015programming], denoted ${\square}$, on the topos of trees, defined as ${\square}X(n) = \lim_k X(k)$. The ${\square}$ functor is the right adjoint of the essential geometric morphism on $\hat{\omega}$ induced by $0 : \omega \to \omega$, the constant map to $0$, and hence it also yields a dependent right adjoint. In , ${\square}$ was used in a simple type theory, employing ‘explicit substitutions’ following . As we will discuss in Section \[sec:discussion\] this approach proved difficult to extend to dependent types, and we wish to use the modal dependent type theory of the present paper to study ${\square}$ in dependent type theory.
An alternative to the constant modality are the *clock quantifiers* of , which unlike the constant modality have already been combined succesfully with dependent types [@Mogelberg:Type; @BirkedalL:gdtt-conf]. They are also slightly more general than the constant modality, as multiple clocks allow coinductive data structures that unroll in multiple dimensions, such as infinitely-wide infinitely-deep trees. The denotational semantics, however, are more complicated, consisting of presheaves over a category of ‘time objects’, restricted to those fulfilling an ‘orthogonality’ condition [@GDTTmodel]. Nevertheless the ${\triangleleft}\dashv {\triangleright}$ adjunction of the topos of trees lifts to this category, and so once again we may construct a CwDRA.
Clocked Type Theory () [@bahr2017clocks] is a recent type theory for guarded recursion that has strongly normalising reduction semantics, and has been shown to have semantics in the category discussed above [@CloTTmodel]. The operator ${\triangleright}$ is refined to a form of dependent function type ${{\triangleright}\, ({\alpha}:\kappa) .} A$ over ticks ${\alpha}$ on clock $\kappa$. Ticks can appear in contexts as $\Gamma, {\alpha}: \kappa$; these are similar to the locks of Fitch-style contexts, except that ticks have names, and can be weakened. The names of ticks play a crucial role in controlling fixed point unfoldings. Finally, the modal operator ${\triangleright}$ on the topos of trees can be generalized to the presheaf topos $\widehat{{{\mathbf{C}}}\times \omega}$ for any category ${{\mathbf{C}}}$, simply by using the identity on ${{\mathbf{C}}}$ to extend the underlying functor (which generates the essential geometric morphism) on $\omega$ to ${{\mathbf{C}}}\times \omega$. In this topos, with ${{\mathbf{C}}}$ the cube category, is used to model guarded *cubical* type theory; an extension of cubical type theory [@CoquandT:cubttc]. In more detail, one uses a CwF where families are certain *fibrations*, and since ${\triangleright}$ preserves fibrations, it does indeed extend to a [CwDRA]{}.
**Cohesive Toposes**Cohesive toposes have also recently been considered as models of a form of modal type theory [@shulman2018brouwer; @2017arXiv170607526R]. Cohesive toposes carry a triple adjunction $\int\dashv\flat\dashv\sharp$ and hence induce two dependent right adjoints. Examples of cohesive toposes include simplicial sets $\hat{\Delta}$ and cubical sets $\hat{{\square}}$; since these are presheaf toposes they also model universes. For example, for simplicial sets, the triple of adjoints are given by the essential geometric morphism induced by the constant functor $0:\Delta\to \Delta$. In the category of cubical sets $\sharp$ has a further right adjoint, used by to reason about parametricity. **Tiny objects** use a ‘tiny’ object $\mathbb{I}$ to construct the fibrant universe in the cubical model of homotopy type theory. By definition, an object $\mathbb{I}$ of a category ${{\mathbf{C}}}$ is *tiny* if the exponentiation functor $(-)^{\mathbb{I}}:{{\mathbf{C}}}\rightarrow{{\mathbf{C}}}$ has a right-adjoint, which they denote by $\surd$. As for ${\triangleleft}$ above, the right adjoint functor $\surd$ exists globally, but not locally; in other words, there is no right adjoint to $(-)^{\top.\mathbb{I}}$ on each category of families over an object $\Gamma\in{{\mathbf{C}}}$, stable under re-indexing $\Gamma$ (except in the trivial case that $\mathbb{I}$ is terminal). Nevertheless our present framework is still applicable: the corresponding *dependent* right adjoint for $(-)^{\mathbb{I}}$, constructed as in Section \[sec:contruction-of-CwDRAs\], plays an important part in the construction of the fibrant universe given in [@licata2018internal].
Universes {#sec:universes}
=========
In this section, we extend our modal dependent type theory with universes. For the semantics, we start from Coquand’s notion of a category with universes [@Coquand:CwU], which covers all presheaf models of dependent type theory with universes. The notion of *category with universes* rests on the observation that in presheaf models one can interpret an inverse ${\ulcorner-\urcorner}$ to the usual function $\El$ from codes to types, and hence obtain a simpler notion of universe than usual .
A [CwU]{} is specified by:
1. A category ${{\mathbf{C}}}$ with a terminal object ${\top}$.
2. For each object $\Gamma\in{{\mathbf{C}}}$ and natural number $n\in{\mathbb{N}}$, a set ${{{\mathbf{C}}}(\Gamma,{n})}$ of *families at universe level $n$* over $\Gamma$.
3. For each object $\Gamma\in{{\mathbf{C}}}$, natural number $n$, and family $A\in{{{\mathbf{C}}}(\Gamma,{n})}$, a set ${{{{\mathbf{C}}}}(\Gamma\mathbin{\scriptstyle\vdash}A)}$ of *elements* (at some level) of the family $A$ over $\Gamma$.
4. For each morphism $\gamma\in{{{{\mathbf{C}}}}(\Delta,\Gamma)}$, *re-indexing* functions $A\in{{{\mathbf{C}}}(\Gamma,{n})} \mapsto A[\gamma]\in{{{\mathbf{C}}}(\Delta,{n})}$ and $a\in{{{{\mathbf{C}}}}(\Gamma\mathbin{\scriptstyle\vdash}A)}\mapsto a[\gamma]\in{{{{\mathbf{C}}}}(\Delta\mathbin{\scriptstyle\vdash}A[\gamma])}$, satisfying equations for associativity and identity as in a CwF.
5. For each object $\Gamma\in{{\mathbf{C}}}$, number $n$ and family $A\in{{{\mathbf{C}}}(\Gamma,{n})}$, a *comprehension object* ${\Gamma{.}A}\in{{\mathbf{C}}}$ equipped with projections and generic elements satisfying equations as in a CwF.
6. For each number $n$, a family ${\mathsf{U}}_{n}\in{{{\mathbf{C}}}({\top},{n+1})}$, the *universe at level $n$*.
7. For each object $\Gamma\in{{\mathbf{C}}}$ and number $n$, a *code* function $A\in{{{\mathbf{C}}}(\Gamma,{n})}\mapsto {\ulcornerA\urcorner}\in{{{{\mathbf{C}}}}(\Gamma\mathbin{\scriptstyle\vdash}{\mathsf{U}}_{n}[!_{\Gamma}])}$, and an *element* function $u\in{{{{\mathbf{C}}}}(\Gamma\mathbin{\scriptstyle\vdash}{\mathsf{U}}_{n}[!_{\Gamma}])} \mapsto
\operatorname{\mathsf{E}}u \in{{{\mathbf{C}}}(\Gamma,{n})}$, satisfying ${\ulcornerA\urcorner}[\gamma] = {\ulcornerA[\gamma]\urcorner}$, $\operatorname{\mathsf{E}}{\ulcornerA\urcorner} = A$, and ${\ulcorner\operatorname{\mathsf{E}}u\urcorner} = u$.
We will of course want the universes to be closed under various type-forming operations, but in this formalisation of universes these definitions are just as for CwFs, without having to explicitly reflect them into the universes.
\[lem:1\] The element function is stable under re-indexing: $(\operatorname{\mathsf{E}}u)[\gamma] = \operatorname{\mathsf{E}}(u[\gamma])$.
$(\operatorname{\mathsf{E}}u)[\gamma] = \operatorname{\mathsf{E}}{\ulcorner(\operatorname{\mathsf{E}}u)[\gamma]\urcorner} = \operatorname{\mathsf{E}}({\ulcorner\operatorname{\mathsf{E}}u\urcorner}[\gamma])
= \operatorname{\mathsf{E}}(u[\gamma])$.
\[cor:generic family CwU\] In a CwU there is a *generic family* ${\mathsf{El}\,\in}{{{\mathbf{C}}}({{\top}{.}{\mathsf{U}}_{n}},{n})}$ of types of level $n$ (for each $n\in{\mathbb{N}}$), with the property that ${\mathsf{El}\,[}{\left(!_{\Gamma},{\ulcornerA\urcorner}\right)}] = A$, for all $A\in{{{\mathbf{C}}}(\Gamma,{n})}$.
Since ${\mathsf{p}}_{{\mathsf{U}}_n} =\ ! :{{\top}{.}}{{\mathsf{U}}_n}\to{\top}$, we have ${\mathsf{q}}\in{{{{\mathbf{C}}}}({{\top}{.}{\mathsf{U}}_n}\mathbin{\scriptstyle\vdash}{\mathsf{U}}_n[!_{{{\top}{.}}{{\mathsf{U}}_n}}])}$ and thus we can define $\El$ to be $\operatorname{\mathsf{E}}{\mathsf{q}}$, and then the required property follows by Lemma \[lem:1\].
For a CwU, there is an underlying CwF with families over $\Gamma$ given as ${{{{\mathbf{C}}}}(\Gamma)} = \bigcup_n {{{\mathbf{C}}}(\Gamma,{n})}$. Using this we can extend the definition of CwDRA to categories with universes in the obvious way, as follows:
A [**category with universe and dependent right adjoint ([CwUDRA]{})**]{} is a CwU with the structure of a [CwDRA]{} such that operation on types preserves universe levels in the sense that $A \in {{{\mathbf{C}}}({\mathsf{L}}\Gamma,{n})}$ implies $\operatorname{\mathsf{R}}_\Gamma A \in {{{\mathbf{C}}}(\Gamma,{n})}$.
Similarly, one can extend the notion of CwF+A from Definition \[def:CwF+A\] to the setting of universes:
A [**weak CwU morphism**]{} $\operatorname{\mathsf{R}}$ is a weak CwF morphism on the underlying CwFs preserving size in the sense that $A \in {{{\mathbf{C}}}(\Gamma,{n})}$ implies $\operatorname{\mathsf{R}}A \in {{{\mathbf{C}}}(\operatorname{\mathsf{R}}\Gamma,{n})}$. A [**CwU+A**]{} consists of a CwU with an adjunction ${\mathsf{L}}\dashv\operatorname{\mathsf{R}}$ on the category of contexts, such that $\operatorname{\mathsf{R}}$ extends to a weak CwU morphism.
The construction of Lemma \[CwDRAfromCwF+A\] extends to a construction of a CwUDRA from a CwU+A. We now show (Lemma \[lem:univ-endo-weak-morphism\]) that the action of the right adjoint on families and elements can be defined by just defining it on the universe as in the following definition.
\[def:amp\] A [**universe endomorphism**]{} on a CwU is a finite limit preserving functor $\operatorname{\mathsf{R}}$ on the category of contexts together with, for each $n$, a family ${\mathsf{Rl}}\in{{{\mathbf{C}}}(\operatorname{\mathsf{R}}({{\top}{.}{\mathsf{U}}_n}),{n})}$ and an element ${\mathsf{r}}\in{{{{\mathbf{C}}}}(\operatorname{\mathsf{R}}({{{\top}{.}{\mathsf{U}}_n}{.}{\mathsf{El}\,}})\mathbin{\scriptstyle\vdash}{\mathsf{Rl}}[\operatorname{\mathsf{R}}{\mathsf{p}}])}$ such that the morphism $$\label{eq:5}
\xymatrix{{\operatorname{\mathsf{R}}({{{\top}{.}{\mathsf{U}}_n}{.}{\mathsf{El}\,}})}
\ar[rr]^{{\left(\operatorname{\mathsf{R}}{\mathsf{p}},{\mathsf{r}}\right)}} \ar[dr]_{\operatorname{\mathsf{R}}{\mathsf{p}}} &&
{{\operatorname{\mathsf{R}}({{\top}{.}{\mathsf{U}}_n}){.}{\mathsf{Rl}}}} \ar[dl]^{{\mathsf{p}}}\\
& {\operatorname{\mathsf{R}}({{\top}{.}{\mathsf{U}}_n})}}$$ over $\operatorname{\mathsf{R}}({{\top}{.}{\mathsf{U}}_n})$ is an isomorphism; in other words there is a morphism ${\ell}: {\operatorname{\mathsf{R}}({{\top}{.}{\mathsf{U}}_n}){.}{\mathsf{Rl}}} \to
\operatorname{\mathsf{R}}({{{\top}{.}{\mathsf{U}}_n}{.}{\mathsf{El}\,}})$ satisfying ${\ell}{\circ}{\left(\operatorname{\mathsf{R}}{\mathsf{p}},{\mathsf{r}}\right)} = {\mathsf{id}}$ and ${\left(\operatorname{\mathsf{R}}{{\mathsf{p}}},{\mathsf{r}}\right)}{\circ}{\ell}= {\mathsf{id}}$.
This means that we have a universe category endomorphism in the sense of : a family ${\mathsf{Rl}}\in{{{\mathbf{C}}}(\operatorname{\mathsf{R}}({{\top}{.}{\mathsf{U}}_n}),{n})}$ gives a pullback square with the morphism ${{{\top}{.}{\mathsf{U}}_n}{.}{\mathsf{El}\,}}\to
{{\top}{.}{\mathsf{U}}_n}$ and the code function. The isomorphism above implies that the universe $\operatorname{\mathsf{R}}({{{\top}{.}{\mathsf{U}}_n}{.}{\mathsf{El}\,}}) \to \operatorname{\mathsf{R}}({{\top}{.}{\mathsf{U}}_n})$ is also pullback of ${{{\top}{.}{\mathsf{U}}_n}{.}{\mathsf{El}\,}}\to
{{\top}{.}{\mathsf{U}}_n}$ along the code function.
Given a CwU with a weak CwU morphism $\operatorname{\mathsf{R}}$, then clearly $\operatorname{\mathsf{R}}$ is a universe endomorphism, with ${\mathsf{Rl}}{\triangleq}\operatorname{\mathsf{R}}({\mathsf{El}\,)}$, ${\mathsf{r}}{\triangleq}\operatorname{\mathsf{R}}{\mathsf{q}}$ and ${\ell}{\triangleq}\nu$. Conversely:
\[lem:univ-endo-weak-morphism\] Any CwU with a universe endomorphism $\operatorname{\mathsf{R}}: {{\mathbf{C}}}\to {{\mathbf{C}}}$ extends to a weak CwU morphism.
Given $A\in{{\mathbf{C}}}(\Gamma,n)$, since we have ${\left(!_{\Gamma},{\ulcornerA\urcorner}\right)}:\Gamma{\rightarrow}{{\top}{.}{\mathsf{U}}_n}$, we can define $$\label{eq:3}
\operatorname{\mathsf{R}}A {\triangleq}{\mathsf{Rl}}[\operatorname{\mathsf{R}}{\left(!_{\Gamma},{\ulcornerA\urcorner}\right)}] \in {{{\mathbf{C}}}(\operatorname{\mathsf{R}}\Gamma,{n})}$$ This is stable under re-indexing, since for $\gamma:\Delta\to \Gamma$ $$\begin{aligned}
\operatorname{\mathsf{R}}(A\,[\gamma]) & {\triangleq}{\mathsf{Rl}}[\operatorname{\mathsf{R}}{\left(!_{\Delta},{\ulcornerA\,[\gamma]\urcorner}\right)}] \\
&= {\mathsf{Rl}}[\operatorname{\mathsf{R}}{\left(!_{\Delta},{\ulcornerA\urcorner}[\gamma]\right)}] \\
& = {\mathsf{Rl}}[\operatorname{\mathsf{R}}({\left(!_{\Gamma},{\ulcornerA\urcorner}\right)}{\circ}\gamma)] \\
& = {\mathsf{Rl}}[\operatorname{\mathsf{R}}{\left(!_{\Gamma},{\ulcornerA\urcorner}\right)} {\circ}\operatorname{\mathsf{R}}\gamma] \\
& = ({\mathsf{Rl}}[\operatorname{\mathsf{R}}{\left(!_{\Gamma},{\ulcornerA\urcorner}\right)}])[\operatorname{\mathsf{R}}\gamma] \\
& {\triangleq}(\operatorname{\mathsf{R}}A)[\operatorname{\mathsf{R}}\gamma]
\end{aligned}$$
Given $a\in {{{{\mathbf{C}}}}(\Gamma\mathbin{\scriptstyle\vdash}A)}$, by Corollary \[cor:generic family CwU\] we have $a \in {{{{\mathbf{C}}}}(\Gamma\mathbin{\scriptstyle\vdash}{\mathsf{El}\,[}{\left(!_{\Gamma},{\ulcornerA\urcorner}\right)}])}$ and hence $${\left({\left(!_{\Gamma},{\ulcornerA\urcorner}\right)},a\right)}:\Gamma{\rightarrow}{\top}.{\mathsf{U}}_n.\El$$ Therefore $${\mathsf{r}}[\operatorname{\mathsf{R}}{\left({\left(!_{\Gamma},{\ulcornerA\urcorner}\right)},a\right)}] \in{{{{\mathbf{C}}}}(\operatorname{\mathsf{R}}\Gamma\mathbin{\scriptstyle\vdash}({\mathsf{Rl}}[\operatorname{\mathsf{R}}{{\mathsf{p}}}])[\operatorname{\mathsf{R}}{\left({\left(!_{\Gamma},{\ulcornerA\urcorner}\right)},a\right)}])}$$ But $({\mathsf{Rl}}[\operatorname{\mathsf{R}}{{\mathsf{p}}}])[\operatorname{\mathsf{R}}{\left({\left(!_{\Gamma},{\ulcornerA\urcorner}\right)},a\right)}] = {\mathsf{Rl}}[\operatorname{\mathsf{R}}({{\mathsf{p}}}{\circ}{\left({\left(!_{\Gamma},{\ulcornerA\urcorner}\right)},a\right)})] = {\mathsf{Rl}}[\operatorname{\mathsf{R}}{\left(!_{\Gamma},{\ulcornerA\urcorner}\right)}) {\triangleq}\operatorname{\mathsf{R}}A$. We can therefore define $$\label{eq:4}
\operatorname{\mathsf{R}}a {\triangleq}{\mathsf{r}}[\operatorname{\mathsf{R}}{\left({\left(!_{\Gamma},{\ulcornerA\urcorner}\right)},a\right)}] \in {{{{\mathbf{C}}}}(\operatorname{\mathsf{R}}\Gamma\mathbin{\scriptstyle\vdash}\operatorname{\mathsf{R}}A)}$$ and this is stable under re-indexing, since for $\gamma:\Delta \to \Gamma$ $$\begin{aligned}
(\operatorname{\mathsf{R}}a)[\operatorname{\mathsf{R}}\gamma] &{\triangleq}{\mathsf{r}}[\operatorname{\mathsf{R}}{\left({\left(!_{\Gamma},{\ulcornerA\urcorner}\right)},a\right)}] [\operatorname{\mathsf{R}}\gamma] \\&= {\mathsf{r}}[\operatorname{\mathsf{R}}{\left({\left(!_{\Delta},{\ulcornerA\urcorner}[\gamma]\right)},a[\gamma]\right)}] \\&=
{\mathsf{r}}[\operatorname{\mathsf{R}}{\left({\left(!_{\Delta},{\ulcornerA[\gamma]\urcorner}\right)},a[\gamma]\right)}] \\
&{\triangleq}\operatorname{\mathsf{R}}(a[\gamma])
\end{aligned}$$
Finally we must show that $\operatorname{\mathsf{R}}$ commutes with comprehension. For this, note that there are pullback squares $$ \xymatrix{{\operatorname{\mathsf{R}}({\Gamma{.}A})} {\save*!/dr-1.4pc/dr:(-1,1)@^{|-}\restore}\ar[rrr]^{\operatorname{\mathsf{R}}{\left({\left(!_{{\Gamma{.}A}},{\ulcornerA\urcorner}[{\mathsf{p}}_A]\right)},{\mathsf{q}}\right)}} \ar[d]_{\operatorname{\mathsf{R}}{\mathsf{p}}}
&&& \operatorname{\mathsf{R}}({{{\top}{.}{\mathsf{U}}_n}{.}{\mathsf{El}\,}}) \ar[d]^{\operatorname{\mathsf{R}}{{\mathsf{p}}}}\\
{\operatorname{\mathsf{R}}\Gamma}\ar[rrr]_{\operatorname{\mathsf{R}}{\left(!_{\Gamma},{\ulcornerA\urcorner}\right)}} &&&
{\operatorname{\mathsf{R}}({\top}.{\mathsf{U}}_n)}}
\qquad
\xymatrix{{\operatorname{\mathsf{R}}\Gamma.\operatorname{\mathsf{R}}A} {\save*!/dr-1.4pc/dr:(-1,1)@^{|-}\restore}\ar[rrr]^{{\left(\operatorname{\mathsf{R}}{\left(!_{\Gamma},{\ulcornerA\urcorner}\right)}\circ{\mathsf{p}},{\mathsf{q}}\right)}} \ar[d]_{{\mathsf{p}}}&&&
{\operatorname{\mathsf{R}}({\top}.{\mathsf{U}}_n).{\mathsf{Rl}}} \ar[d]^{{\mathsf{p}}}\\
{\operatorname{\mathsf{R}}\Gamma}\ar[rrr]_{\operatorname{\mathsf{R}}{\left(!_{\Gamma},{\ulcornerA\urcorner}\right)}} &&&
{\operatorname{\mathsf{R}}({\top}.{\mathsf{U}}_n)}}$$ the former because the functor $\operatorname{\mathsf{R}}$ preserves finite limits and the latter by definition of $\operatorname{\mathsf{R}}A$. Applying with $a={\mathsf{q}}$ we get that the pullback along $\operatorname{\mathsf{R}}{\left(!_{\Gamma},{\ulcornerA\urcorner}\right)}$ of the morphism ${\left(\operatorname{\mathsf{R}}{\mathsf{p}},{\mathsf{r}}\right)}$ in is $$\xymatrix{{\operatorname{\mathsf{R}}(\Gamma. A)} \ar[rr]^{{\left(\operatorname{\mathsf{R}}{\mathsf{p}},\operatorname{\mathsf{R}}{\mathsf{q}}\right)}}
\ar[dr]_{\operatorname{\mathsf{R}}{\mathsf{p}}} &&
{\operatorname{\mathsf{R}}\Gamma.\operatorname{\mathsf{R}}A} \ar[dl]^{{\mathsf{p}}} \\
& {\operatorname{\mathsf{R}}\Gamma}}$$ Then since ${\left(\operatorname{\mathsf{R}}{\mathsf{p}},{\mathsf{r}}\right)}$ is an isomorphism, so is its pullback ${\left(\operatorname{\mathsf{R}}{\mathsf{p}},\operatorname{\mathsf{R}}{\mathsf{q}}\right)}$, as required.
\[sec:codes-image-r\] We observe that for $\operatorname{\mathsf{R}}$ as constructed above, the image under $\operatorname{\mathsf{R}}$ of maps with ${\mathsf{U}}_n$-small fibers is classified by ${\mathsf{Rl}}\in{{{\mathbf{C}}}(\operatorname{\mathsf{R}}({{\top}{.}{\mathsf{U}}_n}),{n})}$. That is to say that ${\left(!_{\operatorname{\mathsf{R}}\Gamma},{\ulcorner\operatorname{\mathsf{R}}A\urcorner}\right)}= {\left(!_{\operatorname{\mathsf{R}}\Gamma},{\ulcorner{\mathsf{Rl}}\urcorner}\right)} \circ \operatorname{\mathsf{R}}{\left(!_\Gamma,{\ulcornerA\urcorner}\right)}$ which is true by our choice of $\operatorname{\mathsf{R}}A = {\mathsf{Rl}}[\operatorname{\mathsf{R}}{\left(!_\Gamma,{\ulcornerA\urcorner}\right)}]$. Hence, the type of codes for such fibers is $\operatorname{\mathsf{R}}$ applied to the codes for types. The same situation occurred for ${\triangleright}$ in , but was not observed at the time.
\[thm:cwudra\] Any CwU equipped with an adjunction on the category of contexts whose right adjoint is a universe endomorphism can be given the structure of a CwUDRA.
Combine Lemmas \[CwDRAfromCwF+A\] and \[lem:univ-endo-weak-morphism\].
For most of the presheaf examples considered in Section \[sec:examples\], the dependent right adjoint is obtained as the direct image of an essential geometric morphism arising from a functor on the category on which the presheaves are defined. We show that in this case, the right adjoint preserves universe levels and hence gives a CwUDRA. For simplicity, we will restrict to one universe and show that the right adjoint preserves smallness with respect to this. Let $U$ be a universe in an ambient set theory. We call the elements of $U$, $U$-sets. A *$U$-small category* is one where both the sets of objects and the set of morphisms are $U$-small. Let us assume that $U$ is $U$-complete — it is closed under limits of $U$-small diagrams. A Grothendieck universe in ZFC would satisfy these conditions.
Let $C,D$ be a $U$-small categories and $f:C\to D$ a functor between them. The direct image $f_*$ of the induced geometric morphism preserves size. In particular, for each endofunctor $f$, the direct image is a weak CwU morphism.
Since $f_*$ is a right adjoint, we know that it induces a weak CwF morphism, and we just need to show that it maps $U$-small families to $U$-small families. Recall first that the direct image $f_*$ is the (pointwise) right Kan extension [@johnstone:elephant A4.1.4] defined on objects by the limit of the diagram $$(\operatorname{\mathrm{Ran}}_f F)d{\triangleq}\lim (f\downarrow d) \stackrel{\pi_1}\to
C^{op}\stackrel{F}\to Uset,$$ for $F\in \widehat{C}$ and $d\in D$. Here $(f\downarrow d)$ denotes the *comma category* consisting of pairs $(c;g:f(c)\to d)$.
A family $\alpha:F\to G$, for $F,G\in \hat{C}$ is $U$-small if for each $c$ and each $x \in G(c)$ the set ${\alpha_c^{-1}}(x)$ is in $U$. Given $(x_g)_{g \in f\downarrow d} \in f_*G(d) = (\operatorname{\mathrm{Ran}}_f G)d$, the preimage ${(f_*\alpha)^{-1}}((x_g)_{g\in f\downarrow d})$ is the set $$\{(y_g)_{g\in f\downarrow d} \in (\operatorname{\mathrm{Ran}}_f G)d \mid \forall g . \alpha_c(y_g) = x_g\}$$ which is the limit of the diagram associating to each $g$ the set ${\alpha^{-1}}_c(x_g)$. Since each of these sets are in $U$ by assumption and since also $f\downarrow d$ is in $U$, by the assumption of $U$ being closed under limits, also ${(f_*\alpha)^{-1}}((x_g)_{g\in f\downarrow d})$ is in $U$ as desired.
**Syntax**At this stage it should hopefully be clear that one can refine and extend the syntax of modal dependent type theory from Section \[sec:syntax\] so that the resulting syntactic type theory can be modelled in a CwU+A. The idea is, of course, to refine the judgement for well-formed types and to include a level $n$, so that it has the form ${\Gamma\vdash_{n}A}$, and likewise for type equality judgements. For example,
In addition to the existing rules for types (indexed with a level) and terms, we then also include:
Finally, we add the following type and term equality rules:
As an example, there is a term $$\widehat{\square}{\triangleq}\lambda x.{\ulcorner{\square}\operatorname{\mathsf{E}}({\ensuremath{\mathsf{open}}}\,x)\urcorner} \;:\; {\square}{\mathsf{U}}_n\to{\mathsf{U}}_n$$ which encodes the ${\square}$ type constructor on the universe in the sense that $$\operatorname{\mathsf{E}}(\widehat{\square}({\ensuremath{\mathsf{shut}}}\, u)) = {\square}(\operatorname{\mathsf{E}}u)$$ This is similar to the $\widehat{{\triangleright}}$ operator of Guarded Dependent Type Theory [@BirkedalL:gdtt-conf], which is essential to defining guarded recursive types. Thus, $\widehat{{\triangleright}}$ arises for general reasons quite unconnected to the specifics of guarded recursion.
Discussion {#sec:discussion}
==========
Related Work
------------
Modal dependent type theory builds on work on the computational interpretation of modal logic with *simple* types. Some of this work involves a standard notion of context; most relevantly to this paper, the calculus for Intuitionistic K of , which employs *explicit substitutions* in terms. Departing from standard contexts, Fitch-style calculi were introduced independently by and . Recent work by argued that Fitch-style calculus can be extended to a variety of different modal logics, and gave a sound categorical interpretation by modelling the modality as a right adjoint. Another non-standard notion of context are the *dual contexts* introduced by for the modal logic Intuituionistic S4 of comonads. Here a context $\Delta;\Gamma$ is understood as meaning ${\square}\Delta\land\Gamma$, so the structure in the context is modelled by the modality itself, not its left adjoint. Recent work by has extended this approach to a variety of modal logics, including Intuitionistic K.
There exists recent work employing variants of dual contexts for modal *dependent* type theory, all involving (co)monads rather than the more basic logic of this paper. Spatial type theory [@shulman2018brouwer], designed for applications in homotopy type theory (see also [@wellen2017formalizing; @licata2018internal]), extends the Davies-Pfenning calculus for a comonad with both dependent types and a second modality, a monad right adjoint to the comonad. Second, the calculus for parametricity of uses *three* zones to extend Davies-Pfenning with a monad *left* adjoint to the comonad. They focus on $\Pi$- and $\Sigma$-types with modalised arguments, but a more standard modality can be extracted by taking the second argument of a modalised $\Sigma$-type to be the unit type. In both the above works the leftmost modality is intended to itself be a right adjoint, so they potentially could also be captured by a Fitch-style calculus. Third, suggest a generalisation of Davies-Pfenning with some unusual properties, as ${\square}$ types carry an auxiliary typed variable and $\Pi$-types may only draw their argument from the modal context. We finally note the dual contexts approach has inspired the mode theories of , but this line of work as yet does not support a term calculus.
We however do not know how to apply the dual context approach to modal logics where the modality is not a (co)monad. For example it is not obvious how to extend Kavvos’s simply-typed calculus for Intuitionistic K. This should be compared to the ease of extending the simply-typed Fitch-style calculus with dependent types. We hope that Fitch-style calculi continue to provide a relatively simple setting for modal dependent type theory as we explore the extensions discussed in the next subsection.
We are not aware of any successful extensions of the explicit substitution approach to dependent types; our own experiments with this while developing Guarded Dependent Type Theory [@clouston2015programming] suggests this is probably possible but becomes unwieldy with real examples. Far more succesful was the Clocked Type Theory [@bahr2017clocks] discussed in Section \[sec:examples\], which can now be seen to have rediscovered the Fitch-style framework, albeit with the innovation of named locks to control fixed-point unfoldings. That work provides the inspiration for the more foundational developments of this paper.
Future work
-----------
We wish to develop operational semantics for dependent Fitch-style calculi, and conjecture that standard techniques for sound normalisation and canonicity can be extended, as was possible for simply-typed Fitch-style calculi [@Borghuis:Coming; @Clouston:fitch-2018], and for Clocked Type Theory [@bahr2017clocks]. Such results should then lead to practical implementation.
The modal axiom Intuitionistic K was used in this paper because it provides a basic notion of modal necessity and holds of many useful models. Nonetheless for particular applications we will want to develop Fitch-style calculi corresponding to more particular logics. There can be no algorithm for converting additional axioms to well-behaved calculi, but we know that Fitch-style calculi are extremely versatile in the simply typed case [@Clouston:fitch-2018], and Clocked Type Theory provides one example of this with dependent types. In particular we are interested in Fitch-style calculi with multiple interacting modalities, each of which is assigned its own lock; we hope to develop guarded type theory with both ${\triangleright}$ and ${\square}$ modalities in this style.
The notion of CwF with a weak CwF endomorphism (Definition \[def:Gir-map\]) is more general than our CwF+A, as it does not require the existence of a left adjoint. Because a weak CwF endomorphism must preserves products, it appears to be a rival candidate for a model of dependent type theory with the K axiom. However we do not know how to capture this class of models in syntax. Understanding this would be valuable because *truncation* [@DBLP:journals/logcom/AwodeyB04], considered as an endofunctor for example on sets, defines such a morphism but is not a right adjoint. Truncation allows one to move between general types and propositions. For example combining it with guarded types would allow us to formalise work in this field that makes that distinction [@birkedal2011first; @clouston2015programming].
[^1]: For an introduction to modal logic, see e.g. .
[^2]: For “Kripke”; not to be confused with Streicher’s K [@streicher1993investigations].
[^3]: This should not be confused with models where there are adjoint functors between different categories which can be composed to define a monad or comonad.
[^4]: This capability was achieved by Bizjak et al. [@BirkedalL:gdtt-conf] via *delayed substitutions*, but this construction does not straightforwardly support an operational semantics [@bahr2017clocks].
|
---
abstract: 'Compositing is one of the most common operations in photo editing. To generate realistic composites, the appearances of foreground and background need to be adjusted to make them compatible. Previous approaches to harmonize composites have focused on learning statistical relationships between hand-crafted appearance features of the foreground and background, which is unreliable especially when the contents in the two layers are vastly different. In this work, we propose an end-to-end deep convolutional neural network for image harmonization, which can capture both the context and semantic information of the composite images during harmonization. We also introduce an efficient way to collect large-scale and high-quality training data that can facilitate the training process. Experiments on the synthesized dataset and real composite images show that the proposed network outperforms previous state-of- the-art methods.'
author:
- |
Yi-Hsuan Tsai$^{1}$ Xiaohui Shen$^{2}$ Zhe Lin$^{2}$ Kalyan Sunkavalli$^{2}$ Xin Lu$^{2}$ Ming-Hsuan Yang$^{1}$\
$^{1}$University of California, Merced $^{2}$Adobe Research\
$^{1}$[{ytsai2, mhyang}@ucmerced.edu]{} $^{2}$[{xshen, zlin, sunkaval, xinl}@adobe.com]{}
bibliography:
- 'mybib.bib'
title: Deep Image Harmonization
---
Introduction
============
Compositing is one of the most common operations in image editing. To generate a composite image, a foreground region in one image is extracted and combined with the background of another image. However, the appearances of the extracted foreground region may not be consistent with the new background, making the composite image unrealistic. Therefore, it is essential to adjust the appearances of the foreground region to make it compatible with the new background (Figure \[fig:intro\]). Previous techniques improve the realism of composite images by transferring statistics of hand-crafted features, including color [@Lalonde_ICCV_2007; @Xue_siggraph_2012] and texture [@Sunkavalli_siggraph_2010], between the foreground and background regions. However, these techniques do not take the contents of the composite images into account, leading to unreliable results when appearances of the foreground and background regions are vastly different.
In this work, we propose a learning-based method by training an end-to-end deep convolutional neural network (CNN) for image harmonization, which can capture both the context and semantic information of the composite images during harmonization. Given a composite image and a foreground mask as the input, our model directly outputs a harmonized image, where the contents are the same as the input but with adjusted appearances on the foreground region.
[@c@ @c@ ]{} ![Our method can adjust the appearances of the composite foreground to make it compatible with the background region. Given a composite image, we show the harmonized images generated by [@Xue_siggraph_2012], [@Zhu_ICCV_2015] and our deep harmonization network. []{data-label="fig:intro"}](figure/real_resize/input/0023.png "fig:"){width="0.48\linewidth"} & ![Our method can adjust the appearances of the composite foreground to make it compatible with the background region. Given a composite image, we show the harmonized images generated by [@Xue_siggraph_2012], [@Zhu_ICCV_2015] and our deep harmonization network. []{data-label="fig:intro"}](figure/real_resize/xue/0023.png "fig:"){width="0.48\linewidth"}\
Composite image & Xue [@Xue_siggraph_2012]\
![Our method can adjust the appearances of the composite foreground to make it compatible with the background region. Given a composite image, we show the harmonized images generated by [@Xue_siggraph_2012], [@Zhu_ICCV_2015] and our deep harmonization network. []{data-label="fig:intro"}](figure/real_resize/zhu/0023.png "fig:"){width="0.48\linewidth"} & ![Our method can adjust the appearances of the composite foreground to make it compatible with the background region. Given a composite image, we show the harmonized images generated by [@Xue_siggraph_2012], [@Zhu_ICCV_2015] and our deep harmonization network. []{data-label="fig:intro"}](figure/real_resize/ours/0023.png "fig:"){width="0.48\linewidth"}\
Zhu [@Zhu_ICCV_2015] & Our harmonization result
Context information has been utilized in several image editing tasks, such as image enhancement [@Hwang_ECCV_2012; @Yan_siggraph_2015], image editing [@Tsai_SIGGRAPH_2016] and image inpainting [@Pathak_CVPR_2016]. For image harmonization, it is critical to understand what it looks like in the surrounding background region near the foreground region. Hence foreground appearances can be adjusted accordingly to generate a realistic composite image. Toward this end, we train a deep CNN model that consists of an encoder to capture the context of the input image and a decoder to reconstruct the harmonized image using the learned representations from the encoder. In addition, semantic information is of great importance to improve image harmonization. For instance, if we know the foreground region to be harmonized is a sky, it is natural to adjust the appearance and color to be blended with the surrounding contents, instead of making the sky green or yellow. However, the above-mentioned encoder-decoder does not explicitly model semantic information without the supervision of high-level semantic labels. Hence, we incorporate another decoder to provide scene parsing of the input image, while sharing the same encoder for learning feature representations. A joint training scheme is adopted to propagate the semantic information to the harmonization decoder. With such semantic guidance, the harmonization process not only captures the image context but also understands semantic cues to better adjust the foreground region. Training an end-to-end deep CNN requires a large-scale training set including various and high-quality samples. However, unlike other image editing tasks such as image colorization [@Zhang_ECCV_2016] and inpainting [@Pathak_CVPR_2016] where unlimited amount of training data can be easily generated, it is relatively difficult to collect a large-scale training set for image harmonization, as generating composite images and ground truth harmonized output requires professional editing skills and a considerable amount of time. To solve this problem, we develop a training data generation method that can synthesize large-scale and high-quality training pairs, which facilitates the learning process. To evaluate the proposed algorithm, we conduct extensive experiments on synthesized and real composite images. We first quantitatively compare our method with different settings to other existing approaches for image harmonization on our synthesized dataset, where the ground truth images are provided. We then perform a user study on real composite images and show that our model trained on the synthesized dataset performs favorably in real cases. The contributions of this work are as follows. First, to the best of our knowledge, this is the first attempt to have an end-to-end learning approach for image harmonization. Second, we demonstrate that our joint CNN model can effectively capture context and semantic information, and can be efficiently trained for both the harmonization and scene parsing tasks. Third, an efficient method to collect large-scale and high-quality training images is developed to facilitate the learning process for image harmonization.
Related Work
============
Our goal is to harmonize a composite image by adjusting its foreground appearances while keeping the same background region. In this section, we discuss existing methods closely related to this setting. In addition, the proposed method adopts a learning-based framework and a joint training scheme. Hence recent image editing methods within this scope are also discussed. Generating realistic composite images requires a good match for both the appearances and contents between foreground and background regions. Existing methods use color and tone matching techniques to ensure consistent appearances, such as transferring global statistics [@Reinhard_CGA_2001; @Pitie_CVMP_2007], applying gradient domain methods [@Perez_siggraph_2003; @Tao_IJCV_2013], matching multi-scale statistics [@Sunkavalli_siggraph_2010] or utilizing semantic information [@Tsai_SIGGRAPH_2016]. While these methods directly match appearances to generate realistic composite images, realism of the image is not considered. Lalonde and Efros [@Lalonde_ICCV_2007] predict the realism of photos by learning color statistics from natural images and use these statistics to adjust foreground appearances to improve the chromatic compatibility. On the other hand, a data-driven method [@Johnson_TVGG_2011] is developed to improve the realism of computer-generated images by retrieving a set of real images with similar global layouts for transferring appearances.
In addition, realism of the image has been studied and used to improve the harmonization results. Xue et al. [@Xue_siggraph_2012] perform human subject experiments to identify most significant statistical measures that determine the realism of composite images and adjust foreground appearances accordingly. Recently, Zhu et al. [@Zhu_ICCV_2015] learn a CNN model to predict the realism of a composite image and incorporate the realism score into a color optimization function for appearance adjustment on the foreground region. Different from the above-mentioned methods, our end-to-end CNN model directly learn from pairs of a composite image as the input and a ground truth image, which ensures the realism of the output results. Recently, neural network based methods for image editing tasks such as image colorization [@Iizuka_SIGGRAPH_2016; @Larsson_ECCV_2016; @Zhang_ECCV_2016], inpainting [@Pathak_CVPR_2016] and filtering [@Liu_ECCV_2016], have drawn much attention due to their efficiency and impressive results. Similar to autoencoders [@Bengio_PAMI_2013], these methods adopt an unsupervised learning scheme that learns feature representations of the input image, where raw data is used for supervision. Although our method shares the similar concept, to the best of our knowledge it is the first end-to-end trainable CNN architecture designed for image harmonization. However, these image editing pipelines may suffer from missing semantic information in the finer level during reconstruction, and such semantics are important cues for understanding image contents. Unlike previous methods that do not explicitly use semantics, we incorporate an additional model to predict pixel-wise scene parsing results and then propagate this information to the harmonization model, where the entire framework is still end-to-end trainable.
Deep Image Harmonization
========================
In this section, we describe the details of our proposed end-to-end CNN model for image harmonization. Given a composite image and a foreground mask as the input, our model outputs a harmonized image by adjusting foreground appearances while retaining the background region. Furthermore, we design a joint training process with scene parsing to understand image semantics and thus improve harmonization results. Figure \[fig:network\] shows an overview of the proposed CNN architecture. Before describing this network, we first introduce a data collection method that allows us to obtain large-scale and high-quality training pairs.
Data Acquisition {#sec:data}
----------------
Data acquisition is an essential step to successfully train a CNN. As described above, an image pair containing the composite and harmonized images is required as the input and ground truth for the network. Unlike other unsupervised learning tasks such as [@Zhang_ECCV_2016; @Pathak_CVPR_2016] that can easily obtain training pairs, image harmonization task requires expertise to generate a high-quality harmonized image from a composite image, which is not feasible to collect large-scale training data. To address this issue, we start from a real image which we treat as the output ground truth of our network. We then select a region (e.g., an object or a scene) and edit its appearances to generate an edited image which we use as the input composite image to the network. The overall process is described in Figure \[fig:data\_coco\_5k\]. This data acquisition method ensures that the ground truth images are always realistic so that the goal of the proposed CNN is to directly reconstruct a realistic output from a composite image. In the following, we introduce the details of how we generate our synthesized dataset.
![Data acquisition methods. We illustrate the approaches for collecting training pairs for the datasets (a) Miscrosoft COCO and Flickr via color transfer, and (b) MIT-Adobe FiveK with different styles. []{data-label="fig:data_coco_5k"}](figure/data_coco.png "fig:"){width="1.0\linewidth"} (a) Miscrosoft COCO & Flickr\
![Data acquisition methods. We illustrate the approaches for collecting training pairs for the datasets (a) Miscrosoft COCO and Flickr via color transfer, and (b) MIT-Adobe FiveK with different styles. []{data-label="fig:data_coco_5k"}](figure/data_5k.png "fig:"){width="1.0\linewidth"} (b) MIT-Adobe FiveK\
{width="1.0\linewidth"}
We first use the Microsoft COCO dataset [@Lin_ECCV_2014], where the object segmentation masks are provided for each image. To generate synthesized composite images, we randomly select an object and edit its appearances via a color transfer method. In order to ensure that the edited images are neither arbitrary nor unrealistic in color and tone, we construct the color transfer functions by searching for proper reference objects. Specifically, given a target image and its corresponding object mask, we search a reference image which contains the object with the same semantics. We then transfer the appearance from the reference object to the target object. As such, we ensure that the edited object still looks plausible but does not match the background context. For color transfer, we compute statistics of the luminance and color temperature, and use the histogram matching method [@Lee_CVPR_2016].
To generate a larger variety of transferred results, we apply different transfer parameters for both the luminance and color temperature on one image, so that our learned network can adapt to different scenarios in real cases. In addition, we apply an aesthetics prediction model [@Kong_ECCV_2016] to filter out low-quality images. An example of generated synthesized input and output pairs are shown in Figure \[fig:data\_coco\_5k\](a). Although the Microsoft COCO dataset provides us with rich object categories, it is still limited to certain objects. To cover more object categories, we augment it with the MIT-Adobe FiveK dataset [@Bychkovsky_CVPR_2011]. In this dataset, each original image has another $5$ different styles that are re-touched by professional photographers using Adobe Lightroom, resulting in $6$ editions of the same image. To edit the original image, we begin with one randomly selected style and manually segment a region. We then crop this segmented region and overlay on the image with another style to generate the synthesized composite image. An example set is presented in Figure \[fig:data\_coco\_5k\](b).
MSCOCO MIT-Adobe Flickr
-------------- -------- ----------- --------
Training set 51187 4086 4720
Test set 3842 68 96
: Number of training and test images on three synthesized datasets.
\[tab:data\]
Since images in the MIT-Adobe FiveK and Microsoft COCO datasets only contain certain scenes and styles, we collect a dataset from Flickr with larger diversity such as images containing different scenes or stylized images. To generate input and ground truth pairs, we apply the same color transfer technique described for the Microsoft COCO dataset. However, since there is no semantic information provided in this dataset to search proper reference objects for transfer, we use a pre-trained scene parsing model [@Zhou_corr_2016] to predict semantic pixel-wise labels. We then compute a spatial-pyramid label histogram [@Lazebnik_CVPR_2006] of the target image and retrieve reference images from the ADE20K dataset [@Zhou_corr_2016] with similar histograms computed from the ground truth annotations. Next, we manually segment a region (e.g., an object or a scene) in the target image. Based on the predicted scene parsing labels within the segmented target region, we find a region in the reference image that shares the same labels as the target region. The composite image is then generated by the color transfer method mentioned above (Figure \[fig:data\_coco\_5k\](a)). With the above-mentioned data acquisition methods on three datasets, we are able to collect large-scale and high-quality training and test pairs (see Table \[tab:data\] for a summarization). This enables us to train an end-to-end CNN for image harmonization with several benefits. First, our data collection method ensures that the ground truth images are realistic, so the network can really capture the image realism and adjust the input image according to the learned representations. Another merit of our method is to enable quantitative evaluations. This is, we can use the synthesized composite image to measure errors by comparing to the ground truth images. Although there should be no single best solution for the image harmonization task, this quantitative measurement can give us a sense of how closer the images generated by different methods are, to a truly realistic image (discussed in Section \[sec:exp\]), which is not addressed by previous approaches.
Context-aware Encoder-decoder
-----------------------------
Motivated by the potential of the Context Encoders [@Pathak_CVPR_2016], our CNN learns feature representations of input images via an encoder and reconstruct the harmonized output results through a decoder. While the proposed deep network bears some resemblance, we add novel components for image harmonization. In the following, we present the objective function and proposed network architecture with discussion of novel components. Given a RGB image $I \in \mathbb{R}^{H \times W \times 3}$ and a provided binary mask $M \in \mathbb{R}^{H \times W \times 1}$ of the composite foreground region, we form the input $X \in \mathbb{R}^{H \times W \times 4}$ by concatenating $I$ and $M$, where $H$ and $W$ are image dimensions. Our objective is to predict an output image $\hat{Y} = \mathcal{F}(X)$ that optimizes the reconstruction ($L2$) loss with respect to the ground truth image $Y$: $$\mathcal{L}_{rec}(X) = \frac{1}{2}\sum_{h,w}\parallel Y_{h,w} - \hat{Y}_{h,w} \parallel_2^2.
\label{eq:l2}$$ Since the $L2$ loss is optimized with the mean of the data distribution, the results are often blurry and thus miss important details and textures from the input image. To overcome these problems, we show that adding skip links from the encoder to the decoder can recover those image details in the proposed network.
Figure \[fig:network\] shows basic components of our network architecture with an encoder and a harmonization decoder. The encoder is a series of convolutional layers and a fully connected layer to learn feature representations from low-level image details to high-level context information. Note that as we do not have any pooling layers, fine details are preserved in the encoder [@Pathak_CVPR_2016]. The decoder is a series of deconvolutional layers which aim to reconstruct the image via up-sampling from the representations learned in the encoder and simultaneously adjust the appearances of the foreground region.
However, image details and textures may be lost during the compression process in the encoder, and thus there is less information to reconstruct the contents of the input image. To retain those details, it is crucial that we add a skip link from each convolutional layer in the encoder to each corresponding deconvolutional layer in the decoder. We show this method is effectively useful without adding additional burdens for training the network. Furthermore, it can alleviate the problem of the $L2$ loss that prefers a blurry image solution. We implement the proposed network in Caffe [@jia2014caffe] and use the stochastic gradient descent solver for optimization with a fixed learning rate $10^{-8}$. In addition, we compute the loss on the entire image rather than the foreground mask to account for the reconstruction differences in the background region. We also try a weighted loss that considers the foreground region more importantly, but the results are similar and thus we use a simple loss function. Since the entire network is trained from scratch, we use the batch normalization [@Ioffe_ICML_2015] followed by a scaling layer and an ELU layer [@Clevert_ICLR_2016] after each convolutional and deconvolutional layers to facilitate the training process. We conduct experiments using the proposed network architecture with different input sizes. Interestingly, we find that the one with larger input size performs better in practice, and thus we use input resolution of $512 \times 512$. This observation also matches our intuition when designing the encoder-decoder architecture with skip links, where the network can learn more context information and details from a larger input image. To generate higher resolution results, we can up-sample the output of the network with joint bilateral filtering [@Petschnigg_SIGGRAPH_2004], in which the input composite image is used as the guidance to keep clear details and sharp textures.
{width="1.0\linewidth"}
Joint Training with Semantics
-----------------------------
In the previous section, we propose an encoder-decoder network architecture for image harmonization. In order to further improve harmonization results, it is natural to consider semantics of the composite foreground region. The ensuing question is how to incorporate such semantics in our CNN, so that the entire network is still end-to-end trainable. In this section, we propose a modified network that can jointly train the image harmonization and scene parsing tasks simultaneously, while propagating semantics to improve harmonization results. The overall architecture is depicted in Figure \[fig:network\], which adds the scene parsing decoder branch. In addition to the reconstruction loss described for image harmonization in , we introduce a pixel-wise cross-entropy loss with the standard softmax function $\mathbb{E}$ for scene parsing: $$\mathcal{L}_{cro}(X) = -\sum_{h,w} \log (\mathbb{E}(X_{h,w};\theta)).
\label{eq:softmax}$$ We then define a combined loss for both tasks and optimize it jointly: $$\mathcal{L} = \lambda_{1} \mathcal{L}_{rec} + \lambda_{2}\mathcal{L}_{cro},
\label{eq:combine}$$ where $\lambda_{i}$ is the weight to control the balance between losses for image harmonization and scene parsing. We design the joint network by inheriting the encoder-decoder architecture described in the previous section. Specifically, we add a decoder to predict scene parsing results, while the encoder is to learn feature representations and is shared for both decoders. To extract semantic knowledge from the scene parsing model and help harmonization process, we concatenate feature maps from each deconvolutional layer of the scene parsing decoder to the harmonization decoder, except for the last layer which focuses on image reconstruction. In addition, skips links [@Long_CVPR_2015] are also connected to the scene parsing decoder to gain more information from the encoder. To enable the training process for the proposed joint network, both the ground truth images for harmonization and scene parsing are required. We then use a subset of the ADE20K dataset [@Zhou_corr_2016], which contains $12080$ training images with the top $25$ frequent labels. Similarly, training pairs for harmonization are obtained in a way described in the data acquisition section via color transfer. To train the joint network, we start with the training data from the ADE20K dataset to obtain an initial solution for both the harmonization and scene parsing by optimizing . We set $\lambda_1 = 1$ and $\lambda_2 = 100$ with a fixed learning rate $10^{-8}$. Next, we fix the scene parsing decoder with $\lambda_2 = 0$ and finetune rest of the network using all the training data introduced in Section \[sec:data\] to achieve the optimal solution for image harmonization. Note that, during this fintuning step, the scene parsing decoder is able to propagate learned semantic information through the links between two decoders. With the incorporated scene parsing model, our network can learn the color distribution of certain semantic categories, e.g., the skin color on human or the sky-like colors. In addition, the learned background semantics can help identify which region to match for better foreground adjustment. During harmonization, it essentially uses these learned semantic priors to improve the realism of output results. Moreover, the incorporation of semantic information through joint training not only helps our image harmonization task, but also can be adopted to benefit other image editing tasks [@Zhang_ECCV_2016; @Pathak_CVPR_2016].
[|c|c|c|c|]{} & MSCOCO & MIT-Adobe & Flickr\
cut-and-paste & 400.5 & 552.5 & 701.6\
Lalonde [@Lalonde_ICCV_2007] & 667.0 & 1207.8 & 2371.0\
Xue [@Xue_siggraph_2012] & 351.6 & 568.3 & 785.1\
Zhu [@Zhu_ICCV_2015] & 322.2 & 360.3 & 475.9\
Ours (w/o semantics) & 80.5 & 168.8 & 491.7\
Ours & **76.1** & **142.8** & **406.8**\
\[tab:mse\]
To validate our scene parsing model, we compare the proposed joint network to a deeplab model [@Chen_ICLR_2015], MSc-COCO-LargeFOV, that has a similar model capacity and size to our model but is initialized from a pre-trained model for semantic segmentation. We evaluate the scene parsing results on the validation set of the ADE20K dataset with the top $25$ frequent labels. The mean intersection-over-union (IoU) accuracy of our joint network is $32.2$, while the MSc-COCO-LargeFOV model achieves IoU as $36.0$. Although our model is not specifically designed for scene parsing and is learned from scratch, it shows that our method performs competitively against a state-of-the-art model for semantic segmentation.
[|c|c|c|c|]{} & MSCOCO & MIT-Adobe & Flickr\
cut-and-paste & 26.3 & 23.9 & 25.9\
Lalonde [@Lalonde_ICCV_2007] & 22.7 & 21.1 & 18.9\
Xue [@Xue_siggraph_2012] & 26.9 & 24.6 & 25.0\
Zhu [@Zhu_ICCV_2015] & 26.9 & 25.8 & 25.4\
Ours (w/o semantics) & 32.2 & 27.5 & 27.2\
Ours & **32.9** & **28.7** & **27.4**\
\[tab:psnr\]
[@c@ @c@ @c@ ]{}
Input & No semantics & With semantics\
![Example results to show the comparison of our network with or without incorporating semantic information. With semantics, our result can recover the skin color and obtain higher PSNR score. []{data-label="fig:semantic"}](figure/synthesize/input/0061_resize.png "fig:"){width="0.32\linewidth"} & ![Example results to show the comparison of our network with or without incorporating semantic information. With semantics, our result can recover the skin color and obtain higher PSNR score. []{data-label="fig:semantic"}](figure/synthesize/ours_no_res/0061_resize.png "fig:"){width="0.32\linewidth"} & ![Example results to show the comparison of our network with or without incorporating semantic information. With semantics, our result can recover the skin color and obtain higher PSNR score. []{data-label="fig:semantic"}](figure/synthesize/ours/0061_resize.png "fig:"){width="0.32\linewidth"}\
18.86 & 28.15 & 33.32\
Experimental Results {#sec:exp}
====================
We present the main results on image harmonization with comparisons to the state-of-the-art methods in this section. More results and analysis can be found in the supplementary material.
We first evaluate the proposed method on our synthesized dataset for quantitative comparisons. Table \[tab:mse\] and \[tab:psnr\] show the results of mean-squared errors (MSE) and PSNR scores between the ground truth and harmonized image. Note that it is the first quantitative evaluation on image harmonization, which reflects how close different results are to realistic images. We show that our joint network consistently achieves better performance compared to the single network without combining scene parsing decoder and other state-of-the-art algorithms [@Lalonde_ICCV_2007; @Xue_siggraph_2012; @Zhu_ICCV_2015] on all three synthesized datasets in terms of MSE and PSNR. In addition, it is also worth noticing that our baseline network without semantics already outperforms other existing methods. In Figure \[fig:synthesize\], we show visual comparisons with respect to PSNR of the harmonization results generated from different methods. Overall, the harmonized images by the proposed methods are more realistic and closer to the ground truth images, with higher PSNR values. In addition, Figure \[fig:semantic\] presents one comparison of our networks with and without incorporating the scene parsing decoder. With semantic understandings, our joint network is able to harmonize foreground regions according to their semantics and produce realistic appearance adjustments, while the one without semantics may generate unsatisfactory results in some cases. To evaluate the effectiveness of the proposed joint network in real scenarios, we create a test set of $52$ real composite images and combine $48$ examples from Xue et al. [@Xue_siggraph_2012], resulting in a total of $100$ high-quality composite images. To cover a variety of real examples, we create composite images including various scenes and stylized images, where the composite foreground region can be an object or a scene.
We follow the same procedure as [@Xue_siggraph_2012; @Zhu_ICCV_2015] to set up a user study on Amazon Mechanical Turk, in which each user sees two randomly selected results at a time and is asked to choose the one that looks more realistic. For sanity checks, we use ground truth images from the synthesized dataset and heavily edited images to create easily distinguishable pairs that are used to filter out bad users. As a result, a total of $225$ subjects participate in this study with a total of $10773$ pairwise results ($10.8$ results for each pair of different methods on average). After obtaining all the pairwise results, we use the Bradley-Terry model (B-T model) [@BradleyTerry; @Lai_CVPR_2016] to calculate the global ranking score for each method.
[@c@ @c@ @c@ ]{} ![Given an input image (a), our network can adjust the foreground region according to the provided mask (b) and produce the output (c). In this example, we invert the mask from the one in the first row to the one in the second row, and generate harmonization results that account for different context and semantic information. []{data-label="fig:guidance"}](figure/guidance/input3.png "fig:"){width="0.32\linewidth"} & ![Given an input image (a), our network can adjust the foreground region according to the provided mask (b) and produce the output (c). In this example, we invert the mask from the one in the first row to the one in the second row, and generate harmonization results that account for different context and semantic information. []{data-label="fig:guidance"}](figure/guidance/mask3.png "fig:"){width="0.32\linewidth"} & ![Given an input image (a), our network can adjust the foreground region according to the provided mask (b) and produce the output (c). In this example, we invert the mask from the one in the first row to the one in the second row, and generate harmonization results that account for different context and semantic information. []{data-label="fig:guidance"}](figure/guidance/output3.png "fig:"){width="0.32\linewidth"}\
(a) Input & &\
& ![Given an input image (a), our network can adjust the foreground region according to the provided mask (b) and produce the output (c). In this example, we invert the mask from the one in the first row to the one in the second row, and generate harmonization results that account for different context and semantic information. []{data-label="fig:guidance"}](figure/guidance/mask_inv3.png "fig:"){width="0.32\linewidth"} & ![Given an input image (a), our network can adjust the foreground region according to the provided mask (b) and produce the output (c). In this example, we invert the mask from the one in the first row to the one in the second row, and generate harmonization results that account for different context and semantic information. []{data-label="fig:guidance"}](figure/guidance/output_inv3.png "fig:"){width="0.32\linewidth"}\
& (b) Mask & (c) Output\
[|c|c|c|c|]{} Dataset & [@Xue_siggraph_2012] & Our test set & Overall\
cut-and-paste & 1.080 & 1.168 & 1.139\
Lalonde [@Lalonde_ICCV_2007] & 0.557 & 0.067 & 0.297\
Xue [@Xue_siggraph_2012] & 1.130 & 0.885 & 1.002\
Zhu [@Zhu_ICCV_2015] & 0.875 & 0.867 & 0.876\
Ours & **1.237** & **1.568** & **1.424**\
\[tab:real\]
{width="1.0\linewidth"}
Table \[tab:real\] shows that our method achieves the highest B-T score in terms of realism compared to state-of-the-art approaches on both our created test set and examples from [@Xue_siggraph_2012]. Interestingly, our method is the only one that can improve the harmonization result with a significant margin from the input image (by cut-and-paste). Figure \[fig:real\] shows sample harmonized images by the evaluated methods. Overall, our joint network produces realistic output images, which validates the effectiveness of using synthesized data to directly learn how to harmonize composite images from realistic ground truth images. The results from [@Xue_siggraph_2012] may be easily affected by the large appearance difference between the background and foreground regions during matching. For the method [@Zhu_ICCV_2015], it may generate unsatisfactory results due to the errors introduced during realism prediction, which may affect the color optimization step. In contrast, our network adopts a single feed-forward scheme learned from a well-constructed training set, and utilizes semantic information to improve harmonization results. The complete results on the real composite test set are presented in the supplementary material. With the provided foreground mask, our network can learn context and semantic information while transforming the composite image to a realistic output image. Therefore, our method can be applied to any foreground masks containing arbitrary objects, scenes or clutter backgrounds. Figure \[fig:guidance\] illustrates one example, where originally the adjusted foreground region is the *child*. Instead, we can invert the mask and focus on harmonizing the region of *inverted child*. The result shows that our network can produce realistic outputs from different foreground masks. Previous image harmonization methods rely on matching statistics [@Lalonde_ICCV_2007; @Xue_siggraph_2012] or optimizing an adjustment function [@Zhu_ICCV_2015], which usually require longer processing time (more than $10$ seconds with a 3.4GHz Core Xeon CPU) on a $512 \times 512$ test image. In contrast, our proposed CNN is able to harmonize an image in 0.1 seconds with a Titan X GPU and 12GB memory, or $3$ seconds with a CPU.
Concluding Remarks
==================
In this paper, we present a novel network that can capture both the context and semantic information for image harmonization. We demonstrate that our joint network can be trained in an end-to-end manner, where the semantic decoder branch can effectively provide semantics to help harmonization. In addition, to facilitate the training process, we develop an efficient method to collect large-scale and high-quality training pairs. Experimental results show that our method performs favorably on both the synthesized datasets and real composite images against other state-of-the-art algorithms.
|
---
abstract: 'It is known that the charged lepton masses obey to high precision an interesting empirical relation (Koide relation). In turn, the light neutrino masses cannot obey such a relation. We note that if neutrinos acquire their mass via the seesaw mechanism, the empirical mass relation could hold for the masses in the Dirac and/or heavy Majorana mass matrix. Examples for the phenomenological consequences are provided. We furthermore modify the mass relation for light neutrino masses including their Majorana phases, and show that it can be fulfilled in this case as well, with interesting predictions for neutrinoless double beta decay. Finally, we remark that while the relation does not hold for the up- and down-quarks, it may be valid for the $u, d, s$ quarks, and for the $c, b, t$ quarks.'
author:
- Werner Rodejohann
- He Zhang
bibliography:
- 'bib.bib'
title: Extended Empirical Fermion Mass Relation
---
Introduction {#sec:intro}
============
The experimentally measured charged-lepton masses reveal that they obey the following empirical mass relation (Koide relation) [@Koide:1982si; @Koide:1983qe] $$\begin{aligned}
\label{eq:koide-relation}
K_\ell=\frac{m_e+m_\mu+m_\tau}{\left(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau}\right)^2}
\simeq \frac{2}{3} \,\end{aligned}$$ with remarkable precision, i.e., the above relation is correct to ${\cal O}(10^{-5})$. A number of authors have tried to understand Eq. based on possible flavor symmetries and phenomenological conjectures [@Koide:1989jq; @Foot:1994yn; @Koide:1995xk; @Koide:1996yf; @Rivero:2005vj; @Krolikowski:2005kr; @Gerard:2005ad; @Koide:2008tr; @Sumino:2008hu; @Sumino:2008hy; @Sumino:2009bt; @Koide:2010np].
Motivated by the idea of grand unification, one may wonder whether a similar mass relation exists for other fermions. Numerical analysis has shown that neither the up-quark masses, down-quark masses nor the light neutrino masses could satisfy such an empirical relation, even if the renormalization group (RG) running effects are taken into account [@Li:2006et; @Xing:2006vk]. Consider for illustration the charged leptons and ignore the small ratios $m_e/m_\tau$, $\sqrt{m_e/m_\tau}$ and $m_\mu/m_\tau$ in Eq. . One approximately obtains $$\begin{aligned}
\label{eq:Kl}
K_\ell \simeq \frac{1}{ 1+2\sqrt{\displaystyle {m_\mu}/{m_\tau}}}
\, .\end{aligned}$$ By using the pole masses of charged leptons given in Particle Data Group [@Nakamura:2010zzi], one can estimate that $\sqrt{m_\mu/m_\tau} \simeq 0.25$, which roughly yields $K_\ell =
2/3$. Obviously, if the mass of the lightest particle can be neglected, the empirical relation relies on the mass ratio between the two heavier masses. In the extreme case of a highly hierarchical mass spectrum, in which the ratio of the second heaviest and the heaviest mass is very small, one has $K \to 1$. This is the case for the up- and down-quarks, for which the values $K_{\rm up} \simeq
0.89 $ and $K_{\rm down} \simeq 0.75$ apply (both evaluated at $M_Z
= 91.2$ GeV [@Xing:2007fb]). The other extreme limit applies when the mass spectrum is nearly degenerate, in which case $K \to
1/3$. In this sense, the empirical mass relation Eq. appears interestingly exactly in the middle of all its possible values[^1]. One can understand also why light neutrinos cannot obey the empirical relation. If they are quasi-degenerate, $K_\nu \simeq \frac 13$, and if they obey a normal hierarchical spectrum, $$K_\nu \simeq \frac{1}{1 + 2\sqrt[4]{{\mbox{$\Delta m^2_{\odot}$}}/{\mbox{$\Delta m^2_{\rm A}$}}}} \ls 0.55 \, ,$$ where ${\mbox{$\Delta m^2_{\odot}$}}$ and ${\mbox{$\Delta m^2_{\rm A}$}}$ are the solar and atmospheric mass-squared differences.
We now extend the expression in Eq. for arbitrary three masses, $m_x$, $m_y$ and $m_z$, i.e., $$\begin{aligned}
\label{eq:K}
K \equiv \frac{m_x + m_y +
m_z}{\left(\sqrt{m_x}+\sqrt{m_y}+\sqrt{m_z}\right)^2} = \frac{1 +
\epsilon_1 +
\epsilon_2}{\left(1+\sqrt{\epsilon_1}+\sqrt{\epsilon_2}\right)^2}
\, ,\end{aligned}$$ where $\epsilon_1=m_x/m_z$ and $\epsilon_2 = m_y / m_z$. The allowed range of $K$ is shown in Fig. \[fig:fig0\].
![\[fig:fig0\] The allowed range of $K$ with respect to the ratios $\epsilon_1$ and $\epsilon_2$. The red and blue curves on the $\epsilon_1$-$\epsilon_2$ plane indicate $K=1/2$ and $K=2/3$, respectively.](fig0.eps){width="7cm" height="5.6cm"}
We have in particular indicated the values of $\epsilon_1$ and $\epsilon_2$, which give $K = \frac 23$. One can read from the plot that the minimum value $K=1/3$ appears at the position $\epsilon_1=\epsilon_2=1$, while the maximum value $K=1$ is obtained for $\epsilon_1=\epsilon_2=0$. In addition, if $\epsilon_1$ ($\epsilon_2$) is vanishing, a lower bound $K \geq 1/2$ can be achieved for $\epsilon_2=1$ ($\epsilon_1=1$).
As mentioned above, in the quark sector both up-type and down-type quark mass spectra are quite hierarchical, and the $K$-value is generally larger than $2/3$. However, if the quarks are divided in light and heavy quarks, instead of up- and down-like quarks (i.e., according to mass instead of their electric charge or isospin), the empirical relation could be well satisfied. For example, considering the pole masses of the heavy quarks at 1$\sigma$ C.L., the ratio $$\begin{aligned}
\label{eq:koide-heavy}
K_{\rm heavy} =
\frac{m_c+m_b+m_t}{\left(\sqrt{m_c}+\sqrt{m_b}+\sqrt{m_t}\right)^2}\end{aligned}$$ is found to be $0.66<K_{\rm heavy}<0.68$. This is in amazing agreement with Eq. . Due to the non-perturbative nature of quantum chromodynamics at low energies, the pole masses of light quarks are not well defined. Therefore, we make use of the running light quark masses at the scale $M_Z=91.2~{\rm GeV}$, and find that a similar relation for light quarks $$\begin{aligned}
\label{eq:koide-light}
K_{\rm light} =
\frac{m_u+m_d+m_s}{\left(\sqrt{m_u}+\sqrt{m_d}+\sqrt{m_s}\right)^2}\end{aligned}$$ is located in the interval $0.49<K_{\rm light}<0.65$, indicating a small deviation from the exact empirical relation at about 1$\sigma$ C.L.
We will focus on the application of the mass relation to the neutrinos in the rest of the paper. As mentioned above, their mass spectrum is not hierarchical enough to reproduce the empirical relation, and $1/3 \ls K_\nu \ls3/5$ typically holds, regardless of the neutrino mass ordering [@Xing:2006vk]. However, we would like to note here that if neutrinos are Majorana particles their mass presumably originates from the seesaw mechanism [@Minkowski:1977sc; @Yanagida:1979as; @GellMann:1980vs; @Mohapatra:1979ia], i.e., the light neutrino masses appear as a combination of Dirac and Majorana mass terms: $$\begin{aligned}
\label{eq:seesaw}
m_\nu =M_{\rm D} M^{-1}_{\rm R} M^T_{\rm D} \, .\end{aligned}$$ In this case, it is not surprising that the empirical mass relation does not directly apply to neutrino masses. Instead, it is much more natural to assume that the relation is fulfilled in $M_{\rm D}$ and/or $M_{\rm R}$. Therefore, in this paper we will extend the empirical mass relation to the seesaw framework, and study some of the phenomenological consequences resulting from this hypothesis. Another crucial aspect of Majorana neutrinos is the presence of Majorana phases, and we will also modify the empirical relation taking this into account. It is shown that the relation can work in this case as well.
The rest of the paper is organized as follows: In Sec. \[sec:seesaw relation\], we introduce the seesaw extended mass relation, and describe four typical scenarios, in which very distinctive predictions on the light and heavy neutrino mass spectrum can be gained. The phenomenological consequences of these four scenarios are figured out, and the parameter spaces are illustrated. In Sec. \[sec:variation\], we generalize the empirical mass relation by including contributions from CP-violating phases, and present the constraints on the Majorana phases as well as the light neutrino masses. In particular, we present the predictions on the effective mass relevant for neutrinoless double beta decay. Finally, in Sec. \[sec:summary\], we summarize our work and state our conclusions.
Mass relation in the seesaw model {#sec:seesaw relation}
=================================
Without loss of generality, one can always work in a basis in which the right-handed neutrino Majorana mass matrix is diagonal, i.e., $M_{\rm R}={\rm diag}(M_1,M_2,M_3)$ with $M_i$ (for $i=1,2,3$) being the masses of right-handed neutrinos. If lepton mixing stems entirely from the charged lepton sector, as possible in flavor symmetry models, the Dirac mass term $M_{\rm D}$ is diagonal, i.e., $M_{\rm D}={\rm
diag}(D_1,D_2,D_3)$. This is a strong assumption, but very helpful for illustrating the point we wish to make in this note. We will comment later on the general case. The light neutrino masses are simply given by $$\begin{aligned}
\label{eq:mass}
m_i = D^2_i/M_i \, .\end{aligned}$$ We stress again that though the empirical relation is shown to be not compatible with the light neutrino masses, the exact relation may exist in the mass matrices $M_{\rm D}$ and/or $M_{\rm R}$. We will discuss 4 different cases.
- Case I: The empirical relation exist in $M_{\rm R}$ whereas $M_{\rm D}$ is an identity matrix, i.e., $D_i = D_0$. In this limit, the right-handed neutrino masses are proportional to the inverse of the light neutrino masses. A hierarchical mass spectrum of right-handed neutrinos can be achieved if one of the light neutrino masses is extremely small. For example, in the normal hierarchy case $m_1 \ll m_2 \ll m_3 $, one obtains $$\begin{aligned}
\label{eq:KRnor}
K_{\rm R} \simeq \frac{1}{1+2\sqrt{m_1/m_2}+2\sqrt{m_1/m_3}} \, ,\end{aligned}$$ which gives $K_{\rm R} \simeq 2/3$ if $m_1/m_2 \simeq 0.03$. In the inverted hierarchy case one has $m_3 \ll m_1 < m_2 $, and $$\begin{aligned}
\label{eq:KRinv}
K_{\rm R} \simeq \frac{1}{1+4\sqrt{m_3/m_2}} \, ,\end{aligned}$$ can be expected. The empirical relation requires $m_3/m_2 \simeq
0.016$.
We show in the upper plot of Fig. \[fig:fig1\] the dependence of $K_{\rm R}$ on the lightest neutrino mass in case I. In the numerical computations, we make use of the values of neutrino mass-squared differences from a global-fit of current neutrino oscillation experiment [@Schwetz:2008er], and allow them to vary in their $1\sigma$ interval.
![\[fig:fig1\] The dependence of $K_{\rm
R}$ (upper plot) and $K_{\rm D}$ (lower plot) on the lightest neutrino mass in case I (case II). The red lines correspond to the normal mass hierarchy, while the blue lines to the inverted hierarchy. The dotted line corresponds to the exact empirical mass relation $K_{\rm R} (K_{\rm D})=2/3$.](fig1a.eps "fig:"){width="7cm"} ![\[fig:fig1\] The dependence of $K_{\rm
R}$ (upper plot) and $K_{\rm D}$ (lower plot) on the lightest neutrino mass in case I (case II). The red lines correspond to the normal mass hierarchy, while the blue lines to the inverted hierarchy. The dotted line corresponds to the exact empirical mass relation $K_{\rm R} (K_{\rm D})=2/3$.](fig1b.eps "fig:"){width="7cm"}
In addition, the dotted line indicates the exact empirical relation, i.e., $K_{\rm R}=2/3$. One directly finds that the empirical relation in right-handed neutrino masses can be achieved for $m_1\simeq 2.5 \times 10^{-4}~{\rm eV}$ in the normal hierarchy case and $m_3\simeq 8\times 10^{-4}~{\rm eV}$ in the inverted hierarchy case. These numerical results are in good agreement with the analytical results.
- Case II: The empirical relation exists in $M_{\rm D}$ whereas $M_{\rm R}$ is an identity matrix, i.e., $M_i = M_0$. According to Eq. , one has $D_i \sim
\sqrt{m_i}$ indicating that the hierarchy of $D_i$ is milder than the light neutrino masses. Therefore, the empirical relation cannot hold, and this case is then ruled out.
In the lower plot of Fig. \[fig:fig1\], we show the value of $K_{\rm D}$ with respect to the lightest neutrino mass in case II. The maximum value of $K_{\rm D}$ cannot exceed 0.5, indicating that case II is incompatible with the experimental data.
- Case III: Both $M_{\rm D}$ and $M_{\rm R}$ fulfill the empirical relation. In this case, we have in total six free parameters in $M_{\rm D}$ and $M_{\rm R}$, out of which five can be fixed by using Eqs. and . One can freely choose one of the parameters, e.g., $M_1$, since the absolute mass scale of right-handed neutrinos cannot be determined from the empirical relation. The light neutrino masses are constrained by the relation, and we will show in what follows that a nearly degenerate spectrum is unfavorable.
In our numerical analysis of case III, we fix the mass of one right-handed neutrino, namely, we take $M_1 =10^9~{\rm GeV}$ in the normal hierarchy case and $M_3 =10^9~{\rm GeV}$ in the inverted hierarchy case. Then, the other two right-handed neutrino masses can be determined from the empirical relation and Eq. for a given set of light neutrino masses. The allowed ratios between right-handed neutrino masses are shown in Fig. \[fig:fig2\] for both the normal and inverted hierarchies.
![\[fig:fig2\] Ratios of heavy right-handed neutrino masses in case III for normal hierarchy (upper plot) and inverted hierarchy (lower plot). The red and green lines correspond to the allowed ranges of $M_2/M_1$ and $M_3/M_1$, respectively.](fig2a.eps "fig:"){width="7cm"} ![\[fig:fig2\] Ratios of heavy right-handed neutrino masses in case III for normal hierarchy (upper plot) and inverted hierarchy (lower plot). The red and green lines correspond to the allowed ranges of $M_2/M_1$ and $M_3/M_1$, respectively.](fig2b.eps "fig:"){width="7cm"}
One reads from the plots that there exist stringent upper bounds on the light neutrino masses, i.e., the mass of the lightest neutrino cannot exceed $4\times10^{-3}~{\rm eV}$. Furthermore, $M_2$ lies below $M_1$ and $M_3$ in the inverted hierarchy case, whereas, in the normal hierarchy case, there exists a flip between $M_2$ and $M_3$ for $m_1 \sim 1.5\times 10^{-4}~{\rm eV}$. A simple estimate shows that, at the flip point, $M_2/M_1 =M_3/M_1 \simeq 0.015$. We stress that the results for this case do not depend on the chosen value of $M_1$.
- Case IV: The eigenvalues of $M_{\rm D}$ are the same as the charged-lepton masses $m_e$, $m_\mu$ and $m_\tau$, respectively, whereas $M_{\rm R}$ remains unconstrained. Such a special structure of $M_{\rm D}$ could arise in some grand unification theories. In this case, the right-handed neutrino masses can be directly obtained according to Eq. once the light neutrino masses are chosen.
The predicted right-handed neutrino masses in case IV are illustrated in Fig. \[fig:fig3\].
![\[fig:fig3\] Ratios of heavy right-handed neutrino masses in case IV for normal hierarchy (upper plot) and inverted hierarchy (lower plot). The red and green lines correspond to the allow ranges of $M_2/M_1$ and $M_3/M_1$, respectively.](fig3a.eps "fig:"){width="7cm"} ![\[fig:fig3\] Ratios of heavy right-handed neutrino masses in case IV for normal hierarchy (upper plot) and inverted hierarchy (lower plot). The red and green lines correspond to the allow ranges of $M_2/M_1$ and $M_3/M_1$, respectively.](fig3b.eps "fig:"){width="7cm"}
In both the normal and inverted hierarchy cases, the right-handed neutrinos possess a hierarchical spectrum with $M_1 \ll M_2 \ll
M_3$. The mass of lightest right-handed neutrino $M_1$ could be located around the TeV scale well within the scope of current colliders, while the heaviest right-handed neutrino mass $M_3$ is greater than $10^{9} ~{\rm GeV}$.
variation of the empirical relation {#sec:variation}
===================================
As shown above, the pure light neutrino masses cannot fulfill the empirical mass relation. However, due to the Majorana nature of light neutrinos, the deviation of light neutrino masses from the empirical relations might be viewed as the effects of non-vanishing Majorana phases. In this sense, it is worth investigating a variation of the empirical relation, in which Majorana CP-violating phases are included, i.e., $$\begin{aligned}
\label{eq:K-phase}
\tilde K_{\nu} = \left|\frac{m_1+m_2 {\rm e}^{i\phi_1}+m_3 {\rm
e}^{i\phi_2}}{\left(\sqrt{m_1}+\sqrt{m_2 }{\rm
e}^{i\frac{\phi_1}{2}}+\sqrt{m_3 }{\rm
e}^{i\frac{\phi_2}{2}}\right)^2} \right|\, .\end{aligned}$$ Allowing $\phi_1$ and $\phi_2$ to vary between 0 and $2\pi$, $\tilde
K_{\nu} = 2/3$ can be easily achieved, since there are two more free parameters entering the relation. In particular, $\tilde K_{\nu}$ could be larger than 1 or close to zero if there exists strong cancellation in the denominator or nominator, which is quite relevant in the case of a nearly degenerate light neutrino mass spectrum.
The allowed parameter spaces of $\phi_1$ and $\phi_2$ required for $\tilde K_{\nu}=2/3$ are shown in Fig. \[fig:fig4\].
![\[fig:fig4\] The allowed parameter spaces of $\phi_1$ and $\phi_2$ for normal hierarchy (upper plot) and inverted hierarchy (lower plot). The red, green, blue and black lines in the plot correspond to the lightest mass $m_1
(m_3)=0,~0.02~{\rm eV}$, and $0.2~{\rm eV}$, respectively.](fig4a.eps "fig:"){width="7.2cm"} ![\[fig:fig4\] The allowed parameter spaces of $\phi_1$ and $\phi_2$ for normal hierarchy (upper plot) and inverted hierarchy (lower plot). The red, green, blue and black lines in the plot correspond to the lightest mass $m_1
(m_3)=0,~0.02~{\rm eV}$, and $0.2~{\rm eV}$, respectively.](fig4b.eps "fig:"){width="7.2cm"}
We see that the two Majorana phases are constrained by the empirical relation, e.g., $\phi_1=\phi_2=0$ is unfavored in both hierarchies. In the nearly degenerate limit, the correlation between $\phi_1$ and $\phi_2$ is basically the same for normal and inverted ordering, since in the limit $m_1\simeq m_2 \simeq m_3$, $$\begin{aligned}
\label{eq:K-deg}
\tilde K_{\nu} \simeq \left| \frac{1+ {\rm e}^{i\phi_1}+{\rm
e}^{i\phi_2}}{\left(1+ {\rm e}^{i\frac{\phi_1}{2}}+{\rm
e}^{i\frac{\phi_2}{2}}\right)^2} \right|\, ,\end{aligned}$$ holds for both cases.
In case of a normal and hierarchical mass spectrum, $\tilde K_\nu$ turns out to be only sensitive to the phase difference $\phi=\phi_2-\phi_1$, and according to the plot $\phi$ should be very close to $-\pi$. This can be understood from Eq. which, in the limit $m_1\sim 0$, can be reduced to $$\begin{aligned}
\label{eq:K-nor}
\tilde K_{\nu} \simeq \left| \frac{1+r^2 {\rm e}^{i\phi}}{1+2r{\rm
e}^{i\frac{\phi}{2}}+r^2 {\rm e}^{i\phi}} \right|\, ,\end{aligned}$$ where $r=\sqrt{m_2/m_3} \simeq 0.4$, and $\phi=\phi_1-\phi_2$. Taking $\phi=-\pi$, one can estimate that $\tilde K_\nu \simeq 0.7$, roughly in agreement with the empirical relation.
An interesting result is found for the inverted hierarchy ($m_2 \simeq m_1 \gg m_3$). One has no dependence on $\phi_2$, and finds $$\begin{aligned}
\label{eq:K-inv}
\tilde K_{\nu} \simeq \left| \frac{1+ {\rm e}^{i\phi_1}}{1+2{\rm
e}^{i\frac{\phi_1}{2}}+{\rm e}^{i\phi_1}} \right|\, .\end{aligned}$$ This expression gives $\tilde K_{\nu} = \frac 23$ for a phase which corresponds to $\sin^2\phi_1/2 = \frac{21}{25}$.
Since the Majorana nature of neutrinos is aimed to be revealed in future neutrinoless double decay experiments, there is a need to address some comments on the connection between the empirical mass relation and the neutrinoless double beta decay process, in which the decay amplitude is proportional to $$\begin{aligned}
\label{eq:meff}
m_{ee} & = & \left| \sum {V}^2_{ei} m_i\right| \\ & = &
\left||V_{e1}|^2 m_1 + |V_{e2}|^2 {\rm e}^{i \phi_1} m_2 +
|V_{e3}|^2 {\rm e}^{i \phi_2} m_3 \right| \nonumber \, .\end{aligned}$$ Here the leptonic flavor mixing matrix $V$ is given by $V_{e1}=\cos\theta_{12}\cos\theta_{13}$, $V_{e2}=\sin\theta_{12}\cos\theta_{13}$, and $V_{e3}=\sin\theta_{13}$, with $\theta_{ij}$ being the lepton mixing angles. We illustrate in Fig. \[fig:fig5\] the constraints on $m_{ee}$ when $\tilde K_{\nu}=2/3$ is satisfied.
![\[fig:fig5\] The allowed range of the effective mass $m_{ee}$ as a function of the lightest neutrino mass in the normal hierarchy (red line) and inverted hierarchy (green line) at 1$\sigma$ C.L. with $\tilde K_{\nu}=2/3$ being satisfied. The dashed lines correspond to the parameter range without assuming $\tilde K_{\nu}=2/3$.](fig5.eps){width="7cm"}
In the normal hierarchy case, the predicted $m_{ee}$ almost saturates all of the experimentally allowed range for $m_1 \lesssim 1$ eV. However, in the inverted hierarchy case, the allowed range of $m_{ee}$ shrinks a lot if $m_3 \lesssim 0.01~{\rm eV}$. This is a consequence of Eq. (\[eq:K-inv\]), and the value $\sin^2\phi_1/2 =
\frac{21}{25}$ following from the requirement $\tilde K_\nu = \frac
23$. Inserting this phase in the expression for the effective mass, whose general value is $$\begin{aligned}
m_{ee} & = & \nonumber
\cos^2 \theta_{13} \sqrt{{\mbox{$\Delta m^2_{\rm A}$}}} \sqrt{1 - \sin^2 2 \theta_{12} \,
\sin^2 \phi_1/2} \, ,\end{aligned}$$ which gives the remarkable result $$\begin{aligned}
m_{ee} & = & \frac 15 \cos^2 \theta_{13} \sqrt{{\mbox{$\Delta m^2_{\rm A}$}}} \sqrt{25 - 21 \sin^2 2
\theta_{12}} \, .\end{aligned}$$ This fixes the effective mass to about 0.025 eV, with little dependence on the oscillation parameters. One should compare this value with the general lower and upper limits of 0.017 and 0.05 eV, respectively.
conclusion {#sec:summary}
==========
In this work we have added several new points regarding the properties of an empirical mass relation (Koide relation), which frequently is discussed in the literature.
We first extended the empirical mass relation of charged leptons to the other fermion sectors. In particular, we have noted that there could exist an universal empirical relation in the quark sectors once the quarks are classified by their mass scales instead of their electric charge or isospin. We then noted that if light neutrinos acquire their mass via the seesaw mechanism, it is not surprising that they fail to obey the empirical relation. Instead, we applied the relation to the Dirac and/or heavy Majorana mass matrix. We illustrated the consequences for neutrino masses in a simplified seesaw framework by assuming all lepton mixing to stem from the charged lepton sector, and analyzed four typical scenarios realizing the empirical relation in different ways. Furthermore, we generalized the empirical mass relation with Majorana phases being included and found that in case of an inverted hierarchy mass spectrum, the effective mass is strongly constrained. The relevant Majorana phase is fixed and a value of about $m_{ee} \simeq
0.025~{\rm eV}$ exists, being a factor of 1.5 larger than the general lower bound.
Let us remark here that in a more general seesaw framework, there will be much more free parameters entering the expression of $M_{\rm
D}$, and there are in principle enough degrees of freedom to reproduce the empirical relation in both $M_{\rm D}$ and $M_{\rm R}$ for any choice of light neutrino masses. In addition, thermal leptogenesis cannot work in our simplified framework even if flavor effects are taken into account. Note that the empirical mass relation discussed in this work could be re-scaled up to a higher and more fundamental scale, i.e., the grand unification scale, by using RG equations. However, as shown in Ref. [@Xing:2006vk], the fermion mass ratios tend to be very stable against radiative corrections even in the supersymmetric model with a large $\tan\beta$. If there are intermediate scales in the RG evolution, the running fermion masses could be modified by the threshold effects, in particular for neutrinos. Such a study could be done in a model-dependent manner and lies beyond the scope of current work.
This work was supported by the ERC under the Starting Grant MANITOP and the Deutsche Forschungsgemeinschaft in the Transregio 27 “Neutrinos and beyond – weakly interacting particles in physics, astrophysics and cosmology”.
[^1]: For $N$ fermion generations the range would be between 1 and $1/N$, with a central value of $(N+1)/2N$.
|
---
abstract: |
We present stellar evolutionary models covering the mass range from 0.4 to 1 M$_{\odot}$ calculated for metallicities Z=0.020 and 0.001 with the MHD equation of state (Hummer & Mihalas, 1988; Mihalas et al. 1988; Däppen et al. 1988). A parallel calculation using the OPAL (Rogers et al. 1996) equation of state has been made to demonstrate the adequacy of the MHD equation of state in the range of 1.0 to 0.8 M$_{\odot}$ (the lower end of the OPAL tables). Below, down to 0.4 M$_{\odot}$, we have justified the use of the MHD equation of state by theoretical arguments and the findings of Chabrier & Baraffe (1997).
We use the radiative opacities by Iglesias & Rogers (1996), completed with the atomic and molecular opacities by Alexander & Fergusson (1994). We follow the evolution from the Hayashi fully convective configuration up to the red giant tip for the most massive stars, and up to an age of 20 Gyr for the less massive ones. We compare our solar-metallicity models with recent models computed by other groups and with observations.
The present stellar models complete the set of grids computed with the same up-to-date input physics by the Geneva group \[Z=0.020 and 0.001, Schaller et al. (1992), Bernasconi (1996), and Charbonnel et al. (1996); Z=0.008, Schaerer et al. (1992); Z=0.004, Charbonnel et al. (1993); Z=0.040, Schaerer et al. (1993); Z=0.10, Mowlavi et al. (1998); enhanced mass loss rate evolutionary tracks, Meynet et al. (1994)\].
author:
- 'C.Charbonnel'
- 'W.Däppen'
- 'D.Schaerer'
- 'P.A.Bernasconi'
- 'A.Maeder'
- 'G.Meynet'
- 'N.Mowlavi'
date: 'Received, accepted October 23, 1998'
subtitle: 'VIII. From 0.4 to 1.0 ${\rm M_{\odot}}$ at Z=0.020 and Z=0.001, with the MHD equation of state [^1] '
title: 'Grids of stellar models.'
---
\#1\#2\#3[=\#3cm ]{}
Introduction
============
In stellar evolution computations, and in particular in the case of stars more massive than the Sun, it is generally sufficient to use a simple equation of state. The plasma of the stellar interior is treated as a mixture of perfect gases of all species (atoms, ions, nuclei and electrons), and the Saha equation is solved to yield the degrees of ionization or molecular formation. In the case of low mass stars however, non ideal effects, such as Coulomb interactions become important. It is then necessary to use a more adequate equation of state than the one employed in the Geneva code for more massive stars. This simple equation of state essentially contains a mixture of ideal gases, ionization of the chemicals is dealt with by the Saha equation, excited states and molecules are neglected, complete pressure ionization is artificially imposed above certain temperatures and pressures, and no Coulomb-pressure correction is included (see Schaller et al. 1992).
For the present grids of models of 0.4 to 1.0 M$_{\odot}$ stars these assumptions are obviously inadequate. For such stars, the most useful equations of state, as far as their smooth realization and versatility are concerned, are (i) the so-called Mihalas-Hummer-Däppen (MHD) equation of state (Hummer and Mihalas 1988; Mihalas et al. 1988; Däppen et al. 1988), and (ii) the OPAL equation of state, the major alternative approach developed at Livermore (Rogers 1986, and references therein; Rogers et al. 1996). A brief description of these two equations of state is given in the next section.
Here, we chose the MHD equation of state. First, we were able to compute very smooth tables specifically for our cases of chemical composition, instead of relying on pre-computed, relatively coarse tables that would require interpolation in the chemical composition. Second, our choice was forced by the fact that the currently available OPAL equation of state tables do not allow to go below stars less massive than $\sim$0.8 M$_{\odot}$. Third, we validated our choice by a comparative calculation with OPAL at its low-mass end. We found results that are virtually indistinguishable from MHD. Fourth, we examined in a parallel theoretical study (Trampedach & Däppen 1998) the arguments about the validity of the MHD equation of state down to the limit of our calculation of 0.4 M$_{\odot}$ (see below). Therefore we do not have to include a harder excluded-volume term such as the one included in the Saumon-Chabrier (SC) equation of state (Saumon & Chabrier 1991, 1992).
Although the MHD equation of state was originally designed to provide the level populations for opacity calculations of stellar [*envelopes*]{}, the associated [*thermodynamic quantities*]{} of MHD can none the less be reliably used also for stellar cores. This is due to the fact that in the deeper interior the plasma becomes virtually fully ionized. Therefore, in practice, it does not matter that the condition to apply the detailed Hummer-Mihalas (1988) occupation formalism for bound species is not fulfilled, because essentially there are no bound species. Other than that, the MHD equation of state includes the usual Coulomb pressure and electron degeneracy, and can therefore be used for low-mass stars and, in principle, even for envelopes of white dwarfs (W. Stolzmann, [*private communication*]{}). The present paper, with its MHD-OPAL comparison (see §3) corroborates this assertion.
This broad applicability of the MHD equation of state for entire stars was specifically demonstrated by its successes in solar modeling and helioseismology (Christensen-Dalsgaard et al. 1988, Charbonnel & Lebreton 1993, Richard et al. 1996, Christensen-Dalsgaard et al. 1996). A solar model that is based on the MHD equation of state from the surface to the center is in all respects very similar to one based on the OPAL equation of state, the major alternative approach developed at Livermore (Rogers 1986, and references therein; Rogers et al. 1996). This similarity even pertains to the theoretical oscillation frequencies that are used in comparisons with the observed helioseismic data.
Although the difference between the MHD and OPAL equations of state is of helioseismological relevance, it has no importance for the lower-mass stellar modeling of the present analysis. This is explicitly validated in the present paper. For the much finer helioseismological analyses, it turned out that in some respect the OPAL model seems to be closer than the MHD model to the one inferred from helioseismological observations (Christensen-Dalsgaard et al. 1996, Basu & Christensen-Dalsgaard 1997). However, we stress that for the present stellar modelling these subtle differences are no compelling reason to abandon the convenience of our ability to compute MHD equation of state tables ourselves, and to go below the range of the available OPAL tables ($\sim$0.8 M$_{\odot}$).
Not only helioseismology, but also fine features in the Hertzsprung-Russell diagram of low- and very low-mass stars impose strong constraints on stellar models (Lebreton & Däppen 1988, D’Antona & Mazzitelli 1994, 1996, Baraffe et al. 1995, Saumon et al. 1995). They have all confirmed the validity of the principal equation-of-state ingredients employed in MHD (Coulomb pressure, partial degeneracy of electrons, pressure ionization). Finally, we have checked that even at the low-mass end of our calculations the physical mechanism for pressure ionization in the MHD equation of state is still achieved by the primary pressure ionization effect of MHD (the reduction of bound-state occupation probabilities due to the electrical microfield; see Hummer & Mihalas 1988). Such a verification was necessary to be sure that our results are not contaminated by the secondary, artificial pressure-ionization device included in MHD for the very low-temperature high-density regime (the so-called $\Psi$ term of Mihalas et al. 1988). A parallel calculation has confirmed that in our models a contamination by this $\Psi$ term can be ruled out (Trampedach & Däppen 1998).
In the present paper, we expand the current mass range of the Geneva evolution models from 0.8 down to 0.4 M$_{\odot}$, by using a specifically calculated set of tables of the MHD equation of state. This work aims to complete the base of extensive grids of stellar models computed by the Geneva group with up-to-date input physics \[Z=0.020 and 0.001, Schaller et al. (1992), Bernasconi (1996), and Charbonnel et al. (1996); Z=0.008, Schaerer et al. (1992); Z=0.004, Charbonnel et al. (1993); Z=0.040, Schaerer et al. (1993); Z=0.10, Mowlavi et al. (1998); enhanced mass loss rate evolutionary tracks, Meynet et al. (1994) \]. In Sect. 2, we present the characteristics of our equation of state and recall the physical ingredients used in our computations. In Sect. 3, we summarize the main characteristics of the present models and discuss the influence of the equation of state on the properties of low mass stars. Finally, we compare our solar-metallicity models with recent models computed by other groups and with observations in §4.
Input physics
=============
The basic physical ingredients used for the complete set of grids of the Geneva group are extensively described in previous papers (see Schaller et al. 1992, hereafter Paper I). With the exception of the equation of state, described in detail in the following subsection, we therefore just mention the main points.
Equation of state
-----------------
We have already justified our choice of the MHD equation of state in the introduction. As mentioned there, MHD is one of the two recent equations of state that have been especially successful in modeling the Sun under the strong constraint of helioseismological data. Historically, the MHD equation of state was developed as part of the international “Opacity Project” \[OP, see Seaton (1987, 1992)\]. It was realized in the so-called [*chemical picture*]{}, where plasma interactions are treated with modifications of atomic states, [*i.e.*]{} the quantum mechanical problem is solved before statistical mechanics is applied. It is based on the so-called free-energy minimization method. This method uses approximate statistical mechanical models (for example the nonrelativistic electron gas, Debye-Hückel theory for ionic species, hard-core atoms to simulate pressure ionization via configurational terms, quantum mechanical models of atoms in perturbed fields, etc). From these models a macroscopic free energy is constructed as a function of temperature $T$, volume $V$, and the concentrations $N_1, \ldots, N_m$ of the $m$ components of the plasma. The free energy is minimized subject to the stoichiometric constraint. The solution of this minimum problem then gives both the equilibrium concentrations and, if inserted in the free energy and its derivatives, the equation of state and the thermodynamic quantities.
The other of these two equations of state is the one underlying the OPAL opacity project (see §2.2). The OPAL equation of state is realized in the so-called [*physical picture*]{}. It starts out from the grand canonical ensemble of a system of the basic constituents (electrons and nuclei), interacting through the Coulomb potential. Configurations corresponding to bound combinations of electrons and nuclei, such as ions, atoms, and molecules, arise in this ensemble naturally as terms in cluster expansions. Any effects of the plasma environment on the internal states are obtained directly from the statistical-mechanical analysis, rather than by assertion as in the chemical picture.
More specifically, in the chemical picture, perturbed atoms must be introduced on a more-or-less [*ad-hoc*]{} basis to avoid the familiar divergence of internal partition functions (see [*e.g.*]{} Ebeling et al. 1976). In other words, the approximation of unperturbed atoms precludes the application of standard statistical mechanics, [*i.e.*]{} the attribution of a Boltzmann-factor to each atomic state. The conventional remedy of the chemical picture against this is a modification of the atomic states, [*e.g.*]{} by cutting off the highly excited states in function of density and temperature of the plasma. Such cut-offs, however, have in general dire consequences due to the discrete nature of the atomic spectrum, [*i.e.*]{} jumps in the number of excited states (and thus in the partition functions and in the free energy) despite smoothly varying external parameters (temperature and density). However, the occupation probability formalism employed by the MHD equation of state avoids these jumps and delivers very smooth thermodynamic quantities. Specifically, the essence of the MHD equation of state is the Hummer-Mihalas (1988) occupation probability formalism, which describes the reduced availability of bound systems immersed in a plasma. Perturbations by charged and neutral particles are taken into account. The neutral contribution is evaluated in a first-order approximation, which is good for stars in which most of the ionization in the interior is achieved by temperature \[the aforementioned study (Trampedach & Däppen 1998) has verified the validity of this assumption down to the lowest mass of our calculation\]. For colder objects (brown dwarfs, giant planets), higher-order excluded-volume effects become very important (Saumon & Chabrier 1991, 1992; Saumon et al. 1995). In the common domain of application of the Saumon et al. (1995) and MHD equations of state, Chabrier & Baraffe (1997) showed that both developments yield very similar results, which strongly validates the use of the MHD equation of state for our mass range of 0.4 to 1.0 M$_{\odot}$.
Despite undeniable advantages of the physical picture, the chemical picture approach leads to smoother thermodynamic quantities, because they can be written as analytical (albeit complicated) expressions of temperature, density and particle abundances. In contrast, the physical picture is normally realized with the unwieldy chemical potential as independent variable, from which density and number abundance follow as dependent quantities. The physical-picture approach involves therefore a numerical inversion before the thermodynamic quantities can be expressed in their “natural” variables temperature, density and particle numbers. This increases computing time greatly, and that is the reason why so far only a limited number of OPAL tables have been produced, only suitable for stars more massive than $\sim$0.8 M$_{\odot}$. Therefore we chose MHD for its smoothness, availability, and the possibility to customize it directly for our calculation, despite the – in principle – sounder conceptual foundation of OPAL.
Opacity tables, treatment of convection, atmosphere and mass loss
-----------------------------------------------------------------
- The OPAL radiative opacities from Iglesias & Rogers (1996) including the spin-orbit interactions in Fe and relative metal abundances based on Grevesse & Noels (1993) are used. These tables are completed at low temperatures below 10000 K with the atomic and molecular opacities by Alexander & Fergusson (1994).
- We use a value of 1.6 for the mixing length parameter $\alpha$. Various observational comparison support this choice. $\alpha = 1.6 \pm 0.1$ leads to the best fit of the red giant branch for a wide range of clusters (see Paper I). It is also the value we get for the calibration of solar models including the same input physics (Richard et al. 1996).
- A grey atmosphere in the Eddington approximation is adopted as boundary condition. Below $\tau = 2/3$, full integration of the structure equations is performed. We discuss the implications of such an approximation in §4.
- Evolution on the pre-main sequence and on the main sequence are calculated at constant mass. On the red giant branch, we take mass loss into account by using the expression by Reimers (1975) : $\dot{M} = 4 \times 10^{-13} \eta L R / M$ (in M$_{\odot}$yr$^{-1}$) where L, M and R are the stellar luminosity, mass and radius respectively (in solar units). At solar metallicity, $\eta=$0.5 is chosen (see Maeder & Meynet 1989). At Z=0.001, the mass loss is lowered by a factor (0.001/0.020)$^{0.5}$ with respect to the models at Z=0.020 for the same stellar parameters.
Nuclear reactions
-----------------
- Nuclear reaction rates are due to Caughlan & Fowler (1988). The screening factors are included according to the prescription of Graboske et al. (1973).
- Deuterium is destroyed on the pre-main sequence at temperatures higher than 10$^6$ K by D(p,$\gamma)^3$He and, to a lower extent, by D(D,p)$^3$H(e$^- \nu)^3$He and D(D,n)$^3$He. We take into account these three reactions. In order to avoid the follow-up of tritium, we consider the last two reactions as a single process, D(D,nucl)$^3$He. The $\beta$ desintegration is considered as instantaneous, which is justified in view of the lifetime of tritium ($\tau _{1/2}$=12.26 yr) compared with the evolutionary timescale. The rate of the D(D,nucl)$^3$He reaction is written as $$<DD>_{nucl} = (1 + {{<DD>_p}\over{<DD>_n}} ) <DD>_n$$ where we take for ${{<DD>_p}\over{<DD>_n}}$ a mean value of 1.065, in agreement with the rates given by Caughlan & Fowler (1988). The corresponding mean branch ratios are I$_p$ = 0.5157 and I$_n$ = 0.4843.
Initial abundances
------------------
- The initial helium content is determined by Y=0.24+($\Delta$Y/$\Delta$Z)Z, where 0.24 corresponds to the current value of the cosmological helium (Audouze 1987). We use the value of 3 for the average relative ratio of helium to metal enrichment ($\Delta$Y/$\Delta$Z) during galactic evolution. This leads to (Y,Z) = (0.300,0.020) and (0.243,0.001). In addition, computations were also performed with (Y,Z) = (0.280,0.020).
- The relative ratios for the heavy elements correspond to the mixture by Grevesse & Noels (1993) used in the opacity computations by Iglesias & Rogers (1996).
- Choosing initial abundance values for D and $^3$He is more complex. Pre-solar abundances for both elements have been reviewed in Geiss (1993), however galactic chemical models face serious problems to describe their evolution (see e.g. Tosi 1996), and no reliable prescription exists to extrapolate their values in time. Deuterium is only destroyed by stellar processing since the Big Bang Nucleosynthesis, so that its abundance decreases with time (i.e. with increasing metallicity). On the other hand, the actual contribution of stars of different masses is still subject to a large debate (Hogan 1995, Charbonnel 1995, Charbonnel & Dias 1998), and observations of ($^3$He/H) in the proto-solar nebulae (Geiss 1993) and in galactic HII regions present a large dispersion. We adopt the same (D/H) and ($^3$He/H) initial values for both metallicities, namely 2.4$\times 10^{-5}$ and 2.0 $\times 10^{-5}$ respectively.
Initial models
--------------
At the low mass range considered in this paper, the observed quasi-static contraction begins rather close to the theoretical deuterium main sequence, once the stars emerge from their parental dense gas and dust. For the present purposes then, we take as starting models polytropic configurations on the Hayashi boundary, neglecting the corrections brought to isochrones and upper tracks by the modern accretion paradigm of star formation (Palla & Stahler 1993, Bernasconi & Maeder 1996). For the mass range considered here, these corrections are likely not to exceed 3$\%$ of the Kelvin-Helmholtz timescale for the pre-main sequence contraction times (Bernasconi 1996). We note, however, that the predicted upper locus for the optical appearance of T Tauri stars in the HR diagram can be as much as half less luminous than the deuterium ignition luminosity on the convective tracks ($\Delta \log L \approx$ 0.3).
Short discussion of the main results
====================================
HR diagram and lifetimes
------------------------
The HR diagrams for pre-main sequence evolution and for the following phases are given in Fig. 1 and 2 respectively for both metallicities. For each stellar mass, Table 1 displays the lifetimes in the contraction phase and in the deuterium- and hydrogen-burning phases. Note that we did not complete the main sequence evolution computations for the less massive stars which have a H-burning phase longer than the age of the universe; for these stars, our last computed model corresponds to an age of 20 Gyr.
- In the stellar mass range we consider, the ignition of deuterium burning (indicated in Fig.1) takes place in a fully convective interior. During pre-main sequence evolution, a radiative core develops. However a proper radiative branch is absent, since these stars maintain a convective envelope all along contraction until the ZAMS has been reached, and further on.
- From Table 1 one can see that the contraction time lasts less than 4 thousandths of the hydrogen-burning stage.
- Due to opacity effects, the entire evolution (pre-main sequence, main sequence and red giant branch) occurs at higher luminosity and effective temperature for a given stellar mass when the initial metallicity is smaller, or when the hydrogen content is lower for the same initial metallicity.
- As a consequence, the contraction phase and the deuterium- and hydrogen-burning phases for a given stellar mass are shorter at lower metallicity (see Table 1), and at lower hydrogen content for the same value of Z.
Influence of the equation of state
----------------------------------
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[ccccc]{}\
& & & &\
\
0.020 & 1.0 & 9059.5 & 8968.1 & 9970.1\
0.020 & 0.8 &22713.3 & 22361.4 & 25151.3\
0.001 & 0.8 &14338.0 & 13986.0 &\
When we first compare the results obtained with the MHD and with the simple Geneva equations of state (see Fig. 3 and 4, and Table 2), we obtain essentially the same results than Lebreton & Däppen (1988). Firstly, the fact that MHD contains ${\rm H}_2$ molecules, and the simple Geneva code does not, is reflected in a shift essentially [*along*]{} the ZAMS. On the other hand, the Coulomb pressure correction, also contained in MHD, causes a slight shift of the ZAMS, clearly visible for higher masses, where there are no hydrogen molecules in the photosphere. This Coulomb effect has been well discussed in the case of helioseismology ([*e.g.*]{} Christensen-Dalsgaard et al. 1996). Conformal to the effect of the MHD equation of state to push the apparent position on the ZAMS upward, it also decreases the lifetime on the ZAMS (see Table 2).
For comparison, we have computed with the OPAL equation of state two 0.8M$_{\odot}$ models (the lowest mass that can be computed with the current OPAL tables), for both metallicities. As can be seen in Fig.3, the corresponding tracks are very close to those obtained with the MHD equation of state, the use of the OPAL equation of state leading to slightly higher effective temperature on the ZAMS. As far as their internal structure is concerned, the models computed with MHD equation of state have slightly deeper convection zones. The main sequence lifetime obtained with the MHD equation of state is slightly higher than the one obtained with the OPAL equation of state (Table 2). The comparison shows that down to 0.8M$_{\odot}$ all is fine with the MHD pressure ionization. As mentioned in the introduction, Trampedach & Däppen (1998) predict a correct functioning of pressure ionization in MHD even for much smaller masses. With the present comparison, we have validated their prediction at least to 0.8M$_{\odot}$.
Comparisons with other sets of models and with observations
===========================================================
We now compare our models with recent models computed by other groups and with various observations.
The strongest approximation in our models lies in the treatment of the atmosphere and of the surface boundary conditions, which are specified by the Eddington approximation. In Figs. 5 and 6 we compare our results with the models of D’Antona & Mazzitelli (1994) and Tout et al. (1996) which also rely on a simple treatment of the boundary conditions. For the mass range considered in our work we obtain predictions very similar to D’Antona & Mazzitelli.
Sophisticated model atmospheres for the computation of very low mass stars have been developed recently (Baraffe et al. 1995, 1998; Brett 1995, Allard et al. 1997). Their use becomes crucial for very low mass stars ($M \la 0.4$ M$_{\odot}$) down to the brown dwarf limit. We refer to Chabrier & Baraffe (1997) and Baraffe et al. (1998) for a discussion of the physical basis of the differences. As can be seen in Fig.5, the Eddington approximation we use results in higher T${\rm eff}$ (from 2 to 5 $\%$ depending on the metallicity) compared to the Chabrier & Baraffe (1997) models at our low mass end; our models also have slightly smaller radius (see Fig.6). In this mass range the predictions agree well with the observations of Popper (1980) and Leggett et al. (1996) and do not allow to disregard one model with respect to the other.
In Fig.7 we show the predicted mass-luminosity relations for the V and K band for different metallicities and ages. The magnitudes were derived from the standard stellar library for evolutionary synthesis of Lejeune et al.(1998), which provides empirically calibrated colours for solar metallicity and a semi-empirical correction for non-solar metallicities. The comparison with the observations of Andersen (1991) and Henry & McCarthy (1993) shows a good agreement with our predictions for both the V and K band (Fig.7). Again a similar agreement is obtained with the models of Brocato et al. (1998) and Baraffe et al. (1998).
From the comparisons in Figs.5 to 8, we conclude (in agreement with Alexander et al. 1997) that for the mass range considered in this work an approximate treatment of the stellar atmosphere leads to a satisphactory agreement between theoretical predictions and observations. Independently ab initio stellar interior and atmosphere models allowing the detailed predictions of all observational properties are of fundamental importance for our understanding of very low mass stars which are out of the scope of the present grids.
Appendix: How to obtain the tables by file-transfer
===================================================
The results of this work and of our previous grids (Papers I to VI) are published by Astronomy and Astrophysics at the Centre de Données Spatiales (CDS at Strasbourg) where the corresponding tables are available in electronic form:
[http://cdsweb.u-strasbg.fr]{}. These data can also be obtained from the Geneva Observatory
[http://obswww.unige.ch/]{} (contact [email protected]).
An ensemble of models were selected to describe each evolutionary track. For each model the tables display the age, actual mass, log $L/L_\odot$, log $T_{eff}$, the surface abundances in mass fraction of H, $^4$He, $^{12}$C, $^{13}$C, $^{14}$N, $^{16}$O, $^{17}$O, $^{18}$O, $^{20}$Ne, $^{22}$Ne, the core mass fraction Qcc, log$(-\dot{M})$ (where $\dot{M}$ is the mass loss rate on the red giant branch), log $\rho _c$ (where $\rho _c$ is the central density), log T$_c$ (where T$_c$ is the central temperature), and the central abundances in mass fraction of the above elements. A detailed description of the models selection and of the tables contents is given in Paper I. We now also provide photometric data in verious systems for all tracks and isochrones (see Schaerer & Lejeune 1998).
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[^1]: Data available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/Abstract.html
|
---
author:
- 'M. Künzer, E. Wirsing'
title: On coefficient valuations of Eisenstein polynomials
---
> [**Abstract**]{}
>
> Let $p\geq 3$ be a prime, let $n > m\geq 1$. Let $\pi_n$ be the norm of $\zeta_{p^n} - 1$ under $C_{p-1}$, so that $\Z_{(p)}[\pi_n]|\Z_{(p)}$ is a purely ramified extension of discrete valuation rings of degree $p^{n-1}$. The minimal polynomial of $\pi_n$ over $\Q(\pi_m)$ is an Eisenstein polynomial; we give lower bounds for its coefficient valuations at $\pi_m$. The function field analogue, as introduced by Carlitz and Hayes, is studied as well.
0.0ex 1.2ex
|
In the highly anisotropic high-temperature superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (2212–BSCCO), a linear array of intrinsic Josephson junctions is formed in the out-of-plane direction due to the almost-insulating layers separating the superconducting copper-oxide bilayers. To investigate conduction across such junctions, mesa structures can be fabricated by lithographic patterning on the surface of almost atomically flat cleaved surfaces of single crystals. Hysteretic multi-branched IV characteristics are observed; each branch corresponds to the expected irreversible characteristics of a junction in the measured array.[@tink]
In principle, the shape of the characteristics can be used to determine the [*c*]{}-axis quasiparticle conductivity and the superconducting energy gap. However, in many circumstances one observes regions in the IV characteristics with negative slope—“backbending”—and even an [*S*]{}-shaped feature. These cannot be described by simple theory, but have been attributed by some authors to nonequilibrium effects and by others to simple heating. In the past, pulsed measurements on 200–500 ns time scales have been used with this geometry in an attempt to circumvent possible problems from sample heating. (See Ref. [@jcf] for corresponding references.)
We have made pulsed four-point IV measurements on a stack of (30 $\mu$m)$^2$ junctions with a system of cryogenically cooled buffer amplifiers with 50 ns time resolution. Square voltage pulses are applied to the sample via a series resistor. For more detail see Ref. [@jcf]. Here we present data for an oxygen-annealed sample (Figs. 1 and 2) and for an as-grown sample from a different batch (Figs. 3 and 4). Both samples have around 30 junctions in the main stack.
Typical IV measurements are shown in Fig. 1. [@os] We identify the drop in voltage with heating—the sample conductivity increases with temperature. Nonequilibrium effects are expected to vary on much shorter timescales. Figure 2 shows a series of IV measurements at various times after the switching-on of different-sized square current pulses. We believe the 50 ns characteristics approximate the intrinsic sample properties up to currents in excess of 20 mA. The 0.3 s characteristics are indistinguishable from dc measurements. The strong time-dependence of the shape of the characteristics shows that the onset of the backbending feature is not associated with the gap voltage. We deduce the temperature on the heated curves by comparison with the 50 ns quasi-intrinsic characteristics at higher base temperatures, e.g., for a current of 15 mA, the sample temperature rises from 33 K to above T$_c$ in $\sim$ 10 $\mu$s.
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Figure 3 shows measurements using high-bias pulse sequences superimposed on a non-zero low-bias level. The increase in low-bias sample voltage following the high-bias pulse is consistent with the mesa cooling back down—the low-bias conductivity decreases steeply from low temperatures towards T$_c$ and varies only slowly around and above T$_c$. Examining the IV data in Figs. 3a–d, respectively 50 $\mu$s and 0.4 ms after the start of the high-bias pulse, we find backbending and an [*S*]{}-shaped feature. In Figs. 3c and 3d, we note the transition from negative to positive slope, most likely associated with sample heating to well above T$_c$; this conclusion is supported by the initial regime of approximately constant voltage after the high-bias pulse, which corresponds to the mesa being heated above T$_c$.
The mesa temperature just after the end of the high-bias pulse can be deduced from the low-bias sample voltage: this is calibrated using measurements made at long times after the high-bias pulse over a range of base temperatures. In Fig. 4, the circle symbols show IV measurements at the end of a 5 $\mu$s high-bias pulse. We label these with the temperature inferred from the low-bias voltage shortly after the end of the high-bias pulse. The values are in good agreement with the temperature inferred from the position of the 5 $\mu$s IV characteristics with respect to the 50 ns characteristics.
These results underline the consistency of invoking a thermal origin for observed backbending and [*S*]{}-shaped features, without positing additional nonequilibrium effects. These measurements and their analysis in terms of thermal models give us confidence in interpreting the short-time IV characteristics, and will enable us to obtain more reliable information on the intrinsic interlayer tunnelling, free from the effects of heating.
This work was supported by the UK EPSRC.
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[9]{} M. Tinkham, [*Introduction to Superconductivity*]{}, 2nd ed. (McGraw–Hill, Singapore,1996). J.C. Fenton, P.J. Thomas, G. Yang, C.E. Gough, Appl. Phys. Lett. [**80**]{} (2002) 2535. The finite rise time and slight overshoot in the current and voltage reflect the “square” pulse shape from the signal generator.
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---
abstract: 'We show that some more results from the literature are particular cases of the so-called ”invariance under twisting” for twisted tensor products of algebras, for instance a result of Beattie–Chen–Zhang that implies the Blattner-Montgomery duality theorem.'
author:
- |
Florin Panaite\
Institute of Mathematics of the Romanian Academy\
PO-Box 1-764, RO-014700 Bucharest, Romania\
e-mail: [email protected]
title: 'More examples of invariance under twisting [^1]'
---
Introduction {#introduction .unnumbered}
============
${\;\;\;\;}$If $A$ and $B$ are (associative unital) algebras and $R:B\ot A\rightarrow A\ot B$ is a linear map satisfying certain axioms (such an $R$ is called a [*twisting map*]{}) then $A\ot B$ becomes an associative unital algebra with a multiplication defined in terms of $R$ and the multiplications of $A$ and $B$; this algebra structure on $A\ot B$ is denoted by $A\ot _RB$ and called the [*twisted tensor product*]{} of $A$ and $B$ afforded by $R$ (cf. [@Cap], [@VanDaele]).
A very general result about twisted tensor products of algebras was obtained in [@jlpvo]. It states that, if $A\ot _RB$ is a twisted tensor product of algebras and on the vector space $A$ we have one more algebra structure denoted by $A'$ and we have also two linear maps $\rho , \lambda :A\rightarrow A\ot B$ satisfying a set of conditions, then one can define a new map $R':B\ot A'\rightarrow A'\ot B$ by a certain formula, this map turns out to be a twisting map and we have an algebra isomorphism $A'\ot _{R'}B\simeq A\ot _RB$. This result was directly inspired by the invariance under twisting of the Hopf smash product (and thus it was called [*invariance under twisting*]{} for twisted tensor products of algebras), but it contains also as particular cases a number of independent and previously unrelated results from Hopf algebra theory, for instance Majid’s theorem stating that the Drinfeld double of a quasitriangular Hopf algebra is isomorphic to an ordinary smash product (cf. [@majid]), a result of Fiore–Steinacker–Wess from [@Fiore03a] concerning a situation where a braided tensor product can be ”unbraided”, and also a result of Fiore from [@fiore] concerning a situation where a smash product can be ”decoupled”.
The aim of this paper is to show that some more results from the literature can be regarded as particular cases of invariance under twisting. Among them is a result from [@bcz] concerning twistings of comodule algebras (which implies the Blattner-Montgomery duality theorem) and a generalization (obtained in [@lgg]) of Majid’s theorem mentioned before, in which quasitriangularity is replaced by a weaker condition, called semiquasitriangularity (a concept introduced in [@gg1]).
Preliminaries
=============
[\[se:1\]]{} ${\;\;\;\;}$ We work over a commutative field $k$. All algebras, linear spaces etc. will be over $k$; unadorned $\ot $ means $\ot_k$. By ”algebra” we always mean an associative unital algebra. We will denote by $\Delta (h)=h_1\ot h_2$ the comultiplication of a Hopf algebra $H$.
We recall from [@Cap], [@VanDaele] that, given two algebras $A$, $B$ and a $k$-linear map $R:B\ot A\rightarrow A\ot B$, with notation $R(b\ot a)=a_R\ot b_R$, for $a\in A$, $b\in B$, satisfying the conditions $a_R\otimes 1_R=a\otimes 1$, $1_R\otimes b_R=1\otimes b$, $(aa')_R\otimes b_R=a_Ra'_r\otimes b_{R_r}$, $a_R\otimes (bb')_R=a_{R_r}\otimes b_rb'_R$, for all $a, a'\in A$ and $b, b'\in B$ (where $r$ is another copy of $R$), if we define on $A\ot B$ a new multiplication, by $(a\ot b)(a'\ot b')=aa'_R\ot b_Rb'$, then this multiplication is associative with unit $1\ot 1$. In this case, the map $R$ is called a [**twisting map**]{} between $A$ and $B$ and the new algebra structure on $A\ot B$ is denoted by $A\ot _RB$ and called the [**twisted tensor product**]{} of $A$ and $B$ afforded by $R$.
([@jlpvo]) \[invtw\] Let $A\ot _RB$ be a twisted tensor product of algebras, and denote the multiplication of $A$ by $a\ot a'\mapsto aa'$. Assume that on the vector space $A$ we have one more algebra structure, denoted by $A'$, with the same unit as $A$ and multiplication denoted by $a\ot
a'\mapsto a*a'$. Assume that we are given two linear maps $\rho ,\lambda :A\rightarrow A\ot B$, with notation $\rho
(a)=a_{(0)}\ot a_{(1)}$ and $\lambda (a)=a_{[0]}\ot a_{[1]}$, such that $\rho $ is an algebra map from $A'$ to $A\ot _RB$, $\lambda
(1)=1\ot 1$ and the following relations hold, for all $a, a'\in A$: $$\begin{aligned}
&&\lambda (aa')=a_{[0]}*(a'_R)_{[0]}\ot (a'_R)_{[1]}(a_{[1]})_R,
\label{4.7} \\
&&a_{(0)_{[0]}}\ot a_{(0)_{[1]}}a_{(1)}=a\ot 1, \label{4.8}\\
&&a_{[0]_{(0)}}\ot a_{[0]_{(1)}}a_{[1]}=a\ot 1. \label{4.9}\end{aligned}$$ Then the map $R':B\ot A'\rightarrow A'\ot B$, $R'(b\ot a)=(a_{(0)_R})_{[0]}\ot
(a_{(0)_R})_{[1]}b_Ra_{(1)}$, is a twisting map and we have an algebra isomorphism $A'\ot _{R'}B\simeq A\ot _RB, \;\;a\ot b\mapsto a_{(0)}\ot
a_{(1)}b$.
Given an algebra $A$, another algebra structure $A'$ on the vector space $A$ (as in Theorem \[invtw\]) may sometimes be obtained by using the following result:
([@jlpvo]) \[prop4.1\] Let $A, B$ be two algebras and $R:B\ot A\rightarrow A\ot B$ a linear map, with notation $R(b\ot a)=a_R\ot b_R$, for all $a\in A$ and $b\in B$. Assume that we are given two linear maps, $\mu :B\ot
A\rightarrow A$, $\mu (b\ot a)=b\cdot a$, and $\rho :A\rightarrow
A\ot B$, $\rho (a)= a_{(0)}\ot a_{(1)}$, and denote $a*a':=a_{(0)}(a_{(1)}\cdot a')$, for all $a, a'\in A$. Assume that the following conditions are satisfied: $$\begin{aligned}
&&\rho (1)=1\ot 1,\;\;\; 1\cdot
a=a, \;\;\; a_{(0)}(a_{(1)}\cdot 1)=a, \label{4.1}\\
&&b\cdot (a*a')=a_{(0)_R}(b_Ra_{(1)}\cdot a'), \label{4.2}\\
&&\rho (a*a')=a_{(0)}a'_{(0)_R}\ot a_{(1)_R}a'_{(1)}, \label{4.3}\end{aligned}$$ for all $a, a'\in A$ and $b\in B$. Then $(A, *, 1)$ is an associative unital algebra.
The examples
============
Twisting comodule algebras
--------------------------
${\;\;\;\;}$Let $H$ be a finite dimensional Hopf algebra and $A$ a right $H$-comodule algebra, with multiplication denoted by $a\ot a'\mapsto aa'$ and comodule structure denoted by $A\rightarrow A\ot H$, $a\mapsto a_{<0>}\ot a_{<1>}$. Let $\nu :H\rightarrow End (A)$ be a convolution invertible linear map, with convolution inverse denoted by $\nu ^{-1}$. For $h\in H$ and $a\in A$, we denote $\nu (h)(a)=a\cdot h\in A$. For $a, a'\in A$ we denote $a*a'=(a\cdot a'_{<1>})a'_{<0>}\in A$. Assume that, for all $a, a'\in A$ and $h\in H$, the following conditions are satisfied: $$\begin{aligned}
&&a\cdot 1_H=a, \;\;\; 1_A\cdot h=\varepsilon (h)1_A, \label{N} \\
&&(a \cdot h_2)_{<0>}\ot (a\cdot h_2)_{<1>}h_1=
a_{<0>}\cdot h_1\ot a_{<1>}h_2, \label{relalpha} \\
&&(a*a')\cdot h=(a\cdot a'_{<1>}h_2)(a'_{<0>}\cdot h_1). \label{relbeta}\end{aligned}$$
Then, by [@bcz], Proposition 2.1, $(A, *, 1_A)$ is also a right $H$-comodule algebra (with the same $H$-comodule structure as for $A$), denoted in what follows by $A_{\nu }$, and moreover $\nu ^{-1}$ satisfies the relations (\[relalpha\]) and (\[relbeta\]) for $A_{\nu }$, that is, for all $a, a'\in A$ and $h\in H$, we have $$\begin{aligned}
&&(\nu ^{-1}(h_2)(a))_{<0>}\ot (\nu ^{-1}(h_2)(a))_{<1>}h_1=
\nu ^{-1}(h_1)(a_{<0>})\ot a_{<1>}h_2, \label{relgamma} \\
&&\nu ^{-1}(h)(aa')=\nu ^{-1}(a'_{<1>}h_2)(a)*\nu ^{-1}(h_1)(a'_{<0>}). \label{reldelta}\end{aligned}$$
([@bcz]) \[dual\] There exists an algebra isomorphism $A_{\nu }\# H^*\simeq A\# H^*$.
We will prove that Theorem \[dual\] is a particular case of Theorem \[invtw\].
We take in Theorem \[invtw\] the algebra $A$ to be the original $H$-comodule algebra $A$, the second algebra structure $A'$ on $A$ to be the comodule algebra $A_{\nu }$, and $B=H^*$. We consider $A\# H^*$ as the twisted tensor product $A\ot _RH^*$, where $R:H^*\ot A\rightarrow A\ot H^*$, $R(\varphi \ot a)=
\varphi _1\cdot a\ot \varphi _2=a_{<0>}\ot \varphi \leftharpoonup a_{<1>}$, for all $\varphi \in H^*$ and $a\in A$, where $\leftharpoonup $ is the right regular action of $H$ on $H^*$. Define the map $\rho :A_{\nu }\rightarrow
A\# H^*$, $\rho (a)=\sum _ia\cdot e_i\# e^i:=a_{(0)}\ot a_{(1)}$, where $\{e_i\}$ and $\{e^i\}$ are dual bases in $H$ and $H^*$. We will prove that $\rho $ is an algebra map. First, by using (\[N\]), it is easy to see that $\rho (1_A)=1_A\# \varepsilon $. We prove that $\rho $ is multiplicative. For $a, a'\in A$, we have: $$\begin{aligned}
\rho (a*a')&=&\sum _i(a*a')\cdot e_i\ot e^i\\
&\overset{(\ref{relbeta})}{=}&\sum _i
(a\cdot a'_{<1>}(e_i)_2)(a'_{<0>}\cdot (e_i)_1)\ot e^i, \end{aligned}$$ which applied on some $h\in H$ on the second component gives $(a\cdot a'_{<1>}h_2)(a'_{<0>}\cdot h_1)$. On the other hand, we have $$\begin{aligned}
\rho (a)\rho (a')&=&\sum _{i, j}(a\cdot e_i\# e^i)(a'\cdot e_j\# e^j)\\
&=&\sum _{i, j} (a\cdot e_i)((e^i)_1\cdot (a'\cdot e_j)\# (e^i)_2e^j),\end{aligned}$$ which applied on some $h\in H$ on the second component gives\
${\;\;\;\;\;\;\;\;\;\;\;}$ $\sum _i (a\cdot e_i)((e^i)_1 (e^i)_2(h_1)\cdot (a'\cdot h_2))$ $$\begin{aligned}
&=&\sum _i (a\cdot e_i)(e^i((a'\cdot h_2)_{<1>}h_1)(a'\cdot h_2)_{<0>})\\
&=&(a\cdot (a'\cdot h_2)_{<1>}h_1)(a'\cdot h_2)_{<0>}\\
&\overset{(\ref{relalpha})}{=}&(a\cdot a'_{<1>}h_2)(a'_{<0>}\cdot h_1), \end{aligned}$$ showing that $\rho $ is indeed multiplicative.
Define now the map $\lambda :A\rightarrow A\ot H^*$, $\lambda (a)=\sum _i
\nu ^{-1}(e_i)(a)\ot e^i:=a_{[0]}\ot a_{[1]}$. First, it is obvious that $\lambda (1_A)=1_A\ot \varepsilon $, because $\nu ^{-1}$ satisfies also the condition (\[N\]). We need to prove now that the relations (\[4.7\]), (\[4.8\]) and (\[4.9\]) are satisfied. It is easy to prove (\[4.8\]) and (\[4.9\]), because $\nu ^{-1}$ is the convolution inverse of $\nu $. We prove now now (\[4.7\]). We have $\lambda (aa')=\sum _i \nu ^{-1}(e_i)(aa')\ot e^i$, which applied on some $h\in H$ on the second component gives $\nu ^{-1}(h)(aa')$. On the other hand, we have $$\begin{aligned}
a_{[0]}*(a'_R)_{[0]}\ot (a'_R)_{[1]}(a_{[1]})_R
&=&a_{[0]}*(a'_{<0>})_{[0]}\ot (a'_{<0>})_{[1]}(a_{[1]}\leftharpoonup
a'_{<1>})\\
&=&\sum _{i, j}\nu ^{-1}(e_i)(a)*\nu ^{-1}(e_j)(a'_{<0>})\ot e^j
(e^i\leftharpoonup a'_{<1>}), \end{aligned}$$ which applied on some $h\in H$ on the second component gives $\nu ^{-1}(a'_{<1>}h_2)(a)*\nu ^{-1}(h_1)(a'_{<0>})$, and this is equal to $\nu ^{-1}(h)(aa')$ because of the relation (\[reldelta\]). Thus, all hypotheses of Theorem \[invtw\] are fulfilled, so we obtain the twisting map $R':H^*\ot A_{\nu }
\rightarrow A_{\nu }\ot H^*$, which looks as follows: $$\begin{aligned}
R'(\varphi \ot a)&=&(a_{(0)_R})_{[0]}\ot (a_{(0)_R})_{[1]}\varphi _Ra_{(1)}\\
&=&a_{(0)_{<0>_{[0]}}}\ot a_{(0)_{<0>_{[1]}}}(\varphi \leftharpoonup
a_{(0)_{<1>}})a_{(1)}\\
&=&\sum _i (a\cdot e_i)_{<0>_{[0]}}\ot (a\cdot e_i)_{<0>_{[1]}}
(\varphi \leftharpoonup (a\cdot e_i)_{<1>})e^i\\
&=&\sum _{i, j}\nu ^{-1}(e_j)((a\cdot e_i)_{<0>})\ot
e^j(\varphi \leftharpoonup (a\cdot e_i)_{<1>})e^i,\end{aligned}$$ which applied on some $h\in H$ on the second component gives\
${\;\;\;\;\;}$ $\sum _i \nu ^{-1}(h_1)((a\cdot e_i)_{<0>})\varphi ((a\cdot e_i)_{<1>}
h_2)e^i(h_3)$ $$\begin{aligned}
&=&\nu ^{-1}(h_1)((a\cdot h_3)_{<0>})\varphi ((a\cdot h_3)_{<1>}h_2)\\
&\overset{(\ref{relalpha})}{=}&\nu ^{-1}(h_1)(a_{<0>}\cdot h_2)
\varphi (a_{<1>}h_3)\\
&=&\nu ^{-1}(h_1)(\nu (h_2)(a_{<0>}))\varphi (a_{<1>}h_3)\\
&=&a_{<0>}\varphi (a_{<1>}h).\end{aligned}$$ Thus, we obtained $R'(\varphi \ot a)=a_{<0>}\ot \varphi \leftharpoonup a_{<1>}$, for all $\varphi \in H^*$ and $a\in A$, that is $R'=R$ and $A_{\nu }\ot _{R'}H^*=A_{\nu }\# H^*$, and so Theorem \[invtw\] provides the algebra isomorphism $A_{\nu }\# H^*\simeq
A\# H^*$, $a\ot \varphi \mapsto a_{(0)}\ot a_{(1)}\varphi=
\sum _i a\cdot e_i\ot e^i\varphi $, which is exactly Theorem \[dual\].
External homogenization
-----------------------
${\;\;\;\;}$Let $H$ be a Hopf algebra and $A$ a right $H$-comodule algebra, with comodule structure denoted by $a\mapsto a_{(0)}\ot a_{(1)}$. We also denote $a_{(0)}\ot a_{(1)}\ot a_{(2)}=a_{(0)_{(0)}}\ot a_{(0)_{(1)}}\ot a_{(1)}=
a_{(0)}\ot a_{(1)_1}\ot a_{(1)_2}$. The external homogenization of $A$, introduced in [@npvo] and denoted by $A[H]$, is an $H$-comodule algebra structure on $A\ot H$, with multiplication $(a\ot h)(a'\ot h')=aa'_{(0)}\ot S(a'_{(1)})ha'_{(2)}h'$. By [@npvo], $A[H]$ is isomorphic as an algebra to the ordinary tensor product $A\ot H$.
We want to obtain this as a consequence of Theorem \[invtw\], actually, we will see that the data in Theorem \[invtw\] lead naturally to the multiplication of $A[H]$. Indeed, we will apply Theorem \[invtw\] to the following data: $A$ is the original comodule algebra we started with, $B=H$, $R$ is the usual flip between $A$ and $H$, $A'=A$ as an algebra, $\rho $ is the comodule structure of $A$ and $\lambda :A\rightarrow A\ot H$ is given by $\lambda (a)=a_{(0)}\ot S(a_{(1)}):=
a_{[0]}\ot a_{[1]}$. It is very easy to see that the hypotheses of Theorem \[invtw\] are fulfilled, so we obtain the twisting map $R':H\ot A\rightarrow A\ot H$ given by $$\begin{aligned}
R'(h\ot a)&=&(a_{(0)})_{[0]}\ot (a_{(0)})_{[1]}h a_{(1)}\\
&=&a_{(0)_{(0)}}\ot S(a_{(0)_{(1)}})ha_{(1)}\\
&=&a_{(0)}\ot S(a_{(1)})ha_{(2)}, \end{aligned}$$ and obviously $A\ot _{R'}H=A[H]$. Thus, as a consequence of Theorem \[invtw\], we obtain the algebra isomorphism from [@npvo]: $A[H]\simeq A\ot H$, $a\ot h\mapsto a_{(0)}\ot a_{(1)}h$.
Doubles of semiquasitriangular Hopf algebras
--------------------------------------------
${\;\;\;\;}$Let $H$ be a finite dimensional Hopf algebra and $r\in H\ot H$ an invertible element, denoted by $r=r^1\ot r^2$, with inverse $r^{-1}=u^1\ot u^2$. Consider the Drinfeld double $D(H)$, which is the tensor product $H^*\ot H$ endowed with the multiplication $(\varphi \ot h)(\varphi '\ot h')=
\varphi (h_1\rightharpoonup \varphi ' \leftharpoonup S^{-1}(h_3))\ot h_2h'$, for all $h, h'\in H$ and $\varphi , \varphi '\in H^*$, where $\rightharpoonup $ and $\leftharpoonup $ are the regular actions of $H$ on $H^*$.
Define the maps $$\begin{aligned}
&&f:D(H)\rightarrow H^*\ot H, \;\;\; f(\varphi \ot h)=\varphi \leftharpoonup
S^{-1}(u^1)\ot u^2h, \\
&&g:H^*\ot H\rightarrow D(H), \;\;\;g(\varphi \ot h)=\varphi \leftharpoonup
S^{-1}(r^1)\ot r^2h.\end{aligned}$$ It is obvious that $f$ and $g$ are linear isomorphisms, inverse to each other, so we can transfer the algebra structure of $D(H)$ to $H^*\ot H$ via these maps. It is natural to ask under what conditions on $r$ this algebra structure on $H^*\ot H$ is a twisted tensor product between $H$ and a certain algebra structure on $H^*$.
We claim that this is the case if $r$ satisfies the following conditions: $$\begin{aligned}
&&\Delta (r^1)\ot r^2={\cal R}^1\ot r^1\ot {\cal R}^2r^2, \label{SQT1} \\
&&r^1\ot \Delta (r^2)={\cal R}^1r^1\ot r^2\ot {\cal R}^2, \label{SQT2} \\
&&{\cal R}^1\ot {\cal R}^2_2r^1\ot {\cal R}^2_1r^2=
{\cal R}^1\ot r^1{\cal R}^2_1\ot r^2{\cal R}^2_2, \label{SQT3}\end{aligned}$$ where ${\cal R}^1\ot {\cal R}^2$ is another copy of $r$. We will obtain this result as a consequence of Theorem \[invtw\], combined with Proposition \[prop4.1\]. Note that the above conditions are part of the axioms of a so-called semiquasitriangular structure (cf. [@gg1]), and that if $r$ satisfies also the other axioms in [@gg1] then it was proved in [@lgg] that $D(H)$ is isomorphic as a Hopf algebra to a Hopf crossed product in the sense of [@gg2].
We take $A=H^*$, with its ordinary algebra structure, $B=H$, and $R:H\ot H^*
\rightarrow H^*\ot H$, $R(h\ot \varphi )=h_1\rightharpoonup \varphi \leftharpoonup
S^{-1}(h_3)\ot h_2$, hence $A\ot _RB=D(H)$. Then define the maps $$\begin{aligned}
&&\mu :H\ot H^*\rightarrow H^*, \;\;\;\mu (h\ot \varphi )=h\cdot
\varphi:=h_1\rightharpoonup \varphi \leftharpoonup S^{-1}(h_2), \\
&&\rho :H^*\rightarrow H^*\ot H, \;\;\;\rho (\varphi )=\varphi _{(0)}
\ot \varphi _{(1)}:=
\varphi \leftharpoonup S^{-1}(r^1)\ot r^2, \\
&&\lambda :H^*\rightarrow H^*\ot H, \;\;\;\lambda (\varphi )=\varphi _{[0]}
\ot \varphi _{[1]}:=\varphi
\leftharpoonup S^{-1}(u^1)\ot u^2.\end{aligned}$$ The corresponding product $*$ on $H^*$ provided by Propositin \[prop4.1\] is given by $$\begin{aligned}
\varphi *\varphi '&=&\varphi _{(0)}(\varphi _{(1)}\cdot \varphi ')\\
&=&(\varphi \leftharpoonup S^{-1}(r^1))(r^2\cdot \varphi ')\\
&=&(\varphi \leftharpoonup S^{-1}(r^1))(r^2_1\rightharpoonup \varphi
' \leftharpoonup S^{-1}(r^2_2)).\end{aligned}$$ We need to prove that the relations (\[4.1\])–(\[4.3\]) hold. We note first that as consequences of (\[SQT1\]) and (\[SQT2\]) we obtain $\varepsilon (r^1)r^2=r^1\varepsilon (r^2)=1=
\varepsilon (u^1)u^2=u^1\varepsilon (u^2)$, hence we have $\rho (\varepsilon )=\lambda (\varepsilon )=\varepsilon \ot 1$ and also we obtain immediately $1\cdot \varphi =\varphi $ and $\varphi _{(0)}(\varphi _{(1)}\cdot \varepsilon )=\varphi $, for all $\varphi \in H^*$, thus (\[4.1\]) holds. We prove now (\[4.2\]). We compute: $$\begin{aligned}
h\cdot (\varphi * \varphi ')&=&h_1\rightharpoonup (\varphi * \varphi ')
\leftharpoonup S^{-1}(h_2)\\
&=&(h_1\rightharpoonup \varphi \leftharpoonup S^{-1}(h_4r^1))
(h_2r^2_1\rightharpoonup \varphi '\leftharpoonup
S^{-1}(h_3r^2_2)),\end{aligned}$$ $$\begin{aligned}
\varphi _{(0)_R}(h_R\varphi _{(1)}\cdot \varphi ')&=&
(\varphi \leftharpoonup S^{-1}(r^1))_R(h_Rr^2\cdot \varphi ')\\
&=&(h_1\rightharpoonup \varphi \leftharpoonup S^{-1}(h_3r^1))
(h_2r^2\cdot \varphi ')\\
&=&(h_1\rightharpoonup \varphi \leftharpoonup S^{-1}(h_4r^1))
(h_2r^2_1\rightharpoonup \varphi '\leftharpoonup
S^{-1}(h_3r^2_2)), \;\;\;q.e.d.\end{aligned}$$ In order to prove (\[4.3\]), we prove first the following relation: $$\begin{aligned}
r^1\ot r^2_1\ot r^2_3{\cal R}^1\ot r^2_2{\cal R}^2=
{\cal R}^1_2r^1\ot r^2_1\ot {\cal R}^1_1r^2_2\ot {\cal R}^2. \label{auxil}\end{aligned}$$ We compute (denoting by $r=\mf R^1\ot \mf R^2=\rho ^1\ot \rho ^2$ two more copies of $r$): $$\begin{aligned}
r^1\ot r^2_1\ot r^2_3{\cal R}^1\ot r^2_2{\cal R}^2
&\overset{(\ref{SQT2})}{=}&\mf R^1r^1\ot r^2\ot \mf R^2_2{\cal R}^1
\ot \mf R^2_1{\cal R}^2\\
&\overset{(\ref{SQT3})}{=}&\mf R^1r^1\ot r^2\ot {\cal R}^1\mf R^2_1
\ot {\cal R}^2\mf R^2_2\\
&\overset{(\ref{SQT2})}{=}&\mf R^1\rho ^1r^1\ot r^2\ot {\cal R}^1\rho ^2
\ot {\cal R}^2\mf R^2, \end{aligned}$$ $$\begin{aligned}
{\cal R}^1_2r^1\ot r^2_1\ot {\cal R}^1_1r^2_2\ot {\cal R}^2
&\overset{(\ref{SQT1})}{=}&
\mf R^1r^1\ot r^2_1\ot {\cal R}^1r^2_2\ot {\cal R}^2\mf R^2\\
&\overset{(\ref{SQT2})}{=}&\mf R^1\rho ^1r^1\ot r^2\ot {\cal R}^1\rho ^2
\ot {\cal R}^2\mf R^2, \end{aligned}$$ and we see that the two terms coincide. Now we prove (\[4.3\]); we compute: $$\begin{aligned}
\rho (\varphi *\varphi ')&=&(\varphi *\varphi ')\leftharpoonup S^{-1}({\cal R}^1)\ot
{\cal R}^2\\
&=&(\varphi \leftharpoonup S^{-1}({\cal R}^1_2r^1))(r^2_1\rightharpoonup
\varphi '\leftharpoonup S^{-1}({\cal R}^1_1r^2_2))\ot {\cal R}^2, \end{aligned}$$ $$\begin{aligned}
\varphi _{(0)}\varphi '_{(0)_R}\ot \varphi _{(1)_R}\varphi '_{(1)}&=&
(\varphi \leftharpoonup S^{-1}(r^1))(\varphi '\leftharpoonup S^{-1}({\cal R}^1))_R
\ot r^2_R{\cal R}^2\\
&=&(\varphi \leftharpoonup S^{-1}(r^1))(r^2_1\rightharpoonup \varphi '\leftharpoonup
S^{-1}(r^2_3{\cal R}^1))\ot r^2_2{\cal R}^2, \end{aligned}$$ and the two terms are equal because of (\[auxil\]).
Thus, we can apply Proposition \[prop4.1\] and we obtain that $(H^*, *, \varepsilon )$ is an associative algebra, which will be denoted in what follows by $\underline{H}^*$.
We will prove now that the hypotheses of Theorem \[invtw\] are fulfilled, for $A'=\underline{H}^*$. Note first that the relations (\[4.1\]) and (\[4.3\]) proved before imply that $\rho $ is an algebra map from $\underline{H}^*$ to $H^*\ot _RH$. We have already seen that $\lambda (\varepsilon )=\varepsilon \ot 1$, so we only have to check the relations (\[4.7\])–(\[4.9\]). To prove (\[4.7\]), we compute (we denote $r^{-1}=U^1\ot U^2=
\mf U^1\ot \mf U^2$ some more copies of $r^{-1}$): $$\begin{aligned}
\lambda (\varphi \varphi ')&=&(\varphi \varphi ')\leftharpoonup S^{-1}(u^1)
\ot u^2\\
&=&(\varphi \leftharpoonup S^{-1}(u^1_2))(\varphi '\leftharpoonup S^{-1}(u^1_1))
\ot u^2\\
&\overset{(\ref{SQT1})}{=}&(\varphi \leftharpoonup S^{-1}(u^1))
(\varphi '\leftharpoonup S^{-1}(U^1))
\ot u^2U^2, \end{aligned}$$ $$\begin{aligned}
\varphi _{[0]}*(\varphi '_R)_{[0]}\ot (\varphi '_R)_{[1]}(\varphi _{[1]})_R
&=&(\varphi \leftharpoonup S^{-1}(u^1))*(\varphi '_R\leftharpoonup
S^{-1}(U^1))\ot U^2u^2_R\\
&=&(\varphi \leftharpoonup S^{-1}(u^1))*
(u^2_1\rightharpoonup \varphi '\leftharpoonup
S^{-1}(U^1u^2_3))\ot U^2u^2_2\\
&=&(\varphi \leftharpoonup S^{-1}(r^1u^1))
(r^2_1u^2_1\rightharpoonup \varphi '\leftharpoonup
S^{-1}(r^2_2U^1u^2_3))\ot U^2u^2_2\\
&\overset{(\ref{SQT2})}{=}&(\varphi \leftharpoonup S^{-1}(r^1u^1\mf U^1))
(r^2_1u^2\rightharpoonup \varphi '\leftharpoonup
S^{-1}(r^2_2U^1\mf U^2_2))\ot U^2\mf U^2_1\\
&\overset{(\ref{SQT3})}{=}&(\varphi \leftharpoonup S^{-1}(r^1u^1\mf U^1))
(r^2_1u^2\rightharpoonup \varphi '\leftharpoonup
S^{-1}(r^2_2\mf U^2_1U^1))\ot \mf U^2_2U^2\\
&\overset{(\ref{SQT2})}{=}&(\varphi \leftharpoonup S^{-1}(r^1\mf U^1))
(\varphi '\leftharpoonup
S^{-1}(r^2\mf U^2_1U^1))\ot \mf U^2_2U^2\\
&\overset{(\ref{SQT2})}{=}&(\varphi \leftharpoonup S^{-1}(u^1))
(\varphi '\leftharpoonup
S^{-1}(U^1))\ot u^2U^2, \end{aligned}$$ and we see that the two terms are equal. The remaining relations (\[4.8\]) and (\[4.9\]) are very easy to prove and are left to the reader. Thus, we can apply Theorem \[invtw\] and we obtain the twisting map $R':H\ot \underline{H}^*\rightarrow \underline{H}^*\ot H$, $$\begin{aligned}
&&R'(h\ot \varphi )=(\varphi _{(0)_R})_{[0]}\ot (\varphi _{(0)_R})_{[1]}
h_R\varphi _{(1)}=h_1\rightharpoonup \varphi \leftharpoonup
S^{-1}(u^1h_3r^1)\ot u^2h_2r^2, \end{aligned}$$ and the algebra isomorphism $\underline{H}^*\ot _{R'}H\simeq
H^*\ot _RH=D(H)$, given by $$\begin{aligned}
&&\varphi \ot h\mapsto \varphi _{(0)}\ot \varphi _{(1)}h=
\varphi \leftharpoonup S^{-1}(r^1)\ot r^2h,\end{aligned}$$ which is exactly the linear isomorphism $g$ defined before. Thus, we have proved that if $r$ satisfies the conditions (\[SQT1\])–(\[SQT3\]) then $D(H)$ is isomorphic as an algebra to a twisted tensor product between $\underline{H}^*$ and $H$.
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[^1]: Research partially supported by the CNCSIS project ”Hopf algebras, cyclic homology and monoidal categories”, contract nr. 560/2009, CNCSIS code $ID_{-}69$.
|
---
abstract: 'We demonstrate a novel dual-beam atom laser formed by outcoupling oppositely polarized components of an $F=1$ spinor Bose-Einstein condensate whose Zeeman sublevel populations have been coherently evolved through spin dynamics. The condensate is formed through all-optical means using a single-beam running-wave dipole trap. We create a condensate in the field-insensitive $m_F=0$ state, and drive coherent spin-mixing evolution through adiabatic compression of the initially weak trap. Such dual beams, number-correlated through the angular momentum-conserving reaction $2m_0\leftrightharpoons m_{+1}+m_{-1}$, have been proposed as tools to explore entanglement and squeezing in Bose-Einstein condensates, and have potential use in precision phase measurements.'
author:
- 'N. Lundblad'
- 'R. J. Thompson'
- 'D. C. Aveline'
- 'L. Maleki'
title: 'Spinor Dynamics-Driven Formation of a Dual-Beam Atom Laser '
---
Since the creation of the first spinor Bose-Einstein condensate (BEC) in 1998[@kettspinor], a considerable body of work has emerged focusing on the properties and dynamics of such condensates, in which the spin degree of freedom has been liberated and the order parameter is vectorial. Metastable states, spin domains, and coreless vortices were observed in sodium[@kettspinor2; @kettspinor3; @kettspinor4], the dynamics of spin mixing in both hyperfine ground states of rubidium have been extensively studied[@chapmanspinor; @hamburg_f2; @hamburg_f1; @hamburg_revivals; @japan_f2], and the coherence of the mixing process has been concretely established[@chapmannature]. Spin mixing has been used to drive the creation of multicomponent condensates at constant temperature[@hamburg_constantT], the spatial magnetization profile of an $F=1$ spinor BEC has been observed[@stamperkurn], and the clock transition has been explored[@kettspinor5]. Considerable theoretical work has been devoted to the understanding of the spin-spin interaction Hamiltonian and the nature of the spinor condensate ground state[@ho; @njp_spinor; @lawpubigelow1998; @japantheory; @chapmantheory; @bigelow1999]. In addition, the spinor condensate has stimulated theoretical proposals regarding schemes to create squeezed and entangled beams of atoms via coherent spin mixing[@duan2000; @pumeystre2000]. Notions of entanglement and squeezing in dual-beam atom lasers are inherently related to the number conservation between the two evolved components, resulting in an Einstein-Podolsky-Rosen relationship between the measured number of each population. In particular, these proposals suggested a scheme using spin-dependent light shifts to create oppositely propagating beams that were each superpositions of $m_F=\pm1$, and explored the correlations between them[@duan2000; @pumeystre2000].
In this article we present observations with an experimental scheme aimed at exploring these ideas: the generation of dual atom laser beams with an inherent number correlation between them due to their spin-mixing origin. The novelty of the scheme lies not in the output coupler (simple magnetic field gradients that tilt the optical potential for the polarizable states) but rather in the correlated nature of the outcoupled populations and the utilization of spinor dynamics to ‘pump’ population into the outcoupled states. In addition, the presence of a true reservoir and the ability to control both the strength of the output coupler and the rate of ‘pumping’ make this a particularly intriguing atom-laser scheme.
In an all-optical experiment, the path to condensation is largely unconcerned with spin, yet once degeneracy (as well as the high densities associated with it) sets in, the spin-spin interactional energy scale becomes relevant. As such, the interparticle interaction can be described by $U({\mathbf r})=\delta(\mathbf{r})(c_0+c_2 \mathbf{F}_1\cdot \mathbf{F}_2)$, where $c_0$ is the traditional scattering length defined as $c_0 = (4\pi\hbar^2/m)(a_0+2a_2)/3$, and $c_2$ is the spin-spin energy scale, which is given by $c_2 = (4\pi\hbar^2/m)(a_2-a_0)/3$[@ho]. Here $a_2$ and $a_0$ are the well-known [$^{87}$Rb]{} scattering lengths in the total spin channels $f=2$ and $f=0$, respectively[@ferro1; @ferro2].
It has been established that the spin-spin interaction of [$^{87}$Rb]{} is ferromagnetic[@chapmanspinor; @hamburg_f2], i.e. $c_2<0$, yet since the energy scale is so small ($c_2=-3.6\times10^{-14}$ Hz cm$^3$) a true ground state of this new Hamiltonian is not observable except in $\mu$G-level magnetically shielded environments[@chapmanspinor]. Nevertheless, a consequence of this interaction at finite field is that spin mixing will occur; in particular, a pure condensate in the field-insensitive $m_F=0$ sublevel will coherently evolve into a superposition condensate through the collision $2m_0\leftrightharpoons m_{+1}+m_{-1}$, which for the condensate atoms is the only allowed spin dynamics[@ho; @chapmanspinor]. This spin mixing should be first observed after a timescale of order $\tau_0=1/(c_2 n)$ at zero field, where $n$ is the mean condensate density; however, the quadratic Zeeman shift is opposed to this evolution, and at a magnetic field where any evolution away from $m_F=0$ is energetically unfavorable we expect no mixing to occur. In addition, despite low fields, if the condensate number is low enough such that the Thomas-Fermi densities force $\tau(B^2)= 1/(c_2 n-\nu_{B^2})$ to be significant compared to the condensate lifetime, mixing will be inaccessible (this restriction is overcome through adiabatic compression, as detailed below). Worth particular mention among the many other investigations of spin dynamics are the observations of number correlation in the evolved components, performed by examining fluctuations in the $\pm 1$ populations of mixed condensates[@chapmanspinor]. Also notable is the observation of number-correlated atom laser beams generated as a result of four-wave mixing in a sodium BEC[@kett4wm].
Our apparatus, similar to that of other groups[@tubingen; @chapman], utilizes a single-beam running-wave dipole trap produced by a focused CO$_2$ laser, which provides at full power a trap depth of approximately 1.6 mK via the DC polarizability of [$^{87}$Rb]{}. We load the dipole trap from an ultra-high vacuum magneto-optical trap (MOT) which is itself loaded by a cold atomic beam provided by an upstream 2-dimensional MOT[@2dmot_amsterdam]. The 2D-MOT exists in a rubidium vapor cell which is differentially pumped from the adjoining science chamber. All 780nm trapping light is provided by a laser system based on a frequency-doubled 1560nm fiber amplifier, described elsewhere[@doubling]. The loading of the dipole trap proceeds according to established technique[@chapman]; we obtain initial populations in the trap of $\sim 2\times10^6$ [$^{87}$Rb]{} atoms at $\sim 120\ \mu K$. The initial trap frequencies (measured via parametric resonance) are approximately 3.2 kHz transversely and 220 Hz longitudinally.
Evaporative cooling proceeds via a programmed rampdown of CO$_2$ laser intensity. We observed the onset of BEC at critical temperatures near 100 nK with around $10^5$ atoms, and typically obtain condensates of $3\times 10^4$ atoms with little or no discernible thermal component. Typical trap frequencies for holding a formed condensate were 175 Hz (transverse) and 12 Hz (longitudinal).
![\[becpalette\] Spinor BEC creation options: a) the default triplet, with repeatable population distribution likely set by initial MOT alignment; b) $m_F=0$ trap, created by selective application of magnetic field gradient along the weak axis of the trap; c) enhanced $m_F=+1$, created via application of a supportive gradient throughout evaporation. Gravity is directed toward the lower right, and the trapping laser is directed toward the upper right. All images are of partially condensed samples at a ballistic expansion time of 17.5 ms. The long axis of the dipole trap is directed toward the upper right. Images are 1.3 mm $\times$ 1 mm. ](fig1a){width="3.375in"}
The spin projections of the condensate are obtained through Stern-Gerlach separation during ballistic expansion. The conveniently located MOT coils are used for this purpose. Absorption imaging is performed on the expanded condensates (or, in the case of the atom lasers, immediately after dipole trap turnoff) using the $F=2\rightarrow F'=3$ transition after optically pumping from the lower ground state.
Fig. \[becpalette\] summarizes the various types of dipole-trap BECs available to us in the current configuration. Typical evaporation yields a condensate with all three components visible; the ratio of these three populations appears to be a constant of the experiment. It has been speculated that this initial population is set by the particular location of the dipole trap within the MOT reservoir during trap loading[@lett].
Application of a magnetic field gradient along the weak axis of the trap during the first few seconds of evaporation preferentially biases out the $m_F=\pm 1$ components, resulting in a BEC solely occupying the field-insensitive $m_F=0$ projection. It should be noted that this process results in nearly the same number of condensed atoms as the gradient-free process, implying a sympathetic cooling process whereby the polarized components remove more than their average share of thermal energy.
Finally, application of a small magnetic field gradient of order a few G/cm in the vertical direction provides a bias for one or the other polarized components. If this supportive gradient is only on for the first few seconds of evaporation, we obtain polarized condensates of number similar to the other options. However, if this gradient is maintained through condensate formation and through ballistic expansion, we observe significant enhancement in condensate number of order 100%; we attribute this to the fact that the trap depth is strongly perturbed by gravity near criticality, and even a gradient small with respect to gravity allows for much more efficient near-critical evaporation.
For the observations reported here, we begin with a nominally pure $m_F=0$ condensate held in a trap whose unperturbed depth is 5 $\mu$K and is approximately a factor of 10 weaker due to gravitational tilt. Typical condensate densities at this point are in the range $4-6\times 10^{13}$ cm$^{-3}$, and even given up to two seconds of observation spin mixing does not occur. We fix background field levels at 60 mG as determined by RF spectroscopy. To coherently mix the condensate we adiabatically compress the trapping field by raising the laser power from 100 mW to 700 mW over 100 ms, holding the compressed condensate for a variable time, adiabatically expanding, and finally ballistically expanding while applying the Stern-Gerlach field, as summarized by Fig. \[nomixing\]. We observe that the fraction of atoms evolved into the polarized projections increases with high-density hold time and eventually reaches a static level of 50%, as expected given previous experiments and theoretical prediction[@chapmanspinor; @njp_spinor]. Oscillations of spin populations are obscured by imaging noise and considerable shot-to-shot number variance. We also observe that the time taken to reach this steady state varies linearly with density, with a possible offset given by the density at which the quadratic Zeeman effect dominates the dynamics. Densities are calculated in the Thomas-Fermi picture wherein the peak condensate density varies as $n_p\propto N^{2/5}U_0^{3/5}$, where $U_0$ is calculated from the trap frequencies alone and is only slightly perturbed by the strong gravitational tilt which so strongly affects evaporation. We observe strong losses at high ($>$ 2 W) compression assumed to be caused by three-body recombination; the loss rates are consistent with projected values of the Thomas-Fermi densities and the measured value of the three-body rate constant $K_{3c}$[@burt].
![\[nomixing\] The effect of adiabatic compression on the spin mixing process; the top row shows condensates held for equivalent durations without compression. a) 100 ms of hold time at 700 mW b) 400 ms of hold time c) 1.2 s of hold time. The slightly fewer overall number in (c) is due to condensate lifetime. The ballistic expansion time for all images is 17.5 ms. Images are 1 mm $\times$ .8 mm. ](fig2){width="3.375in"}
![\[twinbeam\] A typical outcoupling run of the spinor dynamics-driven dual beam atom laser. a) 0 ms: the full condensate, *in situ*. b)+ 20 ms: immediately after outcoupling. The $m_F=-1$ component immediately passes beyond the reach of the dipole trap and experiences ballistic flight and mean-field expansion. The $m_F=+1$ component remains confined in an effective guide and travels in the opposite direction. c) +25 ms: the $m_F=-1$ beam continues to propagate while the $m_F=+1$ beam is turned around and returned toward the origin. d) +45 ms: the $m_F=+1$ beam now falls freely and experiences mean-field expansion, like the $m_F=-1$ component before it. Note a slightly different path than $m_F=-1$. e) +50 ms: continued $m_F=+1$ propagation; note the $m_F=-1$ component has traveled out of the field of view by this point. Images are 1 mm $\times$ .25 mm; gravity is directed toward the lower right and the trapping laser is directed toward the upper right. ](fig3){width="3.375in"}
After compression-driven spin mixing, we implement magnetic outcoupling, similar in principle to the all-optical $m_F=0$ atom laser reported in 2003[@tubingen] but applied in a state-selective fashion whereby applied field gradients nonadiabatically distort the trapping potential and provide a velocity kick for spin-polarized atoms to escape. Fig. \[twinbeam\] illustrates a typical run; here, a magnetic field gradient dominantly directed along the weak axis of the optical trap (toward the upper right) is turned on quickly (over several ms) at a variable delay (5-50 ms) before the optical trap is turned off. In this case ballistic expansion is limited to a minimum time of 100 $\mu$s (nonzero so as to allow for optical pumping), and is thus effectively *in situ*.
Using slightly different velocity kicks we observe several variants of the dual-beam atom laser. Most commonly, we observe immediate outcoupling and ballistic flight of the $m_F=-1$ component while the $m_F=+1$ component first propagates in the opposite direction (as expected), reverses its motion, passes through the parent $m_F=0$ condensate, and finally escapes, as depicted in Fig. \[twinbeam\]. With greater velocity kicks, we also observe the more intuitive case of both polarized components escaping into ballistic flight from opposite ends of the cigar-shaped trap. Rarely, we observe partial exit of an evolved fraction due to insufficient magnetic field tilt compared to the condensate chemical potential $\mu$, which illustrates the fine control of output coupling possible using this scheme. Deviation from horizontal of the trap itself is also assumed to bias the exit paths of the evolved components.
The shape of the outcoupled fraction demands close inspection. To begin with, the transverse atom laser density profile obtained by gravity-directed beams[@bloch_laser1999; @tubingen] is not to be expected here, as we outcouple over timescales short enough such that the combination of gravity and local gradients do not move the outcoupled fraction appreciably over the time that the trap height $U_0$ tilts significantly with respect to $\mu$. This short outcoupling time is helpful in making sure that the broad resulting clouds are visible. We do observe a downward-directed atom laser in the case of the aforementioned ‘supported’ condensate, depicted in Fig. \[downlaser\], outcoupled simply by removing the support in the last few ms before ballistic expansion. This effect is unrelated to the physics of spin mixing or number-correlated dual-beam atom lasers and is essentially analogous to the Tübingen experiment[@tubingen].
The immediate broadening of the outcoupled spinor atom laser pulse compared to the downward-directed one is perhaps counterintuitive, but can be explained in terms of the nature of the condensate and the path it takes via the long axis of the trap, rather than the more traditional transverse outcoupling. The horizontally outcoupled beam experiences preferred mean-field expansion perpendicular to the direction of travel, and thus does not exhibit the tight collimation characteristic of our downward-directed $m_F=+1$ pulse. We have duplicated this behavior using simple simulations combining center-of-mass motion and standard mean-field expansion theory.
![\[downlaser\] Downward outcoupling of the supported $m_F=+1$ condensate (cf. Fig. \[becpalette\]c), generated by removing the supportive gradient that had preferentially created a polarized condensate. This is shown to illustrate the difference in outcoupled beam collimation between the case where mean-field expansion is along the direction of propagation and the case of the spinor dynamics-driven atom laser in Fig. \[twinbeam\], where expansion is perpendicular to travel. Images are 1.2 mm $\times$ .75 mm.](fig4){width="3.375in"}
Since not all of the initial reservoir is used up in the creation of the dual-beam atom laser, in principle our compression and outcoupling process could be repeated as long as a population of $m_F=0$ atoms remained. The current experiment does not produce large condensate numbers, but several groups have produced large-N spinor condensates showing that such quasi-continuous generation of spinor dynamics-driven atom laser pairs is achievable.
The impetus behind this work is the set of proposals that seek to create squeezing and entanglement in oppositely-propagating atom laser beams. Future work will explore the nature of correlations and entanglement in these beams, the possibility of spin-independent outcoupling as in the 2000 proposals, and also explore the possibilities of improving this process into the quasi-continuous regime. Also of interest is the possibility of creating undisturbed number states, which would be useful for Heisenberg-limited precision phase measurements[@kasevichBEC].
We acknowledge helpful conversation with Eric Burt, Nan Yu, James Kohel, and Kenneth Libbrecht. The research described here was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.
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abstract: 'In order to obtain the equation of state and construct hybrid stars, we calculate the thermodynamic potential in the two-flavor Nambu–Jona-Lasinio model with tensor-type four-point interaction between quarks. In addition, we impose the $\beta$ equilibrium and charge neutrality conditions on the system. We show that the tensor condensate appears at large chemical potential, however, it is difficult to hold hybrid stars with two-solar mass by using the equation of state with the tensor interaction. Although we cannot obtain the stars with two-solar mass because of the absence of the repulsive interaction, the estimated magnetic moment density is very large. Therefore, we expect that the tensor interaction describes the magnetic fields of compact stars.'
author:
- Hiroaki Matsuoka
- Yasuhiko Tsue
- João da Providência
- Constança Providência
- Masatoshi Yamamura
title: Hybrid stars from the NJL model with a tensor interaction
---
Introduction
============
One of recent interests is to clarify the phase structure of the world governed by the quantum chromodynamics (QCD). It is well known that at low temperature and small chemical potential the hadronic phase is realized. In this phase, because of the color confinement, we cannot remove a single quark from hadrons. At high temperature and small chemical potential, the quark-gluon plasma phase may be realized. On the other hand, it is very difficult to investigate the phase structure at low temperature and large chemical potential. Many researchers, however, consider that the color superconducting phase may be realized under certain conditions [@Alford:2001].
In order to exhibit the phase structure of QCD, the Nambu–Jona-Lasinio (NJL) model [@Nambu:1961_1; @Nambu:1961_2; @Klevansky:1992; @Hatsuda:1994] has been used by many authors. For example, in Refs.[@Buballa:2002; @Blaschke:2003; @Buballa:2005], the color superconductivity has been discussed in the NJL model. Especially, in Ref.[@Kitazawa:2002] the authors have included the vector interaction as well as the quark-pairing interaction into the NJL model. As a result, it has been shown that the chiral condensate and the color superconducting gap may coexist because of the vector interaction.
The possibility that the spins of quarks may become polarized at large chemical potential has been discussed in Ref.[@Tatsumi:2000]. In addition, since a spin polarization term can be derived from the axial-vector interaction, the authors in Refs.[@Tatsumi:2004; @Nakano:2003; @Maedan:2007] have investigated the possibility of spin polarization by using an axial-vector interaction. A term similar to the spin polarization term can also be obtained from a tensor interaction in the NJL model, which can be interpreted as an anomalous magnetic moment induced dynamically [@Ferrer:2014]. In this paper that term is called the “tensor condensate.” The tensor condensate from the tensor interaction was investigated in Ref.[@Bohr:2012]. It has been shown in Ref. [@Tsue:2012] that the tensor condensed phase may be realized at large chemical potential. Also, in Refs.[@Tsue:2013; @Tsue:2015_1], the relationship between the tensor condensate and color superconductivity has been investigated at zero temperature. Moreover, in our preceding papers [@Matsuoka:2016; @Matsuoka:2017], the relationship between the chiral condensed phase, “tensor condensed phase” and the color superconducting phase has been discussed at finite temperature and finite quark chemical potential. According to these investigations, the chiral condensate, and tensor condensate do not coexist, however, the tensor condensate and two-flavor color superconducting gap may coexist at low temperature and large quark chemical potential. Further, ferromagnetism due to the tensor condensate has been investigated in Ref.[@Tsue:2015_2]. It has been shown that if quarks have an anomalous magnetic moment, the tensor condensate may lead to spontaneous magnetization in high density quark matter.
These conditions, large chemical potential, and low temperature, may be realized in the inner core of compact stars, for e.g., neutron stars and magnetars. It is known that neutron stars have very strong magnetic fields at their surface [@Harding:2006]. However, nobody understands definitely the mechanism that generates such strong magnetic fields. We propose that the tensor condensate is the origin of magnetic fields.
Another interesting topic of research related to neutron stars is the difficulty of describing stars with two-solar mass using equation of state (EOS) that includes hyperons or non-nucleonic degrees of freedom [@Vidana:2011]. Though the calculation in the present investigation does not include strange quarks, we will construct compact stars with a quark core using the EOS obtained from the NJL model with the tensor interaction.
In Sec. \[lagrangian\] we introduce the NJL model with the tensor interaction and then calculate the thermodynamic potential. In Sec. \[numerical\] we discuss numerical results and construct hybrid stars. The last section is devoted to conclusions and remarks.
Lagrangian density and thermodynamic potential {#lagrangian}
==============================================
In this section we introduce the NJL model with the tensor interaction at finite chemical potential. In addition to this, we impose the $\beta$ equilibrium and charge neutrality conditions on the system. The Lagrangian density with flavor $SU(2)$ and color $SU(3)$ symmetry is $$\begin{aligned}
\begin{split}
\mathcal{L}_{\text{total}} &= \mathcal{L}_{\text{NJL}} + \mathcal{L}_T + \mathcal{L}_e + \mathcal{L}_D, \\
\mathcal{L}_{\text{NJL}} &= \bar{\psi} i \gamma^\mu \partial_\mu \psi
+ G_S \big \{ (\bar{\psi} \psi)^2 + (\bar{\psi} i \gamma^5 \vec{\tau} \psi)^2 \big \} \\
\mathcal{L}_T &= -\frac{G_T}{4} \big \{ (\bar{\psi} \gamma^\mu \gamma^\nu \vec{\tau} \psi) \cdot
(\bar{\psi} \gamma_\mu \gamma_\nu \vec{\tau} \psi) \\
& \qquad + (\bar{\psi} i \gamma^5 \gamma^\mu \gamma^\nu \psi)
(\bar{\psi} i \gamma^5 \gamma_\mu \gamma_\nu \psi) \big \}, \\
\mathcal{L}_e &= \bar{\psi}_e i \gamma^\mu \partial_\mu \psi_e, \\
\mathcal{L}_D &= \mu \psi^\dagger \psi
+ \lambda \bigg \{ \psi^{\dagger}_{e} \psi_e + \frac{1}{3} \psi^{\dagger}_{d} \psi_d - \frac{2}{3} \psi^{\dagger}_{u} \psi_u \bigg \},
\end{split}\end{aligned}$$ where $\tau_i \; (i = 1,2,3)$ is the Pauli matrix that operates in the flavor space and $
\psi = \begin{pmatrix} \psi_u \\ \psi_d \end{pmatrix}
$ is the quark fields for up quark $\psi_u$ and down quark $\psi_d$, respectively. The term $\mathcal{L}_T$ represents the tensor interaction. [^1] We have added the term, $\mathcal{L}_e$, for electrons that neutralize stellar matter. For simplicity, in this discussion we ignore the current quark mass and the electron mass. The term $\mathcal{L}_D$ controls densities. The variables $\mu$ and $\lambda$ handle the quark number density and charge density, respectively. Namely, $\mu$ is the quark chemical potential, and $\lambda$ will be identified as the electron chemical potential after optimization of the thermodynamic potential.
In this paper, we pay attention to the term, $(\bar{\psi} \gamma^1 \gamma^2 \tau_3 \psi)^2$ in $\mathcal{L}_T$ since using the Dirac representation, we can derive the spin matrix $\Sigma_z = -i \gamma^1 \gamma^2$. Let us write down the Lagrangian density that we consider in this discussion: $$\begin{aligned}
\begin{split}
\mathcal{L} &= \bar{\psi} i \gamma^\mu \partial_\mu \psi + G_S (\bar{\psi} \psi)^2 + \frac{G_T}{2} (\bar{\psi} \Sigma_z \tau_3 \psi)^2 \\
&+ \bar{\psi}_e i \gamma^\mu \partial_\mu \psi_e + \mu \psi^\dagger \psi \\
&+ \lambda \bigg ( \psi^\dagger_e \psi_e + \frac{1}{3} \psi^\dagger_d \psi_d - \frac{2}{3} \psi^\dagger_u \psi_u \bigg ).
\end{split}\end{aligned}$$
In the mean field approximation, the above Lagrangian density becomes $$\begin{aligned}
\begin{split}
\mathcal{L}_{\text{MFA}} = & \sum_{f = u,d} \bar{\psi}_f \Big[
i \gamma^\mu \partial_\mu - M - \hat{f} F \Sigma_z + \mu_f \gamma^0 \Big] \psi_f \\
& -\frac{M^2}{4 G_S} -\frac{F^2}{2 G_T}
+ \bar{\psi}_e \Big [i \gamma^\mu \partial_\mu + \lambda \gamma^0 \Big ] \psi_e,
\end{split}\end{aligned}$$ where $$\begin{gathered}
M = -2 G_S (\langle \bar{\psi}_u \psi_u \rangle + \langle \bar{\psi}_d \psi_d \rangle), \\
F = -G_T (\langle \bar{\psi}_u \Sigma_z \psi_u \rangle - \langle \bar{\psi}_d \Sigma_z \psi_d \rangle),\end{gathered}$$ and $\langle \cdot \rangle$ means expectation value. Note that the variables, $M$ and $F$, are the order parameters, the chiral condensate, and tensor condensate, respectively. In order to simplify the notation, we have defined $\hat{f}$ in the following way: $$\begin{aligned}
\hat{f} = \begin{cases} 1 & \text{for} \; f = u \\ -1 & \text{for} \; f = d \end{cases},\end{aligned}$$ where $u$ and $d$ represent up quark and down quark, respectively. In addition, we also introduce $$\begin{aligned}
\mu_u = \mu - \frac{2}{3} \lambda, \quad
\mu_d = \mu + \frac{1}{3} \lambda.\end{aligned}$$ We will realize the above variables as chemical potential of the up quark and down quark, respectively. By using the standard technique, we can obtain the single-particle energy as follows: $$\begin{aligned}
\begin{split}
\epsilon^\alpha &= \sqrt{p_z^2 + \Big ( \sqrt{p_x^2+p_y^2+M^2} + \alpha F \Big ) ^2}, \\
\epsilon_e &= p,
\end{split}\end{aligned}$$ where $\alpha = \pm 1$. The first one is for quarks and the second one is for electrons. As we have commented, $\lambda$ is identified as the electron chemical potential. Therefore, we write $\mu_e := \lambda$ in the following discussion. The thermodynamic potential $\Phi$ is obtained as
$$\begin{aligned}
\begin{split}
&\Phi(M,F,\mu,\mu_e) = \Phi_q + \Phi_e, \\
&\Phi_q = N_C \int_{|\vec{p}| \le \Lambda} \frac{d^3 p}{(2 \pi)^3}
\sum_{f = u,d \atop \alpha = \pm 1} \bigg \{
\Big ( \epsilon^\alpha - \mu_f \Big) \theta(\mu_f - \epsilon^\alpha) - \epsilon^\alpha \bigg \}
+ \frac{M^2}{4 G_S} + \frac{F^2}{2 G_T}, \\
&\Phi_e = 2\int_{-\infty}^\infty \frac{d^3 p}{(2 \pi)^3} \Big ( \epsilon_e - \mu_e \Big ) \theta(\mu_e - \epsilon_e) = - \frac{\mu_e^4}{12 \pi^2},
\end{split}\end{aligned}$$
where $\Phi_q$ and $\Phi_e$ are the thermodynamic potentials for the quark and electron, respectively, and $N_C = 3$ is the number of color. We have introduced a three-momentum cutoff parameter $\Lambda$ in the quark thermodynamic potential.
Here we discuss renormalization of the thermodynamic potential briefly. We can derive the pressure $P$ as $P = -\Phi$. However, in this case, $$\begin{aligned}
P(\mu = \mu_e = 0) = -\Phi(M_0,F_0,0,0) \neq 0.\end{aligned}$$ where $M_0$ and $F_0$ minimize $\Phi$ when $\mu = \mu_e = 0$. Namely, the pressure is not zero at zero chemical potentials. Therefore, we redefine a new thermodynamic potential as $$\begin{aligned}
\begin{split}
&\Phi_R (M,F,\mu,\mu_e) \\
&:= \Phi(M,F,\mu,\mu_e) - \Phi(M_0,F_0,0,0).
\end{split}\end{aligned}$$ Then, the renormalized pressure is computed as $P = -\Phi_R$.
We can calculate the quark number density by differentiating the thermodynamic potential with respect to the quark chemical potential: $$\begin{aligned}
\rho = - \frac{\partial \Phi_R}{\partial \mu} = \rho_u + \rho_d,\end{aligned}$$ where $\rho_u$ and $\rho_d$ are the number density of up quark and down quark, respectively. On the other hand, we can obtain the electron number density in the following way: $$\begin{aligned}
\rho_e = -\frac{\partial \Phi_e}{\partial \mu_e} = \frac{\mu_e^3}{3 \pi^2}.\end{aligned}$$
We minimize the thermodynamic potential under the condition $\partial \Phi_R / \partial \mu_e = \partial \Phi_R / \partial \lambda = 0$, namely, $$\begin{aligned}
\rho_e + \frac{1}{3} \rho_d - \frac{2}{3} \rho_u = 0.\end{aligned}$$ It means the charge neutrality condition. Using the above condition and the definition of the quark number density, the number density of up quark and down quark can be written as $$\begin{aligned}
\rho_u = \frac{1}{3} \rho + \rho_e, \quad
\rho_d = \frac{2}{3} \rho - \rho_e.\end{aligned}$$ In addition, the chemical potential of the up quark and down quark can be written as $$\begin{aligned}
\mu_u = \mu - \frac{2}{3} \mu_e, \quad
\mu_d = \mu + \frac{1}{3} \mu_e.\end{aligned}$$ Then we can derive $\mu_d = \mu_u + \mu_e$, and it is the $\beta$ equilibrium condition.
We can write down energy density through the thermodynamic relation as $$E = \Phi_R + \mu \rho = \Phi_R + \mu (\rho_u + \rho_d).$$
Numerical results {#numerical}
=================
We comment on the parameters. The NJL model is not a renormalizable theory, therefore, we have introduced a three-momentum cutoff parameter, $\Lambda$. The parameters, $\Lambda$ and $G_S$, are determined to reproduce the dynamical quark mass and the pion decay constant in the vacuum. The value of the coupling constant with tensor interaction $G_T$ may be determined by experimental values [@Jaminon:1998; @Jaminon:2002] or the Fierz transformation of the scalar and pseudoscalar interaction in the NJL model. In these cases, the sign of $G_T$ becomes opposite to our model. On the other hand, in Ref. [@Battistel:2016], the authors use the opposite and same sign. The value of $G_T$ is not well known, therefore, we treat $G_T$ as a free parameter. The values of parameters are enumerated in the Table. I.
In the following discussion, we do not use the quark chemical potential but baryon chemical potential. It is defined in the following way: $$\begin{aligned}
\mu_B = 3 \mu.\end{aligned}$$ In addition, we also introduce the baryon number density as $$\begin{aligned}
\rho_B = \frac{1}{3} \rho.\end{aligned}$$
Model $\Lambda \; [\text{GeV}]$ $G_S \; [\text{GeV}^{-2}]$ $G_T \; [\text{GeV}^{-2}]$
------- --------------------------- ---------------------------- ----------------------------
GT0 $0.631$ $5.5$ $0$
GT11 $0.631$ $5.5$ $11.0$
GT12 $0.631$ $5.5$ $12.0$
GT13 $0.631$ $5.5$ $13.0$
GT14 $0.631$ $5.5$ $14.0$
: Parameter set
Behavior of $F$
---------------
![The relationship between the baryon chemical potential $\mu_B$ and the tensor condensate $F$ is shown. The horizontal and vertical axes represent the baryon chemical potential and tensor condensate, respectively.[]{data-label="CPvsF"}](CPvsF.eps){width="\columnwidth"}
Figure \[CPvsF\] shows the behavior of the tensor condensate $F$. We vary the value of baryon chemical potential from 0.6 GeV to 1.8 GeV. Since the model GT0 does not have the tensor interaction, the tensor condensate does not occur. Thus, the model GT0 does not appear in this figure. The figure shows that as the $G_T$ becomes larger, $F$ can get nonzero values at smaller chemical potential. Moreover, the value of $F$ becomes larger.
Next we show the competition between the chiral condensate and the tensor condensate; see Fig. \[competition\]. The four graphs represent the competition in models GT11, GT12, GT13, and GT14, respectively. The horizontal axis is the baryon chemical potential, and the vertical axis is the chiral condensate $M$ and the tensor condensate $F$. In all models, while $\mu_B$ is small enough, the chiral condensate is realized. Then, when $\mu_B$ becomes large enough, the tensor condensate is realized. In our calculation, we cannot obtain the situation $M \neq 0$ and $F \neq 0$. Let us summarize the phase transition in our model here. When we use models GT11, GT12, and GT13, the phase transition is of the type $$\begin{aligned}
&\text{Chiral condensed phase} \; (M \neq 0, \; F = 0) \\
& \quad \longrightarrow \text{Chiral symmetric phase} \; (M = 0, \; F = 0) \\
& \qquad \longrightarrow \text{Tensor condensed phase} \; (M = 0, \; F \neq 0),\end{aligned}$$ as the baryon chemical potential becomes larger. The phase transition occurs via the chiral symmetric phase. On the other hand, when we use model GT14, the phase transition is of the type $$\begin{aligned}
&\text{Chiral condensed phase} \; (M \neq 0, \; F = 0)\\
& \quad \longrightarrow \text{Tensor condensed phase} \; (M = 0, \; F \neq 0),\end{aligned}$$ as the baryon chemical potential becomes larger.
Behaviors of $\rho$, $\mu_e$ and etc
------------------------------------
![The relationship between the baryon chemical potential $\mu_B$ and the baryon number density $\rho_B$ is shown. The horizontal and vertical axes represent the baryon chemical potential and baryon number density, respectively.[]{data-label="CPvsD"}](CPvsD.eps){width="\columnwidth"}
![The relationship between the baryon number density $\rho_B$ and the tensor condensate $F$ is shown. The horizontal and vertical axes represent the baryon number density and the tensor condensate, respectively.[]{data-label="DvsF"}](DvsF.eps){width="\columnwidth"}
![The relationship between the baryon number density $\rho_B$ and pressure $P$ is shown. The horizontal and vertical axes represent the baryon number density and pressure, respectively.[]{data-label="DvsP"}](DvsP.eps){width="\columnwidth"}
Figure \[CPvsD\] shows the behavior of the baryon number density of the system. The horizontal and vertical axes are the baryon chemical potential and the baryon number density, respectively. Model GT0 has one discontinuity at $\mu_B \sim 1.0 \; \text{GeV}$. It means that the phase transition occurs from the chiral condensed phase to the chiral symmetric phase. The models GT11, GT12, and GT13 have one discontinuity at $\mu_B \sim 1.0 \; \text{GeV}$ and one sharp rise, but continuous, in the value. The former discontinuity corresponds to the phase transition from the chiral condensed phase to the chiral symmetric phase. The latter sharp rise corresponds to the phase transition from the chiral symmetric phase to the tensor condensed phase. On the other hand, model GT14 has only one discontinuity. The discontinuity corresponds to the phase transition from the chiral condensed phase to the tensor condensed phase.
Next let us see the behavior of the electron chemical potential; see Fig. \[CPvsCPe\]. The horizontal and vertical axes represent the baryon chemical potential and the electron chemical potential, respectively. In all models, the values of $\mu_e$ are nonzero at $\mu_B = 0$ and small baryon chemical potentials. We consider that this is a numerical error, the true values are zero while $\mu_B \lesssim 1.0 \; \text{GeV}$. Although $\mu_e \neq 0$, this occurs at very low baryonic densities, out of the range of the baryonic densities at the transition: hadronic matter $\leftrightarrow$ quark matter, and below the densities needed to build the hybrid star’s EOS. As we have seen in Fig. \[CPvsD\], each of models GT0 and GT14 has one discontinuity, and each of models GT11, GT12, and GT13 has one discontinuity and one continuous sharp rise in the values of $\mu_e$. Here we comment on the discontinuities. It looks like the discontinuities in models GT11, GT12, and GT13 are at smaller baryon chemical potential than that in model GT0. However, the discontinuities in model GT0, ..., and GT13 should be coincident since the tensor condensate has not appeared yet. The differences are due to numerical problem, which, however, will not affect the main conclusions on the hybrid star structure.
Figure \[DvsF\] shows the relationship between the baryon number density and the tensor condensate. The horizontal and vertical axes are $\rho_B$ and $F$, respectively. For the models GT11, GT12, and GT13, $F$ does not have a finite value at small baryon number densities. However, if the baryon number density becomes large enough, $F$ can get nonzero values. Model GT14 does not have points at small baryon number density because $\rho_B$ with such values does not occur (see Fig. \[CPvsD\]). Of course, $F$ is zero at $\rho_B = 0$. If the baryon number density exceeds $\rho_B \sim 0.6 \; \text{fm}^{-3}$, $F$ can obtain finite values.
In Fig. \[DvsDe\], the relation between the baryon number density and the electron number density is shown. The horizontal axis is $\rho_B$, and the vertical axis is $\rho_e$. In model GT0, the value of $\rho_e$ increases linearly as $\rho_B$ becomes larger. On the other hand, in models GT11, ..., and GT14, the curves of $\rho_e$ rise sharply after the tensor condensate is realized. In addition, the value of $\rho_e$ becomes larger at fixed $\rho_B$ as the value of $G_T$ becomes larger. This is due to the fact that the tensor condensate favors larger (smaller) up-quark (down-quark) fractions, see discussion below, and in order to ensure electric charge neutrality the electron number density must increase.
In Figs. \[fraction\_1\] and \[fraction\_2\], we plot up-quark and down-quark fractions, $\rho_u / \rho$ and $\rho_d / \rho$, as a function of the quark number density $\rho$. In model GT0, the two fractions do not change very much. In other models, because of the tensor condensate, $\rho_u / \rho$ increases and $\rho_d / \rho$ decreases. The variations of the two fractions become larger as $G_T$ becomes larger. The tensor interaction reduces the energy of the system and it energetically favors a reduction of the down-quark Fermi momenta. The energy gain compensates the increase of the electric density.
In Fig. \[DvsP\], the baryon number density versus pressure plot is depicted. We have eliminated some isolated points at $P = \rho_B = 0$ for numerical reasons when we compute interpolated functions. Therefore, the baryon number density $\rho_B$ starts from a finite value instead of zero. The pressure of the model GT0 increases monotonically from the origin. [^2] On the other hand, the models GT11, GT12, and GT13 have plateaus. These plateaus indicate the onset of the tensor condensate.
Hybrid star
-----------
First, we explain what we refer to as a “hybrid star.” The hybrid star has an inner core that consists of the quark matter, and the outer core and crust consist of hadrons. In order to obtain the EOS of hadrons, we use the NL3$\omega \rho$ model [@Pereira:2016].
![The relationship between the radius and mass of hybrid stars is shown. We have normalized the mass $m$ by the solar mass $m_\text{sun}$ and the radius $r$ by $r_0 = 10 \; \text{km}$. The horizontal and vertical axes represent the radius $r / r_0$ and mass $m / m_{\text{sun}}$, respectively.[]{data-label="RvsM"}](RvsM.eps){width="\columnwidth"}
Let us discuss the relationship between the radius and mass of hybrid stars by solving the Tolman-Oppenheimer-Volkoff (TOV) equation numerically. In Fig. \[RvsM\], the radius ($r$)-mass ($m$) relation is depicted. We have normalized $r$ and $m$ by $r_0 = 10 \, \text{km}$ and the solar mass $m_\text{sun}$, respectively. See the models GT0, GT11, and GT12. The curves bend at $(r/r_0, m/m_{\text{sun}}) \sim (1.3, 1.3)$. The point corresponds to the appearance of the quark matter at the inner core of the hybrid star; the curves under this point have “hadron cores,” on the other hand, the curves above this point have “quark cores.” The curves of the models GT11 and GT12 bend at $(r/r_0, m/m_{\text{sun}}) \sim (1.2, 1.7)$ and $(r/r_0, m/m_{\text{sun}}) \sim (1.25, 1.5)$ again, respectively. This point means that the tensor condensate appears at the core of the hybrid stars; the curves under this point have cores which are $M = F = 0$. The curves above this point have cores which are $M = 0$ and $F \neq 0$. The curve of the model GT13 snaps off at $(r/r_0, m/m_{\text{sun}}) \sim (1.28, 1.2)$. This point means the onset of the tensor condensate at the core. Since the tensor condensate is realized at smaller chemical potential as $G_T$ becomes larger, the curve of the model GT14 bends earlier than the model GT13. The “bending point” for the model GT14 is at $(r/r_0, m/m_{\text{sun}}) \sim (1.18, 0.7)$. We note that models GT13 and GT14 predict “twin stars” [@Benic:2015; @Alvarez:2017], i.e., stable stars with the same mass but very different radii: a hadronic star and a hybrid star, the first one with a radius is 2 km larger than the hybrid star radius. For more discussion concerning twin stars, please see Refs. [@Benic:2015; @Alvarez:2017].
Estimation of magnetic moment
-----------------------------
We estimate the magnetic moment density by using Eq. (22) in Ref. [@Maruyama:2018]. In our case, the expression of the magnetic moment density becomes $$\begin{aligned}
M_{\text{mag}} = \bigg ( \frac{2}{3} \bar{\rho}_u + \frac{1}{3} \bar{\rho}_d \bigg ) \frac{e}{2 m_q} \frac{\langle \bar{\psi} i \gamma^1 \gamma^2 \tau_3 \psi \rangle}{\langle \psi^\dagger \psi \rangle} 3 \rho_B,\end{aligned}$$ where $\bar{\rho}_u$ and $\bar{\rho}_d$ are the fraction of up and down quark, respectively. We use the current quark mass: $m_q = 0.005 \; \text{GeV}$. In the above equation, $\bar{\rho}_u$ and $\bar{\rho}_d$ satisfy $\bar{\rho}_u + \bar{\rho}_d = 1$; thus, we can write $$\begin{aligned}
\bar{\rho}_u = \frac{\rho_u}{\rho_u + \rho_d}, \;
\bar{\rho}_d = \frac{\rho_d}{\rho_u + \rho_d}.\end{aligned}$$ In Ref. [@Maruyama:2018], a relation $\bar{\rho}_u = \bar{\rho}_d$ is used. However, in our case, this condition is not satisfied because we are imposing the $\beta$ equilibrium and the charge neutrality conditions on the system. Thus, we must use numerical data for $\bar{\rho}_u$ and $\bar{\rho}_d$, namely, $\rho_u$ and $\rho_d$. By using the following relations: $3 \rho_B = \rho$, $\rho_u = \rho / 3 + \rho_e$ and $\rho_d = 2 \rho / 3 - \rho_e$, we can transform $M_{\text{mag}}$ into $$\begin{aligned}
M_{\text{mag}} = (4 \rho_B + \rho_e) \times \frac{e F}{18 m_q G_T \rho_B}.\end{aligned}$$ Here we discuss the dimension. We are using the following unit: $c = \hbar = \mu_0 = 1$, where $\mu_0$ is the vacuum permeability. Therefore, the dimension of $M_{\text{mag}}$ is $[\text{GeV}^{2}]$. We also have the relation: $1 \; \text{Gauss} = 1.955 \times 10^{-20} \; \text{GeV}^2$. Thus, the dimension of $M_{\text{mag}}$ is \[Gauss\].
See Fig. \[CPvsMmag\]. The vertical and horizontal axes are the magnetic moment density and the baryon chemical potential, respectively. As $G_T$ becomes larger, $M_{\text{mag}}$ gets larger and becomes finite at smaller baryon chemical potentials. We can get $M_{\text{mag}} \sim 10^{19} \; \text{G}$.
![The figure shows the relationship between the chemical potential and the magnetic moment. The vertical and horizontal axes represent the magnetic moment density and the baryon chemical potential, respectively.[]{data-label="CPvsMmag"}](CPvsMmag.eps){width="\columnwidth"}
Conclusions and remarks {#conclusions}
=======================
In this paper we have investigated the behavior of tensor condensate and its implication on the properties of the hybrid star by using the NJL model with the tensor interaction under the $\beta$ equilibrium and charge neutrality conditions. As the value of $G_T$ becomes larger, the value of $F$ increases. Moreover, $F$ has nonzero values at a smaller baryon chemical potential.
In addition, we have constructed hybrid stars by using the EOS of this model. When $G_T \le 14.0 \; \text{GeV}^{-2}$, we obtain no hybrid stars with two-solar mass. However, if we include the vector interaction as repulsion, we may obtain compact stars with two-solar mass. It is outside the scope of this paper. We have built stars with a polarized core, and a finite tensor condensate. This could be a mechanism that explains the strong magnetic fields inside magnetars. We expect that our scenario is valid also for hybrid stars with $m_\text{hybrid} > 2 m_\text{sun}$, where $m_\text{hybrid}$ is the mass of the hybrid star.
We must, however, point out some problems that have arisen while calculating the hybrid star families. See the radius-mass curves in Fig. \[RvsM\]. The curves do not reach the maximum mass. In order to depict the curves, we have used numerical EOS data with baryon chemical potential, $\mu_B \lesssim 1.8 \; \text{GeV}$, namely, $\mu \lesssim 0.6 \; \text{GeV}$. Since the quark Fermi momentum should not exceed the value of cutoff parameter $\Lambda$ it is difficult to extend the curves any more. If we use a larger cutoff parameter, we may extend the radius-mass curve, but this requires the determination of a new set of parameters with a large cutoff and this will not be considered in this present work.
In this research, we use the flavor $SU(2)$ NJL model with the tensor interaction. Many researchers consider that the color superconducting phase may be realized in the high density region. In addition, at large chemical potential we should not ignore the contribution from strange quarks. Thus, it is interesting to extend our model to flavor $SU(3)$ case. In our discussion, the renormalization was defined in a such a way that the pressure vanishes when the quark and electron chemical potential are zero. However, a different renormalization procedure of the pressure could have been carried out as done in [@Pereira:2016]. These will be considered in a future work.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
H.M. and Y.T. would like to express their sincere thanks to the members of Many-Body Theory Group of Kochi University. This work was partially supported by “Fundacao para a Ciencia e Tecnologia”, Portugal, under the projects No. UID/FIS/04564/2016, POCI-01-0145-FEDER-029912 \[with financial support from POCI, in its FEDER component, and by the FCT/MCTES budget through national funds (OE)\].
How to construct hybrid stars
=============================
In this Appendix, we discuss how to construct hybrid stars. As we have referred, the hybrid star contains quarks in the inner core while the outer core consists of hadrons. In order to obtain the EOS of hadrons, we use the NL3$\omega \rho$ model. The Lagrangian density is obtained as
$$\begin{aligned}
\mathcal{L}_{\text{NL3$\omega \rho$}} &=
\sum_{N=p,n} \bar{\psi}_N \bigg [
\gamma^\mu (i \partial_\mu - g_{\omega N} \omega_\mu - \frac{1}{2} g_{\rho N} \bm{\tau} \cdot \bm{\rho}_\mu) - (m_N - g_{\sigma N} \sigma)
\bigg ] \psi_N \\
& + \frac{1}{2} \partial_\mu \sigma \partial^\mu \sigma - \frac{1}{2} m^{2}_{\sigma} \sigma^2
-\frac{1}{4} \Omega_{\mu \nu} \Omega^{\mu \nu} + \frac{1}{2} m^{2}_{\omega} \omega^\mu \omega_\mu
-\frac{1}{4} \bm{\rho}^{\mu \nu} \cdot \bm{\rho}_{\mu \nu} + \frac{1}{2} m^{2}_{\rho} \bm{\rho}^\mu \cdot \bm{\rho}_\mu \\
& -\frac{1}{3} b m_N (g_{\sigma N} \sigma)^3 -\frac{1}{4} c (g_{\sigma N} \sigma)^4
+ \Lambda_\omega (g^{2}_{\omega} \omega_\mu \omega^\mu)(g^{2}_{\rho} \bm{\rho}_\mu \cdot \bm{\rho}^\mu),\end{aligned}$$
where $\Omega_{\mu \nu} = \partial_\mu \omega_\nu - \partial_\nu \omega_\mu$ and $\bm{\rho}_{\mu \nu} = \partial_\mu \bm{\rho}_\nu - \partial_\nu \bm{\rho}_\mu$. The Lagrangian density contains the fields of nucleons ($\psi_p$ and $\psi_n$), $\sigma$ meson ($\sigma$), $\rho$ meson ($\bm{\rho}^\mu$) and $\omega$ meson ($\omega^\mu$). We note that the EOS obtained by this model can hold compact stars with two-solar mass.
Our strategy for numerical calculation is as follows:
1. Give an arbitrary value to the central energy density of the hybrid star.
2. If the value is large enough that the quark matter is realized, go to step 3. If the value is not enough, jump to step 5.
3. Solve the TOV equation from the center to the outside of the star by using the EOS of quarks until a certain reference pressure obtained by performing a Maxwell construction.
4. At the reference pressure, switch the EOS to the one of hadrons.
5. Solve the TOV equation to the outside of the star with the EOS of hadrons until the pressure of the star vanishes.
6. Change the value of central energy density and go back to step 2.
In addition to the above strategy, we should discuss the way to switch the EOS from quarks to hadrons. In Fig. \[EOS\], we plot the EOS for the models GT0, GT11, ..., and GT14. The figures in the left column represent baryon chemical potential-pressure plots, and those in the right column represent energy density-pressure plots. The blue curves (NL3$\omega \rho$) are the EOS obtained by the NL3$\omega \rho$ model. The yellow curves (NJL+Tensor) represent the EOS obtained by the NJL model with the tensor interaction. In addition, the green curves (Modified) are the EOS that is used in the construction of hybrid stars. We introduce some functions: $P_i(\mu), \; p_i(E)$ and $(i=Q,H)$. The functions $P_{Q}$ ($P_H$) and $p_{Q}$ ($p_H$) refer to the pressure of NJL+Tensor (NL3$\omega \rho$). The argument of $P_i \; (i=Q,H)$ is the baryon chemical potential. On the other hand, the argument of $p_i \; (i=Q,H)$ is the energy density.
In the left column, there are the cross points, where $P_Q(\mu_0) = P_H(\mu_0) = P_0$. We switch from EOS to the other at the pressure $P_0$. This is the Maxwell construction. In the right column, the horizontal part of the green curve, $E_H \le E \le E_Q$, is defined so that $p_H(E_H) = p_Q(E_Q) = P_0$.
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[^1]: In other papers, the term $\mathcal{L}_T$ is sometimes written by using $\sigma^{\mu \nu} = \frac{i}{2} [\gamma^\mu, \gamma^\nu]$. It means that the interaction does not contain terms $\mu = \nu$. Therefore, in our notation, we do not consider terms $\mu = \nu$.
[^2]: It should be noted that the origin in this graph is not $(\rho_B,P) = (0,0)$.
|
---
abstract: |
Similarity between languages has been suggested to affect quality cross-lingual transfer, albeit only sporadic empirical evidence. In this study we test this hypothesis on a massive scale for the first time, and conclude that there are indeed strong ties between the two. We analyze thousands of language pairs in four different tasks, Bilingual Lexicon Induction, Machine Translation, Dependency parsing and POS tagging, and consistently find that language pairs whose embedding spaces are closer are associated with better mapping, joint and transfer learning properties. We propose a novel distance, Eigenvalue divergence (EVD), to evaluate embedding space similarity, and report that it outperforms existing distances, is computationally more tractable and easier to interpret than previous distance methods. Importantly, EVD can be complemented by typologically-driven linguistic distances to achieve even higher predictions of performance levels in cross-lingual tasks.
Performance in cross-lingual NLP tasks is impacted by the (dis)similarity of languages at hand: e.g., previous work has suggested there is a connection between the expected success of bilingual lexicon induction (BLI) and the assumption of (approximate) isomorphism between monolingual embedding spaces. In this work, we present a large-scale study focused on the correlations between language similarity and task performance, covering thousands of language pairs and four different tasks: BLI, machine translation, parsing, and POS tagging. We propose a novel language distance measure, Eigenvalue Divergence (EVD), which quantifies the degree of isomorphism between two monolingual spaces. We empirically show that **1)** language similarity scores derived from embedding-based EVD distances are strongly associated with performance observed in different cross-lingual tasks, **2)** EVD outperforms other standard embedding-based language distance measures across the board, at the same time being computationally more tractable and easier to interpret. Finally, we demonstrate that EVD captures information which is complementary to typologically driven language distance measures. We report that their combination yields even higher correlations with performance levels in all cross-lingual tasks.
author:
- |
Haim Dubossarsky$^{\mathbf{1}}$, Ivan Vulić$^{\mathbf{1}}$, Roi Reichart$^{\mathbf{2}}$, [Anna Korhonen]{}$^{\mathbf{1}}$\
$^{\mathbf{1}}$ Language Technology Lab, University of Cambridge\
$^{\mathbf{2}}$ Faculty of Industrial Engineering and Management, Technion, IIT\
`{hd423, iv250, alk23}@cam.ac.uk` `[email protected]`
- 'Haim Dubossarsky$^1$ Ivan Vulic$^{1,2}$ Roi Reichart$^3$ Anna Korhonen$^1$'
- '$^1$ Language Technology Lab, University of Cambridge, UK'
bibliography:
- 'acl2020.bib'
title: 'Lost in Embedding Space: Explaining Cross-Lingual Task Performance with Eigenvalue Divergence'
---
Introduction {#s:introduction}
============
EVD: Eigenvalue Divergence {#s:motivation}
==========================
Related Work {#s:related}
============
Experimental setup {#s:experimental}
==================
Analyses and Results {#s:results}
====================
Further Discussion and Conclusion {#s:discussion}
=================================
Acknowledgments {#acknowledgments .unnumbered}
===============
The work of IV and AK is supported by the ERC Consolidator Grant LEXICAL: Lexical Acquisition Across Languages (no 648909) awarded to AK. HD is supported by the Blavatnik Postdoctoral Fellowship Programme.
Appendix {#sec:supplemental}
========
#### Gromov-Hausdorff Distance (GH)
This distance measures the worst case distance between two metric spaces $\mathcal{X}$ and $\mathcal{Y}$ with a distance function $d$, formulated as follows: [$$\begin{aligned}
\begin{split}
\mathcal{H}(\mathcal{X}, \mathcal{Y}) = \max\{& \sup_{x \in \mathcal{X}} \inf_{y \in \mathcal{Y}} d(x, y), \\
& \sup_{y \in \mathcal{Y}} \inf_{x \in \mathcal{X}} d(x, y)\}
\end{split}\end{aligned}$$]{}It measures the distance between the nearest neighbours that are farthest apart. The Gromov-Hausdorff distance (GH) then minimizes this distance over all isometric transforms $\mathcal{X}$ and $\mathcal{Y}$: Computing $\mathcal{GH}$ directly is computationally intractable in practice, but it can be tractably approximated by computing the Bottleneck distance between the metric spaces [@chazal2009gromov]. As shown by , GH also seems to correlate well with the degree of isomorphism between two embedding spaces.
#### Isospectrality (IS)
After length-normalizing the vectors, we compute the nearest neighbour graphs of a subset of $N$ words, and then calculate the Laplacian matrices $L_1$ and $L_2$ of each graph. For $L_1$, the smallest $k_1$ is then sought such that the sum of its $k_1$ largest eigenvalues $\sum^{k_1}_{i=1} {\lambda_1}_i$ is at least 90% of the sum of all its eigenvalues. The same procedure is used to find $k_2$. We then define $k=\min (k_1, k_2)$. The final IS measure $\Delta$ is then the sum of the squared differences of the $k$ largest Laplacian eigenvalues: $\Delta = \sum^k_{i=1}({\lambda_1}_i - {\lambda_2}_i)^2$. The lower $\Delta$, the *more* similar are the graphs and, consequently, the more isomorphic are the two embedding spaces. The IS measure was advocated as a suitable measure of isomorphism in the work of .
|
[*Jorge Bellorín,*]{}$^{a,}$[^1] [*Alvaro Restuccia*]{}$^{a,b,}$[^2] [*and Adrián Sotomayor*]{}$^{c,}$[^3]
$^a$[*Department of Physics, Universidad Simón Bolívar, Valle de Sartenejas,\
1080-A Caracas, Venezuela.*]{}\
$^b$[*Department of Physics*]{}, $^c$[*Department of Mathematics, Universidad de Antofagasta, 1240000 Antofagasta, Chile.*]{}
> [We find the static spherically symmetric solutions (with vanishing shift function) of the complete nonprojectable Hořava theory explicitly, writing the space-time metrics as explicit tensors in local coordinate systems. This completes previous works of other authors that have studied the same configurations. The solutions depend on the coupling constant $\alpha$ of the $(\partial_i \ln N)^2$ term. The $\lambda =1/3$ case of the theory does not possess any extra mode, hence the range of $\alpha$ is in principle not limited by the linear stability of any extra mode. We study the full range of $\alpha$, both in the positive and negative sectors. We find the same wormhole solutions and naked singularities that were found for the Einstein-aether theory in a sector of the space of $\alpha$. There also arise wormholes in other sector of $\alpha$. Our coordinate systems are valid at the throats of the wormholes. We also find the perturbative solutions for small $\alpha$. We give this version of the solutions directly on the original radial coordinate $r$, which is particularly suitable for representing the exterior region of solutions with localized sources. ]{}
Introduction
============
In recent years there have been advances in establishing the consistency of the complete nonprojectable Hořava theory [@Horava:2009uw; @Blas:2009qj][^4], as well as in exploring its viability as a candidate for an ultraviolet completion of general relativity (GR) that can be consistently quantized perturbatively. Most of the early claims about the inconsistency of the theory concern a potentially dangerous extra mode. If one makes only a preliminary comparison between the number of field variables and the number of gauge symmetries, which are less in Hořava theory than in GR, one could conclude that in Hořava theory there is one degree of freedom more than in GR. Although this can be regarded as a generic result, we recently showed [@Bellorin:2013zbp] that *this is not true for the nonprojectable theory with the special value $\lambda = 1/3$*, where $\lambda$ is the coupling constant of the tracelike kinetic term. We showed that at this point the theory has two extra second-class constraints that eliminate the extra mode from the phase space. In turn, the extra constraints protect the value $\lambda = 1/3$ against renormalization running since other values of $\lambda$ violate the constraints. Technically, what makes the value $\lambda = 1/3$ so special is that the time derivative of the spatial metric cannot be completely solved in terms of the canonical momentum from the pure Legendre transformation; instead the extra constraints arise. This result is valid even with the whole $z=3$ potential of the theory, not only for effective Lagrangians [@Bellorin:2013zbp]. Thus, the complete nonprojectable Hořava theory at $\lambda = 1/3$ has a remarkable feature: it possesses exactly the same degrees of freedom of GR. As an immediate consequence, the theory avoids the problem of strong coupling of the extra mode (low-energy divergences of self-interaction coupling constants) [@Charmousis:2009tc; @Papazoglou:2009fj; @Blas:2009ck; @Kimpton:2010xi], since there is no extra mode. Having the same degrees of freedom of GR at all scales, the $\lambda =1/3$ theory does not experience the obstructions other theories of gravitation frequently face in recovering GR: discontinuities or ghosts. Moreover, the linear-order perturbative version of the effective theory for large distances is physically equivalent to the linearized version of GR. This implies that the effective theory propagates gravitational waves exactly in the same way as linearized GR does. Measurements that could signal gravitational waves have been recently obtained [@Ade:2014xna].
In the linear-order perturbative analysis of Ref. [@Park:2009hg], the absence of the extra mode for a particular model of the original nonprojectable Hořava theory at $\lambda = 1/3$ with a Cotton-square term (without the $a_i = \partial_i \ln N$ terms) was found. Since the model considered there acquires conformal symmetry at the UV, one could think that the absence of the extra mode is a consequence of the approximate conformal symmetry, but the analysis of [@Bellorin:2013zbp] shows that this result is actually a consequence of the $\lambda = 1/3$ value. Extra gauge symmetries would lower furthermore the number of degrees of freedom in the $\lambda =1/3$ case.
Even in the theory with $\lambda \neq 1/3$, where the extra mode arises, the disastrous strong coupling could be circumvented, since it arises only if one forces the limit $\lambda \rightarrow 1$ in order to recover GR at low energies [@Charmousis:2009tc]. In Ref. [@Bellorin:2010je] (see also [@Das:2011tx]) two of us showed a way to recover GR for any $\lambda$: the second-order model having only the spatial scalar curvature in the potential is physically equivalent to GR with no need of assuming additional restrictions on the kinetic terms. Thus, the way to recover GR is to neglect all the terms in the potential except the $^{(3)}R$ term, whereas $\lambda$ is left arbitrary. This mechanism avoids the limit $\lambda \rightarrow 1$ for recovering GR.
For both cases, $\lambda = 1/3$ and $\lambda \neq 1/3$, it has been shown the consistency of the Hamiltonian formulation of the complete nonprojectable theory [@Bellorin:2013zbp; @Donnelly:2011df; @Bellorin:2011ff; @Bellorin:2012di] (see also Ref. [@Kluson:2010nf]). It has been fully clarified the set of constraints of the theory, together with their first- or second-class character. There is no evidence leading us to conclude that any of those constraints is an inconsistent equation for the metric variables or their conjugated momenta.[^5] It has been shown that the algebra of constraints closes. All the constraints are preserved in time once consistent equations for the Lagrange multipliers associated to second-class constraints are imposed[^6].
These results, in particular the outstanding properties of the $\lambda = 1/3$ theory, encourage us to deepen on the physical content of the complete nonprojectable Hořava theory. Because of their astrophysical relevance, we devote this paper to study the (vacuum) static spherically symmetric solutions of the complete nonprojectable Hořava theory. Since the interest is in the large-distance physics, we use the lowest-order effective action (second order in derivatives). Unlike the symmetries of GR, the foliation-preserving diffeomorphism symmetry is not enough to eliminate the mixed time-space components of the metric once it is written in spherical coordinates (Schwarzschild coordinates). Thus, those components (the shift function) are left active if only symmetry arguments are used, representing a hair for this class of solutions. For the sake of simplicity, in this paper we only consider static spherically symmetric configurations with vanishing shift function[^7]. The analysis covers simultaneously the $\lambda = 1/3$ and $\lambda \neq 1/3$ cases, since $\lambda$ is a multiplier of a kinetic term. Once the conditions of staticity and vanishing of the shift function are imposed, all kinetic terms vanish, such that $\lambda$ disappears from the field equations. Thus, under these conditions the field equations are the same for $\lambda = 1/3$ and $\lambda \neq 1/3$. The only coupling constant the field equations depend on is the one of the $(\partial_i \ln N )^2$ term, which is denoted by $\alpha$.
Some years before the formulation of the Hořava theory, Eling and Jacobson [@Eling:2006df] (see also [@Eling:2003rd]) studied the static spherically symmetric solutions of the Einstein-aether theory (EA theory), which is also a theory with preferred frame [@Jacobson:2000xp]. It turns out that such solutions are also the static spherically symmetric solutions of the effective action of the complete nonprojectable Hořava theory, since the EA theory becomes equivalent to the large-distance limit of the Hořava theory when the condition of hypersurface orthogonality is imposed on the unit vector of the EA theory [@Blas:2009ck; @Jacobson:2010mx; @Jacobson:2013xta]. This condition is met for the static spherically symmetric configurations [@Eling:2006df]. Eling and Jacobson [@Eling:2006df] studied the solutions in a restricted range for the coupling constant equivalent to $\alpha$. In that range they could find the solutions analytically in terms of a function of the radius. They used such a function as a parameter over which the solution can be explicitly expressed. They found that the solutions have a wormholelike geometry, with two spatial branches joined by a throat (an Einstein-Rosen bridge). These solutions are not black holes.
Directly on the side of the complete nonprojectable Hořava theory, an analysis of the static spherically solutions was done by Kiritsis in Ref. [@Kiritsis:2009vz] (in [@Kiritsis:2009rx] there is a related study on the original theory). Motivated by the linear stability of the (possible) extra mode [@Blas:2009qj], he concentrated the analysis on the $0 < \alpha < 2$ range. This is the same range studied for the solutions of the EA theory in Ref. [@Eling:2006df]. Kiritsis also analyzed the solutions in the $\alpha = 2$ and $\alpha > 2$ ranges. He got the solutions in a similar fashion to [@Eling:2006df]: expressing them in terms of a function of the radius and then using the function as the parameter that controls the configurations. In addition, there is vast literature about solutions of the original Hořava theory, both in the projectable and nonprojectable versions. Most of these studies are perturbative approaches. Exact computations on finding static spherically symmetric solutions can be found in Refs. [@Lu:2009em; @Kehagias:2009is].
In this paper we shall make further developments on analyzing the field equations for static spherically symmetric solutions. Our aim is to find the solutions in a closed way, such that the space-time metric can be explicitly written as a tensor in concrete local coordinate systems. Furthermore, since the $\lambda = 1/3$ theory does not possess any extra mode, the range of $\alpha$ is not restricted *a priori* by physical features associated to the extra mode if one is interested in the dynamics of the $\lambda = 1/3$ theory. Therefore, we extend the analysis of the static spherically solutions to the full range of $\alpha$, including negative values. In Section \[sec:exactsolutions\] we shall show that the field equations can be exactly and explicitly solved in a closed way. We shall perform a managing of the field equations aimed to recover explicitly the constraints of the theory. The central step of the procedure will consist of extracting a purely *algebraic* field equation. This equation can be easily solved according to two different cases of $\alpha$; the solutions arising as one-parameter families on each case. We shall use the corresponding parameter as a transformed radial coordinate, such that the final solution will be explicitly written in terms of the transformed radius. The final exact and explicit expressions will help us to furthermore understand the properties of the solutions.
In order to further explore the solutions, in Section \[sec:perturbations\] we present a different and convenient approach for solving the equations: the perturbative approach. We shall show how the field equations can be solved approximately by assuming an small $\alpha$. Specifically, the analysis will be done at linear order in $\alpha$. The explicit expression we shall find for the perturbative solution has the advantage of being given directly in the original radius $r$ of the spherical coordinates. We shall discuss the spatial ranges of validity of the perturbative solutions. Section 5 contains further discussion and conclusions.
The conditions of staticity and spherical symmetry
==================================================
The action of the complete, nonprojectable Hořava theory is written in terms of the Arnowitt-Deser-Misner (ADM) variables $g_{ij}$, $N$ and $N_i$ as $$S = \int dt d^3x \sqrt{g} N
( G^{ijkl} K_{ij} K_{kl} - \mathcal{V} ),
\label{lagrangianaction}$$ where $$\begin{aligned}
K_{ij} & = & \frac{1}{2N} ( \dot{g}_{ij} - 2 \nabla_{(i} N_{j)} ) \,,
\\[1ex]
G^{ijkl} & = &
\frac{1}{2} ( g^{ik} g^{jl} + g^{il} g^{jk} )
- \lambda g^{ij} g^{kl} \,,\end{aligned}$$ and $\mathcal{V}$ is the potential, which depends explicitly on the curvature tensors, the vector $a_i \equiv \partial_i \ln N$ and derivatives of them. In the nonprojectable theory the lapse function $N$ is regarded as a function of both time and space. We specialize our analysis to the large-distance effective action, which is formed with the quadratic potential $$\mathcal{V}^{(2)} = - R - \alpha a_i a^i \,,
\label{quadraticpotential}$$ where $\alpha$ is a coupling constant. One may put an additional, undetermined coupling constant for the $R$ term since it is covariant by itself under the gauge symmetry of the theory, which is the foliation-preserving diffeomorphisms group, with no need of mixing with the kinetic term. Since such a constant can be always absorbed in the free theory by rescaling the time, we omit it in our discussion.[^8]
The corresponding equations of motion, which are derived by taking variations with respect to $g_{ij}$, $N$ and $N_i$, are given, respectively, by $$\begin{aligned}
\frac{1}{\sqrt{g}} \frac{\partial}{\partial t}
( \sqrt{g} G^{ijkl} K_{kl} )
+ 2 N ( K^{ik} K_k{}^j - \lambda K K^{ij} )
- \frac{1}{2} N g^{ij} G^{klmn} K_{kl} K_{mn}
& & \nonumber \\
+ 2 \nabla_k ( G^{kmn(i} K_{mn} N^{j)} )
- \nabla_k ( G^{mnij} K_{mn} N^k )
+ N (R^{ij} - \frac{1}{2} g^{ij} R )
& & \nonumber \\
- ( \nabla^i \nabla^j N - g^{ij} \nabla^2 N )
+ \alpha N^{-1} ( \nabla^i N \nabla^j N
- \frac{1}{2} g^{ij} \nabla_k N \nabla^k N )
& = & 0 \,,
\label{einstein}
\\
G^{ijkl} K_{ij} K_{kl} - R
+ 2 \alpha N^{-2} ( N \nabla^2 N - \frac{1}{2} \nabla_i N \nabla^i N )
& = & 0 \,,
\label{hamiltonianconstrainlagrange}
\\
\nabla_i ( G^{ijkl} K_{kl} ) &=& 0 \,.
\label{momentunconstrainlagrange}\end{aligned}$$
We proceed to evaluate systematically the field equations for static and spherically symmetric configurations, starting with the condition of staticity. There is an important difference in the role the shift function $N_i$ has for static spherically symmetric configurations of Hořava theory with respect to GR. As it is well known, in GR the only nonzero component of $N_i$ for static spherically symmetric metrics, written in spherical coordinates, can be absorbed by redefining the time in a spatial-dependent way. However, in Hořava theory such a coordinate transformation is not allowed. Once the general static spherically symmetric metrics are written in spherical coordinates, unavoidably the radial component of the shift function remains as an arbitrary function [@Kiritsis:2009rx]. In spite of this, we remark that the shift function $N_i$ *is not a true functional degree of freedom* of Hořava theory. $N_i$ is the Lagrange multiplier of the momentum constraint which generates the spatial diffeomorphisms. It can always be set equal to zero by choosing an appropriated spatial coordinate system. The obstruction in being absorbed in static spherically symmetric metrics is a mere consequence of choosing spherical coordinates. One may write general static spherically symmetric metrics in another coordinate system without shift function but with an arbitrary function in the spatial sector. That is, one may translate the freedom parametrized in $N_i$ to the pure spatial sector.
For the sake of simplicity, in this analysis we shall consider only static configurations with $N_i=0$ (in spherical coordinates). This implies $K_{ij} =0$. Under these considerations Eq. (\[momentunconstrainlagrange\]) is automatically solved, whereas Eqs. (\[einstein\]) and (\[hamiltonianconstrainlagrange\]) reduce to $$\begin{aligned}
N (R^{ij} - \frac{1}{2} g^{ij} R )
- ( \nabla^i \nabla^j N - g^{ij} \nabla^2 N )
+ \alpha N^{-1} ( \nabla^i N \nabla^j N
- \frac{1}{2} g^{ij} \nabla_k N \nabla^k N )
& = & 0 \,, \hspace*{3em}
\label{eisnteinstatic}
\\
R - 2 \alpha N^{-2} ( N \nabla^2 N
- \frac{1}{2} \nabla_i N \nabla^i N )
& = & 0 \,.
\label{hamiltoniancstatic}\end{aligned}$$ Equation (\[hamiltoniancstatic\]) and the trace of (\[eisnteinstatic\]) are equivalent to the system $$\begin{aligned}
R - \left( \alpha + 2 \right) N^{-1} \nabla^2 N
+ \alpha N^{-2} \nabla_i N \nabla^i N & = & 0 \,,
\label{eqtraceprev1}
\\
\left( \alpha - 2 \right) \nabla^2 N & = & 0 \,.
\label{eqtraceprev2}\end{aligned}$$
Let us concentrate first on the case $\alpha \neq 2$, leaving the study of the special case $\alpha = 2$ to the end of Section \[sec:exactsolutions\]. If $\alpha \neq 2$, Eqs. (\[eqtraceprev1\]) and (\[eqtraceprev2\]) are equivalent to $$\begin{aligned}
R + \alpha N^{-2} \nabla_i N \nabla^i N &=& 0 \,,
\label{hamiltonianfin}
\\
\nabla^2 N &=& 0 \,.
\label{nablaN}\end{aligned}$$ The equation of motion (\[eisnteinstatic\]), after including (\[hamiltonianfin\]) and (\[nablaN\]) in it, yields $$R^{ij} - N^{-1} \nabla^i \nabla^j N
+ \alpha N^{-2} \nabla^i N \nabla^j N = 0 \,.
\label{eomstaticlag}$$ The system of equations (\[hamiltonianfin\] - \[eomstaticlag\]) is equivalent to the field equations (\[eisnteinstatic\]) and (\[hamiltoniancstatic\]) (one may derive one of the Eqs. (\[hamiltonianfin\]) or (\[nablaN\]) by combining the other with (\[eomstaticlag\])). Under the Hamiltonian formalism, Eqs. (\[hamiltonianfin\]) and (\[nablaN\]) are deduced directly from constraints in the theory with $\lambda = 1/3$. In the case $\lambda \neq 1/3$, these equations are deduced from combining one constraint with the equations of motion. In Appendix A we show how this works in the Hamiltonian formalism.
Next we evaluate the equations of motion (\[hamiltonianfin\] - \[eomstaticlag\]) for static, spherically symmetric configurations with vanishing shift function given by the following ansatz in spherical coordinates: $$N = N(r)
\,,\hspace{2em}
ds_{(3)}^2 = {\displaystyle\frac{dr^2}{f(r)} + r^2 d\Omega_{(2)}^2} \,.
\label{ansatz}$$ Equations (\[hamiltonianfin\]) and (\[nablaN\]) yield, respectively, $$\begin{aligned}
r f' + f - 1
- \frac{\alpha}{2} r^2 f \left( \frac{ N'}{N} \right)^2
&=& 0 \,,
\label{css2}
\\
( r^2 \sqrt{f} N' )' &=& 0 \,.
\label{css1}\end{aligned}$$ All off-diagonal components of the equation of motion (\[eomstaticlag\]) vanish. The $rr$ and $\theta\theta$ components become, respectively, $$\begin{aligned}
\frac{f'}{r f} + \frac{N''}{N} + \frac{f' N'}{2 f N}
- \alpha \left( \frac{N'}{N} \right)^2 &=& 0 \,,
\label{eom1}
\\
\frac{1}{2} r f' + f - 1 + \frac{ r f N'}{N} &=& 0 \,,
\label{eom2}\end{aligned}$$ and the $\phi\phi$ component is equivalent to the $\theta\theta$ component. We notice that Eq. (\[eom1\]) is a linear combination of Eqs. (\[css2\]), (\[css1\]) and (\[eom2\]). The field equations reduce then exactly to (\[css2\]), (\[css1\]) and (\[eom2\]). The counting of the independent initial data yields that the general solution possesses two arbitrary integration constants. Indeed, the system (\[css2\] - \[eom2\]) is manifestly invariant under scalings of $N$ and $r$. As in GR, the integration constant associated to scalings of $N$ can be absorbed by scaling the time coordinate, which is equivalent to impose the boundary condition $N|_{r=\infty} = 1$. Therefore, the general solution has one physical integration constant.
We close this section by commenting that the field equations (\[css2\] - \[eom2\]) are also valid for a theory that in addition has a term proportional to the square of the Cotton tensor, which was the original $z=3$ term Hořava considered [@Horava:2009uw]. Since three-dimensional spherically symmetric metrics are conformally flat, their Cotton tensor vanishes; hence, in the field equations all terms coming from the $(\mbox{Cotton})^2$ term trivially vanish. This fact gives further relevance to the spherically symmetric solutions in Hořava theory, since they arise in a more complete model in the sense that it has both the lower-order terms dominant at large distances and a high-order term that improves the renormalizability at microscopic scales.
Exact solutions {#sec:exactsolutions}
===============
In this section we focus ourselves on the exact solutions of the field equations (\[css2\]), (\[css1\]) and (\[eom2\]). This will be achieved by appealing to suitable local coordinate systems on each of the cases for $\alpha$ we shall meet.
From Eq. (\[css1\]) we have $$r^2 \sqrt{f} N' = C \,,
\label{rfnprima}$$ where $C$ is an integration constant. By a linear combination of Eqs. (\[css2\]), (\[eom2\]) and (\[rfnprima\]) we obtain the equation $$f - 1 + \frac{2 C \sqrt{f}}{r N}
+ \frac{\alpha}{2} \left( \frac{C}{r N} \right)^2 = 0 \,.
\label{algebraiceq}$$ Notice that this is an algebraic equation and it is quadratic in $\sqrt{f}$ and $(r N)^{-1}$. This is the key algebraic equation we mentioned in the Introduction. If we use the notation $$\beta \equiv \sqrt{\left|1 - \frac{\alpha}{2}\right|} \,,$$ this equation can be rewritten as one of the following two possibilities according to the value of $\alpha$: $$\begin{aligned}
\left( \sqrt{f} + \frac{C}{r N} \right)^2
- \left( \frac{\beta C}{r N} \right)^2 &=& 1
\hspace{2em} \mbox{if} \hspace{2em} \alpha < 2 \,,
\label{bnegative}
\\
\left( \sqrt{f} + \frac{C}{r N} \right)^2
+ \left( \frac{\beta C}{r N} \right)^2 &=& 1
\hspace{2em} \mbox{if} \hspace{2em} \alpha > 2 \,.
\label{bpositive}\end{aligned}$$ In the following subsections we analyze these two possibilities separately and at the end we complete with the $\alpha = 2$ case. We remark that after solving Eqs. (\[rfnprima\]) and (\[algebraiceq\]) it is straightforward to check that the remaining field equation is identically satisfied.
Case $\alpha < 2$
-----------------
We can give the most general solution of Eq. (\[bnegative\]) in terms of one-parameter families of solutions, $$\frac{\beta C}{r N} = s_1 \sinh{\chi} \,,
\hspace{2em}
\sqrt{f} + \frac{C}{r N} = s_2 \cosh{\chi} \,,
\label{parametricsolution}$$ where $\chi$ is an arbitrary parameter and $s_1$, $s_2$ are evaluated on any combination of signs $\pm 1$. Now, our aim is to regard the variable $\chi$ as a new radial coordinate such that the solutions for $N$ and $f$ can be explicitly expressed as functions of $\chi$. $\chi$, as well as $s_1$, $s_2$ and $C$ are restricted by positiveness of the field variables. We first note that, in order to preserve positiveness of $r$ and $N$, we must avoid changes of sign in $\sinh{\chi}$. We take the sector $\chi\in (0,+\infty)$; in Appendix \[apendix:equivalences\] we show that the final solutions in the sector $\chi\in (-\infty,0)$ are equivalent to the ones in $\chi\in (0,+\infty)$. Next, we have the condition $\sqrt{f} > 0$, which translates itself into $$- s_1 \beta^{-1} \sinh{\chi} + s_2 \cosh{\chi} > 0 \,.$$ There are four possibilities in the range $\chi\in (0,+\infty)$, each one valid for a specific sign of $C$ according to (\[parametricsolution\]),
1. $s_1 = +1$, $s_2 = +1$ : is a solution in the range $0 < \tanh{\chi} < \beta$ with $C > 0$.
2. $s_1=+1$, $s_2=-1$ : is not a solution.
3. $s_1=-1$, $s_2=+1$ : is a solution for all $\chi \in (0,+\infty)$ with $C<0$.
4. $s_1=-1$, $s_2=-1$ : is a solution in the range $\beta < \tanh{\chi} < \infty$ with $C < 0$.
Since we are in the $\alpha < 2$ range, we have that the GR limit $\alpha = 0$ corresponds to $\beta = 1$. $\beta < 1$ corresponds to $0 < \alpha < 2$ and $\beta > 1$ to $\alpha < 0$. This implies that for $\alpha \leq 0$ the domain of validity of solution (iv) is an empty set whereas solution (i) is valid in the full range $\chi\in (0,+\infty)$. For $0 < \alpha < 2$ the domains of solutions (i) and (iv) are both nonempty sets. They do not intersect themselves and their union together with the point $\hat{\chi}$ given by $$\tanh{\hat{\chi}} = \beta
\label{throat}$$ constitutes the full range $\chi\in (0,+\infty)$. We then see that there arises a further refinement of the range of $\alpha$: the solutions behave in different ways among the ranges $\alpha < 0$, $\alpha = 0$ and $0 < \alpha < 2$. We stress that these ranges have not arisen as consequence of the physics of any extra mode. They are a feature of the field equations for static spherically symmetric configurations.
Thus, the above list leaves us with two solutions in the $\alpha = 0$ and $\alpha < 0$ cases and three solutions in the $0 < \alpha < 2$ case. For the $0 < \alpha < 2$ range, however, the domains of validity of solutions (i) and (iv) suggest that they can be joined to form a single solution. This is just what happens, as we are going to see shortly. Notice that solutions (i) and (iv) have the same relative sign, $s \equiv s_1 s_2 = +1$, whereas solution (iii) has $s = -1$. We are going to see that the final solutions only depend on the relative sign $s$ and have a global (i.e., everywhere valid), positive, physical integration constant once they are written in the $\chi$ coordinate.
We may obtain a relation between $r$ and $\chi$ valid for any $s_1$, $s_2$ and $C$ in the full range $\alpha <2$. By combining Eqs. (\[rfnprima\]) and (\[parametricsolution\]) we get the equation $$\left( \frac{\cosh\chi}{\sinh\chi} - \frac{s}{\beta} \right)
\frac{d\chi}{d r}
= - \frac{1}{r} \,.
\label{diffeqcoord}$$ This equation can be integrated straightforwardly, yielding $$r = \frac{k e^{s \chi / \beta}}{\sinh{\chi}} \,,
\label{coordinatetransf}$$ where $k$ is an integration constant subject to $k > 0$ by consistency. Relation (\[coordinatetransf\]) allows us to regard $\chi$ as a new radial coordinate. For the two cases of $s$ the value $\chi = 0$ corresponds to the spatial infinity $r = \infty$.
By putting the transformation (\[coordinatetransf\]) back into Eqs. (\[parametricsolution\]) we obtain $N$ and $\sqrt{f}$ as explicit functions of $\chi$, $$N = s_1 \frac{ \beta C }{ k } e^{ - s \chi / \beta} \,,
\hspace{2em}
\sqrt{f} = s_1 ( s \cosh\chi - \beta^{-1} \sinh\chi ) \,.
\label{presolutions}$$ These expressions give the three solutions listed above explicitly in terms of the coordinate $\chi$. Note that for any of the three solutions $s_1 C$ is nothing but $|C|$, since in any case the sign of $C$ must be compensated with $s_1$.
Now, we proceed to show that in the $0 < \alpha < 2$ range solutions (i) and (iv), which have $s = +1$, can be smoothly joined. The joining point is $\hat{\chi}$ (\[throat\]) and at $\hat{\chi}$ the function $f$ given in (\[presolutions\]) is continuous and equal to zero. This is just a coordinate singularity associated to the fact that $f$ is a metric component in the coordinate $r$, as we are going to see. Next, if we adjust both the absolute value of $C$ and the value of $k$ for solutions (i) and (iv), $N$ is also continuous at $\hat{\chi}$ (but not vanishing).
Therefore, there are two solutions of the field equations in the whole $\alpha < 2$ case: the $s=+1$ solution, which is either the joining of solutions (i) and (iv) or just solution (i) if $0 < \alpha < 2$ or $\alpha \leq 0$ respectively, and the $s=-1$ solution, which is solution (iii) for all $\alpha < 2$. Actually, for both solutions functions $N$ and $f$ are given in global expressions in the whole range $\chi \in (0,+\infty)$. Indeed, $|C|$ and $k$ are the unique, global, integration constants the solutions have when they are written in the $\chi$ coordinate. As we discussed above, only one integration constant has physical meaning. The role of the sign $s_1$ in front of the the “unjoined” expression of $\sqrt{f}$ in (\[presolutions\]) for the $s=+1$ solution is equivalent to set $\sqrt{f}$ equal to the absolute value of the combination $\cosh\chi - \beta^{-1} \sinh\chi$ for all $\chi \in (0,+\infty)$. The everywhere valid expressions of the two solutions in $\chi\in (0,+\infty)$ are $$N = \frac{ \beta C }{ k } e^{ - s \chi / \beta} \,,
\hspace{2em}
f = ( \beta^{-1} \sinh\chi - s \cosh\chi )^2\,,
\label{nffinalalphal2}$$ where $C > 0$. Here and in the following we write the global integration constant $|C|$ just as a positive $C$ since we keep the solution in the $\chi$ coordinate, but the reader should keep in mind that this $C$ is different to the local integration constants arising in previous equations.[^9] Thus, the two solutions have a global physical integration constant and each solution is determined by the choice of $s = \pm 1$. Since expressions in (\[nffinalalphal2\]) are $C^{\infty}$ in $\chi\in (0,+\infty)$, the union of solutions (i) and (iv) for $0 < \alpha < 2$ is completely smooth. We impose the boundary condition $N^2|_{r=\infty} = 1$, which fixes the integration constant $k$ to $k = \beta C$.
By combining the functions (\[nffinalalphal2\]) with relations (\[diffeqcoord\]) and (\[coordinatetransf\]) we may write the space-time metric as an explicit tensor in terms of the radial coordinate $\chi$, obtaining the two metrics, $$ds_{(4)}^2 =
- e^{- 2 s \chi / \beta} dt^2
+ \frac{ (\beta C)^2 e^{2 s \chi / \beta}}{\sinh^4{\chi}}
( d\chi^2 + \sinh^2{\chi} d\Omega_{(2)}^2 ) \,,
\label{metricalphalower2}$$ each one determined by the choice of $s$. Both solutions are valid in the range $\chi \in (0,+\infty)$.
Once the spatial part of the metric is written in the $\chi$ coordinate, it arises explicitly as a conformal equivalent of the metric of the 3-hyperboloid in hyperbolic coordinates, $ds^2_{H^3} = d\chi^2 + \sinh^2{\chi} d\Omega_{(2)}^2$. This is the metric induced on the hyperboloid when immersed into a flat Lorentzian $\mathbb{R}^4$ ambient. Since both the hyperboloid and the (spatial) solutions are conformally flat, there always exist local coordinate systems under which one metric can be explicitly and locally written as a conformally transformed of the other one. With relation (\[coordinatetransf\]) we have found a coordinate system that realizes this conformal equivalence explicitly. The conformal factor, however, changes greatly the geometry of the spatial part of the solutions with respect to the 3-hyperboloid, as we are going to see in the following.
From the explicit expression (\[metricalphalower2\]) of the solutions in the coordinate $\chi$ we can compute curvature tensors. The nonzero components of the four-dimensional Riemann tensor are $$\begin{array}{l}
^{(4)}\!R_{t \chi \chi}{}^{t} =
- 2 (\beta \sinh{\chi})^{-1} (\beta^{-1} \sinh{\chi} - s \cosh{\chi}) \,,
\\[2ex]
^{(4)}\!R_{t \theta \theta}{}^{t} =
- {\displaystyle\frac{1}{2}} \sinh^2{\chi} R_{t \chi \chi}{}^{t} \,,
\hspace{2em}
^{(4)}\!R_{t \phi \phi}{}^{t} = \sin^2{\theta}\: {}^{(4)}\!R_{t \theta \theta}{}^{t} \,,
\\[2ex]
^{(4)}\!R_{\chi\theta\theta}{}^{\chi} =
\beta^{-1} \sinh{\chi} (\beta \sinh{\chi} - s\cosh{\chi}) \,,
\hspace{2em}
^{(4)}\!R_{\chi \phi \phi}{}^{\chi} =
\sin^2{\theta}\:{}^{(4)}\!R_{\chi\theta\theta}{}^{\chi} \,, \\[2ex]
^{(4)}\!R_{\theta \phi \theta}{}^{\phi} =
- \beta^{-2}( 1 - (\beta \sinh{\chi} - s\cosh{\chi})^2 ) \,.
\end{array}$$ The 4D and 3D Ricci scalars are equal to $$^{(4)}\!R = R =
- \frac{\alpha}{\beta^4 C^2} e^{-2 s \chi / \beta} \sinh^4{\chi} \,.
\label{ricciscalar1}$$
Let us start the study of the geometry of solutions (\[metricalphalower2\]) by taking the asymptotic limit, which can be done simultaneously for both $s$. The radial coordinate $\chi$ is bad frame to study asymptotic behavior, since the metric components (\[metricalphalower2\]) diverge at $\chi = 0$. However, we can observe the solutions near the spatial infinity by coming back to the original radial coordinate in an approximate way since the coordinate transformation (\[coordinatetransf\]) can be inverted in the limit when $\chi \rightarrow 0$ and simultaneously $r \rightarrow \infty$. The linearized version of the transformation (\[coordinatetransf\]) is $$\frac{1}{r} = \frac{\chi}{\beta C} \,,$$ which is valid for both $s$. Using this transformation in (\[nffinalalphal2\]) we obtain the expanded version of $N^2$ and $f$ at linear order in $r^{-1}$, $$N^2 = f = 1 - s \frac{2C}{r} \,.
\label{expansions}$$ Thus, we have that both solutions are asymptotically flat for any $\alpha$ in the range $\alpha < 2$. The sign of the external mass is given by the choice of $s$. In Section \[sec:perturbations\] we shall present the asymptotic expansion of the exact solutions up to $1/r^3$ order. As a consequence of the asymptotic flatness, the Ricci scalars given in (\[ricciscalar1\]) vanish at $\chi = 0$ for both cases of $s$. Note that this behavior departs greatly from the 3-hyperboloid, which is a manifold of constant, negative, curvature.
![\[fig:rdechialfamenor\] Left: $r$ as a function of $\chi$ with $s = +1$ for the three possibilities of $\alpha$ with respect to $\alpha = 0$ in the $\alpha < 2$ case. $r$ has a critical point only in the case $0 < \alpha < 2$. For $\alpha = 0$ $r$ asymptotes to $r = R_S$ whereas for $\alpha < 0$ asymptotes to $r = 0$. Right: Case $s = -1$ for the same three possibilities of $\alpha$. In this case $r$ decreases monotonically towards $r = 0$ without critical points for all $\alpha < 2$.](./r.eps "fig:") ![\[fig:rdechialfamenor\] Left: $r$ as a function of $\chi$ with $s = +1$ for the three possibilities of $\alpha$ with respect to $\alpha = 0$ in the $\alpha < 2$ case. $r$ has a critical point only in the case $0 < \alpha < 2$. For $\alpha = 0$ $r$ asymptotes to $r = R_S$ whereas for $\alpha < 0$ asymptotes to $r = 0$. Right: Case $s = -1$ for the same three possibilities of $\alpha$. In this case $r$ decreases monotonically towards $r = 0$ without critical points for all $\alpha < 2$.](./rminussign.eps "fig:")
Now we study the full geometry of the solutions (\[metricalphalower2\]). In Fig. \[fig:rdechialfamenor\] we plot $r$ against $\chi$ according to (\[coordinatetransf\]) and considering the two possibilities of $s$. We see that, in all cases, as $\chi$ departs from zero $r$ decreases from infinity; but in the $s = +1$ case $r$ may either reach a critical point for finite $\chi$ or be monotonically decreasing. From (\[coordinatetransf\]) we find that the critical point is just the joining point $\hat{\chi}$ defined in (\[throat\]), which only arises in the range $0 < \alpha < 2$ for the $s = +1$ solution. The other ranges of $\alpha$ in the $s = +1$ case give a monotonically decreasing $r$ with an asymptote for $\chi \rightarrow \infty$. It can be checked from Eq. (\[coordinatetransf\]) that the asymptotes are different for $\alpha = 0$ and $\alpha < 0$. We see again that the qualitative features of the solution $s= +1$ are discontinuously different in the ranges $\alpha = 0$, $0 < \alpha < 2$ and $\alpha < 0$. In the following we describe these solutions separately and also the $s=-1$ solution, for which $r$ is everywhere monotonically decreasing in $\chi$ towards the origin $r=0$.
- [**Schwarzschild solution for the $\alpha = 0$ point**]{}\
For $\alpha = 0$ the coordinate transformation (\[coordinatetransf\]) can be explicitly inverted, yielding $$e^{-2 s \chi} = 1 - s \frac{2 C}{r} = N^2 = f \,.$$ This is the Schwarzschild solution, as expected. The case $s=+1$ represents the positive-mass solution, whereas $s=-1$ is the negative-mass one. For the positive-mass case the radius $r$ asymptotes to the Schwarzschild radius $R_S = 2C$ as $\chi \rightarrow \infty$, as can be seen in Fig. \[fig:rdechialfamenor\]. Thus, (\[metricalphalower2\]) with $s = +1$ in the $\alpha = 0$ case describes the exterior region of the (positive mass) Schwarzschild space-time. For $s= -1$ the coordinate $\chi$ covers the space up to the origin $r=0$, which in this case correspond to $\chi = +\infty$, and at the origin both $N^2$ and $f$ diverge.
- [**$s=+1$ solution of the range $0 < \alpha < 2$**]{}\
The joining point $\hat{\chi}$ corresponds to a lower bound for $r$, which is equal to $$\hat{r} =
C \left(
\frac{ ( 1 - \beta^2 )^{(1+\beta) / 2}}{1 - \beta}
\right)^{1/\beta} \,.
\label{rhat}$$ As $\chi$ continues beyond $\hat{\chi}$ the space extends itself increasing again the values of the radius $r$, as can be seen in Fig. \[fig:rdechialfamenor\]. This kind of space is a wormhole geometry constituted by two spatial branches joined by a throat, which is located at $\hat{\chi}$. This geometry was described by the authors of Ref. [@Eling:2006df] as a solution of the EA theory. At the throat a 2-sphere of minimal area is reached. The spatial infinity we identified above with the value $\chi = 0$ and where we imposed the boundary conditions and obtained the asymptotic flatness corresponds to the infinite end of one of the branches. Let us call branch I to this sector. In the other branch, which we call branch II, $r$ also goes from $\hat{r}$ to the spatial infinity $r = \infty$ (This was called the “interior” branch in [@Eling:2006df]). The original radial coordinate $r$ can be used to cover separately the two branches from their respective infinite boundaries down to the throat $\hat{r}$, but, unlike $\chi$, it fails to cover the throat itself. The $\chi$ coordinate covers the throat and an open set around it that extends itself over the whole wormhole except at the infinite boundary.
Notice that the function $f$ vanishes at the throat. This is just a coordinate singularity since $f$ is a metric component under the $r$ coordinate. It can be seen from (\[metricalphalower2\]) that the metric components in the $\chi$ coordinate are all regular at $\hat{\chi}$. Actually $-N^2$ is one of these components; its value at the throat is given by $-e^{-2 \hat{\chi}/\beta}$, which approaches zero as $\alpha \rightarrow 0$. Moreover, the field equations (\[css2\] - \[eom2\]) themselves and the algebraic equation (\[bnegative\]) cannot be trusted at the throat since these equations are written in the $r$ coordinate. It can be checked that the field equations in the $\chi$ coordinate are exactly solved, even at the throat. We may also see that the Ricci scalar (\[ricciscalar1\]) is regular at the throat $\hat{\chi}$.
At the spatial infinity of the branch II ($\chi \rightarrow \infty$) there also arise singularities. But these are very different since in this case both $N^2$ and $f^{-1}$ vanish. This singularity was characterized in Ref. [@Eling:2006df] in the context of the EA theory. In particular it was studied the finiteness of radial light rays directed to this boundary. However, in Hořava theory light rays do not necessarily represent an upper bound for matter velocity. At the infinity of branch II $^{(4)}R$ diverges if $0 < \alpha < 3/2$ and vanishes if $3/2 < \alpha <2$. Indeed, $^{(4)}R$ has a critical point somewhere in the wormhole when the range is $3/2 < \alpha < 2$; whereas it is monotonically decreasing from the zero value at the boundary of branch I to a negative divergence at the boundary of branch II when $0 < \alpha < 3/2$. To clearly contrast this, in Fig. \[fig:ricci\] we plot the Ricci scalar of the wormhole for these two cases. The critical point of the Ricci scalar is reached at $$\tanh{\chi_c} = 2\beta \,,$$ which is bigger than $\hat{\chi}$. That is, the critical point of $^{(4)}R$ is reached beyond the throat, in branch II.
![\[fig:ricci\] The four-dimensional Ricci scalar on the whole wormhole solution of the range $0 < \alpha <2$. For $0 < \alpha < 3/2$ it decreases monotonically towards a negative divergence at the boundary of branch II. For $3/2 < \alpha < 2$ it has a critical point after passing the throat and reaches again a zero at the boundary of branch II.](./ricci.eps)
For small $\alpha$, $\beta \rightarrow 1^-$, such that $\hat{\chi}$ tends to infinity. Therefore, the location $\hat{r}$ of the throat for $\alpha$ near to zero approximates to the Schwarzschild radius from above. We shall further develop on this connection in Section \[sec:perturbations\]. The perturbative approach for small $\alpha$ we shall perform there yields the perturbative version of branch I of this exact solution.
The location $\hat{r}$ of the throat has also an upper bound in its running with respect to $\alpha$. As $\alpha$ approaches $\alpha = 2$, $\hat{r} \rightarrow e C$ from below ($e$ stands for the Euler number); that is, $\hat{r}$ becomes $e/2$ times the Schwarzschild radius. In general, $\hat{r}$ is monotonically increasing from $2C$ to $eC$ when run in $\alpha \in [0,2]$. We shall see that there is an interesting connection of this upper bound with the $\alpha > 2$ case.
- [**$s=+1$ solution of the $\alpha < 0$ range**]{}\
The case $\alpha < 0$ for the $s=+1$ solution (\[metricalphalower2\]) is qualitatively quite different to the previous two cases. The coordinate transformation (\[coordinatetransf\]) is completely bijective in the full range of $\chi$. $r$ decreases monotonically from $r = \infty$ down to the origin $r=0$ as $\chi$ grows. Thus, there is no analogous for the throat of the $0 < \alpha < 2$ case nor horizon as in the Schwarzschild solution. The discontinuous lacking of a radial lower bound as one passes to the $\alpha < 0$ sector produces a failure in the perturbative solution for the $\alpha < 0$ case (see Section \[sec:perturbations\]). At the point $\coth{\chi} = \beta$ the function $f$ has a critical point and after it increases monotonically towards $r = 0$. Thus, function $f$ is nonzero and regular in the whole domain of the radial coordinate $r$, except at $r = 0$ where it diverges. This divergence is a naked essential singularity (in the terminology appropiated for relativistic matter).[^10] The four-dimensional Ricci scalar $^{(4)}R$ (\[ricciscalar1\]) is monotonically increasing from the zero value at $r = \infty$ towards a divergence at the origin $r=0$.
- [**$s=-1$ solution, full range $\alpha < 2$**]{}\
This is the case of the asymptotically flat solution with negative mass. In Fig. \[fig:rdechialfamenor\] we plot $r$ as a function of $\chi$ for this case. The coordinate transformation behaves in a similar way to the $s=+1$ solution in the $\alpha < 0$ range. The transformation is bijective in its full range, $r$ decreases monotonically from $r=\infty$ to the origin $r=0$ as $\chi$ goes from zero to infinity. There is no lower bound of validity for the radial coordinate $r$, so the whole space can be covered both with it or $\chi$. Unlike the $s=+1$ solution, there are no subranges of $\alpha$ in which the behavior of the solution changes qualitatively; hence the physical features of the solution vary smoothly in the full range $-\infty < \alpha <2$. This includes the $\alpha = 0$ case, which is the negative-mass Schwarzschild solution. The metric is singular at the origin in both coordinate systems, where there is a naked essential singularity. The Ricci scalar (\[ricciscalar1\]) decreases monotonically from the flat spatial infinite towards a negative divergence at the origin, which is labeled by $\chi = \infty$.
Case $\alpha > 2$
-----------------
We take the following four solutions of Eq. (\[bpositive\]), $$\frac{\beta C}{r N} = s_1 \sin{\varphi} \,,
\hspace{2em}
\sqrt{f} + \frac{C}{r N} = s_2 \cos{\varphi} \,.
\label{solution2}$$ In this case $\varphi$ arises as the appropriated transformed radial coordinate. There are four more solutions of Eq. (\[bpositive\]), they have the sine and cosine functions exchanged with respect to (\[solution2\]). It turns out that those solutions lead to configurations that are equivalent to the ones (\[solution2\]) yields, see Appendix \[apendix:equivalences\]. Thus, solutions (\[solution2\]) exhaust all the different solutions of the field equations.
The analysis of the domains of validity and joining of the local solutions in (\[solution2\]) follows the same lines of the $\alpha < 2$ case. Positiveness of $r$ and $N$ leaves us with the domain $\varphi\in (0,\pi)$; in Appendix \[apendix:equivalences\] we show that the domain $\varphi \in (\pi,2\pi)$ leads to the same solutions after a coordinate transformation. In $\varphi\in (0,\pi)$ positiveness of $\sqrt{f}$ leads to the domain of validity
1. $s_1 = +1$, $s_2 = +1$ : solution in $\varphi\in (0,\arctan{\beta})$ with $C>0$.
2. $s_1 = +1$, $s_2 = -1$ : solution in $\varphi\in (\arctan{(-\beta)},\pi)$ with $C>0$.
3. $s_1 = -1$, $s_2 = +1$ : solution in $\varphi\in (0,\arctan{(-\beta)})$ with $C<0$.
4. $s_1 = -1$, $s_2 = -1$ : solution in $\varphi\in (\arctan{\beta},\pi)$ with $C<0$.
In the $\alpha > 2$ case $\beta = \sqrt{\alpha/2 -1}$ and it can take any positive value, $\beta > 0$. Therefore, all of the four local solutions have nonempty domains of validity for any $\alpha > 2$. There are two global solutions, each one identified by the relative sign $s \equiv s_1 s_2$. Solutions (i) and (iv) are joined to form the $s=+1$ solution whereas (ii) and (iii) form the $s=-1$ solution. It turns out that the global $s=-1$ solution is equal to the $s=+1$ one after a coordinate change. We show this in Appendix \[apendix:equivalences\]. Therefore, in the $\alpha > 2$ case there is only one solution. The joining point for solutions (i) and (iv) is given by $$\tan{\hat{\varphi}} = \beta \,.$$
By repeating the same steps of the previous case, for the $s=+1$ solution we find the equations for the coordinates, $$\left( \frac{\cos{\varphi}}{\sin{\varphi}} - \frac{1}{\beta} \right)
\frac{d\varphi}{dr} =
- \frac{1}{r} \,,$$ whose integral is $$r =
\frac{k e^{\varphi / \beta}}{\sin{\varphi}} \,,
\label{coordinatetrans2}$$ where $k > 0$. The value $\varphi = 0$ corresponds to the spatial infinity $r=\infty$. For any $\alpha > 2$ transformation (\[coordinatetrans2\]) has a finite critical point exactly at the joining point $\hat{\varphi}$. This signals that this solution is very similar to the $s=+1$ solution of the $0 < \alpha < 2$ case, it is a wormhole.
The global expression of the solution in $\varphi\in (0,\pi)$ is $$N = \frac{\beta C}{k} e^{-\varphi / \beta} \,,
\hspace{2em}
f = ( \beta^{-1} \sin{\varphi} - \cos{\varphi} )^2 \,,
\label{fngreaterthan2}$$ where $C > 0$. Condition $N|_{\varphi = 0} = 1$ fixes $k = \beta C$. We can again write the metric tensor in terms of the $\varphi$ coordinate. It is $$ds_{(4)}^2 =
- e^{-2\varphi / \beta} dt^2
+ \frac{(\beta C)^2 e^{2\varphi / \beta}}{\sin^4{\varphi}}
(d\varphi^2 + \sin^2{\varphi} d\Omega_{(2)}^2) \,.
\label{solutionalphagreater}$$ The solution is valid in $\varphi \in (0, \pi)$. In this case the coordinate $\varphi$ allows to realize explicitly the local conformal equivalence between the solution and the 3-sphere with its standard metric in 3 spherical coordinates. The nonzero components of the Riemann tensor and the Ricci scalar are $$\begin{aligned}
&&\begin{array}{l}
^{(4)}\!R_{t \varphi \varphi}{}^{t} =
- 2 (\beta \sin{\varphi})^{-1} (\beta^{-1} \sin{\varphi} - \cos{\varphi}) \,,
\\[2ex]
^{(4)}\!R_{t \theta \theta}{}^{t} =
- {\displaystyle\frac{1}{2}} \sin^2{\varphi} R_{t \varphi \varphi}{}^{t} \,,
\hspace{2em}
^{(4)}\!R_{t \phi \phi}{}^{t} = \sin^2{\theta}\: {}^{(4)}\!R_{t \theta \theta}{}^{t} \,,
\\[2ex]
^{(4)}\!R_{\varphi\theta\theta}{}^{\varphi} =
- \beta^{-1} \sin{\varphi} (\beta \sin{\varphi} + \cos{\varphi}) \,,
\hspace{2em}
^{(4)}\!R_{\varphi \phi \phi}{}^{\varphi} =
\sin^2{\theta}\:{}^{(4)}\!R_{\varphi\theta\theta}{}^{\varphi} \,, \\[2ex]
^{(4)}\!R_{\theta \phi \theta}{}^{\phi} =
\beta^{-2}( 1 - (\beta \sin{\varphi} + \cos{\varphi})^2 ) \,,
\end{array}
\\
&&^{(4)}\!R = R =
- {\displaystyle\frac{\alpha}{\beta^4 C^2}} e^{-2 \varphi / \beta} \sin^4{\varphi} \,.
\label{ricciscalar2}\end{aligned}$$
Solution (\[solutionalphagreater\]) also describes a wormhole formed by two spatial branches joined by a throat at $\varphi = \hat{\varphi}$, where a minimal 2-sphere is reached. The coordinate $\varphi$ covers the throat and a open neighborhood of it that extends itself up to infinity. By doing similar considerations as in the previous case, we get that at the spatial infinity labeled by $\varphi = 0$ the solution is asymptotically flat with positive mass, $$N^2 = f = 1 - \frac{2 C}{r} \,.$$ The Ricci scalar (\[ricciscalar2\]) vanishes at the infinity $\varphi = 0$. There is, however, an important difference with respect to the wormhole solution of the $0 < \alpha < 2$ range. At the other infinity $r=\infty$, which is labeled by $\varphi = \pi$, neither $N^2$ nor $f^{-1}$ vanish. On the contrary, they exhibit asymptotically flat behavior with negative mass, $$f = 1 + \frac{ 2 C e^{\pi / \beta}}{r} \,,
\hspace{2em}
N^2 = e^{- 2 \pi / \beta} f \,.$$ Consequently, Ricci scalar (\[ricciscalar2\]) also vanishes at this end. Actually, both in the $0 < \alpha < 2$ and $\alpha > 2$ wormholes $N^2$ is monotonically decreasing as one walks from the infinite boundary of one branch to the infinite boundary of the other one, including the passing through the throat. But the difference arises when the $\varphi$ coordinate in the $\alpha > 2$ case reaches its upper bound, $\varphi = \pi$. There the function $N^2$ ends with a finite nonzero value. In the wormhole of the $0 < \alpha < 2$ case the $\chi$ coordinate runs up to infinity and the function $N^2$ decreases completely to zero.
Thus, we have that both branches of this wormhole solution exhibit asymptotically flat behavior. In one branch the maximum value of $N^2$ is reached at the infinite boundary whereas in the other branch the opposite case occurs: at the infinite boundary the minimum (nonzero) of $N^2$ is reached. This renders as a rather artificial matter the choice of one or the other branch as the reference place to demand asymptotic flatness and hence indicating the sign of the mass of a plausible external source.
As in the wormhole of the $0 < \alpha < 2$ range, function $f$ vanishes at the throat; it can be checked that the field equations are consistently solved once they are written in the $\varphi$ coordinate. Four-dimensional Ricci scalar is regular at the throat and has a critical point at $\tan{\varphi_c} = 2\beta$, which again is a location bigger than the throat. Unlike the wormhole of the $0 < \alpha <2$ range, this solution is completely regular.
![\[fig:rhat\] The location $\hat{r}$ of the throat of the wormhole solutions of the ranges $0 < \alpha <2$ and $\alpha > 2$ varied as a function of the coupling constant $\alpha$. For the $0 < \alpha < 2$ solution the running starts just above $R_S$ and ends just below $e/2$ times the Schwarzschild radius; whereas for the $\alpha > 2$ solution starts just above of that value.](./rhat.eps)
Interestingly, as $\alpha$ approaches $\alpha = 2$ from above, $\beta \rightarrow 0^+$, such that $\hat{\varphi} \rightarrow 0$ and $\hat{r} \rightarrow e C$ from above. Recall that in the $0 < \alpha < 2$ solution $\hat{r}$ goes to the same value from below when $\alpha \rightarrow 2^-$. Thus, the solution (\[solutionalphagreater\]) can be regarded in some sense as the continuation of the $s=+1$ solution of the $0 < \alpha < 2$ case; although the continuation cannot be passed through the $\alpha = 2$ point, as we are going to see below, and the singularity is regularized at the end of branch II. The location $\hat{r}$ of the throat is monotonically increasing when run in $\alpha \in (2,\infty)$, spanning the range $(e C, \infty)$. In Fig. \[fig:rhat\] we put together the runnings of $\hat{r}$ with respect to $\alpha$ for the cases $0 < \alpha < 2$ and $\alpha > 2$.
Case $\alpha = 2$
-----------------
To analyze this special subspace of parameters we need to come back to the original static field equations (\[eisnteinstatic\]) and (\[hamiltoniancstatic\]). When $\alpha = 2$, Eq. (\[hamiltoniancstatic\]) is just the trace of Eq. (\[eisnteinstatic\]), thus in this case the theory lacks one field equation for static configurations. Eq. (\[eisnteinstatic\]) can be rewritten in the form $$R^{ij} - N^{-1} ( \nabla^i \nabla^j N + g^{ij} \nabla^2 N )
+ 2 N^{-2} \nabla^i N \nabla^j N = 0 \,.
\label{eomalpha2}$$ After inserting the static spherically symmetric ansatz with vanishing shift function (\[ansatz\]) into this equation, we get that it leads to two different equations: $$\begin{aligned}
\frac{N''}{N} + \frac{f' N'}{2 f N} + \frac{f'}{2 r f}
+ \frac{N'}{r N} - \left( \frac{N'}{N} \right)^2 &=& 0 \,,
\label{eom1alpha2}
\\
\frac{N''}{N} + \frac{f' N'}{2 f N} + \frac{f'}{2 r f}
+ \frac{3 N'}{r N} + \frac{f - 1}{r^2 f} &=& 0 \,.
\label{eom2alpha2}\end{aligned}$$ A combination of these two equations yields $$\left( \frac{r N'}{N} + 1 \right)^2 = \frac{1}{f} \,.$$ This has the two solutions Kiritsis [@Kiritsis:2009vz] found: $N$ is given in terms of $f$ as $$\frac{r N'}{N} = -1 \pm \frac{1}{\sqrt{f}}
\label{nalpha2}$$ and $f$ is arbitrary. It is straightforward to check that (\[nalpha2\]) solves Eqs. (\[eom1alpha2\]) and (\[eom2alpha2\]) for all $f(r)$. Therefore, we see that in the $\alpha = 2$ case the lacking of one field equation leaves one metric function indeterminate. Because of this, the $\alpha = 2$ case was called degenerated by Kiritsis.
Perturbative solutions {#sec:perturbations}
======================
In this section we proceed to solve the Eqs. (\[css2\] - \[eom2\]) in an approximate way by assuming an small value of $\alpha$. We call such solutions the *perturbative solutions*. Since the approximation consists of expanding the field equations up to linear order in $\alpha$, it is equivalent to do perturbations on the Schwarzschild solution with a scale of the order of $\alpha$ for the perturbations. For solving the field equations the strategy will consist of combining Eqs. (\[css2\]) and (\[eom2\]) to obtain a differential equation for $f$ that can be solved perturbatively in $\alpha$. Then, we shall put the solution for $f$ back into Eq. (\[eom2\]) and manage to find the perturbative solution for $N$. Finally, we shall check that Eq. (\[eom1\]) is solved up to linear order by the $f$ and $N$ found.
One may solve Eq. (\[eom2\]) for $N'/N$ and substitute the resulting expression into Eq. (\[css2\]), obtaining an equation for $f$ that can be written in the form $$8 ( 1 - h ) (r h)'
+ \alpha \left[ \left( (r h)' - h \right)^2
+ 4 (r h)' \right] = 0 \,,
\label{eqh}$$ where $$h \equiv 1 - f \,.$$
Now we start the perturbations. We assume that the functions $f$ and $N$ are linear-order polynomials in $\alpha$. At zeroth order in $\alpha$, Eq. (\[eqh\]) reduces to $(r h^{(0)})' = 0$; its solution is proportional to $r^{-1}$; hence, we obtain the Schwarzschild factor $$f^{(0)} = 1 - \frac{A}{r} \,,$$ where $A$ is an integration constant. By substituting $f^{(0)}$ into Eq. (\[eom2\]) we obtain the zeroth-order lapse function $$N^{(0)} = \left( 1 - \frac{A}{r} \right)^{1/2} \,,$$ where the multiplicative integration constant that arises in this step has been fixed to unity by imposing the boundary condition $N|_\infty = 1$.
The linear-order function $h$ is obtained by expanding $h = h^{(0)} + \alpha h^{(1)}$ and substituting this expansion and the solution for $h^{(0)}$ into Eq. (\[eqh\]). After expanding the resulting equation up to linear order in $\alpha$, we obtain an equation for $h^{(1)}$, $$(r h^{(1)})' =
- \frac{A^2}{8 r^2} \left( 1 - \frac{A}{r} \right)^{-1} \,,$$ which can be integrated straightforwardly, $$h^{(1)} =
-\frac{A}{8 r} \ln{\left( 1 - \frac{A}{r} \right)} + \frac{B}{r}\,,$$ where $B$ is an integration constant.
Similarly, to obtain the linear-order $N$ we expand $N = N^{(0)} + \alpha N^{(1)}$ and substitute into Eq. (\[eom2\]). By expanding the resulting equation up to linear order, we get $$\left( \frac{ N^{(1)} }{ N^{(0)} } \right)' =
- \frac{1}{16 r^2} \left(1 - \frac{A}{r}\right)^{-2}
\left[ \frac{A^2}{r}
+ A \ln{\left(1 - \frac{A}{r}\right)}
- 8 B \right] \,.$$ Its integral is $$\frac{ N^{(1)} }{ N^{(0)} } =
\frac{1}{8}\left( 1 - \frac{A}{r} \right)^{-1}\left[
1 + \left( 1 - \frac{A}{2 r} \right)
\ln{\left( 1 - \frac{A}{r} \right)}
- \frac{4 B}{A} \right]
+ D \,,$$ where $D$ is an integration constant. The boundary condition $N|_\infty = 1$ fixes $D = ( 4 B / A - 1 )/8$; thus, we have $$N^{(1)} =
\frac{1}{8} \left( 1 - \frac{A}{r} \right)^{-1/2}
\left[ \frac{A - 4 B}{r} + \left( 1 - \frac{A}{2 r} \right)
\ln{\left( 1 - \frac{A}{r} \right)} \right] \,.$$ This exhausts the linearized Eqs. (\[css2\]) and (\[eom2\]). It is a matter of straightforward computations to check that the solutions we have found for $f$ and $N$ solve the linearized version of Eq. (\[eom1\]).
We have arrived at the linear-order perturbative solution $$\begin{array}{rcl}
N(r) &=& {\displaystyle
\left( 1 - \frac{A}{r} \right)^{1/2}
+ \frac{\alpha}{8} \left( 1 - \frac{A}{r} \right)^{-1/2}
\left[ \frac{A - 4 B}{r} + \left( 1 - \frac{A}{2 r} \right)
\ln{\left( 1 - \frac{A}{r} \right)} \right] } \,,
\\[2ex]
f(r) &=& {\displaystyle
1 - \frac{A + \alpha B}{r}
+ \frac{\alpha A}{8 r} \ln{\left( 1 - \frac{A}{r} \right)} } \,,
\end{array}
\label{perturbativesolution}$$ which is valid for the range $r > A$. In the $\alpha = 0$ case this solution reproduces the Schwarzschild metric, $A$ being the Schwarzschild radius.
Unlike the Schwarzschild solution of general relativity which has only one arbitrary integration constant, the solution we have obtained has two integration constants: $A$ and $B$. This apparent excess of free parameters has no physical meaning nor it is a mathematical inconsistency; it is just a consequence of the fact that we are not dealing with exact solutions, but with approximated ones. One may check that, when evaluating the Eqs. (\[css2\] - \[eom2\]) on the perturbative solution, the parameter $B$ gets involved only in terms of the quadratic or higher order in $\alpha$. Thus, there is no way to fix it by solving the field equations at linear order in $\alpha$. On the other hand, in Appendix \[appendix:expanding\] we expand directly the exact solution up to linear in $\alpha$, obtaining the value $B = 0$. Hence, for now on we take this value for $B$.
We study the behavior of the perturbative solution (\[perturbativesolution\]) at two limits: near the Schwarzschild radius and the asymptotic limit for large $r$. For simplicity, let us restrict the integration constant $A$ to be positive. This will leads us to the positive mass solutions, which are the most interesting ones physically (the perturbative solution (\[perturbativesolution\]) is valid for both cases).
We start by studying the behavior of the $f$ function, $$f(r) =
1 - \frac{A}{r}
+ \frac{\alpha A}{8 r} \ln{\left( 1 - \frac{A}{r} \right)} \,,$$ for $r \sim A = R_S$. In Fig. \[fig:perturbativefnearhorizon\] we plot the function $f(r)$ at this limit for the three possibilities of $\alpha$. We recall that the domain of validity of the solution falls into $0 < 1 - A/r < 1$ for all $\alpha$. If $\alpha > 0$ function $f$ has a root at some value $\hat{r}$ greater than and near the Schwarzschild radius, $\hat{r} \gtrsim A$, given by the solution of the equation[^11] $$1 - \frac{A}{\hat{r}}
= - \frac{\alpha A}{8 \hat{r}} \ln{\left( 1 - \frac{A}{\hat{r}} \right)} \,.
\label{rootf}$$ Therefore, the perturbative solution is valid up to the value $\hat{r}$ where it has a (coordinate) singularity. This $\hat{r}$ is the perturbative version of the throat we found in the exact solution, which holds for $\alpha > 0$.
![\[fig:perturbativefnearhorizon\]The perturbative function $f$ near the Schwarzschild radius $R_S$ for the three cases of $\alpha$. The curve $\alpha = 0$ is the plot of the Schwarzschild factor which vanishes at $R_S$. The function $f$ for $\alpha > 0$ is monotonically decreasing towards the throat $\hat{r}$ where it vanishes. For $\alpha < 0$ $f$ extends its domain down to $R_S$. Near $R_S$ it reaches a minimum and after it increases monotonically towards a divergence at $R_S$.](./fperturbative.eps)
On the other hand, in the case $\alpha < 0$ the function $f$ has not any root (Eq. (\[rootf\]) has no solution in $A < r < \infty$ with $\alpha < 0$); hence, its domain extends down to the Schwarzschild radius $A$. Instead, near the Schwarzschild radius it has a critical point given by the equation $$1 - \frac{A}{r} =
\frac{\alpha}{8} \left[
\left( 1 - \frac{A}{r} \right) \ln{\left( 1 - \frac{A}{r} \right)}
- \frac{A}{r} \right] \,.$$ As one moves from this critical point towards the Schwarzschild radius the function $f$ grows monotonically without upper bound, exhibiting a divergence at $r = A$. This behavior departs drastically from Schwarzschild solution, which always decreases monotonically towards the Schwarzschild radius. Since the difference between this two functions increases unavoidably as one approaches the Schwarzschild radius (in a region of nonzero measure), the perturbative solution cannot be trusted for the case $\alpha < 0$ if one is interested in the region near the Schwarzschild radius. For $\alpha > 0$ the perturbative solution is totally admissible in the range $\hat{r} > r > \infty$. $\hat{r}$ can get close to the Schwarzschild radius as wish by lowering $\alpha$. The sector $\hat{r} > r > \infty$ corresponds to the branch I of the exact wormhole solution. As in the exact solution, function $N^2$ decreases monotonically as one goes to the throat, reaching a nonzero value there. Notice that the $\alpha < 0$ perturbative solution can still be admissible for values of the radius much bigger than the Schwarzschild radius.
We now study the asymptotic behavior of the perturbative solution (\[perturbativesolution\]). Notice that neither $N^2$ nor $f$ have contributions of order $\alpha$ to the mode $1/r$; indeed, this mode is $A/r$ for both functions, thus we identify the integration constant as $A = 2 G M$, where $M$ is the mass of an external source. The asymptotic expansion, up to $1/r^3$ order, of the perturbative solution is $$\begin{array}{rcl}
N^2 &=&
{\displaystyle 1 - \frac{2 G M}{r} - \frac{\alpha (2 G M)^3}{48 r^3}
+ \mathcal{O}\left(\frac{1}{r^4}\right) } \,,
\\[2ex]
f &=&
{\displaystyle 1 - \frac{2 G M}{r} - \frac{\alpha (2 G M)^2}{8 r^2} - \frac{\alpha (2 G M)^3}{16 r^3} + \mathcal{O}\left(\frac{1}{r^4}\right) } \,.
\end{array}
\label{asymptoticexpansion}$$
One may contrast this asymptotic expansion with the exact solution without need of finding the latter explicitly, since it is easy to check whether an expansion in $1/r$ as (\[asymptoticexpansion\]) solves the field equations up to a given order in $1/r$. Remarkably, it turns out that (\[asymptoticexpansion\]) is the asymptotic solution, up to $1/r^3$ order, of the field equations (\[css2\] - \[eom2\]) *without expanding in $\alpha$*. That is, (\[asymptoticexpansion\]) is precisely the asymptotic expansion of the *exact* solution up to $1/r^3$ order, which turns out to be of linear order in $\alpha$. The asymptotic expansion we show in (\[asymptoticexpansion\]) for the perturbative and exact solutions coincides with the expansion of the exact solution shown in Ref. [@Kiritsis:2009vz], except for the sign of the $1/r^3$ term in the $N^2$ expansion, which we found to be negative.
Discussion and conclusions {#discussion-and-conclusions .unnumbered}
==========================
We have obtained explicitly the static spherically symmetric solutions of the complete nonprojectable Hořava theory, which depends on the vector $\partial_i \ln N$. We have found the components of the space-time metrics explicitly as functions of local coordinates. We have found the solutions for the lowest-order effective action (without cosmological constant) since this kind of configurations are mainly interesting for large-distance physics. We have imposed the condition of vanishing of the shift function in order to simplify the computations. Configurations of the same kind but with nonzero shift function deserve to be further investigated. The only undetermined coupling constant the solutions have is the one of the $(\partial_i \ln N)^2$ term, which is $\alpha$. Indeed, although the $\lambda = 1/3$ and $\lambda \neq 1/3$ theories are in general qualitatively different since their number of propagating degrees of freedom differ (two for the former and three for the latter), the static solutions (with vanishing shift function) are the same for both cases since there is no influence of kinetic terms on such configurations. In Table 1 we show the several solutions found according to the ranges of $\alpha$.
By managing the Lagrangian field equations in order to extract the constraints of the theory (see Appendix \[appendix:hamiltonian\]), we have obtained an algebraic field equation for $N$ and $f$ that could be solved in a closed way. This equation splits out in the cases $\alpha < 2$ and $\alpha > 2$. Its solutions are given in terms of one-parameter families. In all cases we have regarded the free parameter as a transformed radial coordinate. With the transformed coordinate we have established explicit local conformal equivalences between the solutions and standard geometries. In the case $\alpha < 2$ we have ended up with hyperbolic coordinates whereas spherical coordinates arise in $\alpha > 2$.
Although we have not restricted *a priori* the range of $\alpha$ as would be required by the linear stability of an extra mode since it is absent in the $\lambda = 1/3$ case, the qualitative features of the solutions in the $\alpha < 2$ range differ discontinuously among the $\alpha < 0$ and $0 < \alpha < 2$ subranges, the point $\alpha = 0$ being the case of GR. In spite of this, with the hyperbolic coordinates the master expression for the solutions of these three cases can be given in an unified way.
[**Range**]{}
--------------- --- ------------------- ----------- -----------------------
1\. Naked sing. Positive At the origin
2\. Naked sing. Negative At the origin
1\. Schwarzschild Positive At the interior
2\. Schwarzschild Negative At the origin
1\. Wormhole Positive Boundary of branch II
2\. Naked sing. Negative At the origin
$\alpha = 2$
$\alpha > 2$ 1 Wormhole Undefined Completely regular
: \[tableofsolutions\] The several kinds of static spherically symmetric solutions of the complete nonprojectable Hořava theory with vanishing shift function classified according to the values of the coupling constant $\alpha$. We indicate the number of different solutions found in each range. The singularities are the essential ones. All solutions except $\alpha =2$ are unique up to a physical integration constant.
In the ranges $0 < \alpha < 2$ and $\alpha > 2$ there arise wormhole solutions. The coordinate systems we have used are valid at each throat and in a open neighborhood around them. Such carts extend themselves over the whole branches except at infinity.
All solutions (except the degenerated case) are asymptotically flat at least in one sector of the solution. In the wormhole of the $0 < \alpha < 2$ range the asymptotic flatness is manifested with positive mass at the end of branch I, whereas there is a singularity at the end of branch II. Curiously, the wormhole of the $\alpha > 2$ range is asymptotically flat at the ends of its two branches, but, due to the monotonic decreasing of the lapse function over the whole wormhole, the signs of the corresponding masses are different for each branch.
For $\alpha$ near zero and positive, we have that branch I of the wormhole solution tends smoothly to the exterior region of the (positive mass) Schwarzschild space-time. In particular, the location of the throat tends to the Schwarzschild radius for $\alpha \rightarrow 0^+$. Moreover, for locations sufficiently above the Schwarzschild radius, even the negative, small-$\alpha$ solution with positive mass is a small deformation of the exterior Schwarzschild space-time. These results are important for the coupling to stellar matter, where there arise cutoffs for the radial validity of the vacuum solutions. In these scenarios vacuum solutions, as the ones studied here, are of interest only as exterior solutions. In this sense it is interesting to note that the perturbative solution we found is a smooth deformation of the exterior region of the Schwarzschild space-time written directly in the original radial coordinate $r$.
An interesting extension of our work would be the inclusion of a cosmological constant. It is plausible that coordinate transformations similar to the ones we performed here work as well for the case of the field equations of the large-distance effective Hořava theory with a nonvanishing cosmological constant evaluated on static spherically symmetric configurations. In particular, one may elucidate whether there is a minimum for the coordinate transformation, which would signal the presence of a wormhole. We expect to report on this shortly.
We have studied the solutions in the framework of the effective action of the complete nonprojectable Hořava theory. We may briefly compare these solutions with other developments of the original Hořava proposal. All the projectable versions, among them the $f(R)$ models of Refs. [@Kluson:2009xx; @Chaichian:2010yi], automatically exclude these solutions since their lapse function has a nontrivial dependence on the radial coordinate, and there is no allowed coordinate change that can absorb this dependence. On the nonprojectable side, any truncated model allowing the $\partial_i \ln{N}$ terms of Blas, Pujolàs and Sibiryakov [@Blas:2009qj] should possess these solutions as their large-distance approximation of the static spherically symmetric solutions (with vanishing $N_i$), since we are studying the most general effective action (those that do not include the $\partial_i\ln{N}$ terms, as the ones with detailed balance, find the Schwarzschild space-time as their large-distance limit within the space of static spherically symmetric configurations). There is a version of the Hořava theory with a further $U(1)$ gauge symmetry originally proposed for the projectable case [@Horava:2010zj] and later extended to the nonprojectable one [@Zhu:2011xe]. In vacuum, which is the case of the solutions studied here, both of these models have the constraint $^{(3)}R = \mbox{constant}$ forced by the presence of an additional gauge field. This highly restrictive constraint excludes the possibility of having spatial slices of nonconstant curvature, as the wormholes or the naked singularities we have found here.
Finally, we point out that there are more static spherically symmetric solutions once we discard the restriction of vanishing shift function. In particular, it has been shown numerically that static spherically symmetric black holes exist in the EA theory [@Barausse:2011pu; @Eling:2006ec]. They have the aether vector field with both timelike and spacelike components turned on. These black holes must also arise in the the large-distance effective action of the complete Hořava theory since both theories are physically equivalent. In this case the presence of spacelike components of the aether vector field must be equivalent to the activation of the shift function on the side of the Hořava theory.
Acknowledgments {#acknowledgments .unnumbered}
===============
A. R. and A. S. are partially supported by Project Fondecyt No. 1121103, Chile.
The solution in the Hamiltonian formalism
=========================================
\[appendix:hamiltonian\] We focus the solutions in the Hamiltonian formulation of the $\lambda = 1/3$ theory. After this, we shall comment on the $\lambda \neq 1/3$ case.
The bulk part of the Hamiltonian of the complete nonprojectable theory at $\lambda =1/3$ is a sum of local constraints [@Bellorin:2013zbp], $$\begin{aligned}
H = \int d^3x \left(
N \mathcal{H} + N_i \mathcal{H}^i + \sigma \phi + \mu \pi
\right) \,.
\label{hamiltonianfinal}\end{aligned}$$ The shift $N_i$ as well as $\sigma$ and $\mu$ play the role of Lagrange multipliers. The first-class constraint is the momentum constraint $\mathcal{H}^i \equiv - 2 \nabla_j \pi^{ij} + \phi \partial^i N = 0$. The second class ones are $\phi = 0$, $\pi =0$, the Hamiltonian constraint $\mathcal{H} = 0$ and $\mathcal{C}=0$, where $$\begin{aligned}
\mathcal{H} &\equiv&
\frac{1}{\sqrt{g}} \pi^{ij} \pi_{ij} + \sqrt{g} \tilde{\mathcal{V}} \,,
\label{hamiltonianconstraintgeneral}
\\
\mathcal{C} &\equiv&
\frac{3N}{2\sqrt{g}} \pi^{ij} \pi_{ij}
- \sqrt{g} \tilde{\mathcal{V}}\,' \,,
\label{cconstraint}\end{aligned}$$ and we have introduced the modified potential and its derivative $$\begin{aligned}
\tilde{\mathcal{V}} & \equiv &
\mathcal{V}
+ \frac{1}{N} \sum\limits_{r=1} (-1)^r
\nabla_{i_1 \cdots i_r} \left( N
\frac{\partial \mathcal{V}}{\partial ( \nabla_{i_r \cdots i_2} a_{i_1} )}
\right) \,,
\label{modifiedpotential}
\\
\tilde{\mathcal{V}}\,' &\equiv&
\frac{1}{\sqrt{g}} g_{ij} \frac{\delta}{\delta g_{ij}} \int d^3y \sqrt{g} N \tilde{\mathcal{V}} \,.
\label{vprima}\end{aligned}$$
For the large-distance effective action we have $\mathcal{V} = \mathcal{V}^{(2)}$, such that the Hamiltonian and $\mathcal{C}$ constraints become $$\begin{aligned}
\frac{1}{\sqrt{g}} \mathcal{H} & = &
\frac{1}{g} \pi^{ij} \pi_{ij} - R
+ 2 \alpha N^{-1} \nabla^2 N - \alpha a_i a^i \,,
\label{hamiltonianconstrainquadratic}
\\
\frac{1}{\sqrt{g} N} \mathcal{C} & = &
\frac{3}{2 g} \pi^{ij} \pi_{ij}
+ \frac{1}{2} R - 2 N^{-1} \nabla^2 N
+ \frac{\alpha}{2} a_i a^i \,.
\label{cconstraintquadratic}\end{aligned}$$ The system $\mathcal{H}=0$, $\mathcal{C}=0$ can be brought to the form $$\begin{aligned}
g^{-1} \pi^{ij} \pi_{ij}
+ ( \alpha / 2 - 1 ) N^{-1} \nabla^2 N
& = & 0 \,,
\label{c1}
\\
R - ( 1 + 3 \alpha / 2 ) N^{-1} \nabla^2 N
+ \alpha a_i a^i & = & 0 \,.
\label{c2}\end{aligned}$$ The preservation in time of the second-class constraints leads to a system of two equations for the Lagrange multiplier $\sigma$ and $\mu$. This system is $$\begin{aligned}
\beta \left( 2 \nabla^2 \sigma + N a^i \partial_i \mu \right)
- 2 g^{-1} \pi^{ij} \pi_{ij} \sigma
+ \left( \beta \nabla^2 N + 3 g^{-1} N \pi^{ij} \pi_{ij} \right) \mu
= &&
\nonumber \\
- \frac{4 N}{\sqrt{g}} \pi^{ij} ( N R_{ij} -\nabla_i \nabla_j N
+ \alpha N a_i a_j )
+ \frac{4\beta}{\sqrt{g}} \partial_i ( N \partial_j N \pi^{ij} )
\,, &&
\label{sigmaeq}
\\
\nabla^2 \mu
- \frac{\alpha}{N} a^i \partial_i \sigma
- \frac{1}{4} \left(
R + \alpha a_i a^i + (3/\gamma) g^{-1} \pi^{ij} \pi_{ij} \right) \mu
+ \frac{\alpha}{N} a_i a^i \sigma
= \,\,\, \hspace{2em} &&
\nonumber \\
\frac{2\alpha}{\beta \sqrt{g}} \pi^{ij} \left( N R_{ij}
- \nabla_i \nabla_j N + \alpha N a_i a_j \right)
\,, &&
\label{mueq}\end{aligned}$$ where $$\beta \equiv ( 1 - \alpha/2 ) \,,
\hspace{2em}
\gamma \equiv \left( \frac{ 1 - \alpha/2 }{ 1 + 3\alpha/2} \right) \,.$$
Now we move to the canonical equations of motion for the $\lambda = 1/3$ theory. Since $\phi = 0$ is a constraint of the theory, $\dot{\phi}$ vanishes in the totally constrained phase space with no more conditions on the canonical variables. The equations for the evolution of $N$ and $g_{ij}$ are $$\begin{aligned}
\dot{g}_{ij} &=&
\frac{2 N}{\sqrt{g}} \pi_{ij} + 2 \nabla_{(i} N_{j)}
+ \mu g_{ij} \,,
\label{dotg}
\\
\dot{N} &=& \sigma + N^k \nabla_k N \,.
\label{dotn}\end{aligned}$$ Equation (\[dotg\]) and the constraint $\pi = 0$ imply the relation $$g^{kl} \dot{g}_{kl} = 2 \nabla_k N^k + 3 \mu \,.
\label{tracedotg}$$ The last equation of motion is $$\begin{array}{rcl}
\dot{\pi}^{ij} &=& {\displaystyle
-\frac{2 N}{\sqrt{g}} ( \pi^{ik} \pi_k{}^j
-\frac{1}{4} g^{ij} \pi^{kl} \pi_{kl} )
- \sqrt{g} N ( R^{ij} - \frac{1}{2} g^{ij} R ) }
\\[2ex] & & {\displaystyle
+ \sqrt{g} ( \nabla^i \nabla^j N - g^{ij} \nabla^2 N )
- \alpha \sqrt{g} N ( a^i a^j
- \frac{1}{2} g^{ij} a_k a^k ) }
\\[2ex] & &
- 2 \nabla_k N^{(i} \pi^{j)k}
+ \nabla_k ( N^k \pi^{ij})
- \mu \pi^{ij} \,.
\end{array}
\label{dotpi}$$
Let us evaluate all the equations of motion and constraints for static configurations with vanishing shift function. From Eqs. (\[dotn\]) and (\[tracedotg\]) we get that the Lagrange multipliers $\sigma$ and $\mu$ vanish. Putting this information back into Eq. (\[dotg\]) yields that static configurations with vanishing shift function necessarily have vanishing canonical momentum, $\pi^{ij}= 0$. This automatically solves the $\pi = 0$ and the momentum constraints. Also Eqs. (\[sigmaeq\]) and (\[mueq\]) are automatically solved under these conditions. The system of constraints $\mathcal{H} = \mathcal{C} = 0$ given in (\[c1\]) and (\[c2\]), for $\alpha \neq 2$, reduces to $$\begin{aligned}
\nabla^2 N &=& 0 \,,
\label{cstatic1}
\\
R + \alpha a_i a^i &=& 0 \,.
\label{cstatic2}\end{aligned}$$ The equation of motion (\[dotpi\]), after inserting (\[cstatic1\]) and (\[cstatic2\]), yields $$R^{ij} - N^{-1} \nabla^i \nabla^j N + \alpha a^i a^j = 0 \,.
\label{eomstatic}$$ The Eqs. (\[cstatic1\] - \[eomstatic\]) are equal to the Lagrangian equations of motion (\[hamiltonianfin\] - \[eomstaticlag\]). For $\alpha = 2$, Eq. (\[c1\]) gives no information and (\[c2\]) is the trace of Eq. (\[dotpi\]), which in turn matches with the Lagrangian equation of motion (\[eomalpha2\]).
We conclude this appendix by briefly commenting on how the solution arises in the Hamiltonian formulation of the $\lambda \neq 1/3$ case. The main difference the $\lambda \neq 1/3$ theory has with respect to the $\lambda = 1/3$ one is the absence of the $\pi = 0$ and $\mathcal{C} = 0$ constraints. Consequently, the preservation in time of the second-class constraints ($\phi = \mathcal{H} = 0$) leads to only one equation for $\sigma$. Since static configurations with vanishing shift function have again $\pi^{ij}= 0$, the condition $\pi = 0$ holds anyway and the Hamiltonian constraints of both cases become identical since they differ in general by a term proportional to $\pi^2$. The equation for $\sigma$, which can be found in Ref. [@Bellorin:2011ff], is totally solved by $\sigma = 0$, which is a consequence of staticity and $N_i = 0$. When all these conditions are imposed on the time evolution of $\pi^{ij}$, the resulting equation is exactly equal to Eq. (\[dotpi\]) evaluated on $\pi^{ij} = 0$. Moreover, the trace of this equation is just the $\mathcal{C}$ constraint (\[cconstraintquadratic\]). Therefore, the constraints/equations of motion of the Hamiltonian formulation of the $\lambda \neq 1/3$ case evaluated on static configurations with $N_i = 0$ lead to the system of Eqs. (\[hamiltonianconstrainquadratic\]), (\[cconstraintquadratic\]) and (\[dotpi\]) of the $\lambda = 1/3$ case, which is the system of equations we solved in the main body of the paper.
Equivalences between solutions {#apendix:equivalences}
==============================
1. [**Case $\alpha < 2$: solutions in $\chi \in (-\infty,0)$**]{}\
In $\chi\in (-\infty,0)$ one arrives at the same expressions (\[coordinatetransf\]) and (\[nffinalalphal2\]) but with negative integration constants, $$r = \frac{\tilde{k} e^{s\chi/\beta}}{\sinh{\chi}} \,,
\hspace{2em}
N = \frac{\beta \tilde{C}}{\tilde{k}} e^{-s\chi/\beta} \,,
\hspace{2em}
f = (\beta^{-1} \sinh{\chi} - s \cosh{\chi} )^2 \,,$$ where $\tilde{C},\tilde{k} < 0$. After the coordinate transformation $\chi' = - \chi$ and the identification of integration constants $\tilde{C} =-C$ and $\tilde{k} = - k$, one gets the two solutions of (\[coordinatetransf\]) and (\[nffinalalphal2\]) in $\chi' \in (0,+\infty)$.
2. [**Case $\alpha > 2$: Equivalence of the $s=+1$ and $s=-1$ solutions**]{}\
The $s=-1$ solution is $$r = \frac{\tilde{k} e^{-\varphi/\beta}}{\sin{\varphi}} \,,
\hspace{2em}
N = \frac{\beta \tilde{C}}{\tilde{k}} e^{\varphi/\beta} \,,
\hspace{2em}
f = (\beta^{-1} \sin{\varphi} + \cos{\varphi} )^2 \,,$$ with $\varphi\in (0,\pi)$ and $\tilde{C},\tilde{k} > 0$. If one makes the coordinate transformation $\varphi' = - \varphi + \pi$ and the identification $\tilde{C} = C$, $\tilde{k} = e^{\pi/\beta} k$, one then arrives at the $s=+1$ solution given in (\[coordinatetrans2\]) and (\[fngreaterthan2\]).
3. [**Case $\alpha > 2$: Solution in $\varphi \in (\pi,2\pi)$**]{}\
For the $s=+1$ solution in $\varphi \in (\pi,2\pi)$ one arrives at the same expressions (\[coordinatetrans2\]) and (\[fngreaterthan2\]) but with negative integration constants, $$r = \frac{\tilde{k} e^{\varphi/\beta}}{\sin{\varphi}} \,,
\hspace{2em}
N = \frac{\beta \tilde{C}}{\tilde{k}} e^{-\varphi/\beta} \,,
\hspace{2em}
f = (\beta^{-1} \sin{\varphi} - \cos{\varphi} )^2 \,,$$ where $\tilde{C},\tilde{k} < 0$. The appropiated transformation is $\varphi' = \varphi - \pi$ and the identification of constants is $\tilde{C} = - C$, $\tilde{k} = - e^{-\pi/\beta} k$. The proof that the $s=-1$ and $s=+1$ solutions in $\varphi \in (\pi,2\pi)$ are equivalent is done in parallel to equivalence 2.
4. [**Case $\alpha > 2$: the other four solutions**]{}\
The other four solutions Eq. (\[bpositive\]) has are $$\frac{ \beta C' }{r N} = s_1 \cos{\varphi} \,,
\hspace{2em}
\sqrt{f} + \frac{ C'}{rN} = s_2 \sin{\varphi} \,.$$ Among them we first take the $s=+1$ solution in the range $\varphi \in (-\pi/2 , +\pi/2)$. It is $$r = \frac{\tilde{k} e^{-\varphi/\beta}}{\cos{\varphi}} \,,
\hspace{2em}
N = \frac{\beta \tilde{C}}{\tilde{k}} e^{\varphi/\beta} \,,
\hspace{2em}
f = (\beta^{-1} \cos{\varphi} - \sin{\varphi} )^2 \,,
\label{othersolution}$$ with $\tilde{C},\tilde{k} > 0$. The coordinate transformation is $\varphi' = -\varphi + \pi/2$ and the identification is $\tilde{C} = C$, $\tilde{k} = e^{\pi/2\beta} k$. This leads exactly to the solution (\[coordinatetrans2\]) and (\[fngreaterthan2\]). By making similar analysis to the previous equivalences one can show that the $s=-1$ solution in the range $\varphi\in (-\pi/2,+\pi/2)$ and the $s=+1$ and $s=-1$ solutions in $(\pi/2,3\pi/2)$ are all equivalent to (\[othersolution\]).
Expanding in $\alpha$ the exact solution {#appendix:expanding}
========================================
Here we show how the perturbative solution can be obtained from the exact solution when $\alpha \sim 0$. The main point is that the coordinate transformation (\[coordinatetransf\]) can be explicitly inverted at linear order in $\alpha$. For simplicity, we consider only the positive mass solution ($s=+1$). At the end we shall indicate how to recover the expansion of the negative mass solution ($s=-1$).
We start by rewriting the coordinate transformation (\[coordinatetransf\]), $$r = \frac{\beta C e^{\chi / \beta}}{\sinh{\chi}} \,,
\hspace{2em}
\chi > 0 \,,
\hspace{2em}
\beta = \sqrt{ 1 - \alpha / 2 }
\label{app:coordinatetransf}$$ and $C$ is the unique nonfixed integration constant the solution has. We recall that to arrive at (\[app:coordinatetransf\]) it is assumed $C>0$. When expanded up to linear order in $\alpha$, this transformation becomes $${\displaystyle r =
2 C \frac{1 + \alpha ( \chi - 1 ) / 4 }{1 - e^{-2\chi}} } \,.
\label{coordinatetranspertur}$$ We may further refine this expansion if we assume that $\chi$, as a function of $r$, is of linear order in $\alpha$; that is, we use the ansatz $$\chi(r) = \chi^{(0)}(r) + \alpha\: \chi^{(1)}(r) \,.
\label{app:chiexpanded}$$ We first put this expansion back into Eq. (\[coordinatetranspertur\]) and evaluate the resulting equation at zeroth order in $\alpha$. This fixes $\chi^{(0)}(r)$, $$e^{-2 \chi^{(0)}} = 1 - \frac{2C}{r} \,.$$ The second step is to put $\chi^{(0)}(r)$ and the expansion (\[app:chiexpanded\]) back into Eq. (\[coordinatetranspertur\]) and expand up to linear order in $\alpha$. This gives $\chi^{(1)}(r)$ as a combination of $\chi^{(0)}(r)$ and $r$. The whole, linear order function $\chi(r)$ is $$\chi =
- \frac{1}{2} \ln \left( 1 - \frac{2 C}{r} \right)
- \frac{\alpha C}{8 r} \left( 1 - \frac{2 C}{r} \right)^{-1}
\left[ 2 + \ln \left( 1 - \frac{2 C}{r} \right) \right] \,.$$ The final step is to insert this coordinate transformation into the exact solution, which is given in terms of functions of $\chi$ as $$N = e^{ -\chi / \beta} \,,
\hspace{2em}
f = ( \cosh\chi - \beta^{-1} \sinh\chi )^2\,,$$ and expand the functions $N$ and $f$ up to linear order in $\alpha$. We obtain the linear-order solution in terms of the spherical coordinate $r$: $$\begin{array}{rcl}
N &=& {\displaystyle
\left( 1 - \frac{2 C}{r} \right)^{1/2}
+ \frac{\alpha}{8} \left( 1 - \frac{2 C}{r} \right)^{-1/2}
\left[ \frac{2C}{r}
+ \left( 1 - \frac{C}{r} \right) \ln \left( 1 - \frac{2 C}{r} \right)
\right] } \,,
\\[2ex]
f &=& {\displaystyle
1 - \frac{2C}{r}
+ \frac{\alpha C}{4r} \ln\left( 1 - \frac{2 C}{r} \right)
} \,.
\end{array}
\label{finalexpansion}$$ This coincides with the perturbative solution (\[perturbativesolution\]) if we set the value $B = 0$ and identify the free integration constants of both versions according to $A= 2C$. This implies that we take $A>0$ in (\[perturbativesolution\]). The expansion of the negative-mass solution can be obtained from (\[finalexpansion\]) by substituting $C \rightarrow -C$ everywhere in these expressions. This is equivalent to take $A = -2C$, $B = 0$ in (\[perturbativesolution\]), with $C>0$.
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[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
[^4]: The original Hořava theory of gravitation was formulated in Ref. [@Horava:2009uw]. An important class of new terms were incorporated to the potential of the nonprojectable version by Blas, Pujolàs and Sibiryakov in Ref. [@Blas:2009qj]. These terms are needed for a renormalization procedure based on the gauge symmetry; hence we use the word *complete*. We use the term “original theory” to refer to any nonprojectable model without the terms of [@Blas:2009qj], independent of whether or not they impose the detailed balance principle. Previous ideas about the foliation-preserving diffeomorphisms as a group of gauge symmetry were used by Hořava for a membrane action in Ref. [@Horava:2008ih].
[^5]: The equation used in Ref. [@Henneaux:2009zb] to argue that the original nonprojectable Hořava theory necessarily has vanishing lapse function is actually an equation for the trace of the canonical momentum with the physically consistent solution $\pi = 0$ [@Bellorin:2010je].
[^6]: When the original (incomplete) theory is considered with terms of higher order the algebra of constraints also closes, but there arise nonelliptic differential equations [@Bellorin:2010te]. In Ref. [@Li:2009bg] the second-class constraints of the original theory were misinterpreted as first-class ones. That prevents to obtain the closure of their algebra.
[^7]: Static spherically symmetric solutions with active shift function were found for the original Hořava theory in Ref. [@Capasso:2009ks].
[^8]: If this free constant is kept, the only effect it would have on static solutions is to replace the coupling constant $\alpha$ by the ratio of $\alpha$ and it.
[^9]: The constant $C$ arising from (\[rfnprima\]) to (\[presolutions\]) can be regarded as the local integration constant needed to cast the solutions in terms of the coordinate $r$. Necessarily it changes its sign among some sectors of the solutions.
[^10]: There is a proposal to distinguish from an observational point of view between naked singularities and a black holes [@Virbhadra:2002ju].
[^11]: The limit of $\alpha \ln{\alpha}$ when $\alpha \rightarrow 0^+$ is finite and equal to zero.
|
---
abstract: 'We investigate the complexity of the containment problem “Does $L({\mathcal{A}})\subseteq L({\mathcal{B}})$ hold?”, where ${\mathcal{B}}$ is an unambiguous register automaton and ${\mathcal{A}}$ is an arbitrary register automaton. We prove that the problem is decidable and give upper bounds on the computational complexity in the general case, and when ${\mathcal{B}}$ is restricted to have a fixed number of registers.'
author:
- Antoine Mottet
- Karin Quaas
title: The Containment Problem for Unambiguous Register Automata
---
Introduction
============
Register automata [@DBLP:journals/tcs/KaminskiF94] are a widely studied model of computation that extend finite automata with finitely many *registers* that are able to hold values from an infinite domain and perform equality comparisons with data from the input word. This allows register automata to accept *data languages*, i.e., sets of *data words* over $\Sigma\times{\mathbb{D}}$, where $\Sigma$ is a finite alphabet and ${\mathbb{D}}$ is an infinite set called the data domain. The study of register automata is motivated by problems in formal verification and database theory, where the objects under study are accompanied by annotations (identification numbers, labels, parameters, ...), see the survey by Ségoufin [@DBLP:conf/csl/Segoufin06]. One of the central problems in these areas is to check whether a given input document or program complies with a given input specification. In our context, this problem can be formalized as a *containment problem*: given two register automata ${\mathcal{A}}$ and ${\mathcal{B}}$, does $L({\mathcal{A}})\subseteq L({\mathcal{B}})$ hold, i.e., is the data language accepted by ${\mathcal{A}}$ included in the data language accepted by ${\mathcal{B}}$? Here, ${\mathcal{B}}$ is understood as a specification, and one wants to check whether ${\mathcal{A}}$ satisfies the specification. For arbitrary register automata, the containment problem is undecidable [@DBLP:journals/tocl/NevenSV04; @DBLP:journals/tocl/DemriL09]. It is known that one can recover decidability in two different ways. First, the containment problem is known to be -complete when ${\mathcal{B}}$ is a deterministic register automaton [@DBLP:journals/tocl/DemriL09]. This is a severe restriction on the expressive power of ${\mathcal{B}}$, and it is of practical interest to find natural classes of register automata that can be tackled algorithmically and that can express more properties than deterministic register automata. Secondly, one can recover decidability of the containment problem when ${\mathcal{B}}$ is a non-deterministic register automaton with a single register [@DBLP:journals/tcs/KaminskiF94; @DBLP:journals/tocl/DemriL09]. However, in this setting, the problem is Ackermann-complete [@DBLP:conf/lics/FigueiraFSS11]; it can therefore hardly be considered tractable.
This motivates the study of *unambiguous* register automata, which are non-deterministic register automata for which every data word has at most one accepting run. Such automata are strictly more expressive than deterministic register automata [@DBLP:journals/tcs/KaminskiF94; @KaminskiZeitlin].
In the present paper, we investigate the complexity of the containment problem when ${\mathcal{B}}$ is restricted to be an unambiguous register automaton. We prove that the problem is decidable with a [$\textsf{2-EXPSPACE}$]{} complexity, and is even decidable in [$\textsf{EXPSPACE}$]{} if the number of registers of ${\mathcal{B}}$ is a fixed constant. This is a striking difference to the non-deterministic case, where even for a fixed number of registers greater than $1$ the problem is undecidable. Classically, one way to approach the containment problem (for general models of computation) is to reduce it to a reachability problem on an infinite state transition system, called the *synchronized state space of ${\mathcal{A}}$ and ${\mathcal{B}}$*, cf. [@DBLP:conf/lics/OuaknineW04]. Proving decidability or complexity upper bounds for the containment problem then amounts to finding criteria of termination or bounds on the complexity of a reachability algorithm on this space. In this paper, our techniques also rely on the analysis of the synchronized state space of ${\mathcal{A}}$ and ${\mathcal{B}}$, where our main contribution is to provide a bound on the size of synchronized states that one needs to explore before being able to certify that $L({\mathcal{A}})\subseteq L({\mathcal{B}})$ holds. This bound is found by identifying elements of the synchronized state space whose behaviour is similar, and by showing that every element of the synchronized state space is equivalent to a small one. In the general case, where ${\mathcal{B}}$ is unambiguous and ${\mathcal{A}}$ is an arbitrary non-deterministic register automaton, we bound the size of the graph that one needs to inspect by a triple exponential in the size of ${\mathcal{A}}$ and ${\mathcal{B}}$. In the restricted case that ${\mathcal{B}}$ has a fixed number of registers, we proceed to give a better bound that is only doubly exponential in the size of ${\mathcal{A}}$ and ${\mathcal{B}}$.
##### Related Literature {#related-literature .unnumbered}
A thorough study of the current literature on register automata reveals that there exists a variety of different definitions of register automata, partially with significantly different semantics. In this paper, we study register automata as originally introduced by Kaminski and Francez [@DBLP:journals/tcs/KaminskiF94]. Such register automata process data words over an infinite data domain. The registers can take data values that appear in the input data word processed so far. The current input datum can be compared for (in)equality with the data that is stored in the registers. Kaminski and Francez study register automata mainly from a language-theoretic point of view; more results on the connection to logic, as well as the decidability status and computational complexity of classical decision problems like emptiness and containment are presented, e.g., in [@DBLP:journals/tcs/SakamotoI00; @DBLP:journals/tocl/NevenSV04; @DBLP:journals/tocl/DemriL09]. In [@DBLP:journals/corr/abs-1011-6432], register automata over *ordered* data domains are studied.
Kaminski and Zeitlin [@KaminskiZeitlin] define a generalisation of the model in [@DBLP:journals/tcs/KaminskiF94], in the following called *register automata with guessing*. The registers in such automata can non-deterministically reassign, or “guess”, the datum of a register. In particular, such register automata can store data values that have not appeared in the input data word before, in contrast to the register automata in [@DBLP:journals/tcs/KaminskiF94]. Register automata with guessing are strictly more expressive than register automata; for instance, there exists a register automaton with guessing that accepts the complement of the data language accepted by the register automaton in Figure \[fig:ura\] (Example 4 in [@KaminskiZeitlin]). Figueira [@DBLP:journals/corr/abs-1202-3957] studies an alternating version of this model, also over ordered data domains. Colcombet [@DBLP:conf/dcfs/Colcombet15; @DBLP:conf/stacs/Colcombet12] considers *unambiguous* register automata with guessing. In Theorem 12 in [@DBLP:conf/dcfs/Colcombet15], it is claimed that this automata class is effectively closed under complement, so that universality, containment and equivalence are decidable; however, to the best of our knowledge, this claim remains unproved.
Finally, unambiguity has become an important topic in automata theory, as witnessed by the growing body of literature in the recent years [@DBLP:conf/concur/FijalkowR017; @DBLP:conf/icalp/Skrzypczak18; @DBLP:conf/icalp/DaviaudJLMP018; @DBLP:conf/icalp/Raskin18]. In addition to the motivations mentioned above, unambiguous automata form an important model of computation due to their *succinctness* compared to their deterministic counterparts. For example, it is known that unambiguous finite automata can be exponentially smaller than deterministic automata [@DBLP:journals/ijfcs/Leung05] while the fundamental problems (such as emptiness, universality, containment, equivalence) remain tractable.
Main Definitions
================
We study register automata as introduced in the seminal paper by Kaminski and Francez [@DBLP:journals/tcs/KaminskiF94]. Throughout the paper, $\Sigma$ denotes a finite alphabet, and ${\mathbb{D}}$ denotes an infinite set of data values. In our examples, we assume ${\mathbb{D}}={\mathbb N}$, the set of non-negative integers. A *data word* is a finite sequence $(\sigma_1,d_1)\dots(\sigma_k,d_k) \in (\Sigma\times{\mathbb{D}})^*$. A *data language* is a set of data words. We use $\varepsilon$ to denote the *empty data word*. The *length* $k$ of a data word $w$ is denoted by $|w|$. Given a data word $w$ as above and $0\leq i\leq k$, we define the infix $w(i,j]:=(\sigma_{i+1},d_{i+1})\dots(\sigma_j,d_j)$. Note that $w(i,i]=\varepsilon$. We use ${\textup{data}}(w)$ to denote the set $\{d_1,\dots,d_k\}$ of all data occurring in $w$. We use ${\textup{proj}}(w)$ to denote the projection of $w$ onto $\Sigma^*$, i.e., the word $\sigma_1\dots\sigma_k$.
Let ${\mathbb{D}}_\bot$ denote the set ${\mathbb{D}}\cup\{\bot\}$, where $\bot\not\in{\mathbb{D}}$ is a fresh symbol not occurring in ${\mathbb{D}}$. A *partial isomorphism* of ${\mathbb{D}}_\bot$ is an injection $f\colon S\to{\mathbb{D}}_\bot$ with finite domain $S\subset {\mathbb{D}}_\bot$ such that if $\bot\in S$, then $f(\bot)=\bot$. We use boldface lower-case letters like ${\boldsymbol{a}}, {\boldsymbol{b}}, \dots $ to denote tuples in ${\mathbb{D}}_\bot^n$, where $n\in{\mathbb N}$. Given a tuple ${\boldsymbol{a}}\in{\mathbb{D}}_\bot^n$, we write $a_i$ for its $i$-th component, and ${\textup{data}}({\boldsymbol{a}})$ denotes the set $\{a_1,\dots,a_n\}\subseteq{\mathbb{D}}_\bot$ of all data occurring in ${\boldsymbol{a}}$.
Let ${R}=\{{r}_1,\dots,{r}_n\}$ be a finite set of *registers*. A *register valuation* is a mapping ${\boldsymbol{a}}:{R}\to{\mathbb{D}}_\bot$; we may write $a_i$ as shorthand for ${\boldsymbol{a}}({r}_i)$. Let ${\mathbb{D}}_\bot^{R}$ denote the set of all register valuations. Given $\lambda\subseteq{R}$ and $d\in{\mathbb{D}}$, define the register valuation ${\boldsymbol{a}}[\lambda\leftarrow d]$ by $({\boldsymbol{a}}[\lambda\leftarrow d])({r}_i):=d$ if ${r}_i\in\lambda$, and $({\boldsymbol{a}}[\lambda\leftarrow d])({r}_i):=a_i$ otherwise.
A *register constraint* over ${R}$ is defined by the grammar $$\begin{aligned}
\phi ::= {\texttt{true}}\,\mid\, ={r}\,\mid\, \neg\phi \,\mid\,
\phi \wedge \phi \, , \end{aligned}$$ where ${r}\in{R}$. We use $\Phi({R})$ to denote the set of all register constraints over ${R}$. We may use $\neq{r}$ or $\phi_1\vee\phi_2$ as shorthand for $\neg(={r})$ and $\neg(\neg\phi_1\wedge\neg\phi_2)$, respectively. The satisfaction relation $\models$ for $\Phi({R})$ on ${\mathbb{D}}_\bot^{R}\times{\mathbb{D}}$ is defined by structural induction in the obvious way; e.g., ${\boldsymbol{a}},d\models (={r}_1 \, \wedge \, \neq{r}_2)$ if $a_1=d$ and $a_2\neq d$.
A *register automaton over $\Sigma$* is a tuple ${\mathcal{A}}=({R},{\mathcal{L}},{\ell}_{\textup{in}},{\mathcal{L}}_{\textup{acc}},{E})$, where
- ${R}$ is a finite set of registers,
- ${\mathcal{L}}$ is a finite set of *locations*,
- ${\ell}_{\textup{in}}\in{\mathcal{L}}$ is the *initial location*,
- ${\mathcal{L}}_{\textup{acc}}\subseteq{\mathcal{L}}$ is the set of *accepting locations*, and
- ${E}\subseteq {\mathcal{L}}\times \Sigma \times \Phi({R}) \times 2^{R}\times {\mathcal{L}}$ is a finite set of *edges*. We may write ${\ell}\xrightarrow{\sigma,\phi,\lambda}{\ell}'$ to denote an edge $({\ell},\sigma,\phi,\lambda,{\ell}')\in{E}$. Here, $\sigma$ is the label of the edge, $\phi$ is the register constraint of the edge, and $\lambda$ is the set of updated registers of the edge. A register constraint ${\texttt{true}}$ is vacuously true and may be omitted; likewise we may omit $\lambda$ if $\lambda=\emptyset$.
A *state* of ${\mathcal{A}}$ is a pair $({\ell},{\boldsymbol{a}})\in {\mathcal{L}}\times{\mathbb{D}}_\bot^{R}$, where ${\ell}$ is the current location and ${\boldsymbol{a}}$ is the current register valuation. Given two states $({\ell},{\boldsymbol{a}})$ and $({\ell}',{\boldsymbol{a'}})$ and some input letter $(\sigma,d)\in (\Sigma\times{\mathbb{D}})$, we postulate a transition $({\ell},{\boldsymbol{a}})\xrightarrow{\sigma,d}_{\mathcal{A}}({\ell}',{\boldsymbol{a'}})$ if there exists some edge ${\ell}\xrightarrow{\sigma,\phi,\lambda}{\ell}'$ such that ${\boldsymbol{a}},d\models\phi$ and ${\boldsymbol{a'}}={\boldsymbol{a}}[\lambda\leftarrow d]$. If the context is clear, we may omit the index ${\mathcal{A}}$ and write $({\ell},{\boldsymbol{a}})\xrightarrow{\sigma,d}({\ell}',{\boldsymbol{a'}})$ instead of $({\ell},{\boldsymbol{a}})\xrightarrow{\sigma,d}_{\mathcal{A}}({\ell}',{\boldsymbol{a'}})$. We use $\longrightarrow^*$ to denote the reflexive transitive closure of $\longrightarrow$. A *run* of ${\mathcal{A}}$ on the data word $(\sigma_1,d_1)\dots(\sigma_k,d_k)$ is a sequence $({\ell}_0,{\boldsymbol{a^0}}) \xrightarrow{\sigma_1,d_1} ({\ell}_1,{\boldsymbol{a^1}}) \xrightarrow{\sigma_2,d_2} \dots \xrightarrow{\sigma_k,d_k} ({\ell}_k,{\boldsymbol{a^k}})$ of transitions. We say that a run *starts in $({\ell},{\boldsymbol{a}})$* if $({\ell}_0,{\boldsymbol{a^0}})=({\ell},{\boldsymbol{a}})$. A run is *initialized* if it starts in $({\ell}_{\textup{in}},\{\bot\}^{R})$, and a run is *accepting* if ${\ell}_k\in{\mathcal{L}}_{\textup{acc}}$. The data language *accepted* by ${\mathcal{A}}$, denoted by $L({\mathcal{A}})$, is the set of data words $w\in (\Sigma\times{\mathbb{D}})^*$ such that there exists an initialized accepting run of ${\mathcal{A}}$ on $w$.
We classify register automata into *deterministic register automata* (), *unambiguous register automata* (), and *non-deterministic register automata* (). A register automaton is a if for every data word $w$ there is at most one initialized run. A register automaton is a if for every data word $w$ there is at most one initialized accepting run. A register automaton without any restriction is an . We say that a data language $L\subseteq (\Sigma\times{\mathbb{D}})^*$ is -recognizable (-recognizable and -recognizable, respectively), if there exists a ( and , respectively) ${\mathcal{A}}$ over $\Sigma$ such that $L({\mathcal{A}})=L$. We write ${\mathbf{DRA}}$, ${\mathbf{URA}}$, and ${\mathbf{NRA}}$ for the class of -recognizable, -recognizable, and -recognizable, respectively, data languages. Note that ${\mathbf{DRA}}\subseteq {\mathbf{URA}}\subseteq {\mathbf{NRA}}$. Also note that, albeit a semantical property, the unambiguity of a register automaton can be decided using a simple extension of a product construction, cf. [@DBLP:conf/dcfs/Colcombet15].
The *containment problem* is the following decision problem: given two register automata ${\mathcal{A}}$ and ${\mathcal{B}}$, does $L({\mathcal{A}})\subseteq L({\mathcal{B}})$ hold? We consider two more decision problems that stand in a close relation to the containment problem (namely, they both reduce to the containment problem): the *universality problem* is the question whether $L({\mathcal{B}})=(\Sigma\times{\mathbb{D}})^*$ for a given register automaton ${\mathcal{B}}$. The *equivalence problem* is to decide, given two register automata ${\mathcal{A}}$ and ${\mathcal{B}}$, whether $L({\mathcal{A}})=L({\mathcal{B}})$.
Some Facts about Register Automata
==================================
For many computational models, a straightforward approach to solve the containment problem is by a reduction to the emptiness problem using the equivalence: $L({\mathcal{A}})\subseteq L({\mathcal{B}})$ if, and only if, $L({\mathcal{A}})\cap \overline{L({\mathcal{B}})}=\emptyset$. This approach proves useful for ${\mathbf{DRA}}$, which is closed under complementation. Using the decidability of the emptiness problem for ${\textup{NRA}}$, as well as the closure of ${\mathbf{NRA}}$ under intersection [@DBLP:journals/tcs/KaminskiF94], we obtain the decidability of the containment problem for the case where ${\mathcal{A}}$ is an ${\textup{NRA}}$ and ${\mathcal{B}}$ is a ${\textup{DRA}}$. More precisely, and using results in [@DBLP:journals/tocl/DemriL09], the containment problem for this particular case is ${\ensuremath{\textsf{PSPACE}}}$-complete.
In contrast to ${\mathbf{DRA}}$, the class ${\mathbf{NRA}}$ is not closed under complementation [@DBLP:journals/tcs/KaminskiF94] so that the above approach must fail if ${\mathcal{B}}$ is an ${\textup{NRA}}$. Indeed, it is well known that the containment problem for the case where ${\mathcal{B}}$ is an ${\textup{NRA}}$ is undecidable [@DBLP:journals/tocl/DemriL09]. The proof is a reduction from the halting problem for Minsky machines: an ${\textup{NRA}}$ is capable to accept the complement of a set of data words encoding halting computations of a Minsky machine.
In this paper, we are interested in the containment problem for the case where ${\mathcal{A}}$ is an ${\textup{NRA}}$ and ${\mathcal{B}}$ is a ${\textup{URA}}$. When attempting to solve this problem, an obvious idea is to ask whether the class ${\mathbf{URA}}$ is closed under complementation. Kaminski and Francez [@DBLP:journals/tcs/KaminskiF94] proved that ${\mathbf{URA}}$ is *not* closed under complementation, and this even holds for the class of data languages that are accepted by ${\textup{URA}}$ that only use a single register. In Figure \[fig:ura\], we show a standard example of a ${\textup{URA}}$ for which the complement of the accepted data language cannot even be accepted by an ${\textup{NRA}}$ [@KaminskiZeitlin]. Intuitively, this automaton is unambiguous because it is not possible for two different runs of the automaton on some data word to reach the location ${\ell}_1$ with the same register valuation at the same time. Therefore, at any time only one run can proceed to the accepting location ${\ell}_2$. Note that this also implies ${\mathbf{DRA}}\subsetneq {\mathbf{URA}}$.
An alternative approach for solving the containment problem is to explore the (possibly infinite) *synchronized state space of ${\mathcal{A}}$ and ${\mathcal{B}}$*, cf. [@DBLP:conf/lics/OuaknineW04]. Intuitively, the synchronized state space of ${\mathcal{A}}$ and ${\mathcal{B}}$ stores for every state $({\ell},{\boldsymbol{a}})$ that ${\mathcal{A}}$ is in after processing a data word $w$ the *set of states* that ${\mathcal{B}}$ is in after processing the same data word $w$. For an example, see the computation tree on the right side of Figure \[fig:ura\], where the leftmost branch shows the set of states that the URA on the left side of Figure \[fig:ura\] reaches after processing the data word $(\sigma,1)(\sigma,1)(\sigma,1)$, and the rightmost branch shows the set of states that the URA reaches after processing the data word $(\sigma,2)(\sigma,1)(\sigma,3)$. The key property of the synchronized state space of ${\mathcal{A}}$ and ${\mathcal{B}}$ is that it contains sufficient information to decide whether for every data word for which there is an initialized accepting run in ${\mathcal{A}}$ there is also an initialized accepting run in ${\mathcal{B}}$. We formalize this intuition in the following paragraphs.
We start by defining the *state space* of a given ${\textup{NRA}}$. Fix an ${\mathcal{A}}=({R},{\mathcal{L}},{\ell}_{\textup{in}},{\mathcal{L}}_{\textup{acc}},{E})$ over $\Sigma$. A *configuration* of ${\mathcal{A}}$ is a finite set ${C}\subseteq ({\mathcal{L}}\times{\mathbb{D}}^{R}_\bot)$ of states of ${\mathcal{A}}$; if ${C}=\{({\ell},{\boldsymbol{a}})\}$ is a singleton set, in slight abuse of notation and if the context is clear, we may omit the parentheses and write $({\ell},{\boldsymbol{a}})$. Given a configuration ${C}$ and an input letter $(\sigma,d)\in(\Sigma\times{\mathbb{D}})$, we use ${\textup{Succ}}_{\mathcal{A}}({C},(\sigma,d))$ to denote the *successor configuration of ${C}$ on the input $(\sigma, d)$*, formally defined by $$\begin{aligned}
{\textup{Succ}}_{\mathcal{A}}({C},(\sigma,d)) := \{({\ell},{\boldsymbol{a}}) \in ({\mathcal{L}}\times{\mathbb{D}}^{R}_\bot) \mid \exists ({\ell}',{\boldsymbol{a'}})\in {C}. ({\ell}',{\boldsymbol{a'}})\xrightarrow{\sigma,d}_{\mathcal{A}}({\ell},{\boldsymbol{a}})\}.\end{aligned}$$ In order to extend this definition to data words, we define inductively ${\textup{Succ}}_{\mathcal{A}}({C},\varepsilon):={C}$ and ${\textup{Succ}}_{\mathcal{A}}({C}, w \cdot (\sigma,d)) := {\textup{Succ}}_{\mathcal{A}}({\textup{Succ}}_{\mathcal{A}}({C},w),(\sigma,d))$. We say that a configuration ${C}$ is *reachable in ${\mathcal{A}}$* if there exists some data word $w$ such that ${C}= {\textup{Succ}}_{\mathcal{A}}(({\ell}_{\textup{in}},\{\bot\}^{R}), w)$. We say that a configuration ${C}$ is *coverable in ${\mathcal{A}}$* if there exists some configuration ${C}'\supseteq {C}$ such that ${C}'$ is reachable in ${\mathcal{A}}$. We say that a configuration ${C}$ is *accepting* if there exists $({\ell},{\boldsymbol{a}})\in{C}$ such that ${\ell}\in{\mathcal{L}}_{\textup{acc}}$; otherwise we say that ${C}$ is *non-accepting*. We define ${\textup{data}}({C}):=\bigcup_{({\ell},{\boldsymbol{a}})\in C}{\textup{data}}({\boldsymbol{a}})$ as the set of data occurring in configuration ${C}$.
The following proposition follows immediately from the definition of .
\[prop:ura\_implied\_badness\] If ${\mathcal{A}}$ is a and ${C},{C}'$ are two configurations of ${\mathcal{A}}$ such that ${C}\cap{C}'=\emptyset$ and ${C}\cup{C}'$ is coverable, then for every data word $w$ the following holds: if ${\textup{Succ}}_{\mathcal{A}}({C},w)$ is accepting, then ${\textup{Succ}}_{\mathcal{A}}({C}',w)$ is non-accepting.
Let ${C},{C}'$ be two configurations of ${\mathcal{A}}$. Consider two data words $w=(\sigma_1,d_1)\dots(\sigma_k,d_k)$ and $w'=(\sigma_1,d'_1)\dots(\sigma_k,d'_k)$ such that ${\textup{proj}}(w)={\textup{proj}}(w')$. Recall that a partial function $f\colon {\mathbb{D}}_\bot\to{\mathbb{D}}_\bot$ with finite domain is a *partial isomorphism* if it is an injection such that if $\bot\in\operatorname{dom}(f)$ then $f(\bot)=\bot$. Let $f$ be a partial isomorphism of ${\mathbb{D}}_\bot$ and let ${C}$ be a configuration with ${\textup{data}}({C})\subseteq \operatorname{dom}(f)$. We define $f({C}):=\{({\ell},f(d_1),\dots, f(d_{|{R}|})) \mid ({\ell},d_1, \dots, d_{|{R}|})\in {C}\}$; likewise, if $\{d_1,\dots,d_k\}\subseteq\operatorname{dom}(f)$, we define $f((\sigma_1,d_1)\dots (\sigma_k,d_k)):=(\sigma_1,f(d_1))\dots (\sigma_k,f(d_k))$. We say that ${C},w$ and ${C}',w'$ are *equivalent with respect to $f$*, written ${C},w{\sim}_f {C}',w'$, if $$\begin{aligned}
\tag{$\star$}
f({C})={C}' \text{ and } f(w)=w'. \label{ra_equivalence_1} $$ If $w=w'=\varepsilon$, then we may simply write ${C}{\sim}_f{C}'$. We write ${C}{\sim}{C}'$ if ${C}{\sim}_f {C}'$ for some partial isomorphism $f$ of ${\mathbb{D}}_\bot$.
\[prop:ra\_equivalence\] If ${C},w {\sim}{C}',w'$, then ${\textup{Succ}}_{\mathcal{A}}({C},w(0,i]),w(i,k] {\sim}{\textup{Succ}}_{\mathcal{A}}({C}',w'(0,i]),w'(i,k]$ for all $0\leq i\leq k$, where $k=|w|$.
The proof is by induction on $i$. For the induction base, let $i=0$. But then ${\textup{Succ}}_{\mathcal{A}}({C},w(0,0]))={\textup{Succ}}_{\mathcal{A}}({C},\varepsilon)={C}$ and $w(0,k]=w$, and similarly for ${C}'$ and $w'$, so that the statement holds by assumption. For the induction step, let $i>0$. Define ${C}_{i-1}:={\textup{Succ}}_{\mathcal{A}}({C},w(0,i-1])$ and similarly ${C}'_{i-1}$. By induction hypothesis, there exists some bijective mapping $$f_{i-1}:{\textup{data}}({C}_{i-1})\cup {\textup{data}}(w(i-1,k]) \to {\textup{data}}({C}'_{i-1})\cup {\textup{data}}(w'(i-1,k])$$ satisfying (\[ra\_equivalence\_1\]) $f_{i-1}({C}_{i-1})={C}'_{i-1}$ and $f_{i-1}(w(i-1,k])=w'(i-1,k]$. Define ${C}_i := {\textup{Succ}}_{\mathcal{A}}({C}_{i-1},(\sigma_i,d_i))$ and ${C}'_i:={\textup{Succ}}_{\mathcal{A}}({C}'_{i-1},(\sigma_i,d'_i))$. Note that ${\textup{data}}({C}_i)\subseteq{\textup{data}}({C}_{i-1})\cup\{d_i\}$, and similarly for ${\textup{data}}({C}'_i)$. Let $f_i$ be the restriction of $f_{i-1}$ to ${\textup{data}}({C}_i)\cup{\textup{data}}(w(i,k])$. We are going to prove that ${C}_i,w(i,k] {\sim}_{f_i} {C}'_i,w'(i,k]$. Note that $f_i(w(i,k])=w'(i,k]$ holds by definition of $f_i$ and (2). We prove $f_i({C}_i)\subseteq {C}'_i$. Suppose $({\ell},{\boldsymbol{a}})\in {C}_i$. Hence there exists $({\ell}_{i-1},{\boldsymbol{b}})\in {C}_{i-1}$ such that $({\ell}_{i-1},{\boldsymbol{b}})\xrightarrow{\sigma_i,d_i}({\ell},{\boldsymbol{a}})$. Thus there exists an edge ${\ell}_{i-1}\xrightarrow{\sigma_i,\phi,\lambda}{\ell}$ such that ${\boldsymbol{b}},d_i \models \phi$ and ${\boldsymbol{a}}={\boldsymbol{b}}[\lambda\leftarrow d_i]$. By induction hypothesis, there exists $({\ell}_{i-1},{\boldsymbol{b'}})\in{C}'_{i-1}$ such that $f_{i-1}({\boldsymbol{b}})={\boldsymbol{b'}}$. By induction on the structure of $\phi$, one can easily prove that ${\boldsymbol{b}},d_i \models \phi$ if, and only if, ${\boldsymbol{b'}},d'_{i}\models\phi$. Define ${\boldsymbol{a'}}:={\boldsymbol{b'}}[\lambda\leftarrow d'_i]$. We prove $f_i({\boldsymbol{a}})={\boldsymbol{a'}}$: there are two cases: (i) If ${r}\in\lambda$, then $f_i({\boldsymbol{a}}({r})) = f_i(d_i)=d'_i={\boldsymbol{a'}}(r)$. (ii) If ${r}\not\in\lambda$, then $f_i({\boldsymbol{a}}({r})) = f_i({\boldsymbol{b}}({r}))=f_{i-1}({\boldsymbol{b}}({r}))={\boldsymbol{a'}}(r)$. Hence, $f_i({\boldsymbol{a}})={\boldsymbol{a'}}$. Altogether $({\ell}, f_i({\boldsymbol{a}}))\in{C}'_i$, and thus $f_i({C}_i)\subseteq {C}'_i$. The proof for ${C}'_i\subseteq f_i({C}_i)$ is analogous. Altogether, ${C}_i,w(i,k] {\sim}_{f_i} {C}'_i,w'(i,k]$.
As an immediate consequence of Proposition \[prop:ra\_equivalence\], we obtain that $\sim$ preserves the configuration properties of being *accepting* respectively *non-accepting*.
\[corollary:nra\_bad\] Let ${C}$ and ${C}'$ be two configurations of ${\mathcal{A}}$. If ${C},w\sim{C}',w'$ and ${\textup{Succ}}_{\mathcal{A}}({C},w)$ is non-accepting (accepting, respectively), then ${\textup{Succ}}_{\mathcal{A}}({C}',w')$ is non-accepting (accepting, respectively).
Combining the last corollary with Proposition \[prop:ura\_implied\_badness\], we obtain
\[corollary:ura\_bad\] If ${\mathcal{A}}$ is a ${\textup{URA}}$ and ${C},{C}'$ are two configurations such that ${C}\cap{C}'=\emptyset$ and ${C}\cup{C}'$ is coverable in ${\mathcal{A}}$, then for every data word $w$ such that ${C},w\sim{C}',w$, the configurations ${\textup{Succ}}_{\mathcal{A}}({C},w)$ and ${\textup{Succ}}_{\mathcal{A}}({C}',w)$ are non-accepting.
For the rest of this paper, let ${\mathcal{A}}=({R}^{\mathcal{A}},{\mathcal{L}}^{\mathcal{A}},{\ell}^{\mathcal{A}}_{{\textup{in}}},{\mathcal{L}}^{\mathcal{A}}_{{\textup{acc}}},{E}^{\mathcal{A}})$ be an over $\Sigma$, and let ${\mathcal{B}}=({R}^{\mathcal{B}},{\mathcal{L}}^{\mathcal{B}},{\ell}^{\mathcal{B}}_{{\textup{in}}},{\mathcal{L}}^{\mathcal{B}}_{{\textup{acc}}},{E}^{\mathcal{B}})$ be a over $\Sigma$. Without loss of generality, we assume ${R}^{\mathcal{A}}\cap {R}^{\mathcal{B}}=\emptyset$ and ${\mathcal{L}}^{\mathcal{A}}\cap{\mathcal{L}}^{\mathcal{B}}=\emptyset$. We let $m$ be the number of registers of ${\mathcal{A}}$, and we let $n$ be the number of registers of ${\mathcal{B}}$.
A *synchronized configuration of ${\mathcal{A}}$ and ${\mathcal{B}}$* is a pair $(({\ell},{\boldsymbol{d}}),{C})$, where $({\ell},{\boldsymbol{d}})\in ({\mathcal{L}}^{\mathcal{A}}\times{\mathbb{D}}^{{R}^{\mathcal{A}}}_\bot)$ is a single state of ${\mathcal{A}}$, and ${C}\subseteq ({\mathcal{L}}^{\mathcal{B}}\times{\mathbb{D}}^{{R}^{\mathcal{B}}}_\bot)$ is a configuration of ${\mathcal{B}}$. Given a synchronized configuration ${S}$, we use ${\textup{data}}({S})$ to denote the set ${\textup{data}}({\boldsymbol{d}}) \cup {\textup{data}}({C})$ of all data occurring in ${S}$. We define ${S}_{{\textup{in}}}:=(({\ell}_{{\textup{in}}}^{\mathcal{A}},\{\bot\}^m), \{({\ell}_{\textup{in}}^{\mathcal{B}},\{\bot\}^n)\})$ to be the *initial synchronized configuration of ${\mathcal{A}}$ and ${\mathcal{B}}$*. We define the *synchronized state space of ${\mathcal{A}}$ and ${\mathcal{B}}$* to be the (infinite) state transition system $({\mathbb{S}},{\Rightarrow})$, where ${\mathbb{S}}$ is the set of all synchronized configurations of ${\mathcal{A}}$ and ${\mathcal{B}}$, and ${\Rightarrow}$ is defined as follows. If ${S}=(({\ell},{\boldsymbol{d}}),{C})$ and ${S}'=(({\ell}',{\boldsymbol{d'}}),{C}')$, then ${S}{\Rightarrow}{S}'$ if there exists a letter $(\sigma,d)\in(\Sigma\times{\mathbb{D}})$ such that $({\ell},{\boldsymbol{d}})\xrightarrow{\sigma,d}_{\mathcal{A}}({\ell}',{\boldsymbol{d'}})$, and ${\textup{Succ}}_{\mathcal{B}}({C},(\sigma,d))={C}'$. We say that a synchronized configuration *${S}$ reaches a synchronized configuration ${S}'$ in $({\mathbb{S}},{\Rightarrow})$* if there exists a path in $({\mathbb{S}},{\Rightarrow})$ from ${S}$ to ${S}'$. We say that a synchronized configuration ${S}$ is *reachable in $({\mathbb{S}},{\Rightarrow})$* if ${S}_{\textup{in}}$ reaches ${S}$. We say that a synchronized configuration ${S}=(({\ell},{\boldsymbol{d}}),{C})$ is *coverable in $({\mathbb{S}},{\Rightarrow})$* if there exists some synchronized configuration ${S}'=(({\ell},{\boldsymbol{d}}),{C}')$ such that ${C}'\supseteq{C}$ and ${S}'$ is reachable in $({\mathbb{S}},{\Rightarrow})$.
We aim to reduce the containment problem $L({\mathcal{A}})\subseteq L({\mathcal{B}})$ to a reachability problem in $({\mathbb{S}},{\Rightarrow})$. For this, call a synchronized configuration $(({\ell},{\boldsymbol{d}}),{C})$ *bad* if ${\ell}\in{\mathcal{L}}_{\textup{acc}}^{\mathcal{A}}$ is an accepting location and ${C}$ is non-accepting, i.e., ${\ell}'\not\in{\mathcal{L}}_{\textup{acc}}^{\mathcal{B}}$ for all $({\ell}',{\boldsymbol{a}})\in{C}$. The following proposition is easy to prove, cf. [@DBLP:conf/lics/OuaknineW04].
\[prop:reductionToReach\] $L({\mathcal{A}})\subseteq L({\mathcal{B}})$ does not hold if, and only if, some bad synchronized configuration is reachable in $({\mathbb{S}},{\Rightarrow})$.
We extend the equivalence relation $\sim$ defined above to synchronized configurations in a natural manner, i.e, given a partial isomorphism $f$ of ${\mathbb{D}}_\bot$ such that ${\textup{data}}({\boldsymbol{d}})\cup{\textup{data}}({C})\subseteq\operatorname{dom}(f)$, we define $(({\ell},{\boldsymbol{d}}),{C}) \sim_f (({\ell},{\boldsymbol{d}}'),{C}')$ if $f({C})={C}'$ and $f({\boldsymbol{d}})={\boldsymbol{d}}'$. We shortly write ${S}\sim{S}'$ if there exists a partial isomorphism $f$ of ${\mathbb{D}}_\bot$ such that ${S}\sim_f{S}'$. Clearly, an analogon of Proposition \[prop:ra\_equivalence\] holds for this extended relation. In particular, we have the following:
\[prop:equivalence-relation-synch-compatible\] Let ${S},{S}'$ be two synchronized configurations of $({\mathbb{S}},{\Rightarrow})$ such that ${S}\sim{S}'$. If ${S}$ reaches a bad synchronized configuration, so does ${S}'$.
Note that the state transition system $({\mathbb{S}},{\Rightarrow})$ is infinite. First of all, $({\mathbb{S}},{\Rightarrow})$ is not finitely branching: for every synchronized configuration ${S}=(({\ell},{\boldsymbol{d}}),{C})$ in ${\mathbb{S}}$, every datum $d\in{\mathbb{D}}$ may give rise to its own individual synchronized configuration ${S}_d$ such that ${S}\Rightarrow{S}_d$. However, it can be easily seen that for every two different data values $d,d'\in{\mathbb{D}}\backslash{\textup{data}}({S})$, if inputting $(\sigma,d)$ gives rise to a transition ${S}\Rightarrow{S}_d$ and inputting $(\sigma, d')$ gives rise to a transition ${S}\Rightarrow{S}_{d'}$ (for some $\sigma\in\Sigma$), then ${S}_d\sim{S}_{d'}$. Hence there exist synchronized configurations ${S}_1, \dots, {S}_k$ for some $k\in{\mathbb N}$ such that ${S}{\Rightarrow}{S}_i$ for all $i\in\{1,\dots, k\}$, and such that for all ${S}'\in{\mathbb{S}}$ with ${S}{\Rightarrow}{S}'$ there exists $i\in\{1,\dots,k\}$ such that ${S}_i\sim{S}'$. This is why we define in Section \[sect:abstract\] the notion of *abstract configuration*, representing synchronized configurations up to the relation $\sim$. Second, and potentially more harmful for the termination of an algorithm to decide the reachability problem from Proposition \[prop:reductionToReach\], the configuration ${C}$ of ${\mathcal{B}}$ in a synchronized configuration may grow unboundedly. As an example, consider the URA on the left side of Figure \[fig:ura\]. For every $k\geq 1$, the configuration $\{({\ell}_0,\bot),({\ell}_1,d_1),({\ell}_1,d_2)\dots,({\ell}_1,d_k)\}$ with pairwise distinct data values $d_1,\dots,d_k$ is reachable in this URA by inputting the data word $(\sigma,d_1)(\sigma,d_2)\dots(\sigma,d_k)$. In the next section, we prove that we can solve the reachability problem from Proposition \[prop:reductionToReach\] by focussing on a subset of configurations of ${\mathcal{B}}$ that are bounded in size, thus reducing to a reachability problem on a finite graph.
The Containment Problem for Register Automata
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Types
-----
Given $k\in{\mathbb N}$, a $k$-type[^1] of ${\mathbb{D}}_\bot$ is a quantifier-free formula $\varphi(y_1,\dots,y_k)$ formed by a conjunction of (positive or negative) literals of the form $y_i=y_j$ and $y_i=\bot$ that is satisfiable in ${\mathbb{D}}_\bot$. A $k$-type is *complete* if for any other quantifier-free formula $\psi(y_1,\dots,y_k)$, either $\forall y_1,\dots, y_k.(\varphi(y_1,\dots,y_k)\Rightarrow\psi(y_1,\dots,y_k))$ holds or $\varphi\land\psi$ is unsatisfiable. It is easy to see that given ${\boldsymbol{a}}\in{\mathbb{D}}^k$, there is a unique complete $k$-type $\varphi$ such that $\varphi({\boldsymbol{a}})$ holds in ${\mathbb{D}}_\bot$. We call $\varphi$ the *type of ${\boldsymbol{a}}$* and denote it by ${\textup{tp}}({\boldsymbol{a}})$. It may be observed that ${\boldsymbol{a}},{\boldsymbol{b}}\in{\mathbb{D}}_\bot^k$ have the same type if, and only if, there exists a partial isomorphism $f$ of ${\mathbb{D}}_\bot$ such that $f({\boldsymbol{a}})={\boldsymbol{b}}$.
Recall that $m$ and $n$ denote the number of registers of ${\mathcal{A}}$ and ${\mathcal{B}}$. For every ${\boldsymbol{a}}\in{\mathbb{D}}^n_\bot$ and for every complete $(2n+m)$-type $\varphi({\boldsymbol{y}})$, where ${\boldsymbol{y}}=(y_1,\dots,y_{2n+m})$, we define the set $$\begin{aligned}
{\mathcal{L}}_\varphi({\boldsymbol{a}})=\{ {\ell}'\in{\mathcal{L}}^{\mathcal{B}}\mid \exists {\boldsymbol{b}}\in {\mathbb{D}}_\bot^n \text{ such that $({\ell}',{\boldsymbol{b}})\in C$ and } \varphi({\boldsymbol{a}},{\boldsymbol{b}},{\boldsymbol{d}}) \text{ holds in } {\mathbb{D}}_\bot\}.\end{aligned}$$ Let ${S}=(({\ell},{\boldsymbol{d}}),{C})$ be a synchronized configuration and let ${\boldsymbol{a}},{\boldsymbol{b}}\in{\mathbb{D}}^n_\bot$ be two register valuations occurring in ${C}$, i.e., there exist ${\ell}_{{\boldsymbol{a}}},{\ell}_{{\boldsymbol{b}}}\in{\mathcal{L}}^{\mathcal{B}}$ such that $({\ell}_{{\boldsymbol{a}}},{\boldsymbol{a}}),({\ell}_{{\boldsymbol{b}}},{\boldsymbol{b}})\in{C}$. We say that *${\boldsymbol{a}}$ and ${\boldsymbol{b}}$ are indistinguishable in ${S}$*, written ${{{\boldsymbol{a}}} \equiv_{{S}} {{\boldsymbol{b}}}}$, if ${\mathcal{L}}_\varphi({\boldsymbol{a}}) = {\mathcal{L}}_\varphi({\boldsymbol{b}})$ for every complete $(2n+m)$-type $\varphi({\boldsymbol{y}})$.
\[ex:types\_and\_indisc\] Let $({\ell}^{\mathcal{A}},3)$ be a state in some with a single register, and let ${C}' = \{({\ell},1,3), ({\ell},2,3), ({\ell}',1,2)\}$ be a configuration of a with two registers. Let ${S}'=(({\ell}^{\mathcal{A}},3),{C}')$ be the corresponding synchronized configuration of ${\mathcal{A}}$ and ${\mathcal{B}}$. Consider ${\boldsymbol{a}}=(1,3)$ and ${\boldsymbol{b}}=(2,3)$. For the $5$-type $$\begin{aligned}
\varphi_1 = (y_1\neq y_2) \wedge (y_1 \neq y_3) \wedge (y_2=y_4) \wedge (y_4 =y_5) \wedge (y_3\neq y_2)\end{aligned}$$ we have ${\mathcal{L}}_{\varphi_1}({\boldsymbol{a}})=\{{\ell}\}$ as $\varphi_1({\boldsymbol{a}},{\boldsymbol{b}},{\boldsymbol{d}})$ holds in $({\mathbb N},=)$, and similarly, ${\mathcal{L}}_{\varphi_1}({\boldsymbol{b}})=\{{\ell}\}$ as $\varphi_1({\boldsymbol{b}},{\boldsymbol{a}},{\boldsymbol{d}})$ holds in $({\mathbb N},=)$. However, we have ${\mathcal{L}}_{\varphi_2}({\boldsymbol{a}})=\{{\ell}'\}$ and ${\mathcal{L}}_{\varphi_2}({\boldsymbol{b}})=\emptyset$ for the $5$-type $$\begin{aligned}
\varphi_2 = (y_1\neq y_2) \wedge (y_1=y_3) \wedge (y_2\neq y_4) \wedge (y_2=y_5) \wedge (y_4\neq y_1).\end{aligned}$$ Hence ${{{\boldsymbol{a}}} \equiv_{{S}'} {{\boldsymbol{b}}}}$ does *not* hold. However, ${{{\boldsymbol{a}}} \equiv_{{S}} {{\boldsymbol{b}}}}$ for ${S}= (({\ell}^{\mathcal{A}},3),{C})$ with ${C}:={C}' \cup \{({\ell}',2,1)\}$.
\[prop:indiscernible\_implies\_regsim\] Let ${S}=(({\ell}^{\mathcal{A}},{\boldsymbol{d}}), {C})$ be a coverable synchronized configuration of ${\mathcal{A}}$ and ${\mathcal{B}}$. Let ${\boldsymbol{a}},{\boldsymbol{b}}$ be such that ${{{\boldsymbol{a}}} \equiv_{{S}} {{\boldsymbol{b}}}}$. Then the map $f\colon{\textup{data}}({\boldsymbol{a}})\to{\textup{data}}({\boldsymbol{b}})$ defined by $f(a_i):=b_i$ is a partial isomorphism of ${\mathbb{D}}_\bot$. Moreover, if we let ${C}_{{\boldsymbol{a}}} := \{({\ell},{\boldsymbol{a}})\in {C}\mid {\ell}\in{\mathcal{L}}^{\mathcal{B}}\}$ and ${C}_{{\boldsymbol{b}}}:= \{({\ell},{\boldsymbol{b}})\in {C}\mid {\ell}\in{\mathcal{L}}^{\mathcal{B}}\}$, then ${C}_{{\boldsymbol{a}}} {\sim}_f {C}_{{\boldsymbol{b}}}$.
Let $\varphi$ be the complete $(2n+m)$-type of $({\boldsymbol{a}},{\boldsymbol{a}},{\boldsymbol{d}})$. Note that for two vectors ${\boldsymbol{u}},{\boldsymbol{v}}\in{\mathbb{D}}^n_\bot$, $\varphi({\boldsymbol{u}},{\boldsymbol{v}},{\boldsymbol{d}})$ holds in ${\mathbb{D}}_\bot$ iff ${\boldsymbol{u}}={\boldsymbol{v}}$ and ${\textup{tp}}({\boldsymbol{a}},{\boldsymbol{d}})={\textup{tp}}({\boldsymbol{u}}, {\boldsymbol{d}})={\textup{tp}}({\boldsymbol{v}}, {\boldsymbol{d}})$.
Let now $({\ell},{\boldsymbol{a}})$ be in ${C}_{{\boldsymbol{a}}}$. By definition, this means that ${\ell}\in{\mathcal{L}}_\varphi({\boldsymbol{a}})$. By indistinguishibility, ${\ell}\in{\mathcal{L}}_\varphi({\boldsymbol{b}})$ so that $$\begin{aligned}
\label{eq:indiscernible_implies_regsim}
\varphi({\boldsymbol{b}},{\boldsymbol{c}},{\boldsymbol{d}}) \mbox{ holds in } {\mathbb{D}}_\bot\tag{$\dagger$}\end{aligned}$$ for some $({\ell},{\boldsymbol{c}})\in{C}$. Now, (\[eq:indiscernible\_implies\_regsim\]) implies ${\boldsymbol{b}}={\boldsymbol{c}}$ and ${\textup{tp}}({\boldsymbol{b}})={\textup{tp}}({\boldsymbol{a}})$. The former implies that $({\ell},{\boldsymbol{b}})\in{C}_{{\boldsymbol{b}}}$, while the latter implies that $f$ is a partial isomorphism. Conversely, we obtain that $({\ell},{\boldsymbol{b}})\in {C}_{{\boldsymbol{b}}}$ implies $({\ell},{\boldsymbol{a}})\in{C}_{{\boldsymbol{a}}}$. Hence $f({C}_{{\boldsymbol{a}}})={C}_{{\boldsymbol{b}}}$ and thus ${C}_{{\boldsymbol{a}}}{\sim}_f{C}_{{\boldsymbol{b}}}$.
Collapsing Configurations
-------------------------
As we pointed out in the introduction, the crucial ingredient of our algorithm for deciding whether $L({\mathcal{A}})\subseteq L({\mathcal{B}})$ holds is to prevent configurations ${C}$ in a synchronized configuration $(({\ell},{\boldsymbol{d}}),{C})$ to grow unboundedly. We do this by *collapsing two* subconfigurations ${C}_{{\boldsymbol{a}}}, {C}_{{\boldsymbol{b}}}\subseteq{C}$ that behave equivalently with respect to reaching a bad synchronized configuration in $({\mathbb{S}},{\Rightarrow})$ into a *single* subconfiguration. The key notions for deciding when two subconfigurations can be collapsed into a single one are *$k$-types* and *indistinguishability* from the previous subsection.
\[prop:collapse\_URA\] Let ${S}'=(({\ell},{\boldsymbol{d}}),{C}')$ be a coverable synchronized configuration of ${\mathcal{A}}$ and ${\mathcal{B}}$. Let ${\boldsymbol{a}}$ and ${\boldsymbol{b}}$ be two distinct register valuations in ${C}'$ such that ${{{\boldsymbol{a}}} \equiv_{{S}'} {{\boldsymbol{b}}}}$. Let ${C}_{{\boldsymbol{b}}} := \{ ({\ell},{\boldsymbol{b}}) \in {C}' \mid {\ell}\in{\mathcal{L}}^{\mathcal{B}}\}$. Then ${S}:=(({\ell},{\boldsymbol{d}}),{C}'\setminus{C}_{{\boldsymbol{b}}})$ reaches a bad synchronized configuration if, and only if, ${S}'$ reaches a bad synchronized configuration.
The “if” direction follows from the simple observation that for every data word $w$, if ${\textup{Succ}}_{\mathcal{B}}({C}',w)$ is non-accepting, then so is ${\textup{Succ}}_B(D,w)$ for every subset $D\subseteq {C}'$. For the “only if” direction, let ${C}_{{\boldsymbol{a}}} := \{ ({\ell},{\boldsymbol{a}}) \in {C}' \mid {\ell}\in{\mathcal{L}}^{\mathcal{B}}\}$ and ${C}:={C}'\setminus ({C}_{{\boldsymbol{a}}}\cup {C}_{{\boldsymbol{b}}})$. Let $m$ be the number of registers of ${\mathcal{A}}$ and $n$ be the number of registers of ${\mathcal{B}}$. Suppose that there exists a data word $w$ such that there exists an accepting run of ${\mathcal{A}}$ on $w$ that starts in $({\ell},{\boldsymbol{d}})$, and ${\textup{Succ}}_{\mathcal{B}}({C}_{{\boldsymbol{a}}}\cup{C},w)$ is non-accepting. We assume in the following that ${\textup{Succ}}_{\mathcal{B}}({C}_{{\boldsymbol{b}}},w)$ is accepting; otherwise we are done. Without loss of generality, we assume that ${\textup{data}}(w) \cap {\textup{data}}({S}')\subseteq {\textup{data}}({\boldsymbol{b}})\cup{\textup{data}}({\boldsymbol{d}})$. Otherwise, pick for every $d\in {\textup{data}}(w)\cap({\textup{data}}({\boldsymbol{a}})\cup{\textup{data}}({C}))$ such that $d\not\in{\textup{data}}({\boldsymbol{b}})\cup{\textup{data}}({\boldsymbol{d}})$, a fresh datum $d'\in{\mathbb{D}}$ not occurring in ${\textup{data}}(w)\cup{\textup{data}}({S}')$, and simultaneously replace every occurrence of $d$ in $w$ by $d'$. Let $w'$ be the resulting data word. Then $({\ell},{\boldsymbol{d}}),w {\sim}({\ell},{\boldsymbol{d}}),w'$ and ${C}_{{\boldsymbol{b}}},w{\sim}{C}_{{\boldsymbol{b}}},w'$. By Corollary \[corollary:nra\_bad\], ${\textup{Succ}}_{\mathcal{A}}(({\ell},{\boldsymbol{d}}),w')$ is accepting, and ${\textup{Succ}}_{\mathcal{B}}({C}_{{\boldsymbol{b}}},w')$ is accepting, too. Then there must exist some accepting run of ${\mathcal{A}}$ on $w'$ starting in $({\ell},{\boldsymbol{d}})$, and, by Proposition \[prop:ura\_implied\_badness\], ${\textup{Succ}}_{\mathcal{B}}({C}_{{\boldsymbol{a}}}\cup {C},w')$ must be non-accepting. Hence, we could continue the proof with $w'$ instead of $w$. Let us assume henceforth that ${\textup{data}}(w) \cap {\textup{data}}({S}')\subseteq{\textup{data}}({\boldsymbol{b}})\cup{\textup{data}}({\boldsymbol{d}})$ holds.
Let now $w''$ be the data word obtained from $w$ as follows: for every $b_i\in{\textup{data}}(w)$ with $b_i\neq a_i$, pick some fresh datum $e_i\in{\mathbb{D}}$ not occurring in ${\textup{data}}(w)\cup{\textup{data}}({S}')$. Then replace every occurrence of the letter $b_i$ in $w$ by $e_i$.
Note that $({\ell},{\boldsymbol{d}}),w\sim({\ell},{\boldsymbol{d}}),w''$: the key argument for this is that by ${{{\boldsymbol{a}}} \equiv_{{S}'} {{\boldsymbol{b}}}}$ we have $b_i\not\in{\textup{data}}({\boldsymbol{d}})$ whenever $b_i\neq a_i$. By Corollary \[corollary:nra\_bad\], ${\textup{Succ}}_{\mathcal{A}}(({\ell},{\boldsymbol{d}}),w'')$ is accepting. Hence there must exist some accepting run of ${\mathcal{A}}$ on $w''$ starting in $({\ell},{\boldsymbol{d}})$.
Further note that ${C}_{{\boldsymbol{a}}},w''{\sim}{C}_{{\boldsymbol{b}}},w''$: by Proposition \[prop:indiscernible\_implies\_regsim\], ${C}_{{\boldsymbol{a}}}{\sim}_f{C}_{{\boldsymbol{b}}}$, where $f:{\textup{data}}({\boldsymbol{a}})\to{\textup{data}}({\boldsymbol{b}})$ is the bijective mapping defined by $f(a_i)=b_i$ for all $1\leq i \leq n$. Now let $g:{\textup{data}}({\boldsymbol{a}})\cup{\textup{data}}(w'')\to{\textup{data}}({\boldsymbol{b}})\cup{\textup{data}}(w'')$ be the bijective mapping that agrees with $f$ on all data in ${\textup{data}}({\boldsymbol{a}})$, and that maps each datum $d\in{\textup{data}}(w'')\backslash{\textup{data}}({\boldsymbol{a}})$ to $d$. One can easily see that $g$ is a bijection such that $g({C}_{{\boldsymbol{a}}})={C}_{{\boldsymbol{b}}}$ and $g(w'')=w''$ so that indeed ${C}_{{\boldsymbol{a}}},w''{\sim}_g{C}_{{\boldsymbol{b}}},w''$. By Corollary \[corollary:ura\_bad\], ${\textup{Succ}}_{\mathcal{B}}({C}_{{\boldsymbol{a}}},w'')$ and ${\textup{Succ}}_{\mathcal{B}}({C}_{{\boldsymbol{b}}},w'')$ are non-accepting.
Finally, we prove that ${\textup{Succ}}_{\mathcal{B}}(C,w'')$ is non-accepting, too. For this, let $({\ell}',{\boldsymbol{c}})\in C$; we prove that ${\textup{Succ}}_{\mathcal{B}}(({\ell}',{\boldsymbol{c}}),w'')$ is non-accepting. We distinguish the following two cases:
- For all $1\leq i\leq n$ with $a_i\neq b_i$ we have $b_i\not\in{\textup{data}}({\boldsymbol{c}})$. Then $({\ell}',{\boldsymbol{c}}),w {\sim}({\ell}',{\boldsymbol{c}}),w''$, as witnessed by the bijection $f$ such that $f(b_i)=e_i$ for all $b_i\in{\textup{data}}(w)$ such that $b_i\neq a_i$, and that is the identity otherwise. Recall that by assumption ${\textup{Succ}}_{\mathcal{B}}(({\ell}',{\boldsymbol{c}}),w)$ is non-accepting. By Corollary \[corollary:nra\_bad\], ${\textup{Succ}}_{\mathcal{B}}(({\ell}',{\boldsymbol{c}}),w'')$ is non-accepting.
- There exists $1\leq i\leq n$ such that $a_i\neq b_i$ and $b_i\in{\textup{data}}({\boldsymbol{c}})$.
Let $\varphi({\boldsymbol{y}})$ be the $(2n+m)$-type of $({\boldsymbol{b}},{\boldsymbol{c}},{\boldsymbol{d}})$, and note that ${\ell}'\in {\mathcal{L}}_\varphi({\boldsymbol{b}})$. By assumption ${\ell}'\in{\mathcal{L}}_\varphi({\boldsymbol{a}})$ and there exists a state $({\ell}',{\boldsymbol{c'}})\in {C}$ such that $\varphi({\boldsymbol{a}},{\boldsymbol{c'}},{\boldsymbol{d}})$ holds. Note that for all $1\leq j\leq n$ such that $b_i=c_j$ we have $a_i=c'_j$. By assumption, $b_i = c_j$ for some $1\leq j\leq n$. Since $a_i\neq b_i$, we can infer $c_j\neq c'_j$, and hence $({\ell}',{\boldsymbol{c}})\neq({\ell}',{\boldsymbol{c'}})$. Next we prove $({\ell}',{\boldsymbol{c}}),w'' {\sim}({\ell}',{\boldsymbol{c'}}),w''$. We define $f:{\textup{data}}({\boldsymbol{c}})\cup{\textup{data}}(w'') \to {\textup{data}}({\boldsymbol{c'}})\cup{\textup{data}}(w'')$ as follows: $$\begin{aligned}
f\colon \begin{cases}
c_p \mapsto c'_p & 1\leq p\leq n \\
e \mapsto e & e\in{\textup{data}}(w'')
\end{cases}
\end{aligned}$$ We prove below that
1. for all $1\leq p,q\leq n$, $c_p=c_q$ iff $c'_p=c'_q$;
2. for all $1\leq p\leq n$, for all $e\in{\textup{data}}(w'')$, $e=c_p$ iff $e=c'_p$;
note that this implies that $f$ is well-defined and $f$ is a bijective mapping, and hence $({\ell}',{\boldsymbol{c}}),w'' {\sim}_f ({\ell}',{\boldsymbol{c'}}),w''$. By Proposition \[prop:ra\_equivalence\], ${\textup{Succ}}_{\mathcal{B}}(({\ell}',{\boldsymbol{c}}),w'') {\sim}{\textup{Succ}}_{\mathcal{B}}(({\ell}',{\boldsymbol{c'}}),w'')$. By Corollary \[corollary:ura\_bad\], ${\textup{Succ}}_{\mathcal{B}}(({\ell}',{\boldsymbol{c}}),w'')$ and ${\textup{Succ}}_{\mathcal{B}}(({\ell}',{\boldsymbol{c'}}),w'')$ are non-accepting. We now prove the two items from above: (i) Follows directly from the fact that $\varphi({\boldsymbol{a}},{\boldsymbol{c'}},{\boldsymbol{d}})$ and $\varphi({\boldsymbol{b}},{\boldsymbol{c}},{\boldsymbol{d}})$ hold, which implies that ${\boldsymbol{c'}}$ and ${\boldsymbol{c}}$ have the same type. For (ii), recall that ${\textup{data}}(w)\cap{\textup{data}}({S}')\subseteq {\textup{data}}({\boldsymbol{b}})\cup{\textup{data}}({\boldsymbol{d}})$. This, the definition of $w''$, and ${{{\boldsymbol{a}}} \equiv_{{S}'} {{\boldsymbol{b}}}}$ yield the claim.
Altogether, we proved that ${\textup{Succ}}_{\mathcal{B}}({C}',w'')$ is non-accepting, while there exists some accepting run $({\ell},{\boldsymbol{d}})\longrightarrow^*({\ell}'',{\boldsymbol{d''}})$ of ${\mathcal{A}}$ on $w''$. This finishes the proof for the “only if” direction.
When ${S}$ is obtained from ${S}'$ by applying Proposition \[prop:collapse\_URA\] to some pair of register valuations, we say that ${S}'$ *collapses to* ${S}$. We say that ${S}$ is *maximally collapsed* if for all pairs ${\boldsymbol{a}}$ and ${\boldsymbol{b}}$ of distinct register valuations appearing in ${C}$ we have that ${{{\boldsymbol{a}}} \equiv_{{S}} {{\boldsymbol{b}}}}$ does *not* hold. Note that in Proposition \[prop:collapse\_URA\], the synchronized configuration ${S}$ is again coverable. By iterating Proposition \[prop:collapse\_URA\], one obtains that a coverable synchronized configurations reaches a bad synchronized configuration if, and only if, it collapses in finitely many steps to a maximally collapsed synchronized configuration that also reaches a bad synchronized configuration.
Before we present our algorithm for deciding the containment problem, we would like to point out that the intuitive notion of *types* alone is not sufficient for deciding whether synchronized configurations can be collapsed. More precisely, given a coverable synchronized configuration ${S}'=(({\ell}^{\mathcal{A}},{\boldsymbol{d}}),{C}')$ and two register valuations ${\boldsymbol{a}}$ and ${\boldsymbol{b}}$ that occur in ${C}'$ and for which ${\textup{tp}}({\boldsymbol{a}},{\boldsymbol{d}})={\textup{tp}}({\boldsymbol{b}},{\boldsymbol{d}})$, it is in general *not* the case that ${S}'$ reaches a bad synchronized configuration if ${S}:=(({\ell},{\boldsymbol{d}}),{C}'\backslash{C}_{{\boldsymbol{b}}})$, where ${C}_{{\boldsymbol{b}}}:=\{({\ell},{\boldsymbol{b}})\in{C}'\mid{\ell}\in{\mathcal{L}}^{\mathcal{B}}\}$, reaches a bad synchronized configuration. To see that, consider Figure \[fig:ura\_no\_collapse\], where two register automata over a singleton alphabet (we omit the labels at the edges) are depicted: an ${\mathcal{A}}$ with a single register ${r}$ on the left side, and a ${\mathcal{B}}$ with two registers ${r}_1$ and ${r}_2$ on the right side. Note that $L({\mathcal{A}})\subseteq L({\mathcal{B}})$. After processing the input data word $w=(\sigma,1)(\sigma,2)(\sigma,3)$, the synchronized configuration ${S}'=(({\ell}^{\mathcal{A}},3),{C}')$, where ${C}':=\{({\ell},1,3),({\ell},2,3),({\ell}',1,2)\})$, is reached in the synchronized state space of ${\mathcal{A}}$ and ${\mathcal{B}}$. For ${\boldsymbol{a}}=(1,3)$ and ${\boldsymbol{b}}=(2,3)$, we have ${\textup{tp}}({\boldsymbol{a}},{\boldsymbol{d}})={\textup{tp}}({\boldsymbol{b}},{\boldsymbol{d}})$, but ${{{\boldsymbol{a}}} \equiv_{{S}'} {{\boldsymbol{b}}}}$ does not hold (cf. Example \[ex:types\_and\_indisc\]). Indeed, ${\textup{Succ}}_{\mathcal{B}}({C}'\backslash{C}_{{\boldsymbol{b}}},(\sigma,2))$ is non-accepting, while ${C}'$ cannot reach any non-accepting configuration.
Abstract Configurations {#sect:abstract}
-----------------------
In this section, we study synchronized configurations up to the equivalence relation $\sim$. Recall that $m$ is the number of registers of ${\mathcal{A}}$ and $n$ is the number of registers of ${\mathcal{B}}$. An *abstract synchronized configuration of ${\mathcal{A}}$ and ${\mathcal{B}}$* is a tuple $({\ell}, \Gamma, \varphi)$ where $\varphi$ is a complete $(sn+m)$-type for some $s\in{\mathbb N}$, $\Gamma$ is an $s$-tuple of subsets of ${\mathcal{L}}^{\mathcal{B}}$, and ${\ell}\in{\mathcal{L}}^{\mathcal{A}}$.
The *size* of an abstract synchronized configuration is defined to be $(sn+m)\log(sn+m) + s |{\mathcal{L}}^{\mathcal{B}}| + \log(|{\mathcal{L}}^{\mathcal{A}}|)$, which corresponds to the size needed on the tape of a Turing machine to encode an abstract synchronized configuration (where one encodes, for example, an $(sn+m)$-type by giving for each of the $sn+m$ variables, a number in $\{1,\dots,sn+m\}$ in a way that $y_i=y_j$ is a conjunct in $\varphi$ iff $y_i$ and $y_j$ are assigned the same number).
Every synchronized configuration ${S}= (({\ell}^{\mathcal{A}},{\boldsymbol{d}}),{C})$ gives rise to an abstract synchronized configuration in the following way: let ${\boldsymbol{a}}^1,\dots,{\boldsymbol{a}}^s$ be the distinct register valuations in ${C}$, listed in some arbitrary order. Let $\varphi$ be the complete $(sn+m)$-type of $({\boldsymbol{a}}^1,\dots,{\boldsymbol{a}}^s,{\boldsymbol{d}})$. Let $C_{{\boldsymbol{a}}^i}:=\{ {\ell}\in {\mathcal{L}}^{\mathcal{B}}\mid ({\ell},{\boldsymbol{a}}^i)\in C\}$. We obtain an abstract synchronized configuration $({\ell}^{\mathcal{A}}, (C_{{\boldsymbol{a}}^1},\dots,C_{{\boldsymbol{a}}^s}), \varphi)$. Different enumerations of the register valuations of ${C}$ can yield different abstract configurations. We let $\operatorname{abs}({S})$ be the set of all abstract synchronized configurations that can be obtained from ${S}$. Every two abstract synchronized configurations in $\operatorname{abs}({S})$ can be obtained from one another by permuting the variables from the type and the entries from the tuple accordingly. It is easy to prove that ${S}\sim{S}'$ if, and only if, $\operatorname{abs}({S})=\operatorname{abs}({S}')$.
An abstract configuration $({\ell},\Gamma,\varphi)$ is said to be *maximally collapsed* if there exists a synchronized configuration $S$ such that $({\ell},\Gamma,\varphi)\in\operatorname{abs}(S)$ and such that $S$ is maximally collapsed (equivalently, one could ask that *every* $S$ such that $({\ell},\Gamma,\varphi)\in\operatorname{abs}(S)$ is maximally collapsed). The main result of this section is that the number of different register valuations in a maximally collapsed synchronized configuration is bounded. Let $B_r\leq r^r$ be the number of complete $r$-types, which is also called the *Bell number* of order $r$.
\[prop:number\_register\_valuations\] Let ${S}=(({\ell}^{\mathcal{A}},{\boldsymbol{d}}),{C})$ be a maximally collapsed synchronized configuration of ${\mathcal{A}}$ and ${\mathcal{B}}$. The number of different register valuations appearing in ${C}$ is bounded by $(B_{2n+m}\cdot 2^{|{\mathcal{L}}^{\mathcal{B}}|})^{(2n+m)^n}$.
We first prove a slightly worse upper bound, to give an idea of the proof. Let $K:=B_{2n+m}$. We prove that the number of different register valuations is bounded by $2^{|{\mathcal{L}}^{\mathcal{B}}|K}$. Associate with every register valuation ${\boldsymbol{a}}$ appearing in ${C}$ the $K$-tuple $(\mathcal{L}_{\varphi_1}({\boldsymbol{a}}),\dots,\mathcal{L}_{\varphi_{K}}({\boldsymbol{a}}))$ of subsets of ${\mathcal{L}}^{\mathcal{B}}$, where $\varphi_1,\dots,\varphi_{K}$ is an enumeration of all the complete $(2n+m)$-types. Note that there are at most $2^{|{\mathcal{L}}^{\mathcal{B}}|K}$ such tuples. Suppose by contradiction that ${S}$ contains more than $2^{|{\mathcal{L}}^{\mathcal{B}}|K}$ different register valuations. By the pigeonhole principle there are two different register valuations ${\boldsymbol{a}}$ and ${\boldsymbol{b}}$ that have the same associated $K$-tuple. Note that if ${\boldsymbol{a}}$ and ${\boldsymbol{b}}$ share the same $K$-tuple, then ${{{\boldsymbol{a}}} \equiv_{{S}} {{\boldsymbol{b}}}}$. By Proposition \[prop:collapse\_URA\], ${S}$ could be collapsed further, contradiction. Hence, we proved an upper bound of $2^{|{\mathcal{L}}^{\mathcal{B}}|K}$ on the number of different register valuations appearing in a given maximally collapsed synchronized configuration.
We now proceed to prove the actual bound. The important fact is that when ${\boldsymbol{a}}$ and ${\boldsymbol{d}}$ are fixed in ${S}$, then few (i.e., $\leq (2n+m)^n$) entries in the tuple $(\mathcal{L}_{\varphi_1}({\boldsymbol{a}}),\dots,\mathcal{L}_{\varphi_{K}}({\boldsymbol{a}}))$ are non-empty. Indeed, in a given $(2n+m)$-type, each of the variables $y_{n+1},\dots,y_{2n}$ can be constrained to be equal to one of $y_1,\dots,y_n,y_{2n+1},\dots,y_{2n+m}$, or constrained to be different than all of them.
Therefore, it remains to bound the number of $K$-tuples with entries in $2^{{\mathcal{L}}^{\mathcal{B}}}$ and with at most $(2n+m)^n$ non-empty entries. Each such tuple is characterised by the subset $T\subseteq\{1,\dots,K\}$ of entries that are non-empty, together with a $|T|$-tuple of non-empty subsets of ${\mathcal{L}}^{\mathcal{B}}$. Since $|T|$ can be bounded by $(2n+m)^n$, we obtain that there are at most $K^{(2n+m)^n}\cdot 2^{|{\mathcal{L}}^{\mathcal{B}}|(2n+m)^n}$ possible tuples, and thus at most $(B_{2n+m}\cdot 2^{|{\mathcal{L}}^{\mathcal{B}}|})^{(2n+m)^n}$ different register valuations.
Note that the bound in Proposition \[prop:number\_register\_valuations\] is doubly exponential in $n$ and exponential in $|{\mathcal{L}}^{\mathcal{B}}|$ and $m$. As a direct corollary, we obtain a bound on the number of maximally collapsed abstract synchronized configurations.
\[prop:number\_collapsed\_config\] The number of maximally collapsed abstract configurations is bounded by a triple exponential in $|{\mathcal{A}}|$ and $|{\mathcal{B}}|$. If the number of registers of ${\mathcal{B}}$ is fixed, then the number of maximally collapsed abstract configurations is bounded by a double exponential in $|{\mathcal{A}}|$ and $|{\mathcal{B}}|$.
Recall that $m$ is the number of registers of ${\mathcal{A}}$ and $n$ is the number of registers of ${\mathcal{B}}$. By Proposition \[prop:number\_register\_valuations\], a maximally collapsed synchronized configuration ${S}=(({\ell}^{\mathcal{A}},{\boldsymbol{d}}),{C})$ is such that ${C}$ contains at most $K:=(B_{2n+m}\cdot 2^{|{\mathcal{L}}^{\mathcal{B}}|})^{(2n+m)^n}$ different register valuations. Therefore, any abstract synchronized configuration in $\operatorname{abs}({S})$ is described by an $(sn+m)$-type with $s\leq K$. For a given $s$, there are at most $B_{sn+m}\cdot |{\mathcal{L}}^{\mathcal{B}}|^s \cdot |{\mathcal{L}}^{\mathcal{A}}|$ different abstract synchronized configurations. Summing up from $s=0$ to $K$, we obtain that there are at most $$\begin{aligned}
\sum_{s=0}^{K} B_{sn+m}\cdot |{\mathcal{L}}^{\mathcal{B}}|^s \cdot |{\mathcal{L}}^{\mathcal{A}}| &\leq |{\mathcal{L}}^{\mathcal{A}}|\cdot \left(B_m+B_{n+m}|{\mathcal{L}}^{\mathcal{B}}|+\dots+ B_{nK+m}\cdot |{\mathcal{L}}^{\mathcal{B}}|^{K}\right)\\
& \leq |{\mathcal{L}}^{\mathcal{A}}| \cdot (1+K) \cdot B_{nK+m}\cdot |{\mathcal{L}}^{\mathcal{B}}|^{K}\\
&\leq |{\mathcal{L}}^{\mathcal{A}}| \cdot (1+K) \cdot (nK+m)^{(nK+m)}\cdot |{\mathcal{L}}^{\mathcal{B}}|^{K}\end{aligned}$$ maximally collapsed abstract synchronized configurations. Since $K$ is doubly exponential in $|{\mathcal{A}}|$ and $|{\mathcal{B}}|$, this gives the first result. The second result follows from the fact that for fixed $n$, $K$ only depends exponentially on $m$ and $|{\mathcal{L}}^{\mathcal{B}}|$.
Given abstract synchronized configurations $({\ell}^{\mathcal{A}}, \Gamma, \varphi)$ and $({\ell}'^{\mathcal{A}}, \Gamma', \varphi')$, define $({\ell}^{\mathcal{A}}, \Gamma, \varphi)\leadsto ({\ell}'^{\mathcal{A}}, \Gamma', \varphi')$ if there exist synchronized configurations ${S}$ and ${S}'$ such that:
- ${S}{\Rightarrow}{S}'$,
- $({\ell}^{\mathcal{A}}, \Gamma, \varphi)$ is in $\operatorname{abs}({S})$,
- ${S}'$ can be maximally collapsed to some ${S}''$ such that $({\ell}'^{\mathcal{A}}, \Gamma', \varphi')$ is in $\operatorname{abs}(S'')$.
\[lem:abstract\_relation\_pspace\] Given two abstract synchronized configurations $({\ell}^{\mathcal{A}}, \Gamma, \varphi)$ and $({\ell}'^{\mathcal{A}}, \Gamma', \varphi')$, deciding whether $({\ell}^{\mathcal{A}}, \Gamma, \varphi)\leadsto ({\ell}'^{\mathcal{A}}, \Gamma', \varphi')$ holds can be done in polynomial space.
In this proof, we assume without loss of generality that ${\mathbb{D}}={\mathbb N}$. Let $s$ be such that $\varphi$ is an $(sn+m)$-type. Note that there is a synchronized configuration ${S}$ of the form $(({\ell}^{\mathcal{A}},{\boldsymbol{d}}), D)$ such that ${\textup{data}}(D)\cup {\textup{data}}({\boldsymbol{d}})\subseteq\{1,\dots,sn+m\}$ and such that $({\ell}^{\mathcal{A}}, \Gamma, \varphi)\in\operatorname{abs}({S})$. This ${S}$ is moreover computable in polynomial space.
To decide whether $({\ell}^{\mathcal{A}}, \Gamma, \varphi)\leadsto ({\ell}'^{\mathcal{A}}, \Gamma', \varphi')$ holds, one simply:
- guesses a letter $\sigma\in\Sigma$ and a datum $d$ in $\{1,\dots,sn+m+1\}$,
- computes a synchronized configuration ${S}'$ obtained by firing the transition corresponding to $(\sigma,d)$ from ${S}$,
- guesses a sequence $({\boldsymbol{a}}^1,{\boldsymbol{b}}^1),\dots,({\boldsymbol{a}}^r,{\boldsymbol{b}}^r)$ of register valuations such that Proposition \[prop:collapse\_URA\] can be applied $r$ times to obtain a maximally collapsed configuration ${S}''$,
- checks that $({\ell}'^{\mathcal{A}}, \Gamma', \varphi')$ is in $\operatorname{abs}({S}'')$.
At the second step, the size of ${S}'$ is polynomially bounded by the size of ${\mathcal{A}}$, ${\mathcal{B}}$, and of ${S}$. Moreover, the maximal length of a collapsing sequence in the third step is also polynomially bounded, as the number of distinct register valuations decreases after each application of Proposition \[prop:collapse\_URA\]. Therefore, this algorithm uses a polynomial amount of space.
As for synchronized configuration, an abstract synchronized configuration $({\ell}^{\mathcal{A}}, \Gamma, \varphi)$ is called *bad* if ${\ell}^{\mathcal{A}}$ is an accepting location and none of the states in $\Gamma$ contains an accepting location.
\[prop:correctness\_abstraction\] A bad synchronized configuration is reachable in $({\mathbb{S}},{\Rightarrow})$ if, and only if, a bad abstract synchronized configuration is reachable from $\operatorname{abs}({S}_{\textup{in}})$.
We prove that for every coverable synchronized configuration $S$ and every $n\geq 0$, a bad synchronized configuration is reachable in $n$ steps from $S$ if, and only if, a bad abstract synchronized configuration is reachable in $n$ steps from $\operatorname{abs}(S)$. The statement then follows by taking $S:=S_{{\textup{in}}}$. The proof goes by induction on $n$, where the case $n=0$ is trivial in both directions.
Suppose now that $S$ reaches a bad synchronized configuration in $n$ steps. Let $S'$ be such that $S{\Rightarrow}S'$ and such that $S'$ reaches a bad synchronized configuration in $n-1$ steps. Let $S''$ be such that $S'$ can be maximally collapsed to $S''$. By iterating Proposition \[prop:collapse\_URA\], we have that $S''$ reaches a bad synchronized configuration in $n-1$ steps (the fact that the length of the path is unchanged can be seen from the proof of Proposition \[prop:collapse\_URA\]). It follows from the induction hypothesis that some $({\ell}',\Gamma',\varphi')\in\operatorname{abs}(S'')$ reaches a bad abstract synchronized configuration in $n-1$ steps. Let $({\ell},\Gamma,\varphi)$ be an arbitrary abstraction in $\operatorname{abs}(S)$. We have by definition $({\ell},\Gamma,\varphi)\leadsto ({\ell}',\Gamma',\varphi')$, so that $({\ell},\Gamma,\varphi)$ reaches a bad abstract synchronized configuration in $n$ steps. The converse direction is proved similarly.
Finally, we are able to present the main theorem.
\[thm:main\] The containment problem $L({\mathcal{A}})\subseteq L({\mathcal{B}})$, where ${\mathcal{A}}$ is a non-deterministic register automaton and ${\mathcal{B}}$ is an unambiguous register automaton, is in [$\textsf{2-EXPSPACE}$]{}. If the number of registers of ${\mathcal{B}}$ is fixed, the problem is in [$\textsf{EXPSPACE}$]{}.
The algorithm checks whether a bad abstract synchronized configuration is reachable from $\operatorname{abs}({S}_{{\textup{in}}})$, using the classical non-deterministic logspace algorithm for reachability. Every node of the graph can be stored using double-exponential space (see the second paragraph at the beginning of Section \[sect:abstract\]), and the size of the graph is triply exponential in the size of ${\mathcal{A}}$ and ${\mathcal{B}}$ by Proposition \[prop:number\_collapsed\_config\]. Moreover, the relation $\leadsto$ is decidable in polynomial space by Lemma \[lem:abstract\_relation\_pspace\]. Therefore, we obtain that the algorithm uses at most a double-exponential amount of space. In case the number of registers of ${\mathcal{B}}$ is fixed, Proposition \[prop:number\_collapsed\_config\] implies that the size of the graph is doubly exponential in the size of ${\mathcal{A}}$ and ${\mathcal{B}}$. We obtain that the algorithm uses at most an exponential amount of space.
As an immediate corollary of Theorem \[thm:main\], we obtain that the universality problem is in [$\textsf{2-EXPSPACE}$]{} and in [$\textsf{PSPACE}$]{} for fixed number of registers. Similarly, the equivalence problem for unambiguous register automata is in [$\textsf{2-EXPSPACE}$]{}.
Open Problems
=============
The most obvious problem is to figure out the *exact* computational complexity of the containment problem $L({\mathcal{A}})\subseteq L({\mathcal{B}})$, when ${\mathcal{B}}$ is an ${\textup{URA}}$. Finding lower bounds for unambiguous automata is a hard problem. Techniques for proving lower complexity bounds of the containment problem (respectively the universality problem) for the case where ${\mathcal{B}}$ is a non-deterministic automaton rely heavily on non-determinism (cf. Theorem 5.2 in [@DBLP:journals/tocl/DemriL09]); as was already pointed out in [@DBLP:conf/dcfs/Colcombet15], we are lacking techniques for finding lower computational complexity bounds for the case where ${\mathcal{B}}$ is unambiguous, even for the class of finite automata. Concerning the upper bound, computer experiments revealed that maximally collapsed synchronized configurations seem to remain small. Based on these experiments, we believe that the bound in Proposition \[prop:number\_register\_valuations\] is not optimal and can be improved to $O(2^{poly(n,m,|{\mathcal{L}}^{\mathcal{B}}|)})$. If this is correct, we would obtain an [$\textsf{EXPSPACE}$]{} upper-bound for the general containment problem.
We also would like to study to what extent our techniques can be used to solve the containment problem for other computation models. In particular, we are interested in the following:
- One can extend the definition of register automata to work over an ordered domain, where the register constraints are of the form $<r$ and $>r$. Proposition \[prop:collapse\_URA\] turns out to be false in this setting, but it seems plausible that there exists a collapsibility notion that would work for this model.
- An automaton ${\mathcal{B}}$ is said to be $k$-ambiguous if it has at most $k$ accepting runs for every input data word, and polynomially ambiguous if the number of accepting runs for some input data word $w$ is bounded by $p(|w|)$ for some polynomial $p$. Again, it is likely that simple modifications of Proposition \[prop:collapse\_URA\] would give an algorithm for the containment problem for $k$-ambiguous register automata.
- Last but not least, we would like to point out that our techniques cannot directly be applied to the class of unambiguous register automata *with guessing* which we mentioned in the introduction. Thus, the respective containment problem remains open for future research.
[10]{}
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[^1]: Types are a standard notion of model theory (see, e.g., [@Hodges] for a definition). The definition that we give here coincides with the standard notion of types when applied to ${\mathbb{D}}_\bot$.
|
---
abstract: 'The quantum emulation of spin-momentum coupling (SMC), a crucial ingredient for the emergence of topological phases, is currently drawing considerable interest. In previous quantum gas experiments, typically two atomic hyperfine states were chosen as pseudospins. Here, we report the observation of a new kind of SMC achieved by loading a Bose-Einstein condensate (BEC) into periodically driven optical lattices. The $s$- and $p$-bands of a static lattice, which act as pseudospins, are coupled through an additional moving lattice which induces a momentum dependent coupling between the two pseudospins, resulting in $s$-$p$ hybrid Floquet-Bloch bands. We investigate the band structures by measuring the quasimomentum of the BEC for different velocities and strengths of the moving lattice and compare our measurements to theoretical predictions. The realization of SMC with lattice bands as pseudospins paves the way for engineering novel quantum matter using hybrid orbital bands.'
author:
- 'M. A. Khamehchi$^{1}$'
- 'Chunlei Qu$^{2}$'
- 'M. E. Mossman$^{1}$'
- 'Chuanwei Zhang$^{2}$'
- 'P. Engels$^{1}$'
title: 'Spin-momentum coupled Bose-Einstein condensates with lattice band pseudospins'
---
[^1]
[^2]
Spin-momentum coupling (SMC), commonly called spin-orbit coupling, is a crucial ingredient for many important condensed matter phenomena such as topological insulator physics, topological superconductivity, spin Hall effects, etc [@Zutic2004; @Hasan2010; @Qi2011]. In this context, the recent experimental realization of SMC in ultracold atomic gases provides a powerful platform for engineering many interesting and novel quantum phases [@Lin2011; @Pan2012; @Wang2012; @Cheuk2012; @Hamner2014; @Olson2014]. In typical experiments, two atomic hyperfine states act as two pseudospins which are coupled to the momentum of the atoms through stimulated Raman transitions [@Higbie2002; @Spielman2009]. However, ultracold atoms in optical lattice potentials possess other types of degrees of freedom which can also be used to define pseudospins [@Jaksch1998; @Bloch2005]. A natural and important question is whether such new types of pseudospins can be employed to generate SMC.
In optical lattices filled with ultracold atoms, *s*- and *p*-orbital bands are separated by a large energy gap and can be defined as two pseudospin states. One significant difference between hyperfine state pseudospins and lattice band pseudospins lies in the energy dispersion of spin-up and spin-down orientations: the dispersion relations are the same for hyperfine state pseudospins, while they are inverted for lattice band pseudospins. It is well known from topological insulators and superconductor physics that inverted band dispersions, together with SMC, play a central role for topological properties of materials [Bernevig2006,Konig2007,Fu2008]{}. Therefore, it is natural to expect that the inverted band pseudospins, when coupled with the lattice momentum, may lead to interesting topological phenomena in cold atomic optical lattices. Recent experiments with shaken optical lattices (i.e. lattices in which the lattice sites are periodically shifted back and forth in time [@Parker2013]) have realized a simple coupling ($\Omega \sigma _{x}$ coupling, where $%
\Omega $ is the coupling strength and $\sigma _{x}$ a Pauli matrix) between *s*- and *p*-band pseudospins, analogous to Rabi coupling between two regular spins [@Zheng2014]. However, for the exploration of exotic phenomena in optical lattice systems, such as Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phases [@FF64; @LO64] and Majorana fermions [@Fu2008], SMC with $s$- and $p$-bands pseudospins is highly desirable [@Zheng2013; @Wu2013; @Zhang2008; @Sato2009].
In our experiments we realize such $s$-$p$ band SMC for a Bose-Einstein condensate (BEC) using a weak moving lattice to generate Raman coupling between *s*- and *p*-band pseudospins of a static lattice [Muller2007]{}. The moving lattice acts as a periodic driving field [Lignier2007,Eckardt2010,Hauke2012,Goldman2014a,Goldman2014b,Dalessio2014]{} and has previously been used to generate an effective magnetic field in the lowest $s
$-band of a tilted optical lattice [@Aidelsburger2013; @Miyake2013]. In our experiment, the driving frequency of the moving lattice is chosen close to the energy gap between $s$- and $p$-bands at zero quasimomentum, leading to a series of hybrid $s$-$p$ Floquet-Bloch (FB) band structures. FB band structures in optical lattices give rise to interesting and important phenomena in cold atoms and solids [@Sengstock2011; @Struck2013], as is evidenced by the recent experimental realization of a topological Haldane model in a shaken honeycomb optical lattice [@Jotzu2014] and the observation of FB states on the surface of a topological insulator [Wang2013]{}.
Here we show that the moving lattice generates two types of coupling between *s-* and *p*-band pseudospins: a momentum-independent Rabi coupling ($\Omega \sigma _{x}$) and SMC ($\alpha \sigma _{x}\sin (q_{x} d) $, where $q_{x}$ is the quasimomentum and $d$ the lattice period), with strengths of the same order. The coexistence of these two types of coupling leads to asymmetric FB band dispersions [@Zheng2015]. We investigate the FB band structures by measuring the quasimomentum of the BEC. The initial phase of the moving lattice plays a significant role in the Floquet dynamics [@Goldman2014a], the effects of which are explored through a quantum quench induced dynamical coupling of the FB bands. Results are compared to theoretical predictions from a simple two-band model and from numerical simulations of the Gross-Pitaevskii (GP) equation.
**Experimental setup**. To generate the $s$-$p$ band SMC and FB band structures, we begin with a $^{87}$Rb BEC composed of approximately $%
5\times10^{4}$ atoms confined in a crossed dipole trap. A static lattice is generated by two perpendicular laser beams with wavelength $\lambda \approx
810$ nm intersecting at the position of the BEC, as schematically shown in Fig. \[schematic\](a). The harmonic trap frequencies due to the envelope of the static lattice beams and the crossed dipole trap are $(\omega
_{x},\omega _{y},\omega _{z})=2\pi \times (41,159,115)$ Hz, where $\vec{e}%
_{x}$ points along the lattice, $\vec{e}_{y}$ is the horizontal transverse direction, and $\vec{e}_{z}$ is the vertical direction. A weak moving lattice with the same lattice period as the static lattice, $d=\pi /k_{L}$ where $k_{L}=\sqrt{2}\pi /\lambda $, is then overlaid with the static lattice (Fig. \[schematic\](b)). The moving lattice beams are approximately $180$ MHz detuned from the static lattice. A small frequency difference $\Delta _{\omega }$ between the two moving lattice beams determines the velocity of the lattice according to $v_{lattice}=\Delta
_{\omega }/2k_{L}$. To induce $s $-$p$ orbital band coupling, $|\Delta
_{\omega }|$ is chosen close to the energy gap $E_{sp}$ between the $s$- and $p$-bands of the static lattice at quasimomentum $q_{x}=0$.
![**Experimental setup and schematic lattice illustration.** (a) Experimental arrangement. The crossed dipole trap beams propagate in the $%
\vec{e}_x$ and $\vec{e}_z$ direction. The static and moving lattice have overlapping beams propagating along $\vec{e}_x + \vec{e}_y$ and $-\vec{e}_x
+ \vec{e}_y$. (b) Lattice potentials along the $\vec{e}_x$ direction. The lattice period $d$ is identical for the static lattice $V_0$ and the moving lattice $V^{\prime }_x$. The initial offset between lattice sites of the static and moving lattice, $\Delta x$, is given by the initial phase $%
\protect\phi_0$ between the two lattices. (c,d) Illustration of the multi-photon processes for the driven lattice system and the corresponding FB band structure in the first Brillouin zone. The static lattice induces a large energy gap (I) through a 2-photon process and a small energy gap (II) through a 4-photon process. The moving lattice induces an energy gap when the $s$-band and $p$-band are coupled through (III). A smaller energy gap is produced by a combination of the static and moving lattice (IV).[]{data-label="schematic"}](Figure-1_Khamehchi.pdf){width="48.00000%"}
One outstanding feature of the coupling scheme employed in these experiments is the asymmetry of the effective $s$-$p$ FB bands, which exhibit a local minimum located at a finite quasimomentum $q_x \neq 0$. The direction in which the minimum is shifted away from $q_x=0$ is determined by the sign of $%
\Delta_\omega$ (which determines the direction of motion of the moving lattice) and $|\Delta_\omega|-E_{sp}$ (i.e. the detuning of the drive from the bandgap at $q_{x}=0$). Before describing experimental results and a formal derivation of the band structure using Floquet theory [Goldman2014a, Goldman2014b]{}, we lay the groundwork by presenting a multi-photon resonance picture that provides intuitive insights (Fig. [schematic]{}(c,d)). In this picture, one starts with the parabolic dispersion of a free atom in the absence of any external potentials. An optical lattice then induces $2n$-photon couplings (with $n$ being an integer number) between points of the dispersion relation due to absorption and stimulated emission processes. The couplings are centered around pairs of points that fulfill conservation of energy and momentum. At these points, bandgaps open due to avoided crossings. Examples for possible couplings due to the static lattice (red arrows in Fig. \[schematic\](c)) and the moving lattice (blue arrows in Fig. \[schematic\](c)) and the associated bandgaps in the first Brillouin zone are shown in Fig. \[schematic\]. Different coupling strengths lead to different sizes of bandgaps, which result in an asymmetric band structure.
In another pictorial way, the Floquet band structure for the time-periodic system can be constructed by creating multiple copies of the Bloch band structure of the static lattice that are offset in energy by $|\Delta_\omega|
$. The moving lattice couples the $p$-band and the shifted $s$-band (labelled by $s^\prime$ in Fig. \[schematic\](d)) at points where the shifted $s$-band intersects the unshifted $p$-band. The gaps opened by the coupling can formally be calculated using Floquet theory.
**Experimental measurements**. Adiabatic loading of the BEC into an $s $-$p$ FB band is achieved by first ramping on the intensity of the static lattice, followed by adiabatically ramping on the moving lattice intensity. In this procedure, the initial relative phase between the two lattices, $\phi_0$ (Fig. \[schematic\](b)), becomes irrelevant and can effectively be set to zero. As we shall show in the context of Fig. [phase]{}, if the moving lattice is suddenly jumped on instead of adiabatically ramped on, this initial relative phase may manifest itself by drastically changing the dynamics of the system [@Goldman2014a].
![**Effects of the driving frequency.** (a) Band minimum $q_{min}
$ for the upper hybrid band vs. driving frequency $\Delta _{\protect\omega }$. The depth of the moving lattice is $1~E_{R}$. The filled circles are experimental measurements. The black line shows the theoretical prediction of a two-band model. The squares and stars are the results of numerical simulations of the Schrödinger equation and the GP equation, respectively. (b) Upper hybrid $s$-$p$ FB band structure for different driving frequencies $\Delta _{\protect\omega }=4.99~E_{R}$, $5.1~E_{R}$ and $5.22~E_{R}$ from top to bottom. The lowest (black) curve is the $s$ orbital band without the presence of the driving field.[]{data-label="velocity"}](Figure-2_Khamehchi.pdf){width="48.00000%"}
Figure \[velocity\](a) shows the measured position, $q_{min}$, of the band minimum for different driving frequencies, $\Delta_\omega$, after adiabatically loading a BEC into a FB band. The driving frequencies are chosen such that $\hbar \Delta_\omega$ lies in the gap at $q_x = 0$ between the $p$-band ($4.64~E_R$, where $E_{R}=\hbar^2 k_L^2/{2m}=h\times 1749.5$ Hz) and the $d$-band ($5.44~E_R$). After adiabatically loading a BEC into a FB band, the lasers are switched off and the BEC is imaged after 14 ms time-of-flight (TOF). The positional shift of the BEC components is then used to determine the quasimomentum. Each data point is an average over five iterations of the measurement. A shift of the quasimomentum is detected that decreases with increasing driving frequency (Fig. \[velocity\](a)) as the coupling between the $p$-band and shifted $s$-band becomes weaker. The observed shift indicates a shift of the minimum of the upper hybrid band (Fig. \[velocity\](b)) into which the BEC is adiabatically loaded. The solid line in Fig. \[velocity\](a) shows $q_{min}$ calculated from a simple two-band model (see below) and is in reasonable agreement with the data. The symbols are the results from real time simulation of the Schrödinger equation (squares) and the GP equation (stars) with finite nonlinear interaction strength [@note0]. We see that the interaction could modify the single-particle results.
![**Effects of the driving strength.** Band minimum $q_{min}$ vs. driving field strength $V^\prime$ for different driving frequencies of (a) $\lvert\Delta_\protect\omega\rvert=2.92E_R$ and (c) $\lvert\Delta_%
\protect\omega\rvert = 5.21E_R$. The red points are experimental data, the solid lines are the theoretical predictions from a two-band model. $%
sgn(\Delta\protect\omega)$ determines the direction of motion of the moving lattice. (b,d) Corresponding hybrid band structures for different driving field strengths $V_x^\prime = 1.5 E_R$, $0.75 E_R$ and $0 E_R$ (outer to inner curves).[]{data-label="strength"}](Figure-3_Khamehchi.pdf){width="48.00000%"}
Figure \[strength\] presents a complementary data set for which the driving frequency is set to a constant value with $\lvert\Delta_\omega%
\rvert<E_{sp}$ (Fig. \[strength\]a) or $\lvert\Delta_\omega\rvert>E_{sp}$ (Fig. \[strength\]c) and the quasimomentum is determined for various depths of the moving lattice. The sign of $\Delta_\omega$ determines the direction of motion of the moving lattice. For $\lvert\Delta_\omega%
\rvert<E_{sp}$ the BEC resides in the lower hybrid $s$-$p$ FB band (Fig. [strength]{}(b)) while for $\lvert\Delta_\omega\rvert>E_{sp}$ it is in the upper hybrid band (Fig. \[strength\](d)). This leads to a shift of the quasimomentum into opposite directions for the two cases. For a given driving frequency, the coupling of the two bands is stronger for larger driving field strength (i.e. larger depth of the moving lattice) so that the BEC is shifted to a larger absolute value of quasimomentum.
Floquet systems such as the one in our experiment are described by quasienergy bands. They do not have a thermodynamic ground state, and in the presence of many-body interactions their stability can be affected by a variety of factors [@Choudhury2014a; @Choudhury2014b; @Cooper2015]. Experimentally, we study the stability of the system by determining the number of condensed atoms left after the static and the moving lattices are successively and adiabatically ramped on. TOF imaging reveals atom loss and heating of the BEC as shown in Fig. \[stability\]. The dips $\alpha $, $%
\beta $, and $\gamma $ in Fig. \[stability\](a) occur when the driving frequency is chosen such that it leads to a coupling close to the Bloch bands $p$, $d$ and $f$ of the static lattice at $q_{x}=0$ respectively. The lower hybrid band structure, in which the BEC mainly resides, for points $1$, $2$, and $3$ and the corresponding TOF images are shown in panels (b) and (c). Resonance induced collective excitations and modulational instabilities can play a role for the observed losses [@JimenezGarcia2014].
![**Heating of the Floquet system.** (a) Number of atoms remaining after adiabatically loading a BEC into the FB band, normalized to initial atom number determined from independent experimental runs. The static lattice is ramped on to $5.47~E_R$ in $200~ms$. Then the moving lattice is ramped on to a depth of $V^\prime = 0.5~E_R$ in $60~ms$. The dips $\protect\alpha$, $\protect\beta$, and $\protect\gamma$ occur close to the Bloch bands $p$, $d$, and $f$. (b) Effective band structures for the lower hybrid band for data points $1$, $2$, and $3$ of panel (a). (c) TOF images taken at points $1$, $2$, and $3$.[]{data-label="stability"}](Figure-4_Khamehchi.pdf){width="48.00000%"}
**Minimal two-band model**. The dynamics of the BEC are governed by the full time-dependent GP equation, $i\hbar \frac{\partial }{%
\partial t}\psi (\mathbf{r},t)=[H_{0}(t)+V_{trap}+V_{int}]\psi (\mathbf{r},t)
$ where $V_{trap}$ and $V_{int}$ are the external trapping potential and the mean-field interaction, respectively. $H_{0}(t)$ is the single-particle Hamiltonian, $$H_{0}(t)=\frac{p^{2}}{2m}+V_{0}\cos
^{2}(k_{L}x)+V^{\prime}\cos^{2}(k_{L}x+\phi _{0}-\frac{\Delta _{\omega }t}{2}%
),$$ where the second and the third terms describe the static and moving optical lattices, respectively, and $\phi _{0}$ is the initial relative phase between the two sets of lattices.
When the static lattice depth $V_{0}$ is large and when $|\Delta _{\omega }|$ is close to the energy gap $E_{sp}$, higher orbital bands are not significantly populated in the driven process and the system is well described by a simple two-band tight-binding model [@Zheng2015]. Following the standard procedure in Floquet theory, we obtain the effective single-particle Hamiltonian $$H_{0}^{\text{eff}}=\left(
\begin{array}{cc}
\epsilon _{s}(q_{x}) & \Delta _{sp} \\
\Delta _{sp}^{\ast } & \epsilon _{p}(q_{x})-|\Delta _{\omega }|%
\end{array}
\right) , \label{heff}$$ where $$\Delta _{sp}=-i[\Omega -\alpha \sin (q_{x}d)+\beta \cos (q_{x}d)]e^{-i\phi
_{0}}$$ is the coupling between $s$- and $p$-orbital bands that is induced by the moving lattice potential for $\Delta_\omega>0$ [@note1], and $\epsilon
_{s}$ and $\epsilon _{p} $ are the energy dispersions for the uncoupled orbital bands. The three coupling coefficients $\Omega$, $\alpha$ and $\beta$ are given by $\Omega =\frac{V^{\prime }}{4}\langle s_{i}|\sin
(2k_{L}x)|p_{i}\rangle $, $\alpha =\frac{V^{\prime }}{2}\langle s_{i}|\cos
(2k_{L}x)|p_{i+1}\rangle $ and $\beta =\frac{V^{\prime }}{2}\langle
s_{i}|\sin (2k_{L}x)|p_{i+1}\rangle $, where $|s_{i}\rangle $ and $%
|p_{i}\rangle $ are the maximally localized Wannier orbital states in the $i$-th site. $\Omega $ is the coupling between $s$- and $p$- orbital states in the same lattice site, while $\alpha $ and $\beta $ are the couplings between $s$- and $p$-orbital states of nearest neighbouring sites. SMC between $s$-$p$ band psuedospins is represented by $\alpha \sin
(q_{x}d)\sigma _{x}$. This derivation shows that the inversion symmetry of FB band structure is broken due to the coexistence of couplings of different parities. When the moving lattice depth is adiabatically ramped on, the quasimomentum of the BEC gradually shifts away from $q_{x}=0$ in a definite direction following the hybrid band minimum. This is quite different from previous shaken lattice experiments [@Parker2013] where the inversion symmetry of the band was preserved and the BEC could spontaneously choose either side of $%
q_{x}=0$ as its ground state. In that case, the BEC needed to be accelerated to break the inversion symmetry. In our scheme, the position of the true minimum is uniquely determined by the moving velocity direction, moving lattice depth, and driving frequency.
This minimal two-band model captures the essential physics of the driven lattices as we have seen through the comparison of experimental measurements and theoretical values (see Figs. \[velocity\] and \[strength\]), demonstrating the observation of SMC between $s$-$p$ band pseudospins. However, this model may deviate from the experiment when the modulated dynamics involve additional orbital bands or when the nonlinear interaction is strong such that the single-particle band structure will be renormalized by the interaction term.
![**Quench dynamics after suddenly jumping on the coupling between the $s$ and $p$ band.** (a, b) Normalized occupation of the momentum component $-2 \hbar k$ (blue points with solid error bars) and $+2 \hbar k$ (red points with dashed error bars) in (a) and $0~\hbar k$ in (b). The dots are the average of ten experimental measurements for each time. The error bars indicate the spread of the experimental data. The shaded areas are the results of numerical GP simulations calculated for a homogeneous distribution of different initial phases $\protect\phi_0$. The black curve represents the calculation for phase $\protect\phi_0=0$ (c) Bandstructure plot. Jumping on the moving lattice places the BEC (black ellipse) into the gap between two FB bands. (d) Experimental images taken $0.5~ms$ after the quench for the top two images and $0.8~ms$ after the quench for the bottom image. Dashed squares indicate the areas used for counting the atom number in the $-2 \hbar k$ (left square), $0$ (middle square) and $+2 \hbar k$ component (right square). The sum of the atoms in all three boxes is used for the normalization of the experimental data in panels (a) and (b).[]{data-label="phase"}](Figure-5_Khamehchi.pdf){width="48.00000%"}
**Quench dynamics**. Since a Floquet system is generated by a time-periodic Hamiltonian, an important question concerns the role of the initial phase of the driving field [@Goldman2014a]. For the system considered in this work, this phase determines the relative positions between the moving and static lattice sites. Though the relative phase does not change the effective band structure (Eq. \[heff\]), and thus the time-averaged dynamics, it can play a crucial role in the micromotion of the BEC. To demonstrate the effect of the initial relative phase, we study the oscillations in the population of the momentum components $k_{x}=0,\pm
2k_{L} $ after a quantum quench. Figure \[phase\] (a-b) present such quench dynamics after adiabatically ramping on the static lattice to $%
5.47~E_{R}$ followed by a sudden jump on of the moving lattice to $%
V_{x}^{\prime }=1E_{R}$ with an on-resonant driving frequency $|\Delta
_{\omega }|=E_{sp}$ (Fig. \[phase\] (c)). We focus on the evolution during the first 3 ms, during which the BEC mainly stays at $q_{x}=0$ without significant dipole motion in the hybrid bands. The symbols in Fig. [phase]{} (a-b) are experimental data averaged over ten measurements for each time step. There is significant spread in the data for each time step, as indicated by the vertical error bars. This spread is due to the initial phase $\phi _{0}$ between the static and the moving lattice, which is uncontrolled in the experiment, such that each iteration realizes a case with a different, random $\phi _{0}$. The shaded areas represent the result of numerical GP simulations for a homogeneous spread of relative phases. The experimental error bars are in reasonable agreement with the expectation based on these numerics. The numerics reveal that for a fixed initial phase there are two oscillation periods of different timescales (Fig. \[phase\](b)). The fast oscillation (of period $T\approx 0.1ms$) corresponds to the micromotion of particles under the high-frequency periodic driving, whereas the slow oscillation ($T\approx 1.75ms$) corresponds to the time-averaged effective Rabi oscillations between the two hybrid FB bands. For longer holding time, the periodicity is slightly broken due to a small dipole motion.
**Discussion**. We have realized and characterized a new kind of SMC with lattice bands as pseudospins. This not only provides a powerful tool to control orbital states with a driving field, but also enriches the study of novel quantum matter using hybrid orbital bands. There are many directions that can be taken along this route, e.g., the engineering of similar SMC in higher dimensional systems involving different orbital bands, and quantitative analysis and measurements of the effects of strong interactions on the effective bands. The realization of similar SMC for fermionic atoms such as $^{6}$Li and $^{40}$K with tunable interactions may open the door for exploring exotic quantum matters such as FFLO superfluids and Majorana fermions.
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The periodically driven dynamics are simulated using the time-dependent GP equation for a two dimensional system with the same geometry as the experiment. The width of the BEC is around 10 $\mu m$. Since the atom loss is usually large in the experiment, the numerical results are not intended for direct comparison with the measurements. See supplemental material for some examples of GP simulations.
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For $\Delta_\omega<0$, we have $\Delta _{sp}=i[\Omega
+\alpha \sin (q_{x}d)+\beta \cos (q_{x}d)]e^{i\phi _{0}}$. Note that $\beta$ is usually much smaller than $\Omega$ and $\alpha$, however we keep it for completeness.
**Acknowledgements** M.A.K, M.E.M., P.E. are supported by the National Science Foundation (NSF) through Grant No. PHY-1306662. C. Qu and C. Zhang are supported by ARO (W911NF-12-1-0334) and AFOSR (FA9550-13-1-0045).
**Author contributions** M.A.K., C.Q., C.Z. and P.E. conceived the experiment and theoretical modeling; M.A.K., M.E.M. and P.E. performed the experiments; C.Q., C.Z. performed the theoretical calculations; C.Z. and P.E. supervised the project.
**Competing financial interests**: The authors declare no competing financial interests.
[^1]: [email protected]
[^2]: [email protected]
|
---
abstract: |
Convolutional neural networks (CNNs) have enabled significant performance leaps in medical image classification tasks. However, translating neural network models for clinical applications remains challenging due to data privacy issues. Fully Homomorphic Encryption (FHE) has the potential to address this challenge as it enables the use of CNNs on encrypted images. However, current HE technology poses immense computational and memory overheads, particularly for high resolution images such as those seen in the clinical context.
We present CaRENets: **C**ompact and **R**esource-**E**fficient CNNs for high performance and resource-efficient inference on high-resolution encrypted images in practical applications. At the core, CaRENets comprises a new FHE compact packing scheme that is tightly integrated with CNN functions. CaRENets offers dual advantages of memory efficiency (due to compact packing of images and CNN activations) and inference speed (due to reduction in number of ciphertexts created and the associated mathematical operations) over standard interleaved packing schemes.
We apply [CaRENets]{} to perform homomorphic abnormality detection with 80-bit security level in two clinical conditions - Retinopathy of Prematurity (ROP) and Diabetic Retinopathy (DR). The ROP dataset comprises 96 $\times$ 96 grayscale images, while the DR dataset comprises 256 $\times$ 256 RGB images. We demonstrate over 45x improvement in memory efficiency and 4-5x speedup in inference over the interleaved packing schemes. As our approach enables memory-efficient low-latency HE inference without imposing additional communication burden, it has implications for practical and secure deep learning inference in clinical imaging.
author:
- Jin Chao
- Ahmad Al Badawi
- Balagopal Unnikrishnan
- Jie Lin
- Chan Fook Mun
- 'James M. Brown'
- 'J. Peter Campbell'
- Michael Chiang
- 'Jayashree Kalpathy-Cramer'
- Vijay Ramaseshan Chandrasekhar
- Pavitra Krishnaswamy
- |
Khin Mi Mi Aung\
\
{Jin\_Chao, Ahmad\_Al\_Badawi, Mi\_Mi\_Aung}@i2r.a-star.edu.sg
bibliography:
- 'biblio.bib'
title: '[CaRENets]{}: Compact and Resource-Efficient CNN for Homomorphic Inference on Encrypted Medical Images'
---
Introduction {#sec:Introduction}
============
Convolutional neural networks (CNNs) have enabled significant performance leaps in medical image classification tasks. To leverage these advances in clinical practice, several stakeholders are working towards cloud-based health services that can receive patient data, apply previously trained neural network models for inference, and return resulting predictions to the referral source. Although cloud-based inference obviates the need for expensive high-performance hardware at point of care, it involves sharing of sensitive patient data and poses important privacy concerns. Much of the recent work on privacy-preserving deep learning has been focused on developing strategies for training CNNs across multiple institutions without sharing protected health information [@mcmahan2016communication; @chang2018distributed; @ryffel2018generic]. We instead consider prevention of unauthorized breach of patient confidentiality by encrypting the data before transmitting to the cloud. In this scenario, trained CNN models are deployed in cloud infrastructure and applied to encrypted data for inference.
Fully Homomorphic Encryption (FHE) caters to the need for computing on encrypted data, and has been applied to computations involving CNNs on image datasets ranging from MNIST [@MSFT:DGL+16; @badawi2018alexnet] and CIFAR-10 [@hesamifard2017cryptodl] to medical images [@chou2018faster]. However, homomorphic CNN approaches do not easily scale to the image resolution, resource efficiency and latency requirements in typical medical image classification tasks. Most studies using CNNs for medical image classification consider images with resolutions ranging 96 $\times$ 96 to 512 $\times$ 512 [@greenspan2016guest]. Current FHE methods are impractical for such tasks, as they require large amount of memory (e.g., 789 GB$\sim$1183 GB) for mid-range resolution medical images [@chou2018faster]. As such, homomorphic CNN evaluation for practical medical image classification requires strategies to dramatically improve the computational and memory efficiency.
Here, we present a novel resource-efficient method for homomorphic CNN inference on encrypted medical images. Our approach overcomes the need for large cloud memory, yields good latency, and does not impose additional network communication burden.
To motivate specifications for our approach, we consider two ophthalmology application scenarios involving classification of Retinopathy of Prematurity (ROP) and Diabetic Retinopathy (DR).
1. The first application involves ROP screening in low resource settings. ROP is a leading cause of preventable blindness worldwide. As such, several efforts are underway, particularly in rural areas in low and middle income countries, to screen babies for this condition using portable cameras including some with mobile phones. Quantitative screening tools could reduce dependence on expert graders [@li2012telemedicine; @richter2009telemedicine]. Recent work has demonstrated that CNNs are effective for automatic detection of ROP with fundus photos [@brown2018automated]. A cloud-based service for ROP classification with CNNs would benefit from the ability to preserve privacy. At the minimum, such applications would require 96 $\times$ 96 images, and latencies within 30 minutes.
2. The second application involves fundus testing for DR as part of multi-center clinical trials. These applications employ clinical-grade equipment in disparate sites and pool the data for integrated analyses. Several recent studies have demonstrated the use of CNNs for automatically classifying severity of DR [@gulshan2016development]. Clinical trials providers would benefit from the ability to perform automated DR classification while also fulfilling their ethical commitments to maintain patient privacy. At the minimum, such applications would require 256 $\times$ 256 images and latencies within a few hours.
We therefore consider the problem of resource-efficient homomorphic CNN inference for ROP and DR images of resolution 96 $\times$ 96 - 256 $\times$ 256. We assume a secure machine learning service paradigm wherein (a) suitable trained CNN models are available for the use case of interest, and (b) the data referral source has the ability to encrypt images before initiating a request for cloud-based inference. Such approaches are particularly desirable for retinal images as they carry a surprising amount of information about overall patient health [@poplin2018prediction], and also biometric identification information due to unique vascular patterning [@jain2004introduction].
The major contributions of our paper are summarized as follows.
- We propose a new compact packing strategy to pack matrices and vectors into HE-encrypted ciphertexts, and build a new matrix computation library, [A$^\ast$HEMAT]{}, that leverages the SIMD-like execution model associated with modern FHE schemes to facilitate general computations on encrypted matrices such as: addition, multiplication and transposition.
- We further design and develop a compact and resource-efficient homomorphic CNN, [CaRENets]{}, which is built on the tight integration of the optimized matrix-vector multiplication function of [A$^\ast$HEMAT]{} and the CNN layers. We show that [CaRENets]{} are capable of addressing a variety of image classification problems, ranging from standard benchmarking datasets to real-world medical imaging datasets.
- We provide a set of experiments, detailed analysis and comparisons with state-of-the-art solutions to evaluate the performance of our packing technique and [CaRENets]{}, and demonstrate [CaRENets]{} can improve both memory efficiency and inference speed significantly.
**Paper Organization.** Section \[section:Preliminaries\] introduces background and related work. Section \[sec:hemat\] describes A\*HEMAT, our technique to pack matrices into ciphertexts and do the computations on matrices. Section \[sec:carenets\] details our methods and optimizations to develop [CaRENets]{}. We conduct performance evaluation of [CaRENets]{} with MNIST benchmark in Section \[sec:hnet:performance:evaluation\]. In Section \[sec:real:world:usecases\], we introduce two real-world use cases for medical image inference, and evaluate the performance of [CaRENets]{} on the associated image datasets. Finally, in Section \[sec:discussion\] and Section \[sec:conclusion\] we discuss our findings, draw some conclusions and outline directions for future work.
Background and Related Work {#section:Preliminaries}
===========================
In this section, we provide an overview of the core FHE and deep learning concepts used in this work.
Fully Homomorphic Encryption
----------------------------
FHE is an emerging technology that enables computing on encrypted data in the absence of the decryption key. The idea was born in 1978 [@FOSC:RivAdlDer78] and materialized into a secure theoretical scheme in 2009 [@STOC:Gentry09] using the bootstrapping technique that allows ciphertext refreshing. Basically, we start by a noisy encryption scheme that masks the secret message by some noise that can be completely filtered out by the decryption procedure. Moreover, the scheme includes a budget for computation that allows performing a certain number of operations on ciphertexts without decryption. The computation budget decreases as we perform more operations due to the noise growth. Bootstrapping is used to reduce the noise and produce a substitute ciphertext with lower noise and higher computation budget. Unfortunately, bootstrapping is generally too costly for practical applications despite the massive improvements that have been proposed in the HE literature so far. Therefore, HE practitioners opt to use a more efficient variant of FHE known as levelled-FHE [@brakerski2014leveled] that can be configured to support computations with a certain multiplicative depth without bootstrapping.
FHE has gone through a sequence of improvements in functionality and performance [@van2010fully; @brakerski2014leveled; @fan2012somewhat; @chillotti2016faster; @cheon2017homomorphic; @cheonhomomorphic] that enabled the development of interesting privacy-preserving applications [@graepel2012ml; @lauter2014private; @cheon2015homomorphic; @xie2016privlogit; @bonte2018privacy]. In this study, we use the Fan-Vercauteren (FV) [@EPRINT:FanVer12] levelled-FHE that is implemented in Microsoft SEAL [@SEAL] in all our experiments.
In most of current FHE schemes, the plaintext space can be factorized into multiple spaces using the Chinese Remainder Theorem [@DCC:SmaVer14]. This allows one to pack a vector of plaintext messages and operate on them in a SIMD manner. Adding (resp. multiplying) two packed ciphertexts results in component-wise addition (resp. multiplication) of the vectors. Moreover, mature implementations of FHE schemes provide mechanisms to rotate the underlying plaintext vectors inside the ciphertexts to enable inter-slot interaction. For instance, to add two numbers in misaligned slots, one needs to rotate one vector to align the corresponding slots followed by component-wise addition. Moreover, one can extract desired values by multiplying by a mask vector of bits. It should be noted that packing is done at the data encoding phase of the computation right before encryption.
Deep Learning
-------------
Modern Deep Neural Networks, such as Convolutional Neural Networks (CNN), Generative Adversarial Networks (GAN), Long Short-Term Memory networks (LSTM), are composed of linear and nonlinear function blocks. Linear function blocks like Convolution, Full-connected, Shortcut, Element-wise addition, Batch Normalization, Average Pooling, are basically matrix multiplications with simple multiplication and addition operators. Nonlinear function blocks usually contain more complex operations such as exponential in Softmax/Sigmoid or max operation in Max pooling/ReLU layer. The depth of neural networks is increased by repeating the linear and nonlinear blocks in the network architecture. With increasing scale and difficulty of target applications (e.g. from MNIST to CIFAR-10 to ImageNet), the model capacity of neural networks has to be enlarged (e.g. wider and deeper networks) in order to meet the desired prediction accuracies.
Applying homomorphic encryption (HE) in deep neural networks poses unique challenges. The HE algorithms today only support addition-multiplication operations and are extremely slow, often taking tens of minutes to process a single low resolution image given a shallow network. Several specific changes have to be made to the neural network algorithms for it to be compatible with HE algorithms. For instance, reducing numerical precision of network weights to save bitwidth for polynomial coefficients, approximating nonlinear operations (e.g. Max, ReLU, Sigmoid, etc) with low-degree polynomial functions, limiting input size and network width/depth to deal with resource-hungry HE algorithms given devices with limited computational and memory capacity. All these changes would potentially lead to a considerable drop in prediction accuracy.
Related Work
------------
The literature is rich with studies dealing with privacy-preserving prediction as a service. We are more interested in solutions that involve computing on encrypted data using FHE. These related works are introduced in the following paragraphs.
CryptoNets [@MSFT:DGL+16] was the first work that showed how to evaluate the inference phase of CNNs on encrypted images. The authors used a simple 5-layer network to predict the likelihood values of the MNIST [@MNIST] dataset (28 $\times$ 28 $\times$ 1) with high accuracy level (99%) despite the employment of low-precision training, approximate activation functions and shallow network structure. Of particular interest is the packing scheme used in CryptoNets which enabled the prediction of multiple images in a single network evaluation. In CryptoNets, ciphertexts contain 8192 slots that can be used to pack 8192 different pixels. In ciphertext $i$, the authors pack pixel $i$ from 8192 images. Since each image has 784 pixels, they can pack 8192 images in 784 ciphertexts. Running the network on these ciphertexts will generate the predictions for all images simultaneously. This packing scheme is ideal for high-throughput scenarios where the user requests the prediction of a large bundle of images. On the other hand, single-image prediction takes the same time and resources. Hence, this packing scheme is not ideal for single-image (especially for high-resolution images) prediction scenarios as it requires a large number of ciphertexts. Hereafter, we refer to this packing scheme as interleaved packing.
A major limitation of the interleaved packing is the requirement for enormous computing resources. For instance, Faster CryptoNets [@chou2018faster] used a somewhat similar packing scheme (but only for a single image) as CryptoNets and showed how to run deeper networks on high-resolution medical images. For inference of one medical image of resolution (224 $\times$ 224 $\times$ 3), Faster CryptoNets requires 1183.8 GB RAM on a high-end computing platform ( n1-megamem-96). More related to our work is E2DM [@jiang2018secure] which exploits the SIMD-like execution model in modern FHE schemes to facilitate general matrix computations such as addition, multiplication and transposition. As a use case, the authors could homomorphically evaluate a CNN to classify the MNIST dataset. Although E2DM provides efficient computational complexity, it suffers from the following limitations: 1) E2DM assumes the matrices are square and that the ciphertext is large enough to contain the entire matrix, both are not usually the case in real-world applications. 2) This adds extra burden to the user to compute on large matrices as it poses a need to manually decompose large matrices into smaller blocks. 3) More importantly, for deep learning problems, E2DM requires the user to know some information about the model such as filter and stride sizes, which can impose privacy concerns if the model is confidential.
To resolve the limitations above, we propose a new packing scheme. Our packing scheme is compact as it utilizes the number of available slots in ciphertexts. Hence, the proposed scheme is resource-efficient in terms of memory requirement. Moreover, it is flexible and user-friendly as it can be naturally used to perform computation on matrices with arbitrary sizes without zero-padding. Although our scheme requires a relatively larger number of rotations on the ciphertext slots, our experiments demonstrate that this extra cost can be amortized by the SIMD processing our packing scheme exploits.
[A$^\ast$HEMAT]{}- General Computation library for Compactly Packed Matrices {#sec:hemat}
============================================================================
One of the key mathematical computations in neural networks is the matrix (including vector) multiplication. To enable easy efficient integration of neural networks and homomorphic encryption, we have designed and implemented the [A$^\ast$HEMAT]{} library, which can encrypt matrices and vectors into ciphertexts and perform matrix computations such as addition, multiplication and transposition in the encrypted domain. Generally, [A$^\ast$HEMAT]{} encrypts a matrix of $m$ rows and $n$ columns into one of the following four layouts:
1. Row Packing (RP): each row of the matrix is stored into the slots of a separate ciphertext. Thus the encrypted matrix is composed of $m$ ciphertexts.
2. Column Packing (CP): each column of the matrix is stored into the slots of a separate ciphertext. Thus the encrypted matrix is composed of $n$ ciphertexts.
3. Row Compact Packing (RCP): The matrix is stored into the ciphertext slots consecutively in the row-major order. The encrypted matrix is composed of $k$ ciphertexts where $k = \lceil\frac{m \cdot n}{s}\rceil$ and $s$ is the number of slots in ciphertext.
4. Column Compact Packing (CCP): The matrix is stored into the ciphertext slots consecutively in the column-major order. Similar to RCP, the encrypted matrix is composed of $k$ ciphertexts.
[A$^\ast$HEMAT]{} provides functions to convert an encrypted matrix between any of the four layouts. For instance, matrix multiplication can be done most efficiently if the operands are: 1) a RP/RCP matrix and 2) a CP/CCP matrix. Algorithm \[alg:mat:mul\] shows the multiplication of two $d \times d$ square matrices $A$ and $B$, where $A$ is in RCP layout whereas $B$ is in CCP layout. For simplicity, we show the case where the number of ciphertexts required to pack $A$ or $B$ is $k = 1$. Nevertheless, [A$^\ast$HEMAT]{} is flexible to support cases where $k > 1$.
**return** $\bar{C}$
Algorithm \[alg:mat:mul\] requires $O(d)$ Mult (ciphertext-ciphertext multiplication) and $O(d^2)$ CMult (ciphertext-constant multiplication), as well as $O(d^2)$ rotations. The consumed multiplication depth is 1 Mult $+$ 1 CMult. Note that, among all these operations, Mult is the most computationally intensive in terms of runtime and noise growth. Besides the encrypted matrix computation, [A$^\ast$HEMAT]{} also provides the multiplication function of plaintext matrix with ciphertext matrix to enable privacy-preserving Machine Learning as a Service (MLaaS) on encrypted images (provided by the user) using a pre-learned model owned by the cloud or a service provider. The implementation details of this particular scenario are provided in the subsequent section.
CaRENets {#sec:carenets}
========
By leveraging the packing technique and matrix-vector multiplication function in [A$^\ast$HEMAT]{}, we design and implement a compact and resource-efficient homomorphic CNN, called [CaRENets]{}. In this section, we describe how the matrix-vector multiplication function is implemented and how [CaRENets]{}is built by integrating the function with a convolutional neural network. For [CaRENets]{}, We encrypt the input image compactly into ciphertexts whereas the weight vectors are plaintext. [CaRENets]{}is designed to homomorphically evaluate the inference phase of MLaaS on encrypted images.
Image Packing
-------------
In our packing scheme, the pixels of one image are flattened into a big vector, which is subsequently mapped into one or multiple ciphertexts. To make our packing scheme compatible with the traditional CNNs, we have redesigned the procedures of convolutional layers and fully connected layers to use only matrix operations. As a result, we need a few to hold the input or output values for each CNN layer, and the total amount of memory usage of [CaRENets]{} is significantly minimized. Furthermore, by minimizing the number of ciphertexts and the associated addition and multiplication operations, the processing time of one round of prediction, aka, the latency for predicting one single image, is also reduced.
{width=".85\textwidth" height="5.5cm"}
As seen in Figure \[fig:flattened:image\], an image is composed of a 2-D array of pixels, with each pixel having multiple channels such as Red, Green, and Blue. Therefore, each image can be digitized into a 3-dimensional (3-D) tensor. Next, the 3-D tensor is flattened into a 1-D vector according to a certain sequence, e.g., channel by channel. We note that, in the whole processing flow of [CaRENets]{}, the inputs and outputs of each layer are always represented as one virtual 1-D vector.
Each virtual 1-D vector is mapped into one or multiple ciphertexts. In other words, the values in the vector are stored in the slots of the ciphertexts contiguously. [CaRENets]{} uses the parameter , which can be configured by the user, to control how many slots in each ciphertext will be used to store the values. When one ciphertext is filled with the maximum amount of values, it will switch to the next ciphertext.
Design of [CaRENets]{} Layers
-----------------------------
One cannot directly use the traditional deep learning frameworks (e.g., TensorFlow) when running CNN models with homomorphic encryption. Instead, a customized framework built solely on FHE primitives, i.e., additions and multiplications with ciphertexts, is needed. More specifically, we design the convolutional layer and fully connected layer with matrix-based operations, which are represented by the products of the weight matrices and the virtual 1-D vector storing all the inputs of the layer.
The dot product of one row in the weight matrix and the virtual 1-D vector produces exact one element in the output 1-D vector. A bias can also be added into the result if defined by the model. For the convolutional layer, the weight matrix is composed of all the possible shifted locations of all the filters as shown in Figure \[fig:plain:weight:ctxt:vector:mul\]. In each row, the slots where the filter is located are filled with the weights, while all the rest slots are padded with zero. For the fully connected layer, the procedure is similar, except that the number of rows in the weight matrix is equal to the number of output neurons, and the weights in each row corresponds to the input neurons of the fully connected layer.
{width=".75\textwidth" height="5.5cm"}
Parallel Processing
-------------------
In Figure \[fig:plain:weight:ctxt:vector:mul\], the dot products of the rows in the weight matrix and the 1-D vector holding all the inputs composes the output 1-D vector. All the rows can be processed independently, thus it can be naturally paralleled. In our implementation, we use to parallel the multiplication of the weight matrix and input vector for each convolutional or fully connected layer.
There is another level of parallelism which is inside the multiplication of one row with the 1-D input vector. Recall that the 1-D vector is actually mapped into $k$ separate ciphertexts. The weight row can be further divided into $k$ segments, with each segment multiplying with one of the $k$ ciphertexts as shown in Figure \[fig:parallel:processing\]. After getting the $k$ intermediate resultant ciphertexts, we first add them together to get the summed ciphertext, and then perform Halevi and Soup’s [@halevi2014algorithms] to add all the slots in the summed ciphertext, which produces the dot product of the entire row with the input vector.
{width=".95\textwidth" height="5cm"}
There might be all-zero segments in the weight row. When encountering an all-zero segment, we just skip this segment and the corresponding ciphertext. This optimization removes the unnecessary multiplications in the process.
FHE-Friendly Deep Learning
--------------------------
The following are the major considerations to generate a deep learning model that is FHE-Friendly.
1. Simplified CNN Architecture: For CNNs, a major obstacle for translation to the homomorphic domain is the activation functions since they are usually not polynomials, and therefore unsuitable for evaluation with FHE schemes. We use a simplified network structure with square activation function as shown in Table \[tab:network:architecture\].
-- --
-- --
\[tab:network:architecture\]
2. Low-Precision Training: As most of FHE scheme schemes deal only with integer arithmetic we scale the normalized input images and weights to integers by multiplying them by 2- to 4-bit scale factors (integers) and round them to integers similar to fixed-point encoding.
3. Choice of Parameters: FHE parameters need to be configured cautiously to facilitate homomorphic computation, ensure a minimum security level and achieve optimal performance. We follow the scheme used in HCNN [@badawi2018alexnet] to configure the FHE parameters.
MNIST Benchmarking {#sec:hnet:performance:evaluation}
==================
We use the standard MNIST benchmark dataset to evaluate the performance of our packing scheme and [CaRENets]{}. MNIST comprises of 70,000 28 $\times$ 28 grayscale images of handwritten digits (Arabic numerals 0-9). We train a FHE amenable CNN on the standard 60,000 training split, and test on the remaining 10,000 images.
**Hardware Configuration**: All our experiments were performed on a server with an Intel$^\text{\textregistered}$ Xeon$^\text{\textregistered}$ Platinum 8170 CPU @ 2.10 GHz with 26 cores, $188$ GB RAM.
**Efficiency of our Packing Scheme**: We compare performance with the interleaved packing used in CryptoNets [@MSFT:DGL+16]. To ensure fair comparison, we implement the network shown in Table \[tab:network:architecture\] using the two packing schemes: 1) interleaved packing and 2) our compact packing. Both implementations are done in SEAL [v2.3.1]{} and executed on the same machine. The system parameters used in this experiment and performance results are shown in Table \[tab:mnist:performance\]. It can be clearly seen that our compact packing scheme is more efficient than the interleaved packing in terms of runtime (5.1$\times$) and memory usage (5.9$\times$). Although our packing scheme requires higher multiplicative depth due to the rotation operations, we can still achieve better performance due to the low number of ciphertexts used.
Packing scheme [$\log_2{}n$]{} [$\log_2{}q$]{} [$t$]{} Time Memory $\lambda$
---------------------------- ----------------- -------------------- --------------- -------- ---------- --------------
Interleaved [@MSFT:DGL+16] 14 400 4398047232001 155.4 12.1 > 80
Compact 14 540 4398047232001 30.1 2.06 > 80
: Performance on MNIST dataset using interleaved and compact packing schemes for a single image inference. Time, memory and security level $lambda$ units are: sec, GB and bits, respectively.
\[tab:mnist:performance\]
Real-World Use-Case Evaluations {#sec:real:world:usecases}
===============================
We demonstrate the proposed homomorphic evaluation packing strategy on two medical image datasets and evaluate the results for classification accuracy, memory efficiency, and latency. The two datasets have distinct characteristics to showcase generalizability of our method. While the ROP data constitutes posterior pole images segmented for vasculature, the DR data comprises color RGB retinal fundus images.
**Retinopathy of Prematurity (ROP) Data**: We obtained 1,000 posterior pole retinal RGB photographs collected as part of the ongoing Imaging and Informatics in ROP (i-ROP) study . Each image was assessed for the ROP characteristics of retinal arterial tortuosity and venous dilation at the posterior pole [@capone2006standard], and denoted as normal or plus by at least three independent experts. A reference standard diagnosis (RSD) label was assigned to each image based on adjudication among three experts, resulting in 884 normal and 116 plus images. We converted the RGB images to grayscale and applied gamma adjustment and contrast-limited adaptive histogram equalization (CLAHE). Because plus disease predominantly affects the retinal vasculature, we segmented the vessels [@brown2018automated], and downsampled the resulting images to 96 $\times$ 96 for classification experiments.
**Diabetic Retinopathy (DR) Data**: We obtained 249 color retinal fundus images from the IDRiD challenge dataset collected at an eye clinic located in Nanded, Maharashtra, India [@porwal2018indian]. Each image is annotated for features indicating the presence of DR: microaneurysms, hemorrhages, exudates, venous beading, microvascular abnormality, and neovascularization in retinal fundus photographs [@gulshan2016development], and denoted as healthy or disease. The dataset had 168 healthy and 81 disease cases. We resized the original images to 256 $\times$ 256 and normalized them by the maximum intensity value in each image.
{width=".75\textwidth" height="3cm"}
Training procedures
-------------------
We randomly split the pre-processed datasets into training, validation and testing cohorts with split ratios 7:1:2 for ROP images, and 3:1:1 for DR images. Before training, we augmented the training dataset using a combination of flipping and 90 degree rotations, and randomly sampled the augmented dataset to balance the distribution of classes. For each case, we trained the FHE model and a comparable supervised baseline. The splitting, training, and augmentation procedures were identical for both the FHE model and supervised baselines. We tuned the FHE model by varying the dropout rates and number of filters until there was negligible effect on the validation loss.
Accuracy
---------
In each case, we repeated training with 4 different random seeds, and report mean classification AUCs with associated standard deviations. The results are reported in Table 3, and show that the FHE model achieves comparable classification performance (albeit with $\approx 3 \%$ drop in AUC) to the supervised baselines.
Dataset Models AUC Accuracy
---------------------------- ----------------------- ----------------------------- -------------------
HE-Model 0.888 $\pm$ 0.034 0.925 $\pm$ 0.006
Inception-V3 0.915 $\pm$ 0.022 0.905 $\pm$ 0.024
HE-Model 0.799 $\pm$ 0.065 0.725 $\pm$ 0.062
CIFAR-Net 0.829 $\pm$ 0.054 0.765 $\pm$ 0.068
: Classification Results: Clinical Image Use Cases
Performance Evaluation
----------------------
**[CaRENets]{} Performance on ROP**: We evaluate the performance of [CaRENets]{} on the ROP dataset. The same 5-layer network shown in Table \[tab:network:architecture\] is used with both packing schemes. Table \[tab:rop:performance\] shows the system parameters and performance results. Compared with the MNIST results, the results of this experiment demonstrate larger gains with our [CaRENets]{} with compact packing over the interleaved packing. This suggests that the benefits of [CaRENets]{} are even larger as we handle higher resolution images. More specifically, compact packing enables a 4$\times$ improvement over interleaved packing in runtime and a 45.9$\times$ improvement in memory. These results are not counter-intuitive since an image in the ROP dataset is 96 $\times$ 96 $\times$ 1 and requires a large number of ciphertexts to run the 5-layer network using the interleaved packing.
Packing scheme [$\log_2{}n$]{} [$\log_2{}q$]{} [$t$]{} Time Memory $\lambda$
---------------------------- ----------------- -------------------- ------------------ -------- ---------- --------------
Interleaved [@MSFT:DGL+16] 14 450 4503599627763713 3946.4 135 > 80
Compact 14 660 4503599627763713 994.9 2.94 > 80
: Performance on ROP dataset using interleaved and compact packing schemes for a single image inference. Time, memory and security level $lambda$ units are: sec, GB and bits, respectively.
\[tab:rop:performance\]
**[CaRENets]{} Performance on IDRiD**: In this last experiment, we target a more challenging problem and run [CaRENets]{} on the IDRiD dataset, which has images of resolution 256 $\times$ 256 $\times$ 3 pixels. As shown in Table \[tab:idrid:performance\], the memory resources (188 GB) on our test machine were not sufficient to run the 5-layer network using the interleaved packing. On the other hand, we could run the network with the compact packing scheme with evaluation time of less than 2 hours using 17.2 GB memory.
Packing scheme [$\log_2{}n$]{} [$\log_2{}q$]{} [$t$]{} Time Memory $\lambda$
---------------------------- ----------------- -------------------- ------------------ --------- -------------- --------------
Interleaved [@MSFT:DGL+16] 14 450 4503599627763713 [$-$]{} [> 188]{} > 80
Compact 14 660 4503599627763713 6004.7 17.2 > 80
: Performance on IdRID dataset using interleaved and compact packing schemes for a single image inference. Time, memory and security level $lambda$ units are: sec, GB and bits, respectively.
\[tab:idrid:performance\]
Discussions {#sec:discussion}
===========
Medical imaging applications require the ability to compute on encrypted data, in the absence of a decryption key. Homomorphic encryption provides significant security guarantees and ability to compute, but is bottlenecked by the high overhead of CNN computations, especially for high-resolution images. We have described a new resource efficient homomorphic encryption strategy, rooted in novel FHE compact packing strategies, to enable such computations with CNNs for medical image classification tasks. **Tradeoffs**: While we illustrated our method for two use cases, the approach is generalizable to other scenarios with due consideration of tradeoffs between resolution, accuracy, security and performance. FHE technology requires simplification of activation functions, network architectures and network depth. Such simplified networks have limited capacity for accurate classification of very high resolution images. We have outlined some possible tuning strategies to improve accuracy with the limited capacity network in these settings, but these will need to be considered on a case-by-case basis.
**Possible Use Cases**: Given that our current methods lose about 3-5% accuracy, we envision that the first use cases of homomorphic CNN inference will lie primarily in screening and triage applications. This is especially relevant in emergent scenarios where speedy privacy preserving computations would offer significant gains and aid the clinical workflows.
**Extensions**: Our work focused on CPU implementations. Future work will enhance [CaRENets]{} methodologies for practical deployment by scaling to GPU implementations. For instance, HCNN [@badawi2018alexnet] reports 50.51$\times$ improvement on GPU over CryptoNets [@MSFT:DGL+16] on CPU in terms of inference speed. We foresee that our packing strategies can be similarly expanded to GPUs. Future studies will also address the message passing and computational resource aspects to scale with increasing resolution and security level requirements.
Conclusions {#sec:conclusion}
===========
In this work, we first presented [A$^\ast$HEMAT]{}, a library using a new compact packing strategy to encrypt matrices and vectors with homomorphic encryption, and provides general computation functions like additions, multiplications, transpositions on the encrypted matrices. On the basis of [A$^\ast$HEMAT]{}, we further presented [CaRENets]{}, a compact and resource-efficient CNNs for homomorphic inference on encrypted images. [CaRENets]{} leverages the compact packing technique and optimized matrix-vector multiplication function in [A$^\ast$HEMAT]{}, and tightly integrates them into the CNN layers. We evaluated [CaRENets]{} on the standard MNIST benchmark and showed significant gains in both memory efficiency and inference speed over the commonly used interleaved packing. We further evaluated [CaRENets]{} on abnormality detection with high-resolution retinal images. Across experiments, our results showed $4-5\times$ improvement over state-of-the-art in latency and $45-46\times$ gain in memory efficiency.
Acknowledgments
===============
This project was supported by funding from the Deep Learning 2.0 program at the Institute for Infocomm Research (I$^2$R), A\*STAR, Singapore; research grants from the US National Institutes of Health (NIH grants R01EY19474, P30EY010572, and K12EY027720) and the US National Science Foundation (NSF grants SCH-1622679 and SCH-1622542); unrestricted departmental funding from the Oregon Health Sciences University, and a Career Development Award from Research to Prevent Blindness (New York, NY).
|
---
abstract: 'Considerable work has been done on the one-loop effective action in combined electromagnetic and gravitational fields, particularly as a tool for determining the properties of light propagation in curved space. After a short review of previous work, I will present some recent results obtained using the worldline formalism. In particular, I will discuss various ways of generalizing the QED Euler-Heisenberg Lagrangians to the Einstein-Maxwell case.'
author:
- 'F. Bastianelli'
- José Manuel Dávila
-
title: 'The effective action in Einstein-Maxwell theory'
---
[ address=[Dipartimento di Fisica, Università di Bologna and INFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy]{} ]{}
[ address=[Universidad Michoacana de San Nicolás de Hidalgo, Edificio C-3, Apdo. Postal 2-82, C.P. 58040, Morelia, Michoacán, México]{} ]{}
[ address=[Universidad Michoacana de San Nicolás de Hidalgo, Edificio C-3, Apdo. Postal 2-82, C.P. 58040, Morelia, Michoacán, México]{} ]{}
Talk given by C. Schubert at
*XIII Mexican School of Particles and Fields, San Carlos, Mexico, October 2-11, 2008*
The worldline formalism in QED
==============================
Let us start with the “worldline” representation of the one-loop effective action in spinor QED [@feynman; @fradkin]
\(A) &=& - \_0\^[dTT]{}[e]{}\^[-m\^2T]{} [\_[x(T)=x(0)]{}]{}x() [\_[(T)=-(0)]{}]{}() e\^[-\_0\^TdL\[x()\]]{} \[GammaspinorQED\] Here $m$ and $T$ are the mass and proper time of the loop fermion, and the “worldline Lagrangian” $L$ is given by
L&=& x\^2 +ie x\^A\_ +-ie \^F\_\^ \[Lbasic\] The $x(\tau)$ part of the double path integral in (\[GammaspinorQED\]) runs over all closed trajectories in spacetime with fixed periodicity in $T$, and by itself gives the effective action for a scalar loop (up to the normalization). The $\psi(\tau)$ integral represents the spin degree of freedom, and is over antiperiodic Grassmann functions, obeying $\psi(\tau_1)\psi(\tau_2) = - \psi(\tau_2)\psi(\tau_1)$ and $\psi(T) = - \psi(0)$. Similar worldline representations can be written for the effective action with open scalar/spinor lines, at the multiloop level, and for other field theories; see [@review] for a review. However, it is only during the last fifteen years that such representations have gained some popularity for actual state-of-the-art calculations. By now a number of different techniques have been developed for the evaluation of worldline path integrals. We will follow the “string-inspired” approach [@polyakov; @strassler], where one manipulates the path integral into gaussian form, and then performs those gaussian integrals using worldline correlators adapted to the periodicity conditions,
x\^(\_1)x\^(\_2) &=& -G\_B(\_1,\_2) \^,G\_B(\_1,\_2) = \_1 -\_2-[(\_1 -\_2)\^2T]{} -[T6]{}\
&&\
\^(\_1)\^(\_2)&=& G\_F(\_1,\_2) \^, G\_F(\_1,\_2) = [sign]{}(\_1 - \_2)\
\[GBGF\] This procedure leads, for example, with little effort to the following “Bern-Kosower master formula” [@berkos] for the one-loop $N$ - photon amplitude in scalar QED:
$$\begin{aligned}
\Gamma[\lbrace k_i,\varepsilon_i\rbrace]
&=&
{(-ie)}^N
{(2\pi )}^D\delta (\sum k_i)
{\dps\int_{0}^{\infty}}{dT\over T}
{(4\pi T)}^{-{D\over 2}}
e^{-m^2T}
\prod_{i=1}^N \int_0^T
d\tau_i
\nonumber\\
&&\hspace{-80pt}
\times
\exp\biggl\lbrace\sum_{i,j=1}^N
\bigl\lbrack \half G_{Bij} k_i\cdot k_j
+i\dot G_{Bij}k_i\cdot\varepsilon_j
+\half\ddot G_{Bij}\varepsilon_i\cdot\varepsilon_j
\bigr\rbrack\biggr\rbrace
\mid_{{\rm lin}(\varepsilon_1,\ldots,\varepsilon_N)}
\label{bkmaster}\end{aligned}$$
Here $k_i$ and $\varepsilon_i$ are the momentum and polarization of the $i$th photon, and each $\tau_i$ integral represents one photon leg moving around the loop. The notation ${\rm lin}(\varepsilon_1,\ldots,\varepsilon_N)$ means that only terms linear in all polarization vectors are to be kept after expanding the exponential. Apart from the worldline Green’s function $G_B(\tau_i,\tau_j)$, which we abbreviate by $G_{Bij}$, also its first and second derivatives appear, $\dot G_{B12} = {\rm sign}(\tau_1 - \tau_2)
- 2 {{(\tau_1 - \tau_2)}\over T}$, $\ddot G_{B12}
= 2 {\delta}(\tau_1 - \tau_2) - {2\over T}$. The factor ${(4\pi T)}^{-{D\over 2}}$ in (\[bkmaster\]) represents the free path integral determinant in $D$ dimensions.
The corresponding representation of the $N$ photon amplitude for the spinor loop case differs from (\[bkmaster\]) (apart from a factor of $-2$) only by additional terms from the spin path integral in (\[GammaspinorQED\]). Those terms can be inferred from the scalar loop integrand through a certain pattern matching rule [@berkos; @strassler; @review].
A major advantage of the worldline formulation of QED is that it allows one to include an external constant field $F_{\mu\nu}$ in a particularly efficient way [@shaisultanov; @rescsc]. Effectively, the integral representation of a scalar or spinor QED amplitude in such a constant external field is obtained from the corresponding one in vacuum by the following replacements of the worldline Green’s functions and determinants,
G\_B(\_1,\_2) &&\_B(\_1,\_2) = [12[(eF)]{}\^2]{}([eF]{} [e]{}\^[-ieFTG\_[B12]{}]{} + ieFG\_[B12]{} -)\
G\_F(\_1,\_2) &&\_F(\_1,\_2) = G\_[F12]{} [[e]{}\^[-ieFTG\_[B12]{}]{}(eFT)]{}\
\[replacegreen\] (the trigonometric expressions are to be understood as power series in the field strength matrix),
(4T)\^[-[D2]{}]{} && (4T)\^[-[D2]{}]{} [det]{}\^[-]{}\
(4T)\^[-[D2]{}]{} && (4T)\^[-[D2]{}]{} [det]{}\^[-]{}\
\[replacedet\] In particular, applying these changes in (\[bkmaster\]) yields a corresponding master formula for the $N$ - photon amplitudes in a constant field [@shaisultanov; @rescsc]. This master formula, and its extension to spinor QED, have been used for comparatively easy recalculations of the scalar and spinor QED vacuum polarization tensors [@vv], as well as of the photon splitting amplitudes in a magnetic field [@adlsch]. The determinant factors (\[replacedet\]) by themselves (i.e., the $N=0$ case) yield, after renormalization, the well-known effective Lagrangians of Weisskopf and Schwinger [@ws] and Euler-Heisenberg [@eulhei],
\_[scal]{}(F)&=& [116\^2]{} \_0\^[dTT\^3]{} \^[-m\^2T]{}\
[L]{}\_[spin]{}(F)&=& - [18\^2]{} \_0\^[dTT\^3]{} \^[-m\^2T]{} - [e\^23]{}(a\^2-b\^2)T\^2 -1\
\[eulhei\] Here $a,b$ are the two invariants of the Maxwell field, related to $\bf E$, $\bf B$ by $a^2-b^2 = B^2-E^2,\quad ab = {\bf E}\cdot {\bf B}$.
A further extension to the two-loop level has been extensively applied to the study of the two-loop corrections to the effective Lagrangians (\[eulhei\]) [@rescsc; @sd1; @dhrs].
See also [@ss1; @gussho] for the calculation of derivative corrections to the effective Lagrangian at the one-loop level. Here the gaussian form of the path integral is reached by Taylor expanding the background field at the loop center of mass, usually in Fock-Schwinger gauge to achieve manifest covariance.
Generalization to gravitational backgrounds
===========================================
To include an additional background gravitational field, naively one might replace
S\_0=\_0\^T d x\^2 && \_0\^T d x\^g\_(x())x\^ \[replacegrav\] The usual expansion around flat space $g_{\mu\nu} = \delta_{\mu\nu} + \kappa h_{\mu\nu}$ would then yield a graviton vertex operator $\varepsilon_{\mu\nu}\int_0^Td\tau \,\dot x^{\mu}
\dot x^{\nu}\,\e^{ik\cdot x}$. However, using this operator in a formal gaussian integration leads to worldline integrands contaning ill-defined expressions such as $ \delta(0), \delta^2(\tau_i-\tau_j), \ldots $. This comes not unexpected, since path integration in curved space is a subject notorious for its mathematical subtleties even in nonrelativistic quantum mechanics (see., e.g., [@schulman] and refs. therein). Fortunately, during the past decade these issues have been intensively studied, and a consistent formalism has emerged for the calculation of worldline path integrals in general electromagnetic-gravitational backgrounds [@wlgrav]. A detailed account of this recent development has been given in [@basvanbook]. Here we can only mention that the main difficulty arises from the nontriviality of the path integral measure in curved space, which leads to spurious UV divergences. Those can be removed by regularization, but leave an ambiguity which has to be removed by counterterms to the worldline Lagrangian. Those are regularization-dependent, and in general non-covariant, the only known exception being one-dimensional dimensional regularization. A further problem consists in the zero mode which appears in the perturbative expansion of the path integral. In the string-inspired approach this zero mode must be fixed as the loop center-of-mass, but this leads to a nontrivial Fadeev-Popov type determinant in the path integral.
Concerning previous applications of the worldline formalism in curved space, let us mention (i) the calculation of various types of anomalies (see [@basvanbook] and refs therein) (ii) the (re)calculation of the one loop graviton self energy due to a scalar loop [@baszir1], spinor loop [@bacozi], and loops due to vector and arbitrary differential forms [@babegi] (iii) the first calculation of the one loop photon-graviton amplitudes in a constant electromagnetic field [@phograv] (iv) the one loop photon vacuum polarization in a generic gravitational background due to a scalar loop in the semiclassical approximation [@holsho].
The effective action for Einstein-Maxwell theory
================================================
Pure Einstein-Maxwell theory is described by the action
\^[(0)]{}\[g,A\] = d\^4 x ( [12 \^2 ]{} R - [14]{}F\_F\^ ) \[SEM\] (here and in the following we absorb the coupling $e$ into $F$). In 1980, Drummond and Hathrell [@druhat] studied the one-loop corrections $\Gamma^{(1)}_{\rm spin}[g,A]$ to this action due to a spinor loop, and calculated the terms in it quadratic in the electromagnetic field, and linear in the curvature:
\_[spin]{}\^[(DH)]{} &=& ( 5 R F\_\^2 -26 R\_ F\^ F\^\_+2 R\_F\^F\^ +24 (\^F\_)\^2 )\
\[Ldruhat\] The point of singling out these terms is that they contain the information on the modifications of light propagation by weak gravitational fields in the limit of zero photon energies. In the following, our goal is to generalize this result to include the effect of a constant external field nonperturbatively, i.e., we are looking for the gravitational corrections to the Euler-Heisenberg Lagrangians (\[eulhei\]) to linear order in the curvature. Here it must be said that those flat space Lagrangians could be defined in either of two equivalent ways: (i) by the constancy of the background field $F_{\mu\nu}$ (ii) by the property of carrying the full information on the low energy limits of the corresponding $N$ - photon amplitudes. The lowest order gravitational corrections could be defined either by generalizing (i) to covariant constancy, or by generalizing (ii) by requiring that the effective Lagrangians should carry the information on the low energy limits of the amplitudes with $N$ photons and with one graviton. These generalizations are not any more equivalent, and we will adopt (ii) here rather than (i) (for the effective Lagrangian defined by covariant constancy Avramidi has obtained a representation in terms of integrals over the holonomy group [@avramidi]).
With our definition of the generalized Euler-Heisenberg Lagrangian, we have to get all terms involving arbitrary powers of $F_{\mu\nu}$ and one factor of $R_{\mu\nu\kappa\lambda}$ or $\nabla_{\mu}\nabla_{\nu}$. As in the flat space case, the path integrals are gaussianized by a Taylor expansion at the loop center-of-mass $x_0$, made covariant by combining Fock-Schwinger gauge and Riemann normal coordinates [@fsr]
A\_(x\_0+y) &=&-F\_(x\_[0]{})y\^-F\_[; ]{}(x\_[0]{})y\^y\^\
&& -y\^y\^y\^+\
g\_(x\_0+y) &=& g\_(x\_0) + [13]{} R\_(x\_0) y\^y\^+....\
\[fsriemann\] Concentrating on the spinor loop case, the effective Lagrangian then is obtained in the following form,
\_[spin]{} &=& -[18\^2]{} \_0\^[dTT\^3]{} \^[-m\^2T]{} [det]{}\^[-]{}\^[-S\_[int]{}]{}\_[S\_0]{}\
\[Lspinwick\] Here $S_0$ denotes the quadratic part of the worldline action, which is (after a rescaling to the unit circle)
S\_0 &=& \_[0]{}\^[1]{} d([14 T]{}g\_(x\_0) y\^y\^ -[i2]{} F\_(x\_0) y\^y\^ +[12]{}g\_(x\_0) \^\^-iTF\_(x\_0)\^\^)\
\[S0\] It yields again the generalized worldline Green’s functions of (\[replacegreen\]), only that in taking powers of the field strength matrix the lowering and raising of indices involves the metric $g_{\mu\nu}(x_0)$. The interaction part involves the terms coming from the replacement (\[replacegrav\]), as well as a ghost part $S_{\rm gh}$ from the path integral measure, and a term $S_{FP}$ representing the contribution from the Fadeev-Popov determinant mentioned above:
S\_[int]{} &=& S\_[grav]{} +S\_[gh]{} + S\_[em]{} + S\_[em, grav]{} + S\_[FP]{} \[Sint\]
$$\begin{aligned}
S_{grav}+S_{gh}&=&\int ^{1}_{0}\,d\tau\Biggl\lbrace \frac{1}{12T} R_{\mu \alpha \beta \nu} y^{\alpha}y^{\beta}
\biggl\lbrack
\dot{y}^{\mu}\dot{y}^{\nu}+a^\mu a^\nu + b^\mu c^\nu +2\alpha^\mu \alpha^\nu
\biggr\rbrack
\non\\&&
+\frac{1}{6}R_{\mu \alpha \beta \nu}\,y^{\alpha}\,y^{\beta}\,\psi^{\mu}\,\dot{\psi}^{\nu}
+\frac{1}{6} ( R_{\mu \alpha \lambda \beta}+R_{\mu \beta \lambda \alpha} ) \dot{y}^{\alpha}\,y^{\lambda}\,\psi^{\mu}\, \psi^{\beta}\Biggr\rbrace\non\\
S_{em}&=&\int ^{1}_{0}d\tau \bigg[ -\frac{i}{3}F_{\mu \nu ; \alpha} \Big( \dot{y}^{\mu}\, y^{\nu}+3T\psi^{\mu}\,\psi^{\nu}\Big)y^{\alpha}
-\frac{i}{8} F_{\mu \nu; \alpha \beta} \Big( \dot{y}^{\mu}\,y^{\nu}\,+4T\psi^{\mu}\psi^{\nu} \Big)\,y^{\alpha}\,y^{\beta}\bigg] \nonumber\\
S_{em,grav}&=&-\frac{i}{24}\int ^{1}_{0}d\tau
R_{\alpha\mu}{}^{\lambda}{}_{\beta}F_{\lambda\nu}
\Bigl[ \dot{y}^{\mu}\, y^{\nu} +8 T\psi^{\mu}\psi^{\nu} \Bigr]\,y^{\alpha}\,y^{\beta}\non\\
S_{FP}&=&-\third \int ^{1}_{0}d\tau \ \bar{\eta}_\mu
R^{\mu}{}_{\alpha \beta \nu}\,y^\alpha y^\beta
\, \eta^\nu
\label{actions}\end{aligned}$$
It is then a matter of simple combinatorics to arrive at our final result, an integral representation of the leading gravitational correction to the (unrenormalized) Euler-Heisenberg Lagrangian [@badasc]:
$$\begin{aligned}
{\cal L}_{\rm spin} &=&
-{1\over 8\pi^2}
\int^{\infty}_{0} \frac{dT}{T^3}\,\e^{-m^2T}\mbox{det}^{-1/2}\left[ \frac{\tan(FT)}{FT}\right]
\nonumber\\&&\times
\Biggl\lbrace
1+\frac{iT^2}{8}F_{\mu \nu ; \alpha \beta}\,\,{\cal G}^{\alpha \beta}_{B11}\Big(\dot{{\cal G}}^{\mu \nu}_{B11}-2\,{\cal G}^{\mu \nu}_{F11} \Big) \nonumber\\
&&+\frac{i T^2}{8}\left(F_{\mu \nu ; \beta \alpha} + F_{\mu \nu ; \alpha \beta}\right)\dot{{\cal G}}^{\mu \beta}_{B11}{\cal G}^{\nu \alpha}_{B11}+\frac{T}{3}R_{\alpha \beta}\,{\cal G}^{\alpha \beta}_{B11} \nonumber\\
&&-\frac{i T^2}{24}F_{\lambda \nu}R^{\lambda}_{\, \, \, \alpha \beta\mu}\,\left(\dot{{\cal G}}^{\nu \mu}_{B11}\,{\cal G}^{\alpha \beta}_{B11}+\dot{{\cal G}}^{\alpha \mu}_{B11}\,{\cal G}^{\nu \beta}_{B11}+\dot{{\cal G}}^{\beta \mu}_{B11}\,{\cal G}^{\nu \alpha}_{B11}+4\,{\cal G}^{\mu \nu}_{F11}\,{\cal G}^{\alpha \beta}_{B11}\right) \nonumber\\
&&+\frac{T}{12}R_{\mu \alpha \beta \nu}\Big(\dot{{\cal G}}^{\mu \alpha}_{B11}\dot{{\cal G}}^{\beta \nu}_{B11}+\dot{{\cal G}}^{\mu \beta}_{B11}\dot{{\cal G}}^{\alpha\nu}_{B11}
+\Bigl(\ddot{{\cal G}}^{\mu \nu}_{B11}-2g^\mn\delta(0)\Bigr){\cal G}^{\alpha \beta}_{B11}
\non\\
&&+\dot{{\cal G}}^{\alpha \beta}_{B11}\,{\cal G}^{\mu \nu}_{F11}
+\dot{{\cal G}}^{\nu \beta}_{B11}\,{\cal G}^{\mu \alpha}_{F11}
-{\cal G}^{\alpha \beta}_{B11}\,\Bigl(\dot{{\cal G}}^{\mu \nu}_{F11}-2g^\mn\delta(0)\Bigr)
\Big) \nonumber\\
&&-\frac{1}{6}T^{3}F_{\alpha \beta; \gamma}\,F_{\mu \nu ; \eta}\,\int^{1}_{0}d\tau_{1}\Big(\dot{{\cal G}}^{\alpha \nu}_{B12}\,\dot{{\cal G}}^{\beta \mu}_{B12} \, {\cal G}^{\gamma \eta}_{B12}+\dot{{\cal G}}^{\alpha \nu}_{B12}\,{\cal G}^{\beta \eta}_{B12} \, \dot{{\cal G}}^{\gamma \mu}_{B12} \nonumber\\
&&+\frac{3}{2}\,{\cal G}^{\gamma \eta}_{B12}\,{\cal G}^{\alpha \mu}_{F12}\,{\cal G}^{\beta \nu}_{F12}
\Big) \Biggr\rbrace \nonumber\\
\label{resultspin} \end{aligned}$$
($\tau_2 =0$).
As a check on (\[resultspin\]), we have verified that an expansion to order $RFF$ reproduces the result of Drummond-Hathrell up to total derivative terms: \_[spin]{} &=& -[18\^2 m\^2]{} - R F\_\^2 + R\_ F\^F\^\_\
&& + R\_F\^ F\^ - (\_F\_)\^2 + F\_F\^ \_[spin]{} - [L]{}\_[spin]{}\^[(DH)]{} &=& -[18\^2m\^2]{} \^(F\^ F\_[;]{}) + \_(F\_\^\_F\^) -\_(F\_\^\_F\^)\
\[totred\] As to possible applications of the Lagrangian (\[resultspin\]), let us mention that it contains the information on (i) the one graviton - $N$ photon amplitudes in the low energy limit (ii) the modified photon dispersion relations in the background of a strong electromagnetic and weak gravitational field (iii) the Schwinger pair production rate in such a field.
[9]{}
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|
---
abstract: 'We study the cosmological information of weak lensing (WL) peaks, focusing on two other statistics besides their abundance: the stacked tangential-shear profiles and the peak-peak correlation function. We use a large ensemble of simulated WL maps with survey specifications relevant to future missions like $\euclid$ and $\lsst$, to measure and examine the three peak probes. We find that the auto-correlation function of peaks with high signal-to-noise ($\SN$) ratio measured from fields of size 144 $\deg2$ has a maximum of $\sim 0.3$ at an angular scale $\vartheta\sim 10$ arcmin. For peaks with smaller $\SN$, the amplitude of the correlation function decreases, and its maximum occurs on smaller angular scales. The stacked tangential-shear profiles of the peaks also increase with their $\SN$. We compare the peak observables measured with and without shape noise and find that for $\SN\sim3$ only $\sim5\%$ of the peaks are due to large-scale structures, the rest being generated by shape noise. The correlation function of these small peaks is therefore very weak compared to that of small peaks measured from noise-free maps, and also their mean tangential-shear profile is a factor of a few smaller than the noise-free one. The covariance matrix of the probes is examined: the correlation function is only weakly covariant on scales $\vartheta<30$ arcmin, and slightly more on larger scales; the shear profiles are very correlated for $\vartheta>2$ arcmin, with a correlation coefficient as high as 0.7. The cross-covariance of the three probes is relatively weak: the peak abundance and profiles have the largest correlation coefficient $\sim 0.3$. Using the Fisher-matrix formalism, we compute the cosmological constraints for $\{\om,\,\s8,\,w,\,\ns\}$ considering each probe separately, as well as in combination. We find that the peak-peak correlation and shear profiles yield marginalized errors which are larger by a factor of $2-4$ for $\{\om,\,\s8\}$ than the errors yielded by the peak abundance alone, while the errors for $\{w,\,\ns\}$ are similar. By combining the three probes, the marginalized constraints are tightened by a factor of $\sim 2$ compared to the peak abundance alone, the least contributor to the error reduction being the correlation function. This work therefore recommends that future WL surveys use shear peaks beyond their abundance in order to constrain the cosmological model.'
author:
- |
Laura Marian,$^{1,2}$[^1] Robert E. Smith,$^{1}$ Stefan Hilbert$^{1,3}$ and Peter Schneider$^{2}$\
$^2$ Argelander-Institute for Astronomy, Auf dem Hügel 71, D-53121 Bonn, Germany\
$^1$ Max-Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, Garching, D-85748\
$^3$ Kavli Institute of Particle Astrophysics and Cosmology (KIPAC), Stanford University, 452 Lomita Mall, Stanford, CA 94305
title: 'The cosmological information of shear peaks: beyond the abundance '
---
-1.5cm
Introduction {#S1}
============
With the approach of large weak gravitational lensing (WL) missions, such as $\euclid$ and $\lsst$ [@lsst2009; @Euclid2011], it is imperative to optimize the extraction of cosmological information from WL probes. Until now, the WL community has been using mainly the 2-point function of the shear field in order to obtain cosmological constraints [@Jarvisetal2003; @Hoekstraetal2006; @Sembolonietal2006; @Hetterscheidtetal2007; @Kilbingeretal2012], and also to plan and forecast coming missions [@Euclid2011]. Whilst a lot of effort is concentrated on scrutinizing the systematics of WL surveys – a most essential endeavour for the success of the next-generation missions, and indeed for the future of WL as a cosmology probe – the task of selecting the most efficient statistics for parameter estimation is also a crucial aspect that should not be neglected.
One of the possible WL observables involves the ‘peaks’ of the shear field, i.e. regions of high signal-to-noise ($\SN$) of the field, produced by overdense regions of the density field projected along the line of sight. Shear peaks are the imprint of clusters in WL maps, and can be used to detect and measure cluster masses [@Hamanaetal2004; @Wangetal2004; @Maturietal2005; @HennawiSpergel2005; @Dahle2006; @MarianBernstein2006; @Schirmeretal2007; @Maturietal2007; @Abateetal2009]. The peak abundance scales with cosmological parameters in the same way as the halo mass function [@Reblinskyetal1999; @Marianetal2009; @Marianetal2010], and therefore can be used to constrain the cosmological model [@DietrichHartlap2010; @Kratochviletal2010; @Marianetal2012; @Bardetal2013]. The shear-peak abundance can also constrain primordial non-Gaussianity [@Marianetal2011; @Maturietal2011; @Hilbertetal2012], and so far seems to be possibly the most effective WL observable suitable for that purpose – although more work must be done on this subject.
Whilst the abundance of peaks has been investigated by a number of studies, higher-order statistics of the shear peaks have until now been overlooked. Yet, since the abundance of peaks can be used as a cosmological tool, one would expect their clustering to also be valuable. In analogy with 3D observables, just as both the halo correlation function and the halo mass function are very sensitive to cosmology, the same could be true about the shear peaks.
The main goal of this study is to investigate the correlation function of WL peaks, and in particular to quantify the improvements that it yields on cosmological constraints relative to the peak abundance. As a secondary line of inquiry, we shall also pursue the cosmological information contained in the tangential-shear profiles of the peaks. While these statistics – correlations and stacked profiles of overdensities – have been already studied in various theoretical works, as well as by using real and simulated data [@Sheldonetal2004; @Sheldonetal2009; @Mandelbaumetal2010; @OguriTakada2011; @Mandelbaumetal2012; @Cacciatoetal2012], here we explore the characteristics of shear-selected overdensities for which the only known information is the redshift distribution of the source galaxies. This might seem a somewhat unlikely and conservative scenario for future surveys, but here we just examine the intrinsic information contained in the peaks, even if our results will be sub-optimal on this account. We defer to a near-future study a more realistic scenario, where prior information on the peaks is available, and where we can account for various survey uncertainties, such as intrinsic alignments, masked regions, calibration errors of the source galaxies, etc.
We shall present measurements of the peak function, peak-peak correlation function, and peak profiles from an ensemble of WL maps generated with ray-tracing through $N$-body simulations with varying cosmologies, as described in §\[S2\]. We shall use the Fisher-matrix formalism to establish the cosmological sensitivity of the above-mentioned probes. We underscore that our results are drawn from collaborative efforts, which involved generating the $N$-body simulations, the ray-tracing maps, developing the hierarchical algorithm for WL peaks; the present study is part of a larger research program, which has also yielded the results published in [@Marianetal2011; @Marianetal2012] and [@Hilbertetal2012]. Therefore, we shall not give a full description of all our tools, but simply refer to these publications.
The paper is partitioned as follows: in §\[S2\], we provide a description of the ray-tracing maps used in this study; in §\[S3\] we illustrate our measurements of the three peak probes; in §\[S4\] we present the cosmological constraints extracted from the three probes; in §\[S5\] we summarize and conclude.
Simulated WL maps {#S2}
=================
We generated WL maps from ray-tracing through $N$-body simulations. We used 8 simulations which are part of a larger suite performed on the zBOX-2 and supercomputers at the University of Zürich. For all realizations 11 snapshots were output between redshifts $z=[0,2]$; further snapshots are at redshifts $z=\{3,4,5\}$. We shall refer to these simulations as the [ zHORIZON]{} simulations, and they were described in detail in [@Smith2009; @Smithetal2012]. Each of the [zHORIZON]{} simulations was performed using the publicly available [Gadget-2]{} code [@Springel2005], and followed the nonlinear evolution under gravity of $N=750^3$ equal-mass particles in a comoving cube of length $L_{\rm sim}=1500\Mpc$; the softening length was $l_{\rm soft}=60\,
\kpc$. The cosmological model was similar to that determined by the WMAP experiment [@Komatsuetal2009short]. We refer to this cosmology as the fiducial model. The transfer function for the simulations was generated using the publicly available [cmbfast]{} code [@SeljakZaldarriaga1996], with high sampling of the spatial frequencies on large scales. Initial conditions were set at redshift $z=50$ using the serial version of the publicly available [2LPT]{} code [@Scoccimarro1998; @Crocceetal2006]. Table \[tab:zHORIZONcospar\] summarizes the cosmological parameters that we simulated. We also use another series of simulations, identical in every way to the fiducial model, except that we have varied one of the cosmological parameters by a small amount. For each new set we have generated 4 simulations, matching the random realization of the initial Gaussian field with the corresponding one from the fiducial model. The four parameter variations were: $\{n\rightarrow (0.95, 1.05 ),\,\s8\rightarrow (0.7,
0.9),\,\om\rightarrow (0.2, 0.3),\, w\rightarrow (-1.2, -0.8) \}$, and we refer to each of the sets as [zHORIZON-V1a,b]{},…,[ zHORIZON-V4a,b]{}, respectively.
For the WL simulations, we considered a survey similar to $\euclid$ [@Euclid2011] and to [@lsst2009], with: an rms $\sigma_{\gamma}=0.3$ for the intrinsic image ellipticity, a source number density $\nbar =40\,\arcmint^{-2}$, and a redshift distribution of source galaxies given by $
{\mathcal P}(z)={\cal N}(z_0, \beta)\,z^2\exp[-(z/z_0)^{\beta}],
%\label{eq:source_distribution}
$
where the normalization constant ${\cal N}$ ensures that the integral of the source distribution over the source redshift is unity. If this interval extended to infinity, then the normalization could be written analytically as: ${\cal N}=3/(z_0^3\,\Gamma[(3+\beta)/\beta])$. There is a small difference between this value and what we actually used, due to the fact that we considered a source interval of $[0, 3]$. We took $\beta=1.5$, and required that the median redshift of this distribution be $\zmed=0.9$, which fixed $z_0\approx 0.64$, and gave a mean of $z_{\rm mean}=0.95$.
From each $N$-body simulation we generated 16 independent fields of view. Each field had an area of $12 \times 12\,\deg2$ and was tiled by $4096^2$ pixels, yielding an angular resolution $\thetapix
=10\,\arcsect$. For each variational model, the total area was of $\approx 9000\, \deg2$, while for the fiducial model it was of $\approx 18000\, \deg2$. The effective convergence $\kappa$ in each pixel was calculated by tracing a light ray back through the simulation with a multiple-lens-plane ray-tracing algorithm [@Hilbertetal2007; @Hilbertetal2009]. Gaussian shape noise with variance $\sigma_{\gamma}^2 /(\nbar\, \thetapix^{2})$ was then added to each pixel, creating a realistic noise level and correlation in the filtered convergence field [@HilbertMetcalfWhite2007]. We keep the shape noise configuration fixed for each field in different cosmologies, in order to minimize its impact on the comparisons of the peak abundances measured for each cosmology.
WL peaks as a cosmological probe {#S3}
================================
As stated in §\[S1\], we shall consider the cosmological information contained in three shear peak probes: the abundance, the profiles, and the correlation function. Until now, only the abundance of peaks has been examined as a probe for cosmology; in this study we shall investigate the cosmological information contained in the clustering of peaks and their profiles, as well as various combinations of the three probes.
The peaks are detected with an aperture-mass filter, i.e. a compensated filter [@Schneider1996], as points of maxima in the smoothed convergence field. They are assigned a unique $\SN$ and mass using the hierarchical algorithm. This method, as well as the NFW-shaped filter function, were described in detail in our previous work [@Marianetal2012]. The hierarchical algorithm uses a hierarchy of filters of different size, from the largest down to the smallest, to determine the size of the NFW filter that best matches each peak. For every filter used, we write the aperture mass and the $\SN$ at a given point $\btheta_0$ as (\_0) = , \[eq:smoothed\_map2\] (\_0)= , \[eq:SNc\] where $\Mm$ is the mass of the NFW halo used as a model for the filter, and $\theta_{\rm A}$ is the aperture radius of the filter, which in our case was chosen to be the angular size of its NFW radius. $\km$ is the convergence profile of the model, $\bkm$ is the mean convergence inside a certain radius, and $\kappa$ is the measured convergence field. If $\btheta_0$ is the location of a peak generated by an NFW halo matching the filter, then the above equations become (\_0) = , \[eq:Mmax\] (\_0)=. \[eq:SNmax\] The hierarchical method enhances the detection of peaks, their abundance as a function of mass or $\SN$ is independent of any particular choice of filter size, and the cosmological constraints arising from the peak counts are tighter compared to the case when a single-sized filter is used. It is also instructive to present measurements of the peak function, the profiles and the peak-peak correlation function from noise-free maps. In that case, the peaks are selected based on their aperture-mass values, and they are assigned a hierarchical mass through , as described in [@Marianetal2012]. For the purpose of comparison with the results from the noisy maps, we can assign the noise-free peaks a fictitious $\SN$ value, using with the same level of shape noise as in the noisy maps.
The peak-peak correlation function
------------------------------------
We have measured the correlation function of the peaks using the minimum-variance and unbiased estimator introduced by [@LandySzalay1993]. The estimator is: \_[pp]{}=(DD-2DR+RR)/RR, where $DD$, $DR$, and $RR$ represent the peak-peak, peak-random peak, and random peak-random peak pair counts, respectively. The correlation function is measured in 10 angular bins, logarithmically spaced between $\vartheta\in\left[4, 90\right]$ arcmin. The lower bound was chosen to be the smallest possible without discreteness effects contaminating the measurements – the latter appear when the size of the lower bins is only a small multiple of the size of the pixels of the map. The random catalogue that we created in order to build the correlation function estimator contained a million random peaks placed in a field of $12\times12\,\deg2$, i.e. the same size as the simulated maps.
In we present measurements of the correlation function of peaks detected from the maps corresponding to the fiducial cosmology. All symbols denote the average of 128 realizations, the errors being on the mean. The estimator of the average correlation function is defined by ; a similar estimator will also be used to obtain the average tangential-shear profile of the peaks and the average peak function presented later in this section.
The left panel of shows the noise-free results, while in the right one shape noise is included. The solid orange squares/green triangles in the left panel depict the auto-correlation $\corra$ of the peaks with $3.4\geq\SN\geq2.6$ and $\SN\geq3.4$, respectively. The solid red circles, blue squares, and green triangles in the right panel correspond to peaks with $\SN\geq 4.75$, $4.75\geq\SN\geq3.4$, and $3.4\geq\SN\geq2.6$ respectively. The total number of peaks in these bins is $\sim 10,000,\,72,000,\,324,000$ respectively. In both panels, the violet crosses depict the cross-correlation $\corrc$ of the bins $3.4\geq\SN\geq2.6$ and $\SN\geq 4.75$.
The correlation functions are strongly negative for the smallest bins due to an exclusion effect arising from our peak selection: we discard those peaks whose centres are separated from the centre of a larger peak by a distance smaller than approximately one virial radius. The reason for this choice is to avoid classifying substructures as independent peaks.
In general, the peaks with higher/lower $\SN$ have higher/lower correlations, similar to the clustering properties displayed by 3D halos. For the noiseless measurements, the maximum of the auto-correlation of peaks with $\SN\geq3.4$ is reached at $\sim
8\,\arcmint$ and is about 0.35, while for the peaks with $3.4\geq\SN\geq2.6$ the maximum is about 0.25 and it is shifted towards smaller scales $\sim 5\,\arcmint$. The auto-correlation functions decrease monotonically to $0$ with increasing angular scales. The same pattern is followed by the correlation functions of the peaks measured from the noisy maps: the higher the $\SN$ bin, the larger the amplitude of the function and the angular position of its maximum. For the largest peaks, a maximum of $\sim 0.3$ is reached at $\sim 10$ arcmin. Comparing the measurements from noise-free and noisy maps allows us to understand how shape noise impacts the correlation function of the peaks. For example, the green triangles in the left and right panels of are dissimilar, although peaks from the same $\SN$ bin are used for the measurements. The noise-free correlation function is significantly higher, dropping to 0 only at $\sim100\,\arcmint$, unlike its noisy counterpart, which vanishes at $\sim8\,\arcmint$. The vanishing owes to the fact that most peaks in this $\SN$ bin are generated by shape noise – see also – and behave essentially like random points. The genuine signal seen in the left panel of the figure is ‘drowned’ by the large number of shape-noise pair counts. The same is true for the cross-correlation signal depicted by the purple crosses in both panels. The noise-free signal is quite high, similar to the auto-correlation of peaks with $\SN\geq3.4$, and the error bars are significantly smaller than for the auto-correlation of the peaks with $\SN\geq 4.75$ – high-$\SN$ peaks are very scarce in the noise-free maps, hence the errors on their correlation function are quite large, which is the reason why we do not even show it in the figure. Regarding the noisy maps, whilst the cross-correlation function has a smaller signal-to-noise than the auto-correlation of the largest peaks, it can be used to constrain cosmology, as shown in §\[S4\].
The tangential-shear profiles of the peaks
------------------------------------------
We measure the tangential-shear profiles $\gamma_{T}$ around the centres of the peaks, as found through our filtering method. We use 15 angular bins, logarithmically spanning the interval $\vartheta\in[0.3,
26]\,\arcmint$. In we present the average of the stacked profiles, measured from 128 fields corresponding to the fiducial cosmology.
The left panel shows the impact of shape noise on the peak profiles. The solid red pentagons and orange squares depict the stacked profiles of the largest peaks ($\SN\geq4.75$), and all the peaks identified in the noise-free maps. The number of peaks contributing to the two curves are $\sim 4,500$ and $\sim 39,000$ respectively. The noise-free peaks are generated by lensing of large-scale structures (LSS); therefore, they are also present in the noisy maps, albeit with slightly different centre coordinates and amplitude. We match the coordinates of the noise-free peaks to the coordinates of the peaks in the noisy maps. This is done on a field-by-field basis, allowing for a displacement of 5 pixels around the centre of the noise-free peaks. The green empty squares and purple stars indicate the stacked profiles of the same peaks depicted by the solid red pentagons and orange squares, when measured from the noisy maps. The shape noise renders the measured profiles shallower on small scales, $\vartheta\leq\,2\,\arcmint$, due to the shift in the centre coordinates. On scales $\vartheta > \,2\,\arcmint$ there is no significant difference between the noise-free and noisy profiles.
The right panel shows measurements from the noisy maps. The $\SN$-bins are similar to those used for the correlation function: $3.4>\SN\geq
2.6$ – green solid triangles; $4.75>\SN\geq 3.4$ – blue solid squares; and $\SN\geq 4.75$ – red solid circles. For illustration purposes, we also include the stacked profiles of all peaks binned together, represented by the purple empty circles. The more massive the peaks, the higher their profiles: there is about an order of magnitude difference between the average tangential shear corresponding to the highest and lowest $\SN$-bins. Similar to the correlation function, the stacked profiles of the noisy peaks are significantly lower than the noiseless ones. For example, compare the average profile of all peaks from the two panels, i.e. the orange solid squares on the left with the empty purple circles on the right: there is a factor of $\sim\,5$ difference on small scales, and less on larger scales. This is a consequence of several factors: i) most of the low-$\SN$ peaks are caused by shape noise and they represent random points in the shear maps, thereby lowering the profile average of the real peaks; ii) some small genuine LSS peaks are boosted by shape noise above the detection threshold, and so they are included in the right panel, but not in the left one; iii) as shown in the left panel, shape noise does tend to lower the average profiles due to the shifting of the peak centre.
The peak function
-----------------
illustrates the peak function $\Phi$, i.e. the number of peaks per unit $\SN$, in a field with area 144 $\deg2$. We considered 15 logarithmic $\SN$ bins, in the interval $[2.6, 14]$. The red solid circles denote the noiseless peak function measured from 128 fields corresponding to the fiducial cosmology, while the blue solid squares are the same measurement in the presence of shape noise. The two peak functions are very similar at the high-$\SN$ end, and differ by a factor of at least 20 at the low-$\SN$ end, where peaks generated by shape-noise completely dominate the genuine LSS peaks. Modelling the effect of shape noise on the peak function is the subject of a work in progress, so we shall not dwell on it any longer here. We merely mention that the dominance of shape-noise peaks at the low-$\SN$ end of the peak function is in agreement with the behaviour of the peak-peak correlation function, and the tangential-shear profiles displayed in .
Fisher matrix calculations {#S4}
==========================
In this section we employ the Fisher-matrix formalism to derive cosmological constraints from the three peak probes presented in the preceding section. We shall make use of the cosmology dependence of our simulated maps, and measure the peak statistics corresponding to the models varying cosmological parameters around the fiducial values. These measurements can then be used to obtain Fisher predictions.
Throughout this work we shall assume the likelihood function of the peak probes to be a multivariate Gaussian. Consider a vector of measurements of the peak-peak correlation function, peak abundance, and peak profiles $\m=\{\omega_{1},\dots, \omega_{n_{\rm B}^\omega},
\Phi_1 \dots, \Phi_{n_{\rm B}^\Phi}, \gamma_{T}^{1}, \dots,
\gamma_{T}^{\rm n_{\rm B}^{\gamma_T}}\}$, where $n_{\rm
B}^{\omega},\,n_{\rm B}^{\Phi},\,n_{\rm B}^{\gamma_T}$ are the number of bins for the respective measurements. If we similarly define a vector $\bar{\m}$ of the mean of the measurements, we write the likelihood as (||(), ())=\
, where the variable $\p$ indicates the dependence on the cosmological model of the mean and covariance matrix of the measurements, in this case specified by: $\{n,\,\s8,\,\om,\, w\}$. We have also introduced the total number of bins $n_{\rm B}= n_{\rm B}^{\omega} + n_{\rm
B}^{\Phi} + n_{\rm B}^{\gamma_T}$, as well as the total covariance matrix of the measurements in bins $i$ and $j$: C\_[i j]{}=(m\_i-|m\_i)(m\_j-|m\_j), i,j= \[eq:cov\] The Fisher matrix is defined as the ensemble-averaged Hessian of the logarithm of the likelihood function, \_ = - . \[eq:Fisher\_gen\] Note that we shall also use for each probe individually, case in which the vectors $\m,\, \bar{\m}$ and matrix $\C$ will implicitly contain only the measurements of the respective probe, and $n_{\rm B}$ will be the number of bins for that same probe. We shall make no further specification on this subject and rely on the context for clarity.
The general expression of can be rewritten as [@Tegmarketal1997] \_ = + | \_[,p\_]{}\^[T]{} \^[-1]{} |\_[,p\_]{} \[eq:Fisher2\]. Here we have denoted by $x_{,p_\alpha}=\partial x/\partial
p_\alpha$. We shall ignore the first term in for two reasons: i) it is considered to contribute little to the Fisher information – for a discussion on this see [@Tegmarketal1997]; ii) an accurate determination of the derivatives of the covariance with respect to the cosmological parameters would require more realizations of the variational cosmologies than we currently have. Therefore we shall follow the standard approach to Fisher-matrix forecasting in the literature and ignore the trace-term in the above equation.
From the Fisher matrix, one may obtain an estimate of the marginalized errors and covariances of the parameters: \^2\_[p\_p\_]{} = \[\^[-1]{}\]\_, \[eq:marg\_error\] as well as the unmarginalized errors: \_[p\_]{} = \[\_\]\^[-1/2]{} . \[eq:unmarg\_error\] We now turn our attention to the estimation of each element contributing to the simplified expression of the Fisher matrix, discussed above. For each cosmological model, the mean auto- and cross-correlation functions are evaluated as an average of the correlation functions for each field, ()=\_[k=1]{}\^[n]{} \_k(), \[eq:estmean\] where $\hat{\omega}_k$ is the correlation function measured from the $k^{\rm th}$ field, and $n$ is the number of fields considered. Similarly, we compute the mean of the peak function and the tangential-shear profiles. An unbiased, maximum-likelihood estimator for the covariance is \_[i j]{}=\_[k=1]{}\^[n]{} (\_i\^k-\_i) (\_j\^k-\_j). \[eq:cov2\] We use this estimator to compute the fiducial-model-covariance matrix for each of the three probes, as well as for any combination of them. The estimate of the inverse covariance is corrected in the following way [@Anderson1958; @Hartlapetal2007]: =()\^[-1]{}, n\_[B]{} < n-2, \[eq:Cinv\] In order to obtain low-noise estimates of the derivatives with respect to the cosmological parameters, we take advantage of the matched initial conditions of the simulations, and use the double-sided derivative estimator =\_[k=1]{}\^[n]{} , \[eq:derivs\] where $\Delta \alpha$ represents the $\pm$ step in the cosmological parameter $p_\alpha$, e.g. Table \[tab:zHORIZONcospar\]. Note that whilst this approach reduces the cosmic variance in the derivatives, the estimated Fisher matrix is still noisy due to the inverse covariance estimator.
Finally, we mention a technical point regarding the joined constraints presented in the next section. Since we evaluate the combined covariance for the three probes, we are concerned that its elements might be very different, hence introducing numerical instabilities in its inversion, and possibly leading to an inaccurate estimate of the inverse. To prevent this, we apply the same strategy recently used in [@Smithetal2012]: instead of using the covariance matrix, we use the correlation matrix, defined by r\_[i j]{}=C\_[i j]{}/\_i\_j , where $\sigma_{i}=\sqrt{C_{i i}}$ is the rms variance. The inverse correlation and covariance are related by: r\^[-1]{}\_[i j]{}=\_[i]{} \_[j]{} C\^[-1]{}\_[i j]{} We can rewrite the Fisher matrix from as: \_ = \_[i, j=1]{}\^[n\_[B]{}]{}[r]{}\_[i j]{}\^[-1]{} \[eq:Fisher3\], It is this last equation that we shall be using for our forecast, scaling the measured derivatives of the probes by the rms variance of the fiducial functions, and employing the inverse correlation matrix, instead of the inverse covariance.
depicts the derivatives of the auto- and cross-correlation function with respect to the cosmological parameters that we investigate in this work – left and right panels, respectively. The auto-correlation derivatives are quite noisy – primarily due to the relatively small number of peaks with $\SN\geq
4.75$, a characteristic also visible in , the noise being largest on small scales, $\vartheta < 20\,\arcmint$. The $\om$- derivative has the largest amplitude, while $w$ has the smallest. All derivatives tend to 0 on larger scales, signifying that there is no information in the peak-peak correlation function for $\vartheta\geq
2\,{\rm deg}$. Compared to the auto-correlation function, the derivatives of the cross-correlation are less noisy, with the exception of $\ns$. They are also significantly smaller, just like the cross-correlation signal is much smaller than the auto-correlation one. In both panels, the $\om$-derivative is negative for $\vartheta\geq 10\,\arcmint$, whilst for the other parameters, the derivatives cross the 0-line a few times.
presents the shear-profile derivatives with respect to the cosmological parameters: the left panel corresponds to peaks with $\SN\geq 2.6$, while the right one is for $\SN\geq 4.75$, hence the larger measurement noise present there. In both panels, the derivatives decrease to 0 for increasing angular scales, e.g. $\vartheta\sim20\,\arcmint$. In the left panel, all derivatives are positive, with the only exception of $w$, which has a barely-noticeable transition from slightly negative to slightly positive. The $\om$-derivative is the largest, while the $w$-one is the smallest. For the higher-$\SN$ peaks, the derivatives are less smooth, but the general trends are preserved: $\om$ and $\s8$ still have the highest derivatives, followed by $\ns$ and $w$. They all seem to peak on the scales $0.8-1\,\arcmint$, which suggests that a significant amount of information arises from these scales. We shall mostly be using the larger peaks for our constraints.
depicts the derivatives of the peak function with respect to the cosmological parameters, scaled by the peak abundance corresponding to the fiducial cosmology, as a function of $\SN$. Again $\om$ has the largest derivative, followed closely by $\s8$ – both of them positive. The derivatives with respect to $\ns$ and $w$ are much smaller and less smooth.
Since in this study we use numerical derivatives to estimate the Fisher errors, as opposed to analytical ones, we need to ask what the resulting uncertainty on the Fisher errors is. This is particularly needed for our noisiest measurements, i.e. for the correlation function. We address this question in the appendix §\[A1\], and refer the interested reader to that discussion.
One point of interest – useful mainly to develop one’s intuition – is to explore which are the highest contributing scales to the Fisher information. Naturally, we expect a combination of scales to be the most effective at constraining the cosmological model; the most common approach to determine such a combination is to diagonalize the covariance matrix and determine its eigenvalues: the eigenvectors corresponding to the largest eigenvalues will represent the most constraining linear combinations of scales. To see if there is a particular scale which helps to significantly reduce the errors, we shall take two approaches.
First, we compute the cumulative Fisher matrix, i.e. we monitor how the errors evolve when bins are included one-by-one in the calculation of . We choose as starting bins those containing measurements expected to be easier to perform in real surveys. For example, for the peak counts, we start with the highest-$\SN$ bins, and then add progressively smaller-$\SN$ bins. For the profiles and correlation functions, we start with small-scale measurements, and progressively include large angular scales. The results of this exercise are presented in the left panels of : from top to bottom, we show the cumulative unmarginalized Fisher errors arising from $\corra,\,\gamma_{T},$ and $\Phi$, scaled by the final errors – when all bins are included – as a function of bins. For the auto-correlation function, the information seems to saturate at $\sim 40\,\arcmint$: the inclusion of larger-scale measurements does not further improve the constraints. For the profiles, the saturation occurs at $\sim 10\,\arcmint$, while for the peak counts, the inclusion of peaks with $\SN<4$ still improves the constraint on $w$, though not on the other parameters. This is in line with our findings from [@Marianetal2012].
The second approach involves two approximations: i) for each probe we neglect the covariance of the parameters, i.e. we consider only unmarginalized errors – this was also done for the first approach. ii) we disregard the off-diagonal elements of the covariance matrix of each probe. The Fisher matrix can then be written as \_ = \_[i=1]{}\^[n\_[B]{}]{}|[m]{}\_[i, p\_]{}\^2/C\_[i i]{}. \[eq:FishDat\] We can quantify the contribution of each bin $j$ to the total Fisher constraints through the ratio: =. \[eq:FishDat2\] By construction, this ratio works best as a contributing-scales indicator for those probes which have a covariance matrix with as few off-diagonal elements as possible. In our case, this is the auto-correlation function, as can be seen from . The rms of the above ratio is shown in the right panels of , ordered in the same way as the left ones. For the correlation function, the most contributing scales to the $\s8$- and $\om$-constraints are $\sim 30$ and $\sim 40\,\arcmint$ respectively. This is in agreement with the left panel of the figure: the errors on those two parameters are lowered significantly when the respective bins are included in the Fisher estimates. The constraints from stacked profiles receive contributions from all scales below $10\,\arcmint$, while for the counts the smaller bins ($\SN\leq 5$) tighten the errors – though notice that the agreement between the left and right panel is not so good as for the correlation function, due to the high correlations between the low-$\SN$ bins. These correlations are included in the left panel, but not in the right one.
Cosmological constraints {#S5}
========================
Using all the ingredients described in section §\[S4\], we now proceed to constraining the cosmological model using the three peak probes individually, as well as in combination. To this avail, we shall employ only noisy maps, and consider the auto-correlation $\corra$ of peaks with $\SN\geq 4.75$, the cross-correlation $\corrc$ of peaks with $\SN\geq 4.75$ and $3.4\geq\SN\geq2.6$, the stacked profiles $\gamma_T$ of peaks with $\SN\geq4.75$, and the abundance of peaks $\Phi$ with $\SN\geq 2.6$. The Fisher matrix errors are estimated using the covariance on the mean of the fiducial model for the three probes of interest to us here. Thus, our estimated covariance from , which corresponds to an area of $144\,\deg2$, is rescaled to correspond to an area 128 times larger, i.e. $\sim 18000\,\deg2$. Together with the survey specifications given in section §\[S2\], this makes our study representative for two future surveys, $\lsst$ and $\euclid$.
presents the cross-correlation matrix ${\bf r}$ of these probes. By far, $\gamma_T$ has the strongest correlation coefficient of the three: $\sim 0.7$ on scales $2-20\, \arcmint$. For the peak function, the low-$\SN$ bins are the most correlated $\sim
0.5$ for $\SN\leq 5$. This was already established in our earlier work [@Marianetal2012], and it can be explained through the better-known behaviour of halos: small-mass halos are sample-variance dominated, while the large and rare halos follow the Poisson distribution [@HuKravtsov2003; @SmithMarian2011]. Note however how the smallest-$\SN$ bins in seem to be completely uncorrelated: this is most likely due to the overwhelming number of shape-noise peaks, which are random, unclustered, and therefore uncorrelated. $\corra$ displays the smallest correlation coefficient of the three, $\sim 0.3-0.4$ on the scales $20-60\,\arcmint$, with weaker correlations on smaller scales. We further note the weak cross-correlation of $\corra$ and $\Phi$, as well as $\corra$ and $\gamma_T$. There is a visible cross-correlation of $\Phi$ and $\gamma_T$, of $\sim 0.3$ for peaks with $\SN>7$. This is most likely due to the stacked profiles being dominated by the most massive peaks, which also dominate the high-$\SN$ end of the peak function.
\[fisher\_er\]
Table \[fisher\_er\] presents the unmarginalized and marginalized 1-$\sigma$ errors resulting from the three peak probes. Each probe taken by itself, the abundance of peaks has the greatest constraining power, followed by the profiles, and then by the correlation function. Regarding the latter, we note that $\corra$ and $\corrc$ yield very similar constraints, the auto-correlation being more effective for $w$ and $\om$ – a reduction by factors of $\sim 2$ and $\sim 1.5$ respectively in these errors, compared to the cross-correlation. However, when combined with the other two probes, there is little difference between $\corra$ and $\corrc$. The greatest benefit to adding the correlation function or the profiles to the abundance of peaks concerns the time-independent equation-of-state for dark energy: after marginalizing over the other parameters, the errors on $w$ resulting from $\Phi$ and $\corra$ taken individually are similar, while the profiles seem to yield a constraint tighter by a factor of $\sim 1.7$. When all three probes are combined, the constraints on $\om,\,\s8,\,\ns$ improve by a factor of $\sim 1.5-2$ compared to using $\Phi$ alone, while for $w$ the improvement is $\sim
2.5$. Lastly, combining $\Phi$ and $\gamma_T$ is almost as efficient as using all three probes: the contribution of the correlation functions to reducing the errors is negligible, if both the abundance and the profiles are used.
This is further detailed by which shows the forecasted likelihood contours, at 95% confidence level. The largest ellipses – green, dot-dashed – represent the $\corra$ constraints; for many of the parameter combinations, the red, dashed ellipses depicting $\gamma_T$ are almost perpendicularly oriented relative to the $\corra$ ellipses, and also quite tilted with respect to the $\Phi$ ellipses, shown by the blue dotted lines. The greatest gain in combining all three probes is for the case of $w$, where the resulting ellipse is smaller by a factor of $\sim 3$ compared to the $\Phi$ ellipse.
We compare our Fisher constraints to those of [@Hilbertetal2012], derived for an identical survey scenario from the same simulated WL maps. That analysis combines the shear correlation functions, the third moment of the aperture mass, and the peak abundance, and accounts also for local primordial non-Gaussianity. The percentage unmarginalized combined 1-$\sigma$ errors found in [@Hilbertetal2012] for $\{\om,\,\s8,\,w,\,\ns\}$ were $\{0.2,\,0.15,\,0.9,\,0.4\}$. The marginalized errors were $\{0.7,\,0.4,\,2.6,\,0.7\}$. Their constraints are tighter, particularly for $\om$, but the combination of peak probes performs nonetheless very well.
presents the 95%-likelihood contours of the three probes, after we have added cosmic microwave background (CMB) information resulting from an experiment similar to $\planck$. For the latter, we assume that the CMB temperature and polarization spectra can constrain 9 parameters: the dark energy equation-of-state parameters $w_0$ and $w_a$; the density parameter for dark energy $\Omega_{\rm DE}$; the CDM and baryon density parameters scaled by the square of the dimensionless Hubble parameter $\omega_{\rm
CDM}=\Omega_{\rm CDM}\,h^2$ and $\omega_{\rm b}=\Omega_{\rm b}\,h^2$ ($h=H_0/[100\, {\rm km\,s^{-1}\,Mpc^{-1}}]$); the primordial spectral index of scalar perturbations $\ns$; the primordial amplitude of scalar perturbations $A_s$; the running of the spectral index $\alpha$; and the optical depth to the last scattering surface $\tau$. We compute the CMB Fisher matrix as described by [@Eisensteinetal1999]: \_=\_l \_[X,Y]{} \^[-1]{} , where $\{X,Y\}\in \{{\rm TT},\,{\rm EE},\,{\rm TE},\,{\rm BB}\}$, where $C_{l,\rm TT}$ is the temperature power spectrum, $C_{l,\rm EE}$ is the E-mode polarization power spectrum, $C_{l,\rm TE}$ is the temperature-E-mode polarization cross-power spectrum, and $C_{l,\rm
BB}$ is the B-mode polarization power spectrum. The assumed sky coverage is $f_{\rm sky}=0.8$ In order to make the CMB Fisher matrix compatible with our parameters, we rotate it to a new set $
{\bf q}^{T}=\{w_0, w_a, \Omega_{m}, h, f_b, \tau, \ns, \s8, \alpha \} \ ,
$ where for us $w_0=w$. We marginalize over the 5 parameters absent from our analysis, and then add the resulting $4\times4$ matrix to the Fisher matrix for each of the probes, and their combination. This way of implementing the $\planck$ prior follows the same steps taken in our previous studies [@Marianetal2012; @Hilbertetal2012]. shows that combining the peak probes with the CMB dramatically alters all the ellipses involving $\ns$: they are shrunken and almost completely aligned. This is hardly surprising, as the CMB constrains $\ns$ much better than any WL probe, so the ellipses are then dominated by the CMB information. For the other parameters, the changes are small, and there is still a significant gain to be obtained by combining the peak probes, compared to using only one of them, for instance the abundance.
Summary and conclusion {#S6}
======================
In this paper we have investigated cosmological constraints from WL peaks, beyond using just counts. We have measured for the first time the peak-peak correlation function, and estimated its constraining power on the following cosmological parameters: $\{\om,\,\s8,\,w,\,\ns\}$. We have also measured the tangential-shear profiles of the peaks, and explored their cosmological utility. Whilst both the correlation function and stacked profiles of galaxies/clusters are standard tools for cosmology, our study applies them to shear-selected objects, detected only through their gravitational lensing effects. We employed mostly numerical methods, performing measurements of the peak abundance, peak profiles, and peak-peak correlation functions in simulated WL maps. To estimate the cosmological constraints, we used the Fisher-matrix formalism, computing the derivatives with respect to cosmological parameters and the combined covariance of the three peak probes from the simulated maps. The latter have been generated with ray-tracing through a large suite of $N$-body simulations, varying the cosmological model around some fiducial values. Given the distribution of source galaxies and the level of shape noise considered, our simulated maps are relevant for future large WL missions, such as $\euclid$ and $\lsst$.
To measure the correlation function of peaks, we used the [@LandySzalay1993] estimator, known for two essential properties: i) minimum variance; ii) lack of bias due to the unknown mean density of the measured objects. We have shown that the correlation function has its maximum on scales $\vartheta\leq 10\,\arcmint$, with the position and height of the latter depending on the $\SN$ of the considered peaks. For angular scales $\vartheta\geq 8\,\arcmint$, the correlation function of ‘large’ peaks ($\SN\geq 4.75$) has a greater amplitude than that of ‘medium’ and ‘small’ peaks ($4.75>\SN\geq 3.4$ and $\SN<3.4$), mirroring the clustering behaviour of dark-matter halos. We have shown that the low-$\SN$ end of the peak function is completely dominated by shape-noise peaks – only $\sim5\%$ of peaks with $\SN\sim 3$ are genuine LSS peaks. This is the reason why the correlation function of the small peaks quickly declines to $0$: shape-noise peaks are random, and therefore uncorrelated. On the contrary, the correlation function of the peaks measured from noise-free maps is quite high, even with all peaks, small and large, binned together: this is just a statement that peaks due genuinely to projected LSS are quite clustered. The cross-correlation of small- and large-$\SN$ peaks measured from noisy maps is also smaller than its noise-free counterpart, but it can still be used for cosmological constraints.
We have measured the tangential-shear profiles of the peaks using their WL centre, as detected by our hierarchical method. We found that the larger the $\SN$ of the peaks, the higher their average profile. A comparison of the results from noise-free and noisy maps showed that: i) the ratio of the average profile of high-$\SN$ peaks to the average profile of all peaks combined together is $\sim 2$ for noise-free maps, and $\sim 5$ for noisy maps; ii) the profile average of all noisy peaks binned together is a factor of $\sim 5$ smaller than that of the noiseless profiles; iii) the profiles of genuine LSS peaks are shallower on scales $\vartheta<2\,\arcmint$ when the peaks are detected and measured from noisy maps, rather than noise-free maps – this owes to a coordinate shift of the peak centres in the presence of shape noise. The first two findings are mostly due to the overwhelming number of shape-noise peaks which dominate the low-$\SN$ end of the peak function: measurements of the tangential-shear profiles around random points in the map bring down the average of the stacked profiles of genuine LSS structures.
The Fisher-matrix analysis that we performed revealed the forecasted errors on the cosmological parameters from the auto- and cross-correlation functions of the peaks, as well as from the stacked profiles to be larger by at least a factor of $2$ than those obtained using the peak function. Nevertheless, combining the peak function with any of the other three probes reduces the errors significantly, particularly for $w$. The ratio of the marginalized 1-$\sigma$ constraints from $\Phi$ alone to the constraints from $\corra + \Phi$ is $\{1.36,\,1.43,\,1.5,\,1.3\}$, corresponding to $\{\om,\,\s8,\,w,\,\ns\}$; for $\gamma_T + \Phi$ the ratio is $\{1.6,\,1.7,\,2.3,\,1.6\}$; and for $\corra + \gamma_T + \Phi$, the result is $\{1.7,\,2,\,2.5,\,2\}$. Therefore, one of the main conclusions of this study is that future WL surveys should use peak statistics beyond the 1-point function. Note that our $\corra +
\gamma_T + \Phi$ constraints are quite competitive with those derived by [@Hilbertetal2012] from combining the shear correlation functions, the third moment of the aperture mass, and the peak abundance. For the parameters that we have investigated, the stacked tangential-shear profiles seem slightly more constraining than the correlation function. However, if other parameters were to be included in the analysis, particularly those quantifying primordial non-Gaussianities, the peak-peak correlation function might be more powerful. This however, is the subject of an ongoing work, to be presented in the near future.
We have briefly examined the $\SN$ and angular scales contributing most to the Fisher information. The cosmology constraints from the peak-peak correlation function benefit significantly from measurements at scales $\vartheta\sim 20-40\,\arcmint$, and saturate for $\vartheta>50\,\arcmint$. For the tangential-shear profiles, measurements at $\vartheta\sim1-10\,\arcmint$ are important, and the constraints saturate for $\vartheta>10\,\arcmint$. The peak abundance benefits from the inclusion of small-$\SN$ bins.
In this study, we have not exhaustively investigated the properties of the clustering of the peaks. Measuring the correlation function was only the first step of such a study. It would also be interesting to examine the biasing of the peaks with respect to the shear/convergence field, and also to compare the clustering of the peaks with that of the halos in the simulations. Our goal for the near-future is to determine the full constraining power of the shear-selected WL peaks on an expanded set of cosmological parameters, i.e. to extend the recent study of [@Hilbertetal2012] to include the 2-point statistics of the peaks, as well as tomographic techniques.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We are grateful to the Institute for Theoretical Physics of the University of Zürich for its hospitality. We thank V. Springel for making public [Gadget-2]{} and for providing his B-FoF halo finder. LM and PS are supported by the Deutsche Forschungsgemeinschaft (DFG) through the grant MA 4967/1-2, through the Priority Programme 1177 ‘Galaxy Evolution’ (SCHN 342/6 and WH 6/3), and through the Transregio TR33 ‘The Dark Universe’. SH acknowledges support by the National Science Foundation (NSF) grant number AST-0807458-002.
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Estimating the error on the Fisher constraints {#A1}
==============================================
In Fisher forecasts as well as real-data analysis, the derivatives of observables with respect to cosmological parameters are usually estimated analytically. However, due to the lack of analytical predictions for peak observables, in this study we employ numerical derivatives. As discussed in §\[S4\], the latter are estimated as double-sided derivatives (), which reduces the impact of cosmic variance and the dependence on the step $\Delta p_{\alpha}$ in the cosmological parameters around the fiducial value. One natural question is: what is the impact of the uncertainties in the measured derivatives on the resulting Fisher constraints? We shall address this issue in an approximate manner, just for the unmarginalized errors.
The unmarginalized Fisher constraints are computed from . Their variance can be written as \[\_[p\_]{}\]= \_[ ]{}\^[-3]{} [Var]{}\[\_\]. Since we are interested in the impact of the uncertainties in the derivative estimates on the Fisher matrix, we ignore the uncertainties in the covariance matrix of the probes. In any case, we would not be able to reliably estimate the latter from just 128 realizations, so this approximation is quite necessary. Note that the impact of uncertainties in the covariance matrix on Fisher constraints was explored in the recent study of [@Tayloretal2012]. The variance of the Fisher-matrix elements can then be written as \[\_\] = \_[i, j=1]{}\^[n\_[B]{}]{}()\^2 {|[m]{}\_[j, p\_]{}\^2 [Var]{} \[|[m]{}\_[i, p\_]{}\]\
+ |[m]{}\_[i, p\_]{}\^2 [Var]{} \[|[m]{}\_[j, p\_]{}\] +2|[m]{}\_[i, p\_]{}|[m]{}\_[j, p\_]{}[Cov]{} \[|[m]{}\_[i, p\_]{}|[m]{}\_[j, p\_]{}\]}. \[eq:fishvar\] The rms of the variance of the derivatives represents the error bars shown in . The covariance in the above equation must be estimated separately from the data. Since we have only 64 realizations corresponding to each of the variational cosmologies, this estimate is bound to be very noisy. Whilst $\bar{\m}$ can be any of the three peak observables, we actually perform the above calculation for the correlation function alone. The derivatives of the latter have the largest error bars and they also float around the 0 line, which could yield spurious Fisher information. Here we mention that increasing the number of bins to 12 and 15 did not significantly change our parameter constraints for $\corra$. The uncertainties in the unmarginalized 1-$\sigma$ Fisher constraints for $\corra$, expressed as percentages of the values given in Table \[fisher\_er\] are for $\{\om,\,\s8,\,w,\,\ns\}$: $\{26\%,\,31\%,\,17\%,\,26\%\}$ respectively – if we ignore the noisy covariance term in ; and $\{38\%,\,43\%,\,24\%,\,36\%\}$ – if we do take into account the covariance term.
We leave to a future work a more complete treatment of the Fisher-matrix constraints from numerical measurements. We would like to analytically model the peak probes so as not to be vulnerable to numerical effects when computing their derivatives with respect to cosmological parameters.
[^1]: [email protected]
|
---
abstract: 'We present algorithms and experiments for the visualization of directed graphs that focus on displaying their reachability information. Our algorithms are based on the concepts of the path and channel decomposition as proposed in the framework presented in [@ortali2018algorithms] and focus on showing the existence of paths clearly. In this paper we customize these concepts and present experimental results that clearly show the interplay between bends, crossings and clarity. Additionally, our algorithms have direct applications to the important problem of showing and storing transitivity information of very large graphs and databases. Only a subset of the edges is drawn, thus reducing the visual complexity of the resulting drawing, and the memory requirements for storing the transitivity information. Our algorithms require almost linear time, $O(kn+m)$, where $k$ is the number of paths/channels, $n$ and $m$ is the number of vertices and edges, respectively. They produce progressively more abstract drawings of the input graph. No dummy vertices are introduced and the vertices of each path/channel are *vertically aligned*.'
author:
- Panagiotis Lionakis
- Giacomo Ortali
- 'Ioannis G. Tollis'
bibliography:
- 'diffBibBnodesConf.bib'
title: 'Adventures in Abstraction: Reachability in Hierarchical Drawings'
---
Introduction
============
The visualization of directed (sometimes acyclic) graphs has many applications in several areas of science and business. Such graphs often represent hierarchical relationships between objects in a structure (the graph). In several applications, such as graph databases and big data, the graphs are very large and the usual visualization techniques are not applicable. In their seminal paper of 1981, Sugiyama, Tagawa, and Toda [@sugiyama1981methods] proposed a four-phase framework for producing hierarchical drawings of directed graphs. This framework is known in the literature as the [“Sugiyama”]{} framework, or algorithm. Most problems involved in the optimization of various phases of the Sugiyama framework are NP-hard. In [@ortali2018algorithms] a new framework is introduced to visualize directed graphs and their hierarchies which departs from the classical four-phase framework of Sugiyama and computes readable hierarchical visualizations by [“hiding”]{} (*abstracting*) some selected edges while maintaining the complete reachability information of a graph.
In this paper we present several algorithms that follow that framework. Our algorithms reduce the visual complexity of the resulting drawings by (a) drawing the vertices of the graph in some vertical lines, and (b) by progressively *abstracting* some transitive edges thus showing only a subset of the edge set in the output drawing. The process of progressively abstracting the edges gives different visualization results, but they all have the same transitive closure as the input graph. Notice that this type of abstraction has additional applications in storing the transitive closure of huge graphs, which is a significant problem in the area of graph databases and big data [@Jagadish:1990:CTM:99935.99944; @DBLP:conf/sigmod/JinRDY12; @DBLP:conf/sigmod/SchaikM11; @DBLP:conf/edbt/VelosoCJZ14; @DBLP:journals/vldb/YildirimCZ12]. We also present experimental results that show a very interesting interplay between bends, crossings, clarity of the drawings, and the abstraction of edges.
A *path* and a *channel* are both ordered sets of vertices. In a path every vertex is connected by a direct edge to its successor, while in a channel any vertex is connected to it by a directed path (which may be a single edge). The concept of channel can be seen as a generalization of the concept of path. In the literature the channels are also called *chains* [@Jagadish:1990:CTM:99935.99944].
Figure \[teaser\] shows an example of three different hierarchical drawings: part (a) shows the drawing of a directed graph $G$ computed by Tom Sawyer Perspectives [@Tom] (a tool of Tom Sawyer Software) that follows the Sugiyama framework; part (b) shows a hierarchical drawing computed by our first variant algorithm taking $G$ as input; part (c) shows an abstracted hierarchical drawing computed by our final variant that removes all path edges and selected transitive cross edges. Notice that in part (b) the transitive edges within each vertical path are not shown. Part (c) shows a hierarchical drawing where all path edges and transitive cross edges are abstracted. The advantages of the last drawing are (i) clarity of the drawing due to the sparse representation, (ii) all path edges and transitive edges (within a path) are implied by the $x$ and $y$ coordinates, (iii) the drawn graph has the same transitive closure as $G$, (iv) it gives us a technique to store the transitive closure of $G$ in an extremely compact data structure, and (v) a path between vertices that are on different paths (of the decomposition) can be obtained by traversing one cross edge.
Even though the Sugiyama framework is very popular, and many of the (sub)problems for each phase have turned out to be NP-hard, its main limitation is the fact that the heuristic solutions and decisions that are made during previous phases (e.g., crossing reduction) will influence severely the results obtained in later phases. Nevertheless, previous decisions cannot be changed in order to obtain better results. This framework can be viewed as a horizontal decomposition of $G$ into (horizontal) layers. By contrast, the framework of [@ortali2018algorithms] and all variants presented here can be viewed as a vertical decomposition of $G$ into (vertical) paths/channels. Most problems here are [“vertically contained”]{} thus reducing their time complexity. It draws either (a) graph $G$ without the transitive [“path/channel edges”]{} or (b) a condensed form of the transitive closure of $G$. Of course, the [“missing”]{} incident (transitive) edges of a vertex can be drawn interactively on demand. An added advantage of this framework is that it allows (or it even encourages) the user to use his/her own paths as input to the algorithms. This means that paths/channels that are important for specific applications can be easily visualized by vertically aligning their nodes.
The algorithms presented in our paper are variants of the path based algorithm presented in [@ortali2018algorithms]. Namely we present seven variants (including the original one) that progressively remove edges, crossings and bends. Each variant has its own advantages and disadvantages that can exploited in various applications. Furthermore, due to its flexibility, new variants can be created based on the needs of specific applications. We also present experimental results that further demonstrate the power of edge abstraction and their impact on the number of bends, crossings, edge bundling, etc. Notice that the above variants can be easily modified to work using the concepts of channel decomposition of a DAG and of channel graph as described in [@ortali2018algorithms].
Our paper is organized as follows: the next section presents necessary knowledge, including a brief description of the basic concepts of the path based algorithm of [@ortali2018algorithms]. In Section 3 we present the variants that are based on the path based algorithm and the metrics of our experiments. Section 4 presents the experimental results and offers a comparison of the pros and cons of each variant with respect to bends, crossings, and clarity. In Section 5 we present our conclusions and interesting open problems.
Overview of the Path Based Framework {#se:overview}
====================================
The Path Based Hierarchical Drawing Framework exploits a new approach to visualize directed acyclic graphs that focus on their reachability information [@ortali2018algorithms]. This framework is orthogonal to the Sugiyama framework in the sense that it is a vertical decomposition of $G$ into (vertical) paths/channels. Most problems are [“vertically contained”]{} thus reducing their time complexity. The vertices of a graph $G$ are partitioned into paths, called a *path decomposition* and the vertices of each path are drawn *vertically aligned*. It consists of only two steps: (a) the cycle removal step (if the graph contains directed cycles) and (b) the hierarchical drawing step.
For the purposes of reachability we propose that Step (a) follows a simple approach: compute the *Strongly Connected Components (SCC)* of $G$ in linear time and cluster and collapse each SCC into a supernode. Clearly, the resulting graph will be acyclic. This approach has been used in previous papers for various applications, see for example .
Regarding Step (b), the path decomposition may be application defined, user defined or automatically computed by an algorithm. There are several algorithms that compute a path decomposition of minimum cardinality [@DBLP:journals/siamcomp/HopcroftK73; @DBLP:conf/recomb/KuosmanenPGCTM18; @DBLP:conf/stoc/Orlin13; @DBLP:journals/siamcomp/Schnorr78]. For the rest of this paper, we will assume that the path decomposition is an input to the algorithm along with $G$. We use an algorithm that computes a path based hierarchical drawing given a DAG $G=(V,E)$ and a path decomposition $S_p$ of $G$, see [@ortali2018algorithms]. Due to space limitations we describe the algorithm and relevant results in the Appendix.
A *path decomposition* of $G$ is a set of vertex-disjoint paths $S_p= \{P_1,...,P_k\}$ such that every vertex $v\in V$ belongs to exactly one of the paths of $S_p$. A path $P_i\in S_p$ is called a *decomposition path*. The *path decomposition graph* of $G$ associated with a path decomposition $S_p$ is a graph $H=(V,A)$ obtained from $G$ by removing every edge $e=(u,v)$ that connects two vertices on the same decomposition path $P_i\in S_p$ that are not consecutive in the order of $P_i$. An edge of $H$ is a *cross edge* if it is incident to two vertices belonging to two different decomposition paths, else it is a *path edge*. Graph $H$ is obtained from $G$ by removing some transitive edges between vertices in a same path. A *path based hierarchical drawing* of $G$ given $S_p$ is a hierarchical drawing of $H$ where two vertices of $V$ are placed in a same $x$-coordinate if and only if they belong to a same decomposition path $P_i\in S_p$. Algorithm PB-Draw computes a path based hierarchical drawing of $G$. Thus we can read and understand correctly any reachability relation between the vertices of $G$ by visualizing $H$, as shown in Section \[se:thandle\] of the Appendix.
Using a path decomposition with a small cardinality may improve the performance of our algorithm in terms of area, bends, number of crossings and computational time. As discussed at the beginning of this section, computing such a minimum size path decomposition is a well known problem and it provides a great advantage to this framework. Also, the use of the path decomposition concept adds flexibility to the framework, since the paths can be user defined or application specific. The visibility of such important/critical paths is extremely clear in our drawings, since they are all vertically aligned.
Variants, Metrics, and Datasets {#se:variants}
===============================
In this section we present the variants of Algorithm \[algo\] (see Appendix) that we used for our experiments and the metrics that we considered. We performed two types of experiments: (a) based on measurements over datasets with respect to the number of bends and crossings *(Variant 0 and Variant 1)* and (b) based on edge abstraction *(Variant 2, Variant 3, Variant 4, Variant 5, and Variant 6)*. All variants use edge bundling as described by Lemma \[lemma:overlap\] of Section \[se:thandle\] of the Appendix. Refer to Figure \[fig:2\]. Namely, all edges that start from vertices of a decomposition path $P$ and go into the same target vertex $v$ bend at the same point. All such edges use the same straight line segment from the bend to vertex $v$. For example, we bundle edges $(21,30)$ and $(28,30)$ by bending them at the same point and by overlapping them from this point to the target vertex, which is vertex $30$. Similarly we do the same for edges $(4,28)$ and $(20,28)$. This type of edge bundling is very useful in the sense that it reduces the total number of bends and crossings, and it reuses some portions of edges.
Variants
--------
We present now a suite of drawing techniques, our variants, that are based on Algorithm \[algo\]. Our variants can be further customized depending upon the requirements of an application or a user.
- **Variant 0:** This variant is precisely the same as our baseline, Algorithm \[algo\]. See, for example, Figure \[fig:1\].
- **Variant 1:** We denote by *jumping cross edge* an edge $e=(u,v)$ such that $|X(v)-X(u)|>1$. In this variant we place a bend on every jumping cross edge of $\Gamma$. Refer Figure \[fig:2\], where, for example, the jumping cross edge $e=(7,10)$ has a bend.
- **Variant 2:** For every vertex $u$ we abstract edge *$e_1=(u,v)$* if there exists an edge *$e_2=(u,v')$* such that *$v'$* and $v$ are in the same decomposition path $P$ and $v'$ precedes $v$ in the order of $P$ (edges have common source node). Refer to Figure \[fig:3\], where, for example, $e_1=(2,10)$ and $e_2=(2,6)$.
- **Variant 3:** For every vertex $v$ we abstract the edge *$e_1=(u,v)$* if there exists an edge *$e_2=(u',v)$* such that *$u'$* and $u$ are in the same decomposition path $P$ and $u$ precedes $u'$ in the order of $P$ (edges have common target node). Refer to Figure \[fig:4\], where, for example, $e_1=(21,30)$ and $e_2=(28,30)$.
- **Variant 4:** Apply the removal of Variant 2 and Variant 3. Refer to Figure \[fig:5\], where we removed both $(2,10)$ and $(21,30)$.
[“Final Abstraction”]{} {#se:final_ab}
-----------------------
An important aspect of our work is the preservation of the mental map that can be expressed by the reachability information of a DAG. Since the nodes in each path of the decomposition are vertically aligned, drawing the path edges does not add much information to the mental map of the user. Hence their removal from the drawing will reduce the number of crossings and the number of edges drawn. Toward to that, we propose an extended abstraction drawing model generated as a combination of the aforementioned variants as shown in Figure \[fig:6\] and \[fig:7\].
The main purpose of this abstraction is that we want to retain the visual reachability while minimizing the visual complexity of the drawing. For instance, in our variants as stated in previous sections, paths can be either application based e.g., critical paths or user defined. Consider Variant 0 the path edges can be removed from the drawing since their existence is implied by the fact that they share the same $x$-coordinate. We refer to this variant as *Variant 5*. Please notice that we do not remove any number of [“random”]{} edges in order to create less complex drawings of the same graph but rather we use the unique characteristics of the drawing which may also be application depended. We can further reduce the total number of edges drawn, and as a result the number of crossings, by using this abstraction in combination with *Variant 4* to create a more abstracted drawing, called *Variant 6*. Therefore, we define the following two variants:
- **Variant 5**: These drawings are obtained from the drawings of Variant 0 by removing all path edges (see Figure \[fig:6\]).
- **Variant 6**: These drawings are obtained from the drawings of Variant 4 by removing all path edges (see Figure \[fig:7\]).
\
\
\
The following theorem is proved in Section \[se:time\_variants\] of the Appendix.
\[th:1,2,3,4,5,6\] Let $G$ be a DAG with $n$ vertices and $m$ edges, let $S_p$ be a path decomposition of $G$ and let $k$ be the cardinality of $S_p$. It is possible to compute the drawings $\Gamma_1$ according to Variant 1 in $O(n+m)$ time and the drawings $\Gamma_2$, $\Gamma_3$, $\Gamma_4$, $\Gamma_5$, and $\Gamma_6$ according to Variant 2, Variant 3, Variant 4, Variant 5, and Variant 6, respectively, in $O(mk)$ time.
Metrics and Datasets
--------------------
The set of DAGs that was used in the experiments contains five Datasets (DAGs) which were produced in a controlled fashion in order to have a number of nodes and edges, as a factor of the density of the graph. DAG 1 is one of the DAGs that was used to illustrate Algorithm \[algo\] in [@ortali2018algorithms]. Table \[datasets\] in the Appendix gives a summary for each DAG.
### Metrics for the Experimental Results.
Our analysis aims to evaluate the performance of the various variants of the basic algorithm for each of the aforementioned DAGs. To this end, we use the following:
$\bullet$ Number of edges drawn in the drawing.
:
$\bullet$ Number of cross edges drawn in the drawing.
:
$\bullet$ Number of bends.
:
$\bullet$ Number of crossings.
:
$\bullet$ Execution time:
: is the average execution time for producing each drawing.
\[table\_times\]
Analysis of the Performance
===========================
In this section we analyze the results of the experiments presented in this paper. We remark that the variants and experiments are described for the path based framework, but they can be used with the channel based framework as well.
Table \[executiontime\] shows the performance, *Execution time* (ms), of the Java implementation of our suite of drawing solutions as produced by Tom Sawyer Software TS Perspectives [@Tom]. We do not report the execution times of Variant 5 and Variant 6 since they are similar to the execution times of Variant 0 and Variant 4, respectively. We observe that our variants produce hierarchical drawings suitable for large datasets since the reachability information can be seen with little effort while the execution time to produce these results is rather small.
The first figure reflects the *the number of edges drawn* for each of the variants over the five DAGs illustrated in Figure \[m1vd\]. Similar to that, Figures \[m2vd\], \[m3vd\], and \[m4vd\] show the results regarding the *number of cross edges drawn, bends and crossings* respectively for each of the variants.
![Results on *number of cross edges drawn* for each variant over all DAGs.[]{data-label="m1vd"}](metric1VersionsPerDataset.png){width="\textwidth"}
![Results on *number of edges drawn* for each variant over all DAGs.[]{data-label="m2vd"}](metric2VersionsPerDataset.png){width="\textwidth" height="6cm"}
![Results on *number of bends* for each variant over all DAGs.[]{data-label="m3vd"}](metric3VersionsPerDataset.png){width="\textwidth" height="6cm"}
First we discuss the number of edges drawn by our variants. By construction Variant 0 and Variant 1 draw exactly the same set of edges as it is evidenced by Figures \[m1vd\] and \[m2vd\]. The same figures show that Variant 2 and Variant 3 are similar in the number of edges they draw. Clearly, the number of edges drawn by Variant 4 is significantly lower than the number of edges drawn by the other variants. This effect is emphasized in Figure \[m1vd\], where the number of cross edges drawn by Variant 4 for DAG 5 is about one sixth of the number of cross edges drawn by Variant 0 and Variant 1. Finally, we focus on Variant 5 and Variant 6. The sets $E_5$ and $E_6$ of the edges that Variant 5 and Variant 6 draw is a subset of the sets of edges $E_0$ and $E_4$ that Variant 0 and Variant 4 draw, respectively. The cardinality of $E_5$ and $E_6$ is much smaller than the cardinality of $E_0$ and $E_4$ if most of the edges drawn by Variant 0 and Variant 4 are path edges, as shown in Figure \[m2vd\] for DAG 2. Variant 5 and Variant 6 by construction draw the same set of cross edges of respectively Variant 0 and Variant 4.
As can be seen in Figure \[m3vd\] the drawings computed by Variant 2, Variant 3, and Variant 4 have very few bends on the average. For example, DAG 5 in Variant 3 has 270 edges and the corresponding drawing has only 38 bends, i.e., we have 0.14 bends per edge. On the other hand, the drawing computed by Variant 1 is less efficient in placing bends. Refer again to DAG 5: in Variant 1 this DAG has 397 edges and the corresponding drawing has 88 bends, i.e, we have 0.22 bends per edge. The number of bends in drawings computed by Variant 5 and Variant 6 and respectively Variant 0 and Variant 4 is the same, since the path edges are drawn straight line in all our variants.
The number of crossings is influenced heavily by the number of edges drawn and the extent of edge bundling. Figure \[m4vd\] shows that the performance of Variant 1 is slightly better than that of Variant 0. This can be explained by the fact that in Variant 1 there are more bundles of edges and this naturally decreases the number of crossings. The other variants all have much better performance than Variant 0 and Variant 1 because the corresponding drawings contain significantly fewer edges. Figure \[m4vd\] shows that the number of crossings is almost the same in the drawings of Variant 5 and Variant 6 and Variant 0 and Variant 4, respectively. This result is very important, since it is an evidence of the fact that path edges participate in a few crossings and, therefore, the decomposition paths can be visualized very clearly in our drawings.
Conclusions and Open Problems
=============================
We presented a set of variant algorithms that attempt to draw DAGs hierarchically with few bends and crossings, and by abstracting edges in order to improve the clarity of the drawings.
Our study assumes that the path decomposition is given as part of the input, or a minimum size decomposition is computed by one of the known algorithms. However, it is interesting to study the problem of computing a path decomposition and placement of the paths of $G$ which implies the minimum number of jumping cross edges in our drawings. The use of such a decomposition and placement would considerably reduce the number of edges drawn, bends, and crossings in our drawings. Another open problem is the development and implementation of some compaction strategies, which would improve the readability of our drawings and reduce their height. Finally, it would be important to comprehend human understanding issues related to the removal of some transitive edges and increasing reachability comprehension.
|
---
abstract: 'We analyse the algebras generated by free component quantum fields together with the susy generators $Q,\bar Q$. Restricting to hermitian fields we first construct the scalar field algebra from which various scalar superfields can be obtained by exponentiation. Then we study the vector algebra and use it to construct the vector superfield. Surprisingly enough, the result is totally different from the vector multiplet in the literature. It contains two hermitian four-vector components instead of one and a spin-3/2 field similar to the gravitino in supergravity.'
author:
- |
Florin Constantinescu\
Fachbereich Mathematik,\
Johann Wolfgang Goethe Universität Frankfurt,\
Robert-Mayer-Str. 10, D-60054 Frankfurt am Main, Germany
- |
Markus Gut and Günter Scharf\
Institut für Theoretische Physik,\
Universität Zürich,\
Winterthurerstr. 190 , CH-8057 Zürich, Switzerland
title: QUANTIZED HERMITIAN SUPERFIELDS
---
-10mm-5mm \#1\#2
d[[def=]{}]{} \#1\#2 œ
Introduction
============
In this paper we continue our study of quantized free superfields started in \[1\]. In paper \[1\] we have constructed the quantized chiral superfield and we have found that it has the same component expansion as its classical counterpart \[2-5\]. Surprisingly enough, if we go on to more complicated superfields, this is no longer the case !
To understand this interesting fact we can give the following explanations. First of all the infinitesimal susy transformation of a classical superfield $\Phi(x,\theta,\bar\theta)$ is defined by applying the operator $(\xi Q+\bar\xi\bar Q)$ where $Q,\bar Q$ are represented by the usual superspace differential operators. On the other hand, the corresponding transformation of the quantum fields is given by a commutator $$\delta\Phi=[\xi Q+\bar\xi\bar Q,\Phi]\eqno(1.1)$$ where now $Q,\bar Q$ are operators in Fock space. Secondly, the components of the quantum superfield satisfy (anti-)commutation relations. There is an interesting interplay between these relations, the free field equations and the supersymmetric algebra, not existing in the classical case. In fact, the construction of these supersymmetric free field algebras is the main work to be done. First this is carried out for the hermitian scalar superfield in the next section and then for the vector superfield in sect.3. The superfields themselves are obtained from the algebra simply by exponentiation. The vector algebra will be used to construct the vector superfield. The latter comes out totally different from the vector field in the literature \[2-3\]. For example, it contains a spin-3/2 component similar to the gravitino in supergravity and two hermitian four-vector components instead of one. We are obviously driven to another representation of the supersymmetric algebra, when we consider quantized free fields.
The quantized hermitian scalar superfield
=========================================
We want to represent the supersymmetric algebra $$\{Q_a,\bar Q_{\bar b}\}=2\sigma^\mu_{a\bar b}P_\mu=-2i\sigma^\mu_{a\bar
b}\d_\mu,\eqno(2.1)$$ $$\{Q_a, Q_b\}=0=\{\bar Q_a,\bar Q_{\bar b}\}=[Q_a, P_\mu]\eqno(2.2)$$ by operators in Fock space. $Q_a$ and $\bar Q_{\bar b}$ transform according to the $(\eh, 0)$ and $(0, \eh)$ representations of the proper Lorentz group, respectively. As usual one rewrites (2.1) as a Lie-algebra commutator $$[\theta^a Q_a,\bar\theta^{\bar b}\bar Q_{\bar b}]=2\theta\sigma
\bar\theta P=-[\theta Q,\bar\theta\bar Q]\eqno(2.3)$$ by introducing anti-commuting C-numbers $\theta^a, \bar\theta^{\bar b}$. It is our aim to construct a superfield $S$ with the transformation law $$\delta S={\d\over\d\xi}S_\xi\Bigl\vert_{\xi=1}=i[\theta Q+
\bar\theta\bar Q,S]\eqno(2.4)$$ under infinitesimal supersymmetric transformations. Taking as initial value the scalar component field $C(x)$, the solution is given by the finite transformation $$S_\xi(x,\theta,\bar\theta)=e^{i\xi(\theta Q+\bar\theta\bar Q)}C(x)
e^{-i\xi(\theta Q+\bar\theta\bar Q)}.\eqno(2.5)$$ The factor $i$ has been inserted to get a hermitian field $$S_1(x,\theta,\bar\theta)^+=S_1(x,\theta,\bar\theta)\eqno(2.6)$$ by assuming $C(x)$ to be hermitian. Then $C(x)$ must be quantized as a neutral scalar field of mass $m$ $$[C(x), C(x')]=-iD(x-x'),\eqno(2.7)$$ where $D(x)$ is the causal Jordan-Pauli distribution for mass $m$.
We want to calculate (2.5) by means of the Lie series. For the first order terms we need the commutators $$[Q_a,C(x)]=\chi_a(x),\quad [\bar Q_{\bar b},C(x)]=-\bar\chi_{\bar
b}(x), \eqno(2.8)$$ where $\chi$ and $\bar\chi$ are quantized Majorana fields. The second commutation relation follows from the first by taking the adjoint, because (see \[1\] eq.(2.21)) $$Q_a^+=\bar Q_{\bar a}\eqno(2.9)$$ and similarly for $\chi_a$. Next we need the anticommutator $$\{ Q_a,\chi_b(x)\}=\{ Q_a,[Q_b,C]\}=\{ Q_b,[C,Q_a]\}-[C,\{ Q_a,Q_b\}]
=-\{Q_b,\chi_a(x)\},\eqno(2.10)$$ where the Jacobi identity has been used. Due to the antisymmetry in $a,b$, the anticommutator must be of the form $$\{ Q_a,\chi_b(x)\}=-im\eps_{ab}F(x),\eqno(2.11)$$ where $F(x)$ is a scalar field and the mass factor has been introduced for dimensional reasons. Since the $\eps$-tensor is real, the adjoint relation is $$\{\bar Q_{\bar a},\bar\chi_{\bar b}(x)\}=im\eps_{\bar a\bar b}F(x)^+.
\eqno(2.12)$$
The Majorana field $\chi_a$ must be quantized according to $$\{\chi_a(x),\chi_b(x')\}=im\eps_{ab}D(x-x'),$$ $$\{\chi_a(x),\bar\chi_{\bar b}(x')\}=-\sigma^\mu_{a\bar b}\d_\mu
D(x-x').\eqno(2.13)$$ On the other hand, again by the Jacobi identity the first anticommutator is equal to $$=\{\chi_a(x),[Q_b,C(x')]\}=\{Q_b,[C(x'),\chi_a(x)]\}-[C,\{\chi_a,
Q_b\}]=$$ $$=\{Q_b,[C(x'),\chi_a(x)]\}+im\eps_{ba}[C(x'),F(x)].$$ The commutator in the first term must be assumed to be 0, hence we find $$[F(x),C(x')]=D(x-x').$$ We can therefore write $$F(x)=M(x)+iC(x)\eqno(2.14)$$ with $$[M(x),C(x')]=0.\eqno(2.15)$$
We specialize (2.11) to $a=2, b=1$. Using $\eps_{21}=1$ we get $$\{Q_2,\chi_1\}=-imF.\eqno(2.16)$$ This implies$$[Q_a,F(x)]={i\over m}[Q_a,\{Q_2,\chi_1(x)\}]=$$ $$=-{i\over m}([Q_2,\{\chi_1,Q_a\}]+[\chi_1,\{Q_a,Q_2\}])=-\eps_{a1}
[Q_2,F(x)].\eqno(2.17)$$ For $a=1$ this gives $$[Q_1,F(x)]=0$$ and for $a=2$ $$[Q_2,F(x)]=-[Q_2,F(x)]=0.\eqno(2.18)$$ The adjoint relations are $$[\bar Q_{\bar a},F^+(x)]=0.\eqno(2.19)$$
Next we consider $$[F(x),F(x')]={i\over m}[F(x),\{Q_2,\chi_1(x')\}]=$$ $$=-{i\over m}\{Q_2,[\chi_1(x'),F(x)]\}=0=[M(x),M(x')]-[C(x),C(x')].$$ Since the first commutator is $-iD(x-x')$ it follows $$[M(x),M(x')]=-iD(x-x'),\eqno(2.20)$$ so that $M(x)$ is also a hermitian scalar field. This can also be derived from (2.13). To determine the other (anti)commutators we raise the second spinor index in (2.12) with the $\eps$-tensor $$\{\bar Q_{\bar a},\bar\chi^{\bar c}\}=-im\delta_{\bar a}^{\bar c}F^+.
\eqno(2.21)$$ Taking the trace $\bar a=\bar c=1,2$, we have $$F^+={i\over 2m}\{\bar Q_{\bar a},\bar\chi^{\bar a}\}.\eqno(2.22)$$ Let us apply the Dirac operator $i\sigma^\mu_{a\bar c}\d_\mu$ to (2.21) and use the Dirac equation $$\{\bar Q_{\bar a},i\sigma^\mu_{a\bar c}\d_\mu\bar\chi^{\bar c}\}=m\{\bar
Q_{\bar a},\chi_a\}=m\sigma^\mu_{a\bar a}\d_\mu F^+.$$ This leads to $$\{\bar Q_{\bar a},\chi_b\}=\sigma^\mu_{b\bar a}\d_\mu F^+\eqno(2.23)$$ and the adjoint relation $$\{Q_a,\bar\chi_{\bar b}\}=\sigma^\mu_{a\bar b}\d_\mu F.\eqno(2.24)$$
By means of (2.22) and (2.24) we calculate $$[Q_a,F^+(x)]={i\over 2m}[Q_a,\{\bar Q_{\bar a},\bar\chi^{\bar a}\}]=$$ $$=-{i\over 2m}\Bigl([\bar Q_{\bar a},\{\bar\chi^{\bar a},Q_a\}]+
[\bar\chi^{\bar a},\{Q_a,\bar Q_{\bar a}\}]\Bigl)$$ $$={i\over 2m}\sigma^\mu_{a\bar b}[\bar Q^{\bar b},\d_\mu F]+{1\over m}
\sigma^\mu_{a\bar a}\d_\mu\bar\chi^{\bar a}.\eqno(2.25)$$ The adjoint equation is $$[\bar Q_{\bar b},F(x)]=-{i\over 2m}\sigma^\mu_{b\bar b}[Q^b,\d_\mu F^+]-
{1\over m}\sigma^\mu_{a\bar b}\d_\mu\chi^a.\eqno(2.26)$$ Substituting this into (2.25) and using the relations $$\hat\sigma^{\mu\bar aa}=\eps^{\bar a\bar b}\eps^{ab}\sigma^\mu_{b\bar
b}\eqno(2.27)$$ $$(\hat\sigma^\nu\sigma^\mu+\hat\sigma^\mu\sigma^\nu)^{\bar a}_{\bar b}
=2\eta^{\nu\mu}\delta^{\bar a}_{\bar b},\eqno(2.28)$$ we conclude $${3\over 4}[\bar Q_{\bar b},F]=-{i\over 2}\bar\chi_{\bar b}-{1\over m}
\sigma^\mu_{a\bar b}\d_\mu\chi^a.\eqno(2.29)$$ In the second term on the r.h.s. we can use the (adjoint) Dirac equation $$\sigma^\mu_{a\bar b}\d_\mu\chi^a=im\bar\chi_{\bar b}.\eqno(2.30)$$ This finally gives $$[\bar Q_{\bar b},F(x)]=-2i\bar\chi_{\bar b}(x),\eqno(2.31)$$ and the adjoint relation $$[Q_b,F^+]=-2i\chi_b.\eqno(2.32)$$
Now we are ready to evaluate (2.5) because all multiple commutators on the r.h.s. are known. The result is $$S_1(x,\theta,\bar\theta)=C(x)+i(\theta\chi-\bar\theta\bar\chi)+$$ $$-i{m\over 2}(\theta\theta F-\bar\theta\bar\theta
F^+)+\theta\sigma^\mu\bar\theta\d_\mu M+$$ $$+i{m\over 3}(\theta\theta\bar\theta\bar\chi-\theta\chi\bar\theta
\bar\theta)+{1\over 3}\theta\sigma^\mu\bar\theta(\bar\theta\d_\mu
\bar\chi-\theta\d_\mu\chi)+$$ $$+{m^2\over 4}\theta\theta\bar\theta\bar\theta C(x).\eqno(2.33)$$ The third order terms can be rewritten by means of the Dirac equation and the identity $$\bar\theta^{\bar a}\bar\theta^{\bar b}=\eh\eps^{\bar a\bar
b}\bar\theta\bar\theta\eqno(2.34)$$ in the form $$-i{m\over 2}(\bar\theta\bar\theta\theta\chi-\theta\theta\bar\theta
\bar\chi).\eqno(2.35)$$
Another hermitian scalar superfield $S_0$ is obtained as the sum of the chiral plus the anti-chiral superfield $\Phi+\Phi^+$ constructed in \[1\]. This field has a component expansion of the same form as $S_1$ (2.33), but is actually different as can be seen by comparing the coefficients. The difference is not surprising if one realizes the different behaviour under infinitesimal susy-transformation. Indeed, from $$\Phi_\xi=e^{\xi(\theta Q+\bar\theta\bar Q)}A
e^{-\xi(\theta Q+\bar\theta\bar Q)}\eqno(2.36)$$ we find $$\delta_\xi(\Phi+\Phi^+)={\d\over\d\xi}(\Phi_\xi+\Phi^+_\xi)\Bigl\vert
_{\xi=1}=[\theta Q+\bar\theta\bar Q,\Phi]-[\theta Q+\bar\theta\bar Q,\Phi^+]
,\eqno(2.37)$$ in contrast to (2.4). A third scalar field $S_2$ can be constructed as $S_1$ (2.5) but starting with the spinor component $$S_2(x,\theta,\bar\theta)=e^{i(\theta Q+\bar\theta\bar Q)}(i\theta\chi(x)
-i\bar\theta\bar\chi(x))e^{-i(\theta Q+\bar\theta\bar Q)}.\eqno(2.38)$$ All commutators are the same as before so that we immediately find $$S_2=i(\theta\chi-\bar\theta\bar\chi)-im(\theta\theta F-\bar\theta
\bar\theta F^+)
+(\theta\sigma^\mu\bar\theta)(\d_\mu F+\d_\mu F^+)+$$ $$+{3\over 2}im(\theta\chi\bar\theta\bar\theta-\theta\theta\bar\theta
\bar\chi)+{i\over 2}m^2\theta\theta\bar\theta\bar\theta(F^+-F).\eqno(2.39)$$ This scalar field is the most important one because we shall need it when we consider gauge transformations.
Still this is not the whole story because supersymmetry gives further constraints on the component fields. To see this we take the commutator of (2.24) with $\bar Q_{\bar c}$ and use (2.31) $$\sigma^\mu_{a\bar b}[\bar Q_{\bar c},\d_\mu F]=-2i\sigma^\mu_{a\bar b}
\d_\mu\bar\chi_{\bar c}=[\bar Q_{\bar c},\{ Q_a,\bar\chi_{\bar b}\}]=$$ $$=-[Q_a,\{\bar\chi_{\bar b},\bar Q_{\bar c}\}]-[\bar\chi_{\bar b},
\{\bar Q_{\bar c},Q_a\}].$$ Using the relations (2.12),(2.1) and (2.32) the last two commutators can be evaluated. This leads to the following relation $$\sigma^\mu_{a\bar b}\d_\mu\bar\chi_{\bar c}-
\sigma^\mu_{a\bar c}\d_\mu\bar\chi_{\bar b}=-im\eps_{\bar c\bar b}
\chi_a.\eqno(2.40)$$ Here we must only check the nontrivial case $\bar b=1, \bar c=2,$ say. Then the relation is satisfied due to the Dirac equation. The adjoint relation is $$\sigma^\mu_{b\bar a}\d_\mu\chi_c-
\sigma^\mu_{c\bar a}\d_\mu\chi_b=im\eps_{cb}
\bar\chi_{\bar a}.\eqno(2.41)$$
Quantized vector superfield
===========================
We have learnt in the last section that we do not obtain a vector component if we start from a scalar field. Therefore, we now start from a Majorana field $\lambda_a(x)$ as in $S_2$ (2.36), but we do not assume that $\lambda(x)$ is obtained from a scalar field like (2.8). We want to construct the hermitian superfield $$V(x,\theta,\bar\theta)=e^{i(\theta Q+\bar\theta\bar Q)}(i\theta^a
\lambda_a(x)+{\rm h.c.})e^{-i(\theta Q+\bar\theta\bar Q)}.\eqno(3.1)$$ If we omit the $\theta^a$ starting from $i\lambda_a$, we obtain a spinor field $W_a(x,\theta,\bar\theta)$. To determine the necessary commutators we start with the mixed anticommutator $\{Q_a,\bar\lambda_{\bar b}(x)\}$. It must be proportional to $\sigma^\mu_{a\bar b}$ because there is a one-to-one correspondence between spinors of type $(\eh, \eh)$ and four-vectors. Therefore, we introduce two self-conjugate vector fields $v, w$ by requiring $$\{Q_a,\bar\lambda_{\bar b}(x)\}=-m\sigma^\mu_{a\bar b}(v_\mu
+iw_\mu).\eqno(3.2)$$ The adjoint relation is $$\{\bar Q_{\bar a},\lambda_b(x)\}=-m\sigma^\mu_{b\bar a}(v_\mu
-iw_\mu).\eqno(3.3)$$ As in the last section, two other relations follow by means of the Dirac equation. We multiply (3.2) by $\eps^{\bar c\bar b}$ and apply the Dirac operator $i\sigma^\nu_{b\bar c}\d_\nu$. This gives $$\{Q_a,i\sigma^\nu_{b\bar c}\d_\nu\bar\lambda^{\bar c}\}=m\{Q_a,\lambda
_b\}=-im\sigma^\mu_{a\bar b}\eps^{\bar c\bar b}\sigma^\nu_{b\bar c}
\d_\nu (v_\mu+iw_\mu)=$$ $$=-im\sigma^\nu_{b\bar c}\hat\sigma^{\mu\bar c c}\eps_{ac}\d_\nu
(v_\mu+iw_\mu),$$ or $$\{Q_a,\lambda_b\}=-i(\sigma^\nu\hat\sigma^\mu)_{ba}\d_\nu
(v_\mu+iw_\mu).\eqno(3.4)$$ The adjoint relation reads $$\{\bar Q_{\bar a},\bar\lambda_{\bar b}\}=-i(\hat\sigma^\nu\sigma^\mu)
_{\bar b\bar a}\d_\nu(v_\mu-iw_\mu).$$
Next we have to determine the commutators of $v_\mu$ and $w_\mu$. Taking the commutator of (3.2) with $Q_b$ we get $$[Q_b,\{Q_a,\bar\lambda_{\bar b}\}]=-m\sigma^\mu_{a\bar b}
[Q_b,v_\mu+iw_\mu].\eqno(3.5)$$ Since the l.h.s. is antisymmetric in $a,b$ due to the Jacobi identity, we conclude $$m\sigma^\mu_{a\bar b}[Q_b,v_\mu+iw_\mu]=-{1\over 2}\eps_{ab}\bar\Lambda
_{\bar b}\eqno(3.6)$$ where $$\bar\Lambda_{\bar b}=[Q_c,\{Q^c,\bar\lambda_{\bar b}\}].\eqno(3.7)$$ It is easy to see that $\{Q_a,\bar\Lambda_{\bar b}\}=0$ follows and also the adjoint relation $\{\bar Q_{\bar a},\Lambda_{b}\}=0$. Since $\Lambda$ satisfies the Dirac equation as $\lambda$, we also have $\{Q_a,\Lambda_b\}=0$. Using the Jacobi identity again $$[\bar Q_{\bar a},\{Q_a,\Lambda_b\}]=0=-[Q_a,\{\Lambda_b,\bar Q_
{\bar a}\}]-[\Lambda_b,\{\bar Q_{\bar a},Q_a\}]=-2i\sigma^\mu
_{a\bar a}\d_\mu\Lambda_b,\eqno(3.8)$$ we finally conclude $\Lambda=0$. Hence, by (3.6) $$[Q_b,v_\mu+iw_\mu]=0=[\bar Q_{\bar b},v_\mu-iw_\mu],\eqno(3.9)$$ or $$[Q_b,v_\mu]=-i[Q_b,w_\mu],\quad [\bar Q_{\bar b},v_\mu]=
i[\bar Q_{\bar b},w_\mu].\eqno(3.10)$$ This corresponds to the result (2.18-19) in the scalar case.
Let us now consider the commutators $$[Q_a, v^\mu]=-i[Q_a, w^\mu]\=d f^\mu_a,\eqno(3.11)$$ $$[\bar Q_{\bar a}, v^\mu]=i[\bar Q_{\bar a}, w^\mu]\=d -\bar f^\mu
_{\bar a}.\eqno(3.12)$$ From (3.3) we get $$[Q_c,\{\bar Q_{\bar a},\lambda_b(x)\}]=
-m\sigma^\mu_{b\bar a}[Q_c, v_\mu(x)-iw_\mu(x)]=-2m\sigma^\mu_{b\bar a}
f_{\mu c}.$$ By the Jacobi identity this is equal to $$=-[\bar Q_{\bar a},\{\lambda_b,Q_c\}]-[\lambda_b,\{ Q_c,\bar Q_{\bar
a}\}].$$ Using (3.4) and (2.1) we obtain $$=i(\sigma^\nu\hat\sigma^\mu)_{bc}[\bar Q_{\bar a},\d_\nu
(v_\mu+iw_\mu)]-2i\sigma^\mu_{c\bar a}\d_\mu\lambda_b.$$ This gives an inhomogeneous linear field equation $$-i(\sigma^\nu\hat\sigma^\mu)_{bc}\d_\nu\bar f_{\mu\bar a}+
m\sigma^\mu_{b\bar a}f_{\mu c}=i\sigma^\mu_{c\bar a}\d_\mu\lambda_b.
\eqno(3.13)$$ The adjoint equation reads $$-i(\hat\sigma^\nu\sigma^\mu)_{\bar b\bar c}\d_\nu f_{\mu a}+
m\sigma^\mu_{a\bar b}\bar f_{\mu\bar c}=-i\sigma^\mu_{a\bar c}\d_\mu
\bar\lambda_{\bar b}.\eqno(3.14)$$
With the same technique we treat $$\{Q_a,[\bar Q_{\bar b}, v^\mu+iw^\mu]\}=-2\{Q_a,\bar f^\mu_{\bar b}\}$$ $$=-[v^\mu+iw^\mu,\{Q_a,\bar Q_{\bar b}\}]+\{\bar Q_{\bar b},[v^\mu
+iw^\mu, Q_a]\}$$ $$=-2i\sigma^\nu_{a\bar b}\d_\nu (v^\mu +iw^\mu ).$$ This gives the anticommutator $$\{Q_a,\bar f_{\bar b}^\mu\}=i\sigma^\nu_{a\bar b}\d_\nu
(v^\mu+iw^\mu),\eqno(3.15)$$ and the adjoint relation $$\{\bar Q_{\bar a},f_b^\mu\}=-i\sigma^\nu_{b\bar a}\d_\nu
(v^\mu-iw^\mu).$$
To determine the anticommutator of $f$ we use the differential equation (3.13). We multiply it by $\hat\sigma^{\ro\bar a a}$: $$-i\hat\sigma^{\ro\bar a a}(\sigma^\nu\hat\sigma^\mu)_{bc}\d_\nu
\bar f_{\mu\bar a}+m(\sigma^\mu\hat\sigma^\ro)_b^{\, a}f_{\mu c}=
i(\sigma^\mu\hat\sigma^\ro)_c^{\, a}\d_\mu\lambda_b.$$ We put $a=b$ and sum over $b$. Using the trace $\tr (\sigma^\mu
\hat\sigma^\ro)=2\eta^{\mu\ro}$, we obtain $$-i(\hat\sigma^\ro\sigma^\nu\hat\sigma^\mu)^{\bar a}_{\,c}
\d_\nu\bar f_{\mu\bar a}+2m f^\ro_c=i(\sigma^\mu\hat\sigma^\ro)_c^{\, b}
\d_\mu\lambda_b.\eqno(3.16)$$ From (3.14) we find in the same way $$-i(\sigma^\ro\hat\sigma^\nu\sigma^\mu)_{a\bar c}
\d_\nu f_\mu^a+2m\bar f^\ro_{\bar c}=-i(\hat\sigma^\ro\sigma^\mu)^
{\bar b}_{\,\bar c}\d_\mu\bar\lambda_{\bar b}.\eqno(3.17)$$ The two equations (3.16-17) are similar to the Dirac equation, but contain three $\sigma$-matrices instead of one. For brevity we will call the system (3.16-17) the $3\sigma$-equations.
Now we operate with the $3\sigma$-operator $-i(\hat\sigma^\ro\sigma^\alpha
\hat\sigma^\mu)^{\bar b}_{\,c}\d_\alpha$ on (3.15) and substitute (3.16): $$-2m\{Q_a, f^\ro_c\}+i(\sigma^\mu\hat\sigma^\ro)_c^{\,b}\{Q_a,\d_\mu
\lambda_b\}=$$ $$=\sigma^\nu_{a\bar b}(\hat\sigma^\ro\sigma^\alpha\hat\sigma^\mu)^{\bar
b}_{\,c} \d_\alpha\d_\nu(v_\mu+iw_\mu).$$ Using (3.4) we get $$-2m\{Q_a,f^\ro_c\}=-(\sigma^\mu\hat\sigma^\ro)_c^{\,b}
(\sigma^\nu\hat\sigma^\alpha)_{ba}\d_\mu\d_\nu(v_\alpha+iw_\alpha)$$ $$+(\sigma^\nu\hat\sigma^\ro\sigma^\alpha\hat\sigma^\mu)_{ac}\d_\nu
\d_\alpha(v_\mu+iw_\mu).\eqno(3.18)$$ The result is antisymmetric in $a, c$ $$2m\{Q_a,f^\ro_c\}=\Bigl[(\sigma^\mu\hat\sigma^\ro\sigma^\nu
\hat\sigma^\alpha)_{ca}-$$ $$-(\sigma^\mu\hat\sigma^\ro\sigma^\nu\hat\sigma^\alpha)_{ac}
\Bigl]\d_\mu\d_\nu(v_\alpha+iw_\alpha).\eqno(3.19)$$ This antisymmetry is essential for consistency of the algebra because it follows from $\{Q_a, [Q_c, v^\mu]\}$ by means of the Jacobi identity.
It is not hard to simplify the product of four $\sigma$-matrices. Only the antisymmetric terms $\sim\eps_{ac}$ survive so that we finally obtain $$m\{Q_a,f^\ro_c\}=\eps_{ac}\Bigl[-\sq (v^\ro+iw^\ro)+2\d^\ro\d_\mu
(v^\mu+iw^\mu)\Bigl].\eqno(3.20)$$ It is clear that the second derivatives on the r.h.s. must be simplified by means of the wave equations for the vector fields. As in quantum gauge theories \[7\] we assume the massive vector fields $v, w$ to satisfy the Klein-Gordon equation $$\sq v_\nu=-m^2v_\nu,\eqno(3.21)$$ and the same is assumed for $w$. Then we arrive at the result $$\{Q_a, f^\mu_b\}=\eps_{ab}\Bigl[m(v^\mu+iw^\mu)+{2\over m}\d^\mu
\d_\nu(v^\nu+iw^\nu)\Bigl].\eqno(3.22)$$
To exhaust all consequences of supersymmetry we have to analyse the commutator $$[Q_c,\{\bar Q_{\bar a},f^\mu_b\}]=-2i\sigma^\nu_{b\bar a}\d_\nu
f^\mu_c.\eqno(3.23)$$ Using (3.22), (2.1), and the Jacobi identity we find another equation for $f$ $$i(\sigma^\nu_{c\bar a}\d_\nu f^\mu_b-\sigma^\nu_{b\bar a}\d_\nu
f^\mu_c)=
m\eps_{cb}\bar f^\mu_{\bar a}+{2\over m}\eps_{cb}\d^\mu\d_\nu
\bar f^\nu_{\bar a}.\eqno(3.24)$$ This equation has a similar form as (2.41). Therefore, we simplify it in the same way as we have treated (2.40). Due to the antisymmetry in $b,
c$, it is sufficient to consider the case $b=1, c=2$. Then the equation assumes the form $$i(\sigma^\nu_{2\bar a}\d_\nu f^\mu_1-\sigma^\nu_{1\bar a}\d_\nu f^\mu_2
)=m\bar f^\mu_{\bar a}+{2\over m}\d^\mu\d_\nu\bar f^\nu_{\bar a}.
\eqno(3.25)$$ This is a modified Dirac equation $$-i\sigma^\nu_{a\bar a}\d_\nu f^{\mu a}=m\bar f^\mu_{\bar a}+
{2\over m}\d^\mu\d_\nu\bar f^\nu_{\bar a}.\eqno(3.26)$$ But due to the second derivatives it is not a true equation of motion as the $3\sigma$-equations. The adjoint equation reads $$i\sigma^\nu_{a\bar a}\d_\nu\bar f^{\mu\bar a}=m f^\mu_{a}+
{2\over m}\d^\mu\d_\nu f^\nu_{a}.\eqno(3.27)$$
The next step is the decoupling of the fields. For this purpose we set $$f_a^\mu=g_a^\mu+\alpha\d^\mu\lambda_a+\beta\sigma^\mu_{a\bar b}\bar
\lambda^{\bar b},\eqno(3.28)$$ $$\bar f_{\bar a}^\mu=\bar g_{\bar a}^\mu+\alpha^*\d^\mu\bar\lambda_
{\bar a}+\beta^*\sigma^\mu_{b\bar a}\lambda^b,\eqno(3.29)$$ and we choose the constants $\alpha, \beta$ in such a way that the $g$’s satisfy the homogeneous $3\sigma$-equations $$-i(\hat\sigma^\ro\sigma^\nu\hat\sigma^\mu)^{\bar a}_{\,c}
\d_\nu\bar g_{\mu\bar a}+2m g^\ro_c=0,\eqno(3.30)$$ $$-i(\sigma^\ro\hat\sigma^\nu\sigma^\mu)_{a\bar c}
\d_\nu g_\mu^a+2m\bar g^\ro_{\bar c}=0.\eqno(3.31)$$ This implies the relation $$\alpha={i\over m}(1+2\beta^*).\eqno(3.32)$$ On the other hand, if we substitute the same form (3.28-29) into (3.26) we see that all $\lambda$-terms cancel provided $\beta$ is real $$\beta=\beta^ *.\eqno(3.33)$$
We remind the reader that all our fields are free quantum fields. Therefore, the spinor field $\lambda(x)$ satisfies the anticommutation relations (2.13). The situation is not so clear for the vector fields because there exist different possibilities. Calculating the commutator of (3.2) with $v_\mu+iw_\mu$, we find $$[(v_\mu+iw_\mu)(x),(v_\nu+iw_\nu)(x')]=0,\eqno(3.34)$$ so that $$[v_\mu(x),v_\nu(x')]=[w_\mu(x),w_\nu(x')]\=d C_{\mu\nu}(x-x'),\eqno(3.35)$$ assuming that all different fields commute. Similarly, computing the commutators $[v_\mu(x), \{\bar Q_{\bar a},$ $\lambda_b(x')\}]$ and $[v_\mu(x), \{Q_a, \lambda_b(x')\}]$, we obtain the relations $$\{\bar f_{\mu\bar a}(x),\lambda_b(x')\}=-m\sigma^\alpha_{b\bar a}C_
{\mu\alpha}(x-x'),\eqno(3.36)$$ $$\{f_{\mu a}(x),\lambda_b(x')\}=-i(\sigma^\alpha\hat\sigma^\beta)
_{ba}\d_\alpha C_{\mu\beta}(x-x').\eqno(3.37)$$ Substituting (3.29) into (3.36) and assuming that $\bar g^\mu_{\bar a}$ anticommutes with $\lambda_b$ we find $$\alpha^*\sigma^\nu_{b\bar a}\d_\nu\d_\mu D(x-x')+im\beta^*
\sigma_{\mu b\bar a} D(x-x')=-m
\sigma^\nu_{b\bar a}C_{\mu\nu}(x-x').$$ Similarly (3.37) implies $$im\alpha\eps_{ab}\d^\mu D-\beta (\sigma^\nu\hat\sigma^\mu)_{ba}
\d_\nu D=-i(\sigma^\nu\hat\sigma^\beta)_{ba}\d_\nu C^{\mu}_\beta.\eqno(3.38)$$ We postpone the choice of the coefficients $\alpha, \beta$ to the next section.
Since we now have all commutators, we can write down the vector superfield $V$ (3.1) $$V=i\theta\lambda+i\theta\sigma^\nu\hat\sigma^\mu\theta\d_\nu(v_\mu+iw_\mu)
-m\theta\sigma^\mu\bar\theta (v_\mu-iw_\mu)$$ $$+(\theta\sigma^\nu\hat\sigma^\mu\theta)(\bar\theta\d_\nu\bar f_\mu)
-im(\theta\sigma^\mu\bar\theta)(\theta f_\mu)$$ $$-i{m\over 3}(\theta\sigma^\nu\hat\sigma^\mu\theta)(\bar\theta\bar
\theta)\d_\nu (v_\mu-iw_\mu)+i{m\over 3}(\theta\sigma^\mu\bar\theta)
(\theta\sigma^\nu\bar\theta)\d_\nu (v_\mu-iw_\mu)+{\rm h.c.}\eqno(3.39)$$ The last two terms can be rewritten in the form $$i{m\over 2}(\theta\theta)(\bar\theta\bar
\theta)\d^\mu (v_\mu-iw_\mu)+{\rm h.c.}$$ Furthermore, since $\theta\sigma^\nu\hat\sigma^\mu\theta=\theta\theta
\eta^{\nu\mu}$, we get the final result $$V=i\theta\lambda+i(\theta\theta)\d^\mu(v_\mu+iw_\mu)
-m\theta\sigma^\mu\bar\theta (v_\mu-iw_\mu)$$ $$+(\theta\theta)(\bar\theta\d^\mu\bar f_\mu)
-im(\theta\sigma^\mu\bar\theta)(\theta f_\mu)$$ $$+i{m\over 2}(\theta\theta)(\bar\theta\bar
\theta)\d^\mu (v_\mu-iw_\mu)+{\rm h.c.}\eqno(3.40)$$
We want to consider this field $V$ as a gauge superfield. If we add the scalar field $S_2$ (2.39) to $V$, then the component fields are changed consistently according to $$m(v_\mu+iw_\mu)\to m(v_\mu+iw_\mu-{1\over m}\d_\mu F)\eqno(3.41)$$ $$m\d^\mu(v_\mu+iw_\mu)\to m\d^\mu(v_\mu+iw_\mu)-\sq F=
m\d^\mu(v_\mu+iw_\mu)+m^2 F$$ $$\lambda\to\lambda+\chi\eqno(3.42)$$ $$f_{\mu a}\to f_{\mu a}+{i\over m}\d_\mu\chi_a.\eqno(3.43)$$ in all orders. The transformation (3.39) is the usual classical gauge transformation. Therefore, the field $V$ is a good starting point for supergauge theory. We will further investigate the properties of the vector superfield $V$ in the next section by analysing the mixed field $g^\mu_a(x)$. The free quantum superfields are the basis of supergauge theory if considered as operator theory \[6\] in the spirit of a recent monograph \[7\].
Investigation of the mixed field $g^\mu_a$
==========================================
The field $g$ must satisfy the homogeneous $3\sigma$-equations (3.30-31) as well as the modified Dirac equations (3.26-27) $$-i\sigma^\nu_{a\bar a}\d_\nu g^{\mu a}=m\bar g^\mu_{\bar a}+
{2\over m}\d^\mu\d_\nu\bar g^\nu_{\bar a},\eqno(4.1)$$ $$i\sigma^\nu_{a\bar a}\d_\nu\bar g^{\mu\bar a}=m g^\mu_{a}+
{2\over m}\d^\mu\d_\nu g^\nu_{a}.\eqno(4.2)$$ In addition we get constraints from the anticommutators with the supercharges. The anticommutator (3.15) implies $$\{\bar Q_{\bar a},g^\mu_{b}\}=-i\sigma^\nu_{b\bar a}\d_\nu
(v^\mu-iw^\mu)+\alpha m\sigma^ \nu_{b\bar a}
\d^\mu(v_\nu-iw_\nu)$$ $$+i\beta(\sigma^ \mu\hat\sigma^ \alpha\sigma^ \beta)_{b\bar a}
\d_\alpha(v_\beta-iw_\beta)\Bigl].\eqno(4.3)$$ and from (3.22) we obtain $$\{Q_a,g^\mu_b\}=\eps_{ab}\Bigl[m(v^ \mu+iw^ \mu)+{2\over m}
\d^ \mu\d^\nu(v_\nu+iw_\nu)\Bigl]$$ $$+i\alpha(\sigma^\alpha\hat\sigma^\beta)_{ba}\d_\alpha\d^ \mu
(v_\beta+iw_\beta)-\beta m(\sigma^ \mu\hat\sigma^ \nu)_{ba}
(v_\nu+iw_\nu).\eqno(4.4)$$ These relations show that $g$ must be different from zero.
Since the vector index $\mu$ in $g^\mu_a$ corresponds to the spinor representation $(\eh,\eh)$, the mixed field realizes the representation $(\eh,0)\times (\eh,\eh)$ of the proper Lorentz group. By the Clebsch-Gordan decomposition it splits into irreducible representations as follows $$(\eh,0)\times (\eh,\eh)=(0,\eh)+(1,\eh).\eqno(4.6)$$ The dimensions of the representations are $2\times 4=2+6$. We calculate with the most general (reducible) tensor and set $$g^{\mu a}=\sigma^\mu_{b\bar c}g^{ab\bar c}.\eqno(4.7)$$ The l.h.s is the so-called Rarita-Schwinger representation \[8\] of the spinor on the r.h.s. The adjoint field is given by $$\bar g^{\mu\bar a}=\sigma^\mu_{c\bar b}\bar g^{\bar a\bar bc}.\eqno(4.8)$$
If we substitute this representation into the $3\sigma$-equation (3.30) we find the following Dirac equation for higher spin fields \[8\] $$i\sigma^\nu_{a\bar c}\d_\nu\bar g^{\bar b\bar c}_{\;\;\; c}=m g_{ca}
^{\;\;\;\bar b}.\eqno(4.9)$$ Similarly, the other $3\sigma$-equation (3.31) gives the adjoint Dirac equation $$i\hat\sigma^{\nu\bar ab}\d_\nu g_{cb}^{\;\;\;\bar c}=m\bar g^{\bar c
\bar a}_{\;\;\;c}.\eqno(4.10)$$ The indices $c, \bar c$ are simply spectators in this Dirac equation.
Next we have to fulfill the equation (4.1). It boils down to the second order equation $$-i\sigma^\nu_{a\bar a}\d_\nu g^a_{\;\;d\bar d}=m\bar g_{\bar a\bar dd}
+{1\over m}\sigma^\mu_{d\bar d}\sigma^\nu_{c\bar b}\d_\mu\d_\nu
\bar g_{\bar a}^{\;\;\bar bc}.\eqno(4.11)$$ Here we insert (4.9) in the form $$\sigma^\nu_{c\bar b}\d_\nu\bar g^{\;\;\bar bc}_{\bar a}=
-im g^c_{\;\;c\bar a},\eqno(4.12)$$ and get $$-i\sigma^\nu_{a\bar a}\d_\nu g^a_{\;\;d\bar d}
=m\bar g_{\bar a\bar dd}-i\sigma^\mu_{d\bar d}\d_\mu g^c_{\;\;c\bar a}.
\eqno(4.13)$$ Now we decompose $g$ into a symmetric and antisymmetric part $$g_{ad\bar d}=\eh(g_{ad\bar d}+g_{da\bar d})+\eh(g_{ad\bar d}-
g_{da\bar d})\eqno(4.14).$$ Since an antisymmetric second rank tensor has only one independent component, we can write $$g_{ad\bar d}=g'_{ad\bar d}+\eps_{ad}\bar\psi_{\bar d},\eqno(4.15)$$ $$\bar g_{\bar a\bar dd}=\bar g'_{\bar a\bar dd}+\eps_{\bar a\bar d}
\psi_d.$$ This is the decomposition (4.6) into irreducible Lorentz tensors. As we will see, $\bar\psi$ is a true independent spinor field, and $g'$ is symmetric in the indices of the same kind. We use this decompostion in the second Dirac equation (4.10) and obtain $$i\sigma^\nu_{b\bar a}\d_\nu(g^{\prime cb}_{\;\;\;\bar c}+\eps^{cb}
\bar\psi_{\bar c})=-m\bar g_{\bar c\bar a}^{\;\;\;c}.\eqno(4.16)$$ Substituting this into (4.13) we get $$m(\bar g_{\bar a\bar d}^{\;\;\;d}-\bar g_{\bar d\bar a}^{\;\;\;d})=-2i
\sigma^{\nu d}_{\;\;\;\bar a}\d_\nu\bar\psi_{\bar d}+i
\sigma^{\nu d}_{\;\;\;\bar d}\d_\nu g^c_{\;\;c\bar a}.\eqno(4.17)$$
By (4.15) we can express $\bar\psi$ as a contracted tensor $$2\bar\psi_{\bar d}=g^b_{\;\;b\bar d}.\eqno(4.18)$$ Then (4.17) gives the following equation for the spinor field $\psi$: $$m\eps_{\bar a\bar d}\psi^ d=i(\sigma^{\nu d}_{\;\;\;\bar d}\d_\nu
\bar\psi_{\bar a}-\sigma^{\nu d}_{\;\;\;\bar a}\d_\nu\bar\psi
_{\bar d}).\eqno(4.19)$$ This is just the relation (2.40) which shows that $\psi(x)$ is a spin-$\eh$ field satisfying the Dirac equation. Since all steps can be inverted, the equation (4.11) is then satisfied. However, the fields $g'$ and $\psi$ are not completely decoupled. In fact, substituting (4.15) into (4.10) and using (4.19) we obtain an inhomogeneous Dirac equation for $g'$: $$i\hat\sigma_{\bar a}^{\nu b}\d_\nu g'_{bc\bar c}-m\bar g'_{\bar a
\bar cc}=-i\sigma^\mu_{c\bar c}\d_\mu\bar\psi_{\bar a}.\eqno(4.20)$$ Consequently, $g'_{ab\bar c}(x)$, although being symmetric in $a, b$, is not identical with the Rarita - Schwinger field for spin-3/2 particles \[8\].
Finally, there remains to check the consistency of the anticommutators with the supercharge, because this is the point where something goes wrong if there is an incorrect step in the construction. The anticommutator (4.4) gives the corresponding anticommutator for the spinor field: $$\{Q_a, g_{bc\bar c}\}=\eps_{ab}\sigma^\mu_{c\bar c}\Bigl[{m\over 2}
(v_\mu+iw_\mu)+\d_\mu\d^\nu(v_\nu+iw_\nu)({1\over m}+{i\over 2}\alpha)
\Bigl]$$ $$-\beta m\eps_{cb}\hat\sigma^\nu_{\bar c a}(v_\nu+iw_\nu)+\alpha
\sigma^\mu_{c\bar c}\sigma^{\alpha\beta}_{ba}\d_\alpha\d_\mu (v_\beta
+iw_\beta).\eqno(4.21)$$ For any choice of $\alpha$ and $\beta$ the r.h.s. can be decomposed into a symmetric and antisymmetric part with respect to $b, c$. This allows to identify the anticommutators of $g'$ and $\psi$ and shows the consistency of the whole construction. A preferred choice is $$\alpha=0,\quad \beta=-{1\over 2},\eqno(4.22)$$ because by (3.36) this leads to the propagator $$C_{\mu\nu}(x-x')={i\over 2}\eta_{\mu\nu}D(x-x')\eqno(4.23)$$ which has a good ultraviolet behaviour. For this choice let us write down the antisymmetric part of (4.21): $$\{Q_a,\bar\psi_{\bar c}\}=-{1\over 2}\sigma^\mu_{a\bar c}\Bigl[{3\over
2}m(v_\mu+iw_\mu)+{1\over m}\d_\mu\d^\nu(v_\nu+iw_\nu)\Bigl].
\eqno(4.24)$$ Here we have used the simple identity $$\eps_{ab}\sigma^\mu_{c\bar c}-\eps_{ac}\sigma^\mu_{b\bar c}=
\eps_{cb}\sigma^\mu_{a\bar c}.\eqno(4.25)$$
For later use we also calculate the anti-commutation relations of the mixed fields. By means of (3.12) we find $$\{f^\mu_a(x),\bar f^\nu_{\bar b}(x')\}=-\{f^\mu_a(x),[\bar Q_{\bar b}
, v^\nu(x')]\}=[v^\nu(x'),\{f^\mu_a(x),\bar Q_{\bar b}\}]$$ $$=i\sigma^\ro_{a\bar b}\d^x_\ro[v^\mu(x)-iw^\mu(x), v^\nu(x')]=$$ $$=i\sigma^\ro_{a\bar b}\d^x_\ro C^{\mu\nu}(x-x')=-{1\over 2}\eta^
{\mu\nu}\sigma^\ro_{a\bar b}\d^x_\ro D(x-x')\eqno(4.26)$$ for the preferred choice (4.23). This implies the following anti-commutator for the mixed $g$-field: $$\{g^\mu_a(x),\bar g^\nu_{\bar b}(x')\}=-{1\over 2}\eta^
{\mu\nu}\sigma^\ro_{a\bar b}\d^x_\ro D(x-x')-{1\over 4}(\sigma^\mu
\hat\sigma^\ro\sigma^\nu)_{a\bar b}\d_\ro D(x-x').\eqno(4.27)$$ In a similar way we obtain $$\{f^\mu_a(x),f^\nu_b(x')\}=-{i\over 2}\eps_{ab}(m\eta^{\mu\nu}+
{2\over m}\d^\mu\d^\nu)D(x-x'),\eqno(4.28)$$ and $$\{g^\mu_a(x),g^\nu_b(x')\}=-{i\over 4}m(\eps_{ab}\eta^{\mu\nu}-
2i\sigma^{\mu\nu}_{ab})D(x-x')-{i\over m}\eps_{ab}\d^\mu\d^\nu
D(x-x'),\eqno(4.29)$$ $$\{\bar g^\mu_{\bar a}(x),\bar g^\nu_{\bar b}(x')\}={i\over 4}m
(\eps_{\bar a\bar b}\eta^{\mu\nu}+2i\hat\sigma^{\mu\nu}_{\bar a\bar b}
)D(x-x')+{i\over m}\eps_{\bar a\bar b}\d^\mu\d^\nu
D(x-x'),\eqno(4.30)$$ $$\{\bar g^\mu_{\bar a}(x),g^\nu_b(x')\}=-{1\over 2}\eta^
{\mu\nu}\sigma^\ro_{b\bar a}\d^x_\ro D(x-x')-{1\over 4}(\sigma^\nu
\hat\sigma^\ro\sigma^\mu)_{b\bar a}\d_\ro D(x-x').\eqno(4.31)$$
It is interesting to discuss our results from the point of view of representation theory. The massive representations of the $N=1$ supersymmetric algebra have the following spin components:
0.5cm [ ]{} 0.5cm The representation $\Omega_0$ corresponds to the scalar superfield $S_1$ (2.33) because it has 2 scalar and 1 spinor component. The vector superfield $V$ (3.40) may be related to $\Omega_1$, if we consider the mixed field $f^\mu_a$ as the spin-3/2 component. But this is not entirely justified because $f^\mu_a$ contains also a spin-1/2 piece. The surprising result is that it was impossible to realize the representation $\Omega_{1/2}$ with free quantum fields. In fact, if we start with the single scalar component required by this representation, as we did in sect.2, we are driven into the representation $\Omega_0$.
[99]{}
F. Constantinescu, G. Scharf, Causal approach to supersymmetry: chiral superfields
J. Wess, J. Bagger, Supersymmetry and supergravity, 2nd edition, Princeton University Press 1992
P. West, Introduction to supersymmetry and supergravity, 2nd edition, World Scientific, Singapore 1990
O. Piguet, K. Sibold, Renormalized supersymmetry, Birkhäuser, Boston 1986
S. Weinberg, The quantum theory of fields, vol.III, Cambridge University Press 2000
H. Epstein, V. Glaser, Annales Inst. Poincaré [**A 19**]{} (1973) 211
G. Scharf, Quantum gauge theories – a true ghost story, Wiley-Interscience 2001
H. Umezawa, Quantum field theory, North-Holland, Amsterdam, 1956
|
---
abstract: 'Imaging for an occluded object is usually a difficult problem, in this letter, we introduce an imaging scheme based on computational ghost imaging, which can obtain the image of a target object behind an obstacle. According to our theoretical analysis, once the distance between the object and the obstacle is far enough, one can obtain the image of the object by using ghost imaging technique. The wavelength of the light source also affects the quality of the reconstructed image. In addition, if the bucket detector is placed far away from the obstacle, a tiny point-like detector without collecting lens can be applied to realize the imaging. These theoretical results above have been verified with our numerical simulations. Furthermore, the robustness of this imaging scheme is also investigated.'
author:
- Chao Gao
- Xiaoqian Wang
- Lidan Gou
- Yuling Feng
- Hongji Cai
- Zhifeng Wang
- Zhihai Yao
title: Ghost imaging for an occluded object
---
Ghost imaging is a novel imaging technique based on the intensity fluctuation correlations of the light, and it was first proposed with entangled photons[@sov; @pit]. Later, it was found that ghost imaging could also be realized by using classical thermal source[@1stclassical], and there were many discussions about thermal ghost imaging[@thermal1; @thermal2; @thermal3; @Gatti2006; @gatti2006a; @shih2006; @Ferri2008; @thermal4; @thermalA; @visi; @higi; @wulingan1; @wulingan2; @Gao2017].
In 2008, J.H.Shapiro proposed computational ghost imaging[@cgi], and it was verified by experiment in 2009[@cgie]. Different from conventional ghost imaging scheme, computational ghost imaging technique applies a programmable light source, and the experimental setup can be simplified. Ghost imaging displays great potentials in some special situations, such as high lateral resolution imaging[@superresolution], resistance of atmosphere turbulence[@turbulence; @turbulencei] and so on.
In addition to the above features, our recent work shows that ghost imaging may have even more advantages than conventional imaging techniques. Imaging for an occluded object is a difficult problem, in this letter, we proved that, under appropriate condition, one can obtain the image of an occluded object by applying ghost imaging technique, even if the object is blocked by an unknown obstacle.
The schematic diagram of the computational ghost imaging for an occluded object is shown in Fig. \[ns\]. If we view from the bucket detector, the target object is blocked by the obstacle. But when we use computational ghost imaging technique, we can obtain the object’s image.
For simplification, we consider the 1-dimension case. Let $\vec C$ and $\vec D$ represent the transmission functions of the target object and the obstacle, where $\vec C=[c_1,c_2,\cdots,c_N]^T$ and $\vec D=[d_1,d_2,\cdots,d_N]^T$. The light emitted by the programmable light source illuminates the object, and its intensity distribution on the object plane can also be represented by an $1\times N$ vector: $$\vec S(t)=
\left[
\begin{array}{cccccc}
s_1(t)&s_2(t)&...&s_n(t)&...&s_N(t)\\
\end{array}
\right] ^T.$$ The modulated light illuminates and passes through the target object, reached the obstacle plane after $z$ distance of propagating. First, for simplification, we assume that we can collect all the transmitted light by using a bucket detector, and the bucket signal can be written as: $$B(t)=\sum\limits_{m=1}^{N}d_m\sum\limits_{n=1}^{N}A_{mn}c_ns_n(t).$$ which $\{c_n\}$ and $\{d_m\}$ represent the elements of $\vec C$ and $\vec D$. $\{A_{mn}\}$ represent the elements of the propagating matrix $\hat A$. The second-order correlation function[@quantumoptics] of this system is: $$\vec G^{(2)}(n')=\langle \vec s_{n'}(t) B(t)\rangle_t.
\label{g2o}$$ Here, $\langle ...\rangle_t$ represents the ensemble average. In this system, we assume that $\{s_n(t)\}$ are independent and identically distributed. If we take enough measurements, we have[@quantumoptics]: $$\langle s_{n'}(t)s_n(t) \rangle_t=\delta(n',n)l+\langle s \rangle^2.$$ Where $\langle s\rangle$ is the average intensity of the light source, $l$ is the variance of the intensity. Therefore, the non-normalized second-order correlation function of the target object can be expressed by: $$\vec G^{(2)}(n')=lc_{n'}\sum\limits_{m=1}^{N}d_mA_{mn'}+O.
\label{result1}$$
Where $O=\langle s\rangle^2\sum\limits_{m=1}^{N}d_m\sum\limits_{n=1}^{N}A_{mn}c_n$ is a background term which is unrelated to $n'$. In another word, it does not contains the spatial information of the object. So, we focus on term $lc_{n'}\sum\limits_{p=1}^{N}d_mA_{mn'}$. Obviously, $\vec G^{(2)}$ is in proportion to $c_{n'}\sum\limits_{m=1}^{N}d_mA_{mn'}$. In order to find out the relationship between $\vec G^{(2)}$ and the transmission function of the target object, we need to investigate the form of the propagating matrix $\hat A$.
Now, we investigate the propagation progress, and we consider the dispersed case. Let $u_0(n,t')$ be the instantaneous field distribution of the source on the target object plane at time $t'$. For simplification, the dimension and the pixel size at the target object and obstacle planes are equal. For the pixel with transverse size of $\Delta x$, after $z$ distance of traveling, the field distribution on the obstacle plane can be written as[@fddt]: $$u(m,t')=\frac{e^{ikz}(\Delta x)^2}{i\lambda z}\sum\limits_{n=1}^{N}u_0(n,t')e^{\frac{ik}{2z}(m-n)^2(\Delta x)^2}.$$ Where N is the number of the pixels, and $\frac{e^{ikz}(\Delta x)^2}{i\lambda z}\sum\limits_{m=1}^{N}e^{\frac{ik}{2z}(m-n)^2(\Delta x)^2}$ is usually denoted by $h_{z,\lambda}(m-n)$, which is called the point spread function (PSF). We can obtain the intensity distribution on the obstacle plane: $$\begin{aligned}
&I_{obs}(m,t)=\int_{t}^{t+\Delta t} u^*(m,t')u(m,t')dt' \notag \\
&=\int_{t}^{t+\Delta t}[h^*(m-1)u_0^*(1,t')+h^*(m-2)u_0^*(2,t')+... \notag \\
&+h^*(m-n)u_0^*(n,t')+...h^*(m-N)u_0^*(N,t')]\times \notag \\
&[h(m-1)u_0(1,t')+h(m-2)u_0(2,t')+... \notag \\
&+h(m-n)u_0(n,t')+...+h(m-N)u_0(N,t')]dt'.
\label{I2}\end{aligned}$$ Where $\Delta t$ is a short period of time. Here, we assume that the light source is incoherent. In this case, we have: $$\int_{t}^{t+\Delta t} u^*(x_1,t')u(x_2,t') dt'=\delta(x_1,x_2)|u(x_1,t)|^2.$$ And we can simplify Eq. (\[I2\]) into: $$I_{obs}(m,t)=\sum\limits_{n'=1}^{N}\sum\limits_{n=1}^{N}h^*(m-n')h(m-n)I_{obj}(n,t).
\label{obs}$$ Where $I_{obj}(n,t)$ is the intensity distribution through the target object plane. Eq. (\[obs\]) can also be written in the matrix form: $$I_{obs}=
\left[
\begin{array}{cccccc}
A_{11} &A_{12} &\cdots &A_{1n} &\cdots &A_{1N} \\
A_{21} &A_{22} &\cdots &A_{2n} &\cdots &A_{2N} \\
\vdots &\vdots &\ddots &\vdots & &\vdots \\
A_{m1} &A_{m2} &\cdots &A_{mn} &\cdots &A_{mN} \\
\vdots &\vdots & &\vdots &\ddots &\vdots \\
A_{N1} &A_{N2} &\cdots &A_{Nn} &\cdots &A_{NN}
\end{array}
\right]I_{obj}.$$ Where $$A_{mn}=|h(m-n)|^2=\frac{(\Delta x)^4}{\lambda^2 z^2}\sum\limits_{n'=1}^{N}e^{i\frac{\pi (\Delta x)^2 }{\lambda z}[(m-n')^2-(m-n)^2]}
\label{matrixa}$$ is called the intensity point spread function. Fig. \[psf\] gives the intensity point spread function curve of light sources with several typical wavelengthes at different distances of propagating.
Fig. \[psf\] shows that the intensity point spread function is influenced by the distance of propagating $z$ and the wavelength of the light source $\lambda$. When $\lambda$ and/or $z$ is big enough, the intensity point spread function approach to a constant which is unrelated to the spatial coordinates.
The physics behind this progress is: the illuminating light carries the information of the object, propagates a distance of $z$, and reach the obstacle plane. Due to the propagation of the light, the information of the object spread around on the obstacle plane. Every single point on the object plane produces an Airy pattern on the obstacle plane, and the Airy patterns overlap with each other. As a result, every single pixel on the obstacle plane contains the information from multiple points on the object plane. As distance $z$ or wavelength $\lambda$ increases, the area of every Airy pattern increases. While the magnitudes of $z$ and/or $\lambda$ are great enough, we can assume that every pixel on the obstacle plane contains the information from all of the points on the object plane. So that the effective information of the object can always reach the bucket detector via the outside of the obstacle’s border.
The diffraction on the obstacle plane is actually a similar progress: after a distance of traveling, the transmitted light reaches the bucket detector plane. Noticed that, like the situation we discussed above, if this distance is far enough, every pixel on the bucket detector plane contains the information from all of the points on the obstacle plane. Therefore, in this case, it is not necessary to collect all of the transmitted light. Instead, in ideal condition, even a tiny point-like detector can finish the task.
With the assumption of a long distance between the bucket detector and the obstacle, the intensity on the bucket detector plane approaches to be evenly distributed. The bucket signal can be written as: $$B'(t)=\alpha \sum\limits_{m=1}^{N}d_m\sum\limits_{n=1}^{N}A_{mn}c_ns_n(t)$$ Where $\alpha \in (0,1)$ is a constant which depends on the size of the bucket detector. Based on the discussions above, we can now explain why ghost imaging technique can realize the imaging of an occluded object. From Eq. (\[result1\]) we know that $\vec G^{(2)}$ is in proportion to $c_{n'}\sum\limits_{m=1}^{N}d_mA_{mn'}$. Both the information of the target object and the obstacle are contained in $\vec G^{(2)}$. While the magnitude of $\lambda z$ is big enough, the elements in the propagating matrix $\hat A$ approach to a constant which is unrelated to the spatial coordinates. The second-order correlation function of the target object is: $$\vec G^{(2)}(n')\approx l\alpha\bar{d}\bar{A}c_{n'}+\alpha O.$$ Where $\bar{d}=\sum\limits_{n=1}^{N}d_n$, $\bar{A}=\frac{1}{N}\sum\limits_{m=1}^{N}\sum\limits_{n=1}^{N}A_{mn}$. In this case, the spatial information of the obstacle is eliminated. $G^{(2)}$ is now in proportion to the target object’s transmission function, the image of the object can be obtained correctly. Noticed that, one can obtain the image of the target object in this case, even if the shape of the obstacle is unknown. The reason is, different from conventional imaging technique, ghost imaging is a kind of computational imaging scheme which is based on the intensity fluctuation correlations, the imaging quality is only sensitive to the fluctuation of the total (or average) intensity of the transmitted light. When the distance between the object and the obstacle is far enough, the obstacle does very limited effects on the fluctuation of the bucket signal. The result is: in this case, even under the affect of an obstacle, ghost imaging scheme will not fail, we can still obtain the image of the target object.
However, when $\lambda z$ decreases, the curve of the intensity point spread function approaches to $\delta$ function. Thus, the non-opposite angle elements of propagating matrix $\hat A$ approach to zero. In this case, the second-order correlation function of the target object can be written as: $$\vec G^{(2)}(n')\approx l\alpha A_{n'n'}c_{n'}d_{n'}+\alpha O.$$
Obviously, $\vec G^{(2)}$ is in proportion to $c_{n'}d_{n'}$, the product of the transmission function of the target object and the obstacle. We will obtain the mixture image of the target object and the obstacle, we cannot revive the image of the target object correctly.
Thus, to realize the imaging for an occluded object, the distance between the target object and the obstacle should be far enough. Besides, in order to obtain a image with higher quality, we can increase the wavelength of the illuminating light. Furthermore, if we place the bucket detector far away from the obstacle, it is possible to use a tiny detector to realize the imaging.
To verify our theoretical results, the numerical simulations are carried out, and, the robustness of this imaging system is also judged. The schematic diagram of our numerical simulation is shown in Fig. \[ns\]. We take $1,200,000$ measurements for every simulation, and the field distribution of light source is modulated into gaussian randomly distributed. The distance between SLM and the target object is taken to be $0.50$m. The size of the bucket detector is $0.08\times 0.08$mm, and it is placed $10.00$m far from the obstacle, in the center of the bucket detector plane (on the optic axis). As Fig. \[objects\] shows, the target object is an opaque arrow, and the obstacle is a “ghost”-shaped opaque plate, both of them are placed in the center in their planes. The size of the target object is $1.20\times 0.72$ mm, and the size of the obstacle is about $2.08 \times 2.08$ mm. Both the target object and the obstacle plane are pixelated into two $64\times 64$ pixels images, with pixel width $\Delta x=0.04$ mm.
We investigate the influence of the distance between the target object and the obstacle and the wavelength of the light source, respectively.
The influence of the distance between the target object and the obstacle
------------------------------------------------------------------------
In this part, we use a $632.8$ nm laser as the light source. In order to study the influence on the imaging quality of the target object, we change the distance between the target object and the obstacle, and reconstruct the image of the target object by using computational ghost imaging technique, respectively. The results of our numerical simulation are shown in Fig. \[distances\].
It is clear that, while the distance between the target object and the obstacle is far enough, it is possible to realize the imaging for an occluded object by applying computational ghost imaging technique.
The influence of the wavelength of the light source
---------------------------------------------------
In this part, the distance between the target object and the obstacle is taken to be $3.0$ m. In order to find out the influences of the wavelength on the imaging quality, we use light sources with different wavelengthes to implement the computational ghost imaging for the target object. The results of our numerical simulations are shown in Fig. \[lambda\].
Obviously, we can get a clearer view of the target object by applying a light source with longer wavelength. However, the spatial resolution of the reconstructed image is decreased.
The robustness of this imaging system
-------------------------------------
Many computational imaging schemes fail with the affect of noise, thus it is necessary to judge the performance of our imaging scheme under the influence of background noise. We use signal to noise ratio (SNR) to describe the effect of the background noise on the bucket signal, which is defined as: $$\text{SNR}=10\log_{10}\frac{\bar B}{\bar N_b},$$ where $\bar B$ is the average intensity of the bucket signal, $\bar N_b$ is the average intensity of the background noise, the noise is gaussian noise. The reconstructed images of the target object under different SNR are shown in Fig. \[snr\].
The results show that, when SNR of the bucket signal is $6.6570 \text{dB}$, the image of the object can still be recognized, in this case, the average intensity of the noise reached about 22%. The imaging scheme fails when SNR is lower than $5.7093$ dB (namely with about 27% noise). Thus, this imaging scheme can partly resist the effect of noise.
In conclusion, we have proved that ghost imaging can realize the imaging for an occluded object. According to the above discussions, we find that, this unique feature is based on the fact that ghost imaging technique is based on the intensity fluctuation correlations. Due to the diffraction of the light, the bucket detector (with limited size) can always capture the useful fluctuation information which is caused by different illuminating patterns passes through the object. If distance between the target object and the obstacle is far enough, the image of the target object can be reconstructed accurately. While the target object is close to the obstacle, we will obtain the mixture image of the target object and the obstacle, ghost imaging failed in this case. Besides, a better image of the target object can be obtained by using a light source with longer wavelength, but the resolution of the reconstructed image is decreased. In addition, it is possible to realize the imaging by using a tiny point-like bucket detector if the detector is placed far away from the obstacle. The numerical simulations have been carried out, and the results agree with our theoretical analysis.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work is supported by: National Natural Science Foundation of China (11305020); The Science and Technology Research Projects of the Education Department of Jilin Province, China (2016-354); Natural Science Foundation of Jilin Province(20180520165JH).
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|
---
author:
- 'Qian Huang[^1]'
- 'Shuiqing Li[^2]'
- 'Wen-An Yong[^3]'
title: 'Stability Analysis of Quadrature-based Moment Methods for Kinetic Equations[^4]'
---
[^1]: Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China (, <https://www.researchgate.net/profile/Qian_Huang34>).
[^2]: Department of Energy and Power Engineering, Tsinghua University, Beijing 100084, China (, <http://www.thu-lishuiqing.org>).
[^3]: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China (, <https://www.researchgate.net/profile/Wen-An_Yong>).
[^4]: Submitted to the editors DATE.
|
---
abstract: 'Human reading behavior is sensitive to surprisal: more predictable words tend to be read faster. Unexpectedly, this applies not only to the surprisal of the word that is currently being read, but also to the surprisal of upcoming (successor) words that have not been fixated yet. This finding has been interpreted as evidence that readers can extract lexical information parafoveally. Calling this interpretation into question, showed that successor effects appear even in contexts in which those successor words are not yet visible. They hypothesized that successor surprisal predicts reading time because it approximates the reader’s uncertainty about upcoming words. We test this hypothesis on a reading time corpus using an LSTM language model, and find that successor surprisal and entropy are independent predictors of reading time. This independence suggests that entropy alone is unlikely to be the full explanation for successor surprisal effects.'
author:
- |
Marten van Schijndel\
Department of Cognitive Science\
Johns Hopkins University\
[[email protected]]{}\
Tal Linzen\
Department of Cognitive Science\
Johns Hopkins University\
[[email protected]]{}
bibliography:
- 'bibliography.bib'
title: 'Can Entropy Explain Successor Surprisal Effects in Reading?'
---
Introduction
============
One of the most robust findings in the reading literature is that more predictable words are read faster than less predictable words [@ehrlich1981contextual]. Word predictability effects fit into a picture of human cognition in which humans constantly make predictions about upcoming events and test those predictions against their perceptual input [@bar2007proactive].
While the effect of the predictability of the current word ($w_t$) on the reading time at $w_t$ is not controversial, there is a spirited debate in the eye movement literature as to whether reading time at $w_t$ is affected by the predictability of the *successor* word, $w_{t+1}$ [@drieghe11]. Reading is characterized by a series of fixations, which bring a single word into the center of the visual field (the fovea), where visual acuity is highest. Effects of successor predictability have been taken to indicate that readers are able to process words parafoveally, that is, even when those words are not fixated [@kliegletal06]. Such an empirical finding would appear to constitute evidence against serial attention shift models such as E-Z Reader [@reichle2003ez], in which attention is directed at a single word at a time, and in favor of models such as SWIFT [@engbert2002dynamical], in which attention can be distributed over multiple words at the same time.
This interpretation of successor predictability effects was called into question by , who showed that the predictability of word $w_{t+1}$ affected reading time at $w_{t}$ even when $w_{t+1}$ was masked and was not visible until the reader fixated on it directly. A similar result was found by in self-paced reading, a paradigm which similarly precludes parafoveal preview. Short of ascribing psychic abilities to readers, then, the only possible explanation for these findings is that what appears to be an effect of the predictablity of $w_{t+1}$ is a confound driven by the relationship between the predictability of $w_{t+1}$ and an underlying property of $w_t$.
hypothesized that the property of $w_t$ that is confounded with the predictability of $w_{t+1}$ is the reader’s **uncertainty** about the words that could follow $w_t$, but they did not test this hypothesis. The present paper directly evaluates the relation between successor surprisal and uncertainty estimated from a single RNN language model [@gulordavaetal18]. We use a self-paced reading corpus [@futrelletal18], in which parafoveal preview is unavailable. To anticipate our results, we do not find evidence that the effect of successor surprisal can be reduced to uncertainty. We then explore the hypothesis that processing limitations, which lead to uncertainty being calculated over a restricted number of probable words rather than over the entire vocabulary, could account for these conflicting results, with similarly negative results. We conclude that uncertainty is unlikely to be the only explanation for successor surprisal effects.
Surprisal and entropy
=====================
The relationship between the reading time at word $w_t$ and the conditional probability of $w_t$ is logarithmic [@smith2013effect]; in other words, if we use *surprisal* [@hale01] as our probability measure: $$\text{surprisal}(w_t)=-\text{log P}(w_t\mid w_{1\ldots t-1})\label{eqn:surp}$$
then there is a linear correlation between RT($w_t$) and surprisal($w_t$). Surprisal has been shown to be a strong predictor of reading time in linear regression models [e.g., @dembergkeller08; @roarketal09].
Successor surprisal is simply the surprisal of the next observation in a sequence: $$\begin{aligned}
\text{succ.\ surprisal}(w_t)&=-\text{log P}(w_{t+1}\mid w_{1...t})\label{eqn:succsurp}\\
&=\text{surprisal}(w_{t+1})\end{aligned}$$
Finally, the entropy at $w_t$ is defined as follows: $$\begin{aligned}
&H(w_t) = E[\text{surprisal}(w_{t+1})]\label{eqn:esurp}\\
&= -\!\!\!\!\!\!\sum_{w_{t+1}\in V}\!\!\!\text{P}(w_{t+1}\!\mid\! w_{1...t})\ \text{log P}(w_{t+1}\!\mid\! w_{1...t})\label{eqn:fullH}\end{aligned}$$
As mentioned in the introduction, @angeleetal15 hypothesized that the entropy at $w_t$ is the underlying cause for successor ($w_{t+1}$) surprisal effects on $w_t$. This is a plausible hypothesis: the expected successor surprisal in a given context is the entropy at $w_t$ (Equation \[eqn:esurp\]), so in the limit, successor surprisal should be the same as the entropy over possible continuations when averaged over a corpus. In this hypothetical limit-case, we would directly observe Equation \[eqn:fullH\] in the data, as the sequence $w_{1...t+1}$ occurred exactly the expected number of times in the corpus. In practice, with a finite set of observations $T$ which are regressed simultaneously, successor surprisal provides a Monte Carlo estimator of entropy in that corpus: $$\begin{aligned}
\hat H(T)&\approx-\sum^{\mid T\mid}_{t=1}\frac{1}{\mid\!\! T\!\!\mid}\ \text{log P}(w_{t+1}\mid w_{1...t})\\
&=\sum^{\mid T\mid}_{t=1}\frac{1}{\mid\!\! T\!\!\mid}\ \text{surprisal}(w_{t+1})\label{eqn:succsurpapprox}\end{aligned}$$ Therefore, if uncertainty over possible continuations influences reading time, then successor surprisal could be correlated with reading time simply due to its relationship with corpus-level entropy.
Importantly for the present study, if the relationship to uncertainty is the sole underlying reason that successor surprisal can predict reading time, successor surprisal should be a worse predictor of reading time than entropy when the same model distribution $q$ is used to compute both measures. This claim follows directly from Equation \[eqn:succsurpapprox\]: if entropy $H_q$ is the true generator of the data, then it should always be a better predictor than some corpus-level approximation $\hat H_q$ due to noise from the Monte Carlo process.
The syntactic language models used in previous reading studies could not compute successor surprisal and entropy from the same conditional probability distribution, precluding a direct test of this hypothesis; in particular, while the @roarketal09 parser can compute both surprisal and entropy, it estimates them using two different probability distributions due to its use of beam search.
![Successor surprisal plotted against entropy for each word in the Natural Stories Corpus. The Pearson correlation is 0.45, providing empirical validation of the theoretically strong limit-case relation between entropy and successor surprisal.[]{data-label="fig:basecorr"}](top50000.png)
Method {#sec:method}
======
#### Language model:
In contrast with previous work, which used grammar-based language models, we used a single recurrent neural network (RNN) language model to compute entropy and successor surprisal from the same conditional probability distribution. The language model we used was trained by @gulordavaetal18 on 90 million words from the English Wikipedia. The model had two LSTM layers with 650 hidden units each, 650-dimensional word embeddings, a dropout rate of 0.2 and batch size 128, and was trained for 40 epochs (with early stopping).
Unlike grammar-based language models, RNN language models do not explicitly construct syntactic dependencies, which are essential in human sentence comprehension. However, recent work has shown that RNN language models are nevertheless sensitive to the probability of syntactic structures [@linzenetal16; @vanschijndellinzen18; @wilcox2018rnn], tentatively suggesting that they are an adequate substitute for modeling human reading behavior. Importantly, they have the added benefit that all our measures of interest are easy to calculate using Equations \[eqn:surp\], \[eqn:succsurp\], and \[eqn:fullH\] on the model’s softmax layer, which provides a conditional probability distribution over the upcoming word given the preceding words.
#### Data:
The test domain in this work is the Natural Stories Corpus [@futrelletal18]. The corpus is a set of 10 texts (485 sentences) written to sound fluent while still containing many low-frequency and marked syntactic constructions. The sentences within each text were presented in order, and self-paced reading data were collected from 181 native English speakers. We used one third of the sentences for exploration while two thirds were set aside for statistical confirmation. We omit any words consisting of multiple tokens (e.g., *do$\cdot$n’t* and *boar$\cdot$!$\cdot$’*). In this paper, all statistical testing was done on the held-out partition.
Results {#sec:results}
=======
#### Successor surprisal is moderately correlated with entropy:
We first tested the degree to which the Monte Carlo estimation produces a correlation between entropy and successor surprisal when each is computed with the same probability model (i.e. the LSTM language model described in Section \[sec:method\]) and found that the measures were moderately correlated, with Pearson’s $r = 0.454$ (see Figure \[fig:basecorr\]). This moderate correlation between the two measures could plausibly explain the successor surprisal effects on reading time that have been observed in previous studies.
#### Successor surprisal predicts reading time:
Before testing whether entropy can account for the effectiveness of successor surprisal in predicting reading time, we first verified that our successor surprisal measure was positively correlated with reading time as observed with the language models used in previous work [@angeleetal15; @vanschijndelschuler16; @vanschijndelschuler17].
Following previous studies, we used a linear mixed effects regression approach. Unlike linear regression, in which the error term is assumed to come from a single normal distribution, this approach takes into consideration clustered errors that are due to the variability across the particular participants and words in the sample (“random effects”). This makes it possible to estimate the effect of theoretically relevant “fixed effects” in a way that is more likely to generalize to new items and participants. We used the *lme4* R package [@r-lme4] to perform the regression, and included fixed effects for word length, sentence position, unigram frequency, surprisal, and successor surprisal. Unigram frequencies were estimated from the Gigaword corpus [@graffcieri03]. We included random intercepts for each word and subject, and by-subject random slopes for each fixed effect.[^1] All predictors were z-transformed before fitting the models. We compared the log-likelihood of the data under that model to the log-likelihood of one without the fixed effect for successor surprisal to determine the significance of successor surprisal as a fixed effect predictor of reading time.
Successor surprisal was significant as a predictor ($\hat\beta=4.3$, $\hat\sigma=0.52$, $\chi^2(1) = 58$, $p < 0.001$), suggesting that the previously observed relationship between successor surprisal and reading time holds when successor surprisal is computed with our LSTM language model. We note that, in this self-paced reading setting, the regression coefficient of successor surprisal was quite large: it was over half that of the coefficient of $w_t$ surprisal ($\hat\beta=6.0$) and rivaled that of unigram frequency ($\hat\beta = 5.1$).
$\hat\beta$ $\hat\sigma$ $t$
--------------------- ------------- -------------- -------
(Intercept) 331.66 6.31 52.56
Sentence position 0.72 0.51 1.41
Word length 4.74 1.00 4.73
Surprisal 5.67 0.57 9.88
Unigram frequency 4.94 1.18 4.17
Successor surprisal 3.26 0.39 8.34
Entropy 3.12 0.55 5.68
: Fixed effect coefficients from fitting self-paced reading times. Since predictors were z-transformed, the $\hat\beta$ coefficients indicate the change in ms per standard deviation of each predictor.[]{data-label="tab:futsurp_and_entropy"}
#### Entropy and successor surprisal account for different portions of the variance:
If successor surprisal is only predictive of reading time because it approximates entropy as hypothesized by @angeleetal15, then entropy should not only be predictive of reading time, but it should also obviate successor surprisal as a predictor since the approximation (successor surprisal) would only get credit for indirectly modeling part of the influence of entropy. To test this, we added entropy as a fixed effect and as a by-subject random slope to our linear-mixed effects model. Comparing the fit of that model to the fit of a model without each fixed effect of interest, we found that successor surprisal and entropy were both significant predictors of reading time (both $p < 0.001$; see Table \[tab:futsurp\_and\_entropy\]); thus the hypothesis that the effect of entropy should subsume the effect of successor surprisal was not borne out.
Bounded entropy
===============
Entropy and successor surprisal both accounted for independent portions of the variance in reading time in Section \[sec:results\]. Could they both provide indirect approximations of underlying reader uncertainty? So far we computed entropy over the complete distribution of possible upcoming words (*total entropy*). In this section, we explore the possibility that processing limitations cause readers to consider only the best $K$ continuations in the psychological process that causes uncertainty effects [see bounded rationality, @simon82; @jurafsky96]. If this is the case, then total entropy and its successor surprisal approximation could both be predictive of reading time because of their joint correlation with the bounded entropy computed by humans.
$K$ Successor surprisal Total entropy
------- --------------------- ---------------
5 0.212 0.541
50 0.335 0.820
500 0.397 0.947
5000 0.434 0.992
50000 0.454 1
: Correlations between (Center) best-$K$ entropy and successor surprisal and (Right) best-$K$ entropy and total entropy when best-$K$ entropy is computed over the most probable $K$ continuations.[]{data-label="tab:topkcorr"}
To test this hypothesis, we computed entropy over just the best 5, 50, 500, and 5000 continuations in every context. The full vocabulary size of the model was 50000 (plus an UNK token). Successor surprisal was always computed over the full vocabulary so that every observation could be assigned a successor surprisal value.
Entropy was most correlated with successor surprisal when both measures were computed over the entire vocabulary (Table \[tab:topkcorr\]). This is a plausible finding given Equation \[eqn:succsurpapprox\], which indicates that successor surprisal provides a Monte Carlo approximator of the entropy of that same distribution (recall that successor surprisal was calculated over the full vocabulary). It may still be the case, however, that reading time is best predicted by one of the bounded entropy measures. For example, best-50 entropy still has a moderate correlation to successor surprisal (0.335) and a strong correlation to total entropy (0.82); it is possible that total entropy and successor surprisal both predicted reading time thanks to an underlying joint correlation with best-50 entropy.
To test whether that is the case, we used our bounded entropy variants to predict reading time, following the procedure of Section \[sec:results\].[^2] Bounded entropy was a consistently poorer predictor of reading time than total entropy (see Table \[tab:topkspr\]). This suggests that humans may be sensitive to uncertainty over a large number of possible continuations. Moreover, successor surprisal improved as a predictor of reading time as $K$ decreased and the predictive value of bounded entropy weakened. This trade-off indicates that some of the variance in reading time is explained by both measures, which suggests that the predictivity of successor surprisal in previous studies was at least partially driven by reader uncertainty [in line with @angeleetal15]. However, the continued predictivity of successor surprisal in the presence of entropy indicates that there are likely other factors involved as well. For example, it may be that readers make predictions of varying granularity depending on context or attention level. That is, in cases where readers make a prediction based on the best $K$ continuations and $K$ is similar to the bound for computing entropy, then entropy may help predict reading time. Successor surprisal could help absorb variance due to a mismatch between reader $K$ and the model’s $K$.
$K$ $\hat\beta_{H}$ $\hat\sigma_{H}$ $\hat\beta_{s}$ $\hat\sigma_{s}$
------- ----------------- ------------------ ----------------- ------------------
5 3.10 0.70 3.89 0.53
50 3.27 0.71 3.81 0.54
500 3.89 0.70 3.65 0.54
5000 4.39 0.70 3.53 0.54
50000 4.57 0.70 3.48 0.54
: Entropy ($H$) and successor surprisal ($s$) coefficients in the Section \[sec:results\] RT regression model for the exploratory data partition, when $H$ is calculated over the $K$ most probable continuations.[]{data-label="tab:topkspr"}
Related work
============
Previously, @vanschijndelschuler17 performed a similar analysis to the reading time analysis in Section \[sec:results\] in this paper using probabilistic context-free language models. They were forced to compute entropy and successor surprisal with separate models because entropy computation using a grammar-based model requires estimation of uncertainty over both words and parsing actions, and is therefore very computationally expensive. While they also found that entropy and successor surprisal independently predicted reading time, their use of multiple language models means that the independent predictivity in their study could arise from differences in their underlying models instead of from multiple independent reading time influences. In contrast, we wanted to directly compare the measures as estimated by a single model to provide a stronger test of the original hypothesis of @angeleetal15.
@frank13 conducted a related reading time analysis which studied the relationship between entropy reduction [@hale06] and surprisal as computed by neural network language models. Entropy reduction is a measure of how uncertainty about the future changes after an observation compared to before that observation. Since entropy reduction involves the difference between two levels of uncertainty, it is a distinct measure from the amount of uncertainty (entropy) over upcoming observations which we studied in this paper. That is, the fact that uncertainty is reduced after an observation says nothing about the total amount of uncertainty experienced by a reader after that lessening takes place.[^3] @frank13 found that entropy reduction and surprisal are also distinct measures with independent reading time predictivity, similar to the findings of entropy and successor surprisal in the present paper.
@frank13 also tested how the relationship between entropy reduction and surprisal changed when the uncertainty used to estimate entropy reduction was computed over more than just the single next upcoming observation; he found that the predictive value of entropy reduction improves when entropy is computed over multiple future words. However, in the context of the present paper, @angeleetal15 observed a direct relationship between the predictability of a single word ($w_{t+1}$) on the reading time of the preceding word ($w_t$). Further, @vanschijndelschuler16 previously found that successor surprisal best predicts reading time when computed over just the upcoming one or two words even when parafoveal preview is possible, so it seems unlikely that computing entropy over longer upcoming sequences like @frank13 could explain the remaining successor surprisal influence on self-paced reading observed in this study. Therefore, since the goal of the present paper was to test the @angeleetal15 hypothesis that the entropy over $w_{t+1}$ could be the driving influence behind successor surprisal, we focused on testing the relationship between the reading time at $w_t$ and measures of entropy over $w_{t+1}$ and did not explore the influence of uncertainty over words beyond $w_{t+1}$.
Discussion
==========
This paper has used surprisal and entropy estimates from a neural network language model to test the hypothesis that successor surprisal effects in reading can be reduced to reader uncertainty. Successor surprisal and uncertainty accounted for partly non-overlapping portions of the variance in reading time. We interpret our finding of non-overlapping influences as a strong indictation that the predictivity of successor surprisal is not solely driven by uncertainty over the next word.
However, the portions of variance captured by entropy and successor surprisal are not completely disjoint: replacing entropy with bounded variants based on the best $K$ continuations led to weaker predictive power for entropy and a stronger relationship between successor surprisal and reading time, lending support to the hypothesis that entropy is at least a contributing factor in the predictivity of successor surprisal. Finally, the finding that uncertainty was a better predictor of reading time when it was computed over the entire vocabulary rather than just the best $K$ continuations suggests that readers may make a large number of continuation predictions simultaneously.
[^1]: We also ran all of the analyses reported in this paper on the exploratory partition without the random word intercept and obtained qualitatively similar results.
[^2]: For these analyses, we omit by-subject random slopes for sentence position, surprisal, and unigram frequency in order to ensure that all 5 models converge. Leaving all random slopes in the models produces similar qualitative results in those models that do converge.
[^3]: For example, $H(w_t) - 2 = H(w_{t+1})$ does not convey how large $H(w_t)$ or $H(w_{t+1})$ are. This amount of entropy reduction (2) could equally occur in a context of high uncertainty or in one of low uncertainty.
|
---
abstract: 'We have investigated the mechanism of stabilizing the simple-cubic ($sc$) structure in polonium ($\alpha$-Po), based on the phonon dispersion calculations using the first-principles all-electron band method. We have demonstrated that the stable sc structure results from the suppression of the Peierls instability due to the strong spin-orbit coupling (SOC) in $\alpha$-Po. We have also discussed the structural chirality realized in $\beta$-Po, as a consequence of the phonon instability. Further, we have explored the possible superconductivity in $\alpha$-Po, and predicted that it becomes a superconductor with $T_{c} \sim 4$ K. The transverse soft phonon mode at $\bf q \approx \frac{2}{3}$R, which is greatly influenced by the SOC, plays an important role both in the structural stability and the superconductivity in $\alpha$-Po.'
author:
- 'Chang-Jong Kang, Kyoo Kim, B. I. Min'
title: 'Phonon softening and Superconductivity triggered by spin-orbit coupling in simple-cubic $\alpha$-polonium'
---
Polonium (Po), which belongs to chalcogen group in the periodic table, is unique in that it crystallizes in the simple-cubic (sc) structure [@Beamer46]. Po having atomic number Z = 84 was discovered by Marie and Pierre Curie in 1898. Po exists in two metallic allotropes, $\alpha$ and $\beta$-Po. It is $\alpha$-Po that has a sc structure. Above 348 K, $\alpha$-Po transforms to $\beta$-Po, which has the trigonal structure [@Maxwell49]. Selenium (Se) and Tellurium (Te), which are isoelectronic elements to Po in the chalcogen group, also have the trigonal spiral structure [@Beister90]. Note that the trigonal structure can be derived from the $sc$ structure by elongation or contraction along the \[111\] direction [@Kresse94], which is known to occur due to the Peierls distortion in $p$-bonded systems of Se and Te [@Burdett83].
Recently, there have been several reports to explore the origin of the stabilized $sc$ structure in Po [@Min06; @Legut07; @Min09; @Legut09; @Legut10; @Verst10]. General consensus so far is that the large relativistic effects in Po play an important role. The Peierls instability tends to be suppressed in Po by the relativistic effects. However, there was controversy on the role of spin-orbit coupling (SOC) of $6p$ electrons. Min [*et al.*]{}[@Min06] claimed that the SOC in addition to the scalar-relativistic (SR) effects of mass-velocity and Darwin terms is essential, while Legut [*et al.*]{}[@Legut07; @Legut09] claimed that the SR effects are already enough to stabilize $sc$-Po. The refined and comprehensive band calculations by the former group, by considering all the relativistic corrections and different exchange-correlation functionals show that the SOC of $6p$ states is really important in stabilizing the $sc$ phase of Po [@Min09].
In general, the structural transition is induced by the phonon softening instability, and so the phonon dispersion calculations would provide evidence of the structural stability. Verstraete[@Verst10] calculated the phonon dispersion of $sc$-Po based on the pseudo-potential band method, and showed that the phonon softening instability does not occur even without including the SOC. This result seems to support Legut [*et al.*]{}’s claim[@Legut07] that the SR terms are sufficient to stabilize the $sc$ phase. Thereafter, a couple of more studies were reported on the phonon dispersion of the $\alpha$-Po at the ambient pressure[@Belabbes10] and under the pressure[@Zaoui11], by using the pseudo-potential band method. Since Po has a tiny energy difference, order of 1 meV, between the $sc$ and trigonal structures[@Min09], special caution is needed to calculate and analyze the phonon dispersions.
The structural energetics in Po is known to be very sensitive to the volume and the utilized exchange-correlation method in the band calculation[@Min09; @Legut10]. The lattice constant $a_{exp}$ of $\alpha$-Po was first measured by Beamer [*et al.*]{}[@Beamer49] to be 3.345 ${\AA}$. Later, Desando [*et al.*]{}[@Desando66] reported the refined lattice constant of $\alpha$-Po to be 3.359 ${\AA}$. In the above band and phonon dispersion calculations, the old $a^{old}_{exp}=3.345$ ${\AA}$ was referenced. In the present study, we have taken $a_{exp}=3.359$ ${\AA}$ as a reference. We have found that the small difference in $a_{exp}$ is important in analyzing the structural stability.
As mentioned above, the Peierls instability is the key physics in Po [@Min06; @Verst10; @Legut10; @Belabbes10]. Even if the Peierls instability is suppressed by the SOC, $\alpha$-Po in the $sc$ phase would still have the soft phonon mode and so the strong electron-phonon (EP) coupling. Then the natural question is whether $\alpha$-Po would have superconductivity or not. To our knowledge, superconductivity in Po has not been explored yet, probably due to its radioactive and toxic nature.
In this Letter, we have investigated phonon and superconducting properties of $\alpha$-Po, employing the first principles all-electron band methods. The phonon dispersions and superconducting properties of $\alpha$-Po are studied with and without the SOC at different volumes. By means of the phonon dispersion calculations, we have explicitly demonstrated that the stability of the sc-phase arises from the SOC, which concludes the longstanding dispute on the origin of stable sc structure of Po. Further, we have predicted that $\alpha$-Po is a superconductor with $T_{c}$ of $\sim 4$ K.
We have employed the FLAPW band method[@Freeman], implemented in the Elk package [@Elk]. The SOC was included in second-variational scheme and the local-density approximation (LDA) was used for the exchange-correlation energy. Actually, the choice of the LDA is more stringent test for the phonon stability than that of the generalized gradient approximation (GGA), because the GGA always gives more softened phonons than the LDA. For $sc$-Po, $a^0_{th}=3.335 {\AA}$ was obtained in the LDA+SOC scheme[@Band], which is in good agreement with $a_{exp}=3.359 {\AA}$ of $\alpha$-Po. Phonon dispersions were obtained by using the supercell method, implemented in the Elk package. The force constants and the dynamical matrix are obtained from the Hellmann-Feynman forces calculated with small individual displacements of nonequivalent atoms [@Yu91].
![(Color Online) Phonon dispersions of $sc$-Po at $a=a_{exp}$ with SOC (LDA+SOC: blue-solid line) and without it (LDA: red-dotted line). The imaginary phonon frequencies are represented as negative values in the figure.[]{data-label="ph-po1"}](fig1.eps){width="7.5"}
![(Color Online) (a) One of two degenerate normal mode at $\bf q \approx \frac{2}{3}$R in $sc$-Po. All the displacement vectors are placed on (111) planes. (b) The same normal mode shown along the \[111\] viewpoint, which manifests a clockwise helicity. (c) The helical chain structure of $\beta$-Po with definite chirality. Red and Green lines represent short bonds. (d) The difference of charge density of $sc$-Po in the LDA+SOC and that in the LDA (in unit of $e/{\AA}^3$). Notice the charge density depletion in-between neighboring ions. []{data-label="mode"}](fig2.eps){width="8.5"}
Figure \[ph-po1\] shows the phonon dispersions of $\alpha$-Po at $a=a_{exp}$ with and without the SOC [@Phonon]. In the LDA scheme without the SOC (corresponding to the SR band scheme), the phonon softenings occur along all the high symmetry directions. Among those, phonon frequency along $\Gamma$-R ($\bf q \approx \frac{2}{3}$R) is imaginary, which indicates the structural distortion along the \[111\] direction [@GGA]. In contrast, in the LDA+SOC scheme (corresponding to the fully-relativistic band scheme), the phonons become hardened and the imaginary phonon softening disappears. These features reflect that, without the $6p$ SOC, the $sc$ structure of $\alpha$-Po is unstable to a trigonal structure, as in Se and Te.
Note that, for the softened phonon at $\bf q \approx \frac{2}{3}$R, there are two degenerate normal modes having opposite helicities. The one has a clockwise helicity as shown in Fig. \[mode\](b), while the other has a counterclockwise helicity. Thus the softened phonon at $\bf q \approx \frac{2}{3}$R induces the structural transformation to the trigonal $\beta$-Po, which has the helical chain structure with definite chirality (Fig. \[mode\](c)). The chiral structure realized in $\beta$-Po was once proposed in Te [@Fukutome84], which was recently explained in terms of the orbital-ordered chiral charge-density wave (CDW) [@Wezel10; @Wezel11]. This orbital-ordered CDW would be suppressed, if the SOC were larger, as will be discussed in Fig. \[ph-sc-Te\]. The chirality of similar feature was also observed in $1T$-TiSe$_{2}$ [@Ishioka10]. Interestingly, both Te and $1T$-TiSe$_{2}$ exhibit superconductivity under high pressure [@Akahama92; @Mauri96] and Cu intercalation [@Morosan06], respectively, suggesting that the superconductivity emerges when the CDW is suppressed.
The phonon softening in Fig. \[ph-po1\] is closely related to the Peierls mechanism. The phonon softening is described by the renormalization of phonon frequency by the EP interaction (the so-called Kohn anomaly), $$\omega^2({\bf q})=\Omega^2({\bf q})-
|\tilde{g}_{ep}({\bf q})|^2\chi_{0}({\bf q}),$$ where $\omega({\bf q})$ and $\Omega({\bf q})$ correspond to renormalized and bare phonon frequencies, respectively, and $\tilde{g}_{ep}$ and $\chi_{0}({\bf q})$ are the EP coupling parameter and the charge susceptibility. Min [*et al.*]{}[@Min06] obtained that $\chi_{0}({\bf q})$ in $\alpha$-Po has the highest peak at $\bf q \approx \frac{2}{3}$R due to the Fermi surface nesting. They also showed that the SOC suppresses the Fermi surface nesting effect and accordingly the $\chi_{0}({\bf q})$ value, whereby they predicted that the phonon softening is weakened and finally the imaginary phonon softening disappears.
The suppression of the Peierls instability by the SOC occurs not only through the mechanism of Eq. (1) but also through weakening of the bonding strength. This feature is shown clearly in Fig. \[mode\](d), which plots the difference of the charge density in the LDA+SOC and that in the LDA. It is evident that, due to the SOC, the charge density is depleted in-between the neighboring ions, which results in the weakening of the directional bondings of Po chains.
![(Color Online) Phonon dispersion curves of $sc$-Te at the lattice constant of 3.210 ${\AA}$, without the SOC (LDA), with the SOC (LDA+SOC), and with the three times larger SOC (3$\xi^{Te}_{SO}$). []{data-label="ph-sc-Te"}](fig3.eps){width="7.5"}
![(Color Online) (a) The phonon dispersion curves of the $sc$-Po with and without the SOC at the reduced volume $V/V_{exp} = 0.93$ ($a = 3.277\AA$). The filled circle corresponds to the phonon mode at $\bf q \approx \frac{2}{3}$R. The overall phonon modes are hardened at the reduced volume. (b) The behavior of softened TA phonon mode at $\bf q \approx \frac{2}{3}$R in the LDA with varying the lattice constant. Here, $a = 3.335\AA$ ($V/V_{exp} = 0.98$) corresponds to $a^0_{th}$ of the LDA+SOC, while $a = 3.345\AA$ and $a = 3.359\AA$ correspond to the low temperature experimental lattice constants reported by Beamer et al. [@Beamer49] and by Desando et al. [@Desando66], respectively. $a = 3.366\AA$ (filled square) corresponds to the lattice constant at 298 K [@Lide]. []{data-label="ph-po2"}](fig4.eps){width="7.5"}
To verify the role of SOC in the structural stability more convincingly, we have examined the phonon dispersions of Te in the hypothetical $sc$ structure ($sc$-Te) with artificially varying the strength of SOC of Te ($\xi^{Te}_{SO}$). Figure \[ph-sc-Te\] shows that, in the LDA, two imaginary phonon softenings occur along the $\Gamma$-M and $\Gamma$-R directions, while, in the LDA+SOC, that remains only along the $\Gamma$-R direction. However, when we increases the SOC three times ($3\xi^{Te}_{SO}$), even the last imaginary phonon softening disappears eventually. This feature reveals that Te would also have a $sc$ structure, if the SOC of Te becomes larger. The situation of the three times larger SOC in Fig. \[ph-sc-Te\] actually simulates the case in Po, since the SOC of $6p$ electrons in Po ($\xi^{Po}_{SO}= 1.90$ eV) is almost three times stronger than that of $5p$ electrons in Te ($\xi^{Te}_{SO}= 0.72$ eV). The behavior in Fig. \[ph-sc-Te\] provide definite evidence that the SOC really plays a role of suppressing the Peierls instability.
At the reduced volume, phonons tend to be hardened, and so it is expected that the phonon softening observed in Fig. \[ph-po1\] would be reduced. Indeed, the phonon dispersions of $sc$-Po at $a = 3.277\AA$ ($V/V_{exp} = 0.93$) in Fig. \[ph-po2\](a) do not exhibit the imaginary phonon softenings both in the LDA and LDA+SOC schemes. Figure \[ph-po2\](b) shows the behavior of phonon mode at $\bf q \approx \frac{2}{3}$R as a function of the lattice constant. It is seen that, with decreasing the lattice constant, the phonon frequency rises very steeply in the vicinity of $a_{exp}$ (shaded area), and so the imaginary phonon softening disappears for $a < a_{exp}$. This is the reason why the phonon softening instability does not occur at $a^0_{th}=3.335\AA$ ($V/V_{exp} = 0.98$) of the LDA+SOC, as in the Verstraete’s work [@Verst10], but occurs at $a^0_{th}=3.411\AA$ ($V/V_{exp} = 1.05$) of the GGA+SOC [@Min06]. Moreover, since the $\alpha$ to $\beta$ transformation in Po occurs at 348 K, one might have to take into account the thermal expansion ($\alpha_T$) of the lattice constant[@Grabowski07; @Hatt10]. Considering $\alpha_T= 23.5$ $\mu m/m \cdot K$ of $\alpha$-Po at 298 K [@Lide], the lattice elongation amounts to $\sim 0.02$ $\AA$ at 300K, which is large enough to influence the phonon softening instability. In fact, the lattice constant of $\alpha$-Po at 298 K is reported to be 3.366 $\AA$ ($V/V_{exp} = 1.01$) [@Lide]. Then, without the SOC, $\alpha$-Po would be unstable to $\beta$-Po, as indicated by the filled blue square in Fig. \[ph-po2\](b).
We now consider the superconducting properties of the $\alpha$-Po [@superconduc]. In the Eliashberg theory, the average EP coupling constant is given by $$\lambda = 2\int_{0}^{\infty}d\omega\alpha^{2}F(\omega)/\omega$$ with the Eliashberg function expressed as $$\begin{aligned}
\alpha^{2}F(\omega)
& = & \frac{1}{N(E_F)}\sum_{\textbf{q}\nu}\sum_{\textbf{k},n,m}
|g_{\textbf{k}n,\textbf{k}+\textbf{q}m}^{\nu}|^{2} {}
\nonumber \\
& & \times~\delta(\epsilon_{\textbf{k}n})\delta(\epsilon_{\textbf{k}+\textbf{q}m})
\delta(\omega-\omega_{\textbf{q}\nu}).\end{aligned}$$ Here $N(E_F)$ is the DOS at $E_F$, and $\omega_{\textbf{q}\nu}$ denotes the frequency of the phonon mode $\nu$ with momentum $\textbf{q}$. The EP matrix element $g_{\textbf{k}n,\textbf{k}+\textbf{q}m}^{\nu}$ is obtained from $g_{\textbf{k}n,\textbf{k}+\textbf{q}m}^{\nu}
=\langle \textbf{k}n|\textbf{e}_{\textbf{q}\nu}
\cdot\nabla V_{KS}(\textbf{q})|
\textbf{k}+\textbf{q}m\rangle / \sqrt{2M\omega_{\textbf{q}\nu}}$, where $\textbf{e}_{\textbf{q}\nu}$ is the polarization vector and $\nabla V_{KS}(\textbf{q})$ is the gradient of the Kohn-Sham potential with respect to the atomic displacements with the wave vector **q**. Then the critical temperature $T_{c}$ is obtained with the McMillan-Allen-Dynes formula [@Allen75] : $$T_{c} = \frac{\omega_{\texttt{log}}}{1.20}
\texttt{exp}\bigg(-\frac{1.04(1+\lambda)}
{\lambda-\mu^{*}-0.62\lambda\mu^{*}}\bigg),$$ where $\omega_{\texttt{log}}~\Big(\equiv \texttt{exp}\big(\frac{2}{\lambda}
\int_{0}^{\infty}\frac{d\omega}{\omega}\alpha^{2}
F(\omega)\texttt{ln}\omega\big)\Big)$ is the logarithmic average frequency and $\mu^{*}$ is the effective Coulomb repulsion parameter. The estimated superconducting parameters are listed in Table \[Tc-table\].
------ ------------- ------------- -------------- -- ------------------------- ------- ------ --
$V/V_{exp}$ $N(E_F)$ $\theta_{D}$ $\omega_{\texttt{log}}$
(states/eV) (K) (meV)
LDA+ 1 0.63 103.7 6.66 3.38, 2.67
SOC 0.98 0.60 101.5 6.52 4.34, 3.78
0.93 0.57 113.3 6.52 5.57, 5.10
LDA 0.98 0.61 102.5 6.43 5.04, 4.25
0.93 0.57 110.6 6.69 6.28, 5.85
------ ------------- ------------- -------------- -- ------------------------- ------- ------ --
: DOS at $E_{F}$, $N(E_F)$, Debye temperature $\theta_{D}$, and the superconducting parameters of $\alpha$-Po. Two values of $T_{c}$ are for two different values of $\mu^{*}$=0.10 and 0.13.
\[Tc-table\]
Noteworthy in Table \[Tc-table\] is that $\alpha$-Po at the ambient pressure ($V/V_{exp} = 0.98$) would be a superconductor with $\lambda = 0.89$ and $T_c \sim 4 K$. At the reduced volume ($V/V_{exp} = 0.93$), both $\lambda$ and $T_{c}$ increase further to 1.04 and $\sim$ 5 K, respectively. With reducing the volume, $N(E_F)$ decreases, while the Debye temperature $\theta_D$ increases. The effect of the SOC on the superconductivity is found to be not so large but detrimental. For $V/V_{exp} = 0.93$, the SOC reduces $\lambda$ from 1.11 to 1.04, and $T_{c}$ from $\sim$ 6 K to $\sim$ 5 K. It is mainly because of the phonon hardening induced by the SOC, as discussed in Figs. \[ph-po1\] and \[ph-po2\]. This behavior in Po is quite opposite to that in neighboring element Pb, for which the SOC rather softens the phonons and increases $\lambda$ by more than 40% [@Verst08; @Corso08; @Heid10]. This difference is thought to arise from the different Fermi surface nature between two [@Verst08].
![(Color Online) The **q**-dependent EP coupling constants of $\alpha$-Po: (a) in the LDA+SOC for $V/V_{eq} = 0.98$, (b) in the LDA+SOC for $V/V_{eq} = 0.93$, (c) in the LDA for $V/V_{eq} = 0.93$. The Eliashberg functions $\alpha^{2}F(\omega)$ with the SOC at different volumes of $\alpha$-Po: (d) in the LDA+SOC for $V/V_{eq} = 0.98$, (e) in the LDA+SOC for $V/V_{eq} = 0.93$. Integrated Eliashberg functions $\lambda(\omega)$ are shown. []{data-label="elph"}](fig5.eps){width="8.5"}
To examine which phonon contributes largely to superconductivity, we computed the **q**-dependent EP coupling constant $\lambda_{\textbf{q}\nu}$. As shown in Figs. \[elph\](a)-(c), $\lambda_{\textbf{q}\nu}$’s show peaks at **q**’s having phonon softening anomalies along $\Gamma$-X, M-R, and $\Gamma$-R directions. Longitudinal X and transverse $\Gamma$-R phonon modes have fairly large values of $\lambda_{\textbf{q}\nu}$. We found above that $\lambda$ in $sc$-Po increases as the volume is reduced. In view of the phonon hardening at reduced volume, this behavior looks strange, at first glance. The enhanced $\lambda$ can be explained by the behavior of the Eliashberg function $\alpha^{2}F(\omega)$ in Figs. \[elph\](d) and (e). At the reduced volume, the maximum cut-off frequency is larger, and $\alpha^{2}F(\omega)$ is enhanced at low frequency. This leads to enhanced $\lambda$ and higher $T_c$. As shown in Figs. \[elph\](d) and (e), the integrated Eliashberg function $\lambda(\omega)$ has a steep rise at about 4 and 5 meV for $V/V_{exp} = 0.98$ and $V/V_{exp} = 0.93$, respectively. This energy range corresponds to that of the transverse $\Gamma$-R soft phonon mode (Fig. \[ph-po2\]), suggesting that the superconductivity of $sc$-Po comes largely from $\bf q \approx \frac{2}{3}$R phonon mode. In the LDA scheme of Fig. \[elph\](c), the $\bf q \approx \frac{2}{3}$R soft phonon mode produces the larger $\lambda_{\textbf{q}\nu}$, while, in the LDA+SOC scheme of Fig. \[elph\](b), the $\bf q \approx \frac{2}{3}$R phonon mode hardened by the SOC reduces $\lambda_{\textbf{q}\nu}$ by half. This is the reason why the SOC is detrimental to the superconductivity in Po.
In conclusion, based on the phonon dispersion calculations, we have explicitly demonstrated that the SOC is the origin of the stabilized $sc$ structure of $\alpha$-Po. We have also predicted that $\alpha$-Po would be a superconductor with $T_c \sim 4$ K. The soft transverse phonon modes at $\bf q \approx \frac{2}{3}$R in $\alpha$-Po, which are greatly influenced by the SOC, play an important role both in the structural stability of the $sc$ phase and its superconductivity. The experimental verification of the superconductivity in $\alpha$-Po is demanded.
This work was supported by the NRF (No.2009-0079947, No.2011-0025237) and the KISTI supercomputing center (No. KSC-2011-C2-36, KSC-2012-C3-09).
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|
---
abstract: 'By analysing the short-cadence *K2* photometry from the observing Campaign 12 we refine the system parameters of hot Jupiter WASP-28b and hot Saturn WASP-151b. We report the non-detection and corresponding upper limits for transit-timing and transit-duration variations, starspots, rotational and phase-curve modulations and additional transiting planets. We discuss the cause of several background brightening events detected simultaneously in both planetary systems and conclude that they are likely associated with the passage of Mars across the field of view.'
author:
- 'T. Močnik, C. Hellier, and D. R. Anderson'
bibliography:
- 'bibliography.bib'
title: '*K2* Looks Towards WASP-28 and WASP-151'
---
INTRODUCTION
============
Since the failure of the second reaction wheel, the *Kepler* mission was redesigned to observe fields along the ecliptic and renamed into *K2* [@Howell14]. This observing strategy not only minimizes the pointing drift of the spacecraft exerted by the solar radiation pressure, but also enables photometric follow-up observations of exoplanetary systems discovered by the ground-based surveys such as WASP [@Pollacco06]. The *K2* mission has so far observed 10 previously-discovered WASP planetary systems during the first 14 observing campaigns, each with a time-span of around 80d.
In this paper we present the results from analysing the short-cadence *K2* observations of WASP-28 [@Anderson15] and WASP-151 [@Demangeon17], both observed during the Campaign 12. Many close-in planets have been found to possess radii well in excess of the standard planet cooling and contraction model [@Leconte09]. Knowing precise system parameters is crucial for inferring their bulk composition and their dynamical history, and for understanding what causes the observed radius inflation.
The dearth of short-period Neptune-sized planets, also referred to as Neptunian desert, has indicated that hot Jupiters and hot super-Earths might have formed by different mechanisms (e.g. @Mazeh16). WASP-151 was found to lie on the upper boundary of Neptunian desert [@Demangeon17]. In this context, it is particularly beneficial to know parameters for planetary systems that lie close to the Neptunian desert to refine the desert boundaries.
Beside refining the system parameters, we used the *K2* short-cadence data to search for any transit-timing (TTV) and transit-duration variations (TDV), starspot occultations, phase-curve modulations and additional transiting planets. We also provide isochronal age estimates for both systems.
TARGETS
=======
WASP-28 is an inflated hot Jupiter, transiting an F8 $V=12.0$ star with an orbital period of 3.41d. The planet was discovered by @Anderson15 using the WASP photometry, two follow-up transit light curves and two sets of radial-velocity measurements obtained by CORALIE [@Queloz00] and in-transit HARPS [@Mayor03] observations. The system was further photometrically followed-up with ground-based telescopes first by @Petrucci15 who obtained four additional transit light curves and later by @Maciejewski16 with three additional transits. Both follow-up surveys refined the system parameters and reported the absence of any detectable long-term transit-timing variations.
WASP-151 is an inflated hot Saturn, transiting a G1 $V=12.9$ star with an orbital period of 4.53d. The discovery of the planet was recently announced by @Demangeon17. Beside WASP photometry, they used five follow-up transit light curves from ground-based telescopes and the raw *K2* data from the observing Campaign 12. The two accompanying RV datasets were obtained with SOPHIE [@Bouchy09] and CORALIE.
*K2* OBSERVATIONS AND DATA REDUCTION
====================================
WASP-28 and WASP-151 were both observed by *K2* in the 1-min short-cadence mode during the observing Campaign 12, which ran between 2016 December 15 and 2017 March 4.
The *K2* light curve for WASP-151 has already been presented in the discovery paper [@Demangeon17], but was generated only from the raw 30-min long-cadence data, which were made publicly available almost immediately after the downlink. The light curves presented in this paper were generated using the short-cadence target pixel files, provided following the official data release. Not only does the increased cadence of observations enable transit modelling with higher precision, but officially released data also undergo Science Operations Center’s calibration pipeline [@Quintana10]. This level-1 pipeline corrects the data in several ways, such as removing the readout smearing effect, subtracting modelled background, assigning quality flags, etc. Compared to the raw data, the level-1 pipelined target pixel files are used to generate more reliable light curves with lower white noise and to exclude data points which might be affected by cosmic rays or instrumental effects.
We downloaded the short-cadence target pixel files for both systems from the Mikulski Archive for Space Telescopes. We performed the photometric extraction with the PyKE command [@Still12] using a fixed aperture mask of 32 and 30 pixels, centered at WASP-28 and WASP-151, respectively. The pixel mask sizes were chosen by trial and error to extract as much stellar flux as possible while covering fewest background pixels.
The dominant systematic error present in the *K2* data is the sawtooth-shaped flux variations, caused by the pointing drift of the spacecraft and consequent pointing corrections by the thruster firings on roughly 6-h time-scales. To correct for these drift artefacts, we used the modified *K2* Systematic Correction pipeline (; @Aigrain16), optimised for the *K2* short-cadence data [@Mocnik17]. With the removal of drift artefacts, we improved the 1-min photometric precision from 1505 to 370ppm for WASP-28 and from 1317 to 641ppm for WASP-151.
We did not find any evidence of rotational modulations or any other low-frequency periodic light-curve modulations in the drift-corrected version of the light curves. Any other non-periodic low-frequency light-curve modulations were removed with the PyKE command, which divides the light curve with the best-fit second-order polynomial with step and window sizes of 0.3 and 3 days, respectively.
We present in Figures 1 and 2 the normalized and flattened light curves before and after the drift correction, with all the brightening events from Table 1 removed (see Section 3.1). The 5.3-d loss of data collection around BJD 2457790 occurred during the safe mode, which was likely triggered by the spacecraft’s software reset.
{width="8.5cm"}
{width="8.5cm"}
WASP-28 was also observed during the Engineering Campaign between 2014 February 4 and 13. We reduced these data in a similar way as for Campaign 12. Because the time stamps in the target pixel files from the Engineering Campaign are not corrected to the solar system barycenter, we applied the barycentric time correction ourselves with an IDL tool [@Eastman10]. We used the reduced light curve for visual inspection of any potentially interesting events, refinement of the ephemeris, and for the analysis of long-term TTVs. However, we avoided using the light curve for any other aspect of the analysis because the target pixel files from the Engineering Campaign were not fully calibrated by the level-1 pipeline and thus exhibit higher noise and uncertainty which might skew the results.
Background Brightening Events
-----------------------------
Upon visual inspection of the corrected light curves we found five episodes where the intensity of the background increased (these are listed in Table 1).
Two uniform background brightening events occurred at BJD 2457770 and 2457774 in both light curves, each lasting for $\sim$1h. A similar event affected the WASP-28 background at BJD 2457775 while a 4.8-h-long event affected WASP-151 at BJD 2457781. The longest brightening event lasted nearly 0.5d, and was non-uniform and seen only in WASP-28 light curve around BJD 2457776.
We believe that these background brightening events may have been caused by the passage of Mars through the field of view. This is surprising because both targets are located far from the areas on the *K2* detector affected by the direct, spilled or ghost images of Mars (WASP-28 was located on CCD module 9-1 and WASP-151 on module 15-4). All the uniform brightenings coincide within one hour with the centre of Mars entering or leaving certain CCD modules (see Table 1). Although this might be a coincidence, it suggests that the uniform brightening events may have been caused by some sort of instrumental effect such as a video crosstalk. We did not find any correlation between Martian CCD module crossings and the longest and non-uniform background brightening event in WASP-28, which may instead have been caused by a (reflected) ghost image of Mars.
In addition to the above background brightening events, the light curve of WASP-151 exhibited another brightening event centered at BJD 2457799. Inspection of individual images revealed that this was caused by an asteroid flyby, whose path crossed the center of WASP-151.
BJD $-$ 2457700 $F_{max}$ *cause*
----------------- ----------- -------------------------
**WASP-28**
70.022 - 70.058 0.0053 Mars enters module 24-3
74.312 - 74.401 0.0045 Mars leaves module 24-2
75.181 - 75.228 0.0053 Mars enters module 19-3
75.650 - 76.140 0.0097 ghost image of Mars (?)
**WASP-151**
70.006 - 70.064 0.0081 Mars enters module 24-3
74.303 - 74.367 0.0070 Mars leaves module 24-2
80.940 - 81.141 0.0063 Mars enters module 13-1
99.483 - 99.528 0.0075 asteroid flyby
: Light-curve brightening events (listed are the first and last BJD of when the event is taking place, relative peak brightening and a suspected cause for the event)
All the light-curve brightening events combined affected a total of 0.66d (0.9%) and 0.37d (0.5%) of WASP-28 and WASP-151 observations, respectively. We removed the brightening events from the light curves to avoid any potential impact on the following analysis.
SYSTEM PARAMETERS
=================
We obtained the system parameters with the simultaneous Markov chain Monte Carlo (MCMC) analysis of the transit light curves and radial velocity (RV) measurements. The MCMC procedure is presented in @CollierCameron07 and @Pollacco08. The photometric input for each system was the normalized and flattened *K2* Campaign 12 light curve from Section 3. The spectroscopic inputs were the publicly available RV measurements published in the corresponding discovery papers, @Anderson15 for WASP-28 and @Demangeon17 for WASP-151. Because the HARPS RV dataset for WASP-28 covered the planetary transit, we were able to fit also the Rossiter-McLaughlin effect. The four-parameter limb-darkening coefficients were interpolated through the tables of @Sing10.
The stellar masses were determined with the Bayesian mass and age estimator [@Maxted15]. compares the stellar density and stellar spectroscopic effective temperature to stellar evolution models, and applies the MCMC procedure to find best-fitting isochrone. The stellar densities were derived from the transit observables with the initial MCMC system parameter analyses and the stellar effective temperatures were taken from the corresponding discovery papers. The stellar masses obtained with the procedure were fed back into the main MCMC system parameter analysis as a two-step iteration process. The derived isochronal age estimates are given along with the system parameters in Tables 2 and 3.
Because the close-in planets are expected to circularise on very short time-scales, we imposed circular orbits in the main MCMC run as suggested by @Anderson12. In a consequent MCMC run, we then set the eccentricity to be fitted as a free parameter. This allowed us to estimate the eccentricity upper limit from the resulting MCMC chain.
To improve the ephemeris, we performed another MCMC analysis, which employed all the publicly available photometric datasets from the discovery and photometric follow-up papers. We also included the WASP-28 light curve from the *K2* Engineering Campaign from Section 3. As there were several different telescopes with different filters used to obtain these datasets, we sourced the additional limb-darkening coefficients from the appropriate tables of @Claret00 and @Claret04. This approach increased the observations baseline from 79 days to 8.3 years for WASP-28 and 8.6 years for WASP-151, which improved the orbital period precision by factors of 10.8 and 3.4, respectively.
The final MCMC transit models are shown in Figures 3 and 4, plotted over the measured *K2* Campaign 12 light curves of WASP-28 and WASP-151, respectively. The resulting system parameters are given in Tables 2 and 3, which also serve as a comparison to the previous studies. We find system parameters largely in agreement with the previous results, generally with significantly improved precision.
{width="8.5cm"}
{width="8.5cm"}
------------------------------ ------------------------------------ ----------------------------------- --------------------------- --------------------------- ------------------
this work @Anderson15 @Petrucci15 @Maciejewski16
Symbol$^{a}$ Value Value Value Value Unit
$t_{\rm 0}$ $2457642.502126\pm0.000014$ $2455290.40519\pm0.00031$ $2455290.40551\pm0.00102$ $2455290.40595\pm0.00046$ BJD
*P* $3.40883495\pm0.00000015$ $3.4088300\pm0.000006$ $3.408840\pm0.000003$ $3.4088387\pm0.0000016$ d
$(R_{\rm p}/R_{\star})^{2}$ $0.013396\pm0.000026$ $0.01300\pm0.00027$ ... ... ...
$t_{\rm 14}$ $0.13495\pm0.000083$ $0.1349\pm0.0010$ ... ... d
$t_{\rm 12}$, *t*$_{\rm 34}$ $0.01469\pm0.00010$ $0.01441\pm0.00070$ ... ... d
*b* $0.228\pm0.013$ $0.21\pm0.10$ ... $0.25^{+0.14}_{-0.25}$ ...
*i* $88.514\pm0.090$ $88.61\pm0.67$ ... $88.35^{+1.65}_{-0.92}$ $^{\circ}$
*e* 0 (adopted; $<$0.075 at $2\sigma$) 0 (adopted; $<$0.14 at $2\sigma$) ... ... ...
*a* $0.0442\pm0.0010$ $0.04469\pm0.00076$ $0.0445\pm0.0004$ $0.04464\pm0.00073$ au
$M_{\star}$ $0.993\pm0.067$ $1.021\pm0.050$ $1.011\pm0.028$ ... M$_\odot$
$R_{\star}$ $1.083\pm0.025$ $1.094\pm0.031$ $1.123\pm0.052$ $1.083^{+0.020}_{-0.044}$ R$_\odot$
$\rho_{\star}$ $0.7815\pm0.0068$ $0.784\pm0.058$ ... $0.804^{+0.018}_{-0.089}$ $\rho_\odot$
$M_{\rm p}$ $0.889\pm0.058$ $0.907\pm0.043$ $0.899\pm0.035$ $0.927\pm0.049$ $M_{\rm Jup}$
$R_{\rm p}$ $1.219\pm0.028$ $1.213\pm0.042$ $1.354\pm0.166$ $1.250^{+0.029}_{-0.053}$ $R_{\rm Jup}$
$\rho_{\rm p}$ $0.491\pm0.027$ $0.508\pm0.047$ ... $0.474^{+0.041}_{-0.065}$ $\rho_{\rm Jup}$
${T_{\rm p}}^{b}$ $1456\pm40$ $1468\pm37$ $1473\pm30$ ... K
$\lambda^{c}$ $6\pm17$ $8\pm18$ ... ... $^{\circ}$
$\tau_{iso}$ $5.5\pm2.6$ $5^{+3}_{-2}$ $4.2\pm1.0$ ... Gyr
$a_{\rm 1}$, $a_{\rm 2}$ 0.386, 0.602 ... ... ... ...
$a_{\rm 3}$, $a_{\rm 4}$ $-0.356$, 0.057 ... ... ... ...
------------------------------ ------------------------------------ ----------------------------------- --------------------------- --------------------------- ------------------
- Meanings of system parameter symbols are given in Table 3.
- Planet equilibrium temperature is based on assumptions of zero Bond albedo and complete heat redistribution.
- Projected spin-orbit misalignment angle was fitted with Hirano model for the Rossiter-McLaughlin effect.
-------------------------------------- ------------------------------ ----------------------------------- ------------------------------------ ------------------
this work @Demangeon17
Parameter Symbol Value Value Unit
Transit epoch $t_{\rm 0}$ $2457763.676241\pm0.000040$ $2457741.0081^{+0.0001}_{-0.0002}$ BJD
Orbital period *P* $4.5334775\pm0.0000023$ $4.533471\pm0.000004$ d
Area ratio $(R_{\rm p}/R_{\star})^{2}$ $0.010664\pm0.000048$ $0.01021^{+0.00010}_{-0.00007}$ ...
Transit width $t_{\rm 14}$ $0.15268\pm0.00023$ $0.15250^{+0.00083}_{-0.00042}$ d
Ingress and egress duration $t_{\rm 12}$, *t*$_{\rm 34}$ $0.01564\pm0.00027$ ... d
Impact parameter *b* $0.304\pm0.023$ ... ...
Orbital inclination *i* $88.25\pm0.14$ $89.2\pm0.6$ $^{\circ}$
Orbital eccentricity *e* 0 (adopted; $<$0.15 at $2\sigma$) $<$0.003 ...
Orbital separation *a* $0.0550\pm0.00084$ $0.055\pm0.001$ au
Stellar mass $M_{\star}$ $1.081\pm0.049$ $1.077\pm0.081$ M$_\odot$
Stellar radius $R_{\star}$ $1.181\pm0.020$ $1.14\pm0.03$ R$_\odot$
Stellar density $\rho_{\star}$ $0.656\pm0.014$ $0.72^{+0.02}_{-0.04}$ $\rho_\odot$
Planet mass $M_{\rm p}$ $0.316\pm0.031$ $0.31^{+0.04}_{-0.03}$ $M_{\rm Jup}$
Planet radius $R_{\rm p}$ $1.187\pm0.021$ $1.13\pm0.03$ $R_{\rm Jup}$
Planet density $\rho_{\rm p}$ $0.189\pm0.018$ $0.22^{+0.03}_{-0.02}$ $\rho_{\rm Jup}$
Planet equilibrium temperature$^{a}$ $T_{\rm p}$ $1312\pm16$ $1290^{+20}_{-10}$ K
Isochronal age estimate $\tau_{iso}$ $5.5\pm1.3$ $5.1\pm1.3$ Gyr
*K2* limb-darkening coefficients $a_{\rm 1}$, $a_{\rm 2}$ 0.599, $-0.116$ ... ...
$a_{\rm 3}$, $a_{\rm 4}$ 0.601, $-0.336$ ... ...
-------------------------------------- ------------------------------ ----------------------------------- ------------------------------------ ------------------
- Planet equilibrium temperature is based on assumptions of zero Bond albedo and complete heat redistribution.
NO TTV OR TDV
=============
The long-term deviations of transit timings from the expected ephemeris can be caused by an orbital decay due to the tidal interactions between a transiting planet and its host star (e.g. @Murgas14), whereas periodic TTVs are indicative of the gravitational influence of additional planetary or stellar companions in the system [@Holman05]. Similarly, the gravitational interactions can also produce TDVs, albeit at lower amplitudes [@Nesvorny13].
To measure the TTVs and TDVs from the *K2* datasets, we ran an MCMC analysis of system parameters as described in Section 4 on each transit individually, and subtracted the individual transit timings and durations from the ephemeris given in Tables 2 and 3.
@Petrucci15 and @Maciejewski16 have both provided a long-term TTV upper limit for WASP-28 of about 3min. Adding our TTV measurements from the *K2* data confirms the absence of long-term TTVs, with a weighted standard deviation of 27s and a $\chi_{red}^2$ of 1.18. We show in Figure 5 our TTVs from Campaign 12 and Engineering Campaign (listed in Table 4), along with the measurements from the previous surveys. One TTV measurement around BJD 2456510 from @Petrucci15 has been excluded as a $>$3$\sigma$ outlier.
{width="8.5cm"}
[ccc]{}\
WASP-28b&WASP-151b\
$6694.84601\pm0.00038$&$7741.00897\pm0.00025$\
$6698.25471\pm0.00012$&$7745.54219\pm0.00020$\
$6701.66380\pm0.00013$&$7750.07617\pm0.00020$\
$7741.35830\pm0.00010$&$7754.60937\pm0.00022$\
$7744.76713\pm0.00012$&$7759.14279\pm0.00020$\
$7748.17569\pm0.00015$&$7763.67639\pm0.00019$\
$7751.58485\pm0.00011$&$7768.20992\pm0.00023$\
$7754.99364\pm0.00012$&$7772.74284\pm0.00024$\
$7758.40255\pm0.00011$&$7777.27672\pm0.00021$\
$7761.81139\pm0.00010$&$7781.80995\pm0.00024$\
$7765.22007\pm0.00010$&$7795.41064\pm0.00024$\
$7768.62886\pm0.00012$&$7799.94414\pm0.00021$\
$7772.03784\pm0.00010$&$7804.47739\pm0.00023$\
$7775.44647\pm0.00010$&$7809.01082\pm0.00021$\
$7778.85563\pm0.00012$&$7813.54398\pm0.00025$\
$7782.26435\pm0.00012$&\
$7785.67313\pm0.00013$&\
$7792.49103\pm0.00012$&\
$7795.89953\pm0.00013$&\
$7799.30849\pm0.00009$&\
$7802.71741\pm0.00011$&\
$7806.12619\pm0.00010$&\
$7809.53528\pm0.00010$&\
$7812.94372\pm0.00017$&\
$7816.35264\pm0.00010$&\
We also do not find any statistically significant short-term periodicities or drifts in the TTV or TDV measurements during the Campaign 12 data alone in either of the two planetary systems. With the hypothesis of linear transit timings, the $\chi_{red}^2$ for TTVs were 1.09 for WASP-28 and 1.52 for WASP-151, with estimated semi-amplitude upper limits of 20s and 30s for periods shorter than 80d. For TDV, the assumption of constant transit durations yields the $\chi_{red}^2$ values of 1.02 for WASP-28 and 0.70 for WASP-151, with corresponding semi-amplitude upper limits of 50s and 80s. The non-detections of periodic TTVs or TDVs render the existence of any non-transiting close-in massive planets highly unlikely.
WASP-28 will be observed again during the Campaign 19 if the spacecraft’s propellant lasts long enough. Adding to the two existing *K2* observations, this will result in one of the best constraints on the multi-year period change of any hot Jupiter.
NO STARSPOTS
============
If a planet occults a starspot it produces a short in-transit brightening event (e.g. @Silva03). Recurring starspot occultation events can be used to derive a stellar rotational period and a constraint on the spin-orbit misalignment angle. Even the non-recurring occultations can in some configurations constrain the misalignment angle if the transit chord passes the active stellar latitudes at only certain transit phases, such as in the case of a misaligned HAT-P-11 system [@Sanchis11].
@Mocnik16a have demonstrated that the detection starspot occultation events is possible with two-wheeled *K2* observations, as revealed by the detection of several recurring starspot occultations in the *K2* light curve of an aligned WASP-85 system.
To search for occultation events in the *K2* light curves of WASP-28 and WASP-151 we subtracted the best fitting MCMC transit models and carefully visually examined the residual transit light curves for in-transit occultation “bumps”. We found no occultation events with amplitudes above conservative upper limits of 750 and 1000ppm, for WASP-28 and WASP-151, respectively.
The non-detection of occultation events is consistent with the spectral types of F8 for WASP-28 and G1 for WASP-151, and is in agreement with the absence of any detectable rotational modulation in both systems.
NO PHASE-CURVE MODULATIONS
==========================
In addition to the transit, the orbit of a planet around its host star can produce a reflectional modulation, Doppler beaming, ellipsoidal modulation, and secondary eclipse (e.g. @Esteves13), all of which can be detected at optical wavelengths. Because a secondary eclipse is a result of a temporarily-blocked reflectional modulation, its depth equals to the reflectional modulation’s amplitude.
To look for such effects we first removed any low-frequency non-periodic variabilities using the tool with window and step sizes of 3 and 0.3 days, for WASP-28, and 5 and 1 day, for WASP-151, respectively. The window and step sizes were chosen as the best compromise between the efficiency of removing the unwanted non-periodic variability and retaining as much phase-curve modulations signal as possible. Next, we phase-folded the flattened and normalized light curves and binned the phase curves to 200 bins.
We found no phase-curve modulations, and conservatively estimate the semi-amplitude upper limits to be 80 and 90ppm, for WASP-28 and WASP-151, respectively. These upper limits take into account the possible partial removal of the phase-curve signal by the light-curve flattening procedure, which we estimated by signal-injection and recovery tests. However, we are able to provide tighter semi-amplitude upper limits for the reflectional modulations from the absence of secondary eclipses, because the light-curve flattening procedure did not affect phase-curve signals at such short $\sim$3-hour-long time-scales. These are 40ppm for WASP-28 and 80ppm for WASP-151.
Using the system parameters from Section 4, the theoretically expected semi-amplitudes for reflectional modulation, Doppler beaming and ellipsoidal modulations are $166A_{\rm g}$ppm, 1.6ppm and 1.2ppm, respectively for WASP-28, where $A_{\rm g}$ is the planet’s geometric albedo (see the equations of @Mazeh10). For WASP-151 the predicted semi-amplitudes are $88A_{\rm g}$ppm, 0.5ppm and 0.3ppm. We can see that the non-detections of Doppler beamings and ellipsoidal modulations are not surprising, given their small predicted semi-amplitudes. On the other hand, the non-detection of secondary eclipses in WASP-28 system suggests that the planet’s albedo is smaller than 0.24. For WASP-151 the planet’s albedo remains almost unconstrained with an upper limit of around 0.9.
NO ADDITIONAL TRANSITING PLANETS
================================
To search for additional transiting planets, we first removed the transits of the known planet in each of the two light curves by replacing the transits’ normalized and flattened flux with unity. We then used PyKE’s tool, which calculates a box-least-square periodogram as formulated by @Kovacs02. The periodograms did not reveal any significant periodicity peaks in the period range between 0.5 and 30d with transit depths upper limits of 160ppm for WASP-28 and 220ppm for WASP-151.
CONCLUSIONS
===========
We analysed the short-cadence light curves of WASP-28 and WASP-151 which have been observed by the *K2* during Campaign 12. This work significantly refines the system parameters compared to the previous studies, owing to the short cadence and high photometric precision of the drift-corrected *K2* photometry. We did not detect any TTVs, TDVs, starspots, rotational modulations, phase-curve modulations or any additional transiting planets. Instead, we provide tight upper limits.
We observed several episodes where the *K2* background level is brighter than expected. We associated these with the passage of Mars through the *K2*’s field of view during the Campaign 12, although both targets were positioned on the detector far from any of the areas where the contamination by Martian light was expected. Instead, we think that the brightenings might be caused by some instrumental effect such as a video cross talk induced by the presence of Mars in the field. This might also be affecting other Campaign 12 datasets.
|
---
abstract: 'The aim of this paper is the development of consistent tests for the comparison of the distributions of two possibly dependent portfolios. The tests can be used to check whether the two portfolios are risk equivalent. The related testing problem can be endowed into a more general paired data framework by testing marginal homogeneity of bivariate functional data, or even paired random variables taking values in a general Hilbert space. To address this problem, we apply a Cramér-von-Mises type test statistic and suggest a bootstrap as well as permutation procedure to obtain critical values. The usually desired properties of a bootstrap and permutation test can be derived, that are asymptotic exactness under the null hypothesis and consistency under alternatives. Simulations demonstrate the quality of the tests in the finite sample case and confirm the theoretical findings. Finally, we illustrate the application of the approach by comparing real financial time series.'
address:
- '$^{\dagger}$ Department of Statistics, TU Dortmund University, 44221 Dortmund, Germany.'
- |
$^{\ddagger}$ Talanx AG, Group Risk Management, 30659 Hannover, Germany,\
and Institute of Probability and Statistics,\
Leibniz University Hannover, 30167 Hannover, Germany.
author:
- 'Marc Ditzhaus$^{\dagger}$ and Daniel Gaigall$^{\ddagger}$'
title: Tests for detecting risk equivalent portfolios
---
Introduction
============
Due to the availability of high frequency data, economic as well as financial quantities are described and modeled by random functions, so-called stochastic processes. Examples are stock prices, wages or electricity/water/gas consumption. Since classical methods are designed for vector-valued observations rather than for stochastic processes, they usually cannot be applied in this situation. The field of functional data tries to close this gap. For a detailed introduction to functional data we refer to [@ane17], [@ramsil2002; @ramsil2005], [@fervie], and [@hor]. As insinuate above, classical testing problems need to be revisited regarding observations coming from stochastic processes. One popular solution to tackle this problem is to project the random functions to the real line and then apply one of the classical methods. For example, [@cue2006] and [@cue2007] applied the Kolomogorov–Smirnov goodness-of-fit test to randomly projected square integrable functions. [@cuevas] extended this idea to more general spaces. [@DitzhausGaigall2018] did the same by discussing observations with values in a general Hilbert space. In this way, stochastic processes as well as high-dimensional data can be discussed simultaneously. More interaction between these two fields is desirable as stated by [@goiaVieu2016] and [@cuevas14] in the functional data community as well as by [@ahmed2017] from the high-dimensional side. Extending the idea of goodness-of-fit, [@bun] studied the testing problem whether the underlying distribution belongs to a pre-specified parametric family. In this paper, we use also the projection idea but we address a paired-sample testing problem. To be more specific, we suggest a procedure for testing marginal homogeneity. Hereby, we follow the general idea of [@DitzhausGaigall2018] to consider not just a few random projects but all projects from a sufficient large projection space. The advantage of this approach is that no additional randomness has an influence on the result of the test. With a view to the consistency of the testing procedure, we follow the idea in [@DitzhausGaigall2018], where a one-sample goodness-of-fit test is discussed, and apply a test statistic of Cramér-von-Mises type. See [@and] and [@ros] for Cramér-von-Mises tests in the usual cases of real-valued random variables and random vectors with real components. However, the demand for consistency has a price: the distribution of the test statistic under the null hypothesis is unknown and so related quantiles are not available in practice. To solve this problem, we offer a bootstrap as well as a permutation procedure to determine critical values. Permutation tests were already used by [@hal] and [@BugniHorwitz2018] for the unpaired two-sample setting under functional data. The procedure of [@BugniHorwitz2018] even applies to the more general situation of one control group against several treatment groups. In contrast to the unpaired setting, exchangeability is not given in general under the null hypothesis of marginal homogeneity. This is already known for bivariate distributions on ${\mathbb{R}}^2$, see [@gai]. Nevertheless, we can show that the permutation as well as the bootstrap versions of our test keep the nominal level asymptotically under the null hypothesis. Moreover, we prove the consistency of our approach with respect to the bootstrap and permutation procedure under any alternative. While also other applications are possible, for instance to high-dimensional data, we especially focus on functional data in the financial field. In particular, we explain how our Cramér-von-Mises type test can be used to check whether two portfolios are risk equivalent. Thereby, our approach yields also a method to cover seasonality effects in the time series. The paper is structured as follows: We first introduce the model and our general null hypothesis of marginal homogeneity in the paired sample setting. Moreover, we explain how to embed the question of risk equivalent portfolios into this framework. In Section \[sec:stat\], we introduce a Cramér-von-Mises type test for the aforementioned testing problem and derive its asymptotic behavior. The resulting asymptotic law under the null hypothesis can be transfered to a bootstrap as well as a permutation counterpart of the test statistic. In addition to these theoretical findings, we study the small sample performance of the two resampling tests in a numerical simulation study presented in Section \[sec:sim\]. Finally, an application to financial time series, namely historical values of the Nikkei Stock Average, Dow Jones Industrial Average, and Standard & Poor’s 500 is demonstrated in Section \[sec:data\_exam\]. All proofs are conducted in the Appendix.
The model {#sec:model}
=========
Let $H$ be a general separable Hilbert space over ${\mathbb{R}}$ with inner product $\langle \cdot,\cdot \rangle$ and countable orthonormal basis $O=\lbrace e_i;i\in I\rbrace$, where $e_i$ is the $i$-th basis element and the index set $I$ is given by the natural numbers $I={\mathbb{N}}$ or the subset $I=\{1,\dots,|I|\}\subset{\mathbb{N}}$. While the theory is valid for general $H$, we are mainly interested, regarding application to financial and economic observations, on the specific space $H=L^2[0,T]$ containing all measurable and square integrable real-valued functions on the interval $[0,T]$ of length $T\in(0,\infty)$ and equipped with the usual inner product $\langle f, g \rangle = \int_0^T f(x)g(x) \,\mathrm{ d }x$, $f, g\in H$. In that case, we may choose a corresponding orthonormal basis given by normalized Legendre polynomials. Now, let paired observations be given $$\begin{aligned}
X_{j}= (X_{j,1},X_{j,2} ),~j=1,\dots,n,
\end{aligned}$$ that are random variables with values in $H\times H$. In our main application, these observations obtained from the time series of the values of two assets or two portfolios. As detail explained in the following section, we split the time series into equal sized time intervals and obtain by this the $n$ pairs $X_1,\ldots,X_n$ as increment processes or log-return processes. Usual financial models ensure that $X_1,\ldots,X_n$ are independent and identical distributed (i.i.d.), which we suppose throughout the rest of the paper. We suppose that the distribution $P^{X_1}$ of $X_1$ is unknown. While we allow any dependence structure between $X_{1,1}$ and $X_{1,2}$, we like to infer the null hypothesis of marginal homogeneity $$\begin{aligned}
\mathcal H: P^{X_{1,1}} = P^{X_{1,2}} \quad \text{versus}\quad \mathcal K: P^{X_{1,1}} \neq P^{X_{1,2}}.
\end{aligned}$$ In the context of portfolios, $\mathcal H$ may represent the null hypotheses of risk equivalent portfolios as we explain subsequently.
Risk equivalent portfolios {#sec:port}
--------------------------
We consider two portfolios and a time period $[0,T_0]$, where $T_0\in(0,\infty)$ is a time horizon. The related value process of portfolio $i$ is denoted by $$\begin{split}
\Pi_{i}=(\Pi_{i}(t);t\in [0,T_0]),~ i=1,2,
\end{split}$$ where $\Pi_{i}$ is a random variable which takes values in the space of all measurable and square integrable real-valued functions on $[0,T_0]$. For the comparison of $\Pi_1$ and $\Pi_2$, we have only one path for each portfolio available. Statistical inference on the basis of a single observation is difficult, to say the least. Often, additional structural assumptions on the underlying stochastic process we are working with are available. Without such structural assumptions, reasonable statistical analysis is impossible. In the famous random walk model, exponential Lévy model, Black-Scholes model, and Merton model, the structural assumptions mentioned are independence and stationarity of the increments of a Lévy process. Seasonality effects, that are specific trends during certain periods of the time series, can disturb such structural assumptions. Figure \[fig1\] shows the mean monthly indices (open) of the Nikkei Stock Average (Nikkei 225), Dow Jones Industrial Average (DJIA), and Standard & Poor’s 500 (S&P 500) from 01/01/1999 to 01/01/2019. As already mentioned above, we will tackle this problem by splitting up the time horizon into $n$ equal sized time intervals to obtain $n$ observations for each portfolio. To be more specific, let $T_0=nT$ for $T\in(0,\infty)$ then we consider the time periods $[0,T],\dots,[(n-1)T,nT]$ and our observations are the increments during these periods $$\begin{aligned}
X_{j,i} = \Pi_{i}(t+(j-1)T) - \Pi_{i}((j-1)T),~t\in[0,T],~i=1,2,~j=1,\dots,n,
\end{aligned}$$ which are themselves measurable and square integrable real-valued functions on $[0,T]$. The structural assumptions in the popular models mentioned imply an i.i.d. structure of our observations $X_{j,i}$. Under the random walk hypothesis these increments are directly i.i.d.; in the Black-Scholes model and the Merton model, this is also satisfied after applying the one-to-one transformation $$\begin{split}
(\Pi_{1},\Pi_{2})\longmapsto (\log \Pi_{1},\log \Pi_{2}),
\end{split}$$ i.e., if we use log-returns instead of increments. We assume in addition that the increments are independent of the start values of the portfolios. To judge the risk of a portfolio let us introduce the set of law-invariant risk measures $\mathcal R$, where $\rho \in\mathcal R$ is a map from $L^2[0,T_0]$ into ${\mathbb{R}}$. We say two portfolios are risk equivalent if $$\begin{aligned}
\label{eqn:risk_equi}
\rho(\Pi_1|\Pi_1(0)=\pi)=\rho(\Pi_2|\Pi_2(0)=\pi) \text{ for all }\rho \in\mathcal R \text{ and }\pi \in{\mathbb{R}},
\end{aligned}$$ i.e., if they have the same risk when starting at the prize $\pi$. Risk equivalence can be expressed by the increments $X_{j,i}$.
\[thm:marg\] The following statements are equivalent.
\[enu:thm:marg:req\]
(i) \[enu:thm:marg:req\_L(pi)\] We have $P^{\Pi_1|\Pi_1(0)=\pi}= P^{\Pi_2|\Pi_2(0)=\pi}$ for all $\pi\in{\mathbb{R}}$.
(ii) \[enu:thm:marg:req\_req\] The portfolios $\Pi_1$ and $\Pi_2$ are risk equivalent in the sense of .
(iii) \[enu:thm:marg:req\_inc\] The increments of both portfolios follow the same distribution, i.e. $P^{X_{1,1}}=P^{X_{1,2}}$.
![[Mean monthly values from 01/01/1999 to 01/01/2019.]{}[]{data-label="fig1"}](Nikkei_seas.eps){width="1.0\linewidth"}
![[Mean monthly values from 01/01/1999 to 01/01/2019.]{}[]{data-label="fig1"}](DJIA_seas.eps){width="1.0\linewidth"}
![[Mean monthly values from 01/01/1999 to 01/01/2019.]{}[]{data-label="fig1"}](SP500_seas.eps){width="1.0\linewidth"}
Test statistic {#sec:stat}
==============
As postulated in the introduction, we project first the processes $X_{j,i}$ to the real line and then apply a Cramér-von-Mises type test. Projection is done via the inner product, i.e., we consider $\langle X_{j,i},x\rangle$ for $x\in H$. As already done by [@DitzhausGaigall2018], we consider all projections $x$ from a sufficient large projection space $h\subset H$. In fact, as explained in [@DitzhausGaigall2018], the distributions of $X_{1,1}$ and $X_{1,2}$ coincide if and only if $\langle X_{1,1}, x\rangle$ and $\langle X_{1,2}, x \rangle$ have the same distribution for all projections $x\in h$, where $$\begin{aligned}
h = \Bigl\{ \sum_{j=1}^k m_{j} e_{i_j}; k\in I,i_1,\ldots, i_k\in I, i_1< \ldots < i_k, \sum_{j=1}^k m_{j}^2 = 1 \Bigr\}.
\end{aligned}$$ This motivates the following test statistic: $$\label{eqn:CVM}
{{\rm \operatorname {CvM}}}_n=\int D_n(x)\mathcal P(\mathrm d x),$$ where $\mathcal P$ is a suitable probability measure on the projection space $h$ and $D_n(x)$ is the usual two-sample Cramér-von-Mises distance when applying the projection $x\in h$. Let $$\begin{split}
F_{n,i }(x,r)=\frac 1 n\sum_{j=1}^n {{\mathbf{1}}}_{\langle x,X_{j,i}\rangle \le r},~(x,r)\in H\times {\mathbb{R}}, ~i=1,2,
\end{split}$$ be the empirical distribution function of the real-valued random variables $ \langle x,X_{1,i}\rangle,\dots,\langle x,X_{n,i}\rangle$. Then the related two-sample Cramér-von-Mises distance is given by $$\begin{split}
D_n(x)= n \int [ F_{n,1}(x,r)- F_{n,2}(x,r)]^2\bar F_n(x,\mathrm d r),
\end{split}$$ where $\bar F_n = (F_{n,1}+ F_{n,2})/2$. The probability measure $\mathcal P$ can be chosen arbitrarily in advance as long as some regularity assumptions are fulfilled. While [@DitzhausGaigall2018] discussed general conditions on this measure such that the procedure works, we restrict here to the following specific proposal. It is based on two probability measures $\nu_1$ and $\nu_2$ on the index set $I$ such that $\nu_j(\{i\})>0$ for all $i\in I$. In the case of functional data, we can choose shifted Poisson distributions, for instance. In what follows, we specify the probability measure $\mathcal P$ by determining the procedure to generate a realization of $\mathcal P$. This procedure is also useful to obtain the concrete value of the test statistic by Monte-Carlo simulation in applications.
[<span style="font-variant:small-caps;">Step</span> .]{}
Generate a realization $k\in I$ of the distribution $\nu_1$.
Independently of Step 1, generate $i_1,\dots,i_k\in I$ by $k$-times sampling without replacement from the distribution $\nu_2$.
Independently of Steps 1 and 2, generate a realization $(m_1,\dots,m_k)$ of the uniform distribution on the unit circle in ${\mathbb{R}}^{k}$.
Set $x=\sum_{j=1}^k m_je_{i_j}$.
Asymptotics {#sec:asy_uncon}
-----------
For our asymptotic approach, we let $n\to\infty$. In the context of portfolios described in Section \[sec:port\], we suppose that the time horizon $T_0$ tends to $\infty$ leading to a growing number of observed time periods of length $T$. It is well known that the Cramér-von-Mises distance $D_n$ is connected to von Mises’ type functionals, also known as V-Statistics, which are closely related to U-Statistics. For a deeper introduction to these kinds of statistics, we refer the reader to [@kor] and [@serf]. Our statistic ${{\rm \operatorname {CvM}}}_n$ can also be rewritten into a certain V-Statistic and, thus, the same theory can be applied to obtain the following result.
\[theo:asym\_S\] Let $\tau_1,\tau_2,\ldots$ be a sequence of independent standard normal distributed random variables. Under the null hypothesis $\mathcal H$, $$\begin{aligned}
\label{eqn:null_conv}
{{\rm \operatorname {CvM}}}_n \overset{ d}{\to} \sum_{i=1}^\infty \lambda_i (1+\tau_i^2) = Z,
\end{aligned}$$ where $(\lambda_i)_{i\in{\mathbb{N}}}$ is a sequence of non-negative numbers with $\sum_{i=1}^\infty \lambda_i< \infty$ and $\lambda_{i}>0$ for at least one $i\in{\mathbb{N}}$ implying that the distribution function of $Z$ is continuous and strictly increasing on the non-negative half-line.
\[theo:cons\_S\] Under the alternative $\mathcal K$, our statistic ${{\rm \operatorname {CvM}}}_n$ diverges, i.e., ${{\rm \operatorname {CvM}}}_n\overset{p}{\rightarrow}\infty$ as $n\to\infty$.
In general, the test statistic ${{\rm \operatorname {CvM}}}_n $ is not distribution-free under the null hypothesis, i.e., the distribution depends on the unknown distribution of $X_1$. As it can be seen in the proofs, the same applies to $Z$. Given that $\alpha\in(0,1)$ is the significance level, neither a $(1-\alpha)$-quantile $c_{n,1-\alpha}$ of ${{\rm \operatorname {CvM}}}_n $ nor the $(1-\alpha)$-quantile $c_{1-\alpha}$ of $Z$ is available as critical value in applications. To resolve this problem, we propose the estimation of the quantiles via bootstrapping or permutation.
Bootstrap procedure {#sec:boot}
-------------------
We propose a bootstrap procedure in the spirit of [@efron] and follow the idea in [@gai], where the usual two-sample Cramér-von-Mises distance is applied to bivariate random vectors with values in ${\mathbb{R}}^2$. Note that under the null hypothesis $\mathcal H$ the expectations ${ {E}}[ F_{n,1}(x,y)]={ {E}}[ F_{n,2}(x,y)],~(x,y)\in H\times {\mathbb{R}}$, coincide and, thus, we can rewrite our test statistic into $${{\rm \operatorname {CvM}}}_n=n\int \int \{ F_{n,1}(x,y)-{ {E}}[ F_{n,1}(x,y)]+{ {E}}[ F_{n,2}(x,y)]- F_{n,2}(x,y)\}^2\bar F_n(x,\mathrm d y)\mathcal P(\mathrm d x).$$ Denote by $X_{jn}^*= (X_{jn,1}^*,X_{jn,2}^* ),~j=1,\dots,n$, a bootstrap sample from the original observations $X_j$, $j=1,\dots,n$, obtained by $n$-times sampling with replacement. Let $F_{n,i}^*$, $\bar F_{n}^*$ be the bootstrap counterparts of $F_{n,i}$ and $\bar F_n$. Clearly, ${ {E}}[ F_{n,i}^*(x,y)]= F_{n,i}(x,y)$. Consequently, the bootstrap counterpart of our test statistic is $${{\rm \operatorname {CvM}}}_{n}^*=n\int \int\Big( F_{n,1}^*(x,y)- F_{n,1}(x,y)+ F_{n,2}(x,y)- F_{n,2}^*(x,y)\Big)^2\bar F_{n}^*(x,\mathrm d y)\mathcal P(\mathrm d x).$$ Let $c_{n,1-\alpha}^*$ be a $(1-\alpha)$-quantile of ${{\rm \operatorname {CvM}}}_n^*$ given the original observations $X_1,\ldots,X_n$. In applications, concrete values of $c_{n,1-\alpha}^*$ are obtained by Monte-Carlo simulation. In the proofs, we show that the bootstrap statistic mimics asymptotically the limiting null distribution under the null hypothesis $\mathcal H$ implying that $c_{n,1-\alpha}^*$ is an appropriate estimator forthe unknown quantile $c_{n,1-\alpha}$ or $c_{1-\alpha}$, while ${{\rm \operatorname {CvM}}}_n^*$ and $c_{n,1-\alpha}^*$ remain asymptotically finite under general alternatives. This results in an asymptotically exact and consistent bootstrap test $\varphi_n^*={{\mathbf{1}}}_{{{\rm \operatorname {CvM}}}_n>c_{n,1-\alpha}^*}$.
\[theo:boot\] As $n \to \infty$ we have ${ {E}}[\varphi_n^*]={P}({{\rm \operatorname {CvM}}}_n>c_{n,1-\alpha}^*) \to \alpha {{\mathbf{1}}}_{\mathcal H}+ {{\mathbf{1}}}_{\mathcal K}$.
Permutation procedure {#sec:perm}
---------------------
In addition to the bootstrap approach, we provide a permutation procedure for estimating the unknown quantile $c_{n,1-\alpha}$ or $c_{1-\alpha}$. Permutation is a well-established tool for two-sample settings, see for example [@hal] and [@BugniHorwitz2018] for the unpaired two-sample testing problem in the context of functional data analysis. In contrast to the unpaired situation, where the group memberships are randomly mixed up, we here permute just within the pair $X_{j}=(X_{j,1},X_{j,2})$. To be more specific, the permutation observation $X_{j}^\pi = (X_{j,1}^\pi, X_{j,2}^\pi)$ equals either $(X_{j,1},X_{j,2})$ or $(X_{j,2},X_{j,1})$ both with probability 0.5. Let $F_{n,i}^\pi$ be the permutation counterparts of $F_{n,i}$, respectively. Note that $\bar F_n$ is not affected by permuting the pairs. In contrast to the bootstrap approach, we do not need to include the expectations in the test statistics because we clearly have ${ {E}}[ F_{n,1}^\pi(x,y)]={ {E}}[ F_{n,2}^\pi(x,y)] = \bar F_{n}(x,y)$, which remains true even beyond the null hypothesis. Consequently, the permutation version of our test is given by $${{\rm \operatorname {CvM}}}_n^\pi=n\int \int \{ F_{n,1}^\pi(x,y) - F_{n,2}^\pi(x,y)\}^2\bar F_n(x,\mathrm d y)\mathcal P(\mathrm d x).$$ Let $c_{n,1-\alpha}^\pi$ be a $(1-\alpha)$-quantile of ${{\rm \operatorname {CvM}}}_n^\pi$ given the original observations $X_1,\ldots,X_n$. In applications, concrete values of $c_{n,1-\alpha}^*$ are obtained by Monte-Carlo simulation as in the bootstrap procedure. The permutation method results in an asymptotically exact as well as consistent test. Moreover, the permutation test is even finite exact under the additional exchangeability assumption.
\[theo:perm\] As $n\to\infty$ we have ${ {E}}[\varphi_n^\pi]={P}({{\rm \operatorname {CvM}}}_n>c_{n,1-\alpha}^\pi) \to \alpha {{\mathbf{1}}}_{\mathcal H}+ {{\mathbf{1}}}_{{\mathcal K}}$.
Simulations {#sec:sim}
===========
-------------------------- -------------------------- ------ ------ ------ ------ ------
$X_{j,1}(t)$ $X_{j,2}(t)$ $r$ Perm Boot Perm Boot
$B_{j,1}(t)$ $B_{j,2}(t)$ 0 4.9 5.2 9.5 10.7
0.25 4.0 3.4 10.0 9.0
0.5 5.3 3.9 11.0 8.4
$1.5 B_{j,1}(t)$ $1.5 B_{j,2}(t)$ 0 3.8 4.5 8.9 9.6
0.25 4.6 4.4 9.1 9.1
0.5 5.5 4.0 9.6 7.9
$2B_{j,1}(t)$ $2B_{j,2}(t)$ 0 4.8 5.2 10.9 12.0
0.25 3.9 3.9 9.0 8.5
0.5 4.7 3.7 9.0 7.1
$2.5B_{j,1}(t)$ $2.5B_{j,2}(t)$ 0 5.1 5.5 10.3 10.3
0.25 5.9 5.5 10.9 10.3
0.5 5.0 4.3 9.1 7.6
$B_{j,1}(t)+0.5 t (1-t)$ $B_{j,2}(t)+0.5 t (1-t)$ 0 4.8 5.1 9.2 10.0
0.25 5.0 4.7 10.7 9.9
0.5 4.6 3.2 8.6 7.2
$B_{j,1}(t)+t (1-t)$ $B_{j,2}(t)+ t (1-t)$ 0 4.9 5.2 9.4 10.3
0.25 3.8 4.0 9.1 8.3
0.5 3.9 2.6 7.2 6.4
$B_{j,1}(t)+1.5 t (1-t)$ $B_{j,2}(t)+1.5 t (1-t)$ 0 4.7 5.2 9.6 9.9
0.25 5.1 5.1 9.5 9.5
0.5 4.2 2.9 8.7 7.7
$B_{j,1}(t)+2 t (1-t)$ $B_{j,2}(t)+2t (1-t)$ 0 5.0 5.8 9.7 10.7
0.25 5.7 5.2 11.6 11.2
0.5 4.6 3.7 10.4 8.1
-------------------------- -------------------------- ------ ------ ------ ------ ------
: \[tab1\]Empirical sizes.
--------------- -------------------------- ------ ------ ------ ------ ------
$X_{j,1}(t)$ $X_{j,2}(t)$ $r$ Perm Boot Perm Boot
$ B_{j,1}(t)$ $1.5 B_{j,2}(t)$ 0 10.0 10.7 20.3 19.9
0.25 11.2 10.0 22.3 21.0
0.5 17.7 12.0 30.5 25.2
$B_{j,1}(t)$ $2B_{j,2}(t)$ 0 22.9 22.6 43.1 42.9
0.25 43.3 36.3 61.1 58.7
0.5 60.1 50.6 78.1 73.1
$B_{j,1}(t)$ $2.5B_{j,2}(t)$ 0 50.2 47.2 74.9 74.3
0.25 67.9 62.5 86.7 85
0.5 90 84.7 95.8 94.3
$B_{j,1}(t)$ $B_{j,2}(t)+0.5 t (1-t)$ 0 13.9 14.1 20.9 21.2
0.25 13.5 13.0 22.8 21.6
0.5 19.5 17.0 27.8 24.4
$B_{j,1}(t)$ $B_{j,2}(t)+ t (1-t)$ 0 33.4 35.2 44.8 46.9
0.25 45.0 45.1 57.0 57.0
0.5 57.1 52.8 70.0 66.4
$B_{j,1}(t)$ $B_{j,2}(t)+1.5 t (1-t)$ 0 65.9 68.7 77.8 78.8
0.25 77.2 78.7 85.6 86.4
0.5 89.3 87.8 94.3 93.5
$B_{j,1}(t)$ $B_{j,2}(t)+2t (1-t)$ 0 90.4 92.6 95.7 96.0
0.25 95.4 95.8 97.3 97.4
0.5 98.8 98.9 99.4 99.4
--------------- -------------------------- ------ ------ ------ ------ ------
: \[tab2\] Empirical power values.
We aim to verify our theoretical results. Remembering that our tests are suitable for random variables $X_{i,j}$, $i=1,2$, $j=1,\dots,n$, with values in a general separate Hilbert space, we consider the separable Hilbert space $ H$ consisting of all measurable and square integrable functions on the unit interval $[0,1]$. This space is endowed with the usual inner product $\langle \cdot,\cdot \rangle$ and the normalized Legendre polynomials build a corresponding orthonormal basis $O=\lbrace e_i;i\in I\rbrace$, $I={\mathbb{N}}$. We obtain our test statistic by Monte-Carlo simulation based on 500 replications following [<span style="font-variant:small-caps;">Step</span>]{} 1–4 from Section \[sec:stat\]. Thereby, we choose in [<span style="font-variant:small-caps;">Step</span>]{} 1 and [<span style="font-variant:small-caps;">Step</span>]{} 2 a standard Poisson distribution shifted by $1$, i.e., the distribution of $N+1$ for $N\sim \text{Pois}(1)$. In our simulations, the stochastic processes have the form $$\begin{aligned}
X_{j,i}(t)=a_iB_{j,i}(t)+b_it(t-1),~t\in [0,1],~i=1,2,~j=1,\dots,n
\end{aligned}$$ for parameters $a_i\in {\mathbb{R}}\setminus \{0\}$ and $b_i\in{\mathbb{R}}$ and independent bivariate Brownian bridges $B_j=(B_{j,1}, B_{j,2})$ on $[0,1]$, $j=1,\dots,n$, with covariance structure $$\begin{aligned}
{{\rm Cov}}(B_{j,1}(s),B_{j,2}(t))=r(\min(s,t)-st),~s,t\in[0,1],~j=1,\dots,n
\end{aligned}$$ for a dependency parameter $r\in[0,1]$. Each simulation is based on 1000 simulation runs. To obtain the critical values in the permutation or bootstrap procedure, we use Monte-Carlo simulation based on 999 replications. Empirical size and power values of the bootstrap (Boot) and permutation (Perm) tests are displayed in Tables \[tab1\] and \[tab2\], respectively. The simulations are conducted for parameters $r\in\{0,0.25,0.5\}$, $a_i\in\{1,1.5,2,2.5\}$, and $b_i\in\{0,0.5,1,1.5,2\}$, the sample size $n=20$, and significance levels $\alpha\in\{5 \%, 10 \%\}$. The empirical sizes are in almost all cases in a reasonable range around the nominal level $\alpha$. A systematic exception from this observations are the sizes of the bootstrap approach under the strong dependence setting $(r=0.5)$. In this case, the bootstrap decisions are rather conservative with corresponding empirical sizes from $2.6\%$ to $4.3\%$ with an average of $3.5\%$ for $\alpha = 5\%$ as well as values from $6.4\%$ up to $8.4\%$ and an average of $7.6\%$ for $\alpha = 10\%$. This may explain why the power values of the permutation tests are rather higher then those of the bootstrap counterpart under strong dependence $(r=0.5)$ while the simulation results under moderate ($r=0.25$) or no ($r=0$) dependency are almost indistinguishable. Regarding the data example discussed in the upcoming section, we primarily studied here the sample size setting $n=20$. To show that the power values grow for increasing sample sizes $n\in\{20,30,\ldots,70\}$, we conducted additional simulations for two specific alternatives $X_{j,1} (t)= B_{j,1}(t)$ and $X_{j,2} (t)= 1.5 B_{j,2}(t)$ as well as $X_{j,1} (t)= B_{j,1}$ and $X_{j,2}(t) = B_{j,2}(t) +t(1-t)$, $t\in[0,1]$, $j=1,\dots,n$, under moderate $(r=0.25)$ dependency, see Figure \[fig:incrn\] for the results.
[ $X_{j,1} (t)= B_{j,1}(t)$, $X_{j,2} (t)= 1.5 B_{j,2}(t)$ ]{} ![[Power values for increasing sample sizes under moderate ($r=0.25$) dependency. ]{}[]{data-label="fig:incrn"}](Alt1.eps "fig:"){width="1.0\linewidth"}
[ $X_{j,1} (t)= B_{j,1}(t)$, $X_{j,2}(t) = B_{j,2}(t) +t(1-t)$]{}
![[Power values for increasing sample sizes under moderate ($r=0.25$) dependency. ]{}[]{data-label="fig:incrn"}](Alt2.eps){width="1.0\linewidth"}
Application to financial time series {#sec:data_exam}
====================================
We consider the values (open) of the well-known stock market indices Nikkei Stock Average, Dow Jones Industrial Average, and Standard & Poor’s 500 for the time period 01/01/1999 to 01/01/2019. The corresponding time series are presented in Figure \[fig2\] and can be seen as square integrable functions on the interval $[0,T_0]$ for $T_0=20$ (years). We consider the linearly interpolated log-returns of the monthly values. To cover seasonality effects indicated by Figure \[fig1\], we split the time horizon of 20 years into 20 subintervals each representing one year, i.e. $T=1$ and $n=20$. Now, we apply our method to do pairwise comparisons of the indices, where the test statistic is again approximated by 500 random projects following [<span style="font-variant:small-caps;">Step</span>]{} 1–4 and the shifted Poisson distribution is used in [<span style="font-variant:small-caps;">Step</span>]{} 1 and [<span style="font-variant:small-caps;">Step</span>]{} 2 as in Section \[sec:sim\]. The resulting $p$-values for the bootstrap as well as the permutation approach are displayed in Table \[tab3\] for 5000 resampling iterations, respectively. Since DJIA and S&P 500 reflect both the US market, it is not surprising that both tests lead to a very high $p$-value and, thus, do not reject the null hypothesis. Comparisons of each of these US indices with the Japanese Nikkei 225 lead to $p$-values around the typical used $5\%$-benchmark or even significantly below it in case of the permutation approach. This is inline with the first graphically impression, which we get by Figure \[fig2\]. The difference between the $p$-values of the two resampling test may be explained as follows. It is clear that the different stock market indices are not independent. This applies to indices within a national economy and also to indices from different national economies since globalization causes interdependences. Here the national economies of the United States and Japan are dependent from each other. Our simulation results indicate that for strongly dependent observations the bootstrap decisions are slightly conservative partially leading to a small power loss in comparison to the permutation method.
![[Monthly values from 01/01/1999 to 01/01/2019.]{}[]{data-label="fig2"}](Nikkei_full.eps){width="1.0\linewidth"}
![[Monthly values from 01/01/1999 to 01/01/2019.]{}[]{data-label="fig2"}](DJIA_full.eps){width="1.0\linewidth"}
![[Monthly values from 01/01/1999 to 01/01/2019.]{}[]{data-label="fig2"}](SP500_full.eps){width="1.0\linewidth"}
$X_{1,j}(t)$ $X_{2,j}(t)$ $p$-value (perm) $p$-value (boot)
-------------- -------------- ------------------ ------------------
DJIA S&P 500 89.9 73.6
Nikkei 225 S&P 500 2.7 5.5
Nikkei 225 DJIA 1.4 5.6
: \[tab3\]Empirical $p$-values of the test.
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Appendix: Proofs {#appendix-proofs .unnumbered}
================
**Proof of Theorem \[thm:marg\]:** We verify the equivalence of and as well as of and . Clearly, implies . Now, let us suppose that is true. Fix some arbitrary starting point $\pi\in{\mathbb{R}}$. Note that the distribution of a stochastic process is uniquely determined by its finite dimensional marginal distributions, which are in turn uniquely determined by the linear combinations of its components. Hence, it is sufficient for to show that $$\label{eqpr}
\begin{split}
P^{a_1\Pi_1(t_1)+\dots+a_k\Pi_1(t_k)|\Pi_1(0)=\pi}=P^{a_1\Pi_2(t_1)+\dots+a_k \Pi_2(t_k)|\Pi_2(0)=\pi}
\end{split}$$ for fix but arbitrary $0\le t_1\le \dots\le t_k\le T$, $a_1,\dots,a_k\in{\mathbb{R}}$, and $k\in{\mathbb{N}}$. Choosing $\rho$ as the Value-at-Risk at level $\alpha \in(0,1)$ of the linear combination with respect to $0\le t_1\le \dots\le t_k\le T$, $a_1,\dots,a_k$, and $k\in{\mathbb{N}}$, we obtain from that $$\label{eqpr2}
\begin{split}
&{{\rm \operatorname {VaR}}}_\alpha(a_1\Pi_1(t_1)+\dots+a_k\Pi_1(t_k)|\Pi_1(0)=\pi)\\
= &{{\rm \operatorname {VaR}}}_\alpha(a_1\Pi_2(t_1)+\dots+a_k \Pi_2(t_k)|\Pi_2(0)=\pi).
\end{split}$$ Remember that the Value-at-Risk at level $\alpha$ gives the value of the quantile function of the underlying distribution in $1-\alpha$. It is well-known that the quantile function characterize the related distribution. Because $\alpha \in(0,1)$ is arbitrary, (\[eqpr2\]) implies (\[eqpr\]), and follows. To complete the proof, we show the equivalence of and . On the one hand, it is $(X_{1,i},\dots, X_{n,i})$ a function of $\Pi_i$ by definition; denote this function by $ h$ and notice that $ h$ is independent of $i=1,2$. From the independence of the start values it follows for $i=1,2$ $$\begin{split}
\forall \pi\in{\mathbb{R}}: ~P^{(X_{1,i},\dots, X_{n,i})}=P^{(X_{1,i},\dots,X_{n,i})|\Pi_i(0)=\pi}=P^{ h( \Pi_{i})|\Pi_i(0)=\pi}.
\end{split}$$ On the other hand, we obtain $\Pi_i$ as a function of $((X_{,i},\dots, X_{n,i}),\Pi_i(0))$. Denote this function by $d$ and notice that $d$ is independent of $i=1,2$. The independence of the start values implies for $i=1,2$ $$\begin{split}
\forall \pi\in{\mathbb{R}}: ~P^{\Pi_{i}|\Pi_i(0)=\pi}&=P^{ d((X_{1,i},\dots, X_{n,i}),\Pi_i(0))|\Pi_i(0)=\pi}\\
&=P^{ d( (X_{1,i},\dots, X_{n,i}),\pi)|\Pi_i(0)=\pi}=P^{ d( (X_{1,i},\dots, X_{n,i}),\pi)}.
\end{split}$$ Because $X_j$, $j=1,\dots,n,$ are assumed to be independent and identically distributed, this proves the asserted equivalence $\square$\
Now we prove Theorem \[theo:asym\_S\] in a more general way. Instead of ${{\rm \operatorname {CvM}}}_n$ we consider $$\begin{aligned}
S_n= n\int \int\left([ F_{1,n}(x,y)- F_1(x,y) ] - \left [ F_{2,n}(x,y) - F_2(x,y)\right]\right)^2\overline F_n(x,\mathrm d y)\mathcal P(\mathrm d x).
\end{aligned}$$
\[theo:asym\_Sn\] Let $\tau_1,\tau_2,\ldots$ be a sequence of independent standard normal distributed random variables. Under the null hypothesis $\mathcal H$ as well as under any alternative, we have $$\begin{aligned}
\label{eqn:Sn_conv}
S_n \overset{ d}{\rightarrow} \sum_{i=1}^\infty \lambda_i (1+\tau_i^2) = Z,
\end{aligned}$$ where $(\lambda_i)_{i\in{\mathbb{N}}}$ is a sequence of non-negative numbers with $\sum_{i=1}^\infty \lambda_i< \infty$ and $\lambda_{i}>0$ for at least one $i\in{\mathbb{N}}$ implying that the distribution function of $Z$ is continuous and strictly increasing on the non-negative half-line.
**Proof:** Let $x_j=(x_{j,1},x_{j,2})\in H^2,~j\in {\mathbb{N}}$. We introduce the unsymmetric kernel $f$ given by $$\begin{aligned}
\label{eqn:def_unsym_kernel_f}
&f(x_1,x_2,x_3) \\
&= \frac{1}{2}\sum_{i=1}^2 \int \Bigl[ {{\mathbf{1}}}_{ \langle x_{1,1} - x_{3,i}, x \rangle \leq 0} - F_1(x, \langle x_{3,i}, x \rangle ) - {{\mathbf{1}}}_{ \langle x_{1,2} - x_{3,i}, x \rangle \leq 0 } + F_2(x, \langle x_{3,i}, x \rangle ) \Bigr]\nonumber \\
&\times
\Bigl[ {{\mathbf{1}}}_{ \langle x_{2,1} - x_{3,i}, x \rangle \leq 0 } - F_1(x, \langle x_{3,i}, x \rangle ) - {{\mathbf{1}}}_ { \langle x_{2,2} - x_{3,i}, x \rangle \leq 0 } + F_2(x, \langle x_{3,i}, x \rangle ) \Bigr] \mathcal P(\mathrm{ d }x) \nonumber
\end{aligned}$$ as well as its symmetric version $\phi$ defined by $$\begin{aligned}
\phi(x_1,x_2,x_3) = \frac13 \Bigl( f(x_1,x_2,x_3)+f(x_2,x_3,x_1)+ f(x_1,x_3,x_2) \Bigr).
\end{aligned}$$ Clearly, $$\begin{aligned}
S_n&= \frac{1}{n^2}\sum_{i, j, k=1}^n f(X_i,X_{ j},X_{ k}) = \frac{1}{n^2}\sum_{i, j, k=1}^n \phi(X_i,X_{ j},X_{ k}).
\end{aligned}$$ It is easy to check that $$\begin{aligned}
\label{eqn:deg_kernal_1RV}
{ {E}}[f(X_1,x_2,x_3) ] = { {E}}[ f(x_1,X_2,x_3) ] = 0.
\end{aligned}$$ The function $(x_1,x_2)\mapsto { {E}}[\phi(x_1,x_2,X_3)]={ {E}}[ f(x_1,x_2,X_3)] /3$ is not constant with probability one. This can easily been verified and also follows from our considerations below. Moreover, we can deduce from and the independence of the random variables that $$\begin{aligned}
\label{eqn:deg_kernal_2RV}
{ {E}}\big[ \phi(x_1,X_2,X_3) \big] = { {E}}\big\{ { {E}}\big[ \phi(x_1,X_2,X_3) \big | X_3\big] \big\} = 0 = { {E}}\big[ \phi(X_1,X_2,X_3) \big].
\end{aligned}$$ In all, $\phi$ is degenerate of order $1$, see [@AcroneGine1992] for a detailed definition. Hence, we can deduce from Theorem 3.5 of [@AcroneGine1992] that $S_n-{ {E}}[S_n]$ converges in distribution to $Z$ as $n\to \infty$ if and only if $$\begin{aligned}
V_n = 3n^{-1} \sum_{i, j=1}^n \widetilde \phi (X_i,X_{ j})
\end{aligned}$$ converges to $Z$ as well, where $$\begin{aligned}
\widetilde \phi (x_1,x_2) &= { {E}}\big[ \phi(x_1,x_2,X_3) \big] - { {E}}\big[ \phi(x_1,X_2,X_3) \big] - { {E}}\big[ \phi(X_1,x_2,X_3) \big] + { {E}}\big[ \phi(X_1,X_2,X_3) \big].
\end{aligned}$$ The new kernel $\widetilde \phi$ is a projection of $\phi$ to a corresponding function space, details are carried out in [@AcroneGine1992]. By and we can simplify it as follows $$\begin{aligned}
\widetilde \phi (x_1,x_2) = { {E}}\big[ \phi(x_1,x_2,X_3) \big] = \frac{1}{3}{ {E}}\big[ f(x_1,x_2,X_3) \big].
\end{aligned}$$ Thus, we can rewrite $ V_n$ as $$\begin{aligned}
\label{eqn:def_VN+tildef}
V_n = n^{-1} \sum_{i, j=1}^n \widetilde f (X_i,X_{ j}) \text{ with }\widetilde f(x_1,x_2) = { {E}}\big[ f(x_1,x_2,X_3) \big].
\end{aligned}$$ By Lemma \[lem:Mercer\_kernel\], see below, $\widetilde f$ is a degenerated and bounded Mercer kernel. Due to the degeneracy of the kernel, the map $ g(\cdot)\longmapsto { {E}}\big[\widetilde f(X_1,\cdot)g(X_1)\big]$ defines a Hilbert-Schmidt operator in the space of all square integrable functions on $H^2$ with respect to $P^{X_1}$, see also Section 4.3 in [@kor]. In this space, there exists an orthonormal basis of eigenfunctions $(\varphi_i)_{i\in{\mathbb{N}}}$ of this integral operator with corresponding non-negative eigenvalues $(\lambda_i)_{i\in{\mathbb{N}}}$. Since $\widetilde f$ is a bounded Mercer kernel, we obtain, in analogy to the argumentation of [@leuchtneumann] in their proof of Theorem 2.1, from an extension of the Theorem of [@mercer] by [@sun] that for all $x,y$ in the support of $P^{X_1}$ $$\begin{aligned}
\label{eqn:mercer}
\widetilde f(x,y) = \sum_{i=1}^\infty \lambda_i\varphi_i(x)\varphi_i(y),
\end{aligned}$$ where the sum converges absolutely and uniformly on every compact subset of the cartesian square of the support of $P^{X_1}$. In particular, we obtain $$\begin{aligned}
\label{nondegvor}
\sum_{i=1}^\infty \lambda_i^2={ {E}}[ \widetilde f(X_1,X_1)^2]<\infty~\text{and}~\sum_{i=1}^\infty \lambda_i={ {E}}[ \widetilde f(X_1,X_1)]<\infty.
\end{aligned}$$ Regarding we have $$\begin{aligned}
V_n= \sum_{i=1}^\infty \lambda_i\Bigl[ \frac{1}{\sqrt{n}}\sum_{j=1}^n\varphi_i(X_j)\Bigr]^2.
\end{aligned}$$ From the orthogonality of $(\varphi_i)_{i\in{\mathbb{N}}}$ and the multivariate central limit theorem we can conclude that for each fixed $k\in{\mathbb{N}}$ $$\begin{aligned}
\Bigl[ \frac{1}{\sqrt{n}}\sum_{j=1}^n\varphi_1(X_j),\ldots, \frac{1}{\sqrt{n}}\sum_{j=1}^n\varphi_k(X_j) \Bigr] \overset{ d}{\rightarrow} (\tau_1,\ldots,\tau_k) \textrm{ as }n\to\infty,
\end{aligned}$$ where $\tau_1,\tau_2,\ldots$ are independent and standard normal distributed. Combining this, and a standard truncation argument, compare to Theorem 4.3.1 and 4.3.2 of [@kor] as well as the corresponding proofs, yields $$\begin{aligned}
\label{eqn:conv_Tn*}
V_n \overset{ d}{\rightarrow} \sum_{j=1}^\infty \lambda_j \tau_j^2 \textrm{ as }n\to\infty.
\end{aligned}$$ Now, note that by , , and $$\begin{aligned}
&{ {E}}[S_n] \\
= &\frac{n}{n^2} \Big \{ { {E}}[\phi(X_1,X_1,X_1)] + 3(n-1){ {E}}[\phi(X_1,X_1,X_3)] +(n-1)(n-2){ {E}}[\phi(X_1,X_2,X_3)] \Big\} \\
= &\frac{1}{n}{ {E}}[\phi(X_1,X_1,X_1)] + \frac{3(n-1)}{n}{ {E}}[\phi(X_1,X_1,X_3)] \\
&\rightarrow 3{ {E}}[\phi(X_1,X_1,X_3)] = { {E}}[f(X_1,X_1,X_3)] = \sum_{i=1}^\infty \lambda_i~\text{as}~n\to \infty.
\end{aligned}$$ Consequently, follows. Since $ { {E}}[ {{\mathbf{1}}}_{ \langle x, X_{1,l} \rangle \leq \langle x, X_{3,k} \rangle} | X_3 ] = F_l(x, \langle x, X_{3,k} \rangle)$, we can deduce from that $$\begin{aligned}
\label{eqn:sumlambdi_EInt}
\sum_{j=1}^\infty\lambda_i &= { {E}}\big[ f(X_1,X_1,X_3) \big] = { {E}}\big\{ { {E}}\Bigl[ f(X_1,X_1,X_3) \big | X_3 \big] \big\})\\
&= \frac{1}{2}\sum_{k,l=1}^2 \int { {E}}\Big[ F_l(x,\langle x, X_{3,k}\rangle ) - F_l(x,\langle x, X_{3,k}\rangle )^2 \Big]\mathcal P(\mathrm{ d }x).
\end{aligned}$$ Finally, the remaining statement, that $\lambda_i>0$ for at least one, follows from , the non-negativity of the eigenvalues, the trivial inequality $F_l\geq F_l^2$ and $$\begin{aligned}
\label{eqn:E>16}
\int { {E}}\big[ F_k(x,\langle x, X_{3,k}\rangle ) - F_k(x,\langle x, X_{3,k}\rangle )^2 \big]\mathcal P(\mathrm{ d }x)\geq \frac16
\end{aligned}$$ for $k=1,2$, where can be proven in the same way as Lemma A.1 of [@DitzhausGaigall2018]. We omit the a detailed verification of and just want to point out that [@DitzhausGaigall2018] additionally introduced the measure $Q$ (or here we should write $Q_k$), which is the distribution of $\langle P, X_{3,k} \rangle$ for a random variable $P \sim \mathcal P$ independent of $X_3$. $\square$\
\[lem:Mercer\_kernel\] The function $\widetilde f$ is a degenerated and bounded Mercer kernel, i.e., it is continuous, symmetric and positive semidefinite.
**Proof:** It is easy to see that $\widetilde f$ is bounded and symmetric. The degeneracy follows immediately from . For arbitrary $k\in{\mathbb{N}}$ let $c_1,\ldots,c_k\in{\mathbb{R}}$. Then $$\begin{aligned}
\sum_{i,j=1}^k c_ic_j \widetilde f(x_i,x_j)
&= \frac{1}{2}\sum_{\ell=1}^2 { {E}}\Bigl[ \int \Bigl\{ \sum_{i=1}^k c_i \Bigl[ {{\mathbf{1}}}_{ \langle x_{i,1}, x \rangle \leq \langle X_{3,\ell}, x \rangle } - F_1\big(x, \langle X_{3,\ell}, x \rangle \big) \\
&\phantom{=}-{{\mathbf{1}}}_{\langle x_{i,2}, x \rangle \leq \langle X_{3,\ell}, x \rangle} + F_2(x, \langle X_{3,\ell}, x \rangle ) \Big] \Bigr\}^2 \mathcal P(\mathrm{ d }x) \Bigr] \geq 0.
\end{aligned}$$ Hence, $\widetilde f$ is positive semidefinite. For the continuity proof, let $(x_{1n})_{n\in{\mathbb{N}}}$ and $(x_{2n})_{n\in{\mathbb{N}}}$ be sequences in $ H^2$ such that $\lim_{n\to\infty}x_{jn}=x_j\in H^2$, $j=1,2$. By Lemma 3.1 of [@DitzhausGaigall2018] $$\begin{aligned}
\label{eqn:ditz+gaig_intI=0}
{{\mathbf{1}}}_{ \langle x, x_{3,\ell} - y \rangle \neq 0 } = 1
\end{aligned}$$ for $\mathcal P \times P^{X_{1,\ell}}$-almost all $(x,x_{3,\ell})$ and every $y \in H$. This and the continuity of the inner product imply for $\mathcal P$-almost all $x$ and every $\in\{1,2\}$ that $$\begin{aligned}
\lim_{n\to\infty}{{\mathbf{1}}}_{ \langle x, x_{jn,m}\rangle \leq \langle x, X_{3,\ell} \rangle } = {{\mathbf{1}}}_{\langle x, x_{j,m} \rangle \leq \langle x, X_{3,\ell} \rangle }\text{ with probability one}.
\end{aligned}$$ Consequently, $\widetilde f(x_{1n},x_{2n})$ converges to $\widetilde f(x_{1},x_{2})$. $\square$\
**Proof of Theorem \[theo:asym\_S\]:** Since $F_1=F_2$ and, thus, $S_n={{\rm \operatorname {CvM}}}_n$ under the null hypotheses, the statement follows immediately from Theorem \[theo:asym\_Sn\]. $\square$\
**Proof of Theorem \[theo:cons\_S\]:** First, observe that $$\begin{aligned}
&n^{-1}{{\rm \operatorname {CvM}}}_n = n^{-1}S_n \\
&+2 \int \int[ F_{n,1}(x,y)- F_1(x,y) - F_{n,2}(x,y) + F_2(x,y)][ F_1(x,y) - F_2(x,y)]\bar F_n(x,\mathrm d y)\mathcal P(\mathrm d x)\\
& + \int \int[ F_1(x,y) - F_2(x,y)]^2\bar F_n(x,\mathrm d y)\mathcal P(\mathrm d x).
\end{aligned}$$ By Theorem \[theo:asym\_Sn\], $n^{-1}S_n$ converges in probability to $0$. By the Cauchy-Schwarz inequality the absolute value of the second summand is bounded from above by $2\sqrt{n^{-1}S_n}$. In particular, the second summand vanishes in probability as well. The third summand can be rewritten as $$\begin{aligned}
\frac{1}{2n}\sum_{j=1}^n \sum_{k=1}^2 \int \Big[ F_1(x,\langle x, X_{j,k}\rangle) - F_2(x,\langle x, X_{j,k}\rangle)\Big]^2\mathcal P(\mathrm d x) = \frac{1}{2n}\sum_{j=1}^n g(X_j),
\end{aligned}$$ for an appropriate function $g$. By the strong law this sum converges almost surely to $$\begin{aligned}
&\frac{1}{2}\sum_{k=1}^2 \int { {E}}\Bigl[ \Big\{ F_1(x,\langle x, X_{3,k}\rangle) - F_2(x,\langle x, X_{3,k}\rangle)\Big\}^2 \Bigr]\mathcal P(\mathrm d x)\nonumber \\
&= \frac{1}{2}\sum_{k=1}^2 \int \big[ F_1(x,y) - F_2(x,y)\big]^2 Q_k[\mathrm d (x,y)]\label{eqn:cons_simpl_3sum},
\end{aligned}$$ where $Q_k$ is the distribution introduced at the proof’s end of Theorem \[theo:asym\_Sn\]. In analogy to the argumentation of [@DitzhausGaigall2018] in the proof for their Theorem 3.2, we can conclude that each summand from is strictly positive. Finally, we obtain $$\begin{aligned}
n^{-1}{{\rm \operatorname {CvM}}}_n \overset{p}{\rightarrow} \frac{1}{2}\sum_{k=1}^2 \int \big[ F_1(x,y) - F_2(x,y)\big]^2 Q_k(\mathrm d (x, y)) > 0.
\end{aligned}$$ $\square$\
**Proof of Theorem \[theo:boot\]:** From now on, we suppose that the data $X_1,\ldots,X_n$ are fixed. Throughout the whole proof, let $x_j=(x_{j,1},x_{j,2})\in H^2$, $j\in {\mathbb{N}}$. We remark that the distribution of the bootstrap sample depends on the sample size, and that $X_{in}^*$ converges in distribution to $ X_i$ for all $i\in {\mathbb{N}}$. By Theorem 1.10.4 of [@van] we can assume without loss of generality that $X_{in}^*$ converges to $ X_i$ for all $i\in {\mathbb{N}}$ with probability one, and that $ X_r$ is independent from $X_{1n}^*,\ldots,X_{(r-1)n}^*,X_{(r+1)n}^*,\ldots$ for all $r\in{\mathbb{N}}$. Now, define $$\begin{aligned}
&f_{n}^*(x_{1},x_{2},x_{3}) \\
&= \frac{1}{2}\sum_{k=1}^2 \int \Bigl[ {{\mathbf{1}}}_{ \langle x, x_{1,1} -x_{3,k} \rangle \leq 0 } - F_{n,1}(x, \langle x,x_{3,k}\rangle ) - {{\mathbf{1}}}_{\langle x,x_{1,2} - x_{3,k} \rangle \leq 0} + F_{n,2}(x, \langle x, x_{3,k} \rangle ) \Bigr]\\
&\times
\Bigl[ {{\mathbf{1}}}_{ \langle x,x_{2,1} - x_{3,k}\rangle \leq 0 } - F_{n,1}(x, \langle x, x_{3,k} \rangle ) - {{\mathbf{1}}}_{ \langle x, x_{2,2} - x_{3,k} \rangle \leq 0 } + F_{n,2}(x, \langle x, x_{3,k}\rangle ) \Bigr] \mathcal P(\mathrm{ d }x).
\end{aligned}$$ Then we have $$\begin{aligned}
{{\rm \operatorname {CvM}}}_{n}^* = \frac{1}{n^2} \sum_{i,j,k=1}^n f_{n}^*(X_{in}^*,X_{jn}^*,X_{kn}^*).
\end{aligned}$$ Define $$\begin{aligned}
S_n = \frac{1}{n^2} \sum_{i,j,k=1}^n f( X_{i}, X_{j}, X_{k})~\text{and}~\kappa_{n,i,j,k} = f_{n}^*(X_{in}^*,X_{jn}^*,X_{kn}^*) - f( X_i, X_j, X_k),
\end{aligned}$$ where $f$ is defined in . By Theorem \[theo:asym\_Sn\] we already know that $ S_n$ converges in distribution to $Z$. Combining this and $$\begin{aligned}
\label{eqn:boot_null_suff}
{ {E}}[ ({{\rm \operatorname {CvM}}}_{n}^* - S_n)^2] &= n^{-4}\sum_{i_1,\ldots,i_6=1}^n { {E}}[\kappa_{n,i_1,i_2,i_3}\kappa_{n,i_4,i_5,i_6}] \rightarrow 0~\text{as}~n\to \infty
\end{aligned}$$ under the null hypothesis, where the proof of is given later, yields conditional convergence $$\begin{aligned}
{{\rm \operatorname {CvM}}}_{n}^* \overset{ d}{\rightarrow} Z \textrm{ as }n\to\infty
\end{aligned}$$ given the observations $X_1,\ldots,X_n$ under $\mathcal H$ for $Z$ from Theorem \[theo:asym\_S\]. Consequently, we can deduce that $c_{n,1-\alpha}^* \overset{p}{\rightarrow} c_{1-\alpha}$ under $\mathcal H$ and, in particular, the statement under $\mathcal H$ follows, compare to Lemma 1 of [@janssenpauls]. For the statement under the alternative it remains to show that $({{\rm \operatorname {CvM}}}_{n}^*)_{n\in{\mathbb{N}}}$ is a tight sequence of real valued random variables, compare to Theorem 7 of [@janssenpauls], i.e. we have to show $$\begin{aligned}
\limsup_{K\to\infty}\limsup_{n\to\infty} {P}( |{{\rm \operatorname {CvM}}}_{n}^*|\geq K) =0.
\end{aligned}$$ In contrast to , it remains now to show $$\begin{aligned}
\label{eqn:boot_alt_suff}
\limsup_{n\to\infty}{ {E}}[ ( {{\rm \operatorname {CvM}}}_{n}^* - S_n)^2] \leq M < \infty.
\end{aligned}$$ To sum up, we need to verify for the statement under $\mathcal H$ and for the statement under $\mathcal K$. For this purpose, we divide the corresponding sum in into the following six sums $$\begin{aligned}
I_{n,p} = n^{-4}\sum_{i_1,\ldots,i_6=1}^n { {E}}[\kappa_{n,i_1,i_2,i_3}\kappa_{n,i_4,i_5,i_6}] {{\mathbf{1}}}_{ |\{i_1,\ldots,i_6\}| = p },~p=1,\ldots,6.
\end{aligned}$$ First, we will prove that $I_{n,p}$ converges to $0$ for $p\in\{1,2,3,5,6\}$ independently whether the null hypothesis or the alternative is true. At the end, we discuss $I_{n,4}$ separately under the null hypothesis and the alternative. For all considerations below, remind that $\kappa_{n,i,j,m}$ is uniformly bounded by $8$. As a first consequence of this, we obtain that as $n\to \infty$ $$\begin{aligned}
I_{n,1} + I_{2,n} + I_{3,n} \leq \frac{8^2}{n^4}[ n + (2^6-1)n(n-1)+(3^6-2^6)n(n-1)(n-2) ]\rightarrow 0.
\end{aligned}$$ Let us have now a look on all summands with $|\{i_1,\ldots,i_6\}|=5$. Let $r$ be the number that appears twice within the indices $i_1,\ldots,i_6$. Observe that also holds for the bootstrap sample, i.e., $$\begin{aligned}
{ {E}}[ f_{n}^*(X_{1n}^*,x_{2},x_{3}) ] = { {E}}[ f_{n}^*(x_{1},X_{2n}^*,x_{3}) ] = 0.
\end{aligned}$$ Combining this and yields $$\begin{aligned}
{ {E}}[ \kappa_{n,i,j,k} | X_r,X_{r}^* ] = 0
\end{aligned}$$ with probability one whenever $|\{i,j,k\}|=3$. Consequently, $$\begin{aligned}
{ {E}}[ \kappa_{n,i_1,i_2,i_3}\kappa_{n,i_4,i_5,i_6} ] = { {E}}\{ { {E}}[ \kappa_{n,i_1,i_2,i_3} | X_r,X_{r,*} ]{ {E}}[ \kappa_{n,i_4,i_5,i_6} | X_r,X_{r,*}] \} = 0
\end{aligned}$$ Clearly, the same can be shown for the case $|\{i_1,\ldots,i_6\}|=6$. Hence, $I_{n,5}+I_{n,6}=0$.
Now, we consider $I_{n,4}$. Due to the boundedness of $\kappa_{n,i,j,m}$, we always obtain $$\begin{aligned}
I_{n,4} \leq \frac{8^2}{n^4}(4^6-3^6)n(n-1)(n-2)(n-3) \leq 2^{20}.
\end{aligned}$$ From this and the previous considerations we can conclude . Now, let us suppose that the null hypothesis is true. Due to symmetry $\kappa_{n,i,j,k}=\kappa_{n,j,i,k}$ we get $$\begin{aligned}
I_{n,4} \leq \frac{8}{n^4}(4^6-3^6)n(n-1)(n-2)(n-3) \max\Bigl\{ { {E}}[|\kappa_{n,1,1,2}|]+{ {E}}[|\kappa_{n,1,2,2}|] + { {E}}[|\kappa_{n,1,3,2}|] \Bigr\}.
\end{aligned}$$ Consequently, it is sufficient for to prove $$\begin{aligned}
\label{eqn:boot_I4_suff}
\lim_{n\to\infty}\kappa_{n,1,j,2} = 0 ~ \text{in probability}
\end{aligned}$$ for $j=1,2,3$. From the continuity of the inner product, , the underlying independence and the convergence of $X_{1n}^*$, $X_{2n}^*$, $X_{3n}^*$ we obtain that with probability one $$\begin{aligned}
\label{eqn:boot_intI=0}
\lim_{n\to\infty}{{\mathbf{1}}}_{ \langle x, X_{rn,k}^* \rangle \leq \langle x, X_{2n,l}^* \rangle } = {{\mathbf{1}}}_{\langle x, X_{r,k} \rangle \leq \langle x, X_{2,l}\rangle } \text{ for }\mathcal P\text{-almost all }x,
\end{aligned}$$ every $r\in\{1,3\}$ and $k,l\in\{1,2\}$. Analogously, we have $$\begin{aligned}
\label{eqn:boot_intF=0}
&F_{n,k}( x,\langle x, X_{2n,l}^* \rangle ) = { {E}}\Bigl[{{\mathbf{1}}}_{ \langle x, X_{1n,k}^* \rangle \leq \langle x, X_{2n,l}^*\rangle } | X_{2n,l}^* \Bigr] \nonumber \\
&\overset{a.s.}{\rightarrow} { {E}}\Bigl[ {{\mathbf{1}}}_{ \langle x, X_{1,k} \rangle \leq \langle x, X_{2,l} \rangle } | X_{2,l} \Bigr] = F_k( x,\langle x, X_{2,l} \rangle )~\text{as}~n\to \infty.
\end{aligned}$$ Combining and shows for the case $j\in\{1,3\}$. The reason why we need to be more careful in the case $j=2$ is that, in general, is false if $r=j=2$. However, the integrals appearing in the limiting $f( X_1, X_2, X_2)$ vanish when they are restricted to the crucial (random) set $A=\{x:\langle x, X_{2,2} - X_{2,1}\rangle = 0\}$. To be more specific, since the null hypothesis is true and, hence, $F_1=F_2$, we obtain $$\begin{aligned}
&\sum_{k=1}^2 \int_A \Bigl[ {{\mathbf{1}}}_{\langle x, X_{1,1}-X_{2,k}\rangle \leq 0 } - F_1(x, \langle x, X_{2,k}\rangle ) - {{\mathbf{1}}}_{ \langle x, X_{2,2} - X_{2,k}\rangle \leq 0 }+ F_2(x, \langle x, X_{2,k} \rangle ) \Bigr]\\
&\times
\Bigl[ {{\mathbf{1}}}_{\langle x, X_{2,1} - X_{2,k}\rangle \leq 0 } - F_1(x, \langle x, X_{2,k} \rangle ) - {{\mathbf{1}}}_{\langle x, X_{2,2} - X_{2,k} \rangle \leq 0 } + F_2(x, \langle x, X_{2,k} \rangle ) \Bigr] \mathcal P(\mathrm{ d }x)\\
&=2\int_A \Bigl[ {{\mathbf{1}}}_{\langle x, X_{1,1} - X_{2,1} \rangle \leq 0 }- F_1(x, \langle x, X_{2,1} \rangle ) - {{\mathbf{1}}}_{ \langle x, X_{1,2} - X_{2,1} \rangle \leq 0 } + F_1(x, \langle x, X_{2,1} \rangle ) \Bigr]\\
&\times
\Bigl[ F_1(x, \langle x, X_{2,1} \rangle ) - F_1(x, \langle x, X_{2,1} \rangle ) \Bigr] \mathcal P(\mathrm{ d }x) =0.
\end{aligned}$$ Thus, follows again from , , the continuity of the inner product and the convergence of $X_{1n}^*$ and $X_{2n}^*$. $\square$\
**Proof of Theorem \[theo:perm\]:** For fixed $x_1,x_2\in H^2$ with $x_j = (x_{j,1},x_{j,2})$ we define $$\begin{aligned}
\label{eqn:tildef_perm}
f_n^\pi(x_1,x_2) = \frac{1}{2n}\sum_{k=1}^{n}\sum_{i=1}^2 \int &\Bigl[ {{\mathbf{1}}}_{\langle x_{1,1} - X_{k,i}, x \rangle \leq 0 }) - {{\mathbf{1}}}_{ \langle x_{1,2} - X_{k,i}, x \rangle \leq 0 } \Bigr]\nonumber \\
&\times
\Bigl[ {{\mathbf{1}}}_{ \langle x_{2,1} - X_{k,i}, x \rangle \leq 0 } - {{\mathbf{1}}}_{ \langle x_{2,2} - X_{k,i}, x \rangle \leq 0} \Bigr] \mathcal P(\mathrm{ d }x).\end{aligned}$$ By the strong law of large number we have as $n\to\infty$ $$\begin{aligned}
\label{eqn:perm_as_ftilde}
\widetilde f_n^\pi(x_1,x_2) &\overset{a.s.}{\rightarrow} \int \int { {E}}\Big[\Bigl\{ {{\mathbf{1}}}_{\langle x_{1,1} - X_{3,i}, x \rangle \leq 0 } - {{\mathbf{1}}}_{ \langle x_{1,2} - X_{3,i}, x \rangle \leq 0 } \Bigr\}\nonumber \\
&\phantom{\overset{a.s.}{\rightarrow}\int \int}\times
\Bigl\{ {{\mathbf{1}}}_{\langle x_{2,1} - X_{3,i}, x \rangle \leq 0 } - {{\mathbf{1}}}_{ \langle x_{2,2} - X_{3,i}, x \rangle \leq 0 } \Bigr\}\Big] \mathcal P(\mathrm{ d }x)= \widetilde f^\pi(x_1,x_2).\end{aligned}$$ If $\mathcal H$ is true and, thus, $F_1=F_2$, the functions $\widetilde f^\pi$ and $\widetilde f$ from coincide. Since every summand in is obviously bounded by $1$, a single summand has no influence on the asymptotic behavior of $\widetilde f_n^\pi$. That is why we can deduce from , the boundedness of $\widetilde f_n^\pi$ and $\widetilde f^\pi$ by $1$ as well as from the independence of $X_1,X_2$ and $(X_j)_{j\geq 3}$ that for $r,s\in\{1,2\}$ $$\begin{aligned}
{ {E}}[ \{ \widetilde f_n^\pi(X_r,X_s) - \widetilde f^\pi(X_r,X_s) \}^2 ] = { {E}}\{ { {E}}[ \{ \widetilde f_n^\pi(X_r,X_s) - \widetilde f^\pi(X_r,X_s) \}^2 | X_r,X_s ] \} \to 0\end{aligned}$$ as $n\to\infty$. Since convergence in means implies convergence in probability, we obtain $$\begin{aligned}
\label{eqn:perm_sums_widetilde_f}
\frac{1}{n^2}\sum_{i,j=1}^{n} [\widetilde f_n^\pi(X_i,X_j) - \widetilde f^\pi(X_i,X_j)]^2 + \frac{1}{n}\sum_{i=1}^n |\widetilde f_n^\pi(X_i,X_i) - \widetilde f^\pi(X_i,X_i)| \overset{p}{\rightarrow} 0 \end{aligned}$$ as $n\to\infty$. In analogy to Lemma \[lem:Mercer\_kernel\], we obtain that $\widetilde f^\pi$ is a degenerated and bounded Mercer kernel. Thus, there exists an orthonormal basis of eigenfunctions $(\varphi_i^\pi)_{i\in{\mathbb{N}}}$ of the integral operator $g(\cdot)\longmapsto { {E}}\big[\widetilde f^\pi(X_1,\cdot)g(X_1)\big]$ with corresponding non-negative eigenvalues $(\lambda_i^\pi)_{i\in{\mathbb{N}}}$ such that $$\begin{aligned}
\label{eqn:mercer_perm}
\widetilde f^\pi(x,y) = \sum_{i=1}^\infty \lambda_i^\pi\varphi^\pi_i(x)\varphi^\pi_i(y),\end{aligned}$$ where the sum converges absolutely and uniformly on every compact subset of the cartesian square of the support of $P^{X_1}$ and, moreover, $$\begin{aligned}
\label{nondegvor}
\sum_{i=1}^\infty (\lambda_i^\pi)^2={ {E}}[ \widetilde f^\pi(X_1,X_1)^2]<\infty~\text{and}~\sum_{i=1}^\infty \lambda_i^\pi={ {E}}[\widetilde f^\pi(X_1,X_1)]<\infty,\end{aligned}$$ compare to the proof of Theorem \[theo:asym\_Sn\]. In particular, we can deduce from the strong law of large numbers $$\begin{aligned}
\label{eqn:strong_law}
\frac{1}{n}\sum_{i=1}^\infty\varphi_k^\pi(X_i)\varphi_\ell^\pi(X_i) \overset{a.s.}{\rightarrow} {{\mathbf{1}}}_{k=\ell} \textrm{ as }n\to\infty,\,k,\ell \in {\mathbb{N}}.\end{aligned}$$
Now, we fix the observations $X_1,\ldots,X_n$. Without loss of generality, we can assume that holds for these observations. Turning to subsequences, we can, moreover, suppose that holds. In the following, we encode the permutation by random variables $\pi_{i}\in\{0,1\}$, where we set $\pi_{i}=0$ in case of $X_{i}^\pi = X_{i}$ and set $\pi_{i} = 1$ when the pair is actually permuted. We can rewrite the permutation test statistic as follows $$\begin{aligned}
{{\rm \operatorname {CvM}}}_n^\pi = \frac{1}{n} \sum_{i,j=1}^n (-1)^{\pi_i + \pi_j} \widetilde f_n^\pi(X_i,X_j).\end{aligned}$$ To determine the limit distribution of ${{\rm \operatorname {CvM}}}_n^\pi$, we first determine the limit of $$\begin{aligned}
V_n^\pi = \frac{1}{n} \sum_{i,j=1}^n (-1)^{\pi_i + \pi_j} \widetilde f^\pi(X_i,X_j),\end{aligned}$$ where we replaced the kernel $\widetilde f_n^\pi$ by its limit, compare to . Due to , we can rewrite $V_n^\pi$ to $$\begin{aligned}
V_n^\pi = \sum_{i=1}^\infty \lambda_i^\pi \Bigl[ \frac{1}{\sqrt{n}}\sum_{j=1}^n(-1)^{\pi_j}\varphi_i^\pi(X_j)\Bigr]^2.\end{aligned}$$ Note that $\varphi_i^\pi(X_j)$ are constants such that holds. That is why obtain from the Lindeberg-Feller theorem combined with the Cramér-Wold device that for every $k\in{\mathbb{N}}$ $$\begin{aligned}
\Bigl[ \frac{1}{\sqrt{n}}\sum_{j=1}^n(-1)^{\pi_j}\varphi_1^\pi(X_j),\ldots, \frac{1}{\sqrt{n}}\sum_{j=1}^n(-1)^{\pi_j}\varphi_k^\pi(X_j) \Bigr] \overset{ d}{\rightarrow} (\tau_1,\ldots,\tau_k) \textrm{ as }n\to\infty.\end{aligned}$$ Again truncation arguments can be used to obtain $$\begin{aligned}
V_n^\pi \overset{ d}{\rightarrow} \sum_{j=1}^\infty \lambda_j^\pi \tau_j = Z^\pi \textrm{ as }n\to\infty.\end{aligned}$$ Since $\widetilde f_n^{\pi}$ and $\widetilde f^\pi$ are clearly bounded by $1$, we can deduce from that as $n\to\infty$ $$\begin{aligned}
&{ {E}}[ ({{\rm \operatorname {CvM}}}_n^\pi - V_n^\pi)^2 ]\\
&= \frac{1}{n^2}\sum_{i,j,k,\ell=1}^n [\widetilde f_n^\pi(X_i,X_j) - \widetilde f^\pi(X_i,X_j)][\widetilde f_n^\pi(X_k,X_\ell) - \widetilde f^\pi(X_k,X_\ell)] { {E}}[(-1)^{\pi_i+\pi_j+\pi_k + \pi_l}] \\
&= \frac{1}{n^2} \sum_{i=1}^n [\widetilde f_n^\pi(X_i,X_i) - \widetilde f^\pi(X_i,X_i)]^2 + \frac{2}{n^2} \sum_{j,i=1;i\neq j}^n [\widetilde f_n^\pi(X_i,X_j) - \widetilde f^\pi(X_i,X_j)]^2 \\
&\phantom{=}+ \frac{1}{n^2} \sum_{j,i=1;i\neq j}^n [\widetilde f_n^\pi(X_i,X_i) - \widetilde f^\pi(X_i,X_i)][\widetilde f_n^\pi(X_j,X_j) - \widetilde f^\pi(X_j,X_j)] \\
&\leq \frac{4}{n} + \frac{4}{n^2}\sum_{i,j=1}^{n} [\widetilde f_n^\pi(X_i,X_j) - \widetilde f^\pi(X_i,X_j)]^2 + \frac{1}{n}\sum_{i=1}^n |\widetilde f_n^\pi(X_i,X_i) - \widetilde f^\pi(X_i,X_i)| \to 0.\end{aligned}$$ Consequently, our permutation statistic converges to the real-valued limit $Z^\pi$. As already stated above, $\widetilde f$ and $\widetilde f^\pi$ coincide under $\mathcal H$ and so do their corresponding eigenvalues, i.e. $\{\lambda_i;i\in I\}=\{\lambda_i^\pi;i\in I\}$. Finally, $P^{Z^\pi} =P^{Z}$ follows under $\mathcal H$. Finally, we can deduce the permutation test’s asymptotic exactness as well as its consistency (again) from Lemma 1 and Theorem 7 of [@janssenpauls]. $\square$
|
---
author:
- 'M. Bejger$^a$, D. Blaschke$^{b,c,d}$, P. Haensel$^a$, J. L. Zdunik$^a$, M. Fortin$^a$'
date: 'Received ...; accepted ...'
title: Consequences of a strong phase transition in the dense matter equation of state for the rotational evolution of neutron stars
---
[We explore the implications of a strong first-order phase transition region in the dense matter equation of state in the interiors of rotating neutron stars, and the resulting creation of two disjoint families of neutron-star configurations (the so-called high-mass twins).]{} [We numerically obtained rotating, axisymmetric, and stationary stellar configurations in the framework of general relativity, and studied their global parameters and stability.]{} [The instability induced by the equation of state divides stable neutron star configurations into two disjoint families: neutron stars (second family) and hybrid stars (third family), with an overlapping region in mass, the high-mass twin-star region. These two regions are divided by an instability strip. Its existence has interesting astrophysical consequences for rotating neutron stars. We note that it provides a natural explanation for the rotational frequency cutoff in the observed distribution of neutron star spins, and for the apparent lack of back-bending in pulsar timing. It also straightforwardly enables a substantial energy release in a mini-collapse to another neutron-star configuration (core quake), or to a black hole.]{}
Introduction
============
Recent observations of the high-mass pulsars PSR J1614-2230 [@Demorest2010; @Fonseca2016] and PSR J0348+0432 [@Antoniadis2013] with masses $M\, {\approx}2 M_\odot$ has motivated the nuclear and particle physics communities to deepen their understanding of the equation of state (EOS) of high-density matter and of the possible role of exotic states of matter in neutron star (NS) interiors. For further constraints on the high-density EOS from NS and heavy-ion collision experiments see [@Klahn:2006ir] and [@Klahn:2011au].
Measurements of high masses of NSs do not immediately imply that very high densities prevail in their cores, meaning that a transition to exotic forms of matter (hypernuclear matter, quark matter) has to be invoked. Of two models for the high-density EOS with the same stellar mass, the stiffer model will lead to a lower central density but to a larger radius. Therefore, radius measurements for high-mass pulsars are of the utmost importance.
Current radius measurements are controversial. Determinations of the radius $R$ range from about 9 km [@Guillot2013] to 15 km [@Bogdanov2013], but one must be aware of possible systematic flaws (see, e.g., @Heinke2014 [@EH16]); for a recent critical assessment see, for example, [@Fortin2015; @Miller2016; @Haensel2016]. At a gravitational mass of ${\sim}2 M_\odot,$ the range of radius values mentioned above would correspond to a range of central densities of the compact star between $2.5 - 6~n_0$, where $n_0=0.15$ fm$^{-3}$ is the nuclear saturation density.
Several observational programs for simultaneous measuring of pulsar masses and radii are currently in preparation: the Neutron star Interior Composition ExploreR (NICER, @NICER), the Square Kilometer Array (SKA, @SKA), Athena [@Athena+], and possibly, a LOFT-size mission [@LOFT]. Thus there is hope that in the near future it will be possible to reconstruct the cold NS matter EOS $P(\varepsilon)$ (here $P$ is pressure and $\varepsilon$ is energy density) within the measurement errors from the measured $M(R)$ relation by means of inverting the Tolman–Oppenheimer–Volkoff (TOV, @Tolman1939 [@OppenheimerV1939]) equations. In addition, specific proposals to use gravitational waves (GWs) for measuring the NS radius from either the observations of the inspiral (e.g., @Bejger2005 [@DamourNV2012]) or post-merger waveforms (e.g., @BausweinSJ2015) were presented.
The systematic investigation of a wide class of hybrid stars with varying stiffness of hadronic matter at high densities and the possibility of quark matter with varying high-density stiffness has revealed interesting findings [@Alvarez2016]. Possible future measurements of radii of recently discovered high-mass stars [@Demorest2010; @Antoniadis2013] would select a hybrid EOS with a strong first-order phase transition if the outcome of their radius measurement were to show a difference of about 2 km with significance. Such a possibility has been suggested earlier on the basis of a new class of hybrid star EOS that fulfill the generic condition that the baryonic EOS is strongly stiffened at high densities, for instance, by effects of the Pauli exclusion principle (quark exchange interaction between baryons), and a strong first-order deconfinement phase transition requiring sufficiently soft quark matter at the transition between baryonic and quark phases, ${\rm B\longrightarrow Q}$. The quark matter EOS has, however, to stiffen quickly with increasing density so that immediate gravitational collapse that is due to the transition does not occur, allowing stable hybrid stars to exist.
Such a solution for hybrid stars, which form a third family of compact stars that are disconnected from the baryonic branch of compact stars, is very interesting for the possible observational verification of specific features of phase transition to quark matter in NS cores. A sharp first-order phase transition between pure B and Q phases, occurring at constant pressure $P_{_{\rm BQ}}$, is associated with an energy density jump from $\varepsilon_{_{\rm B}}$ to $\varepsilon_{_{\rm Q}}$ ($\varepsilon$ is the energy density including the rest energy of particles). Then, a general necessary condition for the existence of a disconnected family of NSs with Q-phase cores is $\varepsilon_{_{\rm Q}}/\varepsilon_{_{\rm B}}>\lambda_{\rm crit}$, with $\lambda_{\rm crit}=\frac{3}{2}(1+P_{_{\rm BQ}}/\varepsilon_{_{\rm B}})$ [@Seidov1971]. The second term in the brackets comes from general relativity. The condition implies that hybrid stars with small quark cores are unstable to radial perturbations and collapse into black holes (BH), and therefore $\varepsilon_{_{\rm Q}}/\varepsilon_{_{\rm B}}>\lambda_{\rm
crit}>\frac{3}{2}$ implies the existence of a separate hybrid stars branch (a detailed review of the structure and stability of NSs with phase transitions in their cores is given in Sect. 7 of @HaenselPY2007). This corresponds to a separate and small $R$ segment of the $M(R)$ relation. High mass and small radii imply high spacetime curvature and strong gravitational pull, resulting in a relatively flat (nearly horizontal) $M(R)$ segment. This hybrid star branch is characterized by a narrow range of $M$, a broad range of $R$, with $M$ weakly increasing with decreasing $R$, up to a very flat $M$ maximum. These features of the hybrid-star branch result in specific observational signatures of a strong B$\longrightarrow$Q first-order phase transition in NS cores. Moreover, these generic properties of the hybrid-star branch indicate a relative ”softness” of their configurations with respect to perturbations that are due to rotation and oscillations, for example. All these generic features are studied in the present paper, using an illustrative example of an advanced EOS composed of a stiff baryonic segment and a strong first-order phase transition to the quark phase (Sect. 3).
The ${\rm B\longrightarrow Q}$ phase transition might occur smoothed within a finite pressure interval through a mixed BQ phase layer. However, the interplay of the surface tension at the B-Q interface and charge screening of Coulomb interaction (in a mixed state, B and Q phases are electrically charged) make the mixed-state layer very thin [@Endo2006] so that the key features of the hybrid-star branch remain intact [@Alvarez-Castillo2015:2014dva].
Stable branches of static configurations in the $M-R$ plane have very specific generic features (see, e.g., @Benic2014). The high-mass baryon branch is very steep, not only nearly vertical, but even with $R$ increasing with $M$, a feature characteristic of very stiff NS cores. The hybrid stable twin-branch is predicted to be flat, nearly horizontal with very broad maximum. Measuring radii of ${\sim}2 M_\odot$ stars, which span a wide range of values from about $12$ to $15$ km, will clearly indicate a hybrid branch, and if moreover the radii for standard NS masses, $1.2 - 1.6 M_\odot$, are roughly constant at approximately $14$ km, then the evidence for two distinct families that are separated as a result of a strong first-order phase transition would be quite convincing. The range of central pressures for NSs of different radii at $2 M_\odot$, which may bear a connection to the universal hadronization pressure found in heavy-ion collisions, is discussed in @Alvarez2016b.
This possibility of finding observational evidence for a first-order phase transition in NS cores from the phenomenology of $M-R$ characteristics of NS populations offers the chance to answer to the currently controversial question of whether a critical endpoint of first-order phase transitions in the QCD phase diagram exists [@Alvarez-Castillo:2013cxa; @Blaschke:2013ana].
Since the nature of the QCD transition at vanishing baryon density is beyond doubt identified as a crossover in two independent lattice QCD simulations at the physical point [@Bazavov:2014pvz; @Borsanyi:2013bia], the evidence for a first-order phase transition at zero temperature and finite baryon density necessarily implies the existence of a critical endpoint (CEP) of first-order phase transitions in the QCD phase diagram (we note that there are theoretical conjectures about a continuity between hadronic and quark matter phases at low temperatures and finite densities in the QCD phase diagram [@Schafer:1998ef; @Hatsuda:2006ps; @Abuki:2010jq]; if, however, there were observational evidence for a horizontal branch in the $M-R$ diagram of compact stars, these considerations would become obsolete). The very existence of such a CEP is a landmark for identifying the universality class of QCD, and due to its importance for model building and phenomenology, a major target of experimental research programs with ultra-relativistic heavy-ion collisions at BNL RHIC (STAR beam energy scan, @Stephans:2006tg) and CERN SPS (NA49 SHINE, @Gazdzicki:2006), in future at NICA in Dubna [@Sissakian:2006dn] and at FAIR in Darmstadt [@FAIR].
First examples for microscopically founded hybrid star EOS that simultaneously fulfill the above constraints for the existence of a disconnected hybrid star branch and for a high gravitational mass of about $2~M_\odot$ (which implies the existence of so-called high-mass twin stars) have been given in @Blaschke:2013ana. The recent systematic investigation of high-mass twin stars in @Alvarez-Castillo2015:2014dva is based on the EOS developed in @Benic2014, which joins a relativistic density functional for nuclear matter with nucleonic excluded volume stiffening with a Nambu–Jona-Lasinio (NJL) type model for quark matter that provides a high-density stiffening as a result of higher order quark interactions. These examples belong to a new class of EOS that can be considered as a realization of the recently introduced three-window picture for dense QCD matter [@Kojo:2015fua] as a microscopic foundation for the hybrid EOS that was conjectured by @Masuda:2012ed, suggesting a crossover construction between hadronic and quark matter. These three windows depicted in Fig. 1 of @Kojo:2015fua are characterized by the following:
- occasional quark exchange between separated and still well-defined nucleons, leading to quark Pauli blocking effects in dense hadronic matter [@Ropke:1986qs] that can be modeled by a hadronic excluded volume or by repulsive short-range interactions,
- multiple quark exchanges that lead to a partial delocalization of the hadron wavefunctions and to the formation of multi-quark clusters and a softening of the EOS by an attractive mean field,
- baryon wave functions overlap and quarks become delocalized; a Fermi sea for strongly interacting quark matter forms with higher order repulsive short-range interactions, for example, by multipomeron exchange [@Yamamoto:2015lwa] or nonlinear quark interactions [@Benic:2014iaa].
A still open and controversial question in this context is whether chiral symmetry restoration and deconfinement transition (which both coincide on the temperature axis according to lattice QCD simulations) would also occur simultaneously in the dense matter at zero temperature. If this were not the case, then there would be room for a more complex structure of the QCD phase diagram, for instance, with a triple point that is either due to a quarkyonic matter phase (light quarks confined in baryons that form parity doublets, @McLerran:2007qj) or a massive quark matter phase [@Schulz:1987qg] for which there is circumstantial evidence from particle production in ultrarelativistic heavy-ion collision experiments [@Andronic:2009gj]. More circumstantial evidence for a region of strong first-order phase transitions in the QCD phase diagram comes from the baryon stopping signal in the energy dependence of the curvature of the net proton rapidity distribution at midrapidity for energies in the intermediate range between former AGS experiments and the NA49 experiment at the CERN SPS [@Ivanov:2012bh]. This signal has been proven to be quite robust under different experimental constraints [@Ivanov:2015vna] and against hadronic final state interactions [@Batyuk2016].
On the basis of this discussion, the new class of EOS with a three-window structure provides the theoretical background for a [**]{}, the phenomena in the energy scan of heavy-ion collision experiments, and the creation of two disjoint families of NS configurations. Astrophysical observations of the [**]{} with drastically different core compositions may be regarded as a manifestation of these dense matter EOS features in different regions of the QCD phase diagram and under different physical conditions.
In the present work we add to the discussion of high-mass twin stars a detailed investigation of their properties under rigid rotation. This is because we expect a strong response of both branches to rotation. First, the high stiffness of the high-density EOS of the baryonic branch results in large radii: $R\;{\sim}15$ km at $M\;{\sim}2\,M_\odot$, so that the effect of the centrifugal force will be large. Second, the stable static hybrid branch is flat, which makes it particularly susceptible to the effects of rotation: the margin of stability along this branch is narrow.
The neutron star instability induced by a strong phase transition in the EOS was studied in detail by @Zdunik2006, who conjectured that the character of stability is not changed by the rotation rate of the star (disjoint families remain separated at any rotation rate). This question will be of particular interest for discussing compact star phenomena tied to the evolution of their rotational state, which eventually ends by collapse into a black hole (e.g., @FalckeR2014).
The general idea of the present work is to illustrate generic features of a class of high-density quark phase transition EOSs using an exemplary EOS, in order to discuss the regions of stable and unstable configurations related to a strong first-order phase transition. For illustration, we use one of the EOS that was recently developed by @Benic2014.
The article is structured as follows. In Sect. \[sect:methods\] we describe the methods we used to calculate the rotating configurations, stressing particularly the need for high precision of numerical simulations. Precision is particularly important for testing stability criteria of stationary rotating configurations, which are formulated there. Section \[sect:results\] starts with a brief presentation of the EOS that we use to illustrate generic properties of the rotating high-mass twins. Then we construct families of rotating configurations, assess their stability, and classify the regions of instability and their generic features. Discussion of the results in Sect. \[sect:discussion\] involves evolutionary considerations, potential scenarios leading to observational manifestations of massive-twin case, including dynamical phenomena triggered by the instabilities and their possible astrophysical appearances. The final part of Sect. \[sect:discussion\] presents the conclusions.
Methods {#sect:methods}
=======
In order to analyze the astrophysical consequences of the above-mentioned specific type of EOS, we have obtained rigidly rotating, stationary, and axisymmetric NS configurations by means of the numerical library [LORENE]{}[^1] [nrotstar]{} code, using the 3+1 formulation of general relativity of @BonazzolaGSM1993, and employing the multidomain pseudo-spectral decomposition (three domains inside the star). The accuracy is controlled by a 2D general-relativistic virial theorem [@BonazzolaG1994] and for the results presented here is typically on the order of $10^{-7}$. The high accuracy provided by the spectral method implementation of [LORENE]{} and a general low numerical viscosity of spectral methods is particularly suitable for studying the stability of NS models.
The evanescent error behavior of the solutions can be employed by expanding the number of coefficients to obtain the relevant values practically up to machine precision. In this context, we recall that a global parameter that is strictly conserved during the evolution of an isolated NS is its total baryon number (baryon charge) $A_{\rm b}$. Instead of $A_{\rm b}$ it is convenient to use in relativistic astrophysics the (or rest mass of the NS defined by $M_{\rm b}=A_{\rm
b}m_{\rm b}$, where $m_{\rm b}$ is a suitably defined baryon mass. $M_{\rm b}$ is the total mass of $A_{\rm b}$ non-interacting baryons; it is easy to extend these definitions to NS cores built of quarks (three quarks contribute +1 to the baryon number of NS). In our calculations we follow the LORENE unit convention and use a value of the mean baryon mass $m_{\rm b}=1.66\times 10^{-24}$ g. Other definitions of $m_{\rm b}$, suitable for specific applications in NS and supernova physics, are discussed in Sect. 6.2 of @HaenselPY2007. We note that while $M_{\rm b}$ is strictly constant in spinning down or cooling isolated NS, their gravitational mass is changing. We are in general interested in obtaining accurate values of the gravitational mass $M$, the baryon mass $M_{\rm b}$ and, the total angular momentum $J$ that are the properly defined functionals of the stellar structure and EOS suitable to study instabilities (for more details and a discussion comparing the 3+1 formulation with the slow-rotation formulation see the recent review of the stability of rotating NSs with exotic cores, @Haensel2016).
In the following we use the method described in @Zdunik2004 [@Zdunik2006], who studied, among other things, the [**]{} phenomenon proposed for NSs in @Glendenning1997. Back-bending, a temporary spin-up of an isolated NS that is decreasing its total angular momentum $J$ by the dipole radiation, for example, can be robustly quantified by analyzing the extrema of the baryon mass $M_{\rm b}$ along the lines of constant spin frequency $f$, that is, the rate of the baryon mass $M_{\rm
b}$ change with respect to a central EOS parameter $\lambda_c$ (central pressure $P_c$, e.g.,) changing along $f=const.$ lines. The condition for back-bending to occur is $$\left(\frac{\partial M_{\rm b}}{\partial\lambda_c}\right)\bigg\rvert_f < 0
\label{eq:back-bend}$$ (see also @Haensel2016, Sect. 7 for detailed description). Nevertheless, the susceptibility to back-bending [**]{} that the NS is indeed unstable. In order to study the regions of true instability, we use the turning-point theorem formulated for rotating stellar models by [@Sorkin1981; @Sorkin1982; @FriedmanIS1988], which states that the sufficient condition for the change in stability corresponds to an extremum of the gravitational mass $M$, or the baryon mass $M_{\rm b}$ at fixed $J$: $$\left(\frac{\partial M}{\partial\lambda_c}\right)\bigg\rvert_{J}=0,\qquad
\left(\frac{\partial M_{\rm b}}{\partial \lambda_c}\right)\bigg\rvert_{J}=0,
\label{eq:inst_cond1}$$ or equivalently, to an extremum of $J$ at fixed either $M$, or $M_{\rm b}$: $$\left(\frac{\partial J}{\partial \lambda_c}\right)\bigg\rvert_{M} =0,\qquad
\left(\frac{\partial J}{\partial \lambda_c}\right)\bigg\rvert_{M_{\rm b}} =0.
\label{eq:inst_cond2}$$ In the following, we illustrate the generic features of the third family of compact stars with one of the EOSs developed by @Benic2014. The baryon phase EOS is based on the relativistic mean-field (RMF) model DD2 of @Typel2010DD2. It fits the semi-empirical parameters of nuclear matter at saturation well, and features density-dependent coupling constants. In order to obtain a high-density stiffening of B phase, which is necessary to yield a strong B$\longrightarrow$Q phase transition, while fulfilling $M_{\rm max}>2\;M_\odot$ for the hybrid Q branch, the DD2 model is modified using excluded volume (EV) effects that are included to account for the finite size of baryons, resulting in a DD2+EV model for the baryon phase. The EV effect was included without altering the good fit of DD2 to semi-empirical nuclear matter parameter values. The quark phase description is based on the Nambu–Jona-Lasinio density-dependent model, with dimensionless coupling constants $\eta_2=0.12$ and $\eta_4=5$. The complete EOS with a strong first-order phase transition ${\rm B\longrightarrow
Q}$ was obtained using Maxwell construction. We analyze the features of sequences of configurations along the lines of fixed baryon mass $M_{\rm b}$ and total angular momentum $J$.
Results {#sect:results}
=======
The static configurations of NSs for the sample EOS have been studied in @Benic2014. The basic stellar parameters were the gravitational mass $M$ and the circumferential radius $R$. To analyze the stability of rotating stars, it is more convenient to consider the baryon mass $M_{\rm b}$ (also called the rest mass) of the star and the equatorial circular radius $R_{\rm eq}$ (see, e.g., @Haensel2016). Generally, the study of rapidly rotating star configurations offers advantages for the investigation of phase transitions in NS interiors because as a function of the angular momentum (rotation frequency), which is an additional parameter, the profile of the density distribution and therefore also the interior composition would change, which is expected to allow for observational signatures in the course of the rotational evolution of the star (for early examples, see @Glendenning1997 [@Chubarian:1999yn]). The spin frequency range we studied spans the astrophysically relevant range from $f=0$ Hz (static configurations, corresponding to the solutions of the Tolman-Oppenheimer-Volkoff equations), to $f=1000$ Hz (i.e., much higher than the frequency of $716$ Hz of the most rapid pulsar known to date, PSR J1748-2446ad of @Hessels2006).
Stationary uniformly rotating configurations are labeled (determined) by two parameters. In Fig.\[fig:mbr\_regions\] we show various continuous two-parameter curves $[M_{\rm b}(\lambda_{\rm c},\beta), R_{\rm
eq}(\lambda_{\rm c},\beta)]$ in the $M_{\rm b}-R_{\rm eq}$ plane. Here, the parameter $\lambda_{\rm c}$ can for instance be the central pressure $P_{\rm c}$, or central baryon chemical potential $\mu_{\rm b(c)}$ (both behaving continuously and monotonously along the curve). The quantity $\beta$ characterizes the uniform rotation of the star; we chose it to be equal to the frequency of rotation $f$ or the total stellar angular momentum $J$. In Fig. \[fig:mbr\_regions\] we study the regions of back-bending and stability on the baryon mass $M_{\rm b}$-equatorial radius $R_{\rm eq}$ plane. The region in which the back-bending phenomenon is present is marked with the dashed $f=const.$ lines. An isolated NS ($M_{\rm
b}=const.$) that decreases its angular momentum (by electromagnetic dipole radiation, e.g.) crosses the lines of $f=const.$ while moving from the right to the left side of Fig. \[fig:mbr\_regions\]. According to Eq. (\[eq:back-bend\]), in the region of dashed lines, the $M_{\rm b}=const.$ line crosses the $f=const.$ curves such that it results in a [**]{} while the angular momentum is monotonically decreasing, that is, the back-bending. The wavy pattern region denotes the opposite, usually observed behavior, that is, [**]{} with angular momentum loss.
Figure \[fig:mbr\_regions\] also shows two instability regions. The first is the familiar instability with respect to axisymmetric perturbations related to the existence of the maximum mass; the blue line corresponding to maxima of $J$ and denotes its boundary. The second instability is induced by the strong phase transition (red strip between the local minimum and maximum of $J$). The latter divides the space of stable solutions into two disjoint families at any rotation rate [@Zdunik2006]. Assuming that at some moment in its evolution an isolated NS enters the instability strip, it becomes then unstable and collapses to another, more compact configuration along the lines of $M_{\rm
b}=const.$ and $J=const.$ [@Dimmelmeier2009].
Configurations that survive such a mini-collapse are located in the green region. We note that in this idealized picture no mass and angular momentum loss is assumed. Realistically, some mass and angular momentum loss may occur, and so the green region will decrease toward lower spin rates and toward the region that is stable against the back-bending, which is marked with the wavy pattern in Fig. \[fig:mbr\_regions\]. Careful analysis of Fig.\[fig:mbr\_regions\] reveals the existence of a critical angular momentum $J_{\rm crit}$, such that for $J>J_{\rm crit}$ exceeding the maximum mass on the B branch (resulting from accretion) implies in a direct collapse into a rotating BH. This situation is depicted in detail in Fig. \[fig:mbr\_jumps\].
The existence of $J_{\rm crit}$ and its potential observational signatures are discussed in more detail in Sect. \[sect:discussion\]. Here we restrict ourselves to a few comments on how the existence of $J_{\rm crit}$ is due to generic features of the rotating-twin $M(R_{\rm eq})$ curves. Consider first the static ($J=0$) B and Q branches. They have very different $M(R)$ dependence. The maximum mass at $M=M^{\rm (B)stat}_{\rm max}$ (with $P_{\rm c}=P_{_{\rm BQ}}$) is due is to the strong phase transition, and is somewhat lower than the flat maximum on the Q branch, $M^{\rm (Q)stat}_{\rm max}$. We proceed to the case of an increasing $J$, when an NS on the B branch acquires mass and angular momentum from an accretion disk. At some moment it reaches a maximum on the B branch; it has then $J=J_1$, $M^{(\rm B)}_{\rm b}=M_{\rm b1}$. Under further accretion it collapses into a Q configuration of the same $M_{\rm b1}$ and $J_1$, provided $M_{\rm b1}<M_{\rm b,max}^{(\rm Q)}(J_1)$. The equality is reached exactly at $J_{\rm crit}$, and for higher $J$ the B star collapses directly into a BH. A critical $J$ is equivalent to maximum $f,$ however, which can be reached by the baryon stars. This situation is depicted in detail in Fig. \[fig:mbr\_jumps\] and is discussed further in Sect. \[sect:discussion\]. A separate question is related to the way in which the NS reaches the instability line. As shown in @Zdunik2005 and @Bejger2011, the efficiency of transfer of the angular momentum in the process of disk accretion governs the $M(R)$ evolution of a spinning-up NS. To illustrate how an NS can reach the unstable region, we present in Fig. 3 the mass–equatorial radius dependence for evolutionary tracks of accreting NS. A realistic evolutionary path depends on many physical details, such as the configuration and strength of the magnetic field, the accretion rate, and the way the magnetic field interacts with the disk. We assume thin-disk accretion in the presence of the magnetic field by employing a model used previously in @Bejger2011a and @Fortin2016a. Two different efficiencies of angular momentum transfer, $x_l=0.5$ and $x_l=1$, are considered for initially nonrotating configurations of mass $1.4\,M_\odot$ and $1.8\,M_\odot$. The initial magnetic field is $B=10^8$ G. If the accretion stops before the instability is reached, the star evolves by moving horizontally from right to left (secular spin-down). Figure\[fig:mr\] is intended to show that it is possible to reach the instability region at low or high angular momentum corresponding to the two cases considered in Fig.\[fig:mbr\_jumps\].
The effect of the EOS-induced instability strip may also be studied on the angular momentum–spin frequency $J - f$ plane, assuming a fixed baryon mass sequence, see Fig. \[fig:jf\]. The evolution of an isolated NS that decreases its angular momentum $J$ corresponds to a downward movement in this figure. This also means that the central density (and pressure) of such a star increases. A spinning-down NS on the B branch decreases its $J$ until it becomes unstable (point B$_1$), which forces it to collapse (dynamically migrate) to another stable branch (point Q$_1$) with the same $M_{\rm b}$ and $J$. We note here that in principle by adopting a certain EOS we may place constraints on the parameters of massive pulsars with known spin frequency. In the example from Fig. \[fig:jf\], an NS with $M_{\rm b}=2.325\; M_\odot$ and the spin frequency of PSR 1614-2230 (317 Hz) is located on either the upper baryonic branch denoted by B, or on the lower branch (hybrid stars - Q), the two configurations differing greatly in the total angular momentum because of very different moments of inertia. A configuration with a slightly lower $M_{\rm
b}=2.3\; M_\odot$ (blue line) may, for this particular EOS, exist at 317 Hz only in the B phase, since the instability transfers it to frequency ${\simeq}200$ Hz (significantly lower than 317 Hz). A configuration with a slightly higher $M_{\rm b}=2.35\; M_\odot$ (red line) is excluded as a model of PSR 1614-2230; during its evolution, it never spins down to reach 317 Hz: after the dynamical migration and a period of spin-down, it will collapse to a BH.
The time evolution of the rotation period of an isolated NS loosing its energy and angular momentum due to the dipole radiation is presented in Fig. \[fig:tt\]. The assumption of dipole radiation from the pulsar leads to the formula $$\frac{{\rm d}E}{{\rm d}t}=-\frac{\mu^2_B\Omega^4\sin^{2}\alpha}{6c^3},$$ where $E$ is the energy of rotating pulsar and $\mu_B=B\,R^3$ the dipole moment of a star. In the framework of general relativity, we should use the total mass-energy of the star $Mc^2$ in place of $E$. For the evolution of an isolated NS with a fixed total number of baryons, the relation ${\rm d}M=\Omega/c^2\,{\rm d}J$ holds, resulting in the equation for the time evolution, $$\frac{{\rm d}\tilde{J}}{{\rm d}\tilde{t}}=-3.34\times 10^{-9}{B_{12}^2\,R_{6}^6\,f_{_{\rm Hz}}^3~,}$$ where $\tilde{J}$ is the stellar angular momentum in the units of $GM^2_\odot/c$ and $\tilde{t}$ is time in kyr (the same relation in a Newtonian case could be obtained for $E=\frac{1}{2}I\Omega^2$ and $J=I\Omega$, with $I$ being the NS moment of inertia). The timescales of slowing down of a solitary pulsar from milliseconds (close to maximum mass-shedding frequency) down to ${\sim}15$ ms before and after the mini-collapse are clearly comparable - we assumed that the magnetic moment $\mu_B$ does either not change during the evolution, or that the magnetic field decreases.
A mini-collapse is a dynamical process, provided the B$\longrightarrow$Q conversion has detonation character [@Haensel2016]. It involves considerable spin-up and a substantial reorganization of the interior of the star [@Dimmelmeier2009]. It is also exoenergetic: a substantial amount of energy, quantified as the difference between the initial and the final gravitational mass, $\Delta M = M_{\rm ini} - M_{\rm fin}$, is released in the process. The left panel of Fig. \[fig:j\_dm\_f\_df\] shows the relation between the angular momentum $J$ of the star and the energy difference $\Delta M$. The $\Delta M(J)$ relation is approximately quadratic in $J$: $\Delta M(J) = aJ^2 + \Delta
M(0)$. For the particular EOS used in this study, $a=0.106$ and $\Delta
M(0)=0.749$, for $\Delta M$ in $10^{-3}\ M_\odot$ and $J$ in $GM^2_\odot/c$ units. The line ends at a critical $\tilde{J}\approx 2.20,$ where the dynamical collapse forces an NS to the instability region, where it collapses to a BH (see also the right panel of Fig. \[fig:mbr\_jumps\]). The right panel of Fig. \[fig:j\_dm\_f\_df\] shows the amount of spin-up (spin frequency difference) $\Delta f = f_{\rm fin} - f_{\rm ini}$ that is acquired in the mini-collapse as a function of the initial spin frequency $f_{\rm ini}$. The situation presented here corresponds to a specific case studied in detail in @Zdunik2008, where the overpressure of the new metastable phase is set to zero, $\delta\bar{P}=0$.
From the same plot one may estimate the ‘Newtonian’ change of kinetic energy and the luminosity of the process. Assuming that the total angular momentum does not change during the process, we obtain $$\Delta E^{\rm rot} = E^{\rm rot}_{\rm fin} - E^{\rm rot}_{\rm ini} = \frac{1}{2}J\left(\Omega_{\rm fin} - \Omega_{\rm ini}\right).
\label{eq:delta_erot}$$ For the value of $J = 2\,{GM_\odot^2/c}$ the change of the spin frequency is approximately $\Delta f = 240$ Hz ($\Delta \Omega = 1510$ rad/s). For these figures we obtain $\Delta E^{\rm rot}\simeq 2\times 10^{52}$ erg. This value overestimates the difference in total gravitational mass $\Delta M$ by one order of magnitude ($10^{-3}\, M_\odot c^2\simeq 2\times 10^{51}$ erg). The process occurs on a dynamical timescale of a millisecond.
Discussion {#sect:discussion}
==========
The goal of this article is to [**]{} consider a specific class of EOS featuring a substantial phase transition motivated by the theory of dense matter physics. The EOS-induced instability region divides stable NS configurations into two disjoint families (twin families). Its existence has interesting astrophysical consequences for rotating NSs. We note that it facilitates a natural (i.e., not fine-tuned) way for various astrophysical phenomena that we list below. . Even though theoretical models of NSs allow for spin rotation rates much above 1 kHz and although with current observational techniques such rapidly rotating pulsars could be detected (see, e.g., @P10 [@DE11]), so far, the most rapidly rotating NS observed is PSR J1748-2446ad (716 Hz, @Hessels2006). It cannot be excluded [**]{} that some rapidly rotating and massive NSs were created close to their currently observed state, that is, in a specific type of core-collapse supernovæ. If this were the case, then they might appear practically everywhere on the right side of the thick blue line of Fig. \[fig:mbr\_regions\], with the exception of the instability strip (red area), where no stationary axisymmetric solutions are possible. However, as the observations, evolutionary arguments, and numerical simulations tend to suggest, NS that become radio pulsars are not born with ${\sim}$1 - 3 ms periods, but with much longer periods of ${\sim}20-150$ ms, see @FGK06, @Kramer2003, Table 7.6 in @LyneG1998, and references therein (a specific class of NS with millisecond periods at birth in massive core-collapse supernovae are thought to be progenitors of [**]{}, which are observed as soft-gamma ray repeaters or anomalous X-ray pulsars, see e.g., @KargaltsevPavlov2008). Then, after slowing down to a period of a few seconds and entering the pulsar graveyard, they gain their angular momentum, as well as mass, during long-term accretion processes in low-mass binary systems (in the so-called recycling of dead pulsars, see @Alpar1982 [@Radhakrishnan1982; @Wijnands1998]), and it is possible that some of them enter the strong phase transition instability strip sometime in their evolution. Sufficiently massive and sufficiently rapidly rotating NS will then migrate dynamically along the $M_{\rm b}=const.$ track in the direction of the twin branch (see, e.g., @Dimmelmeier2009). Moreover, for some critical angular momentum (critical spin frequency) the value of $M_{\rm b}$ on the right side of the instability strip (the $M_{\rm
b}(R_{\rm eq})$ peaks in Fig. \[fig:mbr\_regions\]) is higher than the corresponding maximum of $M_{\rm b}$ on the twin branch - in that case, the star collapses to a BH. . One of the observational predictions related to substantial dense-matter phase transitions is the detection of the back-bending phenomenon, which occurs at spin frequencies of known pulsars. As we showed in Sect. \[sect:results\], Fig. \[fig:mbr\_regions\], NSs exhibiting an instability that is caused by a strong phase transition avoid the vast majority of the back-bending region for spin frequencies lower than some critical value. The most rapidly rotating currently known pulsar, PSR J1748-2446ad, has a spin period of 716 Hz. From Fig. \[fig:mbr\_regions\] we note that the $f=700$ Hz line is the first dashed line above the no back-bending (wavy pattern) region. When we assume that the EOS used for illustration is the true EOS of dense matter, this means that PSR J1748-2446ad, which does not show the features of back-bending in the timing, still resides on the hadronic branch (does not contain the quark core). Additionally, the most massive stars that are in the back-bending region may not be effective pulsars - they may be electromagnetically exhausted, with their magnetic field dissipated in the violent process of mini-collapse, and therefore not easily detectable.
. The existence of an instability strip creates in a mass- and spin-frequency-dependent radius region of avoidance between the allowed green region and stable baryonic branch on the right of Fig. \[fig:mbr\_regions\], which is broadened with increasing mass (see also Fig. 3 of @Benic2014). For nonrotating $2\;{\rm M}_\odot$ NS, the predicted $R$-gap is ${\sim}1$ km. Small-radius Q-branch twins have $R^{\rm (Q)} = 12.5\;
{\rm km}- 14.5\;{\rm km}$ within a very narrow mass range ${\sim}0.1\;{\rm
M}_\odot$. The measurement of a radius $R>15\;$km for a ${\sim}2\;{\rm M}_\odot$ star indicates a B-branch configuration. If for another ${\sim}2\;{\rm M}_\odot$ NS the radius is determined to be within the range $12.5\; {\rm km}- 14.5\;{\rm km}$, then we obtain strong evidence in favor of distinct B and Q twin branches as a result of a strong B$\longrightarrow$Q phase transition.
. The twins on the B and Q branches have different internal structure. The B-star twin is more compact, and its mass is concentrated in the dense quark core. Consequently, at the same $M$ of twins, one has $I^{\rm (Q)}$ significantly smaller than $I^{\rm (B)}$ , as shown on Fig. \[fig:I-M-R\]. Moreover, the strong first-order phase transition B$\longrightarrow$Q results in an $I$ gap between the B and Q twins (Fig. \[fig:I-M-R\]). The moment of inertia of NS can be measured through the spin-orbit effect contribution to the timing parameters for a binary of two radio pulsars [@DamourSchaefer1988]. The first binary of this type, PSR J0737-3039A,B was discovered more than a decade ago [@Lyne2004]. Pulsar B has become invisible to terrestrial observers in March 2008 because its beam wandered out of our line of sight as a consequence of the geodetic precession effect [@Perera10]. It may reappear as late as in 2034 (or later), depending on the model of the pulsar magnetosphere (see, e.g., @LL14). Since the mass of pulsar A has been accurately determined, a measurement of its moment of inertia through the spin-orbit momentum coupling would allow us to constrain the radius and hence the EOS [@LattimerSchutz2005]. Given the present timing accuracy of the system’s post-Keplerian parameters, that is, the periastron advance, the decrease in the orbital period and the Shapiro shape parameter, from which the spin-orbit coupling contribution is derived, reasonable accuracy may be achieved around the time of pulsar B reappearance in ${\simeq}20$ years [@Kramer09]. However, in the forthcoming era of large radio telescopes (e.g., FAST, SKA) the number of known pulsars will increase by orders of magnitude, including many thousands of millisecond pulsars, out of which we may hopefully expect tens of binary systems with two pulsars suitable for simultaneously measuring $I$ and $M$ of an NS. A sufficiently dense set of pairs $\lbrace I_i,M_i \rbrace$ resulting from these future measurements could then be used to confirm or reject the generic $I(M)$ shape in Fig.\[fig:I-M-R\].
. We assume that the B$\longrightarrow$Q phase transition, mediated by strong interaction, occurs in the detonation regime. A dynamical process involving NS, triggered by the loss of stability, is associated with a substantial energy release (${\sim}10^{52}~{\rm erg}$), heating of dense matter, kinetic energy flow, and some emission of radiation, on both a short time-scale (NS quake, mini-collapse) and long time-scale (surface glowing). The process is probably also associated with a substantial rearrangement of the NS magnetic field. For NS with total angular momenta larger than some critical value, it leads to a direct collapse to a BH. This event is related to the expulsion of the magnetic field and thus the dynamical migration to the high-mass twin branch may be considered a natural extension of the @FalckeR2014 cataclysmic scenario. Alternatively, for NSs below the angular-momentum threshold, the mini-collapse dynamics may influence the magnetic field to such an extent that it becomes a transient source of observable magnetospheric emission after the final configuration ends up on the twin branch.
To conclude, the assumption of the strong phase transition in the NS EOS leads to a number of falsifiable (at least in principle) astrophysical predictions. As we described in Sect. \[sect:results\] and in Fig. \[fig:jf\], a transition to a new exotic phase (deconfined quarks in this example) constrains the range of available $M_{\rm b}$ and $J$. This reasoning can be extended to other EOS functionals, like the moment of inertia $I$; moreover, these constraints may be combined with the observations that are sensitive to the composition of the core, for example, NS cooling studies. The evolutionary scenario in which some of NS collapse to BHs produces a specific NS-BH mass function without a mass gap, which should be possible to test with current and future searches for low-mass BHs with microlensing surveys [@Wyrzykowski2016], for instance. Unless the NS magnetic field is amplified and/or reoriented during the mini-collapse event, it is likely that it is dissipated and disordered during the process - in the latter case, we expect a population of massive ineffective pulsars with a low magnetic field. Moreover, a dynamical mini-collapse creates a characteristic signature of the GW emission, strongly dependent on the EOS and mass of the NS; short transient GW radiation of this type should be detectable by the advanced era interferometric detectors [@Dimmelmeier2009]. If the collapse is not entirely axisymmetric, the final configuration may retain an asymmetry, thus creating a rotating NS that spins down while continuously emitting the almost-monochromatic GWs. Such objects are among the prime astrophysical targets of the Advanced LIGO and Advanced Virgo interferometric detectors (see the demonstration of search pipelines used on the initial LIGO and initial Virgo detector data, e.g., @Aasi2014CQGra [@Aasi2014PhRvD; @Aasi2015ApJ]).
There are several open questions related to the various aspects of the high-mass twin scenario, such as what the conditions are for the NS to reach the instability region via the disk accretion spin-up, what the influence of the possible meta-stability of the quark phase core is on the NS population on the twin branch, and how the electromagnetic and gravitational-wave emission depends on the parameters of the NS. These questions will be addressed in subsequent studies.
This work has been supported in part by the Polish National Science Centre (NCN) under grant No. UMO-2014/13/B/ST9/02621. D.B. has been supported in part by the MEPhI Academic Excellence Project under contract No. 02.a03.21.0005. The authors acknowledge support from the COST Action MP1304 ”NewCompStar” for their networking activities.
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[^1]: http://www.lorene.obspm.fr
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abstract: 'We theoretically analyze a single vortex in a spin polarized 3D trapped atomic Fermi gas near a broad Feshbach resonance. Above a critical polarization the Andreev-like bound states inside the core become occupied by the majority spin component. As a result, the local density difference at the core center suddenly rises at low temperatures. This provides a way to visualize the lowest bound state using phase-contrast imaging. As the polarization increases, the core expands gradually and the energy of the lowest bound state decreases.'
author:
- 'Hui Hu$^{1,2}$, Xia-Ji Liu$^{2}$, and Peter D. Drummond$^{2}$'
title: Visualization of vortex bound states in polarized Fermi gases at unitarity
---
The achievement of superfluidity in trapped ultra-cold atomic $^{6}$Li gases is a landmark advance in the history of physics [@MIT2005]. This is attained by utilizing a broad Feshbach resonance, which is used to tune the inter-atomic interactions. By changing the inverse scattering length $a_{s}$ continuously from negative to positive values, a two-component Fermi gas with *equal* spin populations has a ground state which crosses smoothly from Bardeen-Cooper-Schrieffer (BCS) superfluidity to a Bose-Einstein condensate (BEC) of tightly bound pairs. Of particular interest is the unitarity regime near resonance, where the scattering length diverges ($1/a_{s}\simeq0$). Since the inter-particle spacing is the only relevant length scale, the Fermi gas exhibits a universal behavior [@ho2004].
Quantized vortices are a clear-cut confirmation of superfluidity, and were demonstrated experimentally by Zwierlein *et al.* [@MIT2005]. The equilibrium properties of vortices in a symmetric Fermi superfluid at crossover have been the subject of intense theoretical studies [@gygi; @nygaard; @bulgac; @machida; @mizushima; @levin; @ho2006]. The Andreev-like bound states, which are the fermionic quasiparticle excitations localized in the core, have been widely discussed [@gygi; @machida; @levin; @ho2006]. These bound states are found to play a key role in the structure of vortices.
Most recently, Fermi gases with *unequal* spin populations have been the subject of considerable experimental [@MIT2006; @rice] and theoretical interest [@liu; @sarma; @bedaque; @fflo; @others; @phasediagram; @trap]. The presence of spin polarization leads to exotic forms of pairing, such as breached pairing [@liu] or Sarma superfluidity [@sarma], phase separation [@bedaque], and spatially modulated Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states [@fflo]. An agreement on the *true* ground state of polarized fermionic superfluidity is yet to be reached. However, three recent measurements on the density profiles of polarized $^{6}$Li gases [@MIT2006; @rice], near a Feshbach resonance, indicate a paired superfluid core surrounded by the excess unpaired fermions consistent with a picture of phase separation.
Combining spin polarization with a vortex may help to resolve the issue of the nature of polarized fermion pairing. It is natural to ask how unequal spin populations affect the vortex structure, and how vortex bound states evolve as the polarization increases. This issue arises in the context of pairing and superfluidity in many fields of physics [@rmp]. It is highly relevant to the condensed matter community, where polarized superfluidity is created by applying a magnetic field. There is now strong experimental evidence for the existence of FFLO states in the heavy fermion superconductor CeCoIn$_{5}$ under high fields [@cecoin5]. Strongly interacting polarized Fermi gases have also been under close scrutiny in nuclear matter [@sedrakian], neutron stars [@sedrakian], and high density quark matter [@alford; @rmp], where the spin polarization is created by differences between chemical potentials and/or by mass differences between fermions that form pairs. Polarized vortices of color superfluidity in rotating neutron stars are a possible mechanism for observed glitches in pulsar timing [@rmp].
Here we investigate the properties of a singly quantized vortex in polarized atomic Fermi gases at unitarity, in a cylindrically symmetric trap. Our main results are:
(**A**) We clarify the density profiles of both spin components as a function of polarization. In addition to phase separation, the vortex core suddenly accommodates the excess majority fermions above a critical polarization or a critical chemical potential difference, resulting in a rapid rise of the local density difference inside the core.
(**B**) The local fermionic density of states explains the sudden appearance of an unpaired core of excess majority atoms at the vortex center. The Andreev-like bound states in the core are occupied when the critical chemical potential difference equals the lowest available energy. *This provides a clear visualization of vortex bound states using phase-contrast imaging* [@Shin].
(**C**) With increasing polarization, the vortex core expands while the lowest bound state energy decreases.
The above results are obtained by numerically solving the mean-field Bogoliubov-de Gennes (BdG) equations in a fully self-consistent fashion [@gygi; @BdG], assuming a pairing order parameter that preserves the cylindrical and axially translational symmetries. Symmetry breaking is also possible, *i.e.*, the order parameter may distort cylindrically. This scenario merits further study.
Fermi gases of $^{6}$Li atoms near a broad Feshbach resonance are well characterized using a single channel model [@sc]. The BdG equations describing the quasiparticle wave functions $u_{\eta}\left(\mathbf{r}\right)$ and $v_{\eta}\left(\mathbf{r}\right)$, with excitation energies $E_{\eta}$ read [@gygi]: $$\left[\begin{array}{cc}
{\cal H}_{0}-\mu_{\uparrow} & \Delta(\mathbf{r})\\
\Delta^{*}(\mathbf{r}) & -{\cal H}_{0}+\mu_{\downarrow}\end{array}\right]\left[\begin{array}{c}
u_{\eta}\left(\mathbf{r}\right)\\
v_{\eta}\left(\mathbf{r}\right)\end{array}\right]=E_{\eta}\left[\begin{array}{c}
u_{\eta}\left(\mathbf{r}\right)\\
v_{\eta}\left(\mathbf{r}\right)\end{array}\right],\label{BdG}$$ where ${\cal H}_{0}=-\hbar^{2}\mathbf{\nabla}^{2}/2m+V_{ext}\left(\mathbf{r}\right)$, and $V_{ext}\left(\mathbf{r}\right)=m\omega^{2}\left(x^{2}+y^{2}\right)/2$ is the transverse trapping potential. Along the $z$ axis we instead assume free motion over a length $L$. To account for the unequal spin population $N_{\sigma}$ for $\sigma=\uparrow,\downarrow$, the chemical potentials are shifted as $\mu_{\uparrow,\downarrow}=\mu\pm\delta\mu$, leading to different quasiparticle wave functions for the two components. However, there is a symmetry of the BdG equations under the replacement $u_{\eta\downarrow}^{*}\left(\mathbf{r}\right)\rightarrow v_{\eta\uparrow}\left(\mathbf{r}\right)$, $v_{\eta\downarrow}^{*}\left(\mathbf{r}\right)\rightarrow-u_{\eta\uparrow}\left(\mathbf{r}\right)$, $E_{\eta\downarrow}\rightarrow-E_{\eta\uparrow}$. We can thus retain only $u_{\eta\uparrow}\left(\mathbf{r}\right)$ and $v_{\eta\uparrow}\left(\mathbf{r}\right)$ in Eq. (\[BdG\]), and keep solutions with both positive and negative energies.
The order parameter $\Delta(\mathbf{r})$ and the chemical potentials $\mu_{\uparrow,\downarrow}$ are determined by self-consistency equations for the gap, $\Delta(\mathbf{r})=g\sum_{\eta}u_{\eta}\left(\mathbf{r}\right)v_{\eta}^{*}\left(\mathbf{r}\right)f\left(E_{\eta}\right)$, and the particle density of each component: $n_{\uparrow}\left(\mathbf{r}\right)=\sum_{\eta}\left|u_{\eta}\left(\mathbf{r}\right)\right|^{2}f\left(E_{\eta}\right)$ and $n_{\downarrow}\left(\mathbf{r}\right)=\sum_{\eta}\left|v_{\eta}\left(\mathbf{r}\right)\right|^{2}f\left(-E_{\eta}\right)$. These must be constrained so that $\int d\mathbf{r}n_{\sigma}\left(\mathbf{r}\right)=N_{\sigma}$, where $f\left(x\right)=1/\left(e^{x/k_{B}T}+1\right)$ is the Fermi distribution function, and $g$ ($<0$) is the bare coupling constant, which is related to the $s$-wave scattering length via the regularization prescription: $\left(4\pi\hbar^{2}a_{s}/m\right)^{-1}=1/g+\sum_{\mathbf{k}}1/2\epsilon_{\mathbf{k}}$.
We solve these equations via a *hybrid* procedure, by introducing a high energy cut-off $E_{c}$ above which we use a local density approximation (LDA) for high-lying excitation levels. The standard regularization prescription then yields an effective coupling constant through the self-consistency equation $\Delta(\mathbf{r})=g_{eff}\left(\mathbf{r}\right)\sum'_{\eta}u_{\eta}\left(\mathbf{r}\right)v_{\eta}^{*}\left(\mathbf{r}\right)f\left(E_{\eta}\right)$, where the cut-off summation $\sum'_{\eta}$ is now restricted to $\left|E_{\eta}\right|\leq E_{c}$. Further details of this will be given elsewhere. A clear limitation of the procedure is the use of mean-field factorizations implicit in the BdG equations. From earlier work, we expect this to neglect quantum fluctuations that alter the ground-state energy, while remaining qualitatively correct [@hld].
Below the cut-off, we solve the BdG equations by working in cylindrical coordinates $\left(\rho,\varphi,z\right)$ and taking $\Delta(\mathbf{r})=\Delta(\rho)e^{-i\varphi}$ for a singly quantized vortex. Assuming periodic boundary conditions at $z=\pm L/2$, we write, for the normalized modes, $u_{\eta}\left(\mathbf{r}\right)=u_{nmk_{z}}\left(\rho\right)e^{im\varphi}e^{ik_{z}z}/\sqrt{2\pi L}$ and $v_{\eta}\left(\mathbf{r}\right)=v_{nmk_{z}}\left(\rho\right)e^{i\left(m+1\right)\varphi}e^{ik_{z}z}/\sqrt{2\pi L}$ with $k_{z}=2\pi l/L$. As a consequence, the BdG equations decouple into different $m$ and $l$ sectors [@gygi]. Expanding the radial functions $u_{nmk_{z}}\left(\rho\right)$ and $v_{nmk_{z}}\left(\rho\right)$ in a basis set of 2D harmonic oscillators, we then solve a matrix eigenvalue problem in each sector.
![(Color online). Density profiles of the majority ($\uparrow$-state, solid lines) and minority ($\downarrow$-state, dashed lines) components at $T=0.05T_{F}$ for three typical value of polarization: $p=0.12$ (a), $p=0.35$ (b), and $p=0.75$ (c). Density differences are also plotted in dotted-dashed lines. All the profiles are normalized by $n_{\sigma},_{TF}=\left(1+\beta\right)^{-3/5}\sqrt{15\pi N\lambda/2}/\left(6\pi^{2}\right)\left(\hbar/m\omega\right)^{-3/2}$, which is the peak density for a symmetric gas at unitarity. Panel (d) shows the order parameter profiles. The small oscillations at the edge are a finite size effect.[]{data-label="fig1"}](fig1){width="45.00000%"}
In greater detail, we consider a gas at unitarity with the number of total atoms in the range $N=N_{\uparrow}+N_{\downarrow}=2\times10^{3}-4\times10^{4}$. Two characteristic scales may be defined by considering a symmetric ideal Fermi gas at zero temperature. In the LDA analysis this leads to a Thomas-Fermi (TF) radius $\rho_{TF}^{0}=\left(15\pi N\lambda/2\right)^{1/6}\sqrt{\hbar/m\omega}$, and a Fermi energy $E_{F}=\left(15\pi N\lambda/16\right)^{1/3}\hbar\omega\equiv k_{B}T_{F}$, where we define $\lambda=L/\rho_{TF}^{0}$ as the aspect ratio of the trap. Throughout this Letter, we calculate results at the Feshbach resonance with $1/a_{s}=0$ and use $\lambda=1$ and $E_{c}\simeq2E_{F}$. We also considered coupling constants in the BCS regime but observed no significant changes. Dimensionality effects will be treated elsewhere.
Numerical accuracy was checked by increasing $E_{c}$ up to $4E_{F}$. Due to the high accuracy of our hybrid cut-off procedure, the results were found to be essentially independent of the cut-off energy. We note finally that, for a symmetric gas at unitarity, universality implies a TF radius of $\rho_{TF}=\left(1+\beta\right)^{3/10}\rho_{TF}^{0}$, a chemical potential $\mu=\left(1+\beta\right)^{3/5}E_{F}$, and a maximum order parameter $\Delta_{0}=8\left(1+\beta\right)^{3/5}E_{F}/e^{2}$ [@ho2004], where BCS theory predicts the universal parameter $\beta\simeq-0.41$.
We present in Figs. 1a, 1b and 1c the density profile of each component, as well as the density difference $\delta n\left(\mathbf{r}\right)=$ $n_{\uparrow}\left(\mathbf{r}\right)-n_{\downarrow}\left(\mathbf{r}\right)$, for several polarizations $p=\left(N_{\uparrow}-N_{\downarrow}\right)/N$ at $T=0.05T_{F}$ and $N=10^{4}$. Because of the uniform distribution along the $z$ axis, these profiles are linked to the experimentally observed column densities in the axial direction. Apart from the apparent phase separation at the edge, the most salient feature of the figures is the development of a polarized normal shell inside the vortex core above a certain polarization. This is clearly visible as a prominent peak in the density difference, of width about $0.05\rho_{TF}$. This is observable in the column integrated density difference, which is directly measurable by phase-contrast imaging [@MIT2006; @Shin].
![(Color online). Left panel: centre density difference as a function of polarization at $N=10^{4}$. Inset shows the dependence on the chemical potential difference. Right panel: critical polarization and critical chemical difference as a function of the number of total particles.[]{data-label="fig2"}](fig2){width="45.00000%"}
The onset of a polarized normal shell at the core center is demonstrated by the central density difference as a function of the polarization. This is shown in Fig. 2a, which represents the most important result of this Letter. At a sufficiently low temperature, *i.e.*, $T=0.01T_{F}$, a sudden rise of the center density difference appears at a critical polarization $p_{c}\simeq0.30$. The critical chemical potential difference is $\delta\mu_{c}\simeq0.36E_{F}\sim\Delta_{0}^{2}/2E_{F}$, with a transition width of around $k_{B}T$. This transition is therefore much smoother at finite temperature. The critical polarization is nearly independent of the overall number of atoms $N$, as shown in Fig. 2b for $N$ up to $4\times10^{4}$. We therefore expect that this will apply to current experiments, where the typical number of atoms is around $10^{5}$, and would survive even in the thermodynamic limit.
The appearance of this intriguing shell structure is closely related to the Andreev-like bound states inside the core. In the BCS regime, these states are formed by the spatial variation of the order parameter around the center (see, *i.e.*, Fig. 1d), analogous to a potential well for quasiparticles, of depth $\Delta_{0}$ and of radius equal to the coherence length $\xi=\hbar v_{F}/\Delta_{0}$. Hence, the confinement of the well gives rise to discrete bound levels with spacing of order $\hbar^{2}/m\xi^{2}=\Delta_{0}^{2}/2E_{F}$ [@gygi]. This qualitative picture persists in the strongly interacting unitarity limit [@ho2006].
![(Color online). Local fermionic density of states of spin up (solid lines) and spin down (dashed lines) components inside the vortex core at $N=10^{4}$ and $T=0.05T_{F}$. The thin line in (a) shows the LDOS at $p=0$. Arrows points to the position at the effective energy of the lowest bound state.[]{data-label="fig3"}](fig3){width="8.5cm"}
To provide an intuitive explanation of our results, we calculate the local density of states (LDOS), $$\begin{aligned}
N_{\uparrow}\left(\mathbf{r},E\right) & = & \sum\nolimits _{\eta}\left|u_{\eta}\left(\mathbf{r}\right)\right|^{2}\delta\left(E-E_{\eta}\right),\nonumber \\
N_{\downarrow}\left(\mathbf{r},E\right) & = & \sum\nolimits _{\eta}\left|v_{\eta}\left(\mathbf{r}\right)\right|^{2}\delta\left(E+E_{\eta}\right).\end{aligned}$$ At low temperature, when integrated over negative energy, this leads to the density profiles $n_{\sigma}\left(\mathbf{r}\right)$. In Fig. 3 we show how the LDOS inside the core evolves with increasing the polarization. A small spectral broadening of about $0.01E_{F}$ has been used to regularize the delta function. Without any polarization the LDOS of the two components coincides, leading to a sharp peak located at positive energy $E_{bs}^{0}\simeq\Delta_{0}^{2}/2E_{F}$, associated with the lowest Andreev-like bound state.
In the presence of spin-polarization the peak in the density of states shifts in different directions for the two components. To a good approximation, the energy separation between the two peaks at the vortex center equals $2\delta\mu$. Thus, in the general case of a nonzero polarization one may define an effective energy of the lowest bound state, $E_{bs}$, as the midpoint of these two peaks located at $E_{bs}\mp\delta\mu$. Therefore, a net density difference results precisely when the peak in $N_{\uparrow}\left(\mathbf{r=0},E\right)$ crosses zero energy *i.e.*, $\delta\mu=E_{bs}$. This results in a bound state for the majority spin component, which explains why a polarized normal shell emerges above a critical population chemical potential $\delta\mu_{c}\sim E_{bs}^{0}\simeq\Delta_{0}^{2}/2E_{F}$.
Thus, the integrated column density difference is an indicator of the lowest vortex bound state, and a measurement of the critical polarization $p_{c}$ gives its energy.
We now consider the dependence of the vortex core size on the polarization. We extract the core size from the superfluid density $n_{s}\left(\mathbf{r}\right)$, defined as a ratio of the current density $\mathbf{j}\left(\mathbf{r}\right)=n_{s}\left(\mathbf{r}\right)\mathbf{v}_{s}$ to the superfluid velocity $\mathbf{v}_{s}=\left(\hbar/2m\rho\right)\mathbf{\hat{\varphi}}$ [@ho2006], where, since our normal fluid solutions are non-rotating: $$\mathbf{j}\left(\mathbf{r}\right)=\frac{i\hbar}{m\rho}\sum_{\eta}\left[u_{\eta}^{*}\partial_{\varphi}u_{\eta}f\left(E_{\eta}\right)+v_{\eta}\partial_{\varphi}v_{\eta}^{*}f\left(-E_{\eta}\right)\right]{\bf \hat{\varphi}}.$$ The resulting superfluid density profiles are plotted in the inset of Fig. 4a. The core size may be quantified as the distance from the vortex core at which the superfluid density is 90% of its maximum value, namely, $\xi_{90}$. From Fig. 4a, the core size increases gradually with increasing polarization, and almost doubles at large polarization.
![(Color online). Vortex core size (a) and the effective energy of the lowest bound state (b) as a function of polarization at $N=10^{4}$ and $T=0.05T_{F}$. The core size $\xi_{90}$ at $p=0$ is about $2.5k_{F}^{-1}$, where $k_{F}$ non-interacting Fermi wavelength at center. Solid lines are the scaling relations as described in the text. Inset shows superfluidity density profiles.[]{data-label="fig4"}](fig4){width="8.5cm"}
To explain this, note that while a phase separation occurs at *any* nonzero polarization, only the unpolarized superfluid part can form a vortex. Thus, the vortex core should expand with a scaling of $\xi_{90}\propto\left(2N_{\downarrow}\right)^{-1/3}\propto\left(1-p\right)^{-1/3}$ [@ho2006]. In Fig 4a this scaling is plotted by a solid line, which fits well with our numerical results. Accordingly, one may suspect that the energy of the lowest bound state will decrease as $E_{bs}\propto1/$ $\xi_{90}^{2}\propto\left(1-p\right)^{2/3}$. This is consistent with the effective energy of the lowest bound state shown in Fig. 4b. We expect a phase separation into *multiple* vortex cores in a vortex lattice, as in current non-polarized experiments [@MIT2005].
We have considered an aspect ratio $\lambda=1$, which is closer to the MIT experimental setup [@MIT2006] than the Rice experiment (which has $\lambda=50$ [@rice]). In the opposite limit of $\lambda\ll1$, an interesting aspect of dimensionality would arise. Due to strong phase fluctuations, this quasi-2D geometry would favor the spontaneous formation of vortices at finite temperature [@KBT]. As a result, a lattice of vortex-anti-vortex pairs *without* phase separation may emerge as the ground state. In such a configuration, the spin polarization would be sustained by a polarized normal shell *inside* the vortex cores, analogous to a type-II superconductor in a magnetic field.
In conclusion, we have analyzed vortex structures in a polarized Fermi gas at unitarity. The lowest bound state will be visible via phase-contrast imaging, together with a quantum phase transition at a critical spin polarization.
This work is supported by the Australian Research Council Center of Excellence and by the National Science Foundation of China Grant No. NSFC-10574080 and the National Fundamental Research Program under Grant No. 2006CB921404. One of us (PDD) acknowledges useful discussions with Yong-Il Shin.
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abstract: 'We describe a pump-probe scheme with which the spatial asymmetry of dissociating molecular fragments — as controlled by the carrier-envelope phase of an intense few-cycle laser pulse — can be enhanced by an order of magnitude or more. We illustrate the scheme using extensive, full-dimensional calculations for dissociation of H$_2^+$ and include the averaging necessary for comparison with experiment.'
author:
- 'Fatima Anis and B. D. Esry'
bibliography:
- 'CEPeffects.New.bib'
title: Enhancing the intense field control of molecular fragmentation
---
In recent years, considerable experimental effort has been invested in developing the ability to control chemical reactions with intense, few-cycle laser pulses [@Paulus:Nature:2001:CEPATI; @Roudnev:PRL:2004:CEP; @Kling:Science:2006:ElectronLocalization]. The canonical reaction chosen has been molecular dissociation, and a common measure of the degree of its coherent control is the spatial asymmetry of fragments relative to a linearly polarized laser field. Quantum mechanically, this asymmetry arises from the interference of pathways that lead to even and odd parity states [@PhysRevLett.92.033002; @Roudnev:PRL:2007:CEP; @Hua:JPB:2009]. In strong fields, these pathways can involve many photons, and the relative phase between pathways — and thus the outcome — can be controlled by varying laser parameters such as the carrier envelope phase (CEP) [@Paulus:Nature:2001:CEPATI; @Paulus:PRL.89.153001; @Roudnev:PRL:2004:CEP; @Kling:Science:2006:ElectronLocalization; @Roudnev:PRL:2007:CEP; @Hua:JPB:2009; @Kling.MolPhys.2008; @kremer:213003; @nakajima:213001; @Anis:AtCEP:FTC:2009; @Baltuska:Nature:2003:HHG; @PhysRevLett.105.223001; @PhysRevLett.103.103002] or the relative phase between different colors [@Charron.R641; @Charron.PRL.71.692; @Charron.PRL.75.2815; @he:083002; @he:213002; @RayTwocolorPRL; @SinghPRL.104.023001; @Calvert.JPB.43.011001.2010]. In this Letter, we will focus on control via the CEP.
In the dipole approximation, the CEP $\varphi$ for a Gaussian laser pulse ${\cal E}(t)$ is defined from [@Param] $$\begin{aligned}
{\cal E}(t) = {\cal E}_{0}e^{-t^2/\tau^2}\cos(\omega t+\varphi).
\label{pulse}\end{aligned}$$ Generally, the largest CEP-dependent asymmetries have been observed for ionized electrons [@Paulus:Nature:2001:CEPATI]. The asymmetries for the nuclear fragments resulting from dissociation have, unfortunately, been much smaller [@Kling:Science:2006:ElectronLocalization; @Kling.MolPhys.2008; @kremer:213003]. These weak effects — combined with the ongoing challenge of producing intense, few-cycle, CEP stabilized pulses — greatly limit experimentalists’ abilities to measure and explore this intriguing means of control. One important recent advance is the ability to measure the CEP of each pulse [@Nat.Phys.5.357.2009; @PhysRevA.83.013412], alleviating the need for CEP stability during the measurements.
While CEP-dependent asymmetric break up of H$_2^+$ was predicted a few years ago [@Roudnev:PRL:2004:CEP], successful measurements have not yet been made starting directly from this benchmark system, [*e.g.*]{} in an ion beam experiment [@BenItzik:PRL:2005:H2]. Experiments have instead begun with the more complicated H$_2$ [@Kling:Science:2006:ElectronLocalization; @kremer:213003]. With only one electron, the number of control pathways for H$_2^+$ is smaller than for H$_2$ making the interpretation more straightforward. Moreover, the theory at sub-ionization intensities can be done essentially exactly.
The technical challenges of an ion beam experiment [@wang:043411] are obvious reasons that the H$_2^+$ experiments have not yet been done. A more fundamental problem, however, lies in the fact that H$_2^+$ typically comes in a broad rovibrational distribution in such experiments [@wang:043411]. Unfortunately, dissociation of H$_2^+$ from different initial $v$ by a linearly polarized laser pulse gives fragments with similar energies. Since the asymmetry produced by each $v$ is slightly different, the incoherent averaging over initial $v$ required for an H$_2^+$ beam tends to wash out the overall asymmetry [@Roudnev:PRA:2007:HDp:CEP]. Moreover, one-photon dissociation of higher $v$ dominates the total signal. And, since one-photon dissociation produces a single nuclear parity, its momentum distribution is symmetric, masking the desired asymmetry.
In this Letter, we present a scheme to greatly enhance CEP effects. This enhancement is largely achieved by depleting the undesired higher-$v$ states with a long, weak pump pulse. Subsequent dissociation of this prepared state by a few-cycle probe pulse gives a momentum distribution with an order of magnitude enhanced asymmetry compared to that of an initial incoherent Franck-Condon distribution. In fact, our scheme gives larger asymmetries — at longer pulse lengths — than have been observed so far in H$_2$ experiments [@Kling:Science:2006:ElectronLocalization; @kremer:213003; @PhysRevLett.105.223001; @Kling.MolPhys.2008]. We also propose ways to separate the pump and probe signals. To support our claims, we present theoretical CEP-dependent $p$+H momentum distributions in addition to the up-down asymmetry. Calculating such a differential observable — along with averaging over the intensity distribution of the laser focus — permits us to quantitatively predict the experimental outcome and to provide deeper physical insight.
To obtain the momentum distribution, it is necessary to account for the nuclear rotation. We thus solve the time-dependent Schrödinger equation in the Born-Oppenheimer representation, including nuclear rotation, nuclear vibration, and electronic excitation, but neglecting ionization as well as the Coriolis and all nonadiabatic couplings. The nuclear rotation is included as an expansion of the wave function over the total orbital angular momentum ($J$) basis (see [@anis:033416] for details).
To prepare the system, we use a 785 nm, 45 fs long pump pulse with an intensity of $10^{13}$ W/cm$^2$. This relatively long, weak pump pulse depletes the higher $v$ states, eliminating their spatially symmetric dissociation signal.
![Vibrational state distribution before and after the 45 fs, $10^{13}$ W/cm$^2$ pump pulse.[]{data-label="DPplot"}](DP.jpg){width="2.5in"}
Figure \[DPplot\] shows the $v$-distribution before and after this pump pulse. In an incoherent Franck-Condon $v$-distribution, 9.58% of the population lies in $v\geq8$. Since 90% of this population dissociates in the pump pulse, $v\geq8$ becomes only 1.36% of the total remaining bound population, ensuring that their contribution to the dissociation signal by any subsequent probe pulse will be negligible. Consequently, we performed probe pulse calculations only including $v=$0–7, after verifying for a representative case that $v\geq8$ affected the asymmetry by much less than 1%.
To quantify the enhancement, we compare the results from an initial incoherent Franck-Condon distribution interacting with [*only*]{} the probe pulse (“probe-only”) to the signal from the probe part of our proposed pump-probe scheme (“pump-probe”). We used a 7 fs, 785 nm probe pulse in both cases. In the pump-probe scheme, all calculations were performed at a fixed time delay of 267 fs unless stated otherwise. Since ionization is neglected, we limit the peak intensity to no more than $1.2\times10^{14}$ W/cm$^2$. For the peak intensities above $10^{13}$ W/cm$^2$ required for intensity averaging, our calculations included $p$+H($2l$) manifold in addition to $1s\sigma_{g}$ and $2p\sigma_{u}$ channels. The total population of the $p$+H($2l$) states was less than 5% even for the highest intensity. Consequently, we present momentum distributions based on just the $1s\sigma_{g}$ and $2p\sigma_{u}$ channels.
The fundamental physical observable we focus on is the $p$+H relative momentum distribution $\rho(\bf K)$, which is the most differential observable in recent experiments involving H$_2^+$ dissociation [@McKenna:PRL:2008:H2:10fs:ATD; @wang:043411; @BenItzik:PRL:2005:H2]. To calculate $\rho(\bf K)$, we project the final wave function onto scattering states that behave as $\exp(i{\bf K}\cdot{\bf R}) \phi_{1sA}$ asymptotically, where ${\bf R}$ points from proton $A$ to proton $B$ and $\phi_{1sA}$ is the hydrogen ground state wave function centered on proton $A$. The momentum $\bf K$ thus points from H to $p$. This scattering state, with the nuclear spin included, is then symmetrized to account for the identical nuclei [@Green.PRL.21.1732; @singer:6060; @AnisThesis]. Finally, the momentum distribution \[or its energy-normalized equivalent $\rho(E,\hat{K})$ with $E=K^{2}/2\mu$, $\mu$ the nuclear reduced mass, and $\hat{K}=(\theta_{K},\varphi_{K})$ the direction of $\bf K$ with respect to the polarization direction\] is $$\begin{aligned}
\rho({\bf K})
=&\frac{1}{\mu\sqrt{2\mu E}}\rho(E,\hat{K})
\label{EqMD}
\\
=&\frac{1}{\mu\sqrt{2\mu E}}\biggl|\sum_{J\,{\rm even}}C_{Jg}Y_{JM}(\hat{K})
\!+\!\!\!
\sum_{J\,{\rm odd}}C_{Ju}Y_{JM}(\hat{K})\biggr|^{2}
\nonumber\end{aligned}$$ with $$\begin{aligned}
C_{Jp}= (-i)^{J}e^{-i\delta_{Jp}}\langle EJp|F_{Jp}(t_{f})\rangle,\qquad p=g,u.
\label{EqCoef}\end{aligned}$$ Here, $|F_{Jp}(t_{f})\rangle$ are the $1s\sigma_{g}$ and $2p\sigma_{u}$ nuclear radial wave functions at the final time $t_{f}$, while $|EJp\rangle$ and $\delta_{Jp}$ are the corresponding energy-normalized scattering states and phase shifts, respectively.
Equation (\[EqMD\]) shows that although a linear combination of $1s\sigma_{g}$ and $2p\sigma_{u}$ is necessary to localize the electron as an atomic rather than a molecular state, the spatial asymmetry of $p$+H is due to the interference of even and odd nuclear parity states. This distinction is brought into sharp relief when nuclear rotation is included in the calculation since using simply $1s\sigma_{g}$$\pm$$2p\sigma_{u}$ nuclear wave function— as is done in calculations without rotation — would produce two distinct $p$+H momentum distributions, where clearly only one can be measured. It is the symmetrization requirement that dictates the proper coherent combination to use. This issue is not new, however, and always arises for identical particle scattering where it is known that the primary differences occur for $\theta_K$$\approx$$\pi/2$. Since intense-field dissociation of H$_2^+$ produces very few fragments at this $\theta_K$, the consequences of analyzing incorrectly are less pronounced. For more complicated systems, however, this need no longer be true.
The Franck-Condon-averaged momentum distributions for several CEPs are shown in Figs. \[MomD\](a)–(d) for the probe-only case and in Figs. \[MomD\](e)–(h) for the pump-probe case.
![Franck-Condon-averaged $K^{2}\rho({\bf K})$ (integrated over $\varphi_{K}$) for probe-only \[$\rho(\bf K)$ is reflected to $-K_{\perp}$ for clarity\] for (a) $\varphi =0$, (b) $\varphi=\pi/4$, (c) $\varphi=\pi/2$, and (d) $\varphi=3\pi/4$ ( Gray dotted lines mark $K$ = 7.5, 10, 12.5, and 15 a.u.). (e)–(h) are same as (a)–(d) but for pump-probe. All cases used a 7 fs, $10^{14}$ W/cm$^2$ probe pulse.[]{data-label="MomD"}](phi0p0Sn.jpg "fig:"){height="30mm"} ![Franck-Condon-averaged $K^{2}\rho({\bf K})$ (integrated over $\varphi_{K}$) for probe-only \[$\rho(\bf K)$ is reflected to $-K_{\perp}$ for clarity\] for (a) $\varphi =0$, (b) $\varphi=\pi/4$, (c) $\varphi=\pi/2$, and (d) $\varphi=3\pi/4$ ( Gray dotted lines mark $K$ = 7.5, 10, 12.5, and 15 a.u.). (e)–(h) are same as (a)–(d) but for pump-probe. All cases used a 7 fs, $10^{14}$ W/cm$^2$ probe pulse.[]{data-label="MomD"}](phi0p25Sn.jpg "fig:"){height="30mm"} ![Franck-Condon-averaged $K^{2}\rho({\bf K})$ (integrated over $\varphi_{K}$) for probe-only \[$\rho(\bf K)$ is reflected to $-K_{\perp}$ for clarity\] for (a) $\varphi =0$, (b) $\varphi=\pi/4$, (c) $\varphi=\pi/2$, and (d) $\varphi=3\pi/4$ ( Gray dotted lines mark $K$ = 7.5, 10, 12.5, and 15 a.u.). (e)–(h) are same as (a)–(d) but for pump-probe. All cases used a 7 fs, $10^{14}$ W/cm$^2$ probe pulse.[]{data-label="MomD"}](phi0p5Sn.jpg "fig:"){height="30mm"} ![Franck-Condon-averaged $K^{2}\rho({\bf K})$ (integrated over $\varphi_{K}$) for probe-only \[$\rho(\bf K)$ is reflected to $-K_{\perp}$ for clarity\] for (a) $\varphi =0$, (b) $\varphi=\pi/4$, (c) $\varphi=\pi/2$, and (d) $\varphi=3\pi/4$ ( Gray dotted lines mark $K$ = 7.5, 10, 12.5, and 15 a.u.). (e)–(h) are same as (a)–(d) but for pump-probe. All cases used a 7 fs, $10^{14}$ W/cm$^2$ probe pulse.[]{data-label="MomD"}](phi0p75Sn.jpg "fig:"){height="30mm"}\
![Franck-Condon-averaged $K^{2}\rho({\bf K})$ (integrated over $\varphi_{K}$) for probe-only \[$\rho(\bf K)$ is reflected to $-K_{\perp}$ for clarity\] for (a) $\varphi =0$, (b) $\varphi=\pi/4$, (c) $\varphi=\pi/2$, and (d) $\varphi=3\pi/4$ ( Gray dotted lines mark $K$ = 7.5, 10, 12.5, and 15 a.u.). (e)–(h) are same as (a)–(d) but for pump-probe. All cases used a 7 fs, $10^{14}$ W/cm$^2$ probe pulse.[]{data-label="MomD"}](phi0p0n.jpg "fig:"){height="36.508mm"} ![Franck-Condon-averaged $K^{2}\rho({\bf K})$ (integrated over $\varphi_{K}$) for probe-only \[$\rho(\bf K)$ is reflected to $-K_{\perp}$ for clarity\] for (a) $\varphi =0$, (b) $\varphi=\pi/4$, (c) $\varphi=\pi/2$, and (d) $\varphi=3\pi/4$ ( Gray dotted lines mark $K$ = 7.5, 10, 12.5, and 15 a.u.). (e)–(h) are same as (a)–(d) but for pump-probe. All cases used a 7 fs, $10^{14}$ W/cm$^2$ probe pulse.[]{data-label="MomD"}](phi0p25n.jpg "fig:"){height="36.508mm"} ![Franck-Condon-averaged $K^{2}\rho({\bf K})$ (integrated over $\varphi_{K}$) for probe-only \[$\rho(\bf K)$ is reflected to $-K_{\perp}$ for clarity\] for (a) $\varphi =0$, (b) $\varphi=\pi/4$, (c) $\varphi=\pi/2$, and (d) $\varphi=3\pi/4$ ( Gray dotted lines mark $K$ = 7.5, 10, 12.5, and 15 a.u.). (e)–(h) are same as (a)–(d) but for pump-probe. All cases used a 7 fs, $10^{14}$ W/cm$^2$ probe pulse.[]{data-label="MomD"}](phi0p5n.jpg "fig:"){height="36.508mm"} ![Franck-Condon-averaged $K^{2}\rho({\bf K})$ (integrated over $\varphi_{K}$) for probe-only \[$\rho(\bf K)$ is reflected to $-K_{\perp}$ for clarity\] for (a) $\varphi =0$, (b) $\varphi=\pi/4$, (c) $\varphi=\pi/2$, and (d) $\varphi=3\pi/4$ ( Gray dotted lines mark $K$ = 7.5, 10, 12.5, and 15 a.u.). (e)–(h) are same as (a)–(d) but for pump-probe. All cases used a 7 fs, $10^{14}$ W/cm$^2$ probe pulse.[]{data-label="MomD"}](phi0p75n.jpg "fig:"){height="36.508mm"}
The momentum distributions in all cases exhibit preferential alignment along the laser polarization. Moreover, since the energy distributions $$\begin{aligned}
\rho(E)=\int\rho(E,\hat{K})d\Omega_{K}
\label{EqKER}\end{aligned}$$ for the $1s\sigma_{g}$ and $2p\sigma_{u}$ channels of individual vibrational states overlap roughly in the range 0.5–1.5 eV, we expect spatial asymmetries to appear roughly for 6$\leq$$K$$\leq$10 a.u. The results shown in Fig. \[MomD\] for both experimental scenarios are consistent with this expectation. The momentum distribution for $\varphi=\varphi+\pi$ is the mirror image of the momentum distribution for $\varphi$, as guaranteed by the fact that $\cos(\omega t+\pi)=-\cos\omega t$ in Eq. (\[pulse\]).
While the two experimental scenarios clearly show qualitative differences, the strikingly different distributions make it difficult to judge which produces the larger asymmetry. We thus turn to the quantitative measure of the asymmetry used in previous studies [@Kling:Science:2006:ElectronLocalization; @Hua:JPB:2009; @Kling.MolPhys.2008]: the normalized asymmetry parameter ${\cal A}(E,\varphi)$, $$\begin{aligned}
{\cal A}(E,\varphi) = \rho(E)^{-1}\left[\rho(E)_{\rm Up}
-\rho(E)_{\rm Down}\right].
\label{AsymEq}\end{aligned}$$ For simplicity, we integrate over the whole upper and lower hemispheres in the “Up” and “Down” distributions, respectively, although a narrow angular cut along the laser polarization direction might be chosen to enhance $\cal A$ as in some experimental studies [@Kling:Science:2006:ElectronLocalization; @Kling.MolPhys.2008]. Figure \[MomD\] shows why such cuts are effective since the strongest CEP dependence lies at small $\theta_{K}$. Although the total energy spectrum $\rho(E)$ in principle also depends on CEP [@Hua:JPB:2009], we found negligible CEP-dependence in the Franck-Condon averaged $\rho(E)$ and thus expect essentially no contribution to the CEP dependence from the denominator of $\cal A$.
Figures \[Asymm\](a) and \[Asymm\](b) show ${\cal A}(E,\varphi)$ for the proble-only and the pump-probe, respectively. For the probe-only in Fig. \[Asymm\](a), we can already see reasonable asymmetry in the range 0.2–2.5 eV where it oscillates between $-0.12$ and $0.12$. Comparing Fig. \[Asymm\](a) and Fig. \[Asymm\](b), however, we find a five-fold enhancement of $|{\cal A}(E,\varphi)|$ in the pump-probe case for this intensity.
![(a) Asymmetry defined in Eq. (\[AsymEq\]) for the probe-only case and (b) for the pump-probe case for a $\tau_{\rm FWHM} = 7$ fs and $I=10^{14}$ W/cm$^2$ pulse.[]{data-label="Asymm"}](J02nd.jpg "fig:"){height="1.in"} ![(a) Asymmetry defined in Eq. (\[AsymEq\]) for the probe-only case and (b) for the pump-probe case for a $\tau_{\rm FWHM} = 7$ fs and $I=10^{14}$ W/cm$^2$ pulse.[]{data-label="Asymm"}](Aligned.jpg "fig:"){height="1.in"}
The most crucial factor determining whether an intensity-dependent effect is experimentally observable is whether it survives intensity (or focal volume) averaging. We thus intensity-averaged our results for a 7 fs laser pulse with $\varphi=0$ and a peak intensity of $1.2\times10^{14}$ W/cm$^2$. We used the two-dimensional geometry of Ref. [@McKenna:PRL:2008:H2:10fs:ATD] to perform the intensity-averaging following the procedure described in [@Roudnev:PRA:2007:HDp:CEP]. Figures \[MomD\](a), \[MomD\](e) and \[Asymm\], show a clear up-down asymmetry for $\varphi =0$ in both probe-only and pump-probe cases, and we will check if it survives intensity averaging.
![(a) Asymmetry from the intensity-averaged $\rho(E,\hat{K})$ for the probe-only (thin dashed lines) and the pump-probe (thick dashed lines) cases as well as for a single 7 fs probe pulse with a peak intensity of $1.2\times10^{14}$ W/cm$^2$ (thin and thick solid lines, respectively). (b) KER distributions for the cases shown in (a) normalized to the same peak value.[]{data-label="IntAvg"}](AsymIntAvg.jpg){width="3.3in"}
Figure \[IntAvg\](a) shows ${\cal A}(E,0)$ before and after the intensity averaging for both cases. The corresponding total KER distributions are plotted in Fig. \[IntAvg\](b) to show that the small $\rho(E)$ is the reason for the large ${\cal A}(E,0)$ at higher energies. In the pump-probe case, the intensity averaging has only been performed over the probe-pulse intensity distribution assuming that the weak pump intensity can be made uniform across the probe focal volume.
For the probe-only case, intensity averaging reduces $\cal A$ by more than a factor of three over the entire energy range shown in Fig. \[IntAvg\](a) and makes it 17 times smaller for 0.5 to 1.0 eV, where $\rho(E)$ is large and the single-intensity $\cal A$ is largest. This significant reduction in $\cal A$ is due to the fact that one-photon dissociation — which shows no asymmetry — can occur for $v\geq7$ at very low intensities ($\approx$10$^{10}$ W/cm$^2$). These symmetric contributions are thus amplified by the intensity averaging and swamp any asymmetry because essentially all $v$ contribute to these KER. Figure \[IntAvg\](a) thus shows that intensity averaging makes it very challenging to measure CEP-effects for a single 7 fs or longer pulse in an experiment.
For the pump-probe case, $\cal A$ is also reduced from the single intensity value — but to a much lesser extent than in the probe-only case. In fact, Fig. \[IntAvg\](a) shows that even after intensity averaging $\cal A$ is an order of magnitude larger using the pump-probe scheme compared to the probe-only results. Moreover, we have found that the pump-probe scheme produces a CEP-dependent asymmetry after intensity averaging even for 10 fs pulses. These pulses are much longer than the 6 fs pulses that have been used to date to observe CEP effects [@Kling:Science:2006:ElectronLocalization; @kremer:213003].
Besides depleting the high-lying vibrational states, the pump also impulsively aligns the molecule [@PhysRevA.2011; @PhysRevA.77.053407; @RevModPhys.75.543]). To investigate the sensitivity of $\cal A$ to the alignment, we calculated the asymmetry for three different pump-probe time delays with aligned $\langle\cos^{2}\theta\rangle$=0.56, anti-aligned $\langle\cos^{2}\theta\rangle$=0.22, and dephased $\langle\cos^{2}\theta\rangle$=0.40 angular distributions ($\langle\cos^{2}\theta\rangle$=1/3 for an isotropic distribution). We found that the maximum $\cal A$ was largest for the aligned distribution, followed by the anti-aligned, with the dephased smallest. The enhancement of the aligned $\cal A$ over the dephased was roughly 30%. For this reason we have shown here calculations for the 267 fs delay corresponding to the aligned distribution. This exercise also served to establish that the major source of the ten-fold CEP-dependent asymmetry enhancement is the depletion of the higher-$v$ states.
Another concern for experimentally observing the predicted enhancement is the fact that in our pump-probe scheme the pump pulse already produces fragments. So, separating the probe signal from the pump signal is crucial. Although the dissociating fragments from both pulses overlap in momentum, we expect the asymmetry will still be large in a combined pump-probe signal for two reasons. First, the momentum distribution from the long pump pulse exhibits narrow peaks corresponding to higher vibrational states. Therefore, in the combined pump-probe momentum distribution, the symmetric structure would be very localized in KER, giving small overlap with the broad asymmetric signal resulting in larger asymmetry than the probe-only case. Second, we found that preparing the initial state greatly increased the dissociation probability of the lower vibrational states for aligned (2.33 fold) and dephased (1.74 fold) pump-probe cases compared to the probe-only, thereby enhancing the ratio of the asymmetric signal to the symmetric signal.
The contrast between pump and probe signals can be further improved over the present case using pump pulses longer than 45 fs, thus increasing the depletion of the higher vibrational states and making the pump signal even more structured. A longer pulse will give more alignment, which might also enhance asymmetry. Additionally, instead of using the whole upper and lower hemispheres to define $\cal A$, an angular cut can be used to isolate the aligned asymmetric distribution.
As the dissociating fragments primarily lie along the laser polarization, it might be better to use orthogonal laser polarization directions for the pump and probe pulses to separate their signals [@PhysRevLett.105.223001]. For this, one might want to use the time-delay when the molecules are antialigned relative to the pump polarization to improve the signal. A circularly polarized pump pulse could also be used. Since depletion is the major reason for enhanced CEP effects, we believe the effect will survive using different laser polarizations. An intensity differencing scheme might also be useful to enhance asymmetry [@Wang:OL:2005:IDS].
In this Letter, we have presented a prescription for substantially enhancing CEP effects on the spatial asymmetry of intense-field-induced fragmentation. We have illustrated our proposal with essentially exact calculations for the benchmark system H$_2^+$, including the important averaging implicit in experiments and found a ten-fold increase in the asymmetry. In addition, we have suggested several other steps that could further increase the asymmetry.
While even greater enhancement could be realized by preparing the system in a single initial $v$, our scheme provides a more easily followed experimental avenue yielding a narrow vibrational state distribution. We believe that our scheme is equally applicable to neutral molecules with studies focussed on dissociative ionization channels, where the initial intense long pump pulses can serve to ionize, dissociate undesired vibrational states, and align the molecular ions.
We gratefully acknowledge many useful discussions with I. Ben-Itzhak and J. McKenna regarding experimental limitations. The work was supported by the Chemical Sciences, Geo-Sciences, and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy.
|
---
abstract: 'We explore the possibility that some of the five narrow $\Omega_c$ resonances recently observed at LHCb could correspond to pentaquark states, structured as meson-baryon bound states or molecules. The interaction of the low-lying pseudoscalar mesons with the ground-state baryons in the charm $+1$, strangeness $-2$ and isospin 0 sector is built from t-channel vector meson exchange, using effective Lagrangians. The resulting s-wave coupled-channel unitarized amplitudes show the presence of two structures with similar masses and widths to those of the observed $\Omega_c(3000)^0$ and $\Omega_c(3090)^0$. The identification of these resonances with the meson-baryon bound states found in this work would also imply assigning the values $1/2^-$ for their spin-parity. An experimental determination of the spin-parity of the $\Omega_c(3090)^0$ would help in disentangling its structure, as the quark-based models predict its spin-parity to be either $3/2^-$ or $5/2^-$.'
author:
- Glòria Montaña
- Albert Feijoo
- Àngels Ramos
title: ' A meson-baryon molecular interpretation for some $\Omega_c$ excited baryons.'
---
Introduction
============
The recent observation by the LHCb collaboration of five narrow $\Omega_c^0$ excited resonances decaying into $\Xi_c^+ K^-$ states [@Aaij:2017nav] has triggered a lot of activity in the field of baryon spectroscopy aiming at understanding their inner structure and possibly establishing their unknown values of spin-parity [@Karliner:2017kfm; @Wang:2017vnc; @Wang:2017zjw; @Chen:2017gnu; @Padmanath:2017lng; @Chen:2017sci; @Agaev:2017jyt; @Agaev:2017lip; @Cheng:2017ove; @Wang:2017hej; @Huang:2017dwn; @Yang:2017rpg; @An:2017lwg; @Kim:2017jpx].
In conventional quark models, baryons are composed by three quarks but, in spite of the rather successful description of a wealth of data [@Capstick:1986bm], other more exotic components cannot be ruled out. Within a $css$ quark content picture the presumed spin-parity of the $1/2^+$ and $3/2^+$ $\Omega_c^0$ ground states [@pdg] can be explained naturally, and predictions for the low lying excited states have also been given [@Maltman:1980er; @Migura:2006ep; @Roberts:2007ni; @Valcarce:2008dr; @Ebert:2011kk; @Vijande:2013yxa; @Yoshida:2015tia]. The recent discovery of excited $\Omega^0$ states decaying into $K^-\Xi_c^+$ pairs at LHCb [@Aaij:2017nav] has provided new information against which revisited quark models can be tested. Actually, several interpretations have been proposed which, in general, benefit from the symmetries associated to the presence of a charm quark having a much larger mass than that of their strange companions. Some works interpret all the observed states as P-wave orbital excitations of the $ss$ diquark with respect to the charmed quark [@Karliner:2017kfm; @Wang:2017vnc; @Wang:2017zjw; @Chen:2017gnu], a result which finds support from a recent Lattice QCD simulation reporting the energy spectra of $\Omega_c^0$ baryons with spin up to 7/2 for both positive and negative parity [@Padmanath:2017lng]. In other works, some of the states would be associated to 1P orbital excitations [@Chen:2017sci] and others to radial 2S orbital ones[@Agaev:2017jyt; @Agaev:2017lip; @Cheng:2017ove; @Wang:2017hej]. A pentaquark structure has also been advocated for the excited $\Omega_c^0$ baryons, either from a model that includes the exchange of Goldstone mesons in the interaction between the constituent quarks [@Huang:2017dwn; @Yang:2017rpg; @An:2017lwg] or by employing the quark-soliton model [@Kim:2017jpx].
An alternative scenario is provided by models that can interpret some resonances as being composed by five quarks, structured in the form of a quasi-bound state of an interacting meson-baryon pair. A paradigmatic example is that of the $\Lambda(1405)$ resonance, the mass of which is systematically overpredicted by quark models. Instead, dedicated studies of the meson-baryon interaction in the $I=0$ $S=-1$ sector, employing chiral effective lagrangians and implementing unitarization, predict the $\Lambda(1405)$ as being the superposition of two-poles of the meson-baryon scattering matrix [@pdg; @Oller:2000fj; @Jido:2003cb; @Hyodo:2011ur]. This two-pole structure received support from the simultaneous analysis in [@Magas:2005vu] of different line shapes [@Thomas:1973uh; @Prakhov:2004an] and, more recently, from the analysis of the $\pi \Sigma$ photoproduction data [@Niiyama:2008rt; @Moriya:2013hwg] performed in [@Roca:2013av; @Mai:2014xna]. The existence of pentaquark baryons have been made clearly evident from the recent discovery at LHCb [@Aaij:2015tga] of the excited nucleon resonances $P_c(4380)$ and $P_c(4450)$, seen in the invariant mass distribution of $J/\psi \, p$ pairs from the decay of the $\Lambda_b$. The high mass of these excited nucleons inevitably demands the presence of an additional $c\bar{c}$ pair. Hidden charm baryons having a meson-baryon structure had already been predicted previously [@Wu:2010jy; @Wu:2010vk; @Yang:2011wz; @Xiao:2013yca; @Karliner:2015ina], and later studies confirmed that the narrow pentaquark seen from the $\Lambda_b \to J/\Psi~ K^- p$ decay at CERN could receive a molecular interpretation [@Chen:2015loa; @Roca:2015dva; @He:2015cea; @Meissner:2015mza]. Reactions to find the hidden charm strangeness $S=-1$ partner of the pentaquark have also been proposed recently [@Chen:2015sxa; @Feijoo:2015kts].
The aim of this work is to explore whether some of the observed excited $\Omega_c^0$ resonances admit an interpretation as meson-baryon molecules. Similarly to the $P_c$ pentaquarks, which find more natural having a $c\bar{c}$ pair in its composition rather than being an extremely high energy excitation of the $3q$ system, it is also plausible to expect that some excitations in the $C=1$, $S=-2$ sector can be obtained by adding a $u\bar{u}$ pair to the natural $ssc$ content of the $\Omega_c^0$ [@Yang:2017rpg]. The hadronization of the five quarks could then lead to bound states, generated by the meson baryon interaction in coupled channels. To support this possibility, let us point out that the $\bar K\Xi_c$ and $\bar K\Xi_c^\prime$ thresholds are in the energy range of interest, and that the excited $\Omega_c^0$ baryons under study have been observed from the invariant mass of spectrum of $K^-\Xi^+_c$ pairs.
After the successful description of the $\Lambda(1405)$ as a $\bar{K}N$ quasibound molecular state, many groups devoted efforts to find signs of compositeness in other spin, isospin and flavour sectors, and several well known states and spectral shapes of various reactions have found a more natural explanation in terms of resonances being generated by the interaction of mesons and baryons in coupled channels, see [@Guo:2017jvc] and references therein. In the particular open charm sector with strangeness $S=-2$ approached in the present work, the authors of Ref. [@Hofmann:2005sw] find a rich spectrum of molecules, employing a zero-range exchange of vector mesons as the driving force for the s-wave scattering of pseudo-scalar mesons off the baryon ground states. Similar qualitative findings where obtained in the work of Ref. [@JimenezTejero:2009vq], where finite range effects were explored. Heavy-quark spin symmetry (HQSS) is explicitly considered in the model of Ref. [@Romanets:2012hm], thus treating the pseudoscalar and vector mesons, as well as the ground state $1/2^+$ and $3/2^+$ baryons, on the same footing. In spite of their qualitative differences, the three works predict the existence of $\Omega_c^0$ resonances as poles of the coupled-channel meson baryon scattering amplitude in the complex plane. However, most of the predicted quasibound states are below 3 GeV, too low to explain the observed states. Only the model of Ref. [@JimenezTejero:2009vq] predicts a state at 3117 MeV, but its width turns out to be one order of magnitude larger than that of the closer observed state. In the present work we revisit the original model of Ref. [@Hofmann:2005sw] and find that, after taking an appropriate regularization scheme with physically sound parameters, two $\Omega_c^0$ resonances are generated in the region of interest. Our model is able to reproduce the mass of two of the excited $\Omega_c^0$ baryons seen at CERN, at 3050 MeV and 3090 MeV, as well as their widths. The important observation is that, if these molecular states are identified with the observed ones, their spin-parity would be assigned to be $1/2^-$.
Formalism {#sec:formalism}
=========
The diagrams contributing to the meson-baryon interaction at tree level are depicted in Fig. \[fig:feynmandiagram\_ps\]. For the s-wave amplitude explored here, we will only consider the most important contribution which corresponds to the t-channel vector meson exchange term (Fig. \[fig:feynmandiagram\_ps\](a)). The vertices in this diagram, coupling the vector meson to pseudoscalar mesons ($VPP$) and baryons ($VBB$), are described from effective Lagrangians, that are obtained using the hidden gauge formalism and assuming $SU(4)$ symmetry [@Hofmann:2005sw]: $$\label{eq:vertxVPP}
\mathcal{L}_{VPP}=ig\langle\left[\partial_\mu\phi, \phi\right] V^\mu\rangle,$$ $$\label{eq:vertexBBV}
\mathcal{L}_{VBB}=\frac{g}{2}\sum_{i,j,k,l=1}^4\bar{B}_{ijk}\gamma^\mu\left(V_{\mu,l}^{k}B^{ijl}+2V_{\mu,l}^{j}B^{ilk}\right),$$ where the symbol $\langle~\rangle$ denotes the trace of $SU(4)$ matrices in flavour space, and the factor $g$ is the universal coupling constant, related to the pion decay constant $f=93\rm~MeV$ by: $$\label{eq:g_coup}
g=\frac{m_V}{2f},$$ with $m_V$ being a representative mass of the light (uncharmed) vector mesons from the nonet. This value of $g$ is in accordance with the KSFR[^1] relation [@Kawarabayashi:1966kd; @Riazuddin:1966sw].
![Leading order tree level diagrams contributing to the $PB$ interaction. Baryons and pseudoscalar mesons are depicted by solid and dashed lines, respectively.[]{data-label="fig:feynmandiagram_ps"}](fig1.pdf){width="45.00000%"}
The symbols $\phi$ and $V_\mu$ represent the pseudoscalar fields of the 16-plet of the $\pi$ and the vector fields of the 16-plet of the $\rho$, given by $$\label{eq:matrixphi}
{\scriptsize\phi \! = \!
\begin{pmatrix}
\frac{1}{\sqrt{2}}\pi^0+\frac{1}{\sqrt{6}}\eta+\frac{1}{\sqrt{3}}\eta^\prime & \pi^+ & K^{+} &\ \bar{D}^{0} \\
\pi^- & \!\!\!\!\!\!\! -\frac{1}{\sqrt{2}}\pi^0+\frac{1}{\sqrt{6}}\eta+\frac{1}{\sqrt{3}}\eta^\prime & K^{0} & D^{-} \\
K^{-} & \bar{K}^{0} & \!\!\!\!\!\!\!-\sqrt{\frac{2}{3}}\eta+\frac{1}{\sqrt{3}}\eta^\prime & D_s^{-} \\
D^{0} & D^{+} & D_s^{+} & \eta_c \\
\end{pmatrix}},$$ and $$\label{eq:matrixVmu}
{\scriptsize V_\mu =
\begin{pmatrix}
\frac{1}{\sqrt{2}}(\rho^0+\omega) & \rho^+ & K^{\ast +} & \bar{D}^{\ast 0} \\
\rho^- & \frac{1}{\sqrt{2}}(-\rho^0+\omega) & K^{\ast 0} & D^{\ast -} \\
K^{\ast -} & \bar{K}^{\ast 0} & \phi & D_s^{\ast -} \\
D^{\ast 0} & D^{\ast +} & D_s^{\ast +} & J/\psi \\
\end{pmatrix}_\mu,}$$ respectively, and $B$ is tensor of baryons belonging to the 20-plet of the $p$: $${\scriptsize\begin{tabular}{ll}
$B^{121}=p$, & ~~~~~$B^{122}=n$,
$\vphantom{\sqrt{\frac{2}{3}}}$ \\
$B^{132}=\frac{1}{\sqrt{2}}\Sigma^0-\frac{1}{\sqrt{6}}\Lambda$, & ~~~~~$B^{213}=\sqrt{\frac{2}{3}}\Lambda$,
$\vphantom{\sqrt{\frac{2}{3}}}$\\
$B^{231}=\frac{1}{\sqrt{2}}\Sigma^0+\frac{1}{\sqrt{6}}\Lambda$, & ~~~~~$B^{232}=\Sigma^-$,
$\vphantom{\sqrt{\frac{2}{3}}}$\\
$B^{233}=\Xi^-$, & ~~~~~$B^{311}=\Sigma^+$,
$\vphantom{\sqrt{\frac{2}{3}}}$\\
$B^{313}=\Xi^0$, & ~~~~~$B^{141}=-\Sigma_c^{++}$,
$\vphantom{\sqrt{\frac{2}{3}}}$\\
$B^{142}=\frac{1}{\sqrt{2}}\Sigma_c^++\frac{1}{\sqrt{6}}\Lambda_c$, & ~~~~~$B^{143}=\frac{1}{\sqrt{2}}\Xi_c^{'+}-\frac{1}{\sqrt{6}}\Xi_c^+$,
$\vphantom{\sqrt{\frac{2}{3}}}$\\
$B^{241}=\frac{1}{\sqrt{2}}\Sigma_c^+-\frac{1}{\sqrt{6}}\Lambda_c$, & ~~~~~$B^{242}=\Sigma_c^0$,
$\vphantom{\sqrt{\frac{2}{3}}}$\\
$B^{243}=\frac{1}{\sqrt{2}}\Xi_c^{'0}+\frac{1}{\sqrt{6}}\Xi_c^0$, & ~~~~~$B^{341}=\frac{1}{\sqrt{2}}\Xi_c^{'+}+\frac{1}{\sqrt{6}}\Xi_c^+$,
$\vphantom{\sqrt{\frac{2}{3}}}$\\
$B^{342}=\frac{1}{\sqrt{2}}\Xi_c^{'0}-\frac{1}{\sqrt{6}}\Xi_c^0$, & ~~~~~$B^{343}=\Omega_c^0$,
$\vphantom{\sqrt{\frac{2}{3}}}$\\
$B^{124}=\sqrt{\frac{2}{3}}\Lambda_c$, & ~~~~~$B^{234}=\sqrt{\frac{2}{3}}\Xi_c^0$,
$\vphantom{\sqrt{\frac{2}{3}}}$\\
$B^{314}=\sqrt{\frac{2}{3}}\Xi_c^+$, & ~~~~~$B^{144}=\Xi_{cc}^{++}$,
$\vphantom{\sqrt{\frac{2}{3}}}$\\
$B^{244}=-\Xi_{cc}^+$, & ~~~~~$B^{344}=\Omega_{cc}$,
$\vphantom{\sqrt{\frac{2}{3}}}$\\
\end{tabular}}$$ where the indices $i,j,k$ of $B^{ijk}$ denote the quark content of the baryon fields with the identification $1\leftrightarrow u$, $2\leftrightarrow d$, $3\leftrightarrow s$ and $4\leftrightarrow c$. The phase convention for the isospin states is $\mid{\pi^+}\rangle=-\mid{1,1}\rangle$, $\mid{K^{ -}}\rangle=-\mid{1/2,-1/2}\rangle$ and $\mid{D^{ 0}}\rangle=-\mid{1/2,-1/2}\rangle$ for the pseudoscalar mesons and, analogously, $\mid{\rho^+}\rangle=-\mid{1,1}\rangle$, $\mid{K^{*\, -}}\rangle=-\mid{1/2,-1/2}\rangle$ and $\mid{D^{*\, 0}}\rangle=-\mid{1/2,-1/2}\rangle$ for the vector mesons. For the baryons, we take $\mid{\Sigma^+}\rangle=-\mid{1,1}\rangle$ and $\mid{\Xi^-}\rangle=-\mid{1/2,-1/2}\rangle$. This convention is consistent with the structure of the $\phi$, $V_\mu$ and $B$ fields. It is the one followed in Refs. [@Wu:2010jy; @Gamermann:2006nm] and it differs from that in Ref. [@Hofmann:2005sw] in the sign of the $D^+(D^{\ast +})$ and $D^-(D^{\ast -})$ mesons.
Using the $VPP$ and $VBB$ vertices above one obtains the t-channel Vector-Meson-Exchange (TVME) potential [@Hofmann:2005sw]: $$\begin{aligned}
\label{eq:Vij_1}
V_{ij}&=&g^2 \sum_v C_{ij}^v \bar{u}\left(p_j\right)\gamma^\mu u\left(p_i\right)\frac{1}{t-m_v^2} \nonumber \\
& & \phantom{g^2 \sum_v } \times \left[\left(k_i+k_j\right)_\mu -\frac{k_i^2-k_j^2}{m_v^2}\left(k_i-k_j\right)_\mu\right],\end{aligned}$$ where $p_i$, $p_j$ ($k_i$, $k_j$) are the four-momenta of the baryons (mesons) in the $i$, $j$ channels and $m_v$ is the vector meson mass. Adopting the same mass $m_v=m_V$ for the light vector mesons and accounting for the higher mass of the charmed mesons with a common multiplying factor $\kappa_c=(m_V/m_V^c)^2\approx 1/4$ as in [@Mizutani:2006vq], Eq. (\[eq:Vij\_1\]) simplifies to $$\label{eq:Vij_2}
V_{ij}=-C_{ij}\frac{1}{4f^2}\bar{u}\left(p_j\right)\gamma^\mu u\left(p_i\right)\left(k_i+k_j\right)_\mu,$$ where the limit $t\ll m_V$ has been taken to reduce the t-channel diagram to a contact term. The coefficients $C_{ij}$ are symmetric with respect to the indices and are obtained summing the various vector meson exchange contributions, $ \sum_v C_{ij}^v$ [@Hofmann:2005sw], including the factor $\kappa_c$ in the case of charmed mesons. Working out the Dirac algebra up to order ${\cal O}(p^2/M)$ corrections, one gets the expression: $$\label{eq:Vij}
V_{ij}(\sqrt{s})=-C_{ij}\frac{1}{4f^2}\left(2\sqrt{s}-M_i-M_j\right) N_i N_j$$ where $M_i$, $M_j$ and $E_i$, $E_j$ are the masses and the energies of the baryons and $N=
\sqrt{(E+M)/2M}$. Note that, while $SU(4)$ symmetry is encoded in the values of the coefficients $C^v_{ij}$, the interaction potential is not SU(4) symmetric due to the use of physical masses for the mesons and baryons involved, as well as to the factor $\kappa_c$.
For the isospin $I=0$, charm $C=1$ and strangeness $S=-2$ sector studied here, the available pseudoscalar-baryon channels are $\bar{K}\Xi_c (2964)$, $\bar{K}\Xi'_c (3070)$, $D\Xi (3189)$, $\eta \Omega_c (3246)$, $\eta' \Omega_c (3656)$, $\bar{D}_s \Omega_{cc} (5528)$, and $\eta_c \Omega_c (5678)$, where the values in parentheses indicate their corresponding threshold. The doubly charmed $\bar{D}_s \Omega_{cc}$ and $\eta_c \Omega_c $ channels will be neglected, as their energy is much larger than that of the other channels. We have checked that their inclusion barely influences the results presented here. The matrix of $C_{ij}$ coefficients for the resulting 5-channel interaction is given in Table \[tab:coeff\].
-------------------------------- --------------------- ---------------------------- ------------------------------------------- -------------------------------------------- --------------------------------------------
\[-2.5mm\] [${\bar K}\Xi_c$]{} [${\bar K}\Xi_c^\prime$]{} [ $D\Xi$]{} [ $\eta\Omega_c^0$]{} [$\eta^\prime\Omega_c^0$]{}
\[-2.5mm\] [${\bar K}\Xi_c$]{} $1$ $0$ $\sqrt{\displaystyle\frac{3}{2}}\kappa_c$ $0$ $0$
[${\bar K}\Xi_c^\prime$]{} $1$ $\displaystyle\frac{1}{\sqrt{2}}\kappa_c$ $-\sqrt{6}$ $0$
[ $D\Xi$]{} $2$ $-\displaystyle\frac{1}{\sqrt{3}}\kappa_c$ $-\sqrt{\displaystyle\frac{2}{3}}\kappa_c$
[ $\eta\Omega_c^0$]{} $0$ $0$
[ $\eta^\prime\Omega_c^0$]{} $0$
-------------------------------- --------------------- ---------------------------- ------------------------------------------- -------------------------------------------- --------------------------------------------
: The $C_{ij}$ coefficients for the $I=0$, $C=1$, $S=-2$ sector of the $PB$ interaction.[]{data-label="tab:coeff"}
The interaction of vector mesons with baryons is obtained following the formalism presented in Ref. [@Oset:2009vf], which is extended to SU(4) here. Similarly as for pseudoscalar mesons, we only retain the t-channel vector-exchange term. Employing the effective Lagrangian: $$\label{eq:vertexVVV}
\mathcal{L}_{VVV}=ig\langle {\left[V^\mu,\partial_\nu V_\mu\right] V^\nu}\rangle .$$ for the three-vector $VVV$ vertex and that of Eq. (\[eq:vertexBBV\]) for the $VBB$ one, the resulting interaction kernel for the vector-baryon ($VB$) interaction is identical to that obtained for the pseudoscalar-baryon ($PB$) one (see Eq. (\[eq:Vij\])), multiplied by the product of polarization vectors, $\vec{\epsilon}_i,\;\vec{\epsilon}_j$.
The allowed vector meson-baryon states are $D^*\Xi (3326)$, $\bar{K}^*\Xi_c (3363)$, $\bar{K}^*\Xi'_c (3470)$, $\omega \Omega_c (3480)$, $\phi \Omega_c (3717)$, $\bar{D}_s^* \Omega_{cc} (5662)$ and $J/\psi \Omega_c (5794)$ , where, again, we will neglect the doubly charmed states. The coefficients $C_{ij}$ can be straightforwardly obtained from those for the $PB$ interaction in Table \[tab:coeff\], by considering the following correspondences: $$\begin{matrix}
\pi \rightarrow \rho, & K\rightarrow K^\ast, & \bar{K}\rightarrow\bar{K}^\ast, & D\rightarrow D^\ast, & \bar{D}\rightarrow\bar{D}^\ast,
\end{matrix}$$ $$\begin{matrix}
\frac{1}{\sqrt{3}}\eta+\sqrt{\frac{2}{3}}\eta^\prime\rightarrow\omega {\qquad\rm and\;} & -\sqrt{\frac{2}{3}}\eta+\frac{1}{\sqrt{3}}\eta^\prime\rightarrow\phi\ .
\end{matrix}
\label{eq:eta_phi}$$
The sought resonances will be generated as poles of the scattering amplitude $T_{ij}$, unitarized via the coupled-channel Bethe-Salpeter (B-S) equation, which implements the resummation of loop diagrams to infinite order schematically depicted in Fig. \[fig:BSfeynman\] and has the expression
![Diagrams representing the Bethe-Salpeter equations in meson-baryon ($MB$) scattering. The big empty circle corresponds to the $T_{ij}$ matrix element, the black circles correspond to the potential $V_{ij}$ and the loops represent the propagator $G_{l}$ function. The $i, j, l$ indices stand for the channels of the coupled-channel theory.[]{data-label="fig:BSfeynman"}](fig2.pdf){width="50.00000%"}
$$\label{eq:BSeq}
T_{ij}=V_{ij}+V_{il}G_{l}T_{lj}.$$
Factorizing the $V$ and $T$ matrices on-shell out of the internal integrals, the solution of the former equation $$\label{eq:BSeq2}
T=(1-VG)^{-1}V$$ is purely algebraic. We note that the sum over the polarizations of the internal vector mesons gives $$\sum_{\rm pol}\epsilon_i\epsilon_j=\delta_{ij}+\frac{q_iq_j}{M_V^2} \ ,$$ and, neglecting the correction $\sim q^2/M_V^2$, which is consistent with the approximations done so far, the factor $\vec{\epsilon}_i\vec{\epsilon}_j$ can be factorized out in all the terms of the B-S equation.
The loop function is given by $$\label{eq.Gmatrix}
G_{l}=i\int \frac{d^4q}{(2\pi)^4}\frac{2M_l}{(P-q)^2-M_l^2+i\epsilon}\frac{1}{q^2-m_l^2+i\epsilon},$$ where $M_l$ and $E_l$ correspond to the mass and the energy of the intermediate baryon, $m_l$ is the mass of the intermediate meson, $P=k+p=(\sqrt{s},\vec{0})$ is the total four-momentum of the system in the c.m. frame and $q$ denotes the four-momentum of the meson propagating in the intermediate loop. This function diverges for $\vec{q}\rightarrow\infty$ and it must be regularized with a proper scheme. One may employ the *cut-off* regularization method, which consists in replacing the infinite upper limit of the integral by a large enough cut-off momentum $\Lambda$, $$\label{eq:determine_qcut}
G_{l}^{\rm cut}=\int_{0}^{\Lambda}\frac{d^3q}{(2\pi)^3}\frac{1}{2\omega_l(\vec{q})}\frac{M_l}{E_l(\vec{q})}\frac{1}{\sqrt{s}-\omega_l(\vec{q})-E_l(\vec{q})+i\epsilon},$$ or the alternative *dimensional regularization* (DR) approach, which is the one adopted here: $$\label{eq:GmatrixDR}
\begin{aligned}
G_{l}=&\frac{2M_l}{16\pi^2}\Big\{ a_l(\mu)+\ln\frac{M_l^2}{\mu^2}+\frac{m_l^2-M_l^2+s}{2s}\ln\frac{m_l^2}{M_l^2}+ \\
&+\frac{q_l}{\sqrt{s}}\left[\ln\left(s-(M_l^2-m_l^2\right)+2q_l\sqrt{s})\right. \\
&\phantom{\frac{q_l}{\sqrt{s}}~~}+\ln\left(s+(M_l^2-m_l^2\right)+2q_l\sqrt{s})\\
&\phantom{\frac{q_l}{\sqrt{s}}~~}-\ln\left(-s+(M_l^2-m_l^2\right)+2q_l\sqrt{s})\\
&\phantom{\frac{q_l}{\sqrt{s}}~~}\left.-\ln\left(-s-(M_l^2-m_l^2\right)+2q_l\sqrt{s}) \right] \Big\},
\end{aligned}$$ where $a_l(\mu)$ is the subtraction constant at the regularization scale $\mu$, and $q_l$ is the on-shell three-momentum of the meson in the loop. The choice of the regularization scale $\mu$ and the corresponding subtraction constants $a_l(\mu)$ can be obtained by demanding that, at an energy close to the channel threshold, $G_l$ is similar to $G_l^\text{cut}$ for a certain cut-off $\Lambda$, namely $$\label{eq:a(mu)}
a_l({\mu})= \frac{16\pi^2}{2M_l}\left(G_{l}^\text{cut}(\Lambda)-G_{l}(\mu,a_l=0)\right).$$ The value of $\Lambda$ is usually taken around several hundreds of MeV, which is around the scale that has been integrated out in the zero range approximation of the meson meson-exchange model considered here. Typical values of the DR parameters are $\mu\approx630\rm~MeV$ and $a(\mu)\sim-2.0$ in the case of $SU(3)$ (see Ref. [@Oller:2000fj]) while in $SU(4)$ previous works have taken $\mu=1000\rm~MeV$ and $a(\mu)\sim-2.3$ [@Wu:2010jy; @Wu:2010vk].
The expression for the loop function $G_l$ in Eq. (\[eq:GmatrixDR\]) assumes that the baryon and the meson have fixed masses and no width. When the B-S equation involves channels that include particles with a large width, which is the case of the $\rho$ ($\Gamma_\rho=149.4\rm~MeV$) and $K^\ast$ ($\Gamma_{K^\ast}~=~50.5\rm~MeV$) mesons, this function has to be convoluted with the mass distribution of the particle. Following the method described in [@Oset:2009vf], the loop function in these cases will be replaced by $$\begin{aligned}
\tilde{G}_{l}(s)=-\frac{1}{N}\int_{(m_l-2\Gamma_l)^2}^{(m_l+2\Gamma_l)^2} & \frac{d\tilde{m}_l^2}{\pi}{\rm\, Im\,}\frac{1}{\tilde{m}_l^2-m_l^2+i\,m_l\Gamma(\tilde{m}_l)} \\
& \times G_{l}\left(s,\tilde{m}_l^2,M_l^2\right),
\end{aligned}$$ where we have taken the limits of the integral to extend over a couple of times the width of the meson, and the normalization factor $N$ reads $$N=\int_{(m_l-2\Gamma_l)^2}^{(m_l+2\Gamma_l)^2}d\tilde{m}_l^2\left(-\frac{1}{\pi}\right){\rm\, Im\,}\frac{1}{\tilde{m}_l^2-m_l^2+i\,m_l\Gamma(\tilde{m}_l)} \ .$$ The energy dependent width is given by $$\label{eq:G_kallen}
\Gamma(\tilde{m}_l)=\Gamma_l\frac{m_l^5}{\tilde{m}_l^5}\frac{\lambda^{3/2}(\tilde{m}_l^2,m_1^2,m_2^2)}{\lambda^{3/2}(m_l^2,m_1^2,m_2^2)}\,\theta(\tilde{m}_l-m_1-m_2),$$ where $m_1$ and $m_2$ are the masses of the lighter mesons to which the vector meson in the loop decays, i.e. $m_1=m_2=m_\pi$ for the $\rho$ and $m_1=m_\pi$, $m_2=m_K$ for the $K^\ast$ and $\lambda$ is the Källén function $\lambda(x,y,z)=(x-(\sqrt{y}+\sqrt{z})^2)(x-(\sqrt{y}-\sqrt{z})^2)$.
A resonance generated dynamically from the coupled channel meson-baryon interaction appears as a pole of the scattering amplitude $T$ in the so-called [*second Riemann sheet*]{} (${\rm II}$) of the complex energy plane, which implies performing the calculation of the loop function given in Eq. (\[eq:GmatrixDR\]) with a rotated momentum ($q_l\to -q_l$) or, equivalently, employing [@Roca:2005nm] $$G_{l}^\text{II}(s)=G_{l}(s)+i\,2M_l\frac{q_l}{4\pi\sqrt{s}}.$$ In the case of multiple channels, the loop function of each channel is rotated to the second Riemann sheet only when the real part of the complex energy $z\equiv \sqrt{s}$ is larger than the corresponding channel threshold. In the vicinity of a pole, $z_p$, one may write $$\label{eq:pole}
T_{ij}(s)\sim \frac{g_i g_j} {z-z_p} \ ,$$ and the coupling constants of the resonance to the various channels are obtained from the corresponding residues, calculated from: $$\label{eq:coup_der}
g_i g_j=\left[\frac{\partial}{\partial z}\left.\left(\frac{1}{T_{ij}(z)}\right)\right|_{z_p}\right]^{-1} \ .$$
The amount of $i^{\rm th}$-channel meson-baryon component in a given resonance can be obtained from the real part of: $$\label{eq:coup_der}
X_i = - g_i^2\left.\left(\frac{\partial{G}}{\partial(\sqrt{s})}\right)\right|_{z_p} \ .$$ This expression is based on the model-independent relation between the compositeness of a weakly bound state and the threshold parameters of the interaction generating it, derived in Ref. [@Weinberg:1965zz] . This idea has been reformulated within a field theoretical approach and extended to higher partial waves as well as to unstable (resonance) states [@Gamermann:2009uq; @Hyodo:2011qc; @Aceti:2012dd; @Hyodo:2013nka; @Aceti:2014ala].
Results {#sec:results}
=======
In this section we present the results obtained employing the unitarized model for meson-baryon scattering in coupled channels described above. We first describe the results obtained with the pseudoscalar-baryon interaction kernel of Eq. (\[eq:Vij\]), employing the subtraction constants listed under “Model 1" of Table \[tab:a\_pseudo\] in the loop functions. These subtraction constants are obtained for a regularization scale of $\mu=1000$ GeV and imposing the loop function of each pseudoscalar-baryon channel to coincide, at the corresponding threshold, with the cut-off loop function evaluated for $\Lambda=800$ MeV, see Eq. (\[eq:a(mu)\]). We assume this value to be a natural choice as it roughly corresponds to the mass of the exchanged vector mesons in the t-channel diagram that has been eliminated in favor of the contact interaction employed in this work. In this case, the scattering amplitude $T$ shows two poles having the following properties: $$M_1 = {\rm Re}\,z_1= 3051.6~{\rm MeV},~~\Gamma_1 = -2 {\rm Im} \,z_1= 0.45~{\rm MeV}$$ and $$M_2 = {\rm Re} \,z_2 = 3103.3~{\rm MeV},~~\Gamma_2 = -2 {\rm Im} \,z_1= 17~{\rm MeV} .$$ These resonances have spin-parity $J^P=1/2^-$, as they are obtained from the scattering amplitude of pseudoscalar mesons with baryons of the ground state octet in s-wave. The couplings of each resonance to the various meson-baryon channels are displayed in Table \[tab:pseudo\] under the label “Model 1", where one can also find the corresponding compositeness, given by Eq. (\[eq:coup\_der\]), which measures the amount of each meson-baryon component in the resonance. We observe that the lowest energy state at 3052 MeV couples appreciably to the channels $\bar{K}\Xi'_c$, $D\Xi $ and $\eta \Omega_c^0$. Note that, although the coupling to $\eta \Omega_c^0$ states is the strongest, the compositeness is larger in the $\bar{K}\Xi'_c$ channel, to which the resonance also couples strongly and, in addition, lies closer to the corresponding threshold. The higher energy resonance at 3103 MeV, with a strong coupling to $D\Xi$ and a compositeness in this channel of 0.90, clearly qualifies as being a $D\Xi $ bound state.
[lccccc]{} & $a_{ \bar{K}\Xi_c}$ & $a_{\bar{K}\Xi'_c}$ & $a_{D\Xi }$ & $a_{\eta \Omega_c}$ & $a_{\eta' \Omega_c }$\
\
Model 1 & $-2.19$ & $-2.26$ & $-1.90$ & $-2.31$ & $-2.26$\
$\Lambda$ (MeV) & 800 & 800 & 800 & 800 & 800\
Model 2 & $-1.69$ & $-2.09$ & $-1.93$ & $-2.46$ & $-2.42$\
$\Lambda$ (MeV) & 320 & 620 & 830 & 980 & 980\
[c|cc|cc]{}\
\
&\
\
$M\;\rm[MeV]$ & &\
$\Gamma\;\rm[MeV]$ & &\
\
& $| g_i|$ & $-g_i^2 dG/dE$ & $| g_i|$ & $-g_i^2 dG/dE$\
$\bar{K}\Xi_c (2964)$ & $0.11$ & $0.00+i\,0.00$ & $0.58$ & $0.01+i\,0.03$\
$\bar{K}\Xi'_c (3070)$ & $1.67$ & $0.54+i\,0.01$ & $0.30$ & $0.01-i\,0.01$\
$D\Xi (3189)$ & $1.10$ & $0.05-i\,0.01$ & $4.08$ & $0.90-i\,0.05$\
$\eta \Omega_c (3246)$ & $2.08$ & $0.23+i\,0.00$ & $0.44$ & $0.01+i\,0.01$\
$\eta' \Omega_c (3656)$ & $0.04$ & $0.00+i\,0.00$ & $0.28$ & $0.00+i\,0.00$\
&\
\
$M\;\rm[MeV]$ & &\
$\Gamma\;\rm[MeV]$ & &\
\
& $| g_i|$ & $-g_i^2 dG/dE$ & $| g_i|$ & $-g_i^2 dG/dE$\
$\bar{K}\Xi_c (2964)$ & $0.11$ & $0.00+i\,0.00$ & $0.49$ & $-0.02+i\,0.01$\
$\bar{K}\Xi'_c (3070)$ & $1.80$ & $0.61+i\,0.01$ & $0.35$ & $0.02-i\,0.02$\
$D\Xi (3189)$ & $1.36$ & $0.07-i\,0.01$ & $4.28$ & $0.91-i\,0.01$\
$\eta \Omega_c (3246)$ & $1.63$ & $0.14+i\,0.00$ & $0.39$ & $0.01+i\,0.01$\
$\eta' \Omega_c (3656)$ & $0.06$ & $0.00+i\,0.00$ & $0.28$ & $0.00+i\,0.00$\
We see that our two resonances have energies very similar to the second and fourth $\Omega_c^0$ states discovered by LHCb [@Aaij:2017nav], with properties: $$\begin{aligned}
\Omega_c(3050)^0:~~& M=3050.2\pm0.1\pm0.1^{+0.3}_{-0.5}~{\rm MeV}, \nonumber \\
&\Gamma=0.8\pm0.2\pm0.1~{\rm MeV},\nonumber \\
\Omega_c(3090)^0:~~& M=3090.2\pm0.3\pm0.5^{+0.3}_{-0.5}~{\rm MeV}, \nonumber \\
&\Gamma=8.7\pm1.0\pm0.8~{\rm MeV}.
\label{eq:exp}
\end{aligned}$$ We note that, even if the mass of our heavier state is larger by 10 MeV and its width is about twice the experimental one, our results clearly show the ability of the meson baryon dynamical models for generating states in the energy range of interest.
In an attempt to accommodate better to the data, we relax the condition of forcing that each loop function matches, at the corresponding threshold, the cut-off loop function evaluated for a cut-off $\Lambda=800$ MeV. To this end, we let the values of the five subtracting constants vary freely within a reasonably constrained range and look for sets that reproduce the characteristics of the two observed states, $\Omega_c(3050)^0$ and $\Omega_c(3090)^0$, within $2\sigma$ of the experimental errors \[see Eq. (\[eq:exp\])\]. In order to analyze the correlations, we represent in Fig. \[fig:a\_corr\] the values of each subtraction constant against all the others in the sets that comply with the experimental constraints. We clearly observe an anti-correlation between the subtraction constants $a_{\bar{K}\Xi'_c}$ and $a_{\eta \Omega_c}$. This can be simply understood by noting that the resonance at 3050 MeV couples mostly to these two meson-baryon states, as can be seen from the results in Table \[tab:pseudo\], implying that, if one subtraction constant becomes more negative, favoring a stronger attraction for the pole, the other subtraction constant needs to compensate this effect by being less negative. We also find the subtraction constant $a_{D\Xi}$ to acquire a rather stable value between -1.94 and -1.93. This is clearly a reflection of the resonance at 3090 MeV being essentially a ${D\Xi}$ bound state, which requires a particular value of the subtraction constant $a_{D\Xi}$ to generate the pole at the appropriate experimental energy.
![(Color online) Correlations between the various subtraction constants. The circles represent different configurations of subtraction constants that reproduce the experimental resonances $\Omega_c(3050)^0$ and $\Omega_c(3090)^0$. The red asterisks denote one particular representative set. []{data-label="fig:a_corr"}](subst_const_analysis_2.pdf){width="50.00000%"}
Among all the possible configurations of subtraction constants producing the experimental $\Omega_c^0$ states at 3050 MeV and 3090 MeV represented in Fig. \[fig:a\_corr\], we select a representative set, denoted by red asterisks in the figure, the values of which are listed in Table \[tab:a\_pseudo\] under the label “Model 2". The two poles of the scattering amplitude of “Model 2" have the properties: $$M_1 = {\rm Re}\, z_1 = 3050.3~{\rm MeV},~~~\Gamma_1 = -2 {\rm Im} \,z_1= 0.44~{\rm MeV}$$ $$M_2 = {\rm Re} \,z_2 = 3090.8~{\rm MeV},~~~\Gamma_2 = -2 {\rm Im}\, z_1= 12~{\rm MeV} \ ,$$ which are similar for any of the sets of subtracting constants represented in Fig. \[fig:a\_corr\]. As we see, the stronger changes are found in the higher resonance, which, apart from having been lowered to the experimental energy, its width has been substantially decreased to agree with the experiment at $2\sigma$ level. We see from Table \[tab:a\_pseudo\] that the equivalent values of the cut-off for this new set of subtracting constants now lie in the range $[320-950]\;\rm MeV$. Note that the strongest change corresponds to the subtraction constant $a_{\bar{K}\Xi_c}$ , needed to decrease the width of the $\Omega_c(3090)^0$ towards its experimental value. The equivalent cut-off value of 320 MeV is on the low side of the usually employed values but it still naturally sized.
The five $\Omega_c^0$ states were observed from the $K^-\Xi^+_c$ invariant mass spectrum obtained from a sample of $pp$ collision data at center of mass energies of 7, 8 and 13 TeV, recorded by the LHCb experiment [@Aaij:2017nav]. To model such spectrum from the elementary $pp$ collision reaction is a tremendously difficult task, but we can give a taste of the spectrum that our models would predict by representing, in Fig. \[fig:t2\], the quantity $$q_{K^-} \mid \sum_{i} T_{i\to \bar{K}\Xi_c} \mid^2
\label{eq:t2}$$ versus the $\bar{K}\Xi_c$ center-of-mass energy, where $T_{i\to \bar{K}\Xi_c}$ is the amplitude for the $i \to \bar{K}\Xi_c$ transition obtained here with either “Model 1" (black dashed line) or “Model 2" (red solid line), with $i$ being any of the five coupled channels involved in this sector. The momentum of the $K^-$ in the $\bar{K}\Xi_c$ center-of-mass frame, $q_{K^-}$, acts as a phase-space modulator. We note that, in front of each amplitude $T_{i\to \bar{K}\Xi_c}$ in Eq. (\[eq:t2\]), one should have included a coefficient gauging the strength with which the production mechanism excites the particular meson-baryon channel $i$. Given the limited understanding of the production dynamics, we have assumed all these coefficients to be equal. Therefore, the spectrum displayed in Fig. \[fig:t2\] is merely orientative as it also lacks the background contributions. However, one can still see certain similarities between the spectrum of Fig. 2 in Ref. [@Aaij:2017nav] and the results shown in Fig. \[fig:t2\] (after convoluting appropriately with the experimental energy resolution) in the energy regions of the 3050 MeV and 3090MeV states.
![(Color online) Sum of amplitudes squared times a phase space factor.[]{data-label="fig:t2"}](sum_ampl_i_KXic.pdf){width="45.00000%"}
Finally, we construct the unitarized interaction between vector mesons and baryons in this sector, employing the set of coupling constants of Table \[tab:a\_vector\], which have been obtained for a regularization scale of $\mu=1000$ GeV and imposing the loop function of each vector-baryon channel to coincide, at the corresponding threshold, with the cut-off loop function evaluated for $\Lambda=800$ MeV. The mass and other properties of the resonances found from the vector-baryon interaction in the $S=-2$, $C=1$ and $I=0$ sector are listed in Table \[tab:vector\]. We see a similar pattern as that found for the pseudoscalar-baryon case, one resonance coupling strongly to $D^*\Xi$ and the other coupling strongly to $\bar{K}^*\Xi'_c $ and to $\phi \Omega_c^0$, which mainly takes the role of the $\eta \Omega_c^0$ state of the pseudoscalar case according to the tranformation of Eq. (\[eq:eta\_phi\]). However, the ordering in energies of these resonances appears interchanged with respect to that found in pseudoscalar-baryon scattering, which is simply related to the fact that the energy thresholds of the various vector-meson states have also changed with respect to their pseudoscalar-baryon counterparts. The lower energy resonance at 3231 MeV is mainly a $D^*\Xi$ bound state, while the resonance at 3419 MeV, is mainly a $\bar{K}^*\Xi'_c $ composite state with some admixture of $\omega \Omega_c^0$ and $\phi \Omega_c$ components. These resonances are located at energy values well above the states found by the LHCb collaboration in a region where no narrow structures have been seen [@Aaij:2017nav]. We note, however, that the states found here from the vector-baryon interaction are artificially narrow as they do not couple to, and hence cannot decay into, the pseudoscalar-baryon states that lie at lower energy. In order to account for this possibility in our model one should incorporate the coupling of vector-baryon states to the pseudoscalar-baryon ones, via e.g. box diagrams [@Garzon:2012np; @Liang:2014kra] or employing the methodology of Refs. [@Romanets:2012hm; @GarciaRecio:2008dp] where, on the basis of heavy-quark spin symmetry, the pseudoscalar and vector mesons, as well as the baryons of the octet and those of the decuplet, are treated on the same footing. It would be interesting to perform such calculations in order to see if these structures remain narrow or widen up sufficiently to accommodate to the apparently featureless spectrum (within experimental errors) in this higher energy range. It would also be interesting to explore how the pseudoscalar-baryon resonances studied in the present work would be affected by considering the coupling to the vector-baryon states, a task that goes beyond the scope of the present exploratory study. Note, however, that the energy threshold of the lighter $D^*\Xi$ vector-baryon channel lies above those of the pseudoscalar-meson channels, except for the $\eta^\prime \Omega_c^0$ one which plays a quite irrelevant role in the pseudoscalar-baryon states found here. We therefore expect limited changes in their energy positions and widths, which could anyway be compensated by appropriate changes in the subtraction constants.
[lccccc]{} & $a_{ D^*\Xi}$ & $a_{\bar{K}^*\Xi_c}$ & $a_{\bar{K}^*\Xi'_c}$ & $a_{\omega \Omega_c}$ & $a_{\phi \Omega_c}$\
\
& $-1.97$ & $-2.15$ & $-2.20$ & $-2.27$ & $-2.26$\
[c|cc|cc]{}\
\
$M\;\rm[MeV]$ & &\
$\Gamma\;\rm[MeV]$ & &\
\
& $| g_i|$ & $-g_i^2 dG/dE$ & $| g_i|$ & $-g_i^2 dG/dE$\
$D^*\Xi (3326)$ & $4.30$ & $0.90-i0.00$ & $0.24$ & $0.00+i0.00$\
$\bar{K}^*\Xi_c (3363)$ & $0.64$ & $0.03-i0.00$ & $0.13$ & $0.00+i0.00$\
$\bar{K}^*\Xi'_c (3470)$ & $0.26$ & $0.00-i0.00$ & $1.83$ & $0.42+i0.02$\
$\omega \Omega_c (3480)$ & $0.34$ & $0.01-i0.00$ & $1.56$ & $0.28+i0.00$\
$\phi \Omega_c (3717)$ & $0.00$ & $0.00-i0.00$ & $2.31$ & $0.22+i0.00$\
Conclusions {#sec:conclusions}
===========
In this work we have studied the interaction of the low-lying pseudoscalar mesons with the ground-state baryons in the charm $+1$, strangeness $-2$ and isospin $0$ sector, employing a t-channel vector meson exchange model with effective Lagrangians. We unitarize the amplitude by means of the coupled-channel Bethe-Salpeter equation, paying a especial attention to regulate the loops with natural sized subtraction constants.
The resulting amplitude for the scattering of pseudoscalar mesons with baryons shows the presence of two resonances, having energies and widths very similar to some of the $\Omega_c^0$ states discovered recently at LHCb. By exploring the parameter space of our model we find several cases that can reproduce the mass and width of the $\Omega_c(3050)^0$ and the $\Omega_c(3090)^0$.
Our findings allow us to conclude that two of the five $\Omega_c^0$ states recently observed by the LHCb collaboration could have a meson-baryon molecular origin. The state at 3050 MeV would mostly have a $\bar{K}\Xi'_c $ component (around 50%) with a 20% mixture of $\eta \Omega_c$, while the one at 3090 MeV would be essentially a $D\Xi$ molecule with a 90% strength.
As our model for the scattering of pseudoscalar mesons with baryons in s-wave generates resonances with spin-parity $J^P=1/2^-$, we would anticipate these to be the quantum numbers for the 3050 MeV and 3090 MeV $\Omega_c^0$ states, in contrast to the expectations from quark models which establish either $3/2^-$ or $5/2^-$ for their spin-parity.
An experimental determination of the spin-parity of the $\Omega_c^0$ states would be extremely valuable to disentangle the $3q$ or meson-baryon nature of some of the $\Omega_c^0$ states observed at LHCb. It is also expected that further theoretical studies about the molecular interpretation of baryons in the $S=-2$, $C=1$, $I=0$ sector, including additional components as the ones considered here, can bring new light into this problem.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is partly supported by the Spanish Ministerio de Economia y Competitividad (MINECO) under the project MDM-2014-0369 of ICCUB (Unidad de Excelencia ’María de Maeztu’), and, with additional European FEDER funds, under the contract FIS2014-54762-P. Support has also been received from the Generalitat de Catalunya contract 2014SGR-401.
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[^1]: Kawarabayashi-Suzuki-Fayyazuddin-Riazuddin
|
---
abstract: 'Within the framework of a Beyond Standard Model (BSM) with a local $SU(3)$ family symmetry, we report an updated fit of parameters which account for the known spectrum of quarks and charged lepton masses and the quark mixing in a $4\times 4$ non-unitary $V_{CKM}$. In this scenario, ordinary heavy fermions, top and bottom quarks and tau lepton, become massive at tree level from Dirac See-saw mechanisms implemented by the introduction of a new set of $SU(2)_L$ weak singlet vector-like fermions, $U,D,E,N$, with $N$ a sterile neutrino. The $N_{L,R}$ sterile neutrinos allow the implementation of a $8\times 8$ general See-saw Majorana neutrino mass matrix with four massless eigenvalues at tree level. Hence, light fermions, including neutrinos, obtain masses from loop radiative corrections mediated by the massive $SU(3)$ gauge bosons. $SU(3)$ family symmetry is broken spontaneously in two stages, whose hierarchy of scales yield an approximate $SU(2)$ global symmetry associated with the $Z_1, Y_1^\pm$ gauge boson masses of the order of 2 TeV. A global fit of parameters to include neutrino masses and lepton mixing is in progress.'
author:
- 'Albino Hernández-Galeana'
title: 'Charged Fermion Masses and Mixing from a $SU(3)$ Family Symmetry Model'
---
Introduction
==============
The origen of the hierarchy of fermion masses and mixing is one of the most important open problems in particle physics. Any attempt to account for this hierarchy introduce a mass generation mechanism which distinguish among the different Standard Model (SM) quarks and leptons.
After the discovery of the scalar Higgs boson on 2012, LHC has not found a conclusive evidence of new physics. However, there are theoretical motivations to look for new particles in order to answer some open questions like; neutrino oscillations, dark matter, stability of the Higgs mass against radiative corrections,etc.
In this report, we address the problem of charged fermion masses and quark mixing within the framework of an extension of the SM introduced by the author in [@albinosu32004]. This BSM proposal include a vector gauged $SU(3)$ family symmetry[^1] commuting with the SM group and introduce a hierarchical mass generation mechanism in which the light fermions obtain masses through loop radiative corrections, mediated by the massive bosons associated to the $SU(3)$ family symmetry that is spontaneously broken, while the masses of the top and bottom quarks and that of the tau lepton are generated at tree level from “Dirac See-saw”[@SU3MKhlopov] mechanisms through the introduction of a new set of $SU(2)_L$ weak singlets $U,D,E$ and $N$ vector-like fermions, which do not couple to the $W$ boson, such that the mixing of $U$ and $D$ vector-like quarks with the SM quarks gives rise to and extended $4\times4$ non-unitary CKM quark mixing matrix [@vectorlikepapers].
Model with $SU(3)$ flavor symmetry
==================================
Fermion content
---------------
Before “Electroweak Symmetry Breaking”(EWSB) all ordinary SM fermions remain massless, and the global symmetry in this limit, including R-handed neutrinos, is:
$$\begin{aligned}
SU(3)_{q_L}\otimes SU(3)_{u_R}\otimes SU(3)_{d_R}\otimes
SU(3)_{l_L}\otimes SU(3)_{\nu_R}\otimes SU(3)_{e_R} \nonumber\\ \nonumber\\
\supset SU(3)_{q_L+u_R+d_R+l_L+e_R+\nu_R} \equiv SU(3) \label{su3symmetry}\end{aligned}$$
We define the gauge symmetry group
$$G\equiv SU(3) \otimes G_{SM} \label{gaugegroup}$$
where $SU(3)$ is the gauged family symmetry among families, eq.(\[su3symmetry\]) , and $G_{SM}= SU(3)_C \otimes SU(2)_L \otimes U(1)_Y $ is the “Standard Model” gauge group, with $g_H$, $g_s$, $g$ and $g^\prime$ the corresponding coupling constants. The content of fermions assumes the ordinary quarks and leptons assigned under G as:
$q_{iL}^o=\begin{pmatrix} u_{iL}^o \\ d_{iL}^o \end{pmatrix} \;,\;
l_{iL}^o=\begin{pmatrix} \nu_{iL}^o \\ e_{iL}^o \end{pmatrix} \; , \; Q = T_{3L} + \frac{1}{2} Y$
$$\Psi_q^o = ( 3 , 3 , 2 , \frac{1}{3} )_L=\begin{pmatrix} q_{1L}^o \\ q_{2L}^o \\ q_{3L}^o \end{pmatrix} \quad , \quad \Psi_l^o= ( 3 , 1 , 2 , -1 )_L=\begin{pmatrix} l_{1L}^o \\ l_{2L}^o \\ l_{3L}^o \end{pmatrix}$$
$$\Psi_u^o = ( 3 , 3, 1 , \frac{4}{3} )_R=\begin{pmatrix} u_R^o \\ c_R^o \\ t_R^o \end{pmatrix} \quad ,
\quad \Psi_d^o =(3, 3 , 1 , -\frac{2}{3} )_R=\begin{pmatrix} d_R^o \\ s_R^o \\ b_R^o \end{pmatrix}$$
$$\Psi_e^o = (3 , 1 , 1,-2)_R=\begin{pmatrix} e_R^o \\ \mu_R^o \\ \tau_R^o \end{pmatrix} \: ,$$
where the last entry corresponds to the hypercharge $Y$. The model also includes two types of extra $SU(2)_L$ weak singlet fermions:
[**Right Handed Neutrinos:**]{} $ \Psi_{\nu_R}^o = ( 3 , 1 , 1 , 0 )_R= \begin{pmatrix}
\nu_{e_R} \\ \nu_{\mu_R} \\
\nu_{\tau_R} \end{pmatrix} $ ,
and the vector-like fermions:
[**Sterile Neutrinos:** ]{} $\quad N_L^o, N_R^o = ( 1 , 1 , 1 , 0 ) $ ,
[**The Vector Like quarks:**]{} $$U_L^o, U_R^o = ( 1 , 3 , 1 , \frac{4}{3} ) \quad , \quad
D_L^o, D_R^o = ( 1 , 3 , 1 ,- \frac{2}{3} ) \label{vectorquarks}$$
and
[**The Vector Like electron:**]{} $\quad E_L^o, E_R^o = ( 1 , 1 , 1 , -2 ) $.\
The transformation of these vector-like fermions allows the gauge invariant mass terms
$$M_U \:\bar{U}_L^o \:U_R^o \,+\, M_D \:\bar{D}_L^o \:D_R^o \,+\, M_E \:\bar{E}_L^o \:E_R^o + h.c. \;,$$
and
$$m_D \,\bar{N}_L^o \,N_R^o \,+\, m_L \,\bar{N}_L^o\, (N_L^o)^c \,+\, m_R \,\bar{N}_R^o\, (N_R^o)^c \,+\, h.c$$
The above fermion content make the model anomaly free. After the definition of the gauge symmetry group and the assignment of the ordinary fermions in the usual form under the standard model group and in the fundamental $3$-representation under the $SU(3)$ family symmetry, the introduction of the right-handed neutrinos is required to cancel anomalies [@T.Yanagida1979]. The $SU(2)_L$ weak singlet vector-like fermions have been introduced to give masses at tree level only to the third family of known fermions via Dirac See-saw mechanisms. These vector like fermions, together with the radiative corrections, play a crucial role to implement a hierarchical spectrum for ordinary quarks and charged lepton masses.
$SU(3)$ family symmetry breaking
================================
To implement a hierarchical spectrum for charged fermion masses, and simultaneously to achieve the SSB of $SU(3)$, we introduce the flavon scalar fields: $\eta_i,\;i=2,3$,
$$\eta_i=(3 , 1 , 1 , 0)=\begin{pmatrix} \eta_{i1}^o\\ \eta_{i2}^o\\ \eta_{i3}^o \end{pmatrix} \;, \quad i=2,3,$$
acquiring the “Vacuum ExpectationValues” (VEV’s):
$$\langle \eta_2 \rangle^T = ( 0 , \Lambda_2 , 0) \quad , \quad
\langle \eta_3 \rangle^T = ( 0 , 0, \Lambda_3) \:. \label{veveta2eta3}$$
The corresponding $SU(3)$ gauge bosons are defined in Eq. through their couplings to fermions. Thus, the contribution to the horizontal gauge boson masses from Eq.(\[veveta2eta3\]) read
- $\eta_2:\quad \frac{g_{H_2}^2 \Lambda_2^2}{2} ( Y_1^+ Y_1^- + Y_3^+ Y_3^-) + \frac{g_{H_2}^2 \Lambda_2^2}{4} ( Z_1^2 + \frac{Z_2^2}{3} - 2 Z_1 \frac{Z_2}{ \sqrt{3}} ) $
- $\eta_3:\quad \frac{g_{H_3}^2 \Lambda_3^2}{2} ( Y_2^+ Y_2^- + Y_3^+ Y_3^-) + g_{H_3}^2 \Lambda_3^2 \frac{Z_2^2}{3} $
[*These two scalars in the fundamental representation is the minimal set of scalars to break down completely the $SU(3)$ family symmetry*]{}. Therefore, neglecting tiny contributions from electroweak symmetry breaking, Eq., we obtain the gauge boson mass terms:
$$M_2^2 \,Y_1^+ Y_1^- + M_3^2 \,Y_2^+ Y_2^- + ( M_2^2 + M_3^2) \,Y_3^+ Y_3^-
+ \frac{1}{2} M_2^2 \,Z_1^2 +\frac{1}{2} \frac{M_2^2 + 4 M_3^2}{3} \,Z_2^2
- \frac{1}{2}( M_2^2 ) \frac{2}{\sqrt{3}} \,Z_1 \,Z_2$$
$$M_2^2= \frac{g_{H}^2 \Lambda_2^2}{2} \quad , \quad M_3^2=\frac{g_{H}^2 \Lambda_3^2}{2} \quad , \quad y \equiv \frac{M_3}{M_2}= \frac{\Lambda_3}{\Lambda_2} \label{M23}$$
[ c | c c ]{} & $Z_1$ & $Z_2$\
\
$Z_1$ & $ M_2^2$ & $ - \frac{ M_2^2}{\sqrt{3}}$\
& &\
$Z_2$ & $ - \frac{M_2^2}{\sqrt{3}}$ & $\quad \frac{M_2^2+4 M_3^2}{3}$
Diagonalization of the $Z_1-Z_2$ squared mass matrix yield the eigenvalues
$$\begin{aligned}
M_-^2=\frac{2}{3} \left( M_2^2 + M_3^2 - \sqrt{ (M_3^2 - M_2^2)^2 + M_2^2 M_3^2 } \right)=M_2^2 \,y_-\label{Mm} \\
\nonumber\\
M_+^2=\frac{2}{3} \left( M_2^2 + M_3^2 +\sqrt{ (M_3^2 - M_2^2)^2 + M_2^2 M_3^2 } \right)=M_2^2 \,y_+ \label{Mp}\end{aligned}$$
and the gauge boson mass eigenvalues
$$M_2^2 \,Y_1^+ Y_1^- + M_3^2\,Y_2^+ Y_2^- + ( M_2^2 + M_3^2) \,Y_3^+ Y_3^-
+ M_-^2 \,\frac{Z_-^2}{2} + M_+^2 \,\frac{Z_+^2}{2}$$
or
$$M_2^2 \,Y_1^+ Y_1^- + M_2^2\,y^2\,Y_2^+ Y_2^- + M_2^2 ( 1 + y^2) \,Y_3^+ Y_3^-
+ M_2^2\,y_- \,\frac{Z_-^2}{2} + M_2^2\,y_+ \,\frac{Z_+^2}{2}\, ,$$
where
$$\begin{pmatrix} Z_1 \\ Z_2 \end{pmatrix} = \begin{pmatrix} \cos\phi & - \sin\phi \\
\sin\phi & \cos\phi \end{pmatrix} \begin{pmatrix} Z_- \\ Z_+ \end{pmatrix} \label{z1z2mixing}$$
$$\cos\phi \, \sin\phi=\frac{\sqrt{3}}{4} \,\frac{M_2^2}{\sqrt{ M_2^4 + M_3^2 (M_3^2 - M_2^2) } }$$
*Notice that in the limit $y =\frac{M_3}{M_2} \gg 1$, $\sin\phi \rightarrow 0$, $\cos\phi \rightarrow 1$, and we get an approximate $SU(2)$ global symmetry for the $Z_1, Y_1^\pm$ almost degenerated gauge boson masses of order $M_2$. Thus, the hierarchy of scales in the SSB yields an approximate $SU(2)$ global symmetry in the spectrum of $SU(3)$ gauge boson masses. Actually this approximate $SU(2)$ symmetry may play the role of a custodial symmetry to suppress properly the tree level $\Delta F=2$ “Flavour Changing Neutral Currents” (FCNC) processes mediated by the lower scale of horizontal gauge bosons with masses of few TeV’s*
Electroweak symmetry breaking
=============================
Recently ATLAS [@ATLAS] and CMS [@CMS] at the Large Hadron Collider announced the discovery of a Higgs-like particle, whose properties, couplings to fermions and gauge bosons will determine whether it is the SM Higgs or a member of an extended Higgs sector associated to a BSM theory. The Electroweak Symmetry Breaking (EWSB) in the $SU(3)$ family symmetry model involves the introduction of two triplets of $SU(2)_L$ Higgs doublets, namely;
$$\Phi^u=(3,1,2,-1)=\begin{pmatrix}
\begin{pmatrix} \phi^o\\ \phi^- \end{pmatrix}_1^u \\\\ \begin{pmatrix} \phi^o\\ \phi^- \end{pmatrix}_2^u \\\\
\begin{pmatrix} \phi^o\\ \phi^- \end{pmatrix}_3^u \end{pmatrix} \qquad , \qquad
\Phi^d=(3,1,2,+1)=\begin{pmatrix}
\begin{pmatrix} \phi^+\\ \phi^o \end{pmatrix}_1^d \\\\ \begin{pmatrix} \phi^+\\ \phi^o \end{pmatrix}_2^d \\\\
\begin{pmatrix} \phi^+\\ \phi^o \end{pmatrix}_3^d \end{pmatrix} \, ,$$
with the VEV?s
$$\Phi^u \rangle = \begin{pmatrix} \langle \Phi_1^u \rangle \\ \langle \Phi_2^u \rangle \\ \langle \Phi_3^u \rangle \end{pmatrix} \quad , \quad
\langle \Phi^d \rangle= \begin{pmatrix} \langle \Phi_1^d \rangle \\ \langle \Phi_2^d \rangle
\\ \langle \Phi_3^d \rangle \end{pmatrix} \;,$$
where
$$\Phi_i^u \rangle = \frac{1}{\sqrt[]{2}}
\begin{pmatrix} v_{ui} \\ 0 \end{pmatrix} \quad , \quad
\langle \Phi_i^d \rangle = \frac{1}{\sqrt[]{2}}
\begin{pmatrix} 0 \\ v_{di} \end{pmatrix} \:.$$
The contributions from $\langle \Phi^u \rangle$ and $\langle \Phi^d \rangle$ yield the $W$ and $Z$ gauge boson masses and mixing with the $SU(3)$ gauge bosons
$$\begin{gathered}
\frac{g^2 }{4} \,(v_u^2+v_d^2)\,
W^{+} W^{-} + \frac{ (g^2 + {g^\prime}^2) }{8} \,(v_u^2+v_d^2)\,Z_o^2 \\
\\
+ \frac{1}{4} \sqrt{g^2 + {g^\prime}^2} \,g_H\,Z_o \,
\left[ \,(v_{1u}^2-v_{2u}^2 -v_{1d}^2+v_{2d}^2)\,Z_1 + (v_{1u}^2+v_{2u}^2 -2v_{3u}^2 -v_{1d}^2-v_{2d}^2+2v_{3d}^2)\,\frac{Z_2}{\sqrt{3}} \right. \\
\\
\left. + 2\,(v_{1u} v_{2u}-v_{1d} v_{2d})\,\frac{Y_1^+ + Y_1^-}{\sqrt{2}} + 2\,(v_{1u} v_{3u}-v_{1d} v_{3d})\,\frac{Y_2^+ + Y_2^-}{\sqrt{2}} + 2\,(v_{2u} v_{3u}-v_{2d} v_{3d})\,\frac{Y_3^+ + Y_3^-}{\sqrt{2}} \right] \\
\\
+ \frac{g_H^2}{4} \, \left\{\, \frac{1}{2} \,(v_{1u}^2+v_{2u}^2+v_{1d}^2+v_{2d}^2)\, Z_1^2 +
\frac{1}{2} \,(v_{1u}^2+v_{2u}^2+4 v_{3u}^2+v_{1d}^2+v_{2d}^2+4 v_{3d}^2)\, \frac{Z_2^2}{3} \right. \\
\\
+ (v_{1u}^2+v_{2u}^2+v_{1d}^2+v_{2d}^2)\, Y_1^+ Y_1^- + (v_{1u}^2+v_{3u}^2+v_{1d}^2+v_{3d}^2)\, Y_2^+ Y_2^- +(v_{2u}^2+v_{3u}^2+v_{2d}^2+v_{3d}^2) \,Y_3^+ Y_3^- \\
\\
+ (v_{1u}^2-v_{2u}^2 + v_{1d}^2-v_{2d}^2)\,Z_1 \, \frac{Z_2}{\sqrt{3}}
+ (v_{2u} v_{3u}+v_{2d} v_{3d})\,(Y_1^+ Y_2^- + Y_1^- Y_2^+) \\ \\
+ (v_{1u} v_{2u}+v_{1d} v_{2d})\,(Y_2^+ Y_3^- + Y_2^- Y_3^+) +(v_{1u} v_{3u}+v_{1d} v_{3d})\,(Y_1^+ Y_3^+ + Y_1^- Y_3^-) \\
\\
\left. + 2\,(v_{1u} v_{2u}+v_{1d} v_{2d})\, \frac{Z_2}{\sqrt{3}}\, \frac{Y_1^+ + Y_1^-}{\sqrt{2}} +
(v_{1u} v_{3u}+v_{1d} v_{3d})\, (Z_1 - \frac{Z_2}{\sqrt{3}} )\, \frac{Y_2^+ + Y_2^-}{\sqrt{2}} \right. \\ \left.
- (v_{2u} v_{3u}+v_{2d} v_{3d})\, (Z_1 + \frac{Z_2}{\sqrt{3}} )\, \frac{Y_3^+ + Y_3^-}{\sqrt{2}} \right\} \label{ewyimixcont} \end{gathered}$$
$v_u^2=v_{1u}^2+v_{2u}^2+v_{3u}^2$ , $v_d^2= v_{1d}^2+v_{2d}^2+v_{3d}^2$. Hence, if we define as usual $M_W=\frac{1}{2} g v$, we may write $ v=\sqrt{v_u^2+v_d^2 } \thickapprox
246$ GeV.
$$Y_j^1=\frac{Y_j^+ + Y_j^-}{\sqrt{2}} \quad , \quad Y_j^\pm=\frac{Y_j^1 \mp i Y_j^2}{\sqrt{2}}$$
*The mixing of $Z_o$ neutral gauge boson with the $SU(3)$ gauge bosons modify the couplings of the standard model Z boson with the ordinary quarks and leptons*
Fermion masses
===============
Dirac See-saw mechanisms
------------------------
Now we describe briefly the procedure to get the masses for fermions. The analysis is presented explicitly for the charged lepton sector, with a completely analogous procedure for the $u$ and $d$ quarks and Dirac neutrinos. With the fields of particles introduced in the model, we may write the gauge invariant Yukawa couplings, as
$$h\:\bar{\psi}_l^o \:\Phi^d \:E_R^o \;+\;
h_2 \:\bar{\psi}_e^o \:\eta_2 \:E_L^o \;+\; h_3 \:\bar{\psi}_e^o
\:\eta_3 \:E_L^o \;+\; M \:\bar{E}_L^o \:E_R^o \;+
h.c \label{DiracYC}$$
where $M$ is a free mass parameter because its mass term is gauge invariant and $h$, $h_2$ and $h_3$ are Yukawa coupling constants. When the involved scalar fields acquire VEV’s we get, in the gauge basis ${\psi^{o}_{L,R}}^T = ( e^{o} ,
\mu^{o} , \tau^{o}, E^o )_{L,R}$, the mass terms $\bar{\psi}^{o}_L
{\cal{M}}^o \psi^{o}_R + h.c $, where
$${\cal M}^o = \begin{pmatrix} 0 & 0 & 0 & h \:v_1\\ 0 & 0 & 0 & h \:v_2\\
0 & 0 & 0 & h \:v_3\\
0 & h_2 \Lambda_2 & h_3 \Lambda_3 & M
\end{pmatrix} \equiv \begin{pmatrix} 0 & 0 & 0 & a_1\\ 0 & 0 & 0 & a_2\\
0 & 0 & 0 & a_3\\ 0 & b_2 & b_3 & M
\end{pmatrix} \;. \label{tlmassmatrix}$$
Notice that ${\cal{M}}^o$ has the same structure of a See-saw mass matrix, here for Dirac fermion masses. So, we call ${\cal{M}}^o$ a [**“Dirac See-saw”**]{} mass matrix. ${\cal{M}}^o$ is diagonalized by applying a biunitary transformation $\psi^{o}_{L,R} = V^{o}_{L,R} \;\chi_{L,R}$. The orthogonal matrices $V^{o}_L$ and $V^{o}_R$ are obtained explicitly in Appendix A. From $V_L^o$ and $V_R^o$, and using the relationships defined there, one computes
$$\begin{aligned}
{V^{o}_L}^T {\cal{M}}^{o} \;V^{o}_R =Diag(0,0,-
\lambda_3,\lambda_4) \label{tleigenvalues}\\
\nonumber \\
{V^{o}_L}^T {\cal{M}}^{o} {{\cal{M}}^{o}}^T \;V^{o}_L = {V^{o}_R}^T
{{\cal{M}}^{o}}^T {\cal{M}}^{o} \;V^{o}_R =
Diag(0,0,\lambda_3^2,\lambda_4^2) \:.\label{tlLReigenvalues}\end{aligned}$$
where $\lambda_3^2$ and $\lambda_4^2$ are the nonzero eigenvalues defined in Eqs.(\[nonzerotleigenvalues\]-\[paramtleigenvalues\]), $\lambda_4$ being the fourth heavy fermion mass, and $\lambda_3$ of the order of the top, bottom and tau mass for u, d and e fermions, respectively. We see from Eqs.(\[tleigenvalues\],\[tlLReigenvalues\]) that at tree level the See-saw mechanism yields two massless eigenvalues associated to the light fermions.
One loop contribution to fermion masses
=======================================
Subsequently, the masses for the light fermions arise through one loop radiative corrections. After the breakdown of the electroweak symmetry we can construct the generic one loop mass diagram of Fig. 1. Internal fermion line in this diagram represent the Dirac see-saw mechanism implemented by the couplings in Eq.(\[DiracYC\]). The vertices read from the $SU(3)$ flavor symmetry interaction Lagrangian
$$\begin{gathered}
i {\cal{L}}_{int} = \frac{g_{H}}{2}
\left( \bar{e^{o}}
\gamma_{\mu} e^{o}- \bar{\mu^{o}} \gamma_{\mu} \mu^{o} \right) Z_1^\mu
+ \frac{g_{H}}{2 \sqrt{3}} \left( \bar{e^{o}} \gamma_{\mu} e^{o}+ \bar{\mu^{o}}
\gamma_{\mu} \mu^{o} - 2 \bar{\tau^{o}}
\gamma_{\mu} \tau^{o} \right) Z_2^\mu \\
+ \frac{g_{H}}{\sqrt{2}} \left( \bar{e^{o}} \gamma_{\mu} \mu^{o} Y_1^{+} +
\bar{e^{o}} \gamma_{\mu} \tau^{o} Y_2^{+} + \bar{\mu^{o}} \gamma_{\mu} \tau^{o} Y_3^{+} + h.c.
\right) \:,\label{SU3lagrangian} \end{gathered}$$
{width=".7\textwidth"}
where $g_H$ is the $SU(3)$ coupling constant, $Z_1$, $Z_2$ and $Y_i^j\;,i=1,2,3\;,j=1,2,$ are the eight gauge bosons. The crosses in the internal fermion line mean tree level mixing, and the mass $M$ generated by the Yukawa couplings in Eq.(\[DiracYC\]) after the scalar fields get VEV’s. The one loop diagram of Fig. 1 gives the generic contribution to the mass term $m_{ij}
\:{\bar{e}}_{iL}^o e_{jR}^o$
$$c_Y \frac{\alpha_H}{\pi} \sum_{k=3,4} m_k^o
\:(V_L^o)_{ik}(V_R^o)_{jk} f(M_Y, m_k^o) \qquad , \qquad \alpha_H
\equiv \frac{g_H^2}{4 \pi}$$
where $M_Y$ is the gauge boson mass, $c_Y$ is a factor coupling constant, Eq.(\[SU3lagrangian\]), $m_3^o=-\sqrt{\lambda_3^2}$ and $m_4^o=\lambda_4$ are the See-saw mass eigenvalues, Eq.(\[tleigenvalues\]), and $f(x,y)=\frac{x^2}{x^2-y^2}
\ln{\frac{x^2}{y^2}}$. Using the results of Appendix A, we compute
$$\sum_{k=3,4} m_k^o \:(V_L^o)_{ik}(V_R^o)_{jk} f(M_Y,
m_k^o)= \frac{a_i \:b_j \:M}{\lambda_4^2 - \lambda_3^2}\:F(M_Y) \:,$$
$i=1,2,3$ , $j=2,3$, and $F(M_Y)\equiv
\frac{M_Y^2}{M_Y^2 - \lambda_4^2} \ln{\frac{M_Y^2}{\lambda_4^2}} -
\frac{M_Y^2}{M_Y^2 - \lambda_3^2} \ln{\frac{M_Y^2}{\lambda_3^2}}$. Adding up all the one loop $SU(3)$ gauge boson contributions, we get the mass terms $\bar{\psi^{o}_L} {\cal{M}}_1^o \:\psi^{o}_R + h.c.$,
$${\cal{M}}_1^o = \left( \begin{array}{ccrc} D_{11} & D_{12} & D_{13} & 0\\
0 & D_{22} & D_{23} & 0\\ 0 & D_{32} & D_{33} & 0\\
0 & 0 & 0 & 0
\end{array} \right) \:\frac{\alpha_H}{\pi}\; ,$$
$$\begin{aligned}
D_{11}&=&\frac{1}{2} ( \mu_{22} F_1+\mu_{33} F_2 ) \\
D_{12}&=&\mu_{12} (- \frac{F_{Z_1}}{4}+\frac{F_{Z_2}}{12}) \\
D_{13}&=&- \mu_{13} ( \frac{F_{Z_2}}{6}+ F_m ) \\
D_{22}&=&\mu_{22} (\frac{F_{Z_1}}{4}+\frac{F_{Z_2}}{12} - F_m )+\frac{1}{2} \mu_{33} F_3 \\
D_{23}&=&- \mu_{23} ( \frac{F_{Z_2}}{6} - F_m ) \\
D_{32}&=&- \mu_{32} ( \frac{F_{Z_2}}{6} - F_m ) \\
D_{33}&=& \mu_{33} \frac{F_{Z_2}}{3}+\frac{1}{2} \mu_{22} F_3 \:, \end{aligned}$$
$$F_1 \equiv F(M_{Y_1}) \quad,\quad F_2 \equiv F(M_{Y_2}) \quad,\quad F_3 \equiv F(M_{Y_3})$$
$$M_{Y_1}^2=M_2^2 \quad,\quad M_{Y_2}^2=M_3^2 \quad,\quad M_{Y_3}^2=M_2^2+M_3^2 \,$$
$$F_m=\frac{\cos\phi \sin\phi}{2 \sqrt{3}}\, [\, F(M_-)-F(M_+)\,]$$
with $M_2, M_3, M_-$ and $M_+$ the boson masses defined in Eqs.(\[M23\]-\[Mp\]).
Due to the $Z_1 - Z_2$ mixing, we diagonalize the propagators involving $Z_1$ and $Z_2$ gauge bosons according to Eq.(\[z1z2mixing\])
$$Z_1 = \cos\phi \;Z_- - \sin\phi \;Z_+ \quad , \quad Z_2 = \sin\phi \;Z_- + \cos\phi \;Z_+$$
$$\begin{aligned}
\langle Z_1 Z_1 \rangle &=& \cos^2\phi\; \langle Z_- Z_- \rangle + \sin^2\phi\; \langle Z_+ Z_+ \rangle \\\\
\langle Z_2 Z_2 \rangle &=& \sin^2\phi\; \langle Z_- Z_- \rangle + \cos^2\phi\; \langle Z_+ Z_+ \rangle \\\\
\langle Z_1 Z_2 \rangle &=& \sin\phi \, \cos\phi \;( \langle Z_- Z_- \rangle - \langle Z_+ Z_+ \rangle )\end{aligned}$$
So, in the one loop diagram contributions:
$$F_{Z_1}=\cos^2\phi \,F(M_-) + \sin^2\phi \,F(M_+) \qquad , \qquad F_{Z_2}=\sin^2\phi \,F(M_-) + \cos^2\phi \,F(M_+) \, ,$$
$$\mu_{ij}=\frac{a_i \:b_j \:M}{\lambda_4^2 - \lambda_3^2} = \frac{a_i
\:b_j}{a \:b} \:\lambda_3\:c_{\alpha} \:c_{\beta} \:,$$
and $c_{\alpha} \equiv \cos\alpha \:,\;c_{\beta} \equiv \cos\beta \:,\;
s_{\alpha} \equiv \sin\alpha \:,\;s_{\beta} \equiv \sin\beta$, as defined in the Appendix, Eq.(\[Seesawmixing\]). Therefore, up to one loop corrections we obtain the fermion masses
$$\bar{\psi}^{o}_L {\cal{M}}^{o} \:\psi^{o}_R + \bar{\psi^{o}_L}
{\cal{M}}_1^o \:\psi^{o}_R = \bar{\chi_L} \:{\cal{M}}
\:\chi_R \:,$$
with ${\cal{M}} \equiv \left[ Diag(0,0,-\lambda_3,\lambda_4)+ {V_L^o}^T {\cal{M}}_1^o\:V_R^o \right]$. Using $V_L^o$, $V_R^o$ from Eqs.(\[VoL\]-\[VoR\]) we get the mass matrix
$${\cal{M}}= \begin{pmatrix}
m_{11}&m_{12}&c_\beta \:m_{13}&s_\beta \:m_{13} \\
\\
m_{21}& m_{22} & c_\beta \:m_{23} & s_\beta \:m_{23}\\
\\
c_\alpha \:m_{31}& c_\alpha \:m_{32} & (-\lambda_3+c_\alpha c_\beta
\:m_{33}) & c_\alpha s_\beta \:m_{33} \\
\\
s_\alpha \:m_{31}& s_\alpha \:m_{32} & s_\alpha c_\beta \:m_{33} &
(\lambda_4+s_\alpha s_\beta \:m_{33})
\end{pmatrix} \;,\label{massVI}$$
where
$$\begin{aligned}
m_{11}=\frac{1}{2} \frac{a_2}{a^\prime} \Pi_1 \quad ,& \quad m_{12}= - \frac{1}{2} \frac{a_1 b_3}{a^\prime b} ( \Pi_2 -6 \mu_{22} F_m ) \\
\nonumber \\
m_{21}= \frac{1}{2} \frac{a_1 a_3}{a^\prime a}\Pi_1 \quad ,& \quad
m_{31}=\frac{1}{2} \frac{a_1}{a} \Pi_1 \end{aligned}$$
$$m_{13}=- \frac{1}{2} \frac{a_1 b_2}{a^\prime b} [\Pi_2 +2(2\frac{b_3^2}{b_2^2}-1)
\mu_{22}F_m ]$$
$$m_{22}=\frac{1}{2} \frac{a_3 b_3}{a \, b} \left[\frac{a_2}{a^\prime} ( \Pi_2 -6 \mu_{22} F_m )+
\frac{a^\prime b_2}{a_3 b_3} ( \Pi_3 + \Delta ) \right]$$
[$$m_{23}=\frac{1}{2} \frac{a_3 b_3}{a \, b} \left[\frac{a_2 b_2}{a^\prime b_3} ( \Pi_2 +2(2\frac{b_3^2}{b_2^2}-1 ) \mu_{22} F_m ) - \frac{a^\prime}{a_3} ( \Pi_3 -\frac{b_2^2}{b_3^2} \Delta +2\frac{b^2}{b_3^2}\mu_{33} F_m ) \right]$$ ]{}
$$m_{32}=\frac{1}{2} \frac{a_3 b_3}{a \, b} \left[\frac{a_2}{a_3} ( \Pi_2 -6 \mu_{22} F_m)-\frac{b_2}{b_3} ( \Pi_3 -\frac{{a^\prime}^2 }{a_3^2} \Delta -2\frac{a^2}{a_3^2}\mu_{33} F_m ) \right]$$
$$m_{33}=\frac{1}{2} \frac{a_3 b_3}{a \, b} \left[\frac{a_2 b_2}{a_3 b_3} ( \Pi_2 - 2 \mu_{22} F_m ) + \Pi_3+ \frac{ {a^\prime}^2 b_2^2}{a_3^2 b_3^2} \Delta - \frac{1}{3} \frac{a^2 b^2}{a_3^2 b_3^2}\mu_{33} F_{Z_2}
+ 2 ( \frac{b_2^2}{b_3^2} + 2\frac{a_2^2}{a_3^2}-\frac{{a^\prime}^2}{a_3^2} )\mu_{33} F_m \right]$$
$$\begin{aligned}
\Pi_1 = \mu_{22} F_1 + \mu_{33} F_2 \quad ,& \quad \Pi_2 = \mu_{22} F_{Z_1} + \mu_{33} F_3 \nonumber \\
\nonumber \\
\Pi_3 = \mu_{22} F_3 + \mu_{33} F_{Z_2} \quad ,& \quad \Delta = \frac{1}{2}\mu_{33}(F_{Z_2} - F_{Z_1} )\end{aligned}$$
*Notice that the $m_{ij}$ mass terms depend just on the $\frac{a_i}{a_j}$ and $\frac{b_i}{b_j}$ ratios of the tree level parameters.*
$$a^\prime=\sqrt{a_1^2+a_2^2}\;\; , \;\;a=\sqrt{{a^\prime}^2+a_3^2} \;\; ,
\;\; b=\sqrt{{b_2^2+b_3^2}} \;,$$
The diagonalization of ${\cal{M}}$, Eq.(\[massVI\]) gives the physical masses for u, d, and e charged fermions. Using a new biunitary transformation $\chi_{L,R}=V_{L,R}^{(1)} \;\Psi_{L,R}$; $\bar{\chi}_L \;{\cal{M}} \;\chi_R= \bar{\Psi}_L \:{V_L^{(1)}}^T
{\cal{M}} \; V_R^{(1)} \:\Psi_R $, with ${\Psi_{L,R}}^T = ( f_1 ,
f_2 , f_3 , F )_{L,R}$ the mass eigenfields, that is
$${V^{(1)}_L}^T {\cal{M}} \:{\cal M}^T \;V^{(1)}_L =
{V^{(1)}_R}^T {\cal M}^T \:{\cal{M}} \;V^{(1)}_R =
Diag(m_1^2,m_2^2,m_3^2,M_F^2) \:,$$
$m_1^2=m_e^2$, $m_2^2=m_\mu^2$, $m_3^2=m_\tau^2$ and $M_F^2=M_E^2$ for charged leptons.
Quark $( V_{CKM} )_{4\times 4}$ mixing matrix
----------------------------------------------
Within this $SU(3)$ family symmetry model, the transformations from massless to physical mass fermion eigenfields for quarks and charged leptons are
$$\psi_L^o = V_L^{o} \:V^{(1)}_L \:\Psi_L \qquad \mbox{and}
\qquad \psi_R^o = V_R^{o} \:V^{(1)}_R \:\Psi_R \,.$$
Recall that vector like quarks, Eq.(\[vectorquarks\]), are $SU(2)_L$ weak singlets, and hence they do not couple to the $W$ boson in the interaction basis. In this way, the interaction of L-handed up and down quarks; ${f_{uL}^o}^T=(u^o,c^o,t^o)_L$ and ${f_{dL}^o}^T=(d^o,s^o,b^o)_L$, to the $W$ charged gauge boson may be written as
$$\frac{g}{\sqrt{2}} \,\bar{f^o}_{u L} \gamma_\mu f_{d L}^o
{W^+}^\mu = \frac{g}{\sqrt{2}} \,\bar{\Psi}_{u L}\;
[(V_{u L}^o\,V_{u L}^{(1)})_{3\times 4}]^T \;(V_{d L}^o\,V_{d L}^{(1)})_{3\times 4}\;
\gamma_\mu \Psi_{d L} \;{W^+}^\mu \:,$$
where $g$ is the $SU(2)_L$ gauge coupling. Therefore, the non-unitary $V_{CKM}$ of dimension $4\times4$ is identified as
$$(V_{CKM})_{4\times 4} = [(V_{u L}^o\,V_{u L}^{(1)})_{3\times 4}]^T \;(V_{d L}^o\,V_{d L}^{(1)})_{3\times 4}$$
Numerical results {#numerical}
=================
*To illustrate the spectrum of masses and mixing, let us consider the following fit of space parameters at the $M_Z$ scale [@xingzhang]*
Taking the input values
$$M_2 = 2\,\text{TeV} \quad , \quad M_3 = 2000\,\text{TeV} \quad , \quad \frac{\alpha_H}{\pi}=0.2$$
for the $M_2$, $M_3$ horizontal boson masses, Eq.(\[M23\]), and the $SU(3)$ coupling constant, respectively, and the ratio of the electroweak VEV’s: $v_{iu}$ from $\Phi^u\;$ ($v_{id}$ from $\Phi^d$)
$$v_{1u}=0 \quad , \quad \frac{ v_{2u}}{ v_{3u}} = 0.1 \quad , \quad
\frac{ v_{1d}}{ v_{2d}} = 0.23257 \quad , \quad \frac{ v_{2d}}{ v_{3d}}=0.08373 \: ,$$
we obtain the following mass and mixing matrices, and mass eigenvalues:
Quark masses and mixing
-----------------------
[**u-quarks:**]{}
Tree level see-saw mass matrix:
$${\cal M}_u^o=
\left(
\begin{array}{cccc}
0 & 0 & 0 & 0. \\
0 & 0 & 0 & 29834. \\
0 & 0 & 0 & 298340. \\
0 & 1.49495\times 10^7 & -730572. & 1.58511\times 10^7 \\
\end{array}
\right)
\,\text{MeV} \,,$$
the mass matrix up to one loop corrections:
$${\cal M}_u=
\left(
\begin{array}{cccc}
1.38 & 0. & 0. & 0. \\
0. & -532.587 & -2587.14 & -2442.42 \\
0. & 7064.64 & -172017. & 31927.1 \\
0. & 70.6499 & 338.204 & 2.18023\times 10^7 \\
\end{array}
\right)\,\text{MeV} \, ,$$
and the u-quark masses
$$(\,m_u \;,\; m_c \;,\; m_t \;,\; M_U\,)=
(\,1.38\;,\; 638.22 \;,\;172181\;,\;2.18023\times 10^7\,)\,\text{MeV}$$
[**d-quarks:**]{}
$${\cal M}_d^o=
\left(
\begin{array}{cccc}
0 & 0 & 0 & 13375.7 \\
0 & 0 & 0 & 57510.3 \\
0 & 0 & 0 & 686796. \\
0 & 723708. & -37338.1 & 6.89219\times 10^7 \\
\end{array}
\right)\;\text{MeV}$$
$${\cal M}_d=
\left(
\begin{array}{cccc}
2.82461 & 0.0338487 & -0.656039 & -0.00689715 \\
0.65453 & -25.1814 & -217.369 & -2.28527 \\
0.0562685 & 423.166 & -2820.62 & 46.5371 \\
0.000562713 & 4.23187 & 44.2671 & 6.89291\times 10^7 \\
\end{array}
\right)
\;\text{MeV}$$
$$(\,m_d \;,\; m_s \;,\; m_b \;,\; M_D\,)=
(\, 2.82368 \;,\; 57.0005\;,\; 2860 \;,\; 6.89291\times 10^7 \,)\;\text{MeV}$$
and the quark mixing
$$V_{CKM}=
\left(
\begin{array}{cccc}
0.97362 & 0.225277 & -0.0362485 & 0.000194044 \\
-0.226684 & 0.973105 & -0.040988 & -0.000310055 \\
0.0260403 & 0.0481125 & 0.998387 & -0.00999333 \\
-0.000234396 &- 0.000826552 & -0.011432 & 0.000114632 \\
\end{array}\right)
\label{vckm}$$
Charged leptons:
----------------
$${\cal M}_e^o=
\left(
\begin{array}{cccc}
0 & 0 & 0 & 37956.9 \\
0 & 0 & 0 & 189784. \\
0 & 0 & 0 & 1.93543\times 10^6 \\
0 & 548257. & -30307.4 & 1.94497\times 10^8 \\
\end{array}\right)\;\text{MeV}$$
$${\cal M}_e=
\left(
\begin{array}{cccc}
-0.486368 & -0.00536888 & 0.0971221 & 0.000274163 \\
-0.0967909 & -34.7536 & -250.305 & -0.706579 \\
-0.0096786 & 485.768 & -1661.27 & 10.8107 \\
-0.0000967909 & 4.85792 & 38.2989 & 1.94507\times 10^8 \\
\end{array}
\right)\;\text{MeV}$$
fit the charged lepton masses:
$$( m_e \,,\, m_\mu \,,\, m_\tau \,,\, M_E ) = ( 0.486095 \,,\,102.7\,,\,1746.17\,,\, 3.15956\times 10^8\, )\,\text{MeV}$$
and the charged lepton mixing
$$V_{e \,L}^o\, V_{e \,L}^{(1)}=
\left(
\begin{array}{cccc}
0.973942 & 0.221206 & 0.050052 & 0.000194 \\
-0.226798 & 0.949931 & 0.214927 & 0.0008342 \\
-2.90427\times 10^{-6} & -0.220675 & 0.975296 & 0.009963 \\
2.62189\times 10^{-7} & 0.0013632 & -0.009906& 0.99995 \\
\end{array}
\right)
\label{emix}$$
Conclusions
===========
We reported recent numerical analysis on charged fermion masses and mixing within a BSM with a local $SU(3)$ family symmetry, which combines tree level “Dirac See-saw” mechanisms and radiative corrections to implement a successful hierarchical mass generation mechanism for quarks and charged leptons.
In section \[numerical\] we show a parameter space region where this scenario account for the known hierarchical spectrum of ordinary quarks and charged lepton masses, and the quark mixing in a non-unitary $(V_{CKM})_{4\times 4}$ within allowed values[^2] reported in PDG 2014 [@PDG2014].
*Let me point out here that the solutions for fermion masses and mixing reported in section \[numerical\] suggest that the dominant contribution to EWSB comes from the weak doublets which couple to the third family.*
*It is also worth to comment that fermion content, scalar fields, and their transformation under the gauge group, Eq. , all together, forbid tree level Yukawa couplings between ordinary standard model fermions. Consequently, the flavon scalar fields introduced to break the symmetries: $\Phi^u$, $\Phi^d$, $\eta_2$ and $\eta_3$, couple only ordinary fermions to their corresponding vector like fermion at tree level. Thus, FCNC scalar couplings to ordinary fermions are suppressed by light-heavy mixing angles, which as is shown in the quark mixing $(V_{CKM})_{4 \times 4}$, Eq.(\[vckm\]), and the charged lepton mixing, Eq. , may be small enough to suppress properly the FCNC mediated by the scalar fields within this scenario.*
Acknowledgements {#acknowledgements .unnumbered}
================
It is a pleasure to thank the organizers N.S. Mankoc-Borstnik, H.B. Nielsen, M. Y. Khlopov, and participants for the stimulating Workshop at Bled, Slovenia. This work was partially supported by the “Instituto Politécnico Nacional”, (Grants from EDI and COFAA) and “Sistema Nacional de Investigadores” (SNI) in Mexico.
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Diagonalization of the generic Dirac See-saw mass matrix
========================================================
$${\cal M}^o=
\begin{pmatrix} 0 & 0 & 0 & a_1\\ 0 & 0 & 0 & a_2\\ 0 & 0 & 0 &
a_3\\ 0 & b_2 & b_3 & c \end{pmatrix}$$
Using the biunitary transformations $\psi^{o}_L = V_L^o \:\chi_L$ and $\psi^{o}_R = V_R^o
\:\chi_R $ to diagonalize ${\cal{M}}^o$, the orthogonal matrices $V^{o}_L$ and $V^{o}_R$ may be written explicitly as
$$V^{o}_L = \begin{pmatrix} \frac{a_2}{a^\prime}& \frac{a_1 a_3}{a^\prime
a} & \frac{a_1}{a} \cos\alpha &
\frac{a_1}{a} \sin\alpha\\
\\
- \frac{a_1}{a^\prime} & \frac{a_2 a_3}{a^\prime a} &
\frac{a_2}{a} \cos\alpha &
\frac{a_2}{a} \sin\alpha\\
\\
0 & - \frac{a^\prime}{a} & \frac{a_3}{a} \cos{\alpha}
& \frac{a_3}{a} \sin{\alpha}\\
\\
0 & 0 & -\sin{\alpha} & \cos{\alpha}
\end{pmatrix} \label{VoL}$$
$$V^{o}_R = \begin{pmatrix}
1 & 0 & 0 & 0 \\
\\
0 & \frac{b_3}{b} & \frac{b_2}{b} \cos{\beta} & \frac{b_2}{b} \sin{\beta}\\
\\
0& - \frac{b_2}{b} & \frac{b_3}{b} \cos{\beta} & \frac{b_3}{b} \sin{\beta}\\
\\
0 & 0 & -\sin{\beta} & \cos{\beta}
\end{pmatrix} \label{VoR}$$
where
$$\lambda_3^2 = \frac{1}{2} \left( B - \sqrt{B^2 -4D} \right) \quad , \quad \lambda_4^2 = \frac{1}{2} \left( B + \sqrt{B^2 -4D} \right)
\label{nonzerotleigenvalues}$$
are the nonzero eigenvalues of ${\cal{M}}^{o} {{\cal{M}}^{o}}^T$ (${{\cal{M}}^{o}}^T
{\cal{M}}^{o}$), and
$$\begin{aligned}
B = a^2 + b^2 + c^2 =
\lambda_3^2+\lambda_4^2\quad &, \quad D= a^2
b^2=\lambda_3^2\lambda_4^2 \;,\label{paramtleigenvalues} \end{aligned}$$
$$\cos{\alpha} =\sqrt{\frac{\lambda_4^2 -
a^2}{\lambda_4^2 - \lambda_3^2}} \quad , \quad \sin{\alpha} =
\sqrt{\frac{a^2 - \lambda_3^2}{\lambda_4^2 - \lambda_3^2}} \quad , \quad \cos{\beta} =\sqrt{\frac{\lambda_4^2 - b^2}{\lambda_4^2 - \lambda_3^2}} \quad , \quad \sin{\beta} = \sqrt{\frac{b^2 -
\lambda_3^2}{\lambda_4^2 - \lambda_3^2}} \label{Seesawmixing}$$
[^1]: See [@albinosu32004; @albinosu3bled] and references therein for some other $SU(3)$ family symmetry model proposals.
[^2]: except $(V_{CKM})_{13}$ and $(V_{CKM})_{31}$
|
---
abstract: 'Galaxy disks are characterised by star formation histories that vary systematically along the Hubble sequence. We study global star formation, incorporating supernova feedback, gas accretion and enriched outflows in disks modelled by a multiphase interstellar medium in a fixed gravitational potential. The star formation histories, gas distributions and chemical evolution can be explained in a simple sequence of models which are primarily regulated by the cold gas accretion history.'
author:
- |
Marios Kampakoglou and Joseph Silk\
Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, United Kingdom (e-mail: mariosk,[email protected])\
title: Disk Galaxy Evolution Along the Hubble Sequence
---
Introduction
============
The two most striking characteristics that define the Hubble sequence are morphology and star formation activity. The former is best addressed by numerical simulations [@abadi03; @sharma05; @robertson04; @governato06]. However the latter aspects are so complex that most discussions of disk star formation and chemical evolution are based on analytical calculations [@efstathiou00; @silk01; @ferreras01; @silk03; @matteucci06; @naab06]. This paper extends the analytic approach to study star formation histories that vary systematically along the Hubble sequence. We study global star formation, incorporating supernova feedback, gas accretion and enriched outflows in disks modelled by a multi-phase interstellar medium in a fixed gravitational potential.
One of the current problems afflicting galaxy formation models is the role of gas infall. For both ellipticals [@bower06; @croton06] and massive disks at $z\sim 2$ [@forster06], infall rates are sufficiently high from CDM theory (ellipticals and disks) and observations (disks) that infall must be quenched, otherwise distant ellipticals are too blue and the disks are too massive by the current epoch. The problems may be related via feedback from AGN, but the details are poorly understood with regard to the resulting star formation history, gas infall rates and chemical evolution. In this paper, we focus on disk galaxies, and develop a phenomenological description of disk evolution in which the onset and duration of the gas infall history are found to be the controlling parameters. The star formation histories, gas distributions and chemical evolution can be explained in a simple sequence of models which are primarily regulated by the cold gas accretion history.
One of the main features of galaxies that characterises the significance of the Hubble sequence is the wide range in young stellar content and star formation activity. This variation in stellar content is part of the basis of the Hubble classification itself, and understanding its physical nature and origins is fundamental to understanding galaxy evolution in its broader context. In this paper, we construct a sequence of evolutionary models in which present-day properties of different types of disk galaxies are reproduced.
The first implementation of supernova-driven feedback in the context of CDM was to account for the properties of dwarf galaxies [@dekel86], and subsequent studies have explored the role of supernova feedback in more massive galaxies. Feedback is an important element in our attempts to model galaxy evolution. Energy injection from supernovae is probably the most plausible feedback mechanism for systems with virial temperatures higher than $10^{5}$ K. Winds from quasars might also disrupt galaxy formation or limit the growth of central black holes. Here, we will be concerned exclusively with supernova-driven feedback and we will not consider feedback from an active nucleus. The model presented in this paper is similar to the simple self-regulating model with inflow and outflow developed in , the main difference being in the infall model that we implement.We adopt an exponentially decreasing infall rate normalised in order to reproduce the observed total disk mass density in the solar neighbourhood, whereas in Efstathiou’s model it is the conservation of specific angular momentum that specifies the final radius in the disc for each gas element. Angular momentum conservation is known to be a poor approximation, at least in the numerical simulations, and our more phenomenological model is easily adapted to the chemical evolution constraints.
The main result of this paper is that star formation histories of the different types of disk galaxies can be reproduced using a one parameter model. The key parameter of our model is the time corresponding to the onset of the infall. Using this free parameter, we reproduce the distribution of disk birthrate parameters $b$, the ratio of the current SFR to the average past SFR, for each type of disk galaxy as presented in and summarised in Figure \[fig:FitGauussian\]. We then explore the various implications of the model for the radial and temporal dependences of the gas fraction, star formation rate and metallicity.
The layout of this paper is as follows. Section \[sec:Model\] briefly reviews the main points of our model and presents some basic results. The model is extended in Section \[sec:Refinements\] to include an improved treatment of winds and chemical evolution. Finally, in Section \[sec:discuss\] we discuss our results and present our conclusions.
Model {#sec:Model}
=====
The model described here is a self-regulating model with inflow and outflow. The disk is considered to be a system of independent rings each of $35{\times}{r_{d}}$ pc wide (where $r_{d}$ is in units of kpc)[^1]. Neither radial inflows nor radial outflows are considered. The ring centred at the galactocentric distance $r_{\odot}=~8.5$ kpc is labelled as the solar neighbourhood. For the present-day total disk surface density in the solar neighbourhood, we adopt a value ${\Sigma}_{tot}(r_{\odot},t_{g})=~60$ M$_{\odot}$pc$^{-2}$ ( found $56{\pm}6$ M$_{\odot}$pc$^{-2}$ for their disk model).
Dark Halo and Disk Model
------------------------
The dark halo is assumed to be described by the profile,
$$\rho(r)=~{\frac{{\delta}{\rho}_{c}}{(Cx)(1+Cx)^{2}}}~,$$
where $x{\equiv}r/r_{v}$, ${\rho}_{c}$ is the critical density, $r_v$ is the virial radius at which the halo has mean overdensity of 200 with respect to the background and $C$ is the concentration parameter. For the present model, we adopt a value for $C$ of 10. The circular speed corresponding to this profile is $${v_{H}}^{2}(r)=~v_{v}^{2}{\frac{1}{x}}{\frac{[ln(1+Cx)-Cx/(1+Cx)]}{[ln(1+C)-C/(1+C)]}}~,$$ where $v_{v}^{2}{\equiv}{\frac{GM_{v}}{r_{v}}}$, and $M_{v}$ is the mass of the halo within the virial radius.
For the surface density of the disk at $t=~0$, we assume an exponential profile: $${\Sigma}_{tot}(r,0)=~{\Sigma}_{0}e^{\frac{-r}{r_{d}}},$$ where $M_{D}=~2{\pi}r_{d}^{2}{\Sigma}_{0}$, ${\Sigma}_{tot}(r,0)$ is the total surface density of the ring centred at galactocentric radius $r$ at t= 0, $M_{D}$ is the total disk mass at t= 0 and $r_{d}$ is the scale length. So in our model the entire gas disk has formed instantaneously at $t=~0$. The rotation curve of a cold exponential disk is given by [@freeman70]: $${v_{D}}^{2}(r)=~2v_{c}^{2}y^{2}[I_{0}(y){K}_{0}(y)-I_{1}(y)K_{1}(y)]$$ where $y{\equiv}{\frac{1}{2}}{\frac{r}{r_{d}}}$ and $v_{c}^{2}{\equiv}\frac{GM_{D}}{r_{d}}$. For ratio $\frac{v_{v}}{v_{c}}$ we adopt a value equal to 0.45 that corresponds to the Milky Way. Models using standard disk formation theory with adiabatic contraction within the cuspy halo [@klypin02] reproduce the broad range of observational data available for the Milky Way by adopting a virial mass equal to M$_{v}~{\approx}~10^{12}$M$_{\odot}$, baryonic mass equal to M$_{bar}~{\approx}~4-6{\times}10^{10}$M$_{\odot}$ and virial radius r$_{v}=~258$ kpc. These values in combination with an adopted value for the disk scale length equal to r$_{d}=~3$ kpc [@sackett97] give $\frac{v_{v}}{v_{c}}=~0.45$. In this work, we keep the ratio $\frac{v_{v}}{v_{c}}$ constant for all the models we present.[^2]. The ratio of virial radius of the halo to disk scale length is defined as the collapse factor $f_{coll}{\equiv}{\frac{r_{v}}{r_{d}}}=~50$; such a value is needed to reproduce a median value of $\approx 0.05$ for the dimensionless spin parameter of the halo $\lambda_{H}$ [@efstathiou00]. The disk is truncated at radius $r/r_{d}\approx7.$
### Two-component(stellar and gas) rotating disk
The stellar radial velocity dispersion ${\sigma}_{\star}$ is related to that of the gas clouds: $${\sigma}_{\star}=~{\alpha}{\sigma}_{g},$$ Here we assume ${\alpha}=~5$ [@efstathiou00]. The scaleheight for a two-component rotating disk is [@talbot75]: $$H=~\frac{{{\sigma}_{g}}^{2}}{{\pi}G{\Sigma}_{g}}\frac{1}{(1+\frac{{\Sigma}_{\star}/{\Sigma}_{g}}{{\sigma}_{\star}/{\sigma}_{g}})}$$
Epicyclic frequency Model
-------------------------
The epicyclic frequency is given by the expression $${\kappa}=~2\omega(1+{\frac{1}{2}}{\frac{r}{\omega}}{\frac{d{\omega}}{dr}})^{1/2},$$ in which $\omega$ is the angular velocity of the disk. In the above equation we replace $\omega=~v_{tot}/r$ where $v_{tot}^{2}=~v_{H}^{2}+v_{D}^{2}$. So the epicyclic frequency(in units of $10^{-15}$sec$^{-1}$) is: $${\kappa}=~0.035\sqrt{2}{\frac{v_{tot}}{r}}(1+{\frac{r}{v_{tot}}}{\frac{dv_{tot}}{dr}})^{1/2}$$
Star Formation Rate(SFR)
------------------------
In this work we adopt the star formation law proposed in , to which we refer for a detailed description. The star formation rate (in units of M$_{\odot}$pc$^{-2}$Gyr$^{-1}$) is given by : $${\psi}(r,t)=~\frac{{\epsilon}{\kappa}{\Sigma}_{g}(1-Q^{2})^{1/2}}{Q}$$ where ${\kappa}$ is the epicyclic frequency, ${\Sigma}_{g}$ is the gas surface density in units of M$_{\odot}$pc$^{-2}$, $Q$ is the gravitational instability parameter[@toomre64] and $\epsilon$ is the efficiency of star formation. For the model in this paper we set $\epsilon=~0.02$.
The star formation rate defined above has the following features: (i) star formation can only occur if the disk is gravitationally unstable, $Q<1$; (ii) the degree of instability of the disk, measured by $Q$, is directly linked to the star-formation rate; the smaller $Q$ is, the more rapidly stars are formed.
The stability criterion for a two-component (stellar and gas) rotating disk can be written as [@wang94]: $$Q=~\frac{\kappa}{{\pi}G}(\frac{{\Sigma}_{g}}{{\sigma}_{g}}+\frac{{\Sigma}_{\star}}{{\sigma}_{\star}})^{-1}$$ where ${\Sigma}_{\star}$ and ${\sigma}_{\star}$ are the stellar surface density (in units of M$_{\odot}$pc$^{-2}$) and the radial velocity dispersion (in units of $\rm km\,sec^{-1}$), and ${\kappa}$ is the epicyclic frequency.
IMF
---
The adopted stellar initial mass function is of the standard Salpeter form : $$\frac{dN{\star}}{dm}=~Bm^{-(1+x)},~m_{l}<m<m_{u},~x=~1.35$$ where $m_{l}=~0.1$ M${_{\odot}}$,$m_{u}=~50$M${_{\odot}}$.
For the IMF adopted, one supernova is formed for every $125M{_{\odot}}$ of star formation, assuming that each star of mass greater than 8$M_{\odot}$ releases $10^{51}E_{51}$ ergs in kinetic energy in a supernova explosion. Therefore, the energy injection rate per unit surface area (in units of erg s$^{-1}$pc$^{-2}$) is given by:
$${\dot{E}}_{sn}^{\Omega}=~2.5{\times}10^{32}E_{51}{\epsilon_{c}}{\psi}$$
where the parameter $\epsilon_{c}$ defines the percentage of supernovae energy that goes to the ambient medium. For this model we adopt $\epsilon_{c}=~0.03$.
Energy dissipation due to cloud inelastic collisions
----------------------------------------------------
The rate of energy loss per unit surface area (in units of ergs$^{-1}$pc$^{-2}$) is given by [@efstathiou00]: $${\dot{E}}_{coll}^{\Omega}=~5.{\times}10^{29}(1+\frac{{\Sigma}_{\star}/{\Sigma}_{g}}{{\sigma}_{\star}/{\sigma}_{g}}){\sigma}_{5g}{\Sigma}_{5g}^{3}.$$ Using the above equations we know the energy balance in each ring of the disk. We define the radial velocity dispersion of clouds (in units of $\rm km\,s^{-1}$) as: $${\sigma}_{g}(r,t)=~\sqrt{\frac{2{\times}E(r,t)}{M_{g}(r,t)}}$$ where ${E}(r,t)$ is the energy balance at galactocentric radius $r$ and time $t$, and $M_{g}(r,t)$ is the mass of the cold gas at galactocentric radius $r$ and time $t$.
In summary, the energy balance in each ring of the disk defines the cloud and stellar radial velocity dispersion ${\sigma_{g}}, {\sigma}_{\star}$ through equation (14) and equation (5). Combining this piece of information with the cloud and stellar surface densities ${\Sigma_{g}}, {\Sigma}_{\star},$ we calculate the degree of instability at galactocentric radius $r$ through equation(10). The star formation rate is regulated by the degree of instability, so the final form of the star formation rate is derived by inserting equation (10) into equation (9).
Infall Model {#sec:Infall Model}
------------
The adopted form for the gas infall rate is an exponentially decreasing function in which the rate of gas infall (in units of M$_{\odot}$pc$^{-2}$Gyr$^{-1}$) in each ring is expressed as:
$$f(r,t)=~A(r)e^{-{\frac{t}{{\tau}_{f}}}}$$
where ${\tau}_{f}$ (in units of Gyr) is the infall time scale. The infall rate $f(r,t)$ is normalised to the present-day local disk density, ${{\int}_{t_{low}}^{t_{g}}}f(r,t)dt=~{\Sigma}_{tot}(r,t_{g})$, where ${\Sigma}_{tot}(r,t_{g})$ is the present-day total disk surface density of the ring centred at galactocentric radius $r$, and $t_{g}$ is the age of the galactic disk ($t_{g}=~10$ Gyr). Assuming that the present-day total masses of the different rings exponentially decrease with increasing galactocentric distance with scale-length $r_{d}$, the form of $A(r)$ can be written as [@chang02]:
$$A(r)=~\frac{{\Sigma}_{tot}(r{_{\odot}},t_{g})-{\Sigma}_{tot}(r{_{\odot}},0)}{{{\int}_{t_{low}}^{t_{g}}}e^{-{\frac{t}
{{\tau}_{f}}}} dt}e^{\frac{-(r-r{_{\odot}})}{r_{d}}}$$
where ${\Sigma}_{tot}(r{_{\odot}},0)$ is the total disk surface density in the solar neighbourhood at the epoch which the formation of the disk begins, and ${\Sigma}_{tot}(r{_{\odot}},t_{g})$ is the present-day total disk surface density in the solar neighbourhood. We adopt an exponentially decreasing infall rate normalised in order to reproduce the observed total mass density in the solar neighbourhood. In our infall model, we have introduced a parameter $t_{low}$ that corresponds to the time that infall switches on. Adjusting this parameter we can reproduce the distribution of disk birthrate parameter [*b*]{}, the ratio of the current SFR to the average past SFR, for each type of galaxy presented in . The infall time scale ${\tau}_{f}$ (in units of Gyr) is also a free parameter of the model but we keep it constant, adopting a value of 0.5 Gyr.
According to the adopted infall form, since ${\tau}_{f}$ does not vary with radius, the disk keeps the exponential form with scale length $r_{d}$ at each epoch of evolutionary time.
Hot Phase Model
---------------
An expanding supernova remnant will evaporate a mass of cold gas. The rate of the evaporated mass per unit area is (Efstathiou 2000):
$${\dot{M}}_{ev}^{\Omega}{\approx}1{\times}10^{-7}{\frac{{\sigma}_{5g}^{2}}{{\Sigma}_{5g}(1+\frac{{\Sigma}_{\star}/{\Sigma}_{g}}{{\sigma}_{\star}/{\sigma}_{g}})}}{\times}S_{13}^{0.71}{\gamma}^{0.29}E_{51}^{0.71}f_{\Sigma}^{-0.29}$$
in units of M$_{\odot}$pc$^{-2}$yr$^{-1}$, where ${\Sigma}_{5g}$ is the gas surface density in units of 5 M$_{\odot}$pc$^{-2}$, ${\sigma}_{5g}$ is the cloud radial velocity dispersion in units of 5 kms$^{-1}$ and $S_{13}$ is the supernovae rate in units of $10^{-13}$pc$^{-3}$yr$^{-1}$. The parameter ${\gamma}$ relates the blast wave velocity to the isothermal sound speed (v$_{b}=~{\gamma}c_{h}$, ${\gamma}~{\approx}~2.5$). The parameter $f_{\Sigma}$ is defined through the evaporation parameter (in units of pc$^{2}$): $${\Sigma}^{ev}=~280\frac{ {{\sigma}_{5g}}^{2} } {{{\Sigma}_{5g}}^{2}
}\frac{1}{(1+\frac{ \frac{{\Sigma}_{\star}}{{\Sigma}_{g}}}
{\frac{{\sigma}_{\star}}{{\sigma}_{g}}})}\frac{1}{{\phi}_{\kappa}}=~f_{\Sigma}{{\Sigma}^{ev}}_{\odot}$$ where ${{\Sigma}^{ev}}_{\odot}$ is the evaporation parameter in the local solar neighbourhood and ${\phi}_{\kappa}$ is a parameter that quantifies the effectiveness of classical thermal conductivity. For this model we adopt ${{\Sigma}^{ev}}_{\odot}=~95$ pc$^{2}$ and ${\phi}_{\kappa}=~0.1$. For a system with porosity Q close to unity, the temperature and the age of a supernovae remnant are given by:
$$\begin{aligned}
t_{o}=~5.5{\times}10^{6}{S_{13}}^{-5/11}{\gamma}^{-6/11}E_{51}^{-3/11}n_{h}^{3/11}
{\rm yr}, \\
T_{o}=~1.2{\times}10^{4}S_{13}^{6/11}{\gamma}^{-6/11}E_{51}^{8/11}n_{h}^{-8/11}
{\rm K},\end{aligned}$$
The density of the ambient hot phase is $$n_{h}=~4.3{\times}10^{-3}S_{13}^{0.36}{\gamma}^{-0.36}E_{51}^{0.61}f_{\Sigma}^{-0.393}
{\rm cm^{-3}}.$$ For $10^{5}{\leq}T{\leq}10^{6} K$ we adopt a cooling rate of ${\Lambda}
\approx 2.5{\times}10^{-22}$T$_{5}^{-1.4}$ ergcm$^{3}$s$^{-1}$. So for a gas with primordial composition, the cooling time $t_{cool}$ is $$t_{cool}=~2t_{o}T_{5}^{2.4}f_{\Sigma}^{0.5}$$
Outflow Model {#sec:Outflow Model}
-------------
As mentioned in the previous section, supernova bubbles expand, evaporate cold gas and compress the ambient interstellar medium(ISM). The compressed ISM will also be driven by a form of wind. In order to study the effects of outflow, we assume a simple phenomenological model for galactic winds where the wind mass-loss rate ${\dot{M}}_{W}$ (in units of M$_{\odot}$pc$^{-2}$Gyr$^{-1}$) is assumed to be proportional to the star formation rate. $${\dot{M}}_{W}=~k{\psi}~.$$ We explore many values of k between 0 and 1. The hot gas that escapes from the halo is removed permanently. Superwinds indeed are found around Milky Way-type galaxies [@str07]. This is puzzling from the theoretical perspective because supernovae are considered to be incapable of driving a strong wind from the gravitational potential well of a massive galaxy. Also, at least one example is known of an extended x-ray halo around a normal massive spiral galaxy [@ped06]. In this case, gravitational accretion and shock heating provides the most plausible source for heating the gas.
Galactic Fountain Model
-----------------------
Models of gas flow on the galactic scale were first introduced by and subsequently developed by and others. The galactic fountain originates from the supernovae that warms up the disk gas to temperatures of $10^{6}$K. The upflowing gas cools and condenses into neutral hydrogen clouds that rains onto the disk. The models assume the height to which the hot gas will rise and the expected rate of condensation in the cooling gas depends only on the temperature of the gas at the base of the fountain and the rate of cooling of the upflowing gas. In the model described in this paper the hot gas that is not lost from the disk returns to the disk at the radius from which it was expelled after a time $t_{cool}$.
Chemical Evolution {#sec:Chemical Evolution}
------------------
We include chemical evolution in the model using the instantaneous recycling approximation. So we assume that all processes involving stellar evolution, nucleosynthesis and recycling take place instantaneously on the time scale of galactic evolution. The equation of galactic chemical evolution is $${\Sigma}_{g}dZ=~p{\psi}dt+(Z_{F}-Z)f$$ where f is the infall rate (in units of M$_{\odot}$pc$^{-2}$Gyr$^{-1}$) and p is the yield [@pagel97]. We adopt a yield of $p=~0.02$ and assume that the mass accreted to the disk has zero metallicity ($Z_{F}=~0$).We normalise the metallicities to the solar value for which we adopt $Z_{\odot}=~0.02$.
Different Types of Spiral Galaxies
----------------------------------
To explore the evolution of different types of spiral galaxies we adopt five models in this paper (See Table \[Table.1\]). These models differ only in the mass of the disk and the disk scale length $r_{d}$. All the input parameters that these models have in common are summarized in Table \[Table.Input\].
Input parameters
------------------------------------------- ------------------------------------------------------------
Solar radius $r_{\odot}=~8.5$ kpc
Total disk surface density at $r_{\odot}$ ${\Sigma}_{tot}(r_{\odot},t_{g})=~60$ M$_{\odot}$pc$^{-2}$
Star Formation IMF Salpeter
Solar metallicity $Z_{\odot}=~0.02$
Effective yield $p=~0.02$
Star formation efficiency $\epsilon=~0.02$
Infall timescale $\tau_{f}=~0.5$ Gyr
Galactic disk age $t_{g}=~10$ Gyr
Thermal Conductivity effectiveness $\phi_{\kappa}=~0.1$
Solar evaporation parameter ${{\Sigma}^{ev}}_{\odot}=~95$ pc$^{2}$
Disk to Halo velocity ratio $\frac{v_{v}}{v_{c}}=~0.45$
Collapse factor $f_{coll}=~50$
Wind Model Parameter $k=~0.2$
: Input parameters in common for the five models we examine in this paper.[]{data-label="Table.Input"}
$Type$ Disk Mass $M_{D}$($M_{\odot}$) $r_{d}(kpc)$
-------- -------------------------------- --------------
Sa $5.5\times10^{10}$ 3.
Sb $4.\times10^{10}$ 2.7
Sc $2.7\times10^{10}$ 2.37
Sd $1.5\times10^{10}$ 1.95
Sm $1.\times10^{10}$ 1.7
: Five models adopted in this paper[]{data-label="Table.1"}
Basic Results
-------------
In this section will present the basic results of the model discussed in the previous sections. Unless otherwise stated, we adopt a value 0.2 for the parameter k in the outflow model. For clarification of the plots, when results from models Sc and Sd are similar we define a new type of disk galaxy Sc/d. The data for Sc/d type come from the average between the Sc model data set and the Sd model data set.
### Distribution of disk birth rate b
Values of the birth parameter [*b*]{} were calculated for each type of spiral galaxy. For the calculation of [*b*]{} values we ignore the SFR that corresponds to the first $10^{8}$yr, due to a deficiency in our model. More specifically in the model we assume that the entire gas disk has formed instantaneously at $t=~0$: this is unrealistic and leads to very high rates of star formation for the first $10^{8}$yr. Figure \[fig:birthrate\] shows the disk birthrate parameter [*b*]{} subdivided by galaxy type as a function of our free parameter $t_{low}$ that corresponds to the time that infall switches on. The dotted line indicates the median value of birthrate parameter while the dashed lines show the quartiles of birthrate parameter presented in . In Table \[Table.2\], we present the value of the free parameter $t_{low}$ we adopt in order to reproduce the median value of birthrate parameter for each type of disk galaxy.
$Type$ b(median) $t_{low_{med}}(Gyr)$
-------- ----------- ----------------------
Sa $ 0.07$ 0.10
Sb $ 0.37$ 7.00
Sc $ 1.04$ 8.05
Sd $ 0.63$ 7.80
Sm $ 1.31$ 8.35
: Median value of [*b*]{} and $t_{low}$ value for different models.[]{data-label="Table.2"}
To make our results more useful for observers, we fit Gaussians to the [*b*]{} parameter dispersion for each type of disk galaxy (Figure \[fig:FitGauussian\]). The full width at half maximum for each different Gaussian equals the width of the $t_{low}$ parameter distribution in order to reproduce the quartiles of the [*b*]{} parameter distribution presented in . In Table \[Table.3\], we give the value of ${\sigma}$ for the Gaussian fits as well as the width in $t_{low}$ in order to reproduce the quartiles of the [*b*]{} parameter distribution. The width of the [*b*]{} parameter distrbution between the quartiles for each type of disk galaxy is presented in the last column.
$Type$ ${\sigma}$ ${\Delta}t_{low}(Gyr)$ ${\Delta}b$
-------- ------------ ------------------------ -------------
Sa $ 1.716$ 4.04 0.1
Sb $ 0.386$ 0.91 0.3
Sc $ 0.205$ 0.48 0.6
Sd $ 0.183$ 0.43 0.4
Sm $ 0.270$ 0.64 2.0
: The value of ${\sigma}$ for the Gaussian fits (second column), the width in $t_{low}$ in order to reproduce the quartiles of the [*b*]{} parameter distribution (third column), the width of the [*b*]{} parameter distribution between the quartiles of the distributions (fourth column). The values are shown for different types of disk galaxies.[]{data-label="Table.3"}
### Density Profiles
Figure \[fig:Gas\_Density\_Prof\] and figure \[fig:Stars\_Density\_Prof\] show the evolution of the gas and stellar surface densities, respectively, (in units of M$_{\odot}$pc$^{-2}$) for different ages and different type of disk galaxies. The results are shown for ages of 0, 0.1, 1, 3, 6 (top - bottom) and 10 Gyr ([*dashed*]{} line). The star formation rates are initially high and hence the time scale for star formation is short. Almost the total amount of gas in the central region of disk is transformed to stars in $10^{8}$yr. The star formation rate declines rapidly after 1 Gyr. As figure \[fig:Gas\_Density\_Prof\] and figure \[fig:Stars\_Density\_Prof\] show the star formation at early times is concentrated to the inner parts of the disk which have a high surface density, and hence the gas distribution develops a surface density profile with an inner ’hole’, similar to what is seen in the HI distributions in real galaxies [@deul90].
Model Sa has parameters similar to those of the Milky Way. Gas surface densities from direct observations at the solar neighbourhood is $8{\pm}5$ M$_{\odot}$/pc$^{2}$ [@dame93] in good agreement with the predictions of the model. Slightly higher values of ${\approx}~13-14$ M$_{\odot}$/pc$^{2}$ have been reported by and are in good agreement with Sb model. Taking into account the scatter in the measurements of Milky Way scalelength ($r_{d}\approx2.5-3.5$kpc ) Sb model can also be a good candidate for the Milky Way.
Direct observations of the stars at the solar neighbourhood of Milky Way by and using HST observations of M stars in the Galactic disk determine a surface density of $12.2-14.3$ M$_{\odot}$/pc$^{2}$. Visible stars other than M stars contribute ${\approx}~15$ M$_{\odot}$/pc$^{2}$, resulting in a total stellar surface density in the range of ${\approx}~27-30$ M$_{\odot}$/pc$^{2}$. This value is in good agreement with predictions from Sa and Sb models.
### Gas Velocity Profiles
Figure \[fig:Velocity\_Prof\] shows the radial distribution of the gas velocity at 10 Gyr for different types of disk galaxies, Sa ([*solid*]{} line), Sb ([*thick solid*]{} line), Sc ([*dashed*]{} line), Sd ([*thick dashed*]{} line) and Sm ([*dotted*]{} line). Gas velocity profiles are in good agreement with recent theoretical predictions [@ricotti02] that in a multiphase low metallicity (Z${\approx}5\times10^{-3}Z_{\odot}$) interstellar medium the velocity probability distribution should be close to a Maxwellian with velocity dispersion ${\sigma}{\ge}11$km/sec$^{-1}$. Furthermore studies of nearby face-on galaxies show that the velocity dispersion in the HI layer is decreasing monotonically from about $10-13$ kms$^{-1}$ in the optically bright inner regions to $6-8$ kms$^{-1}$ in the very outer parts [@kamphuis93].
### Star formation Profiles
Figure \[fig:SFR\_Prof\] shows the evolution of the radial distribution of the star formation rate for different ages and different types of disk galaxies. The results are shown for ages of 0.1 ([*solid*]{} line), 1 ([*solid thick*]{} line), 3 ([*dashed*]{} line), 6 ([*dashed thick*]{} line) and 10 Gyr ([*dotted*]{} line).
### Metallicity Profiles
Figure \[fig:Metallicity\_Prof\] shows the evolution of the radial distribution of gas metallicity for different ages and different type of disk galaxies. The results are shown for ages of 0.1 ([*solid*]{} line) and 10 Gyr ([*dotted*]{} line).
Note the very high value of metallicity for Sa type galaxies today. The model predicts that metal-rich winds are needed especially for Sa type galaxies in order to produce reasonable values for metallicity today. This prediction is in agreement with suggestions presented in (for more detailed comments see Section \[sec:Improve Chemical Evolution\])[^3].
### Infall Rate Profiles
The evolution of the radial distribution of the infall rate (in units of M$_{\odot}$pc$^{-2}$Gyr$^{-1}$) is shown in Figure \[fig:Infall\_Prof\]. The results are shown for different ages. As already mentioned the time when the infall switches on is defined by the free parameter $t_{low}$. For the results presented in figure 5, we adopt for $t_{low}$ a value that reproduces the median value of birth parameter [*b*]{} for each type of disk galaxy. For Sa galaxies the results are shown for 3 ([*solid*]{} line) and 6 Gyr ([*dashed*]{} line). Infall rates for Sb, Sc, Sd, and Sm galaxies are presented for ages of 8.5 ([*solid*]{} line) and 10 Gyr ([*dashed*]{} line). Note the different range in the y axis for different types of disk galaxies.
### Disk Global Properties
Figure \[fig:Global\_Prop\] shows the net star formation rates (upper panel) and the evolution of gas fraction and total disk mass, stars plus gas (middle panels), for different types of disk galaxies respectively. Note the very high values for the infall rate for the Sb, Sc, Sd and Sm galaxies. Since these galaxies have smaller mass and the infall is switched on much later in comparison with the Sa type, we need high infall rates in order to reproduce the observed total mass density in the solar neighbourhood.
Star formation rate histories
-----------------------------
In this section, we present results for the star formation rate history at galactocentric radius $r=~1 \rm kpc$ for different types of disk galaxies.
### Galactocentric radius $r=~1$ kpc
In figure \[fig:Global\_Prop\], we show the star formation rate history at galactocentric radius$ $r equal to $r=~1$ kpc (bottom panel) for different types of disk galaxies. Note, that galaxy types which are characterised by large infall rates (Sb, Sc, Sd, Sm) show a peak at the onset of infall. At time $t=~t_{low}$, when the infall switches on, the accreted gas increases the local star formation rate by a bigger factor in comparison with the increase in the mean star formation rate across the disk and causes the peak in Figure \[fig:Global\_Prop\]. This phenomenon is much stronger at smaller radii because of the inner ’hole’ that has developed in the surface density of the gas at late times (see figure \[fig:Gas\_Density\_Prof\]). At late times, the star formation in the inner parts of the disk is stopped due to lack of gas, but the fresh gas comes in at $t=~t_{low}$ to refuel the central regions and increase the local star formation by a big factor. For Sa galaxies, the infall rate is very small so the curve in figure \[fig:Global\_Prop\] is smooth.
Refinements of the model {#sec:Refinements}
========================
The model described in the previous section contain a number of simplifications, which we will attempt to refine in this section. We introduce some simple improvements to the outflow model (Section \[sec:Improve Outflow Model\]) and to the chemical evolution model (Section \[sec:Improve Chemical Evolution\]).
Outflow model {#sec:Improve Outflow Model}
-------------
In Section \[sec:Outflow Model\] we assumed that the [*k*]{} parameter is constant. Instead of keeping the [*k*]{} parameter in the outflow model constant for the different types of disk galaxies, we can assume that [*k*]{} is proportional to the ratio of bulge-to-disk circular velocities. This is a reasonable assumption especially if we believe that AGN feedback contributes to the wind mechanism. For systems with constant density, the bulge-to-disk velocity ratio can be reduced to the following expression: $$\frac{v_{b}}{v_{d}}{\propto}({\frac{r_{e}}{r_{d}}})^{1/a}
\label{eq:outflowprop}$$ where $r_{e}$ is the effective radius of the bulge and $r_{d}$ is the the disk scale length. Following , we can use the following expression for the ${\frac{r_{e}}{r_{disk}}}$ ratio : $${\frac{r_{e}}{r_{d}}}= 0.20 - 0.013(T - 5)~,
\label{eq:outflowmodel}$$ where $T$ defines the galaxy type. The relationship between $T$ and galaxy type is shown in Table \[Table.T\]. Note that eq. \[eq:outflowmodel\] is valid only for galaxies with numerical type $T$ between $1$ and $7$. Initial investigation of equation \[eq:outflowmodel\], using $a=~1$ and $a=~2$, shows that our results are insensitive to this refinement of the model. We plan to investigate the implications of equation (\[eq:outflowmodel\]) in more detail in a future paper.
de Vaucouleurs Sa Sb Sc Sd Sm
---------------- ----- ----- ----- ----- -----
T $1$ $3$ $5$ $7$ $9$
: Definition of numerical type $T$[]{data-label="Table.T"}
Chemical evolution {#sec:Improve Chemical Evolution}
------------------
In Section \[sec:Chemical Evolution\], we assumed that gas ejected from the disk has the same metallicity as the ISM at the time that the gas was ejected. Due to this assumption, the predicted metallicty curve for Sa and Sb galaxies today is very high (see Figure \[fig:Metallicity\_Prof\], top panel). This problem does not appear in smaller galaxies (Sc, Sd, Sm) because the infall rate is higher than the rate of star formation, thus allowing the accreted metal-poor gas to dilute the ISM faster than it can be enriched by evolving stars. A comparison with observational data [@pilyugin04; @kewley05] leaves metal-rich outflows as the only viable mechanism for producing the low effective yields observed in gas-rich galaxies. Following , we parameterise the metallicity $Z_{SN}$ of the SN ejecta as a multiple $\eta$ of the nucleosynthetic yield. For a Salpeter IMF $\eta=~6.2-7.1$. For this paper, we adopt a value $\eta=~4.25$ for Sa type galaxies and a smaller value $\eta=~2$ for the other galaxy types. Furthermore, in this improved model for chemical evolution for the accreted gas, we adopt a metallicity of $Z_{F}=~0.1{\times}$Z$_{\odot}$. The equation of galactic chemical evolution is : $${\Sigma}_{g}dZ=~p{\psi}dt+(Z_{F}-Z)f-Z({\eta}-1){\dot{M}}_{W}$$
In figure \[fig:Improve\_ZModel\] we show the evolution of the radial distribution of gas metallicity for different ages and different type of disk galaxies. The results are shown for ages of 0.1 ([*solid*]{} line) and 10 Gyr ([*dotted*]{} line). Comparison between the model predictions and the observational data is shown in figure \[fig:int\_Z\]. The results come from (Table 2 in this paper). For the oxygen abundance in the solar neighbourhood, we adopt a value $12+0/H=~8.5$ [@pilyugin02]. The model predictions for global metallicities are calculated assuming that the disk is truncated at $r/r_{d}~{\approx}~5.$
Figure \[fig:Z\_gradient\] shows the radial metallicity distribution of the interstellar medium for each of our disk models after $10$ Gyr ([*dashed*]{} line). For the Sa model which is similar to the Milky Way the metallicity gradient is about dlog(Z)/dr= -0.055 dexkpc$^{-1}$ at the solar radius. Observational values for the metallicity gradient of light elements X in the Milky Way are in the range of $-0.04<$dlog(X/H)/dr$<-0.08$ dexkpc$^{-1}$ [@chiappini01]. Our models predict that the metallicity gradient at the solar radius of the disks depends on the type, and increases as we go from early spirals to later types in agreement with observations [@marquez02]. Our predictions for the metallicity gradient of later type disk galaxies are in reasonable agreement with observations [@vila92] although there is a considerable scatter in the published values.
Discussion {#sec:discuss}
==========
We have presented a model of global star formation incorporating supernova feedback, gas accretion and enriched outflows in disks modelled by a multiphase interstellar medium in a fixed gravitational potential.
A key prediction of this model is that star formation histories of different types of disk galaxies can be explained in a simple sequence of models which are primarily regulated by the cold gas accretion history. The distributions of disk birth parameters presented in are reproduced using the parameter $t_{low}$ which varies with the type of disk galaxy. Sa galaxies are characterised by quiescent evolution and a small value for the $t_{low}$ parameter whereas Sb, Sc, Sd and Sm galaxies are characterised by starbursts and relatively large values for the $t_{low}$ parameter.
A description of disk evolution in which the onset and duration of the gas infall history are found to be the controlling parameters is in qualitative agreement with standard ${\Lambda}CDM$ cosmology which predicts that protodisks reside in dark halos with masses $\sim10^{12}\rm M_\odot $ and are in a phase of strong gas accretion with values ${>}100$ M$_{\odot}$/yr [@burkert06].
The mass assembly of galaxies occurs through two main processes: hierarchical merging of smaller entities, and more diffuse gas accretion. The relative importance of the two processes cannot be easily found by cosmological simulations, since many physical parameters such as gas dissipation, star formation and feedback, are still unknown.
There are at least some twenty examples of galaxies which in HI show either signs of interactions and/or have small companions [@sancisi99]. This suggests that galaxies often are in an environment where material for accretion is available. Characteristic examples are the companions NGC 4565-4565A [@rupen91] and NGC 4027-4027A [@phookun92]. These companions have systematic velocities close to those of the main galaxy and HI masses less than 10[%]{} of the main galaxy. The HI picture suggests the capture of a gas rich dwarf by a massive system probably to be followed by tidal disruption and accretion of the dwarf. Such examples have been seen also in the Milky Way. The discovery of the Sagittarius dwarf galaxy [@ibata94] shows that accretion is still taking place at the present time.
The model presented here predicts that metal-rich winds are needed especially for Sa type galaxies in order to produce reasonable values for the metallicity today. This prediction is in agreement with suggestions presented in (for more detailed comments see Section \[sec:Improve Chemical Evolution\]).
The major shortcoming of the present model is the failure of the chemical enrichment model to reproduce the observed values for nuclear metallicities. Our model predicts very high values for metallicities close to the disk centre for Sa and Sb galaxies whereas the predicted nuclear metallicities for Sd and Sm galaxies are quite low. This is probably due to the high star formation rates that the model predicts in the first $10^{8}$yr, due to our implicit assumption that the entire gas disk has formed instantaneously at $t=~0$. The extremely high initial star formation rate result in highly enriched central regions, so enriched that the fresh low metallicity accreted gas cannot dilute the gas efficiently. This is the case especially in galaxies like Sa and Sb characterised by small infall rates. In galaxy types like Sd and Sm, the very high infall rates in the central regions (due to the functional form we assume for the infall model: see Sec \[sec:Infall Model\]), are enough to dilute metals and result in low values for nuclear metallicities. However the chemical evolution model is in good agreement with observations in terms of the integrated metallicities.
In a future paper, we will develop this model further. Along with the improvements mentioned above, we intend to examine the effect of AGN feedback and compare with numerical simulations.
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[^1]: For a disk galaxy with scale length $r_{d}=~3$kpc the width of each zone is equal to $35\times3=~105$ pc
[^2]: The ratio $\frac{v_{v}}{v_{c}}$ defines the disk scale length $r_{d}$. The disk scale length is $r_{d}=~\frac{(\frac{3M_{D}}{4{\pi}200{\rho}_{c}f_{coll}(\frac{v_{v}}{v_{c}})^{2}})^{1/3}}{f_{coll}}$ kpc
[^3]: Note that metallicity today does decline slightly close to the center. This is due to the radial profile we choose for the infall rate since we add most of the fresh low metallicity gas to the central regions.
|
---
abstract: 'Soft x-ray linear and circular dichroism (XLD, XMCD) experiments at the Ce M$_{4,5}$ edges are being used to determine the energy scales characterizing the Ce $4f$ degrees of freedom in the ultrathin ordered surface intermetallic CeAg$_x$/Ag(111). We find that all relevant interactions, i. e. Kondo scattering, crystal field splitting and magnetic exchange coupling occur on small scales. Our study demonstrates the usefulness of combining x-ray absorption experiments probing linear and circular dichroism owing to their strong sensitivity for anisotropies in both charge distribution and paramagnetic response, respectively.'
author:
- 'C. Praetorius'
- 'M. Zinner'
- 'P. Hansmann'
- 'M. W. Haverkort'
- 'K. Fauth'
title: 'Exploring small energy scales with x-ray absorption and dichroism '
---
Rare earth intermetallic compounds display a rich phenomenology of physical properties, encompassing very different kinds of ground states, such as magnetic order, unconventional superconductivity and paramagnetic heavy fermion liquids [@Gege08a; @Gege15a]. The interaction of localized $4f$ electrons with itinerant electronic degrees of freedom may result in the emergence of small characteristic energy scales which produce nontrivial macroscopic behavior at low temperature and complex phase diagrams with competing interactions and orders [@grew91a; @Lohn07a; @Gege08a; @Gege15a; @yang08a]. In a solid environment, the degeneracy of the rare earth $4f$ ground configuration is lifted by the crystal field in general, causing both an anisotropic $4f$ charge distribution and, in conjunction with spin orbit coupling, (single ion) magnetic anisotropy. Unraveling the crystal field induced level structure thus constitutes an essential part of understanding the low temperature physics and of establishing correlations between local $4f$ symmetry at low temperature on the one hand and macroscopic ground state properties on the other.
In this respect, the usefulness of probing the $4f$ configuration with linear polarized soft x-rays [@Sacc92a] has been demonstrated for a variety of Ce compounds in recent years [@Hans08a; @Will09a; @Will10a; @Will11a; @Will12a; @Will12b; @Strig12a; @Will15a], allowing to settle several open issues, where other experiments left room for diverging interpretations. Its magnetic variants, x-ray magnetic linear and circular dichroism (XMLD, XMCD) constitute sensitive element and orbital specific probes of magnetic polarization and anisotropy [@Schu87a; @Laan86b; @Chen95a; @stoe99a; @vanE97a; @vand08a]. XMCD was successfully utilized to reveal the presence of magnetic Kondo screening in CePt$_5$/Pt(111) [@Prae15a].
In the present letter we demonstrate that the combined use of linear and magnetic circular dichroism allows us to determine the crystal field structure without recourse to e. g. inelastic neutron scattering, as in some previous work [@Hans08a; @Will09a; @Will10a]. Our chosen example of an ultrathin Ce-Ag surface intermetallic furthermore highlights a threefold advantage of this approach. First, linear and circular dichroism experiments are frequently both feasible with the same installation and therefore can be performed *in situ* within a single experimental run. Second, the splittings turn out to be of the order of $1$ meV only, making their discrimination from quasi-elastic scattering a difficult task. Last but not least, the sample volume is so small that most alternative methods would face serious sensitivity challenges.
The formation of an ordered intermetallic phase upon depositing minute amounts of Ce onto Ag(111) held at elevated temperature has been reported before [@Schw12a]. In the preparation of our specimens we have adopted a similar procedure. In brief, clean Ag(111) was prepared by cycles of Ar$^+$ ion sputtering ($E_{\text{kin}}$: 1 keV) and subsequent annealing to $920$ K. The crystal was then held at $840$ K while a Ce dose of approx. $1 \times10^{15}$ atoms/cm$^2$ was deposited onto it from a thoroughly outgassed W crucible mounted in a commercial electron beam evaporator.
![\[fig1LEED\] LEED patterns recorded at an electron kinetic energy of $70$ eV from Ag(111) (left) and CeAg$_x$ (right). The satellite reflections in the latter indicate the formation of a $(2\sqrt{3} \times 2\sqrt{3})R30^\circ$ superstructure as in CePt$_5$/Pt(111) at comparable Ce dose [@Kemm14a; @Prae15a]. Ag(111) reflections merge with one of the satellite reflections around $\langle 2 \; \overline{1} \rangle_{\text{CeAg}_x}$ (indicated by arrows), yielding a $(\frac{10}{9} \sqrt{3} \times \frac{10}{9}\sqrt{3})R30^\circ$ relation between CeAg$_x$ and substrate unit cells, respectively. ](CeAgxFig1.png){width="8.5cm"}
Fig. \[fig1LEED\] displays a LEED pattern of a CeAg$_x$ specimen characteristic of this range of Ce coverage next to the one of pristine Ag(111), taken at the same electron kinetic energy. The diffraction pattern is very much reminiscent of our earlier observations for CePt$_5$/Pt(111) at similar Ce dose [@Kemm14a; @Prae15a]. It reveals a combination of two superstructures on two different length scales. The main diffraction features may be attributed to an intermetallic $(1.1\sqrt{3} \times 1.1\sqrt{3})R30^\circ$ surface reconstruction, whereas the satellites indicate the formation of a longer range surface corrugation of $(3\sqrt{3} \times 3\sqrt{3})R30^\circ$ character with respect to this intermetallic phase. This corresponds to a hexagonal surface corrugation in rotational alignment with the substrate lattice and a periodicity of approx. $15$ nm. As indicated by arrows in Fig. \[fig1LEED\] the Ag(111) first order diffraction beams superimpose with the outermost satellite reflections surrounding the $\langle 2\, \overline{1} \rangle$ spots of the surface intermetallic. Like in CePt$_5$ the satellite intensities strongly loose intensity as the initial Ce coverage is increased [@Schw12a; @Zinnerunpub]. Despite the obvious similarities between both systems, there are also some differences. We have so far been unable to determine the exact composition and structure of this ordered Ce-Ag phase which we therefore label as CeAg$_x$. A more extensive account of the properties of CeAg$_x$/Ag(111) and their dependence on Ce coverage shall be given in a separate publication [@Zinnerunpub].
For the purpose of this letter it is sufficient to recognize the formation of a hexagonal structure and we shall therefore analyze our results by assuming sixfold rotational symmetry about the Ce sites. It is a fundamental property of hexagonal crystal fields (CF) to split the atomic Ce $4f^1$ configuration ($j=5/2$) into three Kramers doublets of pure $m_j$ character. Unlike in cubic or tetragonal symmetry [@Hans08a; @Will12a; @Will12b] the CF is therefore fully specified by $|m_j\rangle$ level splittings and ordering. We denote the CF splittings as $\Delta_1 = E_{3/2} - E_{1/2}$ and $\Delta_2 = E_{5/2} - E_{1/2}$ and determine their magnitude and sign from linear and magnetic circular dichroism measurements at the soft x-ray Ce M$_{4,5}$ edges in what follows.
All soft x-ray absorption experiments were carried out at the PM3 bending magnet beamline of BESSY-II, Berlin, using circular polarization ($p\approx 0.93$) [@Kach15a]. X-ray absorption was measured in the total electron yield (TEY) mode and normalized by the TEY captured from a gold mesh. Although with reduced amplitude, linear dichroism can nevertheless be observed by variation of the x-ray angle of incidence $\theta_X$. The polarization averaged, so-called isotropic spectrum is well approximated by oblique incidence data taken at $\theta_X = 60^\circ$ in the present work. Its line shape is independent of the thermal occupation of the CF states, since it is identical for all $| m_j \rangle$ initial states. In contrast, spectra measured at normal incidence (NI, i.e. along the hexagonal symmetry axis) do exhibit temperature dependent line shapes, determined by the fractional occupation of the CF split $|m_j\rangle$ states. Introducing the Boltzmann weights $p_{1,2} = \exp(-\Delta_{1,2}/k_BT)$, the NI spectrum $I^{NI}(T)$ is given by $$\label{NIspecT}
%\begin{displaymath}
I^{NI}(T) = Z^{-1}\left ( I^{NI}_{|1/2\rangle} + p_1 I^{NI}_{|3/2\rangle} + p_2 I^{NI}_{|5/2\rangle} \right ) ,
%\end{displaymath}$$ where $Z=1 + p_1 + p_2$ is the partition function. Evidently, in the limit $k_BT \gg \Delta_1,\Delta_2$ the line shape observed at NI converges to the isotropic one. The two experimental spectra displayed in Fig. \[fig2XAS\]a) demonstrate that this condition is fulfilled in our CeAg$_x$/Ag(111) specimens. The NI spectrum ($\theta_X=0^\circ$) measured at $T=250$ K is hardly distinguishable from the isotropic spectrum.
![\[fig2XAS\] Selection of experimental and simulated Ce M$_{4,5}$ XA spectra for CeAg$_x$/Ag(111). a) Experimental low temperature ($T=15$ K) isotropic and high temperature ($T=250$ K) normal incidence spectrum b) Simulated isotropic spectrum and normal incidence spectra for pure $|m_j\rangle$ initial states c) Experimental normal incidence spectra at various temperatures d) Zoom-in on the main spectral features of both experimental and simulated normal incidence spectra for three temperatures (see text for details). In each panel, individual spectra were displaced along the ordinate for clarity. ](CeAgxFig2.png){width="8.5cm"}
In our analysis we make use of simulated absorption spectra to trace the experimental temperature dependences. These simulations were obtained from full atomic multiplet calculations as implemented in the Quanty Package [@Have12a; @quantylink]. To obtain the best match between calculated and experimental isotropic spectra, the $ff$ $(df)$ Slater Integrals were reduced by 42.5% (17.5%) from their respective Hartree-Fock values, well in accordance with previous work [@Hans08a; @Will10a; @Will12a; @Stri13a; @Will15a]. In addition, the Ce $3d$ core hole spin orbit coupling constant $\zeta_{3d}$ was slightly readjusted to reproduce the experimental separation between the M$_4$ and M$_5$ edges. Theoretical line spectra were convoluted with a Gaussian (FWHM $0.2$ eV) representing the experimental energy resolution as well as with Lorentzian contributions to account for the lifetime of the core excited states. Since the spectral shape of the M$_4$ edge is affected by autoionization decay [@Thol85a], its lifetime broadening was calculated by convolution with a Fano profile ($q\approx 16$) [@Schi11a] rather than a Lorentzian.
The resulting Ce M$_{4,5}$ absorption spectra are displayed in Fig. \[fig2XAS\]b). While the overall agreement of the isotropic spectrum with measured data is very good we note that not all multiplet terms can simultaneously be made to coincide with the experimental features when applying universal scaling factors to the Slater Integrals. This is most apparent for the weak shoulder at $903$ eV, which is not discernible in the calculated spectra since its separation from the main M$_4$ peak (feature C) is too small. The remaining spectra in Fig. \[fig2XAS\]b) demonstrate the different spectral shapes owing to the anisotropic charge distribution in the $|m_j\rangle$ states. In particular, their most prominent peaks (A, B & C) feature considerable variations in their relative intensities. Experimental NI data acquired at $T \gtrsim 100$ K display variations which cannot be resolved on the scale of Fig. \[fig2XAS\]c). In comparison to the isotropic spectra, they nevertheless systematically exhibit a slightly larger ($\approx 1$-$2\%$) C/B peak intensity ratio. Referring to the $|m_j\rangle$ specific spectra of Fig. \[fig2XAS\]b) this observation immediately reveals that $|5/2\rangle$ must be an excited state. As the temperature is lowered, the C/B peak intensity ratio is further enhanced, but in addition the M$_5$ line shape now acquires a noticeable change in spectral appearance. The observed spectral variations restrict the parameter $\Delta_2$ to a relatively narrow energy window of $\Delta_2= 1.1\pm0.2$ meV. The determination of the other CF parameter ($\Delta_1$) on the basis of NI XAS data alone is less obvious. Scenarios with $- 1$ meV $\lesssim \Delta_2 \lesssim 5$ meV can be made to satisfactorily match the sequence of experimental spectra. This is largely owed to the smallness of the linear dichroism associated with the $|3/2\rangle$ fraction of the initial state. The choice of parameters $\Delta_1$ and $\Delta_2$ for the simulations in Fig. \[fig2XAS\]d) therefore already accounts for the information gained from considering the XMCD signal which we shall discuss next.
![\[fig3XMCD\] Selection of experimental and simulated XMCD spectra. a) low temperature XMCD obtained at $\theta_X=60^\circ$ (“isotropic” configuration) alongside with the calculated XMCD spectrum, scaled such as to match the magnitude of the experimental data. b) temperature dependent Ce M$_4$ XMCD in NI geometry c) same as in b) but for $\theta_X=60^\circ$. ](CeAgxFig3.png){width="8.5cm"}
The paramagnetic Ce $4f$ response was probed in an applied magnetic field of $\mu_0H=1.5$ T. While sufficiently small to warrant linear response, it causes an XMCD signal which can reliably be measured over a considerable temperature range. Fig. \[fig3XMCD\]a) displays the dichroic spectrum for the case of largest magnetic polarization obtained in the present work, alongside with the simulated XMCD spectrum. We notice that the spectral appearance of the XMCD is well accounted for by the atomic calculations, which were solely optimized to match the isotropic spectrum. A notable exception is once again the spectral feature at $903$ eV, which produces a small but distinct contribution to magnetic dichroism in the experiment, but is buried in the dichroism produced by the main M$_4$ peak in the calculation. As in our previous work [@Prae15a], we determine the Ce $4f$ polarization by applying the orbital moment XMCD sum rule [@Thol92a] and assuming the atomic relation $m_S = -m_L/4$ between spin and orbital contributions to the total $4f$ magnetic moment to hold. In the case of Fig. \[fig3XMCD\]a), the Ce $4f$ polarization amounts to approx. $0.13$ $\mu_B$/atom. The corresponding asymmetry in the XA spectra is largest at the M$_4$ edge and amounts to about $3.7$% of the TEY signal.
The temperature dependence of the magnetic response at normal and oblique incidence, respectively, is shown in Fig. \[fig3XMCD\]b) and c). Each dichroic spectrum is multiplied by the value of the temperature at which it was obtained. In this way, perfect Curie behavior would be reflected by a constant XMCD magnitude in the plots. At low temperature in particular, the occurrence of single ion magnetic anisotropy is obvious. Its sign and magnitude are directly related to the crystal field splitting scheme. XMCD data therefore provide an independent probe of the CF scheme within the same set of experiments.
![\[fig4ichi\] Inverse magnetic Ce $4f$ susceptibilities of CeAg$_x$, determined from XMCD measurements, along with simulations according to eq. \[fullmodel\]. CF parameters are the same as in Fig. \[fig2XAS\]d). ](CeAgxFig4.png){width="8.5cm"}
A more quantitative evaluation can be performed after extracting the temperature dependent Ce $4f$ susceptibilities from the XMCD data [@Prae15a]. To second order they are given [@Made68a; @Luek79a] by the following expressions for the magnetic field applied along ($\chi_{||}$) and perpendicular ($\chi_\perp$) to the hexagonal axis, respectively: $$\begin{aligned}
\chi^{}_{||} &= &\frac{g^2\mu_B^2}{4k_BTZ} \left ( 1 + 9p_1 + 25p_2 \right ) \\ %9 e^{ -\frac{\Delta_1}{ k_BT} } + 25e^{-\frac{\Delta_2} { k_BT} } \right ) \\
\chi^{}_{\perp} &= & \frac{g^2\mu_B^2}{4k_BTZ} \cdot \left ( 9 + \frac{16k_BT}{\Delta_1} +\right. \nonumber\\%\left ( \frac{10k_BT}{\Delta_2-\Delta_1}-\frac{16k_BT}{\Delta_1} \right ) p_1 - \right. \\ %e^{ -\frac{\Delta_1}{ k_BT} } \right. - \\
& &+\left ( \frac{10k_BT}{\Delta_2-\Delta_1}-\frac{16k_BT}{\Delta_1} \right ) p_1-\left. \frac{10k_BT}{\Delta_2-\Delta_1} p_2 \right ) % e^{-\frac{\Delta_2} { k_BT} } \right )\end{aligned}$$ At an intermediate angle $\theta$ the susceptibility reads $$\label{fullmodel}
\chi^{}_\theta = \frac{\cos^2\theta}{\chi_{||}^{-1}-\lambda}+\frac{\sin^2\theta}{\chi_{\perp}^{-1}-\lambda},$$ where we have additionally allowed for magnetic coupling between Ce sites at the mean field level ($\lambda$).
In Fig. \[fig4ichi\] we show the temperature dependence of the inverse Ce $4f$ susceptibilities. The crystal field splitting induced anisotropy leads to an offset between $\chi_{||}^{-1}$ and $\chi_\perp^{-1}$ which is nearly constant in the temperature range spanned by our experiments. It is notably well perceptible up to high temperatures, where the precise determination of linear dichroism in our experiment is quite challenging. The magnitude of this offset sensitively depends on the CF excitation energies $\Delta_1$ and $\Delta_2$. Nevertheless, the magnetic response in this temperature range is not sufficient to pinpoint the numerical values of $\Delta_1$, $\Delta_2$ and $\lambda$. In conjunction with the restrictions on $\Delta_2$ from above, however, we can determine the combination of parameters which produces the best simultaneous agreement with both linear and circular dichroism.
The outcome of the parameter optimization is displayed in both Fig. \[fig2XAS\]d) and Fig. \[fig4ichi\] for the thermal evolution of the NI XAS and the inverse susceptibilities, respectively. We obtain $\Delta_1 = 0 \pm 0.4$ meV, $\Delta_2 = 1.25 \pm 0.05$ meV and an insignificantly small mean field coupling $\lambda$. These are the parameter values used for the simulations displayed in Figs. \[fig2XAS\]d) and \[fig4ichi\].
The correct slopes in $\chi^{-1}(T)$ in Fig. \[fig4ichi\] are only obtained by allowing for an overall reduction of the Ce magnetic moment by about 13% compared to the value expected for free Ce$^{3+}$ ions, however. From numerical simulations within the simplified NCA scheme proposed by Zwicknagl et al. [@Zwic90a] we estimate that a Kondo temperature of $T_K \gtrsim 20$ K would be required to produce this reduction by Kondo screening. Such a high value for $T_K$ appears quite unlikely, though, considering the small $4f$ hybridization found in the XA spectra and the photoemission results by Schwab et al. [@Schw12a]. It is most likely, therefore, that the discrepancy is mostly due to an underestimation of the Ce $4f$ orbital moment in the sum rule evaluation.
In conclusion, we have presented a soft x-ray absorption study of an ultrathin, ordered intermetallic phase induced by alloying a sub-monolayer quantity of Ce into the surface of Ag(111). Exploring the temperature dependences of both, linear and circular x-ray dichroism, we show that it is possible to explore the energy scales which characterize this material. In addition to the smallness of $T_K$ [@Schw12a], we find that both CF splittings and magnetic exchange coupling occur on energy scales of about 1 meV and below. Our findings highlight both the enormous sensitivity of soft x-ray absorption and the usefulness of scheduling XMCD experiments when characterizing rare earth systems with soft x-rays. This extension comes at little cost, since many soft x-ray end stations provide the means to perform both linear and circular dichroism measurements.
We wish to thank H. Schwab and F. Reinert for helpful discussions as well as H. Kießling and B. Münzing for assistance with experiments. This work was funded by the Deutsche Forschungsgemeinschaft through FOR1162. Access to synchrotron radiation was partially granted by HZB. We also gratefully acknowledge HZB staff for their support during beam time.
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---
abstract: |
-1 Data-driven applications rely on the correctness of their data to function properly and effectively. Errors in data can be incredibly costly and disruptive, leading to loss of revenue, incorrect conclusions, and misguided policy decisions. While data cleaning tools can purge datasets of many errors before the data is used, applications and users interacting with the data can introduce new errors. Subsequent valid updates can obscure these errors and propagate them through the dataset causing more discrepancies. Even when some of these discrepancies are discovered, they are often corrected superficially, on a case-by-case basis, further obscuring the true underlying cause, and making detection of the remaining errors harder.
In this paper, we propose [QFix]{}, a framework that derives explanations and repairs for discrepancies in relational data, by analyzing the effect of queries that operated on the data and identifying potential mistakes in those queries. [QFix]{}is flexible, handling scenarios where only a subset of the true discrepancies is known, and robust to different types of update workloads. We make four important contributions: (a) we formalize the problem of diagnosing the causes of data errors based on the queries that operated on and introduced errors to a dataset; (b) we develop exact methods for deriving diagnoses and fixes for identified errors using state-of-the-art tools; (c) we present several optimization techniques that improve our basic approach without compromising accuracy, and (d) we leverage a tradeoff between accuracy and performance to scale diagnosis to large datasets and query logs, while achieving near-optimal results. We demonstrate the effectiveness of [QFix]{}through extensive evaluation over benchmark and synthetic data.
author:
- |
Xiaolan Wang\
\
\
\
Alexandra Meliou\
\
\
\
Eugene Wu\
\
\
\
title: '[[QFix]{}]{}: Diagnosing errors through query histories'
---
|
---
abstract: 'The remarkable properties and potential applications of Majorana fermions have led to considerable efforts in recent years to realize topological matters that host these excitations. For a number-conserving system, there have been a few proposals, using either coupled-chain models or multi-component system with spin-orbit coupling, to create number fluctuation of fermion pairs in achieving Majorana fermion. In this work, we show that Majorana edge states can occur in a spinless Fermi gas in 1D lattices with tunable $p$-wave interaction. This is facilitated by the conversion between a pair of (open-channel) fermions and a (close-channel) boson, thereby allowing the number fluctuation of fermion pairs in a single chain. This scheme requires neither spin-orbit coupling nor multi-chain setup and can be implemented easily. Using the density-matrix-renormalization-group method, we have identified the Majorana phase in a wide range of parameter regime as well as its associated phase transitions. The topological nature of the Majorana phase manifests itself in a strong edge-edge correlation in an open chain that is robust against disorder, as well as in a non-trivial winding number in the bulk generated by using twisted boundary condition. It is also shown that the Majorana phase in this system can be stable against atom losses due to few-body collisions on the same site, and can be easily identified from the fermion momentum distribution. These results pave the way for probing the intriguing Majorana physics in a simple and stable cold atoms system.'
author:
- 'X. Y. Yin'
- 'Tin-Lun Ho'
- Xiaoling Cui
title: 'Majorana edge state in a number-conserving Fermi gas with tunable $p$-wave interaction '
---
Introduction
============
Majorana fermions, discovered by Majorana in 1937 [@Majorana], has stimulated tremendous research interests over the past decades due to their novel exchange statistics and promise for topological quantum computation [@RMP1; @Alicea]. A Majorana fermion (or a “Majorana" for short) is an equal magnitude superposition of a fermion operator and its adjoint \[$\lambda^1 = f^{\dagger}+f$, or $\lambda^2 = i(f^{\dagger}-f) $\]. It is a mode of excitation rather than a particle in the usual sense. In 2001, Kitaev showed that spinless fermions in a 1D chain coupled to a pairing field will have Majorana fermions at the ends [@Kitaev]. Efforts to simulate this model in solid state matter have led to the proposal of using 1D semiconducting wires with spin-orbit coupling (SOC) in contact with a superconductor [@Oreg; @Lutchyn]. Similar proposals have also been made in the cold atom studies by engineering SOC on an attractive Fermi gas [@Jiang; @Liu; @Wei; @Qu; @Chen; @Wang].
Since Majorana fermions also emerge in the number conserving systems such as the Pfaffian quantum Hall state [@Moore; @Nayak] and Kitaev’s honeycomb spin model [@Kitaev2], there have been questions of whether proximity superconductivity (or lack of number conservation) is necessary for realizing Majaronas in 1D chains. While a single Majorana excitation can not exist in a number conserving system, the correlation of two Majoranas at different locations ($i\langle \lambda^{1}_{i}\lambda^{2}_{j}\rangle$) is well defined. This provides a natural generalization of the presence of Majorana edge modes in a number conserving system, which is defined as a non-zero correlation of the Majoranas at the opposite end of a finite chain, $i\langle \lambda^1_{0}\lambda^2_{N_L}\rangle \neq 0$, in exactly the same way the Majoranas are correlated in the Kitaev model [@Kitaev]. In Refs. [@MCheng; @Das_Sarma; @Fisher], the authors have studied coupled 1D chains with interchain pair hopping using bosonization methods and have concluded the existence of Majorana edge states in these number conserving systems.
Whether Majorana edge states can exist under number conservation is particularly relevant for their realization with cold atoms, as the latter are number conserving [@Zoller; @Diehl; @Buchler; @Altman; @Iemini]. A rigorous proof of their existence was established recently for coupled chain models with inter-chain pair hopping [@Zoller; @Diehl; @Buchler], and for a single chain four-component fermion system with SOC and spin-exchange interaction [@Iemini] which has the similar pair hopping physics as in other coupled chain models. All these studies suggest that the number fluctuation of fermion pairs in a single chain is the key to the emergence of Majoranas in number conserving systems.
In this paper, we propose a much simpler scheme for realizing Majorana edge states that makes use of alkali fermions in a single chain without SOC – by simply tuning a single component Fermi gas in a 1D chain to its $p$-wave resonance. The systematic derivation of this model is given in Ref. [@Cui]. In the two-channel description of $p$-wave resonance, two fermions at neighboring lattice sites can convert to a “close channel" boson in one of the two sites, thereby causing the number fluctuation of fermion pairs in the chain. The close channel bosons play the role of proximity superconductor in the Kitaev model, except that they are now quantum mechanical objects. In this model, neither the number of fermion $N_{f}$ nor the number of boson $N_b$ is conserved. However, the total number $N=2N_{b}+N_{f}$ is \[See Eq. (\[eq\_ham\_full\])\]. An effective fermion-molecule conversion model was also proposed previously using the laser-assisted pair tunneling[@Sylvain]. Here through exact numerical calculations, we confirm the Majorana ground state with strong edge-edge correlations in a broad range of paramters : filling factor, boson detuning, and inter-channel coupling (See yellow regions in Fig. \[fig1\] and Fig. \[fig\_g\]). It is useful to contrast our model with the single-channel model that only consists of spinless fermions with neighboring-site attractions. While both models are number conserving and their mean field theories share the same structure, our exact calculations show that only the resonant two-channel model exhibits strong Majorana correlations. This shows again the essential role played by the number fluctuation of fermion pairs in the chain, which is absent in the single-channel attractive Fermi gas.
Experimentally, a major obstacle for exploring $p$-wave effect in cold atoms is the severe atom loss, as observed in a 3D Fermi gas [@JILA_K40; @Li6_1; @Li6_2; @Toronto_K40]. Recent studies have suggested that the $p$-wave system could be more stable against three-body loss if confined in the quasi-1D geometry [@Cui1; @Cui2]. In our system, an additional optical lattice is applied along the 1D tube and the space is further discretized. In this case, the atom loss comes from the possibility of finding pairs of boson-fermion or boson-boson at the same site and their collisions at close proximity. Here we show that within the two-channel $p$-wave model, the probabilities of finding such pairs are very low in a large region of parameter space for the Majorana phase. This provides promising prospects to realize the Majorana phase and to perform studies with a wide range of cold atom techniques.
Model
=====
Our model is [@footnote_model] $$\begin{aligned}
H=&\sum_{j}\left( -t_b b_j^\dagger b_{j+1}+\text{h.c.} \right)
+\sum_{j}\left(-t_f f_j^\dagger f_{j+1}+\text{h.c.} \right) \nonumber\\
&+\nu\sum_{j} b_j^\dagger b_{j}+g \sum_{j} \left[ b_j^\dagger(f_{j-1}f_j+f_{j}f_{j+1})+\text{h.c.} \right], \label{eq_ham_full}\end{aligned}$$ where $b^\dagger_{j}$ and $f^\dagger_{j}$ create a closed-channel boson and an open-channel fermion at site $j$, with nearest-neighbor hopping $t_b$ and $t_f$ respectively; $g$ is the ($p$-wave) inter-channel coupling and $\nu$ is the boson detuning. As $\nu$ approaches zero, conversion between bosons and fermion pairs for given $g$ will become more frequent due to their energy match. The number of boson ($N_b=\sum_i b^{\dag}_ib_i$) and fermion ($N_f=\sum_i f^{\dag}_if_i$) are not separately conserved, but the sum $N=2N_b+N_f$ is. Since bosons are heavier than fermions, $t_b$ is smaller than $t_f$. In this paper, we will take $t_b=0.2t_f$ and use $t_f$ as the energy unit.
If $b_j$ in Eq. (\[eq\_ham\_full\]) is replaced by a $c$-number, as in mean field approach, Eq. (\[eq\_ham\_full\]) reduces to the Kitaev model [@Kitaev], which has a Majorana phase. The question is whether this phase will survive the quantum fluctuation of the bosons, such that the ground state of Eq. (\[eq\_ham\_full\]) will have the same edge-edge Majorana correlations as in the Kitaev model. This is to be answered in this work.
 to (d).[]{data-label="fig1"}](fig1.pdf){width="65mm"}
Mean-field analysis
===================
We first carry out mean-field analysis to gain a qualitative understanding of the problem. Assuming the bosons fully condense at zero momentum, i.e., $\langle b_{k=0}\rangle=\sqrt{N_b}$, the Hamiltonian $\Omega=H-\mu N$ in grand canonical ensemble can be written as $$\begin{aligned}
\label{eq_ham_mf}
&\Omega=\sum_{k}\left[-2t_f \cos(k d)-\mu\right]f_k^\dagger f_k +N_b(-2t_b +\nu-2\mu) \nonumber\\
&+(\Delta/2)\sum_k \left[ -i
\sin(kd) f_k f_{-k}+\text{h.c.} \right], \label{h_mf}\end{aligned}$$ here $d$ is the lattice spacing; $\Delta=4g\sqrt{n_b}$ with $n_b=N_b/N_L$ the boson filling and $N_L$ the number of lattice sites. Equation (\[h\_mf\]) can be written as $$\frac{\Omega}{N_L} = \frac{1}{N_L} \sum_{k>0} \left( F^{\dagger}_{\mu}(k) {\cal H}_{\mu \nu}(k) F_{\nu} (k)+\xi_k\right)+\frac{|\Delta|^2}{16g^2}(-2t_b+\nu-2\mu),$$ with $\xi_{k} = - 2t_f \cos(kd)-\mu$, $F=(f_{k},f_{-k}^{\dagger})^T$ and $${\cal H}(k) = \left( \begin{array}{cc} \xi_{k} & -i \Delta \sin(kd)
\\ i \Delta \sin(kd) & - \xi_{k} \end{array} \right).
\label{calH}$$
Minimizing the ground state energy of Eq. (\[h\_mf\]) with respect to $\Delta$ and imposing the number constraint $N=2N_{b} + \sum_{k}\langle f^{\dagger}_{k}f^{}_{k}\rangle$, we obtain the gap equation and number equations as: $$\begin{aligned}
\frac{-2t_b+\nu-2\mu}{16g^2} &=& \frac{1}{N_L}\sum_{k>0}\frac{[\sin(kd)]^2}{2E_k}\label{gap_eq},\end{aligned}$$ and $$\begin{aligned}
n(1-c_b)&=&\frac{1}{N_L} \sum_{k>0} \left( 1-\frac{\xi_k}{E_k} \right),
\label{number_eq}
% \hspace{0.3in}\end{aligned}$$ where $E_k=\sqrt{\xi^{2}_{k} + (\Delta\sin(kd))^2 }$ is the excitation spectrum, $c_b=2n_b/n$ is the boson fraction, and $n=N/N_L$ is the total filling. Unlike the Kitaev model where the $p$-wave pairing and the chemical potential are both external inputs [@Kitaev], here $\Delta$ and $\mu$ are determined self-consistently for given $g$, $\nu$ and $n$.
Since the mean-field Hamiltonian Eq. (\[h\_mf\]) is in the form of the Kitaev model, the Majorana phase lies in the region $|\mu|<2t_f$ as pointed out in Ref. [@Kitaev]. The topological character of the ground state is specified by the Berry Phase of the ground state $\chi(k)$ of ${\cal H}(k)$, integrated over the Brillouin Zone, i.e. $\Delta \Phi \equiv i \int^{\pi/d}_{-\pi/d} {\rm d}k \chi^{\dagger}(k) \partial_{k} \chi(k) $. Writing ${\cal H}(k)$ as ${\cal H}(k) = {\bf h}(k)\cdot \vec{\sigma}$, where $${\bf h}(k) = \xi_{k} \hat{\bf z} + \Delta\sin(kd) \hat{\bf y} \equiv E_{k} \left( {\rm cos}\theta_{k} \hat{\bf z} + {\rm sin}\theta_{k}\hat{\bf y}\right), \label{h_k}$$ we have $$\chi(k) = \left( \begin{array}{c} i {\rm sin}(\theta_{k}/2) \\ {\rm cos}(\theta_{k}/2) \end{array}\right) e^{i\gamma(k)}.$$ Here the phase factor $e^{i\gamma(k)}$ is to keep $\chi(k)$ periodic, $$\chi(k) = \chi(k + 2\pi/d).
\label{BC}$$ As seen from Eq. (\[h\_k\]), the tip of ${\bf h}(k)$ traces out a closed curve in the $yz$-plane as $k$ varies from $-\pi/d$ to $\pi/d$. For $|\mu |< 2t_{f}$, this curve encloses the origin, and we have $\theta(-\pi/d) = 0$, and $\theta(\pi/d)= 2\pi$. In order to ensure the periodicity of $\chi(k)$ (Eq. (\[BC\])), $\gamma$ has to satisfy the condition $\gamma(-\pi/d) - \gamma(\pi/d)=\pi$, which gives the Berry phase $\Delta \Phi=\pi$. This is the topologically non-trivial (Majorana) phase. Otherwise for $|\mu |> 2t_{f}$, the curve of ${\bf h}(k)$ does not enclose the origin, and we have $\theta(\pi/d) = \theta(-\pi/d),\ \gamma(\pi/d) = \gamma(-\pi/d)$, and thus $\Delta \Phi=0$. This is the topologically trivial phase.
After $\Delta$ and $\mu$ are determined self-consistently, we obtain the mean field phase diagram in Fig. \[fig1\] over a broad range of $\nu$ and $n$ for a fixed coupling $g=1$. There are three superfluid phases (all with $\Delta\neq 0$): a topological Majorana phase $M=SM + WM$, consisting a “strong Majorana" region $SM$ and a “weak Majorana" region $WM$, and two non-topological superfluid phases $FHP$ (fermion hole pairs) and $FPP$ (fermion particle pairs). The reasons of the nomenclature will be explained further later.
The phase boundaries separating the Majorana phase $M=SM+WM$ and the $FHP$ ($FPP$) phase is determined by the condition $\mu=2t_f$ ($\mu=-2t_f$), as shown by red lines in Fig. \[fig1\]. Our numerical solutions of Eqs. (\[gap\_eq\]) and (\[number\_eq\]) show that the upper boundary ($\mu=2t_f$) takes a sudden upward turn at $n=1$. The red dashed line in Fig. \[fig1\] marks a crossover from SM to WM regime, when the boson fraction $c_b$ continuously decays to a small value $0.01$. Later, we show from exact numerical calculations that these two regions can be distinguished from the behavior of edge-edge Majorana correlations.
The $FPP$ phase exists in sufficiently negative $\nu$, where the system is mostly in the Bose condensate, with a dilute gas of fermion particles forming pair superfluid. In contrast, the $FHP$ phases exists for $n>1$ and for sufficiently high $\nu$. In this regime, the fermion occupation is more favored than bosons, leading to the nearly full fermion filling $n_f\sim1$ with a small fraction of fermion holes. Since the fermion superfluidity essentially relies on the number fluctuations, it can be viewed as the pairing of fermion holes.
DMRG analysis
=============
We have calculated the ground state properties of the Hamiltonian \[Eq. (\[eq\_ham\_full\])\] using density-matrix-renormalization-group (DMRG) method [@alps1; @alps2]. The calculations are done with maximum $800$ truncated states and $30$ sweeps, and the truncation error is $10^{-8}$. Since we consider the filling regime with $n\le 2$, we have set the truncated number of bosons at each site as up to four in our simulation.
Identification of Majorana phase and Majorana edge state
--------------------------------------------------------
In this section, we use DMRG to work out the phase diagram of Eq. (\[eq\_ham\_full\]) \[(I) and (II) below\], and then examine in (III) the presence of Bose condensation (i.e superlfuid nature) in different phases, followed by a study in (IV) of the Majorana correlation in these phases which determines the presence of Majorana edge modes. The section will be ended by a discussion of the momentum distribution which shows the distinct signature of the Majorana phase.
[*(I) Entanglement entropy.*]{} It is known that a thermodynamic phase transition is reflected in a singularity of the entanglement entropy at the transition point [@amico]. Given the many-body ground state $|\psi\rangle$, the reduced density matrix can be written as $\rho_L=\text{Tr}_R |\psi\rangle\langle\psi|$, with $L$, $R$ denoting the left and right half of the lattice. Its eigenvalues $\{\lambda_{\alpha}\}$ determine the entanglement entropy $S=-\sum_{\alpha}\lambda_{\alpha}\ln \lambda_{\alpha}$.
![Entanglement entropy $S$ as functions of $\nu$ for (a) $n=0.75$ and (b) $n=1.25$ with different system sizes $N_{L}$. Green diamonds, red squares and blue circles correspond to $N_L=24, 40$ and $64$, respectively. Here $g=1$. Inset in (a) shows the extrapolation to infinite system for the case of $n=0.75$.[]{data-label="fig_S"}](fig2.pdf){width="65mm"}
We find that the behavior of $S$ as a function of detuning $\nu$ depends on the filling $n$. For $n<1$, we find one cusp in $S$ at $\nu=\nu_{c1}$ (Fig. \[fig\_S\](a)); while for $n>1$, we find two cusps in $S$ at $\nu_{c1},\nu_{c2}$ ($\nu_{c1}<\nu_{c2}$, see Fig. \[fig\_S\](b)). This indicates one or two transitions by tuning $\nu$. Repeating the calculation for different sample sizes, we have obtained the estimate of the critical values of $\nu_{c1},\nu_{c2}$ for infinite systems through extrapolation (see the inset of Fig. \[fig\_S\](a)). These phase boundaries are shown by dots connected by solid lines in the phase diagram in Fig. \[fig1\]. Here we have chosen $g=1$, and we find these phase boundaries are close to those obtained by mean field theory. Accordingly, we adopt the same nomenclature as in mean field theory for the phases obtained from DMRG.\
[*(II) Boson fraction and its variation.*]{} We have also verified the phase boundaries by calculating the boson fraction $c_b$ and its variation $c_b' \equiv \partial c_b/\partial \nu$ as a function of $\nu$. The results are shown in Fig. \[fig\_n\]. One sees that while $c_b$ is continuous in $\nu$, $c_b'$ has one or two sharp cusps depending on whether the filling $n<1$ or $n>1$. The locations of these cusps are consistent with those obtained in (I), thus confirming the phase transitions discussed in (I). In addition, the singularity in $c_b'$ shows that the transitions are of the second order. In Fig. \[fig\_n\], we also compare with the mean-field predictions (thin red lines) and find qualitative agreement.\
![Boson fraction $c_b$ (a, c) and its variation $c_b'$ (b, d) as functions of $\nu$. Panel (a, b) and (c, d) are for fillings $n=0.75$ and $n=1.25$ with $N_L=64$ and $g=1$. Dots and dashed blue lines show the results determined by DMRG. Solid red lines show the results of mean-field predictions.[]{data-label="fig_n"}](fig3.pdf){width="85mm"}
[*(III) Condensation of bosons and fermion pairs:* ]{} The criterion of Bose condensation was generalized to interacting system by Penrose and Onsager [@PO]. It also applies to finite number systems. The Penrose-Onsager criterion makes use of the property of the single particle density matrix $\rho_{ij} \equiv \langle b^{\dagger}_{i} b_{j}\rangle$ evaluated for the state of interest. The state is Bose condensed if $\rho$ has a [*single*]{} maximum eigenvalue $\zeta_{1}$ of order $N_b$, while the ratios $\zeta_{\beta}/\zeta_{1}$ for all other eigenvalues $\zeta_{\beta}$ are much less than 1, and tend to zero as $N_b$ goes to infinity. The eigenfunction associated with $\zeta_{1}$ is referred to as the condensate wavefunction. Similarly, one can also define a fermion pair correlation matrix $\eta_{ij}\equiv \langle f^{\dagger}_{i+1}f^{\dagger}_{i} f^{}_{j} f^{}_{j+1}\rangle$. Condensation of fermion pairs \[as characterized by C.N. Yang as off-diagonal long-range order, see Ref. [@Yang]\] corresponds to a single large eigenvalue of $\eta$ of order $N$, as in the bosonic case.
Using DMRG, we have calculated the eigenvalues of the matrices $\rho_{ij}$ and $\eta_{ij}$ for all the ground states in different phases in Fig. \[fig1\]. To illustrate the Bose condensation, we have shown our results in Fig. \[PO\](a)-(d) as one increases the detuning from $-4$ to $2$ at $n=0.75$. This path takes one from the $FPP$ phase to the $SM$ region and then to the $WM$ region. We see from Fig. \[PO\](a)-(c) that both $FPP$ and $SM$ phases have a distinct large eigenvalue, whereas the large eigenvalue gradually disappears into a continuous (power-law like) distribution of eigenvalues in the $WM$ region, see Fig. \[PO\](d). In contrast, the fermion pair correlation $\eta$ does not show a large distinct eigenvalue in all cases, and appears to be power-law like. The situation at the $FHP$ phase is similar to that of the $FPP$ phase, i.e. there is a distinct eigenvalue for the boson density matrix, and a power-law like distribution for the fermion pair distribution.\
![Red dots show eigenvalues $\zeta_{\beta}$ of single particle density matrix $\rho_{ij} \equiv \langle b^{\dagger}_{i} b_{j}\rangle$. Blue squares show eigenvalues of fermion pair correlation $\eta_{ij}\equiv \langle f^{\dagger}_{i+1}f^{\dagger}_{i} f^{}_{j} f^{}_{j+1}\rangle$. The eigenvalues are sorted from the largest to the smallest. (a), (b), (c), and (d) show the cases with $\nu=-4$, $-2$, $0$, and $2$, respectively, for $n=0.75$. They correspond to FPP, SM, SM, and WM states in Fig. \[fig1\], respectively. Total number of bosons $N_b$ in (a), (b), (c), and (d) are 12.57, 8.80, 5.82, and 3.31, respectively. Here $g=1$ and $N_L=40$.[]{data-label="PO"}](fig4.pdf){width="85mm"}
[*(IV) Edge-edge Majorana correlation.*]{} The emergence of Majorana fermion is associated with long range edge-edge correlations. In the Kitaev chain model [@Kitaev], one defines two sets of Majorana fermion operators $\lambda^1_{i}=f^{\dag}_i+f_i$ and $\lambda^2_{i}=i(f^{\dag}_i-f_i)$, and the Majorana phase can be characterized by the order parameter ${\cal O}\equiv i\langle \lambda^1_1 \lambda^2_{N_L} \rangle$, which directly manifests the correlation between two Majorana modes at different edges [@Kitaev; @Zoller; @Diehl; @Buchler; @Iemini]. For a number conserving system, $|{\cal O}|$ is directly reduced to the edge-edge correlation function $G(1,N_L)$, where $G$ is defined as: $$G(i,j)\equiv | \langle f^\dagger_i f_{j} + h.c.\rangle|.$$
![The behavior of the correlation function $G(1,j)$ in log scale as one passes through the boundaries as shown in Fig. \[fig1\]. (a,b) are for filling $n=0.75$, and (c,d) are for $n=1.25$. The blue solid, red dashed and green dotted lines from (a) to (d) correspond to ($\nu=-3.5, -3, -2.5$), ( $\nu=4, 2, 0$), ($\nu=-3.7, -3.2, -2.7$), and ($\nu=4.4, 3.7, 3$). In (a,c,d), when cross the phase boundaries into SM phase, $G(1,j)$ shows strong revival as $j$ approaches the other edge $j\rightarrow N_L$. In (b), when passing from the $WM$ into the $SM$ region, $G(1,j)$ changes slowly even within a large range of $\nu$. In $WM$ region, $G(1,N_L)$ dropped to a small but non-zero value ($G(1,N_L)/G(1,1)<10\%$). Here we take $g=1$ and $N_L=64$.[]{data-label="fig_G"}](fig5.pdf){width="85mm"}
In Fig. \[fig\_G\], we show the behavior of correlation function $G(1,j)$ as the system passes through different boundaries marked (a) to (d) in Fig. \[fig1\]. For filling $n>1$, as the system enters the $SM$ phase through the phase boundary $\nu_{c1}$ from the $FPP$ phase below, or through the phase boundary $\nu_{c2}$ from the $FHP$ phase above, $G(1,j)$ immediately shows strong revival as $j$ approaches the other edge, which is the hallmark of Majorana edge states. See Fig. \[fig\_G\](c) and Fig. \[fig\_G\](d). In contrast, in the $FPP$ phase ($\nu<\nu_{c1}$) or $FHP$ phase ( $\nu>\nu_{c2}$), $G(1,j)$ decreases exponentially fast as $j$ increases, indicating the absence of Majorana edge correlations.
For $n<1$, similar revival behavior also shows up as $\nu$ across the lower boundary $\nu_{c1}$ (see Fig. \[fig\_G\](a)) . Here, within a small range of $\nu$ from $-3.5$ to $-2.5$, the edge-edge correlation $G(1,N_L)$ (scaled by $G(1,1)$) increases from $10^{-3}$ to as large as $0.6$. Continuously increasing $\nu$, $G(1,j)$ crossovers to a different behavior. It decreases slowly without strong revival but reaching a small yet non-zero value as $j$ approaches the other edge, see Fig. \[fig\_G\](b). Specifically, for a large range of $\nu$ from $0$ to $4$, $G(1,N_L)$ decreases from $0.2$ to $0.05$. Further increasing $\nu$, $G(1,N_L)$ becomes even smaller but still finite (not exponentially small as in the non-Majorana phases $FPP$ and $FHP$). In this sense, in Fig. \[fig1\] we draw a dashed line for filling $n < 1$ to show the crossover from the strong Majorana (SM) to weak Majorana (WM) when $G(1,N_L)$ decays to 10$\%$ of the onsite $G(1,1)$. In this way we highlight the SM region with yellow color in Fig. \[fig1\], where one can find strong Majorana edge-edge correlations.\
[*(V) Momentum distribution:* ]{} The behavior of correlation function in spatial space ($ \sim\langle f^{\dagger}_{i} f_{j}\rangle$) directly determines its Fourier transformation, i.e., the momentum distribution of fermions $n(k) = \langle f^{\dagger}_{k} f_{k}\rangle$ that can be easily measured in cold atoms experiments. In Fig. \[fig7\], we show $n(k)$ from DMRG calculation for three typical values of $\nu$ at filling $n=0.75$ and $1.25$. We can see that in both cases, when $\nu$ stays in the SM region(red dashed line in Fig. \[fig7\]), $n(k)$ features a distinct peak at $k=0$ while decays to zero at the Brillouin edge $k=\pi/d$. We have checked that such property of $n(k)$ holds true for all SM states in Fig. \[fig1\]. In contrast, $n(k)$ is roughly a constant in the $FHP$ phase, and has a hole at $k=0$ in the $FPP$ phase. $n(k)$ for WM state share similar structure as the SM case, while its peak at $k=0$ is not as sharp as the latter.
All these behaviors can be understood from the mean-field theory, where we have the analytical expression $n(k) = [1 + (2t_f \cos(kd) +\mu)/E_k]/(2N_L)$ (see Eq. (\[number\_eq\])). It is easily seen that within the Majorana region ($|\mu|<2t_f$), $n(k)$ is the largest at $k=0$ while it is zero at $k=\pm \pi/d$. In the $FHP$ phase, the detuning $\nu$ is large and positive, which leads to a large and positive $\mu(>2t_f)$ in the mean field theory and a finite $n(k)$ at both $k=0$ and $k=\pm \pi/d$. Such detuning makes it costly to create bosons, making the system more free fermion like. In the $FPP$ phase, $\nu$ is large and negative, forcing most of the particles into the $k=0$ Bose condensate, and leads to a large and negative $\mu(<-2t_f)$ in mean field theory. This leads to $n(k)=0$ at both $k=0$ and $k=\pm \pi/d$. In particular, at small $k$, we have $n(k) = |\Delta\sin(kd)/(2\mu)|^2 \sim k^2 $.
Given the distinct $n(k)$ for different phases, they can be used as experimental indicators of various phases in current system. Note, however, that $n(k)$ cannot be directly related to the edge mode in Majorana physics, because it contains a large contribution from the bulk. This is evidenced by the fact that the behaviors of $n(k)$ are well accounted for by the results of mean field theory.\
![Fermion distribution in momentum space for different $\nu$ at filling (a) $n=0.75$ and (b) $n=1.25$. Blue solid, red dashed, and green dotted lines in (a) show $n(k)$ for $\nu=-4$, $-1$, and $3$, corresponding to FPP, SM, and WM phases, respectively. Blue solid, red dashed, and green dotted lines in (b) show $n(k)$ for $\nu=-4$, $-1$, and $6$, corresponding to FPP, SM, and FHP phases, respectively. Here $g=1$ and $N_L=64$. []{data-label="fig7"}](fig6.pdf){width="65mm"}
Robust topological features of the strong Majorana phase
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In this section, we will show that the strong Majorana (SM) phase has robust topological features, in that the edge-edge correlation survives from disorders, and it can host a non-trivial winding number in the bulk system. We will also show that the condensation of bosons and the edge-edge correlation remain strong for increasing system size.
![The correlation function $G(1,j)$ in the presence of disorder in hopping (a) and disorder in inter-channel coupling (b). Different disorder strengths $D=0$(blue solid), $0.1$(red dashed) and $0.25$(green dotted) are shown. Here $n=0.75$, $g=1$, $\nu=-2$, and the system stays in strong Majorana (SM) phase. []{data-label="fig_disorder"}](fig7.pdf){width="65mm"}
Firstly, we study the robustness of edge-edge correlation in the presence of disorder. Here we impose disorder either in the fermion hopping term ($t_f$) or in the coupling term ($g$) in the Hamiltonian (\[eq\_ham\_full\]), and carry out DMRG simulations with OBC. Specifically, $t$ or $g$ are now site-dependent, $t_f\rightarrow t_f+D\delta_j$ or $g\rightarrow g+D\delta_j$ ($j$ is site index); here $\delta_j\in(-1,1]$ is a random number, and $D$ is the strength of disorder. In Fig. \[fig\_disorder\], we show the behavior of correlation function $G(1,j)$ for a typical SM ground state with different disorders in $t_f$ \[Fig. \[fig\_disorder\](a)\] and $g$ \[Fig. \[fig\_disorder\](b)\]. We see that small disorder cannot change the strong revival character of $G(1,j)$ even for $D$ reaching $0.25$. This shows the edge modes are robust against a fairly large amount of external perturbations.
![Winding number \[Eq. (\[w\])\] as functions of boson detuning for small size systems with twisted boundary. Blue circles, red squares, green diamonds and orange triangles are respectively with $(n,N_L)=(0.25,8),\ (0.5,8), \ (0.75, 8)$, and $(1,6)$. []{data-label="fig_w"}](fig8.pdf){width="65mm"}
Secondly, in order to demonstrate the topological feature of the bulk system, we impose a twisted phase boundary condition and calculated the resulting winding number. Specifically, we turn on the fermion hopping between the two edges ($i=1$ and $L$) as $t_{1L}^*=t_{L1}=t_f e^{i\theta}$, where $\theta\in(0,2\pi]$ is the twist phase. In the single-particle picture, this corresponds to shifting the momentum as $k\rightarrow k+\theta/N$, and when $\theta$ varies from $0$ to $2\pi$, the momentum basis $\{ k\}$ returns to itself and completes a closed loop, so as the Hamiltonian $H$. For the interacting many-body state, we calculate the winding number of the form $$w=i\int_0^{2\pi} d\theta \langle \Psi_{\theta}| \partial_{\theta}| \Psi_{\theta}\rangle/\pi, \label{w}$$ with $\Psi_{\theta}$ is the ground state with twisted phase $\theta$. In Fig. \[fig\_w\] we show $w$ as a function of detuning $\nu$ for several fillings by exactly diagonalizing small size systems. We find that given the filling factor $n$, by increasing the detuning $\nu$ to drive the system from FPP to SM phase, $w$ will have a sudden jump from $0$ to a finite value ($=n\pi$) at a critical $\nu_c$ (in thermodynamic limit $\nu_c$ is expected to recover the lower boundary as shown in Fig. \[fig1\]). This signifies a topological transition between the two phases. The finite $w$ continues to the WM phase when further increasing $\nu$.
Note that here $w$ depends on the filling factor, instead of a constant ($\pi$) Berry phase in the mean-field analysis (see section III). This difference can be attributed to different ways in introducing a closed path in parameter space. Specifically, in the mean-field analysis, the closed path is completed by moving $k$ through the entire Brillouin Zone, which in the single-particle picture corresponds to fermions occupying a Fermi-sea at full filling $n=1$. Here, for interacting many-body system, the closed path is introduced through the twisted boundary and the filling $n$ can be arbitrary. Nevertheless, a remarkable feature of the Majorana phase is that, regardless of the way of introducing closed path, it can always distinguish itself from the trivial phase by producing a non-zero (topological) winding number. Such a non-zero number characterizes the topological nature of the bulk for interacting many-body systems, analogous to the role of $\pi$ Berry phase in the Kitaev chain under mean-field treatment.
Finally, we study the robustness of the Bose condensation and the edge-edge correlation against increasing the system size. We diagonalize the boson single-particle density matrix $\rho_{ij}$ for a typical SM state with different $N_{L}= 24$, $40$, and $64$ in the SM phase. Fig. \[fig\_size\](a) shows that in all three cases, $\rho$ has a distinct largest eigenvalue. We also show the edge-edge correlation function $G(1, N_L)$ for different $N_L$ in Fig. \[fig\_size\](b), and the same strong revival at the edge is found in for all lengths studied.
![(a) Eigenvalues $\zeta_{\beta}$ of single particle density matrix $\rho_{ij} \equiv \langle b^{\dagger}_{i} b_{j}\rangle$ for different system sizes $N_L=24$(Blue dots), $40$(red squares), and $64$(green diamonds), respectively. The eigenvalues are sorted from the largest to the smallest. (b) Correlation function $G(1, j)$ in log scale for $N_L=24$(Blue dots), $40$(red squares), and $64$(green diamonds), respectively. In both panels, $n=0.75$, $g=1$, $\nu=-2$, and the system stays in strong Majorana (SM) phase. []{data-label="fig_size"}](fig9.pdf){width="65mm"}
Comparison with the single-channel model {#single_channel}
----------------------------------------
It is useful to contrast the Hamiltonian (\[eq\_ham\_full\]) with the single-channel fermion model, $$H_{sc}=\sum_{j}\left[ -t_f (f_j^\dagger f_{j+1}+\text{h.c.} )+ UN_{f,i}N_{f,j+1} \right], \label{single_H}$$ where $N_{f,i}= f_i^\dag f_i$ is the fermion number operator at site $i$ and $U<0$ is the attraction between neighboring-site fermions. In this model, the fermion number $N_f=\sum_i N_{f,i}$ is conserved, unlike in the resonance model ($H$ in Eq. (\[eq\_ham\_full\])). Yet this model has the same mean field theory as the resonance model, with the mean field gap defined as $\Delta/2\equiv U\langle f_i f_{i+1}\rangle $. This raises the question of whether the single channel model $H_{sc}$ will also have a Majorana ground state in certain parameter regime.
To compare the ground state of single-channel model $H_{sc}$ \[Eq. (\[single\_H\])\] with that of the resonance model $H$ \[Eq. (\[eq\_ham\_full\]))\], we shall choose the parameters ($\{U,n_{f}\}$ in $H_{sc}$ and $\{\nu,\ n\}$ in $H$) such that both models have the same fermion density $n_{f}$ and the same mean field gap $\Delta$. With this correspondence, we have calculated the ground state of $H_{sc}$ with OBC using DMRG.
We find that for all the detunings $\nu$ in Fig. \[fig1\] that cover the $SM$ and $FPP$ phases, the corresponding $U$ in $H_{sc}$ is so negative that the ground state is a droplet with all fermions packed together in a region of the size of $N_{f}$ sites, see Fig. \[fig\_comparison\](b). Such cluster bound state was also shown previously for few particles[@Berciu]. It can be understood by mapping $H_{sc}$ into a quantum spin chain using the Jordan-Wigner transformation, where the occupied (empty) site is mapped to spin-up (spin-down), and the $U(<0)$ term in Eq. (\[single\_H\]) can be mapped to the ferromagnetic Ising interaction. It is then obvious that for large and negative $U$, the system forms ferromagnetic domains in the ground state, i.e., the occupied and empty sites are spatially well separated as shown in Fig. \[fig\_comparison\](b). Similar ferromagnetic correlation has also been shown in other 1D systems with p-wave attraction [@Cui3; @Gora2]. In our calculations, the droplet may appear in different locations, as the energy difference between droplets at different locations is so small that is below our accuracy of our calculation. Clearly, the droplet phase is not the Majorana phase as found in the resonance model $H$, which exhibits the fermion density distribution as shown in Fig. \[fig\_comparison\](a).
For weaker attraction $U$, corresponding to $WM$ or $FHP$ regions in the large detuning limit in Fig. \[fig1\], the droplet gives way to a gas phase that covers the entire chain, but still there is no strong edge-edge correlations. To conclude, the single-channel model $H_{sc}$ cannot host strong Majorana character for all couplings $U$, distinct from the SM phase in Fig. \[fig1\] of the resonance model. It is also clear from Fig. \[fig1\] that in order to obtain strong Majorana correlations, the detuning $\nu$ should stay in a finite region near the two-channel resonance ($\nu\sim 0$), i.e., when bosons and fermions have comparable proportions and their conversion (or number fluctuation of fermion pairs) is the strongest.
Here we should also remark that to describe $p$-wave Fermi gas in cold atoms experiments, the two-channel model is more realistic than the single channel model. This is because the $p$-wave resonance in these systems are generally very narrow, and the closed-channel bosons can take a sizable proportion as measured in a 3D gas near a $p$-wave resonance[@Toronto_K40].
![(a) and (b) show the fermion density distributions from DMRG calculations for the $p$-wave resonance model, Eq. (\[eq\_ham\_full\]), and the single channel model, Eq. (\[single\_H\]), at the same filling $n_f=0.5$ and the same mean-field gap $\Delta=2.44$. In our $p$-wave resonance model, $\nu=0.5$ and $n=1.25$ (lies inside the SM region in Fig. \[fig1\]). In the single-channel model, $U=-4.66$. Here $N_L=40$.[]{data-label="fig_comparison"}](fig10.pdf){width="65mm"}
Suppressed atom loss in the strong Majorana phase {#atom_loss}
-------------------------------------------------
Experimental realization and detection of Majorana edge state require low atom loss. For $p$-wave fermions in the lattice configuration, a previous study showed that the lattice setup will help to reduce inelastic collisional losses compared to free space [@Gora]. The analysis was based on a single-channel model, and the reduced loss can be attributed to the low probability of finding three fermions close to each other outside the lattice sites [@Gora]. For the present $p$-wave system described by the two-channel lowest-band model, three-fermion collision can be effectively ruled out, while the atom loss is dominantly caused by the fermion-boson or boson-boson collision at the same site. Indeed, previous studies on a continuum gas have shown that the three-body and the four-body loss rates are respectively proportional to the probabilities of finding atom-dimer and dimer-dimer at the same location [@Petrov; @Levinsen], up to a background constant that is determined by the loss rate far from resonance regime. Here, accordingly we examine the probabilities of finding a pair of boson-fermion and boson-boson at the same site, respectively denoted by $P_{bf}$ and $P_{bb}$: $$\begin{aligned}
P_{bf}&=&\frac{1}{N_L}\sum_i \langle N_{b,i} N_{f,i} \rangle,\\
P_{bb}&=&\frac{1}{2N_L}\sum_i \langle N_{b,i}(N_{b,i}-1)\rangle,\end{aligned}$$ with $N_{b,i}= b_i^\dag b_i$ and $N_{f,i}= f_i^\dag f_i$.
![$P_{bf}$ (a) and $P_{bb}$ (b) as functions of filling $n$ for $\nu$ staying in the lower ($\nu=\nu_{c1}$) and upper ($\nu=\nu_{c2}$) boundaries of the SM phase in Fig. \[fig1\]. Here, $g=1$ and $N_L=64$. []{data-label="fig_loss"}](fig11.pdf){width="85mm"}
In Fig. \[fig\_loss\], we show $P_{bf}$ and $P_{bb}$ as functions of filling $n$ for $\nu$ staying in the lower ($\nu_{c1}$) and upper ($\nu_{c2}$) boundaries of the SM phase in Fig. \[fig1\]. We can see that for $n\lesssim 1.25$, both probabilities are less than $10\%$, suggesting the atom loss is well controlled with little atom-dimer and dimer-dimer collisions. The physical reason for these low probabilities is because these configurations do not effectively take advantage of the conversion between boson and fermions ($g$-term in Eq. \[eq\_ham\_full\]) to lower the energy. For example, a boson and a fermion on the same site will stop the boson to convert into a fermion pair due to Pauli blocking. Similarly, if two bosons are at the same site, they cannot both convert to fermion pairs. As a result, in general the ground state does not favor the double occupations of boson-fermion or boson-boson at the same site. However, for large fillings, such double occupations are inevitable, as shown by the increasing $P_{bf}$ and $P_{bb}$ with $n$ in Fig. \[fig\_loss\]. These suggest that the SM phase in Fig. \[fig1\] should be stable enough for lower fillings ($n\lesssim 1.25$).
Now we give an estimation to the loss rate of $^{40}$K and $^{6}$Li fermions in 1D lattices. For $^{40}$K and $^{6}$Li away from Feshbach resonance, the 3D recombination rates are $\alpha_{rec}^{3D}=10^{-25} {\rm cm}^6/s$ [@JILA_K40] and $10^{-24} {\rm cm}^6/s$ [@Li6_1], respectively. For typical transverse confinement length $\sim 50$nm and typical 1D density $\sim 10^{4}{\rm cm}^{-1}$, this leads to the decay time about 1s for $^{40}$K [@Gora2] and 0.1s for $^{6}$Li when the system is out of the resonance regime (non-interacting limit). In the present case, due to the small probability $P_{bf}\lesssim 10\%$ (for filling less than unity), the actual loss rate will be further reduced by one order of magnitude, i.e, the decay time can extend to 10s and 1s, respectively, for $^{40}$K and $^{6}$Li systems. Considering the typical hopping strength $t_f$ about tens to hundreds of Hertz, the time scale for developing the many-body correlation is a few to tens of milliseconds, much shorter than the decay time. We thus expect the Majorana phase can be observed well before severe losses occur in practical cold atoms experiment.
Effect of inter-channel coupling {#effect_g}
--------------------------------
The phase diagram shown in Fig. \[fig1\] is for coupling $g=1$ (in units of hopping $t_f$). To illustrate the situation for different $g$, we have worked out the phase diagrams in the $(g,\nu)$-plane for two different fillings $n=0.75$ and $1.25$ using DMRG. These results together with the mean field predictions are shown in Fig. \[fig\_g\].
![ Phase diagram in the $g-\nu$ plane for two different fillings $n=0.75$ (a) and $1.25$ (b). Dots and black lines are from DMRG simulations after the finite-size scaling while the red lines are from mean field predictions. The solid and dashed lines respectively mark the phase transition and the crossover. The highlighted yellow region denotes the strong Majorana (SM) phase using the same criterion as in Fig. \[fig1\].[]{data-label="fig_g"}](fig12.pdf){width="65mm"}
From Fig. \[fig\_g\], one sees that the difference between DMRG and mean field results grows with increasing $g$, and in large $g$ limit the mean-field theory significantly overestimate the SM region (marked by yellow color) compared to DMRG result. This can be attributed to the enhanced quantum fluctuations as increasing $g$. For filling $n<1$(Fig. \[fig\_g\](a)), the SM phase can always survive in a finite detuning regime, while the lower and upper boundaries both turn upward to higher detunings. In comparison, for filling $n>1$ (Fig. \[fig\_g\](b)), the SM phase finally disappears at a large $g_c$ ($g_c=5$ for $n=1.25)$. For $g>g_c$ the DMRG result suggests that Majorana physics is overwhelmed by certain density waves of bosons and fermions in lattices.
Now we show that it is realistic in experiments to reach the parameter regions $(g,\nu)$ of the Majorana phase. To give an example, we shall consider the 1D $^{40}$K Fermi gas in a lattice with depth $v\equiv V_0/E_L=6$ ($V_0$ is the lattice depth and $E_L=k_L^2/(2m)$ is the recoil energy). The hopping $t_f$ in the lowest band is $t_f/E_L\sim 0.06$. As shown in Ref. [@Cui], $(g,\nu)$ can be expressed by $g= g_{eff} C d^{-3/2}$, $\nu=-2g^{2}/U_{eff}$, where $g_{eff}$ is related to the effective range $r_{eff}=(mg_{eff})^{-2}$, $U_{eff}$ is the effective coupling between fermions (see Eqs. (11) and (13) in Ref. [@Cui]), and $C$ is a constant given by the overlap of Wannier functions ($C=0.06$ for $v=6$, see Fig. 4 in Ref. [@Cui]). Let us consider the regime nearby the first Bloch-wave resonance, where $(l_ok_L)^{-1}\le2$ ($l_o$ is the odd-wave scattering length and $k_L=\pi/d$ is the recoil energy). The range of $U_{eff}$ and $r_{eff}$ are shown in Fig. 3 in Ref. [@Cui], from which one can estimate the range of $(g,\nu)$ in unit of $t_f$. For instance, in the interaction regime of interest, $r_{eff}$ can range from $0.75d$ to $1.5d$, so the ratio between the coupling $g$ and the hopping $t_f$ can range from $0.15$ to $0.25$. Similarly, from the information of $U_{eff}$ one can estimate the range of $\nu/t_f$ as from $-7$ to $42$. Such a broad range of $\nu/t_f$ is facilitated by the small value of $t_f/E_L$, and it well covers the SM region shown in Fig. \[fig\_g\] for $g/t_f\in[0.15,0.25]$. Therefore the strong Majorana correlation can be achieved in a lattice with $v=6$ near a Bloch-wave resonance.
Summary and discussion
======================
In summary, we have shown that the spinless Fermi gas in a 1D optical lattice near a $p$-wave resonance can have Majorana ground state over a sizable range of parameter space that are experimentally accessible. Our scheme makes use of the intrinsic property of cold atoms with double channels and requires neither spin-orbit coupling nor multi-chain setup. Our work, together with other multi-chain studies, show the number fluctuation of fermion pairs are crucial for the formation of Majorana phase. In comparison, we demonstrate that the single-channel fermions with neighboring-site attraction have no strong Majorana features.
In identifying the phase boundaries between the Majorana phase and other trivial superfluid phases, we have examined a number of different physical quantities, including the entanglement entropy, the boson fraction and edge-edge correlation, which give rise to consistent results as shown in Fig.\[fig1\]. In the practical detection of Majorana phase, the low probability of dimer-fermion and dimer-dimer pair at the same site will help to reduce atom loss. In addition, it is proposed to identify various phases in the present system from the momentum distribution of fermions, which can be easily measured experimentally. Our results can be directly tested in the 1D cold atomic gases of $^{40}$K or $^{6}$Li fermions.
Finally, we further summarize our characterization of Majorana edge state in interacting many-body systems. In this work, we show that the phases that we labeled to be Majorana (SM phase) exhibit the following properties identical to the (number non-conserving) Kiteav chain: (i) The ground state in an open chain exhibits a strong edge-edge correlation that is robust against various kind of disorder. (iii) The corresponding ground state in the bulk has a non-zero winding number, distinguished from the nearby phases which has zero winding number. (iii) The phase diagram of our number-conserving Majorana phase is remarkably closed to that of the mean-field theory, which is the Kitaev model (see Fig. \[fig1\]). (iv) The properties of our Majorana state can also be interpreted from the Bosonalization method, which has been carried out in Ref.[@Kane] for a similar boson-fermion model. A second-order topological transition was found, consistent with our findings as shown in Fig. \[fig1\].
All above evidences (i)-(iv) show that much of the essential physics of Majorana state exhibited in the number non-conserving Kitaev chain also appear in our number conserving model. In other studies of Majorana physics in number-conserving models [@Zoller; @Diehl; @Buchler; @Iemini], a ground state degeneracy between different number parity sectors has been established. Our system corresponds to one of the fixed number parity states (i.e. either odd or even fermion number) and we have focused on the Majorana correlation function. We shall explore that the physical effects related to the long range coherence of the Majorana correlation in future studies.
[*Acknowledgment.*]{} We would like to thank Marcello Dalmonte, Wei Yi, Wei Zhang and Miguel Cazallia for helpful discussions. This work is supported by the National Key Research and Development Program of China (2018YFA0307601, 2016YFA0300603) and the Natural Science Foundation of China (No.11622436, No.11374177, No. 11421092, No. 11534014) awarded to X.C. and by NSF Grant DMR-0907366, the MURI Grant FP054294-D, the NASA Grant on Fundamental Physics 1541824, and NSFC grant (No. 11674192) awarded to T.L.H.
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---
abstract: 'We consider the quantum complexity of estimating matrix elements of unitary irreducible representations of groups. For several finite groups including the symmetric group, quantum Fourier transforms yield efficient solutions to this problem. Furthermore, quantum Schur transforms yield efficient solutions for certain irreducible representations of the unitary group. Beyond this, we obtain $\mathrm{poly}(n)$-time quantum algorithms for approximating matrix elements from all the irreducible representations of the alternating group $A_n$, and all the irreducible representations of polynomial highest weight of $U(n)$, $SU(n)$, and $SO(n)$. These quantum algorithms offer exponential speedup in worst case complexity over the fastest known classical algorithms. On the other hand, we show that average case instances are classically easy, and that the techniques analyzed here do not offer a speedup over classical computation for the estimation of group characters.'
author:
- |
Stephen P. Jordan[^1]\
[*Institute for Quantum Information, California Institute of Technology.*]{} `[email protected]`
bibliography:
- 'irreps.bib'
title: Fast quantum algorithms for approximating some irreducible representations of groups
---
Introduction
============
Explicit representations of groups have many uses in physics, chemistry, and mathematics. All representations of finite groups and compact linear groups can be expressed as unitary matrices given an appropriate choice of basis[@Artin9]. This makes them natural candidates for implementation using quantum circuits. Here we show that polynomial size quantum circuits can implement:
- The irreducible representations of any finite group which has an efficient quantum Fourier transform. This includes the symmetric group $S_n$.
- The irreducible representations of the alternating group $A_n$.
- The irreducible representations of polynomial highest weight of the unitary $U(n)$, special unitary $SU(n)$, and special orthogonal $SO(n)$ groups.
Using these quantum circuits one can find a polynomially precise additive approximation to any matrix element of these representations by repeating a simple measurement called the Hadamard test, as described in section \[hadamard\].
More precisely, for the finite groups $S_n$ and $A_n$ we obtain any matrix element of any irreducible representation to within $\pm
\epsilon$ in time that scales polynomially in $1/\epsilon$ and $n$. For the Lie groups $U(n)$, $SU(n)$, and $SO(n)$ we obtain any matrix element of any irreducible representation of polynomial highest weight to within $\pm \epsilon$ in time that scales polynomially in $1/\epsilon$ and $n$. Because the representations considered are of exponentially large dimension, one cannot efficiently find these matrix elements by classically multiplying the matrices representing a set of generators. Note that, many computer science applications use multiplicative approximations. In this case, one computes an estimate $\tilde{x}$ of a quantity $x$ with the requirement that $(1-\epsilon)x
\leq \tilde{x} \leq (1-\epsilon)x$. The approximations obtained in this paper are all additive rather than multiplicative. For some problems, the computational complexity of additive approximations can differ greatly from that of mulitplicative approximations[@Bordewich; @Tutte].
For exponentially large unitary matrices, the typical matrix element is exponentially small. Thus for average instances, a polynomially precise additive approximation provides almost no information. However, it is common that the worst case instances of a problem are hard whereas the average case instances are trivial. In section \[complexity\] I narrow down a class of potentially hard instances for the problem of additively approximating the matrix elements of the irreducible representations of the symmetric group to polynomial precision. I also present a classical randomized algorithm to estimate normalized characters of the symmetric group $S_n$ to within $\pm \epsilon$ in $\mathrm{poly}(n,1/\epsilon)$ time. (The character is normalized by dividing by the dimension of the representation, so that the character of the identity element of the group is 1.) Thus, the techniques described here for evaluating matrix elements of irreducible representations of groups on quantum computers do not provide an obvious quantum speedup for the evaluation of the characters of $S_n$.
Our results on the symmetric group relate closely to the quantum complexity of evaluating Jones polynomials and other topological invariants. Certain problems of approximating Jones and HOMFLY polynomials can be reduced to the approximation of matrix elements or characters of the Jones-Wenzl representation of the braid group, which is a $q$-deformation of certain irreducible representations of the symmetric group [@Aharonov1; @Yard; @Shor_Jordan; @Jordan_Wocjan]. Figure \[complexities\] compares the complexity of estimating matrix elements and characters of the Jones-Wenzl representation of the braid group to the complexity of the corresponding problems for the symmetric group. Exact complexity characterizations (*i.e.* completeness results) are not known for all of these problems, and the exact relationships between the complexity classes referenced in figure \[complexities\] are not rigorously known. Nevertheless, the results seem to suggest that in general the matrix elements are harder to approximate than the normalized characters, and that the Jones-Wenzl representation of braid group is computationally harder than the corresponding irreducible representations of the symmetric group.
symmetric braid
----------------------- ----------- ----------------------------------------------
matrix elements in BQP BQP-complete [@Aharonov1; @Yard]
normalized characters in BPP DQC1-complete [@Shor_Jordan; @Jordan_Wocjan]
Hadamard Test {#hadamard}
=============
The Hadamard test is a standard technique in quantum computation for approximating matrix elements of unitary transformations. Suppose we have an efficient quantum circuit implementing a unitary transformation $U$, and an efficient procedure for preparing the state $\ket{\psi}$. We can then approximate the real part of $\bra{\psi} U
\ket{\psi}$ using the following quantum circuit. $$\mbox{\Qcircuit @C=1em @R=.7em {
\lstick{\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})} & \qw & \ctrl{1} & \gate{H} & \meter \\
\lstick{\ket{\psi}} & {/} \qw & \gate{U} & {/} \qw & \qw & }}$$ The probability of measuring $\ket{0}$ is $$p_0 = \frac{1+\mathrm{Re}(\bra{\psi}U\ket{\psi})}{2}.$$ Thus, one can obtain the real part of $\bra{\psi}U\ket{\psi}$ to precision $\epsilon$ by making $O(1/\epsilon^2)$ measurements and counting what fraction of the measurement outcomes are $\ket{0}$. Similarly, if the control bit is instead initialized to $\frac{1}{\sqrt{2}} (\ket{0} - i \ket{1})$, one can estimate the imaginary part of $\bra{\psi}U\ket{\psi}$. Thus the problem of estimating matrix elements of unitary representations of groups reduces to the problem of implementing these representations with efficient quantum circuits.
Fourier Transforms {#fourier}
==================
Let $G$ be a finite group and let $\hat{G}$ be the set of all irreducible representations of $G$. We choose a basis for the representations such that for any $\rho \in \hat{G}$ and $g \in G$, $g$ is represented by a $d_\rho \times d_\rho$ unitary matrix with entries $\rho_{i,j}(g)$. The quantum Fourier transform over $G$ is the following unitary operator[@Nielsen_Chuang] $$U_{\mathrm{FT}} = \sum_{g \in G} \sum_{\rho \in \hat{G}}
\sum_{i,j = 1}^{d_\rho} \sqrt{\frac{d_\rho}{|G|}}
\rho_{i,j}(g) \ket{\rho,i,j}\bra{g}.$$ Here $\ket{g}$ is a computational basis state (bitstring) indexing the element $g$ of $G$. Similarly, $\ket{\rho,i,j}$ is three bitstrings, one indexing the element $\rho \in \hat{G}$, and two writing out the numbers $i$ and $j$ in binary. The standard discrete Fourier transform is the special case where $G$ is a cyclic group.
The regular representation of any $g \in G$ is $$U_g = \sum_{h \in G} \ket{gh}\bra{h}.$$ A short calculation shows $$U_{\mathrm{FT}} U_g U_{\mathrm{FT}}^{-1} = \sum_{\rho \in \hat{G}}
\sum_{i,j = 1}^{d_\rho} \sum_{i',j'=1}^{d_\rho} \delta_{j,j'}
\rho_{i,i'}(g^{-1}) \ket{\rho,i,j}\bra{\rho,i',j'}.$$ In other words, by conjugating the regular representation of $g$ with the quantum Fourier transform, one recovers the direct sum of all irreducible representations of $g^{-1}$.
Given an efficient quantum circuit implementing $U_{\mathrm{FT}}$ one can thus efficiently estimate any matrix element of any irreducible representation of $G$ using the Hadamard test. Quantum circuits implementing the Fourier transform in $\mathrm{polylog}(|G|)$ time are known for the symmetric group[@Beals] and several other groups[@generalft]. The matrix elements of the representations depend on a choice of basis. The bases used in quantum Fourier transforms are subgroup adapted (see [@generalft]). In particular, the symmetric group Fourier transform described in [@Beals] uses the Young-Yamanouchi basis, also known as Young’s orthogonal form.
In section \[althyp\] we describe a more direct quantum circuit implementation of the irreducible representations of the symmetric group, which generalizes to yield efficient implementations for the alternating group.
Schur Transform {#Schurxform}
===============
Let $\mathcal{H}$ be the Hilbert space of $n$ $d$-dimensional qudits. $$\mathcal{H} = (\mathbb{C}^d)^{\otimes n}.$$ We can act on this Hilbert space by choosing an element $u \in U(d)$ and applying it to each qudit. $$\ket{\psi} \to u^{\otimes n} \ket{\psi}$$ We can also act on this Hilbert space by choosing an element $\pi \in S_n$ and correspondingly permuting the $n$ qudits. $$\ket{\psi} \to M_\pi \ket{\psi}$$ $u^{\otimes n}$ and $M_\pi$ are reducible unitary $nd$-dimensional representations of $U(d)$ and $S_n$, respectively. These two actions on $\mathcal{H}$ commute.
The irreducible representations of $S_n$ are in bijective correspondence with the partitions of $n$. Any partition of $n$ into $d$ parts indexes a unique irreducible representation of $U(d)$. $U(d)$ has infinitely many irreducible representations, so these partitions only index a special subset of them. As discussed in [@Schur], there exists a unitary change of basis $U_{\mathrm{Schur}}$ such that $$U_{\mathrm{Schur}} M_\pi u^{\otimes n} U_{\mathrm{Schur}}^{-1} =
\bigoplus_\lambda \rho_\lambda(\pi) \otimes \nu_\lambda(u),$$ where $\lambda$ ranges over all partitions of $n$ into $d$ parts.
As shown in [@Schur], $U_{\mathrm{Schur}}$ can be implemented by a $\mathrm{poly}(n,d)$ size quantum circuit. Thus, using the Hadamard test, one can efficiently obtain matrix elements of these representations of the symmetric and unitary groups.
Complexity of Symmetric Group Representations {#complexity}
=============================================
As described in section \[fourier\], quantum computers can solve the following problem with probability $1-\delta$ in $\mathrm{poly}(n,1/\epsilon,\log(1/\delta))$ time. Note that standard Young tableaux index the Young-Yamanouchi basis vectors, as discussed in section \[Young\].\
\
**Problem 1:** Approximate a matrix element in the Young-Yamanouchi basis of an irreducible representation for the symmetric group $S_n$.\
**Input:** A Young diagram specifying the irreducible representation, a permutation from $S_n$, a pair of standard Young tableaux indicating the desired matrix element, and a polynomially small parameter $\epsilon$.\
**Output:** The specified matrix element to within $\pm
\epsilon$.\
It appears that no polynomial time classical algorithm for this problem is known. Due mainly to applications in quantum chemistry, many exponential time classical algorithms for the exact computation of entire matrices from representations of the symmetric group have been developed[@Hamermesh; @Boerner; @Wu_Zhang1; @Wu_Zhang2; @Egecioglu; @Clifton; @Rettrup; @Pauncz]. There appears to be no literature on the computation or approximation of individual matrix elements of representations of $S_n$.
On the other hand, the precision of approximation achieved by the quantum algorithm is trivial for average instances. We can see this as follows. Let $\lambda$ be a Young diagram of $n$ boxes, let $\rho_\lambda$ be the corresponding irreducible representation of $S_n$, and let $d_\lambda$ be the dimension of $\rho_\lambda$. For any $\pi \in S_n$, the root mean square of the matrix elements of $\rho_\lambda(\pi)$ is $$\mathrm{RMS}(\rho_\lambda(\pi)) = \sqrt{\frac{1}{d_\lambda^2}
\sum_{a,b \in B} | \bra{a} \rho_\lambda(\pi) \ket{b} |^2},$$ where $B$ is any complete orthonormal basis for the vector space on which $\rho_\lambda$ acts. We see that $$\sum_{a \in B} | \bra{a} \rho_\lambda(\pi) \ket{b} |^2 = 1$$ since, by the unitarity of $\rho_\lambda(\pi)$, this is just the norm of $\ket{b}$. Thus, $$\label{small_rms}
\mathrm{RMS}(\rho_\lambda(\pi)) = \sqrt{\frac{1}{d_\lambda^2}
\sum_{b \in B} 1} = \frac{1}{\sqrt{d_\lambda}}.$$ The interesting instances of problem 1 are those in which $d_\lambda$ is exponentially large. In these instances, the typical matrix element is exponentially small, by equation \[small\_rms\]. Running the quantum algorithm yields polynomial precision, thus one could instead simply guess zero every time, with similar results.
That the average case instances are trivial does not mean that the algorithm is trivial. Hard problems that are trivial on average are a common occurrence. The most relevant example of this is the problem of estimating a knot invariant called the Jones polynomial. A certain problem of estimating the Jones polynomial of knots is BQP-complete[@Freedman; @Aharonov1; @Aharonov2]. The Jones polynomial algorithm is based on estimating matrix elements of certain representations of the braid group to polynomial precision. On average these matrix elements are exponentially small. Nevertheless, the BQP-hardness of the Jones polynomial problem shows that the worst-case instances are as hard as any problem in BQP.
By analogy to the results on Jones polynomials, one might ask ask whether problem 1 is BQP-hard. The existing proofs of BQP-hardness of Jones polynomial estimation rely on the fact that the relevant representations of the braid group are dense in the corresponding unitary group. Thus, one can construct a braid whose representation implements approximately the same unitary as any given quantum circuit. Furthermore, it turns out that the number of crossings needed to achieve a good approximation scales only polynomially with the number of quantum gates in the circuit. Unlike the braid group, the symmetric group is finite. Thus, no representation of it can be dense in a continuous group. Hence, if the problem of estimating matrix elements of the symmetric group is BQP-hard, the proof will have to proceed along very different lines than the BQP-hardness proof for Jones polynomials.
Lacking a hardness proof, the next best thing is to identify a class of instances in which the matrix elements are large enough to make the approximation nontrivial. As shown below, we can do this using the asymptotic character theory of the symmetric group. Note that we need not worry about the matrix elements being too large, because even if we know *a priori* that a given matrix element has magnitude 1, it could still be nontrivial to compute its sign.
Let $\pi$ be a permutation in $S_n$, and let $\lambda$ be a Young diagram of $n$ boxes. The character $$\chi_\lambda(\pi) = {\mathrm{Tr}}(\rho_\lambda(\pi))$$ is clearly independent of the basis in which $\rho_\lambda$ is expressed. Furthermore, the character of a group element depends only on the conjugacy class of the group element, because for any representation $\rho$, $${\mathrm{Tr}}(\rho(h g h^{-1})) = {\mathrm{Tr}}(\rho(h) \rho(g) \rho(h)^{-1}) = {\mathrm{Tr}}(\rho(g)).$$
![\[shapeconverge\] Here is a sequence of Young diagrams, such that as the number of boxes increases, the Young diagram converges asymptotically to some fixed shape, in this case a triangle.](shapeconverge.eps){width="32.00000%"}
To understand the behavior of the characters of $S_n$ as $n$ becomes large, consider a sequence of Young diagrams $\lambda_1,\lambda_2,\lambda_3,\ldots$, where $\lambda_n$ has $n$ boxes. Suppose that the diagram $\lambda_n$, when scaled down by a factor of $1/\sqrt{n}$, converges to a fixed shape $\omega$ in the limit of large $n$, as illustrated in figure \[shapeconverge\]. Let $d_{\lambda_n}$ be the dimension of the irreducible representation corresponding to Young diagram $\lambda_n$. Let $\pi$ be a permutation in $S_k$. We can also consider $\pi$ to be an element of $S_n$ for any $n > k$ which leaves the remaining $n-k$ objects fixed. As shown by Biane[@Biane], $$\label{Bianeformula}
\frac{\chi_{\lambda_n}(\pi)}{d_{\lambda_n}} =
C_\pi(\omega) n^{-|\pi|/2} + O(n^{-|\pi|/2-1}).$$ Here $|\pi|$ denotes the minimum number of transpositions needed to obtain $\pi$. Note that these are general transpositions, not transpositions of neighbors. $C_{\pi}(\omega)$ is a constant that only depends on $\pi \in S_k$ and the shape $\omega$. A precise definition of what it means for the sequence to converge to a fixed shape is given in [@Biane], but for present purposes, the intuitive picture of figure \[shapeconverge\] should be sufficient.
$\chi_{\lambda_n}(\pi)/d_{\lambda_n}$ is the average of the matrix elements on the diagonal of $\rho_{\lambda_n}(\pi)$. In the present setting, where $\pi$ is fixed, $\chi_{\lambda_n}(\pi)/d_{\lambda_n}$ shrinks only polynomially with $n$. Thus polynomial precision is sufficient to provide nontrivial estimates of these matrix elements. Nevertheless, finding diagonal matrix elements of $\rho_{\lambda_n}(\pi)$ for fixed $\pi$ and large $n$ is not computationally hard. This is because, as discussed in section \[althyp\], the Young-Yamanouchi basis is subgroup adapted. Thus, for any $\pi$ which leaves all bit the first $k$ objects fixed, $\rho_{\lambda_n}(\pi)$ is a direct sum of irreducible representations of $\pi$ in $S_k$. Because $k$ is fixed, any irreducible representations of $S_k$ has dimension $O(1)$ and can therefore be computed in $O(1)$ time by multiplying the matrices representing transpositions.
To produce a candidate class of hard instances of problem 1, we recall that the character $\chi_{\lambda_n}(\pi)$ depends only on the conjugacy class of $\pi$. Thus, we consider $\pi'$ conjugate to $\pi$. Like $\pi \in S_n$, $\pi' \in S_n$ leaves at least $n-k$ objects fixed, and the representations $\chi_{\lambda_n}(\pi')$ have diagonal matrix elements with polynomially small average value. However, the objects left fixed by $\pi'$ need not be $k+1,k+2,\ldots,n$. Indeed, $\pi'$ can be chosen so that the object $n$ is not left fixed, in which case $\rho_{\lambda_n}(\pi')$ cannot be written as the direct sum of irreducible representations of $S_m$ for any $m < n$.
There is an additional simple way in which an instance of problem 1 can fail to be hard. Let $r(\pi)$ be the minimal number of transpositions of neighbors needed to construct the permutation $\pi$. If $r(\pi)$ is constant or logarithmic, then the matrix elements of the irreducible representations of $\pi$ can be computed classically in polynomial time by direct recursive application of equation \[rule\]. For a class of hard instances of problem 1 I propose the following.\
\
Let $\pi$ be a permutation in $S_n$. We consider it to permute a series of objects numbered $1,2,3,\ldots,n$. Let $s(\pi)$ be the number of objects that $\pi$ does not leave fixed. Let $l(\pi)$ be the largest numbered object that $\pi$ does not leave fixed. Let $r(\pi)$ be the minimum number of transpositions of neighbors needed to construct $\pi$. Let $\lambda$ be a Young diagram of $n$ boxes, and let $\rho_\lambda$ be the corresponding $d_\lambda$-dimensional irreducible representation of $S_n$. I propose the problems of estimating the diagonal matrix elements of $\rho_\lambda(\pi)$ such that $s(\pi) = O(1)$, $l(\pi) = \Omega(n)$, and $r(\pi) = \Omega(n)$ as a possible class of instances of problem 1 not solvable classically in polynomial time.
Although this hypothesis contains many restrictions on $\pi$, it is clear that permutations satisfying all of these conditions exist. One simple example is the permutation that transposes 1 with $n$.
Characters of the Symmetric Group {#characters}
=================================
Because characters do not depend on a choice of basis, the computational complexity of estimating characters is especially interesting. Hepler[@Hepler] showed that computing the characters of the symmetric group exactly is \#P-hard. It is clear that an algorithm for efficiently approximating matrix elements of a representation can aid in approximating the corresponding character. Specifically, the quantum algorithm for problem 1 yields an efficient solution for the following problem.\
\
**Problem 2:** Approximate a character for the symmetric group $S_n$.\
**Input:** A Young diagram $\lambda$ specifying the irreducible representation, a permutation $\pi$ from $S_n$, and a polynomially small parameter $\epsilon$.\
**Output:** Let $\chi^\lambda(\pi)$ be the character, and let $d_\lambda$ be the dimension of the irreducible representation. The output $\chi_{\mathrm{out}}$ must satisfy $|\chi_{\mathrm{out}} -
\chi^\lambda(\pi)/d_\lambda| \leq \epsilon$ with high probability.\
However, as we show in this section, problem 2 is efficiently solvable using only classical randomized computation. Thus the techniques used for problem 1 do not offer immediate benefit for problem 2. Although this is in some sense a negative result, it provides an interesting illustration of the difference in complexity between estimating individual matrix elements of representations and estimating the characters.
We can reduce problem 2 to problem 1 by sampling uniformly at random from the standard Young tableaux compatible with Young diagram $\lambda$. For each Young tableau sampled we estimate the corresponding diagonal matrix element of $\rho_\lambda(\pi)$, as described in problem 1. By averaging the diagonal matrix elements for polynomially many samples, we obtain the normalized character to polynomial precision. The problem of sampling uniformly at random from the standard Young tableaux of a given shape is nontrivial but it has been solved. Greene, Nijenhuis, and Wilf proved in 1979 that their “hook-walk” algorithm produces the standard Young tableaux of any given shape with uniform probability[@Greene]. Examination of [@Greene] shows that the time needed by the hook-walk algorithm to produce a random standard Young tableaux compatible with a Young diagram of $n$ boxes is upper bounded by $O(n^2)$.
By averaging over diagonal matrix elements we lose some information contained in the individual matrix elements. This observation gives the intuition that it should often be harder to estimate individual matrix elements of a representation than to estimate its trace. Jones polynomials provide an example in which this intuition is confirmed. As discussed in [@Shor_Jordan], computing the Jones polynomial of the trace closure of a braid reduces to computing the normalized character of a certain representation of the braid group. The problem of additively approximating this normalized character is only DQC1-complete. In contrast, the individual matrix elements of this representation yield the Jones polynomial of the plat closure of the braid and are BQP-complete to approximate. We see a very similar phenomenon in the symmetric group; problem 2 is is solvable by a randomized polynomial-time classical algorithm, whereas problem 1 is not, as far as we know.
To construct a classical algorithm for problem 2, first recall that the character of a given group element depends only on the element’s conjugacy class. We can think of any $\pi \in S_n$ as acting on the set $\{1,2,\ldots,n\}$. The sizes of the orbits of the elements of $\{1,2,\ldots,n\}$ under repeated application of $\pi$ form a partition of the integer $n$. For example, consider the permutation $\pi \in S_5$ defined by $$\begin{array}{lllll}
\pi(1) = 2 & \pi(2) = 3 & \pi(3) = 1 & \pi(4) = 5 & \pi(5) = 4.
\end{array}$$ This divides the set $\{1,2,3,4,5\}$ into the orbits $\{1,2,3\}$ and $\{4,5\}$. Thus it corresponds to the partition $(3,2)$ of the integer $5$. Two permutations in $S_n$ are conjugate if and only if they correspond to the same partition. Thus, we can introduce the following notation. For any two partitions $\mu$ and $\lambda$ of $n$ define $\chi_\mu^\lambda$ to be the irreducible character of $S_n$ corresponding to the Young diagram of $\lambda$ evaluated at the conjugacy class corresponding to $\mu$.
To obtain an efficient classical solution to problem 2 we use the following theorem due to Roichman[@Roichman2].
\[Roichman\_rule\] For any partitions $\mu = (\mu_1,\ldots,\mu_l)$ and $\lambda =
(\lambda_1,\ldots,\lambda_k)$ of $n$, the corresponding irreducible character of $S_n$ is given by $$\chi_\mu^\lambda = \sum_\Lambda W_\mu(\Lambda)$$ where the sum is over all standard Young tableaux $\Lambda$ of shape $\lambda$ and $$W_\mu(\Lambda) = \prod_{\stackrel{1 \leq i \leq k}{i \notin B(\mu)}} f_\mu(i,\Lambda)$$ where $
B(\mu) = \{\mu_1 + \ldots + \mu_r | 1 \leq r \leq l \}
$ and $$f_\mu(i,\Lambda) = \left\{ \begin{array}{rl}
-1 & \textrm{box $i+1$ of $\Lambda$ is in the southwest of box $i$} \\
0 & \textrm{$i+1$ is in the northeast of $i$, $i+2$ is in the
southwest of $i+1$, and $i+1 \notin B(\mu)$} \\
1 & \textrm{otherwise}
\end{array} \right.$$
By using the hook walk algorithm we can sample uniformly at random from the standard Young tableaux $\Lambda$ of shape $\lambda$. By inspection of theorem \[Roichman\_rule\] we see that for each $\Lambda$ sampled we can compute $W_\mu(\Lambda)$ classically in $\mathrm{poly}(n)$ time. By averaging the values of $W_\mu(\Lambda)$ obtained during the course of the sampling we can thus obtain a polynomially accurate additive approximation the the normalized character, thereby solving problem 2.
Some readers may notice that theorem \[Roichman\_rule\] is similar in form to the much older and better-known Murnaghan-Nakayama rule. However, the Murnaghan-Nakayama rule is based on a sum over all “rim-hook tableaux” of shape $\lambda$ (see [@Roichman2]). It is not obvious how to sample uniformly at random from the rim-hook tableaux of a given shape. Thus, it is not obvious how to use the Murnaghan-Nakayame rule to obtain a probabilistic classical algorithm for problem 2.
Lie Groups
==========
Introduction {#lieintro}
------------
Because $U(n)$, $SU(n)$ and $SO(n)$ are compact linear groups, all of their representations are unitary given the right choice of basis[@Artin9]. In section \[Schurxform\] we described how to efficiently approximate the matrix elements from certain unitary irreducible representation of $U(n)$. Here we present a more direct approach to this problem, which can handle a larger set of representations of $U(n)$ and also extends to some other compact Lie groups: $SU(n)$ and $SO(n)$.
$U(n)$, $SU(n)$, and $SO(n)$ are subgroups of $GL(n)$, the group of all invertible $n \times n$ matrices. All of the irreducible representations of $U(n)$ and $SU(n)$ can be obtained by restricting the irreducible representations of $GL(n)$ to these subgroups. The best classical algorithms for computing irreducible representations of $GL(n)$ and $U(n)$ appear to be those of [@Burgisser] and [@Grabmeier]. These classical algorithms work by manipulating matrices whose dimension equals the dimension of the representation. Thus, they do not provide a polynomial time algorithm for computing matrix elements from representations whose dimension is exponentially large. The implementation of irreducible representations of $SO(3)$ and $SU(2)$ by quantum circuits has been studied previously by Zalka[@Zalka].
Gel’fand-Tsetlin representation of $U(n)$ {#Gelfand}
-----------------------------------------
The irreducible representations of the Lie group $U(n)$ are most easily described in terms of the corresponding Lie algebra $u(n)$. It is not necessary here delve into the theory of Lie groups and Lie algebras, but those who are interested can see [@Gilmore]. For now it suffices to say that $u(n)$ is the set of all antihermitian $n \times n$ matrices, and for any $u \in U(n)$ there exists $h \in u(n)$ such that $u = e^h$. Given any representation $a:u(n) \to u(m)$ one can construct a representation $A:U(n) \to U(m)$ as follows. For any $u \in U(n)$ find a corresponding $h(u) \in u(n)$ such that $e^h = u$, and set $A(u) = e^{a(h(u))}$. If $a$ is an antihermitian representation of $u(n)$ then $A$ is a unitary representation of $U(n)$. Furthermore, it is clear that $A$ is irreducible if and only if $a$ is irreducible.
It turns out that the irreducible representations of the algebra $gl(n)$ of all $n \times n$ complex matrices remain irreducible when restricted to the subalgebra $u(n)$. Furthermore, all of the irreducible representations of $u(n)$ are obtained this way. Let $E_{ij}$ be the $n \times n$ matrix with all matrix elements equal to zero except for the matrix element in row $i$, column $j$, which is equal to one. The set of all $n^2$ such matrices forms a basis over $\mathbb{C}$ for $gl(n)$. Thus to describe a representation of $gl(n)$ it suffices to describe its action on each of the $E_{ij}$ matrices.
As described in chapter 18, volume 3 of [@Vilenkin], explicit matrix representations of $gl(n)$ were constructed by Gel’fand and Tsetlin. (See also [@Gelfand_works].) In their construction, one thinks of the representation as acting on the formal span of a set of combinatorial objects called Gel’fand patterns. The Gel’fand-Tsetlin representations of $E_{p,p-1}$ and $E_{p-1,p}$ are sparse and simple to compute for all $p \in
\{2,3,\ldots,n\}$. This property makes the Gel’fand-Tsetlin representations particularly useful for quantum computation.
A Gel’fand pattern of width $n$ consists of $n$ rows of integers[^2]. The $j{^\mathrm{th}}$ row (from bottom) has $j$ entries $m_{1,j},m_{2,j},\ldots,m_{j,j}$. (Note that, in contrast to matrix elements, the subscripts on the entries of Gel’fand patterns conventionally indicate column first, then row.) These entries must satisfy $$m_{j,n+1} \geq m_{j,n} \geq m_{j+1,n+1}.$$ Gel’fand patterns are often written out diagrammatically. For example the Gel’fand pattern of width 3 with rows $$\begin{array}{ccc}
m_{1,3} = 4 & m_{2,3} = 1 & m_{3,3} = 0 \\
m_{1,2} = 3 & m_{2,2} = 0 & \\
m_{1,1} = 2 & &
\end{array}$$ is represented by the diagram $$\left( \begin{array}{ccccc}
4 & & 1 & & 0\\
& 3 & & 0 & \\
& & 2 & &
\end{array} \right).$$ This notation has the advantage that the entries that appear directly to the upper left and upper right of a given entry form the upper and lower bounds on the values that entry is allowed to take.
We call the top row of a Gel’fand pattern its weight[^3]. To each weight of width $n$ corresponds one irreducible representation of $gl(n)$. This irreducible representation acts on the formal span of all Gel’fand patterns with that weight (of which there are always finitely many). To describe the action of the representation of $gl(n)$ on these patterns let $$\begin{aligned}
\label{r1}
l_{p,q} & = & m_{p,q}-p \\
\label{r2}
a^j_{p-1} & = & \left| \frac{\prod_{i=1}^p (l_{i,p} - l_{j,p-1})
\prod_{i=1}^{p-2} (l_{i,p-2}-l_{j,p-1}-1)} {\prod_{i \neq j}
(l_{i,p-1} - l_{j,p-1}) (l_{i,p-1} - l_{j,p-1} -1)} \right|^{1/2} \\
\label{r3}
b^j_{p-1} & = & \left| \frac{\prod_{i=1}^p (l_{i,p} - l_{j,p-1} + 1)
\prod_{i=1}^{p-2} (l_{i,p-2}-l_{j,p-1})} {\prod_{i \neq j}
(l_{i,p-1} - l_{j,p-1}) (l_{i,p-1} - l_{j,p-1}+1) } \right|^{1/2}.\end{aligned}$$ Let $M$ be a Gel’fand pattern and let $M_p^{+j}$ be the Gel’fand pattern obtained from $M$ by replacing $m_{j,p}$ with $m_{j,p}+1$. Similarly, let $M_p^{-j}$ be the Gel’fand pattern in which $m_{j,p}$ has been replaced with $m_{j,p}-1$. The representation $a_{\vec{m}}$ of $gl(n)$ corresponding to weight $\vec{m} \in
\mathbb{Z}^n$ is defined by the following rules[^4], known as the Gel’fand-Tsetlin formulas. $$\begin{aligned}
\label{r4}
a_{\vec{m}}(E_{p-1,p})M & = & \sum_{j=1}^{p-1} a^j_{p-1} M_{p-1}^{+j} \\
\label{r5}
a_{\vec{m}}(E_{p,p-1})M & = & \sum_{j=1}^{p-1} b^j_{p-1} M_{p-1}^{-j} \\
\label{r6}
a_{\vec{m}}(E_{p,p})M & = & \left( \sum_{i=1}^p m_{i,p} -
\sum_{j=1}^{p-1} m_{j,p-1} \right) M\end{aligned}$$
These formulas give implicitly a representation for all of $gl(n)$, because any $E_{ij}$ can be obtained from operators of the form $E_{p-1,p}$ and $E_{p,p-1}$ by using the commutation relation $[E_{ik},E_{kl}] = E_{il}$. By restricting the representation $a_{\vec{m}}$ to antihermitian subalgebra of $gl(n)$ and taking the exponential, one obtains an irreducible group representation $A_{\vec{m}}:U(n) \to U(d_{\vec{m}})$, where $d_{\vec{m}}$ is the number of Gel’fand patterns with weight $\vec{m}$.
It should be noted that some references claim that the set of allowed weights for representations of $GL(n)$ is $\mathbb{N}^n$, whereas others identify, as we do, $\mathbb{Z}^n$ as the allowed set of weights. The reason for this is that irreducible representations of $GL(n)$ in which the entries $m_{n,1}, m_{n,2},
\ldots, m_{n,n}$ of the weight are all nonnegative are polynomial invariants[@Konig37]. That is, for any $g \in GL(n)$ and any $\vec{m} \in
\mathrm{N}^n$, each matrix element of the representation $\rho_{\vec{m}}(u)$ is a polynomial function of the $n^2$ matrix elements of $u$. The representations involving negative weights are called holomorphic representations, and many sources choose to neglect them. In the case that $\vec{m} \in \mathbb{N}^n$, the Gel’fand diagrams of width $n$ bijectively correspond to the semistandard Young tableaux of $n$ rows (*cf.* [@Difrancesco], pg. 517).
Quantum Algorithm for U(n) {#ualg}
--------------------------
In this section we obtain an efficient quantum circuit implementation of any irreducible representation of $U(n)$ in which the entries $m_{1,n},\ldots,m_{n,n}$ of the highest weight are all at most polynomially large. The dimension of such representations can grow exponentially with $n$. Unlike the Schur transform, the method here does not require $m_{1,n},\ldots,m_{n,n}$ to be nonnegative. We start by finding a quantum circuit implementing the Gel’fand-Tsetlin representation of an $n \times n$ unitary matrix of the form $$u_0 = \left[ \begin{array}{ccccc}
u_{11} & u_{12} & & & \\
u_{21} & u_{22} & & & \\
& & 1 & & \\
& & & \ddots & \\
& & & & 1
\end{array} \right],$$ where all off-diagonal matrix elements not shown are zero. After that we describe how to extend the construction to arbitrary $n \times n$ unitaries.
For a given weight $\vec{m} \in \mathbb{Z}^n$ we wish to implement the corresponding representation $A_{\vec{m}}(u_0)$ with a quantum circuit. To do this, we first find an $n \times n$ Hermitian matrix $H_0$ such that $e^{i H_0} = u_0$. It is not hard to see that $H_0$ can be computed in polynomial time and takes the form $$H_0 = \left[ \begin{array}{cccccc}
h_{11} & h_{12} & & & \\
h_{12}^* & h_{22} & & & \\
& & 0 & & \\
& & &\ddots & \\
& & & & 0
\end{array} \right].$$ Thus, $$\label{decomp1}
H_0 = h_{11} E_{11} + h_{12} E_{12} + h_{12}^* h_{21} + h_{22} E_{22}.$$ Hence, $$\label{decomp}
a_{\vec{m}}(H_0) = h_{11} a_{\vec{m}}(E_{11}) + h_{12}
a_{\vec{m}}(E_{12}) + h_{12}^* a_{\vec{m}}(E_{21}) + h_{22}
a_{\vec{m}}(E_{22}).$$
To implement $A_{\vec{m}}(u_0)$ with a quantum circuit, we think of $a_{\vec{m}}(H_0)$ as a Hamiltonian and simulate the corresponding unitary time evolution $e^{-i a_{\vec{m}}(H_0)t}$ for $t=-1$. The Hamiltonian $a_{\vec{m}}(H_0)$ has exponentially large dimension in the cases of computational interest. However, examination of equation \[decomp1\] shows that $H_0$ is a linear combination of operators of the form $E_{p,p-1}$ and $E_{p-1,p}$. Thus, by the Gel’fand-Tsetlin rules of section \[Gelfand\], $a_{\vec{m}}(H_0)$ is sparse and that its individual matrix elements are easy to compute. Under this circumstance, one can use the general method for simulating sparse Hamiltonians proposed in [@Aharonov_Tashma].
Define row-sparse Hamiltonians to be those in which each row has at most polynomially many nonzero entries. Further, define row-computable Hamiltonians to be those such that there exists a polynomial time algorithm which, given an index $i$, outputs a list of the nonzero matrix elements in row $i$ and their locations. Clearly, all row computable Hamiltonians are row-sparse. As shown in [@Aharonov_Tashma], the unitary $e^{-iHt}$ induced by any row-computable Hamiltonian can be simulated in polynomial time provided that the spectral norm $\|H\|$ and the time $t$ are at most polynomially large. We have already noted that $a_{\vec{m}}(H_0)$ is row-computable. $a_{\vec{m}}(H_0)$ is row sparse, and because we are considering only polynomial highest weight, the entries of the Gel’fand patterns, and hence the matrix elements of $a_{\vec{m}}(H_0)$ are only polynomially large. Thus, by Gershgorin’s circle theorem $\|a_{\vec{m}}(H_0)\|$ is at most $\mathrm{poly}(n)$.
Having shown that a quantum circuit of $\mathrm{poly}(n)$ gates can implement the Gel’fand-Tsetlin representation of an $n \times n$ unitary of the form $u_0$, the remaining task is to extend this to arbitrary $n \times n$ unitaries. Examination of the preceding construction shows that it works just the same for any unitary of the form $$u_p = {\mathds{1}}_p \oplus u \oplus {\mathds{1}}_{n-p-2},$$ where ${\mathds{1}}_p$ denotes the $p \times p$ identity matrix and $u$ is a $2
\times 2$ unitary. Corresponding to $u_p$ is again an antihermitian matrix of the form $$H_p = 0_p \oplus h \oplus 0_{n-p-2}$$ where $0_p$ is the $p \times p$ matrix of all zeros and $h$ is a $2
\times 2$ antihermitian matrix such that $e^{h} = u$. The only issue to worry about is whether $\|a_{\vec{m}}(H_p)\|$ is at most $\mathrm{poly(n)}$. By symmetry, one expects that $\|a_{\vec{m}}(H_p)\|$ should be independent of $p$. However, this is not obvious from examination of equations \[r1\] through \[r6\]. Nevertheless, it is true, as shown in appendix \[normindep\]. Thus, the norm is no different than in the $p=0$ case, *i.e.* $H_0$.
By concatenating the quantum circuits implementing $A_{\vec{m}}(u_1), A_{\vec{m}}(u_2),\ldots, A_{\vec{m}}(u_L)$, one can implement $A_{\vec{m}}(u_1 u_2 \ldots u_L)$. We next show that any $n
\times n$ unitary can be obtained as a product of $\mathrm{poly}(n)$ matrices, each of the form $u_p$, thus showing that the quantum algorithm is completely general and always runs in polynomial time.
For any $2 \times 2$ matrix $M$, let $\mathcal{E}(M,i,j)$ be the $n
\times n$ matrix in which $M$ acts on the $i{^\mathrm{th}}$ and $j{^\mathrm{th}}$ basis vectors. In other words, the $k,l$ matrix element of $\mathcal{E}(M,i,j)$ is $$\mathcal{E}(M,i,j)_{kl} = \left\{ \begin{array}{ll}
M_{11} & \textrm{if $k=i$ and $l=i$} \\
M_{12} & \textrm{if $k=i$ and $l=j$} \\
M_{21} & \textrm{if $k=j$ and $l=i$} \\
M_{22} & \textrm{if $k=j$ and $l=j$} \\
\delta_{kl} & \textrm{otherwise}
\end{array} \right. .$$ Thus $$u_p = \mathcal{E} \left( \left[ \begin{array}{cccc}
u_{11} & u_{12} \\
u_{21} & u_{22} \end{array} \right], m+1,m+2 \right).$$ Next note that, $$\begin{array}{l}
\mathcal{E} \left( \left[ \begin{array}{cccc}
u_{11} & u_{12} \\
u_{21} & u_{22} \end{array} \right],m+1,m+3 \right) = \vspace{3pt} \\
\mathcal{E} \left( \left[ \begin{array}{cccc}
0 & 1 \\
1 & 0 \end{array} \right],m+2,m+3 \right)
\mathcal{E} \left( \left[ \begin{array}{cccc}
u_{11} & u_{12} \\
u_{21} & u_{22} \end{array} \right], m+1,m+2 \right)
\mathcal{E} \left( \left[ \begin{array}{cccc}
0 & 1 \\
1 & 0 \end{array} \right],m+2,m+3 \right).
\end{array}$$ Thus the matrix $$\mathcal{E} \left( \left[ \begin{array}{cccc}
u_{11} & u_{12} \\
u_{21} & u_{22} \end{array} \right],m+1,m+3 \right)$$ is obtained as a product of three matrices of the form $u_p$. By repeating this conjugation process, one can obtain $$\label{twolevel}
\mathcal{E} \left( \left[ \begin{array}{cccc}
u_{11} & u_{12} \\
u_{21} & u_{22} \end{array} \right],i,j \right)$$ for arbitrary $i,j$ as a product of one matrix of the form $$\mathcal{E} \left( \left[ \begin{array}{cccc}
u_{11} & u_{12} \\
u_{21} & u_{22} \end{array} \right],p+1,p+2 \right)$$ for some $p$ and at most $O(n)$ matrices of the form $$\mathcal{E} \left( \left[ \begin{array}{cccc}
0 & 1 \\
1 & 0 \end{array} \right],q+1,q+2 \right)$$ with various $q$. A matrix of the form shown in equation \[twolevel\] is called a two-level unitary. As shown in section 4.5.1 of [@Nielsen_Chuang], any $n \times n$ unitary is obtainable as a product of $\mathrm{poly}(n)$ two-level unitaries. Thus we obtain $A_{\vec{m}}(U)$ for any $n \times n$ unitary $U$ using $\mathrm{poly}(n)$ quantum gates. One can then obtain any matrix element of $A_{\vec{m}}(U)$ to precision $\pm \epsilon$ by repeating the Hadamard test $O(1/\epsilon^2)$ times.
Special Orthogonal Group
------------------------
The special orthogonal group $SO(n)$ consists of all $n \times n$ real orthogonal matrices with determinant equal to one. The irreducible representations of $SO(n)$ are closely related to those of $U(n)$ and can also be expressed unitarily using a Gel’fand-Tsetlin basis. As discussed in chapter 18, volume 3 of [@Vilenkin], the nature of the representations of $SO(n)$ depends on whether $n$ is even or odd. Following [@Vilenkin] and [@Gelfand_works], we therefore introduce an integer $k$ and consider $SO(2k+1)$ and $SO(2k)$ separately.
The irreducible representations of $SO(2k+1)$ are in bijective correspondence with the set of allowed weight vectors $\vec{m}$ consisting of $k$ entries, each of which is an integer or half-integer. Furthermore, the entries must satisfy $$m_{1,n} \geq m_{2,n} \geq \ldots \geq m_{k,n} \geq 0.$$ The irreducible representations of $SO(2k)$ correspond to the weight vectors $\vec{m}$ with $k-1$ entries, each of which must be an integer or half integer, and which must satisfy $$m_{1,n} \geq m_{2,n} \geq \ldots \geq m_{k-1,n} \geq |m_{k,n}|.$$
As in the case of $U(n)$, the set of allowed Gel’fand patterns is determined by rules for how a row can compare to the one above it. For $SO(n)$ these rules are slightly more complicated, and the rule for the $j{^\mathrm{th}}$ row depends on whether $j$ is odd or even. Specifically the even rule for $j=2k$ is $$m_{1,2k+1} \geq m_{1,2k} \geq m_{2,2k+1} \geq m_{2,3k} \geq \ldots
\geq m_{k,2k+1} \geq m_{k,2k} \geq -m_{k,2k-1},$$ and the odd rule for $j=2k-1$ is $$m_{1,2k} \geq m_{1,2k-1} \geq m_{2,2k} \geq m_{2,2k-1} \geq \ldots
\geq m_{k-1,2k} \geq m_{k-1,2k-1} \geq |m_{k,2k}|.$$ The Lie algebra $so(n)$ corresponding to the Lie group $SO(n)$ is the algebra of all antisymmetric $n \times n$ matrices. For any $G \in SO(n)$ there exists a $g \in so(n)$ such that $e^{g} =
G$. The Lie algebra $so(n)$ is the space of all $n \times n$ real traceless antisymmetric matrices. Thus it is spanned by operators of the form $$I_{k,i} = E_{i,k}-E_{k,i} \quad 1 \leq i < k \leq n.$$ We can fully specify a representation of $so(n)$ by specifying the representations of the operators of the form $I_{q+1,q}$ because these generate $so(n)$. That is, any element of $so(n)$ can be obtained as a linear combination of commutators of such operators. The Gel’fand-Tsetlin representation $b_{\vec{m}}$ of these operators depends on whether $q$ is even or odd, and is given by the following formulas. $$\begin{aligned}
A_{2p}^j(M) & = & \frac{1}{2} \left| \frac{\prod_{r=1}^{p-1}
\left[(l_{r,2p-1}-\frac{1}{2})^2-(l_{j,2p}+\frac{1}{2})^2\right]
\prod_{r=1}^p \left[(l_{r,2p+1}-\frac{1}{2})^2 -
(l_{j,2p}+\frac{1}{2})^2 \right]}
{\prod_{r \neq j} (l_{r,2p}^2 - l_{j,2p}^2)
(l_{r,2p}^2-(l_{j,2p}+1)^2)} \right|^{1/2} \\
B_{2p+1}^j(M) & = & \left| \frac{\prod_{r=1}^p (l_{r,2p}^2-l_{j,2p+1}^2)
\prod_{r=1}^{p+1} (l_{r,2p+2}^2-l_{j,2p+1}^2)}
{l_{j,2p+1}^2 (4l_{j,2p+1}^2-1) \prod_{r \neq j}
(l_{r,2p+1}^2-l_{j,2p+1}^2) (l_{j,2p+1}^2 - (l_{r,2p+1}-1)^2)}
\right|^{1/2} \\
C_{2p}(M) & = & \frac{\prod_{r=1}^p l_{r,2p} \prod_{r=1}^{p+1}
l_{r,2p+2}}{\prod_{r=1}^p l_{r,2p+1} (l_{r,2p+1}-1)} \\
b_{\vec{m}}(I_{2p+1,2p}) M & = & \sum_{j=1}^p A_{2p}^j(M) M_{2p}^{+j} -
\sum_{j=1}^p A_{2p}^j(M_2p^{-j}) M_{2p}^{-j} \\
b_{\vec{m}}(I_{2p+2,2p+1}) M & = & \sum_{j=1}^p B_{2p+1}^j(M)
M_{2p+1}^{+j} - \sum_{j=1}^p B_{2p+1}^j(M_{2p+1}^{-j}) M_{2p+1}^{-j} +
i C_{2p}(M) M\end{aligned}$$
By applying these rules to the set of allowed Gel’fand patterns described above one obtains the irreducible representations of the algebra $so(n)$. By exponentiating these, one then obtains the irreducible representations of the group $SO(n)$. Thus the quantum algorithm for approximating the matrix elements of the irreducible representations of $SO(n)$ is analogous to that for $U(n)$.
Special Unitary Group {#sun}
---------------------
The irreducible representations of $SU(n)$ can be easily constructed from the irreducible representations of $U(n)$, using the following facts taken from chapter 10 of [@BR]. The representations of $U(n)$ can be partitioned into a set of equivalence classes of projectively equivalent representations. Two representations of $U(n)$ with weights $\vec{l} =(l_1,l_2,\ldots,l_n)$ and $\vec{m}=(m_1,m_2,\ldots,m_n)$ are projectively equivalent if and only if there exists some integer $s$ such that $m_i = l_i + s$ for all $1 \leq i \leq n$. Any irreducible representation of $U(n)$ remains irreducible when restricted to $SU(n)$. Furthermore, by choosing one representative from each class of projectively equivalent representations of $U(n)$ and restricting to $SU(n)$ one obtains a complete set of inequivalent irreducible representations of $SU(n)$. The Lie algebra $su(n)$ corresponding to the Lie group $SU(n)$ is easily characterized; it is the space of all traceless $n
\times n$ antihermitian matrices. Thus the matrix elements of the irreducible representations of $SU(n)$ are obtained by essentially the same quantum algorithm given for $U(n)$ in section \[ualg\].
Characters of Lie Groups
------------------------
As always, an algorithm for approximating matrix elements immediately gives us an algorithm for approximating the normalized characters. However, the characters of $U(n)$, $SU(n)$, and $SO(n)$ are classically computable in $\mathrm{poly}(n)$ time. As discussed in [@Fulton_Harris], the characters of any compact Lie group are given by the Weyl character formula. In general this formula may involve sums of exponentially many terms. However, in the special cases of $U(n)$, $SU(n)$, and $SO(n)$ the formula reduces to simpler forms[@Fulton_Harris], given below.
Because characters depend only on conjugacy class, the character $\chi_{\vec{m}}(u)$ depends only on the eigenvalues of $u$. For $u \in
U(n)$ let $\lambda_1,\ldots,\lambda_n$ denote the eigenvalues. Let $\vec{m} = (m_1,m_2,\ldots,m_n) \in \mathbb{Z}^n$ be the weight of a representation of $U(n)$. Let $$\label{l}
l_i = m_i + n - i$$ for each $i \in \{1,2,\ldots,n\}$. The character of the representation of weight $\vec{m}$ is $$\chi^{U(n)}_{\vec{m}}(u) = \frac{\det A}{\det B}$$ where $A$ and $B$ are the following $n \times n$ matrices $$\begin{aligned}
A_{ij} & = & \lambda_i^{l_j} \\
B_{ij} & = & \lambda_i^{n-j}.\end{aligned}$$
This formula breaks down if $u$ has a degenerate spectrum. However, the value of the character for degenerate $u$ can be obtained by taking the limit as some eigenvalues converge to the same value. As shown in [@Weyl7], one can obtain the dimension $d_{\vec{m}}$ of the representation corresponding to a given weight $\vec{m}$ by calculating $\lim_{u \to {\mathds{1}}} \chi_{\vec{m}}(u)$. Specifically, by choosing $\lambda_j = e^{ij \epsilon}$ for each $1 \leq j \leq n$ and taking the limit as $\epsilon \to 0$ one obtains $$d_{\vec{m}} = \frac{\prod_{i<j} (l_j - l_i)}
{\prod_{i<j} (j-i)},$$ where $l_i$ is as defined in equation \[l\].
As discussed in section \[sun\], the irreducible representations of $SU(n)$ are restrictions of irreducible representations of $U(n)$, therefore the characters of $SU(n)$ are given by the same formula as the characters of $U(n)$.
$SO(n)$ consists of real matrices. The characteristic polynomials of these matrices have real coefficients, and thus their roots come in complex conjugate pairs. Thus, the eigenvalues of an element $g \in
SO(2k+1)$ take the form $$\lambda_1, \lambda_2, \ldots, \lambda_k, 1, \lambda_1^*, \lambda_2^*,
\ldots, \lambda_k^*,$$ and for $g \in SO(2k)$, the eigenvalues take the form $$\lambda_1, \lambda_2, \ldots, \lambda_k, \lambda_1^*, \lambda_2^*,
\ldots, \lambda_k^*.$$ As discussed in [@Fulton_Harris], the characters of the special orthogonal group are given by $$\chi_{\vec{m}}^{SO(2k+1)}(g) = \frac{\det C}{\det D}$$ and $$\chi_{\vec{m}}^{SO(2k)}(g) = \frac{\det E + \det F}
{\det G}$$ where $C$ and $D$ are the following $k \times k$ matrices $$\begin{aligned}
C_{ij} & = & \lambda_j^{m_i+n-i+1/2} -
\lambda_j^{-(m_i+n-i+1/2)} \\
D_{ij} & = & \lambda_j^{n-i+1/2} - \lambda_j^{-(n-i+1/2)}\end{aligned}$$ and $E,F,G$ are the following $(k-1) \times (k-1)$ matrices $$\begin{aligned}
E_{ij} & = & \lambda_j^{l_i} + \lambda_j^{-l_i}\\
F_{ij} & = & \lambda_j^{l_i} - \lambda_j^{-l_i}\\
G_{ij} & = & \lambda_j^{n-i} + \lambda_j^{-(n-i)},\end{aligned}$$ where $l_i$ is as defined in equation \[l\].
As with $U(n)$, the character of any element with a degenerate spectrum can be obtained by taking an appropriate limit.
Open Problems Regarding Lie groups
----------------------------------
The quantum circuits presented in the preceeding sections efficiently implement the irreducible representations of $U(n)$, $SU(n)$, and $SO(n)$ that have polynomial highest weight and polynomial $n$. It is an interesting open problem to implement irreducible representations with quantum circuits that scale polynomially in the number of digits used to specify the highest weight. Alternatively, one could try to implement an Schur transform to handle exponential highest weight, which is also an open problem. It is even concievable that Schur-like transforms could be efficiently implemented for exponential $n$. That is, there could exist a quantum circuit of $\mathrm{polylog}(n)$ gates implementing a unitary transform $V$ such that for any $U \in
U(n)$, $V U V^{-1}$ is a direct sum of irreducible representations of $U$. Of course, if $n$ is exponentially large, than we cannot have an explicit description of $U$, rather the group element $U$ could itself be defined by a quantum circuit.
A completely different open problem is presented by the symplectic group. Having constructed quantum circuits for $SO(n)$ and $SU(n)$, the symplectic group is the only “classical” Lie group remaining to be analyzed. Thus it is natural to ask whether its irreducible representations can be efficiently implemented by quantum circuits. Two different groups can go by the name symplectic group depending on the reference. Connected non-compact simple Lie groups have no nontrivial finite-dimensional unitary representations (see [@BR], theorem 8.1.2). This applies to one of the groups that goes by the name of symplectic. On the other hand, the irreducible representations of the compact symplectic group seem promising for implementation by quantum circuits. The main task seems to be finding a basis for these representations that is subgroup adapted and makes the representations unitary. A non-unitary subgroup-adapted basis is given in [@Molev].
Alternating Group {#althyp}
=================
In section \[fourier\], we described a method to approximate matrix elements of the irreducible representations of the symmetric group using the symmetric group quantum Fourier transform. Here we take a more direct approach to this problem, which extends to the alternating group. To do this we must first explicitly describe the Young-Yamanouchi representation of the symmetric group.
Young-Yamanouchi Representation {#Young}
-------------------------------
For a given Young diagram $\lambda$, let $\mathcal{V}_\lambda$ be the vector space formally spanned by all standard Young tableaux compatible with $\lambda$. For example, if $$\lambda = \begin{array}{l} \includegraphics[width=0.3in]{tetris.eps}
\end{array}$$ then $\mathcal{V}_\lambda$ is the 3-dimensional space consisting of all formal linear combinations of $$\begin{array}{l} \includegraphics[width=1.4in]{threetabs.eps}
\end{array}$$ For any given Young diagram $\lambda$, the corresponding irreducible representation in the Young-Yamanouchi basis is a homomorphism $\rho_\lambda$ from $S_n$ to the group of orthogonal linear transformations on $\mathcal{V}_\lambda$. It is not easy to directly compute $\rho_\lambda(\pi)$ for an arbitrary permutation $\pi$. However, it is much easier to compute the representation of a transposition of neighbors. That is, we imagine the elements of $S_n$ as permuting a set of objects $1,2,\ldots,n$, arranged on a line. A neighbor transposition $\sigma_i$ swaps objects $i$ and $i+1$. It is well known that the set $\{ \sigma_1,\sigma_2,\ldots,\sigma_{n-1} \}$ generates $S_n$.
The matrix elements for the Young-Yamanouchi representation of transpositions of neighbors can be obtained using a single simple rule: Let $\Lambda$ be any standard Young tableau compatible with Young diagram $\lambda$ then $$\label{rule}
\rho_\lambda(\sigma_i) \Lambda = \frac{1}{\tau_i^\Lambda} \Lambda +
\sqrt{1-\frac{1}{(\tau_i^\Lambda)^2}} \Lambda',$$ where $\Lambda'$ is the Young tableau obtained from $\Lambda$ by swapping boxes $i$ and $i+1$, and $\tau_i^\Lambda$ is the axial distance from box $i+1$ to box $i$. That is, we are allowed to hop vertically or horizontally to nearest neighbors, and $\tau$ is the number of hops needed to get from box $i+1$ to box $i$, where going down or left counts as $+1$ hop and going up or right counts as $-1$ hop. To illustrate the use of equation \[rule\], some examples are given in figure \[examples\].
![\[examples\] The above matrices are irreducible representations in the Young-Yamanouchi basis with Young diagram {width="0.18in"}. Here $\sigma_i$ is the permutation in $S_4$ that swaps $i$ with $i+1$.](examples_nocap.eps){width="85.00000%"}
In certain cases, starting with a standard Young tableau and swapping boxes $i$ and $i+1$ does not yield a standard Young tableau, as illustrated below. $$\includegraphics[width=2.7in]{validity.eps}$$ Some thought shows that all such cases are of one of the two types shown above. In both of these types, the axial distance is $\pm
1$. By equation \[rule\], the coefficient on the invalid Young tableau is $\sqrt{1-\frac{1}{(\pm 1)^2}} = 0$. Thus the representation lies strictly within the space of standard Young tableaux.
Direct Quantum Algorithm for $S_n$ {#algorithm}
----------------------------------
We can directly implement the irreducible representations of $S_n$ by first decomposing the given permutation into a product of transposition of neighbors. The classical bubblesort algorithm achieves this efficiently. For any permutation in $S_n$, it yields a decomposition consisting of at most $O(n^2)$ transpositions. As seen in the previous section, the Young-Yamanouchi representation of any transposition is a direct sum of $2 \times 2$ and $1 \times 1$ blocks, and the matrix elements of these blocks are easy to compute. As shown in [@Aharonov_Tashma], any unitary with these properties may be implemented by a quantum circuit with polynomially many gates. By concatenating at most $O(n^2)$ such quantum circuits we obtain the representation of any permutation in $S_n$. The Hadamard test allows a measurement to polynomial precision of the matrix elements of this representation.
Algorithm for Alternating Group {#alt}
-------------------------------
Any permutation $\pi$ corresponds to a permutation matrix with matrix element $i,j$ given by $\delta_{\pi(i),j}$. The determinant of any permutation matrix is $\pm 1$, and is known as the sign of the permutation. The permutations of sign $+1$ are called even, and the permutations of sign $-1$ are called odd. This is because a transposition has determinant $-1$, and therefore any product of an odd number of transpositions is odd and any product of an even number of transpositions is even.
The even permutations in $S_n$ form a subgroup called the alternating group $A_n$, which has size $n!/2$. $A_n$ is a simple group (*i.e.* it contains no normal subgroup) and it is the only normal subgroup of $S_n$ other than $\{ {\mathds{1}}\}$ and $S_n$. As one might guess, the irreducible representations of the alternating group are closely related to the irreducible representations of the symmetric group. Consequently, as shown in this section, the quantum algorithm of section \[algorithm\] can be easily adapted to approximate any matrix element of any irreducible representation of $A_n$ to within $\pm \epsilon$ in $\mathrm{poly}(n,1/\epsilon)$ time.
Explicit orthogonal matrix representations of the alternating group are worked out in [@Thrall] and recounted nicely in [@Headley]. Any representation $\rho$ of $S_n$ is automatically also a representation of $A_n$. However an irreducible representation $\rho$ of $S_n$ may no longer be irreducible when restricted to $A_n$. Each irreducible representation of $S_n$ either remains irreducible when restricted to $A_n$ or decomposes into a direct sum of two irreducible representations of $A_n$. All of the irreducible representations of $A_n$ are obtained in this way.
![\[conjugate\] To obtain the conjugate $\hat{\lambda}$ of Young diagram $\lambda$, reflect $\lambda$ about its diagonal. In other words the number of boxes in the $i{^\mathrm{th}}$ column of $\hat{\lambda}$ is equal to the number of boxes in the $i{^\mathrm{th}}$ row of $\lambda$.](conjugate.eps){width="30.00000%"}
The conjugate of Young diagram $\lambda$ is obtained by reflecting $\lambda$ about the main diagonal, as shown in figure \[conjugate\]. If $\lambda$ is not self-conjugate then the representation $\rho_{\lambda}$ of $S_n$ remains irreducible when restricted to $A_n$. In this case we can simply use the algorithm of section \[algorithm\]. If $\lambda$ is self-conjugate then the representation $\rho_\lambda$ of $S_n$ becomes reducible when restricted to $A_n$. It is a direct sum of two irreducible representations of $A_n$, called $\rho_{\lambda+}$ and $\rho_{\lambda-}$. The two corresponding invariant subspaces of the reducible representation are the $+1$ and $-1$ eigenspaces, respectively, of the “associator” operator $S$ defined as follows.
![\[typewriter\] For a given Young diagram, there is a unique Young tableau in “typewriter” order, in which the boxes are numbered from left to right across the top row then from left to right across the next row, and so on, as illustrated in the example above.](typewriter.eps){width="12.00000%"}
Let $\lambda$ be a self-conjugate Young diagram of $n$ boxes. Let $\Lambda_0$ be the “typewriter-order” Young tableau obtained by numbering the boxes from left to right across the first row, then left to right across the second row, and so on, as illustrated in figure \[typewriter\]. For any standard Young tableau $\Lambda$ of shape $\lambda$, let $w_\Lambda \in S_n$ be the permutation that brings the boxes into typewriter order. That is, $w_{\Lambda} \Lambda =
\Lambda_0$. Let $\hat{\Lambda}$ be the conjugate of $\Lambda$, obtained by reflecting $\Lambda$ about the main diagonal. If $\Lambda$ is standard then so is $\hat{\Lambda}$. Let $d(\lambda)$ be the length of the main diagonal of $\lambda$. $S$ is the linear operator on $\mathcal{V}_\lambda$ defined by $$\label{S}
S \Lambda = i^{(n-d(\lambda))/2} \mathrm{sign}(w_\Lambda) \hat{\Lambda}.$$
An orthonormal basis for each of the eigenspaces of $S$ can be easily constructed from the Young-Yamanouchi basis. When $(n-d(\lambda))/2$ is odd, every standard Young tableau $\Lambda$ of shape $\lambda$ has the property $\mathrm{sign}(w_\Lambda) =
-\mathrm{sign}(w_{\hat{\Lambda}})$, and $S$ is a direct sum of $2
\times 2$ blocks of the form $$\left[ \begin{array}{cc} 0 & -i \\
i & 0
\end{array} \right]$$ interchanging $\Lambda$ and $\hat{\Lambda}$. In this case, the linear combinations $
\frac{1}{\sqrt{2}} (\Lambda + i \hat{\Lambda})
$ for each conjugate pair of standard Young tableaux form an orthonormal basis for the $+1$ eigenspace of $S$, and the linear combinations $
\frac{1}{\sqrt{2}}(\Lambda - i \hat{\Lambda})
$ form an orthonormal basis for the $-1$ eigenspace of $S$. Similarly, when $(n-d(\lambda))/2$ is even, $\mathrm{sign}(w_\Lambda) =
\mathrm{sign}(w_{\hat{\Lambda}})$ for all standard Young tableaux $\Lambda$ of shape $\lambda$. Thus $S$ is a direct sum of $2 \times 2$ blocks of the form $$\left[ \begin{array}{cc} 0 & -1 \\
-1 & 0
\end{array} \right]$$ interchanging $\Lambda$ and $\hat{\Lambda}$. In this case the linear combinations $
\frac{1}{\sqrt{2}} (\Lambda - \hat{\Lambda})
$ form an orthonormal basis for the $+1$ eigenspace of $S$ and the linear combinations $
\frac{1}{\sqrt{2}} (\Lambda + \hat{\Lambda})
$ form an orthonormal basis for the $-1$ eigenspace of $S$.
Suppose $\lambda$ is self-conjugate and $(n-d(\lambda))/2$ is even. Any matrix element of the irreducible representation $\rho_{\lambda+}$ of $A_n$ is given by $$\frac{1}{2} (\Lambda + \hat{\Lambda}) \rho_\lambda(\pi) (\Gamma + \hat{\Gamma}),$$ where $\Lambda, \Gamma$ is some pair of standard Young tableaux and $\pi$ is some element of $A_n$. This is a linear combination of only four Young-Yamanouchi matrix elements of $\rho_\lambda(\pi)$. One can use the algorithm of section \[algorithm\] to calculate each of these and then simply add them up with the appropriate coefficients. The cases where $(n-d(\lambda))/2$ is odd and/or we want a matrix element of $\rho_{\lambda-}$ are analogous.
Acknowledgements
================
I thank Greg Kuperberg and anonymous referees for suggesting the approaches described in sections \[fourier\] and \[Schurxform\]. I thank Daniel Rockmore, Cris Moore, Andrew Childs, Aram Harrow, John Preskill, and Jeffrey Goldstone for useful discussions. I thank Isaac Chuang and Vincent Crespi for comments that helped to inspire this work, and anonymous referees for useful comments. Parts of this work were completed the Center for Theoretical physics at MIT, the Digital Materials Laboratory at RIKEN, and the Institute for Quantum Information at Caltech. I thank these institutions as well as the Army Research Office (ARO), the Disruptive Technology Office (DTO), the Department of Energy (DOE), and Franco Nori and Sahel Ashab at RIKEN.
$\|a_{\vec{m}}(H_p)\|$ is independent of $p$ {#normindep}
============================================
As shown in section \[ualg\], the irreducible representation of an arbitrary $u \in U(n)$ with weight $\vec{m}$ can be computed by simulating the time evolution according to a series of Hamiltonians of the form $A_{\vec{m}}(H_p)$, where $A_{\vec{m}}$ is the Gel’fand-Tsetlin representation of the Lie algebra $su(n)$ and $$H_p = 0_p \oplus h \oplus 0_{n-p-2},$$ where $h$ is a $2 \times 2$ antihermitian matrix. The quantum algorithm for simulating these Hamiltonians require that $\|A_{\vec{m}}(H_p)\|$ be at most $\mathrm{poly}(n)$. In section \[ualg\] we showed this to be the case for $p=0$. Here we prove it for all $p$ by showing:
Let $h$ be a fixed $2 \times 2$ antihermitian matrix and let $H_p =
0_p \oplus h \oplus 0_{n-p-2}$. Let $a_{\vec{m}}$ be the Gel’fand-Tsetlin representation of $su(n)$ with weight $\vec{m}$. Then $\|a_{\vec{m}}(H_p)\|$ is independent of $p$.
Let $U_p^k = e^{k H_p}$. Then $$U_p^k = {\mathds{1}}_p \oplus e^{k h} \oplus {\mathds{1}}_{n-p-2}.$$ Thus for any $0 \leq q \leq n$, there exists $V \in U(n)$ such that $$\label{permutebasis}
U_q^k = V U_p^k V^{-1}.$$ Specifically, $V$ is just a permutation matrix. Let $A_{\vec{m}}$ be the Gel’fand-Tsetlin representation of $SU(n)$. That is, $$A_{\vec{m}}(U_q^k) = e^{a_{\vec{m}}(k H_q)}.$$ Thus $$\begin{aligned}
\left\| \frac{{\mathrm{d}}}{{\mathrm{d}}k} A_{\vec{m}}(U_p^k) \right\| & = &
\| a_{\vec{m}}(H_p) e^{k a_{\vec{m}}(H_p)} \| \nonumber \\
\label{normderiv}
& = & \|a_{\vec{m}}(H_p)\|.\end{aligned}$$ Here we have used the fact that $A_{\vec{m}}$ is a unitary representation. Similarly, $$\|a_{\vec{m}}(H_q)\| = \left\| \frac{{\mathrm{d}}}{{\mathrm{d}}k} A_{\vec{m}}(U_q^k) \right\|.$$ Using equation \[permutebasis\], this is equal to $$\left\| \frac{{\mathrm{d}}}{{\mathrm{d}}k} A_{\vec{m}}(V U_p^k V^{-1}) \right\|.$$ Because $A_{\vec{m}}$ is a group homomorphism and $V$ is independent of $k$ this is equal to $$\left\| A_{\vec{m}}(V) \left( \frac{{\mathrm{d}}}{{\mathrm{d}}k} A_{\vec{m}}(U_p^k) \right)
A_{\vec{m}}(V)^{-1} \right\|.$$ Because $A_{\vec{m}}$ is a unitary representation this is equal to $$\left\| \frac{{\mathrm{d}}}{{\mathrm{d}}k} A_{\vec{m}}(U_p^k) \right\|.$$ By equation \[normderiv\] this is equal to $\| a_{\vec{m}}(H_p) \|$.
[^1]: Parts of this work were completed at MIT’s Center for Theoretical Physics and RIKEN’s Digital Materials Laboratory.
[^2]: Some sources omit the top row, as it is left unchanged by the action of the representation.
[^3]: It is actually the *highest* weight of the representation[@Vilenkin], but for brevity I just call it the weight throughout this paper.
[^4]: Warning: [@Vilenkin] contains a misprint, in which the sums in equations \[r4\] and \[r5\] are taken up to $j=p$ instead of $j=p-1$.
|
---
abstract: 'In this paper we study existence, regularity, and approximation of solution to a fractional semilinear elliptic equation of order $s \in (0,1)$. We identify minimal conditions on the nonlinear term and the source which leads to existence of weak solutions and uniform $L^\infty$-bound on the solutions. Next we realize the fractional Laplacian as a Dirichlet-to-Neumann map via the Caffarelli-Silvestre extension. We introduce a first-degree tensor product finite elements space to approximate the truncated problem. We derive a priori error estimates and conclude with an illustrative numerical example.'
author:
- 'Harbir Antil [^1]'
- 'Johannes Pfefferer [^2]'
- 'Mahamadi Warma [^3]'
bibliography:
- 'lit.bib'
title: 'A note on semilinear fractional elliptic equation: analysis and discretization[^4]'
---
35S15, 26A33, 65R20, 65N12, 65N30
Spectral fractional Laplace operator, semi-linear elliptic problems, regularity of weak solutions, discretization, error estimates.
Introduction
============
Let $\Omega\subset\RR^N$ be a bounded open set with boundary $\pOm$. In this paper we investigate the existence, regularity, and finite element approximation of weak solutions of the following semilinear Dirichlet problem
$$\label{ellip-pro}
\begin{cases}
(-\Delta_D)^su +f(x,u)=g\;\;&\mbox{ in }\;\Omega,\\
u=0&\mbox{ on }\;\pOm.
\end{cases}$$
Here, $g$ is a given measurable function on $\Omega$, $f:\;\Omega\times\RR\to\RR$ is measurable and satisfies certain conditions (that we shall specify later), $0<s<1$ and $(-\Delta_D)^s$ denotes the spectral fractional Laplace operator, that is, the fractional $s$ power of the realization in $L^2(\Om)$ of the Laplace operator with zero Dirichlet boundary condition on $\pOm$.
Notice that $(-\Delta_D)^s$ is a nonlocal operator and $f$ is nonlinear with respect to $u$. This makes it challenging to identify the minimum assumptions on $\Omega$, $f$ and $g$ in the study of the existence, uniqueness, regularity and the numerical analysis of the system. The later is the main objective of our paper.
When $f$ is linear in $u$, problems of type have received a great deal of attention. See [@Caf3] for results in $\RR^N$ and [@NALD; @CaSt; @CDDS:11; @ST:10] for results in bounded domains. However, [@Caf3; @CaSt; @ST:10] only deal with the linear problems, on the other hand [@NALD; @CDDS:11] deal with a different class of semilinear problems and assumes $\Omega$ and $f$ to be smooth. We refer to [@NOS] where a numerical scheme to approximate the linear problem was first established. To the best of our knowledge our paper is the first work addressing the existence, regularity, and numerical approximation of with almost minimum conditions on $\Omega$, $f$ and $g$.
We use *Musielak-Orlicz spaces*, endowed with Luxemburg norm, to deal with the nonlinearity. Using the Browder-Minty theorem, we first show the existence and uniqueness of a weak solution. Additional integrability condition on $g$ brings the solution in $L^\infty(\Omega)$. For the latter result, we apply a well-known technique due to Stampacchia. However, when $\Omega$ has a Lipschitz continuous boundary and $f$ is locally Lipschitz continuous we illustrate the regularity shift. For completeness we also derive the Hölder regularity of solution for smooth $\Omega$.
Numerical realization of nonlocal operators poses various challenges for instance, direct discretization of , by using finite elements, requires access to eigenvalues and eigenvectors of $(-\Delta_D)$ which is an intractable problem in general domains. Instead we use the so-called Caffarelli-Silvestre extension to realize the fractional power $(-\Delta_D)^s$. Such an approach is a more suitable choice for numerical methods, see [@NOS] for the linear case. The extension idea was introduced by Caffarelli and Silvestre in $\RR^N$ [@Caf3] and its extensions to bounded domains is given in e.g. [@CDDS:11; @ST:10]. The extension says that fractional powers $(-\Delta_D)^s$ of the spatial operator $-\Delta_D$ can be realized as an operator that maps a Dirichlet boundary condition to a Neumann condition via an extension problem on the semi-infinite cylinder $\mathcal{C} = \Omega \times (0,\infty)$, that is, a Dirichlet-to-Neumann operator. See Section \[s:CS\] for more details.
We derive a priori finite element error estimates for our numerical scheme. Our proof requires the solution to a discrete linearized problem to be uniformly bounded in $L^\infty(\Omega)$, which can be readily derived by using the inverse estimates and under the assumption $s>(N-2)/2$. As a result, when $N \ge 3$, we only have error estimates in case $s>(N-2)/2$. We notice that no restriction on $s$ is needed when $N \le 2$. In summary we are only limited by the $L^\infty(\Omega)$ regularity of the solution to a discrete linearized problem when $N \ge 3$.
Recently, fractional order PDEs have made a remarkable appearance in various scientific disciplines, and have received a great deal of attention. For instance, image processing [@GH:14]; nonlocal electrostatics [@ICH]; biophysics [@bio]; chaotic dynamical systems [@MR1604710]; finance [@MR2064019]; mechanics [@atanackovic2014fractional], where they are used to model viscoelastic behavior [@MR2035411], turbulence [@wow; @NEGRETE] and the hereditary properties of materials [@MR1926470]; diffusion processes [@Abe2005403; @PSSB:PSSB2221330150], in particular processes in disordered media, where the disorder may change the laws of Brownian motion and thus leads to anomalous diffusion [@MR1736459; @MR1081295] and many others [@BV; @MR2025566]. In view of the fact that most of the underlying physics in the aforementioned applications can be described by nonlinear PDEs, it is natural to analyze a prototypical semilinear PDE given in .
The paper is organized as follows: In Section \[s:spec\] we provide definitions of the fractional order Sobolev spaces and the spectral Dirichlet Laplacian. These results are well known. Section \[s:orl\] is devoted to essential properties of Orlicz spaces. We also specify assumptions on $f$ and state several embedding results which are due to Sobolev embedding theorems. Our main results begin in Section \[sec:weaksol\], where we first show existence and uniqueness of a weak solution $u$ to the system in Proposition \[prop-exis\] and later with additional integrability assumption on $g$ we obtain uniform $L^\infty$-bound on $u$ in Theorem \[theo-bound\]. When $\Omega$ is smooth we derive the Hölder regularity of $u$ in Corollary \[cor-212\]. In case $\Omega$ has a Lipschitz continuous boundary and $f$ is locally Lipschitz continuous we deduce regularity shift on $u$ in Corollary \[ellip-regula\]. We state the extension problem in Section \[s:CS\] and show the existence and uniqueness of a solution $\mathcal{U}$ to the extension problem on $\mathcal{C}:=\Omega\times(0,\infty)$ in Lemma \[lem:CS\]. We notice that $u = \mathcal{U}(\cdot,0)$. In Section \[s:disc\] we begin the numerical analysis of our problem. We first derive the energy norm and the $L^2$-norm a priori error estimates for an intermediate linear problem in Lemma \[lemma:linear\]. This is followed by a uniform $L^\infty$-bound on the discrete solution to an intermediate linear problem in Lemma \[lemma:Linftybounded\]. We conclude with the error estimates for our numerical scheme to solve in Theorem \[theorem:semilinear\] and a numerical example.
Analysis of the semilinear elliptic problem {#preli}
===========================================
Throughout this section without any mention, $\Omega\subset\RR^N$ denotes an arbitrary bounded open set with boundary $\pOm$. For each result, if a regularity of $\Omega$ is needed, then we shall specify and if no specification is given, then we mean that the result holds without any regularity assumption on the open set.
The spectral fractional Laplacian {#s:spec}
---------------------------------
Let $H_0^{1}(\Omega)=\overline{\mathcal D(\Omega)}^{H^{1}(\Omega)}$ where $$\begin{aligned}
H^{1}(\Omega)=\{u\in L^2(\Omega):\;\int_{\Omega}|\nabla u|^2\;dx<\infty\}\end{aligned}$$ is the first order Sobolev space endowed with the norm $$\begin{aligned}
\|u\|_{H^{1}(\Omega)}=\left(\int_{\Omega}|u|^2\;dx+\int_{\Omega}|\nabla u|^2\;dx\right)^{\frac 12}.\end{aligned}$$ Let $-\Delta_D$ be the realization on $L^2(\Omega)$ of the Laplace operator with the Dirichlet boundary condition. That is, $-\Delta_D$ is the positive and self-adjoint operator on $L^2(\Omega)$ associated with the closed, bilinear symmetric form $$\begin{aligned}
\mathcal A_D(u,v)=\int_{\Omega}\nabla u\cdot\nabla v\;dx,\;\;u,v\in H_0^{1}(\Omega),\end{aligned}$$ in the sense that $$\begin{cases}
D(\Delta_D)=\{u\in W_0^{1,2}(\Omega):\;\exists\;w\in L^2(\Omega),\; \mathcal A_D(u,v)=(w,v)_{L^2(\Omega)}\;\forall\;v\in H_0^{1}(\Omega)\},\\
-\Delta_Du=w.
\end{cases}$$ For instance if $\Omega$ has a smooth boundary, then $D(\Delta_D)=H^{2}(\Omega)\cap H_0^{1}(\Omega)$, where $$\begin{aligned}
H^2(\Omega):=\{u\in H^1(\Omega),\partial_{x_j}u\in H^1(\Omega),\; j=1,2,\ldots,N\}.\end{aligned}$$ It is well-known that $-\Delta_D$ has a compact resolvent and it eigenvalues form a non-decreasing sequence $0<\lambda_1\le\lambda_2\le\cdots\le\lambda_n\le\cdots$ satisfying $\lim_{n\to\infty}\lambda_n=\infty$. We denote by $\varphi_n$ the orthonormal eigenfunctions associated with $\lambda_n$.
Next, for $0<s<1$, we define the fractional order Sobolev space $$\begin{aligned}
H^s(\Omega):=\left\{u\in L^2(\Omega):\; \int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\;dxdy<\infty\right\},\end{aligned}$$ and we endow it with the norm defined by $$\begin{aligned}
\|u\|_{H^s(\Omega)}=\left(\int_{\Omega}|u|^2\;dx+\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\;dxdy\right)^{\frac 12}.\end{aligned}$$ We also let $$\begin{aligned}
H_0^s(\Omega):=\overline{\mathcal D(\Omega)}^{H^s(\Omega)},\end{aligned}$$ and $$\begin{aligned}
H_{00}^{\frac 12}(\Omega):=\left\{u\in L^2(\Omega):\;\int_{\Omega}\frac{u^2(x)}{\mbox{dist}(x,\pOm)}\;dx<\infty\right\}.\end{aligned}$$ Note that $$\begin{aligned}
\label{norm-sob-es}
\|u\|_{H_0^s(\Omega)}=\left(\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\;dxdy\right)^{\frac 12}\end{aligned}$$ defines a norm on $H_0^s(\Omega)$.
Since $\Omega$ is assumed to be bounded we have the following continuous embedding: $$\label{inj1}
H_0^s(\Omega)\hookrightarrow
\begin{cases}
L^{\frac{2N}{N-2s}}(\Omega)\;\;&\mbox{ if }\; N>2s,\\
L^p(\Omega),\;\;p\in[1,\infty)\;\;&\mbox{ if }\; N=2s,\\
C^{0,s-\frac{N}{2}}(\bOm)\;\;&\mbox{ if }\; N<2s.
\end{cases}$$
We notice that if $N\ge 2$, then $N\ge 2>2s$ for every $0<s<1$, or if $N=1$ and $0<s<\frac 12$, then $N=1>2s$, and thus the first embedding in will be used. If $N=1$ and $s=\frac 12$, then we will use the second embedding. Finally, if $N=1$ and $\frac 12<s<1$, then $N=1<2s$ and hence, the last embedding will be used.
For any $s\geq0$, we also introduce the following fractional order Sobolev space $$\begin{aligned}
\mathbb H^s(\Omega):=\left\{u=\sum_{n=1}^\infty u_n\varphi_n\in L^2(\Omega):\;\;\|u\|_{\mathbb H^s(\Omega)}^2:=\sum_{n=1}^\infty \lambda_n^su_n^2<\infty\right\},\end{aligned}$$ where we recall that $\lambda_n$ are the eigenvalues of $-\Delta_D$ with associated normalized eigenfunctions $\varphi_n$ and $$\begin{aligned}
u_n=(u,\varphi_n)_{L^2(\Omega)}=\int_{\Omega}u\varphi_n\;dx.\end{aligned}$$
It is well-known that $$\label{inf}
\mathbb H^s(\Omega)=
\begin{cases}
H^s(\Omega)=H_0^s(\Omega)\;\;\;&\mbox{ if }\; 0<s<\frac 12,\\
H_{00}^{\frac 12}(\Omega)\;\;&\mbox{ if }\; s=\frac 12,\\
H_0^s(\Omega)\;\;&\mbox{ if }\; \frac 12<s<1.
\end{cases}$$ It follows from that the embedding holds with $H_0^s(\Omega)$ replaced by $\mathbb H^s(\Omega)$.
The spectral fractional Laplacian is defined on the space $\mathbb H^s(\Omega)$ by $$\begin{aligned}
(-\Delta_D)^su=\sum_{n=1}^\infty\lambda_n^su_n\varphi_n\qquad \text{ with } u_n=\int_{\Omega}u\varphi_n.\end{aligned}$$
We notice that in this case we have $$\begin{aligned}
\label{norm-2}
\|u\|_{\mathbb H^s(\Omega)}=\|(-\Delta_D)^{\frac s2}u\|_{L^2(\Omega)}.\end{aligned}$$ Let $\mathcal D(\Omega)$ be the space of test functions on $\Omega$, that is, the space of infinitely continuously differentiable functions with compact support in $\Omega$. Then $\mathcal D(\Omega)\hookrightarrow \mathbb H^s(\Omega)\hookrightarrow L^2(\Omega)$, so, the operator $(-\Delta_D)^s$ is unbounded, densely defined and with bounded inverse $(-\Delta_D)^{-s}$ in $L^2(\Omega)$. The following integral representation of the operator $(-\Delta_D)^s$ given in [@NALD p.2 Formula (3)] will be useful. For a.e. $x\in\Omega$, $$\begin{aligned}
\label{int-rep}
(-\Delta_D)^su(x)=\mbox{P.V.}\int_{\Omega}\left[u(x)-u(y)\right]J(x,y)\;dy+\kappa(x)u(x),\end{aligned}$$ where, letting $K_\Omega(t,x,y)$ denote the heat kernel of the semigroup generated by the operator $-\Delta_D$ on $L^2(\Omega)$, $$\begin{aligned}
J(x,y)=\frac{s}{\Gamma(1-s)}\int_0^\infty\frac{K_\Omega(t,x,y)}{t^{1+s}}\;dt\end{aligned}$$ and $$\begin{aligned}
\kappa(x)=\frac{s}{\Gamma(1-s)}\int_0^\infty\left(1-\int_{\Omega}K_\Omega(t,x,y)\;dy\right)\frac{dt}{t^{1+s}},\end{aligned}$$ where $\Gamma$ denotes the usual Gamma function. We mention that it follows from the properties of the kernel $K_\Omega$ that $J$ is symmetric and nonnegative; i.e. $J(x,y)=J(x,y)\ge 0$ for a.e. $x,y\in\Omega$. In addition we have that $\kappa(x)\ge 0$ for a.e. $x\in\Omega$.
For more details on these topics we refer the reader to [@NALD; @NPV; @Gru2; @LM; @NOS] and their references.
Some results on Orlicz spaces {#s:orl}
-----------------------------
Here we give some important properties of Orlicz type spaces that will be used throughout the paper.
\[assum2\] For a function $f:\Om\times\RR\to\RR$ we consider the following assumption: $$\begin{cases}
f(x,\cdot) \text{ is odd, strictly increasing}&\text{ for a.e. } x\in\Omega,\\
f(x,0)=0 &\text{ for a.e. } x\in \Omega,\\
f(x,\cdot) \text{ is continuous }\;&\text{ for a.e. } x\in \Omega,\\
f(\cdot,t) \text{ is measurable }&\mbox{ for all } t\in\RR,\\
\lim_{t\to\infty}f(x,t)=\infty &\text{ for a.e. } x\in \Omega.
\end{cases}$$
Since $f(x,\cdot)$ is strictly increasing for a.e. $x\in\Omega$, it has an inverse which we denote by $\widetilde{f}(x,\cdot)$. Let $F,\widetilde{F}:\;\Omega\times\RR\to[0,\infty)$ be defined for a.e. $x\in\Omega$ by $$\begin{aligned}
F(x,t):=\int_0^{|t|}f(x,\tau)\;d\tau\;\mbox{ and }\;
\widetilde{F}(x,t):=\int_0^{|t|}\widetilde{f}(x,\tau)\;d\tau.
\end{aligned}$$ The functions $F$ and $\widetilde F$ are complementary Musielak-Orlicz functions such that $F(x,\cdot)$ and $\widetilde F(x,\cdot)$ are complementary $\cN$-functions for a.e. $x\in\Omega$ (in the sense of [@Adam p.229]).
\[assum3\] Under the setting of Assumption \[assum2\], and for a.e. $x\in\Omega$, let both $F(x,\cdot)$ and $\widetilde F(x,\cdot)$ satisfy the global $(\triangle_2)$-condition, that is, there exist two constants $c_1,c_2\in (0,1]$ independent of $x$, such that for a.e. $x\in\Omega$ and for all $t\ge 0$, $$\begin{aligned}
\label{delta-2}
c_1tf(x,t)\le F(x,t)\le tf(x,t)\;\mbox{ and }\; c_2t\widetilde f(x,t)\le \widetilde F(x,t)\le t\widetilde f(x,t).
\end{aligned}$$
First we notice that since the functions $f,\widetilde f$ are odd and $F,\widetilde F$ are even functions, we have that if holds, then it also holds for all $t\in\RR$. Second, Assumption \[assum3\] is equivalent to saying that the Musielak-Orlicz functions $F$ and $\widetilde F$ satisfy the $(\triangle_2^0)$-condition in the sense that there exist two constants $C_1,C_2>0$ such that $$\begin{aligned}
F(x,2t)\le C_1 F(x,t)\;\mbox{ and }\; \widetilde F(x,2t)\le C_2\widetilde F(x,t),\;\forall\;t\in\RR\;\mbox{ and a.e. } x\in\Omega.
\end{aligned}$$ This can be easily verified by following the argument given in the monograph [@Adam p.232]. In that case, we let $$\begin{aligned}
L_F(\Om):=\{u:\Omega\to\RR\text{ measurable}: F(\cdot,u(\cdot))\in L^1(\Omega)\}
\end{aligned}$$ be the Musielak-Orlicz space. The space $L_{\widetilde F}(\Om)$ is defined similarly with $F$ replaced by $\widetilde F$.
[ *If Assumption \[assum3\] holds, then by [@Doman Theorems 1 and 2] (see also [@Adam Theorem 8.19]), $L_F(\Omega)$ endowed with the Luxemburg norm given by $$\begin{aligned}
\norm{u}_{F,\Omega}:=\inf\left\{k>0:\;\int_{\Omega}F \left(x,\frac{u(x)}{k}\right)\;dx\le 1\right\},
\end{aligned}$$ is a reflexive Banach space. The same result also holds for $L_{\widetilde F}(\Omega)$. Moreover, we have the following improved Hölder inequality for Musielak-Orlicz spaces (see e.g. [@Adam Formula (8.11) p.234]): $$\label{hold}
{\left|\int_{\Omega}uv\;dx\right|}\le 2\norm{u}_{F,\Omega}\norm{v}_{\widetilde{F},\Omega},\;\forall\;u\in L_F(\Omega),\; v\in L_{\widetilde F}(\Omega).$$ In addition, by [@Bie Corollary 5.10], we have that $$\begin{aligned}
\label{coer-est}
\lim_{\|u\|_{F,\Omega}\to\infty}\frac{\int_{\Omega}F(x,u)\;dx}{\|u\|_{F,\Omega}}=\infty.
\end{aligned}$$* ]{}
We have the following result.
\[lem:hoelder\] Let Assumption \[assum3\] hold. Then $f(\cdot,u(\cdot))\in L_{\widetilde F}(\Om)$ for all $u\in L_F(\Omega)$.
Assume that Assumption \[assum3\] holds. It follows from the assumptions that there exists a constant $C>0$ such that for all $\xi\in\RR$ and a.e. $x\in\Omega$, $$\begin{aligned}
\widetilde F(x,f(x,\xi)) \le \xi f(x,\xi) \le C F(x,\xi).
\end{aligned}$$ Hence, $$\begin{aligned}
\int_{\Om} \widetilde F(x,f(x,u(x)))\;dx \le C\int_{\Om} F(x,u(x))\;dx <\infty
\end{aligned}$$ and the proof is finished.
Let $0<s<1$. Under Assumption \[assum3\] we can define the Banach space $\cV$ by $$\begin{aligned}
\cV:=\cV(\Omega,F):={\{u\in \mathbb H^{s}(\Om): F(\cdot,u(\cdot))\in L^1(\Om)\}}\end{aligned}$$ and we endow it with the norm defined by $$\begin{aligned}
\norm{u}_{\cV}:=\norm{u}_{\mathbb H^{s}(\Om)}+\norm{u}_{F,\Om}.
\end{aligned}$$
In this case $\cV$ is a reflexive Banach space which is continuously embedded into $\mathbb H^{s}(\Om)$. In addition, it follows from that we have the following continuous embedding $$\begin{aligned}
\label{sobo1}
\cV\hookrightarrow \mathbb H^{s}(\Om)\hookrightarrow L^{2^\star}(\Omega),
\end{aligned}$$ where we recall that $$\begin{aligned}
2^\star=\frac{2N}{N-2s}\;\mbox{ if }\; N\ge 2>2s\;\mbox{ or if }\,N=1\;\mbox{ and }\; 0<s<\frac 12.\end{aligned}$$ If $N=1$ and $s=\frac 12$, then $2^{\star}$ is any number in the interval $[1,\infty)$. If $N=1$ and $\frac 12<s<1$, then we have the continuous embedding $$\begin{aligned}
\label{sobo2}
\cV\hookrightarrow \mathbb H^{s}(\Om)\hookrightarrow C^{0,s-\frac 12}(\bOm).\end{aligned}$$
Weak solutions of the semilinear problem {#sec:weaksol}
----------------------------------------
Now we can introduce our notion of weak solutions to the system .
We recall that we have set $\cV:=\mathbb H^{s}(\Omega)\cap L_F(\Omega)$. We shall denote by $\cV^\star=(\mathbb H^{s}(\Omega)\cap L_F(\Omega))^\star$ the dual of the reflexive Banach space $\cV$ and by $\langle\cdot,\cdot\rangle$ their duality map.
A function $u\in \cV$ is said to be a weak solution of if the identity $$\begin{aligned}
\label{form-ws}
\mathcal F(u,v):=\int_{\Omega}(-\Delta_D)^{\frac s2}u(-\Delta_D)^{\frac s2}v\;dx+\int_{\Omega}f(x,u)v\;dx=\langle g,v\rangle,\end{aligned}$$ holds for every $v\in \cV$ and the right hand side $g\in \cV^\star$.
We have the following result of existence and uniqueness of weak solution.
\[prop-exis\] Let Assumption \[assum3\] hold. Then for every $g\in \cV^\star$, the system has a unique weak solution $u$. In addition, if $g\in \mathbb H^{-s}(\Omega):=(\mathbb H^s(\Omega))^\star$, then there exists a constant $C>0$ such that $$\label{nor-est}
\|u\|_{\mathbb H^s(\Omega)}\le C\|g\|_{\mathbb H^{-s}(\Omega)}.$$
Let $u\in \cV$ be fixed. First we notice that it follows from Lemma \[lem:hoelder\] that $f(\cdot,u(\cdot))\in L_{\widetilde F}(\Omega)$. Next, using the classical Hölder inequality and we have that for all $v\in \cV$, $$\begin{aligned}
\label{ine-bound}
|\mathcal F(u,v)|\le& \|(-\Delta_D)^{\frac s2}u\|_{L^2(\Omega)}\|(-\Delta_D)^{\frac s2}v\|_{L^2(\Omega)}+2\|f(\cdot,u)\|_{\widetilde F,\Omega}\|v\|_{F,\Omega}\notag\\
\le &\left(\|(-\Delta_D)^{\frac s2}u\|_{L^2(\Omega)}+2\|f(\cdot,u)\|_{\widetilde F,\Omega}\right)\|v\|_{\cV}.\end{aligned}$$ Since $\mathcal F(u,\cdot)$ is linear (in the second variable) we have shown that $\mathcal F(u,\cdot)\in \cV^\star$ for every $u\in \cV$. Since $f(x,\cdot)$ is strictly monotone, we have that every $u,v\in \cV$, $u\ne v$, $$\begin{aligned}
\mathcal F(u,u-v)-\mathcal F(v,u-v)>0.\end{aligned}$$ Hence, $\mathcal F$ is strictly monotone. It follows from the continuity of the norm function and the continuity of $f(x,\cdot)$ that $\mathcal F$ is hemi-continuous. It follows also from the $(\Delta_2)$-condition and that $$\begin{aligned}
\lim_{\|u\|_{ F,\Omega}\to\infty}\frac{\int_{\Omega}f(x,u)u\;dx}{\|u\|_{F,\Omega}}=\infty,\end{aligned}$$ and this implies that $$\begin{aligned}
\lim_{\|u\|_{\cV}\to\infty}\frac{\mathcal F(u,u)}{\|u\|_{\cV}}=\infty.\end{aligned}$$ Hence, $\mathcal F$ is coercive. We have shown that for every $u\in \cV$ there exists a unique $A_F\in \cV^\star$ such that $\mathcal F(u,v)=\langle A_F(u),v\rangle$ for every $v\in \cV$. This defines an operator $A_F:\; \cV\to \cV^\star$ which is hemi-continuous, strictly monotone, coercive and bounded (the boundedness follows from ). Therefore $A_F(\cV)=\cV^\star$ and hence, by the Browder-Minty theorem, for every $g\in \cV^\star$, there exists a unique $u\in \cV$ such that $A_F(u)=v$. Now assume that $g\in \mathbb H^{-s}(\Omega)\hookrightarrow \cV^\star$. Then taking $v=u$ in , using the fact that $f(x,u)u\ge 0$ and noticing that $\langle g,u\rangle_{\cV^\star,\cV}=\langle g,u\rangle_{ \mathbb H^{-s}(\Omega), \mathbb H^{s}(\Omega)}$ (recall that $g\in\mathbb H^{-s}(\Omega)$ and $u\in\mathbb H^s(\Omega)$) we get that $$\begin{aligned}
\|u\|_{\mathbb H^s(\Omega)}^2\le |\langle g,u\rangle|\le \|g\|_{\mathbb H^{-s}(\Omega)}\|u\|_{\mathbb H^s(\Omega)}.\end{aligned}$$ We have shown and the proof is finished.
The following theorem is the main result of this section.
\[theo-bound\] Let Assumption \[assum3\] hold and that $g\in L^p(\Omega)$ with $$\label{cond-p}
\begin{cases}
p>\frac{N}{2s}\;\;&\mbox{ if }\; N>2s,\\
p>1 \;\;&\mbox{ if }\; N=2s,\\
p=1\;\;&\mbox{ if }\; N<2s.
\end{cases}$$ Then every weak solution $u$ of belongs to $L^\infty(\Omega)$. Moreover there is a constant $C=C(N,s,p,\Omega)>0$ such that $$\begin{aligned}
\label{inf-norm}
\|u\|_{L^\infty(\Omega)}\le C\|g\|_{L^p(\Omega)}.\end{aligned}$$
We mention that if $N=1$ and $\frac 12<s<1$, then it follows from that the weak solution of is globally Hölder continuous on $\bOm$ and in this case there is nothing to prove. Thus we need to prove the theorem only in the cases $N\ge 2$, or $N=1$ and $0<s\le \frac 12$.
To prove the theorem we need the following lemma which is of analytic nature and will be useful in deriving some a priori estimates of weak solutions of elliptic type equations (see e.g. [@kinderlehrer1980 Lemma B.1.]).
\[lem-01\] Let $\Xi = \Xi(t)$ be a nonnegative, non-increasing function on a half line $t\ge k_0\ge 0$ such that there are positive constants $c, \alpha$ and $\delta$ ($\delta >1$) with $$\Xi(h) \le c(h-k)^{-\alpha}\Xi(k)^{\delta}\mbox{ for } h>k\ge k_0.$$ Then $$\Xi(k_0+d) = 0\quad \mbox{ with }\quad d^{\alpha}= c \Xi(k_0)^{\delta -1}2^{\alpha\delta/(\delta -1)}.$$
Invoking Assumption \[assum3\] and $g\in L^p(\Omega)$ with $p$ satisfying , it follows from that $g\in \cV^\star$. Hence, by Proposition \[prop-exis\], the system has a unique weak solution $u\in \cV$. We prove the result in two steps.
[*Step 1*]{}. Let $u\in \cV$, $k\ge 0$ and set $u_k:=(|u|-k)^+\sgn(u)$. Using [@War Lemma 2.7] we get that $u_k\in \cV$. We claim that $$\begin{aligned}
\label{claim1}
\mathcal F(u_k,u_k)\le\mathcal F(u,u_k).\end{aligned}$$ Indeed, let $A_k:=\{x\in\Om:\;|u(x)|\ge k\}$, $A_k^+:=\{x\in\Om:\;u(x)\ge k\}$ and $A_k^-:=\{x\in\Om:\;u(x)\le -k\}$ so that $A_k=A_k^+\cup A_k^-$. Then $$\label{for-uk}
u_k=
\begin{cases}
u-k\;\;&\mbox{ in }\;A_k^+,\\
u+k &\mbox{ in }\;A_k^-,\\
0 &\mbox{ in }\; \Om\setminus A_k.
\end{cases}$$ Since $f(x,\cdot)$ is odd, monotone increasing and $0\le u_k=u-k\le u$ on $A_k^+$, we have that for a.e. $x\in A_k^+$, $$\begin{aligned}
\label{A1}
f(x,u_k)u_k=f(x,u-k)u_k\le f(x,u)u_k.\end{aligned}$$ Similarly, since $u\le u+k=u_k\le 0$ on $A_k^-$, it follows that for a.e. $x\in A_k^-$, $$\begin{aligned}
\label{A2}
f(x,u_k)u_k=f(x,u+k)u_k\le f(x,u)u_k.\end{aligned}$$ It follows from and that for every $k\ge 0$, $$\begin{aligned}
\label{A1-A2}
\int_{\Omega}f(x,u_k)u_k\;dx\le \int_{\Omega}f(x,u)u_k\;dx.\end{aligned}$$ Next, we show that for every $k\ge 0$, $$\begin{aligned}
\label{pf-claim1-2}
\int_{\Omega}(-\Delta_D)^{\frac s2}u_k(-\Delta_D)^{\frac s2}u_k\;dx\le \int_{\Omega}(-\Delta_D)^{\frac s2}u(-\Delta_D)^{\frac s2}u_k\;dx.\end{aligned}$$ We notice that it follows from the integral representation that $$\begin{aligned}
&\int_{\Omega}(-\Delta_D)^{\frac s2}u_k(-\Delta_D)^{\frac s2}u_k\;dx=\|u_k\|_{\mathbb H^s(\Omega)}^2\\
=&\frac 12\int_{\Omega}\int_{\Omega}|u_k(x)-u_k(y)|^2J(x,y)\;dxdy+\int_{\Omega}\kappa(x)|u_k(x)|^2\;dx.\end{aligned}$$ Calculating and using we get that for every $k\ge 0$, $$\begin{aligned}
\label{pf-claim1}
\int_{\Omega}\int_{\Omega}&|u_k(x)-u_k(y)|^2J(x,y)\;dxdy\\
=&\int_{A_k^+}\int_{A_k^+}(u(x)-u(y))(u_k(x)-u_k(y))J(x,y)\;dxdy\notag\\
&+\int_{A_k^+}\int_{A_k^-}|u(x)-u(y)-2k|^2J(x,y)\;dxdy\notag\\
&+\int_{A_k^-}\int_{A_k^-}(u(x)-u(y))(u_k(x)-u_k(y))J(x,y)\;dxdy\notag\\
&+\int_{A_k^-}\int_{A_k^+}|u(x)-u(y)+2k|^2J(x,y)\;dxdy\notag\\
&+\int_{\Omega\setminus A_k}\int_{A_k}|u_k(y)|^2J(x,y)\;dxdy\notag\\
&+\int_{A_k}\int_{\Omega\setminus A_k}|u_k(x)|^2J(x,y)\;dxdy\notag.\end{aligned}$$ Since $u(x)-u(y)-2k\ge 0$ for a.e. $(x,y)\in A_k^+\times A_k^-$, we have that for a.e. $(x,y)\in A_k^+\times A_k^-$, $$\begin{aligned}
\label{EW1}
(u(x)-u(y)-2k)^2&\le (u(x)-u(y)) (u(x)-u(y)-2k)\\
&=(u(x)-u(y))(u_k(x)-u_k(y)).\notag\end{aligned}$$ Since $u(x)-u(y)+2k\le 0$ for a.e $(x,y)\in A_k^-\times A_k^+$, it follows that for a.e $(x,y)\in A_k^-\times A_k^+$, $$\begin{aligned}
\label{EW2}
(u(x)-u(y)+2k)^2&\le (u(x)-u(y)) (u(x)-u(y)+2k)\\
&=(u(x)-u(y))(u_k(x)-u_k(y)).\notag \end{aligned}$$ For a.e. $(x,y)\in (\Omega \setminus A_k)\times A_k$, we have that (recall that $u_k(x)=0$), $$\begin{aligned}
\label{M}
(u(x)-u(y))(u_k(x)-u_k(y))=-(u(x)-u(y))u_k(y)=(u(y)-u(x))u_k(y).\end{aligned}$$ Using we get the following estimates:
- For a.e. $(x,y)\in (\Omega\setminus A_k)\times A_k^+$ we have that (as $k-u(x)> 0$ and $u(y)-k\ge 0$) $$\begin{aligned}
\label{M1}
(u(x)-u(y))(u_k(x)-u_k(y))=&(u(y)-k+k-u(x))(u(y)-k)\notag\\
=&(u(y)-k)^2+(k-u(x))(u(y)-k)\notag\\
\ge & (u(y)-k)^2=|u_k(y)|^2.\end{aligned}$$
- For a.e. $(x,y)\in (\Omega\setminus A_k)\times A_k^-$ we have that (as $k+u(x) > 0$ and $u(y)+k\le 0$) $$\begin{aligned}
\label{M2}
(u(x)-u(y))(u_k(x)-u_k(y))=&(u(y)+k-k-u(x))(u(y)+k)\notag\\
=&(u(y)+k)^2-(k+u(x))(u(y)+k)\notag\\
\ge& (u(y)+k)^2=|u_k(y)|^2.\end{aligned}$$
Combining and yields for a.e. $(x,y)\in (\Omega\setminus A_k)\times A_k$ $$\begin{aligned}
\label{M12}
(u(x)-u(y))(u_k(x)-u_k(y))\ge |u_k(y)|^2.\end{aligned}$$ Proceeding in the same manner, we also get that for a.e. $(x,y)\in A_k\times (\Omega\setminus A_k)$ (recall that here $u_k(y)=0$), $$\begin{aligned}
\label{M3}
(u(x)-u(y))(u_k(x)-u_k(y))\ge |u_k(x)|^2.\end{aligned}$$ Using , , , and we get from that for every $k\ge 0$ (recall that $J(x,y)\geq 0$ for a.e. $x,y\in \Omega$), $$\begin{aligned}
\label{pf-claim1-0}
\int_{\Omega}\int_{\Omega}&|u_k(x)-u_k(y)|^2J(x,y)\;dxdy\\
\le &\int_{\Omega}\int_{\Omega}(u(x)-u(y))(u_k(x)-u_k(y))J(x,y)\;dxdy.\notag\end{aligned}$$ As for we have that for every $k\ge 0$ (recall that $\kappa(x)\ge 0$ for a.e. $x\in\Omega$), $$\begin{aligned}
\label{kappa}
\int_{\Omega}\kappa(x)|u_k(x)|^2\;dx\le \int_{\Omega}\kappa(x)u(x)u_k(x)\;dx.\end{aligned}$$ Now the estimate follows from and since according to there holds $$\begin{aligned}
\int_{\Omega}(-\Delta_D)^{\frac s2}u(-\Delta_D)^{\frac s2}u_k\;dx&=\frac 12\int_{\Omega}\int_{\Omega}(u(x)-u(y))(u_k(x)-u_k(y))J(x,y)\;dxdy\\
&\quad+\int_{\Omega}\kappa(x)u(x)u_k(x)\;dx.\end{aligned}$$ It follows from and that for every $k\ge 0$, $$\begin{aligned}
\mathcal F(u_k,u_k)=&\int_{\Omega}(-\Delta_D)^{\frac s2}u_k(-\Delta_D)^{\frac s2}u_k\;dx+\int_{\Omega}f(x,u_k)u_k\;dx\\
\le &\int_{\Omega}(-\Delta_D)^{\frac s2}u(-\Delta_D)^{\frac s2}u_k\;dx+\int_{\Omega}f(x,u)u_k\;dx\\
\le &\mathcal F(u,u_k),\end{aligned}$$ and we have proved the claim .
[*Step 2*]{}. Let $u\in\cV$ be the unique weak solution of the system , $k\ge 0$ and let $u_k$ be as above. Let $p_1\in [1,\infty]$ be such that $\frac{1}{p}+\frac{1}{2^\star}+\frac{1}{p_1}=1$ where we recall that $2^\star=\frac{2N}{N-2s}>2$. Since $p>\frac{N}{2s}=\frac{2^\star}{2^\star-2}$, we have that $$\begin{aligned}
\label{eq-B}
\frac{1}{p_1}=1-\frac{1}{2^\star}-\frac{1}{p}>\frac{2^\star}{2^\star}-\frac{1}{2^\star}-\frac{2^\star-2}{2^\star}=\frac{1}{2^\star}\Longrightarrow p_1<2^\star.\end{aligned}$$ Taking $v=u_k$ as a test function in and using the classical Hölder inequality we get that there exists a constant $C=C(N,s,p)>0$ such that $$\begin{aligned}
\label{est}
\mathcal F(u,u_k)=\int_{\Omega}gu_k\;dx\le& \|g\|_{L^p(\Omega)}\|u_k\|_{L^{2^\star}(\Omega)}\|\chi_{A_k}\|_{L^{p_1}(\Omega)},\end{aligned}$$ where $\chi_{A_k}$ denotes the characteristic function of the set $A_k$. Using , , and the fact that $\int_{\Om}f(x,u_k)u_k\;dx\ge 0$, we get that there exist two constants $C,C_1>0$ such that for every $k\ge 0$, $$\begin{aligned}
C\|u_k\|_{L^{2^\star}(\Omega)}^2 &\le \|u_k\|_{\mathbb H^{s}(\Omega)}^2\le \mathcal F(u_k,u_k)\le \mathcal F(u,u_k) \\
&\le C_1 \|g\|_{L^p(\Omega)}\|u_k\|_{L^{2^\star}(\Omega)}\|\chi_{A_k}\|_{L^{p_1}(\Omega)},\end{aligned}$$ and this implies that there exists a constant $C>0$ such that for every $k\ge 0$, $$\begin{aligned}
\label{est2}
\|u_k\|_{L^{2^\star}(\Omega)}\le C \|g\|_{L^p(\Omega)}\|\chi_{A_k}\|_{L^{p_1}(\Omega)}.\end{aligned}$$ Let $h>k$. Then $A_h\subset A_k$ and on $A_h$ we have that $|u_k|\ge h-k$. Therefore, it follows from that for every $h>k\ge 0$, $$\begin{aligned}
\label{B1}
\|\chi_{A_h}\|_{L^{2^\star}(\Omega)}\le C (h-k)^{-1}\|g\|_{L^p(\Omega)}\|\chi_{A_k}\|_{L^{p_1}(\Omega)}.\end{aligned}$$ Let $\delta:=\frac{2^\star}{p_1}>1$ by . Then using the Hölder inequality again we get that there exists a constant $C>0$ such that for every $k\ge 0$, we have $$\begin{aligned}
\label{B2}
\|\chi_{A_k}\|_{L^{p_1}(\Omega)}\le C \|\chi_{A_k}\|_{L^{2^\star}(\Omega)}^\delta.\end{aligned}$$ It follows from and that there exists a constant $C>0$ such that for every $h>k\ge 0$, $$\begin{aligned}
\|\chi_{A_h}\|_{L^{2^\star}(\Omega)}\le C (h-k)^{-1}\|g\|_{L^p(\Omega)}\|\chi_{A_k}\|_{L^{2^\star}(\Omega)}^\delta.\end{aligned}$$ It follows from Lemma \[lem-01\] with $\Xi(k)=\|\chi_{A_k}\|_{L^{2^\star}(\Omega)}$ that there exists a constant $C_1>0$ such that $$\begin{aligned}
\|\chi_{A_K}\|_{L^{2^\star}(\Omega)}=0\;\mbox{ with }\;K=CC_1\|g\|_{L^p(\Omega)}.\end{aligned}$$ We have shown the estimate and the proof is finished.
We have the following improved regularity of weak solutions to the system , in case $\Omega$ is a smooth open set.
\[cor-212\] Let $\Omega\subset\RR^N$ be a bounded open set with smooth boundary. Let Assumption \[assum3\] hold and that $$\begin{aligned}
\label{con-f}
f(\cdot,t)\in L^\infty(\Omega),\;\; \forall\;t\in\RR,\; |t|\le \alpha\;\mbox{ for some constant }\alpha>0. \end{aligned}$$ Then the following assertions hold.
1. Let $g\in L^p(\Omega)$ with $p$ as in . If $2s-\frac Np\ne 1$ (resp. $2s-\frac Np=1$) then the weak solution of belongs to $C^{0,2s-\frac Np}(\bOm)$ (resp. $C_{\star}^1(\bOm)$), where $C_{\star}^1(\bOm)$ is the Hölder-Zygmund space.
2. If $g\in L^\infty(\Omega)$, then $u\in \cap_{\varepsilon>0}C^{0,2s-\varepsilon}(\bOm)$.
Let Assumption \[assum3\] hold and that $f$ satisfies . Let $g\in L^p(\Omega)$ with $p$ as in part (a) or part (b). Then by Theorem \[theo-bound\] the solution $u\in L^\infty(\Omega)$. Hence, by we have that the function $f(\cdot,u(\cdot))\in L^\infty(\Omega)$. Let then $h:=g-f(\cdot,u(\cdot))$. Then $h$ belongs to same space as the function $g$ and $u$ is a weak solution of the Dirichlet problem $$\begin{aligned}
(-\Delta_D)^su=h\;\;\mbox{ in }\;\Omega,\; u=0\;\mbox{ on }\;\pOm.\end{aligned}$$ Now the regularity of $u$ given in part (a) and part (b) follows from [@Gru2 Corollary 3.5].
For all the results presented so far, Assumption \[assum3\] is sufficient. However, to show higher regularity in $\mathbb H^{2s+\beta}(\Omega)$ with $0 \le \beta < 1$ and for the discretization error estimates in the sequel, we need an assumption on the local Lipschitz continuity of the nonlinearity in addition.
\[assum-lip\] For all $M>0$ there exists a constant $L_{M}>0$ such that $f$ satisfies $$\begin{aligned}
|f(x,u_1)-f(y,u_2)|\leq L_{M}|u_1-u_2|
\end{aligned}$$ for a.e. $x, y\in\Omega$ and $u_i\in\mathbb{R}$ with $|u_i|\leq M$, $i=1,2$.
The following result will be frequently used throughout the paper.
\[lem-f-lip\] Let $0\le \beta< 1$ and assume that $f$ satisfies Assumption \[assum-lip\]. Then for every $u\in\mathbb H^\beta(\Omega)\cap L^\infty(\Omega)$, we have that $f(\cdot,u(\cdot))\in\mathbb H^\beta(\Omega)$.
We notice that if $\beta=0$ then there is nothing to prove. Let then $0< \beta< 1$ and $u\in \mathbb H^\beta(\Omega)\cap L^\infty(\Omega)$. Since $f(x,0)=0$, $u\in L^2(\Omega)$, $|u(x)|\le M$ for a.e. $x\in\Omega$, for some constant $M>0$, we have that (by Assumption \[assum-lip\]) $$\begin{aligned}
\label{eq-lip}
|f(x,u(x))|=|f(x,u(x))-f(x,0)|\le L_M|u(x)|\;\;\mbox{ for a.e. }\;x\in\Omega.\end{aligned}$$ This implies that $f(\cdot,u(\cdot))\in L^2(\Omega)$. Assumption \[assum-lip\] also implies that $$\begin{aligned}
\label{eq-lip2}
|f(x,u(x))-f(x,u(y))|\le L_M|u(x)-u(y)|\;\;\mbox{ for a.e. }\;x, y\in\Omega.\end{aligned}$$ We have the following three cases.
- If $\beta=\frac 12$, then using we obtain that $$\begin{aligned}
\int_{\Omega}\frac{|f(x,u(x))|^2}{\mbox{dist}(x,\pOm)}\;dx\le L_M^2\int_{\Omega}\frac{|u(x)|^2}{\mbox{dist}(x,\pOm)}\;dx<\infty.\end{aligned}$$ Hence, $f(\cdot,u(\cdot))\in\mathbb H^{\frac 12}(\Omega)$.
- If $0<\beta<\frac 12$, then it follows from that $$\begin{aligned}
\label{eq-Int}
\int_{\Omega}\int_{\Omega}\frac{|f(x,u(x))-f(y,u(y))|^2}{|x-y|^{N+2\beta}}\;dxdy\le L_M^2\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2\beta}}\;dxdy<\infty,\end{aligned}$$ and this implies that $f(\cdot,u(\cdot))\in H^\beta(\Omega)=\mathbb H^\beta(\Omega)$.
- If $\frac 12<\beta<1$, then the estimate also holds and this implies that $f(\cdot,u(\cdot))\in H^\beta(\Omega)$. Since $f(x,0)=0$ for a.e. $x\in\Omega$, we also get that $f(\cdot,u(\cdot))\in \mathbb H^\beta(\Omega)$ by approximation if necessary.
The proof of the lemma is finished.
We have the following elliptic regularity.
\[ellip-regula\] Let $\Omega\subset\RR^N$ be a bounded open set with Lipschitz continuous boundary. Assume Assumptions \[assum3\] and \[assum-lip\] are fulfilled. In addition, let $0\le \beta< 1$, $g\in \mathbb H^\beta(\Omega)\cap L^p(\Omega)$ with $p$ as in and let $u\in\mathbb H^s(\Omega)$ be the weak solution of . Then $u\in \mathbb H^{2s+\beta}(\Omega)$.
In view of the assumption on $f$ and $g$, it follows from Proposition \[prop-exis\] and Theorem \[theo-bound\] that the system has a unique weak solution $u\in\mathbb H^s(\Omega)\cap L^\infty(\Omega)$. Since $f(\cdot,u(\cdot))\in L^2(\Omega)$ (by Lemma \[lem-f-lip\]) we have that $g-f(\cdot,u(\cdot))\in L^2(\Omega)$ then $$\label{eq:un}
u_n=\lambda_n^{-s}\int_{\Omega}(g-f(\cdot,u))\varphi_n,\quad n\in\N.$$ Using the $\mathbb{H}^{2s}$ norm definition we arrive at $$\|u\|_{\mathbb{H}^{2s}(\Omega)}^2 = \| g - f(\cdot,u) \|_{L^2(\Omega)}^2 ,$$ i.e., $u\in \mathbb H^{2s}(\Omega)\cap L^\infty(\Omega)$ (see also [@CaSt Section 2 pp.772-773]). We have two cases.
- If $2s\ge 1$, then $u\in\mathbb H^\beta(\Omega)$ (recall that $0<\beta< 1$) and hence, $f(\cdot,u(\cdot))\in \mathbb H^\beta(\Om)$ by Lemma \[lem-f-lip\]. We have shown that $g-f(\cdot,u(\cdot))\in \mathbb H^{\beta}(\Omega)$. Since $g-f(\cdot,u(\cdot))\in \mathbb{H}^\beta(\Omega)$, using and the definition of $\mathbb{H}^{2s+\beta}$ we obtain $$\begin{aligned}
\|u\|_{\mathbb{H}^{2s+\beta}(\Omega)}^2&=\sum_{n=1}^\infty u_n^2\lambda_n^{2s+\beta}=\sum_{n=1}^\infty \left(\lambda_n^{-s}\int_{\Omega}(g-f(\cdot,u))\varphi_n\right)^2\lambda_n^{2s+\beta}\\
&=\|g-f(\cdot,u)\|_{\mathbb{H}^{\beta}(\Omega)}^2,\end{aligned}$$ and we have shown that $u\in \mathbb H^{2s+\beta}(\Omega)$ (see also e.g. [@CaSt Section 2]).
- If $2s<1$, then $f(\cdot,u(\cdot))\in \mathbb H^{2s}(\Omega)$ (by Lemma \[lem-f-lip\]) and this implies that $g-f(\cdot,u(\cdot))\in \mathbb H^{\min\{2s,\beta\}}(\Omega)$. As above we then get that $u\in \mathbb H^{2s+\min\{2s,\beta\}}(\Omega)$. Repeating the same argument with $2s+\min\{2s,\beta\}$ in place of $2s$ and so on, we can arrive that in fact $g-f(\cdot,u(\cdot))\in \mathbb H^{\beta}(\Omega)$ and as above this implies that $u\in \mathbb H^{2s+\beta}(\Omega)$.
The proof is finished.
We conclude this section with the following example.
Let $q\in [1,\infty)$ and let $b: \Omega\to (0,\infty)$ be a function in $L^\infty(\Omega)$, that is, $b(x)> 0$ for a.e. $x\in\Omega$. Define the function $f:\Omega\times\mathbb R\to\mathbb R$ by $f(x,t)=b(x)|t|^{q-1}t$. It is clear that $f$ satisfies Assumption \[assum2\] and the associated function $F:\Omega\times\mathbb R\to [0,\infty)$ is given by $F(x,t)=\frac{1}{q+1}b(x)|t|^{q+1}$. For a.e. $x\in\Omega$, the inverse $\widetilde f(x,\cdot)$ of $f(x,\cdot)$ is given by $\widetilde f(x,t)=\left(b(x)\right)^{-\frac 1q}|t|^{\frac{1-q}{q}}t$. Therefore, the complementary function $\widetilde F$ of $F$ is given by $\widetilde F(x,t)=\frac{q}{q+1}\left(b(x)\right)^{-\frac 1q}|t|^{\frac{q+1}{q}}$. Hence, $$\begin{aligned}
tf(x,t)=(q+1) F(x,t)\;\mbox{ and }\; t\widetilde f(x,t)=\frac{q+1}{q}\widetilde F(x,t),\end{aligned}$$ and we have shown that Assumption \[assum3\] is also satisfied. Moreover, we have that $f$ satisfies in Corollary \[cor-212\]. In particular, if $b(x)=C$ for a.e. $x\in\Omega$, for some constant $C>0$, then the function $f$ also satisfies Assumption \[assum-lip\].
The extended problem in the sense of Caffarelli and Silvestre {#s:CS}
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In case that the nonlinearity $f(x,t)$ is identically zero, it is well known that problem can equivalently be posed on a semi-infinite cylinder. This approach is originally due to Caffarelli and Silvestre [@Caf3]. While they assume the unbounded domain $\RR^N$, the restriction to bounded domains was considered in [@CT:10; @CDDS:11; @ST:10]. We mention that for the existence and uniqueness of solutions to the problem on this semi-infinite cylinder it is sufficient to consider an open set with a Lipschitz continuous boundary, see [@CaSt Theorem 2.5] for details. We operate under the same setup in the present section. Since we will send the non-linearity in to its right hand side, it is straightforward to introduce the extended problem in the semi-linear case.
We begin by introducing the required notation. In the following, we denote by $\mathcal{C}$ the aforementioned semi-infinite cylinder with base $\Omega$, i.e., $\mathcal{C}=\Omega\times(0,\infty)$, and its lateral boundary by $\partial_L\mathcal{C}:=\partial\Omega\times[0,\infty)$. For later purposes, we also introduce for any $\mathpzc{Y}>0$ a truncation of the cylinder $\mathcal{C}$ by $\mathcal{C}_\mathpzc{Y}:=\Omega\times (0,\mathpzc{Y})$. Similar to the lateral boundary $\partial_L\mathcal{C}_\mathpzc{Y}$, we set $\partial_L\mathcal{C}_\mathpzc{Y}:=\partial\Omega\times[0,\mathpzc{Y}]$. Consequently, the semi-infinite cylinder and its truncated version are objects defined in $\mathbb{R}^{N+1}$. Throughout the remaining part of the paper, $y$ denotes the extended variable, such that a vector $x'\in\mathbb{R}^{N+1}$ admits the representation $x'=(x_1,\ldots,x_N,x_{N+1})=(x,x_{N+1})=(x,y)$ with $x_i\in \mathbb{R}$ for $i=1,\ldots,N+1$, $x\in\mathbb{R}^N$ and $y\in \mathbb{R}$.
Due to the degenerate/singular nature of the extended problem by Caffarelli and Silvestre, it will be necessary to discuss the solvability of this problem in certain weighted Sobolev spaces with weight function $y^\alpha$, $\alpha\in(-1,1)$, see [@Turesson Section 2.1], [@KO84] and [@GU Theorem 1] for a more sophisticated discussion of such spaces. In this regard, let $\mathcal{D}\subset \mathbb{R}^{N}\times[0,\infty)$ be an open set, such as $\mathcal{C}$ or $\mathcal{C}_\mathpzc{Y}$, then we define the weighted space $L^2(y^\alpha,\mathcal{D})$ as the space of all measurable functions defined on $\mathcal{D}$ with finite norm $\|w\|_{L^2(y^\alpha,\mathcal{D})}:=\|y^{\alpha/2}w\|_{L^2(\mathcal{D})}$. Similarly, using a standard multi-index notation, the space $H^1(y^\alpha,\mathcal{D})$ denotes the space of all measurable functions $w$ on $\mathcal{D}$ whose weak derivatives $D^\delta w$ exist for $|\delta|=1$ and fulfill $$\|w\|_{H^1(y^\alpha,\mathcal{D})}:=\left(\sum_{|\delta|\leq 1}\|D^\delta w\|^2_{L^2(y^\alpha,\mathcal{D})}\right)^{1/2}<\infty.$$ To study the extended problems we also need to introduce the space $$\mathring{H}^1_L(y^\alpha,\mathcal{C}):=\{w\in H^1(y^\alpha,\mathcal{C}):w=0\text{ on } \partial_L\mathcal{C}\}.$$ The space $\mathring{H}^1_L(y^\alpha,\mathcal{C}_\mathpzc{Y})$ is defined in an analogous manner. Formally, we need to indicate the trace of a function on $\Omega$ by introducing the trace mapping on $\Omega$. However, we skip this notation since it will be clear whenever we speak about traces.
Now, the extended problem reads as follows: Given $g\in \mathcal{V}^\star$, find $\mathcal{U}\in \mathring{H}^1_L(y^\alpha,\mathcal{C})$ such that $$\label{eq:extendedweak}
\int_\mathcal{C} y^\alpha \nabla \mathcal{U}\cdot\nabla \Phi\;dxdy+d_s\int_{\Omega}f(x,\mathcal{U})\Phi\;dx=d_s\langle g,\Phi \rangle_{\cV^\star,\cV}\quad \forall \Phi\in\mathring{H}^1_L(y^\alpha,\mathcal{C})$$ with $\alpha=1-2s$ and $d_s=2^\alpha \frac{\Gamma(1-s)}{\Gamma(s)}$, where we recall that $0<s<1$. That is, the function $\mathcal U\in \mathring{H}^1_L(y^\alpha,\mathcal{C})$ is a weak solution of the following problem $$\label{edp}
\begin{cases}
\mbox{div}(y^\alpha\nabla \mathcal U)=0\;\;&\mbox{ in}\;\mathcal C\\
\frac{\partial\mathcal U}{\partial\nu^\alpha}+d_sf(x,\mathcal U)=d_s g\;\;\;&\mbox{ on }\;\Omega\times\{0\},
\end{cases}$$ where we have set $$\begin{aligned}
\frac{\partial\mathcal U}{\partial\nu^\alpha}(x,0)=\lim_{y\to 0}y^\alpha\mathcal U_y(x,y)=\lim_{y\to 0}y^\alpha\frac{\partial\mathcal U(x,y)}{\partial y}.\end{aligned}$$
We have the following result.
\[lem:CS\] Let Assumption \[assum3\] on $f$ be fulfilled and $g\in \mathcal{V}^\star$ with $\mathcal{V}:=\mathbb H^{s}(\Omega)\cap L_F(\Omega)$ as defined at the beginning of Section \[sec:weaksol\]. Then there exists a unique weak solution $\mathcal{U}\in \mathcal V_L:= \left\{v\in \mathring{H}^1_L(y^\alpha,\mathcal{C}): v|_{\Omega\times\{0\}}\in \mathcal{V} \right\}$ of . Furthermore, there holds $\mathcal{U}(\cdot,0)=u\in \mathcal{V}$, where $u$ represents the weak solution of according to .
We already know that if the solution $\mathcal{U}\in\mathring{H}^1_L(y^\alpha,\mathcal{C})$ of exists then $\mathcal U(\cdot,0)=u\in\mathbb H^s(\Omega)$. This is a trivial consequence of the corresponding result for linear problems. Therefore, we just have to prove the existence and uniqueness part. Let us set $$\begin{aligned}
\mathcal E(\mathcal U,\Phi):=\int_\mathcal{C} y^\alpha \nabla \mathcal{U}\cdot\nabla \Phi\;dxdy+d_s\int_{\Omega}f(x,\mathcal{U})\,\Phi\;dx,\;\;\;\mathcal U,\Phi\in\mathcal V_L.\end{aligned}$$ Next, let $\mathcal U\in \mathcal{V}_L$ be fixed. It is clear that $\mathcal E(\mathcal U,\cdot)$ is linear in the second variable. Proceeding exactly as in the proof of Proposition \[prop-exis\], we get that $\mathcal E(\mathcal U,\cdot)\in \mathcal V_L^\star$. In addition, we have that $\mathcal E$ is strictly monotone, hemi-continuous and coercive. This finishes the proof.
In contrast to the nonlocal fractional Dirichlet problem , the extended problem (or equivalently ) is localized such that a discretization by standard finite elements becomes feasible. However, a direct discretization is still challenging due to the semi-infinite computational domain. As remedy, one can employ the exponential decay of the solution $\mathcal{U}$ in certain norms as $y$ tends to infinity, see [@NOS]. In this regard, a truncation of the semi-infinite cylinder is reasonable. This leads to a problem posed on the truncated cylinder $\mathcal{C}_\mathpzc{Y}$: Given $g\in \mathcal{V}^\star$, find $$\begin{aligned}
\mathcal{U}_\mathpzc{Y}\in \mathcal V_{L,\mathpzc{Y}}= \left\{v\in \mathring{H}^1_L(y^\alpha,\mathcal{C}_\mathpzc{Y}): v|_{\Omega\times\{0\}}\in \mathcal{V} \right\}\end{aligned}$$ such that $$\label{eq:truncatedweak}
\int_{\mathcal{C}_\mathpzc{Y}} y^\alpha \nabla \mathcal{U}_\mathpzc{Y}\cdot\nabla \Phi\;dxdy+d_s\int_{\Omega}f(x,\mathcal{U}_\mathpzc{Y})\Phi\;dx=d_s\langle g,\Phi \rangle_{\cV^\star,\cV}\quad \forall \Phi\in\mathring{H}^1_L(y^\alpha,\mathcal{C}_\mathpzc{Y}).$$
In view of the discretization error estimates in the next section, we do not need to estimate the truncation error for the semi-linear problems. Instead, we will use the corresponding results for linear problems.
Discretizing the problem and proof of error estimates {#s:disc}
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The discretization of the linear problem is outlined in [@NOS]. In fact, the theory there will build the basis for the discussion of the semi-linear problems presented in the further course of this section. For the convenience of the reader we will collect the main ingredients from the linear case before we turn towards the treatment of the semi-linear problems. From here on, we assume that the underlying domain $\Omega$ is convex and polyhedral. We notice that such a domain has a Lipschitz continuous boundary, see e.g. [@Chen].
Due to the singular behavior of the solution towards the boundary $\Omega$, anistropically refined meshes are preferable since these can be used to compensate the singular effects. In our context such meshes are defined as follows: Let $\mathscr{T}_\Omega=\{K\}$ be a conforming and quasi-uniform triangulation of $\Omega$, where $K\in \mathbb{R}^N$ is an element that is isoparametrically equivalent either to the unit cube or to the unit simplex in $\mathbb{R}^N$. We assume $\# \mathscr{T}_\Omega \sim M^N$. Thus, the element size $h_{\mathscr{T}_\Omega}$ fulfills $h_{\mathscr{T}_\Omega}\sim M^{-1}$. The collection of all these meshes is denoted by $\mathbb{T}_\Omega$. Furthermore, let $\mathcal{I}_\mathpzc{Y}=\{I\}$ be a graded mesh of the interval $[0,\mathpzc{Y}]$ in the sense that $[0,\mathpzc{Y}]=\bigcup_{k=0}^{M-1}[y_k,y_{k+1}]$ with $$y_k=\left(\frac{k}{M}\right)^\gamma\mathpzc{Y},\quad k=0,\ldots,M,\quad \gamma>\frac{3}{1-\alpha}=\frac{3}{2s}>1.$$ Now, the triangulations $\mathscr{T}_\mathpzc{Y}$ of the cylinder $\mathcal{C}_\mathpzc{Y}$ are constructed as tensor product triangulations by means of $\mathscr{T}_\Omega$ and $\mathcal{I}_\mathpzc{Y}$. The definitions of both imply $\# \mathscr{T}_\mathpzc{Y} \sim M^{N+1}$. Finally, the collection of all those anisotropic meshes $\mathscr{T}_\mathpzc{Y}$ is denoted by $\mathbb{T}$.
Now, we define the finite element spaces posed on the previously introduced meshes. For every $\mathscr{T}_\mathpzc{Y}\in \mathbb{T}$ the finite element spaces $\mathbb{V}(\mathscr{T}_\mathpzc{Y})$ are now defined by $$\mathbb{V}(\mathscr{T}_\mathpzc{Y}):=\{\Phi\in C^0(\overline{ \mathcal{C}_\mathpzc{Y}}):\Phi|_{T}\in\mathcal{P}_1(K)\oplus\mathbb{P}_1(I)\ \forall \;T=K\times I\in \mathscr{T}_\mathpzc{Y},\ \Phi|_{\partial_L\mathcal{C}_\mathpzc{Y}}=0\}.$$ In case that $K$ in the previous definition is a simplex then $\mathcal{P}_1(K)=\mathbb{P}_1(K)$, the set of polynomials of degree at most $1$. If $K$ is a cube then $\mathcal{P}_1(K)$ equals $\mathbb{Q}_1(K)$, the set of polynomials of degree at most 1 in each variable.
Throughout the remainder of the paper, without any mention, $0<s<1$, $\alpha=1-2s$ and $d_s=2^{\alpha}\frac{\Gamma(1-s)}{\Gamma(s)}$.
Using the just introduced notation, the finite element discretization of is given by the function $\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}\in\mathbb{V}(\mathscr{T}_\mathpzc{Y})$ which solves the variational identity $$\label{eq:truncateddiscrete}
\int_{\mathcal{C}_\mathpzc{Y}} y^\alpha \nabla \mathcal{U}_{\mathscr{T}_\mathpzc{Y}}\cdot\nabla \Phi\;dxdy+d_s\int_{\Omega}f(x,\mathcal{U}_{\mathscr{T}_\mathpzc{Y}})\Phi\;dx=d_s\langle g,\Phi \rangle_{\cV^\star,\cV}\quad \forall \Phi\in\mathbb{V}(\mathscr{T}_\mathpzc{Y}).$$
We have the following result.
Let Assumption \[assum3\] on $f$ be fulfilled and $g\in \mathcal{V}^\star$. Then there exists a unique solution $\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}\in\mathbb{V}(\mathscr{T}_\mathpzc{Y})$ of .
The existence of a solution can be proven by means of Browder’s fixed-point theorem employing the monotonicity of the nonlinearity $f$. The uniqueness is a consequence of the $\mathring{H}^1_L(y^\alpha,\mathcal{C}_\mathpzc{Y})$-coercivity of the bilinearform in and the monotonicity of $f$. Indeed, let $\mathcal{U}_1$ and $\mathcal{U}_2$ be two different solutions of . Then we infer that there exists a constant $c>0$ such that $$\begin{aligned}
\|\mathcal{U}_1- & \mathcal{U}_2\|^2_{H^1_L(y^\alpha,\mathcal{C}_\mathpzc{Y})}\leq \\
& c\left(\int_{\mathcal{C}_\mathpzc{Y}} y^\alpha |\nabla (\mathcal{U}_1-\mathcal{U}_2)|^2\;dxdy+d_s\int_{\Omega}(f(x,\mathcal{U}_1)-f(\cdot,\mathcal{U}_2))(\mathcal{U}_1-\mathcal{U}_2)\;dx\right)=0.
\end{aligned}$$ Hence, $\mathcal U_1=\mathcal U_2$ and the proof is finished.
For the error analysis it will be useful to have the intermediate solution $\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}\in\mathbb{V}(\mathscr{T}_\mathpzc{Y})$ which solves the variational identity
$$\label{eq:truncateddiscreteintermediate}
\int_{\mathcal{C}_\mathpzc{Y}} y^\alpha \nabla \tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}\cdot\nabla \Phi\;dxdy=d_s\langle g-f(\cdot,u),\Phi \rangle_{\cV^\star,\cV}\quad \forall \Phi\in\mathbb{V}(\mathscr{T}_\mathpzc{Y}),$$
where $u$ denotes the weak solution of . Since $\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}$ represents the solution of a linear problem, corresponding error estimates are directly applicable.
\[lemma:linear\] Let Assumptions \[assum3\] and \[assum-lip\] on $f$ be fulfilled and $g\in\mathbb{H}^{1-s}(\Omega)\cap L^p(\Omega)$ with $p$ as in . Moreover, let $u$ be the solution of and $\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}$ the solution of . Then there is a constant $c>0$ such that $$\|u-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}\|_{H^s(\Omega)}\leq c|\log(\# \mathscr{T}_\mathpzc{Y})|^s(\# \mathscr{T}_\mathpzc{Y})^{-1/(N+1)}$$ and $$\|u-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}\|_{L^2(\Omega)}\leq c|\log(\# \mathscr{T}_\mathpzc{Y})|^{2s}(\# \mathscr{T}_\mathpzc{Y})^{-(1+s)/(N+1)}$$ provided that $\mathpzc{Y}\sim \log(\# \mathscr{T}_\mathpzc{Y})$.
This is a consequence of [@NOS Theorem 5.4 and Remark 5.5] and [@NOS3 Proposition 4.7] once we know that $f(\cdot,u)\in\mathbb{H}^{1-s}(\Omega)$. Since $g\in \mathbb{H}^{1-s}(\Omega)\cap L^p(\Omega)$ with $p$ as in , it follows from Corollary \[ellip-regula\] that the unique weak solution $u$ belongs to $\mathbb H^{1+s}(\Omega)\hookrightarrow \mathbb H^{1-s}(\Omega)$ and hence $f(\cdot,u(\cdot))\in\mathbb H^{1-s}(\Omega)$ according to Lemma \[lem-f-lip\]. This finishes the proof.
For later purposes, we need to show that $\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}$ is uniformly bounded in $L^\infty(\Omega)$, since we only assume a local Lipschitz condition for the nonlinearity $f$.
\[lemma:Linftybounded\] Let Assumptions \[assum3\] and \[assum-lip\] on $f$ be fulfilled and $g\in \mathbb H^{1-s}(\Omega)\cap L^p(\Omega)$ with $p$ as in . Furthermore, let $s>(N-2)/2$. Then the solution $\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}$ of is uniformly bounded in $L^\infty(\Omega)$.
We denote by $I_{\mathscr{T}_\Omega}u$ the (modified) Clement interpolant of $u$, which is well defined for $u\in\mathbb{H}^{s}(\Omega)$. Next, let $K_*\in \mathscr{T}_\Omega$ be the element where $|u-I_{\mathscr{T}_\Omega}u|$ admits its supremum. By means of an inverse inequality, we deduce $$\begin{aligned}
\|u-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}\|_{L^\infty(\Omega)}&=\|u-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}\|_{L^\infty(K_*)}\leq \|u-I_{\mathscr{T}_\Omega}u\|_{L^\infty(K_*)}+\|I_{\mathscr{T}_\Omega}u-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}\|_{L^\infty(K_*)}\notag\\
&\leq c\left(\|u-I_{\mathscr{T}_\Omega}u\|_{L^\infty(K_*)}+h_{K_*}^{-N/2}\|u-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}\|_{L^2(K_*)}\right),\label{eq:linfty}\end{aligned}$$ where $h_{K_*}$ denotes the diameter of $K$. The first term in is bounded due to Theorem \[theo-bound\]. For the second one, we notice that $h_{K_*}\sim h_{\mathcal{T}_\Omega}\sim M^{-1}$ and $\# \mathscr{T}_\mathpzc{Y}\sim M^{N+1}$. Consequently, the assertion follows from Lemma \[lemma:linear\].
\[lemma:intermediate\] Let Assumptions \[assum3\] and \[assum-lip\] on $f$ be fulfilled, $g\in \mathbb H^{1-s}(\Omega)\cap L^p(\Omega)$ with $p$ as in and $s>(N-2)/2$. Furthermore, let $u$, $\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}$ and $\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}$ be the solutions of , and , respectively. Then there is a constant $c>0$ such that $$\|\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}\|_{L^2(\Omega)}\leq c\|u-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}\|_{L^2(\Omega)}.$$
Due to the $\mathring{H}^1_L(y^\alpha,\mathcal{C}_\mathpzc{Y})$-coercivity of the bilinear form in and , and the monotonicity of $f$, we obtain that there is a constant $c>0$ such that
$$\begin{aligned}
c\|\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}\|^2_{H^1_L(y^\alpha,\mathcal{C}_\mathpzc{Y})}&\leq \int_{\mathcal{C}_\mathpzc{Y}} y^\alpha \nabla (\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}})\cdot\nabla (\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}})\\
&=d_s\int_{\Omega}(f(\cdot,\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}})-f(\cdot,\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}))(\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}) \\
&\quad +d_s\int_{\Omega}(f(\cdot,u)-f(\cdot,\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}))(\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}})\\
&\leq d_s\int_{\Omega}(f(\cdot,u)-f(\cdot,\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}))(\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}).
\end{aligned}$$
Next, observe that both $u$ and $\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}$ are uniformly bounded in $L^\infty(\Omega)$ according to Theorem \[theo-bound\] and Lemma \[lemma:Linftybounded\]. Consequently, the Cauchy-Schwarz inequality and the Lipschitz-continuity of the nonlinearity yield $$\begin{aligned}
\int_{\Omega}(f(\cdot,u)-f(\cdot,\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}))(\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}})&\leq c \|u-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}\|_{L^2(\Omega)}\|\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}\|_{L^2(\Omega)}.\label{eq:lipschitz}
\end{aligned}$$ Finally, the assertion can be deduced by means of the foregoing inequalities and the trace theorem of [@CDDS:11 Proposition 2.1], i.e., $$\|\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}\|_{L^2(\Omega)}\leq c \|\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}-\tilde{\mathcal{U}}_{\mathscr{T}_\mathpzc{Y}}\|_{H^1_L(y^\alpha,\mathcal{C}_\mathpzc{Y})},$$ and the proof is finished.
As a direct consequence of Lemmas \[lemma:linear\] and \[lemma:intermediate\], we obtain the main result of this section.
\[theorem:semilinear\] Let Assumptions \[assum3\] and \[assum-lip\] on $f$ be fulfilled, $g\in\mathbb{H}^{1-s}(\Omega)\cap L^p(\Omega)$ with $p$ as in and let $s>(N-2)/2$. Moreover, let $u$ be the solution of and $\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}$ the solution of . Then there is a constant $c>0$ such that $$\|u-\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}\|_{\mathbb H^s(\Omega)}\leq c|\log(\# \mathscr{T}_\mathpzc{Y})|^s(\# \mathscr{T}_\mathpzc{Y})^{-1/(N+1)}$$ and $$\|u-\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}\|_{L^2(\Omega)}\leq c|\log(\# \mathscr{T}_\mathpzc{Y})|^{2s}(\# \mathscr{T}_\mathpzc{Y})^{-(1+s)/(N+1)}$$ provided that $\mathpzc{Y}\sim \log(\# \mathscr{T}_\mathpzc{Y})$.
We finally illustrate the results of Theorem \[theorem:semilinear\] by a numerical example. Let $N = 2$, $\Omega = (0,1)^2$. Under this setting, the eigenvalues and eigenfunctions of $-\Delta_D$ are: $$\lambda_{k,l} = \pi^2 (k^2 + l^2) , \quad \varphi_{k,l}(x_1,x_2) = \sin(k\pi x_1) \sin(l\pi x_2) \quad k, l \in \mathbb{N} .$$ Let the exact solution to be $$\label{eq:uexact}
u = \lambda_{2,2}^{-s} \sin(2\pi x_1) \sin(2\pi x_2)$$ and nolinearity $f(\cdot,u) = u^3=|u|^2u$. Using we immediately arrive at the expression for datum $g$.
We use Newton’s method to solve the nonlinear problem. The asymptotic relation $\|u-\mathcal{U}_{\mathscr{T}_\mathpzc{Y}}\|_{\mathbb H^s(\Omega)} \approx (\# \mathscr{T}_\mathpzc{Y})^{-1/3}$ is shown in Figure \[f:HsL2\] (left) for different choices of $s = 0.2, 0.4, 0.6$, and $s = 0.8$. We observe a quasi-optimal decay rate which confirms the $\mathbb{H}^s$-estimate in Theorem \[theorem:semilinear\]. We also present the $L^2$-error estimates in Figure \[f:HsL2\] (right), which decays as $(\# \mathscr{T}_\mathpzc{Y})^{-2/3}$ which is better than our theoretical prediction in Theorem \[theorem:semilinear\]. Notice that under the current literature status, theoretically, we cannot expect a better rate than Theorem \[theorem:semilinear\], as we have used the linear result from [@NOS3 Proposition 4.7] to prove Lemma \[lemma:linear\].
![\[f:HsL2\] Rate of convergence on anisotropic meshes for $N = 2$ and $s = 0.2,0.4, 0.6$ and $s = 0.8$ is shown. $U$ is the numerical solution to obtained by using Newton’s method. On the other hand, $u$ is the exact solution given by . The blue line is the reference line. The left panel shows the $\mathbb{H}^s(\Omega)$-error, in all cases we recover $(\# \mathscr{T}_\mathpzc{Y})^{-1/3}$. The right panel shows the $L^2$-error which decays as $(\# \mathscr{T}_\mathpzc{Y})^{-2/3}$.](graded_state_Hs "fig:"){width="48.00000%"} ![\[f:HsL2\] Rate of convergence on anisotropic meshes for $N = 2$ and $s = 0.2,0.4, 0.6$ and $s = 0.8$ is shown. $U$ is the numerical solution to obtained by using Newton’s method. On the other hand, $u$ is the exact solution given by . The blue line is the reference line. The left panel shows the $\mathbb{H}^s(\Omega)$-error, in all cases we recover $(\# \mathscr{T}_\mathpzc{Y})^{-1/3}$. The right panel shows the $L^2$-error which decays as $(\# \mathscr{T}_\mathpzc{Y})^{-2/3}$.](graded_state_L2 "fig:"){width="48.00000%"}
[^1]: Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA, `[email protected]`
[^2]: Chair of Optimal Control, Center of Mathematical Sciences, Technical University of Munich, Boltzmannstra[ß]{}e 3, 85748 Garching by Munich, Germany, `[email protected]`
[^3]: University of Puerto Rico (Rio Piedras Campus), College of Natural Sciences, Department of Mathematics, PO Box 70377 San Juan PR 00936-8377 (USA), `[email protected], [email protected]`
[^4]: The work of the first and second author is partially supported by NSF grant DMS-1521590. The work of the third author is partially supported by the Air Force Office of Scientific Research under the Award No: FA9550-15-1-0027
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abstract: 'We derive total mean curvature integration formulae of a three co-dimensional foliation $\mathcal{F}^{n}$ on a screen integrable half-lightlike submanifold, $M^{n+1}$ in a semi-Riemannian manifold $\overline{M}^{n+3}$. We give generalized differential equations relating to mean curvatures of a totally umbilical half-lightlike submanifold admitting a totally umbilical screen distribution, and show that they are generalizations of those given by [@ds2].'
address:
- ' School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Private Bag X01, Scottsville 3209South Africa'
- ' School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Private Bag X01, Scottsville 3209South Africa'
author:
- 'Fortuné Massamba\*, Samuel Ssekajja\*\*'
title: 'On total mean curvatures of foliated half-lightlike submanifolds in semi-Riemannian manifolds'
---
Introduction
============
The rapidly growing importance of lightlike submanifolds in semi-Riemannian geometry, particularly Lorentzian geometry, and their applications to mathematical physics–like in general relativity and electromagnetism motivated the study of lightlike geometry in semi-Riemannian manifolds. More precisely, lightlike submanifolds have been shown to represent different black hole horizons (see [@db] and [@ds2] for details). Among other motivations for investing in lightlike geometry by many physicists is the idea that the universe we are living in can be viewed as a 4-dimensional hypersurface embedded in $(4+m)$-dimensional spacetime manifold, where $m$ is any arbitrary integer. There are significant differences between lightlike geometry and Riemannian geometry as shown in [@db] and [@ds2], and many more references therein. Some of the pioneering work on this topic is due to Duggal-Bejancu [@db], Duggal-Sahin [@ds2] and Kupeli [@kup]. It is upon those books that many other researchers, including but not limited to [@ca], [@dusa1], [@gup1], [@Jin], [@ma1], [@ma2], [@ms], [@cohen], have extended their theories.
Lightlike geometry rests on a number of operators, like shape and algebraic invariants derived from them, such as trace, determinants, and in general the $r$-th mean curvature $S_{r}$. There is a great deal of work so far on the case $r=1$ (see some in [@db], [@ds2] and many more) and as far as we know, very little has been done for the case $r>1$. This is partly due to the non-linearity of $S_{r}$ for $r>1$, and hence very complicated to study. A great deal of research on higher order mean curvatures $S_{r}$ in Riemannian geometry has been done with numerous applications, for instance see [@krz] and [@woj]. This gap has motivated our introduction of lightlike geometry of $S_{r}$ for $r>1$. In this paper we have considered a half-lightlike submanifold admitting an integrable screen distribution, of a semi-Riemannian manifold. On it we have focused on a codimension 3 foliation of its screen distribution and thus derived integral formulae of its total mean curvatures (see Theorems \[theorem3\] and \[theorem5\]). Furthermore, we have considered totally umbilical half-lightlike submanifolds, with a totally umbilical screen distribution and generalized Theorem 4.3.7 of [@ds2] (see Theorem \[thmy\] and its Corollaries). The paper is organized as follows; In section \[prel\] we summarize the basic notions on lightlike geometry necessary for other sections. In section \[New1\] we give some basic information on Newton transformations of a foliation $\mathcal{F}$ of the screen distribution. Section \[dist\] focuses on integration formulae of $\mathcal{F}$ and their consequences. In section \[screen\] we discus screen umbilical half-lightlike submanifolds and generalizations of some well-known results of [@ds2].
Preliminaries {#prel}
=============
Let $(M^{n+1},g)$ be a two-co-dimensional submanifold of a semi-Riemannian manifold $(\overline{M}^{n+3},\overline{g})$, where $g=\overline{g}|_{TM}$. The submanifold $(M^{n+1},g)$ is called a *half-lightlike* if the radical distribution $\mathrm{Rad}\,T M = T M \cap TM^{\perp}$ is a vector subbundle of the tangent bundle $TM$ and the normal bundle $TM^{\perp}$ of $M$ , with rank one. Let $S(T M)$ be a *screen distribution* which is a semi-Riemannian complementary distribution of $\mathrm{Rad}\,T M$ in $T M$, and also choose a *screen transversal bundle* $S(TM^\perp)$, which is semi-Riemannian and complementary to $\mathrm{Rad}\, TM$ in $TM^\perp$. Then, $$\label{N5}
T M = \mathrm{Rad}\,T M \perp S(T M),\;\;T M^{\perp} = \mathrm{Rad}\,T M \perp S(TM^{\perp}).$$ We will denote by $\Gamma(\Xi)$ the set of smooth sections of the vector bundle $\Xi$. It is well-known from [@db] and [@ds2] that for any null section $E$ of $\mathrm{Rad}\,TM$, there exists a unique null section $N$ of the orthogonal complement of $S(T M^\perp)$ in $S(T M )^\perp$ such that $g(E,N) = 1$, it follows that there exists a lightlike *transversal vector bundle* $l\mathrm{tr}(TM)$ locally spanned by $N$. Let $W\in\Gamma(S(TM^{\perp}))$ be a unit vector field, then $\overline{g}(N,N) = \overline{g}(N, Z)=\overline{g}(N, W) = 0$, for any $Z\in \Gamma(S(TM))$.
Let $\mathrm{tr}(TM)$ be complementary (but not orthogonal) vector bundle to $TM$ in $T\overline{M}$. Then we have the following decompositions of $\mathrm{tr}(TM)$ and $T\overline{M}$ $$\begin{aligned}
\mathrm{tr}(TM)& =l\mathrm{tr}(TM)\perp S(TM^\perp),\label{N8}\\
T\overline{M} & = S(TM)\perp S(TM^\perp)\perp\{\mathrm{Rad}\, TM\oplus l\mathrm{tr}(TM)\}\label{N9} .\end{aligned}$$ It is important to note that the distribution $S(TM)$ is not unique, and is canonically isomorphic to the factor vector bundle $TM/ \mathrm{Rad}\, TM$ [@db]. Let $P$ be the projection of $TM$ on to $S(TM)$. Then the local Gauss-Weingarten equations of $M$ are the following; $$\begin{aligned}
&\overline{\nabla}_{X}Y=\nabla_{X}Y+B(X,Y)N+D(X,Y)W,\label{P1}\\
&\overline{\nabla}_{X}N=-A_{N}X+\tau(X)N+\rho(X)W,\label{P2}\\
&\overline{\nabla}_{X}W=-A_{W}X+\phi(X)N,\label{P3}\\
&\nabla_{X}PY = \nabla^{*}_{X}PY + C(X,PY)E,\label{P4}\\
&\nabla_{X}E =-A^{ * }_{E}X -\tau(X) E,\label{P5}
\end{aligned}$$ for all $E\in\Gamma(\mathrm{Rad}\,T M)$, $N\in\Gamma(l\mathrm{tr}(T M))$ and $W\in\Gamma(S(TM^{\perp}))$, where $\nabla$ and $\nabla^{*}$ are induced linear connections on $TM$ and $S(TM)$, respectively, $B$ and $D$ are called the local second fundamental forms of $M$, $C$ is the local second fundamental form on $S(T M)$. Furthermore, $\{A_{N}, A_{W}\}$ and $A^{*}_{E}$ are the shape operators on $TM$ and $S(TM)$ respectively, and $\tau$, $\rho$, $\phi$ and $\delta$ are differential 1-forms on $TM$. Notice that $\nabla^{*}$ is a metric connection on $S(TM)$ while $\nabla$ is generally not a metric connection. In fact, $\nabla$ satisfies the following relation $$\begin{aligned}
\label{P6}
(\nabla_{X}g)(Y,Z)= B(X,Y)\lambda(Z) + B(X,Z)\lambda(Y),
\end{aligned}$$ for all $X,Y,Z\in\Gamma(TM)$, where $\lambda$ is a 1-form on $TM$ given $\lambda(\cdot)= \overline{g}(\cdot, N)$. It is well-known from [@db] and [@ds2] that $B$ and $D$ are independent of the choice of $S(TM)$ and they satisfy $$\label{P7}
B(X, E) = 0,\;\;\;D(X, E) = −\phi(X),\;\;\forall\, X \in\Gamma(TM).$$ The local second fundamental forms $B$, $D$ and $C$ are related to their shape operators by the following equations $$\begin{aligned}
&g(A^{*}_{E}X,Y)=B(X,Y),\;\;\;\;\; \overline{g}(A^{*}_{E}X,N)=0,\label{P8}\\
&g(A_{W}X,Y) = \varepsilon D(X,Y)+ \phi(X)\lambda(Y),\label{P9}\\
&g(A_{N}X,PY) = C(X,PY ),\;\;\overline{g}(A_{N} X,N) = 0,\label{P10}\\
&\overline{g}(A_{W} X,N) = \varepsilon \rho(X),\;\;\mbox{where}\;\; \varepsilon=\overline{g}(W,W),\label{P11}
\end{aligned}$$ for all $X,Y\in\Gamma(TM)$. From equations (\[P8\]) we deduce that $A^{*}_{E}$ is $S(TM)$-valued, self-adjoint and satisfies $A^{*}_{E}E=0$. Let $\overline{R}$ denote the curvature tensor of $\overline{M}$, then $$\begin{aligned}
\label{P60}
\overline{g}(\overline{R}(X,Y)PZ,N)&=g((\nabla_{X}A_{N})Y, PZ)-g((\nabla_{Y}A_{N})X, PZ)\nonumber\\
&+\tau(Y)C(X,PZ)-\varepsilon\tau(X)C(Y,PZ)\{\rho(Y)D(X,PZ)\nonumber\\
&-\rho(X)D(Y,PZ)\},\;\;\;\forall\, X,Y,Z\in\Gamma(TM).\end{aligned}$$ A half-lightlike submanifold $(M,g)$ of a semi-Riemannian manifold $\overline{M}$ is said to be totally umbilical [@ds2] if on each coordinate neighborhood $\mathcal{U}$ there exist smooth functions $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ on $l\mathrm{tr}(TM)$ and $S(TM^{\perp})$ respect such that $$\begin{aligned}
\label{P61}
B(X,Y)=\mathcal{H}_{1}g(X,Y),\;\;\;D(X,Y)=\mathcal{H}_{2}g(X,Y),\;\;\forall\, X,Y\in\Gamma(TM).\end{aligned}$$ Furthermore, when $M$ is totally umbilical then the following relations follows by straightforward calculations $$\begin{aligned}
\label{P62}
A_{E}^{*}X=\mathcal{H}_{1}PX,\;\;P(A_{W}X)=\varepsilon \mathcal{H}_{2}PX,\;\;D(X,E)=0,\;\;\rho(E)=0,\end{aligned}$$ for all $X,Y\in\Gamma(TM)$.
Next, we suppose that $M$ is a half-lightlike submanifold of $\overline{M}$, with an integrable screen distribution $S(TM)$. Let $M'$ be a leaf of $S(TM)$. Notice that for any screen integrable half-lightlike $M$, the leaf $M'$ of $S(TM)$ is a co-dimension 3 submanifold of $\overline{M}$ whose normal bundle is $\{\mathrm{Rad}\,TM\oplus l\mathrm{tr}(TM)\}\perp S(TM^{\perp})$. Now, using (\[P1\]) and (\[P4\]) we have $$\begin{aligned}
\label{P12}
\overline{\nabla}_{X}Y=\nabla^{*}_{X}Y + C(X,PY)E+B(X,Y)N+D(X,Y)W,\end{aligned}$$ for all $X,Y\in\Gamma(TM')$. Since $S(TM)$ is integrable, then its leave is semi-Riemannian and hence we have $$\begin{aligned}
\label{P13}
\overline{\nabla}_{X}Y=\nabla^{*'}_{X}Y+h'(X,Y), \;\;\; \forall\, X,Y\in\Gamma(TM'),\end{aligned}$$ where $h'$ and $\nabla^{*'}$ are second fundamental form and the Levi-Civita connection of $M'$ in $\overline{M}$. From (\[P12\]) and (\[P13\]) we can see that $$\begin{aligned}
\label{P14}
h'(X,Y)=C(X,PY)E+B(X,Y)N+D(X,Y)W, \end{aligned}$$ for all $X,Y\in\Gamma(TM')$. Since $S(TM)$ is integrable, then it is well-known from [@ds2] that $C$ is symmetric on $S(TM)$ and also $A_{N}$ is self-adjoint on $S(TM)$ (see Theorem 4.1.2 for details). Thus, $h'$ given by (\[P14\]) is symmetric on $TM'$.
Let $L\in \Gamma(\{\mathrm{Rad}\,TM\oplus l\mathrm{tr}(TM)\}\perp S(TM^{\perp}))$, then we can decompose $L$ as $$\begin{aligned}
\label{P15}
L=aE+bN+cW,\end{aligned}$$ for non-vanishing smooth functions on $\overline{M}$ given by $a=\overline{g}(L,N)$, $b=\overline{g}(L,E)$ and $c=\varepsilon \overline{g}(L,W)$. Suppose that $\overline{g}(L,L)>0$, then using (\[P15\]) we obtain a unit normal vector $\widehat{W}$ to $M'$ given by $$\begin{aligned}
\label{P16}
\widehat{W}=\frac{1}{\overline{g}(L,L)}(aE+bN+cW)=\frac{1}{\overline{g}(L,L)}L.\end{aligned}$$ Next we define a (1,1) tensor $\mathcal{A}_{\widehat{W}}$ in terms of the operators $A^{*}_{E}$, $A_{N}$ and $A_{W}$ by $$\begin{aligned}
\label{P17}
\mathcal{A}_{\widehat{W}}X=\frac{1}{\overline{g}(L,L)}(aA^{*}_{E}X+bA_{N}X+cA_{W}X),\end{aligned}$$ for all $X\in\Gamma(TM)$. Notice that $\mathcal{A}_{\widehat{W}}$ is self-adjoint on $S(TM)$. Applying $\overline{\nabla}_{X}$ to $\widehat{W}$ and using equations (\[P17\]) (\[P1\]) and (\[P8\])-(\[P10\]), we have $$\begin{aligned}
\label{P18}
g(\mathcal{A}_{\widehat{W}}X, PY)=-\overline{g}(\overline{\nabla}_{X}\widehat{W}, PY), \;\;\forall\, X,Y\in\Gamma(TM).\end{aligned}$$ Let $\nabla^{*\perp}$ be the connection on the normal bundle $\{\mathrm{Rad}\,TM\oplus l\mathrm{tr}(TM)\}\perp S(TM^{\perp})$. Then from (\[P18\]) we have $$\begin{aligned}
\label{P19}
\overline{\nabla}_{X}\widehat{W}=-\mathcal{A}_{\widehat{W}}X+\nabla^{*\perp}_{X}\widehat{W}, \;\;\forall\, X\in\Gamma(TM),\end{aligned}$$ where $$\begin{aligned}
\nabla^{*\perp}_{X}\widehat{W} & = -\frac{1}{\overline{g}(L,L)}X(\overline{g}(L,L))\widehat{W} + \frac{1}{\overline{g}(L,L)}\left[\{X(a) - a\tau(X)\}E \right.\nonumber\\
&\left. + \{X(b)+b\tau(X) + c\phi(X)\}N + \{X(c) + aD(X,E) + b\rho(X)\}W\right].\nonumber
\end{aligned}$$
[ Let $\overline{M}=(\mathbb{R}^{5}_{1}, \overline{g})$ be a semi-Riemannian manifold, where $\overline{g}$ is of signature $(-,+,+,+,+)$ with respect to canonical basis $(\partial x_{1},\partial x_{2},\partial x_{3},\partial x_{4},\ \partial x_{5})$, where $(x_{1},\cdots,x_{5})$ are the usual coordinates on $\overline{M}$. Let $M$ be a submanifold of $\overline{M}$ and given parametrically by the following equations $$\begin{aligned}
x_{1}=&\varphi_{1},\;\; x_{2}= \sin \varphi_{2}\sin \varphi_{3},\;\; x_{3}=\varphi_{1},\;\; x_{4}=\cos \varphi_{2}\sin \varphi_{3},\\
&x_{5}=\cos \varphi_{3},\;\;\mbox{where}\;\; \varphi_{2}\in[0,2\pi]\;\; \mbox{and}\;\; \varphi_{3}\in(0,\pi/2).
\end{aligned}$$ Then we have $TM=\mathrm{span}\{E,Z_{1},Z_{2}\}$ and $l\mathrm{tr}(TM)=\mathrm{span}\{N\}$, where $$\begin{aligned}
&E=\partial x_{1}+\partial x_{3},\;\; Z_{1}=\cos \varphi_{3}\partial x_{2}-\sin \varphi_{2}\sin \varphi_{3}\partial x_{5},\\
&Z_{2}=\cos \varphi_{3}\partial x_{4}-\cos \varphi_{2}\sin \varphi_{3}\partial x_{5}\;\;\mbox{and}\;\; N=\frac{1}{2}(-\partial x_{1}+\partial x_{3}).\end{aligned}$$ Also, by straightforward calculations, we have $$\begin{aligned}
W=\sin \varphi_{2}\sin \varphi_{3} \partial x_{2}+\cos \varphi_{2}\sin \varphi_{3}\partial x_{4}+\cos \varphi_{3}\partial x_{5}.\end{aligned}$$ Thus, $S(TM^{\perp})=\mathrm{span}\{W\}$ and hence $M$ is a half-lightlike submanifold of $\overline{M}$. Furthermore we have $[Z_{1},Z_{2}]=\cos \varphi_{2} \sin \varphi_{3}\partial x_{2}-\sin \varphi_{2} \sin \varphi_{3}\partial x_{4}$, which leads to $[Z_{1},Z_{2}]=\cos \varphi_{2}\tan \varphi_{3}Z_{1}-\sin \varphi_{2}\tan \varphi_{3}Z_{2}\in\Gamma(S(TM))$. Thus, $M$ is a screen integrable half-lightlike submanifold of $\overline{M}$. Finally, it is easy to see that $A_{N}$ is self-adjoint operator on $S(TM)$. ]{}
In the next sections we shall consider screen integrable half-lightlike submanifolds of semi-Riemannian manifold $\overline{M}$ and derive special integral formulae for a foliation of $S(TM)$, whose normal vector is $\widehat{W}$ and with shape operator $\mathcal{A}_{\widehat{W}}$.
Newton transformations of $\mathcal{A}_{\widehat{W}}$ {#New1}
=====================================================
Let $(\overline{M}^{m+3}, \overline{g})$ be a semi-Riemannian manifold and let $(M^{n+1},g)$ be a screen integrable half-lightlike submanifold of $\overline{M}$. Then $S(TM)$ admits a foliation and let $\mathcal{F}$ be a such foliation. Then, the leaves of $\mathcal{F}$ are co-dimension three submanifolds of $\overline{M}$, whose normal bundle is $S(TM)^{\perp}$. Let $\widehat {W}$ be unit normal vector to $\mathcal{F}$ such that the orientation of $\overline{M}$ coincides with that given by $\mathcal{F}$ and $\widehat{W}$. The Levi-Civita connection $\overline{\nabla}$ on the tangent bundle of $\overline{M}$ induces a metric connection $\nabla'$ on $\mathcal{F}$. Furthermore, $h'$ and $\mathcal{A}_{\widehat{W}}$ are the second fundamental form and shape operator of $\mathcal{F}$. Notice that $\mathcal{A}_{\widehat{W}}$ is self-adjoint on $T\mathcal{F}$ and at each point $p\in \mathcal{F}$ has $n$ real eigenvalues (or principal curvatures) $\kappa_{1}(p),\cdots,\kappa_{n}(p)$. Attached to the shape operator $\mathcal{A}_{\widehat{W}}$ are $n$ algebraic invariants $$S_{r}=\sigma_{r}(\kappa_{1},\cdots, \kappa_{n}),\;\; 1\le r\le n,$$ where $\sigma_{r}: M^{'n}\rightarrow\mathbb{R}$ are symmetric functions given by $$\label{N71}
\sigma_{r}( \kappa_{1},\cdots, \kappa_{n})=\sum_{1\leq i_{1}< \cdots< i_{r}\leq n} \kappa_{i_{1}}\cdots \kappa_{i_{r}}.$$ Then, the characteristic polynomial of $\mathcal{A}_{\widehat{W}}$ is given by $$\det(\mathcal{A}_{\widehat{W}}-t\mathbb{I})=\sum_{\alpha=0}^{n}(-1)^{\alpha}S_{r}t^{n-\alpha},$$ where $\mathbb{I}$ is the identity in $\Gamma(T\mathcal{F})$. The normalized $r$-th mean curvature $H_{r}$ of $M'$ is defined by $$H_{r}=\dbinom{n}{r}^{-1}S_{r}\;\;\;\mbox{and}\;\;\;H_{0}=1. \;\;\mbox{(a constant function 1)}.$$ In particular, when $r=1$ then $H_{1}=\frac{1}{n} \mathrm{tr}(\mathcal{A}_{\widehat{W}})$ which is the *mean curvature* of a $\mathcal{F}$. On the other hand, $H_{2}$ relates directly with the (intrinsic) scalar curvature of $\mathcal{F}$. Moreover, the functions $S_{r}$ ($H_{r}$ respectively) are smooth on the whole $M$ and, for any point $p\in \mathcal{F}$, $S_{r}$ coincides with the $r$-th mean curvature at $p$. In this paper, we shall use $S_{r}$ instead of $H_{r}$.
Next, we introduce the Newton transformations with respect to the operator $\mathcal{A}_{\widehat{W}}$. The Newton transformations $T_{r}:\Gamma(T\mathcal{F})\rightarrow \Gamma(T\mathcal{F})$ of a foliation $\mathcal{F}$ of a screen integrable half-lightlike submanifold $M$ of an $(n+3)$-dimensional semi-Riemannian manifold $\overline{M}$ with respect to $A_{\widehat W}$ are given by by the inductive formula $$\label{N28}
T_{0}=\mathbb{I},\quad T_{r}=(-1)^{r}S_{r}\mathbb{I}+\mathcal{A}_{\widehat{W}}\circ T_{r-1}, \;\;\; 1\leq r\leq n.$$ By Cayley-Hamiliton theorem, we have $T_{n}=0$. Moreover, $T_{r}$ are also self-adjoint and commutes with $\mathcal{A}_{\widehat{W}}$. Furthermore, the following algebraic properties of $T_{r}$ are well-known (see [@krz], [@woj] and references therein for details). $$\begin{aligned}
\mathrm{tr}(T_{r}) & =(-1)^{r}(n-r) S_{r},\label{N31}\\
\mathrm{tr}(\mathcal{A}_{\widehat{W}}\circ T_{r}) & =(-1)^{r}(r+1)S_{r+1},\label{N32}\\
\mathrm{tr}(\mathcal{A}_{\widehat{W}}^{2}\circ T_{r}) & =(-1)^{r+1}(-S_{1} S_{r+1} + (r + 2) S_{r+2}),\label{N33}\\
\mathrm{tr}(T_{r}\circ\nabla'_{X} \mathcal{A}_{\widehat{W}}) & = (-1)^{r}X(S_{r+1})=(-1)^{r}\overline{g}(\nabla'S_{r+1},X),.\label{N34}\end{aligned}$$ for all $X\in\Gamma(T\overline{M})$. We will also need the following divergence formula for the operators $T_{r}$ $$\label{N35}
\mathrm{div}^{\nabla'}(T_{r})=\mathrm{tr}(\nabla' T_{r})=\sum_{\beta=1}^{n}(\nabla'_{Z_{\beta}}T_{r})Z_{\beta},$$ where $\{Z_{1},\cdots,Z_{n}\}$ is a local orthonormal frame field of $T\mathcal{F}$.
Integration formulae for $\mathcal{F}$ {#dist}
======================================
This section is devoted to derivation of integral formulas of foliation $\mathcal{F}$ of $S(TM)$ with a unit normal vector $\widehat{W}$ given by (\[P16\]). By the fact that $\overline{\nabla}$ is a metric connection then $\overline{g}(\overline{\nabla}_{\widehat{W}}\widehat{W}, \widehat{W})=0$. This implies that the vector field $\overline{\nabla}_{\widehat{W}}\widehat{W}$ is always tangent to $\mathcal{F}$. Our main goal will be to compute the divergence of the vectors $T_{r}\overline{\nabla}_{\widehat{W}}\widehat{W}$ and $T_{r}\overline{\nabla}_{\widehat{W}}\widehat{W}+(-1)^{r}S_{r+1}\widehat{W}$. The following technical lemmas are fundamentally important to this paper. Let $\{E,Z_{i},N,W\}$, for $i=1,\cdots,n$ be a quasi-orthonormal field of frame of $T\overline{M}$, such that $S(TM)=\mathrm{span}\{Z_{i}\}$ and $\epsilon_{i}=\overline{g}(Z_{i},Z_{i})$.
\[lemma1\] Let $M$ be a screen integrable half-lightlike submanifold of $\overline{M}^{n+3}$ and let $M'$ be a foliation of $S(TM)$. Let $\mathcal{A}_{\widehat{W}}$ be its shape operator, where $\widehat{W}$ is a unit normal vector to $\mathcal{F}$. Then $$\begin{aligned}
\overline{g}((\nabla'_{X}\mathcal{A}_{\widehat{W}})Y,Z)=\overline{g}(Y,(\nabla'_{X}\mathcal{A}_{\widehat{W}})Z),\;\;\overline{g}((\nabla'_{X}T_{r})Y,Z)=\overline{g}(Y,(\nabla'_{X}T_{r})Z),
\end{aligned}$$ for all $X,Y,Z\in\Gamma(T\mathcal{F})$.
By simple calculations we have $$\begin{aligned}
\label{P20}
\overline{g}((\nabla'_{X}\mathcal{A}_{\widehat{W}})Y,Z)=\overline{g}(\nabla'_{X}(\mathcal{A}_{\widehat{W}}Y),Z)-\overline{g}(\nabla'_{X}Y,\mathcal{A}_{\widehat{W}}Z).
\end{aligned}$$ Using the fact that $\nabla'$ is a metric connection and the symmetry of $\mathcal{A}_{\widehat{W}}$, (\[P20\]) gives $$\begin{aligned}
\label{P21}
\overline{g}((\nabla'_{X}\mathcal{A}_{\widehat{W}})Y,Z)=\overline{g}(Y,\nabla'_{X}(\mathcal{A}_{\widehat{W}}Z))-\overline{g}(Y,\mathcal{A}_{\widehat{W}}(\nabla'_{X}Z)).\end{aligned}$$ Then, from (\[P21\]) we deduce the first relation of the lemma. A proof of the second relation follows in the same way, which completes the proof.
\[lemma2\] Let $M$ be a screen integrable half-lightlike submanifold of $\overline{M}$ and let $\mathcal{F}$ be a co-dimension three foliation of $S(TM)$. Let $\mathcal{A}_{\widehat{W}}$ be its shape operator, where $\widehat{W}$ is a unit normal vector to $\mathcal{F}$. Denote by $\overline{R}$ the curvature tensor of $\overline{M}$. Then $$\begin{aligned}
\mathrm{div}^{\nabla'}(T_{0})&=0,\\
\mathrm{div}^{\nabla'}(T_{r})& =\mathcal{A}_{\widehat{W}}\mathrm{div}^{\nabla'}(T_{r-1}) + \sum_{i=1}^{n}\epsilon_{i}(\overline{R}(\widehat {W},T_{r-1}Z_{i})Z_{i})',
\end{aligned}$$ where $(\overline{R}(\widehat {W},X)Z)'$ denotes the tangential component of $\overline{R}(\widehat {W},X)Z$ for $X,Z\in\Gamma(T\mathcal{F})$. Equivalently, for any $Y\in\Gamma(T\mathcal{F})$ then $$\begin{aligned}
\label{P22}
\overline{g}(\mathrm{div}^{\nabla'}(T_{r}),Y)=\sum_{j=1}^{r}\sum_{i=1}^{n} \epsilon_{i}\overline{g}(\overline{R}(T_{r-1}Z_{i},\widehat{W})(-\mathcal{A}_{\widehat{W}})^{j-1}Y,Z_{i}).\end{aligned}$$
The first equation of the lemma is obvious since $T_{0}=\mathbb{I}$. We turn to the second relation. By direct calculations using the recurrence relation (\[N28\]) we derive $$\begin{aligned}
\label{P23}
\mathrm{div}^{\nabla'}(T_{r})&=(-1)^{r}\mathrm{div}^{\nabla'}(S_{r}\mathbb{I})+\mathrm{div}^{\nabla'}(\mathcal{A}_{\widehat{W}}\circ T_{r-1})\nonumber\\
&=(-1)^{r}\nabla'S_{r}+\mathcal{A}_{\widehat{W}}\mathrm{div}^{\nabla'}(T_{r-1})+\sum_{i=1}^{n}\epsilon_{i}(\nabla'_{Z_{i}}\mathcal{A}_{\widehat{W}})T_{r-1}Z_{i}. \end{aligned}$$ Using Codazzi equation $$\overline{g}(\overline{R}(X,Y)Z,\widehat{W})=\overline{g}((\nabla'_{Y}\mathcal{A}_{\widehat{W}})X,Z)-\overline{g}((\nabla'_{X}\mathcal{A}_{\widehat{W}})Y,Z),$$ for any $X,Y,Z\in\Gamma(T\mathcal{F})$ and Lemma \[lemma1\], we have $$\begin{aligned}
\label{P24}
\overline{g}((\nabla'_{Z_{i}}\mathcal{A}_{\widehat{W}})Y,&T_{r-1}Z_{i})= \overline{g}((\nabla'_{Y}\mathcal{A}_{\widehat{W}})Z_{i},T_{r-1}Z_{i})+\overline{g}(\overline{R}(Y,Z_{i})T_{r-1}Z_{i},\widehat{W})\nonumber\\
&=\overline{g}(T_{r-1}(\nabla'_{Y}\mathcal{A}_{\widehat{W}})Z_{i},Z_{i})+\overline{g}(\overline{R}(\widehat{W},T_{r-1}Z_{i})Z_{i},Y),\end{aligned}$$ for any $Y\in\Gamma(T\mathcal{F})$. Then applying (\[P23\]) and (\[P24\]) we get $$\begin{aligned}
\label{P25}
\overline{g}(\mathrm{div}^{\nabla'}(T_{r}), Y)&=(-1)^{r}\overline{g}(\nabla'S_{r},Y)+\mathrm{tr}(T_{r-1}(\nabla'_{Y}\mathcal{A}_{\widehat{W}}))\nonumber\\
&+\overline{g}(\mathrm{div}^{\nabla'}(T_{r-1}),Y) + \overline{g}(Y, \sum_{i=1}^{n}\epsilon_{i}\overline{R}(\widehat{W},T_{r-1}Z_{i})Z_{i}).\end{aligned}$$ Then, applying (\[P25\]) and (\[N34\]) we get the second equation of the lemma. Finally, (\[P22\]) follows immediately by an induction argument.
Notice that when the ambient manifold is a space form of constant sectional curvature, then $(\overline{R}(\widehat{W}, X )Y)'= 0$ for each $X,Y\in\Gamma(T\mathcal{F})$. Hence, from Lemma (\[lemma2\]) we have $\mathrm{div}^{\nabla'}(T_{r})=0$.
\[lemma3\] Let $M$ be a screen integrable half-lightlike submanifold of $\overline{M}$ and let $\mathcal{F}$ be a co-dimension three foliation of $S(TM)$. Let $\mathcal{A}_{\widehat{W}}$ be its shape operator, where $\widehat{W}$ is a unit normal vector to $\mathcal{F}$. Let $\{Z_{i}\}$ be a local field such $(\nabla'_{X}Z_{i})p=0$, for $i=1,\cdots,n$ and any vector field $X\in\Gamma(T\overline{M})$. Then at $p\in \mathcal{F}$ we have $$\begin{aligned}
g(\nabla'_{Z_{i}}\overline{\nabla}_{\widehat{W}}\widehat{W},Z_{j})&=g(\mathcal{A}_{\widehat{W}}^{2}Z_{i},Z_{j})-\overline{g}(\overline{R}(Z_{i},\widehat{W})Z_{j}, \widehat{W})\\
&-\overline{g}((\nabla'_{\widehat{W}}\mathcal{A}_{\widehat{W}})Z_{i},Z_{j})+g(\overline{\nabla}_{\widehat{W}}\widehat{W},Z_{i})g(Z_{j},\overline{\nabla}_{\widehat{W}}\widehat{W}).
\end{aligned}$$
Applying $\overline{\nabla}_{Z_{i}}$ to $g(\overline{\nabla}_{\widehat{W}}\widehat{W},Z_{j})$ and $\overline{g}(\widehat{W},\overline{\nabla}_{\widehat{W}}Z_{j})$ in turn and then using the two resulting equations, we have $$\begin{aligned}
\label{P26}
-\overline{g}(\overline{\nabla}_{\widehat{W}}\widehat{W},\overline{\nabla}_{Z_{i}}Z_{j})&=g(\overline{\nabla}_{Z_{i}}\overline{\nabla}_{\widehat{W}}\widehat{W},Z_{j})+ \overline{g}(\overline{\nabla}_{Z_{i}}\widehat{W},\overline{\nabla}_{\widehat{W}}Z_{j})\nonumber\\
&+\overline{g}(\widehat{W},\overline{\nabla}_{Z_{i}}\overline{\nabla}_{\widehat{W}}Z_{j}).
\end{aligned}$$ Furthermore, by direct calculations using $(\nabla'_{X}Z_{i})p=0$ we have $$\begin{aligned}
\overline{g}((\nabla'_{\widehat{W}}\mathcal{A}_{\widehat{W}})Z_{i},Z_{j})= \overline{g}(\overline{\nabla}_{\widehat{W}}\widehat{W},\overline{Z_{i}}Z_{j})+\overline{g}(\widehat{W},\overline{\nabla}_{\widehat{W}}\overline{Z_{i}}Z_{j}),\end{aligned}$$ and hence $$\begin{aligned}
\label{P27}
g(\mathcal{A}_{\widehat{W}}^{2}Z_{i},Z_{j})-\overline{g}(\overline{R}&(Z_{i},\widehat{W})Z_{j}, \widehat{W})-\overline{g}((\nabla'_{\widehat{W}}\mathcal{A}_{\widehat{W}})Z_{i},Z_{j})\nonumber\\
&= g(\mathcal{A}_{\widehat{W}}^{2}Z_{i},Z_{j})-\overline{g}(\overline{R}(Z_{i},\widehat{W})Z_{j}, \widehat{W})\nonumber\\
&-\overline{g}(\overline{\nabla}_{\widehat{W}}\widehat{W},\overline{Z_{i}}Z_{j})-\overline{g}(\widehat{W},\overline{\nabla}_{\widehat{W}}\overline{Z_{i}}Z_{j})\nonumber\\
&=g(\mathcal{A}_{\widehat{W}}^{2}Z_{i},Z_{j})-\overline{g}(\overline{\nabla}_{Z_{i}}Z_{j},\overline{\nabla}_{\widehat{W}}\widehat{W})\nonumber\\
&-\overline{g}(\overline{\nabla}_{Z_{i}}\overline{\nabla}_{\widehat{W}}Z_{j},\widehat{W})+\overline{g}(\overline{\nabla}_{[Z_{i},\widehat{W}]}Z_{j},\widehat{W}).\end{aligned}$$ Now, applying (\[P26\]), the condition at $p$ and the following relations $$\begin{aligned}
&\overline{\nabla}_{Z_{i}}\widehat{W}=\sum_{k=1}^{n}\epsilon_{k}\overline{g}( \overline{\nabla}_{Z_{i}}\widehat{W},Z_{k})Z_{k},\;\;\;\overline{\nabla}_{\widehat{W}}Z_{j}=\overline{g}(\overline{\nabla}_{\widehat{W}}Z_{j}, \widehat{W})\widehat{W},\end{aligned}$$ and $
g(\mathcal{A}_{\widehat{W}}^{2}Z_{i},Z_{j})=-\sum_{k=1}^{n}\epsilon_{k}\overline{g}( \overline{\nabla}_{Z_{i}}\widehat{W},Z_{k})\overline{g}(\overline{\nabla}_{Z_{k}}Z_{j},\widehat{W})
$ to the last line of (\[P27\]) and the fact that $S(TM)$ is integrable we get $$\begin{aligned}
&g(\mathcal{A}_{\widehat{W}}^{2}Z_{i},Z_{j})-\overline{g}(\overline{R}(Z_{i},\widehat{W})Z_{j}, \widehat{W})-\overline{g}((\nabla'_{\widehat{W}}\mathcal{A}_{\widehat{W}})Z_{i},Z_{j})\\
&=g(\nabla'_{Z_{i}}\overline{\nabla}_{\widehat{W}}\widehat{W},Z_{j})-g(\overline{\nabla}_{\widehat{W}}\widehat{W},Z_{i})g(Z_{j},\overline{\nabla}_{\widehat{W}}\widehat{W}),\end{aligned}$$ from which the lemma follows by rearrangement.
Notice that, using parallel transport, we can always construct a frame field from the above lemma.
\[proposition1\] Let $M$ be a screen integrable half-lightlike submanifold of an indefinite almost contact manifold $\overline{M}$ and let $\mathcal{F}$ be a foliation of $S(TM)$. Then $$\begin{aligned}
&\mathrm{div}^{\nabla'}(T_{r}\overline{\nabla}_{\widehat{W}}\widehat{W})=\overline{g}( \mathrm{div}^{\nabla'}(T_{r}),\overline{\nabla}_{\widehat{W}}\widehat{W})+(-1)^{r+1}\widehat{W}(S_{r+1})\\
&+(-1)^{r+1}(-S_{1}S_{r+1}+(r+2)S_{r+2})-\sum_{i=1}^{n}\epsilon_{i}\overline{g}(\overline{R}(Z_{i},\widehat{W})T_{r}Z_{i},\widehat{W})\\
&+\overline{g}(\overline{\nabla}_{\widehat{W}}\widehat{W},T_{r}\overline{\nabla}_{\widehat{W}}\widehat{W}),
\end{aligned}$$ where $\{ Z_{i}\}$ is a field of frame tangent to the leaves of $\mathcal{F}$.
From (\[N35\]),we deduce that $$\begin{aligned}
\label{P30}
\mathrm{div}^{\nabla'}(T_{r}Z)=\overline{g}( \mathrm{div}^{\nabla'}(T_{r}),Z)+\sum_{i=1}^{n}\epsilon_{i}\overline{g}(\nabla'_{Z_{i}}Z,T_{r}Z_{i} ),
\end{aligned}$$ for all $Z\in\Gamma(T\mathcal{F})$. Then using (\[P30\]), Lemmas \[lemma2\] and \[lemma3\], we obtain the desired result. Hence the proof.
From Proposition \[proposition1\] we have
\[theorem1\] Let $M$ be a screen integrable half-lightlike submanifold of an indefinite almost contact manifold $\overline{M}$ and let $\mathcal{F}$ be a co-dimension three foliation of $S(TM)$. Then $$\begin{aligned}
&\mathrm{div}^{\overline{\nabla}}(T_{r}\overline{\nabla}_{\widehat{W}}\widehat{W})=\overline{g}( \mathrm{div}^{\nabla'}(T_{r}),\overline{\nabla}_{\widehat{W}}\widehat{W})+(-1)^{r+1}\widehat{W}(S_{r+1})\\
&+(-1)^{r+1}(-S_{1}S_{r+1}+(r+2)S_{r+2})-\sum_{i=1}^{n}\epsilon_{i}\overline{g}(\overline{R}(Z_{i},\widehat{W})T_{r}Z_{i},\widehat{W}).
\end{aligned}$$
A proof follows easily form Proposition \[proposition1\] by recognizing the fact that $$\begin{aligned}
\mathrm{div}^{\overline{\nabla}}(T_{r}\overline{\nabla}_{\widehat{W}}\widehat{W})=\mathrm{div}^{\nabla'}(T_{r}\overline{\nabla}_{\widehat{W}}\widehat{W})-\overline{g}(\overline{\nabla}_{\widehat{W}}\widehat{W},T_{r}\overline{\nabla}_{\widehat{W}}\widehat{W}),
\end{aligned}$$ which completes the proof.
\[theorem2\] Let $M$ be a screen integrable half-lightlike submanifold of $\overline{M}$ and let $\mathcal{F}$ be a co-dimension three foliation of $S(TM)$. Then, $$\begin{aligned}
&\mathrm{div}^{\overline{\nabla}}(T_{r}\overline{\nabla}_{\widehat{W}}\widehat{W}+(-1)^{r}S_{r+1}\widehat{W})=\overline{g}( \mathrm{div}^{\nabla'}(T_{r}),\overline{\nabla}_{\widehat{W}}\widehat{W})\\
&+(-1)^{r+1}(r+2)S_{r+2}-\sum_{i=1}^{n}\epsilon_{i}\overline{g}(\overline{R}(Z_{i},\widehat{W})T_{r}Z_{i},\widehat{W}).
\end{aligned}$$
By straightforward calculations we have $$\begin{aligned}
S_{1}=\mathrm{tr}(\mathcal{A}_{\widehat{W}})=-\sum_{i=1}^{n}\epsilon_{i}\overline{g}(\overline{\nabla}_{Z_{i}}\widehat{W},Z_{i})=-\sum_{i=1}^{n+1}\epsilon_{i}\overline{g}(\overline{\nabla}_{Z_{i}}\widehat{W},Z_{i})=-\mathrm{div}^{\overline{\nabla}}(\widehat{W}),
\end{aligned}$$ where $Z_{n+1}=\widehat{W}$. From this equation we deduce $$\begin{aligned}
\label{P31}
\mathrm{div}^{\overline{\nabla}}(S_{r+1}\widehat{W})=-S_{1}S_{r+1}+\widehat{W}(S_{r+1}).\end{aligned}$$ Then from (\[P31\]) and Theorem \[theorem1\] we get our assertion, hence the proof.
Next, we let $dV$ denote the volume form $\overline{M}$. Then from Theorem \[theorem2\] we
\[corollary1\] Let $M$ be a screen integrable half-lightlike submanifold of a compact semi-Riemannian manifold $\overline{M}$ and let $\mathcal{F}$ be a co-dimension three foliation of $S(TM)$. Then $$\begin{aligned}
&\int_{\overline{M}}\overline{g}( \mathrm{div}^{\nabla'}(T_{r}),\overline{\nabla}_{\widehat{W}}\widehat{W})dV\\
&=\int_{\overline{M}}((-1)^{r}(r+2)S_{r+2}+\sum_{i=1}^{n}\epsilon_{i}\overline{g}(\overline{R}(Z_{i},\widehat{W})T_{r}Z_{i},\widehat{W})dV
\end{aligned}$$
Setting $r=0$ in the above corollary we get
\[corollary3\] Let $M$ be a screen integrable half-lightlike submanifold of a compact semi-Riemannian manifold $\overline{M}$ and let $\mathcal{F}$ be a co-dimension three foliation of $S(TM)$ with mean curvatures $S_{r}$. Then for $r=0$ we have $$\begin{aligned}
\int_{\overline{M}}2S_{2}dV=\int_{\overline{M}}\overline{Ric}(\widehat{W},\widehat{W})dV,\end{aligned}$$ where $\displaystyle\overline{Ric}(\widehat{W},\widehat{W})=\sum_{i=1}^{n}\epsilon_{i}\overline{g}(\overline{R}(Z_{i},\widehat{W})\widehat{W},Z_{i})$.
Notice that the equation in Corollary \[corollary3\] is the lightlike analogue of (3.5) in [@krz] for co-dimension one foliations on Riemannian manifolds.
Next, we will discuss some consequences of the integral formulas developed so far.
A semi-Riemannian manifold $\overline{M}$ of constant sectional curvature $c$ is called a *semi-Riemannian space form* [@db; @ds2] and is denoted by $\overline{M}(c)$. Then, the curvature tensor $\overline{R}$ of $\overline{M}(c)$ is given by $$\label{N24}
\overline{R}(\overline{X},\overline{Y})\overline{Z}=c\{\overline{g}(\overline{Y},\overline{Z})\overline{X}-\overline{g}(\overline{X},\overline{Z})\overline{Y}\},\quad\forall\, \overline{X},\overline{Y},\overline{Z}\in\Gamma(T\overline{M}).$$
\[theorem3\] Let $M$ be a screen integrable half-lightlike submanifold of a compact semi-Riemannian space form $\overline{M}(c)$ of constant sectional curvature $c$. Let $\mathcal{F}$ be a co-dimension three foliation of its screen distribution $S(TM)$. If $V$ is the total volume of $\overline{M}$, then $$\int_{\overline{M}}S_{r}dV=\begin{dcases}
0, & r=2k+1, \\
c^{\frac{r}{2}}\dbinom{\frac{n}{2}}{\frac{r}{2}}V, & r=2k,
\end{dcases}$$ for positive integers $k$.
By setting $\overline{X}=Z_{i}$, $\overline{Y}=\widehat{W}$ and $Z=T_{r}Z_{i}$ in (\[N24\]) we deduce that $ \overline{R}(Z_{i},\widehat{W})T_{r}Z_{i}=-cg(Z_{i},T_{r}Z_{i})\widehat{W}$. Then substituting this equation in Corollary \[corollary1\] we obtain $$\begin{aligned}
\int_{\overline{M}}\overline{g}( \mathrm{div}^{\nabla'}(T_{r}),\overline{\nabla}_{\widehat{W}}\widehat{W})dV=\int_{\overline{M}}((-1)^{r}(r+2)S_{r+2}-c\mathrm{tr}(T_{r}))dV.
\end{aligned}$$ Since $\overline{M}$ is of constant sectional curvature $c$, then Lemma \[lemma2\] implies that $T_{r}=0$ for any $r$ and hence the above equation simplifies to $$\begin{aligned}
\label{P35}
(r+2)\int_{\overline{M}}S_{r+2}dV=c(n-r)\int_{\overline{M}}S_{r}dV.\end{aligned}$$ Since $S_{1}=-\mathrm{div}^{\overline{\nabla}}(\widehat{W})$ and that $\overline{M}$ is compact, then $\int_{\overline{M}}S_{1}dV=0$. Using this fact together with (\[P35\]), mathematical induction gives $\int_{\overline{M}}S_{r}dV=0$ for all $r=2k+1$ (i.e., $r$ odd). For $r$ even we will consider $r=2m$ and $n=2l$ (i.e., both $M$ and $\overline{M}$ are odd dimensional). With these conditions, (\[P35\]) reduces to $$\begin{aligned}
\label{P36}
\int_{\overline{M}}S_{2m+2}dV=c\frac{l-m}{1+m}\int_{\overline{M}}S_{2m}dV.\end{aligned}$$ Now setting $m=0,1,\cdots$ and $S_{0}=1$ in (\[P36\]) we obtain $$\begin{aligned}
\int_{\overline{M}}S_{2}dV=cl V, \;\;\;\; \int_{\overline{M}}S_{4}dV=c^{2}\frac{(l-1)l}{2}V, \end{aligned}$$ and more generally $$\begin{aligned}
\label{P37}
\int_{\overline{M}}S_{2k}dV=c^{k}\frac{(l-k+1)(l-k+2)(l-k+3)\cdots l}{k!}V.\end{aligned}$$ Hence, our assertion follows from \[P37\], which completes the proof.
Next, when $\overline{M}$ is Einstein i.e., $\overline{Ric}=\mu \overline{g}$ we have the following.
\[theorem5\] Let $M$ be a screen integrable half-lightlike submanifold of an Einstein compact semi-Riemannian manifold $\overline{M}$. Let $\mathcal{F}$ be a co-dimension three foliation of its screen distribution $S(TM)$ with totally umbilical leaves. If $V$ is the total volume of $\overline{M}$, then $$\int_{\overline{M}}S_{r}dV=\begin{dcases}
0, & r=2k+1, \\
\left(\frac{\mu}{n}\right)^{\frac{n}{2}}\dbinom{\frac{n}{2}}{\frac{r}{2}}V, & r=2k,
\end{dcases}$$ for positive integers $k$.
Suppose that $\mathcal{A}_{\widehat{W}}=\frac{1}{n}S_{r}\mathbb{I}$. Then by direct calculations using the formula for $T_{r}$ we derive $T_{r}=(-1)^{r+1}\frac{(n-r)}{n}S_{r}\mathbb{I}$. Then, under the assumptions of the theorem we obtain $\overline{Ric}(\widehat {W}, \overline{\nabla}_{\widehat{W}}\widehat{W})=0$ and $\overline{Ric}(\widehat {W},\widehat{W})=\mu$ and hence, Corollary \[corollary1\] reduces to $$\begin{aligned}
\label{P50}
n(r+2)\int_{\overline{M}}S_{r+2}dV=\lambda(n-r)\int_{\overline{M}}S_{r}dV.
\end{aligned}$$ Notice that (\[P50\]) is similar to (\[P35\]) and hence following similar steps as in the previous theorem we get $\int_{\overline{M}}S_{r}dV=0$ for $r$ odd and for $r$ even we get $$\begin{aligned}
\int_{\overline{M}}S_{2k}dV=\left(\frac{\mu}{n}\right)^{k}\frac{(l-k+1)(l-k+2)(l-k+3)\cdots l}{k!}V,\end{aligned}$$ which complete the proof.
Screen umbilical half-lightlike submanifolds {#screen}
============================================
In this section we consider totally umbilical half-lightlike submanifolds of semi-Riemannian manifold, with a totally umbilical screen distribution and thus, give a generalized version of Theorem 4.3.7 of [@ds2] and its Corollaries, via Newton transformations of the operator $A_{N}$.
A screen distribution $S(TM)$ of a half-lightlike submanifold $M$ of a semi-Riemannian manifold $\overline{M}$ is said to be totally umbilical [@ds2] if on any coordinate neighborhood $\mathcal{U}$ there exist a function $K$ such that $$\begin{aligned}
\label{P64}
C(X,PY)=Kg(X,PY),\;\;\;\forall\, X,Y\in\Gamma(TM).\end{aligned}$$ In case $K=0$, we say that $S(TM)$ is totally geodesic. Furthermore, if $S(TM)$ is totally umbilical then by straightforward calculations we have $$\begin{aligned}
\label{P65}
A_{N}X=PX,\;\;\; C(E,PX)=0,\;\;\; \forall\, X\in\Gamma(TM).\end{aligned}$$ Let $\{E,Z_{i}\}$, for $i=1,\cdots,n$, be a quasi-orthonormal frame field of $TM$ which diagonalizes $A_{N}$. Let $l_{0},l_{1},\cdots,l_{n}$ be the respective eigenvalues (or principal curvatures). Then as before, the $r$-th mean curvature $S_{r}$ is given by $$\begin{aligned}
S_{r}=\sigma_{r}(l_{0},\cdots,l_{n}) \;\;\mbox{and}\;\;S_{0}=1.\end{aligned}$$ The characteristic polynomial of $A_{N}$ is given by $$\det(A_{N}-t\mathbb{I})=\sum_{\alpha=0}^{n}(-1)^{\alpha}S_{r}t^{n-\alpha},$$ where $\mathbb{I}$ is the identity in $\Gamma(TM)$. The normalized $r$-th mean curvature $H_{r}$ of $M$ is defined by $\dbinom{n}{r}H_{r}=S_{r}\;\;\;\mbox{and}\;\;\;H_{0}=1$. The Newton transformations $T_{r}:\Gamma(TM)\rightarrow \Gamma(TM)$ of $A_{N}$ are given by the inductive formula $$\label{K28}
T_{0}=\mathbb{I},\quad T_{r}=(-1)^{r}S_{r}\mathbb{I}+A_{N}\circ T_{r-1}, \;\;\; 1\leq r\leq n.$$ By Cayley-Hamiliton theorem, we have $T_{n+1}=0$. Also, $T_{r}$ satisfies the following properties. $$\begin{aligned}
\mathrm{tr}(T_{r}) & =(-1)^{r}(n+1-r) S_{r},\label{K31}\\
\mathrm{tr}(A_{N}\circ T_{r}) & =(-1)^{r}(r+1)S_{r+1},\label{K32}\\
\mathrm{tr}(A_{N}^{2}\circ T_{r}) & =(-1)^{r+1}(-S_{1} S_{r+1} + (r + 2) S_{r+2}),\label{K33}\\
\mathrm{tr}(T_{r}\circ\nabla_{X} A_{N})& = (-1)^{r}X(S_{r+1}).\label{K34}\end{aligned}$$ for all $X\in\Gamma(TM)$.
\[pp1\] Let $(M,g)$ be a totally umbilical half-lightlike submanifold of a semi-Riemannian manifold $\overline{M}$ of constant sectional curvature $c$. Then $$\begin{aligned}
& g(\mathrm{div}^{\nabla}(T_{r}),X) =(-1)^{r-1}\lambda(X)E(S_{r})-\tau(X)\mathrm{tr}(A_{N}\circ T_{r-1})\\
& -c\lambda(X)\mathrm{tr}(T_{r-1}) +g(\mathrm{div}^{\nabla}(T_{r-1}),A_{N}X)+g((\nabla_{E}A_{N})T_{r-1}E,X)\\
&+\sum_{i=1}^{n}\epsilon_{i}\{-\lambda(X)B(Z_{i},A_{N}(T_{r-1}Z_{i}))\\
&+\varepsilon \tau(Z_{i})C(X,T_{r-1}Z_{i})\{\rho(X)D(Z_{i},T_{r-1}Z_{i})-\rho(Z_{i})D(X,T_{r-1}Z_{i})\}\},
\end{aligned}$$ for any $X\in\Gamma(TM)$.
From the recurrence relation (\[K28\]), we derive $$\begin{aligned}
\label{P66}
&g(\mathrm{div}^{\nabla}(T_{r}),X)=(-1)^{r}PX(S_{r})+g((\nabla_{E}A_{N})T_{r-1}E,X)\nonumber\\
&+g(\mathrm{div}^{\nabla}(T_{r-1}),A_{N}X)+\sum_{i=1}^{n}\epsilon_{i}g((\nabla_{Z_{i}}A_{N})T_{r-1}Z_{i},X).\end{aligned}$$ for any $X\in\Gamma(TM)$. But $$\begin{aligned}
\label{P67}
g((\nabla_{Z_{i}}A_{N})T_{r-1}Z_{i},X) &= g(T_{r-1}Z_{i},(\nabla_{Z_{i}}A_{N})X) + g(\nabla_{Z_{i}}A_{N}(T_{r-1}Z_{i}),X)\nonumber\\
& -g(\nabla_{Z_{i}}(A_{N}X),T_{r-1}Z_{i}) + g(A_{N}(\nabla_{Z_{i}}X),T_{r-1}Z_{i})\nonumber\\
&-g(A_{N}(\nabla_{Z_{i}}T_{r-1}Z_{i}),X),
\end{aligned}$$ for all $X\in\Gamma(TM)$.
Then applying (\[P6\]) to (\[P67\]) while considering the fact that $A_{N}$ is screen-valued, we get $$\begin{aligned}
\label{P68}
g((\nabla_{Z_{i}}A_{N})T_{r-1}Z_{i},X)= g(T_{r-1}Z_{i},(\nabla_{Z_{i}}A_{N})X)-\lambda(X)B(Z_{i},A_{N}(T_{r-1}Z_{i}))\end{aligned}$$ Furthermore, using (\[P60\]) and (\[N24\]), the first term on the right hand side of (\[P68\]) reduces to $$\begin{aligned}
\label{P69}
g(T_{r-1}Z_{i},&(\nabla_{Z_{i}}A_{N})X)=-c\lambda(X)g(Z_{i},T_{r-1}Z_{i})+g((\nabla_{X}A_{N})Z_{i},T_{r-1}Z_{i})\nonumber\\
&-\tau(X)C(Z_{i},T_{r-1}Z_{i})+\varepsilon\tau(Z_{i})C(X,T_{r-1}Z_{i})\{\rho(X)D(Z_{i},T_{r-1}Z_{i})\nonumber\\
&-\rho(X)D(X,T_{r-1}Z_{i})\},\end{aligned}$$ for any $X\in\Gamma(TM)$. Finally, replacing (\[P69\]) in (\[P68\]) and then put the resulting equation in (\[P66\]) we get the desired result.
Next, from Proposition \[pp1\] we have the following.
\[thmy\] Let $(M, g)$ be a half-lightlike submanifold of a semi-Riemannian manifold $\overline{M}(c)$ of constant curvature $c$, with a proper totally umbilical screen distribution $S(TM)$. If $M$ is also totally umbilical, then the $r$-th mean curvature $S_{r}$, for $r=0,1,\cdots,n$, with respect to $A_{N}$ are solution of the following differential equation $$\begin{aligned}
E(S_{r+1})-\tau(E)(r+1)S_{r+1}-c(-1)^{r}(n+1-r)S_{r}=\mathcal{H}_{1}(r+1)S_{r+1}.
\end{aligned}$$
Replacing $X$ with $E$ in the Proposition \[pp1\] and then using (\[P61\]) and (\[P65\]) we obtain, for all $r=0,1,\cdots,n$, $$\begin{aligned}
E(S_{r+1})-(-1)^{r}\tau(E)\mathrm{tr}(A_{N}\circ T_{r})-c(-1)^{r}\mathrm{tr}(T_{r})=(-1)^{r}\mathcal{H}_{1}\mathrm{tr}(A_{N}\circ T_{r}),
\end{aligned}$$ from which the result follows by applying (\[K31\]) and (\[K32\]).
\[P71\] Under the hypothesis of Theorem \[thmy\], the induced connection $\nabla$ on $M$ is a metric connection, if and only if, the $r$-th mean curvature $S_{r}$ with respect to $A_{N}$ are solution of the following equation $$\begin{aligned}
E(S_{r+1})-\tau(E)(r+1)S_{r+1}-c(-1)^{r}(n+1-r)S_{r}=0.
\end{aligned}$$
Also the following holds.
\[P72\] Under the hypothesis of Theorem \[thmy\], $\overline{M}(c)$ is a semi-Euclidean space, if and only if, the $r$-th mean curvature $S_{r}$ with respect to $A_{N}$ are solution of the following equation $$\begin{aligned}
E(S_{r+1})-\tau(E)(r+1)S_{r+1}=\mathcal{H}_{1}(r+1)S_{r+1}.
\end{aligned}$$
Notice that Theorem \[thmy\] and Corollary \[P71\] are generalizations of Theorem 4.3.7 and Corollary 4.3.8, respectively, given in [@ds2].
[xxx]{} K. Andrzejewski and Paweł G. Walczak, The Newton transformation and new integral formulae for foliated manifolds, Ann. Glob. Anal. Geom. 37 (2010), no. 2, 103-111. C. Calin, Contributions to geometry of CR-submanifold, Ph.D. thesis, University of Iasi (Romania), 1998. K. L. Duggal and A. Bejancu, Lightlike submanifolds of semi-Riemannian manifolds and applications, Mathematics and Its Applications, Kluwer Academic Publishers, 1996. K. L. Duggal and B. Sahin, Differential geometry of lightlike submanifolds. Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2010. K. L. Duggal and B. Sahin, Screen conformal half-lightlike submanifolds, Int. J. Math. Math. Sci. 2004 (2004), no. 68, 3737-3753. R. S. Gupta and A. Sharfuddin, Generalised Cauchy-Riemann Lightlike Submanifolds of Indefinite Kenmotsu Manifolds, Note Mat. 30 (2010) no. 2, 49-59. D. H. Jin, Non-Existance of lightlike submanifolds of indefinite trans-Sasakian manifolds with non-metric $\theta$-connections, Commun. Korean Math. Soc. 30 (2015), no. 1, 35-43. F. Massamba, Totally contact umbilical lightlike hypersurfaces of indefinite Sasakian manifolds, Kodai Math. J., 31 (2008), 338-358. F. Massamba, On semi-parallel lightlike hypersurfaces of indefinite Kenmotsu manifolds, J. Geom., 95 (2009), 73-89. F. Massamba and S. Ssekajja, Some remarks on quasi generalized CR-null geometry in indefinite nearly cosymplectic manifolds, arXiv: 1604.5436v3. E. Yasar, A. C. Coken, and A. Yucesan, Lightlike hypersurfaces in semi-Riemannian manifold with semi-symmetric non-metric connection, Math. Scand. 102 (2008), no. 2, 253-264. K. Andrzejewski, W. Kozlowski and K. Niedzialomski, Generalized Newton transformation and its applications to extrinsic geometry, Asian J. Math. 20 (2016), no. 2, 293-322. D.N. Kupeli, Singuler semi-Riemannian geometry, Mathematics and Its Applications, Vol. 366, Kluwer Academic Publishers, 1996.
|
---
author:
- 'J.E. Baldwin'
- 'R.N. Tubbs'
- 'G.C. Cox'
- 'C.D. Mackay'
- 'R.W. Wilson'
- 'M.I. Andersen'
bibliography:
- 'bibtex/je\_baldwin.bib'
date: 'Received December 7, 2000 / Accepted January 23, 2000'
title: |
Diffraction-limited 800nm imaging with the\
2.56m Nordic Optical Telescope
---
Introduction
============
The selection of the few sharpest images from a large dataset of short exposures provides one method of obtaining high resolution images through atmospheric seeing at ground-based telescopes. The selected exposures can then be processed using one of the conventional speckle techniques such as shift-and-add. Exposure selection has been applied in many studies of the solar surface and for planetary imaging but has not been extensively tested as a general tool for astronomical imaging at optical wavelengths. Trials by @dantowitz98 and @dantowitz00 using a video camera on the 60-inch Mt.Wilson telescope have shown its promise for reaching close to the diffraction limit. In view of the technical difficulties of adaptive optics at wavelengths shorter than 1 $\mu$m, it seems important to assess this alternative technique quantitatively. In this paper we present such observations, showing the power of the method and argue that new developments in low-noise, fast-readout CCD’s make it attractive for achieving diffraction-limited imaging in the visible waveband for faint astronomical targets using apertures of about 2.5m.
The basis for the method is that the atmosphere behaves as a time-varying random phase screen, with a power spectrum of irregularities characterised by spatial and temporal scales $r_{0}$ and $t_{0}$, whose rms variations are larger than those of a well-adjusted and figured primary mirror. Occasionally the combined phase variations across the telescope aperture, $\delta$, due to the atmosphere and mirror, will be small ($<\sim1$radian). The corresponding image of a star will have a core which is diffraction-limited with a Strehl ratio of $\exp{(-\delta^{2})}$ and angular resolution $1.22\lambda/D$ determined by the aperture diameter $D$. @fried78 calculated the probability, $P$, of “lucky exposures” having phase variations less than 1 radian across an aperture diameter $D$ for seeing defined by $r_{0}$: $$P\simeq5.6\exp{\left(-0.1557\left(D/r_{0}\right)^{2}\right)}$$ This implies, for instance, that for an aperture $D=7r_{0}$, one exposure in 350 would have a Strehl ratio greater than 0.37. For $D=10r_{0}$, the frequency of such good exposures falls to only one in $10^{6}$. This suggests that values of $D$ chosen as $7r_{0}$ may offer the best compromise between high angular resolution and frequency of occurrence.
The technique evidently requires a site with good seeing and a telescope for which it is known that the errors in the mirror figure are small compared with the atmospheric fluctuations on all relevant scales. The primary mirror should ideally match the maximum useful aperture for this technique. If very good seeing is taken to be 0.5 arcsec, and a useful aperture as $7r_{0}$, then D should be 1.4m at 500nm and 2.5m at 800nm. The Nordic Optical Telescope (NOT) matches these criteria very closely.
Observations and Data Reduction
===============================
Observations were made at the Cassegrain focus of the NOT on the nights of 2000 May 12 and 13. The camera was one developed for the JOSE programme of seeing evaluation at the William Herschel Telescope [@stjacques97]. It comprised a 512x512 front illuminated frame-transfer CCD with 15 $\mu$m pixels run by an AstroCam 4100 controller. The controller allows windowing of the area of readout and variable pixel readout rates up to 5.5MHz. The f/11 beam at the focus was converted to f/30 using a single achromat to give an image scale of 41 milliarcsec/pixel (25 pixels/ arcsec). This gives a good match to the full width to half maximum (FWHM) of the diffraction-limited image of 66 milliarcsec for a 2.56m telescope with 0.50m secondary obstruction at 800nm; good signal-to-noise sensitivity per pixel is retained whilst the resolution is degraded only slightly to 77 milliarcsec FWHM by the finite pixel size.
A filter centred at 810nm with a bandwidth of 120nm was used to define the band. All of the exposures of stars were taken at frame rates higher than 150Hz and without autoguiding to ensure that the temporal behaviour of the periods of good seeing was adequately characterised.
Observations of the stars listed in Table \[target\_list\] were made with a variety of frame formats and frame rates. Each run typically comprised between 5000 and 24000 frames over a period of 30-160s. Target stars, both single stars and binaries, were chosen principally lying in the declination range $10\degr$-$20\degr$ and close to the meridian, so that most of the data was taken at zenith angles $<20\degr$. Zenith angles up to $50\degr$ were explored in later observations. The effects of atmospheric dispersion became significant, since no corrective optics were employed.
$$\begin{array}{p{0.25\linewidth}p{0.3\linewidth}l}
\hline
\noalign{\smallskip}
Target & Frame rate / Hz & FWHM / arcseconds \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
v656 Herculis & 185 & 0.49 \\
$\epsilon$ Aquilae & 185 & 0.38 \\
$\gamma$ Aquilae & 185 & 0.46 \\
$\gamma$ Leonis & 159 & 0.57 \\
$\gamma$ Leonis & 182 & 0.46 \\
CN Bo\"otis & 152 & 0.62 \\
$\zeta$ Bo\"otis & 152 & 0.74^{\mathrm{a}} \\
$\zeta$ Bo\"otis & 152 & 0.75^{\mathrm{a}} \\
$\alpha$ Herculis & 191 & 0.38 \\
$\alpha$ Aquilae & 206 & 0.50 \\
$\beta$ Delphini & 373 & 0.42 \\
$\beta$ Delphini & 257 & 0.52 \\
$\beta$ Delphini & 190 & 0.64 \\
$\alpha$ Delphini & 180 & 0.41 \\
$\alpha$ Delphini & 180 & 0.49 \\
\noalign{\smallskip}
\hline
\end{array}$$
Includes some tracking error
The seeing was good, typically 0.5 arcsec (see Table \[target\_list\]), and the short exposure images seen in real time clearly showed a single bright speckle at some instants. The full aperture of the NOT was therefore used for all the observations on both nights.
The analysis of the data was carried out in stages:
1. Some runs in which the detector was saturated during the periods of best seeing and a small number of misrecorded frames in runs taken at the fastest frame rates were excluded from further analysis.
2. For each run, an averaged image was obtained by summing all the remaining frames. The mean sky level was measured from a suitable area of the image and subtracted before determining the FWHM of the seeing disk (Table \[target\_list\]) and the total stellar flux.
3. Each frame was interpolated to give a factor of 4 times as many pixels in each coordinate using sinc interpolation to overcome problems of sampling in the images.
4. The peak brightness and position of the brightest pixel in each frame was measured and stored.
5. The Strehl ratio for each frame was derived using the peak brightness above the sky level measured for that frame, scaled by the total flux from the star and a geometrical factor which relates the pixel scale to the theoretical diffraction response of the NOT.
6. A “good” image was then obtained by taking only those frames with a Strehl ratio greater than some chosen value, shifting the images so that the peak pixels were superposed and adding the frames.
Fig. \[strehl\_histogram\] shows a histogram of the Strehl ratios derived following 1-5 above for individual 5.4 ms exposures in a 32 second run on the star . The substantial improvement in the Strehl of the shift-and-add image and in limiting sensitivity provided by image selection is clear from the spread of the Strehl values. For comparison, a histogram is also plotted of the Strehl ratios obtained for images from simulations of 10,000 realisations of atmospheric irregularities with a Kolmogorov turbulent spectrum over a circular aperture $6.5r_{0}$ in diameter. The close agreement between theory and observation suggests that the conditions corresponded to $6.5r_{0}$ = 2.56m at 810nm for this run. The FWHM of the averaged image from this run was 0.4 arcsec, corresponding to the telescope aperture being only $5.8r_{0}$ for a Kolmogorov spectrum of turbulence. The discrepancy is partly explained by the actual image motion being only 60% of that for a Kolmogorov spectrum, implying an outer scale of turbulence of about 30m.
The timescale on which changes take place in the best images is an important question, setting a limit to the maximum frame exposure time which can be used in this technique, and hence to the faintest limiting magnitude achievable with a given camera. Histograms of the Strehl ratios obtained after averaging together groups of 5 successive frames without correction for image motion are included in Fig. \[strehl\_histogram\]. This shows that exposures as long as 30ms could be used with only a small reduction in the observed Strehl values. The Strehls are reduced substantially if 20 exposures are averaged together. The choice of best exposure time depends on the intended application.
Results
=======
Data from the run on analysed in several different ways illustrate the possible strategies in practice. Fig. \[eaql\_contour\]a shows a 108ms exposure, formed by the summation *without* shifting of 20 consecutive 5.4 ms exposures, during a single period of good atmospheric conditions. The FWHM of the central core in the image is 79$\times$95 milliarcseconds, with a Strehl ratio of 0.29. Fig. \[eaql\_contour\]b was constructed from 12 exposures of 27ms duration, each made by summing *without* shifting five 5.4ms exposures. These 12 exposures were shifted and added to give a final FWHM of 81$\times$96 milliarcseconds and a Strehl ratio of 0.26. Single 5.4ms exposures from 20 widely separated time periods were selected for Fig. \[eaql\_contour\]c. Each of the constituent exposures had a Strehl ratio greater than 0.27, resulting in an image core with 80$\times$93 milliarcseconds FWHM and a Strehl of 0.30 after shifting and adding. For Fig. \[eaql\_contour\]d the exposures with the highest 1% of Strehl ratios are selected, giving a final image with FWHM of 79$\times$94 milliarcseconds and Strehl ratio of 0.30. All of these procedures give images with high Strehl ratios. There is, however, a trade-off between the higher sensitivity for faint reference stars achieved by the long sequence in Fig. \[eaql\_contour\]a and the smoothness of the halo due to averaging many atmospheric configurations as in Fig. 2. b-d. For the remainder of this letter we have used the best 1% of exposures in any observation to produce the final image.
\[eaql\_contour\]
Fig. \[zboo\_bitmap\] shows an example image of generated by selecting exposures from a dataset of 23200 frames. The image shows a diffraction-limited central peak (Strehl = 0.26, FWHM = 83$\times$94 milliarcseconds) and first Airy ring superposed on a faint halo for each star. 300 milliarcseconds from the component stars the surface brightness of the halo reaches only 2% of that in the stellar disks. The 232 frames selected came from a range of different epochs, helping to reduce the level of fluctuation around the stars. The high dynamic range of the technique is evident from the contour plot of the same image (Fig. \[zboo\_contour\]). The fluctuations in the halo reach only 0.1% of the peak brightness 700 milliarcseconds from the stars. The magnitude difference between the two components in this image is 0.048 $\pm$ 0.005, in good agreement with @hipparcos97.
Fig. \[bdel\_bitmap\] shows the result of a similar selection of the 1% of images with the best Strehl ratios from a dataset of 7000 short-exposure CCD images of . In this case the zenith angle of the observation was 50 degrees and the images are blurred by 100milliarcsec due to atmospheric dispersion over the 120nm bandpass of the filter, reducing the Strehl ratio of the final image to 0.25. The magnitude difference between the components is $\Delta$$M=1.070\pm0.005$. This value is in good agreement with those of @barnaby00 of $1.071\pm0.004$ at 798nm and $1.052\pm0.010$ at 884nm made using a 1.5m telescope. The images shown of , and are representative of those for all the stars listed in Table \[target\_list\] with regards to Strehl ratio and core FWHM.
Conclusions and Future Prospects
================================
The observations described here show that selection of short exposure images can reliably provide diffraction-limited images with Strehl ratios of 0.25-0.30 at wavelengths as short as 0.8 microns with 2.5m telescopes. The images are similar in core angular resolution and Strehl ratio to the highest resolution images from adaptive optics at wavelengths shorter than 1 $\mu$m [@graves98]. The faintest stars used in these trial observations were of magnitude +6. The readout noise of the present camera ($\sim100e^{-}$) would have set a limiting magnitude of +11.5 if 30ms exposures had been used. Current development of CCD detectors with effectively zero readout noise and higher quantum efficiency [@mackay01] will provide important advantages in the use of the exposure selection method in the near future. The limiting magnitude for reference stars is expected to be fainter than +15.5, and fainter than +23 for unresolved objects in the same isoplanatic patch. In cases where the seeing is dominated primarily by one turbulent atmospheric layer at a height $H$, the diameter of the isoplanatic patch at the times of the good selected exposures will be $7r_{0}/H$ or say 2.5m/5km = 1.7 arcmin. There would then be $>80\%$ sky coverage by usable reference stars under 0.5 arcsec seeing conditions for 2.5m telescopes.
The Nordic Optical Telescope is operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias.
|
---
abstract: 'We describe an embedding of the quantum circuit language in the Coq proof assistant. This allows programmers to write quantum circuits using high-level abstractions and to prove properties of those circuits using Coq’s theorem proving features. The implementation uses higher-order abstract syntax to represent variable binding and provides a type-checking algorithm for linear wire types, ensuring that quantum circuits are well-formed. We formalize a denotational semantics that interprets circuits as superoperators on density matrices, and prove the correctness of some simple quantum programs.'
author:
- Robert Rand
- Jennifer Paykin
- Steve Zdancewic
title: |
Practice:\
Formal Verification of Quantum Circuits in Coq [^1]
---
Introduction {#sec:introduction}
============
Implementing in Coq {#sec:qwire}
===================
Denotational Semantics {#sec:denotation}
======================
A Taste of Verification {#sec:verification}
=======================
Related and Future Work {#sec:conclusion}
=======================
[^1]: This work is supported in part by ONR MURI No. FA9550-16-1-0082 and NSF Grant No. CCF- 1421193.
|
---
abstract: |
In wireless ad-hoc networks, forwarding data through intermediate relays extends the coverage area and enhances the network throughput. We consider a general wireless multiuser multihop transmission, where each data flow is subject to a constraint on the end-to-end buffering delay and the associated packet drop rate as a quality of service (QoS) requirement. The objective is to maximize the weighted sum-rate between source-destination pairs, while the corresponding QoS requirements are satisfied. We introduce two new distributed cross-layer dynamic route selection schemes in this setting that are designed involving physical, MAC, and network layers. In the proposed opportunistic cross-layer dynamic route selection scheme, routes are assigned dynamically based on the state of network nodes’ buffers and the instantaneous state of fading channels. In the same setting, the proposed time division cross-layer dynamic route selection scheme utilizes the average quality of channels instead for more efficient implementation. Detailed results and comparisons are provided, which demonstrate the superior performance of the proposed cross-layer dynamic route selection schemes.\
author:
- 'Kamal Rahimi Malekshan, and Farshad Lahouti, [^1]'
bibliography:
- 'myrefMSC.bib'
title: 'Distributed Cross-layer Dynamic Route Selection in Wireless Multiuser Multihop Networks'
---
Cross-layer, Dynamic route selection, Multiuser, Multihop, Fading, MANETs, QoS.
Introduction
============
ad-hoc networks (MANETs) are formed by a number of mobile nodes communicating with each other over wireless channels without a pre-established infrastructure. These nodes can dynamically organize arbitrary ad-hoc network topologies. As the transmission range of devices are often limited, the packets may need to be forwarded by one or more intermediate relays before they reach their destinations. This paper focuses on designing distributed route selection strategies that aim at maximizing the quality of service (QoS) constrained network throughput, while taking into account both buffers and channels conditions in a cross-layer manner.\
Research from network and information theory perspectives has led to a number of relaying schemes in MANETs [@Gastpar; @Draves; @Laneman; @Wang]. In [@Laneman] and [@Wang], the authors consider spectral efficiency as a performance measure to select a desired route from one source to a destination in wireless networks without taking into account the effect of other users requesting service in the network. They assume that data is transmitted by the source and forwarded by the intermediate relays without a store and wait stage. Other transmission schemes are proposed in [@Goldsmith2; @Goldsmith3; @Mardani] in which the nodes store data in their buffers, and employ adaptive modulation and coding (AMC) and automatic repeat request (ARQ) to enhance network throughput. The trade-off between delay, diversity and multiplexing in a multihop wireless network with ARQ is studied in [@Goldsmith2]. The problem of finding the path with the minimum end-to-end outage probability from one source to a destination is studied in [@Babaee].\
The selection of a route and corresponding relays for transmission between one specific source and destination pair, may affect the communication of other nodes. Hence, the existence of other source destination pairs in the network and the varying nature of wireless networks indicate that the routes and relays are to be assigned dynamically to guarantee the QoS requirements. Multipath routing decreases the effect of unreliable wireless links in a constantly changing topology and improves QoS of MANETs [@Nasipuri; @Harms; @Mueller; @Dhaka; @Ai]. In fact, starting from single path routing, significant performance gains are achieved by employing a limited number of additional paths, beyond which the potential gain diminishes [@Nasipuri]. In addition, providing alternative paths to intermediate nodes is more advantageous than doing that only to the source node [@Nasipuri]. Multipath routing can also help distribute the transmission load more evenly, which is vital to protect wireless nodes from power depletion [@Harms]. Multipath transmission from one source node to multiple destinations with the aid of multiple relays is considered in [@Melo]. At each instant, the source node chooses a destination for transmission and selects an intermediate relay node to forward the packet. The destination receives both signals transmitted by the source and relay, and uses a selective combining strategy to decode the signal. The authors in [@Ronasi] applied a combination of multipath routing and adaptive channel coding to improve delay and throughput performance of a wireless network. In [@Tsirigos], a multipath routing scheme is proposed for transmission over unreliable wireless ad-hoc links that increases the probability of successful reception for the essential part of information at the destination node.\
In this paper, we propose two new cross-layer dynamic route selection schemes for a multiuser multihop ad-hoc network. The proposed schemes distributedly direct information between source destination pairs through varying scheduled routes. As we demonstrate, the proposed schemes enhance the overall throughput of the network and help meet the QoS requirements. The optimized designs in fact involve physical, MAC, and network layers. To the best of our knowledge, this is the first work which considers cross-layer resource allocation across the three layers in a general multihop ad-hoc network. The opportunistic cross-layer dynamic route selection (OCDR) efficiently and dynamically assigns routes to source-destination pairs based on the state of network nodes’ buffers and the instantaneous condition of fading channels. The proposed time division cross-layer dynamic route selection (TCDR) schedules the assignment of varying routes within the same framework, but utilizing the average link qualities instead. In both schemes, the nodes exchange their average service rate information with one hop neighbors, which also reflect their buffers conditions. The nodes then identify the optimum dynamic route selection plan for forwarding data toward destinations in a distributed manner.\
The remainder of this paper is organized as follows. We introduce the system model in Section II. Section III presents our proposed dynamic route selection schemes. The simulation results are presented in Section IV. Finally, Section V concludes the paper.\
Preliminaries
=============
Consider multiuser multihop transmission in a network consisting of $K$ source-destination pairs that communicate through intermediate relays. The equivalent two-hop model is considered in [@kamal0]. We consider $L$ hop transmission with $M$ relaying nodes at each hop. The source, destination, and relays at hop $l\in\{1,2,...,L\}$ are identified by ${\cal S}=\{S_1,\hdots,S_K\}$, ${\cal D}=\{D_1,\hdots,D_K\}$, and ${\cal R}_l=\{R_{l1},\hdots,R_{lM}\}$, respectively. The source nodes forward their data to relays in the first hop. On the path from source to destination, the relays in hop $l\in\{2,3,...,L-1\}$ receive data from relays in hop $l-1$, store data of each source in a distinct buffer, and forward the data to relays in hop $l+1$. The relays in the last hop ($L$) deliver the data packets of each source node to its destination node. A decode and forward protocol is assumed at the relays. The objective is to maximize the weighted sum-rate of communications between source destination pairs. The QoS requirement is quantified by a probabilistic end-to-end delay constraint and its associated packet loss rate. Figure \[Network\] shows the system model, in which the buffers at the relays are omitted for brevity. For simplified presentation, we define ${\cal R}_i^j\triangleq{\cal R}_i\cup\hdots\cup{\cal R}_j, i,j\in \{1,\hdots,L\}, i<j$.\
Channel Model
-------------
The channels are assumed non-selective, block Rayleigh fading and signals also suffer path loss. Channel time is divided into time slots of duration $T$, where channel gain during a time slot is constant but varies independently between two consequent time slots. At each time slot, the channel state information (CSI) to every next hop neighboring node is perfectly estimated, and reliably fed back to the node without delay. Each node has access to independent bandwidth $W$ in which it transmits with fixed power $P$. It is assumed that given the channel SNR in time slot $t$, $\gamma(t)$, the AMC scheme can achieve the instantaneous capacity of the link. Let $\sqrt{h_{ij}(t)}$ denote the channel gain of the link between nodes $i$ and $j$ at time slot $t$. The capacity between nodes $i$ and $j$ at time slot $t$ is given by $$\label{1}
C_{ij}(t)=W \log_2\left(1+\frac{P\, h_{ij}(t)}{N_0\, W}\right),$$ where $N_0$ is the gaussian noise power density and $\frac{P\, h_{ij}(t)}{N_0\, W}=\gamma_{ij}(t)$ is the instantaneous SNR of received signal at node $j$ from node $i$ at time slot $t$. For Rayleigh fading, the probability density function (PDF) of $\gamma_{ij}$ is $$\label{2}
P_{\gamma_{ij}}(\gamma )=\frac{1}{\bar{\gamma}_{ij}} e^{-\frac{\gamma}{\bar{\gamma}_{ij}}},$$ where $\bar{\gamma}_{ij}$ is the average received SNR of the link between nodes $i$ and $j$.
Queuing Network Model
---------------------
The packets arrive at the source node $k\in \mathcal{S}$ as a Poisson process with average $\rho_k$. We assume that the random service time process in each node, which depends on the state of Rayleigh fading channel, as an approximation to be i.i.d exponential [@Goldsmith2]. This approximation makes the multiuser input, multiuser output, multihop network problem tractable. Packets depart from a node and arrive at the next hop node as Poisson processes [@Bolch]. Thus, we can use an $M/M/1$ queue to model packet delay of each source node at source and any of the intermediate relays. Let $\mu_{ik}$ denote the average service rate at node $i$ for data of source node $k$, and $\mu_{ijk}$ be the average service rate at node $i$ to next hop node $j$ for data of source node $k$. We have $\mu_{ik}=\sum_{j} \mu_{ijk}$. Let $\rho_{ik}$ be the average arrival data rate of source $k$ at node $i$ and $\rho_{ijk}$ be the average data rate of source $k$ that node $i$ forwards to next hop node $j$. Note that $\rho_k=\rho_{kk}, k\in \mathcal{S}$ is the average arrival data rate at source $k$, and $\rho_{ik}=\sum_j \rho_{ijk}$. If the packet loss rate of source $k$ at node $i$ is small, we have $$\label{5}
\rho_{ijk}=\frac{\mu_{ijk}}{\mu_{ik}}\times\rho_{ik}.$$ When the queue is stable, i.e., $\rho_{ik}<\mu_{ik}$, the probability that delay at node $i$ for data of source $k$, $D_{ik}$, exceeds a deadline $D_{ik}^*$ is given by [@Goldsmith2] $$\label{7}
\epsilon_{ik}^*\triangleq P(D_{ik}>D_{ik}^*)=\frac{\rho_{ik}}{\mu_{ik}}e^{-D_{ik}^*(\mu_{ik}-\rho_{ik})}.$$ If $\epsilon_{ik}^*\ll \mu_{ik}/\rho_{ik}$, we have $$\label{8}
D_{ik}^*=\frac{\ln (\frac{\rho_{ik}}{\epsilon_{ik}^*\mu_{ik}})}{\mu_{ik}-\rho_{ik}}\thickapprox \frac{\ln(\frac{1}{\epsilon_{ik}^*})}{\mu_{ik}-\rho_{ik}}.$$
Problem Formulation
-------------------
As a QoS requirement, we consider a limited probability of violating the delay bound. In other words, the probability that end-to-end delay of source $k$ packets exceed $D_k^*$ is to be smaller than a threshold. To achieve this goal, we can limit the delay of source $k$ at each node $i$ (on the path from source to destination) to $D_{ik}^*$, such that $\sum_{i\in{path}} D_{ik}^* \leq D_{k}^*$ [@Goldsmith3]. Thus, the packets of source $k$ are removed from the queue in node $i$ if they are not transmitted before the deadline $D_{ik}^*$. Similarly, to achieve the end-to-end packet loss rate $\epsilon_{k}^*$ for data of source $k$, we restrict packet loss rate due to delay violation at node $i$ (on the path from source to destination) to $\epsilon_{ik}^*$, such that $1-\prod_{i\in path}(1-\epsilon_{ik}^*)\leq \epsilon_{k}^*$.\
The objective in this work is to maximize the weighted sum arrival rate at the source nodes, while the end-to-end delay violation probability (and its associated packet drop rate) is constrained. This design problem may be formulated as follows $$\begin{aligned}
\max F&=\sum_{k\in \mathcal{S}} f_k\rho_k \label{9}\\
\text{s.t.: }& P(D_{ik}>D_{ik}^*)<\epsilon_{ik}^*,\label{91}\\ k\in&\mathcal{S},i\in\mathcal{S}\cup\mathcal{R}_1^L,\nonumber\end{aligned}$$ where $f_k$ is the assigned weight to data rate of source $k$.\
Cross-Layer Dynamic Route Selection
===================================
In cross-layer dynamic route selection data is transmitted between source-destination pairs through dynamic routes. This is in contrast to static routing where a fixed multihop link is established between source-destination pairs. The proposed cross-layer dynamic route selection schemes at the MAC layer schedule the assignment of relays in different hops based on the associated channel and buffer conditions. As we shall demonstrate, the proposed schemes enhance the overall throughput of the network and help meet the diverse QoS requirements.\
Opportunistic Cross-layer Dynamic Route Selection
-------------------------------------------------
In the proposed opportunistic cross-layer dynamic route selection (OCDR), the path for data transmission is dynamically adjusted based on the instantaneous condition of fading channels and the traffic loads of network nodes. In each time slot, each node transmits over the link with the highest instantaneous weighted SNR in comparison with other possible links for transmission. The assigned weights for the SNR of each link are adjusted to meet the QoS requirements of users in the network.\
Let $U_i$ denote the set of next hop nodes for node $i\in{\mathcal{S},\mathcal{R}_1^{L}}$. At time slot $t$, node $i$ chooses to transmit to the next hop node ${\hat j}\in U_i$, if $$\label{10}
{\hat j}=\arg\max_{j\in U_i} \Phi_{ij}(t),$$ where $\Phi_{ij}(t)$ is the priority function in node $i$ for transmission to next hop node $j\in U_i$ at time slot $t$. If multiple links have the same value of $\Phi_{ij}(t)$, one of them is selected randomly. We define $$\label{11}
\Phi_{ij} (t)\triangleq \beta_{ij}\frac{\gamma_{ij}(t)}{\bar{\gamma}_{ij}}$$ where $\beta_{ij}$, are set as described below. While next hop node $j$ is selected for transmission at node $i$, the allocated fraction of time to transmit data of source $k\in \mathcal{S}$ is $\alpha_{ijk}$. Thus, when next hop node $j$ is selected for transmission at node $i$, data of source node $k$ is transmitted with probability $$\label{11a}
\pi_{ijk}=\frac{\alpha_{ijk}}{\sum_{k'\in \mathcal{S}}\alpha_{ijk'}},$$ where $\sum_{k\in \mathcal{S}} \pi_{ijk}=1$. Using (\[1\]), (\[2\]), (\[10\]), (\[11\]), and (\[11a\]), the average data rate that can be transmitted from node $i$ to node $j\in U_i$ for source node $k$ can be computed as $$\begin{gathered}
\label{13}
\hat{\mu}_{ijk}=\frac{\alpha_{ijk}}{\sum_{k'\in\mathcal{S}}\alpha_{ijk'}} \times\\
\int_{\gamma=0}^{\infty}\frac{e^{-\frac{\gamma}{\bar{\gamma}_{ij}}}}{\bar{\gamma}_{ij}}
\prod_{z\in U_i,z\neq j}(1-e^{-\frac{\beta_{ij}\gamma}{\beta_{iz}\bar{\gamma}_{ij}}})W\log_2(1+\gamma)d\gamma.\end{gathered}$$
Based on , (\[8\]), (\[9\]) and (\[91\]), the QoS constraint of source $k$ at node $i$ imposes a limit on the corresponding arrival rate as follows, $$\label{14}
\rho_{ik}\leq \mu_{ik}-\frac{\ln\frac{1}{\epsilon^*_{ik}}}{D^*_{ik}}.$$ Every node in the network restricts its transmitted data rate to each next hop node to avoid QoS violation at all subsequent hops on the path to the destination. These constraints determine the maximum data that node $i$ can forward for source $k\in \mathcal{S}$. Let $\hat{\rho}_{ijk}$ denote the maximum data rate for source $k$ that node $i$ can forward to next hop node $j$ such that all QoS constraints at $j$ and all subsequent hops to destination are satisfied. Also, let $\rho^*_{ik}$ denote the maximum arrival data rate of source $k$ to node $i$ such that QoS constraint at node $i$ and all subsequent hops to destination are satisfied. Therefore, we have $$\label{16b}
\rho^*_{ik}=\sum_{y\in V_i} \hat{\rho}_{yik},\,\, i\in {\cal{R}}_{L},$$ where $V_i$ denotes the set of previous hop nodes of node $i$. At a last hop relay node $i\in \mathcal{R}_L$, $$\label{16}
\mu_{ikk}=\hat{\mu}_{ikk},\,\, i\in {\cal{R}}_{L}$$ and $$\label{16a}
\rho^*_{ik}=\hat{\mu}_{ikk}-\frac{\ln\frac{1}{\epsilon^*_{ik}}}{D^*_{ik}}, \,\, i\in {\cal{R}}_{L}.$$ In any other intermediate relay or source node $i\in\mathcal{S}\cup \mathcal{R}_1^{L-1}$, we have $$\label{17a}
\mu_{ijk}=\min(\hat{\mu}_{ijk},\hat{\rho}_{ijk}), \,\, j\in U_i,$$ and $$\begin{gathered}
\label{17}
\rho^*_{ik}= \sum_{j\in U_i} \min (\hat{\mu}_{ijk},\hat{\rho}_{ijk})-\frac{\ln\frac{1}{\epsilon^*_{ik}}}{D^*_{ik}}, \,\, i\in {\cal S}\cup {\cal R}_1^{L-1}.\end{gathered}$$ The right hand side of (\[17\]) reflects the QoS constraint at node $i$ and all other subsequent hop nodes to destination. Therefore, to satisfy the QoS constraints, the arrival data rate of source $k$ is to be limited to $$\label{18}
\rho_{k}=\rho^*_{kk},\,k\in{\cal S}.$$
Using (\[16a\]), (\[17\]), and (\[18\]), the problem (\[9\]) may be rewritten as an unconstrained optimization problem for maximizing $F$ with respect to design parameters $\{\beta_{ij}\}$ and $\{\alpha_{ijk}\}$ as $$\begin{gathered}
\label{19}
\max_{\beta_{ij},\alpha_{ijk}} F=\sum_{k\in \mathcal{S}} f_k\rho^*_{kk},
\\
\rho^*_{ik}=\left\{
\begin{array}{ll}
\sum_{j\in U_i} \min (\hat{\mu}_{ijk},\hat{\rho}_{ijk})-\frac{\ln\frac{1}{\epsilon^*_{ik}}}{D^*_{ik}} & i\in \mathcal{R}_1^{L-1} \\\\
\hat{\mu}_{ikk}-\frac{\ln\frac{1}{\epsilon^*_{ik}}}{D^*{ik}}& i\in \mathcal{R}_{L}
\end{array}\right.\end{gathered}$$
Every relay periodically (with period $T_1$) calculates the maximum arrival data rate that it can forward for each source using (\[16a\]) or (\[17\]), and restricts the maximum arrival data rate from each node in the previous hop accordingly. Specifically, node $i$ determines the maximum arrival data rate for source $k$ from node $y$ in the previous hop as follows, $$\label{24}
\hat{\rho}_{yik}=\left\{
\begin{array}{ll}
\frac{\rho_{yik}}{\rho_{ik}} \rho^*_{ik} & \text{if }\rho_{ik}\neq 0\\\\
\rho^*_{ik} & \text{if }\rho_{ik}= 0, y\in\mathcal{S}\\\\
\frac{1}{M} \rho^*_{ik} & \text{if }\rho_{ik}= 0, y\not\in\mathcal{S}\\
\end{array}\right.$$ Also, source nodes set their average data rate based on (\[17\]) and (\[18\]), periodically with period $T_1$. We use gradient descent optimization to iteratively optimize $\{\beta_{ij}\}$ and $\{\alpha_{ijk}\}$ at the relays and source nodes. One important advantage of using this optimization method is that it can be implemented in a distributed way, which is appropriate for MANETs without any central control unit. Node $i$ updates $\alpha_{ijk}$ and $\beta_{ij}$ periodically with periods $T_1$ and $T_2$, respectively as follows $$\label{23}
\alpha_{ijk}(n+1)=\alpha_{ijk}(n)+\theta_{\alpha} \frac{\partial F}{\partial \alpha_{ijk}},$$ $$\label{22}
\beta_{ij}(m+1)=\beta_{ij}(m)+\theta_{\beta} \frac{\partial F}{\partial \beta_{ij}},$$ where $n$ and $m$ and $\theta_{\alpha}$ and $\theta_{\beta}$ are the corresponding iteration numbers, and gradient decent step sizes. According to (\[18\]) and (\[19\]), we have $$\label{30}
\frac{\partial F}{\partial \alpha_{ijk}}=\sum_{k\in\mathcal{S}}f_k\frac{\partial \rho_k}{\partial \alpha_{ijk}},$$ $$\label{30}
\frac{\partial F}{\partial \beta_{ij}}=\sum_{k\in\mathcal{S}}f_k\frac{\partial \rho_k}{\partial \beta_{ij}}.$$ According to (\[19\]), at node $i$, $$\label{21}
\frac{\partial \rho_k}{\partial \alpha_{ijk}}=\left\{
\begin{array}{ll}
\frac{\partial\rho_{ik}^*}{\partial \alpha_{ijk}}& \text{if } \rho_{ik}=\rho_{ik}^*\\\\
0 &\text{if } \rho_{ik} < \rho_{ik}^*\\
\end{array},
\right.$$ and $$\label{20}
\frac{\partial \rho_k}{\partial \beta_{ij}}=\left\{
\begin{array}{ll}
\frac{\partial\rho_{ik}^*}{\partial \beta_{ij}}& \text{if } \rho_{ik}=\rho_{ik}^*\\\\
0 &\text{if } \rho_{ik} < \rho_{ik}^*\\
\end{array},
\right.$$ where $\partial\rho_{ik}^*/\partial \alpha_{ijk}$ and $\partial\rho_{ik}^*/\partial \beta_{ij}$ can be easily computed using (\[13\]), (\[16a\]), and (\[17\]). Algorithm \[OCDR\] summarizes the proposed OCDR scheme. We choose $T_2>\!>T_1>\!>T$ to ensure that a node has correctly estimated the arrival rate from previous hop nodes when updating its routing parameters.
- Initially set $\rho_{k}=0$, $\beta_{ij}=1$, $\alpha_{ijk}=1$, $\hat{\rho}_{ijk}=0$, and $\rho^*_{ik}=0$ for all $k\in \mathcal{S}$, $i\in\mathcal{S}\cup\mathcal{R}_1^L$;\
- In each time slot, node $i\in\mathcal{S}\cup\mathcal{R}_1^{L}$ chooses next link $j\in U_i$ for transmission based on (\[10\]), and forwards data of source node $k$ with probability $\pi_{ijk}$ given in (\[11a\]);\
- Node $i\in \mathcal{S}\cup \mathcal{R}_1^L$ updates $\alpha_{ijk}$ for all $j\in U_i$ and $k\in \mathcal{S}$ based on (\[23\]) periodically with period $T_1$;\
- Relay $i\in \mathcal{R}_1^L$ computes $\rho^*_{ik}$ using (\[16a\]) or (\[17\]), determines $\hat{\rho}_{yik}$ for all $y\in V_i$ using (\[24\]), and informs $\hat{\rho}_{yik}$ to previous hop node $y\in V_i$ periodically with period $T_1$;\
- Source node $k \in\mathcal{S}$ computes $\rho^*_{kk}$ using (\[17\]) and sets the data rate $\rho_k=\rho_{kk}^*$ periodically with period $T_1$;\
- Node $i\in \mathcal{S}\cup \mathcal{R}_1^L$ updates $\beta_{ij}$ for all $j\in U_i$ based on (\[22\]) periodically with period $T_2$.\
Time Division Cross-layer Dynamic Route Selection
--------------------------------------------------
In the proposed time division cross-layer dynamic route selection (TCDR), each node assigns a fraction of time for transmission to next hop node based on the average quality of links. Let $\alpha_{ijk}'$ denote the fraction of time that node $i$ forwards data of source $k$ to node $j$ in the next hop. Thus, node $i$ forwards data of source $k$ to its next hop node $j\in U_i$ with probability $$\label{25}
\pi'_{ijk}=\frac{\alpha_{ijk}'}{\sum_{k'\in\mathcal{S}}\sum_{j'\in U_i}\alpha_{ij'k'}'},$$ where $\sum_{k\in\mathcal{S}}\sum_{j\in U_i}\pi'_{ijk}=1$. According to (\[1\]), (\[2\]), and (\[25\]) the average data rate that can be transmitted from node $i$ to node $j\in U_i$ for source node $k$ can be computed as $$\label{26}
\hat{\mu}_{ijk}=\frac{\alpha_{ijk}'}{\sum_{k'\in\mathcal{S}}\sum_{j'\in U_i}\alpha_{ij'k'}'} \times\\
\int_{\gamma=0}^{\infty}\frac{e^{-\frac{\gamma}{\bar{\gamma}_{ij}}}}{\bar{\gamma}_{ij}}
W\log_2(1+\gamma)d\gamma.$$ The service rate at node $i$ for data of source $k$, $\mu_{ik}$ and the maximum arrival data rate of source node $k$ to node $i$ (such that all the QoS constraints at node $i$ and all subsequent hops to destination are satisfied),$\rho^*_{ik}$, are given by (\[16\])-(\[17\]). Using (\[16a\]), (\[17\]), and (\[18\]), the problem (\[9\]) can be rewritten as an unconstrained optimization problem for maximizing $F$ with respect to design parameters $\{\alpha'_{ijk}\}$ as $$\begin{gathered}
\label{39}
\max_{\alpha'_{ijk}} F=\sum_{k\in \mathcal{S}} f_k\rho^*_{kk},
\\
\rho^*_{ik}=\left\{
\begin{array}{ll}
\sum_{j\in U_i} \min (\hat{\mu}_{ijk},\hat{\rho}_{ijk})-\frac{\ln\frac{1}{\epsilon^*_{ik}}}{D^*_{ik}} & i\in \mathcal{R}_1^{L-1} \\\\
\hat{\mu}_{ikk}-\frac{\ln\frac{1}{\epsilon^*_{ik}}}{D^*{ik}}& i\in \mathcal{R}_{L}
\end{array}\right.\end{gathered}$$
A relay node $i\in \mathcal{R}_1^l$ periodically (with period $T_1$) computes the maximum arrival data rate that it can forward for each source node using (\[16a\]) or (\[17\]), and restricts the maximum arrival data rate from each previous hop node $y\in V_i$ according to (\[24\]). A source node $k\in \mathcal{S}$ sets their average data rate based on (\[17\]) and (\[18\]), periodically with period $T_1$. Similar to OCDR, in TCDR, we use gradient decent optimization to iteratively update $\alpha'_{ijk}$ at source nodes and relays. Node $i\in\mathcal{S}\cup\mathcal{R}_1^{L}$ updates $\alpha'_{ijk}$ for all $j\in U_i$ and $k\in \mathcal{S}$, periodically with period $T_1$, as follows $$\label{63}
\alpha'_{ijk}(n+1)=\alpha'_{ijk}(n)+\theta_{\alpha'} \frac{\partial F}{\partial \alpha'_{ijk}},$$ where $n$ is the iteration number and $\theta_{\alpha'}$ is the gradient decent step size. According to (\[18\]) and (\[39\]), we have $$\label{50}
\frac{\partial F}{\partial \alpha'_{ijk}}=\sum_{k\in\mathcal{S}}f_k\frac{\partial \rho_k}{\partial \alpha'_{ijk}}.$$ Based on (\[39\]), at node $i$, $$\label{52}
\frac{\partial \rho_k}{\partial \alpha'_{ijk}}=\left\{
\begin{array}{ll}
\frac{\partial\rho_{ik}^*}{\partial \alpha'_{ijk}}& \text{if } \rho_{ik}=\rho_{ik}^*\\\\
0 &\text{if } \rho_{ik} < \rho_{ik}^*\\
\end{array},
\right.$$ where $\partial\rho_{ik}^*/\partial \alpha'_{ijk}$ can be easily computed using (\[16a\]), (\[17\]), and (\[26\]). Algorithm \[TCDR\] summarizes the proposed TCDR scheme. We choose $T_1>\!>T$ to ensure that a node has enough time to estimate its arrival rate from previous hop nodes before updating its routing parameters. The simulation results show that OCDR provides a higher performance compared to TCDR. However, TCDR does not need access to instantaneous channel SNRs and changes the routes only periodically. Therefore, TCDR has a lower implementation complexity.\
- Initially set $\rho_{k}=0$, $\alpha'_{ijk}=1$, $\hat{\rho}_{ijk}=0$, and $\rho^*_{ik}=0$ for all $k\in \mathcal{S}$, $i\in\mathcal{S}\cup\mathcal{R}_1^L$;\
- In each time slot, node $i\in\mathcal{S}\cup\mathcal{R}_1^{L}$ chooses next link $j\in U_i$ for transmission and forwards data of source node $k$ with probability $\pi'_{ijk}$ given in (\[25\]);\
- Node $i\in \mathcal{S}\cup \mathcal{R}_1^L$ updates $\alpha'_{ijk}$ for all $j\in U_i$ and $k\in \mathcal{S}$ based on (\[63\]) periodically with period $T_1$;\
- Relay $i\in \mathcal{R}_1^L$ computes $\rho^*_{ik}$ using (\[16a\]) or (\[17\]), determines $\hat{\rho}_{yik}$ for all $y\in V_i$ using (\[24\]), and informs $\hat{\rho}_{yik}$ to previous hop node $y\in V_i$, periodically with period $T_1$;\
- Source node $k \in\mathcal{S}$ computes $\rho^*_{kk}$ using (\[17\]) and sets the data rate $\rho_k=\rho_{kk}^*$, periodically with period $T_1$.\
Performance Evaluation
======================
We consider multihop transmission in a one dimensional network. The linear network model for $N=M=2$ and $L=1,2$ is illustrated in Figure \[LN\]. We use standard path loss model to model signal attenuation over each link. Thus the average channel gain between nodes $i$ and $j$ is $\bar{h}_{ij}=cd_{ij}^{-\delta}$, where $c$ is a constant, $d_{ij}$ is the distance between nodes $i$ and $j$, and $\delta$ is the path loss exponent. We set ${cP\over{N_0W}}=1$, $\delta=3$, and $W=1\text{MHz}$. The maximum tolerable delay in each queue, $D_{ik}^*$, and the packet loss rate threshold in node $i$ for data of source $k$, $\epsilon_{ik}^*$, are set at $0.1\text{ms}$ and $10^{-6}$, respectively. The benchmark for comparison is the maximum possible weighted sum-rate performance, by static assignment of one relay in each hop to each source destination pair. This is obtained by examining all possible relay assignment scenarios, while the QoS constraints are satisfied.\
Two hop network
---------------
Consider two-hop transmission in a network consisting of two source-destination pairs and two intermediate relays ($K=M=2, L=1$). Figure \[LN2\] shows the network model. The first source and destination pair are positioned at $d_{S_1}=0$ and $d_{D_1}=1$, respectively. The second source and destination pair are positioned at $d_{S_2}=0.2$ and $d_{D_2}=0.8$, respectively. Relays are positioned at $d_{R_1}$ and $d_{R_2}$. The relays positions vary such that $0.3\leq d_{R_1}\leq 0.7$ and $0.3\leq d_{R_2}\leq 0.7$. We aim to maximize the sum-rate, i.e., we set $f_1=f_2=1$.\
Figure \[2P\] shows the sum-rate performance as a function of the relays positions for OCDR, TCDR, and the benchmark scheme. According to Figure \[2P\], the sum-rate is maximized in each of the schemes (OCDR, TCDR, or benchmark) when relay nodes are positioned at points $d_{R_1}=0.5$ and $d_{R_2}=0.5$, which corresponds to the situation that the channel quality of first and second hops are identical for each source-destination pair. While fixed relay assignment (benchmark) provides very poor performance when channel quality of links are diverse, the proposed OCDR and TCDR schemes provide significantly higher performance by taking advantage of transmitting data through dynamic routes. Figures \[fig6\] and \[fig7\] illustrate the sum-rate performance gains of OCDR and TCDR with respect to the benchmark, respectively. On average, OCDR and TCDR provide $32\%$ and $11\%$ performance gain, however, this gain may reach $60\%$ and $40\%$ depending on the relays positions. The sum-rate performance gain of OCDR with respect to TCDR is shown in Figure \[fig8\]. The OCDR when compared to TCDR enhances the weighted sum-rate performance by up to $32\%$ depending on the relays positions. This comes at the cost of instantaneous channel state information and higher implementation complexity. Indeed, the proposed TCDR scheme only requires the average quality of links and alternates the data transmission path periodically.\
Three hop network
-----------------
In this Subsection, we evaluate the performance of the proposed schemes for three-hop transmission in a linear network consisting of two source-destination pairs and four intermediate relays ($K=M=2,L=2$). The network model is illustrated in Figure \[LN3\]. The source nodes are placed at $d_{S_1}=d_{S_2}=0$ and destination nodes are placed at $d_{D_1}=d_{D_2}=1$. First hop relays are $R_1$ and $R_2$ and second hop relays are $R_3$ and $R_4$, which are positioned at $d_{R_1}$, $d_{R_2}$, $d_{R_3}$, and $d_{R_4}$, respectively.\
Figure \[3P\] shows the sum-rate performance ($f_1=f_2=1$) versus relays position for OCDR, TCDR, and the benchmark scheme, when $d_{R_1}=0.2$, $0.1\leq d_{R_2}\leq 0.5$, $d_{R_3}=0.7$, and $0.5\leq d_{R_4} \leq 0.9$. The maximum sum-rate in each scheme corresponds to the situation that the link quality of hops are identical, i.e., $d_{R_2}=0.1$ and $d_{R_4}=0.7$. When the quality of links are diverse, the static relay assignment (benchmark) provides a very poor performance. However, the proposed OCDR and TCDR schemes provide significantly higher performance by directing data through multiple dynamic routes to destinations. Figures \[fig13\] and \[fig14\] illustrate the sum-rate performance gains of OCDR and TCDR with respect to the benchmark. Also, the sum-rate performance gain of OCDR with respect to TCDR is shown in Figure \[fig14N2\]. According to Figure \[3G\], the performance gain of proposed OCDR and TCDR schemes with respect to the benchmark may reach $55\%$ and $30\%$ depending on relays positions.\
Effect of weights
-----------------
In this Subsection, we evaluate the impact of weights on the average data rate of each source destination pair in a three-hop linear network as illustrated in Figure \[LN3\]. The source nodes are placed at $d_{S_1}=d_{S_2}=0$ and destination nodes are placed at $d_{D_1}=d_{D_2}=1$. The four relays are randomly positioned in the range $[0.1,0.9]$ and in each realization, the two relays closer to source nodes are selected as the first hop relays, $R_1$ and $R_2$, and the other two relays are selected as the second hop relays, $R_3$ and $R_4$. Figures \[fig55\] and \[fig66\] depict respectively, the average rates, $r_1$ and $r_2$, of each source-destination pair and the average weighted sum rate (normalized to the sum of weights), for one hundred random relays positions as a function of the weights assigned to source destination pairs ($f_1$ and $f_2$). One sees that as $f_2/f_1$ increases, the data rate of second source destination pair increases in OCDR, TCDR and the benchmark. The proposed OCDR and TCDR schemes provide significantly higher average sum-rate and better performance differentiation between the users in comparison to the benchmark.\
Conclusions
===========
In this paper, we present two new cross-layer dynamic route selection schemes for multiuser multihop transmission in wireless ad-hoc networks. The proposed schemes set up the routes between source-destination pairs in a cross-layer optimized manner, which takes into account the buffer status of network nodes and the instantaneous condition of fading channels or the average link qualities. The proposed schemes are distributed. Nodes exchange routing information with single hop neighbors and iteratively adjust the routing parameters to optimize the network performance. The simulation results show the effectiveness of our proposed dynamic route selection schemes in comparison with conventional fixed routing.\
[^1]: Kamal Rahimi Malekshan is with the Department of Electrical and Computer Engineering, University of Waterloo, Canada. Email: [email protected].
Farshad Lahouti is with the Center for Wireless Multimedia Communications, Center of Excellence in Applied Electromagnetic Systems, School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran. Email: [email protected].
|
---
abstract: 'We propose a novel end-to-end neural network architecture that, once trained, directly outputs a probabilistic clustering of a batch of input examples in one pass. It estimates a distribution over the number of clusters $k$, and for each $1 \leq k \leq k_\mathrm{max}$, a distribution over the individual cluster assignment for each data point. The network is trained in advance in a supervised fashion on separate data to learn grouping by any perceptual similarity criterion based on pairwise labels (same/different group). It can then be applied to different data containing different groups. We demonstrate promising performance on high-dimensional data like images (COIL-100) and speech (TIMIT). We call this “learning to cluster” and show its conceptual difference to deep metric learning, semi-supervise clustering and other related approaches while having the advantage of performing learnable clustering fully end-to-end.'
author:
- Benjamin Bruno Meier
- Ismail Elezi
- Mohammadreza Amirian
- |
\
Oliver Dürr
- Thilo Stadelmann
bibliography:
- 'annpr2018-2.bib'
title: 'Learning Neural Models for End-to-End Clustering'
---
Introduction
============
Consider the illustrative task of grouping images of cats and dogs by *perceived* similarity: depending on the intention of the user behind the task, the similarity could be defined by animal type (foreground object), environmental nativeness (background landscape, cp. Fig. \[fig:cats\_dogs\]) etc. This is characteristic of clustering perceptual, high-dimensional data like images [@kampffmeyer2017] or sound [@lukic2017speaker]: a user typically has some similarity criterion in mind when thinking about naturally arising groups (e.g., pictures by holiday destination, or persons appearing; songs by mood, or use of solo instrument). As defining such a similarity for every case is difficult, it is desirable to learn it. At the same time, the learned model will in many cases not be a classifier—the task will not be solved by classification—since the number and specific type of groups present at application time are not known in advance (e.g., speakers in TV recordings; persons in front of a surveillance camera; object types in the picture gallery of a large web shop).
Grouping objects with machine learning is usually approached with clustering algorithms [@kaufman1990finding]. Typical ones like K-means [@macqueen1967some], EM [@jin2011expectation], hierarchical clustering [@murtagh1983survey] with chosen distance measure, or DBSCAN [@ester1996density] each have a specific inductive bias towards certain similarity structures present in the data (e.g., K-means: Euclidean distance from a central point; DBSCAN: common point density). Hence, to be applicable to above-mentioned tasks, they need high-level features that already encode the aspired similarity measure. This may be solved by learning salient embeddings [@mikolov2013efficient] with a deep metric learning approach [@hoffer2015deep], followed by an off-line clustering phase using one of the above-mentioned algorithm.
However, it is desirable to combine these distinct phases (learning salient features, and subsequent clustering) into an end-to-end approach that can be trained globally [@lecun1998gradient]: it has the advantage of each phase being perfectly adjusted to the other by optimizing a global criterion, and removes the need of manually fitting parts of the pipeline. Numerous examples have demonstrated the success of neural networks for end-to-end approaches on such diverse tasks as speech recognition [@amodei2016deep], robot control [@levine2016end], scene text recognition [@shi2017end], or music transcription [@sigtia2016end].
![Images of cats (top) and dogs (bottom) in urban (left) and natural (right) environments.[]{data-label="fig:cats_dogs"}](images/cats_dogs_lines.png){width="1.0\columnwidth"}
In this paper, we present a conceptually novel approach that we call *“learning to cluster”* in the above-mentioned sense of grouping high-dimensional data by some perceptually motivated similarity criterion. For this purpose, we define a novel neural network architecture with the following properties: (a) during training, it receives pairs of similar or dissimilar examples to learn the intended similarity function implicitly or explicitly; (b) during application, it is able to group objects of groups never encountered before; (c) it is trained end-to-end in a supervised way to produce a tailor-made clustering model and (d) is applied like a clustering algorithm to find both the number of clusters as well as the cluster membership of test-time objects in a fully probabilistic way.
Our approach builds upon ideas from *deep metric embedding*, namely to learn an embedding of the data into a representational space that allows for specific perceptual similarity evaluation via simple distance computation on feature vectors. However, it goes beyond this by adding the actual clustering step—grouping by similarity—directly to the same model, making it trainable end-to-end. Our approach is also different from *semi-supervised clustering* [@Basu02semi-supervisedclustering], which uses labels for some of the data points in the inference phase to guide the creation of groups. In contrast, our method uses absolutely no labels during inference, and moreover doesn’t expect to have seen any of the groups it encounters during inference already during training (cp. Fig. \[fig:training\_vs\_evaluation\]). Its training stage may be compared to creating K-means, DBSCAN etc. in the first place: it creates a specific clustering model, applicable to data with certain similarity structure, and once created/trained, the model performs “unsupervised learning” in the sense of finding groups. Finally, our approach differs from traditional cluster *analysis* [@kaufman1990finding] in how the clustering algorithm is applied: instead of looking for patterns in the data in an unbiased and exploratory way, as is typically the case in unsupervised learning, our approach is geared towards the use case where users know perceptually what they are looking for, and can make this explicit using examples. We then learn appropriate features and the similarity function simultaneously, taking full advantage of end-to-end learning.
![Training vs. testing: cluster types encountered during application/inference are never seen in training. Exemplary outputs (right-hand side) contain a partition for each $k$ ($1$–$3$ here) and a corresponding probability (best highlighted blue).[]{data-label="fig:training_vs_evaluation"}](images/training_vs_evaluation_02.pdf){width="0.85\columnwidth"}
Our main contribution in this paper is the creation of a neural network architecture that learns to *group* data, i.e., that outputs the same “label” for “similar” objects regardless of (a) it has ever seen this group before; (b) regardless of the actual value of the label (it is hence not a “class”); and (c) regardless of the number of groups it will encounter during a single application run, up to a predefined maximum. This is novel in its concept and generality (i.e., learn to cluster previously unseen groups end-to-end for arbitrary, high-dimensional input without any optimization on test data). Due to this novelty in approach, we focus here on the general idea and experimental demonstration of the principal workings, and leave comprehensive hyperparameter studies and optimizations for future work. In Sec. \[sec:related\_work\], we compare our approach to related work, before presenting the model and training procedure in detail in Sec. \[sec:our\_model\]. We evaluate our approach on different datasets in Sec \[sec:experiments\], showing promising performance and a high degree of generality for data types ranging from 2D points to audio snippets and images, and discuss these results with conclusions for future work in Sec. \[sec:conclusions\].
Related Work {#sec:related_work}
============
Learning to cluster based on neural networks has been approached mostly as a supervised learning problem to extract embeddings for a subsequent off-line clustering phase. The core of all deep metric embedding models is the choice of the loss function. Motivated by the fact that the softmax-cross entropy loss function has been designed as a classification loss and is not suitable for the clustering problem per se, *Chopra et al.* [@DBLP:conf/cvpr/ChopraHL05] developed a “Siamese” architecture, where the loss function is optimized in a way to generate similar features for objects belonging to the same class, and dissimilar features for objects belonging to different classes. A closely related loss function called “triplet loss” has been used by *Schroff et al.* [@schroff2015facenet] to get state-of-the-art accuracy in face detection. The main difference from the Siamese architecture is that in the latter case, the network sees same and different class objects with every example. It is then optimized to jointly learn their feature representation. A problem of both approaches is that they are typically difficult to train compared to a standard cross entropy loss.
*Song et al.* [@DBLP:conf/cvpr/SongXJS16] developed an algorithm for taking full advantage of all the information available in training batches. They later refined the work [@DBLP:conf/cvpr/SongJR017] by proposing a new metric learning scheme based on structured prediction, which is designed to optimize a clustering quality metric (normalized mutual information [@mcdaid2011normalized]). Even better results were achieved by *Wong et al.* [@DBLP:conf/iccv/WangZWLL17], where the authors proposed a novel angular loss, and achieved state-of-the-art results on the challenging real-world datasets *Stanford Cars* [@DBLP:conf/iccvw/Krause13] and *Caltech Birds* [@DBLP:journals/ijcv/BransonHWPB14]. On the other hand, *Lukic et al.* [@lukic2016speaker] showed that for certain problems, a carefully chosen deep neural network can simply be trained with softmax-cross entropy loss and still achieve state-of-the-art performance in challenging problems like speaker clustering. Alternatively, *Wu et al.* [@DBLP:conf/iccv/ManmathaWSK17] showed that state-of-the-art results can be achieved simply by using a traditional margin loss function and being careful on how sampling is performed during the creation of mini-batches.
On the other hand, attempts have been made recently that are more similar to ours in spirit, using deep neural networks only and performing clustering end-to-end [@DBLP:journals/corr/abs-1801-07648]. They are trained in a fully unsupervised fashion, hence solve a different task then the one we motivated above (that is inspired by speaker- or image clustering based on some human notion of similarity). Perhaps first to group objects together in an unsupervised deep learning based manner where *Le et al.* [@DBLP:conf/icml/LeRMDCCDN12], detecting high-level concepts like cats or humans. *Xie et al.* [@xie2016unsupervised] used an autoencoder architecture to do clustering, but experimental evaluated it only simplistic datasets like *MNIST*. CNN-based approaches followed, e.g. by *Yang et al.* [@DBLP:conf/cvpr/YangPB16], where clustering and feature representation are optimized together. *Greff et al.* [@DBLP:conf/nips/GreffSS17] performed perceptual grouping (of pixels within an image into the objects constituting the complete image, hence a different task than ours) fully unsupervised using a neural expectation maximization algorithm. Our work differs from above-mentioned works in several respects: it has no assumption on the type of data, and solves the different task of grouping whole input objects.
A model for end-to-end clustering of arbitrary data {#sec:our_model}
===================================================
![Our complete model, consisting of the embedding network, clustering network (including an optional metric learning part, see Sec. \[sec:metric\_learning\]), cluster-assignment network and cluster-count estimating network.[]{data-label="fig:clustering"}](images/plt_v02.pdf){width="1\columnwidth"}
Our method learns to cluster end-to-end purely ab initio, without the need to explicitly specify a notion of similarity, only providing the information whether two examples belong together. It uses as input $n \ge 2$ examples $x_i$, where $n$ may be different during training and application and constitutes the number of objects that can be clustered at a time, i.e. the maximum number of objects in a partition. The network’s output is two-fold: a probability distribution $P(k)$ over the cluster count $1 \leq k \leq k_\mathrm{max}$; and probability distributions $P(\cdot\mid x_i,k)$ over all possible cluster indexes for each input example $x_i$ and for each $k$.
Network architecture
--------------------
The network architecture (see Fig. \[fig:clustering\]) allows the flexible use of different input types, e.g. images, audio or 2D points. An input $x_i$ is first processed by an embedding network (a) that produces a lower-dimensional representation $z_i=z(x_i)$. The dimension of $z_i$ may vary depending on the data type. For example, 2D points do not require any embedding network. A fully connected layer (FC) with $\mathrm{LeakyReLU}$ activation at the beginning of the clustering network (b) is then used to bring all embeddings to the same size. This approach allows to use the identical subnetworks (b)–(d) and only change the subnet (a) for any data type. The goal of the subnet (b) is to compare each input $z(x_i)$ with all other $z(x_{j\ne i})$, in order to learn an abstract grouping which is then concretized into an estimation of the number of clusters (subnet (d)) and a cluster assignment (subnet (c)).
To be able to process a non-fixed number of examples $n$ as input, we use a recurrent neural network. Specifically, we use stacked residual bi-directional LSTM-layers ($\mathrm{RBDLSTM}$), which are similar to the cells described in [@wu2016google] and visualized in Fig. \[fig:rbdlstm\]. The residual connections allow a much more effective gradient flow during training [@he2016deep] and avoid vanishing gradients. Additionally, the network can learn to use or bypass certain layers using the residual connections, thus reducing the architectural decision on the number of recurrent layers to the simpler one of finding a reasonable upper bound.
![$\mathrm{RBDLSTM}$-layer: A $\mathrm{BDLSTM}$ with residual connections (dashed lines). The variables $x_i$ and $y_i$ are named independently from the notation in Fig. \[sec:our\_model\].[]{data-label="fig:rbdlstm"}](images/Residual_BDLSTM_c_tex.pdf){width="0.5\columnwidth"}
The first of overall two outputs is modeled by the cluster assignment network . It contains a $\mathrm{softmax}$-layer to produce $P(\ell\mid x_i,k)$, which assigns a cluster index $\ell$ to each input $x_i$, given $k$ clusters (i.e., we get a distribution over possible cluster assignments for each input and every possible number of clusters). The second output, produced by the cluster-count estimating network , is built from another $\mathrm{BDLSTM}$-layer. Due to the bi-directionality of the network, we concatenate its first and the last output vector into a fully connected layer of twice as many units using again $\mathrm{LeakyReLUs}$. The subsequent $\mathrm{softmax}$-activation finally models the distribution $P(k)$ for $1\leq k\leq k_\mathrm{max}$. The next subsection shows how this neural network learns to approximate these two complicated probability distributions [@Lee2017OnTA] purely from pairwise constraints on data that is completely separate from any dataset to be clustered. No labels for clustering are needed.
Training and loss
-----------------
In order to define a suitable loss-function, we first define an approximation (assuming independence) of the probability that $x_i$ and $x_j$ are assigned to the same cluster for a given $k$ as $$\begin{aligned}
P_{ij}(k)=\sum_{\ell=1}^{k} P(\ell\mid x_i,k)P(\ell\mid x_j,k).\end{aligned}$$ By marginalizing over $k$, we obtain $P_{ij}$, the probability that $x_i$ and $x_j$ belong to the same cluster: $$\begin{aligned}
P_{ij}=\sum_{k=1}^{k_{\mathrm{max}}} P(k) \sum_{\ell=1}^{k} P(\ell\mid x_i,k)P(\ell\mid x_j,k).\end{aligned}$$ Let $y_{ij}=1$ if $x_i$ and $x_j$ are from the same cluster (e.g., have the same group label) and $0$ otherwise. The loss component for *cluster assignments*, $L_\mathrm{ca}$, is then given by the weighted binary cross entropy as $$\begin{aligned}
L_{\mathrm{ca}}=\frac{-2}{n(n-1)}\sum_{i<j}{\left(\varphi_1 y_{ij} \log(P_{ij})+\varphi_2 (1-y_{ij}) \log(1-P_{ij})\right)}\end{aligned}$$ with weights $\varphi_1$ and $\varphi_2$. The idea behind the weighting is to account for the imbalance in the data due to there being more dissimilar than similar pairs $(x_i,x_j)$ as the number of clusters in the mini batch exceeds $2$. Hence, the weighting is computed using $\varphi_1=c\sqrt{1-\varphi}$ and $\varphi_2=c\sqrt{\varphi}$, with $\varphi$ being the expected value of $y_{ij}$ (i.e., the a priori probability of any two samples in a mini batch coming from the same cluster), and $c$ a normalization factor so that $\varphi_1 + \varphi_2 = 2$. The value $\varphi$ is computed over all possible cluster counts for a fixed input example count $n$, as during training, the cluster count is randomly chosen for each mini batch according to a uniform distribution. The weighting of the cross entropy given by $\varphi$ is then used to make sure that the network does not converge to a sub-optimal and trivial minimum. Intuitively, we thus account for permutations in the sequence of examples by checking rather for pairwise correctness (probability of same/different cluster) than specific indices.
The second loss term, $L_\mathrm{cc}$, penalizes a wrong *number of clusters* and is given by the categorical cross entropy of $P(k)$ for the true number of clusters $k$ in the current mini batch: $$\begin{aligned}
L_\mathrm{cc}&= -\log(P(k)).\end{aligned}$$
The complete loss is given by $L_{\mathrm{tot}}=L_{\mathrm{cc}}+ \lambda L_{\mathrm{ca}}$. During training, we prepare each mini batch with $N$ sets of $n$ input examples, each set with $k=1\ldots k_\mathrm{max}$ clusters chosen uniformly. Note that this training procedure requires only the knowledge of $y_{ij}$ and is thus also possible for weakly labeled data. All input examples are randomly shuffled for training and testing to avoid that the network learns a bias w.r.t. the input order. To demonstrate that the network really learns an intra-class distance and not just classifies objects of a fixed set of classes, it is applied on totally different clusters at evaluation time than seen during training.
Implicit vs. explicit distance learning {#sec:metric_learning}
---------------------------------------
To elucidate the importance and validity of the implicit learning of distances in our subnetwork (b), we also provide a modified version of our network architecture for comparison, in which the calculation of the distances is done explicitly. Therefore, we add an extra component to the network before the RBDLSTM layers, as can be seen in Figure \[fig:clustering\]: the optional metric learning block receives the fixed-size embeddings from the fully connected layer after the embedding network (a) as input and outputs the pairwise distances of the data points. The recurrent layers in block (b) then subsequently cluster the data points based on this pairwise distance information [@chin2010novel; @arias2011clustering] provided by the metric learning block.
We construct a novel metric learning block inspired by the work of *Xing et al.* [@xing2003distance]. In contrast to their work, we optimize it end-to-end with backpropagation. This has been proposed in [@schwenker2001three] for classification alone; we do it here for a clustering task, for the whole covariance matrix, and jointly with the rest of our network. We construct the non-symmetric, non-negative dissimilarity measure $d^2_A$ between two data points $x_i$ and $x_j$ as $$\begin{aligned}
d^2_A(x_i, x_j) = (x_i-x_j)^{T}A(x_i-x_j)
\label{eq:dist}\end{aligned}$$ and let the neural network training optimize $A$ through $L_{\mathrm{tot}}$ without intermediate losses. The matrix $A$ as used in $d^2_A$ can be thought of as a trainable distance metric. In every training step, it is projected into the space of positive semidefinite matrices.
Experimental results {#sec:experiments}
====================
To assess the quality of our model, we perform clustering on three different datasets: for a proof of concept, we test on a set of generated *2D points* with a high variety of shapes, coming from different distributions. For speaker clustering, we use the *TIMIT* [@timit] corpus, a dataset of studio-quality speech recordings frequently used for pure speaker clustering in related work. For image clustering, we test on the *COIL-100* [@nayar1996columbia] dataset, a collection of different isolated objects in various orientations. To compare to related work, we measure the performance with the standard evaluation scores misclassification rate (MR) [@liu2003online] and normalized mutual information (NMI) [@mcdaid2011normalized]. Architecturally, we choose $m=14$ BDLSTM layers and $288$ units in the FC layer of subnetwork (b), $128$ units for the BDLSTM in subnetwork (d), and $\alpha=0.3$ for all $\mathrm{LeakyReLUs}$ in the experiments below. All hyperparameters where chosen based on preliminary experiments to achieve reasonable performance, but not tested nor tweaked extensively. The code and further material and experiments are available online[^1].
We set $k_\mathrm{max}=5$ and $\lambda=5$ for all experiments. For the 2D point data, we use $n=72$ inputs and a batch-size of $N=200$ (We used the batch size of $N=50$ for metric learning with 2D points). For TIMIT, the network input consists of $n=20$ audio snippets with a length of $1.28$ seconds, encoded as mel-spectrograms with $128\times 128$ pixels (identical to [@lukic2017speaker]). For COIL-100, we use $n=20$ inputs with a dimension of $128\times 128\times 3$. For TIMIT and , a simple CNN with 3 conv/max-pooling layers is used as subnetwork (a). For TIMIT, we use $430$ of the $630$ available speakers for training (and $100$ of the remaining ones each for validation and evaluation). For , we train on $80$ of the $100$ classes ($10$ for validation, $10$ for evaluation). For all runs, we optimize using Adadelta [@zeiler2012adadelta] with a learning rate of $5.0$. Example clusterings are shown in Fig. \[fig:output\_2d\_point\_clustering\]. For all configurations, the used hardware set the limit on parameter values: we used the maximum possible batch size and values for $n$ and $k_\mathrm{max}$ that allow reasonable training times. However, values of $n\ge 1000$ where tested and lead to a large decrease in model accuracy. This is a major issue for future work.
[0.31]{} ![Clustering results for (a) 2D point data, (b) COIL-100 objects, and (c) faces from FaceScrub (for illustrative purposes). The color of points / colored borders of images depict true cluster membership.[]{data-label="fig:output_2d_point_clustering"}](images/clusters_example_points_03_v02.png "fig:"){width="0.909\columnwidth"}
[0.31]{} ![Clustering results for (a) 2D point data, (b) COIL-100 objects, and (c) faces from FaceScrub (for illustrative purposes). The color of points / colored borders of images depict true cluster membership.[]{data-label="fig:output_2d_point_clustering"}](images/coil100_clustering.pdf "fig:"){width="0.90\columnwidth"}
[0.31]{} ![Clustering results for (a) 2D point data, (b) COIL-100 objects, and (c) faces from FaceScrub (for illustrative purposes). The color of points / colored borders of images depict true cluster membership.[]{data-label="fig:output_2d_point_clustering"}](images/facescrub_clustering.pdf "fig:"){width="0.90\columnwidth"}
--------------------------- --------- --------- --------- --------- --------- ---------
**MR** **NMI** **MR** **NMI** **MR** **NMI**
L2C ($=$our method) $0.004$ $0.993$ $0.060$ $0.928$ $0.116$ $0.867$
L2C + Euclidean $0.177$ $0.730$ $0.093$ $0.883$ $0.123$ $0.884$
L2C + Mahalanobis $0.185$ $0.725$ $0.104$ $0.882$ $0.093$ $0.890$
L2C + Metric Learning $0.165$ $0.740$ $0.101$ $0.880$ $0.100$ $0.880$
Random cluster assignment $0.485$ $0.232$ $0.435$ $0.346$ $0.435$ $0.346$
Baselines (related work)
--------------------------- --------- --------- --------- --------- --------- ---------
: $\mathrm{NMI} \in [0,1]$ and $\mathrm{MR} \in [0,1]$ averaged over $300$ evaluations of a trained network. We abbreviate our “learning to cluster” method as “L2C”.[]{data-label="tbl:results"}
The results on 2D data as presented in Fig. \[fig:output\_clustering\_1\] demonstrate that our method is able to learn specific and diverse characteristics of intuitive groupings. This is superior to any single traditional method, which only detects a certain class of cluster structure (e.g., defined by distance from a central point). Although [@lukic2017speaker] reach moderately better scores for the speaker clustering task and [@DBLP:conf/cvpr/YangPB16] reach a superior $\mathrm{NMI}$ for , our method finds reasonable clusterings, is more flexible through end-to-end training and is not tuned to a specific kind of data. Hence, we assume, backed by the additional experiments to be found online, that our model works well also for other data types and datasets, given a suitable embedding network. gives the numerical results for said datasets in the row called “L2C” without using the explicit metric learning block. Extensive preliminary experiments on other public datasets like e.g. FaceScrub [@ng2014data] confirm these results: learning to cluster reaches promising performance while not yet being on par with tailor-made state-of-the-art approaches.
We compare the performance of our implicit distance metric learning method to versions enhanced by different explicit schemes for pairwise similarity computation prior to clustering. Specifically, three implementations of the optional metric learning block in subnetwork (b) are evaluated: using a fixed diagonal matrix $A$ (resembling the Euclidean distance), training a diagonal $A$ (resembling Mahalanobis distance), and learning the entire coefficients of the distance matrix $A$. Since we argue above that our approach combines *implicit* deep metric embedding with clustering in an end-to-end architecture, one would not expect that adding *explicit* metric computation changes the results by a large extend. This assumption is largely confirmed by the results in the “L2C$+$…” rows in : for COIL-100, Euclidean gives slightly worse, and the other two slightly better results than L2C alone; for TIMIT, all results are worse but still reasonable. We attribute the considerable performance drop on 2D points using all three explicit schemes to the fact that in this case much more instances are to be compared with each other (as each instance is smaller than e.g. an image, $n$ is larger). This might have needed further adaptations like e.g. larger batch sizes (reduced here to $N=50$ for computational reasons) and longer training times.
Discussion and conclusions {#sec:conclusions}
==========================
We have presented a novel approach to learn neural models that directly output a probabilistic clustering on previously unseen groups of data; this includes a solution to the problem of outputting similar but unspecific “labels” for similar objects of unseen “classes”. A trained model is able to cluster different data types with promising results. This is a complete end-to-end approach to clustering that learns both the relevant features and the “algorithm” by which to produce the clustering itself. It outputs probabilities for cluster membership of all inputs as well as the number of clusters in test data. The learning phase only requires pairwise labels between examples from a separate training set, and no explicit similarity measure needs to be provided. This is especially useful for high-dimensional, perceptual data like images and audio, where similarity is usually semantically defined by humans. Our experiments confirm that our algorithm is able to implicitly learn a metric and directly use it for the included clustering. This is similar in spirit to the very recent work of *Hsu et al.* [@hsu2018learning], but does not need and optimization on the test (clustering) set and finds $k$ autonomously. It is a novel approach to *learn to cluster*, introducing a novel architecture and loss design.
We observe that the clustering accuracy depends on the availability of a large number of different classes during training. We attribute this to the fact that the network needs to learn intra-class distances, a task inherently more difficult than just to distinguish between objects of a fixed amount of classes like in classification problems. We understand the presented work as an early investigation into the new paradigm of learning to cluster by perceptual similarity specified through examples. It is inspired by our work on speaker clustering with deep neural networks, where we increasingly observe the need to go beyond surrogate tasks for learning, training end-to-end specifically for clustering to close a performance leak. While this works satisfactory for initial results, points for improvement revolve around scaling the approach to practical applicability, which foremost means to get rid of the dependency on $n$ for the partition size.
The number $n$ of input examples to assess simultaneously is very relevant in practice: if an input data set has thousands of examples, incoherent single clusterings of subsets of $n$ points would be required to be merged to produce a clustering of the whole dataset based on our model. As the (RBD)LSTM layers responsible for assessing points simultaneously in principle have a long, but still local (short-term) horizon, they are not apt to grasp similarities of thousands of objects. Several ideas exist to change the architecture, including to replace recurrent layers with temporal convolutions, or using our approach to seed some sort of differentiable K-means or EM layer on top of it. Preliminary results on this exist. Increasing $n$ is a prerequisite to also increase the maximum number of clusters $k$, as $k\ll n$. For practical applicability, $k$ needs to be increased by an order of magnitude; we plan to do this in the future. This might open up novel applications of our model in the area of transfer learning and domain adaptation.
#### **Acknowledgements:** We thank the anonymous reviewers for helpful feedback.
[^1]: See <https://github.com/kutoga/learning2cluster>.
|
[|l||r|r||r|r||r|r||r|r||r|r||r|r||r|r|]{} &\
& & & & & & &\
& & & & & & &\
& \#s. & time & \#s. & time & \#s. & time & \#s. & time & \#s. & time & \#s. & time & \#s. & time\
\
& 100 & 386 & 100 & 1854 & 97 & 9396 & 57 & 14051 & 100 & 9637 & 99 & 10670 & 553 & 45995\
& 100 & 317 & 100 & 1584 & 100 & 8100 & 77 & 18046 & 100 &7738 & 100 & 7433 & &\
& 100 & 726 & 100 & 3817 & 88 & 13222 & 38 & 12529 & 92 & 14183 & 90 & 13287 & 508 & 57764\
& 100 & 602 & 100 & 3270 & 97 & 12878 & 54 & 16234 & 96 & 13159 & 96 & 12350 & 543 & 58493\
& 100 & 596 & 100 & 3230 & 97 & 12262 & 53 & 14810 & 96 & 12805 & 96 & 12125 & 542 & 55828\
& 100 & 268 & 100 & 1113 & 100 & 4734 & 87 & 17067 & 100 & 4941 & 100 & 6122 & &\
& 84 & 23830 & 4 & 1596 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 363 & 89 & 25789\
& 100 & 267 & 100 & 1114 & 100 & 4718 & 87 & 17108 & 100 & 4962 & 100 & 6174 & &\
& 84 & 23871 & 4 & 1622 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 338 & 89 & 25831\
\
& 100 & 324 & 100 & 1571 & 100 & 7739 & 74 & 16494 & 100 & 7175 & 100 & 7504 & 574 & 40807\
\
& 100 & 336 & 100 & 1578 & 100 & 7762 & 71 & 16589 & 100 & 7726 & 100 & 7706 & 571 & 41697\
\
& 100 & 320 & 100 & 1533 & 100 & 7623 & 68 & 15120 & 100 & 7216 & 100 & 7598 & 568 & 39410\
\
& 100 & 239 & 100 & 1128 & 100 & 5516 & 84 & 19949 & 100 & 6667 & 100 & 4176 & &\
& 100 & 14527 & 46 & 17887 & 0 & 0 & 0 & 0 & 1 & 497 & 0 & 0 & 147 & 32911\
& 100 & 240 & 100 & 1122 & 100 & 5510 & 83 & 19489 & 100 & 6684 & 100 & 4180 & 583 & 37225\
& 100 & 14465 & 47 & 18206 & 0 & 0 & 0 & 0 & 1 & 495 & 0 & 0 & 148 & 33166\
\
& 100 & 319 & 100 & 1865 & 100 & 12470 & 45 & 15704 & 97 & 13189 & 96 & 15773 & &\
& 95 & 22435 & 18 & 8030 & 2 & 671 & 0 & 0 & 1 & 526 & 3 & 1043 & 119 & 32723\
& 100 & 319 & 100 & 1871 & 100 & 12440 & 45 & 15747 & 98 & 13661 & 95 & 15102 & &\
& 95 & 22401 & 18 & 7991 & 1 & 163 & 0 & 0 & 1 & 437 & 3 & 1020 & 118 & 32012\
|
---
author:
- 'Valeriy G. Bardakov'
title: Braid groups in handlebodies and corresponding Hecke algebras
---
Introduction
============
Let $\mathcal{H}_g$ be a handlebody of genus $g$. The braid group $B_{g,n}$ on $n$ stings in the handlebody $\mathcal{H}_g$ was introduced by A. B. Sossinsky [@S] and independently S. Lambropoulou [@L1]. Properties of this group are studied by V. V. Vershinin [@V; @V1] and by S. Lambropolou [@L1; @L2].
The motivation for studying braids from $B_{g,n}$ comes from studying oriented knots and links in knot complements in compact connected oriented 3-manifolds and in handlebodies, since these spaces may be represented by a fixed braid or a fixed integer-framed braid in $\mathbb{S}^3$ [@LR1; @L1; @HL; @LR2]. Then knots and links in these spaces may be represented by elements of the braid group $B_{g,n}$ or an appropriate cosets of these group [@L1]. In particular, if $M$ denotes the complements of the $g$-unlink or a connected sum of $g$ lens spaces of type $L(p, 1)$ or a handlebody of genus $g$, then knots and links in these spaces may be represented precisely by the braids in $B_{g,n}$ for $n \in \mathbb{N}$. In the case $g=1$, $B_{1,n}$ is the Artin group of type $\mathcal{B}$.
The group $B_{g,n}$ can be considered as a subgroup of the braid group $B_{g+n}$ on $g+n$ strings such that the braids in $B_{g,n}$ leave the first $g$ strings identically fixed. Using this fact V. V. Vershinin [@V] and S. Lambropoulou [@L1] defined a epimorphism of $B_{g,n}$ onto the symmetric group $S_n$ and prove that the kernel of this epimorphism: $P_{g,n}$ is a subgroup of the pure braid group $P_{g+n}$ and is semi-direct products of free groups. On the other side, V. V. Vershinin [@V] noted that there is a decomposition $B_{g,n} = R_{g,n} \leftthreetimes B_n$ for some group $R_{g,n}$ and the braid group $B_n$. S. Lambropoulou proved in [@L2] that $R_{1,n}$ is isomorphic to a free group $F_n$ of rank $n$.
The Hecke algebra of type $\mathrm{A}$, $H_n(q)$ was used by V. F. R. Jones [@J] for the construction of polynomial invariant for classical links, the well known HOMFLYPT polynomial. S. Lambropoulou [@L2] defined a generalization of the Hecke algebra of type $\mathrm{B}$ to construct a HOMFLYPT-type polynomial invariant for links in the solid torus, which was then used in [@DiLaPr] for extending the study to the lens spaces $L(p,1)$.
In the present paper we define some decomposition of $P_{g,n}$ as semi-direct product of free groups, which is different from the decomposition defined in [@L1; @V]. Also, we study the group $R_{g,n}$ and prove that this group is semi-direct products of free groups. Using this decomposition and the decomposition $B_{g,n} = R_{g,n} \leftthreetimes B_n$ we define some algebra, which is a generalization of the Hecke algebra $H_n(q)$.
The paper is organized as follows. In Section 2, we remind some facts on the braid group $B_n$. In particular, we define vertical and horizontal decompositions of the pure braid group $P_n$. In Section 3, we recall some facts on the group $B_{g,n}$, we describe the vertical decomposition of $P_{g,n}$ and we define the horizontal decomposition of $P_{g,n}$. In Section 4, we study $R_{g,n}$ and we construct vertical and horizontal decompositions for this group. In Section 5 we introduce some algebra $H_{g,n}(q)$ which is a generalization of the Hecke algebra $H_{n}(q)$ and we find the quotient of $B_{g,n}$ by the relations $\sigma_i^2 = 1$.
Acknowledgements. {#acknowledgements. .unnumbered}
------------------
The author gratefully acknowledges Prof. Lambropoulou and her students and colleagues: Neslihan Gügümcü, Stathis Antoniou, Dimos Goundaroulis, Ioannis Diamantis, Dimitrios Kodokostas for the kind invitation to the Athens, where this paper was written, for conversation and interesting discussions.
This work was supported by the Russian Foundation for Basic Research (project 16-01-00414).
Braid group
===========
In this section we recall some facts on the braid groups (see [@Bir; @M]).
The braid group $B_m$, $m \geq 2$, on $m$ strings is generated by elements $$\sigma_1, \sigma_2, \ldots, \sigma_{m-1},$$ and is defined by relations $$\begin{array}{ll}
\sigma_i \sigma_j = \sigma_j \sigma_i, & ~\mbox{for}~|i - j| >1, \\
\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}, & ~\mbox{for}~i = 1, 2, \ldots, m-2.
\end{array}$$ A subgroup of $B_m$ which is generated by elements $$a_{ij} = \sigma_{j-1} \sigma_{j-2} \ldots \sigma_{i+1} \sigma_{i}^2 \sigma_{i+1}^{-1} \ldots \sigma_{j-2}^{-1} \sigma_{j-1}^{-1},~~~1 \leq i < j \leq m,$$ is called the [*pure braid group*]{} and is denoted $P_m$. This group is defined by the relations $$\begin{aligned}
& a_{ik} a_{ij} a_{kj} = a_{kj} a_{ik} a_{ij}, \label{re2}\\
& a_{nj} a_{kn} a_{kj} = a_{kj} a_{nj} a_{kn}, ~\mbox{for}~n < j, \label{re3}\\
& (a_{kn} a_{kj} a_{kn}^{-1}) a_{in} = a_{in} (a_{kn} a_{kj} a_{kn}^{-1}), ~\mbox{for}~i < k < n < j, \label{re4}\\
& a_{kj} a_{in} = a_{in} a_{kj}, ~\mbox{for}~k < i < n < j ~\mbox{or}~n < k. \label{re1}\end{aligned}$$ The subgroup $P_m$ is normal in $B_m$, and the quotient $B_m / P_m$ is the symmetric group $S_m$. The generators of $B_m$ act on the generator $a_{ij} \in P_m$ by the rules: $$\begin{aligned}
& \sigma_k^{-1} a_{ij} \sigma_k = a_{ij}, ~\mbox{for}~k \not= i-1, i, j-1, j, \label{c1}\\
& \sigma_{i}^{-1} a_{i,i+1} \sigma_{i} = a_{i,i+1}, \label{c2}\\
& \sigma_{i-1}^{-1} a_{ij} \sigma_{i-1} = a_{i-1,j}, \label{c3}\\
& \sigma_{i}^{-1} a_{ij} \sigma_{i} = a_{i+1,j} [a_{i,i+1}^{-1}, a_{ij}^{-1}], ~\mbox{for}~j \not= i+1 \label{c4}\\
& \sigma_{j-1}^{-1} a_{ij} \sigma_{j-1} = a_{i,j-1}, \label{c5}\\
& \sigma_{j}^{-1} a_{ij} \sigma_{j} = a_{ij} a_{i,j+1} a_{ij}^{-1}, \label{c6}\end{aligned}$$ where $[a, b] = a^{-1} b^{-1} a b$.
Denote by $$U_{i} = \langle a_{1i}, a_{2i}, \ldots, a_{i-1,i} \rangle,~~~i = 2, \ldots, m,$$ a subgroup of $P_m$. It is known that $U_i$ is a free group of rank $i-1$. One can rewrite the relations of $P_m$ as the following conjugation rules (for $\varepsilon = \pm 1$): $$\begin{aligned}
& a_{ik}^{-\varepsilon} a_{kj} a_{ik}^{\varepsilon} = (a_{ij} a_{kj})^{\varepsilon} a_{kj} (a_{ij} a_{kj})^{-\varepsilon}, \label{co1}\\
& a_{kn}^{-\varepsilon} a_{kj} a_{kn}^{\varepsilon} = (a_{kj} a_{nj})^{\varepsilon} a_{kj} (a_{kj} a_{nj})^{-\varepsilon}, ~\mbox{for}~n < j, \label{co2}\\
& a_{in}^{-\varepsilon} a_{kj} a_{in}^{\varepsilon} = [a_{ij}^{-\varepsilon}, a_{nj}^{-\varepsilon}]^{\varepsilon} a_{kj} [a_{ij}^{-\varepsilon}, a_{nj}^{-\varepsilon}]^{-\varepsilon}, ~\mbox{for}~i < k < n, \label{co3}\\
& a_{in}^{-\varepsilon} a_{kj} a_{in}^{\varepsilon} = a_{kj}, ~\mbox{for}~k < i < n < j ~\mbox{or}~ n < k. \label{co4}\end{aligned}$$
From these rules it follows that $U_m$ is normal in $P_m$ and hence $P_m$ has the following decomposition: $P_m = U_m \leftthreetimes P_{m-1}$, where the action of $P_{m-1}$ on $U_m$ is define by the rules (\[co1\]-\[co4\]). By induction on $m$, $P_m$ is the semi-direct product of free groups: $$P_m =U_m \leftthreetimes (U_{m-1} \leftthreetimes (\ldots \leftthreetimes (U_3 \leftthreetimes U_2)\ldots )).$$ We will call this decomposition [*vertical decomposition*]{}.
Let $U_m^{(k)}$, $k = 1, 2, \ldots, m$, be the subgroup of $P_m$ which is generated by $a_{ij}$, where $k < j$. Then $U_m^{(k)} = U_m \leftthreetimes (U_{m-1} \leftthreetimes (\ldots \leftthreetimes (U_{k+2} \leftthreetimes U_{k+1})\ldots ))$. By definition $U_m^{(1)} = P_m$ and these groups form the normal series $$1 = U_m^{(m)} \leq U_m^{(m-1)} \leq \ldots \leq U_m^{(2)} \leq U_m^{(1)} = P_m,$$ where $$U_m^{(r)} / U_m^{(r+1)} \cong F_{r}, ~~~r = 1, 2, \ldots, m-1.$$
On the other side, define the following subgroups of $P_m$: $$V_k = \langle a_{k,k+1}, a_{k,k+2}, \ldots, a_{k,m} \rangle,~~k = 1, 2, \ldots, m-1.$$ This group is free of rank $m-k$. Using the defining relation of $P_m$, it is not difficult to prove the following:
\[l1\] In $P_n$ hold the following conjugation rules (for $\varepsilon = \pm 1$):
1\) $a_{kj}^{-\varepsilon} a_{ik} a_{kj}^{\varepsilon} = (a_{ik} a_{ij})^{\varepsilon} a_{ik} (a_{ik} a_{ij})^{-\varepsilon},$ where $i < k < j$;
2\) $a_{jk}^{-\varepsilon} a_{ik} a_{jk}^{\varepsilon} = (a_{ij} a_{ik})^{\varepsilon} a_{ik} (a_{ij} a_{ik})^{-\varepsilon},$ where $i < j < k$;
3\) $a_{kn}^{-\varepsilon} a_{ij} a_{kn}^{\varepsilon} = [a_{ik}^{-\varepsilon}, a_{in}^{-\varepsilon}]^{\varepsilon} a_{ij} [a_{ik}^{-\varepsilon}, a_{in}^{-\varepsilon}]^{-\varepsilon},$ where $i < k < j < n$;
4\) $a_{in}^{-\varepsilon} a_{kj} a_{in}^{\varepsilon} = a_{kj},$ where $k < i$ and $n < j$;
5\) $a_{kj}^{-\varepsilon} a_{in} a_{kj}^{\varepsilon} = a_{in},$ where $n < k$.
From this lemma it follows that $V_1$ is normal in $P_m$ and we have decomposition $P_m = V_1 \leftthreetimes P_{m-1}$. By induction on $m$, $P_m$ is the semi-direct products of free groups: $$P_m =V_1 \leftthreetimes (V_{2} \leftthreetimes (\ldots \leftthreetimes (V_{m-2} \leftthreetimes V_{m-1})\ldots )).$$ We will call this decomposition [*horizontal decomposition*]{}. Let $V_m^{(k)}$ be a subgroup of $P_m$ which is generated by $a_{ij}$ for $i < k$. Then we have the normal series $$1 = V_m^{(1)} \leq V_m^{(2)} \leq \ldots \leq V_m^{(m-1)} \leq V_m^{(m)} = P_m,$$ where $$V_m^{(r)} / V_m^{(r-1)} \cong F_{m-r+1}, ~~~r = 2, 3, \ldots, m.$$
A motivation for the terms vertical and horizontal is as follows. If we put the generators of $P_m$ in the following table $$\begin{array}{ccccc}
a_{12}, & a_{13}, & \ldots & a_{1,m-1}, & a_{1,m}, \\
& a_{23}, & \ldots & a_{2,m-1}, & a_{2,m}, \\
\ldots & \ldots & \ldots & \ldots & \ldots \\
& & & a_{m-2,m-1}, & a_{m-2,m}, \\
& & & & a_{m-1,m},
\end{array}$$ then the generators from the $k$th row generate $V_k$ and the generators from the $k$th column generate $U_{k+1}$. The group $U_m^{(r)}$ is generated by the last $m-r$ columns of this table and the group $V_m^{(r)}$ is generated by the first $r-1$ rows of this table.
Braid groups in handlebodies
============================
Recall some facts on the braid group $B_{g,n}$ on $n$ strings in the handlebody $\mathcal{H}_g$ (see [@L1; @S; @V]). The group $B_{g,n}$ is generated by elements $$\tau_1, \tau_2, \ldots \tau_g, \sigma_{g+1}, \sigma_{g+2}, \ldots, \sigma_{g+n-1},$$ and is defined by the following list of relations $$\begin{array}{ll}
\sigma_i \sigma_j = \sigma_j \sigma_i, & ~\mbox{for}~|i - j| >1, \\
\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}, & ~\mbox{for}~i = g+1, \ldots, g+n-2, \\
\tau_k \sigma_i = \sigma_i \tau_k, & ~\mbox{for}~k \geq 1, ~~i \geq g+2, \\
\tau_k (\sigma_{g+1} \tau_k \sigma_{g+1}) = (\sigma_{g+1} \tau_k \sigma_{g+1}) \tau_{k}, & ~\mbox{for}~k = 1, \ldots, g, \\
\tau_k (\sigma_{g+1}^{-1} \tau_{k+l} \sigma_{g+1}) = (\sigma_{g+1}^{-1} \tau_{k+l} \sigma_{g+1}) \tau_{k}, & ~\mbox{for}~k = 1, \ldots, g-1, ~~l = 1, \ldots, g-k.
\end{array}$$
The group $B_{g,n}$ can be considered as a subgroup of the classical braid group $B_{g+n}$ on $n+g$ strings such that the braids from $B_{g,n}$ leave the first $g$ strings unbraided. Then $\tau_k = a_{k,g+1}$, in $B_{g+n}$, i. e. $$\tau_k = \sigma_g \sigma_{g-1} \ldots \sigma_{k+1} \sigma_{k}^2 \sigma_{k+1}^{-1} \ldots \sigma_{g-1}^{-1} \sigma_{g}^{-1},~~~k = 1, 2, \ldots, g.$$ The elements $\tau_k$, $k = 1, 2, \ldots, g$, generate a free group of rank $g$ which is isomorphic to $U_{g+1} = \langle a_{1,g+1}, a_{2,g+1}, \ldots, a_{g,g+1} \rangle$ in $B_{g+n}$. Also, we see that some other generators of $P_{g+n}$ lie in $B_{g,n}$. Put them in the table bellow: $$\begin{array}{ccccc}
a_{1,g+1}, & a_{1,g+2}, & \ldots & a_{1,g+n-1}, & a_{1,g+n}, \\
a_{2,g+1}, & a_{2,g+2}, & \ldots & a_{2,g+n-1}, & a_{2,g+n}, \\
\ldots & \ldots & \ldots & \ldots & \ldots \\
a_{g-1,g+1}, & a_{g-1,g+2}, & \ldots & a_{g-1,g+n-1}, & a_{g-1,g+n}, \\
a_{g,g+1}, & a_{g,g+2}, & \ldots & a_{g,g+n-1}, & a_{g,g+n}, \\
\hline
& a_{g+1,g+2}, & \ldots & a_{g+1,g+n-1}, & a_{g+1,g+n}, \\
& \ldots & \ldots & \ldots & \ldots \\
& & & a_{g+n-2,g+n-1}, & a_{g+n-2,g+n}, \\
& & & & a_{g+n-1,g+n}. \\
\end{array}$$
We will denote by $\widetilde{B}_n$ the subgroup of $B_{g,n}$ which is generated by $\sigma_{g+1},$ $\sigma_{g+2},$ $\ldots, \sigma_{g+n-1}$. It is evident that $\widetilde{B}_n$ is isomorphic to $B_n$. The corresponding pure braid group of $\widetilde{B}_n$ will be denote $\widetilde{P}_n$. This group is isomorphic to $P_n$ and is generated by elements from the above table which lie under the horizontal line. The group $\widetilde{P}_n$ has the following vertical decomposition $$\widetilde{P}_n =\widetilde{U}_n \leftthreetimes (\widetilde{U}_{n-1} \leftthreetimes (\ldots \leftthreetimes (\widetilde{U}_3 \leftthreetimes \widetilde{U}_2)\ldots )),$$ where $$\widetilde{U}_i = \langle a_{g+1,g+i}, a_{g+2,g+i}, \ldots, a_{g+i-1,g+i}\rangle,~~~i = 2, 3, \ldots,n.$$
There is an epimorphism: $$\psi_n : B_{g,n} \longrightarrow S_n,$$ which is defined by the rule $$\psi_n(\tau_k) = 1,~k = 1, 2, \ldots, g,~~~\psi_n(\sigma_i) = (i, i+1),~i = g+1, g+2, \ldots, g+n-1.$$ This epimorphism is induced by the standard epimorphism $B_{g+n} \longrightarrow S_{g+n}$. Let $P_{g,n} = \mathrm{Ker} (\psi_n)$. Then $P_{g,n}$ is generated by the element from the table above. In [@L1; @V] it was proved that there exists the following short exact sequence: $$1 \longrightarrow P_{g,n} \longrightarrow B_{g,n} \longrightarrow S_n \longrightarrow 1,$$ and was found the vertical decomposition of $P_{g,n}$: $$P_{g,n} = U_{g+n} \leftthreetimes (U_{g+n-1} \leftthreetimes ( \ldots \leftthreetimes (U_{g+2} \leftthreetimes U_{g+1})\ldots )).$$ If we let $U_{g+n}^{(g+n-i)} = U_{g+n} \leftthreetimes (U_{g+n-1} \leftthreetimes ( \ldots \leftthreetimes (U_{g+n-i+2} \leftthreetimes U_{g+n-i+1})\ldots ))$, then we get the normal series: $$1 = U_{g+n}^{(g+n)} \leq U_{g+n}^{(g+n-1)} \leq \ldots \leq U_{g+n}^{(g)} = P_{g,n},$$ where $$U_{g+n}^{(r)} / U_{g+n}^{(r+1)} \cong F_r, ~~~r = g, g+1, \ldots, g+n-1.$$ The homomorphism which is induced by the embedding $B_{g,n} \longrightarrow B_{g+n}$ sends this normal series to the corresponding normal series for $P_{g+n}$.
Let us construct the horizontal decomposition for $P_{g,n}$. To do this, define the following subgroups in $P_{g,n}$: $$\begin{array}{l}
V_{g,1} = \langle a_{1,g+1}, a_{1,g+2}, \ldots, a_{1,g+n} \rangle, \\
V_{g,2} = \langle a_{2,g+1}, a_{2,g+2}, \ldots, a_{2,g+n} \rangle, \\
.......................................................\\
V_{g,g} = \langle a_{g,g+1}, a_{g,g+2}, \ldots, a_{g,g+n} \rangle, \\
V_{g,g+1} = \langle a_{g+1,g+2}, a_{g+1,g+3}, \ldots, a_{g+1,g+n} \rangle, \\
V_{g,g+2} = \langle a_{g+2,g+3}, a_{g+2,g+4}, \ldots, a_{g+2,g+n} \rangle, \\
.......................................................\\
V_{g,g+n-1} = \langle a_{g+n-1,g+n} \rangle. \\
\end{array}$$ We see that $V_{g,g+i} = V_{g+i}$ for all $i = 1, 2, \ldots, n-1$, these subgroups lie in $\widetilde{P}_n$ and as we know $$\widetilde{P}_n = V_{g,g+1} \leftthreetimes (V_{g,g+2} \leftthreetimes ( \ldots \leftthreetimes (V_{g,g+n-2} \leftthreetimes V_{g,g+n-1}) \ldots ))$$ is the horizontal decomposition of $\widetilde{P}_n$.
We see that the vertical decomposition of $P_{g,n}$ is a part of the vertical decomposition for $P_{g+n}$. For the horizontal decomposition situation is more complicated.
The horizontal decomposition of $P_4$ has the form $$P_4 = V_1 \leftthreetimes (V_2 \leftthreetimes V_3),$$ where $$V_1 = \langle a_{12}, a_{13}, a_{14} \rangle,~~V_2 = \langle a_{23}, a_{24} \rangle,~~V_3 = \langle a_{34} \rangle,$$ and $V_1$ is normal in $P_4$, $V_2$ is normal in $V_2 \leftthreetimes V_3$. The group $P_{2,2}$ contains subgroups $$V_{2,1} = \langle a_{13}, a_{14} \rangle,~~V_{2,2} = V_2 = \langle a_{23}, a_{24} \rangle,~~V_{2,3} = V_3 = \langle a_{34} \rangle,$$ but in this case $V_{2,1}$ is not normal in $P_{2,2}$, Indeed, from Lemma \[l1\] we have the following relations $$a_{23}^{-\varepsilon} a_{13} a_{23}^{\varepsilon} = (a_{12} a_{13})^{\varepsilon} a_{13} (a_{12} a_{13})^{-\varepsilon},~~
a_{24}^{-\varepsilon} a_{14} a_{24}^{\varepsilon} = (a_{12} a_{14})^{\varepsilon} a_{14} (a_{12} a_{14})^{-\varepsilon}.$$ Hence, $P_{2,2}$ contains not only $V_{2,1}$ but its normal closure in $P_{2,2}$ and we get the horizontal decomposition $$P_{2,2} = \overline{V}_{2,1} \leftthreetimes (V_2 \leftthreetimes V_3),$$ where $\overline{V}_{2,1} = \langle V_{2,1} \rangle^{P_{2,2}}$ is the normal closure of $V_{2,1}$ in $P_{2,2}$.
In the general case, let $\overline{V}_{g,i}$ be the normal closure of $V_{g,i}$ in the subgroup $\langle V_{g,i}, V_{g,i+1}, \ldots, V_{g,g}, \widetilde{P}_{n} \rangle$, i.e. $$\overline{V}_{g,i} = \langle V_{g,i} \rangle^{\langle V_{g,i}, V_{g,i+1}, \ldots, V_{g,g}, \widetilde{P}_{n} \rangle}.$$
\[l2\] 1) $\overline{V}_{g,g} = V_{g,g} \cong F_n.$
2\) $\overline{V}_{g,i}$ is a subgroup of $V_i$ for every $i = 1, 2, \ldots, g$ and, in particular, is a free group.
1\) We see that $V_{g,g} = V_g$ and from Lemma \[l1\] it follows that $V_g$ is normal in $\langle V_{g,g}, \widetilde{P}_{n} \rangle$.
2\) The fact that $\overline{V}_{g,i}$ is a subgroup of $V_i$ follows from the conjugation rules of Lemma \[l1\]. The fact that $\overline{V}_{g,i}$ is free follows from the fact that $V_i$ is free.
Since $\widetilde{P}_n \cong P_n$, it has the horizontal decomposition: $$\widetilde{P}_n = V_{g+1} \leftthreetimes (V_{g+2} \leftthreetimes ( \ldots \leftthreetimes (V_{g+n-2} \leftthreetimes V_{g+n-1}) \ldots )).$$
Using Lemma \[l2\], one can construct the horizontal decomposition of $P_{g,n}$.
The group $P_{g,n}$ is the semi-direct products of groups: $$P_{g,n} = \overline{V}_{g,1} \leftthreetimes (\overline{V}_{g,2} \leftthreetimes ( \ldots \leftthreetimes (\overline{V}_{g,g} \leftthreetimes \widetilde{P}_n) \ldots )).$$
Use induction on $g$. If $g=1$, then $P_{1,n} = P_{n+1}$ and the horizontal decomposition for $P_{n+1}$ gives the horizontal decomposition: $P_{1,n} = V_{g} \leftthreetimes \widetilde{P}_n$. As follows from Lemma \[l2\], $V_g = \overline{V}_{g,g}$.
Let $g > 1$. Define a homomorphism of $P_{g,n}$ onto the group $P_{g-1,n}$ which sends all generators of $V_{g,1}$ to the unit and keeps all other generators. The kernel of this homomorphism is the normal closure of $V_{g,1}$ in $P_{g,n}$. Denote this kernel by $\overline{V}_{g,1} = \langle V_{g,1} \rangle^{P_{g,n}}$. Since, $V_{g,1}$ is a subgroup in $V_1$ and $V_1$ is normal in $P_{g+n}$ we get that $\overline{V}_{g,1}$ lies in $V_1$ and hence is a free. We have the decomposition $P_{g,n} = \overline{V}_{g,1} \leftthreetimes P_{g-1,n}$. Using the induction hypothesis we get the required decomposition.
The kernel of the epimorphism $B_{g,n} \to \widetilde{B}_n$
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As was noted in [@V] there is an epimorphism $$\varphi_n : B_{g,n} \longrightarrow \widetilde{B}_n,$$ where the subgroup $\widetilde{B}_n = \langle \sigma_{g+1}, \sigma_{g+2}, \ldots, \sigma_{g+n-1} \rangle$ is isomorphic to $B_n$. This endomorphism is defined by the rule $$\varphi_n(\tau_k) = 1,~k = 1, 2, \ldots, g,~~~\varphi_n(\sigma_i) = \sigma_i,~i = g+1, g+2, \ldots, g+n-1.$$ If we denote $R_{g,n} = \mathrm{Ker}(\varphi_n)$, then $B_{g,n} = R_{g,n} \leftthreetimes \widetilde{B}_n$. The purpose of this section is the description of the group $R_{g,n}$.
Considering the table with the generators of $P_{g,n}$ (see Section 3), we see that all generators, which lie in the fist $g$ rows of this table, are elements of $R_{g,n}$. Denote by $Q_{g,n}$ the subgroup of $R_{g,n}$ that is generated by these elements, i.e. $Q_{g,n} = \langle V_{g,1}, V_{g,2}, \ldots, V_{g,g} \rangle$. Then $R_{g,n} = \langle Q_{g,n} \rangle^{B_{g,n}}$ is the normal closure of $Q_{g,n}$ in $B_{g,n}$.
On the other side, if we denote $$U_{g,g+i} = \langle a_{1,g+i}, a_{2,g+i}, \ldots, a_{g,g+i} \rangle \leq U_{g+i},~~~i = 1, 2, \ldots, n,$$ then $Q_{g,n} = \langle U_{g,g+1}, U_{g,g+2}, \ldots, U_{g,g+n} \rangle$. Note that $U_{g,g+1} = U_{g+1}$. Let $\overline{U}_{g,g+i}$ be the normal closure of $U_{g,g+i}$ in $U_{g+i}$, $i = 1, 2, \ldots, n$. In this notations it holds:
The group $R_{g,n}$ has the following decompositions:
1) $R_{g,n} = \overline{U}_{g,g+n} \leftthreetimes (\overline{U}_{g,g+n-1} \leftthreetimes ( \ldots \leftthreetimes (\overline{U}_{g,g+2} \leftthreetimes \overline{U}_{g,g+1})\ldots )),$\
where $\overline{U}_{g,g+1} = U_{g+1}$.
2) $R_{g,n} = \overline{V}_{g,1} \leftthreetimes (\overline{V}_{g,2} \leftthreetimes ( \ldots \leftthreetimes (\overline{V}_{g,g-1} \leftthreetimes \overline{V}_{g,g})\ldots )),$\
where $\overline{V}_{g,g} = V_g$.
As we know, $R_{g,n}$ is the normal closure of $Q_{g,n}$ in $B_{g,n}$. To find this closure, at first, consider conjugations of the generators of $Q_{g,n}$ by $\sigma_{g+k}$, $k = 1, 2, \ldots, n-1$.
Conjugating generators from the $k$th column of our table by $\sigma_{g+k}$, we have $$\sigma_{g+k}^{-1} a_{i,g+k} \sigma_{g+k} = a_{i,g+k} a_{i,g+k+1} a_{i,g+k}^{-1},~~i = 1, 2, \ldots, g.$$
Conjugating generators from the $(k+1)$st column of our table by $\sigma_{g+k}$, we have $$\sigma_{g+k}^{-1} a_{i,g+k+1} \sigma_{g+k} = a_{i,g+k},~~i = 1, 2, \ldots, g.$$
Generators from all other columns commute with $\sigma_{g+k}$.
Hence, for every generator $a_{ij}$ of $Q_{g,n}$ and every $\sigma_k$, $k = g+1, g+2, \ldots, g+n-1$, the element $\sigma_k^{-1} a_{ij} \sigma_k$ lies in $Q_{g,n}$.
For the group $\widetilde{B}_n$ there exists the following exact sequence $$1 \longrightarrow \widetilde{P}_n \longrightarrow \widetilde{B}_n \longrightarrow S_n \longrightarrow 1.$$ Let $m_{kl} = \sigma_{k-1} \, \sigma_{k-2} \ldots \sigma_l$ for $l < k$ and $m_{kl} = 1$ in other cases. Then the set $$\Lambda_n = \left\{ \prod\limits_{k=g+2}^{g+n} m_{k,j_k}~ \vert ~1 \leq j_k
\leq k \right\}$$ is a Schreier set of coset representatives of $\widetilde{P}_n$ in $\widetilde{B}_n$.
From the previous observations it follows that if $\alpha \in \Lambda$, then for every generator $a_{ij}$ of $Q_{g,n}$ the element $\alpha^{-1} a_{ij} \alpha$ lies in $Q_{g,n}$.
Now we will consider the conjugations of the generators of $Q_{g,n}$ by the generators of $P_{g,n}$.
1\) To prove the first decomposition, take some element $h \in B_{g,n}$ and using the vertical decomposition, write it in the normal form: $$h = u_{g+1} u_{g+2} \ldots u_{g+n} \alpha,~\mbox{where}~u_k \in U_k,~~\alpha \in \Lambda_n.$$ In every group $U_k$, $k = g+1, g+2, \ldots, g+n$, we have two subgroups: $$U_{g,k} = \langle a_{1k}, a_{2k}, \ldots, a_{gk} \rangle,~~\widetilde{U}_{k} = \langle a_{g+1,k}, a_{g+2,k}, \ldots, a_{k-1,k} \rangle \leq \widetilde{P}_n.$$ We see that $U_{g,k}$ is a subgroup of $P_{g,n}$, $\widetilde{U}_{k}$ is a subgroup of $\widetilde{P}_{n}$ and $U_k = \langle U_{g,k}, \widetilde{U}_{k} \rangle$ is a free group. Define the projection $\pi_k : U_k \longrightarrow \widetilde{U}_k$ by the rules $$\pi_k(a_{ik}) = 1,~~i = 1, 2, \ldots, g;$$ $$\pi_k(a_{jk}) = a_{jk},~~j = g+1, g+2, \ldots, k-1.$$ The kernel $\mathrm{Ker}(\pi_k)=\overline{U}_{g,k}$ is the normal closure $\langle U_{g,k} \rangle^{U_k}$ of $U_{g,k}$ into $U_k$. Hence, element $u_k$ can be written in the form $u_k = \overline{u}_k \pi_k(u_k)$ for some $\overline{u}_k \in \overline{U}_{g,k}$. Denote for simplicity $w_{g+i} = \pi_{g+1}(u_{g+i})$, we can rewrite $h$ in the form $$h = (\overline{u}_{g+1} w_{g+1}) (\overline{u}_{g+2} w_{g+2}) \ldots (\overline{u}_{g+n} w_{g+n}) \alpha.$$ Shifting all $w_{g+i}$ to the right we get $$h = \overline{u}_{g+1} \overline{u}_{g+2}^{w_{g+1}^{-1}} \ldots \overline{u}_{g+n}^{w_{g+n-1}^{-1} w_{g+n-2}^{-1} \ldots w_{g+1}^{-1}} w_{g+1} w_{g+2} \ldots w_{g+n} \alpha.$$
The image $\varphi_n(h) = w_{g+1} w_{g+2} \ldots w_{g+n} \alpha$. Hence, the element $$\overline{u}_{g+1} \, \overline{u}_{g+2}^{w_{g+1}^{-1}} \, \ldots \, \overline{u}_{g+n}^{w_{g+n-1}^{-1} w_{g+n-2}^{-1} \ldots w_{g+1}^{-1}}$$ lies in the kernel $\mathrm{Ker}(\varphi_n)=R_{g,n}$. We proved that any element from $R_{g,n}$ lies in the product $$\overline{U}_{g,g+n} \leftthreetimes (\overline{U}_{g,g+n-1} \leftthreetimes ( \ldots \leftthreetimes (\overline{U}_{g,g+2} \leftthreetimes \overline{U}_{g,g+1})\ldots )).$$ On the other side, this product contains $Q_{g,n}$ and is normal in $B_{g,n}$. Hence, $R_{g,n}$ has the required decomposition.
2\) Prove the second decomposition. Denote $G = \overline{V}_{g,1} \leftthreetimes (\overline{V}_{g,2} \leftthreetimes ( \ldots \leftthreetimes (\overline{V}_{g,g-1} \leftthreetimes \overline{V}_{g,g})\ldots ))$ the group from the right side of the decomposition of $R_{g,n}$. Take an arbitrary element $h \in B_{g,n}$. By the theorem on the horizontal decomposition of $B_{g,n}$ it has the following normal form: $$h = v_1 v_2 \ldots v_g b, ~\mbox{where}~v_i \in V_{g,i}, b \in \widetilde{B}_n$$ and $v_1 v_2 \ldots v_g$ lies in $R_{g,n}$. Under the endomorphism $\varphi_n : B_{g,n} \longrightarrow \widetilde{B}_n$ element $h$ goes to the element $b$. Hence, $G$ lies in $R_{g,n}$.
On the other side we must prove that $R_{g,n}$ lies in $G$. We know that $R_{g,n}$ is the normal closure of $Q_{g,n}$ in $B_{g,n}$. As was shown before, if $\alpha \in \Lambda_n$ is a coset representative of $\widetilde{P}_n$ into $\widetilde{B}_n$ and $a_{ij}$ is some generator of $Q_{g,n}$, then $a_{ij}^{\alpha}$ lies in $Q_{g,n}$. Considering the conjugation of $a_{ij} \in Q_{g,n}$ by generators of $P_{g,n}$ and using the Lemma \[l1\] we get an element which lies in $G$.
In the case $g=1$ we have $P_{1,n} = P_{n+1}$ and $R_{1,n} = \langle a_{12}, a_{13},$ $\ldots,$ $a_{1,n+1} \rangle \cong F_n$ and we get the decomposition which was found in [@L2].
$B_{1,n} \cong F_n \leftthreetimes B_n$.
Some analog of the Hecke algebra for the braid group in the handlebody
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Let $q$ be some complex number. Recall that the Hecke algebra $H_{n}(q)$ is an associative $\mathbb{C}$-algebra with unit, which is generated by elements $$s_1, s_2, \ldots, s_{n-1},$$ and is defined by the relations $$\begin{array}{ll}
s_i s_j = s_j s_i, & ~\mbox{for}~|i - j| > 1, \\
s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}, & ~\mbox{for}~1 \leq i \leq n-2, \\
s_i^2 = (q-1) s_i + q, & ~\mbox{for}~i = 1, \ldots, n-1.
\end{array}$$
The algebra $H_{n}(q)$ has the following linear basis: $$S = \{ (s_{i_1} s_{i_1-1} \ldots s_{i_1-k_1}) (s_{i_2} s_{i_2-1} \ldots s_{i_2-k_2}) \ldots (s_{i_p} s_{i_p-1} \ldots s_{i_p-k_p}) \}$$ for $1 \leq i_1 < \ldots < i_p \leq n-1$. The basis $S$ is used in the construction of the Markov trace, leading to the HOMFLYPT or $2$-variable Jones polynomial (see [@J]).
The braid group in the solid torus $B_{1,n}$ is the Artin group of the Coxeter group of type $\mathcal{B}$, which is related to the Hecke algebra of type $\mathcal{B}$. The generalized Hecke algebra of type $\mathcal{B}$, $H_{1,n}(q)$ is defined by S. Lambropoulou in [@L]. $H_{1,n}(q)$ is isomorphic to the affine Hecke algebra of type $\mathcal{A}$, $\widetilde{H}_{n}(q)$. A unique Markov trace is constructed on the algebras $H_{1,n}(q)$ that leads to an invariant for links in the solid torus, the universal analogue of the HOMFLYPT polynomial for the solid torus.
The algebra $H_{1,n}(q)$ is generated by elements $$t, t^{-1}, s_1, s_2, \ldots, s_{n-1},$$ and is defined by the relations $$\begin{array}{ll}
s_i s_j = s_j s_i, & ~\mbox{for}~|i - j| > 1, \\
s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}, & ~\mbox{for}~1 \leq i \leq n-2, \\
s_i^2 = (q-1) s_i + q, & ~\mbox{for}~i = 1, 2, \ldots, n-1,\\
s_1 t s_1 t = t s_1 t s_1, & \\
t s_i = s_i t, & ~\mbox{for}~i > 1.
\end{array}$$ Hence, $$H_{1,n}(q) = \frac{\mathbb{C} [B_{1,n}]}{\langle \sigma_i^2 - (q-1) \sigma_i - q \rangle}.$$ Note that in $H_{1,n}(q)$ the generator $t$ satisfies no polynomial relation, making the algebra $H_{1,n}(q)$ infinite dimensional. If we set $t=0$ in $H_{1,n}(q)$, we obtain the Hecke algebra $H_n(q)$.
In $H_{1,n}(q)$ are defined in [@L2] the elements $$t_i = s_i s_{i-1} \ldots s_1 t s_1 \ldots s_{i-1} s_i,~~~t'_i = s_i s_{i-1} \ldots s_1 t s_1^{-1} \ldots s_{i-1}^{-1} s_i^{-1}.$$ It was then proved that the following sets form linear bases for $H_{1,n}(q)$: $$\Sigma_n = t^{k_1}_{i_1} t^{k_2}_{i_2} \ldots t^{k_r}_{i_r} \sigma, ~\mbox{where}~1 \leq i_1 < \ldots < i_r \leq n-1,$$ $$\Sigma'_n = (t_{i_1}')^{k_1} (t_{i_2}')^{k_2} \ldots (t_{i_r}')^{k_r} \sigma, ~\mbox{where}~1 \leq i_1 < \ldots < i_r \leq n-1,$$ where $k_1, k_2, \ldots, k_r \in \mathbb{Z}$ and $\sigma$ a basis element in $H_{n}(q)$. The basis $\Sigma'_n$ is used in [@L; @L2] for constructing a Markov trace on $\bigcup_{n=1}^{\infty} H_{1,n}(q)$ and a universal HOMFLYPT-type invariant for oriented links in the solid torus.
Let $q \in \mathbb{C}$. The algebra $H_{g,n}(q)$ is an associative algebra over $\mathbb{C}$ with unit that is generated by $$t_1^{\pm 1}, t_2^{\pm 1}, \ldots, t_g^{\pm 1}, s_{g+1}, s_{g+2}, \ldots s_{g+n-1}$$ and is defined by the following relations $$\begin{array}{ll}
s_i s_j = s_j s_i, & ~\mbox{for}~|i - j| >1, \\
s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}, & ~\mbox{for}~i = g+1, g+2, \ldots, g+n-2, \\
s_i^2 = (q-1) s_{i} + q, & ~\mbox{for}~i = g+1, g+2, \ldots, g+n-1, \\
t_k s_i = s_i t_k, & ~\mbox{for}~k \geq 1, i \geq g+2, \\
t_k (s_{g+1} t_k s_{g+1}) = (s_{g+1} t_k s_{g+1}) t_{k}, & ~\mbox{for}~k = 1, 2, \ldots, g, \\
t_k (s_{g+1} t_{k+l} s_{g+1}) = (s_{g+1} t_{k+l} s_{g+1}) t_{k}, & ~\mbox{for}~k = 1, 2, \ldots, g-1; l = 1, 2, \ldots, g-k.
\end{array}$$
More natural to consider the generators $s_{1}, s_{2}, \ldots s_{n-1}$ instead $s_{g+1}, s_{g+2}, \ldots s_{g+n-1}$, but for technical reasons we will use our notation.
We see that $$H_{g,n}(q) = \frac{\mathbb{C} [B_{g,n}]}{\langle \sigma_i^2 - (q-1) \sigma_i - q \rangle}.$$
If we consider the algebra $H_n(q)$ as a vector space over $\mathbb{C}$, then it is isomorphic to the vector space $\mathbb{C}[S_n]$. Thus to study $H_{g,n}(q)$, we define a group $G_{g,n}$ as the quotient of $B_{g,n}$ by the relations $$\sigma_i^2 = 1,~~~i = 1, 2, \ldots, n-1.$$ Denote the natural homomorphism $B_{g,n} \longrightarrow G_{g,n}$ by $\psi$ and let $\psi(a_{ij}) = b_{ij}$.
\[t3\] The group $G_{g,n}$ is the semi-direct product $G_{g,n} = F_g^{n} \leftthreetimes S_n$ of the direct product of $n$ copies of free group $F_g$ of rank $g$ and the symmetric group $S_n$.
We know that $B_{g,n} = R_{g,n} \leftthreetimes \widetilde{B}_n$ and $$R_{g,n} = \overline{U}_{g,g+n} \leftthreetimes (\overline{U}_{g,g+n-1} \leftthreetimes ( \ldots \leftthreetimes (\overline{U}_{g,g+2} \leftthreetimes \overline{U}_{g,g+1})\ldots )).$$ On the other side, $U_k$, $k= g+1, g+2, \ldots, g+n$, is generated by two subgroups: $$U_{g,k} = \langle a_{1k}, a_{2k}, \ldots, a_{gk} \rangle ~\mbox{and}~\widetilde{U}_{k} = \langle a_{g+1,k}, a_{g+2,k}, \ldots, a_{k-1,k} \rangle \leq \widetilde{P}_n.$$ Note that under the homomorphism $\psi$ the subgroups $\widetilde{U}_{k}$ go to 1. Hence, $\psi(\overline{U}_{g,g+k}) = \psi(U_{g,g+k}) \cong F_g$ and $$\psi(R_{g,n}) = \psi(U_{g,g+n}) \leftthreetimes (\psi(U_{g,g+n-1}) \leftthreetimes ( \ldots \leftthreetimes (\psi(U_{g,g+2}) \leftthreetimes \psi(U_{g,g+1}))\ldots ))$$ is the semi-direct product of $n$ copies of free group $F_g$. Also, $G_{g,n} = \psi(R_{g,n}) \leftthreetimes \psi(\widetilde{B}_n) \cong \psi(R_{g,n}) \leftthreetimes S_n$.
Let us show that in fact $\psi(R_{g,n})$ is the direct product of $n$ copies of free group $F_g$. To do it it is enough to prove that $$\psi(R_{g,n}) = \psi(U_{g,g+n}) \times \psi(R_{g,n-1}).$$ Denote $b_{ij} = \psi(a_{ij})$, consider the relations (\[co1\])-(\[co4\]) and find their images under the homomorphism $\psi$. The relation (\[co1\]) goes to the relation $$b_{i,g+j}^{-\varepsilon} b_{g+j,g+n} b_{i,g+j}^{\varepsilon} = (b_{i,g+n} b_{g+j,g+n})^{\varepsilon} b_{g+j,g+n} (b_{i,g+n} b_{g+j,g+n})^{-\varepsilon},$$ but $b_{g+j,g+n} = 1$ and we have the trivial relations.
The relation (\[co2\]) goes to the relation $$b_{l,g+j}^{-\varepsilon} b_{l,g+n} b_{l,g+j}^{\varepsilon} = (b_{l,g+n} b_{g+j,g+n})^{\varepsilon} b_{l,g+n} (b_{l,g+n} b_{g+j,g+n})^{-\varepsilon},$$ but $b_{g+j,g+n} = 1$ and we have the relation $$b_{l,g+j}^{-\varepsilon} b_{l,g+n} b_{l,g+j}^{\varepsilon} = b_{l,g+n}.$$
The relation (\[co3\]) goes to the relation $$b_{i,g+j}^{-\varepsilon} b_{l,g+n} b_{i,g+j}^{\varepsilon} = [b_{i,g+n}^{-\varepsilon}, b_{g+j,g+n}^{-\varepsilon}]^{\varepsilon} b_{l,g+n} [b_{i,g+n}^{-\varepsilon}, b_{g+j,g+n}^{-\varepsilon}]^{-\varepsilon},~~~i < l < g+j,$$ but $b_{g+j,g+n} = 1$ and we have the relation $$b_{i,g+j}^{-\varepsilon} b_{l,g+n} b_{i,g+j}^{\varepsilon} = b_{l,g+n},~~~i < l < g+j.$$
The relation (\[co4\]) goes to the relation $$b_{i,g+j}^{-\varepsilon} b_{l,g+n} b_{i,g+j}^{\varepsilon} = b_{l,g+n},~~~l < i < g+j < g+n.$$
The image of $\widetilde{B}_n \leq B_{g,n}$ is isomorphic to $S_n$. Thus from the decomposition $B_{g,n} = R_{g,n} \leftthreetimes \widetilde{B}_n$ we get the required decomposition for $G_{g,n}$.
From this theorem we have
Every element $h \in G_{g,n}$ has the unique normal form $$h = h_1 h_2 \ldots h_n \alpha,$$ where $h_i$ is a reduced word in the free group $$\psi(U_{g,g+i}) = \langle b_{1, g+i}, b_{2, g+i}, \ldots, b_{g, g+i} \rangle \cong F_g,~~~i = 1, 2, \ldots, n,$$ $\alpha \in \Lambda_n$ is a coset representative of $\psi(R_{g,n})$ in $G_{g,n}$.
Using this normal form and the fact that these normal forms form a linear basis of $\mathbb{C}[G_{g,n}]$, we can try to define a basis of $H_{g,n}(q)$. In the algebra $H_{g,n}(q)$ define the following elements $$\begin{array}{cccc}
t_{1,g+1} = t_1,~~ & t_{1,g+2}, & \ldots, & t_{1,g+n}, \\
t_{2,g+1} = t_2,~~ & t_{2,g+2}, & \ldots, & t_{2,g+n}, \\
\ldots ~~& \ldots & \ldots & \ldots \\
t_{g,g+1} = t_g, ~~& t_{g,g+2}, & \ldots, & t_{g,g+n},
\end{array}$$ where $$t_{ij} = s_{j-1} s_{j-2} \ldots s_1 t_{i,g+1} s_1 \ldots s_{j-2} s_{j-1}, ~~1 \leq i \leq g,~~g+2 \leq j \leq g+n.$$ These elements correspond to the elements $b_{ij}$ from $G_{g,n}$, which were defined in \[t3\] as the images of the elements $a_{ij}$ under the map $\psi$. It is not difficult to see that the elements $$t_{1, g+i}, t_{2, g+i}, \ldots, t_{g, g+i}$$ generate a free group of rank $g$.
The algebra $H_{g,n}(q)$ contains the subalgebra with the set of generators $s_{g+1}, s_{g+2}, \ldots s_{g+n-1}$, which is isomorphic to $H_n(q)$. Let $\Sigma_{g,n}$ be the following set in $H_{g,n}(q)$: $$u_1 u_2 \ldots u_n \sigma,$$ where $u_i$ is a reduced word in the free group $$\langle t_{1, g+i}, t_{2, g+i}, \ldots, t_{g, g+i} \rangle \cong F_g,~~~i = 1, 2, \ldots, n,$$ $\sigma$ is a basis element in $H_{n}(q)$.
There is an isomorphism $H_{g,n}(q) \cong \mathbb{C}[G_{g,n}]$ as $\mathbb{C}$-modules. In particular, the set $\Sigma_{g,n}$ is a basis of the algebra $H_{g,n}(q)$.
The algebra $H_{2,n}(q)$ is the subject of study in [@KL], where one set of elements in $H_{2,n}(q)$, different from our $\Sigma_{2,n}$, is proved to be a spanning set for the algebra.
At the end, we formulate the following question for further investigation.
Is it possible to define a Markov trace on the algebra $\bigcup_{n=1}^{\infty} H_{g,n}(q)$ and construct some analogue of the HOMLYPT polynomial that is an invariant of links in the handlebody $\mathcal{H}_g$?
[HD]{}
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V. F. R. Jones, *Hecke algebra representations of braid groups and links polynomials*, Ann. Math., 1984, **126**, 335–388.
R. Haring-Oldenburg, S. Lambropoulou *Knot theory in handlebodies*, *J. Knot Theory Ramifications*, **11**, N 6 (2002), 921–943.
D. Kodokostas and S. Lambropoulou, *A spanning set and potential basis of the mixed Hecke algebra on two fixed strands*, arXiv: 1704.03676.
S. Lambropoulou, *Solid torus links and Hecke algebras of $B$-type*, Proc. of the Conf. of Quantum Topology, D. N. Yetter ed., World Scientific Press, 1994.
S. Lambropoulou, *Braid structure in knot complements, handlebodies and 3-manifold*, Knots in Hellas ’98 (Delphi), Proc. of International Conference Knot Theory and its Ramifications, WS, 274–289.
S. Lambropoulou, *Knot theory related to generalized and cyclotomic Hecke algebras of type B*, *J. Knot Theory Ramifications*, **8**, N 5 (1999), 621–658.
S. Lambropoulou, C.P. Rourke *Markov’s theorem in 3-manifolds*, *Topology and its Applications*, 1997, **78**, 95–122.
S. Lambropoulou, C.P. Rourke *Algebraic Markov equivalence for links in 3-manifolds*, *Compositio Mathematica*, 2006, **142**, 1039–1062.
A. A. Markov, *Foundations of the algebraic theory of braids*, *Trudy Mat. Inst. Steklov \[Proc. Steklov Inst. Math.\]*, 1945, **16**, 1–54.
A. B. Sossinsky, *Preparation theorem for isotopy invariants of links in 3-manifold*, Quantum Groups. Proc. of the Conf. on Quantum Groups. Berlin a.: Springer-Verl., 1992, 354–362 (Lecture Notes in Math.; N 1510).
V. V. Vershinin, *On braid groups in handlebodies*, *Siberian Math. J*., **39**, N 4 (1998), 645–654. Translation from Sibirsk. Mat. Zh., **39**, N 4 (1998), 755–764.
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---
abstract: 'In this paper, a new type of the discrete fractional Gr[ö]{}nwall inequality is developed, which is applied to analyze the stability and convergence of a Galerkin spectral method for a linear time-fractional subdiffusion equation. Based on the temporal-spatial error splitting argument technique, the discrete fractional Gr[ö]{}nwall inequality is also applied to prove the unconditional convergence of a semi-implicit Galerkin spectral method for a nonlinear time-fractional subdiffusion equation.'
address:
- 'Nanhu College, Jiaxing University, Jiaxing, Zhejiang 314001, China'
- 'Department of Mathematics, National University of Singapore, Singapore 119076, Singapore'
author:
- Yubo Yang
- Fanhai Zeng
title: 'Numerical analysis of linear and nonlinear time-fractional subdiffusion equations'
---
Time-fractional subdiffusion equation ,convolution quadrature ,fractional linear multistep methods ,discrete fractional Gr[ö]{}nwall inequality ,unconditional stability
Introduction
============
Consider the following nonlinear time-fractional subdiffusion equation $$\label{E:1.1}
\left\{\begin{array}{lll}
{_{0}^{C}{\mathcal {D}}}^{\beta}_t u = \mu~\partial^{2}_x u + f(x,t,u), \quad (x,t)\in I\times (0,T), \quad I=(-1,1), \quad T>0,\\ [0.3cm]
u=0,\qquad (x,t)\in \partial I\times (0,T),\\ [0.3cm]
u(x,0)=u_0(x), \qquad x\in I,
\end{array}\right.$$ where ${_{0}^{C}{\mathcal {D}}}^{\beta}_t u$ denotes the Caputo time-fractional derivative of order $0<\beta<1$ defined by (cf. [@Kilbas2006]) $$\label{E:1.2}
{_{0}^{C}{\mathcal {D}}}^{\beta}_t u(x,t) = \frac{1}{\Gamma(1-\beta)} \int_{0}^{t} (t-s)^{-\beta}\partial_s u(x,s) {\rm d}s,$$ in which $\Gamma(z)=\int_{0}^{\infty} s^{z-1}e^{-s} {\rm d}s$ is the Gamma function.
Various time-stepping schemes have been developed for discretizing . The time discretization technique for the time-fractional operator in mainly falls into two categories: interpolation and the fractional linear multistep method (FLMM, which is also called the convolution quadrature (CQ)) based on generating functions that can be derived from the linear multistep method for the initial value problem. For example, piecewise linear interpolation yields the widely applied $L1$ method [@lin2007finite; @liao2018sharp]. High-order interpolation can also be applied, see [@alikhanov2015new; @li2015numerical; @ZhaoSK15]. The FLMM [@Lubich1986Discretized; @Lubich1988Convolution; @Lubich2004Convolution] provides another general framework for constructing high-order methods to discretize the fractional integral and derivative operators. The FLMM inherits the stability properties of linear multistep methods for initial value problem, which greatly facilitates the analysis of the resulting numerical scheme, in a way often strikingly opposed to standard quadrature formulas [@Lubich2004Convolution]. Up to now, the FLMM has been widely applied to discretize the model and its variants.
It is well known that the classical discrete Gr[ö]{}nwall inequality plays an important role in the analysis of the numerical methods for time-dependent partial differential equations (PDEs). Due to the lack of a generalized discrete Gr[ö]{}nwall type inequality for the time-stepping methods of the time-dependent fractional differential equations (FDEs), the analysis of the numerical methods for time-dependent FDEs is more complicated. Recently, a discrete fractional Gr[ö]{}nwall inequality has been established by Liao et al. [@li2016analysis; @liao2018sharp; @liao2018discrete; @liao2018second] for interpolation methods to solve linear and nonlinear time-dependent FDEs. Jin et al. [@jin2018numerical] proposed a criterion for showing the fractional discrete Gr[ö]{}nwall inequality and verified it for the $L1$ scheme and convolution quadrature generated by backward difference formulas.
Till now, there have been some works on the numerical analysis of nonlinear time-dependent FDEs. The stability and convergence of $L1$ finite difference methods were obtained for a time-fractional nonlinear predator-prey model under the restriction $LT^{\beta} < 1/\Gamma(1-\beta)$ in [@yu2015positivity], where $L$ is the Lipschitz constant of the nonlinear function, depending upon an upper bound of numerical solutions [@li2018unconditionally]. Such a condition implied that the numerical results just held locally in time and certain time step restriction condition (see, e.g., [@akrivis1999implicit; @ford2013numerical]) were also required. Similar restrictions also appear in the numerical analysis for the other fractional nonlinear equations (see, e.g. [@li2016efficient; @li2016linear]). In order to avoid such a restriction, the temporal-spatial error splitting argument (see, e.g., [@li2012mathematical]) is extended to the numerical analysis of the nonlinear time-dependent FDEs (see, e.g., [@li2017unconditionally; @li2018unconditionally]). Li et al. proposed unconditionally convergent $L$1-Galerkin finite element methods (FEMs) for nonlinear time-fractional Sch[ö]{}dinger equations [@li2017unconditionally] and nonlinear time-fractional subdiffusion equations[@li2018unconditionally], respectively. In this paper, we follow the idea in [@liao2018sharp] and develop a discrete fractional Gr[ö]{}nwall inequality for analyzing the FLMM that arises from the generalized Newton–Gregory formula (GNGF) of order up to two, see [@zeng2013use]. Compared with the approach based on interpolation in [@liao2018sharp], the discrete kernel $P_{k-j}$ (see Lemmas \[lem2.2\] and \[lem2.3\] below) originates from the generating function that can be obtained exactly, which is much simpler than that in [@liao2018sharp]. Based on the discrete fractional Gr[ö]{}nwall inequality, the numerical analysis of semi-implicit Galerkin spectral method for the time-fractional nonlinear subdiffusion problem is advanced. The temporal-spatial error splitting argument is used to prove the unconditional stability and convergence of the semi-implicit method.
The main task of this work is to establish the discrete fractional Gr[ö]{}nwall type inequality for the stability and convergence analysis of the numerical methods for time-fractional PDEs, the regularity and singularity of the solution at $t=0$ is not considered in detail here; readers can refer to [@stynes2017error; @zeng2017second] for the graded mesh method and correction method for resolving the singularity of the solution of the time-fractional PDEs. The paper is organized as follows. In Section \[sec-2\], the discrete fractional Gr[ö]{}nwall inequality for the CQ is developed, which is applied to the numerical analysis for the linear time-fractional PDE. In Section 3, the unconditional convergence of the semi-implicit Galerkin spectral method is proved by combining the discrete fractional Gr[ö]{}nwall inequality and the temporal-spatial error splitting argument. Some conclusion remarks are given in Section 4.
Numerical analysis for the linear equation {#sec-2}
==========================================
In this section, two numerical schemes are proposed for the linear equation , i.e., $f(x,t,u)=f(x,t)$, in which the time direction is approximated by the fractional linear multistep methods (FLMMs) and the space direction is approximated by the Galerkin spectral method.
Let $\{t_k=k\tau\}_{k=0}^{n_T}$ be a uniform partition of the interval $[0,T]$ with a time step size $\tau=T/n_T$, $n_T$ is a positive integer. For simplicity, the solution of is denoted by $u(t)=u(x,t)$ if no confusion is caused. For function $u(x,t) \in C(0,T;L^2(I))$, denote $u^k=u^k(\cdot)=u(\cdot,t_k)$ and $u^{k-\theta}=(1-\theta) u^k+ \theta u^{k-1}, \theta \in [0,1].$
Let $\mathbb{P}_N(I)$ be the set of all algebraic polynomials of degree at most $N$ on $I$. Define the approximation space as follows: $$\begin{aligned}
V_N^0=\{v:v \in \mathbb{P}_N(I),v(-1)=v(1)=0\}.\end{aligned}$$
As in [@zeng2013use], we present the time discretization for as follows: $$\begin{aligned}
\label{E:2.5}
{D}^{(\beta)}_{\tau} u^k=\mu L_p^{(\beta)} \partial^2_x u^k+L_p^{(\beta)} f^k+R^k,\end{aligned}$$ where $f^k=f(x,t_k)$, $R^k$ is the discretization error in time that will be specified later, $L_p^{(\beta)}$ and ${D}^{(\beta)}_{\tau}$ are defined by $$\begin{aligned}
\label{E:2.3}
L_p^{(\beta)}u^k=\left\{\begin{array}{ll}
u^k, ~~\qquad p=1,\\ [0.3cm]
u^{k-\beta/2}, \quad p=2,
\end{array}\right.\end{aligned}$$ and $$\begin{aligned}
\label{E:2.2}
{D}^{(\beta)}_{\tau} u^k=\frac{1}{{\tau}^{\beta}}\sum_{j=0}^k {\varpi}_{k-j}\left(u^j-u^0\right),\end{aligned}$$ respectively, in which ${\varpi}_{k}$ satisfy the generating function ${\varpi}(z)=(1-z)^{\beta}
=\sum_{k=0}^{\infty}{\varpi}_{k}z^k$.
The fully discrete schemes for are given as follows: find $u_N^k\in V_N^0$ such that $$\label{E:2.6}
\left\{\begin{array}{ll}
\left({D}^{(\beta)}_{\tau} u_N^k,v\right)+ \mu \left(L_p^{(\beta)}\partial_x u_N^{k},\partial_x v\right)= \left(F_p^k,v\right),\quad k=1,2,\cdots,n_T, \forall v \in V_N^0,\\ [0.1cm]
u^0=I_N u_0,
\end{array}\right.$$ where $F_p^k=I_N\left(L_p^{(\beta)} f^k\right)$ and $I_N$ is the Legendre–Gauss–Lobatto (LGL) interpolation operator.
Discrete fractional Gr[ö]{}nwall inequality
-------------------------------------------
In this subsection, we introduce some useful lemmas and present a discrete fractional Gr[ö]{}nwall inequality that is used in the stability and convergence analysis for .
\[lem2.1\] For $0<\beta<1$, let ${\varpi}_{j}$ be given by ${\varpi}(z)=(1-z)^{\beta}=\sum_{k=0}^{\infty}{\varpi}_{k}z^k$. Then one has $$\label{E:2.1.1}
\left\{\begin{array}{llllll}
{\varpi}_{j}=(-1)^j {\beta \choose j}=\frac{\Gamma(j-\beta)}{\Gamma(-\beta)\Gamma(j+1)},\\ [0.5cm]
{\varpi}_{0}=1, {\varpi}_{j}<{\varpi}_{j+1}<0,\qquad j \geq 1,\\ [0.5cm]
\sum_{j=0}^{\infty} {\varpi}_{j}=0,\\ [0.5cm]
{\varpi}_{0}=-\sum_{j=1}^{\infty} {\varpi}_{j}>-\sum_{j=1}^{k} {\varpi}_{j}>0,\\ [0.5cm]
b_{k-1}=\sum_{j=0}^{k-1} {\varpi}_{j}>0, \qquad k \geq 1,\\ [0.5cm]
b_{k-1}=\frac{\Gamma(k-\beta)}{\Gamma(1-\beta)\Gamma(k)}=\frac{k^{-\beta}}{\Gamma(1-\beta)}+O(k^{-1-\beta}),\quad k=1,2,\cdots.
\end{array}\right.$$ Furthermore, $b_{k}-b_{k-1}={\varpi}_{k}<0$ for $k>0$, i.e., $b_{k}<b_{k-1}$.
\[lem2.2\] For $0<\beta<1$, let ${\varpi}_{j}$ be given by ${\varpi}(z)=(1-z)^{\beta}=\sum_{k=0}^{\infty}{\varpi}_{k}z^k$, ${\varrho}_{j}$ be given by $\varrho(z)=(1-z)^{-\beta}=\sum_{k=0}^{\infty}{\varrho}_{k}z^k$, and $${\vartheta}_m:=\sum_{j=0}^{m}{\varpi}_{j}{\varrho}_{m-j},\quad m=0,1,2,\cdots.$$ Then one has $$\label{E:2.1.2}
\left\{\begin{array}{llll}
{\varrho}_{j}=(-1)^j{-\beta \choose j}=\frac{(-1)^j\Gamma(j+\beta)}{\Gamma(\beta)\Gamma(j+1)},\qquad j \geq 0,\\ [0.5cm]
{\varrho}_{0}=1,\quad{\varrho}_{j}>{\varrho}_{j+1}>0,\qquad j \geq 1,\\ [0.5cm]
{\varrho}_{j} \leq (j+1)^{\beta-1},\qquad j \geq 0,\\ [0.5cm]
{\varrho}_{j} \leq j^{\beta-1},\qquad j \geq 1,\\ [0.5cm]
\sum_{j=0}^{k-1} {\varrho}_{j}=\frac{\Gamma(k+\beta)}{\Gamma(1+\beta)\Gamma(k)} \leq \frac{k^{\beta}}{\beta},\quad k=1,2,\cdots,\\ [0.5cm]
{\vartheta}_0=1,\quad{\vartheta}_m=0, \qquad m \geq 1.
\end{array}\right.$$
By the binomial theorem, we can easily get the first two lines in . The middle three lines in can be derived by the technique in [@jin2018numerical p.6]. The last line in can be deduced from the following relation $$1={\varpi}(z)\varrho(z)=\left(\sum_{j=0}^{\infty}{\varrho}_jz^j\right)
\left(\sum_{j=0}^{\infty}{\varpi}_{j}z^j\right)=\sum_{j=0}^{\infty}{\vartheta}_{j}z^j.$$ The proof is complete.
\[lem2.3\] Let $P_{k-j}:={\tau}^{\beta}{\varrho}_{k-j}$. For $0<\beta<1$ and any real $\mu>0$, one has $$\begin{aligned}
\label{E:2.1.3}
\mu \sum_{j=0}^{k-1} P_{k-j} E_{\beta}(\mu t_j^{\beta}) \leq E_{\beta}(\mu t_k^{\beta})-1, \qquad 1\leq k \leq n_T,\end{aligned}$$ where $E_{\beta}$ denotes the Mittag–Leffler function that is defined by $$\begin{aligned}
\label{E:2.1.4}
E_{\beta}(z)=\sum_{l=0}^{\infty} \frac{z^l}{\Gamma(1+l\beta)}.\end{aligned}$$
Denote $v_l(t)=\frac{t^{l\beta}}{\Gamma(1+l\beta)}$. It is easy to obtain $$\begin{aligned}
\sum_{j=0}^{k-1} P_{k-j} \left({_{0}^{C}{\mathcal {D}}}^{\beta}_t v_l\right)(t_j)=\sum_{j=0}^{k-1} P_{k-j} v_{l-1}(t_j).\end{aligned}$$ By the fourth inequality in Lemma \[lem2.2\] and Lemma 3.2 in [@li2013finite], we obtain $$\begin{aligned}
\sum_{j=0}^{k-1} P_{k-j} v_{l-1}(t_j)
&=&\sum_{j=0}^{k-1} {\varrho}_{k-j} \left[\frac{j^{(l-1)\beta}{\tau}^{l\beta}}{\Gamma((l-1)\beta+1)}\right]\\
&=&\sum_{j=0}^{k-1}\left[\frac{{\varrho}_{k-j}j^{(l-1)\beta}}
{\Gamma((l-1)\beta+1)}\cdot\frac{\Gamma(l\beta+1)}{k^{l\beta}}\right] v_l(t_k) \\
&\leq&v_l(t_k)\sum_{j=0}^{k-1}\left[\frac{{(k-j)}^{\beta-1}j^{(l-1)\beta}}
{\Gamma((l-1)\beta+1)}\cdot\frac{\Gamma(l\beta+1)}{k^{l\beta}}\right] \\
&\leq&v_l(t_k)\end{aligned}$$
Therefore, one has $$\begin{aligned}
\sum_{l=1}^{m} {\mu}^l \sum_{j=0}^{k-1} P_{k-j} v_{l-1}(t_j) \leq \sum_{l=1}^{m} {\mu}^l v_l(t_k).\end{aligned}$$ Interchanging the sums on the left-hand side of the above inequality and letting $m\rightarrow \infty$ yields the desired result. The proof is completed.
We now present the discrete fractional Gr[ö]{}nwall inequality in the next theorem.
\[thm:2.1\] (discrete fractional Gr[ö]{}nwall inequality). Let $P_{k-j}:={\tau}^{\beta}{\varrho}_{k-j}$, $0\leq \theta \leq 1$, and $\{g^k\}_{k=0}^{n_T}$ and $\{\lambda_l\}_{l=0}^{n_T-1}$ be given non-negative sequences. Assume that there exists a constant $\lambda$ (independent of the time step size) such that $\lambda \geq \sum_{l=0}^{k-1} \lambda_l$, and that the maximum time step size $\tau$ satisfies $$\begin{aligned}
\label{E:2.1.5}
\tau \leq \frac{1}{\sqrt[\beta]{2\lambda(1+\beta)}}.\end{aligned}$$ Then, for any non-negative sequence $\{v^k\}_{k=0}^N$ satisfying $$\begin{aligned}
\label{E:2.1.6}
{D}^{(\beta)}_{\tau} (v^k)^2 \leq \sum_{l=1}^k \lambda_{k-l} (v^l)^2+v^{k-\theta}g^{k-\theta},
\qquad 1 \leq k \leq n_T,\end{aligned}$$ it holds that $$\begin{aligned}
\label{E:2.1.7}
v^k \leq 2E_{\beta}(2\lambda t_k^{\beta})\left(v^0+\max_{1\leq m \leq k} \sum_{j=0}^m P_{m-j}g^{j-\theta}\right), \qquad 1 \leq k \leq n_T.\end{aligned}$$
By and the last line in Lemma \[lem2.2\], one has $$\begin{aligned}
\label{E:2.1.8}
\sum_{m=0}^{k}P_{k-m}{D}^{(\beta)}_{\tau} (v^m)^2
& = &\sum_{m=0}^{k}{\varrho}_{k-m}\sum_{j=0}^{m}{\varpi}_{m-j}\left[(v^j)^2-(v^0)^2\right] \nonumber \\
& = &\sum_{j=0}^{k}\left[(v^j)^2-(v^0)^2\right]\sum_{m=j}^{k}{\varrho}_{k-m}{\varpi}_{m-j} \nonumber \\
& = &\sum_{j=0}^{k}{\vartheta}_{k-j}\left[(v^j)^2-(v^0)^2\right]=(v^k)^2-(v^0)^2,\end{aligned}$$ where we exchanged the order of summation and rearranged the coefficient $\sum_{m=j}^{k}{\varrho}_{k-m}{\varpi}_{m-j}$ to ${\vartheta}_{k-j}$.
By Lemma \[lem2.3\] and the technique for the proof of Theorem 3.1 in [@liao2018discrete], we derive the desired result, which completes the proof.
We also have an alternative version of the above theorem.
\[cor:2.1\] Theorem \[thm:2.1\] remains valid if the condition is replaced by $$\begin{aligned}
\label{E:2.1.7}
{D}^{(\beta)}_{\tau} v^k \leq \sum_{l=1}^k \lambda_{k-l} v^l+g^{k}, \qquad {\rm for} \quad 1 \leq k \leq n_T.\end{aligned}$$
Similar to the proof of the theorem 3.4 in [@liao2018discrete], and by the proof of Theorem \[thm:2.1\], we can complete our proof.
\[rem:2.1\] If $\lambda_0, \lambda_1,\cdots, \lambda_{k-1}$ are non-positive, a simple deduction will show that both Theorem \[thm:2.1\] and Corollary \[cor:2.1\] hold for any $\tau>0$.
Stability and convergence
-------------------------
We present the stability and convergence result for the scheme .
\[thm:2.2\] Suppose that $u_N^k~(k=1,2,\cdots,n_T)$ are solutions of , $f \in C(0,T;C(\bar I))$. Then for any $\tau>0$, it holds that $$\begin{aligned}
\label{E:2.2.1}
\|u_N^k\| \leq C\left(\|u^0\|+\max_{1\leq m \leq k} \sum_{j=0}^m P_{m-j}\|L_p^{(\beta)}f^{j}\|\right), \qquad 1 \leq k \leq n_T,\end{aligned}$$ where $C$ is a positive constant independent of $\tau$ and $N$.
- For $p=1$, letting $v=2u_N^k$ in , one has $$\begin{aligned}
\label{E:2.2.2}
\left({D}^{(\beta)}_{\tau} u_N^k,2u_N^k\right)+ 2\mu \left(\partial_x u_N^{k},\partial_x u_N^k\right)=\left(I_N f^{k},2u_N^k\right), \qquad 1 \leq k \leq n_T.\end{aligned}$$ By , Lemma \[lem2.1\], and the Young’s inequality, we have $$\begin{aligned}
\label{E:2.2.3}
\left({D}^{(\beta)}_{\tau} u_N^k,2u_N^k\right)
& = &\frac{2}{{\tau}^{\beta}}\left[\sum_{j=0}^k {\varpi}_{k-j}\left(u_N^j,u_N^k\right)-\sum_{j=0}^k {\varpi}_{k-j}\left(u^0,u_N^k\right)\right] \nonumber \\
& = &\frac{1}{{\tau}^{\beta}}\left[2{\varpi}_{0}\left(u_N^k,u_N^k\right)+2\sum_{j=1}^k {\varpi}_{k-j}\left(u_N^j,u_N^k\right)-2b_k\left(u^0,u_N^k\right)\right] \nonumber \\
&\geq&\frac{1}{{\tau}^{\beta}}\left[2{\varpi}_{0}\|u_N^k\|^2+\sum_{j=1}^k {\varpi}_{k-j}\|u_N^j\|^2+\sum_{j=1}^k {\varpi}_{k-j}\|u_N^k\|^2-b_k\|u_N^k\|^2-b_k\|u^0\|^2\right] \nonumber \\
& = &\frac{1}{{\tau}^{\beta}}\left[\sum_{j=0}^k {\varpi}_{k-j}\|u_N^j\|^2-b_n\|u^0\|^2\right]={D}^{(\beta)}_{\tau} \left(\|u_N^k\|^2\right).\end{aligned}$$
Using , the positive-definiteness of $\left(\partial_x u_N^{k},\partial_x u_N^k\right)$, and $\|I_N f^k\| \leq C\|f^k\|$, one has $$\begin{aligned}
\label{E:2.2.4}
{D}^{(\beta)}_{\tau} \left(\|u_N^k\|^2\right)\leq C\|u_N^k\|~\|f^k\|, \qquad {\rm for} \quad 1 \leq k \leq n_T.\end{aligned}$$
Finally, applying the discrete Gr[ö]{}nwall inequality (see Theorem \[thm:2.1\]) and introducing the following notations $$\begin{aligned}
v^k:=\|u_N^k\|,~~v^0:=u^0,~~g^{k-\theta}:=C\|f^{k}\|~({\rm with~\theta=0}),~~\lambda_j:=0~ {\rm for}~~0 \leq j \leq n_T-1,\end{aligned}$$ we immediately get the stability result for $p=1$.
- For $p=2$, letting $v=2u_N^{k-\beta/2}$ in , one has $$\begin{aligned}
\label{E:2.2.5}
\left({D}^{(\beta)}_{\tau} u_N^k,2u_N^{k-\beta/2}\right)+ 2\mu \left(\partial_x u_N^{k-\beta/2},\partial_x u_N^{k-\beta/2}\right)=\left(I_N f^{k-\beta/2},2u_N^{k-\beta/2}\right).\end{aligned}$$
Rearranging the coefficients in , we have $$\begin{aligned}
\label{E:2.2.6}
\left({D}^{(\beta)}_{\tau} u_N^k,2u_N^{k-\beta/2}\right)= \frac{1}{{\tau}^{\beta}}\sum_{j=1}^k {b}_{k-j}\left(u_N^j-u_N^{j-1},2u_N^{k-\beta/2}\right).\end{aligned}$$ Similar to the proof of Lemma 4.1 in [@liao2018discrete] and by Lemma \[lem2.1\], we get $$\begin{aligned}
\label{E:2.2.7}
\left({D}^{(\beta)}_{\tau} u_N^k,2u_N^{k-\beta/2}\right) \geq {D}^{(\beta)}_{\tau} \left(\|u_N^k\|^2\right).\end{aligned}$$ The remaining of the proof is similar to that shown in (i), which is omitted here. The proof is completed.
To obtain the convergence results, we introduce the following two lemmas.
\[lem2.4\] Let $s$ and $r$ be arbitrary real numbers satisfying $0 \leq s \leq r$. There exist a projector $\Pi_N^{1,0}$ and a positive constant $C$ depending only on $r$ such that, for any function $u \in H_0^{s}(I) \cap H^r(I)$, the following estimate holds: $$\|u-\Pi_N^{1,0}u\|_{H^{s}(I)} \leq C N^{s-r} \|u\|_{H^r(I)},$$ where the orthogonal projection operator $\Pi_N^{1,0}: H_0^{1}(I)\rightarrow V_N^0$ is defined as $$\left(\partial_x \left(\Pi_N^{1,0}u-u\right),\partial_x v\right)=0, \qquad \forall v \in V_N^0.$$
\[lem2.5\] Let $r$ be arbitrary real numbers satisfying $r > 1/2$ and $I_N$ be the usual LGL interpolation operator. There exist a positive constant $C$ depending only on $r$ such that, for any function $u \in H^r(I)$, the following estimate holds: $$\|u-I_Nu\| \leq C N^{-r} \|u\|_{H^r(I)}.$$
Next, we consider the convergence analysis for the scheme . We also assume that the solution $u(t)$ satisfies (see, e.g., [@Diethelm-B10; @zeng2017second]) $$\label{solu:u}
u(t)=u_0+c_0t^{\sigma}+\sum_{j=1}^{\infty}c_kt^{\sigma_j},
{\quad}\beta<\sigma<\sigma_j<\sigma_{j+1}.$$ The singularity index $\sigma$ determines the accuracy of the numerical solution if $t^{\sigma}$ is not treated properly. We do not investigate how to deal with the singularity of the solution in this work, the interesting readers can refer to [@Lubich1986Discretized; @stynes2017error; @zeng2017second]. For the time discretization in , the global convergence rate is $\min\{\sigma-\beta,p\}$, that is $q=\min\{\sigma-\beta,p\}$.
Denote $u_*^k=\Pi_N^{1,0}u^k, e^k=u_*^k-u_N^k$ and $\eta^k=u^k-u_*^k$. Noticing that $\left(\partial_x \eta^k,\partial_x v\right)=0$ from Lemma \[lem2.4\], we get the error equation for as follows $$\begin{aligned}
\label{E:2.2.8}
\left({D}^{(\beta)}_{\tau} e^k,v\right)+\mu \left(\partial_x e^k,\partial_x v\right)=\left(G^k,v\right),\end{aligned}$$ where $G^k=\sum_{i=1}^3 G_i^k$ and $$\begin{aligned}
\label{E:2.2.9}
G_1^k=f^k-F_p^k,\quad G_2^k=R^k
=O(\tau^{\sigma-\beta}k^{\sigma-\beta-p}),\quad G_3^k=-{D}^{(\beta)}_{\tau} \eta^k.\end{aligned}$$
By Theorem \[thm:2.2\], Lemmas \[lem2.4\] and \[lem2.5\], we obtain the convergence result for the scheme .
\[thm:2.3\] Suppose that $r \geq 1$, $u$ and $u_N^k~(k=1,2,\cdots,n_T)$ are solutions of and , respectively. If $m \geq r+1, u\in C(0,T; H^m(I)\cap H^1_0(I)), f \in C(0,T;H^m(I))$ and $u_0 \in H^m(I)$. Then for any $\tau > 0$, it holds that $$\begin{aligned}
\label{2.2.10}
\|u^k-u_N^k\| \leq C(\tau^q+N^{-r}),\end{aligned}$$ where $C$ is a positive constant, independent of $\tau, N$, $q=\min\{\sigma-\beta,p\}$.
We consider $p=1$, the case for $p=2$ is similarly proved. By Theorem \[thm:2.2\] and the fifth line of Lemma \[lem2.2\], we need only to evaluate $$\|e^0\|+2C\max_{1\leq k \leq n_T} \left\{\|G_1^k\|+\|G_2^k\|+\|G_3^k\|\right\}$$ to get an error bound. By , Lemmas \[lem2.2\], \[lem2.4\], and \[lem2.5\], we get the error bounds as follows $$\begin{aligned}
\|G_1^k\| &\leq& CN^{-r},\qquad \|G_2^k\| \leq C\tau^q,\qquad
\|G_3^k\|=\frac{1}{{\tau}^{\beta}}\left\|\sum_{j=0}^k {\varpi}_{k-j}\left(\eta^j-\eta^0\right)\right\|
\leq CN^{-r},\\
\|e^0\| &=& \|u_*^0-I_N u_0\| \leq \|u_0-I_Nu_0\|+\|u_0-u_*^0\| \leq CN^{-r}.\end{aligned}$$ The above bounds yield $$\|e^k\| \leq C(\tau^q+N^{-r}).$$ By using Lemma \[lem2.4\] again, one has $$\|u^k-u_N^k\| = \|u^k-u_*^k+u_*^k-u_N^k\| \leq \|\eta^k\|+\|e^k\|\leq C(\tau^q+N^{-r}).$$ The proof is completed.
Numerical analysis for the nonlinear equation
=============================================
In this section, we develop the semi-implicit time-stepping Legendre Galerkin spectral method for the nonlinear problem , i.e., $f(x,t,u)=f(u)$. We then combine the discrete fractional Gr[ö]{}nwall inequality and the temporal-spatial error splitting argument (see, e.g., [@li2012mathematical; @li2017unconditionally]) to prove the stability and convergence of the numerical scheme. We assume that the solution of problem satisfies the following condition $$\begin{aligned}
\label{E:3.1}
\|u_0\|_{H^{r+1}}+\|u\|_{L^{\infty}(0,T;H^{r+1})} \leq K,\end{aligned}$$ where $K$ is a positive constant independent of $N$ and $\tau$.
Throughout this section, $C$ denotes a generic positive constant, which may vary at different occurrences, but is independent of $N$ and time step size $\tau$. Denote $C_i~(i=1,2,\cdots)$ as positive constants independent of $N$ and $\tau$.
The following inverse inequalities (see, e.g., [@Adams03; @li2003legendre; @brenner2007mathematical]) will be used in the numerical analysis $$\begin{aligned}
&&\|v\|_{L^{\infty}} \leq \frac{N+1}{\sqrt{2}} \|v\|, \qquad \forall~ v \in V_N^0,\label{E:3.2}\\
&&\|v\|_{L^{\infty}} \leq C_I \|v\|_{H^2}, \qquad \forall~ v \in H^1_0(I)\cap H^2(I),\label{E:3.3}\end{aligned}$$ where $C_I$ is a positive constant depending only on the interval $I$.
We extend the method with $p=1$ to the nonlinear equation , while the nonlinear term $f(u_N^k)$ is approximated by $f(u_N^{k-1})$. We obtain the following semi-implicit Galerkin spectral method: find $u_N^k\in V_N^0$ such that $$\label{E:3.7}
\left\{\begin{array}{ll}
\left({D}^{(\beta)}_{\tau} u_N^k,v\right)+ \mu \left(\partial_x u_N^{k},\partial_x v\right)= \left(f(u_N^{k-1}),v\right),\quad \forall v \in V_N^0, \quad k=1,2,\cdots,n_T,\\ [0.1cm]
u^0=I_N u_0,
\end{array}\right.$$ where $I_N$ is the LGL interpolation operator.
We also assume that the nonlinear term $f(u)$ satisfies the local Lipschitz condition $$\|f(u^{k})-f(u^{k-1})\| \leq \|f'(\xi)\|~\|u^{k}-u^{k-1}\|\leq C \|u^{k}-u^{k-1}\|,
{\quad} |\xi|\leq K_1,$$ where $K_1$ is a positive constant that is suitably large.
An error estimate of the time discrete system
---------------------------------------------
In order to obtain the unconditionally stability of , we now introduce a time-discrete system $$\label{E:3.1.1}
\left\{\begin{array}{lll}
{D}^{(\beta)}_{\tau} U^k=\mu \partial^2_x U^{k}+f(U^{k-1}),\qquad k=1,2,\cdots,n_T,\\
U^k(x)=0,\qquad {\rm for} \quad x \in \partial I, \quad k=1,2,\cdots,n_T,\\
U^0(x)=u_0(x),\qquad {\rm for} \quad x \in I.
\end{array}\right.$$
Let $R_1^k$ be the time discretization error of . Then we can obtain $$\begin{aligned}
\label{E:3.4}
{D}^{(\beta)}_{\tau} u^k=\mu \partial^2_x u^{k}+f(u^{k-1})+R_1^k,\qquad k=1,2,\cdots,n_T,\end{aligned}$$ where $$\begin{aligned}
\label{E:3.5}
R_1^k = {D}^{(\beta)}_{\tau} u^k-{_{0}^{C}{\mathcal {D}}}^{\beta}_t u^k+f(u^{k})-f(u^{k-1})
=O(\tau^{\tilde{q}}),\end{aligned}$$ in which $\tilde{q}=\min\{\sigma-\beta,1\}$ when the first-order extrapolation is applied.
Letting ${\varepsilon}^k=u^k-U^k$ and subtracting from gives $$\begin{aligned}
\label{E:3.1.4}
{D}^{(\beta)}_{\tau} {\varepsilon}^k=\mu \partial^2_x {\varepsilon}^{k}+f(u^{k-1})-f(U^{k-1})+R_1^k,\qquad k=1,2,\cdots,n_T.\end{aligned}$$
Define $$K_1=\max_{1\leq k \leq n_T} \|u^k\|_{L^{\infty}}+\max_{1\leq k \leq n_T} \|u^k\|_{H^2}+\max_{1\leq k \leq n_T} \|{D}^{(\beta)}_{\tau}u^k\|_{H^2}+1.$$ We now present an error bound of ${\varepsilon}^k=u^k-U^k$ as follows.
\[thm:3.1\] Suppose that $r \geq 1$, $u$ and $U^k~(k=1,2,\cdots,n_T)$ are solutions of and , respectively. If $u\in C(0,T; H^2(I)\cap H^1_0(I))$, $|f'(z)|$ is bounded for $|z|\leq K_1$, $f\in H^1(I)$, and $u_0 \in H^m(I)$. Then there exists a suitable constant $\tau_0^*>0$ such that when $\tau \leq \tau_0^*$, it holds $$\begin{aligned}
\|{\varepsilon}^k\|_{H^2} &\leq& C_*\tau^{\tilde{q}},\label{E:3.1.5}\\
\|U^k\|_{L^{\infty}}+\|{D}^{(\beta)}_{\tau}U^k\|_{H^2} &\leq& 2K_1,\label{E:3.1.6}\end{aligned}$$ where $C_*$ is a constant independent of $N$ and $\tau$, and $\tilde{q}=\min\{\sigma-\beta,1\}$.
We use the mathematical induction method to prove . Obviously, holds for $k=0$. Assume holds for $k \leq n-1$. Then, by and , one has $$\begin{aligned}
\label{E:3.1.7}
\|U^k\|_{\infty}
\leq \|u^k\|_{L^{\infty}}+C_I\|{\varepsilon}^k\|_{H^2}
\leq \|u^k\|_{L^{\infty}}+C_IC_*\tau^{\tilde{q}}
\leq K_1\qquad k \leq n-1,\end{aligned}$$ provided $\tau \leq (C_IC_*)^{-1/\tilde{q}}$. Moreover, for $\tau\leq (C_*)^{-1/\tilde{q}}$, we have $$\begin{aligned}
\label{E:3.1.8}
\|U^k\|_{H^2} \leq \|u^k\|_{H^2}+\|{\varepsilon}^k\|_{H^2} \leq \|u^k\|_{H^2}+C_*\tau^{\tilde{q}}
\leq K_1.\end{aligned}$$
The following estimates are easily obtained: $$\begin{aligned}
\|f(u^{k-1})-f(U^{k-1})\| &\leq& C\|{\varepsilon}^{k-1}\|, \label{E:3.1.9}\\
\left\|\partial_x \left(f(u^{k-1})-f(U^{k-1})\right)\right\|
&\leq&C\|{\varepsilon}^{k-1}\|_{H^1}.\label{E:3.1.10}\end{aligned}$$
Next, we prove that holds for $k\leq n$ in . Multiplying both sides of by $2{\varepsilon}^{n}$ and integrating the result over $I$ yields $$\begin{aligned}
\label{E:3.1.12}
{D}^{(\beta)}_{\tau} \left(\|{\varepsilon}^n\|^2\right)
& \leq &2\left(f(u^{k-1})-f(U^{k-1}),{\varepsilon}^{n}\right)
+2\left(R_1^n,{\varepsilon}^{n}\right) \nonumber \\
& \leq & C\|{\varepsilon}^{n}\|^2+C\|{\varepsilon}^{n-1}\|^2
+2\|{\varepsilon}^{n}\|~\|R_1^n\|,\end{aligned}$$ where is used.
Applying and Theorem \[thm:2.1\] with $$\begin{aligned}
v^n:=\|{\varepsilon}^{n}\|,\quad v^0:=0,\quad g^{n-\theta}:=2\|R_1^n\|~({\rm with~\theta=0}), \\
\lambda_0=\lambda_1:=C,\quad \lambda_j:=0\quad {\rm for}\quad 2 \leq j \leq n_T-1,\end{aligned}$$ one has $$\begin{aligned}
\label{E:3.1.13}
\|{\varepsilon}^{n}\| \leq C_1 \tau^{\tilde{q}},{\quad} \text{if} \quad
\tau\leq \frac{1}{\sqrt[\beta]{C(1+\beta)}}.\end{aligned}$$
To derive an estimate of $\|\partial_x {\varepsilon}^{n}\|$, we multiply by $2{D}^{(\beta)}_{\tau}{\varepsilon}^{n}$ and integrate the result over $I$ to obtain $$\begin{aligned}
\label{E:3.1.14}
& &2\|{D}^{(\beta)}_{\tau} {\varepsilon}^n\|^2+\mu {D}^{(\beta)}_{\tau} \left(\|\partial_x{\varepsilon}^n\|^2\right)=2\left(f(u^{n-1})-f(U^{n-1}),{D}^{(\beta)}_{\tau}{\varepsilon}^{n}\right)
+2\left(R_1^n,{D}^{(\beta)}_{\tau}{\varepsilon}^{n}\right).\end{aligned}$$
By Young inequality, , , and , we can derive $$\begin{aligned}
\left|\left(f(u^{n-1})-f(U^{n-1}),{D}^{(\beta)}_{\tau}{\varepsilon}^{n}\right)\right|
&\leq& \frac{3}{2}\left\|f(u^{n-1})-f(U^{n-1})\right\|^2+\frac{2}{3}\|{D}^{(\beta)}_{\tau} {\varepsilon}^n\|^2
\leq \frac{2}{3}\|{D}^{(\beta)}_{\tau} {\varepsilon}^n\|^2 +C{\tau}^{2\tilde{q}},\\
\left|\left(R_1^n,{D}^{(\beta)}_{\tau}{\varepsilon}^{n}\right)\right|
&\leq& \frac{3}{2}\left\|R_1^n\right\|^2+\frac{2}{3}\|{D}^{(\beta)}_{\tau} {\varepsilon}^n\|^2
\leq \frac{2}{3}\|{D}^{(\beta)}_{\tau} {\varepsilon}^n\|^2 +C{\tau}^{2\tilde{q}}.\end{aligned}$$ Substituting the above estimates into , we have
$$\begin{aligned}
\label{E:3.1.15}
{D}^{(\beta)}_{\tau} \left(\|\partial_x{\varepsilon}^n\|^2\right) \leq C {\tau}^{2\tilde{q}}.\end{aligned}$$
Applying Corollary \[cor:2.1\] with $$\begin{aligned}
v^n:=\|\partial_x {\varepsilon}^{n}\|^2,\quad v^0:=0,\quad g^{n}:=C {\tau}^{2\tilde{q}},
\quad \lambda_j:=0\quad {\rm for}\quad 0 \leq j \leq n_T-1,\end{aligned}$$ one has $$\begin{aligned}
\label{E:3.1.16}
\|\partial_x{\varepsilon}^{n}\| \leq C_2 \tau^{\tilde{q}}.\end{aligned}$$
We can similarly derive an estimate of $\|\partial^2_x{\varepsilon}^{n}\|$ by multiplying by $-2{D}^{(\beta)}_{\tau}\left(\partial_x^2{\varepsilon}^{n}\right)$ and integrating the result over $I$. Similar to , by Young inequality, , , , and , we can get $$\begin{aligned}
\label{E:3.1.17}
\|\partial^2_x{\varepsilon}^{n}\| \leq C_3 \tau^{\tilde{q}}.\end{aligned}$$
Combing , and , we obtain $$\begin{aligned}
\label{E:E:3.1.18}
\|{\varepsilon}^{n}\|_{H^2} \leq C_*\tau^{\tilde{q}},\end{aligned}$$ where $C_*=\sqrt{C_1^2+C_2^2+C_3^2}$ is a constant independent of $N$ and $\tau$.
Moreover, we can derive that $$\begin{aligned}
\|U^{n}\|_{L^{\infty}} &\leq& \|u^{n}\|_{L^{\infty}}+C_I\|\varepsilon^{n}\|_{H^2} \leq \|u^{n}\|_{L^{\infty}}+C_*C_I\tau \leq K_1,\\
\|{D}^{(\beta)}_{\tau} U^{n}\|_{H^2} &\leq& \|{D}^{(\beta)}_{\tau} u^{n}\|_{H^2}+\|{D}^{(\beta)}_{\tau} \varepsilon^{n}\|_{H^2} \leq \|{D}^{(\beta)}_{\tau} u^{n}\|_{H^2}+C_*{\tau}^{\tilde{q}-\beta}\leq K_1,\end{aligned}$$ where $\tau \leq \tau_0= \min \left\{(C_*C_I)^{-1},C_*^{\frac{1}{1-\beta}}\right\}$, and is used. Thus, the proof is completed.
An error estimate of the space discrete system
----------------------------------------------
The weak form of time-discrete system satisfies $$\begin{aligned}
\label{E:3.2.1}
\left({D}^{(\beta)}_{\tau} U^k,v\right)=\mu \left(\partial^2_x U^{k},v\right)+\left(f(U^{k-1}),v\right),\qquad v \in H^2(I).\end{aligned}$$ Let $$U^k_*=\Pi_N^{1,0}U^k, \quad {\bar e}^k=U^k_*-u_N^k, \quad k=1,2,\cdots,n_T.$$ Subtracting from , we have $$\begin{aligned}
\label{E:3.2.2}
\left({D}^{(\beta)}_{\tau} {\bar e}^k,v\right)+\mu \left(\partial_x {\bar e}^{k},\partial_x v\right)=\left(f(U^{k-1})-f(u_N^{k-1}),v\right)+\left(R_2^k,v\right),\end{aligned}$$ where $$R_2^k={D}^{(\beta)}_{\tau} (U_*^k-U^k).$$
It is easy to obtain $\left\|\Pi_N^{1,0}v\right\|_{L^{\infty}}\leq C\|v\|_{H^2}$ for any $v \in H^2(I)$. By Theorem \[thm:3.1\], one has $$\|U_*^k\|_{L^{\infty}}\leq C\|U^k\|_{H^2} \leq C,\qquad k=1,2,\cdots,n_T.$$ Then, we define $$K_2=\max_{1\leq k \leq n_T}\|U_*^k\|_{L^{\infty}}+1.$$
Now, we are ready to give an error estimate of $\|U^k-u_N^k\|$.
\[thm:3.2\] Suppose that $r \geq 1$, $u_N^k$ and $U^k~(k=1,2,\cdots,n_T)$ are solutions of and , respectively. Assume that $U^k\in H^2(I)\cap H^1_0(I)$, $|f'(z)|$ is bounded for $|z|\leq K_1$, $f\in H^1(I)$, and $U^0 \in H^2(I)$. Then there exists a positive constant $N_0^*$ such that when $N \geq N_0^*$, it holds $$\begin{aligned}
\|U^k-u_N^k\| &\leq& N^{-\frac{3}{2}},\label{E:3.2.3}\\
\|u_N^k\|_{L^{\infty}} &\leq& K_2.\label{E:3.2.4}\end{aligned}$$
We prove by using the mathematic induction method. From Lemma \[lem2.5\], one has $$\|U^0-u_N^0\|=\|u_0-I_Nu_0\| \leq C_4N^{-2} \leq N^{-\frac{3}{2}}$$ when $N\geq (C_4)^2$. Assume that holds for $k \leq n-1$. By the inverse inequality , we have $$\begin{aligned}
\|u_N^k\|_{L^{\infty}}\leq \|U_*^k\|_{L^{\infty}}+\|U_*^k-u_N^k\|_{L^{\infty}} \leq \|U_*^k\|_{L^{\infty}}+\frac{N+1}{\sqrt{2}}\|{\bar e}^k\| \leq \|U_*^k\|_{L^{\infty}}+C_5N^{-\frac{1}{2}} \leq K_2,\end{aligned}$$ when $N\geq (C_5)^2$.
By and the assumption, $\|U^{k}\|_{L^{\infty}}$ and $\|u_N^{k}\|_{L^{\infty}}$ are bounded for $k\leq n-1$. Therefore, $f'(\xi)$ is bounded when $|\xi|\leq \max\{2K_1,K_2\}$. Combining Lemma \[lem2.4\] and the boundedess of $f'(\xi)$ yields $$\begin{aligned}
\label{E:3.2.5}
\left\|f(U^{k-1})-f(u_N^{k-1})\right\|= \|f'(\xi)(u_N^{k-1}-U^{k-1})\|
\leq C\left\|U^{k-1}-u_N^{k-1}\right\|
\leq C\|{\bar e}^{k-1}\|+CN^{-2}.\end{aligned}$$ By Lemma \[lem2.4\] and Theorem \[thm:3.1\], we obtain $$\begin{aligned}
\label{E:3.2.6}
\|R_2^n\|^2 \leq CN^{-4}\left\|{D}^{(\beta)}_{\tau}U^k \right\|_{H^2} \leq CN^{-4}.\end{aligned}$$
Letting $k\leq n$ and $v=2{\bar e}^{k}$ in , we have $$\begin{aligned}
\left({D}^{(\beta)}_{\tau} {\bar e}^k,2{\bar e}^k\right)+2\mu
\left(\partial_x {\bar e}^{k},\partial_x {\bar e}^k\right)
=2\left(f(U^{k-1})-f(u_N^{k-1}),
{\bar e}^k\right)+2\left(R_2^k,{\bar e}^k\right).\end{aligned}$$ Combing , , and the above inequality yields $$\begin{aligned}
\label{E:3.2.7}
{D}^{(\beta)}_{\tau} \left(\|{\bar e}^k\|^2\right)
& \leq & 2\left(f(U^{k-1})-f(u_N^{k-1}),
{\bar e}^k\right)+2\left(R_2^k,{\bar e}^k\right) \nonumber \\
& \leq & \left(\left\|f(U^{k-1})-f(u_N^{k-1})\right\|^2
+\|{\bar e}^k\|^2\right)+\left(\|R_2^k\|^2+\|{\bar e}^k\|^2\right) \nonumber \\
& \leq &2\|{\bar e}^k\|^2+C\|{\bar e}^{k-1}\|^2+CN^{-4},{\quad} k\leq n.\end{aligned}$$ Applying Theorem \[thm:2.1\] yields $\|{\bar e}^k\| \leq CN^{-2}(k\leq n)$, which leads to $$\begin{aligned}
\label{E:3.2.9}
\|U^n-u_N^n\|\leq \|U^n-U_*^n\|+\|{\bar e}^n\| \leq C_6N^{-2} \leq N^{-\frac{3}{2}},\end{aligned}$$ when $N \geq (C_6)^2$. That is to say, holds for $k=n$. Furthermore, we have $$\begin{aligned}
\label{E:3.2.10}
\|u_N^n\|_{L^{\infty}}
\leq \|U_*^n\|_{L^{\infty}}+\|{\bar e}^n\|_{L^{\infty}}
\leq \|U_*^k\|_{L^{\infty}}+\frac{N+1}{\sqrt{2}}\|{\bar e}^n\|
\leq \|U_*^k\|_{L^{\infty}}+C_5N^{-\frac{1}{2}}\leq K_2,\end{aligned}$$ when $N \geq (C_5)^2$. Letting $N_0^*=\left\lceil\max\left\{(C_4)^2,(C_5)^2,(C_6)^2\right\}\right\rceil$ completes the proof.
Error estimate of the fully discrete system
-------------------------------------------
By the boundedness of $u_N^k$ and Theorem \[thm:2.2\], we immediately obtain the following result.
\[thm:3.3\] Suppose that $r \geq 1$, $u$ and $u_N^k~(k=1,2,\cdots,n_T)$ are solutions of and , respectively. If $m \geq r+1, u\in C(0,T; H^m(I)\cap H^1_0(I))$ and satisfies , $|f'(z)|$ is bounded for $|z|\leq K_1$, $f\in H^1(I)$, and $u_0 \in H^m(I)$. Then, there exist two positive constants $\tau^*_0$ and $N^*_0$ such that when $\tau \leq \tau^*_0$ and $N \geq N^*_0$, it holds $$\begin{aligned}
\label{E:3.3.1}
\|u^k-u_N^k\| \leq C\left(\tau^{\min\{\sigma-\beta,1\}}+N^{-r}\right).\end{aligned}$$
\[rem:3.1\] We can also extend the method with $p=2$ to the nonlinear equation , and the nonlinear term $f(u_N^k)$ is approximated by a second-order extrapolation $f(2u_N^{k-1}-u_N^{k-2})$. The numerical method is given by: find $u_N^k\in V_N^0$ for $k\geq2$ such that $$\label{E:3.3.5}
\left\{\begin{array}{ll}
\left({D}^{(\beta)}_{\tau} u_N^k,v\right)+ \mu \left(\partial_x u_N^{k-\beta/2},\partial_x v\right)=\left((1-\frac{\beta}{2})f(2u_N^{k-1}-u_N^{k-2})+\frac{\beta}{2}f(u_N^{k-1}),v\right),\quad \forall v \in V_N^0,\\ [0.1cm]
u^0=I_N u_0,
\end{array}\right.$$ where $u_N^1$ can be derived by the fully implicit method or with a smaller step size.
The stability and convergence analysis of the method is similar to that of .
\[thm:3.4\] Suppose that $r \geq 1$, $u$ and $u_N^k~(k=1,2,\cdots,n_T)$ are solutions of and , respectively. Assume that $m \geq r+1, u\in C(0,T; H^m(I)\cap H^1_0(I))$ and satisfies , $|f'(z)|$ and $|f''(z)|$ are bounded for $|z|\leq K_1$, $f\in H^1(I)$, and $u_0 \in H^m(I)$. Then, there exist two positive constants $\tau_1^*, N^*_1$ such that when $\tau \leq \tau^*_1$ and $N \geq N^*_1$, it holds $$\begin{aligned}
\|u^k-u_N^k\| \leq C\left(\tau^{\min\{\sigma-\beta,2\}}+N^{-r}\right).\label{E:3.3.6}\end{aligned}$$
Numerical results
=================
In this section, a numerical example is presented to illustrate the proposed method.
Consider the model problem with $\mu=1$ and $f(x,t,u)=u+u^2$. The initial condition is chosen as $$u_0(x)=\sin(2\pi x).$$
Since the exact solution of the problem is unknown, the reference solutions are derived by setting $N=2^9, \tau=1/2^{12}$. In Table \[s4:tb1\], we list the $L^2$-errors and convergence rates of the method in temporal direction with $N=2^9$ and different $\beta$. From Table \[s4:tb1\], we can observe the first-order accuracy in time at $t=1$ for $\beta=0.2$ and $\beta=0.9$. In Table \[s4:tb2\], we show the $L^2$-errors and convergence rates of the method in temporal direction for $N=2^9$. From Table \[s4:tb2\], a second convergence order is obtained for $\beta=0.9$ due to relatively good regularity of the solution. However, we do not observe second-order accuracy for $\beta=0.2$ due to slightly stronger singularity of the solution, but second-order convergence can be recovered by adding the correction terms, which is not investigated here; see [@Lubich1986Discretized; @zeng2017second].
$\tau$ $\beta=0.2$ Order $\beta=0.9$ Order
---------- ------------- ------- ------------- ------- -- -- -- -- -- -- -- --
$2^{-5}$ 3.6747e-3 1.85544e-2
$2^{-6}$ 1.7904e-3 1.03 9.22270e-3 1.00
$2^{-7}$ 8.7440e-4 1.03 4.54197e-3 1.02
$2^{-8}$ 4.2187e-4 1.05 2.19854e-3 1.04
$2^{-9}$ 1.9670e-4 1.10 1.02610e-3 1.09
: The $L^2$-errors and convergence rate of the method in time.[]{data-label="s4:tb1"}
$\tau$ $\beta=0.2$ Order $\beta=0.9$ Order
---------- ------------- ------- ------------- ------- -- -- -- -- -- -- -- --
$2^{-5}$ 7.9765e-4 4.9601e-03
$2^{-6}$ 3.6951e-4 1.11 1.2854e-03 1.94
$2^{-7}$ 1.7264e-4 1.09 3.3088e-04 1.95
$2^{-8}$ 7.9981e-5 1.11 8.4300e-05 1.97
$2^{-9}$ 3.5914e-5 1.15 2.1181e-05 1.99
: The $L^2$-errors and convergence rate of the method in time.[]{data-label="s4:tb2"}
Conclusion
==========
A discrete fractional Gr[ö]{}nwall inequality for convolution quadrature with the convolution coefficients generated by the generating function is developed. We illustrate its use through the stability and convergence analysis of the Galerkin spectral method for the linear time-fractional subdiffusion equations. We then combined the discrete fractional Gr[ö]{}nwall inequality and the temporal-spatial error splitting argument [@li2012mathematical] to prove the unconditional convergence of the Galerkin spectral method for the nonlinear time-fractional subdiffusion equation.
We only developed a discrete fractional Gr[ö]{}nwall inequality for the convolution quadrature with the coefficients generated by the generalized Newton–Gregory formula of order up to order two [@zeng2013use]. How to construct a discrete fractional Gr[ö]{}nwall inequality for other convolution quadratures (see, e.g., [@Lubich1986Discretized]) of high-order accuracy will be considered in our future work. It will be interesting to consider the discrete fractional Gr[ö]{}nwall inequality for analyzing the numerical methods for multi-term nonlinear time-fractional differential equations [@zeng2017second].
Acknowledgements {#acknowledgements .unnumbered}
================
The authors wish to thank the referees for their constructive comments and suggestions, which greatly improved the quality of this paper.
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abstract: '[*Chandra*]{} observations of the nearby, Lyman-continuum (LyC) emitting galaxy Tol 1247-232 resolve the X-ray emission and show that it is dominated by a point-like source with a hard spectrum ($\Gamma = 1.6 \pm 0.5$) and a high luminosity ($(9 \pm 2) \times 10^{40} \rm \, erg \, s^{-1}$). Comparison with an earlier [*XMM-Newton*]{} observation shows flux variation of a factor of 2. Hence the X-ray emission likely arises from an accreting X-ray source: a low-luminosity AGN or one or a few X-ray binaries. The [*Chandra*]{} X-ray source is similar to the point-like, hard spectrum ($\Gamma = 1.2 \pm 0.2$), high luminosity ($10^{41} \rm \, erg \, s^{-1}$) source seen in Haro 11, which is the only other confirmed LyC-emitting galaxy that has been resolved in X-rays. We discuss the possibility that accreting X-ray sources contribute to LyC escape.'
author:
- |
P. Kaaret$^{1}$ [^1], M. Brorby$^{1}$, L. Casella$^{1}$, A.H. Prestwich$^{2}$\
$^{1}$Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52245, USA\
$^{2}$Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, 02318, USA\
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'Resolving the X-ray emission from the Lyman continuum emitting galaxy Tol 1247-232'
---
\[firstpage\]
galaxies: star formation – galaxies: individual: Tol 1247-232, Haro 11 – X-rays: galaxies – X-rays: binaries
Introduction
============
Between 100 million and 1 billion years after the Big Bang, the intergalactic medium (IGM) changed from being cold and neutral to being warm and ionized. This reionization of the universe required sources of radiation energetic enough to ionize neutral hydrogen. Massive stars produce copious Lyman radiation and are often considered the most likely sources of reionization [@Loeb2010]. However, Lyman radiation is efficiently absorbed by the gas from which stars form and the details of how it escapes from a galaxy are poorly understood. Lyman continuum (LyC) and line emission is absorbed by dust and the Ly$\alpha$ is resonantly scattered by neutral hydrogen. Both reduce the escape fraction below values predicted by simple models of HII regions (Hayes et al. 2010). It appears that some source of feedback is required to blow the neutral gas and dust away from the starburst to prevent scattering and allow the Lyman emission to escape [@Wofford2013; @Orsi2012]. Obvious sources of mechanical power are stellar winds and supernovae ejecta [@Tenorio1999; @Hayes2010; @Heckman2011]. Another potential source of mechanical power is outflows from accreting compact objects. The power in the outflow from an accreting object is often comparable to the radiative luminosity [@Gallo2005; @Justham2012]. In some systems, particularly those found in actively star forming galaxies [@Kaaret2017], there is evidence that the mechanical power exceeds the radiative luminosity by large factors [@Pakull2010].
Most searches for LyC radiation from local galaxies have come up empty handed [@Leitherer1995; @Grimes2007]. There are only three known nearby (closer than 1000 Mpc) galaxies from which LyC emission has been directly detected: Haro 11 [@Bergvall2006], Tololo 1247-232 [@Leitet2013 hereafter Tol1247], and Mrk 54 [@Leitherer2016]. Remarkably, the two that have been observed in X-rays are very luminous with $L_X \sim 10^{41}$ erg/s in the 0.5–10 keV band for both Tol1247 [@Rosa2009] and Haro 11 [@Prestwich2015].
High resolution Chandra imaging of Haro 11 shows diffuse thermal emission ($kT \sim 0.7$ keV) and two bright point sources: one with a luminosity of $\sim 10^{41}$ erg/s and a very hard spectrum ($\Gamma = 1.2 \pm 0.3$) located in a star-forming knot and second with a luminosity of $5 \times 10^{40}$ erg/s and a softer spectrum ($\Gamma \sim 2.2$) located 3 arcseconds away in another star-forming knot [@Prestwich2015]. The sources are likely accreting objects. Modelling of the mechanical power from the central star-forming knots shows that the total mechanical power in supernovae plus stellar winds is comparable to the luminosity of the X-ray sources [@Prestwich2015]. Thus, if the X-ray sources produce outflows with power comparable to their luminosities, then feedback from accreting X-ray sources may help enable Lyman escape.
We recently obtained [*Chandra*]{} observations of the nearby LyC-emitting galaxy, Tol1247, that permit us to resolve the X-ray emission on physical scales of 500 pc, smaller than the 10 kpc optical extent of the galaxy. We describe the observations and analysis in section \[sec:obs\] then discuss our results and their interpretation in section \[sec:results\]. We adopt a distance to Tol1247 of 207 Mpc and a redshift of 0.048 [@Leitherer2016].
Observations and Analysis {#observations}
=========================
\[sec:obs\]
Figure \[optical\_image\] shows a Hubble Space Telescope (HST) image of Tol1247 obtained with the WFC3 using the F438W filter (B-band). Using the Graphical Astronomy and Image Analysis Tool ([*GAIA*]{}), we aligned the HST image to stars in the 2mass catalogue that has an astrometric accuracy of $0.2\arcsec$ [@2mass]. We found 10 coincidences with bright, isolated stars in the HST image and the root mean square deviation between the stellar centroids and catalogue positions was $0.22\arcsec$.
![Optical image of Tol 1247-232. An HST image in the F438W filter that shows the optical extent of the irregular galaxy used to define the red contours in Fig. \[xray\_images\]. The black arrow points North and has a length of $4\arcsec$.[]{data-label="optical_image"}](f438w_asp.pdf){width="2.75in"}
Our approved 30 ks [*Chandra*]{} observation was divided into two 15 ks observations for operational reasons. The first began at 07:26:01 UTC on 2016-05-13 (ObsID 16971, hereafter observation A) and the second began at 06:52:32 UTC on 2016-05-14 (ObsID 18845, hereafter observation B), about one day later. We used CIAO version 4.8 with data processing version of 10.4.3.1 and CALDB version 4.7.1.
There is strong emission coincident with the galaxy in both observations. Using a source extraction region with a radius of 3$\arcsec$ encompassing the whole galaxy, we found a net count rate in the 0.3–8 keV band of $2.5 \pm 0.4$ c/ks for A and $3.1 \pm 0.5$ c/ks for B. There is no evidence for variability between the two observations, so we chose to combine the two observations for further analysis. To align the observations, we used wavdetect to search for sources on the S3 chip (where the target galaxy is located) and then searched for matches within $1.0\arcsec$ between X-rays sources and objects in the USNO B1 catalog, excluding the target galaxy. We shifted the astrometry of the individual observations in RA and DEC and then merged the two observations. We estimate that the astrometry is uncertain at the level of $0.5\arcsec$. X-ray images from the merged observations are shown in Fig. \[xray\_images\]. There is diffuse X-ray emission spread across the central regions of the galaxy and about half of the emission is concentrated in a single unresolved source near the peak of the optical emission. The harder X-ray emission, at energies above 1.5 keV, is concentrated in the single unresolved source, near the center of the optical emission and consistent, within the astrometric uncertainties, with the positions of the star forming knots visible in Figure \[optical\_image\].
![X-ray images of Tol 1247-232. Top – full band (0.3–8 keV) X-ray image showing a bright point source near the peak of the optical emission and extended diffuse emission. The green circle shows the $3\arcsec$ radius extraction region used to obtain the spectrum of the whole galaxy. Bottom – hard band (1.5–8 keV) X-ray image showing that the hard X-ray emission arises from a single unresolved source. The green circle shows the $0.85\arcsec$ radius extraction region used to obtain the spectrum of the unresolved source. The X-ray pixels are 0.492 arcseconds or 494 pc at the distance of Tol 1247-232. The red contours are from the optical image in Fig. \[optical\_image\]. The black arrow points North and has a length of $4\arcsec$.[]{data-label="xray_images"}](xray_broad.pdf){width="2.75in"}
![X-ray images of Tol 1247-232. Top – full band (0.3–8 keV) X-ray image showing a bright point source near the peak of the optical emission and extended diffuse emission. The green circle shows the $3\arcsec$ radius extraction region used to obtain the spectrum of the whole galaxy. Bottom – hard band (1.5–8 keV) X-ray image showing that the hard X-ray emission arises from a single unresolved source. The green circle shows the $0.85\arcsec$ radius extraction region used to obtain the spectrum of the unresolved source. The X-ray pixels are 0.492 arcseconds or 494 pc at the distance of Tol 1247-232. The red contours are from the optical image in Fig. \[optical\_image\]. The black arrow points North and has a length of $4\arcsec$.[]{data-label="xray_images"}](xray_hard.pdf){width="2.75in"}
---------- ------------- ------------------------ ------------------------ ------------------------
Spectrum $\Gamma$ Flux
0.5-8 keV 0.5-8 keV 0.5-2 keV
Chandra 2.1$\pm$0.4 $2.40^{+0.59}_{-0.47}$ $2.67^{+0.60}_{-0.50}$ $1.42^{+0.37}_{-0.31}$
Point 1.6$\pm$0.5 $1.41^{+0.55}_{-0.40}$ $1.63^{+0.60}_{-0.44}$ $0.57^{+0.18}_{-0.23}$
XMM 2.2$\pm$0.3 $4.41^{+1.15}_{-0.88}$
---------- ------------- ------------------------ ------------------------ ------------------------
: X-ray spectral fits.
Note: Fluxes are in units of $10^{-14} \rm \, erg \, cm^2 \, s^{-1}$. \[xspec\]
We extracted a spectrum from the whole galaxy, using a region with a radius of $3\arcsec$ shown in Figure \[xray\_images\], and fitted it with an absorbed, redshifted powerlaw ($z = 0.048$) using the Cash statistic with the absorption column fixed to the Milky Way value of $6.6 \times 10^{20} \rm \, cm^{-2}$. The results, including the photon index ($\Gamma$), the observed flux (uncorrected for absorption), and the intrinsic flux (corrected for the assumed absorption), are given in Table \[xspec\] as “Chandra”. Uncertainties on the spectral parameters are quoted at the 90% confidence level. We also extracted a spectrum for a region with a radius of $0.85\arcsec$ centred on the hard X-ray emission in Figure \[xray\_images\]. We fitted using the same model and the results are given in the table as “Point”. In calculating the intrinsic point source flux, we corrected for the fraction of the point spread function outside the extraction region.
We re-analysed the XMM-Newton observations of Tol1247 obtained on 2005-06-22 [@Rosa2009]. We extracted a spectrum for the EPIC-PN using a circular region with a radius of $20\arcsec$ in order to encompass nearly all source counts and an annular background region with radii of $40\arcsec$ and $70\arcsec$ centred on the source. We used the same spectral model as for the Chandra data and fixed the absorption column to the Milky Way value. The results are in Table \[xspec\] as “XMM”. Our measured flux is compatible with that reported by @Rosa2009, while our photon index is marginally softer, possibly due to the fact that we did not include any absorption intrinsic to Tol1247. We also extracted background-subtracted light curves and grouped the data into 1500 s time bins, obtaining at least 23 net counts in each of the 7 time bins. Following the timing analysis of @Sutton2012, we find a $3 \sigma$ upper limit on the fractional variability 0.26.
Results and Discussion {#results}
======================
\[sec:results\] The Chandra images, Fig. \[xray\_images\], show that Tol1247 produces soft emission spread across the central part of the galaxy and point-like emission with a luminosity of $(9 \pm 2) \times 10^{40} \rm \, erg \, s^{-1}$ and a hard spectrum with a photon index of $1.56 \pm 0.46$. Comparing the flux measured with [*XMM-Newton*]{} to the total flux measured with [*Chandra*]{} shows evidence for a decrease in luminosity of $1.0 \times 10^{41} \rm \, erg \, s^{-1}$ from 2005 to 2016. The variability is strong evidence for the presence of at least one highly luminous accreting source in Tol1247. The source region for XMM-Newton is larger than the galaxy, but it is unlikely that the extra flux originates outside the galaxy since the centroid measured with [*XMM-Newton*]{} is well centred on the galaxy. The [*Chandra*]{} point source may be the source of the excess [*XMM-Newton*]{} flux, in which case only a single accreting object is required. Otherwise, Tol1247 may contain a small number of luminous accreting sources that coincidentally decreased in luminosity together. The presence of one or more highly luminous accreting sources and the similarity of the X-ray emission from Tol1247 to that found from Haro 11 [@Prestwich2015], the only other LyC-emitting galaxy that has been resolved in X-rays, may suggest a relation between accreting sources and LyC escape. Below, we discuss possible origins of the X-ray emission and the implications for the presence of accretion-powered outflows, the origin of the diffuse X-ray emission, and the relation of accretion-powered sources to LyC escape.
Nature of the Compact X-ray Source(s)
-------------------------------------
The variable X-ray emission from Tol1247 may be due an active galactic nucleus (AGN). Tol1247 lies within the pure starforming region of the Baldwin-Phillips-Terlevich (BPT) diagram [@Baldwin1981] and there are no indications of an active nucleus at other wavelengths [@Leitet2013]. However, a minor AGN contribution cannot be excluded. The X-ray luminosity would place an AGN in the low-luminosity (LLAGN) regime where the accretion flow is sub-Eddington and thought to be radiatively inefficient and produce powerful outflows with mechanical energy comparable to or larger than the X-ray luminosity [@Ho2008].
Another possible origin of the X-ray emission is from an ultraluminous or hyper-luminous X-ray source (ULX or HLX), for a review, see @Kaaret2017. ULXs are thought to be X-ray binaries that contain stellar-mass black holes or neutron stars in super-Eddington accretion states. HLXs may be super-Eddington accretors or intermediate mass black holes (IMBHs) with masses in the range $100-10^{5} M_{\sun}$. If the compact object is a stellar-mass black hole or a neutron star, then the accretion flow would be highly super-Eddington and strong outflows would be expected [@Kaaret2017]. If the compact object is an IMBH with a mass above $10^{4} M_{\sun}$, then the accretion flow would be sub-Eddington ($L_X/L_{\rm Edd} < 0.05$) and, hence, radiatively inefficient and the source of a powerful outflow, similar to that of a LLAGN. For a compact object mass near $10^{3} M_{\sun}$, the source could be in the near-Eddington X-ray thermal dominant state, where strong outflows are not observed.
The emission could, instead, arise from a small number of ULXs that happened to all decrease in flux from 2005 to 2016. The Chandra point source extraction region corresponds to 900 pc at the distance to Tol1247. This encompasses the whole of the central starburst in M82 with approximately 20 X-ray sources and emission dominated by two particularly bright objects, M82 X-1 and X-2. As noted above, ULXs are thought to be X-ray binaries in super-Eddington accretion states with strong outflows.
Repeated X-ray imaging of Tol1247 with [*Chandra*]{} could reveal whether the X-ray emission arises from a single source or multiple sources. Detection of strong X-ray variability, with rms of 10–20 per cent on timescales of 100s to 1000s of seconds as seen from several other HLXs [@Sutton2012], would suggest that the source is in the hard X-ray state and that the compact object is an IMBH. Radio emission consistent with the fundamental plane relation between radio luminosity ($L_{\rm R}$), $L_X$, and black hole mass ($M_{\rm BH}$) for accreting black holes in the hard state [@Merloni2003] would positively identify the source as in the hard state and could place constraints on the black hole mass. We estimate that the radio flux at 5 GHz would be 14 $\mu$Jy for a $M_{\rm BH} = 10^{6} M_{\sun}$ and 2.4 $\mu$Jy for a $M_{\rm BH} = 10^{5} M_{\sun}$. Conversely, detection of X-ray pulsations would identify the compact object as a neutron star. One possible constraint on the nature of the X-ray emitter(s) is the age of star formation in the host galaxy. @Buat2002 fitted the far ultraviolet spectrum of Tol1247 with spectral synthesis models and found the best fits for a recent starburst (1 Myr) or continuous star formation over at most 5 Myrs. The observed spectrum is bluer than any of the synthesis models. @Rosa2007 concluded that Tol1247 contains a single, young star burst with an age of less than 4 Myr, based on the galaxy’s high H$\beta$ equivalent width, high [Oiii]{} to H$\beta$ ratio, shallow radio spectral index, and low ratio of 1.4 GHz versus H$\alpha$ flux. The optical results indicate that the stellar population is dominated by massive, young stars while the radio properties show that the starburst has not yet had time to produce a large number of type II supernovae that would power radio synchrotron emission. @Puschnig2017 suggest that the galaxy has been forming stars for the past 30 Myr, but that the central region is dominated by very young stars with a mean age of 3 Myr. Therefore, the accreting object(s) responsible for the X-ray emission in Tol1247 likely evolved on a very rapid time scale, less than 4 Myr.
Accretion from the interstellar medium onto a pre-existing massive black hole could occur on time scales as rapid as the gravitational infall of the gas leading to star formation. Thus, a low-luminosity AGN intepretation for the X-ray emission is compatible with the young age of the star forming activity. The leading model for IMBH formation is dynamical interactions in star clusters. Simulations indicate that it is possible to form compact objects with masses above $1000 M_{\sun}$ in sufficiently massive and compact star clusters in less than 4 Myr [@Portegies2004]. This is compatible with the young age of the starburst. Indeed, a key requirement for runaway collisions of massive stars in a cluster is that they can reach the cluster center via dynamical interactions before exploding as supernovae. Conversely, formation of a high-mass X-ray binary is thought to occur after evolution of one member of the binary into the supergiant phase, requiring $\sim 12$ Myr [@Tauris2006]. This time scale is incompatible with the young age of the starburst in Tol1247.
Diffuse X-ray emission
----------------------
The Chandra images, Fig. \[xray\_images\], show that Tol1247 produces soft emission spread across the central part of the galaxy. Subtracting the aperture-corrected flux for the point source from the flux measured for the whole galaxy, we find that the absorption-corrected diffuse flux for the whole galaxy corresponds to an X-ray luminosity of $L_X = (4 \pm 2) \times 10^{40} \rm \, erg \, s^{-1}$ in the 0.5–2 keV band. The limited number of X-ray counts do not permit a detailed spectral analysis and the bolometric luminosity may be higher if the spectrum contains absorbed thermal emission at low temperatures.
Such soft diffuse emission is thought to be produced by energy injected into the interstellar medium by supernovae, stellar winds, and X-ray binaries, and, hence, proportional to the SFR. Different SFR indicators probe star formation on different times scales. As noted above, the starburst in Tol1247 is quite young. Using the SFR estimate from @Rosa2007 of $47 \rm \, M_{\odot} \, yr^{-1}$ based on the H$\alpha$ luminosity and therefore appropriate for a young starburst and the relation between SFR and diffuse soft X-ray luminosity from @Mineo2012b, predicts $L_X = 4 \times 10^{40} \rm \, erg \, s^{-1}$ in the 0.5–2 keV band in agreement with the measured value. However, the $L_X-$SFR relation of @Mineo2012b was derived using IR and FUV as SFR indicators and may not be applicable for SFRs derived from H$\alpha$, also SFR estimates for Tol1247 vary significantly, e.g. @Puschnig2017 find $35 \rm \, M_{\odot} \, yr^{-1}$ from H$\alpha$ and $96 \rm \, M_{\odot} \, yr^{-1}$ from 1.4 GHz radio emission.
The weak radio emission suggests that Tol1247 has not yet produced a large number of type II supernovae [@Rosa2007]. Hence, the diffuse emission must be powered by other means. @Justham2012 suggest that X-ray binaries may dominate the energy input to the ISM for starburst ages less than 6 Myr and that the relative importance of X-ray binaries versus supernovae will be greatest for near solar metallicity galaxies with starburst masses near $10^8 \rm \, M_{\odot}$. Hence, Tol 1247-232, with a starburst mass of $1.3 \times 10^8 \rm \, M_{\odot}$ [@Rosa2007], may be a good candidate for a galaxy in which feedback and heating of the ISM is dominated by X-ray binaries.
![Mechanical power from an instantaneous burst with a total stellar mass of $10^7 \rm \, M_{\odot}$ versus age of the burst, following @Prestwich2015. The solid line is the power from stellar winds. The dashed line is the power from supernovae. The dotted line is at $10^{41} \rm \, erg \, s^{-1}$, representative of the level of accretion power present in Tol 1247-232.[]{data-label="power"}](power.pdf){width="2.75in"}
Accretion-powered outflows and Lyman-continuum escape
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The similarity of the X-ray emission from Tol1247 to that found from Haro 11, the only other LyC emitting galaxy that has been resolved in X-rays, may suggest a role for highly luminous X-ray sources in enabling LyC escape. Most of the possible interpretations of the source(s) of the X-ray emission observed from Tol1247 suggest the presence of at least one accretion-powered object that produces an X-ray luminosity near $10^{41} \rm \, erg \, s^{-1}$ and an outflow with a comparable mechanical energy. LyC photons may escape a galaxy via ionized or empty channels through the ISM leading out from the star-forming region. X-rays may photoionize the ISM. An outflow may either entrain matter, creating an empty channel, or deposit energy, ionizing the ISM. Any of these processes may help enable LyC escape.
To compare the stellar- versus accretion-powered mechanical energies, we used Starburst99 to model the mechanical power from an instantaneous star-forming burst with a Kroupa initial mass function with mass limits of 0.1 and 100 $M_{\odot}$, upper and lower exponents of 2.3 and 1.3, and a turnover mass of 0.5 $M_{\odot}$. Fig. \[power\] show the power for a total stellar mass of $10^7 \rm \, M_{\odot}$, similar to that measured in the bright knots B and C in Haro 11 [@Adamo2010] that appear similar to the knots in Tol1247, and at a metallicity of $Z = 0.004$, close to the metallicity of Tol1247 [@Terlevich1993]. The power output from the accreting X-ray source dominates until 3.3 Myr. Winds from Wolf-Rayet stars briefly dominate, reaching $5 \times 10^{41} \rm \, erg \, s^{-1}$. After 3.8 Myr, supernovae provide mechanical power similar to or somewhat larger than that of the accreting source. Hence, the accreting source may dominate the mechanical power at early times, depending on when it was formed, and provides a significant contribution at later times.
H$\alpha$ imaging of Tol1247 shows an arc with a diameter of $\sim 300$ pc near the location of the brighter star cluster [@Puschnig2017], near the X-ray point-source. @Puschnig2017 interpret this as evidence that feedback from the star formation in the star cluster produces outflows or a highly turbulent ionized medium that create at least one clear sightline for LyC to leak. They also find low extinction along the light of sight to the central star clusters. The young age of the central star clusters and deficit of radio emission argue against the feedback being due to supernova, but is consistent with feedback due to high-mass X-ray binaries.
It is not possible to draw robust conclusions from a sample of two objects, and an important question is whether some other factor drives LyC leakage and the presence of highly luminous X-ray sources is merely a by product. Many properties of Tol1247 and Haro 11 are similar, including the young age of their star clusters and their high central SFR surface density. @Verhamme2017 have suggested that SFR surface density is the key determinant of LyC escape and that Lyman break analog galaxies (LBAs) and Green Pea galaxies (GPs) may be LyC emitters. Interestingly, both LBAs and GPs produce strong X-ray emission [@Basu2013; @Brorby2017]. Conversely, there are HLX host galaxies that are not known to be LyC emitters. M82 is the closest example, however the strong absorption seen from the X-ray sources [@Kaaret2009] and the edge-on orientation of M82 are not favourable to observing LyC; it has been suggested that LyC escapes from M82 along other lines of sight [@Devine1999]. Inclination effects may be important if LyC escapes only along specific lines of sight. Searches for LyC emission or its proxies from HLX host galaxies and for X-ray emission from additional LyC emitters would be excellent tests of the relation between accretion-powered objects and LyC escape.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the referee for thoughtful comments that significantly improved the manuscript, particularly the discussion section. Support for this work was provided by the National Aeronautics and Space Administration through Chandra Award Number GO5-16081X issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. We acknowledge the use of the HyperLeda database.
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